diff --git "a/data_tmp/process_19/tokenized_finally.jsonl" "b/data_tmp/process_19/tokenized_finally.jsonl"
deleted file mode 100644--- "a/data_tmp/process_19/tokenized_finally.jsonl"
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@@ -1,10067 +0,0 @@
-{"id": "6987.png", "formula": "\\begin{align*} \\begin{array} { l } ( p - 2 ) \\otimes ( p - 1 ) = ( p - 2 ) ^ 3 / ( p - 4 ) ^ 4 / \\cdots / 1 ^ 4 , \\\\ ( p - 1 ) \\otimes ( p - 1 ) = ( p - 1 ) ^ 2 / ( p - 3 ) ^ 4 / \\cdots / 2 ^ 4 / 0 ^ 3 . \\end{array} \\end{align*}"}
-{"id": "8134.png", "formula": "\\begin{align*} h \\psi = \\psi ( ? h ' ) h '' \\end{align*}"}
-{"id": "2597.png", "formula": "\\begin{align*} I _ 4 ( t , \\theta , x ) : = \\Big \\| \\| J ' _ g ( t , r , \\xi ) - J _ g ( t , r , \\xi ) \\| _ { \\Pi _ x ^ { ( \\phi ) } ; \\frac { 1 } { \\gamma _ 0 - 1 } } \\Big \\| _ { \\mathbf 1 _ { 0 \\leq r \\leq \\theta } d r ; \\frac { 1 } { \\gamma _ 0 - 1 } } . \\end{align*}"}
-{"id": "8863.png", "formula": "\\begin{align*} c _ 2 & = 3 \\nu ^ { 7 } - 1 2 3 \\nu ^ { 6 } + 1 3 3 0 \\nu ^ 5 - 1 9 1 8 \\nu ^ 4 - 2 8 8 9 7 { \\nu } ^ 3 + 6 5 1 7 7 \\nu ^ 2 + 9 3 1 0 0 \\nu + 1 2 0 5 4 4 \\\\ & + \\left ( 3 6 \\nu ^ { 7 } - 6 2 4 \\nu ^ { 6 } + 3 4 8 \\nu ^ 5 + 2 5 6 1 6 \\nu ^ 4 - 7 3 3 2 \\nu ^ 3 - 2 7 2 3 6 8 \\nu ^ 2 - 1 3 4 5 5 6 \\nu + 6 8 8 7 8 4 \\right ) \\\\ & + \\left ( 1 0 8 \\nu ^ { 7 } - 7 2 \\nu ^ { 6 } - 4 8 4 8 \\nu ^ 5 - 3 5 9 1 6 \\nu ^ 4 + 2 4 7 5 4 8 \\nu ^ 3 - 2 5 2 7 2 0 \\nu ^ 2 + 1 4 4 4 5 6 \\nu - 6 4 7 6 7 6 \\right ) { \\mu } ^ 2 \\end{align*}"}
-{"id": "3419.png", "formula": "\\begin{align*} | f ( T ) - f ( T ^ M ) | \\le \\log \\frac { \\rho _ { 0 , \\inf } ^ M ( T ) e ^ { \\tau _ 0 ^ M ( T ) } + \\rho _ { * , \\inf } ^ M ( T ) e ^ { \\tau _ * ^ M ( T ) } } { \\rho _ { 0 , \\inf } ^ M ( T ) + \\rho _ { * , \\inf } ^ M ( T ) } \\le \\log e ^ { \\eta ^ M ( T ) } = \\eta ^ M ( T ) , \\end{align*}"}
-{"id": "5746.png", "formula": "\\begin{align*} | f _ { m , n } ( t ) | ^ 2 & \\leq \\sum _ { m , n \\in \\N } | f _ { m , n } ( t ) | ^ 2 & & \\\\ & = \\int _ 0 ^ 1 \\int _ 0 ^ 1 x | f ( x , y , t ) | ^ 2 d x d y \\\\ & \\leq \\int _ 0 ^ 1 \\int _ 0 ^ 1 | | f | | _ \\infty ^ 2 d x d y \\\\ & = | | f | | _ \\infty ^ 2 . \\end{align*}"}
-{"id": "2460.png", "formula": "\\begin{align*} Q _ 1 = [ 0 : i : 1 ] , Q _ 2 = [ 0 : - i : 1 ] . \\end{align*}"}
-{"id": "1651.png", "formula": "\\begin{align*} G _ { - k } : = \\sigma \\big ( G _ k \\big ) = \\frac { 1 } { \\varkappa _ k } \\begin{pmatrix} a _ k - i \\sqrt [ + ] { b ^ 2 - a _ k ^ 2 } \\\\ - b \\end{pmatrix} e ^ { - 2 \\pi i k x } \\end{align*}"}
-{"id": "1556.png", "formula": "\\begin{align*} \\pi _ z \\circ \\widetilde { F } ^ 0 ( z , x ) = \\pi _ z \\circ F ^ 0 ( z , x ) = z + q ( z ) = f ( z ) \\end{align*}"}
-{"id": "860.png", "formula": "\\begin{align*} \\min \\frac 1 2 \\norm { B } _ F ^ 2 \\mbox { s u b j e c t t o } B ^ T v ^ * = u ^ \\ast , \\ ; B u = v . \\end{align*}"}
-{"id": "3576.png", "formula": "\\begin{align*} \\begin{matrix} \\begin{aligned} & C _ { 0 } ^ { \\vee } = \\mathrm { C o n e } \\left \\{ \\begin{matrix} ( 0 , 1 , 0 , 0 ) , ( 1 , 0 , 0 , 0 ) , ( 0 , 0 , 1 , 0 ) , ( 0 , 0 , 0 , 1 ) \\\\ ( 1 , 1 , 0 , - 1 ) , ( - 1 , 0 , 1 , 1 ) \\end{matrix} \\right \\} , \\\\ & C _ { N E } ^ { \\vee } = \\mathrm { C o n e } \\left \\{ ( 1 , 1 , 0 , 0 ) , ( 0 , 1 , 1 , 0 ) , ( 0 , 0 , 1 , 1 ) , ( 1 , 1 , 1 , 0 ) , ( 0 , 1 , 1 , 1 ) \\right \\} . \\end{aligned} \\end{matrix} \\end{align*}"}
-{"id": "3440.png", "formula": "\\begin{align*} \\lambda _ { \\mathrm { m a x } } : = \\lim _ { M \\rightarrow \\infty } \\frac { 1 } { M } \\log \\| \\Pi _ { M } \\| , \\end{align*}"}
-{"id": "473.png", "formula": "\\begin{align*} g ( x _ 1 , \\dots , x _ n ) \\circ F ( z _ 1 , \\dots , z _ n ) : = g ( \\partial / \\partial z _ 1 , \\dots , \\partial / \\partial z _ 1 ) F ( z _ 1 , \\dots , z _ n ) . \\end{align*}"}
-{"id": "801.png", "formula": "\\begin{align*} ( A _ d / P ^ \\alpha ) ^ * = ( A _ d / P ) ^ * \\oplus \\left ( A _ d / P \\right ) ^ { \\alpha - 1 } . \\end{align*}"}
-{"id": "751.png", "formula": "\\begin{align*} S _ { a } = - \\sum _ { i = 1 } ^ { m } \\lambda _ { i } \\ln ^ { a } \\lambda _ { i } \\end{align*}"}
-{"id": "1831.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } | f ( x ) | ^ 2 d x = \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } | \\hat { f } ( \\alpha ) | ^ 2 d \\alpha . \\end{align*}"}
-{"id": "2802.png", "formula": "\\begin{align*} U ^ t = \\sum _ { r = 0 } ^ \\infty Y _ r . \\end{align*}"}
-{"id": "476.png", "formula": "\\begin{align*} W ^ { \\perp } : = \\{ f \\in S \\mid f \\circ g = 0 , \\} \\subset S . \\end{align*}"}
-{"id": "3255.png", "formula": "\\begin{align*} \\mathcal { P } _ n ( z ) = z \\mathcal { P } _ { n - 1 } ( z ) - a b \\mathcal { P } _ { n - 2 } ( z ) \\end{align*}"}
-{"id": "5412.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\| u _ n ^ 1 - u _ n ^ 2 \\| ^ 2 & \\leq \\limsup _ { n \\to \\infty } ( 1 + C _ 0 ) \\varepsilon \\| u _ n ^ 1 - u _ n ^ 2 \\| \\\\ & + \\limsup _ { n \\to \\infty } D _ \\varepsilon | u _ n ^ 1 - u _ n ^ 2 | _ p \\\\ & + \\limsup _ { n \\to \\infty } C _ 1 | K | _ \\infty | u _ n ^ 1 - u _ n ^ 2 | _ q ^ q \\\\ & = ( 1 + C _ 0 ) \\varepsilon \\limsup _ { n \\to \\infty } \\| u _ n ^ 1 - u _ n ^ 2 \\| \\end{align*}"}
-{"id": "111.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { N - 1 } B ( 0 ) ^ { 2 j } = \\sum _ { j = 1 } ^ { N } B ( 0 ) ^ { 2 j } = \\widehat P \\ , . \\end{align*}"}
-{"id": "2363.png", "formula": "\\begin{align*} a f & = \\sum \\frac { 1 } { n ! } n f _ { n + 1 } \\Phi _ n \\\\ a ^ + f & = \\sum \\frac { 1 } { n ! } f _ { n - 1 } \\Phi _ n \\\\ \\langle f | g \\rangle & = \\sum \\frac { 1 } { n ! } \\overline { f } _ n g _ n \\end{align*}"}
-{"id": "8322.png", "formula": "\\begin{align*} & \\int _ { - \\infty } ^ { \\infty } \\frac { 1 } { | z _ j ( t ) - z ( \\beta , t ) | ^ k } d \\beta \\\\ = & \\int _ { | z ( 0 , t ) - z ( \\beta , t ) | \\leq 2 d _ I ( t ) } \\frac { 1 } { | z _ j ( t ) - z ( \\beta , t ) | ^ k } d \\beta + \\int _ { | z ( 0 , t ) - z ( \\beta , t ) | \\geq 2 d _ I ( t ) } \\frac { 1 } { | z _ j ( t ) - z ( \\beta , t ) | ^ k } d \\beta \\\\ : = & I + \\it { I I } . \\end{align*}"}
-{"id": "4350.png", "formula": "\\begin{align*} \\mathcal P _ a : = & \\big \\{ P \\in { [ 2 , n ] \\choose k - 1 } \\ : \\ P \\cap [ i ] = [ 2 , i ] \\cap S \\big \\} , \\\\ \\mathcal P _ b : = & \\big \\{ P \\in { [ 2 , n ] \\choose k } \\ : \\ P \\cap [ i ] = [ i ] \\setminus S \\big \\} . \\end{align*}"}
-{"id": "9156.png", "formula": "\\begin{align*} \\rho ^ { ( - ) \\left ( N \\right ) } ( \\mathbf { x } ^ { \\left ( - \\right ) } ( t _ { i } ) , t _ { i } ) = \\rho ^ { ( + ) \\left ( N \\right ) } ( \\mathbf { x } ^ { ( - ) } ( t _ { i } ) , t _ { i } ) . \\end{align*}"}
-{"id": "5564.png", "formula": "\\begin{align*} \\lambda _ { \\sigma } ^ { T } = \\lambda _ { \\sigma ( a ) } ^ { T } \\cdots \\lambda _ { \\sigma ( b ) } ^ { T } . \\end{align*}"}
-{"id": "2161.png", "formula": "\\begin{align*} g _ { L + 1 } ( \\mathbf { x } ) T _ { j } p ( \\mathbf { x } ) & = \\sum _ { \\ell = 0 } ^ { d - 1 } \\sum _ { \\| \\alpha \\| \\leq \\ell } c _ { \\ell , \\alpha } T ^ { \\alpha } ( g _ { L + 1 + \\ell } ) ( \\mathbf { x } ) = \\sum _ { \\ell = 0 } ^ { d + 1 } \\sum _ { \\| \\alpha \\| \\leq \\ell } c _ { \\ell , \\alpha } ' T ^ { \\alpha } ( g _ { L + \\ell } ) ( \\mathbf { x } ) . \\end{align*}"}
-{"id": "5634.png", "formula": "\\begin{align*} 1 & > \\frac { 8 } { 2 7 } ( \\frac { 1 } { b _ { 1 3 } + x _ 1 } + \\frac { 1 } { b _ { 1 2 } + x _ 1 } ) \\\\ & \\ge \\frac { 8 } { 2 7 } \\frac { ( 1 + 1 ) ^ 2 } { b _ { 1 3 } + b _ { 2 3 } + 2 x _ 1 } \\\\ & = \\frac { 3 2 } { 2 7 } \\frac { 1 } { 1 + \\frac { 1 } { 2 } x _ 1 } \\ge \\frac { 3 2 } { 2 7 } \\frac { 1 } { 1 + \\frac { 1 } { 1 1 } } = \\frac { 8 8 } { 8 1 } , \\end{align*}"}
-{"id": "8016.png", "formula": "\\begin{align*} S = \\mathbb { C } [ t , u : u ^ 2 = t ( t - \\alpha _ 1 ) \\cdots ( t - \\alpha _ { 2 n } ) ] \\end{align*}"}
-{"id": "2404.png", "formula": "\\begin{align*} \\dim \\mathcal T _ { F _ \\upsilon } ( \\mathcal F _ \\sigma ) & = \\dim \\mathcal F _ \\upsilon + ( k + 2 ) ( 2 r + 4 ) = \\dim F _ \\sigma - ( 4 + 2 ( k + r ) ) + ( k + 2 ) ( 2 r + 4 ) \\\\ & = \\dim \\mathcal F _ \\sigma + ( k + 1 ) ( 2 r + 2 ) + 2 . \\end{align*}"}
-{"id": "2506.png", "formula": "\\begin{align*} h _ r = r + 1 = { r + 1 \\choose r } . \\end{align*}"}
-{"id": "5852.png", "formula": "\\begin{align*} \\dd _ * x = | \\Delta ( G ) | _ F ^ { \\frac 1 { 2 } } \\dd _ { G } x . \\end{align*}"}
-{"id": "6051.png", "formula": "\\begin{align*} T _ C = \\left \\langle ( X _ 0 , y _ 0 ) , ( X _ 1 , y _ 1 ) , \\ldots , ( X _ { \\ell - 1 } , y _ { \\ell - 1 } ) , ( X _ { \\ell } , y _ { \\ell } ) \\right \\rangle \\end{align*}"}
-{"id": "4727.png", "formula": "\\begin{align*} ( \\rho , u , p ) ( x , 0 ) = \\left \\{ \\begin{array} { l l } ( 1 . 0 , 0 . 0 , 1 . 0 ) ~ ~ ~ ~ x < 0 . 5 , \\\\ ( 0 . 1 2 5 , 0 . 0 , 0 . 1 ) ~ x > 0 . 5 . \\end{array} \\right . \\end{align*}"}
-{"id": "1177.png", "formula": "\\begin{align*} | g ( 1 ) - g ( 0 ) - g ' ( 0 ) | = \\left | \\int _ 0 ^ 1 g '' ( t ) ( 1 - t ) d t \\right | \\leq \\frac { 1 } { 2 } \\sup _ { 0 \\leq t \\leq 1 } | g '' ( t ) | . \\end{align*}"}
-{"id": "7110.png", "formula": "\\begin{align*} g _ { m } = ( 1 + \\psi \\bar { \\psi } ) e _ { 0 } + ( \\psi + \\bar { \\psi } ) e _ { 1 } - i ( \\psi - \\bar { \\psi } ) e _ { 2 } + ( - 1 + \\psi \\bar { \\psi } ) e _ { 3 } \\end{align*}"}
-{"id": "4515.png", "formula": "\\begin{align*} d _ { \\mathcal { H } } ( A , B ) = \\inf \\{ \\delta > 0 : A \\subset B _ { \\delta } , B \\subset A _ { \\delta } \\} , \\end{align*}"}
-{"id": "4651.png", "formula": "\\begin{align*} \\chi _ 3 = & \\chi _ { \\varphi } ^ 3 & \\chi _ 2 = & 3 \\chi _ { \\varphi } ^ 2 & \\chi _ 1 = & 3 \\chi _ { \\varphi } . \\end{align*}"}
-{"id": "4151.png", "formula": "\\begin{align*} f _ 1 ( 0 ) = 3 \\sqrt { 3 } \\pi c _ { 0 , V } > 0 . \\end{align*}"}
-{"id": "3891.png", "formula": "\\begin{align*} u ^ \\gamma ( x ) = \\gamma ^ { \\frac { 5 } { 2 } } u ( \\gamma x ) , v ^ \\gamma ( x ) = \\gamma ^ { \\frac { 5 } { 2 } } v ( \\gamma x ) , \\end{align*}"}
-{"id": "572.png", "formula": "\\begin{align*} \\xi ^ { s + a _ 1 } \\left ( 1 + p \\left ( x _ s + a _ 2 \\right ) - \\left ( 1 + p \\left ( x _ s + b _ 2 \\right ) \\right ) \\right ) . \\end{align*}"}
-{"id": "9730.png", "formula": "\\begin{align*} v _ \\ast : = u - p _ { \\ast } , \\end{align*}"}
-{"id": "4242.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\delta d \\theta & = f & & \\Omega , \\\\ \\delta \\theta & = 0 & & \\Omega , \\\\ \\nu \\wedge \\theta & = 0 & & \\partial \\Omega , \\end{aligned} \\right . \\end{align*}"}
-{"id": "615.png", "formula": "\\begin{align*} \\begin{multlined} \\gamma _ { \\mathcal M \\rtimes \\mathcal G } \\bigl ( ( f , x ) ; ( f _ 1 , x _ 1 ) , \\dots , ( f _ n , x _ n ) \\bigr ) \\\\ = \\left ( \\gamma _ { \\mathcal M } ( f ; f _ { x ^ { - 1 } ( 1 ) } , \\dots , f _ { x ^ { - 1 } ( n ) } ) , \\gamma _ { \\mathcal G } ( x ; x _ 1 , \\dots , x _ n ) \\right ) \\ . \\end{multlined} \\end{align*}"}
-{"id": "7726.png", "formula": "\\begin{align*} \\norm { F _ { t } } = \\sup _ { \\norm { u } _ { 2 } \\leq 1 } { \\norm { \\sum _ { i = 1 } ^ l F _ { t } ^ i ( \\rho _ { i } u ) } _ { 2 } } \\leq \\sum _ { i = 1 } ^ l \\sup _ { \\norm { u } _ { 2 } \\leq 1 } { \\norm { F _ { t } ^ i ( \\rho _ { i } u ) } _ { 2 } } \\leq \\sum _ { i = 1 } ^ l \\norm { F _ { t } ^ i } \\leq l \\cdot L = : C . \\end{align*}"}
-{"id": "4109.png", "formula": "\\begin{align*} X ( z ) = \\mathcal { O } \\begin{pmatrix} 1 & z ^ { - \\frac { 1 } { 4 } } h _ { \\alpha + \\frac { 1 } { 2 } } ( z ) & z ^ { - \\frac { 1 } { 4 } } h _ { \\alpha + \\frac { 1 } { 2 } } ( z ) \\\\ 1 & z ^ { - \\frac { 1 } { 4 } } h _ { \\alpha + \\frac { 1 } { 2 } } ( z ) & z ^ { - \\frac { 1 } { 4 } } h _ { \\alpha + \\frac { 1 } { 2 } } ( z ) \\\\ 1 & z ^ { - \\frac { 1 } { 4 } } h _ { \\alpha + \\frac { 1 } { 2 } } ( z ) & z ^ { - \\frac { 1 } { 4 } } h _ { \\alpha + \\frac { 1 } { 2 } } ( z ) \\end{pmatrix} . \\end{align*}"}
-{"id": "4355.png", "formula": "\\begin{align*} \\mathcal P _ a ( i ) : = \\ & \\Big \\{ P \\ : \\ i \\in P \\in { [ 2 , n ] \\choose k - 1 } \\Big \\} , \\\\ \\mathcal P _ b ( i ) : = \\ & \\Big \\{ P \\ : \\ i \\notin P \\in { [ 2 , n ] \\choose k } \\Big \\} , \\end{align*}"}
-{"id": "341.png", "formula": "\\begin{align*} M _ { \\partial } = k L \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 0 & n - 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "3584.png", "formula": "\\begin{align*} V ^ { ( a ) } _ { \\Lambda } = \\bigoplus _ { \\lambda \\in \\Lambda } V ^ { ( a ) } _ { \\lambda } \\end{align*}"}
-{"id": "3374.png", "formula": "\\begin{align*} t \\geq t _ 0 = e ^ { - \\bar { c } N / d } . \\end{align*}"}
-{"id": "8763.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x ^ a } \\left ( \\log { \\frac { \\overline { F } } { F } } \\right ) = 2 \\psi _ a \\ , . \\end{align*}"}
-{"id": "8892.png", "formula": "\\begin{align*} S ( \\sigma ) ( N ) & = \\int _ 1 ^ N \\sigma ( x ) d x + \\frac { 1 } { 2 } \\left ( \\sigma ( N ) + \\sigma ( 1 ) \\right ) \\\\ & + \\sum _ { k = 2 } ^ K \\frac { B _ k } { k ! } \\ , \\left ( \\sigma ^ { ( k - 1 ) } ( N ) - \\sigma ^ { ( k - 1 ) } ( 1 ) \\right ) + \\frac { ( - 1 ) ^ { K + 1 } } { K ! } \\int _ 1 ^ N \\overline { B _ { K } } ( x ) \\ , \\sigma ^ { ( K ) } ( x ) \\ , d x \\end{align*}"}
-{"id": "3032.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 2 } ^ \\infty \\frac { \\lambda _ k [ k ] ^ m _ { p , q } - \\alpha u _ k [ k ] ^ n _ { p , q } } { 1 - \\alpha } | a _ { k } | + \\sum \\limits _ { k = 1 } ^ \\infty \\frac { \\mu _ k [ k ] ^ m _ { p , q } - ( - 1 ) ^ { n + j - ( m + i ) } \\alpha v _ k [ k ] ^ n _ { p , q } } { 1 - \\alpha } | b _ { k } | \\leq 1 , \\end{align*}"}
-{"id": "9535.png", "formula": "\\begin{align*} F ( a , b ; t ) : = \\sum _ { n = 0 } ^ { \\infty } \\frac { ( a q ) _ n } { ( b q ) _ n } t ^ n . \\end{align*}"}
-{"id": "1932.png", "formula": "\\begin{align*} ( x ^ 2 + x ) F _ { p + 2 } ( x ) + F _ p ( x ) = x ^ 2 \\left ( \\frac { 1 } { Z _ 1 ^ p } + \\sum _ { j = 2 } ^ \\infty \\frac { 1 } { Z _ j ^ p } \\right ) . \\end{align*}"}
-{"id": "9796.png", "formula": "\\begin{align*} L _ * = \\{ x _ n = y = 0 \\} . \\end{align*}"}
-{"id": "8546.png", "formula": "\\begin{align*} \\Big | \\sum _ { k = 0 } ^ s \\int ( | D _ t \\theta _ k | ^ 2 + | D _ t \\sigma _ k | ^ 2 ) d \\alpha - 4 ( \\norm { \\partial _ { \\alpha } ^ k D _ t \\tilde { \\theta } } _ { L ^ 2 } ^ 2 + \\norm { \\partial _ { \\alpha } ^ k D _ t \\tilde { \\sigma } } _ { L ^ 2 } ^ 2 ) \\Big | \\leq C \\epsilon ^ 3 . \\end{align*}"}
-{"id": "4789.png", "formula": "\\begin{align*} f = & \\ | \\psi | ^ 2 + \\operatorname { d i v } \\ ! \\left [ \\psi ^ * \\Bbb { J } \\left ( { \\mathcal { A } } - i \\hbar \\nabla \\right ) \\psi \\right ] \\\\ = & \\ | \\psi | ^ 2 + \\operatorname { d i v } ( | \\psi | ^ 2 \\Bbb { J } { \\mathcal { A } } ) + i \\hbar \\{ \\psi , \\psi ^ * \\} \\ , . \\end{align*}"}
-{"id": "8170.png", "formula": "\\begin{align*} B _ 1 = \\{ X _ { \\gamma } \\mid \\gamma \\in \\Gamma \\} = \\{ X _ { i j } \\mid 1 \\leq i \\leq j \\leq \\ell \\} \\subset B \\end{align*}"}
-{"id": "2480.png", "formula": "\\begin{align*} | P | ( x ) = \\sup \\Bigl \\{ \\ , \\sum _ { i _ 1 , \\dots , i _ n } | A ( u ^ 1 _ { i _ 1 } , \\dots , u ^ n _ { i _ n } ) | : u ^ 1 , \\dots , u ^ n \\in \\Pi ( x ) \\Bigr \\} \\end{align*}"}
-{"id": "3088.png", "formula": "\\begin{align*} \\| g \\| _ 2 ^ 2 = & \\int _ G | f | ^ 2 d x + \\sum _ l \\int _ { K _ l } ( \\frac { 1 } { | K _ l | } \\int _ { K _ l } f ) ^ 2 d x \\\\ \\le & \\int _ G | f | ^ 2 d x + \\sum _ l \\frac { 1 } { | K _ l | } ( \\int _ { K _ l } f d x ) ^ 2 \\\\ \\le & \\int _ G | f | ^ 2 d x + \\sum _ l \\int _ { K _ l } | f | ^ 2 d x = \\| f \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "3144.png", "formula": "\\begin{align*} R ( u , v ) \\left ( T ^ \\kappa ( u ) \\otimes \\mathbf { I } \\right ) \\left ( \\mathbf { I } \\otimes T ^ \\nu ( v ) \\right ) = \\left ( \\mathbf { I } \\otimes T ^ \\nu ( v ) \\right ) \\left ( T ^ \\kappa ( u ) \\otimes \\mathbf { I } \\right ) R ( u , v ) , \\end{align*}"}
-{"id": "1855.png", "formula": "\\begin{align*} \\phi ( x , 0 ) = \\psi ( x , 0 ) \\qquad \\mbox { a n d } \\partial _ t \\phi ( x , 0 ) = \\partial _ t \\psi ( x , 0 ) . \\end{align*}"}
-{"id": "2003.png", "formula": "\\begin{align*} I _ + ( q , z ) : = I _ 0 ( q ) z \\phi _ 1 + I _ 1 ( q ) \\phi _ 2 , \\end{align*}"}
-{"id": "9765.png", "formula": "\\begin{align*} D ^ \\alpha q _ \\infty ( 0 ) = 0 \\quad \\alpha = ( \\alpha ' , 0 ) | \\alpha | \\leq \\kappa - 2 , \\end{align*}"}
-{"id": "1165.png", "formula": "\\begin{align*} & \\lim _ { n \\to + \\infty } \\dfrac { C _ { \\beta , n - 2 k , k } } { C _ { \\beta , n } n ^ { k \\beta } } = A _ { \\beta } ^ k \\in ( 0 , + \\infty ) , \\end{align*}"}
-{"id": "6891.png", "formula": "\\begin{align*} \\begin{cases} w ^ 1 _ { 1 0 } v ^ 2 _ { 0 1 } - w ^ 1 _ { 0 1 } v ^ 2 _ { 1 0 } - w ^ 2 _ { 1 0 } v ^ 1 _ { 0 1 } + w ^ 2 _ { 0 1 } v ^ 1 _ { 1 0 } = 0 \\ , , \\\\ w ^ 2 _ { 1 0 } v ^ 2 _ { 0 1 } - w ^ 2 _ { 0 1 } v ^ 2 _ { 1 0 } + w ^ 1 _ { 1 0 } v ^ 1 _ { 1 0 } - w ^ 1 _ { 0 1 } v ^ 1 _ { 1 0 } = 0 \\ , , \\end{cases} \\end{align*}"}
-{"id": "9624.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\frac { ( - q ) _ n } { ( q ) _ n } \\frac { q ^ n } { 1 - q ^ n } - 2 \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { q ^ { n ( n + 3 ) / 2 } ( - q ) _ { n - 1 } } { ( q ) _ n ( 1 - q ^ n ) } = \\sum _ { n = 1 } ^ { N } \\frac { q ^ n } { 1 - q ^ n } . \\end{align*}"}
-{"id": "3300.png", "formula": "\\begin{align*} \\alpha _ { d } \\colon \\mathcal { M } \\to \\mathbb { C } , \\alpha _ { d } ( f ) = \\begin{cases} 1 & \\mbox { i f $ f $ h a s a d i v i s o r o f d e g r e e $ d $ } , \\\\ 0 & \\mbox { o t h e r w i s e , } \\end{cases} \\end{align*}"}
-{"id": "898.png", "formula": "\\begin{align*} \\Gamma = q ^ { \\sum m _ u m _ v \\lambda ( \\phi _ u , \\phi _ v ) } \\end{align*}"}
-{"id": "9768.png", "formula": "\\begin{align*} q ^ { \\rm e v e n } ( X ) = \\frac 1 2 \\left [ q ( X ) + q ( X - 2 ( \\boldsymbol { e } _ \\ast \\cdot X ) \\boldsymbol { e } _ \\ast ) \\right ] \\end{align*}"}
-{"id": "2322.png", "formula": "\\begin{align*} \\frac { 1 } { z - H _ g } = R _ g ( z ) = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 0 & R _ { \\Omega } ( z ) \\end{array} \\right ) + \\left ( \\begin{array} { c } 1 \\\\ R _ { \\Omega } ( z ) | g \\rangle \\end{array} \\right ) \\frac { 1 } { C _ g ( z ) } \\left ( 1 , \\langle g | R _ { \\Omega } ( z ) \\right ) \\end{align*}"}
-{"id": "9026.png", "formula": "\\begin{align*} f _ { \\{ b _ { i + 1 } , c _ { i + 1 } \\} , c _ i } ( x _ 1 , x _ 2 ) = \\begin{cases} c _ i & \\{ x _ 1 , x _ 2 \\} = \\{ b _ { i + 1 } , c _ { i + 1 } \\} , \\\\ x _ 1 & \\end{cases} \\end{align*}"}
-{"id": "9161.png", "formula": "\\begin{align*} R = \\sum _ { i = 1 } ^ n ( e ^ i \\# 1 _ H ) \\otimes ( \\varepsilon \\# e _ i ) \\in D ^ { \\omega } ( H ) \\otimes D ^ { \\omega } ( H ) , \\end{align*}"}
-{"id": "1106.png", "formula": "\\begin{align*} & \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) = n ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } \\times \\\\ & \\mathbb { P } ( \\xi ^ { ( n ) } ( I _ { n , k } ) = 0 , \\ \\xi ^ { ( n ) } ( J _ { n , k , j } ) > 0 , \\ \\forall \\ 0 \\leq j \\leq k ) . \\end{align*}"}
-{"id": "2515.png", "formula": "\\begin{align*} f ( s ) = s + ( 1 - s ) ^ { 1 + \\alpha } l ( 1 - s ) , s \\geq 0 , \\end{align*}"}
-{"id": "9030.png", "formula": "\\begin{align*} f _ { \\{ 1 \\} , c _ 1 } ( x ) = \\begin{cases} c _ 1 & x = 1 , \\\\ x & \\end{cases} \\end{align*}"}
-{"id": "9100.png", "formula": "\\begin{align*} | ( - \\Delta _ p ) ^ s \\Upsilon ( x ) | \\le \\begin{cases} \\dfrac { c _ 1 } { | x | ^ { \\alpha ( p - 1 ) + p s } } & \\alpha ( p - 1 ) < N , \\\\ [ 1 0 p t ] \\dfrac { c _ 2 \\log ( | x | ) } { | x | ^ { N + p s } } & \\alpha ( p - 1 ) = N , \\\\ [ 1 0 p t ] \\dfrac { c _ 3 } { | x | ^ { N + p s } } & \\alpha ( p - 1 ) > N , \\end{cases} \\end{align*}"}
-{"id": "609.png", "formula": "\\begin{align*} \\varphi ^ { } _ \\ast ( x _ 1 , \\dots , x _ m ) : = \\left ( \\prod _ { \\varphi ( i ) = 1 } x _ i , \\dots , \\prod _ { \\varphi ( i ) = n } x _ i \\right ) \\ , \\end{align*}"}
-{"id": "529.png", "formula": "\\begin{align*} \\frac { 1 } { h } \\big ( A ^ h ( x ' , h x _ 3 ) ^ { - 1 } - \\bar A ( x ' ) ^ { - 1 } \\big ) = \\displaystyle { - \\bar A ( x ' ) ^ { - 1 } A _ 1 ( x ' , x _ 3 ) \\bar A ( x ' ) ^ { - 1 } + \\mathcal { O } ( h ) } \\end{align*}"}
-{"id": "6115.png", "formula": "\\begin{align*} S _ { \\mu _ 1 , \\mu _ 2 , d } = \\langle V _ 0 ^ i V _ 1 ^ j V _ 2 ^ k \\mid ( i , j , k ) \\ \\in ( \\Z _ { \\geq 0 } ) ^ 3 \\ , \\mbox { w i t h } \\ , i + \\mu _ 1 j + \\mu _ 2 k \\geq d - \\mu \\rangle . \\end{align*}"}
-{"id": "3287.png", "formula": "\\begin{align*} \\alpha = \\frac { 1 } { 1 0 m + 7 } , \\qquad \\beta = \\frac { 2 m + 1 } { 1 6 ( 2 0 m ^ 2 + 4 m + 1 3 ) } \\end{align*}"}
-{"id": "3097.png", "formula": "\\begin{align*} s ^ * \\Omega _ { g _ 1 + g _ 2 , n _ 1 + n _ 2 } ( \\tau _ { \\mathbf n _ 1 } , \\tau _ { \\mathbf n _ 2 } ) = & \\ , { \\textstyle \\sum _ k } \\ \\Omega _ { g _ 1 , n _ 1 + 1 } ( \\tau _ { \\mathbf n _ 1 } , e ^ k ) \\cdot \\Omega _ { g _ 2 , 1 + n _ 2 } ( e _ k , \\tau _ { \\mathbf n _ 2 } ) , \\\\ r ^ * \\Omega _ { g + 1 , n } ( \\tau _ { \\mathbf n } ) = & \\ , { \\textstyle \\sum _ k } \\ \\Omega _ { g , n + 2 } ( \\tau _ { \\mathbf n } , e ^ k , e _ k ) . \\end{align*}"}
-{"id": "3140.png", "formula": "\\begin{align*} \\hat \\alpha _ i ( u ) = \\frac { \\hat \\lambda _ { i } ( u ) } { \\hat \\lambda _ { i + 1 } ( u ) } = \\alpha _ { N - i } ( u - ( N - i ) c ) . \\end{align*}"}
-{"id": "47.png", "formula": "\\begin{align*} d \\mathcal { Y } _ t ^ { v } = & \\ - f ^ { \\alpha _ { t - } } ( V _ t ^ { v } , \\mathcal { Z } _ t ^ { v } ) d t - \\sum _ { k \\in I } q ^ { \\alpha _ { t - } k } \\left [ e ^ { \\mathcal { U } _ t ^ { v } ( \\alpha _ { t - } , k ) } - 1 - \\mathcal { U } _ t ^ { v } ( \\alpha _ { t - } , k ) \\right ] d t + \\lambda d t \\\\ & + ( \\mathcal { Z } _ t ^ { v } ) ^ { t r } d W _ t + \\sum _ { k , k ' \\in I } \\mathcal { U } _ t ^ { v } ( k ' , k ) \\chi _ { \\{ \\alpha _ { t - } = k ' \\} } d \\tilde { N } _ t ^ { k ' k } . \\end{align*}"}
-{"id": "7332.png", "formula": "\\begin{align*} \\begin{cases} \\rho ( x ) u _ { t t } - \\operatorname { d i v } ( K ( x ) \\nabla u ) + v ^ 2 u = 0 & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ \\rho ( x ) v _ { t t } - \\operatorname { d i v } ( K ( x ) \\nabla v ) + u ^ 2 v = 0 & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ u _ t = v _ t = 0 & ~ \\hbox { i n } ~ \\omega \\times ( 0 , T ) \\end{cases} \\end{align*}"}
-{"id": "9979.png", "formula": "\\begin{align*} & u ^ { 1 } = v ^ { 1 } , u ^ { 2 } = v ^ { 2 } , \\\\ & u ^ { 3 } = v ^ { 3 } , u ^ { 4 } = q ^ { 4 } , \\\\ & u ^ { 5 } = q ^ { 5 } , u ^ { 6 } = 2 q ^ { 3 } - ( v ^ { 1 } + v ^ { 2 } + v ^ { 3 } ) . \\end{align*}"}
-{"id": "3926.png", "formula": "\\begin{align*} H ( s , \\phi ) = \\left ( M _ j ( Q _ { i j } ( s , \\phi ) ) \\right ) _ { i , j } \\end{align*}"}
-{"id": "5992.png", "formula": "\\begin{align*} u _ n ( t , \\omega ) & : = \\rho ^ { g ^ { ( t ) } _ n } \\left ( F \\left ( \\omega \\otimes _ t \\frac { W } { \\sqrt { n } } \\right ) \\right ) . \\end{align*}"}
-{"id": "1918.png", "formula": "\\begin{align*} A _ j ^ 2 = \\frac { \\cos ^ 2 Z _ j } { 1 + \\frac { \\sin Z _ j \\cos Z _ j } { Z _ j } } = \\frac { Z _ j ^ 2 } { Z _ j ^ 2 + x ^ 2 + x } . \\end{align*}"}
-{"id": "8449.png", "formula": "\\begin{align*} \\begin{cases} ( D _ t ^ 2 - i A \\partial _ { \\alpha } ) \\tilde { \\theta } = G \\\\ ( D _ t ^ 2 - i A \\partial _ { \\alpha } ) \\tilde { \\sigma } = \\tilde { G } \\\\ \\end{cases} \\end{align*}"}
-{"id": "6214.png", "formula": "\\begin{align*} f ( x ) = \\int _ 0 ^ { \\infty } e ^ { - \\frac { \\| x \\| ^ 2 } { 2 \\lambda } } \\frac { \\nu _ { \\mu } ( d \\lambda ) } { \\sqrt { 2 \\pi \\lambda } } . \\end{align*}"}
-{"id": "8677.png", "formula": "\\begin{align*} f _ \\delta ( x ) : = ( f \\ast \\rho _ \\delta ) ( x ) = \\int _ { B _ \\delta ( x ) } f ( y ) \\rho _ \\delta ( x - y ) \\ , d y = \\int _ { B _ \\delta ( 0 ) } f ( x - y ) \\rho _ \\delta ( y ) \\ , d y \\ , . \\end{align*}"}
-{"id": "8363.png", "formula": "\\begin{align*} | c _ 0 ^ j | = | ( \\Phi ^ { - 1 } ) _ z ( \\Phi ( z _ j ) ) | \\geq \\inf _ { \\alpha \\in \\mathbb { R } } | Z _ { \\alpha } | \\geq \\beta _ 0 . \\end{align*}"}
-{"id": "9935.png", "formula": "\\begin{align*} E ^ { \\nu } \\left [ \\left . \\frac { d \\mu } { d \\nu } ( X _ { 0 } ) \\right | \\bigcap _ { n \\geq 1 } \\mathcal { F } _ { 0 , \\infty } ^ { \\mathcal { Y } } \\vee \\mathcal { F } _ { n , \\infty } ^ { \\mathcal { X } } \\right ] & = E ^ { \\nu } \\left [ \\left . \\frac { d \\mu } { d \\nu } ( X _ { 0 } ) \\right | { F } _ { 0 , \\infty } ^ { \\mathcal { Y } } \\right ] ~ P ^ { \\nu } ~ a . s . \\end{align*}"}
-{"id": "1343.png", "formula": "\\begin{align*} \\sup _ { s \\in { \\mathbb { R } ^ { \\ge 0 } } , \\Delta s \\in [ 0 , \\eta L ] } \\left | \\sqrt { s + \\Delta s } - \\sqrt { s } \\right | \\cdot \\left \\| b ( t ) \\right \\| _ { { C } _ { 0 } ( Y ) } = & \\ \\sqrt { \\eta L } \\cdot \\left \\| b ( t ) \\right \\| _ { { C } _ { 0 } ( Y ) } \\ ; . \\end{align*}"}
-{"id": "5540.png", "formula": "\\begin{align*} L ^ * m = 0 ~ \\iff ~ m \\in \\mathcal K = \\{ c \\mu : c \\in \\mathbb R \\} , \\end{align*}"}
-{"id": "424.png", "formula": "\\begin{align*} G _ B ( x , x ' ) = \\int _ { 0 } ^ { \\infty } p _ B ( t , x , x ' ) \\dd t . \\end{align*}"}
-{"id": "5471.png", "formula": "\\begin{align*} \\hat I _ s ^ { t , a } \\ = \\ \\sum _ { n \\geq 1 } a \\ , 1 _ { [ t , \\hat T _ n ) } ( s ) + \\sum _ { \\substack { n \\geq 1 \\\\ t < \\hat T _ n } } \\hat \\eta _ n \\ , 1 _ { [ \\hat T _ n , \\hat T _ { n + 1 } ) } ( s ) , \\qquad t \\leq s \\leq T , \\end{align*}"}
-{"id": "8601.png", "formula": "\\begin{align*} \\bar { u } = \\sqrt { \\frac { 1 - \\sqrt { 1 - \\frac { 4 \\omega ^ 2 } { ( \\Lambda - 1 ) ^ 2 } } } { 2 } } \\ \\ \\textrm { a n d } \\ \\ \\bar { v } = \\sqrt { \\frac { 1 + \\sqrt { 1 - \\frac { 4 \\omega ^ 2 } { ( \\Lambda - 1 ) ^ 2 } } } { 2 } } . \\end{align*}"}
-{"id": "5638.png", "formula": "\\begin{align*} 3 + \\frac { A } { \\sqrt { A ^ 2 + \\frac { 3 2 } { 2 7 } a _ 2 } } = \\frac { 5 } { 4 } + \\frac { 5 } { 4 } \\frac { C } { \\sqrt { C ^ 2 + \\frac { 3 2 } { 2 7 } a _ 2 } } , \\end{align*}"}
-{"id": "1847.png", "formula": "\\begin{align*} v ^ { \\rm t o p . } ( x ) = \\frac { 1 } { \\sqrt { L } } . \\end{align*}"}
-{"id": "8842.png", "formula": "\\begin{align*} \\tau _ L & = \\inf \\{ t > 0 : \\ x _ { ( N - 1 ) } ( t ) < 1 / L \\} ; \\\\ \\eta _ L & = \\inf \\{ t > \\tau _ L : \\ x _ { ( N - 1 ) } ( t ) \\ge 1 / 2 \\} , \\end{align*}"}
-{"id": "8834.png", "formula": "\\begin{align*} \\mathcal G _ { \\beta , \\gamma } ( x , y , \\zeta , \\omega ) = \\frac 1 2 x ^ \\top Q x + f ( A ^ f x ) + G ( x ) + \\max _ { y ' } \\langle A ^ h x , y ' \\rangle - H ^ * ( y ' ) - \\frac { \\beta } { 2 } \\norm { y - y ' } ^ 2 \\\\ + H ^ * ( y ) + \\frac 1 2 \\omega ^ \\top Q ^ { \\dagger } \\omega + f ^ * ( \\zeta ) + \\max _ { x ' } \\langle - ( A ^ h ) ^ \\top y - ( A ^ f ) ^ \\top \\zeta - \\omega , x ' \\rangle - G ( x ' ) - \\frac { \\gamma } { 2 } \\norm { x - x ' } ^ 2 \\end{align*}"}
-{"id": "3215.png", "formula": "\\begin{align*} & \\Phi _ { \\mu _ k } ( \\tilde { w } ^ { k + j } ) + Q ( \\tilde { w } ^ { k + j } ) \\Delta _ { j + \\frac { 1 } { 2 } } \\tilde { w } = 0 , \\end{align*}"}
-{"id": "6170.png", "formula": "\\begin{align*} \\min \\left \\{ \\sum _ { i = 1 } ^ k \\int _ { \\R ^ { 2 n } } \\dfrac { | u _ i ( x ) - u _ i ( y ) | ^ 2 } { | x - y | ^ { n + 2 s } } \\ ; : \\ ; \\begin{array} { l l } u _ i ( x , 0 ) \\cdot u _ j ( x , 0 ) \\equiv 0 \\ ; \\textrm { a . e . i n } \\ ; \\R ^ n \\ ; { \\rm f o r } \\ ; \\ ; i \\neq j , \\\\ u _ i \\equiv \\varphi _ i , { \\rm o n } \\ ; \\R ^ n \\setminus \\tilde \\Omega \\ ; \\textrm { f o r } \\ ; \\ ; i = 1 , \\dots , k , \\end{array} \\right \\} \\end{align*}"}
-{"id": "7826.png", "formula": "\\begin{align*} 1 + \\| T \\| \\geqslant \\| T + \\lambda I \\| & \\geqslant \\| T x _ n + \\lambda x _ n \\| \\geqslant \\| \\lambda x _ n + \\lambda \\| T \\| x _ n - ( - T x _ n + \\lambda \\| T \\| x _ n ) \\| \\\\ & \\geqslant \\| \\lambda x _ n + \\lambda \\| T \\| x _ n \\| - \\| T x _ n - \\lambda \\| T \\| x _ n \\| = 1 + \\| T \\| - \\| T x _ n - \\lambda \\| T \\| x _ n \\| . \\end{align*}"}
-{"id": "4037.png", "formula": "\\begin{align*} I ( P , V ) & = O \\left ( m + m ^ { 2 / 3 } \\left ( \\sum _ { \\substack { \\\\ v _ 1 , v _ 2 , v _ 3 } } | S _ { v _ 1 } \\cap S _ { v _ 2 } \\cap S _ { v _ 1 } | \\right ) ^ { 1 / 3 } \\right ) \\\\ & = O \\left ( m + m ^ { 2 / 3 } \\left ( M ^ { 2 } n ^ { 2 } + n ^ { 3 } \\right ) ^ { 1 / 3 } \\right ) \\\\ & = O \\left ( m + M ^ { 2 / 3 } m ^ { 2 / 3 } n ^ { 2 / 3 } + m ^ { 2 / 3 } n \\right ) . \\end{align*}"}
-{"id": "548.png", "formula": "\\begin{align*} \\big [ \\partial _ { i j } y _ 0 , ~ \\partial _ i \\vec b _ 0 , ~ \\tilde d _ 0 \\big ] ( x ' ) = \\big [ \\partial _ 1 y _ 0 , ~ \\partial _ 2 \\vec y _ 0 , ~ \\vec b _ 0 \\big ] ( x ' ) \\cdot \\left [ \\begin{array} { c c c } \\Gamma _ { i j } ^ 1 & \\Gamma _ { i 3 } ^ 1 & \\Gamma _ { 3 3 } ^ 1 \\\\ \\Gamma _ { i j } ^ 2 & \\Gamma _ { i 3 } ^ 2 & \\Gamma _ { 3 3 } ^ 2 \\\\ \\Gamma _ { i j } ^ 3 & \\Gamma _ { i 3 } ^ 3 & \\Gamma _ { 3 3 } ^ 3 \\\\ \\end{array} \\right ] ( x ' , 0 ) \\qquad \\mbox { f o r } \\ ; i , j = 1 , 2 , \\end{align*}"}
-{"id": "8781.png", "formula": "\\begin{align*} L _ { \\overline { R } } = - \\frac { D + 4 q C } { 4 p } = \\frac { \\overline { \\widehat { \\kappa } } } { 2 } = \\overline { \\widehat { \\kappa } } _ G \\ , . \\end{align*}"}
-{"id": "212.png", "formula": "\\begin{align*} L u = 0 \\end{align*}"}
-{"id": "5327.png", "formula": "\\begin{align*} [ i H , B ] = 2 A , [ A , i H ] = 2 B \\quad \\mbox { a n d } \\quad [ B , A ] = - 2 i H . \\end{align*}"}
-{"id": "5646.png", "formula": "\\begin{align*} \\mathcal { G } _ { p } ^ { 2 , \\ell } ( H ; \\mathbb { R } ) = \\Big \\{ \\Phi : \\ , H \\to \\mathbb { R } , \\ ; | \\Phi | _ { \\mathcal { G } _ { p } ^ { 2 , \\ell } ( H ; \\mathbb { R } ) } = \\sup _ { u \\in H } \\frac { \\| \\Phi ^ { ( 2 ) } ( u ) \\| _ { \\mathcal { B } ( H ; \\mathbb { R } ) } } { ( 1 + \\| u \\| _ { H } ^ { \\ell - 2 } ) } < \\infty \\Big \\} , \\end{align*}"}
-{"id": "6049.png", "formula": "\\begin{align*} d e g _ { H ( w ) } ( \\lambda m , y ' ) & = d e g _ { H ^ * + ( \\lambda m - m , 0 ) } ( \\lambda m , y ' ) + d e g _ { H ^ * + ( \\lambda m , 0 ) } ( \\lambda m , y ' ) \\\\ & = d e g _ { H ^ * } ( m , y ' ) + d e g _ { H ^ * } ( 0 , y ' ) = d . \\end{align*}"}
-{"id": "9111.png", "formula": "\\begin{align*} \\mathrm { S } ^ k V \\simeq \\bigoplus \\limits _ { l = 0 } ^ { [ k / 2 ] } R _ { G } ( ( k - 2 l ) \\pi _ 1 ) . \\end{align*}"}
-{"id": "1130.png", "formula": "\\begin{align*} & B = \\sum \\limits _ { j = 1 } ^ k B _ j . \\end{align*}"}
-{"id": "5603.png", "formula": "\\begin{align*} 2 \\int _ { - \\infty } ^ 0 u \\bar { u } _ x d x = \\mathcal { Q } _ 0 ^ - + 2 \\int _ 0 ^ t | u _ x ( 0 , s ) | ^ 2 d s - \\frac { \\alpha } { \\gamma } \\int _ { - \\infty } ^ 0 v ^ 2 d x \\end{align*}"}
-{"id": "5485.png", "formula": "\\begin{align*} d X _ t = b ( X _ t ) d t + d W _ t , X _ 0 = x _ 0 \\in \\R ^ d , ~ t \\geq 0 . \\end{align*}"}
-{"id": "6480.png", "formula": "\\begin{align*} t _ { B } = \\frac { t + 2 ( a ^ { 2 } + b ^ { 2 } - 1 ) } { b ^ { 2 } } . \\end{align*}"}
-{"id": "2283.png", "formula": "\\begin{align*} R ( z ) = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 0 & R _ { \\Omega } ( z ) \\end{array} \\right ) + \\left ( \\begin{array} { c } 1 \\\\ R _ { \\Omega } ( z ) | E ) \\end{array} \\right ) \\frac { 1 } { C ( z ) } \\left ( 1 , ( E | R _ { \\Omega } ( z ) \\right ) . \\end{align*}"}
-{"id": "7302.png", "formula": "\\begin{align*} x \\wedge y = \\Pi ^ { \\deg x + \\deg y } x y , & & x \\vdash y = \\Pi ^ { \\deg y - \\deg x } x y . \\end{align*}"}
-{"id": "7467.png", "formula": "\\begin{gather*} ( f \\otimes g ) ( m \\otimes l ) \\colon = ( - 1 ) ^ { | m | \\cdot | g | } f ( m ) \\otimes g ( l ) , \\end{gather*}"}
-{"id": "6686.png", "formula": "\\begin{align*} C _ { n } & = \\prod _ { m = 1 } ^ { n } ( 1 - 2 ^ { \\frac { 1 } { 2 } - i ( m \\omega - \\lambda _ { * } ) } ) \\zeta ( \\frac { 1 } { 2 } + i ( m \\omega - \\lambda _ { * } ) ) , \\\\ \\tilde { C } _ { n } & = | ( 1 - 2 ^ { \\frac { 1 } { 2 } - i ( n \\omega - \\lambda _ { * } ) } ) \\zeta ( \\frac { 1 } { 2 } + i ( \\lambda _ { * } - n \\omega ) ) | ^ { 2 } C _ { n - 1 } . \\end{align*}"}
-{"id": "970.png", "formula": "\\begin{align*} \\psi ( x ) = \\textrm { d i s t } ( x , \\partial M ) , x \\in M _ \\epsilon . \\end{align*}"}
-{"id": "7898.png", "formula": "\\begin{align*} R [ F ] : = R \\{ F \\} / \\langle F r - r ^ p F \\ | \\ r \\in R \\rangle . \\end{align*}"}
-{"id": "2532.png", "formula": "\\begin{align*} \\psi ( x , z ) & = - \\beta ( x ) z + \\kappa ( x ) \\int _ 0 ^ \\infty ( e ^ { - z y } - 1 + z y ) \\frac { d y } { \\Gamma ( - \\gamma ( x ) ) y ^ { 1 + \\gamma ( x ) } } \\\\ & = - \\beta ( x ) z + \\kappa ( x ) z ^ { \\gamma ( x ) } , x \\in E , z \\geq 0 , \\end{align*}"}
-{"id": "3675.png", "formula": "\\begin{align*} \\sum _ { i \\in S } d _ i H ( \\delta _ i ; \\eta ) = 0 \\end{align*}"}
-{"id": "5813.png", "formula": "\\begin{align*} a _ n r ^ n = \\frac { 1 } { \\pi } \\int _ 0 ^ { 2 \\pi } \\Re ( f ( r e ^ { i \\theta } ) ) e ^ { - i n \\theta } d \\theta , \\end{align*}"}
-{"id": "9171.png", "formula": "\\begin{align*} q _ E \\ : C ^ * ( G ) \\to C ^ * _ E ( G ) = C ^ * ( G ) / I \\end{align*}"}
-{"id": "5523.png", "formula": "\\begin{align*} \\Lambda _ T ( u ) = \\frac { u } { \\sqrt { T } } \\int _ 0 ^ T \\gamma ( X _ t ) . d W _ t + \\frac { u ^ 2 } { 2 T } \\int _ 0 ^ T \\| \\gamma ( X _ t ) \\| ^ 2 d t + u r _ T , ~ u \\in \\mathbb R , \\end{align*}"}
-{"id": "582.png", "formula": "\\begin{align*} \\beta & = \\alpha _ 0 ^ { - 1 } ( \\alpha _ 0 + \\alpha _ 1 ) ^ 2 , \\\\ \\beta ' & = \\alpha _ 0 ^ { - 1 } \\alpha _ 1 ^ 2 . \\end{align*}"}
-{"id": "2345.png", "formula": "\\begin{align*} \\kappa : X & \\to \\mathcal { M } _ + ( Y \\times Z ) \\\\ \\kappa ( x ; d y , d z ) & = \\kappa _ 1 ( x , d y ) \\kappa _ 2 ( x , d z ) \\end{align*}"}
-{"id": "9962.png", "formula": "\\begin{align*} \\mathcal { D } _ { e q } ^ { X } ( t ) = \\exp \\left ( \\sum _ { g = 0 } ^ { \\infty } \\hbar ^ { g - 1 } F _ g ^ { X } ( t ) \\right ) \\end{align*}"}
-{"id": "2031.png", "formula": "\\begin{align*} h _ { \\delta } ( t ) = 0 \\end{align*}"}
-{"id": "5443.png", "formula": "\\begin{align*} \\sigma ( T ^ n ) = \\{ \\lambda _ i ^ n : i \\in \\N \\} . \\end{align*}"}
-{"id": "1425.png", "formula": "\\begin{align*} u _ { t } + \\Delta u + \\mathcal { H } ( t , x , m , D u ) = 0 , \\end{align*}"}
-{"id": "2119.png", "formula": "\\begin{align*} \\overline { a } _ { k k } : = \\lim _ { t \\to \\infty } \\frac { 1 } { t } \\int _ { 0 } ^ t a _ { k k } ( s ) d s , \\ ; \\ ; k = 1 , \\dots , d , \\end{align*}"}
-{"id": "6719.png", "formula": "\\begin{align*} \\bar { q } _ 1 + \\lambda \\dot { \\bar { q } } _ 2 - \\lambda ^ 2 \\ddot { \\bar { q } } _ 1 = 0 , \\bar { q } _ 2 - \\lambda \\dot { \\bar { q } } _ 1 - \\lambda ^ 2 \\ddot { \\bar { q } } _ 2 = 0 , \\end{align*}"}
-{"id": "9429.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\rightarrow 0 } \\int _ { \\gamma _ { \\tau } ( \\varepsilon ) } \\frac { 1 } { 4 8 \\pi } ( \\log _ { e } | F ( z ) | ^ 2 ) \\overline { g ( z ) } d \\overline { z } = 0 . \\end{align*}"}
-{"id": "8046.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & - 1 \\\\ - 1 & 0 \\end{pmatrix} \\begin{pmatrix} \\zeta ^ h & 0 \\\\ 0 & ( - 1 ) ^ h \\zeta ^ { - h } \\end{pmatrix} \\begin{pmatrix} 0 & - 1 \\\\ - 1 & 0 \\end{pmatrix} & = \\begin{pmatrix} ( - 1 ) ^ h \\zeta ^ { - h } & 0 \\\\ 0 & \\zeta ^ { h } \\end{pmatrix} = \\begin{pmatrix} \\zeta ^ { n - h } & 0 \\\\ 0 & \\zeta ^ { - n + h } \\end{pmatrix} . \\end{align*}"}
-{"id": "7295.png", "formula": "\\begin{align*} ( 1 - h _ q ^ - ) \\Phi _ p ( c ) k _ q \\approx _ { \\frac { \\gamma } { 1 8 M } } ( 1 - h _ q ^ - ) \\Phi _ q ( c ) k _ q = 0 \\end{align*}"}
-{"id": "3397.png", "formula": "\\begin{align*} \\rho ^ M _ { \\sup } ( T ) = \\frac { 1 } { 1 + \\prod _ { i } \\rho ^ { M - 1 } _ { \\inf } ( T _ i ) } \\ , \\ , \\rho ^ M _ { \\inf } ( T ) = \\frac { 1 } { 1 + \\prod _ { i } \\rho ^ { M - 1 } _ { \\sup } ( T _ i ) } . \\end{align*}"}
-{"id": "857.png", "formula": "\\begin{align*} \\norm { h ( x _ 1 ) - h ( x _ 2 ) } & \\leq ( \\norm { v } \\norm { \\hat u ^ \\ast } _ * + \\norm { \\hat v } \\norm { u ^ \\ast } _ * ) \\norm { x _ 1 - x _ 2 } = ( \\norm { v } + \\norm { u ^ \\ast } _ * ) \\norm { x _ 1 - x _ 2 } . \\end{align*}"}
-{"id": "7247.png", "formula": "\\begin{align*} \\phi = \\mu - \\sqrt { \\mu ^ { 2 } + \\hat { \\phi ^ 2 } ( 0 ) - 2 L _ r \\phi } = : \\tilde { F } ( \\phi , \\mu ) . \\end{align*}"}
-{"id": "6795.png", "formula": "\\begin{align*} \\mathcal { R } _ 1 \\quad \\equiv \\left \\{ \\partial _ j u ^ i + \\partial _ i u ^ j = 0 \\ , , \\right . 1 \\leq i , j \\leq n \\ , . \\end{align*}"}
-{"id": "2855.png", "formula": "\\begin{align*} \\alpha = \\frac { 1 } { d } \\left ( a + x \\xi + y \\xi ^ 2 + z \\xi ^ 3 \\right ) \\end{align*}"}
-{"id": "3683.png", "formula": "\\begin{align*} \\hat { Y } = \\sum _ { i \\in S } d _ i E \\big ( x _ i ; \\widehat { \\beta } ( B ) \\big ) \\end{align*}"}
-{"id": "2059.png", "formula": "\\begin{align*} A _ h ^ \\pm ( M ) = \\sum _ { n \\geq 1 } ( \\widehat { h } _ { \\delta } ( - n ) \\pm \\widehat { h } _ { \\delta } ( n ) ) \\frac { a ( n ) } { n } + \\mathcal { O } \\bigg ( ( M q ) ^ { \\epsilon } M ^ { - \\frac { 1 } { 4 } } q ^ { \\frac { 1 } { 4 } } \\prod _ { \\substack { p \\mid \\gcd ( q , M ) \\\\ p ^ 2 \\mid q } } p ^ { \\frac { 1 } { 2 } } \\bigg ) . \\end{align*}"}
-{"id": "7618.png", "formula": "\\begin{align*} ( L u ) ( x ) \\ = \\ \\int \\limits _ { \\mathbb R ^ d } a ( x - y ) \\mu ( x , y ) ( u ( y ) - u ( x ) ) d y \\end{align*}"}
-{"id": "6673.png", "formula": "\\begin{align*} & | \\psi _ { n } \\rangle = ( A ^ { \\dagger } ) ^ { n } | \\psi _ { 0 } \\rangle , | \\tilde { \\psi } _ { n } \\rangle = A | \\psi _ { n } \\rangle , | \\tilde { \\psi } _ { 0 } \\rangle = 0 , \\\\ & H _ { - } | \\psi _ { n } \\rangle = E _ { n } | \\psi _ { n } \\rangle , \\ H _ { + } | \\tilde { \\psi } _ { n } \\rangle = E _ { n } | \\tilde { \\psi } _ { n } \\rangle . \\end{align*}"}
-{"id": "7999.png", "formula": "\\begin{align*} \\max _ { x \\in Q _ 2 ( 0 ) } e ^ { - a | x - y | ^ 2 } & = \\max _ { x \\in Q _ 2 ( 0 ) } \\prod _ { i = 1 } ^ d e ^ { - a | x _ i - y _ i | ^ 2 } \\\\ & = \\prod _ { i = 1 } ^ d \\max _ { - 1 \\leq x _ i \\leq 1 } e ^ { - a | x _ i - y _ i | ^ 2 } = \\prod _ { i = 1 } ^ d e ^ { - a | x _ i ^ * - y _ i | ^ 2 } , \\end{align*}"}
-{"id": "467.png", "formula": "\\begin{align*} g _ k = & ( \\sum _ { s = 0 } ^ { d - a _ i } ( - 1 ) ^ s \\binom { d - a _ i } { s } y ^ s _ \\alpha y ^ { d - a _ i - s } _ \\beta ) ( \\sum _ { j = 0 } ^ { a _ i - k } \\mu _ j y ^ j _ \\alpha y ^ { a _ i - k - j } _ \\beta ) \\\\ = & \\sum _ { s = 0 } ^ { d - a _ i } \\sum _ { j = 0 } ^ { a _ i - k } ( - 1 ) ^ s \\mu _ j \\binom { d - a _ i } { s } y ^ { s + j } _ { \\alpha } y ^ { d - k - s - j } _ { \\beta } \\\\ = & \\sum _ { j = 0 } ^ { a _ i - k } \\sum _ { l = j } ^ { j + d - a _ i } ( - 1 ) ^ { l - j } \\mu _ j \\binom { d - a _ i } { l - j } y ^ l _ \\alpha y ^ { d - k - l } _ \\beta . \\end{align*}"}
-{"id": "9862.png", "formula": "\\begin{align*} C _ \\tau \\cap y _ 1 C _ \\tau = C _ \\tau \\cap y _ 1 C _ \\tau \\cap y _ 2 C _ \\tau = l _ \\gamma \\cup l _ \\gamma ^ { - 1 } , \\end{align*}"}
-{"id": "9679.png", "formula": "\\begin{align*} \\xi : = \\sum _ { n \\in \\mathbb N } \\xi _ n ^ 2 < \\infty , \\ \\ \\ \\ \\ \\ \\sum _ { n \\geq 2 } \\xi _ n ^ 2 < 1 \\end{align*}"}
-{"id": "9199.png", "formula": "\\begin{align*} \\mathrm { i n j } \\left ( \\sqrt { \\frac { 1 2 } { 9 } } \\mathbb { O } P ^ { 2 } \\right ) = \\frac { \\pi } { 2 } \\frac { \\sqrt { 1 2 } } { \\sqrt { 9 } } = \\frac { \\sqrt { 3 } } { 3 } \\pi > \\frac { \\pi } { 2 } . \\end{align*}"}
-{"id": "8079.png", "formula": "\\begin{align*} z M ( a + 1 , 2 , z ) = \\lim _ { b \\to 0 } \\ , \\frac { b } { a } M ( a , b , z ) . \\end{align*}"}
-{"id": "2653.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\Big ( ( \\delta I _ { n } + { C _ { 2 , 2 } } ) - { C _ { 2 , 1 } } ( \\delta I _ { N } + C _ { 1 , 1 } ) ^ { - 1 } { C _ { 1 , 2 } } \\Big ) ^ { - 1 } = \\alpha . \\end{align*}"}
-{"id": "6786.png", "formula": "\\begin{align*} \\varphi ( t , \\alpha ) = \\varphi ^ t ( \\alpha ) = M _ 1 ( t ) v ( \\alpha ) + M _ 2 ( t ) w ( \\alpha ) \\ , , \\end{align*}"}
-{"id": "9341.png", "formula": "\\begin{align*} \\int _ { M } \\langle \\psi _ { a } , \\mathbf { H } _ { a } \\rangle = - \\mathrm { V o l } ( M ) - \\frac { 1 } { n } \\int _ { M } \\| a ^ { T } \\| ^ 2 d V . \\end{align*}"}
-{"id": "1410.png", "formula": "\\begin{align*} { \\rm r e s } ( \\tilde { r } , a ) = \\frac { 1 } { z - \\bar { a } } u ( z ) | _ { z = a } = \\frac 1 { a - \\bar { a } } u ( a ) = \\overline { \\frac 1 { \\bar { a } - a } \\overline { u ( a ) } } = \\overline { \\frac { 1 } { \\bar { a } - a } { u ( \\bar { a } ) } } = \\overline { { \\rm r e s } ( \\tilde { r } , \\overline { a } ) } . \\end{align*}"}
-{"id": "9202.png", "formula": "\\begin{align*} \\tilde { N } \\equiv \\left \\{ \\left . \\left ( z _ { 1 } , 0 \\right ) \\in \\mathbb { C } ^ { 2 } \\right \\vert \\left \\vert z _ { 1 } \\right \\vert = 1 \\right \\} \\end{align*}"}
-{"id": "1573.png", "formula": "\\begin{align*} \\sup \\left | \\frac { \\partial \\psi _ w ^ { - 1 } } { \\partial z } ( z ) \\right | = O ( 1 / \\sqrt { w } ) \\end{align*}"}
-{"id": "5349.png", "formula": "\\begin{align*} A _ { \\alpha } = X _ { \\alpha } - Y _ { \\alpha } \\quad \\mbox { a n d } B _ { \\alpha } = i ( X _ { \\alpha } + Y _ { \\alpha } ) \\end{align*}"}
-{"id": "4445.png", "formula": "\\begin{align*} C _ 3 \\left [ \\left ( \\frac { m ^ 2 } { j } + m \\right ) \\left ( \\frac { m ^ 3 } { ( \\beta j ) ^ { 5 } } + \\frac { m } { \\beta j } \\right ) \\right ] = C _ 3 \\left ( \\frac { m ^ 5 } { j ^ { 6 } } + \\frac { m ^ 4 } { j ^ 5 } + \\frac { m ^ 3 } { j ^ 2 } + \\frac { m ^ 2 } { j } \\right ) \\end{align*}"}
-{"id": "4036.png", "formula": "\\begin{align*} I ( P , V ) & = O \\left ( m ^ { 2 / 3 } \\left ( m + \\sum _ { x \\in P } i ( x ) ( i ( x ) - 1 ) ( i ( x ) - 2 ) \\right ) ^ { 1 / 3 } \\right ) \\\\ & = O \\left ( m + m ^ { 2 / 3 } \\left ( \\sum _ { x \\in P } i ( x ) ( i ( x ) - 1 ) ( i ( x ) - 2 ) \\right ) ^ { 1 / 3 } \\right ) . \\end{align*}"}
-{"id": "226.png", "formula": "\\begin{align*} & X _ i = \\partial _ { z _ i } , \\ i = 1 , . . . , N . \\\\ & Y _ j = z _ j \\partial _ t , \\ j = 1 , . . . , N . \\end{align*}"}
-{"id": "5854.png", "formula": "\\begin{align*} R ( f , f ' ) & = ( - 1 ) ^ { \\frac 1 2 n ( n - 1 ) } \\Delta ( f ) ; \\\\ R ( f , g _ 1 g _ 2 ) & = R ( f , g _ 1 ) R ( f , g _ 2 ) ; \\\\ R ( f _ 1 f _ 2 , g ) & = R ( f _ 1 , g ) R ( f _ 2 , g ) . \\end{align*}"}
-{"id": "1099.png", "formula": "\\begin{align*} & \\sum _ { i _ 1 , \\cdots , i _ { k } \\in \\Lambda ( I ) \\ } \\prod _ { j = 1 } ^ k ( { \\tau } ^ * _ { i _ j } - x _ j ) _ + \\\\ = & ( n S ( I ) / 4 ) ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } \\sum _ { i _ 1 , \\cdots , i _ { k } \\in \\Lambda ( I ) \\ } \\prod _ { j = 1 } ^ k ( { m } ^ * _ { i _ j } - G _ n ( x _ j ) / S ( I ) ) _ + \\\\ = & ( n S ( I ) / 4 ) ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } | \\Sigma _ k ( G _ n ( x _ 1 ) / S ( I ) , \\cdots , G _ n ( x _ k ) / S ( I ) ) | . \\end{align*}"}
-{"id": "3029.png", "formula": "\\begin{align*} ( L ^ m _ { p , q } f \\ast \\Phi _ i ) ( z ) = z + \\sum \\limits _ { k = 2 } ^ \\infty \\lambda _ k [ k ] _ { p , q } ^ m a _ { k } { z ^ k } + ( - 1 ) ^ { m + i } \\sum \\limits _ { k = 1 } ^ \\infty \\mu _ k [ k ] _ { p , q } ^ m b _ { k } { \\overline { z } ^ k } , \\end{align*}"}
-{"id": "3277.png", "formula": "\\begin{align*} \\frac { d } { d X } V _ n ( X , Y ) = n U _ n ( X , Y ) . \\end{align*}"}
-{"id": "8041.png", "formula": "\\begin{align*} y ^ r x y ^ k y ^ { - r } = x y ^ { k + r ( n - 2 ) } = x y ^ { k + 4 r s - 2 r } \\end{align*}"}
-{"id": "3499.png", "formula": "\\begin{align*} f _ { k } ( x ) = \\oint _ { \\Sigma _ { - \\infty } } \\frac { d t } { 2 \\pi i } \\frac { \\Gamma ( t ) } { t + k } e ^ { - \\frac { \\gamma } { 2 } t ^ { 2 } + x t } , g _ { k } ( x ) = \\int ^ { 1 + i \\infty } _ { 1 - i \\infty } \\frac { d s } { 2 \\pi i } \\frac { 1 } { ( s + k ) \\Gamma ( s ) } e ^ { \\frac { \\gamma } { 2 } s ^ { 2 } - x s } , \\end{align*}"}
-{"id": "8262.png", "formula": "\\begin{align*} \\begin{cases} z _ { t t } - i a z _ { \\alpha } = - i \\\\ \\frac { d } { d t } z _ j ( t ) = ( v - \\frac { \\lambda _ j i } { 2 \\pi ( \\overline { z - z _ j } ) } ) \\Big | _ { z = z _ j } \\\\ ( I - \\mathfrak { H } ) \\Big ( \\bar { z } _ t + \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i } { 2 \\pi ( z ( \\alpha , t ) - z _ j ( t ) ) } \\Big ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "4897.png", "formula": "\\begin{align*} t \\circ f \\circ c _ s ^ I = t \\circ i _ { s , t } = s = s _ U \\circ c _ s ^ I \\end{align*}"}
-{"id": "7577.png", "formula": "\\begin{align*} ( \\Delta g _ { \\nu ( t - s ) } ) = ( 4 \\pi \\nu ( t - s ) ) ^ { - ( \\frac { d } { 2 } + 1 ) } S \\left ( \\frac { x } { ( 4 ( t - s ) ) ^ { \\frac { 1 } { 2 } } } \\right ) . \\end{align*}"}
-{"id": "697.png", "formula": "\\begin{align*} D e t ( g _ { i \\bar { j } } ) = \\frac { Z ( X ) } { ( 1 - 4 p \\pi _ 1 ^ \\mathbb { R } ) ^ { \\frac { 3 K } { p } } } , \\end{align*}"}
-{"id": "9821.png", "formula": "\\begin{align*} D ' ( r ) = \\frac { n + a - 1 } { r } D ( r ) + \\frac { 2 } { r } \\int _ { B _ r } \\nabla u \\cdot \\nabla ( X \\cdot \\nabla u ) | y | ^ a . \\end{align*}"}
-{"id": "3228.png", "formula": "\\begin{align*} \\| \\Delta _ 1 w ^ k \\| = O ( \\| { w } ^ { k + \\frac { 1 } { 2 } } - { w } ^ { \\ast } \\| ^ { 1 + \\tilde { c } \\alpha } ) . \\end{align*}"}
-{"id": "5001.png", "formula": "\\begin{align*} & { \\rm A v g } _ { \\rm p r i m e } \\Phi ( \\log L _ q ( { \\rm S y m } _ f ^ { \\mu } , \\sigma ) - \\log L _ q ( { \\rm S y m } _ f ^ { \\nu } , \\sigma ) ) \\\\ & = { \\rm A v g } _ { \\rm p o w e r } \\Phi ( \\log L _ q ( { \\rm S y m } _ f ^ { \\mu } , \\sigma ) - \\log L _ q ( { \\rm S y m } _ f ^ { \\nu } , \\sigma ) ) \\\\ & = \\int _ { \\mathbb { R } } \\mathcal { M } _ { \\sigma } ( u ) \\Phi ( u ) \\frac { d u } { \\sqrt { 2 \\pi } } \\end{align*}"}
-{"id": "31.png", "formula": "\\begin{align*} \\pi _ t ^ { i , \\ast } = _ { \\Pi ^ i } \\left ( \\frac { \\mathcal { Z } _ t ^ { i } + \\theta ^ { i } ( V _ t ) } { 1 - \\delta } \\right ) \\end{align*}"}
-{"id": "9803.png", "formula": "\\begin{align*} \\bigcup _ { j = 1 } ^ { \\infty } K _ j = E _ { h , m } \\setminus \\bigcap _ { j = 1 } ^ { \\infty } \\mathcal { O } _ j . \\end{align*}"}
-{"id": "4673.png", "formula": "\\begin{align*} \\bar \\pi _ { n , g , h } = \\sum _ { j = 0 } ^ { g } \\sum _ { \\substack { k _ { 2 } ' , \\dots , k _ { h } ' , k _ { 2 } '' , \\ldots , k _ { n - h } '' \\\\ \\sum _ { i } i k _ { i } ' = j , \\ \\sum _ { i } i k _ { i } '' = g - j } } \\ ! \\ ! \\ ! \\ ! ( { e _ { 2 } ' } ^ { k _ { 2 } ' } \\cdots { e _ { h } ' } ^ { k _ { h } ' } \\otimes { e _ { 2 } '' } ^ { k _ { 2 } '' } \\cdots { e _ { n - h } '' } ^ { k _ { n - h } '' } ) U _ { k _ { 2 } ' \\ldots , k _ { h } ' } U _ { k _ { 2 } '' , \\ldots , k _ { n - h } '' } \\end{align*}"}
-{"id": "21.png", "formula": "\\begin{align*} | \\mathbf { y } ^ { i } ( v ) | & \\leq C _ y ( 1 + | v | ) , \\\\ | \\mathbf { z } ^ { i } ( v ) | & \\leq K _ z = \\frac { C _ v } { C _ { \\eta } - C _ v } , \\\\ | \\mathbf { y } ^ { i } ( v ) - \\mathbf { y } ^ { j } ( v ) | & \\leq \\frac { 1 } { q ^ { \\min } } \\left ( K _ f + \\frac { C _ v C _ { \\eta } C _ z } { ( C _ { \\eta } - C _ v ) ^ 2 } \\right ) , \\end{align*}"}
-{"id": "1673.png", "formula": "\\begin{align*} \\rho _ A ( X ) ( f g ) = \\varphi ^ * ( g ) \\rho _ A ( X ) f + \\varphi ^ * ( f ) \\rho _ A ( X ) g . \\end{align*}"}
-{"id": "4931.png", "formula": "\\begin{align*} x ^ { n + 1 } + y z ^ n = ( x ^ { n + 1 } - x z ^ { n ^ 4 - n } + y z ^ n ) + z ^ { n ^ 4 - 4 n } ( x z ^ { 3 n } ) \\end{align*}"}
-{"id": "2648.png", "formula": "\\begin{align*} g _ { j k } = \\begin{cases} \\int _ { \\Omega } \\phi _ { j } ( x ) \\epsilon \\phi _ { k } ( x ) d \\nu ( x ) , & 1 \\leq j \\leq N , 1 \\leq k \\leq M , \\\\ - \\int _ { \\Omega } \\epsilon \\phi _ { j } ( x ) \\phi _ { k } ( x ) d \\nu ( x ) , & N < j \\leq M , 1 \\leq k \\leq N , \\\\ \\alpha _ { j k } , & N < j , k \\leq M . \\end{cases} \\end{align*}"}
-{"id": "2738.png", "formula": "\\begin{align*} \\max _ { z \\in \\mathcal { Q } } \\Re { f _ { 1 } ( z ) } - \\Re { f ( z _ { 0 } ) } = \\Re { f ( z _ { 4 } ) } - \\Re { f ( z _ { 0 } ) } \\leq - 2 N ^ { - \\frac { 1 } { 6 } } . \\end{align*}"}
-{"id": "5615.png", "formula": "\\begin{align*} \\int _ 0 ^ { + \\infty } v _ { x x x } v d x = - \\int _ 0 ^ { + \\infty } v _ { x x } v _ x d x - v _ { x x } ( 0 , t ) v ( 0 , t ) = \\frac 1 2 v _ x ^ 2 ( 0 , t ) - v _ { x x } ( 0 , t ) v ( 0 , t ) \\end{align*}"}
-{"id": "2701.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\phi } \\Re { f ( 4 ; R e ^ { i \\phi } ) } & = - \\Big ( \\frac { 4 \\tau } { 1 - \\tau ^ { 2 } } + \\frac { \\tau } { 1 - 2 \\tau R \\cos \\phi + \\tau ^ 2 R ^ { 2 } } - \\frac { \\tau } { \\tau ^ { 2 } - 2 \\tau R \\cos \\phi + R ^ { 2 } } \\Big ) R \\sin \\phi \\\\ & \\begin{cases} < 0 , \\phi > 0 ; \\\\ > 0 , \\phi < 0 . \\end{cases} \\end{align*}"}
-{"id": "8822.png", "formula": "\\begin{align*} \\mathbb P ( i _ { k + 1 } = i ) = \\begin{cases} \\frac { 1 } { n } & | I _ { \\mathrm { k i n k } } | = n \\\\ \\frac { 1 } { 2 n } & | I _ { \\mathrm { k i n k } } | < n i \\in I _ { \\mathrm { k i n k } } \\\\ \\frac { 1 } { 2 n } + \\frac { 1 } { 2 ( n - | I _ { \\mathrm { k i n k } } | ) } & | I _ { \\mathrm { k i n k } } | < n i \\not \\in I _ { \\mathrm { k i n k } } \\end{cases} \\end{align*}"}
-{"id": "6372.png", "formula": "\\begin{align*} w ( t , x ) : = ( 1 - \\delta ) \\varphi ( t , x ) + \\delta \\phi ( x ) - \\psi ( t , x ) - 3 \\varepsilon t , \\ t \\in [ 0 , T ' ] , x \\in \\Omega . \\end{align*}"}
-{"id": "8130.png", "formula": "\\begin{align*} u = S ( b _ i ) a _ i = b _ i S ^ { - 1 } ( a _ i ) , \\ : \\ : \\ : \\ : \\ : u ^ { - 1 } = S ^ { - 2 } ( b _ i ) a _ i = S ^ { - 1 } ( b _ i ) S ( a _ i ) = b _ i S ^ 2 ( a _ i ) . \\end{align*}"}
-{"id": "1885.png", "formula": "\\begin{align*} g ( t ) = \\left ( \\frac { 1 } { 2 } t - 1 \\right ) + \\left ( \\frac { 1 } { 2 } t + 1 \\right ) e ^ { - t } \\end{align*}"}
-{"id": "6909.png", "formula": "\\begin{align*} \\det ( d \\varphi ) = g _ 1 + g _ 2 + g _ 3 - g _ 4 + p _ 5 ( g _ 4 + g _ 5 ) + g _ 6 \\ , . \\end{align*}"}
-{"id": "4723.png", "formula": "\\begin{align*} u ( x , t ) = \\left \\{ \\begin{array} { l l } 0 & ~ ~ ~ x < \\dfrac { ( ( \\sqrt { 6 } - 2 ) t + 1 ) } { 4 } , \\medskip \\\\ \\dfrac { ( x - 0 . 2 5 ) } { t } + \\dfrac { 1 } { 2 } & ~ ~ ~ \\dfrac { ( ( \\sqrt { 6 } - 2 ) t + 1 ) } { 4 } < x < \\dfrac { 2 t + 1 } { 4 } , \\medskip \\\\ 1 & ~ ~ ~ x > \\dfrac { 2 t + 1 } { 4 } . \\end{array} \\right . \\end{align*}"}
-{"id": "471.png", "formula": "\\begin{align*} H & = \\prod ^ { 2 d - 1 } _ { j = 0 } ( \\xi ^ { j } x + \\xi ^ { - j } y + ( - 1 ) ^ { j } z ) \\\\ & = \\prod ^ { d - 1 } _ { j = 0 } ( \\xi ^ { j } x + \\xi ^ { - j } y + ( - 1 ) ^ { j } z ) ( \\xi ^ { j + d } x + \\xi ^ { - j + d } y + ( - 1 ) ^ { j + d } z ) \\\\ & = \\prod ^ { d - 1 } _ { j = 0 } ( \\xi ^ { j } x + \\xi ^ { - j } y + ( - 1 ) ^ { j } z ) ( \\xi ^ { j } x + \\xi ^ { - j } y + ( - 1 ) ^ { j + d } z ) \\\\ & = \\prod ^ { d - 1 } _ { j = 0 } ( \\xi ^ { j } x + \\xi ^ { - j } y - z ) ( \\xi ^ { j } x + \\xi ^ { - j } y + z ) = F . \\end{align*}"}
-{"id": "4368.png", "formula": "\\begin{align*} \\frac { { n - t - k - 1 \\choose k - t - 1 } } { { n - t - 1 \\choose k - t - 1 } } = \\frac { n - 2 k + 1 } { n - t - k } \\cdot \\frac { { n - t - k \\choose k - t - 1 } } { { n - t - 1 \\choose k - t - 1 } } = \\ldots = \\prod _ { i = 1 } ^ { k } \\frac { n - k + 1 - i } { n - t - i } . \\end{align*}"}
-{"id": "4589.png", "formula": "\\begin{align*} \\hat { f } _ { g , \\delta , j } = \\hat { \\nu } _ g \\hat { \\phi ^ D _ \\delta } \\psi ( 2 ^ { - j } . ) \\end{align*}"}
-{"id": "9548.png", "formula": "\\begin{align*} \\mathrm { s p t } ( n , N ) = \\sum _ { j = 0 } ^ { n - 1 } p ( j , N ) \\sigma ( n - j , N ) - \\frac { 1 } { 2 } N _ { 2 , N } ( n ) , \\end{align*}"}
-{"id": "714.png", "formula": "\\begin{align*} & R _ { 1 \\bar { 1 } 1 \\bar { 1 } } = - 3 2 p ^ 3 K , \\\\ & R _ { 1 \\bar { 2 } 1 \\bar { 2 } } = ( p - 1 ) f ^ { ( 1 ) } ( 0 ) , \\\\ & R _ { 1 \\bar { 1 } 2 \\bar { 2 } } = - p f ^ { ( 1 ) } ( 0 ) , \\\\ & R _ { 2 \\bar { 2 } 2 \\bar { 2 } } = \\left ( - 3 + \\frac { 1 } { K } \\right ) \\frac { f ^ { ( 1 ) } ( 0 ) ^ 2 } { 1 6 } , \\end{align*}"}
-{"id": "5890.png", "formula": "\\begin{align*} g = \\sum _ { w \\in \\tilde { W } } ( - 1 ) ^ { \\ell ( w ) } q ^ { - \\ell ( w ) } T _ w = \\sum _ { w \\in \\tilde { W } } ( - 1 ) ^ { \\ell ( w ) } q ^ { - \\ell ( w ) } \\left ( \\sum _ { y \\leq w } q ^ { \\frac { \\ell ( y ) } { 2 } } ( - 1 ) ^ { \\ell ( w ) - \\ell ( y ) } C ' _ y \\right ) , \\end{align*}"}
-{"id": "2511.png", "formula": "\\begin{align*} H _ { n } = { \\displaystyle \\int \\limits _ { 0 } ^ { 1 } } \\frac { 1 - x ^ { n } } { 1 - x } d x . \\end{align*}"}
-{"id": "3808.png", "formula": "\\begin{align*} \\mathfrak { I } = \\int W ( w ) I ( m , N _ 0 w , q ) \\overline { I ( m ' , N _ 0 w , q ' ) } \\ : e \\left ( - \\frac { N _ 0 n _ 1 n _ 2 w } { q _ 2 q _ 2 ' q _ 1 r } \\right ) \\mathrm { d } w . \\end{align*}"}
-{"id": "8501.png", "formula": "\\begin{align*} \\mathcal { T } = [ 0 , T _ 0 ] , \\end{align*}"}
-{"id": "4942.png", "formula": "\\begin{align*} v _ 1 z ^ { 3 n } \\equiv \\left ( \\sum _ { j = 2 } ^ { d - 3 } ( - 1 ) ^ { j } v _ j z ^ { 3 n } \\right ) + ( - 1 ) ^ { d } ( x z ^ { 3 n } ) \\end{align*}"}
-{"id": "9825.png", "formula": "\\begin{align*} \\liminf _ { \\ell \\to \\infty } W _ \\mu ( r _ \\ell , u ) \\geq \\liminf _ { \\ell \\to \\infty } - \\mu H _ \\mu ( r _ \\ell , u ) = 0 . \\end{align*}"}
-{"id": "8605.png", "formula": "\\begin{align*} \\left | u _ \\Lambda ( x ) - \\bar { u } \\right | \\cdot \\left | u _ \\Lambda ( x ) - \\bar { v } \\right | + \\left | u _ \\Lambda ' ( x ) \\right | = O ( 1 ) e ^ { - \\left ( \\sqrt { 2 } \\sqrt { 1 - 4 c _ 0 ^ 2 } + O \\left ( \\frac { 1 } { \\sqrt { \\Lambda } } \\right ) \\right ) | x | } , \\end{align*}"}
-{"id": "2938.png", "formula": "\\begin{align*} \\| \\widetilde { U } _ { 1 } ^ { \\ast } U _ { 2 } \\| _ { F } ^ { 2 } = \\| E B ^ { \\dagger } \\| _ { F } ^ { 2 } - \\| A A ^ { \\dagger } E B ^ { \\dagger } \\| _ { F } ^ { 2 } . \\end{align*}"}
-{"id": "9809.png", "formula": "\\begin{align*} \\mathcal { I } _ a ( v ) : = \\mathcal { F } _ a ( \\bar { v } ) . \\end{align*}"}
-{"id": "2572.png", "formula": "\\begin{align*} \\dot { \\mathbf P } _ { \\delta _ x } ^ { ( \\phi ) } [ Y _ t ( \\phi ) ^ { - 1 } ] = \\dot { \\mathbf P } _ \\nu ^ { ( \\phi ) } [ Y ^ { ( t _ 0 , t ] } _ t ( \\phi ) ^ { - 1 } ] + \\epsilon _ x ^ 1 ( t _ 0 , t ) + \\epsilon _ x ^ 2 ( t _ 0 , t ) , \\end{align*}"}
-{"id": "7188.png", "formula": "\\begin{gather*} \\sup \\left \\{ \\mathrm { e } ^ { \\alpha t } \\left \\Vert \\frac { \\partial y _ { 1 } ( t , \\eta ) } { \\partial \\eta } a - \\frac { \\partial y _ { 2 } ( t , \\eta ) } { \\partial \\eta } a \\right \\Vert : ( t , \\eta , a ) \\in \\mathbb { R } _ { + } \\times \\mathbb { L } _ { \\xi } ^ { - } \\times \\mathbb { L } _ { \\xi } ^ { - } , \\left \\Vert \\eta \\right \\Vert \\le r , \\left \\Vert a \\right \\Vert = r \\right \\} . \\end{gather*}"}
-{"id": "3711.png", "formula": "\\begin{align*} \\begin{cases} E ( z _ B ) = \\sum _ { i \\in U } p _ i z _ i = \\sum _ x p ( x ; \\lambda ) \\sum _ { i \\in U _ x } z _ i = \\sum _ x p ( x ; \\lambda ) N _ z \\bar { Z } _ x \\\\ Z = E ( \\sum _ { i \\in U } \\delta _ i z _ i / p _ i ) = E [ \\sum _ x n _ { x B } \\bar { z } _ { x B } / p ( x ; \\lambda ) ] \\end{cases} \\end{align*}"}
-{"id": "6141.png", "formula": "\\begin{align*} ( 2 a ^ 2 c - b ^ 2 ( 2 m - 2 ) ) \\phi ( v ) = 2 a ^ 2 u + b ^ 2 ( 2 m - 2 ) v + 2 a b \\delta . \\end{align*}"}
-{"id": "1568.png", "formula": "\\begin{align*} \\sup \\left | \\Phi _ { w _ { m + 1 } } \\circ \\widetilde { F } _ { w _ m } ( z , x ) - \\Phi _ { w _ m } ( z , x ) - \\frac { \\sqrt { w _ m } } { 2 } \\right | = o ( w _ m ) . \\end{align*}"}
-{"id": "7720.png", "formula": "\\begin{align*} \\varphi ( K ) = \\bigcap _ { k = 1 } ^ { \\infty } \\tilde { V } _ { k } , \\tilde { V } _ { k } : = \\bigcup _ { x \\in \\varphi ( K ) } B _ { \\frac { 1 } { k } } ( x ) . \\end{align*}"}
-{"id": "580.png", "formula": "\\begin{align*} \\alpha _ 0 + 2 \\alpha _ 1 = \\beta - \\beta ' . \\end{align*}"}
-{"id": "1602.png", "formula": "\\begin{align*} m ' ( x ) = \\frac { 2 } { \\sigma ^ 2 ( x ) S ' ( x ) } . \\end{align*}"}
-{"id": "5203.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow 0 } u ( \\gamma s ) / u ( s ) = \\gamma ^ { \\rho } . \\end{align*}"}
-{"id": "7197.png", "formula": "\\begin{gather*} \\left \\Vert \\chi ^ { t } \\circ \\chi ^ { s } \\left ( \\xi + \\eta + h ( \\xi , \\eta ) \\right ) - \\chi ^ { t } \\circ \\chi ^ { s } ( \\xi ) \\right \\Vert = O \\left ( \\mathrm { e } ^ { - \\alpha ( t + s ) } \\right ) , t \\to \\infty , \\end{gather*}"}
-{"id": "8354.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j } { \\pi } \\frac { 1 } { ( 0 - z _ j ( t ) ) ^ 2 } = \\frac { \\lambda } { \\pi } \\frac { 4 x y i } { ( x ^ 2 + y ^ 2 ) ^ 2 } . \\end{align*}"}
-{"id": "9487.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } ( - \\Delta _ { p } ) ^ { s } u + ( - \\Delta _ { q } ) ^ { s } u = f \\ , \\ , \\ , \\ , \\mbox { i n } \\ , \\ , \\ , \\ , \\mathbb { R } ^ { N } \\\\ u ( x ) \\geq 0 , \\ , \\ , \\ , \\ , x \\in \\mathbb { R } ^ { N } \\end{array} \\right . \\end{align*}"}
-{"id": "6833.png", "formula": "\\begin{align*} \\det ( d \\varphi ) = u ^ 1 _ { 1 0 } u ^ 2 _ { 0 1 } - u ^ 1 _ { 0 1 } u ^ 2 _ { 1 0 } + u ^ 3 _ { 1 0 } u ^ 4 _ { 0 1 } - u ^ 3 _ { 0 1 } u ^ 4 _ { 1 0 } + ( c _ 1 c _ 2 + s _ 1 s _ 2 ) q _ 1 + ( s _ 1 c _ 2 - s _ 2 c _ 1 ) q _ 2 \\ , , \\end{align*}"}
-{"id": "4107.png", "formula": "\\begin{align*} \\beta = \\alpha + \\frac { 1 } { 4 } . \\end{align*}"}
-{"id": "7812.png", "formula": "\\begin{align*} | [ \\langle x , y \\rangle \\xi , \\xi ] | = \\| x \\| \\| y \\| . \\end{align*}"}
-{"id": "6367.png", "formula": "\\begin{align*} ( \\varepsilon _ 1 \\theta + d d ^ c \\rho ) ^ n = e ^ { c _ 1 } g d V , \\end{align*}"}
-{"id": "2908.png", "formula": "\\begin{align*} \\Psi _ \\mu ( F _ n \\ , | \\ , F _ n ^ { \\complement } ) & = \\int \\Big [ H _ { K _ { F _ n } ( ( \\theta , x ) , \\cdot ) } ( \\xi ^ { F _ n } ) - K _ { F _ n } \\big ( ( \\theta , x ) , E _ { F _ n | F _ n ^ \\complement } \\big ) \\Big ] \\dd \\mu ( \\theta , x ) \\\\ & = \\int \\log Z _ { F _ n | F _ n ^ \\complement } ( \\theta , x ) \\ ; \\dd \\mu ( \\theta , x ) \\ ; . \\end{align*}"}
-{"id": "3780.png", "formula": "\\begin{align*} p ' ( \\alpha ) & = p ' _ 1 p ' _ 2 \\cdots p ' _ { R - k } p ' _ { R - k + 1 } = ( p _ 1 - 1 ) p _ 2 \\cdots p _ { R - k } ( p _ { R - k + 1 } - 1 ) \\hbox { a n d } \\\\ q ' ( \\alpha ) & = q ' _ 0 q ' _ 1 \\cdots , \\hbox { w h e r e } q ' _ j = | \\{ i : \\ p ' _ i = j \\} | . \\end{align*}"}
-{"id": "10057.png", "formula": "\\begin{align*} \\Phi _ t ( K _ { \\theta _ n ^ 0 , G _ n ^ 0 } ( u , \\varphi ) ) = K _ { \\theta _ n ^ 0 , G _ n ^ 0 } ( \\tilde \\Phi _ t ( u ; \\theta _ n ^ 0 , G _ n ^ 0 ) , \\varphi + \\omega t ) , t \\ge 0 , \\end{align*}"}
-{"id": "3461.png", "formula": "\\begin{align*} \\Re ( M + 1 ) \\big ( F ( s _ { \\pm } ; x _ { N } ( k ) ) - F ( 1 - k ; x _ { N } ( k ) ) \\big ) = \\frac { 1 } { 2 } N ^ { 1 / 4 } ( M + 1 ) ^ { 1 / 4 } ( 1 + o ( 1 ) ) . \\end{align*}"}
-{"id": "2974.png", "formula": "\\begin{align*} ( q ^ 5 ; q ^ 5 ) _ \\infty & = 1 + \\sum _ { n = 1 } ^ \\infty ( - 1 ) ^ n ( q ^ { 5 P _ { 5 , n } } + q ^ { 5 Q _ { 5 , n } } ) . \\end{align*}"}
-{"id": "3429.png", "formula": "\\begin{gather*} \\tfrac { 1 } { 2 } ( \\omega ( u , v ) - \\omega ( v , u ) ) = \\omega ( u , v ) - \\tfrac { 1 } { 2 } ( \\omega ( u + v , u + v ) - \\omega ( u , u ) - \\omega ( v , v ) ) \\end{gather*}"}
-{"id": "4370.png", "formula": "\\begin{align*} { k \\choose t } { n - k + t + 2 \\choose t + 2 } = & \\ \\frac { k ! ( n - k + t + 2 ) ! } { ( k - t ) ! t ! ( t + 2 ) ! ( n - k ) ! } \\\\ \\le & \\ \\Big ( \\frac { n - k - 2 } { n - k - t - 4 } \\Big ) ^ { t + 4 } \\cdot \\frac { k ! ( n - k - 2 ) ! } { ( k - t ) ! ( 2 t + 2 ) ! ( n - k - t - 4 ) ! } \\cdot { 2 t + 2 \\choose t } \\\\ \\le & \\ \\Big ( \\frac 5 4 \\Big ) ^ { t + 4 } \\frac { ( n - k - 2 ) ! } { ( 2 t + 2 ) ! ( n - k - 2 t - 4 ) ! } \\cdot 2 ^ { 2 t } \\\\ \\le & \\ 5 ^ { t + 1 } { n - k - 2 \\choose 2 t + 2 } . \\end{align*}"}
-{"id": "4408.png", "formula": "\\begin{align*} \\mu ( x , y ) = \\frac { 1 } { \\pi r ^ 2 } \\int _ { D _ r ( x ) } \\log ( 1 + | 2 \\xi y | ^ 2 + | y ^ 2 | ^ 2 ) d A ( \\xi ) . \\end{align*}"}
-{"id": "4210.png", "formula": "\\begin{align*} R ( x _ n ) - R ( y _ n ) & = \\frac { 1 } { 2 \\pi i } \\oint _ { | z | = r _ n } \\left ( \\frac { R ( z ) - \\mathbb { I } } { z - x _ n } - \\frac { R ( z ) - \\mathbb { I } } { z - y _ n } \\right ) d z \\\\ & = \\frac { x _ n - y _ n } { 2 \\pi i } \\oint _ { | z | = r _ n } \\frac { R ( z ) - \\mathbb { I } } { ( z - x _ n ) ( z - y _ n ) } d z \\end{align*}"}
-{"id": "4821.png", "formula": "\\begin{align*} \\frac { d } { d t } \\lambda _ 1 ( \\mathcal B ) \\Big | _ { t = 0 } = \\int _ { M } | \\Pi _ { g } | ^ 2 d \\sigma > 0 , \\end{align*}"}
-{"id": "1364.png", "formula": "\\begin{align*} \\partial _ { n } \\otimes \\operatorname { i d } ( \\lambda , a ) = \\sum _ { i = 1 } ^ n ( - 1 ) ^ { i } \\left ( ( F _ i ^ 0 ( \\lambda ) , S _ i ( \\lambda ) \\cdot a ) - ( F _ i ^ 1 ( \\lambda ) , a ) \\right ) , \\end{align*}"}
-{"id": "543.png", "formula": "\\begin{align*} \\begin{aligned} u ^ h ( x ' , x _ 3 ) = & \\ , \\ , y _ 0 ( x ' ) + h v ^ h ( x ' ) + h ^ 2 w ^ h ( x ' ) + x _ 3 \\vec b _ 0 ( x ' ) + h ^ 2 \\vec d _ 0 \\big ( x ' , \\frac { x _ 3 } { h } \\big ) \\\\ & + h ^ 3 \\vec k _ 0 \\big ( x ' , \\frac { x _ 3 } { h } \\big ) + h x _ 3 \\vec p ^ { \\ , h } ( x ' ) + h ^ 2 x _ 3 \\vec q ^ { \\ , h } ( x ' ) + h ^ 3 \\vec r ^ { \\ , h } \\big ( x ' , \\frac { x _ 3 } { h } \\big ) \\mbox { f o r a l l } \\ , ( x ' , x _ 3 ) \\in \\Omega ^ h . \\end{aligned} \\end{align*}"}
-{"id": "7328.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\frac { 1 } { \\rho } \\operatorname { d i v } ( K ( x ) \\nabla u + a ( x ) \\nabla w ) = { } & u - p - r \\in L ^ 2 ( \\Omega ) , \\\\ \\frac { 1 } { \\rho } \\operatorname { d i v } ( K ( x ) \\nabla v + b ( x ) \\nabla z ) = { } & v - q - s \\in L ^ 2 ( \\Omega ) , \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "5487.png", "formula": "\\begin{align*} \\{ \\Phi _ { l , r } : r = 0 , & . . , \\max ( 0 , 2 ^ { l d } - 1 ) , ~ l = \\{ - 1 , 0 \\} \\cup \\mathbb N \\} , \\\\ & V _ J \\equiv \\textrm { s p a n } ( \\Phi _ { l , r } : r , l \\le J ) , \\end{align*}"}
-{"id": "3211.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\displaystyle { \\mathop { \\rm M a x i m i z e } _ { X \\in S ^ m } } & & A _ 0 \\bullet X \\\\ \\mbox { s u b j e c t t o } & & A ( \\tau ) \\bullet X \\ge 0 \\ ( \\tau \\in T ) \\\\ & & I \\bullet X = 1 \\\\ & & X \\in S ^ m _ + , \\end{array} \\end{align*}"}
-{"id": "5921.png", "formula": "\\begin{align*} { \\rm c o e f f } _ { D _ i } ( \\Gamma _ i ) = { \\rm g l c t } ( K _ { X _ i } + \\Gamma ' _ i + M _ i \\mid D _ i ) , \\end{align*}"}
-{"id": "6498.png", "formula": "\\begin{align*} r _ { n , p } = { \\pi ^ { - \\frac { n } { 2 p } } \\left ( \\frac { 2 \\Gamma \\left ( \\frac { 1 } { 2 } ( n + p + 2 ) \\right ) } { \\Gamma \\left ( \\frac { n + 2 } { 2 } \\right ) \\Gamma \\left ( \\frac { p + 1 } { 2 } \\right ) } \\right ) ^ { - \\frac { n } { p } } } . \\end{align*}"}
-{"id": "109.png", "formula": "\\begin{align*} \\boxed { S _ k ( B ( 0 ) , B _ 1 ) = \\sum _ { j = \\frac { k - 1 } 2 } ^ { 2 N - \\frac { k + 1 } 2 } \\begin{pmatrix} j \\\\ \\frac { k - 1 } 2 \\end{pmatrix} \\begin{pmatrix} 2 N - j - 1 \\\\ \\frac { k - 1 } 2 \\end{pmatrix} B ( 0 ) ^ { 2 j } B _ 2 ( 0 ) k = 1 , 3 , \\dots , 2 N - 1 \\ , . } \\end{align*}"}
-{"id": "5873.png", "formula": "\\begin{align*} & a _ 1 = - a b a - ( a y + b ' ) ( X + a ' a ) , \\\\ & X _ 1 = X - ( X b + a ' ) a - ( X Y + a ' b ' + 1 ) ( X + a a ' ) \\\\ & a _ 1 ' = X b - ( X Y + a ' b ' ) a ' . \\end{align*}"}
-{"id": "1916.png", "formula": "\\begin{align*} F _ p ( x ) \\equiv x ^ 2 \\sum _ { j = 1 } ^ \\infty \\frac { A _ j ^ 2 } { Z _ j ^ p } , \\end{align*}"}
-{"id": "3345.png", "formula": "\\begin{align*} \\left | [ u ^ n ] \\widetilde { Z } ( u ) \\right | \\le \\left | [ u ^ n ] \\exp ( \\sum _ { k \\ge 1 } 2 0 ( r + 1 ) \\frac { u ^ k } { k } ) \\right | = [ u ^ n ] ( 1 - u ) ^ { - 2 0 ( r + 1 ) } , \\end{align*}"}
-{"id": "3736.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( | x | ^ { - \\mu } * F ( u _ k ) ) f ( u _ k ) u _ k \\phi & = \\lim _ { k \\to \\infty } m ( \\| u _ k \\| ^ n ) \\int _ { \\Omega } | \\nabla u _ k | ^ n - \\nabla u _ k | ^ { n - 2 } \\nabla u _ k . \\nabla u \\phi ~ d x \\\\ & + \\int _ { \\Omega } ( | x | ^ { - \\mu } * F ( u ) ) f ( u ) u \\phi ~ d x + o _ k ( 1 ) . \\end{align*}"}
-{"id": "1170.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\mathbb { E } \\chi ^ { ( n ) } ( ( x , + \\infty ) ) = ( \\pi / 2 ) e ^ { c _ 0 - x } . \\end{align*}"}
-{"id": "9323.png", "formula": "\\begin{align*} \\int _ { M } ( 1 + \\langle \\psi , \\mathbf { H } \\rangle ) d V = 0 . \\end{align*}"}
-{"id": "2492.png", "formula": "\\begin{align*} T ^ c M = D _ 1 \\varsubsetneq D _ 2 \\varsubsetneq \\cdots \\varsubsetneq D _ \\rho = T M , \\end{align*}"}
-{"id": "6778.png", "formula": "\\begin{align*} u _ t + u \\nabla u + \\nabla _ { \\ ! x } p = 0 \\ , , \\end{align*}"}
-{"id": "9329.png", "formula": "\\begin{align*} \\widehat { W } = W - \\frac { 1 } { \\mathrm { V o l } ( M ) } c \\end{align*}"}
-{"id": "617.png", "formula": "\\begin{align*} \\begin{gathered} \\varphi ^ \\ast ( x y ) = ( \\varphi ^ y ) ^ \\ast ( x ) \\varphi ^ \\ast ( y ) \\\\ ( \\varphi \\psi ) ^ x = \\varphi ^ x \\psi ^ { \\varphi ^ \\ast ( x ) } \\end{gathered} \\end{align*}"}
-{"id": "7172.png", "formula": "\\begin{gather*} X ^ { t + s } ( x ) = X ^ { t } \\left ( \\chi ^ { s } ( x ) \\right ) X ^ { s } ( x ) , X ^ { - s } \\left ( \\chi ^ { s } ( x ) \\right ) = \\left [ X ^ { s } ( x ) \\right ] ^ { - 1 } . \\end{gather*}"}
-{"id": "5672.png", "formula": "\\begin{align*} u \\sim v \\end{align*}"}
-{"id": "2470.png", "formula": "\\begin{align*} ( ( a x ) ^ { q } - ( a x ) ) ^ { q - 1 } & = ( a ^ { q } x ^ { q } - a x ) ^ { q - 1 } = a ^ { q - 1 } ( x ^ { q } - x ) ^ { q - 1 } , \\\\ ( ( x + b ) ^ { q } - ( x + b ) ) ^ { q - 1 } & = ( x ^ { q } + b ^ { q } - x + b ) ^ { q - 1 } = ( x ^ { q } - x ) ^ { q - 1 } . \\end{align*}"}
-{"id": "7587.png", "formula": "\\begin{align*} f ( x , z ) = \\int _ 0 ^ 1 \\nabla \\eta ( x - ( 1 - \\lambda ) z , t ) d \\lambda \\sigma ( x - z , s ) \\end{align*}"}
-{"id": "8802.png", "formula": "\\begin{align*} \\Vert \\overline { u } ^ { \\varepsilon _ n } _ { \\tau _ n + h _ n } - \\overline { u } ^ { \\varepsilon _ n } _ { \\tau _ n } \\Vert _ { - r } ^ 2 = \\sum _ { k = 1 } ^ { \\infty } ( 1 + k ^ 2 ) ^ { - r } \\left ( \\langle \\overline { u } ^ { \\varepsilon _ n } _ { \\tau _ n + h _ n } , \\phi _ { k } \\rangle - \\langle \\overline { u } ^ { \\varepsilon _ n } _ { \\tau _ n } , \\phi _ { k } \\rangle \\right ) ^ 2 . \\end{align*}"}
-{"id": "4689.png", "formula": "\\begin{align*} u ( x ) = \\sin ( \\pi x ) , ~ x \\in [ - 1 , 1 ] , \\end{align*}"}
-{"id": "3591.png", "formula": "\\begin{align*} \\tau ( a ) : v \\mapsto \\begin{cases} v & \\textrm { i f } v \\in V _ 1 ^ { ( a ) } \\oplus V _ 0 ^ { ( a ) } \\oplus V _ { \\frac { 1 } { 4 } } ^ { ( a ) } \\\\ - v & \\textrm { i f } v \\in V _ { \\frac { 1 } { 3 2 } } ^ { ( a ) } . \\end{cases} \\end{align*}"}
-{"id": "1979.png", "formula": "\\begin{align*} e ( V \\otimes \\O _ \\alpha ) = c _ r ( V ) + c _ { r - 1 } ( V ) \\alpha + \\cdots + c _ 1 ( V ) \\alpha ^ { r - 1 } + \\alpha ^ r \\in H _ T ^ { 2 r } ( X ) . \\end{align*}"}
-{"id": "1357.png", "formula": "\\begin{align*} C ^ n ( \\Lambda , \\mathcal { M } ) : = \\prod _ { ( \\lambda _ 0 , \\ldots , \\lambda _ { n - 1 } ) \\in \\Lambda ^ { * n } } \\mathcal { M } ( s ( \\lambda _ { n - 1 } ) ) \\ ; , \\end{align*}"}
-{"id": "6844.png", "formula": "\\begin{align*} N _ { 1 2 } = \\sum _ { i = 1 } ^ 6 f _ i g _ i \\ , , \\end{align*}"}
-{"id": "6283.png", "formula": "\\begin{align*} \\min _ { \\mathcal { A } \\epsilon - } \\max _ { m \\in [ \\frac { 1 } { 3 } \\binom { n } { 2 } , \\frac { 2 } { 3 } \\binom { n } { 2 } ] } & \\mathbb { E } _ { G \\sim G ( n , m ) } [ ( A ( G ) - \\frac { m } { \\binom { n } { 2 } } ) ^ 2 ] = \\Omega \\left ( \\frac { 1 } { n ^ 3 \\epsilon ^ 2 } \\right ) . \\end{align*}"}
-{"id": "6015.png", "formula": "\\begin{align*} I ( \\omega ) = \\begin{cases} \\int _ 0 ^ 1 g ( t , \\dot { \\omega } ( t ) ) d t & \\omega \\\\ \\infty & . \\end{cases} \\end{align*}"}
-{"id": "6528.png", "formula": "\\begin{align*} f ( t , x ( t ) ) = \\bigl [ g ( t , x ( t ) ) - g ( t , y ( t ) ) \\bigr ] + g ( t , y ( t ) ) + q ( t , x ( t ) ) , t \\geq 0 , \\end{align*}"}
-{"id": "8340.png", "formula": "\\begin{align*} A _ 1 = 1 + \\frac { 1 } { 2 \\pi } \\int \\frac { | D _ t Z ( \\alpha , t ) - D _ t Z ( \\beta , t ) | ^ 2 } { ( \\alpha - \\beta ) ^ 2 } d \\beta - \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j } { \\pi } R e \\Big \\{ \\frac { D _ t Z - \\dot { z } _ j } { c _ 0 ^ j ( \\alpha - w _ 0 ^ j ) ^ 2 } \\Big \\} , \\end{align*}"}
-{"id": "7871.png", "formula": "\\begin{align*} a _ S ( \\xi ) = \\prod _ { j \\in I \\setminus S } \\cos ^ 2 ( \\pi \\xi _ j ) \\cdot \\prod _ { i \\in S } \\sin ^ 2 ( \\pi \\xi _ i ) , \\end{align*}"}
-{"id": "3391.png", "formula": "\\begin{align*} | \\gamma _ M - \\gamma _ N | & \\ll \\left ( \\sup _ { k \\geq N } a _ k + \\sum _ { k = N } ^ { \\infty } \\frac { a _ k } { k } \\right ) ^ { 1 / 2 } + \\sup _ { k \\geq N } | \\mu _ k | + \\sum _ { k = N } ^ \\infty \\frac { | \\mu _ k | } { k } \\\\ & \\ll \\left ( \\sup _ { k \\geq N } a _ k + \\sum _ { k = N } ^ { \\infty } \\frac { a _ k } { k } \\right ) ^ { 1 / 2 } + \\sup _ { k \\geq N } a _ k + \\sum _ { k = N } ^ \\infty \\frac { a _ k } { k } . \\end{align*}"}
-{"id": "216.png", "formula": "\\begin{align*} \\mathcal { B } _ \\beta u = \\Delta _ { z } u + \\frac { | z | ^ { 2 \\beta } } { 4 } \\Delta _ { t } u , \\ z \\in \\mathbb { R } ^ N , \\ t \\in \\mathbb { R } ^ m , \\ \\beta > 0 \\end{align*}"}
-{"id": "9887.png", "formula": "\\begin{align*} A = \\begin{pmatrix} Q H _ { 1 } \\\\ \\vdots \\\\ Q H _ { 4 } \\end{pmatrix} & = \\begin{pmatrix} 0 & 1 \\\\ 0 & 1 \\\\ 1 & 0 \\\\ 1 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "993.png", "formula": "\\begin{align*} \\chi _ \\C ( A , B ) = \\chi ( A , B ) + \\chi ( B , A ) - \\varepsilon ( A , B ) , \\end{align*}"}
-{"id": "724.png", "formula": "\\begin{align*} \\lambda _ { i } = \\frac { \\theta _ { i } } { r } , ~ ~ ~ ~ i = 1 , \\ldots , m , \\end{align*}"}
-{"id": "5789.png", "formula": "\\begin{align*} R _ 2 ( \\Gamma , q ) = T ( \\Gamma ; 2 , q + 1 ) \\end{align*}"}
-{"id": "5981.png", "formula": "\\begin{align*} \\rho ^ { G _ n } \\left ( F \\left ( \\frac 1 n \\sum _ { k = 1 } ^ n \\delta _ { W _ { ( n , k ) } } \\right ) \\right ) & = \\frac { 1 } { n } \\rho ^ g _ n ( n F \\circ L _ n ) . \\end{align*}"}
-{"id": "6235.png", "formula": "\\begin{align*} \\sum _ { \\substack { e \\in E \\\\ s ( e ) = v } } \\mu ( e ) \\geq 1 \\end{align*}"}
-{"id": "3414.png", "formula": "\\begin{align*} \\rho ^ M _ { * , \\inf } ( T ) : = \\inf \\{ \\rho _ * ( S ) : S ^ { ( M ) } = T ^ { ( M ) } \\} , \\rho ^ M _ { * , \\sup } ( T ) : = \\sup \\{ \\rho _ * ( S ) : S ^ { ( M ) } = T ^ { ( M ) } \\} . \\end{align*}"}
-{"id": "4666.png", "formula": "\\begin{gather*} \\lambda \\cdot a _ { 0 } = a _ 0 , \\ \\ \\lambda \\cdot a _ { 1 } = a _ 0 \\lambda + a _ 1 , \\ \\ \\lambda \\cdot a _ { 2 } = a _ 0 \\lambda ^ { [ 2 ] } + a _ 1 \\lambda + a _ 2 , \\\\ \\lambda \\cdot a _ { 3 } = a _ 0 \\lambda ^ { [ 3 ] } + a _ 1 \\lambda ^ { [ 2 ] } + a _ 2 \\lambda + a _ 3 , \\ldots . \\end{gather*}"}
-{"id": "7387.png", "formula": "\\begin{align*} \\begin{bmatrix} u ^ t & \\beta \\end{bmatrix} \\begin{bmatrix} D f ( x , \\lambda _ 1 , \\lambda _ 2 ) & \\bar { p } \\\\ \\bar { q } ^ t & 0 \\end{bmatrix} & = \\begin{bmatrix} p ^ t & 0 \\end{bmatrix} , & \\begin{bmatrix} D f ( x , \\lambda _ 1 , \\lambda _ 2 ) & \\bar { p } \\\\ \\bar { q } ^ t & 0 \\end{bmatrix} \\begin{bmatrix} v \\\\ \\beta \\end{bmatrix} & = \\begin{bmatrix} q \\\\ 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "6951.png", "formula": "\\begin{align*} - \\infty \\ , < \\ , c \\ , \\leq \\ , \\inf \\mathcal F _ { h _ j } \\ , \\leq \\ , \\mathcal F _ { h _ j } ( \\mathbf 0 ) \\ , = \\ , 0 . \\end{align*}"}
-{"id": "3944.png", "formula": "\\begin{align*} \\dfrac { \\partial f _ { i } ( x , m ^ { \\hat v } _ { t } , \\hat { v } _ { i } ( x , m ^ { \\hat v } _ { t } ) ) } { \\partial v _ { i } } + D u ^ { \\hat v } _ { i } ( x , t ) . \\dfrac { \\partial g _ { i } ( x , m ^ { \\hat v } _ { t } , \\hat { v } _ { i } ( x , m ^ { \\hat v } _ { t } ) ) } { \\partial v _ { i } } = 0 , \\ , x , t \\ , . \\end{align*}"}
-{"id": "2420.png", "formula": "\\begin{align*} \\begin{cases} V W = W V = I , \\\\ V F = F W ^ t \\end{cases} \\Leftrightarrow \\begin{cases} V W = W V = I , \\\\ V F V ^ t = F , \\end{cases} \\end{align*}"}
-{"id": "4168.png", "formula": "\\begin{align*} L _ { \\alpha } ( n ^ { 3 / 2 } z ) & = \\begin{pmatrix} n ^ { - \\frac { 1 } { 2 } } & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & n ^ { \\frac { 1 } { 2 } } \\end{pmatrix} L _ { \\alpha } ( z ) , L _ { \\alpha } ( n ^ { 3 / 2 } f ( z ) ) = \\begin{pmatrix} n ^ { \\frac { 1 } { 2 } } & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & n ^ { - \\frac { 1 } { 2 } } \\end{pmatrix} L _ { \\alpha } ( n f ( z ) ) \\end{align*}"}
-{"id": "3508.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to \\infty } \\sqrt { \\gamma } K _ { \\mathrm { c r i t } } \\left ( \\sqrt { \\gamma } x , \\sqrt { \\gamma } y ; \\gamma \\right ) = \\frac { 1 } { \\sqrt { 2 \\pi } } e ^ { - \\frac { 1 } { 2 } y ^ 2 } \\end{align*}"}
-{"id": "3970.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } \\Delta ( e ^ { - 4 \\pi m \\frac { y ^ 2 } { v } } ) d y = \\left \\{ \\begin{array} { c c l } \\sqrt { \\frac { v } { 4 m } } & \\mbox { i f } & \\Delta = i d \\\\ 0 & \\mbox { i f } & \\Delta = \\frac { \\partial } { \\partial y } \\ , \\mbox { o r } \\ , \\Delta = y \\frac { \\partial ^ 2 } { \\partial y ^ 2 } \\end{array} \\right . \\end{align*}"}
-{"id": "1744.png", "formula": "\\begin{align*} D B _ { t } ( s ) = \\int _ { 0 } ^ { s \\wedge t } K ( t , u ) d u . \\end{align*}"}
-{"id": "7683.png", "formula": "\\begin{align*} 2 \\xi = ( \\mathbf { i } \\eta - U \\mathbf { i } \\eta - U \\xi ) + \\xi = ( 1 - U ) ( \\mathbf { i } \\eta + \\xi ) , \\end{align*}"}
-{"id": "9470.png", "formula": "\\begin{align*} S _ { s , m } = \\displaystyle \\inf _ { u \\in D ^ { s , m } ( \\mathbb { R } ^ N ) \\backslash \\{ 0 \\} } \\frac { [ u ] _ { s , m } ^ m } { \\Vert u \\Vert _ { m ^ * _ { s } } ^ m } . \\end{align*}"}
-{"id": "688.png", "formula": "\\begin{align*} ( \\widehat \\rho ^ { ( \\vec l ) } _ i ) _ { \\vec V } \\circ \\varpi ( f _ 1 , \\dots , f _ n ) = f _ i \\circ ( \\widehat \\rho ^ { ( \\vec k ) } _ i ) _ { \\vec W } \\ . \\end{align*}"}
-{"id": "9589.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d z ^ 2 } \\left ( \\sum _ { j = 1 } ^ N \\left [ \\begin{matrix} N \\\\ j \\end{matrix} \\right ] \\frac { q ^ { j ^ 2 } ( q ) _ j } { ( z q ) _ j ( z ^ { - 1 } q ) _ j } \\right ) _ { z = 1 } = \\sum _ { n = 1 } ^ { \\infty } N _ { 2 , N } ( n ) q ^ n . \\end{align*}"}
-{"id": "5220.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow + \\infty } U ( p t ) / U ( t ) = p ^ { \\rho } . \\end{align*}"}
-{"id": "9982.png", "formula": "\\begin{align*} & g _ { i j } = g _ { j i } , \\\\ & c _ { n k m } = \\frac { 1 } { 3 } ( g _ { m n , k } - g _ { k n , m } ) \\\\ & g _ { i j , k } + g _ { j k , i } + g _ { k i , j } = 0 , \\\\ & c _ { n m l , k } + c ^ s _ { m l } c _ { s n k } = 0 . \\end{align*}"}
-{"id": "6399.png", "formula": "\\begin{align*} \\gamma _ { k + 1 } ^ { { \\scriptscriptstyle ( \\mu ) } } = \\frac { \\left ( \\gamma _ { k } ^ { { \\scriptscriptstyle ( \\mu ) } } - \\gamma _ { k } \\right ) } { \\mu \\left ( \\gamma _ { k } ^ { { \\scriptscriptstyle ( \\mu ) } } - \\gamma _ { k } \\right ) + \\delta _ { k + 1 } } , \\qquad \\gamma _ { 0 } ^ { { \\scriptscriptstyle ( \\mu ) } } = \\frac { 1 } { \\mu } , \\end{align*}"}
-{"id": "6130.png", "formula": "\\begin{align*} X ^ { a _ k } Y ^ { b _ k } ( \\alpha _ { d - \\mu _ 1 } Y - \\beta _ { d - \\mu _ 2 } X ) = ( \\alpha _ { d - \\mu _ 1 } Y - \\beta _ { d - \\mu _ 2 } X ) \\bigg ( \\sum _ { \\ell = 1 } ^ m \\Omega ( R _ \\ell ) G ^ * _ \\ell \\bigg ) , \\end{align*}"}
-{"id": "1682.png", "formula": "\\begin{align*} & \\frac { \\partial w } { \\partial \\overline { z } } = - w \\left ( n { { \\overline { z } } ^ { n - 1 } } + { { a } _ { 1 } } \\left ( n - 1 \\right ) { { \\overline { z } } ^ { n - 2 } } + { { a } _ { 2 } } \\left ( n - 2 \\right ) { { \\overline { z } } ^ { n - 3 } } + \\cdots + 2 { { a } _ { n - 2 } } \\overline { z } + { { a } _ { n - 1 } } \\right ) \\end{align*}"}
-{"id": "9484.png", "formula": "\\begin{align*} \\lambda = \\frac { 4 ^ { \\alpha + 1 } } { \\sigma } C \\left [ K _ { 0 } ^ { \\frac { 1 } { p - 1 } } R _ { 0 } ^ { \\frac { s ( p - q ) } { p - 1 } - \\alpha } + \\frac { Q ( u ; R _ 0 ) } { R _ 0 ^ { \\alpha } } \\right ] . \\end{align*}"}
-{"id": "3734.png", "formula": "\\begin{align*} ( 2 n - 1 - q ) a \\| u \\| ^ { 2 n } + ( n - 1 - q ) & b \\| u \\| ^ n + q \\int _ { \\Omega } ( | x | ^ { - \\mu } * F ( u ) ) f ( u ) . u \\\\ & - \\left ( \\int _ { \\Omega } ( | x | ^ { - \\mu } * f ( u ) u ) f ( u ) . u + \\int _ { \\Omega } ( | x | ^ { - \\mu } * F ( u ) ) f ' ( u ) u ^ 2 \\right ) < 0 . \\end{align*}"}
-{"id": "1259.png", "formula": "\\begin{align*} T _ { 4 , 4 } & = \\sum _ { j \\in \\{ \\pm \\} } B \\int e ^ { i \\Theta ( \\eta , \\nu ) } m _ 2 ( \\eta , \\nu ) \\theta _ j ( \\nu / \\eta ) d \\nu \\\\ & + B \\int e ^ { i \\Theta ( \\eta , \\nu ) } m _ 2 ( \\eta , \\nu ) ( 1 - \\theta _ + - \\theta _ - ) ( \\nu / \\eta ) d \\nu . \\end{align*}"}
-{"id": "4091.png", "formula": "\\begin{align*} T _ { \\alpha } \\Phi _ { \\alpha } ( z ) \\begin{pmatrix} 0 \\\\ 0 \\\\ 1 \\end{pmatrix} = \\frac { 2 \\pi } { \\sqrt { 3 } } z ^ { - \\gamma + \\frac { 1 } { 3 } } \\left ( \\mathbb { I } + \\mathcal O ( z ^ { - 1 } ) \\right ) L _ { \\alpha } ( z ) \\begin{pmatrix} 0 \\\\ 0 \\\\ 1 \\end{pmatrix} e ^ { - 3 z ^ { \\frac { 1 } { 3 } } } \\end{align*}"}
-{"id": "4059.png", "formula": "\\begin{align*} v _ n v ' _ { n ' } w = \\sum _ { i = 0 } ^ { M } \\sum _ { j = 0 } ^ L \\binom { n - L } { i } \\binom { L } { j } ( v _ { n - L - i + j } v ' ) _ { n ' + L + i - j } w \\end{align*}"}
-{"id": "3751.png", "formula": "\\begin{align*} q _ { l } ^ { ( j ) } = \\frac { i } { 2 \\chi } \\left [ \\left ( \\mathcal { D } _ { l , m } ^ { ( j ) } , \\mathcal { D } _ { l , m } ^ { ( j ) } \\right ) \\left \\vert \\left ( \\mathcal { B } _ { l , m } ^ { ( j ) } , \\mathcal { D } _ { l , m } ^ { ( j ) } \\right ) \\right \\vert ^ { - 2 } + \\gamma _ { l , m } ^ { ( j ) } \\right ] . \\end{align*}"}
-{"id": "7491.png", "formula": "\\begin{align*} f ^ { s - 2 } = e ^ { ( s - 2 ) \\log f } \\end{align*}"}
-{"id": "1354.png", "formula": "\\begin{align*} \\mathcal F ( w ) = \\bigoplus _ { v \\in \\operatorname { O b j } \\Lambda } \\bigoplus _ { b \\in B _ v } \\Z v \\Lambda w & \\to \\mathcal { M } ( w ) \\\\ ( v , b , \\lambda ) & \\mapsto b \\cdot \\lambda \\ ; , \\end{align*}"}
-{"id": "9213.png", "formula": "\\begin{align*} e ( G ' ) \\leq e ( S ) + \\sum _ { x \\in \\bar { S } } d _ { G ' } ( x ) \\leq \\binom { s } { 2 } + ( k - s ) ( n - s ) . \\end{align*}"}
-{"id": "3476.png", "formula": "\\begin{align*} G _ N ( x , r ) - G _ { \\infty , \\gamma } ( x , r ) = \\oint _ { \\Sigma _ - ( - 1 / 2 ) } g _ N ( t ; x , r ) \\frac { d t } { 2 \\pi i } - \\oint _ { \\Sigma ' _ - { ( - 1 / 2 ) } } g _ { \\infty } ( t ; x , r ) \\frac { d t } { 2 \\pi i } . \\end{align*}"}
-{"id": "3011.png", "formula": "\\begin{align*} B _ n ( x _ 1 \\dots x _ n ) & = \\sum _ { k = 1 } ^ n B _ { n , k } ( x _ 1 , \\dots , x _ { n - k + 1 } ) . \\end{align*}"}
-{"id": "9418.png", "formula": "\\begin{align*} ( f , g ) _ { r e g } : = \\lim _ { \\varepsilon \\rightarrow 0 } \\int _ { \\mathcal { F } _ N ( f , \\varepsilon ) } f ( z ) \\overline { g ( z ) } \\frac { d x d y } { y ^ { k - 2 } } . \\end{align*}"}
-{"id": "9758.png", "formula": "\\begin{align*} \\bar { q } _ \\ell : = \\frac { q _ \\ell } { \\sigma _ \\ell } , \\end{align*}"}
-{"id": "7276.png", "formula": "\\begin{align*} p = ( B _ p , n _ p , \\psi _ p ) \\end{align*}"}
-{"id": "4406.png", "formula": "\\begin{align*} ( \\omega _ o | _ S ) ^ 2 = ( 1 + | 2 s t | ^ 2 + | t ^ 2 | ^ 2 ) d d ^ c | s | ^ 2 \\wedge d d ^ c | t | ^ 2 . \\end{align*}"}
-{"id": "1617.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow \\infty } \\frac { 1 } { T } \\int _ 0 ^ T \\mu \\left ( X _ t ^ { Z ^ { b } } \\right ) X _ t ^ { Z ^ { b } } \\ , d t = \\int _ 0 ^ { b } \\mu ( x ) x \\frac { m ' ( x ) } { m ( ( 0 , b ) ) } \\ , d x = \\frac { 1 } { S ' ( b ) m ( ( 0 , b ) ) } . \\end{align*}"}
-{"id": "8680.png", "formula": "\\begin{align*} u _ { \\lambda , \\nu } ^ A & = \\frac { ( - 1 ) ^ k k ! \\pi ^ { \\frac { p + q } { 2 } } } { \\Gamma ( \\frac { \\nu + \\rho ' } { 2 } ) } \\times \\begin{cases} \\frac { \\Gamma ( \\frac { 2 \\nu + p ' } { 4 } ) } { \\Gamma ( \\frac { 2 \\nu + p ' } { 2 } ) } u ^ C _ { \\lambda , \\nu } & \\mbox { f o r $ m > 0 $ , } \\\\ \\frac { \\Gamma ( \\frac { \\nu } { 2 } - \\lfloor \\frac { k } { 2 } \\rfloor ) } { 2 ^ k \\Gamma ( \\nu - k ) } u ^ C _ { \\lambda , \\nu } & \\mbox { f o r $ m = 0 $ . } \\end{cases} \\end{align*}"}
-{"id": "1397.png", "formula": "\\begin{align*} \\frac { \\tfrac { \\partial f } { \\partial x _ 1 } ( x _ 1 , \\ldots , x _ d ) } { r _ 1 ( x _ 1 ) } = \\cdots = \\frac { \\tfrac { \\partial f } { \\partial x _ d } ( x _ 1 , \\ldots , x _ d ) } { r _ d ( x _ d ) } \\end{align*}"}
-{"id": "3119.png", "formula": "\\begin{align*} M ^ { ( 2 ) } = \\begin{pmatrix} 1 & 8 & 0 & 0 \\\\ 1 & 9 & 0 & 0 \\\\ 1 & 9 & 9 & 9 \\\\ 1 & 9 & - 9 & 9 \\end{pmatrix} \\end{align*}"}
-{"id": "2380.png", "formula": "\\begin{align*} l = & 0 , 1 , 2 , \\cdots , N / 2 \\\\ l = & 1 / 2 , 3 / 2 , \\cdots , N / 2 . \\end{align*}"}
-{"id": "3084.png", "formula": "\\begin{align*} g ( x ) = \\left \\{ \\begin{array} { l l } & f ( x ) , x \\in G , \\\\ & \\frac { 1 } { m ( K _ l ) } \\int _ { K _ l } f , x \\in K _ l , l = 1 , 2 , \\cdots \\end{array} \\right . \\end{align*}"}
-{"id": "7519.png", "formula": "\\begin{align*} \\theta _ i : = \\frac { \\partial \\Psi } { \\partial { u ^ i } } . \\end{align*}"}
-{"id": "4570.png", "formula": "\\begin{align*} \\mathcal { H } ^ s _ \\delta ( F ) = \\inf \\left \\{ \\sum _ { i = 1 } ^ { \\infty } ( \\mathrm { d i a m } ( U _ i ) ) ^ s : \\bigcup _ i U _ i \\supset F , \\forall i \\geq 1 , U _ i \\subset \\mathbb { R } ^ n , \\mathrm { d i a m } ( U _ i ) < \\delta \\right \\} . \\end{align*}"}
-{"id": "4520.png", "formula": "\\begin{align*} D _ k = \\{ i \\in \\{ 1 , \\dots , N \\} : m _ i \\in [ 2 ^ { k - 1 } , 2 ^ { k } ) \\} . \\end{align*}"}
-{"id": "6087.png", "formula": "\\begin{align*} f ( x ) = ( x - a _ 1 ) ^ { e ^ 1 - 1 } \\ldots ( x - a _ k ) ^ { e ^ k - 1 } \\Big ( ( x - a _ 1 ) \\ldots ( x - a _ k ) h ' ( x ) + \\\\ e ^ 1 ( \\widehat { x - a _ 1 } ) ( x - a _ 2 ) \\ldots ( x - a _ k ) h ( x ) + \\\\ e ^ 2 ( x - a _ 1 ) ( \\widehat { x - a _ 2 } ) \\ldots ( x - a _ k ) h ( x ) + \\\\ e ^ k ( x - a _ 1 ) \\ldots ( x - a _ { k - 1 } ) ( \\widehat { x - a _ k } ) \\Big ) \\end{align*}"}
-{"id": "819.png", "formula": "\\begin{align*} \\left ( U _ j ^ { ( i ) } \\right ) ^ p - U _ j ^ { ( i ) } = U _ { j - 1 } ^ { ( i ) } . \\end{align*}"}
-{"id": "5152.png", "formula": "\\begin{align*} \\langle \\psi , H _ { m a x } \\widetilde { \\psi } \\rangle _ { L ^ 2 ( \\R ^ 3 ) ^ 4 } & - \\langle H _ { m a x } \\psi , \\widetilde { \\psi } \\rangle _ { L ^ 2 ( \\R ^ 3 ) ^ 4 } \\\\ & = \\sum _ { ( j , m _ j , k _ j ) \\in I } \\Gamma ^ + _ { m _ j , k _ j } ( f ) \\cdot \\overline { \\Gamma _ { m _ j , k _ j } ^ - ( \\tilde { f } ) } - \\Gamma ^ - _ { m _ j , k _ j } ( f ) \\cdot \\overline { \\Gamma _ { m _ j , k _ j } ^ + ( \\tilde { f } ) } , \\end{align*}"}
-{"id": "8012.png", "formula": "\\begin{align*} u ( t , x ) & \\geq \\int _ { \\mathbb { R } ^ d } t ^ { - \\frac { d } { 2 } } e ^ { - c _ 1 ( t + 1 ) } e ^ { - \\frac { | x - y | ^ 2 } { c _ 2 t } } u _ 0 ( y ) \\ , \\mathrm d y \\\\ & = ( c _ 2 \\pi ) ^ { \\frac { d } { 2 } } e ^ { - c _ 1 ( t + 1 ) } ( e ^ { \\frac { c _ 2 } { 4 } t \\Delta } u _ 0 ) ( x ) . \\end{align*}"}
-{"id": "3370.png", "formula": "\\begin{align*} N _ { 1 1 } ^ 2 D _ 1 & \\ , = ( d _ 1 - N _ { 1 2 } ^ 2 ) \\ , D _ 2 = ( d _ 1 - N _ { 2 1 } ^ 2 ) \\ , D _ 2 \\ , , \\\\ p & \\ , = ( d _ 1 - N _ { 2 1 } ^ 2 ) \\ , D _ 1 + ( 2 d _ 2 - N _ { 2 2 } ^ 2 ) \\ , D _ 2 . \\end{align*}"}
-{"id": "4991.png", "formula": "\\begin{gather*} \\begin{cases} f : \\ \\mathbb { Z } ^ 2 \\rightarrow \\mathbb { Z } ^ 2 \\mid f ( x , y ) \\mapsto ( y , x ) \\\\ g : \\ \\mathbb { Z } ^ 2 \\rightarrow \\mathbb { Z } ^ 2 \\mid g ( x , y ) \\mapsto ( - x , - y ) \\\\ \\varphi : \\ \\mathbb { Z } ^ 2 \\rightarrow \\mathbb { Z } ^ 2 \\mid \\varphi ( x , y ) \\mapsto ( x , x - y ) \\\\ I : \\ \\mathbb { Z } ^ 2 \\rightarrow \\mathbb { Z } ^ 2 \\mid I ( x , y ) \\mapsto ( x , y ) \\end{cases} \\end{gather*}"}
-{"id": "7244.png", "formula": "\\begin{align*} \\ker D _ \\phi ( 0 , \\mu ) = \\mbox { s p a n } \\{ \\phi ^ * _ k \\} \\mbox { w i t h } \\phi ^ * _ k ( x ) : = \\cos \\left ( x k \\right ) . \\end{align*}"}
-{"id": "6578.png", "formula": "\\begin{gather*} | I _ 1 | + t _ 1 + \\cdots + | I _ p | + t _ p - q = k . \\end{gather*}"}
-{"id": "6462.png", "formula": "\\begin{align*} \\ell ( e ) ^ { * } & p _ { v } = 0 \\\\ \\ell ( e ) ^ { * } & e _ { 1 } \\otimes \\cdots \\otimes e _ { n } = \\langle e | e _ { 1 } \\rangle _ { A } e _ { 2 } \\otimes \\cdots \\otimes e _ { n } . \\end{align*}"}
-{"id": "416.png", "formula": "\\begin{align*} \\Phi ( x ) : = - \\frac { 2 } { \\mathcal { G } _ { g } ^ \\gamma ( M ) } \\int G _ g ( x , y ) \\mathcal { G } _ { g } ^ \\gamma ( \\dd y ) . \\end{align*}"}
-{"id": "5897.png", "formula": "\\begin{align*} p ( g ) = p ( k _ 1 a k _ 2 ) = \\sum _ { i = - N _ 1 } ^ { N _ 2 } ( \\pi ^ \\lambda ) ^ i p _ i ( k _ 1 , k _ 2 ) \\end{align*}"}
-{"id": "2800.png", "formula": "\\begin{align*} C ( \\alpha ) = \\log \\left ( 1 - e _ 0 + e _ 0 e ^ { \\alpha } \\right ) = \\log \\left ( q + ( 1 - q ) e ^ { \\alpha } \\right ) . \\end{align*}"}
-{"id": "5505.png", "formula": "\\begin{align*} \\Gamma _ T = \\{ \\tilde { \\Phi } _ { \\lambda , k , j } : \\lambda \\leq J , k , 1 \\leq j \\leq d \\} \\subset V _ J ^ { \\otimes d } . \\end{align*}"}
-{"id": "7211.png", "formula": "\\begin{align*} \\mu _ k : = ( k + 1 ) ^ { - r } \\mbox { f o r } k \\in \\N _ 0 . \\end{align*}"}
-{"id": "4425.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ t \\Bigl { \\langle } \\frac { \\dd ( w - z ^ i ) } { \\dd t } ( \\tau ) , \\chi _ { \\varepsilon } ^ { \\pm } ( w ( \\tau ) - z ^ i ) \\Bigl { \\rangle } \\dd \\tau & = \\int _ { { \\mathcal { Q } } _ t } \\frac { \\dd } { \\dd t } ( w - z ^ i ) _ { \\varepsilon } ^ { \\pm } \\dd x \\dd \\tau \\\\ & = \\int _ { \\Omega } \\bigl ( ( w ( t ) - z ^ i ) _ { \\varepsilon } ^ { \\pm } - ( w ( 0 , \\cdot ) - z ^ i ) _ { \\varepsilon } ^ { \\pm } \\bigr ) \\dd x . \\end{aligned} \\end{align*}"}
-{"id": "7232.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } [ ( \\mu - \\phi ) ^ 2 ] ^ { \\prime \\prime } ( \\xi ) = f ( \\xi ) \\geq c _ 0 , \\mbox { f o r a l l } \\xi \\in ( 0 , | x _ 0 | ) . \\end{align*}"}
-{"id": "9515.png", "formula": "\\begin{align*} h _ { \\beta , M } ( t ) = \\left \\{ \\begin{array} { r c } t ^ { \\beta } , & \\mbox { s e } \\ t \\leq M , \\\\ t M ^ { \\beta - 1 } , & \\mbox { s e } \\ t \\geq M . \\end{array} \\right . \\end{align*}"}
-{"id": "3194.png", "formula": "\\begin{align*} U \\varphi _ 0 = \\varGamma C \\varphi _ 0 = - \\varGamma \\varphi _ 0 . \\end{align*}"}
-{"id": "4787.png", "formula": "\\begin{align*} \\dot { \\bar { \\mathbf { q } } } _ k = \\frac 1 { w _ k } \\frac { \\partial H } { \\partial { \\bar { \\bf p } _ k } } \\ , , \\qquad \\dot { \\bar { \\mathbf { p } } } _ k = - \\frac 1 { w _ k } \\frac { \\partial H } { \\partial { \\bar { \\bf q } _ k } } \\ , , \\end{align*}"}
-{"id": "6149.png", "formula": "\\begin{align*} \\kappa _ X ^ 2 = 2 p ( 4 p a - k ^ 2 ) , \\kappa _ X \\cdot h _ X = 0 , 2 p \\mid \\div _ { H ^ 2 ( X , \\Z ) } ( \\kappa _ X ) . \\end{align*}"}
-{"id": "5693.png", "formula": "\\begin{align*} \\| x ^ { A B } & - ( \\bar x + h _ 0 v _ 0 ) \\| = \\| x ^ { A B } - ( \\bar x + h _ 0 v _ 0 ^ A ) - h _ 0 ( v _ 0 - v _ 0 ^ A ) \\| \\\\ & \\leq \\| x ^ { A B } - x ^ A \\| + h _ 0 \\| v _ 0 - v _ 0 ^ A \\| \\le ( K + 1 ) \\left \\| x ^ A - x ^ B \\right \\| \\ + h _ 0 \\eta \\\\ & \\leq ( K + 1 ) 2 \\eta h _ 0 + h _ 0 \\eta = h _ 0 \\eta \\left ( 2 K + 3 \\right ) = \\varepsilon h _ 0 \\ , . \\end{align*}"}
-{"id": "2274.png", "formula": "\\begin{align*} a & = H = z - R ( z ) ^ { - 1 } , & H : & D = R ( z ) V \\to V \\end{align*}"}
-{"id": "7165.png", "formula": "\\begin{gather*} \\lim _ { t \\to \\infty } \\left | \\varphi ( t ; \\varphi _ { 0 } , r _ { 0 } ) - \\varphi ( t ; \\varphi _ { \\ast } , 1 ) \\right | = \\infty \\quad \\forall \\left \\{ \\varphi _ { 0 } , \\varphi _ { \\ast } \\right \\} \\subset [ 0 , 2 \\pi ) . \\end{gather*}"}
-{"id": "931.png", "formula": "\\begin{align*} E _ 3 ^ q = I _ 3 - \\sum _ { j _ 3 , j _ 2 , j _ 1 = 0 } ^ q C _ { j _ 3 j _ 2 j _ 1 } ^ 2 - \\sum _ { j _ 3 , j _ 2 , j _ 1 = 0 } ^ q C _ { j _ 3 j _ 1 j _ 2 } C _ { j _ 3 j _ 2 j _ 1 } \\ \\ \\ ( i _ 1 = i _ 2 \\ne i _ 3 ) , \\end{align*}"}
-{"id": "8687.png", "formula": "\\begin{align*} c _ { h , i , j } ( \\lambda , \\nu ) = \\frac { 2 ^ { - 2 i - 2 h } \\Gamma ( \\frac { 2 \\nu + p ' + 2 } { 4 } ) \\Gamma ( \\frac { \\lambda + \\rho + \\nu - \\rho ' } { 2 } + i ) } { h ! i ! j ! \\Gamma ( \\frac { 2 \\nu + p ' + 2 } { 4 } - j ) \\Gamma ( \\frac { p '' } { 2 } + i ) \\Gamma ( \\frac { \\lambda + \\rho + \\nu - \\rho ' } { 2 } ) } \\end{align*}"}
-{"id": "101.png", "formula": "\\begin{align*} B ( 0 ) ^ { - \\ell } B _ 1 & = \\underbrace { ( B _ 1 B _ 2 ( 0 ) ) \\cdots ( B _ 1 B _ 2 ( 0 ) ) } _ { \\ell \\ \\mbox { t i m e s } } \\cdot B _ 1 \\\\ & = B _ 1 \\cdot \\underbrace { ( B _ 2 ( 0 ) B _ 1 ) \\cdots ( B _ 2 ( 0 ) B _ 1 ) } _ { \\ell \\ \\mbox { t i m e s } } \\\\ & = B _ 1 \\cdot B ( 0 ) ^ { \\ell } \\ , . \\end{align*}"}
-{"id": "1978.png", "formula": "\\begin{align*} e ( V ) : = c _ r ( V ) \\in H ^ { 2 r } ( X ) \\subset H _ T ^ { 2 r } ( X ) . \\end{align*}"}
-{"id": "8201.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty p ( n ) q ^ n = \\frac { 1 } { ( q ; q ) _ \\infty } . \\end{align*}"}
-{"id": "6220.png", "formula": "\\begin{align*} \\int _ 0 ^ { + \\infty } \\frac { e ^ { - \\frac { | | x | | ^ 2 } { 2 \\lambda } } } { ( 2 \\pi \\lambda ) ^ { n / 2 } } k _ n ( \\lambda ) d \\lambda = g _ n ( x ) . \\end{align*}"}
-{"id": "7714.png", "formula": "\\begin{align*} h _ { s } ( p ) : = f \\bigl ( g ( p ) + c s \\bigr ) , L _ { s } : = \\{ p \\in M \\ , \\vert \\ , h _ { s } ( p ) = 1 \\} . \\end{align*}"}
-{"id": "3409.png", "formula": "\\begin{align*} \\rho _ * ( T ) : = \\frac { d _ * ( T ) } { d ( T ) } \\rho _ 0 ( T ) : = \\frac { d _ 0 ( T ) } { d ( T ) } , \\end{align*}"}
-{"id": "1956.png", "formula": "\\begin{align*} a ^ c = d , b ^ c = c . \\end{align*}"}
-{"id": "9061.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\xi _ 1 + \\xi _ 2 + \\xi _ 3 - \\xi _ 1 ' - \\xi _ 2 ' - \\xi _ 3 ' = 0 \\\\ } } | a _ { \\xi _ 1 } \\overline { a _ { \\xi _ 1 ' } } a _ { \\xi _ 2 } \\overline { a _ { \\xi _ 2 ' } } a _ { \\xi _ 3 } \\overline { a _ { \\xi _ 3 ' } } | \\leq ( u ( N ) / N ) ^ 3 \\cdot N ^ { \\gamma 6 } = O ( 1 ) . \\end{align*}"}
-{"id": "9304.png", "formula": "\\begin{align*} & { \\bf a } _ i ^ T { \\bf x } \\geq b _ i , & i \\in M _ 1 ; \\\\ & { \\bf a } _ i ^ T { \\bf x } \\leq b _ i , & i \\in M _ 2 ; \\\\ & { \\bf a } _ i ^ T { \\bf x } = b _ i , & i \\in M _ 3 ; \\\\ & x _ j \\geq 0 , & j \\in N _ 1 ; \\\\ & x _ j \\leq 0 , & j \\in N _ 2 , \\end{align*}"}
-{"id": "5696.png", "formula": "\\begin{align*} \\overline { \\hat T _ { C _ 1 } ( x _ 0 , f _ 1 ( x _ 0 ) , f _ 2 ( x _ 0 ) ) - \\hat T _ { C _ 2 } ( x _ 0 , f _ 1 ( x _ 0 ) , f _ 2 ( x _ 0 ) ) } = X \\times \\mathbb { R } \\times \\mathbb { R } \\ , . \\end{align*}"}
-{"id": "5175.png", "formula": "\\begin{align*} F ^ { - 1 } ( 1 - u ) = c ( 1 + a ( u ) ) u ^ { - \\gamma } \\exp ( \\int _ { u } ^ { 1 } \\frac { \\ell ( t ) } { t } d t ) . \\end{align*}"}
-{"id": "90.png", "formula": "\\begin{align*} E _ - \\dot = < e , v _ - > ^ \\perp \\ , . \\end{align*}"}
-{"id": "53.png", "formula": "\\begin{align*} g \\in C ^ 1 ( \\R ) \\ , , g ( 0 ) = 0 \\ , , g ' ( J ) > 0 \\forall \\ , J \\ , . \\end{align*}"}
-{"id": "4460.png", "formula": "\\begin{align*} W _ k = \\psi \\circ \\eta \\end{align*}"}
-{"id": "6877.png", "formula": "\\begin{align*} \\varphi ( t , \\alpha ) = \\left ( A ( t ) A ^ { - 1 } ( 0 ) \\right ) \\ , \\left ( A ( 0 ) v ( \\alpha ) \\right ) = \\widetilde { A } ( t ) V ( \\alpha ) \\ , , \\end{align*}"}
-{"id": "2813.png", "formula": "\\begin{align*} \\displaystyle { \\int _ 0 ^ { \\infty } K _ { i j } ( s ) \\ , d s = 1 , \\ \\int _ 0 ^ \\infty G _ i ( s ) \\ , d s = 1 , \\ \\mbox { f o r } \\ i , j = 1 , \\ldots , n } . \\end{align*}"}
-{"id": "3928.png", "formula": "\\begin{align*} \\varepsilon _ M : { } ^ H ( \\Bbbk [ G ] \\otimes M ) \\to M \\quad , \\varepsilon _ M ( \\sum f _ i \\otimes m _ i ) = \\sum f _ i ( 1 ) m _ i \\ , \\sum f _ i \\otimes m _ i \\in { } ^ H ( \\Bbbk [ G ] \\otimes M ) . \\end{align*}"}
-{"id": "5795.png", "formula": "\\begin{align*} \\gamma \\ = \\ \\omega ^ { \\beta _ 1 } + \\omega ^ { \\beta _ 2 } + \\cdots + \\omega ^ { \\beta _ k } \\ \\qquad ( k \\in \\N ) \\end{align*}"}
-{"id": "8701.png", "formula": "\\begin{align*} u ^ C _ { \\lambda , \\nu } = \\sum _ { i + 2 j = k } \\frac { 2 ^ { - i } \\Gamma ( \\frac { \\nu } { 2 } - j ) } { i ! j ! \\Gamma ( \\frac { \\nu } { 2 } - \\lfloor \\frac { k } { 2 } \\rfloor ) \\Gamma ( \\frac { p } { 2 } + i ) } \\Delta _ { \\mathfrak { v } } ^ i \\square ^ j \\delta . \\end{align*}"}
-{"id": "4248.png", "formula": "\\begin{align*} \\int _ { B _ { R } } \\left \\langle a ( x ) \\left \\lvert d u \\right \\rvert ^ { p - 2 } d u ; d \\psi \\right \\rangle = \\int _ { B _ { R } } \\left \\langle f ; \\psi \\right \\rangle \\psi \\in W _ { 0 } ^ { 1 , p } \\left ( B _ { R } ; \\Lambda ^ { k } \\mathbb { R } ^ { n } \\otimes \\mathbb { R } ^ { N } \\right ) . \\end{align*}"}
-{"id": "2695.png", "formula": "\\begin{align*} \\mathcal { Q } _ { \\tau , z _ 2 } & = \\left \\{ z \\in \\mathbb { C } : \\Big | \\frac { \\tau z - 1 } { z - \\tau } \\Big | = \\Big | \\frac { \\tau z _ { 2 } - 1 } { z _ { 2 } - \\tau } \\Big | \\right \\} . \\end{align*}"}
-{"id": "4275.png", "formula": "\\begin{align*} \\textup { s a t } ( n , C _ 4 , K _ 4 - e ) = 0 . \\end{align*}"}
-{"id": "5468.png", "formula": "\\begin{align*} K _ t ( \\hat \\omega ) \\ < \\ \\sup _ { 0 \\leq s \\leq t } K _ s ( \\hat \\omega ) \\ = \\ \\sup _ { 0 \\leq s \\leq \\hat \\tau ( \\hat \\omega ) } K _ s ( \\hat \\omega ) . \\end{align*}"}
-{"id": "5478.png", "formula": "\\begin{align*} v ( \\hat \\tau , \\hat X _ { \\hat \\tau } ^ { t , x , a } ) \\ = \\ Y _ { \\hat \\tau } ^ { t , x , a } , \\hat \\P \\end{align*}"}
-{"id": "1021.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to + \\infty } \\left ( n ^ 2 \\ln \\cos \\frac { \\alpha _ n } { 2 } + \\ln n + x + \\frac { 3 \\ln ( 2 \\ln n ) } { 8 } \\right ) = 0 . \\end{align*}"}
-{"id": "3655.png", "formula": "\\begin{align*} B = A \\cup \\{ v _ { ( 1 , 2 ) } , v _ { ( 1 , 3 ) } , v _ { ( 2 , 3 ) } \\} \\cup \\{ a _ 1 \\cdot v _ { ( 2 , 3 ) } , a _ 2 \\cdot v _ { ( 1 , 3 ) } , a _ 3 \\cdot v _ { ( 1 , 2 ) } \\} . \\end{align*}"}
-{"id": "2092.png", "formula": "\\begin{align*} { y } _ R & = { s _ 0 ( t ) * p _ { \\rm o b s } ( t ) } + \\sum _ { i = 1 } ^ { M } { s _ i ( i T + t ) * p _ { \\rm o b s } ( i T + t ) } = \\sum _ { i = 0 } ^ { M } { s _ i ( i T + t ) * p _ { \\rm o b s } ( i T + t ) } . \\end{align*}"}
-{"id": "6984.png", "formula": "\\begin{align*} a \\otimes 1 = \\begin{cases} 1 & \\ \\ a = 0 , \\\\ ( a + 1 ) / ( a - 1 ) & \\ \\ 1 \\le a \\le p - 2 , \\\\ ( p - 2 ) ^ 2 / 1 & \\ \\ a = p - 1 . \\end{cases} \\end{align*}"}
-{"id": "5198.png", "formula": "\\begin{align*} H ^ { t } ( x ) = H ( x - c \\log t ) , t > 0 . \\end{align*}"}
-{"id": "5367.png", "formula": "\\begin{align*} \\frac { a _ { n m } d _ { n + 1 \\ , m + 3 } } { a _ { n - 1 \\ , m + 3 } d _ { n \\ , m + 6 } } = \\frac { b _ { n \\ , m + 6 } c _ { n + 1 \\ , m + 3 } } { b _ { n - 1 \\ , m + 3 } c _ { n m } } = \\pm \\frac { n } { n + 1 } . \\end{align*}"}
-{"id": "934.png", "formula": "\\begin{align*} I _ { { 0 0 } _ { \\tau _ { p + 1 } , \\tau _ p } } ^ { ( i _ 1 i _ 2 ) q } = \\frac { \\Delta } { 2 } \\left ( \\zeta _ 0 ^ { ( i _ 1 ) } \\zeta _ 0 ^ { ( i _ 2 ) } + \\sum _ { i = 1 } ^ { q } \\frac { 1 } { \\sqrt { 4 i ^ 2 - 1 } } \\left ( \\zeta _ { i - 1 } ^ { ( i _ 1 ) } \\zeta _ { i } ^ { ( i _ 2 ) } - \\zeta _ i ^ { ( i _ 1 ) } \\zeta _ { i - 1 } ^ { ( i _ 2 ) } \\right ) - { \\bf 1 } _ { \\{ i _ 1 = i _ 2 \\} } \\right ) , \\end{align*}"}
-{"id": "4083.png", "formula": "\\begin{align*} \\vartheta ( \\vartheta + \\alpha ) ( \\vartheta + \\alpha + \\tfrac { 1 } { 2 } ) \\phi + z \\phi = 0 , \\vartheta = z \\frac { d } { d z } . \\end{align*}"}
-{"id": "5569.png", "formula": "\\begin{align*} \\mathcal { M } ^ { \\pm } ( t ) : = \\int _ { \\R ^ { \\pm } } | u ( x , t ) | ^ 2 d x = \\mathcal { M } ^ { \\pm } ( 0 ) \\pm 2 \\ , \\int _ 0 ^ t u _ x ( 0 , s ) \\bar { u } ( 0 , s ) d s ; \\end{align*}"}
-{"id": "8961.png", "formula": "\\begin{align*} A = \\bigcap _ { t \\in ( 0 , 1 ] \\cap \\mathbb { Q } } A _ t , \\end{align*}"}
-{"id": "9710.png", "formula": "\\begin{align*} T _ { \\textbf { d } } ( \\partial D ) = - I _ { \\textbf { d } } ( P _ i ) _ { c } + 1 = - 1 \\Rightarrow I _ { \\textbf { d } } ( P _ i ) _ { c } = 2 . \\end{align*}"}
-{"id": "280.png", "formula": "\\begin{align*} Q ( t ) = \\sum _ { j = l } ^ m a _ j t ^ j . \\end{align*}"}
-{"id": "9873.png", "formula": "\\begin{align*} \\tau | S _ \\tau | \\le | \\{ p _ 1 - p _ 2 = s ~ : ~ p _ 1 , p _ 2 \\in P , \\ , s \\in S _ \\tau \\} | \\le | A | ^ { - 1 } | \\{ q a ^ { - 1 } - p _ 2 = s ~ : ~ p _ 2 \\in P , \\ , s \\in S _ \\tau , \\ , q \\in P A , \\ , a \\in A \\} | \\end{align*}"}
-{"id": "3577.png", "formula": "\\begin{align*} \\begin{matrix} \\begin{aligned} z _ { 1 } z _ { 2 } z _ { 3 } z _ { 4 } \\times \\prod _ { i = 1 } ^ { 3 } ( 1 + z _ { i } ) ( 1 + z _ { i } z _ { 4 } ) \\times \\prod _ { 1 \\leq i < j \\leq 3 } ( 1 - z _ { i } z _ { j } z _ { 4 } ) \\\\ \\ ; \\ ; \\times \\left ( 1 - ( z _ { 1 } z _ { 2 } + z _ { 1 } z _ { 3 } + z _ { 2 } z _ { 3 } + z _ { 1 } z _ { 2 } z _ { 3 } ) z _ { 4 } - z _ { 1 } z _ { 2 } z _ { 3 } z _ { 4 } ^ { 2 } \\right ) . \\end{aligned} \\end{matrix} \\end{align*}"}
-{"id": "6697.png", "formula": "\\begin{align*} q _ i = \\alpha _ i ( \\bar { q } ) + \\beta _ { i k } ( \\bar { q } ) \\ , \\dot { \\bar { q } } _ k , \\end{align*}"}
-{"id": "2405.png", "formula": "\\begin{align*} \\upsilon ' & = ( a , b ) ( i , d ) ( c , j ) \\phi _ { ( i , j ) } ^ - , \\\\ \\upsilon '' & = ( a , b ) ( i , c ) ( j , d ) \\phi _ { ( i , j ) } ^ - \\end{align*}"}
-{"id": "7082.png", "formula": "\\begin{align*} m = \\frac { 2 . 1 6 . 4 } { 8 . 8 } = 2 \\end{align*}"}
-{"id": "7893.png", "formula": "\\begin{align*} \\dim \\mathcal { M } ( L ) \\leq \\frac { 1 } { 2 } ( n - m - 1 ) ( n + m ) - \\sum \\limits _ { i = 2 } ^ { m i n \\lbrace n - m , c \\rbrace } n - m - i , \\end{align*}"}
-{"id": "9153.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\overline { \\Theta } _ { i } ( \\overline { \\mathbf { r } } ) \\equiv \\prod \\limits _ { \\substack { j = 1 , N ; \\\\ i < j } } \\overline { \\Theta } _ { i j } ( \\overline { \\mathbf { r } } ) , \\\\ \\overline { \\Theta } _ { i j } ( \\overline { \\mathbf { r } } ) \\equiv \\overline { \\Theta } \\left ( \\left \\vert \\mathbf { r } _ { i } - \\mathbf { r } _ { j } \\right \\vert - \\sigma \\right ) . \\end{array} \\right . \\end{align*}"}
-{"id": "9981.png", "formula": "\\begin{align*} u _ { t } ^ { i } = g ^ { i k } \\partial _ { x } \\frac { \\partial H } { \\partial u ^ { k } } , \\quad g ^ { i k } = - \\begin{pmatrix} 0 & 1 & 1 & 0 & 0 & 0 \\\\ 1 & 0 & 1 & 0 & 0 & 0 \\\\ 1 & 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 2 \\end{pmatrix} , \\end{align*}"}
-{"id": "2663.png", "formula": "\\begin{align*} f _ { j } ( z ) = \\sum _ { l = 1 } ^ { N + n } \\frac { z - \\varrho _ { l } } { 1 - z - \\varrho _ { l } } \\tilde { c } _ { l j } , j = 1 , \\ldots , N + n . \\end{align*}"}
-{"id": "4143.png", "formula": "\\begin{align*} \\varphi _ { 1 , \\pm } ( x ) = \\pm \\pi i \\mu ^ * ( [ 0 , x ] ) = \\pm \\pi i ( 1 - \\mu ^ * ( [ x , \\infty ) ) \\end{align*}"}
-{"id": "4282.png", "formula": "\\begin{align*} n \\binom { | V ( F ) | - 2 } { r - 1 } \\end{align*}"}
-{"id": "4468.png", "formula": "\\begin{align*} G ( \\nabla ^ 2 u , \\nabla u , u ) = F \\big ( A [ u ] \\big ) \\quad \\Psi ( \\nabla u , u , x ) = \\psi ^ { 1 / k } ( \\eta ) . \\end{align*}"}
-{"id": "249.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 2 n } X _ i ^ 2 u = | \\nabla _ z u | ^ 2 + | z | ^ 2 ( \\partial _ t u ) ^ 2 \\end{align*}"}
-{"id": "7160.png", "formula": "\\begin{align*} \\mathbb { F } = \\mathbb { F } _ { \\kappa _ { p ( t ) } } \\mathbb { G } . \\end{align*}"}
-{"id": "6524.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\bigl \\| { \\bf q } ( t , \\cdot ) \\bigr \\| _ { W ^ { p } } = 0 , \\mbox { i . e . , } \\lim _ { t \\rightarrow \\infty } \\ , \\lim _ { l \\rightarrow \\infty } \\ \\ \\sup _ { x \\geq 0 } \\Biggl [ \\frac { 1 } { l } \\int _ { x } ^ { x + l } \\bigl \\| q ( t + s ) \\bigr \\| ^ { p } \\ , d s \\Biggr ] ^ { 1 / p } = 0 . \\end{align*}"}
-{"id": "3101.png", "formula": "\\begin{align*} & \\textstyle \\sum _ v \\big ( 3 g _ v - 3 + n _ v - \\sum _ { l \\in L _ v } ( k _ l + b _ l ) - \\sum _ { e \\in E _ v } k _ { ( e , v ) } \\big ) + \\sum _ { l \\in L } k _ l \\\\ & \\textstyle + \\sum _ { e = ( v _ 1 , v _ 2 ) \\in E } ( k _ { ( e , v _ 1 ) } + k _ { ( e , v _ 2 ) } + 1 ) = 3 g - 3 + n - \\sum _ { l \\in L } b _ l . \\end{align*}"}
-{"id": "9926.png", "formula": "\\begin{align*} & \\lim _ { n \\to \\infty } D ( P ^ { \\mu } | _ { \\mathcal { F } ^ { \\mathcal { Y } } _ { 0 , n } } \\| P ^ { \\nu } | _ { \\mathcal { F } ^ { \\mathcal { Y } } _ { 0 , n } } ) = D ( P ^ { \\mu } | _ { \\mathcal { F } ^ { \\mathcal { Y } } _ { 0 , \\infty } } \\| P ^ { \\nu } | _ { \\mathcal { F } ^ { \\mathcal { Y } } _ { 0 , \\infty } } ) \\end{align*}"}
-{"id": "3814.png", "formula": "\\begin{align*} \\| g \\| _ { S , \\pi } : = \\inf \\left \\{ \\left | w \\right | _ \\pi : w \\in S , w = _ G g \\right \\} . \\end{align*}"}
-{"id": "426.png", "formula": "\\begin{align*} 2 \\pi G _ B + K _ H = \\int ^ \\infty _ 0 Q _ u \\dd u , \\end{align*}"}
-{"id": "7077.png", "formula": "\\begin{align*} c ( \\vec { e } _ i , \\vec { e } _ j ) = \\begin{cases} \\frac { 1 } { 2 ^ { k - 1 } } , & i = j , \\\\ \\frac { 1 } { 2 ^ k } , & i \\neq j . \\end{cases} \\end{align*}"}
-{"id": "8801.png", "formula": "\\begin{align*} H _ { i j } ( s , y ) = \\int _ { s } ^ { T } \\overline { \\phi } _ i ( t ) \\left ( \\int _ { 0 } ^ { \\pi } G _ { t - s } ( x , y ) \\phi _ j ( x ) \\ , \\textrm { d } x \\right ) \\ , \\textrm { d } t = \\int _ { s } ^ { T } \\overline { \\phi } _ i ( t ) \\phi _ j ( y ) e ^ { - j ^ 2 ( t - s ) } \\ , \\textrm { d } t . \\end{align*}"}
-{"id": "8167.png", "formula": "\\begin{align*} x _ \\theta ( z ) ^ { k + 1 } = \\sum _ { n \\in \\mathbb Z } \\Big ( \\sum _ { j _ 1 + \\dots + j _ { k + 1 } = n } x _ \\theta ( j _ 1 ) \\dots x _ \\theta ( j _ { k + 1 } ) \\Big ) z ^ { - n - k - 1 } = 0 . \\end{align*}"}
-{"id": "5533.png", "formula": "\\begin{align*} & \\sqrt { T } E ^ { \\Pi ^ { D _ T } } [ \\max _ { \\lambda \\leq J , k , j } | Z _ { \\lambda , k , j } | | X ^ T ] \\\\ & \\lesssim \\sqrt { 2 \\log 2 ( V _ J ^ { \\otimes d } ) } ( C _ T ' + 1 ) \\max _ { \\lambda \\leq J , k , j } ( \\| \\tilde { \\Phi } _ { \\lambda , k , j } \\| _ { \\mu _ 0 } ^ 2 + c _ T ' ) = O _ { P _ { b _ 0 } } ( \\sqrt { J } ) \\end{align*}"}
-{"id": "8504.png", "formula": "\\begin{align*} & | \\ddot { z } _ 1 ( t ) - \\ddot { z } _ 2 ( t ) | \\\\ \\leq & 1 0 \\epsilon ( 2 x ( t ) ) ( \\frac { | \\lambda | } { 4 \\pi x ( t ) } + 5 \\epsilon ) + ( 6 \\epsilon ) 2 0 \\epsilon x ( t ) + ( 1 0 \\epsilon ) 2 x ( t ) \\\\ = & 2 2 0 \\epsilon ^ 2 x ( t ) + \\epsilon ( 2 0 x ( t ) + \\frac { 5 | \\lambda | } { \\pi } ) . \\end{align*}"}
-{"id": "8.png", "formula": "\\begin{align*} \\delta Y _ t ^ { i , r } ( m ) \\leq \\bar { Y } _ t ^ r & = \\frac { C _ v C _ { \\eta } } { C _ { \\eta } - C _ v } \\frac { e ^ { \\rho t } ( e ^ { - ( \\rho t + C _ { \\eta } ( t - r ) ) } - e ^ { - ( \\rho m + C _ { \\eta } ( m - r ) ) } ) } { \\rho + C _ { \\eta } } | v - \\bar { v } | \\\\ & \\leq \\frac { C _ v } { C _ { \\eta } - C _ v } | v - \\bar { v } | \\end{align*}"}
-{"id": "3674.png", "formula": "\\begin{align*} \\sum _ { i \\in U } H ( \\delta _ i ; \\eta ) = 0 \\end{align*}"}
-{"id": "2167.png", "formula": "\\begin{align*} \\int \\tau _ { \\mathbf { x } } f ( \\mathbf { z } ) g ( \\mathbf { z } ) \\ , d w ( \\mathbf { z } ) = \\int f ( \\mathbf { z } ) \\tau _ { - \\mathbf { x } } g ( \\mathbf { z } ) \\ , d w ( \\mathbf { z } ) . \\end{align*}"}
-{"id": "5007.png", "formula": "\\begin{align*} A = A ^ { ( 0 ) } \\cup \\bigcup _ { k = 0 } ^ \\infty A ^ { ( k ) } , \\end{align*}"}
-{"id": "1379.png", "formula": "\\begin{align*} m ( J , q ) = l J _ { ( l ) } \\leq q < J _ { ( l + 1 ) } \\ ; . \\end{align*}"}
-{"id": "8732.png", "formula": "\\begin{align*} \\sum _ { k = j } ^ m \\binom { m } { k } \\binom { k } { j } ( - 1 ) ^ { k - j } = \\left \\{ \\begin{aligned} & 0 , { \\qquad } 0 \\leq j \\leq m - 1 , \\\\ & 1 , { \\qquad } j = m . \\end{aligned} \\right . \\end{align*}"}
-{"id": "1826.png", "formula": "\\begin{align*} \\psi ^ { \\rm t o p . } ( x , t ) = \\sqrt { \\frac { \\ell } { 2 L } } \\left ( 1 - i \\frac { t } { \\ell } \\right ) \\end{align*}"}
-{"id": "2208.png", "formula": "\\begin{align*} \\boldsymbol { u } ^ * = - \\frac { 1 } { 2 } { R } ^ { - 1 } B ^ T \\left ( P ^ T + P \\right ) \\boldsymbol { x } \\end{align*}"}
-{"id": "7596.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { I _ 3 } _ { L ^ \\infty ( 0 , T ; C ^ { \\alpha , p } ) } \\le \\frac { C } { \\nu } \\left ( \\frac { T } { \\nu } \\right ) ^ { \\frac { 1 } { 2 } } M _ X ^ { 1 + 4 \\alpha } \\norm { X - \\mathrm { I d } } _ { L i p ( 0 , T ; C ^ { 1 + \\alpha } ) } \\\\ \\norm { X ' } _ { L i p ( 0 , T ; C ^ { 1 + \\alpha } ) } \\norm { \\tau } _ { L ^ \\infty ( 0 , T ; C ^ { \\alpha , p } ) } . \\end{gathered} \\end{align*}"}
-{"id": "3411.png", "formula": "\\begin{align*} \\rho _ * ( T ) = \\frac { \\prod _ i \\rho _ 0 ( T _ i ) } { 1 - \\prod _ i \\rho _ 0 ( T _ i ) + \\prod _ i \\left ( 1 + \\rho _ * ( T _ i ) \\right ) } , \\end{align*}"}
-{"id": "1047.png", "formula": "\\begin{align*} & \\lim _ { n \\to + \\infty } \\det ( - A ) / \\det ( - B ) = 1 . \\end{align*}"}
-{"id": "9816.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } L _ a \\hat { w } & = & | y | ^ a \\hat { f } & \\R ^ { n + 1 } \\setminus \\{ x _ n = y = 0 \\} \\\\ \\hat { w } ( x ' , 0 , 0 ) & = & 0 & \\R ^ { n - 1 } \\\\ \\lim _ { | X | \\to \\infty } \\hat { w } ( X ) & = & 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "5136.png", "formula": "\\begin{align*} \\begin{pmatrix} \\Gamma _ { m _ j , k _ j } ^ + ( f _ { m _ j , k _ j } ) \\\\ \\Gamma _ { m _ j , k _ j } ^ - ( f _ { m _ j , k _ j } ) \\end{pmatrix} : = E _ { k _ j } \\begin{pmatrix} A ^ + \\\\ A ^ - \\end{pmatrix} . \\end{align*}"}
-{"id": "8633.png", "formula": "\\begin{align*} z _ \\varepsilon ^ \\pm - z _ 0 ( 0 ) = O ( \\varepsilon ) \\ \\ \\textrm { a s } \\ \\varepsilon \\to 0 , \\end{align*}"}
-{"id": "3393.png", "formula": "\\begin{align*} f ( T ) = F ( T ) - \\sum _ i F ( T _ i ) = \\log \\left ( \\frac { I ( T ) } { \\prod _ i I ( T _ i ) } \\right ) = \\log \\left ( \\frac { I ( T ) } { I _ 0 ( T ) } \\right ) = \\log \\left ( 1 + \\prod _ i \\frac { I _ 0 ( T _ i ) } { I ( T _ i ) } \\right ) . \\end{align*}"}
-{"id": "2958.png", "formula": "\\begin{align*} \\theta _ 1 & = \\alpha _ 1 \\nabla _ 1 + \\alpha _ 2 \\nabla _ 2 + \\alpha _ 3 \\nabla _ 3 + \\alpha _ 4 \\nabla _ 4 , \\\\ \\theta _ 2 & = \\beta _ 1 \\nabla _ 1 + \\beta _ 2 \\nabla _ 2 + \\beta _ 3 \\nabla _ 3 . \\end{align*}"}
-{"id": "1101.png", "formula": "\\begin{align*} & \\mathbb { E } \\sum _ { i _ 1 , \\cdots , i _ { k } \\ } \\prod _ { j = 1 } ^ k ( { \\tau } ^ * _ { i _ j } - x _ j ) _ + \\\\ = & \\mathbb { E } ( n S ( I ) / 4 ) ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } | \\Sigma _ k ( G _ n ( x _ 1 ) / S ( I ) , \\cdots , G _ n ( x _ k ) / S ( I ) ) | \\\\ = & ( S ( I ) / 4 ) ^ k \\int _ { I ^ k } \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) d y _ 1 \\cdots d y _ k . \\end{align*}"}
-{"id": "4747.png", "formula": "\\begin{align*} e ^ { \\alpha \\xi _ i + \\beta ( A \\xi ) _ i } e ^ { W ( \\xi ) } = e ^ { W \\left ( \\xi + { \\bf e } _ i \\right ) } , \\end{align*}"}
-{"id": "9413.png", "formula": "\\begin{align*} \\mathrm { R e s } _ { Q _ { \\tau } } f \\ ; d z = \\begin{cases} \\frac { 1 } { e _ { \\tau } } \\mathrm { R e s } _ { \\tau } f , & \\tau \\in \\mathbb { H } , \\\\ \\frac { 1 } { 2 \\pi i } \\alpha _ { \\tau } a _ { \\tau } ( 0 ) , & \\tau \\in \\mathcal { C } _ N . \\end{cases} \\end{align*}"}
-{"id": "523.png", "formula": "\\begin{align*} \\phi ( y ) = g ^ { - 1 } ( y ^ m ) = x . \\end{align*}"}
-{"id": "6612.png", "formula": "\\begin{align*} | v _ n | & = 2 \\left | \\sin \\frac { \\pi } { 2 } ( - \\varphi ) ^ { n } ( 1 - 2 \\varepsilon ( - \\varphi ) ^ { - n } ) \\right | \\\\ & = \\pi \\varphi ^ { n } \\ , | 1 - 2 \\varepsilon ( - \\varphi ) ^ { - n } | \\ , ( 1 + \\mathcal { O } ( \\varphi ^ { 2 n } ) ) \\\\ & \\leq \\pi \\varphi ^ n \\sqrt { 5 } \\left ( 1 + \\mathcal { O } ( \\varphi ^ { 2 n } ) \\right ) . \\end{align*}"}
-{"id": "7423.png", "formula": "\\begin{align*} \\displaystyle Z _ + ( s , \\lambda ) = Z _ - ( - s , \\lambda ) , \\ ; \\ ; s \\in \\C , \\lambda \\in U . \\end{align*}"}
-{"id": "9369.png", "formula": "\\begin{align*} & D _ 1 ( G _ k ) = \\langle x , y \\rangle D _ 2 ( G _ k ) , \\\\ & D _ 2 ( G _ k ) = \\langle x ^ 2 , y ^ 2 , [ y , x ] \\rangle D _ { 3 } ( G _ k ) , \\\\ & D _ i ( G _ k ) = \\langle x ^ { 2 ^ l } , [ y , x , \\overset { i - 3 } \\ldots , x , y ] , [ y , x , \\overset { i - 1 } { \\ldots } , x ] \\rangle D _ { i + 1 } ( G _ k ) \\end{align*}"}
-{"id": "4541.png", "formula": "\\begin{align*} \\int \\int | \\hat { \\nu } _ g ( \\omega ) | ^ 2 | \\psi ( 2 ^ { - j } \\omega ) | ^ 2 d \\omega d g = \\int \\int | \\hat { \\mu } ( \\omega ) | ^ 2 | \\hat { \\mu } ( g \\omega ) | ^ 2 | \\psi ( 2 ^ { - j } \\omega ) | ^ 2 d \\omega d g . \\end{align*}"}
-{"id": "9427.png", "formula": "\\begin{align*} F ( z ) = ( z - \\tau ) ^ { 1 2 \\nu _ \\tau ^ { ( N ) } ( f ) } F _ 0 ( z ) , \\end{align*}"}
-{"id": "7935.png", "formula": "\\begin{align*} \\abs { T _ i } \\leq 2 \\abs { T _ i \\cap R } \\leq \\sum _ { j = 1 } ^ m ( d _ j ^ { T _ i \\cap R } - 1 ) + 1 \\leq \\sum _ { j = 1 } ^ m ( d _ j ^ { T _ i } - 1 ) + 1 , \\end{align*}"}
-{"id": "7879.png", "formula": "\\begin{align*} | \\Bbbk _ { l } ^ { ( m ) } ( | S | ) | = | \\Bbbk _ { l } ^ { ( m ) } ( m - | S | ) | \\le e ^ { - \\frac { c l ( m - | S | ) } { m } } . \\end{align*}"}
-{"id": "4347.png", "formula": "\\begin{align*} H _ \\tau = \\frac { H \\times T } { Z ( \\widetilde { G } ) } \\end{align*}"}
-{"id": "9846.png", "formula": "\\begin{align*} \\sum _ { i = 2 } ^ { r } ( - t ) ^ { 4 - i } ( 1 - x ) ^ r _ { [ i ] } \\end{align*}"}
-{"id": "2049.png", "formula": "\\begin{align*} R _ * ' \\frac { q } { R _ * } = q ' \\end{align*}"}
-{"id": "6175.png", "formula": "\\begin{align*} \\int _ { B _ 1 } ( U _ n ^ 2 \\nabla g _ 1 \\cdot \\nabla \\varphi _ 1 + \\bar U _ n ^ 2 \\nabla g _ 2 \\cdot \\nabla \\varphi _ 2 ) d x = 0 , \\forall \\varphi _ i \\in C _ 0 ^ \\infty ( B _ 1 ) , i = 1 , 2 , \\end{align*}"}
-{"id": "9881.png", "formula": "\\begin{align*} \\tilde { \\pi } ^ { \\nu } _ { n - } ( \\cdot ) : = \\pi _ { n - } ^ { \\nu } ( \\cdot + a _ { n } ) \\end{align*}"}
-{"id": "5083.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j \\in \\mathbb { S } } c _ j y _ j \\right \\| \\leq \\left \\| \\sum _ { j \\in \\mathbb { S } } c _ j ( x _ j - y _ j ) \\right \\| + \\left \\| \\sum _ { j \\in \\mathbb { S } } c _ j x _ j \\right \\| = ( 1 + \\alpha ) \\left \\| \\sum _ { j \\in \\mathbb { S } } c _ j x _ j \\right \\| + \\beta \\left \\| \\sum _ { j \\in \\mathbb { S } } c _ j y _ j \\right \\| + \\gamma \\left ( \\sum \\limits _ { j \\in \\mathbb { S } } | c _ j | ^ 2 \\right ) ^ \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "6234.png", "formula": "\\begin{align*} ( p _ { s ( e ) } W ^ * ( Y _ { e } Y _ { e } ^ { * } ) p _ { s ( e ) } , \\phi ^ p ) \\cong \\begin{cases} ( L ( \\Z ) , \\tau ) & \\mu ( e ) \\geq 1 \\\\ \\underset { \\phi ( p _ { s ( e ) } ) \\mu ( e ) } { ( L ( \\Z ) , \\tau ) } \\oplus \\underset { \\phi ( p _ { s ( e ) } ) ( 1 - \\mu ( e ) ) } { \\C } & \\mu ( e ) < 1 \\end{cases} . \\end{align*}"}
-{"id": "8866.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 e _ 3 } { \\partial a ^ 2 } & = 3 \\nu ^ 4 - 5 5 \\nu ^ 3 - 2 1 \\nu ^ 2 + 4 7 1 \\nu + 1 0 1 0 > 0 \\end{align*}"}
-{"id": "7306.png", "formula": "\\begin{align*} \\frac { \\delta S } { \\delta \\omega } = & \\Pi ^ { n - 2 } \\left ( ( - 1 ) ^ { n - 1 } d \\theta ^ { n - 2 } + \\theta ^ { n - 2 } \\omega + ( - 1 ) ^ { n - 1 } \\omega \\theta ^ { n - 2 } \\right ) = \\\\ = & \\Pi ^ { n - 2 } ( \\theta ^ { n - 3 } ( d \\theta + \\omega \\theta + \\theta \\omega ) ) . \\end{align*}"}
-{"id": "8339.png", "formula": "\\begin{align*} ( I - \\mathbb { H } ) \\frac { Z _ { \\alpha } } { ( Z ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } = - & \\partial _ { \\alpha } ( I - \\mathbb { H } ) \\frac { 1 } { Z ( \\alpha , t ) - z _ j ( t ) } \\\\ = & - \\partial _ { \\alpha } \\frac { 2 } { ( \\Phi ^ { - 1 } ) _ z ( \\Phi ( z _ j ( t ) ) ) ( \\alpha - \\Phi ( z _ j ( t ) ) ) } \\\\ = & \\frac { 2 } { ( \\Phi ^ { - 1 } ) _ z ( \\Phi ( z _ j ( t ) ) ) ( \\alpha - \\Phi ( z _ j ( t ) ) ) ^ 2 } . \\end{align*}"}
-{"id": "1396.png", "formula": "\\begin{align*} f ( x _ 1 , \\ldots , x _ d ) & = h ( p _ 1 ( x _ 1 ) + \\cdots + p _ d ( x _ d ) ) ~ ~ \\\\ f ( x _ 1 , \\ldots , x _ d ) & = h ( p _ 1 ( x _ 1 ) \\cdot \\ldots \\cdot p _ d ( x _ d ) ) , \\end{align*}"}
-{"id": "707.png", "formula": "\\begin{align*} \\\\ \\lim _ { x \\to 1 ^ - } ( 1 - x ) ^ { k + 4 } Z ^ { ( k + 1 ) } ( x ) & = 3 \\sum _ { l = 0 } ^ k \\binom { k } { l } \\lim _ { x \\to 1 ^ - } \\left ( ( 1 - x ) ^ { l + 1 } f ^ { ( l ) } ( x ) \\right ) \\lim _ { x \\to 1 ^ - } \\left ( ( 1 - x ) ^ { k - l + 3 } Z ^ { ( k - l ) } ( x ) \\right ) , \\\\ & = 3 \\frac { 2 p - 1 } { 8 } \\sum _ { l = 0 } ^ k \\binom { k } { l } l ! ( k + 2 - l ) ! , \\\\ & = 3 \\frac { 2 p - 1 } { 8 } k ! \\sum _ { l = 0 } ^ k ( k + 2 - l ) ( k + 1 - l ) , \\\\ & = 3 \\frac { 2 p - 1 } { 8 } k ! \\sum _ { l = 1 } ^ { k + 1 } l ( l + 1 ) = \\frac { 2 p - 1 } { 8 } ( k + 3 ) ! . \\end{align*}"}
-{"id": "2762.png", "formula": "\\begin{align*} \\frac { z + w } { z - w } \\prod _ { k = 1 } ^ { m } \\frac { z - \\pi _ { k } } { z + \\pi _ { k } } \\frac { w + \\pi _ { k } } { w - \\pi _ { k } } = \\frac { z + w } { z - w } + \\sum _ { p = 1 } ^ { m } \\Big ( \\frac { w + \\pi _ { p } } { w - \\pi _ { p } } - \\frac { z - \\pi _ { p } } { z + \\pi _ { p } } \\Big ) \\prod _ { k = 1 } ^ { p - 1 } \\frac { z - \\pi _ { k } } { z + \\pi _ { k } } \\frac { w + \\pi _ { k } } { w - \\pi _ { k } } , \\end{align*}"}
-{"id": "1151.png", "formula": "\\begin{align*} | y _ 0 - y _ { k + 1 } | > \\max ( | y _ 0 - y _ i | , | y _ i - y _ { k + 1 } | ) \\geq \\min ( | y _ 0 - y _ i | , | y _ i - y _ { k + 1 } | ) > \\\\ \\min \\left ( \\left | y _ 0 ^ 2 - y _ i ^ 2 \\right | / 4 , \\left | y _ i ^ 2 - y _ { k + 1 } ^ 2 \\right | / 4 \\right ) = \\left | ( 1 + | u _ i | / \\ln n ) ^ 2 - 1 \\right | S ( I ) ^ 2 / 4 \\geq \\varepsilon _ 0 ( \\ln n ) ^ { - 1 } . \\end{align*}"}
-{"id": "5481.png", "formula": "\\begin{align*} v ( t , x ) + \\psi ( t , x ) \\ = \\ 0 , \\end{align*}"}
-{"id": "2745.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\phi } \\Re { f \\Big ( \\tau + \\frac { R } { \\eta _ { - } } e ^ { i \\phi } \\Big ) } & = \\Big ( - x + \\frac { 1 } { 1 + R ^ 2 + 2 R \\cos \\phi } \\Big ) R \\sin \\phi \\\\ & < - \\Big ( x - \\frac { 1 } { ( R - 1 ) ^ 2 } \\Big ) R \\sin \\phi < 0 , \\end{align*}"}
-{"id": "7800.png", "formula": "\\begin{align*} \\varphi ^ G _ { \\zeta } ( u ) = { \\rm V o l } _ { d - 1 } ( A _ { \\zeta } ( u ) ) , \\end{align*}"}
-{"id": "9898.png", "formula": "\\begin{align*} g ( y ) & = \\begin{cases} 0 & y < - M + a + 1 \\\\ 2 f ( y - 1 ) & y \\in [ - M + a + 1 , - M + a + 3 ) \\\\ 2 f ( y - 1 ) - g ( y - 2 ) & y \\in [ - M + a + 3 , M + a + 1 ] \\\\ - g ( y - 2 ) & y > M + a + 1 \\end{cases} \\end{align*}"}
-{"id": "1670.png", "formula": "\\begin{align*} & B _ { t } ( m , \\ell ^ { 2 n + 1 } d ) - \\ell ^ { \\lambda - 1 } \\left ( \\frac { ( - 1 ) ^ { \\lambda } m } { \\ell } \\right ) B _ { t } ( m , \\ell ^ { 2 n - 1 } d ) \\\\ = & B _ { t } ( m , \\ell ^ { 2 n + 2 } ( d / \\ell ) ) - \\ell ^ { \\lambda - 1 } \\left ( \\frac { ( - 1 ) ^ { \\lambda } m } { \\ell } \\right ) B _ { t } ( m , \\ell ^ { 2 n } ( d / \\ell ) ) \\\\ = & \\ell ^ { ( k - 2 ) n } \\ , ( B _ { t } ( \\ell ^ { 2 n } m , \\ell d ) - B _ { t } ( \\ell ^ { 2 n - 2 } m , d / \\ell ) ) \\equiv 0 \\pmod { \\ell ^ { ( k - 2 ) n } } . \\end{align*}"}
-{"id": "8370.png", "formula": "\\begin{align*} | \\bar { F } ( z _ j ( t ) , t ) | \\leq \\norm { F ( \\cdot , t ) } _ { L ^ { \\infty } ( \\Sigma ( t ) ) } = \\norm { z _ t - \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i } { 2 \\pi } \\frac { 1 } { \\overline { z ( \\alpha , t ) - z _ j ( t ) ) } } } _ { \\infty } . \\end{align*}"}
-{"id": "6326.png", "formula": "\\begin{align*} \\tfrac { d } { d t } \\mathcal { E } ( q ( t ) ) + \\int _ 0 ^ 1 { q ^ 2 ( q _ { x x } - \\nu ^ { - \\frac { 1 } { 2 } } f _ 0 ( x ) ) ^ 2 \\ , d x } = 0 , \\end{align*}"}
-{"id": "7329.png", "formula": "\\begin{align*} E _ { u , v } ( t ) \\geq { } & \\frac { 1 } { 2 } \\int _ { \\Omega } \\rho ( x ) | u _ t ( x , t ) | ^ { 2 } + \\rho ( x ) | v _ t ( x , t ) | ^ { 2 } \\ , d x \\\\ & { } + { } \\frac { 1 } { 2 } \\int _ { \\Omega } \\nabla u ( x , t ) ^ { \\top } \\cdot K ( x ) \\cdot \\nabla u ( x , t ) + \\nabla v ( x , t ) ^ { \\top } \\cdot K ( x ) \\cdot \\nabla v ( x , t ) \\ , d x \\\\ = { } & \\frac { 1 } { 2 } \\| ( u , v , u _ t , v _ t ) \\| _ { \\mathcal { H } } ^ { 2 } . \\end{align*}"}
-{"id": "3475.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\iint _ { ( C , \\infty ) ^ 2 } d x d r \\left \\lvert G _ N ( x , r ) - G _ { \\infty , \\gamma } ( x , r ) \\right \\rvert ^ 2 = { } & 0 , \\\\ \\lim _ { N \\to \\infty } \\iint _ { ( C , \\infty ) ^ 2 } d r d y \\left \\lvert H _ N ( r , y ) - H _ { \\infty , \\gamma } ( r , y ) \\right \\rvert ^ 2 = { } & 0 . \\end{align*}"}
-{"id": "7027.png", "formula": "\\begin{align*} f _ { n } ( x ) = n ^ { 3 } \\left [ \\left ( \\frac { x } { n } \\right ) ^ { 3 } + a \\left ( \\frac { x } { n } \\right ) + b \\right ] \\end{align*}"}
-{"id": "7394.png", "formula": "\\begin{align*} \\overline { P } ^ S = \\alpha ^ S \\frac { \\sum _ t \\overline { P } ^ D _ t } { \\sum _ t \\rho ^ S _ t } \\overline { P } ^ W = \\alpha ^ W \\frac { \\sum _ t \\overline { P } ^ D _ t } { \\sum _ t \\rho ^ W _ t } \\end{align*}"}
-{"id": "3773.png", "formula": "\\begin{align*} \\tau = \\frac { ( 2 r _ 1 / \\lambda ) ^ { - s } - r _ 2 ^ { - s } } { r _ 0 ^ { - s } - r _ 2 ^ { - s } } . \\end{align*}"}
-{"id": "8762.png", "formula": "\\begin{align*} - \\frac { \\overline { F } } { 2 F } \\ , F _ a + \\frac 1 2 F ^ c \\overline { g } _ { c a } = \\overline { F } \\psi _ a \\ , . \\end{align*}"}
-{"id": "8428.png", "formula": "\\begin{align*} g _ c : = - 2 [ \\bar { f } , \\mathfrak { H } \\frac { 1 } { z _ { \\alpha } } + \\bar { \\mathfrak { H } } \\frac { 1 } { \\bar { z } _ { \\alpha } } ] \\bar { f } _ { \\alpha } + \\frac { 1 } { \\pi i } \\int \\Big ( \\frac { z _ t ( \\alpha , t ) - z _ t ( \\beta , t ) } { z ( \\alpha , t ) - z ( \\beta , t ) } \\Big ) ^ 2 ( z - \\bar { z } ) _ { \\beta } d \\beta . \\end{align*}"}
-{"id": "8521.png", "formula": "\\begin{align*} \\norm { \\frac { 1 } { \\pi } \\sum _ { j = 1 } ^ 2 \\frac { 2 \\lambda _ j D _ t ^ 2 \\zeta } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } } _ { H ^ s } \\leq \\| D _ t ^ 2 \\zeta \\| _ { H ^ s } \\norm { \\frac { 2 } { \\pi } \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } } _ { H ^ s } \\end{align*}"}
-{"id": "2327.png", "formula": "\\begin{align*} a _ x & \\mapsto a ^ T _ x , & a ^ + _ x \\mapsto ( a ^ + _ x ) ^ T & = a _ x \\end{align*}"}
-{"id": "7839.png", "formula": "\\begin{align*} \\| f \\| _ { \\ell ^ p ( \\mathcal Z ) } = \\Big ( \\sum _ { m \\in \\mathcal Z } | f ( m ) | ^ p \\Big ) ^ { 1 / p } \\| f \\| _ { \\ell ^ { \\infty } ( \\mathcal Z ) } = \\sup _ { m \\in \\mathcal Z } | f ( m ) | . \\end{align*}"}
-{"id": "1020.png", "formula": "\\begin{align*} \\frac { n ^ 2 \\alpha _ n ^ 2 } { 8 } = \\frac { n ^ 2 F _ n ^ 2 ( x ) } { 3 2 } & = \\frac { 3 2 \\ln n } { 3 2 } + \\frac { 8 x + 3 \\ln ( 2 \\ln n ) } { ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } } \\frac { ( 3 2 \\ln n ) ^ { \\frac { 1 } { 2 } } } { 3 2 } + \\frac { ( 8 x + 3 \\ln ( 2 \\ln n ) ) ^ 2 } { 3 2 \\cdot 4 \\cdot ( 2 \\ln n ) } \\\\ & = \\ln n + \\frac { 8 x + 3 \\ln ( 2 \\ln n ) } { 8 } + o ( 1 ) \\end{align*}"}
-{"id": "296.png", "formula": "\\begin{align*} \\Lambda _ { [ \\beta ] } \\ ; = \\ ; [ \\beta _ 1 ] _ * ( \\Lambda ) \\circ \\cdots \\circ [ \\beta _ \\ell ] _ * ( \\Lambda ) \\textnormal { f o r p a r t i t i o n s } \\beta \\ ; = \\ ; ( \\beta _ 1 , \\dots , \\beta _ \\ell ) . \\end{align*}"}
-{"id": "7560.png", "formula": "\\begin{align*} \\begin{gathered} \\left [ \\tau \\circ X ^ { - 1 } ( t ) \\right ] _ \\alpha \\le \\left [ \\tau ( t ) \\right ] _ \\alpha \\norm { \\nabla _ x X ^ { - 1 } ( t ) } _ { L ^ \\infty } ^ { \\alpha } \\le \\left [ \\tau ( t ) \\right ] _ \\alpha ( 1 + \\norm { X - \\mathrm { I d } } _ { L ^ \\infty ( 0 , T ; C ^ { 1 + \\alpha } ) } ) ^ \\alpha . \\end{gathered} \\end{align*}"}
-{"id": "2279.png", "formula": "\\begin{align*} 1 = & \\mathcal { L } \\delta & z = & i \\mathcal { L } \\delta ' \\end{align*}"}
-{"id": "4711.png", "formula": "\\begin{align*} q _ 0 ^ 2 = \\frac { 1 } { 3 } f _ { j - 2 } ^ { + } - \\frac { 7 } { 6 } f _ { j - 1 } ^ { + } + \\frac { 1 1 } { 6 } f _ j ^ { + } , ~ q _ 1 ^ 2 = - \\frac { 1 } { 6 } f _ { j - 1 } ^ { + } + \\frac { 5 } { 6 } f _ { j } ^ { + } + \\frac { 1 } { 3 } f _ { j + 1 } ^ { + } , ~ q _ 2 ^ 2 = \\frac { 1 } { 3 } f _ { j } ^ { + } + \\frac { 5 } { 6 } f _ { j + 1 } ^ { + } - \\frac { 1 } { 6 } f _ { j + 2 } ^ { + } , \\end{align*}"}
-{"id": "1079.png", "formula": "\\begin{align*} \\frac { n ^ 2 \\alpha _ n ^ 2 } { 4 \\ln n } = \\frac { n ^ 2 G _ n ^ 2 ( x ) } { 1 6 \\ln n } \\to 2 \\end{align*}"}
-{"id": "4457.png", "formula": "\\begin{align*} ( \\phi _ { i k } ) _ { Z _ { i j k } } = ( \\phi _ { i j } ) _ { Z _ { i j k } } \\circ ( \\phi _ { j k } ) _ { Z _ { i j k } } \\end{align*}"}
-{"id": "6796.png", "formula": "\\begin{align*} \\partial ^ 2 _ { j k } u ^ i + \\partial ^ 2 _ { i k } u ^ j = 0 \\ , , 1 \\leq i , j , k \\leq n \\ , . \\end{align*}"}
-{"id": "5986.png", "formula": "\\begin{align*} d Y ( t ) = - g ^ * ( t , Z ( t ) ) d t + Z ( t ) d W ( t ) , Y ( 1 ) = F ( W ) . \\end{align*}"}
-{"id": "4761.png", "formula": "\\begin{align*} x = \\sum _ { j = M } ^ { \\infty } x _ j t ^ j \\end{align*}"}
-{"id": "4811.png", "formula": "\\begin{align*} \\tilde u _ { i , n } : = \\frac { \\sigma _ i } { \\sigma _ 1 ^ 2 + \\cdots + \\sigma _ d ^ 2 } ( b _ { 1 , n } + \\tilde b _ 2 ) = : b _ { 1 , n } ^ { \\sigma _ i } + \\tilde b _ 2 ^ { \\sigma _ i } \\tilde u _ { i } = \\frac { \\sigma _ i } { \\sigma _ 1 ^ 2 + \\cdots + \\sigma _ d ^ 2 } ( b _ { 1 } + \\tilde b _ 2 ) = ; b _ 1 ^ { \\sigma _ i } + b _ 2 ^ { \\sigma _ i } . \\end{align*}"}
-{"id": "1502.png", "formula": "\\begin{align*} m ( t , \\cdot ) = e ^ { - \\Delta t } m _ { 0 } - \\varepsilon \\int _ { 0 } ^ { t } e ^ { - \\Delta ( t - s ) } \\mathrm { d i v } ( m \\bar { H } _ { p } ( s , \\cdot , m , D u ) \\ d s . \\end{align*}"}
-{"id": "4859.png", "formula": "\\begin{align*} \\alpha ( \\delta ( \\mu ) ) = \\frac { a } { 2 } \\rho _ { j } [ w _ { ( j , r _ s ) } , w ^ * _ { ( j , r _ s ) } ] + \\frac { a } { 2 } \\sum _ { i \\neq j } ^ { } \\rho _ { i } [ w _ { ( i , r _ s ) } , w ^ * _ { ( j , r _ s ) } ] + \\frac { a } { 2 } \\ , \\ , \\sum _ { t \\neq j } [ \\alpha ( w _ { ( t , r _ s ) } ) , w ^ * _ { ( t , r _ s ) } ] . \\end{align*}"}
-{"id": "6862.png", "formula": "\\begin{align*} \\pi _ k \\left ( ( x _ 1 , \\dots , x _ n ) \\right ) = ( x _ { k + 1 } , \\dots , x _ n ) . \\end{align*}"}
-{"id": "7875.png", "formula": "\\begin{align*} \\frac 1 { { m \\choose l } } \\sum _ { \\substack { J \\subseteq I \\\\ | J | = l } } w _ S ( \\varepsilon ( J ) ) & = \\frac { 1 } { { m \\choose l } } \\sum _ { j = 0 } ^ { l } ( - 1 ) ^ j \\binom { | S | } { j } \\binom { m - | S | } { l - j } \\\\ & = \\Bbbk _ { l } ^ { ( m ) } ( | S | ) , \\end{align*}"}
-{"id": "5938.png", "formula": "\\begin{align*} { \\rm c o e f f } _ P ( B _ S ) = 1 - \\frac { 1 } { m } + \\sum _ { i = 1 } ^ n \\frac { \\alpha _ i \\lambda _ i } { m } + \\frac { \\beta } { m ' p } , \\end{align*}"}
-{"id": "8017.png", "formula": "\\begin{align*} \\Omega _ R / d R = \\mathbb C \\omega _ 0 \\oplus \\ldots \\oplus \\mathbb C \\omega _ { 2 n } \\end{align*}"}
-{"id": "1265.png", "formula": "\\begin{align*} I ( S , S , S ) ( \\xi ) = \\frac { e ^ { - i a \\ln | \\xi | } } { | \\xi | } \\left ( \\bar E + \\bar F e ^ { - 2 i a \\ln | \\xi | } e ^ { 8 i \\xi ^ 3 / 9 } \\right ) + O ( | \\xi | ^ { - 2 + \\gamma / 2 } ) \\end{align*}"}
-{"id": "4384.png", "formula": "\\begin{align*} | I - p _ { \\pm } | = | ( \\delta ^ 3 / 2 , \\pm \\delta , - \\delta / 2 ) | < 2 \\delta . \\end{align*}"}
-{"id": "7604.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { \\nabla _ x X ^ { - 1 } ( s ) - \\nabla _ x X ^ { - 1 } ( t ) } _ { C ^ { \\alpha } } \\le | t - s | \\norm { X - \\mathrm { I d } } _ { L i p ( 0 , T ; C ^ { 1 + \\alpha } ) } M _ X ^ { 1 + 2 \\alpha } . \\end{gathered} \\end{align*}"}
-{"id": "8297.png", "formula": "\\begin{align*} \\begin{cases} ( D _ t ^ 2 - i A \\partial _ { \\alpha } ) \\zeta = - i \\\\ ( I - \\mathcal { H } ) D _ t \\bar { \\zeta } = 0 , \\end{cases} \\end{align*}"}
-{"id": "3750.png", "formula": "\\begin{align*} \\mathcal { P = } \\sum _ { j , l , m } q _ { l } ^ { ( j ) } | a _ { l , m } ^ { ( j ) } | ^ { 2 } , \\end{align*}"}
-{"id": "9791.png", "formula": "\\begin{align*} { \\rm I } + { \\rm I I } : = \\| u ( r \\ , \\cdot \\ , ) - P _ { X _ \\circ } ( r \\ , \\cdot \\ , ) \\| _ { L ^ 2 ( B _ { 1 / 2 } ( r ^ { - 1 } X _ \\circ ) , | y | ^ a ) } + \\| u ( r \\ , \\cdot \\ , ) - P _ { Z _ \\circ } ( r \\ , \\cdot \\ , ) \\| _ { L ^ 2 ( B _ { 1 / 2 } ( r ^ { - 1 } X _ \\circ ) , | y | ^ a ) } . \\end{align*}"}
-{"id": "5914.png", "formula": "\\begin{align*} K _ { X ' } + B ' + M ' = \\pi ^ * ( K _ X + B + M ) , \\end{align*}"}
-{"id": "1420.png", "formula": "\\begin{align*} y _ { t + 1 } \\ \\ \\ = \\ \\ \\ E \\big [ \\min \\big ( U ^ 2 , y _ t \\big ) \\big ] \\ \\ \\ = \\ \\ \\ y _ t - \\frac { 2 } { 3 } y _ t ^ { \\frac { 3 } { 2 } } . \\end{align*}"}
-{"id": "384.png", "formula": "\\begin{align*} x _ { 1 , 2 } & = \\pm \\alpha _ 2 , & x _ { 5 , 6 } = \\pm \\sqrt { | s | ^ 2 - | a c | + \\sqrt { ( | s | ^ 2 - | a c | ) ^ 2 - 1 } } , \\\\ x _ { 3 , 4 } & = \\pm | s - \\sqrt { a c } | , & x _ { 7 , 8 } = \\pm \\sqrt { | s | ^ 2 - | a c | - \\sqrt { ( | s | ^ 2 - | a c | ) ^ 2 - 1 } } , \\end{align*}"}
-{"id": "6402.png", "formula": "\\begin{align*} \\left \\Vert p _ { k } \\right \\Vert ^ { 2 } = \\left \\Vert r _ { k } \\right \\Vert ^ { 2 } \\left ( 1 + \\delta _ { k } \\frac { \\left \\Vert p _ { k - 1 } \\right \\Vert ^ { 2 } } { \\| r _ { k - 1 } \\| ^ { 2 } } \\right ) , \\end{align*}"}
-{"id": "4149.png", "formula": "\\begin{align*} \\varphi _ { 1 , - } ( x ) = \\varphi _ { 1 , + } ( x ) + \\varphi _ { 2 , + } ( x ) , x \\in \\Delta _ 2 \\cap O _ V . \\end{align*}"}
-{"id": "9398.png", "formula": "\\begin{align*} \\langle I ^ { \\prime } ( u _ { n } ) , \\xi _ { j , \\delta } u _ { n } \\rangle = & \\int _ { \\mathbb { R } ^ { N } } \\int _ { \\mathbb { R } ^ { N } } \\Phi ( u _ { n } ( x ) - u _ { n } ( y ) ) ( \\xi _ { j , \\delta } u _ { n } ( x ) - \\xi _ { j , \\delta } u _ { n } ( y ) ) K ( x , y ) d x d y \\\\ & - \\int _ { \\Omega } | u _ { n } | ^ { p _ { s } ^ { \\ast } - 2 } u _ { n } \\xi _ { j , \\delta } d x - \\lambda \\int _ { \\Omega } f ( x , u _ { n } ) \\xi _ { j , \\delta } u _ { n } d x \\end{align*}"}
-{"id": "4486.png", "formula": "\\begin{align*} \\lambda ( \\underline { u } \\sigma + \\nabla ^ 2 \\underline { u } + N \\nabla ^ 2 d ^ 2 - 2 \\varepsilon \\sigma ) \\geq \\lambda ( \\underline { u } \\sigma + \\nabla ^ 2 \\underline { u } + 2 N \\nabla d \\otimes \\nabla d - 3 \\varepsilon \\sigma ) \\end{align*}"}
-{"id": "647.png", "formula": "\\begin{align*} \\psi ^ { - 1 } \\{ s \\} : = \\{ j ^ s _ 1 < \\dots < j ^ s _ r \\} \\ . \\end{align*}"}
-{"id": "3778.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ \\Omega B _ 0 ( v _ 0 - v _ 1 ) \\cdot ( v _ 0 - v _ 1 ) = & \\int _ \\Omega ( B _ 0 v _ 0 \\cdot v _ 0 - 2 B _ 0 v _ 0 \\cdot v _ 1 + B _ 1 v _ 1 \\cdot v _ 1 ) \\\\ & + \\int _ \\Omega ( B _ 0 - B _ 1 ) v _ 1 \\cdot v _ 1 \\\\ = & \\Re \\int _ { \\partial \\Omega } ( g _ 1 - g _ 0 ) h + \\int _ D ( B _ 0 - B _ 1 ) v _ 1 \\cdot v _ 1 . \\end{aligned} \\end{align*}"}
-{"id": "1646.png", "formula": "\\begin{align*} G _ { - k } : = \\sigma ( F _ { - k } ) = \\frac { 1 } { \\varkappa _ k } \\begin{pmatrix} a _ k + \\sqrt [ + ] { a _ k ^ 2 - b ^ 2 } \\\\ - b \\end{pmatrix} e ^ { - 2 \\pi i k x } \\end{align*}"}
-{"id": "5949.png", "formula": "\\begin{align*} \\omega _ { ( n , k ) } ( t ) : = \\sqrt { n } \\left ( \\omega \\Big ( \\frac { k - 1 + t } { n } \\Big ) - \\omega \\Big ( \\frac { k - 1 } { n } \\Big ) \\right ) , t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "6492.png", "formula": "\\begin{align*} & \\int _ { \\Q _ { \\ell } ^ { \\times } } | y | ^ { s - 1 } \\gamma ^ { - 1 } ( y ) ( f _ 1 ( y ) \\chi _ 1 ( y ) | y | ^ { 1 / 2 } + g _ 1 ( y ) \\psi _ 1 ( y ) | y | ^ { 1 / 2 } ) ( f _ 2 ( y ) \\chi _ 2 ( y ) | y | ^ { 1 / 2 } + g _ 2 ( y ) \\psi _ 2 ( y ) | y | ^ { 1 / 2 } ) d ^ { \\times } y \\\\ & = P _ 1 ( s ) L ( \\mathbf { 1 } , s ) + P _ 2 ( s ) L ( \\chi _ 1 \\psi _ 2 \\gamma ^ { - 1 } , s ) + P _ 3 ( s ) L ( \\psi _ 1 \\chi _ 2 \\gamma ^ { - 1 } , s ) , \\end{align*}"}
-{"id": "6418.png", "formula": "\\begin{align*} R _ { k + 1 } = \\left [ \\begin{array} { c c } R _ { k } & v _ { k } \\\\ & \\eta _ { k } \\end{array} \\right ] , v _ { k } \\in \\mathbb { R } ^ { k } , \\quad \\eta _ { k } \\in \\mathbb { R } , \\end{align*}"}
-{"id": "5838.png", "formula": "\\begin{align*} \\rho ( c ) : = \\frac { \\int f ( x ) c ( x ) | x | _ F ^ { - 1 } d x } { \\int \\mathcal { F } f ( x ) c ^ { - 1 } ( x ) d x } . \\end{align*}"}
-{"id": "9882.png", "formula": "\\begin{align*} P ^ { \\mu } | _ { \\mathcal { F } ^ { \\mathcal { Y } } _ { 0 , \\infty } } = P ^ { \\nu } | _ { \\mathcal { F } ^ { \\mathcal { Y } } _ { 0 , \\infty } } \\implies \\mu = \\nu . \\end{align*}"}
-{"id": "787.png", "formula": "\\begin{align*} \\iota ( g ) = \\prod _ { i = 1 } ^ { { s } } \\left ( \\sigma _ i ^ * \\right ) ^ { \\frac { q ^ d - 1 } { q ^ s - 1 } b _ i } . \\end{align*}"}
-{"id": "6167.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u _ i = 0 & B ^ + _ 1 \\\\ u _ i \\partial _ { \\nu } u _ i = f _ { i } ( u _ i ) u _ i & \\mathcal { B } _ 1 . \\end{cases} \\end{align*}"}
-{"id": "6975.png", "formula": "\\begin{align*} A ( x _ 0 , x _ 2 , \\dots ) = \\ln f ' ( x _ 0 ) + c f ( x _ 0 ) + H \\left ( f ( x _ 0 ) , \\frac { x _ 2 } { f ' ( x _ 0 ) } + \\frac { f '' ( x _ 0 ) } { ( f ' ( x _ 0 ) ) ^ 2 } , \\dots \\right ) \\end{align*}"}
-{"id": "1281.png", "formula": "\\begin{align*} \\sqrt { \\eta } K ( z , w ) ( \\eta ) & = \\int _ { | \\mu | \\le | \\eta | ^ { 3 / 2 } / 2 } e ^ { 3 i \\mu ^ 2 / 4 } z \\left ( \\eta + \\frac { \\mu } { \\sqrt { \\eta } } \\right ) w \\left ( \\eta - \\frac { \\mu } { \\sqrt { \\eta } } \\right ) d \\mu \\\\ & + \\int _ { | \\mu | \\le | \\eta | ^ { 3 / 2 } / 2 } e ^ { 3 i \\mu ^ 2 / 4 } z \\left ( \\eta + \\frac { \\mu } { \\sqrt { \\eta } } \\right ) w \\left ( \\eta - \\frac { \\mu } { \\sqrt { \\eta } } \\right ) d \\mu . \\end{align*}"}
-{"id": "3001.png", "formula": "\\begin{align*} \\frac { 1 } { G _ m ( x ) } & = \\sum _ { n = 0 } ^ \\infty p ' _ m ( n ) x ^ n . \\end{align*}"}
-{"id": "1805.png", "formula": "\\begin{align*} A \\subset \\bigcap _ { i = 1 } ^ { d } \\left \\{ \\left | B _ { s _ { 1 } } ^ { i } - B _ { t _ { 0 } } ^ { i } \\right | < 2 \\eta \\right \\} \\cup \\bigcup _ { i = 1 } ^ { d } \\left \\{ \\sup _ { s \\in I } \\left | B _ { s _ { 1 } } ^ { i } - B _ { s } ^ { i } \\right | > \\eta \\right \\} \\cup \\bigcup _ { i = 1 } ^ { d } \\left \\{ \\sup _ { t \\in J } \\left | B _ { t } ^ { i } - B _ { t _ { 0 } } ^ { i } \\right | > \\eta \\right \\} , \\end{align*}"}
-{"id": "905.png", "formula": "\\begin{align*} s _ m & = \\sum _ { j = 1 } ^ \\ell \\sum _ { k = 1 } ^ { r _ j } m ^ j _ k \\left ( \\sum _ { i \\in Q _ 0 ^ j } d ^ i _ k - 1 \\right ) & w _ m & = \\sum _ { j = 1 } ^ \\ell w ^ j \\end{align*}"}
-{"id": "4122.png", "formula": "\\begin{align*} T ( z ) = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & i n m _ { \\frac { 1 } { 2 } } \\\\ 0 & 0 & 1 \\end{pmatrix} L ^ { - 1 } X ( z ) \\begin{pmatrix} e ^ { - n g _ { 1 } ( z ) } & 0 & 0 \\\\ 0 & e ^ { n ( g _ { 1 } ( z ) - g _ { 2 } ( z ) ) } & 0 \\\\ 0 & 0 & e ^ { n g _ { 2 } ( z ) } \\end{pmatrix} L , \\end{align*}"}
-{"id": "4440.png", "formula": "\\begin{align*} N _ L ^ { \\pi } = k _ L N _ L ^ H , N _ X ^ { \\pi } = k _ X N _ X ^ H , 2 N _ X ^ { \\pi } = m _ { \\pi } N _ L ^ { \\pi } , 2 N _ X ^ H = m _ H N _ L ^ H . \\end{align*}"}
-{"id": "1197.png", "formula": "\\begin{align*} E ^ { \\gamma _ - } _ { \\tau } ( x , z ' ) & = \\frac { \\tau } { 2 ( 2 \\pi ) ^ 2 \\gamma _ - ^ { 3 / 2 } } \\int _ { \\Bbb R } I _ { \\tilde \\tau , 0 } ( x - \\tilde { z } ' , \\zeta _ 2 ) d \\zeta _ 2 , \\\\ \\partial _ { x _ k } E ^ { \\gamma _ - } _ { \\tau } ( x , z ' ) & = \\frac { \\tau ^ 2 } { 2 ( 2 \\pi ) ^ 2 \\gamma _ - ^ { 2 } } \\int _ { \\Bbb R } I _ { \\tilde \\tau , k } ( x - \\tilde { z } ' , \\zeta _ 2 ) d \\zeta _ 2 \\Theta _ k ( x , z ' ) \\qquad ( k = 1 , 2 , 3 ) , \\end{align*}"}
-{"id": "9598.png", "formula": "\\begin{align*} \\textup { s p t } ( n , N ) = \\frac { A \\cdot ( N + 1 ) _ n } { n ! } + \\frac { B \\cdot ( N ) _ n } { n ! } + \\cdots . \\end{align*}"}
-{"id": "8970.png", "formula": "\\begin{align*} \\eta _ v \\left ( B ( x , r ) \\setminus v u ^ { - 1 } B ( x , r ) \\right ) & = \\eta _ { x ^ { - 1 } v } \\left ( B ( 0 , r ) \\setminus x ^ { - 1 } v u ^ { - 1 } B ( x , r ) \\right ) \\\\ & = \\eta _ { x ^ { - 1 } v } \\left ( B ( 0 , r ) \\setminus z B ( 0 , r ) \\right ) \\ , , \\end{align*}"}
-{"id": "6036.png", "formula": "\\begin{align*} P ( G ) & = P ( \\deg ( v _ 1 ) , \\deg ( v _ 2 ) , \\dots , \\deg ( v _ n ) ) = \\sum _ { 1 \\leq i < j \\leq n } \\min \\{ \\deg ( v _ i ) , \\deg ( v _ j ) \\} . \\end{align*}"}
-{"id": "2708.png", "formula": "\\begin{align*} \\varphi ( x ) = \\frac { 1 } { 2 \\pi } \\sqrt { \\frac { 4 - x } { x } } , 0 < x < 4 , \\end{align*}"}
-{"id": "7063.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } | _ { t = 0 } E ( t ) ^ c \\geq c E ^ { c - 1 } \\frac { d ^ 2 E } { d t ^ 2 } | _ { t = 0 } > 0 \\end{align*}"}
-{"id": "7695.png", "formula": "\\begin{align*} \\norm { u - u _ { k } } ^ 2 _ { 2 } & = \\int \\limits _ { K } \\norm { u ( p ) - u _ { k } ( p ) } ^ 2 _ { S _ { p } } \\ , \\mathrm { d } \\mu + \\int \\limits _ { M \\setminus K } \\norm { u _ { k } ( p ) } ^ 2 _ { S _ { p } } \\ , \\mathrm { d } \\mu \\\\ & \\leq \\int \\limits _ { K } \\norm { u ( p ) - w _ { k } ( p ) } ^ 2 _ { S _ { p } } \\ , \\mathrm { d } \\mu + \\int \\limits _ { M \\setminus K } \\norm { w _ { k } ( p ) } ^ 2 _ { S _ { p } } \\ , \\mathrm { d } \\mu = \\norm { u - w _ { k } } ^ 2 _ { 2 } , \\end{align*}"}
-{"id": "3251.png", "formula": "\\begin{align*} 2 ^ { n - 1 } \\prod _ { j = 1 } ^ { n - 1 } \\sin \\pi j / n . \\end{align*}"}
-{"id": "5440.png", "formula": "\\begin{align*} \\log \\hat { \\rho } _ s ( A ) \\ge \\max _ { f ^ k ( p ) = p , k \\in \\N } \\left \\lbrace \\frac { s - \\lfloor s \\rfloor } { k } r _ { \\lfloor s \\rfloor + 1 } ( A ^ k ( p ) ) + \\frac { 1 - s + \\lfloor s \\rfloor } { k } r _ { \\lfloor s \\rfloor } ( A ^ k ( p ) ) \\right \\rbrace . \\end{align*}"}
-{"id": "5723.png", "formula": "\\begin{align*} \\frac { d } { d z } E _ { \\alpha , 1 } ( \\lambda ( x - z ) ^ { \\alpha } ) & = - \\lambda ( x - z ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } ( \\lambda ( x - z ) ^ { \\alpha } ) \\end{align*}"}
-{"id": "1984.png", "formula": "\\begin{align*} H : = c _ 1 ( \\O _ { \\P ^ n } ( 1 ) ) \\in H _ T ^ * ( \\P ^ n ) . \\end{align*}"}
-{"id": "6900.png", "formula": "\\begin{align*} \\det ( d \\varphi ) = \\sum _ { i = 1 } ^ 6 p _ i g _ i \\ , , \\end{align*}"}
-{"id": "3596.png", "formula": "\\begin{align*} \\tau ( a _ 1 ) = \\tau ( a _ { - 1 } ) : ( a _ 2 , \\ , a _ { - 2 } ) ( a _ 3 , \\ , a _ { - 3 } ) ; \\\\ \\tau ( a _ 2 ) = \\tau ( a _ { - 2 } ) : ( a _ 1 , \\ , a _ { - 1 } ) ( a _ 3 , \\ , a _ { - 3 } ) ; \\\\ \\tau ( a _ 3 ) = \\tau ( a _ { - 3 } ) : ( a _ 1 , \\ , a _ { - 1 } ) ( a _ 2 , \\ , a _ { - 2 } ) . \\end{align*}"}
-{"id": "7206.png", "formula": "\\begin{align*} L f = \\sum _ { k \\neq 0 } \\sigma ( k ) \\hat f ( k ) e ^ { i k ( \\cdot ) } \\end{align*}"}
-{"id": "9258.png", "formula": "\\begin{align*} \\mathcal { L } : = F ^ { i j } [ D _ { i j } - A _ { i j } ^ k ( \\cdot , u , D u ) D _ k ] - ( D _ { p _ k } B ( \\cdot , u , D u ) ) D _ k , \\end{align*}"}
-{"id": "3379.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { k _ 0 } G ( c k ) - G ( c ( k - 1 ) ) = G ( c k _ 0 ) - G ( 0 ) = \\delta . \\end{align*}"}
-{"id": "1841.png", "formula": "\\begin{align*} A _ j = \\cos ( Z _ j ) \\left [ 1 + \\frac { \\sin ( Z _ j ) \\cos ( Z _ j ) } { Z _ j } \\right ] ^ { - 1 / 2 } . \\end{align*}"}
-{"id": "3880.png", "formula": "\\begin{align*} M ( u ( t ) , v ( t ) ) & : = \\| u ( t ) \\| ^ 2 _ { L ^ 2 } + c \\| v ( t ) \\| ^ 2 _ { L ^ 2 } = M ( u _ 0 , v _ 0 ) , \\\\ E ( u ( t ) , v ( t ) ) & : = \\frac { 1 } { 2 m } \\| \\nabla u ( t ) \\| ^ 2 _ { L ^ 2 } + \\frac { c } { 4 M } \\| \\nabla v ( t ) \\| ^ 2 _ { L ^ 2 } + \\emph { R e } ( \\lambda \\langle v ( t ) , u ^ 2 ( t ) \\rangle ) = E ( u _ 0 , v _ 0 ) , \\end{align*}"}
-{"id": "1539.png", "formula": "\\begin{align*} \\max _ { y \\in I } { \\{ | g ( \\sqrt { s i } , x , y ) | \\} } = | g ( \\sqrt { s i } , x , x ) | \\end{align*}"}
-{"id": "412.png", "formula": "\\begin{align*} S ^ { { \\rm c l } , 0 } _ { { \\rm L } } ( g _ 3 , g _ 1 ) = S ^ { { \\rm c l } , 0 } _ { { \\rm L } } ( g _ 3 , g _ 2 ) + S ^ { { \\rm c l } , 0 } _ { { \\rm L } } ( g _ 2 , g _ 1 ) S ^ { \\rm c l } _ { \\rm M } ( g _ 3 , g _ 1 ) = S ^ { \\rm c l } _ { \\rm M } ( g _ 3 , g _ 2 ) + S ^ { \\rm c l } _ { \\rm M } ( g _ 2 , g _ 1 ) . \\end{align*}"}
-{"id": "1326.png", "formula": "\\begin{align*} \\rho [ Z , P ] = \\sup _ { \\lambda \\in \\varLambda _ \\rho } \\int _ 0 ^ 1 \\textup { A V a R } _ s [ Z , P ] \\ ; \\lambda ( d s ) . \\end{align*}"}
-{"id": "922.png", "formula": "\\begin{align*} - \\Biggl . { \\bf 1 } _ { \\{ i _ 1 = i _ 2 \\ne 0 \\} } { \\bf 1 } _ { \\{ j _ 1 = j _ 2 \\} } \\zeta _ { j _ 3 } ^ { ( i _ 3 ) } - { \\bf 1 } _ { \\{ i _ 2 = i _ 3 \\ne 0 \\} } { \\bf 1 } _ { \\{ j _ 2 = j _ 3 \\} } \\zeta _ { j _ 1 } ^ { ( i _ 1 ) } - { \\bf 1 } _ { \\{ i _ 1 = i _ 3 \\ne 0 \\} } { \\bf 1 } _ { \\{ j _ 1 = j _ 3 \\} } \\zeta _ { j _ 2 } ^ { ( i _ 2 ) } \\Biggr ) , \\end{align*}"}
-{"id": "1697.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\varphi _ n ( x ) = \\chi _ V ( x ) \\ \\ \\ \\forall \\ x \\in X . \\end{align*}"}
-{"id": "1108.png", "formula": "\\begin{align*} & \\int _ { I ^ k } \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) d y _ 1 \\cdots d y _ k \\\\ \\leq & \\int _ { I ^ k } n ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } \\prod _ { j = 1 } ^ k \\mathbb { P } ( \\xi ^ { ( n ) } ( [ y _ j , y _ j + G _ n ( x _ j ) / S ( I ) ] ) = 0 ) d y _ 1 \\cdots d y _ k \\\\ = & \\prod _ { j = 1 } ^ k \\left [ n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } \\int _ { I } \\mathbb { P } ( \\xi ^ { ( n ) } ( [ y _ j , y _ j + G _ n ( x _ j ) / S ( I ) ] ) = 0 ) d y _ j \\right ] . \\end{align*}"}
-{"id": "9294.png", "formula": "\\begin{align*} J _ K ^ { 2 , + } u ( x ) : = & \\{ ( p , X ) \\in \\mathbb { R } ^ n \\times \\mathbb { S } ^ n | \\ u ( y ) \\le u ( x ) + p \\cdot ( y - x ) \\\\ & + \\frac { 1 } { 2 } X ( y - x ) \\cdot ( y - x ) + o ( | y - x | ^ 2 ) \\ { \\rm a s } \\ y \\in K \\rightarrow x \\} , \\end{align*}"}
-{"id": "6961.png", "formula": "\\begin{align*} \\alpha _ 0 = h ( z _ 0 ) , \\alpha _ n = n ! \\sum _ { k = 1 } ^ n \\frac { h ^ { ( k ) } ( z _ 0 ) } { k ! } \\sum _ { i _ 1 + \\dots + i _ k = n } \\frac { z _ { i _ 1 } } { i _ 1 ! } \\dots \\frac { z _ { i _ k } } { i _ k ! } , n \\geq 1 . \\end{align*}"}
-{"id": "4407.png", "formula": "\\begin{align*} { \\rm R i c c i } ( \\omega _ o | _ S ) = \\Xi ^ * ( d d ^ c ( - \\log ( 1 + | z ^ 1 | ^ 2 + | z ^ 2 | ^ 2 ) ) ) \\end{align*}"}
-{"id": "2162.png", "formula": "\\begin{align*} p ( \\mathbf { x } ) g _ { L + 1 } ( \\mathbf { x } ) = \\sum _ { \\ell = 0 } ^ { d } \\sum _ { \\| \\alpha \\| \\leq \\ell } d _ { \\ell , \\alpha } T ^ { \\alpha } ( g _ { L + 1 + \\ell } ) ( \\mathbf { x } ) , \\end{align*}"}
-{"id": "6539.png", "formula": "\\begin{align*} m ^ P _ \\beta = m ^ { P ' } _ \\beta . \\end{align*}"}
-{"id": "612.png", "formula": "\\begin{align*} \\gamma _ { \\mathfrak S } ( \\sigma \\tau ; \\sigma _ 1 \\tau _ 1 , \\dots , \\sigma _ n \\tau _ n ) = \\gamma _ { \\mathfrak S } ( \\sigma ; \\sigma _ { \\tau ^ { - 1 } ( 1 ) } , \\dots , \\sigma _ { \\tau ^ { - 1 } ( n ) } ) \\gamma ( \\tau ; \\tau _ 1 , \\dots , \\tau _ n ) \\ , \\end{align*}"}
-{"id": "6644.png", "formula": "\\begin{gather*} \\alpha _ j = \\lVert \\mathbf 1 _ { 2 ^ { j } \\leq \\lvert \\xi \\rvert \\leq 2 ^ { j + 1 } } m ( \\xi ) \\rVert _ { \\infty } \\textup { a n d } \\beta _ j = \\lVert \\mathbf 1 _ { 2 ^ { j } \\leq \\lvert \\xi \\rvert \\leq 2 ^ { j + 1 } } \\nabla m ( \\xi ) \\cdot \\xi \\rVert _ { \\infty } . \\end{gather*}"}
-{"id": "749.png", "formula": "\\begin{align*} \\mathbb { E } _ { g } \\ ! \\left [ R \\right ] = m n \\psi _ { 0 } ( n ) + \\frac { 1 } { 2 } m ( m + 1 ) \\end{align*}"}
-{"id": "6465.png", "formula": "\\begin{align*} Y _ \\lambda : = L ^ \\xi _ \\lambda + \\sqrt { \\lambda } ( L ^ \\eta _ \\lambda ) ^ * . \\end{align*}"}
-{"id": "2749.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\left ( \\frac { 2 } { N } \\right ) ^ k R _ { N } ^ { ( k ) } \\left ( \\frac { 2 u _ 1 } { N } , \\ldots , \\frac { 2 u _ k } { N } \\right ) = \\mathrm { P f } { \\begin{bmatrix} D S ^ { ( \\mathrm { h a r d } ) } ( u _ i , u _ j ) & S ^ { ( \\mathrm { h a r d } ) } ( u _ i , u _ j ) \\\\ - S ^ { ( \\mathrm { h a r d } ) } ( u _ j , u _ i ) & I S ^ { ( \\mathrm { h a r d } ) } ( u _ i , u _ j ) \\end{bmatrix} } _ { i , j = 1 } ^ { k } \\end{align*}"}
-{"id": "1324.png", "formula": "\\begin{align*} R \\big ( \\varPsi _ { ( a , b ] } \\big ) = \\rho [ Z , P ] \\le \\rho [ V , Q ] = R \\big ( \\varPsi \\big ) . \\end{align*}"}
-{"id": "9565.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - 1 ) ^ { n - 1 } z ^ n q ^ { \\frac { n ( n + 1 ) } { 2 } } ( q ) _ { n } } { ( 1 - q ^ { n } ) ( z q ) _ n } = \\sum _ { n = 1 } ^ N \\frac { z q ^ n } { 1 - z q ^ n } . \\end{align*}"}
-{"id": "783.png", "formula": "\\begin{align*} Z _ { i _ 0 } = Z _ 1 ^ { q ^ { i _ 0 - 1 } } . \\end{align*}"}
-{"id": "5082.png", "formula": "\\begin{align*} & \\langle h , x _ U \\rangle = A _ U h = A _ I \\pi ( U ) ^ { - 1 } h = \\langle \\pi ( U ) ^ { - 1 } h , x _ U \\rangle = \\langle h , \\pi ( U ) x _ U \\rangle , \\\\ & \\langle h , \\tau _ U \\rangle = \\Psi _ U h = \\Psi _ I \\pi ( U ) ^ { - 1 } h = \\langle \\pi ( U ) ^ { - 1 } h , \\tau _ U \\rangle = \\langle h , \\pi ( U ) \\tau _ U \\rangle , \\end{align*}"}
-{"id": "7030.png", "formula": "\\begin{align*} E _ { \\alpha , \\beta } ( z ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ { n } } { \\Gamma ( \\alpha n + \\beta ) } , ( z , \\alpha , \\beta \\in \\mathbb { C } , \\mathfrak { R ( \\alpha ) } > 0 \\ ; ) . \\end{align*}"}
-{"id": "9334.png", "formula": "\\begin{align*} \\quad = \\| \\nabla W ^ { T } \\| ^ { 2 } ( x ) + \\mathrm { t r a c e } ( A ^ 2 _ { W ^ { \\perp } } ) ( x ) - 2 \\mathrm { t r a c e } ( A _ { W ^ { \\perp } } \\nabla W ^ { T } ) ( x ) , \\end{align*}"}
-{"id": "6884.png", "formula": "\\begin{align*} A ( t ) = \\cosh ( s ( t ) ) \\begin{pmatrix} \\cos ( \\mu ( t ) ) & - \\sin ( \\mu ( t ) ) \\\\ \\sin ( \\mu ( t ) ) & \\cos ( \\mu ( t ) ) \\end{pmatrix} + \\sinh ( s ( t ) ) \\begin{pmatrix} \\cos ( \\theta ( t ) ) & \\sin ( \\theta ( t ) ) \\\\ \\sin ( \\theta ( t ) ) & - \\cos ( \\theta ( t ) ) \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "2871.png", "formula": "\\begin{align*} R ( 0 ) \\left ( [ R ^ * _ 2 , R _ 2 ] [ R ^ * _ 1 , R _ 1 ] \\right ) = R ( 0 ) [ R ^ * _ 1 , R _ 1 ] - R ( 0 ) R _ 2 R ^ * _ 2 [ R ^ * _ 1 , R _ 1 ] = R ( 0 ) [ R ^ * _ 1 , R _ 1 ] . \\end{align*}"}
-{"id": "2676.png", "formula": "\\begin{align*} I _ 2 & = M ( \\frac { 1 } { 2 } , 1 , 2 v ) + ( z + 1 ) \\int _ { 0 } ^ { v } d s \\ , e ^ { ( v - s ) ( z + 1 ) } M ( \\frac { 1 } { 2 } , 1 , 2 s ) . \\end{align*}"}
-{"id": "5122.png", "formula": "\\begin{align*} \\psi ^ { m _ j } _ { j - 1 / 2 } & : = \\frac { 1 } { \\sqrt { 2 j } } \\left ( \\begin{array} { c } \\sqrt { j + m _ j } \\ , Y ^ { m _ j - 1 / 2 } _ { j - 1 / 2 } \\\\ \\sqrt { j - m _ j } \\ , Y ^ { m _ j + 1 / 2 } _ { j - 1 / 2 } \\\\ \\end{array} \\right ) , \\\\ \\psi ^ { m _ j } _ { j + 1 / 2 } & : = \\frac { 1 } { \\sqrt { 2 j + 2 } } \\left ( \\begin{array} { c } \\sqrt { j + 1 - m _ j } \\ , Y ^ { m _ j - 1 / 2 } _ { j + 1 / 2 } \\\\ - \\sqrt { j + 1 + m _ j } \\ , Y ^ { m _ j + 1 / 2 } _ { j + 1 / 2 } \\\\ \\end{array} \\right ) ; \\end{align*}"}
-{"id": "9345.png", "formula": "\\begin{align*} \\textup { h s p e c } ^ { \\mathcal { P } } _ { \\trianglelefteq } ( G ) & = \\textup { h s p e c } ^ { \\mathcal { D } } _ { \\trianglelefteq } ( G ) = \\textup { h s p e c } ^ { \\mathcal { F } } _ { \\trianglelefteq } ( G ) = [ 0 , 1 / 3 ] \\cup \\{ 1 \\} \\\\ \\textup { h s p e c } ^ { \\mathcal { L } } _ { \\trianglelefteq } ( G ) & = [ 0 , 1 / 5 ] \\cup \\{ 3 / 5 \\} \\cup \\{ 1 \\} \\end{align*}"}
-{"id": "7629.png", "formula": "\\begin{align*} \\int \\limits _ { \\mathbb R ^ d } a ( \\xi - q ) \\mu ( \\xi , q ) ( \\varkappa _ 1 ( q ) - \\varkappa _ 1 ( \\xi ) ) \\ d q \\ = \\ \\int \\limits _ { \\mathbb R ^ d } a ( \\xi - q ) \\mu ( \\xi , q ) ( \\xi - q ) \\ d q \\ - \\ b , \\end{align*}"}
-{"id": "5728.png", "formula": "\\begin{align*} \\frac { d } { d x } ( b _ 1 x E _ { \\alpha , 2 } ( \\lambda x ^ { \\alpha } ) ) | _ { x = 0 } & = b _ 1 x \\frac { d } { d x } ( E _ { \\alpha , 2 } ( \\lambda x ^ { \\alpha } ) ) | _ { x = 0 } + b _ 1 E _ { \\alpha , 2 } ( \\lambda x ^ { \\alpha } ) | _ { x = 0 } = b _ 1 . \\end{align*}"}
-{"id": "5011.png", "formula": "\\begin{align*} \\lim _ { \\mathfrak N \\ni N \\to \\infty } \\frac { \\# ( \\omega _ N \\cap A ^ { ( i ) } ) } { N } = \\beta ^ { ( i ) } , i = 1 , 2 . \\end{align*}"}
-{"id": "1024.png", "formula": "\\begin{align*} & \\lim _ { n \\to + \\infty } n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } D _ n ( F _ n ( x ) / 2 ) = e ^ { c _ 0 - x } , \\end{align*}"}
-{"id": "9310.png", "formula": "\\begin{align*} \\max _ { j = 1 , \\dots , n } x _ { i j } \\leq s _ i , ~ i = 1 , \\dots , m ; \\end{align*}"}
-{"id": "7731.png", "formula": "\\begin{align*} Y _ { I , J } = \\{ ( A _ 1 , \\ldots , A _ l , B _ 1 , \\ldots , B _ l ) \\in Y \\mid A _ i = 0 \\iff i \\in I , \\ B _ j = 0 \\iff j \\in J \\} . \\end{align*}"}
-{"id": "1554.png", "formula": "\\begin{align*} U _ { 3 , j } ^ \\delta \\left ( \\begin{array} { c } z \\\\ x \\end{array} \\right ) = \\left ( \\begin{array} { c } z + O ( z ^ j x ) \\\\ x \\end{array} \\right ) . \\end{align*}"}
-{"id": "2734.png", "formula": "\\begin{align*} \\min _ { w \\in \\Sigma _ { + } ^ { 4 } } & \\Re { f ( w ) } - \\Re { f ( z _ { 0 } ) } \\geq \\Re { f ( w _ { 5 } ) } - \\Re { f ( z _ { 0 } ) } \\\\ & = 4 \\eta _ { - } ( w _ 5 - z _ { 0 } ) + \\log \\big ( 1 - \\frac { 1 } { w _ 5 - \\tau } \\big ( \\frac { 1 } { \\tau } - \\tau \\big ) \\big ) \\\\ & > 4 - \\log 3 . \\end{align*}"}
-{"id": "10033.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } a ^ { - n } f ( z q ^ { - n } ) = 0 . \\end{align*}"}
-{"id": "2843.png", "formula": "\\begin{align*} \\mathcal { T } ^ { \\prime } = \\mathcal { R } \\backslash \\bigl ( \\mathcal { R } _ { t f } \\cup \\bigcup \\nolimits _ { i \\in I } N ( X ^ { \\prime } , r _ { i } ^ { \\prime } ) \\bigr ) \\ , \\end{align*}"}
-{"id": "1039.png", "formula": "\\begin{align*} & \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) = 0 \\leq ( n / 4 ) ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } \\prod _ { j = 1 } ^ k D _ n ( F _ n ( x _ j ) / 2 ) . \\end{align*}"}
-{"id": "5398.png", "formula": "\\begin{align*} c ( 2 k ) = - \\frac { k ^ 2 + 1 } { 2 } \\end{align*}"}
-{"id": "5130.png", "formula": "\\begin{align*} \\begin{pmatrix} \\Gamma _ { m _ j , k _ j } ^ + ( f _ { m _ j , k _ j } ) \\\\ \\Gamma _ { m _ j , k _ j } ^ - ( f _ { m _ j , k _ j } ) \\end{pmatrix} : = D _ { k _ j } \\begin{pmatrix} A ^ + \\\\ A ^ - \\end{pmatrix} ; \\end{align*}"}
-{"id": "4864.png", "formula": "\\begin{align*} { \\rm S t } _ \\chi ( | \\sigma | ) = \\underset { \\substack { \\sigma \\subset \\overline { \\tau } \\\\ \\tau \\in \\Delta ( \\chi ) } } { \\sqcup } | \\tau | . \\end{align*}"}
-{"id": "3666.png", "formula": "\\begin{align*} p _ i = p \\end{align*}"}
-{"id": "10016.png", "formula": "\\begin{align*} S ( \\ell _ { 1 } ( X ) , \\mathcal { D } _ { \\infty } ( X ) ) = 1 - \\frac { 1 } { \\cot ( X ) } \\ , \\end{align*}"}
-{"id": "874.png", "formula": "\\begin{gather*} \\varphi ( t ) : = \\begin{cases} t - a _ { k + 1 } & \\mbox { i f } a _ { k + 1 } \\le t < b _ k , \\\\ 1 & \\mbox { i f } b _ k \\le t < a _ k , \\end{cases} \\end{gather*}"}
-{"id": "2353.png", "formula": "\\begin{align*} \\Gamma _ 1 '' & = \\Gamma _ 1 \\setminus \\{ b , ( b _ { c _ 1 } , b _ { c _ 2 } ) , \\cdots \\cdots ( b _ { c _ { k 1 } } , b _ { c _ k } ) \\} \\\\ \\Gamma _ 2 '' & = \\Gamma _ 2 + \\{ ( c , b ) , ( b _ { c _ 1 } , b _ { c _ 2 } ) , \\cdots \\cdots ( b _ { c _ { k 1 } } , b _ { c _ k } ) \\} \\end{align*}"}
-{"id": "4333.png", "formula": "\\begin{align*} ( m - ( k - \\ell ) ) x _ { i _ 1 } \\ldots x _ { i _ \\ell } e _ { k - \\ell } ( B ) = & x _ 1 \\ldots x _ { i _ { \\ell - 1 } } \\sum _ { b \\in B } e _ { k - \\ell + 1 } ( \\{ i _ \\ell \\} \\sqcup B \\setminus \\{ b \\} ) - \\\\ & x _ 1 \\ldots x _ { i _ { \\ell - 1 } } ( m - ( k - \\ell + 1 ) ) e _ { k - \\ell + 1 } ( B ) . \\end{align*}"}
-{"id": "1947.png", "formula": "\\begin{align*} 0 > \\lim _ { p \\rightarrow \\infty } \\sum _ { j = 1 } ^ p ( \\epsilon _ j - \\tan \\epsilon _ j ) > \\sum _ { j = 1 } ^ { j ^ * - 1 } ( \\epsilon _ j - \\tan \\epsilon _ j ) + ( 1 - \\tan 1 ) \\left ( \\frac { \\chi } { \\pi } \\right ) ^ 3 \\zeta ( 3 ) . \\end{align*}"}
-{"id": "2086.png", "formula": "\\begin{align*} C ( \\bar r , t | { { \\bar r } _ { t x } } , { t _ 0 } ) & = \\frac { 1 } { { { \\sqrt { ( 4 \\pi D ( t - { t _ 0 } ) ) } } } } { e ^ { \\frac { { - { { ( z - { z _ { \\rm t x } } - v ( t - { t _ 0 } ) ) } ^ 2 } } } { { 4 D ( t - { t _ 0 } ) } } - { k _ d } ( t - { t _ 0 } ) } } \\\\ & \\times \\sum \\limits _ { m = 1 } ^ \\infty { \\frac { { { J _ 0 } ( { \\lambda _ { 0 m } } { \\rho _ { t x } } ) { J _ 0 } ( { \\lambda _ { 0 m } } \\rho ) } } { { { N _ { 0 m } } } } } { e ^ { - \\gamma _ { 0 m } ^ 2 ( t - { t _ 0 } ) } } u ( t - t _ 0 ) . \\end{align*}"}
-{"id": "6281.png", "formula": "\\begin{align*} R _ k ( \\rho , \\epsilon , n ) = \\Omega \\left ( \\min \\{ \\sqrt { \\frac { k } { n } } + \\left ( \\frac { \\log k } { \\rho n } + \\frac { k ^ 2 } { \\rho n ^ 2 } \\right ) , 1 \\} \\right ) \\ \\end{align*}"}
-{"id": "2731.png", "formula": "\\begin{align*} a = \\frac { \\eta - } { \\varphi ( 4 ) } ( z _ { 0 } - z _ { 1 } ) = 2 - \\pi _ { * } , b = \\frac { \\eta - } { \\varphi ( 4 ) } ( z _ { 0 } - w _ { 1 } ) = 1 - \\pi _ { * } . \\end{align*}"}
-{"id": "5619.png", "formula": "\\begin{align*} \\alpha \\ , \\int _ 0 ^ { + \\infty } u v \\bar { u } _ x d x = \\int _ 0 ^ { + \\infty } ( i u _ t \\bar { u } _ x + u _ { x x } \\bar { u } _ x ) d x - \\beta \\ , \\int _ 0 ^ { + \\infty } | u | ^ 2 u \\bar { u } _ x d x . \\end{align*}"}
-{"id": "6677.png", "formula": "\\begin{align*} A ( \\omega ) = | x | ^ { - \\frac { i \\omega } { 2 } } O \\ , | x | ^ { - \\frac { i \\omega } { 2 } } , A ^ { \\dagger } ( \\omega ) = | x | ^ { \\frac { i \\omega } { 2 } } O ^ { \\dagger } \\ , | x | ^ { \\frac { i \\omega } { 2 } } , \\end{align*}"}
-{"id": "6443.png", "formula": "\\begin{align*} \\xi _ { k } = \\| r _ { 0 } \\| ^ { 2 } e _ { 1 } ^ { T } T _ { k } ^ { - 2 } e _ { 1 } = \\| \\| r _ { 0 } \\| L _ { k } ^ { - T } L _ { k } ^ { - 1 } e _ { 1 } \\| ^ { 2 } \\equiv \\| y \\| ^ { 2 } \\end{align*}"}
-{"id": "458.png", "formula": "\\begin{align*} \\Pr ( \\pi ' ( \\lambda ) = i , \\ , & \\xi ( \\lambda ) = i + j ) \\\\ & = \\Pr ( \\exists \\ , t < \\lambda : \\pi ' ( t ) = i , \\ , \\xi ( t ) = i + j ) \\ , \\frac { b + c ( i + j ) } { 1 + b + c ( i + j ) } \\\\ & = [ b + c ( i + j ) ] v _ { i , i + j } . \\end{align*}"}
-{"id": "6889.png", "formula": "\\begin{align*} M _ 1 ( t ) = \\begin{pmatrix} \\cos ( \\mu t ) & - \\sin ( \\mu t ) \\\\ \\sin ( \\mu t ) & \\cos ( \\mu t ) \\end{pmatrix} \\quad { \\rm a n d } M _ 2 ( t ) = \\begin{pmatrix} \\cos ( \\theta t ) & - \\sin ( \\theta t ) \\\\ \\sin ( \\theta t ) & \\cos ( \\theta t ) \\end{pmatrix} \\end{align*}"}
-{"id": "396.png", "formula": "\\begin{align*} e ^ { \\omega } = \\frac { V _ { \\hat g } } { V _ { g } } + \\frac { V _ { \\hat g } } { 2 } \\Delta _ { g } \\phi \\end{align*}"}
-{"id": "1856.png", "formula": "\\begin{align*} \\phi ( x , 0 ) = \\sqrt { \\frac { L } { 2 } } v ^ { \\rm t o p } ( x ) ( a + b ) + \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { \\sqrt { 2 k _ n } } \\left [ v ^ { \\rm o d d } ( n , x ) ( a _ n + b _ n ) + v ^ { \\rm e v e n } ( n , x ) ( c _ n + d _ n ) \\right ] \\end{align*}"}
-{"id": "9643.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N + 1 } \\frac { ( q ^ 2 ; q ^ 2 ) _ { n - 1 } ( q ^ 2 ; q ^ 2 ) _ N ( z q ; q ^ 2 ) _ { N - n + 1 } ( z q ) ^ { n - 1 } } { ( z q ^ 2 ; q ^ 2 ) _ { n } ( q ^ 2 ; q ^ 2 ) _ { N - n + 1 } ( z q ; q ^ 2 ) _ { N + 1 } } = \\sum _ { n = 1 } ^ { N + 1 } \\frac { ( q ; q ^ 2 ) _ { n - 1 } ( q ^ 2 ; q ^ 2 ) _ { N } ( z q ^ 2 ; q ^ 2 ) _ { N - n + 1 } ( z q ^ 2 ) ^ { n - 1 } } { ( z q ; q ^ 2 ) _ { n } ( q ^ 2 ; q ^ 2 ) _ { N - n + 1 } ( z q ^ 2 ; q ^ 2 ) _ { N + 1 } } , \\end{align*}"}
-{"id": "6395.png", "formula": "\\begin{align*} \\widetilde { \\beta } _ { k } = \\frac { \\sqrt { \\delta _ { k } } } { \\gamma _ { k - 1 } } , \\quad \\widetilde { \\alpha } _ { k } = \\frac { 1 } { \\gamma _ { k - 1 } } + \\frac { \\delta _ { k - 1 } } { \\gamma _ { k - 2 } } , \\quad \\delta _ { 0 } = 0 , \\quad \\gamma _ { - 1 } = 1 . \\end{align*}"}
-{"id": "4207.png", "formula": "\\begin{align*} R ( z ) = \\mathbb { I } + \\mathcal { O } \\left ( \\frac { 1 } { n } \\right ) n \\to \\infty \\end{align*}"}
-{"id": "8333.png", "formula": "\\begin{align*} - I m [ D _ t Z , \\mathbb { H } ] \\partial _ { \\alpha } D _ t \\bar { Z } = \\frac { 1 } { 2 \\pi } \\int \\frac { | D _ t Z ( \\alpha , t ) - D _ t Z ( \\beta , t ) | ^ 2 } { ( \\alpha - \\beta ) ^ 2 } d \\beta \\geq 0 . \\end{align*}"}
-{"id": "6307.png", "formula": "\\begin{align*} | E _ G ( S _ 0 , S _ 0 ) + E _ G ( S _ 0 , S ^ c _ 0 ) - e ( G ) \\left ( k ( n - k ) + \\binom { k } { 2 } \\right ) | \\leq C \\max \\{ \\sqrt { \\rho } , \\sqrt { \\frac { \\log n } { n } } \\} k \\sqrt { n \\log n } \\end{align*}"}
-{"id": "3907.png", "formula": "\\begin{align*} \\partial _ r ( t _ j ) = \\left \\{ \\begin{array} { l l } t _ j & \\mbox { i f $ j < r $ } \\\\ 0 & \\mbox { i f $ j = r $ } \\\\ t _ { j - 1 } & \\mbox { i f $ j > r $ } \\end{array} \\right . \\delta _ r ( t _ j ) = \\left \\{ \\begin{array} { l l } t _ j & \\mbox { i f $ j < r $ } \\\\ t _ j + t _ { j + 1 } & \\mbox { i f $ j = r $ } \\\\ t _ { j + 1 } & \\mbox { i f $ j > r $ . } \\end{array} \\right . \\end{align*}"}
-{"id": "1778.png", "formula": "\\begin{align*} \\beta = \\eta - \\frac { \\alpha } { 2 } ( t - s ) ^ { 2 H } . \\end{align*}"}
-{"id": "7189.png", "formula": "\\begin{gather*} h ( \\xi , \\eta ) = - \\intop _ { 0 } ^ { \\infty } \\left [ P _ { \\xi } ^ { + } + P _ { \\xi } ^ { 0 } \\right ] \\left [ X ^ { s } ( \\xi ) \\right ] ^ { - 1 } w ( s , y _ { \\ast } ( s , \\xi , \\eta ) , \\xi ) \\mathrm { d } s \\in \\mathbb { L } _ { \\xi } ^ { + } \\oplus \\mathbb { L } _ { \\xi } ^ { 0 } , \\end{gather*}"}
-{"id": "4840.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big ( t ^ { \\rho } \\frac { d u } { d t } \\Big ) + t ^ { \\sigma } u ^ { \\gamma } = 0 , t \\geq 0 , \\end{align*}"}
-{"id": "1084.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } \\int _ { I _ 2 } D _ n ( \\sqrt { 4 - y ^ 2 } / S ( I ) \\cdot G _ n ( x ) / 2 ) d y = 0 . \\end{align*}"}
-{"id": "2586.png", "formula": "\\begin{align*} V _ T ( \\theta f ) ( x ) = \\phi ( x ) \\int _ 0 ^ \\theta \\Pi _ x ^ { ( \\phi ) } \\Big [ \\frac { f ( \\xi _ T ) } { \\phi ( \\xi _ T ) } \\exp \\Big \\{ - \\int _ 0 ^ T \\big ( \\kappa \\gamma V _ { T - s } ( r f ) ^ { \\gamma - 1 } \\big ) ( \\xi _ s ) d s \\Big \\} \\Big ] d r . \\end{align*}"}
-{"id": "7715.png", "formula": "\\begin{align*} \\int \\limits _ M \\norm { u _ { s } ( p ) } ^ 2 _ { S _ { p } } \\ , \\mathrm { d } \\mu = \\int \\limits _ M \\norm { \\sqrt { h _ { s } } ( p ) u _ { s } ( p ) } ^ 2 _ { S _ { p } } \\ , \\mathrm { d } \\mu . \\end{align*}"}
-{"id": "470.png", "formula": "\\begin{align*} F & = L ^ d - \\overline { L } ^ d \\\\ & = \\left ( r _ 1 \\xi ^ { \\frac { l } { 2 } } x _ 1 + r _ 2 \\xi ^ { \\frac { 2 l + k + 2 } { 2 } } x _ 2 + r _ 3 \\xi ^ { \\frac { l + k + 1 } { 2 } } x _ 3 \\right ) ^ d - \\left ( r _ 1 \\xi ^ { - \\frac { l } { 2 } } x _ 1 + r _ 2 \\xi ^ { - \\frac { 2 l + k + 2 } { 2 } } x _ 2 + r _ 3 \\xi ^ { - \\frac { l + k + 1 } { 2 } } x _ 3 \\right ) ^ d . \\end{align*}"}
-{"id": "3444.png", "formula": "\\begin{align*} R ^ { ( n ) } _ { N } ( x _ 1 , \\ldots , x _ { n } ) : = \\frac { N ! } { ( N - n ) ! } \\idotsint \\mathcal { P } _ { N } ( x ) \\ , d x _ { n + 1 } \\cdots d x _ N = \\det [ K _ N ( x _ i , x _ j ) ] _ { i , j = 1 } ^ n , \\end{align*}"}
-{"id": "7912.png", "formula": "\\begin{align*} w _ k = w _ { - k } , k = - \\hat { n } , \\ldots , \\hat { n } \\end{align*}"}
-{"id": "6013.png", "formula": "\\begin{align*} \\sup _ { Q \\in \\mathcal P ^ * } \\left ( F \\left ( \\int _ { \\C } \\bar { \\omega } \\ , Q ( d \\bar { \\omega } ) \\right ) - \\tilde { \\alpha } ^ g ( Q ) \\right ) = \\sup _ { \\omega \\in \\C _ 0 } \\left ( F ( \\omega ) - I ( \\omega ) \\right ) , \\end{align*}"}
-{"id": "6725.png", "formula": "\\begin{align*} \\lambda \\dot { \\varphi } _ 4 = \\varphi _ 3 - \\varphi _ 1 , \\end{align*}"}
-{"id": "2856.png", "formula": "\\begin{align*} N _ { M / \\Q } ( F ( u , v ) ) = \\pm \\frac { d ^ { 6 m } } { i _ 0 } \\end{align*}"}
-{"id": "914.png", "formula": "\\begin{align*} J ^ { * } [ \\psi ^ { ( k ) } ] _ { T , t } = \\int \\limits _ t ^ { * T } \\psi _ k ( t _ k ) \\ldots \\int \\limits _ t ^ { * t _ { 2 } } \\psi _ 1 ( t _ 1 ) d { \\bf w } _ { t _ 1 } ^ { ( i _ 1 ) } \\ldots d { \\bf w } _ { t _ k } ^ { ( i _ k ) } , \\end{align*}"}
-{"id": "1267.png", "formula": "\\begin{align*} f ( \\xi ) = K ( S , S ) ( \\xi ) , g ( \\xi ) = \\bar S ( \\xi ) . \\end{align*}"}
-{"id": "834.png", "formula": "\\begin{align*} \\omega _ i ^ \\sigma = \\sum _ { \\nu = 1 } ^ g \\rho ( \\sigma ) _ { \\nu , i } \\omega _ \\nu . \\end{align*}"}
-{"id": "3519.png", "formula": "\\begin{align*} e _ { \\varnothing } , e _ a , e ^ a , e _ { a b } = e _ { b a } . \\end{align*}"}
-{"id": "2547.png", "formula": "\\begin{align*} d \\mu ^ F = \\frac { F } { \\mu ( F ) } d \\mu . \\end{align*}"}
-{"id": "3767.png", "formula": "\\begin{align*} \\gamma _ 1 ( \\nabla u ) = ( \\sigma _ 1 ( x ) + i \\epsilon _ 1 ( x ) ) \\nabla u + \\zeta _ 1 \\overline { \\nabla u } , \\end{align*}"}
-{"id": "7711.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\chi ( x ) = \\lim _ { x \\to \\infty } \\int \\limits _ { } ^ { } \\frac { \\sin ( t ) } { t } f ( \\frac { t } { x } ) \\ , \\mathrm { d } t \\stackrel { } { = } \\int \\limits _ { } ^ { } \\frac { \\sin ( t ) } { t } f ( 0 ) \\ , \\mathrm { d } t = 1 , \\end{align*}"}
-{"id": "3631.png", "formula": "\\begin{align*} \\alpha _ { - 2 } & : = \\alpha _ 0 \\cdot \\alpha _ 1 - \\frac { 1 } { 2 ^ 2 } ( \\alpha _ 0 + \\alpha _ 1 ) - \\frac { 8 } { 3 } \\alpha _ { - 1 } \\\\ & = - \\frac { 2 t } { 3 } a _ 1 - \\frac { 1 } { 3 } v _ { ( 2 , 3 ) } + \\frac { 4 } { 3 } ( a _ 2 + a _ { - 2 } ) \\cdot v _ { ( 1 , 3 ) } + \\frac { 4 } { 3 } ( a _ 3 + a _ { - 3 } ) \\cdot v _ { ( 1 , 2 ) } \\in M _ 0 ^ { ( a _ 1 ) } . \\end{align*}"}
-{"id": "449.png", "formula": "\\begin{align*} h _ T = \\psi _ T ^ * ( e ^ { \\omega _ T } g _ { \\tau _ T } ) . \\end{align*}"}
-{"id": "8625.png", "formula": "\\begin{align*} u ' = \\pm \\frac { u ^ 2 - u ^ 4 - c _ 0 ^ 2 } { \\sqrt { 2 } \\sqrt { u ^ 4 + c _ 0 ^ 2 } } . \\end{align*}"}
-{"id": "9747.png", "formula": "\\begin{align*} \\begin{cases} q \\ge 0 & L _ * \\\\ L _ a q \\le 0 & \\R ^ { n + 1 } \\\\ L _ a q = 0 & \\R ^ { n + 1 } \\setminus L _ * \\\\ q L _ a q = 0 & \\R ^ { n + 1 } . \\end{cases} \\end{align*}"}
-{"id": "4074.png", "formula": "\\begin{align*} Y ( z ) = \\left ( \\mathbb { I } + \\mathcal { O } \\left ( \\frac { 1 } { z } \\right ) \\right ) \\begin{pmatrix} z ^ { n } & 0 & 0 \\\\ 0 & z ^ { - \\lceil \\frac { n } { 2 } \\rceil } & 0 \\\\ 0 & 0 & z ^ { - \\lfloor \\frac { n } { 2 } \\rfloor } \\end{pmatrix} . \\end{align*}"}
-{"id": "5031.png", "formula": "\\begin{align*} x \\vee y = x \\vee z \\quad x \\vee y = x \\vee ( y \\wedge z ) . \\end{align*}"}
-{"id": "3302.png", "formula": "\\begin{align*} \\Delta ( n ) : = \\max _ { u > 0 } \\sum _ { d \\mid n , \\ , d \\in ( u , e u ] } 1 , \\end{align*}"}
-{"id": "8933.png", "formula": "\\begin{align*} R _ { \\mathrm { G o F } } ( n , s , a , b ) : = \\adjustlimits \\inf _ { \\phi } \\sup _ { x _ 0 } \\Big \\{ P ( \\textrm { F A } ) + \\ , \\sup _ { { x } } \\ , P ( \\textrm { M D } ( x ) ) \\Big \\} \\end{align*}"}
-{"id": "9681.png", "formula": "\\begin{align*} \\kappa _ { n t } ( \\zeta ) = \\frac { \\eta _ n ( \\zeta ) } { \\psi _ { n t } ( \\zeta ) } , \\ \\ \\ \\zeta \\in \\mathbb T , \\ n \\in \\mathbb N , \\ t > 0 . \\end{align*}"}
-{"id": "266.png", "formula": "\\begin{align*} \\widetilde { \\alpha } _ { f \\oplus g , ( x , y ) } = \\widetilde { \\alpha } _ { f , x } + \\widetilde { \\alpha } _ { g , y } , \\end{align*}"}
-{"id": "1275.png", "formula": "\\begin{align*} \\MoveEqLeft \\int e ^ { - 3 i \\Phi ( \\xi , \\eta ) } \\frac { 1 } { 3 \\partial _ { \\eta } \\Phi ( \\xi , \\eta ) } \\partial _ \\eta K ( S , S ) ( \\eta ) \\bar S ( \\eta - \\xi ) ( \\varphi _ 1 ( \\eta / \\xi ) - \\phi ( \\xi ^ { \\gamma } \\eta ) ) d \\eta \\\\ & = \\int _ { \\xi ^ { - \\gamma } / 2 } ^ { 2 \\xi } O \\left ( \\frac { 1 } { \\xi ^ 2 | \\eta | ^ { 3 / 2 } } \\right ) = O ( | \\xi | ^ { - 2 + \\gamma / 2 } ) . \\end{align*}"}
-{"id": "1484.png", "formula": "\\begin{align*} W ^ { j } _ { 3 } = - \\varepsilon \\int _ { \\mathbb { T } ^ { d } } ( \\partial _ { x _ { j } } w ^ { 1 } - \\partial _ { x _ { j } } w ^ { 2 } ) \\left ( \\Theta _ { q } ( t , x , \\mu ^ { 1 } , D w ^ { 1 } ) \\mu ^ { 1 } _ { x _ { j } } - \\Theta _ { q } ( t , x , \\mu ^ { 2 } , D w ^ { 2 } ) \\mu ^ { 1 } _ { x _ { j } } \\right ) \\ d x , \\end{align*}"}
-{"id": "5018.png", "formula": "\\begin{align*} \\lambda _ 1 ( t , x ) + \\lambda _ 2 ( t , x ) + \\lambda _ 3 ( t , x ) = 0 . \\end{align*}"}
-{"id": "1429.png", "formula": "\\begin{align*} H ^ { s } ( \\mathbb { T } ^ { d } ) = \\left \\{ f \\in L ^ { 2 } ( \\mathbb { T } ^ { d } ) : \\| f \\| _ { s } < \\infty \\right \\} , \\end{align*}"}
-{"id": "9249.png", "formula": "\\begin{align*} w _ { \\tau } = u _ { \\tau \\tau } + \\frac { c _ 1 } { 2 } | u _ \\tau | ^ 2 , \\end{align*}"}
-{"id": "6907.png", "formula": "\\begin{align*} I = & \\langle a _ { 1 1 } a _ { 2 2 } - a _ { 1 2 } a _ { 2 1 } - 1 , a _ { 1 3 } a _ { 2 4 } - a _ { 1 4 } a _ { 2 3 } - 1 , a _ { 1 4 } - a _ { 1 2 } + a _ { 1 1 } , a _ { 2 4 } - a _ { 2 2 } + a _ { 2 1 } \\rangle \\\\ \\subset & \\ \\mathbb { A } = \\mathbb { Q } [ a _ { 1 1 } , a _ { 1 2 } , a _ { 1 3 } , a _ { 1 4 } , a _ { 2 1 } , a _ { 2 2 } , a _ { 2 3 } , a _ { 2 4 } ] \\ , . \\end{align*}"}
-{"id": "2925.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\frac { \\omega _ k } { \\omega _ { k + 1 } } = 1 . \\end{align*}"}
-{"id": "6726.png", "formula": "\\begin{align*} A _ { \\kappa \\lambda \\nu } = \\frac { 1 } { 2 } \\left ( \\frac { \\partial h _ { \\lambda \\nu } } { \\partial x ^ \\kappa } - \\frac { \\partial h _ { \\kappa \\nu } } { \\partial x ^ \\lambda } \\right ) , \\end{align*}"}
-{"id": "6050.png", "formula": "\\begin{align*} ( x _ 1 , y _ 1 ) ( x _ 2 , y _ 2 ) = ( x ' _ 1 + \\lambda m , y ' _ 1 ) ( x ' _ 2 + \\lambda m , y ' _ 2 ) . \\end{align*}"}
-{"id": "7784.png", "formula": "\\begin{align*} U _ { p } f = | \\det U | ^ { - 1 / p } f \\circ U ^ { - 1 } , p \\ge 1 , \\end{align*}"}
-{"id": "8090.png", "formula": "\\begin{align*} a _ n = \\sum _ { k = 1 } ^ n \\frac { 1 } { k ! } \\ , \\binom { n - 1 } { k - 1 } . \\end{align*}"}
-{"id": "4580.png", "formula": "\\begin{align*} \\int _ { 2 ^ j \\leq | \\omega | \\leq 2 ^ { j + 1 } } | \\hat { \\mu } ( \\omega ) | ^ 2 | \\omega | ^ { t - n } d \\omega \\leq A ( \\mu , 2 ^ { j + 1 } ) 2 ^ { j ( t - n ) } \\leq C 2 ^ { ( j + 1 ) s } 2 ^ { j ( t - n ) } = C 2 ^ { s } 2 ^ { j ( s + t - n ) } . \\end{align*}"}
-{"id": "5829.png", "formula": "\\begin{align*} s _ n ^ 2 = \\frac { \\sum _ { i = 1 } ^ n X _ i ^ 2 } { n } - \\frac { \\sum _ { i \\neq j } X _ i X _ j } { n ( n - 1 ) } \\end{align*}"}
-{"id": "8965.png", "formula": "\\begin{align*} \\widetilde { E } _ n = \\{ \\mbox { T h e r e e x i s t s } \\ \\alpha \\in \\mathbb { R } ^ { p _ n } : \\ , \\alpha ' x _ i ( T _ n ) > 0 , \\ \\mbox { i f } \\ y _ i = 1 ; \\ \\ \\alpha ' x _ i ( T _ n ) < 0 , \\ \\mbox { i f } \\ y _ i = 0 \\} , \\end{align*}"}
-{"id": "8561.png", "formula": "\\begin{align*} \\norm { \\frac { i } { \\pi } \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j \\zeta _ { \\alpha } } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } } _ { H ^ 1 } \\leq K _ s ^ { - 1 } d _ I ( t ) ^ { - 5 / 2 } \\epsilon , \\end{align*}"}
-{"id": "7626.png", "formula": "\\begin{align*} \\frac { \\partial w ^ { \\varepsilon } ( x , t ) } { \\partial t } = \\big ( - \\frac { b } { \\varepsilon } \\big ) \\cdot \\nabla u ( x ^ \\varepsilon , t ) + \\frac { \\partial u } { \\partial t } ( x ^ \\varepsilon , t ) + \\varepsilon \\varkappa _ 1 \\big ( \\frac { x } { \\varepsilon } \\big ) \\otimes \\big ( - \\frac { b } { \\varepsilon } \\big ) \\cdot \\nabla \\nabla u ( x ^ \\varepsilon , t ) + \\phi _ \\varepsilon ^ { ( 0 ) } ( x , t ) , \\end{align*}"}
-{"id": "10009.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { n = M + 1 } ^ { N } { \\frac { a _ { n } } { n ^ { \\sigma + \\varepsilon } } n ^ { - s } } \\Big \\| _ { \\mathfrak { X } ( X ) } \\leq \\frac { C } { N ^ { \\varepsilon } } + \\frac { C } { M ^ { \\varepsilon } } + \\sum _ { n = M } ^ { N - 1 } { C \\Big [ \\frac { 1 } { n ^ { \\varepsilon } } - \\frac { 1 } { ( n + 1 ) ^ { \\varepsilon } } \\Big ] } = \\frac { 2 C } { M ^ { \\varepsilon } } \\end{align*}"}
-{"id": "9781.png", "formula": "\\begin{align*} \\mathcal { H } ^ \\beta ( E ) : = \\lim _ { \\delta \\downarrow 0 } \\mathcal { H } ^ \\beta _ \\delta ( E ) . \\end{align*}"}
-{"id": "7229.png", "formula": "\\begin{align*} ( \\mu - \\phi ) ^ 2 ( x ) = [ ( \\mu - \\phi ) ^ 2 ] ^ { \\prime } ( 0 ) x + \\frac { 1 } { 2 } [ ( \\mu - \\phi ) ^ 2 ] ^ { \\prime \\prime } ( \\xi ) x ^ 2 \\end{align*}"}
-{"id": "7926.png", "formula": "\\begin{align*} R _ N ^ 3 & = - \\frac { 1 7 } { 5 7 6 0 } \\frac { ( b - a ) ^ 5 f ^ { ( 4 ) } ( \\xi ) } { ( N - 2 ) ^ 4 } & \\xi \\in [ a - h / 2 , b + h / 2 ] \\\\ R _ N ^ 5 & = \\frac { 3 6 7 } { 9 6 7 6 8 0 } \\frac { ( b - a ) ^ 7 f ^ { ( 6 ) } ( \\xi ) } { ( N - 4 ) ^ 6 } & \\xi \\in [ a - 3 h / 2 , b + 3 h / 2 ] \\end{align*}"}
-{"id": "3529.png", "formula": "\\begin{align*} \\mathcal { M } _ { 6 } \\simeq \\left \\{ Y _ { 5 } ^ { 2 } = F _ { 4 } ( Y _ { 0 } , . . . , Y _ { 4 } ) \\right \\} \\subset \\mathbb { P } ( 1 ^ { 5 } , 2 ) , \\end{align*}"}
-{"id": "6055.png", "formula": "\\begin{align*} \\alpha _ i = \\left \\{ \\begin{array} { l l } n , & \\mbox { i f } 1 \\leq i \\leq x \\\\ y - z , & \\mbox { i f } i = x + 1 \\\\ z , & \\mbox { i f } i = x + 2 \\\\ 0 , & \\mbox { i f } x + 3 \\leq i \\leq r , \\end{array} \\right . \\end{align*}"}
-{"id": "7651.png", "formula": "\\begin{align*} I ^ { i j } _ 3 \\ = \\ \\int \\limits _ { \\mathbb T ^ d } \\ ! \\int \\limits _ { \\mathbb R ^ d } \\ ! a ( \\xi \\ ! - \\ ! q ) \\mu ( \\xi , q ) ( \\varkappa _ 1 ( \\xi ) - \\varkappa _ 1 ( q ) ) ^ i ( \\varkappa _ 1 ( \\xi ) - \\varkappa _ 1 ( q ) ) ^ j v _ 0 ( \\xi ) d q d \\xi . \\end{align*}"}
-{"id": "9905.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } } g ( y ) h _ { x } Q ( d y ) ) = \\int _ { \\mathcal { Z } } g ( h _ { x } ( z ) ) Q ( d z ) . \\end{align*}"}
-{"id": "210.png", "formula": "\\begin{align*} \\Delta _ z u + | z | ^ 2 \\partial _ t ^ 2 u = V u , ( z , t ) \\in \\mathbb { R } ^ N \\times \\mathbb { R } \\end{align*}"}
-{"id": "7049.png", "formula": "\\begin{align*} W = P _ { \\mathcal V } U _ * ( \\frac { \\partial } { \\partial z } ) = u _ z ^ v \\frac { \\partial } { \\partial v } + \\phi _ { \\bar v z } \\phi ^ { v \\bar v } \\frac { \\partial } { \\partial v } = ( u _ z ^ v - a _ z ^ v ) \\frac { \\partial } { \\partial v } \\end{align*}"}
-{"id": "1220.png", "formula": "\\begin{gather*} I ( f , g , h ) ( \\xi ) : = e ^ { - i \\xi ^ 3 } \\iint _ { \\eta _ 1 + \\eta _ 2 + \\eta _ 3 = \\xi } e ^ { i ( \\eta _ 1 ^ 3 + \\eta _ 2 ^ 3 + \\eta _ 3 ^ 3 ) } f ( \\eta _ 1 ) g ( \\eta _ 2 ) \\bar h ( - \\eta _ 3 ) d \\eta _ 1 d \\eta _ 2 . \\end{gather*}"}
-{"id": "7239.png", "formula": "\\begin{align*} \\liminf _ { \\xi \\to 0 } \\phi ^ \\prime ( \\xi ) \\gtrsim 2 \\int _ { - \\pi } ^ { 0 } K _ r ^ \\prime ( y ) \\phi ^ \\prime ( y ) \\ , d y = c \\end{align*}"}
-{"id": "7119.png", "formula": "\\begin{align*} \\hat { Q } _ { \\lambda ^ * } = \\inf \\{ Q ( q ) : \\ , \\ , \\lambda ^ * = \\lambda _ 2 ( q ) , \\ , \\ , q \\in L ^ 2 ( 0 , \\pi ) \\} , \\end{align*}"}
-{"id": "6759.png", "formula": "\\begin{align*} \\begin{array} { r l } = \\ { \\strut \\displaystyle - 2 a _ { 1 } \\over \\displaystyle 2 a _ { 1 } + a _ { 0 } ( s + 1 ) - { \\strut \\displaystyle 3 a _ { 0 } a _ { 2 } ( s + 1 ) \\over \\displaystyle 3 a _ { 2 } + a _ { 1 } ( s + 2 ) - { \\strut \\displaystyle 4 a _ { 1 } a _ { 3 } ( s + 2 ) \\over \\displaystyle 4 a _ { 3 } + a _ { 2 } ( s + 3 ) } } } & \\\\ & \\ddots \\end{array} \\end{align*}"}
-{"id": "1771.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ e ^ { \\alpha M _ { s , t } ^ { ( n ) } } \\right ] = \\mathbb { E } \\left [ e ^ { \\alpha \\max _ { 1 \\leq i \\leq n } Z _ { i } } \\right ] \\leq \\mathbb { E } \\left [ e ^ { \\alpha \\max _ { 1 \\leq i \\leq n } Y _ { i } } \\right ] = \\mathbb { E } \\left [ e ^ { \\alpha N _ { s , t } ^ { ( n ) } } \\right ] , \\end{align*}"}
-{"id": "6325.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 { q ^ { - 1 } \\ , d x } = \\int _ 0 ^ 1 { q ^ { - 1 } _ 0 \\ , d x } = \\nu ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "3636.png", "formula": "\\begin{align*} \\alpha _ { - 1 } & = - \\frac { 3 t } { 2 ^ 2 } a _ 1 - \\frac { 1 } { 2 ^ 2 } v _ { ( 2 , 3 ) } + a _ 1 \\cdot v _ { ( 2 , 3 ) } \\in M _ 0 ^ { ( a _ 1 ) } \\\\ \\beta _ { - 1 } & = - t a _ 1 + a _ 1 \\cdot v _ { ( 2 , 3 ) } \\in M _ { \\frac { 1 } { 2 ^ 2 } } ^ { ( a _ 1 ) } \\end{align*}"}
-{"id": "4020.png", "formula": "\\begin{align*} \\phi ^ 2 X = - X + \\eta ( X ) \\xi , g ( \\phi X , \\phi Y ) = g ( X , Y ) - \\eta ( X ) \\eta ( Y ) , \\end{align*}"}
-{"id": "4136.png", "formula": "\\begin{align*} A ( 0 ; r _ n , R ) = \\{ z \\in \\mathbb C \\mid r _ n < | z | < R \\} \\end{align*}"}
-{"id": "3704.png", "formula": "\\begin{align*} \\sum _ { i \\in B } ( w _ i - N / n _ B ) ^ 2 = \\sum _ t \\big ( \\sum _ { i \\in B _ t } w _ i ^ 2 - 2 ( N / n _ B ) \\sum _ { i \\in B _ t } w _ i + n _ { t B } ( N / n _ { t B } ) ^ 2 \\big ) \\end{align*}"}
-{"id": "4102.png", "formula": "\\begin{align*} T _ { \\alpha } \\Phi _ { \\alpha } ( z ) \\Psi _ { \\alpha } ^ T ( z ) \\tilde { T } _ { \\alpha } ^ T = - 4 \\pi ^ 2 \\mathbb { I } \\end{align*}"}
-{"id": "387.png", "formula": "\\begin{align*} \\hat { \\mu } = ( - n , \\lambda + \\beta , \\kappa ' ) \\end{align*}"}
-{"id": "2175.png", "formula": "\\begin{align*} | \\Phi _ { t } ( \\mathbf { x } , \\mathbf { y } ) | \\leq \\int \\Big | g _ t ( \\mathbf { x } , \\mathbf { z } ) ( \\Phi ^ { \\{ 1 \\} } _ t * g _ t ) ( \\mathbf { z } , \\mathbf { y } ) \\Big | \\ , d w ( \\mathbf { z } ) \\leq \\int _ { d ( \\mathbf { x } , \\mathbf { y } ) \\leq 2 d ( \\mathbf { x } , \\mathbf { z } ) } + \\int _ { d ( \\mathbf { x } , \\mathbf { y } ) \\leq 2 d ( \\mathbf { y } , \\mathbf { z } ) } = I _ 1 + I _ 2 . \\end{align*}"}
-{"id": "5715.png", "formula": "\\begin{align*} \\frac { 1 } { \\Gamma ( 2 - \\alpha ) } \\int _ 0 ^ x \\frac { 1 } { ( x - t ) ^ { \\alpha - 1 } } t ^ { \\alpha - 2 } E _ { \\alpha , \\alpha - 1 } ( \\lambda t ^ { \\alpha } ) f ( 0 ) d t & = f ( 0 ) E _ { \\alpha , 1 } ( \\lambda x ^ { \\alpha } ) . \\end{align*}"}
-{"id": "81.png", "formula": "\\begin{align*} C _ 1 \\cdot \\delta _ j < 1 \\ , , C _ 1 = \\sup g ' ( J ) \\ , , j = 1 , \\ldots , N - 1 \\ , , \\end{align*}"}
-{"id": "9986.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } E = 0 , \\\\ \\ell _ { E } ^ { \\ast } ( \\mathbf { p } ) = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "5700.png", "formula": "\\begin{align*} \\frac { d } { d x } E _ { \\alpha , 1 } ( \\lambda x ^ { \\alpha } ) = \\lambda x ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } ( \\lambda x ^ { \\alpha } ) . \\end{align*}"}
-{"id": "3155.png", "formula": "\\begin{align*} B ^ g ( u ) = T _ { 2 3 } ( u ) \\widehat { T } _ { 1 3 } ( u ) - T _ { 1 3 } ( u ) \\widehat { T } _ { 1 2 } ( u ) \\end{align*}"}
-{"id": "8076.png", "formula": "\\begin{align*} W _ k ^ n ( 1 ) - 2 \\sqrt { n } \\ge \\lambda _ k ( H ^ c ) \\ge \\lambda _ k ( H _ k ) = \\min _ { i = 1 } ^ k \\langle v _ i H _ k , v _ i \\rangle . \\end{align*}"}
-{"id": "3368.png", "formula": "\\begin{align*} p = \\sum _ { j = 0 } ^ 2 d _ j D _ j \\end{align*}"}
-{"id": "6585.png", "formula": "\\begin{align*} V \\coloneqq \\{ v \\in H ^ 1 ( \\Omega ) : v = g \\Gamma _ D \\} . \\end{align*}"}
-{"id": "189.png", "formula": "\\begin{align*} & \\pi _ { X _ { 1 , C } , X _ { 2 , V } , X _ { 3 , V } , X _ { 4 , V } } = X _ { 1 , C } \\wedge X _ { 2 , V } + X _ { 3 , V } \\wedge X _ { 4 , V } \\end{align*}"}
-{"id": "1836.png", "formula": "\\begin{align*} T _ { t t } ( t ) + \\lambda T ( t ) = 0 , \\end{align*}"}
-{"id": "2632.png", "formula": "\\begin{align*} \\sigma _ k = \\frac { 1 } { 2 } ( 1 + \\tau ^ 2 ) + N ^ { - \\frac { 1 } { 3 } } 2 ^ { - \\frac { 4 } { 3 } } ( 1 - \\tau ^ 2 ) ( \\pi _ { k } - \\kappa ) , k = 1 , \\ldots , m , \\end{align*}"}
-{"id": "5306.png", "formula": "\\begin{align*} ( \\alpha _ 0 + \\alpha _ 1 I _ 1 ) ( \\beta _ 0 + \\beta _ 1 I _ 2 ) = & 0 \\end{align*}"}
-{"id": "8096.png", "formula": "\\begin{align*} ( r _ k ) _ { k \\ge 0 } = ( 1 , - 1 4 , 8 6 , - 3 6 6 0 , - 1 0 4 2 2 0 2 , - 2 4 7 9 4 8 2 6 0 , - 1 0 8 4 4 8 5 4 0 4 2 0 , \\ldots ) . \\end{align*}"}
-{"id": "9716.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } \\bar u & \\geq & \\varphi & \\R ^ n \\\\ ( - \\Delta ) ^ s \\bar u & \\geq & 0 & \\R ^ n \\\\ ( - \\Delta ) ^ s \\bar u & = & 0 & \\{ \\bar u > \\varphi \\} \\\\ \\lim _ { | x | \\to \\infty } \\bar u ( x ) & = & 0 \\end{array} \\right . \\quad s : = \\frac { 1 - a } { 2 } \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "194.png", "formula": "\\begin{align*} \\pi _ { T M } ( { \\bf x } , { \\bf y } ) = \\left ( \\begin{array} { c c c | c c c } 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & x ^ 1 \\\\ 0 & 0 & 0 & 0 & - x ^ 1 & 0 \\\\ \\hline 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & x ^ 1 & 0 & 0 & y ^ 1 \\\\ 0 & - x ^ 1 & 0 & 0 & - y ^ 1 & 0 \\end{array} \\right ) , \\end{align*}"}
-{"id": "6185.png", "formula": "\\begin{align*} U ( X ) = v _ 1 ^ R ( X - \\tilde v _ 1 ^ R ( X ) e _ n ) , , \\bar U ( X ) = v _ 2 ^ R ( X - \\tilde { \\bar v } _ 2 ^ R ( X ) e _ n ) , \\end{align*}"}
-{"id": "815.png", "formula": "\\begin{align*} U ^ { q ^ d } \\lambda ^ { q ^ d } - \\lambda ^ { q ^ d - 1 } \\lambda U = \\lambda , \\end{align*}"}
-{"id": "7761.png", "formula": "\\begin{align*} 0 \\in W ( U ^ * ( \\| B \\| _ { \\infty } A + \\lambda \\| A \\| _ { \\infty } B ) V ) & = W ( \\| B \\| _ { \\infty } \\| A \\| _ { \\infty } V ^ * V + \\lambda \\| A \\| _ { \\infty } U ^ * B V ) \\\\ & = W ( \\| B \\| _ { \\infty } \\| A \\| _ { \\infty } I + \\lambda \\| A \\| _ { \\infty } U ^ * B V ) . \\end{align*}"}
-{"id": "8881.png", "formula": "\\begin{align*} V ^ r _ d ( \\Gamma ) = Z ( \\wedge ^ { d - r + 1 } \\phi _ { \\Gamma } ) . \\end{align*}"}
-{"id": "9260.png", "formula": "\\begin{align*} \\sup _ { | \\tau | = 1 , \\tau \\cdot \\nu = 0 } u _ { \\tau \\tau } ( x _ 0 ) \\ ( { \\rm o r } \\ \\sup _ { | \\tau | = 1 , \\tau \\cdot \\nu = 0 } | u _ { \\tau \\tau } ( x _ 0 ) | ) \\le \\epsilon M _ 2 ( R ^ \\prime ) + C _ \\epsilon ( 1 + \\frac { 1 } { ( R ^ \\prime ) ^ 2 } ) . \\end{align*}"}
-{"id": "4332.png", "formula": "\\begin{align*} & e _ 2 ( \\{ 1 , 2 , 3 \\} ) + e _ 2 ( \\{ 1 , 4 , 5 \\} ) + e _ 2 ( \\{ 6 , 3 , 4 \\} ) + e _ 2 ( \\{ 6 , 2 , 5 \\} ) \\\\ = & e _ 2 ( \\{ 1 , 3 , 4 \\} ) + e _ 2 ( \\{ 1 , 2 , 5 \\} ) + e _ 2 ( \\{ 6 , 2 , 3 \\} ) + e _ 2 ( \\{ 6 , 4 , 5 \\} ) . \\end{align*}"}
-{"id": "949.png", "formula": "\\begin{align*} J ^ { * } [ \\psi ^ { ( k ) } ] _ { T , t } = \\hbox { \\vtop { \\offinterlineskip \\halign { \\hfil # \\hfil \\cr { \\rm l . i . m . } \\cr $ \\stackrel { } { { } _ { p \\to \\infty } } $ \\cr } } } \\sum \\limits _ { j _ 1 , \\ldots j _ k = 0 } ^ { p } C _ { j _ k \\ldots j _ 1 } \\zeta _ { j _ 1 } ^ { ( i _ 1 ) } \\ldots \\zeta _ { j _ k } ^ { ( i _ k ) } \\end{align*}"}
-{"id": "2573.png", "formula": "\\begin{align*} \\epsilon _ x ^ 1 ( t _ 0 , t ) & : = \\dot { \\mathbf P } _ { \\delta _ x } ^ { ( \\phi ) } [ Y ^ { ( t _ 0 , t ] } _ t ( \\phi ) ^ { - 1 } ] - \\dot { \\mathbf P } _ \\nu ^ { ( \\phi ) } [ Y ^ { ( t _ 0 , t ] } _ t ( \\phi ) ^ { - 1 } ] ; \\\\ \\epsilon _ x ^ 2 ( t _ 0 , t ) & : = \\dot { \\mathbf P } _ { \\delta _ x } ^ { ( \\phi ) } [ Y _ t ( \\phi ) ^ { - 1 } - Y ^ { ( t _ 0 , t ] } _ t ( \\phi ) ^ { - 1 } ] . \\end{align*}"}
-{"id": "5170.png", "formula": "\\begin{align*} \\lim _ { u \\rightarrow 0 } \\frac { U ( \\lambda u ) - U ( u ) } { s ( x u ) - s ( u ) } = \\frac { \\log \\mu } { \\log x } . \\end{align*}"}
-{"id": "3317.png", "formula": "\\begin{align*} | \\sum _ { f \\in \\mathcal { M } _ { n } } \\Lambda ( f ) \\chi ( f ) | = | \\sum _ { i = 1 } ^ { \\deg L ( u , \\chi ) } \\gamma _ i ( \\chi ) ^ n | \\le ( \\ell + \\deg ( M ) - 1 ) q ^ { \\frac { n } { 2 } } , \\end{align*}"}
-{"id": "3523.png", "formula": "\\begin{align*} M _ { 3 , 6 } ^ { o } : = \\left \\{ A \\in M _ { 3 , 6 } \\mid D ( i _ { 1 } , i _ { 2 } , i _ { 3 } ) \\not = 0 \\ , ( 1 \\leq i _ { 1 } < i _ { 2 } < i _ { 3 } \\leq 6 ) \\right \\} \\end{align*}"}
-{"id": "4299.png", "formula": "\\begin{align*} \\textup { s a t } ( n , C _ t , C _ k ) = 0 . \\end{align*}"}
-{"id": "3862.png", "formula": "\\begin{align*} \\mathbf { E } [ \\mathcal { N } _ { t , x } f ( Z _ t ^ x ) ] & = \\sum _ { ( s , y ) \\in \\mathcal { S } ( t , x ) } \\mathcal { N } _ { t , x } f ( y - s e _ d ) P ( Z _ t ^ x = y - s e _ d ) \\\\ & = \\sum _ { ( s , y ) \\in \\mathcal { S } ( t , x ) } \\mathcal { N } f ( y - s e _ d ) P ( Z _ t ^ x = y - s e _ d ) . \\\\ & = \\mathbf { E } [ \\mathcal { N } f ( Z _ t ^ x ) ] \\end{align*}"}
-{"id": "42.png", "formula": "\\begin{align*} | \\delta \\mathcal { Y } _ 0 ^ { v } ( T ) - L | & = \\lim _ { T ' \\rightarrow \\infty } | \\delta \\mathcal { Y } _ 0 ^ { v } ( T ) - \\delta \\mathcal { Y } _ 0 ^ { v } ( T ' ) | \\\\ & = \\lim _ { T ' \\rightarrow \\infty } \\left | \\delta \\mathcal { Y } _ 0 ^ { v } ( T ) - \\mathbb { E } ^ { \\mathbb { Q } } \\left [ h ^ { \\alpha ^ { m ( T ' ) } _ { T ' } } ( V _ { T ' } ^ v ) - \\mathbf { y } ^ { \\alpha ^ { m ( T ' ) } _ { T ' } } ( V _ { T ' } ^ { v } ) \\right ] \\right | , \\end{align*}"}
-{"id": "9591.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { n q ^ n ( 1 - q ^ { N n } ) } { 1 - q ^ n } & = \\sum _ { n = 1 } ^ { \\infty } \\sum _ { m = 1 } ^ { \\infty } n q ^ { m n } - \\sum _ { n = 1 } ^ { \\infty } \\sum _ { m = N + 1 } ^ { \\infty } n q ^ { m n } . \\end{align*}"}
-{"id": "1194.png", "formula": "\\begin{align*} \\lim _ { \\tau \\longrightarrow \\infty } \\frac { 1 } { \\tau } \\log \\left \\vert I _ { f } ( \\tau , T ) \\right \\vert = - 2 l ( D , B ) . \\end{align*}"}
-{"id": "7753.png", "formula": "\\begin{align*} [ \\langle x , x \\rangle \\xi , \\xi ] = \\Big [ \\sum _ { \\alpha \\in \\Lambda } \\langle x , x _ { \\alpha } \\rangle \\langle x _ { \\alpha } , x \\rangle \\xi , \\xi \\Big ] = \\sum _ { \\alpha \\in \\Lambda } \\| x \\| ^ 2 . \\end{align*}"}
-{"id": "7934.png", "formula": "\\begin{align*} T _ 1 \\cup \\dots \\cup T _ t = R \\cup Q , \\end{align*}"}
-{"id": "308.png", "formula": "\\begin{align*} G _ \\Theta \\ ; = \\ ; [ G ( \\delta _ \\Theta ) ^ \\circ , G ( \\delta _ \\Theta ) ^ \\circ ] \\end{align*}"}
-{"id": "9051.png", "formula": "\\begin{align*} \\frac { 1 } { \\pi R ^ 2 } \\int _ { B ( R ) } | F _ { \\mu } | ^ 2 & = \\alpha \\frac { 1 } { \\pi R ^ 2 } \\int _ { B ( R ) } | F _ { \\mu _ { A } } | ^ 2 + \\beta \\frac { 1 } { \\pi R ^ 2 } \\int _ { B ( R ) } | F _ { \\mu _ B } | ^ 2 \\\\ & + 2 \\sqrt { \\alpha \\beta } \\frac { 1 } { \\pi R ^ 2 } \\int _ { B ( R ) } ( F _ { \\mu _ { A } } \\cdot F _ { \\mu _ B } ) . \\end{align*}"}
-{"id": "4285.png", "formula": "\\begin{align*} \\frac { 2 e ( G ) ^ 4 } { n ^ 4 } - \\frac { 3 } { 4 } e ( G ) n = \\left ( 2 n ^ { 3 \\delta } - \\frac { 3 } { 4 } \\right ) n ^ { 8 / 3 + \\delta } > \\frac { n ^ 2 } { 2 } \\end{align*}"}
-{"id": "5756.png", "formula": "\\begin{align*} \\bigg | \\int _ 0 ^ t ( t - \\xi ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } ( - \\mu _ { m , n } ( t - \\xi ) ^ \\alpha ) F _ { m , n } ( \\xi ) d \\xi \\bigg | & = \\frac { 1 } { \\mu _ { m , n } } \\bigg | \\int _ 0 ^ t ( t - \\xi ) ^ { - 1 } E _ { \\alpha , 0 } ( - \\mu _ { m , n } ( t - \\xi ) ^ \\alpha ) F _ { m , n } ( \\xi ) d \\xi \\bigg | \\\\ & \\leq \\frac { | | f | | _ \\infty } { \\mu _ { m , n } } \\int _ 0 ^ t \\bigg | ( t - \\xi ) ^ { - 1 } E _ { \\alpha , 0 } ( - \\mu _ { m , n } ( t - \\xi ) ^ \\alpha ) \\bigg | d \\xi \\end{align*}"}
-{"id": "3313.png", "formula": "\\begin{align*} L ( u , \\chi ) = \\prod _ { P \\in \\mathcal { P } } ( 1 - \\chi ( P ) u ^ { \\deg ( P ) } ) ^ { - 1 } . \\end{align*}"}
-{"id": "5760.png", "formula": "\\begin{align*} \\left \\langle \\varrho _ \\R ( x ) ( g \\cdot x _ 1 ) , g \\cdot x _ 3 \\right \\rangle = \\mu _ \\R ( g \\cdot x _ 1 , g \\cdot x _ 3 ) ( x ) = \\mu _ \\R ( x _ 1 , x _ 3 ) ( { \\rm A d } ( g ^ { - 1 } ) ( x ) ) , \\end{align*}"}
-{"id": "4349.png", "formula": "\\begin{align*} \\mathcal T _ 2 ( k ) : = \\big \\{ [ k ] \\big \\} \\cup \\big \\{ \\{ 1 \\} \\cup [ k + 1 , 2 k - 1 ] \\big \\} \\cup \\big \\{ \\{ 2 \\} \\cup [ k + 1 , 2 k - 1 ] \\big \\} . \\end{align*}"}
-{"id": "4782.png", "formula": "\\begin{align*} M _ 2 = \\Big \\{ z _ 1 ( T ) \\in Y \\ \\Big | \\ z _ 1 \\mbox { i s t h e s o l u t i o n t o } ( \\ref { 6 1 8 } ) \\mbox { w i t h s o m e } v ( \\cdot ) \\in \\mathcal { U } _ 2 \\mbox { s a t i s f y i n g } | v | _ { \\mathcal { U } _ 2 } \\leq 1 \\Big \\} . \\end{align*}"}
-{"id": "2323.png", "formula": "\\begin{align*} | \\alpha _ x \\rangle = \\frac { 1 } { \\sqrt { C _ R ( x ) ^ 2 + \\pi ^ 2 | g ( x ) | ^ 4 } } \\left ( C _ R ( x ) \\left ( \\begin{array} { c } 0 \\\\ | \\varepsilon _ x \\rangle \\end{array} \\right ) + \\overline { g } ( x ) \\left ( \\begin{array} { c } 1 \\\\ \\frac { \\mathcal { P } } { x \\Omega } | g \\rangle \\end{array} \\right ) \\right ) . \\end{align*}"}
-{"id": "2756.png", "formula": "\\begin{align*} \\phi _ { 1 } ''' ( \\kappa ; u ) - \\kappa \\big ( \\phi _ { 1 } '' ( \\kappa ; u ) - u \\phi _ { 1 } ( \\kappa ; u ) \\big ) - u \\phi _ { 1 } ' ( \\kappa ; u ) - \\frac { 1 } { 2 } \\phi _ { 1 } ( \\kappa ; u ) & = 0 , \\\\ \\phi _ { 2 } ''' ( \\kappa ; v ) + \\kappa \\big ( \\phi _ { 2 } '' ( \\kappa ; v ) - v \\phi _ { 2 } ( \\kappa ; v ) \\big ) - v \\phi _ { 2 } ' ( \\kappa ; v ) - \\frac { 1 } { 2 } \\phi _ { 2 } ( \\kappa ; v ) & = 0 . \\end{align*}"}
-{"id": "8315.png", "formula": "\\begin{align*} ( I - \\mathfrak { H } ) \\bar { z } _ t = & - \\frac { i } { \\pi } \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j } { z ( \\alpha , t ) - z _ j ( t ) } . \\end{align*}"}
-{"id": "9459.png", "formula": "\\begin{align*} g _ \\epsilon ( z ) : = \\sum _ { \\gamma \\in \\Gamma _ 0 ( N ) _ \\infty \\backslash \\Gamma _ 0 ( N ) } \\phi _ \\epsilon ( \\mathrm { I m } \\gamma z ) , \\end{align*}"}
-{"id": "5194.png", "formula": "\\begin{align*} H ^ { s t } ( x ) = ( H ^ { s } ( x ) ) ^ { t } = H ( \\alpha ( s ) x + \\beta ( s ) ) ^ { t } . \\end{align*}"}
-{"id": "4595.png", "formula": "\\begin{align*} \\hat { \\phi ^ D _ \\delta } ( \\omega ) = 0 . \\end{align*}"}
-{"id": "8748.png", "formula": "\\begin{align*} \\phi _ n ( x ) = ( 1 + e ^ x \\tau ) ^ { - 1 - n } \\phi ( x ) , { \\qquad } \\phi ( x ) = - \\frac { \\sin ( \\alpha \\pi ) } { \\pi } e ^ { ( 1 + \\alpha ) x } F _ { \\omega } ( - e ^ x ) . \\end{align*}"}
-{"id": "9974.png", "formula": "\\begin{align*} \\frac { | p ' ( x ) | } { | p ( x ) | } & = \\frac { | \\sum _ { j = 1 } ^ k j a _ j i ^ j x ^ { j - 1 } | } { | p ( x ) | } \\ < 2 \\frac { \\sum _ { j = 1 } ^ k j | a _ j | | x ^ { j - 1 } | } { | a _ k x ^ k | } = 2 \\sum _ { j = 1 } ^ k \\frac { j | a _ j | } { | a _ k | | x ^ { k - j + 1 } | } \\ < \\frac { 2 R } { | a _ k | } \\sum _ { j = 1 } ^ k \\frac { j } { | x | } \\\\ & = \\frac { k ( k + 1 ) R } { | a _ k | | x | } . \\end{align*}"}
-{"id": "7058.png", "formula": "\\begin{align*} \\alpha = - ( \\Delta - 2 ) ^ { - 1 } \\frac { 2 | q | ^ 2 } { \\phi ^ 2 _ 0 } \\geq \\frac { 1 } { 3 } \\frac { | q | ^ 2 } { \\phi _ 0 ^ 2 } > 0 a . e . , \\end{align*}"}
-{"id": "8600.png", "formula": "\\begin{align*} u v = \\frac { \\omega } { \\Lambda - 1 } , \\ u ^ 2 + v ^ 2 = 1 . \\end{align*}"}
-{"id": "1498.png", "formula": "\\begin{align*} \\mathcal { H } ( t , x , m , D u ) = \\phi ( m ) H _ { 1 } ( t , x , m , D u ) + H _ { 2 } ( t , x , m , D u ) , \\end{align*}"}
-{"id": "9971.png", "formula": "\\begin{align*} \\begin{dcases*} \\partial _ t u ( 0 , x ) = \\mathring { u } _ 2 & o n $ \\R { n } $ \\\\ u ( 0 , x ) = \\mathring { u } _ 1 & o n $ \\R { n } $ , \\end{dcases*} \\end{align*}"}
-{"id": "9050.png", "formula": "\\begin{align*} \\lim \\limits _ { R \\rightarrow \\infty } \\frac { 1 } { \\pi R ^ 2 } \\int _ { B ( R ) } H ( F ( x ) ) d x = \\mathbb { E } [ H ( F ) ] . \\end{align*}"}
-{"id": "9901.png", "formula": "\\begin{align*} \\| g \\| _ { \\infty } & \\leq 4 M \\| f \\| _ { \\infty } \\\\ \\int g ( h ( x , z ) Q ( d z ) & = f ( x ) & x \\in [ - M + a , M + a ] \\\\ \\left | \\int g ( h ( x , z ) Q ( d z ) \\right | & = 0 & x \\not \\in [ - M + a , M + a ] \\end{align*}"}
-{"id": "2339.png", "formula": "\\begin{align*} [ a _ x , a ^ + _ y ] & = \\delta _ { x , y } & [ a _ x , a _ y ] & = [ a ^ + _ x , a ^ + _ y ] = 0 \\end{align*}"}
-{"id": "6127.png", "formula": "\\begin{align*} \\Omega \\left ( D _ A ^ a D _ B ^ b ( F _ { i , j } ) \\right ) = ( - 1 ) ^ b X ^ a Y ^ b \\Omega \\left ( F _ { i , j } \\right ) . \\end{align*}"}
-{"id": "4458.png", "formula": "\\begin{align*} ( \\eta _ i ) _ { V ' _ { i j } } \\circ \\phi ' _ { i j } = ( \\eta _ j ) _ { V ' _ { i j } } \\end{align*}"}
-{"id": "2455.png", "formula": "\\begin{align*} P ( R , t ) \\ = \\ \\frac { 1 } { 1 - t ^ d } \\left ( \\frac { 1 } { 1 - t } - p ( t ) \\right ) , \\end{align*}"}
-{"id": "8471.png", "formula": "\\begin{align*} \\begin{cases} \\sum _ { l = 1 } ^ n k _ l = k \\\\ \\sum _ { l = 1 } ^ n l k _ l = n . \\end{cases} \\end{align*}"}
-{"id": "2228.png", "formula": "\\begin{align*} \\# \\left ( J \\cap J _ { k } \\right ) = 1 \\forall k \\in \\left [ n \\right ] \\end{align*}"}
-{"id": "208.png", "formula": "\\begin{align*} & X _ { 1 , C } = \\dfrac { \\partial } { \\partial x ^ 3 } , & & Y _ { 1 , V } = x ^ 1 \\dfrac { \\partial } { \\partial y ^ 2 } , \\\\ & X _ { 2 , C } = \\dfrac { \\partial } { \\partial x ^ 2 } , & & Y _ { 2 , V } = x ^ 3 \\dfrac { \\partial } { \\partial y ^ 2 } . \\end{align*}"}
-{"id": "921.png", "formula": "\\begin{align*} J [ \\psi ^ { ( 2 ) } ] _ { T , t } = \\hbox { \\vtop { \\offinterlineskip \\halign { \\hfil # \\hfil \\cr { \\rm l . i . m . } \\cr $ \\stackrel { } { { } _ { p _ 1 , p _ 2 \\to \\infty } } $ \\cr } } } \\sum _ { j _ 1 = 0 } ^ { p _ 1 } \\sum _ { j _ 2 = 0 } ^ { p _ 2 } C _ { j _ 2 j _ 1 } \\Biggl ( \\zeta _ { j _ 1 } ^ { ( i _ 1 ) } \\zeta _ { j _ 2 } ^ { ( i _ 2 ) } - { \\bf 1 } _ { \\{ i _ 1 = i _ 2 \\ne 0 \\} } { \\bf 1 } _ { \\{ j _ 1 = j _ 2 \\} } \\Biggr ) , \\end{align*}"}
-{"id": "8074.png", "formula": "\\begin{align*} s _ i \\cdot f ( x _ 1 , \\ldots , x _ i , x _ { i + 1 } , \\ldots , x _ n ) = f ( x _ 1 , \\ldots , x _ { i + 1 } , x _ { i } , \\ldots , x _ n ) . \\end{align*}"}
-{"id": "3965.png", "formula": "\\begin{align*} \\phi ( \\tau , z ) = \\sum _ { \\mu \\bmod { 2 m } } h _ { m , \\mu } ( \\tau ) \\theta _ { m , \\mu } ( \\tau , z ) , \\end{align*}"}
-{"id": "5280.png", "formula": "\\begin{align*} r ( x ) = r ( x _ 0 ) \\exp \\left ( \\int _ { x _ 0 } ^ { x } \\frac { - 1 + \\ell ( t ) } { r ( t ) } d t \\right ) , \\end{align*}"}
-{"id": "5769.png", "formula": "\\begin{align*} A _ d ( Q , q ) : = \\sum _ { ( \\Gamma , r ) } ( q - 1 ) ^ { b _ 1 ( \\Gamma ) } q ^ { \\delta ( \\Gamma , r ) } , \\end{align*}"}
-{"id": "1870.png", "formula": "\\begin{align*} G ^ + _ { \\rm O U T } ( x , t ; x ' , t ' ) = G ^ + _ { \\rm I N } ( x , t ; x ' , t ' ) . \\end{align*}"}
-{"id": "5816.png", "formula": "\\begin{align*} | a _ n | r ^ n + 2 \\Re ( a _ 0 ) \\leq 4 \\ , \\ , \\underset { | z | = r } { m a x } ( | \\sum _ { k = 0 } ^ d b _ k X ^ k | , 0 ) \\end{align*}"}
-{"id": "5688.png", "formula": "\\begin{align*} \\| x _ m ^ A - x _ m ^ B \\| = t _ m \\| v _ m ^ A - v _ m ^ B \\| \\le \\frac { \\| x _ m ^ A - x _ 0 \\| } { 1 - \\| v _ m ^ A - v ^ A \\| } \\left ( \\| v _ m ^ A - v ^ A \\| + \\| v ^ A - v ^ B \\| + \\| v ^ B - v _ m ^ B \\| \\right ) \\ , . \\end{align*}"}
-{"id": "5038.png", "formula": "\\begin{align*} U _ n ( x ) = n ^ { - m / 2 } \\sum _ { ( i _ 1 , \\dots , i _ m ) \\in I _ m ^ n } u _ { n x } ( V _ { i _ 1 } , \\dots , V _ { i _ m } ) , x \\in \\R , \\end{align*}"}
-{"id": "2636.png", "formula": "\\begin{align*} [ d X ] = C _ { N , M } \\Delta _ { N } ^ { 2 } ( \\lambda _ { 1 } , \\ldots , \\lambda _ { N } ) \\big ( \\prod _ { k = 1 } ^ { N } \\lambda _ { k } ^ { n } d \\lambda _ { k } \\big ) \\ , d \\mu _ { M } ( U ) d \\mu _ { N } ( V ) , \\end{align*}"}
-{"id": "445.png", "formula": "\\begin{align*} \\int _ D ( \\tilde { X } _ { t _ 0 } ( x ) ) _ + d x = & \\int _ D ( \\tilde { X } _ { t _ 0 } ( x ) ) _ { - } d x \\\\ \\geq & \\int _ { D \\cap \\mathcal { A } } ( \\tilde { X } _ { t _ 0 } ( x ) ) _ { - } d x \\ge 1 - | \\log s | ^ { - \\kappa } . \\end{align*}"}
-{"id": "7397.png", "formula": "\\begin{align*} G _ { x } = \\{ h \\in G \\mid h x = x \\} \\end{align*}"}
-{"id": "6090.png", "formula": "\\begin{align*} g ( x ) = ( x - a ^ 1 _ j ) \\ldots ( x - a ^ { k _ j } _ j ) h ' _ j ( x ) + e _ 1 ( \\widehat { x - a ^ 1 _ j } ) ( x - a ^ 2 _ j ) \\ldots ( x - a ^ { k _ j } _ j ) h _ j ( x ) + \\\\ e ^ 2 _ j ( x - a ^ 1 _ j ) ( \\widehat { x - a ^ 2 _ j } ) \\ldots ( x - a ^ { k _ j } _ j ) h _ j ( x ) + \\\\ e _ k ( x - a ^ 1 _ j ) \\ldots ( x - a ^ { k _ j - 1 } _ j ) ( \\widehat { x - a ^ { k _ j } _ j } ) \\end{align*}"}
-{"id": "9069.png", "formula": "\\begin{align*} & \\mathcal { K } = \\{ - 1 + K \\leq k \\leq K : \\mu _ { f } ( I _ k ) \\geq \\delta \\} & \\mathcal { G } = \\cup _ { k \\not \\in \\mathcal { K } } \\mathcal { E } ^ { ( k ) } . \\end{align*}"}
-{"id": "5098.png", "formula": "\\begin{align*} x = \\sum \\limits _ { j \\in \\mathbb { J } } \\langle x , x _ j \\rangle \\tau _ j \\iff \\sum \\limits _ { j \\in \\mathbb { J } } ( 2 e - c _ j ) \\langle x , x _ j \\rangle \\langle \\tau _ j , x \\rangle = \\langle x , x \\rangle \\iff \\sum \\limits _ { j \\in \\mathbb { J } } c _ j ^ 2 \\langle x , x _ j \\rangle \\langle x _ j , x \\rangle = \\langle x , x \\rangle . \\end{align*}"}
-{"id": "9479.png", "formula": "\\begin{align*} \\psi = u - v = ( u - v ) ^ { + } - ( u - v ) ^ { - } , \\varphi = ( u - v ) ^ { + } = \\psi ^ { + } . \\end{align*}"}
-{"id": "7296.png", "formula": "\\begin{align*} p = p _ 0 > p _ 1 > \\dots > p _ n > \\dots \\end{align*}"}
-{"id": "9968.png", "formula": "\\begin{align*} \\varphi ( L ) : = \\sup _ { t \\in I } e ^ { - L t } \\int _ 0 ^ t e ^ { L s } \\eta ( s ) \\ , d s \\xrightarrow [ L \\to + \\infty ] { } 0 , \\end{align*}"}
-{"id": "1768.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\left ( B _ { t _ { i } } - B _ { s } \\right ) \\left ( B _ { t _ { j } } - B _ { t _ { i } } \\right ) \\right ] = \\frac { 1 } { 2 } \\left [ ( t _ { j } - s ) ^ { 2 H } - ( t _ { j } - t _ { i } ) ^ { 2 H } - ( t _ { i } - s ) ^ { 2 H } \\right ] \\leq \\frac { 1 } { 2 } ( t - s ) ^ { 2 H } , \\end{align*}"}
-{"id": "10074.png", "formula": "\\begin{align*} f : \\begin{pmatrix} x \\\\ y \\\\ \\theta \\end{pmatrix} \\mapsto \\begin{pmatrix} x + \\O ( \\| ( x , y ) \\| ^ N ) \\\\ y + \\O ( \\| ( x , y ) \\| ^ N ) \\\\ \\theta + \\omega + \\O ( \\| ( x , y ) \\| ^ L ) \\end{pmatrix} , \\end{align*}"}
-{"id": "4567.png", "formula": "\\begin{align*} \\gamma _ s = \\begin{cases} s & s \\in ( 0 , ( n - 1 ) / 2 ] ; \\\\ ( n - 1 ) / 2 & s \\in [ ( n - 1 ) / 2 , n / 2 ] ; \\\\ ( n + 2 s - 2 ) / 4 & s \\in [ n / 2 , ( n + 2 ) / 2 ] ; \\\\ s - 1 & s \\in [ ( n + 2 ) / 2 , n ) . \\end{cases} \\end{align*}"}
-{"id": "1720.png", "formula": "\\begin{align*} R ( t , s ) = \\mathbb { E } \\left [ B _ { t } B _ { s } \\right ] = \\frac { 1 } { 2 } \\left ( t ^ { 2 H } + s ^ { 2 H } - | t - s | ^ { 2 H } \\right ) \\end{align*}"}
-{"id": "709.png", "formula": "\\begin{align*} \\exists ( \\eta _ k ) _ { k \\in \\mathbb { N } } \\in \\mathcal { C } ^ \\infty \\left ( [ 0 , 1 ] \\right ) ^ \\mathbb { N } , e ^ { - F ( x ) } \\underset { x \\to 1 ^ - } { \\sim } ( 1 - x ) \\sum _ { k = 0 } ^ { + \\infty } \\eta _ k \\left ( ( 1 - x ) ^ 3 L o g ( 1 - x ) \\right ) ^ k , \\end{align*}"}
-{"id": "2584.png", "formula": "\\begin{align*} P ^ \\beta _ t f ( x ) = \\int _ E p _ t ^ \\beta ( x , y ) f ( y ) d y \\xrightarrow [ t \\to \\infty ] { } \\phi ( x ) \\langle f , \\phi ^ * \\rangle _ m . \\end{align*}"}
-{"id": "6019.png", "formula": "\\begin{align*} \\rho ^ \\mu _ \\epsilon ( F ) = \\int _ { \\R ^ d } \\rho ^ { \\delta _ x } _ \\epsilon ( F ) \\ , \\mu ( d x ) . \\end{align*}"}
-{"id": "3471.png", "formula": "\\begin{align*} \\frac { d } { d x } \\Re N F ( x \\pm i / 4 ; w ) \\big \\rvert _ { x = - 1 } = 1 + o ( 1 ) > 0 , \\frac { d ^ 2 } { d x ^ 2 } \\Re F ( x \\pm i / 4 ; w ) < 0 , \\end{align*}"}
-{"id": "967.png", "formula": "\\begin{align*} D D ^ { ( - 1 ) } = ( k - \\lambda ) 1 _ { G } + \\lambda G \\end{align*}"}
-{"id": "3124.png", "formula": "\\begin{align*} A ^ { ( 2 ) } & = \\frac { 1 } { y ^ { - \\frac { 1 } { 2 } } + y ^ \\frac { 1 } { 2 } } A ^ { ( 2 ) } _ { \\underline { \\chi ( \\O _ S ) } } ( y ) \\ , , \\\\ B ^ { ( 2 ) } & = A ^ { ( 2 ) } _ { \\underline { K _ S ^ 2 } } ( y ) \\ , , \\\\ C ^ { ( 2 ) } _ { 1 1 } & = - q ^ { - \\frac { 1 } { 4 } } \\frac { 1 } { y ^ { - \\frac { 1 } { 2 } } + y ^ \\frac { 1 } { 2 } } A ^ { ( 2 ) } _ { \\underline { \\beta ^ 1 \\beta ^ 1 } } ( y ) A ^ { ( 2 ) } _ { \\underline { \\beta ^ 1 K _ S } } ( y ) \\ , , \\end{align*}"}
-{"id": "1271.png", "formula": "\\begin{align*} \\int _ 0 ^ { 2 / | \\xi | ^ { \\gamma } } e ^ { - 3 i \\Phi ( \\xi , \\eta ) } \\phi ( | \\xi | ^ { \\gamma } \\eta ) d \\eta & = \\int _ 0 ^ { 2 / | \\xi | ^ { \\gamma } } e ^ { - 3 i \\eta \\xi ^ 2 } \\phi ( | \\xi | ^ { \\gamma } \\eta ) + e ^ { - 3 i \\eta \\xi ^ 2 } \\left ( e ^ { 3 i \\xi \\eta ^ 2 - 3 i \\eta ^ 3 / 4 } - 1 \\right ) \\phi ( | \\xi | ^ { \\gamma } \\eta ) d \\eta \\\\ & = O ( | \\xi | ^ { - 2 } ) + O \\left ( \\int _ 0 ^ { | \\xi | ^ { - \\gamma } } | \\eta | ^ 2 | \\xi | d \\eta \\right ) = O ( | \\xi | ^ { 1 - 3 \\gamma } ) \\end{align*}"}
-{"id": "3338.png", "formula": "\\begin{align*} g ( j ) = \\begin{cases} z _ 1 & \\mbox { i f $ m \\nmid j $ , } \\\\ z _ 2 & \\mbox { o t h e r w i s e , } \\end{cases} \\end{align*}"}
-{"id": "6623.png", "formula": "\\begin{align*} P _ N ( \\varphi ) \\geq P _ 1 ( \\varphi ) = 1 . 8 6 \\ldots , \\end{align*}"}
-{"id": "6743.png", "formula": "\\begin{align*} L _ j = \\dfrac { \\partial } { \\partial t _ j } + ( a _ j + i b _ j ) ( t _ j ) \\dfrac { \\partial } { \\partial x } , \\ ; j = 1 , \\ldots , n , \\end{align*}"}
-{"id": "2083.png", "formula": "\\begin{align*} { C _ z } ( z , t \\mid { z _ { \\rm t x } } , { t _ 0 } ) = \\frac { 1 } { { 2 \\pi } } \\int _ { - \\infty } ^ \\infty { \\tilde C _ z ( \\beta , t \\mid z _ { \\rm t x } , { t _ 0 } ) } { e ^ { j \\beta z } } d \\beta . \\end{align*}"}
-{"id": "8815.png", "formula": "\\begin{align*} \\overline { X } _ t ^ k & = \\langle \\overline { v } ^ k _ t , \\phi \\rangle - \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { \\pi } v ^ k ( s , x ) \\phi '' ( x ) \\ , \\textrm { d } s \\ , \\textrm { d } x , \\\\ \\overline { X } _ t & = \\langle \\overline { v } _ t , \\phi \\rangle - \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { \\pi } v ( s , x ) \\phi '' ( x ) \\ , \\textrm { d } s \\ , \\textrm { d } x \\end{align*}"}
-{"id": "4079.png", "formula": "\\begin{align*} \\Psi _ 0 ( z ) + \\Psi _ 1 ( z ) + \\Psi _ 2 ( z ) & = 1 + \\mathcal { O } ( z ^ { - 3 / 2 } ) , \\\\ \\Psi _ 0 ( z ) \\Psi _ 1 ( z ) + \\Psi _ 0 ( z ) \\Psi _ 2 ( z ) + \\Psi _ 1 ( z ) \\Psi _ 2 ( z ) & = \\frac { 1 } { z } + \\mathcal { O } ( z ^ { - 2 } ) \\\\ \\Psi _ 0 ( z ) \\Psi _ 1 ( z ) \\Psi _ 2 ( z ) & = \\frac { 1 } { 4 z ^ 2 } + \\mathcal O ( z ^ { - 3 } ) \\end{align*}"}
-{"id": "5953.png", "formula": "\\begin{align*} d Y ( t ) = - { g ^ * \\left ( n t - \\lfloor n t \\rfloor , \\sqrt { n } Z ( t ) \\right ) } \\ , d t + Z ( t ) \\ , d W ( t ) , Y ( 1 ) = F \\left ( \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\delta _ { W _ { ( n , k ) } } \\right ) . \\end{align*}"}
-{"id": "6222.png", "formula": "\\begin{align*} Q \\left ( \\lim _ { k \\to \\infty } F ^ \\textnormal { E M } ( \\theta _ k ' ) , \\theta ' \\right ) = \\max _ { \\theta \\in \\Theta } Q ( \\theta , \\theta ' ) , \\end{align*}"}
-{"id": "5621.png", "formula": "\\begin{align*} 2 \\int _ 0 ^ { + \\infty } u _ t \\bar { u } _ x d x = \\int _ 0 ^ { + \\infty } \\frac { d } { d t } ( u \\bar { u } _ x ) d x + ( u ( 0 , t ) \\bar { u } _ t ( 0 , t ) ) . \\end{align*}"}
-{"id": "8478.png", "formula": "\\begin{align*} \\| q \\| _ { H ^ s } \\leq & \\Big ( \\sum _ { n = 0 } ^ s \\| \\partial _ { \\alpha } ^ n q \\| _ { L ^ 2 } ^ 2 \\Big ) ^ { 1 / 2 } \\\\ \\leq & \\Big ( \\sum _ { n = 0 } ^ s ( 4 0 0 | \\lambda x ( 0 ) | n ! ( n + 1 ) ! d _ I ( t ) ^ { - 3 / 2 } ) ^ 2 \\Big ) ^ { 1 / 2 } \\\\ \\leq & 4 0 0 ( ( s + 2 ) ! ) ^ 2 | \\lambda x ( 0 ) | d _ I ( t ) ^ { - 3 / 2 } \\\\ \\leq & K _ s ^ { - 1 } \\epsilon d _ I ( t ) ^ { - 3 / 2 } . \\end{align*}"}
-{"id": "8577.png", "formula": "\\begin{align*} \\{ E _ 2 ^ + , E _ 2 ^ - \\} = \\sum _ j 1 \\otimes ( x _ j y _ j - \\tfrac { 1 } { 2 } ) - \\sum _ { \\alpha > 0 } c _ \\alpha s _ \\alpha \\otimes ( \\alpha \\alpha ^ \\vee - 1 ) + \\sum _ { j } ( x _ j y _ j + \\tfrac { 1 } { 2 } ) \\otimes 1 - \\sum _ { \\alpha > 0 } c _ \\alpha s _ \\alpha \\otimes 1 = Z _ 0 + \\Omega _ c + H . \\end{align*}"}
-{"id": "1624.png", "formula": "\\begin{align*} u _ r ( x ) : = \\frac { \\psi _ r ' ( x ) } { \\psi _ r ( x ) } - x = \\frac { S ' ( x ) } { \\psi _ r ' ( x ) } \\int _ 0 ^ x \\psi _ r ( t ) \\theta _ r ( t ) m ' ( t ) d t . \\end{align*}"}
-{"id": "4536.png", "formula": "\\begin{align*} \\hat { f } _ { g , \\delta , j } = \\hat { \\nu } _ g \\hat { \\phi ^ D _ \\delta } \\psi ( 2 ^ { - j } . ) \\end{align*}"}
-{"id": "5156.png", "formula": "\\begin{align*} A _ { s s } : = \\mu + \\nu , B _ { s s } : = - ( k _ j + \\lambda ) , s \\sim ( j , m _ j , k _ j ) \\in I _ 3 . \\end{align*}"}
-{"id": "4376.png", "formula": "\\begin{align*} \\mathrm { A r e a } _ { \\mathcal { P } } ( w ( X ) ) = \\mathrm { m i n } \\left \\{ k \\mid w ( X ) = _ { F r e e ( X ) } \\prod _ { i = 1 } ^ k \\theta _ i ( X ) r _ i \\theta _ i ( X ) ^ { - 1 } , r _ i \\in R ^ { \\pm 1 } , \\theta _ i ( X ) \\mbox { w o r d s i n } X \\right \\} . \\end{align*}"}
-{"id": "1526.png", "formula": "\\begin{align*} [ [ x , y ] , z ] = [ [ x , z ] , y ] + [ x , [ y , z ] ] \\end{align*}"}
-{"id": "7918.png", "formula": "\\begin{align*} f ^ { ( n + 1 ) } ( \\xi ) = \\sum _ { i = 1 } ^ { M } \\frac { f ^ { ( n + 1 ) } ( \\xi _ i ) } { M } \\end{align*}"}
-{"id": "5140.png", "formula": "\\begin{align*} ( k _ j + \\lambda + \\gamma _ { k _ j } ) \\left ( \\Gamma ^ + ( \\psi ) \\right ) _ { m _ j , k _ j } = ( \\mu - \\nu ) \\left ( \\Gamma ^ - ( \\psi ) \\right ) _ { m _ j , k _ j } ; \\end{align*}"}
-{"id": "309.png", "formula": "\\begin{align*} c _ { M , i } ( \\Lambda ) \\ ; = \\ ; \\begin{cases} 6 & \\textnormal { f o r $ i = 0 $ } , \\\\ \\tfrac { 2 } { 3 } [ \\Theta ] ^ 3 & \\textnormal { f o r $ i = 1 $ } , \\\\ 0 & \\textnormal { f o r $ i > 1 $ } , \\end{cases} \\end{align*}"}
-{"id": "7443.png", "formula": "\\begin{align*} \\displaystyle R _ q ( Z _ 1 , \\ldots , Z _ q ) = \\frac { 1 } { \\lvert W ' \\rvert } \\sum _ { w \\in W } w \\cdot \\left ( \\frac { T _ q ( Z _ 1 , \\ldots , Z _ q ) } { \\prod \\limits _ { 1 \\leqslant j \\leqslant q } 2 Z _ j } \\right ) \\end{align*}"}
-{"id": "2696.png", "formula": "\\begin{align*} \\mathcal { \\tilde { C } } = \\left \\{ z \\in \\mathcal { Q } _ { \\tau , z _ 2 } : \\mathrm { A r g } ( z _ { 3 } ) \\leq | \\mathrm { A r g } ( z ) | \\leq \\pi \\right \\} \\cup \\left \\{ z = z _ { 1 } + i y : - \\Im { z _ { 3 } } \\leq y \\leq \\Im { z _ { 3 } } \\right \\} . \\end{align*}"}
-{"id": "892.png", "formula": "\\begin{align*} \\Phi ^ 1 = \\{ \\alpha _ 3 \\prec \\alpha _ 1 + \\alpha _ 3 \\prec \\alpha _ 1 \\} \\quad \\Phi ^ 2 = \\{ \\alpha _ 2 \\} . \\end{align*}"}
-{"id": "6660.png", "formula": "\\begin{align*} H _ 1 ( \\{ 1 , 2 , 3 \\} ) = - & | K _ 1 ( \\{ \\{ 1 , 2 , 3 \\} \\} ) | + | K _ 1 ( \\{ \\{ 1 , 2 \\} , \\{ 3 \\} \\} ) | + | K _ 1 ( \\{ \\{ 2 , 3 \\} , \\{ 1 \\} \\} ) | \\\\ + & | K _ 1 ( \\{ \\{ 1 , 3 \\} , \\{ 2 \\} \\} ) | - | K _ 1 ( \\{ \\{ 1 \\} , \\{ 2 \\} , \\{ 3 \\} \\} ) | \\\\ = - & 1 + 1 + 1 - 1 = 0 . \\end{align*}"}
-{"id": "7109.png", "formula": "\\begin{align*} x = \\Psi _ { m } \\mathfrak { c } ( \\Psi _ { m } ) ^ { * } \\end{align*}"}
-{"id": "551.png", "formula": "\\begin{align*} \\Box \\phi = - S ( \\phi ) ( \\partial ^ \\alpha \\phi , \\partial _ \\alpha \\phi ) , \\end{align*}"}
-{"id": "7682.png", "formula": "\\begin{align*} \\xi + U \\xi = ( 1 + U ) \\xi = ( 1 - U ) \\mathbf { i } \\eta = \\mathbf { i } \\eta - U \\mathbf { i } \\eta . \\end{align*}"}
-{"id": "9109.png", "formula": "\\begin{align*} R = \\left ( \\dfrac { c _ 1 } { \\lambda _ 1 ( g ) \\| g _ 2 \\| _ { \\infty } } \\right ) ^ { \\frac 1 { s p } } \\quad k _ 1 = \\dfrac { k } { R } , \\end{align*}"}
-{"id": "5078.png", "formula": "\\begin{align*} & \\langle h , x _ g \\rangle = A _ g h = A _ e \\pi _ { g ^ { - 1 } } h = \\langle \\pi _ { g ^ { - 1 } } h , x _ e \\rangle = \\langle h , \\pi _ g x _ e \\rangle , \\\\ & \\langle h , \\tau _ g \\rangle = \\Psi _ g h = \\Psi _ e \\pi _ { g ^ { - 1 } } h = \\langle \\pi _ { g ^ { - 1 } } h , \\tau _ g \\rangle = \\langle h , \\pi _ g \\tau _ e \\rangle , ~ \\forall g \\in G , \\forall h \\in \\mathcal { H } \\\\ & \\iff x _ g = \\pi _ g x _ e , ~ \\tau _ g = \\pi _ g \\tau _ e , ~ \\forall g \\in G . \\end{align*}"}
-{"id": "3136.png", "formula": "\\begin{align*} f ( u , v ) = 1 + g ( u , v ) = \\frac { u - v + c } { u - v } . \\end{align*}"}
-{"id": "2740.png", "formula": "\\begin{align*} z _ { - } = \\theta , z _ { + } = \\frac { 1 } { \\tau } - \\tau - \\theta . \\end{align*}"}
-{"id": "9376.png", "formula": "\\begin{align*} c _ i = [ y , x , \\overset { i - 1 } { \\ldots } , x ] z _ i = [ c _ { i - 1 } , y ] = [ y , x , \\overset { i - 2 } { \\ldots } , x , y ] \\end{align*}"}
-{"id": "5307.png", "formula": "\\begin{align*} \\alpha _ 0 + \\alpha _ 1 I _ 1 + \\alpha _ 2 I _ 2 = & 0 , \\end{align*}"}
-{"id": "1204.png", "formula": "\\begin{align*} I _ { \\tilde \\tau , k } ( x - \\tilde { z } ' , \\zeta _ 2 ) & = \\int _ { \\R } { e } ^ { - \\tilde \\tau r \\lambda } Q _ k ( \\zeta _ 1 , \\zeta _ 2 ) \\frac { d \\zeta _ 1 } { \\sqrt { 1 + \\zeta _ 1 ^ 2 } } , \\end{align*}"}
-{"id": "3567.png", "formula": "\\begin{align*} W _ { 2 } = U V z , \\ ; \\ ; W _ { 3 } = U V y , \\ ; \\ ; x = V u \\end{align*}"}
-{"id": "1103.png", "formula": "\\begin{align*} & \\liminf _ { n \\to + \\infty } \\int _ { I ^ k } \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) d y _ 1 \\cdots d y _ k \\geq ( M ( I ) ) ^ k \\prod _ { j = 1 } ^ k \\left ( e ^ { c _ 0 - x _ j } \\right ) , \\end{align*}"}
-{"id": "5781.png", "formula": "\\begin{align*} R ( \\Gamma , q , T ) = \\epsilon ( \\Gamma ) + T \\sum _ { A \\subseteq E ( \\Gamma ) } R ( \\Gamma / A , q , q ^ { b _ 1 ( \\Gamma [ A ] ) } T ) , \\end{align*}"}
-{"id": "3746.png", "formula": "\\begin{align*} \\mathcal { D } _ { l , m } ^ { ( j ) } ( \\mathbf { r } ) = \\left \\{ \\begin{array} [ c ] { l } - \\frac { \\eta _ { 0 } } { l ( l + 1 ) } \\mbox { \\boldmath $ { \\nabla } $ } \\times \\lbrack j _ { l } ( K r ) \\mathbf { Y } _ { l , m } ] \\quad ; j = 1 \\\\ - \\frac { i \\eta _ { 0 } K } { l ( l + 1 ) } j _ { l } ( K r ) \\mathbf { Y } _ { l , m } ( \\hat { \\mathbf { r } } ) \\quad ; j = 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "4990.png", "formula": "\\begin{align*} N _ { a _ 1 , a _ 2 , n _ 1 , n _ 2 } : x ^ { n _ 1 } y ^ { a _ 1 } + k _ 1 y ^ { n _ 2 } + k _ 2 x ^ { a _ 2 } = 0 \\end{align*}"}
-{"id": "6599.png", "formula": "\\begin{align*} P _ { F _ n } ( \\varphi , \\varepsilon ) = \\prod _ { r = 1 } ^ { F _ n } \\left | 2 \\sin \\pi ( r \\varphi + \\varepsilon ) \\right | , \\end{align*}"}
-{"id": "7428.png", "formula": "\\begin{align*} \\displaystyle \\int _ { \\mathcal { U } } \\varphi ( \\pi ) d \\pi = \\frac { 1 } { \\lvert W ( G , \\sigma ) \\rvert } \\int _ { \\mathcal { V } } \\varphi ( \\pi _ \\lambda ) d \\lambda , \\ ; \\ ; \\ ; \\varphi \\in C _ c ( \\mathcal { U } ) \\end{align*}"}
-{"id": "9146.png", "formula": "\\begin{align*} L _ { 1 \\left ( 1 \\right ) } \\widehat { \\rho } _ { 1 } ^ { ( N ) } ( \\mathbf { x } _ { 1 } , t ) = 0 , \\end{align*}"}
-{"id": "6259.png", "formula": "\\begin{align*} A _ 1 : = \\bigoplus _ { i \\in 1 } ^ \\infty A _ { 0 , i } \\oplus \\bigoplus _ { i = 1 } ^ \\infty A _ { 1 , i } ' \\oplus \\bigoplus _ { i = 1 } ^ { \\infty } M _ { n _ { i } } ( \\C ) B _ 1 = \\bigoplus _ { j \\in 1 } ^ \\infty B _ { 0 , j } \\oplus \\bigoplus _ { j = 1 } ^ \\infty B _ { 1 , j } ' \\oplus \\bigoplus _ { j = 1 } ^ { \\infty } M _ { m _ { j } } ( \\C ) \\end{align*}"}
-{"id": "5616.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { + \\infty } v ^ 2 v _ x d x = \\int _ 0 ^ { + \\infty } \\frac { d } { d x } \\frac { v ^ 3 } { 3 } d x = - \\frac { v ^ 3 } { 3 } ( 0 , t ) . \\end{align*}"}
-{"id": "3758.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { 0 } ^ { \\infty } \\left | j _ { l } ( \\alpha r ) \\right | ^ { 2 } d r = \\frac { \\pi } { 2 ( 2 l + 1 ) } \\frac { 1 } { \\alpha } < \\infty ; \\alpha \\in \\mathbb { R } ^ { * } \\end{aligned} \\end{align*}"}
-{"id": "9136.png", "formula": "\\begin{align*} \\lim _ { \\tau \\rightarrow + \\infty } \\frac { \\partial } { \\partial t } I _ { M } ( \\rho _ { 1 } ^ { ( N ) } ( t ) , \\mathbf { b } ) = 0 . \\end{align*}"}
-{"id": "5023.png", "formula": "\\begin{align*} \\mathbf { u } \\cdot \\nabla { g } & = 0 , \\\\ \\omega ^ { i k } Q ^ { k j } \\vec { w } ^ i \\vec { w } ^ j = \\omega ^ { i k } \\big ( Q ^ { k j } \\vec { w } ^ j \\big ) \\vec { w } ^ i = \\lambda _ 1 \\omega ^ { i k } \\vec { w } ^ k \\vec { w } ^ j & = 0 , \\\\ Q ^ { i k } \\omega ^ { k j } \\vec { w } ^ i \\vec { w } ^ j = \\big ( Q ^ { i k } \\vec { w } ^ i \\big ) \\omega ^ { k j } \\vec { w } ^ j = \\lambda _ 1 \\omega ^ { k j } \\vec { w } ^ k \\vec { w } ^ j & = 0 . \\end{align*}"}
-{"id": "5887.png", "formula": "\\begin{align*} T _ w = \\sum _ { y \\leq w } q ^ { \\frac { \\ell ( y ) } { 2 } } ( - 1 ) ^ { \\ell ( w ) - \\ell ( y ) } C _ y ' . \\end{align*}"}
-{"id": "2902.png", "formula": "\\begin{align*} [ a ] \\eta = [ ( \\overline { a } ) \\bullet \\overline { m } ] \\stackrel { } { = } [ \\overline { T } \\circ ( \\overline { m } \\bullet ( \\overline { a } ) ) ] \\stackrel { p r o p \\ref { p r o p : c o m p o s a s h o r t } } { = } [ ( \\overline { m } \\bullet \\overline { ( a ) } ) \\circ \\overline { T } ] \\stackrel { } { = } [ ( \\overline { m } \\circ \\overline { T } ) \\bullet \\overline { ( a ) } ] \\stackrel { r e m \\ref { r m : T m } } { = } [ \\overline { m } \\bullet \\overline { ( a ) } ] = \\eta [ a ] \\end{align*}"}
-{"id": "3043.png", "formula": "\\begin{align*} X ( t ) & = X ( 0 ) + \\int \\limits _ { 0 } ^ { t } b ( X ( u ) ) d u + \\int \\limits _ { 0 } ^ { t } \\sigma ( X ( t ) ) d W ( t ) + \\int \\limits _ { 0 } ^ { t } \\int \\limits _ { E _ { 0 } } \\sigma ^ { 0 } ( X ( { u - } ) , z ) \\widetilde { N } ( d u , d z ) \\\\ & \\ \\ \\ + \\int \\limits _ { 0 } ^ { t } \\int \\limits _ { E _ { 1 } } \\sigma ^ { 1 } ( X ( { u - } ) , z ) N ( d u , d z ) + \\int \\limits _ { 0 } ^ { t } \\int \\limits _ { E _ { 2 } } \\sigma ^ { 2 } ( X ( { u - } ) , z ) N ( d u , d z ) , \\end{align*}"}
-{"id": "9619.png", "formula": "\\begin{align*} U ( q , N + 1 ) - U ( q , N ) = \\frac { q ^ { N + 1 } ( - q ) _ N } { 1 - q ^ { N + 1 } } . \\end{align*}"}
-{"id": "1266.png", "formula": "\\begin{gather*} E = \\pi | A | ^ 2 A , F = i \\frac { \\sqrt { 2 } \\pi } { 3 } e ^ { i a \\ln 3 } | A | ^ 2 A . \\end{gather*}"}
-{"id": "9419.png", "formula": "\\begin{align*} ( f _ { \\theta } , g ) _ { r e g } = 0 . \\end{align*}"}
-{"id": "6769.png", "formula": "\\begin{align*} [ f , \\alpha ] ( { i , j } ) = \\left \\{ \\begin{array} { c c } \\frac { f ( \\alpha _ i ) - f ( \\alpha _ j ) } { \\alpha _ i - \\alpha _ j } & \\alpha _ i \\neq \\alpha _ j \\\\ \\partial _ i { f } ( \\alpha _ i ) & \\alpha _ i = \\alpha _ j \\end{array} \\right . \\end{align*}"}
-{"id": "9551.png", "formula": "\\begin{align*} a ( n , N ) = \\textup { s s p t d } _ { o } ( n , N ) - \\textup { s s p t d } _ { o } ( n - N , N ) . \\end{align*}"}
-{"id": "9634.png", "formula": "\\begin{align*} S _ 1 ( z , q , N ) = \\frac { z q ( 1 - q ^ { 2 N } ) } { ( 1 - z q ^ 2 ) ( 1 - z q ^ { 2 N - 1 } ) } F _ { N - 1 } ( 1 , z q ^ 2 , z q : q ^ 2 ) . \\end{align*}"}
-{"id": "3426.png", "formula": "\\begin{gather*} \\tfrac { p _ 1 } 2 = 2 s ^ 2 - \\sum _ { i < j } e _ i e _ j . \\end{gather*}"}
-{"id": "4115.png", "formula": "\\begin{align*} \\begin{aligned} g _ { 1 + } ( x ) - g _ { 1 - } ( x ) & = 2 \\pi i , \\\\ g _ { 2 + } ( x ) - g _ { 2 - } ( x ) & = 2 \\pi i \\nu ^ * ( [ x , 0 ] ) , \\\\ - g _ { 1 - } ( x ) + g _ { 2 - } ( x ) + g _ { 2 + } ( x ) & = \\pi i . \\end{aligned} \\end{align*}"}
-{"id": "2950.png", "formula": "\\begin{align*} X _ { | V } = \\{ s _ { | V } : s \\in X \\} \\end{align*}"}
-{"id": "2859.png", "formula": "\\begin{align*} y ^ 2 - x z - i t z ^ 2 = 0 . \\end{align*}"}
-{"id": "4877.png", "formula": "\\begin{align*} \\widehat { A } = A \\cup N _ \\infty ( A ) . \\end{align*}"}
-{"id": "712.png", "formula": "\\begin{align*} \\forall v , w \\in \\mathbb { C } ^ 2 \\setminus \\{ 0 \\} , B i s ( v , w ) = B i s \\left ( \\frac { v } { \\lvert v \\rvert } , \\frac { w } { \\lvert w \\rvert } \\right ) . \\end{align*}"}
-{"id": "4687.png", "formula": "\\begin{align*} ( Q u ) ^ { ( l ) } ( x _ j ) = \\frac { 1 } { \\Delta x ^ l } \\displaystyle { \\sum _ { k = j - d } ^ { j + d } b _ { k , l } u _ k } , ~ ~ ~ ~ ~ ~ l = 0 , \\ldots , d - 1 , \\end{align*}"}
-{"id": "381.png", "formula": "\\begin{align*} | | P ' | - | Q ' | | & \\leqslant \\left | \\frac { c - a } { c } ( P ' ) ^ { - 1 } - \\frac { b - a } { b } ( Q ' ) ^ { - 1 } \\right | \\leqslant \\left | 1 - \\frac { a } { c } \\right | \\frac { 1 } { | P ' | } + \\left | 1 - \\frac { a } { b } \\right | \\frac { 1 } { | Q ' | } \\\\ & \\leqslant \\frac { 1 } { 1 2 } \\left | 1 - \\frac { a } { c } \\right | + \\frac { 1 } { 1 1 } \\frac { 2 } { | b | } < \\frac { 5 } { 4 8 } + \\frac { 5 } { 4 8 } = \\frac { 5 } { 2 4 } , \\end{align*}"}
-{"id": "5942.png", "formula": "\\begin{align*} ( B _ H + M _ H ) = ( B _ Z + M _ Z ) | _ H . \\end{align*}"}
-{"id": "3485.png", "formula": "\\begin{align*} \\Im f ' _ { M , N } ( t ) = ( M + 1 ) \\Im \\psi ( \\sigma ) - \\Im \\psi ( \\sigma - N ) = - M \\left ( \\sum ^ { \\infty } _ { n = 0 } \\Im \\frac { 1 } { \\sigma + n } \\right ) + \\sum ^ { N } _ { n = 1 } \\Im \\frac { 1 } { \\sigma - 1 } , \\end{align*}"}
-{"id": "3996.png", "formula": "\\begin{align*} a _ { \\mu } ( n - T ^ { - 1 } [ \\mu / 2 ] ) = a _ F ( \\left ( \\begin{smallmatrix} n & \\mu / 2 \\\\ \\mu / 2 ^ t & T \\end{smallmatrix} \\right ) ) \\end{align*}"}
-{"id": "8644.png", "formula": "\\begin{align*} \\bar R = 1 \\ \\ \\textrm { a n d } \\ \\ \\sin \\bar \\varphi = \\frac { 2 \\omega } { \\Lambda - 1 } . \\end{align*}"}
-{"id": "5386.png", "formula": "\\begin{align*} c _ { n + 1 \\ , m - 3 } | | v _ { n m } ^ 1 | | ^ 2 = - \\overline { b _ { n m } } | | v _ { n + 1 \\ , m - 3 } ^ { 1 } | | ^ 2 . \\end{align*}"}
-{"id": "1559.png", "formula": "\\begin{align*} \\Phi _ { \\widetilde { F } ^ 0 } ( z , x ) = \\phi _ f ( z ) \\end{align*}"}
-{"id": "2774.png", "formula": "\\begin{align*} e _ i : = 1 - \\sum _ { j = 1 } ^ S q _ { i , j } . \\end{align*}"}
-{"id": "3400.png", "formula": "\\begin{align*} f ( T ) = F ( T ) - \\sum _ i F ( T _ i ) = \\log m ( T ) - \\sum _ i \\log m ( T _ i ) = - \\log \\left ( \\frac { m _ 0 ( T ) } { m ( T ) } \\right ) . \\end{align*}"}
-{"id": "5499.png", "formula": "\\begin{align*} P _ { b _ 0 } \\big ( | Z _ T ( f ) | \\geq \\sqrt { T } x \\big ) = P _ { b _ 0 } \\big ( | Z _ T ( f ) | \\geq \\sqrt { T } x , [ Z _ \\cdot ( f ) ] _ T \\leq T d _ L ^ 2 ( f , 0 ) \\big ) \\leq 2 e ^ { - \\frac { x ^ 2 } { 2 d _ L ^ 2 ( f , 0 ) } } . \\end{align*}"}
-{"id": "7404.png", "formula": "\\begin{align*} d _ x ( z , w ) = l _ x ( \\gamma ) = & l _ x ( \\gamma _ { | _ { [ 0 , a ] } } ) + l _ x ( \\gamma _ { | _ { [ a , 1 ] } } ) \\\\ = & d _ x ( z , y ) + d _ x ( y , w ) . \\end{align*}"}
-{"id": "4544.png", "formula": "\\begin{align*} P = \\bigsqcup _ { \\lambda \\in \\Lambda } \\lambda ^ \\circ . \\end{align*}"}
-{"id": "6057.png", "formula": "\\begin{align*} \\Phi ^ + = \\{ \\beta _ 1 : = \\alpha _ { i _ 1 } , \\beta _ 2 : = s _ { i _ 1 } ( \\alpha _ { i _ 2 } ) , \\ldots , \\beta _ N : = s _ { i _ 1 } \\cdots s _ { i _ { N - 1 } } ( \\alpha _ { i _ N } ) \\} \\end{align*}"}
-{"id": "5368.png", "formula": "\\begin{align*} \\frac { a _ { n m } d _ { n + 1 \\ , m + 3 } } { a _ { n - 1 \\ , m + 3 } d _ { n \\ , m + 6 } } = \\frac { b _ { n \\ , m + 6 } c _ { n + 1 \\ , m + 3 } } { b _ { n - 1 \\ , m + 3 } c _ { n m } } = \\frac { n } { n + 1 } . \\end{align*}"}
-{"id": "298.png", "formula": "\\begin{align*} \\delta _ Z \\ ; \\hookrightarrow \\ ; \\delta _ X * \\delta _ Y \\ ; \\in \\ ; \\langle \\delta _ { X Y } \\rangle \\textnormal { w h e r e } \\delta _ { X Y } \\ ; : = \\ ; \\delta _ X \\oplus \\delta _ Y . \\end{align*}"}
-{"id": "2748.png", "formula": "\\begin{align*} \\sigma _ { n + 1 } = \\cdots = \\sigma _ { N } = 1 \\ \\mathrm { a n d } \\ \\sigma _ i \\in ( 1 , \\infty ) , i = 1 , \\ldots , n . \\end{align*}"}
-{"id": "8932.png", "formula": "\\begin{align*} \\gamma & \\le s \\frac { \\binom { n / 2 } { s / 2 } ^ 2 } { \\binom { n } { n / 2 } } \\le s \\frac { 1 } { 2 \\pi } \\frac { n / 2 } { s / 2 ( n - s ) / 2 } 2 ^ { n h _ 2 ( s / n ) } \\left ( \\sqrt { \\frac { n } { 8 ( n / 2 ) ^ 2 } } 2 ^ n \\right ) ^ { - 1 } \\\\ & \\le \\sqrt { \\frac { 2 n } { \\pi ^ 2 } } 2 ^ { - n ( 1 - h _ 2 ( s / n ) ) } , \\end{align*}"}
-{"id": "4337.png", "formula": "\\begin{align*} ^ L ( \\widetilde { G } _ \\tau ) \\cong \\frac { \\widetilde { ^ L G } \\times { ^ L ( T / Z ( \\widetilde { G } ) ) } } { Z ( \\widetilde { ^ L G } ) } = ( \\widetilde { ^ L G } ) _ { ^ L \\tau } . \\end{align*}"}
-{"id": "3843.png", "formula": "\\begin{align*} K \\dfrac { \\varepsilon ^ 2 h _ { 0 0 } ^ { m + 1 } } { ( 2 W _ 0 ) ^ { 2 m } } { h ^ i } _ l = ^ { h } { { { R _ 0 } ^ i } _ { 0 l } } + 2 \\Phi ^ i _ { \\parallel l } - \\Phi ^ i _ { l \\parallel 0 } + 2 \\Phi ^ r { { \\Phi _ r } ^ i } _ l - \\Phi ^ r _ l \\Phi ^ i _ r . \\end{align*}"}
-{"id": "6286.png", "formula": "\\begin{align*} \\tilde { R } _ k ( \\epsilon , n ) = \\min _ { \\mathcal { A } \\epsilon - } \\max _ { W \\in \\tilde { \\mathcal { W } } [ k ] } \\mathbb { E } _ { G \\sim G _ n ( W ) , \\hat { B } \\sim \\mathcal { A } _ G } [ \\delta _ 2 ( \\hat { B } , W ) ^ 2 ] . \\end{align*}"}
-{"id": "3607.png", "formula": "\\begin{align*} \\left ( \\sum _ { \\hat { x } \\in \\hat { X } } \\alpha _ { \\hat { x } } \\hat { x } \\right ) ^ { \\hat { \\psi } _ { \\sigma } } = \\sum _ { \\hat { x } \\in \\hat { X } } \\alpha _ { \\hat { x } } ^ { \\hat { \\phi } _ { \\sigma } } \\hat { x } ^ { \\hat { \\psi } _ { \\sigma } } . \\end{align*}"}
-{"id": "168.png", "formula": "\\begin{align*} \\pi _ { T M , X _ { C } , X _ { V } , c } ( { \\bf x } , { \\bf y } ) = \\end{align*}"}
-{"id": "2880.png", "formula": "\\begin{align*} S = \\frac { 1 } { 2 } \\left ( [ J _ - , J _ + ] - 1 \\right ) \\end{align*}"}
-{"id": "5606.png", "formula": "\\begin{align*} v _ { \\delta , \\epsilon } ( x , t ) = \\int \\int \\tilde { v } ( y , s ) \\theta _ { \\epsilon } ( x - y ) \\theta _ { \\delta } ( t - s ) d y d s . \\end{align*}"}
-{"id": "5411.png", "formula": "\\begin{align*} \\| m ^ { - 1 } ( u ) - m ^ { - 1 } ( v ) \\| & = \\left \\| \\frac { u - v } { \\| u \\| } + \\frac { v \\| v \\| - v \\| u \\| } { \\| u \\| \\cdot \\| v \\| } \\right \\| = \\left \\| \\frac { u - v } { \\| u \\| } + \\frac { v ( \\| v \\| - \\| u \\| ) } { \\| u \\| \\cdot \\| v \\| } \\right \\| \\leq \\\\ & \\leq \\frac { \\| u - v \\| } { \\| u \\| } + \\frac { \\left | \\| v \\| - \\| u \\| \\right | } { \\| u \\| } \\leq \\frac { 2 \\| u - v \\| } { \\| u \\| } \\leq \\frac { 2 } { \\beta } \\| u - v \\| , \\end{align*}"}
-{"id": "5648.png", "formula": "\\begin{align*} _ 0 I _ t ^ \\gamma v ( t _ n ) \\approx \\tau ^ \\gamma \\sum _ { k = 0 } ^ n b _ { n - k } ^ { ( - \\gamma ) } v ^ k , \\end{align*}"}
-{"id": "6908.png", "formula": "\\begin{align*} \\det ( d \\varphi ) = \\sum _ { i = 1 } ^ 6 p _ i g _ i \\ , , \\end{align*}"}
-{"id": "4459.png", "formula": "\\begin{align*} K _ { M } = \\psi \\circ \\eta \\end{align*}"}
-{"id": "1938.png", "formula": "\\begin{align*} \\mathcal { A } = \\sum _ { j = 1 } ^ \\infty Z _ j A _ j ^ 2 \\left ( 1 + \\frac { \\chi ^ 2 } { Z _ j ^ 2 } \\right ) - \\sum _ { j = 1 } ^ \\infty ( j - 1 ) \\pi \\end{align*}"}
-{"id": "4953.png", "formula": "\\begin{align*} f _ 3 = y ^ n + v _ 1 z ^ n - v _ 2 z ^ n + \\cdots + ( - 1 ) ^ { i + 1 } v _ i z ^ n + \\cdots + ( - 1 ) ^ { d - 2 } v _ { d - 3 } z ^ n + ( - 1 ) ^ { d - 3 } ( x - p ) z ^ n , \\end{align*}"}
-{"id": "3957.png", "formula": "\\begin{align*} & f _ i ( x , m , v _ i ) = \\chi ( v _ i ) + \\varphi _ { i , \\lambda } ( m ( x ) ) \\ , , h _ i ( x , m ) = \\psi _ i ( x ) \\ , , \\\\ & \\chi ( v ) = \\frac { | v | ^ 2 } { 2 } \\ , , \\varphi _ { 1 , \\lambda } ( \\mu _ 1 , \\mu _ 2 ) = \\mu _ 1 + \\lambda \\mu _ { 2 } , \\varphi _ { 2 , \\lambda } ( \\mu _ 1 , \\mu _ 2 ) = \\mu _ 2 + \\lambda \\mu _ { 1 } \\ , , \\end{align*}"}
-{"id": "6288.png", "formula": "\\begin{align*} \\| Q ( n , \\pi ) _ { \\pi } - Q \\| ^ 2 _ { 2 } \\leq 2 \\min _ { B \\in \\mathbb { R } ^ { k \\times k } , \\pi } \\| B _ { \\pi } - Q \\| _ 2 ^ 2 + \\frac { 2 } { n ^ 2 } = 2 \\hat { \\epsilon } _ k ^ { ( O ) } ( Q ) ^ 2 + \\frac { 2 } { n ^ 2 } . \\end{align*}"}
-{"id": "9226.png", "formula": "\\begin{align*} y _ { u _ 1 u _ 2 } ^ { } = \\frac { \\sum _ { i = 1 } ^ { K } \\frac { 1 } { d ( b _ { i 1 } b _ { i 2 } , u _ 1 u _ 2 ) } y _ { b _ { i 1 } b _ { i 2 } } } { \\sum _ { i = 1 } ^ { K } \\frac { 1 } { d ( b _ { i 1 } b _ { i 2 } , u _ 1 u _ 2 ) } } , \\end{align*}"}
-{"id": "5371.png", "formula": "\\begin{align*} a _ { 1 + p + q \\ , 2 t + 3 p - 3 q } d _ { 2 + p + q \\ , 2 t + 3 p - 3 q + 3 } = \\frac { p + 1 } { p + q + 2 } \\left ( 2 c - ( p + 1 ) t - p ( p + 2 ) \\right ) \\end{align*}"}
-{"id": "1549.png", "formula": "\\begin{align*} U _ 2 ^ \\delta \\left ( \\begin{array} { c } z \\\\ x \\end{array} \\right ) = \\left ( \\begin{array} { c } z + O ( z x ) \\\\ x \\end{array} \\right ) . \\end{align*}"}
-{"id": "8353.png", "formula": "\\begin{align*} D _ t Z ( \\alpha , t ) = \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { 2 \\pi } \\frac { 1 } { \\overline { \\alpha - z _ j ( t ) } } = \\frac { \\lambda i } { 2 \\pi } \\frac { \\overline { z _ 1 ( t ) - z _ 2 ( t ) } } { \\overline { ( \\alpha - z _ 1 ( t ) ) ( \\alpha - z _ 2 ( t ) ) } } \\end{align*}"}
-{"id": "6542.png", "formula": "\\begin{align*} n _ \\beta = ( - 1 ) ^ { w - 1 } \\ , w \\ , m ^ P _ \\beta . \\end{align*}"}
-{"id": "2298.png", "formula": "\\begin{align*} \\hat { H } : \\mathbb { C } \\oplus \\mathcal { L } & \\rightarrow \\mathbb { C } \\oplus \\mathcal { L } ^ * \\\\ \\hat { H } & = \\left ( \\begin{array} { c c } 0 & ( \\hat { E } | \\\\ - | \\hat { E } ) & \\hat { \\Omega } \\end{array} \\right ) , \\end{align*}"}
-{"id": "9734.png", "formula": "\\begin{align*} N ( r , v ) = \\frac { D ( r , v ) } { H ( r , v ) } , \\end{align*}"}
-{"id": "4487.png", "formula": "\\begin{align*} M = \\min _ { x \\in \\partial \\Omega } \\min _ { \\xi \\in T _ x ( \\partial \\Omega ) , | \\xi | = 1 } ( u + \\nabla _ { \\xi \\xi } u ) \\geq c _ 0 \\end{align*}"}
-{"id": "5729.png", "formula": "\\begin{align*} \\frac { d } { d x } \\left ( \\int _ 0 ^ x ( x - t ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } [ \\lambda ( x - ) ^ { \\alpha } ] f ( t ) d t \\right ) \\Big | _ { x = 0 } & = ( x - x ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } [ \\lambda ( x - x ) ^ { \\alpha } ] f ( x ) \\Big | _ { x = 0 } = 0 . \\end{align*}"}
-{"id": "3606.png", "formula": "\\begin{align*} \\hat { \\sigma } = ( \\hat { a } _ 1 , \\ , \\hat { a } _ 2 , \\ , \\hat { a } _ 3 ) ( \\hat { a } _ { - 1 } , \\ , \\hat { a } _ { - 2 } , \\ , \\hat { a } _ { - 3 } ) . \\end{align*}"}
-{"id": "6680.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } d \\rho \\ , \\psi _ { \\rho } ( x ) \\psi ^ { * } _ { \\rho } ( y ) & = \\int _ { - \\infty } ^ { \\infty } d \\rho \\ , x ^ { - \\frac { 1 } { 2 } + i \\rho } \\ , y ^ { - \\frac { 1 } { 2 } - i \\rho } \\\\ & = 2 \\pi \\delta ( x - y ) . \\end{align*}"}
-{"id": "8865.png", "formula": "\\begin{align*} c _ 2 | _ { \\mu = 0 } & = 3 \\nu ^ 7 + 1 3 3 0 \\nu ^ 5 + 6 5 1 7 7 \\nu ^ 2 + 9 3 1 0 0 \\nu + ( 1 2 0 5 4 4 - 1 2 3 \\nu ^ 6 - 1 9 1 8 \\nu ^ 4 - 2 8 8 9 7 \\nu ^ 3 ) > 0 , \\\\ c _ 2 | _ { \\mu = 1 } & = ( 1 - \\nu ) ( 1 6 1 6 5 2 - 1 4 7 \\nu ^ 6 + 6 7 2 \\nu ^ 5 + 3 8 4 2 \\nu ^ 4 + 1 6 0 6 0 \\nu ^ 3 + ( 2 6 4 6 5 2 - 1 9 5 2 5 9 \\nu ) \\nu ) \\ge 0 , \\end{align*}"}
-{"id": "2548.png", "formula": "\\begin{align*} \\{ ( X _ t ) _ { t \\geq 0 } ; \\mathbf P _ \\mu ^ { X _ T ( g ) } \\} \\overset { f . d . d . } { = } \\{ ( X _ t + W _ t ) _ { t \\geq 0 } ; \\mathbf P _ \\mu \\otimes \\mathbb N ^ { W _ T ( g ) } _ \\mu \\} . \\end{align*}"}
-{"id": "4004.png", "formula": "\\begin{align*} A _ { U ( \\beta ) f _ j } & = a _ { \\kappa } \\nu _ { d ^ 2 \\beta } \\beta ^ { 1 - \\kappa } Q _ \\beta ( f _ j ) \\left \\langle f _ j , f _ j \\right \\rangle _ { d ^ 2 \\beta } = a _ { \\kappa } \\nu _ { d ^ 2 } \\beta ^ { 1 - \\kappa } Q _ \\beta ( f _ j ) \\left \\langle f _ j , f _ j \\right \\rangle _ { d ^ 2 } = \\beta ^ { 1 - \\kappa } Q _ \\beta ( f _ j ) \\cdot A _ { f _ j } . \\end{align*}"}
-{"id": "3342.png", "formula": "\\begin{align*} \\left | [ u ^ n ] Z ( u ) \\right | \\le t ^ { \\frac { n } { 2 } } \\binom { 2 0 ( r + 1 ) + n - 1 } { n } \\end{align*}"}
-{"id": "3375.png", "formula": "\\begin{align*} W : = P A | _ { \\C ^ J } . \\end{align*}"}
-{"id": "1891.png", "formula": "\\begin{align*} v ^ { \\rm e v e n } ( n , x ) ^ 2 + v ^ { \\rm o d d } ( n , x ) ^ 2 = \\frac { 2 } { L } , \\end{align*}"}
-{"id": "8705.png", "formula": "\\begin{align*} ( x _ 1 + \\cdots + x _ n ) ^ m = \\sum _ { \\substack { \\alpha \\in \\Z _ { \\geq 0 } ^ n \\\\ \\abs { \\alpha } = m } } \\frac { m ! } { \\alpha _ 1 ! \\cdots \\alpha _ n ! } x _ 1 ^ { \\alpha _ 1 } \\cdots x _ n ^ { \\alpha _ n } . \\end{align*}"}
-{"id": "8380.png", "formula": "\\begin{align*} g _ 2 : = \\frac { i } { \\pi } \\sum _ { j = 1 } ^ N \\lambda _ j \\Big ( \\frac { 2 z _ { t t } + i - \\ddot { z } _ j } { ( z ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } - 2 \\frac { ( z _ t - \\dot { z } _ j ( t ) ) ^ 2 } { ( z ( \\alpha , t ) - z _ j ( t ) ) ^ 3 } \\Big ) . \\end{align*}"}
-{"id": "4353.png", "formula": "\\begin{align*} \\mathcal P _ a : = & \\big \\{ P \\in { [ 2 , n ] \\choose k - 1 } \\ : \\ P \\cap [ x ] = [ 2 , x ] \\cap S \\big \\} , \\\\ \\mathcal P _ b : = & \\big \\{ P \\in { [ 2 , n ] \\choose k } \\ : \\ P \\cap [ x ] = [ x ] \\setminus S \\big \\} , \\end{align*}"}
-{"id": "9893.png", "formula": "\\begin{align*} N ( n , k ) & = \\binom { n } { k } E ( a ( z ) ^ { k } b ( z ) ^ { n - k } ) = \\binom { n } { k } E ( z ^ { n + k } ) \\\\ & = \\binom { n } { k } \\frac { 1 } { n + k + 1 } < \\infty \\end{align*}"}
-{"id": "3749.png", "formula": "\\begin{align*} \\mathcal { E } _ { \\mathcal { E } } = \\sum _ { j , l , m } \\frac { | a _ { l , m } ^ { ( j ) } | ^ { 2 } } { \\left ( \\mathcal { B } _ { l , m } ^ { ( j ) } , \\mathcal { B } _ { l , m } ^ { ( j ) } \\right ) } . \\end{align*}"}
-{"id": "1064.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } n ( 2 \\ln n ) ^ { - \\frac { 1 } { 2 } } D _ n ( ( 1 + z / \\ln n ) G _ n ( x ) / 2 ) = e ^ { c _ 0 - x - 2 z } . \\end{align*}"}
-{"id": "57.png", "formula": "\\begin{align*} d _ 1 = k _ 1 \\min _ { J \\in D _ J } g ' ( J ) > 0 \\ , , d _ 2 = k _ 2 \\max _ { J \\in D _ J } g ' ( J ) \\end{align*}"}
-{"id": "3853.png", "formula": "\\begin{align*} \\mathbf { E } [ f ( X ^ x _ t ) 1 _ { \\{ \\tau _ y > t \\} } ] = E [ g ( X ^ x _ t ) ] . \\end{align*}"}
-{"id": "4924.png", "formula": "\\begin{align*} \\sup \\{ \\ell ( H ^ i ( ( f _ 1 , \\ldots , f _ d ) ; M ) ) \\mid f _ 1 , \\ldots , f _ d \\mbox { p a r a m e t e r s o n $ M $ } \\} \\leq \\sum _ { j = 0 } ^ { i } { d \\choose i - j } \\ell ( H ^ j _ m ( M ) ^ \\vee ) . \\end{align*}"}
-{"id": "6600.png", "formula": "\\begin{align*} P _ { F _ n } ( \\varphi , \\varepsilon ) = \\overline { A } _ { n } ( \\varepsilon ) B _ { n } \\overline { C } _ n ( \\varepsilon ) , \\end{align*}"}
-{"id": "2226.png", "formula": "\\begin{align*} | \\ ! \\ ! | \\nabla E _ d * u | \\ ! \\ ! | _ { M ^ r } \\leq C | \\ ! \\ ! | u | \\ ! \\ ! | _ { M ^ p } \\frac { 1 } { r } = \\frac { 1 } { p } - \\frac { 1 } { d } \\end{align*}"}
-{"id": "7594.png", "formula": "\\begin{align*} \\begin{gathered} \\left [ \\eta \\cdot \\nabla , \\Gamma \\right ] ( \\tau \\circ X ^ { - 1 } ) ( t ) = I _ 1 + I _ 2 + I _ 3 + I _ 4 + I _ 5 + I _ 6 , \\end{gathered} \\end{align*}"}
-{"id": "6783.png", "formula": "\\begin{align*} ( d \\varphi ^ t ) ^ T \\varphi '' + \\nabla _ { \\ ! \\alpha } p = 0 \\ , . \\end{align*}"}
-{"id": "3784.png", "formula": "\\begin{align*} r _ \\alpha = s _ { \\lambda ( \\alpha ) } + \\sum _ { \\mu > _ { l e x } \\lambda ( \\alpha ) } b _ \\mu s _ \\mu . \\end{align*}"}
-{"id": "2002.png", "formula": "\\begin{align*} I ( q , z ) = I _ 0 ( q ) z \\phi _ 1 + I _ 1 ( q ) \\phi _ 2 + I _ 2 ( q ) \\frac { \\phi _ 3 } { z } + I _ 3 ( q ) \\frac { \\phi _ 4 } { z ^ 2 } , \\end{align*}"}
-{"id": "114.png", "formula": "\\begin{align*} f _ 0 ( d ) & ~ \\dot = ~ \\sum _ { \\ell = 1 } ^ { \\infty } \\frac { d ^ { 2 \\ell + 1 } } { ( \\ell ! ) ^ 2 } = d \\left [ I _ 0 ( 2 d ) - 1 \\right ] \\\\ f _ 1 ( d ) & ~ \\dot = ~ \\sum _ { h = 1 } ^ { \\infty } \\frac { d ^ { 2 h } } { h ! ( h - 1 ) ! } = d I _ 1 ( 2 d ) \\ , , \\end{align*}"}
-{"id": "6548.png", "formula": "\\begin{align*} E ( \\beta ) : = \\left \\{ P \\in E \\ ; \\big { | } \\ ; \\beta | _ E \\sim w P \\right \\} . \\end{align*}"}
-{"id": "2252.png", "formula": "\\begin{align*} & \\langle \\alpha _ { \\vartheta } | \\alpha _ { \\vartheta ' } \\rangle = \\delta _ { \\vartheta , \\vartheta ' } & & \\langle \\alpha _ x | \\alpha _ x ' \\rangle = \\delta ( x - x ' ) \\\\ & \\langle \\alpha _ \\vartheta ' | \\alpha _ x \\rangle = 0 & & \\langle \\alpha _ x ' | \\alpha _ { \\vartheta } \\rangle = 0 \\end{align*}"}
-{"id": "7770.png", "formula": "\\begin{align*} T _ { * , \\mathbb I } f = \\sup _ { t \\in \\mathbb I } | T _ t f | , f \\in L ^ p ( X ) . \\end{align*}"}
-{"id": "4264.png", "formula": "\\begin{align*} s : = \\left \\lbrace \\begin{aligned} & p ^ { ' } & & p \\geq 2 \\\\ & p & & 1 < p < 2 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "9340.png", "formula": "\\begin{align*} \\int _ { M } \\Big ( 1 + \\langle \\psi _ { a } , \\mathbf { H } _ { a } \\rangle - \\langle \\psi , a \\rangle \\langle \\mathbf { H } , a \\rangle \\Big ) d V = 0 , \\end{align*}"}
-{"id": "6860.png", "formula": "\\begin{align*} \\begin{cases} a '' + \\lambda \\nu = 0 \\ , , \\\\ \\langle a ' , \\nu \\rangle = 0 \\ , , \\\\ a ( t ) \\in M \\ , . \\end{cases} \\end{align*}"}
-{"id": "7401.png", "formula": "\\begin{align*} \\mathcal { F } ( r ) & = \\frac { 2 \\frac { r } { 1 - r } } { \\left ( 1 - \\left ( \\frac { r } { 1 - r } \\right ) ^ 2 \\right ) ^ 2 } \\left ( \\frac { r } { ( 1 - r ) ^ 3 } + \\frac { r } { ( 1 - r ) ^ 4 } \\right ) + \\frac { \\left ( \\frac { r } { 1 - r } \\right ) ^ 2 } { 1 - 2 r } \\\\ & \\quad + 2 \\frac { \\left ( \\frac { r } { 1 - r } \\right ) ^ 2 } { 1 - \\left ( \\frac { r } { 1 - r } \\right ) ^ 2 } \\frac { 1 } { ( 1 - r ) ^ 3 } + \\frac { r } { 1 - r } + \\frac { r } { ( 1 - r ) ^ 2 } . \\end{align*}"}
-{"id": "6061.png", "formula": "\\begin{align*} t _ k = \\min _ { \\pi _ 1 } \\left \\{ \\sum _ { m = 1 } ^ { N } d _ m ( \\pi _ 1 ) c _ m \\right \\} - \\min _ { \\pi _ 2 } \\left \\{ \\sum _ { m = 1 } ^ { N } d _ m ( \\pi _ 2 ) c _ m \\right \\} , \\end{align*}"}
-{"id": "8116.png", "formula": "\\begin{align*} f _ n ( t ) \\ = \\ \\sum _ { i = 0 } ^ { \\lfloor n / 2 \\rfloor } \\xi _ { n , i } \\ , t ^ i ( 1 + t ) ^ { n - 2 i } \\end{align*}"}
-{"id": "8147.png", "formula": "\\begin{align*} & d _ 1 = a _ 1 , \\ : \\ : \\ : d _ i = a _ { i - 1 } b _ { i } a _ { i } ^ { - 1 } b _ { i } ^ { - 1 } \\ : \\ : \\ : 2 \\leq i \\leq g , \\\\ & e _ 1 = a _ 1 , \\ : \\ : \\ : e _ i = b _ 1 a _ 1 ^ { - 1 } b _ 1 ^ { - 1 } a _ 1 \\ldots b _ { i - 1 } a _ { i - 1 } ^ { - 1 } b _ { i - 1 } ^ { - 1 } a _ { i - 1 } b _ i a _ i ^ { - 1 } b _ i ^ { - 1 } \\ : \\ : \\ : 2 \\leq i \\leq g . \\end{align*}"}
-{"id": "4335.png", "formula": "\\begin{align*} A ^ \\bullet ( X ^ m ) _ { \\mathbb { Q } } = & \\bigoplus _ { B \\subset \\{ 1 , \\ldots , m \\} } I m ( \\Gamma ^ { \\boxtimes \\{ 1 , \\ldots , m \\} \\setminus B } \\boxtimes ( \\Delta _ 2 - \\Gamma ) ^ { \\boxtimes B } ) \\\\ A ^ \\bullet ( X ^ n ) _ { \\mathbb { Q } } = & \\bigoplus _ { B \\subset \\{ 1 , \\ldots , n \\} } I m ( \\Gamma ^ { \\boxtimes \\{ 1 , \\ldots , n \\} \\setminus B } \\boxtimes ( \\Delta _ 2 - \\Gamma ) ^ { \\boxtimes B } ) . \\end{align*}"}
-{"id": "7661.png", "formula": "\\begin{align*} \\int \\limits _ { \\mathbb { R } ^ d } a _ { \\rm s y m } ( \\xi - q ) \\mu ( \\xi , q ) \\big ( \\tilde \\varphi ^ i _ { 0 } ( q ) - \\tilde \\varphi ^ i _ { 0 } ( \\xi ) \\big ) \\ , d q \\ = \\ 2 \\ , \\int \\limits _ { \\mathbb { R } ^ d } c ^ i ( \\xi - q ) \\mu ( \\xi , q ) \\ , d q \\end{align*}"}
-{"id": "9410.png", "formula": "\\begin{align*} \\mathcal { E } _ 2 ( z ) = \\sum _ { 1 \\leq j \\leq D ( N ) - 1 } \\frac { \\det ( A _ { f , j } ) } { \\det ( A _ N ) } ( E _ 2 ( z ) - d _ j E _ 2 ( d _ j z ) ) . \\end{align*}"}
-{"id": "6423.png", "formula": "\\begin{align*} w _ { k + 1 } = \\frac { 1 } { \\alpha _ { k + 1 } } \\left [ \\begin{array} { c c c c c } \\left ( - 1 \\right ) ^ { k } \\prod _ { i = 1 } ^ { k } \\frac { \\beta _ { i } } { \\alpha _ { i } } , & \\dots , & \\frac { \\beta _ { k - 1 } } { \\alpha _ { k - 1 } } \\frac { \\beta _ { k } } { \\alpha _ { k } } , & - \\frac { \\beta _ { k } } { \\alpha _ { k } } , & 1 \\end{array} \\right ] ^ { T } . \\end{align*}"}
-{"id": "6080.png", "formula": "\\begin{align*} V '' ( t ) ( V ' ) ^ \\frac { 1 } { n - k - 1 } & \\leq a ( V ' ) ^ { 1 - \\delta _ 1 } + c b t ^ \\frac { k - 1 } { k } V ^ { \\delta _ 2 + \\delta _ 3 } ( V ' ) ^ { 1 - \\delta _ 2 } + \\tfrac { c b } { { 1 + \\delta _ 3 } } t ^ \\frac { n - k - 2 } { n - k - 1 } ( V ^ { 1 + \\delta _ 3 } ) ' \\\\ & + c | H | t ^ \\frac { k - 1 } { k } V ^ { \\frac { 1 } { k } } ( V ' ) ^ { 1 - \\delta _ 2 } + \\alpha c | H | t ^ \\frac { n - k - 2 } { n - k - 1 } ( V ^ { 1 / \\alpha } ) ' \\end{align*}"}
-{"id": "15.png", "formula": "\\begin{align*} F _ s ^ { i j } ( z ) = & \\ f ^ i ( V _ s , z + Z _ s ^ { j } ( m ) ) - f ^ j ( V _ s , Z _ s ^ j ( m ) ) , \\end{align*}"}
-{"id": "3599.png", "formula": "\\begin{align*} \\left \\langle \\sum _ { \\hat { x } \\in \\hat { X } } \\alpha _ { \\hat { x } } \\hat { x } , \\sum _ { \\hat { y } \\in \\hat { X } } \\beta _ { \\hat { y } } \\hat { y } \\right \\rangle = \\sum _ { \\hat { x } , \\hat { y } \\in \\hat { X } } \\alpha _ { \\hat { x } } \\beta _ { \\hat { y } } \\lambda _ { [ \\hat { x } , \\hat { y } ] } . \\end{align*}"}
-{"id": "2710.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } R _ { N } ^ { ( 1 ) } \\big ( N x , N x \\big ) d x = \\varphi ( x ) d x , 0 < x < 4 . \\end{align*}"}
-{"id": "4227.png", "formula": "\\begin{align*} \\pi _ { n } ^ { - 1 } ( N \\cdot w b _ { n } ) = ( N \\cdot w b _ { 0 } ) \\ , \\dot { \\cup } \\ , ( N \\cdot v b _ { 0 } ) \\end{align*}"}
-{"id": "8790.png", "formula": "\\begin{align*} 0 < \\sigma ^ 2 ( \\varepsilon ) = \\int _ { \\mathbb { R } } z ^ 2 \\ , Q ^ { \\varepsilon } ( \\textrm { d } z ) < \\infty . \\end{align*}"}
-{"id": "1238.png", "formula": "\\begin{align*} K ( S , S ) ( \\eta ) & = \\frac { 1 } { \\sqrt { | \\eta | } } \\int e ^ { 3 i \\mu ^ 2 / 4 } \\tilde S \\left ( \\eta , \\frac { \\mu } { \\sqrt { | \\eta | } } \\right ) d \\mu \\\\ \\sqrt { | \\eta | } K ( S , S ) ( \\eta ) & = \\int _ { | \\mu | \\le | \\eta | ^ { 3 / 2 } / 2 } + \\int _ { | \\eta | ^ { 3 / 2 } / 2 \\le | \\mu | } = T _ { 2 , 1 } + T _ { 2 , 2 } . \\end{align*}"}
-{"id": "5832.png", "formula": "\\begin{align*} \\frac { \\partial \\phi _ { X } ( u ) } { \\partial u } & = \\int i x e ^ { i x u } f ( x ) d x \\end{align*}"}
-{"id": "5204.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow + \\infty } \\frac { U ^ { \\ast } ( \\gamma x ) } { U ^ { \\ast } ( x ) } = \\gamma \\lim _ { x \\rightarrow + \\infty } \\frac { U ( \\gamma x ) } { U ( x ) } = \\gamma ^ { \\rho + 1 } . \\end{align*}"}
-{"id": "8646.png", "formula": "\\begin{align*} w _ 1 = w , \\ w _ 2 = \\varepsilon w _ 1 ' , \\ \\varphi _ 1 = \\varphi , \\ \\varphi _ 2 = \\varphi _ 1 ' , \\end{align*}"}
-{"id": "9890.png", "formula": "\\begin{align*} A ' = \\begin{pmatrix} 0 & 0 & \\frac { 3 } { 4 } & \\frac { 1 } { 4 } \\\\ 0 & 0 & \\frac { 1 } { 2 } & \\frac { 1 } { 2 } \\\\ \\frac { 3 } { 4 } & \\frac { 1 } { 4 } & 0 & 0 \\\\ \\frac { 1 } { 2 } & \\frac { 1 } { 2 } & 0 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "8156.png", "formula": "\\begin{align*} m _ K ( \\varphi \\otimes \\psi ) = m _ K \\circ ( i _ { H _ { \\mathrm { r e g } } } \\otimes i _ { H _ { \\mathrm { r e g } } } ) \\ ! \\left ( \\varphi \\otimes 1 \\otimes \\psi \\otimes 1 \\right ) = d _ { H _ { \\mathrm { r e g } } , H _ { \\mathrm { r e g } } } ( \\varphi \\otimes 1 \\otimes \\psi \\otimes 1 ) & = \\psi ( S ( b _ i ) ? b _ j ) \\varphi ( ? a _ j a _ i ) \\\\ & = \\varphi ( a _ j ? a _ i ) \\psi ( S ( b _ i ) b _ j ? ) \\end{align*}"}
-{"id": "7400.png", "formula": "\\begin{align*} u ( x , 0 ) = u _ 0 ( x ) . \\end{align*}"}
-{"id": "5260.png", "formula": "\\begin{align*} H ( x , F ) = : \\frac { R ( F _ 1 , t ) } { R ( F , t ) } = \\frac { ( 1 - F ( x ) ) \\left ( \\int _ { x } ^ { u e p ( F ) } \\int _ { u } ^ { u e p ( F ) } 1 - F ( t ) d t d u \\right ) } { \\left ( \\int _ { x } ^ { u e p ( F ) } 1 - F ( t ) d t \\right ) ^ 2 } . \\end{align*}"}
-{"id": "8149.png", "formula": "\\begin{align*} f ( \\overset { I } { U } ( i ) ) = \\overset { I } { v } { ^ N } \\overset { I } { A } ( i _ 1 ) ^ { m _ 1 } \\overset { I } { B } ( j _ 1 ) ^ { n _ 1 } \\ldots \\overset { I } { A } ( i _ k ) ^ { m _ k } \\overset { I } { B } ( j _ k ) ^ { n _ k } \\end{align*}"}
-{"id": "9458.png", "formula": "\\begin{align*} J _ { N , 1 } ( T _ m . D _ f ) = \\sum _ { \\tau \\in \\mathcal { F } _ N } \\nu _ { \\tau } ^ { ( N ) } ( f ) \\sum _ { a d = m \\atop b \\pmod { d } } J _ { N , 1 } \\left ( \\frac { a \\tau + b } { d } \\right ) \\end{align*}"}
-{"id": "9399.png", "formula": "\\begin{align*} I & = \\int _ { \\mathbb { R } ^ { N } } \\int _ { \\mathbb { R } ^ { N } } \\Phi ( u _ { n } ( x ) - u _ { n } ( y ) ) ( u _ { n } ( x ) - u _ { n } ( y ) ) \\xi _ { j , \\delta } ( x ) K ( x , y ) d x d y \\\\ & \\geq \\Lambda ^ { - 2 } \\int _ { \\mathbb { R } ^ N } \\int _ { \\mathbb { R } ^ N } \\frac { | u _ { n } ( x ) - u _ { n } ( y ) | ^ { p } } { | x - y | ^ { N + p s } } \\xi _ { j , \\delta } ( x ) d x d y \\\\ & = \\Lambda ^ { - 2 } \\int _ { \\mathbb { R } ^ { N } } | D ^ { s } u _ { n } | ^ { p } \\xi _ { j , \\delta } ( x ) d x \\end{align*}"}
-{"id": "2342.png", "formula": "\\begin{align*} \\langle a ^ + ( \\nu ) \\mu | f \\rangle & = \\langle \\mu | a ( \\nu ) f \\rangle \\\\ \\langle a ( \\varphi ) \\mu | f \\rangle & = \\langle \\mu | a ( \\varphi ) ^ + f \\rangle \\end{align*}"}
-{"id": "2652.png", "formula": "\\begin{align*} G & = \\lim _ { \\delta \\to 0 } \\begin{bmatrix} { \\delta I _ { N } + C _ { 1 , 1 } } & { C _ { 1 , 2 } } \\\\ { C _ { 2 , 1 } } & \\delta I _ { n } + { C _ { 2 , 2 } } \\end{bmatrix} ^ { - 1 } \\\\ & = \\begin{bmatrix} G _ { 1 1 } & G _ { 1 2 } \\\\ G _ { 2 1 } & \\lim _ { \\delta \\to 0 } \\big ( ( \\delta I _ { n } + { C _ { 2 , 2 } } ) - { C _ { 2 , 1 } } ( \\delta I _ { N } + C _ { 1 , 1 } ) ^ { - 1 } { C _ { 1 , 2 } } \\big ) ^ { - 1 } \\end{bmatrix} . \\end{align*}"}
-{"id": "2198.png", "formula": "\\begin{align*} \\dot { \\boldsymbol { x } } ( t ) & = A ( t ) \\boldsymbol { x } ( t ) + B ( t ) \\boldsymbol { u } ( t ) \\\\ \\boldsymbol { y } ( t ) & = C ( t ) \\boldsymbol { x } ( t ) \\end{align*}"}
-{"id": "4338.png", "formula": "\\begin{align*} s _ \\alpha ( \\lambda ) = \\lambda - \\lambda ( H _ \\alpha ) d \\alpha , \\end{align*}"}
-{"id": "1971.png", "formula": "\\begin{align*} 2 d _ { 3 3 } = d _ { 1 2 } + d _ { 2 2 } + \\sum _ { \\substack { \\ell \\in \\mathcal { N } ( 3 ) \\\\ \\ell \\neq 1 , 2 } } d _ { \\ell 2 } . \\end{align*}"}
-{"id": "4005.png", "formula": "\\begin{align*} \\left \\langle U ( \\beta ) f , U ( \\beta ) f \\right \\rangle _ { d ^ 2 \\beta } = \\sum _ j | c _ j | ^ 2 \\left \\langle U ( \\beta ) f _ j , U ( \\beta ) f _ j \\right \\rangle _ { d ^ 2 \\beta } . \\end{align*}"}
-{"id": "8857.png", "formula": "\\begin{align*} e _ 1 \\ge e _ 1 | _ { a = a _ { c r } } = \\frac 1 { 3 2 } \\cdot \\frac { 3 ( 1 + \\nu ) ^ 2 c _ 1 } { 9 \\nu ^ 4 - 6 6 \\nu ^ 3 + 7 6 \\nu ^ 2 + 2 \\nu + 2 3 5 } \\end{align*}"}
-{"id": "4352.png", "formula": "\\begin{align*} \\mathcal P _ a ^ i : = & \\big \\{ P \\in { [ 2 , n ] \\choose k - 1 } \\ : \\ P \\cap [ i ] = S _ i \\cap [ 2 , i ] \\big \\} , \\\\ \\mathcal P _ b ^ i : = & \\big \\{ P \\in { [ 2 , n ] \\choose k } \\ : \\ P \\cap [ i ] = [ i ] \\setminus S _ i \\big \\} , \\end{align*}"}
-{"id": "530.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 } \\frac { 1 } { h ^ 2 } \\mathcal E ^ h ( u ^ h ) = \\displaystyle { \\mathcal { I } _ 2 ^ O ( y ) } \\end{align*}"}
-{"id": "2176.png", "formula": "\\begin{align*} m _ { \\ell } ( \\xi ) = m ( 2 ^ { \\ell } \\xi ) \\phi ( \\xi ) = m _ { \\ell , 1 } ( \\xi ) e ^ { - \\| \\xi \\| ^ 2 } = m _ { \\ell , 2 } ( \\xi ) e ^ { - \\| \\xi \\| ^ 2 } e ^ { - \\| \\xi \\| ^ 2 } . \\end{align*}"}
-{"id": "3691.png", "formula": "\\begin{align*} p ( x ) = \\sum _ { i \\in U _ x } p _ i / N _ x \\end{align*}"}
-{"id": "7335.png", "formula": "\\begin{align*} E _ { w ^ n , z ^ n } ( t ) = \\frac { 1 } { \\alpha _ n ^ 2 } E _ { u ^ n , v ^ n } ( t ) . \\end{align*}"}
-{"id": "756.png", "formula": "\\begin{align*} C = V M _ { \\gamma } V ^ * , \\end{align*}"}
-{"id": "8361.png", "formula": "\\begin{align*} A _ 1 = 1 + \\frac { 1 } { 2 \\pi } \\int \\frac { | D _ t Z ( \\alpha , t ) - D _ t Z ( \\beta , t ) | ^ 2 } { ( \\alpha - \\beta ) ^ 2 } d \\beta - \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j } { \\pi } R e \\Big \\{ \\frac { D _ t Z - \\dot { z } _ j } { c _ 0 ^ j ( \\alpha - \\omega _ 0 ^ j ) ^ 2 } \\Big \\} \\end{align*}"}
-{"id": "155.png", "formula": "\\begin{align*} \\left . + \\sum _ { s = 1 } ^ { N } \\dfrac { \\partial v ^ { i _ 1 \\dots i _ k } } { \\partial x ^ { s } } ( { \\bf x } ) y ^ s \\dfrac { \\partial } { \\partial y ^ { i _ 1 } } \\wedge \\dots \\wedge \\dfrac { \\partial } { \\partial y ^ { i _ k } } \\right ) . \\end{align*}"}
-{"id": "692.png", "formula": "\\begin{align*} P ( [ \\varphi ; f _ 1 , \\dots , f _ n ; x ] ) : = \\varpi ( f _ 1 , \\dots , f _ n ) \\circ \\theta ^ { - 1 } \\circ \\widehat { [ \\rho , x ] } ^ { } _ { X _ 1 \\dots X _ m } \\ , \\end{align*}"}
-{"id": "4603.png", "formula": "\\begin{align*} S _ 2 ( S _ 1 ( x ) + s _ 1 ) + s + 2 = S _ 2 S _ 1 ( x ) + S _ 2 ( s _ 1 ) + s _ 2 , \\end{align*}"}
-{"id": "6029.png", "formula": "\\begin{align*} A ^ n ( t ) = \\int _ 0 ^ t q ^ { Q _ n } ( s ) d s = W ( t ) - W ^ { Q _ n } ( t ) , \\end{align*}"}
-{"id": "2293.png", "formula": "\\begin{align*} g ( \\omega , \\mathbf { n } , i ) & = \\i c ( \\omega ) \\sum _ l e \\frac { \\sqrt { \\omega _ 0 } } { 2 \\pi } \\Pi ( \\mathbf { n } ) _ { i , l } ( \\psi _ { 0 } | \\mathbf { x } | \\psi _ 1 ) _ l \\\\ ( K f ) ( \\omega , \\mathbf { n } , i ) & = \\omega \\sum _ l \\Pi ( n ) _ { i l } f ( \\omega , \\mathbf { n } , l ) . \\end{align*}"}
-{"id": "123.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 ( 1 - x ^ 2 ) ^ \\ell \\ , d x = \\frac { 2 ^ { \\ell + 1 } \\cdot \\ell ! } { ( 1 + 2 \\ell ) ! ! } = \\frac { 2 ^ { 2 \\ell + 1 } \\cdot ( \\ell ! ) ^ 2 } { ( 1 + 2 \\ell ) ! } \\ell \\ge 1 \\ , . \\end{align*}"}
-{"id": "3062.png", "formula": "\\begin{align*} = \\pi ^ { - ( 1 - s ) } \\Gamma ( 1 - s ) \\frac { \\Gamma ( s ) ^ { 2 } } { \\Gamma ( s + 2 q ) \\Gamma ( s - 2 q ) } ( a b ) ^ { 2 s - 3 / 2 } c ^ { - ( 1 - s ) } \\sum _ { j = 0 } ^ { \\infty } { 2 j + 4 q \\choose j } \\frac { d ^ { j + 2 q } \\Gamma ( j + 2 q - 1 + s ) } { 2 ^ { 2 q - 1 } c ^ { j + 2 q } \\Gamma ( - 1 + s ) } ( 1 / 2 ) ^ { j } . \\end{align*}"}
-{"id": "3525.png", "formula": "\\begin{align*} \\bar { \\omega } ( g a ) = ( \\det g ) ^ { - 1 } \\bar { \\omega } ( a ) , \\end{align*}"}
-{"id": "8648.png", "formula": "\\begin{align*} \\hat { H } = 0 \\ \\ \\textrm { o n } \\ \\ W ^ u ( 0 , 0 , \\pi - \\bar { \\varphi } , 0 ) \\cup W ^ s ( 0 , 0 , \\bar { \\varphi } , 0 ) . \\end{align*}"}
-{"id": "7104.png", "formula": "\\begin{align*} \\textbf { H } = - \\frac { 1 } { 2 } ( ( d _ { X } d _ { X } x + d _ { Y } d _ { Y } x , g ) \\tilde { g } + ( d _ { X } d _ { X } x + d _ { Y } d _ { Y } x , \\tilde { g } ) g ) = \\frac { 1 } { 2 } ( \\alpha + \\beta ) \\tilde { g } \\ , m o d \\ , G . \\end{align*}"}
-{"id": "5530.png", "formula": "\\begin{align*} E ^ { \\Pi ^ { D _ T } } [ e ^ { u \\sqrt { T } \\langle b _ j - b _ { 0 , j } , a _ \\lambda \\Phi _ { \\lambda , k } \\rangle _ { L ^ 2 } } | X ^ { T } ] & = e ^ { \\frac { u } { \\sqrt { T } } \\int _ 0 ^ T \\tilde { \\Phi } _ { \\lambda , k , j } ( X _ t ) . d W _ t + \\frac { u ^ 2 } { 2 } \\| \\tilde { \\Phi } _ { \\lambda , k , j } \\| _ { \\mu _ 0 } ^ 2 + u r _ T + u ^ 2 \\tilde { r } _ T } \\\\ & \\times \\frac { \\int _ { D _ T } e ^ { S _ T ( b ) + \\ell _ T ( b _ u ) } d \\Pi ( b ) } { \\int _ { D _ T } e ^ { \\ell _ T ( b ) } d \\Pi ( b ) } , \\end{align*}"}
-{"id": "1701.png", "formula": "\\begin{align*} I _ n : = \\int _ { E _ n } \\varphi \\ , d \\mu \\geq 0 \\ \\ \\forall \\ n \\geq 1 \\ \\ \\ \\ \\ \\ \\ \\mbox { ( t o b e p r o v e d ) } . \\end{align*}"}
-{"id": "4770.png", "formula": "\\begin{align*} \\nabla \\Phi _ i = { \\mathcal O } \\left ( \\frac { 1 } { | x | ^ { 2 } } \\right ) \\ \\nabla ^ { 2 } \\Phi _ i = { \\mathcal O } \\left ( \\frac { 1 } { | x | ^ { 3 } } \\right ) \\ x \\rightarrow \\infty . \\end{align*}"}
-{"id": "5528.png", "formula": "\\begin{align*} & a _ \\lambda | \\langle \\mu _ 0 ( b _ j - P _ { V _ J } b _ { 0 , j } ) , \\Phi _ { \\lambda , k } / \\mu _ 0 - P _ { V _ J } [ \\Phi _ { \\lambda , k } / \\mu _ 0 ] \\rangle _ { L ^ 2 } | \\\\ & + a _ \\lambda | \\langle \\mu _ 0 ( b _ { 0 , j } - P _ { V _ J } b _ { 0 , j } ) , \\Phi _ { \\lambda , k } / \\mu _ 0 - P _ { V _ J } [ \\Phi _ { \\lambda , k } / \\mu _ 0 ] \\rangle _ { L ^ 2 } | = : ( I ) + ( I I ) . \\end{align*}"}
-{"id": "1422.png", "formula": "\\begin{align*} \\sup _ { \\tau \\in { \\mathcal T } } E [ Z ^ 1 _ { U , \\tau } ] = 1 - \\inf _ { \\tau \\in { \\mathcal T } } E [ Z ^ { 1 , - } _ { U , \\tau } ] ; \\end{align*}"}
-{"id": "829.png", "formula": "\\begin{align*} \\mathrm { d i v } _ Y ( d T ) = \\prod _ { i = 1 } ^ { s } \\prod _ { \\mathfrak { P } | P _ i } { \\mathfrak { P } _ i } ^ S \\prod _ { Q \\mid \\mathfrak { p } _ \\infty } Q ^ { q - 2 } \\cdot _ { \\mathbb { F } _ q ( T ) / \\mathbb { F } _ { q ^ d } K _ { q , P ^ \\alpha } } ( \\mathfrak { p } _ \\infty ^ { - 2 } ) , \\end{align*}"}
-{"id": "5655.png", "formula": "\\begin{align*} \\widetilde { U } _ i ( \\zeta ) & = \\tau \\frac { \\widetilde { B } ( \\zeta ) - 1 } { \\zeta } \\widetilde f ( \\zeta ) = \\tau \\sum _ { n = 1 } ^ { \\infty } \\Big ( \\sum _ { k = 0 } ^ { n - 1 } B _ { n - k } f ^ { k + 1 } \\Big ) \\zeta ^ { n } = \\tau \\sum _ { n = 1 } ^ { \\infty } \\Big ( \\sum _ { k = 1 } ^ { n } B _ { n - ( k - 1 ) } f ^ { k } \\Big ) \\zeta ^ { n } , \\end{align*}"}
-{"id": "8190.png", "formula": "\\begin{align*} \\frac { F - \\mathbb { E } ( F ) } { \\sqrt { \\mathbb { V } ( F ) } } & = \\frac { { \\sqrt { \\mathbb { V } ( F ' ) } } } { { \\sqrt { \\mathbb { V } ( F ) } } } \\frac { F ' - \\mathbb { E } ( F ' ) } { \\sqrt { \\mathbb { V } ( F ' ) } } + \\frac { F - F ' - \\mathbb { E } ( F - F ' ) } { { \\sqrt { \\mathbb { V } ( F ) } } } , \\end{align*}"}
-{"id": "5583.png", "formula": "\\begin{align*} \\begin{cases} ( \\partial _ t + \\partial _ x ^ 3 ) e ^ { - t \\partial _ x ^ 3 } \\phi ( x , t ) = 0 & ( x , t ) \\in \\mathbb { R } \\times \\mathbb { R } , \\\\ e ^ { - t \\partial _ x ^ 3 } \\phi ( x ) \\big | _ { t = 0 } = \\phi ( x ) & x \\in \\mathbb { R } . \\end{cases} \\end{align*}"}
-{"id": "9732.png", "formula": "\\begin{align*} v \\ , L _ a v = - p \\ , L _ a u \\geq 0 . \\end{align*}"}
-{"id": "8852.png", "formula": "\\begin{align*} \\frac { \\partial s _ 4 } { \\partial \\mu } = - M \\left ( ( 3 - w ) M ^ 2 + ( 1 + w ) M - 2 + 2 ( M - 1 ) ^ 2 w ^ 2 \\right ) < 0 \\end{align*}"}
-{"id": "1808.png", "formula": "\\begin{align*} F _ { i } ^ { t , ( n ) } = \\frac { 2 ^ { n } } { t } \\left ( \\int _ { i 2 ^ { - n } t } ^ { ( i + 1 ) 2 ^ { - n } t } u _ { t } ( r ) d r \\right ) , \\quad 0 \\leq i \\leq 2 ^ { n } - 1 . \\end{align*}"}
-{"id": "5201.png", "formula": "\\begin{align*} H _ { 1 } ^ { t } ( x _ { 1 } ) = H _ { 1 } ( x _ { 1 } t ^ { \\rho } ) , t > 0 , t \\neq 1 . \\end{align*}"}
-{"id": "558.png", "formula": "\\begin{align*} ( 1 + i | \\nabla | ^ { - 1 } \\partial _ t ) ( u _ { \\lambda _ 1 } v _ { \\lambda _ 2 } ) = { } & \\big ( ( 1 + i | \\nabla | ^ { - 1 } \\partial _ t ) u _ { \\lambda _ 1 } \\big ) v _ { \\lambda _ 2 } - i | \\nabla | ^ { - 1 } \\mathcal { M } ( | \\nabla | ^ { - 1 } \\partial _ t u _ { \\lambda _ 1 } , v _ { \\lambda _ 2 } ) \\\\ & { } + i | \\nabla | ^ { - 1 } ( u _ { \\lambda _ 1 } \\partial _ t v _ { \\lambda _ 2 } ) - i | \\nabla | ^ { - 1 } ( | \\nabla | ^ { - 1 } \\partial _ t u _ { \\lambda _ 1 } | \\nabla | v _ { \\lambda _ 2 } ) . \\end{align*}"}
-{"id": "2417.png", "formula": "\\begin{align*} h \\left ( u _ { i j } ^ { ( s ) } ( u _ { k l } ^ { ( t ) } ) ^ * \\right ) = \\delta _ { s t } \\delta _ { i k } \\frac { ( Q _ s ) _ { l , j } } { d _ s } , \\ ; \\ ; h \\left ( ( u _ { i j } ^ { ( s ) } ) ^ * u _ { k l } ^ { ( t ) } \\right ) = \\delta _ { s t } \\delta _ { j l } \\frac { ( Q _ s ^ { - 1 } ) _ { k , i } } { d _ s } . \\end{align*}"}
-{"id": "6923.png", "formula": "\\begin{align*} \\langle X _ { \\sigma ( i ) } , D ^ * P P _ i Z \\rangle & = \\langle P \\widetilde V _ { \\sigma ( i ) } , P _ i Z \\rangle = \\langle \\widetilde V _ { \\sigma ( i ) } , P _ i Z \\rangle - \\kappa ( P _ i Z ) _ { \\sigma ( i ) } \\\\ & = \\langle \\widetilde V _ { \\sigma ( i ) } , Z \\rangle - \\langle ( { \\rm I } - P _ i ) \\widetilde V _ { \\sigma ( i ) } , Z \\rangle - \\kappa ( P _ i Z ) _ { \\sigma ( i ) } . \\end{align*}"}
-{"id": "8036.png", "formula": "\\begin{gather*} \\{ 1 \\} , \\{ y ^ { 2 s + 1 } \\} , \\quad \\{ y ^ n \\} , \\{ y ^ { 6 s + 3 } \\} , \\\\ \\{ y ^ { 2 j } , y ^ { - 2 j } \\} 1 \\leq j < 2 s + 1 = n / 2 , \\{ y ^ { 2 j + 1 } , y ^ { n - 2 j - 1 } \\} , \\enspace \\{ y ^ { n + 2 j + 1 } , y ^ { 2 n - 2 j - 1 } \\} 1 \\leq 2 j + 1 < l = n / 2 , \\\\ \\{ x , x y ^ 4 , \\dots , x y ^ { 2 n - 4 } \\} , \\{ x y , x y ^ 5 , \\dots , x y ^ { 2 n - 3 } \\} , \\{ x y ^ 2 , x y ^ 6 , \\dots , x y ^ { 2 n - 2 } \\} , \\\\ \\{ x ^ 3 , x y ^ 7 , \\dots , x y ^ { 2 n - 1 } \\} \\end{gather*}"}
-{"id": "3531.png", "formula": "\\begin{align*} \\mathbb { H } _ { 2 } = \\left \\{ W \\in M a t ( 2 , \\mathbb { C } ) \\mid ( W ^ { \\dagger } - W ) / 2 i > 0 \\right \\} , \\end{align*}"}
-{"id": "369.png", "formula": "\\begin{align*} w _ 0 ^ { ( j ) } = z _ 1 ^ { ( j ) } , w _ 1 ^ { ( j ) } = t z _ 1 ^ { ( j ) } + c y _ 1 ^ { ( j ) } , w _ { n + 2 } ^ { ( i ) } = 2 t w _ { n + 1 } ^ { ( j ) } - w _ n ^ { ( j ) } , j = 1 , \\dotsc , j _ 0 . \\end{align*}"}
-{"id": "2704.png", "formula": "\\begin{align*} \\varphi ( 4 ) = ( 1 6 N ) ^ { - \\frac { 1 } { 3 } } , h ( u ) = e ^ { \\frac { \\eta _ { - } } { \\varphi ( 4 ) } ( z _ { 0 } + 1 ) u } , T _ { n } = e ^ { - 2 N f ( z _ { 0 } ) } \\prod _ { k = 1 } ^ { n } \\big ( \\frac { \\sigma _ k } { \\tau } \\big ) ^ { 2 } . \\end{align*}"}
-{"id": "8303.png", "formula": "\\begin{align*} ( D _ t ^ 2 - i A \\partial _ { \\alpha } ) \\tilde { \\theta } = G _ c + G _ d , \\end{align*}"}
-{"id": "4618.png", "formula": "\\begin{align*} \\mathcal { Q } = \\begin{bmatrix} A _ 0 ^ 1 & & & A _ 1 \\\\ A _ 1 & A _ 0 ^ 2 & & \\\\ & \\ddots & \\ddots & \\\\ & & A _ 1 & A _ 0 ^ { \\ell } \\end{bmatrix} \\end{align*}"}
-{"id": "6119.png", "formula": "\\begin{align*} \\begin{pmatrix} Q _ 1 & Q _ 1 \\\\ Q _ 2 & 0 \\\\ 0 & Q _ 3 \\end{pmatrix} , \\end{align*}"}
-{"id": "1176.png", "formula": "\\begin{align*} 0 & \\leq 1 - \\det ( - A ) / \\det ( - B ) \\\\ & \\leq \\left | ( - B ) ^ { - 1 / 2 } ( B - A ) ( - B ) ^ { - 1 / 2 } \\right | _ 2 ^ 2 \\\\ & \\leq \\left | ( - B ) ^ { - 1 / 2 } \\right | _ { \\infty } ^ 4 | B - A | _ 2 ^ 2 = \\left | ( - B ) ^ { - 1 } \\right | _ { \\infty } ^ 2 | B - A | _ 2 ^ 2 , \\end{align*}"}
-{"id": "743.png", "formula": "\\begin{align*} \\psi _ { 1 } ( x ) = \\frac { \\dd ^ { 2 } \\ln \\Gamma ( x ) } { \\dd x ^ { 2 } } \\end{align*}"}
-{"id": "8452.png", "formula": "\\begin{align*} ( D _ t ^ 2 - i A \\partial _ { \\alpha } ) \\theta _ k = G _ k ^ { \\theta } \\\\ ( D _ t ^ 2 - i A \\partial _ { \\alpha } ) \\sigma _ k = G _ k ^ { \\sigma } , \\\\ \\end{align*}"}
-{"id": "1717.png", "formula": "\\begin{align*} | f ( \\varphi ( h , x ) ) - f ( x ) | & = | a _ i \\varphi ( h , x ) + b _ i - ( a _ n x + b _ n ) | \\leq | \\varphi ( h , x ) - x | \\cdot | a _ i | + | ( a _ i - a _ n ) x - ( b _ n - b _ i ) | \\\\ & \\leq \\dfrac { \\varepsilon } { 4 } \\cdot u ( x ) \\cdot | a _ i | + | a _ i - a _ n | \\cdot | x - j _ i | . \\end{align*}"}
-{"id": "1974.png", "formula": "\\begin{align*} 2 \\deg ( i ) d _ { i i } = \\sum _ { k \\in \\mathcal { N } ( i ) } d _ { k k } = \\deg ( i ) d _ { \\ell \\ell } . \\end{align*}"}
-{"id": "2324.png", "formula": "\\begin{align*} R ( z ) = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 0 & R _ { \\Omega } ( z ) \\end{array} \\right ) + \\left ( \\begin{array} { c } 1 \\\\ R _ { \\Omega } ( z ) | 1 \\rangle \\end{array} \\right ) \\frac { 1 } { C ( z ) } \\left ( 1 , \\langle 1 | R _ { \\Omega } ( z ) \\right ) \\end{align*}"}
-{"id": "1023.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to + \\infty } \\ln \\left ( n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } D _ n ( \\alpha _ n ) \\right ) = c _ 0 - x . \\end{align*}"}
-{"id": "8986.png", "formula": "\\begin{align*} \\mathbb { \\hat { E } } _ { \\sigma } [ M _ { t } ] = M _ { \\sigma } . \\end{align*}"}
-{"id": "6250.png", "formula": "\\begin{align*} Y _ i ^ { ( d ) } = \\frac { 1 } { \\sqrt { b ^ d } } \\sum _ { j , k = 1 } ^ { b ^ d } \\left [ \\ell ( \\xi ^ i _ { j , k } ) + \\sqrt { \\lambda } \\ell ( \\eta ^ i _ { j , k } ) ^ * \\right ] \\otimes e _ { j , k } \\end{align*}"}
-{"id": "9222.png", "formula": "\\begin{align*} C _ { ( i - 1 ) W + j , ( l - 1 ) W + m } = \\begin{cases} \\sigma _ \\psi ^ 2 + \\sigma _ \\zeta ^ 2 & i = l , j = m , \\\\ \\sigma _ \\psi ^ 2 \\exp \\left ( - \\frac { d ( i j , l m ) } { d _ 0 } \\right ) & e l s e . \\end{cases} \\end{align*}"}
-{"id": "2333.png", "formula": "\\begin{align*} & ( a _ y ^ + ) | x _ 1 , \\cdots , x _ n \\rangle = | y , x _ 1 , \\cdots x _ n ) \\rangle \\\\ & a _ y | x _ 1 , \\cdots , x _ n \\rangle \\\\ & = \\delta _ { y , x _ 1 } | x _ 2 , \\cdots , x _ n \\rangle + \\delta _ { y , x _ 2 } | x _ 1 , x _ 3 , \\cdots , x _ n \\rangle + \\cdots + \\delta _ { y , x _ n } | x _ 1 , \\cdots , x _ { n - 1 } \\rangle \\end{align*}"}
-{"id": "2995.png", "formula": "\\begin{align*} \\Big ( \\sum _ { n = 0 } ^ \\infty p ' _ m ( n ) q ^ n \\Big ) \\Big ( \\sum _ { n = 0 } ^ \\infty e _ { m + 2 , n } q ^ n \\Big ) & = 1 \\end{align*}"}
-{"id": "3993.png", "formula": "\\begin{align*} \\varrho ( n ) = \\begin{cases} 3 / 2 & n , \\\\ 1 & n . \\end{cases} \\end{align*}"}
-{"id": "4287.png", "formula": "\\begin{align*} \\frac { 1 } { r } \\left ( \\binom { s - 2 } { r - 1 } + \\binom { s - 3 } { r - 2 } \\right ) n \\end{align*}"}
-{"id": "4096.png", "formula": "\\begin{align*} \\Phi _ { \\alpha } ( z ) \\Psi _ { \\alpha } ^ T ( z ) = \\mathcal O \\left ( z ^ { \\min ( \\alpha , - \\frac { 1 } { 2 } ) } \\right ) \\alpha \\in ( - 1 , 0 ) \\setminus \\{ - \\tfrac { 1 } { 2 } \\} . \\end{align*}"}
-{"id": "5991.png", "formula": "\\begin{align*} \\omega \\oplus _ t \\overline \\omega ( s ) = \\omega ( s \\wedge t ) + \\overline \\omega ( s ) 1 _ { [ t , 1 ] } ( s ) . \\end{align*}"}
-{"id": "4748.png", "formula": "\\begin{align*} C _ { \\xi - { \\bf e } _ i , \\xi } : = e ^ { W ( \\xi ) } . \\end{align*}"}
-{"id": "8208.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty c ( n ) q ^ n : = \\frac { 2 E _ 2 ( 2 \\tau ) - E _ 2 ( \\tau ) } { ( q ^ 2 ; q ^ 2 ) _ \\infty } . \\end{align*}"}
-{"id": "8685.png", "formula": "\\begin{align*} \\mathcal { F } ( E _ { \\mathfrak { v } } ) & = - E _ { \\mathfrak { v } } - p , & \\mathcal { F } ( \\abs { X } ^ 4 \\partial _ T ) & = \\langle Z , T \\rangle \\Delta _ { \\mathfrak { v } } ^ 2 , \\\\ \\mathcal { F } ( \\abs { X } ^ 2 \\partial _ { J _ T X } ) & = \\partial _ { J _ T X } \\Delta _ { \\mathfrak { v } } , & \\mathcal { F } ( \\abs { Z } ^ 2 \\partial _ T ) & = \\langle Z , T \\rangle \\Box + 2 \\partial _ T . \\end{align*}"}
-{"id": "9542.png", "formula": "\\begin{align*} \\frac { ( q ) _ { N } } { ( z q ) _ { N } ( z ^ { - 1 } q ) _ { N } } = \\frac { 1 } { ( q ) _ N } + ( 1 - z ) \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - 1 ) ^ n ( q ) _ n q ^ { n ( n + 1 ) / 2 } } { ( q ) _ { n + N } } \\left ( \\frac { 1 } { 1 - z q ^ n } - \\frac { 1 } { z - q ^ n } \\right ) . \\end{align*}"}
-{"id": "896.png", "formula": "\\begin{align*} Y _ m = ( - 1 ) ^ { s _ m } \\cdot q ^ { w _ m } \\cdot y _ { e _ 1 } ^ { \\gamma ( 1 ) } \\cdots y _ { e _ n } ^ { \\gamma ( n ) } \\end{align*}"}
-{"id": "1043.png", "formula": "\\begin{align*} & \\mathbb { P } ( \\xi ( A ) = 0 ) = \\det ( - K _ A ) . \\end{align*}"}
-{"id": "1044.png", "formula": "\\begin{align*} K _ n ( x , y ) : & = K ^ { C U E ( n ) } ( x , y ) = \\dfrac { 1 } { 2 \\pi } \\dfrac { \\sin ( n ( x - y ) / 2 ) } { \\sin ( ( x - y ) / 2 ) } \\\\ & = \\dfrac { 1 } { 2 \\pi } \\sum \\limits _ { k = 0 } ^ { n - 1 } e ^ { ( k - ( n - 1 ) / 2 ) i ( x - y ) } . \\end{align*}"}
-{"id": "545.png", "formula": "\\begin{align*} \\begin{aligned} \\nabla u ^ h ( x ' , h x _ 3 ) = & \\ , Q _ 0 + h \\Big ( \\big [ \\nabla v ^ h , ~ \\vec p ^ { \\ , h } \\big ] + P _ 0 \\Big ) + h ^ 2 \\Big ( \\big [ \\nabla w ^ h , ~ \\vec q ^ { \\ , h } \\big ] + \\big [ x _ 3 \\nabla \\vec p ^ { \\ , h } , ~ \\partial _ 3 \\vec r ^ { \\ , h } \\big ] + \\big ( \\nabla _ { \\rm t a n } \\vec d _ 0 , ~ \\partial _ 3 \\vec k _ 0 \\big ] \\Big ) \\\\ & + \\mathcal { O } ( h ^ 3 ) \\Big ( | \\nabla \\vec k _ 0 | + | \\nabla \\vec q ^ { \\ , h } | + | \\nabla \\vec r ^ { \\ , h } | \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "8033.png", "formula": "\\begin{align*} p ( t ) = \\sum _ { r = 1 } ^ { 2 n + 1 } a _ r t ^ r , \\mbox { w h e r e } a _ k = \\pm c ^ { 2 n - 2 k + 2 } a _ { 2 n + 2 - k } \\end{align*}"}
-{"id": "6819.png", "formula": "\\begin{align*} \\begin{cases} 2 \\cosh ( 2 s ) s '' + \\sinh ( 2 s ) \\big ( 2 ( s ' ) ^ 2 - ( \\theta ' ) ^ 2 - ( \\mu ' ) ^ 2 \\big ) = 0 \\ , , \\\\ \\cosh ( s ) \\mu '' + 2 \\sinh ( s ) s ' \\mu ' = 0 \\ , , \\\\ \\sinh ( s ) \\theta '' + 2 \\cosh ( s ) s ' \\theta ' = 0 \\ , . \\end{cases} \\end{align*}"}
-{"id": "3006.png", "formula": "\\begin{align*} - \\frac { G _ m ' ( x ) } { G _ m ( x ) } & = \\sum _ { n = 1 } ^ \\infty \\sigma ' _ m ( n ) x ^ { n - 1 } . \\end{align*}"}
-{"id": "6190.png", "formula": "\\begin{align*} \\tilde u ^ - _ \\infty ( X _ 0 ) - q _ 1 ( X _ 0 ) = 0 , \\tilde u ^ + _ \\infty ( X _ 0 ) - q _ 2 ( X _ 0 ) = 0 . \\end{align*}"}
-{"id": "317.png", "formula": "\\begin{align*} a ( \\tilde { \\gamma } ) = \\tilde { \\gamma } ^ * ( a _ { 2 n } ) \\oplus \\tilde { \\gamma } ^ * ( a _ { 2 n + 2 } ) = u ^ 2 v ^ { n - 2 } \\oplus u ^ 2 v ^ { n - 1 } . \\end{align*}"}
-{"id": "4857.png", "formula": "\\begin{align*} \\alpha ( w ^ * _ { ( j , r _ q ) } ) = a w ^ * _ { ( j , r _ q ) } , \\end{align*}"}
-{"id": "1703.png", "formula": "\\begin{align*} \\int _ { \\varphi _ n > 0 , \\ { \\mathcal L } \\varphi _ n \\leq 0 } \\varphi \\ , d \\mu = \\int _ { \\varphi _ n > 0 , \\ { \\mathcal L } \\varphi _ n \\leq 0 } \\varphi _ n \\ , d \\mu . \\end{align*}"}
-{"id": "982.png", "formula": "\\begin{align*} E ( G _ { r , d } ) = \\{ & \\{ x _ { i } , y _ { j } \\} \\mid 1 \\leq i \\leq 2 , 1 \\leq j \\leq d \\} \\cup \\{ \\{ x _ { 1 } , z _ { 1 } \\} , \\{ x _ { 2 } , z _ { 2 r - 3 } \\} \\} \\\\ & \\cup \\{ \\{ z _ { i } , z _ { i + 1 } \\} \\mid 1 \\leq i \\leq 2 r - 4 \\} . \\end{align*}"}
-{"id": "5026.png", "formula": "\\begin{align*} r s _ d ( m , n , I ) = O _ d \\left ( \\left ( \\frac { I } { m n } \\right ) ^ { \\frac { d + 1 } { 2 } } m n \\right ) . \\end{align*}"}
-{"id": "5016.png", "formula": "\\begin{align*} \\left | \\{ \\log _ M N _ i \\} - a \\right | < \\epsilon , N _ 0 \\leq N _ i \\in \\mathfrak N _ i , i = 1 , 2 . \\end{align*}"}
-{"id": "3679.png", "formula": "\\begin{align*} p _ i \\propto \\mbox { P r } ( S _ i = 1 | x _ i , i \\in U ) \\frac { \\mbox { P r } ( \\delta _ i = 1 | x _ i , i \\in B \\cup S ) } { \\mbox { P r } ( S _ i = 1 | x _ i , i \\in B \\cup S ) } \\end{align*}"}
-{"id": "2639.png", "formula": "\\begin{align*} \\epsilon _ { i , j } : = \\int _ { \\Omega ^ { 2 } } \\epsilon ( x , y ) \\det \\ ! \\begin{bmatrix} \\phi _ { i } ( x ) & \\phi _ { i } ( y ) \\\\ \\phi _ { j } ( x ) & \\phi _ { j } ( y ) \\end{bmatrix} d \\nu ( x ) d \\nu ( y ) \\end{align*}"}
-{"id": "7358.png", "formula": "\\begin{align*} \\leq \\max \\big [ d ( x , y ) + \\frac 1 m + \\frac 1 n , d ( x , y ) + \\frac 1 m + \\frac 1 n ] = d ( x , y ) + \\frac 1 m + \\frac 1 n \\end{align*}"}
-{"id": "4558.png", "formula": "\\begin{align*} \\varphi _ 1 \\cdots \\varphi _ { m _ k } . ( u \\otimes ( v _ p ) _ z ) = ( \\tau _ 1 ( x _ 1 - x _ 2 ) + 1 ) \\tau _ 2 \\cdots \\tau _ { i - 1 } ( \\tau _ i ( x _ i - x _ { i + 1 } ) + 1 ) \\varphi _ { i + 1 } \\cdots \\varphi _ { m _ k } . ( u \\otimes ( v _ p ) _ z ) . \\end{align*}"}
-{"id": "5586.png", "formula": "\\begin{align*} \\mathcal { L } f ( x , t ) = \\mathcal { L } _ { - 1 } f ( x , t ) + \\mathcal { L } _ 1 f ( x , t ) , \\end{align*}"}
-{"id": "883.png", "formula": "\\begin{align*} \\gamma = \\sum _ { v = 1 } ^ N m _ v \\beta _ v \\in D _ Q . \\end{align*}"}
-{"id": "4465.png", "formula": "\\begin{align*} \\gamma _ { i j } = \\frac { 1 } { u } \\sigma _ { i j } + \\frac { \\nabla _ i u \\nabla _ j u } { u ^ 2 ( u + w ) } . \\end{align*}"}
-{"id": "7601.png", "formula": "\\begin{align*} \\begin{gathered} \\nabla \\cdot \\left ( \\eta ( s ) - \\eta ( t ) \\right ) = \\nabla _ x X ^ { - 1 } ( s ) : ( \\Delta _ 1 \\nabla _ a X ' ( s , t ) ) + \\nabla _ x X ^ { - 1 } ( s ) : ( \\Delta _ 2 \\nabla _ a X ' ( s , t ) ) \\\\ + \\left ( \\nabla _ x X ^ { - 1 } ( s ) - \\nabla _ x X ^ { - 1 } ( t ) \\right ) : \\left ( \\nabla _ a X ' \\circ X ^ { - 1 } \\right ) ( t ) , \\end{gathered} \\end{align*}"}
-{"id": "2139.png", "formula": "\\begin{align*} u _ t - \\Delta u + \\nabla \\cdot ( u \\nabla v ) & = 0 , \\ \\ & x \\in { \\mathbb R } ^ d , \\ t > 0 , \\\\ \\Delta v + u & = 0 , \\ \\ & x \\in { \\mathbb R } ^ d , \\ t > 0 , \\\\ u ( x , 0 ) & = u _ 0 ( x ) \\ge 0 , \\ \\ & x \\in { \\mathbb R } ^ d . \\end{align*}"}
-{"id": "555.png", "formula": "\\begin{align*} 2 Q _ 0 ( u , v ) = \\Box ( u v ) - \\Box u v - u \\Box v . \\end{align*}"}
-{"id": "4796.png", "formula": "\\begin{align*} \\begin{matrix} - \\tfrac { \\nu } { | x | } ( 1 + \\sqrt { 1 - \\nu ^ 2 } ) \\leq \\lambda _ j ^ - ( x ) \\leq 0 \\leq \\lambda _ j ^ + ( x ) \\leq \\tfrac { \\nu } { | x | } ( 1 + \\sqrt { 1 - \\nu ^ 2 } ) , \\end{matrix} \\quad \\ j = 1 , 2 . \\end{align*}"}
-{"id": "8197.png", "formula": "\\begin{align*} X ^ { k l } = | \\Gamma ^ { l } _ { \\tau ^ { p } _ { k } } - \\Gamma ^ { l } _ { \\tau ^ { p } _ { k - 1 } } | . \\end{align*}"}
-{"id": "422.png", "formula": "\\begin{align*} K ( x , y ) : = \\int _ { 0 } ^ { \\infty } Q _ u ( x , y ) \\dd u , \\ \\end{align*}"}
-{"id": "6808.png", "formula": "\\begin{align*} p = - \\tfrac { 1 } { 2 } \\ , \\langle v , A ^ T A '' v \\rangle \\ , . \\end{align*}"}
-{"id": "1762.png", "formula": "\\begin{align*} g ' ( x ) = H \\left [ \\left ( ( x + 1 ) ^ { 2 H - 1 } - x ^ { 2 H - 1 } \\right ) - \\left ( x ^ { 2 H - 1 } - ( x - 1 ) ^ { 2 H - 1 } \\right ) \\right ] . \\end{align*}"}
-{"id": "7038.png", "formula": "\\begin{align*} 1 + \\frac { z \\mathbb { F } _ { \\alpha _ { j } , \\beta _ { j } , \\lambda _ { j } } ^ { \\prime \\prime } ( z ) } { \\mathbb { F } _ { \\alpha _ { j } , \\beta _ { j } , \\lambda _ { j } } ^ { \\prime } ( z ) } = \\sum _ { j = 1 } ^ { n } \\frac { 1 } { \\lambda _ { j } } \\left ( \\frac { z \\mathbb { E } _ { \\alpha _ { j } , \\beta _ { j } } ^ { \\prime } ( z ) } { \\mathbb { E } _ { \\alpha _ { j } , \\beta _ { j } } ( z ) } \\right ) + 1 - \\sum _ { j = 1 } ^ { n } \\frac { 1 } { \\lambda _ { j } } . \\end{align*}"}
-{"id": "6570.png", "formula": "\\begin{gather*} D _ a = \\partial _ { x ^ a } + \\sum _ I p _ { I a } \\partial _ { p _ I } . \\end{gather*}"}
-{"id": "763.png", "formula": "\\begin{align*} M = \\sum a _ i T ^ i \\in \\mathbb { F } _ { q ^ d } [ T ] , C _ { q ^ d } ( M ) ( u ) = M { * } _ d u = \\sum a _ i C _ { q ^ d } ^ i ( T ) ( u ) . \\end{align*}"}
-{"id": "4988.png", "formula": "\\begin{align*} \\mathcal { F } _ m : u ^ m + v ^ m + 1 & = 0 \\\\ H _ { n , l } : x ^ n y ^ \\ell + y ^ n + x ^ \\ell & = 0 . \\\\ \\end{align*}"}
-{"id": "9820.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } U & \\ge & 0 & \\R ^ { n - 1 } , \\\\ ( - \\Delta ) ^ { - \\frac { a } { 2 } } U & = & - C \\mathcal { F } _ a ( \\hat { w } ) & \\{ x ' : U ( x ' ) > 0 \\} , \\\\ ( - \\Delta ) ^ { - \\frac { a } { 2 } } U & \\le & - C \\mathcal { F } _ a ( \\hat { w } ) & \\R ^ { n - 1 } \\\\ \\lim _ { | x ' | \\to \\infty } U ( x ' ) & = & 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "499.png", "formula": "\\begin{align*} Q : = x _ 1 t _ 1 \\odot x _ 2 t _ 2 - x _ 1 t _ 2 \\odot x _ 2 t _ 1 \\in I _ 2 ( K _ C ) = I _ 2 ( K _ C ) ^ + . \\end{align*}"}
-{"id": "2255.png", "formula": "\\begin{align*} D _ l & = \\mathfrak { H } R ( z ) = \\bigg \\{ \\xi \\in \\mathbb { C } \\oplus \\mathcal { L } : \\xi = c ( 1 , ( E | R ( z ) ) + ( 0 , ( f | ) \\bigg \\} \\\\ D _ r & = R ( z ) \\mathfrak { H } = \\bigg \\{ \\xi \\in \\mathbb { C } \\oplus \\mathcal { L } : \\xi = c \\binom { 1 } { - R ( z ) | E ) } + \\binom { 0 } { R ( z ) f } \\bigg \\} \\end{align*}"}
-{"id": "7246.png", "formula": "\\begin{align*} D _ { \\phi } F ( \\phi , \\mu ) = ( \\mu - \\phi ) { \\rm i d } - L _ r , \\end{align*}"}
-{"id": "948.png", "formula": "\\begin{align*} I _ { T , t } ^ { * ( i _ 1 \\ldots i _ 6 ) } = { \\int \\limits _ t ^ { * } } ^ T \\ldots { \\int \\limits _ t ^ { * } } ^ { t _ 2 } d { \\bf w } _ { t _ 1 } ^ { ( i _ 1 ) } \\ldots d { \\bf w } _ { t _ 6 } ^ { ( i _ 6 ) } \\end{align*}"}
-{"id": "4417.png", "formula": "\\begin{align*} & \\partial _ t u = \\nabla \\cdot ( \\nabla u + u _ { \\alpha } \\otimes \\nabla \\mathcal { V } ) , ( t , x ) \\in ( 0 , T ) \\times \\Omega , \\\\ & u ( 0 ) = u _ 0 , \\end{align*}"}
-{"id": "745.png", "formula": "\\begin{align*} \\mathbb { E } _ { f } \\ ! \\left [ S ^ { 2 } \\right ] = \\lim _ { q \\to 1 } \\mathbb { E } _ { f } \\ ! \\left [ T ^ { 2 } \\right ] = \\frac { \\Gamma ( m n ) } { 2 } \\left ( - \\frac { 2 } { \\Gamma ( m n + q ) } \\mathbb { E } _ { g } \\ ! \\left [ L \\right ] + \\frac { 1 } { \\Gamma ( m n + 2 q ) } \\mathbb { E } _ { g } \\ ! \\left [ L ^ { 2 } \\right ] \\right ) ^ { \\prime \\prime } \\bigg | _ { q = 1 } , \\end{align*}"}
-{"id": "2466.png", "formula": "\\begin{align*} n = { \\displaystyle \\sum \\limits _ { \\pi ( P ) = Q } } e _ { \\pi } ( P ) \\end{align*}"}
-{"id": "654.png", "formula": "\\begin{align*} \\mathcal M ( ( x _ \\ast \\vec a ) ^ \\varphi _ j ; b _ j ) = \\mathcal M ( ( x ' _ \\ast \\vec a ) ^ { \\varphi ' } _ j ; b _ j ) \\end{align*}"}
-{"id": "8451.png", "formula": "\\begin{align*} G _ d : = - 2 [ \\bar { q } , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } \\bar { \\mathfrak { F } } } { \\zeta _ { \\alpha } } - 2 [ \\bar { \\mathfrak { F } } , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } \\bar { q } } { \\zeta _ { \\alpha } } - 2 [ \\bar { q } , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } \\bar { q } } { \\zeta _ { \\alpha } } - 4 D _ t q , \\end{align*}"}
-{"id": "6084.png", "formula": "\\begin{align*} \\begin{gathered} \\iota _ n ' \\bigg \\vert _ { [ \\P _ n ] _ m / \\mathfrak { S } _ n } : { [ \\P _ n ] _ { m } / \\mathfrak { S } _ n } \\longrightarrow { [ \\P _ { n + 1 } ] _ { m } / \\mathfrak { S } _ { n + 1 } } \\end{gathered} \\end{align*}"}
-{"id": "4195.png", "formula": "\\begin{align*} E _ { i n } ^ { ( 4 ) } ( z ) & = \\left ( \\mathbb { I } - \\frac { A _ n ^ { ( 3 ) } ( z ) - A _ n ^ { ( 3 ) } ( 0 ) } { n ^ 9 z } \\right ) E _ { i n } ^ { ( 3 ) } ( z ) \\\\ E _ { o u t } ^ { ( 4 ) } ( z ) & = \\left ( \\mathbb { I } - \\frac { A _ n ^ { ( 3 ) } ( 0 ) } { n ^ 9 z } \\right ) ^ { - 1 } E _ { o u t } ^ { ( 3 ) } ( z ) \\end{align*}"}
-{"id": "3898.png", "formula": "\\begin{align*} T _ { s , \\zeta } = \\begin{pmatrix} 1 & 0 & 0 \\\\ s & 1 & { \\zeta } ^ { \\ast } \\\\ \\zeta & 0 & I \\end{pmatrix} , \\end{align*}"}
-{"id": "9813.png", "formula": "\\begin{align*} \\bar { h } _ \\beta ( x ' , x _ n , y ) : = \\int _ { \\R ^ { n - 1 } } h _ \\beta ( z ' ) P _ a ( x ' - z ' , x _ n , y ) . \\end{align*}"}
-{"id": "5592.png", "formula": "\\begin{align*} \\varphi ^ + ( t ) : = \\| v ( \\cdot , t ) \\| ^ 2 _ { L ^ 2 ( \\R ^ + ) } + \\| u _ x ( \\cdot , t ) \\| ^ 2 _ { L ^ 2 ( \\R ^ + ) } + \\| v _ x ( \\cdot , t ) \\| ^ 2 _ { L ^ 2 ( \\R ^ + ) } , \\end{align*}"}
-{"id": "690.png", "formula": "\\begin{align*} \\gamma ( f ; f _ 1 , \\dots , f _ n ) = f \\circ \\varpi ( f _ 1 , \\dots , f _ n ) \\circ \\theta ^ { - 1 } \\ . \\end{align*}"}
-{"id": "10026.png", "formula": "\\begin{align*} \\theta ( x v , x / v , y w , y / w ) - \\theta ( x w , x / w , y v , y / v ) = \\frac { y } { v } \\theta ( x y , x / y , v w , v / w ) . \\end{align*}"}
-{"id": "1086.png", "formula": "\\begin{align*} & n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } \\int _ { I _ 1 } D _ n ( \\sqrt { 4 - y ^ 2 } / S ( I ) \\cdot G _ n ( x ) / 2 ) d y = n ( 2 \\ln n ) ^ { - \\frac { 1 } { 2 } } D _ n ( G _ n ( x ) / 2 ) \\times \\\\ & 2 S ( I ) \\int _ { 0 } ^ { ( b _ 1 / a _ 1 - 1 ) \\ln n } \\frac { D _ n ( ( 1 + z / \\ln n ) G _ n ( x ) / 2 ) } { D _ n ( G _ n ( x ) / 2 ) } \\frac { ( 1 + z / \\ln n ) a _ 1 } { \\sqrt { 4 - ( 1 + z / \\ln n ) ^ 2 a _ 1 ^ 2 } } d z . \\end{align*}"}
-{"id": "987.png", "formula": "\\begin{align*} I ( p ) = \\langle a _ { d } b _ { d - 1 } , a _ { d } b _ { d - 2 } , a _ { d - 1 } b _ { d - 2 } , \\dots , a _ { d } b _ { j } , \\dots , a _ { j + 2 } b _ { j } \\rangle : \\langle a _ { j + 1 } b _ j \\rangle = \\langle a _ { j + 2 } , \\dots , a _ d \\rangle . \\end{align*}"}
-{"id": "403.png", "formula": "\\begin{align*} \\mathcal { S } _ { \\rm M } ( \\varphi , g ) = \\int _ M \\Big ( 2 \\pi ( 1 - \\textbf { h } ) \\phi \\Delta _ g \\phi + ( \\frac { 8 \\pi ( 1 - \\textbf { h } ) } { V _ { g } } - K _ { g } ) \\phi + \\frac { 2 } { 1 - \\frac { \\gamma ^ 2 } { 4 } } \\frac { 1 } { V _ { \\hat g } } ( \\gamma \\varphi ) e ^ { \\gamma \\varphi } \\Big ) { \\rm d v } _ { g } . \\end{align*}"}
-{"id": "3541.png", "formula": "\\begin{align*} \\begin{aligned} & \\rho _ { 1 } = ( 1 , 1 , 0 , 0 ) , & & \\rho _ { 2 } = ( 0 , 1 , 1 , 0 ) , & & \\rho _ { 3 } = ( 0 , 0 , 1 , 1 ) , \\\\ & \\rho _ { 4 } = ( 1 , 1 , 1 , 0 ) , & & \\rho _ { 5 } = ( 0 , 1 , 1 , 1 ) . \\end{aligned} \\end{align*}"}
-{"id": "3521.png", "formula": "\\begin{align*} ( a _ { i j } ( g \\cdot u ) ) = \\frac { B _ 0 ^ 0 k _ \\varnothing } { b ^ 2 } ( \\mathring { B } ^ { - 1 } ) _ i ^ k ( a _ { k l } ( u ) ) ( \\mathring { B } ^ { - 1 } ) _ j ^ l - D ^ 0 _ { i j } \\end{align*}"}
-{"id": "693.png", "formula": "\\begin{align*} h = h ' \\circ \\widehat { [ \\rho , x ] } _ { X _ 1 \\dots X _ m } \\end{align*}"}
-{"id": "9642.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\frac { ( q ^ 2 ; q ^ 2 ) _ N } { ( q ^ 2 ; q ^ 2 ) _ { N - n } } \\frac { ( - 1 ) ^ { n - 1 } q ^ { n ^ 2 } } { ( 1 - q ^ { 2 n } ) ( q ; q ^ 2 ) _ n } = \\sum _ { m = 1 } ^ { \\infty } \\left ( \\sum _ { n = 1 } ^ { N } \\sum _ { \\substack { \\vec { \\pi } \\in S _ 3 \\\\ | \\vec { \\pi } | = m } } w ( \\vec { \\pi } ) \\right ) q ^ m . \\end{align*}"}
-{"id": "7085.png", "formula": "\\begin{align*} \\begin{cases} - \\partial _ { t t } u + \\Delta u = | \\nabla u | ^ 2 , \\qquad ~ ( t , x ) \\in \\mathbb { R } ^ { 1 + d } \\\\ u | _ { t = 0 } = f _ 0 , ~ \\partial _ t u | _ { t = 0 } = f _ 1 \\end{cases} ~ , \\end{align*}"}
-{"id": "9422.png", "formula": "\\begin{align*} \\frac { \\theta ( \\Delta ) } { \\Delta } = E _ 2 . \\end{align*}"}
-{"id": "3638.png", "formula": "\\begin{align*} ( \\alpha _ { - 1 } - \\beta _ { - 1 } ) \\cdot \\alpha _ { - 1 } = - \\frac { ( 2 t - 1 ) t } { 2 ^ 4 } ( a _ 1 - a _ { - 1 } ) - \\frac { t - 1 } { 2 ^ 4 } a _ 1 \\cdot v _ { ( 2 , 3 ) } + \\frac { 3 t - 2 } { 2 ^ 4 } a _ { - 1 } \\cdot v _ { ( 2 , 3 ) } , \\end{align*}"}
-{"id": "1300.png", "formula": "\\begin{align*} h = g _ 1 x _ 1 ^ { \\varepsilon _ 1 } y _ 1 ^ { - \\varepsilon _ 1 } g _ 1 ^ { - 1 } g _ 2 x _ 2 ^ { \\varepsilon _ 2 } y _ 2 ^ { - \\varepsilon _ 2 } g _ 2 ^ { - 1 } \\cdots g _ n x _ n ^ { \\varepsilon _ n } y _ n ^ { - \\varepsilon _ n } g _ n ^ { - 1 } \\end{align*}"}
-{"id": "6304.png", "formula": "\\begin{align*} \\mathbb { P } _ { G _ 1 \\sim G ( n , m ) } \\left [ \\mathcal { H } ^ c _ { \\rho , C } \\right ] \\mathbb { P } _ { G \\sim G _ { n , p } } \\left [ E ( G ) = m \\right ] \\leq O \\left ( \\frac { 1 } { n ^ { \\frac { C } { 1 6 } - 2 } } \\right ) . \\end{align*}"}
-{"id": "5850.png", "formula": "\\begin{align*} { \\cal F } _ D f ( y ) = \\int _ { F ^ n } f ( x ) \\psi \\left ( x ^ T D y \\right ) d _ D x \\end{align*}"}
-{"id": "941.png", "formula": "\\begin{align*} { \\sf M } \\left \\{ \\left ( J ^ { * } [ \\psi ^ { ( 3 ) } ] _ { T , t } - \\sum \\limits _ { j _ 1 , j _ 2 , j _ 3 = 0 } ^ { p } C _ { j _ 3 j _ 2 j _ 1 } \\zeta _ { j _ 1 } ^ { ( i _ 1 ) } \\zeta _ { j _ 2 } ^ { ( i _ 2 ) } \\zeta _ { j _ 3 } ^ { ( i _ 3 ) } \\right ) ^ 2 \\right \\} \\le \\frac { C } { p } \\end{align*}"}
-{"id": "9017.png", "formula": "\\begin{align*} f _ { \\{ 1 ^ * \\} , 1 } ( x ) = \\begin{cases} 1 & 1 ^ * = x , \\\\ x & \\end{cases} \\end{align*}"}
-{"id": "6914.png", "formula": "\\begin{align*} I _ 2 = I _ 1 + \\langle a _ { 2 1 } '' a _ { 2 2 } - a _ { 2 2 } '' a _ { 2 1 } + a _ { 1 1 } '' a _ { 1 2 } - a _ { 1 2 } '' a _ { 1 1 } \\rangle \\end{align*}"}
-{"id": "6493.png", "formula": "\\begin{align*} & \\int _ { \\Q _ { \\ell } ^ { \\times } } | y | ^ { s - 1 } \\gamma ^ { - 1 } ( y ) ( f _ 1 ( y ) \\chi _ 1 ( y ) | y | ^ { 1 / 2 } + g _ 1 ( y ) \\chi _ 1 ( y ) v ( y ) | y | ^ { 1 / 2 } ) ( f _ 2 ( y ) \\gamma \\chi _ 1 ^ { - 1 } ( y ) | y | ^ { 1 / 2 } + g _ 2 ( y ) \\gamma \\chi _ 1 ^ { - 1 } ( y ) v ( y ) | y | ^ { 1 / 2 } ) d ^ { \\times } y \\\\ & = P _ 1 ( s ) L ( \\mathbf { 1 } , s ) + P _ 2 ( s ) L ( \\mathbf { 1 } , s ) ^ 2 + P _ 3 ( s ) L ( \\mathbf { 1 } , s ) ^ 3 . \\end{align*}"}
-{"id": "8063.png", "formula": "\\begin{align*} \\psi _ c ^ + ( \\omega _ { i } ) = - c ^ { n + 3 - 2 i } \\omega _ { n + 3 - i } , \\mbox { w h e r e } n + 3 \\leq i \\leq 2 n . \\end{align*}"}
-{"id": "2354.png", "formula": "\\begin{align*} \\mathfrak { p } & = \\{ \\mathfrak { p } _ + , \\mathfrak { p } _ - , \\{ q _ i , r _ i \\} _ { i \\in I } \\} \\\\ [ 1 , n ] & = \\mathfrak { p } _ + + \\mathfrak { p } _ - + \\sum _ { i \\in I } \\{ q _ i , r _ i \\} \\\\ \\mathfrak { p } _ + & \\subset \\ ; \\omega _ + , \\ ; \\mathfrak { p } _ - \\subset \\omega _ - , \\ ; q _ i \\in \\omega _ - , \\ ; r _ i \\in \\omega _ + , \\ ; q _ i > r _ i \\end{align*}"}
-{"id": "8508.png", "formula": "\\begin{align*} | I | = \\Big | \\frac { \\lambda i ( \\dot { z } _ 1 ) ^ 2 } { 2 \\pi } 2 x ( t ) \\Big \\{ & \\frac { 1 } { ( \\zeta ( \\alpha , t ) - z _ 1 ( t ) ) ^ 3 ( \\zeta ( \\alpha , t ) - z _ 2 ( t ) ) } + \\frac { 1 } { ( \\zeta ( \\alpha , t ) - z _ 1 ( t ) ) ^ 2 ( \\zeta ( \\alpha , t ) - z _ 2 ( t ) ^ 2 ) } \\\\ & + \\frac { 1 } { ( \\zeta ( \\alpha , t ) - z _ 1 ( t ) ) ( \\zeta ( \\alpha , t ) - z _ 2 ( t ) ) ^ 3 } \\Big \\} \\Big | , \\end{align*}"}
-{"id": "1223.png", "formula": "\\begin{gather*} K ( f , g ) ( \\eta ) = \\int _ \\nu e ^ { \\frac { 3 i } 4 \\eta \\nu ^ 2 } f \\left ( \\frac { \\eta + \\nu } { 2 } \\right ) g \\left ( \\frac { \\eta - \\nu } { 2 } \\right ) d \\nu , \\end{gather*}"}
-{"id": "4284.png", "formula": "\\begin{align*} f _ { h _ 1 , h _ 2 } ( H ) \\left ( \\binom { n } { 2 } - e ( G ) \\right ) \\geq f _ { h _ 1 , h _ 2 } ( H ) \\left ( \\binom { n } { 2 } - \\textup { e x } ( n , K _ s ) \\right ) . \\end{align*}"}
-{"id": "789.png", "formula": "\\begin{align*} \\sigma ( L ) = L ^ { \\mu } a ^ { q ^ { { s } } - 1 } , \\end{align*}"}
-{"id": "9702.png", "formula": "\\begin{align*} T _ { \\textbf { d } } ( \\gamma ) = \\frac { 1 } { 2 \\pi } \\int _ { \\gamma } ( \\kappa _ { \\textbf { d } } - \\kappa _ { \\gamma } ) d s . \\end{align*}"}
-{"id": "1931.png", "formula": "\\begin{align*} F _ 2 ( x ) = \\frac { 1 } { 2 } x + c x ^ 2 , \\end{align*}"}
-{"id": "9242.png", "formula": "\\begin{align*} A ( x , z , p ) = \\frac { 1 } { 2 } a _ { k l } ( x , z ) p _ k p _ l I - a _ 0 ( x , z ) p \\otimes p , \\end{align*}"}
-{"id": "6198.png", "formula": "\\begin{align*} \\tilde \\Omega = \\bigcup _ { i = 1 } ^ k \\mathcal N ( u _ i ) , \\mathcal N ^ o ( u _ i ) \\cap \\mathcal N ^ o ( \\sum _ { j \\neq i } u _ j ) = \\emptyset \\ ; \\forall i \\ ; , \\end{align*}"}
-{"id": "2207.png", "formula": "\\begin{align*} P A + A ^ T P + Q - \\frac { 1 } { 2 } P B R ^ { - 1 } B ^ { T } P - \\frac { 1 } { 4 } P B R ^ { - 1 } B ^ { T } P ^ T - \\frac { 1 } { 4 } P ^ T B R ^ { - 1 } B ^ { T } P \\ = 0 \\end{align*}"}
-{"id": "4790.png", "formula": "\\begin{align*} H _ 0 : = - i \\alpha \\cdot \\nabla + m \\beta = - i \\sum _ { j = 1 } ^ 3 \\alpha _ j \\partial _ j + m \\beta , \\end{align*}"}
-{"id": "3197.png", "formula": "\\begin{align*} { \\rm i n d } _ \\varGamma ( U ) = 4 - 2 N . \\end{align*}"}
-{"id": "6913.png", "formula": "\\begin{align*} \\mathsf { N F } ( f _ 2 , I _ 1 ) = \\mathsf { N F } ( f _ 3 , I _ 1 ) = \\mathsf { N F } ( f _ 6 , I _ 1 ) = a _ { 2 1 } '' a _ { 2 2 } - a _ { 2 2 } '' a _ { 2 1 } + a _ { 1 1 } '' a _ { 1 2 } - a _ { 1 2 } '' a _ { 1 1 } \\ , . \\end{align*}"}
-{"id": "6003.png", "formula": "\\begin{align*} \\widetilde { F } ^ n ( \\omega _ 1 , \\ldots , \\omega _ n ) = F ^ n ( \\omega _ 1 ( 1 ) , \\ldots , \\omega _ n ( 1 ) ) = n \\ , F \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\delta _ { \\omega _ i ( 1 ) } \\right ) , \\end{align*}"}
-{"id": "9366.png", "formula": "\\begin{align*} [ x ^ { 2 ^ { i - 1 } } , y ] \\equiv [ x , y ] ^ { 2 ^ { i - 1 } } [ x , y , x ] ^ { \\binom { 2 ^ { i - 1 } } { 2 } } \\ , \\cdots \\ , [ x , y , x , \\overset { i - 2 } \\ldots , x ] ^ { \\binom { 2 ^ { i - 1 } } { i - 1 } } \\gamma _ { i + 1 } ( G _ k ) . \\end{align*}"}
-{"id": "5032.png", "formula": "\\begin{align*} \\d X ( t ) = b ( X ( t ) ) \\d t + \\sigma ( X ( t ) ) \\d B _ t ^ H , \\ \\ X ( 0 ) = x , \\ \\ t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "9879.png", "formula": "\\begin{align*} \\left \\| f ( \\cdot ) - \\int _ { \\mathcal { Y } ^ { N } } g ( y _ { [ 1 , N ] } ) P ( d y _ { [ 1 , N ] } | X _ { 1 } = \\cdot ) \\right \\| _ { \\infty } < \\epsilon , \\end{align*}"}
-{"id": "8113.png", "formula": "\\begin{align*} \\widetilde { A } _ n ( t ) \\ = \\ \\sum _ { m = 0 } ^ n { n \\choose m } t ^ { n - m } A _ m ( t ) , \\end{align*}"}
-{"id": "4806.png", "formula": "\\begin{align*} u _ i = \\frac { \\sigma _ i } { \\sigma _ 1 ^ 2 + \\ldots + \\sigma _ d ^ 2 } b . \\end{align*}"}
-{"id": "4436.png", "formula": "\\begin{align*} u ( 0 ) = u _ 0 . \\end{align*}"}
-{"id": "5636.png", "formula": "\\begin{align*} 2 = & q _ 1 + q _ 2 + q _ 3 \\\\ > & \\frac { 1 6 } { 2 7 } \\frac { 1 } { A + \\sqrt { A ^ 2 + \\frac { 3 2 } { 2 7 } ( a _ 2 + a _ 3 ) } } + \\frac { 1 } { B + \\sqrt { B ^ 2 + \\frac { 3 2 } { 2 7 } ( a _ 1 + a _ 3 ) } } + \\frac { 1 } { C + \\sqrt { C ^ 2 + \\frac { 3 2 } { 2 7 } ( a _ 1 + a _ 2 ) } } \\\\ \\ge & \\frac { 1 6 } { 3 } \\frac { 1 } { A + B + C + \\sqrt { A ^ 2 + \\frac { 3 2 } { 2 7 } ( a _ 2 + a _ 3 ) } + \\sqrt { B ^ 2 + \\frac { 3 2 } { 2 7 } ( a _ 1 + a _ 3 ) } + \\sqrt { C ^ 2 + \\frac { 3 2 } { 2 7 } ( a _ 1 + a _ 2 ) } } \\end{align*}"}
-{"id": "4464.png", "formula": "\\begin{align*} \\gamma ^ { i j } = u \\sigma _ { i j } - u \\frac { \\nabla _ i u \\nabla _ j u } { w ( u + w ) } \\end{align*}"}
-{"id": "1989.png", "formula": "\\begin{align*} ( i _ * ) ^ { - 1 } = \\frac { i ^ * } { e ( N _ { F / X } ) } . \\end{align*}"}
-{"id": "1352.png", "formula": "\\begin{align*} a \\cdot ( n \\lambda ) = \\begin{cases} n ( a \\cdot \\lambda ) \\in \\mathcal { M } ( s ( \\lambda ) ) \\ , , & v = r ( \\lambda ) \\\\ 0 \\ , , & v \\not = r ( \\lambda ) \\end{cases} \\ ; . \\end{align*}"}
-{"id": "6942.png", "formula": "\\begin{align*} \\| f \\| _ { B M O } = \\| f \\| _ { L ^ 2 } + [ f ] _ { B M O } \\end{align*}"}
-{"id": "4976.png", "formula": "\\begin{align*} F = F i x ( U _ { m + \\ell } \\cdots U _ { m + 1 } ) m \\in \\N . \\end{align*}"}
-{"id": "7539.png", "formula": "\\begin{align*} \\left \\{ \\begin{gathered} \\mathcal { V } ( X , \\tau , a , s ) = \\mathbb { L } _ \\nu ( u _ 0 ) \\circ X ( a , s ) + ( \\mathbb { U } \\left ( \\left ( \\tau - v \\otimes v ) \\circ X ^ { - 1 } \\right ) \\right ) \\circ X ( a , s ) , \\\\ \\mathcal { T } ( X , \\tau , a , s ) = \\left ( g \\tau + \\tau g ^ T - 2 k \\tau + 2 \\rho K ( g + g ^ T ) \\right ) ( a , s ) , \\end{gathered} \\right . \\end{align*}"}
-{"id": "1273.png", "formula": "\\begin{align*} T _ { 5 , 1 } & = e ^ { i \\pi / 4 } \\sqrt { \\frac { 4 \\pi } { 3 } } A | A | ^ 2 \\frac { e ^ { i a \\ln | \\xi | } } { | \\xi | } \\int _ { \\nu > 0 } e ^ { - 3 i \\nu } \\frac { 1 } { \\sqrt { | \\nu | } } d \\nu + O ( | \\xi | ^ { - 2 \\gamma } ) + O ( | \\xi | ^ { 1 - 3 \\gamma } ) + O ( | \\xi | ^ { - 2 + \\gamma / 2 } ) \\\\ & = \\frac { 2 \\pi } { 3 } A | A | ^ 2 \\frac { e ^ { i a \\ln | \\xi | } } { | \\xi | } + O ( | \\xi | ^ { - 2 + \\gamma / 2 } ) \\end{align*}"}
-{"id": "6451.png", "formula": "\\begin{align*} ( A + \\widehat \\Delta _ A ) x _ k = b + \\widehat \\Delta _ b , \\widehat \\Delta _ A \\equiv L ( \\Delta \\widehat A ) L ^ { T } , \\widehat \\Delta _ b \\equiv L ( \\Delta \\widehat b ) \\ , . \\end{align*}"}
-{"id": "3103.png", "formula": "\\begin{align*} \\partial _ { \\bar q } \\mathcal F ^ Z _ g ( q , \\bar q ) = \\frac { 1 } { 2 } C ^ { q q } _ { \\bar q } \\Big ( D _ { q } ^ 2 \\mathcal F ^ Z _ { g - 1 } ( q , \\bar q ) + \\sum _ { g _ 1 + g _ 2 = g } D _ { q } \\mathcal F ^ Z _ { g _ 1 } ( q , \\bar q ) D _ { q } \\mathcal F ^ Z _ { g _ 2 } ( q , \\bar q ) \\Big ) , \\end{align*}"}
-{"id": "9308.png", "formula": "\\begin{align*} \\min \\sum _ { i = 1 } ^ m \\sum _ { j = 1 } ^ n c _ { i j } x _ { i j } \\end{align*}"}
-{"id": "2321.png", "formula": "\\begin{align*} \\sup _ { | z | = r } \\left | z \\langle \\varphi | R ( z ) | \\psi \\rangle - \\langle \\varphi | \\psi \\rangle \\right | \\longrightarrow 0 \\mbox { f o r } r \\rightarrow \\infty . \\end{align*}"}
-{"id": "6581.png", "formula": "\\begin{gather*} A _ i = - D _ i A + A a _ i . \\end{gather*}"}
-{"id": "3496.png", "formula": "\\begin{align*} \\left . \\frac { d ^ { 2 } } { d y ^ { 2 } } \\Re { f _ { M } ( \\Re { q _ { M } ( \\theta ) } + i y ; \\theta ) } \\right | _ { y = y _ { 0 } } = 0 . \\end{align*}"}
-{"id": "1297.png", "formula": "\\begin{align*} R ( \\xi _ 0 ) = - 2 | y / 3 | ^ { 3 / 2 } , R '' ( \\xi _ 0 ) = 2 \\sqrt { 3 | y | } . \\end{align*}"}
-{"id": "4896.png", "formula": "\\begin{align*} \\| f _ { U _ n } ( a _ n ) - f _ { U _ m } ( a _ m ) \\| = \\| f _ { U _ m } ( a _ n - a _ m ) \\| \\leq \\| a _ n - a _ m \\| \\end{align*}"}
-{"id": "4028.png", "formula": "\\begin{align*} J \\circ \\pi _ { \\ast } = \\pi _ { \\ast } \\circ \\phi , J ' \\circ \\pi ' _ { \\ast } = \\pi ' _ { \\ast } \\circ \\phi ' . \\end{align*}"}
-{"id": "4280.png", "formula": "\\begin{align*} \\textup { s a t } ( n , C _ 4 , C _ k ) = 0 . \\end{align*}"}
-{"id": "3206.png", "formula": "\\begin{align*} 0 = \\lim _ { r \\to \\infty } \\frac { \\sqrt { \\sum _ { i = p + 1 } ^ { m } \\left ( e ^ r _ i \\tilde { y } ^ r _ { i i } - \\mu _ r \\right ) ^ 2 } } { \\mu _ r } = \\lim _ { r \\to \\infty } \\sqrt { \\sum _ { i = p + 1 } ^ { m } \\left ( \\frac { e ^ r _ i } { \\mu _ r } \\tilde { y } ^ r _ { i i } - 1 \\right ) ^ 2 } , \\end{align*}"}
-{"id": "1708.png", "formula": "\\begin{align*} | ( T _ l ( s + h ) & f ) ( x ) - ( T _ l ( s ) f ) ( x ) | = | f ( s + h + x ) - f ( s + x ) | \\\\ & \\leq \\sum _ { n = 1 } ^ { | N | } | f ( n \\cdot \\delta + r ) - f ( ( n - 1 ) \\cdot \\delta + r ) | + | f ( r ) - f ( 0 ) | + | f ( 0 ) | \\end{align*}"}
-{"id": "602.png", "formula": "\\begin{align*} \\mathcal { D } _ { R } & : = \\big \\{ \\overline { k } = \\big ( k _ 1 m ^ { - R } , \\cdots , k _ d m ^ { - R } \\big ) \\ , \\big | \\ , k _ i = 0 , \\ldots , m ^ { R } - 1 , \\ i = 1 , \\ldots , d \\big \\} , \\\\ \\\\ \\Omega _ n & : = \\bigg \\{ \\big ( \\overline { k } + [ 0 , m ^ { - j } ) ^ d \\big ) \\cap [ 0 , 1 ) ^ d \\ , \\bigg | \\ , j = 0 , \\ldots , J - 1 , \\ , \\overline { k } \\in \\mathcal { D } _ { R } \\bigg \\} ; \\end{align*}"}
-{"id": "105.png", "formula": "\\begin{align*} \\begin{pmatrix} \\alpha + \\frac k 2 \\\\ \\frac k 2 \\end{pmatrix} \\begin{pmatrix} 2 N - \\alpha - 1 - \\frac k 2 \\\\ \\frac k 2 - 1 \\end{pmatrix} \\ , , \\alpha = 0 , \\ldots , 2 N - k \\ , . \\end{align*}"}
-{"id": "3020.png", "formula": "\\begin{align*} D _ { p , q } f ( z ) = 1 + \\sum _ { { k = 2 } } ^ { \\infty } [ k ] _ { p , q } a _ { k } z ^ { k - 1 } . \\end{align*}"}
-{"id": "7989.png", "formula": "\\begin{align*} \\sum _ { p \\leq n } \\log p = n + O \\left ( \\frac { n } { \\log n } \\right ) \\ , , \\end{align*}"}
-{"id": "6083.png", "formula": "\\begin{gather*} l ( \\lambda ) : = ( \\overline { S _ { \\lambda } } ) = \\sum ( \\lvert R ( \\lambda ) _ i \\rvert + 1 ) - \\sum \\lvert F ( \\lambda ) _ j \\rvert - \\lvert F ( \\lambda ) \\rvert . \\end{gather*}"}
-{"id": "3247.png", "formula": "\\begin{align*} & [ \\ ! [ T _ { q , k q + p } \\# - T _ { p , q } \\# - k T _ { q , q + 1 } ] \\ ! ] \\\\ = & [ \\ ! [ T _ { q , k q + p } ] \\ ! ] - [ \\ ! [ T _ { p , q } ] \\ ! ] - k [ \\ ! [ T _ { q , q + 1 } ] \\ ! ] \\\\ = & ( k [ 1 , q - 1 , q - 1 , 1 ] + O _ 1 ) - ( \\lfloor q / p \\rfloor [ 1 , p - 1 , p - 1 , 1 ] + O _ 2 ) - k ( [ 1 , q - 1 , q - 1 , 1 ] + O _ 3 ) \\\\ = & O _ 1 - \\lfloor q / p \\rfloor [ 1 , p - 1 , p - 1 , 1 ] - O _ 2 - k O _ 3 . \\end{align*}"}
-{"id": "28.png", "formula": "\\begin{align*} U ( X _ { t } ( { \\pi ^ { \\ast } } ) , t ) = \\mathbb { E } \\left [ U ( X _ { s } ( { \\pi ^ { \\ast } } ) , s ) | \\mathcal { G } _ { t } \\right ] , \\end{align*}"}
-{"id": "8324.png", "formula": "\\begin{align*} [ D _ t ^ 2 , \\partial _ { \\alpha } ] = - D _ t ( b _ { \\alpha } ) \\partial _ { \\alpha } - b _ { \\alpha } D _ t \\partial _ { \\alpha } - b _ { \\alpha } \\partial _ { \\alpha } D _ t \\end{align*}"}
-{"id": "5654.png", "formula": "\\begin{align*} U ^ { n } = U _ h ^ { n } + \\tau \\sum _ { k = 1 } ^ { n } B _ { n - ( k - 1 ) } f ^ { k } , \\mbox { w i t h } \\ ; n = 1 , 2 , \\ldots , \\end{align*}"}
-{"id": "2681.png", "formula": "\\begin{align*} z _ { \\pm } = \\frac { 1 } { 2 } \\Big ( \\frac { 1 } { \\tau } + \\tau \\pm i \\big ( \\frac { 1 } { \\tau } - \\tau \\big ) \\sqrt { \\frac { 4 } { x } - 1 } \\Big ) . \\end{align*}"}
-{"id": "4238.png", "formula": "\\begin{align*} \\mathcal { H } _ { T } \\left ( \\Omega ; \\Lambda ^ { k } \\right ) = \\left \\{ \\omega \\in W _ { T } ^ { 1 , 2 } \\left ( \\Omega ; \\Lambda ^ { k } \\right ) : d \\omega = 0 \\delta \\omega = 0 \\Omega \\right \\} \\end{align*}"}
-{"id": "3319.png", "formula": "\\begin{align*} S ( n , \\alpha ) : = \\sum _ { f \\in \\mathcal { M } _ { n } } \\alpha ( f ) . \\end{align*}"}
-{"id": "4099.png", "formula": "\\begin{align*} \\tilde { t } _ { 1 } = \\tilde { \\gamma } + \\tilde M _ { 1 } , \\tilde { t } _ { 2 } = \\tilde \\gamma ( \\tilde \\gamma - \\tfrac { 1 } { 3 } ) + \\tilde M _ { 1 } ( \\tilde M _ { 1 } + \\tilde \\gamma - \\tfrac { 2 } { 3 } ) - \\tilde { M } _ { 2 } , \\tilde { t } _ { 3 } = 2 \\tilde { \\gamma } - \\tfrac { 1 } { 3 } + \\tilde M _ { 1 } , \\end{align*}"}
-{"id": "5904.png", "formula": "\\begin{align*} \\begin{pmatrix} a _ 1 & b _ 1 \\\\ c _ 1 & d _ 1 \\end{pmatrix} \\begin{pmatrix} & \\pi ^ { - 1 } \\\\ - \\pi & \\end{pmatrix} \\begin{pmatrix} a _ 2 & b _ 2 \\\\ c _ 2 & d _ 2 \\end{pmatrix} \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} = \\begin{pmatrix} \\pi ^ { - 1 } a _ 1 d _ 1 - \\pi b _ 1 b _ 2 \\\\ \\pi ^ { - 1 } c _ 1 d _ 2 - \\pi b _ 2 d _ 1 \\end{pmatrix} \\end{align*}"}
-{"id": "4197.png", "formula": "\\begin{align*} \\frac { A _ n ^ { ( 3 ) } ( z ) } { n ^ 9 z } \\frac { A _ n ^ { ( 3 ) } ( 0 ) } { n ^ 9 z } & = \\mathcal { O } \\left ( n ^ { - 2 } \\right ) \\frac { A _ n ^ { ( 3 ) } ( 0 ) } { n ^ 9 z } = \\mathcal { O } \\left ( n ^ { - \\frac { 1 } { 2 } } \\right ) . \\end{align*}"}
-{"id": "9855.png", "formula": "\\begin{align*} B = C , \\end{align*}"}
-{"id": "8564.png", "formula": "\\begin{align*} \\mathcal { T } _ 1 : = \\Big \\{ T \\in [ 0 , \\delta _ 1 \\epsilon ^ { - 2 } ] : \\sup _ { t \\in [ 0 , T ] } \\norm { \\kappa _ { \\alpha } ( \\cdot , t ) - \\kappa _ { \\alpha } ( \\cdot , 0 ) } _ { \\infty } \\leq \\frac { 1 } { 1 0 } \\Big \\} \\end{align*}"}
-{"id": "6704.png", "formula": "\\begin{align*} P _ { 2 k } = \\frac { \\partial L } { \\partial \\ddot { \\bar { q } } _ k } = ( m \\dot { q } _ i + u _ i ) \\beta _ { i k } . \\end{align*}"}
-{"id": "8440.png", "formula": "\\begin{align*} ( I - \\mathcal { H } ) \\frac { 1 } { \\zeta ( \\alpha , t ) - z _ j ( t ) } = \\frac { 2 } { \\zeta ( \\alpha , t ) - z _ j ( t ) } . \\end{align*}"}
-{"id": "9058.png", "formula": "\\begin{align*} \\int _ { \\mathbb { T } ^ 2 } d x \\left ( \\frac { 1 } { \\pi r ^ 2 } \\int _ { B ( x , r ) } | f ( y ) | ^ 2 d y \\right ) ^ 3 = O ( 1 ) . \\end{align*}"}
-{"id": "8579.png", "formula": "\\begin{align*} ( \\alpha \\alpha ^ \\vee ) ^ * = \\sum _ { j k } ( \\alpha , y _ j ) ( \\alpha ^ \\vee , x _ k ) ( x _ j y _ k ) ^ * = \\sum _ { j k } \\tfrac { 2 } { ( \\alpha , \\alpha ) } ( \\alpha , y _ j ) ( \\alpha , x _ k ) x _ k y _ j = \\alpha \\alpha ^ \\vee , \\end{align*}"}
-{"id": "3615.png", "formula": "\\begin{align*} \\alpha _ 0 \\cdot \\beta _ 0 + \\alpha _ 1 \\cdot \\beta _ 1 = 2 v _ { ( 1 , 2 ) } \\cdot v _ { ( 1 , 3 ) } & - \\frac { 1 } { 2 ^ 2 } \\sum _ { i = 1 } ^ 6 a _ i - \\frac { 1 } { 2 ^ 3 } ( v _ { ( 1 , 2 ) } + v _ { ( 1 , 3 ) } - 4 v _ { ( 2 , 3 ) } ) \\\\ & - \\frac { 4 } { 3 } ( ( a _ 2 + a _ { - 2 } ) \\cdot v _ { ( 1 , 3 ) } + ( a _ 3 + a _ { - 3 } ) \\cdot v _ { ( 1 , 2 ) } ) . \\end{align*}"}
-{"id": "7256.png", "formula": "\\begin{align*} D _ { \\phi \\phi \\phi } ^ 3 \\Phi ( 0 , \\mu ^ * _ { k } ) [ \\phi ^ * _ { k } , \\phi ^ * _ { k } , \\phi ^ * _ { k } ] = - \\frac { 3 } { 2 } \\phi ^ * _ { k } \\left ( \\frac { 1 } { \\mu ^ * _ { k } } + \\frac { 1 } { 2 ( \\mu ^ * _ { k } - ( 2 k ) ^ { - r } ) } \\right ) . \\end{align*}"}
-{"id": "9969.png", "formula": "\\begin{align*} d ( x , y ) : = \\sum _ { n = 1 } ^ \\infty 2 ^ { - n } | \\langle x ^ * _ n , x - y \\rangle | . \\end{align*}"}
-{"id": "1049.png", "formula": "\\begin{align*} & 0 \\leq \\sum _ { j = 1 } ^ k \\langle B _ j f , f \\rangle = \\langle B f , f \\rangle \\leq \\sum _ { j = 1 } ^ k \\| \\chi _ { I ( y _ j , F _ n ( x _ j ) ) } f \\| _ { L ^ 2 } ^ 2 \\leq \\| f \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "7937.png", "formula": "\\begin{align*} X _ { 1 } ^ \\tau = ( x _ { \\tau ( 1 ) , 1 } , \\dots , x _ { \\tau ( n ) , n } ) \\end{align*}"}
-{"id": "7925.png", "formula": "\\begin{align*} R _ 3 ( f ( x ) ) & = \\frac { - 1 7 } { 5 7 6 0 } \\ , h ^ { n + 2 } f ^ { ( n + 1 ) } ( \\xi ) & \\xi \\in [ - h , h ] \\\\ R _ 5 ( f ( x ) ) & = \\frac { 3 6 7 } { 9 6 7 6 8 0 } \\ , h ^ { n + 2 } f ^ { ( n + 1 ) } ( \\xi ) & \\xi \\in [ - h , h ] \\end{align*}"}
-{"id": "5278.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow u e p ( F ) ^ { - } } q ( F , x ) = - 1 . . \\end{align*}"}
-{"id": "2189.png", "formula": "\\begin{align*} \\mathcal { H } ( \\boldsymbol { x } , \\boldsymbol { u } , { \\boldsymbol { \\lambda } } ) = \\frac { 1 } { 2 } \\boldsymbol { x } ^ { T } ( t ) Q ( t ) \\boldsymbol { x } ( t ) + \\frac { 1 } { 2 } \\boldsymbol { u } ^ { T } ( t ) R ( t ) \\boldsymbol { u } ( t ) + \\boldsymbol { \\lambda } ^ { T } ( t ) [ A ( t ) \\boldsymbol { x } ( t ) + B ( t ) \\boldsymbol { u } ( t ) ] \\end{align*}"}
-{"id": "3643.png", "formula": "\\begin{align*} \\beta _ { - 2 } & : = \\alpha _ 0 \\cdot \\beta _ 1 - \\frac { 1 } { 2 ^ 2 } \\beta _ 0 - \\frac { 1 } { 2 ^ 3 } \\beta _ 1 - \\frac { 8 } { 3 } \\beta _ { - 1 } = \\frac { 4 } { 3 } ( ( a _ 2 + a _ { - 2 } ) \\cdot v _ { ( 1 , 3 ) } - ( a _ 3 , a _ { - 3 } ) \\cdot v _ { ( 1 , 2 ) } ) \\in M _ { \\frac { 1 } { 2 ^ 2 } } ^ { ( a _ 1 ) } . \\end{align*}"}
-{"id": "9510.png", "formula": "\\begin{align*} \\overline { u } _ { \\delta , R } = \\left \\{ \\begin{array} { r c } U _ { \\delta } ( r ) , & \\mbox { s e } \\ r \\leq R , \\\\ 0 , & \\mbox { s e } \\ r \\geq \\theta R . \\end{array} \\right . \\end{align*}"}
-{"id": "5074.png", "formula": "\\begin{align*} ( \\{ y _ j = S _ { x , \\tau } ^ { - 1 } x _ j + V e _ j - V \\theta _ \\tau S _ { x , \\tau } ^ { - 1 } x _ j \\} _ { j \\in \\mathbb { J } } , \\{ \\omega _ j = S _ { x , \\tau } ^ { - 1 } \\tau _ j + U e _ j - U \\theta _ x S _ { x , \\tau } ^ { - 1 } \\tau _ j \\} _ { j \\in \\mathbb { J } } ) \\end{align*}"}
-{"id": "5987.png", "formula": "\\begin{align*} Y ( t , \\omega ) = \\rho ^ { g ^ { ( t ) } } ( F ( \\omega \\otimes _ t \\cdot ) ) . \\end{align*}"}
-{"id": "3408.png", "formula": "\\begin{align*} f ( T ) = F ( T ) - \\sum _ i F ( T _ i ) = \\log \\frac { d ( T ) } { \\prod _ i d ( T _ i ) } = \\log \\frac { d ( T ) } { d _ 0 ( T ) + \\prod _ i d _ 0 ( T _ i ) } = - \\log \\frac { d _ 0 ( T ) + d _ * ( T ) } { d ( T ) } . \\end{align*}"}
-{"id": "4899.png", "formula": "\\begin{align*} s _ X = \\eta _ { S \\otimes X } ( s \\otimes X ) \\colon S \\otimes X \\to I \\end{align*}"}
-{"id": "6910.png", "formula": "\\begin{align*} g _ 4 + g _ 5 = - u ^ 3 _ { 1 0 } u ^ 2 _ { 0 1 } + u ^ 3 _ { 0 1 } u ^ 2 _ { 1 0 } - u ^ 3 _ { 1 0 } u ^ 1 _ { 0 1 } + u ^ 3 _ { 0 1 } u ^ 1 _ { 1 0 } \\ , . \\end{align*}"}
-{"id": "3597.png", "formula": "\\begin{align*} \\langle \\langle a _ 1 , a _ 2 \\rangle \\rangle = \\langle \\langle a _ 1 , a _ { - 2 } \\rangle \\rangle = \\langle \\langle a _ { - 1 } , a _ 2 \\rangle \\rangle = \\langle \\langle a _ { - 1 } , a _ { - 2 } \\rangle \\rangle . \\end{align*}"}
-{"id": "1729.png", "formula": "\\begin{align*} c _ { p , r } \\left ( O \\right ) = \\inf \\left \\{ \\lVert \\varphi \\rVert _ { \\mathbb { D } _ { r } ^ { p } } : \\varphi \\in \\mathbb { D } _ { r } ^ { p } , \\ \\varphi \\geq 1 \\ O , \\ \\varphi \\geq 0 \\ \\boldsymbol { W } \\right \\} , \\end{align*}"}
-{"id": "3805.png", "formula": "\\begin{align*} \\frac { N ^ { 2 / 3 + i v } } { q r ^ { 2 / 3 } } \\sum _ \\pm \\sum _ { n _ 1 | q r } n _ 1 ^ { 1 / 3 } & \\sum _ { n _ 2 = 1 } ^ \\infty \\frac { \\lambda _ \\pi ( n _ 1 , n _ 2 ) } { n _ 2 ^ { 1 / 3 } } S ( r \\bar a , \\pm n _ 2 ; q r / n _ 1 ) \\\\ & \\times \\int _ 0 ^ \\infty V ( z ) z ^ { i v } e \\left ( \\frac { N x z } { q Q } \\pm \\frac { 3 ( N n _ 1 ^ 2 n _ 2 z ) ^ { 1 / 3 } } { q r ^ { 1 / 3 } } \\right ) \\mathrm { d } z . \\end{align*}"}
-{"id": "4579.png", "formula": "\\begin{align*} I _ t ( \\mu ) = \\int | \\hat { \\mu } ( \\omega ) | ^ 2 | \\omega | ^ { t - n } d \\omega = \\int _ { | \\omega | \\leq 1 } | \\hat { \\mu } ( \\omega ) | ^ 2 | \\omega | ^ { t - n } d \\omega + \\sum _ { j \\geq 0 } \\int _ { 2 ^ j \\leq | \\omega | \\leq 2 ^ { j + 1 } } | \\hat { \\mu } ( \\omega ) | ^ 2 | \\omega | ^ { t - n } d \\omega . \\end{align*}"}
-{"id": "1718.png", "formula": "\\begin{align*} | f ( \\varphi ( h , x ) ) - f ( x ) | & \\leq \\sum _ { m = 1 } ^ { N _ n } | f ( \\varphi ( s _ n + t _ m , x ) ) - f ( \\varphi ( t _ m , x ) ) | + | f ( \\varphi ( r _ n , x ) ) - f ( x ) | \\\\ & \\leq N _ n s _ n \\cdot c _ n + \\dfrac { \\varepsilon } { 2 } \\cdot v ( x ) \\leq \\varepsilon \\cdot v ( x ) . \\end{align*}"}
-{"id": "1594.png", "formula": "\\begin{align*} \\varphi _ { v ^ o _ n } ( z ^ o _ n ) = 1 - \\frac { k _ n } { 2 n } + o \\left ( \\frac { k _ n } { n } \\right ) , \\end{align*}"}
-{"id": "3866.png", "formula": "\\begin{align*} A _ 0 = \\left \\{ \\left ( \\begin{array} { c c } 1 & 1 \\\\ 0 & 1 \\end{array} \\right ) , \\left ( \\begin{array} { c c } 1 & 0 \\\\ 1 & 1 \\end{array} \\right ) \\right \\} , \\end{align*}"}
-{"id": "4132.png", "formula": "\\begin{align*} N _ { \\alpha , + } ( x ) & = N _ { \\alpha , - } ( x ) \\begin{pmatrix} 0 & x ^ \\alpha & 0 \\\\ - x ^ { - \\alpha } & 0 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} , x \\in ( 0 , q ) , \\\\ N _ { \\alpha , + } ( x ) & = N _ { \\alpha , - } ( x ) \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 0 & - | x | ^ { - \\alpha } \\\\ 0 & | x | ^ { \\alpha } & 0 \\end{pmatrix} , x < 0 , \\end{align*}"}
-{"id": "4200.png", "formula": "\\begin{align*} P ( z ) = \\left ( \\mathbb { I } - \\frac { A _ n ^ { ( 3 ) } ( 0 ) } { n ^ 9 z } \\right ) ^ { - 1 } \\left ( \\mathbb { I } - \\frac { A _ n ^ { ( 2 ) } ( 0 ) } { n ^ 6 z } \\right ) ^ { - 1 } \\left ( \\mathbb { I } - \\frac { A _ n ^ { ( 1 ) } ( 0 ) } { n ^ 3 z } \\right ) ^ { - 1 } N ( z ) , z \\in \\partial D ( 0 , R ) . \\end{align*}"}
-{"id": "7068.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta ( x ) & = 1 \\otimes x + x \\otimes 1 - \\frac { 3 } { 2 } x \\otimes x + \\frac { 1 } { 2 } y \\otimes y , \\\\ \\Delta ( y ) & = 1 \\otimes y - \\frac { 3 } { 2 } x \\otimes y + y \\otimes 1 - \\frac { 3 } { 2 } y \\otimes x . \\end{aligned} \\end{align*}"}
-{"id": "3571.png", "formula": "\\begin{align*} \\mathcal { M } _ { 6 } = \\bigcup _ { k = 0 } ^ { 4 } \\phi _ { \\sigma _ { k } } \\left ( \\mathcal { M } _ { 3 , 3 } \\setminus D _ { 0 } \\right ) . \\end{align*}"}
-{"id": "8639.png", "formula": "\\begin{align*} u _ \\Lambda ( 0 ) = v _ \\Lambda ( 0 ) . \\end{align*}"}
-{"id": "2994.png", "formula": "\\begin{align*} p ' _ { m } ( n ) = \\sum _ { k = 1 } ^ \\infty ( - 1 ) ^ { k + 1 } \\big ( p ' _ { m } ( n - P _ { m + 2 , k } ) + p ' _ m ( n - Q _ { m + 2 , k } ) \\big ) . \\end{align*}"}
-{"id": "1237.png", "formula": "\\begin{align*} | T _ { 1 , 1 } | + | T _ { 1 , 2 } | \\lesssim \\int _ 0 ^ { | \\xi | ^ { - 2 } } | \\eta | ^ { - 1 / 2 } | \\xi | ^ { - k } d \\eta = O ( | \\xi | ^ { - k - 1 } ) \\end{align*}"}
-{"id": "8874.png", "formula": "\\begin{align*} e _ 8 & = 9 6 + ( 6 b + 1 2 3 6 ) p + ( - 3 6 0 \\mu ^ 2 + 1 6 2 b - 5 4 2 4 \\mu - 3 0 0 ) p ^ 2 \\\\ & + ( 2 4 b \\mu ^ 2 - 1 2 b ^ 2 + 5 8 6 8 \\mu ^ 2 - 2 4 b + 1 6 5 6 \\mu ) p ^ 3 + ( 4 8 b ^ 2 \\mu - 6 9 6 b \\mu ^ 2 - 4 8 \\mu ^ 3 - 2 0 8 8 \\mu ^ 2 - 7 2 \\mu ) p ^ 4 \\\\ & + ( 4 8 b \\mu ^ 2 + 5 1 2 \\mu ^ 3 + 1 4 4 \\mu ^ 2 ) p ^ 5 - 6 4 \\mu ^ 3 p ^ 6 \\end{align*}"}
-{"id": "2223.png", "formula": "\\begin{align*} u _ \\lambda ( x , t ) = \\lambda ^ 2 u ( \\lambda x , \\lambda ^ 2 t ) \\ \\ { \\rm f o r \\ e a c h \\ \\ } \\lambda > 0 . \\end{align*}"}
-{"id": "10044.png", "formula": "\\begin{align*} V ( r , \\theta , t ) = \\frac { 1 } { r } \\sum _ { j = 1 } ^ n \\frac { m _ j } { | 1 - e ^ { - i \\theta } r ^ { - 1 } q _ j ( \\omega t ) | } = \\frac { \\sum _ { j = 1 } ^ n m _ j } { r } + \\O \\left ( \\frac { 1 } { r ^ 3 } \\right ) , \\end{align*}"}
-{"id": "7608.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { g _ \\epsilon } _ { L ^ \\infty ( 0 , T ; C ^ { \\alpha , p } ) } \\le M _ X ^ { \\alpha } \\norm { u _ 0 } _ { 1 + \\alpha , p } + C _ 1 \\norm { X - \\mathrm { I d } } _ { L i p ( 0 , T ; C ^ { 1 + \\alpha } ) } ^ { \\alpha } \\norm { \\sigma _ { \\epsilon , 0 } } _ { \\alpha , p } + S _ 2 ( T ) Q _ 2 , \\end{gathered} \\end{align*}"}
-{"id": "4325.png", "formula": "\\begin{align*} \\inf _ { q \\in \\mathbb { P } _ l } | u - q | _ { H ^ m ( D ) } \\leq C ( \\rho _ D ) h _ D ^ { l + 1 - m } | u | _ { H ^ { l + 1 } ( D ) } , \\forall u \\in H ^ { l + 1 } ( D ) , l = 0 , 1 , . . . , k , m \\leq l \\end{align*}"}
-{"id": "1590.png", "formula": "\\begin{align*} \\Phi _ { v ^ \\iota _ n } ( z ^ \\iota _ n , x ^ \\iota _ n ) = \\frac { \\sqrt { v ^ \\iota _ n } } { 2 } ( \\Phi _ { \\widetilde { F } } ( z ^ \\iota _ n , x ^ \\iota _ n ) + o ( 1 ) ) = \\frac { k _ n } { 2 n } + o \\left ( \\frac { k _ n } { n } \\right ) . \\end{align*}"}
-{"id": "9321.png", "formula": "\\begin{align*} \\langle \\ ; , \\ ; \\rangle = - d x _ { 1 } ^ { 2 } + d x _ { 2 } ^ { 2 } + \\cdots + d x _ { m } ^ { 2 } , \\end{align*}"}
-{"id": "6091.png", "formula": "\\begin{align*} \\begin{gathered} \\Big ( \\bigcap \\limits _ { j } V _ j \\big ( ( a ^ 1 _ { 1 } , \\ldots , a ^ { k _ 1 } _ 1 ) , \\ldots , ( a ^ 1 _ j , \\ldots , a ^ { k _ j } _ j ) , \\ldots , ( a ^ 1 _ r , \\ldots , a ^ { k _ r } _ r ) \\big ) \\Big ) \\\\ \\leq \\sum \\limits _ { i , j } e ^ i _ j - r \\end{gathered} \\end{align*}"}
-{"id": "4078.png", "formula": "\\begin{align*} \\Psi _ 0 ( z ) = 1 - \\frac { 1 } { z } + \\mathcal { O } ( z ^ { - 2 } ) , \\Psi _ 1 ( z ) = \\frac { 1 } { 2 z } + \\mathcal { O } ( z ^ { - 3 / 2 } ) , \\Psi _ 2 ( z ) = \\frac { 1 } { 2 z } + \\mathcal { O } ( z ^ { - 3 / 2 } ) , \\end{align*}"}
-{"id": "1654.png", "formula": "\\begin{align*} \\omega \\big ( F _ k , G _ k \\big ) = \\omega \\big ( F _ { - k } , G _ { - k } \\big ) = 0 \\quad \\omega \\big ( F _ k , G _ { - k } \\big ) = \\omega \\big ( F _ { - k } , G _ k \\big ) = 2 \\end{align*}"}
-{"id": "3158.png", "formula": "\\begin{align*} f ^ q ( x , t ) = \\frac { q x - q ^ { - 1 } t } { x - t } . \\end{align*}"}
-{"id": "1802.png", "formula": "\\begin{align*} c _ { 2 , 1 } \\left ( \\liminf _ { n \\to \\infty } G _ { n } \\right ) = 0 . \\end{align*}"}
-{"id": "9091.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\mathbb { R } ^ N } g ( x ) | u _ { n _ k } ( x ) | ^ p d x & \\to \\int _ { \\mathbb { R } ^ N } g ( x ) | u ( x ) | ^ p d x , \\\\ \\int _ { \\mathbb { R } ^ N } g ( x ) | u _ { n _ k } ( x ) | ^ { p - 2 } u _ { n _ k } ( x ) u ( x ) d x & \\to \\int _ { \\mathbb { R } ^ N } g ( x ) | u ( x ) | ^ p d x . \\end{aligned} \\end{align*}"}
-{"id": "5247.png", "formula": "\\begin{align*} F ^ { - 1 } ( 1 - u ) = d - s ( u ) + \\int _ { u } ^ { 1 } t ^ { - 1 } s ( t ) d t . \\end{align*}"}
-{"id": "148.png", "formula": "\\begin{align*} & \\{ f \\circ q _ M , g \\circ q _ M \\} _ { T M } = 0 , \\\\ & \\{ f \\circ q _ M , l _ { d g } \\} _ { T M } = - a ( d g ) ( f ) \\circ q _ M , \\\\ & \\{ l _ { d f } , l _ { d g } \\} _ { T M } = l _ { [ d f , d g ] } \\end{align*}"}
-{"id": "6240.png", "formula": "\\begin{align*} \\gamma : = \\sum _ { i = 1 } ^ n \\frac { 1 - \\beta } { \\alpha _ i } \\geq ( 1 - \\beta ) \\left [ \\frac { n - 1 } { \\alpha _ 1 } + \\frac { 1 } { 1 - \\alpha _ 1 } \\right ] \\geq ( 1 - \\beta ) [ n + 2 \\sqrt { n ( n - 1 ) } ] , \\end{align*}"}
-{"id": "691.png", "formula": "\\begin{align*} P ( X _ 1 \\dots X _ m ) : = \\varpi ( X _ 1 \\dots X _ m ) \\ . \\end{align*}"}
-{"id": "4710.png", "formula": "\\begin{align*} S _ 0 = \\{ x _ { j - 2 } , x _ { j - 1 } , x _ { j } \\} , S _ 1 = \\{ x _ { j - 1 } , x _ { j } , x _ { j + 1 } \\} , S _ 2 = \\{ x _ { j } , x _ { j + 1 } , x _ { j + 2 } \\} , \\end{align*}"}
-{"id": "6431.png", "formula": "\\begin{align*} ( B _ { k + 1 } B _ { k + 1 } ^ { T } - \\rho _ { k + 1 } ^ { - 1 } I ) y = z _ { k + 1 } . \\end{align*}"}
-{"id": "595.png", "formula": "\\begin{align*} \\xi ^ x _ { t _ 1 , t _ 2 } : = - \\overline { \\alpha } ( t _ 2 - t ) - \\overline { \\sigma } \\frac { \\beta ^ x _ { t _ 1 - t } + \\tilde { \\beta } ^ x _ { t _ 1 - t } } { 2 } - \\min _ { 0 \\le s \\le \\underline { \\sigma } ( t _ 2 - t _ 1 ) } \\hat { \\beta } _ s , \\end{align*}"}
-{"id": "4914.png", "formula": "\\begin{gather*} \\bar S = \\int _ { t _ 1 } ^ { t _ 2 } \\bar L ( \\dot \\eta , \\eta ) \\ , \\d t \\end{gather*}"}
-{"id": "9566.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { q ^ { \\frac { n ( n + 1 ) } { 2 } } ( q ) _ { n } } { ( 1 - c q ^ { n } ) ( - q ) _ n } = \\frac { 1 } { c } \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - q / c ) _ { n - 1 } ( q ) _ { n } ( c q ) _ { N - n } ( c q ) ^ { n } } { ( - q ) _ { n } ( c q ) _ { N } } . \\end{align*}"}
-{"id": "3836.png", "formula": "\\begin{align*} 2 \\Phi ^ i : & = - k _ 0 y ^ i + \\frac { 1 } { 2 } h _ { 0 0 } \\bar { k } ^ i - \\dfrac { 2 m \\pi } { m + 1 } \\frac { h _ { 0 0 } ^ { \\frac { m + 1 } { 2 } } } { W _ 0 ^ m } s _ 0 ^ i \\\\ & - \\frac { 2 m s } { s ^ 2 - m s ^ 2 + m b ^ 2 } \\Bigl ( r _ { 0 0 } + \\dfrac { 2 m \\pi } { m + 1 } \\frac { h _ { 0 0 } ^ { \\frac { m + 1 } { 2 } } } { W _ 0 ^ m } s _ 0 \\Bigr ) , \\end{align*}"}
-{"id": "202.png", "formula": "\\begin{align*} & X _ { 1 , V } = x ^ 1 \\dfrac { \\partial } { \\partial y ^ 1 } + x ^ 3 \\dfrac { \\partial } { \\partial y ^ 3 } , & & Y _ { 1 , V } = - \\dfrac { x ^ 2 } { x ^ 1 } \\dfrac { \\partial } { \\partial y ^ 3 } , \\\\ & X _ { 2 , V } = x ^ 1 \\dfrac { \\partial } { \\partial y ^ 2 } , & & Y _ { 2 , V } = \\dfrac { \\partial } { \\partial y ^ 3 } . \\end{align*}"}
-{"id": "4296.png", "formula": "\\begin{align*} \\displaystyle \\# ( \\{ x , y , z \\} : x y , z y , z x \\notin E ( G ) ) & \\geq \\frac { m ( 4 m - n ^ 2 ) } { 3 n } \\\\ & = \\frac { ( \\binom { n } { 2 } - C ' n ^ { 3 / 2 } ) ( 4 \\binom { n } { 2 } - 4 C ' n ^ { 3 / 2 } - n ^ 2 ) } { 3 n } \\\\ & = \\frac { n ^ 4 - o ( n ^ 4 ) } { 6 n } \\\\ & = \\frac { n ^ { 3 } } { 6 } - o ( n ^ 3 ) . \\end{align*}"}
-{"id": "2910.png", "formula": "\\begin{align*} \\Psi _ { \\mu _ + } \\Big ( A _ { \\ell _ i } \\ , \\Big | \\ , g F _ n \\setminus \\big [ R ^ g _ n \\cup \\bigcup _ { j = i } ^ m A _ { \\ell _ j } \\big ] \\Big ) & \\geq \\Psi _ \\mu \\Big ( A _ { \\ell _ i } \\ , \\Big | \\ , g F _ n \\setminus \\big [ R ^ g _ n \\cup \\bigcup _ { j = i } ^ m A _ { \\ell _ j } \\big ] \\Big ) + \\varepsilon \\ ; . \\end{align*}"}
-{"id": "1838.png", "formula": "\\begin{align*} u ^ { \\rm o d d } ( n , x ) = \\sqrt { \\frac { 2 } { L } } \\sin ( k _ n x ) , \\end{align*}"}
-{"id": "2340.png", "formula": "\\begin{align*} [ a _ x , a ^ + _ y ] & = \\delta ( x - y ) & [ a _ x , a _ y ] & = [ a ^ + _ x , a ^ + _ y ] = 0 \\end{align*}"}
-{"id": "10052.png", "formula": "\\begin{align*} \\widetilde H ( \\tilde q _ 1 , \\dots , \\tilde q _ n , \\tilde p _ 1 , \\dots , \\tilde p _ n ) = \\sum _ { 1 \\le j \\le n } \\frac { 1 } { 2 \\mu _ j } \\| \\tilde p _ j \\| ^ 2 - \\widetilde U ( \\tilde q _ 1 , \\dots , \\tilde q _ n ) , \\end{align*}"}
-{"id": "7083.png", "formula": "\\begin{align*} M _ { 2 , 2 } \\boxtimes _ { D \\mathbb { Z } _ 2 } M _ { 2 , 2 } & = 2 M _ { 2 , 2 } \\\\ M _ { 2 , 1 } \\boxtimes _ { D \\mathbb { Z } _ 2 } M _ { 2 , 1 } & = \\mathcal { M } ( ( \\mathbb { Z } _ 2 \\times 0 , \\mathbb { Z } _ 2 \\times 0 ) , \\psi _ { t r i v i a l } ) . \\end{align*}"}
-{"id": "8331.png", "formula": "\\begin{align*} A _ 1 = 1 - I m \\Big \\{ ( I - \\mathbb { H } ) ( F _ z \\circ Z Z _ { \\alpha } D _ t Z ) + ( I - \\mathbb { H } ) \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i ( D _ t Z Z _ { \\alpha } - \\dot { z } _ j ( t ) Z _ { \\alpha } ) } { 2 \\pi ( Z ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } \\Big \\} \\end{align*}"}
-{"id": "2976.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\frac { q ^ { n ^ 2 } } { ( q ; q ) _ n } & = \\frac { 1 } { ( q ; q ) _ \\infty ( q ^ 4 ; q ^ 5 ) _ \\infty } . \\end{align*}"}
-{"id": "6453.png", "formula": "\\begin{align*} \\psi _ \\lambda ( e _ { i , j } ) : = \\begin{cases} \\lambda ^ i ( 1 - \\lambda ) & i = j \\\\ 0 & \\end{cases} . \\end{align*}"}
-{"id": "6230.png", "formula": "\\begin{align*} \\psi _ \\lambda ( e _ { i , j } ) : = \\begin{cases} \\lambda ^ i ( 1 - \\lambda ) & i = j \\\\ 0 & \\end{cases} . \\end{align*}"}
-{"id": "6868.png", "formula": "\\begin{align*} \\mathcal { R } _ 2 \\quad \\equiv \\begin{cases} f _ 1 = y ^ 1 _ { 1 0 } y ^ 2 _ { 0 1 } - y ^ 1 _ { 0 1 } y ^ 2 _ { 1 0 } - 1 = 0 \\ , , \\\\ f _ 2 = y ^ 1 _ { 2 0 } y ^ 2 _ { 0 1 } + y ^ 1 _ { 1 0 } y ^ 2 _ { 1 1 } - y ^ 1 _ { 1 1 } y ^ 2 _ { 1 0 } - y ^ 1 _ { 0 1 } y ^ 2 _ { 2 0 } = 0 \\ , , \\\\ f _ 3 = y ^ 1 _ { 1 1 } y ^ 2 _ { 0 1 } + y ^ 1 _ { 1 0 } y ^ 2 _ { 0 2 } - y ^ 1 _ { 0 2 } y ^ 2 _ { 1 0 } - y ^ 1 _ { 0 1 } y ^ 2 _ { 1 1 } = 0 \\ , , \\\\ f _ 4 = y ^ 1 _ { 2 0 } + y ^ 1 _ { 0 2 } = 0 \\ , , \\\\ f _ 5 = y ^ 2 _ { 2 0 } + y ^ 2 _ { 0 2 } = 0 \\ , . \\end{cases} \\end{align*}"}
-{"id": "1711.png", "formula": "\\begin{align*} T ( s - h ) y - T ( s ) y = T ( s - h ) ( y - T ( h ) y ) \\in V _ 1 \\subset V _ 0 \\end{align*}"}
-{"id": "9723.png", "formula": "\\begin{align*} \\Delta _ x \\varphi _ g \\leq - c < 0 \\quad B _ 1 \\cap \\{ y = 0 \\} . \\end{align*}"}
-{"id": "4867.png", "formula": "\\begin{align*} b _ { W _ 1 } ( f ) = b _ { W _ 2 } ( f | _ { W _ 2 } ) . \\end{align*}"}
-{"id": "6760.png", "formula": "\\begin{align*} c _ { m , k } = ( m + 1 ) ( - 1 ) ^ { k - 1 } \\sum _ { j \\leq m / 2 } a _ { m , 2 j } ( 1 - 2 j ) B _ { 2 j } { j \\choose k - 1 } . \\end{align*}"}
-{"id": "4053.png", "formula": "\\begin{align*} \\delta ( f ) = - ( - 1 ) ^ n f \\circ b \\end{align*}"}
-{"id": "1906.png", "formula": "\\begin{align*} f ( x ) \\equiv \\cos ( \\kappa _ j x ) + \\frac { \\xi } { \\kappa _ j } \\sin ( \\kappa _ j | x | ) = \\frac { \\xi L } { 2 } \\left [ \\frac { 1 } { Z _ j ^ 2 } + 2 \\sum _ { n = 1 } ^ \\infty \\frac { \\cos ( k _ n x ) } { Z _ j ^ 2 - ( \\pi n ) ^ 2 } \\right ] . \\end{align*}"}
-{"id": "9678.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\varphi ( T ) \\omega _ n ( T ) \\varphi _ n ( T ) x _ 0 \\| \\leq & K C _ { { \\rm p o l } , T } ( 1 + \\varepsilon _ 0 ) \\frac { ( 2 ( 1 + c _ 1 ^ 2 ) ) ^ { 1 / 2 } } { 1 - c _ 1 } \\Bigl ( \\int _ { \\mathbb T } | \\varphi | ^ 2 | \\psi _ { 1 n } | ^ 2 m \\Bigr ) ^ { 1 / 2 } \\\\ & \\ \\varphi \\in H ^ \\infty \\ n \\in \\mathbb N . \\end{aligned} \\end{align*}"}
-{"id": "891.png", "formula": "\\begin{align*} \\phi _ 1 & = \\alpha _ 1 & \\phi _ 2 & = \\alpha _ 3 & \\phi _ 3 & = \\alpha _ 2 + \\alpha _ 3 & \\phi _ 4 & = \\alpha _ 2 . \\end{align*}"}
-{"id": "1852.png", "formula": "\\begin{align*} \\psi ^ { \\rm t o p . } ( x , t ) = \\sqrt { \\frac { \\ell } { 2 } } \\ , v ^ { \\rm t o p . } ( x ) \\left ( 1 - i \\frac { t } { \\ell } \\right ) = \\sqrt { \\frac { \\ell } { 2 L } } \\left ( 1 - i \\frac { t } { \\ell } \\right ) , \\end{align*}"}
-{"id": "4764.png", "formula": "\\begin{align*} \\mathcal { Y } = \\big \\{ v \\in L ^ { 2 } _ { \\sigma } ( \\mathbb { R } ^ 2 ) \\big | v _ F \\in C ^ { \\infty } _ { \\sigma , c } ( \\mathbb { R } ^ 2 ) v _ R \\in \\mathcal { R } v | _ { \\mathcal { F } } = v _ F | _ { \\mathcal { F } } v | _ { \\mathcal { S } _ 0 } = v _ S | _ { \\mathcal { S } _ 0 } \\big \\} , \\end{align*}"}
-{"id": "9948.png", "formula": "\\begin{align*} \\norm { B ( \\psi _ k ) } = O ( k ^ { - \\infty } ) \\end{align*}"}
-{"id": "2878.png", "formula": "\\begin{align*} \\begin{aligned} { } [ K _ 1 , K _ 2 ] = K _ 3 \\qquad \\quad \\ [ K _ 2 , K _ 3 ] & = { K _ 2 } ^ 2 + \\{ K _ 1 , K _ 2 \\} + d K _ 2 + e _ 1 \\quad \\ \\\\ [ K _ 3 , K _ 1 ] & = { K _ 1 } ^ 2 + \\{ K _ 1 , K _ 2 \\} + d K _ 1 + e _ 2 \\end{aligned} \\end{align*}"}
-{"id": "2528.png", "formula": "\\begin{align*} \\mathbf P _ \\mu [ e ^ { - X _ t ( f ) } ] = e ^ { - \\mu ( V _ t f ) } , \\end{align*}"}
-{"id": "4698.png", "formula": "\\begin{align*} v ^ { ( 0 ) } & = u ^ n , \\\\ v ^ { ( 1 ) } & = v ^ { ( 0 ) } + \\Delta t L ( v ^ { ( 0 ) } ) , \\\\ v ^ { ( 2 ) } & = \\frac { 3 } { 4 } v ^ { ( 0 ) } + \\frac { 1 } { 4 } ( v ^ { ( 1 ) } + \\Delta t L ( v ^ { ( 1 ) } ) ) , \\\\ v ^ { ( 3 ) } & = \\frac { 1 } { 3 } v ^ { ( 0 ) } + \\frac { 2 } { 3 } ( v ^ { ( 2 ) } + \\Delta t L ( v ^ { ( 2 ) } ) ) , \\\\ u ^ { n + 1 } & = v ^ { ( 3 ) } . \\end{align*}"}
-{"id": "5538.png", "formula": "\\begin{align*} \\mu _ b - \\mu _ { b + h } = v _ { b , h } + w _ { b , h } = ( L _ b ^ * ) ^ { - 1 } [ f _ h ] + ( L _ { b + h } ^ * ) ^ { - 1 } [ \\bar f _ h ] , \\end{align*}"}
-{"id": "4344.png", "formula": "\\begin{align*} H _ { \\tilde { C } _ \\sigma } ( C ) & = \\{ h Z \\in ( \\widetilde { H } / Z ) ^ W \\ , | \\ , \\alpha ( h Z ) = + 1 , \\ , \\forall \\alpha \\in R s _ \\alpha \\tilde { C } _ \\sigma \\} \\\\ & = \\{ h Z \\in ( \\widetilde { H } / Z ) ^ W \\ , | \\ , s _ \\alpha ( h Z ) = h z , \\ , \\forall h z \\in h Z , \\ , \\forall \\alpha \\in R s _ \\alpha \\tilde { C } _ \\sigma \\} , \\end{align*}"}
-{"id": "3424.png", "formula": "\\begin{gather*} e = \\frac 1 { | Z ( J ) | } \\sum _ { z \\in Z ( J ) } \\chi ( z ) ^ { - 1 } z \\end{gather*}"}
-{"id": "7158.png", "formula": "\\begin{align*} e ( b ) = a _ 1 ( b - 1 ) - a _ 2 \\log b , \\end{align*}"}
-{"id": "5930.png", "formula": "\\begin{align*} \\pi ^ * ( K _ X + \\Delta _ s + t ( G + \\Delta - \\Delta _ s ) ) = K _ Y + \\sum _ i ( e _ i + ( 1 - s + t s ) f _ i + t g _ i ) E _ i + M _ Y \\end{align*}"}
-{"id": "2723.png", "formula": "\\begin{align*} \\Sigma _ { + } ^ { 1 } = \\left \\{ w : \\Re w = w _ { 1 } , 0 \\leq \\Im w \\leq \\Im w _ 2 \\right \\} . \\end{align*}"}
-{"id": "2044.png", "formula": "\\begin{align*} c _ { \\chi } ( m r _ 2 ) = r _ 2 \\chi _ * ( r _ n ) \\tau ( \\chi _ * ) \\overline { \\chi _ * } ( m ) \\mu ( ( r _ n , m ) ) \\varphi ( ( r _ n , m ) ) . \\end{align*}"}
-{"id": "5244.png", "formula": "\\begin{align*} F ^ { - 1 } ( 1 - u ) = d + s ( u ) + \\int _ { u } ^ { 1 } t ^ { - 1 } s ( t ) d t , 0 < u < 1 , \\end{align*}"}
-{"id": "3189.png", "formula": "\\begin{align*} U \\varphi = U ^ * \\varphi = C \\varGamma \\varphi = C \\varphi . \\end{align*}"}
-{"id": "1560.png", "formula": "\\begin{align*} \\pi _ z \\circ \\Psi _ { \\widetilde { F } ^ 0 } ( Z ) = \\psi _ f ( Z ) . \\end{align*}"}
-{"id": "9077.png", "formula": "\\begin{align*} \\mu _ K ( A ) & = \\sum _ { \\substack { \\zeta ^ { ( k ) } \\in A \\\\ k \\in \\mathcal { K } } } \\mu _ { f } ( I _ k ) \\Big / \\sum _ { k \\in \\mathcal { K } } \\mu _ { f } ( I _ k ) = \\sum _ { \\substack { \\zeta ^ { ( k ) } \\in A \\\\ k \\in \\mathcal { K } } } \\sum _ { \\xi \\in I _ k } | a _ { \\xi } | ^ 2 \\Big / \\sum _ { k \\in \\mathcal { K } } \\mu _ { f } ( I _ k ) . \\end{align*}"}
-{"id": "9254.png", "formula": "\\begin{align*} \\begin{array} { r l } D _ \\beta [ \\tilde v _ 1 - \\zeta ^ 2 ( 0 ) w _ 1 ( 0 ) f ] = \\ ! \\ ! & \\ ! \\ ! \\displaystyle D _ \\beta ( \\zeta ^ 2 \\tilde w _ 1 ) - \\zeta ^ 2 ( 0 ) w _ 1 ( 0 ) D _ \\beta f \\\\ = \\ ! \\ ! & \\ ! \\ ! \\displaystyle \\zeta ^ 2 D _ \\beta \\tilde w _ 1 - \\zeta ^ 2 ( 0 ) w _ 1 ( 0 ) D _ \\beta f , { \\rm o n } \\ B _ R \\cap \\partial \\Omega , \\end{array} \\end{align*}"}
-{"id": "5916.png", "formula": "\\begin{align*} \\mathcal { G } { \\rm l c t } _ { d , p , \\Lambda } \\coloneqq \\{ { \\rm g l c t } ( K _ X + B + M \\mid P + N ) \\mid B + M \\in \\mathcal { G } \\mathcal { B } ( X ) _ { p , \\Lambda } , P + N \\in \\mathcal { G } \\mathcal { B } ( X ) _ { p , \\Lambda } , \\dim X = d \\} \\end{align*}"}
-{"id": "7451.png", "formula": "\\begin{align*} \\displaystyle I ( f _ 1 , f _ 2 ) = \\int _ { H _ 1 ( F ) \\backslash G ' _ { r s } ( F ) / H _ 2 ( F ) } O _ \\eta ( \\gamma , f _ 1 ) \\overline { O _ \\eta ( \\gamma , f _ 2 ) } d \\gamma . \\end{align*}"}
-{"id": "1803.png", "formula": "\\begin{align*} B _ { t } = \\int _ { 0 } ^ { t } K ( t , s ) d \\omega ( s ) , \\end{align*}"}
-{"id": "1887.png", "formula": "\\begin{align*} h ( t ) - h ' ( t ) = \\frac { t ^ 4 } { ( 1 2 - 6 t + t ^ 2 ) ^ 2 } \\geq 0 . \\end{align*}"}
-{"id": "488.png", "formula": "\\begin{align*} \\lambda _ i + \\mu _ i = 0 . \\end{align*}"}
-{"id": "8548.png", "formula": "\\begin{align*} \\norm { | D | ^ { 1 / 2 } \\frac { 1 } { 2 } ( I - \\mathbb { H } ) \\tilde { \\theta } } _ { H ^ s } ^ 2 = i \\sum _ { k = 0 } ^ s \\int \\partial _ { \\alpha } ^ k \\frac { 1 } { 2 } ( I - \\mathbb { H } ) \\tilde { \\theta } \\overline { \\partial _ { \\alpha } ^ { k + 1 } \\frac { 1 } { 2 } ( I - \\mathbb { H } ) \\tilde { \\theta } } d \\alpha \\end{align*}"}
-{"id": "8520.png", "formula": "\\begin{align*} & \\norm { D _ t \\tilde { \\sigma } - 4 ( D _ t \\bar { \\mathfrak { F } } - D _ t q ) } _ { H ^ s } \\\\ = & \\norm { D _ t ( I - \\mathcal { H } ) \\Big \\{ - ( \\mathcal { H } + \\bar { \\mathcal { H } } ) D _ t \\zeta - [ D _ t \\zeta , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } ( \\zeta - \\bar { \\zeta } ) } { \\zeta _ { \\alpha } } \\Big \\} - D _ t ( \\bar { \\mathcal { H } } + \\mathcal { H } ) \\bar { \\mathfrak { F } } } _ { H ^ s } \\\\ \\leq & C \\epsilon ^ 2 . \\end{align*}"}
-{"id": "2637.png", "formula": "\\begin{align*} f _ { N } ( \\lambda ) = \\frac { C _ { N , M } } { Z _ { N } } \\Delta _ { N } ^ { 2 } ( \\lambda ) \\big ( \\prod _ { k = 1 } ^ { N } \\lambda _ { k } ^ { n } \\big ) \\ , I _ { 1 } I _ { 2 } , \\end{align*}"}
-{"id": "6822.png", "formula": "\\begin{align*} M _ 2 = \\begin{pmatrix} \\cos ( k t ) & - \\sin ( k t ) \\\\ \\sin ( k t ) & \\cos ( k t ) \\end{pmatrix} \\ , , { \\rm a n d } w ( \\alpha ) = \\frac { e ^ { k \\alpha _ 2 } } { k } \\begin{pmatrix} \\sin ( k \\alpha _ 1 ) \\\\ - \\cos ( k \\alpha _ 1 ) \\end{pmatrix} \\ , , \\end{align*}"}
-{"id": "3951.png", "formula": "\\begin{align*} \\dfrac { \\partial f _ { 1 } ( x , m ^ { \\hat v } _ { t } , \\hat { v } _ { 1 } ( x , m ^ { \\hat v } _ { t } ) ) } { \\partial v _ { 1 } } + D u ^ { \\hat v } _ { 1 } ( x , t ) . \\dfrac { \\partial g _ { 1 } ( x , m ^ { \\hat v } _ { t } , \\hat { v } _ { 1 } ( x , m ^ { \\hat v } _ { t } ) ) } { \\partial v _ { 1 } } = 0 \\ , . \\end{align*}"}
-{"id": "8490.png", "formula": "\\begin{align*} \\dot { y } ( 0 ) = - F _ 2 ( z _ 2 ( 0 ) , 0 ) - \\frac { | \\lambda | } { 4 \\pi x ( 0 ) } \\leq - \\frac { 9 | \\lambda | } { 4 0 \\pi x ( 0 ) } . \\end{align*}"}
-{"id": "9842.png", "formula": "\\begin{align*} \\Gamma ( c _ 1 c _ 2 ) = \\{ c _ 1 ' c _ 2 ' \\in E ( H ^ * ) \\ , : \\ , c _ 1 ' \\in L ( u _ 1 ) c _ 2 ' \\in L ( u _ 2 ) \\} . \\end{align*}"}
-{"id": "8487.png", "formula": "\\begin{align*} \\begin{cases} \\dot { z } _ 1 ( t ) = \\bar { F } ( z _ 1 ( t ) , t ) + \\frac { \\lambda _ 2 i } { 2 \\pi } \\frac { 1 } { \\overline { z _ 1 ( t ) - z _ 2 ( t ) } } = \\bar { F } ( z _ 1 ( t ) ) - \\frac { | \\lambda | i } { 4 \\pi x ( t ) } \\\\ \\dot { z } _ 2 ( t ) = \\bar { F } ( z _ 2 ( t ) , t ) + \\frac { \\lambda _ 1 i } { 2 \\pi } \\frac { 1 } { \\overline { z _ 2 ( t ) - z _ 1 ( t ) } } = \\bar { F } ( z _ 2 ( t ) ) - \\frac { | \\lambda | i } { 4 \\pi x ( t ) } . \\end{cases} \\end{align*}"}
-{"id": "710.png", "formula": "\\begin{align*} B i s ( v , w ) = \\displaystyle \\frac { \\displaystyle \\sum _ { 1 \\leq i , j , k , l \\leq 2 } R _ { i \\bar { j } k \\bar { l } } v _ i \\bar { v _ j } w _ k \\bar { w _ l } } { \\left ( \\sum _ { 1 \\leq i , j \\leq 2 } g _ { i \\bar { j } } v _ i \\overline { v _ j } \\right ) \\left ( \\sum _ { 1 \\leq i , j \\leq 2 } g _ { i \\bar { j } } w _ i \\overline { w _ j } \\right ) } , \\end{align*}"}
-{"id": "9495.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { n \\rightarrow \\infty } \\frac { \\beta _ n } { \\gamma ^ n } = \\beta _ 0 + ( p - 1 ) \\displaystyle \\lim _ { n \\rightarrow \\infty } \\frac { \\gamma ^ { n - 1 } - \\gamma } { \\gamma ^ n ( \\gamma - 1 ) } = \\beta _ 0 + ( p - 1 ) \\frac { \\gamma } { \\gamma - 1 } = \\frac { p _ { s } ^ * ( p _ { s } ^ * - \\theta ' ) } { \\theta ' ( p _ { s } ^ * - \\theta ' p ) } . \\end{align*}"}
-{"id": "6935.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow T ^ { * } } \\left ( \\| \\rho \\| _ { L ^ \\infty ( 0 , T ; L ^ \\infty ) } + \\| \\mathbf { u } \\| _ { L ^ s ( 0 , T ; L ^ r ) } \\right ) = \\infty , \\ \\ \\frac 2 s + \\frac 3 r \\leq 1 , \\ 3 < r \\leq \\infty . \\end{align*}"}
-{"id": "8940.png", "formula": "\\begin{align*} B _ { { q p } _ { b l } } ( 2 ; \\epsilon ) & = \\{ y \\in X : q p _ { b l } ( 2 , y ) < q p _ { b l } ( 2 , 2 ) + \\epsilon \\thinspace \\mbox { a n d } \\thinspace q p _ { b l } ( y , 2 ) < q p _ { b l } ( 2 , 2 ) + \\epsilon \\} \\\\ & = \\{ 0 , 1 , 2 \\} . \\end{align*}"}
-{"id": "1920.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\frac { A _ j ^ 2 } { Z _ j ^ p } = \\frac { \\cos ^ 2 ( Z _ 1 ) } { Z _ 1 ^ { p - 1 } \\left [ Z _ 1 + \\sin ( Z _ 1 ) \\cos ( Z _ 1 ) \\right ] } + \\sum _ { j = 2 } ^ \\infty \\frac { \\cos ^ 2 ( Z _ j ) } { Z _ j ^ { p - 1 } \\left [ Z _ j + \\sin ( Z _ j ) \\cos ( Z _ j ) \\right ] } . \\end{align*}"}
-{"id": "1731.png", "formula": "\\begin{align*} T _ { t } u ( x ) = \\int _ { \\boldsymbol { W } } u \\left ( e ^ { - t } x + \\sqrt { 1 - e ^ { - 2 t } } \\omega \\right ) P ( d \\omega ) . \\end{align*}"}
-{"id": "7783.png", "formula": "\\begin{align*} C _ p ( d , G ) = C _ p ( d , U ( G ) ) . \\end{align*}"}
-{"id": "8656.png", "formula": "\\begin{align*} \\varphi _ { i , \\varepsilon } = \\varphi _ { i , 0 } + O ( \\varepsilon ) , \\ \\ i = 1 , 2 , \\end{align*}"}
-{"id": "8213.png", "formula": "\\begin{align*} U _ m ( f ) ( \\tau ) : = \\frac { 1 } { m } \\sum _ { \\lambda = 0 } ^ { m - 1 } f \\left ( \\frac { \\tau + \\lambda } { m } \\right ) , \\tau \\in \\mathbb { H } . \\end{align*}"}
-{"id": "1344.png", "formula": "\\begin{align*} \\pi _ * ( f ) ( y ) = \\sum _ { x \\in \\pi ^ { - 1 } ( y ) } f ( x ) f \\in C _ c ( G ) y \\in H \\end{align*}"}
-{"id": "8606.png", "formula": "\\begin{align*} u ' = \\frac { u ^ 2 - u ^ 4 - c _ 0 ^ 2 } { \\sqrt { 2 } \\sqrt { u ^ 4 + c _ 0 ^ 2 } } , \\ \\ u ( 0 ) = \\sqrt { c _ 0 } . \\end{align*}"}
-{"id": "8258.png", "formula": "\\begin{align*} z _ t ( \\alpha , t ) = v ( z ( \\alpha , t ) , t ) . \\end{align*}"}
-{"id": "2230.png", "formula": "\\begin{align*} I _ { n _ { j } } \\cap \\left ( \\bigcup _ { k = 1 } ^ { j } J _ { k } \\right ) = I _ { \\infty } \\cap \\left ( \\bigcup _ { k = 1 } ^ { j } J _ { k } \\right ) \\forall j . \\end{align*}"}
-{"id": "3834.png", "formula": "\\begin{align*} 2 G ^ i = ^ { h } { \\gamma _ { 0 } ^ { i } } _ 0 - k _ 0 y ^ i + \\frac { 1 } { 2 } h _ { 0 0 } \\bar { k } ^ i - F s _ 0 ^ i - \\frac { 1 } { b ^ 2 } ( r _ { 0 0 } + F s _ 0 ) \\Bigl ( \\frac { 2 } { F } y ^ i - b ^ i \\Bigr ) . \\end{align*}"}
-{"id": "8489.png", "formula": "\\begin{align*} | F _ 2 ( z _ 2 ( t ) , t ) | \\leq | F ( z _ 2 ( t ) , t ) | \\leq \\| F ( \\cdot , t ) \\| _ { L ^ { \\infty } ( \\Omega ( t ) ) } = \\| \\mathfrak { F } ( \\cdot , t ) \\| _ { L ^ { \\infty } ( \\mathbb { R } ) } \\leq 5 \\epsilon . \\end{align*}"}
-{"id": "3434.png", "formula": "\\begin{gather*} c _ 2 ( V ) = - b ^ 2 , c _ 2 \\big ( S ^ 2 ( V ) \\big ) = - 4 b ^ 2 , c _ 2 \\big ( S ^ 3 ( V ) \\big ) = - 1 0 b ^ 2 , \\\\ c _ 2 \\big ( S ^ 4 ( V ) \\big ) = - 2 0 b ^ 2 , c _ 2 \\big ( S ^ 5 ( V ) \\big ) = - 3 5 b ^ 2 . \\end{gather*}"}
-{"id": "49.png", "formula": "\\begin{align*} \\delta \\lambda = \\frac { \\mathbb { E } ^ { \\mathbb { Q } } \\left [ \\delta \\mathcal { Y } _ T ^ { v } - \\delta \\mathcal { Y } _ 0 ^ { v } \\right ] } { T } = \\frac { \\mathbb { E } ^ { \\mathbb { Q } } \\left [ \\mathbf { y } ^ { \\alpha _ T } ( V _ T ^ v ) - \\bar { \\mathbf { y } } ^ { \\alpha _ T } ( V _ T ^ v ) \\right ] - \\left [ \\mathbf { y } ^ i ( v ) - \\bar { \\mathbf { y } } ^ i ( v ) \\right ] } { T } . \\end{align*}"}
-{"id": "5096.png", "formula": "\\begin{align*} \\| U x \\| ^ 2 = \\left \\| \\sum _ { j \\in \\mathbb { J } } c _ j x _ j \\right \\| ^ 2 = \\left \\| \\sum _ { j \\in \\mathbb { J } } c _ j c _ j ^ * \\right \\| = \\| x \\| ^ 2 \\end{align*}"}
-{"id": "8426.png", "formula": "\\begin{align*} - 2 [ z _ t , \\mathfrak { H } ] \\frac { \\partial _ { \\alpha } z _ t } { z _ { \\alpha } } = & - 2 [ \\bar { f } , \\mathfrak { H } ] \\frac { \\partial _ { \\alpha } \\bar { f } } { z _ { \\alpha } } - 2 [ \\bar { p } , \\mathfrak { H } ] \\frac { \\partial _ { \\alpha } \\bar { f } } { z _ { \\alpha } } - 2 [ \\bar { f } , \\mathfrak { H } ] \\frac { \\partial _ { \\alpha } \\bar { p } } { z _ { \\alpha } } - 2 [ \\bar { p } , \\mathfrak { H } ] \\frac { \\partial _ { \\alpha } \\bar { p } } { z _ { \\alpha } } \\end{align*}"}
-{"id": "9425.png", "formula": "\\begin{align*} \\gamma _ { \\tau } ( \\varepsilon ) : = \\left \\{ \\begin{array} { l l } \\{ z \\in \\mathbb { H } \\ ; : \\ ; | z - \\tau | = \\varepsilon \\} & \\tau \\in \\mathbb { H } , \\\\ \\{ z \\in \\mathcal { F } _ N \\ ; : \\ ; \\mathrm { I m } ( \\sigma _ { \\tau } z ) = 1 / \\varepsilon \\} & \\tau \\in \\{ i \\infty \\} \\cup \\mathbb { Q } . \\end{array} \\right . \\end{align*}"}
-{"id": "9040.png", "formula": "\\begin{align*} \\mathbb { E } [ F ( x ) F ( y ) ] = \\int e \\left ( \\langle x - y , \\lambda \\rangle \\right ) d \\mu ( \\lambda ) . \\end{align*}"}
-{"id": "1506.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r @ { \\ , } c @ { \\ , } c @ { \\ } l @ { \\quad } l @ { \\quad } l @ { \\ , } c } n _ { t } & + & u \\cdot \\ ! \\nabla n & = \\Delta n ^ m - \\nabla \\ ! \\cdot ( n S ( x , n , c ) \\nabla c ) , \\ & x \\in \\Omega , & t > 0 , \\\\ c _ { t } & + & u \\cdot \\ ! \\nabla c & = \\Delta c - c + n , \\ & x \\in \\Omega , & t > 0 , \\\\ u _ { t } & + & ( u \\cdot \\nabla ) u & = \\Delta u + \\nabla P + n \\nabla \\phi , \\ & x \\in \\Omega , & t > 0 , \\\\ & & \\nabla \\cdot u & = 0 , \\ & x \\in \\Omega , & t > 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "8212.png", "formula": "\\begin{align*} & Z = Z ( \\tau ) : = \\frac { \\eta ( 5 0 \\tau ) } { \\eta ( 2 \\tau ) } , \\\\ & F = F ( \\tau ) : = \\frac { 1 } { 2 4 } \\left ( 5 0 E _ 2 ( 1 0 \\tau ) - 2 5 E _ 2 ( 5 \\tau ) - 2 E _ 2 ( 2 \\tau ) + E _ 2 ( \\tau ) \\right ) . \\end{align*}"}
-{"id": "3246.png", "formula": "\\begin{align*} V _ { F } ^ * = \\left \\{ \\mathbf { z } \\in \\mathbb { C } ^ n : \\nabla F ( \\mathbf { z } ) = \\mathbf { 0 } \\right \\} , \\end{align*}"}
-{"id": "1924.png", "formula": "\\begin{align*} ( x ^ 2 + x ) \\frac { A _ j ^ 2 } { Z _ j ^ 2 } = 1 - A _ j ^ 2 , \\end{align*}"}
-{"id": "4571.png", "formula": "\\begin{align*} \\mathcal { H } ^ s ( F ) = \\lim _ { \\delta \\to 0 } \\mathcal { H } ^ s _ { \\delta } ( F ) . \\end{align*}"}
-{"id": "9098.png", "formula": "\\begin{align*} u _ + \\in L ^ { p \\gamma _ n } ( \\mathbb { R } ^ N ) , \\gamma _ n = \\left ( \\frac { N } { N - s p } \\right ) ^ n \\end{align*}"}
-{"id": "2236.png", "formula": "\\begin{align*} \\overline { \\partial } f ( x + \\i y ) = ( \\i / 2 ) ( f ( x + \\i 0 ) - f ( x - \\i 0 ) ) \\delta ( y ) \\end{align*}"}
-{"id": "2259.png", "formula": "\\begin{align*} D f ( k ) & = k f ( k ) \\\\ \\eta ( k ) & = \\mathrm { s i g n } ( k ) | k | ^ { - 1 / 2 } \\\\ \\chi ( k ) & = \\tfrac { 1 } { 2 } | k | ^ { - 1 / 2 } \\end{align*}"}
-{"id": "1302.png", "formula": "\\begin{align*} h _ n = g _ { n , 1 } x _ { n , 1 } ^ { \\varepsilon _ { n , 1 } } y _ { n , 1 } ^ { - \\varepsilon _ { n , 1 } } g _ { n , 1 } ^ { - 1 } g _ { n , 2 } x _ { n , 2 } ^ { \\varepsilon _ { n , 2 } } y _ { n , 2 } ^ { - \\varepsilon _ { n , 2 } } g _ { n , 2 } ^ { - 1 } \\cdots g _ { n , m _ n } x _ { n , m _ n } ^ { \\varepsilon _ { n , m _ n } } y _ { n , m _ n } ^ { - \\varepsilon _ { n , m _ n } } g _ { n , m _ n } ^ { - 1 } \\sum \\limits _ { i = 1 } ^ { m _ n } \\rho _ { g _ { n , i } } ( x _ { n , i } , y _ { n , i } ) < \\delta . \\end{align*}"}
-{"id": "7340.png", "formula": "\\begin{align*} \\begin{cases} \\rho ( x ) \\varphi _ { t t } - \\operatorname { d i v } ( K ( x ) \\nabla w ) + V _ 1 ( x , t ) \\varphi = V _ 3 ( x , t ) \\psi & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ \\rho ( x ) \\psi _ { t t } - \\operatorname { d i v } ( K ( x ) \\nabla \\psi ) + V _ 2 ( x , t ) \\psi = V _ 3 ( x , t ) \\varphi & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ \\varphi = \\psi = 0 & ~ \\hbox { i n } ~ \\omega \\times ( 0 , T ) . \\end{cases} \\end{align*}"}
-{"id": "5748.png", "formula": "\\begin{align*} | | J _ 0 ( \\gamma _ m x ) \\sin ( n \\pi y ) | | _ 2 ^ 2 = \\int _ 0 ^ 1 \\int _ 0 ^ 1 x | J _ 0 ( \\gamma _ m x ) \\sin ( n \\pi y ) | ^ 2 d x d y = \\int _ 0 ^ 1 | s i n ( n \\pi y ) | ^ 2 \\left ( \\int _ 0 ^ 1 x J _ 0 ^ 2 ( \\gamma _ m x ) d x \\right ) d y . \\end{align*}"}
-{"id": "6318.png", "formula": "\\begin{align*} \\mathrm { T V } ( R , P ) = o ( 1 ) . \\end{align*}"}
-{"id": "4293.png", "formula": "\\begin{align*} \\frac { 2 \\binom { s - 2 } { r - 1 } e ( \\overline { G } ) } { n - s + 2 } \\geq \\frac { 2 \\binom { s - 2 } { r - 1 } } { n } \\left ( \\binom { n } { 2 } - e ( G ) \\right ) \\geq \\frac { 2 \\binom { s - 2 } { r - 1 } } { n } \\left ( \\frac { n ^ 2 } { 2 ( r - 1 ) } - \\frac { n } { 2 } - \\alpha _ { r , s } \\right ) . \\end{align*}"}
-{"id": "3449.png", "formula": "\\begin{align*} v _ M ( \\theta ) = \\begin{cases} ( M + 1 ) \\log \\sin \\theta - M \\log \\sin \\frac { M \\theta } { M + 1 } - \\log \\sin \\frac { \\theta } { M + 1 } - \\log ( M + 1 ) , & \\theta \\in ( 0 , \\pi ) , \\\\ M \\log ( 1 + M ^ { - 1 } ) , & \\theta = 0 , \\end{cases} \\end{align*}"}
-{"id": "5406.png", "formula": "\\begin{align*} H ( \\phi _ t ( x ) , t ) : = \\alpha \\left ( \\tfrac { d } { d t } \\phi _ t ( x ) \\right ) \\ge 0 \\end{align*}"}
-{"id": "2037.png", "formula": "\\begin{align*} \\sum _ { \\substack { a \\mod d \\\\ \\gcd ( a , d ) = 1 } } \\left < \\frac { a } { d } \\right > ^ { \\pm } e ^ { 2 \\pi i n \\frac { a } { d } } = \\pi \\sum _ { \\substack { a \\mod d \\\\ \\gcd ( a , d ) = 1 } } \\left ( \\Lambda ( 1 , f , \\frac { a } { d } ) \\pm \\Lambda ( 1 , f , - \\frac { a } { d } ) \\right ) e ^ { 2 \\pi i n \\frac { a } { d } } . \\end{align*}"}
-{"id": "3500.png", "formula": "\\begin{align*} \\frac { 1 } { s - t } = \\int _ { 0 } ^ { \\infty } e ^ { - ( s - t ) u } d u , \\Re ( s - t ) > 0 , \\end{align*}"}
-{"id": "6233.png", "formula": "\\begin{align*} y _ { \\lambda } = \\ell ( u ) + \\sqrt { \\lambda } \\ell ( v ) ^ { * } \\end{align*}"}
-{"id": "295.png", "formula": "\\begin{align*} \\varpi _ * ( \\alpha ) & = \\sum _ i c _ { M , i } ( \\Lambda _ 1 \\circ \\Lambda _ 2 ) \\otimes H ^ i \\\\ \\varpi _ * ( \\beta ) & = \\sum _ { i , j } ( c _ { M , i } ( \\Lambda _ 1 ) * c _ { M , j } ( \\Lambda _ 2 ) ) \\otimes H ^ { i + j } \\end{align*}"}
-{"id": "4596.png", "formula": "\\begin{align*} \\int \\int | \\hat { \\mu } ( \\omega ) | ^ 2 | \\hat { \\mu } ( g \\omega ) | ^ 2 \\hat { \\phi ^ D _ \\delta } ( \\omega ) \\psi ( 2 ^ { - j } \\omega ) d \\omega d g = C ( n ) \\int \\left ( \\int _ { S ^ { n - 1 } } | \\hat { \\mu } ( t \\sigma ) | ^ 2 d \\sigma \\right ) ^ 2 \\psi ( 2 ^ { - j } t ) \\hat { \\phi } ^ D _ \\delta ( t ) t ^ { n - 1 } d t , \\end{align*}"}
-{"id": "4671.png", "formula": "\\begin{align*} \\pi _ { n , g } ( \\lambda _ { 1 } , \\ldots , \\lambda _ { n } ; a ) = \\sum _ { j = 0 } ^ { g } \\pi _ { h , j } ( \\lambda _ { 1 } , \\ldots , \\lambda _ { h } ; a ) \\cdot \\pi _ { n - h , g - j } ( \\lambda _ { h + 1 } , \\ldots , \\lambda _ { n } ; a ) . \\end{align*}"}
-{"id": "4201.png", "formula": "\\begin{align*} R ( z ) = \\begin{cases} S ( z ) N ^ { - 1 } ( z ) , & z \\in \\mathbb { C } \\setminus \\left ( \\Sigma _ S \\cup \\overline { D ( 0 , R ) } \\cup \\overline { D ( q , r _ q ) } \\right ) , \\\\ S ( z ) P ^ { - 1 } ( z ) , & z \\in D ( 0 , R ) \\setminus \\left ( \\Sigma _ S \\cup \\partial D ( 0 , r _ n ) \\right ) , \\\\ S ( z ) Q ^ { - 1 } ( z ) , & z \\in D ( q , r _ q ) \\setminus \\Sigma _ S . \\end{cases} \\end{align*}"}
-{"id": "466.png", "formula": "\\begin{align*} g _ k = & ( y _ \\alpha - y _ \\beta ) ^ { d - a _ i } \\sum _ { j = 0 } ^ { a _ i - k } \\lambda _ j ( y _ \\alpha - y _ \\beta ) ^ j ( y _ \\alpha + y _ \\beta ) ^ { a _ i - k - j } \\\\ = & ( \\sum _ { s = 0 } ^ { d - a _ i } ( - 1 ) ^ s \\binom { d - a _ i } { s } y ^ s _ \\alpha y ^ { d - a _ i - s } _ \\beta ) ( \\sum _ { t = 0 } ^ { a _ i - k } \\lambda _ t ( y _ \\alpha - y _ \\beta ) ^ t ( y _ \\alpha + y _ \\beta ) ^ { a _ i - k - t } ) \\end{align*}"}
-{"id": "3347.png", "formula": "\\begin{align*} - n \\log R = n \\frac { \\log \\log r } { \\log r } \\log t , R ^ L \\le t ^ { - \\frac { \\log \\log r } { \\log r } ( 2 \\log _ t r - 1 ) } = \\frac { t ^ { \\frac { \\log \\log r } { \\log r } } } { ( \\log r ) ^ 2 } \\le \\frac { t } { ( \\log r ) ^ 2 } . \\end{align*}"}
-{"id": "4209.png", "formula": "\\begin{align*} R ^ { - 1 } ( y _ n ) R ( x _ n ) = \\mathbb { I } + \\mathcal { O } \\left ( ( x - y ) n ^ { - \\frac { 5 } { 2 } } \\right ) \\end{align*}"}
-{"id": "9675.png", "formula": "\\begin{align*} f = X x _ 0 . \\end{align*}"}
-{"id": "5546.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\partial _ t \\| \\rho ^ 2 \\phi _ { \\varepsilon } \\| _ { L ^ { 2 , \\varepsilon } } ^ 2 + \\lambda \\| \\rho \\phi _ { \\varepsilon } \\| _ { L ^ { 4 , \\varepsilon } } ^ 4 + m ^ { 2 } \\| \\rho ^ 2 \\psi _ { \\varepsilon } \\| _ { L ^ { 2 , \\varepsilon } } ^ 2 + \\| \\rho ^ 2 \\nabla _ { \\varepsilon } \\psi _ { \\varepsilon } \\| _ { L ^ { 2 , \\varepsilon } } ^ 2 = \\Theta _ { \\rho ^ 4 , \\varepsilon } + \\Psi _ { \\rho ^ 4 , \\varepsilon } \\end{align*}"}
-{"id": "6966.png", "formula": "\\begin{align*} \\theta _ 0 = - \\frac { d y _ 0 } { y _ 1 } , \\quad \\theta _ 1 = - \\frac { d y _ 1 } { y _ 1 } + y _ 2 d y _ 0 , \\quad \\theta _ 2 = - y _ 1 d y _ 2 + y _ 1 \\big ( y _ 3 - 2 y _ 2 ^ 2 \\big ) d y _ 0 . \\end{align*}"}
-{"id": "8067.png", "formula": "\\begin{align*} M _ { o d d } \\begin{pmatrix} n _ 1 \\\\ n _ 2 \\\\ m _ 1 \\\\ m _ 2 \\\\ \\vdots \\\\ m _ h \\\\ \\vdots \\\\ m _ { ( n - 1 ) / 2 } \\end{pmatrix} = \\begin{pmatrix} 2 n \\\\ - 1 - \\displaystyle { \\sum _ { i = n + 3 } ^ { 2 n } } c ^ { n + 3 - 2 i } P _ { i - n - 3 , - i } \\\\ 0 \\\\ 0 \\\\ \\vdots \\\\ 0 \\\\ 0 \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "7209.png", "formula": "\\begin{align*} L _ r f ( x ) = \\mathcal { F } ^ { - 1 } ( m _ r \\hat f ) ( x ) , \\end{align*}"}
-{"id": "3672.png", "formula": "\\begin{align*} E ( \\sum _ { i \\in B } w _ i y _ i - Y | x _ U ) = E ( \\sum _ { i \\in B } w _ i x _ i ^ { \\top } \\beta ) - X ^ { \\top } \\beta = 0 \\end{align*}"}
-{"id": "7689.png", "formula": "\\begin{align*} \\psi _ { + } ^ { \\mathrm { e } } = \\psi _ { - } ^ { \\mathrm { e } } \\quad \\ ; \\ ; \\psi _ { + } ^ { \\mathrm { o } } = - \\psi _ { - } ^ { \\mathrm { o } } . \\end{align*}"}
-{"id": "6527.png", "formula": "\\begin{align*} \\lim _ { l \\rightarrow \\infty } \\ , \\lim _ { t \\rightarrow \\infty } \\ \\ \\sup _ { \\xi \\geq 0 , u \\in K } \\Biggl [ \\frac { 1 } { l } \\int _ { \\xi } ^ { \\xi + l } \\bigl \\| q ( t + s , u ) \\bigr \\| ^ { p } \\ , d s \\Biggr ] ^ { 1 / p } = 0 . \\end{align*}"}
-{"id": "8277.png", "formula": "\\begin{align*} w _ 0 - i a _ 0 ( \\partial _ { \\alpha } \\xi _ 0 + 1 ) = - i , \\end{align*}"}
-{"id": "6183.png", "formula": "\\begin{align*} 2 r H _ r ( t , z ) = s \\tilde H _ s ( s , y ) + y \\tilde H _ y ( s , y ) . \\end{align*}"}
-{"id": "6509.png", "formula": "\\begin{align*} \\bar { f } ( x ) = \\int _ { H } f ( x , y ) \\mu ^ { x } ( d y ) , x \\in { H } , \\end{align*}"}
-{"id": "7861.png", "formula": "\\begin{align*} | \\mathfrak m _ N ( \\xi ) - \\lambda _ N ^ 2 ( \\xi ) | \\le 1 7 \\min \\Big \\{ e ^ { - \\frac { c \\kappa ( d , N ) ^ 2 } { 4 0 0 } \\sum _ { i = 1 } ^ d \\cos ^ 2 ( \\pi \\xi _ i ) } , \\kappa ( d , N ) ^ 2 \\sum _ { i = 1 } ^ d \\cos ^ 2 ( \\pi \\xi _ i ) \\Big \\} . \\end{align*}"}
-{"id": "2307.png", "formula": "\\begin{align*} H R ( z ) | f ) & = - | f ) + z R ( z ) | f ) \\\\ ( f | R ( z ) H f & = - ( f | + z ( f | R ( z ) f . \\end{align*}"}
-{"id": "6767.png", "formula": "\\begin{align*} \\phi ( \\gamma x s ^ { - 1 } ) = \\rho ( \\gamma ) \\phi ( x ) \\tau ( s ) ^ { - 1 } . \\end{align*}"}
-{"id": "8253.png", "formula": "\\begin{align*} & U _ 5 ( F ) = F ( 1 + 4 \\cdot 5 t ) , \\\\ & U _ 5 ( F t ^ { - 1 } ) = F ( - 3 - 3 \\cdot 5 ^ 2 t - 5 ^ 3 t ^ 2 ) , \\\\ & U _ 5 ( F t ^ { - 2 } ) = F ( 1 + 5 ^ 2 t + 0 \\cdot t ^ 2 - 5 ^ 5 t ^ 3 ) , \\\\ & U _ 5 ( F t ^ { - 3 } ) = F ( 9 \\cdot 5 + 9 \\cdot 5 ^ 3 t + 0 \\cdot t ^ 2 + 0 \\cdot t ^ 3 - 5 ^ 7 t ^ 4 ) , \\\\ & U _ 5 ( F t ^ { - 4 } ) = F ( - 5 1 \\cdot 5 - 5 1 \\cdot 5 ^ 3 t + 0 \\cdot t ^ 2 + 0 \\cdot t ^ 3 + 0 \\cdot t ^ 4 - 5 ^ 9 t ^ 5 ) . \\end{align*}"}
-{"id": "9720.png", "formula": "\\begin{align*} \\lambda _ { X _ \\circ } \\in [ 1 + s , \\infty ) \\quad s : = \\frac { 1 - a } { 2 } . \\end{align*}"}
-{"id": "4573.png", "formula": "\\begin{align*} I _ t ( \\mu ) = \\gamma ( n , s ) \\int | \\hat { \\mu } ( \\omega ) | ^ 2 | \\omega | ^ { t - n } d \\omega , \\end{align*}"}
-{"id": "1774.png", "formula": "\\begin{align*} \\begin{aligned} D e ^ { \\frac { \\alpha } { 2 } M _ { s , t } ^ { ( n ) } } & = \\frac { \\alpha } { 2 } \\exp \\left ( \\frac { \\alpha } { 2 } M _ { s ; t _ { 1 } , \\cdots , t _ { n } } \\right ) D M _ { s , t } ^ { ( n ) } . \\end{aligned} \\end{align*}"}
-{"id": "3923.png", "formula": "\\begin{align*} \\rho _ r ( x , y ) = \\left ( \\sum _ i \\left | x _ i - y _ i \\right | ^ r \\right ) ^ { 1 / r } \\end{align*}"}
-{"id": "5684.png", "formula": "\\begin{align*} \\| x _ \\alpha ^ A - x _ 0 \\| \\le & \\| x _ 1 ^ A - x _ 0 \\| + t _ \\alpha M \\le \\delta - \\frac { M } { \\eta } \\| x _ 1 ^ A - x _ 1 ^ B \\| + \\frac { \\| x _ 1 ^ A - x _ 1 ^ B \\| } { \\eta } M = \\delta \\ , \\end{align*}"}
-{"id": "7964.png", "formula": "\\begin{align*} S _ 1 ^ A \\sqcup \\dots \\sqcup S _ { s _ A } ^ A = [ n ] \\setminus A , \\end{align*}"}
-{"id": "1759.png", "formula": "\\begin{align*} C _ { H } = \\max \\left \\{ 2 ^ { 2 H - 1 } - 1 , 1 \\right \\} \\leq 1 \\end{align*}"}
-{"id": "7497.png", "formula": "\\begin{align*} E P _ { \\nu } = \\left ( \\frac { 1 - d } { 2 } + \\nu \\right ) P _ { \\nu } + \\mathrm { c o n s t . } \\end{align*}"}
-{"id": "6285.png", "formula": "\\begin{align*} & R _ k ( \\rho , \\epsilon , \\Lambda , n ) = O \\left ( \\Lambda ^ 2 \\sqrt { \\frac { k } { n } } + \\Lambda \\left ( \\frac { \\log k } { \\rho n } + \\frac { k ^ 2 } { \\rho n ^ 2 } \\right ) \\right ) + O \\left ( \\Lambda ^ 2 \\frac { k ^ 2 \\log n } { n \\epsilon } + \\frac { \\Lambda ^ 2 } { n ^ 2 \\rho ^ 2 \\epsilon ^ 2 } \\right ) , \\end{align*}"}
-{"id": "6781.png", "formula": "\\begin{align*} \\Phi '' ( t , \\Phi ( t , \\beta ) ) = \\left ( u _ t + u \\nabla u \\right ) ( t , \\Phi ( t , \\beta ) ) = - \\nabla _ { \\ ! x } p ( t , \\Phi ( t , \\beta ) ) \\ , . \\end{align*}"}
-{"id": "7996.png", "formula": "\\begin{align*} \\mathcal { K } ^ { p , q } ( \\mathcal { U } , U _ { 0 } ) : = \\left \\lbrace T \\in \\mathcal { K } ^ { p , q } ( \\mathcal { U } ) \\ , \\mid \\ , T _ 0 = 0 \\right \\rbrace = \\mathcal { K } ^ { p , q } ( U _ { 1 } ) \\oplus \\mathcal { K } ^ { p , q - 1 } ( U _ { 0 1 } ) . \\end{align*}"}
-{"id": "4966.png", "formula": "\\begin{align*} F S ( ( a _ i ) _ { i \\in \\Z } ) = \\left \\{ \\sum _ { i \\in I } a _ i \\ , : \\ , I \\subseteq \\Z , 0 < \\vert I \\vert < \\infty \\right \\} \\end{align*}"}
-{"id": "2029.png", "formula": "\\begin{align*} \\frac { 1 } { M } \\sum _ { 0 \\le a \\le M } \\left < \\frac { a } { M } \\right > ^ { \\pm } h \\left ( \\frac { a } { M } \\right ) = \\end{align*}"}
-{"id": "3896.png", "formula": "\\begin{align*} g = \\mathrm { d i a g } ( \\lambda _ 1 , \\bar \\lambda _ 1 ^ { - 1 } , \\lambda _ 3 , \\ldots , \\lambda _ { n + 1 } ) . \\end{align*}"}
-{"id": "6481.png", "formula": "\\begin{align*} t _ { B } = \\frac { 1 + 2 a b + 2 ( a ^ { 2 } + b ^ { 2 } - 1 ) } { b ^ { 2 } } = \\frac { ( a + b ) ^ { 2 } - 1 + a ^ { 2 } + b ^ { 2 } } { b ^ { 2 } } = \\frac { a ^ { 2 } + b ^ { 2 } } { b ^ { 2 } } > 1 . \\end{align*}"}
-{"id": "7333.png", "formula": "\\begin{align*} \\begin{cases} \\rho ( x ) w _ { t t } - \\operatorname { d i v } ( K ( x ) \\nabla w ) + 2 v u z + v ^ 2 w = 0 & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ \\rho ( x ) z _ { t t } - \\operatorname { d i v } ( K ( x ) \\nabla z ) + 2 u v w + u ^ 2 z = 0 & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ w = z = 0 & ~ \\hbox { i n } ~ \\omega \\times ( 0 , T ) , \\end{cases} \\end{align*}"}
-{"id": "186.png", "formula": "\\begin{align*} & \\pi _ { X _ { 1 , C } , X _ { 2 , V } , X _ { 3 , C } , X _ { 4 , V } } = X _ { 1 , C } \\wedge X _ { 2 , V } + X _ { 3 , C } \\wedge X _ { 4 , V } \\end{align*}"}
-{"id": "2429.png", "formula": "\\begin{align*} H _ 1 = \\left \\langle \\frac { 1 } { 2 } \\begin{pmatrix} 1 + \\sqrt { - 1 } & 1 + \\sqrt { - 1 } \\\\ 1 + \\sqrt { - 1 } & 1 - \\sqrt { - 1 } \\end{pmatrix} , \\begin{pmatrix} 1 & 0 \\\\ 0 & \\sqrt { - 1 } \\end{pmatrix} \\right \\rangle . \\end{align*}"}
-{"id": "7987.png", "formula": "\\begin{align*} \\log g ( n ) & = \\sum _ { p \\leq \\sqrt { n } } { \\frac { \\xi _ { p } ( 1 + \\xi _ { p } ) } { 2 } } \\log p + \\sum _ { \\sqrt { n } < p \\leq n } \\log p \\\\ & = \\sum _ { p \\leq \\sqrt { n } } { \\frac { \\xi _ { p } ( 1 + \\xi _ { p } ) } { 2 } } \\log p + \\sum _ { \\sqrt { n } < p \\leq \\frac { n } { 2 } } \\log p + \\sum _ { \\frac { n } { 2 } < p \\leq n } \\log p \\ , . \\end{align*}"}
-{"id": "8192.png", "formula": "\\begin{align*} \\frac { 1 } { \\mathbb { V } ( D _ n ) } \\Sigma _ n & = \\frac { 1 } { \\mathbb { V } ( D _ n ) } \\begin{pmatrix} \\mathbb { V } ( D _ n ) & & ( D _ n , D ' _ n ) \\\\ ( D _ n , D ' _ n ) & & \\mathbb { V } ( D ' _ n ) \\\\ \\end{pmatrix} \\\\ & \\stackrel { a . s . } { \\rightarrow } I , \\end{align*}"}
-{"id": "5397.png", "formula": "\\begin{align*} c ( 1 ) = - \\frac 1 2 . \\end{align*}"}
-{"id": "5843.png", "formula": "\\begin{align*} \\Gamma ( c ) = c ( - 1 ) \\rho ( c ) = \\int _ { F } \\psi ( x ) c ( x ) | x | _ F ^ { - 1 } d x , \\end{align*}"}
-{"id": "1342.png", "formula": "\\begin{align*} \\left ( \\sum _ { i \\in I } \\varphi _ i \\right ) \\circ \\left ( \\bigoplus _ { i \\in I } \\psi _ i \\right ) \\left ( e ^ \\frac { 1 } { 2 } b e ^ \\frac { 1 } { 2 } \\right ) = \\sum _ { i \\in I } e _ i \\circ \\kappa _ i \\left ( e ^ \\frac { 1 } { 2 } b e ^ \\frac { 1 } { 2 } \\right ) = \\sum _ { i \\in I } \\kappa _ i \\left ( e ^ \\frac { 1 } { 2 } b e ^ \\frac { 1 } { 2 } \\right ) \\end{align*}"}
-{"id": "8757.png", "formula": "\\begin{align*} D ^ { \\sigma , \\alpha } _ { 0 , t } ( u ( t ) - u ( 0 ) ) = - u ( t ) + f ( u , t ) , { \\quad } u ( 0 ) = u _ 0 , { \\quad } t \\in ( 0 , T ] , \\end{align*}"}
-{"id": "5461.png", "formula": "\\begin{align*} & K _ t - \\tilde K _ t - \\int _ 0 ^ t \\big ( Z _ s - \\tilde Z _ s \\big ) \\ , d \\hat W _ s + \\int _ 0 ^ t \\int _ U \\big ( L _ s ( z ) - \\tilde L _ s ( z ) \\big ) \\ , \\lambda _ \\pi ( d z ) d s \\\\ & = \\int _ 0 ^ t \\int _ U \\big ( L _ s ( z ) - \\tilde L _ s ( z ) \\big ) \\ , \\hat \\pi ( d s \\ , d z ) + \\int _ 0 ^ t \\int _ \\Lambda \\big ( R _ s ( b ) - \\tilde R _ s ( b ) \\big ) \\ , \\hat \\theta ( d s \\ , d b ) , \\end{align*}"}
-{"id": "4738.png", "formula": "\\begin{align*} T _ p = 0 , T _ { p - n } = M p = 1 , \\dots , n . \\end{align*}"}
-{"id": "4396.png", "formula": "\\begin{align*} g _ { \\delta } ( t , t ^ { - 1 } ) : = f ( t ) \\end{align*}"}
-{"id": "3737.png", "formula": "\\begin{align*} f ( \\mathbf { r } ) = f \\theta ( a - r ) + f _ { 0 } \\theta ( r - a ) . \\end{align*}"}
-{"id": "7347.png", "formula": "\\begin{align*} \\phi ( x ) = \\left \\{ \\begin{array} { c } \\eta ( x ) \\phi _ i ( x ) x \\in V _ i \\\\ 0 \\end{array} \\right . \\end{align*}"}
-{"id": "3261.png", "formula": "\\begin{align*} z ^ 3 + p z + q = 0 . \\end{align*}"}
-{"id": "9289.png", "formula": "\\begin{align*} B ( z - r \\nu ( z ) , r ) \\cap \\Omega = \\emptyset , \\end{align*}"}
-{"id": "7092.png", "formula": "\\begin{align*} \\begin{cases} - \\partial _ { t t } w _ n + \\Delta w _ n = | \\nabla F _ n ^ \\omega + \\nabla w _ n | ^ 2 + 2 \\nabla u _ { n - 1 } \\nabla w _ n + 2 \\nabla P _ { > N ^ \\gamma } u _ { n - 1 } \\cdot \\nabla F _ n ^ \\omega ~ , \\\\ w _ n | _ { t = 0 } = 0 ~ , ~ ~ \\partial _ t w _ n | _ { t = 0 } = 0 ~ . \\end{cases} \\end{align*}"}
-{"id": "959.png", "formula": "\\begin{align*} \\Pr \\left ( \\liminf _ { j \\rightarrow \\infty } \\| \\nabla f ( X _ j ) \\| = 0 \\right ) = 1 . \\end{align*}"}
-{"id": "8181.png", "formula": "\\begin{align*} 0 = \\sum _ { i \\in I _ n } \\Lambda _ i ^ * g _ i = \\sum _ { i \\in I _ n } \\Lambda _ i ^ * ( \\alpha _ i e _ 1 ) = \\sum _ { i \\in I _ n } \\alpha _ i \\Lambda _ i ^ * e _ 1 , \\end{align*}"}
-{"id": "5057.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j \\in \\mathbb { S } } \\Psi _ j ^ * A _ j h \\right \\| ^ 2 & = \\left \\langle \\sum _ { j \\in \\mathbb { S } } c _ j A _ j ^ * A _ j h , \\sum _ { k \\in \\mathbb { S } } c _ k A _ k ^ * A _ k h \\right \\rangle = \\sum _ { j \\in \\mathbb { S } } c _ j ^ 2 \\| A _ j h \\| ^ 2 \\\\ & \\leq \\left ( \\sup \\{ c _ j ^ 2 \\} _ { j \\in \\mathbb { J } } \\right ) \\sum \\limits _ { j \\in \\mathbb { S } } \\| A _ j h \\| ^ 2 , ~ \\end{align*}"}
-{"id": "6128.png", "formula": "\\begin{align*} \\Omega \\big ( D _ A ^ a D _ B ^ b ( q ) \\big ) = ( - 1 ) ^ { b + 1 } X ^ a Y ^ b \\big ( \\alpha _ { d - \\mu _ 1 } Y - \\beta _ { d - \\mu _ 2 } X \\big ) . \\end{align*}"}
-{"id": "565.png", "formula": "\\begin{align*} ( 0 , 0 ) _ { p ^ n + 1 } & = p ^ n - 2 , \\\\ ( 0 , i ) _ { p ^ n + 1 } = ( i , 0 ) _ { p ^ n + 1 } = ( i , i ) _ { p ^ n + 1 } & = p ^ n ( p ^ n - 1 ) & & \\textrm { f o r } i \\ne 0 , \\\\ ( i , j ) _ { p ^ n + 1 } & = ( p ^ n - 1 ) ^ 2 & & \\textrm { f o r } i \\ne j \\textrm { a n d } i , j \\ne 0 , \\end{align*}"}
-{"id": "305.png", "formula": "\\begin{align*} \\Lambda _ { Y ^ \\pm } \\ ; \\subset \\ ; \\Lambda _ { Z _ 1 } \\circ \\Lambda _ { Z _ 2 } \\textnormal { w h e r e } Y ^ \\pm \\ ; = \\ ; ( g _ 1 \\pm g _ 2 ) ( Z ) . \\end{align*}"}
-{"id": "8021.png", "formula": "\\begin{align*} \\Omega _ R / d R \\cong \\mathbb { C } \\omega _ 0 \\oplus \\bigoplus _ { i = 3 } ^ { 4 } U _ i ^ { \\frac { ( 1 - ( - 1 ) ^ k ) n } { 2 k } } \\oplus \\bigoplus _ { h = 1 } ^ { k - 1 } V _ { h } ^ { \\oplus \\frac { ( 1 - ( - 1 ) ^ h ) n } { k } } . \\end{align*}"}
-{"id": "6602.png", "formula": "\\begin{align*} v _ n : = 2 \\sin \\pi \\left ( \\frac { ( - \\varphi ) ^ { n } } { 2 } - \\varepsilon \\right ) . \\end{align*}"}
-{"id": "4982.png", "formula": "\\begin{align*} { Z } ( C / \\mathbb { F } _ q , T ) = \\frac { L ( C / \\mathbb { F } _ q , T ) } { ( 1 - T ) ( 1 - q T ) } . \\end{align*}"}
-{"id": "1246.png", "formula": "\\begin{align*} e ^ { i a \\ln | \\frac { \\eta + \\nu } { \\eta - \\nu } | } & = e ^ { i a \\ln | 1 + \\eta / \\nu | } e ^ { - i a \\ln | 1 - \\eta / \\nu | } = \\left ( 1 + i a \\frac { \\eta } { \\nu } + O ( \\eta ^ 2 / \\nu ^ 2 ) \\right ) \\left ( 1 + i a \\frac { \\eta } { \\nu } + O ( \\eta ^ 2 / \\nu ^ 2 ) \\right ) \\\\ & = 1 + 2 i a \\frac { \\eta } { \\nu } + O ( \\eta ^ 2 / \\nu ^ 2 ) , \\end{align*}"}
-{"id": "6665.png", "formula": "\\begin{align*} \\zeta ( s ) & = \\frac { 1 } { 1 - 2 ^ { 1 - s } } \\ , \\eta ( s ) \\\\ & = \\frac { 1 } { 1 - 2 ^ { 1 - s } } \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { n + 1 } } { n ^ { s } } , { \\rm R e } \\ , s > 0 , s \\neq 1 . \\end{align*}"}
-{"id": "4007.png", "formula": "\\begin{align*} a ' _ j ( p ^ 2 n ) = a ' _ j ( n ) a ' _ j ( p ^ 2 ) - \\chi ( p ) a ' _ j ( n ) \\delta _ { p \\mid n } - \\chi ( p ) a ' _ j ( n / p ^ 2 ) , \\end{align*}"}
-{"id": "10020.png", "formula": "\\begin{align*} \\mho ( q , p , N ) = \\exp { \\left ( \\frac { \\log { N } } { \\log { \\log { N } } } \\left ( \\log { \\sqrt { \\frac { q } { p } } } + o ( 1 ) \\right ) \\right ) } . \\end{align*}"}
-{"id": "7807.png", "formula": "\\begin{align*} A ( q , 2 ) \\le A ( q , 1 ) ^ { 1 / 2 } A ( q , \\infty ) ^ { 1 / 2 } = A ( q ' , \\infty ) ^ { 1 / 2 } A ( q , \\infty ) ^ { 1 / 2 } \\le C _ { q ' } ^ { 1 / 2 } C _ { q } ^ { 1 / 2 } , \\end{align*}"}
-{"id": "4949.png", "formula": "\\begin{align*} ( y ^ n ) ^ { 3 n ^ { d - 3 } + n ^ { d - 4 } + \\cdots + n + 1 } \\equiv \\left ( \\left ( \\sum _ { j = 1 } ^ { d - 3 } ( - 1 ) ^ { j } v _ i z ^ n \\right ) + ( - 1 ) ^ { d } x z ^ n \\right ) ^ { 3 n ^ { d - 3 } + n ^ { d - 4 } + \\cdots + n + 1 } \\equiv 0 . \\end{align*}"}
-{"id": "9721.png", "formula": "\\begin{align*} \\Gamma ( u ) = { \\rm R e g } ( u ) \\cup { \\rm S i n g } ( u ) \\cup { \\rm O t h e r } ( u ) , \\end{align*}"}
-{"id": "3922.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k { b _ i } ^ u \\leq \\sum _ { i = 1 } ^ k { b _ i } ^ v { b _ i } ^ l \\leq b ^ v \\sum _ { i = 1 } ^ k { b _ i } ^ l \\leq \\left ( \\sum _ { i = 1 } ^ k b _ i \\right ) ^ v \\left ( \\sum _ { i = 1 } ^ k b _ i \\right ) ^ l \\leq \\left ( \\sum _ { i = 1 } ^ k b _ i \\right ) ^ u \\end{align*}"}
-{"id": "5731.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow 0 } x w _ x & = 0 & & w ( 1 , y , t ) = 0 \\\\ w ( x , 0 , t ) & = 0 & & w ( x , 1 , t ) = 0 \\\\ w ( x , y , 0 ) & = 0 & & w _ t ( x , y , 0 ) = 0 . \\end{align*}"}
-{"id": "209.png", "formula": "\\begin{align*} \\pi _ { T M , X _ { 1 , C } , Y _ { 1 , V } , X _ { 2 , C } , Y _ { 2 , V } , c } ( { \\bf x } , { \\bf y } ) = \\left ( \\begin{array} { c c c | c c c } 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & x ^ 3 & x ^ 1 \\\\ 0 & 0 & 0 & 0 & 0 & 0 \\\\ \\hline 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & - x ^ 3 & 0 & 0 & 0 & y ^ 1 \\\\ 0 & - x ^ 1 & 0 & 0 & - y ^ 1 & 0 \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "1384.png", "formula": "\\begin{align*} J _ { ( i ) } = \\begin{cases} K _ { ( i ) } \\ , , & 1 \\leq i < l \\\\ K _ { ( i + 1 ) } \\ , , & l \\leq i \\leq n - 1 \\end{cases} \\ ; . \\end{align*}"}
-{"id": "2093.png", "formula": "\\begin{align*} & { \\rm P r } ( \\textbf { y } _ R = y | b _ 0 , { b } _ 1 , \\cdots , { b } _ M ) = \\frac { e ^ { - \\mathbb { E } ( \\textbf { y } _ R | b _ 0 , { b } _ 1 , \\cdots , { b } _ M ) } ( \\mathbb { E } ( \\textbf { y } _ R | b _ 0 , { b } _ 1 , \\cdots , { b } _ M ) ) ^ y } { y ! } , \\end{align*}"}
-{"id": "6177.png", "formula": "\\begin{align*} | g _ i ( x ' , x _ n , z ) - g _ i ( x ' , 0 , 0 ) - b _ i ( x ' ) r | = O ( r ^ { 3 / 2 } ) , ( x ' , 0 , 0 ) \\in B _ { 1 / 2 } \\cap L , \\end{align*}"}
-{"id": "5165.png", "formula": "\\begin{align*} \\psi _ { \\beta } ( x ) = \\exp ( - ( - x ) ^ { \\beta } ) 1 _ { ( x < 0 ) } + 1 _ { ( x \\geq 0 ) } . \\end{align*}"}
-{"id": "4106.png", "formula": "\\begin{align*} X ( z ) = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & i \\end{pmatrix} Y ( z ) \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & z ^ { \\frac { 1 } { 4 } } & 0 \\\\ 0 & 0 & z ^ { - \\frac { 1 } { 4 } } \\end{pmatrix} \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & \\frac { 1 } { \\sqrt { 2 } } & \\frac { 1 } { \\sqrt { 2 } } \\\\ 0 & \\frac { 1 } { \\sqrt { 2 } } & - \\frac { 1 } { \\sqrt { 2 } } \\end{pmatrix} \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & i \\end{pmatrix} , \\end{align*}"}
-{"id": "9394.png", "formula": "\\begin{align*} I ( u ) & = \\int _ { \\mathbb { R } ^ N } \\int _ { \\mathbb { R } ^ N } \\mathcal { P } \\Phi ( u ( x ) - u ( y ) ) K ( x , y ) d x d y - \\frac { 1 } { p _ { s } ^ { \\ast } } \\int _ \\Omega | u | ^ { p _ { s } ^ { \\ast } } - \\lambda \\int _ { \\Omega } F ( x , u ) d x \\\\ & \\geq \\frac { \\Lambda ^ { - 2 } } { p } [ u ] _ { s , p } ^ { p } - \\frac { 1 } { p ^ { \\ast } _ { s } } \\int _ { \\Omega } | u | ^ { p _ { s } ^ { \\ast } } - \\frac { 1 } { c _ { 2 } } \\lambda \\int _ { \\Omega } f ( x , u ) u d x \\end{align*}"}
-{"id": "5199.png", "formula": "\\begin{align*} H _ { 1 } ^ { t } ( x ) = H _ { 1 } ( x t ^ { \\rho } ) , x \\in \\mathbb { R } . \\end{align*}"}
-{"id": "4945.png", "formula": "\\begin{align*} \\left ( \\sum _ { j = 2 } ^ { d - 3 } ( - 1 ) ^ { j } v _ j z ^ { 3 n ^ 2 + n } \\right ) + ( - 1 ) ^ { d } ( x z ^ { 3 n ^ 2 + n } ) = - z ^ { 3 n ^ 2 } f _ 3 + v _ 1 z ^ { 3 n ^ 2 + n } + y ^ n z ^ { 3 n ^ 2 } \\in I _ n . \\end{align*}"}
-{"id": "6145.png", "formula": "\\begin{align*} a ^ 2 - e b ^ 2 = t , \\end{align*}"}
-{"id": "2598.png", "formula": "\\begin{align*} G ( \\theta ) ^ { \\gamma _ 0 - 1 } = \\| e ^ { - J _ G ( r ) } \\| _ { \\mathbf 1 _ { 0 \\leq r \\leq \\theta } d r ; \\frac { 1 } { \\gamma _ 0 - 1 } } , \\theta \\geq 0 , \\end{align*}"}
-{"id": "7910.png", "formula": "\\begin{align*} \\Delta _ 3 ' = \\frac { h ^ 2 } { 2 4 } ( f ' ( b ) - f ' ( a ) ) \\end{align*}"}
-{"id": "5362.png", "formula": "\\begin{align*} v _ { n + 1 \\ , m + 3 } ^ 1 = \\frac 1 { a _ { n m } } X _ { \\alpha + \\beta } . v _ { n m } ^ 1 \\end{align*}"}
-{"id": "6459.png", "formula": "\\begin{align*} g : = \\frac { \\sqrt { 1 + \\lambda } } { 2 } ( v + i w ) h : = \\frac { \\sqrt { 1 + \\lambda } } { 2 \\sqrt { \\lambda } } ( v - i w ) . \\end{align*}"}
-{"id": "1180.png", "formula": "\\begin{align*} | B + \\lambda ( A - B ) | _ 2 & \\leq | \\lambda | | A | _ 2 + | \\lambda - 1 | | B | _ 2 \\leq ( | \\lambda - 1 / 2 | + 1 / 2 ) ( | A | _ 2 + | B | _ 2 ) \\\\ & \\leq ( \\rho + 1 ) ( | A | _ 2 + | B | _ 2 ) = | A | _ 2 + | B | _ 2 + 1 . \\end{align*}"}
-{"id": "9560.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - \\alpha ) _ n ( \\tau q ) ^ n } { ( \\tau q ^ { N + 1 - n } ) _ n } = \\frac { ( - \\alpha \\tau q ) _ N } { ( \\tau q ) _ N } . \\end{align*}"}
-{"id": "8727.png", "formula": "\\begin{align*} \\rho _ j ( i _ j ) = { \\displaystyle \\sum _ { n _ j = 1 } ^ { i _ j } \\prod _ { r = 1 } ^ { n _ j - 1 } \\left ( { q _ j ( r ) \\over p _ j ( r ) } \\right ) \\over \\displaystyle \\sum _ { n _ j = 1 } ^ { N _ j } \\prod _ { r = 1 } ^ { n _ j - 1 } \\left ( { q _ j ( r ) \\over p _ j ( r ) } \\right ) } . \\end{align*}"}
-{"id": "7838.png", "formula": "\\begin{align*} x \\cdot \\xi = \\langle x , \\xi \\rangle = \\sum _ { k = 1 } ^ d x _ k \\xi _ k \\end{align*}"}
-{"id": "7471.png", "formula": "\\begin{gather*} \\gamma ^ * \\xi = q _ s ^ { 2 m ( \\gamma ) } \\xi \\end{gather*}"}
-{"id": "9623.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { q ^ { n ( n + 1 ) / 2 } } { 1 - q ^ { n } } = \\sum _ { n = 1 } ^ { \\infty } \\left ( \\textup { s s p t d } ( n , N ) - \\textup { s s p t d } ( n - N , N ) \\right ) q ^ n . \\end{align*}"}
-{"id": "1155.png", "formula": "\\begin{align*} p _ { n , k } : = \\sup _ { z \\in I } \\mathbb { P } ( \\xi ^ { ( n ) } ( [ z , z + d _ 0 ] ) = 0 ) = \\sup _ { z \\in I } \\mathbb { P } ( \\xi ^ { ( n ) } ( [ z , z + G _ n ( - C _ 0 ) / S ( I ) ] ) = 0 ) . \\end{align*}"}
-{"id": "2221.png", "formula": "\\begin{align*} \\boldsymbol { u } ^ { * } & = - \\frac { 1 } { 2 } R ^ { - 1 } B ^ { T } ( P + P ^ T ) \\boldsymbol { x } ^ { * } \\\\ & = \\begin{bmatrix} - 1 . 2 8 8 7 x _ 1 ^ { * } + 0 . 4 2 6 7 x _ 2 ^ { * } \\\\ - 1 . 7 2 4 0 x _ 1 ^ { * } + - 5 . 0 7 8 7 x _ 2 ^ { * } \\end{bmatrix} \\end{align*}"}
-{"id": "3829.png", "formula": "\\begin{align*} { g ^ * } _ { i j } = e ^ { 2 \\rho } g _ { i j } , g ^ { * i j } = e ^ { - 2 \\rho } g ^ { i j } \\end{align*}"}
-{"id": "3752.png", "formula": "\\begin{align*} \\Im \\left [ \\mathcal { P } \\right ] = \\sum _ { j , l , m } g _ { l } ^ { \\left ( j \\right ) } | a _ { l , m } ^ { ( j ) } | ^ { 2 } , \\end{align*}"}
-{"id": "6582.png", "formula": "\\begin{align*} \\| u - g \\| _ { L ^ 2 ( \\Gamma _ D ) } ^ 2 = \\| u \\| _ { L ^ 2 ( \\Gamma _ D ) } ^ 2 + \\| g \\| _ { L ^ 2 ( \\Gamma _ D ) } ^ 2 - 2 \\int _ { \\Gamma _ D } u g \\ , d \\mathcal { H } ^ { N - 1 } \\ge \\frac { 1 } { 2 } \\| u \\| _ { L ^ 2 ( \\Gamma _ D ) } ^ 2 - 7 \\| g \\| _ { L ^ 2 ( \\Gamma _ D ) } ^ 2 , \\end{align*}"}
-{"id": "4756.png", "formula": "\\begin{align*} { \\sf L } f ( \\xi ) & = \\left [ \\log \\left ( ( a + 1 ) ^ 2 + \\frac 1 2 \\right ) + \\log \\left ( ( a - 1 ) ^ 2 + \\frac 1 2 \\right ) - 2 \\log \\left ( a ^ 2 + \\frac 1 2 \\right ) \\right ] \\\\ & + m \\left [ \\log \\left ( a ^ 2 - \\frac 1 2 \\right ) - \\log \\left ( a ^ 2 + \\frac 1 2 \\right ) \\right ] \\\\ & = \\log \\left ( \\frac { 4 a ^ 4 - 4 a ^ 2 + 9 } { 4 a ^ 4 + 4 a ^ 2 + 1 } \\right ) + m \\left [ \\log \\left ( \\frac { 2 a ^ 2 - 1 } { 2 a ^ 2 + 1 } \\right ) \\right ] \\leq 0 . \\end{align*}"}
-{"id": "4749.png", "formula": "\\begin{align*} W ( y _ k ) \\geq - \\frac { \\alpha } { 2 } S ( y _ k ) + O ( 1 ) = \\frac { | \\alpha | } { 2 } k + O ( 1 ) . \\end{align*}"}
-{"id": "9579.png", "formula": "\\begin{align*} & \\sum _ { n = 1 } ^ { N } \\frac { ( - 1 ) ^ { n - 1 } q ^ { \\frac { n ( n + 1 ) } { 2 } } } { ( 1 - c q ^ n ) ( q ) _ n ( q ) _ { N - n } } \\left ( n + \\sum _ { k = 1 } ^ n \\frac { q ^ k } { 1 - q ^ k } \\right ) \\\\ & = \\frac { - c } { ( 1 - c ) ^ 2 ( q ) _ N } \\left ( 1 - \\frac { ( q ) _ N } { ( c q ) _ N } \\right ) + \\frac { 1 } { ( 1 - c ) ( c q ) _ N } \\sum _ { k = 1 } ^ N \\frac { ( c q ) _ k q ^ k } { ( q ) _ k ( 1 - q ^ k ) } . \\end{align*}"}
-{"id": "417.png", "formula": "\\begin{align*} \\langle F \\rangle _ { { \\rm M L } , \\hat g } = \\langle F ( \\cdot - \\tfrac { Q } { 2 } \\omega ) \\rangle _ { { \\rm M L } , g } \\exp \\big ( \\tfrac { 1 + 6 Q ^ 2 } { 9 6 \\pi } S _ { \\rm L } ^ { { \\rm c l } , 0 } ( \\hat g , g ) + \\beta S ^ { { \\rm c l } } _ { \\rm M } ( \\hat g , g ) \\big ) \\end{align*}"}
-{"id": "2384.png", "formula": "\\begin{align*} \\delta : \\mathfrak { F } & \\to \\mathbb { C } , & \\delta ( 1 ) & = 1 , & \\delta ( X _ i ) & = 0 . \\end{align*}"}
-{"id": "9943.png", "formula": "\\begin{align*} P _ k = \\frac { 1 } { 2 \\pi \\imath } \\oint _ { { \\mathcal C } _ k } \\left ( z - { H } \\right ) ^ { - 1 } d z \\end{align*}"}
-{"id": "4750.png", "formula": "\\begin{align*} R _ k = e ^ { - W ( y _ k ) } \\leq C e ^ { - \\frac { | \\alpha | } { 2 } k } , \\end{align*}"}
-{"id": "1035.png", "formula": "\\begin{align*} & \\mathbb { P } ( \\xi ^ { ( n ) } ( I ( x , a ) ) = 0 ) = D _ n ( a / 2 ) . \\end{align*}"}
-{"id": "4566.png", "formula": "\\begin{align*} \\gamma _ s = \\begin{cases} s & s \\in ( 0 , ( n - 1 ) / 2 ] ; \\\\ ( n - 1 ) / 2 & s \\in [ ( n - 1 ) / 2 , n / 2 ] ; \\\\ ( n + 2 s - 2 ) / 4 & s \\in [ n / 2 , ( n + 2 ) / 2 ] ; \\\\ s - 1 & s \\in [ ( n + 2 ) / 2 , n ) . \\end{cases} \\end{align*}"}
-{"id": "6263.png", "formula": "\\begin{align*} \\pi _ g \\left ( \\sum _ { h \\in G } a _ h \\right ) = a _ g , \\mbox { w h e r e } a _ h \\in A _ h , \\mbox { f o r a n y } h \\in G . \\end{align*}"}
-{"id": "163.png", "formula": "\\begin{align*} \\pi _ { X _ { C } , X _ { V } } = X _ { C } \\wedge X _ { V } \\end{align*}"}
-{"id": "495.png", "formula": "\\begin{align*} \\mu _ 2 ( \\sum _ { i , j } a _ { i j } \\ , \\omega _ i \\otimes \\omega _ j ) = \\sum _ { i , j } a _ { i j } f _ i ' ( z ) f _ j ' ( z ) ( d z ) ^ 4 . \\end{align*}"}
-{"id": "4060.png", "formula": "\\begin{align*} J ^ { M _ \\psi } _ { M _ 1 , M _ 2 } ( f ) = \\eta \\circ \\left ( \\sum _ { \\chi , i } f _ i ' \\otimes J ^ { M _ \\chi } _ { M _ 1 , M _ 2 } ( f _ i ) \\right ) \\end{align*}"}
-{"id": "1844.png", "formula": "\\begin{align*} - v _ { x x } ( x ) = \\lambda v ( x ) . \\end{align*}"}
-{"id": "514.png", "formula": "\\begin{align*} \\mu _ 2 ( Q _ { i , j } ) = \\mu _ { 1 , | L | } ( x \\wedge y ) \\mu _ { 1 , | K - L | } ( t _ i \\wedge t _ j ) . \\end{align*}"}
-{"id": "1996.png", "formula": "\\begin{align*} | L | : = L \\otimes \\O \\left ( - \\sum m _ i [ q _ i ] \\right ) . \\end{align*}"}
-{"id": "1936.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\frac { 1 } { Z _ j ^ 2 } = \\frac { 1 } { 2 } + \\frac { 1 } { x } \\qquad \\mbox { a n d } \\sum _ { j = 1 } ^ \\infty \\frac { 1 } { Z _ j ^ 4 } = \\frac { 1 } { 6 } \\left ( 1 + \\frac { 4 } { x } + \\frac { 6 } { x ^ 2 } \\right ) . \\end{align*}"}
-{"id": "2468.png", "formula": "\\begin{align*} G _ { P } = \\left \\{ g \\in G : g P = P \\right \\} , \\end{align*}"}
-{"id": "5548.png", "formula": "\\begin{align*} | S _ n ( f _ 1 \\otimes \\ldots \\otimes f _ n ) | \\leqslant ( n ! ) ^ { \\beta } \\prod _ { i = 1 } ^ n \\| f _ i \\| _ s . \\end{align*}"}
-{"id": "7927.png", "formula": "\\begin{align*} R _ N = - \\frac { 7 } { 5 7 6 0 } \\frac { ( b - a ) ^ 5 f ^ { ( 4 ) } ( \\xi ) } { ( N - 2 ) ^ 4 } \\xi \\in [ a , b ] \\end{align*}"}
-{"id": "9143.png", "formula": "\\begin{align*} \\rho _ { 1 } ^ { \\left ( N \\right ) } ( \\mathbf { x } _ { 1 } ^ { ( + ) } ( t _ { 1 } ) , t _ { i } ) = \\rho ^ { \\left ( N \\right ) } ( \\mathbf { x } _ { 1 } ^ { ( - ) } ( t _ { i } ) , t _ { i } ) , \\end{align*}"}
-{"id": "2921.png", "formula": "\\begin{align*} \\Pi _ 0 ( \\psi \\cdot f ) = \\psi | _ M \\ , \\Pi _ 0 ( f ) \\ , , \\quad \\forall \\ , \\psi \\in C ( X ) \\ , , \\ , f \\in C _ { \\rm c } ( \\Xi ) \\ , . \\end{align*}"}
-{"id": "5131.png", "formula": "\\begin{align*} & \\lim _ { r \\to 0 } \\abs * { \\begin{pmatrix} f _ { m _ j , k _ j } ^ + ( r ) \\\\ f _ { m _ j , k _ j } ^ - ( r ) \\end{pmatrix} - ( M _ { k _ j } \\log r + \\mathbb { I } _ 2 ) \\begin{pmatrix} A ^ + \\\\ A ^ - \\end{pmatrix} } r ^ { - 1 / 2 } = 0 , \\end{align*}"}
-{"id": "5183.png", "formula": "\\begin{align*} F _ { X _ n } ( x ) = \\mathbb { P } ( X _ n \\leq x ) \\rightarrow F _ { X } ( x ) = \\mathbb { P } ( X \\leq x ) n \\rightarrow + \\infty . \\end{align*}"}
-{"id": "8153.png", "formula": "\\begin{align*} \\left ( v _ { A } ^ { - 1 } v _ { B } ^ { - 1 } v _ { A } ^ { - 1 } \\right ) ^ 2 \\triangleright \\varphi & = \\frac { \\mu ^ l ( v ^ { - 1 } ) } { \\mu ^ l ( v ) } \\varphi \\ ! \\left ( S ^ { - 1 } ( a _ i ) g ^ { - 1 } v ^ { - 1 } S ( ? ) b _ i \\right ) \\\\ \\left ( v _ { A } ^ { - 1 } v _ { B } ^ { - 1 } v _ { A } ^ { - 1 } \\right ) ^ { - 2 } \\triangleright \\varphi & = \\frac { \\mu ^ l ( v ) } { \\mu ^ l ( v ^ { - 1 } ) } \\varphi \\ ! \\left ( b _ j S ^ { - 1 } ( ? ) a _ j g ^ { - 1 } v \\right ) \\end{align*}"}
-{"id": "9041.png", "formula": "\\begin{align*} \\xi _ 1 + \\xi _ 2 + . . . + \\xi _ { 2 l } = 0 \\end{align*}"}
-{"id": "1548.png", "formula": "\\begin{align*} F _ 1 ^ \\delta \\left ( \\begin{array} { c } z \\\\ x \\end{array} \\right ) = \\left ( \\begin{array} { c } a _ 1 ^ \\delta ( x ) z + a ^ \\delta _ 2 ( x ) z ^ 2 + \\dots \\\\ b ^ \\delta _ 0 ( x ) + b ^ \\delta _ 1 ( x ) z + b ^ \\delta _ 2 ( x ) z ^ 2 + \\dots \\end{array} \\right ) , \\end{align*}"}
-{"id": "1209.png", "formula": "\\begin{align*} \\sigma _ 2 \\int _ 0 ^ 1 \\partial _ { \\sigma _ 2 } \\tilde { P } ( 0 , t \\sigma _ 2 , x , z ' ) \\ , d t & = \\int _ 0 ^ 1 \\frac { ( 1 - a _ 0 ^ 2 ) t \\sigma _ 2 ^ 2 } { ( 1 + t ^ 2 \\sigma _ 2 ^ 2 ) ^ { 3 / 2 } \\sqrt { 1 - \\sin ^ 2 \\theta } \\sqrt { t ^ 2 \\sigma _ 2 ^ 2 - s ( \\theta ) } } \\ , d t \\\\ & = \\frac { 1 - a _ 0 ^ 2 } { 2 \\sqrt { 1 - \\sin ^ 2 \\theta } } \\int _ 0 ^ { \\sigma _ 2 ^ 2 } \\frac { d \\tau } { ( 1 + \\tau ) ^ { 3 / 2 } ( \\tau - s ( \\theta ) ) ^ { 1 / 2 } } . \\end{align*}"}
-{"id": "8178.png", "formula": "\\begin{align*} \\| S f \\| ^ 2 = \\big \\| \\sum _ { i \\in { I _ n } , j \\in { J _ m } } c _ { i j } \\Lambda _ { i + 1 } ^ * e _ j \\big \\| ^ 2 & = \\big \\| \\sum _ { i \\in { I _ n } , j \\in { J _ m } } r _ { i j } \\Lambda _ { i + 1 } ^ * e _ j \\big \\| ^ 2 \\\\ & \\leq B _ { \\Lambda } \\sum _ { i \\in { I _ n } , j \\in { J _ m } } | r _ { i j } | ^ 2 . \\end{align*}"}
-{"id": "5842.png", "formula": "\\begin{align*} \\rho ( c ) = c ( - 1 ) \\int _ { F } \\psi ( x ) c ( x ) | x | _ F ^ { - 1 } d x : = c ( - 1 ) \\lim _ { r \\to \\infty } \\int _ { | x | \\leq r } \\psi ( x ) c ( x ) | x | _ F ^ { - 1 } d x . \\end{align*}"}
-{"id": "8018.png", "formula": "\\begin{align*} ( 2 k + r + 3 ) P _ { k , i } = - \\sum _ { j = 1 } ^ { r } ( 3 j + 2 k - 2 r ) a _ j P _ { k - r + j - 1 , i } \\mbox { f o r a l l } k \\geq 0 \\end{align*}"}
-{"id": "8281.png", "formula": "\\begin{align*} \\mathcal { F } | D | ^ { \\delta } f ( \\xi ) = ( 2 \\pi | \\xi | ) ^ { \\delta } \\mathcal { F } f ( \\xi ) . \\end{align*}"}
-{"id": "1603.png", "formula": "\\begin{align*} \\mathcal { A } : = \\frac { 1 } { 2 } \\sigma ^ 2 ( x ) \\frac { d ^ 2 } { d x ^ 2 } + \\mu ( x ) x \\frac { d } { d x } = \\frac { 1 } { 2 } \\frac { d } { d m } \\frac { d } { d S } \\end{align*}"}
-{"id": "1747.png", "formula": "\\begin{align*} \\psi _ { N } ( x ) = \\begin{cases} 1 , & | x | \\leq N \\\\ 0 & | x | \\geq N + 1 , \\end{cases} \\end{align*}"}
-{"id": "2698.png", "formula": "\\begin{align*} \\Sigma _ { \\pm } ^ { \\mathrm { l o c a l } } & = \\left \\{ w _ { 1 } + r e ^ { \\pm i \\frac { 1 } { 3 } \\pi } : r \\in [ 0 , 2 \\kappa N ^ { - \\frac { 1 } { 3 } } ) \\right \\} , \\\\ \\Sigma _ { \\pm } ^ { 1 } & = \\left \\{ w _ { 1 } + \\kappa N ^ { - \\frac { 1 } { 3 } } \\pm i y : y \\in [ \\sqrt { 3 } \\kappa N ^ { - \\frac { 1 } { 3 } } , \\Im { w _ { 2 } } ] \\right \\} , \\\\ \\Sigma _ { \\pm } ^ { 2 } & = \\left \\{ R e ^ { \\pm i \\phi } : \\phi \\in [ 0 , \\mathrm { A r g } ( w _ { 2 } ) ) \\right \\} , \\end{align*}"}
-{"id": "8670.png", "formula": "\\begin{align*} \\widehat { \\beta _ d ^ { ( 2 ) } ( \\nu ) } : = \\lim _ { L } \\lim _ { L ' \\geq L } \\nabla _ d ( R ^ { L } ( \\Theta ) , R ^ { L ' } ( \\Theta ) ) . \\end{align*}"}
-{"id": "5628.png", "formula": "\\begin{align*} \\beta \\ , \\int _ 0 ^ { + \\infty } | u | ^ 2 u \\bar { u } _ t \\ , d x = \\frac { \\beta } { 4 } \\frac { d } { d t } \\int _ 0 ^ { + \\infty } | u | ^ 4 d x \\end{align*}"}
-{"id": "7418.png", "formula": "\\begin{align*} \\displaystyle Z _ + ( s , t ) = Z _ - ( - s , t ) \\end{align*}"}
-{"id": "150.png", "formula": "\\begin{align*} \\pi _ { T M } ( { \\bf x } , { \\bf y } ) = \\sum _ { i , j = 1 } ^ { N } \\pi ^ { i j } ( { \\bf x } ) \\frac { \\partial } { \\partial x ^ i } \\wedge \\frac { \\partial } { \\partial y ^ j } + \\dfrac { 1 } { 2 } \\sum _ { i , j , s = 1 } ^ { N } \\dfrac { \\partial \\pi ^ { i j } } { \\partial x ^ s } ( { \\bf x } ) y ^ s \\frac { \\partial } { \\partial y ^ i } \\wedge \\frac { \\partial } { \\partial y ^ j } \\end{align*}"}
-{"id": "8020.png", "formula": "\\begin{align*} D _ { 2 k } & : = \\langle x , y \\ , | \\ , x ^ { 2 k } = 1 , y ^ { 2 } = 1 , x y x = y \\rangle , \\\\ D i c _ k & : = \\langle x , y \\ , | \\ , x ^ { 2 k } = 1 , y ^ 2 = x ^ k , x y x = y \\rangle , \\\\ \\mathbb U _ k & : = \\langle x , y \\ , | \\ , x ^ { 2 k } = 1 , y ^ { 2 } = 1 , x y x = y x ^ k \\rangle . \\\\ \\end{align*}"}
-{"id": "5718.png", "formula": "\\begin{align*} & \\frac { 1 } { \\Gamma ( 2 - \\alpha ) } \\int _ z ^ x { ( x - t ) ^ { 1 - \\alpha } } ( t - z ) ^ { \\alpha - 2 } E _ { \\alpha , \\alpha - 1 } ( \\lambda ( t - z ) ^ { \\alpha } ) d t \\\\ & = \\frac { 1 } { \\Gamma ( 2 - \\alpha ) } \\int _ 0 ^ { x - z } { ( x - ( \\xi + z ) ) ^ { 1 - \\alpha } } \\xi ^ { \\alpha - 2 } E _ { \\alpha , \\alpha - 1 } ( \\lambda \\xi ^ { \\alpha } ) d \\xi . \\end{align*}"}
-{"id": "7050.png", "formula": "\\begin{align*} g ^ { i j } \\partial _ z \\Gamma _ { v v } ^ v u ^ v _ i u ^ v _ j = g ^ { i j } \\phi _ { v z } u ^ v _ i u ^ v _ j . \\end{align*}"}
-{"id": "9856.png", "formula": "\\begin{align*} \\int _ M { { L _ A } ( { { \\left | { \\nabla u } \\right | } ^ 2 } ) d v o { l _ g } } = \\int _ { \\partial M } { \\left \\langle { \\nabla ( { { \\left | { \\nabla u } \\right | } ^ 2 } ) , A \\overrightarrow { n } } \\right \\rangle d v o { l _ { \\bar g } } } , \\end{align*}"}
-{"id": "4943.png", "formula": "\\begin{align*} ( v _ 1 z ^ { 3 n } ) ^ n \\equiv \\left ( \\left ( \\sum _ { j = 2 } ^ { d - 3 } ( - 1 ) ^ { j } v _ i z ^ { 3 n } \\right ) + ( - 1 ) ^ { d } ( x z ^ { 3 n } ) \\right ) ^ n . \\end{align*}"}
-{"id": "8626.png", "formula": "\\begin{align*} u _ 0 ( 0 ) = \\sqrt { c _ 0 } , \\end{align*}"}
-{"id": "1262.png", "formula": "\\begin{align*} Q ( \\lambda _ - ( \\mu ) ) = Q _ - + \\frac { Q '' _ - } { 2 } \\mu ^ 2 , \\lambda _ - ( 0 ) = r _ - , \\lambda _ - ' ( 0 ) = 1 , | \\lambda _ - ' | > 1 / 2 . \\end{align*}"}
-{"id": "4002.png", "formula": "\\begin{align*} H _ \\mu ( \\tau ) = \\begin{cases} h _ \\mu ( d _ T \\tau ) n \\\\ h _ \\mu ( 4 d _ T \\tau ) n . \\end{cases} \\end{align*}"}
-{"id": "2971.png", "formula": "\\begin{align*} n p ( n ) & = \\sum _ { k = 0 } ^ \\infty \\sigma ( k ) p ( n - k ) . \\end{align*}"}
-{"id": "2583.png", "formula": "\\begin{align*} \\mathbf P _ { \\delta _ x } [ \\eta _ t X _ t ( f ) | \\| X _ t \\| \\neq 0 ] = \\frac { \\mathbf P _ { \\delta _ x } [ \\eta _ t X _ t ( f ) \\mathbf 1 _ { \\| X _ t \\| \\neq 0 } ] } { \\mathbf P _ { \\delta _ x } ( \\| X _ t \\| \\neq 0 ) } = \\frac { \\eta _ t } { \\mathbf P _ { \\delta _ x } ( \\| X _ t \\| \\neq 0 ) } P ^ \\beta _ t f ( x ) \\stackrel [ t \\to \\infty ] { } { \\sim } \\langle f , \\phi ^ * \\rangle _ m . \\end{align*}"}
-{"id": "2311.png", "formula": "\\begin{align*} a ^ + & = \\begin{pmatrix} 1 , & ( 1 | \\frac { \\mathcal { P } } { z \\Omega } \\end{pmatrix} & b ^ + & = \\begin{pmatrix} 0 , & ( \\delta _ x | \\end{pmatrix} . \\end{align*}"}
-{"id": "936.png", "formula": "\\begin{align*} \\Biggl . \\Biggl . + \\frac { ( i ^ 2 + 3 i - 1 ) \\zeta _ { i + 1 } ^ { ( i _ 2 ) } \\zeta _ { i } ^ { ( i _ 1 ) } - ( i ^ 2 + i - 3 ) \\zeta _ { i } ^ { ( i _ 2 ) } \\zeta _ { i + 1 } ^ { ( i _ 1 ) } } { \\sqrt { ( 2 i + 1 ) ( 2 i + 3 ) } ( 2 i - 1 ) ( 2 i + 5 ) } \\Biggr ) \\Biggr ] - \\frac { 1 } { 2 4 } { \\bf 1 } _ { \\{ i _ 1 = i _ 2 \\} } { \\Delta ^ 3 } , \\end{align*}"}
-{"id": "3887.png", "formula": "\\begin{align*} K ( \\phi _ \\omega , \\psi _ \\omega ) + 2 \\omega M ( \\phi _ \\omega , \\psi _ \\omega ) = 3 P ( \\phi _ \\omega , \\psi _ \\omega ) . \\end{align*}"}
-{"id": "3748.png", "formula": "\\begin{align*} \\mathcal { E } _ { \\mathcal { E } , \\mathcal { P } } = \\sum _ { j , l , m } R _ { l , m } ^ { ( j ) } | a _ { l , m } ^ { ( j ) } | ^ { 2 } , \\end{align*}"}
-{"id": "9970.png", "formula": "\\begin{align*} u ( t ) = S ( t ) x _ 0 + \\int _ 0 ^ \\tau S ( t - s ) f ( s ) \\ , d s + \\int _ \\tau ^ t S ( t - s ) f _ k ( s , u ( s ) ) \\ , d s \\ ; \\ ; t \\in [ \\tau , T ] , \\end{align*}"}
-{"id": "3299.png", "formula": "\\begin{align*} \\mathrm { V a r } ( \\alpha ; n , h ) : = \\frac { 1 } { \\left | \\mathcal { M } _ n \\right | } \\sum _ { f _ 0 \\in \\mathcal { M } _ n } \\left | \\sum _ { f \\in I ( f _ 0 , h ) } \\alpha ( f ) - q ^ { h + 1 } \\langle \\alpha \\rangle _ { \\mathcal { M } _ n } \\right | ^ 2 , \\end{align*}"}
-{"id": "6892.png", "formula": "\\begin{align*} \\begin{cases} v ^ 1 _ { 1 0 } \\big ( w ^ 1 _ { 1 0 } - w ^ 2 _ { 0 1 } \\big ) + v ^ 1 _ { 0 1 } \\big ( w ^ 1 _ { 0 1 } + w ^ 2 _ { 1 0 } \\big ) = 0 \\ , , \\\\ v ^ 1 _ { 1 0 } \\big ( w ^ 1 _ { 0 1 } + w ^ 2 _ { 1 0 } \\big ) + v ^ 1 _ { 0 1 } \\big ( w ^ 1 _ { 1 0 } - w ^ 2 _ { 0 1 } \\big ) = 0 \\ , . \\end{cases} \\end{align*}"}
-{"id": "2463.png", "formula": "\\begin{align*} \\begin{pmatrix} w & x & z \\\\ x & y & w ^ 3 \\end{pmatrix} . \\end{align*}"}
-{"id": "9481.png", "formula": "\\begin{align*} ( u - v ) ^ { + } ( x ) = C \\geq 0 , \\ \\ \\forall x \\in \\mathbb { R } ^ { N } . \\end{align*}"}
-{"id": "1909.png", "formula": "\\begin{align*} \\sin [ \\kappa _ j ( | x | - t ) ] + \\sin [ \\kappa _ j ( | x | + t ) ] \\stackrel { ? } { = } \\sin ( \\kappa _ j | x - t | ) + \\sin ( \\kappa _ j | x + t | ) . \\end{align*}"}
-{"id": "8058.png", "formula": "\\begin{align*} \\Upsilon _ i ( \\epsilon _ i , \\nu _ i ) & = \\frac { ( 1 - ( - 1 ) ^ k ) n } { 2 k } ( \\delta _ { i , 3 } + \\delta _ { i , 4 } ) + \\frac { 1 - ( - 1 ) ^ n } { 4 } ( \\delta _ { i , 4 } - \\delta _ { i , 3 } ) + \\frac { 1 } { 2 } ( - 1 ) ^ i \\displaystyle { \\sum _ { i = n + 3 } ^ { 2 n } } c ^ { n + 3 - 2 i } P _ { i - n - 3 , - i } . \\end{align*}"}
-{"id": "428.png", "formula": "\\begin{align*} \\mathcal { G } _ t ( \\dd x ) : = e ^ { \\gamma X _ t ( x ) - \\frac { \\gamma ^ 2 } { 2 } K _ t ( x , x ) } \\ , \\mu ( \\dd x ) \\end{align*}"}
-{"id": "9042.png", "formula": "\\begin{align*} \\mu = \\alpha \\mu _ { A } + \\beta \\mu _ B \\end{align*}"}
-{"id": "3680.png", "formula": "\\begin{align*} \\| E ( y _ i | x _ i , i \\in U ) - E ( y _ j | x _ j , j \\in U ) \\| = O \\big ( \\| x _ i - x _ j \\| \\big ) \\end{align*}"}
-{"id": "791.png", "formula": "\\begin{align*} b _ { i - 1 } = \\mu b _ i \\mod q ^ { { s } } - 1 \\end{align*}"}
-{"id": "7650.png", "formula": "\\begin{align*} I ^ { i j } _ 2 \\ = \\ \\int \\limits _ { \\mathbb T ^ d } \\ ! \\int \\limits _ { \\mathbb R ^ d } \\ ! a ( \\xi \\ ! - \\ ! q ) \\mu ( \\xi , q ) \\big ( ( \\xi - q ) ^ i ( \\varkappa _ 1 ( \\xi ) - \\varkappa _ 1 ( q ) ) ^ j + ( \\varkappa _ 1 ( \\xi ) - \\varkappa _ 1 ( q ) ) ^ i ( \\xi - q ) ^ j \\big ) v _ 0 ( \\xi ) d q d \\xi , \\end{align*}"}
-{"id": "3314.png", "formula": "\\begin{align*} L ( u , \\chi ) = \\frac { \\prod _ { P \\mid M } ( 1 - u ^ { \\deg ( P ) } ) } { 1 - q u } . \\end{align*}"}
-{"id": "5069.png", "formula": "\\begin{align*} h - h _ n & = h - h _ { n - 1 } - \\frac { 2 } { a + b } S _ { x , \\tau } ( h - h _ { n - 1 } ) = \\left ( I _ \\mathcal { H } - \\frac { 2 } { b + a } S _ { x , \\tau } \\right ) ( h - h _ { n - 1 } ) \\\\ & = \\cdots = \\left ( I _ \\mathcal { H } - \\frac { 2 } { b + a } S _ { x , \\tau } \\right ) ^ n h , ~ \\forall h \\in \\mathcal { H } , ~ \\forall n \\geq 1 . \\end{align*}"}
-{"id": "5377.png", "formula": "\\begin{align*} \\langle v _ { n m } ^ k , A _ { \\alpha } ^ * . v _ { n - 2 k + 2 l \\ , m } ^ { l - 1 } \\rangle = \\langle v _ { n m } ^ k , ( l - 2 ) v _ { n - 2 k + 2 l \\ , m } ^ { l - 2 } + ( n - 2 k + l + 1 ) v _ { n - 2 k + 2 l \\ , m } ^ { l } \\rangle \\neq 0 . \\end{align*}"}
-{"id": "4175.png", "formula": "\\begin{align*} A _ n ^ { ( 1 ) } ( z _ 1 ) E _ n ( z _ 2 ) & = \\mathcal { O } ( n ^ 2 ) \\\\ E _ n ^ { - 1 } ( z _ 1 ) A _ n ^ { ( 1 ) } ( z _ 2 ) & = \\mathcal { O } ( n ^ 2 ) . \\end{align*}"}
-{"id": "9172.png", "formula": "\\begin{align*} H _ { 1 } \\left ( t \\right ) & = \\gamma \\left ( t \\right ) , \\\\ H _ { s } ( 0 ) , H _ { s } ( b ) & \\in N \\quad s \\in \\left [ 0 , 1 \\right ] , \\\\ H _ { 0 } \\left ( t \\right ) & \\in N \\quad t \\in \\left [ 0 , b \\right ] . \\end{align*}"}
-{"id": "6621.png", "formula": "\\begin{align*} P _ { F _ { n _ j } } ( \\varphi , \\varepsilon _ j ) = P _ { F _ { n _ j } } ( \\varphi ) \\cdot \\frac { \\overline { A } _ { n _ j } ( \\varepsilon _ j ) \\overline { C } _ { n _ j } ( \\varepsilon _ j ) } { A _ { n _ j } C _ { n _ j } } , \\end{align*}"}
-{"id": "4124.png", "formula": "\\begin{align*} \\varphi ( z ) = - g _ { 1 } ( z ) + \\tfrac { 1 } { 2 } g _ { 2 } ( z ) + \\tfrac { 1 } { 2 } ( V ( z ) + \\ell ) , \\end{align*}"}
-{"id": "1591.png", "formula": "\\begin{align*} & \\Phi _ { w _ { m + 1 } } ( z _ { m + 1 } , x _ { m + 1 } ) \\\\ = & \\Phi _ { v ^ \\iota _ n } ( z ^ \\iota _ n , x ^ \\iota _ n ) + \\sum _ { j = n ^ 2 + k _ n } ^ { m - 1 } \\left ( \\frac { \\sqrt { w _ j } } { 2 } + o \\left ( \\frac { 1 } { n ^ 2 } \\right ) \\right ) + \\frac { \\sqrt { w _ m } } { 2 } + o \\left ( \\frac { 1 } { n ^ 2 } \\right ) \\\\ = & \\Phi _ { v ^ \\iota _ n } ( z ^ \\iota _ n , x ^ \\iota _ n ) + \\sum _ { j = n ^ 2 + k _ n } ^ { m } \\left ( \\frac { \\sqrt { w _ j } } { 2 } + o \\left ( \\frac { 1 } { n ^ 2 } \\right ) \\right ) , \\end{align*}"}
-{"id": "410.png", "formula": "\\begin{align*} S ^ { \\rm c l } _ { \\rm M } ( \\hat g , g ) = \\int _ M \\Big ( 2 \\pi ( 1 - \\textbf { h } ) \\phi \\Delta _ g \\phi + ( \\frac { 8 \\pi ( 1 - \\textbf { h } ) } { V _ { g } } - K _ { g } ) \\phi + \\frac { 2 } { V _ { \\hat g } } \\omega e ^ { \\omega } \\Big ) { \\rm d v } _ { g } . \\end{align*}"}
-{"id": "9101.png", "formula": "\\begin{align*} \\begin{aligned} 0 \\le I _ 1 & \\le | \\Upsilon ( x ) | ^ { p - 1 } \\int _ { | y | > \\frac { 3 } { 2 } | x | } \\dfrac { d y } { | x - y | ^ { N + s p } } \\\\ & = C ( N , s , p ) \\dfrac { | \\Upsilon ( x ) | ^ { p - 1 } } { | x | ^ { p s } } \\le \\dfrac { C ( N , s , p , C _ 1 , \\alpha ) } { | x | ^ { \\alpha ( p - 1 ) + p s } } . \\end{aligned} \\end{align*}"}
-{"id": "7814.png", "formula": "\\begin{align*} \\Big | [ \\langle x , y \\rangle \\xi , \\xi ] \\Big | = \\| x \\| \\| y \\| , \\end{align*}"}
-{"id": "6260.png", "formula": "\\begin{align*} 1 - \\psi \\left ( \\sum _ { j = 1 } ^ \\infty 1 _ { B _ { 0 , j } } \\right ) . \\end{align*}"}
-{"id": "6802.png", "formula": "\\begin{align*} I _ 3 = I _ 2 + \\langle f _ 6 , \\dots , f _ { 1 2 } \\rangle \\subset \\mathbb { Q } [ y _ 1 , y _ 2 , y _ 3 ] \\ , . \\end{align*}"}
-{"id": "6276.png", "formula": "\\begin{align*} f _ n = \\sum _ { j = 0 } ^ { { k _ n } - 1 } f ( t _ j ^ n ) \\mathbf { 1 } _ { [ t _ j ^ n , t _ { j + 1 } ^ n ) } . \\end{align*}"}
-{"id": "7164.png", "formula": "\\begin{gather*} \\dot { \\varphi } = 1 + r ^ { 2 } , \\quad \\dot { r } = r \\left ( 1 - r ^ { 2 } \\right ) { } ^ { 3 } . \\end{gather*}"}
-{"id": "563.png", "formula": "\\begin{align*} C _ i = \\{ \\alpha ^ { t } \\ | \\ t \\equiv i \\pmod e \\} , i = 0 , 1 , \\dots , e - 1 , \\end{align*}"}
-{"id": "6062.png", "formula": "\\begin{align*} \\lambda _ { 2 l - 1 } ( { \\bf c } ) = \\varepsilon _ { j _ { k _ l } } \\left ( \\tilde { e } _ { j _ { { k _ l } - 1 } } ^ { m a x } \\cdots \\tilde { e } _ { j _ 2 } ^ { m a x } \\tilde { e } _ { j _ 1 } ^ { m a x } { \\bf c } \\right ) , \\end{align*}"}
-{"id": "7233.png", "formula": "\\begin{align*} K _ r * \\phi ( x + h ) & - K _ r * \\phi ( x - h ) \\\\ & = \\int _ { - \\pi } ^ { 0 } \\left ( K _ r ( x - y ) - K _ r ( x + y ) \\right ) \\left ( \\phi ( y + h ) - \\phi ( y - h ) \\right ) \\ , d y . \\end{align*}"}
-{"id": "8930.png", "formula": "\\begin{align*} \\mathbb { E } [ \\zeta | d ( x , \\hat { x } ) = k ] & = s ( n - s ) \\left ( 1 - 4 \\frac { k } { n } + 4 \\frac { k ^ 2 } { n ^ 2 } - 4 \\frac { k ( n - k ) } { n ^ 2 ( n - 1 ) } \\right ) \\\\ & = s ( n - s ) \\left ( \\left ( 1 - 2 \\frac { k } { n } \\right ) ^ 2 - 4 \\frac { k ( n - k ) } { n ^ 2 ( n - 1 ) } \\right ) \\end{align*}"}
-{"id": "2975.png", "formula": "\\begin{align*} ( q ^ 5 ; q ^ 5 ) _ \\infty & = \\Big ( \\frac { 1 } { ( q ; q ) _ \\infty ( q ^ 4 ; q ^ 5 ) _ \\infty } \\Big ) \\Big ( ( q ; q ^ 5 ) _ \\infty ( q ^ 4 ; q ^ 5 ) _ \\infty ( q ^ 5 ; q ^ 5 ) _ \\infty \\Big ) . \\end{align*}"}
-{"id": "6197.png", "formula": "\\begin{align*} \\Gamma \\left ( \\frac { \\pi } { 2 } \\right ) = \\frac 1 2 \\ ; . \\end{align*}"}
-{"id": "1129.png", "formula": "\\begin{align*} & B _ j ( u , v ) = { 2 \\pi } \\rho _ { s c } ( y _ j ) K _ { ( y _ j , y _ j + a _ j ) } ^ { C U E ( n ) } \\left ( { 2 \\pi } \\rho _ { s c } ( y _ j ) u , { 2 \\pi } \\rho _ { s c } ( y _ j ) v \\right ) , \\end{align*}"}
-{"id": "6370.png", "formula": "\\begin{align*} u ^ { \\varepsilon } ( t , x ) : = \\int _ { \\mathbb { R } } v _ s ( t , x ) \\chi _ { \\varepsilon } ( s - 1 ) d s - B \\varepsilon ( t + 1 ) \\end{align*}"}
-{"id": "1222.png", "formula": "\\begin{gather*} \\Phi ( \\xi , \\eta ) = \\eta \\xi ^ 2 - \\xi \\eta ^ 2 + \\frac 1 4 \\eta ^ 3 , \\end{gather*}"}
-{"id": "2442.png", "formula": "\\begin{gather*} z ' ( x ) = F ( x , y , y ' , y '' , z ) \\end{gather*}"}
-{"id": "8141.png", "formula": "\\begin{align*} \\forall \\ , x \\in A , \\forall \\ , y \\in B , \\ : \\ : y x = ( x \\cdot a _ i ) ( y \\cdot b _ i ) . \\end{align*}"}
-{"id": "9387.png", "formula": "\\begin{align*} D = \\{ j \\in \\mathbb { N } _ 0 \\mid \\exists g \\in L : g \\equiv _ { P _ { j + 1 } ( G ) Z } c _ j \\} . \\end{align*}"}
-{"id": "7864.png", "formula": "\\begin{align*} \\Bbbk _ k ^ { ( n ) } ( x ) = \\frac { 1 } { \\binom { n } { k } } \\sum _ { j = 0 } ^ k ( - 1 ) ^ j \\binom { x } { j } \\binom { n - x } { k - j } . \\end{align*}"}
-{"id": "8136.png", "formula": "\\begin{align*} \\psi \\triangleright \\varphi = \\psi \\varphi , \\ : \\ : \\ : \\ : \\ : h \\triangleright \\varphi = \\varphi ( ? h ) . \\end{align*}"}
-{"id": "8104.png", "formula": "\\begin{align*} \\alpha _ { j , k } = & \\ ; \\ ; ( - 1 + 3 \\cdot 2 ^ { m - 1 } - 2 \\cdot 3 ^ m ) ( \\tau ) _ { m - 1 } / ( m - 1 ) ! \\\\ & + ( 7 - 1 7 \\cdot 2 ^ m + 1 7 \\cdot 3 ^ m ) ( \\tau ) _ { m } / m ! \\\\ & + ( - 1 3 + 3 8 \\cdot 2 ^ m - 3 3 \\cdot 3 ^ m ) ( \\tau ) _ { m + 1 } / ( m + 1 ) ! \\\\ & + 6 ( 1 - 4 \\cdot 2 ^ m + 3 \\cdot 3 ^ m ) ( \\tau ) _ { m + 2 } / ( m + 2 ) ! , \\end{align*}"}
-{"id": "5554.png", "formula": "\\begin{align*} { \\rm l i m } _ { q \\to \\infty } P _ { G ( q ) } ( G ( q ) , G ( q ) ) = 1 . \\end{align*}"}
-{"id": "468.png", "formula": "\\begin{align*} \\frac { d ^ 2 + \\sum ^ 3 _ { i = 1 } \\gcd ( a _ i - a _ 4 , d ) + \\sum _ { 1 \\leq i < j \\leq 3 } \\gcd ( a _ i - a _ j , d ) + 1 } { 2 } . \\end{align*}"}
-{"id": "546.png", "formula": "\\begin{align*} \\big ( ( \\nabla u ^ h ) ( A ^ h ) ^ { - 1 } \\big ) ( x ' , h x _ 3 ) = Q _ 0 \\bar A ^ { - 1 } \\Big ( I d _ 3 + h \\bar A ^ { - 1 } J _ 1 ^ h \\bar A ^ { - 1 } + h ^ 2 \\bar A ^ { - 1 } J _ 2 ^ h \\bar A ^ { - 1 } + J _ h ^ 3 \\Big ) , \\end{align*}"}
-{"id": "776.png", "formula": "\\begin{align*} \\sigma ( \\lambda _ i ) = \\begin{cases} \\zeta _ { \\sigma , i } \\lambda _ { i + 1 } & \\mbox { i f } 1 \\leq i < { s } \\\\ \\zeta _ { \\sigma , d } \\lambda _ 1 & \\mbox { i f } i = { s } , \\end{cases} \\end{align*}"}
-{"id": "6652.png", "formula": "\\begin{align*} \\ell ( E ) \\geq \\tau _ { \\mathcal { Q } _ E } ( x ) = \\tau ( x ) \\geq \\max \\{ 1 , \\ell ( Q ) \\mathbf 1 _ { \\frac { 1 } 3 Q } \\ ; : \\ ; Q \\in \\mathcal Q _ E \\} , x \\in E . \\end{align*}"}
-{"id": "7402.png", "formula": "\\begin{align*} \\widetilde { E } _ { q , k } = ( 0 , 1 ) \\times B ( q , 1 / k ) \\subset \\mathbb { R } \\times \\mathbb { R } ^ { n - 1 } , q \\in B ^ { n - 1 } _ 1 ( 0 ) \\cap \\mathbb { Q } ^ { n - 1 } , \\ , k \\in \\mathbb { N } . \\end{align*}"}
-{"id": "7010.png", "formula": "\\begin{align*} a _ j = \\frac { \\lambda _ i ^ \\ell } { d _ j } \\left ( { \\textsl { \\footnotesize R } } _ j ' ( x ) + \\frac { \\ell { \\textsl { \\footnotesize R } } _ j ( x ) } { \\lambda _ j } - { \\textsl { \\footnotesize R } } _ j ( x ) \\frac { d _ j ' } { d _ j } \\right ) . \\end{align*}"}
-{"id": "6653.png", "formula": "\\begin{align*} M _ { 1 , 1 } = \\mathbf 1 _ { \\tau ( x ) \\leq 1 0 0 N } A _ { \\tau } f , \\end{align*}"}
-{"id": "7486.png", "formula": "\\begin{align*} \\kappa ( t ) : = \\frac { \\dot { x } \\ddot { y } - \\ddot { x } \\dot { y } } { ( \\dot { x } ^ 2 + \\dot { y } ^ 2 ) ^ { \\frac { 3 } { 2 } } } , \\end{align*}"}
-{"id": "3494.png", "formula": "\\begin{align*} \\frac { d } { d y } \\Re { f _ { M } ( x + i y ; \\theta ) } = { } & - \\left . \\Im { \\frac { d } { d z } f _ { M } ( z ; \\theta ) } \\right | _ { z = x + i y } \\\\ = { } & - ( M + 1 ) \\arctan \\frac { y } { M + 1 + x } + \\arctan \\frac { y } { x } \\end{align*}"}
-{"id": "7162.png", "formula": "\\begin{align*} p _ { \\textrm { f l } } ( \\varrho ) : = \\varrho ^ 2 \\psi _ 0 ' ( \\varrho ) \\textrm { a n d } p _ { \\textrm { e l } } ( b ) : = - e ( b ) - \\frac { 2 } { 3 } b e ' ( b ) , \\end{align*}"}
-{"id": "8302.png", "formula": "\\begin{align*} ( I - \\mathcal { H } ) ( \\bar { \\zeta } - \\alpha ) = 0 . \\end{align*}"}
-{"id": "8044.png", "formula": "\\begin{align*} R _ h ( x ) = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} , R _ h ( y ) = \\begin{pmatrix} \\zeta ^ h & 0 \\\\ 0 & ( - 1 ) ^ h \\zeta ^ { - h } \\end{pmatrix} \\end{align*}"}
-{"id": "7379.png", "formula": "\\begin{align*} \\frac { d N } { d t } & = ( N _ r - N ) D - I _ { N R } \\frac { N } { \\kappa _ { N R } + N } R , \\\\ \\frac { d R } { d t } & = \\bigg ( \\mu _ { N R } \\frac { N } { \\kappa _ { N R } + N } - ( D + m _ R ( R ) ) \\bigg ) R - I _ { R P } \\frac { R } { \\kappa _ { R P } + R } P , \\\\ \\frac { d P } { d t } & = \\bigg ( \\mu _ { R P } \\frac { R } { \\kappa _ { R P } + R } - ( D + m _ { P 0 } ) \\bigg ) P , \\end{align*}"}
-{"id": "6707.png", "formula": "\\begin{align*} \\beta _ { i k } \\ , d ( m \\dot { q } _ i + u _ i ) = d P _ { 2 k } - ( m \\dot { q } _ i + u _ i ) \\beta ' _ { i k l } \\ , d Q _ { 1 l } , \\end{align*}"}
-{"id": "3316.png", "formula": "\\begin{align*} \\left | \\sum _ { P \\in \\mathcal { P } _ { n } } \\chi ( P ) \\right | \\le \\begin{cases} \\min \\{ \\frac { q ^ { \\frac { n } { 2 } } } { n } ( \\ell + \\deg ( M ) + 1 ) , \\frac { q ^ n } { n } \\} & \\mbox { i f $ \\chi \\neq \\chi _ 0 $ , } \\\\ \\frac { q ^ n } { n } & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "260.png", "formula": "\\begin{align*} \\frac { g } { f ^ { p + 1 } } f ^ { \\beta } \\otimes \\delta = \\sum _ { i = 0 } ^ p \\partial _ t ^ i u ^ { ( i ) } , \\end{align*}"}
-{"id": "1700.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\chi _ { K _ n } ( x ) = \\chi _ { ( \\bigcup _ { n \\geq 1 } K _ n ) } ( x ) \\leq \\chi _ A ( x ) \\leq \\chi _ { ( \\bigcap _ { n \\geq 1 } V _ n ) } ( x ) = \\lim _ { n \\rightarrow + \\infty } \\chi _ { V _ n } ( x ) \\ \\ \\ \\ \\ \\forall \\ x \\in X . \\end{align*}"}
-{"id": "2627.png", "formula": "\\begin{align*} S ^ { ( \\mathrm { s o f t } ) } ( \\kappa , \\pi ; u , v ) = & \\int _ { \\mathcal { C } _ { > } } \\frac { d z } { 2 \\pi i } \\int _ { \\mathcal { C } _ { < } } \\frac { d w } { 2 \\pi i } e ^ { \\frac { 1 } { 3 } ( w - \\kappa ) ^ 3 - u ( w - \\kappa ) - \\frac { 1 } { 3 } ( z - \\kappa ) ^ 3 + v ( z - \\kappa ) } \\\\ & \\times \\frac { 1 } { \\sqrt { 4 z w } } \\frac { z + w } { z - w } \\prod _ { k = 1 } ^ { m } \\frac { z - \\pi _ { k } } { z + \\pi _ { k } } \\frac { w + \\pi _ { k } } { w - \\pi _ { k } } , \\end{align*}"}
-{"id": "4029.png", "formula": "\\begin{align*} \\eta = - i \\partial h _ { * _ | S } \\end{align*}"}
-{"id": "6668.png", "formula": "\\begin{align*} [ Q , Q ^ { \\dagger } ] _ { + } = H , [ Q , H ] = 0 = [ Q ^ { \\dagger } , H ] , \\end{align*}"}
-{"id": "2488.png", "formula": "\\begin{align*} ( T x ) ( \\alpha ) & = x ( a ) \\ , , \\\\ ( T x ) ( \\beta ) & = x ( a ) - x ( b ) \\ , . \\end{align*}"}
-{"id": "3872.png", "formula": "\\begin{align*} \\phi ( v _ 1 , \\dotsc , v _ { r } ) = v _ 1 g _ 1 v _ 2 g _ 2 \\dotsb v _ { r - 1 } g _ { r - 1 } v _ r , \\end{align*}"}
-{"id": "2089.png", "formula": "\\begin{align*} y ( t ) = \\frac { 4 \\pi } { 3 } R _ { \\rm r x } ^ 3 ( s ( t ) * { C ( \\bar r _ { \\rm r x } , t | { \\bar r _ { \\rm t x } } , { t _ 0 } ) } ) . \\end{align*}"}
-{"id": "7466.png", "formula": "\\begin{gather*} \\sum _ { i = 1 } ^ b l _ i = 4 - 4 g . \\end{gather*}"}
-{"id": "4172.png", "formula": "\\begin{align*} \\left ( A _ n ^ { ( 1 ) } ( z ) \\right ) ^ k = \\mathcal { O } ( n ^ 2 ) . \\end{align*}"}
-{"id": "7665.png", "formula": "\\begin{align*} \\int \\limits _ { \\mathbb { R } ^ d } a _ { \\rm s y m } ( \\xi - q ) \\mu ( \\xi , q ) \\big ( \\tilde \\varphi ^ i _ { 0 } ( q ) - \\tilde \\varphi ^ i _ { 0 } ( \\xi ) \\big ) \\ , d q \\ = \\ 2 \\ , \\int \\limits _ { \\mathbb { R } ^ d } c ^ i ( \\xi - q ) \\mu ( \\xi , q ) \\ , d q . \\end{align*}"}
-{"id": "5314.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\alpha _ 0 = H _ { q _ 1 } + H _ { p _ 2 } - E _ { 3 , 4 , 9 , 1 0 } , & \\alpha _ 1 = H _ { q _ 2 } - E _ { 1 5 , 1 6 } , & \\alpha _ 2 = H _ { p _ 2 } - E _ { 5 , 6 } , \\\\ \\alpha _ 3 = H _ { p _ 1 } + H _ { q _ 2 } - E _ { 1 , 2 , 1 1 , 1 2 } , & \\alpha _ 4 = H _ { q _ 1 } - E _ { 7 , 8 } , & \\alpha _ 5 = H _ { p _ 1 } - E _ { 1 3 , 1 4 } \\end{array} \\end{align*}"}
-{"id": "4545.png", "formula": "\\begin{align*} z _ { 2 j - 1 } = w _ { 2 j - 1 } + i \\cdot w _ { 2 j } ; z _ { 2 j } = w _ { 2 j - 1 } - i \\cdot w _ { 2 j } j = 1 , \\ldots , k \\end{align*}"}
-{"id": "6541.png", "formula": "\\begin{align*} n _ \\beta : = ( - 1 ) ^ { w - 1 } e ( \\mathcal { M } _ \\beta ) , \\end{align*}"}
-{"id": "1258.png", "formula": "\\begin{align*} m _ 2 ( \\eta , \\nu ) & = \\frac { \\eta - 2 \\nu } { \\eta + \\nu } e ^ { 2 i a \\ln | ( \\eta + \\nu ) / 2 | } \\left ( 1 + \\frac { 2 i a } { ( \\eta + \\nu ) ^ 3 } \\right ) \\chi \\left ( \\frac { \\eta + \\nu } { 2 } \\right ) S \\left ( \\frac { \\eta - \\nu } { 2 } \\right ) \\\\ & + \\frac { \\eta - 2 \\nu } { \\eta + \\nu } e ^ { - 2 i a \\ln | ( \\eta + \\nu ) / 2 | } \\left ( 1 - \\frac { 2 i a } { ( \\eta + \\nu ) ^ 3 } \\right ) \\chi \\left ( - \\frac { \\eta + \\nu } { 2 } \\right ) S \\left ( \\frac { \\eta - \\nu } { 2 } \\right ) \\end{align*}"}
-{"id": "6271.png", "formula": "\\begin{align*} \\hat { \\theta } _ T = - \\frac { \\int _ 0 ^ T X _ t \\mathrm { d } X _ t } { \\int _ 0 ^ T X _ t ^ 2 \\mathrm { d } t } = \\theta - \\frac { \\int _ 0 ^ T X _ t \\mathrm { d } B ^ { H } _ t } { \\int _ 0 ^ T X _ t ^ 2 \\mathrm { d } t } , \\end{align*}"}
-{"id": "7789.png", "formula": "\\begin{align*} \\mathcal D _ t ^ { \\alpha } m ^ G ( t \\xi ) = \\int _ { G } ( 2 \\pi i x \\cdot \\xi ) ^ { \\alpha } e ^ { - 2 \\pi i t x \\cdot \\xi } { \\rm d } x , t > 0 , \\end{align*}"}
-{"id": "771.png", "formula": "\\begin{align*} H _ { q ^ d , M } = { \\bigtimes } _ { j = 1 } ^ r H _ { q ^ d , M _ j ^ { a _ j } } . \\end{align*}"}
-{"id": "7913.png", "formula": "\\begin{align*} x _ k = - x _ { - k } , \\quad \\ k = - \\hat { n } , \\ldots , \\hat { n } \\end{align*}"}
-{"id": "6265.png", "formula": "\\begin{align*} \\pi _ g ( \\pi _ h ( x ) ) = \\left \\{ \\begin{array} { l } \\pi _ g ( x ) , , \\\\ 0 , . \\end{array} \\right . \\end{align*}"}
-{"id": "10063.png", "formula": "\\begin{align*} \\tilde x = ( m _ n M _ m ) ^ { - 1 / 4 } x _ n , \\tilde y = ( m _ n M _ m ) ^ { 1 / 2 } \\mu _ n ^ { - 1 } y _ n , \\tilde G = \\mu _ n ^ { - 1 } G _ n . \\end{align*}"}
-{"id": "4278.png", "formula": "\\begin{align*} \\textup { s a t } ( n , C _ { 2 t + 1 } , C _ { 2 k + 1 } ) = 0 \\end{align*}"}
-{"id": "9513.png", "formula": "\\begin{align*} \\frac { a } { p } + \\frac { b } { q } - \\frac { a + b } { p _ { s } ^ * } = c _ { \\lambda } > 0 \\end{align*}"}
-{"id": "5860.png", "formula": "\\begin{align*} & \\int _ { f \\in M _ { n } } \\alpha \\chi _ 0 ^ { - 1 } ( f ( 0 ) ) \\beta \\chi _ 0 ^ { - 1 } ( f ( 1 ) ) \\gamma \\chi _ 0 ^ { - 1 } ( R ( G , f ) ) \\dd f \\\\ = & \\int _ { f \\in M _ { n } } \\alpha ( f ( 0 ) ) | f ( 0 ) | _ F ^ { - 1 } \\beta ( f ( 1 ) ) | f ( 1 ) | _ { F } ^ { - 1 } \\prod _ { i = 1 } ^ k \\chi ( \\N _ { g _ i } \\varphi _ i ( f ) ) | \\varphi _ i ( f ) | _ { F _ { g _ i } } ^ { - 1 } \\dd f \\end{align*}"}
-{"id": "5857.png", "formula": "\\begin{align*} \\int _ { f \\in M _ { n - 1 } } \\chi \\chi _ 0 ^ { - 1 } ( R ( G , f ) ) \\dd f = \\int _ { f \\in M _ { n - 1 } } \\prod _ { i = 1 } ^ k \\chi ( \\N _ { g _ i } \\varphi _ i ( f ) ) | \\varphi _ i ( f ) | _ { F _ { g _ i } } ^ { - 1 } \\dd f . \\end{align*}"}
-{"id": "483.png", "formula": "\\begin{align*} f = \\sum _ { i = 1 } ^ k f _ i ( \\mathbf { x } ^ i ) , \\end{align*}"}
-{"id": "4289.png", "formula": "\\begin{align*} ( n - s - 1 ) \\left ( \\binom { s - 2 } { r - 1 } + \\binom { s - 3 } { r - 1 } \\right ) + \\binom { s - 3 } { r } + 4 \\binom { s - 3 } { r - 1 } + 3 \\binom { s - 3 } { r - 2 } . \\end{align*}"}
-{"id": "8264.png", "formula": "\\begin{align*} Z ( \\alpha , t ) : = z \\circ h ^ { - 1 } ( \\alpha , t ) , b = h _ t \\circ h ^ { - 1 } , D _ t : = \\partial _ t + b \\partial _ { \\alpha } , \\end{align*}"}
-{"id": "4090.png", "formula": "\\begin{align*} \\phi _ 3 ( z ) = \\frac { 2 \\pi } { \\sqrt { 3 } } z ^ { - \\gamma } e ^ { - 3 z ^ { \\frac { 1 } { 3 } } } \\left ( 1 + M _ 1 z ^ { - \\frac { 1 } { 3 } } + M _ 2 z ^ { - \\frac { 2 } { 3 } } + \\mathcal { O } ( z ^ { - 1 } ) \\right ) \\end{align*}"}
-{"id": "5627.png", "formula": "\\begin{align*} \\int _ 0 ^ { + \\infty } u _ { x x } \\bar { u } _ t d x = - \\frac 1 2 \\frac { d } { d t } \\int _ 0 ^ { + \\infty } | u _ x | ^ 2 d x - ( u _ x ( 0 , t ) \\bar { u } _ t ( 0 , t ) ) . \\end{align*}"}
-{"id": "8347.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int \\frac { | D _ t Z ( \\alpha , t ) - D _ t Z ( \\beta , t ) | ^ 2 } { ( \\alpha - \\beta ) ^ 2 } d \\beta = & \\sum _ { 1 \\leq j , k \\leq N } \\frac { \\lambda _ j \\lambda _ k } { ( 2 \\pi ) ^ 3 } \\frac { 1 } { ( \\alpha - z _ j ) \\overline { ( \\alpha - z _ k ) } } \\frac { 2 \\pi i } { \\overline { z _ k } - z _ j } \\\\ = & \\sum _ { 1 \\leq j , k \\leq N } \\frac { \\lambda _ j \\lambda _ k } { ( 2 \\pi ) ^ 2 } \\frac { 1 } { ( \\alpha - z _ j ) \\overline { ( \\alpha - z _ k ) } } \\frac { i } { \\overline { z _ k } - z _ j } . \\end{align*}"}
-{"id": "3743.png", "formula": "\\begin{gather*} \\min \\limits _ { \\mathbf { J } \\in X \\cap \\overline { B _ { \\Gamma } \\left ( \\mathbf { J } _ { 0 } \\right ) } } \\mathcal { E } \\left ( \\mathbf { J } \\right ) , \\\\ \\overline { B _ { \\Gamma } \\left ( \\mathbf { J } _ { 0 } \\right ) } = \\left \\{ \\mathbf { J } \\in L ^ { 2 } \\left ( V ; \\mathbb { C } ^ { 3 } \\right ) : \\| \\mathbf { J } - \\mathbf { J } _ { 0 } \\| \\leq \\Gamma \\right \\} . \\end{gather*}"}
-{"id": "5254.png", "formula": "\\begin{align*} r ( x ) = - \\int _ { x } ^ { a } b ( t ) d t . \\end{align*}"}
-{"id": "7624.png", "formula": "\\begin{align*} H ^ { \\varepsilon } w ^ { \\varepsilon } ( x , t ) \\ : = \\frac { \\partial w ^ \\varepsilon } { \\partial t } - L ^ { \\varepsilon } w ^ \\varepsilon \\ = \\ \\Big ( \\frac { \\partial u } { \\partial t } ( x ^ \\varepsilon , t ) - \\Theta \\cdot \\nabla \\nabla u ( x ^ \\varepsilon , t ) \\ + \\ \\phi ^ \\varepsilon ( x ^ \\varepsilon , t ) \\Big ) | _ { _ { \\ , x ^ \\varepsilon = x - \\frac { b } { \\varepsilon } \\ , t } } , \\end{align*}"}
-{"id": "7663.png", "formula": "\\begin{align*} v _ 0 ^ { \\boldsymbol { \\ell } } ( \\xi ) \\ = \\ \\mathbf { 1 } \\ + \\ { \\boldsymbol { \\ell } } \\ , \\tilde \\varphi _ 0 ( \\xi ) \\ + \\ O ( { | \\boldsymbol { \\ell } | } ^ 2 ) , \\tilde \\varphi _ 0 \\in ( L ^ 2 ( \\mathbb T ^ d ) ) ^ d , \\end{align*}"}
-{"id": "453.png", "formula": "\\begin{align*} \\mathcal { Z } _ { q = 0 , m = 0 } ( g ) : = \\Big ( \\frac { { \\det } ' ( - \\Delta _ g ) } { V _ g } \\Big ) ^ { - \\mathbf { c } _ { \\rm m a t } / 2 } \\end{align*}"}
-{"id": "6637.png", "formula": "\\begin{align*} \\Lambda _ { S , r , s } ( f , g ) = \\sum _ { S \\in \\mathcal S } \\lvert S \\rvert \\langle f \\rangle _ { S , r } \\langle g \\rangle _ { S , s } . \\end{align*}"}
-{"id": "7200.png", "formula": "\\begin{align*} B = \\frac { 1 } { 4 \\pi } \\int _ { - \\pi } ^ \\pi \\phi ^ 2 ( x ) \\ , d x . \\end{align*}"}
-{"id": "1677.png", "formula": "\\begin{align*} [ X , f Y ] = \\varphi ^ * ( f ) [ X , Y ] + \\rho _ A ( X ) f \\alpha _ A ( Y ) . \\end{align*}"}
-{"id": "3809.png", "formula": "\\begin{align*} \\mathfrak { C } _ 1 = \\mathop { \\sum \\sum } _ { \\substack { d _ 1 , d _ 1 ' | q _ 1 } } d _ 1 d _ 1 ' \\mathop { \\mathop { \\sideset { } { ^ \\star } \\sum } _ { \\substack { \\alpha \\bmod { q _ 1 r / n _ 1 } \\\\ n _ 1 \\alpha \\equiv - m \\bmod { d _ 1 } } } \\ ; \\mathop { \\sideset { } { ^ \\star } \\sum } _ { \\substack { \\alpha ' \\bmod { q _ 1 r / n _ 1 } \\\\ n _ 1 \\alpha ' \\equiv - m ' \\bmod { d _ 1 ' } } } } _ { \\bar { \\alpha } q _ 2 ' - \\bar { \\alpha } ' q _ 2 \\equiv n _ 2 \\bmod { q _ 1 r / n _ 1 } } \\ ; 1 , \\end{align*}"}
-{"id": "1679.png", "formula": "\\begin{align*} \\left ( \\begin{matrix} \\frac { \\partial } { \\partial x } & - \\frac { \\partial } { \\partial y } \\\\ \\frac { \\partial } { \\partial y } & \\frac { \\partial } { \\partial x } \\\\ \\end{matrix} \\right ) \\left ( \\begin{matrix} u \\\\ v \\\\ \\end{matrix} \\right ) = 0 \\Leftrightarrow \\frac { \\partial w } { \\partial \\overline { z } } = 0 \\end{align*}"}
-{"id": "8922.png", "formula": "\\begin{align*} \\mathbb { E } [ N _ w ^ { \\hat { x } } ( G ' [ + ] ) - N _ w ^ { \\hat { x } } ( H [ + ] ) \\mid \\hat { x } ] = \\frac { a - b } { n } \\left ( ( n ^ { + + } - n _ C ^ { + + } ) - ( n ^ { + - } - n _ C ^ { + - } ) \\right ) ( n _ C ^ { + + } - n _ C ^ { + - } ) \\end{align*}"}
-{"id": "6587.png", "formula": "\\begin{align*} - \\frac { 1 } { 2 } \\bar { \\varphi } ( r _ i ) r _ i ^ { - 1 / 2 } \\bar { \\psi } ^ { ( i ) } ( r _ i , 0 ) = \\bar { \\Psi } ( r _ i , \\pi ) - \\bar { \\Psi } ( r _ i , 0 ) = & \\ \\int _ 0 ^ { \\pi } \\partial _ { \\theta _ i } \\bar { \\Psi } ( r _ i , \\theta _ i ) \\ , d \\theta _ i \\\\ = & \\ \\int _ 0 ^ { \\pi } r _ i ^ { - 1 } \\left ( \\partial _ { \\theta _ i } ^ 2 \\bar { S } _ i \\bar { \\psi } ^ { ( i ) } + \\partial _ { \\theta _ i } \\bar { S } _ i \\partial _ { \\theta _ i } \\bar { \\psi } ^ { ( i ) } \\right ) \\ , d \\theta _ i \\end{align*}"}
-{"id": "1212.png", "formula": "\\begin{align*} \\nabla _ x ^ k \\Phi _ \\tau ( x , y ) = \\frac { \\tau } { 4 \\pi \\gamma _ + } \\big ( I _ { k , \\tau } ( x , y ) + J _ { k , \\tau } ^ { ( 1 ) } ( x , y ) + J _ { k , \\tau } ^ { ( 2 ) } ( x , y ) \\big ) \\qquad ( k = 0 , 1 ) , \\end{align*}"}
-{"id": "361.png", "formula": "\\begin{align*} f ( t ) = t \\cdot \\frac { f ( t ) } { t } \\le t _ { \\ast } \\cdot 1 / \\eta < M \\quad \\forall t \\in [ 1 / \\eta , t _ { \\ast } ] \\end{align*}"}
-{"id": "7737.png", "formula": "\\begin{align*} \\bigg [ \\Big ( \\| \\langle x , y \\rangle \\| \\langle x , x \\rangle + \\lambda \\| x \\| ^ 2 \\langle x , y \\rangle \\Big ) \\xi , \\langle x , x \\rangle \\xi \\bigg ] = 0 . \\end{align*}"}
-{"id": "4827.png", "formula": "\\begin{align*} \\mathrm { d } X _ t = & V _ t \\mathrm { d } t , \\\\ \\mathrm { d } V _ t = & - ( V _ t + \\nabla _ x U ( X _ t ) ) \\mathrm { d } t + \\sqrt { 2 } \\mathrm { d } W _ t . \\end{align*}"}
-{"id": "4213.png", "formula": "\\begin{align*} E _ { i n } ( z ) = \\frac { \\sqrt { 3 } } { 2 \\pi } n ^ { 2 \\beta } \\left ( \\frac { f ( z ) } { z } \\right ) ^ { \\frac { 2 \\beta } { 3 } } \\left ( \\mathbb I - B _ n ^ { ( 3 ) } ( z ) \\right ) \\left ( \\mathbb I - B _ n ^ { ( 2 ) } ( z ) \\right ) \\left ( \\mathbb I - B _ n ^ { ( 1 ) } ( z ) \\right ) E _ n ( z ) \\end{align*}"}
-{"id": "1536.png", "formula": "\\begin{align*} u _ 1 ( x , t ) = P . V . \\frac { 1 } { 2 \\ , \\pi } \\int _ { - \\infty } ^ { \\infty } \\int _ { 0 } ^ { 1 } { e ^ { s i t } g ( \\sqrt { i s } , x , y ) f ( y ) d y \\ , d s } . \\end{align*}"}
-{"id": "5105.png", "formula": "\\begin{align*} \\langle x _ j , x _ k \\rangle & = \\frac { \\| x _ j + x _ k \\| ^ 2 - \\| x _ j - x _ k \\| ^ 2 + i \\| x _ j + i x _ k \\| ^ 2 - i \\| x _ j - i x _ k \\| ^ 2 } { 4 } \\\\ & = \\frac { ( \\| x _ j \\| ^ 2 + \\| x _ k \\| ^ 2 ) - ( \\| x _ j \\| ^ 2 + \\| - x _ k \\| ^ 2 ) + i ( \\| x _ j \\| ^ 2 + \\| i x _ k \\| ^ 2 ) - i ( \\| x _ j \\| ^ 2 + \\| - i x _ k \\| ^ 2 ) } { 4 } = 0 . \\end{align*}"}
-{"id": "4964.png", "formula": "\\begin{align*} ( p - 1 ) ( q - 1 ) + \\dfrac { q - 1 } { 2 } = \\dfrac { 2 p q - 2 p - q + 1 } { 2 } = \\dfrac { ( 2 p - 1 ) ( q - 1 ) } { 2 } . \\end{align*}"}
-{"id": "1183.png", "formula": "\\begin{align*} & O ( ( \\ln n ) ^ { 2 } / n ) \\det ( + B ) = O ( ( \\ln n ) ^ { 2 } e ^ { - \\pi ^ 2 c _ 0 ^ 2 ( \\ln n ) / 8 } / n ) = O ( ( n \\ln n ) ^ { - 1 } ) . \\end{align*}"}
-{"id": "4940.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ { d - 3 } ( - 1 ) ^ { i + 1 } v _ i z ^ { 3 n } \\right ) + ( - 1 ) ^ { d - 1 } ( x z ^ { 3 n } ) = z ^ { 2 n } f _ 3 - y z ^ { 2 n } \\in I _ n . \\end{align*}"}
-{"id": "9954.png", "formula": "\\begin{align*} = ( x _ z F _ x + y _ z F _ y ) ( x _ \\theta G _ x + y _ \\theta G _ y ) - ( x _ \\theta F _ x + y _ \\theta F _ y ) ( x _ z G _ x + y _ z G _ y ) . \\end{align*}"}
-{"id": "6710.png", "formula": "\\begin{align*} \\frac { \\partial V ( q ) } { \\partial q _ i } ( \\alpha ' _ { i k } + \\beta ' _ { i l k } \\ , \\dot { \\bar { q } } _ l ) = \\frac { \\partial \\bar { V } ( Q _ 1 , Q _ 2 ) } { \\partial Q _ 1 } , \\end{align*}"}
-{"id": "8172.png", "formula": "\\begin{align*} \\begin{aligned} & X _ { { 1 1 } } ( - j - 1 ) \\sp { b _ { 1 1 } } X _ { { 1 2 } } ( - j ) \\sp { a _ { 1 2 } } X _ { { 1 1 } } ( - j ) \\sp { a _ { 1 1 } } \\ , , \\ \\ , b _ { 1 1 } + a _ { 1 2 } + a _ { 1 1 } = k + 1 , \\\\ & X _ { { 1 1 } } ( - j - 1 ) \\sp { b _ { 1 1 } } X _ { 2 2 } ( - j ) \\sp { a _ { { 2 2 } } } X _ { { 1 2 } } ( - j ) \\sp { a _ { 1 2 } } \\ , , \\ \\ , b _ { { 1 1 } } + a _ { { 2 2 } } + a _ { { 1 2 } } = k + 1 , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "6008.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow \\infty } \\sup _ { Q \\in \\Lambda _ a } Q \\left ( \\int _ 0 ^ 1 | q ^ Q ( t ) | d t \\ge r \\right ) = 0 . \\end{align*}"}
-{"id": "4981.png", "formula": "\\begin{align*} { Z } ( C / \\mathbb { F } _ q , T ) = \\bigg ( \\sum _ { s = 1 } ^ { \\infty } \\frac { { N } _ { s } T ^ s } { s } \\bigg ) . \\end{align*}"}
-{"id": "5566.png", "formula": "\\begin{align*} \\omega = \\sum _ { \\substack { \\alpha \\in A ( r - 1 , n ) } } \\sum _ { \\substack { \\rho \\in \\Sigma _ { 0 } ( k , n ) \\\\ \\lfloor \\alpha \\rfloor \\geq \\lfloor \\rho \\rfloor } } \\omega _ { \\alpha \\rho } \\lambda _ T ^ { \\alpha } \\phi ^ T _ { \\rho } . \\end{align*}"}
-{"id": "8510.png", "formula": "\\begin{align*} \\norm { \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { 2 \\pi } \\frac { ( \\dot { z } _ j ) ^ 2 } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 3 } } _ { H ^ s } \\leq K _ s ^ { - 1 } \\epsilon ^ 2 + K _ s ^ { - 1 } \\epsilon \\frac { | \\lambda | } { x ( 0 ) } d _ I ( t ) ^ { - 5 / 2 } . \\end{align*}"}
-{"id": "7178.png", "formula": "\\begin{gather*} \\mathcal { U } _ { r } ( \\mathcal { M } ) \\subset \\mathcal { N } _ { r } ( \\mathcal { M } ) : = \\left \\{ ( \\xi , \\eta ) : \\xi \\in \\mathcal { M } , \\ ; \\eta \\in \\mathbb { L } _ { \\xi } ^ { - } , \\ ; \\left \\Vert \\eta \\right \\Vert < r \\right \\} . \\end{gather*}"}
-{"id": "3031.png", "formula": "\\begin{align*} h ( z ) = z - \\sum \\limits _ { k = 2 } ^ \\infty | a _ { k } | { z ^ k } \\quad g _ m ( z ) = ( - 1 ) ^ { m + i - 1 } \\sum \\limits _ { k = 1 } ^ \\infty | b _ { k } | { z ^ k } , | b _ 1 | < 1 . \\end{align*}"}
-{"id": "5118.png", "formula": "\\begin{align*} \\| T x \\| ^ p = \\left \\| \\sum _ { j \\in \\mathbb { J } } c _ j x _ j \\right \\| ^ p = \\sum _ { j \\in \\mathbb { J } } | c _ j | ^ p = \\left \\| \\sum _ { j \\in \\mathbb { J } } c _ j \\tau _ j \\right \\| ^ p = \\| x \\| ^ p . \\end{align*}"}
-{"id": "2954.png", "formula": "\\begin{align*} r _ { ( k ) \\ , \\sigma \\left ( i _ { 1 } \\right ) . . . \\sigma \\left ( i _ { k } \\right ) } = \\dfrac { c _ { j } \\alpha _ { ( j ) \\ , i _ { 1 } . . . i _ { j } } } { k ! } = \\dfrac { j ! } { k ! } c _ { j } a _ { ( j ) \\ , i _ { 1 } . . . i _ { j } } \\end{align*}"}
-{"id": "4925.png", "formula": "\\begin{align*} \\ell ( H ^ i ( f _ 1 , \\ldots , f _ d ; M ) ) \\leq \\sum _ { j = 0 } ^ { i } { d \\choose i - j } \\ell ( H ^ j _ m ( M ) ^ \\vee ) . \\end{align*}"}
-{"id": "4380.png", "formula": "\\begin{align*} \\left [ a ^ n , w ( a , b ) \\right ] = a ^ { n + k } b ^ { \\pm 1 } u ( a , b ) a ^ { - n } u ( a , b ) ^ { - 1 } b ^ { \\mp 1 } a ^ { - k } \\end{align*}"}
-{"id": "6954.png", "formula": "\\begin{align*} \\textbf { S } _ 1 \\left ( \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ n y _ i \\right ) = \\textbf { S } _ 1 ( x _ 1 ) + ( n - 1 ) c ( x _ 1 ) . \\end{align*}"}
-{"id": "3038.png", "formula": "\\begin{align*} \\| f \\| _ { B _ { 1 , \\infty } ^ { \\lambda , a } } : = \\| f \\| _ { L ^ 1 ( \\R ^ d ) } + \\sum \\limits _ { k = 1 } ^ { d } \\sup \\limits _ { h \\in [ - 1 , 1 ] } | h | ^ { - \\lambda / a _ k } \\| \\Delta _ { h e _ k } f \\| _ { L ^ 1 ( \\R ^ d ) } , \\end{align*}"}
-{"id": "6497.png", "formula": "\\begin{align*} R _ { n , p } = { \\pi ^ { \\frac { n } { 2 p } + n } { \\Gamma \\left ( \\frac { n + 2 } { 2 } \\right ) ^ { - 2 } } \\left ( \\frac { 2 \\Gamma \\left ( \\frac { 1 } { 2 } ( n + p + 2 ) \\right ) } { \\Gamma \\left ( \\frac { n + 2 } { 2 } \\right ) \\Gamma \\left ( \\frac { p + 1 } { 2 } \\right ) } \\right ) ^ { \\frac { n } { p } } } \\end{align*}"}
-{"id": "5141.png", "formula": "\\begin{align*} ( \\mu + \\nu ) \\left ( \\Gamma ^ + ( \\psi ) \\right ) _ { m _ j , k _ j } = - ( k _ j + \\lambda - \\gamma _ { k _ j } ) \\left ( \\Gamma ^ - ( \\psi ) \\right ) _ { m _ j , k _ j } ; \\end{align*}"}
-{"id": "1712.png", "formula": "\\begin{align*} | V ^ { - 1 } T ( h ) V y - y | _ Y & = V ^ { - 1 } | T ( h ) V y - V y | _ X \\leq \\varepsilon \\cdot ( V ^ { - 1 } u ) \\end{align*}"}
-{"id": "2989.png", "formula": "\\begin{align*} q ( 1 5 ) & = 1 + ( q ( 1 5 - 2 P _ { 5 , 1 } ) + q ( 1 5 - 2 Q _ { 5 , 1 } ) ) - ( q ( 1 5 - 2 P _ { 5 , 2 } ) + q ( 1 5 - 2 Q _ { 5 , 2 } ) ) \\\\ & = 1 + q ( 1 3 ) + q ( 1 1 ) - q ( 5 ) - q ( 1 ) = 1 + 1 8 + 1 2 - 3 - 1 = 2 7 . \\end{align*}"}
-{"id": "26.png", "formula": "\\begin{align*} d X _ { t } ( \\pi ) = X _ t ( \\pi ) \\pi _ { t } ^ { t r } \\left ( \\theta ^ { \\alpha _ { t - } } ( V _ { t } ) d t + d W _ { t } \\right ) . \\end{align*}"}
-{"id": "5139.png", "formula": "\\begin{gather*} A B ^ * = B A ^ * , \\\\ \\begin{pmatrix} A & - B \\\\ B & A \\end{pmatrix} = 0 . \\end{gather*}"}
-{"id": "7552.png", "formula": "\\begin{align*} \\norm { f } _ { L ^ \\infty ( 0 , T ; Y ) } = \\sup _ { t \\in [ 0 , T ] } \\norm { f ( t ) } _ Y , \\end{align*}"}
-{"id": "2102.png", "formula": "\\begin{align*} \\int _ a ^ b x ( t ) d \\omega ( t ) : = \\lim \\limits _ { | \\Pi | \\to 0 } \\sum _ { t _ i \\in \\Pi } x ( t _ i ) ( \\omega ( t _ { i + 1 } ) - \\omega ( t _ i ) ) , \\end{align*}"}
-{"id": "2262.png", "formula": "\\begin{align*} p _ { \\pm \\i \\pi } & = | \\alpha _ { \\pm \\i \\pi } \\rangle \\langle \\beta _ { \\pm \\i \\pi } | \\\\ | \\alpha _ { \\pm \\i \\pi } \\rangle & = \\binom { 1 } { - \\displaystyle \\frac { 1 } { \\pm \\i \\pi - \\Omega } | E \\rangle } & \\langle \\beta _ { \\pm \\i \\pi } | & = \\left ( 1 , \\langle E | \\frac { 1 } { \\pm \\i \\pi - \\Omega } \\right ) \\end{align*}"}
-{"id": "9631.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\mathcal { D } ( n , m ) } ( - 1 ) ^ { \\# ( \\pi ) } \\sum _ { j = \\xi } ^ { s ( \\pi ) } ( - 1 ) ^ { j - 1 } = \\frac { 1 } { 2 } \\sum _ { \\substack { \\pi \\in \\overline { \\mathcal { P } } ( n ) \\\\ l ( \\pi ) \\leq m } } ( - 1 ) ^ { \\# ( \\pi ) } , \\end{align*}"}
-{"id": "9115.png", "formula": "\\begin{align*} j _ \\ast ( E ) = \\ker \\left ( ( j _ 1 ) _ \\ast ( E _ 1 ) \\oplus ( j _ 2 ) _ \\ast ( E _ 2 ) \\to ( j _ { 1 2 } ) _ \\ast ( F ) \\right ) . \\end{align*}"}
-{"id": "75.png", "formula": "\\begin{align*} v = \\rho - \\widetilde \\rho , w = J - J _ b \\ , , \\end{align*}"}
-{"id": "965.png", "formula": "\\begin{align*} \\frac { d x } { d t } \\ ; = \\ ; - \\sigma ( x - y ) , \\frac { d y } { d t } \\ ; = \\ ; \\rho x - y - x z , \\frac { d z } { d t } \\ ; = \\ ; x y - \\beta z , \\end{align*}"}
-{"id": "725.png", "formula": "\\begin{align*} L = \\sum _ { i = 1 } ^ { m } \\theta _ { i } ^ { q } \\end{align*}"}
-{"id": "4263.png", "formula": "\\begin{align*} q : = \\left \\lbrace \\begin{aligned} & \\frac { n p } { n + p } & & n > 2 p > 2 \\\\ & \\frac { 3 } { 2 } & & n = p = 2 \\\\ & \\frac { n p } { n p - n + p } & & 1 < p < 2 \\end{aligned} \\right . \\end{align*}"}
-{"id": "2576.png", "formula": "\\begin{align*} v _ t ( x ) = \\big ( 1 + \\epsilon _ { R ( \\langle v _ t , \\phi ^ * \\rangle _ m ) } ( x ) \\big ) \\langle v _ t , \\phi ^ * \\rangle _ m \\phi ( x ) , t > 0 , x \\in E . \\end{align*}"}
-{"id": "6779.png", "formula": "\\begin{align*} \\beta \\mapsto x ( t , \\beta ) = \\Phi ( t , \\beta ) = \\Phi ^ t ( \\beta ) \\end{align*}"}
-{"id": "8635.png", "formula": "\\begin{align*} p _ \\varepsilon ^ \\pm - p _ 0 = O ( \\varepsilon ) \\ \\textrm { a n d } \\ q _ \\varepsilon ^ \\pm - q _ 0 = O ( \\varepsilon ) \\ \\textrm { a s } \\ \\varepsilon \\to 0 , \\end{align*}"}
-{"id": "903.png", "formula": "\\begin{align*} Y ^ j _ m = ( - 1 ) ^ { \\sum _ { k = 1 } ^ { r _ j } m _ k ^ j ( \\sum _ { i \\in Q ^ j _ 0 } d ^ i _ k - 1 ) } \\cdot q ^ { w ^ j } \\cdot \\prod ^ { \\to } _ { i \\in Q ^ j _ 0 } y _ { e _ i } ^ { \\gamma ^ j ( i ) } \\end{align*}"}
-{"id": "3018.png", "formula": "\\begin{align*} \\ln ( F _ m ( x ) ) & = - \\sum _ { k = 1 } ^ \\infty \\big ( \\ln ( 1 - x ^ { k m } ) + \\ln ( 1 - x ^ { k m - ( m - 1 ) } ) + \\ln ( 1 - x ^ { k m - 1 } ) \\big ) \\\\ & = \\sum _ { k = 1 } ^ \\infty \\big ( ( n - 1 ) ! \\cdot \\sigma ' _ m ( n ) \\big ) \\frac { x ^ n } { n ! } . \\end{align*}"}
-{"id": "3811.png", "formula": "\\begin{align*} a = \\langle e , e \\rangle ( 0 , 1 ) , b = \\langle a , c \\rangle , c = \\langle a , d \\rangle , d = \\langle e , b \\rangle . \\end{align*}"}
-{"id": "1231.png", "formula": "\\begin{align*} P ( 2 ) = 0 , P ' ( 2 ) = 0 , P '' ( 2 ) = 1 , \\end{align*}"}
-{"id": "4639.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } h _ { f _ { 1 , \\lambda _ i } } ( 0 ) = \\lim _ { i \\to \\infty } h _ { f _ { 2 , \\lambda _ i } } ( 0 ) = 0 . \\end{align*}"}
-{"id": "3839.png", "formula": "\\begin{align*} 2 \\Phi ^ i h _ { 0 0 } ^ { \\frac { m + 1 } { 2 } } W _ 0 ^ m = A ^ i _ { ( 1 ) } h _ { 0 0 } ^ { m + 1 } + A ^ i _ { ( 2 ) } h _ { 0 0 } ^ { \\frac { m + 1 } { 2 } } W _ 0 ^ m + A ^ i _ { ( 3 ) } W _ 0 ^ { 2 m } , \\end{align*}"}
-{"id": "6174.png", "formula": "\\begin{align*} \\varphi _ 1 = \\varphi _ 2 \\end{align*}"}
-{"id": "5488.png", "formula": "\\begin{align*} L = L _ b = \\frac { 1 } { 2 } \\Delta + b . \\nabla = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ d \\frac { \\partial ^ 2 } { \\partial x _ i ^ 2 } + \\sum _ { i = 1 } ^ d b _ i ( \\cdot ) \\frac { \\partial } { \\partial x _ i } , \\end{align*}"}
-{"id": "679.png", "formula": "\\begin{align*} \\mathcal M ( a _ 1 \\dots a _ n ; a ) & \\cong ( \\mathcal M \\wr \\widetilde { \\mathbb E } _ { \\mathcal G } ) ( a _ 1 \\dots a _ n , a ) _ { \\mu _ n } \\\\ & \\xrightarrow [ \\cong ] { H } \\mathcal C ( H ( a _ 1 \\dots a _ n ) , H ( a ) ) _ { \\mu _ n } \\ . \\end{align*}"}
-{"id": "1320.png", "formula": "\\begin{align*} P _ 0 ( Z _ 1 \\leq z ) = \\varPsi _ 1 ^ { - 1 } ( z ) \\ge \\varPsi _ 2 ^ { - 1 } ( z ) = P _ 0 ( Z _ 2 \\le z ) \\end{align*}"}
-{"id": "6520.png", "formula": "\\begin{align*} \\mu _ n & : = \\prod _ { j = 1 } ^ m d ^ { k ( n _ j + n q ) } \\cdot ( r _ n ) ^ * \\left ( T _ { f , a _ 1 } ^ k \\wedge \\cdots \\wedge T _ { f , a _ m } ^ k \\right ) \\\\ & = \\bigwedge _ { j = 1 } ^ m \\left ( d d ^ c G _ { r _ n ( x ) } \\left ( \\sigma _ j \\circ a ^ { n , j } ( x ) \\right ) \\right ) ^ k . \\end{align*}"}
-{"id": "6199.png", "formula": "\\begin{align*} \\{ x \\in \\mathcal { B } _ 2 : x _ n \\leq - \\delta \\} \\subset \\{ x \\in \\mathcal { B } _ 2 : g ( x , 0 ) = 0 \\} \\subset \\{ x \\in \\mathcal { B } _ 2 : x _ n \\leq \\delta \\} , \\end{align*}"}
-{"id": "5.png", "formula": "\\begin{align*} \\delta Y _ t ^ { i , r } ( m ) : = Y _ t ^ { i , r , v } ( m ) - Y _ t ^ { i , r , \\bar { v } } ( m ) \\ \\ \\ \\ \\delta Z _ t ^ { i , r } ( m ) : = Z _ t ^ { i , r , v } ( m ) - Z _ t ^ { i , r , \\bar { v } } ( m ) . \\end{align*}"}
-{"id": "7533.png", "formula": "\\begin{align*} g _ { \\nu t } ( x ) = \\frac { 1 } { ( 4 \\pi \\nu t ) ^ { \\frac { d } { 2 } } } e ^ { - \\frac { | x | ^ 2 } { 4 \\nu t } } . \\end{align*}"}
-{"id": "4913.png", "formula": "\\begin{align*} L _ 2 = - \\int _ { \\Omega } g ( \\Delta _ R u , u ) \\ , \\mu ( x ) \\ , , \\end{align*}"}
-{"id": "4042.png", "formula": "\\begin{align*} I _ M ( P _ 1 , V ) = \\sum _ { x \\in P _ 1 } i ( x ) \\leq C M | P _ 1 | \\leq C M m . \\end{align*}"}
-{"id": "8992.png", "formula": "\\begin{align*} \\rho _ + : = \\frac { X ( 0 ) ( 1 - \\beta \\gamma _ g ) + \\sqrt { \\beta ^ 2 \\gamma _ g ^ 2 X ( 0 ) ^ 2 + 2 \\beta \\gamma _ g - 1 } } { 2 \\beta \\gamma _ g - 1 } . \\end{align*}"}
-{"id": "192.png", "formula": "\\begin{align*} \\pi ( { \\bf x } ) = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ 0 & 0 & x ^ 1 \\\\ 0 & - x ^ 1 & 0 \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "2411.png", "formula": "\\begin{align*} e q ( X ) = \\Delta \\cdot q ( Y ) \\end{align*}"}
-{"id": "8358.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j } { \\pi } \\frac { 1 } { ( 0 - z _ j ) ^ 2 } \\dot { z } _ j ( t ) = \\frac { \\lambda i } { 4 \\pi } \\frac { 1 } { x } \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j } { \\pi } \\frac { 1 } { ( 0 - z _ j ) ^ 2 } = \\frac { \\lambda i } { 4 \\pi x } \\frac { \\lambda } { \\pi } \\frac { 4 x y i } { ( x ^ 2 + y ^ 2 ) ^ 2 } = - \\frac { \\lambda ^ 2 y } { \\pi ^ 2 ( x ^ 2 + y ^ 2 ) ^ 2 } . \\end{align*}"}
-{"id": "446.png", "formula": "\\begin{align*} A & : = \\left \\{ \\ , | \\{ x \\in D \\ , : \\ , \\tilde { X } _ { t _ 0 } ( x ) \\ge - 1 ) \\} | \\ge ( \\log s ) ^ { - \\kappa } \\ , \\right \\} , \\\\ B _ n & : = \\left \\{ \\ , | \\{ x \\in D \\ , : \\ , \\tilde { X } _ { t _ 0 } ( x ) \\ge n ) \\} | \\ge n ^ { - 1 } ( \\log n ) ^ { - 2 } \\ , \\right \\} , \\\\ \\bar B _ n & : = B _ n \\setminus B _ { n - 1 } , \\ n \\ge n _ 0 + 1 \\bar B _ { n _ 0 } = B _ { n _ 0 } \\setminus A . \\end{align*}"}
-{"id": "5705.png", "formula": "\\begin{align*} Y _ 0 ( x ) = \\frac { 2 } { \\pi } J _ 0 ( x ) \\left ( \\ln \\frac { x } { 2 } + C \\right ) - \\frac { 2 } { \\pi } \\sum _ { m = 1 } ^ { \\infty } \\left [ \\frac { ( - 1 ) ^ m } { ( m ! ) ^ 2 } \\left ( \\frac { x } { 2 } \\right ) ^ { 2 m } \\sum _ { k = 0 } ^ m \\frac { 1 } { m } \\right ] , \\end{align*}"}
-{"id": "9445.png", "formula": "\\begin{align*} ( f _ { \\theta } - \\mathcal { E } _ 2 , \\xi _ 0 ( j _ { N , m } ) ) _ { r e g } = \\beta _ { N , m } ( a _ { \\theta } ( 0 ) - b ( 0 ) ) + a _ \\theta ( m ) - b ( m ) + \\sum _ { \\tau \\in \\mathcal { F } _ N } \\nu _ \\tau ^ { ( N ) } ( f ) j _ { N , m } ( \\tau ) , \\end{align*}"}
-{"id": "2686.png", "formula": "\\begin{align*} H ( z , w ) = \\frac { 1 } { \\sqrt { w ^ { 2 } - 1 } \\sqrt { z ^ { 2 } - 1 } } \\frac { 1 - z w } { w - z } \\Big ( \\frac { z - \\tau } { \\tau z - 1 } \\frac { \\tau w - 1 } { w - \\tau } \\Big ) ^ { n } \\prod _ { k = 1 } ^ { n } \\frac { \\sigma _ { k } w - \\tau } { \\tau w - \\sigma _ { k } } \\frac { \\tau z - \\sigma _ { k } } { \\sigma _ { k } z - \\tau } . \\end{align*}"}
-{"id": "9192.png", "formula": "\\begin{align*} \\mathcal { K } & \\equiv \\mathrm { s p a n } \\left \\{ \\left . J \\in \\Lambda _ { N } \\right \\vert J \\left ( t \\right ) = 0 t \\in \\left ( 0 , b \\right ] \\right \\} , \\\\ \\mathcal { K } _ { b } & \\equiv \\left \\{ \\left . J \\in \\Lambda _ { N } \\right \\vert J \\left ( b \\right ) = 0 \\right \\} . \\end{align*}"}
-{"id": "7802.png", "formula": "\\begin{align*} | m ^ G ( \\xi ) | & \\le ( \\pi | \\xi | ) ^ { - 1 } \\varphi _ { \\zeta } ^ G ( 0 ) + ( 2 \\pi | \\xi | ) ^ { - 1 } \\int _ { - u _ { \\zeta } } ^ { u _ { \\zeta } } | ( \\varphi _ { \\zeta } ^ G ) ' ( u ) | { \\rm d } u \\\\ & = ( \\pi | \\xi | ) ^ { - 1 } \\varphi _ { \\zeta } ^ G ( 0 ) - ( \\pi | \\xi | ) ^ { - 1 } \\int _ { 0 } ^ { u _ { \\zeta } } ( \\varphi _ { \\zeta } ^ G ) ' ( u ) { \\rm d } u \\\\ & \\le 6 \\pi ^ { - 1 } ( L | \\xi | ) ^ { - 1 } . \\end{align*}"}
-{"id": "9255.png", "formula": "\\begin{align*} v : = \\tilde v _ 1 - \\zeta ^ 2 ( 0 ) w _ 1 ( 0 ) f - K ( 1 + M _ 2 ( R ) ) \\zeta ^ 2 \\phi , \\end{align*}"}
-{"id": "7708.png", "formula": "\\begin{align*} \\chi ( x ) : = \\int \\limits _ { - \\infty } ^ { \\infty } \\frac { \\sin ( x t ) } { t } f ( t ) \\ , \\mathrm { d } t , \\end{align*}"}
-{"id": "9342.png", "formula": "\\begin{align*} \\Big ( ( 1 + n - t ^ 2 ) \\cosh ( t ) - n t \\sinh ( t ) , ( 1 + n - t ^ 2 ) \\sinh ( t ) - n t \\cosh ( t ) , 0 , \\cdots , 0 \\Big ) + n b = \\mu _ { a } a . \\end{align*}"}
-{"id": "5255.png", "formula": "\\begin{align*} R ( F , x ) = ( 1 - F ( x ) ) ^ { - 1 } \\int _ { x } ^ { u e p ( F ) } 1 - F ( t ) \\ d t , \\ x < u e p ( F ) . \\end{align*}"}
-{"id": "2204.png", "formula": "\\begin{align*} s _ f & = \\boldsymbol { x } ^ T ( t _ f ) F ( t _ f ) \\boldsymbol { x } ( t _ f ) - \\boldsymbol { x } ^ T ( t _ f ) P ( t _ f ) \\boldsymbol { x } ( t _ f ) \\\\ & = \\boldsymbol { x } ^ T ( t _ f ) \\big [ F ( t _ f ) - P ( t _ f ) \\big ] \\boldsymbol { x } ( t _ f ) \\end{align*}"}
-{"id": "8209.png", "formula": "\\begin{align*} \\eta ( \\tau ) : = q ^ { 1 / 2 4 } ( q ; q ) _ \\infty . \\end{align*}"}
-{"id": "6733.png", "formula": "\\begin{align*} P _ { 1 k l } - \\frac { 1 } { 8 c ^ 2 } \\frac { \\partial } { \\partial x _ n } \\left ( 2 \\frac { \\partial Q _ { 2 k l } } { \\partial x ^ n } - \\frac { \\partial Q _ { 2 n l } } { \\partial x ^ k } - \\frac { \\partial Q _ { 2 k n } } { \\partial x ^ l } \\right ) = 0 , \\end{align*}"}
-{"id": "6397.png", "formula": "\\begin{align*} \\omega _ { i } \\equiv \\frac { ( r _ { 0 } , q _ { i } ) ^ { 2 } } { \\| r _ 0 \\| ^ 2 } \\qquad \\mbox { s o t h a t } \\qquad \\sum _ { i = 1 } ^ { N } \\omega _ { i } = 1 \\ , , \\end{align*}"}
-{"id": "2285.png", "formula": "\\begin{align*} ( \\hat { E } | C ( z ) | E ) & = ( E | C ( z ) | \\hat { E } ) = - i \\pi \\sigma ( z ) \\\\ \\hat { \\Omega } R ( z ) | E ) & = - | \\hat { E } ) + z R ( z ) | E ) \\\\ ( E | R ( z ) \\hat { \\Omega } & = - ( \\hat { E } | + z ( E 1 | R ( z ) \\end{align*}"}
-{"id": "4393.png", "formula": "\\begin{align*} | f _ { \\delta } ( \\tfrac { 1 } { \\delta } , \\delta ) | ^ { - ( \\tfrac { 1 } { \\delta ^ 2 } + \\delta ^ 2 ) } \\sim 1 , | f _ { \\delta } ( \\tfrac { 1 } { \\delta } , \\delta ) | ^ { - ( \\tfrac { 1 } { \\delta ^ 2 } + \\delta ^ 2 ) } = 0 \\int _ { C _ 1 } | f _ { \\delta } | ^ 2 e ^ { - | \\cdot | ^ 2 } \\omega _ o \\le C / \\sqrt { \\delta } , \\end{align*}"}
-{"id": "506.png", "formula": "\\begin{align*} 2 g - 4 g ' + 2 = r _ { 4 } + 2 r _ { 2 } . \\end{align*}"}
-{"id": "7706.png", "formula": "\\begin{align*} \\int \\limits _ { - \\infty } ^ { - \\epsilon } \\frac { \\cos ( x t ) } { t } h \\left ( \\frac { t } { 2 \\pi } \\right ) \\ , \\mathrm { d } t = - \\int \\limits _ { \\epsilon } ^ { \\infty } \\frac { \\cos ( x t ) } { t } h \\left ( \\frac { t } { 2 \\pi } \\right ) \\ , \\mathrm { d } t . \\end{align*}"}
-{"id": "750.png", "formula": "\\begin{align*} \\mathbb { E } _ { f } \\ ! \\left [ S \\right ] = \\psi _ { 0 } ( m n + 1 ) - \\psi _ { 0 } ( n ) - \\frac { m + 1 } { 2 n } . \\end{align*}"}
-{"id": "9987.png", "formula": "\\begin{align*} B ^ { i j } = - ( g ^ { i j } ( \\mathbf { b } _ x ) \\partial _ x ^ { } + c ^ { i j } _ k ( \\mathbf { b } _ x ) b ^ k _ { x x } + c ^ \\alpha w ^ i _ { \\alpha k } ( \\mathbf { b } _ x ) b ^ k _ { x x } \\partial _ x ^ { - 1 } w ^ j _ { \\alpha h } ( \\mathbf { b } _ x ) b ^ h _ { x x } ) \\end{align*}"}
-{"id": "3964.png", "formula": "\\begin{align*} M = \\left ( \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} , \\left [ \\lambda , \\mu \\right ] \\right ) \\longmapsto \\widetilde { M } : = \\begin{pmatrix} a & 0 & b & \\mu \\\\ \\lambda ' & 1 & \\mu ' & 0 \\\\ c & 0 & d & - \\lambda \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "3840.png", "formula": "\\begin{align*} A ^ i _ { ( 1 ) } = & - 2 \\sigma _ 0 S ^ i _ 0 - \\sigma _ 0 W _ 0 \\bar { k } ^ i + \\sigma _ 0 k _ 0 W ^ i + 2 \\sigma _ 0 \\sigma _ 1 S _ 0 W ^ i , \\\\ A ^ i _ { ( 2 ) } = & - k _ 0 y ^ i + \\frac { 1 } { 2 } h _ { 0 0 } \\bar { k } ^ i + \\sigma _ 1 R _ { 0 0 } b ^ i , \\\\ A ^ i _ { ( 3 ) } = & - 4 \\sigma _ 1 R _ { 0 0 } y ^ i + 2 \\sigma _ 1 h _ { 0 0 } W _ r \\bar { k } ^ r y ^ i + \\frac { \\sigma _ 1 } { 2 } h _ { 0 0 } ^ { \\frac { m + 3 } { 2 } } W _ r \\bar { k } ^ r b ^ i . \\end{align*}"}
-{"id": "570.png", "formula": "\\begin{align*} \\Delta \\left ( \\xi ^ s ( 1 + p x _ s ) R _ n ^ * \\right ) = \\left \\lbrace \\xi ^ s ( 1 + p x _ s ) a - \\xi ^ s ( 1 + p x _ s ) b : a , b \\in R _ n ^ * , a \\ne b \\right \\rbrace . \\end{align*}"}
-{"id": "1988.png", "formula": "\\begin{align*} i _ j ^ * T \\P ^ n = \\bigoplus _ { k \\neq j } \\C \\left \\{ \\frac { \\partial } { \\partial z _ k } \\right \\} = \\bigoplus _ { k \\neq j } \\O _ { \\alpha _ j - \\alpha _ k } . \\end{align*}"}
-{"id": "3078.png", "formula": "\\begin{align*} | I _ 1 | & \\leq ( n + 2 - \\beta ) | z | \\int _ { | z | \\leq | x - z | < \\frac { 1 } { \\beta } < \\infty } \\frac { 1 } { | x - z | ^ { n + 1 - \\beta } } \\ , d x + C \\sum _ { k = 2 } ^ \\infty \\int _ { | z | \\leq | x - z | < \\frac { 1 } { \\beta } < \\infty } \\frac { | z | ^ k } { k ! | x - z | ^ { n + k - \\beta } } \\ , d x \\\\ & \\leq C \\frac { | z | ^ \\beta } { 1 - \\beta } \\leq C \\frac { \\beta ^ { - \\beta } } { 1 - \\beta } . \\end{align*}"}
-{"id": "4733.png", "formula": "\\begin{align*} \\mathbf { T } = ( T _ { p - q } ) _ { p , q = 0 } ^ { n - 1 } , & A T _ { j } = B T _ { j - n } j = 1 , 2 , \\cdots , n - 1 , \\\\ \\mathbf { U } = ( U _ { p - q } ) _ { p , q = 0 } ^ { n - 1 } , & A U _ { j } = B U _ { j - n } j = 1 , 2 , \\cdots , n - 1 . \\end{align*}"}
-{"id": "4034.png", "formula": "\\begin{align*} i ( x ) = | \\{ S _ { v } \\in V : x \\in S _ { v } \\} | \\end{align*}"}
-{"id": "535.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 } \\frac { 1 } { h ^ 2 } \\inf { \\mathcal E ^ h } = 0 . \\end{align*}"}
-{"id": "1967.png", "formula": "\\begin{align*} 2 d _ { j j } = \\sum _ { k \\in \\mathcal { N } ( j ) } d _ { k i } = d _ { i i } . \\end{align*}"}
-{"id": "1946.png", "formula": "\\begin{align*} \\lim _ { p \\rightarrow \\infty } \\sum _ { j = 1 } ^ p ( \\epsilon _ j - \\tan \\epsilon _ j ) > \\sum _ { j = 1 } ^ { j ^ * - 1 } ( \\epsilon _ j - \\tan \\epsilon _ j ) + ( 1 - \\tan 1 ) \\left ( \\frac { \\chi } { \\pi } \\right ) ^ 3 \\lim _ { p \\rightarrow \\infty } \\sum _ { j = j * } ^ p \\frac { 1 } { ( j - 1 ) ^ 3 } . \\end{align*}"}
-{"id": "5974.png", "formula": "\\begin{align*} X ^ { n , q } ( t , \\omega _ 1 , \\ldots , \\omega _ n ) : = \\int _ 0 ^ t q ( s , \\omega _ 1 , \\ldots , \\omega _ n ) d s + B ^ n ( t , \\omega _ 1 , \\ldots , \\omega _ n ) . \\end{align*}"}
-{"id": "1025.png", "formula": "\\begin{align*} & \\lim _ { n \\to + \\infty } \\mathbb { E } \\sum _ { i _ 1 , \\cdots , i _ { k } \\ } \\prod _ { j = 1 } ^ k ( \\tau _ { i _ j } - x _ j ) _ + = ( 2 \\pi ) ^ k \\prod _ { j = 1 } ^ k \\left ( e ^ { c _ 0 - x _ j } / 4 \\right ) , \\end{align*}"}
-{"id": "9988.png", "formula": "\\begin{align*} \\ell _ F ( \\boldsymbol { \\varphi } ) = \\partial _ t \\varphi ^ i - V ^ i _ j \\partial _ x \\varphi ^ j , \\ell _ F ^ * ( \\boldsymbol { \\psi } ) = - \\partial _ t \\psi _ k + \\partial _ x ( V ^ i _ k \\psi _ i ) . \\end{align*}"}
-{"id": "8028.png", "formula": "\\begin{align*} \\phi ( t ) = \\zeta t , \\phi ( u ) = \\pm \\xi u , \\end{align*}"}
-{"id": "6289.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\| \\hat { B } _ { \\hat { \\pi } } - Q \\| _ 2 ^ 2 | \\mathcal { E } \\right ] = O \\left ( \\hat { \\epsilon } _ k ^ { ( O ) } ( Q ) ^ 2 + r \\left ( \\frac { \\log k } { n } + \\frac { k ^ 2 } { n ^ 2 } \\right ) + \\tilde { \\nu } ^ 2 \\right ) . \\end{align*}"}
-{"id": "2425.png", "formula": "\\begin{align*} u ^ { ( t ) } _ { n m } & = \\binom { 2 t } { t - n } ^ { \\frac { 1 } { 2 } } _ { q ^ { - 2 } } \\binom { 2 t } { t - m } ^ { - \\frac { 1 } { 2 } } _ { q ^ { - 2 } } \\sum ^ { ( t - n ) \\land ( t + m ) } _ { i = 0 \\lor ( m - n ) } q ^ { ( t - n - i ) ( n - m + 2 i ) } q ^ { - i ( n - m + i ) } \\\\ & \\ ; \\ ; \\ ; \\ ; \\times \\binom { t - n } { i } _ { q ^ { - 2 } } \\binom { t + n } { t + m - i } _ { q ^ { - 2 } } ( - q c ^ * _ q ) ^ i c ^ { n - m + i } _ q a ^ { t - n - i } _ q ( a ^ * _ q ) ^ { t + m - i } , \\end{align*}"}
-{"id": "5861.png", "formula": "\\begin{align*} & \\varphi _ i ( h ) \\mapsto \\varphi _ i ( h ) / \\varphi _ i ( x ( x - 1 ) G ' ) , 1 \\le i \\le k ; \\\\ & h ( 0 ) \\mapsto - h ( 0 ) / G ( 0 ) ; \\\\ & h ( 1 ) \\mapsto h ( 1 ) / G ( 1 ) . \\end{align*}"}
-{"id": "3612.png", "formula": "\\begin{align*} \\alpha _ 0 & : = - \\frac { 1 } { 2 } a _ 1 + 2 a _ 2 + 2 a _ { - 2 } + v _ { ( 1 , 2 ) } \\in M _ 0 ^ { ( a _ 1 ) } \\\\ \\beta _ 0 & : = - \\frac { 1 } { 3 } ( a _ 1 + a _ { - 1 } + 2 a _ 2 + 2 a _ { - 2 } ) + v _ { ( 1 , 2 ) } \\in M _ { \\frac { 1 } { 2 ^ 2 } } ^ { ( a _ 1 ) } \\\\ \\alpha _ 1 & : = - \\frac { 1 } { 2 } a _ 1 + 2 a _ 3 + 2 a _ { - 3 } + v _ { ( 1 , 3 ) } \\in M _ 0 ^ { ( a _ 1 ) } \\\\ \\beta _ 1 & : = - \\frac { 1 } { 3 } ( a _ 1 + a _ { - 1 } + 2 a _ { - 2 } + 2 a _ { - 3 } ) + v _ { ( 1 , 3 ) } \\in M _ { \\frac { 1 } { 2 ^ 2 } } ^ { ( a _ 1 ) } . \\end{align*}"}
-{"id": "1969.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\ell \\in \\mathcal { N } ( 3 ) \\\\ \\ell \\neq 1 , 2 } } d _ { \\ell 1 } = 0 . \\end{align*}"}
-{"id": "4215.png", "formula": "\\begin{align*} E _ n ^ { - 1 } ( z ) A _ n ^ { ( k ) } ( s ) E _ n ( z ) & = O \\left ( n ^ { \\frac { 5 ( k - 1 ) } { 2 } } \\right ) , \\\\ E _ n ^ { - 1 } ( z ) B _ n ^ { ( k ) } ( s ) E _ n ( z ) & = O \\left ( n ^ { - \\frac { k } { 2 } } \\right ) , \\end{align*}"}
-{"id": "8234.png", "formula": "\\begin{align*} a ( 2 i , k ) & = \\sum _ { j = 1 } ^ \\infty \\left ( a ( 2 i - 1 , j ) m ( j , k ) + b ( 2 i - 1 , j ) x _ 0 ( j , k ) \\right ) , \\\\ b ( 2 i , k ) & = \\sum _ { j = 1 } ^ \\infty b ( 2 i - 1 , j ) x _ 1 ( j , k ) , \\\\ a ( 2 i + 1 , k ) & = \\sum _ { j = 1 } ^ \\infty \\left ( a ( 2 i , j ) y _ 0 ( j , k ) + b ( 2 i , j ) z _ 0 ( j , k ) \\right ) , \\\\ b ( 2 i + 1 , k ) & = \\sum _ { j = 1 } ^ \\infty \\left ( a ( 2 i , j ) y _ 1 ( j , k ) + b ( 2 i , j ) z _ 1 ( j , k ) \\right ) , \\end{align*}"}
-{"id": "6187.png", "formula": "\\begin{align*} v _ R ( x , z ) = U ( x _ n , z ) + o ( | ( x , z ) | ^ { 1 / 2 } ) , \\end{align*}"}
-{"id": "1665.png", "formula": "\\begin{align*} G _ { n } ^ { ( t ) } : = F _ { m } ^ { ( t ) } \\mid T _ { k / 2 , 4 N } ( \\ell ^ { 2 n } ) - \\ell ^ { \\lambda - 1 } \\left ( \\frac { ( - 1 ) ^ { \\lambda } m } { \\ell } \\right ) F _ { m } ^ { ( t ) } \\mid T _ { k / 2 , 4 N } ( \\ell ^ { 2 n - 2 } ) . \\end{align*}"}
-{"id": "5469.png", "formula": "\\begin{align*} \\langle x , y \\rangle _ { - 1 } : = \\big \\langle B ^ { 1 / 2 } x , B ^ { 1 / 2 } y \\big \\rangle . \\end{align*}"}
-{"id": "7966.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ m ( d _ j - d _ { q , j } ^ A ) \\geq n - \\abs { S _ q ^ A } + 1 . \\end{align*}"}
-{"id": "1208.png", "formula": "\\begin{align*} { E } _ { \\tau } ^ { \\gamma _ - } ( x , z ' ) & = { E } _ { \\tau , 0 } ^ { s , \\gamma _ - } ( x , z ' ) + { E } _ { \\tau , 0 } ^ { m , \\gamma _ - } ( x , z ' ) , \\\\ \\partial _ { x _ k } { E } _ { \\tau } ^ { \\gamma _ - } ( x , z ' ) & = { E } _ { \\tau , k } ^ { s , \\gamma _ - } ( x , z ' ) + { E } _ { \\tau , k } ^ { m , \\gamma _ - } ( x , z ' ) \\qquad ( k = 1 , 2 , 3 ) , \\end{align*}"}
-{"id": "7954.png", "formula": "\\begin{align*} v = \\sum _ { a \\in S } \\alpha _ a x _ a = \\sum _ { b \\in T } \\beta _ b x _ b \\end{align*}"}
-{"id": "607.png", "formula": "\\begin{align*} \\| g \\| _ { L ^ 3 } = \\bigg \\| \\sum _ { j \\in \\mathbb { N } } \\sum _ { k , e } c _ { j , k , e } \\psi _ { j , k , e } \\bigg \\| _ { L ^ 3 } \\leq \\bigg \\| \\sum _ { j \\leq J } \\sum _ { k , e } c _ { j , k , e } \\psi _ { j , k , e } \\bigg \\| _ { L ^ 3 } + \\bigg \\| \\sum _ { j > J } \\sum _ { k , e } c _ { j , k , e } \\psi _ { j , k , e } \\bigg \\| _ { L ^ 3 } \\end{align*}"}
-{"id": "10002.png", "formula": "\\begin{align*} ( F _ { 2 m } A ) ( 2 n ) = ( S O _ { 2 n } ) _ { + } \\wedge _ { S O _ { 2 n - 2 m } } ( A \\wedge T ^ { 2 n - 2 m } ) . \\end{align*}"}
-{"id": "7654.png", "formula": "\\begin{align*} \\frac { \\partial w ^ { \\varepsilon } } { \\partial t } - L ^ { \\varepsilon } w ^ { \\varepsilon } = \\frac { \\partial u ^ 0 } { \\partial t } ( x ^ \\varepsilon , t ) - \\Theta \\cdot \\nabla \\nabla u ^ 0 ( x ^ \\varepsilon , t ) \\ + \\ \\phi ^ \\varepsilon ( x ^ \\varepsilon , t ) = \\phi ^ \\varepsilon ( x ^ \\varepsilon , t ) , w ^ { \\varepsilon } ( x , 0 ) = \\varphi ( x ) + \\psi ^ \\varepsilon ( x ) \\end{align*}"}
-{"id": "7147.png", "formula": "\\begin{align*} y _ { k , i } = \\theta ^ { \\tau } \\phi _ { k , i } + \\varepsilon _ { k , i } , k \\geq 0 , \\ , \\ , i = 1 , \\ldots , n , \\end{align*}"}
-{"id": "299.png", "formula": "\\begin{align*} G _ S \\ ; : = \\ ; G ( \\delta _ S ) ^ \\circ \\textnormal { \\em i s s e m i s i m p l e f o r } S \\ ; \\in \\ ; \\{ X , Y , Z , X Y \\} . \\end{align*}"}
-{"id": "3832.png", "formula": "\\begin{align*} ^ { h } { \\gamma _ { 0 } ^ { i } } _ 0 = ^ { \\alpha } { \\gamma _ { 0 } ^ { i } } _ 0 + k _ 0 y ^ i - \\frac { 1 } { 2 } h _ { 0 0 } \\bar { k } ^ i , \\end{align*}"}
-{"id": "7485.png", "formula": "\\begin{align*} 2 \\widehat { g } - 2 = 2 ( 2 g - 2 ) + b _ 1 , \\end{align*}"}
-{"id": "1540.png", "formula": "\\begin{align*} u _ 1 ( x , t ) = \\frac { 1 } { \\pi } \\int _ 0 ^ \\infty \\int _ 0 ^ 1 \\left ( g _ 1 ( s , x , y ) \\cos { s \\ , t } - g _ 2 ( s , x , y ) \\sin { s \\ , t } \\right ) f ( y ) d y d s \\end{align*}"}
-{"id": "3395.png", "formula": "\\begin{align*} f ( T ) = - \\log \\rho ( T ) . \\end{align*}"}
-{"id": "272.png", "formula": "\\begin{align*} u _ i v _ i = 0 \\quad 1 \\leq i \\leq n . \\end{align*}"}
-{"id": "8407.png", "formula": "\\begin{align*} & \\norm { \\Big ( \\frac { | z _ { \\alpha } | ^ { - 2 k } } { a } \\Big ) _ t } _ { \\infty } + \\norm { \\frac { | z _ { \\alpha } | ^ { - 2 k } } { a } } _ { \\infty } \\\\ \\leq & C ( \\| \\partial _ { \\alpha } \\xi _ 0 \\| _ { H ^ { s - 1 } } , \\| v _ 0 \\| _ { H ^ s } , \\| w _ 0 \\| _ { H ^ s } , \\mathcal { E } ( t ) , d _ I ( t ) ^ { - 1 } , d _ P ( t ) ^ { - 1 } , N \\lambda _ { m a x } , C _ 1 , C _ 2 , \\alpha _ 0 ) \\end{align*}"}
-{"id": "2476.png", "formula": "\\begin{align*} & y ^ 2 + y = \\left \\{ \\begin{array} { l } { \\alpha } _ 1 x + { \\alpha } _ 2 x ^ { - 1 } + { \\alpha } _ 3 ( x - 1 ) ^ { - 1 } + { \\alpha } _ 4 ( x - \\lambda ) ^ { - 1 } \\\\ x ^ 3 + \\alpha x + \\beta x ^ { - 1 } + \\gamma ( x - 1 ) ^ { - 1 } \\\\ x ^ 3 + \\alpha x + \\beta x ^ { - 3 } + \\gamma x ^ { - 1 } \\\\ x ^ 5 + \\alpha x ^ 3 + \\beta x ^ { - 1 } \\\\ x ^ 7 + \\alpha x ^ 5 + \\beta x ^ 3 \\end{array} \\right . \\\\ \\end{align*}"}
-{"id": "2118.png", "formula": "\\begin{align*} d x ( t ) = [ a ( t ) x ( t ) + h _ 1 ( t ) ] d t + [ c ( t ) x ( t ) + h _ 2 ( t ) ] d \\omega ( t ) \\end{align*}"}
-{"id": "4223.png", "formula": "\\begin{align*} E _ { i n } ^ { - 1 } ( y _ n ) R ^ { - 1 } ( y _ n ) R ( x _ n ) E _ { i n } ( x _ n ) = \\mathbb I + \\mathcal { O } \\left ( ( x - y ) n ^ { - \\frac { 1 } { 2 } } \\right ) \\end{align*}"}
-{"id": "89.png", "formula": "\\begin{align*} \\begin{aligned} \\lambda _ - = - 1 \\ , , & v _ { - } = ( 1 , - 1 , - 1 , 1 , \\ldots , 1 , - 1 , - 1 , 1 ) \\ , , \\\\ \\lambda _ + = 1 \\ , , & e = ( 1 , 1 , \\ldots , 1 , 1 ) \\ , . \\end{aligned} \\end{align*}"}
-{"id": "9773.png", "formula": "\\begin{align*} p _ { * , X _ \\ell } ( x , 0 ) = c _ \\ell ( \\boldsymbol { e } _ \\ell \\cdot x ) ^ \\kappa \\quad p _ { * } ( x , 0 ) = c _ \\ast ( \\boldsymbol { e } _ \\ast \\cdot x ) ^ \\kappa . \\end{align*}"}
-{"id": "2184.png", "formula": "\\begin{align*} \\int | K _ \\ell ( \\mathbf x , \\mathbf y ) & - K _ { \\ell } ( \\mathbf x , \\mathbf y ' ) | \\ , d w ( \\mathbf x ) \\leq C M 2 ^ \\ell \\| \\mathbf y - \\mathbf y ' \\| . \\end{align*}"}
-{"id": "3682.png", "formula": "\\begin{align*} E \\big [ \\sum _ { i \\in S } d _ i E ( \\hat { y } _ i | x _ i ) \\big ] & = E \\big [ \\sum _ { i \\in S } d _ i E ( y _ i | x _ i , i \\in B ) \\big ] = E \\big [ \\sum _ { i \\in S } d _ i E ( y _ i | x _ i , i \\in U ) \\big ] \\\\ & = \\sum _ { i \\in U } E ( y _ i | x _ i , i \\in U ) = E ( Y | x _ U ) \\end{align*}"}
-{"id": "2378.png", "formula": "\\begin{align*} \\varphi ( \\frac { 1 } { 2 } ) & = \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} , & \\varphi ( + \\frac { 1 } { 2 } ) & = \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} \\end{align*}"}
-{"id": "8583.png", "formula": "\\begin{align*} D = E _ 2 ^ + + E _ 2 ^ - \\in \\mathfrak g . \\end{align*}"}
-{"id": "6379.png", "formula": "\\begin{align*} \\delta _ j : = e ^ { C _ 1 + M _ F } \\| g _ j - g \\| _ p , \\ \\end{align*}"}
-{"id": "3235.png", "formula": "\\begin{align*} S _ { \\delta } ( x ) : = \\{ \\tau \\in S ( x ) \\mid g ( x , \\tau ) > \\max _ { \\tau \\in T } g ( x , \\tau ) - \\delta \\} \\end{align*}"}
-{"id": "286.png", "formula": "\\begin{align*} T _ { \\theta , h } ( P ) \\ ( x ) = \\ \\frac { 1 } { n ! } \\ \\sum _ { k = 0 } ^ n \\ P ^ { ( k ) } ( 0 ) \\ G _ n ^ { ( n - k ) } ( x , \\theta , h ) \\ = \\ \\frac { 1 } { n ! } \\ P \\boxplus G _ n ( x , \\theta , h ) . \\end{align*}"}
-{"id": "4628.png", "formula": "\\begin{align*} \\mathcal { P } = \\mathbb { I } _ { \\ell } \\otimes A _ 0 + \\Sigma \\otimes A _ 1 , \\end{align*}"}
-{"id": "1511.png", "formula": "\\begin{align*} \\frac { \\norm { \\langle \\cdot , \\theta \\rangle } _ { L ^ p ( \\mu _ K ) } } { p } = \\Bigl ( \\frac { n + p } { n } \\Bigr ) ^ { 1 / p } \\frac { \\norm { \\langle \\cdot , \\theta \\rangle } _ { L ^ p ( \\nu _ K ) } } { p } , \\end{align*}"}
-{"id": "9327.png", "formula": "\\begin{align*} F _ { v } : M \\to \\R , F _ { v } ( x ) = \\langle v , W ( x ) \\rangle . \\end{align*}"}
-{"id": "3998.png", "formula": "\\begin{align*} e ( \\frac { 1 } { 2 } \\nu ^ t M ^ { - 1 } \\mu ) = e ( \\frac { 1 } { d } \\nu _ n m _ { n , n } \\mu _ n ) \\end{align*}"}
-{"id": "1434.png", "formula": "\\begin{align*} w _ { t } + \\Delta w + \\varepsilon P \\Theta ( t , x , \\mu , D w ) = 0 , \\end{align*}"}
-{"id": "5661.png", "formula": "\\begin{align*} { \\rm I } _ 3 & = \\sum _ { j = 0 } ^ { n - 1 } \\int _ { t _ j } ^ { t _ { j + 1 } } \\| A ^ { \\frac { s } { 2 } } ( \\bar E _ h ( t _ n - t _ j ) - B _ { n - j } ) \\| ^ p \\ , \\d t \\\\ & \\leq c \\tau ^ { p + 1 } \\sum _ { j = 0 } ^ { n - 1 } ( t _ { n + 1 } - t _ { j } ) ^ { ( ( 1 - \\frac { s } { 2 } ) \\alpha + \\gamma - 2 ) p } \\leq c \\left \\{ \\begin{array} { l l } \\tau ^ { p \\eta } , & \\eta < 1 , \\\\ \\tau ^ { p } \\ell _ n , & \\eta = 1 , \\\\ \\tau ^ { p } t _ n ^ { p ( \\eta - 1 ) } , & \\eta > 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "1057.png", "formula": "\\begin{align*} \\| ( - B ) ^ { - 1 } \\| & = ( 1 - \\lambda _ 1 ( B ) ) ^ { - 1 } \\leq 1 + n ( 4 \\ln n ) ^ { \\frac { 1 } { 2 } } e ^ { \\max \\{ x _ j | 1 \\leq j \\leq k \\} - c _ 0 } e ^ { 1 - ( \\ln n ) ^ { \\frac { 1 } { 2 } } / 2 } \\\\ & \\leq 1 + O ( n ( \\ln n ) ^ { \\frac { 1 } { 2 } } e ^ { - ( \\ln n ) ^ { \\frac { 1 } { 2 } } / 2 } ) = O ( n ( \\ln n ) ^ { \\frac { 1 } { 2 } } e ^ { - ( \\ln n ) ^ { \\frac { 1 } { 2 } } / 2 } ) , \\end{align*}"}
-{"id": "5736.png", "formula": "\\begin{align*} E _ { \\alpha , 2 } ( 0 ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { ( 0 ^ \\alpha ) ^ k } { \\Gamma ( k \\alpha + 2 ) } = \\frac { 1 } { \\Gamma ( 2 ) } = 1 . \\end{align*}"}
-{"id": "7039.png", "formula": "\\begin{align*} \\mathbb { F } _ { \\alpha , \\beta , \\lambda } ( z ) = \\int \\limits _ { 0 } ^ { z } \\left ( \\frac { \\mathbb { E } _ { \\alpha , \\beta } ( t ) } { t } \\right ) ^ { 1 / \\lambda } d t , \\end{align*}"}
-{"id": "961.png", "formula": "\\begin{align*} p q - \\tfrac { 1 } { \\lambda ^ 2 } ( p ( 1 - q ) + ( 1 - p ) q ) + ( 1 - p ) ( 1 - q ) \\le ( p + ( 1 - p ) ) ( q + ( 1 - q ) ) = 1 . \\end{align*}"}
-{"id": "6.png", "formula": "\\begin{align*} F _ s ^ { i , r } ( z ) = & \\ f ^ i ( V _ s ^ { r , v } , q ( Z _ s ^ { i , r , v } ( m ) ) ) - f ^ i ( V _ s ^ { r , \\bar { v } } , q ( Z _ s ^ { i , r , v } ( m ) ) ) \\\\ & \\ + f ^ i ( V _ s ^ { r , \\bar { v } } , q ( z + Z _ s ^ { i , r , \\bar { v } } ( m ) ) ) - f ^ i ( V _ s ^ { r , \\bar { v } } , q ( Z _ s ^ { i , r , \\bar { v } } ( m ) ) ) \\end{align*}"}
-{"id": "5849.png", "formula": "\\begin{align*} \\varphi ( f ) = ( \\varphi _ 1 ( f ) , \\varphi _ 2 ( f ) , \\cdots , \\varphi _ k ( f ) ) = ( f { g _ 1 } , f { g _ 2 } , \\cdots , f { g _ k } ) . \\end{align*}"}
-{"id": "5433.png", "formula": "\\begin{align*} \\xi _ j ( T ) = \\log \\lvert \\lambda _ k \\rvert , \\end{align*}"}
-{"id": "4888.png", "formula": "\\begin{align*} C _ 0 ( X ) = \\{ f \\colon X \\to \\mathbb { C } \\mid \\forall \\varepsilon > 0 \\ ; \\exists K \\subseteq X \\colon | f ( X \\setminus K ) | < \\varepsilon \\} \\text . \\end{align*}"}
-{"id": "145.png", "formula": "\\begin{align*} [ X _ { A } , f Y _ { A } ] = f [ X _ { A } , Y _ { A } ] + a ( X _ A ) ( f ) Y _ A \\end{align*}"}
-{"id": "9193.png", "formula": "\\begin{align*} \\mathcal { K = K } _ { b } . \\end{align*}"}
-{"id": "5690.png", "formula": "\\begin{align*} \\| x ^ { A B } _ m & - ( x _ 0 + t _ m v _ 0 ) \\| = \\| x ^ { A B } _ m - ( x _ 0 + t _ m v _ m ^ A ) - t _ m ( v _ 0 - v _ m ^ A ) \\| \\\\ & \\leq \\| x ^ { A B } _ m - x ^ A _ m \\| + t _ m \\| v _ 0 - v _ m ^ A \\| \\le ( K + 1 ) \\left \\| x ^ A _ m - x ^ B _ m \\right \\| + t _ m \\frac { \\varepsilon } { 2 K + 3 } \\\\ & \\leq ( K + 1 ) \\cdot \\frac { 2 \\varepsilon } { 2 K + 3 } \\ , t _ m + \\frac { \\varepsilon } { 2 K + 3 } \\ , t _ m = \\varepsilon t _ m \\ , . \\end{align*}"}
-{"id": "2655.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\mathrm { P f } [ K _ { N } ( x _ { j } , x _ { k } ) ] _ { j , k = 1 } ^ { l } d \\nu ( x _ { l } ) & = ( N - l + 1 ) \\mathrm { P f } [ K _ { N } ( x _ { j } , x _ { k } ) ] _ { j , k = 1 } ^ { l - 1 } . \\end{align*}"}
-{"id": "2498.png", "formula": "\\begin{align*} \\rho \\ , b _ 2 - b _ 3 + a _ 1 \\ , C ( \\rho , 0 ) = 0 \\ \\ \\ \\ { \\rm a n d } \\ \\ \\ \\rho \\ , b _ 3 - b _ 2 + b _ 4 \\ , C ( \\rho , 0 ) = 0 , \\end{align*}"}
-{"id": "9526.png", "formula": "\\begin{align*} & \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - 1 ) ^ { n - 1 } q ^ { n ( n + 1 ) / 2 } } { ( 1 - q ^ n ) } \\\\ & = \\sum _ { n = 1 } ^ { N } \\sum _ { k = 1 } ^ { \\infty } \\left [ \\begin{matrix} N - 1 \\\\ n - 1 \\end{matrix} \\right ] k ( - 1 ) ^ { n - 1 } q ^ { n k + n ( n - 1 ) / 2 } - q ^ N \\sum _ { n = 1 } ^ { N } \\sum _ { k = 1 } ^ { \\infty } \\left [ \\begin{matrix} N - 1 \\\\ n - 1 \\end{matrix} \\right ] k ( - 1 ) ^ { n - 1 } q ^ { n k + n ( n - 1 ) / 2 } . \\end{align*}"}
-{"id": "5408.png", "formula": "\\begin{align*} \\alpha _ M ( \\mathbf { v } ) = \\frac { \\bigl \\langle \\dot \\gamma _ 0 ( 0 ) , \\dot \\beta ( t _ 0 ) \\bigr \\rangle } { \\bigl \\langle \\dot \\gamma _ 0 ( 0 ) , n _ M ( x ) \\bigr \\rangle } , \\end{align*}"}
-{"id": "5127.png", "formula": "\\begin{gather*} A B ^ * = B A ^ * , \\\\ \\begin{pmatrix} A & - B \\\\ B & A \\end{pmatrix} = 0 . \\end{gather*}"}
-{"id": "1555.png", "formula": "\\begin{align*} \\widetilde { F } ^ \\delta \\begin{pmatrix} z \\\\ x \\end{pmatrix} = \\begin{pmatrix} z + z ^ 2 + a _ 3 ^ \\delta z ^ 3 + a _ 4 ^ \\delta ( x ) z ^ 4 + \\dots \\\\ b _ 0 ( x ) + b _ 1 ^ \\delta ( x ) z + b _ 2 ^ \\delta ( x ) z ^ 2 + \\dots \\end{pmatrix} , \\end{align*}"}
-{"id": "7908.png", "formula": "\\begin{align*} p _ 2 ( x ) = \\frac { x ^ 2 - h x } { - 2 h } f ' ( - h / 2 ) + f ( 0 ) + \\frac { x ^ 2 + h x } { 2 h } f ' ( h / 2 ) \\end{align*}"}
-{"id": "2683.png", "formula": "\\begin{align*} e ^ { \\frac { \\eta _ { - } } { \\varphi ( x ) } ( \\Re { z _ { - } } + 1 ) ( u - v ) } \\frac { 1 } { \\varphi ( x ) } S _ { N } & \\Big ( N \\big ( x + \\frac { u } { N \\varphi ( x ) } \\big ) , N \\big ( x + \\frac { v } { N \\varphi ( x ) } \\big ) \\Big ) = : I _ { N } + J _ { N } \\end{align*}"}
-{"id": "3548.png", "formula": "\\begin{align*} \\begin{matrix} \\ell ^ { ( 1 ) } = ( - 1 , \\ ; \\ ; \\ , 0 , \\ ; \\ ; \\ , 0 , \\ ; \\ ; \\ , 1 , \\ ; \\ ; \\ , 0 , \\ ; \\ ; \\ , 0 , \\ ; \\ ; \\ , 0 , \\ ; \\ ; \\ , 1 , - 1 ) , \\\\ \\ell ^ { ( 2 ) } = ( \\ ; \\ ; \\ , 0 , - 1 , \\ ; \\ ; \\ , 0 , \\ ; \\ ; \\ , 1 , - 1 , \\ ; \\ ; \\ , 1 , \\ ; \\ ; \\ , 0 , \\ ; \\ ; \\ , 0 , \\ ; \\ ; \\ , 0 ) , \\\\ \\ell ^ { ( 3 ) } = ( \\ ; \\ ; \\ , 0 , \\ ; \\ ; \\ , 0 , - 1 , \\ ; \\ ; \\ , 0 , \\ ; \\ ; \\ , 0 , \\ ; \\ ; \\ , 1 , - 1 , \\ ; \\ ; \\ , 1 , \\ ; \\ ; \\ , 0 ) , \\\\ \\ell ^ { ( 4 ) } = ( \\ ; \\ ; \\ , 0 , \\ ; \\ ; \\ , 0 , \\ ; \\ ; \\ , 0 , - 1 , \\ ; \\ ; \\ , 1 , - 1 , \\ ; \\ ; \\ , 1 , - 1 , \\ ; \\ ; \\ , 1 ) . \\end{matrix} \\end{align*}"}
-{"id": "9715.png", "formula": "\\begin{align*} \\Lambda ( u ) : = \\{ ( x , 0 ) : u ( x , 0 ) = \\varphi ( x ) \\} . \\end{align*}"}
-{"id": "8756.png", "formula": "\\begin{align*} { } _ F D _ { \\tau } ^ { \\alpha , \\sigma , \\gamma , m , n } u = { } _ F u ^ { ( \\alpha , \\sigma ) } _ n + \\tau ^ { - \\alpha } \\sum _ { k = 1 } ^ m w ^ { ( \\alpha , \\sigma ) } _ { n , k } ( u _ k - u _ 0 ) - b _ n ^ { ( \\alpha , \\sigma ) } u _ 0 , \\end{align*}"}
-{"id": "1215.png", "formula": "\\begin{align*} \\Sigma _ r ^ M Y _ r ^ M = \\mathbb { X } _ r ^ { x , u , \\mathbb { Z } } + \\eta _ r ^ M . \\end{align*}"}
-{"id": "1513.png", "formula": "\\begin{align*} g = \\alpha P _ { 1 } ^ { e _ { 1 } } \\cdots P _ { r } ^ { e _ { r } } , \\end{align*}"}
-{"id": "8345.png", "formula": "\\begin{align*} & \\Big | \\frac { D _ t Z ( \\alpha , t ) - D _ t ( \\beta , t ) } { \\alpha - \\beta } \\Big | ^ 2 = \\Big | \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j } { 2 \\pi } \\frac { 1 } { ( \\alpha - z _ j ) ( \\beta - z _ j ) } \\Big | ^ 2 \\\\ = & \\sum _ { j = 1 } ^ N \\sum _ { k = 1 } ^ N \\frac { \\lambda _ j \\lambda _ k } { ( 2 \\pi ) ^ 2 } \\frac { 1 } { ( \\alpha - z _ j ) ( \\beta - z _ j ) \\overline { ( \\alpha - z _ k ) ( \\beta - z _ k ) } } \\\\ \\end{align*}"}
-{"id": "1007.png", "formula": "\\begin{align*} \\varphi _ k \\left ( \\lambda \\right ) = \\int _ 0 ^ { \\lambda } \\varphi _ k ' \\left ( s \\right ) d s = \\int _ 0 ^ { \\lambda } \\frac { s ^ { k - 1 } } { ( k - 1 ) ! } e ^ { - s } d s . \\end{align*}"}
-{"id": "861.png", "formula": "\\begin{gather*} { u ^ * } ^ T u = { v ^ * } ^ T v , \\quad \\norm { u } _ 2 = \\norm { v ^ \\ast } _ 2 = 1 , \\end{gather*}"}
-{"id": "3045.png", "formula": "\\begin{align*} L ^ { x } ( t ) : = \\sigma ( x ) W ( t ) + \\int \\limits _ { 0 } ^ { t } \\int \\limits _ { E _ { 0 } } \\sigma ^ { 0 } ( x , z ) \\widetilde { N } ( d u , d z ) + \\int \\limits _ { 0 } ^ { t } \\int \\limits _ { E _ { 1 } } \\sigma ^ { 1 } ( x , z ) N ( d u , d z ) . \\end{align*}"}
-{"id": "7447.png", "formula": "\\begin{align*} \\displaystyle ( \\varphi _ 1 , \\varphi _ 2 ) _ { Y _ n } = \\int _ { Y _ n } \\int _ { G _ n ( F ) \\times G _ n ( F ) } f _ 1 ( h _ 1 x ) \\overline { f _ 2 ( h _ 2 x ) } d h _ 1 d h _ 2 d x = \\int _ { G _ n ( F ) } f ( h ) d h \\end{align*}"}
-{"id": "6417.png", "formula": "\\begin{align*} s = \\sqrt { 1 - c ^ { 2 } } , c = | c | \\mathrm { s i g n } ( \\sigma ) . \\end{align*}"}
-{"id": "6880.png", "formula": "\\begin{align*} u _ t + u \\nabla u = A '' A ^ { - 1 } x = - \\nabla p \\ , . \\end{align*}"}
-{"id": "5780.png", "formula": "\\begin{align*} A ( C _ n , q ^ { - 1 } , T ^ { - 1 } ) = ( - 1 ) ^ n A ( C _ n , q , T ) , n > 1 , \\end{align*}"}
-{"id": "2467.png", "formula": "\\begin{align*} n = e _ { \\pi } ( P ) \\frac { \\left \\vert G \\right \\vert } { \\left \\vert G _ { P } \\right \\vert } , \\end{align*}"}
-{"id": "9604.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\frac { q ^ n ( q ^ { n + 1 } ) _ { N - n } } { ( q ^ { 2 n } ; q ^ 2 ) _ { N - n + 1 } } = \\frac { 1 } { ( q ) _ N } \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - 1 ) ^ { n - 1 } n q ^ { n ( n + 1 ) / 2 } } { 1 + q ^ n } . \\end{align*}"}
-{"id": "9022.png", "formula": "\\begin{align*} f _ { \\{ b _ { i + 1 } \\} , c _ i } ( x ) = \\begin{cases} c _ i & x = b _ { i + 1 } , \\\\ x & \\end{cases} \\end{align*}"}
-{"id": "9915.png", "formula": "\\begin{align*} E _ { \\pi ^ m _ { n _ - } } [ f ( x _ n ) | Y _ n = y _ n ] & = \\int f ( x _ { n } ) \\frac { g ( x _ { n } , y _ { n } ) \\pi ^ m _ { n _ - } ( d x _ n ) } { \\int _ { \\mathcal { X } } g ( x _ { n } , y _ { n } ) \\pi ^ m _ { n _ - } ( d x _ n ) } \\\\ & = \\frac { \\int f ( x _ { n } ) g ( x _ { n } , y _ { n } ) \\pi ^ m _ { n _ - } ( d x _ n ) } { \\int _ { \\mathcal { X } } g ( x _ { n } , y _ { n } ) \\pi ^ m _ { n _ - } ( d x _ n ) } \\end{align*}"}
-{"id": "1663.png", "formula": "\\begin{align*} \\epsilon _ { 2 } \\chi _ { 2 } ' ( - 1 ) ( - 1 ) ^ { \\frac { k - 1 } { 2 } } & = t \\chi _ { 2 } ( - 1 ) e _ { 4 } ( k - 1 ) \\\\ & = \\chi _ { 2 } ( - 1 ) e _ { 4 } ( k - 1 ) e _ { 4 } ( 1 - t ) \\\\ & = \\chi _ { 2 } ( - 1 ) e _ { 4 } ( r - t ) = 1 , \\end{align*}"}
-{"id": "4962.png", "formula": "\\begin{align*} \\Delta _ { \\widetilde { K } } ( t ) = \\widetilde { a } _ 0 + \\sum _ { \\widetilde { s } > 0 } \\widetilde { a } _ { \\widetilde { s } } ( t ^ { \\widetilde { s } } + t ^ { - \\widetilde { s } } ) , \\mbox { a n d } \\Delta _ { K ( t ) } = a _ 0 + \\sum _ { s > 0 } a _ s ( t ^ s + t ^ { - s } ) , \\end{align*}"}
-{"id": "620.png", "formula": "\\begin{align*} \\varphi ^ \\ast ( y ) \\cdot K _ \\varphi \\cdot \\varphi ^ \\ast ( y ) ^ { - 1 } = K _ { \\varphi ^ y } \\ . \\end{align*}"}
-{"id": "1116.png", "formula": "\\begin{align*} & \\det ( + B ) = D _ n ( \\pi \\delta _ n ) \\geq D _ n ( \\alpha _ n ) = ( n \\ln n ) ^ { - 1 } . \\end{align*}"}
-{"id": "8319.png", "formula": "\\begin{align*} \\mathfrak { K } ^ { \\ast } f ( \\alpha ) : = p . v . \\int _ { - \\infty } ^ { \\infty } R e \\{ - \\frac { 1 } { \\pi i } \\frac { z _ { \\alpha } } { | z _ { \\alpha } | } \\frac { | z _ { \\beta } | } { z ( \\alpha , t ) - z ( \\beta , t ) } \\} f ( \\beta ) d \\beta . \\end{align*}"}
-{"id": "3657.png", "formula": "\\begin{align*} \\left \\{ t a _ i - a _ i \\cdot v _ { ( j , k ) } \\mid \\{ i , j , k \\} = \\{ 1 , 2 , 3 \\} \\right \\} \\cup \\left \\{ t a _ { - i } + a _ i \\cdot v _ { ( j , k ) } \\mid \\{ i , j , k \\} = \\{ 1 , 2 , 3 \\} \\right \\} . \\end{align*}"}
-{"id": "5157.png", "formula": "\\begin{align*} H ^ { - 1 } ( u ) = \\inf \\{ x \\in \\mathbb { R } , F ( x ) \\geq u \\} , 0 \\leq u \\leq 1 . \\end{align*}"}
-{"id": "2088.png", "formula": "\\begin{align*} y ( t ) & = \\int _ { 0 } ^ { T } { s ( \\tau ) p _ { \\rm o b s } ( t - \\tau ) d \\tau } = s ( t ) * p _ { \\rm o b s } ( t ) \\\\ & = \\iiint _ { \\Omega _ { \\rm r x } } { { s ( t ) * C ( \\bar { r } , t | { { \\bar { r } } _ { \\rm t x } } , { t _ 0 = 0 } ) } \\rho d \\rho \\varphi d z } , \\end{align*}"}
-{"id": "7210.png", "formula": "\\begin{align*} \\mu _ k = \\int _ 0 ^ 1 t ^ k d \\sigma ( t ) , \\end{align*}"}
-{"id": "3824.png", "formula": "\\begin{align*} & \\langle G _ m , \\dots , G _ { s + 1 } , \\psi ( H _ q ) , G _ { s - 1 } , \\dots , G _ 1 ; \\alpha _ { \\bar f } \\bar f \\rangle \\\\ = & \\alpha _ { \\bar f } \\sum \\langle V _ m , G _ m , \\dots , V _ { s + 1 } , G _ { s + 1 } , V _ { s } , \\psi ( H _ q ) , V _ { s - 1 } , G _ { s - 1 } , \\dots , V _ 1 , G _ 1 , V _ 0 ; x _ i \\rangle + \\sum _ l \\alpha _ l g _ l , \\end{align*}"}
-{"id": "5958.png", "formula": "\\begin{align*} d Y ( t ) = - g ^ * ( t , Z ( t ) ) d t + Z ( t ) d W ^ \\epsilon ( t ) , Y ( 1 ) = F ( W ^ \\epsilon ) , \\end{align*}"}
-{"id": "4898.png", "formula": "\\begin{align*} ( a , b ) + ( c , d ) = \\begin{cases} ( a + c , b + d ) & a + c < 1 \\\\ ( a + c - 1 , b + d + 1 ) & a + c \\geq 1 \\end{cases} \\end{align*}"}
-{"id": "1075.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to + \\infty } \\sup _ { w \\in [ 1 , A ] } \\frac { 1 } { n \\sin ( w \\alpha _ n / 2 ) } = \\lim \\limits _ { n \\to + \\infty } \\frac { 1 } { n \\sin ( \\alpha _ n / 2 ) } = \\lim \\limits _ { n \\to + \\infty } \\frac { 2 } { n \\alpha _ n } = 0 . \\end{align*}"}
-{"id": "4418.png", "formula": "\\begin{align*} \\Bigl ( \\frac { \\partial u ^ i } { \\partial \\nu } + \\alpha ^ i u ^ i \\frac { \\partial \\mathcal { V } } { \\partial \\nu } \\Bigr ) ( t , x ) = \\sigma ^ i ( t , x , u ^ i _ { | _ { \\partial \\Omega } } ( t , x ) , \\mathcal { V } _ { | _ { \\partial \\Omega } } ( t , x ) ) , ( t , x ) \\in [ 0 , T ] \\times \\partial \\Omega . \\end{align*}"}
-{"id": "4739.png", "formula": "\\begin{align*} T _ p = M , T _ { p - n } = 0 p = 1 , \\dots , n , \\end{align*}"}
-{"id": "7801.png", "formula": "\\begin{align*} \\frac 1 2 = \\int _ 0 ^ { \\infty } \\varphi ^ G _ { \\zeta } ( u ) { \\rm d } u \\ge \\varphi ^ G _ { \\zeta } ( 0 ) \\int _ { 0 } ^ { u _ 0 } e ^ { - u \\varphi ^ G _ { \\zeta } ( 0 ) } { \\rm d } u = \\int _ { 0 } ^ { u _ 0 \\varphi ^ G _ { \\zeta } ( 0 ) } e ^ { - u } { \\rm d } u = 1 - e ^ { - u _ 0 \\varphi ^ G _ { \\zeta } ( 0 ) } , \\end{align*}"}
-{"id": "8163.png", "formula": "\\begin{align*} \\sum _ { j _ 1 + \\cdots + j _ { k + 1 } = m } x ( j _ 1 ) \\cdots x ( j _ { k + 1 } ) = 0 , \\end{align*}"}
-{"id": "5393.png", "formula": "\\begin{align*} | | v _ { n \\ , m + 6 } ^ 1 | | ^ 2 = - \\frac { \\overline { d _ { n \\ , m + 6 } } } { a _ { n - 1 \\ , m + 3 } } | | v _ { n - 1 \\ , m + 3 } ^ 1 | | ^ 2 = \\frac { \\overline { d _ { n \\ , m + 6 } } } { a _ { n - 1 \\ , m + 3 } } \\cdot \\frac { \\overline { b _ { n - 1 \\ , m + 3 } } } { c _ { n m } } | | v _ { n m } ^ 1 | | ^ 2 . \\end{align*}"}
-{"id": "4023.png", "formula": "\\begin{align*} \\textrm { R i c } = \\lambda g + \\nu \\eta \\otimes \\eta \\end{align*}"}
-{"id": "7931.png", "formula": "\\begin{align*} v = \\sum _ { a \\in [ n ] } y _ { a , 1 } \\otimes \\dots \\otimes y _ { a , m } \\end{align*}"}
-{"id": "2609.png", "formula": "\\begin{align*} M ( \\theta ) = \\sup _ { x \\in E } M ( r , x ) \\leq c _ f \\gamma _ 0 ^ { \\frac { 1 } { \\gamma _ 0 - 1 } } \\int _ 0 ^ \\theta \\big \\| M ( r u ^ { \\frac { 1 } { \\gamma _ 0 - 1 } } ) \\big \\| _ { \\mathbf 1 _ { 0 \\leq u \\leq 1 } \\frac { d u } { u } ; \\gamma _ 0 - 1 } d r . \\end{align*}"}
-{"id": "1096.png", "formula": "\\begin{align*} & \\lim _ { n \\to + \\infty } \\mathbb { E } \\sum _ { i _ 1 , \\cdots , i _ { k } \\ } \\prod _ { j = 1 } ^ k ( { \\tau } ^ * _ { i _ j } - x _ j ) _ + = ( M ( I ) S ( I ) ) ^ k \\prod _ { j = 1 } ^ k \\left ( e ^ { c _ 0 - x _ j } / 4 \\right ) , \\end{align*}"}
-{"id": "549.png", "formula": "\\begin{align*} \\langle \\partial _ { i j } \\vec y _ 0 , \\tilde d _ 0 \\rangle = G _ { n p } \\Gamma _ { i j } ^ n \\Gamma _ { 3 3 } ^ { p } , \\langle \\partial _ { i } \\vec b _ 0 , \\partial _ j \\vec b _ 0 \\rangle = G _ { n p } \\Gamma _ { i 3 } ^ n \\Gamma _ { j 3 } ^ { p } , \\end{align*}"}
-{"id": "9090.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta _ p ) ^ s u ( x ) = \\mu g ( x ) | u ( x ) | ^ { p - 2 } u ( x ) & \\Omega , \\\\ u ( x ) = 0 & \\mathbb { R } ^ N \\setminus \\Omega , \\\\ \\end{cases} \\end{align*}"}
-{"id": "627.png", "formula": "\\begin{align*} \\varphi \\psi = q ( \\varphi , x ) \\psi = q ( \\varphi \\psi ^ x , \\psi ^ \\ast ( x ) ) \\ , \\end{align*}"}
-{"id": "8837.png", "formula": "\\begin{align*} p \\ge \\frac { N } { 2 ( N - 1 ) } = \\frac { M + 1 } { 2 M } = \\frac 1 2 + \\frac 1 { 2 M } . \\end{align*}"}
-{"id": "7261.png", "formula": "\\begin{align*} \\left [ \\frac { 1 } { 2 } \\phi ^ { 2 } - \\mu \\phi \\right ] _ { x x } - \\phi = 0 . \\end{align*}"}
-{"id": "5957.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } Y _ n ( t ) = u ( t , 0 ) . \\end{align*}"}
-{"id": "2302.png", "formula": "\\begin{align*} a & = \\begin{pmatrix} 1 \\\\ - \\frac { \\mathcal { P } } { x \\Omega } | E ) \\end{pmatrix} & b & = \\begin{pmatrix} 0 \\\\ | \\delta _ x ) \\end{pmatrix} \\end{align*}"}
-{"id": "3253.png", "formula": "\\begin{align*} \\Omega _ n ( z ) = ( z - ( a + b ) ) \\Pi _ n ^ { } ( z ) , \\end{align*}"}
-{"id": "9683.png", "formula": "\\begin{align*} \\beta _ t = \\sum _ { n \\in \\mathbb N } \\eta _ n ^ 2 \\frac { \\omega _ n \\varphi _ n } { \\psi _ { n t } } \\ \\mathbb T \\setminus \\sigma . \\end{align*}"}
-{"id": "4682.png", "formula": "\\begin{align*} \\mu _ j ( u ) = \\frac { 1 } { 8 } ( - u _ { j - 1 } + 1 0 u _ j - u _ { j + 1 } ) ~ ~ ~ \\mbox { f o r } ~ ~ j = 3 , \\ldots , m , \\end{align*}"}
-{"id": "7894.png", "formula": "\\begin{align*} \\dim L \\wedge L + \\dim \\mathrm { I m } \\gamma ^ { \\prime } _ { 2 } & \\leq \\dim L \\wedge L + \\sum \\limits _ { i = 2 } ^ { c } \\dim \\ker \\alpha _ { i } \\cr & = \\dim L / L ^ { 2 } \\wedge L / L ^ { 2 } + \\sum \\limits _ { i = 2 } ^ { c } \\dim ( L ^ { i } / L ^ { i + 1 } \\otimes _ { m o d } ( L / Z ( L ) ) ^ { a b } ) . \\end{align*}"}
-{"id": "2443.png", "formula": "\\begin{gather*} \\partial _ q \\textrm { a n d } D _ x : = \\partial _ x + p \\partial _ y + q \\partial _ p + F ( x , y , p , q , z ) \\partial _ z \\end{gather*}"}
-{"id": "7095.png", "formula": "\\begin{align*} c _ { \\leq N , D } ( M ) : = \\max \\left ( 1 , \\frac { M } { N } \\right ) ^ { D } ~ . \\end{align*}"}
-{"id": "5128.png", "formula": "\\begin{align*} \\gamma _ { k _ j } : = \\sqrt { \\abs { \\delta _ { k _ j } } } , j = \\frac 1 2 , \\frac 3 2 , \\dots . \\end{align*}"}
-{"id": "5545.png", "formula": "\\begin{align*} \\rho ( x ) = \\langle h x \\rangle ^ { - \\nu } = ( 1 + | h x | ^ 2 ) ^ { - \\nu / 2 } \\end{align*}"}
-{"id": "855.png", "formula": "\\begin{gather*} \\chi _ \\tau ( t ) = \\sum _ { j = k } ^ \\infty ( b _ j - a _ { j + 1 } ) = \\sum _ { j = k } ^ \\infty ( \\tau _ { j } - \\tau _ { j + 1 } ) \\left ( 1 - \\frac { 1 } { j + 1 } \\right ) = \\tau _ k - \\sum _ { j = k } ^ \\infty \\frac { \\tau _ j - \\tau _ { j + 1 } } { j + 1 } , \\end{gather*}"}
-{"id": "4870.png", "formula": "\\begin{align*} { } { \\rm E x t } ^ \\ell ( \\mathbb { C } _ { \\overline { \\sigma } } , \\mathbb { C } _ { \\overline { \\tau } } ) = 0 ( \\ell \\neq 0 ) . \\end{align*}"}
-{"id": "890.png", "formula": "\\begin{align*} \\phi _ 1 = \\beta ^ 1 _ 1 & = \\alpha _ 1 & \\phi _ 2 = \\beta ^ 2 _ 1 & = \\alpha _ 3 & \\phi _ 3 = \\beta ^ 2 _ 2 & = \\alpha _ 2 + \\alpha _ 3 & \\phi _ 4 = \\beta ^ 2 _ 3 & = \\alpha _ 2 \\end{align*}"}
-{"id": "5771.png", "formula": "\\begin{align*} A _ d ( C _ n , q ) = q ^ d + \\sum _ { k = 1 } ^ { d - 1 } \\left [ ( d - k + 1 ) ^ n - 2 ( d - k ) ^ n + ( d - k - 1 ) ^ n \\right ] q ^ k + \\left [ - d ^ n + ( d - 1 ) ^ n + n d ^ { n - 1 } \\right ] , \\end{align*}"}
-{"id": "7660.png", "formula": "\\begin{align*} b ^ i ( { \\boldsymbol { \\ell } } ) \\ = \\ \\ell ^ j \\ , \\int \\limits _ { \\mathbb R ^ d } \\int \\limits _ { \\mathbb T ^ d } z ^ i \\ , c ^ j ( z ) \\ , \\mu ( \\xi , \\xi - z ) d z d \\xi \\ + \\ \\ell ^ j \\int \\limits _ { \\mathbb R ^ d } \\int \\limits _ { \\mathbb T ^ d } z ^ i \\ , a _ { \\rm s y m } ( z ) \\mu ( \\xi , \\xi - z ) \\ , \\tilde \\varphi ^ j _ { 0 } ( \\xi ) d z d \\xi + O ( | { \\boldsymbol { \\ell } } | ^ 2 ) , \\end{align*}"}
-{"id": "7564.png", "formula": "\\begin{align*} \\partial _ t X ^ { - 1 } = - \\left ( \\left ( \\partial _ t X \\right ) \\circ X ^ { - 1 } \\right ) \\left ( ( \\nabla _ a X ) ^ { - 1 } \\circ X ^ { - 1 } \\right ) \\end{align*}"}
-{"id": "8338.png", "formula": "\\begin{align*} ( I - \\mathbb { H } ) \\frac { Z _ { \\alpha } } { ( Z ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } = \\frac { 2 } { ( \\Phi ^ { - 1 } ) _ z ( \\Phi ( z _ j ( t ) ) ) ( \\alpha - \\Phi ( z _ j ( t ) ) ) ^ 2 } \\end{align*}"}
-{"id": "7895.png", "formula": "\\begin{align*} \\dim \\mathcal { M } ( L ) + \\dim ( L ^ { 2 } \\cap I ) & \\leq \\dim \\mathcal { M } ( L / I ) + \\dim \\mathcal { M } ( I ) + \\dim ( ( L / I ) ^ { a b } \\otimes I ) \\cr & = \\dim \\mathcal { M } ( L / I ) + \\dim ( ( L ) ^ { a b } \\otimes I ) . \\end{align*}"}
-{"id": "8115.png", "formula": "\\begin{align*} f ( t ) \\ = \\ \\sum _ { i = 0 } ^ { \\lfloor n / 2 \\rfloor } \\gamma _ i t ^ i ( 1 + t ) ^ { n - 2 i } \\end{align*}"}
-{"id": "5800.png", "formula": "\\begin{align*} ( f \\circ g ) ' \\ = \\ \\big ( T _ g ( f ^ { \\uparrow _ 3 } ) \\big ) ' \\ = \\ T _ g \\big ( ( f ^ { \\uparrow _ 3 } ) ' \\big ) \\cdot \\log _ 3 ( g ) ' \\ = \\ T _ g \\big ( ( f ^ { \\uparrow 3 } ) ' \\big ) \\cdot ( \\ell _ 3 ' \\circ g ) \\cdot g ' . \\end{align*}"}
-{"id": "8828.png", "formula": "\\begin{align*} r = \\sqrt { \\frac { 2 \\mathrm { G a p } ( x , \\bar \\zeta , \\bar \\omega ) } { L _ { f , Q } } } \\end{align*}"}
-{"id": "7749.png", "formula": "\\begin{align*} { \\rm R e } [ \\langle x , \\lambda y \\rangle \\xi , \\xi ] = \\| x \\| \\left \\| y \\right \\| . \\end{align*}"}
-{"id": "1671.png", "formula": "\\begin{align*} \\gamma _ G U ( n ) ^ A \\simeq \\bigsqcup _ { k = 0 } ^ n \\gamma _ { W _ A } U ( n - k ) \\times C _ { N _ A , A } ( k ) \\end{align*}"}
-{"id": "5671.png", "formula": "\\begin{align*} e _ { a b } ( t ) ( [ c , d ] ) = \\left \\{ \\begin{array} { l l } 1 & \\\\ t & \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "2387.png", "formula": "\\begin{align*} \\deg { X _ i } & = 0 i \\in I _ 0 \\\\ \\deg { X _ i } & = 1 i \\in I _ 1 \\\\ \\deg ( X _ { i _ 1 } \\cdots X _ { i _ k } ) & = \\deg ( X _ { i _ 1 } ) + \\cdots + \\deg ( X _ { i _ k } ) \\end{align*}"}
-{"id": "7607.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { X _ \\epsilon ' } _ { L ^ \\infty ( 0 , T ; C ^ { 1 + \\alpha } ) } \\le T \\norm { X _ \\epsilon ' } _ { L i p ( 0 , T ; C ^ { 1 + \\alpha , p } ) } , \\\\ \\norm { \\eta _ \\epsilon ( t ) } _ { C ^ { \\alpha } } \\le t \\norm { X ' } _ { L i p ( 0 , T ; C ^ { 1 + \\alpha , p } ) } M _ \\epsilon ^ { \\alpha } . \\end{gathered} \\end{align*}"}
-{"id": "9392.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } \\int _ { \\mathbb { R } ^ N } \\Phi ( u ( x ) - u ( y ) ) ( u ( x ) - u ( y ) ) K ( x , y ) d x d y = \\int _ \\Omega | u | ^ { p ^ { \\ast } _ { s } } d x + \\lambda \\int _ { \\Omega } f ( x , u ) u d x \\end{align*}"}
-{"id": "277.png", "formula": "\\begin{align*} B _ { h , \\alpha } ( f ) ( x ) : = e ^ { i \\alpha } f ( x + i h ) + e ^ { - i \\alpha } f ( x - i h ) \\end{align*}"}
-{"id": "6046.png", "formula": "\\begin{align*} \\sum _ { b \\in V ( B ) } \\lambda _ G ( v , b ) = \\lambda _ G ( v , b _ v ) + 2 ( p - 1 ) \\leq \\lambda _ { G ' } ( v , \\beta ) + \\sum _ { i = 1 } ^ { p - 1 } \\lambda _ { G ' } ( v , z _ i ) \\end{align*}"}
-{"id": "100.png", "formula": "\\begin{align*} B ( 0 ) ^ { \\pm \\ell } B _ 1 = B _ 1 B ( 0 ) ^ { \\mp \\ell } . \\end{align*}"}
-{"id": "7953.png", "formula": "\\begin{align*} \\sum _ { a \\in T } x _ a = 0 . \\end{align*}"}
-{"id": "9112.png", "formula": "\\begin{align*} I _ + : = ( \\bigoplus _ { n > 0 } A _ n ) \\end{align*}"}
-{"id": "9210.png", "formula": "\\begin{align*} e ( \\bar { S } ) + e ( \\bar { S } , S ) & = \\sum _ { x \\in \\bar { S } } d _ { S } ( x ) + \\frac { 1 } { 2 } \\sum _ { x \\in \\bar { S } } d _ { \\bar { S } } ( x ) \\\\ & = \\frac { 1 } { 2 } \\sum _ { x \\in \\bar { S } } d _ { S } ( x ) + \\frac { 1 } { 2 } \\sum _ { x \\in \\bar { S } } \\left ( d _ S ( x ) + d _ { \\bar { S } } ( x ) \\right ) \\\\ & = \\frac { 1 } { 2 } \\sum _ { x \\in \\bar { S } } \\left ( d _ S ( x ) + d _ { G ' } ( x ) \\right ) . \\end{align*}"}
-{"id": "2301.png", "formula": "\\begin{align*} p _ { \\pm \\i \\pi } & = | \\alpha _ { \\pm \\i \\pi } ) ( \\beta _ { \\pm \\i \\pi } | \\\\ | \\alpha _ { \\pm \\i \\pi } ) & = \\binom { 1 } { - \\displaystyle \\frac { 1 } { \\pm \\i \\pi - \\Omega } | E ) } & ( \\beta _ { \\pm \\i \\pi } | & = \\left ( 1 , ( E | \\frac { 1 } { \\pm \\i \\pi - \\Omega } \\right ) \\end{align*}"}
-{"id": "7817.png", "formula": "\\begin{align*} \\langle x , x \\rangle \\langle ( \\| y \\| \\langle x , x \\rangle x + \\lambda \\| x \\| \\langle y , x \\rangle x ) , x \\rangle = 0 , \\end{align*}"}
-{"id": "7876.png", "formula": "\\begin{align*} \\{ J \\subseteq I \\colon | J | = l \\} = \\bigcup _ { j = 0 } ^ { l } \\{ J \\subseteq I \\colon | J | = l , \\ \\ | J \\cap S | = j \\} . \\end{align*}"}
-{"id": "8778.png", "formula": "\\begin{align*} \\overline { S } _ { a \\alpha } = 0 \\ , . \\end{align*}"}
-{"id": "3879.png", "formula": "\\begin{align*} Y ( I ) = ( C \\cap L ^ \\infty ) ( I , H ^ 1 ) \\cap L ^ 2 ( I , W ^ { 1 , \\frac { 2 d } { d - 2 } } ) & d \\geq 3 . \\end{align*}"}
-{"id": "1050.png", "formula": "\\begin{align*} & ( - B ) ^ p f = \\sum ( 1 - \\lambda _ k ( B ) ) ^ p \\langle f , e _ k \\rangle e _ k = f + \\sum ( ( 1 - \\lambda _ k ( B ) ) ^ p - 1 ) \\langle f , e _ k \\rangle e _ k . \\end{align*}"}
-{"id": "3427.png", "formula": "\\begin{gather*} \\langle v _ 1 \\otimes \\cdots \\otimes v _ n , w _ 1 \\otimes \\cdots \\otimes w _ n \\ > = \\sum _ { \\sigma \\in S _ n } ( - 1 ) ^ \\sigma \\langle v _ 1 , w _ { \\sigma ( 1 ) } \\rangle \\cdots \\langle v _ n , w _ { \\sigma ( n ) } \\rangle , \\end{gather*}"}
-{"id": "63.png", "formula": "\\begin{align*} U ( x , 0 ) = \\begin{cases} U _ \\ell & x < 0 \\\\ U _ r & x > 0 \\end{cases} \\end{align*}"}
-{"id": "2070.png", "formula": "\\begin{align*} D R ' ( \\rho | \\rho _ { \\rm t x } ) \\mid _ { \\rho = \\rho _ c } = - { k _ { f } } R ( \\rho _ c | \\rho _ { \\rm t x } ) , \\end{align*}"}
-{"id": "1085.png", "formula": "\\begin{align*} & \\int _ { I _ 1 } D _ n ( \\sqrt { 4 - y ^ 2 } / S ( I ) \\cdot G _ n ( x ) / 2 ) d y = \\int _ { a _ 1 } ^ { b _ 1 } D _ n ( z / S ( I ) \\cdot G _ n ( x ) / 2 ) \\frac { z } { \\sqrt { 4 - z ^ 2 } } d z \\\\ = & S ( I ) ( \\ln n ) ^ { - 1 } \\int _ { 0 } ^ { ( b _ 1 / a _ 1 - 1 ) \\ln n } D _ n ( ( 1 + z / \\ln n ) G _ n ( x ) / 2 ) \\frac { ( 1 + z / \\ln n ) a _ 1 } { \\sqrt { 4 - ( 1 + z / \\ln n ) ^ 2 a _ 1 ^ 2 } } d z , \\end{align*}"}
-{"id": "5409.png", "formula": "\\begin{align*} ( - \\varDelta ) ^ { \\alpha / 2 } u ( x ) + V ( x ) u ( x ) = f ( x , u ( x ) ) - K ( x ) | u ( x ) | ^ { p - 2 } u ( x ) , x \\in \\R ^ N , \\end{align*}"}
-{"id": "6433.png", "formula": "\\begin{align*} \\sigma _ { k } ^ { 2 } = \\frac { \\delta _ { k } } { \\gamma _ { k - 1 } ^ { 2 } } c _ { k - 1 } ^ { 2 } , \\qquad \\tau _ { k } = \\frac { 1 } { \\gamma _ { j - 1 } } + \\frac { \\delta _ { k } } { \\gamma _ { j - 1 } } , \\end{align*}"}
-{"id": "4419.png", "formula": "\\begin{align*} & \\forall i \\in \\{ 1 , \\ldots , n \\} : \\ ; \\partial _ t u ^ i - \\sum _ { j = 1 } ^ d \\partial _ j \\big ( \\partial _ j u ^ i + \\alpha ^ i u ^ i \\partial _ j \\mathcal { V } \\big ) = 0 , ( t , x ) \\in ( 0 , T ) \\times \\Omega , \\\\ & \\mathcal { V } ( t ) = \\mathcal { B } ( t , u ( t ) ) \\textrm { f o r a . e } t \\in ( 0 , T ) . \\end{align*}"}
-{"id": "3797.png", "formula": "\\begin{align*} a b 1 ^ d = \\lambda ( ( \\alpha _ i + \\alpha _ j ) \\alpha _ 1 \\cdots \\alpha _ { i - 1 } \\alpha _ { i + 1 } \\cdots \\alpha _ { j - 1 } \\alpha _ { j + 1 } \\cdots \\alpha _ R ) \\end{align*}"}
-{"id": "10038.png", "formula": "\\begin{align*} \\psi _ \\lambda ( x ; s ) = b _ z ( \\lambda ; s ) \\Psi _ \\lambda ( x ) + b _ z ( 1 / \\lambda ; s ) \\Psi _ { 1 / \\lambda } ( x ) , \\end{align*}"}
-{"id": "4959.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { n \\to \\infty } \\dfrac { \\ell ( R / ( ( a , x _ 2 ) + I _ n ) ) } { \\ell ( R / I _ n ) } \\geq \\dfrac { n ^ 3 } { ( s + 1 ) ( n ^ 3 + n ^ 2 ) } = \\frac { 1 } { s + 1 } \\neq 0 , \\end{align*}"}
-{"id": "9235.png", "formula": "\\begin{align*} \\norm { q + x _ 0 r } ^ 2 = ( p _ { j _ 1 0 } - x _ 0 ) ^ 2 . \\end{align*}"}
-{"id": "2021.png", "formula": "\\begin{align*} T _ j ^ { d , j ' } = q ^ d F ( j ) E ( d , j , j ' ) F ( j ' ) J _ { j ' } \\left ( \\tau , z = \\frac { \\alpha _ j - \\alpha _ { j ' } } { d } \\right ) . \\end{align*}"}
-{"id": "6738.png", "formula": "\\begin{align*} p _ n ( 2 m _ i - p _ n ) & = p _ n ( 2 ( p _ { n } + 1 ) - p _ n ) . \\\\ & = p _ n ( p _ n + 2 ) . \\\\ { p _ n } ^ 2 & < p _ n ( p _ n + 2 ) < ( p _ n + 1 ) ^ 2 . \\end{align*}"}
-{"id": "9599.png", "formula": "\\begin{align*} \\frac { ( N + 1 ) _ n } { n ! } = \\frac { ( N + n ) ! } { N ! n ! } & = \\frac { ( n + 1 ) ( n + 2 ) \\cdots ( n + N ) } { N ! } \\\\ & = \\frac { n ^ N } { N ! } + O ( n ^ { N - 1 } ) . \\end{align*}"}
-{"id": "10080.png", "formula": "\\begin{align*} \\begin{aligned} \\varphi ( t ; z ) & = \\varphi ( 0 ; z ) + \\dot { \\varphi } ( 0 ; z ) t + \\int _ { 0 } ^ t ( t - s ) \\ddot { \\varphi } ( s ; z ) \\ , d s \\\\ & = z + X ( z ) t + \\int _ { 0 } ^ t ( t - s ) D X ( \\varphi ( s ; z ) ) X ( \\varphi ( s ; z ) ) \\ , d s . \\end{aligned} \\end{align*}"}
-{"id": "5801.png", "formula": "\\begin{align*} T _ g \\big ( ( f ^ { \\uparrow _ 3 } ) ' \\big ) \\ = \\ ( f ' / \\ell _ 3 ' ) \\circ g \\ = \\ ( f ' \\circ g ) / ( \\ell _ 3 ' \\circ g ) . \\end{align*}"}
-{"id": "3792.png", "formula": "\\begin{align*} m _ { \\mathcal { M } ( \\alpha ) } ( \\mu ) = \\begin{cases} m _ { \\mathcal { M } ( \\beta ) } ( \\mu ) + ( - 1 ) ^ { R + d + 1 } & \\mu = ( a + d + 1 ) ( b - 1 ) \\\\ m _ { \\mathcal { M } ( \\beta ) } ( \\mu ) & \\mu \\neq ( a + d + 1 ) ( b - 1 ) . \\end{cases} \\end{align*}"}
-{"id": "9205.png", "formula": "\\begin{align*} R _ { s } : [ h ( s ) , b ] \\rightarrow M , R _ { s } ( t ) = H ( t , s ) . \\end{align*}"}
-{"id": "5208.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { x } U ( t ) d \\lambda ( t ) = c \\exp \\left ( \\int _ { 1 } ^ { x } t ^ { - 1 } b ( t ) d t \\right ) , a . e . , \\end{align*}"}
-{"id": "9600.png", "formula": "\\begin{align*} \\frac { ( N ) _ n } { n ! } = \\frac { n ^ { N - 1 } } { ( N - 1 ) ! } + O ( n ^ { N - 2 } ) . \\end{align*}"}
-{"id": "2713.png", "formula": "\\begin{align*} F _ m & ( \\pi ; x ) = 1 + \\sum _ { k = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ k } { k ! } \\int _ { x } ^ { \\infty } \\cdots \\int _ { x } ^ { \\infty } \\det [ { K _ { \\mathrm { A i r y } } ( \\pi ; u _ i , u _ j ) } ] _ { i , j = 1 } ^ { k } d u _ 1 \\cdots d u _ k , \\end{align*}"}
-{"id": "7697.png", "formula": "\\begin{align*} \\int \\limits _ { M \\setminus K } \\norm { D ^ * u ( p ) } ^ 2 _ { S _ { p } } \\ , \\mathrm { d } \\mu \\leq \\sum _ { k } \\int \\limits _ { M \\setminus V _ { k } } \\norm { D ^ * u ( p ) } ^ 2 _ { S _ { p } } \\ , \\mathrm { d } \\mu = 0 . \\end{align*}"}
-{"id": "7641.png", "formula": "\\begin{align*} a ( z ) \\ge 0 , \\int _ { \\mathbb { R } ^ d } a ( z ) \\ , d z = 1 , \\end{align*}"}
-{"id": "5389.png", "formula": "\\begin{align*} | | v _ { n + 2 \\ , m } ^ 1 | | ^ 2 = - \\frac { c _ { n + 2 \\ , m } } { \\overline { b _ { n + 1 \\ , m + 3 } } } | | v _ { n + 1 \\ , m + 3 } ^ 1 | | ^ 2 = \\frac { c _ { n + 2 \\ , m } } { \\overline { b _ { n + 1 \\ , m + 3 } } } \\cdot \\frac { \\overline { d _ { n + 1 \\ , m + 3 } } } { a _ { n m } } | | v _ { n m } ^ 1 | | ^ 2 . \\end{align*}"}
-{"id": "7501.png", "formula": "\\begin{align*} \\check { \\nabla } _ X Y : = \\nabla _ X Y + \\nu X * Y \\end{align*}"}
-{"id": "7371.png", "formula": "\\begin{align*} | S _ 1 ( A ) | \\leq | \\overline { E _ { A p } } \\times [ 0 , r ] | = | \\overline { E _ { A p } } | 2 ^ \\omega \\leq | E _ { A p } | ^ \\omega 2 ^ \\omega = \\kappa ^ \\omega 2 ^ \\omega = \\kappa ^ \\omega = \\kappa \\end{align*}"}
-{"id": "5426.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\lambda _ k ( A , \\mu ) = \\lambda _ \\infty ( A , \\mu ) , \\end{align*}"}
-{"id": "5351.png", "formula": "\\begin{align*} A _ { \\alpha + \\beta } = X _ { \\alpha + \\beta } + Y _ { \\alpha + \\beta } \\quad \\mbox { a n d } B _ { \\alpha + \\beta } = i ( X _ { \\alpha + \\beta } - Y _ { \\alpha + \\beta } ) \\end{align*}"}
-{"id": "3560.png", "formula": "\\begin{align*} \\mathcal { M } _ { 3 , 3 } ^ { L o c } = \\mathrm { S p e c } \\ , \\mathbb { C } [ C _ { 0 } \\cap L ] \\simeq \\left \\{ ( x , y , z , u , v ) \\mid x y z = u v \\right \\} , \\end{align*}"}
-{"id": "1876.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\frac { A _ j ^ 2 } { Z _ j ^ 2 \\left [ Z _ j ^ 2 - ( \\pi m ) ^ 2 \\right ] } = 0 . \\end{align*}"}
-{"id": "2926.png", "formula": "\\begin{align*} \\| f \\| _ { \\omega } ^ 2 \\coloneqq \\sum _ { k = 0 } ^ { \\infty } | a _ k | ^ 2 w _ k < \\infty . \\end{align*}"}
-{"id": "3179.png", "formula": "\\begin{align*} { \\rm i n d } _ \\varGamma ( U ^ { - 1 } ) = { \\rm i n d } _ \\varGamma ( U ) . \\end{align*}"}
-{"id": "3225.png", "formula": "\\begin{align*} \\| \\Phi _ { \\mu _ k } ( w ^ k ) \\| = O ( \\mu _ k ^ { 1 + \\tilde { c } \\alpha } ) \\end{align*}"}
-{"id": "7703.png", "formula": "\\begin{align*} \\bigl ( \\tau _ { x } f \\bigr ) ( y ) : = f ( y - x ) \\quad \\ ; \\ ; \\tilde { f } ( y ) : = f ( - y ) . \\end{align*}"}
-{"id": "3282.png", "formula": "\\begin{align*} a , b = \\sqrt [ n ] { \\frac { q \\pm \\sqrt { q ^ 2 - 4 p ^ n } } { 2 } } , \\end{align*}"}
-{"id": "5874.png", "formula": "\\begin{align*} P = & - A X A - A Y B - A Z C - Y ' C , \\\\ Q = & - B X A - B Y B - B Z C - X ' C , \\\\ R = & - C X A - B ' A - C Y B - A ' B - C Z C - B ' Y ' C - A ' X ' C . \\end{align*}"}
-{"id": "1475.png", "formula": "\\begin{align*} V _ { 1 2 } = \\frac { \\varepsilon \\bar { m } } { 2 } \\sum _ { i = 1 } ^ { d } \\int _ { \\mathbb { T } ^ { d } } ( \\mu ^ { 1 } - \\mu ^ { 2 } ) ^ { 2 } \\frac { \\partial } { \\partial x _ { i } } \\left ( \\Theta _ { p _ { i } q } ( t , x , \\mu ^ { 1 } , D w ^ { 1 } ) \\right ) \\ d x . \\end{align*}"}
-{"id": "8883.png", "formula": "\\begin{align*} s _ { \\alpha } = \\frac { \\gamma _ { \\alpha , \\beta } } { g _ { \\alpha , \\beta } } s _ { \\beta } o n U _ { \\alpha } \\cap U _ { \\beta } , \\end{align*}"}
-{"id": "9857.png", "formula": "\\begin{align*} \\int _ M { \\left \\langle { \\nabla u , \\nabla ( { L _ A } u ) } \\right \\rangle d v o { l _ g } } = - \\int _ M { ( { L _ A } u ) ( \\Delta u ) d v o { l _ g } } + \\int _ { \\partial M } { ( { L _ A } u ) \\left \\langle { \\nabla u , \\overrightarrow { n } } \\right \\rangle d v o { l _ { \\bar g } } } . \\end{align*}"}
-{"id": "9435.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq j \\leq D ( N ) - 1 } a _ j ( E _ 2 ( z ) - d _ j E _ 2 ( d _ j z ) ) = 0 , \\end{align*}"}
-{"id": "3968.png", "formula": "\\begin{align*} h ( \\tau , z ) : = f ( \\tau , z ) \\cdot v ^ k \\cdot e ^ { - 4 \\pi N \\frac { y ^ 2 } { v } } \\end{align*}"}
-{"id": "6176.png", "formula": "\\begin{align*} J ( M _ 1 , M _ 2 ) + J ( m _ 1 , m _ 2 ) = J ( g _ 1 , g _ 2 ) + J ( h _ 1 , h _ 2 ) . \\end{align*}"}
-{"id": "1538.png", "formula": "\\begin{align*} u _ 2 ( x , t ) = P . V . \\frac { 1 } { 2 \\ , \\pi } \\int _ { 0 } ^ { t } \\int _ { - \\infty } ^ { \\infty } \\int _ { 0 } ^ { 1 } { e ^ { r i ( t - s ) } g ( \\sqrt { i r } , x , y ) F ( y , s ) d y \\ , d r \\ , d s } . \\end{align*}"}
-{"id": "1416.png", "formula": "\\begin{align*} O P T ( T ) - E _ k ( \\eta ) = \\inf _ { \\tau \\in \\mathcal { T } } E [ Z ^ { k + 1 } _ { \\eta , \\tau } ] . \\end{align*}"}
-{"id": "8640.png", "formula": "\\begin{align*} \\varepsilon = \\sqrt { \\Lambda - 1 } , \\end{align*}"}
-{"id": "4189.png", "formula": "\\begin{align*} \\frac { A _ n ^ { ( 3 ) } ( z ) } { n ^ 9 z } = \\mathcal { O } \\left ( n ^ { - \\frac { 1 } { 2 } } \\right ) . \\end{align*}"}
-{"id": "2985.png", "formula": "\\begin{align*} L & = \\begin{cases} - 1 , & 2 n = \\Delta _ { 4 k + 1 } , \\Delta _ { 4 k + 2 } k ; \\\\ 1 , & 2 n = \\Delta _ { 4 k + 3 } , \\Delta _ { 4 k + 4 } k ; \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "9194.png", "formula": "\\begin{align*} \\mathrm { d i s t } ( N , \\tilde { N } ) & = b \\\\ & \\leq r + \\tilde { r } , \\end{align*}"}
-{"id": "450.png", "formula": "\\begin{align*} \\mathcal { Z } _ { N , \\tau } = \\sum \\limits _ { T \\in \\mathcal { T } _ { N , \\tau } } \\mathcal { Z } ( T ) . \\end{align*}"}
-{"id": "1992.png", "formula": "\\begin{align*} \\int _ { \\P ^ n } e ( T \\P ^ n ) = \\sum _ { j = 0 } ^ n \\frac { e ( i _ j ^ * T \\P ^ n ) } { e ( N _ { p _ j / X } ) } = \\sum _ { j = 0 } ^ n \\frac { e ( i _ j ^ * T \\P ^ n ) } { e ( i _ j ^ * T \\P ^ n ) } = n + 1 . \\end{align*}"}
-{"id": "8988.png", "formula": "\\begin{align*} x _ i = x _ i ( \\mathbf { w } ) = - w _ i ^ { - 1 } ( \\mathbf { w } _ { R } \\cdot \\mathbf { x } _ { R } ) . \\end{align*}"}
-{"id": "5319.png", "formula": "\\begin{align*} H _ { { \\rm I V } } ( q , p ; \\alpha , \\beta ; t ) = & q p ( p - q - t ) + \\alpha q + \\beta p \\end{align*}"}
-{"id": "8847.png", "formula": "\\begin{align*} \\Lambda = & \\Lambda ( a , b ) = \\int _ { ( 2 a - b ) ^ + } ^ a ( h ( x , a ) - h ( a , b ) ) { \\rm d } x + \\int _ a ^ { \\frac { a + b } 2 } ( h ( a , x ) - h ( a , b ) ) { \\rm d } x \\\\ & + \\int _ { \\frac { a + b } 2 } ^ b ( h ( x , b ) - h ( a , b ) ) { \\rm d } x + \\int _ { b } ^ { 2 b - a } ( h ( b , x ) - h ( a , b ) ) { \\rm d } x . \\end{align*}"}
-{"id": "7819.png", "formula": "\\begin{align*} \\| x \\| ^ 4 \\| y \\| x \\otimes x + \\lambda [ y , x ] \\| x \\| ^ 3 x \\otimes x = 0 . \\end{align*}"}
-{"id": "4956.png", "formula": "\\begin{align*} N = \\dfrac { \\kappa [ [ x _ 1 , \\ldots , x _ d ] ] } { ( x _ d ) } \\cong \\kappa [ [ x _ 1 , \\cdots , x _ { d - 1 } ] ] , \\end{align*}"}
-{"id": "8840.png", "formula": "\\begin{align*} 6 - t _ { z a } & = \\frac { - 2 p ( M \\mu + 6 ) + M a + 6 M + a + 6 } { M + 1 - 2 p } > \\frac { - 2 M \\mu + M a + 6 M + a - 6 } { M + 1 - 2 p } \\\\ & > \\frac { - 2 M \\mu + 6 M - 6 } { M + 1 - 2 p } = \\frac { 2 M ( 1 - \\mu ) + 4 M - 6 } { M + 1 - 2 p } > \\frac 2 { M + 1 - 2 p } > 0 . \\end{align*}"}
-{"id": "4535.png", "formula": "\\begin{align*} \\nu _ g ( B _ { 2 . 5 \\delta } ( z ) ) = \\int _ { B _ { 2 . 5 \\delta } ( z ) } d \\nu _ g ( x ) \\leq \\int \\phi ^ D ( ( z - x ) / \\delta ) d \\nu _ g ( x ) = h _ { g , \\delta } ( z ) . \\end{align*}"}
-{"id": "2055.png", "formula": "\\begin{align*} \\alpha _ { n , M } ( t ) = \\frac { 1 } { M } \\sum _ { d | M } \\sum _ { \\substack { a \\mod d \\\\ \\gcd ( a , d ) = 1 } } e ^ { - 2 \\pi i n \\frac { a } { d } } L \\big ( f , t , \\frac { a } { d } \\big ) . \\end{align*}"}
-{"id": "9524.png", "formula": "\\begin{align*} d ( n , N ) = t ( n , N ) - t ( n - N , N ) , \\end{align*}"}
-{"id": "3212.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\displaystyle { \\mathop { \\rm M i n i m i z e } _ { x \\in \\R ^ { \\frac { m ( m + 1 ) } { 2 } } } } & & \\frac { 1 } { 2 } x ^ { \\top } M x + c ^ { \\top } x + \\omega { \\| x \\| ^ 4 } \\\\ \\mbox { s u b j e c t t o } & & \\sum _ { i = 1 } ^ { n } \\tau ^ { i - 1 } x _ i \\le \\sum _ { i = 1 } ^ n \\tau ^ { 2 i } + \\sin ( 9 \\pi \\tau ) + 2 \\ \\ ( \\tau \\in T ) \\\\ & & X + \\kappa I \\in S ^ m _ + \\end{array} \\end{align*}"}
-{"id": "9154.png", "formula": "\\begin{align*} F _ { s } \\equiv \\int \\limits _ { \\Gamma _ { N } } d \\overline { \\mathbf { x } } \\overline { \\Theta } ^ { ( N ) } ( \\overline { \\mathbf { r } } ) \\prod \\limits _ { i = 1 , s } \\delta ( \\mathbf { x } _ { i } - \\overline { \\mathbf { x } } _ { i } ) . \\end{align*}"}
-{"id": "2821.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\displaystyle { x _ i ^ { \\prime } ( t ) = x _ i ( t ) \\biggl ( b _ i - \\mu _ i x _ i ( t ) - \\sum _ { j = 1 } ^ p a _ { i j } \\int _ 0 ^ { \\infty } K _ { i j } ( s ) x _ j ( t - s ) \\ , d s } \\\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\displaystyle { - c _ i \\int _ 0 ^ { \\infty } G _ i ( s ) u _ i ( t - s ) \\ , d s \\biggr ) } , \\\\ \\displaystyle { u _ i ^ { \\prime } ( t ) = - e _ i u _ i ( t ) + d _ i x _ i ( t ) , i = 1 , \\ldots , p } . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "6315.png", "formula": "\\begin{align*} \\liminf _ { n \\rightarrow + \\infty } \\mathbb { E } _ { G _ 0 \\sim \\mathbb { P } _ 1 } \\left [ \\log \\frac { \\mathbb { P } _ 2 [ G = G _ 0 ] } { \\mathbb { P } _ 1 [ G = G _ 0 ] } | \\mathcal { E } \\right ] \\geq 0 . \\end{align*}"}
-{"id": "5970.png", "formula": "\\begin{align*} X ( t ) = W ( t ) + \\int _ 0 ^ t q ( s ) d s = W ( t ) + \\int _ 0 ^ t q ( s , W ) d s . \\end{align*}"}
-{"id": "9936.png", "formula": "\\begin{align*} \\frac { \\norm { ( \\Delta + M _ V - \\lambda _ k I ) \\psi _ k } } { \\norm { \\psi _ k } } = O ( k ^ { - \\infty } ) \\quad \\lim _ { k \\to \\infty } | \\lambda _ k | = + \\infty . \\end{align*}"}
-{"id": "6232.png", "formula": "\\begin{align*} \\phi ^ p ( \\ \\cdot \\ ) : = \\frac { 1 } { \\phi ( p ) } \\phi ( p \\ \\cdot \\ p ) . \\end{align*}"}
-{"id": "9488.png", "formula": "\\begin{align*} & \\displaystyle \\int _ { \\mathbb { R } ^ N } \\ ! \\ ! \\int _ { \\mathbb { R } ^ { N } } \\ ! \\ ! \\left ( \\frac { \\vert u ( x ) - u ( y ) \\vert ^ { p - 2 } } { \\vert x - y \\vert ^ { N + s p } } + \\frac { \\vert u ( x ) - u ( y ) \\vert ^ { q - 2 } } { \\vert x - y \\vert ^ { N + s q } } \\right ) \\ ! \\ ! ( u ( x ) - u ( y ) ) ( \\varphi ( x ) - \\varphi ( y ) ) \\dd x \\dd y \\\\ & = \\displaystyle \\int _ { \\mathbb { R } ^ N } f \\varphi \\dd x , \\ \\ \\ \\mbox { f o r a l l } \\ \\varphi \\in \\mathcal { W } . \\end{align*}"}
-{"id": "5344.png", "formula": "\\begin{align*} | | v ^ { k + 1 } | | ^ 2 = \\frac { k } { n - k } | | v ^ k | | ^ 2 . \\end{align*}"}
-{"id": "1051.png", "formula": "\\begin{align*} & \\sum _ j \\lambda _ j ( B _ 1 ) ^ 2 = | B _ 1 | _ 2 ^ 2 = | ( - B ) ^ { - 1 / 2 } ( A - B ) ( - B ) ^ { - 1 / 2 } | _ 2 ^ 2 \\\\ \\leq & \\| ( - B ) ^ { - 1 / 2 } \\| ^ 2 | A - B | _ 2 ^ 2 \\| ( - B ) ^ { - 1 / 2 } \\| ^ 2 = \\| ( - B ) ^ { - 1 } \\| ^ 2 | A - B | _ 2 ^ 2 . \\end{align*}"}
-{"id": "5872.png", "formula": "\\begin{align*} B = ( w _ i : - n \\leq i \\leq n ) \\end{align*}"}
-{"id": "6508.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\displaystyle d \\bar { X } _ { t } = A \\bar { X } _ { t } d t - B ( \\bar { X } _ { t } ) d t + \\bar { f } ( \\bar { X } _ { t } ) d t + \\sigma _ 1 ( \\bar { X } _ { t } ) d W ^ { Q _ { 1 } } _ t , \\\\ \\bar { X } _ { 0 } = x , \\end{array} \\right . \\end{align*}"}
-{"id": "8023.png", "formula": "\\begin{align*} \\Omega _ R / d R \\cong \\mathbb { C } \\omega _ 0 \\oplus U _ 1 ^ { \\Xi _ 1 } \\oplus U _ 2 ^ { \\Xi _ 2 } \\oplus \\bigoplus _ { j = 1 } ^ { \\frac { n - 1 } { 2 } } V _ { j } , \\end{align*}"}
-{"id": "8597.png", "formula": "\\begin{align*} \\min _ { a \\in [ n ] , \\binom { a - 1 } { k - 1 } \\leq \\ell - 2 } \\left \\lceil \\frac { ( \\ell - 1 ) ( n - a ) } k \\right \\rceil + \\binom { a } k = \\left \\lceil \\frac { ( \\ell - 1 ) ( n - c ) } k \\right \\rceil + \\binom { c } k . \\end{align*}"}
-{"id": "7019.png", "formula": "\\begin{align*} \\varphi _ 2 = \\varphi _ 1 \\circ \\alpha \\quad \\alpha \\circ T _ g = T _ g \\circ \\alpha , g \\in G . \\end{align*}"}
-{"id": "3420.png", "formula": "\\begin{align*} \\log \\frac { A _ { \\sup } } { A _ { \\inf } } = \\sum _ i \\left ( - \\log \\left ( \\frac { 1 + \\rho _ i e ^ { - \\tau _ i } } { 1 + \\rho _ i } \\right ) \\right ) \\le \\sum _ i \\frac { \\rho _ i } { 1 + \\rho _ i } \\tau _ i \\le \\frac { 1 } { 3 } \\sum _ i \\tau _ i = \\frac { 1 } { 3 } \\Sigma _ * , \\end{align*}"}
-{"id": "2838.png", "formula": "\\begin{align*} \\mu _ i \\to \\mu = \\left | F _ { A _ \\infty } \\right | ^ 2 \\frac { \\omega ^ n } { n ! } + \\nu \\ , \\end{align*}"}
-{"id": "2743.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\phi } \\Re { f \\Big ( \\frac { 1 } { \\tau } + \\frac { R } { \\eta _ { - } } e ^ { i \\phi } \\Big ) } & = - \\Big ( x - \\frac { 1 } { 1 + R ^ 2 + 2 R \\cos \\phi } \\Big ) R \\sin \\phi \\\\ & < - \\Big ( x - \\frac { 1 } { ( R - 1 ) ^ 2 } \\Big ) R \\sin \\phi < 0 . \\end{align*}"}
-{"id": "1684.png", "formula": "\\begin{align*} \\left | w \\right | \\left ( { { k } _ { 1 } } \\right ) = \\left | \\Phi \\right | { { e } ^ { - { { k } _ { 1 } } } } \\le { { \\left | w \\right | } _ { M a x } } \\left ( 0 \\right ) = \\left | \\Phi \\right | \\end{align*}"}
-{"id": "4813.png", "formula": "\\begin{align*} \\mathcal { S } _ g ^ * ( f ) = ( A _ g ^ * f , B _ \\gamma ^ * f ) , \\end{align*}"}
-{"id": "6155.png", "formula": "\\begin{align*} ( \\omega _ + ) _ n = 4 \\int _ 0 ^ 1 r ^ { 2 n + 1 } \\int _ r ^ 1 \\frac { \\omega ( s ) } { s } d s d r = 4 \\int _ 0 ^ 1 \\frac { \\omega ( s ) } { s } \\int _ 0 ^ s r ^ { 2 n + 1 } d r d s = \\frac { \\omega _ n } { n + 1 } . \\end{align*}"}
-{"id": "9095.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ M } | v ( x ) | ^ p \\ , d x = 1 . \\end{align*}"}
-{"id": "1256.png", "formula": "\\begin{align*} T _ { 4 , 5 } & = 8 B \\int e ^ { 3 i \\eta \\nu ^ 2 } \\left ( \\eta - \\frac { \\nu } { 2 } \\right ) \\frac { e ^ { i \\beta ( \\eta + \\nu ) ^ 3 / 8 } } { ( \\eta + \\nu ) ^ 3 } \\Bigg ( e ^ { 2 i a \\ln | ( \\eta + \\nu ) / 2 | } \\chi ' \\left ( \\frac { \\eta + \\nu } { 2 } \\right ) \\\\ & \\qquad \\qquad - e ^ { - 2 i a \\ln | ( \\eta + \\nu ) / 2 | } \\chi ' \\left ( - \\frac { \\eta + \\nu } { 2 } \\right ) \\Bigg ) S \\left ( \\frac { \\eta - \\nu } { 2 } \\right ) d \\nu . \\end{align*}"}
-{"id": "5410.png", "formula": "\\begin{align*} \\mathcal { O } ( u ) : = \\left \\{ u ( \\cdot - k ) \\ : \\ k \\in \\mathbb { Z } ^ N \\right \\} \\end{align*}"}
-{"id": "6687.png", "formula": "\\begin{align*} E _ { n } = | ( 1 - 2 ^ { \\frac { 1 } { 2 } - i ( n \\omega - \\lambda _ { * } ) } ) | ^ { 2 } | \\zeta ( \\frac { 1 } { 2 } + i ( \\lambda _ { * } - n \\omega ) ) | ^ { 2 } , \\end{align*}"}
-{"id": "8065.png", "formula": "\\begin{align*} 2 n & = n _ 1 + n _ 2 + \\sum _ { h = 1 } ^ { \\frac { n - 1 } { 2 } } 2 m _ h , \\\\ \\sum _ { i = 1 } ^ { 2 n } \\zeta ^ { \\frac { ( 2 n + 3 - 2 i ) k } { 2 } } & = n _ 1 + n _ 2 + \\sum _ { h = 1 } ^ { \\frac { n - 1 } { 2 } } 2 m _ h \\cos ( 2 \\pi h k / n ) \\mbox { f o r } 1 \\leq k \\leq \\widetilde { m } = ( n - 1 ) / 2 , \\\\ - 1 - \\sum _ { i = n + 3 } ^ { 2 n } c ^ { n + 3 - 2 i } P _ { i - n - 3 , - i } & = n _ 1 - n _ 2 \\quad \\psi _ c ^ + . \\\\ \\end{align*}"}
-{"id": "7293.png", "formula": "\\begin{align*} \\psi _ q ( a ) ^ * ( h _ p ^ - - h _ q ^ - ) = ( ( h _ p ^ - - h _ q ^ - ) \\psi _ q ( a ) ) ^ * & = ( ( h _ p ^ - - h _ q ^ - ) \\psi _ q ( a ) h _ q ^ + ) ^ * \\\\ & = ( ( h _ p ^ - - h _ q ^ - ) \\psi _ q ( a ) ( h _ q ^ + - k _ q ) ) ^ * , \\end{align*}"}
-{"id": "7142.png", "formula": "\\begin{align*} y = \\int _ 0 ^ x \\rho ( s , t ) d s , \\ \\ t = t ; v = \\frac { 1 } { \\rho } . \\end{align*}"}
-{"id": "9360.png", "formula": "\\begin{align*} \\mathcal { I } \\colon I _ 0 ( G ) = G , I _ j ( G ) = I _ { j - 1 } ( G ) ^ 2 \\end{align*}"}
-{"id": "9633.png", "formula": "\\begin{align*} S _ 1 ( z , q , N ) : = \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] _ { q ^ 2 } \\frac { ( q ^ 2 ; q ^ 2 ) _ { n } ( q ^ 2 ; q ^ 2 ) _ { n - 1 } ( z q ; q ^ 2 ) _ { N - n } ( z q ) ^ { n } } { ( z q ^ 2 ; q ^ 2 ) _ { n } ( z q ; q ^ 2 ) _ { N } } . \\end{align*}"}
-{"id": "376.png", "formula": "\\begin{align*} \\beta = A + B \\alpha + C \\bar { \\alpha } + D | \\alpha | ^ 2 , \\end{align*}"}
-{"id": "1015.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } D _ n ( F _ n ( x ) / 2 ) = e ^ { c _ 0 - x } . \\end{align*}"}
-{"id": "3537.png", "formula": "\\begin{align*} C o n e ( \\mathcal { A } ) \\cap \\left \\{ x \\mid \\langle m , x \\rangle = 1 \\right \\} = \\mathcal { A } \\end{align*}"}
-{"id": "9472.png", "formula": "\\begin{align*} T a i l _ { m } ( u ; x ; R ) = \\left ( R ^ { s m } \\displaystyle \\int _ { B _ { R } ^ { c } ( x ) } \\frac { \\vert u ( y ) \\vert ^ { m - 1 } } { \\vert x - y \\vert ^ { N + s m } } \\dd y \\right ) ^ { 1 / ( m - 1 ) } . \\end{align*}"}
-{"id": "3232.png", "formula": "\\begin{align*} s _ j ^ { k } = \\min ( t ^ { k } _ j , u ^ { k } _ j ) , \\end{align*}"}
-{"id": "66.png", "formula": "\\begin{align*} \\sigma _ { - 1 } & = J _ * - J _ \\ell \\ , = \\ , - \\left ( f ^ - _ * - f ^ - _ \\ell \\right ) \\ , = \\ , - \\left ( \\rho _ * - \\rho _ \\ell \\right ) , \\\\ \\sigma _ { 1 } & = J _ r - J _ * \\ , = \\ , f ^ + _ r - f ^ + _ * \\ , = \\ , \\rho _ r - \\rho _ * . \\end{align*}"}
-{"id": "3288.png", "formula": "\\begin{align*} [ C ] = \\pi ^ * H - m E , [ D _ 0 ] = \\pi ^ * H ' - ( i m + 1 ) E , i = 1 , 2 . \\end{align*}"}
-{"id": "2013.png", "formula": "\\begin{align*} J \\left ( \\frac { I _ 1 ( q ) } { I _ 0 ( q ) } \\phi _ 2 , z \\right ) = \\frac { I ( q , z ) } { I _ 0 ( q ) } , \\end{align*}"}
-{"id": "8372.png", "formula": "\\begin{align*} \\ddot { z } _ j ( t ) = - \\sum _ { k : k \\neq j } \\frac { \\lambda _ k i \\overline { \\dot { z } _ j ( t ) - \\dot { z } _ k ( t ) } } { 2 \\pi ( \\overline { z _ j ( t ) - z _ k ( t ) } ) ^ 2 } + \\bar { F } _ z ( z _ j ( t ) , t ) \\dot { z } _ j ( t ) + \\bar { F } _ t ( z _ j ( t ) , t ) \\end{align*}"}
-{"id": "5625.png", "formula": "\\begin{align*} 2 \\alpha \\int _ { 0 } ^ { + \\infty } ( | u | ^ 2 ) _ x v d x = - \\frac { d } { d t } \\int _ 0 ^ { + \\infty } 2 ( u \\bar { u } _ x ) d x - \\mathcal { Q } ^ { u } ( t ) , \\end{align*}"}
-{"id": "1367.png", "formula": "\\begin{align*} ( \\mu , x ) \\sim ( \\nu , y ) \\Longleftrightarrow \\ & \\mu ( \\lfloor x \\rfloor , \\lceil x \\rceil ) = \\nu ( \\lfloor y \\rfloor , \\lceil y \\rceil ) x - \\lfloor x \\rfloor = y - \\lfloor y \\rfloor \\end{align*}"}
-{"id": "6001.png", "formula": "\\begin{align*} F ^ n ( x _ 1 , \\dots , x _ n ) : = n F \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\delta _ { x _ i } \\right ) . \\end{align*}"}
-{"id": "3689.png", "formula": "\\begin{align*} \\mu ( x ) = \\sum _ { i \\in U _ x } \\mu _ i / N _ x \\end{align*}"}
-{"id": "7422.png", "formula": "\\begin{align*} \\displaystyle \\lambda _ { E / F } ( \\psi ' _ \\lambda ) = \\eta _ { E / F } ( \\lambda ) \\lambda _ { E / F } ( \\psi ' ) \\end{align*}"}
-{"id": "2772.png", "formula": "\\begin{align*} ( \\mathbf { A } _ { i } ^ t ) _ { j } = \\left \\{ \\begin{array} { c @ { \\mbox { i f } \\quad } l } B _ { i , j } ^ t & 1 \\leq j \\leq S \\\\ 1 - \\sum _ { j = 1 } ^ S B _ { i , j } ^ t & j = S + 1 , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "2109.png", "formula": "\\begin{align*} M _ 0 : = \\max \\{ \\hat { A } , 2 K \\hat { C } \\} \\end{align*}"}
-{"id": "4052.png", "formula": "\\begin{align*} b _ i ( a _ 0 , \\ldots , a _ { n } ) = ( a _ 0 , \\ldots , a _ i a _ { i + 1 } , \\ldots , a _ n ) . \\end{align*}"}
-{"id": "5786.png", "formula": "\\begin{align*} \\sum _ { d \\geq 1 } \\tilde R _ d ( \\Gamma , q ) T ^ { - d } = \\epsilon ( \\Gamma ) + T \\sum _ { d \\geq 1 } \\sum _ { A \\subseteq E ( \\Gamma ) } \\tilde R _ d ( \\Gamma / A , q ) \\left ( q ^ { b _ 1 ( \\Gamma [ A ] ) } T \\right ) ^ { - d } . \\end{align*}"}
-{"id": "4076.png", "formula": "\\begin{align*} \\begin{aligned} F _ { 1 , + } ( x ) + F _ { 1 , - } ( x ) - F _ 2 ( x ) & = V ' ( x ) = 1 , & & x \\in ( 0 , q ) , \\\\ F _ { 2 , + } ( x ) + F _ { 2 , - } ( x ) - F _ 1 ( x ) & = 0 , & & x \\in ( - \\infty , 0 ) . \\end{aligned} \\end{align*}"}
-{"id": "9092.png", "formula": "\\begin{align*} \\liminf _ { R \\to \\infty } \\int _ { \\mathbb { R } ^ N } g _ { + } ( x ) | \\varphi _ R ( x ) | ^ p d x \\ge \\int _ { \\mathbb { R } ^ N } g _ + ( x ) d x = \\infty . \\end{align*}"}
-{"id": "1485.png", "formula": "\\begin{align*} W ^ { j } _ { 4 } = - \\varepsilon \\int _ { \\mathbb { T } ^ { d } } ( \\partial _ { x _ { j } } w ^ { 1 } - \\partial _ { x _ { j } } w ^ { 2 } ) \\left ( \\Theta _ { p } ( t , x , \\mu ^ { 2 } , D w ^ { 2 } ) \\mu ^ { 1 } _ { x _ { j } } - \\Theta _ { q } ( t , x , \\mu ^ { 2 } , D w ^ { 2 } ) \\mu ^ { 2 } _ { x _ { j } } \\right ) \\ d x , \\end{align*}"}
-{"id": "2552.png", "formula": "\\begin{align*} \\Pi _ { \\mu } ^ { ( T , g ) } [ f ( \\xi _ 0 ) ] & = \\frac { 1 } { \\mu ( P ^ \\beta _ T g ) } \\Pi _ { \\mu } \\Big [ g ( \\xi _ T ) \\exp \\Big \\{ \\int _ 0 ^ T \\beta ( \\xi _ s ) d s \\Big \\} f ( \\xi _ 0 ) \\Big ] \\\\ & = \\frac { 1 } { \\mu ( P ^ \\beta _ T g ) } \\int _ E ( P ^ \\beta _ T g ) ( x ) \\cdot f ( x ) \\mu ( d x ) , \\end{align*}"}
-{"id": "7476.png", "formula": "\\begin{align*} m ( 4 g - 4 ) = m \\aleph ( s - 2 ) - m \\sum _ { i = 1 } ^ b ( k _ i ( s - 2 ) + l _ i ) \\end{align*}"}
-{"id": "1139.png", "formula": "\\begin{align*} & | A _ { j , j } - B _ j | _ 2 ^ 2 = O ( a _ j ^ 2 + n ^ 4 a _ j ^ 6 ) = O \\left ( \\frac { \\ln n } { n ^ 2 } + \\frac { ( \\ln n ) ^ 3 } { n ^ 2 } \\right ) = O \\left ( \\frac { ( \\ln n ) ^ 3 } { n ^ 2 } \\right ) , \\end{align*}"}
-{"id": "3765.png", "formula": "\\begin{align*} \\begin{aligned} f _ { 0 } ^ { ( 1 ) } & \\coloneqq \\frac { c _ { 0 } } { d _ { 0 } ^ 2 } \\\\ f _ { 1 } ^ { ( 1 ) } & \\coloneqq \\frac { c _ { 1 } d _ { 0 } - 2 c _ { 0 } d _ { 1 } } { d _ { 0 } ^ 3 } \\\\ f _ { 2 } ^ { ( 1 ) } & \\coloneqq \\frac { 3 c _ { 0 } d _ { 1 } ^ { 2 } + c _ { 2 } d _ { 0 } ^ { 2 } - 2 c _ { 0 } d _ { 0 } d _ { 2 } - 2 c _ { 1 } d _ { 0 } d _ { 1 } } { d _ { 0 } ^ 4 } . \\end{aligned} \\end{align*}"}
-{"id": "7906.png", "formula": "\\begin{align*} \\Delta _ 3 = \\frac { h ^ 2 } { 2 4 } \\left [ \\frac { f ( b + h / 2 ) - f ( b - h / 2 ) } { h } - \\frac { f ( a + h / 2 ) - f ( a - h / 2 ) } { h } \\right ] \\end{align*}"}
-{"id": "6479.png", "formula": "\\begin{align*} t = ( 1 - a ^ { 2 } - b ^ { 2 } ) + ( 1 + ( t _ { B } - 1 ) b ^ { 2 } - a ^ { 2 } ) ; \\end{align*}"}
-{"id": "6339.png", "formula": "\\begin{align*} G _ 1 = & \\frac { m _ { r _ 1 } } { m _ s h _ { s r } + D ( m _ { b r _ 1 } + n _ { s p } ) } , \\\\ G _ 2 = & \\frac { m _ { r _ 2 } } { m _ { r _ 1 } h _ { r r } + D ( m _ { b r _ 2 } + n _ { s p } ) } , \\end{align*}"}
-{"id": "3873.png", "formula": "\\begin{align*} \\frac { | G | | A | } { | A | ^ 2 } = \\sum _ j \\nu _ j ^ 2 \\geq \\frac { q - 1 } { 2 } \\nu _ j ^ 2 , \\end{align*}"}
-{"id": "3356.png", "formula": "\\begin{align*} [ u ^ k ] \\log \\widetilde { H _ { \\chi } } ( u ) \\le \\frac { 1 } { k } \\Big ( 1 . 4 ^ k q + \\sum _ { d i = k , \\ , d \\neq 1 } 2 ^ i q ^ { d } \\Big ) \\le 1 0 \\frac { q ^ { k } } { k } , \\end{align*}"}
-{"id": "2659.png", "formula": "\\begin{align*} g _ { j k } : & = { \\alpha _ { j - N , k - N } = } 4 \\pi \\eta _ { - } ^ { j + k - 2 N - 2 } \\frac { \\varrho _ { j } - \\varrho _ { k } } { ( 1 - \\varrho _ { j } - \\varrho _ { k } ) \\sqrt { 1 - 2 \\varrho _ { j } } \\sqrt { 1 - 2 \\varrho _ { k } } } . \\end{align*}"}
-{"id": "3962.png", "formula": "\\begin{align*} E _ { 2 , r } ^ k ( Z ) = \\underset { g \\in C _ { 2 , r } \\backslash \\Gamma _ 2 } \\sum \\ f ( g \\langle Z \\rangle ^ * ) j ( g , Z ) ^ { - k } \\end{align*}"}
-{"id": "7500.png", "formula": "\\begin{align*} P ^ i \\mapsto \\sum _ { j = 1 } ^ n A ^ i _ j P ^ j + B ^ i , \\end{align*}"}
-{"id": "2325.png", "formula": "\\begin{align*} H R ( z ) \\xi = \\xi + z R ( z ) \\xi . \\end{align*}"}
-{"id": "9866.png", "formula": "\\begin{align*} \\frac { | A | | S | } { 2 K } \\le \\sum _ { l \\in S } \\sum _ { x \\in A } A ( l x ) = \\sum _ { \\beta \\in B } \\sum _ { x \\in A } A ( \\alpha x + \\beta ) = \\sum _ { \\beta \\in B } r _ { A - \\alpha A } ( \\beta ) \\ , . \\end{align*}"}
-{"id": "4569.png", "formula": "\\begin{align*} \\hat { \\mu } ( \\omega ) = \\int e ^ { - 2 \\pi i ( \\omega , x ) } d \\mu ( x ) . \\end{align*}"}
-{"id": "136.png", "formula": "\\begin{align*} \\left ( \\mathcal { L } _ { X } \\pi \\right ) ( d f , d g ) = X \\left ( \\{ f , g \\} \\right ) - \\{ X \\left ( f \\right ) , g \\} - \\{ f , X \\left ( g \\right ) \\} , \\end{align*}"}
-{"id": "533.png", "formula": "\\begin{align*} \\langle \\partial _ { i j } y , \\partial _ k y \\rangle = \\frac { 1 } { 2 } \\big ( \\partial _ i G _ { k j } + \\partial _ j G _ { i k } - \\partial _ k G _ { i j } \\big ) \\end{align*}"}
-{"id": "2297.png", "formula": "\\begin{align*} R ( z ) = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 0 & R _ { \\Omega } ( z ) \\end{array} \\right ) + \\left ( \\begin{array} { c } 1 \\\\ - R _ { \\Omega } ( z ) | E ) \\end{array} \\right ) \\frac { 1 } { z - \\i \\pi \\sigma ( z ) } \\left ( 1 , ( E | R _ { \\Omega } ( z ) \\right ) . \\end{align*}"}
-{"id": "5719.png", "formula": "\\begin{align*} \\frac { 1 } { \\Gamma ( 2 - \\alpha ) } \\int _ z ^ x { ( x - t ) ^ { 1 - \\alpha } } ( t - z ) ^ { \\alpha - 2 } E _ { \\alpha , \\alpha - 1 } ( \\lambda ( t - z ) ^ { \\alpha } ) d t & = E _ { \\alpha , 1 } ( \\lambda ( x - z ) ^ { \\alpha } ) \\end{align*}"}
-{"id": "1586.png", "formula": "\\begin{align*} \\Phi _ { \\widetilde { F } } ( z ^ \\iota _ n , x ^ \\iota _ n ) & = \\Phi _ { \\widetilde { F } } \\circ \\widetilde { \\boldsymbol { F } } _ { n ^ 2 + k _ n , n ^ 2 + \\kappa _ 0 } ( z _ 0 , x _ 0 ) \\\\ & = \\Phi _ { \\widetilde { F } } ( z _ 0 , x _ 0 ) + k _ n - \\kappa _ 0 + o ( 1 ) \\\\ & = \\Phi _ F ( z , x ) + k _ n + o ( 1 ) , \\end{align*}"}
-{"id": "9752.png", "formula": "\\begin{align*} \\int _ { B _ R } q \\ , L _ a q = 0 \\quad R < 1 , \\end{align*}"}
-{"id": "7403.png", "formula": "\\begin{align*} l _ x ( \\gamma [ 0 , a ] ) + l _ x ( \\gamma [ a , 1 ] ) = l _ x ( \\gamma [ 0 , 1 ] ) , \\end{align*}"}
-{"id": "99.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} S _ k ( B ( 0 ) , B _ 1 ) = & \\sum _ { ( \\ell _ 1 , \\ldots , \\ell _ { k + 1 } ) } B ( 0 ) ^ { \\ell _ 1 } \\cdot B _ 1 \\cdot B ( 0 ) ^ { \\ell _ 2 } \\cdot B _ 1 \\cdots B ( 0 ) ^ { \\ell _ k } \\cdot B _ 1 \\cdot B ( 0 ) ^ { \\ell _ { k + 1 } } \\\\ & 0 \\le \\ell _ j \\le 2 N - k \\ , , \\sum _ { j = 1 } ^ { k + 1 } \\ell _ j = 2 N - k \\ , . \\end{aligned} \\right . \\end{align*}"}
-{"id": "5172.png", "formula": "\\begin{align*} C < \\lim \\inf _ { h \\rightarrow + \\infty } A ( h ) / B ( h ) = A > 0 , \\lim \\sup _ { h \\rightarrow + \\infty } B ( h ) / A ( h ) < B . \\end{align*}"}
-{"id": "9532.png", "formula": "\\begin{align*} \\sigma ( q ) : = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ( n + 1 ) / 2 } } { ( - q ) _ n } , \\end{align*}"}
-{"id": "5837.png", "formula": "\\begin{align*} | c ( x ) | = | x | _ F ^ { { \\rm r e } \\ , c } . \\end{align*}"}
-{"id": "4769.png", "formula": "\\begin{align*} ( \\tilde { u } _ R ( t ) , \\varphi ( t ) ) _ { \\mathcal { H } } - ( \\tilde { u } _ { R 0 } , \\varphi ( 0 ) ) _ { \\mathcal { H } } = \\int _ 0 ^ t \\Big [ & ( \\tilde { u } _ R , \\partial _ t \\varphi ) _ { \\mathcal { H } } + 2 \\nu a ( \\tilde { u } _ R , \\varphi ) + 2 \\nu a ( H , \\varphi ) \\\\ - b _ R ( \\tilde { u } _ R & , \\varphi , \\tilde { u } _ R ) - b ( H , \\varphi , \\tilde { u } _ R ) - b ( \\tilde { u } _ R , \\varphi , H ) \\Big ] d t . \\end{align*}"}
-{"id": "3082.png", "formula": "\\begin{align*} | L | & \\leq \\int _ { \\frac { 3 } { 2 \\beta } \\leq | x | \\leq \\frac { 2 } { \\beta } } \\frac { 1 } { | x | ^ { n - \\beta } } \\ , d x \\\\ & \\leq \\frac { 1 } { \\beta } [ ( \\frac { 3 } { 2 \\beta } ) ^ \\beta - ( \\frac { 2 } { \\beta } ) ^ \\beta ] . \\end{align*}"}
-{"id": "660.png", "formula": "\\begin{align*} k ^ { ( \\varphi ' _ j ) } _ s = k ^ { ( \\varphi ^ y ) } _ { \\delta ^ { ( \\psi ) } _ j ( s ) } = k ^ { ( \\varphi ) } _ { y ^ { - 1 } ( \\delta ^ { ( \\psi ) } _ j ( s ) ) } \\end{align*}"}
-{"id": "7385.png", "formula": "\\begin{align*} \\dot { x } = a + b x + x ^ 3 . \\end{align*}"}
-{"id": "2618.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta _ p u ( z ) = f ( z , u ( z ) , D u ( z ) ) & \\mbox { i n } \\ \\Omega , \\\\ \\frac { \\partial u } { \\partial n _ p } + \\beta ( z ) | u | ^ { p - 2 } u = 0 & \\mbox { o n } \\ \\partial \\Omega . \\end{array} \\right \\} \\end{align*}"}
-{"id": "9960.png", "formula": "\\begin{align*} \\begin{aligned} ( X \\in \\C , i \\in \\abs X , f \\colon Y \\rightarrow X ) & \\mapsto \\abs { f } ^ { - 1 } ( i ) \\\\ ( X , i , f g ) \\xrightarrow { g } ( X , i , f ) & \\mapsto \\abs { f g } ^ { - 1 } ( i ) \\xrightarrow { \\smash { \\abs { g } ^ { \\abs f } _ i } } \\abs { f } ^ { - 1 } ( i ) \\rlap { . } \\end{aligned} \\end{align*}"}
-{"id": "4772.png", "formula": "\\begin{align*} W = \\sum _ { i = 1 } ^ N ( \\alpha _ i \\beta _ i + \\alpha _ i p _ i + q _ i \\beta _ i ) + W _ 0 \\end{align*}"}
-{"id": "5228.png", "formula": "\\begin{align*} x _ { 0 } = \\frac { t _ { n } x _ { 0 } } { t _ { n } } \\geq \\frac { U ( V ( t _ { n } x _ { 0 } ) - ) } { U ( V ( t _ { n } ) + ) } \\leq \\frac { U ( V ( t _ { n } x _ { 0 } ) - 1 ) } { U ( V ( t _ { n } ) + 1 ) } . \\end{align*}"}
-{"id": "7535.png", "formula": "\\begin{align*} \\left \\{ \\begin{gathered} \\mathbb { U } ( f ) ( x , t ) = \\int _ 0 ^ t ( g _ { \\nu ( t - s ) } * \\mathbb { H } \\mathrm { d i v } \\ , \\ , f ) ( x , s ) d s , \\\\ \\mathbb { G } ( f ) ( x , t ) = \\int _ 0 ^ t ( g _ { \\nu ( t - s ) } * \\mathbb { H } \\nabla \\mathrm { d i v } \\ , \\ , f ) ( x , s ) d s . \\end{gathered} \\right . \\end{align*}"}
-{"id": "4342.png", "formula": "\\begin{align*} \\Phi ( b ) \\Phi ( a ) = \\omega ( a , b ) \\Phi ( a ) \\Phi ( b ) . \\end{align*}"}
-{"id": "6290.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\| \\hat { B } _ { \\hat { \\pi } } - Q \\| _ 2 ^ 2 | \\mathcal { E } \\right ] = O \\left ( \\hat { \\epsilon } _ k ^ { ( O ) } ( Q ) ^ 2 + r \\left ( \\frac { \\log k } { n } + \\frac { k ^ 2 } { n ^ 2 } \\right ) + \\nu ^ 2 \\right ) . \\end{align*}"}
-{"id": "6940.png", "formula": "\\begin{align*} \\begin{cases} \\rho \\geq 0 , \\ \\rho \\in C ( [ 0 , T ] ; W ^ { 1 , q } ) , \\ \\rho _ t \\in C ( [ 0 , T ] ; L ^ { q } ) , \\\\ \\nabla \\mathbf { u } \\in L ^ { \\infty } ( 0 , T ; H ^ 1 ) \\cap L ^ 2 ( 0 , T ; W ^ { 1 , q } ) , \\\\ \\sqrt { \\rho } \\mathbf { u } , \\ \\sqrt { \\rho } \\dot { \\mathbf { u } } \\in L ^ { \\infty } ( 0 , T ; L ^ 2 ) , \\\\ P \\geq 0 , \\ P \\in C ( [ 0 , T ] ; W ^ { 1 , q } ) , \\ P _ t \\in C ( [ 0 , T ] ; L ^ { q } ) , \\end{cases} \\end{align*}"}
-{"id": "5567.png", "formula": "\\begin{align*} \\begin{cases} i u _ t + u _ { x x } = \\alpha u v + \\beta u | u | ^ 2 , & ( x , t ) \\in \\R ^ + \\times ( 0 , T ) , \\\\ v _ t + v _ { x x x } + v v _ x = \\gamma ( | u | ^ 2 ) _ x , & ( x , t ) \\in \\R ^ + \\times ( 0 , T ) , \\\\ u ( x , 0 ) = u _ 0 ( x ) , \\ v ( x , 0 ) = v _ 0 ( x ) , & x \\in \\R ^ + , \\\\ u ( 0 , t ) = f ( t ) , \\ v ( 0 , t ) = g ( t ) , & t \\in ( 0 , T ) , \\end{cases} \\end{align*}"}
-{"id": "9845.png", "formula": "\\begin{align*} [ T _ i ] = \\frac { ( - t ) ^ { 4 - i } [ ( 1 - x ) ^ r _ { [ i ] } - x ^ { 1 - i } ( 1 - x ^ 2 ) ^ r _ { [ i ] } ] } { ( 1 - x ) ^ { r + 1 } } , \\end{align*}"}
-{"id": "154.png", "formula": "\\begin{align*} X = \\sum _ { i _ 1 , \\dots , i _ k = 1 } ^ { N } v ^ { i _ 1 \\dots i _ k } ( { \\bf x } ) \\dfrac { \\partial } { \\partial x ^ { i _ 1 } } \\wedge \\dots \\wedge \\dfrac { \\partial } { \\partial x ^ { i _ k } } \\in \\mathcal { X } ^ { k } ( M ) \\end{align*}"}
-{"id": "406.png", "formula": "\\begin{align*} \\int _ { M } K _ g { \\rm d v } _ g = 4 \\pi \\chi ( M ) \\end{align*}"}
-{"id": "5113.png", "formula": "\\begin{align*} & ( \\widehat { S } _ { f , \\tau } ^ { - 1 } \\widehat { \\theta } _ \\tau + V ( I _ { \\ell ^ p ( \\mathbb { J } ) } - \\theta _ f \\widehat { S } _ { f , \\tau } ^ { - 1 } \\widehat { \\theta } _ \\tau ) ) ( \\theta _ f \\widehat { S } _ { f , \\tau } ^ { - 1 } + ( I _ { \\ell ^ p ( \\mathbb { J } ) } - \\theta _ f \\widehat { S } _ { f , \\tau } ^ { - 1 } \\widehat { \\theta } _ \\tau ) U ) \\\\ & = \\widehat { S } _ { f , \\tau } ^ { - 1 } + V U - V \\theta _ f \\widehat { S } _ { f , \\tau } ^ { - 1 } \\widehat { \\theta } _ \\tau U . \\end{align*}"}
-{"id": "6903.png", "formula": "\\begin{align*} f = \\big ( \\mu ^ 2 - \\theta ^ 2 \\big ) \\big ( ( s _ 2 c _ 1 - c _ 2 s _ 1 ) q _ 1 + ( c _ 1 c _ 2 + s _ 1 s _ 2 ) q _ 2 \\big ) = 0 \\end{align*}"}
-{"id": "3340.png", "formula": "\\begin{align*} \\sum _ { d _ 1 d _ 2 = n , \\ , d _ 1 \\neq n } 2 ^ { d _ 1 } x ^ { d _ 2 } \\le 7 x ^ { n } . \\end{align*}"}
-{"id": "5326.png", "formula": "\\begin{align*} Y = \\frac 1 2 \\left ( \\left [ \\begin{array} { c c } 1 & 0 \\\\ 0 & - 1 \\end{array} \\right ] - i \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & 0 \\end{array} \\right ] \\right ) = \\frac 1 2 \\left ( A - i B \\right ) . \\end{align*}"}
-{"id": "3315.png", "formula": "\\begin{align*} L ( u , \\chi ) = \\prod _ { i = 1 } ^ { \\deg L ( u , \\chi ) } ( 1 - \\gamma _ i ( \\chi ) u ) , \\end{align*}"}
-{"id": "8306.png", "formula": "\\begin{align*} ( I - \\mathcal { H } ) b = - [ D _ t \\zeta , \\mathcal { H } ] \\frac { \\bar { \\zeta } _ { \\alpha } - 1 } { \\zeta _ { \\alpha } } - \\frac { i } { \\pi } \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j } { \\zeta ( \\alpha , t ) - z _ j ( t ) } . \\end{align*}"}
-{"id": "6209.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } | \\hat { g } ( s ) | ^ 2 d s = \\int _ { - \\infty } ^ { \\infty } | g ( x ) | ^ 2 d x . \\end{align*}"}
-{"id": "765.png", "formula": "\\begin{align*} \\sigma \\rho _ D \\sigma ^ { - 1 } = \\rho _ { \\sigma ( D ) } , \\end{align*}"}
-{"id": "7003.png", "formula": "\\begin{align*} \\textstyle { 2 \\cos ( \\frac { 2 \\pi j } { n } ) } \\eta _ a = ( \\xi ^ j + \\xi ^ { - j } ) \\eta _ a = \\eta _ { a - 1 } + \\eta _ { a + 1 } . \\end{align*}"}
-{"id": "6010.png", "formula": "\\begin{align*} \\sup _ { Q \\in \\P _ 1 ( \\C ) } ( F ( Q ) - \\tilde { \\alpha } ^ g ( Q ) ) = \\sup _ { Q \\in \\Q \\cap \\P _ 1 ( \\C ) } ( F ( Q ) - \\alpha ^ g ( Q ) ) \\end{align*}"}
-{"id": "9731.png", "formula": "\\begin{align*} \\kappa _ \\ast : = \\kappa _ 0 , L _ \\ast : = L ( p _ { \\ast } ) , \\quad m _ \\ast : = m _ { 0 } , \\end{align*}"}
-{"id": "7187.png", "formula": "\\begin{gather*} \\max \\left \\{ \\left \\Vert y ( t , \\eta ) \\right \\Vert , \\left \\Vert y _ { \\eta } ^ { \\prime } ( t , \\eta ) a \\right \\Vert \\right \\} \\le R \\mathrm { e } ^ { - \\alpha t } \\quad \\forall ( t , \\eta ) \\in \\mathbb { R } _ { + } \\times \\mathbb { L } _ { \\xi } ^ { - } , \\quad \\forall a \\in \\mathbb { L } _ { \\xi } ^ { - } : \\left \\Vert a \\right \\Vert = r . \\end{gather*}"}
-{"id": "7719.png", "formula": "\\begin{align*} k _ { t } ( x , y ) = \\frac { 1 } { t ^ { n + 1 } } \\left . { \\frac { \\partial \\phi } { \\partial x _ { k } } } \\right | _ { \\frac { x - y } { t } } \\bigl ( f ( x ) - f ( y ) \\bigr ) + \\frac { 1 } { t ^ { n } } \\phi \\left ( \\frac { x - y } { t } \\right ) \\left . { \\frac { \\partial f } { \\partial x _ { k } } } \\right | _ { y } . \\end{align*}"}
-{"id": "8526.png", "formula": "\\begin{align*} ( I - \\mathcal { H } ) A = & 1 + i [ D _ t \\zeta , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } \\mathfrak { F } } { \\zeta _ { \\alpha } } + i [ D _ t ^ 2 \\zeta , \\mathcal { H } ] \\frac { \\bar { \\zeta } _ { \\alpha } - 1 } { \\zeta _ { \\alpha } } - ( I - \\mathcal { H } ) \\frac { 1 } { 2 \\pi } \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j ( D _ t \\zeta ( \\alpha , t ) - \\dot { z } _ j ( t ) ) } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } . \\end{align*}"}
-{"id": "4490.png", "formula": "\\begin{align*} \\underline { u } + \\nabla _ { 1 1 } \\underline { u } - \\nabla _ n ( u - \\underline { u } ) \\Pi _ { 1 1 } = u + \\nabla _ { 1 1 } u \\geq M . \\end{align*}"}
-{"id": "9393.png", "formula": "\\begin{align*} \\left | \\lambda \\int _ { \\Omega } F ( x , u ) d x \\right | & = \\lambda \\int _ { \\Omega } F ( x , u ) d x \\leq \\lambda \\frac { 1 } { c _ { 2 } } \\int _ { \\Omega } f ( x , u ) u d x \\\\ & = \\frac { 1 } { c _ { 2 } } \\left ( \\Phi ( u ( x ) - u ( y ) ) ( u ( x ) - u ( y ) ) K ( x , y ) d x d y - \\int _ \\Omega | u | ^ { p ^ { \\ast } _ { s } } d x \\right ) \\\\ & \\leq \\frac { \\Lambda ^ { 2 } } { c _ { 2 } } [ u ] ^ { p } _ { s , p } - \\frac { 1 } { c _ { 2 } } \\int _ { \\Omega } | u | ^ { p _ { s } ^ { \\ast } } \\end{align*}"}
-{"id": "566.png", "formula": "\\begin{align*} U = \\bigcup _ { j = 0 } ^ { p ^ n - 1 } \\xi ^ j ( 1 + p x _ j ) R _ n ^ * \\mbox { a n d } V = \\bigcup _ { j = 0 } ^ { p ^ n - 1 } ( 1 + p x _ j ) R _ n ^ * . \\end{align*}"}
-{"id": "2928.png", "formula": "\\begin{align*} f _ Z ( z ) = \\begin{vmatrix} 1 & 1 \\ldots 1 \\\\ ( k _ { z _ j } ( z ) ) _ { j = 1 } ^ t & ( k _ { z _ j } ( z _ k ) ) _ { j , k = 1 } ^ t \\end{vmatrix} . \\end{align*}"}
-{"id": "7537.png", "formula": "\\begin{align*} \\begin{gathered} g ( a , t ) = ( \\nabla u ) ( X ( a , t ) , t ) = \\mathbb { L } _ \\nu ( \\nabla u _ 0 ) \\circ X ( a , t ) \\\\ + \\mathbb { G } \\left ( \\tau \\circ X ^ { - 1 } \\right ) \\circ X ( a , t ) - \\mathbb { U } \\left ( \\nabla _ x \\left ( ( v \\otimes v ) \\circ X ^ { - 1 } \\right ) \\right ) \\circ X ( a , t ) . \\end{gathered} \\end{align*}"}
-{"id": "350.png", "formula": "\\begin{align*} | I m ( \\partial ' _ k ) _ * | = \\frac { \\frac { 1 } { 2 } n ( n ^ 2 - 1 ) } { ( \\frac { 1 } { 2 } n ( n ^ 2 - 1 ) , k ) } \\cdot \\frac { n } { ( n , k ) } , \\end{align*}"}
-{"id": "9086.png", "formula": "\\begin{align*} 0 \\le \\varphi \\le 1 \\mathbb { R } ^ N , \\varphi ( x ) = 1 | x | \\le 1 \\varphi ( x ) = 0 | x | \\ge 2 . \\end{align*}"}
-{"id": "2634.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\mathbb { P } \\Big ( 2 ^ { - \\frac { 4 } { 3 } } N ^ { - \\frac { 1 } { 3 } } \\big ( \\lambda _ { \\mathrm { m a x } } - 4 N \\big ) \\leq x \\Big ) = F ^ { ( \\mathrm { s o f t } ) } ( \\kappa , \\pi ; x ) . \\end{align*}"}
-{"id": "5231.png", "formula": "\\begin{align*} S ( u ) = c ( 1 + a ( u ) ) \\exp ( \\int _ { u } ^ { 1 } \\frac { b ( t ) } { t } d t ) . \\end{align*}"}
-{"id": "1288.png", "formula": "\\begin{align*} \\left | \\int _ { | \\nu | \\le | \\eta | / 2 } \\right | \\lesssim \\int _ { | \\nu | \\le | \\eta | / 2 } | 2 \\eta - \\nu | | \\eta | ^ { - 2 k - 1 } d \\nu = O ( | \\eta | ^ { 1 - 2 k } ) . \\end{align*}"}
-{"id": "9036.png", "formula": "\\begin{align*} \\overline { a _ { \\xi } } = a _ { - \\xi } . \\end{align*}"}
-{"id": "6464.png", "formula": "\\begin{align*} \\mathbb { E } ( Y _ { e _ 1 } \\cdots Y _ { e _ { j - 1 } } ) = \\alpha _ { j - 1 } p _ { s ( e _ j ) } \\mathbb { E } ( Y _ { e _ { j + 1 } } \\cdots Y _ { e _ { n } } ) = \\beta _ { j + 1 } p _ { t ( e _ j ) } \\end{align*}"}
-{"id": "8394.png", "formula": "\\begin{align*} \\mathcal { E } ( t ) : = \\max _ { \\tau \\in [ 0 , t ] } E ( \\tau ) . \\end{align*}"}
-{"id": "5232.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { x } t ^ { - 1 } g ( t ) d t = \\int _ { 1 } ^ { x } t ^ { - 1 } U ( t ) d t - \\int _ { 1 } ^ { x } \\left ( \\int _ { 1 } ^ { t } U ( s ) d s \\right ) t ^ { - 2 } d t . \\end{align*}"}
-{"id": "8818.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathbb R ^ N } \\sum _ { j = 1 } ^ J f ( A _ j x - b _ j ) + \\sum _ { i = 1 } ^ I g ( x ^ { ( i ) } ) \\end{align*}"}
-{"id": "1500.png", "formula": "\\begin{align*} u _ { t } + \\Delta u + \\varepsilon \\bar { H } ( t , x , m , D u ) = 0 , \\end{align*}"}
-{"id": "8279.png", "formula": "\\begin{align*} ( I - \\mathfrak { H } _ 0 ) \\bar { v } _ 0 = - \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i } { \\pi } \\frac { 1 } { \\xi _ 0 ( \\alpha ) + \\alpha - z _ j ( 0 ) } , \\end{align*}"}
-{"id": "9145.png", "formula": "\\begin{align*} \\rho _ { 1 } ^ { \\left ( N \\right ) ( + ) } ( \\mathbf { x } _ { 1 } ^ { ( + ) } ( t _ { 1 } ) , t _ { i } ) = \\rho ^ { \\left ( N \\right ) } ( \\mathbf { x } _ { 1 } ^ { ( + ) } ( t _ { i } ) , t _ { i } ) . \\end{align*}"}
-{"id": "2503.png", "formula": "\\begin{align*} a _ 0 = 1 , \\ , a _ 1 = \\cdots = a _ { s - 1 } = r + 2 , \\ , a _ s = \\cdots = a _ r = \\cdots = r + 1 . \\end{align*}"}
-{"id": "2296.png", "formula": "\\begin{align*} \\eta _ t ( b ^ + ) & = \\exp { ( \\i H _ 0 ' t ) } b ^ + \\exp { ( - \\i H _ 0 ' t ) } \\\\ \\eta _ t ( a _ \\lambda ) & = \\exp { ( \\i H _ 0 ' t ) } a _ \\lambda \\exp { ( - \\i H _ 0 ' t ) } \\end{align*}"}
-{"id": "9424.png", "formula": "\\begin{align*} d ( ( \\log _ { e } | F ( z ) | ^ 2 ) \\overline { g ( z ) } d \\overline { z } ) = ( 4 8 \\pi ) f _ { \\theta } ( z ) \\overline { g ( z ) } d x d y . \\end{align*}"}
-{"id": "3335.png", "formula": "\\begin{align*} \\mathrm { E } _ { \\pi \\in S _ n } f ( \\pi ) = [ u ^ n ] ( 1 - u ) ^ { - z _ 1 } ( 1 - u ^ m ) ^ { ( - z _ 2 + z _ 1 ) / m } . \\end{align*}"}
-{"id": "4399.png", "formula": "\\begin{align*} \\Psi _ 2 ^ * \\frac { \\omega _ o ^ 2 } { 2 } = ( 1 + 4 | x y | ^ 2 + | y | ^ 4 ) d V ( x , y ) , \\end{align*}"}
-{"id": "5507.png", "formula": "\\begin{align*} E ^ { \\Pi ^ { D _ T } } \\big [ \\max _ { k , j } \\sqrt T | \\langle b _ j - b _ { 0 , j } , a _ \\lambda \\Phi _ { \\lambda , k } \\rangle _ { L ^ 2 } | | X ^ T \\big ] = O _ { P _ { b _ 0 } } ( \\sqrt \\lambda ) , \\end{align*}"}
-{"id": "4700.png", "formula": "\\begin{align*} u ( x , 0 ) = \\sin ( x ) , \\ ; \\ ; \\ ; x \\in [ 0 , 2 \\pi ] . \\end{align*}"}
-{"id": "6815.png", "formula": "\\begin{align*} R _ 1 = \\begin{pmatrix} \\cos ( \\mu ) & - \\sin ( \\mu ) \\\\ \\sin ( \\mu ) & \\cos ( \\mu ) \\end{pmatrix} \\quad { \\rm a n d } R _ 2 = \\begin{pmatrix} \\cos ( \\theta ) & \\sin ( \\theta ) \\\\ \\sin ( \\theta ) & - \\cos ( \\theta ) \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "4868.png", "formula": "\\begin{align*} \\widetilde { \\Delta } _ k ( \\chi , x ) = \\{ \\tau \\in \\Delta _ k ( \\chi ) \\mid \\tau \\subset \\overline { { \\rm S t } ( | \\sigma _ x | ) } \\} , \\end{align*}"}
-{"id": "9071.png", "formula": "\\begin{align*} b _ k ( x ) = \\frac { 1 } { \\mu _ { f } ( I _ k ) ^ { 1 / 2 } } \\sum _ { \\xi \\in \\mathcal { E } ^ { ( k ) } } a _ { \\xi } e ( \\langle \\xi , x \\rangle ) \\end{align*}"}
-{"id": "6204.png", "formula": "\\begin{align*} f ( x ) = \\frac { e ^ { x } } { ( 1 + e ^ x ) ^ 2 } = \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n + 1 } n e ^ { - n | x | } . \\end{align*}"}
-{"id": "4893.png", "formula": "\\begin{align*} f ( a ) = f _ U ( a _ U ) + f _ V ( a _ V ) \\end{align*}"}
-{"id": "7662.png", "formula": "\\begin{align*} \\frac { \\partial b ^ i ( \\boldsymbol { \\ell } ) } { \\partial { \\ell } ^ j } \\Big | _ { \\boldsymbol { \\ell } = 0 } \\ = \\ 2 \\Theta _ { \\rm s y m } ^ { i j } , \\end{align*}"}
-{"id": "5549.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\| ( 1 - H _ { \\delta , \\varepsilon } ) \\mathcal { E } ^ { \\varepsilon , * } f \\| _ { H ^ { 1 / 2 + \\kappa , \\varepsilon } ( \\rho ^ { - 2 } ) } = 0 \\end{align*}"}
-{"id": "7456.png", "formula": "\\begin{align*} \\displaystyle \\int _ X z ( \\chi ) \\sum _ { \\Pi \\in p ^ { - 1 } ( \\chi ) } L _ \\Pi ( f ) \\mu _ \\chi ( \\Pi ) \\overline { \\mu } ( \\chi ) = 0 \\end{align*}"}
-{"id": "1912.png", "formula": "\\begin{align*} \\frac { 2 } { \\kappa _ j } \\left [ \\cos ( \\kappa _ j x ) + \\left ( \\frac { \\xi } { \\kappa _ j } \\right ) \\sin ( \\kappa _ j | x | ) \\right ] \\sin ( \\kappa _ j t ) = 2 \\left ( \\frac { \\xi L } { 2 } \\right ) \\left [ \\frac { t } { Z _ j ^ 2 } + 2 \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { k _ n } \\frac { \\cos ( k _ n x ) \\sin ( k _ n t ) } { Z _ j ^ 2 - ( \\pi n ) ^ 2 } \\right ] \\end{align*}"}
-{"id": "8039.png", "formula": "\\begin{align*} y ^ r x y ^ k y ^ { - r } = x y ^ { k + ( 2 q + 1 ) ( n - 2 ) } = x y ^ { k - 4 q + n - 2 } . \\end{align*}"}
-{"id": "405.png", "formula": "\\begin{align*} \\langle F \\rangle _ { { \\rm M L } , \\hat g } = \\langle F ( \\cdot - \\tfrac { Q } { 2 } \\omega ) \\rangle _ { { \\rm M L } , g } \\times \\exp \\big ( \\tfrac { 1 + 6 Q ^ 2 } { 9 6 \\pi } S ^ { { \\rm c l } , 0 } _ { { \\rm L } } ( \\hat g , g ) + \\beta S ^ { { \\rm c l } } _ { \\rm M } ( \\hat g , g ) \\big ) \\end{align*}"}
-{"id": "7843.png", "formula": "\\begin{align*} \\frac { \\pi ^ { d / 2 } } { \\Gamma ( d / 2 + 1 ) } = \\frac { 2 ^ { 2 m + 1 } \\pi ^ m m ! } { ( 2 m + 1 ) ! } \\le \\frac { 2 ^ { d } \\pi ^ { ( d - 1 ) / 2 } m ^ { m + 1 / 2 } e ^ { d - m } e ^ { { 1 } / { 1 2 } } } { d ^ { d + 1 / 2 } } \\le \\bigg ( \\frac { e ^ { 1 / 6 } } { 3 \\pi } \\bigg ) ^ { 1 / 2 } \\frac { ( 2 \\pi e ) ^ { d / 2 } } { d ^ { d / 2 } } \\le \\frac { ( 2 \\pi e ) ^ { d / 2 } } { 2 d ^ { d / 2 } } . \\end{align*}"}
-{"id": "1763.png", "formula": "\\begin{align*} \\gamma _ { H } = \\begin{cases} 1 , & H \\leq \\frac { 1 } { 2 } , \\\\ \\frac { 3 } { 2 } , & H > \\frac { 1 } { 2 } , \\end{cases} \\end{align*}"}
-{"id": "1445.png", "formula": "\\begin{align*} I I I _ { A } = - \\varepsilon \\int _ { 0 } ^ { t } \\int _ { \\mathbb { T } ^ { d } } \\left ( \\partial ^ { \\alpha } \\mu ^ { n + 1 } \\right ) ( \\mu ^ { n } + \\bar { m } ) \\sum _ { i = 1 } ^ { d } \\left [ \\left ( \\Theta _ { q p _ { i } } ( \\tau , x , \\mu ^ { n } , D w ^ { n } ) \\right ) \\left ( \\partial ^ { \\alpha } \\partial _ { x _ { i } } \\mu ^ { n } \\right ) \\right ] \\ d x d \\tau , \\end{align*}"}
-{"id": "7421.png", "formula": "\\begin{align*} \\displaystyle Z ( 1 - s , \\widetilde { W } , \\widehat { \\phi } ^ { \\psi ' _ \\lambda } ) = \\lvert \\lambda \\rvert _ F ^ { n ( s - 1 / 2 ) } \\omega _ { \\pi } ( \\lambda ) Z ( 1 - s , \\widetilde { W } , \\widehat { \\phi } ) . \\end{align*}"}
-{"id": "9276.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l l } \\mathcal { F } ^ \\epsilon [ u ] \\ ! \\ ! & \\ ! \\ ! = B ( \\cdot , u , D u ) , \\ ! \\ ! & \\ ! \\ ! { \\rm i n } \\ \\Omega , \\\\ \\mathcal { G } [ u ] \\ ! \\ ! & \\ ! \\ ! = 0 , \\ ! \\ ! & \\ ! \\ ! { \\rm o n } \\ \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "1309.png", "formula": "\\begin{align*} O _ d & = \\bigl \\{ \\bigl ( ( x , d ) , ( y , d ) \\bigr ) : ( x , y ) \\in V _ { n _ d } \\bigr \\} \\cup \\bigl \\{ \\bigl ( - ( x , d ) , - ( y , d ) \\bigr ) : ( x , y ) \\in V _ { n _ d } \\bigr \\} & & \\\\ O _ d & = \\bigl \\{ \\bigl ( ( x , d ) , ( y , d ) \\bigr ) : ( x , y ) \\in Z \\times Z \\bigr \\} \\cup \\bigl \\{ \\bigl ( - ( x , d ) , - ( y , d ) \\bigr ) : ( x , y ) \\in Z \\times Z \\bigr \\} & & \\end{align*}"}
-{"id": "8880.png", "formula": "\\begin{align*} & \\frac { 3 9 9 6 p ^ 5 - 2 8 4 p ^ 6 - 1 9 9 5 6 p ^ 4 + 3 7 3 2 9 p ^ 3 - 3 2 9 1 p ^ 2 + 9 9 p - 1 + ( 4 0 0 p ^ 3 - 4 4 p ^ 4 - 1 1 0 5 p ^ 2 + 6 6 p - 1 ) R } { 2 p ^ 4 } \\\\ & \\ge 2 2 1 2 0 . 5 - 1 5 7 6 \\sqrt { 1 9 7 } = 0 . 2 8 5 8 9 6 \\dot > 0 \\end{align*}"}
-{"id": "8518.png", "formula": "\\begin{align*} \\norm { D _ t \\tilde { \\theta } - 2 ( \\bar { \\mathfrak { F } } - q ) } _ { H ^ { s + 1 / 2 } } \\leq & \\norm { - ( \\mathcal { H } + \\bar { \\mathcal { H } } ) D _ t \\zeta - [ D _ t \\zeta , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } ( \\zeta - \\bar { \\zeta } ) } { \\zeta _ { \\alpha } } } _ { H ^ { s + 1 / 2 } } \\\\ \\leq & C \\epsilon ^ 2 . \\end{align*}"}
-{"id": "4477.png", "formula": "\\begin{align*} H = \\textrm { t r a c e } ( \\nabla ^ 2 u + u \\sigma ) = \\Delta u + n u \\end{align*}"}
-{"id": "398.png", "formula": "\\begin{align*} \\langle F \\rangle _ { { \\rm L } , g } = \\int F ( \\varphi ) e ^ { - \\mathcal { S } _ { \\rm L } ( \\varphi , g ) } \\mathcal { D } \\varphi \\end{align*}"}
-{"id": "5660.png", "formula": "\\begin{align*} { \\rm I } _ { 2 , 1 } & \\leq c \\left \\{ \\begin{array} { l l } \\tau ^ { p \\eta } , & \\eta < 1 , \\\\ \\tau ^ { p } \\ell _ n , & \\eta = 1 , \\\\ \\tau ^ { p } t _ n ^ { p ( \\eta - 1 ) } , & \\eta > 1 , \\end{array} \\right . \\end{align*}"}
-{"id": "7863.png", "formula": "\\begin{align*} \\sum _ { \\substack { i = 1 \\\\ | x _ i | \\ge 2 } } ^ d x _ i ^ 2 \\le m \\end{align*}"}
-{"id": "9735.png", "formula": "\\begin{align*} I : = \\int _ { B _ 1 } v \\ , L _ a v \\end{align*}"}
-{"id": "3297.png", "formula": "\\begin{align*} \\mathcal { M } _ { n ; M } : = \\{ f \\in \\mathcal { M } _ { n } : \\gcd ( f , M ) = 1 \\} . \\end{align*}"}
-{"id": "409.png", "formula": "\\begin{align*} \\phi = - \\tfrac { 2 } { V _ { \\hat g } } \\int G _ { g } ( \\cdot , y ) { \\rm v } _ { \\hat g } ( \\dd y ) . \\end{align*}"}
-{"id": "6483.png", "formula": "\\begin{align*} t _ { B } = \\frac { 1 + 2 a b - a ^ { 2 } + 2 ( a ^ { 2 } + b ^ { 2 } - 1 ) } { b ^ { 2 } } = \\frac { b ^ { 2 } } { b ^ { 2 } } = 1 . \\end{align*}"}
-{"id": "5822.png", "formula": "\\begin{align*} \\ln ( \\phi _ { ( \\bar X _ n , s _ n ^ 2 ) } ( t _ 1 , t _ 2 ) ) = \\ln \\left ( \\iiint e ^ { i t _ 1 \\bar x _ n + i t _ 2 s _ n ^ 2 } \\prod _ { i = 1 } ^ n f ( x _ i ) d x _ i \\right ) . \\end{align*}"}
-{"id": "6375.png", "formula": "\\begin{align*} u _ s ( t , x ) : = \\alpha _ s \\varphi ( t , x ) + ( 1 - \\alpha _ s ) \\rho - C s t - C s , \\end{align*}"}
-{"id": "604.png", "formula": "\\begin{align*} A _ n : = \\bigg \\{ \\max _ { \\omega \\in \\Omega _ n } \\bigg | \\int _ { \\mathbb { T } ^ d } \\phi _ { \\omega } ( x ) \\ , d W ( x ) \\bigg | \\leq \\frac { \\sqrt { n } } { \\sigma } \\ , \\gamma _ n \\bigg \\} . \\end{align*}"}
-{"id": "8010.png", "formula": "\\begin{align*} u _ M ( t , 0 ) & = ( 4 \\pi ( t + 1 ) ) ^ { - \\frac { d - 1 } { 2 } } ( 4 \\pi t ) ^ { - \\frac { 1 } { 2 } } \\int _ M ^ { M + 1 } e ^ { - \\frac { | y _ 1 | ^ 2 } { 4 t } } \\ , \\mathrm d y _ 1 \\\\ & \\leq ( 4 \\pi ( t + 1 ) ) ^ { - \\frac { d - 1 } { 2 } } ( 4 \\pi t ) ^ { - \\frac { 1 } { 2 } } e ^ { - \\frac { M ^ 2 } { 4 t } } . \\end{align*}"}
-{"id": "5290.png", "formula": "\\begin{align*} A _ 5 ^ { ( 1 ) } : & \\left \\{ \\begin{array} { r c l } \\bar { q } _ 1 & = & - q _ 1 - p _ 2 + a q _ 2 ^ { - 1 } + b _ 1 \\\\ \\bar { p } _ 1 & = & q _ 2 \\\\ \\bar { q } _ 2 & = & - q _ 2 - p _ 1 + a q _ 1 ^ { - 1 } + b _ 2 \\\\ \\bar { p } _ 2 & = & q _ 1 \\\\ \\end{array} \\right . , \\end{align*}"}
-{"id": "6484.png", "formula": "\\begin{align*} q _ v = p _ { v } \\cdot \\bigvee _ { f \\in E _ 1 } u _ f u _ f ^ * , \\end{align*}"}
-{"id": "3008.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty n p _ m ' ( n ) x ^ n & = \\Big ( \\sum _ { n = 0 } ^ \\infty \\sigma _ m ' ( n ) x ^ n \\Big ) \\Big ( \\sum _ { n = 0 } ^ \\infty p _ m ' ( n ) x ^ n \\Big ) . \\end{align*}"}
-{"id": "2640.png", "formula": "\\begin{align*} \\int _ { \\Omega ^ { m } } \\mathrm { P f } \\begin{bmatrix} \\epsilon ( x _ { i } , x _ { j } ) \\end{bmatrix} _ { i , j = 1 } ^ { m } \\det \\left [ \\phi _ { i } ( x _ { j } ) \\right ] _ { i , j = 1 } ^ { m } \\prod _ { i = 1 } ^ { m } d \\nu ( x _ { i } ) = \\frac { m ! } { 2 ^ { \\frac { m } { 2 } } } \\mathrm { P f } \\begin{bmatrix} \\epsilon _ { i , j } \\end{bmatrix} _ { i , j = 1 } ^ { m } . \\end{align*}"}
-{"id": "2.png", "formula": "\\begin{align*} | Y ^ i _ t | \\leq K _ y : = \\frac { K _ f } { \\rho } \\ \\ \\ \\ | Z _ t ^ i | \\leq K _ z : = \\frac { C _ v } { C _ { \\eta } - C _ v } . \\end{align*}"}
-{"id": "2076.png", "formula": "\\begin{align*} D { \\lambda _ { n } } { J _ n } ' ( { \\lambda _ { n } } \\rho _ c ) = - k _ { f } J _ n ( { \\lambda _ { n } } \\rho _ c ) . \\end{align*}"}
-{"id": "4801.png", "formula": "\\begin{align*} \\psi ^ { m _ j } _ { j - 1 / 2 } & : = \\frac { 1 } { \\sqrt { 2 j } } \\left ( \\begin{array} { c } \\sqrt { j + m _ j } \\ , Y ^ { m _ j - 1 / 2 } _ { j - 1 / 2 } \\\\ \\sqrt { j - m _ j } \\ , Y ^ { m _ j + 1 / 2 } _ { j - 1 / 2 } \\\\ \\end{array} \\right ) , \\\\ \\psi ^ { m _ j } _ { j + 1 / 2 } & : = \\frac { 1 } { \\sqrt { 2 j + 2 } } \\left ( \\begin{array} { c } \\sqrt { j + 1 - m _ j } \\ , Y ^ { m _ j - 1 / 2 } _ { j + 1 / 2 } \\\\ - \\sqrt { j + 1 + m _ j } \\ , Y ^ { m _ j + 1 / 2 } _ { j + 1 / 2 } \\\\ \\end{array} \\right ) ; \\end{align*}"}
-{"id": "247.png", "formula": "\\begin{align*} \\Delta _ { \\mathbb { H } } u = V u , \\ \\ B _ { r _ 0 } \\end{align*}"}
-{"id": "6376.png", "formula": "\\begin{align*} \\beta _ s : = \\frac { ( 1 - A s ) s } { ( 1 - \\alpha _ s ) } \\geq \\varepsilon _ 1 > 0 \\end{align*}"}
-{"id": "3849.png", "formula": "\\begin{align*} K \\dfrac { \\varepsilon ^ 2 h _ { 0 0 } ^ { m + 1 } } { ( 2 W _ 0 ) ^ { 2 m } } { h ^ i } _ l = ^ { h } { { { R _ 0 } ^ i } _ { 0 l } } + 2 \\Phi ^ i _ { \\parallel l } - \\Phi ^ i _ { l \\parallel 0 } + 2 \\Phi ^ r { { \\Phi _ r } ^ i } _ l - \\Phi ^ r _ l \\Phi ^ i _ r . \\end{align*}"}
-{"id": "527.png", "formula": "\\begin{align*} & \\mathcal { Q } _ 2 ( x ' , F _ { 2 \\times 2 } ) = \\min \\left \\{ \\mathcal { Q } _ 3 \\big ( \\bar A ( x ' ) ^ { - 1 } \\tilde F \\bar A ( x ' ) ^ { - 1 } \\big ) ; ~ \\tilde F \\in \\mathbb { R } ^ { 3 \\times 3 } \\mbox { w i t h } \\tilde F _ { 2 \\times 2 } = F _ { 2 \\times 2 } \\right \\} , \\\\ & \\mathcal { Q } _ 3 ( F ) = D ^ 2 W ( I d _ 3 ) ( F , F ) . \\end{align*}"}
-{"id": "5512.png", "formula": "\\begin{align*} & E \\| Z _ 2 - P _ { ( \\kappa ) } ( Z _ 2 ) \\| _ { ( B ^ \\rho _ { 1 \\infty } ) ^ * } = E \\sup _ { \\| \\eta \\| _ { B ^ \\rho _ { 1 \\infty } } \\le 1 } \\left | Z _ 2 ( \\eta - P _ { V ^ { \\otimes d } _ \\kappa } [ \\eta ] ) \\right | \\\\ & \\lesssim \\sum _ { \\kappa < \\lambda } 2 ^ { - \\lambda ( \\rho - d / 2 ) } E \\max _ { k , j } \\left | Z _ 2 ( \\Phi _ { \\lambda , k , j } ) \\right | \\lesssim \\sum _ { \\kappa < \\lambda } 2 ^ { - \\lambda ( \\rho - d / 2 ) } \\sqrt \\lambda = _ { \\kappa \\to \\infty } o ( 1 ) . \\end{align*}"}
-{"id": "6251.png", "formula": "\\begin{align*} t : = \\frac { ( a + 1 ) \\tau ( p _ 1 ) } { a \\tau ( p _ 1 ) + \\tau ( q _ 1 ) } . \\end{align*}"}
-{"id": "8848.png", "formula": "\\begin{align*} h ( x , y ) = - 2 \\log \\left ( \\max \\left \\{ x , \\frac { y } 2 \\right \\} \\right ) \\ge 0 ; \\end{align*}"}
-{"id": "8657.png", "formula": "\\begin{align*} | \\varphi _ { 1 , \\varepsilon } - \\pi + \\bar { \\varphi } | \\cdot | \\varphi _ { 1 , \\varepsilon } - \\bar { \\varphi } | + | \\varphi _ { 2 , \\varepsilon } | = O ( 1 ) e ^ { - \\left ( \\sqrt { 1 - 4 c _ 0 ^ 2 } + O ( \\varepsilon ) \\right ) | x | } , \\end{align*}"}
-{"id": "5550.png", "formula": "\\begin{align*} [ ( 1 - H _ { \\delta , \\varepsilon } ) \\mathcal { E } ^ { \\varepsilon , * } f ] ( x ) = ( 1 - H _ { \\delta , \\varepsilon } ) ( x ) \\int _ { y \\in \\mathbb { R } ^ { 3 } : y _ { i } > \\delta } w ^ { \\varepsilon } ( x - y ) f ( y ) \\mathrm { d } y , \\end{align*}"}
-{"id": "5614.png", "formula": "\\begin{align*} \\frac { \\alpha } { 2 \\gamma } \\frac { d } { d t } \\int _ { 0 } ^ { + \\infty } v ^ 2 d x + \\frac { \\alpha } { \\gamma } \\int _ { 0 } ^ { + \\infty } v _ { x x x } v d x + \\frac { \\alpha } { \\gamma } \\int _ { 0 } ^ { + \\infty } v ^ 2 v _ x d x = \\alpha \\int _ { 0 } ^ { + \\infty } ( | u | ^ 2 ) _ x v d x \\end{align*}"}
-{"id": "7369.png", "formula": "\\begin{align*} d ( b _ s , b _ t ) = d ( b _ s , Y _ s ) + d ( Y _ s , Y _ y ) + d ( Y _ t , b _ s ) = s + d ( Y _ s , Y _ t ) + t > 2 t \\end{align*}"}
-{"id": "7975.png", "formula": "\\begin{align*} \\lvert O _ { 2 n } ^ { - } ( q ) \\rvert = \\frac { 1 } { \\gcd ( 4 , q ^ n + 1 ) } q ^ { n ( n - 1 ) } ( q ^ n + 1 ) \\prod _ { i = 1 } ^ { n - 1 } ( q ^ { 2 i } - 1 ) \\ , . \\end{align*}"}
-{"id": "9357.png", "formula": "\\begin{align*} \\log _ 2 \\lvert G : G ^ { 2 ^ k } \\rvert = \\log _ 2 \\lvert G _ k : G _ k ^ { \\ , 2 ^ k } \\rvert = 2 ^ k + 2 ^ { k - 1 } + k - 1 . \\end{align*}"}
-{"id": "3629.png", "formula": "\\begin{align*} a _ 1 \\cdot ( \\alpha _ { - 1 } \\cdot \\alpha _ 2 ) = - \\frac { t } { 2 ^ 8 } ( a _ 3 - a _ { - 3 } ) + \\frac { 1 } { 2 ^ 8 } ( a _ 3 - a _ { - 3 } ) \\cdot v _ { ( 1 , 2 ) } \\end{align*}"}
-{"id": "6692.png", "formula": "\\begin{align*} \\phi _ { n } ( z ) = z ^ { \\frac { 1 } { \\omega } ( \\frac { i } { 2 } + C ) } z ^ { n } , \\end{align*}"}
-{"id": "9253.png", "formula": "\\begin{align*} \\tilde w _ 1 : = w _ 1 - g ( 0 ) \\nu _ 1 - C _ 1 ( 1 + M _ 2 ( R ) ) | x | ^ 2 , \\end{align*}"}
-{"id": "3566.png", "formula": "\\begin{align*} \\begin{aligned} \\pi _ { 1 } ^ { - 1 } ( o ) & = \\left \\{ o \\times [ U , V , W _ { 1 } , 0 , 0 ] \\right \\} \\cup \\left \\{ o \\times [ U , V , 0 , W _ { 2 } , 0 ] \\right \\} \\cup \\left \\{ o \\times [ U , V , 0 , 0 , W _ { 3 } ] \\right \\} \\\\ & = : \\mathbb { P } _ { 1 } ^ { 2 } \\cup \\mathbb { P } _ { 2 } ^ { 2 } \\cup \\mathbb { P } _ { 3 } ^ { 2 } . \\end{aligned} \\end{align*}"}
-{"id": "7556.png", "formula": "\\begin{align*} M _ X = 1 + \\norm { X - \\mathrm { I d } } _ { L ^ \\infty ( 0 , T ; C ^ { 1 + \\alpha } ) } . \\end{align*}"}
-{"id": "8040.png", "formula": "\\begin{align*} x ( x y ^ k ) x = y ^ { k } x = x y ^ { k ( n - 1 ) } = x y ^ { k ( 4 a + 1 ) } = x y ^ { k + 4 a k } . \\end{align*}"}
-{"id": "4862.png", "formula": "\\begin{align*} \\displaystyle h ( x ) = \\sum _ { W \\in \\mathcal { W } } f _ W ( x + \\sqrt { - 1 } y ) , \\end{align*}"}
-{"id": "2549.png", "formula": "\\begin{align*} \\mathbf m ^ \\xi _ T ( d s , d w ) : = 2 \\alpha ( \\xi _ s ) d s \\cdot \\mathbb N _ { \\xi _ s } ( d w ) + d s \\cdot \\int _ { ( 0 , \\infty ) } y \\mathbf P _ { y \\delta _ { \\xi _ s } } ( X \\in d w ) \\pi ( \\xi _ s , d y ) ; \\end{align*}"}
-{"id": "7574.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\Delta g _ { \\nu ( t - s ) } * \\Delta _ 2 \\tau ( s , t ) d s = \\int _ 0 ^ t \\int _ { \\mathbb { R } ^ d } K ( x , z , t , s ) \\tau ( z , t ) d z d s , \\end{align*}"}
-{"id": "9034.png", "formula": "\\begin{align*} M _ f ( x , r ) : = \\frac { 1 } { V o l ( B ( x , r ) ) } \\int _ { B ( x , r ) } | f _ { E } | ^ 2 \\end{align*}"}
-{"id": "1858.png", "formula": "\\begin{align*} \\Phi ^ { \\rm o d d } ( n , x , t ) = ( 2 k _ n ) ^ { - 1 / 2 } \\ , u ^ { \\rm o d d } ( n , x ) \\ , e ^ { - i k _ n t } \\qquad \\mbox { w i t h } k _ n = \\frac { 2 \\pi n } { L } \\mbox { a n d } n = 1 , 2 , 3 , \\dots . \\end{align*}"}
-{"id": "2580.png", "formula": "\\begin{align*} & \\big \\langle \\psi _ 0 \\big ( \\cdot , ( 1 + \\epsilon _ { R ( u ) } ) u \\phi \\big ) , \\phi ^ * \\big \\rangle _ m \\\\ & = \\big \\langle \\kappa ( x ) \\big ( 1 + \\epsilon _ { R ( u ) } ( x ) \\big ) ^ { \\gamma ( x ) } u ^ { \\gamma ( x ) } \\phi ( x ) ^ { \\gamma ( x ) } , \\phi ^ * ( x ) \\big \\rangle _ { m ( d x ) } \\\\ & \\stackrel [ u \\to 0 ] { } { \\sim } \\langle u ^ { \\gamma ( x ) } , \\kappa ( x ) \\phi ( x ) ^ { \\gamma ( x ) } \\phi ^ * ( x ) \\rangle _ { m ( d x ) } . \\end{align*}"}
-{"id": "5834.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 \\Psi ( u ) } { \\partial ^ 2 u } = - \\sigma ^ 2 \\end{align*}"}
-{"id": "1418.png", "formula": "\\begin{align*} \\textrm { O P T } ( T + 1 ) \\ \\ \\ = \\ \\ \\ E \\big [ \\min \\big ( U , \\textrm { O P T } ( T ) \\big ) \\big ] \\ \\ \\ = \\ \\ \\ \\textrm { O P T } ( T ) - \\frac { 1 } { 2 } \\big ( \\textrm { O P T } ( T ) \\big ) ^ 2 . \\end{align*}"}
-{"id": "7280.png", "formula": "\\begin{align*} p = ( F _ p , \\epsilon _ p , h _ p , R _ p , \\psi _ p ) \\end{align*}"}
-{"id": "7292.png", "formula": "\\begin{align*} p = ( F _ p , \\epsilon _ p , h _ p , R _ p , \\psi _ p ) \\end{align*}"}
-{"id": "9966.png", "formula": "\\begin{align*} \\begin{dcases*} \\dot { u } ( t ) = A u ( t ) + f ( t ) & o n $ I $ , \\\\ u ( 0 ) = x _ 0 , \\end{dcases*} \\end{align*}"}
-{"id": "9020.png", "formula": "\\begin{align*} f _ { \\{ b _ 2 \\} , b _ 1 } ( x ) = \\begin{cases} b _ 1 & x = b _ 2 , \\\\ x & \\end{cases} f _ { \\{ b _ 2 ' \\} , b _ 1 } ( x ) = \\begin{cases} b _ 1 & x = b _ 1 ' , \\\\ x & \\end{cases} \\end{align*}"}
-{"id": "9483.png", "formula": "\\begin{align*} m _ j \\leq \\displaystyle \\inf _ { B _ j } u \\leq \\displaystyle \\sup _ { B _ j } \\leq M _ j , \\ M _ j - m _ j = \\lambda R _ { j } ^ { \\alpha } , \\mbox { f o r a n y } j \\geq 0 . \\end{align*}"}
-{"id": "7782.png", "formula": "\\begin{align*} \\int _ { U ( G ) } \\langle x , \\xi \\rangle ^ 2 { \\rm d } x & = c ( G , S ) ^ 2 | G | ^ { - 1 } \\int _ { G } \\langle S ^ { - 1 } x , \\xi \\rangle ^ 2 { \\rm d } x \\\\ & = c ( G , S ) ^ 2 | G | ^ { - 1 } \\langle M ( S ^ { - 1 } \\xi ) , S ^ { - 1 } \\xi \\rangle \\\\ & = c ( G , S ) ^ 2 | G | ^ { - 1 } | \\xi | ^ 2 . \\end{align*}"}
-{"id": "8468.png", "formula": "\\begin{align*} \\norm { \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { 2 \\pi } \\frac { 1 } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } } _ { H ^ s } \\leq K _ s ^ { - 1 } \\epsilon d _ I ( t ) ^ { - 5 / 2 } . \\end{align*}"}
-{"id": "4169.png", "formula": "\\begin{align*} \\mathcal L _ n ( z ) : = L _ { \\alpha } ( n ^ { 3 / 2 } z ) D _ 2 ^ { - n } ( z ) L _ { \\alpha } ^ { - 1 } ( n ^ { 3 / 2 } f ( z ) ) . \\end{align*}"}
-{"id": "8453.png", "formula": "\\begin{align*} \\theta _ k = ( I - \\mathcal { H } ) \\partial _ { \\alpha } ^ k \\tilde { \\theta } , \\sigma _ k = ( I - \\mathcal { H } ) \\partial _ { \\alpha } ^ k \\tilde { \\sigma } . \\end{align*}"}
-{"id": "65.png", "formula": "\\begin{align*} J _ * + g ( J _ * ) \\delta = f ^ + _ \\ell - f ^ - _ r \\ , , \\rho _ { * , r } - \\rho _ { * , \\ell } = - 2 g ( J _ * ) \\delta \\ , , \\end{align*}"}
-{"id": "1066.png", "formula": "\\begin{align*} & \\lim \\limits _ { n \\to + \\infty } \\frac { ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } } { n \\sin ( \\alpha _ n / 2 ) } = \\lim \\limits _ { n \\to + \\infty } \\frac { ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } } { n \\alpha _ n / 2 } \\\\ & = \\lim \\limits _ { n \\to + \\infty } \\frac { ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } } { n ( 1 + z / \\ln n ) G _ n ( x ) / 4 } = \\lim \\limits _ { n \\to + \\infty } \\frac { ( 3 2 \\ln n ) ^ { \\frac { 1 } { 2 } } } { n G _ n ( x ) } = 1 , \\end{align*}"}
-{"id": "8895.png", "formula": "\\begin{align*} \\beta ( 1 , F ) : = B _ + ^ y ( F ) \\\\ \\beta ( n , F ) : = \\left ( B _ + ^ x \\right ) ^ { n - 1 } \\left ( B _ + ^ y ( F ) \\right ) \\end{align*}"}
-{"id": "4994.png", "formula": "\\begin{align*} q = \\begin{cases} p , & Q / 2 < p \\le Q , \\\\ p Q / ( 2 p - Q ) , & p \\ge Q . \\end{cases} \\end{align*}"}
-{"id": "3558.png", "formula": "\\begin{align*} C _ { 0 } : = \\mathrm { C o n e } \\left \\{ \\begin{aligned} ( - 1 , \\ ; \\ ; \\ , 0 , - 1 , 0 , 1 , 0 , 0 , 1 , 0 ) , \\\\ ( - 1 , - 1 , \\ , \\ ; \\ ; 0 , 1 , 0 , 0 , 1 , 0 , 0 ) , \\\\ ( \\ ; \\ ; \\ , 0 , - 1 , - 1 , 0 , 0 , 1 , 0 , 0 , 1 ) , \\\\ ( - 1 , - 1 , - 1 , 1 , 0 , 1 , 0 , 1 , 0 ) , \\\\ ( - 1 , - 1 , - 1 , 0 , 1 , 0 , 1 , 0 , 1 ) \\ , \\end{aligned} \\right \\} . \\end{align*}"}
-{"id": "4511.png", "formula": "\\begin{align*} \\Delta ( F ) = \\left \\{ r _ 1 , r _ 2 , r _ 2 \\in \\mathbb { R } _ { + } ^ 3 : \\exists x , y , z \\in F , r _ 1 = | x - y | , r _ 2 = | z - y | , r _ 3 = | x - z | \\right \\} . \\end{align*}"}
-{"id": "9610.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } a ( n , N ) q ^ n = \\sum _ { n = 1 } ^ { N } \\frac { q ^ n ( - q ) _ { n - 1 } } { 1 - q ^ n } . \\end{align*}"}
-{"id": "10068.png", "formula": "\\begin{align*} K ^ s ( u , \\varphi , \\theta _ n ^ 0 , G _ n ^ 0 ) = \\begin{pmatrix} \\varphi + \\O ( u ) \\\\ ( m _ n M _ n ) ^ { 1 / 4 } u + \\O ^ * ( u ^ 2 ) + \\O ( u ^ 3 ) \\\\ \\theta _ n ^ 0 + \\O ( u ^ 3 ) \\\\ \\O ( u ^ 8 ) \\\\ ( m _ n M _ n ) ^ { - 1 / 2 } \\mu _ n u + \\O ^ * ( u ^ 2 ) + \\O ( u ^ 3 ) \\\\ G _ n ^ 0 + \\O ( u ^ 3 ) \\end{pmatrix} , \\end{align*}"}
-{"id": "2251.png", "formula": "\\begin{align*} & \\langle \\alpha _ \\vartheta ' | \\alpha _ { \\vartheta ' } \\rangle = \\delta _ { \\vartheta , \\vartheta ' } & & \\langle \\alpha _ x ' | \\alpha _ \\vartheta \\rangle = \\langle \\alpha _ \\vartheta | \\alpha _ x \\rangle = 0 & & \\langle \\alpha _ x | \\alpha _ { x ' } \\rangle = \\delta ( x - x ' ) \\end{align*}"}
-{"id": "5935.png", "formula": "\\begin{align*} B _ { Z , X } = B _ { Z , Y } + N . \\end{align*}"}
-{"id": "981.png", "formula": "\\begin{align*} V ( G _ { r , d } ) = \\{ x _ { 1 } , x _ { 2 } , y _ { 1 } , \\ldots , y _ { d } , z _ { 1 } , \\ldots , z _ { 2 r - 3 } \\} \\end{align*}"}
-{"id": "7900.png", "formula": "\\begin{align*} \\tilde { \\kappa } ' \\colon F _ * ( N / I N \\otimes \\omega _ R / I \\omega _ R ) & \\to ( N / I N \\otimes \\omega _ R / I \\omega _ R ) \\\\ \\overline { n } \\otimes \\overline { m } & \\mapsto \\sum _ { i = 1 } ^ n \\overline { \\phi _ i ( 1 ) } \\cdot \\overline { \\gamma _ N ( n ) } \\otimes \\overline { \\kappa _ R ( r _ i f ^ { p - 1 } m ) } \\\\ & = \\sum _ { i = 1 } ^ t \\phi _ i ( 1 ) \\gamma _ N ( n ) \\otimes \\kappa _ R ( r _ i f ^ { p - 1 } m ) + I ( N \\otimes \\omega _ R ) \\end{align*}"}
-{"id": "2656.png", "formula": "\\begin{align*} \\varrho _ { k } = \\frac { \\tau } { \\sigma _ { k } + \\tau } , k = 1 , 2 , \\ldots , N , \\end{align*}"}
-{"id": "9073.png", "formula": "\\begin{align*} \\mu _ { K } = \\sum _ { k \\in \\mathcal { K } } \\sum _ { \\xi \\in \\mathcal { E } ^ { ( k ) } } \\mu _ { f } ( I _ k ) \\delta _ { \\zeta ^ k } \\Big / \\sum _ { k \\in \\mathcal { K } } \\sum _ { \\xi \\in \\mathcal { E } ^ { ( k ) } } \\mu _ { f } ( I _ k ) . \\end{align*}"}
-{"id": "8511.png", "formula": "\\begin{align*} \\norm { \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { 2 \\pi } \\frac { \\ddot { z } _ j } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } } _ { H ^ s } \\leq K _ s ^ { - 1 } \\epsilon ^ 2 d _ I ( t ) ^ { - 5 / 2 } . \\end{align*}"}
-{"id": "8725.png", "formula": "\\begin{align*} \\rho ^ { ( k ) } _ j ( i ) = P ( \\tau _ { N _ j } < \\tau _ 0 | Z _ j ^ { ( k ) } ( 0 ) = i ) = \\rho _ j ( i ) . \\end{align*}"}
-{"id": "9540.png", "formula": "\\begin{align*} \\frac { ( q ) _ N } { ( z q ) _ N ( z ^ { - 1 } q ) _ N } = \\sum _ { n = 0 } ^ \\infty \\sum _ { m = - \\infty } ^ { \\infty } M _ { S _ 2 } ( m , n ) z ^ m q ^ n . \\end{align*}"}
-{"id": "4720.png", "formula": "\\begin{align*} f ( u ) = \\frac { u ^ 2 } { ( u ^ 2 + ( 1 - u ^ 2 ) ) } , \\end{align*}"}
-{"id": "9517.png", "formula": "\\begin{align*} G _ { \\beta , M } ( t ) = \\displaystyle \\int _ { 0 } ^ { t } ( h _ { \\beta , M } ' ( \\tau ) ) ^ { \\frac { 1 } { p } } \\dd \\tau \\geq \\frac { p } { \\beta + p - 1 } t ( \\min \\{ t , M \\} ) ^ \\frac { \\beta - 1 } { p } . \\end{align*}"}
-{"id": "6165.png", "formula": "\\begin{align*} K ( x , x , \\lambda ) \\sim \\sum _ { j = 0 } ^ \\infty c _ j ( x ) \\lambda ^ { \\frac { n + m - j } { m _ 0 } - k } + \\sum _ { l = 0 } ^ \\infty \\left ( c ' _ l ( x ) \\log \\lambda + c '' _ l ( x ) \\right ) \\lambda ^ { - l - k } , \\end{align*}"}
-{"id": "8299.png", "formula": "\\begin{align*} \\mathcal { H } f ( \\alpha ) : = \\frac { 1 } { \\pi i } p . v . \\int _ { - \\infty } ^ { \\infty } \\frac { \\zeta _ { \\beta } } { \\zeta ( \\alpha , t ) - \\zeta ( \\beta , t ) } f ( \\beta ) d \\beta . \\end{align*}"}
-{"id": "2250.png", "formula": "\\begin{align*} & C _ 1 ( x ) = = 1 - \\int \\d y g ( y ) h ( y ) \\frac { \\mathcal { P } } { x - y } & & C _ 2 ( x ) = g ( x ) h ( x ) \\\\ & A ( x ) = \\frac { \\mathcal { P } } { x - \\Omega } | g \\rangle & & A ' ( x ) = \\langle h | \\frac { \\mathcal { P } } { x - \\Omega } \\\\ & B ( x ) = g ( x ) | \\delta _ x \\rangle & & B ' ( x ) = h ( x ) \\langle \\delta _ x | \\end{align*}"}
-{"id": "469.png", "formula": "\\begin{align*} \\sum ^ 3 _ { i = 1 } \\gcd ( a _ i - a _ 4 , d ) + \\sum _ { 1 \\leq i < j \\leq 3 } \\gcd ( a _ i - a _ j , d ) = \\sum ^ { } _ { 1 \\leq i \\leq j \\leq 4 } \\gcd ( a _ i - a _ j , d ) \\leq 3 d + 3 . \\end{align*}"}
-{"id": "6487.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { e \\colon s ( e ) = v } \\mu ( e ) = \\delta \\forall v \\in V . \\end{align*}"}
-{"id": "4291.png", "formula": "\\begin{align*} \\frac { 1 } { r } \\sum _ { v \\in V ( G ) } ( \\mbox { n u m b e r o f $ K _ { r - 1 } $ ' s i n $ N ( v ) $ } ) \\geq \\frac { 1 } { r } \\left ( \\binom { s - 2 } { r - 1 } + \\binom { s - 3 } { r - 2 } \\right ) n . \\end{align*}"}
-{"id": "3474.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\iint _ { ( C , \\infty ) ^ 2 } d x d y \\lvert X _ n ( x , y ) - X _ { \\infty } ( x , y ) \\rvert ^ 2 = 0 , X = G H . \\end{align*}"}
-{"id": "9605.png", "formula": "\\begin{align*} f ( q , N + 1 ) = \\frac { f ( q , N ) } { 1 + q ^ { N + 1 } } + \\frac { q ^ { N + 1 } } { 1 - q ^ { 2 N + 2 } } , \\end{align*}"}
-{"id": "2820.png", "formula": "\\begin{align*} E ^ { * , p } = ( x _ 1 ^ { * , p } , u _ 1 ^ { * , p } , \\ldots , x _ p ^ { * , p } , u _ p ^ { * , p } , 0 , 0 , \\ldots , 0 , 0 ) , \\end{align*}"}
-{"id": "5350.png", "formula": "\\begin{align*} A _ { \\beta } = X _ { \\beta } + Y _ { \\beta } , \\quad B _ { \\beta } = i ( X _ { \\beta } - Y _ { \\beta } ) \\end{align*}"}
-{"id": "2099.png", "formula": "\\begin{align*} d x ( t ) = A ( t ) x ( t ) d t + C ( t ) x ( t ) d \\omega ( t ) , \\ x ( t _ 0 ) = x _ 0 \\in \\R ^ d , t \\geq t _ 0 , \\end{align*}"}
-{"id": "1369.png", "formula": "\\begin{align*} \\left ( \\alpha _ \\lambda , x + t ( d ( \\alpha _ \\lambda ) - x ) \\right ) = \\left ( \\lambda ' \\alpha _ \\eta , d ( \\lambda ' ) + z + t ( d ( \\alpha _ \\eta ) - z ) \\right ) \\sim \\left ( \\alpha _ \\eta , z + t ( d ( \\alpha _ \\eta ) - z ) \\right ) \\ ; , \\end{align*}"}
-{"id": "1721.png", "formula": "\\begin{align*} B _ { t } = \\frac { 1 } { \\sqrt { C ( H ) } } \\left \\{ \\int _ { - \\infty } ^ { 0 } \\left [ ( t - s ) ^ { H - \\frac { 1 } { 2 } } - ( - s ) ^ { H - \\frac { 1 } { 2 } } \\right ] d W _ { s } + \\int _ { 0 } ^ { t } ( t - s ) ^ { H - \\frac { 1 } { 2 } } d W _ { s } \\right \\} , \\end{align*}"}
-{"id": "919.png", "formula": "\\begin{align*} \\zeta _ { j } ^ { ( i ) } = \\int \\limits _ t ^ T \\phi _ { j } ( s ) d { \\bf w } _ s ^ { ( i ) } \\end{align*}"}
-{"id": "9694.png", "formula": "\\begin{align*} \\hat { z } _ { i + 1 } & = \\scriptstyle { \\cal \\prod } \\left ( \\hat { z _ { i } } - \\eta \\cdot \\frac { \\partial | | y - z . G | | ^ 2 } { \\partial z } \\vert _ { z = \\hat { z } _ { i } } \\right ) = \\scriptstyle { \\cal \\prod } \\left ( \\hat { z _ { i } } - 2 \\eta \\cdot y G ^ { T } + \\eta \\cdot \\hat { z } _ { i } G G ^ { T } \\right ) , \\end{align*}"}
-{"id": "9332.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\| \\nabla F _ { e _ { i } } \\| ^ { 2 } = \\sum _ { i , k = 1 } ^ { n } \\langle \\nabla F _ { e _ { i } } , e _ { k } \\rangle ^ 2 = \\sum _ { i , k = 1 } ^ { n } [ e _ { k } \\langle e _ { i } , W \\rangle ] ^ { 2 } \\end{align*}"}
-{"id": "7562.png", "formula": "\\begin{align*} \\begin{gathered} \\frac { 1 } { t - s } \\left ( X ' \\left ( X ^ { - 1 } ( x , t ) , t \\right ) - X ' \\left ( X ^ { - 1 } ( x , s ) , s \\right ) \\right ) \\\\ = \\int _ 0 ^ 1 \\left ( ( \\partial _ t X ' ) \\left ( X ^ { - 1 } ( x , \\beta _ \\tau ) , \\beta _ \\tau \\right ) + \\left ( \\partial _ t X ^ { - 1 } \\right ) ( x , \\beta _ \\tau ) ( \\nabla _ a X ' ) \\left ( X ^ { - 1 } ( x , \\beta _ \\tau ) , \\beta _ \\tau \\right ) \\right ) d \\tau , \\end{gathered} \\end{align*}"}
-{"id": "812.png", "formula": "\\begin{align*} ( T - \\rho ) * _ d u & = u ^ { q ^ d } + ( T - \\rho ) u \\\\ ( T - \\rho ) ^ 2 * _ d u & = u ^ { q ^ { 2 d } } + \\left ( ( T - \\rho ) ^ { q ^ d } + ( T - \\rho ) \\right ) u ^ { q ^ d } + ( T - \\rho ) ^ 2 u . \\end{align*}"}
-{"id": "5147.png", "formula": "\\begin{align*} \\abs * { \\begin{pmatrix} a \\\\ c \\end{pmatrix} \\begin{pmatrix} b \\\\ d \\end{pmatrix} } : = \\begin{vmatrix} a & b \\\\ c & d \\end{vmatrix} . \\end{align*}"}
-{"id": "5580.png", "formula": "\\begin{align*} \\| w \\| _ { Y ^ { \\kappa , b , \\alpha } } = \\big \\| \\big ( 1 + | \\xi | + | \\tau | ^ { \\frac 1 3 } \\big ) ^ { \\kappa } \\mu ( \\xi , \\tau ) \\hat { w } ( \\xi , \\tau ) \\big \\| _ { L _ { \\tau } ^ 2 L ^ 2 _ { \\xi } } , \\end{align*}"}
-{"id": "36.png", "formula": "\\begin{align*} e ^ { \\mathcal { Y } _ T ^ { \\alpha _ T } } - e ^ { \\mathcal { Y } _ 0 ^ { i } } = & \\ \\sum _ { j \\geq 1 } \\left [ e ^ { \\mathcal { Y } _ { T \\wedge T _ { j } - } ^ { \\alpha ^ { j - 1 } } } - e ^ { \\mathcal { Y } _ { T \\wedge T _ { j - 1 } } ^ { \\alpha ^ { j - 1 } } } \\right ] + \\sum _ { j \\geq 1 } \\left [ e ^ { \\mathcal { Y } _ { T _ { j } } ^ { \\alpha ^ { j } } } - e ^ { \\mathcal { Y } _ { T _ { j } - } ^ { \\alpha ^ { j - 1 } } } \\right ] \\chi _ { \\{ T _ { j } \\leq T \\} } \\\\ = & \\ ( I ) + ( I I ) . \\end{align*}"}
-{"id": "5053.png", "formula": "\\begin{align*} [ L _ b ^ { \\mathcal P } , [ L _ a ^ { \\mathcal P } , { \\mathcal P } ] ] = - [ L _ { { \\mathcal F } ( a , b ) } ^ { \\mathcal P } , { \\mathcal P } ] + \\phi ( a , b ) \\mathcal P \\textrm { , f o r a l l $ a , b \\in \\mathbb V $ . } \\end{align*}"}
-{"id": "3661.png", "formula": "\\begin{align*} \\widehat { Y } = N \\bar { y } _ B \\end{align*}"}
-{"id": "6574.png", "formula": "\\begin{gather*} e _ i e _ { I , t } = e _ { I i , t } , \\\\ e _ 0 e _ { I , t } = e _ { I i , t + 1 } . \\end{gather*}"}
-{"id": "8672.png", "formula": "\\begin{align*} \\omega \\cdot ( u \\cdot \\nabla ) u + u \\cdot ( u \\cdot \\nabla ) \\omega & = \\div \\big ( u ( u \\cdot \\omega ) \\big ) \\\\ u \\cdot ( \\omega \\cdot \\nabla ) u & = \\frac { 1 } { 2 } \\div \\big ( | u | ^ 2 \\omega \\big ) \\\\ \\omega \\cdot \\nabla p & = \\div ( p \\ , \\omega ) \\ , . \\end{align*}"}
-{"id": "7044.png", "formula": "\\begin{align*} \\langle \\cdot , \\cdot \\rangle = \\int _ M ( \\cdot , \\cdot ) d \\mu _ g . \\end{align*}"}
-{"id": "3028.png", "formula": "\\begin{align*} \\Psi _ j ( z ) = z + \\sum \\limits _ { k = 2 } ^ \\infty u _ { k } { z ^ k } + ( - 1 ) ^ { j } \\sum \\limits _ { k = 1 } ^ \\infty v _ { k } { \\overline { z } ^ k } \\end{align*}"}
-{"id": "3250.png", "formula": "\\begin{align*} e ^ { i \\theta } = \\cos \\theta + i \\sin \\theta \\end{align*}"}
-{"id": "6685.png", "formula": "\\begin{align*} \\psi _ { n } ( x ) & = C _ { n } | x | ^ { - \\frac { 1 } { 2 } + i ( n \\omega + ( \\frac { \\omega } { 2 } - \\lambda _ { * } ) ) } , \\ n = 1 , 2 , \\cdots , \\\\ \\tilde { \\psi } _ { n } ( x ) & = \\tilde { C } _ { n } | x | ^ { - \\frac { 1 } { 2 } + i ( n \\omega - ( \\frac { \\omega } { 2 } + \\lambda _ { * } ) ) } , \\tilde { \\psi } _ { 0 } ( x ) = 0 , \\end{align*}"}
-{"id": "9116.png", "formula": "\\begin{align*} \\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix} \\mapsto \\begin{bmatrix} 1 & B \\\\ 0 & 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "3681.png", "formula": "\\begin{align*} \\widehat { Y } = \\sum \\limits _ { i \\in S } d _ i \\hat { y } _ i \\end{align*}"}
-{"id": "7009.png", "formula": "\\begin{align*} { \\textsl { \\footnotesize L } } _ i { \\textsl { \\footnotesize R } } _ i ' = { \\textsl { \\footnotesize L } } _ i ' { \\textsl { \\footnotesize R } } _ i = d _ i { \\textsl { \\footnotesize L } } _ i ' { \\textsl { \\footnotesize R } } _ i ' = d _ i ' . \\end{align*}"}
-{"id": "9685.png", "formula": "\\begin{align*} X _ * ^ * \\Bigl ( \\eta _ n ^ 2 \\frac { \\omega _ n \\varphi _ n } { \\psi _ { n t } } \\varphi F _ { \\tau , \\mathbb T \\setminus \\sigma } \\oplus \\eta _ n ^ 2 \\frac { \\omega _ n \\varphi _ n } { \\psi _ { n t } } \\varphi F _ { \\mathbb T , \\mathbb T \\setminus \\sigma } \\Bigr ) = ( \\varphi \\eta _ n ) ( T ) y _ { n t } . \\end{align*}"}
-{"id": "7420.png", "formula": "\\begin{align*} \\displaystyle Z ( 1 - s , \\widetilde { W _ \\lambda } , \\widehat { \\phi } ) = \\omega _ { \\pi } ( \\lambda ) ^ { n - 1 } \\delta _ n ( a ( \\lambda ) ) \\lvert \\det a ( \\lambda ) \\rvert _ F ^ { s - 1 } Z ( 1 - s , \\widetilde { W } , \\widehat { \\phi } ) . \\end{align*}"}
-{"id": "6021.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\downarrow 0 } \\ , \\ , \\ , \\inf \\left \\{ \\tilde { \\alpha } ^ \\mu _ \\epsilon ( Q ) \\ , : \\ , Q \\in \\mathcal { P } ( \\C ) , \\ H ( Q ) = \\nu _ \\epsilon \\right \\} & = \\inf \\left \\{ \\alpha ^ \\mu _ 0 ( Q ) \\ , : \\ , Q \\in \\mathcal { P } ( \\C ) , \\ H ( Q ) = \\nu \\right \\} . \\end{align*}"}
-{"id": "7741.png", "formula": "\\begin{align*} \\| \\langle x , x \\rangle \\xi \\| = \\| x \\| ^ 2 . \\end{align*}"}
-{"id": "176.png", "formula": "\\begin{align*} & \\pi _ { X _ { 1 , V } , X _ { 2 , V } , X _ { 3 , V } , X _ { 4 , V } } = X _ { 1 , V } \\wedge X _ { 2 , V } + X _ { 3 , V } \\wedge X _ { 4 , V } \\end{align*}"}
-{"id": "4751.png", "formula": "\\begin{align*} a _ { k , 1 } = t _ k + O ( 1 ) = c k + O ( 1 ) . \\end{align*}"}
-{"id": "1029.png", "formula": "\\begin{align*} & \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) : = ( n / 4 ) ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } \\times \\\\ & \\mathbb { P } \\Big ( ( y _ 1 , \\cdots , y _ k ) \\in \\Sigma _ k ( F _ n ( x _ 1 ) , \\cdots , F _ n ( x _ k ) ) \\Big ) , \\end{align*}"}
-{"id": "3133.png", "formula": "\\begin{align*} \\tilde { T } ( u ) T ( u ) = \\mathbf { I } , \\end{align*}"}
-{"id": "2782.png", "formula": "\\begin{align*} H ( \\alpha ) = \\log ( 1 - q + q e ^ { \\alpha } ) . \\end{align*}"}
-{"id": "7991.png", "formula": "\\begin{align*} \\log g ( n ) - \\log h ( n ) = \\sum _ { p \\leq \\sqrt { n } } { \\frac { \\xi _ { p } ( 1 + \\xi _ { p } ) } { 2 } } \\log p + \\frac { n } { 2 } + O \\left ( \\frac { n } { \\log n } \\right ) \\ , . \\end{align*}"}
-{"id": "3776.png", "formula": "\\begin{align*} A _ \\ell = \\{ x \\in A : ( x , \\partial A ) > \\ell \\} . \\end{align*}"}
-{"id": "8207.png", "formula": "\\begin{align*} E _ 2 ( \\tau ) : = 1 - 2 4 \\sum _ { n = 1 } ^ \\infty \\frac { n q ^ n } { 1 - q ^ n } , q = e ^ { 2 \\pi i \\tau } , \\mathrm { I m } \\tau > 0 . \\end{align*}"}
-{"id": "118.png", "formula": "\\begin{align*} \\widetilde \\zeta _ { j , N } = \\gamma + \\zeta _ { j , N } \\ , , \\zeta _ { j , N } \\dot = \\sum _ { \\ell = 1 } ^ { \\min \\{ j , 2 N - j - 1 \\} } \\gamma ^ { 2 \\ell + 1 } \\begin{pmatrix} j \\\\ \\ell \\end{pmatrix} \\begin{pmatrix} 2 N - j - 1 \\\\ \\ell \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "5845.png", "formula": "\\begin{align*} \\phi ( x _ 1 , \\dots , x _ k ) = \\sum _ { i = 1 } ^ k { \\rm T r } _ i ( x _ i ) . \\end{align*}"}
-{"id": "3961.png", "formula": "\\begin{align*} u _ i ( x , t ) = \\frac { 1 } { 2 } x ^ * P ^ i _ t x + x ^ * \\nu ^ i _ t + \\tau ^ i _ t . \\end{align*}"}
-{"id": "6506.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\displaystyle d \\bar { X } _ { t } = A \\bar { X } _ { t } d t - B ( \\bar { X } _ t ) d t + \\bar { f } ( \\bar { X } _ { t } ) d t + \\sigma _ 1 ( \\bar { X } _ t ) d W ^ { Q _ { 1 } } _ { t } , \\\\ \\bar { X } _ { 0 } = x , \\end{array} \\right . \\end{align*}"}
-{"id": "5673.png", "formula": "\\begin{align*} x ^ n - m y ^ n = \\pm 1 \\ ; \\ ; { \\rm i n } \\ ; \\ ; x , y \\in \\Z . \\end{align*}"}
-{"id": "8035.png", "formula": "\\begin{align*} \\mathbb U _ n = \\langle x , y \\ , | \\ , x ^ 2 , y ^ { 2 n } , x y x y ^ { n + 1 } \\rangle . \\end{align*}"}
-{"id": "6831.png", "formula": "\\begin{align*} I = \\langle c _ 1 ^ 2 + s _ 1 ^ 2 - 1 , c _ 2 ^ 2 + s _ 2 ^ 2 - 1 \\rangle \\subset \\mathbb { A } = \\mathbb { Q } ( \\mu , \\theta ) [ c _ 1 , s _ 1 , c _ 2 , s _ 2 ] \\ , , \\end{align*}"}
-{"id": "1942.png", "formula": "\\begin{align*} \\mathcal { A } _ p = \\sum _ { j = 1 } ^ p \\left [ ( j - 1 ) \\pi + \\epsilon _ j - \\tan \\epsilon _ j + ( \\chi + 1 ) \\chi ^ 2 \\frac { A _ j ^ 2 } { Z _ j ^ 3 } \\right ] - \\sum _ { j = 1 } ^ p ( j - 1 ) \\pi , \\end{align*}"}
-{"id": "2530.png", "formula": "\\begin{align*} P ^ \\beta _ t f ( x ) = \\int _ E p _ t ^ \\beta ( x , y ) f ( y ) m ( d y ) . \\end{align*}"}
-{"id": "5764.png", "formula": "\\begin{align*} ( X _ { \\mathcal { R } / A } \\times _ A K ) ( T ) & = X _ { \\mathcal { R } / A } ( T ) \\\\ & = X / _ \\mathcal { R } ( \\mathcal { R } \\otimes _ A T ) \\\\ & = X / _ A ( \\mathcal { R } \\otimes _ A T ) , \\end{align*}"}
-{"id": "8676.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ q _ t ( W _ x ^ { \\alpha , p } ) } & : = \\biggl ( \\int _ 0 ^ { + \\infty } \\| f ( s ) \\| _ { W ^ { \\alpha , p } } \\ , d s \\biggl ) ^ \\frac { 1 } { q } \\ , , \\\\ \\| f \\| _ { L ^ q _ t ( C _ x ^ { \\theta } ) } & : = \\biggl ( \\int _ 0 ^ { + \\infty } \\| f ( s ) \\| _ { C ^ { \\theta } } \\ , d s \\biggl ) ^ \\frac { 1 } { q } \\ , . \\end{align*}"}
-{"id": "4032.png", "formula": "\\begin{align*} \\begin{aligned} 0 \\longrightarrow P ( n - N ) ^ { \\oplus \\alpha _ { N - n } } & \\longrightarrow P ( n - N + 1 ) ^ { \\oplus \\alpha _ { N - n - 1 } } \\longrightarrow \\dots \\\\ & \\longrightarrow P ( - 1 ) ^ { \\oplus \\alpha _ 1 } \\longrightarrow P ^ { \\oplus \\alpha _ 0 } \\longrightarrow E \\longrightarrow 0 . \\end{aligned} \\end{align*}"}
-{"id": "3651.png", "formula": "\\begin{align*} f _ \\Lambda ( a d _ b ) ( a ) = \\langle a , b \\rangle f _ \\Lambda ( a d _ b ) ( b ) + f _ \\Lambda ( a d _ b ) ( a _ 0 + a _ { \\frac { 1 } { 4 } } + a _ { \\frac { 1 } { 3 2 } } ) = f _ \\Lambda ( 1 ) \\langle a , b \\rangle b \\end{align*}"}
-{"id": "7388.png", "formula": "\\begin{align*} \\dot { x } & = y \\\\ \\dot { y } & = a x + k _ 1 b y + b x ^ 2 + k _ 2 x y + x ^ 2 y + \\epsilon x ^ 3 + k _ 3 x ^ 4 , \\end{align*}"}
-{"id": "5698.png", "formula": "\\begin{align*} u ( x , y , 0 ) & = \\varphi ( x , y ) & & u _ t ( x , y , 0 ) = \\psi ( x , y ) . \\end{align*}"}
-{"id": "8185.png", "formula": "\\begin{align*} T ^ * \\Lambda _ k ^ * g = T ^ * \\Big ( \\sum _ { i \\in \\mathbb { N } } \\Lambda _ i ^ * \\Theta _ i \\Lambda _ k ^ * g \\Big ) & = \\sum _ { i \\in \\mathbb { N } } T ^ * \\Lambda _ i ^ * \\Theta _ i \\Lambda _ k ^ * g \\\\ & = \\sum _ { i \\in \\mathbb { N } } \\Lambda _ { i + 1 } ^ * \\Theta _ i \\Lambda _ k ^ * g = \\sum _ { i \\in \\mathbb { N } } \\big ( \\Lambda _ k \\Theta _ i ^ * \\Lambda _ { i + 1 } \\big ) ^ * g , \\end{align*}"}
-{"id": "22.png", "formula": "\\begin{align*} \\mathbf { y } ( v ) & = \\chi _ { \\{ v \\geq 0 \\} } \\int _ 0 ^ { v } e ^ { \\frac { y ^ 2 } { 2 } } [ \\frac 1 2 e ^ { - y ^ 2 } + N ( y ) - 1 ] d y + \\chi _ { \\{ v < 0 \\} } \\int _ 0 ^ { v } e ^ { \\frac { y ^ 2 } { 2 } } [ - \\frac 1 2 e ^ { - y ^ 2 } + N ( y ) ] d y ; \\\\ \\mathbf { z } ( v ) & = \\chi _ { \\{ v \\geq 0 \\} } e ^ { \\frac { v ^ 2 } { 2 } } [ \\frac 1 2 e ^ { - v ^ 2 } + N ( v ) - 1 ] + \\chi _ { \\{ v < 0 \\} } e ^ { \\frac { v ^ 2 } { 2 } } [ - \\frac 1 2 e ^ { - v ^ 2 } + N ( v ) ] . \\end{align*}"}
-{"id": "6378.png", "formula": "\\begin{align*} ( \\theta + d d ^ c \\rho _ j ) ^ n = \\left ( a _ j + \\frac { | g _ j - g | } { \\| g _ j - g \\| _ p } \\right ) d V , \\end{align*}"}
-{"id": "4794.png", "formula": "\\begin{align*} \\left ( H _ 0 - \\frac { \\nu } { | x | } - a _ 1 \\right ) \\psi ^ { a _ 1 } _ C = 0 , \\end{align*}"}
-{"id": "1249.png", "formula": "\\begin{align*} K ( S , S ) ( \\eta ) = \\frac { 1 } { \\sqrt { | \\eta | } } \\int e ^ { 3 i \\mu ^ 2 / 4 } \\tilde { S } ( \\eta , \\mu / \\sqrt { | \\eta | } ) d \\mu , \\end{align*}"}
-{"id": "6863.png", "formula": "\\begin{align*} \\mathsf { V } ( I ) = \\big \\{ a \\in \\mathbb { L } ^ { n } \\colon f ( a ) = 0 \\ { \\rm f o r \\ a l l \\ } ~ f \\in I \\big \\} \\subset \\mathbb { L } ^ { n } . \\end{align*}"}
-{"id": "4165.png", "formula": "\\begin{align*} A _ n ^ { ( 1 ) } ( z ) = \\frac { 1 } { f ' ( 0 ) } E _ n ( z ) T _ { \\alpha } ^ { - 1 } A _ { \\alpha } T _ { \\alpha } E _ n ^ { - 1 } ( z ) . \\end{align*}"}
-{"id": "2671.png", "formula": "\\begin{align*} I _ { 0 } ( 2 \\sqrt { z } ) = \\int _ { \\mathcal { C } _ { \\{ 0 \\} } } \\frac { d s } { 2 \\pi i s } e ^ { s z + \\frac { 1 } { s } } , \\end{align*}"}
-{"id": "8448.png", "formula": "\\begin{align*} q = - \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { 2 \\pi } \\frac { 1 } { \\zeta ( \\alpha , t ) - z _ j ( t ) } . \\end{align*}"}
-{"id": "7960.png", "formula": "\\begin{align*} \\bigsqcup _ { q \\in [ s ] } S _ q = \\bigsqcup _ { r \\in [ t ] } T _ r = [ n ] . \\end{align*}"}
-{"id": "569.png", "formula": "\\begin{align*} \\xi ^ s ( 1 + p x _ s ) R _ n ^ * - \\xi ^ t ( 1 + p x _ t ) R _ n ^ * = R _ { 2 n } ^ * \\setminus \\left ( \\xi ^ s V \\cup \\xi ^ t V \\right ) . \\end{align*}"}
-{"id": "6693.png", "formula": "\\begin{align*} z ^ { \\frac { 1 } { \\omega } ( \\frac { i } { 2 } + C ) } f ( z ) = \\sum _ { n = 0 } ^ { \\infty } c _ { n } \\phi _ { n } ( z ) , \\end{align*}"}
-{"id": "7707.png", "formula": "\\begin{align*} \\varphi ( t ) = \\hat { \\check { \\varphi } } ( t ) = \\int \\limits _ { - \\infty } ^ { \\infty } \\mathsf { e } ^ { - 2 \\pi \\mathbf { i } x t } \\check { \\varphi } ( x ) \\ , \\mathrm { d } x . \\end{align*}"}
-{"id": "9416.png", "formula": "\\begin{align*} \\mathrm { R e s } _ { Q _ { \\tau } } f _ { \\theta } d z = \\begin{cases} \\frac { 1 } { 2 \\pi i } \\nu _ { \\tau } ^ { ( N ) } ( f ) & \\tau \\in \\mathbb { H } , \\\\ \\frac { \\alpha _ \\tau } { 2 \\pi i } \\left ( \\nu _ { \\tau } ^ { ( N ) } ( f ) - \\frac { k } { 1 2 } \\right ) & \\tau \\in \\mathcal { C } _ N . \\end{cases} \\end{align*}"}
-{"id": "2218.png", "formula": "\\begin{align*} s = & \\frac { \\partial q } { \\partial t } + \\frac { \\partial q } { \\partial x } \\left [ \\left ( - \\frac { 1 } { t + 1 } \\right ) x + u \\right ] + 1 0 0 0 e ^ 2 + u ^ 2 \\\\ & s _ f = 1 0 e ^ 2 ( t _ f ) - q ( x ( t _ f ) , t _ f ) \\end{align*}"}
-{"id": "7919.png", "formula": "\\begin{align*} R _ N = \\hat { R } _ n M h ^ { n + 2 } f ^ { ( n + 1 ) } ( \\xi ) \\end{align*}"}
-{"id": "370.png", "formula": "\\begin{align*} x _ { m + 1 } \\sqrt { c } + z _ { m + 1 } \\sqrt { a } & = ( x \\sqrt { c } + z \\sqrt { a } ) ( s + \\sqrt { a c } ) ^ { m + 1 } \\\\ & = ( x \\sqrt { c } + z \\sqrt { a } ) ( s + \\sqrt { a c } ) ^ { m } ( s + \\sqrt { a c } ) \\\\ & = ( x _ { m } \\sqrt { c } + z _ { m } \\sqrt { a } ) ( s + \\sqrt { a c } ) \\\\ & = ( s x _ m + a z _ m ) \\sqrt { c } + ( s z _ m + c x _ m ) \\sqrt { a } , \\end{align*}"}
-{"id": "9491.png", "formula": "\\begin{align*} G _ { \\beta , M } ( t ) = \\displaystyle \\int _ { 0 } ^ { t } ( g _ { \\beta , M } ' ( \\tau ) ) ^ { \\frac { 1 } { m } } \\dd \\tau = \\frac { \\beta ^ { \\frac { 1 } { m } } m } { \\beta + m - 1 } ( \\min \\{ t , M \\} ) ^ { \\frac { \\beta + m - 1 } { m } } \\end{align*}"}
-{"id": "7848.png", "formula": "\\begin{align*} U _ k = ( d - ( k + 1 ) m , d - k m ] \\end{align*}"}
-{"id": "2578.png", "formula": "\\begin{align*} s = \\int _ { \\langle v _ s , \\phi ^ * \\rangle } ^ \\infty \\big \\langle \\psi _ 0 \\big ( \\cdot , ( 1 + \\epsilon _ { R ( u ) } ) u \\phi \\big ) , \\phi ^ * \\big \\rangle _ m ^ { - 1 } d u , s \\in ( 0 , \\infty ) . \\end{align*}"}
-{"id": "9216.png", "formula": "\\begin{align*} e ( G ' ) & \\leq \\max \\left \\{ \\binom { k } { 2 } , \\binom { \\frac { k } { 2 } + 1 } { 2 } + \\left ( \\frac { k } { 2 } - 1 \\right ) \\left ( n - \\frac { k } { 2 } - 1 \\right ) \\right \\} \\\\ & = \\max \\left \\{ \\binom { k } { 2 } , \\binom { \\frac { k } { 2 } - 1 } { 2 } + \\left ( \\frac { k } { 2 } - 1 \\right ) \\left ( n - \\frac { k } { 2 } + 1 \\right ) + 1 \\right \\} . \\end{align*}"}
-{"id": "3735.png", "formula": "\\begin{align*} \\left | a \\| v _ k \\| ^ { 2 n } + b \\| v _ k \\| ^ n - \\lambda H ( v _ k ) - \\int _ { \\Omega } ( | x | ^ { - \\mu } * F ( v _ k ) ) f ( v _ k ) v _ k ~ d x \\right | = o ( \\| v _ k \\| ) \\end{align*}"}
-{"id": "6296.png", "formula": "\\begin{align*} \\mathbb { P } _ { G \\sim G _ { n , p } } \\left [ \\mathcal { H } ^ c _ { \\rho , C } \\right ] \\leq \\mathbb { P } _ { G \\sim G _ { n , p } } \\left ( | e ( G ) - p | > \\frac { C \\sqrt { \\rho \\log n } } { 2 \\sqrt { n } } \\right ) + \\mathbb { P } _ { G \\sim G _ { n , p } } \\left ( \\bigcup _ { \\emptyset \\not = S \\subseteq [ n ] } A ^ c _ { \\rho , S , \\frac { C } { 2 } } ( p ) \\right ) . \\end{align*}"}
-{"id": "4011.png", "formula": "\\begin{align*} H ( \\tau ) = \\sum _ i \\sum _ \\delta \\alpha _ { i , \\delta } \\ , f _ { i } ( \\delta \\tau ) . \\end{align*}"}
-{"id": "1583.png", "formula": "\\begin{align*} ( z _ 0 , x _ 0 ) : = U \\circ \\boldsymbol { F } _ { n ^ 2 + \\kappa _ 0 , n ^ 2 } ( z , x ) , \\end{align*}"}
-{"id": "1552.png", "formula": "\\begin{align*} F _ 2 ^ \\delta \\circ U _ 2 ^ \\delta \\left ( \\begin{array} { c } z \\\\ x \\end{array} \\right ) & = \\left ( \\begin{array} { c } P ( x ) z + \\dots \\\\ b _ 0 ^ \\delta ( x ) + \\dots \\end{array} \\right ) \\\\ & = \\left ( \\begin{array} { c } \\left [ \\prod _ { n = 0 } ^ \\infty a _ 1 ^ \\delta ( ( b _ 0 ^ \\delta ) ^ { n } ( x ) ) \\right ] z + \\dots \\\\ b _ 0 ^ \\delta ( x ) + \\dots \\end{array} \\right ) \\end{align*}"}
-{"id": "191.png", "formula": "\\begin{align*} \\pi ( { \\bf x } ) = x ^ 1 \\frac { \\partial } { \\partial x ^ 2 } \\wedge \\frac { \\partial } { \\partial x ^ 3 } , \\end{align*}"}
-{"id": "4834.png", "formula": "\\begin{align*} \\nu ^ 2 - ( \\lambda ^ 2 + \\mu ^ 2 ) = \\frac { 1 } { 2 } | \\mu | \\Delta \\ln | \\mu | , \\end{align*}"}
-{"id": "9365.png", "formula": "\\begin{align*} P _ i ( G _ k ) / P _ { i + 1 } ( G _ k ) \\cong \\begin{cases} C _ 2 \\times C _ 2 & \\\\ C _ 2 \\times C _ 2 \\times C _ 2 & \\\\ C _ 2 & \\\\ C _ 2 \\times C _ 2 & \\\\ C _ 2 & . \\end{cases} \\end{align*}"}
-{"id": "614.png", "formula": "\\begin{align*} \\begin{multlined} ( \\mathcal M \\rtimes \\mathcal G ) ( a _ 1 \\dots a _ n ; a ) \\\\ : = \\left \\{ ( f , x ) \\ ; \\middle | \\ ; x \\in \\mathcal G ( n ) , \\ f \\in \\mathcal M ( a _ { x ^ { - 1 } ( 1 ) } \\dots a _ { x ^ { - 1 } ( n ) } ; a ) \\right \\} \\ ; \\end{multlined} \\end{align*}"}
-{"id": "9486.png", "formula": "\\begin{align*} T a i l _ { p } ( u ; x ; R _ { 0 } ) ^ { p - 1 } & = R _ { 0 } ^ { s p } \\displaystyle \\int _ { B _ { R _ { 0 } } ^ { c } ( x ) } \\frac { \\vert u ( y ) \\vert ^ { p - 1 } } { \\vert x - y \\vert ^ { N + s p } } \\dd y \\\\ & \\leq C \\left \\Vert u \\right \\Vert _ { L ^ { \\infty } ( B _ { 2 R _ { 0 } } ( x _ 0 ) ) } ^ { p - 1 } + C R _ { 0 } ^ { s p } \\displaystyle \\int _ { B _ { 2 R _ { 0 } } ^ { c } ( x _ { 0 } ) } \\frac { \\vert u ( y ) \\vert ^ { p - 1 } } { \\vert x _ { 0 } - y \\vert ^ { N + s p } } \\dd y \\end{align*}"}
-{"id": "8867.png", "formula": "\\begin{align*} c _ { 3 a } & = - 3 \\nu ^ 6 + 1 2 \\nu ^ 5 + 6 3 2 \\nu ^ 4 + 1 7 9 4 \\nu ^ 3 - 3 7 6 2 4 \\nu ^ 2 + 6 5 2 4 4 \\nu + 6 4 8 7 7 > 0 \\\\ c _ { 3 b } & = 2 ( 1 - \\nu ) ( - 3 \\nu ^ 7 + 1 2 \\nu ^ 6 + 6 5 2 \\nu ^ 5 + 2 4 1 7 \\nu ^ 4 - 4 2 5 6 1 \\nu ^ 3 + 7 3 8 6 4 \\nu ^ 2 + 4 1 3 3 6 \\nu + 9 1 3 2 3 ) \\\\ & + ( 1 - \\mu ) ( 3 \\nu ^ 6 ( 2 2 0 - \\nu ^ 2 + 4 \\nu ) + 2 \\nu ( 1 4 9 0 \\nu ^ 4 - 2 2 9 9 3 \\nu ^ 3 + 3 9 8 9 8 \\nu ^ 2 + 8 9 0 \\nu + 1 0 9 2 6 2 ) + 8 4 7 7 ) \\ge 0 \\end{align*}"}
-{"id": "2166.png", "formula": "\\begin{align*} \\tau _ { \\mathbf { x } } ( p g _ { 1 } ) ( - \\mathbf { y } ) & = \\tau _ { \\mathbf { x } } \\left ( \\sum _ { \\ell = 0 } ^ { d } \\sum _ { \\| \\alpha \\| \\leq \\ell } c _ { \\ell , \\alpha } T ^ { \\alpha } ( g _ { 1 + \\ell } ) ( - \\mathbf { y } ) \\right ) = \\sum _ { \\ell = 0 } ^ { d } \\sum _ { \\| \\alpha \\| \\leq \\ell } c _ { \\ell , \\alpha } T ^ { \\alpha } \\tau _ { \\mathbf { x } } ( g _ { 1 + \\ell } ) ( - \\mathbf { y } ) , \\end{align*}"}
-{"id": "6603.png", "formula": "\\begin{align*} ( P _ { F _ { n } } ( \\varphi , \\varepsilon ) ) ^ 2 = & ( 2 \\sin \\pi ( F _ { n } \\varphi + \\varepsilon ) ) ^ 2 \\\\ & \\times \\prod _ { r = 1 } ^ { F _ { n } - 1 } 4 \\left ( \\sin ^ 2 \\pi \\left ( r \\varphi + \\frac { ( - \\varphi ) ^ { n } } { 2 } \\right ) - \\sin ^ 2 \\pi \\left ( \\frac { ( - \\varphi ) ^ { n } } { 2 } - \\varepsilon \\right ) \\right ) . \\end{align*}"}
-{"id": "5405.png", "formula": "\\begin{align*} W ' = A \\cup B _ + ' \\cup B _ - ' \\cup C _ + ' \\cup C _ - ' \\cup D ' \\cup E _ + \\cup E _ - \\cup F . \\end{align*}"}
-{"id": "586.png", "formula": "\\begin{align*} \\varrho : = \\lim _ { n \\to \\infty } \\ , \\frac { 1 } { n } \\log \\int _ { \\mathcal { X } ^ n } \\prod _ { m = 0 } ^ { n - 1 } ( C ( x _ m ) c ^ { x _ { m + 1 } } ) \\ , \\kappa ( x _ 0 , \\mathrm { d } x _ 1 ) \\ , \\kappa ( x _ 1 , \\mathrm { d } x _ 2 ) \\ , \\ldots \\ , \\kappa ( x _ { n - 1 } , \\mathrm { d } x _ n ) \\end{align*}"}
-{"id": "3653.png", "formula": "\\begin{align*} u = \\sum _ { i = 1 } ^ n \\lambda _ i v _ i \\textrm { a n d } v = \\sum _ { j = 1 } ^ n \\mu _ j v _ j \\end{align*}"}
-{"id": "1505.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r @ { \\ , } c @ { \\ , } c @ { \\ } l @ { \\quad } l } n _ { t } & + & u \\cdot \\ ! \\nabla n & = \\nabla \\cdot \\big ( D ( n ) \\nabla n - n S ( x , n , c ) \\nabla c \\big ) , \\\\ c _ { t } & + & u \\cdot \\ ! \\nabla c & = \\Delta c - c + n , \\\\ u _ { t } & + & \\kappa ( u \\cdot \\nabla ) u & = \\Delta u + \\nabla P + n \\nabla \\phi , \\\\ & & \\nabla \\cdot u & = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "4084.png", "formula": "\\begin{align*} \\phi _ 4 ( z ) = - \\frac { 4 \\pi ^ 2 } { \\Gamma ( 1 + \\alpha ) \\Gamma ( \\frac { 3 } { 2 } + \\alpha ) } \\phi _ 0 ( z ) \\end{align*}"}
-{"id": "2005.png", "formula": "\\begin{align*} c _ 1 ( T \\P ^ 1 | _ 0 ) = z = - c _ 1 ( T \\P ^ 1 | _ \\infty ) , \\end{align*}"}
-{"id": "5495.png", "formula": "\\begin{align*} \\Pi _ T \\Big ( b : \\sum _ { j = 1 } ^ d \\| b _ j - b _ { 0 , j } \\| _ { \\infty } \\geq ( \\log T ) ^ \\delta T ^ { - \\frac { s \\wedge [ a - ( d / 2 - 2 ) _ + ] } { 2 a + d } } \\big | X ^ T \\Big ) \\rightarrow ^ { P _ { b _ 0 } } 0 ~ T \\to \\infty . \\end{align*}"}
-{"id": "9295.png", "formula": "\\begin{align*} J _ K ^ { 2 , - } u ( x ) : = & \\{ ( p , X ) \\in \\mathbb { R } ^ n \\times \\mathbb { S } ^ n | \\ u ( y ) \\ge u ( x ) + p \\cdot ( y - x ) \\\\ & + \\frac { 1 } { 2 } X ( y - x ) \\cdot ( y - x ) + o ( | y - x | ^ 2 ) \\ { \\rm a s } \\ y \\in K \\rightarrow x \\} , \\end{align*}"}
-{"id": "3042.png", "formula": "\\begin{align*} \\mu _ { k } ( d z ) = \\widetilde { \\mu } _ { k } ( d z _ { k } ) \\otimes \\prod \\limits _ { j \\neq k } \\delta _ { 0 } ( d z _ { j } ) , \\ \\ k \\in \\{ 1 , \\dots , d \\} , \\end{align*}"}
-{"id": "5294.png", "formula": "\\begin{align*} \\{ I _ 1 , I _ 2 \\} = \\left ( \\frac { \\partial I _ 1 } { \\partial q _ 1 } \\frac { \\partial I _ 2 } { \\partial p _ 1 } - \\frac { \\partial I _ 1 } { \\partial p _ 1 } \\frac { \\partial I _ 2 } { \\partial q _ 1 } \\right ) + \\left ( \\frac { \\partial I _ 1 } { \\partial q _ 2 } \\frac { \\partial I _ 2 } { \\partial p _ 2 } - \\frac { \\partial I _ 1 } { \\partial p _ 2 } \\frac { \\partial I _ 2 } { \\partial q _ 2 } \\right ) = 0 . \\end{align*}"}
-{"id": "6864.png", "formula": "\\begin{align*} L u = \\sum _ { | \\nu | \\le q } c _ \\nu \\partial ^ \\nu u = f \\ , , \\end{align*}"}
-{"id": "1639.png", "formula": "\\begin{align*} \\lambda _ k = \\left \\{ \\begin{array} { l c } 4 \\pi k \\sqrt [ + ] { | c | ^ 2 - \\pi ^ 2 k ^ 2 } , & 0 < \\pi | k | < | c | , \\\\ 4 \\pi i k \\sqrt [ + ] { \\pi ^ 2 k ^ 2 - | c | ^ 2 } , & \\pi | k | > | c | , \\end{array} \\right . \\end{align*}"}
-{"id": "1898.png", "formula": "\\begin{align*} \\eta _ \\pm ^ \\mu ( \\lambda ) = ( \\lambda , \\pm \\lambda ) + ( t _ 0 , x _ 0 ) \\qquad \\mbox { w i t h } K _ \\pm ^ \\mu = \\frac { d } { d \\lambda } \\eta _ \\pm ^ \\mu ( \\lambda ) = ( 1 , \\pm 1 ) , \\end{align*}"}
-{"id": "5211.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { s } t ^ { - 1 } b ( t x ) d t = \\log \\left \\{ \\frac { h ( s x ) } { h ( x ) } \\right \\} \\rightarrow \\log s ^ { ( 1 + \\rho ) } x \\rightarrow \\infty . \\end{align*}"}
-{"id": "9574.png", "formula": "\\begin{align*} S _ 2 & = \\frac { 1 } { c } \\sum _ { n = 1 } ^ N \\frac { \\left ( q / c \\right ) _ { n - 1 } ( c q ) _ { N - n } ( c q ) ^ n } { ( q ) _ n ( c q ) _ N ( q ) _ { N - n } } \\left ( \\frac { 1 } { c - 1 } - \\frac { 1 } { ( 1 / c ) _ n } \\sum _ { k = 1 } ^ n \\left [ \\begin{matrix} n \\\\ k \\end{matrix} \\right ] \\frac { ( q c ) _ k ( 1 / c ) _ { n - k } c ^ { - k } } { 1 - q ^ k } \\right ) \\\\ & = \\frac { - 1 } { ( 1 - c ) ^ 2 ( q ) _ N } \\left ( 1 - \\frac { ( q ) _ N } { ( c q ) _ N } \\right ) + S _ { 2 } ^ { * } , \\end{align*}"}
-{"id": "357.png", "formula": "\\begin{align*} u _ { \\alpha } \\geq z _ { \\alpha } : = \\psi ^ { - 1 } ( \\lambda _ { 1 } a ( \\alpha ) ) e _ { 1 } , \\ \\forall \\ \\alpha \\in ( t _ { k - 1 } , t _ { k } ) . \\end{align*}"}
-{"id": "7701.png", "formula": "\\begin{align*} \\norm { \\Psi _ { i } ^ * ( v ) } _ { 2 } ^ 2 \\geq \\frac { 1 } { L } \\int \\limits _ { U _ { i } } \\norm { v ( p ) } _ { S _ { p } } ^ 2 \\ , \\mathrm { d } \\mu = \\frac { 1 } { L } \\norm { v } _ { 2 } ^ 2 . \\end{align*}"}
-{"id": "348.png", "formula": "\\begin{align*} \\nu _ p ( n ( n ^ 2 - 1 ) , k ) = \\nu _ p ( n ( n ^ 2 - 1 ) , l ) \\end{align*}"}
-{"id": "9687.png", "formula": "\\begin{align*} \\begin{aligned} & W _ t \\in \\mathcal L ( H ^ 2 , \\mathcal N ) \\ W _ t S = T | _ { \\mathcal N } W _ t \\\\ & \\ \\| W _ t \\| \\leq C _ 2 \\ t > 0 . \\end{aligned} \\end{align*}"}
-{"id": "4688.png", "formula": "\\begin{align*} ( Q u ) ^ { ( l ) } ( x _ j ) = \\frac { 1 } { \\Delta x ^ l } \\displaystyle { \\sum _ { k = j - ( d - 1 ) } ^ { j + ( d - 1 ) } b _ { k , l } u _ k } , ~ ~ ~ ~ ~ ~ l = 0 , \\dots , d - 1 , \\end{align*}"}
-{"id": "4927.png", "formula": "\\begin{align*} \\dfrac { e _ I ( R ) } { \\ell ( R / I ) } & = \\dfrac { e _ { ( f _ 1 ^ N , \\ldots , f _ d ^ N ) } ( R ) } { N ^ d \\cdot \\ell ( R / I ) } \\geq \\dfrac { e _ { ( f _ 1 ^ N , \\ldots , f _ d ^ N ) } ( R ) } { N ^ d \\cdot \\ell ( R / ( f _ 1 ^ N , \\ldots , f _ d ^ N ) ) } \\\\ & \\geq \\dfrac { 1 } { N ^ d } \\left ( 1 - \\dfrac { \\ell ( H ^ { d - 1 } ( f _ 1 ^ N , \\ldots , f _ d ^ N ; R ) ) } { \\ell ( R / ( f _ 1 ^ N , \\ldots , f _ d ^ N ) ) } \\right ) \\geq \\frac { 1 } { 2 N ^ d } , \\end{align*}"}
-{"id": "7522.png", "formula": "\\begin{align*} \\mathfrak { A } ( \\eta , \\xi ) : = R _ { g ^ D } ( \\xi , J \\eta , \\xi , J \\eta ) - R _ { g ^ D } ( \\eta , \\xi , \\eta , \\xi ) \\end{align*}"}
-{"id": "7145.png", "formula": "\\begin{align*} \\chi _ t = - \\frac { 1 } { \\epsilon } v ( \\chi ^ 3 - \\chi ) + \\frac { \\chi _ { y y } } { v } - \\frac { 2 \\chi _ y v _ y } { v ^ 2 } , \\end{align*}"}
-{"id": "8316.png", "formula": "\\begin{align*} ( I - \\mathfrak { H } ) \\bar { z } _ t = - \\sum _ { j = 1 } ^ N ( I - \\mathfrak { H } ) \\frac { \\lambda _ j i } { 2 \\pi ( z ( \\alpha , t ) - z _ j ( t ) ) } . \\end{align*}"}
-{"id": "8219.png", "formula": "\\begin{align*} X ^ 5 + \\sum _ { j = 0 } ^ 4 a _ j ( t ) X ^ j = - t \\prod \\limits _ { \\lambda = 0 } ^ 4 ( 1 - X t _ { \\lambda } ^ { - 1 } ) . \\end{align*}"}
-{"id": "537.png", "formula": "\\begin{align*} u ^ h ( x ' , x _ 3 ) = y _ 0 ( x ' ) + x _ 3 \\vec b _ 0 ( x ' ) + \\displaystyle { h ^ 2 \\vec d _ 0 \\big ( x ' , \\frac { x _ 3 } { h } \\big ) } \\end{align*}"}
-{"id": "7818.png", "formula": "\\begin{align*} 0 & = ( x \\otimes x ) \\bigg ( ( \\| y \\| ( x \\otimes x ) ( x ) + \\lambda \\| x \\| ( y \\otimes x ) ( x ) ) \\otimes x \\bigg ) = ( x \\otimes x ) \\bigg ( ( \\| y \\| \\| x \\| ^ 2 x + \\lambda \\| x \\| ^ 3 y ) \\otimes x \\bigg ) \\\\ & = ( x \\otimes x ) ( \\| y \\| \\| x \\| ^ 2 x \\otimes x + \\lambda \\| x \\| ^ 3 y \\otimes x ) . \\end{align*}"}
-{"id": "1289.png", "formula": "\\begin{align*} | \\partial _ \\eta K ( z , w ) | = O ( | \\eta | ^ { - k } ) , | \\eta | > 1 . \\end{align*}"}
-{"id": "6876.png", "formula": "\\begin{align*} p = - \\tfrac { 1 } { 2 } \\ , \\langle v , A ^ T A '' v \\rangle \\ , . \\end{align*}"}
-{"id": "2651.png", "formula": "\\begin{align*} \\det \\begin{bmatrix} A _ { 1 } & A _ { 2 } \\\\ A _ { 3 } & A _ { 4 } \\end{bmatrix} = \\det ( A _ { 1 } ) \\det ( A _ { 4 } - A _ { 3 } A _ { 1 } ^ { - 1 } A _ { 2 } ) = \\det ( A _ { 4 } ) \\det ( A _ { 1 } - A _ { 2 } A _ { 4 } ^ { - 1 } A _ { 3 } ) , \\end{align*}"}
-{"id": "9298.png", "formula": "\\begin{align*} \\tilde \\Phi ( x , y ) : = v ( x ) - \\tilde u ( y ) - \\phi ( x , y ) , \\end{align*}"}
-{"id": "10076.png", "formula": "\\begin{align*} E ^ { \\leq } : = F \\circ K ^ { \\leq } - K ^ { \\leq } \\circ R ^ { \\leq } \\end{align*}"}
-{"id": "276.png", "formula": "\\begin{align*} f ( z ) = c z ^ n e ^ { - a z ^ 2 + b z } \\prod _ { k = 1 } ^ \\infty \\left ( 1 - \\frac { z } { x _ k } \\right ) e ^ { \\tfrac { z } { x _ k } } , \\end{align*}"}
-{"id": "2315.png", "formula": "\\begin{align*} H _ { \\mathrm { r a d } } & = \\sum _ \\mathbf { m } \\omega _ \\mathbf { m } \\sum _ { i , l } \\Pi ( \\mathbf { m } ) _ { i l } b ^ + _ { \\mathbf { m } , i } b _ { \\mathbf { m } , l } \\\\ H _ { \\mathrm { i n t } } & = \\sum _ { \\mathbf { m } , i , l } ( ( e a / 3 ) \\sqrt { 2 \\pi \\omega _ 0 } L ^ { - 3 / 2 } ) c ( \\omega _ \\mathbf { m } - \\omega _ 0 ) \\Pi ( \\mathbf { m } ) _ { i , l } ( E _ { 1 i , 0 } b _ { \\mathbf { m } , l } + E _ { 0 , 1 i } b ^ + _ { \\mathbf { m } , l } ) \\end{align*}"}
-{"id": "4946.png", "formula": "\\begin{align*} & - v _ { i + 1 } z ^ { 3 n ^ { i + 2 } + n ^ { i + 1 } + \\cdots + n ^ 2 + n } \\\\ & = v _ { i + 1 } z ^ { 3 n ^ { i + 2 } + n ^ { i + 1 } + \\cdots + n ^ 2 - n } ( z ^ t - z ^ n x ^ n ) - z ^ { 3 n ^ { i + 2 } + n ^ { i + 1 } + \\cdots + n ^ 2 } f _ { i + 4 } + ( v _ { i + 1 } z ^ { 3 n ^ { i + 1 } + n ^ { i } + \\cdots + n ^ 2 + n } ) ^ n \\\\ & - \\left ( \\left ( \\sum _ { j = i + 2 } ^ { d - 3 } v _ j z ^ { 3 n ^ { i + 1 } + n ^ { i } + \\cdots + n ^ 2 + n } \\right ) + ( - 1 ) ^ { d - 1 } x z ^ { 3 n ^ { i + 1 } + n ^ { i } + \\cdots + n ^ 2 + n } \\right ) ^ n \\in I _ n , \\end{align*}"}
-{"id": "9871.png", "formula": "\\begin{align*} \\sigma : = \\sum _ { x } r ^ 2 _ { A ( A + A ) } ( x ) = \\sum _ x \\sum _ { l _ 1 , l _ 2 \\in L } A ( l ^ { - 1 } _ 1 x ) A ( l ^ { - 1 } _ 2 x ) \\le 2 \\sum _ { h \\in \\Omega } r _ { L ^ { - 1 } L } ( h ) \\sum _ { a \\in A } A ( h a ) \\ , , \\end{align*}"}
-{"id": "7840.png", "formula": "\\begin{align*} \\kappa ( d , N ) = N d ^ { - 1 / 2 } . \\end{align*}"}
-{"id": "1959.png", "formula": "\\begin{align*} T ' : = \\frac { R [ y ] } { ( F ( y ) , y \\cdot G ( y ) ) } , \\end{align*}"}
-{"id": "5107.png", "formula": "\\begin{align*} \\langle x _ j , x _ k \\rangle & = \\frac { \\| x _ j + x _ k \\| ^ 2 - \\| x _ j - x _ k \\| ^ 2 + i \\| x _ j + i x _ k \\| ^ 2 - i \\| x _ j - i x _ k \\| ^ 2 } { 4 } \\\\ & = \\frac { ( | 1 | ^ 2 + | 1 | ^ 2 ) - ( | 1 | ^ 2 + | - 1 | ^ 2 ) + i ( | 1 | ^ 2 + | i | ^ 2 ) - i ( | 1 | ^ 2 + | - i | ^ 2 ) } { 4 } = 0 . \\end{align*}"}
-{"id": "2042.png", "formula": "\\begin{align*} f \\big ( \\frac { a } { M } + i y \\big ) & = \\xi _ R ( - M ) \\overline { \\xi _ { q / R } } ( a ) \\big ( \\tilde { f } _ R \\mid V _ { q , R } ^ { M , a } \\big ) \\left ( \\frac { a } { M } + i y \\right ) \\\\ & = \\xi _ R ( - M ) \\overline { \\xi _ { q / R } } ( a ) R ^ { \\frac { k } { 2 } } ( - i M R y ) ^ { - k } \\tilde { f } _ R \\big ( - \\frac { \\overline { R a } } { M } + i \\frac { 1 } { M ^ 2 R y } \\big ) . \\end{align*}"}
-{"id": "1662.png", "formula": "\\begin{align*} \\ell ^ { ( k - 2 ) n } & B ^ { ( N ) } _ { t } ( \\ell ^ { 2 n } m , d ) = B ^ { ( N ) } _ { t } ( m , \\ell ^ { 2 n } d ) \\\\ & \\quad + \\left [ \\left ( \\frac { ( - 1 ) ^ { \\lambda } d } { \\ell } \\right ) - \\left ( \\frac { ( - 1 ) ^ { \\lambda } m } { \\ell } \\right ) \\right ] \\cdot \\sum _ { k = 1 } ^ { n } \\ell ^ { ( \\lambda - 1 ) k } \\left ( \\frac { ( - 1 ) ^ { \\lambda } d } { \\ell } \\right ) ^ { k - 1 } B ^ { ( N ) } _ { t } ( m , \\ell ^ { 2 n - 2 k } d ) . \\end{align*}"}
-{"id": "321.png", "formula": "\\begin{align*} \\begin{array} { c c c } S q ^ 2 ( u ) = u ^ 2 , & S q ^ 2 ( v _ { 2 n - 4 } ) = v _ { 2 n - 2 } , & S q ^ 2 ( a _ { 2 n } ) = a _ { 2 n + 2 } . \\end{array} \\end{align*}"}
-{"id": "3473.png", "formula": "\\begin{align*} K _ * ( x , y ) = \\int ^ { \\infty } _ C G _ * ( x , r ) H _ * ( r , y ) d r . \\end{align*}"}
-{"id": "5501.png", "formula": "\\begin{align*} T h _ T ^ 2 ( b _ 1 , b _ 2 ) \\equiv \\int _ 0 ^ T \\| b _ 1 ( X _ s ) - b _ 2 ( X _ s ) \\| ^ 2 d s = \\sum _ { j = 1 } ^ d \\int _ 0 ^ T | b _ { 1 , j } ( X _ s ) - b _ { 2 , j } ( X _ s ) | ^ 2 d s . \\end{align*}"}
-{"id": "8114.png", "formula": "\\begin{align*} \\widetilde { A } _ { n , r } ( t ) \\ = \\ \\widetilde { A } ^ + _ { n , r } ( t ) \\ , + \\ , \\widetilde { A } ^ - _ { n , r } ( t ) , \\end{align*}"}
-{"id": "7566.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { \\mathbb { G } ( \\tau \\circ X ^ { - 1 } ) } _ { L ^ \\infty ( 0 , T ; C ^ { \\alpha , p } ) } \\le C _ 1 \\norm { X - \\mathrm { I d } } _ { L i p ( 0 , T ; C ^ { 1 + \\alpha } ) } ^ \\alpha \\norm { \\tau ( 0 ) } _ { \\alpha , p } ( 1 + C _ 3 ( T , X ) ) \\\\ + C _ 2 \\norm { \\tau } _ { L i p ( 0 , T ; C ^ { \\alpha , p } ) } C _ 4 ( T , X ) \\end{gathered} \\end{align*}"}
-{"id": "5482.png", "formula": "\\begin{align*} B ( t , x ; \\eta ) \\ & = \\ \\big \\{ ( s , y ) \\in ( 0 , T ) \\times H : \\max \\{ | x - y | , | t - s | \\} < \\eta \\big \\} , \\\\ \\partial B ( t , x ; \\eta ) \\ & = \\ \\big \\{ ( s , y ) \\in ( 0 , T ) \\times H : \\max \\{ | x - y | , | t - s | \\} = \\eta \\big \\} . \\end{align*}"}
-{"id": "2197.png", "formula": "\\begin{align*} J = 0 . 5 \\Big [ \\boldsymbol { e } ^ T ( t _ f ) F ( t _ f ) \\boldsymbol { e } ( t _ f ) \\Big ] + 0 . 5 \\Big [ \\int _ { t _ 0 } ^ { t _ f } \\boldsymbol { e } ^ T ( t ) Q ( t ) \\boldsymbol { e } ( t ) + \\boldsymbol { u } ^ T ( t ) R ( t ) \\boldsymbol { u } ( t ) d t \\Big ] \\end{align*}"}
-{"id": "2419.png", "formula": "\\begin{align*} \\begin{cases} U U ^ * = U ^ * U = I , \\\\ U F = F \\overline { U } , \\end{cases} \\end{align*}"}
-{"id": "6136.png", "formula": "\\begin{align*} \\Psi ^ t _ { { i _ \\ell , j _ \\ell , k _ \\ell + 1 } } = A ^ t _ { { i _ \\ell , j _ \\ell , k _ \\ell + 1 } } - B ^ t _ { i _ \\ell , j _ \\ell + 1 , k _ \\ell } , \\end{align*}"}
-{"id": "244.png", "formula": "\\begin{align*} \\Delta _ { \\mathbb { H } } u = \\Delta _ z u + \\frac { | z | ^ 2 } { 4 } \\partial _ { t } ^ 2 u \\end{align*}"}
-{"id": "7123.png", "formula": "\\begin{align*} \\lambda ^ * _ m = \\lambda _ { i _ m } ( \\hat { q } ) , ~ ~ ~ ~ \\phi ^ * _ m = \\phi _ { i _ m } ( \\hat { q } ) , ~ ~ ~ m = 1 , 2 , \\end{align*}"}
-{"id": "1864.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty Y _ { j , m } Y _ { j , n } = \\delta _ { m n } , \\end{align*}"}
-{"id": "346.png", "formula": "\\begin{align*} | I m ( \\partial ' _ k ) _ * | = \\frac { \\frac { 1 } { 2 } n ( n - 1 ) } { ( \\frac { 1 } { 2 } n ( n - 1 ) , k ) } \\cdot \\frac { n ( n + 1 ) } { ( n ( n + 1 ) , k ) } , \\end{align*}"}
-{"id": "8832.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathbb R ^ n } \\sum _ { i = 1 } ^ m \\log ( 1 + \\exp ( b _ i ( A x ) _ i ) ) + \\lambda \\norm { x } _ 1 \\end{align*}"}
-{"id": "5184.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } F _ { n } ( a _ { n } x + b _ { n } ) = H ( x ) . \\end{align*}"}
-{"id": "7132.png", "formula": "\\begin{align*} L ^ 2 _ { \\mathrm { p e r } } = \\Big \\{ g ( x ) \\big | g ( x + L ) = g ( x ) \\ \\mathrm { f o r \\ a l l } \\ x \\in \\mathbb { R } , \\ { \\mathrm { a n d } \\ } g ( x ) \\in L ^ 2 ( 0 , L ) \\Big \\} , \\end{align*}"}
-{"id": "2418.png", "formula": "\\begin{align*} v _ \\varphi = ( i d \\otimes \\varphi ) \\mathbb { U } \\end{align*}"}
-{"id": "4829.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } s } J _ { s , t } A _ 1 & = J _ { s , t } C _ 1 , \\\\ \\frac { \\mathrm { d } } { \\mathrm { d } s } J _ { s , t } C _ 1 & = J _ { s , t } C _ 1 - U '' ( X _ s ) J _ { s , t } A _ 1 \\end{align*}"}
-{"id": "5976.png", "formula": "\\begin{align*} q ( t , \\omega _ 1 , \\omega _ 2 ) = \\beta ( t , \\omega _ 1 ) 1 _ { [ 0 , 1 ] } ( t ) + \\widehat q [ \\omega _ 1 ] ( t - 1 , \\omega _ 2 ) 1 _ { ( 1 , 2 ] } ( t ) . \\end{align*}"}
-{"id": "9766.png", "formula": "\\begin{align*} u ( X ) = x _ 1 ^ 2 x _ 2 ^ 2 - ( x _ 1 ^ 2 + x _ 2 ^ 2 ) y ^ 2 + \\frac { 1 } { 3 } y ^ 4 . \\end{align*}"}
-{"id": "4167.png", "formula": "\\begin{align*} E _ n ^ { - 1 } ( z _ 1 ) E _ n ( z _ 2 ) = \\mathbb { I } + \\mathcal { O } \\left ( n ^ \\frac { 5 } { 2 } ( z _ 1 - z _ 2 ) \\right ) \\end{align*}"}
-{"id": "2034.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } A _ h ^ \\pm ( M ) = \\frac { 1 } { M } \\sum _ { 0 \\le a \\le M } \\left < \\frac { a } { M } \\right > ^ { \\pm } h \\left ( \\frac { a } { M } \\right ) . \\end{align*}"}
-{"id": "856.png", "formula": "\\begin{align*} \\tau _ k > \\chi _ \\tau ( \\tau _ k ) > \\tau _ k - \\frac { 1 } { k + 1 } \\sum _ { j = k } ^ \\infty ( \\tau _ j - \\tau _ { j + 1 } ) = \\tau _ k \\left ( 1 - \\frac 1 { k + 1 } \\right ) , \\end{align*}"}
-{"id": "5689.png", "formula": "\\begin{align*} \\| v _ 1 - v _ 2 \\| < \\frac \\varepsilon 2 \\mbox { a n d } | - 1 - ( r _ 1 - r _ 2 ) | = | r _ 1 - ( r _ 2 - 1 ) | < \\frac \\varepsilon 2 \\ , . \\end{align*}"}
-{"id": "8894.png", "formula": "\\begin{align*} C h ( \\emptyset ) & : = \\left ( x \\mapsto 1 ~ \\forall x \\in I \\right ) = : { \\bf 1 } , \\\\ C h ( ( f _ i ) \\sqcup w ) & : = \\left ( x \\mapsto \\int _ a ^ x f _ i ( t ) C h ( w ) ( t ) d t ~ \\forall x \\in I \\right ) \\end{align*}"}
-{"id": "624.png", "formula": "\\begin{align*} \\varphi ^ \\ast ( y ) \\cdot \\operatorname { D e c } ^ G _ \\varphi \\cdot \\varphi ^ \\ast ( y ) ^ { - 1 } = \\operatorname { D e c } ^ G _ { \\varphi ^ y } \\ . \\end{align*}"}
-{"id": "8138.png", "formula": "\\begin{align*} \\forall \\ , g \\in H , \\ : \\forall \\ , \\psi \\in \\mathcal { O } ( H ) , \\ : \\ : \\ : \\ : \\widetilde { g } \\widetilde { h } = \\widetilde { g h } , \\ : \\ : \\ : g \\widetilde { h } = \\widetilde { h } g , \\ : \\ : \\ : \\widetilde { h } \\psi = \\psi \\ ! \\left ( S ^ { - 1 } ( h '' ) ? \\right ) h ' . \\end{align*}"}
-{"id": "1862.png", "formula": "\\begin{align*} Y _ { j , n } = \\left ( u ^ { \\rm e v e n } ( j , x ) , v ^ { \\rm e v e n } ( n , x ) \\right ) _ { L ^ 2 } = \\frac { \\xi L A _ j } { Z _ j ^ 2 - ( \\pi n ) ^ 2 } . \\end{align*}"}
-{"id": "6915.png", "formula": "\\begin{align*} f = \\hat f _ 1 g _ 1 + \\hat f _ 4 g _ 4 + \\hat f _ 5 g _ 5 \\ , , \\end{align*}"}
-{"id": "9644.png", "formula": "\\begin{align*} \\frac { ( b ; q ^ 2 ) _ N } { ( a ; q ^ 2 ) _ N } \\frac { ( a ; q ^ 2 ) _ { N - n } } { ( b ; q ^ 2 ) _ { N - n } } \\left ( \\frac { a } { b } \\right ) ^ n = \\frac { ( q ^ { 2 - 2 N } / b ; q ^ 2 ) _ n } { ( q ^ { 2 - 2 N } / a ; q ^ 2 ) _ n } \\end{align*}"}
-{"id": "8157.png", "formula": "\\begin{align*} h \\cdot \\varphi _ 1 \\otimes \\ldots \\otimes \\varphi _ g = \\varphi _ 1 \\ ! \\left ( S ( h ^ { ( 1 ) } ) ? h ^ { ( 2 ) } \\right ) \\otimes \\ldots \\otimes \\varphi _ g \\ ! \\left ( S ( h ^ { ( 2 g - 1 ) } ) ? h ^ { ( 2 g ) } \\right ) . \\end{align*}"}
-{"id": "9533.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { n - 1 } z ^ n q ^ { n ^ 2 } } { ( z q ; q ^ 2 ) _ { n } ( 1 - z q ^ { 2 n } ) } = \\sum _ { n = 1 } ^ { \\infty } \\frac { z ^ n q ^ { \\frac { n ( n + 1 ) } { 2 } } ( q ; q ) _ { n - 1 } } { ( z q ; q ) _ n } . \\end{align*}"}
-{"id": "7437.png", "formula": "\\begin{align*} \\displaystyle \\beta ( \\widetilde { W } ) = c _ 1 ( \\pi ) \\beta ( W ) \\end{align*}"}
-{"id": "3747.png", "formula": "\\begin{align*} \\mbox { \\boldmath $ { \\nabla } $ } \\times \\mbox { \\boldmath $ { \\nabla } $ } \\times \\mathbf { J } _ { \\mathcal { E } , \\mathcal { P } } ( \\mathbf { r } ) - K ^ { 2 } \\mathbf { J } _ { \\mathcal { E } , \\mathcal { P } } ( \\mathbf { r } ) = \\mathbf { 0 } . \\end{align*}"}
-{"id": "4521.png", "formula": "\\begin{align*} S ( K ) = \\{ ( c _ 1 , c _ 2 , c _ 3 ) \\in C _ 1 \\times C _ 2 \\times C _ 3 : ( | c _ 1 - c _ 2 | , | c _ 2 - c _ 3 | , | c _ 3 - c _ 1 | ) \\in K \\} . \\end{align*}"}
-{"id": "8374.png", "formula": "\\begin{align*} u _ { t t } + i a u _ { \\alpha } = - i a _ t \\bar { z } _ { \\alpha } = - \\frac { \\bar { z } _ { t t } - i } { | z _ { t t } + i | } a _ t | z _ { \\alpha } | : = g . \\end{align*}"}
-{"id": "768.png", "formula": "\\begin{align*} M = \\prod _ { j = 1 } ^ r M _ j ^ { \\alpha _ j } , \\end{align*}"}
-{"id": "1141.png", "formula": "\\begin{align*} & | B | _ 2 ^ 2 = \\sum \\limits _ { j = 1 } ^ k | B _ j | _ 2 ^ 2 = O \\left ( ( \\ln n ) ^ { 1 / 2 } \\right ) . \\end{align*}"}
-{"id": "4257.png", "formula": "\\begin{align*} \\sum _ { j = 2 } ^ { i + 1 } E _ { 2 } ( V ( d w ) , B _ { j } ) \\leq \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { i } E _ { 2 } ( V ( d w ) , B _ { j } ) + \\frac { 2 c _ { 4 } \\lambda ^ { \\frac { p } { 2 } } } { \\sigma ^ { n } } \\sum _ { j = 1 } ^ { i } \\omega \\left ( R _ { j } \\right ) . \\end{align*}"}
-{"id": "1972.png", "formula": "\\begin{align*} 2 d _ { \\ell \\ell } = \\sum _ { k = 1 } ^ { m } d _ { k i } , \\end{align*}"}
-{"id": "7370.png", "formula": "\\begin{align*} | E _ { A p } | \\leq | A \\times A | 2 ^ \\omega = \\kappa ^ 2 2 ^ \\omega \\leq \\kappa ^ \\omega 2 ^ \\omega = \\kappa ^ \\omega = \\kappa \\end{align*}"}
-{"id": "9279.png", "formula": "\\begin{align*} L ^ \\epsilon : = ( F ^ \\epsilon ) ^ { i j } [ ( D _ { i j } - A _ { i j } ^ k ( \\cdot , u _ \\epsilon , D u _ \\epsilon ) D _ k ) + \\epsilon \\delta _ { i j } \\sum _ { l = 1 } ^ n ( D _ { l l } - A _ { l l } ^ k ( \\cdot , u _ \\epsilon , D u _ \\epsilon ) D _ k ) ] , \\end{align*}"}
-{"id": "8152.png", "formula": "\\begin{align*} v _ A ^ { - 1 } \\triangleright \\varphi & = \\varphi ^ { v ^ { - 1 } } = \\varphi ( v ^ { - 1 } ? ) , \\\\ v _ B ^ { - 1 } \\triangleright \\varphi & = \\mu ^ l ( v ) ^ { - 1 } \\left ( \\mu ^ l \\ ! \\left ( g ^ { - 1 } v \\ , ? \\right ) \\varphi ^ v \\right ) ^ { v ^ { - 1 } } \\end{align*}"}
-{"id": "17.png", "formula": "\\begin{align*} G _ s ^ { i } ( \\bar { Y } _ s , \\bar { Y } _ s ^ { - i } ) = - ( q ^ { i j } + q ^ { j i } ) ( e ^ { \\bar { Y } _ s } - e ^ { - \\bar { Y } _ s } ) - \\rho \\bar { Y } _ s , \\end{align*}"}
-{"id": "9217.png", "formula": "\\begin{align*} e ( G ' ) & = e ( S ) + e ( \\bar { S } ) + e ( \\bar { S } , S ) \\leq \\binom { s } { 2 } + \\frac { 1 } { 2 } \\sum _ { x \\in \\bar { S } } \\left ( d _ S ( x ) + d _ { G ' } ( x ) \\right ) \\leq \\binom { s } { 2 } + \\frac { 1 } { 2 } ( s - 1 + k ) ( n - s ) . \\end{align*}"}
-{"id": "2665.png", "formula": "\\begin{align*} f _ { j } ( z ) & = \\prod _ { k \\neq j , k = 1 } ^ { N + n } \\frac { z - \\varrho _ { k } } { 1 - z - \\varrho _ { k } } \\frac { 1 - \\varrho _ { j } - \\varrho _ { k } } { \\varrho _ { j } - \\varrho _ { k } } , j = 1 , \\ldots , N + n . \\end{align*}"}
-{"id": "2331.png", "formula": "\\begin{align*} & f \\in \\mathfrak { W } ( X ) \\mapsto T _ r ( f ) : \\mathfrak { W } ( X ) / \\mathfrak { I } _ r \\to \\mathfrak { W } ( X ) / \\mathfrak { I } _ r \\\\ & T _ r ( f ) ( g + \\mathfrak { I } _ r ) = g f + \\mathfrak { I } _ r \\end{align*}"}
-{"id": "270.png", "formula": "\\begin{align*} { { m + j } \\choose j } = \\sum _ { i = 0 } ^ j { { m + j - i - 1 } \\choose { j - i } } . \\end{align*}"}
-{"id": "241.png", "formula": "\\begin{align*} \\Delta _ \\mathbb { H } u = \\sum _ { i = 1 } ^ { 2 n } X _ i ^ 2 u \\end{align*}"}
-{"id": "7767.png", "formula": "\\begin{align*} \\| A + \\mu ( \\| B \\| _ { \\nu } A + \\lambda \\| A \\| _ { \\nu } B ) \\| _ { \\nu } \\geq \\nu \\bigg ( A y + \\mu ( \\nu ( B y ) A y + \\lambda \\nu ( A y ) B y ) \\bigg ) \\geq \\nu ( A y ) = \\| A \\| _ { \\nu } \\end{align*}"}
-{"id": "8850.png", "formula": "\\begin{align*} s _ 3 & = M \\left ( - 2 { M } ^ 2 \\mu { w } ^ 2 + { M } ^ 2 \\mu w + 4 M \\mu { w } ^ 2 + { M } ^ { 3 } - 3 { M } ^ 2 \\mu - M \\mu w - 2 \\mu { w } ^ 2 + { M } ^ 2 - M \\mu + 2 \\mu \\right ) \\\\ & - a \\cdot ( M - 1 ) \\left [ M \\left ( \\left ( M - 1 \\right ) ^ 2 - \\left ( w - 2 \\right ) ^ 2 \\right ) + ( 1 - w ) \\left ( M ^ 2 - w - 1 \\right ) \\right ] \\end{align*}"}
-{"id": "9536.png", "formula": "\\begin{align*} \\frac { 1 } { ( q ) _ N } \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - 1 ) ^ { n - 1 } n q ^ { n ( n + 1 ) / 2 } } { 1 - q ^ n } & + \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { q ^ { n ^ 2 } } { ( q ) _ n } \\sum _ { k = 1 } ^ n \\frac { q ^ k } { ( 1 - q ^ k ) ^ 2 } = \\frac { 1 } { ( q ) _ N } \\sum _ { n = 1 } ^ { \\infty } \\frac { n q ^ n ( 1 - q ^ { N n } ) } { 1 - q ^ n } . \\end{align*}"}
-{"id": "3127.png", "formula": "\\begin{align*} M ^ + _ { \\varphi } & : = \\{ x \\in A ^ + \\ | \\ \\varphi ( x ) < \\infty \\} , \\ M _ { \\varphi } : = \\mathrm { s p a n } \\ ; M ^ + _ { \\varphi } , \\\\ N _ { \\varphi } & : = \\{ x \\in A \\ | \\ \\varphi ( x ^ * x ) < \\infty \\} . \\end{align*}"}
-{"id": "4474.png", "formula": "\\begin{align*} u ( x ) = \\langle x , \\eta ^ { - 1 } ( x ) \\rangle , x \\in \\eta ( M ) \\subset \\mathbb { S } ^ n . \\end{align*}"}
-{"id": "8124.png", "formula": "\\begin{align*} h ( \\Gamma , t ) \\ = \\sum _ { F \\subseteq V } \\ell _ F ( \\Gamma _ F , t ) . \\end{align*}"}
-{"id": "2668.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { R } ^ { 2 } } d x d y \\ , e ^ { - \\frac { 1 } { 2 } x ^ { 2 } - \\frac { 1 } { 2 } y ^ { 2 } } \\frac { x - y } { x + y } \\Big ( e ^ { \\alpha x ^ { 2 } + \\beta y ^ { 2 } } - e ^ { \\alpha y ^ { 2 } + \\beta x ^ { 2 } } \\Big ) = \\frac { 4 \\pi ( \\alpha - \\beta ) } { ( 1 - \\alpha - \\beta ) \\sqrt { 1 - 2 \\alpha } \\sqrt { 1 - 2 \\beta } } . \\end{align*}"}
-{"id": "3633.png", "formula": "\\begin{align*} ( \\alpha _ 0 - \\beta _ 0 ) \\cdot \\alpha _ { - 2 } = & - \\frac { 1 } { 2 \\cdot 3 } ( a _ 1 - a _ { - 1 } ) + \\frac { 2 3 t - 2 } { 3 ^ 2 } ( a _ 2 + a _ { - 2 } ) + \\frac { 2 } { 3 ^ 2 } ( a _ 3 + a _ { - 3 } ) + \\frac { t } { 2 \\cdot 3 } v _ { ( 1 , 2 ) } - \\frac { 1 } { 3 } v _ { ( 2 , 3 ) } \\\\ & - \\frac { 2 ^ 3 } { 3 ^ 2 } ( a _ 1 + a _ { - 1 } ) \\cdot v _ { ( 2 , 3 ) } + \\frac { 2 0 } { 3 ^ 2 } ( a _ 2 + a _ { - 2 } ) \\cdot v _ { ( 1 , 3 ) } + \\frac { 2 ^ 3 } { 3 ^ 2 } ( a _ 3 + a _ { - 3 } ) \\cdot v _ { ( 1 , 2 ) } \\end{align*}"}
-{"id": "988.png", "formula": "\\begin{align*} I ( p ) = & \\langle a _ { d } b _ { d - 1 } , a _ { d } b _ { d - 2 } , \\dots , a _ { d } b _ { j } , \\dots , a _ { i + 1 } b _ { j } \\rangle : \\langle a _ { i } b _ { j } \\rangle = \\langle a _ d , \\dots , a _ { i + 1 } , b _ { i - 1 } , \\dots , b _ { j + 1 } \\rangle . \\end{align*}"}
-{"id": "6108.png", "formula": "\\begin{align*} \\xi _ { i i } & = \\log \\theta _ { i i } i = 1 , \\dots , p , \\\\ \\xi _ { i j } & = \\frac { \\theta _ { i j } } { \\theta _ { i i } } , 1 \\leq j < i \\leq p \\end{align*}"}
-{"id": "3909.png", "formula": "\\begin{align*} a / b = x / y = z / ( 1 - x ) . \\end{align*}"}
-{"id": "9237.png", "formula": "\\begin{align*} \\mathcal { F } [ u ] : = F ( D ^ 2 u - A ( \\cdot , u , D u ) ) = B ( \\cdot , u , D u ) , \\mbox { i n } \\ \\Omega , \\end{align*}"}
-{"id": "8128.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } \\widetilde { A } _ { n , r } ( t ) \\ , \\frac { z ^ n } { n ! } \\ = \\ \\frac { ( 1 - t ) e ^ { ( r t + 1 ) z } } { e ^ { r t z } - t e ^ { r z } } \\end{align*}"}
-{"id": "1968.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\ell \\in \\mathcal { N } ( 3 ) \\\\ \\ell \\neq 1 , 2 } } d _ { \\ell 2 } = 0 . \\end{align*}"}
-{"id": "5115.png", "formula": "\\begin{align*} \\operatorname { T r a c e } ( T ) & = \\sum _ { j = 1 } ^ m g _ j ( \\omega _ j ) = \\sum _ { j = 1 } ^ m g _ j \\left ( \\sum _ { k = 1 } ^ n f _ k ( \\omega _ j ) \\tau _ k \\right ) = \\sum _ { k = 1 } ^ n \\sum _ { j = 1 } ^ m f _ k ( \\omega _ j ) g _ j ( \\tau _ k ) \\\\ & = \\sum _ { k = 1 } ^ n f _ k \\left ( \\sum _ { j = 1 } ^ m g _ j ( \\tau _ k ) \\omega _ j \\right ) = \\sum _ { k = 1 } ^ n f _ k ( T \\tau _ k ) . \\end{align*}"}
-{"id": "5425.png", "formula": "\\begin{align*} \\lambda _ 1 ( A , \\mu ) > \\lambda _ 2 ( A , \\mu ) > \\cdots > \\lambda _ k ( A , \\mu ) > \\ldots > \\lambda _ \\infty ( A , \\mu ) = - \\infty , \\end{align*}"}
-{"id": "5582.png", "formula": "\\begin{align*} \\begin{cases} ( i \\partial _ t + \\partial _ x ^ 2 ) e ^ { i t \\partial _ x ^ 2 } \\phi ( x ) = 0 , & ( x , t ) \\in \\mathbb { R } \\times \\mathbb { R } , \\\\ e ^ { i t \\partial _ x ^ 2 } \\phi ( x ) \\big | _ { t = 0 } = \\phi ( x ) , & x \\in \\mathbb { R } . \\end{cases} \\end{align*}"}
-{"id": "8132.png", "formula": "\\begin{align*} \\overset { I } { T _ 1 } \\overset { J } { T _ 2 } = \\overset { I \\otimes J } { T } \\ ! \\ ! \\ ! _ { 1 2 } , \\ : \\ : 1 _ { H ^ * } = \\overset { \\mathbb { C } } { T } , \\ : \\ : \\Delta ( \\overset { I } { T ^ { \\ , a } _ { \\ , b } } ) = \\sum _ i \\overset { I } { T ^ { \\ , a } _ { \\ , i } } \\otimes \\overset { I } { T ^ { \\ , i } _ { \\ , b } } , \\ : \\ : \\varepsilon ( \\overset { I } { T } ) = I _ { \\dim ( I ) } , \\ : \\ : S ( \\overset { I } { T } ) = \\overset { I } { T } { ^ { - 1 } } \\end{align*}"}
-{"id": "7521.png", "formula": "\\begin{align*} \\mathfrak { S } ( \\xi , \\eta ) = 2 R _ { g ^ D } ( \\xi , J \\eta ^ \\sharp , \\xi , J \\eta ^ \\sharp ) - 2 R _ { g ^ D } ( \\eta ^ \\sharp , \\xi , \\eta ^ \\sharp , \\xi ) . \\end{align*}"}
-{"id": "8953.png", "formula": "\\begin{align*} \\mathbb { P } ( Y = 1 \\ , | \\ , X = x ) = \\frac { 1 } { 1 + \\exp \\left \\{ - \\beta _ 0 - \\sum _ { j = 1 } ^ p \\beta _ j ( x ( t _ j ) - m ( t _ j ) ) \\right \\} } \\end{align*}"}
-{"id": "9746.png", "formula": "\\begin{align*} \\frac { H _ \\lambda ' } { H _ \\lambda } ( \\rho , q ) = \\frac { 2 } { \\rho } ( N ( \\rho , q ) - \\lambda ) . \\end{align*}"}
-{"id": "4001.png", "formula": "\\begin{align*} \\begin{pmatrix} e ( \\frac { \\nu _ 0 } { 2 p } ) & e ( - \\frac { \\nu _ 0 } { 2 p } ) \\\\ e ( \\frac { \\mu \\nu _ 0 } { 2 p } ) & e ( - \\frac { \\mu \\nu _ 0 } { 2 p } ) \\end{pmatrix} . \\end{align*}"}
-{"id": "3826.png", "formula": "\\begin{align*} B ( T ) = \\int _ 0 ^ T S _ \\pm ( t , 0 ) ^ * B _ 0 S _ \\mp ( t , 0 ) d t , \\end{align*}"}
-{"id": "8776.png", "formula": "\\begin{align*} \\psi _ { 1 2 } = 0 \\ , , \\ \\ \\psi _ { a \\alpha } = 0 \\ , . \\end{align*}"}
-{"id": "6419.png", "formula": "\\begin{align*} B _ { k } = \\left [ \\begin{array} { c c c c c } \\alpha _ { 1 } & \\beta _ { 1 } \\\\ & \\alpha _ { 2 } & \\beta _ { 2 } \\\\ & & \\ddots & \\ddots \\\\ & & & \\ddots & \\beta _ { k - 1 } \\\\ & & & & \\alpha _ { k } \\end{array} \\right ] . \\end{align*}"}
-{"id": "9618.png", "formula": "\\begin{align*} U ( q , N ) & : = \\sum _ { n = 1 } ^ { N } \\frac { q ^ n ( - q ) _ { n - 1 } } { 1 - q ^ n } , \\\\ V ( q , N ) & : = \\frac { 1 - q ^ N } { 2 } \\left ( \\sum _ { n = 1 } ^ { \\infty } n q ^ n ( - q ^ { n + 1 } ) _ { N - 1 } + \\sum _ { n = 1 } ^ { \\infty } n q ^ n ( q ^ { n + 1 } ) _ { N - 1 } \\right ) . \\end{align*}"}
-{"id": "2064.png", "formula": "\\begin{align*} D \\frac { { \\partial C ( \\bar r , t | { { \\bar r } _ { \\rm t x } } , { t _ 0 } ) } } { { \\partial \\rho } } \\mid _ { \\bar r = ( \\rho _ c , z , \\varphi ) } = - k _ { f } C ( \\rho _ c , z , \\varphi , t | { { \\bar r } _ { \\rm t x } } , { t _ 0 } ) . \\end{align*}"}
-{"id": "193.png", "formula": "\\begin{align*} X = x ^ 1 \\left ( \\frac { \\partial v ^ 2 } { \\partial x ^ 2 } ( { \\bf x } ) + \\frac { \\partial v ^ 3 } { \\partial x ^ 3 } ( { \\bf x } ) \\right ) \\frac { \\partial } { \\partial x ^ 1 } + v ^ 2 ( { \\bf x } ) \\frac { \\partial } { \\partial x ^ 2 } + v ^ 3 ( { \\bf x } ) \\frac { \\partial } { \\partial x ^ 3 } , \\end{align*}"}
-{"id": "84.png", "formula": "\\begin{align*} L _ { \\pm } ( t ) & = \\sum _ { ( \\pm 1 ) - w a v e s } | \\Delta f ^ \\pm | \\ , , \\\\ L _ 0 ( t ) & = \\frac 1 2 \\left ( \\sum _ { 0 - w a v e s } | \\Delta f ^ + | + | \\Delta f ^ - | \\right ) \\end{align*}"}
-{"id": "10073.png", "formula": "\\begin{align*} \\omega ( \\partial _ u K ( u , \\theta _ n ^ 0 , \\varphi ) , \\partial _ { \\theta _ n ^ 0 } K ( u , \\theta _ n ^ 0 , \\varphi ) ) = \\lim _ { t \\to \\infty } \\Psi ^ * _ t \\omega ( \\partial _ u K ( u , \\theta _ n ^ 0 , \\varphi ) , \\partial _ { \\theta _ n ^ 0 } K ( u , \\theta _ n ^ 0 , \\varphi ) ) = 0 . \\end{align*}"}
-{"id": "1535.png", "formula": "\\begin{align*} - v '' ( x ) + m ^ 2 \\ , v ( x ) = f ( x ) , \\ ; x \\in I , \\ ; m \\ge 0 , v ( 0 ) = v ( 1 ) = 0 . \\end{align*}"}
-{"id": "3551.png", "formula": "\\begin{align*} \\mathtt { a } ^ { \\ell _ { + } } \\square _ { l } = \\mathtt { a } ^ { \\ell _ { + } } \\left ( \\frac { \\partial \\ ; } { \\partial \\mathtt { a } } \\right ) ^ { \\ell _ { + } } - \\mathtt { a } ^ { \\ell } \\cdot \\mathtt { a } ^ { \\ell _ { - } } \\left ( \\frac { \\partial \\ ; } { \\partial \\mathtt { a } } \\right ) ^ { \\ell _ { - } } , \\end{align*}"}
-{"id": "1045.png", "formula": "\\begin{align*} A = \\chi _ { I _ { n , k } } P _ n \\chi _ { I _ { n , k } } , \\ B = \\sum \\limits _ { j = 1 } ^ k B _ j , \\ B _ j = \\chi _ { I ( y _ j , F _ n ( x _ j ) ) } P _ n \\chi _ { I ( y _ j , F _ n ( x _ j ) ) } , \\end{align*}"}
-{"id": "5714.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\left ( \\int _ 0 ^ t ( t - z ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } [ \\lambda ( t - z ) ^ { \\alpha } ] f ( z ) d z \\right ) & = \\frac { d } { d t } \\int _ 0 ^ t ( t - z ) ^ { \\alpha - 2 } E _ { \\alpha , \\alpha - 1 } ( \\lambda ( t - z ) ^ { \\alpha } ) f ( z ) d z . \\end{align*}"}
-{"id": "1687.png", "formula": "\\begin{align*} 2 ^ q \\vert 2 ^ { 2 ^ q - 1 } \\vert 2 ^ { 2 ^ q - 1 } \\cdot 2 ^ { 2 ^ { q + 1 } p - 2 ^ q } = 2 ^ { 2 ^ { q + 1 } p - 1 } = 2 ^ { 2 n - 1 } . \\end{align*}"}
-{"id": "6387.png", "formula": "\\begin{align*} P ( X _ D = i _ D ) = & P ( X _ { C _ 1 } = i _ { C _ 1 } ) \\\\ \\times & P ( X _ { C _ 2 } = i _ { C _ 2 } ) \\cdots P ( X _ { C _ r } = i _ { C _ r } ) , \\end{align*}"}
-{"id": "6896.png", "formula": "\\begin{align*} N = d y - d y ^ T = 0 \\ , . \\end{align*}"}
-{"id": "9594.png", "formula": "\\begin{align*} & 1 + ( 1 - z ) ( 1 - z ^ { - 1 } ) \\sum _ { n = 1 } ^ { N } \\frac { ( - 1 ) ^ n ( 1 + q ^ n ) q ^ { \\frac { n ( 3 n + 1 ) } { 2 } } ( q ) _ { N } ^ 2 } { ( 1 - z q ^ n ) ( 1 - z ^ { - 1 } q ^ n ) ( q ) _ { N - n } ( q ) _ { N + n } } \\\\ & = \\frac { ( q ) _ N ^ { 2 } } { ( z q ) _ N ( z ^ { - 1 } q ) _ N } \\sum _ { n = 0 } ^ { N } \\frac { ( z ) _ { n } ( z ^ { - 1 } ) _ { n } q ^ n } { ( q ) _ n } . \\end{align*}"}
-{"id": "4395.png", "formula": "\\begin{align*} f : = \\chi f _ o - u \\end{align*}"}
-{"id": "7415.png", "formula": "\\begin{align*} \\mu ^ U _ x ( \\Phi ^ U _ { 1 / N } ( \\mathcal { Z } _ U \\cap \\mathcal F | U ( x ) ) ) = 0 . \\end{align*}"}
-{"id": "8158.png", "formula": "\\begin{align*} \\varphi \\ast \\psi = \\varphi \\ ! \\left ( ? b _ j b _ i \\right ) \\star \\psi \\ ! \\left ( S ^ { - 1 } ( a _ i ) ? a _ j \\right ) . \\end{align*}"}
-{"id": "2780.png", "formula": "\\begin{align*} \\mathbb { P } ( \\mathbf { J } ^ t = \\mathbf { j } ; \\mathbf { R } ^ t = \\mathbf { r } ) = \\mathbb { P } ( \\mathbf { J } ^ t = \\mathbf { j } ) \\mathbb { P } ( \\mathbf { R } ^ t = \\mathbf { r } ) . \\end{align*}"}
-{"id": "4484.png", "formula": "\\begin{align*} G ^ { i j } = \\frac { \\partial G } { \\partial \\nabla _ { i j } u } \\quad \\textrm { a n d } \\tilde G ^ i = \\frac { \\partial G } { \\partial \\nabla _ i u } + g ^ { i k } \\sigma \\left ( \\nabla \\psi ( N ) , \\frac { 1 } { u ^ 2 w } e _ k - \\frac { \\nabla _ l u } { u ^ 3 w } x \\right ) \\end{align*}"}
-{"id": "5489.png", "formula": "\\begin{align*} L ^ * = L _ b ^ * = \\frac { 1 } { 2 } \\Delta - b . \\nabla - d i v ( b ) , ~ ~ ~ d i v ( b ) = \\sum _ { j = 1 } ^ d \\frac { \\partial b _ j } { \\partial x _ j } , \\end{align*}"}
-{"id": "1274.png", "formula": "\\begin{align*} T _ { 5 , 2 } = \\frac { 2 \\pi } { 3 } A | A | ^ 2 \\frac { e ^ { i a \\ln | \\xi | } } { | \\xi | } + O ( | \\xi | ^ { - 2 + \\gamma / 2 } ) . \\end{align*}"}
-{"id": "2781.png", "formula": "\\begin{align*} \\mathcal { F } ( \\alpha ) = 1 - p + p e ^ { \\alpha } . \\end{align*}"}
-{"id": "3012.png", "formula": "\\begin{align*} \\Big ( \\sum _ { m = 1 } ^ \\infty x _ m \\frac { t ^ m } { m ! } \\Big ) & = \\sum _ { n = 0 } ^ \\infty B _ n ( x _ 1 , \\dots , x _ n ) \\frac { t ^ n } { n ! } , \\end{align*}"}
-{"id": "8476.png", "formula": "\\begin{align*} \\| \\partial _ { \\alpha } ^ n q \\| _ { L ^ 2 } \\leq & \\sum _ { k = 1 } ^ n \\sum \\frac { n ! } { ( k _ 1 ) ! . . . ( k _ n ) ! } \\prod _ { l = 1 } ^ n \\frac { ( 1 + 5 \\epsilon ) ^ { k _ l } } { ( l ! ) ^ { k _ l } } \\times ( 1 0 0 ( k + 1 ) ! | \\lambda x ( t ) | d _ I ( t ) ^ { - 3 / 2 } ) \\\\ \\leq & 4 0 0 S ( n ) | \\lambda | x ( 0 ) ( n + 1 ) ! d _ I ( t ) ^ { - 3 / 2 } , \\end{align*}"}
-{"id": "3603.png", "formula": "\\begin{align*} u ^ { \\psi _ { 4 X } } \\cdot v ^ { \\psi _ { 4 X } } = ( u \\cdot v ) ^ { \\psi _ { 4 X } } \\textrm { a n d } \\langle u ^ { \\psi _ { 4 X } } , v ^ { \\psi _ { 4 X } } \\rangle = \\langle u , v \\rangle ^ { \\phi _ { 4 X } } \\end{align*}"}
-{"id": "1449.png", "formula": "\\begin{align*} V I I _ { A } \\leq \\sum _ { i = 1 } ^ { d } \\varepsilon F ( M _ { n } ) \\int _ { t } ^ { T } \\int _ { \\mathbb { T } ^ { d } } \\frac { 1 } { 2 \\sigma _ { 1 0 } } | \\partial ^ { \\alpha } \\partial _ { x _ { j } } w ^ { n + 1 } | ^ { 2 } + \\frac { \\sigma _ { 1 0 } } { 2 } | \\partial ^ { \\alpha } \\partial ^ { 2 } _ { x _ { i } x _ { j } } w ^ { n } | \\ d x d \\tau . \\end{align*}"}
-{"id": "5005.png", "formula": "\\begin{align*} \\bigcup _ { m = 1 } ^ M \\psi _ m ( V ) \\subset V , \\end{align*}"}
-{"id": "5233.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow + \\infty } \\frac { \\left \\vert g ( t x ) - g ( t ) \\right \\vert } { g ( t ) } = 0 , \\end{align*}"}
-{"id": "330.png", "formula": "\\begin{align*} M _ { \\Phi } = ( 4 m + 3 ) L \\begin{pmatrix} 2 m + 1 & 2 m + 1 & 2 m ^ 2 + m \\\\ 0 & 4 m + 2 & 4 m ^ 2 + 4 m + 1 \\\\ 4 m + 2 & 0 & 4 m ^ 2 + 4 m + 1 \\\\ 0 & 0 & 8 m ^ 2 + 1 2 m + 4 \\end{pmatrix} \\end{align*}"}
-{"id": "3709.png", "formula": "\\begin{align*} \\begin{cases} \\sum \\limits _ { i \\in B } 1 / \\hat { p } _ i = N \\\\ \\sum \\limits _ { i \\in U } \\hat { p } _ i = n _ B \\end{cases} \\end{align*}"}
-{"id": "8721.png", "formula": "\\begin{align*} E ( T ^ * _ { m , N } ) = ( N - 1 ) \\sum _ { j \\leq m } \\hat { \\nu } ( j ) \\sum _ { k = j } ^ { N - 1 } { 1 \\over N - k } . \\end{align*}"}
-{"id": "5726.png", "formula": "\\begin{align*} y ' ( 0 ) & = \\frac { d } { d x } ( b _ 0 E _ { \\alpha , 1 } ( \\lambda x ^ { \\alpha } ) ) | _ { x = 0 } + \\frac { d } { d x } ( b _ 1 x E _ { \\alpha , 2 } ( \\lambda x ^ { \\alpha } ) ) | _ { x = 0 } + \\frac { d } { d x } \\left ( \\int _ 0 ^ x ( x - t ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } [ \\lambda ( x - t ) ^ { \\alpha } ] f ( t ) d t \\right ) \\Big | _ { x = 0 } . \\end{align*}"}
-{"id": "5907.png", "formula": "\\begin{align*} T _ { s _ 0 ( s _ 1 s _ 0 ) ^ n } \\star \\varphi _ m = q ^ { 2 n + 1 } \\varphi _ { m + n } + ( q - 1 ) \\sum _ { k = 0 } ^ { 2 n } q ^ { 2 n - k } \\varphi _ { m + n - k } ; \\end{align*}"}
-{"id": "7981.png", "formula": "\\begin{align*} \\left | ^ 2 B _ 2 ( q ) \\right | = q ^ 2 ( q ^ 2 + 1 ) ( q - 1 ) = q ^ 2 ( q - 1 ) ( q + \\sqrt { 2 q } + 1 ) ( q - \\sqrt { 2 q } + 1 ) \\ , , \\end{align*}"}
-{"id": "345.png", "formula": "\\begin{align*} 2 t = m i n ( r - 1 , s ) + r . \\end{align*}"}
-{"id": "4929.png", "formula": "\\begin{align*} y z ^ { 2 n } = z ^ n ( x ^ { n + 1 } - x z ^ { n ^ 4 - n } + y z ^ n ) + x ( z ^ { n ^ 4 } - z ^ n x ^ n ) \\end{align*}"}
-{"id": "7568.png", "formula": "\\begin{align*} \\begin{gathered} \\Gamma ( \\tau \\circ X ^ { - 1 } ) ( t ) = \\int _ 0 ^ t \\Delta g _ { \\nu ( t - s ) } * \\left ( \\left ( \\tau \\circ X ^ { - 1 } \\right ) ( s ) - \\left ( \\tau \\circ X ^ { - 1 } \\right ) ( t ) \\right ) d s \\\\ + \\int _ 0 ^ t \\Delta g _ { \\nu ( t - s ) } * \\left ( \\tau \\circ X ^ { - 1 } \\right ) ( t ) d s . \\end{gathered} \\end{align*}"}
-{"id": "4891.png", "formula": "\\begin{align*} C _ 0 ( U ) + C _ 0 ( V ) = \\{ g _ U + g _ v \\in C _ 0 ( X ) \\mid g _ U \\in C _ 0 ( U ) , g _ V \\in C _ 0 ( V ) \\} \\end{align*}"}
-{"id": "338.png", "formula": "\\begin{align*} M _ { \\Phi } = \\frac { n ( n - 1 ) } { 2 } L \\begin{pmatrix} 1 & 0 & 0 \\\\ 2 & 0 & 0 \\\\ 0 & 1 & n + 1 \\\\ 0 & 0 & 2 n + 2 \\end{pmatrix} \\end{align*}"}
-{"id": "562.png", "formula": "\\begin{align*} \\Delta A & : = \\{ a - a ' : a , a ' \\in A , a \\ne a ' \\} , \\\\ \\Delta _ + A & : = \\{ a + a ' : a , a ' \\in A , a \\ne - a ' \\} , \\\\ A - B & : = \\{ a - b : a \\in A , b \\in B , a \\ne b \\} , \\\\ A + B & : = \\{ a + b : a \\in A , b \\in B , a \\ne - b \\} , \\\\ A + g & : = \\{ a + g : a \\in A \\} . \\end{align*}"}
-{"id": "8733.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { k = j } ^ m \\binom { m } { k } \\binom { k } { j } ( - 1 ) ^ { k - j } = & \\sum _ { k = j } ^ m ( - 1 ) ^ { k - j } \\frac { m ! } { k ! ( m - k ) ! } \\frac { k ! } { ( k - j ) ! j ! } \\\\ = & \\frac { m ( m - 1 ) \\cdots ( m - j + 1 ) } { j ! } \\sum _ { k = j } ^ m \\frac { ( m - j ) ! ( - 1 ) ^ { k - j } } { ( m - k ) ! ( k - j ) ! } \\\\ = & \\frac { m ( m - 1 ) \\cdots ( m - j + 1 ) } { j ! } ( 1 - 1 ) ^ { m - j } = 0 , { \\quad } j < m . \\end{aligned} \\end{align*}"}
-{"id": "8268.png", "formula": "\\begin{align*} A _ 1 : = A | Z _ { \\alpha } | ^ 2 . \\end{align*}"}
-{"id": "8125.png", "formula": "\\begin{align*} h ( \\Delta ( \\Gamma ) , t ) \\ = \\ \\sum _ { F \\subseteq V } t ^ { n - | F | } \\ , h ( \\Gamma _ F , t ) . \\end{align*}"}
-{"id": "5955.png", "formula": "\\begin{align*} d Y ( t ) = - g ^ * \\left ( t , \\sqrt { n } Z ( t ) \\right ) \\ , d t + Z ( t ) \\ , d W ( t ) , Y ( 1 ) = F \\left ( \\frac { W } { \\sqrt { n } } \\right ) . \\end{align*}"}
-{"id": "6597.png", "formula": "\\begin{align*} A _ n & = | 2 F _ n \\sin ( \\pi \\varphi ^ n ) | , \\\\ B _ n & = \\prod _ { t = 1 } ^ { F _ n - 1 } \\left | \\frac { s _ { n t } } { 2 \\sin ( \\pi t / F _ n ) } \\right | , \\\\ C _ n & = \\prod _ { t = 1 } ^ { ( F _ n - 1 ) / 2 } \\left ( 1 - \\frac { s _ { n 0 } ^ 2 } { s _ { n t } ^ 2 } \\right ) , \\end{align*}"}
-{"id": "8967.png", "formula": "\\begin{align*} \\mu _ n = \\beta _ n \\textbf { 1 } _ { E _ n \\cap \\mathcal { Z } } \\ , , \\end{align*}"}
-{"id": "625.png", "formula": "\\begin{align*} \\varphi ^ \\ast ( v y ) \\cdot u x = ( \\varphi ^ y ) ^ \\ast ( v ) \\cdot ( \\varphi ^ \\ast ( y ) \\cdot u \\cdot \\varphi ^ \\ast ( y ) ^ { - 1 } ) \\cdot \\varphi ^ \\ast ( y ) x \\ . \\end{align*}"}
-{"id": "5287.png", "formula": "\\begin{align*} & \\phi _ 0 ( \\sigma ) = \\{ ( I , I ) \\ , , \\ , ( 0 , I ) \\} \\ ; , \\\\ & \\phi _ 1 ( \\sigma ) = \\{ ( I \\oplus I ^ { ( 1 ) } , I ) \\} \\ ; , \\\\ & \\phi _ 2 ( \\sigma ) = \\emptyset \\ ; , \\\\ & \\phi _ 3 ( \\sigma ) = \\{ ( I ^ { ( 3 ) } , I \\oplus I ^ { ( 3 ) } \\} ) \\ ; , \\\\ & \\phi _ k ( \\sigma ) = \\emptyset \\end{align*}"}
-{"id": "6669.png", "formula": "\\begin{align*} E _ { n } = \\frac { \\langle \\psi _ { n } | H | \\psi _ { n } \\rangle } { \\langle \\psi _ { n } | \\psi _ { n } \\rangle } = \\frac { ( | Q | \\psi _ { n } \\rangle | ^ { 2 } + | Q ^ { \\dagger } | \\psi _ { n } \\rangle | ^ { 2 } ) } { \\langle \\psi _ { n } | \\psi _ { n } \\rangle } \\geq 0 . \\end{align*}"}
-{"id": "1582.png", "formula": "\\begin{align*} w _ { n ^ 2 + k } = \\frac { 1 } { n ^ 2 + k + O ( \\log n ) } , \\end{align*}"}
-{"id": "2542.png", "formula": "\\begin{align*} \\lim _ { ( 0 , \\infty ) \\ni t \\to 0 } \\frac { \\int _ E f ( x , u t ) \\rho ( d x ) } { \\int _ E f ( x , t ) \\rho ( d x ) } = \\lim _ { ( 0 , \\infty ) \\ni t \\to 0 } \\frac { \\int _ E f ( x , t ) \\rho ( d x ) } { \\int _ E f ( x , u ^ { - 1 } t ) \\rho ( d x ) } = \\big ( ( u ^ { - 1 } ) ^ { \\alpha _ 0 } \\big ) ^ { - 1 } = u ^ { \\alpha _ 0 } . \\end{align*}"}
-{"id": "1179.png", "formula": "\\begin{align*} | { \\det } _ 2 ( + A ) - { \\det } _ 2 ( + B ) | = | g ( 1 ) - g ( 0 ) | \\leq | g ( 1 ) - g ( 0 ) - g ' ( 0 ) | + | g ' ( 0 ) | \\\\ \\leq \\frac { 1 } { 2 } \\sup _ { 0 \\leq t \\leq 1 } | g '' ( t ) | + | g ' ( 0 ) | \\leq \\frac { 1 } { \\rho ^ 2 } \\sup _ { | \\lambda - 1 / 2 | \\leq \\rho + 1 / 2 } | g ( \\lambda ) | + | g ' ( 0 ) | . \\end{align*}"}
-{"id": "7671.png", "formula": "\\begin{align*} \\int \\limits _ { \\mathbb T ^ d } \\hat a _ { \\rm s y m } ( \\xi - \\eta ) \\ , \\mu ( \\xi , \\eta ) \\ , ( r _ { \\boldsymbol { \\ell } } ( \\eta ) - r _ { \\boldsymbol { \\ell } } ( \\xi ) ) d \\eta \\ = \\ 2 g _ { \\boldsymbol { \\ell } } ( \\xi ) , r _ { \\boldsymbol { \\ell } } \\in ( L ^ 2 ( \\mathbb { T } ^ d ) ) ^ d . \\end{align*}"}
-{"id": "6079.png", "formula": "\\begin{align*} \\delta _ 1 = \\frac { 1 } { n - m - 1 } - \\frac { 1 } { n - k - 1 } , \\qquad \\delta _ 2 = \\frac { 1 } { k } - \\frac { 1 } { n - k - 1 } , \\qquad \\delta _ 3 = \\frac { 1 } { n - k - 1 } - \\frac { 1 } { p } , \\end{align*}"}
-{"id": "6830.png", "formula": "\\begin{align*} N _ { 1 2 } = \\sum _ { i = 1 } ^ 6 f _ i g _ i = B _ { \\ell k } \\big ( u ^ \\ell _ { ; 1 } u ^ k _ { ; 2 } - u ^ \\ell _ { ; 2 } u ^ k _ { ; 1 } \\big ) \\ , , \\end{align*}"}
-{"id": "8227.png", "formula": "\\begin{align*} U _ 5 ( F \\rho t ^ i ) = F \\Big ( ( \\sum _ { j = \\left \\lceil \\frac { i } { 5 } \\right \\rceil } ^ \\infty x _ 0 ( i , j ) t ^ j ) + \\rho ( \\sum _ { j = \\left \\lceil \\frac { i } { 5 } \\right \\rceil } ^ \\infty x _ 1 ( i , j ) t ^ j ) \\Big ) , \\end{align*}"}
-{"id": "3649.png", "formula": "\\begin{align*} B = \\{ a _ 1 , a _ { - 1 } , a _ 2 , a _ { - 2 } , a _ 3 , a _ { - 3 } , v _ { ( 1 , 2 ) } , v _ { ( 1 , 3 ) } , v _ { ( 2 , 3 ) } , a _ 1 \\cdot v _ { ( 2 , 3 ) } , a _ 2 \\cdot v _ { ( 1 , 3 ) } , a _ 3 \\cdot v _ { ( 1 , 2 ) } \\} . \\end{align*}"}
-{"id": "8813.png", "formula": "\\begin{align*} v ( t , x ) = \\int _ 0 ^ t \\int _ { 0 } ^ { \\pi } G _ { t - s } ( x , y ) f ( v ( s , y ) ) \\ , \\widetilde { W } ( \\textrm { d } s , \\textrm { d } y ) \\widetilde { \\mathbb { P } } \\textrm { - a l m o s t s u r e l y } \\end{align*}"}
-{"id": "3888.png", "formula": "\\begin{align*} \\frac { 3 } { 2 } K ( \\phi _ \\omega , \\psi _ \\omega ) + \\frac { 5 } { 2 } \\omega M ( \\phi _ \\omega , \\psi _ \\omega ) = 5 P ( \\phi _ \\omega , \\psi _ \\omega ) . \\end{align*}"}
-{"id": "3026.png", "formula": "\\begin{align*} L ^ m _ { p , q } f ( z ) & = L ^ m _ { p , q } h ( z ) + ( - 1 ) ^ m \\overline { L ^ m _ { p , q } g ( z ) } = z + \\sum _ { { k = 2 } } ^ { \\infty } [ k ] ^ m _ { p , q } a _ { k } z ^ { k } + ( - 1 ) ^ m \\sum _ { { k = 1 } } ^ { \\infty } [ k ] ^ m _ { p , q } \\overline { b _ { k } z ^ { k } } . \\end{align*}"}
-{"id": "8827.png", "formula": "\\begin{align*} \\max _ { ( \\zeta , \\omega ) \\in \\mathcal R } \\sigma ^ \\circ _ { \\partial G _ i ( \\hat x ^ { ( i ) } ) } ( ( A ^ f ) _ i ^ \\top \\zeta + \\omega ^ { ( i ) } ) < 1 \\Rightarrow x ^ { ( i ) } _ \\star = \\hat x ^ { ( i ) } \\ ; . \\end{align*}"}
-{"id": "1943.png", "formula": "\\begin{align*} \\mathcal { A } _ p = \\sum _ { j = 1 } ^ p ( \\epsilon _ j - \\tan \\epsilon _ j ) + ( \\chi + 1 ) \\chi ^ 2 \\sum _ { j = 1 } ^ p \\frac { A _ j ^ 2 } { Z _ j ^ 3 } \\end{align*}"}
-{"id": "7821.png", "formula": "\\begin{align*} \\lim _ n | [ \\langle T x _ n , S x _ n \\rangle \\xi _ n , \\xi _ n ] | ^ 2 & \\leq \\lim _ n \\| \\langle T x _ n , S x _ n \\rangle \\xi _ n \\| ^ 2 \\\\ & = \\lim _ n [ \\langle T x _ n , S x _ n \\rangle ^ * \\langle T x _ n , S x _ n \\rangle \\xi _ n , \\xi _ n ] \\\\ & \\leq \\| T \\| ^ 2 \\lim _ n | [ \\langle S x _ n , S x _ n \\rangle \\xi _ n , \\xi _ n ] | \\\\ & \\leq \\| T \\| ^ 2 \\| S \\| ^ 2 , \\end{align*}"}
-{"id": "8885.png", "formula": "\\begin{align*} s _ { \\alpha } = \\bar { g } _ { \\alpha , \\beta } \\cdot s _ { \\beta } o n U _ { \\alpha } \\cap U _ { \\beta } , \\end{align*}"}
-{"id": "5281.png", "formula": "\\begin{align*} 1 - F ( x ) = k ( x _ 0 ) r ( x _ 0 ) \\exp \\left ( \\int _ { x _ 0 } ^ { x } \\frac { \\ell ( x ) } { r ( t ) } d t \\right ) , \\ x _ 0 \\leq x < u e p ( F ) . \\end{align*}"}
-{"id": "8663.png", "formula": "\\begin{align*} R _ 0 ( \\l ) = - \\frac { 1 } { m _ + ( \\l ) + m _ - ( \\l ) } , R _ 1 ( \\l ) = \\frac { m _ + ( \\l ) m _ - ( \\l ) } { m _ + ( \\l ) + m _ - ( \\l ) } \\end{align*}"}
-{"id": "4527.png", "formula": "\\begin{align*} L ( A ^ { k } _ 1 , A ^ { k } _ 2 ) = \\bigcup ^ { l \\cap A ^ k _ 1 \\neq \\emptyset , l \\cap A ^ k _ 2 \\neq \\emptyset } _ { l } l \\cap F . \\end{align*}"}
-{"id": "6416.png", "formula": "\\begin{align*} c ^ { 2 } = \\frac { 2 \\sigma ^ { 2 } } { \\chi ^ { 2 } + \\left ( \\rho - \\tau \\right ) \\chi } = \\frac { 1 } { 2 } \\frac { \\chi ^ { 2 } - \\left ( \\rho - \\tau \\right ) ^ { 2 } } { \\chi ^ { 2 } + \\left ( \\rho - \\tau \\right ) \\chi } = \\frac { 1 } { 2 } \\left ( 1 - \\frac { \\rho - \\tau } { \\chi } \\right ) . \\end{align*}"}
-{"id": "4272.png", "formula": "\\begin{align*} \\textup { s a t } ( n , K _ s ) = ( s - 2 ) ( n - s + 2 ) + \\binom { s - 2 } { 2 } . \\end{align*}"}
-{"id": "3175.png", "formula": "\\begin{align*} \\mathcal { T } _ \\pm = d ^ * \\ker ( T \\mp 1 ) , \\mathcal { B } _ \\pm = \\ker ( \\varGamma \\pm 1 ) \\cap \\ker d \\end{align*}"}
-{"id": "4609.png", "formula": "\\begin{align*} \\angle x _ 1 x _ 0 x _ 3 = \\angle z _ 1 x _ 0 z _ 3 . \\end{align*}"}
-{"id": "6434.png", "formula": "\\begin{align*} \\sigma _ { k } = - \\sqrt { \\gamma _ { k } \\frac { \\delta _ { k } } { \\gamma _ { k - 1 } } } \\left ( s _ { k - 1 } \\sigma _ { k - 1 } + c _ { k - 1 } \\tau _ { k - 1 } \\right ) , \\qquad \\tau _ { k } = \\gamma _ { k } \\left ( \\delta _ { k } \\frac { \\tau _ { k - 1 } } { \\gamma _ { k - 1 } } + 1 \\right ) . \\end{align*}"}
-{"id": "8334.png", "formula": "\\begin{align*} ( I - \\mathbb { H } ) \\frac { 1 } { Z ( \\alpha , t ) - z _ 0 } = \\frac { 2 } { c _ 1 ( \\alpha - w _ 0 ) } , c _ 1 = ( \\Phi ^ { - 1 } ) _ z ( w _ 0 ) , w _ 0 = \\Phi ( z _ 0 , t ) . \\end{align*}"}
-{"id": "8799.png", "formula": "\\begin{align*} \\left [ \\begin{array} { l l } \\partial _ t \\widetilde { v } ( t , x ) = \\partial _ { x x } \\widetilde { v } ( t , x ) + f ( \\widetilde { v } ( t , x ) ) \\dot { \\widetilde { W } } ( t , x ) , & ( t , x ) \\in [ 0 , T ] \\times [ 0 , \\pi ] , \\\\ \\widetilde { v } ( t , 0 ) = \\widetilde { v } ( t , \\pi ) = 0 , & \\textrm { f o r a l l } \\ , \\ , t \\in [ 0 , T ] , \\\\ \\widetilde { v } ( 0 , x ) = 0 , & \\textrm { f o r a l l } \\ , \\ , x \\in [ 0 , \\pi ] , \\end{array} \\right . \\end{align*}"}
-{"id": "2143.png", "formula": "\\begin{align*} \\mathcal { T } _ m f = \\mathcal { F } ^ { - 1 } ( m \\mathcal { F } f ) , \\end{align*}"}
-{"id": "6302.png", "formula": "\\begin{align*} \\mathbb { P } \\left [ A ^ c _ { S , \\rho , \\frac { C } { 2 } } \\right ] \\leq 2 \\exp \\left ( - C k \\log n / 1 6 \\right ) = 2 n ^ { - \\frac { C k } { 1 6 } } . \\end{align*}"}
-{"id": "1467.png", "formula": "\\begin{align*} V _ { 9 } = - \\varepsilon \\int _ { \\mathbb { T } ^ { d } } ( \\mu ^ { 1 } - \\mu ^ { 2 } ) ( \\mu ^ { 2 } ) \\sum _ { i = 1 } ^ { d } \\sum _ { j = 1 } ^ { d } \\left [ \\Theta _ { p _ { i } p _ { j } } ( t , x , \\mu ^ { 2 } , D w ^ { 2 } ) \\left ( \\frac { \\partial ^ { 2 } ( w ^ { 1 } - w ^ { 2 } ) } { \\partial x _ { i } \\partial x _ { j } } \\right ) \\right ] \\ d x , \\end{align*}"}
-{"id": "8441.png", "formula": "\\begin{align*} ( I - \\mathfrak { { H } } ) b \\circ \\kappa = - [ z _ t , \\mathfrak { H } ] \\frac { \\bar { z } _ { \\alpha } - 1 } { z _ { \\alpha } } - \\frac { i } { \\pi } \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j } { z ( \\alpha , t ) - z _ j ( t ) } . \\end{align*}"}
-{"id": "407.png", "formula": "\\begin{align*} K _ { \\hat { g } } = e ^ { - \\omega } ( K _ { g } - \\Delta _ g \\omega ) . \\end{align*}"}
-{"id": "6676.png", "formula": "\\begin{align*} O & = \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n + 1 } \\exp \\left ( ( \\ln n ) \\ , x \\frac { d } { d x } \\right ) , \\\\ O ^ { \\dagger } & = \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { n + 1 } } { n } \\exp \\left ( ( \\ln n ^ { - 1 } ) \\ , x \\frac { d } { d x } \\right ) , \\end{align*}"}
-{"id": "650.png", "formula": "\\begin{align*} ( x _ \\ast \\vec a ) ^ \\varphi _ j = ( x ' _ \\ast \\vec a ) ^ { \\varphi ' } _ j \\ . \\end{align*}"}
-{"id": "8859.png", "formula": "\\begin{align*} c _ { 1 b } & = 6 1 4 4 0 ( 1 - \\mu ) + ( 1 - \\nu ) [ c _ { 1 b 1 } + c _ { 1 b 2 } \\mu ] \\end{align*}"}
-{"id": "8127.png", "formula": "\\begin{align*} \\sum _ { k \\ge 0 } \\chi _ { k P } ( w ) t ^ k \\ = \\ \\frac { \\varphi ^ \\ast _ P ( t ) ( w ) } { ( 1 - t ) \\prod _ { i \\ge 1 } ( 1 - t ^ { \\lambda _ i ( w ) } ) } \\end{align*}"}
-{"id": "5573.png", "formula": "\\begin{align*} & \\mathcal { E } _ 1 ( s ) = 2 \\ , ( u _ x ( 0 , s ) \\bar { u } _ s ( 0 , s ) ) + \\tfrac { \\alpha } { \\gamma } v _ x ( 0 , s ) v _ s ( 0 , s ) , \\\\ & \\mathcal { E } _ 2 ( s ) = \\tfrac { \\alpha } { 2 \\gamma } \\big [ \\ , v _ { x x } ( 0 , s ) + \\tfrac 1 2 v ^ 2 ( 0 , s ) - \\gamma | u ( 0 , s ) | ^ 2 \\ , \\big ] ^ 2 . \\end{align*}"}
-{"id": "1095.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } | \\mathbb P ( \\tau _ k ^ * \\in I _ 1 ) - \\mathbb P ( \\widetilde { \\tau } _ k \\in I _ 1 ) | = 0 . \\end{align*}"}
-{"id": "3182.png", "formula": "\\begin{align*} & \\mathcal { B } _ \\pm ( - U , \\varGamma ) = \\ker ( \\varGamma \\pm 1 ) \\cap \\ker ( - C + 1 ) = \\mathcal { T } _ \\mp ( U , \\varGamma ) , \\\\ & \\mathcal { T } _ \\pm ( - U , \\varGamma ) = \\ker ( \\varGamma \\mp 1 ) \\cap \\ker ( - C - 1 ) = \\mathcal { B } _ \\mp ( U , \\varGamma ) , \\end{align*}"}
-{"id": "7498.png", "formula": "\\begin{align*} E P _ { \\nu } = \\left ( \\frac { 1 - d } { 2 } + \\nu \\right ) P _ { \\nu } \\end{align*}"}
-{"id": "3328.png", "formula": "\\begin{align*} w ( B _ { \\lambda , \\mu } ) : = \\sum _ { T \\in B _ { \\lambda , \\mu } } w ( T ) . \\end{align*}"}
-{"id": "652.png", "formula": "\\begin{align*} \\left ( ( \\chi ; \\vec h ; [ z ] ) \\circ ( \\psi ; \\vec g ; [ y ] ) \\right ) \\circ ( \\varphi ; \\vec f ; [ x ] ) = ( \\chi ; \\vec h ; [ z ] ) \\circ \\left ( ( \\psi ; \\vec g ; [ y ] ) \\circ ( \\chi ; \\vec h ; [ z ] ) \\right ) \\ . \\end{align*}"}
-{"id": "7429.png", "formula": "\\begin{align*} \\displaystyle \\lim \\limits _ { z \\to z _ 0 } \\lVert \\ell _ z ( f ) - \\ell _ { z _ 0 } ( f ) \\rVert _ { M , d , D } = 0 . \\end{align*}"}
-{"id": "9162.png", "formula": "\\begin{align*} A _ s = \\{ a \\in A : \\delta ( a ) = a \\otimes s \\} \\end{align*}"}
-{"id": "274.png", "formula": "\\begin{align*} \\mathbf { H } ^ j ( X , C ^ { \\bullet } ) = 0 \\quad j \\ge k + 1 . \\end{align*}"}
-{"id": "9864.png", "formula": "\\begin{align*} \\Big \\{ \\begin{array} { c c } a _ 1 a _ 2 = a ' _ 1 a ' _ 2 \\\\ a _ 1 b _ 2 + b _ 1 = a ' _ 1 b ' _ 2 + b ' _ 1 \\\\ \\end{array} \\ , , ( a _ 1 , b _ 1 ) , ( a _ 2 , b _ 2 ) , ( a ' _ 1 , b ' _ 1 ) , ( a ' _ 2 , b ' _ 2 ) \\in A ' \\ , . \\end{align*}"}
-{"id": "5197.png", "formula": "\\begin{align*} \\forall ( s , t ) \\in ( \\mathbb { R } _ { + } \\setminus { 0 } ) ^ { 2 } , \\beta ( s t ) = \\alpha ( t ) \\beta ( s ) + \\beta ( t ) . \\end{align*}"}
-{"id": "7477.png", "formula": "\\begin{gather*} \\Pi ^ { - 1 } ( D _ p ^ * ) = \\coprod _ { i = 1 } ^ { k _ i } \\widetilde { D ^ * _ p } ^ i \\end{gather*}"}
-{"id": "2294.png", "formula": "\\begin{align*} \\sigma ^ { ( N ) } _ 3 \\psi _ m & = m \\psi _ m & \\sigma ^ { ( N ) } _ \\pm \\psi _ m & = \\bigg ( \\frac { N } { 2 } ( \\frac { N } { 2 } + 1 ) m ( m \\pm 1 ) \\bigg ) ^ { 1 / 2 } \\psi _ { m \\pm 1 } . \\end{align*}"}
-{"id": "4625.png", "formula": "\\begin{align*} M u _ { n - 2 } - 2 M u _ { n - 1 } + ( M + \\tau ^ 2 K ) u _ n = \\mathbf { 0 } , \\end{align*}"}
-{"id": "8446.png", "formula": "\\begin{align*} ( I - \\mathcal { H } ) A = & 1 + i [ D _ t \\zeta , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } \\mathfrak { F } } { \\zeta _ { \\alpha } } + i [ D _ t ^ 2 \\zeta , \\mathcal { H } ] \\frac { \\bar { \\zeta } _ { \\alpha } - 1 } { \\zeta _ { \\alpha } } - ( I - \\mathcal { H } ) \\frac { 1 } { 2 \\pi } \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j ( D _ t \\zeta ( \\alpha , t ) - \\dot { z } _ j ( t ) ) } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } . \\end{align*}"}
-{"id": "3950.png", "formula": "\\begin{align*} \\partial _ { m _ { j } } U _ { i } ( x , m , t ) ( \\xi ) = \\partial _ { m _ { i } } U _ { j } ( \\xi , m , t ) ( x ) = \\partial _ { m _ { i } m _ { j } } ^ { 2 } V ( m , t ) ( x , \\xi ) \\ , . \\end{align*}"}
-{"id": "7678.png", "formula": "\\begin{align*} ( V \\oplus W ) ^ + : = V ^ + \\oplus W ^ + \\ ; \\ ; ( V \\oplus W ) ^ - : = V ^ - \\oplus W ^ - . \\end{align*}"}
-{"id": "7928.png", "formula": "\\begin{align*} p _ 2 ( x ) = \\frac { x ( x - h ) } { 2 h ^ 2 } f ( - h ) + \\frac { ( x - h ) ( x + h ) } { - h ^ 2 } f ( 0 ) + \\frac { x ( x + h ) } { 2 h ^ 2 } f ( h ) \\end{align*}"}
-{"id": "5024.png", "formula": "\\begin{align*} \\lambda _ 1 ( t _ 0 , x _ 0 ) = \\displaystyle \\max _ { ( t , x ) \\in [ 0 , T ] \\times { \\bar \\Omega } } \\lambda _ 1 ( t , x ) . \\end{align*}"}
-{"id": "5989.png", "formula": "\\begin{align*} \\widetilde { q } _ \\omega ( s , \\overline \\omega ) : = \\sqrt { 1 - t } q ^ Q ( t + s ( 1 - t ) , \\omega \\otimes _ t \\overline \\omega ) . \\end{align*}"}
-{"id": "8955.png", "formula": "\\begin{align*} \\mathbb { P } ( Y = 1 \\ , | \\ , X = x ) = \\frac { 1 } { 1 + \\exp \\left \\{ - \\beta _ 0 - \\sum _ { j = 1 } ^ p \\beta _ j \\langle x - m , u _ j \\rangle _ 2 \\right \\} } , \\end{align*}"}
-{"id": "2170.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\infty 2 ^ { j ( \\mathbf N \\slash 2 + \\delta ) } \\| f ( \\cdot ) \\Psi _ n ( \\| \\cdot \\| ) \\| _ { L ^ 1 ( d w ) } = B < \\infty . \\end{align*}"}
-{"id": "10053.png", "formula": "\\begin{align*} | \\tilde p _ j | ^ 2 = y _ j ^ 2 + \\frac { G _ j ^ 2 } { r _ j ^ 2 } . \\end{align*}"}
-{"id": "8855.png", "formula": "\\begin{align*} e _ 5 & = - 2 4 p ^ 4 \\mu ^ 2 + 2 4 a \\mu p ^ 3 + 2 0 4 \\mu ^ 2 p ^ 3 - 4 p ^ 2 a ^ 2 - 1 9 2 a \\mu p ^ 2 - 1 2 \\mu ^ 2 p ^ 2 + 2 4 \\mu p ^ 3 + 4 5 p a ^ 2 - 1 6 p ^ 2 a \\\\ & - 1 9 2 p ^ 2 \\mu + 3 a ^ 2 + 9 0 a p - 4 p ^ 2 + 6 a + 4 5 p + 3 \\\\ e _ 6 & = 3 6 0 p ( a + 1 - 2 \\mu p ) ^ 2 \\ge 0 . \\end{align*}"}
-{"id": "5865.png", "formula": "\\begin{align*} \\Phi _ n : \\C ^ n \\rightarrow M _ n , \\ ( z _ 1 , \\cdots , z _ n ) \\mapsto f ( z ) = \\prod _ { i = 1 } ^ n ( z - z _ i ) = z ^ n + b _ { n - 1 } z ^ { n - 1 } + \\dots + b _ 0 \\end{align*}"}
-{"id": "172.png", "formula": "\\begin{align*} [ \\pi _ { X _ { V } , Y _ { V } } , \\pi _ { X _ { V } , Y _ { V } } ] = 2 [ Y _ { V } , X _ { V } ] \\wedge X _ { V } \\wedge Y _ { V } = 0 \\end{align*}"}
-{"id": "9130.png", "formula": "\\begin{align*} M ( \\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } , \\mathbf { b } ) = \\frac { 1 } { 2 } \\left [ M ( \\mathbf { v } _ { 1 } , \\mathbf { b } ) + M ( \\mathbf { v } _ { 2 } , \\mathbf { b } ) \\right ] \\end{align*}"}
-{"id": "6322.png", "formula": "\\begin{align*} q _ t = q ^ { 2 } ( q _ { x x } - \\nu ^ { - \\frac { 1 } { 2 } } f _ 0 ) Q _ T , \\end{align*}"}
-{"id": "3430.png", "formula": "\\begin{gather*} V | _ { 2 D _ 8 } = 1 5 V _ 0 \\oplus 1 5 V _ 1 \\oplus 1 5 V _ 2 \\oplus 1 5 V _ 3 \\oplus 3 0 V _ 4 \\oplus 3 2 V _ 5 \\oplus 3 2 V _ 6 . \\end{gather*}"}
-{"id": "4140.png", "formula": "\\begin{align*} \\varphi _ { 2 } ( z ) = - g _ { 2 } ( z ) + \\tfrac { 1 } { 2 } g _ { 1 } ( z ) + \\begin{cases} - \\frac { \\pi i } { 2 } , & \\Im ( z ) > 0 , \\\\ \\frac { \\pi i } { 2 } , & \\Im ( z ) < 0 , \\end{cases} \\end{align*}"}
-{"id": "6531.png", "formula": "\\begin{align*} & \\frac { 1 } { l } \\Biggl ( \\int ^ { t + l } _ { t } L _ { f } ^ { r } ( s + \\tau ) \\ , d s \\Biggr ) ^ { 1 / r } \\Biggl ( \\int ^ { t + l } _ { t } \\bigl \\| x ( s + \\tau ) - x ( s ) \\bigr \\| ^ { p } \\ , d t \\Biggr ) ^ { 1 / p } \\\\ \\leq & M l ^ { ( 1 / p ) + ( 1 / r ) - 1 } \\| L _ { f } \\| _ { W ^ { r } } = l ^ { ( 1 / q ) - 1 } \\| L _ { f } \\| _ { W ^ { r } } \\leq \\| L _ { f } \\| _ { W ^ { r } } , t \\in I . \\end{align*}"}
-{"id": "8002.png", "formula": "\\begin{align*} x _ i ^ * = \\begin{cases} 1 \\textmd { o r } - 1 , j + 1 \\leq i \\leq d , \\\\ y _ i , i \\leq j . \\end{cases} \\end{align*}"}
-{"id": "200.png", "formula": "\\begin{align*} \\pi _ { T M , X _ V , Y _ V , c } ( { \\bf x } , { \\bf y } ) = \\left ( \\begin{array} { c c c | c c c } 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & x ^ 1 \\\\ 0 & 0 & 0 & 0 & - x ^ 1 & 0 \\\\ \\hline 0 & 0 & 0 & 0 & 0 & x ^ 1 \\\\ 0 & 0 & x ^ 1 & 0 & 0 & y ^ 1 + x ^ 2 \\\\ 0 & - x ^ 1 & 0 & - x ^ 1 & - y ^ 1 - x ^ 2 & 0 \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "490.png", "formula": "\\begin{align*} p _ i = \\{ x _ 1 = \\cdots = \\widehat { x _ i } = \\cdots = x _ n = 0 \\} , i = 1 , \\dots , k , \\end{align*}"}
-{"id": "4062.png", "formula": "\\begin{align*} w ( x ) = x ^ { \\alpha } e ^ { - n V ( x ) } \\end{align*}"}
-{"id": "1563.png", "formula": "\\begin{align*} ( \\Psi _ { \\widetilde { F } ^ \\delta } ) ^ { - 1 } ( z , x ) = - \\frac { 1 } { z } - b \\log \\left ( \\frac { 1 } { z } \\right ) + o ( 1 ) . \\end{align*}"}
-{"id": "7268.png", "formula": "\\begin{align*} \\mathcal { F } \\left ( \\left [ \\frac { 1 } { 2 } \\phi ^ { 2 } - \\mu \\phi \\right ] \\psi _ { x x } \\right ) ( 0 ) = \\mathcal { F } \\left ( \\phi \\psi \\right ) ( 0 ) . \\end{align*}"}
-{"id": "5364.png", "formula": "\\begin{gather*} ( n + 1 ) ^ 2 a _ { n m } d _ { n + 1 \\ , m + 3 } b _ { n \\ , m + 6 } c _ { n + 1 \\ , m + 3 } \\\\ = n ^ 2 a _ { n - 1 \\ , m + 3 } d _ { n \\ , m + 6 } b _ { n - 1 \\ , m + 3 } c _ { n m } \\end{gather*}"}
-{"id": "1986.png", "formula": "\\begin{align*} \\prod _ { i \\in I } ( H - \\alpha _ i ) = [ \\{ x _ i = 0 : \\forall i \\in I \\} ] , \\end{align*}"}
-{"id": "7712.png", "formula": "\\begin{align*} c \\cdot \\norm { \\psi ( \\overline { D } ) u } _ { 1 } = c \\cdot \\norm { v } _ { 1 } \\leq \\norm { v } + \\norm { \\overline { D } v } \\leq \\sqrt { 2 } \\norm { ( \\mathbf { i } + \\overline { D } ) v } = \\sqrt { 2 } \\norm { u } . \\end{align*}"}
-{"id": "7591.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { \\left [ X ' \\circ X ^ { - 1 } \\cdot \\nabla , \\mathbb { G } \\right ] ( \\tau \\circ X ^ { - 1 } ) } _ { L ^ \\infty ( 0 , T ; C ^ { \\alpha , p } ) } \\\\ \\le ( \\norm { X ' } _ { L ^ \\infty ( 0 , T ; C ^ { 1 + \\alpha } ) } + \\norm { X ' } _ { L i p ( 0 , T ; C ^ { 1 + \\alpha } ) } T ^ { \\frac { 1 } { 2 } } ) R \\end{gathered} \\end{align*}"}
-{"id": "9096.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } | f ( x ) | | \\varphi ( x ) | ^ p \\ , d x = \\iint _ { \\mathbb { R } ^ N \\times \\mathbb { R } ^ M } | h ( x , y ) | | \\tau ( x , y ) | ^ p d x d y . \\end{align*}"}
-{"id": "9170.png", "formula": "\\begin{align*} \\Psi ( V _ s ) = 1 \\otimes W _ s . \\end{align*}"}
-{"id": "6679.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } d x \\ , \\psi _ { \\rho } ^ { * } ( x ) \\psi _ { \\rho ' } ( x ) & = \\int _ { 0 } ^ { \\infty } d x \\ , x ^ { - 1 - i ( \\rho - \\rho ' ) } \\\\ & = 2 \\pi \\delta ( \\rho - \\rho ' ) . \\end{align*}"}
-{"id": "1411.png", "formula": "\\begin{align*} \\inf _ { \\tau \\in { \\mathcal T } } E [ Z _ { \\tau } ] = E [ \\min _ { t \\in [ 1 , T ] } Z _ t ] + \\inf _ { \\tau \\in { \\mathcal T } } E \\big [ Z _ { \\tau } - E [ \\min _ { i \\in [ 1 , T ] } Z _ i | { \\mathcal F } _ { \\tau } ] \\big ] . \\end{align*}"}
-{"id": "9928.png", "formula": "\\begin{align*} & \\frac { d P ^ { \\mu } | _ { \\mathcal { G } _ { 1 } } | \\mathcal { G } _ { 2 } } { d P ^ { \\nu } | _ { \\mathcal { G } _ { 1 } } | \\mathcal { G } _ { 2 } } = \\frac { E ^ { \\nu } [ \\frac { d \\mu } { d \\nu } ( X _ { 0 } ) | \\mathcal { G } _ { 1 } \\vee \\mathcal { G } _ { 2 } ] } { E ^ { \\nu } [ \\frac { d \\mu } { d \\nu } ( X _ { 0 } ) | \\mathcal { G } _ { 2 } ] } ~ ~ ~ P ^ { \\mu } ~ a . s . \\end{align*}"}
-{"id": "1893.png", "formula": "\\begin{align*} \\gamma ^ \\mu ( \\tau ) = \\frac { 1 } { \\sqrt { 1 - v ^ 2 } } ( \\tau , v \\tau ) + ( t _ 0 , x _ 0 ) , \\end{align*}"}
-{"id": "7378.png", "formula": "\\begin{align*} m _ { R } ( c _ { R } ) = m _ { R 0 } \\bigg ( 1 + \\frac { \\max \\{ c _ { R } - c _ { R M 0 } , 0 \\} } { c _ { R M } } \\bigg ) . \\end{align*}"}
-{"id": "4591.png", "formula": "\\begin{align*} \\int \\int f _ { g , \\delta } ( z ) d \\nu _ g ( z ) d g = \\sum _ { j \\in \\mathbb { Z } } \\int \\int \\hat { f } _ { g , \\delta } ( - \\omega ) \\hat { \\nu } _ { g } ( \\omega ) \\psi ( 2 ^ { - j } \\omega ) d \\omega d g . \\end{align*}"}
-{"id": "9629.png", "formula": "\\begin{align*} \\frac { 1 } { ( q ) _ n ( 1 - q ^ n ) } = \\ \\frac { 1 } { ( 1 - q ) } \\cdot \\frac { 1 } { ( 1 - q ^ 2 ) } \\cdots \\frac { 1 } { ( 1 - q ^ { n - 1 } ) } \\sum _ { \\nu ( n ) = 0 } ^ { \\infty } ( \\nu ( n ) + 1 ) q ^ { \\nu ( n ) n } , \\end{align*}"}
-{"id": "5514.png", "formula": "\\begin{align*} \\beta _ { ( B ^ \\rho _ { 1 \\infty } ) ^ * } ( \\nu _ T , \\mathcal N _ { b _ 0 } ) \\le \\beta _ { V ^ { \\otimes d } _ \\kappa } ( \\nu _ { T , \\kappa } , \\mathcal N _ { b _ 0 , \\kappa } ) + \\sum _ { i = 1 } ^ 2 E \\| \\tilde Z _ i - P _ { ( \\kappa ) } ( \\tilde Z _ i ) \\| _ { ( B ^ \\rho _ { 1 \\infty } ) ^ * } \\end{align*}"}
-{"id": "7450.png", "formula": "\\begin{align*} \\displaystyle J ( f _ 1 , f _ 2 ) = \\int _ { H ^ V ( F ) \\backslash G ^ V _ { r s } ( F ) / H ^ V ( F ) } O ( \\delta , f _ 1 ) \\overline { O ( \\delta , f _ 2 ) } d \\delta . \\end{align*}"}
-{"id": "1650.png", "formula": "\\begin{align*} G _ k : = - \\frac { 1 } { \\overline { \\varkappa _ k } } \\begin{pmatrix} - b \\\\ a _ k + i \\sqrt [ + ] { b ^ 2 - a _ k ^ 2 } \\end{pmatrix} e ^ { 2 \\pi i k x } \\end{align*}"}
-{"id": "9568.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\frac { q ^ n } { 1 - q ^ { 2 n } } = \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] ( - 1 ) ^ { n - 1 } \\frac { ( - q ) _ { n - 1 } ( - q ) _ { N - n } } { ( - q ) _ { N } } \\frac { q ^ n } { 1 - q ^ n } . \\end{align*}"}
-{"id": "2012.png", "formula": "\\begin{align*} I _ + ( q , z ) : = I _ 0 ( q ) z + I _ 1 ( q ) H . \\end{align*}"}
-{"id": "3399.png", "formula": "\\begin{align*} m _ 0 ( T ) & = \\prod _ { i } m ( T _ i ) , \\\\ m ( T ) & = m _ 0 ( T ) + \\sum _ { i } m _ 0 ( T _ i ) \\prod _ { j \\neq i } m ( T _ j ) . \\end{align*}"}
-{"id": "999.png", "formula": "\\begin{align*} & \\lim _ { n \\to + \\infty } \\mathbb { E } \\sum _ { i _ 1 , \\cdots , i _ { k } \\ } \\prod _ { j = 1 } ^ k ( \\tau _ { i _ j } ^ { ( n ) } - x _ j ) _ + = \\prod _ { j = 1 } ^ k f ( x _ j ) . \\end{align*}"}
-{"id": "27.png", "formula": "\\begin{align*} U ( x , t ) : = \\sum _ { j \\geq 1 } U ^ { \\alpha ^ { j - 1 } } ( x , t ) \\chi _ { [ T _ { j - 1 } , T _ { j } ) } ( t ) . \\end{align*}"}
-{"id": "9059.png", "formula": "\\begin{align*} & \\sum _ { \\substack { \\xi _ 1 , \\xi _ 2 , \\xi _ 3 \\\\ \\xi _ 1 ' , \\xi _ 2 ' , \\xi _ 3 ' } } \\int _ { \\mathbb { T } ^ 2 } d x e ( \\langle \\xi _ 1 + \\xi _ 2 + \\xi _ 3 - \\xi _ 1 ' - \\xi _ 2 ' - \\xi _ 3 ' , x \\rangle ) \\\\ & = | \\{ \\xi _ i \\in \\mathcal { E } ( E ) : \\xi _ 1 + \\xi _ 2 + \\xi _ 3 + \\xi _ 4 + \\xi _ 5 + \\xi _ 6 = 0 \\} | \\end{align*}"}
-{"id": "8975.png", "formula": "\\begin{align*} \\left | \\eta _ L \\left ( \\bigcup _ { i = 1 } ^ N B _ i \\right ) - \\eta _ { L ' } \\left ( \\bigcup _ { i = 1 } ^ N B _ i \\right ) \\right | \\lesssim N \\cdot d ( L , L ' ) ^ { 1 / 4 } \\end{align*}"}
-{"id": "4441.png", "formula": "\\begin{align*} k _ L m _ { \\pi } = k _ X m _ H . \\end{align*}"}
-{"id": "1929.png", "formula": "\\begin{align*} ( x ^ 2 + x ) F _ { p + 2 } ' ( x ) + ( p - 1 ) F _ { p + 2 } ( x ) + F _ p ' ( x ) - \\frac { 2 } { x } F _ p ( x ) = 0 . \\end{align*}"}
-{"id": "8599.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } u ( u ^ 2 + \\Lambda v ^ 2 - 1 ) - \\omega v = 0 , \\\\ v ( v ^ 2 + \\Lambda u ^ 2 - 1 ) - \\omega u = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "7166.png", "formula": "\\begin{gather*} \\lim _ { t \\to \\infty } \\rho \\left ( g ^ { t } ( p ) , \\mathcal { A } \\right ) = 0 \\quad \\forall p \\in \\mathfrak { B } . \\end{gather*}"}
-{"id": "8068.png", "formula": "\\begin{align*} \\begin{pmatrix} n _ 1 \\\\ n _ 2 \\\\ m _ 1 \\\\ m _ 2 \\\\ \\vdots \\\\ m _ h \\\\ \\vdots \\\\ m _ { ( n - 1 ) / 2 } \\end{pmatrix} = \\begin{pmatrix} \\frac { 1 } { 2 } - \\frac { 1 } { 2 } \\displaystyle { \\sum _ { i = n + 3 } ^ { 2 n } } c ^ { n + 3 - 2 i } P _ { i - n - 3 , - i } \\\\ \\frac { 3 } { 2 } + \\frac { 1 } { 2 } \\displaystyle { \\sum _ { i = n + 3 } ^ { 2 n } } c ^ { n + 3 - 2 i } P _ { i - n - 3 , - i } \\\\ 2 \\\\ 2 \\\\ \\vdots \\\\ 2 \\\\ \\end{pmatrix} . \\\\ \\end{align*}"}
-{"id": "8500.png", "formula": "\\begin{align*} \\hat { d } _ I ( t ) = \\min _ { j = 1 , 2 } \\inf _ { \\alpha \\in \\mathbb { R } } I m \\{ \\zeta ( \\alpha , t ) - z _ j ( t ) \\} \\geq 1 + \\frac { | \\lambda | } { 1 8 \\pi x ( 0 ) } t , \\quad \\forall ~ t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "1692.png", "formula": "\\begin{align*} 2 ^ q \\vert 2 ^ { 2 ^ q - 1 } \\vert 2 ^ { 2 ^ q - 1 } \\cdot 2 ^ { 2 ^ q \\cdot p - 2 ^ q } = 2 ^ { 2 ^ q \\cdot p - 1 } = 2 ^ { n - 1 } . \\end{align*}"}
-{"id": "3592.png", "formula": "\\begin{align*} \\langle u ^ { \\tau ( a ) } , v ^ { \\tau ( a ) } \\rangle & = \\langle u _ 1 + u _ 0 + u _ { \\frac { 1 } { 4 } } - u _ { \\frac { 1 } { 3 2 } } , v _ 1 + v _ 0 + v _ { \\frac { 1 } { 4 } } - v _ { \\frac { 1 } { 3 2 } } \\rangle \\\\ & = \\langle u _ 1 , v _ 1 \\rangle + \\langle u _ 0 , v _ 0 \\rangle + \\langle u _ { \\frac { 1 } { 4 } } , v _ { \\frac { 1 } { 4 } } \\rangle + \\langle u _ { \\frac { 1 } { 3 2 } } , v _ { \\frac { 1 } { 3 2 } } \\rangle \\end{align*}"}
-{"id": "9602.png", "formula": "\\begin{align*} N _ { \\textup { S C } } ( n , N ) = \\sum _ { \\vec { \\pi } \\in S _ N , | \\vec { \\pi } | = n \\atop \\imath ( \\vec { \\pi } ) = \\vec { \\pi } } w _ { \\textup { S C } } ( \\vec { \\pi } ) . \\end{align*}"}
-{"id": "8654.png", "formula": "\\begin{align*} \\varphi _ { 2 , 0 } = \\varphi _ { 1 , 0 } ' < 0 , \\end{align*}"}
-{"id": "4547.png", "formula": "\\begin{align*} z _ n ^ 2 + \\sum _ { j = 1 } ^ k z _ { 2 j - 1 } z _ { 2 j } = 0 , \\end{align*}"}
-{"id": "434.png", "formula": "\\begin{align*} C ( A , \\eta , \\gamma ) : = \\int _ { [ 0 , t ] \\times D ^ 2 } [ \\gamma ( A + \\eta u ) + 1 ] ^ 2 e ^ { 2 \\gamma A + ( 2 \\gamma \\eta + \\gamma ^ 2 ) u } Q _ u ( x , y ) \\mu ( \\dd x ) \\mu ( \\dd y ) . \\end{align*}"}
-{"id": "8290.png", "formula": "\\begin{align*} - i ( I - \\mathfrak { H } ) a _ t \\bar { z } _ { \\alpha } = g _ 1 + g _ 2 , \\end{align*}"}
-{"id": "501.png", "formula": "\\begin{align*} \\mu _ 2 ( Q ) = \\mu _ { 1 , F } ( x _ 1 \\wedge x _ 2 ) \\mu _ { 1 , K - F } ( t _ 1 \\wedge t _ 2 ) . \\end{align*}"}
-{"id": "3719.png", "formula": "\\begin{align*} \\hat { \\gamma } = \\min ( W _ B + ( 1 - W _ B ) \\widehat { V } ( \\bar { y } _ w ) / ( \\bar { y } _ B - \\bar { y } _ w ) ^ 2 , 1 ) \\end{align*}"}
-{"id": "2408.png", "formula": "\\begin{align*} q ( a x ) = a ^ 2 q ( x ) \\end{align*}"}
-{"id": "8845.png", "formula": "\\begin{align*} C _ t & = \\left \\{ x _ { ( k ) } ( t ) < \\frac a 3 x _ { ( k + 1 ) } ( t ) > \\frac { 2 a } 3 \\right \\} , \\bar C _ T = \\bigcap _ { t \\ge T } C _ t , \\end{align*}"}
-{"id": "7837.png", "formula": "\\begin{align*} | \\mathfrak m _ N ^ { B ^ 2 } ( \\xi ) | \\le C e ^ { - \\frac { c \\kappa ( d , N ) ^ 2 } { 1 0 0 } \\sum _ { i = 1 } ^ d \\sin ^ 2 ( \\pi \\xi _ i ) } + C e ^ { - \\frac { c \\kappa ( d , N ) ^ 2 } { 1 0 0 } \\sum _ { i = 1 } ^ d \\cos ^ 2 ( \\pi \\xi _ i ) } . \\end{align*}"}
-{"id": "7915.png", "formula": "\\begin{align*} R _ { n } ( f ( x ) ) = \\hat { R } _ n h ^ { n + 2 } f ^ { ( n + 1 ) } ( \\xi ) \\xi \\in [ - h , h ] \\end{align*}"}
-{"id": "6619.png", "formula": "\\begin{align*} \\overline { C } _ n ( \\varepsilon ) & = \\prod _ { t = 1 } ^ { \\eta } \\left ( 1 - \\frac { v _ n ^ 2 } { s _ { n t ^ 2 } } \\right ) \\prod _ { t = \\eta + 1 } ^ { ( F _ { n } - 1 ) / 2 } \\left ( 1 - \\frac { v _ n ^ 2 } { s _ { n t ^ 2 } } \\right ) \\\\ & \\geq \\prod _ { t = 1 } ^ { \\infty } \\left ( 1 - \\frac { ( 1 - 2 \\varepsilon ( - \\varphi ) ^ { - n } ) ^ 2 } { u _ t ^ 2 } \\right ) - \\mathcal { O } ( \\varphi ^ { n / 5 } ) . \\end{align*}"}
-{"id": "7265.png", "formula": "\\begin{align*} - \\mu \\phi + L _ 2 \\phi + \\frac { 1 } { 2 } \\left ( \\phi ^ { 2 } - \\widehat { \\phi ^ 2 } ( 0 ) \\right ) = 0 , \\end{align*}"}
-{"id": "10004.png", "formula": "\\begin{align*} \\sigma _ { u } ( D ) = \\inf \\big \\{ \\sigma \\ , \\colon \\ , \\big ( \\sum _ { n = 1 } ^ { N } \\frac { a _ { n } } { n ^ { \\sigma } } n ^ { - s } \\big ) _ { N } \\mathcal { D } _ { \\infty } ( X ) \\big \\} \\ , . \\end{align*}"}
-{"id": "8004.png", "formula": "\\begin{align*} \\sum _ { n \\in Q _ 2 ( 0 ) \\bigcap \\mathbb { N } ^ d } e ^ { - a | n - y | ^ 2 } & = \\sum _ { \\substack { n = ( n _ 1 , n _ 2 , \\cdots , n _ d ) \\\\ n _ i \\in \\{ - 1 , 0 , 1 \\} , i = 1 , 2 , \\cdots , d } } e ^ { - a | n - y | ^ 2 } \\geq \\Theta \\sup _ { x \\in Q _ 2 ( 0 ) } e ^ { - a | x - y | ^ 2 } , \\end{align*}"}
-{"id": "9497.png", "formula": "\\begin{align*} \\Vert u \\Vert _ { s , m } = \\left ( \\displaystyle \\int _ { \\mathbb { R } ^ N } \\displaystyle \\int _ { \\mathbb { R } ^ { N } } \\frac { \\vert u ( x ) - u ( y ) \\vert ^ { m } } { \\vert x - y \\vert ^ { N + s m } } \\dd x \\dd y \\right ) ^ { \\frac { 1 } { m } } . \\end{align*}"}
-{"id": "451.png", "formula": "\\begin{align*} \\P _ { N , \\tau } ( T ) : = \\frac { \\mathcal { Z } ( T ) } { \\mathcal { Z } _ { N , \\tau } } . \\end{align*}"}
-{"id": "6998.png", "formula": "\\begin{align*} k v _ j = \\xi ^ { r - 2 j } v _ j , e v _ j = [ r - j + 1 ] v _ { j - 1 } , f v _ j = [ j + 1 ] v _ { j + 1 } , \\end{align*}"}
-{"id": "1636.png", "formula": "\\begin{align*} \\xi _ k : = \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} e ^ { 2 \\pi i k x } \\quad \\eta _ k : = \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} e ^ { 2 \\pi i k x } . \\end{align*}"}
-{"id": "58.png", "formula": "\\begin{align*} \\rho = f ^ + + f ^ - \\ , , J = f ^ + - f ^ - \\end{align*}"}
-{"id": "5951.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\ , \\rho ^ { g _ n } \\left ( F \\left ( \\frac { W } { \\sqrt { n } } \\right ) \\right ) = \\sup _ { \\omega \\in \\C _ 0 } \\left ( F ( \\omega ) - \\int _ 0 ^ 1 g ( t , \\dot { \\omega } ( t ) ) d t \\right ) . \\end{align*}"}
-{"id": "5117.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j \\in \\mathbb { S } } c _ j x _ j \\right \\| ^ p = \\sum _ { j \\in \\mathbb { S } } | c _ j | ^ p , ~ \\forall c _ j \\in \\mathbb { K } , \\forall j \\in \\mathbb { S } . \\end{align*}"}
-{"id": "9860.png", "formula": "\\begin{align*} b ^ { - 1 } a g = g b ^ { - 1 } a \\ , . \\end{align*}"}
-{"id": "9914.png", "formula": "\\begin{align*} ( \\pi _ { n _ - } , y ) \\mapsto \\pi _ { n } \\quad : \\pi _ { n } ( \\cdot ) : = E _ { \\pi _ { n _ - } } [ 1 _ { X _ n \\in \\cdot } | Y _ n = y ] \\end{align*}"}
-{"id": "7269.png", "formula": "\\begin{align*} \\mathcal { F } \\left ( \\frac { 1 } { 2 } \\phi ^ { 2 } - \\mu \\phi \\right ) * \\mathcal { F } \\left ( \\psi _ { x x } \\right ) ( 0 ) = \\hat \\phi * \\hat \\psi ( 0 ) . \\end{align*}"}
-{"id": "3501.png", "formula": "\\begin{align*} ( x - y ) e ^ { x t - y s } = \\Big ( \\frac { \\partial } { \\partial t } + \\frac { \\partial } { \\partial s } \\Big ) e ^ { x t - y s } , \\end{align*}"}
-{"id": "9627.png", "formula": "\\begin{align*} w ( m , N ) : = \\sum _ { n = 1 } ^ { N } \\sum _ { \\substack { \\pi \\in \\mathcal { P } ( m , N + 1 ) \\\\ > n } } \\left \\{ \\nu ( n ) + 1 \\right \\} , \\end{align*}"}
-{"id": "8321.png", "formula": "\\begin{align*} S _ 2 ( A , f ) = \\int \\prod _ { j = 1 } ^ m \\frac { A _ j ( \\alpha ) - A _ j ( \\beta ) } { \\gamma _ j ( \\alpha ) - \\gamma _ j ( \\beta ) } f _ { \\beta } ( \\beta ) d \\beta . \\end{align*}"}
-{"id": "9615.png", "formula": "\\begin{align*} & ( 1 - q ^ { N } ) ( - q ) _ { N - 1 } \\cdot \\left . G ( z , q , N ) \\right | _ { z = q } \\\\ & = \\frac { q ^ N } { 1 - q ^ N } ( - 1 ) _ N - \\sum _ { m = N } ^ { \\infty } \\frac { 2 q ^ m } { 1 - q ^ { 2 m } } + \\sum _ { m = 1 } ^ { \\infty } \\frac { q ^ { m } } { 1 - q ^ { m } } \\frac { ( - 1 ) _ m } { ( - q ^ N ) _ m } - \\sum _ { m = 1 } ^ { N } \\frac { q ^ m } { 1 - q ^ m } . \\end{align*}"}
-{"id": "3707.png", "formula": "\\begin{align*} \\begin{cases} C o v _ { N _ t } ( \\delta _ i , \\epsilon _ i ) \\rightarrow 0 \\\\ E _ { N _ t } ( \\delta _ i ) = \\sum _ { i \\in U _ t } \\delta _ i / N _ t \\rightarrow p _ t > 0 \\end{cases} \\end{align*}"}
-{"id": "2316.png", "formula": "\\begin{align*} H _ g = \\left ( \\begin{array} { c c } 0 & \\langle g | \\\\ | g \\rangle & \\Omega \\end{array} \\right ) . \\end{align*}"}
-{"id": "8979.png", "formula": "\\begin{align*} K h ( L ) ( q , t ) = q ^ { - \\sigma ( L ) } \\left ( ( q + q ^ { - 1 } ) \\left ( \\sum _ { E \\subset \\{ 2 , \\dots n \\} } ( t q ^ 2 ) ^ { \\sum _ { j \\in E , k \\notin E } 2 l _ { j k } } \\right ) + \\left ( q ^ { - 1 } + t q ^ 2 \\cdot q \\right ) K h ' ( L ) ( t q ^ 2 ) \\right ) \\end{align*}"}
-{"id": "1474.png", "formula": "\\begin{align*} V _ { 7 } = \\frac { \\varepsilon } { 2 } \\sum _ { i = 1 } ^ { d } \\int _ { \\mathbb { T } ^ { d } } ( \\mu ^ { 1 } - \\mu ^ { 2 } ) ^ { 2 } \\frac { \\partial } { \\partial x _ { i } } \\left ( ( \\mu ^ { 2 } ) \\Theta _ { p _ { i } q } ( t , x , \\mu ^ { 1 } , D w ^ { 1 } ) \\right ) \\ d x , \\end{align*}"}
-{"id": "6947.png", "formula": "\\begin{align*} \\exp ( \\vartheta \\ , \\mathbf W ) \\ , = \\ , \\mathbf R \\ , = \\ , \\mathbf I \\ , + \\ , \\sin \\vartheta \\ , \\mathbf W \\ , + \\ , ( 1 - \\cos \\vartheta ) \\ , \\mathbf W ^ { 2 } \\ , . \\end{align*}"}
-{"id": "6180.png", "formula": "\\begin{align*} \\lim _ { ( t , z ) \\rightarrow ( 0 , 0 ) } \\frac { H - U _ t h } { U _ t } = \\lim _ { ( t , z ) \\rightarrow ( 0 , 0 ) } \\frac { H } { U _ t } = 0 . \\end{align*}"}
-{"id": "134.png", "formula": "\\begin{align*} \\mathcal { L } _ { X } \\pi = 0 \\end{align*}"}
-{"id": "2337.png", "formula": "\\begin{align*} \\mathcal { V } ( f ) & = \\sum _ \\alpha ( \\Delta \\alpha ) f ( x _ \\alpha ) | x _ \\alpha \\rangle \\\\ \\langle f | g \\rangle & = \\sum _ \\alpha ( \\Delta \\alpha ) f ( x _ \\alpha ) g ( x _ \\alpha ) \\end{align*}"}
-{"id": "4830.png", "formula": "\\begin{align*} \\mathbb { P } ( Z ^ 1 _ t - Z ^ 2 _ t \\in B ( 0 , \\delta ) , Z ^ 1 _ t , Z ^ 2 _ t \\in B ( 0 , R ' ) ) = & \\mathbb { P } ( Z _ t ^ 1 , Z _ t ^ 2 \\in B ( 0 , R ' ) ) \\\\ & - \\mathbb { P } ( Z ^ 1 _ t - Z ^ 2 _ t \\notin B ( 0 , \\delta ) , Z ^ 1 _ t , Z _ t ^ 1 \\in B ( 0 , R ' ) ) \\end{align*}"}
-{"id": "3173.png", "formula": "\\begin{align*} \\mathcal { B } _ \\pm = \\ker ( \\varGamma \\pm 1 ) \\cap \\ker ( C + 1 ) . \\end{align*}"}
-{"id": "6837.png", "formula": "\\begin{align*} \\begin{cases} a _ { 1 1 } a _ { 2 2 } - a _ { 1 2 } a _ { 2 1 } - 1 = 0 \\ , , \\\\ a _ { 1 3 } a _ { 2 4 } - a _ { 1 4 } a _ { 2 3 } - 1 = 0 \\ , , \\\\ a _ { 1 4 } - a _ { 1 2 } + a _ { 1 1 } = 0 \\ , , \\\\ a _ { 2 4 } - a _ { 2 2 } + a _ { 2 1 } = 0 \\ , , \\\\ a _ { 2 1 } '' a _ { 2 2 } - a _ { 2 2 } '' a _ { 2 1 } + a _ { 1 1 } '' a _ { 1 2 } - a _ { 1 2 } '' a _ { 1 1 } = 0 \\ , , \\end{cases} \\end{align*}"}
-{"id": "9992.png", "formula": "\\begin{align*} \\begin{pmatrix} \\psi \\\\ \\psi ^ 1 \\\\ \\psi ^ 2 \\end{pmatrix} _ x = \\lambda \\begin{pmatrix} 0 & 1 & 0 \\\\ q ^ 1 & q ^ 3 & q ^ 5 \\\\ q ^ 2 & q ^ 4 & q ^ 6 \\end{pmatrix} \\begin{pmatrix} \\psi \\\\ \\psi ^ 1 \\\\ \\psi ^ 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "7614.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { \\mathcal { T } } _ { L ^ \\infty ( 0 , T ; C ^ { \\alpha , p } ) } \\le K _ 2 ( \\norm { u _ 0 } _ { 1 + \\alpha , p } ( \\rho K + \\norm { \\sigma _ 0 } _ { \\alpha , p } ) + \\norm { \\sigma _ 0 } _ { \\alpha , p } ( \\Gamma ^ { \\alpha } \\norm { \\sigma _ 0 } _ { \\alpha , p } + \\rho K \\Gamma ^ { \\alpha } + k ) ) \\\\ + A _ 4 B _ 4 , \\end{gathered} \\end{align*}"}
-{"id": "3859.png", "formula": "\\begin{align*} \\mathbf { E } [ h ^ { \\pm } ( Z ^ x _ t ) ] = W _ x ^ { \\pm } h ^ { \\pm } ( t ) = W _ { t , x } ^ { \\pm } h ^ { \\pm } ( t , x ) . \\end{align*}"}
-{"id": "2098.png", "formula": "\\begin{align*} & { \\rm P r } ( E | b _ 0 , b _ 1 , \\cdots , b _ M ) = \\\\ & \\sum _ { y \\underset { b _ 0 = 0 } { \\overset { b _ 0 = 1 } { \\lessgtr } } { \\rm T h r } } { \\frac { e ^ { - \\mathbb { E } ( \\textbf { y } _ R | b _ 0 , b _ 1 , \\cdots , b _ M ) } ( \\mathbb { E } ( \\textbf { y } _ R | b _ 0 , b _ 1 , \\cdots , b _ M ) ) ^ y } { y ! } } . \\end{align*}"}
-{"id": "4645.png", "formula": "\\begin{align*} M _ j \\cap \\left ( \\prod _ { k \\neq j } M _ j \\right ) = K _ { n - 1 } ( \\beta ) . \\end{align*}"}
-{"id": "4181.png", "formula": "\\begin{align*} P ^ { ( 2 ) } _ + ( z ) \\left ( P ^ { ( 2 ) } _ - ( z ) \\right ) ^ { - 1 } = \\mathbb { I } + \\frac { A _ n ^ { ( 2 ) } ( z ) } { n ^ 6 z } - \\frac { A _ n ^ { ( 1 ) } ( 0 ) A _ n ^ { ( 1 ) } ( z ) A _ n ^ { ( 1 ) } ( 0 ) } { n ^ 9 z ^ 3 } + \\mathcal { O } ( n ^ { - 1 } ) \\end{align*}"}
-{"id": "6438.png", "formula": "\\begin{align*} \\eta = \\min \\left \\{ \\delta : \\ ( A + \\Delta A ) \\ , \\widetilde x = b + \\Delta b , \\ \\| \\Delta A \\| \\leq \\delta \\| A \\| , \\ \\| \\Delta b \\| \\leq \\delta \\| b \\| \\right \\} \\ , . \\end{align*}"}
-{"id": "8048.png", "formula": "\\begin{align*} \\tau _ k ( a ) = \\begin{pmatrix} e ^ { \\pi \\imath k / n } & 0 \\\\ 0 & e ^ { - \\pi \\imath k / n } \\end{pmatrix} , \\tau _ k ( x ) = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "9506.png", "formula": "\\begin{align*} \\displaystyle \\max _ { t \\geq 0 } I _ { \\lambda } ( t u _ 0 ) = I _ { \\lambda } ( t _ { \\lambda } u _ 0 ) \\end{align*}"}
-{"id": "1437.png", "formula": "\\begin{align*} \\mathcal { F } \\left ( \\mathbb { P } _ { \\delta } f \\right ) ( k ) = \\left \\{ \\begin{array} { l l } \\mathcal { F } f ( k ) , & | k | \\leq 1 / \\delta , \\\\ 0 , & | k | > 1 / \\delta . \\end{array} \\right . \\end{align*}"}
-{"id": "4211.png", "formula": "\\begin{align*} R ^ { - 1 } ( y _ n ) R ( x _ n ) & = \\mathbb { I } + R ^ { - 1 } ( y _ n ) ( R ( x _ n ) - R ( y _ n ) ) = \\mathbb { I } + \\mathcal { O } \\left ( ( x - y ) n ^ { - \\frac { 5 } { 2 } } \\right ) . \\end{align*}"}
-{"id": "6577.png", "formula": "\\begin{gather*} E ^ { p , q } _ 1 = 0 q < n , \\end{gather*}"}
-{"id": "1569.png", "formula": "\\begin{align*} \\frac { \\partial \\psi _ w ^ { - 1 } } { \\partial z } ( z ) & = \\frac { 1 } { 2 \\pi i } \\frac { z - \\zeta ^ - ( w ) } { z - \\zeta ^ + ( w ) } \\frac { ( z - \\zeta ^ - ( w ) ) - ( z - \\zeta ^ + ( w ) ) } { ( z - \\zeta ^ - ( w ) ) ^ 2 } \\\\ & = \\frac { 1 } { 2 \\pi i } \\frac { \\zeta ^ + ( w ) - \\zeta ^ - ( w ) } { ( z - \\zeta ^ + ( w ) ) ( z - \\zeta ^ - ( w ) ) } . \\end{align*}"}
-{"id": "7060.png", "formula": "\\begin{align*} \\Phi _ 0 = \\phi _ 0 d v d \\b { v } = \\frac { \\phi _ 0 } { 2 } ( d v \\otimes d \\b { v } + d \\b { v } \\otimes d v ) , \\quad \\dot { \\Phi } _ 0 = q d v ^ 2 + \\o { q d v ^ 2 } , \\ddot { \\Phi } _ 0 = \\left ( \\frac { 2 | q | ^ 2 } { \\phi ^ 2 _ 0 } + 2 \\alpha \\right ) \\Phi _ 0 . \\end{align*}"}
-{"id": "6040.png", "formula": "\\begin{align*} g _ k ( s ) = ( 4 k - 2 s ) ( s - 1 ) - r ( s - r ) / 2 \\leq 2 k ^ 2 - 2 k = \\tfrac { n ^ 2 - 4 n } { 8 } , \\end{align*}"}
-{"id": "374.png", "formula": "\\begin{align*} v _ m ^ { ( i ) } & = \\frac { 1 } { 2 \\sqrt { a } } \\left ( ( z _ 0 ^ { ( i ) } \\sqrt { a } + x _ 0 ^ { ( i ) } \\sqrt { c } ) ( s + \\sqrt { a c } ) ^ m + ( z _ 0 ^ { ( i ) } \\sqrt { a } - x _ 0 ^ { ( i ) } \\sqrt { c } ) ( s - \\sqrt { a c } ) ^ m \\right ) \\\\ w _ n ^ { ( j ) } & = \\frac { 1 } { 2 \\sqrt { b } } \\left ( ( z _ 1 ^ { ( j ) } \\sqrt { b } + y _ 1 ^ { ( j ) } \\sqrt { c } ) ( t + \\sqrt { b c } ) ^ n + ( z _ 1 ^ { ( j ) } \\sqrt { b } - y _ 1 ^ { ( j ) } \\sqrt { c } ) ( t - \\sqrt { b c } ) ^ n \\right ) . \\end{align*}"}
-{"id": "4799.png", "formula": "\\begin{align*} \\begin{pmatrix} f ^ + _ { m _ { 1 / 2 } , k _ { 1 / 2 } } \\\\ f ^ - _ { m _ { 1 / 2 } , k _ { 1 / 2 } } \\end{pmatrix} = \\begin{pmatrix} e ^ { - \\sqrt { m ^ 2 - a ^ 2 } r } r ^ { - a k _ { 1 / 2 } / m } \\\\ - \\sqrt { \\frac { m + a } { m - a } } e ^ { - \\sqrt { m ^ 2 - a ^ 2 } r } r ^ { - a k _ { 1 / 2 } / m } \\end{pmatrix} . \\end{align*}"}
-{"id": "5111.png", "formula": "\\begin{align*} \\left \\| \\frac { x + y } { 2 } \\right \\| ^ 4 & \\leq 2 \\left \\| \\frac { x } { 2 } \\right \\| ^ 4 + 2 \\left \\| \\frac { x } { 2 } \\right \\| ^ 4 + 1 2 \\left \\| \\frac { x } { 2 } \\right \\| ^ 2 \\left \\| \\frac { y } { 2 } \\right \\| ^ 2 - \\left \\| \\frac { x - y } { 2 } \\right \\| ^ 4 \\leq \\frac { 1 } { 8 } + \\frac { 1 } { 8 } + \\frac { 3 } { 4 } - \\frac { \\epsilon ^ 4 } { 1 6 } \\\\ & = 1 - \\frac { \\epsilon ^ 4 } { 1 6 } \\leq ( 1 - \\delta ) ^ 4 . \\end{align*}"}
-{"id": "2922.png", "formula": "\\begin{align*} \\begin{aligned} & \\underset { \\{ \\rho _ { l , k } \\geq 0 \\} } { \\textrm { m a x i m i z e } } & & \\min _ { k } \\mathrm { S E } _ k ^ { \\mathrm { f p Z F } } \\\\ & \\textrm { s u b j e c t t o } & & \\sum _ { k = 1 } ^ { K } \\rho _ { l , k } \\leq P _ { \\mathrm { m a x } , l } , \\ ; \\forall l , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "10056.png", "formula": "\\begin{align*} \\omega = \\sum _ { j = 1 } ^ { n - 1 } ( d r _ j \\wedge d y _ j + d \\theta _ j \\wedge d G _ j ) - \\frac { 4 } { x _ n ^ 3 } \\ , d x _ n \\wedge d y _ n + d \\theta _ n \\wedge d G _ n , \\end{align*}"}
-{"id": "6293.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\hat { \\mathcal { A } } ( D ) \\in S \\right ) = \\int _ S f _ D d \\mu . \\end{align*}"}
-{"id": "3048.png", "formula": "\\begin{align*} \\int \\limits _ { E _ { 2 } } | \\sigma _ { i } ^ { 2 } ( x , \\xi ) - \\sigma _ { i } ^ { 2 } ( y , \\xi ) | ^ { 1 + \\tau } m ( d \\xi ) \\leq \\max \\limits _ { j \\in \\{ 1 , \\dots , d \\} } \\int \\limits _ { | z | > 1 } | z | ^ { 1 + \\tau } \\mu _ { j } ( d z ) \\sum \\limits _ { k = 1 } ^ { d } | x _ { k } - y _ { k } | , \\end{align*}"}
-{"id": "6184.png", "formula": "\\begin{align*} v _ 1 ^ R = ( 1 + \\frac { \\beta } { R } ) v _ R , v _ 2 ^ R = ( 1 + \\frac { \\beta } { R } ) { \\bar v } _ R . \\end{align*}"}
-{"id": "6422.png", "formula": "\\begin{align*} ( B _ { k + 1 } ^ { T } B _ { k + 1 } - \\rho _ { k + 1 } I ) y = z _ { k + 1 } . \\end{align*}"}
-{"id": "7963.png", "formula": "\\begin{align*} \\sum _ { a \\in [ n ] } x _ a = 0 , \\end{align*}"}
-{"id": "1626.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow 0 + } r V _ r ( x ) = \\begin{cases} \\mu ( b ^ \\ast ) b ^ \\ast , & x \\geq x _ r ^ \\ast , \\\\ \\mu ( b ^ \\ast ) b ^ \\ast , & x < x _ r ^ \\ast . \\end{cases} \\end{align*}"}
-{"id": "4793.png", "formula": "\\begin{align*} \\sigma _ { e s s } ( H _ D ) = \\sigma _ { e s s } ( H _ 0 ) = ( - \\infty , - m ] \\cup [ m , + \\infty ) , \\end{align*}"}
-{"id": "6313.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\mathrm { T V } ( \\mathbb { P } _ 1 , \\mathbb { P } _ 2 ) = 0 . \\end{align*}"}
-{"id": "4546.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ k z _ { 2 j - 1 } z _ { 2 j } = 0 \\end{align*}"}
-{"id": "3641.png", "formula": "\\begin{align*} ( a _ 2 \\cdot v _ { ( 1 , 3 ) } ) \\cdot ( a _ 1 \\cdot v _ { ( 2 , 3 ) } ) = \\frac { t ^ 2 } { 2 ^ 5 } ( a _ 1 - a _ { - 1 } + a _ 2 - a _ { - 2 } ) + \\frac { ( 2 t - 1 ) t } { 2 ^ 5 } a _ 3 - \\frac { ( 2 t + 1 ) t } { 2 ^ 5 } a _ { - 3 } + \\frac { t } { 2 ^ 5 } v _ { ( 1 , 2 ) } & \\\\ + \\frac { t } { 2 ^ 6 } ( v _ { ( 1 , 3 ) } + v _ { ( 2 , 3 ) } ) + \\frac { t } { 2 ^ 3 } ( a _ 1 \\cdot v _ { ( 1 , 2 ) } + a _ 2 \\cdot v _ { ( 1 , 3 ) } - a _ 3 \\cdot v _ { ( 1 , 2 ) } ) & . \\end{align*}"}
-{"id": "269.png", "formula": "\\begin{align*} { { k + j - 1 } \\choose j } = \\sum _ { i = 0 } ^ j { { \\ell + j - i - 1 } \\choose { j - i } } \\cdot { { k - \\ell + i - 1 } \\choose i } . \\end{align*}"}
-{"id": "3733.png", "formula": "\\begin{align*} \\frac { 1 } { \\rho } \\| \\eta _ { \\rho } - u _ k \\| & = \\| \\frac { ( \\xi _ k ( w _ \\rho ) - 1 ) } { \\rho } u _ k - \\xi _ k ( w _ \\rho ) f \\| \\rightarrow \\| u _ k \\langle \\xi ^ { ' } _ k ( 0 ) , f \\rangle - f \\| \\ \\ \\ \\ \\rho \\rightarrow 0 . \\end{align*}"}
-{"id": "567.png", "formula": "\\begin{align*} R _ n ^ * = - R _ n ^ * , \\qquad \\textrm { a n d } P = - P . \\end{align*}"}
-{"id": "4551.png", "formula": "\\begin{align*} \\tilde { \\Psi } ( { \\bf x } ^ { \\boldsymbol { \\alpha } } ) & = { \\boldsymbol { \\mu } } ^ { \\boldsymbol { \\alpha } } = \\bigodot _ { 1 \\leq i \\leq m } { \\mu _ i } ^ { \\alpha _ i } \\\\ & = \\bigodot _ { 1 \\leq i \\leq m } { \\tilde { \\Psi } \\left ( \\phi ( [ L ( \\mu _ i ) ] ) \\right ) } ^ { \\alpha _ i } \\\\ & = \\bigodot _ { 1 \\leq i \\leq m } { \\tilde { \\Psi } ( x _ i ) } ^ { \\alpha _ i } \\end{align*}"}
-{"id": "4131.png", "formula": "\\begin{align*} N ( z ) = \\left ( \\mathbb { I } + \\mathcal { O } \\left ( \\frac { 1 } { z } \\right ) \\right ) \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & z ^ { \\frac { 1 } { 4 } } & 0 \\\\ 0 & 0 & z ^ { - \\frac { 1 } { 4 } } \\end{pmatrix} \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & \\frac { 1 } { \\sqrt { 2 } } & \\frac { i } { \\sqrt { 2 } } \\\\ 0 & \\frac { i } { \\sqrt { 2 } } & \\frac { 1 } { \\sqrt { 2 } } \\end{pmatrix} . \\end{align*}"}
-{"id": "5682.png", "formula": "\\begin{align*} \\| x _ \\alpha ^ A - x _ \\alpha ^ B \\| & = \\| x _ \\beta ^ A - x _ \\beta ^ B + h _ \\beta ( w ^ A _ \\beta - w ^ B _ \\beta ) \\| \\le \\| x _ \\beta ^ A - x _ \\beta ^ B \\| - h _ \\beta \\eta \\\\ & \\le \\| x _ 1 ^ A - x _ 1 ^ B \\| - ( t _ \\beta + h _ \\beta ) \\eta \\ , . \\end{align*}"}
-{"id": "4528.png", "formula": "\\begin{align*} 2 A ^ { k } _ 3 \\cap L ( A ^ { k } _ 1 , A ^ { k } _ 2 ) = \\emptyset , \\end{align*}"}
-{"id": "2751.png", "formula": "\\begin{align*} D S _ { N } ( u , v ) & = \\Big ( \\int _ { \\widetilde { \\mathcal { C } } } d z \\int _ { \\widetilde { \\Sigma } } d w - \\int _ { \\widetilde { \\mathcal { C } } _ { + } } d z \\int _ { \\widetilde { \\Sigma } } d w - \\int _ { \\widetilde { \\mathcal { C } } _ { - } } d z \\int _ { \\widetilde { \\Sigma } } d w \\Big ) \\Big ( \\cdot \\Big ) \\\\ & = : I _ { 1 } ( u , v ) - I _ { 2 } ( u , v ) - I _ { 3 } ( u , v ) . \\end{align*}"}
-{"id": "2269.png", "formula": "\\begin{align*} ( z - a ) R ( z ) f & = f f \\in V \\\\ R ( z ) ( z - a ) f & = f f \\in D \\end{align*}"}
-{"id": "864.png", "formula": "\\begin{align*} \\norm { \\overline { B } } _ F ^ 2 & = \\norm { u ^ * } _ 2 ^ 2 - ( { u ^ \\ast } ^ T u ) ^ 2 + ( { v ^ \\ast } ^ T v ) ( { u ^ \\ast } ^ T u ) + \\norm { v } _ 2 ^ 2 - ( { u ^ \\ast } ^ T u ) ( { v ^ \\ast } ^ T v ) \\\\ & = \\norm { u ^ * } _ 2 ^ 2 - ( { u ^ \\ast } ^ T u ) ^ 2 + \\norm { v } _ 2 ^ 2 . \\end{align*}"}
-{"id": "4760.png", "formula": "\\begin{align*} x = \\sum _ { j = 0 } ^ { \\infty } x _ j t ^ j \\end{align*}"}
-{"id": "8723.png", "formula": "\\begin{align*} \\hat { \\nu } ( j ) = \\frac { 2 ^ { j - 1 } ( - 1 ) ^ { m + j } ( m - j + 1 ) { { N - 1 } \\choose { m } } { { m } \\choose { j - 1 } } } { ( N - j ) \\sum _ { k = 0 } ^ { m - 1 } { { N - 1 } \\choose { k } } } a n d p g f _ { \\hat { T } _ { j , N } } ( s ) = \\prod _ { k = j } ^ { N - 1 } \\left [ { ( 1 - \\frac { k - 1 } { N - 1 } ) s \\over 1 - \\frac { k - 1 } { N - 1 } s } \\right ] . \\end{align*}"}
-{"id": "9729.png", "formula": "\\begin{align*} p _ { \\ast , X _ \\circ } ( X ) : = \\lim _ { r \\downarrow 0 } \\frac { u ( X _ \\circ + r X ) } { r ^ { \\kappa _ { X _ \\circ } } } \\end{align*}"}
-{"id": "7128.png", "formula": "\\begin{align*} f ( \\rho , \\chi ) = - 3 \\rho + \\frac { 8 \\Theta } { 3 } \\ln \\frac { \\rho } { 3 - \\rho } + \\frac { 1 } { 4 } \\big ( \\chi ^ 2 - 1 \\big ) ^ 2 , \\end{align*}"}
-{"id": "388.png", "formula": "\\begin{align*} b _ k & = p \\cdot ( \\hat { \\alpha } _ k , \\hat { \\mu } _ k + ( 0 , \\rho , h ^ { \\vee } ) ) \\\\ & = p \\cdot ( ( m _ k , \\alpha _ k , 0 ) , ( n _ k , \\mu _ k + \\rho , \\kappa ' + h ^ { \\vee } ) ) \\\\ & = p \\cdot ( \\kappa ' + h ^ { \\vee } ) m _ k + p \\cdot ( \\mu _ k + \\rho , \\alpha _ k ) , \\end{align*}"}
-{"id": "323.png", "formula": "\\begin{align*} M _ { \\Phi } = L \\begin{pmatrix} \\frac { n ( n - 1 ) } { 2 } & \\frac { n ( n - 1 ) } { 2 } & \\frac { n ( n - 1 ) ( n - 3 ) } { 8 } \\\\ 0 & n ( n - 1 ) & \\frac { n ( n - 1 ) ^ 2 } { 4 } \\\\ n ( n - 1 ) & 0 & \\frac { n ( n - 1 ) ^ 2 } { 4 } \\\\ 0 & 0 & \\frac { n ( n ^ 2 - 1 ) } { 2 } \\end{pmatrix} , \\end{align*}"}
-{"id": "4654.png", "formula": "\\begin{align*} g _ 2 : = g _ 1 h _ 1 ^ { b } h _ 2 : = g _ 1 ^ { - b } h _ 1 ^ { - 1 } ( \\mbox { a n d } k _ 2 = g _ 2 ^ { - 1 } h _ 2 ^ { - 2 } = g _ 1 ^ { b - 1 } h _ 1 ^ { 1 - b } ) . \\end{align*}"}
-{"id": "5212.png", "formula": "\\begin{align*} \\normalsize \\int _ { x } ^ { + \\infty } U ( t ) d \\lambda ( t ) = A - \\int _ { 0 } ^ { x } U ( t ) d \\lambda ( t ) = c \\exp \\left ( - \\int _ { 1 } ^ { x } t ^ { - 1 } B ( t ) d t \\right ) , a . e . \\end{align*}"}
-{"id": "895.png", "formula": "\\begin{align*} \\frac { 1 } { ( 1 - q ) ^ 2 ( 1 - q ^ 2 ) ^ 2 } = \\frac { 1 } { ( 1 - q ) ( 1 - q ^ 2 ) } + \\frac { q } { ( 1 - q ) ^ 3 } + \\frac { q ^ 4 } { ( 1 - q ) ^ 2 ( 1 - q ^ 2 ) ^ 2 } . \\end{align*}"}
-{"id": "7025.png", "formula": "\\begin{align*} n Y ^ { 2 } = X ^ { 3 } + a n ^ { 2 } X + b n ^ { 3 } \\end{align*}"}
-{"id": "3298.png", "formula": "\\begin{align*} \\langle \\alpha \\rangle _ { S } : = \\frac { 1 } { | S | } \\sum _ { f \\in S } \\alpha ( S ) . \\end{align*}"}
-{"id": "2142.png", "formula": "\\begin{align*} M = \\sup _ { t > 0 } \\| \\psi ( \\cdot ) m ( t \\cdot ) \\| _ { W ^ { s } _ 2 } < \\infty \\end{align*}"}
-{"id": "3522.png", "formula": "\\begin{align*} A = \\left ( \\begin{matrix} a _ { 0 1 } & a _ { 0 2 } & a _ { 0 3 } & a _ { 0 4 } & a _ { 0 5 } & a _ { 0 6 } \\\\ a _ { 1 1 } & a _ { 1 2 } & a _ { 1 3 } & a _ { 1 4 } & a _ { 1 5 } & a _ { 1 6 } \\\\ a _ { 2 1 } & a _ { 2 2 } & a _ { 2 3 } & a _ { 2 4 } & a _ { 2 5 } & a _ { 2 6 } \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "9056.png", "formula": "\\begin{align*} \\int _ { | | t - w | | \\leq R ^ { - 1 / 2 } } d \\mu _ { B } ( t ) = o ( 1 ) \\end{align*}"}
-{"id": "8057.png", "formula": "\\begin{align*} \\Upsilon _ i ( \\epsilon _ i , \\nu _ i ) & = \\frac { n } { 2 k } + \\frac { \\epsilon _ i } { 4 } \\displaystyle { \\sum _ { i = n + 3 } ^ { 2 n } } c ^ { \\frac { n + 3 - 2 i } { 2 } } P _ { i - n - 3 , - i } + \\frac { \\nu _ i } { 4 } \\displaystyle { \\sum _ { i = n + 3 } ^ { 2 n } } \\zeta ^ { \\frac { 2 n + 3 - 2 i } { 2 } } c ^ { \\frac { n + 3 - 2 i } { 2 } } P _ { i - n - 3 , - i } \\\\ \\end{align*}"}
-{"id": "8642.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\varepsilon ^ 2 u '' & = & u ^ 3 - u + v ^ 2 u + \\varepsilon ^ 2 v ^ 2 u - \\omega v , \\\\ & & \\\\ \\varepsilon ^ 2 v '' & = & v ^ 3 - v + u ^ 2 v + \\varepsilon ^ 2 u ^ 2 v - \\omega u , \\end{array} \\right . \\end{align*}"}
-{"id": "4530.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } f ( z ) d \\nu _ g ( z ) = \\int _ F \\int _ F f ( u - g v ) d \\mu ( u ) d \\mu ( v ) , f \\in C _ 0 ( \\mathbb { R } ^ n ) . \\end{align*}"}
-{"id": "1260.png", "formula": "\\begin{align*} m _ 3 ( \\eta , \\nu ) = O ( 1 ) , \\partial _ \\nu m _ 3 ( \\eta , \\nu ) = O ( | \\nu | ^ { - 1 } ) , \\partial _ { \\nu \\nu } ^ 2 m _ 3 ( \\eta , \\nu ) = O ( | \\nu | ^ { - 2 } ) , | \\nu | > 2 | \\eta | \\end{align*}"}
-{"id": "1257.png", "formula": "\\begin{align*} Q ( X ) = \\frac { 3 } { 4 } X ^ 2 + \\frac { \\beta } { 8 } ( 1 + X ) ^ 3 \\mbox { s o t h a t } \\Theta ( \\eta , \\nu ) = \\eta ^ 3 Q ( \\nu / \\eta ) . \\end{align*}"}
-{"id": "7029.png", "formula": "\\begin{align*} f ( z ) = z + \\sum \\limits _ { n = 2 } ^ { \\infty } a _ { n } z ^ { n } , \\end{align*}"}
-{"id": "4882.png", "formula": "\\begin{align*} U ( x ^ * , c ) = \\{ z = x + \\sqrt { - 1 } y \\in \\mathbb { C } ^ n \\ , ; \\ , | y | ^ 2 < | x - x ^ * | ^ 2 + c ^ 2 \\} . \\end{align*}"}
-{"id": "8444.png", "formula": "\\begin{align*} i A = i - D _ t \\zeta \\frac { \\partial _ { \\alpha } \\mathfrak { F } } { \\zeta _ { \\alpha } } - \\frac { i } { 2 \\pi } \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j ( D _ t \\zeta ( \\alpha , t ) - \\dot { z } _ j ( t ) ) } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } - ( D _ t ^ 2 \\zeta + i ) \\Psi _ { \\zeta } \\circ \\zeta . \\end{align*}"}
-{"id": "2699.png", "formula": "\\begin{align*} f ( 4 ; z _ 0 ) = \\log ( - \\tau ) + 2 ( 1 + \\tau ) / ( 1 - \\tau ) , f ''' ( 4 ; z _ 0 ) = - 3 2 \\eta _ { - } ^ { 3 } . \\end{align*}"}
-{"id": "7988.png", "formula": "\\begin{align*} \\log g ( n ) - \\log h ( n ) = \\sum _ { p \\leq \\sqrt { n } } { \\frac { \\xi _ { p } ( 1 + \\xi _ { p } ) } { 2 } } \\log p + \\sum _ { \\sqrt { n } < p \\leq \\frac { n } { 2 } } \\log p \\ , . \\end{align*}"}
-{"id": "8811.png", "formula": "\\begin{align*} Q _ M ( A \\times B \\times [ s , t ] ) = \\int _ { s } ^ { t } \\int _ { A \\cap B } f ^ 2 ( v ( r , x ) ) \\ , \\textrm { d } r \\ , \\textrm { d } x \\end{align*}"}
-{"id": "2232.png", "formula": "\\begin{align*} & \\iiint \\delta ( x - y ) \\delta ( y - z ) \\varphi ( x , y , z ) \\d x \\d y \\d z = \\int \\varphi ( x , x , x ) \\d x \\\\ & \\frac { \\mathcal { P } } { x - \\omega } \\frac { \\mathcal { P } } { y - \\omega } = \\frac { 1 } { y - x } \\left ( \\frac { \\mathcal { P } } { x - \\omega } - \\frac { \\mathcal { P } } { y - \\omega } \\right ) + \\pi ^ 2 \\delta ( x - \\omega ) \\delta ( y - \\omega ) \\end{align*}"}
-{"id": "10031.png", "formula": "\\begin{align*} D ( f , g ) ( x ) & = \\Big ( f ( x ) g ( q x ) - f ( q x ) g ( x ) \\Big ) a ^ { - 1 } ( 1 - q ) ( 1 + a ^ 2 x ) w ( x ) , \\end{align*}"}
-{"id": "9496.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\infty } \\frac { 1 } { \\beta _ j } < \\infty , \\ \\ \\mbox { a n d } \\ \\ \\prod _ { j = 1 } ^ { + \\infty } \\beta _ { j } ^ { \\frac { 1 } { \\beta _ j } } < \\infty . \\end{align*}"}
-{"id": "5337.png", "formula": "\\begin{align*} B ^ * . w ^ k = i \\left ( - \\overline { b _ { k + 2 } } \\frac { | | w ^ k | | ^ 2 } { | | w ^ { k + 2 } | | ^ 2 } w ^ { k + 2 } + \\overline { a _ { k - 2 } } \\frac { | | w ^ k | | ^ 2 } { | | w ^ { k - 2 } | | ^ 2 } w ^ { k - 2 } \\right ) . \\end{align*}"}
-{"id": "4844.png", "formula": "\\begin{align*} F ( w ) = \\beta ( w ) , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , F \\circ e ^ { \\theta } ( w ) = \\alpha ( w ) , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\forall w \\in W _ { \\leq n - 1 } . \\end{align*}"}
-{"id": "5446.png", "formula": "\\begin{align*} X _ t \\ & = \\ e ^ { t A } \\ , x _ 0 + \\int _ 0 ^ t e ^ { ( t - s ) A } \\ , b _ s ( X , \\alpha _ s ) \\ , d s + \\int _ 0 ^ t e ^ { ( t - s ) A } \\ , \\sigma _ s ( X , \\alpha _ s ) \\ , d W _ s \\\\ & \\ + \\int _ 0 ^ t \\int _ { U \\setminus \\{ 0 \\} } e ^ { ( t - s ) A } \\ , \\gamma _ s ( X , \\alpha _ s , z ) \\ , \\big ( \\pi ( d s \\ , d z ) - \\lambda _ \\pi ( d z ) \\ , d s \\big ) , \\qquad 0 \\leq t \\leq T . \\end{align*}"}
-{"id": "8734.png", "formula": "\\begin{align*} D _ { \\tau } ^ { \\alpha , \\sigma , \\gamma , m , n } u = \\tau ^ { - \\alpha } \\sum _ { k = 0 } ^ n \\omega ^ { ( \\alpha , \\sigma ) } _ { n - k } ( u ( t _ k ) - u _ 0 ) + \\tau ^ { - \\alpha } \\sum _ { k = 1 } ^ m w ^ { ( \\alpha , \\sigma ) } _ { n , k } ( u ( t _ k ) - u _ 0 ) , \\end{align*}"}
-{"id": "3587.png", "formula": "\\begin{align*} \\langle u , v \\rangle = \\frac { 1 } { \\nu } \\langle u , a \\cdot v \\rangle = \\frac { 1 } { \\nu } \\langle a \\cdot u , v \\rangle = 0 \\end{align*}"}
-{"id": "9597.png", "formula": "\\begin{align*} \\frac { 1 } { ( 1 - q ) ^ { N + 1 } } & = \\sum _ { n = 0 } ^ \\infty \\frac { ( N + 1 ) _ n } { n ! } q ^ n , \\\\ \\frac { 1 } { ( 1 - q ) ^ N } & = \\sum _ { n = 0 } ^ \\infty \\frac { ( N ) _ n } { n ! } q ^ n , \\end{align*}"}
-{"id": "2589.png", "formula": "\\begin{align*} g ( t , \\theta , x ) : = \\frac { V _ t ( \\theta \\eta _ t f ) ( x ) } { \\eta _ t \\phi ( x ) } \\xrightarrow [ t \\to \\infty ] { } G ( \\theta ) : = \\Big ( \\frac { 1 } { 1 + \\theta ^ { - ( \\gamma _ 0 - 1 ) } } \\Big ) ^ { \\frac { 1 } { \\gamma _ 0 - 1 } } , x \\in E , \\theta \\geq 0 . \\end{align*}"}
-{"id": "1861.png", "formula": "\\begin{align*} c _ n = \\frac { 1 } { 2 } \\sqrt { \\frac { k _ n } { \\kappa _ j } } \\left ( 1 + \\frac { \\kappa _ j } { k _ n } \\right ) Y _ { j , n } \\qquad \\mbox { a n d } d _ n = \\frac { 1 } { 2 } \\sqrt { \\frac { k _ n } { \\kappa _ j } } \\left ( 1 - \\frac { \\kappa _ j } { k _ n } \\right ) Y _ { j , n } , \\end{align*}"}
-{"id": "3952.png", "formula": "\\begin{align*} f _ { 1 } ( x , m ^ { \\hat v } _ { t } , \\hat { v } _ { 1 } ( x , m ^ { \\hat v } _ { t } ) ) + D u ^ { \\hat v } _ { 1 } ( x , t ) . g _ { 1 } ( x , m ^ { \\hat v } _ { t } , \\hat { v } _ { 1 } ( x , m ^ { \\hat v } _ { t } ) ) = H _ { 1 } ( x , m ^ { \\hat v } _ { t } , D u ^ { \\hat v } _ { 1 } ( x , t ) ) \\ , , \\end{align*}"}
-{"id": "8031.png", "formula": "\\begin{align*} \\psi _ c ^ \\pm ( u ) = \\pm t ^ { - n - 1 } c ^ { n + 1 } u a _ 1 = \\prod _ { i = 1 } ^ { 2 n } \\alpha _ i = c ^ { 2 n } , \\end{align*}"}
-{"id": "5081.png", "formula": "\\begin{align*} A _ { \\sigma ( U V ) } A _ { \\sigma ( U W ) } ^ * & = f ( U V ) \\overline { f ( U W ) } A _ V A _ W ^ * , A _ { \\sigma ( U V ) } \\Psi _ { \\sigma ( U W ) } ^ * = f ( U V ) \\overline { f ( U W ) } A _ V \\Psi _ W ^ * , \\\\ \\Psi _ { \\sigma ( U V ) } \\Psi _ { \\sigma ( U W ) } ^ * & = f ( U V ) \\overline { f ( U W ) } \\Psi _ V \\Psi _ W ^ * , ~ \\forall U , V , W \\in \\mathcal { U } . \\end{align*}"}
-{"id": "4203.png", "formula": "\\begin{align*} R ( z ) = S ( z ) \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & z ^ { - \\beta } & 0 \\\\ 0 & 0 & z ^ { - \\beta } \\end{pmatrix} D _ 0 ^ { - n } ( z ) \\Psi _ { \\alpha } ^ T ( n ^ 3 f ( z ) ) \\frac { - 1 } { 4 \\pi ^ 2 } \\begin{pmatrix} 1 & 0 & 0 \\\\ - 2 \\alpha - \\frac { 1 } { 2 } & - 1 & 0 \\\\ \\alpha ( \\alpha + \\frac { 1 } { 2 } ) & 2 \\alpha + \\frac { 1 } { 2 } & 1 \\end{pmatrix} E _ { i n } ^ { - 1 } ( z ) \\end{align*}"}
-{"id": "7744.png", "formula": "\\begin{align*} \\langle x , x \\rangle \\Big \\langle \\Big ( \\| y \\| \\langle x , x \\rangle x + \\mu \\langle y , x \\rangle x \\Big ) , x \\Big \\rangle = 0 , \\end{align*}"}
-{"id": "6515.png", "formula": "\\begin{align*} h ( \\lambda ) : = & \\phi _ \\lambda ^ { - 1 } \\circ \\mathfrak { a } _ { \\underline { n } } ( \\lambda ) \\\\ = & \\left ( \\phi _ { \\lambda , 1 } ^ { - 1 } \\left ( f _ \\lambda ^ { n _ 1 } ( a _ 1 ( \\lambda ) ) \\right ) , \\ldots , \\phi _ { \\lambda , m } ^ { - 1 } \\left ( f _ \\lambda ^ { n _ m } ( a _ m ( \\lambda ) ) \\right ) \\right ) . \\end{align*}"}
-{"id": "8047.png", "formula": "\\begin{align*} \\sigma _ k ( a ) = \\begin{pmatrix} e ^ { \\pi \\imath k / n } & 0 \\\\ 0 & e ^ { - \\pi \\imath k / n } \\end{pmatrix} , \\sigma _ k ( x ) = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "6663.png", "formula": "\\begin{align*} H _ { r } ( T ) = ( - 1 ) ^ { | T | } | \\hat { K } _ r ( T ) | = ( - 1 ) ^ { m } \\binom { n } { ( k - r m ) d - m } _ { k - r m } . \\end{align*}"}
-{"id": "8097.png", "formula": "\\begin{align*} e _ { k , \\nu } = ( - 1 ) ^ k 2 ^ { - 2 k } ( \\nu - k + 1 / 2 ) _ { 2 k } \\binom { 2 k } { k } , \\end{align*}"}
-{"id": "9207.png", "formula": "\\begin{align*} A : = \\{ s \\in \\lbrack 0 , 1 ] \\ , : \\ , \\bar { L } ( h ( s ^ { \\prime } ) , s ^ { \\prime } ) = \\bar { R } ( h ( s ^ { \\prime } ) , s ^ { \\prime } ) s ^ { \\prime } \\leq s \\ , \\} , \\end{align*}"}
-{"id": "5867.png", "formula": "\\begin{align*} \\int _ { \\C ^ n } \\prod _ { i = 1 } ^ n \\alpha \\chi _ 0 ^ { - 1 } ( z _ i ) \\beta \\chi _ 0 ^ { - 1 } ( 1 - z _ i ) \\prod _ { i \\neq j } \\gamma ( ( z _ i - z _ j ) ) \\textstyle \\prod \\limits _ { i = 0 } ^ n \\dd z _ i \\displaystyle = n ! \\prod _ { j = 0 } ^ { n - 1 } \\frac { \\Gamma _ { \\C } ( \\alpha \\gamma ^ { j } ) \\Gamma _ { \\C } ( \\beta \\gamma ^ { j } ) \\Gamma _ { \\C } ( \\gamma ^ { j + 1 } ) } { \\Gamma _ { \\C } ( \\alpha \\beta \\gamma ^ { n + j - 1 } ) \\Gamma _ { \\C } ( \\gamma ) } . \\end{align*}"}
-{"id": "1886.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } - \\frac { 1 } { 2 } ( t + 1 ) e ^ { - t } = 0 . \\end{align*}"}
-{"id": "8595.png", "formula": "\\begin{align*} \\left ( | E ( H ) | - \\binom { | A | } k \\right ) k \\geq ( \\ell - 1 ) | B | , \\end{align*}"}
-{"id": "9280.png", "formula": "\\begin{align*} A = o ( | p | ^ 2 ) I , p \\cdot D _ p A \\le o ( | p | ^ 2 ) I , p \\cdot D _ p B \\le o ( | p | ^ 2 ) , \\end{align*}"}
-{"id": "2592.png", "formula": "\\begin{align*} g ( t , \\theta , x ) = \\int _ 0 ^ { \\theta } \\Pi _ x ^ { ( \\phi ) } [ ( \\phi ^ { - 1 } f ) ( \\xi _ t ) e ^ { - \\frac { 1 } { \\gamma _ 0 - 1 } J _ g ( t , r , \\xi ) } ] d r , t \\geq 0 , \\theta \\geq 0 , x \\in E , \\end{align*}"}
-{"id": "9328.png", "formula": "\\begin{align*} \\int _ { x \\in M } F _ { v } ( x ) d V = 0 . \\end{align*}"}
-{"id": "5045.png", "formula": "\\begin{align*} h _ { n x } ( v ) = ( n c _ n / \\log n ) ^ { 1 / 2 } w _ { n m } ( v - x ) . \\end{align*}"}
-{"id": "7551.png", "formula": "\\begin{align*} \\norm { f } _ { 1 + \\alpha , p } = \\norm { f } _ { C ^ { 1 + \\alpha } ( \\mathbb { R } ^ d ) } + \\norm { f } _ { W ^ { 1 , p } ( \\mathbb { R } ^ d ) } . \\end{align*}"}
-{"id": "6082.png", "formula": "\\begin{align*} \\int _ 0 ^ t s ^ \\frac { k - 1 } { k } V ^ { \\delta _ 2 + \\delta _ 3 } ( V ' ) ^ { 1 - \\delta _ 2 } d s & \\leq t ^ \\frac { k - 1 } { k } \\int _ 0 ^ t V ^ { \\delta _ 2 + \\delta _ 3 } ( V ' ) ^ { 1 - \\delta _ 2 } d s \\\\ & \\leq t ^ { \\frac { k - 1 } { k } } t ^ { \\delta _ 2 } \\left ( \\int _ 0 ^ t V ^ { ( \\delta _ 2 + \\delta _ 3 ) / ( 1 - \\delta _ 2 ) } ( V ' ) d s \\right ) ^ { 1 - \\delta _ 2 } \\\\ & = t ^ \\frac { n - k - 2 } { n - k - 1 } \\left ( \\frac { 1 - \\delta _ 2 } { 1 + \\delta _ 3 } \\right ) ^ { 1 - \\delta _ 2 } V ^ { 1 + \\delta _ 3 } . \\end{align*}"}
-{"id": "3724.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\int _ { | u | \\geq M } \\frac { F ( y , u ) } { | x - y | ^ { \\mu } } F ( x , u ) d y ~ d x = o ( M ) , \\ ; \\int _ { \\Omega } \\int _ { | u _ k | \\geq M } \\frac { F ( y , u _ k ) } { | x - y | ^ { \\mu } } F ( x , u _ k ) d y ~ d x = o ( M ) , \\end{align*}"}
-{"id": "7740.png", "formula": "\\begin{align*} \\bigg [ \\Big ( \\| y \\| \\langle x , x \\rangle ^ 2 + \\bar { \\lambda } \\| x \\| \\langle x , x \\rangle \\langle x , y \\rangle \\Big ) \\xi , \\langle x , x \\rangle \\xi \\bigg ] = 0 \\end{align*}"}
-{"id": "5798.png", "formula": "\\begin{align*} T _ g ( \\phi ) \\ : = \\ \\sum _ { n = 0 } ^ \\infty \\frac { \\phi ^ { ( n ) } \\circ \\ell _ { \\lambda _ g + 3 } } { n ! } \\varepsilon ^ n . \\end{align*}"}
-{"id": "3484.png", "formula": "\\begin{align*} \\frac { d } { d y } \\Re f _ M ( x _ 0 + i y ; 0 ) \\begin{cases} < 0 , & y > 0 , \\\\ > 0 , & y < 0 . \\end{cases} \\end{align*}"}
-{"id": "3555.png", "formula": "\\begin{align*} c ( n , m , k , l ) : = \\frac { 1 } { \\Gamma ( \\frac { 1 } { 2 } ) ^ { 3 } } \\frac { \\Gamma ( m + n + l + \\frac { 1 } { 2 } ) \\Gamma ( n + k + l + \\frac { 1 } { 2 } ) \\Gamma ( m + k + l + \\frac { 1 } { 2 } ) } { m ! \\ , n ! \\ , k ! \\ , ( m + l ) ! \\ , ( n + l ) ! \\ , ( k + l ) ! } . \\end{align*}"}
-{"id": "1380.png", "formula": "\\begin{align*} m ( J , 0 ) & = 0 \\ ; , \\\\ m ( J , k ) & = n - 1 \\ ; , \\\\ m ( J , q - 1 ) & = \\begin{cases} m ( J , q ) & q \\not \\in J \\ ; , \\\\ m ( J , q ) - 1 & q \\in J \\ ; . \\end{cases} \\end{align*}"}
-{"id": "8278.png", "formula": "\\begin{align*} a _ 0 | \\partial _ { \\alpha } \\xi _ 0 + 1 | = | w _ 0 + i | . \\end{align*}"}
-{"id": "10019.png", "formula": "\\begin{align*} \\Vert M _ { \\lambda } \\Vert = \\Vert \\mathfrak { B } T _ { r } \\mathfrak { B } _ { X } ^ { - 1 } \\Vert = \\Vert T _ { r } \\Vert \\leq \\sup _ { n } \\prod _ { k = 1 } ^ { n } \\Vert T _ { r _ { k } } \\Vert \\ , . \\end{align*}"}
-{"id": "9106.png", "formula": "\\begin{align*} \\Upsilon ( x ) = \\dfrac { 1 } { | x | ^ \\alpha } \\quad \\forall | x | > 1 , \\end{align*}"}
-{"id": "7325.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\rho u - \\operatorname { d i v } ( K ( x ) \\nabla u + a ( x ) \\nabla u ) = { } & - \\operatorname { d i v } ( a ( x ) \\nabla p ) + \\rho ( p + r ) , \\\\ \\rho v - \\operatorname { d i v } ( K ( x ) \\nabla v + b ( x ) \\nabla v ) = { } & - \\operatorname { d i v } ( b ( x ) \\nabla q ) + \\rho ( q + s ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "3204.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\frac { 1 } { \\mu _ r } E _ { X _ r } = D _ { Y _ { \\ast } } ^ { - 1 } . \\end{align*}"}
-{"id": "894.png", "formula": "\\begin{align*} \\phi _ 1 & = e _ 2 & \\phi _ 2 & = e _ 1 + e _ 2 & \\phi _ 3 & = e _ 1 \\end{align*}"}
-{"id": "8816.png", "formula": "\\begin{align*} \\overline { X } ^ k ( \\varrho _ h ) = \\overline { M } ^ { k , h } + \\overline { B } ^ { k , h } , \\end{align*}"}
-{"id": "9519.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { n - 1 } q ^ { \\frac { n ( n + 1 ) } { 2 } } } { ( 1 - q ^ n ) ( q ) _ n } = \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ n } { 1 - q ^ n } , \\end{align*}"}
-{"id": "4777.png", "formula": "\\begin{align*} \\max \\{ 1 , C \\} ( b + l ) \\log _ 2 n = & \\ ( 1 + o ( 1 ) ) \\max \\{ 1 , C \\} C ''^ { - 2 } ( b + l ) \\frac { h _ 1 ^ { 2 } } { \\log _ 2 \\log _ 2 n } < \\frac 1 4 C _ 2 \\frac { ( b + l ) b ^ { 2 } } { \\log _ 2 \\log _ 2 n } = \\frac { M ( 2 ) } 4 . \\end{align*}"}
-{"id": "747.png", "formula": "\\begin{align*} \\mathbb { E } _ { r } \\ ! \\left [ r \\right ] = m n , ~ ~ ~ ~ \\mathbb { E } _ { r } \\ ! \\left [ r ^ { 2 } \\right ] = m n ( m n + 1 ) , \\end{align*}"}
-{"id": "7971.png", "formula": "\\begin{align*} \\bigsqcup _ { a \\in A } \\{ a \\} \\sqcup \\bigsqcup _ { q \\in [ s _ A ] } S ^ A _ q = \\bigsqcup _ { b \\in B } \\{ b \\} \\sqcup \\bigsqcup _ { r \\in [ s _ B ] } S ^ B _ r = [ n ] . \\end{align*}"}
-{"id": "9475.png", "formula": "\\begin{align*} W ^ { s , m } _ { 0 } ( \\Omega ) \\ni \\varphi \\mapsto ( u , \\varphi ) : = \\displaystyle \\int _ { \\mathbb { R } ^ N } \\displaystyle \\int _ { \\mathbb { R } ^ N } \\frac { J _ { m } ( u ( x ) - u ( y ) ) ( \\varphi ( x ) - \\varphi ( y ) ) } { | x - y | ^ { N + s m } } \\dd x \\dd y \\end{align*}"}
-{"id": "4616.png", "formula": "\\begin{align*} \\mathcal { P } = \\begin{bmatrix} A _ 0 & & & A _ 1 \\\\ A _ 1 & A _ 0 & & \\\\ & \\ddots & \\ddots & \\\\ & & A _ 1 & A _ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "5465.png", "formula": "\\begin{align*} K _ { \\hat \\tau } ^ n \\ & = \\ Y _ { \\hat \\tau } ^ n - Y _ 0 ^ n - g ( \\hat X ) - \\int _ 0 ^ { \\hat \\tau } f _ s ( \\hat X , \\hat I _ s ) \\ , d s + \\int _ 0 ^ { \\hat \\tau } \\int _ \\Lambda R _ s ^ n ( b ) \\ , \\hat \\theta ( d s \\ , d b ) \\\\ & \\ + \\int _ 0 ^ { \\hat \\tau } Z _ s ^ n \\ , d \\hat W _ s + \\int _ 0 ^ { \\hat \\tau } \\int _ U L _ s ^ n ( z ) \\ , ( \\hat \\pi ( d s \\ , d z ) - \\lambda _ \\pi ( d z ) d s ) . \\end{align*}"}
-{"id": "3311.png", "formula": "\\begin{align*} \\frac { 1 } { q ^ { \\ell } \\phi ( M ) } \\sum _ { \\chi \\in G ( R _ { \\ell , M } ) } \\chi ( A ) \\overline { \\chi } ( B ) = \\begin{cases} 1 & A \\equiv B \\bmod R _ { \\ell , M } , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "4164.png", "formula": "\\begin{align*} P ^ { ( 1 ) } _ + ( z ) P ^ { ( 1 ) } _ - ( z ) ^ { - 1 } = \\mathbb { I } + \\frac { A _ n ^ { ( 1 ) } ( z ) } { n ^ 3 z } + E _ n ( z ) \\mathcal { O } ( n ^ { - 3 } ) E _ n ^ { - 1 } ( z ) \\end{align*}"}
-{"id": "6755.png", "formula": "\\begin{align*} \\mathbf v \\approx x ^ p \\ , \\overline x \\ , ^ q \\end{align*}"}
-{"id": "2069.png", "formula": "\\begin{align*} \\frac { \\rho ^ 2 R '' ( \\rho | \\rho _ { \\rm t x } ) } { R ( \\rho | \\rho _ { \\rm t x } ) } + \\frac { \\rho R ' ( \\rho | \\rho _ { \\rm t x } ) } { R ( \\rho | \\rho _ { \\rm t x } ) } - \\frac { \\rho ^ 2 T ' ( t | t _ 0 ) } { D T ( t | t 0 ) } = \\frac { \\Phi '' ( \\varphi | \\varphi _ { \\rm t x } ) } { - \\Phi ( \\varphi | \\varphi _ { \\rm t x } ) } \\overset { ( a ) } { = } \\alpha , \\end{align*}"}
-{"id": "2393.png", "formula": "\\begin{align*} M _ 0 = & \\{ \\mathfrak { p } : ( l , l - 1 ) \\in \\mathfrak { p } \\} , & M _ { r s } = & \\{ \\mathfrak { p } : \\mathfrak { p } ' _ r , \\mathfrak { p } '' _ s \\in \\mathfrak { p } \\} , \\end{align*}"}
-{"id": "808.png", "formula": "\\begin{align*} G _ 0 & > G _ 1 = \\cdots = G _ { q ^ { d } - 1 } = N _ 1 \\\\ & > G _ { q ^ { d } } = \\cdots = G _ { q ^ { 2 d } - 1 } = N _ 2 \\\\ & > G _ { q ^ { 2 d s } } = \\cdots = G _ { q ^ { 3 d } - 1 } = N _ 3 \\\\ & > \\cdots \\\\ & > G _ { q ^ { d ( \\alpha - 2 ) } } = \\cdots = G _ { q ^ { d ( \\alpha - 1 ) } } = N _ { \\alpha - 1 } \\\\ & > \\{ 1 \\} . \\end{align*}"}
-{"id": "4637.png", "formula": "\\begin{align*} u _ 0 ^ { n s } ( x ) = \\begin{cases} \\cos ^ 2 4 \\pi \\big ( x - \\frac { 1 } { 2 } \\big ) , & x \\in \\big ( \\frac { 3 } { 8 } , \\frac { 5 } { 8 } \\big ) , \\\\ 0 , & x \\in [ 0 , 1 ] \\backslash \\big ( \\frac { 3 } { 8 } , \\frac { 5 } { 8 } \\big ) , \\end{cases} \\end{align*}"}
-{"id": "7779.png", "formula": "\\begin{align*} M _ { t } ^ { B ^ 2 } f ( x ) = d \\int _ { 0 } ^ { 1 } r ^ { d - 1 } { A _ { t r } ^ d } f ( x ) { \\rm d } r , \\end{align*}"}
-{"id": "6502.png", "formula": "\\begin{align*} K ^ \\circ : = \\{ y \\in \\R ^ n : \\ \\langle y , z \\rangle \\leq 1 \\forall z \\in K \\} \\ , . \\end{align*}"}
-{"id": "7866.png", "formula": "\\begin{align*} | \\mathfrak m _ N ( \\xi ) | \\le 8 e ^ { - \\frac { c \\kappa ( d , N ) ^ 2 } { 1 0 0 } \\sum _ { i = 1 } ^ d \\sin ^ 2 ( \\pi \\xi _ i ) } + 8 e ^ { - \\frac { c \\kappa ( d , N ) ^ 2 } { 1 0 0 } \\sum _ { i = 1 } ^ d \\cos ^ 2 ( \\pi \\xi _ i ) } , \\end{align*}"}
-{"id": "8876.png", "formula": "\\begin{align*} e _ 9 & = 4 p ^ 2 \\mu ( 4 \\mu ^ 2 p ^ 3 - 1 8 \\mu p ^ 2 + 9 9 \\mu p - 3 + 1 5 p - 9 6 ) - 1 2 p ^ 2 + 9 3 p + 3 + [ 6 p ^ 2 ( 1 - 2 \\mu p ) ( 2 \\mu p + 1 ) ] b , \\end{align*}"}
-{"id": "3562.png", "formula": "\\begin{align*} u : v : y z : z x : x y = U : V : W _ { 1 } : W _ { 2 } : W _ { 3 } , \\end{align*}"}
-{"id": "1235.png", "formula": "\\begin{align*} J _ j ( f , g ) = \\int e ^ { - 3 i \\Phi ( \\xi , \\eta ) } \\partial _ \\eta \\left ( \\frac { 1 } { - 3 i \\partial _ \\eta \\Phi ( \\xi , \\eta ) } f ( \\eta ) g ( \\eta - \\xi ) \\varphi _ j ( \\eta / \\xi ) \\right ) d \\eta . \\end{align*}"}
-{"id": "5901.png", "formula": "\\begin{align*} \\bar { \\psi } _ n : = \\sum _ { k = n } ^ { \\infty } \\psi _ k + \\varphi _ { k + 1 } . \\end{align*}"}
-{"id": "7590.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { \\left ( \\eta ( t ) \\cdot ( \\nabla \\nabla g _ { \\nu ( t - s ) } ) * \\sigma ( s ) - ( \\nabla \\nabla g _ { \\nu ( t - s ) } ) * ( \\eta ( t ) \\sigma ( s ) ) \\right ) } _ { \\alpha , p } \\\\ \\le \\frac { C } { \\left ( \\nu ( t - s ) \\right ) ^ { \\frac { 1 } { 2 } } } \\norm { \\eta ( t ) } _ { C ^ { 1 + \\alpha } } \\norm { \\sigma ( s ) } _ { \\alpha , p } \\end{gathered} \\end{align*}"}
-{"id": "9823.png", "formula": "\\begin{align*} W _ \\lambda ( r , u ) : = H _ \\lambda ( r , u ) ( N ( r , u ) - \\lambda ) . \\end{align*}"}
-{"id": "1427.png", "formula": "\\begin{align*} m ( 0 , x ) = m _ { 0 } ( x ) , u ( T , x ) = u _ { T } ( x ) . \\end{align*}"}
-{"id": "5076.png", "formula": "\\begin{align*} A _ g = A _ e \\pi _ { g ^ { - 1 } } , ~ \\Psi _ g = \\Psi _ e \\pi _ { g ^ { - 1 } } , ~ \\forall g \\in G \\end{align*}"}
-{"id": "4190.png", "formula": "\\begin{align*} \\frac { A _ n ^ { ( 2 ) } ( z ) } { n ^ 6 z } \\frac { A _ n ^ { ( 2 ) } ( 0 ) } { n ^ 6 z } & = \\mathcal { O } \\left ( n ^ { - 1 } \\right ) \\\\ \\frac { A _ n ^ { ( 1 ) } ( 0 ) A _ n ^ { ( 1 ) } ( z ) A _ n ^ { ( 1 ) } ( 0 ) } { n ^ 9 z ^ 3 } \\frac { A _ n ^ { ( 2 ) } ( 0 ) } { n ^ 6 z } & = \\mathcal { O } \\left ( n ^ { - \\frac { 3 } { 2 } } \\right ) \\\\ \\frac { A _ n ^ { ( 2 ) } ( 0 ) } { n ^ 6 z } & = \\mathcal { O } ( 1 ) . \\end{align*}"}
-{"id": "5458.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow + \\infty } \\bar J ^ { \\mathcal R , \\alpha , k } ( \\bar \\nu ^ { \\alpha , k } ) \\ = \\ J ( \\alpha ) , \\end{align*}"}
-{"id": "830.png", "formula": "\\begin{align*} \\omega = \\prod _ { i = 1 } ^ { { s } } \\prod _ { k = 2 } ^ \\alpha \\lambda _ { i , k } ^ { \\mu _ { i , k } } \\lambda _ { i , 1 } ^ { - \\mu _ { i , 1 } } d T , \\end{align*}"}
-{"id": "2202.png", "formula": "\\begin{align*} s ( \\boldsymbol { x } ^ * ( t ) , \\boldsymbol { u } ^ * ( t ) , t ) = \\min _ { \\boldsymbol { x } \\in \\mathbb { X } , \\boldsymbol { u } \\in \\mathbb { U } } s ( \\boldsymbol { x } ( t ) , \\boldsymbol { u } ( t ) , t ) , \\forall t \\in [ t _ 0 , t _ f ) \\end{align*}"}
-{"id": "5035.png", "formula": "\\begin{align*} \\tilde J _ b ( f ) = \\begin{pmatrix} \\alpha _ f ( b ) & 0 \\\\ \\beta _ f ( b ) & \\alpha _ f ( b ) \\end{pmatrix} . \\end{align*}"}
-{"id": "973.png", "formula": "\\begin{align*} \\mathtt E ( A ) = \\left | \\left \\{ ( i , j ) \\ , | \\ , 2 \\leq i \\leq j + 1 \\leq n \\quad \\textrm { a n d } r _ { i , j } = r _ { i - 1 , j + 1 } \\right \\} \\right | \\ , , \\end{align*}"}
-{"id": "2306.png", "formula": "\\begin{align*} ( \\hat { E } | C ( z ) | E ) & = ( E | C ( z ) | \\hat { E } ) = - i \\pi \\sigma ( z ) \\\\ \\hat { \\Omega } R ( z ) | E ) & = - | \\hat { E } ) + z R ( z ) | E ) \\\\ ( E | R ( z ) \\hat { \\Omega } & = - ( \\hat { E } | + z ( E 1 | R ( z ) \\end{align*}"}
-{"id": "9113.png", "formula": "\\begin{align*} \\frac { m } { d } : = \\min _ i \\left \\{ \\frac { n _ i } { d _ i } \\left | f _ R ^ { \\# } ( a _ i ) \\neq 0 \\right . \\right \\} \\end{align*}"}
-{"id": "6867.png", "formula": "\\begin{align*} F = \\det ( d y ) - 1 = y ^ 1 _ { 1 0 } y ^ 2 _ { 0 1 } - y ^ 1 _ { 0 1 } y ^ 2 _ { 1 0 } - 1 \\ , , \\end{align*}"}
-{"id": "4800.png", "formula": "\\begin{align*} \\left ( - \\nu + | x | \\lambda _ j ^ + ( x ) ( N ^ 2 + 1 ) \\right ) ^ 2 \\leq \\nu ^ 2 , \\quad \\ j = 1 , 2 . \\end{align*}"}
-{"id": "805.png", "formula": "\\begin{align*} \\frac { H ^ i } { H ^ { i + 1 } } = \\{ \\sigma ( D ) D ^ { - 1 } \\mod H ^ { i + 1 } : D \\in H ^ i \\} , \\end{align*}"}
-{"id": "2861.png", "formula": "\\begin{align*} x \\odot y = ( x \\vee y ' ) \\wedge y , \\end{align*}"}
-{"id": "7688.png", "formula": "\\begin{align*} T \\psi _ { \\mp } ( D ) = \\psi _ { \\pm } ( D ) T . \\end{align*}"}
-{"id": "9891.png", "formula": "\\begin{align*} \\begin{pmatrix} \\beta _ { 0 } \\\\ \\beta _ { 1 } \\\\ \\beta _ { 2 } \\\\ \\vdots \\\\ \\beta _ { n } \\end{pmatrix} & = \\begin{pmatrix} N ( 0 , 0 ) & N ( 1 , 0 ) & \\cdots & N ( n , 0 ) \\\\ 0 & N ( 1 , 1 ) & \\cdots & N ( n , 1 ) \\\\ \\vdots & \\ddots & \\ddots & \\vdots \\\\ 0 & \\cdots & 0 & N ( n , n ) \\end{pmatrix} \\begin{pmatrix} \\alpha _ { 0 } \\\\ \\alpha _ { 1 } \\\\ \\alpha _ { 2 } \\\\ \\vdots \\\\ \\alpha _ { n } \\end{pmatrix} \\end{align*}"}
-{"id": "2901.png", "formula": "\\begin{align*} \\widetilde { H } _ { } = - \\frac { \\partial ^ { 2 } } { \\partial \\rho ^ { 2 } } + \\frac { k ( k - \\sigma _ 3 ) } { \\rho ^ { 2 } } + \\rho ^ { 2 } , \\end{align*}"}
-{"id": "2747.png", "formula": "\\begin{align*} \\psi _ { n , \\alpha , \\tau } ( z , w ) & = \\frac { 1 - z w } { w - z } \\Big ( \\frac { w } { z } \\Big ) ^ { \\alpha } \\prod _ { k = 1 } ^ { n } \\frac { \\tau z - \\sigma _ { k } } { \\sigma _ { k } z - \\tau } \\frac { \\sigma _ { k } w - \\tau } { \\tau w - \\sigma _ { k } } . \\end{align*}"}
-{"id": "4068.png", "formula": "\\begin{align*} \\frac { d \\mu ^ * _ { V , \\theta } ( s ) } { d s } = \\begin{cases} c _ { 0 , V } s ^ { - \\frac { 1 } { \\theta + 1 } } \\left ( 1 + o ( 1 ) \\right ) , & s \\to 0 + , \\\\ [ 5 p t ] c _ { 1 , V } ( q - s ) ^ { 1 / 2 } \\left ( 1 + o ( 1 ) \\right ) , & s \\to q - , \\end{cases} \\end{align*}"}
-{"id": "734.png", "formula": "\\begin{align*} \\mathcal { A } _ { s , t } = A _ { s , t } ^ { ( n - m + 1 , n - m + 1 ) } ( n - m + q ) , \\end{align*}"}
-{"id": "8078.png", "formula": "\\begin{align*} \\frac { r n ^ { 1 / 6 } } { 1 6 \\sqrt { s ( 1 + r ) } } \\ge \\frac { r n ^ { 1 / 6 } } { 1 6 \\sqrt { s } ( 1 + r / 2 ) } = \\frac { m n ^ { 1 / 6 } } { 3 2 s \\sqrt { n } + 8 m \\sqrt { s } } . \\end{align*}"}
-{"id": "6854.png", "formula": "\\begin{align*} \\Phi '' ( t , \\Phi ( t , \\beta ) ) = u _ t + u \\nabla u = - \\nabla p ( t , \\Phi ( t , \\beta ) ) \\ , , \\end{align*}"}
-{"id": "7625.png", "formula": "\\begin{align*} \\lim \\limits _ { \\varepsilon \\to 0 } \\| \\phi ^ \\varepsilon \\| _ { L ^ { \\infty } ( ( 0 , T ) , \\ L ^ 2 ( \\mathbb R ^ d ) ) } = 0 . \\end{align*}"}
-{"id": "4313.png", "formula": "\\begin{align*} \\big ( \\nabla ( \\Pi _ D ^ k v - v ) , \\nabla p \\big ) _ { D } = 0 , \\forall p \\in \\mathbb { P } _ k ( D ) . \\end{align*}"}
-{"id": "3305.png", "formula": "\\begin{align*} \\Delta _ q \\colon \\mathcal { M } \\to \\mathbb { C } , \\Delta _ q ( f ) : = \\max _ { 0 \\le i \\le \\deg ( f ) } \\sum _ { d \\mid f , \\ , \\deg ( d ) = i } 1 . \\end{align*}"}
-{"id": "5168.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow + \\infty } \\frac { S ( \\mu x ) } { S ( x ) } = \\mu ^ { \\rho } . \\end{align*}"}
-{"id": "1339.png", "formula": "\\begin{align*} e _ i \\left ( \\kappa _ i \\left ( e ^ \\frac { 1 } { 2 } b e ^ \\frac { 1 } { 2 } \\right ) \\right ) = \\kappa _ i \\left ( e ^ \\frac { 1 } { 2 } b e ^ \\frac { 1 } { 2 } \\right ) \\end{align*}"}
-{"id": "3002.png", "formula": "\\begin{align*} x G ' _ m ( x ) & = \\sum _ { n = 0 } ^ \\infty n e _ { m + 2 , n } x ^ { n } . \\end{align*}"}
-{"id": "2278.png", "formula": "\\begin{align*} Y ( t ) & = \\ 1 _ { t > 0 } & \\check { Y } ( t ) & = Y ( - t ) = \\ 1 _ { t < 0 } \\end{align*}"}
-{"id": "8368.png", "formula": "\\begin{align*} \\lambda _ { m a x } = \\max _ { 1 \\leq j \\leq N } | \\lambda _ j | , \\end{align*}"}
-{"id": "3559.png", "formula": "\\begin{align*} C _ { 0 } = \\mathrm { C o n e } \\left \\{ ( 0 , 1 , 0 , 0 ) , ( 1 , 0 , 0 , 1 ) , ( 0 , 0 , 1 , 0 ) , ( 1 , 0 , 1 , 0 ) , ( 0 , 1 , 0 , 1 \\} \\right \\} \\subset \\mathbb { R } ^ { 4 } . \\end{align*}"}
-{"id": "7816.png", "formula": "\\begin{align*} [ \\langle x + \\lambda y , x + \\lambda y \\rangle \\xi , \\xi ] = ( \\| x \\| + \\| y \\| ) ^ 2 , \\end{align*}"}
-{"id": "8913.png", "formula": "\\begin{align*} s \\Lambda & = \\omega ( 1 ) \\\\ \\mu & \\gtrsim \\log ( 1 + n / s ^ 2 ) . \\end{align*}"}
-{"id": "7122.png", "formula": "\\begin{align*} - \\frac { d ^ 2 } { d x ^ 2 } \\phi ^ * _ m ( x ) + \\hat { q } ( x ) \\phi ^ * _ m ( x ) = \\lambda ^ * _ m \\phi ^ * _ m ( x ) , \\ , \\ , \\ , \\ , x \\in ( 0 , \\pi ) , ~ ~ ~ m = 1 , 2 . \\end{align*}"}
-{"id": "1483.png", "formula": "\\begin{align*} W ^ { j } _ { 2 } = - \\varepsilon \\int _ { \\mathbb { T } ^ { d } } ( \\partial _ { x _ { j } } w ^ { 1 } - \\partial _ { x _ { j } } w _ { 2 } ) \\left ( \\Theta _ { x _ { j } } ( t , x , \\mu ^ { 1 } , D w ^ { 1 } ) - \\Theta _ { x _ { j } } ( t , x , \\mu ^ { 2 } , D w ^ { 2 } ) \\right ) \\ d x , \\end{align*}"}
-{"id": "5624.png", "formula": "\\begin{align*} - \\beta \\ , \\int _ 0 ^ { + \\infty } | u | ^ 2 u \\bar { u } _ x d x & = - \\frac { \\beta } { 2 } \\int _ 0 ^ { + \\infty } | u | ^ 2 ( u \\bar { u } _ x + \\bar { u } u _ x ) d x \\\\ & = - \\frac { \\beta } { 4 } \\int _ 0 ^ { + \\infty } \\frac { d } { d x } | u | ^ 4 d x = \\frac { \\beta } { 4 } | u ( 0 , t ) | ^ 4 . \\end{align*}"}
-{"id": "2660.png", "formula": "\\begin{align*} G & = 4 \\pi D _ { 1 } D _ { 2 } \\widetilde { G } D _ { 2 } D _ { 1 } , \\end{align*}"}
-{"id": "9520.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } n q ^ n ( q ^ { n + 1 } ) _ \\infty = \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { n - 1 } q ^ { \\frac { n ( n + 1 ) } { 2 } } } { ( 1 - q ^ n ) ( q ) _ n } = \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ n } { 1 - q ^ n } . \\end{align*}"}
-{"id": "7495.png", "formula": "\\begin{align*} ( X * Y , Z ) = ( X , Y * Z ) \\end{align*}"}
-{"id": "2544.png", "formula": "\\begin{align*} v _ t ( x ) : = \\lim _ { \\lambda \\to \\infty } V _ t ( \\lambda \\mathbf 1 _ E ) ( x ) = - \\log \\mathbf P _ { \\delta _ x } ( \\| X _ t \\| = 0 ) = \\mathbb N _ x ( \\| W _ t \\| \\neq 0 ) . \\end{align*}"}
-{"id": "4482.png", "formula": "\\begin{align*} \\nabla _ { i j } ( u - \\underline { u } ) = - \\nabla _ { n } ( u - \\underline { u } ) \\Pi _ { i j } \\quad \\textrm { f o r a n y } i , j < n , \\end{align*}"}
-{"id": "6589.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\int _ 0 ^ 1 x ^ { - 1 / 2 } w _ n ( x , 0 ) \\ , d x = \\int _ 0 ^ 1 x ^ { - 1 / 2 } w ( x , 0 ) \\ , d x . \\end{align*}"}
-{"id": "3910.png", "formula": "\\begin{align*} \\mathbf { M } = \\begin{pmatrix} \\alpha & 0 \\\\ 0 & \\beta \\end{pmatrix} \\mathbf { T } ( r _ 1 ) \\cdots \\mathbf { T } ( r _ k ) , \\end{align*}"}
-{"id": "9516.png", "formula": "\\begin{align*} J _ { 1 } & : = \\lambda \\displaystyle \\int _ { \\mathbb { R } ^ N } g u ^ { r } u _ M ^ { \\beta - 1 } \\dd x \\\\ J _ { 2 } & : = \\displaystyle \\int _ { \\mathbb { R } ^ N } u ^ { p _ { s } ^ * } u _ M ^ { \\beta - 1 } \\dd x \\end{align*}"}
-{"id": "8578.png", "formula": "\\begin{align*} ( y _ j \\otimes 1 ) ^ * = x _ j \\otimes 1 \\qquad \\textup { a n d } ( 1 \\otimes y _ j ) ^ * = 1 \\otimes x _ j . \\end{align*}"}
-{"id": "6133.png", "formula": "\\begin{align*} \\Omega ( \\rho ^ A _ i ) = X \\Omega ( p ^ A _ i ) . \\end{align*}"}
-{"id": "7270.png", "formula": "\\begin{align*} - \\sum _ { k \\neq 0 } \\mathcal { F } \\left ( \\frac { 1 } { 2 } \\phi ^ { 2 } - \\mu \\phi \\right ) ( - k ) k ^ 2 \\hat \\psi ( k ) = \\sum _ { k \\neq 0 } \\hat \\phi ( - k ) \\hat \\psi ( k ) \\end{align*}"}
-{"id": "1362.png", "formula": "\\begin{align*} \\partial _ n ( \\lambda ) = \\sum _ { j = 1 } ^ n \\sum _ { l = 0 } ^ 1 ( - 1 ) ^ { j + l } F ^ l _ j ( \\lambda ) , \\end{align*}"}
-{"id": "6043.png", "formula": "\\begin{align*} g _ k ( k + i ) = 2 6 i ^ 2 + ( - 1 2 - 4 k ) i + 2 k ^ 2 + 1 \\end{align*}"}
-{"id": "5823.png", "formula": "\\begin{align*} \\ln ( \\phi _ { \\bar X _ n } ( t _ 1 ) ) = \\ln \\left ( \\iiint e ^ { i t _ 1 \\bar x _ n } \\prod _ { i = 1 } ^ n f ( x _ i ) d x _ i \\right ) , \\end{align*}"}
-{"id": "7303.png", "formula": "\\begin{align*} 2 E x = & \\theta x + ( - 1 ) ^ { p + q } x \\theta = \\xi _ i a \\otimes \\xi ^ i \\alpha + ( - 1 ) ^ { p + q } a \\xi _ i \\otimes \\alpha \\xi ^ i = \\\\ = & \\xi _ i a \\otimes \\xi ^ i \\alpha + ( - 1 ) ^ p a \\xi _ i \\otimes \\xi ^ i \\alpha = \\xi _ i \\wedge a \\otimes \\xi ^ i \\alpha , \\end{align*}"}
-{"id": "6690.png", "formula": "\\begin{align*} \\int d x \\ , \\phi _ { n } ^ { * } ( x ) \\phi _ { n ' } ( x ) = \\int d x \\ , x ^ { - 1 - i ( n - n ' ) \\omega } . \\end{align*}"}
-{"id": "4786.png", "formula": "\\begin{align*} \\bar { \\omega } ( f ) ( u _ f , v _ f ) = \\int _ S \\omega ( f ( x ) ) ( u _ f ( x ) , v _ f ( x ) ) \\mu _ S . \\end{align*}"}
-{"id": "2190.png", "formula": "\\begin{align*} \\frac { \\partial \\mathcal { H } } { \\partial \\boldsymbol { u } } = 0 \\implies \\boldsymbol { u } ^ \\ast ( t ) = - R ^ { - 1 } ( t ) B ^ { T } ( t ) \\boldsymbol { \\lambda } ^ \\ast ( t ) \\end{align*}"}
-{"id": "6869.png", "formula": "\\begin{align*} \\mathcal { R } _ 3 \\quad \\equiv \\begin{cases} \\mathcal { R } _ 2 \\ , , \\\\ \\partial ^ \\nu f _ 1 = 0 \\ , , & \\ | \\nu | = 2 \\ , , \\\\ \\partial ^ \\nu f _ 4 = 0 \\ , , & \\ | \\nu | = 1 \\ , , \\\\ \\partial ^ \\nu f _ 5 = 0 \\ , , & \\ | \\nu | = 1 \\ , . \\end{cases} \\end{align*}"}
-{"id": "8755.png", "formula": "\\begin{align*} { } _ F u ^ { ( \\alpha , \\sigma ) } _ n = { } _ L u _ { n , n _ 0 } ^ { ( \\alpha , \\sigma ) } + { } _ H ^ F { u } _ { n , n _ 0 } ^ { ( \\alpha , \\sigma ) } . \\end{align*}"}
-{"id": "2205.png", "formula": "\\begin{align*} \\dot { P } + P A + A ^ T P + Q - \\frac { 1 } { 2 } P B R ^ { - 1 } B ^ { T } P - \\frac { 1 } { 4 } P B R ^ { - 1 } B ^ { T } P ^ T - \\frac { 1 } { 4 } P ^ T B R ^ { - 1 } B ^ { T } P \\ = 0 \\end{align*}"}
-{"id": "732.png", "formula": "\\begin{align*} C _ { k } ( x ) = ( - 1 ) ^ { k } k ! L _ { k } ^ { ( n - m ) } ( x ) \\end{align*}"}
-{"id": "1436.png", "formula": "\\begin{align*} \\mu ( 0 , x ) = \\mu _ { 0 } ( x ) : = m ( 0 , x ) - \\bar { m } . \\end{align*}"}
-{"id": "6670.png", "formula": "\\begin{align*} Q & = \\begin{pmatrix} 0 & 0 \\\\ A & 0 \\end{pmatrix} , Q ^ { \\dagger } = \\begin{pmatrix} 0 & A ^ { \\dagger } \\\\ 0 & 0 \\end{pmatrix} , \\\\ H & = \\begin{pmatrix} H _ { - } & 0 \\\\ 0 & H _ { + } \\end{pmatrix} = \\begin{pmatrix} A ^ { \\dagger } A & 0 \\\\ 0 & A A ^ { \\dagger } \\end{pmatrix} , \\end{align*}"}
-{"id": "8698.png", "formula": "\\begin{align*} 2 ( j + 1 ) ( \\nu + \\rho ' - 2 j - 5 ) q _ \\ell ^ { i , j + 1 } = 1 6 ( i + 1 ) ( i + 2 ) ( i + 2 \\ell + 2 ) ( i + 2 \\ell + 3 ) q _ \\ell ^ { i + 2 , j } . \\end{align*}"}
-{"id": "3785.png", "formula": "\\begin{align*} r _ \\alpha - r _ \\beta = s _ \\nu . \\end{align*}"}
-{"id": "5230.png", "formula": "\\begin{align*} \\lim \\sup _ { h \\rightarrow + \\infty } \\lim _ { \\lambda \\rightarrow + \\infty } \\inf \\mathbb { P } ( A ^ { \\ast } ( h ) / B ^ { \\ast } ( h ) < 1 / \\lambda ) = 0 . \\end{align*}"}
-{"id": "7956.png", "formula": "\\begin{align*} x _ { a , 2 } = \\sum _ { b = d _ 1 + 1 } ^ { n } \\alpha _ { a , b } x _ { b , 2 } \\end{align*}"}
-{"id": "1205.png", "formula": "\\begin{align*} & \\Gamma _ { \\varepsilon } : \\ , \\zeta _ 1 = i b _ 0 ( \\zeta _ 2 ) + \\varepsilon e ^ { i \\phi } , \\\\ & \\Gamma _ { + , \\varepsilon } : \\ , \\zeta _ 1 = e ^ { i \\pi / 2 } w + \\varepsilon , \\\\ & \\Gamma _ { - , \\varepsilon } : \\ , \\zeta _ 1 = e ^ { i \\pi / 2 } w - \\varepsilon . \\end{align*}"}
-{"id": "1329.png", "formula": "\\begin{align*} \\begin{aligned} \\min _ { u _ 1 , \\pi } & \\ \\rho \\big [ Z ^ { u _ 1 , \\pi } , P \\big ] \\\\ & \\ u _ 1 \\in U _ 1 , \\\\ & \\ \\pi ( \\cdot ) \\lessdot U _ 2 ( \\cdot , u _ 1 ) . \\end{aligned} \\end{align*}"}
-{"id": "1060.png", "formula": "\\begin{align*} & 0 \\leq \\limsup _ { n \\to + \\infty } ( n / 4 ) ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } \\mathbb { P } ( \\xi ^ { ( n ) } ( I _ { n , k } ) = \\xi ^ { ( n ) } ( J _ { n , k , j } ) = 0 ) \\\\ & \\leq \\lim _ { n \\to + \\infty } ( n / 4 ) ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } D _ n ( ( z _ { j + 1 } - z _ j ) / 2 ) \\left ( \\lim _ { n \\to + \\infty } ( n / 4 ) ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } D _ n ( F _ n ( x _ 0 ) / 2 ) \\right ) ^ { k - 1 } \\\\ & = 0 \\cdot \\left ( e ^ { c _ 0 - x _ 0 } / 4 \\right ) ^ { k - 1 } = 0 , \\ , \\ , \\ , \\ , \\forall \\ , \\ , 1 \\leq j \\leq k , \\end{align*}"}
-{"id": "6028.png", "formula": "\\begin{align*} W ^ Q ( t ) : = W ( t ) - \\int _ 0 ^ t q ^ Q ( s ) d s , t \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "1094.png", "formula": "\\begin{align*} | \\mathbb P ( \\tau _ k ^ * \\in I _ 1 ) - \\mathbb P ( \\widetilde { \\tau } _ k \\in I _ 1 ) | \\leq & \\mathbb P ^ { G U E ( n ) } ( \\xi ^ { ( n ) } ( L _ k ^ { ( 1 ) } ) = 0 ) + \\mathbb P ^ { G U E ( n ) } ( \\xi ^ { ( n ) } ( L _ k ^ { ( 0 ) } ) = 0 ) \\\\ \\leq & n ^ { - 1 / 4 + o ( 1 ) } + n ^ { - 1 / 4 + o ( 1 ) } = n ^ { - 1 / 4 + o ( 1 ) } , \\end{align*}"}
-{"id": "4204.png", "formula": "\\begin{align*} S ( z ) \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & z ^ { - \\beta } & 0 \\\\ 0 & 0 & z ^ { - \\beta } \\end{pmatrix} = \\mathcal { O } \\begin{pmatrix} 1 & h _ { - \\alpha - \\frac { 1 } { 2 } } ( z ) & h _ { - \\alpha - \\frac { 1 } { 2 } } ( z ) \\\\ 1 & h _ { - \\alpha - \\frac { 1 } { 2 } } ( z ) & h _ { - \\alpha - \\frac { 1 } { 2 } } ( z ) \\\\ 1 & h _ { - \\alpha - \\frac { 1 } { 2 } } ( z ) & h _ { - \\alpha - \\frac { 1 } { 2 } } ( z ) \\end{pmatrix} , \\end{align*}"}
-{"id": "6476.png", "formula": "\\begin{align*} t ( a + b ) ^ { 2 } + 2 ( a + b ) ( 1 - ( a + b ) ) = 2 a ( 1 - a ) + 2 b ( 1 - b ) + t _ { B } b ^ { 2 } , \\end{align*}"}
-{"id": "9189.png", "formula": "\\begin{align*} \\mathcal { K } : = \\mathrm { s p a n } \\left \\{ \\left . J \\in \\Lambda _ { N } \\left ( 0 \\right ) \\right \\vert J \\left ( t \\right ) = 0 t \\in \\left ( 0 , b \\right ] \\ , \\right \\} . \\end{align*}"}
-{"id": "9267.png", "formula": "\\begin{align*} \\begin{array} { l l } c \\ ! \\ ! & \\ ! \\ ! \\displaystyle = \\frac { 2 [ C ( b + \\frac { 1 } { a } ) + ( \\epsilon _ 1 M _ 2 ( R ) + C _ { \\epsilon _ 1 } ) ] } { \\sigma } + b \\\\ \\ ! \\ ! & \\ ! \\ ! \\displaystyle \\le C [ ( \\epsilon _ 1 + a ) M _ 2 ( R ) + a ( M ^ \\prime _ 2 ( R ) ) ^ 2 + \\frac { 1 } { a } ] + C _ { \\epsilon _ 1 } + \\frac { C ^ \\prime } { R ^ 2 } , \\end{array} \\end{align*}"}
-{"id": "4442.png", "formula": "\\begin{align*} \\langle w _ 1 ( \\det ( D _ \\bullet ) ) , v \\rangle _ { C ^ { \\infty } _ { X , L } } = \\langle w _ 2 ( T L ) , v _ * [ S ^ 1 \\times \\partial D ] \\rangle _ L + ( I _ { \\mu _ L } ( A ) - 1 ) \\langle w _ 1 ( T L ) , v _ * [ S ^ 1 \\times \\{ 1 \\} ] \\rangle _ L . \\end{align*}"}
-{"id": "9044.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 } { \\sqrt { N } } \\sum _ { | \\xi | ^ 2 = E } e ( \\langle \\xi , x \\rangle ) . \\end{align*}"}
-{"id": "461.png", "formula": "\\begin{align*} \\int _ 0 ^ t E ( s ) \\ , d H ( s ) & \\sim \\int _ 0 ^ t e ^ { - \\alpha ( t - s ) } E ( t ) \\ , d H ( s ) \\\\ & = E ( t ) \\bigg ( \\Big [ e ^ { - \\alpha ( t - s ) } H ( s ) \\Big ] _ { s = 0 } ^ t - \\int _ 0 ^ t \\alpha e ^ { - \\alpha ( t - s ) } H ( s ) \\ , d s \\bigg ) \\\\ & \\sim E ( t ) \\bigg ( H ( t ) - \\int _ 0 ^ t \\alpha e ^ { - 2 \\alpha ( t - s ) } H ( t ) \\ , d s \\bigg ) \\\\ & \\sim \\tfrac { 1 } { 2 } E ( t ) H ( t ) , \\end{align*}"}
-{"id": "9122.png", "formula": "\\begin{align*} \\rho _ { 1 } ( \\mathbf { x } _ { 1 } , t ) = \\mathcal { L } _ { B G } \\rho _ { 1 } ^ { ( N ) } ( \\mathbf { x } _ { 1 } , t ) , \\end{align*}"}
-{"id": "9654.png", "formula": "\\begin{align*} \\int _ \\sigma | \\varphi | ^ 2 | f | ^ 2 m & = \\| \\varphi ( U _ \\sigma ) X x \\| ^ 2 = \\| X \\varphi ( T ) x \\| ^ 2 \\leq \\| X \\| ^ 2 \\| \\varphi ( T ) x \\| ^ 2 \\\\ & \\leq \\| X \\| ^ 2 K ^ 2 C _ { { \\rm p o l } , T } ^ 2 \\| x \\| ^ 2 \\int _ { \\mathbb T } | \\varphi | ^ 2 | \\psi | ^ 2 m \\end{align*}"}
-{"id": "6412.png", "formula": "\\begin{align*} \\| \\widehat { R } \\widehat { z } \\| ^ { 2 } = \\left [ \\begin{array} { c } s \\\\ c \\end{array} \\right ] ^ { T } \\left [ \\begin{array} { c c } \\rho & \\sigma \\\\ \\sigma & \\tau \\end{array} \\right ] \\left [ \\begin{array} { c } s \\\\ c \\end{array} \\right ] \\end{align*}"}
-{"id": "6888.png", "formula": "\\begin{align*} \\varphi ( t , \\alpha ) = M _ 1 ( t ) v ( \\alpha ) + M _ 2 ( t ) w ( \\alpha ) \\ , , \\end{align*}"}
-{"id": "1145.png", "formula": "\\begin{align*} y _ j = \\gamma _ n ( u _ j ) , \\ a _ j = G _ n ( x _ j ) / S ( I ) , \\ \\ \\forall \\ n > N _ { 6 , 0 } . \\end{align*}"}
-{"id": "8692.png", "formula": "\\begin{align*} \\partial _ { [ S , X ] } \\abs { X ' } ^ { 2 h } \\abs { X '' } ^ { 2 i } \\abs { Z } ^ { 2 j } & = 2 j \\langle [ S , X ] , Z \\rangle \\abs { X ' } ^ { 2 h } \\abs { X '' } ^ { 2 i } \\abs { Z } ^ { 2 j - 2 } \\\\ & = 2 j \\langle J _ Z S , X \\rangle \\abs { X ' } ^ { 2 h } \\abs { X '' } ^ { 2 i } \\abs { Z } ^ { 2 j - 2 } \\\\ & = \\frac { j } { h + 1 } \\partial _ { J _ Z S } \\abs { X ' } ^ { 2 h + 2 } \\abs { X '' } ^ { 2 i } \\abs { Z } ^ { 2 j - 2 } . \\end{align*}"}
-{"id": "7250.png", "formula": "\\begin{align*} F ( \\phi , \\mu ) = 0 \\mbox { f o r } ( \\phi , \\mu ) \\in \\mathcal { O } \\times Y \\end{align*}"}
-{"id": "5775.png", "formula": "\\begin{align*} A ( Q , q , T ) : = \\sum _ { d \\geq 0 } A _ d ( Q , q ) T ^ d , R ( \\Gamma , q , T ) : = \\sum _ { d \\geq 0 } R _ d ( \\Gamma , q ) T ^ d . \\end{align*}"}
-{"id": "4820.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\Delta _ g u & = & 0 \\ ; \\ ; \\ ; \\mbox { i n } \\ ; \\ ; \\ ; M \\\\ \\frac { 2 } { n - 2 } \\frac { \\partial u } { \\partial \\nu } - H _ { g } u & = & - u ^ { \\beta } \\ ; \\ ; \\ ; \\mbox { o n } \\ ; \\ ; \\ ; \\partial M , \\end{array} \\right . \\end{align*}"}
-{"id": "8841.png", "formula": "\\begin{align*} X _ 1 & = p - p _ 1 = \\frac { M ( 2 \\mu p - 1 ) - 1 } { 2 M \\mu } , \\\\ X _ 2 & = p - p _ 2 = \\frac { 2 a p - a - 1 + ( 2 \\mu p - a - 1 ) M } { 2 ( M \\mu + a ) } , \\\\ X _ 3 & = p - p _ 3 = \\frac { 2 p - a - 1 + ( 2 \\mu p - a - 1 ) M } { 2 ( M \\mu + 1 ) } . \\end{align*}"}
-{"id": "994.png", "formula": "\\begin{align*} G \\star G _ T ( x ) = \\sum _ { y \\in \\Z ^ d } G ( x - y ) G _ T ( y ) = \\sum _ { k = 1 } ^ T \\mathbb E [ G ( x - S _ k ) ] . \\end{align*}"}
-{"id": "3063.png", "formula": "\\begin{align*} Z ^ { \\ast } _ { r } ( s ) = \\pi ^ { - s } \\Gamma ( s ) Z _ { r } ( s ) . \\end{align*}"}
-{"id": "1913.png", "formula": "\\begin{align*} \\frac { L } { 4 } \\left [ 1 - \\frac { 2 } { L } \\left ( x \\bmod L \\right ) \\right ] = \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { k _ n } \\sin ( k _ n x ) , \\end{align*}"}
-{"id": "2948.png", "formula": "\\begin{align*} ( \\tilde { H } _ k ) : \\forall \\Gamma \\in C _ { p + k , p - k + 1 } ^ \\infty ( X , \\mathbb { C } ) \\mbox { s u c h t h a t } d \\Gamma = 0 , \\mbox { w e h a v e } \\Gamma \\in I m \\ , \\partial \\Rightarrow \\Gamma \\in I m \\ , \\partial \\bar { \\partial } . \\end{align*}"}
-{"id": "7613.png", "formula": "\\begin{align*} M _ X = 1 + \\norm { X - \\mathrm { I d } } _ { L ^ \\infty ( 0 , T ; C ^ { 1 + \\alpha } ) } \\le 1 + T \\Gamma . \\end{align*}"}
-{"id": "1900.png", "formula": "\\begin{align*} \\mathcal { E } ( t _ s ) = - \\frac { \\pi } { 6 L } + \\frac { \\mathcal { B } - \\mathcal { C } } { L } + \\frac { \\mathcal { C } } { 2 L } \\int _ { 0 } ^ { L } \\sum _ { n = - \\infty } ^ \\infty \\left [ \\delta \\left ( t _ s - x - n L \\right ) + \\delta \\left ( t _ s + x - n L \\right ) \\right ] d x . \\end{align*}"}
-{"id": "4701.png", "formula": "\\begin{align*} r _ l = \\log ( e _ { l + 1 } / e _ { l } ) / \\log ( N _ { l + 1 } / N _ l ) , \\ ; \\ ; l = 0 , 1 , 2 , \\ldots , \\end{align*}"}
-{"id": "7941.png", "formula": "\\begin{align*} \\sum _ { a \\in [ n ] \\setminus \\{ 1 \\} } ( \\Pi _ 1 x _ { a , 1 } ) \\otimes x _ { a , 2 } \\otimes x _ { a , 3 } = \\sum _ { a \\in [ n ] \\setminus \\{ 1 \\} } ( \\alpha _ { a } \\Pi _ 1 x _ { a , 1 } ) \\otimes y _ { \\sigma ( a ) , 2 } \\otimes y _ { \\sigma ( a ) , 3 } . \\end{align*}"}
-{"id": "9682.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { n \\in \\mathbb N } \\| \\varphi ( T ) x _ { n t } \\| ^ 2 & = \\sum _ { n \\in \\mathbb N } \\| ( \\varphi \\kappa _ { n t } \\omega _ n \\varphi _ n ) ( T ) x _ 0 \\| ^ 2 \\\\ & \\leq \\Bigl ( K C _ { { \\rm p o l } , T } ( 1 + \\varepsilon _ 0 ) \\frac { ( 2 ( 1 + c _ 1 ^ 2 ) ) ^ { 1 / 2 } } { 1 - c _ 1 } \\Bigr ) ^ 2 \\sum _ { n \\in \\mathbb N } \\int _ { \\mathbb T } | \\varphi \\kappa _ { n t } | ^ 2 | \\psi _ { n t } | ^ 2 m , \\end{aligned} \\end{align*}"}
-{"id": "809.png", "formula": "\\begin{align*} H _ i : = G _ i \\cap H _ { q ^ d , P ^ \\alpha } , | H _ i | = \\begin{cases} ( q ^ { d } - 1 ) ^ { s - 1 } q ^ { d ( s - 1 ) ( \\alpha - 1 ) } & i = 0 \\\\ q ^ { d ( s - 1 ) ( \\alpha - i ) } & i \\geq 1 . \\end{cases} \\end{align*}"}
-{"id": "1200.png", "formula": "\\begin{align*} Q _ 0 ( \\zeta _ 1 , \\zeta _ 2 ) = \\frac { 4 \\sqrt { \\gamma _ - } \\sqrt { 1 + \\zeta _ 2 ^ 2 } \\sqrt { 1 + \\zeta _ 1 ^ 2 } P ( \\zeta _ 1 , \\zeta _ 2 ) } { P ( \\zeta _ 1 , \\zeta _ 2 ) + a _ 0 ^ 2 \\sqrt { 1 + \\zeta _ 1 ^ 2 } } , \\end{align*}"}
-{"id": "1689.png", "formula": "\\begin{align*} \\begin{aligned} s _ b ( N _ k ^ { n } ) & = \\frac { n + 1 } { 2 } ( b - 1 ) ( 2 ^ { k } - 1 ) \\\\ & + ( b - C _ { n } ^ { n - 1 } ) + ( b - C _ { n } ^ { n - 3 } ) + \\cdots + ( b - C _ { n } ^ { 0 } ) \\\\ & + ( C _ { n } ^ n - 1 ) + ( C _ { n } ^ { n - 2 } - 1 ) + \\cdots + ( C _ { n } ^ { 1 } - 1 ) , \\end{aligned} \\end{align*}"}
-{"id": "6849.png", "formula": "\\begin{align*} \\nabla \\cdot u = 0 \\end{align*}"}
-{"id": "2211.png", "formula": "\\begin{align*} \\tilde { R } \\boldsymbol { u } + \\frac { 1 } { 2 } \\tilde { R } ^ { - 1 } B ^ { T } ( P + P ^ T ) \\boldsymbol { x } - \\tilde { R } ^ { - 1 } B ^ { T } \\boldsymbol { g } = \\boldsymbol { 0 } \\\\ \\implies \\boldsymbol { u } = - \\frac { 1 } { 2 } { R } ^ { - 1 } B ^ T \\left ( P ^ T + P \\right ) \\boldsymbol { x } + R ^ { - 1 } B ^ T \\boldsymbol { g } \\end{align*}"}
-{"id": "3052.png", "formula": "\\begin{align*} h _ i ( \\Delta ) \\leq \\binom { d } { i } . \\end{align*}"}
-{"id": "4746.png", "formula": "\\begin{align*} S ( \\xi ) : = \\sum \\limits _ { i = 1 } ^ n \\xi _ i . \\end{align*}"}
-{"id": "10012.png", "formula": "\\begin{align*} \\Big \\Vert \\sum _ { n = M + 1 } ^ { N } \\frac { a _ { n } } { n ^ { L + \\varepsilon } } n ^ { - s } \\Big \\Vert _ { \\mathfrak { X } ( X ) } \\leq \\frac { 1 } { N ^ { \\frac { \\varepsilon } { 2 } } } + \\frac { 1 } { M ^ { \\frac { \\varepsilon } { 2 } } } + \\vert L + \\varepsilon \\vert \\sum _ { n = M } ^ { N - 1 } \\frac { 1 } { n ^ { 1 + \\frac { \\varepsilon } { 2 } } } \\ , \\end{align*}"}
-{"id": "4936.png", "formula": "\\begin{align*} x ^ { 2 n + 1 } = x ^ n ( x ^ { n + 1 } - x z ^ { n ^ 4 - n } + y z ^ n ) + x ^ { n + 1 } z ^ { n ^ 4 - n } - x ^ n y z ^ n \\end{align*}"}
-{"id": "8075.png", "formula": "\\begin{align*} \\begin{array} { r c r @ { } l } \\sigma _ k ( F ) ( x ^ j ) & = & j & x ^ { j - 1 } \\\\ \\sigma _ k ( H ) ( x ^ j ) & = & ( 2 j - k ) & x ^ j \\\\ \\sigma _ k ( E ) ( x ^ j ) & = & ( k - j ) & x ^ { j + 1 } \\\\ \\rho _ k ( J ) ( x ^ j ) & = & ( - 1 ) ^ j & x ^ { k - j } . \\\\ \\end{array} \\end{align*}"}
-{"id": "9191.png", "formula": "\\begin{align*} U _ { 0 } : = \\mathcal { K } \\left ( 0 \\right ) ^ { \\perp } \\cap T _ { \\gamma \\left ( 0 \\right ) } N , \\end{align*}"}
-{"id": "5250.png", "formula": "\\begin{align*} \\left ( Q ( 1 - s ) \\right ) ^ { \\prime } = - s ^ { - 1 } ( 2 \\log 1 / s ) ^ { 1 / 2 } ( 1 + o ( 1 ) ) . \\end{align*}"}
-{"id": "799.png", "formula": "\\begin{align*} P ( T ) = \\prod _ { i = 1 } ^ { s } ( T - \\zeta _ 1 ^ { q ^ { i - 1 } } ) , \\end{align*}"}
-{"id": "6343.png", "formula": "\\begin{align*} m ' _ d & = \\eta G _ 1 G _ 2 m _ s h _ { s r } h _ { r r } h _ { r d } \\\\ & ~ ~ ~ + \\eta D G _ 1 G _ 2 ( m _ { b r _ 1 } + n _ { s p } ) h _ { r r } h _ { r d } \\\\ & ~ ~ ~ + \\eta D G _ 2 ( m _ { b r _ 2 } + n _ { s p } ) h _ { r d } + \\eta D m _ { b d } , \\end{align*}"}
-{"id": "3658.png", "formula": "\\begin{align*} n \\cdot e ^ { - c _ r ( \\log \\log n ) ^ { r } } \\ , < \\ , p _ { r } ( n ) \\ , = \\ , O _ { r } \\left ( n \\frac { ( \\log \\log n ) ^ { r - 1 } } { ( \\log n ) ^ { r } } \\right ) . \\end{align*}"}
-{"id": "6383.png", "formula": "\\begin{align*} \\mathbb { P } ( \\{ i , j \\} \\in Y ) = \\det ( K _ { \\{ i , j \\} } ) = K _ { \\{ i \\} } K _ { \\{ j \\} } - K ^ 2 _ { \\{ i , j \\} } , \\end{align*}"}
-{"id": "8091.png", "formula": "\\begin{align*} ( 1 - z ) ^ 2 f _ 1 ' ( z ) - f _ 1 ( z ) = z - 1 , \\end{align*}"}
-{"id": "5510.png", "formula": "\\begin{align*} \\tilde { \\varepsilon } _ T ^ { - 1 } \\sum _ j E ^ { \\Pi ^ { D _ T } } [ \\| b _ j - P _ { V _ J } b _ { 0 , j } \\| _ \\infty | X ^ T ] = O _ { P _ { b _ 0 } } \\Big ( \\frac { J ^ { \\frac { 3 } { 2 } } 2 ^ { J ( d - 2 ) } ( \\log T ) ^ \\eta } { \\tilde { \\varepsilon } _ T \\sqrt { T } } \\Big ) = o _ { P _ { b _ 0 } } ( 1 ) . \\end{align*}"}
-{"id": "2080.png", "formula": "\\begin{align*} N _ { n m } ^ { } = \\int _ { 0 } ^ { \\rho _ c } { \\rho J _ n ^ 2 } ( { \\lambda _ { n m } } \\rho ) d \\rho = \\frac { { { { \\rho _ c } ^ 2 } } } { 2 } ( J _ n ^ 2 ( { \\lambda _ { n m } } \\rho _ c ) - J _ { n - 1 } ^ { } ( { \\lambda _ { n m } } \\rho _ c ) J _ { n + 1 } ^ { } ( { \\lambda _ { n m } } \\rho _ c ) ) . \\end{align*}"}
-{"id": "2626.png", "formula": "\\begin{align*} K _ { N } ( u , v ) = \\begin{bmatrix} D S _ { N } ( u , v ) & S _ { N } ( u , v ) \\\\ - S _ { N } ( v , u ) & I S _ { N } ( u , v ) \\end{bmatrix} , \\end{align*}"}
-{"id": "4423.png", "formula": "\\begin{align*} & 0 \\leq \\chi ^ { \\pm } _ { \\varepsilon } \\chi ^ { \\pm } \\leq 1 \\quad \\chi ^ { \\pm } _ { \\varepsilon } \\chi ^ { \\pm 2 } = \\chi ^ { \\pm } _ { \\varepsilon } , \\\\ & \\forall ( f , w ) \\in \\L ^ 1 ( U ) \\times \\L ^ 1 ( U ) : \\ ; \\chi ^ { \\pm } _ { \\varepsilon } ( w ) f \\underset { \\varepsilon \\rightarrow 0 } { \\longrightarrow } \\chi ^ { \\pm } ( w ) f \\quad \\quad \\L ^ 1 ( U ) . \\end{align*}"}
-{"id": "5010.png", "formula": "\\begin{align*} R : = \\frac r \\sigma ( 1 + r ^ d ) ^ { 1 / d } < 1 , \\end{align*}"}
-{"id": "9653.png", "formula": "\\begin{align*} | \\alpha _ t & - \\alpha | ^ 2 \\leq C ^ 2 \\Bigl ( 2 N \\Bigl ( \\Bigl | \\frac { \\psi _ n } { \\psi _ { n t } } - 1 \\Bigr | ^ 2 + \\sum _ { k \\neq n , k \\leq N } \\xi _ k ^ 4 \\Bigl | \\frac { \\psi _ k } { \\psi _ { k t } } - 1 \\Bigr | ^ 2 \\Bigr ) + 8 \\varepsilon ^ 2 \\Bigr ) \\\\ & \\leq 2 N C ^ 2 \\max \\bigl ( 1 , \\sup _ { k \\in \\mathbb N } \\xi _ k ^ 4 \\bigr ) \\sum _ { k = 1 } ^ N \\Bigl | \\frac { \\psi _ k } { \\psi _ { k t } } - 1 \\Bigr | ^ 2 + 8 C ^ 2 \\varepsilon ^ 2 \\ \\bigcup _ { n = 1 } ^ N \\tau _ n . \\end{align*}"}
-{"id": "8629.png", "formula": "\\begin{align*} h _ i \\left ( \\bar { u } , 0 , \\varepsilon \\right ) = 0 , \\ \\ h _ i \\left ( \\bar { v } , 0 , \\varepsilon \\right ) = 0 , \\ \\ i = 1 , 2 , \\ \\ \\varepsilon \\in [ 0 , \\varepsilon _ 0 ) . \\end{align*}"}
-{"id": "3956.png", "formula": "\\begin{align*} \\partial _ { m _ { 2 } } U _ { 1 } ( x , m , t ) ( \\xi ) = \\partial _ { m _ { 1 } } U _ { - 1 } ( \\xi , m , t ) ( x ) \\ , , \\partial _ { m _ { 1 } } U _ { 2 } ( x , m , t ) ( \\xi ) = \\partial _ { m _ { 2 } } U _ { - 1 } ( \\xi , m , t ) ( x ) \\ , . \\end{align*}"}
-{"id": "4593.png", "formula": "\\begin{align*} \\hat { \\nu } _ g ( \\omega ) = \\hat { \\mu } ( \\omega ) \\hat { \\mu } ( g \\omega ) . \\end{align*}"}
-{"id": "3542.png", "formula": "\\begin{align*} C _ { N E } ^ { \\vee } = \\sigma _ { 1 } ^ { ( 1 ) } \\cup \\sigma _ { 2 } ^ { ( 1 ) } = \\sigma _ { 1 } ^ { ( 2 ) } \\cup \\sigma _ { 2 } ^ { ( 2 ) } \\cup \\sigma _ { 3 } ^ { ( 2 ) } \\end{align*}"}
-{"id": "6840.png", "formula": "\\begin{align*} \\det ( d \\varphi ) = \\sum _ { i = 1 } ^ 6 p _ i g _ i \\ , , \\end{align*}"}
-{"id": "9009.png", "formula": "\\begin{align*} \\sum _ { i \\ , = \\ , 1 } ^ { m } \\overline { D } _ { t } E _ { i } & = \\sum _ { i \\ , = \\ , 1 } ^ { m } D _ { t } E _ { i } + E _ { m + 1 } \\sum _ { i \\ , = \\ , 1 } ^ { m } \\tau _ { i } \\\\ & = \\sum _ { i \\ , = \\ , 1 } ^ { m } \\left \\{ ( D _ { t } E _ { i } \\cdot E _ { 1 } ) E _ { 1 } + \\dotsb + ( D _ { t } E _ { i } \\cdot E _ { m } ) E _ { m } \\right \\} + E _ { m + 1 } \\sum _ { i \\ , = \\ , 1 } ^ { m } \\tau _ { i } \\ , . \\end{align*}"}
-{"id": "1962.png", "formula": "\\begin{align*} \\sum _ { k \\in \\mathcal { T } _ { \\ell } } d _ { k i } = 0 , i \\in \\mathcal { T } _ { \\ell } . \\end{align*}"}
-{"id": "4706.png", "formula": "\\begin{align*} \\frac { d u _ j } { d t } & = - \\Phi _ j \\frac { 1 } { \\Delta x } \\Big ( F _ { j + \\frac { 1 } { 2 } } ^ { \\mbox { W E N O } } - F _ { j - \\frac { 1 } { 2 } } ^ { \\mbox { W E N O } } \\Big ) - ( 1 - \\Phi _ j ) \\frac { 1 } { \\Delta x } \\Big ( F _ { j + \\frac { 1 } { 2 } } ^ { \\mbox { B S Q I } } - F _ { j - \\frac { 1 } { 2 } } ^ { \\mbox { B S Q I } } \\Big ) . \\end{align*}"}
-{"id": "349.png", "formula": "\\begin{align*} \\nu _ p ( n ( n ^ 2 - 1 ) , k ) = \\nu _ p ( n - 1 , k ) \\cdot \\nu _ p ( n , k ) \\cdot \\nu _ p ( n + 1 , k ) , \\end{align*}"}
-{"id": "2309.png", "formula": "\\begin{align*} R ( z ) = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 0 & R _ { \\Omega } ( z ) \\end{array} \\right ) + \\left ( \\begin{array} { c } 1 \\\\ R _ { \\Omega } ( z ) | 1 \\rangle \\end{array} \\right ) \\frac { 1 } { z + \\i \\sigma ( z ) } \\left ( 1 , \\langle 1 | R _ { \\Omega } ( z ) \\right ) . \\end{align*}"}
-{"id": "8713.png", "formula": "\\begin{align*} 0 < \\lambda _ 1 ^ { ( j ) } \\leq \\lambda _ { N _ j } ^ { ( j ) } = 1 , \\end{align*}"}
-{"id": "9023.png", "formula": "\\begin{align*} f _ { \\{ c _ { i + 1 } \\} , c _ i } ( x ) = \\begin{cases} c _ { i + 1 } & x = c _ i , \\\\ x & \\end{cases} \\end{align*}"}
-{"id": "4737.png", "formula": "\\begin{align*} T _ { p } U _ { q - n } = T _ { p - n } U _ { q } = 0 ( p , q = 1 , 2 , \\cdots n - 1 ) , \\end{align*}"}
-{"id": "5926.png", "formula": "\\begin{align*} G _ { S ' } = n R _ { S ' } + n \\Delta _ { S ' } - \\lfloor ( n + 1 ) \\Delta _ { S ' } \\rfloor + P _ { S ' } , \\end{align*}"}
-{"id": "8860.png", "formula": "\\begin{align*} c _ { 1 b 1 } & = ( - 1 3 5 \\nu ^ 7 + 6 9 3 \\nu ^ 6 + 4 1 7 \\nu ^ 5 + 4 8 2 1 \\nu ^ 4 ) - 6 4 3 4 9 \\nu ^ 3 + 1 3 4 0 2 3 \\nu ^ 2 - 4 8 0 6 1 \\nu + 1 2 5 2 7 \\\\ & \\ge - 6 4 3 4 9 \\nu ^ 3 + 1 3 4 0 2 3 \\nu ^ 2 - 4 8 0 6 1 \\nu + 1 2 5 2 7 \\ge 1 0 0 0 ( - 6 7 \\nu ^ 3 + 1 3 4 \\nu ^ 2 - 6 7 \\nu + 1 2 ) \\\\ & = \\frac { 1 0 0 0 } { 2 7 } \\left [ 5 6 + 6 7 ( 4 - 3 \\nu ) ( 1 - 3 \\nu ) ^ 2 \\right ] > 0 \\end{align*}"}
-{"id": "3874.png", "formula": "\\begin{align*} \\int _ { r < \\left \\vert y \\right \\vert < R } y \\nu \\left ( d y \\right ) = 0 \\quad 0 < r < R < \\infty . \\end{align*}"}
-{"id": "1680.png", "formula": "\\begin{align*} \\widehat { O } \\left ( \\begin{matrix} u \\\\ v \\\\ \\end{matrix} \\right ) = 0 \\end{align*}"}
-{"id": "8538.png", "formula": "\\begin{align*} g _ d : = - 2 [ \\bar { p } , \\mathfrak { H } ] \\frac { \\partial _ { \\alpha } \\bar { f } } { z _ { \\alpha } } - 2 [ \\bar { f } , \\mathfrak { H } ] \\frac { \\partial _ { \\alpha } \\bar { p } } { z _ { \\alpha } } - 2 [ \\bar { p } , \\mathfrak { H } ] \\frac { \\partial _ { \\alpha } \\bar { p } } { z _ { \\alpha } } - 4 p _ t . \\end{align*}"}
-{"id": "5517.png", "formula": "\\begin{align*} \\int _ 0 ^ T f _ h ( X _ t ) d t & = \\int _ 0 ^ T L L ^ { - 1 } [ f _ h ] ( X _ t ) d t \\\\ & = L ^ { - 1 } [ f _ h ] ( X _ T ) - L ^ { - 1 } [ f _ h ] ( X _ 0 ) - \\int _ 0 ^ T \\nabla L ^ { - 1 } [ f _ h ] ( X _ t ) . d W _ t . \\end{align*}"}
-{"id": "6454.png", "formula": "\\begin{align*} \\psi _ \\lambda ( e _ { i , j } ) : = \\begin{cases} \\lambda ^ i \\frac { ( 1 - \\lambda ) } { ( 1 - \\lambda ^ n ) } & i = j \\\\ 0 & \\end{cases} . \\end{align*}"}
-{"id": "3949.png", "formula": "\\begin{align*} U _ { i } ( x , m , t ) = \\partial _ { m _ { i } } V ( m , t ) ( x ) \\ , , \\end{align*}"}
-{"id": "3441.png", "formula": "\\begin{align*} \\lambda _ { k , M } : = \\frac { 1 } { 2 M } \\log \\left ( k \\mathrm { ^ { t h } \\ , l a r g e s t \\ , e i g e n v a l u e \\ , o f \\ , } \\Pi _ { M } ^ { * } \\Pi _ { M } \\right ) , k = 1 , 2 , \\ldots , N , \\end{align*}"}
-{"id": "1981.png", "formula": "\\begin{align*} X = \\mathrm { T o t } ( V ) ^ T , \\end{align*}"}
-{"id": "9802.png", "formula": "\\begin{align*} E _ { h , m } : = \\Sigma _ { \\kappa } ^ m \\cap E _ h \\quad E _ { h , m } ^ { { \\rm n x t } } : = \\Sigma _ { \\kappa } ^ { m , { \\rm n x t } } \\cap E _ h . \\end{align*}"}
-{"id": "7540.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { X _ 2 - X _ 1 } _ { L i p ( 0 , T _ 0 ; C ^ { 1 + \\alpha , p } ) } + \\norm { \\tau _ 2 - \\tau _ 1 } _ { L i p ( 0 , T _ 0 ; C ^ { \\alpha , p } ) } + \\norm { v _ 2 - v _ 1 } _ { L ^ \\infty ( 0 , T _ 0 , C ^ { 1 + \\alpha , p } ) } \\\\ \\le C ( \\norm { u _ 2 ( 0 ) - u _ 1 ( 0 ) } _ { 1 + \\alpha , p } + \\norm { \\tau _ 2 ( 0 ) - \\tau _ 1 ( 0 ) } _ { \\alpha , p } ) \\end{gathered} \\end{align*}"}
-{"id": "6957.png", "formula": "\\begin{align*} \\textbf { S } _ 1 ( x _ 1 ) + 2 \\{ \\textbf { S } _ 2 ( \\textbf { x } ) \\} _ { 1 2 } + \\textbf { S } _ 1 ( x _ 2 ) = \\textbf { S } _ 1 ( y _ 1 ) + 2 \\{ \\textbf { S } _ 2 ( \\textbf { y } ) \\} _ { 1 2 } + \\textbf { S } _ 1 ( y _ 2 ) . \\end{align*}"}
-{"id": "5543.png", "formula": "\\begin{align*} \\| u \\| _ { B ^ t _ { 1 \\infty } } \\lesssim \\| \\Delta u \\| _ { B ^ { t - 2 } _ { 1 \\infty } } \\forall u \\in B ^ t _ { 1 \\infty } \\cap \\{ u : \\langle u , e _ 0 \\rangle = 0 \\} , \\end{align*}"}
-{"id": "2120.png", "formula": "\\begin{align*} d y ( t ) = - A ^ T ( t ) y ( t ) d t - C ^ T ( t ) y ( t ) d \\omega ( t ) . \\end{align*}"}
-{"id": "7670.png", "formula": "\\begin{align*} \\max _ { i = 1 , \\ldots , d } \\ \\max _ { \\xi \\in \\mathbb { T } ^ d } | g ^ i _ { \\boldsymbol { \\ell } } ( \\xi ) | \\ \\to \\ 0 \\ ; \\ ; \\mbox { a s } \\ ; { \\boldsymbol { \\ell } } \\to 0 , \\mbox { a n d c o n s e q u e n t l y } \\ ; \\| g _ { \\boldsymbol { \\ell } } \\| _ { ( L ^ 2 ( \\mathbb { T } ^ d ) ) ^ d } \\ \\to \\ 0 \\ ; \\ ; \\mbox { a s } \\ ; { \\boldsymbol { \\ell } } \\to 0 . \\end{align*}"}
-{"id": "6992.png", "formula": "\\begin{align*} ( a , b ) \\otimes ( 1 , 0 ) = ( a - 1 , b + 1 ) / ( a + 1 , b ) / ( a , b - 1 ) , \\end{align*}"}
-{"id": "4322.png", "formula": "\\begin{align*} \\begin{aligned} S _ K ( u , v ) : = \\ ; & h _ K ^ { - 1 } \\ ; \\sum _ { F \\subset \\partial K } \\Bigg [ ( Q _ K u - Q _ F u , Q _ K v - Q _ F v ) _ { F } \\\\ & + \\epsilon _ F h _ F \\sum _ { e \\subset \\partial F } ( ( u - Q _ F u ) , ( v - Q _ F v ) ) _ e \\Bigg ] , \\end{aligned} \\end{align*}"}
-{"id": "7885.png", "formula": "\\begin{align*} \\Big | \\mathfrak m _ N ( \\xi ) - e ^ { - \\kappa ( d , N ) ^ 2 \\sum _ { i = 1 } ^ d \\sin ^ 2 ( \\pi \\xi _ i ) } \\Big | \\le 3 \\kappa ( d , N ) ^ 2 \\sum _ { i = 1 } ^ d \\sin ^ 2 ( \\pi \\xi _ i ) . \\end{align*}"}
-{"id": "9378.png", "formula": "\\begin{align*} ( x y ) ^ { 2 ^ i } = x ^ { 2 ^ i } [ y , x , x ^ 2 , \\ldots , x ^ { 2 ^ { i - 2 } } ] ^ 2 [ y , x , x ^ 2 , \\ldots , x ^ { 2 ^ { i - 1 } } ] i \\ge 2 , \\end{align*}"}
-{"id": "1926.png", "formula": "\\begin{align*} ( x ^ 2 + x ) F _ { p + 2 } ( x ) + F _ p ( x ) = x ^ 2 \\sum _ { j = 1 } ^ \\infty \\frac { 1 } { Z _ j ^ p } . \\end{align*}"}
-{"id": "1371.png", "formula": "\\begin{align*} H \\big ( 0 , [ ( \\lambda , x ) ] \\big ) = \\left [ \\left ( \\alpha _ \\lambda , x \\right ) \\right ] = \\left [ \\left ( \\lambda , x \\right ) \\right ] \\ ; , \\end{align*}"}
-{"id": "8869.png", "formula": "\\begin{align*} c _ { 4 a } & = 8 \\nu ^ 4 + 4 0 \\nu ^ 3 - 1 3 9 5 \\nu ^ 2 + 4 3 5 4 \\nu + 4 7 5 7 > 0 \\\\ c _ { 4 b } & = 4 ( 1 - \\nu ) ( 4 \\nu ^ 5 + 2 1 \\nu ^ 4 - 7 1 2 \\nu ^ 3 + 2 0 1 1 \\nu ^ 2 + 3 1 0 2 \\nu + 3 0 5 0 ) \\\\ & + ( 1 - \\mu ) ( 2 \\nu ^ 6 + 1 1 \\nu ^ 5 - 3 6 0 \\nu ^ 4 + 9 1 2 \\nu ^ 3 + 1 7 0 5 \\nu ^ 2 + 3 6 5 5 \\nu + 5 4 3 ) \\ge 0 . \\end{align*}"}
-{"id": "850.png", "formula": "\\begin{gather*} \\inf _ { B \\in L ( \\R ^ n , \\R ^ n ) } \\{ \\| B \\| \\mid A + B \\mbox { s i n g u l a r } \\} = \\frac { 1 } { \\| A ^ { - 1 } \\| } , \\end{gather*}"}
-{"id": "519.png", "formula": "\\begin{align*} W _ 1 = \\langle x \\ , \\omega _ { 1 , 0 } , \\dots , x ^ { d _ 1 - 1 } \\ , \\omega _ { 1 , 0 } \\rangle = \\langle v _ 1 , \\dots , v _ { d _ 1 - 1 } \\rangle , \\\\ W _ 3 = \\langle x \\ , \\omega _ { 3 , 0 } , \\dots , x ^ { d _ 3 - 1 } \\ , \\omega _ { 3 , 0 } \\rangle = \\langle w _ 1 , \\dots , w _ { d _ 3 - 1 } \\rangle , \\end{align*}"}
-{"id": "9620.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } q ^ { N + 1 } \\left \\{ \\sum _ { n = 1 } ^ { \\infty } n q ^ { n - 1 } ( - q ^ { n } ) _ { N } - \\sum _ { n = 1 } ^ { \\infty } n q ^ n ( - q ^ { n + 1 } ) _ { N } \\right \\} \\\\ & = \\frac { 1 } { 2 } q ^ { N + 1 } \\left \\{ \\sum _ { n = 1 } ^ { \\infty } n q ^ { n - 1 } ( - q ^ { n } ) _ { N } - \\sum _ { n = 1 } ^ { \\infty } ( n + 1 ) q ^ n ( - q ^ { n + 1 } ) _ { N } + \\sum _ { n = 1 } ^ { \\infty } q ^ n ( - q ^ { n + 1 } ) _ N \\right \\} \\\\ & = \\frac { 1 } { 2 } q ^ { N + 1 } \\sum _ { n = 0 } ^ { \\infty } q ^ n ( - q ^ { n + 1 } ) _ N . \\end{align*}"}
-{"id": "2209.png", "formula": "\\begin{align*} \\dot { \\boldsymbol { g } } + A ^ T \\boldsymbol { g } + C ^ T Q \\boldsymbol { z } - \\frac { 1 } { 2 } ( P + P ^ T ) B R ^ { - 1 } B ^ T \\boldsymbol { g } = 0 . \\end{align*}"}
-{"id": "10024.png", "formula": "\\begin{align*} \\sup \\{ \\sigma _ { a } ( D ) - \\sigma _ { u } ( D ) \\} = 1 - \\frac { 1 } { \\cot ( X ) } \\ , , \\end{align*}"}
-{"id": "1558.png", "formula": "\\begin{align*} \\Psi _ { T } ( Z ) : = \\lim _ { n \\rightarrow \\infty } T ^ n ( ( \\omega ^ o ) ^ { - 1 } ( Z - n ) , 0 ) . \\end{align*}"}
-{"id": "7237.png", "formula": "\\begin{align*} ( \\mu - \\phi ) \\phi ^ \\prime ( \\xi ) = K _ r ^ \\prime * \\phi ( \\xi ) = \\int _ { - \\pi } ^ 0 \\left ( K _ r ( \\xi - y ) - K _ r ( \\xi + y ) \\right ) \\phi ^ \\prime ( y ) \\ , d y . \\end{align*}"}
-{"id": "9261.png", "formula": "\\begin{align*} M _ 2 ^ \\prime ( R ) = \\sup \\limits _ { B _ { R } \\cap \\partial \\Omega } \\sup \\limits _ { | \\tau | = 1 , \\tau \\cdot \\nu = 0 } | u _ { \\tau \\tau } | , \\end{align*}"}
-{"id": "8608.png", "formula": "\\begin{align*} H ( u , v ) = \\frac { ( u ' ) ^ 2 } { 2 } + \\frac { ( v ' ) ^ 2 } { 2 } - \\frac { ( 1 - u ^ 2 - v ^ 2 ) ^ 2 } { 4 } - \\frac { \\Lambda - 1 } { 2 } \\left ( u v - \\frac { \\omega } { \\Lambda - 1 } \\right ) ^ 2 \\end{align*}"}
-{"id": "4051.png", "formula": "\\begin{align*} d ( t ) = \\sum _ \\alpha [ \\alpha , \\alpha ^ * ] . \\end{align*}"}
-{"id": "742.png", "formula": "\\begin{align*} \\mathbb { V } \\ ! _ { f } \\ ! \\left [ S \\right ] = - \\psi _ { 1 } \\left ( m n + 1 \\right ) + \\frac { m + n } { m n + 1 } \\psi _ { 1 } \\left ( n \\right ) - \\frac { ( m + 1 ) ( m + 2 n + 1 ) } { 4 n ^ { 2 } ( m n + 1 ) } , \\end{align*}"}
-{"id": "2048.png", "formula": "\\begin{align*} \\widetilde { f ^ \\chi } _ { R ' } ( z ) = f ^ \\chi \\mid W _ { R ' } ( z ) = \\sum _ { \\ell \\mid r _ { * 0 } } \\beta _ { F _ \\chi } ( \\ell ) \\ell ^ { - \\frac { k } { 2 } } \\left ( Q _ * \\frac { r _ { * 0 } } { \\ell } \\right ) ^ { \\frac { k } { 2 } } \\widetilde { F _ \\chi } _ { \\frac { R R _ * ' } { R _ * } } \\left ( Q _ * \\frac { r _ { * 0 } } { \\ell } z \\right ) . \\end{align*}"}
-{"id": "8505.png", "formula": "\\begin{align*} \\norm { \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { 2 \\pi } \\frac { \\dot { z } _ j } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } } _ { H ^ s } \\leq K _ s ^ { - 1 } \\epsilon \\frac { | \\lambda | } { x ( 0 ) } d _ I ( t ) ^ { - 5 / 2 } + C \\epsilon ^ 2 . \\end{align*}"}
-{"id": "9941.png", "formula": "\\begin{align*} \\Delta + M _ V = \\frac { 4 \\pi ^ 2 } { T ^ 2 } \\left ( A + \\frac { \\beta } { 4 } \\right ) ^ 2 + Q _ 0 + M _ V . \\end{align*}"}
-{"id": "8274.png", "formula": "\\begin{align*} \\bar { z } _ t ( \\alpha , t ) = f + p , p = - \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i } { 2 \\pi } \\frac { 1 } { z ( \\alpha , t ) - z _ j ( t ) } . \\end{align*}"}
-{"id": "9423.png", "formula": "\\begin{align*} \\frac { \\partial _ z F ( z ) } { F ( z ) } = 1 2 \\frac { \\partial _ z f ( z ) } { f ( z ) } - k \\frac { \\partial _ z \\Delta ( z ) } { \\Delta ( z ) } = 1 2 \\frac { \\partial _ z f ( z ) } { f ( z ) } - k ( 2 \\pi i ) E _ 2 ( z ) = ( 2 4 \\pi i ) f _ { \\theta } ( z ) . \\end{align*}"}
-{"id": "4865.png", "formula": "\\begin{align*} M * K = \\{ ( x , t y ) \\in M \\times D ^ m \\ , ; \\ , x \\in M , y \\in K , 0 < t < 1 \\} . \\end{align*}"}
-{"id": "6373.png", "formula": "\\begin{align*} u ( x _ 0 ) - v ( x _ 0 ) + \\min _ { \\partial D } ( v - u ) = \\tilde { u } ( x ) - v ( x ) \\leq 0 , \\end{align*}"}
-{"id": "9582.png", "formula": "\\begin{align*} F ( a , b ; t ) = \\frac { 1 } { 1 - t } \\sum _ { n = 0 } ^ { \\infty } \\frac { ( b / a ) _ n ( - a t ) ^ n q ^ { n ( n + 1 ) / 2 } } { ( b q ) _ n ( t q ) _ n } . \\end{align*}"}
-{"id": "1278.png", "formula": "\\begin{gather*} S _ A ( \\xi ) : = \\chi ( \\xi ) e ^ { i a \\ln \\xi } \\left ( A + B e ^ { 2 i a \\ln | \\xi | } \\frac { e ^ { - i \\frac { 8 } { 9 } \\xi ^ 3 } } { \\xi ^ 3 } \\right ) , S _ A ( - \\xi ) = \\overline { S _ A ( \\xi ) } , \\end{gather*}"}
-{"id": "848.png", "formula": "\\begin{align*} a _ j ^ { - 1 } f ( t _ j ) = a _ j ^ { - 1 } [ f , G _ { s a m p } ( t _ j , \\cdot ) ] _ { s a m p } = [ f , \\overline { a _ j } ^ { - 1 } G _ { s a m p } ( t _ j , \\cdot ) ] _ { s a m p } = [ f , M _ j ] _ { s a m p } \\end{align*}"}
-{"id": "2778.png", "formula": "\\begin{align*} K ( \\mathbf { k } , \\mathbf { n } ) = \\mathbb { P } ( \\mathbf { J } ^ { t } + \\mathbf { R } ^ { t } = \\mathbf { n } ) . \\end{align*}"}
-{"id": "1459.png", "formula": "\\begin{align*} \\frac { d E _ { \\mu } } { d t } = \\sum _ { \\ell = 1 } ^ { 1 4 } V _ { \\ell } , \\end{align*}"}
-{"id": "2330.png", "formula": "\\begin{align*} a _ x ^ + | \\mathfrak { m } \\rangle & = | \\mathfrak { m } + \\ 1 _ x \\rangle & a _ x | \\mathfrak { m } \\rangle & = \\sum _ { y \\in X } \\delta _ { x , y } | \\mathfrak { m } - \\ 1 _ x \\rangle . \\end{align*}"}
-{"id": "8336.png", "formula": "\\begin{align*} ( I - \\mathbb { H } ) ( \\frac { 1 } { Z ( \\alpha , t ) - z _ 0 } - \\frac { 1 } { c _ 1 ( \\alpha - w _ 0 ) } ) = 0 . \\end{align*}"}
-{"id": "2368.png", "formula": "\\begin{align*} S _ i \\ne \\emptyset & \\Leftrightarrow i \\in \\{ i _ 1 , \\cdots , i _ p \\} , & S _ { j _ i } & = T _ i \\end{align*}"}
-{"id": "9774.png", "formula": "\\begin{align*} L _ a q ^ { \\rm o d d } ( X ) = L _ a q ( X ) - L _ a q ( X - 2 ( \\boldsymbol { e } _ \\ast \\cdot X ) \\boldsymbol { e } _ \\ast ) = 0 \\quad X \\in \\R ^ { n + 1 } \\setminus L _ * \\end{align*}"}
-{"id": "3396.png", "formula": "\\begin{align*} \\rho ^ M _ { \\inf } ( T ) : = \\inf \\{ \\rho ( S ) : S ^ { ( M ) } = T ^ { ( M ) } \\} , \\rho ^ M _ { \\sup } ( T ) : = \\sup \\{ \\rho ( S ) : S ^ { ( M ) } = T ^ { ( M ) } \\} . \\end{align*}"}
-{"id": "2970.png", "formula": "\\begin{align*} P ( \\beta ) = h ( \\beta ) + \\beta \\partial P ( \\beta ) , \\end{align*}"}
-{"id": "7994.png", "formula": "\\begin{align*} A ^ { p , q } ( \\mathcal { U } ) : = A ^ { p , q } ( U _ { 0 } ) \\oplus A ^ { p , q } ( U _ { 1 } ) \\oplus A ^ { p , q - 1 } ( U _ { 0 1 } ) \\end{align*}"}
-{"id": "1842.png", "formula": "\\begin{align*} \\phi ^ { \\rm o d d } ( n , x , t ) = ( 2 k _ n ) ^ { - 1 / 2 } u ^ { \\rm o d d } ( n , x ) \\ , e ^ { - i k _ n t } \\end{align*}"}
-{"id": "4422.png", "formula": "\\begin{align*} \\partial _ t u = \\nabla \\cdot ( u _ { \\alpha } \\otimes \\nabla \\mathcal { V } + \\nabla u ) , ( t , x ) \\in ( 0 , T ) \\times \\Omega . \\end{align*}"}
-{"id": "9534.png", "formula": "\\begin{align*} & \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] _ { q ^ 2 } \\frac { ( - 1 ) ^ { n - 1 } z ^ n q ^ { n ^ 2 } ( q ^ 2 ; q ^ 2 ) _ { n } } { ( z q ; q ^ 2 ) _ n ( 1 - z q ^ { 2 n } ) } \\\\ & = \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] _ { q ^ 2 } \\left ( \\frac { ( q ; q ) _ { 2 n - 2 } z ^ { 2 n - 1 } q ^ { n ( 2 n - 1 ) } } { ( z q ; q ) _ { 2 n - 1 } } + \\frac { ( q ; q ) _ { 2 n - 1 } z ^ { 2 n } q ^ { n ( 2 n + 1 ) } } { ( z q ; q ) _ { 2 n } } \\right ) \\frac { ( q ^ { 2 } ; q ^ 2 ) _ n } { ( z q ^ { 2 N + 1 } ; q ^ 2 ) _ { n } } . \\end{align*}"}
-{"id": "4600.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } f ( z ) d \\nu _ g ( z ) = \\int _ { F _ 1 } \\int _ { F _ 2 } f ( u - g v ) d \\mu _ 1 ( u ) d \\mu _ 2 ( v ) , f \\in C _ 0 ( \\mathbb { R } ^ n ) . \\end{align*}"}
-{"id": "5694.png", "formula": "\\begin{align*} \\tilde x ^ A + t w ^ A = x ^ A + t ( \\varepsilon v - k ) + t w ^ A = x ^ A + t ( \\varepsilon v - k + w ^ A ) \\in A \\end{align*}"}
-{"id": "7377.png", "formula": "\\begin{align*} c _ R = B C F _ { W R } \\ , c _ W . \\end{align*}"}
-{"id": "9836.png", "formula": "\\begin{align*} & \\left \\{ \\begin{array} { r c l l } w _ { 1 } ( x ' , 0 , 0 ) & \\geq & \\psi ( x ' ) & \\R ^ { n - 1 } \\\\ L _ a w _ 1 & = & 0 & \\R ^ { n + 1 } \\setminus \\{ ( x ' , 0 , 0 ) : w _ 1 ( x ' , 0 , 0 ) = \\psi ( x ' ) \\} \\\\ L _ a w _ 1 & \\leq & 0 & \\R ^ { n + 1 } \\\\ \\lim _ { | X | \\to \\infty } w _ 1 ( X ) & = & 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "2883.png", "formula": "\\begin{align*} J _ { \\mu \\nu } & = - i L _ { \\mu \\nu } + \\frac { 1 } { 2 } \\Sigma _ { \\mu \\nu } , \\end{align*}"}
-{"id": "408.png", "formula": "\\begin{align*} G _ { g } ( x , x ' ) = - \\frac { 1 } { 2 \\pi } \\log ( d _ { g } ( x , x ' ) ) + m _ { g } ( x , x ' ) \\end{align*}"}
-{"id": "3254.png", "formula": "\\begin{align*} \\Omega _ n ' ( a + b ) = \\Pi _ n ^ ( a + b ) ; \\end{align*}"}
-{"id": "2377.png", "formula": "\\begin{align*} S _ 3 \\psi _ { l , m , j } & = m \\psi _ { l , m , j } \\\\ S _ + \\psi _ { l , m , j } & = \\sqrt { l ( l + 1 ) m ( m + 1 ) } \\psi _ { l , m + 1 , j } \\\\ S _ - \\psi _ { l , m , j } & = \\sqrt { l ( l + 1 ) - m ( m - 1 ) } \\psi _ { l , m - 1 , j } \\\\ S ^ 2 \\psi _ { l , m , j } & = l ( l + 1 ) \\psi _ { l , m , j } \\end{align*}"}
-{"id": "8387.png", "formula": "\\begin{align*} A : = a | z _ { \\alpha } | , D : = \\frac { \\partial _ { \\alpha } } { z _ { \\alpha } } . \\end{align*}"}
-{"id": "9580.png", "formula": "\\begin{align*} { } & \\frac { 1 } { ( q ) _ N } \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - 1 ) ^ { n - 1 } n q ^ { n ( n + 1 ) / 2 } } { 1 - q ^ n } + \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { q ^ { n ( n + 1 ) } } { ( q ) _ n ( 1 - q ^ n ) } F ( q ^ N , q ^ n ; q ^ n ) = L ( q , N ) , \\end{align*}"}
-{"id": "96.png", "formula": "\\begin{align*} 0 = \\bigl \\| \\textbf { x } ^ { ( N - 1 ) } \\bigr \\| _ { \\ell _ 1 } = \\bigl \\| \\textbf { x } \\bigr \\| _ { \\ell _ 1 } - 2 \\left ( \\sum _ { i = 1 } ^ { N - 1 } | x _ { k _ i } | \\right ) . \\end{align*}"}
-{"id": "1628.png", "formula": "\\begin{align*} d X _ t = \\mu X _ t ( 1 - \\gamma X _ t ) d t + \\sigma X _ t ( 1 - \\gamma X _ t ) d W _ t , X _ 0 = x \\in ( 0 , 1 / \\gamma ) . \\end{align*}"}
-{"id": "4039.png", "formula": "\\begin{align*} i ( ( b , d ) ) = | \\{ S _ { a c } \\in V : ( b , d ) \\in S _ { a c } , \\ , ( a , b ) \\in R _ M \\} | . \\end{align*}"}
-{"id": "5146.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\begin{vmatrix} f ^ + ( r ) & \\overline { \\widetilde { f } ^ + ( r ) } \\\\ f ^ - ( r ) & \\overline { \\widetilde { f } ^ - ( r ) } \\end{vmatrix} = \\begin{vmatrix} D \\begin{pmatrix} A ^ + \\\\ A ^ - \\end{pmatrix} \\overline { D \\begin{pmatrix} \\widetilde A ^ + \\\\ \\widetilde A ^ - \\end{pmatrix} } \\end{vmatrix} , \\end{align*}"}
-{"id": "8214.png", "formula": "\\begin{align*} U _ m \\left ( \\sum _ { n = - \\infty } ^ \\infty f _ n q ^ n \\right ) = \\sum _ { n = - \\infty } ^ \\infty f _ { m n } q ^ n . \\end{align*}"}
-{"id": "5271.png", "formula": "\\begin{align*} f _ { n } 1 _ { C _ { N } } = \\sum _ { j = 1 } ^ { m ( n ) } \\alpha _ { j , n } 1 _ { A _ { n , j } } \\end{align*}"}
-{"id": "7406.png", "formula": "\\begin{align*} \\Phi _ t ^ U ( A ) : = \\bigcup _ { \\substack { q \\in \\mathbb { Q } \\\\ q < t } } \\phi _ q ^ U ( A \\cap U _ { \\mathfrak r / 2 } ) , 0 \\leq t < \\mathfrak r / 2 . \\end{align*}"}
-{"id": "978.png", "formula": "\\begin{align*} X ( \\mu , b ) \\cong \\bigsqcup _ { P ' = M ' N ' } X ^ { M ' } ( \\mu _ { P ' } , b _ { P ' } ) , \\end{align*}"}
-{"id": "6261.png", "formula": "\\begin{align*} \\varepsilon = i \\pi ; \\end{align*}"}
-{"id": "5015.png", "formula": "\\begin{align*} \\nabla L _ { \\beta _ m } = ( 1 + s / d ) \\left ( \\frac { \\beta _ m } { R _ m } \\right ) ^ { s / d } - \\lambda , 1 \\leq m \\leq M , \\end{align*}"}
-{"id": "2022.png", "formula": "\\begin{align*} I _ j ( q , z ) : = \\frac { i _ j ^ * i _ * J ( \\tau , z ) } { e ( N _ { p _ j / \\P ^ 4 } ) } = z \\sum _ { d \\geq 0 } q ^ d \\frac { \\prod \\limits _ { k = 0 } ^ { 5 d } ( 5 \\alpha _ j + k z ) } { \\prod \\limits _ { l = 0 } ^ 4 \\prod \\limits _ { k = 0 \\atop ( l , k ) \\neq ( j , 0 ) } ^ d ( \\alpha _ j - \\alpha _ l + k z ) } \\end{align*}"}
-{"id": "5711.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle { _ c D _ { 0 t } ^ { \\alpha } \\left ( \\int _ 0 ^ x ( x - t ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } [ \\lambda ( x - t ) ^ { \\alpha } ] f ( t ) d t \\right ) = } \\\\ \\displaystyle { \\frac { 1 } { \\Gamma ( 2 - \\alpha ) } \\int _ 0 ^ x \\frac { 1 } { ( x - t ) ^ { \\alpha - 1 } } \\frac { d ^ 2 } { d t ^ 2 } \\left ( \\int _ 0 ^ t ( t - z ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } [ \\lambda ( t - z ) ^ { \\alpha } ] f ( z ) d z \\right ) d t . } \\end{array} \\end{align*}"}
-{"id": "4863.png", "formula": "\\begin{align*} \\sigma = \\varphi ( \\cap H _ i ) \\cap S ^ { m - 1 } . \\end{align*}"}
-{"id": "1383.png", "formula": "\\begin{align*} \\widetilde { d } ( \\widehat { \\Xi } ( \\lambda _ 0 , \\ldots , \\lambda _ n ; J , q ) ) = \\sum _ { i = 1 } ^ { m } d _ { J _ { ( i ) } } ( \\lambda _ { i - 1 } ) e _ { J _ { ( i ) } } + \\sum _ { i = m + 1 } ^ { n - 1 } d _ { J _ { ( i ) } } ( \\lambda _ { i } ) e _ { J _ { ( i ) } } \\end{align*}"}
-{"id": "2509.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } B _ { n , p } \\frac { t ^ { n } } { n ! } = \\frac { ( p + 1 ) ( t - H _ { p } ) e ^ { p t } } { ( e ^ { t } - 1 ) ^ { p + 1 } } + ( p + 1 ) \\sum _ { k = 1 } ^ { p } \\binom { p } { k } \\frac { H _ { k } } { ( e ^ { t } - 1 ) ^ { k + 1 } } , \\end{align*}"}
-{"id": "8231.png", "formula": "\\begin{align*} & L _ { 2 i - 1 } ( \\tau ) : = U _ 5 ( Z ( \\tau ) L _ { 2 i - 2 } ( \\tau ) ) , \\\\ & L _ { 2 i } ( \\tau ) : = U _ 5 ( L _ { 2 i - 1 } ( \\tau ) ) . \\end{align*}"}
-{"id": "7392.png", "formula": "\\begin{align*} \\overline { \\mathbf { x } } _ I = \\frac { 1 } { N _ I } \\sum _ { i \\in I } \\mathbf { x } _ i \\end{align*}"}
-{"id": "1545.png", "formula": "\\begin{align*} F ^ \\delta \\left ( \\begin{array} { c } z \\\\ x \\end{array} \\right ) = \\left ( \\begin{array} { c } z + q ( z + \\delta x ) \\\\ \\delta x - q ( z + \\delta x ) \\end{array} \\right ) , \\end{align*}"}
-{"id": "5553.png", "formula": "\\begin{align*} \\tilde { C } _ { \\varepsilon } ( f , g , h ) = h \\circ \\Q _ { \\varepsilon } ^ { - 1 } ( f \\prec g ) - f \\left ( h \\circ \\Q _ { \\varepsilon } ^ { - 1 } g \\right ) , \\end{align*}"}
-{"id": "8179.png", "formula": "\\begin{align*} \\big \\| \\sum _ { i \\in { I _ n } } \\Lambda _ i ^ * g _ i \\big \\| ^ 2 = \\big \\langle \\sum _ { i \\in { I _ n } } \\Lambda _ i ^ * g _ i , \\sum _ { j \\in { I _ n } } \\Lambda _ j ^ * g _ j \\big \\rangle & = \\sum _ { i \\in { I _ n } } \\sum _ { j \\in { I _ n } } \\langle \\Lambda _ i ^ * g _ i , \\Lambda _ j ^ * g _ j \\rangle \\\\ & = \\sum _ { i \\in { I _ n } } \\langle g _ i , g _ i \\rangle = \\sum _ { i \\in { I _ n } } \\| g _ i \\| ^ 2 . \\end{align*}"}
-{"id": "2081.png", "formula": "\\begin{align*} H _ { n m } = { \\frac { { { J _ n } ( { \\lambda _ { n m } } { \\rho _ { \\rm t x } } ) } } { { { N _ { n m } } } } } L _ n , \\ ; \\ ; n \\geq 0 , m \\geq 1 . \\end{align*}"}
-{"id": "3236.png", "formula": "\\begin{align*} & F _ P ( x ) : = P F ( x ) P ^ { \\top } = F _ 0 + \\sum _ { i = 1 } ^ n x _ i F _ P ^ i , \\ { V } _ P : = P ^ { - \\top } V P ^ { - 1 } , \\end{align*}"}
-{"id": "663.png", "formula": "\\begin{align*} \\rho _ i ( j ) = \\begin{cases} - \\infty & j < i \\ , \\\\ 1 & j = i \\ , \\\\ \\infty & j > i \\ . \\end{cases} \\end{align*}"}
-{"id": "7766.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ k t _ { j } \\langle ( \\| B \\| _ { \\nu } A + \\lambda \\| A \\| _ { \\nu } B ) , x _ { j } y _ { j } ^ * \\rangle = 0 . \\end{align*}"}
-{"id": "147.png", "formula": "\\begin{align*} [ d f , d g ] = d \\{ f , g \\} , \\\\ a ( d f ) ( g ) = \\{ f , g \\} \\end{align*}"}
-{"id": "6474.png", "formula": "\\begin{align*} p _ 0 = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & 0 \\end{array} \\right ) \\oplus 0 p _ { n - 1 } = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 0 & 1 \\end{array} \\right ) \\oplus 1 . \\end{align*}"}
-{"id": "4199.png", "formula": "\\begin{align*} P _ + ( z ) P _ - ^ { - 1 } ( z ) & = \\mathbb { I } + \\mathcal { O } ( n ^ { - 1 } ) n \\to \\infty , & & z \\in \\partial D \\left ( 0 , r _ n \\right ) , \\\\ P ( z ) N ^ { - 1 } ( z ) & = \\mathbb { I } + \\mathcal { O } ( n ^ { - 1 } ) n \\to \\infty , & & z \\in \\partial D \\left ( 0 , R \\right ) . \\end{align*}"}
-{"id": "5190.png", "formula": "\\begin{align*} \\phi _ { \\gamma } ( x ) = \\exp ( - x ^ { - \\gamma } ) \\mathbb { I } _ { \\left [ 0 , + \\infty \\right [ } ( x ) , x \\in \\mathbb { R } \\ \\end{align*}"}
-{"id": "5431.png", "formula": "\\begin{align*} | \\lambda _ 1 | \\geq | \\lambda _ 2 | \\geq | \\lambda _ 3 | \\geq \\ldots \\lim _ { n \\to \\infty } \\lambda _ n = 0 \\end{align*}"}
-{"id": "7350.png", "formula": "\\begin{align*} T _ k \\ast U _ k + S _ k \\ast V _ k = \\delta _ { \\scriptscriptstyle { 0 } } . \\end{align*}"}
-{"id": "2011.png", "formula": "\\begin{align*} I ( q , z ) = I _ 0 ( q ) z + I _ 1 ( q ) H + I _ 2 ( q ) \\frac { H ^ 2 } { z } + I _ 3 ( q ) \\frac { H ^ 3 } { z ^ 3 } , \\end{align*}"}
-{"id": "1134.png", "formula": "\\begin{align*} & | K ^ { G U E ( n ) } ( x , y ) | = \\sqrt { n } \\frac { O ( n ^ { - 1 / 4 } ) O ( n ^ { - 1 / 4 } ) } { | x - y | } = \\frac { O ( 1 ) } { | x - y | } \\leq \\frac { O ( 1 ) } { \\varepsilon _ 0 ( 2 \\ln n ) ^ { - 1 } } = O ( \\ln n ) . \\end{align*}"}
-{"id": "9596.png", "formula": "\\begin{align*} F ( q , N ) = \\frac { A } { ( 1 - q ) ^ { N + 1 } } + \\frac { B } { ( 1 - q ) ^ N } + \\cdots , \\end{align*}"}
-{"id": "9613.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { \\infty } \\frac { q ^ { N + m } } { 1 + q ^ { N + m } } \\frac { ( - 1 ) _ m } { ( - q ^ N ) _ m } & = \\frac { q ^ N } { 1 + q ^ N } \\left ( F \\left ( \\frac { - 1 } { q } , - q ^ N ; q \\right ) - 1 \\right ) \\\\ & = \\frac { ( - 1 ) _ N q ^ N } { 1 - q ^ N } - \\frac { 2 q ^ N } { 1 - q ^ { 2 N } } , \\end{align*}"}
-{"id": "9165.png", "formula": "\\begin{align*} \\tau ^ D = \\sigma ^ D \\circ . \\end{align*}"}
-{"id": "6120.png", "formula": "\\begin{align*} p & = ( Q _ 1 , Q _ 2 , 0 ) \\\\ p & = ( 2 Q _ 1 , Q _ 2 , Q _ 3 ) \\end{align*}"}
-{"id": "8137.png", "formula": "\\begin{align*} \\widetilde { h } \\triangleright \\varphi = \\varphi ( S ^ { - 1 } ( h ) ? ) . \\end{align*}"}
-{"id": "4989.png", "formula": "\\begin{align*} y '^ m = x '^ { \\lambda } ( x ' - 1 ) \\end{align*}"}
-{"id": "1398.png", "formula": "\\begin{align*} f ( x _ 1 , \\ldots , x _ d ) & = h ( p _ 1 ( x _ 1 ) + \\cdots + p _ d ( x _ d ) ) ~ ~ \\\\ f ( x _ 1 , \\ldots , x _ d ) & = h ( p _ 1 ( x _ 1 ) \\cdot \\ldots \\cdot p _ d ( x _ d ) ) , \\end{align*}"}
-{"id": "1492.png", "formula": "\\begin{align*} E _ { w } ( t ) = E _ { w } ( T ) - \\int _ { t } ^ { T } \\sum _ { j = 1 } ^ { d } \\sum _ { \\ell = 1 } ^ { 6 } W ^ { j } _ { \\ell } \\ d \\tau . \\end{align*}"}
-{"id": "7674.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ n \\sum _ { p = 1 } ^ n \\sum _ { q = 1 } ^ n \\sum _ { r = 1 } ^ n \\sum _ { s = 1 } ^ n ( b _ { p l } b _ { l i } b _ { q j } a _ { r k } c _ { q r } ^ s c _ { p s } ^ t - b _ { p l } b _ { l k } b _ { q i } a _ { r j } c _ { q r } ^ s c _ { s p } ^ t - b _ { p l } b _ { l j } b _ { q i } a _ { r k } c _ { q r } ^ s c _ { p s } ^ t ) = 0 , & \\\\ \\forall i , j , k , t : 1 \\leq i , j , k , t \\leq n . & \\end{align*}"}
-{"id": "9967.png", "formula": "\\begin{align*} V ( f ) ( t ) : = \\int _ 0 ^ t S ( t - s ) f ( s ) \\ , d s \\ ; \\ ; t \\in I . \\end{align*}"}
-{"id": "893.png", "formula": "\\begin{align*} \\lambda _ { Q ^ 1 _ 1 } ( \\beta ' , \\beta '' ) = 1 > 0 , \\lambda _ { Q _ 1 \\setminus Q ^ 1 _ 1 } ( \\beta ' , \\beta '' ) = 1 > 0 . \\end{align*}"}
-{"id": "5643.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\sin ( n x ) = \\frac { \\sin x } { 2 ( 1 - \\cos x ) } , \\end{align*}"}
-{"id": "2712.png", "formula": "\\begin{align*} K _ { \\mathrm { G U E } } ( \\pi ; u , v ) = \\int _ { - i \\infty } ^ { i \\infty } \\frac { d z } { 2 i \\pi } \\int _ { \\mathcal { C } _ { \\{ \\pi _ { 1 } , \\ldots , \\pi _ m \\} } } \\frac { d w } { 2 i \\pi } \\frac { 1 } { z - w } e ^ { \\frac { 1 } { 2 } z ^ 2 - \\frac { 1 } { 2 } w ^ 2 } e ^ { v z - u w } \\prod _ { k = 1 } ^ { m } \\frac { z - \\pi _ { k } } { w - \\pi _ { k } } \\end{align*}"}
-{"id": "9374.png", "formula": "\\begin{align*} & G _ k = \\Phi _ 0 ( G _ k ) = \\langle x , y \\rangle \\Phi _ 1 ( G _ k ) G _ k / \\Phi _ 1 ( G _ k ) \\cong C _ 2 \\times C _ 2 , \\\\ & \\Phi _ 1 ( G _ k ) = \\langle x ^ 2 , y ^ 2 , [ y , x ] , [ y , x , x ] \\rangle \\Phi _ 2 ( G _ k ) \\\\ & \\Phi _ 1 ( G _ k ) / \\Phi _ 2 ( G _ k ) \\cong C _ 2 ^ { \\ , 4 } , \\end{align*}"}
-{"id": "4331.png", "formula": "\\begin{align*} \\begin{aligned} | u - Q _ F u _ I | _ { 1 , F } & \\lesssim | u - \\Pi _ F u _ I | _ { 1 , F } + | \\Pi _ F u _ I - Q _ F u _ I | _ { 1 , F } \\\\ & \\lesssim | u - \\Pi _ F u _ I | _ { 1 , F } + \\epsilon _ F ^ { - 1 } h _ F ^ { - 1 } \\| \\Pi _ F u _ I - Q _ F u _ I \\| _ { 0 , F } \\\\ & \\leq | u - \\Pi _ F u _ I | _ { 1 , F } + \\epsilon _ F ^ { - 1 } h _ F ^ { - 1 } \\| \\Pi _ F u _ I - u _ I \\| _ { 0 , F } \\\\ & + \\epsilon _ F ^ { - 1 } h _ F ^ { - 1 } \\| u _ I - Q _ F u _ I \\| _ { 0 , F } \\end{aligned} \\end{align*}"}
-{"id": "264.png", "formula": "\\begin{align*} \\sum _ { j = 2 } ^ p ( j - 1 ) Q _ { j - 1 } ( \\alpha ) f ^ { p - j + 1 } v _ j \\in I _ p ( D ) . \\end{align*}"}
-{"id": "8839.png", "formula": "\\begin{align*} p _ 2 - p _ 1 & = \\frac { a ( M + 1 ) ( M \\mu - 1 ) } { 2 M \\mu ( M \\mu + a ) } = \\frac { a ( M + 1 ) ( a + ( M - 2 ) f ) } { 2 M \\mu ( M \\mu + a ) } > 0 , \\\\ p _ 2 - p _ 3 & = \\frac { ( 1 - a ) ^ 2 ( M + 1 ) } { 2 ( M \\mu + 1 ) ( M \\mu + a ) } > 0 , \\end{align*}"}
-{"id": "7479.png", "formula": "\\begin{align*} \\Im \\rho ^ { \\frac { 1 } { 2 } } = \\langle q \\rangle \\langle q , - q \\rangle . \\end{align*}"}
-{"id": "778.png", "formula": "\\begin{align*} \\sigma _ f \\lambda _ i = f * _ d \\lambda _ i = f ( \\rho _ i ) \\cdot \\lambda _ i . \\end{align*}"}
-{"id": "7299.png", "formula": "\\begin{align*} \\phi ( a ) = \\psi _ q ( a ) + ( 1 - h ^ - ) \\Phi ' _ q ( a ) ( h _ s ^ - - h ^ + ) + ( 1 - h ^ - ) R _ s \\Phi ' _ q ( a ) ( h _ s ^ + - h _ s ^ - ) . \\end{align*}"}
-{"id": "344.png", "formula": "\\begin{align*} m i n ( r - 1 , t ) + m i n ( r , t ) = m i n ( r - 1 , s ) + m i n ( r , s ) . \\end{align*}"}
-{"id": "2642.png", "formula": "\\begin{align*} \\det \\left [ \\phi _ { i } ( x _ { j } ) \\right ] & = \\sum _ { 1 \\leq i < j \\leq m } ( - 1 ) ^ { i + j + k + 1 } \\det \\begin{bmatrix} \\phi _ { i } ( x _ { 1 } ) & \\phi _ { i } ( x _ { k } ) \\\\ \\phi _ { j } ( x _ { 1 } ) & \\phi _ { j } ( x _ { k } ) \\end{bmatrix} \\det \\begin{bmatrix} \\phi _ { r } ( x _ { s } ) \\end{bmatrix} _ { r \\notin \\{ i , j \\} , s \\notin \\{ 1 , k \\} } . \\end{align*}"}
-{"id": "2110.png", "formula": "\\begin{align*} \\lambda _ k ( \\omega ) : = \\inf _ { V \\in { \\mathcal G } _ { d - k + 1 } } \\sup _ { y \\in V } \\limsup _ { t \\rightarrow \\infty } \\frac { 1 } { t } \\log | \\Phi _ \\omega ( t _ 0 , t ) y | , k = 1 , \\ldots , d , \\end{align*}"}
-{"id": "2771.png", "formula": "\\begin{align*} R ^ t _ j : = \\sum _ { i = 1 } ^ S B _ { i , j } ^ t . \\end{align*}"}
-{"id": "5235.png", "formula": "\\begin{align*} u e p ( F ) = + \\infty . \\end{align*}"}
-{"id": "4081.png", "formula": "\\begin{align*} \\frac { d \\mu ^ * } { d s } = - \\frac { 1 } { \\pi } \\Im F _ { 1 , + } ( s ) = \\frac { 1 } { \\pi } \\Im \\Psi _ { 0 , + } ( s ) \\end{align*}"}
-{"id": "7904.png", "formula": "\\begin{align*} \\hat { R } _ N ^ n = \\frac { \\hat { R } _ n } { M ^ { n + 1 } } \\end{align*}"}
-{"id": "3185.png", "formula": "\\begin{align*} { \\rm i n d } _ { - \\varGamma } ( U ) & = ( \\dim \\mathcal { T } _ - ( U , \\varGamma ) - \\dim \\mathcal { B } _ - ( U , \\varGamma ) ) \\\\ & - ( \\dim \\mathcal { T } _ + ( U , \\varGamma ) - \\dim \\mathcal { B } _ + ( U , \\varGamma ) ) \\\\ & = - { \\rm i n d } _ \\varGamma ( U ) . \\end{align*}"}
-{"id": "7161.png", "formula": "\\begin{align*} \\mathbb { B } _ { \\kappa _ { p ( t ) } } : = \\mathbb { F } _ { \\kappa _ { p ( t ) } } \\mathbb { F } _ { \\kappa _ { p ( t ) } } ^ T , \\mathbb { C } _ { \\kappa _ { p ( t ) } } : = \\mathbb { F } _ { \\kappa _ { p ( t ) } } ^ T \\mathbb { F } _ { \\kappa _ { p ( t ) } } . \\end{align*}"}
-{"id": "2399.png", "formula": "\\begin{align*} \\gamma _ { Q ( \\alpha ) } ( a ^ 2 ) & = \\gamma _ { Q ( \\alpha ) } ( ( a ^ \\dag ) ^ 2 ) = 0 \\\\ \\gamma _ { Q ( \\alpha ) } ( a a ^ \\dag ) & = \\alpha + 1 / 2 & \\gamma _ { Q ( \\alpha ) } ( a ^ \\dag a ) & = \\alpha - 1 / 2 \\end{align*}"}
-{"id": "7733.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } { \\rm R e } [ \\langle x , \\lambda y \\rangle \\xi _ { n } , \\xi _ { n } ] = \\left \\| x \\right \\| \\left \\| y \\right \\| . \\end{align*}"}
-{"id": "4669.png", "formula": "\\begin{align*} \\tilde U _ { k _ { 1 } , \\ldots , k _ { n } } = \\sum _ { h _ 1 , \\dots , h _ n } \\alpha _ { h _ 1 , \\dots , h _ n , k _ 1 , \\dots , k _ n } \\prod _ { j = 1 } ^ n a _ { h _ j } , \\end{align*}"}
-{"id": "3947.png", "formula": "\\begin{align*} \\dfrac { \\partial H _ { i } } { \\partial q _ { i } } ( x , m ^ { \\hat v } _ { t } , D u ^ { \\hat v } _ { i } ( x , t ) ) & = g _ { i } ( x , m ^ { \\hat v } _ { t } , \\hat { v } _ { i } ( x , m ^ { \\hat v } _ { t } ) ) \\ , . \\end{align*}"}
-{"id": "9912.png", "formula": "\\begin{align*} & \\lim _ { n \\to \\infty } \\left | \\int _ { \\mathcal { X } } \\tilde { f } _ { n } ( x ) \\pi _ { n - } ^ { \\mu } ( d x ) - \\int _ { \\mathcal { X } } \\tilde { f } _ { n } ( x ) \\pi _ { n - } ^ { \\nu } ( d x ) \\right | + \\frac { 2 } { 3 } \\epsilon \\\\ = & \\lim _ { n \\to \\infty } \\bigg | \\int _ { \\mathcal { Y } } g _ { n } ( y _ { n } ) P ^ { \\mu } ( d y _ { n } | y _ { [ 0 , n - 1 ] } ) \\\\ & - \\int _ { \\mathcal { Y } } g _ { n } ( y _ { n } ) P ^ { \\nu } ( d y _ { n } | y _ { 0 , n - 1 ] } ) \\bigg | + \\frac { 2 } { 3 } \\epsilon \\end{align*}"}
-{"id": "6401.png", "formula": "\\begin{align*} \\sum _ { j = k } ^ { k + d - 1 } \\gamma _ { j } \\| r _ { j } \\| ^ { 2 } + \\gamma _ { k + d } \\Vert r _ { k + d } \\Vert ^ 2 < \\| x - x _ { k } \\| _ { A } ^ { 2 } \\end{align*}"}
-{"id": "8445.png", "formula": "\\begin{align*} & i ( I - \\mathcal { H } ) A = i - [ D _ t \\zeta , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } \\mathfrak { F } } { \\zeta _ { \\alpha } } - [ D _ t ^ 2 \\zeta , \\mathcal { H } ] \\frac { \\bar { \\zeta } _ { \\alpha } - 1 } { \\zeta _ { \\alpha } } - ( I - \\mathcal { H } ) \\frac { i } { 2 \\pi } \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j ( D _ t \\zeta ( \\alpha , t ) - \\dot { z } _ j ( t ) ) } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } \\\\ \\end{align*}"}
-{"id": "7813.png", "formula": "\\begin{align*} \\lim _ n | [ \\langle T x _ n , S x _ n \\rangle \\xi _ n , \\xi _ n ] | = \\| T \\| \\| S \\| . \\end{align*}"}
-{"id": "51.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\rho + \\partial _ x J = 0 , & \\\\ \\partial _ t J + \\partial _ x \\rho = - 2 k ( x ) g ( J ) , & \\end{cases} \\end{align*}"}
-{"id": "4194.png", "formula": "\\begin{align*} P ^ { ( 3 ) } _ + ( z ) \\left ( P ^ { ( 3 ) } _ - ( z ) \\right ) ^ { - 1 } = \\mathbb { I } + \\mathcal { O } ( n ^ { - \\frac { 1 } { 2 } } ) \\end{align*}"}
-{"id": "3454.png", "formula": "\\begin{align*} F ' ( t _ w ; w ) = 0 t _ w \\in ( - N , + \\infty ) , \\end{align*}"}
-{"id": "9814.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } L _ { 1 + a } \\bar { h } _ \\beta ( x ' , 0 , y ) & = & 0 & \\R ^ { n } \\setminus \\{ y = 0 \\} \\\\ \\bar { h } _ \\beta ( x ' , 0 , 0 ) & = & h _ \\beta ( x ' ) & \\R ^ { n - 1 } \\\\ \\lim _ { | ( x ' , y ) | \\to \\infty } \\bar { h } _ \\beta ( x ' , 0 , y ) & = & 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "8382.png", "formula": "\\begin{align*} \\mathfrak { K } ^ { \\ast } f ( \\alpha ) = p . v . \\int R e \\Big \\{ - \\frac { 1 } { \\pi i } \\frac { z _ { \\alpha } } { | z _ { \\alpha } | } \\frac { | z _ { \\beta } ( \\beta , t ) | } { z ( \\alpha ) - z ( \\beta ) } \\Big \\} f ( \\beta ) d \\beta . \\end{align*}"}
-{"id": "1174.png", "formula": "\\begin{align*} & \\lim _ { n \\to + \\infty } ( n / 4 ) ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } D _ n ( ( z _ { j + 1 } - z _ j ) / 2 ) = 0 , \\end{align*}"}
-{"id": "3017.png", "formula": "\\begin{align*} G _ m ( x ) & = \\sum _ { n = 0 } ^ \\infty e _ { m + 2 , n } x ^ n . \\end{align*}"}
-{"id": "2601.png", "formula": "\\begin{align*} & e ^ { - J _ G ( r ) } - \\| ( \\phi ^ { - 1 } f ) ( \\xi _ t ) ^ { \\gamma _ 0 - 1 } e ^ { - J _ G ( r ) } \\| _ { \\Pi _ x ^ { ( \\phi ) } ; \\frac { 1 } { \\gamma _ 0 - 1 } } \\\\ & = e ^ { - J _ G ( r ) } \\Big ( 1 - \\Pi _ x ^ { ( \\phi ) } [ ( \\phi ^ { - 1 } f ) ( \\xi _ t ) ] ^ { \\gamma _ 0 - 1 } \\Big ) \\xrightarrow [ t \\to \\infty ] { } 0 , x \\in E , r \\geq 0 . \\end{align*}"}
-{"id": "1675.png", "formula": "\\begin{align*} \\xi ( [ X , Y ] ) = \\rho _ A ( X ) \\varphi ^ * \\xi ( \\alpha _ A ( Y ) ) - \\rho _ A ( Y ) \\varphi ^ * \\xi ( \\alpha _ A ( X ) ) - d ^ 0 \\xi ( X , Y ) . \\end{align*}"}
-{"id": "6194.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } r } \\log H ( r ) = \\frac { 2 N ( r ) } { r } . \\end{align*}"}
-{"id": "4471.png", "formula": "\\begin{align*} G ( \\nabla ^ 2 u , \\nabla u , u ) & = \\Upsilon ( \\nabla u , u , x ) \\quad \\textrm { i n } \\ , \\Omega \\\\ u & = \\varphi \\quad \\textrm { o n } \\ , \\partial \\Omega \\end{align*}"}
-{"id": "9504.png", "formula": "\\begin{align*} \\overline { c } _ { \\lambda } : = \\displaystyle \\inf _ { u \\in \\mathcal { W } \\backslash \\{ 0 \\} } \\displaystyle \\max _ { t \\geq 0 } I _ { \\lambda } ( t u ) , \\end{align*}"}
-{"id": "8109.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k ( { \\rm S } ) } ( u _ i ( { \\rm S } ) - u _ i ( { \\rm S } _ 0 ) ) - 2 k ( { \\rm S } ) \\end{align*}"}
-{"id": "5581.png", "formula": "\\begin{align*} \\| w \\| _ { W ^ { \\kappa , b , \\alpha } } = \\big \\| \\big ( 1 + | \\xi | + | \\tau | ^ { \\frac 1 3 } \\big ) ^ { \\kappa } \\nu ( \\xi , \\tau ) \\hat { w } ( \\xi , \\tau ) \\big \\| _ { L _ { \\tau } ^ 2 L ^ 2 _ { \\xi } } , \\end{align*}"}
-{"id": "8038.png", "formula": "\\begin{align*} y ^ r x y ^ k y ^ { - r } = x y ^ { k + r ( n - 1 ) - l } = x y ^ { k + r ( n - 2 ) } \\end{align*}"}
-{"id": "5623.png", "formula": "\\begin{align*} \\int _ 0 ^ { + \\infty } u _ { x x } \\bar { u } _ x d x & = \\frac 1 2 \\int _ 0 ^ { + \\infty } ( u _ { x x } \\bar { u } _ x + \\bar { u } _ { x x } u _ x ) d x \\\\ & = \\frac 1 2 \\int _ 0 ^ { + \\infty } \\frac { d } { d x } | u _ x | ^ 2 = - \\frac 1 2 | u _ x ( 0 , t ) | ^ 2 . \\end{align*}"}
-{"id": "5378.png", "formula": "\\begin{align*} \\langle A _ { \\alpha + \\beta } . v _ { n - 1 \\ , m - 3 } ^ k , v _ { n m } ^ k \\rangle = \\langle v _ { n - 1 \\ , m - 3 } ^ k , A _ { \\alpha + \\beta } ^ * . v _ { n m } ^ k \\rangle , \\end{align*}"}
-{"id": "7051.png", "formula": "\\begin{align*} | u ^ v _ i | ^ 2 | u ^ v _ j | ^ 2 - ( ( \\o { u ^ v _ i } ) ^ 2 ( u ^ v _ j ) ^ 2 ) = 2 ( a d - b c ) ^ 2 \\geq 0 , \\end{align*}"}
-{"id": "3480.png", "formula": "\\begin{align*} f ' _ { M , N } ( t ; \\theta ) = ( M + 1 ) \\psi ( N + \\frac { N } { M + 1 } t ) - \\psi ( \\frac { N } { M + 1 } t ) - \\big ( M \\log N + \\log ( M + 1 ) + v _ { M } ( \\theta ) \\big ) , \\end{align*}"}
-{"id": "167.png", "formula": "\\begin{align*} \\pi _ { T M , X _ { C } , X _ { V } , c } = \\pi _ { T M } + \\lambda c ( { \\bf x } ) \\pi _ { X _ { C } , X _ { V } } \\end{align*}"}
-{"id": "6357.png", "formula": "\\begin{align*} \\mathcal { P } _ { \\rm o u t } ^ { S T } & = { \\rm P r o b } \\left \\{ \\mathbb { P } _ { e , S T | p , h } > \\mathbb { P } _ { e , t } \\right \\} \\\\ & \\simeq { \\rm P r o b } \\left \\{ g _ { 1 1 } h _ { 1 1 } - g _ { 2 1 } h _ { 2 1 } / 2 < \\mathcal { A } _ { t h 3 } \\right \\} , \\end{align*}"}
-{"id": "9676.png", "formula": "\\begin{align*} \\begin{aligned} | \\varphi _ n f | & \\leq \\| X \\| K C _ { { \\rm p o l } , T } \\| \\varphi _ n ( T ) x _ 0 \\| | \\psi _ n | \\\\ & \\leq ( 1 + \\varepsilon _ 0 ) \\| X \\| K C _ { { \\rm p o l } , T } | \\psi _ n | \\ \\ \\sigma n \\in \\mathbb N . \\end{aligned} \\end{align*}"}
-{"id": "1201.png", "formula": "\\begin{align*} I _ { \\tilde { \\tau } , k } ( x - \\tilde { z } ' , \\zeta _ 2 ) = \\int _ { \\Gamma _ { x , z ' } } e ^ { - \\tilde { \\tau } r \\lambda } Q _ k ( \\zeta _ 1 , \\zeta _ 2 ) \\frac { d \\zeta _ 1 } { \\sqrt { 1 + \\zeta _ 1 ^ 2 } } . \\end{align*}"}
-{"id": "9865.png", "formula": "\\begin{align*} \\frac { | A | | { L } | } { K } \\le I ( A \\times A , { L } ) = \\sum _ { l \\in { L } } \\sum _ { x \\in A } A ( l x ) \\le 2 \\sum _ { l \\in { L } _ * } \\sum _ { x \\in A } A ( l x ) \\ , , \\end{align*}"}
-{"id": "6088.png", "formula": "\\begin{align*} g ( x ) = ( x - a _ 1 ) \\ldots ( x - a _ k ) h ' ( x ) + e ^ 1 ( \\widehat { x - a _ 1 } ) ( x - a _ 2 ) \\ldots ( x - a _ k ) h ( x ) + \\\\ e ^ 2 ( x - a _ 1 ) ( \\widehat { x - a _ 2 } ) \\ldots ( x - a _ k ) h ( x ) + \\\\ e ^ k ( x - a _ 1 ) \\ldots ( x - a _ { k - 1 } ) ( \\widehat { x - a _ k } ) \\end{align*}"}
-{"id": "1067.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to + \\infty } \\left ( \\frac { 1 } { 8 } \\ln ( 2 \\ln n ) - \\frac { 1 } { 4 } \\ln \\left ( n \\sin \\frac { \\alpha _ n } { 2 } \\right ) \\right ) = 0 . \\end{align*}"}
-{"id": "2744.png", "formula": "\\begin{align*} T & = \\log \\Big ( 1 + \\frac { \\frac { 1 } { \\tau } + \\tau - 2 \\theta } { \\theta - \\tau } \\Big ) ^ 2 - \\log \\Big ( 1 + \\frac { \\eta _ { - } } { R ^ 2 } \\big ( \\tau + \\frac { 1 } { \\tau } - 2 \\theta \\big ) \\Big ) . \\end{align*}"}
-{"id": "2134.png", "formula": "\\begin{align*} ( \\inf _ { z \\in L ^ 0 ( Z ) } \\Phi ^ \\ast ( \\cdot , z ) ) ^ \\ast ( x ) = \\Phi ( x , 0 ) , x \\in X , \\end{align*}"}
-{"id": "7526.png", "formula": "\\begin{align*} \\tau = \\sigma \\circ X \\end{align*}"}
-{"id": "4045.png", "formula": "\\begin{align*} \\sum _ r m _ r = | A | ^ 2 = N ^ 2 . \\end{align*}"}
-{"id": "3322.png", "formula": "\\begin{align*} \\mathrm { V a r } ( \\alpha ; n , h ) = \\frac { \\sum _ { \\chi _ 0 \\neq \\chi \\in G ( R _ { n - h - 1 , 1 } ) } \\left | S ( n , \\alpha \\cdot \\chi ) \\right | ^ 2 } { q ^ { 2 ( n - h - 1 ) } } . \\end{align*}"}
-{"id": "5917.png", "formula": "\\begin{align*} \\overline { \\Lambda } = \\bigcup _ { m \\in \\mathbb { N } } \\mathcal { C } _ m \\end{align*}"}
-{"id": "677.png", "formula": "\\begin{align*} \\begin{gathered} \\left \\{ ( \\widehat \\rho _ j ) _ { X ' } : X ' \\to X _ { x ^ { - 1 } ( \\delta ( j ) ) } \\right \\} _ { j = 1 } ^ n \\ , \\\\ \\left \\{ ( \\widehat \\rho _ j ) _ { X '' } : X '' \\to X _ { x ^ { - 1 } ( \\delta ( j ) ) } \\right \\} _ { j = 1 } ^ n \\ , \\end{gathered} \\end{align*}"}
-{"id": "9922.png", "formula": "\\begin{align*} D ( P ( X , Y ) \\| Q ( X , Y ) ) = D ( P ( X ) \\| Q ( X ) ) + D ( P ( Y | X ) \\| Q ( Y | X ) ) \\end{align*}"}
-{"id": "3238.png", "formula": "\\begin{align*} \\left ( { H } _ P ( x , V ) \\right ) _ { i , j } : = \\frac { 1 } { 2 } { F _ P ^ i \\bullet \\left ( \\mathcal { L } _ { F _ P ( x ) } ^ { - 1 } \\mathcal { L } _ { V _ P } + \\mathcal { L } _ { V _ P } \\mathcal { L } _ { F _ P ( x ) } ^ { - 1 } \\right ) F _ P ^ j } \\end{align*}"}
-{"id": "4636.png", "formula": "\\begin{align*} t _ j = \\begin{cases} 0 , & j = 0 , \\\\ \\frac { 1 } { n } ( j + \\delta ( \\texttt { R a n d ( 0 , 1 , j ) } - 0 . 5 ) ) , & j \\in [ 1 , \\ell - 1 ] , \\\\ 1 , & j = \\ell , \\end{cases} \\end{align*}"}
-{"id": "8829.png", "formula": "\\begin{align*} L _ { f , Q } = \\max ( \\max _ { 1 \\leq j \\leq J } L ( \\nabla f _ j ) , \\max _ { 1 \\leq i \\leq I } \\rho ( Q _ { i , i } ) ) \\end{align*}"}
-{"id": "8416.png", "formula": "\\begin{align*} \\| z _ { t t } ( \\cdot , t ) \\| _ { H ^ s } \\leq \\sqrt { 2 } \\frac { ( 2 \\| w _ 0 \\| _ { H ^ s } + 1 ) ^ { 1 / 2 } } { ( 2 C _ 2 ) ^ { - s + 1 / 2 } } \\mathcal { E } ( 0 ) ^ { 1 / 2 } : = M _ 1 . \\end{align*}"}
-{"id": "2254.png", "formula": "\\begin{align*} M ( x + \\i y ) & = \\mu ( x ) \\delta ( y ) \\\\ \\mu ( x ) & = \\frac { 1 } { 2 \\pi \\i } ( R ( x - \\i 0 ) - R ( x + \\i 0 ) ) = | \\alpha _ x \\rangle \\langle \\alpha _ x | \\\\ | \\alpha _ x \\rangle & = ( 1 + \\pi ^ 2 ) ^ { - 1 / 2 } \\left ( \\frac { \\mathcal { P } } { x - \\Omega } | E \\rangle + | \\delta _ x \\rangle \\right ) \\end{align*}"}
-{"id": "6319.png", "formula": "\\begin{align*} u _ x ( 0 , t ) = u _ x ( 1 , t ) = 0 \\quad t \\in ( 0 , \\ , T ) , \\end{align*}"}
-{"id": "4542.png", "formula": "\\begin{align*} \\int | \\hat { \\nu } ( \\omega ) \\hat { \\phi } _ \\delta ( \\omega ) | ^ 2 d \\omega = \\int | \\hat { \\nu _ \\delta } ( \\omega ) | ^ 2 d \\omega = \\int \\nu ^ 2 _ \\delta ( z ) d z . \\end{align*}"}
-{"id": "3982.png", "formula": "\\begin{align*} \\langle \\phi \\mid \\eta , \\psi \\rangle = \\langle \\phi , \\psi \\mid \\eta ^ { - 1 } \\rangle , \\end{align*}"}
-{"id": "1635.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot { \\varphi } _ 1 = i \\varphi _ { 1 x x } - 2 i \\varphi _ 1 ^ 2 \\varphi _ 2 , \\\\ \\dot { \\varphi } _ 2 = - i \\varphi _ { 2 x x } + 2 i \\varphi _ 1 \\varphi _ 2 ^ 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "9806.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } u ( x ' , x _ n , y ) & \\geq & 0 & B _ 1 \\cap \\{ x _ n = y = 0 \\} \\\\ L _ a u & \\leq & 0 & B _ 1 \\\\ u \\ , L _ a u & = & 0 & B _ 1 \\\\ L _ a u & = & 0 & B _ 1 \\setminus \\Lambda ( u ) \\\\ u ( x , y ) & = & u ( x , - y ) & B _ 1 \\end{array} \\right . \\end{align*}"}
-{"id": "1532.png", "formula": "\\begin{align*} { \\cal L } ( f ' ) ( t ) = t \\ , { \\cal L } ( f ) ( t ) - f ( 0 ^ + ) , \\mbox { f o r a l l $ t \\ge 0 $ } . \\end{align*}"}
-{"id": "5611.png", "formula": "\\begin{align*} \\sup _ { \\delta \\leq y \\leq - 2 \\delta + L } \\| u _ { \\delta , \\epsilon } v _ { \\delta , \\epsilon } \\| _ { L ^ 2 ( 0 , T ) } & \\leq \\sup _ { \\delta \\leq y \\leq - 2 \\delta + L } \\| v _ { \\delta , \\epsilon } \\| _ { L ^ 2 ( 0 , T ) } \\| u _ { \\delta , \\epsilon } \\| _ { L ^ { \\infty } ( 0 , T ) } \\\\ & \\leq \\sup _ { \\delta \\leq y \\leq - 2 \\delta + L } \\| v _ { \\delta , \\epsilon } \\| _ { L ^ 2 ( 0 , T ) } \\| u _ { \\delta , \\epsilon } \\| _ { H ^ 1 ( 0 , T ) } \\\\ & \\leq 1 + \\delta ^ { - 1 } . \\end{align*}"}
-{"id": "2911.png", "formula": "\\begin{align*} h _ \\mu ( X ) - \\mu ( f _ \\Phi ) & = h _ \\nu ( Y ) + h _ \\mu ( \\Omega \\ , | \\ , Y ) - \\mu ( f _ \\Phi ) \\ ; . \\end{align*}"}
-{"id": "7845.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { d _ 0 } e ^ { \\varphi ( m ) } \\le e ^ { \\varphi ( d _ 0 ) } \\sum _ { m = 0 } ^ { d _ 0 } e ^ { - ( d _ 0 - m ) \\log 3 } \\le \\frac { 3 } { 2 } e ^ { \\varphi ( d _ 0 ) } \\le \\frac { 3 } { 2 } e ^ { \\frac { 3 \\log ( 2 ^ { 1 / 3 } \\cdot 1 0 \\cdot e ) d } { 2 0 } } \\le \\frac { 3 } { 2 } e ^ { \\frac { 3 d } { 5 } } , \\end{align*}"}
-{"id": "5657.png", "formula": "\\begin{align*} \\Big ( \\sum _ { j = 0 } ^ { n - 1 } \\int _ { t _ { j } } ^ { t _ { j + 1 } } \\| A ^ { \\frac { s } { 2 } } \\big ( \\bar E ( t _ { n } - t ) - B _ { n - j } P _ { h } \\big ) \\| ^ p \\ , \\d s \\Big ) ^ { 1 / p } \\leq c ( t _ n ^ { \\frac { r \\alpha } { 2 } + \\gamma - \\frac 1 { p ' } } h ^ { 2 - s - r } + t _ n ^ { \\max ( \\eta - 1 , 0 ) } \\tau ^ \\mu ) , \\end{align*}"}
-{"id": "489.png", "formula": "\\begin{align*} \\nabla ( [ f ] ) = \\nabla ( [ f _ 1 ] ) \\oplus \\cdots \\oplus \\nabla ( [ f _ k ] ) \\end{align*}"}
-{"id": "3755.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { 0 } ^ { a } \\left | r j _ { l } ( \\alpha r ) \\right | ^ 2 d r & = \\frac { a ^ 3 } { 2 } \\left [ j _ { l } ^ { 2 } ( \\alpha a ) - j _ { l + 1 } ( \\alpha a ) j _ { l - 1 } ( \\alpha a ) \\right ] ; \\alpha \\in \\mathbb { R } \\\\ & < \\infty . \\end{aligned} \\end{align*}"}
-{"id": "287.png", "formula": "\\begin{align*} t _ j ( h , \\theta ) = x _ j + O \\left ( \\frac { 1 } { h } \\right ) . \\end{align*}"}
-{"id": "2649.png", "formula": "\\begin{align*} K _ { N } ( x , y ) = \\begin{bmatrix} D S _ { N } ( x , y ) & S _ { N } ( x , y ) \\\\ - S _ { N } ( y , x ) & I S _ { N } ( x , y ) \\end{bmatrix} , \\end{align*}"}
-{"id": "674.png", "formula": "\\begin{align*} ( \\widehat \\rho ^ { ( \\vec m ) } _ i ) _ Z \\circ \\varpi ( g _ 1 , \\dots , g _ n ) \\circ \\varpi ( f _ 1 , \\dots , f _ n ) & = g _ i \\circ ( \\widehat \\rho ^ { ( \\vec l ) } _ i ) _ Y \\circ \\varpi ( f _ 1 , \\dots , f _ n ) \\\\ & = g _ i f _ i \\circ ( \\widehat \\rho ^ { ( \\vec k ) } _ i ) _ X \\ , \\end{align*}"}
-{"id": "3549.png", "formula": "\\begin{align*} \\omega _ { 0 } ( z ) = \\sum _ { n _ { 1 } , n _ { 2 } , n _ { 3 } , n _ { 4 } \\geq 0 } c ( n _ { 1 } , n _ { 2 } , n _ { 3 } , n _ { 4 } ) z _ { 1 } ^ { n _ { 1 } } z _ { 2 } ^ { n _ { 2 } } z _ { 3 } ^ { n _ { 3 } } z _ { 4 } ^ { n _ { 4 } } \\end{align*}"}
-{"id": "9315.png", "formula": "\\begin{align*} \\min \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ n d _ { i j } x _ { i j } \\end{align*}"}
-{"id": "821.png", "formula": "\\begin{align*} \\sigma ( U _ { j } ^ { ( i ) } ) = U _ { j } ^ { ( i + 1 ) } \\qquad j = 2 , \\ldots , \\alpha . \\end{align*}"}
-{"id": "592.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\ , \\min _ { s \\in [ 0 , t ] } \\big ( Z ^ x _ s + \\tilde { \\Lambda } ^ { x , k } _ s \\big ) = \\min _ { s \\in [ 0 , t ] } \\big ( Z ^ x _ s + \\tilde { \\Lambda } ^ { x , \\infty } _ s \\big ) \\end{align*}"}
-{"id": "7648.png", "formula": "\\begin{align*} I ^ { i j } \\ = \\ \\Theta ^ { i j } + \\Theta ^ { j i } . \\end{align*}"}
-{"id": "7167.png", "formula": "\\begin{gather*} \\chi ^ { t } ( \\mathcal { M } ) = \\mathcal { M } \\quad \\forall t \\in \\mathbb { R } ; \\quad \\lim _ { t \\to \\infty } \\inf _ { \\xi \\in M } \\left \\Vert \\chi ^ { t } ( x ) - \\xi \\right \\Vert = 0 \\quad \\forall x \\in \\mathcal { D } . \\end{gather*}"}
-{"id": "7075.png", "formula": "\\begin{align*} c ( x , y ) = \\frac { u x y } { 2 ^ { k + 1 } } , \\ \\ \\ \\ \\ \\omega ( x , y , z ) = \\begin{cases} \\frac { 1 } { 2 } , & x = y = z = 2 ^ { k - 1 } , \\\\ 0 . & \\end{cases} \\end{align*}"}
-{"id": "3286.png", "formula": "\\begin{align*} \\alpha = \\frac { 3 } { 4 m ^ 2 + 1 1 m + 1 0 } , \\qquad \\beta = \\frac { 4 m + 5 } { ( 2 m + 3 ) ^ 2 } \\end{align*}"}
-{"id": "8422.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } C _ 1 | \\alpha - \\beta | \\leq | z ( \\alpha , t ) - z ( \\beta , t ) | \\leq 2 | \\alpha - \\beta | , \\\\ \\| u _ t ( \\cdot , t ) \\| _ { H ^ 1 } ^ 2 \\leq 2 \\| u _ t ( \\cdot , 0 ) \\| _ { H ^ 1 } ^ 2 , \\\\ \\inf _ { \\alpha \\in \\mathbb { R } } a | z _ { \\alpha } | ( \\alpha , t ) \\geq \\frac { 1 } { 2 } \\alpha _ 0 . \\end{align*}"}
-{"id": "5414.png", "formula": "\\begin{align*} \\| u \\| ^ 2 : & = \\langle ( - \\varDelta ) ^ { \\alpha / 2 } u , u \\rangle + \\int _ { \\R ^ N } V ( x ) u ^ 2 \\ , d x \\\\ & = c _ { N , \\alpha } \\iint _ { \\R ^ N \\times \\R ^ N } \\frac { | u ( x ) - u ( y ) | ^ 2 } { | x - y | ^ { N + \\alpha } } \\ , d x \\ , d y + \\int _ { \\R ^ N } V ( x ) u ^ 2 \\ , d x . \\end{align*}"}
-{"id": "1752.png", "formula": "\\begin{align*} \\left [ c _ { p , r } \\left ( M _ { s , t } - \\frac { \\alpha } { 2 } ( t - s ) ^ { 2 H } > \\beta \\right ) \\right ] ^ { p } \\leq \\left [ \\sum _ { l = 0 } ^ { r } \\left ( \\frac { \\alpha } { p } \\right ) ^ { l p } ( t - s ) ^ { l H p } \\right ] e ^ { - \\alpha \\beta } . \\end{align*}"}
-{"id": "9790.png", "formula": "\\begin{align*} P _ { X _ \\circ } ( X ) : = p _ { * , X _ \\circ } ( X - X _ \\circ ) . \\end{align*}"}
-{"id": "463.png", "formula": "\\begin{align*} \\ell \\circ f = \\sum _ { k = 0 } ^ { d } { k y _ j ^ { k - 1 } g _ k + y _ j ^ k \\ell ^ { \\prime } \\circ g _ k } = \\sum _ { k = 0 } ^ { d - 1 } { ( k + 1 ) y _ j ^ k g _ { k + 1 } + y _ j ^ k \\ell ^ { \\prime } \\circ g _ k } . \\end{align*}"}
-{"id": "2040.png", "formula": "\\begin{align*} \\tilde { f } _ R = f \\mid W _ R \\in S _ k ( q , \\overline { \\xi _ R } \\xi _ { q / R } ) . \\end{align*}"}
-{"id": "302.png", "formula": "\\begin{align*} \\dim V _ { S , 1 } \\ ; = \\ ; \\deg ( \\Lambda _ { S , 1 } ) \\ ; \\geq \\ ; \\deg ( \\Lambda _ { S _ 1 } ) \\ ; > \\ ; 1 . \\end{align*}"}
-{"id": "7431.png", "formula": "\\begin{align*} \\displaystyle \\lvert S _ \\pi \\rvert = 2 ^ k n _ 1 \\ldots n _ k \\end{align*}"}
-{"id": "5483.png", "formula": "\\begin{align*} \\hat \\tau \\ : = \\ \\inf \\big \\{ s \\in [ t , T ] : ( s , \\hat X _ s ^ { t , x , a } ) \\notin B ( t , x ; \\eta / 2 ) \\big \\} , \\hat \\theta \\ : = \\ \\hat \\tau \\wedge T , \\end{align*}"}
-{"id": "8479.png", "formula": "\\begin{align*} \\norm { ( I - \\mathcal { H } ) b } _ { H ^ s } \\leq & \\norm { [ D _ t \\zeta , \\mathcal { H } ] \\frac { \\bar { \\zeta } _ { \\alpha } - 1 } { \\zeta _ { \\alpha } } } _ { H ^ s } + \\norm { \\frac { i } { \\pi } \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j } { \\zeta ( \\alpha , t ) - z _ j ( t ) } } _ { H ^ s } \\\\ \\leq & C \\epsilon ^ 2 + K _ s ^ { - 1 } \\epsilon d _ I ( t ) ^ { - 3 / 2 } . \\end{align*}"}
-{"id": "2562.png", "formula": "\\begin{align*} \\inf _ { x \\in E } \\widetilde \\psi ( x , z ) \\geq \\widehat \\psi ( z ) , z \\geq 0 ; \\int _ 1 ^ \\infty \\frac { 1 } { \\widehat \\psi ( z ) } d z < \\infty ; \\widehat \\psi ( + \\infty ) = + \\infty . \\end{align*}"}
-{"id": "2913.png", "formula": "\\begin{align*} \\mu \\big ( [ u ] \\ , | \\ , \\xi ^ { A ^ \\complement } \\big ) & = \\mu \\big ( [ w ^ { - 1 } u ] \\ , | \\ , \\xi ^ { A ^ \\complement } \\big ) \\end{align*}"}
-{"id": "5864.png", "formula": "\\begin{align*} S _ n ( \\alpha , \\beta , \\gamma ) = S _ { n - 1 } ( \\alpha \\gamma , \\beta \\gamma , \\gamma ) \\frac { \\Gamma ( \\alpha ) \\Gamma ( \\beta ) \\Gamma ( \\gamma ^ n ) } { \\Gamma ( \\alpha \\beta \\gamma ^ { n - 1 } ) \\Gamma ( \\gamma ) } = \\prod _ { j = 0 } ^ { n - 1 } \\frac { \\Gamma ( \\alpha \\gamma ^ { j } ) \\Gamma ( \\beta \\gamma ^ { j } ) \\Gamma ( \\gamma ^ { j + 1 } ) } { \\Gamma ( \\alpha \\beta \\gamma ^ { n + j - 1 } ) \\Gamma ( \\gamma ) } . \\end{align*}"}
-{"id": "8810.png", "formula": "\\begin{align*} M _ t ( \\phi ) = \\langle \\overline { v } _ t , \\phi \\rangle - \\int _ { 0 } ^ { t } \\langle v ( s , \\cdot ) , \\phi '' \\rangle \\ , \\textrm { d } s , t \\leq T , \\end{align*}"}
-{"id": "6817.png", "formula": "\\begin{align*} \\sinh ^ 2 ( s ) \\theta '' - \\cosh ^ 2 ( s ) \\mu '' + 2 \\sinh ( s ) \\cosh ( s ) s ' ( \\theta ' - \\mu ' ) = 0 \\ , , \\end{align*}"}
-{"id": "736.png", "formula": "\\begin{align*} \\Gamma ( m - k ) = \\frac { ( - 1 ) ^ { k } } { ( 1 - m ) _ { k } } \\Gamma ( m ) \\end{align*}"}
-{"id": "1515.png", "formula": "\\begin{align*} M ( t , P ^ e ) = | P | ^ { e q } , \\end{align*}"}
-{"id": "7399.png", "formula": "\\begin{align*} f ( x ) = 2 + \\sin x . \\end{align*}"}
-{"id": "5641.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\frac { \\cos ( n x ) } { n ^ { 2 m } } = \\frac { ( - 1 ) ^ { m - 1 } ( 2 \\pi ) ^ { 2 m } } { 2 ( 2 m ) ! } \\ , B _ { 2 m } \\left ( \\frac { x } { 2 \\pi } \\right ) \\end{align*}"}
-{"id": "5055.png", "formula": "\\begin{align*} L _ j ^ * L _ k = \\left \\{ \\begin{array} { l l } I _ { \\mathcal { H } _ 0 } & \\mbox { i f } j = k \\\\ 0 & \\mbox { i f } j \\neq k \\end{array} \\right . ~ \\sum \\limits _ { j \\in \\mathbb { J } } L _ j L _ j ^ * = I _ { \\ell ^ 2 ( \\mathbb { J } ) } \\otimes I _ { \\mathcal { H } _ 0 } \\end{align*}"}
-{"id": "6790.png", "formula": "\\begin{align*} \\pi _ k \\left ( ( x _ 1 , \\dots , x _ n ) \\right ) = ( x _ { k + 1 } , \\dots , x _ n ) . \\end{align*}"}
-{"id": "2047.png", "formula": "\\begin{align*} f ^ \\chi = \\sum _ { \\ell \\mid { r _ { * 0 } } } \\beta _ { F _ \\chi } ( \\ell ) F _ \\chi \\mid B _ \\ell \\in S _ k ( [ q , r ^ 2 ] , \\chi ^ 2 ) . \\end{align*}"}
-{"id": "4110.png", "formula": "\\begin{align*} \\tilde { Y } ( z ) = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 1 & 0 \\end{pmatrix} Y ( z ) \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 1 & 0 \\end{pmatrix} \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & z ^ { - 1 / 2 } & 0 \\\\ 0 & 0 & z ^ { 1 / 2 } \\end{pmatrix} \\end{align*}"}
-{"id": "946.png", "formula": "\\begin{align*} J ^ { * } [ \\psi ^ { ( 5 ) } ] _ { T , t } = \\hbox { \\vtop { \\offinterlineskip \\halign { \\hfil # \\hfil \\cr { \\rm l . i . m . } \\cr $ \\stackrel { } { { } _ { p \\to \\infty } } $ \\cr } } } \\sum \\limits _ { j _ 1 , \\ldots , j _ 5 = 0 } ^ { p } C _ { j _ 5 \\ldots j _ 1 } \\zeta _ { j _ 1 } ^ { ( i _ 1 ) } \\ldots \\zeta _ { j _ 5 } ^ { ( i _ 5 ) } , \\end{align*}"}
-{"id": "8174.png", "formula": "\\begin{align*} \\theta _ q \\cdot g ( 1 _ G ) & = \\theta ( q g ) \\\\ [ 1 m m ] & = ( \\alpha ( q g ) , \\gamma ( q g ) ) . \\end{align*}"}
-{"id": "8187.png", "formula": "\\begin{align*} T ^ * S _ \\Lambda T = T ^ * T _ \\Lambda T _ \\Lambda ^ * T = T ^ * T _ \\Lambda ( T ^ * T _ \\Lambda ) ^ * = T _ \\Lambda \\mathcal { T } \\mathcal { T } ^ * T _ \\Lambda ^ * = T _ \\Lambda T _ \\Lambda ^ * = S _ \\Lambda . \\end{align*}"}
-{"id": "3917.png", "formula": "\\begin{align*} \\hat \\rho _ Z ( x ) : = Z ^ { - 2 } \\rho _ Z ( Z ^ { - 1 / 3 } x ) \\end{align*}"}
-{"id": "9293.png", "formula": "\\begin{align*} \\Theta ( \\hat x , \\hat y ) = & F ^ { i j } ( M [ u ] ( \\hat y ) ) [ D _ z A _ { i j } ( \\hat x , \\bar z , D u ( \\hat y ) ) ( u ( \\hat y ) - v ( \\hat x ) ) + R ^ A _ { i j } ( \\hat x , \\hat y ) ] \\\\ & + D _ z B ( \\hat x , \\tilde z , D u ( \\hat y ) ) ( u ( \\hat y ) - v ( \\hat x ) ) + R ^ B ( \\hat x , \\hat y ) , \\end{align*}"}
-{"id": "3893.png", "formula": "\\begin{align*} S _ \\omega ( u ( t ) , v ( t ) ) = S _ \\omega ( u _ 0 , v _ 0 ) < S _ \\omega ( \\phi _ \\omega , \\psi _ \\omega ) \\end{align*}"}
-{"id": "1990.png", "formula": "\\begin{align*} \\int _ X \\phi = \\int _ X i _ * \\left ( \\frac { i ^ * \\phi } { e ( N _ { F / X } ) } \\right ) = \\int _ F \\frac { i ^ * \\phi } { e ( N _ { F / X } ) } , \\end{align*}"}
-{"id": "7936.png", "formula": "\\begin{align*} & \\min \\{ k _ 2 , k _ 3 \\} \\geq n - d _ 1 + 2 , \\\\ & \\min \\{ k _ 1 , k _ 3 \\} \\geq n - d _ 2 + 2 , \\\\ & \\min \\{ k _ 1 , k _ 2 \\} \\geq n - d _ 3 + 2 . \\end{align*}"}
-{"id": "2854.png", "formula": "\\begin{align*} f ( x ) = x ^ 4 - i t x ^ 2 + 1 \\in \\Z _ M [ x ] \\end{align*}"}
-{"id": "6359.png", "formula": "\\begin{align*} x \\doteq J _ { i _ 0 } \\cap \\left \\{ \\begin{array} { l c l } \\cup _ { j = 1 } ^ q ( X ^ { t _ j } \\cap S ^ { t _ j } _ x ) , \\\\ X ^ { i } \\cap S ^ i _ x , \\end{array} \\right . \\end{align*}"}
-{"id": "9706.png", "formula": "\\begin{align*} I _ { \\textbf { d } } ( P _ i ) = T _ { \\textbf { d } } ( \\partial \\Omega ( P _ i ) ) + 1 . \\end{align*}"}
-{"id": "9389.png", "formula": "\\begin{align*} & \\geq \\Lambda ^ { - 2 } \\int _ { \\mathbb { R } ^ N } \\int _ { \\mathbb { R } ^ N } \\frac { | t w _ { 0 } ( x ) - t w _ { 0 } ( y ) | ^ { p } } { | x - y | ^ { N + p s } } d x d y - t ^ { p _ { s } ^ { \\ast } } \\int _ \\Omega | w _ { 0 } | ^ { p ^ { \\ast } _ { s } } d x - \\lambda \\int _ { \\Omega } f ( x , t w _ { 0 } ) t w _ { 0 } d x \\\\ & = \\Lambda ^ { - 2 } t ^ { p } [ w _ { 0 } ] _ { s , p } ^ { p } - t ^ { p _ { s } ^ { \\ast } } \\int _ \\Omega | w _ { 0 } | ^ { p ^ { \\ast } _ { s } } d x - \\lambda \\int _ { \\Omega } f ( x , t w _ { 0 } ) t w _ { 0 } d x \\end{align*}"}
-{"id": "8702.png", "formula": "\\begin{align*} d _ \\mu = d _ 0 + \\mu d _ 1 + \\mu ^ 2 d _ 2 + \\cdots \\mbox { a n d } v _ \\mu = \\mu ^ { - 1 } v _ { - 1 } + v _ 0 + \\mu v _ 1 + \\cdots . \\end{align*}"}
-{"id": "5812.png", "formula": "\\begin{align*} a _ 0 = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } f ( r e ^ { i \\theta } ) d \\theta . \\end{align*}"}
-{"id": "7317.png", "formula": "\\begin{align*} \\begin{cases} P _ 1 ( x , D ) u + V _ 1 u = V _ 3 v \\\\ P _ 2 ( x , D ) v + V _ 2 v = V _ 3 u , \\end{cases} \\end{align*}"}
-{"id": "2677.png", "formula": "\\begin{align*} \\frac { s - t } { 1 - s - t } = \\int _ { \\mathcal { C } _ { r _ 4 } } \\frac { d w } { 2 \\pi i } \\frac { 1 } { w ^ 2 - 1 } \\frac { ( w + 1 ) s - w } { 1 - ( w + 1 ) s } \\frac { ( \\frac { 1 } { w } + 1 ) s - \\frac { 1 } { w } } { 1 - ( \\frac { 1 } { w } + 1 ) s } . \\end{align*}"}
-{"id": "2550.png", "formula": "\\begin{align*} Y _ t : = \\int _ { ( 0 , t ] \\times \\mathbb W } w _ { t - s } \\mathbf n _ T ( d s , d w ) , 0 \\leq t \\leq T . \\end{align*}"}
-{"id": "3284.png", "formula": "\\begin{align*} \\sqrt { - Y } ( \\zeta ^ j \\xi + \\zeta ^ { - j } \\xi ^ { - 1 } ) = 2 \\sqrt { - Y } \\cos \\frac { ( 1 + 4 j ) \\pi } { 2 n } \\end{align*}"}
-{"id": "504.png", "formula": "\\begin{align*} \\theta = \\frac { \\bar { \\partial } b _ { p _ 1 } } { z } \\cdot \\frac { \\partial } { \\partial z } , \\end{align*}"}
-{"id": "7396.png", "formula": "\\begin{align*} G = { \\rm I s o m } ( X ) \\end{align*}"}
-{"id": "8994.png", "formula": "\\begin{align*} \\begin{aligned} & \\mbox { C a s e A } : ~ \\exists ~ t ^ * \\in [ t _ 2 , t _ 2 + \\tau + 1 ] \\mbox { s u c h t h a t } ~ v ^ k _ { j } ( t ^ * ) \\leq \\Delta ( t _ 1 ) . \\\\ & \\mbox { C a s e B } : ~ \\forall ~ t \\in [ t _ 2 , t _ 2 + \\tau + 1 ] , v ^ k _ { j } ( t ) > \\Delta ( t _ 1 ) . \\end{aligned} \\end{align*}"}
-{"id": "4432.png", "formula": "\\begin{align*} \\forall i \\in \\{ 1 , \\ldots , n \\} : \\ ; u ^ i ( 0 , x ) = u _ 0 ^ i , x \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "2819.png", "formula": "\\begin{align*} x _ i ^ * = \\frac { R _ i ^ n } { R _ 0 ^ n } > 0 , u _ i ^ * = \\frac { d _ i } { e _ i } x _ i ^ * > 0 , i = 1 , \\ldots , n . \\end{align*}"}
-{"id": "3698.png", "formula": "\\begin{align*} V ( \\widetilde { Y } ) = \\sum _ x \\sum _ { i \\in U _ x } p _ i y _ i ^ 2 / p _ x ^ 2 - \\sum _ x \\sum _ { i \\in U _ x } p _ i ^ 2 y _ i ^ 2 / p _ x ^ 2 \\end{align*}"}
-{"id": "3953.png", "formula": "\\begin{align*} f _ { 1 } ( x , m ^ { \\hat v } _ t , \\hat { v } _ { 1 } ( x , m ^ { \\hat v } _ t ) ) + D u ^ { \\hat v } _ { 1 } ( x , t ) . g _ { 1 } ( x , m ^ { \\hat v } _ t , \\hat { v } _ { 1 } ( x , m ^ { \\hat v } _ t ) ) = H _ { 1 } ( x , m ^ { \\hat v } _ t , D U _ { 1 } ( x , m ^ { \\hat v } _ t , t ) ) \\ , , \\end{align*}"}
-{"id": "6930.png", "formula": "\\begin{align*} \\dim _ K d ^ Q ( \\C ( G , e ) ) = \\sum _ { x \\in Q ^ { c l } / G } \\ell ( k C _ G ( [ x ] ) e ) . \\end{align*}"}
-{"id": "2043.png", "formula": "\\begin{align*} f \\big ( \\frac { \\alpha } { M r } + i y \\big ) = \\sum _ { m = 1 } ^ \\infty a ( m ) e ^ { 2 \\pi i m \\frac { u } { r } } e ^ { 2 \\pi i m \\frac { a } { M } + i y } = \\frac { 1 } { \\varphi ( r ) } \\sum _ { \\chi \\mod { r } } \\overline { \\chi } ( u ) \\sum _ { m = 1 } ^ \\infty a ( m ) c _ { \\chi } ( m ) e ^ { 2 \\pi i m \\left ( \\frac { a } { M } + i y \\right ) } . \\end{align*}"}
-{"id": "1987.png", "formula": "\\begin{align*} i _ j ^ * H = \\alpha _ j . \\end{align*}"}
-{"id": "849.png", "formula": "\\begin{align*} [ a _ j ^ { - 1 } S _ j , G _ { t _ k } ] _ { s a m p } = \\bigg [ \\frac { S _ j } { a _ j } , \\overline { a _ k } \\frac { G _ { t _ k } } { \\overline { a _ k } } \\bigg ] _ { s a m p } = \\frac { a _ k } { a _ j } [ S _ j , M _ k ] _ { s a m p } = \\frac { a _ k } { a _ j } \\delta _ { j , k } = \\delta _ { j , k } \\end{align*}"}
-{"id": "6093.png", "formula": "\\begin{gather*} d i m \\Bigg ( \\varphi ^ { - 1 } \\big ( ( a ^ 1 _ { 1 } , \\ldots , a ^ { k _ 1 } _ 1 ) , \\ldots , ( a ^ 1 _ j , \\ldots , a ^ { k _ j } _ j ) , \\ldots , ( a ^ 1 _ r , \\ldots , a ^ { k _ r } _ r ) \\big ) \\Bigg ) = n + r - \\sum \\limits _ { i , j } e ^ i _ j \\\\ = n + \\lvert F ( \\lambda ) \\rvert - \\sum _ { i } \\lvert R ( \\lambda ) _ i \\rvert \\end{gather*}"}
-{"id": "8720.png", "formula": "\\begin{align*} \\hat { \\nu } ( j ) = \\frac { 2 ^ { j - 1 } ( - 1 ) ^ { m + j } ( m - j + 1 ) { { N - 1 } \\choose { m } } { { m } \\choose { j - 1 } } } { ( N - j ) \\sum _ { k = 0 } ^ { m - 1 } { { N - 1 } \\choose { k } } } a n d p g f _ { \\hat { T } _ { j , N } } ( s ) = \\prod _ { k = j } ^ { N - 1 } \\left [ { ( 1 - \\frac { k - 1 } { N - 1 } ) s \\over 1 - \\frac { k - 1 } { N - 1 } s } \\right ] . \\end{align*}"}
-{"id": "8228.png", "formula": "\\begin{align*} U _ 5 ( F Z t ^ i ) = F \\Big ( ( \\sum _ { j = \\left \\lceil \\frac { i + 1 } { 5 } \\right \\rceil } ^ \\infty y _ 0 ( i , j ) t ^ j ) + \\rho ( \\sum _ { j = \\left \\lceil \\frac { i + 1 } { 5 } \\right \\rceil } ^ \\infty y _ 1 ( i , j ) t ^ j ) \\Big ) , \\end{align*}"}
-{"id": "5100.png", "formula": "\\begin{align*} \\widehat { S } _ { A , \\Psi } = I _ \\mathcal { X } \\end{align*}"}
-{"id": "2491.png", "formula": "\\begin{align*} \\varphi ( x ) = 1 \\end{align*}"}
-{"id": "8092.png", "formula": "\\begin{align*} n b _ n - ( 2 n - 1 ) b _ { n - 1 } + ( n - 2 ) b _ { n - 2 } = \\begin{cases} 1 , & \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "7538.png", "formula": "\\begin{align*} \\left \\{ \\begin{gathered} X ( a , t ) = a + \\int _ 0 ^ t \\mathcal { V } ( X , \\tau , a , s ) d s , \\\\ \\tau ( a , t ) = \\sigma _ 0 ( a ) + \\int _ 0 ^ t \\mathcal { T } ( X , \\tau , a , s ) d s , \\\\ v ( a , t ) = \\mathcal { V } ( X , \\tau , t ) \\end{gathered} \\right . \\end{align*}"}
-{"id": "2194.png", "formula": "\\begin{align*} J ^ \\ast ( \\boldsymbol { x } ( t ) , t ) = \\frac { 1 } { 2 } \\boldsymbol { x } ^ { \\ast T } ( t ) P ( t ) \\boldsymbol { x } ^ \\ast ( t ) \\end{align*}"}
-{"id": "7336.png", "formula": "\\begin{align*} \\begin{cases} \\rho ( x ) w _ { t t } - \\operatorname { d i v } ( K ( x ) \\nabla w ) = 0 & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ \\rho ( x ) z _ { t t } - \\operatorname { d i v } ( K ( x ) \\nabla z ) = 0 & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ w _ t = z _ t = 0 & ~ \\hbox { i n } ~ \\omega \\times ( 0 , T ) . \\end{cases} \\end{align*}"}
-{"id": "7079.png", "formula": "\\begin{align*} \\xi ( x , y ) ^ 2 = \\xi ( x , 2 y ) = \\xi ( x , 0 ) = 1 \\implies \\xi ( x , y ) = \\pm 1 \\end{align*}"}
-{"id": "9733.png", "formula": "\\begin{align*} W _ \\lambda ( r , u ) : = \\frac { 1 } { r ^ { 2 \\lambda } } D ( r , u ) - \\frac { \\lambda } { r ^ { 2 \\lambda } } H ( r , u ) \\end{align*}"}
-{"id": "4243.png", "formula": "\\begin{align*} d : = \\left \\lbrace \\begin{aligned} & \\frac { n p } { n p - n + p } & & 1 < p < n , \\\\ & 1 + \\varepsilon & & p \\geq n , \\varepsilon > 0 . \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "2624.png", "formula": "\\begin{align*} R _ { N } ^ { ( k ) } ( \\lambda _ 1 , \\ldots , \\lambda _ k ) : = \\frac { N ! } { ( N - k ) ! } \\int \\cdots \\int f _ { N } ( \\lambda _ 1 , \\ldots , \\lambda _ N ) d \\lambda _ { k + 1 } \\cdots d \\lambda _ { N } , \\end{align*}"}
-{"id": "6140.png", "formula": "\\begin{align*} \\kappa ^ 2 = - e s ^ 2 / 2 n = - 2 ( n a ^ 2 + b ^ 2 + r c ^ 2 ) \\quad \\textnormal { a n d } \\kappa _ * = ( 2 n a / s , 2 b / s ) \\in \\Z / 2 n \\Z \\times \\Z / 2 \\Z . \\end{align*}"}
-{"id": "2773.png", "formula": "\\begin{align*} ( \\mathbf { z } _ i ) _ j = \\left \\{ \\begin{array} { c @ { \\mbox { i f } \\quad } l } q _ { i , j } & 1 \\leq j \\leq S \\\\ e _ i & j = S + 1 , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "5519.png", "formula": "\\begin{align*} b _ u = b _ u ( T , \\gamma ) = b - \\tfrac { u } { \\sqrt { T } } \\gamma \\in V _ J ^ { \\otimes d } . \\end{align*}"}
-{"id": "366.png", "formula": "\\begin{align*} 5 5 ( | s | - 1 ) \\sqrt { | a c | } & \\geqslant 5 5 ( \\sqrt { | a c | } - 2 ) \\sqrt { | a c | } \\\\ & = 5 5 | a c | - 1 1 0 \\sqrt { | a c | } \\\\ & = 4 0 | a c | + 1 5 | a c | - 1 1 0 \\sqrt { | a c | } , \\end{align*}"}
-{"id": "8655.png", "formula": "\\begin{align*} w _ 1 = \\frac { 1 } { 8 } { \\varphi _ 2 ^ 2 + \\frac { 1 } { 4 } \\sin ^ 2 \\varphi _ 1 - \\frac { c _ 0 } { 2 } \\sin \\varphi _ 1 } + \\varepsilon h _ 1 ( \\varphi _ 1 , \\varphi _ 2 , \\varepsilon ) , \\ \\ w _ 2 = \\varepsilon h _ 2 ( \\varphi _ 1 , \\varphi _ 2 , \\varepsilon ) , \\ \\ ( \\varphi _ 1 , \\varphi _ 2 ) \\in \\mathcal { K } , \\end{align*}"}
-{"id": "9793.png", "formula": "\\begin{align*} { \\rm I I } & = \\| u ( Z _ \\circ + r \\ , \\cdot \\ , ) - p _ { * , Z _ \\circ } ( r \\ , \\cdot \\ , ) \\| _ { L ^ 2 ( B _ { 1 / 2 } ( r ^ { - 1 } ( X _ \\circ - Z _ \\circ ) ) , | y | ^ a ) } \\\\ & \\leq \\| u ( Z _ \\circ + r \\ , \\cdot \\ , ) - p _ { * , Z _ \\circ } ( r \\ , \\cdot \\ , ) \\| _ { L ^ 2 ( B _ { 1 } , | y | ^ a ) } \\leq C _ h r ^ \\lambda . \\end{align*}"}
-{"id": "7026.png", "formula": "\\begin{align*} n y ^ { 2 } = x ^ { 3 } + a n ^ { 2 } x + b n ^ { 3 } \\end{align*}"}
-{"id": "2873.png", "formula": "\\begin{align*} F f = P _ { M _ 1 } z _ 2 f , f \\in M _ 1 . \\end{align*}"}
-{"id": "7589.png", "formula": "\\begin{align*} \\begin{gathered} \\left ( \\int | f ( x , z ) | ^ p d x \\right ) ^ { \\frac { 1 } { p } } \\le \\norm { \\eta ( t ) } _ { L i p } \\norm { \\sigma ( s ) } _ { L ^ p } , \\\\ \\mathrm { e s s s u p } _ { x , z } | f ( x , z ) | \\le \\norm { \\eta ( t ) } _ { L i p } \\norm { \\sigma ( s ) } _ { L ^ \\infty } , \\\\ \\mathrm { e s s s u p } _ { x , z } | f ( x + h , z ) - f ( x , z ) | \\le | h | ^ { \\alpha } C \\norm { \\eta ( t ) } _ { C ^ { 1 + \\alpha } } \\norm { \\sigma ( s ) } _ { C ^ { \\alpha } } . \\end{gathered} \\end{align*}"}
-{"id": "616.png", "formula": "\\begin{align*} \\begin{gathered} M : \\mathcal M \\rtimes \\mathcal G \\rtimes \\mathcal G \\to \\mathcal M \\rtimes \\mathcal G \\ ; ( f , x , y ) \\mapsto ( f , x y ) \\\\ H : \\mathcal M \\to \\mathcal M \\rtimes \\mathcal G \\ ; f \\mapsto ( f , e ) \\ , \\end{gathered} \\end{align*}"}
-{"id": "3660.png", "formula": "\\begin{align*} Y = \\sum _ { i \\in U } y _ i \\end{align*}"}
-{"id": "1541.png", "formula": "\\begin{align*} u _ 2 ( x , t ) = \\frac { 1 } { \\pi } \\int _ 0 ^ t \\int _ 0 ^ \\infty \\int _ 0 ^ 1 \\left ( g _ 1 ( r , x , y ) \\cos { r \\ , ( t - s ) } - g _ 2 ( r , x , y ) \\sin { r \\ , ( t - s ) } \\right ) F ( y , s ) d y d r d s \\end{align*}"}
-{"id": "5346.png", "formula": "\\begin{align*} | | v ^ 2 | | = \\frac { 1 } { 2 } , \\quad | | v ^ 3 | | = \\frac { 1 } { \\sqrt { 6 } } , | | v ^ 4 | | = \\frac { 1 } { 2 } , \\quad | | v ^ 5 | | = 1 . \\end{align*}"}
-{"id": "1471.png", "formula": "\\begin{align*} V _ { 1 4 } = - \\varepsilon \\bar { m } \\int _ { \\mathbb { T } ^ { d } } ( \\mu ^ { 1 } - \\mu ^ { 2 } ) \\sum _ { i = 1 } ^ { d } \\sum _ { j = 1 } ^ { d } \\left [ \\Theta _ { p _ { i } p _ { j } } ( t , x , \\mu ^ { 2 } , D w ^ { 2 } ) \\left ( \\frac { \\partial ^ { 2 } ( w ^ { 1 } - w ^ { 2 } ) } { \\partial x _ { i } \\partial x _ { j } } \\right ) \\right ] \\ d x . \\end{align*}"}
-{"id": "5841.png", "formula": "\\begin{align*} S _ n ( \\alpha , \\beta , \\gamma ) = \\prod _ { j = 0 } ^ { n - 1 } \\frac { \\Gamma ( \\alpha \\gamma ^ { j } ) \\Gamma ( \\beta \\gamma ^ { j } ) \\Gamma ( \\gamma ^ { j + 1 } ) } { \\Gamma ( \\alpha \\beta \\gamma ^ { n + j - 1 } ) \\Gamma ( \\gamma ) } . \\end{align*}"}
-{"id": "1493.png", "formula": "\\begin{align*} w ^ { n + 1 } ( T , x ) = \\mathbb { P } _ { \\delta } P G ( x , m ^ { n } ( T , x ) ) . \\end{align*}"}
-{"id": "3351.png", "formula": "\\begin{align*} G _ d ( u ) : = 1 + \\sum _ { n \\ge 1 } \\Big ( \\sum _ { i = 0 } ^ { | \\mathcal { P } _ d | } \\binom { n } { i } \\Big ) \\mathrm { E } _ { \\pi \\in S _ n } f ( \\pi ) u ^ n . \\end{align*}"}
-{"id": "322.png", "formula": "\\begin{align*} \\begin{array} { c c c } a ( f _ 1 ) = ( 1 , 0 , 1 ) & a ( f _ 2 ) = ( 0 , 1 , 1 ) & a ( f _ 3 ) = ( 0 , 0 , 2 ) \\end{array} \\end{align*}"}
-{"id": "8896.png", "formula": "\\begin{align*} \\chi _ N = \\frac 1 2 \\cdot ( \\chi _ { \\{ a , b , c , d \\} } + \\chi _ { \\{ a , b \\} } + \\chi _ { \\{ c , e \\} } + \\chi _ { \\{ d , e \\} } ) . \\end{align*}"}
-{"id": "8957.png", "formula": "\\begin{align*} \\langle X , \\beta \\rangle _ K = \\sum _ { j = 1 } ^ p \\beta _ j \\langle X , k ( \\cdot , t _ j ) \\rangle _ K = \\sum _ { j = 1 } ^ p \\beta _ j ( X ( t _ j ) - m ( t _ j ) ) . \\end{align*}"}
-{"id": "1517.png", "formula": "\\begin{align*} ( q - 1 ) \\sum _ { k = 1 } ^ { n - 1 } q ^ { k } \\min \\{ e , \\lfloor \\frac { k } { d } \\rfloor \\} & = \\sum _ { k = 1 } ^ { n - 1 } \\sum _ { j = q ^ { k } } ^ { q ^ { k + 1 } - 1 } \\min \\{ e , \\lfloor \\frac { \\log _ { q } j } { d } \\rfloor \\} \\\\ & = \\sum _ { k = 1 } ^ { q ^ { n } - 1 } \\min \\{ e , \\lfloor \\frac { \\log _ { q } k } { d } \\rfloor \\} . \\end{align*}"}
-{"id": "3790.png", "formula": "\\begin{align*} r _ { 1 1 1 6 3 1 1 } - r _ { 1 1 1 3 6 1 1 } & = \\sum _ { T \\in A _ { 5 , 3 } ( \\alpha ) } s _ { ( T ) } - \\sum _ { U \\in B _ { 5 , 3 } ( \\alpha ) } s _ { ( U ) } \\\\ & = ( s _ { 6 4 2 2 } + s _ { 6 4 2 1 1 } + s _ { 6 4 1 1 1 1 } ) - ( s _ { 6 4 2 1 1 } + s _ { 6 4 1 1 1 1 } ) = s _ { 6 4 2 2 } . \\end{align*}"}
-{"id": "5093.png", "formula": "\\begin{align*} \\left \\langle \\sum _ { j \\in \\mathbb { S } } F _ j ^ * A _ j x , \\sum _ { k \\in \\mathbb { S } } F _ k ^ * A _ k x \\right \\rangle = \\sum _ { j \\in \\mathbb { S } } \\left \\langle A _ j x , F _ j \\left ( \\sum _ { k \\in \\mathbb { S } } F _ k ^ * A _ k x \\right ) \\right \\rangle = \\sum _ { j \\in \\mathbb { S } } \\langle A _ j x , A _ j x \\rangle \\end{align*}"}
-{"id": "2273.png", "formula": "\\begin{align*} & R ( z ) ^ * = \\i \\int _ 0 ^ \\infty e ^ { - \\i \\overline { z } t } T ( t ) ^ * \\d t \\\\ & R ( \\overline { z } ) = \\i \\int _ { - \\infty } ^ 0 e ^ { \\i \\overline { z } t } T ( t ) \\d t = \\i \\int _ 0 ^ \\infty e ^ { - \\i \\overline { z } t } T ( - t ) \\d t . \\end{align*}"}
-{"id": "7457.png", "formula": "\\begin{align*} \\displaystyle N ( \\pi ) ^ k \\left \\lvert D ( \\lambda \\mapsto ( R ( u ) L ( v ) f _ { \\pi _ \\lambda } ) ( g ) ) _ { \\lambda = 0 } \\right \\rvert \\ll \\Xi ^ G ( g ) \\sigma ( g ) ^ { \\deg ( D ) } \\end{align*}"}
-{"id": "356.png", "formula": "\\begin{align*} - \\Delta ( u _ { n } - u _ { \\ast } ) = \\frac { a ( \\alpha _ { n } ) - a ( \\alpha _ { \\ast } ) } { a ( \\alpha _ { n } ) } \\Delta u _ { \\ast } + \\frac { f _ { \\ast } ( u _ { n } ) - f _ { \\ast } ( u _ { \\ast } ) } { a ( \\alpha _ { n } ) } = : g _ { n } ( x ) , \\ \\forall \\ n \\in \\N . \\end{align*}"}
-{"id": "2195.png", "formula": "\\begin{align*} \\dot { P } ( t ) + Q ( t ) + P ( t ) A ( t ) + A ^ { T } ( t ) P ( t ) - P ( t ) B ( t ) R ^ { - 1 } ( t ) B ^ { T } ( t ) P ( t ) = 0 \\end{align*}"}
-{"id": "1739.png", "formula": "\\begin{align*} R ( t , s ) = \\mathbb { E } \\left [ B _ { t } B _ { s } \\right ] = \\frac { 1 } { 2 } \\left ( t ^ { 2 H } + s ^ { 2 H } - | t - s | ^ { 2 H } \\right ) . \\end{align*}"}
-{"id": "7502.png", "formula": "\\begin{align*} d = 1 - \\frac { 2 } { h } , \\end{align*}"}
-{"id": "8746.png", "formula": "\\begin{align*} ( y ^ { ( \\ell ) } ) ' ( t ) = \\lambda y ^ { ( \\ell ) } ( t ) + u ( t ) , { \\quad } y ^ { ( \\ell ) } ( b ^ { ( n ) } _ { \\ell } \\tau ) = 0 . \\end{align*}"}
-{"id": "9348.png", "formula": "\\begin{align*} G _ k ^ { \\ , 2 ^ k } \\cong C _ 2 \\times C _ 2 \\times C _ 2 , \\log _ 2 \\vert G _ k : G _ k ^ { \\ , 2 ^ k } \\rvert = \\log _ 2 \\lvert G _ k \\rvert - 3 \\end{align*}"}
-{"id": "1823.png", "formula": "\\begin{align*} \\phi ^ { \\rm e v e n } ( j , x , t ) = ( \\kappa _ j L ) ^ { - 1 / 2 } A _ j \\left [ \\cos ( \\kappa _ j x ) + \\frac { \\xi } { \\kappa _ j } \\sin ( \\kappa _ j | x | ) \\right ] \\ , e ^ { - i \\kappa _ j t } , \\end{align*}"}
-{"id": "1272.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { | \\xi | ^ { - \\gamma } } | \\xi | ^ { - 3 } | \\eta | ^ { - 1 / 2 } d \\eta = O ( | \\xi | ^ { - 3 + \\gamma / 2 } ) \\end{align*}"}
-{"id": "3713.png", "formula": "\\begin{align*} \\eta = ( \\bar { y } _ B - \\widehat { \\bar { Y } _ B ^ c } ) ^ 2 / \\widehat { V } ( \\widehat { \\bar { Y } _ B ^ c } ) \\end{align*}"}
-{"id": "5181.png", "formula": "\\begin{align*} \\underset { 1 / n \\leq s \\leq 1 - 1 / n } { \\sup } \\frac { \\left \\vert \\sqrt { n } ( U _ { n } ( s ) - s ) - B _ { n } ( s ) \\right \\vert } { ( s ( 1 - s ) ) ^ { 1 / 2 - \\nu } } = O _ { p } ( n ^ { - \\nu } ) \\end{align*}"}
-{"id": "6030.png", "formula": "\\begin{align*} X ( t ) = B ( t ) + A ( t ) = B ( t ) + \\int _ 0 ^ t q ( s ) d s , \\end{align*}"}
-{"id": "246.png", "formula": "\\begin{align*} \\int _ { B _ r } u ^ 2 \\psi = O ( e x p ( - A r ^ { - \\alpha } ) ) , \\ \\end{align*}"}
-{"id": "2112.png", "formula": "\\begin{align*} \\Phi _ \\omega ( s , t ) E _ k ^ s ( \\omega ) = E _ k ^ t ( \\omega ) , \\qquad \\hbox { f o r a l l } \\ ; s , t \\in [ t _ 0 , \\infty ) , k = 1 , \\ldots , d . \\end{align*}"}
-{"id": "5180.png", "formula": "\\begin{align*} \\phi ^ { - 1 } ( 1 - s ) = \\left \\{ ( 2 \\log ( 1 / s ) ) ^ { 1 / 2 } - \\frac { \\log 4 \\pi + \\log \\log ( 1 / s ) } { 2 ( 2 \\log ( 1 / s ) ) ^ { 1 / 2 } } + O ( ( \\log \\log ( 1 / s ) ^ { 2 } ( \\log 1 / s ) ^ { - 1 / 2 } ) ) \\right \\} . \\end{align*}"}
-{"id": "3722.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { f ( x , t ) t } { ( f ( x , t ) ) ^ r } = 0 \\ ; x \\in \\Omega , \\ ; \\ ; r > 1 . \\end{align*}"}
-{"id": "8831.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathbb R ^ n } \\sum _ { i = 1 } ^ m \\log ( 1 + \\exp ( b _ i ( A x ) _ i ) ) + \\frac { \\lambda } { 2 } \\norm { x } _ 2 ^ 2 \\end{align*}"}
-{"id": "3207.png", "formula": "\\begin{align*} \\| \\mathcal { L } _ { X _ r } \\mathcal { L } _ { Y _ r } \\mathcal { L } _ { X _ r } ^ { - 1 } - \\mathcal { L } _ { Y _ r } \\| _ 2 & \\le \\| \\mathcal { L } _ { Y _ r } \\mathcal { L } _ { X _ r } - \\mathcal { L } _ { X _ r } \\mathcal { L } _ { Y _ r } \\| _ 2 \\| \\mathcal { L } _ { X _ r } ^ { - 1 } \\| _ 2 \\\\ & \\le \\mu _ r \\| \\mathcal { L } _ { X _ r } ^ { - 1 } \\| _ 2 \\frac { \\| X _ r \\circ Y _ r - \\mu _ r I \\| _ F } { \\mu _ r } \\\\ & = \\frac { \\mu _ r } { \\lambda _ r } \\frac { \\| X _ r \\circ Y _ r - \\mu _ r I \\| _ F } { \\mu _ r } , \\end{align*}"}
-{"id": "5925.png", "formula": "\\begin{align*} { \\rm c o e f f } _ D ( P ' ) = \\left \\{ \\begin{array} { l l } 0 & \\mbox { i f } D = S \\\\ - { \\rm c o e f f } _ D \\lfloor \\Gamma ' - D ' + \\{ ( n + 1 ) \\Delta \\} \\rfloor & \\end{array} \\right . \\end{align*}"}
-{"id": "3538.png", "formula": "\\begin{align*} \\mathcal { M } _ { S e c P } ^ { L o c } = \\mathrm { S p e c } \\ , \\mathbb { C } [ C _ { N E } \\cap L ] \\simeq \\left \\{ ( x _ { i j } ) \\in \\mathbb { A } ^ { 2 \\times 3 } \\mid \\mathrm { r k } \\left ( \\begin{matrix} x _ { 1 1 } & x _ { 1 2 } & x _ { 1 3 } \\\\ x _ { 2 1 } & x _ { 2 2 } & x _ { 2 3 } \\end{matrix} \\right ) \\leq 1 \\right \\} , \\end{align*}"}
-{"id": "4554.png", "formula": "\\begin{align*} k , k + 1 , \\ldots , k + p , k ' & \\leq k , k + 1 , \\ldots , k + p , k \\\\ & < k , k + 1 , \\ldots , k + p , k + p + 1 \\\\ & \\leq k , k + 1 , \\ldots , k + d _ 1 \\\\ & = { \\bf i } ^ { ( 1 ) } \\end{align*}"}
-{"id": "5491.png", "formula": "\\begin{align*} e ^ { \\ell _ T ( b ) } = \\frac { d P _ b ^ T } { d P _ 0 ^ T } ( X ^ T ) = \\exp \\Big ( - \\frac { 1 } { 2 } \\int _ 0 ^ T \\| b ( X _ t ) \\| ^ 2 d t + \\int _ 0 ^ T b ( X _ t ) . d X _ t \\Big ) , \\end{align*}"}
-{"id": "675.png", "formula": "\\begin{align*} \\varpi ( g _ 1 f _ 1 , \\dots , g _ n f _ n ) = \\varpi ( g _ 1 , \\dots , g _ n ) \\circ \\varpi ( f _ 1 , \\dots , f _ n ) \\ . \\end{align*}"}
-{"id": "8108.png", "formula": "\\begin{align*} R : = \\sum _ { i = 1 } ^ g ( m _ i - i ) = \\sum _ { i = 1 } ^ g m _ i - \\binom { g + 1 } { 2 } \\end{align*}"}
-{"id": "4303.png", "formula": "\\begin{align*} Y ' = Y \\backslash \\{ y _ k \\} . \\end{align*}"}
-{"id": "1995.png", "formula": "\\begin{align*} f : L ^ { \\otimes 5 } \\stackrel { \\cong } { \\longrightarrow } \\omega _ { C , \\log } : = \\omega _ C \\left ( \\sum _ i 5 [ q _ i ] \\right ) , \\end{align*}"}
-{"id": "4129.png", "formula": "\\begin{align*} \\varphi ( z ) = \\begin{cases} - \\pi i + 3 \\pi i c _ { 0 , V } z ^ { \\frac { 1 } { 3 } } + \\mathcal { O } \\left ( z ^ \\frac { 2 } { 3 } \\right ) , & , \\\\ [ 5 p t ] \\pi i - 3 \\pi i c _ { 0 , V } z ^ { \\frac { 1 } { 3 } } + \\mathcal { O } \\left ( z ^ \\frac { 2 } { 3 } \\right ) , & , \\end{cases} \\end{align*}"}
-{"id": "3186.png", "formula": "\\begin{align*} \\inf \\sigma ( H ) \\setminus \\{ 0 \\} = \\inf \\sigma ( 1 - T ^ 2 ) \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "2730.png", "formula": "\\begin{align*} \\mathcal { C } _ { + } ^ { \\mathrm { l o c a l } } & = \\left \\{ z = z _ { 1 } + r e ^ { i \\frac { 2 \\pi } { 3 } } : 0 \\leq r < \\eta _ { - } ^ { - 1 } \\delta \\right \\} , \\\\ \\mathcal { C } _ { + } ^ { 1 } & = \\left \\{ z = \\Re { z _ { 2 } } + i y : \\Im { z _ { 2 } } < y < \\Im { z _ { 3 } } \\right \\} , \\\\ \\mathcal { C } _ { + } ^ { 2 } & = \\left \\{ z \\in \\mathcal { C } _ { \\tau , 0 } : 1 < \\Re { z } < \\Re { z _ { 2 } } ~ ~ \\Im z \\geq 0 \\right \\} . \\end{align*}"}
-{"id": "7517.png", "formula": "\\begin{align*} \\mathfrak { S } ( \\xi , \\eta ) : = \\sum _ { i , j , k , l , p , q , r , s } ( c _ { i j , p } c ^ { p , q } c _ { q , r s } - c _ { i j , r s } ) c ^ { r , k } c ^ { s , l } \\xi ^ i \\xi ^ j \\eta ^ k \\eta ^ l \\geq 0 \\end{align*}"}
-{"id": "1878.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\frac { A _ j ^ 2 } { Z _ j ^ 2 - ( \\pi m ) ^ 2 } = \\sum _ { j = 1 } ^ \\infty \\frac { A _ j ^ 2 } { Z _ j ^ 2 } . \\end{align*}"}
-{"id": "5297.png", "formula": "\\begin{align*} \\langle { \\bf f } , { \\bf g } \\rangle = { \\bf f } ^ T J { \\bf g } , J = \\begin{bmatrix} I _ n & 0 \\\\ 0 & - I _ K \\end{bmatrix} , \\end{align*}"}
-{"id": "3070.png", "formula": "\\begin{align*} T = T _ 1 + T _ 2 . \\end{align*}"}
-{"id": "678.png", "formula": "\\begin{align*} \\theta \\circ \\widehat { [ \\rho , x ] } _ X = \\widehat { [ \\rho , x ] \\mathrlap ' } _ X \\ . \\end{align*}"}
-{"id": "3908.png", "formula": "\\begin{align*} \\begin{cases} x p + z q = a \\\\ y p + ( 1 - x ) q = b \\end{cases} \\end{align*}"}
-{"id": "7337.png", "formula": "\\begin{align*} \\begin{cases} \\rho ( x ) \\varphi _ { t t } - \\operatorname { d i v } ( K ( x ) \\nabla \\varphi ) = 0 & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ \\rho ( x ) \\psi _ { t t } - \\operatorname { d i v } ( K ( x ) \\nabla \\psi ) = 0 & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ \\varphi = \\psi = 0 & ~ \\hbox { i n } ~ \\omega \\times ( 0 , T ) . \\end{cases} \\end{align*}"}
-{"id": "8566.png", "formula": "\\begin{align*} \\{ v , v ' \\} = v v ' + v ' v = B ( v , v ' ) , \\end{align*}"}
-{"id": "7881.png", "formula": "\\begin{align*} \\xi ' _ i = \\begin{cases} \\xi _ i & i \\not \\in V _ { \\xi } , \\\\ \\xi _ i - \\frac { 1 } { 2 } & i \\in V _ { \\xi } \\frac { 1 } { 4 } < \\xi _ i \\le \\frac { 1 } { 2 } , \\\\ \\xi _ i + \\frac { 1 } { 2 } & i \\in V _ { \\xi } - \\frac { 1 } { 2 } \\le \\xi _ i < - \\frac { 1 } { 4 } , \\end{cases} \\end{align*}"}
-{"id": "3609.png", "formula": "\\begin{align*} B : = A \\cup \\{ v _ { ( 1 , 2 ) } , v _ { ( 1 , 3 ) } , v _ { ( 2 , 3 ) } \\} \\cup \\{ a _ 1 \\cdot v _ { ( 2 , 3 ) } , a _ 2 \\cdot v _ { ( 1 , 3 ) } , a _ 3 \\cdot v _ { ( 1 , 2 ) } \\} \\end{align*}"}
-{"id": "6037.png", "formula": "\\begin{align*} P ( d _ 1 , d _ 2 , \\dots , d _ n ) & = \\sum _ { i = 2 } ^ n \\min \\{ d _ 1 , d _ i \\} + \\sum _ { i = 2 } ^ { n - 1 } \\min \\{ d _ i , d _ n \\} + P ( d _ 2 , \\dots , d _ { n - 1 } ) \\\\ & = ( n - 1 ) d _ 1 + \\sum _ { i = 2 } ^ { n - 1 } d _ i + P ( d _ 2 , \\dots , d _ { n - 1 } ) . \\end{align*}"}
-{"id": "1203.png", "formula": "\\begin{align*} \\tilde { P } ( \\sigma , x , z ' ) & = \\sqrt { a _ 0 ^ 2 - \\sin ^ 2 \\theta + \\sigma _ 1 ^ 2 \\cos 2 \\theta + \\frac { ( 1 - a _ 0 ^ 2 ) \\sigma _ 2 ^ 2 } { 1 + \\sigma _ 2 ^ 2 } + i \\sigma _ 1 \\sqrt { 1 + \\sigma _ 1 ^ 2 } \\sin 2 \\theta } . \\end{align*}"}
-{"id": "6847.png", "formula": "\\begin{align*} N _ { 1 2 } = \\hat f _ 1 g _ 1 + \\hat f _ 4 g _ 4 + \\hat f _ 5 g _ 5 \\ , , \\end{align*}"}
-{"id": "1188.png", "formula": "\\begin{align*} Z ( 0 ) = Z ( 1 ) . \\end{align*}"}
-{"id": "8960.png", "formula": "\\begin{align*} A = \\{ \\mbox { f o r a n y g i v e n } t \\mbox { t h e r e e x i s t s } 0 < t _ 0 < t \\mbox { s . t . } \\mbox { s i g n } ( B _ j ( t _ 0 ) ) = s _ j , \\ j = 1 , \\ldots , n \\} \\end{align*}"}
-{"id": "9099.png", "formula": "\\begin{align*} v _ k ( x ) = \\min \\{ k , u _ + ( x ) \\} . \\end{align*}"}
-{"id": "3161.png", "formula": "\\begin{align*} \\varGamma U \\varGamma = U ^ { - 1 } \\end{align*}"}
-{"id": "7380.png", "formula": "\\begin{align*} m _ R ( R ) = m _ { R 0 } + \\frac { m _ { R 0 } } { c _ { R M } } \\max \\left \\{ \\frac { B C F _ { C W } c _ r } { 1 + B C F _ { C W } R } - c _ { R M 0 } , 0 \\right \\} . \\end{align*}"}
-{"id": "6354.png", "formula": "\\begin{align*} \\mathbb { P } _ { e , D | p , h } = Q \\left ( \\frac { R P _ t \\sqrt { T _ s } \\sum _ { i = 1 } ^ 2 \\sum _ { j = 1 } ^ 2 g _ { i j } h _ { i j } } { \\sqrt { 2 N _ 0 } } \\right ) . \\end{align*}"}
-{"id": "2398.png", "formula": "\\begin{align*} \\mathfrak { W } & \\cong \\mathfrak { W } ^ { \\otimes r } \\otimes \\mathbb { C } [ x ] ^ { \\otimes ( n - 2 r ) } \\\\ \\gamma _ Q & = \\gamma _ { Q ( \\alpha _ 1 ) } \\otimes \\cdots \\otimes \\gamma _ { Q ( \\alpha _ r ) } \\otimes ( \\gamma _ 1 ) ^ { \\otimes ( p - 2 r ) } \\otimes ( \\gamma _ 0 ) ^ { \\otimes ( n - p ) } . \\end{align*}"}
-{"id": "3304.png", "formula": "\\begin{align*} \\frac { 1 } { x } \\sum _ { n \\le x } \\Delta ( n ) = \\Omega ( \\log \\log x ) . \\end{align*}"}
-{"id": "6435.png", "formula": "\\begin{align*} \\frac { \\tau _ { k } } { \\gamma _ { k } } = \\left [ 1 + \\delta _ { k } ( 1 + \\delta _ { k - 1 } ( 1 + \\dots + \\delta _ { 2 } ( 1 + \\delta _ { 1 } ) \\dots ) ) \\right ] = \\frac { \\left \\Vert p _ { k } \\right \\Vert ^ { 2 } } { \\left \\Vert r _ { k } \\right \\Vert ^ { 2 } } . \\end{align*}"}
-{"id": "7579.png", "formula": "\\begin{align*} \\begin{gathered} | ( \\nabla S ) ( x ) | \\le \\frac { C _ d } { ( 1 + 2 C ^ 2 T + | x | ^ 2 ) ^ d } \\end{gathered} \\end{align*}"}
-{"id": "6667.png", "formula": "\\begin{align*} \\zeta ( { \\sc \\frac { 1 } { 2 } } + i \\lambda _ { * } ) = 0 , \\end{align*}"}
-{"id": "5664.png", "formula": "\\begin{align*} D [ u ( t ) ] ( \\sigma ) & = \\left \\{ \\begin{array} { l l } D [ E ( t ) u _ { 0 } ] ( \\sigma ) + D [ \\int _ { 0 } ^ { t } \\bar { E } ( t - s ) \\ , \\d W ( s ) ] ( \\sigma ) , & \\sigma \\leq t \\leq T , \\\\ 0 , & 0 < t < \\sigma , \\end{array} \\right . \\\\ & = \\left \\{ \\begin{array} { l l } \\bar { E } ( t - \\sigma ) , & \\sigma \\leq t \\leq T , \\\\ 0 , & 0 < t < \\sigma . \\end{array} \\right . \\end{align*}"}
-{"id": "464.png", "formula": "\\begin{align*} 0 = \\ell \\circ f = \\ell \\circ ( \\sum _ { k = 0 } ^ { a _ i } { y ^ { k } _ i g _ k } ) = \\sum _ { k = 0 } ^ { a _ i } k y ^ { k - 1 } _ i g _ k + y ^ k _ i \\ell ^ \\prime \\circ g _ k \\end{align*}"}
-{"id": "4230.png", "formula": "\\begin{align*} c ( w , w ^ { \\prime } ) = \\deg \\left ( \\Phi _ { w ^ { \\prime } } ^ { - 1 } \\circ \\Psi _ { w ^ { \\prime } } \\right ) ( - 1 ) ^ { I } ( 1 + ( - 1 ) ^ { \\kappa ( w , w ' ) } ) \\end{align*}"}
-{"id": "3930.png", "formula": "\\begin{align*} \\sum y \\cdot \\lambda ( S n _ 1 ) \\otimes y \\cdot n _ 0 = & \\ \\sum \\lambda \\bigl ( S ( n _ 1 \\cdot y ^ { - 1 } ) \\bigr ) \\otimes y \\cdot n _ 0 = \\\\ & \\ \\sum \\lambda ( S n _ 1 ) \\otimes n _ 0 \\ , . \\end{align*}"}
-{"id": "8257.png", "formula": "\\begin{align*} \\begin{cases} \\begin{rcases} & v _ t + v \\cdot \\nabla v = - \\nabla P - ( 0 , 1 ) \\\\ & d i v ~ v = 0 , ~ ~ c u r l ~ v = 0 \\\\ \\end{rcases} \\Omega ( t ) \\\\ P \\equiv 0 \\quad \\quad \\quad \\quad \\quad ~ ~ \\quad \\Sigma ( t ) \\\\ ( 1 , v ) ( t , \\Sigma ( t ) ) . \\end{cases} \\end{align*}"}
-{"id": "5761.png", "formula": "\\begin{align*} \\mathcal { F } \\left ( ( x , g ) ^ { - 1 } \\cdot f \\right ) ( x _ 1 , x _ 3 ) & = \\sum _ { x _ 2 } \\varpi ( \\langle x _ 2 , x _ 3 \\rangle ) ( ( x , g ) ^ { - 1 } \\cdot f ) ( x _ 1 , x _ 2 ) \\\\ & = \\sum _ { x _ 2 } \\varpi ( \\langle x _ 2 , x _ 3 \\rangle ) f ( g \\cdot x _ 1 , \\varrho _ \\R ( x ) ( g \\cdot x _ 1 ) + g \\cdot x _ 2 ) \\end{align*}"}
-{"id": "2962.png", "formula": "\\begin{align*} \\theta _ 1 & = \\alpha _ 1 \\nabla _ 1 + \\alpha _ 2 \\nabla _ 2 + \\alpha _ 3 \\nabla _ 3 , \\\\ \\theta _ 2 & = \\beta _ 1 \\nabla _ 1 + \\beta _ 2 \\nabla _ 2 . \\end{align*}"}
-{"id": "8477.png", "formula": "\\begin{align*} S ( n ) = \\sum _ { k = 1 } ^ n \\sum \\frac { n ! } { ( k _ 1 ) ! . . . ( k _ n ) ! } \\prod _ { l = 1 } ^ n \\frac { 1 } { ( l ! ) ^ { k _ l } } \\end{align*}"}
-{"id": "802.png", "formula": "\\begin{align*} ( A _ d / P ^ \\alpha ) = A _ d / P \\oplus P / P ^ 2 \\oplus P ^ 2 / P ^ 3 \\oplus \\cdots \\oplus P ^ { \\alpha - 1 } / P ^ \\alpha . \\end{align*}"}
-{"id": "6363.png", "formula": "\\begin{align*} \\int _ X \\dot { \\varphi } _ t g d V \\leq C : = C ' \\mu ( X ) + \\mu ( X ) \\log V _ 2 - \\mu ( X ) \\log \\mu ( X ) . \\end{align*}"}
-{"id": "2421.png", "formula": "\\begin{align*} \\begin{cases} U U ^ * = U ^ * U = I , \\\\ ( F \\overline { U } F ^ { - 1 } ) ^ * F \\overline { U } F ^ { - 1 } = I \\end{cases} \\Leftrightarrow \\begin{cases} U U ^ * = U ^ * U = I , \\\\ U ^ t F ^ * F \\overline { U } = F ^ * F , \\end{cases} \\end{align*}"}
-{"id": "2130.png", "formula": "\\begin{align*} { \\rm ( { \\textbf H } _ 2 ^ \\prime ) } \\| C \\| _ { \\infty , \\R } = \\sup _ { t \\in \\R } | C ( t ) | < \\infty \\lim \\limits _ { \\delta \\to 0 } \\sup _ { - \\infty < s < t < \\infty , | t - s | \\leq \\delta } \\frac { | C ( t ) - C ( s ) | } { | t - s | ^ \\alpha } = 0 . \\end{align*}"}
-{"id": "315.png", "formula": "\\begin{align*} ( n ( n - 1 ) , 0 , 0 ) = ( s B _ 2 + t B _ 3 ) \\Phi ( L ' _ 1 ) - s \\Phi ( L ' _ 2 ) - t \\Phi ( L ' _ 3 ) . \\end{align*}"}
-{"id": "3868.png", "formula": "\\begin{align*} P _ i ( x _ 1 , \\dotsc , x _ n ) = 0 , \\ ; \\ ; \\ ; \\ ; 1 \\leq i \\leq k , \\end{align*}"}
-{"id": "4267.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\delta ( a ( x ) \\lvert d u \\rvert ^ { p - 2 } d u ) ) & = \\delta F & & B _ { R } , \\\\ \\delta u & = 0 & & B _ { R } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "7978.png", "formula": "\\begin{align*} | F _ 4 ( q ) | & = q ^ { 2 4 } ( q ^ { 1 2 } - 1 ) ( q ^ 8 - 1 ) ( q ^ 6 - 1 ) ( q ^ 2 - 1 ) \\\\ & = q ^ { 2 4 } ( q - 1 ) ^ 4 ( q + 1 ) ^ 4 ( q ^ 2 + 1 ) ^ 2 ( q ^ 2 - q + 1 ) ^ 2 ( q ^ 2 + q + 1 ) ^ 2 ( q ^ 4 + 1 ) ( q ^ 4 - q ^ 2 + 1 ) \\ , , \\end{align*}"}
-{"id": "2431.png", "formula": "\\begin{align*} f ( x _ 0 , x _ 1 ) = x _ 0 ^ n + \\sum _ { i = d } ^ n A _ i x _ 0 ^ { n - i } x _ 1 ^ i \\ ( A _ d \\neq 0 ) , \\end{align*}"}
-{"id": "2608.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\big \\| f _ n \\big \\| _ { \\mu ; p } = \\Big ( \\limsup _ { n \\to \\infty } \\int f ^ p _ n d \\mu \\Big ) ^ { \\frac { 1 } { p } } \\leq \\Big ( \\int \\limsup _ { n \\to \\infty } f ^ p _ n d \\mu \\Big ) ^ { \\frac { 1 } { p } } = \\big \\| \\limsup _ { n \\to \\infty } f _ n \\big \\| _ { \\mu ; p } . \\end{align*}"}
-{"id": "6829.png", "formula": "\\begin{align*} N _ { i j } = & y _ { i ; j } - y _ { j ; i } = u ^ \\ell _ { ; i } B _ { \\ell k } u ^ k _ { ; j } - u ^ \\ell _ { ; j } B _ { \\ell k } u ^ k _ { ; i } \\\\ = & B _ { \\ell k } \\big ( u ^ \\ell _ { ; i } u ^ k _ { ; j } - u ^ \\ell _ { ; j } u ^ k _ { ; i } \\big ) = u ^ \\ell _ { ; i } \\big ( B _ { \\ell k } - B _ { k \\ell } \\big ) u ^ k _ { ; j } \\ , . \\end{align*}"}
-{"id": "7205.png", "formula": "\\begin{align*} \\| f \\| _ { B ^ s _ { \\infty , \\infty } } = \\sup _ { j \\geq 0 } 2 ^ { s j } \\left \\| \\sum _ { k \\in \\Z } e ^ { i k ( \\cdot ) } \\varphi _ j ( k ) \\hat f ( k ) \\right \\| _ \\infty < \\infty \\end{align*}"}
-{"id": "6718.png", "formula": "\\begin{align*} q _ i = c _ i \\cos \\omega _ i t + c ' _ i \\sin \\omega _ i t , \\end{align*}"}
-{"id": "10040.png", "formula": "\\begin{align*} H ( q , p , t ) = \\frac { 1 } { 2 } \\| p \\| ^ 2 - U ( q , t ) , ( q , p ) \\in \\R ^ 2 \\times \\R ^ 2 , \\end{align*}"}
-{"id": "8609.png", "formula": "\\begin{align*} \\frac { \\omega } { \\Lambda - 1 } = c _ 0 + \\tilde { \\omega } \\left ( \\sqrt { \\Lambda - 1 } \\right ) \\end{align*}"}
-{"id": "8584.png", "formula": "\\begin{align*} \\mu = [ ( i _ 1 , j _ 1 ) , \\ldots , ( i _ p , j _ p ) ] \\mapsto ( [ i _ 1 , \\ldots , i _ p ] , [ j _ 1 , \\ldots , j _ p ] ) = ( \\alpha ( \\mu ) , \\beta ( \\mu ) ) . \\end{align*}"}
-{"id": "2623.png", "formula": "\\begin{align*} \\eta _ { + } = \\frac { 1 } { 1 - \\tau ^ 2 } , \\eta _ { - } = \\frac { \\tau } { 1 - \\tau ^ 2 } , \\tau \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "4965.png", "formula": "\\begin{align*} F U ( ( A _ i ) _ { i \\in \\Z } ) = \\left \\{ \\bigcup _ { i \\in I } A _ i \\ , : \\ , I \\subseteq \\Z , 0 < \\vert I \\vert < \\infty \\right \\} \\end{align*}"}
-{"id": "4412.png", "formula": "\\begin{align*} \\omega _ o | _ { \\Sigma } = \\Psi ^ * \\omega _ o = d d ^ c ( | s | ^ 2 + | t | ^ 2 + | s ^ 2 t ^ 2 | ) , \\end{align*}"}
-{"id": "8664.png", "formula": "\\begin{align*} R _ i ( \\l ) = R _ i ( \\l _ * ) \\frac { \\l - \\l ^ { ( i ) } _ 0 } { \\l _ * - \\l ^ { ( i ) } _ 0 } \\sqrt { \\frac { \\l _ * } { \\l } } \\prod _ { j \\ge 1 } \\frac { ( \\l - \\l ^ { ( i ) } _ j ) \\sqrt { ( \\l _ * - a _ j ) ( \\l _ * - b _ j ) } } { ( \\l _ * - \\l ^ { ( i ) } _ j ) \\sqrt { ( \\l - a _ j ) ( \\l - b _ j ) } } , \\end{align*}"}
-{"id": "1113.png", "formula": "\\begin{align*} & B ( u , v ) = - \\frac { 2 \\pi } { n } K _ { ( 0 , n \\delta _ n ) } ^ { C U E ( n ) } \\left ( \\frac { 2 \\pi } { n } u , \\frac { 2 \\pi } { n } v \\right ) . \\end{align*}"}
-{"id": "686.png", "formula": "\\begin{align*} \\xi _ { \\vec a } = \\lambda ^ { ( K ) } _ { \\vec a } \\circ [ \\mathrm { i d } ; \\xi _ { a _ 1 } , \\dots , \\xi _ { a _ n } ; e _ n ] \\circ \\lambda ^ { ( H ) - 1 } _ { \\vec a } \\ , \\end{align*}"}
-{"id": "3111.png", "formula": "\\begin{align*} T _ L = N _ L \\ , . \\end{align*}"}
-{"id": "3039.png", "formula": "\\begin{align*} \\Psi _ { x } ( \\lambda ) & = \\sum \\limits _ { j = 1 } ^ { d } 2 c _ { j } x _ { j } \\ 1 _ { \\R _ { + } } ( x _ { j } ) \\lambda _ { j } ^ { 2 } + \\int \\limits _ { \\R _ { + } ^ { d } } \\left ( 1 - e ^ { i \\lambda \\cdot z } \\right ) \\nu ( d z ) \\\\ & \\ \\ \\ + \\sum \\limits _ { j = 1 } ^ { d } \\ 1 _ { \\R _ { + } } ( x _ { j } ) x _ { j } \\int \\limits _ { | z | \\leq 1 } \\left ( 1 + i \\lambda \\cdot z - e ^ { i \\lambda \\cdot z } \\right ) \\mu _ { j } ( d z ) . \\end{align*}"}
-{"id": "585.png", "formula": "\\begin{align*} Z ^ x _ s = Z _ 0 ^ x + \\int _ 0 ^ s \\alpha ^ x _ r \\ , \\mathrm { d } r + \\int _ 0 ^ s \\sigma ^ x _ r \\ , \\mathrm { d } B _ r , \\ ; \\ ; s \\in [ 0 , T ] , x \\in \\mathcal { X } , \\end{align*}"}
-{"id": "4993.png", "formula": "\\begin{align*} \\begin{cases} 1 < p < Q / ( 2 - Q ) , & 1 < Q < 2 , \\\\ p > Q / 2 , & Q \\ge 2 , \\end{cases} \\end{align*}"}
-{"id": "5807.png", "formula": "\\begin{align*} \\Omega _ t = \\bigcup _ { s < t } \\Omega _ s \\ \\ \\ \\ \\ \\ \\Omega _ t = i n t \\Bigl ( \\bigcap _ { u > t } \\Omega _ u \\Bigr ) _ 0 . \\end{align*}"}
-{"id": "1982.png", "formula": "\\begin{align*} H _ T ^ * ( \\P ^ n ) = \\frac { \\Q [ H , \\alpha _ 0 , \\dots , \\alpha _ n ] } { ( H - \\alpha _ 0 ) \\cdots ( H - \\alpha _ n ) } . \\end{align*}"}
-{"id": "6882.png", "formula": "\\begin{align*} \\psi ( s , \\mu , \\theta ) = \\cosh ( s ) R _ 1 + \\sinh ( s ) R _ 2 \\ , , \\end{align*}"}
-{"id": "7031.png", "formula": "\\begin{align*} \\mathbb { F ( } z ) = \\mathbb { F } _ { \\alpha _ { j } , \\beta _ { j } , \\lambda _ { j } , \\zeta } ( z ) = \\left \\{ \\zeta \\int \\limits _ { 0 } ^ { z } t ^ { \\zeta - 1 } \\prod _ { j = 1 } ^ { n } \\left ( \\frac { \\mathbb { E } _ { \\alpha _ { j } , \\beta _ { j } } ( t ) } { t } \\right ) ^ { 1 / \\lambda _ { j } } d t \\right \\} ^ { 1 / \\zeta } , \\end{align*}"}
-{"id": "7171.png", "formula": "\\begin{gather*} \\dot { y } = v ^ { \\prime } \\left ( \\chi ^ { t } ( x ) \\right ) y , \\end{gather*}"}
-{"id": "7149.png", "formula": "\\begin{align*} p _ { 1 } = p _ { 1 } ^ { \\nu } = \\lim _ { \\varepsilon \\rightarrow 0 } \\frac { \\log r _ { 1 } \\left ( \\varepsilon \\right ) } { \\log \\varepsilon } , q _ { 1 } = q _ { 1 } ^ { \\nu } = \\lim _ { \\varepsilon \\rightarrow \\infty } \\frac { \\log r _ { 1 } \\left ( \\varepsilon \\right ) } { \\log \\varepsilon } , \\end{align*}"}
-{"id": "599.png", "formula": "\\begin{align*} \\| g \\| _ { B ^ s _ { p , q } } : = \\bigg ( \\sum _ { j \\in \\mathbb { N } _ 0 } 2 ^ { j q \\big ( s + d ( \\frac { 1 } { 2 } - \\frac { 1 } { p } ) \\big ) } \\bigg ( \\sum _ { k \\in P _ j ^ d } \\sum _ { e \\in E _ j } | \\langle \\psi _ { j , k , e } , g \\rangle | ^ p \\bigg ) ^ { q / p } \\bigg ) ^ { 1 / q } , \\end{align*}"}
-{"id": "3005.png", "formula": "\\begin{align*} F _ m ' ( x ) & = - \\frac { G _ m ' ( x ) } { ( G _ m ( x ) ) ^ 2 } = - \\frac { G _ m ' ( x ) } { G _ m ( x ) } \\cdot \\frac { 1 } { G _ m ( x ) } = - \\frac { G _ m ' ( x ) } { G _ m ( x ) } \\cdot F _ m ( x ) . \\end{align*}"}
-{"id": "6901.png", "formula": "\\begin{align*} \\det ( d \\varphi ) = u ^ 1 _ { 1 0 } u ^ 2 _ { 0 1 } - u ^ 1 _ { 0 1 } u ^ 2 _ { 1 0 } + u ^ 3 _ { 1 0 } u ^ 4 _ { 0 1 } - u ^ 3 _ { 0 1 } u ^ 4 _ { 1 0 } + ( c _ 1 c _ 2 + s _ 1 s _ 2 ) q _ 1 + ( s _ 1 c _ 2 - s _ 2 c _ 1 ) q _ 2 \\ , , \\end{align*}"}
-{"id": "72.png", "formula": "\\begin{align*} \\sigma ^ + _ 1 - \\sigma ^ + _ { - 1 } & = \\sigma _ { 1 } ^ - - \\sigma ^ - _ { - 1 } + 2 \\left [ g ( J ^ + _ * ) - g ( J ^ - _ * ) \\right ] \\delta \\\\ & = \\sigma _ { 1 } ^ - - \\sigma ^ - _ { - 1 } + 2 g ' ( s ) \\left [ J ^ + _ * - J ^ - _ * \\right ] \\delta \\end{align*}"}
-{"id": "9186.png", "formula": "\\begin{align*} m _ { T } = \\dim \\mathrm { P r o j } _ { \\tilde { N } } \\mathcal { K } _ { b } \\left ( b \\right ) , \\end{align*}"}
-{"id": "3108.png", "formula": "\\begin{align*} c ( E ) & = 1 + c _ 1 + | n | \\cdot p t + \\sum _ { 0 \\leq i < j \\leq s } c _ 1 ( L _ i ) c _ 1 ( L _ j ) \\\\ & = 1 + ( s + 1 ) c _ 1 ( L _ s ) + \\sum _ { i = 1 } ^ s i \\beta ^ i \\\\ & + | n | \\cdot p t + \\frac { s } { 2 ( s + 1 ) } c _ 1 ^ 2 - \\sum _ { 1 \\leq i < j \\leq s } \\frac { i ( s + 1 - j ) } { s + 1 } \\beta ^ i \\beta ^ j - \\sum _ { 1 \\leq i \\leq s } \\frac { i ( s + 1 - i ) } { 2 ( s + 1 ) } ( \\beta ^ i ) ^ 2 \\ , , \\end{align*}"}
-{"id": "974.png", "formula": "\\begin{align*} X _ n ( q ) = q ^ { { n \\choose 2 } } Y _ n \\left ( \\frac { 1 } { q } \\right ) \\ , . \\end{align*}"}
-{"id": "636.png", "formula": "\\begin{align*} [ \\varphi ^ \\ast ( u ) x \\varphi ^ \\ast ( u ) ^ { - 1 } ] = [ x ] \\in \\operatorname { I n r } ^ G _ { \\psi \\varphi } \\backslash K _ { \\psi \\varphi } \\ . \\end{align*}"}
-{"id": "2394.png", "formula": "\\begin{align*} \\Delta : \\mathfrak { F } & \\to \\mathbb { C } & X _ i & \\mapsto 0 & \\delta ( 1 ) & = 1 . \\end{align*}"}
-{"id": "8614.png", "formula": "\\begin{align*} h = u v - \\frac { \\omega } { \\Lambda - 1 } . \\end{align*}"}
-{"id": "85.png", "formula": "\\begin{align*} & \\int _ 0 ^ 1 \\int _ { 0 } ^ { \\infty } \\left \\{ \\rho \\partial _ t \\phi + J \\partial _ x \\phi \\right \\} \\ , d x d t + \\int _ 0 ^ 1 \\rho _ 0 ( x ) \\phi ( x , 0 ) \\ , d x = 0 \\\\ & \\int _ 0 ^ 1 \\int _ { 0 } ^ \\infty \\left \\{ J \\partial _ t \\phi + \\rho \\partial _ x \\phi - 2 k ( x ) g ( J ) \\right \\} \\ , d x d t + \\int _ 0 ^ 1 J _ 0 ( x ) \\phi ( x , 0 ) \\ , d x = 0 \\end{align*}"}
-{"id": "3024.png", "formula": "\\begin{align*} L ^ m _ { p , q } h ( z ) & = z + \\sum _ { { k = 2 } } ^ { \\infty } [ k ] ^ m _ { p , q } a _ { k } z ^ { k } . \\end{align*}"}
-{"id": "5667.png", "formula": "\\begin{align*} & \\| D [ U ^ { n } ] \\| _ { L ^ p ( \\Omega ; L ^ q ( 0 , T ; \\mathcal { L } _ 2 ^ 0 ) ) } ^ q = \\| \\sum _ { j = 0 } ^ { n - 1 } \\chi _ { [ t _ { j } , t _ { j + 1 } ) } ( s ) B _ { n - j } P _ h \\| _ { L ^ p ( \\Omega ; L ^ q ( 0 , T ; \\mathcal { L } _ 2 ^ 0 ) ) } ^ q \\\\ = & \\| \\sum _ { j = 0 } ^ { n - 1 } \\chi _ { [ t _ { j } , t _ { j + 1 } ) } ( s ) B _ { n - j } P _ h \\| _ { L ^ q ( 0 , T ; \\mathcal { L } _ 2 ^ 0 ) } ^ q = \\tau \\sum _ { j = 0 } ^ { n - 1 } \\| B _ { n - j } P _ h \\| _ { \\mathcal { L } _ 2 ^ 0 } ^ q < \\infty . \\end{align*}"}
-{"id": "2264.png", "formula": "\\begin{align*} a ' & = ( 1 , \\langle E | \\frac { \\mathcal { P } } { x - \\Omega } ) & b ' & = ( 0 , \\langle \\delta _ x | ) . \\end{align*}"}
-{"id": "8458.png", "formula": "\\begin{align*} E _ k ^ { \\sigma } : = \\int \\frac { 1 } { A } | D _ t \\sigma _ k | ^ 2 + i \\sigma _ k \\overline { \\partial _ { \\alpha } \\sigma _ k } d \\alpha . \\end{align*}"}
-{"id": "1732.png", "formula": "\\begin{align*} \\mathcal { D } ( L ) & = \\left \\{ u \\in L ^ { p } ( \\boldsymbol { W } , P ) : \\lim _ { t \\downarrow 0 } \\frac { T _ { t } u - u } { t } \\textrm { e x i s t s i n } L ^ { p } \\textrm { - s p a c e } \\right \\} \\end{align*}"}
-{"id": "9265.png", "formula": "\\begin{align*} \\begin{array} { l l } | D u ( x ) - D u ( x _ 0 ) | \\ ! \\ ! & \\ ! \\ ! \\displaystyle \\le ( \\sup _ { B _ R ( x _ 0 ) \\cap \\partial \\Omega } \\sup _ { | \\tau | = 1 , \\tau \\cdot \\nu = 0 } | D _ \\tau D u | ) | x - x _ 0 | \\\\ \\ ! \\ ! & \\ ! \\ ! \\displaystyle \\le C ( 1 + M ^ \\prime _ 2 ( R ) ) | x - x _ 0 | , \\end{array} \\end{align*}"}
-{"id": "6734.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\left ( \\Box h _ { k l } - \\frac { 1 } { 2 } \\frac { \\partial ^ 2 h _ { n l } } { \\partial x ^ k \\partial x _ n } - \\frac { 1 } { 2 } \\frac { \\partial ^ 2 h _ { k n } } { \\partial x _ n \\partial x ^ l } \\right ) = 0 . \\end{align*}"}
-{"id": "3992.png", "formula": "\\begin{align*} \\partial ^ { [ 2 ] } \\bar { \\partial ^ { [ 2 ] } } = \\partial _ 1 \\bar { \\partial _ 1 } \\partial _ 4 \\bar { \\partial _ 4 } - \\partial _ 1 \\partial _ 4 \\bar { \\partial _ 2 } ^ 2 - \\bar { \\partial _ 1 } \\bar { \\partial _ 4 } \\partial _ 2 ^ 2 + \\partial _ 2 ^ 2 \\bar { \\partial _ 2 ^ 2 } \\end{align*}"}
-{"id": "1993.png", "formula": "\\begin{align*} \\int _ { \\P ^ 2 } H ^ 2 = \\sum _ { j = 0 } ^ 2 \\frac { i _ j ^ * H ^ 2 } { e ( i _ j ^ * T \\P ^ 2 ) } = \\frac { \\alpha _ 0 ^ 2 } { ( \\alpha _ 0 - \\alpha _ 1 ) ( \\alpha _ 0 - \\alpha _ 2 ) } + \\frac { \\alpha _ 1 ^ 2 } { ( \\alpha _ 1 - \\alpha _ 0 ) ( \\alpha _ 1 - \\alpha _ 2 ) } + \\frac { \\alpha _ 2 ^ 2 } { ( \\alpha _ 2 - \\alpha _ 0 ) ( \\alpha _ 2 - \\alpha _ 1 ) } = \\dots = 1 , \\end{align*}"}
-{"id": "3955.png", "formula": "\\begin{align*} & U _ { 1 } ( x , m , t ) = \\partial _ { m _ { 1 } } V _ { 1 } ( m , t ) ( x ) \\ , , U _ { - 1 } ( x , m , t ) = \\partial _ { m _ { 2 } } V _ { 1 } ( m , t ) ( x ) \\ , , \\\\ & U _ { 2 } ( x , m , t ) = \\partial _ { m _ { 2 } } V _ { 2 } ( m , t ) ( x ) \\ , , U _ { - 2 } ( x , m , t ) = \\partial _ { m _ { 1 } } V _ { 2 } ( m , t ) ( x ) \\ , , \\end{align*}"}
-{"id": "3209.png", "formula": "\\begin{align*} \\| ( W _ { \\ast } , p ^ { \\ast } ) \\| ^ 2 & = \\| W _ { \\ast } \\| ^ 2 + \\| p ^ { \\ast } \\| ^ 2 \\\\ & = \\lim _ { k \\in K \\to \\infty } \\left ( \\| W _ { k } \\| ^ 2 + \\| p ^ { k } \\| ^ 2 \\right ) \\\\ & = 1 . \\end{align*}"}
-{"id": "8983.png", "formula": "\\begin{align*} & D _ { y } h ( y ) = - 2 k ( y - z ) e ^ { - k | y - z | ^ { 2 } } , \\\\ & D _ { y y } ^ { 2 } h ( y ) = ( 4 k ^ { 2 } ( y _ { i } - z _ { i } ) ( y _ { j } - z _ { j } ) - 2 k \\delta _ { i j } ) e ^ { - k | y - z | ^ { 2 } } = ( 4 k ^ { 2 } ( y - z ) ( y - z ) ^ { T } - 2 k I _ { d \\times d } ) e ^ { - k | y - z | ^ { 2 } } . \\end{align*}"}
-{"id": "4722.png", "formula": "\\begin{align*} u ( x , 0 ) = \\left \\{ \\begin{array} { l l } 1 ~ ~ ~ ~ x \\leq 0 . 2 5 , \\\\ 0 ~ ~ ~ ~ x > 0 . 2 5 . \\end{array} \\right . \\end{align*}"}
-{"id": "658.png", "formula": "\\begin{align*} \\Phi ( \\varphi ; f _ 1 , \\dots , f _ m ; [ u ] , [ x ] ) = ( \\varphi ; ( f _ 1 , u _ 1 ) , \\dots , ( f _ m , u _ m ) ; [ x ] ) \\ . \\end{align*}"}
-{"id": "7615.png", "formula": "\\begin{align*} N = \\max \\{ \\norm { u _ 0 } _ { 1 + \\alpha , p } , \\norm { \\sigma _ 0 } _ { \\alpha , p } \\} , \\end{align*}"}
-{"id": "18.png", "formula": "\\begin{align*} \\bar { Y } _ t ^ { i , \\rho , v } = & \\ \\bar { Y } _ T ^ { i , \\rho , v } + \\int _ t ^ T \\left [ \\sum _ { k \\in I } q ^ { i k } ( e ^ { \\bar { Y } _ s ^ { k , \\rho , v } - \\bar { Y } _ s ^ { i , \\rho , v } } - 1 ) - \\rho \\bar { Y } _ s ^ { i , \\rho , v } + \\rho { Y } _ 0 ^ { m ^ { 0 } , \\rho , v _ 0 } \\right ] d s \\\\ & + \\int _ t ^ T f ^ i ( V _ s ^ { v } , Z _ s ^ { i , \\rho , v } ) d s - \\int _ t ^ T ( Z _ s ^ { i , \\rho , v } ) ^ { t r } d W _ s , \\end{align*}"}
-{"id": "2798.png", "formula": "\\begin{align*} r _ t ( \\mathbf { n } ) : = \\mathbb { P } \\left ( \\mathbf { U } ^ t = \\mathbf { m } \\right ) \\end{align*}"}
-{"id": "6098.png", "formula": "\\begin{gather*} \\Big ( \\widetilde { H } _ { p - 2 } ( \\widehat { 0 } , \\lambda ) \\Big ) ^ { \\mathfrak { S } _ n } = 0 \\lambda \\in \\P ^ m _ n , l ^ { m } ( \\lambda ) \\geq 2 . \\end{gather*}"}
-{"id": "7129.png", "formula": "\\begin{align*} p ( \\rho ) = \\left \\{ \\begin{array} { l l l l } \\displaystyle \\rho ^ 2 \\frac { \\partial f } { \\partial \\rho } = - 3 \\rho ^ 2 + \\frac { 8 \\Theta \\rho } { 3 - \\rho } & \\mathrm { i f } \\ 0 \\leq \\rho < 3 , \\\\ \\displaystyle + \\infty , & \\mathrm { i f } \\ \\rho \\geq 3 . \\end{array} \\right . \\end{align*}"}
-{"id": "328.png", "formula": "\\begin{align*} M _ { \\partial } = k L \\begin{pmatrix} 1 & 0 & \\frac { n - 3 } { 4 } \\\\ 0 & 0 & \\frac { n - 1 } { 2 } \\end{pmatrix} . \\end{align*}"}
-{"id": "1042.png", "formula": "\\begin{align*} & | T | _ 2 ^ 2 = \\int \\int | K ( x , y ) | ^ 2 d x d y \\end{align*}"}
-{"id": "6938.png", "formula": "\\begin{align*} ( \\rho , \\rho \\mathbf { u } , P ) ( x , 0 ) = ( \\rho _ 0 , \\rho _ 0 \\mathbf { u } _ 0 , P _ 0 ) ( x ) , \\ \\ x \\in \\Omega , \\end{align*}"}
-{"id": "5486.png", "formula": "\\begin{align*} \\mu _ b ( A ) = ^ { a . s . } \\lim _ { T \\to \\infty } \\frac { 1 } { T } \\int _ 0 ^ T 1 _ A ( X _ t ) d t \\end{align*}"}
-{"id": "7489.png", "formula": "\\begin{align*} \\Pi _ s = q _ s \\circ ( \\Pi _ \\infty \\otimes 1 ) \\colon \\xi \\mapsto \\displaystyle \\int \\sqrt { \\xi } \\end{align*}"}
-{"id": "1185.png", "formula": "\\begin{align*} - c U + \\frac { U ^ 2 } { 2 } + \\left ( 1 - \\partial _ x ^ 2 \\right ) ^ { - 1 } U = 0 . \\end{align*}"}
-{"id": "2046.png", "formula": "\\begin{align*} \\beta _ { F _ \\chi } ( p ^ j ) = \\begin{cases} 1 & j = 0 \\\\ - a _ \\chi ( p ) & j = 1 \\\\ - p ^ { k - 1 } \\chi ^ 2 ( p ) & j = 2 p \\nmid q ' , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "4258.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { i + 1 } E _ { 2 } ( V ( d w ) , B _ { j } ) \\leq 2 E _ { 2 } ( V ( d w ) , B _ { 1 } ) + \\frac { 2 c _ { 4 } \\lambda ^ { \\frac { p } { 2 } } } { \\sigma ^ { n } } \\sum _ { j = 1 } ^ { i } \\omega \\left ( R _ { j } \\right ) \\leq \\frac { \\sigma ^ { n } \\lambda ^ { \\frac { p } { 2 } } } { 1 0 0 } . \\end{align*}"}
-{"id": "6468.png", "formula": "\\begin{align*} u _ i : = 1 \\otimes \\left [ e _ { i ( k + 1 ) , \\ell } \\otimes e _ { 1 , 1 } \\right . & + \\left ( e _ { i ( k + 1 ) + 1 , \\ell + 1 } + \\cdots + e _ { ( i + 1 ) ( k + 1 ) - 1 , \\ell + k } \\right ) \\otimes 1 \\\\ & \\left . + e _ { ( i + 1 ) ( k + 1 ) , \\ell + k + 1 } \\otimes e _ { 0 , 0 } \\right ] \\end{align*}"}
-{"id": "4108.png", "formula": "\\begin{align*} X _ { + } ( x ) & = X _ { - } ( x ) \\begin{pmatrix} 1 & \\sqrt { 2 } x ^ { \\beta } e ^ { - n V ( x ) } & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} , & & x > 0 , \\\\ X _ { + } ( x ) & = X _ { - } ( x ) \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 0 & ( - 1 ) ^ { n + 1 } \\\\ 0 & ( - 1 ) ^ n & 0 \\end{pmatrix} , & & x < 0 , \\end{align*}"}
-{"id": "3361.png", "formula": "\\begin{align*} \\dim H = \\bigoplus _ { V \\in ( H ) } \\dim V \\cdot \\dim P ( V ) . \\end{align*}"}
-{"id": "3812.png", "formula": "\\begin{align*} a & = \\langle e , e , e , e , e , e \\rangle ( 1 , 2 ) ( 3 , 4 ) ( 5 , 6 ) \\\\ b & = \\langle e , e , e , e , e , b \\rangle ( 4 , 5 ) \\\\ c & = \\langle a , e , e , e , e , c \\rangle ( 2 , 3 ) \\\\ d & = \\langle a , e , e , e , e , d \\rangle ( 2 , 3 ) ( 4 , 5 ) \\end{align*}"}
-{"id": "8593.png", "formula": "\\begin{align*} \\zeta ( 3 ) = 1 . 2 0 2 0 5 6 9 0 3 2 . . . . . \\end{align*}"}
-{"id": "2990.png", "formula": "\\begin{align*} q ( 1 5 ) & = 0 + q \\Big ( \\frac { 3 0 - 2 } { 2 } \\Big ) + q \\Big ( \\frac { 3 0 - 1 2 } { 2 } \\Big ) - q \\Big ( \\frac { 3 0 - 2 2 } { 2 } \\Big ) - q \\Big ( \\frac { 3 0 - 2 6 } { 2 } \\Big ) \\\\ & = q ( 1 4 ) + q ( 9 ) - q ( 4 ) - q ( 2 ) = 2 2 + 8 - 2 - 1 = 2 7 . \\end{align*}"}
-{"id": "8060.png", "formula": "\\begin{align*} \\Xi _ 1 = \\frac { 1 } { 2 } - \\frac { 1 } { 2 } \\sum _ { i = n + 3 } ^ { 2 n } c ^ { \\frac { n + 3 - 2 i } { 2 } } P _ { i - n - 3 , - i } \\mbox { a n d } \\Xi _ 2 = \\frac { 3 } { 2 } + \\frac { 1 } { 2 } \\sum _ { i = n + 3 } ^ { 2 n } c ^ { \\frac { n + 3 - 2 i } { 2 } } P _ { i - n - 3 , - i } , \\end{align*}"}
-{"id": "8244.png", "formula": "\\begin{align*} \\frac { F '' ( z ) F ( z ) - ( F ' ( z ) ) ^ 2 } { ( F ( z ) ) ^ 2 } = \\sum _ { n = 0 } ^ \\infty \\frac { q ^ { 2 n + 1 } ( - 2 z ^ { - 3 } + z ^ { - 4 } q ^ { 2 n + 1 } ) } { ( 1 - z ^ { - 1 } q ^ { 2 n + 1 } ) ^ 2 } - \\sum _ { n = 0 } ^ \\infty \\frac { q ^ { 4 n + 2 } } { ( 1 - z q ^ { 2 n + 1 } ) ^ 2 } . \\end{align*}"}
-{"id": "7425.png", "formula": "\\begin{align*} p = m t = n i \\geq 2 q = m s = n j \\geq 1 . \\end{align*}"}
-{"id": "8969.png", "formula": "\\begin{align*} \\mathrm { H } ^ 2 ( L _ u \\cap B ( 0 , r ) ) = \\left \\{ \\begin{array} { c c c } c _ 1 \\sqrt { r ^ 4 - | u | ^ 4 } & \\mathrm { i f } & | u | \\leq r \\\\ 0 & \\mathrm { o t h e r w i s e } & \\end{array} \\right . \\ , , \\end{align*}"}
-{"id": "779.png", "formula": "\\begin{align*} \\sigma _ k ^ * ( \\sigma _ \\ell ) = \\begin{cases} 1 & \\mbox { i f } k \\neq \\ell \\\\ \\zeta & \\mbox { i f } k = \\ell . \\end{cases} \\end{align*}"}
-{"id": "4803.png", "formula": "\\begin{align*} \\mathbf { S } = \\frac { 1 } { 2 } \\left ( \\begin{array} { c c } \\sigma & 0 \\\\ 0 & \\sigma \\end{array} \\right ) L : = - i x \\wedge \\nabla . \\end{align*}"}
-{"id": "9886.png", "formula": "\\begin{align*} & f ( x ) = \\sum _ { i = 1 } ^ { n } P ^ { \\mu } ( y _ { i } \\in \\cdot | X _ { 1 } = x ) \\begin{pmatrix} g _ { i } ( c _ { 1 } ) \\\\ \\vdots \\\\ g _ { i } ( c _ { K } ) \\end{pmatrix} \\\\ & = \\begin{pmatrix} Q H _ { x } & T ( x | : ) A & \\cdots & T ( x | : ) T ^ { n - 2 } A \\end{pmatrix} \\alpha \\end{align*}"}
-{"id": "2620.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta _ p u ( z ) + | u ( z ) | ^ { p - 2 } u ( z ) = \\hat { f } ( z , u ( z ) , D u ( z ) ) + \\epsilon e ( z ) & \\mbox { i n } \\ \\Omega , \\\\ \\frac { \\partial u } { \\partial n _ p } + \\beta ( z ) | u | ^ { p - 2 } u = 0 & \\mbox { o n } \\ \\partial \\Omega . \\end{array} \\right \\} \\end{align*}"}
-{"id": "9296.png", "formula": "\\begin{align*} \\bar J _ K ^ { 2 , \\pm } u ( x ) : = & \\{ ( p , X ) \\in \\mathbb { R } ^ n \\times \\mathbb { S } ^ n | \\ \\exists x _ k \\in K \\ { \\rm a n d } \\ ( p _ k , X _ k ) \\in J _ K ^ { 2 , \\pm } u ( x ) \\ { \\rm s u c h \\ t h a t } \\\\ & ( x _ k , u ( x _ k ) , p _ k , X _ k ) \\rightarrow ( x , u ( x ) , p , X ) \\ { \\rm a s } \\ k \\rightarrow + \\infty \\} , \\end{align*}"}
-{"id": "9713.png", "formula": "\\begin{align*} T _ { \\textbf { d } } ( \\partial \\Omega ( P _ i ) ) = I _ { \\textbf { d } } ^ 1 ( P _ { i } ) _ { c } - 1 T _ { \\textbf { d } } ( \\partial v ) = I _ { \\textbf { d } } ( v ) - 1 . \\end{align*}"}
-{"id": "93.png", "formula": "\\begin{align*} H _ 1 \\doteq \\{ ( x _ 1 , \\dots , x _ { 2 N } ) \\in \\R ^ { 2 N } & : x _ 1 + x _ 4 + \\dots + x _ { 2 N - 3 } + x _ { 2 N } = 0 \\} , \\\\ H _ 2 \\doteq \\{ ( x _ 1 , \\dots , x _ { 2 N } ) \\in \\R ^ { 2 N } & : x _ 2 + x _ 3 + \\dots + x _ { 2 N - 2 } + x _ { 2 N - 1 } = 0 \\} \\ , . \\end{align*}"}
-{"id": "6101.png", "formula": "\\begin{align*} \\phi ( \\mu ) = \\frac { 1 } { \\sqrt { 2 \\pi } } \\exp ( - \\mu ^ 2 / 2 ) \\phi _ k ( \\mu ) = \\frac { 1 } { k } \\phi ( \\mu / k ) . \\end{align*}"}
-{"id": "3857.png", "formula": "\\begin{align*} \\det W ^ { \\pm } _ { t , x } = \\prod _ { ( s , y ) \\in \\mathcal { S } ( t , x ) } P ( Z _ s ^ y = y \\pm s e _ d ) > 0 . \\end{align*}"}
-{"id": "8792.png", "formula": "\\begin{align*} { \\Vert \\phi ' \\Vert } _ { - r } ^ 2 = \\sum _ { k = 1 } ^ { \\infty } \\langle \\phi ' , \\phi _ { r , k } \\rangle ^ 2 = \\sum _ { k = 1 } ^ { \\infty } ( 1 + k ^ 2 ) ^ { - r } \\langle \\phi ' , \\phi _ { k } \\rangle ^ 2 , \\phi ' \\in H _ { - r } ( [ 0 , \\pi ] ) . \\end{align*}"}
-{"id": "9707.png", "formula": "\\begin{align*} T _ { \\textbf { d } } ( \\partial M ) = T _ { \\textbf { d } } ( \\partial M _ { c } ) - \\sum _ i ^ b T _ { \\textbf { d } } ^ i ( \\partial \\Omega ( P _ i ) ) = - \\sum _ i ^ b ( I _ { \\textbf { d } } ^ i ( P _ i ) - 1 ) = - \\chi ( M _ c ) + b . \\end{align*}"}
-{"id": "9007.png", "formula": "\\begin{align*} & \\dot { \\gamma } \\cdot \\overline { D } _ { t } X _ { j } \\times X _ { 1 } \\times \\dotsb \\times X _ { m - 1 } = 0 \\\\ & \\dot { \\gamma } ( t ) \\times X _ { 1 } ( t ) \\times \\dotsb \\times X _ { m - 1 } ( t ) \\neq 0 \\end{align*}"}
-{"id": "2557.png", "formula": "\\begin{align*} \\{ ( Y _ t ) _ { t \\geq 0 } ; \\dot { \\mathbf P } ^ { ( \\phi ) } _ \\mu \\} \\overset { f . d . d . } { = } \\{ ( W _ t ) _ { t \\geq 0 } ; \\mathbb N _ \\mu ^ { ( \\phi ) } \\} . \\end{align*}"}
-{"id": "6601.png", "formula": "\\begin{align*} \\overline { A } _ n ( \\varepsilon ) = 2 F _ { n } | \\sin \\pi ( ( - \\varphi ) ^ { n } - \\varepsilon ) | , \\\\ \\overline { C } _ n ( \\varepsilon ) = \\prod _ { t = 1 } ^ { ( F _ { n } - 1 ) / 2 } \\left ( 1 - \\frac { v _ { n } ^ 2 } { s _ { n t } ^ 2 } \\right ) , \\end{align*}"}
-{"id": "9593.png", "formula": "\\begin{align*} \\frac { 1 } { ( q ) _ N } \\sum _ { n = 1 } ^ { \\infty } \\frac { n q ^ n ( 1 - q ^ { N n } ) } { 1 - q ^ n } = \\sum _ { n = 1 } ^ { \\infty } \\left ( \\sum _ { j = 0 } ^ { n - 1 } p ( j , n ) \\sigma ( n - j , N ) \\right ) q ^ n . \\end{align*}"}
-{"id": "8343.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int \\frac { | D _ t Z ( \\alpha , t ) - D _ t Z ( \\beta , t ) | ^ 2 } { ( \\alpha - \\beta ) ^ 2 } d \\beta = \\sum _ { 1 \\leq j , k \\leq N } \\frac { \\lambda _ j \\lambda _ k } { ( 2 \\pi ) ^ 2 } \\frac { 1 } { ( \\alpha - z _ j ) \\overline { ( \\alpha - z _ k ) } } \\frac { i } { \\overline { z _ k } - z _ j } . \\end{align*}"}
-{"id": "3248.png", "formula": "\\begin{align*} P A _ 0 \\cdot P A _ 1 \\cdot \\dots \\cdot P A _ { n - 1 } = 1 - x ^ n . \\end{align*}"}
-{"id": "1664.png", "formula": "\\begin{align*} f ( \\tau ) \\mid T _ { k / 2 , 4 N } ( \\ell ^ { 2 } ) = \\sum _ { n } \\left ( a ( \\ell ^ { 2 } n ) + \\ell ^ { \\lambda - 1 } \\left ( \\frac { ( - 1 ) ^ { \\lambda } n } { \\ell } \\right ) a ( n ) + \\ell ^ { 2 \\lambda - 1 } a ( n / \\ell ^ { 2 } ) \\right ) q ^ { n } , \\end{align*}"}
-{"id": "5742.png", "formula": "\\begin{align*} & \\frac { d } { d t } \\Bigg ( \\int _ 0 ^ t ( t - \\xi ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } [ - \\mu _ { m , n } ( t - \\xi ) ^ { \\alpha } ] F _ { m , n } ( \\xi ) d \\xi \\Bigg ) _ { t = 0 } \\\\ & = ( t - t ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } ( - \\mu _ { m , n } ( t - t ) ^ \\alpha ) F _ { m , n } ( t ) \\Big | _ { t = 0 } + \\Bigg ( \\int _ 0 ^ t \\frac { \\partial } { \\partial t } ( t - \\xi ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } ( - \\mu _ { m , n } ( t - \\xi ) ^ { \\alpha } ) F ( \\xi ) d \\xi \\Bigg ) _ { t = 0 } = 0 . \\end{align*}"}
-{"id": "6421.png", "formula": "\\begin{align*} z _ { k + 1 } = \\left [ \\begin{array} { c } s _ { k } z _ { k } \\\\ c _ { k } \\end{array} \\right ] \\end{align*}"}
-{"id": "6297.png", "formula": "\\begin{align*} E ( S , S ^ c ) + E ( S ) - e ( G ) \\left [ k \\left ( n - k \\right ) + \\binom { k } { 2 } \\right ] | > C \\max \\{ \\sqrt { \\rho } , \\sqrt { \\frac { \\log n } { n } } \\} k \\sqrt { n \\log n } | . \\end{align*}"}
-{"id": "2912.png", "formula": "\\begin{align*} \\eta ( x ) _ g = M \\big ( ( g ^ { - 1 } x ) _ { F } \\big ) \\end{align*}"}
-{"id": "6590.png", "formula": "\\begin{align*} P _ N ( \\alpha ) : = \\prod _ { r = 1 } ^ { N } | 2 \\sin ( \\pi r \\alpha ) | , \\end{align*}"}
-{"id": "806.png", "formula": "\\begin{align*} x _ 1 \\in \\mathbb { F } _ { q ^ d } : x _ i = x _ 1 ^ { q ^ { i - 1 } } , \\ ; i = 2 , \\ldots , s . \\end{align*}"}
-{"id": "670.png", "formula": "\\begin{align*} \\mu _ { \\vec k } : = \\mu _ { k _ 1 } \\diamond \\dots \\diamond \\mu _ { k _ n } \\ . \\end{align*}"}
-{"id": "3068.png", "formula": "\\begin{align*} u ( x ) = \\int _ { \\R ^ 2 } \\frac { ( x - y ) ^ \\perp } { | x - y | ^ { 2 + 2 \\alpha } } \\omega ( y ) d y . \\end{align*}"}
-{"id": "5361.png", "formula": "\\begin{align*} v _ { n + 2 \\ , m } ^ 1 = \\frac 1 { b _ { n + 1 \\ , m + 3 } } Y _ \\beta . \\frac 1 { a _ { n m } } X _ { \\alpha + \\beta } . v _ { n m } ^ 1 \\end{align*}"}
-{"id": "4590.png", "formula": "\\begin{align*} \\int f _ { g , \\delta } ( z ) d \\nu _ g ( z ) = \\int \\hat { \\nu } _ g ( \\omega ) \\overline { \\hat { \\overline { f } } } _ { g , \\delta } ( \\omega ) d \\omega . \\end{align*}"}
-{"id": "6349.png", "formula": "\\begin{align*} g _ { j i } = A _ 0 \\exp \\left ( - 2 \\frac { \\left ( x _ p + d \\right ) ^ 2 + \\left ( y _ p \\right ) ^ 2 } { w _ { z _ { \\rm e q } } ^ 2 } \\right ) , \\end{align*}"}
-{"id": "5207.png", "formula": "\\begin{align*} U ( x ) = c x ^ { - 1 } B ( x ) \\exp \\left ( - \\int _ { 1 } ^ { x } t ^ { - 1 } B ( t ) d t \\right ) . \\end{align*}"}
-{"id": "7011.png", "formula": "\\begin{align*} a _ j ' = \\frac { \\lambda _ j ^ \\ell { \\textsl { \\footnotesize R } } _ j ( 0 ) } { d _ j } a _ j = \\frac { \\lambda _ j ^ \\ell { \\textsl { \\footnotesize R } } _ j ( 0 ) } { d _ j } \\left ( \\frac { \\ell } { \\lambda _ j } - \\frac { d _ j ' } { d _ j } \\right ) . \\end{align*}"}
-{"id": "2152.png", "formula": "\\begin{align*} 0 < h _ t ( \\mathbf x , \\mathbf y ) = h _ t ( \\mathbf y , \\mathbf x ) , \\end{align*}"}
-{"id": "5437.png", "formula": "\\begin{align*} \\gamma _ 1 ( A , \\mu _ p ) + \\gamma _ 2 ( A , \\mu _ p ) + \\cdots + \\gamma _ s ( A , \\mu _ p ) & = \\frac { 1 } { k } r _ s ( A ^ k ( p ) ) . \\end{align*}"}
-{"id": "8211.png", "formula": "\\begin{align*} \\rho ^ 2 = \\rho + 5 \\rho t - t . \\end{align*}"}
-{"id": "1296.png", "formula": "\\begin{align*} V ( y ) = \\frac { 1 } { \\pi } \\int _ 0 ^ \\infty A e ^ { i y \\xi + \\xi ^ 3 } e ^ { i a \\ln \\xi } \\chi ( \\xi ) d \\xi \\ + \\ L ^ 2 . \\end{align*}"}
-{"id": "355.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } a ( \\alpha _ { n } ) \\| u _ { n } \\| ^ { 2 } - \\int _ { \\Omega } F _ { \\ast } ( u _ { n } ) d x = I _ { k } ( u _ { n } ) < 0 , \\end{align*}"}
-{"id": "1741.png", "formula": "\\begin{align*} K ( t , s ) = \\sqrt { \\frac { H ( 2 H - 1 ) } { \\beta ( 2 - 2 H , H - \\frac { 1 } { 2 } ) } } s ^ { \\frac { 1 } { 2 } - H } \\int _ { s } ^ { t } ( u - s ) ^ { H - \\frac { 3 } { 2 } } u ^ { H - \\frac { 1 } { 2 } } d u , \\end{align*}"}
-{"id": "3787.png", "formula": "\\begin{align*} m _ { \\mathcal { M } ( \\alpha ) } ( \\mu ) - m _ { \\mathcal { M } ( \\beta ) } ( \\mu ) = ( - 1 ) ^ { R - \\ell ( \\mu ) } c _ \\mu . \\end{align*}"}
-{"id": "9844.png", "formula": "\\begin{align*} H _ I = H - N _ H [ I ] \\qquad L _ I ( u ) = L ( u ) \\setminus N _ H ( I ) u \\in V ( G _ I ) . \\end{align*}"}
-{"id": "6516.png", "formula": "\\begin{align*} \\mathfrak { a } _ { ( n ) } \\circ r _ n ( x ) & = ( f _ { r _ n ( x ) } ^ { q n } , \\ldots , f _ { r _ n ( x ) } ^ { q n } ) \\left ( \\mathfrak { a } _ { ( 0 ) } \\circ r \\circ L ^ { - n } _ 0 ( x ) \\right ) \\\\ & = ( f _ { r _ n ( x ) } ^ { q n } , \\ldots , f _ { r _ n ( x ) } ^ { q n } ) \\circ \\phi _ { r _ n ( x ) } \\left ( L ^ { - n } _ 0 ( x ) \\right ) \\\\ & = \\phi _ { r _ n ( x ) } \\left ( L ^ n _ { r _ n ( x ) } \\circ L ^ { - n } _ 0 ( x ) \\right ) \\end{align*}"}
-{"id": "8129.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } \\widetilde { B } _ n ( t ) \\ , \\frac { z ^ n } { n ! } \\ = \\ \\frac { ( 1 - t ) e ^ { ( 2 t + 1 ) z } } { e ^ { 2 t z } - t e ^ { 2 z } } . \\end{align*}"}
-{"id": "5848.png", "formula": "\\begin{align*} \\eta ' : F _ g \\rightarrow F ^ { n } , f ( x ) = b _ { n - 1 } x ^ { n - 1 } + b _ { n - 2 } x ^ { n - 2 } + \\cdots + b _ 0 \\mapsto ( b _ { n - 1 } , b _ { n - 2 } , \\cdots , b _ 0 ) . \\end{align*}"}
-{"id": "9240.png", "formula": "\\begin{align*} D _ x A , D _ x B , D _ z A , D _ z B = O ( | p | ^ 2 ) , D _ p A , D _ p B = O ( | p | ) , \\end{align*}"}
-{"id": "653.png", "formula": "\\begin{align*} F ^ { \\mathcal G } ( a _ 1 \\dots a _ m ) : = F ( a _ 1 ) \\dots F ( a _ m ) \\ ; \\end{align*}"}
-{"id": "1456.png", "formula": "\\begin{align*} \\left | \\partial ^ { \\beta } \\Theta ( t , x , p ^ { 1 } , q ^ { 1 } ) - \\partial ^ { \\beta } \\Theta ( t , x , p ^ { 2 } , q ^ { 2 } ) \\right | \\leq c \\left ( | p ^ { 1 } - p ^ { 2 } | + \\sum _ { i = 1 } ^ { d } | q ^ { 1 } _ { i } - q ^ { 2 } _ { i } | \\right ) . \\end{align*}"}
-{"id": "2739.png", "formula": "\\begin{align*} \\left | \\frac { 1 } { ( \\varphi ( 4 ) ) ^ k } \\mathrm { P f } \\Big [ K _ { N } \\Big ( 4 N + \\frac { u _ i } { \\varphi ( 4 ) } , 4 N + \\frac { u _ j } { \\varphi ( 4 ) } \\Big ) \\Big ] _ { i , j = 1 } ^ { k } \\right | \\leq ( 2 k ) ^ { \\frac { k } { 2 } } C ^ { k } \\prod _ { j = 1 } ^ { k } e ^ { - ( a - v ) u _ { j } } . \\end{align*}"}
-{"id": "1010.png", "formula": "\\begin{align*} \\ln D _ n ( \\alpha ) = n ^ 2 \\ln \\cos \\frac { \\alpha } { 2 } - \\frac { 1 } { 4 } \\ln \\left ( n \\sin \\frac { \\alpha } { 2 } \\right ) + c _ 0 + O \\left ( \\frac { 1 } { n \\sin ( \\alpha / 2 ) } \\right ) , \\end{align*}"}
-{"id": "4508.png", "formula": "\\begin{align*} \\theta _ ! ( x \\otimes y ) q = \\theta _ ! ( x \\otimes q ^ * x ) \\ ; . \\end{align*}"}
-{"id": "9350.png", "formula": "\\begin{align*} [ y _ 0 , y _ { 2 ^ k - i } ] ^ { - 1 } = [ y _ { 2 ^ k - i } , y _ 0 ] = [ y _ 0 , y _ i ] ^ { x ^ { - i } } = [ y _ 0 , y _ i ] \\end{align*}"}
-{"id": "6264.png", "formula": "\\begin{align*} \\pi _ g ( \\pi _ h ( x ) \\pi _ k ( y ) ) = \\left \\{ \\begin{array} { l } \\pi _ h ( x ) \\pi _ k ( y ) , , \\\\ 0 , . \\end{array} \\right . \\end{align*}"}
-{"id": "9057.png", "formula": "\\begin{align*} \\xi _ 1 + \\xi _ 2 + \\xi _ 3 + \\xi _ 4 + \\xi _ 5 + \\xi _ 6 = 0 \\end{align*}"}
-{"id": "6350.png", "formula": "\\begin{align*} f _ { h } ( h ) = \\frac { 2 ( \\alpha \\beta ) ^ { \\frac { \\alpha + \\beta } { 2 } } } { \\Gamma ( \\alpha ) \\Gamma ( \\beta ) } h ^ { \\frac { \\alpha + \\beta } { 2 } - 1 } k _ { \\alpha - \\beta } ( 2 \\sqrt { \\alpha \\beta h } ) , \\end{align*}"}
-{"id": "2151.png", "formula": "\\begin{align*} e ^ { t \\Delta } f ( \\mathbf x ) = \\mathcal F ^ { - 1 } ( e ^ { - t \\| \\xi \\| ^ 2 } \\mathcal F f ( \\xi ) ) ( \\mathbf x ) = \\int _ { \\mathbb R ^ N } h _ t ( \\mathbf x , \\mathbf y ) f ( \\mathbf y ) \\ , d w ( \\mathbf y ) , \\end{align*}"}
-{"id": "2795.png", "formula": "\\begin{align*} T ( \\mathbf { n } ; \\mathbf { m } ) : = \\mathbb { P } \\left ( \\mathbf { U } ^ t = \\mathbf { m } | \\mathbf { N } ^ t = \\mathbf { n } \\right ) , \\end{align*}"}
-{"id": "7865.png", "formula": "\\begin{align*} \\sum _ { x = 0 } ^ n \\binom { n } { x } \\Bbbk _ k ^ { ( n ) } ( x ) \\Bbbk _ m ^ { ( n ) } ( x ) = 2 ^ n \\binom { n } { k } ^ { - 1 } \\delta _ k ( m ) . \\end{align*}"}
-{"id": "6253.png", "formula": "\\begin{align*} y ^ \\epsilon _ { s , t } : = u _ j ^ * y _ i u _ k . \\end{align*}"}
-{"id": "4283.png", "formula": "\\begin{align*} n \\binom { | V ( F ) | - 2 } { r - 1 } + \\binom { | V ( F ) | - \\alpha ( F ) - 1 } { r } \\end{align*}"}
-{"id": "9370.png", "formula": "\\begin{align*} D _ i ( G _ k ) / D _ { i + 1 } ( G _ k ) \\cong \\begin{cases} C _ 2 \\times C _ 2 & \\\\ C _ 2 \\times C _ 2 \\times C _ 2 & \\\\ C _ 2 \\times C _ 2 \\times C _ 2 & \\end{cases} \\end{align*}"}
-{"id": "6331.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 { q ^ { - 1 } ( \\cdot , \\ , t ) \\ , d x } = \\int _ 0 ^ 1 { q ^ { - 1 } _ 0 \\ , d x } = \\nu ^ { - \\frac { 1 } { 2 } } \\quad t > 0 . \\end{align*}"}
-{"id": "601.png", "formula": "\\begin{align*} \\Omega _ n : = \\big \\{ ( j , k , e ) \\in \\Omega \\ , \\big | \\ , j = 0 , \\ldots , J - 1 \\big \\} , \\end{align*}"}
-{"id": "2196.png", "formula": "\\begin{align*} \\dot { P } ( t ) + P ( t ) A ( t ) + A ^ { T } ( t ) P ( t ) + Q ( t ) - P ( t ) B ( t ) R ^ { - 1 } ( t ) B ^ { T } ( t ) P ( t ) = 0 \\end{align*}"}
-{"id": "9377.png", "formula": "\\begin{align*} L _ i = \\langle x ^ { 2 ^ i } \\rangle M _ i M _ i = \\langle c _ { \\nu ( i ) + 1 } \\rangle \\gamma _ { \\nu ( i ) + 2 } ( G _ k ) C _ i \\trianglelefteq G _ k , \\end{align*}"}
-{"id": "8706.png", "formula": "\\begin{align*} \\sum _ { | \\alpha | = m } \\frac { ( x ) _ \\alpha } { \\alpha ! } & = \\sum _ { k = 0 } ^ m \\frac { ( x _ n ) _ k } { k ! } \\sum _ { | \\beta | = m - k } \\frac { ( x ' ) _ { \\beta } } { \\beta ! } \\\\ & = \\sum _ { k = 0 } ^ m \\frac { ( x _ n ) _ k ( x _ 1 + \\cdots + x _ { n - 1 } ) _ { m - k } } { k ! ( m - k ) ! } \\\\ & = \\frac { ( x _ 1 + \\cdots + x _ { n - 1 } + x _ n ) _ m } { m ! } , \\end{align*}"}
-{"id": "5570.png", "formula": "\\begin{align*} \\mathcal { Q } ^ { \\pm } ( t ) : = \\int _ { \\R ^ { \\pm } } \\Big ( \\frac { \\alpha } { \\gamma } v ^ 2 + 2 \\ , \\big ( u \\bar { u } _ x ) \\Big ) d x = \\mathcal { Q } ^ { \\pm } ( 0 ) \\mp \\int _ 0 ^ t \\big ( \\mathcal { Q } ^ u ( s ) + \\mathcal { Q } ^ v ( s ) \\big ) d s , \\end{align*}"}
-{"id": "4050.png", "formula": "\\begin{align*} d f = \\{ w , f \\} _ { \\omega _ \\eta } \\end{align*}"}
-{"id": "7752.png", "formula": "\\begin{align*} E \\langle x , x _ { \\alpha } \\rangle \\langle x _ { \\alpha } , x \\rangle E = \\| x \\| ^ 2 E \\ , \\ , \\ , \\ , \\ , ( \\alpha \\in \\Lambda ) . \\end{align*}"}
-{"id": "9292.png", "formula": "\\begin{align*} & B ( \\hat y , u ( \\hat y ) , D u ( \\hat y ) ) - B ( \\hat x , v ( \\hat x ) , D v ( \\hat x ) ) \\\\ = & D _ x B ( \\tilde x , u ( \\hat y ) , D u ( \\hat y ) ) ( \\hat y - \\hat x ) + D _ z B ( \\hat x , \\tilde z , D u ( \\hat y ) ) ( u ( \\hat y ) - v ( \\hat x ) ) \\\\ & + \\delta D _ p B ( \\hat x , v ( \\hat x ) , \\tilde p ) [ D d ( \\hat x ) + D d ( \\hat y ) - 2 ( \\hat x - z ) ] , \\end{align*}"}
-{"id": "7464.png", "formula": "\\begin{gather*} Z ( X _ \\eta ) = \\int _ { \\widetilde { \\eta } } \\sqrt { \\xi } . \\end{gather*}"}
-{"id": "3506.png", "formula": "\\begin{align*} \\left . \\frac { d } { d x } \\Re f _ { N } \\big ( \\xi , x \\pm \\frac { i } { 2 } \\big ) \\right \\rvert _ { x = N ^ { 1 / 4 } } > 0 , \\left . \\frac { d } { d x } \\Re f _ { N } \\big ( \\xi , x \\pm \\frac { i } { 2 } \\big ) \\right \\rvert _ { x = - N ^ { 1 / 4 } } < 0 . \\end{align*}"}
-{"id": "913.png", "formula": "\\begin{align*} J [ \\psi ^ { ( k ) } ] _ { T , t } = \\int \\limits _ t ^ T \\psi _ k ( t _ k ) \\ldots \\int \\limits _ t ^ { t _ { 2 } } \\psi _ 1 ( t _ 1 ) d { \\bf w } _ { t _ 1 } ^ { ( i _ 1 ) } \\ldots d { \\bf w } _ { t _ k } ^ { ( i _ k ) } , \\end{align*}"}
-{"id": "234.png", "formula": "\\begin{align*} \\int _ { B _ r } | u | d z d t = O ( r ^ k ) , \\ \\end{align*}"}
-{"id": "822.png", "formula": "\\begin{align*} y ^ { q ^ d } - y = \\sum _ { i = 1 } ^ { { s } } \\frac { b _ i } { U _ 1 ^ { ( i ) } } , U _ 1 ^ { ( i ) } = T - \\rho _ i . \\end{align*}"}
-{"id": "2829.png", "formula": "\\begin{align*} M _ 0 = \\left [ \\begin{array} { c c } M _ { 0 , 1 1 } & A _ { 1 2 } \\\\ A _ { 2 1 } & { M } _ { 0 , 2 2 } \\\\ \\end{array} \\right ] , \\end{align*}"}
-{"id": "3152.png", "formula": "\\begin{align*} \\psi ( A B ) = \\psi ( B ) \\psi ( A ) . \\end{align*}"}
-{"id": "1207.png", "formula": "\\begin{align*} I _ { \\tilde \\tau , k } ( x - \\tilde { z } ' , \\zeta _ 2 ) = I ^ s _ { \\tilde { \\tau } , k } ( x - \\tilde { z } ' , \\zeta _ 2 ) + I ^ m _ { \\tilde { \\tau } , k } ( x - \\tilde { z } ' , \\zeta _ 2 ) . \\end{align*}"}
-{"id": "4117.png", "formula": "\\begin{align*} m _ { 1 } = \\int _ { 0 } ^ { q } s \\ , d \\mu ^ * ( s ) m _ { \\frac { 1 } { 2 } } = \\int _ { 0 } ^ { q } \\sqrt { s } d \\mu ^ * ( s ) . \\end{align*}"}
-{"id": "2454.png", "formula": "\\begin{align*} P ( R / x R , t ) \\ = \\ \\frac { 1 } { 1 - t } - p ( t ) \\end{align*}"}
-{"id": "8413.png", "formula": "\\begin{align*} C = C ( \\| \\partial _ { \\alpha } \\xi _ 0 \\| _ { H ^ { s - 1 } } , \\| v _ 0 \\| _ { H ^ s } , \\| w _ 0 \\| _ { H ^ s } , \\mathcal { E } ( t ) , d _ I ( t ) ^ { - 1 } , d _ P ( t ) ^ { - 1 } , N \\lambda _ { m a x } , C _ 1 , C _ 2 , \\alpha _ 0 ) \\end{align*}"}
-{"id": "2520.png", "formula": "\\begin{align*} n l ( a _ n ) \\operatorname { P } ( \\mathbf Z _ n \\neq \\mathbf 0 | Z _ 0 = \\mathbf i ) ^ \\alpha \\xrightarrow [ n \\to \\infty ] { } \\frac { ( \\mathbf i \\cdot \\mathbf u ) ^ \\alpha } \\alpha , \\end{align*}"}
-{"id": "3233.png", "formula": "\\begin{align*} \\bar { s } & : = { \\sup \\left \\{ s \\mid \\lambda _ { \\rm m i n } ( X + s \\Delta X ) \\ge 0 , s \\ge 0 \\right \\} } \\\\ & = \\begin{cases} - \\displaystyle { \\frac { 1 } { \\lambda _ { \\rm m i n } ( X ^ { - 1 } \\Delta X ) } } \\ & \\mbox { i f } \\lambda _ { \\rm m i n } ( X ^ { - 1 } \\Delta X ) < 0 \\\\ { \\infty } \\ & \\mbox { o t h e r w i s e } \\end{cases} \\end{align*}"}
-{"id": "3553.png", "formula": "\\begin{align*} x : = \\frac { a _ { 2 } c _ { 1 } } { a _ { 0 } c _ { 0 } } , y : = \\frac { a _ { 1 } b _ { 2 } } { a _ { 0 } b _ { 0 } } , z : = \\frac { b _ { 1 } c _ { 2 } } { b _ { 0 } c _ { 0 } } , u : = - \\frac { a _ { 1 } b _ { 1 } c _ { 1 } } { a _ { 0 } b _ { 0 } c _ { 0 } } , v : = - \\frac { a _ { 2 } b _ { 2 } c _ { 2 } } { a _ { 0 } b _ { 0 } c _ { 0 } } \\end{align*}"}
-{"id": "6857.png", "formula": "\\begin{align*} \\varphi ( t , \\alpha ) = \\varphi ^ t ( \\alpha ) = A ( t ) v ( \\alpha ) \\ , , \\end{align*}"}
-{"id": "6254.png", "formula": "\\begin{align*} y ^ \\ell _ { s , 0 } : = u _ j ^ * y _ i u _ { a + 1 } \\qquad y ^ r _ { s , 0 } : = u _ { a + 1 } ^ * y _ i u _ j . \\end{align*}"}
-{"id": "772.png", "formula": "\\begin{align*} P _ { q ^ d , P ^ \\alpha } = \\{ D \\in \\mathbb { F } _ { q ^ d } [ T ] \\mod P ^ \\alpha , D \\equiv 1 \\mod P \\} . \\end{align*}"}
-{"id": "6311.png", "formula": "\\begin{align*} Z _ { \\hat { \\mathcal { A } } ( G ) } = \\Omega ( \\frac { 1 } { r _ n \\epsilon } ) . \\end{align*}"}
-{"id": "2440.png", "formula": "\\begin{align*} \\widetilde { \\varphi _ { \\ell } ^ { H _ 1 } } ( x _ 0 , x _ 1 ) = ( x _ 0 ^ \\ell + x _ 1 ^ \\ell ) + \\sum _ { 0 < j < \\ell , j \\equiv 0 \\pmod { 4 } } \\frac { ( - 1 ) ^ { \\ell / 4 } \\binom { \\ell } { j } } { ( ( - 1 ) ^ { \\ell / 4 } + 2 ^ { \\ell / 2 - 2 } ) } x _ 0 ^ { \\ell - j } x _ 1 ^ j . \\end{align*}"}
-{"id": "4766.png", "formula": "\\begin{align*} ( \\partial _ t \\tilde { u } _ N , w _ j ) _ { \\mathcal { H } } = 2 \\nu a ( \\tilde { u } _ N , w _ j ) + 2 \\nu a ( H , w _ j ) + b _ R ( \\tilde { u } _ N , \\tilde { u } _ N , w _ j ) - b ( H , w _ j , \\tilde { u } _ N ) - b ( \\tilde { u } _ N , w _ j , H ) , \\tilde { u } _ N | _ { t = 0 } = \\tilde { u } _ { N 0 } . \\end{align*}"}
-{"id": "1486.png", "formula": "\\begin{align*} W ^ { j } _ { 5 } = - \\varepsilon \\sum _ { i = 1 } ^ { d } \\int _ { \\mathbb { T } ^ { d } } ( \\partial _ { x _ { j } } w ^ { 1 } - \\partial _ { x _ { j } } w ^ { 2 } ) \\left ( \\Theta _ { p _ { i } } ( t , x , \\mu ^ { 1 } , D w ^ { 1 } ) \\partial ^ { 2 } _ { x _ { i } x _ { j } } w ^ { 1 } - \\Theta _ { p _ { i } } ( t , x , \\mu ^ { 2 } , D w ^ { 2 } ) \\partial ^ { 2 } _ { x _ { i } x _ { j } } w ^ { 1 } \\right ) \\ d x , \\end{align*}"}
-{"id": "303.png", "formula": "\\begin{align*} \\widehat { G } \\ ; = \\ ; G _ 1 \\times G _ 2 \\times Z \\ ; \\twoheadrightarrow \\ ; G ^ \\circ \\end{align*}"}
-{"id": "1040.png", "formula": "\\begin{align*} & \\sup _ { y _ 1 , \\cdots , y _ k \\in [ 0 , 2 \\pi ) } \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) \\leq ( n / 4 ) ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } \\prod _ { j = 1 } ^ k D _ n ( F _ n ( x _ j ) / 2 ) \\\\ & = \\prod _ { j = 1 } ^ k ( n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } D _ n ( F _ n ( x _ j ) / 2 ) / 4 ) \\to \\prod _ { j = 1 } ^ k \\left ( e ^ { c _ 0 - x _ j } / 4 \\right ) , \\ , \\ , \\ , n \\to + \\infty , \\end{align*}"}
-{"id": "5466.png", "formula": "\\begin{align*} K _ t \\ & : = \\ Y _ t - Y _ 0 - g ( \\hat X ) - \\int _ 0 ^ t f _ s ( \\hat X , \\hat I _ s ) \\ , d s + \\int _ 0 ^ t \\int _ \\Lambda R _ s ( b ) \\ , \\hat \\theta ( d s \\ , d b ) \\\\ & \\ + \\int _ 0 ^ t Z _ s \\ , d \\hat W _ s + \\int _ 0 ^ t \\int _ U L _ s ( z ) \\ , ( \\hat \\pi ( d s \\ , d z ) - \\lambda _ \\pi ( d z ) d s ) , \\qquad 0 \\leq t \\leq T . \\end{align*}"}
-{"id": "9543.png", "formula": "\\begin{align*} N _ { k , N } ( n ) & : = \\sum _ { m = - \\infty } ^ { \\infty } m ^ k N _ { S _ 1 } ( m , n ) , \\\\ M _ { k , N } ( n ) & : = \\sum _ { m = - \\infty } ^ { \\infty } m ^ k M _ { S _ 2 } ( m , n ) , \\end{align*}"}
-{"id": "7592.png", "formula": "\\begin{align*} \\begin{gathered} \\left [ \\eta \\cdot \\nabla , \\Gamma \\right ] \\left ( \\tau \\circ X ^ { - 1 } \\right ) = \\eta ( t ) \\cdot \\nabla \\Gamma \\left ( \\tau \\circ X ^ { - 1 } \\right ) - \\Gamma \\left ( \\eta \\cdot \\nabla \\left ( \\tau \\circ X ^ { - 1 } \\right ) \\right ) \\end{gathered} \\end{align*}"}
-{"id": "432.png", "formula": "\\begin{align*} \\tilde { \\mathcal { D } } _ t : = \\int _ D W _ t ( x ) \\ , \\mu ( \\dd x ) . \\end{align*}"}
-{"id": "3257.png", "formula": "\\begin{align*} \\mathcal { P } _ n ( e ^ { i \\theta } a + e ^ { - i \\theta } b ) & = ( e ^ { i \\theta } a + e ^ { - i \\theta } b ) ( e ^ { i ( n - 1 ) \\theta } a ^ { n - 1 } + e ^ { - i ( n - 1 ) \\theta } b ^ { n - 1 } ) \\\\ & \\quad \\quad \\quad - a b ( e ^ { i ( n - 2 ) \\theta } a ^ { n - 2 } + e ^ { - i ( n - 2 ) \\theta } b ^ { n - 2 } ) \\\\ & = e ^ { i n \\theta } a ^ n + e ^ { - i n \\theta } b ^ n . \\end{align*}"}
-{"id": "8632.png", "formula": "\\begin{align*} \\begin{array} { c } p _ \\varepsilon = 2 c _ 0 \\frac { u _ \\varepsilon ^ 2 + z _ \\varepsilon ^ 2 } { u _ \\varepsilon ^ 4 + c _ 0 ^ 2 } - 2 c _ 0 + O ( \\varepsilon ) e ^ { - \\left ( \\sqrt { 2 } \\sqrt { 1 - 4 c _ 0 ^ 2 } + O ( \\varepsilon ) \\right ) | x | } , \\\\ \\\\ q _ \\varepsilon = O ( \\varepsilon ) e ^ { - \\left ( \\sqrt { 2 } \\sqrt { 1 - 4 c _ 0 ^ 2 } + O ( \\varepsilon ) \\right ) | x | } , \\end{array} \\end{align*}"}
-{"id": "9197.png", "formula": "\\begin{align*} \\dim ( H _ { b } \\cap T _ { \\gamma \\left ( b \\right ) } \\tilde { N } ) & = \\dim H _ { b } + \\dim T _ { \\gamma \\left ( b \\right ) } \\tilde { N } - \\dim ( H _ { b } + T _ { \\gamma \\left ( b \\right ) } \\tilde { N } ) \\\\ & \\geq n - \\dim \\left ( \\mathcal { K } \\right ) + \\tilde { n } - \\left ( m - 1 - \\dim \\left ( \\mathcal { K } \\right ) \\right ) \\\\ & \\geq n + \\tilde { n } - m + 1 \\\\ & \\geq k , \\end{align*}"}
-{"id": "5509.png", "formula": "\\begin{align*} & \\tilde { \\varepsilon } _ T ^ { - 1 } \\frac { J 2 ^ { J d / 2 } ( \\log T ) ^ \\eta } { \\sqrt { T } } E ^ { \\Pi ^ { D _ T } } \\Big [ \\max _ { \\lambda \\leq J , k , j } \\sqrt { T } | \\langle b _ j - b _ { 0 , j } , a _ \\lambda \\Phi _ { \\lambda , k } \\rangle _ { L ^ 2 } | \\big | X ^ T \\Big ] \\\\ & = O _ { P _ { b _ 0 } } \\Big ( \\tilde { \\varepsilon } _ T ^ { - 1 } \\frac { J ^ { 3 / 2 } 2 ^ { J d / 2 } ( \\log T ) ^ \\eta } { \\sqrt { T } } \\Big ) = O _ { P _ { b _ 0 } } \\Big ( ( \\log T ) ^ { 3 / 2 + \\eta - \\delta } \\Big ) . \\end{align*}"}
-{"id": "1681.png", "formula": "\\begin{align*} \\frac { { \\rm { D } } } { \\partial \\overline { z ^ { i } } } w = \\frac { \\partial w } { \\partial \\overline { z ^ { i } } } + w \\frac { \\partial K } { \\partial \\overline { z ^ { i } } } = 0 , ~ ~ i = 1 , 2 , \\cdots , n \\end{align*}"}
-{"id": "5659.png", "formula": "\\begin{align*} { \\rm I } _ { 2 } \\leq & \\sum _ { j = 0 } ^ { n - 2 } \\int _ { t _ j } ^ { t _ { j + 1 } } \\| A _ h ^ { \\frac { s } { 2 } } ( \\bar E _ h ( t _ n - t ) - \\bar E _ h ( t _ n - t _ j ) ) P _ h \\| ^ p \\d t \\\\ & + \\int _ { t _ { n - 1 } } ^ { t _ n } \\| A _ h ^ { \\frac { s } { 2 } } ( \\bar E _ h ( t _ n - t ) - \\bar E _ h ( \\tau ) ) P _ h \\| ^ p \\d t : = { \\rm I } _ { 2 , 1 } + { \\rm I } _ { 2 , 2 } . \\end{align*}"}
-{"id": "9174.png", "formula": "\\begin{align*} \\left \\{ J ( t ) \\ | \\ J \\in \\Lambda \\right \\} = \\dot { \\gamma } \\left ( t \\right ) ^ { \\perp } \\end{align*}"}
-{"id": "5383.png", "formula": "\\begin{align*} \\frac { k } { n } c _ { n - 1 \\ , m - 3 } | | v _ { n m } ^ k | | ^ 2 = - \\overline { b _ { n m } } | | v _ { n + 1 \\ , m - 3 } ^ { k + 1 } | | ^ 2 . \\end{align*}"}
-{"id": "4604.png", "formula": "\\begin{align*} S _ 1 ( S _ 1 ^ { - 1 } ( x ) - S _ 1 ^ { - 1 } ( s _ 1 ) ) + s _ 1 = x _ 1 . \\end{align*}"}
-{"id": "4220.png", "formula": "\\begin{align*} E _ { i n } ( z ) = \\frac { \\sqrt { 3 } } { 2 \\pi } n ^ { 2 \\beta } \\left ( \\frac { f ( z ) } { z } \\right ) ^ { \\frac { 2 \\beta } { 3 } } E _ n ( z ) \\left ( \\mathbb { I } + C _ n ( z ) \\right ) , \\end{align*}"}
-{"id": "3094.png", "formula": "\\begin{align*} m \\{ x : | T _ { 1 1 } f | > t \\} & \\le \\frac { C ( n ) \\| f \\| _ 1 } { t } , \\end{align*}"}
-{"id": "7271.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\widehat { \\phi ^ { 2 } } ( k ) - \\mu \\hat \\phi ( k ) + k ^ { - 2 } \\hat \\phi ( k ) = 0 \\mbox { f o r a l l } k \\neq 0 , \\end{align*}"}
-{"id": "3504.png", "formula": "\\begin{align*} \\frac { \\sin \\pi s } { \\sin \\pi t } = \\frac { \\sin \\pi ( s - t ) } { \\sin \\pi t } e ^ { i \\pi t } + e ^ { - i \\pi ( s - t ) } , \\end{align*}"}
-{"id": "3222.png", "formula": "\\begin{align*} \\| F ( x ^ k ) \\circ V _ k - \\mu _ { k - 1 } I \\| _ F = O ( \\mu _ { k - 1 } ^ { 1 + \\alpha } ) . \\end{align*}"}
-{"id": "6173.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ( U _ n g _ 1 ) = 0 , \\\\ \\Delta ( - \\bar U _ n g _ 2 ) = 0 , \\\\ g _ 1 = g _ 2 , ( g _ 1 ) _ r + ( g _ 2 ) _ r = 0 , \\end{cases} \\end{align*}"}
-{"id": "5049.png", "formula": "\\begin{align*} [ L _ b ^ { \\mathcal P } , [ L _ a ^ { \\mathcal P } , { \\mathcal P } ] ] = - [ L _ { { \\mathcal F } ( a , b ) } ^ { \\mathcal P } , { \\mathcal P } ] \\textrm { , f o r a l l $ a , b \\in \\mathbb V $ . } \\end{align*}"}
-{"id": "6272.png", "formula": "\\begin{align*} \\hat { \\theta } _ { n } : = - \\frac { \\sum _ { i = 1 } ^ n X _ { t _ { i - 1 } } ( X _ { t _ i } - X _ { t _ { i - 1 } } ) } { \\Delta _ n \\sum _ { i = 1 } ^ n X _ { t _ i } ^ 2 } , \\end{align*}"}
-{"id": "2630.png", "formula": "\\begin{align*} K ^ { ( \\mathrm { s o f t } ) } ( \\kappa , \\pi ; u , v ) = \\begin{bmatrix} D S ^ { ( \\mathrm { s o f t } ) } ( \\kappa , \\pi ; u , v ) & S ^ { ( \\mathrm { s o f t } ) } ( \\kappa , \\pi ; u , v ) \\\\ - S ^ { ( \\mathrm { s o f t } ) } ( \\kappa , \\pi ; v , u ) & I S ^ { ( \\mathrm { s o f t } ) } ( \\kappa , \\pi ; u , v ) \\end{bmatrix} . \\end{align*}"}
-{"id": "5532.png", "formula": "\\begin{align*} \\sup _ { b \\in D _ T } | S _ T ( b ) | \\lesssim | u | \\sqrt { T } ( \\log T ) ^ { - \\eta } ( 2 ^ { - 2 J } M _ T \\varepsilon _ T + 2 ^ { - J ( s + 1 + d / 2 ) } ) = | u | \\times o ( 1 ) , \\end{align*}"}
-{"id": "2574.png", "formula": "\\begin{align*} \\langle v _ { t } , \\phi ^ * \\rangle _ m = \\langle \\phi ^ { - 1 } v _ t \\phi , \\phi ^ * \\rangle _ m \\leq \\| \\phi ^ { - 1 } v _ t \\| _ \\infty \\cdot \\langle \\phi , \\phi ^ * \\rangle _ m \\xrightarrow [ t \\to \\infty ] { } 0 . \\end{align*}"}
-{"id": "1625.png", "formula": "\\begin{align*} r V _ r ( x ) = \\begin{cases} \\mu ( x _ r ^ \\ast ) x _ r ^ \\ast + r ( x - x _ r ^ \\ast ) , & x \\geq x _ r ^ \\ast , \\\\ \\mu ( x _ r ^ \\ast ) x _ r ^ \\ast - r \\int _ { x } ^ { x _ r ^ \\ast } \\frac { \\psi _ r ' ( t ) } { \\psi _ r ' ( x _ r ^ \\ast ) } d t , & x < x _ r ^ \\ast . \\end{cases} \\end{align*}"}
-{"id": "7353.png", "formula": "\\begin{align*} E _ A = \\bigcup \\{ [ a _ 1 , a _ 2 ] \\mid a _ 1 , a _ 2 \\in A \\} \\end{align*}"}
-{"id": "6018.png", "formula": "\\begin{align*} Q ( \\cdot ) = \\int _ { \\R ^ d } Q ^ { \\omega ( 0 ) = x } ( \\cdot ) Q ^ 0 ( d x ) , \\end{align*}"}
-{"id": "5381.png", "formula": "\\begin{align*} a _ { n - 1 \\ , m - 3 } \\binom { n - 1 } { k - 1 } | | v _ { n m } ^ k | | ^ 2 = - \\overline { d _ { n m } } \\binom { n - 2 } { k - 1 } | | v _ { n - 1 \\ , m - 3 } ^ k | | ^ 2 \\end{align*}"}
-{"id": "1443.png", "formula": "\\begin{align*} M _ { n } = \\sup _ { t \\in [ 0 , T ] } \\left ( \\| D w ^ { n } \\| _ { s - 1 } ^ { 2 } + \\| \\mu ^ { n } \\| _ { s - 1 } ^ { 2 } \\right ) , \\end{align*}"}
-{"id": "9961.png", "formula": "\\begin{align*} \\begin{aligned} ( X , i , f , j , g ) & \\mapsto ( Y , \\varepsilon j , g ) \\\\ ( X , i , f , j , g h ) \\xrightarrow { \\smash h } ( X , i , f , j , g ) & \\mapsto ( Y , \\varepsilon j , g h ) \\xrightarrow { \\smash h } ( Y , \\varepsilon j , g ) \\rlap { , } \\end{aligned} \\end{align*}"}
-{"id": "4226.png", "formula": "\\begin{align*} \\mathrm { c l } ( N ^ { w } b _ { 0 } ) = \\gamma _ { 1 } \\cdots \\gamma _ { n } \\left ( \\mathrm { c l } \\left ( w N w ^ { - 1 } w b _ { 0 } \\right ) \\right ) = \\gamma _ { 1 } \\cdots \\gamma _ { n } \\{ w b _ { 0 } \\} \\end{align*}"}
-{"id": "512.png", "formula": "\\begin{align*} & ( a ) a = ( [ 1 ] ^ 6 , [ 2 ] ^ { g - 6 } ) \\mbox { o r } \\\\ & ( b ) a = ( [ 1 ] ^ 4 , [ 3 ] ^ 2 , [ 2 ] ^ { g - 6 } ) , \\end{align*}"}
-{"id": "4718.png", "formula": "\\begin{align*} \\Phi ( x _ { j \\pm a } ) = 1 , \\ ; \\ ; a = 1 , 2 , \\ldots , M , \\end{align*}"}
-{"id": "1534.png", "formula": "\\begin{align*} v ( x ) = \\int _ { 0 } ^ { 1 } { g ( m , x , y ) f ( y ) d y } , x \\in I , \\end{align*}"}
-{"id": "8918.png", "formula": "\\begin{align*} n ( \\textrm { U n c h a n g e d } ) & = | \\textrm { U n c h a n g e d } | \\\\ n ( \\textrm { I n c o r r e c t , U n c h a n g e d } ) & = | \\textrm { I n c o r r e c t } \\cap \\textrm { U n c h a n g e d } | , \\end{align*}"}
-{"id": "3799.png", "formula": "\\begin{align*} m _ { a p ( \\alpha ) } ( \\{ b , a - b + 1 \\} ) & = m _ { a p ( \\beta ) } ( \\{ b , a - b + 1 \\} ) + 1 \\\\ m _ { a p ( \\beta ) } ( \\{ b - 1 , a - b + 1 \\} ) & = m _ { a p ( \\alpha ) } ( \\{ b - 1 , a - b + 1 \\} ) + 1 \\\\ m _ { a p ( \\alpha ) } ( \\{ x , y \\} ) & = m _ { a p ( \\beta ) } ( \\{ x , y \\} ) \\hbox { o t h e r w i s e . } \\end{align*}"}
-{"id": "7258.png", "formula": "\\begin{align*} \\ddot \\phi _ k ( 0 ) = D ^ 2 _ { \\phi \\phi } \\psi ( 0 , \\mu ^ * _ { k } ) [ \\phi ^ * _ k , \\phi ^ * _ k ] . \\end{align*}"}
-{"id": "9347.png", "formula": "\\begin{align*} \\log _ 2 \\lvert G _ k \\rvert = 2 ^ k + 2 ^ { k - 1 } + k + 2 . \\end{align*}"}
-{"id": "6123.png", "formula": "\\begin{align*} \\begin{aligned} \\alpha _ { d - \\mu _ 1 } & = - ( q _ { 0 , d - \\mu } B _ { 0 , \\mu _ 2 } + q _ { 1 , d - \\mu } B _ { 1 , \\mu _ 2 } + q _ { 2 , d - \\mu } B _ { 2 , \\mu _ 2 } ) \\\\ \\beta _ { d - \\mu _ 2 } & = q _ { 0 , d - \\mu } A _ { 0 , \\mu _ 1 } + q _ { 1 , d - \\mu } A _ { 1 , \\mu _ 1 } + q _ { 2 , d - \\mu } A _ { 2 , \\mu _ 1 } . \\end{aligned} \\end{align*}"}
-{"id": "6615.png", "formula": "\\begin{align*} 0 & \\leq \\prod _ { t = \\eta + 1 } ^ { \\infty } \\left ( 1 - \\frac { ( 1 - 2 \\varepsilon ( - \\varphi ) ^ { - n } ) ^ 2 } { u _ t ^ 2 } \\right ) \\\\ & \\leq \\left ( 1 - ( 1 - 2 \\varepsilon ( - \\varphi ) ^ { - n } ) ^ 2 \\sum _ { t = 1 } ^ { \\infty } \\frac { 1 } { u _ { t + \\eta } ^ 2 } \\right ) ^ { - 1 } \\\\ & \\leq \\left ( 1 - \\frac { 1 } { 4 } \\sum _ { t = 1 } ^ { \\infty } \\frac { 1 } { ( t + \\eta - 1 ) ^ 2 } \\right ) ^ { - 1 } = \\left ( 1 - \\mathcal { O } ( \\eta ^ { - 1 } ) \\right ) ^ { - 1 } , \\\\ \\end{align*}"}
-{"id": "871.png", "formula": "\\begin{gather*} u ^ * + v ^ * \\in \\overline { N } _ D ( u ) , - v ^ * \\in \\overline { N } _ K ( u + v ) , \\norm { u } = \\norm { v ^ * } _ * = 1 . \\end{gather*}"}
-{"id": "2866.png", "formula": "\\begin{align*} 2 \\pi \\norm { \\beta ^ * } _ 1 = \\inf _ g \\ \\norm { s ( g ) } _ { H _ g } \\norm { \\beta } _ { H _ g } \\end{align*}"}
-{"id": "7066.png", "formula": "\\begin{align*} \\sum _ { f , g \\in L } T _ { f a } T _ { g b } N _ { f g } ^ c & = \\sum _ { d , e \\in L } T _ { d b } T _ { c e } N _ { a d } ^ e . \\end{align*}"}
-{"id": "6308.png", "formula": "\\begin{align*} | E _ { G ' } ( S _ 0 , S _ 0 ) + E _ { G ' } ( S _ 0 , S ^ c _ 0 ) - e ( G ' ) \\left ( k ( n - k ) + \\binom { k } { 2 } \\right ) | \\leq C \\max \\{ \\sqrt { \\rho } , \\sqrt { \\frac { \\log n } { n } } \\} k \\sqrt { n \\log n } . \\end{align*}"}
-{"id": "2529.png", "formula": "\\begin{align*} V _ t f ( x ) + \\Pi _ x \\Big [ \\int _ 0 ^ { t \\wedge \\zeta } \\psi ( \\xi _ s , V _ { t - s } f ) d s \\Big ] = \\Pi _ x [ f ( \\xi _ t ) \\mathbf 1 _ { t < \\zeta } ] , t \\geq 0 , x \\in E . \\end{align*}"}
-{"id": "6598.png", "formula": "\\begin{align*} s _ { n t } : = 2 \\sin \\pi \\left ( \\frac { t } { F _ n } - \\varphi ^ n \\left ( \\left \\{ \\frac { t F _ { n - 1 } } { F _ n } \\right \\} - 1 / 2 \\right ) \\right ) . \\end{align*}"}
-{"id": "1765.png", "formula": "\\begin{align*} M _ { s ; t _ { 1 } , \\cdots , t _ { n } } = g ( B _ { s ; t _ { 1 } , \\cdots , t _ { n } } ) = \\max _ { 1 \\leq i \\leq n } \\left ( B _ { t _ { i } } - B _ { s } \\right ) . \\end{align*}"}
-{"id": "3036.png", "formula": "\\begin{align*} \\int \\limits _ { \\R _ + ^ d } \\left ( | z | \\wedge | z | ^ 2 + \\sum \\limits _ { j \\in \\{ 1 , \\dots , d \\} \\backslash \\{ i \\} } ( 1 \\wedge z _ j ) \\right ) \\mu _ i ( d z ) < \\infty , \\ \\ \\mu _ i ( \\{ 0 \\} ) = 0 . \\end{align*}"}
-{"id": "2178.png", "formula": "\\begin{align*} \\tilde K _ \\ell ( \\mathbf x , \\mathbf y ) = \\tau _ { - \\mathbf y } ( \\mathcal F ^ { - 1 } m _ \\ell ) ( \\mathbf x ) , \\end{align*}"}
-{"id": "7905.png", "formula": "\\begin{align*} N ^ * = \\left \\lceil \\frac { 1 - ( n - 1 ) ^ 2 / ( \\hat { R } _ { r 0 } ) ^ { 1 / ( n - 1 ) } } { 1 - ( n - 1 ) / ( \\hat { R } _ { r 0 } ) ^ { 1 / ( n - 1 ) } } \\right \\rceil \\end{align*}"}
-{"id": "1090.png", "formula": "\\begin{align*} & \\lim _ { n \\to + \\infty } n ( 2 \\ln n ) ^ { - \\frac { 1 } { 2 } } D _ n ( G _ n ( x ) / 2 ) = e ^ { c _ 0 - x } , \\end{align*}"}
-{"id": "9206.png", "formula": "\\begin{align*} \\bar { H } ( t , s ) : = \\begin{cases} \\bar { L } ( t , s ) , \\quad 0 \\leq t \\leq h ( s ) , \\\\ \\bar { R } ( t , s ) , \\quad h ( s ) \\leq t \\leq 1 , \\end{cases} \\end{align*}"}
-{"id": "3619.png", "formula": "\\begin{align*} a _ 1 \\cdot ( ( a _ 2 - a _ { - 2 } ) \\cdot v _ { ( 1 , 3 ) } ) = \\frac { 1 } { 2 ^ 5 } ( a _ 2 - a _ { - 2 } ) \\cdot v _ { ( 1 , 3 ) } . \\end{align*}"}
-{"id": "4390.png", "formula": "\\begin{align*} T ^ { \\pm } _ 1 ( x , y ) = x y \\mp 1 , T _ 1 ( x , y ) = T _ 1 ^ - ( x , y ) T _ 1 ^ + ( x , y ) = x ^ 2 y ^ 2 - 1 , \\end{align*}"}
-{"id": "1735.png", "formula": "\\begin{align*} \\lVert u \\rVert _ { r , p } = \\lVert ( I - L ) ^ { - \\frac { r } { 2 } } u \\rVert _ { p } . \\end{align*}"}
-{"id": "6186.png", "formula": "\\begin{align*} \\gamma _ R ( X ) = - \\frac { | x ' | ^ 2 } { 2 R } + 2 ( n - 1 ) \\frac { x _ n r } { R } + 2 \\beta \\frac { r } { R } , \\bar \\gamma _ R ( X ) = - \\frac { | x ' | ^ 2 } { 2 R } - 2 ( n - 1 ) \\frac { x _ n r } { R } - 2 \\beta \\frac { r } { R } \\end{align*}"}
-{"id": "4494.png", "formula": "\\begin{align*} w = \\sqrt { 1 + | \\nabla v | ^ 2 } \\textrm { a n d } \\gamma ^ { i j } = \\sigma _ { i j } - \\frac { \\nabla _ i v \\nabla _ j v } { w ( 1 + w ) } . \\end{align*}"}
-{"id": "7069.png", "formula": "\\begin{align*} \\epsilon ( x ) = \\epsilon ( y ) = 0 . \\end{align*}"}
-{"id": "3942.png", "formula": "\\begin{align*} \\frac { d } { d \\theta } f ( m _ i + \\theta \\tilde m _ i , m _ { - i } ) \\big | _ { \\theta = 0 } = \\int _ { \\R ^ n } \\frac { \\partial f } { \\partial m _ i } ( m ) ( \\xi ) \\ , \\tilde m _ i ( \\xi ) \\ , d \\xi , \\end{align*}"}
-{"id": "4971.png", "formula": "\\begin{align*} \\forall y \\in C \\left ( U ( y ) = y \\to \\langle \\tilde { x } - v _ 0 , \\tilde { x } - y \\rangle < \\frac { 1 } { k + 1 } \\right ) . \\end{align*}"}
-{"id": "4641.png", "formula": "\\begin{align*} f _ { 1 , \\lambda _ i } ^ { m _ i } ( 0 ) = c _ 1 ( \\lambda _ i ) f _ { 2 , \\lambda _ i } ^ { n _ i } ( 0 ) = c _ 2 ( \\lambda _ i ) . \\end{align*}"}
-{"id": "5288.png", "formula": "\\begin{align*} \\mathrm { g r } ( \\mu ) \\circ \\mathrm { c a n } \\circ \\big ( \\mathrm { g r } \\ , E ( \\iota _ V ) \\otimes \\mathrm { g r } \\ , E ( \\iota _ W ) \\big ) & = \\mathrm { g r } ( \\mu ) \\circ \\mathrm { g r } ( E ( \\iota _ V ) \\otimes E ( \\iota _ W ) ) \\circ \\mathrm { c a n } \\\\ & = \\mathrm { g r } ( \\phi ) \\circ \\mathrm { c a n } \\end{align*}"}
-{"id": "169.png", "formula": "\\begin{align*} c _ i \\circ q _ M \\textrm { a n d } l _ { d c _ i } = \\sum _ { s = 1 } ^ { N } \\dfrac { \\partial c _ i } { \\partial x _ s } ( { \\bf x } ) y _ s , i = 1 , \\dots r , \\end{align*}"}
-{"id": "5846.png", "formula": "\\begin{align*} \\frac { \\prod _ { i = 1 } ^ k \\Gamma _ { E _ i } ( c _ i ) } { \\Gamma ( c ) } \\end{align*}"}
-{"id": "2956.png", "formula": "\\begin{align*} \\mathcal { O } _ { \\chi ^ * } = \\mathcal { O } _ { \\chi } ^ * . \\end{align*}"}
-{"id": "917.png", "formula": "\\begin{align*} C _ { j _ k \\ldots j _ 1 } = \\int \\limits _ { [ t , T ] ^ k } K ( t _ 1 , \\ldots , t _ k ) \\prod _ { l = 1 } ^ { k } \\phi _ { j _ l } ( t _ l ) d t _ 1 \\ldots d t _ k , \\end{align*}"}
-{"id": "7355.png", "formula": "\\begin{align*} d ( x , x ' ) & = d ( x , Y ) + d ( Y , Y ' ) + d ( Y ' , x ' ) \\\\ \\tag * { a n d } d ( y , y ' ) & = d ( y , Y ) + d ( Y , Y ' ) + d ( Y ' , y ' ) \\end{align*}"}
-{"id": "4646.png", "formula": "\\begin{align*} \\binom { d } { 2 } K ^ 2 _ S = P _ d ( S ) + q ( S ) - p _ g ( S ) - 1 . \\end{align*}"}
-{"id": "2600.png", "formula": "\\begin{align*} \\Pi _ x ^ { ( \\phi ) } [ ( \\phi ^ { - 1 } f ) ( \\xi _ t ) ] = \\phi ( x ) ^ { - 1 } \\Pi _ x [ f ( \\xi _ t ) e ^ { - \\int _ 0 ^ t \\beta ( \\xi _ s ) d s } ] = \\phi ( x ) ^ { - 1 } P ^ \\beta _ t f ( x ) \\xrightarrow [ t \\to \\infty ] { } 1 , x \\in E . \\end{align*}"}
-{"id": "7610.png", "formula": "\\begin{align*} \\begin{gathered} \\mathcal { P } = C ^ { \\beta } ( 0 , T ; C ^ { 1 + \\alpha , p } ) \\times C ^ { \\beta } ( 0 , T ; C ^ { \\alpha , p } ) \\times L ^ \\infty ( 0 , T ; C ^ { 1 + \\alpha , p } ) , \\\\ \\norm { ( X - \\mathrm { I d } , \\tau , v ) } _ { \\mathcal { P } } = \\norm { X - \\mathrm { I d } } _ { C ^ \\beta ( 0 , T ; C ^ { 1 + \\alpha , p } ) } + \\norm { \\tau } _ { C ^ \\beta ( 0 , T ; C ^ { \\alpha , p } ) } + \\delta \\norm { v } _ { L ^ \\infty ( 0 , T ; C ^ { 1 + \\alpha , p } ) } \\end{gathered} \\end{align*}"}
-{"id": "735.png", "formula": "\\begin{align*} \\binom { q - 1 } { - 1 } \\binom { q - 1 } { 1 } \\frac { \\Gamma ( n + q ) } { ( m - 1 ) ! } = 0 , ~ ~ ~ ~ q > - 1 \\backslash \\{ 0 \\} , \\end{align*}"}
-{"id": "1809.png", "formula": "\\begin{align*} u _ { t } ^ { ( n ) } ( s ) = \\frac { \\int _ { 0 } ^ { ( i + 1 ) 2 ^ { - n } t } u _ { t } ( r ) d r - \\int _ { 0 } ^ { i 2 ^ { - n } t } u _ { t } ( r ) d r } { 2 ^ { - n } t } \\to u _ { t } ( s ) \\end{align*}"}
-{"id": "1529.png", "formula": "\\begin{align*} u _ 1 ( x , t ) = 2 \\sum _ { n = 1 } ^ { \\infty } { \\left ( \\int _ { 0 } ^ { 1 } { f ( y ) \\sin { ( n \\pi y ) } d y } \\right ) e ^ { - n ^ 2 \\pi ^ 2 t } \\sin { ( n \\pi x ) } } , \\ ; x \\in I , \\ , t \\ge 0 . \\end{align*}"}
-{"id": "3256.png", "formula": "\\begin{align*} \\mathcal { P } _ n ( e ^ { i \\theta } a + e ^ { - i \\theta } b ) = e ^ { i n \\theta } a ^ n + e ^ { - i n \\theta } b ^ n . \\end{align*}"}
-{"id": "1284.png", "formula": "\\begin{align*} | K ( z , w ) | = O ( | \\eta | ^ { - 1 - k } ) , | \\eta | > 1 . \\end{align*}"}
-{"id": "6352.png", "formula": "\\begin{align*} \\mathbb { P } _ { e , M | p } = \\int _ 0 ^ \\infty \\int _ 0 ^ \\infty \\mathbb { P } _ { e , M | p , h } f _ { h _ { 1 1 } } ( h _ { 1 1 } ) f _ { h _ { 2 1 } } ( h _ { 2 1 } ) d h _ { 1 1 } d h _ { 2 1 } , \\end{align*}"}
-{"id": "6850.png", "formula": "\\begin{align*} u _ t + u \\nabla u + \\nabla p = 0 \\ , , \\end{align*}"}
-{"id": "7234.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\left ( 2 \\mu - \\phi ( x ) - \\phi ( y ) \\right ) \\left ( \\phi ( y ) - \\phi ( x ) \\right ) = K _ r * \\phi ( x ) - K _ r * \\phi ( y ) \\end{align*}"}
-{"id": "3128.png", "formula": "\\begin{align*} R ( u , v ) = \\mathbf { I } \\otimes \\mathbf { I } + g ( u , v ) \\mathbf { P } , g ( u , v ) = \\frac { c } { u - v } . \\end{align*}"}
-{"id": "8935.png", "formula": "\\begin{align*} R _ { \\mathrm { T S T } } ( n , s , a , b ) : = \\adjustlimits \\inf _ { \\phi } \\sup _ { x , y } P \\big ( \\phi ( G , H ) \\neq \\mathbf { 1 } \\{ x = y \\} \\ , | \\ , x , y \\big ) , \\end{align*}"}
-{"id": "418.png", "formula": "\\begin{align*} P _ \\gamma : = - 8 \\pi ( 1 - \\mathbf { h } ) \\frac { 1 } { \\mathcal { G } _ { g } ^ \\gamma ( M ) ^ 2 } \\iint _ { M ^ 2 } G _ g ( x , x ' ) d \\mathcal { G } _ { g } ^ \\gamma ( \\dd x ) \\mathcal { G } _ { g } ^ \\gamma ( \\dd x ' ) + \\frac { 2 } { 1 - \\frac { \\gamma ^ 2 } { 4 } } \\frac { \\mathcal { D } _ { g } ^ \\gamma ( M ) } { \\mathcal { G } _ { g } ^ \\gamma ( M ) } . \\end{align*}"}
-{"id": "121.png", "formula": "\\begin{align*} 0 \\le \\zeta _ { j , N } & \\le \\sum _ { { \\ell = 1 } } ^ { \\infty } \\frac { d ^ { 2 \\ell + 1 } } { ( \\ell ! ) ^ 2 } { ( 1 + x _ j ) } ^ \\ell \\left ( 1 - x _ j - \\frac 1 N \\right ) ^ \\ell \\\\ & \\le \\sum _ { { \\ell = 1 } } ^ { \\infty } \\frac { d ^ { 2 \\ell + 1 } } { ( \\ell ! ) ^ 2 } { ( 1 - x _ j ^ 2 ) } ^ \\ell \\ , . \\end{align*}"}
-{"id": "55.png", "formula": "\\begin{align*} \\widetilde J ( x ) = J _ b \\ , , \\widetilde \\rho ( x ) = - 2 g ( J _ b ) \\int _ 0 ^ x k ( y ) \\ , d y + C \\ , , \\end{align*}"}
-{"id": "6887.png", "formula": "\\begin{align*} \\begin{cases} 2 \\cosh ( 2 s ) s '' + \\sinh ( 2 s ) \\big ( 2 ( s ' ) ^ 2 - ( \\theta ' ) ^ 2 - ( \\mu ' ) ^ 2 \\big ) = 0 \\ , , \\\\ \\cosh ( s ) \\mu '' + 2 \\sinh ( s ) s ' \\mu ' = 0 \\ , , \\\\ \\sinh ( s ) \\theta '' + 2 \\cosh ( s ) s ' \\theta ' = 0 \\ , . \\end{cases} \\end{align*}"}
-{"id": "2964.png", "formula": "\\begin{align*} H ( z , w ) = \\frac { 1 } { 2 \\pi i } \\int _ { \\sigma _ { r } } H ( z , v ) \\frac { 1 } { v - w } d v ( w \\in \\Delta _ { r } z \\in \\Delta _ { s } , v \\in \\sigma _ { r } ) . \\end{align*}"}
-{"id": "9590.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d z ^ 2 } \\left ( \\frac { ( q ) _ N } { ( z q ) _ N ( z ^ { - 1 } q ) _ N } \\right ) _ { z = 1 } = \\sum _ { n = 1 } ^ { \\infty } M _ { 2 , N } ( n ) q ^ n . \\end{align*}"}
-{"id": "4410.png", "formula": "\\begin{align*} \\tilde F | _ { S } = g . \\end{align*}"}
-{"id": "2107.png", "formula": "\\begin{align*} \\Phi _ \\omega ( s , t ) \\circ \\Phi _ \\omega ( \\tau , s ) = \\Phi _ \\omega ( \\tau , t ) , \\forall t _ 0 \\leq \\tau \\leq s \\leq t \\leq t _ 0 + T . \\end{align*}"}
-{"id": "2062.png", "formula": "\\begin{align*} \\rho = \\rho _ c , \\ ; \\ ; \\ ; - \\infty < z < + \\infty , \\ ; \\ ; \\ ; 0 \\leq \\varphi < 2 \\pi . \\end{align*}"}
-{"id": "2607.png", "formula": "\\begin{align*} ( y , u , t ) \\mapsto \\gamma _ 0 C _ X ^ { - 1 } \\mathbf 1 _ { \\gamma ( y ) = \\gamma _ 0 } \\kappa ( y ) \\phi ( y ) ^ { \\gamma _ 0 - 1 } u ^ { - 1 } M ( u t , r u ^ { \\frac { 1 } { \\gamma _ 0 - 1 } } , y ) ^ { \\gamma _ 0 - 1 } \\end{align*}"}
-{"id": "1224.png", "formula": "\\begin{gather*} \\forall \\xi > 0 , S ( \\xi ) = \\chi ( \\xi ) e ^ { i a \\ln | \\xi | } \\left ( A + B e ^ { 2 i a \\ln | \\xi | } \\frac { e ^ { i \\beta \\xi ^ 3 } } { \\xi ^ 3 } \\right ) , \\end{gather*}"}
-{"id": "8507.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { 2 \\pi } \\frac { ( \\dot { z } _ j ) ^ 2 } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 3 } = & \\frac { \\lambda i ( \\dot { z } _ 1 ) ^ 2 } { 2 \\pi } ( \\frac { 1 } { ( \\zeta ( \\alpha , t ) - z _ 1 ( t ) ) ^ 3 } - \\frac { 1 } { ( \\zeta ( \\alpha , t ) - z _ 2 ( t ) ) ^ 3 } ) \\\\ & + \\frac { \\lambda i ( ( \\dot { z } _ 1 ) ^ 2 - ( \\dot { z } _ 2 ) ^ 2 ) } { 2 \\pi } \\frac { 1 } { ( \\zeta ( \\alpha , t ) - z _ 2 ( t ) ) ^ 3 } : = I + \\it { I I } . \\end{align*}"}
-{"id": "5968.png", "formula": "\\begin{align*} d X ( t ) = q ( t , X ) d t + d W ( t ) , X ( 0 ) = 0 , \\end{align*}"}
-{"id": "7225.png", "formula": "\\begin{align*} ( \\mu - \\phi ) ^ 2 ( x ) = \\left [ ( \\mu - \\phi ) ^ 2 \\right ] ^ \\prime ( \\xi ) ( x - x _ 0 ) + \\mu ^ 2 \\end{align*}"}
-{"id": "3940.png", "formula": "\\begin{align*} ( \\phi \\psi ) ( f ) = \\sum f _ { 1 } \\sigma ( S f _ { 2 } | _ { H } ) \\sigma ( f _ { 3 } | _ { H } ) = \\sum f _ { 1 } \\sigma \\bigl ( \\bigl ( S ( f _ { 2 } ) f _ { 3 } \\bigr ) | _ { H } \\bigr ) = f \\ , . \\end{align*}"}
-{"id": "2565.png", "formula": "\\begin{align*} v _ { s + t } ( x ) + \\int _ 0 ^ t P ^ \\beta _ { t - r } \\psi _ 0 ( x , v _ { s + r } ) d r = P ^ \\beta _ t v _ s ( x ) \\in [ 0 , \\infty ) , s > 0 , t \\geq 0 , x \\in E . \\end{align*}"}
-{"id": "9508.png", "formula": "\\begin{align*} & ( i ) \\ ( - \\Delta _ p ) ^ s U = U ^ { p _ s ^ * - 1 } \\ \\mbox { i n } \\ \\mathbb { R } ^ { N } \\\\ & ( i i ) \\ \\vert \\vert U \\vert \\vert _ { s , p } ^ p = \\left \\Vert U \\right \\Vert _ { p _ s ^ * } ^ { p _ s ^ * } = S ^ { \\frac { N } { s p } } . \\end{align*}"}
-{"id": "6432.png", "formula": "\\begin{align*} \\alpha _ { j } = \\frac { 1 } { \\sqrt { \\gamma _ { j - 1 } } } , \\ j = 1 , \\dots , k , \\qquad \\beta _ { j } = \\sqrt { \\frac { \\delta _ { j } } { \\gamma _ { j - 1 } } } , \\ j = 1 , \\dots , k - 1 , \\end{align*}"}
-{"id": "503.png", "formula": "\\begin{align*} \\begin{aligned} & | L | = | \\phi ^ * ( w ) | = | 2 p + q | , & | K - L | = | \\psi ^ * ( \\bar { \\sigma } ( w ) ) | = | 2 p + q ' | , \\\\ & | L | = | \\phi ^ * ( w ' ) | = | 2 p ' + r | , & | K - L | = | \\psi ^ * ( \\bar { \\sigma } ( w ' ) ) | = | 2 p ' + r ' | . \\end{aligned} \\end{align*}"}
-{"id": "7827.png", "formula": "\\begin{align*} [ \\langle T x , S x \\rangle \\xi , \\xi ] = \\lambda \\| T \\| \\| S \\| \\end{align*}"}
-{"id": "3959.png", "formula": "\\begin{align*} \\phi ( x ) = \\gamma _ { \\delta } * \\mathbb { I } _ { B _ r } ( x ) \\ , , \\gamma _ { \\delta } ( x ) = \\gamma ( x / \\delta ) / \\delta \\ , , \\end{align*}"}
-{"id": "6918.png", "formula": "\\begin{align*} I _ \\theta \\subset \\big \\{ i \\leq n : \\ ; | u _ i | \\leq \\theta \\big \\} \\quad J _ \\beta \\subset \\big \\{ i \\leq n : \\ ; | u _ i | \\geq \\beta \\big \\} \\end{align*}"}
-{"id": "5979.png", "formula": "\\begin{align*} \\int _ 0 ^ n g ( t - \\lfloor t \\rfloor , q ( t ) ) d t = n \\int _ 0 ^ 1 g \\left ( n t - \\lfloor n t \\rfloor , \\frac { \\overline { q } ( t ) } { \\sqrt { n } } \\right ) d t . \\end{align*}"}
-{"id": "1157.png", "formula": "\\begin{align*} & \\widehat { \\varphi ( x ) } ( s ) = e ^ { - J ( x ) s ^ 2 / 2 } \\prod _ { j = 1 } ^ { + \\infty } \\cos s x _ j 2 . \\end{align*}"}
-{"id": "6737.png", "formula": "\\begin{align*} ( m _ i - b ) ( m _ i + b ) & = a c . \\\\ { m _ i } ^ 2 - b ^ 2 & = a c . \\\\ { m _ i } ^ 2 - a c & = b ^ 2 . \\\\ { m _ i } ^ 2 - a c & > 0 . \\\\ a c & < { m _ i } ^ 2 . \\end{align*}"}
-{"id": "6338.png", "formula": "\\begin{align*} & m _ { { \\rm L a g } } ( a , b , D ) = a + D b , \\\\ { \\it V a r } _ { { \\rm L a g } } & ( a , b , D ) = a + D ( b + b ^ 2 ) + 2 a b . \\end{align*}"}
-{"id": "4524.png", "formula": "\\begin{align*} \\sum _ { c _ 1 \\in C _ 1 } \\# S ( K , c _ 1 ) = \\# S ( K ) . \\end{align*}"}
-{"id": "4798.png", "formula": "\\begin{align*} \\begin{pmatrix} \\partial _ r + \\dfrac { a k _ { 1 / 2 } } { m r } & - ( m + a ) \\\\ - ( m - a ) & \\partial _ r + \\dfrac { a k _ { 1 / 2 } } { m r } \\end{pmatrix} \\cdot \\begin{pmatrix} f ^ + _ { m _ { 1 / 2 } , k _ { 1 / 2 } } \\\\ f ^ - _ { m _ { 1 / 2 } , k _ { 1 / 2 } } \\end{pmatrix} = 0 . \\end{align*}"}
-{"id": "5438.png", "formula": "\\begin{align*} \\varphi _ c ^ s ( T ) = ( \\varphi _ c ^ { \\lfloor s \\rfloor + 1 } ( T ) ) ^ { s - \\lfloor s \\rfloor } ( \\varphi _ c ^ { \\lfloor s \\rfloor } ( T ) ) ^ { 1 - s + \\lfloor s \\rfloor } , \\end{align*}"}
-{"id": "2095.png", "formula": "\\begin{align*} \\hat { B } _ 0 = \\mathrm { a r g } \\max _ { b _ 0 \\in \\{ 0 , 1 \\} } { { \\rm P r } ( \\textbf { y } _ R = y | b _ 0 , { b } _ 1 , \\cdots , { b } _ M ) } , \\end{align*}"}
-{"id": "4086.png", "formula": "\\begin{align*} \\mathbb K ^ { ( \\alpha , \\frac { 1 } { 2 } ) } ( x , y ) = \\frac { 1 } { 2 \\pi i ( x - y ) } \\begin{pmatrix} - 1 & 1 & 0 \\end{pmatrix} \\Phi _ { \\alpha , + } ^ { - 1 } ( y ) \\Phi _ { \\alpha , + } ( x ) \\begin{pmatrix} 1 \\\\ 1 \\\\ 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "1736.png", "formula": "\\begin{align*} C _ { r , p } \\left ( O \\right ) = \\inf \\left \\{ \\lVert \\phi \\rVert _ { p } ^ { p } : \\ ( I - L ) ^ { - \\frac { r } { 2 } } \\phi \\geq 1 \\ O , \\ ( I - L ) ^ { - \\frac { r } { 2 } } \\phi \\geq 0 \\ \\boldsymbol { W } \\right \\} , \\end{align*}"}
-{"id": "5984.png", "formula": "\\begin{align*} g ^ { ( t ) } ( s , q ) : = ( 1 - t ) g \\left ( t + s ( 1 - t ) , \\frac { q } { \\sqrt { 1 - t } } \\right ) . \\end{align*}"}
-{"id": "4678.png", "formula": "\\begin{align*} { B } _ j ^ d ( \\xi ) = \\Big ( \\frac { x _ j - \\xi } { x _ j - x _ { j - d } } \\Big ) { B } _ j ^ { d - 1 } ( \\xi ) + \\Big ( \\frac { \\xi - x _ { j - d - 1 } } { x _ { j - 1 } - x _ { j - d - 1 } } \\Big ) { B } _ { j - 1 } ^ { d - 1 } ( \\xi ) , \\ ; \\ ; \\xi \\in \\mathbb { R } . \\end{align*}"}
-{"id": "6125.png", "formula": "\\begin{align*} \\dim I _ i & = 3 ( i - \\mu + 1 ) - ( i - \\mu - \\mu _ 1 + 1 ) - ( i - \\mu - \\mu _ 1 + 1 ) \\\\ & = i - \\mu + \\mu _ 1 + \\mu _ 2 + 1 = i + 1 . \\end{align*}"}
-{"id": "3510.png", "formula": "\\begin{align*} f _ { \\gamma } ( t ) = \\log \\Gamma ( t _ 0 + k ^ { - 1 } t ) + \\frac { \\gamma t _ 0 - \\psi ( t _ 0 ) - \\gamma t _ 0 } { k } t - \\frac { \\gamma } { 2 k ^ 2 } t ^ 2 , \\end{align*}"}
-{"id": "7487.png", "formula": "\\begin{align*} \\gamma \\colon [ 0 , 1 ] \\to U , t \\mapsto \\gamma ( t ) = z ( t ) , \\end{align*}"}
-{"id": "944.png", "formula": "\\begin{align*} { \\sf M } \\left \\{ \\left ( J ^ { * } [ \\psi ^ { ( 4 ) } ] _ { T , t } - \\sum \\limits _ { j _ 1 , j _ 2 , j _ 3 , j _ 4 = 0 } ^ { p } C _ { j _ 4 j _ 3 j _ 2 j _ 1 } \\zeta _ { j _ 1 } ^ { ( i _ 1 ) } \\zeta _ { j _ 2 } ^ { ( i _ 2 ) } \\zeta _ { j _ 3 } ^ { ( i _ 3 ) } \\zeta _ { j _ 4 } ^ { ( i _ 4 ) } \\right ) ^ 2 \\right \\} \\le \\frac { C } { p ^ { 1 - \\varepsilon } } \\end{align*}"}
-{"id": "4774.png", "formula": "\\begin{align*} \\chi ( G ) > & ~ h _ 0 - O _ k \\big ( \\sqrt [ 2 k - 2 ] { \\log _ 2 n } \\big ) & k \\ge 3 , \\\\ \\chi ( G ) > & ~ h _ 0 - O \\big ( \\sqrt { \\log _ 2 n \\cdot \\log _ 2 \\log _ 2 n } \\big ) & k = 2 . \\end{align*}"}
-{"id": "2231.png", "formula": "\\begin{align*} \\delta ( x - y ) T ( y - z ) = \\delta ( x - y ) T ( x - z ) . \\end{align*}"}
-{"id": "2063.png", "formula": "\\begin{align*} & D { \\nabla ^ 2 } C ( \\bar r , t | { { \\bar r } _ { \\rm t x } } , { t _ 0 } ) - \\bar v ( \\bar r ) \\cdot { \\bar \\nabla C } ( \\bar r , t | { { \\bar r } _ { \\rm t x } } , { t _ 0 } ) - { k _ { d } } C ( \\bar r , t | { { \\bar r } _ { \\rm t x } } , { t _ 0 } ) + S ( \\bar r , t , { \\bar { r } _ { \\rm t x } } , t _ 0 ) \\\\ & = \\frac { { \\partial C ( \\bar r , t | { { \\bar r } _ { \\rm t x } } , { t _ 0 } ) } } { { \\partial t } } , \\end{align*}"}
-{"id": "5187.png", "formula": "\\begin{align*} H _ { 2 } ( x ) = H _ { 1 } ( A x + B ) . \\end{align*}"}
-{"id": "6626.png", "formula": "\\begin{align*} f ( y _ 1 , \\ldots , y _ v ) & = y _ 1 \\cdots y _ { m - \\mu } \\cdot ( y _ { m - \\mu + 1 } \\cdots y _ m \\oplus y _ { m + 1 } \\cdots y _ { m + \\mu } ) , \\\\ f ( y _ 1 , \\ldots , y _ v ) & = y _ 1 \\cdots y _ { m - 2 } \\cdot ( y _ { m - 1 } \\cdot y _ m \\oplus y _ { m + 1 } \\cdot y _ { m + 2 } \\oplus \\cdots \\oplus y _ { m + 2 \\nu - 3 } \\cdot y _ { m + 2 \\nu - 2 } ) , \\end{align*}"}
-{"id": "2373.png", "formula": "\\begin{align*} \\omega ( \\sigma _ 1 ) = \\omega ( \\sigma _ 2 ) & = 0 & \\omega ( \\sigma _ 3 ) & = - z . \\end{align*}"}
-{"id": "8100.png", "formula": "\\begin{align*} 8 k d _ k = - \\ , [ h ^ { k + 2 } ] \\ , & \\Bigg ( \\sum _ { j = 0 } ^ { k - 1 } d _ j h ^ j \\bigg ( B ( h ) ( 1 - h ) ^ { - ( j + 1 / 2 ) } \\\\ & + C ( h ) ( 1 - 2 h ) ^ { - ( j + 1 / 2 ) } + D ( h ) ( 1 - 3 h ) ^ { - ( j + 1 / 2 ) } \\bigg ) \\Bigg ) , \\end{align*}"}
-{"id": "331.png", "formula": "\\begin{align*} M _ { \\partial } = k L \\begin{pmatrix} 1 & 0 & m \\\\ 0 & 0 & 2 m + 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "2350.png", "formula": "\\begin{align*} \\xi _ s & = \\{ s , c _ { s , 1 } , \\cdots , c _ { s , k _ s } \\} s \\in \\sigma \\\\ \\pi _ t & = \\{ c _ { t , 1 } , \\cdots , c ( t , k _ t ) \\} t \\in S _ - \\setminus ( S _ + + \\sigma ) \\\\ \\pi & = S _ + + \\sigma \\\\ \\varrho & = S _ - \\setminus ( S _ + + \\sigma ) \\end{align*}"}
-{"id": "3948.png", "formula": "\\begin{align*} u _ { i } ( x , t ) = U _ { i } ( x , m _ { t } , t ) . \\end{align*}"}
-{"id": "6360.png", "formula": "\\begin{align*} v = \\textup { c h } ( \\_ ) \\sqrt { \\textup { t d } ( X ) } \\colon K _ { \\textup { n u m } } ( X ) \\rightarrow H ^ * ( X , \\Q ) . \\end{align*}"}
-{"id": "3055.png", "formula": "\\begin{align*} Z _ { r } ( s ) = \\sum _ { ( m , n ) \\neq ( 0 , 0 ) } \\frac { r ^ { 2 s } ( \\theta ( m , n ) ) } { ( m ^ { 2 } + n ^ { 2 } ) ^ { s } } . \\end{align*}"}
-{"id": "9853.png", "formula": "\\begin{align*} ( a - 1 ) b = w ^ 2 ( 1 - w ) . \\end{align*}"}
-{"id": "8774.png", "formula": "\\begin{align*} L _ R = - \\frac D 4 = \\frac { \\widehat { \\kappa } } { 2 } = \\widehat { \\kappa } _ G \\ , . \\end{align*}"}
-{"id": "9074.png", "formula": "\\begin{align*} \\int _ { \\mathbb { T } ^ 2 } \\left | \\sum _ { \\xi } e ( \\langle \\xi , x \\rangle ) \\right | ^ { 2 l } d x = \\sum _ { \\xi _ 1 , . . . \\xi _ { 2 k } } \\int _ { \\mathbb { T } ^ 2 } e ( \\xi _ 1 - \\xi _ 2 + . . . - \\xi _ { 2 l } ) d x . \\end{align*}"}
-{"id": "6525.png", "formula": "\\begin{align*} \\lim _ { l \\rightarrow \\infty } \\ , \\lim _ { t \\rightarrow \\infty } \\ \\ \\sup _ { x \\geq 0 } \\Biggl [ \\frac { 1 } { l } \\int _ { x } ^ { x + l } \\bigl \\| q ( t + s ) \\bigr \\| ^ { p } \\ , d s \\Biggr ] ^ { 1 / p } = 0 . \\end{align*}"}
-{"id": "4892.png", "formula": "\\begin{align*} A | _ U + A | _ V = \\{ a _ U + a _ V \\mid a _ U \\in A | _ U , a _ V \\in A | _ V \\} \\end{align*}"}
-{"id": "6622.png", "formula": "\\begin{align*} P _ N ( \\varphi ) = \\left ( \\prod _ { j = 1 } ^ J P _ { F _ { n _ j } } ( \\varphi , \\varepsilon _ j ) \\right ) \\left ( \\prod _ { j = J + 1 } ^ m P _ { F _ { n _ j } } ( \\varphi , \\varepsilon _ j ) \\right ) \\geq K _ 1 ^ J \\cdot 1 ^ { m - J } \\geq K _ 1 ^ J . \\end{align*}"}
-{"id": "6851.png", "formula": "\\begin{align*} \\beta \\mapsto x ( t , \\beta ) = \\Phi ( t , \\beta ) = \\Phi ^ t ( \\beta ) \\end{align*}"}
-{"id": "3346.png", "formula": "\\begin{align*} | [ u ^ n ] \\widetilde { Z } ( u ) | \\le \\sum _ { i \\ge 0 } R ^ { i - n } [ u ^ i ] \\widetilde { Z } ( u ) = \\frac { \\widetilde { Z } ( R ) } { R ^ n } \\end{align*}"}
-{"id": "5716.png", "formula": "\\begin{align*} & \\frac { 1 } { \\Gamma ( 2 - \\alpha ) } \\int _ 0 ^ x \\frac { 1 } { ( x - t ) ^ { \\alpha - 1 } } \\int _ 0 ^ t z ^ { \\alpha - 2 } E _ { \\alpha , \\alpha - 1 } ( \\lambda z ^ { \\alpha } ) \\frac { d } { d t } f ( t - z ) d z d t \\\\ & = \\frac { 1 } { \\Gamma ( 2 - \\alpha ) } \\int _ 0 ^ x { ( x - t ) ^ { 1 - \\alpha } } \\int _ 0 ^ t ( t - z ) ^ { \\alpha - 2 } E _ { \\alpha , \\alpha - 1 } ( \\lambda ( t - z ) ^ { \\alpha } ) f ' ( z ) d z d t \\end{align*}"}
-{"id": "6566.png", "formula": "\\begin{align*} \\begin{array} { l l l l } m ^ P _ h \\ , = 1 & P \\in E ( h ) , & m ^ P _ { 2 h } = 1 & P \\in E ( 2 h ) , \\\\ m ^ P _ { 3 h } = 3 & P \\in E ( 3 h ) , & m ^ P _ { 4 h } = 1 6 & P \\in E ( 4 h ) . \\end{array} \\end{align*}"}
-{"id": "9754.png", "formula": "\\begin{align*} N ( 0 ^ + , q , X _ \\circ ) = m _ \\circ \\end{align*}"}
-{"id": "9411.png", "formula": "\\begin{align*} D _ { \\ ; > r } = \\sum _ { z \\in S \\atop \\mathrm { I m } ( \\tilde { z } ) > r } n _ z Q _ { \\tilde { z } } . \\end{align*}"}
-{"id": "6469.png", "formula": "\\begin{align*} u _ i u _ i ^ * = q _ { i ( k + 1 ) } + 1 \\otimes \\left ( e _ { i ( k + 1 ) + 1 , i ( k + 1 ) + 1 } + \\cdots + e _ { ( i + 1 ) ( k + 1 ) - 1 , ( i + 1 ) ( k + 1 ) - 1 } \\right ) \\otimes 1 + p _ { ( i + 1 ) ( k + 1 ) } . \\end{align*}"}
-{"id": "9382.png", "formula": "\\begin{align*} K \\varrho _ k \\cap H _ k = \\langle \\overline { y _ j } \\mid 0 \\le j < 2 ^ k j \\equiv _ { 2 ^ n } 0 , 1 , \\ldots , m - 1 \\rangle . \\end{align*}"}
-{"id": "7567.png", "formula": "\\begin{align*} \\Gamma ( \\tau \\circ X ^ { - 1 } ) = \\int _ 0 ^ t \\Delta g _ { \\nu ( t - s ) } * ( \\tau \\circ X ^ { - 1 } ( s ) ) d s , \\end{align*}"}
-{"id": "3436.png", "formula": "\\begin{align*} \\dot { v } ( t ) = X _ t v , v ( 0 ) = v _ 0 \\in \\mathbb { R } ^ N , \\end{align*}"}
-{"id": "2789.png", "formula": "\\begin{align*} ( \\mathbf { Q } ^ T ) ^ k = \\mathbf { Q } ^ k , \\end{align*}"}
-{"id": "7304.png", "formula": "\\begin{align*} 2 E ' x & = \\theta x - ( - 1 ) ^ { p + q } x \\theta = \\xi _ i a \\otimes \\xi ^ i \\alpha - ( - 1 ) ^ { p + q } a \\xi _ i \\otimes \\alpha \\xi ^ i = \\\\ & = \\xi _ i a \\otimes \\xi ^ i \\alpha - ( - 1 ) ^ p a \\xi _ i \\otimes \\xi ^ i \\alpha = \\xi _ i \\vdash a \\otimes \\xi ^ i \\alpha = \\eta _ { i j } \\frac { \\partial } { \\partial \\xi _ j } a \\otimes \\xi ^ i \\alpha . \\end{align*}"}
-{"id": "9725.png", "formula": "\\begin{align*} \\Gamma ( u ) = { \\rm R e g } ( u ) \\cup { \\rm S i n g } ( u ) = { \\rm R e g } ( u ) \\cup \\Sigma _ 2 ( u ) = { \\rm R e g } ( u ) \\cup \\bigcup _ { m = 0 } ^ { n - 1 } \\Sigma ^ m _ 2 ( u ) . \\end{align*}"}
-{"id": "3463.png", "formula": "\\begin{align*} \\lvert q _ k ( s , t ; \\xi , \\eta ) \\rvert = | e ^ { \\xi \\tau - \\eta \\sigma } | \\leq e ^ { \\xi - 2 \\eta } , . \\end{align*}"}
-{"id": "3402.png", "formula": "\\begin{align*} \\rho ^ M _ { \\inf } ( T ) : = \\inf \\{ \\rho ( S ) : S ^ { ( M ) } = T ^ { ( M ) } \\} , \\rho ^ M _ { \\sup } ( T ) : = \\sup \\{ \\rho ( S ) : S ^ { ( M ) } = T ^ { ( M ) } \\} , \\end{align*}"}
-{"id": "5527.png", "formula": "\\begin{align*} e ^ { \\tilde { r } _ T u ^ 2 + \\tilde { r } _ T ' | u | } \\frac { \\int _ { D _ { T , u } } e ^ { \\ell _ T ( g ) } d \\Pi ( g ) } { \\int _ { D _ T } e ^ { \\ell _ T ( g ) } d \\Pi ( g ) } = e ^ { \\tilde { r } _ T u ^ 2 + \\tilde { r } _ T ' | u | } \\frac { \\Pi ( D _ { T , u } | X ^ T ) } { \\Pi ( D _ T | X ^ T ) } , \\end{align*}"}
-{"id": "6292.png", "formula": "\\begin{align*} \\int _ S f _ { \\mathcal { A ' } ( D ) } d \\mu & = \\mathbb { P } ( \\mathcal { A ' } ( D ) \\in S ) \\\\ \\ & \\leq \\exp \\left ( \\epsilon d ( D , D ' ) \\right ) \\mathbb { P } ( \\mathcal { A ' } ( D ' ) \\in S ) \\\\ & = \\exp \\left ( \\epsilon d ( D , D ' ) \\right ) \\int _ S f _ { \\mathcal { A } ' ( D ' ) } d \\mu , \\end{align*}"}
-{"id": "2056.png", "formula": "\\begin{align*} A _ h ^ \\pm ( M ) = \\sum _ { n \\in \\mathbb { Z } } \\widehat { h } ( n ) ( \\alpha _ { - n , M } ( 1 ) \\pm \\alpha _ { n , M } ( 1 ) ) , \\end{align*}"}
-{"id": "6447.png", "formula": "\\begin{align*} \\| x _ k \\| ^ 2 = \\| x _ 0 \\| ^ 2 + 2 \\| r _ { 0 } \\| x _ 0 ^ T V _ { k } T _ { k } ^ { - 1 } e _ { 1 } + \\xi _ k . \\end{align*}"}
-{"id": "2314.png", "formula": "\\begin{align*} a _ { \\mathbf { m } , j } = & \\sum _ { j = 1 , 2 } ( \\mathbf { e } _ { \\mathbf { m } , j } ) _ i b _ { \\mathbf { m } , i } & a _ { \\mathbf { m } , j } ^ + & = \\sum _ { j = 1 , 2 } ( \\mathbf { e } _ { \\mathbf { m } , j } ) _ i b _ { \\mathbf { m } , i } ^ + . \\end{align*}"}
-{"id": "8598.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } { u } '' & = & u ^ 3 - u + \\Lambda v ^ 2 u - \\omega v , \\\\ & & \\\\ { v } '' & = & v ^ 3 - v + \\Lambda u ^ 2 v - \\omega u , \\end{array} \\right . \\end{align*}"}
-{"id": "6740.png", "formula": "\\begin{align*} L ( p _ n , { m _ i } ^ 2 ) & = p _ n ( 2 m _ i - p _ n ) . \\end{align*}"}
-{"id": "4123.png", "formula": "\\begin{align*} L = \\begin{pmatrix} \\sqrt { 2 } e ^ { \\frac { 2 n \\ell } { 3 } } & 0 & 0 \\\\ 0 & e ^ { - \\frac { n \\ell } { 3 } } & 0 \\\\ 0 & 0 & e ^ { - \\frac { n \\ell } { 3 } } \\end{pmatrix} . \\end{align*}"}
-{"id": "2560.png", "formula": "\\begin{align*} V _ t f ( x ) + \\Pi _ x \\Big [ \\int _ 0 ^ { t \\wedge \\zeta } \\widetilde \\psi ( \\xi _ s , V _ { t - s } f ) d s \\Big ] = \\Pi _ x \\big [ f ( \\xi _ t ) \\mathbf 1 _ { t < \\zeta } \\big ] . \\end{align*}"}
-{"id": "7135.png", "formula": "\\begin{align*} \\Phi ( \\rho ) _ t + \\big ( \\Phi ( \\rho ) u \\big ) _ x + \\big ( p ( \\rho ) - p ( \\tilde { \\rho } ) \\big ) u _ x = 0 . \\end{align*}"}
-{"id": "1219.png", "formula": "\\begin{align*} | A | ^ 2 = 2 \\ln \\left ( \\frac { 1 } { 1 - \\kappa ^ 2 } \\right ) , \\mathop { \\mathrm { R e } } A \\kappa \\end{align*}"}
-{"id": "6611.png", "formula": "\\begin{align*} | 1 - 2 \\varepsilon ( - \\varphi ) ^ { - n } | \\leq 1 + 2 \\varphi = \\sqrt { 5 } . \\end{align*}"}
-{"id": "869.png", "formula": "\\begin{gather*} F ( x ) : = K - g ( x ) , x \\in \\R ^ n . \\end{gather*}"}
-{"id": "5980.png", "formula": "\\begin{align*} X ^ n ( t ) = W ^ Q ( t ) + \\int _ 0 ^ t q _ n ( s ) d s . \\end{align*}"}
-{"id": "1254.png", "formula": "\\begin{align*} T _ { 4 , 2 } = O ( | \\eta | ^ { - 2 } ) . \\end{align*}"}
-{"id": "2879.png", "formula": "\\begin{gather*} [ J _ 0 , J _ \\pm ] = \\pm J _ \\pm , \\qquad \\{ J _ + , J _ - \\} = 2 J _ 0 . \\end{gather*}"}
-{"id": "1214.png", "formula": "\\begin{align*} \\abs e = k \\textnormal { s u c h t h a t } t _ { \\abs e } = t _ { k } , \\textnormal { a n d } \\gamma _ { \\abs e } : = \\gamma _ { k } , \\textnormal { f o r s u c h e l e m e n t s } e \\in E _ { T } ^ { k + 1 } . \\end{align*}"}
-{"id": "9728.png", "formula": "\\begin{align*} B _ r ( X _ \\circ ) & : = \\{ X \\in \\R ^ { n + 1 } : | X - X _ \\circ | < r \\} , \\\\ B _ r ^ * ( x _ \\circ ) & : = \\{ x \\in \\R ^ n : | x - x _ \\circ | < r \\} , \\\\ B _ r ' ( x _ \\circ ' ) & : = \\{ x ' \\in \\R ^ { n - 1 } : | x ' - x _ \\circ ' | < r \\} , \\end{align*}"}
-{"id": "2106.png", "formula": "\\begin{align*} \\Phi ( t ) = I + \\int _ { t _ 0 } ^ t A ( s ) \\Phi ( s ) d s + \\int _ { t _ 0 } ^ t C ( s ) \\Phi ( s ) d \\omega ( s ) , \\ t \\in [ t _ 0 , t _ 0 + T ] . \\end{align*}"}
-{"id": "343.png", "formula": "\\begin{align*} \\nu _ p ( \\frac { 1 } { 2 } n , k ) \\cdot \\nu _ p ( n , k ) = \\nu _ p ( \\frac { 1 } { 2 } n , l ) \\cdot \\nu _ p ( n , l ) , \\end{align*}"}
-{"id": "8373.png", "formula": "\\begin{align*} F _ z ( z ( \\alpha , t ) , t ) = \\frac { \\partial _ { \\alpha } F ( z ( \\alpha , t ) , t ) } { z _ { \\alpha } } = \\frac { \\bar { z } _ { t \\alpha } } { z _ { \\alpha } } - \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i } { 2 \\pi } \\frac { 1 } { ( z ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } . \\end{align*}"}
-{"id": "440.png", "formula": "\\begin{align*} T ^ t _ k : = & \\inf \\{ s \\geq R ^ t _ { k - 1 } \\ : \\ B _ s = A + ( \\eta + \\gamma ) ( t + s ) \\} \\\\ R ^ t _ k : = & \\inf \\{ s \\geq T ^ t _ { k } \\ : \\ B _ s = \\gamma ( t + s ) \\} . \\end{align*}"}
-{"id": "1150.png", "formula": "\\begin{align*} & | y _ 0 - y _ { k + 1 } | > | y _ 0 - y _ i | > b _ 0 - a = b - b _ 0 > | y _ i - y _ { k + 1 } | > \\varepsilon _ 0 ( \\ln n ) ^ { - 1 } . \\end{align*}"}
-{"id": "7846.png", "formula": "\\begin{align*} e ^ { - \\frac { 1 6 d } { 1 9 } } + e ^ { - \\frac { 7 2 d } { 9 5 } } \\sum _ { m = 1 } ^ { d _ 0 } e ^ { \\varphi ( m ) } \\le e ^ { - \\frac { 1 6 d } { 1 9 } } + \\frac { 3 } { 2 } e ^ { - \\frac { 3 d } { 1 9 } } \\le e ^ { - \\frac { d } { 1 0 } } \\Big ( e ^ { - \\frac { 1 4 1 } { 1 9 0 } } + \\frac { 3 } { 2 } \\Big ) \\le 2 e ^ { - \\frac { d } { 1 0 } } . \\end{align*}"}
-{"id": "7785.png", "formula": "\\begin{align*} U _ { p } \\circ M _ t ^ { G } = M _ t ^ { U ( G ) } \\circ U _ { p } . \\end{align*}"}
-{"id": "6794.png", "formula": "\\begin{align*} \\mathcal { R } _ { q + 1 } = \\rho _ 1 ( \\mathcal { R } _ q ) \\quad \\equiv \\begin{cases} F = 0 \\ , , \\\\ \\frac { \\partial F } { \\partial x _ j } = 0 , & \\ 1 \\le j \\le n \\ , . \\end{cases} \\end{align*}"}
-{"id": "9241.png", "formula": "\\begin{align*} K _ A ( \\partial \\Omega ) ( x , z , p ) = - \\delta \\nu ( x ) + P ( D _ p A ( x , z , p ) \\cdot \\nu ( x ) ) P \\end{align*}"}
-{"id": "2497.png", "formula": "\\begin{align*} \\aligned \\left \\{ \\begin{array} { l } ( i ) : \\ \\ \\ \\ \\big [ { \\sf x } _ i , { \\sf x } _ j \\big ] , \\ \\ \\ i < j , \\\\ \\\\ ( i i ) : \\ \\ \\ \\big [ { \\sf x } _ i , J { \\sf x } _ j \\big ] , \\ \\ \\ i , j = 1 , \\ldots , n , \\\\ \\\\ ( i i i ) : \\ \\ \\big [ J { \\sf x } _ i , J { \\sf x } _ j \\big ] , \\ \\ \\ i < j , \\end{array} \\right . \\endaligned \\end{align*}"}
-{"id": "1868.png", "formula": "\\begin{align*} \\phi ^ { \\rm e v e n } _ { \\rm O U T } ( j , x , t ) = \\sum _ { n = 0 } ^ \\infty \\left [ \\overline { \\alpha _ { n j } } \\psi ^ { \\rm e v e n } ( n , x , t ) - \\beta _ { n j } \\overline { \\psi ^ { \\rm e v e n } ( n , x , t ) } \\right ] , \\end{align*}"}
-{"id": "9636.png", "formula": "\\begin{align*} \\frac { ( b ; q ^ 2 ) _ N } { ( a ; q ^ 2 ) _ N } \\frac { ( a ; q ^ 2 ) _ { N - n } } { ( b ; q ^ 2 ) _ { N - n } } \\left ( \\frac { a } { b } \\right ) ^ n = \\frac { ( q ^ { 2 - 2 N } / b ; q ^ 2 ) _ n } { ( q ^ { 2 - 2 N } / a ; q ^ 2 ) _ n } \\end{align*}"}
-{"id": "9322.png", "formula": "\\begin{align*} \\Delta \\psi = n \\mathbf { H } , \\end{align*}"}
-{"id": "7419.png", "formula": "\\begin{align*} \\displaystyle Z ( s , W _ \\lambda , \\phi ) = \\delta _ n ( a ( \\lambda ) ) \\lvert \\det a ( \\lambda ) \\rvert _ F ^ { - s } Z ( s , W , \\phi ) , \\ ; \\ ; \\ ; \\Re ( s ) \\gg 1 . \\end{align*}"}
-{"id": "2126.png", "formula": "\\begin{align*} E | \\lambda _ k ( \\omega ) | ^ r \\leq \\eta ^ r E \\Big [ 2 + \\Big ( \\frac { 2 M _ 0 } { \\mu } \\Big ) ^ { p } ( 1 + \\Gamma _ p ( \\omega ) ) \\Big ] ^ r , \\forall k = 1 , \\dots , d . \\end{align*}"}
-{"id": "8779.png", "formula": "\\begin{align*} \\overline { \\widehat { \\kappa } } = - \\frac { D + 4 q C } { 2 p } \\ , . \\end{align*}"}
-{"id": "4741.png", "formula": "\\begin{align*} : t S \\cap \\mathbb { Z } ^ d | = q ( t ) + O \\left ( t ^ { \\frac { ( d - 1 ) ( d - 2 ) } { 2 d - 3 } + \\epsilon } \\right ) . \\end{align*}"}
-{"id": "2158.png", "formula": "\\begin{align*} T _ j ^ { 0 } = I , \\ \\ T ^ { \\alpha } : = T _ { 1 } ^ { \\alpha _ 1 } \\circ T _ 2 ^ { \\alpha _ 2 } \\circ \\ldots \\circ T _ { N } ^ { \\alpha _ N } . \\end{align*}"}
-{"id": "7219.png", "formula": "\\begin{align*} L _ r f ( x _ 0 ) - L _ r g ( x _ 0 ) = \\int _ { - \\pi } ^ \\pi ( K _ r ( x _ 0 - y ) - \\min K _ r ) \\left ( f ( y ) - g ( y ) \\right ) \\ , d y > 0 , \\end{align*}"}
-{"id": "671.png", "formula": "\\begin{align*} \\rho ^ { ( \\vec k ) } _ i ( j ) = \\begin{cases} - \\infty & j \\le \\sum _ { s < i } k _ s \\ , \\\\ [ . 5 e x ] j - \\sum _ { s < i } k _ s & \\sum _ { s < i } k _ s < j \\le \\sum _ { s \\le i } k _ s \\ , \\\\ [ . 5 e x ] \\infty & j > \\sum _ { s \\le i } k _ s \\ . \\end{cases} \\end{align*}"}
-{"id": "1125.png", "formula": "\\begin{align*} & \\int _ { I ^ k } \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) d y _ 1 \\cdots d y _ k \\geq \\int _ { I _ 1 ^ k } \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) d y _ 1 \\cdots d y _ k \\\\ = & S ( I ) ^ k ( \\ln n ) ^ { - k } \\int _ { ( - l _ n , l _ n ) ^ k } \\phi _ { k , n } \\big ( \\gamma _ n ( u _ 1 ) , \\cdots , \\gamma _ n ( u _ k ) \\big ) \\prod _ { j = 1 } ^ k \\beta _ n ( u _ j ) d u _ 1 \\cdots d u _ k . \\end{align*}"}
-{"id": "6843.png", "formula": "\\begin{align*} I _ 1 = I + \\langle a _ { 1 4 } '' - a _ { 1 2 } '' + a _ { 1 1 } '' , a _ { 2 4 } '' - a _ { 2 2 } '' + a _ { 2 1 } '' \\rangle \\\\ \\subset \\ \\mathbb { A } _ 1 = \\mathbb { Q } [ a , a '' ] \\ , . \\end{align*}"}
-{"id": "8146.png", "formula": "\\begin{align*} h \\cdot v = h _ { C _ { g , n } } \\triangleright v . \\end{align*}"}
-{"id": "1168.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\mathbb { E } \\chi ^ { ( n ) } ( ( x , + \\infty ) ) = ( \\pi / 2 ) e ^ { c _ 0 - x } = e ^ { c _ 1 - x } . \\end{align*}"}
-{"id": "484.png", "formula": "\\begin{align*} \\ker ( d \\nabla _ { [ f ] } ) = { \\mathbf T } _ { [ f ] } ( \\Theta ( f ) \\cdot [ f ] ) , \\end{align*}"}
-{"id": "1854.png", "formula": "\\begin{align*} \\psi ( x , t ) = a \\ , \\psi ^ { \\rm t o p . } ( x , t ) + b \\ , \\overline { \\psi ^ { \\rm t o p . } ( x , t ) } + \\sum _ { n = 1 } ^ \\infty \\left [ a _ n \\psi ^ { \\rm o d d } ( n , x , t ) + b _ n \\overline { \\psi ^ { \\rm o d d } ( n , x , t ) } + c _ n \\psi ^ { \\rm e v e n } ( n , x , t ) + d _ n \\overline { \\psi ^ { \\rm e v e n } ( n , x , t ) } \\right ] , \\end{align*}"}
-{"id": "781.png", "formula": "\\begin{align*} \\sigma _ { f _ { i _ 0 } } * _ d \\lambda _ j = \\left ( 1 - \\prod _ { \\substack { i = 1 \\\\ i \\neq i _ 0 } } ^ { { s } } ( \\rho _ { j } - \\rho _ i ) \\right ) \\cdot \\lambda _ j . \\end{align*}"}
-{"id": "794.png", "formula": "\\begin{align*} e _ i = \\frac { q ^ { { s } } - 1 } { ( q ^ { { s } } - 1 , b _ i ) } , \\end{align*}"}
-{"id": "3623.png", "formula": "\\begin{align*} ( \\alpha _ { - 1 } - \\beta _ { - 1 } ) \\cdot a _ 0 = & \\left ( \\frac { 2 t } { 3 } - \\frac { 1 } { 2 ^ 3 } \\right ) a _ 2 - \\frac { 1 } { 2 ^ 3 } ( a _ { - 2 } - a _ 3 - a _ { - 3 } ) - \\frac { 1 } { 2 ^ 4 } ( v _ { ( 1 , 2 ) } - v _ { ( 1 , 3 ) } + 4 v _ { ( 2 , 3 ) } ) \\\\ & + \\frac { 1 } { 2 ^ 3 \\cdot 3 } ( 7 a _ 1 + 4 a _ { - 1 } ) \\cdot v _ { ( 2 , 3 ) } - \\frac { 2 } { 3 } a _ 2 \\cdot v _ { ( 1 , 3 ) } + \\frac { 1 } { 2 \\cdot 3 } ( a _ 3 + a _ { - 3 } ) \\cdot v _ { ( 1 , 2 ) } . \\end{align*}"}
-{"id": "1746.png", "formula": "\\begin{align*} D ^ { l } e ^ { \\frac { \\alpha } { p } M _ { s , t } } = \\left ( \\frac { \\alpha } { p } \\right ) ^ { l } e ^ { \\frac { \\alpha } { p } M _ { s , t } } D M _ { s , t } \\otimes \\cdots \\otimes D M _ { s , t } \\in L ^ { p } ( \\boldsymbol { W } ; \\mathcal { H } ^ { \\otimes l } ) \\end{align*}"}
-{"id": "485.png", "formula": "\\begin{align*} A ( f ) = \\AA ( \\nabla f ) = \\prod _ { i = 1 } ^ k \\AA ( \\nabla f _ i ) = A ( f _ 1 ) \\cdots A ( f _ k ) , \\end{align*}"}
-{"id": "2814.png", "formula": "\\begin{align*} \\hat { M } = \\Big [ \\delta _ { i j } \\mu _ i - | a _ { i j } | - \\delta _ { i j } \\frac { c _ i d _ i } { e _ i } \\Big ] \\end{align*}"}
-{"id": "7407.png", "formula": "\\begin{align*} \\Phi _ t ( A ) = \\bigcup _ { U \\in \\mathcal U } \\Phi _ t ^ U ( A ) , \\end{align*}"}
-{"id": "7084.png", "formula": "\\begin{align*} - \\big [ P A + A ^ \\top P + Q - \\L D ( R + D ^ \\top P D ) ^ { - 1 } D ^ \\top \\L \\big ] = \\begin{pmatrix} - 2 t & 1 \\\\ 1 & 2 t \\end{pmatrix} . \\end{align*}"}
-{"id": "6723.png", "formula": "\\begin{align*} \\dot { Q } _ { 2 1 } = \\frac { P _ { 2 1 } } { \\lambda ^ 2 m } , \\dot { Q } _ { 2 2 } = \\frac { P _ { 2 2 } } { \\lambda ^ 2 m } - \\frac { Q _ { 2 1 } } { \\lambda } , \\end{align*}"}
-{"id": "8998.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta & = 4 X ( 0 ) ^ 2 ( 1 - \\beta \\gamma _ g ) ^ 2 - 4 ( 1 - 2 \\beta \\gamma _ g ) ( 1 + X ( 0 ) ^ 2 ) \\\\ & = 4 \\Big ( \\beta ^ 2 \\gamma _ g ^ 2 X ( 0 ) ^ 2 + 2 \\beta \\gamma _ g - 1 \\Big ) > 0 . \\end{aligned} \\end{align*}"}
-{"id": "5647.png", "formula": "\\begin{align*} \\partial _ t ^ \\alpha u _ h ( t ) + A _ h u _ h ( t ) = { _ 0 I _ t ^ \\gamma } P _ h \\dot { W } ( t ) , \\forall 0 < t \\leq T . \\end{align*}"}
-{"id": "1062.png", "formula": "\\begin{align*} y - x = ( G _ n ( y ) - G _ n ( x ) ) ( n / 4 ) ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } , \\end{align*}"}
-{"id": "2596.png", "formula": "\\begin{align*} \\| f \\| _ { \\mu ; p } : = \\Big ( \\int _ { S } | f | ^ p d \\mu \\Big ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "1612.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow \\infty } \\frac { \\mathbb { E } _ x \\left [ Z _ T ^ { b ^ \\ast } \\right ] } { T } = \\lim _ { T \\rightarrow \\infty } \\frac { 1 } { T } \\mathbb { E } _ x \\int _ 0 ^ T \\mu \\left ( X _ t ^ { Z ^ { b ^ \\ast } } \\right ) X _ t ^ { Z ^ { b ^ \\ast } } \\ , d t . \\end{align*}"}
-{"id": "5791.png", "formula": "\\begin{align*} ( x , g ) \\cdot ( x _ 1 , x _ 2 ) = ( \\varrho _ { \\R ^ \\vee } ( x ) ( g \\cdot x _ 2 ) + g \\cdot x _ 1 , g \\cdot x _ 2 ) . \\end{align*}"}
-{"id": "6205.png", "formula": "\\begin{align*} \\mu ( d v ) = \\left ( \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n + 1 } n ^ 2 e ^ { - \\frac { n ^ 2 } { 2 } v } \\right ) { \\bf 1 } _ { ( 0 , + \\infty ) } ( v ) d v \\end{align*}"}
-{"id": "9527.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { n - 1 } z ^ n q ^ { \\frac { n ( n + 1 ) } { 2 } } } { ( 1 - q ^ n ) ( z q ) _ n } = \\sum _ { n = 1 } ^ { \\infty } \\frac { z ^ n q ^ n } { 1 - q ^ n } , \\end{align*}"}
-{"id": "3007.png", "formula": "\\begin{align*} F _ m ' ( x ) & = \\sum _ { n = 0 } ^ \\infty n p _ m ' ( n ) x ^ { n - 1 } . \\end{align*}"}
-{"id": "6047.png", "formula": "\\begin{align*} \\mathcal { E } _ 1 & = E ( H ) \\ \\cap \\ E ( C _ { w m } [ n ] ) , \\\\ \\mathcal { E } _ 2 & = \\big \\{ ( m - 1 , y _ 1 ) ( m , y _ 2 ) \\mid ( 0 , y _ 2 ) ( m - 1 , y _ 1 ) \\in E ( H ) \\big \\} , \\end{align*}"}
-{"id": "7461.png", "formula": "\\begin{align*} \\displaystyle \\left \\lvert D \\left ( \\lambda \\mapsto ( \\pi _ \\lambda ( g ) e , e ) \\right ) _ { \\lambda = 0 } \\right \\rvert \\ll N ( \\rho ) ^ r \\Xi ^ G ( g ) \\sigma ( g ) ^ { \\deg ( D ) } \\end{align*}"}
-{"id": "8161.png", "formula": "\\begin{align*} \\dot W ( \\boldsymbol { x } , \\boldsymbol { k } ) \\le - \\left \\| \\begin{bmatrix} \\| ( P _ N \\otimes I _ { M } ) \\boldsymbol { x } \\| \\\\ \\| { \\rm a v g } ( \\boldsymbol { x } ) - { x } ^ * \\| \\end{bmatrix} \\right \\| _ { { \\mathcal { M } } } ^ { 2 } \\end{align*}"}
-{"id": "9940.png", "formula": "\\begin{align*} \\beta = 2 ( n - 1 ) \\quad Q _ 0 = - \\frac { ( n - 1 ) ^ 2 } { 4 } I . \\end{align*}"}
-{"id": "3892.png", "formula": "\\begin{align*} S _ \\omega ( u ^ \\gamma , v ^ \\gamma ) & = \\frac { 1 } { 2 } K ( u ^ \\gamma , v ^ \\gamma ) + \\frac { \\omega } { 2 } M ( u ^ \\gamma , v ^ \\gamma ) - P ( u ^ \\gamma , v ^ \\gamma ) \\\\ & = \\frac { \\gamma ^ 2 } { 2 } K ( u , v ) + \\frac { \\omega } { 2 } M ( u , v ) - \\gamma ^ { \\frac { 5 } { 2 } } P ( u , v ) . \\end{align*}"}
-{"id": "9824.png", "formula": "\\begin{align*} \\lim _ { r \\downarrow 0 } H _ \\lambda ( r , u ) = + \\infty . \\end{align*}"}
-{"id": "2603.png", "formula": "\\begin{align*} J _ g ( t , r , \\xi ) - J ' _ g ( t , r , \\xi ) = ( \\gamma _ 0 - 1 ) t \\int _ 0 ^ 1 \\big ( \\mathbf 1 _ { \\gamma ( \\cdot ) > \\gamma _ 0 } \\kappa \\gamma \\cdot ( \\phi \\eta _ { u t } ) ^ { \\gamma - 1 } g ( u t , r u ^ { \\frac { 1 } { \\gamma _ 0 - 1 } } , \\cdot ) ^ { \\gamma - 1 } \\big ) \\big ( \\xi _ { ( 1 - u ) t } \\big ) d u . \\end{align*}"}
-{"id": "7572.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { \\Delta _ 1 \\tau ( s , t ) } _ { C ^ { \\alpha , p } } \\le | t - s | M _ X ^ { \\alpha } \\norm { \\tau } _ { L i p ( 0 , T ; C ^ { \\alpha , p } ) } , \\\\ \\end{gathered} \\end{align*}"}
-{"id": "1065.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to + \\infty } \\left ( \\ln D _ n ( \\alpha _ n ) - n ^ 2 \\ln \\cos \\frac { \\alpha _ n } { 2 } + \\frac { 1 } { 4 } \\ln \\left ( n \\sin \\frac { \\alpha _ n } { 2 } \\right ) - c _ 0 \\right ) = 0 . \\end{align*}"}
-{"id": "9918.png", "formula": "\\begin{align*} | f _ { n - } ^ { \\mu } ( x _ { 2 } ) - f _ { n - } ^ { \\mu } ( x ^ { * } ) | & = \\left | \\int _ { \\mathcal { X } } t ( x _ { 2 } | x _ { 1 } ) - t ( x ^ { * } | x _ { 1 } ) d \\pi _ { n - } ^ { \\mu } ( d x _ { 1 } ) \\right | \\\\ & \\leq \\int _ { \\mathcal { X } } | t ( x _ { 2 } | x _ { 1 } ) - t ( x ^ { * } | x _ { 1 } ) | d \\pi _ { n - } ^ { \\mu } ( x _ { 1 } ) \\leq \\epsilon \\end{align*}"}
-{"id": "4015.png", "formula": "\\begin{align*} a _ { \\Phi ( F ) } ( M ) = a _ F ( \\begin{psmallmatrix} M & 0 \\\\ 0 & 0 \\end{psmallmatrix} ) \\neq 0 . \\end{align*}"}
-{"id": "3122.png", "formula": "\\begin{align*} A ^ { ( r ) } ( y ) & = F _ 0 ^ { ( r ) } ( y ) A ^ { ( r ) } _ { \\underline { \\chi ( \\O _ S ) } } ( y ) ( - 1 ) ^ { r } \\\\ & = \\frac { 1 } { [ r ] _ y } \\prod _ { k \\geq 1 } \\frac { 1 } { ( 1 - q ^ { r k } ) ^ { 1 0 } ( 1 - y ^ { - r } q ^ { r k } ) ( 1 - y ^ r q ^ { r k } ) } \\\\ & = \\frac { y ^ \\frac { 1 } { 2 } - y ^ { - \\frac { 1 } { 2 } } } { \\phi _ { - 2 , 1 } ( q ^ r , y ^ r ) ^ \\frac { 1 } { 2 } \\ , \\tilde \\Delta ( q ^ r ) ^ \\frac { 1 } { 2 } } \\ , , \\end{align*}"}
-{"id": "5357.png", "formula": "\\begin{align*} H _ \\beta . v _ { n m } ^ k = \\frac { m - n - 1 + 2 k } 2 v _ { n m } ^ k . \\end{align*}"}
-{"id": "6994.png", "formula": "\\begin{align*} { \\pi ( j ) = \\textstyle { \\frac { 2 ( j + 1 ) } { n ^ 2 } } , \\ \\ 0 \\leq j \\le n - 2 , \\pi ( n - 1 ) = \\frac { 1 } { n } } ; \\end{align*}"}
-{"id": "8029.png", "formula": "\\begin{align*} p ( t ) & = t ( t - \\alpha _ 1 ) ( t - \\zeta \\alpha _ 1 ) \\cdots ( t - \\zeta ^ { k - 1 } \\alpha _ 1 ) \\cdots ( t - \\alpha _ { 2 n / k } ) \\cdots ( t - \\zeta ^ { k - 1 } \\alpha _ { 2 n / k } ) \\\\ & = t ( t ^ k - \\alpha _ 1 ^ k ) ( t ^ k - \\alpha _ 2 ^ k ) \\cdots ( t ^ k - \\alpha _ { 2 n / k } ^ k ) \\\\ & = \\sum _ { q = 0 } ^ { \\frac { 2 n } { k } } ( - 1 ) ^ q e _ q ( \\alpha _ 1 ^ k , \\dots , \\alpha _ { 2 n / k } ^ k ) t ^ { 2 n - q k + 1 } , \\\\ \\end{align*}"}
-{"id": "2769.png", "formula": "\\begin{align*} \\lim _ { \\kappa \\to 0 } K ^ { ( \\mathrm { s o f t } ) } ( \\kappa , \\pi ; u , v ) = K ^ { ( \\mathrm { s o f t } ) } ( 0 , \\pi ; u , v ) , \\end{align*}"}
-{"id": "8325.png", "formula": "\\begin{align*} \\bar { z } _ { t t } = & \\partial _ t \\bar { z } _ t = \\partial _ t \\Big ( F ( z ( \\alpha , t ) , t ) - \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i } { 2 \\pi ( z ( \\alpha , t ) - z _ j ( t ) ) } \\Big ) \\\\ = & F _ z ( z ( \\alpha , t ) , t ) z _ t + F _ t + \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i ( z _ t ( \\alpha , t ) - \\dot { z } _ j ( t ) ) } { 2 \\pi ( z ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } \\end{align*}"}
-{"id": "7595.png", "formula": "\\begin{align*} \\begin{gathered} I _ 1 + I _ 6 = \\frac { 1 } { \\nu } \\left ( \\eta ( t ) \\cdot \\nabla \\left ( g _ { \\nu t } * \\left ( \\tau \\circ X ^ { - 1 } ( t ) \\right ) \\right ) - \\nabla \\left ( g _ { \\nu t } * \\left ( \\eta ( t ) \\tau \\circ X ^ { - 1 } ( t ) \\right ) \\right ) \\right ) \\\\ - \\frac { 1 } { \\nu } g _ { \\nu t } * \\left ( \\nabla \\cdot \\eta ( t ) \\left ( \\tau \\circ X ^ { - 1 } ( t ) \\right ) \\right ) \\end{gathered} \\end{align*}"}
-{"id": "6717.png", "formula": "\\begin{align*} m \\ddot { q } _ i + h _ i q _ i = 0 , \\end{align*}"}
-{"id": "8751.png", "formula": "\\begin{align*} u ^ { ( \\alpha , \\sigma ) } _ n = { } _ L u _ { n , n _ 0 } ^ { ( \\alpha , \\sigma ) } + { } _ H u _ { n , n _ 0 } ^ { ( \\alpha , \\sigma ) } \\equiv \\tau ^ { - \\alpha } \\sum _ { k = n - n _ 0 } ^ { n } \\omega ^ { ( \\alpha , \\sigma ) } _ { n - k } u _ { k } + \\tau ^ { - \\alpha } \\sum _ { k = 0 } ^ { n - n _ 0 - 1 } \\omega ^ { ( \\alpha , \\sigma ) } _ { n - k } u _ { k } . \\end{align*}"}
-{"id": "7283.png", "formula": "\\begin{align*} ( \\psi _ p ( a b ) - \\psi _ p ( a ) \\psi _ p ( b ) ) ( h _ q ^ - - h _ s ^ - ) = ( \\psi _ q ( a b ) - \\psi _ p ( a ) \\psi _ q ( b ) ) ( h _ q ^ - - h _ s ^ - ) \\end{align*}"}
-{"id": "1973.png", "formula": "\\begin{align*} 2 m d _ { \\ell \\ell } = \\sum _ { k = 1 } ^ m d _ { k k } = m d _ { 1 1 } . \\end{align*}"}
-{"id": "6134.png", "formula": "\\begin{align*} F _ { i , j } = h _ 1 p ^ A _ 1 + h _ 2 p ^ A _ 2 , \\end{align*}"}
-{"id": "3967.png", "formula": "\\begin{align*} { \\mathcal K } ( f , g ) = 4 f ' \\cdot \\bar { g ' } y ^ { k + 2 } + 2 i k ( f ' \\cdot \\bar { g } - f \\cdot \\bar { g ' } ) y ^ { k + 1 } . \\end{align*}"}
-{"id": "6106.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } k \\eta ^ k \\pi _ k ( \\eta ^ k y ) = \\frac { 1 } { y } \\phi ( \\log \\eta ) \\end{align*}"}
-{"id": "8355.png", "formula": "\\begin{align*} - \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j } { \\pi } \\frac { 1 } { ( 0 - z _ j ( t ) ) ^ 2 } D _ t Z ( 0 ) = - \\frac { \\lambda i } { \\pi } \\frac { x } { x ^ 2 + y ^ 2 } \\frac { \\lambda } { \\pi } \\frac { 4 x y i } { ( x ^ 2 + y ^ 2 ) ^ 2 } = \\frac { 4 \\lambda ^ 2 x ^ 2 y } { \\pi ^ 2 ( x ^ 2 + y ^ 2 ) ^ 3 } \\end{align*}"}
-{"id": "3163.png", "formula": "\\begin{align*} { \\rm i n d } _ \\varGamma ( U ) = ( M _ - - m _ - ) - ( M _ + - m _ + ) . \\end{align*}"}
-{"id": "3394.png", "formula": "\\begin{align*} \\rho ( T ) = \\frac { 1 } { 1 + \\prod _ { i } \\rho ( T _ i ) } . \\end{align*}"}
-{"id": "7.png", "formula": "\\begin{align*} | F _ s ^ { i , r } ( 0 ) | = & \\ | f ^ i ( V _ s ^ { r , v } , q ( Z _ s ^ { i , r , v } ( m ) ) ) - f ^ i ( V _ s ^ { r , \\bar { v } } , q ( Z _ s ^ { i , r , v } ( m ) ) ) | \\\\ \\leq & \\ \\frac { C _ v C _ { \\eta } } { C _ { \\eta } - C _ v } | V ^ { r , v } _ s - V ^ { r , \\bar { v } } _ s | \\leq \\frac { C _ v C _ { \\eta } } { C _ { \\eta } - C _ v } e ^ { - C _ { \\eta } ( s - r ) } | v - \\bar { v } | , s \\in [ r , T ] , \\end{align*}"}
-{"id": "6606.png", "formula": "\\begin{align*} \\left | \\frac { \\overline { A } _ { n _ j } ( \\varepsilon _ j ) } { A _ { n _ j } } \\right | = 1 + p _ j + \\mathcal { O } ( \\varphi ^ { 2 n _ j } ) , \\end{align*}"}
-{"id": "2247.png", "formula": "\\begin{align*} & C ( 0 ) = 1 - \\int _ 1 ^ c \\d y \\frac { 2 g ( y ) ^ 2 } { y } , & & C ( \\i u ) = 1 - \\int _ 1 ^ c \\d y \\frac { 2 y g ( y ) ^ 2 } { u ^ 2 + y ^ 2 } \\end{align*}"}
-{"id": "2621.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta _ p u _ { \\epsilon } ( z ) = f ( z , u _ { \\epsilon } ( z ) , D u _ { \\epsilon } ( z ) ) + \\epsilon e ( z ) \\ \\mbox { f o r a l m o s t a l l } \\ z \\in \\Omega , & \\\\ \\frac { \\partial u _ { \\epsilon } } { \\partial n _ p } + \\beta ( z ) u ^ { p - 1 } _ { \\epsilon } = 0 \\ \\mbox { o n } \\ \\partial \\Omega & \\end{array} \\right \\} \\end{align*}"}
-{"id": "10045.png", "formula": "\\begin{align*} \\dot x = - \\frac { x ^ 3 } { 4 } \\frac { \\partial \\widetilde H } { \\partial y } , \\dot y = - \\frac { x ^ 3 } { 4 } \\left ( - \\frac { \\partial \\widetilde H } { \\partial x } \\right ) , \\dot \\theta = \\frac { \\partial \\widetilde H } { \\partial G } , \\dot G = - \\frac { \\partial \\widetilde H } { \\partial \\theta } . \\end{align*}"}
-{"id": "5329.png", "formula": "\\begin{align*} Y . w ^ k = \\frac 1 2 \\left ( \\lambda - ( k - 1 ) \\right ) w ^ { k - 2 } = b _ k w ^ { k - 2 } \\end{align*}"}
-{"id": "9408.png", "formula": "\\begin{align*} - \\sum _ { \\tau \\in \\mathcal { F } _ N } \\nu _ \\tau ^ { ( N ) } ( f ) - \\sum _ { m = 1 } ^ \\infty \\biggl ( \\sum _ { \\tau \\in \\mathcal { F } _ N } \\nu _ \\tau ^ { ( N ) } ( f ) J _ { N , m } ( \\tau ) \\biggr ) q ^ m = f _ { \\theta } ( z ) - \\mathcal { E } _ 2 ( z ) , \\end{align*}"}
-{"id": "3162.png", "formula": "\\begin{align*} Q : = \\frac { 1 } { 2 i } [ \\varGamma , C ] \\end{align*}"}
-{"id": "3795.png", "formula": "\\begin{align*} \\epsilon ( \\alpha ) & = \\epsilon _ 0 ( \\alpha ) \\cdots \\epsilon _ { R - k - 1 } ( \\alpha ) \\\\ & = ( z _ 1 - 2 + \\chi ( p _ 1 = 0 ) ) ( z _ 2 - 2 ) \\cdots ( z _ { R - k - 1 } - 2 ) ( z _ { R - k } - 2 + \\chi ( p _ { R - k + 1 } = 0 ) ) . \\end{align*}"}
-{"id": "5345.png", "formula": "\\begin{align*} | | v ^ k | | ^ 2 = \\frac { ( k - 1 ) ! ( n - k ) ! } { ( n - 1 ) ! } | | v ^ 1 | | ^ 2 = \\frac { 1 } { \\binom { n - 1 } { k - 1 } } | | v ^ 1 | | ^ 2 \\end{align*}"}
-{"id": "178.png", "formula": "\\begin{align*} & \\pi _ { X _ { C } , Y _ { V } } = X _ { C } \\wedge Y _ { V } , \\\\ & \\pi _ { X _ { C } , Y _ { C } } = X _ { C } \\wedge Y _ { C } \\end{align*}"}
-{"id": "1123.png", "formula": "\\begin{align*} & \\gamma _ n ( u ) = \\sqrt { 4 - ( 1 - u / \\ln n ) ^ 2 S ( I ) ^ 2 } , \\ \\ \\beta _ n ( u ) = \\frac { ( 1 - u / \\ln n ) S ( I ) } { \\sqrt { 4 - ( 1 - u / \\ln n ) ^ 2 S ( I ) ^ 2 } } . \\end{align*}"}
-{"id": "4222.png", "formula": "\\begin{align*} \\left ( \\frac { f ( x _ n ) } { x _ n } \\right ) ^ { \\frac { 2 \\beta } { 3 } } \\left ( \\frac { f ( y _ n ) } { y _ n } \\right ) ^ { - \\frac { 2 \\beta } { 3 } } = 1 + \\mathcal O \\left ( ( x - y ) n ^ { - 3 } \\right ) \\end{align*}"}
-{"id": "1133.png", "formula": "\\begin{align*} & | x - y | \\geq | y _ i - y _ j | - \\max ( a _ i , a _ j ) \\geq \\varepsilon _ 0 ( \\ln n ) ^ { - 1 } - \\varepsilon _ 0 ( 2 \\ln n ) ^ { - 1 } = \\varepsilon _ 0 ( 2 \\ln n ) ^ { - 1 } . \\end{align*}"}
-{"id": "5472.png", "formula": "\\begin{align*} v ( s , \\hat X _ s ^ { t , x , a } ) \\ = \\ Y _ s ^ { t , x , a } , \\hat \\P , \\ , t \\leq s \\leq T \\end{align*}"}
-{"id": "7584.png", "formula": "\\begin{align*} \\begin{gathered} \\eta = X ' \\circ X ^ { - 1 } . \\end{gathered} \\end{align*}"}
-{"id": "3380.png", "formula": "\\begin{align*} N ( c , k ) = ( 1 + o ( 1 ) ) f ( c , k ) = ( 1 + o ( 1 ) ) 2 n ( G ( c k ) - G ( c ( k - 1 ) ) ) \\end{align*}"}
-{"id": "6068.png", "formula": "\\begin{align*} 1 \\ , = \\ , \\| \\epsilon _ i \\wedge \\epsilon _ j \\| _ x \\ , = \\ , \\lambda _ i \\lambda _ j \\| f _ \\ast ( e _ i ) \\wedge f _ \\ast ( e _ j ) \\| _ x \\ , \\leq \\ , \\lambda _ i \\lambda _ j \\epsilon \\phi ^ { - 4 } \\| e _ i \\wedge e _ j \\| _ x \\ , = \\ , \\lambda _ i \\lambda _ j \\epsilon \\phi ^ { - 4 } \\ , , \\end{align*}"}
-{"id": "9079.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\zeta ^ { ( k ) } \\in A \\\\ k \\in \\mathcal { K } } } \\sum _ { \\xi \\in I _ k } | a _ { \\xi } | ^ 2 = \\sum _ { \\xi \\in A } | a _ { \\xi } | ^ 2 + O ( \\delta K ) . \\end{align*}"}
-{"id": "2818.png", "formula": "\\begin{align*} \\displaystyle { b _ i - \\lambda _ i x _ i ^ * - \\sum _ { j = 1 } ^ n a _ { i j } x _ j ^ * = 0 , - e _ i u _ i ^ * + d _ i x _ i ^ * = 0 , i = 1 , \\ldots , n } . \\end{align*}"}
-{"id": "4816.png", "formula": "\\begin{align*} Q _ g ^ { a , b } ( M , \\partial M ) = \\inf \\{ Q _ g ^ { a , b } ( u ) ; \\textrm { $ u > 0 $ i n $ C ^ { \\infty } ( M ) $ } \\} \\end{align*}"}
-{"id": "3453.png", "formula": "\\begin{align*} F ' ( t ; w ) = ( \\log N + w / N ) - \\psi ( t + N ) , F '' ( t ; w ) = - \\psi ' ( t + N ) . \\end{align*}"}
-{"id": "9176.png", "formula": "\\begin{align*} S _ { t } \\left ( J \\right ) = J ^ { \\prime } , S _ { t } ^ { \\prime } + S _ { t } ^ { 2 } + R = 0 , \\end{align*}"}
-{"id": "2595.png", "formula": "\\begin{align*} J ' _ g ( t , r , \\xi ) : = \\gamma _ 0 ( \\gamma _ 0 - 1 ) t \\int _ 0 ^ 1 \\big ( \\mathbf 1 _ { \\gamma ( \\cdot ) = \\gamma _ 0 } \\kappa \\cdot ( \\phi \\eta _ { u t } ) ^ { \\gamma _ 0 - 1 } g \\big ( u t , r u ^ { \\frac { 1 } { \\gamma _ 0 - 1 } } , \\cdot \\big ) ^ { \\gamma _ 0 - 1 } \\big ) ( \\xi _ { ( 1 - u ) t } ) d u . \\end{align*}"}
-{"id": "8456.png", "formula": "\\begin{align*} E _ k ^ { \\theta } : = \\int \\frac { 1 } { A } | D _ t \\theta _ k | ^ 2 + i \\theta _ k \\overline { \\partial _ { \\alpha } \\theta _ k } d \\alpha . \\end{align*}"}
-{"id": "1901.png", "formula": "\\begin{align*} \\mathcal { E } ( t _ s ) = - \\frac { \\pi } { 6 L } + \\frac { \\mathcal { B } } { L } , \\qquad \\mbox { f o r } t _ s > 0 . \\end{align*}"}
-{"id": "733.png", "formula": "\\begin{align*} L _ { k } ^ { ( n - m ) } ( x ) = \\sum _ { i = 0 } ^ { k } ( - 1 ) ^ { i } \\binom { n - m + k } { k - i } \\frac { x ^ i } { i ! } \\end{align*}"}
-{"id": "3410.png", "formula": "\\begin{align*} f ( T ) = - \\log \\Big ( \\rho _ * ( T ) + \\rho _ 0 ( T ) \\Big ) . \\end{align*}"}
-{"id": "5421.png", "formula": "\\begin{align*} \\lambda _ 1 ( A , \\mu ) > \\lambda _ 2 ( A , \\mu ) > \\cdots > \\lambda _ k ( A , \\mu ) > \\lambda _ \\infty ( A , \\mu ) = - \\infty \\end{align*}"}
-{"id": "9767.png", "formula": "\\begin{align*} \\lambda _ { * , X _ \\ell } : = N ( 0 ^ + , u ( X _ \\ell + \\ , \\cdot \\ , ) - p _ { * , X _ \\ell } ) , \\end{align*}"}
-{"id": "367.png", "formula": "\\begin{align*} 3 2 | a c - a ^ 2 | & \\leqslant 3 2 | a c | + 3 2 | a | ^ 2 \\\\ & \\leqslant 3 2 | a c | + 3 2 | a | \\cdot \\frac { | c | } { 4 } \\\\ & = 4 0 | a c | , \\end{align*}"}
-{"id": "4329.png", "formula": "\\begin{align*} \\begin{aligned} h _ K ^ { - 1 / 2 } \\| Q _ K u _ I - Q _ F u _ I \\| _ { 0 , F } & = h _ K ^ { - 1 / 2 } \\| Q _ F ( Q _ K u _ I - u _ I ) \\| _ { 0 , F } \\\\ & \\lesssim h _ K ^ { - 1 / 2 } \\| Q _ K u _ I - u _ I \\| _ { 0 , F } \\\\ & \\lesssim h _ K ^ { - 1 / 2 } ( \\| Q _ K u _ I - u \\| _ { 0 , F } + \\| u - u _ I \\| _ { 0 , F } ) \\\\ & \\lesssim h _ K ^ { - 1 } ( \\| Q _ K u _ I - u \\| _ { 0 , K } + \\| u - u _ I \\| _ { 0 , K } ) \\\\ & + ( | Q _ K u _ I - u | _ { 1 , K } + | u - u _ I | _ { 1 , K } ) \\\\ & \\lesssim h _ K ^ { k } | u | _ { k + 1 , K } \\end{aligned} \\end{align*}"}
-{"id": "4309.png", "formula": "\\begin{align*} S ^ { } _ K ( u , v ) = \\sum _ { r = 1 } ^ { N _ K } \\chi _ r ( u - \\Pi _ K u ) \\chi _ r ( v - \\Pi _ K v ) , \\ ; u , v \\in V _ h , \\end{align*}"}
-{"id": "584.png", "formula": "\\begin{align*} Z ^ x _ t = Z ^ x _ 0 + \\int _ 0 ^ t \\alpha ^ x _ s \\ , \\mathrm { d } s + \\int _ 0 ^ t \\sigma ^ x _ s \\ , \\mathrm { d } B _ s , \\ ; \\ ; t \\in [ 0 , T ] , x \\in \\mathcal { X } , \\end{align*}"}
-{"id": "3373.png", "formula": "\\begin{align*} \\inf _ { x \\in S } \\| B x \\| _ 2 > 0 H ^ \\perp \\cap S = \\emptyset . \\end{align*}"}
-{"id": "3457.png", "formula": "\\begin{align*} Q _ k ( s , t ) = \\frac { \\exp \\Big ( ( M + 1 ) F ( t ; x _ N ( k ) ) \\Big ) } { \\exp \\Big ( ( M + 1 ) F ( s ; x _ N ( k ) ) \\Big ) } \\frac { \\Gamma ( t ) } { \\Gamma ( s ) } , q _ k ( s , t ; \\xi , \\eta ) = \\frac { \\exp ( ( t + k - 1 ) \\xi \\sqrt { \\frac { M + 1 } { N } } ) } { \\exp ( ( s + k - 1 ) \\eta \\sqrt { \\frac { M + 1 } { N } } ) } = \\frac { e ^ { \\xi \\tau } } { e ^ { \\eta \\sigma } } , \\end{align*}"}
-{"id": "4365.png", "formula": "\\begin{align*} G _ { X , \\omega } ( s ) ( A ) : = e ^ { \\omega s } X ^ s A X ^ { - s } + e ^ { \\omega ( 1 - s ) } X ^ { 1 - s } A X ^ { - ( 1 - s ) } . \\end{align*}"}
-{"id": "5416.png", "formula": "\\begin{align*} & \\int _ { \\R ^ N } \\frac { 1 } { | x | ^ \\alpha } | u ( \\cdot - x _ n ) | ^ 2 \\ , d x \\\\ & \\leq \\int _ { \\R ^ N } \\frac { 1 } { | x | ^ \\alpha } | u ( \\cdot - x _ n ) - \\varphi _ m ( \\cdot - x _ n ) | ^ 2 \\ , d x + \\int _ { \\R ^ N } \\frac { 1 } { | x | ^ \\alpha } | \\varphi _ m ( \\cdot - x _ n ) | ^ 2 \\ , d x \\\\ & = \\int _ { \\R ^ N } \\frac { 1 } { | x | ^ \\alpha } | u ( \\cdot - x _ n ) - \\varphi _ m ( \\cdot - x _ n ) | ^ 2 \\ , d x + o ( 1 ) . \\end{align*}"}
-{"id": "67.png", "formula": "\\begin{align*} \\sigma _ { \\pm 1 } = \\Delta J = \\pm \\Delta f ^ \\pm = \\pm \\Delta \\rho \\ , . \\end{align*}"}
-{"id": "7836.png", "formula": "\\begin{align*} [ \\langle v , x \\rangle \\langle x , y \\rangle \\langle y , v \\rangle \\xi , \\xi ] = \\lambda \\| x \\| ^ 2 \\| y \\| ^ 2 \\end{align*}"}
-{"id": "4354.png", "formula": "\\begin{align*} \\frac { { n - i \\choose z _ a } } { { n - i \\choose z _ b } } \\ge \\frac { ( n - i - z _ b ) } { z _ b + 1 } = \\frac { ( n - 2 k + z _ a ) } { z _ b + 1 } \\ge \\frac { ( n - 2 k + z _ b + 1 ) } { z _ b + 1 } \\ge \\frac { ( n - k - 2 ) } { k - 2 } . \\end{align*}"}
-{"id": "2489.png", "formula": "\\begin{align*} ( T _ 1 x ) ( t ) & = x ( p ) \\\\ ( T _ 1 x ) ( s ) & = x ( p ) - x ( \\varphi ( s ) ) \\qquad \\end{align*}"}
-{"id": "9277.png", "formula": "\\begin{align*} \\ \\bar u : = c _ 0 + c _ 1 \\psi \\end{align*}"}
-{"id": "8307.png", "formula": "\\begin{align*} ( I - \\mathcal { H } ) A = & 1 + i [ D _ t \\zeta , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } \\mathfrak { F } } { \\zeta _ { \\alpha } } + i [ D _ t ^ 2 \\zeta , \\mathcal { H } ] \\frac { \\bar { \\zeta } _ { \\alpha } - 1 } { \\zeta _ { \\alpha } } - ( I - \\mathcal { H } ) \\frac { 1 } { 2 \\pi } \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j ( D _ t \\zeta ( \\alpha , t ) - \\dot { z } _ j ( t ) ) } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } . \\end{align*}"}
-{"id": "7561.png", "formula": "\\begin{align*} \\nabla _ x ( X ' \\circ X ^ { - 1 } ) = \\left ( ( \\nabla _ a X ) \\circ X ^ { - 1 } \\right ) ^ { - 1 } \\left ( ( \\nabla _ a X ' ) \\circ X ^ { - 1 } \\right ) \\end{align*}"}
-{"id": "4105.png", "formula": "\\begin{align*} \\mathbb K ^ { ( \\alpha , \\frac { 1 } { 2 } ) } ( x , y ) = \\frac { 1 } { 2 \\pi i ( x - y ) } \\begin{pmatrix} 1 & - 1 & 0 \\end{pmatrix} \\frac { 1 } { 4 \\pi ^ 2 } \\Psi _ { \\alpha , + } ^ T ( y ) \\begin{pmatrix} 1 & 0 & 0 \\\\ - 2 \\alpha - \\frac { 1 } { 2 } & - 1 & 0 \\\\ \\alpha ( \\alpha + \\frac { 1 } { 2 } ) & 2 \\alpha + \\frac { 1 } { 2 } & 1 \\end{pmatrix} \\Phi _ { \\alpha , + } ( x ) \\begin{pmatrix} 1 \\\\ 1 \\\\ 0 \\end{pmatrix} \\end{align*}"}
-{"id": "7623.png", "formula": "\\begin{align*} w ^ { \\varepsilon } ( x , t ) \\ = \\ u ( x - \\frac { b } { \\varepsilon } \\ , t , t ) + \\varepsilon \\varkappa _ 1 ( \\frac { x } { \\varepsilon } ) \\cdot \\nabla u ( x - \\frac { b } { \\varepsilon } \\ , t , t ) + \\varepsilon ^ 2 \\varkappa _ 2 ( \\frac { x } { \\varepsilon } ) \\cdot \\nabla \\nabla u ( x - \\frac { b } { \\varepsilon } \\ , t , t ) , \\end{align*}"}
-{"id": "2952.png", "formula": "\\begin{align*} & \\{ s : s ( x ) \\leq s ( y ) s ( z ) = \\infty s ( u ) \\leq 1 \\} \\end{align*}"}
-{"id": "6741.png", "formula": "\\begin{align*} L ( p _ n , { ( m _ i + 1 ) } ^ 2 ) & = p _ n ( 2 ( m _ i + 1 ) - p _ n ) . \\end{align*}"}
-{"id": "608.png", "formula": "\\begin{align*} \\mathcal M ^ \\triangledown : = \\Delta \\wr \\mathcal M \\to \\nabla \\ . \\end{align*}"}
-{"id": "7778.png", "formula": "\\begin{align*} { A _ r ^ d } f ( x ) = \\int _ { \\mathbb S ^ { d - 1 } } f ( x - r \\theta ) { \\rm d } \\sigma _ { d - 1 } ( \\theta ) , \\end{align*}"}
-{"id": "5021.png", "formula": "\\begin{align*} \\lambda _ 3 ( \\tilde { t } , \\tilde { x } ) = \\displaystyle \\min _ { ( t , x ) \\in [ 0 , T ] \\times { \\bar \\Omega } } \\lambda _ 3 ( t , x ) . \\end{align*}"}
-{"id": "5755.png", "formula": "\\begin{align*} | | u ( \\cdot , \\cdot , t ) | | _ 2 ^ 2 & \\leq 2 \\left [ \\sum _ { m , n \\in \\N } | C _ { m , n } | ^ 2 + \\sum _ { m , n \\in \\N } | D _ { m , n } | ^ 2 \\right ] \\\\ & = 2 | | \\varphi | | _ 2 ^ 2 + | | \\psi | | _ 2 ^ 2 . \\end{align*}"}
-{"id": "8645.png", "formula": "\\begin{align*} R = 1 - \\varepsilon ^ 2 w , \\end{align*}"}
-{"id": "3295.png", "formula": "\\begin{align*} \\mu ( f ) = \\begin{cases} ( - 1 ) ^ { k } & \\mbox { i f $ f = P _ 1 P _ 2 \\cdots P _ k $ f o r d i s t i n c t $ P _ i \\in \\mathcal { P } $ , } \\\\ 0 & \\mbox { o t h e r w i s e } , \\end{cases} \\end{align*}"}
-{"id": "1835.png", "formula": "\\begin{align*} \\partial _ t ^ 2 \\phi - \\partial _ x ^ 2 \\phi + 2 \\xi \\ , \\delta ( x ) \\phi = 0 . \\end{align*}"}
-{"id": "2559.png", "formula": "\\begin{align*} F ( x , u ) = \\lim _ { t \\to \\infty } \\bar F ( x , u , t ) , x \\in E , u \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "10.png", "formula": "\\begin{align*} | \\mathbf { y } ^ { i } ( t , v _ 1 ; m ) - \\mathbf { y } ^ { i } ( t , v _ 2 ; m ) | \\leq K _ z | v _ 1 - v _ 2 | K _ z : = \\frac { C _ v } { C _ { \\eta } - C _ v } . \\end{align*}"}
-{"id": "95.png", "formula": "\\begin{align*} \\bigl \\| \\textbf { x } ^ { ( 2 ) } \\bigr \\| _ { \\ell _ 1 } & = \\bigl \\| \\textbf { x } ^ { ( 1 ) } \\bigr \\| _ { \\ell _ 1 } - 2 | \\bigl ( \\textbf { x } ^ { ( 1 ) } \\bigr ) _ { k _ 2 } | \\\\ & = \\bigl \\| \\textbf { x } \\bigr \\| _ { \\ell _ 1 } - 2 \\left ( | x _ { k _ 1 } | + \\bigl | \\bigl ( \\textbf { x } ^ { ( 1 ) } \\bigr ) _ { k _ 2 } \\bigr | \\right ) \\ , . \\end{align*}"}
-{"id": "5920.png", "formula": "\\begin{align*} \\beta _ i ^ { ( j _ i ) } = { \\rm g l c t } ( K _ { X _ i } + B _ i + M _ i \\mid B _ i ^ { ( j _ i ) } ) < b _ { i , i } ^ { ( j _ i ) } - b _ { i } ^ { ( j _ i ) } , \\end{align*}"}
-{"id": "7677.png", "formula": "\\begin{align*} ( V ^ { \\mathrm { o p } } ) ^ \\pm : = V ^ \\mp . \\end{align*}"}
-{"id": "1304.png", "formula": "\\begin{align*} U _ n = V _ n \\cup \\{ ( x ^ { - 1 } , y ^ { - 1 } ) \\in X ^ { - 1 } \\times X ^ { - 1 } : ( x , y ) \\in V _ n \\} \\cup \\varDelta _ { \\widetilde { X } } , \\end{align*}"}
-{"id": "1392.png", "formula": "\\begin{align*} \\triangledown _ n ^ { \\mathcal { N } } \\left ( \\eta , a \\right ) = \\sum _ { \\sigma \\in \\Sigma _ { n } } \\operatorname { s g n } ( \\sigma ) \\left ( ( \\eta _ 1 ^ \\sigma , \\ldots , \\eta _ n ^ \\sigma ) , a \\right ) \\end{align*}"}
-{"id": "9549.png", "formula": "\\begin{align*} \\sigma ( k , N ) = \\sum _ { d | k \\atop d \\geq k / N } d . \\end{align*}"}
-{"id": "1119.png", "formula": "\\begin{align*} & \\mathbb { P } ( \\xi ^ { ( n ) } ( [ y , y + G _ n ( x ) / S ( I ) ] ) = 0 ) - D _ n ( \\sqrt { 4 - y ^ 2 } / S ( I ) \\cdot G _ n ( x ) / 2 ) \\\\ = & \\mathbb { P } ( \\xi ^ { ( n ) } ( [ y , y + \\delta _ n / \\rho _ { s c } ( y ) ] ) = 0 ) - D _ n ( \\pi \\delta _ n ) \\\\ = & \\mathbb { P } ^ { G U E ( n ) } ( \\lambda _ i \\not \\in [ y , y + \\delta _ n / \\rho _ { s c } ( y ) ] , 1 \\leq i \\leq n ) \\\\ & - \\mathbb { P } ^ { C U E ( n ) } ( \\theta _ i \\not \\in [ 0 , 2 \\pi \\delta _ n ] , 1 \\leq i \\leq n ) \\leq O ( ( n \\ln n ) ^ { - 1 } ) , \\end{align*}"}
-{"id": "6747.png", "formula": "\\begin{align*} \\widehat { u } ( t , \\xi ) = \\dfrac { 1 } { 1 - e ^ { - i 2 \\pi \\xi c _ { j 0 } } } \\int _ { 0 } ^ { 2 \\pi } e ^ { - i \\xi H ( t _ j , \\tau ) } \\widehat { f } _ j ( t _ 1 , \\ldots , t _ j - \\tau , \\ldots , t _ n , \\xi ) d \\tau , \\end{align*}"}
-{"id": "4889.png", "formula": "\\begin{align*} \\{ f \\in C _ 0 ( X ) \\mid f ( X \\setminus U ) = 0 \\} \\simeq C _ 0 ( U ) \\end{align*}"}
-{"id": "9996.png", "formula": "\\begin{align*} c ^ 1 = 2 , c ^ 2 = 1 , c ^ 3 = 2 . \\end{align*}"}
-{"id": "1767.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } \\left [ \\left ( B _ { t _ { i } } - B _ { s } \\right ) \\left ( B _ { t _ { j } } - B _ { s } \\right ) \\right ] & \\leq \\mathbb { E } \\left [ \\left ( B _ { t _ { i } } - B _ { s } \\right ) ^ { 2 } \\right ] \\\\ & = ( t _ { i } - s ) ^ { 2 H } \\\\ & \\leq ( t - s ) ^ { 2 H } . \\end{aligned} \\end{align*}"}
-{"id": "5044.png", "formula": "\\begin{align*} v ^ \\top \\bar Q _ n ( x ) v = \\int \\Big ( \\sum _ { i = 0 } ^ d v _ i t ^ i \\Big ) ^ 2 g ( x + c _ n t ) w ( t ) \\ , d t . \\end{align*}"}
-{"id": "8861.png", "formula": "\\begin{align*} c _ { 1 b 2 } & = ( - 8 1 \\nu ^ 7 + 2 7 \\nu ^ 6 + 1 2 8 7 \\nu ^ 5 + 1 3 5 6 3 \\nu ^ 4 ) - 7 1 2 1 1 \\nu ^ 3 + 8 5 9 2 9 \\nu ^ 2 - 6 6 6 9 9 \\nu + 8 9 4 \\\\ & \\ge - 7 1 2 1 1 \\nu ^ 3 + 8 5 9 2 9 \\nu ^ 2 - 6 6 6 9 9 \\nu + 8 9 4 0 9 > 8 0 0 0 0 ( - \\nu ^ 3 + \\nu ^ 2 - \\nu + 1 ) \\ge 0 . \\end{align*}"}
-{"id": "9917.png", "formula": "\\begin{align*} f _ { n - } ^ { \\mu } ( x _ { n } ) = \\frac { d \\pi _ { n - } ^ { \\mu } } { d \\phi } ( x _ { n } ) = \\int _ { \\mathcal { X } } t ( x _ { n } | x _ { n - 1 } ) \\pi _ { n - 1 } ^ { \\mu } ( d x _ { n - 1 } ) \\end{align*}"}
-{"id": "8696.png", "formula": "\\begin{align*} j ( \\nu + \\rho ' - q - j ) c _ { h , i , j } = 4 ( i + 1 ) ( i + 2 ) ( 2 i + p ) ( 2 i + p + 2 ) c _ { h , i + 2 , j - 2 } \\end{align*}"}
-{"id": "7578.png", "formula": "\\begin{align*} \\tilde { S } ( u ) = \\sup _ { z \\le C T ^ { \\frac { 1 } { 2 } } } | ( \\nabla S ) ( u - z ) | \\end{align*}"}
-{"id": "3178.png", "formula": "\\begin{align*} { \\rm i n d } _ { \\varGamma ^ \\prime } ( U ^ \\prime ) = { \\rm i n d } _ \\varGamma ( U ) . \\end{align*}"}
-{"id": "3214.png", "formula": "\\begin{align*} J _ a ^ k : = \\left \\{ i \\in \\{ 1 , 2 , \\ldots , p ( x ^ { \\ast } ) \\} \\mid \\hat { g } _ i ( x ^ k ) + \\nabla \\hat { g } _ i ( x ^ k ) ^ { \\top } \\Delta _ { \\frac { 1 } { 2 } } x ^ k = 0 \\right \\} . \\end{align*}"}
-{"id": "7769.png", "formula": "\\begin{align*} \\| f \\| _ { \\ell ^ p ( \\mathcal Z ) } = \\Big ( \\sum _ { m \\in \\mathcal Z } | f ( m ) | ^ p \\Big ) ^ { 1 / p } \\| f \\| _ { \\ell ^ { \\infty } ( \\mathcal Z ) } = \\sup _ { m \\in \\mathcal Z } | f ( m ) | . \\end{align*}"}
-{"id": "1236.png", "formula": "\\begin{align*} \\MoveEqLeft \\int e ^ { - 3 i \\Phi ( \\xi , \\eta ) } f ( \\eta ) g ( \\eta - \\xi ) \\varphi _ 4 ( \\eta / \\xi ) d \\eta \\\\ & = \\int e ^ { - 3 i \\xi \\mu ^ 2 } f ( \\psi _ 1 ( \\mu / \\xi ) \\xi ) g ( ( \\psi _ 1 ( \\mu / \\xi ) - 1 ) \\xi ) \\varphi _ 4 ( \\psi _ 1 ( \\mu / \\xi ) ) \\psi ' _ 1 ( \\mu / \\xi ) d \\mu \\\\ & = : \\int e ^ { - 3 i \\xi \\mu ^ 2 } h ( \\xi , \\mu ) d \\mu . \\end{align*}"}
-{"id": "1738.png", "formula": "\\begin{align*} c _ { p , r } \\left ( \\bigcup _ { n = 1 } ^ { \\infty } A _ { n } \\right ) = \\lim _ { n \\to \\infty } c _ { p , r } ( A _ { n } ) . \\end{align*}"}
-{"id": "3341.png", "formula": "\\begin{align*} Z ( u ) : = \\exp \\Big ( \\sum _ { 1 \\le k \\le L } \\frac { 2 0 t ^ k } { k } u ^ k + \\sum _ { k > L } 2 0 ( r + 1 ) \\frac { t ^ { \\frac { k } { 2 } } } { k } u ^ k \\Big ) . \\end{align*}"}
-{"id": "10021.png", "formula": "\\begin{align*} S ( \\ell _ { 1 } ( X ) , \\mathcal { H } _ { p } ( X ) ) = \\sup \\{ \\sigma _ { a } ( D ) - \\sigma _ { \\mathcal { H } _ { p } ( X ) } ( D ) \\ , \\colon \\ , D \\in \\mathfrak { D } ( X ) \\} = 1 - \\frac { 1 } { \\cot ( X ) } \\ , . \\end{align*}"}
-{"id": "8853.png", "formula": "\\begin{align*} s _ 4 \\ge s _ 4 | _ { \\mu = 1 } & = ( 1 - w ) ( M - 1 ) ( w M ( 2 M - 3 ) + M + w + 1 ) \\ge 0 . \\end{align*}"}
-{"id": "7593.png", "formula": "\\begin{align*} \\left [ \\eta \\cdot \\nabla , \\mathbb { G } \\right ] = ( R \\otimes R ) \\mathbb { H } \\left [ \\eta \\cdot \\nabla , \\Gamma \\right ] + \\left [ \\eta ( t ) \\cdot \\nabla , ( R \\otimes R ) \\mathbb { H } \\right ] \\Gamma \\end{align*}"}
-{"id": "9220.png", "formula": "\\begin{align*} y _ { i j } = G _ 0 - 1 0 \\eta \\log _ { 1 0 } L _ { i j } + \\psi _ { i j } + \\zeta _ { i j } , \\end{align*}"}
-{"id": "6752.png", "formula": "\\begin{align*} \\begin{bmatrix} 1 & & \\\\ * & 1 & \\\\ & & 1 \\end{bmatrix} \\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\end{bmatrix} , \\begin{bmatrix} 1 & & \\\\ & 1 & \\\\ * & & 1 \\end{bmatrix} \\begin{bmatrix} 0 \\\\ 0 \\\\ * \\end{bmatrix} , \\begin{bmatrix} 1 & & \\\\ & 1 & \\\\ & * & 1 \\end{bmatrix} \\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "4430.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ 0 ^ t \\Bigl { \\langle } \\frac { \\dd u _ p } { \\dd t } ( \\tau ) , w ( \\tau ) \\Bigr { \\rangle } \\dd \\tau = - \\int _ { \\mathcal { Q } _ t } u _ p \\cdot \\partial _ t w \\dd x \\dd \\tau - \\int _ { \\Omega } u _ 0 \\cdot w ( 0 ) \\dd x + \\int _ { \\Omega } u _ p ( t ) \\cdot w ( t ) \\dd x . \\end{aligned} \\end{align*}"}
-{"id": "8990.png", "formula": "\\begin{align*} \\mathbf { z } _ R = \\mbox { e v } _ { \\mathcal { R } } ( ( x - \\alpha _ i ) F ( x ) ) \\ ; \\mbox { , } \\ ; \\mathbf { w } _ { \\overline { R } } = \\mbox { e v } _ { \\overline { \\mathcal { R } } } ( F ( x ) ) . \\end{align*}"}
-{"id": "9662.png", "formula": "\\begin{align*} g = \\begin{cases} ( | g _ 1 | ^ 2 + | g _ 2 | ^ 2 ) ^ { 1 / 2 } & \\tau , \\\\ g _ 1 & \\mathbb T \\setminus \\tau . \\end{cases} \\end{align*}"}
-{"id": "5941.png", "formula": "\\begin{align*} q ( K _ X + B + M ) \\sim q L = q f ^ * L _ Z = q f ^ * ( K _ Z + B _ Z + M _ Z ) . \\end{align*}"}
-{"id": "2827.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\displaystyle { x _ i ^ { \\prime } ( t ) \\le x _ i ( t ) \\biggl ( b _ i - \\mu _ i x _ i ( t ) - \\sum _ { j = 1 } ^ n a _ { i j } \\int _ 0 ^ { \\infty } K _ { i j } ( s ) x _ j ( t - s ) \\ , d s } \\\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\displaystyle { - c _ i \\int _ 0 ^ { \\infty } G _ i ( s ) u _ i ( t - s ) \\ , d s \\biggr ) } , \\\\ \\displaystyle { u _ i ^ { \\prime } ( t ) \\le - e _ i u _ i ( t ) + d _ i x _ i ( t ) , i = 1 , \\ldots , n } , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "938.png", "formula": "\\begin{align*} + \\sum _ { i _ { 1 } , i _ { 2 } , i _ { 3 } , i _ { 4 } = 1 } ^ { m } G _ 0 ^ { ( i _ { 4 } ) } G _ 0 ^ { ( i _ { 3 } ) } G _ 0 ^ { ( i _ { 2 } ) } \\Sigma _ { i _ { 1 } } \\hat I _ { 0 0 0 0 _ { \\tau _ { p + 1 } , \\tau _ p } } ^ { * ( i _ { 4 } i _ { 3 } i _ { 2 } i _ { 1 } ) } + { \\bf q } _ { p + 1 , p } + { \\bf r } _ { p + 1 , p } , \\end{align*}"}
-{"id": "9541.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( q ) _ n q ^ { n ^ 2 } } { ( z q ) _ n ( z ^ { - 1 } q ) _ n } = \\frac { 1 } { ( q ) _ N } + ( 1 - z ) \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - 1 ) ^ n ( q ) _ n q ^ { n ( 3 n + 1 ) / 2 } } { ( q ) _ { n + N } } \\left ( \\frac { 1 } { 1 - z q ^ n } - \\frac { 1 } { z - q ^ n } \\right ) , \\end{align*}"}
-{"id": "402.png", "formula": "\\begin{align*} S ^ { { \\rm c l } , 0 } _ { { \\rm L } } ( \\hat g , g ) : = \\int _ M \\big ( | d \\omega | ^ 2 _ { g } + 2 K _ { g } \\omega \\big ) { \\rm d v } _ { g } , \\end{align*}"}
-{"id": "4378.png", "formula": "\\begin{align*} H _ r = \\left \\langle X _ r \\right \\rangle \\leq F _ 2 ^ { ( 1 ) } \\times \\dots \\times F _ 2 ^ { ( r ) } , \\end{align*}"}
-{"id": "4610.png", "formula": "\\begin{align*} \\| \\nu _ \\delta \\| _ 2 ^ 2 \\lesssim \\delta ^ { - 4 } \\sum _ { j } 2 ^ { 2 j } \\delta ^ 3 \\delta ^ { - 2 s } / ( 2 ^ j \\delta ^ { - 2 s } ) ^ 3 = \\sum _ j \\delta ^ { - 1 + 2 s } 2 ^ { - j } . \\end{align*}"}
-{"id": "257.png", "formula": "\\begin{align*} \\frac { h _ i } { f ^ i } = \\sum _ { j = i } ^ p { j \\choose i } Q _ { j - i } ( - \\beta ) \\frac { g _ j } { f ^ j } . \\end{align*}"}
-{"id": "7884.png", "formula": "\\begin{align*} \\Big | \\mathfrak m _ N ( \\xi ) - e ^ { - \\kappa ( d , N ) ^ 2 \\sum _ { i = 1 } ^ d \\sin ^ 2 ( \\pi \\xi _ i ) } \\Big | \\le 1 7 e ^ { - \\frac { c \\kappa ( d , N ) ^ 2 } { 4 0 0 } \\sum _ { i = 1 } ^ d \\sin ^ 2 ( \\pi \\xi _ i ) } . \\end{align*}"}
-{"id": "3586.png", "formula": "\\begin{align*} \\langle u , v \\cdot w \\rangle = \\langle u \\cdot v , w \\rangle . \\end{align*}"}
-{"id": "214.png", "formula": "\\begin{align*} \\Delta u = V u \\end{align*}"}
-{"id": "8987.png", "formula": "\\begin{align*} c ( A _ { \\varepsilon } ) = 1 , \\varepsilon > 0 , \\ A _ { \\varepsilon } : = \\{ \\omega \\in A : \\rho ( \\omega , \\omega ^ { 0 } ) \\leq \\varepsilon \\} . \\end{align*}"}
-{"id": "3076.png", "formula": "\\begin{align*} | I _ 3 | & \\leq \\int _ { \\frac { 1 } { 2 \\beta } \\leq | x | \\leq \\frac { 1 } { \\beta } } \\frac { 1 } { | x | ^ { n - \\beta } } \\ , d x + \\int _ { \\frac { 1 } { \\beta } \\leq | x - z | \\leq \\frac { 2 } { \\beta } } \\frac { 1 } { | x - z | ^ { n - \\beta } } \\ , d x \\\\ & \\leq C \\frac { 1 - 2 ^ { - \\beta } } { \\beta } \\beta ^ { - \\beta } + C \\frac { 2 ^ \\beta - 1 } { \\beta } \\beta ^ { - \\beta } . \\end{align*}"}
-{"id": "6595.png", "formula": "\\begin{align*} K _ 1 \\leq \\prod _ { r = 1 } ^ { F _ { n _ j } } | 2 \\sin \\pi ( r \\varphi + k _ j \\varphi ) | \\leq K _ 2 , \\end{align*}"}
-{"id": "5454.png", "formula": "\\begin{align*} \\hat V _ 0 ^ { \\mathcal R } \\ = \\ \\bar V _ 0 ^ { \\mathcal R } . \\end{align*}"}
-{"id": "1934.png", "formula": "\\begin{align*} F _ 4 ( x ) \\equiv x ^ 2 \\sum _ { j = 1 } ^ \\infty \\frac { A _ j ^ 2 } { Z _ j ^ 4 } = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "835.png", "formula": "\\begin{align*} \\omega _ i ^ { \\sigma _ 1 \\sigma _ 2 } = \\sum _ { \\nu , \\mu = 1 } ^ { g } \\rho ( \\sigma _ 1 ) ^ { \\sigma _ 2 } _ { \\nu , i } \\rho ( \\sigma _ 2 ) _ { \\mu , \\nu } \\omega _ \\mu \\end{align*}"}
-{"id": "5881.png", "formula": "\\begin{align*} C _ w C _ s = \\begin{cases} - \\left ( q ^ { \\frac { 1 } { 2 } } + q ^ { - \\frac { 1 } { 2 } } \\right ) C _ w & ~ s \\in \\mathcal { R } ( w ) \\\\ \\sum _ { \\substack { | \\ell ( w ) - \\ell ( y ) | = 1 \\\\ y s < y } } C _ y & \\textit { i f } ~ s \\not \\in \\mathcal { R } ( w ) \\end{cases} . \\end{align*}"}
-{"id": "3511.png", "formula": "\\begin{align*} f _ { k } ( x , t ) = \\log \\Gamma ( 1 + k - t ) + \\frac { 1 } { 2 } \\gamma t ^ 2 - ( x - \\log k ) t , \\end{align*}"}
-{"id": "2432.png", "formula": "\\begin{align*} [ T ^ { n - d } ] \\frac { P _ f ( T ) } { ( 1 - T ) ( 1 - q T ) } ( x _ 0 T + x _ 1 ( 1 - T ) ) ^ n = \\frac { f ( x _ 0 , x _ 1 ) - x _ 0 ^ n } { q - 1 } . \\end{align*}"}
-{"id": "2163.png", "formula": "\\begin{align*} T _ j ( p g _ { L + 1 } ) ( \\mathbf { x } ) & = \\sum _ { \\ell = 0 } ^ { d } \\sum _ { \\| \\alpha \\| \\leq \\ell } d _ { \\ell , \\alpha } T _ j \\circ T ^ { \\alpha } ( g _ { L + 1 + \\ell } ) ( \\mathbf { x } ) \\\\ & = \\sum _ { \\ell = 0 } ^ { d } \\sum _ { \\| \\alpha \\| \\leq \\ell } d _ { \\ell , \\alpha } T ^ { \\alpha + e _ j } ( g _ { L + 1 + \\ell } ) ( \\mathbf { x } ) = \\sum _ { \\ell = 0 } ^ { d + 1 } \\sum _ { \\| \\alpha \\| \\leq \\ell } d _ { \\ell , \\alpha } ' T ^ { \\alpha } ( g _ { L + \\ell } ) ( \\mathbf { x } ) . \\end{align*}"}
-{"id": "6651.png", "formula": "\\begin{align*} P _ { \\tau , Q } & = \\sum _ { j = 0 } ^ { 2 } P _ { \\tau , Q } ^ { j } , \\\\ \\textup { w h e r e } P _ { \\tau , Q } ^ { 0 } ( n ) & = P _ { \\tau , Q } ( n ) \\mathbf 1 _ { \\mathbb S _ { \\tau } } ( n ) , \\\\ \\textup { a n d } P _ { \\tau , Q } ^ { 1 } ( n ) & = P _ { \\tau , Q } ( n ) \\mathbf 1 _ { 0 < \\textup { d i s t } ( n , \\mathbb S _ { \\tau } ) < \\tau Q ^ { 1 + \\epsilon } / N } , \\end{align*}"}
-{"id": "6246.png", "formula": "\\begin{align*} ( A , \\psi ) : = \\underset { \\alpha _ { 1 } , \\cdots , \\alpha _ { n } } { \\overset { e _ { 1 1 } , \\cdots , e _ { n n } } { M _ { n } ( \\C ) } } \\oplus \\underset { \\gamma } { \\C } , \\end{align*}"}
-{"id": "5316.png", "formula": "\\begin{align*} \\varphi = w _ { \\alpha _ 5 } \\circ w _ { \\alpha _ 2 } \\circ \\sigma _ { 0 1 } \\circ \\sigma _ { 1 2 } . \\end{align*}"}
-{"id": "6624.png", "formula": "\\begin{align*} T = \\sum _ { X \\in 2 ^ V } \\tau _ X X , \\end{align*}"}
-{"id": "182.png", "formula": "\\begin{align*} [ \\pi _ { T M , X _ { C } , Y _ { V } , c } , \\pi _ { T M , X _ { C } , Y _ { V } , c } ] = 2 \\lambda c ( { \\bf x } ) [ \\pi _ { T M } , X _ { C } \\wedge Y _ { V } ] = 0 , \\end{align*}"}
-{"id": "8177.png", "formula": "\\begin{align*} S ( \\Lambda _ { i } ^ * e _ { j } ) = \\Lambda _ { i + 1 } ^ * e _ { j } . \\end{align*}"}
-{"id": "7192.png", "formula": "\\begin{gather*} \\xi ^ { \\prime } ( 0 ) q + \\nu ( 0 ) p = H ( q , p ) \\end{gather*}"}
-{"id": "1058.png", "formula": "\\begin{align*} & \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) \\geq ( n / 4 ) ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } \\mathbb { P } ( \\xi ^ { ( n ) } ( I _ { n , k } ) = 0 ) \\\\ & - ( n / 4 ) ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } \\sum _ { j = 1 } ^ k \\mathbb { P } ( \\xi ^ { ( n ) } ( I _ { n , k } ) = \\xi ^ { ( n ) } ( J _ { n , k , j } ) = 0 ) . \\end{align*}"}
-{"id": "9699.png", "formula": "\\begin{align*} I _ { \\textbf { d } } ( P _ i ) = \\frac { 1 } { 2 \\pi } \\int _ { \\partial \\Omega ( P _ i ) } d \\theta . \\end{align*}"}
-{"id": "7040.png", "formula": "\\begin{align*} E ( z ) = E ( u ( z ) ) = \\frac { 1 } { 2 } \\int _ M | d u ( z ) | ^ 2 d \\mu _ g \\end{align*}"}
-{"id": "4773.png", "formula": "\\begin{align*} \\lambda _ 2 ( Q _ 1 ) = \\lambda _ 1 ( \\widetilde { Q _ 1 } ) \\geq \\lambda _ 1 ( \\widetilde { Q ' _ 1 } ) . \\end{align*}"}
-{"id": "9178.png", "formula": "\\begin{align*} \\mathcal { K } \\equiv \\left \\{ \\left . J \\in \\Lambda \\right \\vert J \\left ( t \\right ) = 0 t \\in \\left ( 0 , \\frac { \\pi } { 2 } + r \\right ] \\right \\} . \\end{align*}"}
-{"id": "5806.png", "formula": "\\begin{align*} f _ t = \\Psi \\circ g _ t . \\end{align*}"}
-{"id": "4498.png", "formula": "\\begin{align*} H _ v - \\Theta ^ t _ v = \\frac { \\partial } { \\partial v } \\big ( H - \\Theta ^ t \\big ) = F ^ { i j } a _ { i j } - 2 \\Theta ^ t = - \\Theta ^ t \\leq 0 \\end{align*}"}
-{"id": "7182.png", "formula": "\\begin{gather*} G ( t , s , \\xi ) : = \\begin{cases} - X ^ { t } ( \\xi ) \\left [ P _ { \\xi } ^ { + } + P _ { \\xi } ^ { 0 } \\right ] \\left [ X ^ { s } ( \\xi ) \\right ] ^ { - 1 } & t \\le s \\\\ X ^ { t } ( \\xi ) P _ { \\xi } ^ { - } \\left [ X ^ { s } ( \\xi ) \\right ] ^ { - 1 } & t > s . \\end{cases} \\end{gather*}"}
-{"id": "9799.png", "formula": "\\begin{align*} w : = v _ { \\ast } - q _ \\circ . \\end{align*}"}
-{"id": "5061.png", "formula": "\\begin{align*} \\left \\| \\sum \\limits _ { j \\in \\mathbb { J } } L _ j A _ j h \\right \\| ^ 2 = \\sum \\limits _ { j \\in \\mathbb { J } } \\| A _ j h \\| ^ 2 = \\sum \\limits _ { j \\in \\mathbb { J } } \\sum \\limits _ { k \\in \\mathbb { L } _ j } | \\langle h , u _ { j , k } \\rangle | ^ 2 ; ~ \\left \\| \\sum \\limits _ { j \\in \\mathbb { J } } L _ j \\Psi _ j h \\right \\| ^ 2 = \\sum \\limits _ { j \\in \\mathbb { J } } \\sum \\limits _ { k \\in \\mathbb { L } _ j } | \\langle h , v _ { j , k } \\rangle | ^ 2 . \\end{align*}"}
-{"id": "5376.png", "formula": "\\begin{align*} \\langle A _ { \\alpha } . v _ { n m } ^ k , v _ { n - 2 k + 2 l \\ , m } ^ { l - 1 } \\rangle = \\langle - ( k - 1 ) v _ { n m } ^ { k - 1 } + ( n - k ) v _ { n m } ^ { k + 1 } , v _ { n - 2 k + 2 l \\ , m } ^ { l - 1 } \\rangle = 0 . \\end{align*}"}
-{"id": "6803.png", "formula": "\\begin{align*} y ^ 1 _ { 1 1 } = y ^ 1 _ { 2 0 } = y ^ 2 _ { 2 0 } = y ^ 2 _ { 1 1 } = 0 \\ , . \\end{align*}"}
-{"id": "3193.png", "formula": "\\begin{align*} U \\varphi _ 0 = U C _ - \\varphi = - C _ - \\varphi = - \\varphi _ 0 , \\end{align*}"}
-{"id": "6238.png", "formula": "\\begin{align*} \\left ( p _ { 3 } W ^ { * } ( u _ { 4 } Y _ { e _ 1 } Y _ { e _ 1 } ^ { * } u _ { 4 } ^ { * } ) p _ { 3 } , \\phi ^ { p _ 3 } \\right ) = \\begin{cases} \\underset { \\alpha _ { n } } { ( L ( \\Z ) , \\tau ) } & \\gamma ' \\geq 1 \\\\ \\underset { \\alpha _ { n } \\gamma ' } { ( L ( \\Z ) , \\tau ) } \\oplus \\underset { \\alpha _ { n } ( 1 - \\gamma ' ) } { \\C } & \\gamma ' < 1 \\end{cases} . \\end{align*}"}
-{"id": "8722.png", "formula": "\\begin{align*} p g f _ { T _ { m , d } ^ * } ( s ) = \\sum _ { j \\leq m } \\hat { \\nu } ( j ) p g f _ { \\hat { T } _ { j , d } } ( s ) , \\end{align*}"}
-{"id": "220.png", "formula": "\\begin{align*} L u = \\psi \\left ( f '' ( \\rho ) + \\frac { Q - 1 } { \\rho } f ' ( \\rho ) \\right ) , \\ . \\end{align*}"}
-{"id": "5262.png", "formula": "\\begin{align*} 1 - F ( x ) = c ( x ) \\exp \\left ( - \\int _ { x _ 1 } ^ { x } \\frac { a ( t ) } { r ( t ) } d t \\right ) , \\ x _ 1 \\leq x < u e p ( F ) , \\end{align*}"}
-{"id": "5713.png", "formula": "\\begin{align*} \\frac { d } { d t } \\left ( \\int _ 0 ^ t ( t - z ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } [ \\lambda ( t - z ) ^ { \\alpha } ] f ( z ) d z \\right ) & = \\int _ 0 ^ t ( t - z ) ^ { \\alpha - 2 } E _ { \\alpha , \\alpha - 1 } ( \\lambda ( t - z ) ^ { \\alpha } ) f ( z ) d z . \\end{align*}"}
-{"id": "2025.png", "formula": "\\begin{align*} { \\mathrm { R e s } } _ { z = \\frac { \\alpha _ { j ' } - \\alpha _ j } { d } } I _ j ( q , z ) = q ^ d E ( d , j , j ' ) F ( j ' ) I _ { j ' } \\left ( q , z = \\frac { \\alpha _ { j ' } - \\alpha _ j } { d } \\right ) . \\end{align*}"}
-{"id": "3754.png", "formula": "\\begin{align*} \\left [ \\int _ { 0 } ^ { a } \\left | r j _ { l } ( k r ) \\right | ^ 2 d r \\right ] \\left [ \\int _ { 0 } ^ { a } \\left | r j _ { l } ( K r ) \\right | ^ 2 d r \\right ] \\geq \\left | \\int _ { 0 } ^ { a } j _ { l } ( k r ) j _ { l } ( K r ) r ^ 2 d r \\right | ^ 2 ; j = 2 , \\end{align*}"}
-{"id": "8550.png", "formula": "\\begin{align*} \\Big | \\mathcal { E } _ s - 4 \\sum _ { k = 0 } ^ s \\Big \\{ \\norm { \\partial _ { \\alpha } ^ k D _ t \\tilde { \\theta } } _ { L ^ 2 } ^ 2 + \\norm { \\partial _ { \\alpha } ^ k D _ t \\tilde { \\sigma } } _ { L ^ 2 } ^ 2 + \\norm { \\partial _ { \\alpha } ^ k | D | ^ { 1 / 2 } \\tilde { \\theta } } _ { L ^ 2 } ^ 2 + \\norm { \\partial _ { \\alpha } ^ k | D | ^ { 1 / 2 } \\tilde { \\sigma } } _ { L ^ 2 } ^ 2 \\Big \\} \\Big | \\leq C \\epsilon ^ 3 . \\end{align*}"}
-{"id": "2521.png", "formula": "\\begin{align*} \\{ a _ n \\mathbf Z _ n \\cdot \\mathbf j ; P ( \\cdot | \\mathbf Z _ n \\neq \\mathbf 0 , \\mathbf Z _ 0 = \\mathbf i ) \\} \\xrightarrow [ n \\to \\infty ] { \\operatorname { l a w } } ( \\mathbf v \\cdot \\mathbf j ) \\mathbf z ^ { ( \\alpha ) } , \\end{align*}"}
-{"id": "7175.png", "formula": "\\begin{gather*} \\mathbb { L } _ { x } ^ { - } = \\mathbb { J } _ { x } ^ { - } \\oplus \\mathbb { K } _ { x } ^ { - } \\end{gather*}"}
-{"id": "293.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty e _ n X ^ n & \\ ; = \\ ; \\exp \\Bigl ( \\ , \\sum _ { \\nu = 1 } ^ \\infty \\tfrac { ( - 1 ) ^ { \\nu + 1 } } { \\nu } \\ , p _ \\nu \\ , X ^ \\nu \\Bigr ) . \\end{align*}"}
-{"id": "5906.png", "formula": "\\begin{align*} T _ { ( s _ 1 s _ 0 ) ^ n } \\star \\varphi _ m = q ^ { 2 n } \\varphi _ { m + n } + ( q - 1 ) \\sum _ { k = 1 } ^ { 2 n } q ^ { 2 n - k } \\psi _ { m + n - k } ; \\end{align*}"}
-{"id": "1018.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to + \\infty } \\left ( \\frac { 1 } { 8 } \\ln ( 2 \\ln n ) - \\frac { 1 } { 4 } \\ln \\left ( n \\sin \\frac { \\alpha _ n } { 2 } \\right ) \\right ) = 0 . \\end{align*}"}
-{"id": "4225.png", "formula": "\\begin{align*} E _ { i n } ^ { - 1 } ( y _ n ) \\mathcal { O } \\left ( ( x - y ) n ^ { - \\frac { 5 } { 2 } } ) \\right ) E _ { i n } ( x _ n ) & = \\left ( \\mathbb { I } + C _ n ( y _ n ) \\right ) ^ { - 1 } E _ n ^ { - 1 } ( y _ n ) \\mathcal { O } \\left ( ( x - y ) n ^ { - \\frac { 5 } { 2 } } \\right ) E _ n ( x _ n ) \\left ( \\mathbb { I } + C _ n ( x _ n ) \\right ) \\\\ & = E _ n ^ { - 1 } ( y _ n ) \\mathcal { O } \\left ( ( x - y ) n ^ { - \\frac { 5 } { 2 } } \\right ) E _ n ( x _ n ) \\end{align*}"}
-{"id": "8703.png", "formula": "\\begin{align*} \\int _ { S ^ { p - 1 } } \\omega _ { 1 } ^ { 2 \\alpha _ 1 } \\dots \\omega _ { p } ^ { 2 \\alpha _ { p } } \\ , d \\omega = 2 ^ { - 2 \\abs { \\alpha } + 1 } \\frac { ( 2 \\alpha ) ! \\pi ^ { \\frac { p } { 2 } } } { \\alpha ! \\Gamma ( \\frac { p } { 2 } + \\abs { \\alpha } ) } . \\end{align*}"}
-{"id": "6390.png", "formula": "\\begin{align*} \\mathcal { K } _ { k } ( A , v ) \\equiv \\mathrm { s p a n } \\{ v , A v , \\dots , A ^ { k - 1 } v \\} , k = 1 , 2 , \\dots , \\end{align*}"}
-{"id": "7448.png", "formula": "\\begin{align*} \\displaystyle & \\int _ { N _ n ( F ) \\backslash P _ n ( F ) } \\int _ { N _ { n } ( F ) \\backslash G _ { n } ( F ) } W _ f ( p , h ) d h d p = \\\\ & \\int _ { P _ { n } ( F ) \\backslash G _ { n } ( F ) } \\int _ { P _ { n - 1 } ( F ) U _ n ( F ) \\backslash P _ n ( F ) } \\int _ { N _ { n - 1 } ( F ) \\backslash P _ { n - 1 } ( F ) } \\int _ { N _ { n - 1 } ( F ) \\backslash G _ { n - 1 } ( F ) } W _ f ( p ' p , h ' h ) \\lvert \\det ( p ' h ' ) \\rvert ^ { - 1 } d h ' d p ' d p d h \\end{align*}"}
-{"id": "980.png", "formula": "\\begin{align*} \\beta _ { i , i + j } ( I ) = \\sum _ { 1 \\leq p \\leq m , { \\rm d e g } ( f _ { p } ) = j } \\binom { n _ { p } } { i } . \\end{align*}"}
-{"id": "2006.png", "formula": "\\begin{align*} L ^ { \\otimes 5 } = \\omega _ { C , \\log } \\otimes \\O ( - d ) \\cong \\O ( - d - 1 ) , \\end{align*}"}
-{"id": "4463.png", "formula": "\\begin{align*} & g _ { i j } = \\frac { 1 } { u ^ 2 } ( \\sigma _ { i j } + \\frac { 1 } { u ^ 2 } \\nabla _ i u \\nabla _ j u ) \\\\ & \\eta = - \\frac { 1 } { ( u ^ 2 + | \\nabla u | ^ 2 ) ^ { 1 / 2 } } ( \\nabla u - u x ) \\\\ & h _ { i j } = \\frac { 1 } { u \\sqrt { u ^ 2 + | \\nabla u | ^ 2 } } ( u \\sigma _ { i j } + \\nabla _ { i j } u ) \\end{align*}"}
-{"id": "6628.png", "formula": "\\begin{align*} f ( y _ 1 , \\ldots , y _ v ) = y _ 1 \\cdots y _ { m - 2 } \\cdot ( y _ { m - 1 } \\cdot y _ m \\oplus y _ { m + 1 } \\cdot y _ { m + 2 } \\oplus \\cdots \\oplus y _ { m + 2 \\nu - 3 } \\cdot y _ { m + 2 \\nu - 2 } \\oplus a ) \\end{align*}"}
-{"id": "8043.png", "formula": "\\begin{gather*} \\rho _ 2 ( x ) = 1 , \\enspace \\rho _ 2 ( y ) = - 1 , \\rho _ 3 ( x ) = - 1 , \\enspace \\rho _ 3 ( y ) = - 1 , \\rho _ 4 ( x ) = - 1 , \\enspace \\rho _ 4 ( y ) = 1 , \\\\ \\rho _ 5 ( x ) = 1 , \\enspace \\rho _ 5 ( y ) = \\imath , \\rho _ 6 ( x ) = 1 , \\enspace \\rho _ 6 ( y ) = - \\imath , \\rho _ 7 ( x ) = - 1 , \\\\ \\rho _ 7 ( y ) = - \\imath , \\rho _ 8 ( x ) = - 1 , \\enspace \\rho _ 8 ( y ) = \\imath , \\end{gather*}"}
-{"id": "5895.png", "formula": "\\begin{align*} \\tilde { f } = \\sum _ { n = 1 } ^ { \\infty } q ^ { - n } ( - 1 - q ) C ' _ { ( s _ 0 s _ 1 ) ^ n } + q ^ { - n } \\left ( q ^ { \\frac { 1 } { 2 } } + q ^ { - \\frac { 1 } { 2 } } \\right ) C ' _ { s _ 0 ( s _ 1 s _ 0 ) ^ n } . \\end{align*}"}
-{"id": "8736.png", "formula": "\\begin{align*} \\omega ^ { ( \\alpha , \\sigma ) } ( z ) = \\left ( { 1 - z e ^ { - \\sigma \\tau } } \\right ) ^ { \\alpha } \\sum _ { k = 1 } ^ p g _ { k - 1 } ( 1 - z e ^ { - \\sigma \\tau } ) ^ { k - 1 } = \\sum _ { k = 0 } ^ { \\infty } \\omega ^ { ( \\alpha , \\sigma ) } _ { k } z ^ k , \\end{align*}"}
-{"id": "7282.png", "formula": "\\begin{align*} \\lVert \\Delta ^ { p , * } _ a ( h _ p ^ - - h _ q ^ - ) \\rVert + \\lVert \\Delta ^ { p , * } _ a ( h _ q ^ - - h _ s ^ - ) \\rVert & = \\lVert \\Delta ^ { p , * } _ a ( h _ p ^ - - h _ q ^ - ) \\rVert + \\lVert \\Delta ^ { q , * } _ a ( h _ q ^ - - h _ s ^ - ) \\rVert \\\\ & < \\epsilon _ s - \\epsilon _ p , \\end{align*}"}
-{"id": "639.png", "formula": "\\begin{align*} & \\varphi ^ \\ast ( \\psi ^ \\ast ( w z ) v \\psi ^ \\ast ( z ) ^ { - 1 } \\psi ^ \\ast ( z ) y ) u \\varphi ^ \\ast ( \\psi ^ \\ast ( z ) y ) ^ { - 1 } \\\\ & = ( \\psi \\varphi ^ { v y } ) ^ \\ast ( w z ) \\varphi ^ \\ast ( v y ) u \\left ( ( \\psi \\varphi ^ y ) ^ \\ast ( z ) \\varphi ^ \\ast ( y ) \\right ) ^ { - 1 } \\\\ & = ( \\psi \\varphi ^ y ) ^ \\ast ( w z ) \\varphi ^ \\ast ( v y ) u \\varphi ^ \\ast ( y ) ^ { - 1 } ( \\psi \\varphi ^ y ) ^ \\ast ( z ) ^ { - 1 } \\ , \\end{align*}"}
-{"id": "9949.png", "formula": "\\begin{align*} ( e ^ { - F } ) ^ * e ^ { - F } = I + T \\end{align*}"}
-{"id": "9977.png", "formula": "\\begin{align*} & u _ { x x } = v _ { x t } w _ { x x } - v _ { x x } w _ { x t } + w _ { x t } ^ { 2 } - w _ { x x } w _ { t t } , \\\\ & u _ { x t } = v _ { t t } w _ { x x } - v _ { x t } w _ { x t } , \\\\ & u _ { t t } = v _ { x t } ^ { 2 } - v _ { x x } v _ { t t } + v _ { t t } w _ { x t } - v _ { x t } w _ { t t } , \\end{align*}"}
-{"id": "6366.png", "formula": "\\begin{align*} \\gamma _ s : = \\frac { \\lambda _ s } { 1 - \\alpha _ s } \\geq \\varepsilon _ 1 , \\end{align*}"}
-{"id": "5712.png", "formula": "\\begin{align*} \\frac { d } { d x } \\int _ 0 ^ x f ( x , t ) d t = f ( x , x ) + \\int _ 0 ^ x \\frac { d } { d x } f ( x , t ) d t . \\end{align*}"}
-{"id": "3384.png", "formula": "\\begin{align*} p _ k = \\Theta \\left ( \\frac { \\delta } { n } 2 ^ { k - 1 } \\right ) . \\end{align*}"}
-{"id": "6616.png", "formula": "\\begin{align*} \\prod _ { t = 1 } ^ { \\eta } \\left ( 1 - \\frac { ( 1 - 2 \\varepsilon ( - \\varphi ) ^ { - n } ) ^ 2 } { u _ t ^ 2 } \\right ) & \\geq \\prod _ { t = 1 } ^ { \\infty } \\left ( 1 - \\frac { ( 1 - 2 \\varepsilon ( - \\varphi ) ^ { - n } ) ^ 2 } { u _ t ^ 2 } \\right ) \\big ( 1 - \\mathcal { O } ( \\eta ^ { - 1 } ) \\big ) . \\end{align*}"}
-{"id": "3249.png", "formula": "\\begin{align*} ( \\cos \\theta + i \\sin \\theta ) ^ n = \\cos n \\theta + i \\sin n \\theta . \\end{align*}"}
-{"id": "6798.png", "formula": "\\begin{align*} \\mathcal { R } _ 2 \\quad \\equiv \\begin{cases} \\partial _ j u ^ i + \\partial _ i u ^ j = 0 \\ , , \\\\ \\partial ^ 2 _ { j k } u ^ i = 0 \\ , , \\end{cases} \\end{align*}"}
-{"id": "4064.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } p _ j ( x ) q _ k ( x ^ { \\theta } ) w ( x ) d x = \\delta _ { j , k } \\end{align*}"}
-{"id": "3727.png", "formula": "\\begin{align*} M ( \\tau ^ n ) = n l ^ * - n E ( u _ 0 ) + M ( \\| u _ 0 \\| ^ n ) < M \\left ( \\left ( \\frac { 2 n - \\mu } { 2 n } \\alpha _ n \\right ) ^ { n - 1 } \\right ) + M ( \\| u _ 0 \\| ^ n ) \\end{align*}"}
-{"id": "2222.png", "formula": "\\begin{align*} u _ t - \\Delta u + \\nabla \\cdot ( u \\nabla v ) & = 0 , \\ \\ & x \\in { \\mathbb R } ^ d , \\ t > 0 , \\\\ \\Delta v + u & = 0 , \\ \\ & x \\in { \\mathbb R } ^ d , \\ t > 0 , \\\\ u ( x , 0 ) & = u _ 0 ( x ) , \\ \\ & x \\in { \\mathbb R } ^ d . \\end{align*}"}
-{"id": "2882.png", "formula": "\\begin{align*} [ \\ell _ { \\mu \\nu } , \\ell _ { \\rho \\sigma } ] = \\delta _ { \\nu \\rho } \\ell _ { \\mu \\sigma } - \\delta _ { \\nu \\sigma } \\ell _ { \\mu \\rho } - \\delta _ { \\mu \\rho } \\ell _ { \\nu \\sigma } + \\delta _ { \\mu \\sigma } \\ell _ { \\nu \\rho } . \\end{align*}"}
-{"id": "8196.png", "formula": "\\begin{align*} G ^ { \\varphi } = ( G ^ { \\varphi ( i ) \\varphi ( j ) } ) _ { i , j \\in [ n ] } . \\end{align*}"}
-{"id": "4046.png", "formula": "\\begin{align*} \\sum _ r m _ r ^ 2 = O ( \\log ( N ) N ^ 3 ) . \\end{align*}"}
-{"id": "7319.png", "formula": "\\begin{align*} \\mathcal { A } = \\left ( \\begin{array} { c c c c } 0 & 0 & I & 0 \\\\ 0 & 0 & 0 & I \\\\ \\frac { 1 } { \\rho } \\operatorname { d i v } ( K ( x ) \\nabla ( \\cdot ) ) & 0 & \\frac { 1 } { \\rho } \\operatorname { d i v } ( a ( x ) \\nabla ( \\cdot ) ) & 0 \\\\ 0 & \\frac { 1 } { \\rho } \\operatorname { d i v } ( K ( x ) \\nabla ( \\cdot ) ) & 0 & \\frac { 1 } { \\rho } \\operatorname { d i v } ( b ( x ) \\nabla ( \\cdot ) ) \\end{array} \\right ) , \\end{align*}"}
-{"id": "3845.png", "formula": "\\begin{align*} 2 h _ { 0 0 } ^ { \\frac { m + 1 } { 2 } } W _ { 0 } ^ m \\Phi ^ { i } _ { \\parallel l } & = & h _ { 0 0 } ^ { m + 1 } B ^ { i } _ { ( 1 ) \\parallel l } + m h _ { 0 0 } ^ { m + 1 } B ^ i _ { ( 2 1 ) l } + h _ { 0 0 } ^ { \\frac { m + 1 } { 2 } } W _ { 0 } ^ m B ^ { i } _ { ( 2 2 ) l } + \\\\ & & W _ { 0 } ^ { 2 m } B ^ i _ { ( 3 ) l } + m W _ 0 ^ { 2 m - 1 } B ^ i _ { ( 4 ) l } , \\end{align*}"}
-{"id": "7748.png", "formula": "\\begin{align*} \\Big | [ \\langle x , y \\rangle \\xi , \\xi ] \\Big | = \\| y \\| . \\end{align*}"}
-{"id": "8474.png", "formula": "\\begin{align*} \\norm { \\sum _ { j = 1 } ^ 2 \\partial _ { \\alpha } ^ k f _ j ( \\cdot , t ) \\circ g } _ { L ^ 2 } \\leq 1 0 0 ( k + 1 ) ! | \\lambda x ( t ) | d _ I ( t ) ^ { - 3 / 2 } . \\end{align*}"}
-{"id": "6832.png", "formula": "\\begin{align*} \\det ( d \\varphi ) = \\sum _ { i = 1 } ^ 6 p _ i g _ i \\ , , \\end{align*}"}
-{"id": "5352.png", "formula": "\\begin{align*} V | _ K = \\bigoplus _ { n \\in \\N , \\ , m \\in \\Z } V _ { n m } . \\end{align*}"}
-{"id": "7666.png", "formula": "\\begin{align*} \\int \\limits _ { \\mathbb R ^ d } \\int \\limits _ { \\mathbb T ^ d } \\alpha ( \\xi - q ) \\mu ( \\xi , q ) \\ , d \\xi d q = \\int \\limits _ { \\mathbb R ^ d } \\int \\limits _ { \\mathbb T ^ d } \\alpha ( \\xi - q ) \\mu ( q , \\xi ) \\ , d \\xi d q ; \\end{align*}"}
-{"id": "8768.png", "formula": "\\begin{align*} \\rho _ 0 = \\frac { \\widehat { \\kappa } } { 2 } + \\frac { \\widetilde { \\kappa } } { ( n - 3 ) ( n - 2 ) F } + \\frac { t r ( T ) } { 2 F } - \\frac { \\Delta _ 1 F } { 4 F ^ 2 } \\ , . \\end{align*}"}
-{"id": "8567.png", "formula": "\\begin{align*} \\sigma ( x ) ( \\omega ) = \\mu _ x ( \\omega ) = x \\wedge \\omega , \\sigma ( y ) ( \\omega ) = \\partial _ y ( \\omega ) , \\end{align*}"}
-{"id": "2943.png", "formula": "\\begin{align*} & \\| \\widetilde { V } _ { 2 } ^ { \\ast } V _ { 1 } \\Sigma _ { 1 } \\| _ { F } ^ { 2 } = \\| E \\| _ { F } ^ { 2 } - \\| E B ^ { \\dagger } B \\| _ { F } ^ { 2 } , \\\\ & \\| \\widetilde { \\Sigma } _ { 1 } \\widetilde { U } _ { 1 } ^ { \\ast } U _ { 2 } \\| _ { F } ^ { 2 } = \\| E \\| _ { F } ^ { 2 } - \\| A A ^ { \\dagger } E \\| _ { F } ^ { 2 } . \\end{align*}"}
-{"id": "1985.png", "formula": "\\begin{align*} H - \\alpha _ i = [ \\{ x _ i = 0 \\} ] . \\end{align*}"}
-{"id": "4519.png", "formula": "\\begin{align*} A _ { r _ 1 , \\dots , r _ n } = \\{ ( a _ 1 , a _ 2 , \\dots , a _ n ) \\in A : i \\in [ 1 , n ] , a _ i \\equiv r _ i \\mod 1 0 0 \\} . \\end{align*}"}
-{"id": "667.png", "formula": "\\begin{align*} [ \\psi ; g ; v , y ] \\circ [ \\varphi ; f ; u , x ] = [ \\psi \\varphi ^ y ; g f ; \\varphi ^ \\ast ( v y ) u \\varphi ^ \\ast ( y ) ^ { - 1 } , \\varphi ^ \\ast ( y ) x ] \\ ; \\end{align*}"}
-{"id": "7652.png", "formula": "\\begin{align*} D ^ { i j } _ 2 = \\ b ^ i \\int \\limits _ { \\mathbb T ^ d } \\varkappa ^ j _ 1 ( \\xi ) v _ 0 ( \\xi ) d \\xi \\ + \\ b ^ j \\int \\limits _ { \\mathbb T ^ d } \\varkappa ^ i _ 1 ( \\xi ) v _ 0 ( \\xi ) d \\xi , \\end{align*}"}
-{"id": "8981.png", "formula": "\\begin{align*} \\mathbb { \\hat { E } } [ X ] : = \\sup _ { P \\in \\mathcal { P } } E _ { P } [ X ] \\in [ - \\infty , \\infty ] , \\ \\ \\ \\ X \\in \\mathcal { L } ( \\Omega ) . \\end{align*}"}
-{"id": "7435.png", "formula": "\\begin{align*} \\displaystyle \\gamma ^ * ( 0 , \\mathbf { 1 } _ E , \\psi ' _ E ) = \\lim \\limits _ { s \\to 0 ^ + } \\zeta _ E ( s ) \\gamma ( s , \\mathbf { 1 } _ E , \\psi ' _ E ) = ( \\frac { \\log ( q _ F ) } { \\log ( q _ E ) } ) \\lim \\limits _ { s \\to 0 ^ + } \\zeta _ F ( s ) \\gamma ( s , \\mathbf { 1 } _ E , \\psi ' _ E ) \\end{align*}"}
-{"id": "3763.png", "formula": "\\begin{align*} \\frac { \\int _ { 0 } ^ { a } \\left | r j _ { l } ( K r ) \\right | ^ { 2 } d r } { \\left | \\int _ { 0 } ^ { a } j _ { l } ( k r ) j _ { l } ( K r ) r ^ 2 d r \\right | ^ 2 } = f _ { 0 } ^ { ( 2 ) } + f _ { 2 } ^ { ( 2 ) } \\ , \\chi ^ 2 + O [ \\chi ^ 3 ] \\end{align*}"}
-{"id": "3678.png", "formula": "\\begin{align*} \\widehat { Y } = \\sum _ { i \\in B } y _ i / \\hat { p } _ i \\end{align*}"}
-{"id": "3290.png", "formula": "\\begin{align*} \\frac { \\sum _ { x \\le n \\le x + x ^ { \\varepsilon } } \\alpha ( n ) } { x ^ { \\varepsilon } } = \\frac { \\sum _ { n \\le x } \\alpha ( n ) } { x } ( 1 + o ( 1 ) ) , x \\to \\infty \\end{align*}"}
-{"id": "2992.png", "formula": "\\begin{align*} e _ { g , n } & = \\begin{cases} 1 , & n = 0 ; \\\\ ( - 1 ) ^ k , & n = P _ { g , k } Q _ { g , k } ; \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "1994.png", "formula": "\\begin{align*} \\int _ { \\P ^ 2 } H ^ 2 = \\dots = \\frac { ( \\alpha _ 0 - \\alpha _ 2 ) ^ 2 } { ( \\alpha _ 0 - \\alpha _ 1 ) ( \\alpha _ 0 - \\alpha _ 2 ) } + \\frac { ( \\alpha _ 1 - \\alpha _ 2 ) ^ 2 } { ( \\alpha _ 1 - \\alpha _ 0 ) ( \\alpha _ 1 - \\alpha _ 2 ) } = \\dots = 1 , \\end{align*}"}
-{"id": "6112.png", "formula": "\\begin{align*} \\begin{aligned} & T _ 0 ^ { \\mu _ 2 } Y \\ \\ T _ 0 ^ { \\mu _ 2 - 1 } T _ 1 Y \\ \\ \\cdots \\ \\ T _ 1 ^ { \\mu _ 2 } Y \\\\ [ 5 p t ] & T _ 0 ^ { \\mu _ 1 } X \\ \\ \\cdots \\ \\ T _ 1 ^ { \\mu _ 1 } X \\end{aligned} \\end{align*}"}
-{"id": "4188.png", "formula": "\\begin{align*} A _ n ^ { ( 3 ) } ( z ) = - \\frac { \\left ( A _ n ^ { ( 1 ) } ( z ) - A _ n ^ { ( 1 ) } ( 0 ) \\right ) A _ n ^ { ( 1 ) } ( z ) \\left ( A _ n ^ { ( 1 ) } ( z ) - A _ n ^ { ( 1 ) } ( 0 ) \\right ) } { z ^ 2 } , \\end{align*}"}
-{"id": "7948.png", "formula": "\\begin{align*} \\sum _ { a \\in Q } x _ a = \\bigg [ \\sum _ { a \\in Q \\setminus S } x _ a \\bigg ] - \\bigg [ \\sum _ { b \\in T } x _ b \\bigg ] , \\end{align*}"}
-{"id": "2960.png", "formula": "\\begin{align*} \\theta _ 1 & = \\alpha _ 1 \\nabla _ 1 + \\alpha _ 2 \\nabla _ 2 + \\alpha _ 3 \\nabla _ 3 , \\\\ \\theta _ 2 & = \\beta _ 1 \\nabla _ 1 + \\beta _ 2 \\nabla _ 2 . \\end{align*}"}
-{"id": "8442.png", "formula": "\\begin{align*} F _ { \\zeta } \\circ \\zeta = \\frac { \\partial _ { \\alpha } \\mathfrak { F } } { \\zeta _ { \\alpha } } . \\end{align*}"}
-{"id": "4409.png", "formula": "\\begin{align*} g | _ { C _ 2 } = f . \\end{align*}"}
-{"id": "6332.png", "formula": "\\begin{align*} \\mathcal { E } ( q ( t ) ) + \\int _ 0 ^ t \\int _ 0 ^ 1 { q ^ 2 ( q _ { x x } - \\nu ^ { - \\frac { 1 } { 2 } } f ) ^ 2 \\ , d x } + \\nu ^ { - 1 / 2 } \\int _ 0 ^ t \\int _ 0 ^ 1 q _ x \\left [ \\int _ 0 ^ x f _ \\tau \\ , d s \\right ] \\ , d x d \\tau = \\mathcal { E } ( q _ 0 ) , \\end{align*}"}
-{"id": "2567.png", "formula": "\\begin{align*} - \\log \\mathbf P _ \\mu ( \\| X _ t \\| = 0 ) = \\mu ( v _ t ) \\leq \\mu ( \\phi ) \\| \\phi ^ { - 1 } v _ t \\| _ { \\infty } \\xrightarrow [ t \\to \\infty ] { } 0 . \\end{align*}"}
-{"id": "9827.png", "formula": "\\begin{align*} \\tilde u _ { X _ \\circ , 0 } L _ a \\tilde u _ { X _ \\circ , 0 } = 0 \\quad \\R ^ { n + 1 } . \\end{align*}"}
-{"id": "3022.png", "formula": "\\begin{align*} ( D _ { q } f ) ( z ) = \\frac { f ( z ) - f ( q z ) } { ( 1 - q ) z } . \\end{align*}"}
-{"id": "3497.png", "formula": "\\begin{align*} \\left . \\frac { d } { d y } \\Re { f _ { M } ( x + i y ; \\theta ) } \\right | _ { y = 0 } = \\left . \\frac { d } { d y } \\Re { f _ { M } ( x + i y ; \\theta ) } \\right | _ { y = \\Im { q _ { M } ( \\theta ) } } = 0 , \\end{align*}"}
-{"id": "7586.png", "formula": "\\begin{align*} \\begin{gathered} \\left ( \\eta ( t ) \\cdot ( \\nabla \\nabla g _ { \\nu ( t - s ) } ) * \\sigma ( s ) - ( \\nabla \\nabla g _ { \\nu ( t - s ) } ) * ( \\eta ( t ) \\sigma ( s ) ) \\right ) ( x ) \\\\ = \\int _ { \\mathbb { R } ^ d } \\nabla \\nabla g _ { \\nu ( t - s ) } ( z ) z \\cdot \\left ( \\int _ 0 ^ 1 \\nabla \\eta ( x - ( 1 - \\lambda ) z , t ) d \\lambda \\right ) \\sigma ( x - z , s ) d z \\end{gathered} \\end{align*}"}
-{"id": "8007.png", "formula": "\\begin{align*} A _ i & \\geq e ^ { - a | 1 - y _ i | ^ 2 } + e ^ { - a y _ i ^ 2 } \\geq \\frac { 1 } { 2 } e ^ { - \\frac { a } { 2 } ( | 1 - y _ i | ^ 2 + y _ i ^ 2 ) } \\\\ & = \\frac { 1 } { 2 } e ^ { a ( y _ i - y _ i ^ 2 - \\frac { 1 } { 2 } ) } \\geq \\frac { 1 } { 2 } e ^ { - \\frac { a } { 2 } } . \\end{align*}"}
-{"id": "8789.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\rightarrow 0 } \\frac { 1 } { \\sigma ^ 2 ( \\varepsilon ) } \\int _ { \\vert z \\vert > \\kappa \\sigma ( \\varepsilon ) } z ^ 2 \\ , Q ^ { \\varepsilon } ( \\textrm { d } z ) = 0 \\end{align*}"}
-{"id": "6866.png", "formula": "\\begin{align*} \\mathcal { R } _ { q + 1 } = \\rho _ 1 ( \\mathcal { R } _ q ) \\quad \\equiv \\begin{cases} F = 0 \\ , , \\\\ \\frac { \\partial F } { \\partial x _ j } = 0 , & \\ 1 \\le j \\le n \\ , . \\end{cases} \\end{align*}"}
-{"id": "3289.png", "formula": "\\begin{align*} ( H '' - n E ) \\cdot C & = ( D _ 0 - C ) \\cdot C = - C \\cdot C > 0 , \\\\ ( H '' - n E ) \\cdot E & = n > 0 . \\end{align*}"}
-{"id": "4500.png", "formula": "\\begin{align*} \\mathcal { O } = \\{ & z \\in C _ 0 ^ { 4 , \\alpha } ( \\bar \\Omega ) : z > 0 \\textrm { i n } \\Omega , \\ , \\nabla _ n z > 0 \\textrm { o n } \\partial \\Omega , \\\\ & z + \\underline { v } \\textrm { i s a d m i s s i b l e } \\textrm { a n d } \\| z \\| _ { C ^ { 4 , \\alpha } ( \\bar \\Omega ) } \\leq C + \\| \\underline { v } \\| _ { C ^ { 4 , \\alpha } ( \\bar \\Omega ) } \\} , \\end{align*}"}
-{"id": "9586.png", "formula": "\\begin{align*} f ( q , N + 1 ) = \\frac { f ( q , N ) } { 1 - q ^ { N + 1 } } + \\frac { q ^ { N + 1 } } { ( 1 - q ^ { N + 1 } ) ^ 2 } . \\end{align*}"}
-{"id": "454.png", "formula": "\\begin{align*} \\ln \\frac { \\mathcal { Z } _ { \\rm m a t } ( \\hat g ) } { \\mathcal { Z } _ { \\rm m a t } ( g ) } = \\frac { \\mathbf { c } _ { \\rm m a t } } { 9 6 \\pi } S ^ { { \\rm c l } , 0 } _ { { \\rm L } } ( \\hat g , g ) . \\end{align*}"}
-{"id": "4444.png", "formula": "\\begin{gather*} g ( x ' ) = x ' + N ^ { j - i } v ' + \\sum _ { k \\geq 1 } a _ k N ^ { k ( j - i ) } x ' \\\\ g ( v ' ) = v ' + \\sum _ { k \\geq 0 } b _ k N ^ { k ( j - i ) } x ' + \\sum _ { k \\geq 1 } c _ k N ^ { k ( j - i ) } v ' \\end{gather*}"}
-{"id": "1527.png", "formula": "\\begin{align*} \\displaystyle L = ( \\oplus _ { i = 1 } ^ m \\mathfrak { g } _ i ) \\ltimes ( \\oplus _ { k = 1 } ^ n I _ k ) \\end{align*}"}
-{"id": "8716.png", "formula": "\\begin{align*} \\rho ( s ) = \\sum _ { k \\leq s } \\pi ( s ) = { \\displaystyle \\sum _ { n = 1 } ^ s \\prod _ { r = 1 } ^ n \\left ( { q ( r ) \\over p ( r ) } \\right ) \\over \\displaystyle \\sum _ { n = 1 } ^ M \\prod _ { r = 1 } ^ n \\left ( { q ( r ) \\over p ( r ) } \\right ) } . \\end{align*}"}
-{"id": "1261.png", "formula": "\\begin{align*} m _ 3 ( \\eta , \\nu ) = O ( | \\eta | ) , \\partial _ \\nu m _ 3 ( \\eta , \\nu ) = O ( | \\eta | ) , \\partial _ { \\nu \\nu } ^ 2 m _ 3 ( \\eta , \\nu ) = O ( | \\eta | ) , | \\nu | < 2 | \\eta | \\end{align*}"}
-{"id": "4901.png", "formula": "\\begin{align*} \\alpha _ A \\big ( ( s \\otimes X ) \\circ g \\big ) = c _ s \\circ g \\end{align*}"}
-{"id": "1758.png", "formula": "\\begin{align*} X _ { i } = \\frac { B _ { t _ { i } } - B _ { t _ { i - 1 } } } { L ^ { H } } \\sim N ( 0 , 1 ) , \\end{align*}"}
-{"id": "5578.png", "formula": "\\begin{align*} \\mathcal { E } ^ { \\pm } ( t ) : = \\mathcal { E } ^ { \\pm } ( 0 ) \\mp \\frac { \\alpha } { 2 \\gamma } \\int _ 0 ^ { t } v ^ 2 _ { x x } ( 0 , s ) d s . \\end{align*}"}
-{"id": "3447.png", "formula": "\\begin{align*} \\Sigma ^ \\infty _ > = { } & \\{ - 1 + r e ^ { 2 \\pi i / 3 } \\mid 0 \\leq r < \\infty \\} \\cup \\{ - 1 + r e ^ { \\pi i / 3 } \\mid - \\infty < r \\leq 0 \\} , \\\\ \\mathcal { C } ^ \\infty _ < = { } & \\{ 1 + r e ^ { 4 \\pi i / 3 } \\mid - \\infty < r \\leq 0 \\} \\cup \\{ 1 + r e ^ { 5 \\pi i / 3 } \\mid 0 \\leq r < \\infty \\} . \\end{align*}"}
-{"id": "5853.png", "formula": "\\begin{align*} R ( f , g ) : = a ^ m b ^ n \\prod _ { i = 1 } ^ n \\prod _ { j = 1 } ^ m ( \\alpha _ i - \\beta _ j ) = a ^ m \\prod _ { i = 1 } ^ n g ( \\alpha _ i ) = ( - 1 ) ^ { m n } b ^ n \\prod _ { j = 1 } ^ m f ( \\beta _ j ) . \\end{align*}"}
-{"id": "6172.png", "formula": "\\begin{align*} \\mathcal B _ 1 = \\mathcal N ( u _ 1 ) \\cup \\mathcal N ( u _ 2 ) , \\mathcal N ^ o ( u _ 1 ) \\cap \\mathcal N ^ o ( u _ 2 ) = \\emptyset , \\end{align*}"}
-{"id": "4998.png", "formula": "\\begin{align*} { \\rm A v g } _ { \\chi } \\Phi \\left ( \\log L ( s , \\chi ) \\right ) = \\int _ { \\mathbb { C } } \\mathcal { M } _ { \\sigma } ( w ) \\Phi ( w ) | d w | . \\end{align*}"}
-{"id": "2837.png", "formula": "\\begin{align*} S ( A _ \\infty ) : = \\biggl \\{ x \\in X \\ , \\biggr | \\ , \\lim _ { \\sigma \\downarrow 0 } \\sigma ^ { 4 - 2 n } \\int _ { B _ { \\sigma } ( x ) } \\left \\vert F _ { A _ \\infty } \\right \\vert ^ { 2 } \\frac { \\omega ^ n } { n ! } \\neq 0 \\ \\biggr \\} . \\end{align*}"}
-{"id": "4127.png", "formula": "\\begin{align*} T ( z ) = \\mathcal { O } \\begin{pmatrix} 1 & z ^ { - \\frac { 1 } { 4 } } h _ { \\alpha + \\frac { 1 } { 2 } } ( z ) & z ^ { - \\frac { 1 } { 4 } } h _ { \\alpha + \\frac { 1 } { 2 } } ( z ) \\\\ 1 & z ^ { - \\frac { 1 } { 4 } } h _ { \\alpha + \\frac { 1 } { 2 } } ( z ) & z ^ { - \\frac { 1 } { 4 } } h _ { \\alpha + \\frac { 1 } { 2 } } ( z ) \\\\ 1 & z ^ { - \\frac { 1 } { 4 } } h _ { \\alpha + \\frac { 1 } { 2 } } ( z ) & z ^ { - \\frac { 1 } { 4 } } h _ { \\alpha + \\frac { 1 } { 2 } } ( z ) \\end{pmatrix} . \\end{align*}"}
-{"id": "8527.png", "formula": "\\begin{align*} G _ c : = - 2 [ \\bar { \\mathfrak { F } } , \\mathcal { H } \\frac { 1 } { \\zeta _ { \\alpha } } + \\bar { \\mathcal { H } } \\frac { 1 } { \\bar { \\zeta } _ { \\alpha } } ] \\bar { \\mathfrak { F } } _ { \\alpha } + \\frac { 1 } { \\pi i } \\int \\Big ( \\frac { D _ t \\zeta ( \\alpha , t ) - D _ t \\zeta ( \\beta , t ) } { \\zeta ( \\alpha , t ) - \\zeta ( \\beta , t ) } \\Big ) ^ 2 ( \\zeta - \\bar { \\zeta } ) _ { \\beta } d \\beta : = G _ { c 1 } + G _ { c 2 } . \\end{align*}"}
-{"id": "9795.png", "formula": "\\begin{align*} D _ { \\kappa - 2 } ( q ) \\cap L _ \\ast = D _ { \\kappa - 2 } ( q _ \\circ ) \\cap L _ \\ast \\subsetneq L _ \\ast . \\end{align*}"}
-{"id": "8838.png", "formula": "\\begin{align*} F ( \\tilde y ) - F ( y ) & = \\frac { z - y _ i } { M } \\left [ ( M - 1 ) z + ( M + 1 ) y _ i - 2 M \\mu ( y ) \\right ] \\\\ \\mu ( \\tilde y ) ^ 2 - \\mu ( y ) ^ 2 & = \\frac { z - y _ i } { M ^ 2 } \\left [ z - y _ i + 2 M \\mu ( y ) \\right ] \\end{align*}"}
-{"id": "2770.png", "formula": "\\begin{align*} \\mathbf { N } ^ { t + 1 } : = \\mathbf { J } ^ t + \\mathbf { R } ^ t , \\end{align*}"}
-{"id": "7346.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } \\phi ( \\gamma ( t ) ) = { } & \\gamma ^ \\prime ( t ) \\gamma ^ \\prime ( t ) \\phi \\\\ = { } & \\tilde \\nabla ^ 2 \\phi ( \\gamma ^ \\prime ( t ) , \\gamma ^ \\prime ( t ) ) + ( \\tilde \\nabla _ { \\gamma ^ \\prime ( t ) } \\gamma ^ \\prime ( t ) ) ( \\phi ) \\\\ = { } & \\tilde \\nabla ^ 2 \\phi ( \\gamma ^ \\prime ( t ) , \\gamma ^ \\prime ( t ) ) \\geq c \\Vert \\gamma ^ \\prime ( t ) \\Vert ^ 2 \\end{align*}"}
-{"id": "8973.png", "formula": "\\begin{align*} d ( u , v ) = \\sqrt { \\frac { | \\langle u , v \\rangle | } { \\| u \\| \\cdot \\| v \\| } } \\end{align*}"}
-{"id": "6974.png", "formula": "\\begin{align*} \\lim _ { t \\to \\pm 1 } \\frac { f ^ { ( n ) } ( t ) } { ( f ' ( t ) ) ^ n } = 0 \\quad \\lim _ { t \\to \\pm 1 } \\frac { d } { d t } \\frac { f ^ { ( n ) } ( t ) } { ( f ' ( t ) ) ^ n } = 0 . \\end{align*}"}
-{"id": "6579.png", "formula": "\\begin{gather*} G ^ { ( J , s ) } _ { ( I i , t ) , j } = G ^ { ( J , s ) } _ { ( I j , t ) , i } G ^ { ( J , s ) } _ { ( I i , t ) , 0 } = 0 . \\end{gather*}"}
-{"id": "8061.png", "formula": "\\begin{align*} \\phi _ { \\zeta } ^ + | _ { \\{ \\omega _ i , \\omega _ { n + 3 - i } \\} } = \\left ( \\begin{array} { c c } \\zeta ^ { \\frac { 2 n + 3 - 2 i } { 2 } } & 0 \\\\ 0 & \\zeta ^ { \\frac { 2 i - 3 } { 2 } } \\\\ \\end{array} \\right ) \\mbox { a n d } \\psi _ c ^ + | _ { \\{ \\omega _ i , \\omega _ { n + 3 - i } \\} } = \\left ( \\begin{array} { c c } 0 & - c ^ { - \\frac { n + 3 - 2 i } { 2 } } \\\\ - c ^ { n + 3 - 2 i } & 0 \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "4232.png", "formula": "\\begin{align*} \\delta \\left ( a ( x ) \\lvert d u \\rvert ^ { p - 2 } d u \\right ) = f \\delta u = 0 \\Omega . \\end{align*}"}
-{"id": "6038.png", "formula": "\\begin{align*} \\overline { \\kappa } ( G ) \\leq 2 + \\tfrac { n ^ 2 - 4 n } { 4 n ( n - 1 ) } = 2 + \\tfrac { n - 4 } { 4 ( n - 1 ) } , \\end{align*}"}
-{"id": "7023.png", "formula": "\\begin{align*} \\chi ( A , B ) = \\sum \\limits ^ { K _ { 1 } } _ { k = 1 } \\frac { ( - 1 ) ^ { N + i _ { k } } } { ( i _ { k } - M ) ! ( N - i _ { k } ) ! } \\sum \\limits ^ { N - M } _ { l = 0 } \\sum \\limits ^ { l } _ { m = 0 } ( - 1 ) ^ { m } \\sigma _ { m } ( M , N ) i ^ { l - m } _ { k } T _ { N - M - l } \\end{align*}"}
-{"id": "460.png", "formula": "\\begin{align*} \\Delta _ n = \\sum _ { i = 1 } ^ { n } \\delta _ i , \\end{align*}"}
-{"id": "6503.png", "formula": "\\begin{align*} c _ { n , p } = \\frac { \\omega _ { n + p } } { \\omega _ 2 \\omega _ n \\omega _ { p - 1 } } \\ , . \\end{align*}"}
-{"id": "9477.png", "formula": "\\begin{align*} J _ { m } ( b ) - J _ m ( a ) = ( m - 1 ) ( b - a ) \\displaystyle \\int _ { 0 } ^ { 1 } \\vert a + t ( b - a ) \\vert ^ { m - 2 } \\dd t \\end{align*}"}
-{"id": "215.png", "formula": "\\begin{align*} \\Delta _ { \\mathbb { H } } u = V u \\end{align*}"}
-{"id": "141.png", "formula": "\\begin{align*} H ^ { k } _ { \\pi } ( M ) = \\dfrac { k e r \\left ( \\delta _ { \\pi } : \\mathcal { X } ^ k ( M ) \\longrightarrow \\mathcal { X } ^ { k + 1 } ( M ) \\right ) } { I m \\left ( \\delta _ { \\pi } : \\mathcal { X } ^ { k - 1 } ( M ) \\longrightarrow \\mathcal { X } ^ k ( M ) \\right ) } . \\end{align*}"}
-{"id": "481.png", "formula": "\\begin{align*} I _ { \\nu - \\ell } \\cap \\mathcal { C } _ { \\nu - \\ell } = ( 0 ) . \\end{align*}"}
-{"id": "5382.png", "formula": "\\begin{align*} \\langle A _ { \\alpha + \\beta } . v _ { n + 1 \\ , m - 3 } ^ { k + 1 } , v _ { n m } ^ k \\rangle = \\langle v _ { n + 1 \\ , m - 3 } ^ { k + 1 } , A _ { \\alpha + \\beta } ^ * . v _ { n m } ^ k \\rangle , \\end{align*}"}
-{"id": "4334.png", "formula": "\\begin{align*} \\sum _ { n \\in A } \\lambda _ A A = & \\sum _ i \\sum _ { ( \\epsilon _ 1 , \\ldots , \\epsilon _ { k + 1 } ) \\in \\{ 0 , 1 \\} ^ { k + 1 } } ( - 1 ) ^ { \\sum \\epsilon _ i } B ^ i \\bigsqcup \\{ c ^ i _ { 1 , \\epsilon _ 1 } , \\ldots , c ^ i _ { k + 1 , \\epsilon _ { k + 1 } } \\} \\\\ & + \\sum _ i \\sum _ { ( \\epsilon _ 1 , \\ldots , \\epsilon _ k ) \\in \\{ 0 , 1 \\} ^ k } ( - 1 ) ^ { \\sum \\epsilon _ i } \\{ r _ i \\} \\cup B ^ i \\bigsqcup \\{ c ^ i _ { 1 , \\epsilon _ 1 } , \\ldots , c ^ i _ { k , \\epsilon _ k } \\} , \\end{align*}"}
-{"id": "6301.png", "formula": "\\begin{align*} \\frac { c _ 0 ^ 2 k n / 2 } { \\left ( k ( n - k ) + \\binom { k } { 2 } \\right ) p + C c _ 0 k \\sqrt { n \\log n } / 6 } & \\geq \\frac { c _ 0 ^ 2 k n / 2 } { 2 C \\max \\{ k n \\rho , k \\sqrt { \\rho n \\log n } , k \\log n \\} } \\\\ & \\geq \\frac { \\max \\{ \\rho k n , k \\log n \\} / 2 } { 2 C \\max \\{ k n \\rho , k \\sqrt { \\rho n \\log n } , k \\log n \\} } , c _ 0 , \\\\ & \\geq \\frac { 1 } { 4 C } . \\end{align*}"}
-{"id": "1316.png", "formula": "\\begin{align*} \\varPhi { [ Z , P ] } ( p ) = \\inf \\big \\{ \\eta : P [ Z \\le \\eta ] \\ge p \\big \\} , p \\in ( 0 , 1 ] . \\end{align*}"}
-{"id": "7017.png", "formula": "\\begin{align*} \\sigma _ \\omega = \\zeta _ \\omega . \\end{align*}"}
-{"id": "4475.png", "formula": "\\begin{align*} S _ k ( \\kappa _ 1 , \\ldots , \\kappa _ n ) = [ S _ { n , n - k } ( \\kappa _ 1 ^ { - 1 } , \\ldots , \\kappa _ n ^ { - 1 } ) ] ^ { - 1 } \\end{align*}"}
-{"id": "2234.png", "formula": "\\begin{align*} ( z - A ) R ( z ) f & = f f \\in V & R ( z ) ( z - A ) f & = f f \\in D \\end{align*}"}
-{"id": "7433.png", "formula": "\\begin{align*} \\displaystyle \\mu _ { G _ n ( E ) } ^ E ( \\pi ) = d ^ E ( \\sigma ) j ( \\sigma ) ^ { - 1 } \\end{align*}"}
-{"id": "4180.png", "formula": "\\begin{align*} E ^ { ( 2 ) } _ { i n } ( z ) & = \\left ( \\mathbb { I } - \\frac { A _ n ^ { ( 1 ) } ( z ) - A _ n ^ { ( 1 ) } ( 0 ) } { n ^ 3 z } \\right ) E _ { i n } ^ { ( 1 ) } ( z ) , & & z \\in D ( 0 , r _ n ) \\\\ E ^ { ( 2 ) } _ { o u t } ( z ) & = \\left ( \\mathbb { I } - \\frac { A _ n ^ { ( 1 ) } ( 0 ) } { n ^ 3 z } \\right ) ^ { - 1 } , & & z \\in A ( 0 ; r _ n , R ) , \\end{align*}"}
-{"id": "3115.png", "formula": "\\begin{align*} \\beta = \\beta ( L ) & = ( \\beta _ 1 , \\ldots , \\beta _ s ) \\\\ & = ( K _ S - \\beta ^ 1 , \\ldots , K _ S - \\beta ^ s ) \\end{align*}"}
-{"id": "8557.png", "formula": "\\begin{align*} \\kappa _ t = b \\circ \\kappa \\end{align*}"}
-{"id": "2512.png", "formula": "\\begin{align*} { \\displaystyle \\sum \\limits _ { n \\geq 0 } } B _ { n , p } \\frac { t ^ { n } } { n ! } = \\frac { \\left ( p + 1 \\right ) e ^ { p t } } { \\left ( e ^ { t } - 1 \\right ) ^ { p + 1 } } \\left ( t + { \\displaystyle \\sum \\limits _ { s = 1 } ^ { p } } \\left ( - 1 \\right ) ^ { s } \\frac { 1 } { s } \\dbinom { p } { s } \\right ) + \\left ( p + 1 \\right ) { \\displaystyle \\sum \\limits _ { k = 0 } ^ { p } } \\dbinom { p } { k } \\frac { H _ { k } } { \\left ( e ^ { t } - 1 \\right ) ^ { k + 1 } } . \\end{align*}"}
-{"id": "8310.png", "formula": "\\begin{align*} | \\zeta ( \\alpha , t ) - z _ j ( t ) | ^ { - 1 } = O ( \\frac { 1 } { \\alpha + i \\frac { | \\lambda | } { x ( 0 ) } t } ) , \\end{align*}"}
-{"id": "8622.png", "formula": "\\begin{align*} \\begin{array} { r c l } u ' & = & z , \\\\ & & \\\\ z ' & = & u ^ 3 - u + \\frac { c _ 0 ^ 2 } { u } \\left ( 2 \\frac { u ^ 2 + z ^ 2 } { u ^ 4 + c _ 0 ^ 2 } - 1 \\right ) . \\end{array} \\end{align*}"}
-{"id": "2719.png", "formula": "\\begin{align*} \\sigma _ k = \\tau \\theta + \\tau h ( \\theta ) \\frac { \\pi _ { k } } { \\sqrt { N } } , k = 1 , \\ldots , m . \\end{align*}"}
-{"id": "1476.png", "formula": "\\begin{align*} V _ { 9 } \\leq \\varepsilon \\mathcal { G } _ { 1 } ( K ) \\sum _ { i = 1 } ^ { d } \\sum _ { j = 1 } ^ { d } \\int _ { \\mathbb { T } ^ { d } } ( \\mu ^ { 1 } - \\mu ^ { 2 } ) \\frac { \\partial ^ { 2 } ( w ^ { 1 } - w ^ { 2 } ) } { \\partial _ { x _ { i } } \\partial _ { x _ { j } } } \\ d x . \\end{align*}"}
-{"id": "4330.png", "formula": "\\begin{align*} \\begin{aligned} \\| u _ I - Q _ F u _ I \\| _ { 0 , e } & \\lesssim \\epsilon _ F ^ { - 1 } \\big ( h _ F ^ { - 1 / 2 } \\| u _ I - Q _ F u _ I \\| _ { 0 , F } + h _ F ^ { 1 / 2 } | u _ I - Q _ F u _ I | _ { 1 , F } \\big ) \\\\ & \\lesssim \\epsilon _ F ^ { - 1 } \\big ( h _ F ^ { - 1 / 2 } \\| u _ I - u \\| _ { 0 , F } + h _ F ^ { - 1 / 2 } \\| u - Q _ F u _ I \\| _ { 0 , F } \\\\ & + h _ F ^ { 1 / 2 } | u _ I - u | _ { 1 , F } + h _ F ^ { 1 / 2 } | u - Q _ F u _ I | _ { 1 , F } \\big ) \\end{aligned} \\end{align*}"}
-{"id": "1580.png", "formula": "\\begin{align*} \\boldsymbol { F } _ { ( n + 1 ) ^ 2 , n ^ 2 } ( z , x ) & = F ^ { \\kappa _ 1 } \\circ U ^ { - 1 } \\circ \\widetilde { \\boldsymbol { F } } _ { ( n + 1 ) ^ 2 - \\kappa _ 1 , n ^ 2 + \\kappa _ 0 } \\circ U \\circ \\boldsymbol { F } _ { n ^ 2 + \\kappa _ 0 , n ^ 2 } ( z , x ) + o ( 1 ) \\\\ & = F ^ { \\kappa _ 1 } \\circ U ^ { - 1 } \\circ \\widetilde { \\boldsymbol { F } } _ { ( n + 1 ) ^ 2 - \\kappa _ 1 , ( n + 1 ) ^ 2 - k _ n } ( z ^ o _ n , x ^ o _ n ) + o ( 1 ) . \\end{align*}"}
-{"id": "685.png", "formula": "\\begin{align*} \\Theta ( \\xi ) : = \\left \\{ \\xi _ a : H ( a ) \\to K ( a ) \\right \\} _ { a \\in \\mathcal M } \\ . \\end{align*}"}
-{"id": "9830.png", "formula": "\\begin{align*} f ( x _ \\circ ' ) : = \\begin{cases} N ( 0 ^ + , u ( X _ \\circ + \\ , \\cdot \\ , ) ) & X _ \\circ \\in \\mathcal { N } ( u ) \\\\ 0 & X _ \\circ \\notin \\mathcal { N } ( u ) , \\end{cases} \\end{align*}"}
-{"id": "5666.png", "formula": "\\begin{align*} { \\rm I } _ { 2 } & \\le c \\| \\sum _ { j = 0 } ^ { n - 1 } \\chi _ { [ t _ { j } , t _ { j + 1 } ) } ( s ) B _ { n - j } P _ { h } \\| _ { L ^ 2 ( 0 , T ; \\mathcal { L } _ 2 ^ 0 ) } = c \\big ( \\tau \\sum _ { j = 0 } ^ { n - 1 } \\| B _ { n - j } P _ { h } \\| _ { \\mathcal { L } _ 2 ^ 0 } ^ 2 \\big ) ^ { \\frac { 1 } { 2 } } < \\infty . \\end{align*}"}
-{"id": "117.png", "formula": "\\begin{align*} \\widetilde \\zeta _ { j , N } & = \\sum _ { \\substack { k = 1 \\\\ k \\ , } } ^ { \\min \\{ 2 j + 1 , 4 N - 2 j - 1 \\} } \\gamma ^ k \\begin{pmatrix} j \\\\ \\frac { k - 1 } { 2 } \\end{pmatrix} \\begin{pmatrix} 2 N - j - 1 \\\\ \\frac { k - 1 } { 2 } \\end{pmatrix} \\\\ & = \\sum _ { \\substack { \\ell = 0 } } ^ { \\min \\{ j , 2 N - j - 1 \\} } \\gamma ^ { 2 \\ell + 1 } \\begin{pmatrix} j \\\\ \\ell \\end{pmatrix} \\begin{pmatrix} 2 N - j - 1 \\\\ \\ell \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "8536.png", "formula": "\\begin{align*} \\norm { [ \\frac { 1 } { 2 } ( I + \\mathcal { H } ) \\bar { q } , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } \\frac { 1 } { 2 } ( I - \\mathcal { H } ) \\partial _ { \\alpha } ^ k \\tilde { \\theta } } { \\zeta _ { \\alpha } } } _ { L ^ 2 } \\leq & C \\| q \\| _ { H ^ { k } } \\| \\partial _ { \\alpha } \\theta \\| _ { H ^ { k - 1 } } \\leq K _ s ^ { - 1 } \\epsilon ^ 2 d _ I ( t ) ^ { - 3 / 2 } . \\end{align*}"}
-{"id": "6169.png", "formula": "\\begin{align*} \\min \\left \\{ \\sum _ { i = 1 } ^ k \\int _ { \\Omega } | \\nabla u _ i | ^ 2 \\ ; : \\ ; \\begin{array} { l l } u _ i ( x , 0 ) \\cdot u _ j ( x , 0 ) \\equiv 0 \\ ; \\R ^ n \\textrm { - a . e . } \\ ; { \\rm f o r } \\ ; \\ ; i \\neq j , \\\\ u _ i = \\varphi _ i , { \\rm o n } \\ ; \\partial \\Omega \\ ; \\textrm { f o r } \\ ; \\ ; i = 1 , \\dots , k , \\end{array} \\right \\} \\end{align*}"}
-{"id": "3015.png", "formula": "\\begin{align*} x _ n & = \\sum _ { k = 1 } ^ n ( - 1 ) ^ { k - 1 } ( k - 1 ) ! \\cdot B _ { n , k } ( y _ 1 , \\dots , y _ { n - k + 1 } ) . \\end{align*}"}
-{"id": "4934.png", "formula": "\\begin{align*} y ^ { 2 n + 1 } = y ^ { n + 1 } ( y ^ n - z ^ n x ) + y ^ n x ( y z ^ n + x ^ { n + 1 } ) - x ^ 2 ( x ^ n y ^ n ) \\end{align*}"}
-{"id": "6206.png", "formula": "\\begin{align*} \\prod _ { n = 1 } ^ { \\infty } ( 1 - q ^ { 2 n - 1 } ) ^ 2 ( 1 - q ^ { 2 n } ) = \\sum _ { n = - \\infty } ^ { \\infty } ( - 1 ) ^ n q ^ { n ^ 2 } . \\end{align*}"}
-{"id": "1218.png", "formula": "\\begin{align*} \\forall \\xi > 0 , e ^ { - i \\xi ^ 3 } \\hat V ( \\xi ) = \\chi ( \\xi ) e ^ { i a \\ln | \\xi | } \\left ( A + B e ^ { 2 i a \\ln | \\xi | } \\frac { e ^ { - i \\frac { 8 } { 9 } \\xi ^ 3 } } { \\xi ^ 3 } \\right ) + z ( \\xi ) , \\end{align*}"}
-{"id": "46.png", "formula": "\\begin{align*} & G _ s ^ { i } ( Y _ s ^ i , Y _ s ^ { - i } ) - \\bar { G } _ s ^ { i } ( \\bar { Y } _ s ^ i , \\bar { Y } _ s ^ { - i } ) \\\\ = & \\ G _ s ^ { i } ( Y _ s ^ i , Y _ s ^ { - i } ) - { G } _ s ^ { i } ( \\bar { Y } _ s ^ i , \\bar { Y } _ s ^ { - i } ) + G _ s ^ { i } ( \\bar { Y } _ s ^ i , \\bar { Y } _ s ^ { - i } ) - \\bar { G } _ s ^ { i } ( \\bar { Y } _ s ^ i , \\bar { Y } _ s ^ { - i } ) \\\\ \\leq & \\ C _ g \\left ( | \\delta Y _ s ^ i | + \\sum _ { k \\neq i } \\delta Y _ s ^ { k + } \\right ) . \\end{align*}"}
-{"id": "2587.png", "formula": "\\begin{align*} \\frac { \\mathbf P _ { \\delta _ x } [ X _ T ( f ) e ^ { - \\theta X _ T ( f ) } ] } { \\mathbf P _ { \\delta _ x } [ X _ T ( f ) ] } = \\mathbf P _ { \\delta _ x } ^ { X _ T ( f ) } [ e ^ { - \\theta X _ T ( f ) } ] = \\mathbf P _ { \\delta _ x } [ e ^ { - \\theta X _ T ( f ) } ] \\dot { \\mathbf P } _ x ^ { ( T , f ) } [ e ^ { - \\theta Y _ T ( f ) } ] , \\end{align*}"}
-{"id": "6546.png", "formula": "\\begin{align*} V _ \\eta = V _ { \\eta ' } + u _ { \\eta ' , q } e _ q . \\end{align*}"}
-{"id": "766.png", "formula": "\\begin{align*} H _ { q ^ d , M } = ( \\sigma - 1 ) G _ { q ^ d , M } , \\end{align*}"}
-{"id": "7735.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } { \\rm R e } [ \\langle x , \\lambda y \\rangle \\xi _ { n } , \\xi _ { n } ] = \\left \\| x \\right \\| \\left \\| y \\right \\| . \\end{align*}"}
-{"id": "8421.png", "formula": "\\begin{align*} \\frac { d } { d t } a | z _ { \\alpha } | = \\frac { ( z _ { t t } + i ) \\cdot z _ { t t t } } { | z _ { t t } + i | } = \\frac { ( z _ { t t } + i ) \\cdot ( i a z _ { t \\alpha } + i a _ t z _ { \\alpha } ) } { | z _ { t t } + i | } . \\end{align*}"}
-{"id": "2310.png", "formula": "\\begin{align*} a & = \\begin{pmatrix} 1 \\\\ \\frac { \\mathcal { P } } { z \\Omega } | 1 ) \\end{pmatrix} & b & = \\begin{pmatrix} 0 \\\\ | \\delta _ x ) \\end{pmatrix} \\end{align*}"}
-{"id": "3894.png", "formula": "\\begin{align*} ( \\gamma f ' ( \\gamma ) ) ' & = 2 \\gamma K ( u , v ) - \\frac { 2 5 } { 4 } \\gamma ^ { \\frac { 3 } { 2 } } P ( u , v ) \\\\ & = 2 \\left ( \\gamma K ( u , v ) - \\frac { 5 } { 2 } \\gamma ^ { \\frac { 3 } { 2 } } P ( u , v ) \\right ) - \\frac { 5 } { 4 } \\gamma ^ { \\frac { 3 } { 2 } } P ( u , v ) \\\\ & \\leq 2 f ' ( \\gamma ) \\end{align*}"}
-{"id": "8171.png", "formula": "\\begin{align*} U ( { \\mathfrak g } _ 0 ) \\cdot x _ \\theta ( z ) \\sp { k + 1 } \\cong L _ { A _ { \\ell - 1 } } ( ( k + 1 ) \\theta \\vert _ { \\mathfrak h ' _ 0 } ) = L _ { A _ { \\ell - 1 } } ( 2 ( k + 1 ) \\omega _ 1 ) . \\end{align*}"}
-{"id": "9570.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\frac { ( - 1 ) ^ { n - 1 } q ^ { \\frac { n ( n + 1 ) } { 2 } } } { ( 1 - c q ^ n ) ( q ) _ n ( q ) _ { N - n } } \\left ( \\sum _ { k = 1 } ^ n \\frac { q ^ k } { 1 - q ^ k } \\right ) = \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { q ^ { n ( n + 1 ) } } { ( q ) _ n ( 1 - q ^ n ) } F ( q ^ N , q ^ n ; c q ^ n ) , \\end{align*}"}
-{"id": "1668.png", "formula": "\\begin{align*} \\ell ^ { ( k - 2 ) n } B _ { t } ( \\ell ^ { 2 n } m , d ) & = C _ { n } ( d ) = C _ { 0 } ( \\ell ^ { 2 n } d ) - \\ell ^ { \\lambda - 1 } \\left ( \\frac { ( - 1 ) ^ { \\lambda } m } { \\ell } \\right ) C _ { 0 } ( \\ell ^ { 2 n - 2 } d ) \\\\ & = B _ { t } ( m , \\ell ^ { 2 n } d ) - \\ell ^ { \\lambda - 1 } \\left ( \\frac { ( - 1 ) ^ { \\lambda } m } { \\ell } \\right ) B _ { t } ( m , \\ell ^ { 2 n - 2 } d ) . \\end{align*}"}
-{"id": "3700.png", "formula": "\\begin{align*} \\hat { \\tau } _ 2 = \\sum _ x p _ x ^ { - 2 } \\sum _ { i \\in U _ x } \\delta _ i p _ i y _ i ^ 2 = \\sum _ x p _ x ^ { - 1 } \\sum _ { i \\in U _ x } \\delta _ i y _ i ^ 2 = \\sum _ x p _ x ^ { - 1 } \\sum _ { i \\in B _ x } y _ i ^ 2 \\end{align*}"}
-{"id": "6507.png", "formula": "\\begin{align*} \\hat { X } _ { t } ^ { \\varepsilon } = x + \\int _ { 0 } ^ { t } A \\hat { X } _ { s } ^ { \\varepsilon } d s - \\int _ { 0 } ^ { t } B ( \\hat { X } _ { s } ^ { \\varepsilon } ) d s + \\int _ { 0 } ^ { t } f ( X _ { s ( \\delta ) } ^ { \\varepsilon } , \\hat { Y } _ { s } ^ { \\varepsilon } ) d s + \\int _ { 0 } ^ { t } \\sigma _ 1 ( \\hat { X } _ { s } ^ { \\varepsilon } ) d W ^ { Q _ { 1 } } _ s , \\end{align*}"}
-{"id": "7524.png", "formula": "\\begin{align*} u = \\mathbb { U } ( \\sigma ) \\end{align*}"}
-{"id": "6513.png", "formula": "\\begin{align*} \\| \\Psi ^ { n } ( d d ^ { c } \\varphi ) \\| _ U & = \\left ( \\frac { d ^ { ( k - 1 ) m } } { N } \\right ) ^ n \\int _ U d d ^ { c } \\varphi \\wedge ( \\omega + d ^ { - n m } d d ^ { c } u _ { n m } ) ^ { k - 1 } \\\\ & \\leq \\left ( \\frac { d ^ { ( k - 1 ) m } } { N } \\right ) ^ n C ' \\| d d ^ c \\varphi \\| _ W . \\end{align*}"}
-{"id": "8283.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow T _ 0 ^ * - } \\| ( z _ t , z _ { t t } ) \\| _ { C ( [ 0 , T ] ; H ^ s \\times H ^ s ) } + \\sup _ { t \\rightarrow T _ 0 ^ * } ( d _ I ( t ) ^ { - 1 } + d _ P ( t ) ^ { - 1 } ) = \\infty . \\end{align*}"}
-{"id": "6348.png", "formula": "\\begin{align*} r _ i = \\left [ \\begin{array} { r l } r _ i ^ { ( 1 ) } \\\\ r _ i ^ { ( 2 ) } \\end{array} \\right ] = \\left [ \\begin{array} { r l } \\sum ^ 2 _ { j = 1 } R g _ { j i } h _ { j i } P _ t T _ s s _ j ^ { ( 1 ) } + n _ i ^ { ( 1 ) } ~ ~ ~ ~ ~ ~ ~ \\\\ \\sum ^ 2 _ { j = 1 } R g _ { j i } h _ { j i } P _ t T _ s ( 1 - s _ j ^ { ( 1 ) } ) + n _ i ^ { ( 2 ) } \\end{array} \\right ] , \\end{align*}"}
-{"id": "7616.png", "formula": "\\begin{align*} \\begin{gathered} X _ \\epsilon = ( 2 - \\epsilon ) X _ 1 + ( \\epsilon - 1 ) X _ 2 , \\\\ \\tau _ \\epsilon = ( 2 - \\epsilon ) \\tau _ 1 + ( \\epsilon - 1 ) \\tau _ 2 , 1 \\le \\epsilon \\le 2 . \\end{gathered} \\end{align*}"}
-{"id": "8927.png", "formula": "\\begin{align*} \\zeta : = & ( n ( \\textrm { U n c h a n g e d } ) - 2 n ( \\textrm { I n c o r r e c t , U n c h a n g e d } ) ) ( n ( \\textrm { C h a n g e d } ) - 2 n ( \\textrm { I n c o r r e c t , C h a n g e d } ) ) \\\\ = & ( n - s - 2 \\eta _ 1 ) ( s - 2 \\eta _ 2 ) . \\end{align*}"}
-{"id": "4523.png", "formula": "\\begin{align*} S ( K , c _ 1 ) = \\{ ( c _ 2 , c _ 3 ) \\in C _ 2 \\times C _ 3 : ( c _ 1 , c _ 2 , c _ 3 ) \\in S ( K ) \\} . \\end{align*}"}
-{"id": "8726.png", "formula": "\\begin{align*} \\rho ( i _ 1 , \\ldots , i _ d ) = \\prod _ { j = 1 } ^ d \\rho _ j ( i _ j ) . \\end{align*}"}
-{"id": "8944.png", "formula": "\\begin{align*} x _ i ( t ) = x _ i \\ , \\ , \\forall i = 1 , \\dots , n \\ , . \\end{align*}"}
-{"id": "7238.png", "formula": "\\begin{align*} \\lim _ { \\xi \\to - 0 } & \\frac { K _ r ( \\xi - y ) - K _ r ( \\xi + y ) } { | \\xi | } \\\\ & = \\lim _ { \\xi \\to - 0 } \\left ( \\frac { K _ r ( y - \\xi ) - K _ r ( y ) } { \\xi } + \\frac { K _ r ( y ) - K _ r ( y + \\xi ) } { \\xi } \\right ) \\frac { \\xi } { | \\xi | } = 2 K _ r ^ \\prime ( y ) \\end{align*}"}
-{"id": "5148.png", "formula": "\\begin{align*} f ( r ) = \\begin{cases} D \\begin{pmatrix} A ^ + r ^ \\gamma \\\\ A ^ - r ^ { - \\gamma } \\end{pmatrix} & \\ r < 1 , \\\\ 0 & \\ r > 2 . \\end{cases} \\end{align*}"}
-{"id": "478.png", "formula": "\\begin{align*} \\omega \\bigl ( ( a _ 1 x _ 1 + \\cdots + a _ n x _ n ) ^ m / m ! \\bigr ) & = \\frac { ( a _ 1 x _ 1 + \\cdots + a _ n x _ n ) ^ m } { m ! } \\circ { \\mathfrak D } _ { \\omega } \\\\ & = \\frac { ( a _ 1 \\partial / \\partial z _ 1 + \\cdots + a _ n \\partial / \\partial z _ n ) ^ m } { m ! } { \\mathfrak D } _ { \\omega } = { \\mathfrak D } _ { \\omega } ( a _ 1 , \\dots , a _ n ) , \\end{align*}"}
-{"id": "7327.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} w = { } & { } u - p \\in H _ { 0 } ^ { 1 } ( \\Omega ) , \\\\ z = { } & { } v - q \\in H _ { 0 } ^ { 1 } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "7231.png", "formula": "\\begin{align*} f ( 0 ) = - K _ r ^ \\prime * \\phi ^ \\prime ( 0 ) = 2 \\int _ 0 ^ \\pi K _ r ^ \\prime ( y ) \\phi ^ \\prime ( y ) \\ , d y = c > 0 \\end{align*}"}
-{"id": "7951.png", "formula": "\\begin{align*} \\sum _ { a \\in S } x _ a = \\sum _ { a \\in S } y _ { \\sigma ( a ) } . \\end{align*}"}
-{"id": "1892.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { n - 1 } \\sum _ { j = 1 } ^ \\infty ( - k _ m + k _ { n - m } ) Y _ { j , m } Y _ { j , n - m } = 0 , \\end{align*}"}
-{"id": "1458.png", "formula": "\\begin{align*} \\frac { d E _ { \\mu } } { d t } = \\int _ { \\mathbb { T } ^ { d } } ( \\mu ^ { 1 } - \\mu ^ { 2 } ) ( \\mu ^ { 1 } _ { t } - \\mu ^ { 2 } _ { t } ) \\ d x . \\end{align*}"}
-{"id": "853.png", "formula": "\\begin{gather*} B : = v { z ^ * } ^ T + w { u ^ * } ^ T - ( { u ^ * } ^ T u ) w { z ^ * } ^ T . \\end{gather*}"}
-{"id": "5591.png", "formula": "\\begin{align*} & Z _ 1 = \\mathcal { C } \\big ( \\R _ t ; \\ , H ^ s ( \\R _ x ) \\big ) \\cap \\mathcal { C } \\big ( \\R _ x ; \\ , H ^ { ( 2 s + 1 ) / 4 } ( \\R _ t ) \\big ) \\cap X ^ { s , b } , \\\\ & Z _ 2 = \\mathcal { C } \\big ( \\R _ t ; \\ , H ^ { \\kappa } ( \\R _ x ) \\big ) \\cap \\mathcal { C } \\big ( \\R _ x ; \\ , H ^ { ( \\kappa + 1 ) / 3 } ( \\R _ t ) \\big ) \\cap Y ^ { \\kappa , b , \\alpha } , \\end{align*}"}
-{"id": "7163.png", "formula": "\\begin{gather*} \\begin{cases} \\dot { x } = x ( 1 - x ^ { 2 } - y ^ { 2 } ) ^ { 3 } - y ( 1 + x ^ { 2 } + y ^ { 2 } ) , \\\\ \\dot { y } = x ( 1 + x ^ { 2 } + y ^ { 2 } ) + y ( 1 - x ^ { 2 } - y ^ { 2 } ) ^ { 3 } \\end{cases} \\end{gather*}"}
-{"id": "9179.png", "formula": "\\begin{align*} \\dim ( U _ { 0 } \\cap \\mathcal { K } \\left ( 0 \\right ) ^ { \\perp } ) & = \\dim U _ { 0 } + \\dim \\mathcal { K } \\left ( 0 \\right ) ^ { \\perp } - \\dim ( U _ { 0 } + \\mathcal { K } \\left ( 0 \\right ) ^ { \\perp } ) \\\\ & \\geq \\dim U _ { 0 } + n - 1 - \\dim \\mathcal { K } ( 0 ) - \\left ( n - 1 \\right ) \\\\ & \\geq \\dim U _ { 0 } - \\dim \\mathcal { K } \\\\ & > \\ell - \\left ( \\ell - k + 1 \\right ) \\\\ & = k - 1 . \\end{align*}"}
-{"id": "5145.png", "formula": "\\begin{align*} \\int _ { \\R ^ 3 } \\abs { x } \\abs * { - i \\alpha \\cdot \\nabla \\psi ( x ) } ^ 2 \\ , d x \\geq \\int _ { \\R ^ 3 } \\frac { \\abs { \\psi ( x ) } ^ 2 } { \\abs { x } } \\ , d x + \\frac { 1 } { 4 } \\int _ { \\R ^ 3 } \\frac { \\abs * { \\psi ( x ) - \\frac { R } { \\abs { x } } \\psi \\big ( R \\frac { x } { \\abs { x } } \\big ) } ^ 2 } { \\abs { x } \\log ^ 2 ( \\abs { x } / R ) } \\ , d x . \\end{align*}"}
-{"id": "2965.png", "formula": "\\begin{align*} H ( z , v ) = \\frac { 1 } { 2 \\pi i } \\int _ { \\sigma _ { s } } H ( u , v ) \\frac { 1 } { u - z } d u ( z \\in \\Delta _ { s } , v \\in \\Delta _ { R } , u \\in \\sigma _ { s } ) . \\end{align*}"}
-{"id": "7459.png", "formula": "\\begin{align*} \\displaystyle D ( \\lambda \\mapsto ( z ^ k f ) _ { \\pi _ \\lambda } ( g ) ) _ { \\lambda = 0 } = D ( \\lambda \\mapsto \\chi _ { \\pi _ \\lambda } ( z ) ^ k f _ { \\pi _ \\lambda } ( g ) ) _ { \\lambda = 0 } \\end{align*}"}
-{"id": "7156.png", "formula": "\\begin{align*} \\inf _ { \\left \\vert \\hat { \\xi } \\right \\vert = 1 } \\int _ { S _ { d - 1 } } \\left \\vert \\hat { \\xi } \\cdot w \\right \\vert ^ { 2 } \\Pi \\left ( r , d w \\right ) \\geq c _ { 0 } > 0 . \\end{align*}"}
-{"id": "143.png", "formula": "\\begin{align*} [ \\pi _ { \\lambda } , \\pi _ { \\lambda } ] & = [ \\pi + \\lambda X , \\pi + \\lambda X ] = 2 \\lambda [ \\pi , X ] + \\lambda ^ 2 [ X , X ] = 0 . \\end{align*}"}
-{"id": "6818.png", "formula": "\\begin{align*} \\cosh ^ 2 ( s ) \\mu ' - \\sinh ^ 2 ( s ) \\theta ' = \\mathrm { c o n s t a n t } \\ , . \\end{align*}"}
-{"id": "2932.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} , B = \\begin{pmatrix} \\frac { 1 } { 1 + 2 \\tau } & 0 \\\\ 0 & \\tau \\end{pmatrix} , \\end{align*}"}
-{"id": "3547.png", "formula": "\\begin{align*} \\omega _ { 0 } ( z ) = \\sum _ { n _ { 1 } , . . . , n _ { 4 } \\geq 0 } \\frac { 1 } { \\Gamma ( \\frac { 1 } { 2 } ) ^ { 3 } } \\frac { \\prod _ { i = 1 } ^ { 3 } \\Gamma ( n \\cdot \\ell _ { i } + \\frac { 1 } { 2 } ) } { \\prod _ { i = 4 } ^ { 9 } \\Gamma ( n \\cdot \\ell _ { i } + 1 ) } z _ { 1 } ^ { n _ { 1 } } z _ { 2 } ^ { n _ { 2 } } z _ { 3 } ^ { n _ { 3 } } z _ { 4 } ^ { n _ { 4 } } , \\end{align*}"}
-{"id": "3876.png", "formula": "\\begin{align*} M = 2 m . \\end{align*}"}
-{"id": "8069.png", "formula": "\\begin{align*} \\begin{aligned} \\psi _ c ^ + ( \\omega _ 1 ) = - c ^ { \\frac { 3 } { 2 } } \\omega _ 4 , \\psi _ c ^ + ( \\omega _ 2 ) = - c ^ { \\frac { 1 } { 2 } } \\omega _ 3 , \\psi _ c ^ + ( \\omega _ 3 ) = - c ^ { - \\frac { 1 } { 2 } } \\omega _ 2 , \\psi _ c ^ + ( \\omega _ 4 ) = - c ^ { - \\frac { 3 } { 2 } } \\omega _ 1 . \\end{aligned} \\end{align*}"}
-{"id": "3165.png", "formula": "\\begin{align*} \\varGamma U \\varGamma = U ^ { - 1 } . \\end{align*}"}
-{"id": "6999.png", "formula": "\\begin{align*} \\textsl { \\footnotesize X } _ j = [ \\xi ^ { j } - \\xi ^ { - j } , \\xi ^ { 2 j } - \\xi ^ { - 2 j } , \\ldots , \\xi ^ { ( n - 1 ) j } - \\xi ^ { - ( n - 1 ) j } , 0 ] ^ { \\tt T } , \\end{align*}"}
-{"id": "10006.png", "formula": "\\begin{align*} \\sigma _ { \\mathfrak { X } ( X ) } ( D ) = \\inf \\big \\{ \\sigma \\in \\mathbb { R } \\colon \\sum _ { n } \\frac { a _ { n } } { n ^ { \\sigma } } n ^ { - s } \\mathfrak { X } ( X ) \\big \\} \\ , \\end{align*}"}
-{"id": "6617.png", "formula": "\\begin{align*} \\prod _ { t = 1 } ^ { \\eta } \\left ( 1 - \\frac { \\mathcal { O } ( \\varphi ^ { n / 5 } ) } { u _ t ^ 2 - ( 1 - 2 \\varepsilon ( - \\varphi ) ^ { - n } ) ^ 2 } \\right ) & > 1 - \\mathcal { O } ( \\varphi ^ { n / 5 } ) \\sum _ { t = 1 } ^ { \\infty } \\frac { 1 } { u _ t ^ 2 - ( 1 - 2 \\varepsilon ( - \\varphi ) ^ { - n } ) ^ 2 } \\\\ & = 1 - \\mathcal { O } ( \\varphi ^ { n / 5 } ) . \\end{align*}"}
-{"id": "3702.png", "formula": "\\begin{align*} t _ i = ( t _ { 1 i } , t _ { 2 i } , . . . t _ { K i } ) ^ { \\top } = \\big ( t _ 1 ( x _ i ) , t _ 2 ( x _ i ) , . . . t _ K ( x _ i ) \\big ) ^ { \\top } = t ( x _ i ) \\end{align*}"}
-{"id": "8893.png", "formula": "\\begin{align*} P ( \\sigma ) ( x ) & = \\int _ 1 ^ x \\sigma ( t ) d t + \\frac { 1 } { 2 } \\left ( \\sigma ( t ) + \\sigma ( 1 ) \\right ) \\\\ & + \\sum _ { k = 2 } ^ K \\frac { B _ k } { k ! } \\ , \\left ( \\sigma ^ { ( k - 1 ) } ( t ) - \\sigma ^ { ( k - 1 ) } ( 1 ) \\right ) + \\frac { ( - 1 ) ^ { K + 1 } } { K ! } \\int _ 1 ^ x \\overline { B _ { K } } ( t ) \\ , \\sigma ^ { ( K ) } ( t ) \\ , d t . \\end{align*}"}
-{"id": "5014.png", "formula": "\\begin{align*} L ( \\beta _ 1 , \\ldots , \\beta _ M , \\lambda ) : = \\sum _ { m = 1 } ^ M \\beta _ m ^ { 1 + s / d } R _ m ^ { - s / d } - \\lambda \\sum _ { m = 1 } ^ M \\beta _ m , \\end{align*}"}
-{"id": "2392.png", "formula": "\\begin{align*} \\lfloor X _ A \\rfloor _ \\mathfrak { p } & = \\prod _ { i = 1 } ^ m Q ( A ( \\mathfrak { p } _ i ) ) \\\\ Q ( A ( \\mathfrak { p } _ i ) ) & = Q ( A ( r _ i ) , A ( s _ i ) ) \\mathfrak { p } _ i = ( r _ i , s _ i ) , r _ i > s _ i \\end{align*}"}
-{"id": "2612.png", "formula": "\\begin{align*} \\mu ( d x ) = c _ 1 x ^ { - 1 - \\beta } 1 _ { \\{ x > 0 \\} } + c _ 2 | x | ^ { - 1 - \\beta } 1 _ { \\{ x < 0 \\} } \\end{align*}"}
-{"id": "9282.png", "formula": "\\begin{align*} \\partial ^ + F ( r _ 0 ) = \\{ s \\in \\mathbb { S } ^ n | \\ F ( r ) \\le F ( r _ 0 ) + s \\cdot ( r - r _ 0 ) + o ( | r - r _ 0 | ) \\ { \\rm h o l d s } \\ { \\rm f o r } \\ r \\ { \\rm n e a r } \\ r _ 0 \\} . \\end{align*}"}
-{"id": "1589.png", "formula": "\\begin{align*} \\Phi _ { w _ m } ( z _ m , x _ m ) = \\Phi _ { v ^ \\iota _ n } ( z ^ \\iota _ n , x ^ \\iota _ n ) + \\sum _ { j = n ^ 2 + k _ n } ^ { m - 1 } \\left ( \\frac { \\sqrt { w _ j } } { 2 } + o \\left ( \\frac { 1 } { n ^ 2 } \\right ) \\right ) . \\end{align*}"}
-{"id": "7208.png", "formula": "\\begin{align*} - \\mu \\phi + L _ r \\phi + \\frac { 1 } { 2 } \\phi - \\frac { 1 } { 2 } \\widehat { \\phi ^ 2 } ( 0 ) = 0 , r > 1 . \\end{align*}"}
-{"id": "7709.png", "formula": "\\begin{align*} \\frac { \\sin ( 2 \\pi a t ) } { \\pi t } = \\hat { \\mathbb { 1 } } _ { [ - a , a ] } ( t ) . \\end{align*}"}
-{"id": "9177.png", "formula": "\\begin{align*} \\mathcal { V } ( t ) = \\left \\{ \\left . J \\left ( t \\right ) \\right \\vert J \\in \\mathcal { V } \\right \\} \\oplus \\left \\{ \\left . J ^ { \\prime } \\left ( t \\right ) \\right \\vert J \\in \\mathcal { V } J \\left ( t \\right ) = 0 \\right \\} . \\end{align*}"}
-{"id": "1372.png", "formula": "\\begin{align*} s _ { \\lambda } ^ * s _ { \\lambda } = p _ { s ( \\lambda ) } \\ , , \\sum _ { \\lambda \\in v \\Lambda ^ n } s _ { \\lambda } s _ { \\lambda } ^ * = p _ { v } \\ , , s _ { \\lambda } s _ { \\mu } = \\delta _ { s ( \\lambda ) , r ( \\mu ) } s _ { \\lambda \\mu } \\ ; . \\end{align*}"}
-{"id": "7096.png", "formula": "\\begin{align*} c ^ { \\rho , \\gamma } _ { k , D } ( M ) : = M ^ \\rho \\max \\left ( 1 , \\frac { M } { N ^ \\gamma } \\right ) ^ { D } ~ . \\end{align*}"}
-{"id": "4717.png", "formula": "\\begin{align*} \\Phi ( x _ j ) = \\left \\{ \\begin{array} { l l } 1 ~ ~ ~ ~ | E _ j | > K \\Delta x ^ 4 , \\\\ 0 ~ ~ ~ ~ ~ ~ , \\end{array} \\right . \\end{align*}"}
-{"id": "5840.png", "formula": "\\begin{align*} S _ n ( \\alpha , \\beta , \\gamma ) = \\alpha ^ n \\gamma ^ { \\frac { n ( n - 1 ) } { 2 } } ( - 1 ) \\int _ { f \\in M _ { n } } \\alpha \\chi _ 0 ^ { - 1 } ( f ( 0 ) ) \\beta \\chi _ 0 ^ { - 1 } ( f ( 1 ) ) \\gamma \\chi _ 0 ^ { - \\frac 1 { 2 } } ( \\Delta ( f ) ) \\prod _ { i = 1 } ^ l \\frac { \\Gamma _ { h _ i } ( \\gamma ) } { \\Gamma ( \\gamma ) ^ { \\deg h _ i } } \\dd f \\end{align*}"}
-{"id": "1414.png", "formula": "\\begin{align*} Z ^ 1 _ t - \\textrm { M A R } _ t = Z ^ { \\infty } _ t \\ \\textrm { f o r a l l } \\ t \\in [ 1 , T ] . \\end{align*}"}
-{"id": "1510.png", "formula": "\\begin{align*} \\P \\biggl ( \\max _ { \\{ X _ { i _ 1 } , \\ldots , X _ { i _ n } \\} \\subset \\{ \\pm X _ 1 , \\ldots , \\pm X _ N \\} } \\max _ { \\epsilon \\in \\{ - 1 , + 1 \\} ^ n } & \\norm { \\epsilon _ 1 X _ { i _ 1 } + \\ldots + \\epsilon _ n X _ { i _ n } } _ 1 > C n ^ { 3 / 2 } \\sqrt { \\log ( 2 N / n ) } \\biggr ) \\\\ & \\le \\exp \\big ( ( 2 n + 1 ) \\log 2 - ( C ^ 2 / 7 2 - 1 ) n \\log ( 2 N / n ) \\big ) \\\\ & \\le \\exp \\big ( - c n \\log ( 2 N / n ) \\big ) , \\end{align*}"}
-{"id": "7930.png", "formula": "\\begin{align*} v = \\sum _ { a \\in [ n ] } x _ { a , 1 } \\otimes \\dots \\otimes x _ { a , m } \\end{align*}"}
-{"id": "5336.png", "formula": "\\begin{align*} A ^ * . w ^ k = \\overline { b _ { k + 2 } } \\frac { | | w ^ k | | ^ 2 } { | | w ^ { k + 2 } | | ^ 2 } w ^ { k + 2 } + \\overline { a _ { k - 2 } } \\frac { | | w ^ k | | ^ 2 } { | | w ^ { k - 2 } | | ^ 2 } w ^ { k - 2 } \\end{align*}"}
-{"id": "5703.png", "formula": "\\begin{align*} \\frac { 1 } { \\Gamma ( \\nu ) } \\int _ 0 ^ z ( z - t ) ^ { \\nu - 1 } E _ { \\alpha , \\beta } ( \\lambda t ^ \\alpha ) t ^ { \\beta - 1 } d t = z ^ { \\beta + \\nu - 1 } E _ { \\alpha , \\beta + \\nu } ( \\lambda z ^ { \\alpha } ) \\end{align*}"}
-{"id": "6978.png", "formula": "\\begin{align*} v ^ \\prime ( 0 ) = 0 , \\end{align*}"}
-{"id": "4359.png", "formula": "\\begin{align*} | \\mathcal C _ 3 ( n , k ) | \\ = \\ 3 \\ + \\ & { n - 2 \\choose k - 2 } - { n - k - 2 \\choose k - 2 } + \\\\ & { n - 3 \\choose k - 2 } - { n - k - 2 \\choose k - 2 } + \\\\ & { n - 4 \\choose k - 2 } - { n - k - 3 \\choose k - 2 } + \\\\ & \\cdots \\\\ & { n - k - 1 \\choose k - 2 } - { n - 2 k \\choose k - 2 } . \\end{align*}"}
-{"id": "6148.png", "formula": "\\begin{align*} e _ 1 a ^ 2 - e _ 2 b ^ 2 = t . \\end{align*}"}
-{"id": "1999.png", "formula": "\\begin{align*} \\left \\langle \\phi _ { m _ 1 } \\psi ^ { a _ 1 } \\cdots \\phi _ { m _ n } \\psi ^ { a _ n } \\cdot \\phi _ 1 \\right \\rangle _ { 0 , n + 1 } ^ \\mathrm { F J R W } = \\sum _ { i = 1 } ^ n \\left \\langle \\phi _ { m _ 1 } \\psi ^ { a _ 1 } \\cdots \\phi _ { m _ i } \\psi ^ { a _ i - 1 } \\cdots \\phi _ { m _ n } \\psi ^ { a _ n } \\right \\rangle _ { 0 , n } ^ \\mathrm { F J R W } \\end{align*}"}
-{"id": "866.png", "formula": "\\begin{gather*} D ^ * F ( x , y ) ( { y ^ * } ' ) = \\nabla f _ 1 ( x ) ^ T { y ^ * } ' + D ^ * F _ 2 ( x , y - f _ 1 ( x ) ) ( { y ^ * } ' ) \\end{gather*}"}
-{"id": "1159.png", "formula": "\\begin{align*} & { C _ { \\beta , n _ 1 , 1 } } = ( 2 \\pi ) ^ { n _ 1 + 1 } M _ { n _ 1 } ( \\beta , \\beta , \\beta / 2 ) \\\\ = & ( 2 \\pi ) ^ { n _ 1 + 1 } \\frac { \\Gamma ( \\beta ( n _ 1 + 3 ) / 2 + 1 ) \\Gamma ( \\beta ( n _ 1 + 2 ) / 2 + 1 ) } { \\Gamma ( 3 \\beta / 2 + 1 ) \\Gamma ( \\beta + 1 ) \\Gamma ( \\beta ( n _ 1 + 1 ) / 2 + 1 ) ( \\Gamma ( \\beta / 2 + 1 ) ) ^ { n _ 1 - 1 } } . \\end{align*}"}
-{"id": "9923.png", "formula": "\\begin{align*} D ( P | _ { \\mathcal { F } \\vee \\mathcal { G } } \\| Q | _ { \\mathcal { F } \\vee \\mathcal { G } } ) & = D ( P | _ { \\mathcal { F } } \\| Q | _ { \\mathcal { F } } ) + D ( P | _ { \\mathcal { G } } | \\mathcal { F } \\| Q | _ { \\mathcal { G } } | \\mathcal { F } ) \\end{align*}"}
-{"id": "5149.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\begin{vmatrix} f ^ + ( r ) & \\overline { \\widetilde { f } ^ + ( r ) } \\\\ f ^ - ( r ) & \\overline { \\widetilde { f } ^ - ( r ) } \\end{vmatrix} = \\begin{vmatrix} \\Gamma ^ + ( f ) & \\overline { \\Gamma ^ + ( \\widetilde f ) } \\\\ \\Gamma ^ - ( f ) & \\overline { \\Gamma ^ - ( \\widetilde f ) } \\end{vmatrix} . \\end{align*}"}
-{"id": "2934.png", "formula": "\\begin{align*} & A ^ { \\dagger } = V \\begin{pmatrix} \\Sigma _ { 1 } ^ { - 1 } & 0 \\\\ 0 & 0 \\end{pmatrix} U ^ { \\ast } = V _ { 1 } \\Sigma _ { 1 } ^ { - 1 } U _ { 1 } ^ { \\ast } , \\\\ & B ^ { \\dagger } = \\widetilde { V } \\begin{pmatrix} \\widetilde { \\Sigma } _ { 1 } ^ { - 1 } & 0 \\\\ 0 & 0 \\end{pmatrix} \\widetilde { U } ^ { \\ast } = \\widetilde { V } _ { 1 } \\widetilde { \\Sigma } _ { 1 } ^ { - 1 } \\widetilde { U } _ { 1 } ^ { \\ast } . \\end{align*}"}
-{"id": "6979.png", "formula": "\\begin{align*} A ( x _ 0 ) = \\ln | f ^ \\prime ( x _ 0 ) | + H \\left ( f ( x _ 0 ) - [ f ( x _ 0 ) ] , \\frac { f ^ { \\prime \\prime } ( x _ 0 ) } { ( f ^ \\prime ( x _ 0 ) ) ^ 2 } , \\ldots , \\frac { f ^ { ( q ) } ( x _ 0 ) } { ( f ^ \\prime ( x _ 0 ) ) ^ q } \\right ) + c _ 0 [ f ( x _ 0 ) ] \\end{align*}"}
-{"id": "9209.png", "formula": "\\begin{align*} e ( H ) & \\leq e x ( n ; \\mathcal { L } _ { n , 2 k + 1 } ) = \\max \\left \\{ \\binom { 2 k + 1 } { 2 } , \\binom { n } { 2 } - \\binom { n - k } { 2 } \\right \\} . \\end{align*}"}
-{"id": "4633.png", "formula": "\\begin{align*} \\mathbf { y } _ i = \\left [ \\begin{array} { c | c | c } U _ { i 1 } \\mathbb { I } _ n & \\dots & U _ { i \\ell } \\mathbb { I } _ n \\end{array} \\right ] \\begin{bmatrix} \\mathbf { z } _ 1 \\\\ \\vdots \\\\ \\mathbf { z } _ \\ell \\end{bmatrix} \\end{align*}"}
-{"id": "9035.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { | \\xi | ^ 2 = E } a _ { \\xi } e ( \\langle x , \\xi \\rangle ) \\end{align*}"}
-{"id": "1965.png", "formula": "\\begin{align*} 2 \\deg ( i ) d _ { i i } = \\sum _ { k \\in \\mathcal { N } ( i ) } d _ { k k } + \\sum _ { \\substack { j , k \\in \\mathcal { N } ( i ) \\\\ j \\neq k } } d _ { k j } . \\end{align*}"}
-{"id": "2643.png", "formula": "\\begin{align*} \\beta _ { r , s } = \\int _ { \\mathbb { R } ^ { 2 } } d x d y \\frac { x - y } { x + y } e ^ { - \\frac { 1 } { 2 } ( x ^ { 2 } + y ^ { 2 } ) } \\frac { 1 } { ( r ! s ! ) ^ { 2 } } ( x ^ { 2 r } y ^ { 2 s } - x ^ { 2 s } y ^ { 2 r } ) . \\end{align*}"}
-{"id": "7656.png", "formula": "\\begin{align*} \\frac { \\partial v _ \\psi } { \\partial t } - L ^ { \\varepsilon } v _ \\psi = 0 , v _ \\psi ( x , 0 ) = \\psi ( x ) , \\end{align*}"}
-{"id": "5924.png", "formula": "\\begin{align*} N ' \\coloneqq - ( K _ { X ' } + B ' + M ' ) = - \\phi ^ * ( K _ X + B + M ) , A ' \\coloneqq - ( K _ { X ' } + \\Gamma ' + \\beta M ' ) = - \\phi ^ * ( K _ X + \\Gamma + \\beta M ) . \\end{align*}"}
-{"id": "313.png", "formula": "\\begin{align*} \\Phi ( f ) = n ! c h _ { 2 n } ( f ) \\oplus ( n + 1 ) ! c h _ { 2 n + 2 } ( f ) , \\end{align*}"}
-{"id": "3563.png", "formula": "\\begin{align*} \\begin{matrix} \\begin{aligned} W _ { 1 } W _ { 2 } & = U V z , & & & W _ { 1 } W _ { 3 } & = U V y , & & & W _ { 2 } W _ { 3 } & = U V x , \\\\ W _ { 1 } x & = U v & & & W _ { 2 } y & = U v & & & W _ { 3 } z & = U v \\\\ W _ { 1 } x & = V u , & & & W _ { 2 } y & = V u , & & & W _ { 3 } z & = V u \\end{aligned} \\end{matrix} \\end{align*}"}
-{"id": "2375.png", "formula": "\\begin{align*} \\gamma _ { Q _ 1 } ( a ^ 2 ) & = \\gamma _ { Q _ 1 } ( ( a ^ + ) ^ 2 ) = 0 , & \\gamma _ { Q _ 1 } ( a a ^ + ) & = \\frac { 1 } { 4 z } + \\frac { 1 } { 2 } , & \\gamma _ { Q _ 1 } ( a ^ + a ) & = \\frac { 1 } { 4 z } \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "5317.png", "formula": "\\begin{align*} \\frac { d q _ 1 } { d t } = \\frac { \\partial I } { \\partial p _ 1 } , & \\frac { d p _ 1 } { d t } = - \\frac { \\partial I } { \\partial q _ 1 } , & \\frac { d q _ 2 } { d t } = \\frac { \\partial I } { \\partial p _ 2 } , & \\frac { d p _ 2 } { d t } = - \\frac { \\partial I } { \\partial q _ 2 } \\end{align*}"}
-{"id": "7844.png", "formula": "\\begin{align*} \\varphi ' ( x ) = \\frac { 3 } { 2 } \\log \\bigg ( \\frac { 2 ^ { 1 / 3 } e d } { x } \\bigg ) - \\frac { 3 } { 2 } \\ge \\log 3 . \\end{align*}"}
-{"id": "3244.png", "formula": "\\begin{align*} \\mathcal { H } ^ { 2 k } ( B _ { 4 r } ( a ) \\cap ( \\cup ^ m _ { j = 1 } A _ { i _ j } ) ) & = \\mathcal { H } ^ { 2 k } ( \\cup ^ m _ { j = 1 } ( B _ { 4 r } ( a ) \\cap A _ { i _ j } ) ) = \\sum ^ m _ { j = 1 } \\mathcal { H } ^ { 2 k } ( B _ { 4 r } ( a ) \\cap A _ { i _ j } ) \\\\ & \\geq \\sum _ { j = 1 } ^ { m } \\Omega ( 2 k ) r ^ { 2 k } = m \\Omega ( 2 k ) r ^ { 2 k } \\end{align*}"}
-{"id": "9456.png", "formula": "\\begin{align*} \\int _ { \\mathcal { F } _ N } F _ { n , \\epsilon } ( z ) \\ ; d \\mu ( z ) = \\lim _ { \\epsilon '' \\to 0 } \\int _ { \\mathcal { F } _ N ( \\epsilon '' ) } J _ { N , n } ( z ) \\ ; d \\mu ( z ) . \\end{align*}"}
-{"id": "3452.png", "formula": "\\begin{align*} F ( t ; w ) = ( \\log N + w / N ) t - \\log \\Gamma ( t + N ) + \\log \\Gamma ( N ) , \\end{align*}"}
-{"id": "7939.png", "formula": "\\begin{align*} \\Pi _ a ( x _ { \\tau ( b ) , 1 } ) = 0 b \\in [ a - 1 ] , \\end{align*}"}
-{"id": "7320.png", "formula": "\\begin{align*} D ( \\mathcal { A } ) = \\{ ( u , v , x , z ) \\in \\mathcal { H } : w , z \\in H _ { 0 } ^ { 1 } ( \\Omega ) , \\operatorname { d i v } ( K ( x ) \\nabla u + a ( x ) \\nabla w ) , \\operatorname { d i v } ( K ( x ) \\nabla v + a ( x ) \\nabla z ) \\in L ^ { 2 } ( \\Omega ) \\} \\end{align*}"}
-{"id": "9689.png", "formula": "\\begin{align*} X W _ t h = \\alpha _ t h \\ \\ \\ t > 0 \\ \\ h \\in H ^ 2 . \\end{align*}"}
-{"id": "4685.png", "formula": "\\begin{align*} \\mu _ j ( u ) = \\frac { 1 3 } { 2 4 0 } ( u _ { j - 5 } + u _ { j - 1 } ) - \\frac { 7 } { 1 5 } ( u _ { j - 4 } + u _ { j - 2 } ) + \\frac { 7 3 } { 4 0 } u _ { j - 3 } ~ ~ ~ \\mbox { f o r } ~ ~ j = 5 , \\ldots , m . \\end{align*}"}
-{"id": "2553.png", "formula": "\\begin{align*} \\Pi _ { \\mu } ^ { ( T , g ) } ( \\xi _ 0 \\in d x ) = \\frac { 1 } { \\mu ( P ^ \\beta _ T g ) } ( P ^ \\beta _ T g ) ( x ) \\mu ( d x ) , x \\in E . \\end{align*}"}
-{"id": "9195.png", "formula": "\\begin{align*} m = \\dim M , n = \\dim N , \\quad \\quad \\tilde { n } = \\dim \\tilde { N } , \\end{align*}"}
-{"id": "8745.png", "formula": "\\begin{align*} \\widehat { u } _ n ^ { ( \\ell ) } = 2 \\mathbf { I m } \\bigg \\{ \\sum _ { j = 0 } ^ { N - 1 } w _ j ^ { ( \\ell ) } F _ \\omega ( \\lambda _ j ^ { ( \\ell ) } ) ( 1 - \\tau \\lambda _ j ^ { ( \\ell ) } ) ^ { - [ n - b _ { \\ell - 1 } ^ { ( n ) } - 1 ] } y ^ { ( \\ell ) } ( \\tau \\lambda _ j ^ { ( \\ell ) } ) \\bigg \\} , \\end{align*}"}
-{"id": "1442.png", "formula": "\\begin{align*} w ^ { n + 1 } ( t , \\cdot ) = e ^ { \\Delta ( T - t ) } \\mathbb { P } _ { \\delta } w _ { T } - \\varepsilon P \\int _ { t } ^ { T } e ^ { \\Delta ( s - t ) } \\Theta ( s , \\cdot , \\mu ^ { n } ( s , \\cdot ) , D w ^ { n } ( s , \\cdot ) ) \\ d s . \\end{align*}"}
-{"id": "1175.png", "formula": "\\begin{align*} & \\det ( - A ) / \\det ( - B ) = \\det ( - B ) ^ { - 1 / 2 } ( - A ) ( - B ) ^ { - 1 / 2 } \\\\ & = \\det \\left ( + ( - B ) ^ { - 1 / 2 } ( B - A ) ( - B ) ^ { - 1 / 2 } \\right ) , \\end{align*}"}
-{"id": "4852.png", "formula": "\\begin{align*} \\delta ( w ^ * _ { ( k , r _ { \\sigma } ) } ) = \\beta _ { ( k , r _ { \\sigma } ) } [ w _ { ( s _ 1 , r _ p ) } , w ^ * _ { ( j , r _ s ) } ] + \\Theta _ { ( k , r _ { \\sigma } ) } , \\end{align*}"}
-{"id": "5772.png", "formula": "\\begin{align*} A _ d ( C _ 2 , q ) = q ^ d + 2 q ^ { d - 1 } + \\cdots + 2 q + 1 , \\end{align*}"}
-{"id": "7314.png", "formula": "\\begin{align*} \\begin{cases} u _ { t t } - \\Delta u + m _ { 1 } ^ 2 u + g v ^ 2 u = 0 & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ v _ { t t } - \\Delta v + m _ { 2 } ^ 2 v + h u ^ 2 v = 0 & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\end{cases} \\end{align*}"}
-{"id": "885.png", "formula": "\\begin{align*} \\Phi ^ j = \\{ \\beta ^ j _ 1 , \\beta ^ j _ 2 , \\ldots , \\beta ^ j _ { r _ j } \\} \\end{align*}"}
-{"id": "6287.png", "formula": "\\begin{align*} \\hat { \\delta } _ 2 ( \\hat { B } , Q ) ^ 2 & = \\min _ { \\pi } \\| \\hat { B } _ { \\pi } - Q \\| _ 2 ^ 2 \\geq \\frac { 1 } { 2 } \\| \\hat { B } _ { \\pi } \\| _ 2 ^ 2 - \\| Q \\| _ 2 ^ 2 = \\frac { 1 } { 2 } \\| \\hat { B } \\| _ 2 ^ 2 - \\| Q \\| ^ 2 . \\end{align*}"}
-{"id": "634.png", "formula": "\\begin{align*} \\delta ^ \\ast ( \\rho ^ \\ast ( \\delta ^ \\ast ( x ) ) \\cdot x ^ { - 1 } ) = ( \\delta \\rho \\delta ^ { x ^ { - 1 } } ) ^ \\ast ( x ) \\cdot \\delta ^ \\ast ( x ^ { - 1 } ) \\ . \\end{align*}"}
-{"id": "10000.png", "formula": "\\begin{align*} ( F _ { 2 m } A ) ( 2 n ) = ( S p _ { 2 n } ) _ { + } \\wedge _ { S p _ { 2 n - 2 m } } ( A \\wedge T ^ { 2 n - 2 m } ) . \\end{align*}"}
-{"id": "2678.png", "formula": "\\begin{align*} I = \\int _ { 0 } ^ { v } d s \\ , e ^ { - ( z + 1 ) s } M ( \\frac { 1 } { 2 } , 1 , 2 s ) . \\end{align*}"}
-{"id": "4321.png", "formula": "\\begin{align*} V _ h = \\Tilde { W } _ h : = \\{ v _ h \\in H ^ 1 _ 0 ( \\Omega ) \\cap C ^ 0 ( \\overline { \\Omega } ) : v _ h \\vert _ K \\in \\Tilde { W } _ k ( K ) , \\ ; \\forall K \\in \\mathcal { T } _ h \\} \\end{align*}"}
-{"id": "8254.png", "formula": "\\begin{align*} K _ 1 = \\dots = K _ s = { [ N + 1 ] \\choose \\leqslant k + 2 } , K _ { s + 1 } = \\dots = K _ r = { [ N + 1 ] \\choose \\leqslant k + 1 } \\ , . \\end{align*}"}
-{"id": "2817.png", "formula": "\\begin{align*} R _ q ^ { p + 1 , q } = \\left | \\begin{array} { c c c c c } \\hat { a } _ { 1 1 } & a _ { 1 2 } & \\cdots & a _ { 1 p } & b _ 1 \\\\ a _ { 2 1 } & \\hat { a } _ { 2 2 } & \\cdots & a _ { 2 p } & b _ 2 \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\ddots \\\\ a _ { p 1 } & a _ { p 2 } & \\cdots & \\hat { a } _ { p p } & b _ p \\\\ a _ { q 1 } & a _ { q 2 } & \\cdots & a _ { q p } & b _ q \\\\ \\end{array} \\right | , \\end{align*}"}
-{"id": "3877.png", "formula": "\\begin{align*} \\exists ~ c \\in \\mathbb { R } \\backslash \\{ 0 \\} \\ : \\ \\lambda = c \\overline { \\mu } . \\end{align*}"}
-{"id": "5444.png", "formula": "\\begin{align*} \\rho _ s ( T ) = \\rho _ { \\lfloor s \\rfloor + 1 } ( T ) ^ { s - \\lfloor s \\rfloor } \\rho _ { \\lfloor s \\rfloor } ( T ) ^ { 1 - s + \\lfloor s \\rfloor } , \\end{align*}"}
-{"id": "284.png", "formula": "\\begin{align*} P \\boxplus Q \\ ( x ) \\ = \\ \\sum _ { k = 0 } ^ { n } P ^ { ( k ) } ( 0 ) \\cdot Q ^ { ( n - k ) } ( x ) . \\end{align*}"}
-{"id": "5773.png", "formula": "\\begin{align*} \\epsilon ( \\Gamma ) : = \\begin{cases} 1 & \\Gamma = \\bullet \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "2646.png", "formula": "\\begin{align*} \\widetilde { V } = \\begin{bmatrix} \\begin{array} { c c } ( \\lambda _ { i } ^ { j - 1 } ) _ { i = 1 , \\ldots , N ; j = 1 , \\ldots , N + n } & \\\\ \\mathrm { d i a g } ( 1 , 1 ! , \\cdots , ( n - 1 ) ! ) & 0 _ { n \\times N } \\end{array} \\end{bmatrix} . \\end{align*}"}
-{"id": "3412.png", "formula": "\\begin{align*} \\rho _ 0 ( T ) = \\frac { 1 - \\prod _ i \\rho _ 0 ( T _ i ) } { 1 - \\prod _ i \\rho _ 0 ( T _ i ) + \\prod _ i \\left ( 1 + \\rho _ * ( T _ i ) \\right ) } . \\end{align*}"}
-{"id": "6226.png", "formula": "\\begin{align*} \\mu = \\lim _ { k \\to \\infty } \\frac { \\| \\theta _ { k + 1 } - \\theta ^ \\star \\| } { \\| \\theta _ k - \\theta ^ \\star \\| } , \\end{align*}"}
-{"id": "6688.png", "formula": "\\begin{align*} x \\frac { d \\psi _ { 0 } ( x ) } { d x } = ( - \\frac { 1 } { 2 } + i ( \\frac { \\omega } { 2 } - \\lambda _ { * } ) ) \\psi _ { 0 } ( x ) . \\end{align*}"}
-{"id": "2477.png", "formula": "\\begin{align*} P ( x ) = \\int _ 0 ^ 1 \\xi x ^ n \\ , d \\mu \\end{align*}"}
-{"id": "9798.png", "formula": "\\begin{align*} D _ { \\kappa - 2 } ( q ) \\cap L _ \\ast = D _ { \\kappa - 2 } ( q _ \\circ ) \\cap L _ \\ast \\subsetneq L _ \\ast , \\end{align*}"}
-{"id": "8817.png", "formula": "\\begin{align*} X \\coprod X = \\underset { K \\subset X \\atop K } { c o l i m } \\left ( X \\coprod K \\right ) \\end{align*}"}
-{"id": "1091.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } \\int _ { I _ 1 } D _ n ( \\sqrt { 4 - y ^ 2 } / S ( I ) \\cdot G _ n ( x ) / 2 ) d y = e ^ { c _ 0 - x } S ( I ) \\frac { \\sqrt { 4 - a ^ 2 } } { | a | } , \\end{align*}"}
-{"id": "7074.png", "formula": "\\begin{align*} ( X \\otimes Y ) . [ A ] = ( X . [ A ] ) . Y = Y . ( X . [ A ] ) = y + x + [ A ] \\end{align*}"}
-{"id": "9454.png", "formula": "\\begin{align*} P _ { n , \\epsilon } ( z ) : = \\sum _ { \\gamma \\in \\Gamma _ 0 ( N ) _ \\infty \\backslash \\Gamma _ 0 ( N ) } \\psi _ \\epsilon ( \\mathrm { I m } ( \\gamma z ) ) e ( - n ( \\gamma z ) ) . \\end{align*}"}
-{"id": "3644.png", "formula": "\\begin{align*} a _ 1 \\cdot ( ( \\alpha _ { - 1 } - \\beta _ { - 1 } ) \\cdot ( \\alpha _ { - 2 } - \\beta _ { - 2 } ) ) = - \\frac { 1 } { 2 ^ 2 } ( \\alpha _ { - 1 } \\cdot \\beta _ { - 2 } + \\alpha _ { - 2 } \\cdot \\beta _ { - 1 } ) - \\frac { 1 } { 2 ^ 2 } \\langle \\beta _ { - 1 } , \\beta _ { - 2 } \\rangle a _ 1 . \\end{align*}"}
-{"id": "7245.png", "formula": "\\begin{align*} S : = \\{ ( \\phi , \\mu ) \\in U : F ( \\phi , \\mu ) = 0 \\} , \\end{align*}"}
-{"id": "6356.png", "formula": "\\begin{align*} \\mathcal { P } _ { \\rm o u t } ^ D & = { \\rm P r o b } \\left \\{ \\mathbb { P } _ { e , D | p , h } > \\mathbb { P } _ { e , t } \\right \\} \\\\ & = { \\rm P r o b } \\left \\{ \\sum _ { i = 1 } ^ 2 \\sum _ { j = 1 } ^ 2 g _ { i j } h _ { i j } < \\mathcal { A } _ { t h 2 } \\right \\} , \\end{align*}"}
-{"id": "1076.png", "formula": "\\begin{align*} & \\ln ( D _ n ( w \\alpha _ n ) / D _ n ( \\alpha _ n ) ) = \\ln D _ n ( w \\alpha _ n ) - \\ln D _ n ( \\alpha _ n ) \\\\ \\leq & n ^ 2 \\ln \\cos \\frac { w \\alpha _ n } { 2 } - \\frac { 1 } { 4 } \\ln \\left ( n \\sin \\frac { w \\alpha _ n } { 2 } \\right ) - n ^ 2 \\ln \\cos \\frac { \\alpha _ n } { 2 } + \\frac { 1 } { 4 } \\ln \\left ( n \\sin \\frac { \\alpha _ n } { 2 } \\right ) + 1 . \\end{align*}"}
-{"id": "1645.png", "formula": "\\begin{align*} G _ k : = \\sigma ( F _ k ) = \\frac { 1 } { \\varkappa _ k } \\begin{pmatrix} a _ k + \\sqrt [ + ] { a _ k ^ 2 - b ^ 2 } \\\\ - b \\end{pmatrix} e ^ { 2 \\pi i k x } \\end{align*}"}
-{"id": "4481.png", "formula": "\\begin{align*} S _ { n } ( \\kappa _ 1 ^ { - 1 } , \\ldots , \\kappa _ n ^ { - 1 } ) & = \\tilde \\psi ^ k S _ { n - k } ( \\kappa _ 1 ^ { - 1 } , \\ldots , \\kappa _ n ^ { - 1 } ) \\\\ & \\geq \\tilde \\psi ^ k S _ { n } ( \\kappa _ 1 ^ { - 1 } , \\ldots , \\kappa _ n ^ { - 1 } ) ^ { \\frac { n - k } { n } } . \\end{align*}"}
-{"id": "9655.png", "formula": "\\begin{align*} \\delta _ 0 ^ 2 \\| \\varphi _ 0 ( T ) x \\| ^ 2 & \\leq \\| X \\varphi _ 0 ( T ) x \\| ^ 2 = \\int _ \\sigma | \\varphi _ 0 f | ^ 2 m \\\\ & = \\int _ { \\sigma \\setminus \\tau } | \\varphi _ 0 f | ^ 2 m + \\int _ \\tau | \\varphi _ 0 f | ^ 2 m \\\\ & \\leq \\varepsilon ^ 2 \\| X \\| ^ 2 + \\int _ \\tau \\gamma ^ 2 \\delta _ 0 ^ 2 \\| \\varphi _ 0 ( T ) x \\| ^ 2 | \\psi _ 0 | ^ 2 m \\\\ & \\leq \\varepsilon ^ 2 \\| X \\| ^ 2 + \\gamma ^ 2 \\delta _ 0 ^ 2 \\| \\varphi _ 0 ( T ) x \\| ^ 2 \\end{align*}"}
-{"id": "2716.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } ( 2 ^ { \\frac { 4 } { 3 } } N ^ { \\frac { 1 } { 3 } } ) ^ k R _ { N } ^ { ( k ) } \\Big ( 4 N + 2 ^ { \\frac { 4 } { 3 } } N ^ { \\frac { 1 } { 3 } } u _ 1 , \\ldots , 4 N + 2 ^ { \\frac { 4 } { 3 } } N ^ { \\frac { 1 } { 3 } } u _ k \\Big ) = \\det \\big [ K _ { \\mathrm { A i r y } } ( \\pi ; u _ { i } , u _ { j } ) \\big ] _ { i , j = 1 } ^ { k } \\end{align*}"}
-{"id": "1382.png", "formula": "\\begin{align*} \\lambda \\left ( \\sum _ { i = 1 } ^ { m } \\sum _ { j = J _ { ( i ) } } ^ { k } d _ { j } ( \\lambda _ { i - 1 } ) e _ j + \\sum _ { j = q + 1 } ^ k d _ j ( \\lambda _ { m } ) e _ j + \\sum _ { i = m + 1 } ^ { n - 1 } \\sum _ { j = J _ { ( i ) } } ^ { k } d _ { j } ( \\lambda _ { i } ) e _ j \\ , , ~ d ( \\lambda _ 0 \\cdots \\lambda _ n ) \\right ) \\ ; . \\end{align*}"}
-{"id": "1142.png", "formula": "\\begin{align*} & \\| ( - B ) ^ { - 1 } \\| = ( 1 - \\lambda _ 1 ( B ) ) ^ { - 1 } \\leq \\max _ { 1 \\leq i \\leq k } ( 1 - \\lambda _ 1 ( B _ i ) ) ^ { - 1 } + 1 \\\\ \\leq & n ( \\ln n ) ^ { - \\frac { 1 } { 2 } } e ^ { 3 C _ 0 - c _ 0 - ( \\ln n ) ^ { \\frac { 1 } { 2 } } / 2 + 1 } + 1 = O \\left ( n ( \\ln n ) ^ { - \\frac { 1 } { 2 } } e ^ { - ( \\ln n ) ^ { \\frac { 1 } { 2 } } / 2 } \\right ) . \\end{align*}"}
-{"id": "8624.png", "formula": "\\begin{align*} H _ 0 = \\left ( 1 + \\frac { c _ 0 ^ 2 } { u ^ 4 } \\right ) \\frac { z ^ 2 } { 2 } - \\frac { 1 } { 4 } \\left ( 1 - u ^ 2 - \\frac { c _ 0 ^ 2 } { u ^ 2 } \\right ) ^ 2 . \\end{align*}"}
-{"id": "717.png", "formula": "\\begin{align*} & R _ { 1 \\bar { 1 } 1 \\bar { 1 } } = - 3 2 p ^ 3 K , \\\\ & R _ { 1 \\bar { 2 } 1 \\bar { 2 } } = ( p - 1 ) f ^ { ( 1 ) } ( 0 ) , \\\\ & R _ { 1 \\bar { 1 } 2 \\bar { 2 } } = - p f ^ { ( 1 ) } ( 0 ) , \\\\ & R _ { 2 \\bar { 2 } 2 \\bar { 2 } } = - \\frac { f ^ { ( 3 ) } ( 0 ) } { 1 6 } + \\frac { f ^ { ( 1 ) } ( 0 ) ^ 2 } { 1 6 p K } . \\end{align*}"}
-{"id": "3009.png", "formula": "\\begin{align*} B _ { n , k } ( x _ 1 , x _ 2 , \\dots , x _ { n - k + 1 } ) & = \\sum _ { \\pi ( n , k ) } \\frac { n ! } { j _ 1 ! j _ 2 ! \\dots j _ { n - k + 1 } ! } \\Big ( \\frac { x _ 1 } { 1 ! } \\Big ) ^ { j _ 1 } \\Big ( \\frac { x _ 2 } { 2 ! } \\Big ) ^ { j _ 2 } \\dots \\Big ( \\frac { x _ { n - k + 1 } } { ( n - k + 1 ) ! } \\Big ) ^ { j _ { n - k + 1 } } \\end{align*}"}
-{"id": "7692.png", "formula": "\\begin{align*} \\Gamma ( M ; S ) : = \\left \\{ v \\colon M \\to S \\ ; \\vert \\ ; v _ p \\in S _ p p \\in M \\right \\} , \\end{align*}"}
-{"id": "9414.png", "formula": "\\begin{align*} ( c z + d ) ^ { - 2 } E _ 2 \\left ( \\frac { a z + b } { c z + d } \\right ) = E _ 2 ( z ) + \\frac { 1 2 } { 2 \\pi i } \\cdot \\frac { c } { c z + d } \\end{align*}"}
-{"id": "5605.png", "formula": "\\begin{align*} u _ { \\delta , \\epsilon } ( x , t ) = \\int \\int \\tilde { u } ( y , s ) \\theta _ { \\epsilon } ( x - y ) \\theta _ { \\delta } ( t - s ) d y d s , \\end{align*}"}
-{"id": "4470.png", "formula": "\\begin{align*} \\underline { \\psi } ( x ) : = G ( \\nabla ^ 2 \\underline u , \\nabla \\underline u , \\underline u ) & \\geq \\Psi ( \\nabla \\underline u , \\underline u , x ) \\quad \\textrm { i n } \\ , \\Omega \\\\ \\underline u & = \\varphi \\quad \\textrm { o n } \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "4154.png", "formula": "\\begin{align*} \\sqrt { 3 } i x ^ { 1 / 3 } f _ { 1 } ( x ) - \\sqrt { 3 } i x ^ { 2 / 3 } f _ { 2 } ( x ) = 3 \\varphi _ { 1 + } ( x ) = 3 \\pi i \\mu ^ { * } ( [ 0 , x ] ) \\sim 9 \\pi i c _ { 0 , V } x ^ { 1 / 3 } x \\downarrow 0 . \\end{align*}"}
-{"id": "2434.png", "formula": "\\begin{align*} N _ f ( t ) = \\frac { 1 } { q - 1 } \\sum _ { i = d } ^ { n } A _ i / \\binom { n } { i } t ^ { i - d } . \\end{align*}"}
-{"id": "7528.png", "formula": "\\begin{align*} \\frac { d } { d t } \\tau = F ( g , \\tau ) \\end{align*}"}
-{"id": "5268.png", "formula": "\\begin{align*} \\lambda ^ { 0 } ( A ) = \\inf \\{ \\sum _ { n = 0 } ^ { \\infty } \\lambda ( A _ { n } ) , A \\subset \\bigcup \\limits _ { n = 0 } ^ { \\infty } A _ { n } , A _ { n } \\in \\mathcal { C } \\} . \\end{align*}"}
-{"id": "8782.png", "formula": "\\begin{align*} \\frac { ( n - 2 ) ^ 2 } { n } \\ , L _ C = L _ R - \\frac { \\kappa } { n ( n - 1 ) } \\ , . \\end{align*}"}
-{"id": "4987.png", "formula": "\\begin{align*} Z ( C / \\mathbb { F } _ q , T ) = ( 1 + C _ 1 T + . . . + C _ { 2 g } T ^ { 2 g } ) ( 1 + T + T ^ 2 + \\ldots ) ( 1 + q T + q ^ 2 T ^ 2 + \\ldots ) . \\end{align*}"}
-{"id": "2967.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { \\partial \\Delta } \\frac { 1 } { w - z } \\sum _ { m , n = 0 } ^ { \\infty } a _ { m n } z ^ { m } w ^ { n } f ( w ) d w = \\sum _ { m , n = 0 } ^ { \\infty } a _ { m n } z ^ { m } \\frac { 1 } { 2 \\pi i } \\int _ { \\partial \\Delta } \\frac { 1 } { w - z } w ^ { n } f ( w ) d w \\end{align*}"}
-{"id": "2346.png", "formula": "\\begin{align*} \\delta ( x - y ) d y & = \\varepsilon _ x ( d y ) \\\\ \\delta ( x - y ) d x & = \\varepsilon _ y ( d x ) \\\\ \\delta ( x - y ) d x d y & = \\Lambda ( d x , d y ) , \\end{align*}"}
-{"id": "9756.png", "formula": "\\begin{align*} q _ { Z _ \\infty } ( x + \\xi , 0 ) = q _ { Z _ \\infty } ( x , 0 ) \\quad ( \\xi , x ) \\in ( L _ \\ast + L ( q ) ) \\times \\R ^ n . \\end{align*}"}
-{"id": "2799.png", "formula": "\\begin{align*} r _ t ( \\mathbf { m } ) = \\sum _ { n \\in \\mathbb { N } _ 0 ^ S } p _ t ( \\mathbf { n } ) T ( \\mathbf { n } ; \\mathbf { m } ) . \\end{align*}"}
-{"id": "7112.png", "formula": "\\begin{align*} ( \\mu + 1 ) K - 2 \\mu H + \\mu - 1 = 0 \\end{align*}"}
-{"id": "4713.png", "formula": "\\begin{align*} I S _ 0 & = \\frac { 1 3 } { 1 2 } ( f ^ { + } _ { j - 2 } - 2 f ^ { + } _ { j - 1 } + f ^ { + } _ { j } ) ^ 2 + \\frac { 1 } { 4 } ( f ^ { + } _ { j - 2 } - 4 f ^ { + } _ { j - 1 } + 3 f ^ { + } _ { j } ) ^ 2 , \\\\ I S _ 1 & = \\frac { 1 3 } { 1 2 } ( f ^ { + } _ { j - 1 } - 2 f ^ { + } _ { j } + f ^ { + } _ { j + 1 } ) ^ 2 + \\frac { 1 } { 4 } ( f ^ { + } _ { j - 1 } - f ^ { + } _ { j + 1 } ) ^ 2 , \\\\ I S _ 2 & = \\frac { 1 3 } { 1 2 } ( f ^ { + } _ { j } - 2 f ^ { + } _ { j + 1 } + f ^ { + } _ { j + 2 } ) ^ 2 + \\frac { 1 } { 4 } ( 3 f ^ { + } _ { j } - 4 f ^ { + } _ { j + 1 } + f ^ { + } _ { j + 2 } ) ^ 2 , \\end{align*}"}
-{"id": "2505.png", "formula": "\\begin{align*} h _ { j } ^ { \\langle j \\rangle } = { j + 1 \\choose j + 1 } + { j \\choose j } + \\cdots + { j - r + 2 \\choose j - r + 2 } = r + 1 = h _ { j + 1 } . \\end{align*}"}
-{"id": "7445.png", "formula": "\\begin{align*} \\displaystyle \\lim \\limits _ { s \\to 0 ^ + } s \\widetilde { D } _ s ( \\varphi ) = \\frac { \\prod \\limits _ { i = 1 } ^ N h _ \\ell } { \\sum _ { \\ell = 1 } ^ N h _ \\ell } ( 2 \\pi ) ^ { N - 1 } \\psi _ 0 ( 0 ) = \\frac { D } { n } ( 2 \\pi ) ^ { N - 1 } 2 ^ { 1 - c } \\psi _ 0 ( 0 ) \\end{align*}"}
-{"id": "841.png", "formula": "\\begin{align*} f ( t ) = \\sum _ { n = - \\infty } ^ { \\infty } f ( t _ n ) S _ { n } ( t ) \\textrm { w i t h } S _ { n } ( t ) = \\frac { \\int _ { I } \\kappa ( t _ n , x ) \\overline { \\kappa ( t , x ) } d x } { \\int _ { I } \\vert \\kappa ( t _ n , x ) \\vert ^ 2 d x } t \\in \\mathbb { R } \\end{align*}"}
-{"id": "1477.png", "formula": "\\begin{align*} V _ { 1 4 } \\leq \\varepsilon ^ { 2 } \\mathcal { G } _ { 2 } ( K ) E _ { \\mu } + \\frac { 1 } { 8 } \\sum _ { j = 1 } ^ { d } \\int _ { \\mathbb { T } ^ { d } } ( \\partial _ { x _ { j } } ( D w ^ { 1 } - D w ^ { 2 } ) ) ^ { 2 } \\ d x . \\end{align*}"}
-{"id": "5753.png", "formula": "\\begin{align*} \\left | \\int \\limits _ 0 ^ 1 x f _ { y y y } ( x , y , \\xi ) J _ 0 ( \\gamma _ m x ) d x \\right | = \\left | \\int \\limits _ 0 ^ 1 \\sqrt { x } ( \\sqrt x \\ , f _ { y y y } ( x , y \\xi ) ) J _ 0 ( \\gamma _ m x ) d x \\right | \\leq \\frac { C _ { 2 7 } } { ( \\gamma _ m ) ^ { 5 / 2 } } \\max ( \\sqrt { x } f _ { y y y x x } ) \\leq \\frac { C _ { 2 8 } } { m ^ { 5 / 2 } } . \\end{align*}"}
-{"id": "1361.png", "formula": "\\begin{align*} F ^ 0 _ j ( \\lambda ) = \\lambda ( 0 , d ( \\lambda ) - e _ { c _ j } ) , F ^ 1 _ j ( \\lambda ) = \\lambda ( e _ { c _ j } , d ( \\lambda ) ) . \\end{align*}"}
-{"id": "2687.png", "formula": "\\begin{align*} f _ { \\tau , \\theta } ( z ) = \\frac { 4 \\tau \\cos ^ { 2 } \\theta } { 1 - \\tau ^ { 2 } } ( z + 1 ) + \\log \\frac { \\tau z - 1 } { z - \\tau } , z \\in \\mathbb { C } , \\end{align*}"}
-{"id": "7405.png", "formula": "\\begin{align*} \\Phi _ t ( A ) : = \\{ x \\in M : d _ x ( x , A ) < t \\} , \\end{align*}"}
-{"id": "3701.png", "formula": "\\begin{align*} \\widehat { V } ( \\widetilde { Y } ) = \\sum _ x \\big ( p _ x ^ { - 1 } - 1 \\big ) p _ x ^ { - 1 } \\sum _ { i \\in B _ x } y _ i ^ 2 \\end{align*}"}
-{"id": "4576.png", "formula": "\\begin{align*} E ^ k ( \\mu , g , \\delta ) = \\mu ^ { 2 k } \\{ ( x _ 1 , \\dots , x _ k , y _ 1 , \\dots , y _ k ) \\in \\mathbb { R } ^ { 2 k n } : | ( x _ i - g y _ i ) - ( x _ j - g y _ j ) | \\leq \\delta \\} . \\end{align*}"}
-{"id": "3900.png", "formula": "\\begin{align*} \\P ( 1 , 0 , \\ , \\ldots , \\ , 0 , 0 ) ^ t = \\infty . \\end{align*}"}
-{"id": "7874.png", "formula": "\\begin{align*} \\frac { 1 } { m ! } \\sum _ { \\sigma \\in { \\rm S y m } ( I ) } \\prod _ { j \\in I } \\cos ( 2 \\pi y _ { \\sigma ^ { - 1 } ( j ) } \\xi _ j ) & = \\frac { 1 } { { m \\choose l } } \\sum _ { \\substack { J \\subseteq I \\\\ | J | = l } } \\prod _ { j \\in J } \\cos ( 2 \\pi \\xi _ j ) \\\\ & = \\sum _ { S \\subseteq I } a _ S ( \\xi ) \\frac 1 { { m \\choose l } } \\sum _ { \\substack { J \\subseteq I \\\\ | J | = l } } w _ S ( \\varepsilon ( J ) ) . \\end{align*}"}
-{"id": "253.png", "formula": "\\begin{align*} R _ { T _ 0 } = b _ 1 + \\cdots + b _ R . \\end{align*}"}
-{"id": "2213.png", "formula": "\\begin{align*} \\boldsymbol { g } ( t ) = - \\int _ { t } ^ { \\infty } e x p ^ { \\left [ A ^ { T } - \\frac { 1 } { 2 } ( P + P ^ T ) B R ^ { - 1 } B ^ { T } ( \\tau - t ) \\right ] } C ^ { T } Q \\boldsymbol { z } ( \\tau ) d \\tau \\end{align*}"}
-{"id": "5572.png", "formula": "\\begin{align*} \\mathcal { E } ^ { \\pm } ( t ) & : = \\int _ { \\R ^ { \\pm } } \\Big ( \\alpha v | u | ^ 2 - \\frac { \\alpha } { 6 \\gamma } v ^ 3 + \\frac { \\beta } { 2 } | u | ^ 4 + \\frac { \\alpha } { 2 \\gamma } v _ x ^ 2 + | u _ x | ^ 2 \\Big ) d x \\\\ & = \\mathcal { E } ^ { \\pm } ( 0 ) \\mp \\int _ 0 ^ t \\big ( \\mathcal { E } _ 1 ( s ) + \\mathcal { E } _ 2 ( s ) \\big ) d s , \\end{align*}"}
-{"id": "2171.png", "formula": "\\begin{align*} \\tau _ { - \\mathbf y } ( f * \\phi ) = \\sum _ { j , n \\in \\mathbb { N } _ 0 } \\tau _ { - \\mathbf y } ( f _ j * \\phi _ n ) = \\sum _ { j , n \\in \\mathbb { N } _ 0 } f _ j * ( \\tau _ { - \\mathbf y } \\phi _ n ) \\end{align*}"}
-{"id": "201.png", "formula": "\\begin{align*} & X _ 1 = x ^ 1 \\dfrac { \\partial } { \\partial x ^ 1 } + x ^ 3 \\dfrac { \\partial } { \\partial x ^ 3 } , Y _ 1 = - \\dfrac { x ^ 2 } { x ^ 1 } \\dfrac { \\partial } { \\partial x ^ 3 } , \\\\ & X _ 2 = x ^ 1 \\dfrac { \\partial } { \\partial x ^ 2 } , Y _ 2 = \\dfrac { \\partial } { \\partial x ^ 3 } . \\end{align*}"}
-{"id": "3365.png", "formula": "\\begin{align*} d _ 1 D _ 1 = \\dim ( V ^ * \\o P ( V ) ) = D _ 0 + m D _ 1 \\end{align*}"}
-{"id": "7203.png", "formula": "\\begin{align*} \\phi ( x ) = \\frac { 2 \\pi ^ 2 - x ^ 2 } { 1 8 } , \\mbox { f o r } \\mu = \\frac { \\pi ^ 2 } { 9 } \\end{align*}"}
-{"id": "3004.png", "formula": "\\begin{align*} - \\sum _ { n = 0 } ^ \\infty n e _ { m + 2 , n } x ^ n & = \\Big ( \\sum _ { n = 0 } ^ \\infty e _ { m + 2 , n } x ^ n \\Big ) \\Big ( \\sum _ { n = 0 } ^ \\infty \\sigma ' _ m ( n ) x ^ n \\Big ) . \\end{align*}"}
-{"id": "479.png", "formula": "\\begin{align*} \\nabla ^ { p } ( F ) = \\{ g \\circ F \\mid g \\in S _ { p } \\} \\end{align*}"}
-{"id": "509.png", "formula": "\\begin{align*} \\mu _ 2 ( Q ) & = f ' ( z ) ( k ( z ) ^ 2 f ( z ) ) ' - ( k ( z ) f ( z ) ) ' ( k ( z ) f ( z ) ) ' = \\\\ & - k ' ( z ) ^ 2 f ( z ) ^ 2 \\neq 0 . \\end{align*}"}
-{"id": "5759.png", "formula": "\\begin{align*} ( x , g ) \\cdot ( x _ 1 , x _ 2 ) = ( g \\cdot x _ 1 , \\varrho _ \\R ( x ) ( g \\cdot x _ 1 ) + g \\cdot x _ 2 ) , \\end{align*}"}
-{"id": "8323.png", "formula": "\\begin{align*} \\it { I I } \\leq & \\frac { 2 ^ k } { C _ 0 ^ k } \\int _ { | \\beta | \\geq \\frac { 2 } { C _ 1 } d _ I ( t ) } | \\beta | ^ { - k } d \\beta = 2 \\frac { 2 ^ k } { ( k - 1 ) C _ 0 ^ k } \\frac { C _ 1 ^ { k - 1 } } { 2 ^ { k - 1 } } d _ I ( t ) ^ { - k + 1 } = \\frac { 4 C _ 1 ^ { k - 1 } } { ( k - 1 ) C _ 0 ^ { k - 1 } } d _ I ( t ) ^ { - k + 1 } . \\end{align*}"}
-{"id": "1282.png", "formula": "\\begin{align*} \\int _ { | \\mu | \\le | \\eta | ^ { 3 / 2 } / 2 } e ^ { 3 i \\mu ^ 2 / 4 } z \\left ( \\eta + \\frac { \\mu } { \\sqrt { \\eta } } \\right ) w \\left ( \\eta - \\frac { \\mu } { \\sqrt { \\eta } } \\right ) = | \\eta | ^ { 3 / 2 } \\int _ { | \\nu | < 1 / 2 } e ^ { 3 i \\eta ^ 3 \\nu ^ 2 / 4 } z ( \\eta ( 1 + \\nu ) ) w ( \\eta ( 1 - \\nu ) ) d \\nu \\end{align*}"}
-{"id": "4664.png", "formula": "\\begin{align*} \\lambda \\cdot f ( x ) : = \\lambda \\cdot \\sum _ { j = 0 } ^ n a _ j x ^ { [ n - j ] } & = \\sum _ { j = 0 } ^ n a _ j \\sum _ { h = 0 } ^ { n - j } ( - 1 ) ^ h \\lambda ^ { [ h ] } x ^ { [ n - j - h ] } \\\\ & = \\sum _ { k = 0 } ^ n \\left ( \\sum _ { j + h = k } ( - 1 ) ^ h a _ j \\lambda ^ { [ h ] } \\right ) x ^ { [ n - k ] } . \\end{align*}"}
-{"id": "4292.png", "formula": "\\begin{align*} \\frac { 1 } { r - 1 } \\binom { s - 2 } { r - 1 } n - \\binom { s - 2 } { r - 1 } - o _ n ( 1 ) \\end{align*}"}
-{"id": "8909.png", "formula": "\\begin{align*} \\mathbb { E } _ { P _ 0 } [ L ^ 2 ( G ) ] & \\le \\frac { e ^ { \\frac { 4 t ^ 2 } { n } \\nu } } { \\binom { m } { t } } \\sum _ { k = 0 } ^ t \\binom { t } { k } \\binom { m - t } { t - k } \\exp { 2 k \\nu ( 1 - 4 t / n ) } , \\\\ & = e ^ { \\frac { 4 t ^ 2 } { n } \\nu } \\mathbb { E } [ \\xi ^ Z ] , \\end{align*}"}
-{"id": "9137.png", "formula": "\\begin{align*} \\lim _ { \\tau \\rightarrow + \\infty } I _ { M } ( \\rho _ { 1 } ^ { ( N ) } ( t ) , \\mathbf { b } ) \\leq - \\lim _ { \\tau \\rightarrow + \\infty } ( t - t _ { o } ) k ^ { 2 } = - \\infty , \\end{align*}"}
-{"id": "4592.png", "formula": "\\begin{align*} \\hat { f } _ { g , \\delta } = \\hat { \\nu } _ g \\hat { \\phi ^ D _ \\delta } . \\end{align*}"}
-{"id": "7108.png", "formula": "\\begin{align*} d \\Psi _ { m } = - m \\xi \\Psi _ { m } = m \\begin{pmatrix} \\phi & - \\phi ^ { 2 } \\\\ 1 & - \\phi \\end{pmatrix} \\omega \\Psi _ { m } . \\end{align*}"}
-{"id": "2366.png", "formula": "\\begin{align*} a ^ + _ { x _ c } \\Phi ( x _ \\alpha ) & = \\Phi ( x _ { \\alpha + \\{ c \\} } ) \\\\ a _ { x _ c } \\Phi ( x _ \\alpha ) & = \\sum _ { b \\in \\alpha } \\delta _ { b , c } \\Phi ( x _ { \\alpha \\setminus \\{ c \\} } ) \\end{align*}"}
-{"id": "7527.png", "formula": "\\begin{align*} \\left \\{ \\begin{gathered} \\partial _ t X = \\mathcal { U } ( X , \\tau ) , \\\\ \\partial _ t \\tau = \\mathcal { T } ( X , \\tau ) . \\end{gathered} \\right . \\end{align*}"}
-{"id": "9208.png", "formula": "\\begin{align*} \\tilde { \\gamma } ( t ) = t \\gamma ^ { \\prime } ( 0 ) \\end{align*}"}
-{"id": "435.png", "formula": "\\begin{align*} \\max _ { u \\le A + ( \\gamma + \\eta ) t } \\left | [ \\gamma ( u - \\gamma t ) + 1 ] e ^ { \\gamma u - \\frac { \\gamma ^ 2 t } { 2 } } \\right | = [ \\gamma ( \\eta t + A ) + 1 ] e ^ { \\frac { \\gamma ^ 2 t } { 2 } + \\gamma [ \\eta t + A ] } , \\end{align*}"}
-{"id": "1562.png", "formula": "\\begin{align*} \\Phi _ { \\widetilde { F } ^ \\delta } ( z , x ) = - \\frac { 1 } { z } - b \\log \\left ( - \\frac { 1 } { z } \\right ) + o ( 1 ) . \\end{align*}"}
-{"id": "74.png", "formula": "\\begin{align*} \\sigma ^ + _ 1 + \\sigma ^ + _ { - 1 } & = \\sigma _ 1 ^ - + \\sigma ^ - _ { - 1 } \\\\ \\sigma ^ + _ { 1 } - \\sigma ^ + _ { - 1 } & = \\frac { 1 - g ' ( s ) \\delta } { 1 + g ' ( s ) \\delta } \\left ( \\sigma _ 1 ^ - - \\sigma _ { - 1 } ^ - \\right ) = ( 1 - 2 c ) \\left ( \\sigma _ 1 ^ - - \\sigma _ { - 1 } ^ - \\right ) . \\end{align*}"}
-{"id": "1806.png", "formula": "\\begin{align*} \\begin{aligned} c _ { 2 , 1 } ( A ) \\leq & \\ c _ { 2 , 1 } \\left ( \\bigcap _ { i = 1 } ^ { d } \\left \\{ \\left | B _ { s _ { 1 } } ^ { i } - B _ { t _ { 0 } } ^ { i } \\right | < 2 \\eta \\right \\} \\right ) \\\\ & + \\sum _ { i = 1 } ^ { d } c _ { 2 , 1 } \\left ( \\sup _ { s \\in I } \\left | B _ { s _ { 1 } } ^ { i } - B _ { s } ^ { i } \\right | > \\eta \\right ) \\\\ & + \\sum _ { i = 1 } ^ { d } c _ { 2 , 1 } \\left ( \\sup _ { t \\in J } \\left | B _ { t } ^ { i } - B _ { t _ { 0 } } ^ { i } \\right | > \\eta \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "4100.png", "formula": "\\begin{align*} \\tilde { T } _ { \\alpha } = \\begin{pmatrix} 1 & \\tilde { t } _ 3 & \\tilde { t } _ 2 \\\\ 0 & 1 & \\tilde { t } _ 1 \\\\ 0 & 0 & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "4826.png", "formula": "\\begin{align*} \\Pi _ { \\widehat { g } } = e ^ f \\Pi _ { g } + \\frac { \\partial } { \\partial \\nu } ( e ^ f ) g , \\end{align*}"}
-{"id": "5086.png", "formula": "\\begin{align*} & \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n \\langle \\tau _ j , x _ k \\rangle \\langle \\tau _ k , x _ j \\rangle = \\frac { 1 } { \\dim \\mathcal { H } } \\left ( \\sum _ { j = 1 } ^ n \\langle x _ j , \\tau _ j \\rangle \\right ) ^ 2 \\\\ & = \\frac { 1 } { \\dim \\mathcal { H } } \\left ( \\sum _ { j = 1 } ^ n \\langle \\tau _ j , x _ j \\rangle \\right ) ^ 2 = \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n \\langle \\tau _ j , \\tau _ k \\rangle \\langle x _ k , x _ j \\rangle . \\end{align*}"}
-{"id": "2308.png", "formula": "\\begin{align*} ( \\delta _ x | \\delta _ y ) & = \\delta ( x - y ) & \\int \\d x | \\delta _ x ) ( \\delta _ x | & = 1 . \\end{align*}"}
-{"id": "9514.png", "formula": "\\begin{align*} c _ { \\lambda } & = a \\left ( \\frac { 1 } { p } - \\frac { N - s p } { N p } \\right ) + b \\left ( \\frac { 1 } { q } - \\frac { 1 } { p _ { s } ^ * } \\right ) \\\\ & \\geq a \\frac { s } { N } . \\end{align*}"}
-{"id": "5706.png", "formula": "\\begin{align*} J _ { p - 1 } ( x ) - J _ { p + 1 } ( x ) & = 2 \\frac { d } { d x } J _ p ( x ) , \\\\ J _ { p - 1 } ( x ) + J _ { p + 1 } ( x ) & = \\frac { 2 p } { x } J _ p ( x ) . \\end{align*}"}
-{"id": "7398.png", "formula": "\\begin{align*} r = \\mathop { \\sum } _ { i = 0 } ^ { k - 1 } ( { \\rm d i m } \\ , X _ { i } - { \\rm d i m } \\ , X _ { i + 1 } ) + r _ k = { \\rm d i m } \\ , X - { \\rm d i m } \\ , X _ { k } + r _ k \\ ; . \\end{align*}"}
-{"id": "3927.png", "formula": "\\begin{align*} \\Lambda ( r \\otimes x \\cdot f ) = & \\ \\sum r _ { 1 } \\sigma \\bigl ( S ( \\pi ( r _ { 2 } ) ) ( x \\cdot f ) \\bigr ) = \\sum ( x \\cdot r _ { 1 } ) \\sigma \\bigl ( S ( \\pi ( r _ { 2 } ) \\cdot x ^ { - 1 } ) ( x \\cdot f ) \\bigr ) = \\\\ & \\ \\sum ( x \\cdot r _ { 1 } ) \\sigma \\bigl ( x \\cdot \\bigl ( S ( \\pi ( r _ { 2 } ) ) f \\bigr ) \\bigr ) = x \\cdot \\sum r _ { 1 } \\sigma \\bigl ( S ( \\pi ( r _ { 2 } ) ) f \\bigr ) = \\\\ & \\ x \\cdot \\Lambda ( r \\otimes f ) \\end{align*}"}
-{"id": "10025.png", "formula": "\\begin{align*} \\Big | \\sum _ { n = N } ^ { M } \\frac { \\varepsilon _ { n } \\lambda _ { n } } { n ^ { \\sigma } } \\Big | \\leq \\Big \\| \\sum _ { n = N } ^ { M } \\frac { \\varepsilon _ { n } a _ { n } } { n ^ { \\sigma } } + x _ { 0 } \\sum _ { n = N } ^ { M } \\frac { \\lambda _ { n } \\varepsilon _ { n } } { n ^ { \\sigma } } \\Big \\| = \\Big \\| \\sum _ { n = N } ^ { M } \\varepsilon _ { n } \\frac { a _ { n } + \\lambda _ { n } x _ { 0 } } { n ^ { \\sigma } } \\Big \\| \\ , . \\end{align*}"}
-{"id": "8454.png", "formula": "\\begin{align*} G _ k ^ { \\theta } = ( I - \\mathcal { H } ) ( \\partial _ { \\alpha } ^ k G + [ D _ t ^ 2 - i A \\partial _ { \\alpha } , \\partial _ { \\alpha } ^ k ] \\tilde { \\theta } ) - [ D _ t ^ 2 - i A \\partial _ { \\alpha } , \\mathcal { H } ] \\partial _ { \\alpha } ^ k \\tilde { \\theta } , \\end{align*}"}
-{"id": "1093.png", "formula": "\\begin{align*} & n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } \\int _ I D _ n ( \\sqrt { 4 - y ^ 2 } / S ( I ) \\cdot G _ n ( x ) / 2 ) d y \\\\ = & \\sum _ { j = 1 } ^ 2 n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } \\int _ { I _ j } D _ n ( \\sqrt { 4 - y ^ 2 } / S ( I _ j ) \\cdot G _ n ( x ) / 2 ) d y \\\\ \\to & M ( I _ 1 ) e ^ { c _ 0 - x } + M ( I _ 2 ) e ^ { c _ 0 - x } = M ( I ) e ^ { c _ 0 - x } , \\ , \\ , \\ , n \\to + \\infty , \\end{align*}"}
-{"id": "9251.png", "formula": "\\begin{align*} v _ 1 = v _ \\tau + \\zeta ^ 2 [ b ( 2 u _ { \\beta \\tau } + c _ 1 u _ \\beta u _ \\tau ) + b ^ 2 ( u _ { \\beta \\beta } + \\frac { c _ 1 } { 2 } | u _ \\beta | ^ 2 ) ] , \\end{align*}"}
-{"id": "4805.png", "formula": "\\begin{align*} b _ 2 ^ { \\mathrm { e x p } } : = E \\left [ e ^ { C | M _ 2 ( \\omega ) | ^ 2 } \\right ] < \\infty \\end{align*}"}
-{"id": "7899.png", "formula": "\\begin{align*} \\kappa ' \\colon F _ * ( ( N \\otimes \\omega _ R ) / I ( N \\otimes \\omega _ R ) ) & \\to ( N \\otimes \\omega _ R ) / I ( N \\otimes \\omega _ R ) \\\\ n \\otimes m + I ( N \\otimes \\omega _ R ) & \\mapsto \\sum _ { i = 1 } ^ t \\phi _ i ( 1 ) \\gamma _ N ( n ) \\otimes \\kappa _ R ( r _ i f ^ { p - 1 } m ) + I ( N \\otimes \\omega _ R ) \\end{align*}"}
-{"id": "2243.png", "formula": "\\begin{align*} A = & A ( x ) = \\frac { \\mathcal { P } } { x - \\Omega } | g \\rangle & A ' = & A ' ( x ) = \\langle h | \\frac { \\mathcal { P } } { x - \\Omega } \\\\ B = & B ( x ) = \\delta ( x - \\Omega ) | g \\rangle & B ' = & B ' ( x ) = \\langle h | \\delta ( x - \\Omega ) \\end{align*}"}
-{"id": "1766.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\left ( B _ { t _ { i } } - B _ { s } \\right ) \\left ( B _ { t _ { j } } - B _ { s } \\right ) \\right ] = \\mathbb { E } \\left [ \\left ( B _ { t _ { i } } - B _ { s } \\right ) ^ { 2 } + \\left ( B _ { t _ { i } } - B _ { s } \\right ) \\left ( B _ { t _ { j } } - B _ { t _ { i } } \\right ) \\right ] . \\end{align*}"}
-{"id": "4881.png", "formula": "\\begin{align*} r ^ * ( \\omega ) = \\inf _ { t \\ge 1 } \\dfrac { r ( t \\omega ) } { t } ( \\omega \\in S ^ { n - 1 } ) . \\end{align*}"}
-{"id": "9075.png", "formula": "\\begin{align*} \\mathbb { E } [ F ^ R _ { \\mu _ { K } } ( y ) F ^ R _ { \\mu _ { K } } ( x ) ] = \\int _ { \\mathbb { S } ^ 1 } e ( \\langle \\lambda , R ( x - y ) \\rangle ) d \\mu _ { K } ( \\lambda ) . \\end{align*}"}
-{"id": "7274.png", "formula": "\\begin{align*} \\bar \\phi _ { k } ( 0 ) = \\bar \\mu _ { k } . \\end{align*}"}
-{"id": "7186.png", "formula": "\\begin{gather*} \\tilde { \\rho } ( y _ { 1 } ( \\cdot , \\star , \\ast ) , y _ { 2 } ( \\cdot , \\star , \\ast ) ) : = \\sup \\left \\{ \\mathrm { e } ^ { a t } \\left \\Vert y _ { 1 } ( t , \\xi , \\eta ) - y _ { 2 } ( t , \\xi , \\eta ) \\right \\Vert : ( t , \\xi , \\eta ) \\in \\mathbb { R } _ { + } \\times \\mathcal { N } _ { r } ( \\mathcal { M } ) \\right \\} . \\end{gather*}"}
-{"id": "362.png", "formula": "\\begin{align*} a z ^ 2 - c x ^ 2 & = a - c \\\\ b z ^ 2 - c y ^ 2 & = b - c . \\end{align*}"}
-{"id": "3741.png", "formula": "\\begin{align*} x _ { 0 } & \\coloneqq k _ { 0 } a \\coloneqq \\omega \\sqrt { \\epsilon _ { 0 } \\mu _ { 0 } } a = 2 \\pi \\frac { a } { \\lambda _ { 0 } } , \\\\ x & \\coloneqq k a \\coloneqq \\omega \\sqrt { \\epsilon } \\sqrt { \\mu } a \\coloneqq x \\sqrt { \\epsilon _ { r } } \\sqrt { \\mu _ { r } } = 2 \\pi \\frac { a } { \\lambda } , \\end{align*}"}
-{"id": "7940.png", "formula": "\\begin{align*} \\Pi _ a = \\Pi _ a P ^ { - 1 } D _ a P . \\end{align*}"}
-{"id": "7148.png", "formula": "\\begin{align*} Z _ { t } ^ { \\nu } = \\int _ { 0 } ^ { t } \\int _ { \\mathbf { R } _ { 0 } ^ { d } } \\chi _ { \\alpha } \\left ( y \\right ) y \\tilde { J } \\left ( d s , d y \\right ) + \\int _ { 0 } ^ { t } \\int _ { \\mathbf { R } _ { 0 } ^ { d } } \\left ( 1 - \\chi _ { \\alpha } \\left ( y \\right ) \\right ) y J \\left ( d s , d y \\right ) , ~ t \\geq 0 , \\end{align*}"}
-{"id": "6452.png", "formula": "\\begin{align*} \\overset { p } { \\underset { t } { ( A , \\phi ) } } : = ( A , t \\phi ) . \\end{align*}"}
-{"id": "5651.png", "formula": "\\begin{align*} E ( t ) = \\frac { 1 } { 2 \\pi \\rm i } \\int _ \\Gamma e ^ { z t } z ^ { \\alpha - 1 } ( z ^ \\alpha + A ) ^ { - 1 } \\d z \\quad \\mbox { a n d } \\bar E ( t ) = \\frac { 1 } { 2 \\pi \\rm i } \\int _ \\Gamma e ^ { z t } z ^ { - \\gamma } ( z ^ \\alpha + A ) ^ { - 1 } \\d z . \\end{align*}"}
-{"id": "5135.png", "formula": "\\begin{align*} E _ { k _ j } : = \\frac { 1 } { 2 i \\gamma _ { k _ j } ( \\lambda + k - i \\gamma _ { k _ j } ) } \\begin{pmatrix} \\lambda + k - i \\gamma _ { k _ j } & \\nu - \\mu \\\\ - ( \\nu + \\mu ) & - ( \\lambda + k - i \\gamma _ { k _ j } ) \\end{pmatrix} ; \\end{align*}"}
-{"id": "2245.png", "formula": "\\begin{align*} & H \\kappa ( x ) = \\kappa ( x ) H = x \\kappa ( x ) & & H p ( x ) = p ( x ) H = x p ( x ) \\\\ & H | \\alpha ( x ) \\rangle = x | \\alpha ( x ) \\rangle & & \\langle \\alpha ' ( x ) | H = x \\langle \\alpha ' ( x ) | \\end{align*}"}
-{"id": "6404.png", "formula": "\\begin{align*} \\left \\Vert r _ { k } \\right \\Vert ^ { 2 } \\phi _ { k } = \\left ( \\sum _ { j = 0 } ^ { k } \\| r _ { j } \\| ^ { - 2 } \\right ) ^ { - 1 } ; \\end{align*}"}
-{"id": "5245.png", "formula": "\\begin{align*} \\lim _ { u \\downarrow 0 } \\frac { F ^ { - 1 } ( 1 - s u ) - F ^ { - 1 } ( 1 - u ) } { F ^ { - 1 } ( 1 - t u ) - F ^ { - 1 } ( 1 - u ) } = \\frac { \\log s } { \\log t } . ) \\end{align*}"}
-{"id": "6945.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } \\mathbf { X } ( x , t ) = \\mathbf { u } ( \\mathbf { X } ( x , t ) , t ) , \\\\ \\mathbf { X } ( x , 0 ) = x . \\end{cases} \\end{align*}"}
-{"id": "9854.png", "formula": "\\begin{align*} b ^ r ( 1 - a b ) & = ( 1 - w ) ^ r [ 1 - ( 1 + w ^ 2 ) ( 1 - w ) ] \\\\ & = ( 1 - w ) ^ r [ w - w ^ 2 + w ^ 3 ] \\\\ & = w ( 1 - w ) ^ r - w ^ 2 ( 1 - w ) ^ { r + 1 } . \\end{align*}"}
-{"id": "5142.png", "formula": "\\begin{align*} ( k _ j + \\lambda ) \\left ( \\Gamma ^ + ( \\psi ) \\right ) _ { m _ j , k _ j } = ( \\mu - \\nu ) \\left ( \\Gamma ^ - ( \\psi ) \\right ) _ { m _ j , k _ j } , \\end{align*}"}
-{"id": "334.png", "formula": "\\begin{align*} M _ { \\partial } \\cdot \\begin{pmatrix} 1 & - m & - ( 2 m + 1 ) \\\\ 0 & 0 & 1 \\\\ 0 & 1 & 2 \\end{pmatrix} = \\begin{pmatrix} k L & 0 & - k L \\\\ 0 & ( 2 m + 1 ) k L & ( 4 m + 2 ) k L \\end{pmatrix} . \\end{align*}"}
-{"id": "3762.png", "formula": "\\begin{align*} \\frac { \\int _ { 0 } ^ { a } \\left | r j _ { l } ( K r ) \\right | ^ { 2 } d r } { \\left | \\int _ { 0 } ^ { a } j _ { l } ( k r ) j _ { l } ( K r ) r ^ 2 d r \\right | ^ 2 } \\geq \\frac { \\int _ { 0 } ^ { a } \\left | r j _ { l } ( K _ { 0 } r ) \\right | ^ { 2 } d r } { \\left | \\int _ { 0 } ^ { a } j _ { l } ( k r ) j _ { l } ( K _ { 0 } r ) r ^ 2 d r \\right | ^ 2 } ; j = 2 . \\end{align*}"}
-{"id": "3470.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d x ^ 2 } \\Re F ( x \\pm i / 4 ; x _ N ( k ) ) = - \\Re \\psi ' ( x + N \\pm i / 4 ) = - \\sum ^ { \\infty } _ { n = 0 } \\frac { ( n + N + x ) ^ 2 - \\frac { 1 } { 1 6 } } { ( ( n + N + x ) ^ 2 + \\frac { 1 } { 1 6 } ) ^ 2 } < 0 . \\end{align*}"}
-{"id": "9750.png", "formula": "\\begin{align*} \\Gamma _ a ( X ) : = C _ { n , a } | X | ^ { - n + 1 - a } . \\end{align*}"}
-{"id": "1643.png", "formula": "\\begin{align*} \\sigma ( \\alpha + i \\beta ) = \\alpha - i \\beta . \\end{align*}"}
-{"id": "1070.png", "formula": "\\begin{align*} & \\frac { n ^ 2 \\alpha _ n ^ 2 } { 8 } - \\frac { n ^ 2 G _ n ^ 2 ( x ) } { 3 2 } = \\frac { n ^ 2 ( 1 + z / \\ln n ) ^ 2 G _ n ( x ) ^ 2 } { 3 2 } - \\frac { n ^ 2 G _ n ^ 2 ( x ) } { 3 2 } \\\\ = & \\frac { ( z / \\ln n ) ( 2 + z / \\ln n ) n ^ 2 G _ n ( x ) ^ 2 } { 3 2 } \\\\ = & ( z / \\ln n ) ( 2 + z / \\ln n ) ( \\ln n + o ( \\ln n ) ) \\to 2 z \\end{align*}"}
-{"id": "8107.png", "formula": "\\begin{align*} W _ { \\rm S } : = \\sum _ { i = 1 } ^ { g } ( \\ell _ i - i ) . \\end{align*}"}
-{"id": "9031.png", "formula": "\\begin{align*} f _ { \\{ 1 \\} , b _ 1 } ( x ) = \\begin{cases} b _ 1 & x = 1 , \\\\ x & \\end{cases} \\end{align*}"}
-{"id": "455.png", "formula": "\\begin{align*} \\ln \\frac { \\mathcal { Z } _ { q , m = 0 } ( \\hat g ) } { \\mathcal { Z } _ { q , m = 0 } ( g ) } = \\frac { \\mathbf { c } _ { \\rm m a t } } { 9 6 \\pi } S ^ { { \\rm c l } , 0 } _ { { \\rm L } } ( \\hat g , g ) + \\frac { q ^ 2 ( 1 - \\textbf { h } ) } { 4 \\pi } S ^ { \\rm c l } _ { \\rm M } ( \\hat g , g ) . \\end{align*}"}
-{"id": "6618.png", "formula": "\\begin{align*} \\prod _ { t = 1 } ^ { \\eta } \\left ( 1 - \\frac { v _ n ^ 2 } { s _ { n t } ^ 2 } \\right ) \\geq \\prod _ { t = 1 } ^ { \\infty } \\left ( 1 - \\frac { ( 1 - 2 \\varepsilon ( - \\varphi ) ^ { - n } ) ^ 2 } { u _ t ^ 2 } \\right ) \\big ( 1 - \\mathcal { O } ( \\eta ^ { - 1 } ) \\big ) \\big ( 1 - \\mathcal { O } ( \\varphi ^ { n / 5 } ) \\big ) . \\end{align*}"}
-{"id": "5439.png", "formula": "\\begin{align*} \\log \\hat { \\rho } _ s ( A ) = \\max _ { \\mu \\in \\mathcal { M } _ f } \\left \\lbrace \\inf _ n \\frac { 1 } { n } \\int \\log V _ s ( A ^ n ( x ) ) d \\mu \\right \\rbrace = \\max _ { \\mu \\in \\mathcal { M } _ f } \\left \\lbrace \\lim _ n \\frac { 1 } { n } \\int \\log \\varphi ^ s _ c ( A ^ n ( x ) ) d \\mu \\right \\rbrace . \\end{align*}"}
-{"id": "713.png", "formula": "\\begin{align*} \\forall z \\in T _ p , \\forall \\psi \\in A u t ( T _ p ) , \\forall v , w \\in \\mathbb { C } ^ 2 \\setminus \\{ 0 \\} , B i s _ { \\psi ( z ) } ( \\partial \\psi _ z ( v ) , \\partial \\psi _ z ( w ) ) = B i s _ z ( v , w ) . \\end{align*}"}
-{"id": "3131.png", "formula": "\\begin{align*} T _ { i j } ( u ) = \\delta _ { i j } \\mathbf { 1 } + \\sum _ { \\ell \\geq 0 } T _ { i j } [ \\ell ] u ^ { - \\ell - 1 } , \\end{align*}"}
-{"id": "656.png", "formula": "\\begin{align*} & ( \\psi ; g _ 1 , \\dots , g _ l ; [ v ] , [ y ] ) \\circ ( \\varphi ; f _ 1 , \\dots , f _ n ; [ u ] , [ x ] ) \\\\ & = ( \\psi \\varphi ^ y ; \\gamma ( g _ 1 ; ( ( v y ) _ \\ast \\vec f ) ^ \\psi _ 1 ) , \\dots , \\gamma ( g _ l ; ( v y ) _ \\ast \\vec f ) ^ \\psi _ l ; [ \\varphi ^ \\ast ( v y ) u \\varphi ^ \\ast ( y ) ^ { - 1 } ] , [ \\varphi ^ \\ast ( y ) x ] ) \\ , \\end{align*}"}
-{"id": "3130.png", "formula": "\\begin{align*} \\left [ T _ { i j } ( u ) , T _ { k l } ( v ) \\right ] = g ( u , v ) \\left ( T _ { i l } ( u ) T _ { k j } ( v ) - T _ { i l } ( v ) T _ { k j } ( u ) \\right ) . \\end{align*}"}
-{"id": "3545.png", "formula": "\\begin{align*} ( \\sigma _ { i } ^ { ( k ) } ) ^ { \\vee } = \\mathrm { C o n e } \\left \\{ \\ell ^ { ( 1 ) } , \\ell ^ { ( 2 ) } , \\ell ^ { ( 3 ) } , \\ell ^ { ( 4 ) } \\right \\} . \\end{align*}"}
-{"id": "7791.png", "formula": "\\begin{align*} ( \\xi \\cdot \\nabla ) ^ { \\alpha } m ^ G ( \\xi ) = \\mathcal D _ t ^ { \\alpha } m ^ G ( t \\xi ) | _ { t = 1 } . \\end{align*}"}
-{"id": "3918.png", "formula": "\\begin{align*} y _ i ( t _ 1 , \\ldots , t _ r , z , v ) : = v \\sum _ { k = 0 } ^ { d - 1 } p _ { i , k } ( t _ 1 , \\ldots , t _ { r } ) z ^ k \\end{align*}"}
-{"id": "4918.png", "formula": "\\begin{align*} \\kappa = w \\otimes w \\ , , \\end{align*}"}
-{"id": "5417.png", "formula": "\\begin{align*} \\| u \\| _ \\mu ^ 2 & = \\frac 1 2 \\| u \\| _ \\mu ^ 2 + \\frac 1 2 \\| u \\| _ \\mu ^ 2 \\\\ & \\geq \\frac 1 2 \\left ( 1 - \\frac { \\mu } { c _ { N , \\alpha } H _ { N , \\alpha } } \\right ) c _ { N , \\alpha } \\iint _ { \\R ^ N \\times \\R ^ N } \\frac { | u ( x ) - u ( y ) | ^ 2 } { | x - y | ^ { N + \\alpha } } \\ , d x \\ , d y + \\frac 1 2 \\int _ { \\R ^ N } V ( x ) u ^ 2 \\ , d x \\\\ & \\geq \\frac 1 2 \\left ( 1 - \\frac { \\mu } { c _ { N , \\alpha } H _ { N , \\alpha } } \\right ) \\| u \\| ^ 2 \\end{align*}"}
-{"id": "3690.png", "formula": "\\begin{align*} \\sum _ { i \\in U _ x } [ E ( y _ i | \\delta _ i ) - \\mu ( x ) ] = \\sum _ { i \\in U _ x } [ \\mu _ i - \\mu ( x ) ] = 0 \\end{align*}"}
-{"id": "9518.png", "formula": "\\begin{align*} ( \\beta - 1 ) \\frac { p _ { s } ^ * } { p } + p _ { s } ^ * = \\theta ( p _ { s } ^ * - 1 ) \\ \\mbox { i . e . } \\ \\beta = p \\theta \\frac { ( p _ { s } ^ * - 1 ) } { p _ { s } ^ * } - ( p - 1 ) . \\end{align*}"}
-{"id": "7503.png", "formula": "\\begin{align*} \\mathcal { E } Z ^ i = \\left ( \\frac { 1 - d } { 2 } + \\frac { s - 2 } { 2 } \\right ) Z ^ i . \\end{align*}"}
-{"id": "1781.png", "formula": "\\begin{align*} \\xi & = P \\left ( Y _ { j } > ( 1 - \\theta + c ' ) ^ { 1 / 2 } \\sqrt { 2 \\log ( 2 ^ { n } ) } \\right ) \\\\ & \\geq \\frac { ( 1 - \\theta + c ' ) ^ { 1 / 2 } \\sqrt { 2 \\log ( 2 ^ { n } ) } } { 1 + 2 ( 1 - \\theta + c ' ) \\log ( 2 ^ { n } ) } \\exp \\left ( - ( 1 - \\theta + c ' ) \\log ( 2 ^ { n } ) \\right ) \\\\ & \\geq C 2 ^ { - n ( 1 - \\theta + c ' ) } \\end{align*}"}
-{"id": "312.png", "formula": "\\begin{align*} h ^ * ( \\alpha + \\beta ) = ( \\alpha + \\beta ) \\circ h = \\triangledown \\circ ( \\alpha \\vee \\beta ) \\circ \\sigma \\circ h , \\end{align*}"}
-{"id": "3530.png", "formula": "\\begin{align*} F _ { 4 } = ( Y _ { 0 } Y _ { s } + Y _ { 2 } Y _ { 3 } - Y _ { 1 } Y _ { 4 } ) ^ { 2 } + 4 Y _ { 0 } Y _ { 1 } Y _ { 4 } Y _ { s } \\end{align*}"}
-{"id": "1276.png", "formula": "\\begin{align*} \\tilde I ( v ) : = I ( v , v , v ) . \\end{align*}"}
-{"id": "9950.png", "formula": "\\begin{align*} u _ k ( x ) = a _ k \\ , ( x _ 1 + i x _ 2 ) ^ k \\end{align*}"}
-{"id": "3210.png", "formula": "\\begin{align*} y ^ { \\ast } ( A ) : = \\sum _ { j = 1 } ^ p \\xi _ j ^ { \\ast } \\delta _ A ( s _ j ) \\ \\ \\ ( A \\in \\mathcal { B } ) , \\end{align*}"}
-{"id": "2784.png", "formula": "\\begin{align*} N ^ t = \\sum _ { r = 0 } ^ \\infty X _ r . \\end{align*}"}
-{"id": "6249.png", "formula": "\\begin{align*} P : = e _ { 1 , 1 } + \\cdots + e _ { a , a } . \\end{align*}"}
-{"id": "4817.png", "formula": "\\begin{align*} \\left \\{ \\begin{matrix} \\mathcal L \\varphi = \\lambda _ 1 ( \\mathcal L ) \\varphi & \\textrm { i n $ M $ } \\\\ \\mathcal B \\varphi = 0 & \\textrm { o n $ \\partial M $ } \\end{matrix} \\right . \\end{align*}"}
-{"id": "5677.png", "formula": "\\begin{align*} X _ 0 : = \\overline { s p a n } \\{ X _ 1 , x _ 1 , x _ 2 , \\dots , x _ k \\} \\ , . \\end{align*}"}
-{"id": "8539.png", "formula": "\\begin{align*} D _ t ^ 2 q = & \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { 2 \\pi } \\frac { D _ t ^ 2 \\zeta - \\ddot { z } _ j ( t ) } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } - \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { \\pi } \\frac { ( D _ t \\zeta ) ^ 2 - 2 D _ t \\zeta \\dot { z } _ j } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 3 } - \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { \\pi } \\frac { \\dot { z } _ j ^ 2 } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 3 } . \\end{align*}"}
-{"id": "9247.png", "formula": "\\begin{align*} 0 \\le \\zeta \\le 1 \\ { \\rm i n } \\ \\bar B _ R ( x _ 0 ) \\cap S _ \\zeta , \\zeta ( x _ 0 ) = 1 , \\end{align*}"}
-{"id": "5259.png", "formula": "\\begin{align*} R ( F _ 1 , x ) = \\left ( \\int _ { x } ^ { u e p ( F ) } \\int _ { u } ^ { u e p ( F ) } 1 - F ( t ) d t d u \\right ) / \\left ( \\int _ { x } ^ { u e p ( F ) } 1 - F ( t ) d t \\right ) . \\end{align*}"}
-{"id": "7305.png", "formula": "\\begin{align*} \\frac { \\delta S } { \\delta \\theta } = & ( - 1 ) ^ { ( n ^ 2 + n ) / 2 } ( n - 2 ) E ^ { n - 3 } ( d \\omega + \\omega ^ 2 ) \\\\ \\frac { \\delta S } { \\delta \\omega } = & ( - 1 ) ^ { ( n ^ 2 - n ) / 2 } E ^ { n - 3 } ( d \\theta + \\theta \\omega + \\omega \\theta ) . \\end{align*}"}
-{"id": "5601.png", "formula": "\\begin{align*} B & = \\left ( i \\int _ { - \\infty } ^ 0 u _ { x x } \\bar { u } d x - i \\alpha \\int _ { - \\infty } ^ 0 | u | ^ 2 v - \\beta i \\int _ { - \\infty } ^ 0 | u | ^ 4 d x \\right ) \\\\ & = - \\int _ { - \\infty } ^ 0 | u _ x | ^ 2 d x - \\alpha \\int _ { - \\infty } ^ 0 | u | ^ 2 v - \\beta \\int _ { - \\infty } ^ 0 | u | ^ 4 d x . \\end{align*}"}
-{"id": "1772.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ e ^ { \\alpha N _ { s , t } ^ { ( n ) } } \\right ] & = \\mathbb { E } \\left [ e ^ { \\alpha \\xi _ { s , t } } \\right ] \\mathbb { E } \\left [ \\exp \\left ( \\alpha \\max _ { 1 \\leq i \\leq n } \\left ( \\omega _ { t _ { i } } - \\omega _ { s } \\right ) \\right ) \\right ] \\\\ & \\leq \\exp \\left ( \\frac { \\alpha ^ { 2 } } { 2 } \\gamma _ { H } ( t - s ) ^ { 2 H } \\right ) \\mathbb { E } \\left [ \\exp \\left ( \\alpha \\sup _ { s \\leq u \\leq t } \\left ( \\omega _ { u } - \\omega _ { s } \\right ) \\right ) \\right ] . \\end{align*}"}
-{"id": "9722.png", "formula": "\\begin{align*} ( u ) : = \\bigg \\{ X _ \\circ \\in \\Gamma ( u ) : \\lim _ { r \\downarrow 0 } \\frac { \\mathcal { H } ^ { n } ( \\Lambda ( u ) \\cap B _ r ( X _ \\circ ) ) } { r ^ n } = 0 \\bigg \\} . \\end{align*}"}
-{"id": "5777.png", "formula": "\\begin{align*} A ( C _ 3 , q , T ) = & ( q - 1 ) \\left ( \\frac { 6 T ^ 3 } { ( 1 - T ) ^ 3 ( 1 - q T ) } + \\frac { 6 T ^ 2 } { ( 1 - T ) ^ 2 ( 1 - q T ) } + \\frac { T } { ( 1 - T ) ( 1 - q T ) } \\right ) + \\frac { 6 T ^ 2 } { ( 1 - T ) ^ 3 } + \\frac { 3 T } { ( 1 - T ) ^ 2 } \\\\ = & ( q - 1 ) \\frac { T ( T ^ 2 + 4 T + 1 ) } { ( 1 - T ) ^ 3 ( 1 - q T ) } + \\frac { 3 T ( T + 1 ) } { ( 1 - T ) ^ 3 } \\\\ = & \\frac { ( 2 q + 1 ) T ^ 2 + ( q + 2 ) T } { ( 1 - T ) ^ 2 ( 1 - q T ) } . \\end{align*}"}
-{"id": "9234.png", "formula": "\\begin{align*} & \\alpha _ { 2 : s } ( \\alpha ^ * ) = u + x _ 0 ^ j v + \\alpha ^ * w . \\\\ & \\alpha _ 1 ( \\alpha ^ * ) = 1 - \\alpha ^ * - 1 ^ T ( u + x _ 0 ^ j v + \\alpha ^ * w ) , \\end{align*}"}
-{"id": "4411.png", "formula": "\\begin{align*} F | _ { C _ 2 } = \\tilde F | _ { C _ 2 \\times \\{ 1 \\} } = g | _ { C _ 2 \\times \\{ 1 \\} } = f , \\end{align*}"}
-{"id": "4784.png", "formula": "\\begin{align*} i \\hbar \\partial _ t \\psi = H \\psi \\ , . \\end{align*}"}
-{"id": "9070.png", "formula": "\\begin{align*} & \\tilde { F } _ x ( y ) = \\sum _ { k \\in \\mathcal { K } } \\sum _ { \\xi \\in \\mathcal { E } ^ { ( k ) } } a _ { \\xi } e ( \\langle \\xi , x \\rangle ) e \\left ( \\left \\langle \\frac { \\xi } { \\sqrt { E } } - \\zeta ^ k , R y \\right \\rangle \\right ) e ( \\zeta ^ k , R y ) \\\\ & \\tilde { \\psi } _ x ( y ) = F _ x ( y ) - \\tilde { F } _ x ( y ) . \\end{align*}"}
-{"id": "3359.png", "formula": "\\begin{align*} ( h f ) ( v ) = f \\left ( S ( h ) v \\right ) , \\ h \\in H , \\ f \\in V ^ * , \\ v \\in V . \\end{align*}"}
-{"id": "8938.png", "formula": "\\begin{align*} B _ { { q p } _ { b l } } ( 0 ; \\epsilon ) & = \\{ y \\in X : q p _ { b l } ( 0 , y ) < q p _ { b l } ( 0 , 0 ) + \\epsilon \\thinspace \\mbox { a n d } \\thinspace q p _ { b l } ( y , 0 ) < q p _ { b l } ( 0 , 0 ) + \\epsilon \\} \\\\ & = \\left \\{ \\begin{array} { l l } \\{ 0 \\} , & \\mbox { i f } \\thinspace \\thinspace \\epsilon \\leq 1 , \\\\ \\{ 0 , 1 , 2 \\} , & \\mbox { i f } \\thinspace \\thinspace \\epsilon > 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "7676.png", "formula": "\\begin{align*} \\partial v = \\left \\{ \\begin{array} { l l } 0 & \\mbox { i f } v \\in V ^ + , \\\\ 1 & \\mbox { i f } v \\in V ^ - . \\end{array} \\right . \\end{align*}"}
-{"id": "152.png", "formula": "\\begin{align*} c _ i \\circ q _ M \\textrm { a n d } l _ { d c _ i } = \\sum _ { s = 1 } ^ { N } \\dfrac { \\partial c _ i } { \\partial x ^ s } ( { \\bf x } ) y ^ s , i = 1 , \\dots r , \\end{align*}"}
-{"id": "3726.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\int _ { | u _ k | \\geq M } \\frac { F ( y , u _ k ) } { | x - y | ^ { \\mu } } F ( x , u ) d y ~ d x = o ( M ) \\ \\ M \\to \\infty . \\end{align*}"}
-{"id": "6142.png", "formula": "\\begin{align*} 7 L _ 3 - 1 0 \\delta \\qquad \\textnormal { a n d } 1 3 L _ 3 - 2 0 \\delta = \\iota ^ * ( 7 L _ 3 - 1 0 \\delta ) . \\end{align*}"}
-{"id": "7382.png", "formula": "\\begin{align*} \\begin{bmatrix} p ^ t & \\gamma \\end{bmatrix} \\begin{bmatrix} D f ( x , \\lambda _ 1 , \\lambda _ 2 ) & \\bar { p } \\\\ \\bar { q } ^ t & 0 \\end{bmatrix} & = \\begin{bmatrix} 0 & 1 \\end{bmatrix} , & \\begin{bmatrix} D f ( x , \\lambda _ 1 , \\lambda _ 2 ) & \\bar { p } \\\\ \\bar { q } ^ t & 0 \\end{bmatrix} \\begin{bmatrix} q \\\\ \\gamma \\end{bmatrix} & = \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "7980.png", "formula": "\\begin{align*} \\left | ^ 3 D _ 4 ( q ) \\right | = q ^ { 1 2 } ( q - 1 ) ^ 2 ( q + 1 ) ^ 2 ( q ^ 2 - q + 1 ) ^ 2 ( q ^ 2 + q + 1 ) ^ 2 ( q ^ 4 - q ^ 2 + 1 ) \\ , . \\end{align*}"}
-{"id": "757.png", "formula": "\\begin{align*} R = U M _ { \\sqrt { \\mu } } V ^ * , R ^ * = V M _ { \\sqrt { \\mu } } U ^ * , \\end{align*}"}
-{"id": "7833.png", "formula": "\\begin{align*} \\lim _ n [ \\langle x _ n , x \\rangle \\langle x , y \\rangle \\langle y , x _ n \\rangle \\xi _ n , \\xi _ n ] = \\lambda \\| x \\| ^ 2 \\| y \\| ^ 2 \\end{align*}"}
-{"id": "5763.png", "formula": "\\begin{align*} X _ { \\mathcal { R } / A } \\times _ A K = X _ { \\mathcal { R } ^ \\varphi / K } \\end{align*}"}
-{"id": "2768.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\mathbb { P } \\Big ( 2 ^ { - \\frac { 4 } { 3 } } N ^ { - \\frac { 1 } { 3 } } \\big ( \\lambda _ { \\mathrm { m a x } } ( H ^ 2 ) - 4 N \\big ) \\leq x \\Big ) = F _ { \\mathrm { G U E } } ( 2 ^ { - \\frac { 2 } { 3 } } x ) F _ { \\mathrm { G U E } } ( 2 ^ { - \\frac { 2 } { 3 } } x ) . \\end{align*}"}
-{"id": "7035.png", "formula": "\\begin{align*} \\delta = \\frac { - \\left ( \\frac { 2 } { \\lambda } - 2 \\zeta + 1 \\right ) + \\sqrt { \\left ( \\frac { 2 } { \\lambda } - 2 \\zeta + 1 \\right ) ^ { 2 } + 8 \\zeta } } { 4 \\zeta } , 0 \\leq \\delta < 1 . \\end{align*}"}
-{"id": "7570.png", "formula": "\\begin{align*} \\tau \\circ X ^ { - 1 } ( x , s ) - \\tau \\circ X ^ { - 1 } ( x , t ) = \\Delta _ 1 \\tau ( x , s , t ) + \\Delta _ 2 \\tau ( x , s , t ) , \\end{align*}"}
-{"id": "1452.png", "formula": "\\begin{align*} \\mu ( t , \\cdot ) = \\mu _ { 0 } + \\int _ { 0 } ^ { t } \\left [ \\Delta \\mu ( \\tau , \\cdot ) - \\varepsilon \\mathrm { d i v } \\left ( \\left ( \\bar { m } + \\mu ( \\tau , \\cdot ) \\right ) \\Theta _ { p } ( \\tau , \\cdot , \\mu , D w ) \\right ) \\right ] \\ d \\tau , \\end{align*}"}
-{"id": "3802.png", "formula": "\\begin{align*} Q = t ^ \\varepsilon \\left ( \\frac { N } { K } \\right ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "4156.png", "formula": "\\begin{align*} D _ 1 ( z ) = \\begin{cases} \\begin{pmatrix} e ^ { - 3 \\omega z ^ { 1 / 3 } } & 0 & 0 \\\\ 0 & e ^ { - 3 \\omega ^ 2 z ^ { 1 / 3 } } & 0 \\\\ 0 & 0 & e ^ { - 3 z ^ { 1 / 3 } } \\end{pmatrix} , & \\Im z > 0 , \\\\ \\begin{pmatrix} e ^ { - 3 \\omega ^ 2 z ^ { 1 / 3 } } & 0 & 0 \\\\ 0 & e ^ { - 3 \\omega z ^ { 1 / 3 } } & 0 \\\\ 0 & 0 & e ^ { - 3 z ^ { 1 / 3 } } \\end{pmatrix} , & \\Im z < 0 , \\end{cases} \\end{align*}"}
-{"id": "9052.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\frac { 1 } { \\pi R ^ 2 } \\int _ { B ( R ) } ( F _ { \\mu _ A } \\cdot F _ { \\mu _ B } ) d x \\right ] = 0 \\end{align*}"}
-{"id": "6151.png", "formula": "\\begin{align*} R ^ { \\omega , \\nu } f _ r ( z ) = \\int _ { \\mathbb { D } } f _ r ( \\xi ) \\overline { B _ z ^ \\nu ( \\xi ) } \\omega ( \\xi ) d A ( \\xi ) . \\end{align*}"}
-{"id": "6912.png", "formula": "\\begin{align*} f = \\sum _ { i = 1 } ^ 6 f _ i g _ i \\ , , \\end{align*}"}
-{"id": "6405.png", "formula": "\\begin{align*} \\mu \\gamma _ { k + 1 } ^ { { \\scriptscriptstyle ( \\mu ) } } = \\frac { \\mu \\left ( \\gamma _ { k } ^ { { \\scriptscriptstyle ( \\mu ) } } - \\gamma _ { k } \\right ) } { \\mu \\left ( \\gamma _ { k } ^ { { \\scriptscriptstyle ( \\mu ) } } - \\gamma _ { k } \\right ) + \\delta _ { k + 1 } } < \\frac { \\mu \\gamma _ { k } ^ { { \\scriptscriptstyle ( \\mu ) } } } { \\mu \\gamma _ { k } ^ { { \\scriptscriptstyle ( \\mu ) } } + \\delta _ { k + 1 } } \\leq \\frac { \\phi _ { k } } { \\phi _ { k } + \\delta _ { k + 1 } } = \\phi _ { k + 1 } . \\end{align*}"}
-{"id": "6558.png", "formula": "\\begin{align*} m ^ P _ \\beta = \\# \\left \\{ \\right \\} + 2 \\cdot \\# \\left \\{ \\right \\} = e ( S ' ) = e ( S ) - \\eta . \\end{align*}"}
-{"id": "2556.png", "formula": "\\begin{align*} Y _ t : = \\int _ { ( 0 , t ] \\times \\mathbb W } w _ { t - s } \\mathbf n ( d s , d w ) , t \\geq 0 . \\end{align*}"}
-{"id": "4069.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { ( c _ V n ) ^ 3 } K ^ { \\alpha , \\frac { 1 } { 2 } } _ { V , n } \\left ( \\frac { x } { ( c _ V n ) ^ 3 } , \\frac { y } { ( c _ V n ) ^ 3 } \\right ) = \\mathbb K ^ { ( \\alpha , \\frac { 1 } { 2 } ) } ( x , y ) \\end{align*}"}
-{"id": "1791.png", "formula": "\\begin{align*} \\begin{aligned} P \\left ( \\bigcap _ { j = 1 } ^ { N } \\{ | X _ { j } | \\leq \\alpha _ { j } + c \\} \\right ) & = 2 ^ { N } \\int _ { 0 } ^ { \\alpha _ { N } + c } \\cdots \\int _ { 0 } ^ { \\alpha _ { 1 } + c } \\frac { 1 } { \\sqrt { 2 \\pi | \\Sigma | } } \\exp \\left ( - \\frac { 1 } { 2 } \\bold { x } ^ { T } \\Sigma ^ { - 1 } \\bold { x } \\right ) d x _ { 1 } \\cdots d x _ { N } \\\\ & \\leq 2 ^ { N } \\frac { 1 } { \\sqrt { 2 \\pi | \\Sigma | } } \\prod _ { j = 1 } ^ { N } ( \\alpha _ { j } + c ) \\\\ & = O ( n ^ { - N ( 1 - H ) } ) , \\end{aligned} \\end{align*}"}
-{"id": "4009.png", "formula": "\\begin{align*} T _ f ( Y , M ) & : = \\underset { n \\ge 1 , \\ , ( n , M ) = 1 } { \\sum } | a ' ( f , n ) | ^ { 2 } e ^ { - n / Y } \\end{align*}"}
-{"id": "8330.png", "formula": "\\begin{align*} F _ z \\circ Z Z _ { \\alpha } D _ t Z + F _ t \\circ Z Z _ { \\alpha } + \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i D _ t Z Z _ { \\alpha } - \\lambda _ j i \\dot { z } _ j ( t ) Z _ { \\alpha } } { 2 \\pi ( Z ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } + i A _ 1 = i Z _ { \\alpha } . \\end{align*}"}
-{"id": "3205.png", "formula": "\\begin{align*} \\| X _ r \\circ Y _ r - \\mu _ { r } I \\| _ F & = \\left \\| \\begin{pmatrix} D _ { X _ { r } } & O \\\\ O & E _ { X _ { r } } \\end{pmatrix} \\circ \\tilde { Y } _ r - \\mu _ r I \\right \\| _ F \\\\ & \\ge \\sqrt { \\sum _ { i = p + 1 } ^ { m } ( e ^ r _ { i } \\tilde { y } _ { i i } ^ r - \\mu _ r ) ^ 2 } , \\end{align*}"}
-{"id": "6671.png", "formula": "\\begin{align*} H _ { - } = A ^ { \\dagger } A , H _ { + } = A A ^ { \\dagger } , \\end{align*}"}
-{"id": "728.png", "formula": "\\begin{align*} \\mathbb { E } _ { f } \\ ! \\left [ T ^ { k } \\right ] = \\frac { \\Gamma ( m n ) } { ( q - 1 ) ^ { k } } \\sum _ { i = 0 } ^ { k } \\binom { k } { i } \\frac { ( - 1 ) ^ { i } } { \\Gamma ( m n + q i ) } \\mathbb { E } _ { g } \\ ! \\left [ L ^ { i } \\right ] . \\end{align*}"}
-{"id": "7381.png", "formula": "\\begin{align*} f ( x , \\lambda _ 1 , \\lambda _ 2 ) = 0 , \\ D _ x f ( x , \\lambda _ 1 , \\lambda _ 2 ) q = 0 , \\end{align*}"}
-{"id": "3367.png", "formula": "\\begin{align*} d _ i D _ k \\ , = \\ , \\sum _ { j = 0 } ^ 2 \\ , N _ { i j } ^ k \\ , D _ j 0 \\le i , k \\le 2 . \\end{align*}"}
-{"id": "8030.png", "formula": "\\begin{align*} e _ q ( x _ 1 , x _ 2 , \\dots , x _ { 2 n / k } ) & = \\sum _ { 1 \\leq j _ 1 < j _ 2 < \\ldots < j _ q \\leq 2 n / k } x _ { j _ 1 } x _ { j _ 2 } \\cdots x _ { j _ q } . \\end{align*}"}
-{"id": "1593.png", "formula": "\\begin{align*} \\Phi _ { v ^ o _ n } ( z ^ o _ n , x ^ o _ n ) = \\frac { k _ n } { 2 n } + o \\left ( \\frac { k _ n } { n } \\right ) + 1 - \\frac { k _ n } { n } + o \\left ( \\frac { 1 } { n } \\right ) = 1 - \\frac { k _ n } { 2 n } + o \\left ( \\frac { k _ n } { n } \\right ) . \\end{align*}"}
-{"id": "8761.png", "formula": "\\begin{align*} R _ { h i j k } \\ = \\ g _ { h s } R ^ { s } _ { \\ , i j k } \\ = \\ g _ { h s } ( \\partial _ k \\Gamma ^ { s } _ { i j } - \\partial _ j \\Gamma ^ { s } _ { i k } + \\Gamma ^ { r } _ { i j } \\Gamma ^ { s } _ { r k } - \\Gamma ^ { r } _ { i k } \\Gamma ^ { s } _ { r j } ) \\end{align*}"}
-{"id": "8095.png", "formula": "\\begin{align*} \\rho _ n = - e ^ { - 1 } \\Gamma ( n ) M ( n + 1 , 2 , 1 ) U ( n , 0 , 1 ) . \\end{align*}"}
-{"id": "7194.png", "formula": "\\begin{gather*} \\mu ( \\epsilon ) : = \\max _ { \\left \\Vert q \\right \\Vert \\le \\epsilon } \\left [ \\left ( \\intop _ { 0 } ^ { 1 } \\left \\Vert \\xi ^ { \\prime } ( s q ) - \\xi ^ { \\prime } ( 0 ) \\right \\Vert \\mathrm { d } s \\right ) ^ { 2 } + \\left \\Vert \\nu ( q ) - \\nu ( 0 ) \\right \\Vert ^ { 2 } \\right ] ^ { 1 / 2 } \\to 0 , \\quad \\epsilon \\to 0 , \\end{gather*}"}
-{"id": "1706.png", "formula": "\\begin{align*} \\chi _ { E _ { \\alpha + ( 1 / n ) } } ( x ) = P ( x , E _ { \\alpha + ( 1 / n ) } ) \\ \\ \\mu \\mbox { - a . e . } x \\in X . \\end{align*}"}
-{"id": "325.png", "formula": "\\begin{align*} a ( \\tilde { \\alpha } _ { i , k } ) = k u ^ 2 \\otimes \\Sigma ^ { - 2 } c _ { n - 1 } ( \\alpha _ i ) \\oplus k u ^ 2 \\otimes \\Sigma ^ { - 2 } c _ { n } ( \\alpha _ i ) . \\end{align*}"}
-{"id": "9149.png", "formula": "\\begin{align*} \\overline { \\Theta } ^ { ( N ) } ( \\overline { \\mathbf { r } } ) = 1 \\end{align*}"}
-{"id": "6152.png", "formula": "\\begin{align*} R ^ { \\omega , \\nu } f ( z ) = \\int _ { \\mathbb { D } } f ( \\xi ) \\overline { B _ z ^ \\nu ( \\xi ) } \\omega ( \\xi ) d A ( \\xi ) = \\sum _ { k = 0 } ^ \\infty \\left ( \\frac { \\omega _ k } { \\nu _ k } \\right ) f _ k z ^ k . \\end{align*}"}
-{"id": "7007.png", "formula": "\\begin{align*} a _ j ' & = \\cos ^ \\ell ( \\theta _ j ) \\frac { 1 6 \\sin ^ 4 ( \\theta _ j ) } { n ^ 2 } \\left ( 1 + O \\left ( \\frac { 1 } { n } \\right ) \\right ) \\\\ a _ j & = \\cos ^ \\ell ( \\theta _ j ) \\frac { 1 6 \\sin ^ 4 ( \\theta _ j ) } { n ^ 2 } \\left ( \\frac { \\ell } { \\cos ( \\theta _ j ) } + O \\left ( n ^ 3 \\sin ^ 3 ( \\theta _ j ) \\right ) \\right ) \\left ( 1 + O \\left ( \\frac { 1 } { n } \\right ) \\right ) \\end{align*}"}
-{"id": "838.png", "formula": "\\begin{align*} A _ D \\begin{pmatrix} a _ 1 ^ q \\\\ \\vdots \\\\ a _ g ^ q \\end{pmatrix} = A _ D ^ q \\begin{pmatrix} a _ 1 ^ q \\\\ \\vdots \\\\ a _ g ^ q \\end{pmatrix} \\Leftrightarrow A _ D ^ { q ^ { d - 1 } } \\begin{pmatrix} a _ 1 \\\\ \\vdots \\\\ a _ g \\end{pmatrix} = A _ D \\begin{pmatrix} a _ 1 \\\\ \\vdots \\\\ a _ g \\end{pmatrix} . \\end{align*}"}
-{"id": "8364.png", "formula": "\\begin{align*} Z ( \\alpha , t ) - z _ j ( t ) = \\Phi ^ { - 1 } ( \\alpha , t ) - \\Phi ^ { - 1 } ( \\omega _ 0 ^ j ) = \\Phi ^ { - 1 } _ z ( z ' ) ( \\alpha - \\omega _ 0 ^ j ) \\end{align*}"}
-{"id": "1198.png", "formula": "\\begin{align*} \\lambda = \\lambda ( \\zeta _ 1 , x , z ' ) = - i \\sin \\theta \\zeta _ 1 + \\cos \\theta \\sqrt { 1 + \\zeta _ 1 ^ 2 } . \\end{align*}"}
-{"id": "2382.png", "formula": "\\begin{align*} h _ 1 ( x _ 1 ) & = x _ 1 \\otimes 1 \\cdots \\otimes 1 , & \\cdots & , & h _ k ( x _ k ) & = 1 \\otimes \\cdots \\otimes 1 \\otimes x _ k \\end{align*}"}
-{"id": "2300.png", "formula": "\\begin{align*} \\mu ( x ) & = \\frac { 1 } { 2 \\pi \\i } ( R ( x - \\i 0 ) - R ( x + \\i 0 ) ) = | \\alpha _ x ) ( \\beta _ x | \\\\ | \\alpha _ x ) & = ( x ^ 2 + \\pi ^ 2 ) ^ { - 1 / 2 } \\left ( \\binom { 1 } { \\frac { \\mathcal { P } } { x - \\Omega } | E ) } + x \\binom { 0 } { | \\delta _ x ) } \\right ) \\\\ ( \\beta _ x | & = ( x ^ 2 + \\pi ^ 2 ) ^ { - 1 / 2 } \\bigg ( - ( 1 , ( E | \\frac { \\mathcal { P } } { x - \\Omega } ) + x ( 0 , ( \\delta _ x | ) \\bigg ) \\end{align*}"}
-{"id": "6391.png", "formula": "\\begin{align*} A V _ { k } = V _ { k } T _ { k } + \\widetilde { \\beta } _ { k } v _ { k + 1 } e _ { k } ^ { T } \\end{align*}"}
-{"id": "2563.png", "formula": "\\begin{align*} \\inf _ { x \\in E } \\mathbf P _ { \\delta _ x } ( \\| X _ t \\| = 0 ) > 0 , \\end{align*}"}
-{"id": "8198.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\frac { 1 } { n } \\sum _ { l = 1 } ^ { n } \\sum _ { k = 1 } ^ { N _ { p } - 1 } X ^ { k l } \\geq \\sum _ { k = 1 } ^ { N _ { p } - 1 } \\liminf _ { n \\to \\infty } \\frac { 1 } { n } \\sum _ { l = 1 } ^ { n } X ^ { k l } = p N _ p - p . \\end{align*}"}
-{"id": "4555.png", "formula": "\\begin{align*} \\mu _ k \\odot \\mu ' _ k & = m a x \\left ( c h _ q ( M _ k ) \\circ c h _ q ( M ' _ k ) \\right ) \\\\ & = m a x \\left ( m a x \\left ( c h _ q ( L ( \\mu _ { + } ) ) \\right ) , m a x \\left ( c h _ q ( L ( \\mu _ { - } ) ) \\right ) \\right ) . \\end{align*}"}
-{"id": "9929.png", "formula": "\\begin{align*} \\frac { d \\pi _ { n } ^ { \\mu } } { d \\pi _ { n } ^ { \\nu } } ( x ) & = \\frac { E ^ { \\nu } [ \\frac { d \\mu } { d \\nu } ( X _ { 0 } ) | Y _ { [ 0 , n ] } , X _ { n } = x ] } { E ^ { \\nu } [ \\frac { d \\mu } { d \\nu } ( X _ { 0 } ) | Y _ { [ 0 , n ] } ] } ~ ~ ~ ~ ~ P ^ { \\mu } ~ a . s . \\end{align*}"}
-{"id": "9363.png", "formula": "\\begin{align*} j _ 0 = \\begin{cases} 2 ^ { k - 1 } - \\frac { m } { 2 } & \\\\ 2 ^ { k - 1 } + 1 - \\frac { m } { 2 } & \\end{cases} \\end{align*}"}
-{"id": "1030.png", "formula": "\\begin{align*} & \\mathbb { E } \\sum _ { i _ 1 , \\cdots , i _ { k } \\ } \\prod _ { j = 1 } ^ k ( \\tau _ { i _ j } - x _ j ) _ + \\\\ = & \\mathbb { E } ( n / 4 ) ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } | \\Sigma _ k ( F _ n ( x _ 1 ) , \\cdots , F _ n ( x _ k ) ) | \\\\ = & \\int _ { [ 0 , 2 \\pi ) ^ k } \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) d y _ 1 \\cdots d y _ k . \\end{align*}"}
-{"id": "4177.png", "formula": "\\begin{align*} E _ n ^ { - 1 } ( z _ j ) E _ n ( z _ { j + 1 } ) = \\mathcal { O } ( n ) \\end{align*}"}
-{"id": "6358.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } X ^ 1 = \\{ 0 \\} _ { \\mu _ 1 } \\times \\{ 0 , 1 \\} ^ { \\Lambda - \\{ \\mu _ 1 \\} } , \\\\ X ^ 2 = \\{ 1 \\} _ { \\mu _ 1 } \\times \\{ 0 \\} _ { \\mu _ 2 } \\times \\{ 0 , 1 \\} ^ { \\Lambda - \\{ \\mu _ 1 , \\mu _ 2 \\} } , \\\\ X ^ 3 = \\{ 1 \\} _ { \\mu _ 1 } \\times \\{ 1 \\} _ { \\mu _ 2 } \\times \\{ 0 , 1 \\} ^ { \\Lambda - \\{ \\mu _ 1 , \\mu _ 2 \\} } , \\end{array} \\right . \\end{align*}"}
-{"id": "5858.png", "formula": "\\begin{align*} h ( x ) = \\sum _ { i = 1 } ^ k \\sum _ { j = 1 } ^ { d _ i } h ( \\alpha _ { i j } ) \\frac { \\prod _ { ( k , l ) \\neq ( i , j ) } ( x - \\alpha _ { k l } ) } { \\prod _ { ( k , l ) \\neq ( i , j ) } ( \\alpha _ { i j } - \\alpha _ { k l } ) } \\Rightarrow a = \\sum _ { i = 1 } ^ k \\sum _ { j = 1 } ^ { d _ i } \\frac { h ( \\alpha _ { i j } ) } { G ' ( \\alpha _ { i j } ) } \\end{align*}"}
-{"id": "9709.png", "formula": "\\begin{align*} T _ { \\textbf { d } } ( \\partial S ) = 0 \\neq - \\chi ( S ) = - \\chi ( S _ c ) + b = - 2 . \\end{align*}"}
-{"id": "6691.png", "formula": "\\begin{align*} \\int d x \\ , \\phi _ { n } ^ { * } ( x ) \\phi _ { n ' } ( x ) = - \\frac { i } { \\omega } \\oint \\frac { d z } { z ^ { 1 + ( n - n ' ) } } = \\frac { 2 \\pi } { \\omega } \\delta _ { n n ' } , \\end{align*}"}
-{"id": "3585.png", "formula": "\\begin{align*} V = V _ { \\mathcal { F } } ^ { ( a ) } = \\bigoplus _ { \\lambda \\in \\mathcal { F } } V ^ { ( a ) } _ { \\lambda } \\end{align*}"}
-{"id": "2500.png", "formula": "\\begin{align*} { \\rm d i m } \\ , \\frak { a u t } _ { C R } ( \\mathbb M _ \\rho ) = { \\rm d i m } \\ , \\frak g _ - + 2 \\ , n ^ 2 + { \\rm d i m } \\ , \\frak g _ + . \\end{align*}"}
-{"id": "8357.png", "formula": "\\begin{align*} \\dot { z } _ 2 = \\frac { \\lambda i } { 2 \\pi } \\frac { 1 } { \\overline { z _ 2 - z _ 1 } } = \\frac { \\lambda i } { 4 \\pi } \\frac { 1 } { x } . \\end{align*}"}
-{"id": "8463.png", "formula": "\\begin{align*} \\| F _ { \\zeta } ( \\zeta ( \\alpha , t ) , t ) \\| _ { \\infty } = \\norm { \\frac { \\partial _ { \\alpha } \\mathfrak { F } ( \\alpha , t ) } { \\zeta _ { \\alpha } } } _ { \\infty } \\leq \\frac { \\| \\mathfrak { F } \\| _ { H ^ s } } { \\| \\zeta _ { \\alpha } \\| _ { \\infty } } \\leq \\frac { 5 \\epsilon } { 1 - 5 \\epsilon } \\leq 6 \\epsilon . \\end{align*}"}
-{"id": "1530.png", "formula": "\\begin{align*} u _ 2 ( x , t ) = \\int _ { 0 } ^ { t } { u _ s ( x , t - s ) d s } , \\end{align*}"}
-{"id": "5738.png", "formula": "\\begin{align*} f ( x , y , t ) & = \\sum _ { m , n \\in \\N } F _ { m , n } ( t ) J _ 0 ( \\gamma _ m x ) \\sin ( n \\pi y ) , \\\\ u ( x , y , t ) & = \\sum _ { m , n \\in \\N } T _ { m , n } ( t ) J _ 0 ( \\gamma _ m x ) \\sin ( n \\pi y ) . \\end{align*}"}
-{"id": "7523.png", "formula": "\\begin{align*} c ( x , y ) : = \\log \\left ( 1 + \\sum _ { i = 1 } ^ { n - 1 } e ^ { x ^ i - y ^ i } \\right ) - \\log ( n ) - \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n - 1 } x ^ i - y ^ i . \\end{align*}"}
-{"id": "9140.png", "formula": "\\begin{align*} \\lim _ { \\tau \\rightarrow + \\infty } I _ { M } ( \\rho _ { 1 } ^ { ( N ) } ( t ) , \\mathbf { b } ) = I _ { M } ( \\rho _ { 1 \\infty } ^ { ( N ) } ( \\mathbf { x } _ { 1 } ) , \\mathbf { b } ) = 0 . \\end{align*}"}
-{"id": "9302.png", "formula": "\\begin{align*} \\tilde \\Theta ( \\hat x , \\hat y ) = & - \\tau K \\eta ( \\hat y ) \\{ F ^ { i j } [ D _ { i j } \\underline u ( \\hat y ) - Y _ { i j } ^ u + K ( D _ i \\underline u ( \\hat y ) - p ^ u _ i ) ( D _ j \\underline u ( \\hat y ) - p ^ u _ j ) ] \\\\ & - [ F ^ { i j } D _ { p _ k } A _ { i j } ( \\hat y , u ( \\hat y ) , p ^ u ) + D _ { p _ k } B ( \\hat y , u ( \\hat y ) , p ^ u ) ] ( D _ k \\underline u ( \\hat y ) - p ^ u _ k ) ) \\} . \\end{align*}"}
-{"id": "9467.png", "formula": "\\begin{align*} \\| u \\| _ { s , m } = \\| u \\| _ { W ^ { s , m } ( \\Omega ) } = \\| u \\| _ { L ^ { m } ( \\Omega ) } + [ u ] _ { s , m , \\Omega } , \\end{align*}"}
-{"id": "8901.png", "formula": "\\begin{align*} T _ { } & = T ^ { \\hat x } ( \\hat C ) - T ^ { \\hat x } ( \\hat C ' ) \\\\ T _ { } & = T ^ { \\hat x } ( \\hat C ) - T ^ { \\hat x } ( \\hat D ) , \\end{align*}"}
-{"id": "9076.png", "formula": "\\begin{align*} \\mu _ { f } ( A ) = \\sum _ { \\xi \\in A } | a _ { \\xi } | ^ 2 . \\end{align*}"}
-{"id": "2590.png", "formula": "\\begin{align*} G ( \\theta ) = \\int _ 0 ^ \\theta e ^ { - \\frac { 1 } { \\gamma _ 0 - 1 } J _ G ( r ) } d r , \\theta \\geq 0 , \\end{align*}"}
-{"id": "722.png", "formula": "\\begin{align*} S = - \\sum _ { i = 1 } ^ { m } \\lambda _ { i } \\ln \\lambda _ { i } , \\end{align*}"}
-{"id": "8412.png", "formula": "\\begin{align*} & \\frac { d } { d t } \\Big ( d _ P ( t ) ^ { - 1 } + d _ I ( t ) ^ { - 1 } + \\mathcal { E } ( t ) \\Big ) \\leq C , \\end{align*}"}
-{"id": "4763.png", "formula": "\\begin{align*} ( \\tilde { u } ( t ) , \\varphi ( t ) ) _ { \\mathcal { H } } - ( \\tilde { u } _ 0 , \\varphi ( 0 ) ) _ { \\mathcal { H } } = \\int _ 0 ^ t \\Big [ & ( \\tilde { u } , \\partial _ t \\varphi ) _ { \\mathcal { H } } + 2 \\nu a ( \\tilde { u } , \\varphi ) + 2 \\beta \\nu a ( H , \\varphi ) \\\\ & - b ( \\tilde { u } , \\varphi , \\tilde { u } ) - \\beta b ( H , \\varphi , \\tilde { u } ) - \\beta b ( \\tilde { u } , \\varphi , H ) \\Big ] d t . \\end{align*}"}
-{"id": "9808.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } L _ a \\bar { v } & = & 0 & \\R ^ { n + 1 } \\setminus \\{ x _ n = y = 0 \\} \\\\ \\bar { v } ( x ' , 0 , 0 ) & = & v ( x ' ) & \\R ^ { n - 1 } \\\\ \\lim _ { | X | \\to \\infty } \\bar { v } ( X ) & = & 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "3905.png", "formula": "\\begin{align*} h _ m g h _ m ^ { - 1 } = ( a _ { i , j } ^ { ( m ) } ) = { \\begin{pmatrix} a _ { 1 , 1 } ^ { ( m ) } & \\beta ^ { ( m ) } \\\\ \\alpha ^ { ( m ) } & A ^ { ( m ) } \\end{pmatrix} } . \\end{align*}"}
-{"id": "5177.png", "formula": "\\begin{align*} F ^ { - 1 } ( 1 - u ) = d + s ( u ) + \\int _ { u } ^ { 1 } \\frac { s ( t ) } { t } d t , 0 < u < 1 , \\end{align*}"}
-{"id": "7916.png", "formula": "\\begin{align*} R _ N = \\sum _ { i = 1 } ^ { M } \\hat { R } _ n h ^ { n + 2 } f ^ { ( n + 1 ) } ( \\xi _ i ) \\end{align*}"}
-{"id": "9777.png", "formula": "\\begin{align*} q _ { Z _ \\infty } = q . \\end{align*}"}
-{"id": "6614.png", "formula": "\\begin{align*} s _ { n t } = \\pi \\varphi ^ { n } ( u _ t + \\mathcal { O } ( \\varphi ^ { n / 5 } ) ) , \\end{align*}"}
-{"id": "818.png", "formula": "\\begin{align*} U _ 3 ^ { q ^ d } - U _ 3 = - { \\frac { U _ 2 } { T - \\rho } } . \\end{align*}"}
-{"id": "2215.png", "formula": "\\begin{align*} g ( t ) & = \\alpha c m \\int _ { t } ^ { \\infty } e ^ { ( a - \\frac { p b ^ 2 } { n } ) ( \\tau - t ) } s i n ( \\omega \\tau ) d \\tau \\\\ & = \\beta \\left [ - ( a n - { p b ^ 2 } ) s i n ( \\omega t ) + n \\omega c o s ( \\omega t ) \\right ] \\end{align*}"}
-{"id": "7511.png", "formula": "\\begin{align*} p _ { t } ( x , E ) = \\left \\{ \\begin{array} { l l } Z _ { t } ( x ) \\ast 1 _ { E } ( x ) & t > 0 x \\in \\mathbf { \\mathbb { Q } } _ { p } ^ { n } \\\\ & \\\\ 1 _ { E } ( x ) , & t = 0 x \\in \\mathbf { \\mathbb { Q } } _ { p } ^ { n } . \\end{array} \\right . \\end{align*}"}
-{"id": "3265.png", "formula": "\\begin{align*} L _ n ( X , Y ) = X L _ { n - 1 } ( X , Y ) - Y L _ { n - 2 } ( X , Y ) \\end{align*}"}
-{"id": "6835.png", "formula": "\\begin{align*} N _ { 1 2 } = \\big ( \\mu ^ 2 - \\theta ^ 2 \\big ) \\big ( ( s _ 2 c _ 1 - c _ 2 s _ 1 ) q _ 1 + ( c _ 1 c _ 2 + s _ 1 s _ 2 ) q _ 2 \\big ) = 0 \\end{align*}"}
-{"id": "8304.png", "formula": "\\begin{align*} G _ c : = - 2 [ \\bar { \\mathfrak { F } } , \\mathcal { H } \\frac { 1 } { \\zeta _ { \\alpha } } + \\bar { \\mathcal { H } } \\frac { 1 } { \\bar { \\zeta } _ { \\alpha } } ] \\bar { \\mathfrak { F } } _ { \\alpha } + \\frac { 1 } { \\pi i } \\int \\Big ( \\frac { D _ t \\zeta ( \\alpha , t ) - D _ t \\zeta ( \\beta , t ) } { \\zeta ( \\alpha , t ) - \\zeta ( \\beta , t ) } \\Big ) ^ 2 ( \\zeta - \\bar { \\zeta } ) _ { \\beta } d \\beta . \\end{align*}"}
-{"id": "7983.png", "formula": "\\begin{align*} \\sqrt { n } < p \\leq n \\Rightarrow \\xi _ { p } = 1 . \\end{align*}"}
-{"id": "1394.png", "formula": "\\begin{align*} \\widetilde { f } ( \\lambda ) = \\sum _ { i = 1 } ^ { k } \\sum _ { j = 1 } ^ { d _ i ( \\lambda ) } f ( \\underline { \\lambda } _ { i , j } ) \\end{align*}"}
-{"id": "8737.png", "formula": "\\begin{align*} \\omega ^ { ( \\alpha , \\sigma ) } ( z ) = \\left ( \\frac { ( 1 - z e ^ { - \\sigma \\tau } ) } { 2 ( 1 + z e ^ { - \\sigma \\tau } ) } \\right ) ^ { \\alpha } = \\sum _ { k = 0 } ^ { \\infty } \\omega ^ { ( \\alpha , \\sigma ) } _ { k } z ^ k . \\end{align*}"}
-{"id": "1171.png", "formula": "\\begin{align*} & \\rho ^ { ( n , k ) } ( ( x , + \\infty ) ^ k ) = \\frac { ( { \\chi } ^ { ( n ) } ( ( x , + \\infty ) ) ) ! } { ( { \\chi } ^ { ( n ) } ( ( x , + \\infty ) ) - k ) ! } , \\end{align*}"}
-{"id": "3753.png", "formula": "\\begin{align*} g _ { l } ^ { \\left ( j \\right ) } = \\frac { 1 } { 2 \\chi } \\left \\{ \\left ( \\mathcal { D } _ { l , m } ^ { ( j ) } , \\mathcal { D } _ { l , m } ^ { ( j ) } \\right ) \\left \\vert \\left ( \\mathcal { B } _ { l , m } ^ { ( j ) } , \\mathcal { D } _ { l , m } ^ { ( j ) } \\right ) \\right \\vert ^ { - 2 } + \\Re \\left [ \\gamma _ { l , m } ^ { ( j ) } \\right ] \\right \\} . \\end{align*}"}
-{"id": "7903.png", "formula": "\\begin{align*} L _ n \\left ( f ( x ) \\right ) = h \\sum _ { k = - \\hat { n } } ^ { \\hat { n } } w ^ n _ k f ( x _ k ) \\end{align*}"}
-{"id": "7473.png", "formula": "\\begin{gather*} \\Re ( k _ i ( s - 2 ) + l _ i ) > 2 , \\qquad \\forall i = 1 , \\dots , b ; \\end{gather*}"}
-{"id": "8846.png", "formula": "\\begin{align*} \\{ \\hat y _ 1 , \\dots , \\hat y _ { N - 1 } \\} & = \\{ x _ 1 , \\dots , x _ N \\} ' _ { \\hat p } , \\\\ \\{ \\tilde y _ 1 , \\dots , \\tilde y _ { N - 1 } \\} & = \\{ x _ 1 , \\dots , x _ N \\} ' _ { \\tilde p } \\end{align*}"}
-{"id": "4372.png", "formula": "\\begin{align*} { n - u - 1 \\choose k - 1 } - \\frac { k } { k - t } { n - u - 1 \\choose n - k - 1 } - \\prod _ { i = 1 } ^ { k } \\frac { n - k + 1 - i } { n - t - i } { n - 1 \\choose k - 1 } \\ge 0 . \\end{align*}"}
-{"id": "9637.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ n ( - q ; q ) _ n ( - \\frac { a q } { b } ; q ) _ n \\ , b ^ n } { ( a q ; q ^ 2 ) _ { n + 1 } } = \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ n ( - \\frac { a q } { b } ; q ) _ n \\ , b ^ n q ^ { \\frac { n ( n + 1 ) } { 2 } } } { ( - b ; q ) _ { n + 1 } } . \\end{align*}"}
-{"id": "7505.png", "formula": "\\begin{align*} ( \\mathcal { F } ( \\mathcal { F } f ) ) ( \\xi ) = f ( - \\xi ) , \\end{align*}"}
-{"id": "6000.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t v ( t , x ) + \\frac 1 2 \\Delta v ( t , x ) + g ( t , \\nabla v ( t , x ) ) = 0 [ 0 , 1 ] \\times \\R ^ { d } \\\\ v ( 1 , x ) = F ( x _ 1 , \\ldots , x _ { n - 1 } , x ) , x \\in \\R ^ d . \\end{cases} \\end{align*}"}
-{"id": "5480.png", "formula": "\\begin{align*} ( u + \\psi ) ( t , x ) = \\max _ { ( 0 , T ) \\times H } ( u + \\psi ) \\end{align*}"}
-{"id": "82.png", "formula": "\\begin{align*} \\sigma _ 1 = \\Delta J = ( f _ 0 ^ + - f _ 0 ^ - ) = J _ 0 ( 0 + ) \\ , . \\end{align*}"}
-{"id": "230.png", "formula": "\\begin{align*} \\int _ { B _ r } u ^ 2 \\psi d z d t = O ( r ^ { l } ) , \\ \\end{align*}"}
-{"id": "2939.png", "formula": "\\begin{align*} \\| \\widetilde { V } _ { 2 } ^ { \\ast } V _ { 1 } \\| _ { F } ^ { 2 } = \\| A ^ { \\dagger } E \\| _ { F } ^ { 2 } - \\| A ^ { \\dagger } E B ^ { \\dagger } B \\| _ { F } ^ { 2 } . \\end{align*}"}
-{"id": "6682.png", "formula": "\\begin{align*} A ( \\omega ) | x | ^ { - \\frac { 1 } { 2 } + i \\rho } & = ( 1 - 2 ^ { \\frac { 1 } { 2 } + i ( \\rho - \\frac { \\omega } { 2 } ) } ) \\zeta ( \\frac { 1 } { 2 } - i ( \\rho - \\frac { \\omega } { 2 } ) ) | x | ^ { - \\frac { 1 } { 2 } + i ( \\rho - \\omega ) } = 0 , \\\\ H _ { - } | x | ^ { - \\frac { 1 } { 2 } + i \\rho } & = | 1 - 2 ^ { \\frac { 1 } { 2 } - i ( \\rho - \\frac { \\omega } { 2 } ) } | ^ { 2 } | \\zeta ( \\frac { 1 } { 2 } - i ( \\rho - \\frac { \\omega } { 2 } ) ) | ^ { 2 } | x | ^ { - \\frac { 1 } { 2 } + i \\rho } = 0 . \\end{align*}"}
-{"id": "7645.png", "formula": "\\begin{align*} \\int \\limits _ { \\mathbb R ^ d } a ( z ) \\mu ( \\xi , \\xi - z ) ( \\varkappa ^ { i j } _ 2 ( \\xi - z ) - \\varkappa ^ { i j } _ 2 ( \\xi ) ) d z + b ^ i \\varkappa _ 1 ^ j ( \\xi ) + \\int \\limits _ { \\mathbb R ^ d } a ( z ) \\mu ( \\xi , \\xi - z ) \\big ( \\frac 1 2 z ^ i z ^ j - z ^ i \\varkappa ^ { j } _ 1 ( \\xi - z ) \\big ) d z \\ = \\ \\Theta ^ { i j } . \\end{align*}"}
-{"id": "1599.png", "formula": "\\begin{align*} \\Re ( \\Psi ^ { - 1 } _ { \\widetilde { F } } ( z ^ o _ { n , k + 1 } , u ^ o _ { n , k + 1 } ) ) & = \\Re ( \\Psi ^ { - 1 } _ { \\widetilde { F } } ( z ^ o _ { n } , u ^ o _ { n } ) ) + k + 1 + o ( 1 ) \\\\ & < \\kappa _ 1 - k _ n - R - 1 + k + 1 + o ( 1 ) \\leq - R - 1 + o ( 1 ) < - R . \\end{align*}"}
-{"id": "2282.png", "formula": "\\begin{align*} H | + ) & = \\omega _ 0 | + ) & H | - ) & = 0 \\end{align*}"}
-{"id": "2352.png", "formula": "\\begin{align*} \\Gamma _ 1 ' & = \\Gamma _ 1 + \\{ t , ( t , c _ { t , 1 } ) , ( c _ { t , 1 } , c _ { t , 2 } ) , \\cdots , ( c _ { t , k _ t - 1 } , c _ { t , k _ t } ) \\} \\\\ \\Gamma _ 2 ' & = \\Gamma _ 2 \\setminus \\{ ( t , c _ { t , 1 } ) , ( c _ { t , 1 } , c _ { t , 2 } ) , \\cdots , ( c _ { t , k _ t 1 } , c _ { t , k _ t } ) \\} \\end{align*}"}
-{"id": "5513.png", "formula": "\\begin{align*} \\sqrt T ( E ^ { \\Pi _ T } [ b | X ^ T ] - \\hat G _ J ) = o _ { P _ { b _ 0 } } ( 1 ) ( B ^ \\rho _ { 1 \\infty } ) ^ * . \\end{align*}"}
-{"id": "6279.png", "formula": "\\begin{align*} \\delta _ 2 ( W , W ' ) = \\inf _ { \\phi : [ 0 , 1 ] \\to [ 0 , 1 ] } \\| W ^ \\phi - W ' \\| _ 2 \\ , , \\end{align*}"}
-{"id": "7857.png", "formula": "\\begin{align*} \\frac { 1 } { M } \\Big | \\sum _ { s = 1 } ^ M e ^ { 2 \\pi i ( x + s e _ 1 ) \\cdot \\eta } \\Big | \\le M ^ { - 1 } \\| \\eta _ 1 \\| ^ { - 1 } \\le 2 \\kappa ( r , R ) ^ { - \\frac { 1 } { 3 } + \\frac { 2 \\delta } { 3 } } . \\end{align*}"}
-{"id": "6926.png", "formula": "\\begin{align*} \\bar { c } _ \\alpha ( \\rho ) = ( 1 < \\bar { c } _ \\alpha ( X _ 1 ) < \\cdots < \\bar { c } _ \\alpha ( X _ m ) ) . \\end{align*}"}
-{"id": "9657.png", "formula": "\\begin{align*} g \\neq 0 \\ \\ \\sigma \\ X Y h = g h \\ h \\in L ^ 2 ( \\sigma ) . \\end{align*}"}
-{"id": "9911.png", "formula": "\\begin{align*} \\tilde { f } _ { n } ( x ) = \\int _ { \\mathcal { Z } } g _ { n } ( h ( x , z ) ) Q ( d z ) \\\\ \\sup _ { x \\in K _ { n } } \\left | f ( x ) - \\tilde { f } _ { n } ( x ) \\right | \\leq \\frac { \\epsilon } { 3 } \\end{align*}"}
-{"id": "5521.png", "formula": "\\begin{align*} S _ T ( b ) = u \\sqrt { T } \\left ( G ( b ) - \\langle b - b _ 0 , \\gamma \\rangle _ { \\mu _ 0 } \\right ) . \\end{align*}"}
-{"id": "5348.png", "formula": "\\begin{align*} X _ { \\alpha } = \\left [ \\begin{array} { c c c } 0 & 1 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right ] , X _ { \\beta } = \\left [ \\begin{array} { c c c } 0 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 0 & 0 \\end{array} \\right ] \\quad \\mbox { a n d } X _ { \\alpha + \\beta } = \\left [ \\begin{array} { c c c } 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right ] . \\end{align*}"}
-{"id": "7136.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ 0 ^ L \\big ( \\frac { \\rho u ^ 2 } { 2 } + \\frac { \\epsilon \\chi _ x ^ 2 } { 2 } + \\frac { \\rho ( \\chi ^ 2 - 1 ) ^ 2 } { 4 \\epsilon } \\big ) d x + \\int _ 0 ^ L \\Big ( \\mu ^ 2 + \\nu u _ x ^ 2 + u p _ x ( \\rho ) \\Big ) d x d t = 0 . \\end{align*}"}
-{"id": "8094.png", "formula": "\\begin{align*} G - a _ n ' / a _ n = b _ n / a _ n \\sim - 2 \\pi e ^ { 1 - 4 \\sqrt { n } } n \\to \\infty . \\end{align*}"}
-{"id": "8902.png", "formula": "\\begin{align*} \\nu : = ( a - b ) ^ 2 \\left ( \\frac { 1 } { a ( 1 - a / n ) } + \\frac { 1 } { b ( 1 - b / n ) } \\right ) . \\end{align*}"}
-{"id": "4855.png", "formula": "\\begin{align*} \\delta ( \\mu ) = \\frac { 1 } { 2 } \\ , \\ , \\sum _ { j } [ w _ { ( j , r _ 1 ) } , w ^ * _ { ( j , r _ 1 ) } ] + \\frac { 1 } { 2 } \\sum _ { r _ 1 \\neq r _ s } [ w _ { ( t , r _ s ) } , w ^ * _ { ( t , r _ s ) } ] . \\end{align*}"}
-{"id": "6440.png", "formula": "\\begin{align*} x _ { k } = \\| r _ { 0 } \\| V _ { k } T _ { k } ^ { - 1 } e _ { 1 } , \\quad \\mbox { a n d } \\| x _ { k } \\| ^ { 2 } = \\| r _ { 0 } \\| ^ { 2 } e _ { 1 } ^ { T } T _ { k } ^ { - 1 } V _ { k } ^ { T } V _ { k } T _ { k } ^ { - 1 } e _ { 1 } . \\end{align*}"}
-{"id": "3869.png", "formula": "\\begin{align*} Q _ j ( x _ 1 , \\dotsc , x _ n ) = 0 , \\ ; \\ ; \\ ; \\ ; 1 \\leq j \\leq k ' , \\end{align*}"}
-{"id": "7922.png", "formula": "\\begin{align*} M = N - ( n - 1 ) \\end{align*}"}
-{"id": "5539.png", "formula": "\\begin{align*} \\| [ ( L _ { b } ^ * ) ^ { - 1 } - ( L _ { b _ 0 } ^ * ) ^ { - 1 } ] [ f _ h ] \\| _ { L ^ 2 } \\lesssim \\| b - b _ 0 \\| _ \\infty ( \\| \\hat b _ T - b _ 0 \\| _ { \\infty } + \\| b - b _ 0 \\| _ { \\infty } ) = o _ { P _ { b _ 0 } } ( 1 / \\sqrt T ) . \\end{align*}"}
-{"id": "8111.png", "formula": "\\begin{align*} \\widetilde { A } _ n ( t ) \\ = \\ \\sum _ { i = 0 } ^ { \\lfloor n / 2 \\rfloor } \\widetilde { \\gamma } _ { n , i } \\ , t ^ i ( 1 + t ) ^ { n - 2 i } , \\end{align*}"}
-{"id": "2237.png", "formula": "\\begin{align*} & M ( \\varphi _ 1 ) M ( \\varphi _ 2 ) = M ( \\varphi _ 1 \\varphi _ 2 ) & & M ( z _ 1 ) M ( z _ 2 ) = \\delta ( z _ 1 - z _ 2 ) M ( z _ 1 ) . \\end{align*}"}
-{"id": "3091.png", "formula": "\\begin{align*} m \\{ x : | T _ { 1 1 } f | > t \\} & \\le \\frac { C ( n ) \\| f \\| _ 1 } { t } , \\end{align*}"}
-{"id": "3669.png", "formula": "\\begin{align*} a _ i = 1 / p _ i \\end{align*}"}
-{"id": "4315.png", "formula": "\\begin{align*} \\int _ { \\partial D } ( \\Pi _ D ^ k v - v ) = 0 , k = 1 . \\end{align*}"}
-{"id": "3416.png", "formula": "\\begin{align*} \\tau _ 0 ^ M ( T ) = \\inf \\left \\{ t \\ge 0 : \\rho _ { 0 , \\sup } ^ M ( T ) \\le e ^ t \\rho _ { 0 , \\inf } ^ M ( T ) \\right \\} , \\tau _ * ^ M ( T ) = \\inf \\left \\{ t \\ge 0 : \\rho _ { * , \\sup } ^ M ( T ) \\le e ^ t \\rho _ { * , \\inf } ^ M ( T ) \\right \\} \\end{align*}"}
-{"id": "9628.png", "formula": "\\begin{align*} d ( m , N ) + w ( m , N ) = \\textup { l p t } ( m , N ) . \\end{align*}"}
-{"id": "9512.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { n \\rightarrow \\infty } \\Vert u _ n \\Vert _ { s , p } ^ p = : a \\geq 0 \\ \\mbox { a n d } \\ \\displaystyle \\lim _ { n \\rightarrow \\infty } \\Vert u _ n \\Vert _ { s , q } ^ q = : b \\geq 0 \\end{align*}"}
-{"id": "1137.png", "formula": "\\begin{align*} & | A _ { j , j } - B _ j | _ 2 ^ 2 = O ( ( 1 + n ^ 2 a _ j ^ 2 ) ^ 2 ) a _ j ^ 2 = O ( a _ j ^ 2 + n ^ 4 a _ j ^ 6 ) ; \\end{align*}"}
-{"id": "4665.png", "formula": "\\begin{align*} \\lambda \\cdot a _ k = \\sum _ { j + h = k } a _ j \\lambda ^ { [ h ] } = \\sum _ { j = 0 } ^ k a _ j \\lambda ^ { [ k - j ] } , \\end{align*}"}
-{"id": "904.png", "formula": "\\begin{align*} Y _ m = \\Gamma \\cdot Y ^ 1 _ m \\cdots Y ^ \\ell _ m = \\Gamma \\cdot ( - 1 ) ^ { s _ m } \\cdot q ^ { w _ m } \\cdot \\prod _ { 1 \\leq j \\leq \\ell } ^ { \\to } \\left ( \\prod _ { i \\in Q _ 0 ^ j } ^ { \\to } y _ { e _ i } ^ { \\gamma ( i ) } \\right ) \\end{align*}"}
-{"id": "1264.png", "formula": "\\begin{align*} I ( S , S , S ) ( \\xi ) = \\frac { e ^ { i a \\ln | \\xi | } } { | \\xi | } \\left ( E + F e ^ { 2 i a \\ln | \\xi | } e ^ { - 8 i \\xi ^ 3 / 9 } \\right ) + O ( | \\xi | ^ { - 2 + \\gamma / 2 } ) \\end{align*}"}
-{"id": "5051.png", "formula": "\\begin{align*} [ [ [ { \\mathcal P } , a ] , { \\mathcal P } ] , { \\mathcal P } ] = 0 . \\end{align*}"}
-{"id": "8613.png", "formula": "\\begin{align*} u = v \\ \\ \\textrm { o r } \\ \\ u ^ 2 + v ^ 2 = 1 . \\end{align*}"}
-{"id": "8620.png", "formula": "\\begin{align*} \\begin{array} { r c l } 0 & = & q , \\\\ & & \\\\ 0 & = & ( p + 2 c _ 0 ) \\left ( u ^ 2 + \\frac { c _ 0 ^ 2 } { u ^ 2 } \\right ) - 2 c _ 0 - 2 c _ 0 \\frac { z ^ 2 } { u ^ 2 } , \\\\ & & \\\\ u ' & = & z , \\\\ & & \\\\ z ' & = & u ^ 3 - u + \\frac { c _ 0 } { u } ( p + c _ 0 ) . \\end{array} \\end{align*}"}
-{"id": "9159.png", "formula": "\\begin{align*} \\gamma ( g , h ; x ) = \\omega ( g , h , x ) \\omega ( x , g \\triangleleft x , h \\triangleleft x ) \\omega ^ { - 1 } ( g , x , h \\triangleleft x ) , \\end{align*}"}
-{"id": "6312.png", "formula": "\\begin{align*} \\binom { a } { k } / \\binom { b } { k } & = \\left ( a ! ( b - k ) ! \\right ) / \\left ( b ! ( a - k ) ! \\right ) \\\\ & = \\Theta ( \\frac { a ^ a ( b - k ) ^ { b - k } } { b ^ b ( a - k ) ^ { a - k } } \\sqrt { \\frac { ( a - k ) b } { a ( b - k ) } } ) \\\\ & = \\Theta \\left ( ( \\frac { a } { b } ) ^ k ( 1 + \\frac { k } { a - k } ) ^ { a - k - \\frac { 1 } { 2 } } ( 1 + \\frac { k } { b - k } ) ^ { - ( b - k ) - \\frac { 1 } { 2 } } \\right ) . \\end{align*}"}
-{"id": "9738.png", "formula": "\\begin{align*} p ( x + \\xi ) = p ( x ) + \\int _ 0 ^ 1 \\xi \\cdot \\nabla p ( x + t \\xi ) \\ , \\d t = p ( x ) . \\end{align*}"}
-{"id": "2693.png", "formula": "\\begin{align*} r ^ { 2 } - r ( \\tau + \\frac { 1 } { \\tau } ) + 1 - y ^ { 2 } = \\frac { 1 } { 4 } \\big ( \\frac { 1 } { \\tau } - \\tau \\big ) ^ { 2 } \\frac { 2 \\cos \\theta \\cos \\phi + \\cos 2 \\phi } { \\cos ^ 2 \\theta } , \\end{align*}"}
-{"id": "1078.png", "formula": "\\begin{align*} & \\ln \\frac { D _ n ( w \\alpha _ n ) } { D _ n ( \\alpha _ n ) } \\leq - n ^ 2 ( w \\alpha _ n - \\alpha _ n ) \\alpha _ n / 4 + 1 = - ( w - 1 ) \\frac { n ^ 2 \\alpha _ n ^ 2 } { 4 } + 1 \\end{align*}"}
-{"id": "5662.png", "formula": "\\begin{align*} \\| u ( t _ n ) & - U ^ { n } \\| _ { L ^ { p } ( \\Omega ; H ) } \\leq \\| ( \\bar E ( t _ n ) - B _ n P _ h ) u _ 0 \\| _ { L ^ p ( \\Omega ; H ) } \\\\ & + \\| \\sum _ { j = 0 } ^ { n - 1 } \\int _ { t _ j } ^ { t _ { j + 1 } } ( \\bar { E } ( t _ n - t ) - B _ { n - j } P _ h ) \\d W ( t ) \\| _ { L ^ p ( \\Omega ; H ) } : = { \\rm I } + { \\rm I I } . \\end{align*}"}
-{"id": "5300.png", "formula": "\\begin{align*} & f _ * : h \\mapsto 3 h - e _ { 0 , 1 , 2 , 3 } , e _ i \\mapsto 2 h - e _ { i + 1 , i + 2 , i + 3 } ( i = 0 , 1 , 2 , 3 \\mod 4 ) . \\end{align*}"}
-{"id": "8765.png", "formula": "\\begin{align*} \\frac { B _ { 2 2 } } { B } = C = c o n s t . \\end{align*}"}
-{"id": "4785.png", "formula": "\\begin{align*} \\dot { \\bar { \\mathbf { q } } } = \\partial _ { \\bar { \\bf p } } H ( \\bar { \\mathbf { q } } , \\bar { \\mathbf { p } } ) \\ , , \\qquad \\dot { \\bar { \\mathbf { p } } } = - \\partial _ { \\bar { \\bf q } } H ( \\bar { \\mathbf { q } } , \\bar { \\mathbf { p } } ) \\ , , \\end{align*}"}
-{"id": "2641.png", "formula": "\\begin{align*} \\mathrm { P f } \\begin{bmatrix} \\epsilon ( x _ { i } , x _ { j } ) \\end{bmatrix} _ { 1 \\leq i , j \\leq m } & = \\sum _ { k = 2 } ^ { m } ( - 1 ) ^ { k } \\epsilon ( x _ { 1 } , x _ { k } ) \\mathrm { P f } \\begin{bmatrix} \\epsilon ( x _ { i } , x _ { j } ) \\end{bmatrix} _ { i , j \\notin \\{ 1 , k \\} } . \\end{align*}"}
-{"id": "3143.png", "formula": "\\begin{align*} T ( u ) = \\mathbf { F } ( u ) \\cdot \\mathbf { D } ( u ) \\cdot \\mathbf { E } ( u ) \\ , . \\end{align*}"}
-{"id": "3583.png", "formula": "\\begin{align*} V ^ { ( a ) } _ { \\lambda } = \\{ v \\in V \\mid a \\cdot v = \\lambda v \\} . \\end{align*}"}
-{"id": "4694.png", "formula": "\\begin{align*} \\frac { d u _ j } { d t } = - \\frac { 1 } { \\Delta x } \\Big ( \\frac { 1 } { 1 2 } f _ { j - 2 } - \\frac { 2 } { 3 } f _ { j - 1 } + \\frac { 2 } { 3 } f _ { j + 1 } - \\frac { 1 } { 1 2 } f _ { j - 2 } \\Big ) , \\end{align*}"}
-{"id": "4404.png", "formula": "\\begin{align*} \\lambda _ r ( x , y , z ) : = \\frac { 1 } { \\pi r ^ 2 } \\int _ { D _ r ( z ) } \\log | \\zeta - 1 | ^ 2 d A ( \\zeta ) . \\end{align*}"}
-{"id": "7468.png", "formula": "\\begin{gather*} W _ D = \\sum _ { 1 \\leq i < j < k \\leq m } a _ { i j } a _ { j k } a ^ * _ { i k } ; \\end{gather*}"}
-{"id": "7207.png", "formula": "\\begin{align*} L _ r f : = \\sum _ { k \\neq 0 } | k | ^ { - r } \\hat f ( k ) e ^ { i k ( \\cdot ) } , r > 1 , \\end{align*}"}
-{"id": "7383.png", "formula": "\\begin{align*} \\dot { y } & = \\alpha a + \\beta y ^ 2 \\\\ \\alpha & = \\langle p , \\left . D _ s f ( x , \\lambda _ 1 ( s ) , \\lambda _ 2 ( s ) ) \\right | _ { s = 0 } \\rangle \\\\ \\beta & = \\langle p , ( D _ { x , x } f ) ( q , q ) \\rangle \\end{align*}"}
-{"id": "1248.png", "formula": "\\begin{align*} m _ 1 ( \\eta , \\nu ) : = e ^ { 2 i a \\ln | ( \\eta + \\nu ) / 2 | } \\frac { e ^ { i a \\ln | \\frac { \\eta + \\nu } { \\eta - \\nu } | } } { ( \\eta + \\nu ) ^ 3 } , \\end{align*}"}
-{"id": "2504.png", "formula": "\\begin{align*} h _ j = r + 1 = { j \\choose j } + { j - 1 \\choose j - 1 } + \\cdots + { j - r + 1 \\choose j - r + 1 } . \\end{align*}"}
-{"id": "2959.png", "formula": "\\begin{align*} \\theta _ 1 & = \\alpha _ 1 \\nabla _ 1 + \\alpha _ 2 \\nabla _ 2 + \\alpha _ 3 \\nabla _ 3 + \\alpha _ 4 \\nabla _ 4 , \\\\ \\theta _ 2 & = \\beta _ 1 \\nabla _ 1 + \\beta _ 2 \\nabla _ 2 + \\beta _ 3 \\nabla _ 3 , \\\\ \\theta _ 3 & = \\gamma _ 1 \\nabla _ 1 + \\gamma _ 2 \\nabla _ 2 . \\end{align*}"}
-{"id": "6188.png", "formula": "\\begin{align*} \\bar v _ R ( x , z ) = \\bar U ( x _ n , z ) + o ( | ( x , z ) | ^ { 1 / 2 } ) , \\end{align*}"}
-{"id": "2808.png", "formula": "\\begin{align*} \\tilde { C } _ { i , j } ( s ) : = \\frac { \\left ( \\mathrm { C o v } ( \\mathbf { N } ^ t , \\mathbf { N } ^ { t + s } ) \\right ) _ { i , j } } { \\sqrt { \\overline { \\Sigma } _ { i , i } \\overline { \\Sigma } _ { j , j } } } \\end{align*}"}
-{"id": "3086.png", "formula": "\\begin{align*} T _ 1 f = & T _ 1 g + T _ 1 b \\mathrm { I } _ { G ^ * } + T _ 1 b \\mathrm { I } _ { F ^ * } \\\\ = & ( T _ 1 g + T _ 1 b \\mathrm { I } _ { F ^ * } ) + T _ 1 b \\mathrm { I } _ { G ^ * } \\\\ \\equiv & T _ { 1 1 } f + T _ { 1 2 } f , \\end{align*}"}
-{"id": "7965.png", "formula": "\\begin{align*} \\abs { S ^ A _ q } \\geq \\sum _ { j = 1 } ^ m ( d ^ A _ { q , j } - 1 ) + 2 . \\end{align*}"}
-{"id": "1695.png", "formula": "\\begin{align*} ( { \\mathcal L } \\varphi ) ( x ) : = \\int \\varphi ( y ) \\ , P ( x , d y ) , \\ \\ \\forall \\ x \\in X , \\ \\ \\forall \\ \\varphi \\in L _ { \\infty } , \\end{align*}"}
-{"id": "9897.png", "formula": "\\begin{align*} X _ { n + 1 } & = X _ { n } + N ( 1 , 1 ) \\\\ Y _ { n } & = \\begin{cases} X _ { n } + 1 & w . p . ~ \\frac { 1 } { 2 } \\\\ X _ { n } - 1 & w . p . ~ \\frac { 1 } { 2 } \\end{cases} \\end{align*}"}
-{"id": "9420.png", "formula": "\\begin{align*} F ( z ) : = \\frac { f ( z ) ^ { 1 2 } } { \\Delta ( z ) ^ { k } } . \\end{align*}"}
-{"id": "4126.png", "formula": "\\begin{align*} T ( z ) = \\left ( \\mathbb { I } + \\mathcal { O } \\left ( \\frac { 1 } { z } \\right ) \\right ) \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & z ^ { \\frac { 1 } { 4 } } & 0 \\\\ 0 & 0 & z ^ { - \\frac { 1 } { 4 } } \\end{pmatrix} \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & \\frac { 1 } { \\sqrt { 2 } } & \\frac { i } { \\sqrt { 2 } } \\\\ 0 & \\frac { i } { \\sqrt { 2 } } & \\frac { 1 } { \\sqrt { 2 } } \\end{pmatrix} . \\end{align*}"}
-{"id": "9907.png", "formula": "\\begin{align*} & \\left | \\int \\tilde { f } d \\pi ^ { \\mu } _ { n - } - \\int \\tilde { f } d \\pi _ { n - } ^ { \\nu } \\right | + \\frac { 2 } { 3 } \\epsilon \\\\ & = | \\int _ { \\mathcal { Y } ^ { N ' + 1 } } g ( y _ { [ n , n + N ' ] } ) P ^ { \\mu } ( d y _ { [ n , n + N ' ] } | Y _ { [ 0 , n - 1 ] } ) \\\\ & - \\int _ { \\mathcal { Y } ^ { N ' + 1 } } g ( y _ { [ n , n + N ' ] } ) P ^ { \\nu } ( d y _ { [ n , n + N ' ] } | Y _ { [ 0 , n - 1 ] } ) | + \\frac { 2 } { 3 } \\epsilon \\end{align*}"}
-{"id": "6560.png", "formula": "\\begin{align*} n ( d ; ( a _ i ) ; ) = \\deg M _ 0 ( S , E , P , \\beta ) = \\big { | } M _ 0 ( S _ r , E , P , \\beta ) \\big { | } = m ^ P _ \\beta , \\end{align*}"}
-{"id": "3242.png", "formula": "\\begin{align*} H ^ m ( X \\times Y , F ) \\simeq \\bigoplus _ { r + s = m } H ^ r ( X , F ) \\otimes H ^ s ( Y , F ) \\end{align*}"}
-{"id": "5919.png", "formula": "\\begin{align*} B _ i = \\sum _ j b _ i ^ { ( j ) } B _ { i } ^ { ( j ) } , \\end{align*}"}
-{"id": "7745.png", "formula": "\\begin{align*} \\| y \\| \\langle x , x \\rangle + \\mu \\langle x , x \\rangle \\langle y , x \\rangle \\langle x , x \\rangle = 0 . \\end{align*}"}
-{"id": "7170.png", "formula": "\\begin{gather*} X ^ { t } ( x ) : = \\frac { \\partial \\chi ^ { t } ( x ) } { \\partial x } \\end{gather*}"}
-{"id": "8728.png", "formula": "\\begin{align*} p g f _ { T _ { m , d } ^ * } ( s ) = \\sum _ { j \\leq m } \\hat { \\nu } ( j ) p g f _ { \\hat { T } _ { j , d } } ( s ) , \\end{align*}"}
-{"id": "8919.png", "formula": "\\begin{align*} C ^ { \\mathfrak { i } , \\mathfrak { j } } & : = S ^ { \\mathfrak { i } , \\mathfrak { j } } \\cap \\{ u \\in [ 1 : n ] : x ( u ) \\neq y ( u ) \\} \\\\ n _ C ^ { \\mathfrak { i } , \\mathfrak { j } } & : = | C ^ { \\mathfrak { i } , \\mathfrak { j } } | \\end{align*}"}
-{"id": "6787.png", "formula": "\\begin{align*} ( a '' ) ^ k + \\Gamma _ { i j } ^ k ( a ' ) ^ i ( a ' ) ^ j = 0 \\ , . \\end{align*}"}
-{"id": "9769.png", "formula": "\\begin{align*} q ^ { \\rm o d d } ( X ) = \\frac 1 2 \\left [ q ( X ) - q ( X - 2 ( \\boldsymbol { e } _ \\ast \\cdot X ) \\boldsymbol { e } _ \\ast ) \\right ] . \\end{align*}"}
-{"id": "3190.png", "formula": "\\begin{align*} U C \\varphi = U ^ 2 \\varphi = \\varphi . \\end{align*}"}
-{"id": "233.png", "formula": "\\begin{align*} \\int _ { B _ r } \\psi ^ { - \\frac { 1 } { q - 1 } } d z d t < \\infty , \\ \\end{align*}"}
-{"id": "1295.png", "formula": "\\begin{align*} v : = S _ A + z _ A = S _ A + \\tilde { \\Psi } _ A ( z ) = S _ A + \\Psi _ A ( S + z ) - S _ A = \\Psi _ A ( v ) . \\end{align*}"}
-{"id": "5322.png", "formula": "\\begin{align*} \\frac { d q _ 1 } { d t } = & \\frac { f _ 1 ( q _ 1 , v _ 2 , u _ 4 , v _ 4 ) } { 2 ( - 1 + v _ 2 u _ 4 ) } , & \\frac { d v _ 2 } { d t } = & \\frac { f _ 2 ( q _ 1 , v _ 2 , u _ 4 , v _ 4 ) } { 2 ( - 1 + v _ 2 u _ 4 ) } , \\\\ \\frac { d u _ 4 } { d t } = & \\frac { f _ 3 ( q _ 1 , v _ 2 , u _ 4 , v _ 4 ) } { 2 ( - 1 + v _ 2 u _ 4 ) } , & \\frac { d v _ 4 } { d t } = & \\frac { f _ 4 ( q _ 1 , v _ 2 , u _ 4 , v _ 4 ) } { 2 ( - 1 + v _ 2 u _ 4 ) } , \\end{align*}"}
-{"id": "1749.png", "formula": "\\begin{align*} D ^ { l } F = \\left ( \\frac { \\alpha } { p } \\right ) ^ { l } e ^ { \\frac { \\alpha } { p } M _ { s , t } } D M _ { s , t } \\otimes \\cdots \\otimes D M _ { s , t } \\end{align*}"}
-{"id": "137.png", "formula": "\\begin{align*} X \\left ( \\{ f , g \\} \\right ) - \\{ X \\left ( f \\right ) , g \\} - \\{ f , X \\left ( g \\right ) \\} = 0 . \\end{align*}"}
-{"id": "5928.png", "formula": "\\begin{align*} K _ S + B ^ { + } _ S + M _ S = ( K _ X + B ^ { + } + M ) | _ S , \\end{align*}"}
-{"id": "5013.png", "formula": "\\begin{align*} \\lim _ { \\mathfrak N \\ni N \\to \\infty } \\mu _ N ( A _ { m _ 1 \\ldots m _ l } ) = \\mu ( A _ { m _ 1 \\ldots m _ l } ) , 1 \\leq m _ 1 , \\ldots , m _ l \\leq M , \\end{align*}"}
-{"id": "4659.png", "formula": "\\begin{align*} \\begin{aligned} F _ { p , s } ( a + b + c ) & \\ge t ( F _ { p , r } ( a + b ) , F _ { r , s } ( c ) ) \\ge \\\\ & \\ge t ( t ( F _ { p , q } ( a ) , F _ { q , r } ( b ) ) , F _ { r , s } ( c ) ) = \\\\ & = t _ * ( F _ { p , q } ( a ) , F _ { q , r } ( b ) , F _ { r , s } ( c ) ) . \\end{aligned} \\end{align*}"}
-{"id": "9521.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\mathcal { D } ( n ) } ( - 1 ) ^ { \\# ( \\pi ) - 1 } s ( \\pi ) = d ( n ) , \\end{align*}"}
-{"id": "1777.png", "formula": "\\begin{align*} \\alpha = \\frac { \\eta } { \\gamma _ { H } ( t - s ) ^ { 2 H } + ( t - s ) } , \\end{align*}"}
-{"id": "4526.png", "formula": "\\begin{align*} b _ i \\in \\mathbb { R } ^ 2 , r _ i > 0 , \\lim _ { i \\to \\infty } r _ i = \\infty . \\end{align*}"}
-{"id": "7460.png", "formula": "\\begin{align*} \\displaystyle \\left \\lvert D \\left ( \\lambda \\mapsto ( \\pi _ \\lambda ( g ) \\pi _ \\lambda ( f ^ \\vee ) e , e ) \\right ) _ { \\lambda = 0 } \\right \\rvert \\ll N ( \\rho ) ^ r \\Xi ^ G ( g ) \\sigma ( g ) ^ { \\deg ( D ) } \\end{align*}"}
-{"id": "1533.png", "formula": "\\begin{align*} g ( 0 , x , y ) = \\left \\{ \\begin{array} { c c } ( 1 - x ) y , & 0 \\leq y \\leq x \\leq 1 , \\\\ \\\\ x ( 1 - y ) , & 0 < x < y \\leq 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "2272.png", "formula": "\\begin{align*} & T ( 0 ) = 1 \\\\ & T ( s + t ) = T ( s ) T ( t ) s , t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "7323.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} w = { } & u - p , \\\\ z = { } & v - q . \\end{aligned} \\right . \\end{align*}"}
-{"id": "6788.png", "formula": "\\begin{align*} \\begin{cases} a '' + \\lambda b = 0 \\ , , \\\\ \\langle a ' , b \\rangle = 0 \\ , , \\\\ a ( t ) \\in M \\ , . \\end{cases} \\end{align*}"}
-{"id": "1917.png", "formula": "\\begin{align*} Z = x \\cot ( Z ) , \\end{align*}"}
-{"id": "5000.png", "formula": "\\begin{align*} { \\rm A v g } _ { \\rm p r i m e } = \\lim _ { q \\to \\infty } \\ ; \\ ; ( m : { \\rm f i x e d } ) , \\ ; \\ ; { \\rm o r } \\ ; \\ ; { \\rm A v g } _ { \\rm p o w e r } = \\lim _ { m \\to \\infty } \\ ; \\ ; ( q : { \\rm f i x e d } ) . \\end{align*}"}
-{"id": "2758.png", "formula": "\\begin{align*} \\phi _ { 1 } ' ( \\kappa ; u ) = \\kappa \\phi _ { 1 } ( \\kappa ; u ) - \\frac { 1 } { 2 } \\int _ { \\mathcal { C } _ { < } } \\frac { d w } { 2 \\pi i } \\sqrt { 2 w } e ^ { \\frac { 1 } { 3 } ( w - \\kappa ) ^ 3 - u ( w - \\kappa ) } , \\\\ \\phi _ { 1 } ''' ( \\kappa ; u ) = \\kappa \\phi _ { 1 } '' ( \\kappa ; u ) - \\frac { 1 } { 2 } \\int _ { \\mathcal { C } _ { < } } \\frac { d w } { 2 \\pi i } \\sqrt { 2 w } ( w - \\kappa ) ^ 2 e ^ { \\frac { 1 } { 3 } ( w - \\kappa ) ^ 3 - u ( w - \\kappa ) } . \\end{align*}"}
-{"id": "6415.png", "formula": "\\begin{align*} \\| u \\| ^ { 2 } = 2 ( \\chi ^ { 2 } + \\left ( \\rho - \\tau \\right ) \\chi ) , \\end{align*}"}
-{"id": "8929.png", "formula": "\\begin{align*} \\mathbb { E } [ \\eta _ 1 | d ( x , \\hat { x } ) = k ] & = \\frac { k } { n } ( n - s ) \\\\ \\mathbb { E } [ \\eta _ 2 | d ( x , \\hat { x } ) = k ] & = \\frac { k } { n } ( s ) \\\\ \\mathbb { E } [ \\eta _ 1 \\eta _ 2 | d ( x , \\hat { x } ) = k ] & = ( n - s ) ( s ) \\frac { k ( k - 1 ) } { n ( n - 1 ) } = s ( n - s ) \\left ( \\frac { k ^ 2 } { n ^ 2 } - \\frac { k ( n - k ) } { n ^ 2 ( n - 1 ) } \\right ) . \\end{align*}"}
-{"id": "5707.png", "formula": "\\begin{align*} \\frac { d } { d x } \\left ( x ^ p J _ p ( x ) \\right ) & = x ^ p J _ { p - 1 } ( x ) , \\\\ \\frac { d } { d x } \\left ( x ^ { - p } J _ p ( x ) \\right ) & = - x ^ { - p } J _ { p + 1 } ( x ) , \\end{align*}"}
-{"id": "9989.png", "formula": "\\begin{align*} & b _ { t } ^ { i } = V ^ { i } ( \\mathbf { b } _ { x } ) \\\\ & p _ { k , t } = V ^ i _ { k h } b ^ h _ { x x } p _ i + V ^ i _ { k } p _ { i , x } \\end{align*}"}
-{"id": "6572.png", "formula": "\\begin{gather*} W '' = W / W ' \\cong \\R \\{ e _ 0 \\} . \\end{gather*}"}
-{"id": "2127.png", "formula": "\\begin{align*} \\Phi ^ \\prime ( t + s , \\omega ) = \\Phi ^ \\prime ( t , \\theta _ s \\omega ) \\circ \\Phi ^ \\prime ( s , \\omega ) . \\end{align*}"}
-{"id": "9737.png", "formula": "\\begin{align*} f ^ { + } : = \\max \\{ f , 0 \\} \\quad f ^ { - } : = \\max \\{ - f , 0 \\} = - \\min \\{ f , 0 \\} . \\end{align*}"}
-{"id": "4661.png", "formula": "\\begin{align*} S = \\bigcup _ { n = 0 } ^ \\infty S ( n ) , \\end{align*}"}
-{"id": "9585.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\textup { s p t } ( n , N ) q ^ n = \\sum _ { n = 1 } ^ N \\frac { q ^ n } { ( 1 - q ^ n ) ^ 2 ( 1 - q ^ { n + 1 } ) \\cdots ( 1 - q ^ N ) } . \\end{align*}"}
-{"id": "2065.png", "formula": "\\begin{align*} { C ( \\bar r , t | { { \\bar r } _ { \\rm t x } } , { t _ 0 } ) } = C _ { \\rho \\varphi } ( \\rho , \\varphi , t | \\rho _ { \\rm t x } , \\varphi _ { \\rm t x } , { t _ 0 } ) C _ z ( z , t | z _ { \\rm t x } , { t _ 0 } ) , \\end{align*}"}
-{"id": "5645.png", "formula": "\\begin{align*} W ( t ) = \\sum _ { \\ell = 1 } ^ { \\infty } \\gamma _ { \\ell } ^ \\frac { 1 } { 2 } e _ { \\ell } \\beta _ { \\ell } ( t ) , \\end{align*}"}
-{"id": "8383.png", "formula": "\\begin{align*} a _ t | z _ { \\alpha } | = ( I + \\mathfrak { K } ^ { \\ast } ) ^ { - 1 } \\Big \\{ R e ( \\frac { i z _ { \\alpha } } { | z _ { \\alpha } | } ( g _ 1 + g _ 2 ) ) \\Big \\} . \\end{align*}"}
-{"id": "4224.png", "formula": "\\begin{align*} E _ { i n } ^ { - 1 } ( y _ n ) R ^ { - 1 } ( y _ n ) R ( x _ n ) E _ { i n } ( x _ n ) & = E _ { i n } ^ { - 1 } ( y _ n ) \\left ( \\mathbb { I } + \\mathcal { O } ( ( x - y ) n ^ { - 5 / 2 } ) \\right ) E _ { i n } ( x _ n ) \\\\ & = \\mathbb { I } + \\mathcal { O } \\left ( ( x - y ) n ^ { - \\frac { 1 } { 2 } } \\right ) + E _ { i n } ^ { - 1 } ( y _ n ) \\mathcal { O } \\left ( ( x - y ) n ^ { - \\frac { 5 } { 2 } } \\right ) E _ { i n } ( x _ n ) . \\end{align*}"}
-{"id": "4835.png", "formula": "\\begin{align*} x _ \\alpha & = x _ { \\alpha } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , i f \\ , \\ , \\alpha \\in \\{ \\alpha _ 1 , \\alpha _ 2 \\} \\\\ & = 0 \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , i f \\ , \\ , \\alpha \\notin \\{ \\alpha _ 1 , \\alpha _ 2 \\} \\end{align*}"}
-{"id": "6181.png", "formula": "\\begin{align*} | H ( t , z ) - a U ( t , z ) | \\leq C _ 0 r ^ { 1 / 2 } U ( t , z ) , r ^ 2 = t ^ 2 + z ^ 2 , a \\in \\R , \\end{align*}"}
-{"id": "8891.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { l = 0 } ^ k f _ l ( x ) \\log ^ l ( x ) \\end{align*}"}
-{"id": "9224.png", "formula": "\\begin{align*} \\epsilon _ { T } = \\frac { 1 } { V } \\sum _ { i = 1 } ^ { V } \\left ( \\hat { \\psi } _ { v _ { i 1 } v _ { i 2 } } - \\psi _ { v _ { i 1 } v _ { i 2 } } \\right ) ^ 2 , \\end{align*}"}
-{"id": "1915.png", "formula": "\\begin{align*} \\phi ( x , t ) = - \\int _ { S ^ 1 } \\left [ \\left ( \\partial _ t E ( x , t ; x ' , 0 ) \\right ) f ( x ' ) + E ( x , t ; x ' , 0 ) g ( x ' ) \\right ] d x ' . \\end{align*}"}
-{"id": "1322.png", "formula": "\\begin{align*} \\varPhi \\big [ \\lambda Z _ 1 + ( 1 - \\lambda ) Z _ 2 , P \\big ] = \\lambda \\varPhi [ Z _ 1 , P ] + ( 1 - \\lambda ) \\varPhi [ Z _ 2 , P ] . \\end{align*}"}
-{"id": "2009.png", "formula": "\\begin{align*} t = \\sum _ { 1 \\leq m \\leq 4 } t ^ m \\phi _ m \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\tau = I _ + ( q , - z ) + \\phi _ 1 z . \\end{align*}"}
-{"id": "1816.png", "formula": "\\begin{align*} \\left [ - \\partial _ x ^ 2 + V _ \\xi ( x ) \\right ] \\chi _ { \\xi , j } ( x ) = \\omega _ { \\xi , j } ^ 2 \\ , \\chi _ { \\xi , j } ( x ) \\end{align*}"}
-{"id": "4979.png", "formula": "\\begin{align*} & ( 1 - \\lambda _ { m + 1 } ) + ( 1 - \\lambda _ { m + 1 } ) B _ m = B _ { m + 1 } \\\\ & ( 1 - \\lambda _ { m + 1 } ) C _ m = C _ { m + 1 } \\end{align*}"}
-{"id": "1648.png", "formula": "\\begin{align*} F _ k : = \\frac { 1 } { \\varkappa _ k } \\begin{pmatrix} - b \\\\ a _ k - i \\sqrt [ + ] { b ^ 2 - a _ k ^ 2 } \\end{pmatrix} e ^ { 2 \\pi i k x } \\end{align*}"}
-{"id": "6709.png", "formula": "\\begin{align*} \\frac { \\partial H } { \\partial P _ { 2 k } } = \\ddot { \\bar { q } } _ k , \\end{align*}"}
-{"id": "7235.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\left ( 2 \\mu - \\phi ( x ) - \\phi ( y ) \\right ) \\left ( \\phi ( y ) - \\phi ( x ) \\right ) = L _ r \\phi ( x ) - L _ r \\phi ( y ) . \\end{align*}"}
-{"id": "8836.png", "formula": "\\begin{align*} h ( y ) = F ( y ) + k \\mu ( y ) ^ 2 , \\end{align*}"}
-{"id": "3627.png", "formula": "\\begin{align*} \\alpha _ { - 1 } & = - \\frac { 3 t } { 2 ^ 2 } a _ 1 - \\frac { 1 } { 2 ^ 2 } v _ { ( 2 , 3 ) } + a _ 1 \\cdot v _ { ( 2 , 3 ) } \\in M _ 0 ^ { ( a _ 1 ) } \\\\ \\alpha _ 2 & = a _ { - 1 } \\in M _ 0 ^ { ( a _ 1 ) } . \\end{align*}"}
-{"id": "5982.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } \\tilde { \\rho } ^ g _ n ( n F \\circ L _ n ) = \\sup _ { Q \\in \\P ( \\C ) } ( F ( Q ) - \\tilde { \\alpha } ^ g ( Q ) ) = \\sup _ { Q \\in \\P ^ * } ( F ( Q ) - \\tilde { \\alpha } ^ g ( Q ) ) . \\end{align*}"}
-{"id": "9278.png", "formula": "\\begin{align*} \\mathcal { L } ^ \\epsilon : = L ^ \\epsilon - D _ { p _ k } B ( \\cdot , u _ \\epsilon , D u _ \\epsilon ) D _ k , \\end{align*}"}
-{"id": "399.png", "formula": "\\begin{align*} \\mathcal { S } _ { \\rm L } ( \\varphi , g ) : = \\frac { 1 } { 4 \\pi } \\int _ M \\big ( | d \\varphi | ^ 2 _ g + Q K _ g \\varphi + 4 \\pi \\mu e ^ { \\gamma \\varphi } \\big ) \\ , { \\rm d v } _ g \\end{align*}"}
-{"id": "2707.png", "formula": "\\begin{align*} \\sigma _ { n + 1 } = \\cdots = \\sigma _ { N } = 1 \\ \\mathrm { a n d } \\ \\sigma _ i \\in ( \\tau , \\infty ) , i = 1 , \\ldots , n . \\end{align*}"}
-{"id": "8270.png", "formula": "\\begin{align*} \\beta _ 0 ( t ) : = \\inf _ { \\alpha \\in \\mathbb { R } } | Z _ { \\alpha } ( \\alpha , t ) | , M _ 0 ( t ) : = \\| F ( \\cdot , t ) \\| _ { \\infty } \\end{align*}"}
-{"id": "4437.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ 0 ^ T \\Bigl { \\langle } \\frac { { \\dd } u _ p } { { \\dd } \\tau } ( \\tau ) , w ( \\tau ) \\Bigr { \\rangle } \\dd \\tau + \\int _ { \\mathcal { Q } _ T } ( \\nabla u _ p + u _ { p } \\nabla \\mathcal { V } _ p ) ( \\tau , x ) \\cdot \\nabla w ( \\tau , x ) \\dd x d \\tau \\end{aligned} \\end{align*}"}
-{"id": "2403.png", "formula": "\\begin{align*} b ( \\upsilon ) + c ( \\upsilon ) & = b _ \\upsilon ( i , j ' ) + b _ \\upsilon ( 1 , n ) + \\sum \\limits _ { ( s , t ) \\in \\sigma ^ - _ { ( i , j ) ( i ' , j ' ) } } b _ \\upsilon ( s , t ) + \\sum \\limits _ { ( s , t ) \\in \\sigma ^ - _ { ( i , j ) ( i ' , j ' ) } } c _ \\upsilon ^ \\ell ( s , t ) \\\\ & = 2 ( k + 2 ) + b ( \\sigma ) + 2 r + c ( \\sigma ) = b ( \\sigma ) + c ( \\sigma ) + 4 + 2 ( k + r ) , \\end{align*}"}
-{"id": "1660.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 H ^ c } { \\partial p _ n \\partial q _ l } \\ , p _ l - \\frac { \\partial ^ 2 H ^ c } { \\partial p _ n \\partial p _ l } \\ , q _ l = 0 \\quad \\frac { \\partial ^ 2 H ^ c } { \\partial q _ n \\partial q _ l } \\ , p _ l - \\frac { \\partial ^ 2 H ^ c } { \\partial q _ n \\partial p _ l } \\ , q _ l = 0 \\end{align*}"}
-{"id": "4724.png", "formula": "\\begin{align*} u ( x , 0 ) = \\left \\{ \\begin{array} { l l } 1 ~ ~ ~ ~ x > 0 . 2 5 , \\\\ 0 ~ ~ ~ ~ x \\leq 0 . 2 5 \\end{array} \\right . \\end{align*}"}
-{"id": "2611.png", "formula": "\\begin{align*} \\Phi = \\frac { d \\eta } { d \\sigma } \\kappa ^ { - 1 } \\le \\Phi ( z ) \\le \\kappa S \\end{align*}"}
-{"id": "6309.png", "formula": "\\begin{align*} | E ( G ) - E ( G ' ) | \\leq | e ( G ) - e ( G ' ) | \\left ( k ( n - k ) + \\binom { k } { 2 } \\right ) + 2 C \\max \\{ \\sqrt { \\rho } , \\sqrt { \\frac { \\log n } { n } } \\} k \\sqrt { n \\log n } , \\end{align*}"}
-{"id": "6841.png", "formula": "\\begin{align*} \\det ( d \\varphi ) = g _ 1 + g _ 2 + g _ 3 - g _ 4 + p _ 5 ( g _ 4 + g _ 5 ) + g _ 6 \\ , . \\end{align*}"}
-{"id": "3561.png", "formula": "\\begin{align*} \\Gamma : = \\{ u = v = x y = y z = z x = 0 \\} . \\end{align*}"}
-{"id": "1053.png", "formula": "\\begin{align*} & \\min ( | x - y | , 2 \\pi - | x - y | ) \\geq \\min ( | y _ i - y _ j | , 2 \\pi - | y _ i - y _ j | ) - \\max ( F _ n ( x _ i ) , F _ n ( x _ j ) ) \\\\ & \\geq \\min \\{ z _ { i + 1 } - z _ i | 1 \\leq i \\leq k \\} - \\min \\{ z _ { i + 1 } - z _ i | 1 \\leq i \\leq k \\} / 2 \\\\ & = \\min \\{ z _ { i + 1 } - z _ i | 1 \\leq i \\leq k \\} / 2 : = a _ 0 \\in ( 0 , 2 \\pi ) , \\end{align*}"}
-{"id": "2335.png", "formula": "\\begin{align*} & ( a _ x f ) ( x _ 1 , \\cdots , x _ n ) = f ( x , x _ 1 , \\cdots x _ n ) \\\\ & ( a _ x ^ + f ) ( x _ 1 , \\cdots , x _ n ) \\\\ & = \\delta _ { x , x _ 1 } f ( x _ 2 , \\cdots , x _ n ) + \\delta _ { x , x _ 2 } f ( x _ 1 , x _ 3 , \\cdots , x _ n ) + \\cdots + \\delta _ { x , x _ n } f ( x _ 1 , \\cdots , x _ { n - 1 } ) \\end{align*}"}
-{"id": "3308.png", "formula": "\\begin{align*} \\left | \\left ( \\mathcal { M } / R _ { \\ell , M } \\right ) ^ { \\times } \\right | = q ^ { \\ell } \\phi ( M ) . \\end{align*}"}
-{"id": "847.png", "formula": "\\begin{align*} T \\Phi ^ * ( s ) = [ \\Phi ^ * ( s ) , \\Phi ^ * ( \\cdot ) ] _ { E ^ * } = [ \\Phi ( \\cdot ) , \\Phi ( s ) ) ] _ { E } = G ( \\cdot , s ) = ( G ( s , \\cdot ) ) ^ * : = G _ s ^ * s \\in \\Omega \\end{align*}"}
-{"id": "8559.png", "formula": "\\begin{align*} ( I - \\mathcal { H } ) b = - [ D _ t \\zeta , \\mathcal { H } ] \\frac { \\bar { \\zeta } _ { \\alpha } - 1 } { \\zeta _ { \\alpha } } - \\frac { i } { \\pi } \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j } { \\zeta ( \\alpha , t ) - z _ j ( t ) } . \\end{align*}"}
-{"id": "1501.png", "formula": "\\begin{align*} m _ { t } - \\Delta m + \\varepsilon \\mathrm { d i v } ( m \\bar { H } _ { p } ( t , x , m , D u ) ) = 0 , \\end{align*}"}
-{"id": "8752.png", "formula": "\\begin{align*} \\begin{aligned} { } _ H u _ { n , n _ 0 } ^ { ( \\alpha , \\sigma ) } = & \\tau \\sum _ { k = 0 } ^ { n - n _ 0 - 1 } { e ^ { - ( n - k ) \\sigma \\tau } } u _ { k } \\int _ { - \\infty } ^ { \\infty } ( 1 + e ^ x \\tau ) ^ { - 1 - ( n - k ) } \\phi ( x ) d x . \\end{aligned} \\end{align*}"}
-{"id": "4814.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\Delta f + \\frac { R _ g } { n - 1 } f & = & 0 \\ ; \\ ; \\ ; \\mbox { i n } \\ ; \\ ; \\ ; M \\\\ \\frac { \\partial f } { \\partial \\nu } - \\frac { H _ { \\gamma } } { n - 1 } f & = & 0 \\ ; \\ ; \\ ; \\mbox { o n } \\ ; \\ ; \\ ; \\partial M . \\end{array} \\right . \\end{align*}"}
-{"id": "3595.png", "formula": "\\begin{align*} w _ { \\rho ( a _ 0 , a _ 1 ) } = - w _ { \\rho ( a _ 0 , a _ 2 ) } = - w _ { \\rho ( a _ 0 , a _ { - 2 } ) } = w _ { \\rho ( a _ 0 , a _ { - 1 } ) } . \\end{align*}"}
-{"id": "9335.png", "formula": "\\begin{align*} \\mathrm { t r a c e } ( A _ { Q _ { 1 } ^ { x } } ) = \\mathrm { t r a c e } ( A ^ 2 _ { \\xi } ) ( x ) + \\| \\nabla ^ { \\perp } \\xi \\| ^ { 2 } ( x ) \\end{align*}"}
-{"id": "352.png", "formula": "\\begin{align*} \\nu _ p ( \\frac { 1 } { 2 } n ( n ^ 2 - 1 ) , k ) = \\nu _ p ( \\frac { 1 } { 2 } n ( n ^ 2 - 1 ) , l ) . \\end{align*}"}
-{"id": "5103.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j \\in \\mathbb { S } } \\widehat { F } _ j A _ j x \\right \\| ^ p = \\left \\| \\sum _ { j \\in \\mathbb { S } } \\widehat { F } _ j \\widehat { L } _ j \\left ( \\sum _ { k \\in \\mathbb { S } } L _ k A _ k x \\right ) \\right \\| ^ p = \\sum _ { j \\in \\mathbb { S } } \\left \\| \\widehat { L } _ j \\left ( \\sum _ { k \\in \\mathbb { S } } L _ k A _ k x \\right ) \\right \\| ^ p = \\sum _ { j \\in \\mathbb { S } } \\| A _ j x \\| ^ p . \\end{align*}"}
-{"id": "4501.png", "formula": "\\begin{align*} M _ { 0 , z ^ 0 } ( h ) = H ^ { i j } | _ { v ^ 0 } \\nabla _ { i j } h + H ^ i | _ { v ^ 0 } \\nabla _ i h + ( H _ { v } | _ { v ^ 0 } - \\Xi ^ 0 _ { v } | _ { v ^ 0 } ) h \\end{align*}"}
-{"id": "4752.png", "formula": "\\begin{align*} n ! \\ , L \\sum _ { u = 1 } ^ L ( u + 1 ) ^ { n - 3 } e ^ { - b u ^ 2 } \\le C ( n , b ) L , \\end{align*}"}
-{"id": "1819.png", "formula": "\\begin{align*} V _ \\xi ( x ) = \\xi \\left [ \\delta ( x - a / 2 ) + \\delta ( x + a / 2 ) \\right ] , \\end{align*}"}
-{"id": "9135.png", "formula": "\\begin{align*} \\lim _ { \\tau \\rightarrow + \\infty } \\rho _ { 1 } ^ { ( N ) } ( \\mathbf { x } _ { 1 } , t ) = \\rho _ { 1 M } ^ { ( N ) } ( \\mathbf { v } _ { 1 } ) . \\end{align*}"}
-{"id": "5969.png", "formula": "\\begin{align*} q _ n ( t ) : = q ^ Q ( t ) 1 _ { \\{ | q ^ Q ( t ) | \\le n \\} } + \\frac { n } { | q ^ Q ( t ) | } q ^ Q ( t ) 1 _ { \\{ | q ^ Q ( t ) | > n \\} } . \\end{align*}"}
-{"id": "4926.png", "formula": "\\begin{align*} \\ell ( H ^ { d - 1 } ( J + ( x ^ t ) ; M ) ) = \\ell ( H ^ { d - 2 } ( J ; M / ( x ^ t ) M ) ) \\geq \\ell \\left ( \\dfrac { H ^ { d - 2 } ( J ; M ) } { x ^ t \\cdot H ^ { d - 2 } ( J ; M ) } \\right ) . \\end{align*}"}
-{"id": "680.png", "formula": "\\begin{align*} ( \\mathcal M \\rtimes \\mathcal G ) ( a _ 1 \\dots a _ n ; a ) & \\cong ( \\mathcal M \\wr \\widetilde { \\mathbb G } _ { \\mathcal G } ) ( a _ 1 \\dots a _ n , a ) _ { \\mu _ n } \\\\ & \\xrightarrow [ \\cong ] { H _ { { \\widetilde { \\mathbb G } } _ { \\mathcal G } } } ( \\mathcal C \\times _ { \\widetilde { \\mathbb E } _ { \\mathcal G } } \\widetilde { \\mathbb G } _ { \\mathcal G } ) ( H ( a _ 1 \\dots a _ n ) , H ( a ) ) _ { \\mu _ n } \\ . \\end{align*}"}
-{"id": "9380.png", "formula": "\\begin{align*} \\lvert H _ k : H _ k \\cap \\underbrace { P _ i ( G _ k ) } _ { = 1 } \\rvert = \\lvert H _ k \\rvert = \\lvert H : H \\cap P _ i ( G ) \\rvert \\end{align*}"}
-{"id": "283.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } ( - 1 ) ^ k \\ P ^ { ( k ) } ( 0 ) \\cdot Q ^ { ( n - k ) } ( 0 ) = 0 . \\end{align*}"}
-{"id": "711.png", "formula": "\\begin{align*} \\forall 1 \\leq i , j , k , l \\leq 2 , R _ { i \\bar { j } k \\bar { l } } = - g _ { i \\bar { j } k \\bar { l } } + \\sum _ { 1 \\leq \\alpha , \\beta \\leq 2 } g _ { i k \\bar { \\alpha } } g ^ { \\bar { \\alpha } \\beta } g _ { \\beta \\bar { j } \\bar { l } } , \\end{align*}"}
-{"id": "696.png", "formula": "\\begin{align*} \\forall \\psi = ( \\psi _ 1 , \\psi _ 2 ) \\in A u t ( T _ p ) , g = g \\circ \\psi + \\frac { 2 } { 3 } L o g \\left \\lvert D e t ( J a c _ \\mathbb { C } \\left ( \\psi \\right ) \\right \\rvert . \\end{align*}"}
-{"id": "7124.png", "formula": "\\begin{align*} \\mu _ 1 D Q ( \\hat { q } ) ( h ) + \\mu _ 2 D \\lambda _ 2 ( \\hat { q } ) ( h ) = 0 , ~ ~ \\forall h \\in L ^ 2 , \\end{align*}"}
-{"id": "8689.png", "formula": "\\begin{align*} h ( \\lambda + \\rho + \\nu - \\rho ' + 2 i ) c _ { h , i , j } = ( i + 1 ) ( 2 i + p '' ) c _ { h - 1 , i + 1 , j } , \\end{align*}"}
-{"id": "9740.png", "formula": "\\begin{align*} L _ a ( v ^ 2 ) = 2 v \\ , L _ a v + 2 | \\nabla v | ^ 2 | y | ^ { a } \\geq 2 | \\nabla v | ^ 2 | y | ^ a . \\end{align*}"}
-{"id": "9883.png", "formula": "\\begin{align*} E ^ { \\nu } [ \\frac { d \\mu } { d \\nu } ( X _ { 0 } ) | \\mathcal { F } ^ { \\mathcal { Y } } _ { 0 , \\infty } ] = E ^ { \\nu } [ \\frac { d \\mu } { d \\nu } ( X _ { 0 } ) | \\bigcap _ { n \\geq 0 } \\mathcal { F } ^ { \\mathcal { X } } _ { n , \\infty } \\vee \\mathcal { F } ^ { \\mathcal { Y } } _ { 0 , \\infty } ] ~ ~ ~ P ^ { \\mu } ~ a . s . \\end{align*}"}
-{"id": "5101.png", "formula": "\\begin{align*} ~ \\operatorname { I n e q u a l i t y } ~ ( \\ref { F R A M E I N E Q U A L I T Y B A N A C H } ) . \\end{align*}"}
-{"id": "4955.png", "formula": "\\begin{align*} 1 \\geq \\lim _ { n \\rightarrow \\infty } \\dfrac { \\ell ( N / I _ n N ) } { \\ell ( R / I _ n ) } \\geq \\lim _ { n \\rightarrow \\infty } \\dfrac { t } { \\ell ( R / I _ n ) \\leq 3 ( { t + ( 3 n ^ { d - 2 } + n ^ { d - 3 } + \\cdots + n ^ 2 ) } ) + ( 8 ( 2 n ) ^ { d - 2 } ) ^ d } = 1 / 3 \\end{align*}"}
-{"id": "4358.png", "formula": "\\begin{align*} \\mathcal A \\ : = & \\ \\Big \\{ F \\in { [ m ] \\choose k - 1 } \\ : \\ j _ 3 \\in F , \\ F \\cap ( H _ 1 \\cup H _ 2 ) = \\{ j _ 2 \\} \\Big \\} , \\\\ \\mathcal A ' \\ : = & \\ \\Big \\{ F \\in { [ m ] \\choose k - 1 } \\ : \\ j _ 3 \\in F , \\ F \\cap ( H _ 1 \\cup H _ 2 ) = \\{ j _ 1 \\} \\Big \\} . \\end{align*}"}
-{"id": "9368.png", "formula": "\\begin{align*} D _ i ( G _ k ) = \\begin{cases} \\langle [ y , x , \\overset { i - 1 } { \\ldots } , x ] \\rangle D _ { i + 1 } ( G _ k ) & \\\\ \\langle [ y , x , \\overset { i - 3 } { \\ldots } , x , y ] , [ y , x , \\overset { i - 1 } \\ldots , x ] \\rangle D _ { i + 1 } ( G _ k ) & \\end{cases} \\end{align*}"}
-{"id": "2188.png", "formula": "\\begin{align*} J ( \\boldsymbol { x } ( t ) , \\boldsymbol { u } ( t ) , t ) = 0 . 5 \\Big [ \\boldsymbol { x } ^ T ( t _ f ) F ( t _ f ) \\boldsymbol { x } ( t _ f ) \\Big ] + 0 . 5 \\Big [ \\int _ { t _ 0 } ^ { t _ f } \\boldsymbol { x } ^ T ( t ) Q ( t ) { \\boldsymbol { x } } ( t ) + \\boldsymbol { u } ^ T ( t ) R ( t ) \\boldsymbol { u } ( t ) d t \\Big ] \\end{align*}"}
-{"id": "5022.png", "formula": "\\begin{align*} g ( \\tilde { t } , \\tilde { x } ) = \\displaystyle \\min _ { ( t , x ) \\in [ 0 , T ] \\times { \\bar \\Omega } } g ( t , x ) \\end{align*}"}
-{"id": "7544.png", "formula": "\\begin{align*} \\begin{gathered} \\left ( \\frac { d } { d \\epsilon } \\left ( \\mathbb { U } ( \\tau _ \\epsilon \\circ X _ \\epsilon ^ { - 1 } ) \\circ X _ \\epsilon \\right ) \\right ) \\circ X _ \\epsilon ^ { - 1 } = [ \\eta _ \\epsilon \\cdot \\nabla _ x , \\mathbb { U } ] ( \\sigma _ \\epsilon ) + \\mathbb { U } ( \\delta _ \\epsilon ) , \\end{gathered} \\end{align*}"}
-{"id": "928.png", "formula": "\\begin{align*} E _ k ^ q = I _ k - \\sum _ { j _ 1 , \\ldots , j _ k = 0 } ^ { q } C _ { j _ k \\ldots j _ 1 } { \\sf M } \\left \\{ J [ \\psi ^ { ( k ) } ] _ { T , t } \\sum \\limits _ { ( j _ 1 , \\ldots , j _ k ) } \\int \\limits _ t ^ T \\phi _ { j _ k } ( t _ k ) \\ldots \\int \\limits _ t ^ { t _ { 2 } } \\phi _ { j _ { 1 } } ( t _ { 1 } ) d { \\bf f } _ { t _ 1 } ^ { ( i _ 1 ) } \\ldots d { \\bf f } _ { t _ k } ^ { ( i _ k ) } \\right \\} , \\end{align*}"}
-{"id": "7052.png", "formula": "\\begin{align*} ( g ^ { t t } \\phi _ { v \\b { v } } u ^ v _ t u ^ v _ t \\o { e ^ v } ) _ { ; t t } = g ^ { t t } \\phi _ { v \\b { v } } u ^ v _ t u ^ v _ t \\o { e ^ v } _ { ; t t } , \\end{align*}"}
-{"id": "7499.png", "formula": "\\begin{align*} \\nabla _ { X } Y - \\nabla _ Y X - [ X , Y ] = 0 \\end{align*}"}
-{"id": "5110.png", "formula": "\\begin{align*} \\| y - y ^ * \\| ^ 4 & = \\| ( y - x ) - ( y ^ * - x ) \\| ^ 4 \\\\ & \\leq 2 \\| y - x \\| ^ 4 + 2 \\| y ^ * - x \\| ^ 4 + 1 2 \\| y - x \\| ^ 2 \\| y ^ * - x \\| ^ 2 - \\| ( y - x ) + ( y ^ * - x ) \\| ^ 4 \\\\ & = 2 d ^ 4 + 2 d ^ 4 + 1 2 d ^ 2 d ^ 2 - \\| y + y ^ * - 2 x \\| ^ 4 \\leq 1 6 d ^ 4 - 1 6 d ^ 4 = 0 . \\end{align*}"}
-{"id": "9478.png", "formula": "\\begin{align*} J _ { m } ( u ( x ) - u ( y ) ) - J _ { m } ( v ( x ) - v ( y ) ) = ( m - 1 ) \\left [ ( u - v ) ( x ) - ( u - v ) ( y ) \\right ] Q _ { m } ( x , y ) , \\end{align*}"}
-{"id": "4119.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ 0 \\log ( z - s ) d \\nu _ t ( s ) = \\log ( z - t ) - \\log \\left ( z ^ { \\frac { 1 } { 2 } } - \\sqrt { t } \\right ) = \\log \\left ( z ^ { \\frac { 1 } { 2 } } + \\sqrt { t } \\right ) . \\end{align*}"}
-{"id": "6012.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\ , \\rho ^ { G _ n } \\left ( F \\left ( \\frac 1 n \\sum _ { k = 1 } ^ n { W _ { ( n , k ) } } \\right ) \\right ) = \\sup _ { Q \\in \\P _ 1 ( \\C ) } \\left ( F \\left ( \\int _ { \\C } \\omega \\ , Q ( d \\omega ) \\right ) - { \\alpha } ^ g ( Q ) \\right ) . \\end{align*}"}
-{"id": "2577.png", "formula": "\\begin{align*} \\frac { d \\langle v _ r , \\phi ^ * \\rangle _ m } { d r } = - \\langle \\psi _ 0 ( \\cdot , v _ r ) , \\phi ^ * \\rangle _ m > 0 a . e . . \\end{align*}"}
-{"id": "4693.png", "formula": "\\begin{align*} \\frac { d u _ j } { d t } = - \\displaystyle { \\sum _ { k = 1 } ^ { m + d } \\mu _ k ( f ) \\Big ( B ^ d _ k ( x _ j ) \\Big ) ' } , ~ j = d - 1 , \\ldots , m - d + 1 . \\end{align*}"}
-{"id": "9665.png", "formula": "\\begin{align*} h _ { 0 1 } = ( \\theta _ 2 \\theta _ 0 + \\theta _ 1 ) h _ 0 \\chi _ \\tau + h _ 0 \\chi _ { \\mathbb T \\setminus \\tau } \\ \\ \\ \\ \\ \\ h _ { 0 2 } = ( - \\overline \\theta _ 2 + \\overline \\theta _ 1 \\theta _ 0 ) h _ 0 | _ \\tau . \\end{align*}"}
-{"id": "3783.png", "formula": "\\begin{align*} \\lambda ( \\alpha ^ t ) = \\lambda ( 1 ^ { N - 2 R + k + 2 - \\delta _ \\alpha } 2 ^ { q ' _ 0 } 3 ^ { q ' _ 1 } \\cdots ) . \\end{align*}"}
-{"id": "9695.png", "formula": "\\begin{align*} \\sigma _ { c } ( t ) = \\sum _ { i = 1 } ^ M \\sigma ( t - \\tau _ i ) + A , \\end{align*}"}
-{"id": "7872.png", "formula": "\\begin{align*} \\sum _ { S \\subseteq I } a _ S = \\prod _ { j \\in I } \\big ( \\cos ^ 2 ( \\pi \\xi _ j ) + \\sin ^ 2 ( \\pi \\xi _ j ) \\big ) = 1 . \\end{align*}"}
-{"id": "8404.png", "formula": "\\begin{align*} \\partial _ { \\alpha } ^ { k - 1 } \\partial _ t z _ { \\alpha } ^ { - 1 } = - \\frac { \\partial _ { \\alpha } ^ k z _ t } { z _ { \\alpha } ^ 2 } + H _ k , \\end{align*}"}
-{"id": "9123.png", "formula": "\\begin{align*} M _ { X _ { E } } ( \\rho _ { 1 } ( t ) ) \\equiv \\int \\limits _ { \\Gamma _ { 1 } } d \\mathbf { x } _ { 1 } \\rho _ { 1 } ( \\mathbf { x } _ { 1 } , t ) X _ { E } ( \\mathbf { x } _ { 1 } , t ) = - \\int \\limits _ { \\Gamma _ { 1 } } d \\mathbf { x } _ { 1 } \\rho _ { 1 } ( \\mathbf { x } _ { 1 } , t ) \\ln \\frac { \\rho _ { 1 } ( \\mathbf { x } _ { 1 } , t ) } { A _ { 1 } } \\equiv S ( \\rho _ { 1 } ( t ) ) . \\end{align*}"}
-{"id": "9049.png", "formula": "\\begin{align*} F _ { \\mu _ A } ( x ) = \\sum _ { \\xi \\in ( \\mu _ A ) } X _ { \\xi } e ( \\langle \\xi , x \\rangle ) \\end{align*}"}
-{"id": "6809.png", "formula": "\\begin{align*} \\varphi ( t , \\alpha ) = \\left ( A ( t ) A ^ { - 1 } ( 0 ) \\right ) \\ , \\left ( A ( 0 ) v ( \\alpha ) \\right ) = \\widetilde { A } ( t ) V ( \\alpha ) \\ , , \\end{align*}"}
-{"id": "2400.png", "formula": "\\begin{align*} \\frac { \\sum _ { j = r + 1 } ^ d \\sigma _ j ^ 2 } { \\sum _ { j = 1 } ^ d \\sigma _ j ^ 2 } < \\epsilon \\end{align*}"}
-{"id": "5496.png", "formula": "\\begin{align*} d ^ 2 _ L ( f , g ) = \\sum _ { i = 1 } ^ d \\left \\| \\partial _ { x _ i } L _ { b _ 0 } ^ { - 1 } [ f - g ] \\right \\| _ \\infty ^ 2 , \\end{align*}"}
-{"id": "5644.png", "formula": "\\begin{align*} \\partial _ t ^ \\alpha u ( t ) + A u ( t ) = { _ 0 I _ t ^ \\gamma } \\dot { W } ( t ) , \\forall 0 < t \\leq T , \\mbox { w i t h } u ( 0 ) = u _ { 0 } , \\end{align*}"}
-{"id": "1317.png", "formula": "\\begin{align*} \\rho [ Z , P ] = \\sup _ { \\mu \\in \\mathcal { D } _ \\rho } \\bigg \\{ \\int _ 0 ^ 1 \\varPhi [ Z , P ] ( p ) \\ ; \\mu ( d p ) - \\rho ^ * ( \\mu ) \\bigg \\} . \\end{align*}"}
-{"id": "8844.png", "formula": "\\begin{align*} 2 N \\left ( p \\mu - \\frac { x _ 1 + x _ N } 2 \\right ) & = 2 p ( x _ 2 + \\dots + x _ { N - 1 } ) - ( N - 2 p ) x _ 1 - ( N - 2 p ) x _ N \\\\ & \\ge 2 p ( x _ k + \\dots + x _ { N - 1 } ) - ( N - 2 p ) \\delta - ( N - 2 p ) a \\\\ & \\ge 2 p ( N - k ) a ( 1 - \\Delta ) - ( N - 2 p ) \\delta - ( N - 2 p ) a \\\\ & = a \\left [ 2 p ( N - k ) ( 1 - \\Delta ) - ( N - 2 p ) \\right ] - ( N - 2 p ) \\delta \\end{align*}"}
-{"id": "2078.png", "formula": "\\begin{align*} \\delta ( \\varphi - { \\varphi _ { \\rm t x } } ) = \\sum \\limits _ { n = 0 } ^ \\infty { L _ n \\cos ( n ( \\varphi - \\varphi _ { \\rm t x } ) ) } , \\end{align*}"}
-{"id": "6689.png", "formula": "\\begin{align*} B \\psi _ { 0 } ( x ) = \\lambda _ { * } \\psi _ { 0 } ( x ) , \\ \\ B = i | x | ^ { - \\frac { ( 1 - i \\omega ) } { 2 } } x \\frac { d } { d x } | x | ^ { \\frac { ( 1 - i \\omega ) } { 2 } } . \\end{align*}"}
-{"id": "4405.png", "formula": "\\begin{align*} \\omega _ o | _ S = \\Phi ^ * \\omega _ o = d d ^ c ( | s | ^ 2 + | t | ^ 2 + | s t ^ 2 | ^ 2 ) . \\end{align*}"}
-{"id": "2157.png", "formula": "\\begin{align*} g _ { L } ( \\mathbf { x } ) = \\max \\{ 0 , ( 1 - \\| \\mathbf { x } \\| ^ 2 ) \\} ^ { L } . \\end{align*}"}
-{"id": "4791.png", "formula": "\\begin{align*} \\quad { \\sigma } _ 1 = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} , { \\sigma } _ 2 = \\begin{pmatrix} 0 & - i \\\\ i & 0 \\end{pmatrix} , \\quad { \\sigma } _ 3 = \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "8724.png", "formula": "\\begin{align*} E ( T ^ * _ { m , N } ) = ( N - 1 ) \\sum _ { j \\leq m } \\hat { \\nu } ( j ) \\sum _ { k = j } ^ { N - 1 } { 1 \\over N - k } . \\end{align*}"}
-{"id": "8535.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } ( I + \\mathcal { H } ) D _ t \\zeta = \\frac { 1 } { 2 } ( I + \\mathcal { H } ) \\bar { q } + \\frac { 1 } { 2 } ( \\mathcal { H } + \\bar { \\mathcal { H } } ) \\bar { \\mathfrak { F } } , \\end{align*}"}
-{"id": "9910.png", "formula": "\\begin{align*} I ^ { + } ( y _ { [ 0 , n - 1 ] } ) & = \\{ y _ { n } \\in \\mathbb { Y } | \\int _ { \\mathcal { X } } f ( x ) F ( \\pi _ { n - 1 } ^ { \\mu } , y _ { n } ) ( d x ) \\\\ & > \\int _ { X } f ( x ) F ( \\pi _ { n - 1 } ^ { \\nu } , y _ { n } ) ( d x ) \\} \\end{align*}"}
-{"id": "8496.png", "formula": "\\begin{align*} y ( t ) = y ( 0 ) + \\int _ 0 ^ t \\dot { y } ( \\tau ) d \\tau . \\end{align*}"}
-{"id": "4120.png", "formula": "\\begin{align*} g _ { 2 } ( z ) & = \\int _ 0 ^ q \\left ( \\int _ { - \\infty } ^ 0 \\log ( z - s ) d \\nu _ t ( s ) \\right ) d \\mu ^ * ( t ) = \\int _ { 0 } ^ { q } \\log \\left ( z ^ { \\frac { 1 } { 2 } } + \\sqrt { t } \\right ) d \\mu ^ * ( t ) . \\end{align*}"}
-{"id": "5295.png", "formula": "\\begin{align*} H ^ 2 ( \\mathcal { X } , \\Z ) = \\bigoplus _ { i = 1 } ^ n \\Z H _ i \\oplus \\bigoplus _ { k = 1 } ^ { K } \\Z E _ k , H _ 2 ( \\mathcal { X } , \\Z ) = \\bigoplus _ { i = 1 } ^ n \\Z h _ i \\oplus \\bigoplus _ { k = 1 } ^ { K } \\Z e _ k \\end{align*}"}
-{"id": "5293.png", "formula": "\\begin{align*} \\frac { d q _ 1 } { d t } = \\frac { \\partial I _ i } { \\partial p _ 1 } , & \\frac { d p _ 1 } { d t } = - \\frac { \\partial I _ i } { \\partial q _ 1 } \\\\ \\frac { d q _ 2 } { d t } = \\frac { \\partial I _ i } { \\partial p _ 2 } , & \\frac { d p _ 2 } { d t } = - \\frac { \\partial I _ i } { \\partial q _ 2 } \\end{align*}"}
-{"id": "916.png", "formula": "\\begin{align*} K ( t _ 1 , \\ldots , t _ k ) = \\begin{cases} \\psi _ 1 ( t _ 1 ) \\ldots \\psi _ k ( t _ k ) \\ & \\hbox { f o r } \\ \\ t _ 1 < \\ldots < t _ k \\\\ ~ \\\\ ~ \\\\ 0 \\ & \\hbox { o t h e r w i s e } \\end{cases} , \\ \\ \\ \\ t _ 1 , \\ldots , t _ k \\in [ t , T ] , \\ \\ \\ \\ k \\ge 2 , \\end{align*}"}
-{"id": "6025.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\tilde { \\alpha } ^ \\mu _ { \\epsilon _ n } ( Q _ n ) = \\liminf _ { \\epsilon \\downarrow 0 } \\ , \\ , \\ , \\inf \\left \\{ \\tilde { \\alpha } ^ \\mu _ \\epsilon ( Q ) \\ , : \\ , Q \\in \\mathcal { P } ( \\C ) , \\ Q \\circ H ^ { - 1 } = \\nu _ \\epsilon \\right \\} \\end{align*}"}
-{"id": "9110.png", "formula": "\\begin{align*} H ^ 0 ( X _ I , L ( \\lambda ) ) \\simeq \\begin{cases} R _ G ( \\lambda ) & \\ \\lambda \\in \\Lambda ^ + _ I ( G ) ; \\\\ 0 & ; \\end{cases} \\end{align*}"}
-{"id": "2303.png", "formula": "\\begin{align*} a ' & = ( 1 , ( E | \\frac { \\mathbb { P } } { x - \\Omega } ) & b ' & = ( 0 , ( \\delta _ x | ) . \\end{align*}"}
-{"id": "5221.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + 1 } U ( n + 1 ) / U ( n ) = 1 . \\end{align*}"}
-{"id": "6102.png", "formula": "\\begin{align*} r _ k ( \\phi _ k , \\delta ^ \\phi _ k ) \\leq r _ k ( \\phi _ k , \\hat { \\mu } _ 0 ) = R _ 0 , \\end{align*}"}
-{"id": "5291.png", "formula": "\\begin{align*} & \\begin{array} { r c l } I _ 1 & = & q _ 1 p _ 1 ( q _ 1 + p _ 1 - b ) - a ( q _ 1 + p _ 1 ) \\\\ I _ 2 & = & q _ 2 p _ 2 ( q _ 2 + p _ 2 - b ) - a ( q _ 2 + p _ 2 ) . \\\\ \\end{array} \\end{align*}"}
-{"id": "3371.png", "formula": "\\begin{align*} Q ( s ) : = \\inf \\{ x \\in \\mathbb { R } : F ( x ) \\geq s \\} , 0 < s \\leq 1 , Q ( 0 ) : = \\lim _ { s \\searrow 0 } Q ( s ) . \\end{align*}"}
-{"id": "5150.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\begin{vmatrix} f ^ + ( r ) & \\overline { \\widetilde { f } ^ + ( r ) } \\\\ f ^ - ( r ) & \\overline { \\widetilde { f } ^ - ( r ) } \\end{vmatrix} = \\begin{vmatrix} E \\begin{pmatrix} A ^ + \\\\ A ^ - \\end{pmatrix} \\overline { E \\begin{pmatrix} \\widetilde A ^ + \\\\ \\widetilde A ^ - \\end{pmatrix} } \\end{vmatrix} , \\end{align*}"}
-{"id": "2522.png", "formula": "\\begin{align*} \\psi ( x , z ) : = - \\beta ( x ) z + \\sigma ( x ) ^ 2 z ^ 2 + \\int _ { ( 0 , \\infty ) } ( e ^ { - z y } - 1 + z y ) \\pi ( x , d y ) , x \\in E , z \\geq 0 , \\end{align*}"}
-{"id": "754.png", "formula": "\\begin{align*} C = \\sum _ m \\lambda _ m ( v _ m \\otimes v _ m ) . \\end{align*}"}
-{"id": "6364.png", "formula": "\\begin{align*} ( \\omega _ t + d d ^ c \\varphi _ t ) ^ n \\wedge d t = e ^ { \\dot { \\varphi } _ t + F ( t , x , \\varphi _ t ) } g ( x ) d V ( x ) \\wedge d t . \\end{align*}"}
-{"id": "7585.png", "formula": "\\begin{align*} \\begin{gathered} \\sum _ { i , j } ( \\partial _ i \\partial _ j g _ { \\nu ( t - s ) } * ) ( \\eta _ i ( s ) - \\eta _ i ( t ) ) \\sigma _ { j k } ( s ) , \\\\ \\sum _ { i , j } \\eta _ i ( t ) \\left ( \\partial _ i \\partial _ j g _ { \\nu ( t - s ) } \\right ) * \\sigma _ { j k } ( s ) , \\ , \\ , \\mathrm { a n d \\ ; r e s p e c t i v e l y } \\sum _ { i , j } \\left ( \\partial _ i \\partial _ j g _ { \\nu ( t - s ) } \\right ) * ( \\eta _ i ( s ) \\sigma _ { j k } ( s ) ) . \\end{gathered} \\end{align*}"}
-{"id": "9849.png", "formula": "\\begin{align*} \\frac { 1 } { ( 1 - u ^ 2 ) ^ { r + 1 } } \\begin{bmatrix} \\frac { u ^ 4 } { 1 - u } ( ( 1 - u ) ^ r - 1 ) - \\frac { u ^ 3 } { 1 - u } ( ( 1 - u ^ 2 ) ^ r - 1 ) \\\\ - \\frac { u ^ 6 } { 1 - u ^ 3 } ( ( 1 - u ) ^ r - 1 ) + \\frac { u ^ 3 } { 1 - u ^ 3 } ( ( 1 - u ^ 4 ) ^ r - 1 ) \\end{bmatrix} \\end{align*}"}
-{"id": "5360.png", "formula": "\\begin{align*} v _ { n + 2 \\ , m } ^ 1 = \\frac 1 { a _ { n + 1 \\ , m - 3 } } X _ { \\alpha + \\beta } . \\frac 1 { b _ { n m } } Y _ \\beta . v _ { n m } ^ 1 \\end{align*}"}
-{"id": "7118.png", "formula": "\\begin{align*} & u '' + ( - q _ 0 + \\lambda + u ^ 2 ) u = 0 \\\\ & \\phi _ k '' + ( - q _ 0 + \\lambda _ k ) \\phi _ k = 0 \\end{align*}"}
-{"id": "2991.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty p ' _ m ( n ) q ^ n & = \\frac { 1 } { ( q ; q ^ m ) _ \\infty ( q ^ { m - 1 } ; q ^ m ) _ \\infty ( q ^ m ; q ^ m ) _ \\infty } . \\end{align*}"}
-{"id": "10064.png", "formula": "\\begin{align*} q = \\frac { 1 } { 2 } ( \\tilde x + \\tilde y ) , p = \\frac { 1 } { 2 } ( \\tilde x - \\tilde y ) . \\end{align*}"}
-{"id": "324.png", "formula": "\\begin{align*} \\begin{array} { l l } c _ { n - 1 } ( \\alpha _ 1 ) = ( n - 2 ) ! \\Sigma ^ 2 v _ { 2 n - 4 } & c _ n ( \\alpha _ 1 ) = \\frac { ( n - 1 ) ! } { 2 } \\Sigma ^ 2 v _ { 2 n - 2 } \\\\ c _ { n - 1 } ( \\alpha _ 2 ) = 0 & c _ n ( \\alpha _ 2 ) = ( n - 1 ) ! \\Sigma ^ 2 v _ { 2 n - 2 } . \\end{array} \\end{align*}"}
-{"id": "5455.png", "formula": "\\begin{align*} \\bar J ^ { \\mathcal R } ( \\bar \\nu ) \\ = \\ \\bar J ( \\bar \\alpha ^ { \\bar \\nu } ) . \\end{align*}"}
-{"id": "8402.png", "formula": "\\begin{align*} & \\norm { \\frac { 1 } { z _ { \\alpha } ^ { k - 2 } } \\partial _ { \\alpha } ^ { k - 1 } ( \\frac { 1 } { z _ { \\alpha } } ) \\partial _ { \\alpha } D u } _ { L ^ 2 } \\\\ \\leq & C ( \\| \\partial _ { \\alpha } \\xi _ 0 \\| _ { H ^ { s - 1 } } , \\| v _ 0 \\| _ { H ^ s } , \\| w _ 0 \\| _ { H ^ s } , \\mathcal { E } ( t ) , d _ I ( t ) ^ { - 1 } , d _ P ( t ) ^ { - 1 } , N \\lambda _ { m a x } , C _ 1 , C _ 2 ) . \\end{align*}"}
-{"id": "7635.png", "formula": "\\begin{align*} K \\varphi ( \\xi ) \\ = \\ \\int \\limits _ { \\mathbb R ^ d } a ( \\xi - q ) \\mu ( \\xi , q ) \\varphi ( q ) \\ d q \\ = \\ \\int \\limits _ { \\mathbb T ^ d } \\hat a ( \\xi - \\eta ) \\mu ( \\xi , \\eta ) \\varphi ( \\eta ) \\ d \\eta , \\varphi \\in L ^ 2 ( \\mathbb T ^ d ) , \\end{align*}"}
-{"id": "8371.png", "formula": "\\begin{align*} | \\dot { z } _ j ( t ) | \\leq & \\| z _ t \\| _ { \\infty } + \\norm { \\sum _ { k \\neq j } \\frac { \\lambda _ k i } { 2 \\pi ( \\overline { z _ j ( t ) - z _ k ( t ) } ) } } _ { \\infty } + \\norm { \\sum _ { k = 1 } ^ N \\frac { \\lambda _ k i } { 2 \\pi \\overline { z ( \\alpha , t ) - z _ k ( t ) } } } _ { \\infty } \\\\ \\leq & \\| z _ t \\| _ { H ^ 1 } + N \\lambda _ { m a x } ( d _ P ( t ) ^ { - 1 } + d _ I ( t ) ^ { - 1 } ) . \\end{align*}"}
-{"id": "7307.png", "formula": "\\begin{align*} 2 E _ 0 x & = \\theta x + ( - 1 ) ^ { ( p + \\bar { p } + q + \\bar { q } ) } x \\theta , & 2 F _ 1 x & = \\theta x - ( - 1 ) ^ { ( p + \\bar { p } + q + \\bar { q } ) } x \\theta \\\\ 2 F _ 2 x & = \\bar { \\theta } x + ( - 1 ) ^ { ( p + \\bar { p } + q + \\bar { q } ) } x \\bar { \\theta } , & 2 E _ 3 & = \\bar { \\theta } x - ( - 1 ) ^ { ( p + \\bar { p } + q + \\bar { q } ) } x \\bar { \\theta } \\end{align*}"}
-{"id": "3708.png", "formula": "\\begin{align*} \\widehat { V } ( \\widehat { Y } ) = \\sum _ t \\big ( \\hat { p } _ t ^ { - 1 } - 1 \\big ) \\hat { p } _ t ^ { - 1 } \\sum _ { i \\in B _ t } ( y _ i - t _ i ^ { \\top } \\hat { \\beta } ) ^ 2 \\end{align*}"}
-{"id": "3935.png", "formula": "\\begin{align*} \\Lambda ( r \\otimes x \\cdot f ) = & \\ \\sum r _ { 0 } \\sigma \\bigl ( S ( r _ { 1 } ) ( x \\cdot f ) \\bigr ) = \\sum ( x \\cdot r _ { 0 } ) \\sigma \\bigl ( S ( r _ { 1 } \\cdot x ^ { - 1 } ) ( x \\cdot f ) \\bigr ) = \\\\ & \\ \\sum ( x \\cdot r _ { 0 } ) \\sigma \\bigl ( x \\cdot \\bigl ( S ( r _ { 1 } ) f \\bigr ) \\bigr ) = x \\cdot \\sum r _ { 0 } \\sigma \\bigl ( S ( r _ { 1 } ) f \\bigr ) = \\\\ & \\ x \\cdot \\Lambda ( r \\otimes f ) \\end{align*}"}
-{"id": "3333.png", "formula": "\\begin{align*} B ( u ) & = \\prod _ { i = 1 } ^ { r } ( \\mu _ i u + \\sum _ { y = 1 } ^ { \\mu _ i - 1 } \\sum _ { b = 2 } ^ { \\mu _ i - y + 1 } \\binom { \\mu _ i - y - 1 } { b - 2 } y u ^ { b } ) \\\\ & = \\prod _ { i = 1 } ^ { r } ( \\mu _ i u + \\sum _ { y = 1 } ^ { \\mu _ i - 1 } y u ^ 2 ( 1 + u ) ^ { \\mu _ i - y - 1 } ) \\\\ & = \\prod _ { i = 1 } ^ { r } ( ( 1 + u ) ^ { \\mu _ i } - 1 ) , \\end{align*}"}
-{"id": "4716.png", "formula": "\\begin{align*} B ^ { n } ( t ) = \\left \\{ \\begin{array} { l l } \\left ( \\dfrac { t - t ^ { n - 3 / 2 } } { \\Delta t } \\right ) & \\mbox { i f } t ^ { n - 3 / 2 } < t \\le t ^ { n - 1 / 2 } , \\medskip \\\\ \\left ( \\dfrac { t ^ { n + 1 / 2 } - t } { \\Delta t } \\right ) & \\mbox { i f } t ^ { n - 1 / 2 } < t \\le t ^ { n + 1 / 2 } , \\medskip \\\\ 0 & \\mbox { o t h e r w i s e } \\end{array} \\right . \\end{align*}"}
-{"id": "7375.png", "formula": "\\begin{align*} \\frac { d N } { d t } & = ( N _ r - N ) D - I _ { N R } \\frac { N } { \\kappa _ { N R } + N } R , \\\\ \\frac { d R } { d t } & = \\bigg ( \\mu _ { N R } \\frac { N } { \\kappa _ { N R } + N } - ( D + m _ R ( c _ R ) ) \\bigg ) R - I _ { R P } \\frac { R } { \\kappa _ { R P } + R } P , \\\\ \\frac { d P } { d t } & = \\bigg ( \\mu _ { R P } \\frac { R } { \\kappa _ { R P } + R } - ( D + m _ { P 0 } ) \\bigg ) P , \\\\ \\frac { d c _ { T } } { d t } & = ( c _ { r } - c _ { T } ) D , \\end{align*}"}
-{"id": "8636.png", "formula": "\\begin{align*} \\hat { H } ( p , q , u , z ) = 0 . \\end{align*}"}
-{"id": "5878.png", "formula": "\\begin{align*} C _ w = \\sum _ { y \\leq w } ( - 1 ) ^ { \\ell ( w ) - \\ell ( y ) } q ^ { \\frac { \\ell ( W ) - \\ell ( y ) } { 2 } } P _ { y , w } ( q ^ { - 1 } ) q ^ { - \\frac { \\ell ( y ) } { 2 } } T _ y \\end{align*}"}
-{"id": "4240.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} d u & = f & & B _ { R } , \\\\ \\delta u & = g & & B _ { R } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "4912.png", "formula": "\\begin{align*} \\langle w ^ i w ^ j \\rangle = g ^ { i j } \\ , , \\end{align*}"}
-{"id": "2835.png", "formula": "\\begin{align*} \\sqrt { - 1 } \\Lambda F _ A = \\mu \\cdot { \\bf I } _ { E } \\ , \\end{align*}"}
-{"id": "9060.png", "formula": "\\begin{align*} 1 5 \\pi ^ 3 \\sum _ { \\xi _ 1 , \\xi _ 2 , \\xi _ 3 } | a _ { \\xi _ 1 } | ^ 2 | a _ { \\xi _ 2 } | ^ 2 | a _ { \\xi _ 3 } | ^ 2 = 1 5 \\left ( \\sum _ { \\xi } | a _ { \\xi } | ^ 2 \\right ) ^ 3 = 1 5 \\pi ^ 3 . \\end{align*}"}
-{"id": "5084.png", "formula": "\\begin{align*} \\| h - S _ { y , \\tau } S _ { x , \\tau } ^ { - 1 } h \\| & = \\left \\| \\sum _ { j \\in \\mathbb { J } } \\langle h , x _ j \\rangle S _ { x , \\tau } ^ { - 1 } \\tau _ j - \\sum _ { j \\in \\mathbb { J } } \\langle h , y _ j \\rangle S _ { x , \\tau } ^ { - 1 } \\tau _ j \\right \\| = \\left \\| \\sum _ { j \\in \\mathbb { J } } \\langle h , x _ j - y _ j \\rangle S _ { x , \\tau } ^ { - 1 } \\tau _ j \\right \\| \\\\ & \\leq \\sum _ { j \\in \\mathbb { J } } \\| h \\| \\| x _ j - y _ j \\| \\| S _ { x , \\tau } ^ { - 1 } \\tau _ j \\| = \\beta \\| h \\| . \\end{align*}"}
-{"id": "5270.png", "formula": "\\begin{align*} 0 \\leq \\ \\sum _ { k = 1 } ^ { \\infty } \\lambda ( I _ { k } ) - \\ \\sum _ { k = 1 } ^ { k _ { 0 } } \\lambda ( I _ { k } ) = \\sum _ { k \\geq k _ { 0 } + 1 } \\lambda ( I _ { k } ) < \\varepsilon / 2 . \\end{align*}"}
-{"id": "1122.png", "formula": "\\begin{align*} & \\int _ { I ^ k } \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) d y _ 1 \\cdots d y _ k \\\\ \\geq & S ( I ) ^ k ( \\ln n ) ^ { - k } \\int _ { ( 0 , l _ n ) ^ k } \\phi _ { k , n } \\big ( \\gamma _ n ( u _ 1 ) , \\cdots , \\gamma _ n ( u _ k ) \\big ) \\prod _ { j = 1 } ^ k \\beta _ n ( u _ j ) d u _ 1 \\cdots d u _ k . \\end{align*}"}
-{"id": "6924.png", "formula": "\\begin{align*} Q _ 0 = 1 < Q _ 1 \\cdots < Q _ m \\end{align*}"}
-{"id": "5559.png", "formula": "\\begin{align*} \\left | \\bigcup _ { g \\in G ( q ) } X ( q _ 0 ) ^ g \\right | \\le | X ( q _ 0 ) | \\ , | G ( q ) : G ( q _ 0 ) | \\le c _ { 1 0 } q _ 0 ^ { s / 2 } q ^ { d / 2 } q _ 0 ^ { - d / 2 } = c _ { 1 0 } q ^ { \\frac { d } { 2 } + \\frac { s - d } { 2 k } } \\le c _ { 1 0 } q ^ { \\frac { s } { 2 } - \\frac { 1 } { 2 } + \\frac { 1 } { 2 k } } . \\end{align*}"}
-{"id": "826.png", "formula": "\\begin{align*} R _ { X / Y } = \\prod _ { i = 1 } ^ s \\prod _ { Q _ { \\nu , i } \\mapsto P _ i } Q _ { \\nu , i } ^ { \\delta _ { i } } \\end{align*}"}
-{"id": "5492.png", "formula": "\\begin{align*} \\hat b _ T = \\hat b ( X ^ T ) = _ { b \\in V _ J ^ { \\otimes d } } \\big [ - \\ell _ T ( b ) + \\frac { 1 } { 2 } \\| b \\| _ { \\mathbb H } ^ 2 \\big ] , \\end{align*}"}
-{"id": "3502.png", "formula": "\\begin{align*} \\frac { \\psi ( s ) - \\psi ( t ) } { s - t } = \\sum ^ { \\infty } _ { n = 0 } \\frac { - 1 } { s - t } \\Big ( \\frac { 1 } { n + s } - \\frac { 1 } { n + t } \\Big ) = \\sum ^ { \\infty } _ { n = 0 } \\frac { 1 } { ( n + s ) ( n + t ) } , \\end{align*}"}
-{"id": "320.png", "formula": "\\begin{align*} \\begin{array} { c c c } f ^ * ( a _ { 2 n } ) = x u ^ 2 v _ { 2 n - 4 } + y u v _ { 2 n - 2 } & & f ^ * ( a _ { 2 n + 2 } ) = z u ^ 2 v _ { 2 n - 2 } . \\end{array} \\end{align*}"}
-{"id": "7272.png", "formula": "\\begin{align*} \\sum _ { k \\neq 0 } \\left ( \\frac { 1 } { 2 } \\widehat { \\phi ^ { 2 } } ( k ) - \\mu \\hat \\phi ( k ) + k ^ { - 2 } \\hat \\phi ( k ) \\right ) e ^ { i x k } = 0 . \\end{align*}"}
-{"id": "7765.png", "formula": "\\begin{align*} | \\sum _ { j = 1 } ^ k t _ { j } \\langle B , x _ { j } y _ { j } ^ * \\rangle | = \\| B \\| _ { \\nu } . \\end{align*}"}
-{"id": "9362.png", "formula": "\\begin{align*} \\gamma _ { 2 i } ( G _ k ) / \\gamma _ { 2 i + 1 } ( G _ k ) \\cong C _ 2 \\gamma _ { 2 i + 1 } ( G _ k ) / \\gamma _ { 2 i + 2 } ( G _ k ) \\cong \\left \\{ \\begin{array} { c c } C _ 2 \\times C _ 2 & i \\ne 2 ^ { k - 1 } \\\\ C _ 2 & i = 2 ^ { k - 1 } . \\end{array} \\right . \\end{align*}"}
-{"id": "6182.png", "formula": "\\begin{align*} U _ t h _ r = H _ r + \\frac { 1 } { 2 } \\frac { H } { r } . \\end{align*}"}
-{"id": "5649.png", "formula": "\\begin{align*} \\tau ^ { - \\alpha } \\sum _ { k = 0 } ^ { n } b _ { n - k } ^ { ( \\alpha ) } ( U ^ { k } - U ^ 0 ) + A _ { h } U ^ { n } = \\tau ^ { \\gamma } \\sum _ { k = 0 } ^ { n } b _ { n - k } ^ { ( - \\gamma ) } f ^ { k } , \\ ; n = 1 , 2 , \\dots , N , \\end{align*}"}
-{"id": "5066.png", "formula": "\\begin{align*} A _ { \\sigma ( U V ) } A _ { \\sigma ( U W ) } ^ * & = A _ I \\pi ( \\sigma ( U V ) ) ^ { - 1 } ( A _ I \\pi ( \\sigma ( U W ) ) ^ { - 1 } ) ^ * \\\\ & = A _ I ( \\overline { f ( U V ) } \\pi ( U ) \\pi ( V ) ) ^ { - 1 } \\overline { f ( U W ) } \\pi ( U ) \\pi ( W ) A ^ * _ I \\\\ & = f ( U V ) \\overline { f ( U W ) } A _ I \\pi ( V ) ^ { - 1 } ( A _ I \\pi ( W ) ^ { - 1 } ) ^ * \\\\ & = f ( U V ) \\overline { f ( U W ) } A _ V A _ W ^ * , \\end{align*}"}
-{"id": "156.png", "formula": "\\begin{align*} X _ { V } = \\sum _ { i _ 1 , \\dots , i _ k = 1 } ^ { N } v ^ { i _ 1 \\dots i _ k } ( { \\bf x } ) \\dfrac { \\partial } { \\partial y ^ { i _ 1 } } \\wedge \\dots \\wedge \\dfrac { \\partial } { \\partial y ^ { i _ k } } \\in \\mathcal { X } ^ { k } ( T M ) . \\end{align*}"}
-{"id": "4762.png", "formula": "\\begin{align*} x = \\sum _ { j = 0 } ^ { \\infty } x _ j t ^ j , \\end{align*}"}
-{"id": "9047.png", "formula": "\\begin{align*} & | X ( \\mu _ { f } ) - X ( \\mu ) | = \\frac { 1 } { \\pi R ^ 2 } \\int _ { B ( R ) } | F _ { \\mu } - ( F _ { \\mu } - F _ { \\mu _ { f } } ) | ^ 2 - | F _ { \\mu } | ^ 2 \\\\ & \\leq 2 \\underset { B ( R ) } { \\sup } | F _ { \\mu _ f } - F _ { \\mu } | | F _ \\mu | + \\underset { B ( R ) } { \\sup } | F _ { \\mu _ f } - F _ { \\mu } | ^ 2 \\leq 2 M \\alpha + \\alpha ^ 2 \\leq \\epsilon / 4 t \\end{align*}"}
-{"id": "9132.png", "formula": "\\begin{align*} K _ { M } ( \\rho _ { 1 } ^ { ( N ) } ( t ) , \\mathbf { b } ) = 0 \\Leftrightarrow \\rho _ { 1 } ^ { ( N ) } ( \\mathbf { x } _ { 1 } , t ) \\equiv \\rho _ { 1 M } ^ { ( N ) } ( \\mathbf { v } _ { 1 } ) . \\end{align*}"}
-{"id": "2372.png", "formula": "\\begin{align*} 0 & \\le \\varrho _ i \\le 1 , & \\varrho _ 1 + \\varrho _ 2 & = 1 , & \\varrho _ 1 & \\ge \\varrho _ 2 , & 0 & \\le z \\le 1 / 2 \\end{align*}"}
-{"id": "8754.png", "formula": "\\begin{align*} y ^ { ( j ) } _ { n } = \\frac { e ^ { - \\tau \\sigma } } { 1 + \\lambda _ j \\tau } \\left ( y ^ { ( j ) } _ { n - 1 } + \\tau u _ { n - 1 } \\right ) , { \\quad } y ^ { ( j ) } _ 0 = 0 . \\end{align*}"}
-{"id": "6478.png", "formula": "\\begin{align*} t _ { B } & = \\frac { 1 } { b ^ { 2 } } \\left ( \\left [ 1 + \\frac { b ^ { 2 } } { ( a + b ) ^ { 2 } } - \\frac { ( b - a ) ^ { 2 } } { ( a + b ) ^ { 2 } } \\right ] ( a + b ) ^ { 2 } - 4 a b \\right ) \\\\ & = \\frac { 1 } { b ^ { 2 } } ( ( a + b ) ^ { 2 } + b ^ { 2 } - ( b - a ) ^ { 2 } - 4 a b ) = 1 , \\end{align*}"}
-{"id": "3622.png", "formula": "\\begin{align*} a _ 1 \\cdot ( ( \\alpha _ { - 1 } - \\beta _ { - 1 } ) \\cdot ( \\alpha _ 0 + \\beta _ 0 + \\frac { 5 } { 3 } \\alpha _ 2 ) ) = \\frac { 1 } { 2 ^ 2 } ( \\alpha _ { - 1 } \\cdot \\beta _ 0 - ( \\alpha _ 0 + \\frac { 5 } { 3 } \\alpha _ 2 ) \\cdot \\beta _ { - 1 } ) + \\frac { 1 } { 2 ^ 2 } \\langle \\beta _ 0 , \\beta _ { - 1 } \\rangle . \\end{align*}"}
-{"id": "2761.png", "formula": "\\begin{align*} I S ^ { \\mathrm { s o f t } } ( \\kappa , \\pi ; u , v ) & = I S ^ { ( \\mathrm { s o f t } ) } ( \\kappa ; u , v ) \\\\ & + \\sum _ { p = 1 } ^ { m } \\Big ( \\phi _ { 2 , p - 1 } ( \\kappa , \\pi ; u ) \\phi _ { 2 , p } ( \\kappa , \\pi ; v ) - \\phi _ { 2 , p } ( \\kappa , \\pi ; u ) \\phi _ { 2 , p - 1 } ( \\kappa , \\pi ; v ) \\Big ) . \\end{align*}"}
-{"id": "8235.png", "formula": "\\begin{align*} L _ i = F \\left ( \\sum _ { j = 1 } ^ \\infty a ( i , j ) t ^ j + \\rho \\sum _ { j = 1 } ^ \\infty b ( i , j ) t ^ j \\right ) . \\end{align*}"}
-{"id": "1882.png", "formula": "\\begin{align*} \\psi ^ { ( 1 ) } ( y ) = \\int _ 0 ^ \\infty \\frac { t e ^ { - y t } } { 1 - e ^ { - t } } d t , \\end{align*}"}
-{"id": "3406.png", "formula": "\\begin{align*} \\tau ^ { M } ( T ) \\leq 2 ^ { - ( M - 1 ) } \\sum _ { i = 1 } ^ { w _ { M - 1 } ( T ) } \\tau ^ 1 ( T _ { v _ i } ) \\leq 2 ^ { - ( M - 1 ) } \\sum _ { i = 1 } ^ { w _ { M - 1 } ( T ) } \\deg ( T _ { v _ i } ) \\le 2 ^ { - ( M - 1 ) } w _ M ( T ) . \\end{align*}"}
-{"id": "9259.png", "formula": "\\begin{align*} v _ 1 ( 0 ) = \\zeta ^ 2 ( 0 ) w _ 1 ( 0 ) \\le \\epsilon M _ 2 ( R ) + C _ \\epsilon ( 1 + \\frac { 1 } { R ^ 2 } ) , \\end{align*}"}
-{"id": "3120.png", "formula": "\\begin{align*} M ^ { ( 3 ) } = \\begin{pmatrix} 1 & 8 & 0 & 0 & 0 & 0 & 0 \\\\ 1 & 9 & 0 & 0 & 0 & 0 & 0 \\\\ 1 & 9 & 9 & 9 & 0 & 0 & 0 \\\\ 1 & 9 & - 9 & 9 & 0 & 0 & 0 \\\\ 1 & 9 & 0 & 0 & 9 & 9 & 0 \\\\ 1 & 9 & 0 & 0 & - 9 & 9 & 0 \\\\ 1 & 9 & 9 & 9 & 9 & 9 & 9 \\end{pmatrix} \\end{align*}"}
-{"id": "9819.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } \\mathcal { F } _ a ( \\hat { u } - \\hat { w } ) ( x ' ) & = & - \\mathcal { F } _ a ( \\hat { w } ) ( x ' ) & ( \\hat { u } - \\hat { w } ) ( x ' , 0 , 0 ) > 0 \\\\ \\mathcal { F } _ a ( \\hat { u } - \\hat { w } ) ( x ' ) & \\leq & - \\mathcal { F } _ a ( \\hat { w } ) ( x ' ) & ( \\hat { u } - \\hat { w } ) ( x ' , 0 , 0 ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "1167.png", "formula": "\\begin{align*} \\mathbb { E } \\sum _ { k = 1 } ^ n ( \\tau _ k - x ) _ + = \\mathbb { E } \\sum _ { k = 1 } ^ n ( m _ k - F _ n ( x ) ) _ + ( n / 4 ) ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } \\\\ = 2 \\pi ( n / 4 ) ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } D _ n ( F _ n ( x ) / 2 ) \\to ( \\pi / 2 ) e ^ { c _ 0 - x } . \\end{align*}"}
-{"id": "2830.png", "formula": "\\begin{align*} U _ 1 ( t ) = \\sum _ { i = 1 } ^ p \\eta _ i \\biggl ( G ( x _ i ( t ) , x _ i ^ * ) + \\frac { c _ i } { d _ i } \\frac { ( u _ i ( t ) - u _ i ^ * ) ^ 2 } { 2 } \\biggr ) . \\end{align*}"}
-{"id": "8887.png", "formula": "\\begin{align*} \\widehat \\phi ( \\emptyset ) & = 1 _ \\Omega , \\\\ \\widehat \\phi ( F _ 1 F _ 2 ) & = \\widehat \\phi ( F _ 1 ) \\widehat \\phi ( F _ 2 ) , \\\\ \\widehat \\phi ( B _ + ^ \\omega ( F ) ) & = \\phi \\left ( \\omega \\widehat \\phi ( F ) \\right ) . \\end{align*}"}
-{"id": "4640.png", "formula": "\\begin{align*} h _ { f _ { 1 , \\lambda } } ( 0 ) = 0 h _ { f _ { 2 , \\lambda } } ( 0 ) = 0 . \\end{align*}"}
-{"id": "1769.png", "formula": "\\begin{align*} N _ { s , t } ^ { ( n ) } = \\max _ { 1 \\leq i \\leq n } Y _ { i } = \\max _ { 1 \\leq i \\leq n } \\left ( \\omega _ { t _ { i } } - \\omega _ { s } \\right ) + \\xi _ { s , t } . \\end{align*}"}
-{"id": "591.png", "formula": "\\begin{align*} \\underline { \\tilde { \\Lambda } } ^ { x , \\infty } _ t ( \\omega ) = \\begin{cases} \\Lambda ^ { x , \\infty } _ t , \\ ; \\ ; t \\in [ 0 , \\underline { \\tau } ^ { x , \\infty } ( \\omega ) ) , \\\\ - \\lim _ { k \\to \\infty } \\ , \\underline { Z } ^ { x , k } _ { \\underline { \\tau } ^ { x , k } ( \\omega ) } ( \\omega ) , \\ ; \\ ; t \\in ( \\underline { \\tau } ^ { x , \\infty } ( \\omega ) , T ] , \\end{cases} \\end{align*}"}
-{"id": "5273.png", "formula": "\\begin{align*} \\forall x \\in \\mathbb { R } , h ( x ) = x h ( 1 ) . \\end{align*}"}
-{"id": "2760.png", "formula": "\\begin{align*} S ^ { \\mathrm { s o f t } } ( \\kappa , \\pi ; u , v ) & = S ^ { ( \\mathrm { s o f t } ) } ( \\kappa ; u , v ) \\\\ & + \\sum _ { p = 1 } ^ { m } \\Big ( \\phi _ { 1 , p } ( \\kappa , \\pi ; u ) \\phi _ { 2 , p - 1 } ( \\kappa , \\pi ; v ) - \\phi _ { 1 , p - 1 } ( \\kappa , \\pi ; u ) \\phi _ { 2 , p } ( \\kappa , \\pi ; v ) \\Big ) , \\end{align*}"}
-{"id": "4483.png", "formula": "\\begin{align*} L = G ^ { i j } \\nabla _ { i j } + \\tilde { G } ^ { i } \\nabla _ i \\end{align*}"}
-{"id": "3875.png", "formula": "\\begin{align*} \\nu \\left ( B \\right ) = \\int _ 0 ^ { \\infty } \\int _ { \\left \\vert w \\right \\vert = 1 } 1 _ B \\left ( r w \\right ) a \\left ( r , w \\right ) j \\left ( r \\right ) r ^ { d - 1 } \\Sigma \\left ( d w \\right ) d r , \\forall B \\in \\mathcal { B } \\left ( \\mathbf { R } ^ d _ 0 \\right ) , \\end{align*}"}
-{"id": "6659.png", "formula": "\\begin{align*} h ^ { * } _ d ( I _ { r , k } ^ { n } ) = \\sum \\limits _ { i \\geq 0 } ( - 1 ) ^ i \\binom { n } { i } \\binom { n } { ( k - r i ) d - i } _ { k - r i } . \\end{align*}"}
-{"id": "7312.png", "formula": "\\begin{align*} E _ { u , v } ( t ) = { } & \\frac { 1 } { 2 } \\int _ \\Omega \\rho ( x ) | u _ t ( x , t ) | ^ 2 + \\rho ( x ) | v _ t ( x , t ) | ^ 2 + \\nabla u ( x , t ) ^ { \\top } \\cdot K ( x ) \\cdot \\nabla u ( x , t ) \\ , d x \\\\ & { } + { } \\frac { 1 } { 2 } \\int _ { \\Omega } \\nabla v ( x , t ) ^ { \\top } \\cdot K ( x ) \\cdot \\nabla v ( x , t ) + ( u v ) ^ 2 ( x , t ) \\ , d x , \\end{align*}"}
-{"id": "3439.png", "formula": "\\begin{align*} \\Pi _ { M } ^ { } = X _ { M } \\cdots X _ { 2 } X _ { 1 } . \\end{align*}"}
-{"id": "8245.png", "formula": "\\begin{align*} F '' ( 1 ) = - 2 ( q ; q ) _ \\infty \\sum _ { n = 0 } ^ \\infty \\frac { q ^ { 2 n + 1 } } { ( 1 - q ^ { 2 n + 1 } ) ^ 2 } . \\end{align*}"}
-{"id": "3933.png", "formula": "\\begin{align*} \\sigma _ N ( n ) = \\sum g _ i \\otimes n _ i = \\sum S \\bigl ( \\bigl ( \\iota \\circ \\sigma _ { \\Bbbk [ H ] } \\bigr ) ( n _ 1 ) \\bigr ) \\otimes n _ 0 = \\nu \\bigl ( \\eta ( \\sigma ) \\bigr ) _ N ( n ) \\ , , \\end{align*}"}
-{"id": "5910.png", "formula": "\\begin{align*} T _ { s _ 1 s _ 0 } \\star \\varphi _ m = T _ { s _ 1 } T _ { s _ 0 } \\star \\varphi _ m = q ^ 2 \\varphi _ { m + 1 } + ( q - 1 ) ( q \\psi _ m + \\psi _ { m - 1 } ) , \\end{align*}"}
-{"id": "1054.png", "formula": "\\begin{align*} & | A - B | _ 2 ^ 2 = \\sum _ { i \\neq j } \\int _ { I ( y _ i , F _ n ( x _ i ) ) } d x \\int _ { I ( y _ j , F _ n ( x _ j ) ) } | K _ n ( x , y ) | ^ 2 d y \\\\ = & \\sum _ { i \\neq j } \\int _ { I ( y _ i , F _ n ( x _ i ) ) } d x \\int _ { I ( y _ j , F _ n ( x _ j ) ) } O ( 1 ) d y = \\sum _ { i \\neq j } F _ n ( x _ i ) F _ n ( x _ j ) O ( 1 ) \\\\ = & \\sum _ { i \\neq j } O \\left ( \\frac { \\ln n } { n ^ 2 } \\right ) O ( 1 ) = k ( k - 1 ) O \\left ( \\frac { \\ln n } { n ^ 2 } \\right ) = O \\left ( \\frac { \\ln n } { n ^ 2 } \\right ) , \\end{align*}"}
-{"id": "2485.png", "formula": "\\begin{align*} \\| P \\| _ r & = | a _ 1 | + | a _ 2 | \\ , , \\\\ \\| P \\| _ \\infty & = \\begin{cases} | a _ 1 | + | a _ 2 | \\ , , \\quad \\\\ \\max \\{ | a _ 1 | , | a _ 2 | , | a _ 1 + a _ 2 | \\} \\ , , \\quad \\end{cases} \\end{align*}"}
-{"id": "7141.png", "formula": "\\begin{align*} ( \\rho u ) _ t + ( \\rho u ^ 2 ) _ x + p ' ( \\rho ) \\rho _ x = \\nu \\big [ \\rho \\frac { d } { d t } ( \\frac { 1 } { \\rho } ) _ x + \\rho u ( \\frac { 1 } { \\rho } ) _ { x x } \\big ] - \\frac { \\epsilon } { 2 } \\big ( \\chi _ x ^ 2 \\big ) _ x , \\end{align*}"}
-{"id": "2072.png", "formula": "\\begin{align*} \\frac { D R _ n '' ( \\rho | \\rho _ { \\rm t x } ) } { R _ n ( \\rho | \\rho _ { \\rm t x } ) } + \\frac { D R _ n ' ( \\rho | \\rho _ { \\rm t x } ) } { \\rho R _ n ( \\rho | \\rho _ { \\rm t x } ) } - \\frac { D n ^ 2 } { \\rho ^ 2 } = \\frac { T ' _ n ( t | t _ 0 ) } { T _ n ( t | t _ 0 ) } \\overset { ( b ) } { = } - \\gamma _ n ^ 2 , \\end{align*}"}
-{"id": "8259.png", "formula": "\\begin{align*} z _ { t t } - i a z _ { \\alpha } = - i . \\end{align*}"}
-{"id": "7810.png", "formula": "\\begin{align*} | y + z | ^ 2 = | y | ^ 2 + | z | ^ 2 + 2 \\langle z , y \\rangle \\le N _ 1 ^ 2 \\end{align*}"}
-{"id": "9841.png", "formula": "\\begin{align*} \\alpha _ 0 ( v ) \\hat { w } _ { i + 1 } ( v ) + \\sum _ { j = 1 } ^ r \\alpha _ j ( v ) | N ^ j _ { G } ( v ) | & \\geq \\gamma ( v ) = \\alpha _ 0 ( v ) + \\sum _ { j = 1 } ^ r \\alpha _ j ( v ) | N _ { G } ^ j ( v ) | , \\end{align*}"}
-{"id": "1578.png", "formula": "\\begin{align*} ( z ^ o _ n , x ^ o _ n ) : = \\widetilde { \\boldsymbol { F } } _ { ( n + 1 ) ^ 2 - k _ n , n ^ 2 + k _ n } ( z ^ \\iota _ n , x ^ \\iota _ n ) . \\end{align*}"}
-{"id": "8568.png", "formula": "\\begin{align*} \\begin{array} { r c l c r c l c r c l c r c l } h & : = & E _ { 1 1 } - E _ { 2 2 } , & & e _ 1 ^ + & : = & E _ { 1 2 } , & & e _ 2 ^ + & : = & E _ { 3 1 } - E _ { 2 4 } , & & e _ 3 ^ + & : = & E _ { 1 4 } + E _ { 3 2 } , \\\\ z & : = & E _ { 3 3 } - E _ { 4 4 } , & & e _ 1 ^ - & : = & E _ { 2 1 } , & & e _ 2 ^ - & : = & E _ { 1 3 } + E _ { 4 2 } , & & e _ 3 ^ - & : = & E _ { 4 1 } - E _ { 2 3 } . \\end{array} \\end{align*}"}
-{"id": "9157.png", "formula": "\\begin{align*} \\rho ^ { ( + ) \\left ( N \\right ) } ( \\mathbf { x } ^ { \\left ( + \\right ) } , t ) = \\rho ^ { ( - ) \\left ( N \\right ) } ( \\mathbf { x } ^ { ( + ) } , t ) , \\end{align*}"}
-{"id": "5059.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j \\in \\mathbb { S } } F _ j ^ * A _ j h \\right \\| ^ 2 = \\left \\langle \\sum _ { j \\in \\mathbb { S } } F _ j ^ * A _ j h , \\sum _ { k \\in \\mathbb { S } } F _ k ^ * A _ k h \\right \\rangle = \\sum _ { j \\in \\mathbb { S } } \\left \\langle A _ j h , F _ j \\left ( \\sum _ { k \\in \\mathbb { S } } F _ k ^ * A _ k h \\right ) \\right \\rangle = \\sum _ { j \\in \\mathbb { S } } \\| A _ j h \\| ^ 2 . \\end{align*}"}
-{"id": "1621.png", "formula": "\\begin{align*} V _ r ( x ) = \\begin{cases} x + \\frac { 1 } { r } \\theta _ r ( x _ r ^ \\ast ) , & x \\geq x _ r ^ \\ast , \\\\ \\frac { \\psi _ r ( x ) } { \\psi _ r ' ( x _ r ^ \\ast ) } , & x < x _ r ^ \\ast . \\end{cases} \\end{align*}"}
-{"id": "7772.png", "formula": "\\begin{align*} p _ t ( \\xi ) = e ^ { - 2 \\pi t L | \\xi | } , \\end{align*}"}
-{"id": "5284.png", "formula": "\\begin{align*} & P E ^ { 2 n p ^ i } = { ^ { ( i ) } } ( \\mathrm { K e r } \\ , V _ { i - 1 } ) \\ ; , & Q E ^ { 2 n p ^ s } = { ^ { ( i ) } } ( \\mathrm { C o k e r } \\ , F _ { i - 1 } ) \\ ; . \\end{align*}"}
-{"id": "166.png", "formula": "\\begin{align*} & X _ C ( f \\circ q _ M ) = X ( f ) = 0 , \\\\ & X _ V ( f \\circ q _ M ) = \\sum _ { s = 1 } ^ { N } y ^ s \\dfrac { \\partial } { \\partial x ^ s } X ( f ) = 0 , \\\\ & X _ V ( f \\circ q _ M ) = 0 , \\\\ & X _ V ( l _ { d f } ) = X ( f ) = 0 . \\end{align*}"}
-{"id": "6223.png", "formula": "\\begin{align*} \\mathcal { V } ( \\theta ) = \\mathcal { L } ( \\theta ^ \\star ) - \\mathcal { L } ( \\theta ) \\end{align*}"}
-{"id": "2168.png", "formula": "\\begin{align*} 1 = \\Psi _ { 0 } ( \\| \\mathbf { x } \\| ) + \\sum _ { n = 1 } ^ { \\infty } \\Psi ( 2 ^ { - n } \\| \\mathbf { x } \\| ) = \\sum _ { n = 0 } ^ { \\infty } \\Psi _ { n } ( \\| \\mathbf { x } \\| ) \\mathbf { x } \\in \\mathbb { R } ^ N . \\end{align*}"}
-{"id": "2146.png", "formula": "\\begin{align*} w ( B ( t \\mathbf x , t r ) ) = t ^ { \\mathbf N } w ( B ( \\mathbf x , r ) ) \\ \\ \\mathbf x \\in \\mathbb R ^ N , \\ t , r > 0 , \\end{align*}"}
-{"id": "4638.png", "formula": "\\begin{align*} h _ f ( z ) = \\lim _ { n \\to \\infty } \\frac { h ( f ^ n ( z ) ) } { d ^ n } . \\end{align*}"}
-{"id": "1156.png", "formula": "\\begin{align*} & n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } \\sup _ { z \\in I } \\Big ( \\mathbb { P } ( \\xi ^ { ( n ) } ( [ z , z + G _ n ( - C _ 0 ) / S ( I ) ] ) = 0 ) \\\\ & - D _ n ( \\sqrt { 4 - z ^ 2 } / S ( I ) \\cdot G _ n ( - C _ 0 ) / 2 ) \\Big ) < 1 , \\end{align*}"}
-{"id": "2004.png", "formula": "\\begin{align*} J \\left ( \\frac { I _ 1 ( q ) } { I _ 0 ( q ) } \\phi _ 2 , z \\right ) = \\frac { I ( q , z ) } { I _ 0 ( q ) } . \\end{align*}"}
-{"id": "1144.png", "formula": "\\begin{align*} C _ 0 = \\max \\limits _ { 1 \\leq j \\leq k } ( | x _ j | + | u _ j | ) , \\ \\varepsilon _ 1 = \\min \\limits _ { 0 \\leq i < j \\leq k } \\big | | u _ i | - | u _ j | \\big | , \\ \\varepsilon _ 0 = \\varepsilon _ 1 S ( I ) ^ 2 / ( 2 + 4 \\varepsilon _ 1 ) . \\end{align*}"}
-{"id": "9089.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\mathbb { R } ^ N } g ( x ) | u _ { n _ k } ( x ) | ^ p d x & \\to \\int _ { \\mathbb { R } ^ N } g ( x ) | u ( x ) | ^ p d x , \\\\ \\int _ { \\mathbb { R } ^ N } g ( x ) | u _ { n _ k } ( x ) | ^ { p - 2 } u _ { n _ k } ( x ) u ( x ) d x & \\to \\int _ { \\mathbb { R } ^ N } g ( x ) | u ( x ) | ^ p d x . \\end{aligned} \\end{align*}"}
-{"id": "4360.png", "formula": "\\begin{align*} \\Gamma _ { \\sigma } ( A ) : = \\sigma ^ { \\frac { 1 } { 2 } } A \\sigma ^ { \\frac { 1 } { 2 } } , \\end{align*}"}
-{"id": "1444.png", "formula": "\\begin{align*} N _ { n } = \\sum _ { 1 \\leq | \\alpha | \\leq s } \\int _ { 0 } ^ { T } \\| \\partial ^ { \\alpha } D w ^ { n } \\| _ { 0 } ^ { 2 } \\ d \\tau + \\sum _ { 0 \\leq | \\alpha | \\leq s - 1 } \\int _ { 0 } ^ { T } \\| \\partial ^ { \\alpha } D \\mu ^ { n } \\| _ { 0 } ^ { 2 } \\ d \\tau . \\end{align*}"}
-{"id": "2715.png", "formula": "\\begin{align*} \\sigma _ k = \\frac { 1 } { 2 } ( 1 + \\tau ^ 2 ) + N ^ { - \\frac { 1 } { 3 } } 2 ^ { - \\frac { 4 } { 3 } } ( 1 - \\tau ^ 2 ) \\pi _ { k } , k = 1 , \\ldots , m . \\end{align*}"}
-{"id": "9834.png", "formula": "\\begin{align*} g ( 0 ) = f ( x _ \\circ ' ) . \\end{align*}"}
-{"id": "3714.png", "formula": "\\begin{align*} \\bar { Y } _ B ^ c = \\sum _ { i \\in U \\setminus B } y _ i / ( N - n _ B ) \\end{align*}"}
-{"id": "3676.png", "formula": "\\begin{align*} \\mbox { P r } ( \\delta _ i = 1 | x _ i , i \\in S ) = \\mbox { P r } ( \\delta _ i = 1 | x _ i , i \\in U ) \\end{align*}"}
-{"id": "800.png", "formula": "\\begin{align*} A _ d / P ^ \\alpha = \\left ( A _ 1 / P \\right ) ^ \\alpha . \\end{align*}"}
-{"id": "4854.png", "formula": "\\begin{align*} \\alpha ( w ^ * _ { ( j , r _ 1 ) } ) = \\sum _ { i \\geq 1 } ^ { } \\lambda _ { i } w ^ * _ { ( i , r _ 1 ) } , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\lambda _ { i } \\in \\Q . \\end{align*}"}
-{"id": "2212.png", "formula": "\\begin{align*} J ( \\boldsymbol { e } ( t ) , \\boldsymbol { u } ( t ) , t ) = 0 . 5 \\Big [ \\int _ { t _ 0 } ^ { \\infty } \\boldsymbol { e } ^ T ( t ) Q \\boldsymbol { e } ( t ) + \\boldsymbol { u } ^ T ( t ) R \\boldsymbol { u } ( t ) d t \\Big ] \\end{align*}"}
-{"id": "7690.png", "formula": "\\begin{align*} T \\psi _ { + } ^ \\mathsf { e } ( D ) = T \\bigl ( \\psi _ + + \\psi _ - \\bigr ) ( D ) = \\bigl ( \\psi _ - + \\psi _ + \\bigr ) ( D ) T = \\psi _ { + } ^ \\mathsf { e } ( D ) T \\end{align*}"}
-{"id": "3518.png", "formula": "\\begin{align*} \\hat \\theta _ { \\varnothing } , \\hat \\theta _ a , \\hat \\omega ^ a : = \\d x ^ a , \\hat \\pi _ { a b } : = \\d p _ { a b } \\end{align*}"}
-{"id": "651.png", "formula": "\\begin{align*} ( \\psi ; \\vec g ; [ y ] ) \\circ ( \\varphi ; \\vec f ; [ x ] ) : = \\left ( \\psi \\varphi ^ y ; \\gamma ( g _ 1 ; ( y _ \\ast \\vec f ) ^ \\psi _ 1 ) , \\dots , \\gamma ( g _ n ; ( y _ \\ast \\vec f ) ^ \\psi _ n ) ; [ \\varphi ^ \\ast ( y ) x ] \\right ) \\ . \\end{align*}"}
-{"id": "5560.png", "formula": "\\begin{align*} c ' q ^ { \\frac { 1 } { b } ( \\dim X + e ) } \\cdot q ^ { ( 1 - \\frac { 1 } { b } ) \\dim X } = c ' q ^ { \\dim X + \\frac { e } { b } } \\end{align*}"}
-{"id": "7576.png", "formula": "\\begin{align*} S ( x ) = 4 \\pi e ^ { - | x | ^ 2 } \\left ( | x | ^ 2 - \\frac { d } { 2 } \\right ) \\end{align*}"}
-{"id": "2433.png", "formula": "\\begin{align*} \\varphi _ \\ell ^ { H _ 1 } ( x _ 0 , x _ 1 ) = ( ( - 1 ) ^ { \\ell / 4 } + 2 ^ { ( \\ell - 4 ) / 2 } ) ( x _ 0 ^ \\ell + x _ 1 ^ \\ell ) + \\sum _ { 0 < j < \\ell , j \\equiv 0 \\pmod { 4 } } ( - 1 ) ^ { \\ell / 4 } \\binom { \\ell } { j } x _ 0 ^ { \\ell - j } x _ 1 ^ j . \\end{align*}"}
-{"id": "9611.png", "formula": "\\begin{align*} H ( z , q , N ) = z \\frac { \\partial } { \\partial z } \\frac { ( q ^ { N } z ) _ { \\infty } } { ( z ) _ { \\infty } } = z \\frac { \\partial } { \\partial z } \\frac { 1 } { ( z ) _ N } = \\frac { z } { ( z ) _ N } \\sum _ { r = 0 } ^ { N - 1 } \\frac { q ^ r } { 1 - z q ^ r } \\end{align*}"}
-{"id": "6634.png", "formula": "\\begin{align*} T ' = ( 1 - x _ s X ) \\bar \\sigma , T ' = ( x _ s - X ) \\bar \\sigma , \\end{align*}"}
-{"id": "5429.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac 1 n \\log \\lVert A ^ n ( x ) v \\rVert = \\lambda _ i ( A , \\mu ) , \\end{align*}"}
-{"id": "9906.png", "formula": "\\begin{align*} \\tilde { f } ( x ) = \\int _ { \\mathcal { Y } ^ { N ' + 1 } } g ( y _ { [ n , n + N ' ] } ) P ( d y _ { [ n , n + N ' ] } | X _ { n } = x ) \\end{align*}"}
-{"id": "8616.png", "formula": "\\begin{align*} \\varepsilon = \\frac { 1 } { \\sqrt { \\Lambda - 1 } } , \\end{align*}"}
-{"id": "1751.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } \\left [ \\left | \\lVert D ^ { l } e ^ { \\frac { \\alpha } { p } M _ { s , t } } \\rVert _ { \\mathcal { H } ^ { \\otimes l } } \\right | ^ { p } \\right ] & = \\mathbb { E } \\left [ \\left ( \\frac { \\alpha } { p } \\right ) ^ { l p } e ^ { \\alpha M _ { s , t } } ( t - s ) ^ { l H p } \\right ] \\\\ & = \\left ( \\frac { \\alpha } { p } \\right ) ^ { l p } ( t - s ) ^ { l H p } e ^ { \\frac { \\alpha ^ { 2 } } { 2 } ( t - s ) ^ { 2 H } } , \\end{aligned} \\end{align*}"}
-{"id": "7308.png", "formula": "\\begin{align*} \\deg { E _ 0 } & = ( 1 , 0 , 1 , 0 ) , & \\deg { E _ 1 } & = ( 0 , 1 , - 1 , 0 ) , & \\deg { E _ 2 } & = ( 0 , - 1 , 0 , - 1 ) , \\\\ \\deg { E _ 3 } & = ( - 1 , 0 , 0 , 1 ) , & \\deg { F _ 0 } & = ( - 1 , 0 , - 1 , 0 ) , & \\deg { F _ 1 } & = ( 0 , - 1 , 1 , 0 ) , \\\\ \\deg { F _ 2 } & = ( 0 , 1 , 0 , 1 ) , & \\deg { F _ 3 } & = ( 1 , 0 , 0 , - 1 ) , & \\deg H _ i & = ( 0 , 0 , 0 , 0 ) . \\end{align*}"}
-{"id": "9612.png", "formula": "\\begin{align*} ( 1 - q ^ { N } ) ( q ) _ { N - 1 } \\cdot \\left . H ( z , q , N ) \\right | _ { z = q } = \\sum _ { r = 1 } ^ { N } \\frac { q ^ r } { 1 - q ^ r } . \\end{align*}"}
-{"id": "8576.png", "formula": "\\begin{align*} [ y _ i , x _ j ] - [ y _ j , x _ i ] = - \\sum _ { \\alpha > 0 } \\frac { 2 c _ \\alpha } { ( \\alpha , \\alpha ) } ( ( \\alpha , x _ i ) ( x _ j , \\alpha ) - ( \\alpha , x _ j ) ( x _ i , \\alpha ) ) s _ \\alpha = 0 , \\end{align*}"}
-{"id": "4381.png", "formula": "\\begin{align*} \\omega _ o = d d ^ c | z | ^ 2 \\end{align*}"}
-{"id": "2319.png", "formula": "\\begin{align*} ( \\Omega f ) ( \\omega ) = \\omega f ( \\omega ) . \\end{align*}"}
-{"id": "5119.png", "formula": "\\begin{align*} H _ 0 : = - i \\alpha \\cdot \\nabla + m \\beta , \\end{align*}"}
-{"id": "1003.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\mathbb { P } ( \\chi ^ { ( n ) } ( A ) = k ) = \\mathbb { P } ( \\chi ( A ) = k ) = \\left ( - f ' ( x ) \\right ) ^ k e ^ { f ' ( x ) } / k ! . \\end{align*}"}
-{"id": "3013.png", "formula": "\\begin{align*} B _ { n + 1 } ( x _ 1 , \\dots , x _ { n + 1 } ) & = \\sum _ { i = 0 } ^ n \\binom { n } { i } B _ { n - i } ( x _ 1 , \\dots , x _ { n - i } ) x _ { i + 1 } . \\end{align*}"}
-{"id": "6449.png", "formula": "\\begin{align*} \\widetilde \\eta = \\frac { \\Vert \\widetilde r \\Vert } { \\Vert \\widehat { A } \\Vert \\ , \\Vert \\widetilde x \\Vert + \\Vert \\widehat { b } \\Vert } , \\end{align*}"}
-{"id": "7062.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } E ( t ) ^ c = c E ^ { c - 2 } \\left ( \\left ( c - 1 \\right ) \\left ( \\frac { d E } { d t } \\right ) ^ 2 + E \\frac { d ^ 2 E } { d t ^ 2 } \\right ) . \\end{align*}"}
-{"id": "5710.png", "formula": "\\begin{align*} _ c D _ { 0 t } ^ { \\alpha } y ( x ) = \\ , _ c D _ { 0 t } ^ { \\alpha } \\left ( b _ 0 E _ { \\alpha , 1 } ( \\lambda x ^ { \\alpha } ) \\right ) + _ c D _ { 0 t } ^ { \\alpha } \\left ( b _ 1 x E _ { \\alpha , 2 } ( \\lambda x ^ { \\alpha } ) \\right ) + _ c D _ { 0 t } ^ { \\alpha } \\left ( \\int _ 0 ^ x ( x - t ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } [ \\lambda ( x - t ) ^ { \\alpha } ] f ( t ) d t \\right ) . \\end{align*}"}
-{"id": "1107.png", "formula": "\\begin{align*} & \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) \\leq n ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } \\mathbb { P } ( \\xi ^ { ( n ) } ( I _ { n , k } ) = 0 ) \\\\ & \\leq n ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } \\prod _ { j = 1 } ^ k \\mathbb { P } ( \\xi ^ { ( n ) } ( [ y _ j , y _ j + G _ n ( x _ j ) / S ( I ) ] ) = 0 ) . \\end{align*}"}
-{"id": "6917.png", "formula": "\\begin{align*} H ^ { d - 2 } ( Y _ 1 , \\Omega ^ { d - 2 } _ { Y _ 1 } \\otimes _ { \\mathcal { O } _ { Y _ 1 } } \\mathcal { O } _ { X _ 1 } ( Y _ 1 ) | _ { \\mathcal { O } _ { Y _ 1 } } ) \\rightarrow H ^ { d - 1 } ( Y _ 1 , \\mathcal { F } _ i ) \\rightarrow H ^ { d - 1 } ( Y _ 1 , \\mathcal { F } _ { i + 1 } ) \\\\ \\rightarrow H ^ { d - 1 } ( Y _ 1 , \\Omega ^ { d - 2 } _ { Y _ 1 } \\otimes _ { \\mathcal { O } _ { Y _ 1 } } \\mathcal { O } _ { X _ 1 } ( Y _ 1 ) | _ { \\mathcal { O } _ { Y _ 1 } } ) \\end{align*}"}
-{"id": "5387.png", "formula": "\\begin{align*} \\langle A _ { \\alpha + \\beta } . v _ { n - 1 \\ , m + 3 } ^ { k - 1 } , v _ { n m } ^ k \\rangle = \\langle v _ { n - 1 \\ , m + 3 } ^ { k - 1 } , A _ { \\alpha + \\beta } ^ * . v _ { n m } ^ k \\rangle , \\end{align*}"}
-{"id": "5041.png", "formula": "\\begin{align*} w _ { n m } ( x ) = \\frac { x ^ m } { c _ n ^ { m + 1 } } \\ , w \\Big ( \\frac { x } { c _ n } \\Big ) , \\end{align*}"}
-{"id": "5422.png", "formula": "\\begin{align*} \\mathcal B = E _ 1 ( x ) \\oplus \\cdots \\oplus E _ k ( x ) \\oplus E _ \\infty ( x ) \\end{align*}"}
-{"id": "4111.png", "formula": "\\begin{align*} \\tilde { Y } ( z ) = \\mathcal { O } \\begin{pmatrix} 1 & z ^ { - 1 / 2 } h _ { \\alpha + \\frac { 1 } { 2 } } & z ^ { 1 / 2 } h _ { \\alpha } ( z ) \\\\ 1 & z ^ { - 1 / 2 } h _ { \\alpha + \\frac { 1 } { 2 } } & z ^ { 1 / 2 } h _ { \\alpha } ( z ) \\\\ 1 & z ^ { - 1 / 2 } h _ { \\alpha + \\frac { 1 } { 2 } } & z ^ { 1 / 2 } h _ { \\alpha } ( z ) \\end{pmatrix} . \\end{align*}"}
-{"id": "9712.png", "formula": "\\begin{align*} I _ { \\textbf { d } } ^ 1 ( P _ { i } ) _ { c } \\in S I _ { \\textbf { d } } ^ 2 ( P _ { i } ) _ { c } = I _ { \\textbf { d } } ( v ) \\in D . \\end{align*}"}
-{"id": "6753.png", "formula": "\\begin{align*} \\mathbf { u = u } _ 0 , \\mathbf u _ 1 , \\dots , \\mathbf u _ k = \\mathbf v \\end{align*}"}
-{"id": "5460.png", "formula": "\\begin{align*} Y _ t \\ & = \\ g ( \\hat X ) + \\int _ t ^ T f ( \\hat X , \\hat I _ s ) \\ , d s + K _ T - K _ t - \\int _ t ^ T \\int _ \\Lambda R _ s ( b ) \\ , \\hat \\theta ( d s \\ , d b ) \\\\ & \\ - \\int _ t ^ T Z _ s \\ , d \\hat W _ s - \\int _ t ^ T \\int _ U L _ s ( z ) \\ , ( \\hat \\pi ( d s \\ , d z ) - \\lambda _ \\pi ( d z ) \\ , d s ) , 0 \\leq t \\leq T , \\ , \\ , \\hat \\P \\\\ R _ t ( b ) \\ & \\leq \\ 0 , d t \\otimes d \\hat \\P \\otimes \\lambda _ 0 ( d b ) \\hat \\Omega \\times [ 0 , T ] \\times \\Lambda . \\end{align*}"}
-{"id": "3116.png", "formula": "\\begin{align*} q ^ { \\frac { 1 - r } { 2 r } c _ 1 ( E ) ^ 2 } q ^ { c _ 2 ( E ) } = q ^ { | n | + d ( \\beta ) } \\ , . \\end{align*}"}
-{"id": "2536.png", "formula": "\\begin{align*} L ( t ) = c ( t ) \\exp \\Big \\{ \\int _ { t _ 0 } ^ t \\epsilon ( u ) \\frac { d u } { u } \\Big \\} , t \\geq t _ 0 , \\end{align*}"}
-{"id": "7093.png", "formula": "\\begin{align*} \\begin{cases} i \\partial _ t u + \\Delta u + ( V * | u | ^ 2 ) u = 0 , ( t , x ) \\in \\mathbb { R } \\times \\mathbb { T } ^ 3 \\\\ u | _ { t = 0 } = \\phi ^ \\omega . \\end{cases} \\end{align*}"}
-{"id": "6748.png", "formula": "\\begin{align*} \\widehat { u } ( t , \\xi ) = \\dfrac { 1 } { e ^ { i 2 \\pi \\xi c _ { j 0 } } - 1 } \\int _ { 0 } ^ { 2 \\pi } e ^ { i \\xi \\widetilde { H } ( t _ j , \\tau ) } \\widehat { f } _ j ( t _ 1 , \\ldots , t _ j + \\tau , \\ldots , t _ n , \\xi ) d \\tau , \\end{align*}"}
-{"id": "7041.png", "formula": "\\begin{align*} e ^ { \\phi } = \\phi _ { v \\b { v } } . \\end{align*}"}
-{"id": "5144.png", "formula": "\\begin{align*} \\mathcal { D } ( T _ { A _ \\nu , \\mathbb { I } _ 4 } ) = \\{ \\psi \\in \\mathcal { D } ( H _ { m a x } ) : A _ \\nu \\Gamma ^ + ( \\psi ) = \\Gamma ^ - ( \\psi ) \\} , \\end{align*}"}
-{"id": "8574.png", "formula": "\\begin{align*} \\begin{array} { r c l c r c l } \\ ; \\ ! [ H , x \\otimes \\omega ] & = & x \\otimes \\omega , & \\quad & [ H , y \\otimes \\omega ] & = & - y \\otimes \\omega , \\\\ \\ ; \\ ! [ Z _ 0 , p \\otimes x ] & = & p \\otimes x , & \\quad & [ Z _ 0 , 1 \\otimes y ] & = & - p \\otimes y . \\end{array} \\end{align*}"}
-{"id": "8203.png", "formula": "\\begin{align*} ( a ; q ) _ n = \\frac { ( a ; q ) _ \\infty } { ( a q ^ n ; q ) _ \\infty } . \\end{align*}"}
-{"id": "6511.png", "formula": "\\begin{align*} T _ { f , a _ i } ^ { k + 1 } = 0 \\ \\ \\Lambda . \\end{align*}"}
-{"id": "7184.png", "formula": "\\begin{gather*} \\mathcal { Y } _ { r , R } : = \\left \\{ y ( \\cdot ) \\in \\mathrm { C } \\left ( \\mathbb { R } _ { + } \\ ! \\mapsto \\ ! \\mathbb { R } ^ { n } ; \\alpha \\right ) : \\left \\Vert y ( t ) \\right \\Vert \\le R \\mathrm { e } ^ { - \\alpha t } \\ ; \\forall t \\ge 0 \\right \\} . \\end{gather*}"}
-{"id": "2224.png", "formula": "\\begin{align*} u _ t + ( - \\Delta ) ^ { \\alpha / 2 } u + \\nabla \\cdot ( u \\nabla v ) & = 0 , \\ \\ & x \\in { \\mathbb R } ^ d , \\ t > 0 , \\\\ \\Delta v + u & = 0 , \\ \\ & x \\in { \\mathbb R } ^ d , \\ t > 0 , \\end{align*}"}
-{"id": "7620.png", "formula": "\\begin{align*} \\| a \\| _ { L ^ 1 ( \\mathbb R ^ d ) } = \\int \\limits _ { \\mathbb R ^ d } a ( z ) \\ , d z = a _ 1 > 0 ; \\int \\limits _ { \\mathbb R ^ d } | z | ^ 2 a ( z ) \\ , d z < \\infty . \\end{align*}"}
-{"id": "1596.png", "formula": "\\begin{align*} \\begin{aligned} | u ^ o _ { n , k + 1 } - \\pi _ x \\circ \\widetilde { F } ( z ^ o _ { n , k } , u ^ o _ { n , k } ) | & < | z ^ o _ { n , k + 1 } - \\pi _ z \\circ \\widetilde { F } ( z ^ o _ { n , k } , u ^ o _ { n , k } ) | \\\\ & < \\frac { | u ^ o _ { n , k } - x ^ o _ { n , k } | } { 4 } + \\frac { D } { n ^ 2 } , \\end{aligned} \\end{align*}"}
-{"id": "5985.png", "formula": "\\begin{align*} \\omega \\otimes _ t \\overline \\omega ( s ) : = \\omega ( s \\wedge t ) + \\sqrt { 1 - t } \\ , \\overline \\omega \\left ( \\frac { s - t } { 1 - t } \\right ) 1 _ { [ t , 1 ] } ( s ) . \\end{align*}"}
-{"id": "7201.png", "formula": "\\begin{align*} ( u _ t + u u _ x ) _ x = u . \\end{align*}"}
-{"id": "7259.png", "formula": "\\begin{align*} 0 = \\lim _ { n \\to \\infty } \\left ( \\mu ( s _ n ) - \\phi ( s _ n ) ( \\pi ) \\right ) \\geq \\lambda \\pi > 0 . \\end{align*}"}
-{"id": "8603.png", "formula": "\\begin{align*} \\tilde { \\omega } ( 0 ) = 0 . \\end{align*}"}
-{"id": "4101.png", "formula": "\\begin{align*} L _ { \\alpha } ( z ) \\tilde { L } _ { \\alpha } ^ T ( z ) = 3 \\mathbb I . \\end{align*}"}
-{"id": "184.png", "formula": "\\begin{align*} \\pi _ { T M , X _ { C } , Y _ { C } , c } ( { \\bf x } , { \\bf y } ) = \\end{align*}"}
-{"id": "4615.png", "formula": "\\begin{align*} ( K + \\tau M ) \\mathbf { u } _ k = M \\mathbf { u } _ { k - 1 } , k \\in [ 1 , \\ell ] , \\end{align*}"}
-{"id": "1753.png", "formula": "\\begin{align*} \\begin{aligned} \\left [ c _ { p , r } \\left ( M _ { s , t } > \\eta \\right ) \\right ] ^ { p } & \\leq \\left [ \\sum _ { l = 0 } ^ { r } \\left ( \\frac { \\eta } { p ( t - s ) ^ { 2 H } } \\right ) ^ { l p } ( t - s ) ^ { l H p } \\right ] e ^ { - \\frac { \\eta ^ { 2 } } { 2 ( t - s ) ^ { 2 H } } } \\\\ & = \\sum _ { l = 0 } ^ { r } \\left ( \\frac { \\eta } { p ( t - s ) ^ { H } } \\right ) ^ { l p } e ^ { - \\frac { \\eta ^ { 2 } } { 2 ( t - s ) ^ { 2 H } } } . \\end{aligned} \\end{align*}"}
-{"id": "9815.png", "formula": "\\begin{align*} L _ a \\hat { u } = u L _ a \\xi + | y | ^ a \\nabla \\xi \\cdot \\nabla u = : | y | ^ a \\hat { f } ( X ) \\quad \\R ^ { n + 1 } \\setminus \\{ x _ n = y = 0 \\} . \\end{align*}"}
-{"id": "5125.png", "formula": "\\begin{align*} \\delta _ { k _ j } = \\delta _ { k _ j } ( \\lambda , \\mu , \\nu ) : = ( k _ j + \\lambda ) ^ 2 + \\mu ^ 2 - \\nu ^ 2 . \\end{align*}"}
-{"id": "423.png", "formula": "\\begin{align*} \\tilde Q _ u ( x , y ) : = Q _ u ( x , y ) - \\int _ D \\left ( Q _ u ( x , z ) + Q _ u ( y , z ) \\right ) \\bar h ( z ) \\dd z \\int _ D Q _ u ( z , z ' ) \\bar h ( z ) \\bar h ( z ' ) \\dd z \\dd z ' . \\end{align*}"}
-{"id": "630.png", "formula": "\\begin{align*} \\psi ^ \\ast ( x ^ { - 1 } y ) = ( \\psi ^ { x ^ { - 1 } y } ) ^ \\ast ( x ) ^ { - 1 } \\psi ^ \\ast ( y ) \\ . \\end{align*}"}
-{"id": "3425.png", "formula": "\\begin{gather*} V _ 2 = \\alpha \\circ ( \\phi , V _ 1 ) . \\end{gather*}"}
-{"id": "4822.png", "formula": "\\begin{align*} \\left \\{ \\begin{matrix} \\mathcal L \\varphi = \\lambda _ 1 ( \\mathcal L ) \\varphi & \\textrm { i n $ M $ } \\\\ \\mathcal B \\varphi = 0 & \\textrm { o n $ \\partial M $ } \\end{matrix} \\right . \\end{align*}"}
-{"id": "3600.png", "formula": "\\begin{align*} v \\in \\bigoplus _ { \\lambda \\in \\Lambda } V _ \\lambda ^ { ( u ) } \\Leftrightarrow f _ \\Lambda ( a d _ u ) ( v ) = 0 \\end{align*}"}
-{"id": "448.png", "formula": "\\begin{align*} - \\Delta _ g \\tilde G ( \\cdot , y ) = \\delta _ y ( \\cdot ) \\quad D , \\tilde G ( \\cdot , y ) _ { | \\mathcal { C } } = 0 . \\end{align*}"}
-{"id": "7760.png", "formula": "\\begin{align*} \\| A \\| _ { ( k ) } = \\sum _ { j = 1 } ^ k s _ { j } ( A ) , \\end{align*}"}
-{"id": "9661.png", "formula": "\\begin{align*} X _ 0 T _ 0 = U _ { \\mathbb T } X _ 0 \\ \\ \\ \\ \\bigvee _ { n \\geq 0 } U _ { \\mathbb T } ^ { - n } X _ 0 \\mathcal N = L ^ 2 ( \\mathbb T ) . \\end{align*}"}
-{"id": "1016.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to + \\infty } \\frac { 1 } { n \\sin ( \\alpha _ n / 2 ) } = \\lim \\limits _ { n \\to + \\infty } \\frac { 2 } { n \\alpha _ n } \\lim \\limits _ { n \\to + \\infty } \\frac { \\alpha _ n / 2 } { \\sin ( \\alpha _ n / 2 ) } = 0 . \\end{align*}"}
-{"id": "1202.png", "formula": "\\begin{align*} f ( \\sigma ) & = \\sqrt { 1 + \\sigma _ 1 ^ 2 } \\sqrt { 1 + \\sigma _ 2 ^ 2 } , F _ k ( \\sigma , x , z ' ) = Q _ k ( \\zeta _ 1 ( \\sigma _ 1 , x , z ' ) , \\sigma _ 2 ) \\frac { 1 } { \\sqrt { 1 + \\sigma _ 1 ^ 2 } } . \\end{align*}"}
-{"id": "2159.png", "formula": "\\begin{align*} p ( \\mathbf { x } ) g _ { L } ( \\mathbf { x } ) = \\sum _ { \\ell = 0 } ^ { d } \\sum _ { \\| \\alpha \\| \\leq \\ell } c _ { \\ell , \\alpha } T ^ { \\alpha } ( g _ { L + \\ell } ) ( \\mathbf { x } ) \\end{align*}"}
-{"id": "7151.png", "formula": "\\begin{align*} \\left \\vert u \\right \\vert _ { \\beta , \\infty } : = \\sup _ { j } w \\left ( N ^ { - j } \\right ) ^ { - \\beta } \\left \\vert u \\ast \\varphi _ { j } \\right \\vert _ { 0 } < \\infty , \\end{align*}"}
-{"id": "2374.png", "formula": "\\begin{align*} [ x _ 1 , x _ 2 ] & = - i z , & [ x _ 1 , x _ 3 ] & = [ x _ 2 , x _ 3 ] = 0 \\end{align*}"}
-{"id": "2996.png", "formula": "\\begin{align*} p ' _ m ( n ) + \\sum _ { k = 1 } ^ \\infty ( - 1 ) ^ k \\big ( p ' _ m ( n - P _ { m + 2 , k } ) + p ' _ m ( n - Q _ { m + 2 , k } ) \\big ) & = 0 . \\end{align*}"}
-{"id": "77.png", "formula": "\\begin{align*} ( v , w ) ( \\cdot , 0 ) = ( \\rho _ 0 - \\widetilde \\rho , J _ 0 - J _ b ) ( \\cdot ) \\ , , w ( 0 , t ) = w ( 1 , t ) = 0 \\end{align*}"}
-{"id": "1927.png", "formula": "\\begin{align*} ( x ^ 2 + x ) F _ { p + 2 } ' ( x ) + ( 2 x + 1 ) F _ { p + 2 } ( x ) + F _ p ' ( x ) = 2 x \\sum _ { j = 1 } ^ \\infty \\frac { 1 } { Z _ j ^ p } - p x ^ 2 \\sum _ { j = 1 } ^ \\infty \\frac { 1 } { Z _ j ^ { p + 1 } } \\left ( \\frac { d Z _ j } { d x } \\right ) . \\end{align*}"}
-{"id": "9645.png", "formula": "\\begin{align*} S _ 1 ( z , q , N ) & = \\frac { ( 1 - z q ^ 2 ) ( 1 - z q ^ { 2 N + 1 } ) } { ( 1 - z q ) ( 1 - z q ^ { 2 N + 2 } ) } \\sum _ { n = 0 } ^ { N } \\frac { ( q ; q ^ 2 ) _ n ( q ^ { - 2 N } ; q ^ 2 ) _ n q ^ { 2 n } } { ( z q ^ 3 ; q ^ 2 ) _ n ( q ^ { - 2 N } / z ; q ^ 2 ) _ n } \\\\ & = \\sum _ { n = 0 } ^ { N } \\frac { ( q ; q ^ 2 ) _ n ( z q ^ 2 ; q ^ 2 ) _ { N - n } ( q ^ 2 ; q ^ 2 ) _ N ( z q ^ 2 ) ^ n } { ( z q ; q ^ 2 ) _ { n + 1 } ( z q ^ 2 ; q ^ 2 ) _ { N + 1 } ( q ^ 2 ; q ^ 2 ) _ { N - n } } , \\end{align*}"}
-{"id": "7372.png", "formula": "\\begin{align*} d ( c _ i , a ) & = d ( c _ i , e _ i ) + d ( e _ i , a ) \\\\ & = d ( c _ i , e _ i ) + d ( f ( e _ i ) , f ( a ) ) \\\\ & = d ( c _ i , e _ i ) + d ( e _ i , f ( a ) ) = d ( c _ i , f ( a ) ) \\end{align*}"}
-{"id": "7159.png", "formula": "\\begin{align*} b ^ { \\frac { \\alpha } { 2 } } & \\in L ^ { \\frac { 1 0 } { 3 } } ( 0 , T ; L ^ { \\frac { 1 0 } { 3 } } ( \\Omega ) ) , \\\\ \\log b & \\in L ^ { \\frac { 8 } { 3 } } ( 0 , T ; L ^ { \\frac { 8 } { 3 } } ( \\Omega ) ) . \\end{align*}"}
-{"id": "205.png", "formula": "\\begin{align*} X _ { C } = x ^ 3 \\dfrac { \\partial } { \\partial x ^ 2 } + x ^ 1 \\dfrac { \\partial } { \\partial x ^ 3 } + y ^ 3 \\dfrac { \\partial } { \\partial y ^ 2 } + y ^ 1 \\dfrac { \\partial } { \\partial y ^ 3 } Y _ { V } = \\dfrac { \\partial } { \\partial y ^ 2 } . \\end{align*}"}
-{"id": "1872.png", "formula": "\\begin{align*} \\mathcal { B } \\equiv L \\left ( \\sum _ { n = 1 } ^ \\infty k _ n \\sum _ { j = 1 } ^ \\infty | \\beta _ { n j } | ^ 2 + \\frac { 1 } { 4 \\ell } \\sum _ { j = 1 } ^ \\infty | \\beta _ { 0 j } + \\overline { \\alpha _ { 0 j } } | ^ 2 \\right ) . \\end{align*}"}
-{"id": "3828.png", "formula": "\\begin{align*} F = \\dfrac { \\alpha ^ { m + 1 } } { \\beta ^ m } = \\pi \\frac { h _ { 0 0 } ^ { \\frac { m + 1 } { 2 } } } { W _ { 0 } ^ m } , \\end{align*}"}
-{"id": "4690.png", "formula": "\\begin{align*} \\| e ^ k \\| _ { \\infty } = \\displaystyle { \\max _ { j } | u ^ { ( k ) } _ j - ( Q _ d u ) ^ { ( k ) } ( x _ j ) | } , \\ ; \\ ; k = 0 , 1 , \\end{align*}"}
-{"id": "3016.png", "formula": "\\begin{align*} G _ m ( x ) & = \\exp \\Big ( \\sum _ { k = 1 } ^ \\infty d _ n \\frac { x ^ n } { n ! } \\Big ) = \\sum _ { n = 0 } ^ \\infty B _ n ( d _ 1 , \\dots , d _ n ) \\frac { x ^ n } { n ! } \\end{align*}"}
-{"id": "7796.png", "formula": "\\begin{align*} \\Big \\| \\big ( \\sum _ { j = 1 } ^ d | \\mu _ j * f | ^ 2 \\big ) ^ { 1 / 2 } \\Big \\| _ { L ^ p } \\le C _ { p } ( \\sigma ( G ) , Q ( G ) ) \\| f \\| _ { L ^ p } , \\end{align*}"}
-{"id": "1216.png", "formula": "\\begin{align*} Q ( \\overline { X } , \\overline { Y } , \\overline { Z } , \\overline { W } ) : = g ( \\widehat { P } ( \\overline { X } , \\overline { W } ) , \\widehat { P } ( \\overline { Y } , \\overline { Z } ) ) - g ( \\widehat { P } ( \\overline { X } , \\overline { Z } ) , \\widehat { P } ( \\overline { Y } , \\overline { W } ) ) . \\end{align*}"}
-{"id": "2735.png", "formula": "\\begin{align*} \\max _ { z \\in \\mathcal { C } _ { + } ^ { 1 } } \\Re { f ( z ) } - \\Re { f ( z _ { 0 } ) } = \\Re { f ( z _ { 2 } ) } - \\Re { f ( z _ { 0 } ) } \\leq - 2 N ^ { - \\frac { 1 } { 6 } } . \\end{align*}"}
-{"id": "1017.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to + \\infty } \\left ( \\ln D _ n ( \\alpha _ n ) - n ^ 2 \\ln \\cos \\frac { \\alpha _ n } { 2 } + \\frac { 1 } { 4 } \\ln \\left ( n \\sin \\frac { \\alpha _ n } { 2 } \\right ) - c _ 0 \\right ) = 0 . \\end{align*}"}
-{"id": "5033.png", "formula": "\\begin{align*} | B + C | = | C | + ( a f - b d ) . \\end{align*}"}
-{"id": "9861.png", "formula": "\\begin{align*} ( d - a ) x - c y - b u + \\tau ( a - 1 ) = 0 \\ , . \\end{align*}"}
-{"id": "2450.png", "formula": "\\begin{gather*} Y = \\begin{pmatrix} Y _ 1 & Y _ 2 \\end{pmatrix} , X = \\begin{pmatrix} X _ 1 \\\\ X _ 2 \\end{pmatrix} , A = \\begin{pmatrix} A _ { 1 1 } & A _ { 1 2 } \\\\ A _ { 2 1 } & A _ { 2 2 } \\end{pmatrix} , \\\\ Z = \\begin{pmatrix} Z _ 1 & Z _ 2 \\end{pmatrix} , W = \\begin{pmatrix} W _ 1 \\\\ W _ 2 \\end{pmatrix} , \\end{gather*}"}
-{"id": "4804.png", "formula": "\\begin{align*} b ( t , x , \\omega ) = b _ 1 ( t , x ) + b _ 2 ( t , x , \\omega ) \\end{align*}"}
-{"id": "6078.png", "formula": "\\begin{align*} C _ 1 & = m + \\frac { 2 p - 1 } { m ( p - n + k ) } , \\\\ C _ 2 & = ( n - m - 1 ) + \\frac { 2 p - 1 } { ( n - m - 1 ) ( p - n + k ) } . \\end{align*}"}
-{"id": "5160.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } F _ { n } ( \\alpha _ { n } x + \\beta _ { n } ) = H _ { 2 } ( x ) , x \\in C ( H _ { 2 } ) . \\end{align*}"}
-{"id": "5040.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\Big ( Y _ j - \\sum _ { m = 0 } ^ d \\beta _ m \\Big ( \\frac { Z _ j - x } { c _ n } \\Big ) ^ m \\Big ) ^ 2 \\frac { 1 } { c _ n } w \\Big ( \\frac { Z _ j - x } { c _ n } \\Big ) . \\end{align*}"}
-{"id": "6247.png", "formula": "\\begin{align*} Y _ i : = \\frac { 1 } { \\sqrt { N } } \\sum _ { j , k = 1 } ^ N \\left [ \\ell ( \\xi ^ i _ { j , k } ) + \\sqrt { \\lambda _ i } \\ell ( \\eta ^ i _ { k , j } ) ^ * \\right ] \\otimes e _ { j , k } \\end{align*}"}
-{"id": "7823.png", "formula": "\\begin{align*} T \\| S & \\Longleftrightarrow \\pi ( T ) \\| \\pi ( S ) \\\\ & \\Longleftrightarrow ( \\pi ( T ) ^ * \\pi ( T ) \\| \\pi ( T ) ^ * \\pi ( S ) ~ ~ ~ ~ ~ ~ \\| \\pi ( T ) ^ * \\pi ( S ) \\| = \\| \\pi ( T ) \\| \\| \\pi ( S ) \\| ) \\\\ & \\Longleftrightarrow ( T ^ * T \\| T ^ * S \\| T ^ * S \\| = \\| T \\| \\| S \\| ) . \\end{align*}"}
-{"id": "6384.png", "formula": "\\begin{align*} \\mathbb { P } ( A \\subseteq \\bar { Y } ) = \\det ( I - K _ A ) , A \\subseteq V . \\end{align*}"}
-{"id": "2475.png", "formula": "\\begin{align*} U _ m : = \\{ a \\in k \\ , | \\ , ( a \\prod _ { j = 0 } ^ { \\frac { p ^ t - 1 } { m } - 1 } ( a ^ m - b _ j ) ) = 0 \\} \\end{align*}"}
-{"id": "9447.png", "formula": "\\begin{align*} ( f _ { \\theta } - \\mathcal { E } _ 2 , \\xi _ 0 ( J _ { N , m } ) ) _ { r e g } = a _ \\theta ( m ) - b ( m ) + \\sum _ { \\tau \\in \\mathcal { F } _ N } \\nu _ \\tau ^ { ( N ) } ( f ) J _ { N , m } ( \\tau ) . \\end{align*}"}
-{"id": "7869.png", "formula": "\\begin{align*} { \\rm S y m } ( I ) = \\bigcup _ { \\substack { J \\subseteq I \\\\ | J | = l } } \\{ \\sigma \\in { \\rm S y m } ( I ) \\colon | y _ { \\sigma ^ { - 1 } ( j ) } | = 1 \\textrm { e x a c t l y f o r } j \\in J \\} . \\end{align*}"}
-{"id": "6532.png", "formula": "\\begin{align*} \\frac { 1 } { l } & \\Biggl ( \\int ^ { t + l } _ { t } \\Bigl ( \\sup _ { u \\in K } \\bigl \\| f ( s + \\tau , u ) - f ( s , u ) \\bigr \\| \\Bigr ) ^ { q } \\ , d s \\Biggr ) ^ { 1 / q } \\\\ \\leq & \\frac { c _ { q } } { l } \\Biggl [ \\epsilon \\Biggl ( \\int ^ { t + l } _ { t } \\bigl [ L _ { f } ^ { q } ( s + \\tau ) + L _ { f } ^ { q } ( s ) \\bigr ] \\ , d s \\Biggr ) ^ { 1 / q } + \\sum _ { i = 1 } ^ { k } \\Biggl ( \\int ^ { t + l } _ { t } \\bigl \\| f ( s + \\tau , x _ { i } ) - f ( s , x _ { i } ) \\bigr \\| ^ { q } \\ , d s \\Biggr ) ^ { 1 / q } \\Biggr ] . \\end{align*}"}
-{"id": "2909.png", "formula": "\\begin{align*} \\Psi _ { \\mu _ + } \\Big ( A _ { \\ell _ i } \\ , \\Big | \\ , g F _ n \\setminus \\big [ R ^ g _ n \\cup \\bigcup _ { j = i } ^ m A _ { \\ell _ j } \\big ] \\Big ) & = \\Psi _ { \\mu ^ { ( \\ell _ i - 1 ) } K _ { A _ { \\ell _ i } } } \\Big ( A _ { \\ell _ i } \\ , \\Big | \\ , g F _ n \\setminus \\big [ R ^ g _ n \\cup \\bigcup _ { j = i } ^ m A _ { \\ell _ j } \\big ] \\Big ) \\ ; . \\end{align*}"}
-{"id": "5678.png", "formula": "\\begin{align*} c _ 0 ( \\mathbb { N } ) \\oplus B = l _ \\infty ( \\mathbb { N } ) \\ , . \\end{align*}"}
-{"id": "6848.png", "formula": "\\begin{align*} \\hat A = \\hat A ( t ) = \\begin{pmatrix} a _ { 1 1 } & a _ { 1 2 } \\\\ a _ { 2 1 } & a _ { 2 2 } \\end{pmatrix} \\in \\mathbb { S L } ( 2 ) \\end{align*}"}
-{"id": "9263.png", "formula": "\\begin{align*} L : = F ^ { i j } [ D _ { i j } - A _ { i j } ^ k ( \\cdot , u , D u ) D _ k ] , \\end{align*}"}
-{"id": "162.png", "formula": "\\begin{align*} X _ { V } \\left ( \\{ x ^ i , y ^ j \\} _ { T M } \\right ) & - \\{ X _ { V } ( x ^ i ) , y ^ j \\} _ { T M } - \\{ x ^ i , X _ { V } ( y ^ j ) \\} _ { T M } = \\\\ & = X _ { V } \\left ( \\pi ^ { i j } ( { \\bf x } ) \\right ) - \\{ x ^ i , v ^ j ( { \\bf x } ) \\} _ { T M } = 0 , \\end{align*}"}
-{"id": "3590.png", "formula": "\\begin{align*} \\langle a , v \\rangle = \\langle a , \\phi a \\rangle = \\phi \\end{align*}"}
-{"id": "2797.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\mathbf { U } ^ t = \\mathbf { m } \\right ) = \\sum _ { \\mathbf { n } \\in \\mathbb { N } _ 0 ^ S } \\mathbb { P } \\left ( \\mathbf { U } ^ t = \\mathbf { m } | \\mathbf { N } ^ t = \\mathbf { n } \\right ) \\mathbb { P } \\left ( \\mathbf { N } ^ t = \\mathbf { n } \\right ) . \\end{align*}"}
-{"id": "10066.png", "formula": "\\begin{align*} \\tilde z = \\frac { 1 } { q + p } ( z - z ^ 0 ) , \\tilde \\rho = \\frac { \\rho } { 6 ( q + p ) ^ 6 } . \\end{align*}"}
-{"id": "8772.png", "formula": "\\begin{align*} \\mu _ 2 - \\mu _ 1 = \\frac 1 F \\left ( \\frac { \\widetilde { \\kappa } } { n - 2 } - ( n - 3 ) ( B ^ 2 _ 2 - C B ^ 2 ) \\right ) \\ , , \\end{align*}"}
-{"id": "7696.png", "formula": "\\begin{align*} \\frac { 1 } { k } & \\stackrel { ( * ) } { > } \\norm { D ^ * u - D u _ { k } } _ { 2 } ^ 2 = \\int \\limits _ { V _ { k } } \\norm { D ^ * u ( p ) - D u _ { k } ( p ) } ^ 2 _ { S _ { p } } \\ , \\mathrm { d } \\mu + \\int \\limits _ { M \\setminus V _ { k } } \\norm { D ^ * u ( p ) } ^ 2 _ { S _ { p } } \\ , \\mathrm { d } \\mu \\\\ & \\geq \\int \\limits _ { M \\setminus V _ { k } } \\norm { D ^ * u ( p ) } ^ 2 _ { S _ { p } } \\ , \\mathrm { d } \\mu . \\end{align*}"}
-{"id": "3303.png", "formula": "\\begin{align*} \\frac { 1 } { x } \\sum _ { n \\le x } \\Delta ( n ) = O ( \\log ^ { A _ 0 } x \\sqrt { \\log \\log x } ) \\end{align*}"}
-{"id": "5931.png", "formula": "\\begin{align*} t _ s = \\min \\left \\{ t _ { i } ( s ) \\mid E _ i \\right \\} , \\end{align*}"}
-{"id": "5674.png", "formula": "\\begin{align*} \\beta _ 1 \\ldots \\beta _ n = \\pm 1 . \\end{align*}"}
-{"id": "6674.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n + 1 } ( n x ) ^ { - s } = ( 1 - 2 ^ { 1 - s } ) \\zeta ( s ) x ^ { - s } , \\ { \\rm R e } \\ , s > 0 , \\end{align*}"}
-{"id": "4660.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } F _ { p _ n , p } ( \\varepsilon ) = 1 , \\qquad \\lim \\limits _ { n \\to \\infty } F _ { q _ n , q } ( \\varepsilon ) = 1 . \\end{align*}"}
-{"id": "1508.png", "formula": "\\begin{align*} a = \\frac { m + \\alpha - 1 - \\frac { m + \\alpha - 1 } { p } } { m + \\alpha - 1 + \\frac { 1 } { 3 } - \\frac { 1 } { 2 } } = \\frac { p - 1 } { p } \\cdot \\frac { 6 ( m + \\alpha - 1 ) } { 6 m + 6 \\alpha - 7 } \\in ( 0 , 1 ) \\end{align*}"}
-{"id": "1957.png", "formula": "\\begin{align*} \\pi \\cdot \\frac { R [ y ] } { ( F ( y ) , G ( y ) ) } = 0 . \\end{align*}"}
-{"id": "3142.png", "formula": "\\begin{align*} \\mu ( \\bar t ) \\equiv \\mu ( \\{ \\bar t ^ 1 , \\bar t ^ 2 , \\dots , \\bar t ^ { N - 1 } \\} ) = \\{ \\bar t ^ { N - 1 } - c , \\bar t ^ { N - 2 } - 2 c , \\dots , \\bar t ^ { 1 } - ( N - 1 ) c \\} . \\end{align*}"}
-{"id": "5733.png", "formula": "\\begin{align*} \\varphi ( x , y ) & = u ( x , y , 0 ) \\\\ & = \\sum _ { m , n \\in \\N } [ C _ { m , n } E _ { \\alpha , 1 } ( 0 ) + ( 0 ) D _ { m , n } E _ { \\alpha , 2 } ( 0 ) ] J _ 0 ( \\gamma _ m x ) \\sin ( n \\pi y ) \\\\ & = \\sum _ { m , n \\in \\N } C _ { m , n } J _ 0 ( \\gamma _ m x ) \\sin ( n \\pi y ) . \\end{align*}"}
-{"id": "8795.png", "formula": "\\begin{align*} \\tau ( ( f _ 1 , x _ 1 ) , ( f _ 2 , x _ 2 ) ) = d _ 1 ( f _ 1 , f _ 2 ) + d _ 2 ( x _ 1 , x _ 2 ) \\end{align*}"}
-{"id": "9268.png", "formula": "\\begin{align*} M _ 2 ^ * ( R ) = \\sup _ { B _ R ( x _ 0 ) \\cap \\Omega } ( d _ x ^ 2 | D ^ 2 u ( x ) | ) , \\end{align*}"}
-{"id": "1839.png", "formula": "\\begin{align*} u ^ { \\rm e v e n } ( j , x ) = \\sqrt { \\frac { 2 } { L } } A _ j \\left [ \\cos ( \\kappa _ j x ) + \\frac { \\xi } { \\kappa _ j } \\sin ( \\kappa _ j | x | ) \\right ] , \\end{align*}"}
-{"id": "2968.png", "formula": "\\begin{align*} \\mu _ x ( \\phi ) = \\frac { 1 } { n } \\sum _ { i = 0 } ^ { n - 1 } \\phi ( f ^ i ( x ) ) . \\end{align*}"}
-{"id": "9801.png", "formula": "\\begin{align*} - \\int _ { B _ { 1 / 2 } } | L _ a u _ { 2 r } | = - \\| v _ { 2 r } \\| _ { L ^ 2 ( \\partial B _ 1 ) } \\int _ { B _ { 1 / 2 } } | L _ a \\tilde v _ { 2 r } | \\geq - C \\| v _ { 2 r } \\| _ { L ^ 2 ( \\partial B _ 1 ) } . \\end{align*}"}
-{"id": "420.png", "formula": "\\begin{align*} A _ 2 = & \\frac { 2 } { \\mathcal { G } _ { g } ^ \\gamma ( M ) } \\iint _ { M ^ 2 } K _ { \\hat g } ( x ) G _ { \\hat g } ( x , x ' ) \\ , { \\rm v } _ { \\hat g } ( \\dd x ) \\mathcal { G } _ { g } ^ \\gamma ( \\dd x ' ) \\\\ = & \\frac { 2 } { \\mathcal { G } _ { g } ^ \\gamma ( M ) } \\iint _ { M ^ 2 } \\big ( K _ g ( x ) - \\Delta _ g \\omega ( x ) \\big ) \\Big ( G _ g ( x , x ' ) + \\tfrac { 1 } { 2 } ( \\phi ( x ) + \\phi ( x ' ) ) - S ^ { \\rm c l } _ { { \\rm A Y } } ( \\hat g , g ) \\Big ) \\ , { \\rm v } _ g ( \\dd x ) \\mathcal { G } _ { g } ^ \\gamma ( \\dd x ' ) . \\end{align*}"}
-{"id": "4455.png", "formula": "\\begin{align*} ( \\phi _ { i k } ) _ { U _ { i j k } } = ( \\phi _ { i j } ) _ { U _ { i j k } } \\circ ( \\phi _ { j k } ) _ { U _ { i j k } } . \\end{align*}"}
-{"id": "3366.png", "formula": "\\begin{align*} N _ { i j } ^ k = [ V _ i \\otimes V _ j : V _ k ] = [ ( V _ i \\otimes V _ j ) ^ * : V _ k ^ * ] = [ V _ j ^ * \\otimes V _ i ^ * : V _ k ^ * ] = [ V _ j \\otimes V _ i : V _ k ] = N _ { j i } ^ k . \\end{align*}"}
-{"id": "817.png", "formula": "\\begin{align*} u _ 3 ^ { q ^ d } + ( T - \\rho ) u _ 3 = u _ 2 . \\end{align*}"}
-{"id": "5338.png", "formula": "\\begin{align*} a _ k + \\overline { b _ { k + 2 } } \\frac { | | w ^ k | | ^ 2 } { | | w ^ { k + 2 } | | ^ 2 } = 0 , \\quad \\forall k . \\end{align*}"}
-{"id": "8691.png", "formula": "\\begin{align*} j ( \\nu + \\rho ' - q - 2 j ) c _ { h , i , j } = 2 ( i + 1 ) ( i + 2 ) ( 2 i + p ) ( 2 i + p + 2 ) c _ { h , i + 2 , j - 1 } \\end{align*}"}
-{"id": "3376.png", "formula": "\\begin{align*} F ( x ) & = \\frac { 1 } { 2 \\pi } \\int \\int _ { z ^ 2 + z '^ 2 \\leq 2 x } e ^ { - z ^ 2 / 2 } e ^ { - z '^ 2 / 2 } d z d z ' \\\\ & = \\int _ 0 ^ { \\sqrt { 2 x } } e ^ { - r ^ 2 / 2 } r d r \\\\ & = 1 - e ^ { - x } \\end{align*}"}
-{"id": "1939.png", "formula": "\\begin{align*} \\mathcal { A } = \\sum _ { j = 1 } ^ \\infty \\left [ Z _ j - \\tan Z _ j + ( \\chi + 1 ) \\chi ^ 2 \\frac { A _ j ^ 2 } { Z _ j ^ 3 } \\right ] - \\sum _ { j = 1 } ^ \\infty ( j - 1 ) \\pi . \\end{align*}"}
-{"id": "1363.png", "formula": "\\begin{align*} C _ n ^ { \\operatorname { c u b } } ( \\Lambda , \\mathcal { N } ) = \\bigoplus _ { \\lambda \\in Q _ n ( \\Lambda ) } \\mathcal { N } ( { s ( \\lambda ) } ) \\end{align*}"}
-{"id": "9880.png", "formula": "\\begin{align*} \\pi _ { n - } ^ { \\nu } ( K + a _ { n } ) = 1 ~ P ^ { \\mu } ~ a . s . \\end{align*}"}
-{"id": "3588.png", "formula": "\\begin{align*} \\frac { 1 } { \\mu } \\langle a \\cdot u , v \\rangle = \\langle u , v \\rangle = \\frac { 1 } { \\nu } \\langle u , a \\cdot v \\rangle = \\frac { 1 } { \\nu } \\langle a \\cdot u , v \\rangle \\end{align*}"}
-{"id": "5037.png", "formula": "\\begin{align*} P \\Big ( \\max _ { k = 0 , \\dots , n ^ m } | H _ n ( x _ { n k } ) | > \\eta \\Big ) \\leq \\sum _ { k = 0 } ^ { n ^ m } P ( | H _ n ( x _ { n k } ) | > \\eta ) = o ( 1 ) . \\end{align*}"}
-{"id": "9181.png", "formula": "\\begin{align*} f \\left ( 0 , s \\right ) \\in N , f \\left ( b , s \\right ) \\in \\tilde { N } , \\quad V ( t ) : = \\frac { \\partial f } { \\partial s } ( t , 0 ) . \\label { e n d s i n N a n d N t i l d e } \\end{align*}"}
-{"id": "2718.png", "formula": "\\begin{align*} h ( \\theta ) = \\frac { ( \\theta - \\tau ) ( \\frac { 1 } { \\tau } - \\theta ) } { \\sqrt { ( \\frac { 1 } { \\tau } - \\tau ) ( \\tau + \\frac { 1 } { \\tau } - 2 \\theta ) } } , x ( \\theta ) = \\frac { ( \\frac { 1 } { \\tau } - \\tau ) ^ 2 } { ( \\theta - \\tau ) ( \\frac { 1 } { \\tau } - \\theta ) } , \\end{align*}"}
-{"id": "4932.png", "formula": "\\begin{align*} z ^ n x ^ { n + 1 } = z ^ n ( x ^ { n + 1 } + y z ^ n ) - y z ^ { 2 n } \\end{align*}"}
-{"id": "3966.png", "formula": "\\begin{align*} \\mathbb { D } ^ J \\left ( v ^ k e ^ { - 4 m \\frac { y } { v ^ 2 } } \\cdot e ^ { 2 \\pi i ( ( n - n ' ) u + ( r - r ' ) x ) ) } \\cdot e ^ { - 2 \\pi ( ( n + n ' ) v + ( r + r ' ) y } \\right ) = 0 \\end{align*}"}
-{"id": "4906.png", "formula": "\\begin{align*} ( u \\circ \\eta , v \\circ \\eta ) _ 0 = \\int _ { \\Omega } g ( u , v ) \\ , \\mu ( x ) \\ , . \\end{align*}"}
-{"id": "3885.png", "formula": "\\begin{align*} K _ \\omega ( u _ n , v _ n ) & = H _ \\omega ( u _ n , v _ n ) - 3 P ( u _ n , v _ n ) \\\\ & = \\sum _ { j = 1 } ^ l H _ \\omega ( U ^ j , V ^ j ) + H _ \\omega ( u ^ l _ n , v ^ l _ n ) - 3 P ( u _ n , v _ n ) + o _ n ( 1 ) \\\\ & = \\sum _ { j = 1 } ^ l K _ \\omega ( U ^ j , V ^ j ) + 3 \\sum _ { j = 1 } ^ l P ( U ^ j , V ^ j ) - 3 P ( u _ n , v _ n ) + H _ \\omega ( u ^ l _ n , v ^ l _ n ) + o _ n ( 1 ) . \\end{align*}"}
-{"id": "1851.png", "formula": "\\begin{align*} \\psi ^ { \\rm e v e n } ( n , x , t ) = ( 2 k _ n ) ^ { - 1 / 2 } v ^ { \\rm e v e n } ( n , x ) e ^ { - i k _ n t } , \\end{align*}"}
-{"id": "360.png", "formula": "\\begin{align*} f ( t ) : = \\gamma t \\cdot \\frac { 1 - t / t _ { \\ast } } { 1 + c t } . \\end{align*}"}
-{"id": "5862.png", "formula": "\\begin{align*} S _ { n - 1 } ( \\alpha , \\beta , \\gamma ) = \\prod _ { j = 0 } ^ { n - 2 } \\frac { \\Gamma ( \\alpha \\gamma ^ { j } ) \\Gamma ( \\beta \\gamma ^ { j } ) \\Gamma ( \\gamma ^ { j + 1 } ) } { \\Gamma ( \\alpha \\beta \\gamma ^ { n + j - 1 } ) \\Gamma ( \\gamma ) } . \\end{align*}"}
-{"id": "6369.png", "formula": "\\begin{align*} \\beta _ { s } : = \\frac { 1 - \\lambda _ s } { \\alpha _ s } = 1 + O ( | s - 1 | ) . \\end{align*}"}
-{"id": "3745.png", "formula": "\\begin{align*} \\mathbf { J } _ { \\mathcal { E } , \\mathcal { P } } ( \\mathbf { r } ) = \\sum _ { j , l , m } \\frac { a _ { l , m } ^ { ( j ) } } { ( \\mathcal { B } _ { l , m } ^ { ( j ) } , \\mathcal { D } _ { l , m } ^ { ( j ) } ) } \\mathcal { D } _ { l , m } ^ { ( j ) } ( \\mathbf { r } ) , \\end{align*}"}
-{"id": "401.png", "formula": "\\begin{align*} \\langle F \\rangle _ { { \\rm L } , \\hat g } = \\langle F ( \\cdot \\ , - \\tfrac { Q } { 2 } \\omega ) \\rangle _ { { \\rm L } , g } \\exp \\big ( \\frac { \\mathbf { c } _ { \\rm L } } { 9 6 \\pi } S ^ { { \\rm c l } , 0 } _ { { \\rm L } } ( \\hat g , g ) \\big ) \\end{align*}"}
-{"id": "7762.png", "formula": "\\begin{align*} \\| A \\| _ { \\nu } = \\max \\{ \\nu ( A x ) : x \\in \\mathcal { M } _ { n \\times 1 } , \\nu ( x ) \\leq 1 \\} . \\end{align*}"}
-{"id": "6799.png", "formula": "\\begin{align*} f _ 1 = \\det ( d y ) - 1 = y ^ 1 _ { 1 0 } y ^ 2 _ { 0 1 } - y ^ 1 _ { 0 1 } y ^ 2 _ { 1 0 } - 1 \\ , , \\end{align*}"}
-{"id": "3870.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k \\deg ( W _ k ) \\leq \\deg ( V _ 1 ) \\deg ( V _ 2 ) . \\end{align*}"}
-{"id": "5613.png", "formula": "\\begin{align*} \\mathcal { M } ( t ) = \\int _ 0 ^ { + \\infty } | u ( x , t ) | ^ 2 d x = \\mathcal { M } ( 0 ) . \\end{align*}"}
-{"id": "7831.png", "formula": "\\begin{align*} \\| T + \\lambda S \\| = \\lim _ n \\| ( T + \\lambda S ) ( x _ n ) \\| = \\| ( T + \\lambda S ) ( x ) \\| . \\end{align*}"}
-{"id": "4848.png", "formula": "\\begin{align*} W _ { r _ i } = \\big \\langle w _ { ( 1 , r _ i ) } , \\dots , w _ { ( n _ i , r _ { i } ) } \\big \\rangle , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , 1 \\leq i \\leq m , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , W _ { M - 1 } = \\langle \\mu \\rangle . \\end{align*}"}
-{"id": "1619.png", "formula": "\\begin{align*} \\begin{aligned} g ( x ) = \\sigma ( x ) u ' ( x ) & = \\begin{cases} \\sigma ( x ) \\ell ^ \\ast S ' ( x ) m ( ( 0 , x ) ) , & \\mbox { i f $ x \\in ( 0 , b ^ * ) $ } \\\\ \\sigma ( x ) & \\mbox { i f $ x \\geq b ^ * $ } , \\end{cases} \\end{aligned} \\end{align*}"}
-{"id": "6217.png", "formula": "\\begin{align*} t \\mapsto I ( t ) = \\int _ { \\R ^ n } \\left [ f ( x ) - \\frac { 1 } { \\sqrt { ( 2 \\pi ) ^ n \\det t } } e ^ { - \\d x ^ * t ^ { - 1 } x } \\right ] ^ 2 d x . \\end{align*}"}
-{"id": "8719.png", "formula": "\\begin{align*} p g f _ { T _ { m , d } ^ * } ( s ) = \\sum _ { j \\leq m } \\hat { \\nu } ( j ) p g f _ { \\hat { T } _ { j , d } } ( s ) , \\end{align*}"}
-{"id": "7157.png", "formula": "\\begin{align*} p _ { \\rm e l } ( b ) = - e ( b ) - \\frac { 2 } { 3 } b e ' ( b ) . \\end{align*}"}
-{"id": "1550.png", "formula": "\\begin{align*} F _ 2 ^ \\delta \\left ( \\begin{array} { c } z \\\\ x \\end{array} \\right ) = \\left ( \\begin{array} { c } z + a ^ \\delta _ 2 ( x ) z ^ 2 + \\dots \\\\ b ^ \\delta _ 0 ( x ) + b ^ \\delta _ 1 ( x ) z + b ^ \\delta _ 2 ( x ) z ^ 2 + \\dots \\end{array} \\right ) . \\end{align*}"}
-{"id": "1871.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\left [ | \\beta _ { 0 j } | ^ 2 + \\Re ( \\alpha _ { 0 j } \\beta _ { 0 j } ) \\right ] = - \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ \\infty \\left | \\beta _ { 0 j } + \\overline { \\alpha _ { 0 j } } \\right | ^ 2 , \\end{align*}"}
-{"id": "9038.png", "formula": "\\begin{align*} \\mu _ { f } = \\sum _ { | \\xi | ^ 2 = E } | a _ { \\xi } | ^ 2 \\delta _ { \\xi / \\sqrt { E } } \\end{align*}"}
-{"id": "1252.png", "formula": "\\begin{align*} | T _ { 3 , 1 2 } | + | T _ { 3 , 2 2 } | \\lesssim \\int _ { | \\nu | > 2 0 } | \\nu | ^ { - 3 } d \\nu = O ( 1 ) . \\end{align*}"}
-{"id": "5104.png", "formula": "\\begin{align*} \\| U x \\| ^ p = \\left \\| \\sum _ { j \\in \\mathbb { J } } \\widehat { F } _ j A _ j x \\right \\| ^ p & = \\left \\| \\sum _ { j \\in \\mathbb { J } } \\widehat { F } _ j \\widehat { L } _ j \\left ( \\sum _ { k \\in \\mathbb { J } } L _ k A _ k x \\right ) \\right \\| ^ p = \\sum _ { j \\in \\mathbb { J } } \\left \\| \\widehat { L } _ j \\left ( \\sum _ { k \\in \\mathbb { J } } L _ k A _ k x \\right ) \\right \\| ^ p \\\\ & = \\sum _ { j \\in \\mathbb { J } } \\| A _ j x \\| ^ p = \\| x \\| ^ p , \\end{align*}"}
-{"id": "5794.png", "formula": "\\begin{align*} \\sum _ { i } \\log ( f _ i ) \\ = \\ \\sum _ { \\beta < \\alpha } s _ \\beta \\ell _ { \\beta + 1 } + c + \\epsilon \\end{align*}"}
-{"id": "4539.png", "formula": "\\begin{align*} \\int | f _ { g , \\delta , j } ( x ) | ^ 2 d x = \\int | \\hat { \\nu } _ { g , j } ( \\omega ) | ^ 2 | \\hat { \\phi } ^ D _ \\delta ( \\omega ) | ^ 2 d \\omega = 0 . \\end{align*}"}
-{"id": "9691.png", "formula": "\\begin{align*} \\sigma = \\{ \\zeta \\in \\mathbb T \\ : \\ | g _ 1 ( \\zeta ) | < c _ 3 \\} . \\end{align*}"}
-{"id": "9415.png", "formula": "\\begin{align*} ( f _ { \\theta } | _ 2 { \\sigma _ t } ) ( z ) = \\frac { \\theta ( f | _ k { \\sigma _ t } ) ( z ) } { ( f | _ k { \\sigma _ t } ) ( z ) } - \\frac { k } { 1 2 } E _ 2 ( z ) \\end{align*}"}
-{"id": "9831.png", "formula": "\\begin{align*} g ( z ' ) : = \\begin{cases} N ( 0 ^ + , \\tilde u _ { X _ \\circ , 0 } ( Z + \\ , \\cdot \\ , ) ) & X _ \\circ \\in \\mathcal { N } ( u ) \\\\ 0 & X _ \\circ \\notin \\mathcal { N } ( u ) \\end{cases} \\end{align*}"}
-{"id": "4601.png", "formula": "\\begin{align*} \\int f ( t ) d \\nu ( t ) = \\int f ( | x _ 1 - x _ 2 | ) d \\mu _ 1 ( x _ 1 ) d \\mu _ 2 ( x _ 2 ) . \\end{align*}"}
-{"id": "1148.png", "formula": "\\begin{align*} & \\sqrt { 4 - y _ i ^ 2 } / S ( I ) = ( 1 + | u _ i | / \\ln n ) S ( I ) / S ( I ) = 1 + | u _ i | / \\ln n \\leq 1 + C _ 0 / \\ln n . \\end{align*}"}
-{"id": "689.png", "formula": "\\begin{align*} \\mathcal M _ { \\mathcal C } ( X _ 1 \\dots X _ m ; X ) : = \\mathcal C ( \\varpi ( X _ 1 \\dots X _ m ) , X ) _ { \\mu _ m } \\ ; \\end{align*}"}
-{"id": "1528.png", "formula": "\\begin{align*} u ( x , t ) = u _ 1 ( x , t ) + u _ 2 ( x , t ) , x \\in I , \\ , t \\ge 0 , \\end{align*}"}
-{"id": "1945.png", "formula": "\\begin{align*} \\tan \\epsilon _ j - \\epsilon _ j = \\frac { \\epsilon _ j ^ 3 } { 3 } + \\frac { 2 \\epsilon _ j ^ 5 } { 1 5 } + \\frac { 1 7 \\epsilon _ j ^ 7 } { 3 1 5 } + \\cdots . \\end{align*}"}
-{"id": "6973.png", "formula": "\\begin{align*} & f ( 0 ) = 0 , \\ , f ( x ) \\ge 0 , \\ , f ( - x ) = f ( x ) , \\\\ \\lim _ { x \\to \\pm 1 } & \\frac { d ^ p f } { ( d x ) ^ p } ( x ) = \\infty , \\ , \\lim _ { x \\to \\pm 1 } \\frac { d ^ p } { ( d x ) ^ p } \\frac 1 { f ' ( x ) } = 0 , \\ , \\mbox { f o r $ p = 0 , 1 , \\dots $ } \\end{align*}"}
-{"id": "301.png", "formula": "\\begin{align*} n Z \\ ; = \\ ; X _ 1 + Y _ 1 \\textnormal { a n d } \\dim X _ 1 \\ ; = \\ ; \\dim X , \\dim Y _ 1 \\ ; = \\ ; \\dim Y . \\end{align*}"}
-{"id": "1902.png", "formula": "\\begin{align*} \\phi ^ { \\rm e v e n } ( j , x , t ) = \\phi ^ { \\rm e v e n } _ { \\rm O U T } ( j , x , t ) \\end{align*}"}
-{"id": "7565.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { X ' \\circ X ^ { - 1 } } _ { L i p ( 0 , T ; C ^ { \\alpha } ) } \\le \\norm { X ' } _ { L i p ( 0 , T ; C ^ { 1 + \\alpha } ) } \\norm { X - \\mathrm { I d } } _ { L i p ( 0 , T ; C ^ { 1 + \\alpha } ) } \\left ( 1 + \\norm { X - \\mathrm { I d } } _ { L ^ \\infty ( 0 , T ; C ^ { 1 + \\alpha } ) } \\right ) ^ { 1 + 3 \\alpha } . \\end{gathered} \\end{align*}"}
-{"id": "7886.png", "formula": "\\begin{align*} | \\mathfrak m _ N ( \\xi ) | \\le 1 6 e ^ { - \\frac { c \\kappa ( d , N ) ^ 2 } { 4 0 0 } \\sum _ { i = 1 } ^ d \\cos ^ 2 ( \\pi \\xi _ i ) } , \\end{align*}"}
-{"id": "2028.png", "formula": "\\begin{align*} \\lim _ { M \\to \\infty } G _ M ^ + ( x ) & = \\frac { 1 } { 2 \\pi } \\sum _ { n \\ge 1 } \\frac { a ( n ) \\sin ( 2 \\pi n x ) } { n ^ 2 } ; \\\\ \\lim _ { M \\to \\infty } G _ M ^ - ( x ) & = \\frac { 1 } { 2 \\pi i } \\sum _ { n \\ge 1 } \\frac { a ( n ) ( \\cos ( 2 \\pi n x ) - 1 ) } { n ^ 2 } . \\end{align*}"}
-{"id": "6501.png", "formula": "\\begin{align*} r ( K , y ) : = \\max \\{ \\lambda > 0 : \\ \\lambda y \\in K \\} \\ , , y \\in \\R ^ n \\setminus \\{ 0 \\} \\ , . \\end{align*}"}
-{"id": "1373.png", "formula": "\\begin{align*} \\lambda _ i ^ { \\sigma } = \\lambda \\left ( \\sum _ { j = 1 } ^ { i - 1 } d _ { C _ { ( \\sigma ( j ) ) } } ( \\lambda ) e _ { C _ { ( \\sigma ( j ) ) } } , \\sum _ { j = 1 } ^ { i } d _ { C _ { ( \\sigma ( j ) ) } } ( \\lambda ) e _ { C _ { ( \\sigma ( j ) ) } } \\right ) \\ ; . \\end{align*}"}
-{"id": "7071.png", "formula": "\\begin{align*} \\rho \\otimes \\rho & = y _ 1 ^ 3 \\oplus y _ 1 \\oplus y _ 1 ^ 2 \\oplus 2 \\rho \\\\ \\rho \\otimes \\alpha & = \\rho , \\ \\ \\forall \\alpha \\in \\{ y _ 1 ^ 3 , y _ 1 , y _ 1 ^ 2 \\} \\end{align*}"}
-{"id": "8420.png", "formula": "\\begin{align*} \\| u _ t ( \\cdot , t ) \\| _ { H ^ 1 } ^ 2 = \\| u _ t ( \\cdot , 0 ) \\| _ { H ^ 1 } ^ 2 + \\int _ 0 ^ t \\frac { d } { d \\tau } \\| u _ t ( \\cdot , \\tau ) \\| _ { H ^ 1 } ^ 2 d \\tau \\leq \\| u _ t ( \\cdot , 0 ) \\| _ { H ^ 1 } ^ 2 + T _ 0 C . \\end{align*}"}
-{"id": "5264.png", "formula": "\\begin{align*} r ( x ) = - \\int _ { x } ^ { u e p ( F ) } b ( t ) d t , \\ x < u e p ( F ) . \\end{align*}"}
-{"id": "7619.png", "formula": "\\begin{align*} ( L u ) ( x ) \\ = \\ \\lambda ( x ) \\int \\limits _ { \\mathbb R ^ d } a ( x - y ) \\mu ( y ) ( u ( y ) - u ( x ) ) d y \\end{align*}"}
-{"id": "8498.png", "formula": "\\begin{align*} I m \\{ \\zeta ( \\alpha , t ) - z _ j ( t ) \\} = & I m \\Big \\{ \\zeta ( \\alpha , 0 ) - z _ j ( 0 ) - \\int _ 0 ^ t \\dot { z } _ j ( \\tau ) d \\tau \\Big \\} + I m \\int _ 0 ^ t D _ { \\tau } \\zeta ( \\alpha , \\tau ) d \\tau \\\\ & - \\int _ 0 ^ t b ( \\alpha , \\tau ) I m \\{ \\partial _ { \\alpha } \\zeta ( \\alpha , \\tau ) \\} d \\tau . \\end{align*}"}
-{"id": "4058.png", "formula": "\\begin{align*} \\langle v ' , Y ( Y ( v _ 1 , & z _ 1 - z _ 2 ) v _ 2 , z _ 2 ) v _ 3 \\rangle \\\\ & = F ( \\alpha _ 1 , \\alpha _ 2 , \\alpha _ 3 ) ^ { - 1 } ( z _ 1 - z _ 2 ) ^ { - \\hat { b } ( \\alpha _ 1 , \\alpha _ 2 ) } z _ 2 ^ { - \\hat { b } ( \\alpha _ 1 , \\alpha _ 3 ) - \\hat { b } ( \\alpha _ 2 , \\alpha _ 3 ) } \\left ( 1 + \\frac { z _ 1 - z _ 2 } { z _ 2 } \\right ) ^ { - \\hat { b } ( \\alpha _ 1 , \\alpha _ 3 ) } f ( z _ 1 , z _ 2 ) \\end{align*}"}
-{"id": "7102.png", "formula": "\\begin{align*} ( s , ~ \\nu , ~ \\sigma ^ \\prime , ~ \\gamma ) = ( 1 . 9 8 4 0 , ~ 2 . 1 0 0 1 , ~ 1 . 1 3 2 0 5 , ~ 0 . 8 8 ) ~ . \\end{align*}"}
-{"id": "7202.png", "formula": "\\begin{align*} \\bar \\phi _ { k } ( 0 ) = \\bar \\mu _ { k } . \\end{align*}"}
-{"id": "3170.png", "formula": "\\begin{align*} \\ker ( \\gamma \\pm 1 ) \\cap \\ker ( c + 1 ) = \\ker ( \\gamma \\pm 1 ) \\cap \\ker \\partial . \\end{align*}"}
-{"id": "4629.png", "formula": "\\begin{align*} \\Sigma = \\begin{bmatrix} & & & & 1 \\\\ 1 & & & & \\\\ & 1 & & & \\\\ & & \\ddots & & \\\\ & & & 1 & \\end{bmatrix} \\in \\mathbb { R } ^ { n \\times n } , \\end{align*}"}
-{"id": "342.png", "formula": "\\begin{align*} | I m ( \\partial ' _ k ) _ * | = \\frac { \\frac { 1 } { 2 } n ( n - 1 ) } { ( \\frac { 1 } { 2 } n ( n - 1 ) , k ) } \\cdot \\frac { n ( n + 1 ) } { ( n ( n + 1 ) , k ) } . \\end{align*}"}
-{"id": "7759.png", "formula": "\\begin{align*} \\| A \\| _ p = \\bigg \\{ \\sum _ { i = 1 } ^ n s _ { i } ( A ) ^ p \\bigg \\} ^ \\frac { 1 } { p } , \\end{align*}"}
-{"id": "5094.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathbb { J } } \\langle F _ j x , F _ j x \\rangle = \\sum _ { j \\in \\mathbb { J } } \\left \\langle \\sum _ { k \\in \\mathbb { L } _ j } a _ { j , k } e _ { j , k } , \\sum _ { m \\in \\mathbb { L } _ j } a _ { j , m } e _ { j , m } \\right \\rangle = \\sum _ { j \\in \\mathbb { J } } \\sum _ { k \\in \\mathbb { L } _ j } a _ { j , k } a _ { j , k } ^ * = \\langle x , x \\rangle . \\end{align*}"}
-{"id": "5793.png", "formula": "\\begin{align*} \\log T ( f ) \\ & = \\ \\Phi ( \\log f ) + \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { n ! } L _ n \\left ( \\Phi \\left ( \\frac { f ' } { f } \\right ) , \\ldots , \\Phi \\left ( \\frac { f ^ { ( n ) } } { f } \\right ) \\right ) \\epsilon ^ n \\\\ & = \\ \\Phi ( \\log f ) + \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { n ! } \\Phi \\left ( L _ n \\left ( \\frac { f ' } { f } , \\ldots , \\frac { f ^ { ( n ) } } { f } \\right ) \\right ) \\epsilon ^ n . \\end{align*}"}
-{"id": "8977.png", "formula": "\\begin{align*} P ^ { \\mathcal { K } } ( n ) = \\sum _ { \\lambda \\vdash m } C ^ { \\mathcal { K } } ( \\lambda ) \\left ( ( \\pm ) _ \\lambda q ^ { ( \\lambda ) } \\right ) ^ n \\end{align*}"}
-{"id": "7215.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 f ( t , x ) \\ , d \\sigma _ r ( t ) = \\sum _ { k \\neq 0 } | k | ^ { - r } e ^ { i x k } = K _ r ( x ) . \\end{align*}"}
-{"id": "5373.png", "formula": "\\begin{align*} & a _ { 1 + p + q \\ , 2 t + 3 p - 3 q } = 2 c - ( p + 1 ) t - p ( p + 2 ) , \\\\ & b _ { 1 + p + q \\ , 2 t + 3 p - 3 q } = 2 c + ( q + 1 ) t - q ( q + 2 ) , \\\\ & c _ { 2 + p + q \\ , 2 t + 3 p - 3 q - 3 } = \\frac { q + 1 } { p + q + 2 } , \\\\ & d _ { 2 + p + q \\ , 2 t + 3 p - 3 q + 3 } = \\frac { p + 1 } { p + q + 2 } . \\end{align*}"}
-{"id": "6538.png", "formula": "\\begin{align*} m ^ { P } _ \\beta = \\sum _ { k | \\beta } \\ ; \\frac { ( - 1 ) ^ { ( k - 1 ) w / k } } { k ^ 2 } \\ , \\mu ( k ) \\ , \\mathcal N ^ { P } _ { \\beta / k } ( S , E ) , \\end{align*}"}
-{"id": "10029.png", "formula": "\\begin{align*} \\langle f , g \\rangle _ { k , l , m } & = \\left ( \\int _ { - 1 } ^ { - q ^ { k + 1 } } + \\int _ { z q ^ { m + 1 } } ^ { z q ^ { - l } } \\right ) f ( x ) \\overline { g ( x ) } w ( x ) d _ q x \\\\ & = ( 1 - q ) \\sum _ { n = 0 } ^ k f ( - q ^ n ) \\overline { g ( - q ^ n ) } w ( - q ^ n ) q ^ n \\\\ & + ( 1 - q ) \\sum _ { n = - l } ^ m f ( z q ^ n ) \\overline { g ( z q ^ n ) } w ( z q ^ n ) z q ^ n . \\end{align*}"}
-{"id": "6810.png", "formula": "\\begin{align*} x = \\Phi ^ t ( \\beta ) = \\varphi ^ t \\circ v ^ { - 1 } = A ( t ) \\beta \\ , , \\end{align*}"}
-{"id": "704.png", "formula": "\\begin{align*} \\left ( ( 2 p - 1 ) x + \\frac { 4 p K } { f ( x ) } \\right ) \\frac { f ^ { ( 1 ) } ( x ) } { f ( x ) ^ 2 } = ( 2 p - 1 ) x + \\frac { 6 p K + 1 } { f ( x ) } - 2 ( p + 1 ) \\frac { \\displaystyle \\int _ 0 ^ x f ( t ) ^ 3 \\ ; d t } { f ( x ) ^ 3 } + \\frac { 4 e ^ { 3 F ( 0 ) } } { f ( x ) ^ 3 } . \\end{align*}"}
-{"id": "9124.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } S ( \\rho _ { 1 } ( t ) ) = 0 \\Leftrightarrow \\rho _ { 1 } ( \\mathbf { x } _ { 1 } , t ) = \\rho _ { 1 M } ^ { ( N ) } ( \\mathbf { v } _ { 1 } ) \\end{align*}"}
-{"id": "8985.png", "formula": "\\begin{align*} P ( \\{ \\sup _ { 0 \\leq t \\leq T } | \\mathbb { \\hat { E } } _ { t } [ \\xi ^ { n } ] - M _ { t } | > \\varepsilon \\} ) & = P ( \\{ \\sup _ { 0 \\leq t \\leq T } | \\mathbb { \\hat { E } } _ { t } [ \\xi ^ { n } ] - \\mathbb { \\hat { E } } _ { t } [ M _ { T } ] | > \\varepsilon \\} ) \\\\ & \\leq P ( \\{ \\sup _ { 0 \\leq t \\leq T } \\mathbb { \\hat { E } } _ { t } [ | \\xi ^ { n } - M _ { T } | ] > \\varepsilon \\} ) \\\\ & \\leq \\frac { 1 } { \\varepsilon } \\mathbb { \\hat { E } } [ | \\xi ^ { n } - M _ { T } | ] . \\end{align*}"}
-{"id": "2015.png", "formula": "\\begin{align*} E ( d , j , j ' ) = \\frac { 1 } { d } \\frac { \\prod \\limits _ { k = 0 } ^ { 5 d } \\left ( 5 \\alpha _ j + k \\frac { \\alpha _ { j ' } - \\alpha _ j } { d } \\right ) } { \\prod \\limits _ { l = 0 } ^ 4 \\prod \\limits _ { k = 0 \\atop ( l , k ) \\neq ( j , 0 ) } ^ d \\left ( \\alpha _ j - \\alpha _ l + k \\frac { \\alpha _ { j ' } - \\alpha _ j } { d } \\right ) } , \\end{align*}"}
-{"id": "4000.png", "formula": "\\begin{align*} \\begin{pmatrix} e ( \\frac { \\nu _ 0 } { p } ) & e ( - \\frac { \\nu _ 0 } { p } ) \\\\ e ( \\frac { \\mu \\nu _ 0 } { p } ) & e ( - \\frac { \\mu \\nu _ 0 } { p } ) \\end{pmatrix} \\end{align*}"}
-{"id": "2108.png", "formula": "\\begin{align*} d \\Psi ( t _ 0 , t ) = - A ^ T ( t ) \\Psi ( t _ 0 , t ) d t - C ^ T ( t ) \\Psi ( t _ 0 , t ) d \\omega ( t ) \\end{align*}"}
-{"id": "6166.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta + m _ i ^ 2 ) ^ { 1 / 2 } u _ i + V ( x ) u _ i = \\omega _ i u _ i ^ 3 - \\beta u _ i \\sum _ { j \\neq i } a _ { i j } u ^ 2 _ j \\ ; \\\\ u _ i \\in H ^ { 1 / 2 } ( \\R ^ N ) , \\end{cases} \\end{align*}"}
-{"id": "4560.png", "formula": "\\begin{align*} \\tau _ 1 . A ( z ) & = { \\tau _ 1 } ^ 2 \\tau _ 2 \\cdots \\tau _ { c - 1 } Q ( z ) \\\\ & = { \\tau _ 1 } ^ 2 e ( h _ 1 , h _ c , h _ 2 , \\ldots , h _ { c - 1 } , q , h _ { c + 1 } , \\ldots , h _ { m _ k } ) \\tau _ 2 \\cdots \\tau _ { c - 1 } Q ( z ) \\\\ & = \\tau _ 2 \\cdots \\tau _ { c - 1 } Q ( z ) \\end{align*}"}
-{"id": "2779.png", "formula": "\\begin{align*} p _ { t + 1 } ( \\mathbf { n } ) = \\sum _ { \\mathbf { k } \\in \\mathbb { N } _ 0 ^ S } p _ t ( \\mathbf { k } ) K ( \\mathbf { k } , \\mathbf { n } ) . \\end{align*}"}
-{"id": "5459.png", "formula": "\\begin{align*} \\| r \\| _ { _ { { \\bf L ^ p ( \\lambda _ 0 ) } } } ^ p \\ : = \\ \\int _ \\Lambda | r ( b ) | ^ p \\ , \\lambda _ 0 ( d b ) \\ < \\ \\infty . \\end{align*}"}
-{"id": "6114.png", "formula": "\\begin{align*} \\alpha _ { d - \\mu _ 1 } Y = \\beta _ { d - \\mu _ 2 } X \\end{align*}"}
-{"id": "6143.png", "formula": "\\begin{align*} H = 2 u _ 1 + \\frac { n + 1 } { 2 } v _ 1 + \\ell L = u _ 2 - e ' v _ 2 . \\end{align*}"}
-{"id": "4963.png", "formula": "\\begin{align*} \\Delta _ { T ( p , q ) } ( t ) = \\dfrac { ( t ^ { p q } - 1 ) ( t - 1 ) } { ( t ^ p - 1 ) ( t ^ q - 1 ) } \\end{align*}"}
-{"id": "5493.png", "formula": "\\begin{align*} d \\Pi ( b | X ^ T ) = \\frac { e ^ { \\ell _ T ( b ) } d \\Pi ( b ) } { \\int e ^ { \\ell _ T ( b ) } d \\Pi ( b ) } \\propto e ^ { \\ell _ T ( b ) - \\frac { 1 } { 2 } \\| b \\| _ { \\mathbb H } ^ 2 } , ~ ~ b \\in V _ J ^ { \\otimes d } . \\end{align*}"}
-{"id": "3415.png", "formula": "\\begin{align*} \\tau _ 0 ^ M ( T ) : = \\log \\frac { \\rho _ { 0 , \\sup } ^ M ( T ) } { \\rho _ { 0 , \\inf } ^ M ( T ) } , \\tau _ * ^ M ( T ) = \\log \\frac { \\rho _ { * , \\sup } ^ M ( T ) } { \\rho _ { * , \\inf } ^ M ( T ) } , \\end{align*}"}
-{"id": "6967.png", "formula": "\\begin{align*} g v l = \\theta _ 1 \\wedge d \\theta _ 1 . \\end{align*}"}
-{"id": "5504.png", "formula": "\\begin{align*} \\tilde { \\Phi } _ { \\lambda , k , j , i } = \\begin{cases} a _ \\lambda P _ { V _ J } [ \\Phi _ { \\lambda , k } / \\mu _ 0 ] & i = j , \\\\ 0 & i \\neq j . \\end{cases} \\end{align*}"}
-{"id": "5792.png", "formula": "\\begin{align*} A _ n ( q ) & = \\frac { 1 } { ( q - 1 ) q ^ { 2 n + 1 } } \\left ( ( q - 2 ) q ^ { 2 n + 1 } + q ^ { 3 n } ( q ^ n - 1 ) + q ^ { 2 n + 1 } ( q ^ n - 1 ) + q ^ { 2 n + 2 } + ( q - 1 ) q ^ { 3 n + 1 } \\right ) \\\\ & = 2 + q ^ n + \\sum _ { i = 0 } ^ { n - 1 } q ^ { i + n - 1 } + \\sum _ { i = 0 } ^ { n - 1 } q ^ i . \\end{align*}"}
-{"id": "7969.png", "formula": "\\begin{align*} \\abs { J _ q ^ A } & \\geq \\sum _ { j = 2 } ^ m ( d _ j - d _ { q , j } ^ A ) \\\\ & \\geq n - \\abs { S _ q ^ A } + d _ { q , 1 } ^ A - d _ 1 + 1 , \\end{align*}"}
-{"id": "5721.png", "formula": "\\begin{align*} _ c D _ { 0 t } ^ { \\alpha } \\left ( \\int _ 0 ^ x ( x - t ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } [ \\lambda ( x - t ) ^ { \\alpha } ] f ( t ) d t \\right ) & = f ( x ) - \\int _ 0 ^ x f ( z ) \\frac { d } { d z } [ E _ { \\alpha , 1 } ( \\lambda ( x - z ) ^ { \\alpha } ) ] d z . \\end{align*}"}
-{"id": "8926.png", "formula": "\\begin{align*} \\eta _ 1 & : = n ( \\textrm { U n c h a n g e d , I n c o r r e c t } ) \\\\ \\eta _ 2 & : = n ( \\textrm { C h a n g e d , I n c o r r e c t } ) \\end{align*}"}
-{"id": "960.png", "formula": "\\begin{align*} y = \\mathcal { H } ( x ^ * ) + \\zeta , \\end{align*}"}
-{"id": "5186.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } F _ { n } ( \\alpha _ { n } x + \\beta _ { n } ) = H _ { 2 } ( x ) , x \\in C ( H _ { 2 } ) . \\end{align*}"}
-{"id": "8911.png", "formula": "\\begin{align*} \\mathrm { B C } ( P _ { y } , P _ { x _ 0 } ) & \\ge \\left ( 1 - \\frac { a + b } { 2 n } + \\frac { a b } { n } - 2 \\frac { ( a - b ) ^ 2 } { n ^ 2 } \\right ) ^ { s ( n - s ) } \\\\ & = \\left ( 1 - \\frac { ( \\sqrt { a } - \\sqrt { b } ) ^ 2 } { 2 n } - 2 \\frac { ( a - b ) ^ 2 } { n ^ 2 } \\right ) ^ { s ( n - s ) } \\\\ & \\ge \\left ( 1 - \\frac { ( \\sqrt { a } - \\sqrt { b } ) ^ 2 } { n } \\right ) ^ { s ( n - s ) } \\\\ & \\ge \\exp { - 2 s ( \\sqrt { a } - \\sqrt { b } ) ^ 2 } , \\end{align*}"}
-{"id": "715.png", "formula": "\\begin{align*} & g _ { 1 2 } = g _ { 2 1 } = 0 , \\\\ & g _ { 1 1 2 } = g _ { 1 2 1 } = g _ { 2 1 1 } = 0 , \\\\ & g _ { 2 2 2 } = 0 , \\\\ & g _ { 1 1 1 2 } = g _ { 1 1 2 1 } = g _ { 1 2 1 1 } = g _ { 2 1 1 1 } = 0 , \\\\ & g _ { 1 2 2 2 } = g _ { 2 1 2 2 } = g _ { 2 2 1 2 } = g _ { 2 2 2 1 } = 0 . \\end{align*}"}
-{"id": "3806.png", "formula": "\\begin{align*} I ( m , n _ 1 ^ 2 n _ 2 , q ) : = \\int _ 0 ^ \\infty U ( y ) y ^ { - i t } e \\left ( \\pm \\frac { 2 \\sqrt { m N y } } { q } \\pm \\frac { 3 ( N n _ 1 ^ 2 n _ 2 ( y + u ) ) ^ { 1 / 3 } } { q r ^ { 1 / 3 } } \\right ) \\mathrm { d } y . \\end{align*}"}
-{"id": "8941.png", "formula": "\\begin{align*} q p _ { b l } ( y _ n , y _ m ) & \\leq \\sum \\limits _ { k = n } ^ { m - 1 } s ^ k q p _ { b l } ( y _ k , y _ { k + 1 } ) \\\\ & \\leq \\sum \\limits _ { k = n } ^ { m - 1 } ( 2 s \\lambda ) ^ k \\Big ( \\frac { q p _ { b l } ( y _ 0 , y _ 1 ) + q p _ { b l } ( y _ 1 , y _ 0 ) } { 2 } \\Big ) \\\\ & \\leq \\frac { ( 2 s \\lambda ) ^ n } { 1 - 2 s \\lambda } \\Big ( \\frac { q p _ { b l } ( y _ 0 , y _ 1 ) + q p _ { b l } ( y _ 1 , y _ 0 ) } { 2 } \\Big ) . \\end{align*}"}
-{"id": "7790.png", "formula": "\\begin{align*} M _ t ^ G f ( x ) = \\mathcal F ^ { - 1 } ( m ^ G ( t \\xi ) \\mathcal F f ) ( x ) = \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ t ^ { \\infty } \\frac { t } { u } \\bigg ( 1 - \\frac { t } { u } \\bigg ) ^ { \\alpha - 1 } \\mathcal P _ { u } ^ { \\alpha } f ( x ) \\frac { { \\rm d } u } { u } . \\end{align*}"}
-{"id": "2412.png", "formula": "\\begin{align*} S ^ { \\uparrow } = \\bigcup \\limits _ { X \\in S } X ^ { \\uparrow } , S ^ { \\downarrow } = \\bigcup \\limits _ { X \\in S } X ^ { \\downarrow } . \\end{align*}"}
-{"id": "3498.png", "formula": "\\begin{align*} f _ { - 1 } ( x ) = \\oint _ { \\Sigma _ { - \\infty } } \\frac { d t } { 2 \\pi i } \\Gamma ( t ) e ^ { - \\frac { \\gamma } { 2 } t ^ { 2 } + x t } , g _ { - 1 } ( x ) = \\int ^ { 1 + i \\infty } _ { 1 - i \\infty } \\frac { d s } { 2 \\pi i } \\frac { 1 } { \\Gamma ( s ) } e ^ { \\frac { \\gamma } { 2 } s ^ { 2 } - x s } \\end{align*}"}
-{"id": "5222.png", "formula": "\\begin{align*} V ( y ) = \\inf \\{ x > 0 , U ( x ) \\geq x \\} \\end{align*}"}
-{"id": "880.png", "formula": "\\begin{align*} y _ { \\gamma _ 1 + \\gamma _ 2 } = - q ^ { - \\frac { 1 } { 2 } \\lambda ( \\gamma _ 1 , \\gamma _ 2 ) } y _ { \\gamma _ 1 } y _ { \\gamma _ 2 } . \\end{align*}"}
-{"id": "5188.png", "formula": "\\begin{align*} \\frac { F _ { n } ^ { - 1 } ( u _ { i } ) - b _ { n } } { a _ { n } } \\rightarrow H _ { 1 } ^ { - 1 } ( u _ { i } ) , i = 1 , 2 . \\end{align*}"}
-{"id": "5681.png", "formula": "\\begin{align*} \\overline { \\hat T _ { C _ 1 } ( x _ 0 , f _ 1 ( x _ 0 ) , f _ 2 ( x _ 0 ) ) - \\hat T _ { C _ 2 } ( x _ 0 , f _ 1 ( x _ 0 ) , f _ 2 ( x _ 0 ) ) } = X \\times \\mathbb { R } \\times \\mathbb { R } \\ , . \\end{align*}"}
-{"id": "6368.png", "formula": "\\begin{align*} ( t , z ) \\mapsto v _ s ( t , z ) : = \\frac { \\alpha _ s } { s } u ( t s , x ) + ( 1 - \\alpha _ s ) \\rho ( x ) - C | s - 1 | t \\end{align*}"}
-{"id": "4428.png", "formula": "\\begin{align*} q ' < \\Bigl ( \\frac { \\mathcal { H } } { \\mathcal { H } - b } \\Bigr ) ' = \\frac { \\mathcal { H } } { b } . \\end{align*}"}
-{"id": "655.png", "formula": "\\begin{align*} \\widetilde F ^ { \\mathcal G } ( [ \\varphi ; f _ 1 , \\dots , f _ n ; x ] ) = [ \\varphi ; F ( f _ 1 ) , \\dots , F ( f _ n ) ; x ] \\ . \\end{align*}"}
-{"id": "5571.png", "formula": "\\begin{align*} & \\mathcal { Q } ^ u ( s ) = 2 | u _ x ( 0 , s ) | ^ 2 + 2 \\big ( u ( 0 , s ) \\bar { u } _ s ( 0 , s ) \\big ) - \\beta | u ( 0 , s ) | ^ 4 , \\\\ & \\mathcal { Q } ^ v ( s ) = \\frac { \\alpha } { \\gamma } v _ x ^ 2 ( 0 , s ) - \\frac { 2 \\alpha } { \\gamma } v _ { x x } ( 0 , s ) v ( 0 , s ) - \\frac { 2 \\alpha } { 3 \\gamma } v ^ 3 ( 0 , s ) . \\end{align*}"}
-{"id": "161.png", "formula": "\\begin{align*} X _ { V } \\left ( \\{ x ^ i , x ^ j \\} _ { T M } \\right ) & - \\{ X _ { V } ( x ^ i ) , x ^ j \\} _ { T M } - \\{ x ^ i , X _ { V } ( x ^ j ) \\} _ { T M } = 0 \\end{align*}"}
-{"id": "6858.png", "formula": "\\begin{align*} \\varphi ( t , \\alpha ) = \\varphi ^ t ( \\alpha ) = M _ 1 ( t ) v ( \\alpha ) + M _ 2 ( t ) w ( \\alpha ) \\ , , \\end{align*}"}
-{"id": "9647.png", "formula": "\\begin{align*} & \\sum _ { n = 1 } ^ { N + 1 } \\frac { ( q ^ 2 ; q ^ 2 ) _ { n - 1 } ( q ^ 2 ; q ^ 2 ) _ N ( z q ; q ^ 2 ) _ { N - n + 1 } ( z q ) ^ { n } } { ( z q ^ 2 ; q ^ 2 ) _ { n } ( q ^ 2 ; q ^ 2 ) _ { N - n + 1 } ( z q ; q ^ 2 ) _ { N + 1 } } \\\\ & = \\frac { 1 } { ( 1 - q ^ { 2 N + 2 } ) } \\sum _ { n = 1 } ^ { N + 1 } \\left ( \\frac { ( q ; q ) _ { 2 n - 2 } z ^ { 2 n - 1 } q ^ { n ( 2 n - 1 ) } } { ( z q ; q ) _ { 2 n - 1 } } + \\frac { ( q ; q ) _ { 2 n - 1 } z ^ { 2 n } q ^ { n ( 2 n + 1 ) } } { ( z q ; q ) _ { 2 n } } \\right ) \\frac { \\left ( q ^ { 2 N - 2 n + 4 } ; q ^ 2 \\right ) _ n } { \\left ( z q ^ { 2 N + 3 } ; q ^ 2 \\right ) _ n } . \\end{align*}"}
-{"id": "3451.png", "formula": "\\begin{align*} \\psi ( z ) = - \\gamma _ 0 + \\sum ^ { \\infty } _ { n = 0 } \\left ( \\frac { 1 } { n + 1 } - \\frac { 1 } { n + z } \\right ) , \\psi ' ( z ) = \\sum ^ { \\infty } _ { n = 0 } \\frac { 1 } { ( n + z ) ^ 2 } , \\end{align*}"}
-{"id": "10051.png", "formula": "\\begin{align*} T ( B ^ { - 1 } \\tilde p ) = \\frac { 1 } { 2 } \\tilde p ^ { \\top } ( B ^ { - 1 } ) ^ { \\top } M B ^ { - 1 } \\tilde p = \\frac { 1 } { 2 } \\tilde p ^ { \\top } A M A ^ { \\top } \\tilde p , \\end{align*}"}
-{"id": "1542.png", "formula": "\\begin{align*} F \\left ( \\begin{array} { c } z \\\\ x \\end{array} \\right ) = \\left ( \\begin{array} { c } z + q _ 1 ( z + \\delta x ) \\\\ \\delta x - q _ 1 ( z + \\delta x ) \\end{array} \\right ) \\ ; \\ ; \\mathrm { a n d } \\ ; \\ ; G \\left ( \\begin{array} { c } w \\\\ y \\end{array} \\right ) = \\left ( \\begin{array} { c } w + q _ 2 ( w + \\delta y ) \\\\ \\delta y - q _ 2 ( w + \\delta y ) \\end{array} \\right ) . \\end{align*}"}
-{"id": "8249.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty n p ( n ) q ^ n = \\frac { 1 } { ( q ; q ) _ \\infty } \\sum _ { n = 1 } ^ \\infty \\frac { n q ^ n } { 1 - q ^ n } = \\frac { 1 - E _ 2 ( \\tau ) } { 2 4 ( q ; q ) _ \\infty } . \\end{align*}"}
-{"id": "2149.png", "formula": "\\begin{align*} d ( \\mathbf x , \\mathbf y ) = \\inf _ { \\sigma \\in G } \\| \\sigma ( \\mathbf x ) - \\mathbf y \\| \\end{align*}"}
-{"id": "7611.png", "formula": "\\begin{align*} ( X , \\tau , v ) \\rightarrow \\mathcal { S } ( X , \\tau , v ) = ( X ^ { n e w } , \\tau ^ { n e w } , v ^ { n e w } ) \\end{align*}"}
-{"id": "4216.png", "formula": "\\begin{align*} E _ n ^ { - 1 } ( z ) \\frac { A _ n ^ { ( 1 ) } ( s ) - A _ n ^ { ( 1 ) } ( 0 ) } { s } E _ n ( z ) = \\frac { 1 } { 2 \\pi i } \\oint _ { | t | = 2 C n ^ { - \\frac { 5 } { 2 } } } \\frac { E _ n ^ { - 1 } ( z ) A _ n ^ { ( 1 ) } ( t ) E _ n ( z ) } { t ( t - s ) } d t \\end{align*}"}
-{"id": "8150.png", "formula": "\\begin{align*} \\forall \\ , x \\in \\mathcal { L } _ { g , 0 } ( H ) , \\ : \\ : \\ : \\widetilde { f } ( x ) = \\widehat { f } x \\widehat { f } ^ { - 1 } . \\end{align*}"}
-{"id": "2517.png", "formula": "\\begin{align*} E [ e ^ { - u \\mathbf z ^ { ( \\alpha ) } } ] = 1 - ( 1 + u ^ { - \\alpha } ) ^ { - 1 / \\alpha } , u \\geq 0 . \\end{align*}"}
-{"id": "7529.png", "formula": "\\begin{align*} \\left \\{ \\begin{gathered} \\partial _ t u - \\nu \\Delta u = \\mathbb { H } \\left ( \\mathrm { d i v } \\ , \\ , ( \\sigma - u \\otimes u ) \\right ) , \\\\ \\nabla \\cdot u = 0 , \\\\ \\partial _ t \\sigma + u \\cdot \\nabla \\sigma = ( \\nabla u ) \\sigma + \\sigma ( \\nabla u ) ^ T - 2 k \\sigma + 2 \\rho K ( ( \\nabla u ) + ( \\nabla u ) ^ T ) , \\\\ u ( x , 0 ) = u _ 0 ( x ) , \\sigma ( x , 0 ) = \\sigma _ 0 ( x ) . \\end{gathered} \\right . \\end{align*}"}
-{"id": "5599.png", "formula": "\\begin{align*} A _ 2 = \\frac { \\alpha } { 4 \\gamma } \\frac { d } { d t } \\int _ { - \\infty } ^ 0 \\ ! \\ ! \\ ! x v ^ 2 d x + \\frac { 3 \\alpha } { 4 \\gamma } \\int _ { - \\infty } ^ 0 \\ ! \\ ! \\ ! v _ x ^ 2 d x - \\frac { \\alpha } { 6 \\gamma } \\int _ { - \\infty } ^ 0 \\ ! \\ ! \\ ! v ^ 3 d x \\end{align*}"}
-{"id": "7768.png", "formula": "\\begin{align*} x \\cdot \\xi = \\langle x , \\xi \\rangle = \\sum _ { k = 1 } ^ d x _ k \\xi _ k \\end{align*}"}
-{"id": "2602.png", "formula": "\\begin{align*} J _ G ( r ) - J ' _ G ( t , r , \\xi ) \\xrightarrow [ t \\to \\infty ] { L ^ 2 ( \\Pi _ x ^ { ( \\phi ) } ) } \\int _ 0 ^ 1 \\frac { \\gamma _ 0 } { u } G \\big ( r u ^ { \\frac { 1 } { \\gamma _ 0 - 1 } } \\big ) ^ { \\gamma _ 0 - 1 } \\big \\langle 1 - C _ X ^ { - 1 } \\mathbf 1 _ { \\gamma ( \\cdot ) = \\gamma _ 0 } \\kappa \\phi ^ { \\gamma _ 0 - 1 } , \\phi \\phi ^ * \\big \\rangle _ m d u = 0 . \\end{align*}"}
-{"id": "3200.png", "formula": "\\begin{align*} { \\rm i n d } _ \\varGamma ( - U ) = M _ - - M _ + \\end{align*}"}
-{"id": "2291.png", "formula": "\\begin{align*} \\mathbf { k } _ \\lambda & = \\mathbf { k } _ { \\mathbf { m } , j } = \\frac { 2 \\pi } { L } \\mathbf { m } , & \\omega _ \\lambda = | \\mathbf { k } _ \\lambda | & = \\frac { 2 \\pi } { L } | \\mathbf { m } | & \\mathbf { e } _ \\lambda & = \\mathbf { e } _ j ( \\mathbf { k } _ \\lambda ) . \\end{align*}"}
-{"id": "6684.png", "formula": "\\begin{align*} \\psi _ { 0 } ( x ) = | x | ^ { - \\frac { 1 } { 2 } + i ( \\frac { \\omega } { 2 } - \\lambda _ { * } ) } , \\ A \\psi _ { 0 } ( x ) = 0 , \\ E _ { 0 } = 0 . \\end{align*}"}
-{"id": "2446.png", "formula": "\\begin{gather*} \\begin{alignedat} { 3 } \\phi ( e _ 1 ) & = \\alpha _ 1 e _ 1 + \\alpha _ 2 e _ 2 , & \\phi ( e _ 2 ) & = \\beta _ 1 e _ 1 + \\beta _ 2 e _ 2 , & \\phi ( e _ 3 ) & = \\gamma e _ 3 , \\\\ \\phi ( e _ 4 ) & = \\delta _ 4 e _ 4 + \\delta _ 5 e _ 5 , & \\phi ( e _ 5 ) & = \\epsilon _ 4 e _ 4 + \\epsilon _ 5 e _ 5 , & \\phi ( e _ 6 ) & = \\zeta e _ 6 , \\end{alignedat} \\end{gather*}"}
-{"id": "2554.png", "formula": "\\begin{align*} \\{ ( X _ t ) _ { t \\geq 0 } ; \\mathbf P _ \\mu ^ { ( \\phi ) } \\} \\overset { f . d . d . } { = } \\{ ( X _ t + W _ t ) _ { t \\geq 0 } ; \\mathbf P _ \\mu \\otimes \\mathbb N ^ { ( \\phi ) } _ \\mu \\} . \\end{align*}"}
-{"id": "7851.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { k _ 0 - 1 } u _ { d - k m } \\ge \\frac { 1 } { m } \\sum _ { k = 1 } ^ { k _ 0 - 1 } \\sum _ { j \\in U _ { k - 1 } } u _ j \\ge \\frac { 1 } { m } \\Big ( \\bar { m } - \\sum _ { j \\in U _ { k _ 0 } } u _ j - \\sum _ { j \\in U _ { k _ 0 - 1 } } u _ j \\Big ) \\ge \\frac { 1 } { m } \\big ( m - ( 1 - \\delta _ 0 ) m \\big ) \\ge \\delta _ 0 . \\end{align*}"}
-{"id": "631.png", "formula": "\\begin{align*} ( \\varphi ^ y \\psi ) ^ \\ast ( y ^ { - 1 } ) ^ { - 1 } \\cdot K _ { ( \\varphi ^ y \\psi ) ^ { y ^ { - 1 } } } \\cdot ( \\varphi ^ y \\psi ) ^ \\ast ( y ^ { - 1 } ) = K _ { \\varphi ^ y \\psi } \\ . \\end{align*}"}
-{"id": "1472.png", "formula": "\\begin{align*} V _ { 1 } = - \\int _ { \\mathbb { T } ^ { d } } \\left | \\nabla \\left ( \\mu ^ { 1 } - \\mu ^ { 2 } \\right ) \\right | ^ { 2 } \\ d x . \\end{align*}"}
-{"id": "1797.png", "formula": "\\begin{align*} X _ { n } = \\frac { B _ { \\theta ^ { n } } - B _ { \\theta ^ { n + 1 } } } { ( \\theta ^ { n } - \\theta ^ { n + 1 } ) ^ { H } } \\sim N ( 0 , 1 ) , \\end{align*}"}
-{"id": "6911.png", "formula": "\\begin{align*} I _ 1 = I + \\langle a _ { 1 4 } '' - a _ { 1 2 } '' + a _ { 1 1 } '' , a _ { 2 4 } '' - a _ { 2 2 } '' + a _ { 2 1 } '' \\rangle \\\\ \\subset \\ \\mathbb { A } _ 1 = \\mathbb { Q } [ a , a '' ] \\ , . \\end{align*}"}
-{"id": "3838.png", "formula": "\\begin{align*} \\sigma _ 0 = & \\frac { m } { m + 1 } \\frac { e ^ { \\frac { m - 1 } { m } k ( x ) } } { 2 ^ { m - 1 } } , \\\\ \\sigma _ 1 = & \\frac { 2 m s } { s ^ 2 - m s ^ 2 + m b ^ 2 } . \\end{align*}"}
-{"id": "8200.png", "formula": "\\begin{align*} \\lim _ { p \\to 0 } \\sum _ { k = 1 } ^ { N _ p } \\abs { t ( F ; \\Gamma _ { \\tau ^ p _ k } ) - t ( F ; \\Gamma _ { \\tau ^ p _ { k - 1 } } ) } \\leq { n \\choose 2 } \\limsup _ { p \\to 0 } p N _ p < \\infty \\end{align*}"}
-{"id": "5819.png", "formula": "\\begin{align*} \\phi _ { X _ 1 } ( 2 t ) = \\phi _ { X _ 1 } ( t ) ^ 3 \\overline { \\phi _ { X _ 1 } } ( t ) \\end{align*}"}
-{"id": "8241.png", "formula": "\\begin{align*} S _ { C 5 } ( z , q ) = \\sum _ { n = 1 } ^ \\infty \\frac { q ^ { ( n ^ 2 + n ) / 2 } ( q ^ { 2 n + 1 } ; q ^ 2 ) _ \\infty ( q ^ { n + 1 } ; q ) _ \\infty } { ( z q ^ n , z ^ { - 1 } q ^ n ; q ) _ \\infty } . \\end{align*}"}
-{"id": "6929.png", "formula": "\\begin{align*} \\dim _ K d ^ x ( \\C ( G , e ) ) = \\ell ( k C _ G ( x ) e ) . \\end{align*}"}
-{"id": "404.png", "formula": "\\begin{align*} \\langle F \\rangle _ { { \\rm M L } , g } = \\int F ( \\varphi ) e ^ { - \\beta \\mathcal { S } _ { \\rm M } ( \\varphi , g ) - \\mathcal { S } _ { \\rm L } ( \\varphi , g ) } \\mathcal { D } \\varphi . \\end{align*}"}
-{"id": "8457.png", "formula": "\\begin{align*} \\frac { d } { d t } E _ k ^ { \\theta } = \\int \\frac { 2 } { A } R e D _ t \\theta _ k \\bar { G } _ k - \\int \\frac { 1 } { A } \\frac { a _ t } { a } \\circ \\kappa ^ { - 1 } | D _ t \\theta _ k | ^ 2 \\end{align*}"}
-{"id": "9457.png", "formula": "\\begin{align*} Q _ { 1 , \\epsilon } ( z ) : = \\psi _ { \\epsilon } ( \\mathrm { I m } ( \\tilde { z } ) ) e ( - \\tilde { z } ) \\end{align*}"}
-{"id": "4273.png", "formula": "\\begin{align*} \\textup { s a t } ( n , K _ 3 , K _ 4 ) = n - 2 . \\end{align*}"}
-{"id": "4147.png", "formula": "\\begin{align*} \\varphi _ { 2 , \\pm } ( x ) = \\mp \\pi i \\nu ^ * ( [ x , 0 ] ) , x \\in \\Delta _ 2 , \\end{align*}"}
-{"id": "5092.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathbb { J } } \\langle F _ j U x , F _ j U x \\rangle = \\left \\langle \\sum _ { j \\in \\mathbb { J } } F _ j ^ * F _ j U x , U x \\right \\rangle = \\langle U x , U x \\rangle = \\langle x , x \\rangle , \\end{align*}"}
-{"id": "1923.png", "formula": "\\begin{align*} F _ { p + 2 } ( x ) = x ^ 2 \\sum _ { j = 1 } ^ \\infty \\frac { 1 } { Z _ j ^ { p } } \\left ( \\frac { A _ j ^ 2 } { Z _ j ^ 2 } \\right ) . \\end{align*}"}
-{"id": "6216.png", "formula": "\\begin{align*} \\int _ { \\mathcal { P } } \\frac { e ^ { - \\frac { x ^ * v ^ { - 1 } x } { 2 } } } { ( 2 \\pi ) ^ { n / 2 } ( \\det v ) ^ { 1 / 2 } } \\mu _ i ( d v ) = f ( x ) \\end{align*}"}
-{"id": "3645.png", "formula": "\\begin{align*} \\alpha _ { - 2 } & : = - \\frac { 2 t } { 3 } a _ 1 - \\frac { 4 t } { 3 } ( a _ 2 - a _ { - 2 } + a _ 3 - a _ { - 3 } ) - \\frac { 1 } { 3 } v _ { ( 2 , 3 ) } + \\frac { 8 } { 3 } ( a _ 2 \\cdot v _ { ( 1 , 3 ) } + a _ 3 \\cdot v _ { ( 1 , 2 ) } ) \\\\ \\beta _ { - 2 } & : = - \\frac { 4 t } { 3 } ( a _ 2 - a _ { - 2 } + a _ 3 - a _ { - 3 } ) + \\frac { 8 } { 3 } ( a _ 2 \\cdot v _ { ( 1 , 3 ) } - a _ 3 \\cdot v _ { ( 1 , 2 ) } ) . \\end{align*}"}
-{"id": "4133.png", "formula": "\\begin{align*} N ( z ) = N _ { 2 \\beta } ( z ) \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & z ^ { - \\beta } & 0 \\\\ 0 & 0 & z ^ { \\beta } \\end{pmatrix} \\end{align*}"}
-{"id": "6792.png", "formula": "\\begin{align*} L u = \\sum _ { | \\nu | \\le q } c _ \\nu \\partial ^ \\nu u = f \\ , , \\end{align*}"}
-{"id": "4394.png", "formula": "\\begin{align*} f _ o ( t ) = e ^ { t ^ 2 / 2 } , \\end{align*}"}
-{"id": "3132.png", "formula": "\\begin{align*} \\alpha _ i ( u ) = \\frac { \\lambda _ { i } ( u ) } { \\lambda _ { i + 1 } ( u ) } , i = 1 , \\ldots , N - 1 . \\end{align*}"}
-{"id": "3772.png", "formula": "\\begin{align*} W = \\int _ { \\partial \\Omega } u h , W _ 0 = \\int _ { \\partial \\Omega } u _ 0 h . \\end{align*}"}
-{"id": "6931.png", "formula": "\\begin{align*} \\dim _ K d ^ Q ( \\C ( G , \\mu , e ) ) = \\ell ( k G e ) . \\end{align*}"}
-{"id": "8735.png", "formula": "\\begin{align*} \\omega ^ { ( \\alpha , \\sigma ) } ( z ) = \\left ( \\sum _ { k = 1 } ^ p \\frac { 1 } { k } ( 1 - z e ^ { - \\sigma \\tau } ) ^ k \\right ) ^ { \\alpha } = \\sum _ { k = 0 } ^ { \\infty } \\omega ^ { ( \\alpha , \\sigma ) } _ { k } z ^ k . \\end{align*}"}
-{"id": "38.png", "formula": "\\begin{align*} - \\partial _ t \\mathbf { y } ^ i ( t , v ) & + \\frac 1 2 ( \\kappa ^ { t r } \\kappa \\nabla _ v ^ 2 \\mathbf { y } ^ i ( t , v ) ) + \\eta ( v ) ^ { t r } \\nabla _ v \\mathbf { y } ^ i ( t , v ) \\\\ & + f ^ i ( v , \\kappa ^ { t r } \\nabla _ v \\mathbf { y } ^ i ( t , v ) ) + \\sum _ { k \\in I } q ^ { i k } \\left ( e ^ { ( \\mathbf { y } ^ k - \\mathbf { y } ^ i ) ( t , v ) } - 1 \\right ) = 0 , \\end{align*}"}
-{"id": "5162.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } F ^ { n } ( a _ { n } x + b _ { n } ) = H ( x ) , \\end{align*}"}
-{"id": "3939.png", "formula": "\\begin{align*} \\psi ( x \\cdot f ) = \\sum ( x \\cdot f ) _ { 1 } \\sigma \\bigl ( S \\bigl ( ( x \\cdot f ) _ { 2 } | _ { H } ) \\bigr ) \\bigr ) \\otimes ( x \\cdot f ) _ { 3 } | _ { H } = \\sum f _ { 1 } \\sigma ( S f _ { 2 } | _ { H } ) \\otimes x \\cdot f _ { 3 } | _ { H } , \\end{align*}"}
-{"id": "8352.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\rightarrow \\pm \\infty } A _ 1 ( \\alpha , t ) = 1 . \\end{align*}"}
-{"id": "5155.png", "formula": "\\begin{align*} A _ { s s } : = k _ j + \\lambda , B _ { s s } : = \\mu - \\nu , s s \\sim ( j , m _ j , k _ j ) \\in I _ 3 , \\end{align*}"}
-{"id": "6122.png", "formula": "\\begin{align*} Z _ j \\stackrel { \\Gamma } { \\longmapsto } \\sum _ { k = 0 } ^ { \\mu _ 1 } a _ { j , k } X _ { k } + \\sum _ { k = 0 } ^ { \\mu _ 2 } b _ { j , k } Y _ { k } \\stackrel { \\Phi ' } { \\longmapsto } \\sum _ { k = 0 } ^ { \\mu _ 1 } a _ { j , k } ( T _ 0 ^ { \\mu _ 1 - k } T _ 1 ^ k X ) + \\sum _ { k = 0 } ^ { \\mu _ 2 } b _ { j , k } ( T _ 0 ^ { \\mu _ 2 - k } T _ 1 ^ k Y ) . \\end{align*}"}
-{"id": "1432.png", "formula": "\\begin{align*} m _ { t } - \\Delta m + \\varepsilon \\mathrm { d i v } ( m \\mathcal { H } _ { p } ( t , x , m , D u ) ) = 0 , \\end{align*}"}
-{"id": "1702.png", "formula": "\\begin{align*} I _ n = \\int _ { \\varphi _ n > 0 } \\varphi \\ , d \\mu = \\int _ { \\varphi _ n > 0 , \\ { \\mathcal L } \\varphi _ n \\leq 0 } \\varphi \\ , d \\mu + \\int _ { \\varphi _ n > 0 , \\ { \\mathcal L } \\varphi _ n > 0 } \\varphi \\ , d \\mu . \\end{align*}"}
-{"id": "1905.png", "formula": "\\begin{align*} - \\left [ \\cos ( \\kappa _ j x ) + \\frac { \\xi } { \\kappa _ j } \\sin ( \\kappa _ j | x | ) \\right ] \\sin ( \\kappa _ j t ) = - \\frac { \\xi L } { 2 } \\left [ \\frac { \\kappa _ j t } { Z _ j ^ 2 } + 2 \\sum _ { n = 1 } ^ \\infty \\left ( \\frac { \\kappa _ j } { k _ n } \\right ) \\frac { \\cos ( k _ n x ) \\sin ( k _ n t ) } { Z _ j ^ 2 - ( \\pi n ) ^ 2 } \\right ] \\end{align*}"}
-{"id": "7514.png", "formula": "\\begin{align*} J _ N ( z _ 0 , r ) = \\frac { 1 } { N } \\sum _ { j = 0 } ^ { N - 1 } \\chi _ { B ( z _ 0 , r ) } ( x _ j ) \\simeq \\mu ( B ( z _ 0 , r ) ) . \\end{align*}"}
-{"id": "7057.png", "formula": "\\begin{align*} \\Phi = \\left ( \\begin{matrix} { } \\Phi _ { v v } & \\Phi _ { v \\b { v } } \\\\ \\Phi _ { v \\b { v } } & \\Phi _ { \\b { v } \\b { v } } \\end{matrix} \\right ) = \\left ( \\begin{matrix} { } t q & \\frac { \\phi _ 0 } { 2 } + \\frac { t ^ 2 } { 2 } \\left ( \\frac { | q | ^ 2 } { \\phi ^ 2 _ 0 } + \\alpha \\right ) \\phi _ 0 \\\\ \\frac { \\phi _ 0 } { 2 } + \\frac { t ^ 2 } { 2 } \\left ( \\frac { | q | ^ 2 } { \\phi ^ 2 _ 0 } + \\alpha \\right ) \\phi _ 0 & t \\o { q } \\end{matrix} \\right ) + O ( t ^ 4 ) , \\end{align*}"}
-{"id": "5315.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\check { \\alpha } _ 0 = h _ { p _ 1 } + h _ { q _ 2 } - e _ { 1 , 2 , 3 , 4 } , & \\check { \\alpha } _ 1 = h _ { p _ 2 } - e _ { 1 5 , 1 6 } , & \\check { \\alpha } _ 2 = h _ { q _ 2 } - e _ { 5 , 6 } , \\\\ \\check { \\alpha } _ 3 = h _ { q _ 1 } + h _ { p _ 2 } - e _ { 9 , 1 0 , 1 1 , 1 2 } , & \\check { \\alpha } _ 4 = h _ { p _ 1 } - e _ { 7 , 8 } , & \\check { \\alpha } _ 5 = h _ { q _ 1 } - e _ { 1 3 , 1 4 } . \\end{array} . \\end{align*}"}
-{"id": "5223.png", "formula": "\\begin{align*} V ( y ) = \\inf \\{ x > 0 , U ( x ) \\leq x \\} \\end{align*}"}
-{"id": "4909.png", "formula": "\\begin{align*} L _ \\varepsilon & = \\tfrac 1 2 \\int _ \\Omega \\bigl [ \\lvert u \\rvert ^ 2 + 2 \\ , \\varepsilon \\ , g ( u , u _ \\beta ' ) + \\varepsilon ^ 2 \\ , ( \\lvert u ' \\rvert ^ 2 + g ( u , u _ \\beta '' ) \\bigr ] \\ , \\mu ( x ) + O ( \\varepsilon ^ 3 ) \\\\ & \\equiv L _ 0 + \\varepsilon \\ , L _ 1 + \\tfrac 1 2 \\ , \\varepsilon ^ 2 \\ , L _ 2 + O ( \\varepsilon ^ 3 ) \\ , . \\end{align*}"}
-{"id": "9668.png", "formula": "\\begin{align*} F _ { \\tau , \\mathbb T \\setminus \\sigma } \\oplus F _ { \\mathbb T , \\mathbb T \\setminus \\sigma } = J h _ { 0 1 } | _ { \\mathbb T \\setminus \\sigma } , \\end{align*}"}
-{"id": "1954.png", "formula": "\\begin{align*} a ^ c ( \\gamma ) = a ( c \\gamma c ) \\ ; \\gamma \\in G _ { K , S } . \\end{align*}"}
-{"id": "1611.png", "formula": "\\begin{align*} \\frac { \\mathbb { E } _ x \\left [ Z _ T ^ { b ^ \\ast } \\right ] } { T } = \\frac { x - \\mathbb { E } _ x \\left [ X _ T ^ { Z ^ { b ^ \\ast } } \\right ] } { T } + \\frac { 1 } { T } \\mathbb { E } _ x \\int _ 0 ^ T \\mu \\left ( X _ t ^ { Z ^ { b ^ \\ast } } \\right ) X _ t ^ { Z ^ { b ^ \\ast } } \\ , d t . \\end{align*}"}
-{"id": "3490.png", "formula": "\\begin{align*} \\Sigma ^ R _ > = { } & \\{ - 1 + r e ^ { 2 \\pi i / 3 } \\mid 0 \\leq r \\leq R \\} \\cup \\{ - 1 + r e ^ { \\pi i / 3 } \\mid - R \\leq r \\leq 0 \\} , \\\\ \\mathcal { C } ^ R _ < = { } & \\{ 1 + r e ^ { 4 \\pi i / 3 } \\mid - R \\leq r \\leq 0 \\} \\cup \\{ 1 + r e ^ { 5 \\pi i / 3 } \\mid 0 \\leq r \\leq R \\} . \\end{align*}"}
-{"id": "6540.png", "formula": "\\begin{align*} m ^ { t o t } _ \\beta = w ^ 2 \\ , m ^ P _ \\beta . \\end{align*}"}
-{"id": "7686.png", "formula": "\\begin{align*} T f ( D ) h = T \\bigl ( \\underset { n \\rightarrow \\infty } { \\mathrm { l i m } } { f _ { n } ( D ) h } \\bigr ) = \\underset { n \\rightarrow \\infty } { \\mathrm { l i m } } { T f _ { n } ( D ) h } = \\underset { n \\rightarrow \\infty } { \\mathrm { l i m } } { \\tilde { f _ { n } } ( D ) T h } = { \\tilde { f } ( D ) T h } , \\end{align*}"}
-{"id": "2733.png", "formula": "\\begin{align*} \\min _ { w \\in \\Sigma _ { + } ^ { 2 } \\cup \\Sigma _ { + } ^ { 3 } } & \\Re { f ( w ) } - \\Re { f ( z _ { 0 } ) } = \\Re { f ( w _ { 3 } ) } - \\Re { f ( z _ { 0 } ) } \\\\ & \\geq \\Re { f ( w _ { 2 } ) } - \\Re { f ( z _ { 0 } ) } \\geq 2 N ^ { - \\frac { 1 } { 6 } } , \\end{align*}"}
-{"id": "4421.png", "formula": "\\begin{align*} \\forall i \\in \\{ 1 , \\ldots , n \\} : \\ ; u ^ i ( 0 ) = u _ 0 ^ i . \\end{align*}"}
-{"id": "6977.png", "formula": "\\begin{align*} \\frac { d } { d t } \\varphi _ t ( x ) = v ( \\varphi _ t ( x ) ) . \\end{align*}"}
-{"id": "329.png", "formula": "\\begin{align*} | I m ( \\partial ' _ k ) _ * | = \\frac { \\frac { 1 } { 2 } n ( n ^ 2 - 1 ) } { ( \\frac { 1 } { 2 } n ( n ^ 2 - 1 ) , k ) } \\cdot \\frac { n } { ( n , k ) } . \\end{align*}"}
-{"id": "8408.png", "formula": "\\begin{align*} E ( t & ) \\leq E ( 0 ) \\\\ & + \\int _ 0 ^ t C ( \\| \\partial _ { \\alpha } \\xi _ 0 \\| _ { H ^ { s - 1 } } , \\| v _ 0 \\| _ { H ^ s } , \\| w _ 0 \\| _ { H ^ s } , \\mathcal { E } ( \\tau ) , d _ I ( \\tau ) ^ { - 1 } , d _ P ( \\tau ) ^ { - 1 } , N \\lambda _ { m a x } , C _ 1 , C _ 2 , \\alpha _ 0 ) d s . \\end{align*}"}
-{"id": "414.png", "formula": "\\begin{align*} { \\det } ' ( - \\Delta _ g ) = \\exp ( - \\partial _ s \\zeta ( s ) | _ { s = 0 } ) \\end{align*}"}
-{"id": "3135.png", "formula": "\\begin{align*} \\widehat { T } _ { i j } ( u ) = \\tilde { T } _ { N + 1 - j , N + 1 - i } ( u ) , \\end{align*}"}
-{"id": "851.png", "formula": "\\begin{gather*} { \\rm r a d } = { \\rm r g } . \\end{gather*}"}
-{"id": "5770.png", "formula": "\\begin{align*} A _ d ( Q , q ) = \\sum _ \\Gamma ( q - 1 ) ^ { b _ 1 ( \\Gamma ) } R _ d ( \\Gamma , q ) , \\end{align*}"}
-{"id": "5353.png", "formula": "\\begin{align*} X _ { \\alpha + \\beta } . v _ { n m } ^ 1 = a _ { n m } v _ { n + 1 \\ , m + 3 } ^ 1 \\end{align*}"}
-{"id": "597.png", "formula": "\\begin{align*} \\gamma _ n ( \\kappa ) = \\kappa \\sigma \\sqrt { \\frac { 2 \\log \\# \\Omega _ n } { n } } \\ \\ \\textup { f o r } \\ \\ \\kappa > \\kappa * \\end{align*}"}
-{"id": "8903.png", "formula": "\\begin{align*} y _ T ( + ) & = [ 1 : m ] \\cup ( m + T ) \\sim T \\\\ y _ T ( - ) & = [ m + 1 : 2 m ] \\cup T \\sim ( m + T ) , \\end{align*}"}
-{"id": "5962.png", "formula": "\\begin{align*} u ( t , x ) = \\sup _ { y \\in \\R ^ d } \\left ( f ( y ) - ( 1 - t ) g \\left ( \\frac { y - x } { 1 - t } \\right ) \\right ) . \\end{align*}"}
-{"id": "9185.png", "formula": "\\begin{align*} A & : \\Lambda _ { N , \\tilde { N } } \\times \\Lambda _ { N , \\tilde { N } } \\longrightarrow \\mathbb { R } , \\\\ A \\left ( J _ { 1 } , J _ { 2 } \\right ) & = \\left \\langle J _ { 1 } ^ { \\prime } \\left ( b \\right ) - S _ { \\dot { \\gamma } \\left ( b \\right ) } J _ { 1 } , J _ { 2 } \\left ( b \\right ) \\right \\rangle . \\end{align*}"}
-{"id": "4114.png", "formula": "\\begin{align*} \\begin{aligned} g _ { 1 + } ( x ) - g _ { 1 - } ( x ) & = 2 \\pi i \\mu ^ * ( [ x , \\infty ) ) , \\\\ g _ { 1 - } ( x ) + g _ { 1 + } ( x ) - g _ { 2 } ( x ) - V ( x ) & \\begin{cases} = \\ell & [ 0 , q ] , \\\\ < \\ell & ( q , \\infty ) , \\end{cases} \\end{aligned} \\end{align*}"}
-{"id": "3816.png", "formula": "\\begin{align*} \\gamma _ { G , S , \\pi } ( \\ell ) = \\# B _ { G , S , \\pi } ( x , \\ell ) \\ : . \\end{align*}"}
-{"id": "1825.png", "formula": "\\begin{align*} \\psi ^ { \\rm e v e n } ( n , x , t ) = ( k _ n L ) ^ { - 1 / 2 } \\ , \\cos ( k _ n x ) \\ , e ^ { - i k _ n t } , \\end{align*}"}
-{"id": "1728.png", "formula": "\\begin{align*} \\lVert F \\rVert _ { \\mathbb { D } _ { r } ^ { p } } = \\left ( \\mathbb { E } \\left [ | F | ^ { p } \\right ] + \\sum _ { k = 1 } ^ { r } \\mathbb { E } \\left [ \\left | \\lVert D ^ { k } F \\rVert _ { \\mathcal { H } ^ { \\otimes k } } \\right | ^ { p } \\right ] \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "6941.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow T ^ { * } } \\left ( \\| \\rho \\| _ { L ^ { \\infty } ( 0 , T ; L ^ \\infty ) } + \\| P \\| _ { L ^ { \\infty } ( 0 , T ; L ^ \\infty ) } \\right ) = \\infty . \\end{align*}"}
-{"id": "5814.png", "formula": "\\begin{align*} 2 \\Re ( a _ 0 ) = \\frac { 1 } { \\pi } \\int _ 0 ^ { 2 \\pi } \\Re ( f ( r e ^ { i \\theta } ) ) d \\theta . \\end{align*}"}
-{"id": "5943.png", "formula": "\\begin{align*} \\Theta = B + \\sum _ { z \\in Z \\setminus V } t _ z f ^ * z , \\end{align*}"}
-{"id": "8506.png", "formula": "\\begin{align*} \\norm { \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { 2 \\pi } \\frac { ( \\dot { z } _ j ) ^ 2 } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 3 } } _ { H ^ s } \\leq K _ s ^ { - 1 } \\epsilon ^ 2 + K _ s ^ { - 1 } \\epsilon \\frac { | \\lambda | } { x ( 0 ) } d _ I ( t ) ^ { - 5 / 2 } . \\end{align*}"}
-{"id": "6683.png", "formula": "\\begin{align*} \\zeta ( \\frac { 1 } { 2 } - i ( \\rho - \\frac { \\omega } { 2 } ) ) = \\zeta ( \\frac { 1 } { 2 } + i \\lambda _ { * } ) = 0 , \\end{align*}"}
-{"id": "8032.png", "formula": "\\begin{align*} \\psi _ c ^ \\pm ( u ) = \\pm t ^ { - n - 1 } \\imath c ^ { n + 1 } u a _ 1 = \\prod _ { i = 1 } ^ { 2 n } \\alpha _ i = - c ^ { 2 n } . \\end{align*}"}
-{"id": "6672.png", "formula": "\\begin{align*} A | \\psi _ { 0 } \\rangle = 0 , H _ { - } | \\psi _ { 0 } \\rangle = A ^ { \\dagger } A | \\psi _ { 0 } \\rangle = 0 . \\end{align*}"}
-{"id": "4255.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\delta ( a ( x _ { 0 } ) \\lvert d v _ { i } \\rvert ^ { p - 2 } d v _ { i } ) ) & = 0 & & B _ { i } , \\\\ \\delta v _ { i } & = 0 & & B _ { i } , \\\\ \\nu \\wedge v _ { i } & = \\nu \\wedge w & & \\partial B _ { i } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "7411.png", "formula": "\\begin{align*} \\mu _ { y _ n } ( B _ { d _ x } ( y _ n , r ) ) = \\frac { \\mu _ { y _ n } ^ U ( B _ { d _ x } ( y _ n , r ) ) } { \\mu _ { y _ n } ^ U ( B _ { d _ x } ( y _ n , \\mathfrak r ) ) } = \\frac { \\mu _ y ^ U ( B _ { d _ x } ( y _ n , r ) ) } { \\mu _ { y } ^ U ( B _ { d _ x } ( y _ n , \\mathfrak r ) ) } . \\end{align*}"}
-{"id": "6762.png", "formula": "\\begin{align*} \\begin{array} { r l } = - \\frac { 2 } { y } + 1 - \\sqrt { 1 - z } + \\ { \\strut \\displaystyle z y \\over \\displaystyle 3 ( 2 - y ) - { \\strut \\displaystyle ( 3 + z ) y ^ 2 \\over \\displaystyle 5 ( 2 - y ) - { \\strut \\displaystyle ( 8 + z ) y ^ 2 \\over \\displaystyle 7 ( 2 - y ) } } } & \\\\ & \\ddots \\end{array} \\end{align*}"}
-{"id": "6814.png", "formula": "\\begin{align*} \\psi ( s , \\mu , \\theta ) = \\cosh ( s ) R _ 1 + \\sinh ( s ) R _ 2 \\ , , \\end{align*}"}
-{"id": "1048.png", "formula": "\\begin{align*} & \\det ( - B ) = \\prod _ { j = 1 } ^ k \\mathbb { P } ( \\xi ^ { ( n ) } ( I ( y _ j , F _ n ( x _ j ) ) ) = 0 ) = \\prod _ { j = 1 } ^ k D _ n ( F _ n ( x _ j ) / 2 ) > 0 , \\end{align*}"}
-{"id": "6647.png", "formula": "\\begin{align*} \\widehat { M _ { \\zeta , q } } ( \\xi ) = \\sum _ { \\ell \\in \\mathbb Z ^ { d } _ q } G ( a / q , \\ell ) \\widetilde \\zeta ( \\xi - \\ell / q ) . \\end{align*}"}
-{"id": "1950.png", "formula": "\\begin{align*} \\bar { \\psi } ^ - : = \\bar \\psi \\cdot ( \\bar \\psi ^ c ) ^ { - 1 } , \\end{align*}"}
-{"id": "5598.png", "formula": "\\begin{align*} A _ 2 = \\frac { \\alpha } { 2 } \\int _ { - \\infty } ^ 0 x v \\big ( \\bar { u } u _ x + \\bar { u } _ x u \\big ) d x = \\frac { \\alpha } { 2 } \\int _ { - \\infty } ^ 0 x v \\big ( | u | ^ 2 \\big ) _ x d x . \\end{align*}"}
-{"id": "9744.png", "formula": "\\begin{align*} \\lambda _ * = N ( \\rho , q ) \\quad \\textrm { f o r a l l } \\rho \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "130.png", "formula": "\\begin{align*} \\zeta _ { j , N } & = \\sum _ { \\substack { \\ell = 1 } } ^ { \\infty } \\frac { \\gamma ^ { 2 \\ell + 1 } } { \\ell ! } \\frac { ( - j ) _ \\ell ( - 2 N + j + 1 ) _ \\ell } { ( 1 ) _ \\ell } \\ , . \\end{align*}"}
-{"id": "7126.png", "formula": "\\begin{align*} \\hat { q } = q _ 0 - \\nu \\phi _ 2 ^ 2 ( \\hat { q } ) ~ ~ \\mbox { a . e . i n } ~ ~ \\Omega , \\end{align*}"}
-{"id": "4348.png", "formula": "\\begin{align*} \\mathcal H _ u : = \\ \\Big \\{ A \\in { [ n ] \\choose k } \\ : \\ [ 2 , u + 1 ] \\subset A \\Big \\} \\cup \\Big \\{ A \\in { [ n ] \\choose k } \\ : \\ 1 \\in A , [ 2 , u + 1 ] \\cap A \\ne \\emptyset \\Big \\} . \\end{align*}"}
-{"id": "5744.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow 0 } x s _ x & = 0 & & s ( 1 , y , t ) = 0 \\\\ s ( x , 0 , t ) & = 0 & & s ( x , 1 , t ) = 0 \\\\ s ( x , y , 0 ) & = 0 & & s _ t ( x , y , 0 ) = 0 . \\end{align*}"}
-{"id": "8699.png", "formula": "\\begin{align*} ( \\nu + \\rho ' - 4 j - 2 \\ell - 4 ) \\partial _ T q _ \\ell ^ { i , j } ( p _ 1 , p _ 2 , p _ 3 ) = 0 . \\end{align*}"}
-{"id": "3854.png", "formula": "\\begin{align*} P ( Z _ 1 ^ x = x + e _ i ) , \\ P ( Z _ 1 ^ x = x - e _ i ) > 0 \\ ( i = 1 , \\cdots , d ) \\end{align*}"}
-{"id": "9775.png", "formula": "\\begin{align*} L _ a q ^ { \\rm o d d } ( X ) = L _ a q ( X ) - L _ a q ( X ) = 0 \\quad X \\in L _ \\ast . \\end{align*}"}
-{"id": "8159.png", "formula": "\\begin{align*} \\rho ( v _ B ^ { - 1 } ) \\circ ( F \\circ S ) ( \\varphi ) & = \\mu ^ l ( v ) ^ { - 1 } \\mu ^ r \\ ! \\left ( g v ^ { - 1 } v ' ? \\right ) \\varphi \\ ! \\left ( v b _ i g v '' a _ i \\right ) = \\mu ^ l ( v ) ^ { - 1 } \\mu ^ r \\ ! \\left ( g \\overline { Y } _ j ? \\right ) \\varphi \\ ! \\left ( v ^ 2 b _ i g \\overline { X } _ j a _ i \\right ) \\\\ & = \\mu ^ l ( v ) ^ { - 1 } \\mu ^ r \\ ! \\left ( g S ( a _ j ) S ^ { - 1 } ( b _ k ) ? \\right ) \\varphi \\ ! \\left ( v ^ 2 g S ^ { - 2 } ( b _ i ) b _ j a _ k a _ i \\right ) = ( \\star ) \\end{align*}"}
-{"id": "8389.png", "formula": "\\begin{align*} [ \\partial _ { \\alpha } ^ k , A \\bold { n } ] D u = & \\sum _ { m = 1 } ^ k c _ { m , k } \\partial _ { \\alpha } ^ m ( A \\bold { n } ) \\partial _ { \\alpha } ^ { k - m } D u , \\end{align*}"}
-{"id": "5653.png", "formula": "\\begin{align*} \\widetilde { B } ( \\zeta ) = \\sum _ { j = 0 } ^ { \\infty } B _ { j } \\zeta ^ { j } \\mbox { w i t h } \\widetilde { B } ( \\zeta ) = 1 + \\zeta \\big ( \\tau ^ { - \\alpha } \\delta ( \\zeta ) ^ { \\alpha } + A _ h \\big ) ^ { - 1 } \\tau ^ { \\gamma - 1 } \\delta ( \\zeta ) ^ { - \\gamma } . \\end{align*}"}
-{"id": "443.png", "formula": "\\begin{align*} \\tilde { X } _ t ( z ) = X _ t ( z ) - \\int _ D X _ t ( x ) \\bar h ( x ) \\dd x , \\end{align*}"}
-{"id": "4937.png", "formula": "\\begin{align*} z ^ { n ^ 4 + 2 n } = z ^ { 2 n } ( z ^ { n ^ 4 } - z ^ n x ^ n ) - x ^ n z ^ { 3 n } \\end{align*}"}
-{"id": "9743.png", "formula": "\\begin{align*} a \\in ( - 1 , 0 ) \\dim _ { \\mathcal { H } } \\mathcal { N } _ \\ast = \\dim _ { \\mathcal { H } } L _ \\ast = n - 1 . \\end{align*}"}
-{"id": "6944.png", "formula": "\\begin{align*} f _ { \\Omega _ r } ( x ) = \\frac { 1 } { | \\Omega _ r ( x ) | } \\int _ { \\Omega _ r ( x ) } f ( y ) d y , \\end{align*}"}
-{"id": "7944.png", "formula": "\\begin{align*} { 2 n \\leq ( d _ 1 - 1 ) + \\sum _ { j = 2 } ^ m ( k _ j - 1 ) + 1 } , \\end{align*}"}
-{"id": "7042.png", "formula": "\\begin{align*} a ^ v _ { \\alpha } = - \\phi _ { \\alpha \\b { v } } \\phi ^ { v \\b { v } } , \\end{align*}"}
-{"id": "1318.png", "formula": "\\begin{align*} R \\big ( \\varPhi { [ Z , P ] } \\big ) = \\rho [ Z , P ] . \\end{align*}"}
-{"id": "5990.png", "formula": "\\begin{align*} q ^ { Q _ \\omega } ( s , \\overline \\omega ) = \\sqrt { 1 - t } q ^ Q ( t + s ( 1 - t ) , \\omega \\otimes _ t \\overline \\omega ) , \\end{align*}"}
-{"id": "5597.png", "formula": "\\begin{align*} A _ 1 = - \\frac { 1 } { 2 } \\int _ { - \\infty } ^ 0 x \\big ( \\bar { u } _ { x x } u _ x + u _ { x x } \\bar { u } _ x \\big ) d x = - \\frac { 1 } { 2 } \\int _ { - \\infty } ^ 0 x \\big ( | u _ x | ^ 2 \\big ) _ x d x = \\frac { 1 } { 2 } \\int _ { - \\infty } ^ 0 | u _ x | ^ 2 d x . \\end{align*}"}
-{"id": "9748.png", "formula": "\\begin{align*} w : = \\tilde v _ { r } . \\end{align*}"}
-{"id": "2702.png", "formula": "\\begin{align*} \\Re \\{ f _ { 1 } ( 4 ; z ) - f ( 4 ; z _ { 0 } ) \\} & = \\Re \\{ f ( 4 ; z _ 2 ) - f ( 4 ; z _ { 0 } ) \\} \\leq - \\frac { 1 } { 4 } \\kappa N ^ { - \\frac { 1 } { 3 } } . \\end{align*}"}
-{"id": "9978.png", "formula": "\\begin{align*} \\partial _ { t } v ^ { k } = \\partial _ { x } \\frac { ( v ^ { k } ) ^ { 2 } - q ^ { 3 } v ^ { k } - q ^ { 1 } } { q ^ { 5 } } , k = 1 , 2 , 3 , \\end{align*}"}
-{"id": "3220.png", "formula": "\\begin{align*} & \\Delta _ { \\frac { 1 } { 2 } } { w } ^ k = - Q _ k ^ { - 1 } \\Phi _ { \\mu _ k } ( { w } ^ k ) , \\\\ & \\| \\Delta _ { \\frac { 1 } { 2 } } { w } ^ k \\| \\le \\| Q _ k ^ { - 1 } \\| _ F \\| \\Phi _ { \\mu _ k } ( { w } ^ k ) \\| = O ( \\| \\Phi _ { \\mu _ k } ( { w } ^ k ) \\| ) , \\end{align*}"}
-{"id": "98.png", "formula": "\\begin{align*} \\left [ B ( 0 ) + \\gamma B _ 1 \\right ] ^ { 2 N } = \\sum _ { k = 0 } ^ { 2 N } \\gamma ^ k S _ k ( B ( 0 ) , B _ 1 ) , \\end{align*}"}
-{"id": "2131.png", "formula": "\\begin{align*} \\Theta _ t ( \\tilde { A } , \\tilde { C } , \\omega ) : = ( S ^ A _ t ( \\tilde { A } ) , S ^ C _ t ( \\tilde { C } ) , \\theta _ t \\omega ) . \\end{align*}"}
-{"id": "1325.png", "formula": "\\begin{align*} R ( \\varPsi ) & = \\sup _ { \\mu \\in \\mathcal { D } _ { \\rho } } \\bigg \\{ \\int _ 0 ^ 1 \\varPsi ( p ) \\ ; \\mu ( d p ) - R ^ * ( \\mu ) \\bigg \\} , \\\\ \\intertext { w i t h } R ^ * ( \\mu ) & = \\sup _ { \\varPsi \\in \\mathbb { Q } _ { } } \\bigg \\{ \\int _ 0 ^ 1 \\varPsi ( p ) \\ ; \\mu ( d p ) - R ( \\varPsi ) \\bigg \\} . \\end{align*}"}
-{"id": "4621.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 u } { \\partial t ^ 2 } & = \\frac { \\partial ^ 2 u } { \\partial x ^ 2 } , \\Omega \\times ( 0 , T ] , \\\\ u & = 0 , \\mathbf { x } \\in \\partial \\Omega , \\\\ \\frac { \\partial u } { \\partial t } ( 0 , \\mathbf { x } ) & = 0 , \\\\ u ( 0 , \\mathbf { x } ) & = s ( \\mathbf { x } ) . \\end{align*}"}
-{"id": "8678.png", "formula": "\\begin{align*} ( f \\star g ) _ \\delta ( x ) - f _ \\delta ( x ) \\star g _ \\delta ( x ) & = \\int _ { B _ \\delta ( 0 ) } \\big ( f ( x - y ) - f ( x ) \\big ) \\star \\big ( g ( x - y ) - g ( x ) \\big ) \\rho _ \\delta ( y ) \\ , d y \\\\ & - \\int _ { B _ \\delta ( 0 ) } \\big ( f ( x - y ) - f ( x ) \\big ) \\rho _ \\delta ( y ) \\ , d y \\star \\int _ { B _ \\delta ( 0 ) } \\big ( g ( x - y ) - g ( x ) \\big ) \\rho _ \\delta ( y ) \\ , d y , \\end{align*}"}
-{"id": "6989.png", "formula": "\\begin{align*} \\begin{array} { l } \\mathsf { p } _ { ( 0 , 0 ) } ( 1 ) = 3 p ^ 2 , \\mathsf { p } _ { ( 0 , 0 ) } ( x ^ r ) = 4 \\cos ^ 2 \\left ( \\frac { 2 \\pi r } { p + 1 } \\right ) - 1 , \\\\ \\mathsf { p } _ { ( 0 , 0 ) } ( y ^ s ) = 1 - 4 \\cos \\left ( \\frac { 2 ( p - 1 ) \\pi s } { p ^ 2 + 1 } \\right ) \\cos \\left ( \\frac { 2 ( p + 1 ) \\pi s } { p ^ 2 + 1 } \\right ) . \\end{array} \\end{align*}"}
-{"id": "1191.png", "formula": "\\begin{align*} u ( t ) = e ^ { - t A } u _ 0 + \\int _ 0 ^ t e ^ { - ( t - s ) A } \\left [ - u ( s ) u _ x ( s ) - ( 1 - \\partial _ x ^ 2 ) ^ { - 1 } u _ x ( s ) \\right ] d s = : F ( u ) ( t ) \\end{align*}"}
-{"id": "9066.png", "formula": "\\begin{align*} F _ x ( y ) = f \\left ( x + \\frac { R } { \\sqrt { E } } y \\right ) = \\sum _ { | \\xi | ^ 2 = E } a _ { \\xi } e ( \\langle \\xi , x \\rangle ) e \\left ( \\left \\langle \\frac { \\xi } { \\sqrt { E } } , R y \\right \\rangle \\right ) . \\end{align*}"}
-{"id": "5675.png", "formula": "\\begin{align*} A ^ \\circ + B ^ \\circ & = \\{ x ^ * \\in X ^ * \\ | \\ \\mbox { t h e r e e x i s t } a ^ * , \\ , b ^ * \\in X ^ * \\mbox { s . t . } x ^ * = a ^ * + b ^ * \\mbox { a n d } \\\\ & \\langle a ^ * , a \\rangle \\le 1 \\mbox { a n d } \\langle b ^ * , b \\rangle \\le 1 \\mbox { f o r a l l } a \\in A , \\ , b \\in B \\} \\ , , \\end{align*}"}
-{"id": "6950.png", "formula": "\\begin{align*} \\left | 1 - \\cos \\vartheta _ j \\right | = \\frac { 1 } { 2 } \\left | ( 1 - \\cos \\vartheta _ j ) \\mathbf W _ j ^ 2 \\right | \\leq L { h _ { j } } \\end{align*}"}
-{"id": "5339.png", "formula": "\\begin{align*} \\frac { a _ k } { \\overline { b _ { k + 2 } } } = - \\frac { | | w ^ k | | ^ 2 } { | | w ^ { k + 2 } | | ^ 2 } \\in \\left ( - \\infty , 0 \\right ) , \\quad \\forall k . \\end{align*}"}
-{"id": "5880.png", "formula": "\\begin{align*} C _ { s _ 0 s _ 1 s _ 0 } ' = q ^ { - \\frac { 3 } { 2 } } \\left ( T _ { s _ 0 s _ 1 s _ 0 } + T _ { s _ 1 s _ 0 } + T _ { s _ 0 s _ 1 } + T _ { s _ 0 } + T _ { s _ 1 } + 1 \\right ) . \\end{align*}"}
-{"id": "5379.png", "formula": "\\begin{align*} a _ { n - 1 \\ , m - 3 } | | v _ { n m } ^ k | | ^ 2 = - \\frac { n - k } { n - 1 } \\overline { d _ { n m } } | | v _ { n - 1 \\ , m - 3 } ^ k | | ^ 2 . \\end{align*}"}
-{"id": "12.png", "formula": "\\begin{align*} \\delta Y _ t ^ { i } ( m , n ) \\leq \\bar { Y } _ t \\leq K _ f \\int _ { n } ^ { m } e ^ { - \\rho ( s - t ) } d s = \\frac { K _ f } { \\rho } e ^ { \\rho t } ( e ^ { - \\rho n } - e ^ { - \\rho m } ) , \\end{align*}"}
-{"id": "9696.png", "formula": "\\begin{align*} u = u ' + \\Sigma _ { i } t _ { i } \\frac { b _ { i } \\cdot e _ { n } } { | | e _ { n } | | ^ { 2 } } , \\end{align*}"}
-{"id": "2038.png", "formula": "\\begin{align*} \\alpha _ { n , M } ( t ) = \\frac { 1 } { M } \\sum _ { a \\bmod { M } } e ^ { - 2 \\pi i n \\frac { a } { M } } L ( t , f , \\frac { a } { M } ) = \\frac { 1 } { M } \\sum _ { d \\mid M } \\sum _ { \\substack { a \\bmod { d } \\\\ \\gcd ( a , d ) = 1 } } e ^ { - 2 \\pi i n \\frac { a } { d } } L ( t , f , \\frac { a } { d } ) . \\end{align*}"}
-{"id": "436.png", "formula": "\\begin{align*} Q ^ x \\le \\sum _ { n = 1 } ^ \\infty \\sum _ { i = 1 } ^ { \\infty } \\big ( A + \\eta n \\big ) e ^ { ( \\gamma \\eta + \\frac { \\gamma ^ 2 } { 2 } ) n } \\mathbf { 1 } _ { \\{ T ^ x _ i \\in ( n - 1 , n ] \\} } = : \\sum _ { n = 1 } ^ \\infty Q ^ { x , n } . \\end{align*}"}
-{"id": "2705.png", "formula": "\\begin{align*} \\left | \\frac { 1 } { ( \\varphi ( 4 ) ) ^ k } \\mathrm { P f } \\Big [ K _ { N } \\Big ( 4 N + \\frac { u _ i } { \\varphi ( 4 ) } , 4 N + \\frac { u _ j } { \\varphi ( 4 ) } \\Big ) \\Big ] _ { i , j = 1 } ^ { k } \\right | \\leq ( 2 k ) ^ { \\frac { k } { 2 } } C ^ { k } \\prod _ { j = 1 } ^ { k } e ^ { - ( a - b ) u _ { j } } . \\end{align*}"}
-{"id": "4196.png", "formula": "\\begin{align*} P ^ { ( 4 ) } _ + ( z ) \\left ( P ^ { ( 4 ) } _ - ( z ) \\right ) ^ { - 1 } = \\mathbb { I } + \\mathcal { O } ( n ^ { - 1 } ) \\end{align*}"}
-{"id": "1652.png", "formula": "\\begin{align*} F _ k = \\alpha _ k + i \\alpha _ { - k } \\quad G _ k = \\beta _ k + i \\beta _ { - k } , \\end{align*}"}
-{"id": "1949.png", "formula": "\\begin{align*} a _ n ( f ) \\bigl ( \\textstyle { \\frac { K / \\Q } { n } } \\bigr ) = a _ n ( f ) , \\quad \\textrm { f o r a l m o s t a l l $ n \\geq 1 $ } . \\end{align*}"}
-{"id": "5428.png", "formula": "\\begin{align*} A ( x ) E _ i ( x ) = E _ i ( f ( x ) ) , i \\in \\N , \\end{align*}"}
-{"id": "8484.png", "formula": "\\begin{align*} | F ( z _ 1 ( t ) , t ) ^ 2 - F ( z _ 2 ( t ) , t ) ^ 2 | = & | ( F ( z _ 1 ( t ) , t ) + F ( z _ 2 ( t ) , t ) ) F _ { \\zeta } ( \\tilde { x } + i y ( t ) , t ) ( z _ 1 ( t ) - z _ 2 ( t ) ) | \\\\ \\leq & 1 2 0 \\epsilon ^ 2 x ( t ) . \\end{align*}"}
-{"id": "5506.png", "formula": "\\begin{align*} | \\Gamma _ T | _ 2 = C ( d , \\mu _ 0 ) \\max _ { \\lambda \\le J } a _ \\lambda , \\sigma _ { \\Gamma _ T } = C ( d , \\mu _ 0 , \\Phi ) 2 ^ { J ( \\alpha + d / 2 ) } \\max _ { \\lambda \\le J } a _ \\lambda . \\end{align*}"}
-{"id": "5695.png", "formula": "\\begin{align*} \\{ \\partial _ C ^ \\infty f _ 1 ( x _ 0 ) \\} \\cap \\{ - \\partial _ C ^ \\infty f _ 2 ( x _ 0 ) \\} = \\{ \\mathbf { 0 } \\} \\ , . \\end{align*}"}
-{"id": "1153.png", "formula": "\\begin{align*} & \\min _ { 0 \\leq i \\leq k + 1 } | z _ j ' - z _ i | = z _ { j + 1 } - z _ j ' = z _ j ' - z _ j = ( z _ { j + 1 } - z _ j ) / 2 \\\\ & \\geq \\min _ { 0 \\leq i < l \\leq k + 1 } | y _ i - y _ l | / 2 \\geq \\varepsilon _ 0 ( 2 \\ln n ) ^ { - 1 } > d _ 0 \\end{align*}"}
-{"id": "3975.png", "formula": "\\begin{align*} E _ { k , m } \\mid U _ \\ell = \\underset { \\substack { 1 \\leq s \\leq b \\ell \\\\ b \\mid s } } \\sum E _ { k , m \\ell ^ 2 , s } . \\end{align*}"}
-{"id": "3833.png", "formula": "\\begin{align*} 2 G ^ i = ^ { \\alpha } { \\gamma _ { 0 } ^ { i } } _ 0 + 2 \\omega \\alpha s _ 0 ^ i + 2 \\Theta ( r _ { 0 0 } - 2 \\alpha \\omega s _ 0 ) \\Bigl ( \\frac { y ^ i } { \\alpha } + \\frac { \\omega ' } { \\omega - s \\omega ' } b ^ i \\Bigr ) , \\end{align*}"}
-{"id": "8544.png", "formula": "\\begin{align*} \\mathcal { E } _ s = \\sum _ { k = 0 } ^ s \\Big \\{ \\int | D _ t \\theta _ k | ^ 2 + i \\theta _ k \\overline { \\partial _ { \\alpha } \\theta _ k } d \\alpha + \\int | D _ t \\sigma _ k | ^ 2 + i \\sigma _ k \\overline { \\partial _ { \\alpha } \\sigma _ k } d \\alpha \\Big \\} + O ( \\epsilon ^ 3 ) . \\end{align*}"}
-{"id": "6035.png", "formula": "\\begin{align*} P ( d _ 1 , d _ 2 , \\dots , d _ n ) = \\sum _ { 1 \\leq i < j \\leq n } \\min \\{ d _ i , d _ j \\} . \\end{align*}"}
-{"id": "8667.png", "formula": "\\begin{align*} m _ + ( \\l ) = i \\sqrt { \\l } \\frac { V _ 2 ( \\l ) } { V _ { 1 } ( \\l ) } , \\end{align*}"}
-{"id": "642.png", "formula": "\\begin{align*} \\rho _ i ( j ) = \\begin{cases} - \\infty & j < i \\ , \\\\ 1 & j = i \\ , \\\\ \\infty & j > i \\ , \\end{cases} \\end{align*}"}
-{"id": "6895.png", "formula": "\\begin{align*} \\det ( d \\varphi ) = \\sum _ { i = 1 } ^ 6 p _ i g _ i \\ , , \\end{align*}"}
-{"id": "236.png", "formula": "\\begin{align*} \\mathcal { L } u = V u \\ B _ { r _ 0 } , \\end{align*}"}
-{"id": "3526.png", "formula": "\\begin{align*} \\mathcal { D } _ { K 3 } : = \\left \\{ [ \\omega ] \\in \\mathbb { P } ( ( U ( 2 ) ^ { \\oplus 2 } \\oplus A _ { 1 } ^ { \\oplus 2 } ) \\otimes \\mathbb { C } ) \\mid \\omega . \\omega = 0 , \\omega . \\bar { \\omega } > 0 \\right \\} ^ { + } , \\end{align*}"}
-{"id": "9714.png", "formula": "\\begin{align*} T _ { \\textbf { d } } ( \\partial v ) = I _ { \\textbf { d } } ( v ) - 1 = - ( I _ { \\textbf { d } } ^ 1 ( P _ { i } ) _ { c } - 1 ) = - T _ { \\textbf { d } } ( \\partial \\Omega ( P _ i ) ) . \\end{align*}"}
-{"id": "3134.png", "formula": "\\begin{align*} \\tilde { T } _ { i j } ( u ) = ( - 1 ) ^ { i + j } t ^ { 1 \\ldots \\hat \\jmath \\ldots N } _ { 1 \\ldots \\hat \\imath \\ldots N } ( u - c ) \\ \\mbox { q d e t } ( T ( u ) ) ^ { - 1 } . \\end{align*}"}
-{"id": "4696.png", "formula": "\\begin{align*} F _ { j + \\frac { 1 } { 2 } } = - \\frac { 1 } { 1 2 } f _ { j - 1 } + \\frac { 7 } { 1 2 } f _ { j } + \\frac { 7 } { 1 2 } f _ { j + 1 } - \\frac { 1 } { 1 2 } f _ { j + 2 } = : { F } ^ { \\rm C B S Q I } _ { j + \\frac { 1 } { 2 } } ( t ) , \\end{align*}"}
-{"id": "2389.png", "formula": "\\begin{align*} x _ c \\mathcal { O } ( x _ A ) = & x _ c x _ { A ( \\pi ( k ) ) } \\cdots x _ { A ( \\pi ( j ) ) } x _ { A ( \\pi ( j - 1 ) ) } \\cdots x _ { ( A ( \\pi ( 1 ) ) } \\\\ \\mathcal { O } ( x _ c x _ A ) = & x _ { A ( \\pi ( k ) ) } \\cdots x _ { A ( \\pi ( j ) ) } x _ c x _ { A ( \\pi ( j - 1 ) ) } \\cdots x _ { ( A ( \\pi ( 1 ) ) } \\\\ x _ c \\mathcal { O } ( x _ A ) \\mathcal { O } ( x _ c x _ A ) = & [ x _ c , x _ { A ( \\pi ( k ) ) } \\cdots x _ { A ( \\pi ( j ) ) } ] x _ { A ( \\pi ( j 1 ) ) } \\cdots x _ { ( A ( \\pi ( 1 ) ) } \\end{align*}"}
-{"id": "3151.png", "formula": "\\begin{align*} [ E _ i ( u ) , F _ j ( v ) ] & = c \\ \\delta _ { i , j } \\ \\delta ( u , v ) \\Big ( k ^ + _ { i } ( u ) \\cdot k ^ + _ { i + 1 } ( u ) ^ { - 1 } - k ^ - _ { i } ( v ) \\cdot k ^ - _ { i + 1 } ( v ) ^ { - 1 } \\Big ) , \\end{align*}"}
-{"id": "8425.png", "formula": "\\begin{align*} - 2 \\partial _ t ( I - \\mathfrak { H } ) \\bar { z } _ t = - 2 \\partial _ t ( I - \\mathfrak { H } ) p = & - 4 p _ t , \\end{align*}"}
-{"id": "4726.png", "formula": "\\begin{align*} e = \\frac { p } { ( \\gamma - 1 ) } + \\frac { 1 } { 2 } \\rho u ^ 2 , \\end{align*}"}
-{"id": "3262.png", "formula": "\\begin{align*} a , b = \\sqrt [ 3 ] { - \\frac { q } { 2 } \\pm \\sqrt { \\frac { q ^ 2 } { 4 } + \\frac { p ^ 3 } { 2 7 } } } . \\end{align*}"}
-{"id": "6720.png", "formula": "\\begin{align*} Q _ { 1 1 } = \\bar { q } _ 1 , Q _ { 2 1 } = \\dot { \\bar { q } } _ 1 , P _ { 2 1 } = \\lambda ^ 2 m \\ddot { \\bar { q } } _ 1 , \\end{align*}"}
-{"id": "4725.png", "formula": "\\begin{align*} u ( x , t ) = \\left \\{ \\begin{array} { l l } 1 & ~ ~ ~ ~ x < \\dfrac { ( ( \\sqrt { 3 } - 1 ) t + 1 ) } { 4 } , \\medskip \\\\ \\dfrac { 1 } { 2 } \\Big ( \\dfrac { t - 4 x + 1 } { t } \\Big ) & ~ ~ ~ \\dfrac { ( ( \\sqrt { 3 } - 1 ) t + 1 ) } { 4 } < x < \\dfrac { t + 1 } { 4 } , \\medskip \\\\ 0 & ~ ~ ~ x > \\dfrac { t + 1 } { 4 } . \\end{array} \\right . \\end{align*}"}
-{"id": "975.png", "formula": "\\begin{align*} H _ 0 ( q , t ) & = 1 , \\\\ H _ 1 ( q , t ) & = S _ q ( 1 , 0 ) + S _ q ( 1 , 1 ) t = t , \\\\ H _ 2 ( q , t ) & = S _ q ( 2 , 0 ) + S _ q ( 2 , 1 ) t + S _ q ( 2 , 2 ) t ^ 2 = t + q t ^ 2 . \\end{align*}"}
-{"id": "3884.png", "formula": "\\begin{align*} M ( u _ n , v _ n ) & = \\sum _ { j = 1 } ^ l M ( U ^ j _ n , V ^ j _ n ) + M ( u ^ l _ n , v ^ l _ n ) + o _ n ( 1 ) , \\\\ K ( u _ n , v _ n ) & = \\sum _ { j = 1 } ^ l K ( U ^ j , V ^ j ) + K ( u ^ l _ n , v ^ l _ n ) + o _ n ( 1 ) , \\\\ P ( u _ n , v _ n ) & = \\sum _ { j = 1 } ^ l P ( U ^ j , V ^ j ) + P ( u ^ l _ n , v ^ l _ n ) + o _ n ( 1 ) , \\end{align*}"}
-{"id": "2267.png", "formula": "\\begin{align*} \\langle \\hat { E } R ( z ) | E \\rangle & = - \\i \\pi \\sigma ( z ) \\\\ \\hat { \\Omega } R ( z ) | E \\rangle & = - | \\hat { E } \\rangle + z R ( z ) | E \\rangle \\end{align*}"}
-{"id": "7967.png", "formula": "\\begin{align*} J _ q ^ A = \\bigsqcup _ { j = 2 } ^ m \\begin{cases} \\{ j _ 1 , \\dots , j _ { d _ j - d _ { q , j } ^ A } \\} , & d _ j - d _ { q , j } ^ A \\geq 1 , \\\\ \\{ \\} , & d _ j - d _ { q , j } ^ A \\leq 0 . \\end{cases} \\end{align*}"}
-{"id": "378.png", "formula": "\\begin{align*} \\bar { \\beta } ^ 2 = & q _ 1 ^ 2 + q _ 2 ^ 2 \\alpha ^ 2 + q _ 3 ^ 2 \\bar { \\alpha } ^ 2 + q _ 4 | \\alpha | ^ 4 + q _ 5 ^ 2 + 2 ( q _ 1 q _ 2 + q _ 3 q _ 4 \\bar { \\alpha } ^ 2 ) \\alpha + 2 ( q _ 1 q _ 3 + q _ 2 q _ 4 \\alpha ^ 2 ) \\bar { \\alpha } + 2 ( q _ 1 q _ 4 + q _ 2 q _ 3 ) | \\alpha | ^ 2 + \\\\ & \\ , + 2 q _ 1 q _ 5 \\beta + 2 ( q _ 2 q _ 5 \\alpha + q _ 3 q _ 5 \\bar { \\alpha } ) \\beta + 2 q _ 4 q _ 5 | \\alpha | ^ 2 \\beta , \\end{align*}"}
-{"id": "7453.png", "formula": "\\begin{align*} \\displaystyle \\omega ( \\xi _ { - } ) = \\eta ' ( ( - 1 ) ^ { n - 1 } \\tau ) ^ { n ( n + 1 ) / 2 } \\end{align*}"}
-{"id": "2803.png", "formula": "\\begin{align*} { C } _ { i , j } ( t , t + s ) : = \\mathbb { E } [ N ^ { t } _ i N ^ { t + s } _ j ] - \\overline { \\mu } _ i \\overline { \\mu } _ j , \\end{align*}"}
-{"id": "4921.png", "formula": "\\begin{align*} I _ n = ( z ^ { n ^ 4 } - z ^ n x ^ n , y ^ n - z ^ n x , x ^ { n + 1 } - x z ^ { n ^ 4 - n } + y z ^ n ) . \\end{align*}"}
-{"id": "9396.png", "formula": "\\begin{align*} \\Lambda ^ { - 2 } [ u _ { \\pm } ] _ { s , p } ^ { p } & \\leq { \\| u _ { \\pm } \\| } ^ { p ^ { \\ast } _ { s } } _ { p ^ { \\ast } _ { s } } + c _ { 4 } \\| u _ { \\pm } \\| ^ { q } _ { q } \\leq \\tilde { C } [ u _ { \\pm } ] _ { s , p } ^ { r } \\end{align*}"}
-{"id": "6852.png", "formula": "\\begin{align*} \\Phi ' ( t , \\beta ) = u ( t , \\Phi ( t , \\beta ) ) \\ , , \\end{align*}"}
-{"id": "9546.png", "formula": "\\begin{align*} \\mathrm { s p t } ( n ) = \\frac { 1 } { 2 } \\left ( M _ { 2 } ( n ) - N _ { 2 } ( n ) \\right ) . \\end{align*}"}
-{"id": "2984.png", "formula": "\\begin{align*} ( q ^ 2 ; q ^ 2 ) _ \\infty & = 1 + \\sum _ { n = 1 } ^ \\infty ( - 1 ) ^ n ( q ^ { 2 P _ { 5 , n } } + q ^ { 2 Q _ { 5 , n } } ) . \\end{align*}"}
-{"id": "1848.png", "formula": "\\begin{align*} f ( x ) = c \\ , v ^ { \\rm t o p . } ( x ) + \\sum _ { n = 1 } ^ \\infty \\left ( a _ n v ^ { \\rm o d d } ( n , x ) + b _ n v ^ { \\rm e v e n } ( n , x ) \\right ) , \\end{align*}"}
-{"id": "3060.png", "formula": "\\begin{align*} \\pi ^ { - s } \\Gamma ( s ) Z ( r , s ) = \\pi ^ { - ( 1 - s ) } \\Gamma ( 1 - s ) ( a b ) ^ { - 1 / 2 } Z ( r ^ { \\ast } , 1 - s ) \\end{align*}"}
-{"id": "7002.png", "formula": "\\begin{align*} \\delta _ b \\ = \\ \\gamma _ { b - 1 } + \\gamma _ { b - 3 } + \\cdots + \\gamma _ { b - 1 - 2 \\lfloor \\frac { b - 1 } { 2 } \\rfloor } . \\end{align*}"}
-{"id": "721.png", "formula": "\\begin{align*} c = \\prod _ { i = 1 } ^ { m } \\Gamma ( n - i + 1 ) \\Gamma ( i ) . \\end{align*}"}
-{"id": "8823.png", "formula": "\\begin{align*} & \\tilde { x } _ { k + 1 } ^ { ( i ) } = \\overline { x } _ { k + 1 } ^ { ( i ) } \\\\ & \\hat x _ { k + 1 } ^ { ( i ) } = \\hat x _ { k } ^ { ( i ) } - \\frac { 1 - \\theta _ k / \\theta _ 0 } { c _ k } ( \\tilde { x } _ { k + 1 } ^ { ( i ) } - \\tilde { x } _ { k } ^ { ( i ) } ) \\end{align*}"}
-{"id": "5810.png", "formula": "\\begin{align*} a _ n r ^ n = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } f ( r e ^ { i \\theta } ) e ^ { - i n \\theta } d \\theta , \\end{align*}"}
-{"id": "9141.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } L _ { B G } k _ { 1 } ^ { ( N ) } ( \\mathbf { r } _ { 1 } , t ) = 1 \\\\ L _ { B G } k _ { 2 } ^ { ( N ) } ( \\mathbf { r } _ { 1 } , \\mathbf { r } _ { 2 } , t ) = 1 \\end{array} . \\right . \\end{align*}"}
-{"id": "8221.png", "formula": "\\begin{align*} p _ j = \\sum \\limits _ { \\lambda = 0 } ^ 4 t _ { \\lambda } ^ { - j } , j \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "6771.png", "formula": "\\begin{align*} \\| Z ^ { 1 / p } - { A } ^ { 1 / p } \\| _ 2 = \\left ( 2 \\sqrt { n } + \\frac { 2 ^ { 1 / q } } { p } \\right ) { \\| Z - A \\| _ 2 } ^ { 1 / p } . \\end{align*}"}
-{"id": "7627.png", "formula": "\\begin{align*} \\phi ^ \\varepsilon \\ = \\ \\phi ^ { ( 0 ) } _ \\varepsilon \\ + \\ \\phi ^ { ( L ) } _ \\varepsilon . \\end{align*}"}
-{"id": "521.png", "formula": "\\begin{align*} \\dim I _ 2 ( K _ C ) = \\frac { ( g - 2 ) ( g - 3 ) } { 2 } , \\dim H ^ 0 ( 4 K _ C ) = 7 g - 7 . \\end{align*}"}
-{"id": "8513.png", "formula": "\\begin{align*} I \\leq & | \\ddot { z } _ 1 ( t ) | \\| \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { 2 \\pi } \\frac { 1 } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } \\| _ { H ^ s } \\\\ \\leq & ( 1 0 \\epsilon + \\frac { 6 | \\lambda | } { x ( t ) } \\epsilon ) | \\lambda x ( 0 ) | ( ( s + 6 ) ! ) ^ 2 d _ I ( t ) ^ { - 5 / 2 } \\\\ \\leq & K _ s ^ { - 1 } \\epsilon ^ 2 d _ I ( t ) ^ { - 5 / 2 } . \\end{align*}"}
-{"id": "3640.png", "formula": "\\begin{align*} ( a _ { - 1 } \\cdot v _ { ( 2 , 3 ) } ) \\cdot ( a _ 1 \\cdot v _ { ( 2 , 3 ) } ) = ( a _ 1 \\cdot v _ { ( 2 , 3 ) } ) \\cdot ( a _ 1 \\cdot v _ { ( 2 , 3 ) } ) - t ( a _ 1 - a _ { - 1 } ) \\cdot ( a _ 1 \\cdot v _ { ( 2 , 3 ) } ) . \\end{align*}"}
-{"id": "8398.png", "formula": "\\begin{align*} D ^ m \\frac { 1 } { ( z ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } = \\frac { ( - 1 ) ^ { m } ( m + 1 ) ! } { ( z ( \\alpha , t ) - z _ j ( t ) ) ^ { m + 2 } } , \\end{align*}"}
-{"id": "8443.png", "formula": "\\begin{align*} i A \\bar { \\zeta } _ { \\alpha } = i A + i A \\partial _ { \\alpha } ( \\bar { \\zeta } - \\alpha ) = i A + i A \\zeta _ { \\alpha } \\Psi _ { \\zeta } \\circ \\zeta = i A + ( D _ t ^ 2 \\zeta + i ) \\Psi _ { \\zeta } \\circ \\zeta . \\end{align*}"}
-{"id": "3912.png", "formula": "\\begin{align*} \\mathbf { M } = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} = \\begin{pmatrix} \\alpha \\ , p _ k ( r _ 1 , \\dots , r _ k ) & \\alpha \\ , p _ { k - 1 } ( r _ 1 , \\dots , r _ { k - 1 } ) \\\\ \\beta \\ , p _ { k - 1 } ( r _ 2 , \\dots , r _ k ) & \\beta \\ , p _ { k - 2 } ( r _ 2 , \\dots , r _ { k - 1 } ) \\end{pmatrix} , \\end{align*}"}
-{"id": "3601.png", "formula": "\\begin{align*} V _ { \\mathcal { P } } = V _ { ( \\mathcal { M } , * ) } \\textrm { a n d } U _ { \\mathcal { P } } = U _ { ( \\mathcal { M } , * ) } \\cup \\left \\{ f _ { \\{ \\frac { 1 } { 3 2 } \\} } ( a d _ { \\hat { a } _ { \\pm i } } ) ( \\hat { a } _ j - \\hat { a } _ { - j } ) \\right \\} \\cup \\left \\{ f _ { \\{ 1 , 0 , \\frac { 1 } { 4 } \\} } ( a d _ { \\hat { a } _ { \\pm i } } ) ( \\hat { a } _ j + \\hat { a } _ { - j } ) \\right \\} \\end{align*}"}
-{"id": "2452.png", "formula": "\\begin{align*} P ( R , t ) \\ = \\ \\frac { 1 } { 1 - t ^ d } P ( R / x R , t ) . \\end{align*}"}
-{"id": "3624.png", "formula": "\\begin{align*} a _ 1 \\cdot ( ( \\alpha _ { - 1 } - \\beta _ { - 1 } ) \\cdot a _ 0 ) = - \\frac { 1 } { 2 ^ 2 } \\beta _ { - 1 } \\cdot \\alpha _ 0 \\end{align*}"}
-{"id": "6467.png", "formula": "\\begin{align*} \\Phi ( r _ \\ell ) = ( 1 - t ) \\lambda _ 0 ^ \\ell ( 1 - \\lambda _ 0 ) + \\lambda _ 0 ^ { \\ell + 1 } ( 1 - \\lambda _ 0 ) + \\cdots + \\lambda _ 0 ^ { \\ell + k } ( 1 - \\lambda _ 0 ) + t \\lambda _ 0 ^ { \\ell + k + 1 } ( 1 - \\lambda _ 0 ) . \\end{align*}"}
-{"id": "393.png", "formula": "\\begin{align*} \\Psi _ { \\mu ^ T } ( G ) : = L _ { \\mu ^ T } ( Q ) \\otimes p ^ * G . \\end{align*}"}
-{"id": "6342.png", "formula": "\\begin{align*} f ' _ { n , d } ( n ) = & L a g \\Big \\{ n , \\eta G _ 1 G _ 2 m _ s h _ { s r } h _ { r r } h _ { r d } , \\eta G _ 1 G _ 2 ( m _ { b r _ 1 } + n _ { s p } ) \\\\ & \\times h _ { r r } h _ { r d } + \\eta G _ 2 ( m _ { b r _ 2 } + n _ { s p } ) h _ { r d } + \\eta m _ { b d } , D \\Big \\} . \\end{align*}"}
-{"id": "1164.png", "formula": "\\begin{align*} & \\mathbb { P } ( a \\geq k ) \\leq n \\left ( C n ^ { - \\frac { 1 } { \\beta + 1 } } \\right ) ^ { \\frac { \\beta k ( k - 1 ) } { 2 } + k - 1 } , \\ \\lim _ { n \\to + \\infty } \\mathbb { E } ( a - 1 ) _ + ^ p = 0 , \\end{align*}"}
-{"id": "4650.png", "formula": "\\begin{align*} p _ 0 = & ( 1 : 0 : 1 ) & p _ 1 = & ( 1 : 0 : - 1 ) & p _ 2 = & ( 0 : 1 : 1 ) & p _ 3 = & ( 0 : 1 : - 1 ) \\\\ \\end{align*}"}
-{"id": "43.png", "formula": "\\begin{align*} \\delta Y _ t ^ i : = Y _ t ^ i - \\bar { Y } _ t ^ i , \\ \\ \\delta Z _ t ^ i : = Z _ t ^ i - \\bar { Z } _ t ^ i \\ \\ \\ \\ \\delta \\xi ^ i : = \\xi ^ i - \\bar { \\xi } ^ i . \\end{align*}"}
-{"id": "351.png", "formula": "\\begin{align*} ( \\frac { 1 } { 2 } n ( n ^ 2 - 1 ) , k ) \\cdot ( n , k ) = ( \\frac { 1 } { 2 } n ( n ^ 2 - 1 ) , l ) \\cdot ( n , l ) . \\end{align*}"}
-{"id": "3692.png", "formula": "\\begin{align*} E \\big ( \\sum _ { i \\in U _ x } \\frac { \\delta _ i y _ i } { p ( x ) } \\big ) - \\sum _ { i \\in U _ x } y _ i = p ( x ) ^ { - 1 } \\sum _ { i \\in U _ x } \\big ( p _ i - p ( x ) \\big ) y _ i \\neq 0 \\end{align*}"}
-{"id": "540.png", "formula": "\\begin{align*} A ^ h ( x ) ^ { - 1 } = \\bar A ( x ' ) ^ { - 1 } + \\bar A ( x ' ) ^ { - 1 } \\Big ( - h A _ 1 \\big ( x ' , \\frac { x _ 3 } { h } \\big ) & + h ^ 2 A _ 1 \\big ( x ' , \\frac { x _ 3 } { h } \\big ) \\bar A ( x ' ) ^ { - 1 } A _ 1 \\big ( x ' , \\frac { x _ 3 } { h } \\big ) \\\\ & - \\frac { h ^ 2 } { 2 } A _ 2 \\big ( x ' , \\frac { x _ 3 } { h } \\big ) \\Big ) \\bar A ( x ' ) ^ { - 1 } + o ( h ^ 2 ) , \\end{align*}"}
-{"id": "568.png", "formula": "\\begin{align*} \\xi ^ s ( 1 + p x _ s ) R _ n ^ * - \\xi ^ t ( 1 + p x _ t ) R _ n ^ * & = \\xi ^ s ( 1 + p x _ s ) R _ n ^ * + \\xi ^ t ( 1 + p x _ t ) R _ n ^ * . \\end{align*}"}
-{"id": "8766.png", "formula": "\\begin{align*} \\kappa _ G = - \\frac D 4 \\ , . \\end{align*}"}
-{"id": "7518.png", "formula": "\\begin{align*} \\mathcal { X } ( g ( \\mathcal { Y } , \\mathcal { Z } ) ) = g ( D _ \\mathcal { X } \\mathcal { Y } , \\mathcal { Z } ) + g ( \\mathcal { Y } , D ^ * _ \\mathcal { X } \\mathcal { Z } ) \\end{align*}"}
-{"id": "2872.png", "formula": "\\begin{align*} \\operatorname { i n d } ( A _ { 1 } , A _ { 2 } ) : = & \\operatorname { d i m } \\operatorname { k e r } ( d _ 1 ) - \\operatorname { d i m } ( \\operatorname { k e r } ( d _ 2 ) \\ominus d _ { 1 } ( { \\mathcal H } ) ) + \\operatorname { d i m } ( { \\mathcal H } \\ominus d _ { 2 } ( { \\mathcal H \\oplus \\mathcal H } ) ) . \\end{align*}"}
-{"id": "5936.png", "formula": "\\begin{align*} { \\rm c o e f f } _ P ( \\bar { B } _ S ) = 1 - \\frac { 1 } { m } + \\sum _ { i = 1 } ^ n \\frac { \\alpha _ i \\lambda _ i } { m } , \\end{align*}"}
-{"id": "3245.png", "formula": "\\begin{align*} ( \\mathbf { x } , \\mathbf { y } ) & = ( x _ 0 \\cdots x _ j - \\cdots - , - \\cdots - y _ j \\cdots y _ n ) \\\\ & = ( - \\cdots - x _ k \\cdots x _ n , y _ 0 \\cdots y _ k - \\cdots - ) . \\end{align*}"}
-{"id": "1567.png", "formula": "\\begin{align*} \\varphi _ { w _ m } ^ { - 1 } ( \\mathcal { R } _ { w _ m } ) = \\pi _ z ( \\Phi _ { w _ m } ^ { - 1 } ( \\mathcal { R } _ { w _ m } ) ) = \\pi _ z ( \\Phi _ { w _ m } ^ { - 1 } ( \\mathcal { R } _ { w _ m } ) \\cap \\Omega _ { w _ m } ) . \\end{align*}"}
-{"id": "4588.png", "formula": "\\begin{align*} \\nu _ g ( B _ { 2 . 5 \\delta } ( z ) ) = \\int _ { B _ { 2 . 5 \\delta } ( z ) } d \\nu _ g ( x ) \\leq \\int \\phi ^ D ( ( z - x ) / \\delta ) d \\nu _ g ( x ) = h _ { g , \\delta } ( z ) . \\end{align*}"}
-{"id": "7191.png", "formula": "\\begin{gather*} \\xi ( q ) + \\nu ( q ) z + h ( \\xi ( q ) , \\nu ( q ) z ) = \\xi ( 0 ) + \\nu ( 0 ) z _ { 0 } , \\end{gather*}"}
-{"id": "4048.png", "formula": "\\begin{align*} f \\circ \\alpha _ X ( b ) = \\alpha _ Y ( b ) \\circ f \\end{align*}"}
-{"id": "8915.png", "formula": "\\begin{align*} \\mathbb { E } [ \\# G _ 1 & ] \\le \\binom { n } { 2 } \\frac { a } { 2 n } \\le \\frac { n a } { 4 } \\\\ P ( \\# G _ 1 \\ge \\mathbb { E } [ \\# G _ 1 & ] + \\sqrt { n a \\log ( 3 n ) } ) \\le 1 / 3 n , \\end{align*}"}
-{"id": "531.png", "formula": "\\begin{align*} Q = \\big [ \\partial _ 1 y , ~ \\partial _ 2 y , ~ \\vec b \\big ] \\in W ^ { 1 , 2 } \\cap L ^ \\infty ( \\omega , \\mathbb { R } ^ { 3 \\times 3 } ) . \\end{align*}"}
-{"id": "5073.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathbb { J } } \\langle h , x _ j \\rangle \\tau _ j & = \\sum _ { j \\in \\mathbb { J } } \\langle h , U f _ j \\rangle V f _ j = V \\left ( \\sum _ { j \\in \\mathbb { J } } \\langle U ^ * h , f _ j \\rangle f _ j \\right ) = V U ^ * h = U V ^ * h \\\\ & = U \\left ( \\sum _ { j \\in \\mathbb { J } } \\langle V ^ * h , f _ j \\rangle f _ j \\right ) = \\sum _ { j \\in \\mathbb { J } } \\langle h , V f _ j \\rangle U f _ j = \\sum _ { j \\in \\mathbb { J } } \\langle h , \\tau _ j \\rangle x _ j , ~ \\forall h \\in \\mathcal { H } . \\end{align*}"}
-{"id": "4449.png", "formula": "\\begin{align*} q - p = d _ { q } + d _ { p } . \\end{align*}"}
-{"id": "6757.png", "formula": "\\begin{align*} x ^ m \\approx x ^ m x ^ \\omega . \\end{align*}"}
-{"id": "8405.png", "formula": "\\begin{align*} & \\norm { \\frac { 1 } { z _ { \\alpha } ^ { k - 1 } } \\partial _ { \\alpha } ^ { k - 1 } \\partial _ t ( \\frac { 1 } { z _ { \\alpha } } ) u _ { \\alpha } } _ { L ^ 2 } \\\\ \\leq & C ( \\| \\partial _ { \\alpha } \\xi _ 0 \\| _ { H ^ { s - 1 } } , \\| v _ 0 \\| _ { H ^ s } , \\| w _ 0 \\| _ { H ^ s } , \\mathcal { E } ( t ) , d _ I ( t ) ^ { - 1 } , d _ P ( t ) ^ { - 1 } , N \\lambda _ { m a x } , C _ 1 , C _ 2 ) . \\end{align*}"}
-{"id": "242.png", "formula": "\\begin{align*} \\Delta _ { \\mathbb { H } } u = \\Delta _ z u + \\frac { | z | ^ 2 } { 4 } \\partial _ { t } ^ 2 u + \\partial _ t T u \\end{align*}"}
-{"id": "4235.png", "formula": "\\begin{align*} \\operatorname * { c u r l } ( a ( x ) \\lvert \\operatorname * { c u r l } u \\rvert ^ { p - 2 } \\operatorname * { c u r l } u ) = f & & \\Omega . \\end{align*}"}
-{"id": "4377.png", "formula": "\\begin{align*} \\delta _ G ( n ) = \\mathrm { m a x } \\left \\{ \\mathrm { A r e a } _ { \\mathcal { P } } ( w ( X ) ) \\mid w ( X ) \\mbox { n u l l - h o m o t o p i c , } ~ l ( w ( X ) ) \\leq n \\right \\} . \\end{align*}"}
-{"id": "4030.png", "formula": "\\begin{align*} \\textrm { R i c } _ { C ( S ) } \\left ( \\frac { d } { d r } , \\cdot \\right ) = 0 , \\textrm { R i c } _ { C ( S ) } ( X , Y ) = - \\nu \\left ( g ( X , Y ) - \\eta ( X ) \\eta ( Y ) \\right ) \\end{align*}"}
-{"id": "7078.png", "formula": "\\begin{align*} \\xi ( x , y ) \\xi ( x , 0 ) = \\xi ( x , y + 0 ) \\implies \\xi ( x , 0 ) = 1 , \\xi ( 0 , x ) = 1 \\end{align*}"}
-{"id": "80.png", "formula": "\\begin{align*} D _ J = [ J _ { { \\rm m i n } } , J _ { { \\rm m a x } } ] \\end{align*}"}
-{"id": "8391.png", "formula": "\\begin{align*} E ( t ) : = \\sum _ { k = 0 } ^ s \\Big \\{ \\int \\frac { | z _ { \\alpha } | ^ { - 2 k + 1 } } { a | z _ { \\alpha } | } | \\partial _ { \\alpha } ^ k u _ t | ^ 2 d \\alpha + R e \\int \\bold { n } | z _ { \\alpha } | D ^ { k + 1 } f \\overline { D ^ k f } d \\alpha \\Big \\} \\end{align*}"}
-{"id": "1313.png", "formula": "\\begin{align*} g _ { n _ k } = a _ { g _ { n _ k } } ^ { \\gamma _ { g _ { n _ k } } } b _ { g _ { n _ k } } ^ { \\delta _ { g _ { n _ k } } } c _ { g _ { n _ k } } ^ { \\varepsilon _ { g _ { n _ k } } } a _ { g _ { n _ k } } ^ { - \\gamma _ { g _ { n _ k } } } , \\ ( b _ { g _ { n _ k } } , c _ { g _ { n _ k } } ) \\in V _ { n _ k } a _ { g _ { n _ k } } \\in W _ { n _ k } . \\end{align*}"}
-{"id": "5531.png", "formula": "\\begin{align*} S _ T ( b ) & = u \\sqrt { T } \\langle b _ j - b _ { 0 , j } , a _ \\lambda \\Phi _ { \\lambda , k } \\rangle _ { L ^ 2 } - u \\sqrt { T } \\langle b _ j - b _ { 0 , j } , a _ \\lambda P _ { V _ J } [ \\Phi _ { \\lambda , k } / \\mu _ 0 ] \\rangle _ { \\mu _ 0 } \\\\ & = u \\sqrt { T } a _ \\lambda \\langle \\mu _ 0 ( b _ j - b _ { 0 , j } ) , \\Phi _ { \\lambda , k } / \\mu _ 0 - P _ { V _ J } [ \\Phi _ { \\lambda , k } / \\mu _ 0 ] \\rangle _ { L ^ 2 } . \\end{align*}"}
-{"id": "2792.png", "formula": "\\begin{align*} e _ i : = 1 - \\sum _ { j = 1 } ^ S q _ { i , j } . \\end{align*}"}
-{"id": "996.png", "formula": "\\begin{align*} \\widetilde a ( m , n ) = ( n \\cdot m ) ^ { \\frac { 1 } { d - 1 } } . \\end{align*}"}
-{"id": "10028.png", "formula": "\\begin{align*} w ( x ) = w ( x ; a ; q ) = \\frac { ( - q x ; q ) _ \\infty } { ( - a ^ 2 x ; q ) _ \\infty } . \\end{align*}"}
-{"id": "8886.png", "formula": "\\begin{align*} \\bar { g } _ { \\alpha , \\beta } . \\acute { s } _ { \\beta } - \\acute { s } _ { \\alpha } = s _ { \\alpha } . \\phi _ { \\alpha , \\beta } , o n U _ { \\alpha } \\cap U _ { \\beta } . \\end{align*}"}
-{"id": "3312.png", "formula": "\\begin{align*} L ( u , \\chi ) = \\sum _ { f \\in \\mathcal { M } } \\chi ( f ) u ^ { \\deg ( f ) } , \\end{align*}"}
-{"id": "9755.png", "formula": "\\begin{align*} Z _ \\infty = ( z _ \\infty , 0 ) \\in L _ \\ast \\qquad D ^ \\alpha q ( Z _ \\infty ) = 0 \\quad \\alpha = ( \\alpha ' , 0 ) | \\alpha | \\leq \\kappa - 2 . \\end{align*}"}
-{"id": "8569.png", "formula": "\\begin{align*} [ h , e _ 1 ^ \\pm ] = \\pm 2 e _ 1 ^ \\pm , \\qquad [ e _ 1 ^ + , e _ 1 ^ - ] = h , [ h , z ] = 0 = [ e _ 1 ^ \\pm , z ] . \\end{align*}"}
-{"id": "7180.png", "formula": "\\begin{gather*} X ^ { t - s } \\left ( \\chi ^ { s } ( x ) \\right ) = X ^ { t } ( x ) X ^ { - s } \\left ( \\chi ^ { s } ( x ) \\right ) = \\left [ X ^ { t } ( x ) \\right ] \\left [ X ^ { s } ( x ) \\right ] ^ { - 1 } \\end{gather*}"}
-{"id": "5240.png", "formula": "\\begin{align*} x \\hookrightarrow F ^ { \\ast } ( x ) = F ( u e p ( F ) - \\frac { 1 } { x } ) \\end{align*}"}
-{"id": "267.png", "formula": "\\begin{align*} \\partial _ t \\cdot Q ( \\partial _ t t ) = Q ( \\partial _ t t + 1 ) \\cdot \\partial _ t \\quad t \\cdot Q ( \\partial _ t t ) = Q ( \\partial _ t t - 1 ) \\cdot t . \\end{align*}"}
-{"id": "5612.png", "formula": "\\begin{align*} \\int _ 0 ^ { + \\infty } | u _ { \\delta } ( x , t ) | ^ 2 d x = \\int _ 0 ^ { + \\infty } | u _ { \\delta } ( x , 0 ) | ^ 2 d x . \\end{align*}"}
-{"id": "5973.png", "formula": "\\begin{align*} B ^ n ( t , \\omega _ 1 , \\ldots , \\omega _ n ) = \\begin{cases} \\omega _ 1 ( t ) - \\omega _ 1 ( 0 ) & t \\in [ 0 , 1 ] \\\\ \\omega _ { k + 1 } ( t - k ) - \\omega _ { k + 1 } ( 0 ) + { \\sum _ { i = 1 } ^ k [ \\omega _ i ( 1 ) - \\omega _ i ( 0 ) ] } & t \\in [ k , k + 1 ] , \\ k \\leq n - 1 . \\end{cases} \\end{align*}"}
-{"id": "9724.png", "formula": "\\begin{align*} m _ { X _ \\circ } : = \\dim L ( p _ { * , X _ \\circ } ) . \\end{align*}"}
-{"id": "7301.png", "formula": "\\begin{align*} F = \\bigcup _ { \\alpha < \\aleph _ 1 } F _ \\alpha . \\end{align*}"}
-{"id": "5879.png", "formula": "\\begin{align*} C ' _ w = q ^ { - \\frac { \\ell ( w ) } { 2 } } \\sum _ { y \\leq w } T _ y , \\end{align*}"}
-{"id": "2347.png", "formula": "\\begin{align*} a _ \\alpha ^ + & = a ^ + _ { b _ 1 } \\cdots a ^ + _ { b _ n } \\\\ a _ \\alpha & = a _ { b _ 1 } \\cdots a _ { b _ n } \\end{align*}"}
-{"id": "1381.png", "formula": "\\begin{align*} \\lambda \\left ( \\sum _ { i = 1 } ^ { m } \\sum _ { j = J _ { ( i ) } + 1 } ^ { k } d _ { j } ( \\lambda _ { i - 1 } ) e _ j + \\sum _ { j = q + 1 } ^ k d _ j ( \\lambda _ { m } ) e _ j + \\sum _ { i = m + 1 } ^ { n - 1 } \\sum _ { j = J _ { ( i ) } + 1 } ^ { k } d _ { j } ( \\lambda _ { i } ) e _ j \\ , , ~ d ( \\lambda _ 0 \\cdots \\lambda _ n ) \\right ) \\end{align*}"}
-{"id": "9959.png", "formula": "\\begin{align*} D ( \\C ) = \\textstyle \\sum _ { X \\in \\C } \\C / X \\rlap { ; } \\end{align*}"}
-{"id": "1685.png", "formula": "\\begin{align*} \\begin{gathered} N _ k ^ { 2 n } = ( b ^ { 2 ^ { k } } - 1 ) ^ { 2 n } = \\sum _ { i = 1 } ^ { n } \\Big ( C _ { 2 n } ^ { 2 i } b ^ { 2 i \\cdot 2 ^ { k } } - C _ { 2 n } ^ { 2 i - 1 } b ^ { ( 2 i - 1 ) \\cdot 2 ^ { k } } \\Big ) + 1 . \\end{gathered} \\end{align*}"}
-{"id": "1115.png", "formula": "\\begin{align*} & \\det ( + B ) = \\mathbb { P } ^ { C U E ( n ) } ( \\theta _ i \\not \\in [ 0 , 2 \\pi \\delta _ n ] , 1 \\leq i \\leq n ) = D _ n ( \\pi \\delta _ n ) . \\end{align*}"}
-{"id": "1097.png", "formula": "\\begin{align*} & \\Sigma _ k ( a _ 1 , \\cdots , a _ k ) : = \\bigcup _ { i _ 1 , \\cdots , i _ { k } \\in \\Lambda ( I ) \\ } \\prod _ { j = 1 } ^ k J _ { i _ j } ( a _ j ) \\subset ( a , b ) ^ k , \\end{align*}"}
-{"id": "6751.png", "formula": "\\begin{align*} | \\widehat { u } ( t _ 0 , \\xi ) | \\geqslant \\int _ { - \\frac { \\delta } { 2 } } ^ { \\frac { \\delta } { 2 } } e ^ { \\xi \\psi ( r _ 0 + h ) } d h & \\geqslant \\int _ { - \\frac { \\delta } { 2 } } ^ { \\frac { \\delta } { 2 } } e ^ { - \\xi A h ^ 2 } d h = \\int _ { - \\frac { \\delta \\sqrt { A } } { 2 } } ^ { \\frac { \\delta \\sqrt { A } } { 2 } } e ^ { - \\xi w ^ 2 } d w \\geqslant \\dfrac { C } { \\sqrt { \\xi } } , \\end{align*}"}
-{"id": "2826.png", "formula": "\\begin{align*} x _ i ^ { \\prime } ( t ) & \\leq x _ i ( t ) \\biggl ( - \\mu _ i x _ i ( t ) - \\sum _ { j = 1 } ^ { p } a _ { i j } \\int _ 0 ^ { \\infty } K _ { i j } ( s ) ( x _ j ( t - s ) - x _ j ^ * ) \\ , d s \\\\ & - c _ i \\int _ 0 ^ { \\infty } G _ { i } ( s ) u _ i ( t - s ) \\ , d s - \\sum _ { j = p + 1 } ^ n a _ { i j } \\int _ 0 ^ { \\infty } K _ { i j } ( s ) x _ j ( t - s ) \\ , d s \\biggr ) . \\end{align*}"}
-{"id": "4905.png", "formula": "\\begin{gather*} S = \\int _ { t _ 1 } ^ { t _ 2 } L ( \\dot \\eta , \\eta ) \\ , \\d t \\end{gather*}"}
-{"id": "8287.png", "formula": "\\begin{align*} ( I - \\mathfrak { H } _ 0 ) ( \\bar { v } _ 0 + \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { 2 \\pi } \\frac { 1 } { z _ 0 ( \\alpha ) - z _ j ( 0 ) } ) = 0 . \\end{align*}"}
-{"id": "3160.png", "formula": "\\begin{align*} \\sigma \\left ( U _ 0 \\right ) = \\varphi ^ { - 1 } \\left ( \\sigma \\left ( T _ 0 \\right ) \\right ) , \\end{align*}"}
-{"id": "4522.png", "formula": "\\begin{align*} \\sum _ { K \\in \\mathcal { N } _ { \\delta } ( \\Delta ( F ) ) } \\# S ( K ) = \\# C _ 1 \\# C _ 2 \\# C _ 3 . \\end{align*}"}
-{"id": "7777.png", "formula": "\\begin{align*} d \\int _ { 0 } ^ { 1 } r ^ { d - 1 } { \\rm d } r = 1 , \\lim _ { d \\to \\infty } d \\int _ { 0 } ^ { 1 - \\varepsilon } r ^ { d - 1 } { \\rm d } r = 0 . \\end{align*}"}
-{"id": "120.png", "formula": "\\begin{align*} x _ j = - 1 + \\frac j N \\ , , \\frac { j } { N } = ( 1 + x _ j ) \\ , , j = 0 , \\ldots , 2 N - 1 \\ , . \\end{align*}"}
-{"id": "3705.png", "formula": "\\begin{align*} \\widehat { Y } - Y = \\sum _ { i \\in B } w _ i ( t _ i ^ { \\top } \\beta + \\epsilon _ i ) - \\sum _ { i \\in U } t _ i ^ { \\top } ( \\beta + \\epsilon _ i ) = \\sum _ { i \\in B } w _ i \\epsilon _ i - \\sum _ { i \\in U } \\epsilon _ i \\\\ \\end{align*}"}
-{"id": "2087.png", "formula": "\\begin{align*} p _ { \\rm o b s } ( t ) = \\iiint _ { \\Omega _ { \\rm r x } } { { C ( \\bar { r } , t | { { \\bar { r } } _ { \\rm t x } } , { t _ 0 = 0 } ) } \\rho d \\rho \\varphi d z } , \\end{align*}"}
-{"id": "4895.png", "formula": "\\begin{align*} \\bigcup _ { U \\in W } C _ 0 ( U ) = \\{ g \\in C _ 0 ( X ) \\mid g \\in C _ 0 ( U ) U \\in W \\} \\end{align*}"}
-{"id": "3267.png", "formula": "\\begin{align*} x a ^ n + y b ^ n - ( a + b ) ( x a ^ { n - 1 } + y b ^ { n - 1 } ) + a b ( x a ^ { n - 2 } + y b ^ { n - 2 } ) = 0 , \\ ; n \\geq 2 , \\end{align*}"}
-{"id": "3794.png", "formula": "\\begin{align*} \\sum _ { \\mu : \\ \\mu _ 1 = a + d + 1 } m _ { \\mathcal { M } ( \\alpha ) } ( \\mu ) \\ - & \\sum _ { \\mu : \\ \\mu _ 1 = a + d + 1 } m _ { \\mathcal { M } ( \\beta ) } ( \\mu ) \\\\ = & \\sum _ { \\substack { ( i , j ) \\in S _ \\alpha : \\\\ i \\geq 2 j \\geq 2 } } 2 ^ { \\max \\{ i - 1 , 0 \\} + \\max \\{ j - 1 , 0 \\} } - \\sum _ { \\substack { ( i , j ) \\in S _ \\beta : \\\\ i \\geq 2 j \\geq 2 } } 2 ^ { \\max \\{ i - 1 , 0 \\} + \\max \\{ j - 1 , 0 \\} } , \\end{align*}"}
-{"id": "5998.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } Y _ n ( t _ n , \\omega ) & = \\lim _ { n \\to \\infty } \\rho ^ { G ^ { ( t _ n ) } _ n } \\left ( F \\circ L _ n \\left ( \\omega \\otimes _ { t _ n } W \\right ) \\right ) , \\end{align*}"}
-{"id": "8694.png", "formula": "\\begin{align*} 2 h ( \\lambda + \\rho + \\nu - \\rho ' + 2 i ) c _ { h , i , j } = ( i + 1 ) ( 2 i + p '' ) c _ { h - 1 , i + 1 , j } \\end{align*}"}
-{"id": "6939.png", "formula": "\\begin{align*} \\mathbf { u } = \\mathbf { 0 } , \\ \\ x \\in \\partial \\Omega . \\end{align*}"}
-{"id": "6298.png", "formula": "\\begin{align*} ( p - e ( G ) ) \\left [ k \\left ( n - k \\right ) + \\binom { k } { 2 } \\right ] & \\leq \\frac { C \\sqrt { \\rho \\log n } } { 2 \\sqrt { n } } \\left [ k \\left ( n - k \\right ) + \\binom { k } { 2 } \\right ] \\\\ & \\leq \\frac { C \\sqrt { \\rho \\log n } } { 2 \\sqrt { n } } k n \\\\ & = \\frac { C } { 2 } k \\sqrt { \\rho } \\sqrt { n \\log n } \\\\ & \\leq \\frac { C } { 2 } \\max \\{ \\sqrt { \\rho } , \\sqrt { \\frac { \\log n } { n } } \\} k \\sqrt { n \\log n } . \\end{align*}"}
-{"id": "2946.png", "formula": "\\begin{align*} \\mathcal { C } _ p ( X ) = \\{ [ \\Omega ] _ A / \\Omega \\mbox { s t r i c t l y w e a k l y p o s i t i v e s u c h t h a t } \\Omega \\mbox { i s } p \\mbox { - H S } \\} \\subset H _ A ^ { p , p } ( X , \\mathbb { R } ) \\subset H _ A ^ { p , p } ( X , \\mathbb { C } ) . \\end{align*}"}
-{"id": "7494.png", "formula": "\\begin{align*} ( \\omega _ { n + 1 } \\cdot f ) ( z ) : = f ( \\omega _ { n + 1 } ^ { - 1 } z ) . \\end{align*}"}
-{"id": "4307.png", "formula": "\\begin{align*} a ( u , v ) : = ( \\nabla u , \\nabla v ) = ( f , v ) \\forall v \\in H _ 0 ^ 1 ( \\Omega ) , \\end{align*}"}
-{"id": "6706.png", "formula": "\\begin{align*} \\ddot { \\bar { q } } _ k = \\left \\{ \\frac { 1 } { m } P _ { 2 m } - \\Big [ \\frac { u _ i } { m } + ( \\alpha ' _ { i n } + \\beta ' _ { i l n } \\ , \\dot { \\bar { q } } _ l ) \\dot { \\bar { q } } _ n \\Big ] \\beta _ { i m } \\right \\} B _ { m k } , \\end{align*}"}
-{"id": "6504.png", "formula": "\\begin{align*} \\langle x , \\alpha \\rangle & = T ^ 2 + \\langle x - \\alpha , \\alpha \\rangle \\\\ & \\geq T ^ 2 - T D \\\\ & > a T ( T + D ) \\\\ & \\geq a \\| x \\| \\| \\alpha \\| \\ , , \\end{align*}"}
-{"id": "4680.png", "formula": "\\begin{align*} \\mathcal { N } _ j ^ d : = \\{ x _ { j - d } , x _ { j - d + 1 } , \\ldots , x _ { j - 1 } \\} , \\ ; \\ ; j = 0 , 1 , \\ldots , m + d \\end{align*}"}
-{"id": "3921.png", "formula": "\\begin{align*} d _ \\infty ( \\phi , \\psi ) = \\sup _ { i , j } \\left | \\phi _ { i j } - \\psi _ { i j } \\right | \\end{align*}"}
-{"id": "4948.png", "formula": "\\begin{align*} & z ^ { t + ( 3 n ^ { d - 2 } + n ^ { d - 3 } + \\cdots + n ^ 2 ) } \\\\ & = z ^ { ( 3 n ^ { d - 2 } + n ^ { d - 3 } + \\cdots + n ^ 2 ) } ( z ^ t - z ^ n x ^ n ) + ( x ^ { n - 1 } ) ( x z ^ { 3 n ^ { d - 2 } + n ^ { d - 3 } + \\cdots + n ^ 2 + n } ) \\in I _ n . \\end{align*}"}
-{"id": "2742.png", "formula": "\\begin{align*} f ' ( z ) & = \\Big ( \\frac { 1 } { \\tau } - \\tau \\Big ) \\Big ( \\frac { 1 } { ( \\frac { 1 } { \\tau } - \\theta ) ( \\theta - \\tau ) } - \\frac { 1 } { ( \\frac { 1 } { \\tau } - z ) ( z - \\tau ) } \\Big ) \\end{align*}"}
-{"id": "3218.png", "formula": "\\begin{align*} \\mu _ k = \\gamma _ 2 \\mu _ { k - 1 } ^ { 1 + c \\alpha } , \\ \\epsilon _ k = \\gamma _ 1 \\gamma _ 2 ^ { 1 + \\alpha } \\mu _ { k - 1 } ^ { ( 1 + c \\alpha ) ( 1 + \\alpha ) } \\end{align*}"}
-{"id": "1182.png", "formula": "\\begin{align*} 0 & \\leq 1 - \\det ( - A ) / \\det ( - B ) \\leq \\left | ( - B ) ^ { - 1 } \\right | _ { \\infty } ^ 2 | B - A | _ 2 ^ 2 \\\\ & = C k ( k - 1 ) e ^ { O ( ( \\ln n ) ^ { 1 / 2 } ) } O \\left ( \\frac { \\ln n } { n ^ 2 } \\right ) \\to 0 , \\end{align*}"}
-{"id": "3825.png", "formula": "\\begin{align*} & \\overline { \\langle V _ m , G _ m , \\dots , V _ { s + 1 } , G _ { s + 1 } , V _ { s } , \\psi ( H _ q ) , V _ { s - 1 } , G _ { s - 1 } , \\dots , V _ 1 , G _ 1 , V _ 0 ; x _ i \\rangle } \\\\ \\leq & \\overline { \\langle G _ m , \\dots , G _ { s + 1 } , \\psi ( H _ q ) , G _ { s - 1 } , \\dots , G _ 1 ; \\alpha _ { \\bar f } \\bar f \\rangle } = \\bar g = \\bar h . \\end{align*}"}
-{"id": "6192.png", "formula": "\\begin{align*} \\omega = C _ { l , r } = Q _ l \\times B _ r , \\end{align*}"}
-{"id": "4184.png", "formula": "\\begin{align*} E _ n ( z ) \\mathcal { O } ( n ^ { - 3 } ) E _ n ^ { - 1 } ( z ) \\frac { A _ n ^ { ( 1 ) } ( 0 ) } { n ^ 3 z } = \\mathcal { O } ( n ^ { - \\frac { 3 } { 2 } } ) , | z | = r _ n , \\end{align*}"}
-{"id": "1241.png", "formula": "\\begin{align*} e ^ { i a \\ln ( \\eta ^ 2 - \\mu ^ 2 / | \\eta | ) } & = e ^ { i a \\ln | \\eta | ^ 2 } \\exp \\left ( i a \\ln \\left ( 1 - \\frac { \\mu ^ 2 } { | \\eta | ^ 3 } \\right ) \\right ) = e ^ { i a \\ln | \\eta | ^ 2 } \\left ( 1 + \\frac { \\mu ^ 2 } { | \\eta | ^ 3 } \\phi \\left ( \\frac { \\mu } { | \\eta | ^ { 3 / 2 } } \\right ) \\right ) , \\end{align*}"}
-{"id": "243.png", "formula": "\\begin{align*} T u = \\sum _ { i = 1 } ^ n ( y _ j \\partial _ { x _ j } u - x _ j \\partial _ { y _ j } u ) \\end{align*}"}
-{"id": "6549.png", "formula": "\\begin{align*} ( - 1 ) ^ { w - 1 } w \\ , n _ \\beta = \\sum _ { k | \\beta } \\frac { ( - 1 ) ^ { ( k - 1 ) w / k } } { k ^ 2 } \\mu ( k ) \\sum _ { P \\in E ( \\beta / k ) } \\mathcal N ^ P _ { \\beta / k } ( S , E ) . \\end{align*}"}
-{"id": "5358.png", "formula": "\\begin{gather*} - \\frac k n a _ { n m } d _ { n + 1 \\ ; m + 3 } + \\frac { n + 1 - k } n b _ { n m } c _ { n + 1 \\ , m - 3 } \\\\ - \\frac { n - k } { n - 1 } c _ { n m } b _ { n - 1 \\ , m + 3 } + \\frac { k - 1 } { n - 1 } d _ { n m } a _ { n - 1 \\ , m - 3 } = \\frac { m - n - 1 + 2 k } 2 . \\end{gather*}"}
-{"id": "2132.png", "formula": "\\begin{align*} d x ( t ) = \\tilde { A } ( t ) x ( t ) d t + \\tilde { C } ( t ) x ( t ) d \\omega ( t ) , x ( 0 ) = x _ 0 \\in R ^ d , \\end{align*}"}
-{"id": "581.png", "formula": "\\begin{align*} \\alpha _ 0 ^ 2 + 2 \\alpha _ 0 \\alpha _ 1 + \\alpha _ 1 ^ 2 & = \\alpha _ 0 \\beta \\\\ \\alpha _ 1 ^ 2 & = \\alpha _ 0 \\beta ' \\\\ \\beta & = \\beta ' , \\alpha _ 0 = 0 . \\end{align*}"}
-{"id": "1810.png", "formula": "\\begin{align*} B _ { t } ^ { ( n ) } ( \\omega ) = \\begin{cases} \\sum _ { i = 0 } ^ { 2 ^ { n } - 1 } F _ { i } ^ { t , ( n ) } \\left ( \\omega _ { ( i + 1 ) 2 ^ { - n } t } - \\omega _ { i 2 ^ { - n } t } \\right ) , & 0 < t \\leq 1 , \\\\ 0 , \\vphantom { \\sum _ { i = 0 } ^ { 2 ^ { n } - 1 } } & t = 0 . \\end{cases} \\end{align*}"}
-{"id": "8377.png", "formula": "\\begin{align*} ( \\partial _ t ^ 2 + i a \\partial _ { \\alpha } ) ( I - \\mathfrak { H } ) u = & - \\frac { i } { \\pi } \\sum _ { j = 1 } ^ N ( \\partial _ t ^ 2 + i a \\partial _ { \\alpha } ) \\frac { \\lambda _ j } { z ( \\alpha , t ) - z _ j ( t ) } \\\\ = & \\frac { i } { \\pi } \\sum _ { j = 1 } ^ N \\lambda _ j \\Big ( \\frac { 2 z _ { t t } + i - \\ddot { z } _ j } { ( z ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } - 2 \\frac { ( z _ t - \\dot { z } _ j ( t ) ) ^ 2 } { ( z ( \\alpha , t ) - z _ j ( t ) ) ^ 3 } \\Big ) \\end{align*}"}
-{"id": "8872.png", "formula": "\\begin{align*} s _ 8 & = 5 1 2 b ^ 3 p ^ 8 - 2 6 8 8 b ^ 3 p ^ 7 + 5 7 6 0 b ^ 3 p ^ 6 + 3 4 5 6 b ^ 2 p ^ 7 - 6 9 1 2 b ^ 3 p ^ 5 - 1 2 6 7 2 b ^ 2 p ^ 6 + 5 1 8 4 b ^ 3 p ^ 4 \\\\ & + 1 6 4 1 6 b ^ 2 p ^ 5 + 5 1 8 4 b p ^ 6 - 1 9 4 4 b ^ 3 p ^ 3 - 1 1 6 6 4 b ^ 2 p ^ 4 - 1 0 3 6 8 b p ^ 5 + 7 7 7 6 b ^ 2 p ^ 3 + 1 7 2 8 p ^ 5 - 2 9 1 6 b ^ 2 p ^ 2 \\\\ & + 1 1 6 6 4 b p ^ 3 - 1 1 6 6 4 b p ^ 2 - 7 7 7 6 p ^ 3 + 4 3 7 4 b p + 1 7 4 9 6 p ^ 2 - 1 7 4 9 6 p + 6 5 6 1 \\end{align*}"}
-{"id": "7281.png", "formula": "\\begin{align*} \\lVert \\Delta ^ { p , + } _ { a , b , \\lambda , \\mu } ( h _ p ^ - - h _ q ^ - ) \\rVert + \\lVert \\Delta ^ { p , + } _ { a , b , \\lambda , \\mu } ( h _ q ^ - - h _ s ^ - ) \\rVert < \\epsilon _ q - \\epsilon _ p + \\epsilon _ s - \\epsilon _ q = \\epsilon _ s - \\epsilon _ p , \\end{align*}"}
-{"id": "5944.png", "formula": "\\begin{align*} \\Theta _ Z = B _ Z + \\sum _ { z \\in Z \\setminus V } t _ z z . \\end{align*}"}
-{"id": "2344.png", "formula": "\\begin{align*} \\kappa : X & \\to \\mathcal { M } _ + ( Y \\times Z ) \\\\ \\kappa ( x ; d y , d z ) & = \\kappa _ 1 ( x , d y ) \\kappa _ 2 ( y , d z ) \\end{align*}"}
-{"id": "3146.png", "formula": "\\begin{align*} k ^ { \\pm } _ i ( u ) ^ { - 1 } E _ i ( v ) k ^ { \\pm } _ i ( u ) & = f ( v , u ) \\ E _ i ( v ) , \\\\ k ^ { \\pm } _ { i + 1 } ( u ) ^ { - 1 } E _ i ( v ) k ^ { \\pm } _ { i + 1 } ( u ) & = f ( u , v ) \\ E _ i ( v ) , \\end{align*}"}
-{"id": "1642.png", "formula": "\\begin{align*} F _ { - k } : = - \\frac { 1 } { \\varkappa _ k } \\begin{pmatrix} - b \\\\ a _ k + \\sqrt [ + ] { a _ k ^ 2 - b ^ 2 } \\end{pmatrix} e ^ { 2 \\pi i k x } \\end{align*}"}
-{"id": "5709.png", "formula": "\\begin{align*} y ( x ) = b _ 0 E _ { \\alpha , 1 } ( \\lambda x ^ { \\alpha } ) + b _ 1 x E _ { \\alpha , 2 } ( \\lambda x ^ { \\alpha } ) + \\int _ 0 ^ x ( x - t ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } [ \\lambda ( x - t ) ^ { \\alpha } ] f ( t ) d t \\end{align*}"}
-{"id": "6031.png", "formula": "\\begin{align*} \\overline { \\kappa } ( G ) = \\displaystyle \\sum _ { \\{ u , v \\} \\subseteq V ( G ) } \\kappa _ G ( u , v ) / \\tbinom { n } { 2 } , \\end{align*}"}
-{"id": "8052.png", "formula": "\\begin{align*} \\omega _ 1 ( y ) & = - 1 , \\omega _ 1 ( x ) = 1 , \\\\ \\omega _ 2 ( y ) & = - 1 , \\omega _ 1 ( x ) = \\imath , \\\\ \\omega _ 3 ( y ) & = - 1 , \\omega _ 2 ( x ) = - \\imath . \\end{align*}"}
-{"id": "4768.png", "formula": "\\begin{align*} ( \\tilde { u } _ N ( t ) , \\varphi ( t ) ) _ { \\mathcal { H } } - ( \\tilde { u } _ { N 0 } , \\varphi ( 0 ) ) _ { \\mathcal { H } } = \\int _ 0 ^ t \\Big [ & ( \\tilde { u } _ N , \\partial _ t \\varphi ) _ { \\mathcal { H } } + 2 \\nu a ( \\tilde { u } _ N , \\varphi ) + 2 \\nu a ( H , \\varphi ) \\\\ - b _ R ( \\tilde { u } _ N & , \\varphi , \\tilde { u } _ N ) - b ( H , \\varphi , \\tilde { u } _ N ) - b ( \\tilde { u } _ N , \\varphi , H ) \\Big ] d t . \\end{align*}"}
-{"id": "9311.png", "formula": "\\begin{align*} x _ { i j } \\geq 0 , ~ i = 1 , \\dots , n , ~ j = 1 , \\dots , m , \\end{align*}"}
-{"id": "998.png", "formula": "\\begin{align*} \\rho _ { s c } ( x ) = \\sqrt { ( 4 - x ^ 2 ) _ + } / ( 2 \\pi ) , \\end{align*}"}
-{"id": "1775.png", "formula": "\\begin{align*} D F _ { N } = f _ { N } ' ( M _ { s , t } ^ { ( n ) } ) D M _ { s , t } ^ { ( n ) } . \\end{align*}"}
-{"id": "6518.png", "formula": "\\begin{align*} d ^ { n q + n _ j } G _ { r _ n ( x ) } \\left ( \\tilde { a } _ j \\circ r _ n ( x ) \\right ) & = G _ { r _ n ( x ) } \\left ( F _ { r _ n ( x ) } ^ { n _ j + n q } \\left ( \\tilde { a } _ j \\circ r _ n ( x ) \\right ) \\right ) \\\\ & = G _ { r _ n ( x ) } \\left ( \\sigma _ j \\circ a ^ { n , j } ( x ) \\right ) + \\log | u _ { n , j } ( x ) | , \\end{align*}"}
-{"id": "3325.png", "formula": "\\begin{align*} m _ { \\lambda } ( X _ i : 1 \\le i \\le m ) = \\sum _ { \\mu \\vdash n } c _ { \\lambda , \\mu } p _ { \\mu } ( X _ i : 1 \\le i \\le m ) \\end{align*}"}
-{"id": "6804.png", "formula": "\\begin{align*} \\mathcal { R } _ 2 ^ { ( 1 ) } \\quad \\equiv \\begin{cases} y ^ 1 _ { 1 0 } y ^ 2 _ { 0 1 } - y ^ 1 _ { 0 1 } y ^ 2 _ { 1 0 } - 1 = 0 \\ , , \\\\ y ^ k _ \\nu = 0 \\ , , & \\ | \\nu | = 2 \\ , , k = 1 , 2 \\ , . \\end{cases} \\end{align*}"}
-{"id": "5602.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { - \\infty } ^ 0 x | u | ^ 2 d x = - 2 \\ , \\int _ { - \\infty } ^ 0 u \\bar { u } _ x d x . \\end{align*}"}
-{"id": "5330.png", "formula": "\\begin{align*} A . w ^ k = \\left ( X + Y \\right ) . w ^ k = a _ k w ^ { k + 2 } + b _ k w ^ { k - 2 } \\end{align*}"}
-{"id": "390.png", "formula": "\\begin{align*} \\mathbb { V } ^ { \\kappa } _ { \\lambda } \\hookrightarrow ( \\mathbb { M } ^ { \\kappa } _ { \\lambda } ) ^ { \\vee } \\to \\dotsb \\to \\bigoplus _ { \\ell ( w ) = i } ( \\mathbb { M } ^ { \\kappa } _ { w \\cdot \\lambda } ) ^ { \\vee } \\to \\dotsb \\to ( \\mathbb { M } ^ { \\kappa } _ { w _ 0 \\cdot \\lambda } ) ^ { \\vee } \\to 0 . \\end{align*}"}
-{"id": "6521.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ m d ^ { k ( n _ j + n q ) } h \\circ r _ n \\cdot ( r _ n ) ^ * { \\mathrm { L e b } } = & \\prod _ { j = 1 } ^ m d ^ { k ( n _ j + n q ) } \\cdot ( r _ n ) ^ * \\mu \\\\ & \\underset { n \\rightarrow \\infty } { \\longrightarrow } ( \\phi _ 0 ) ^ * \\left ( \\bigwedge _ { j = 1 } ^ m ( \\pi _ j ) ^ * \\mu _ { f _ 0 } \\right ) . \\end{align*}"}
-{"id": "7362.png", "formula": "\\begin{align*} \\inf _ { y _ 1 y _ 2 y _ 3 } \\max \\left \\{ \\max _ { i = 1 , 2 , 3 } \\{ | d ( x , y _ i ) - ( r - d ( p , x ) ) | \\} , \\max _ { 1 \\leq i < j \\leq 3 } \\{ d ( x , y _ i ) + d ( x , y _ j ) - d ( y _ i , y _ j ) \\} \\right \\} \\end{align*}"}
-{"id": "4351.png", "formula": "\\begin{align*} \\mathcal P _ a : = & \\big \\{ P \\in { [ 2 , n ] \\choose k - 1 } \\ : \\ P \\cap [ i ] = [ 2 , i ] \\setminus T \\big \\} , \\\\ \\mathcal P _ b : = & \\big \\{ P \\in { [ 2 , n ] \\choose k } \\ : \\ P \\cap [ i ] = [ i ] \\cap T \\big \\} , \\end{align*}"}
-{"id": "8328.png", "formula": "\\begin{align*} A : = ( a h _ { \\alpha } ) \\circ h ^ { - 1 } . \\end{align*}"}
-{"id": "8301.png", "formula": "\\begin{align*} ( I - \\mathcal { H } ) ( A - 1 ) = i [ D _ t \\zeta , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } D _ t \\bar { \\zeta } } { \\zeta _ { \\alpha } } + i [ D _ t ^ 2 \\zeta , \\mathcal { H } ] \\frac { \\bar { \\zeta } _ { \\alpha } - 1 } { \\zeta _ { \\alpha } } . \\end{align*}"}
-{"id": "6444.png", "formula": "\\begin{align*} y _ { j } = \\left ( - 1 \\right ) ^ { j + 1 } \\frac { 1 } { \\| r _ { j - 1 } \\| } \\left ( \\sum _ { i = j - 1 } ^ { k - 1 } \\psi _ { i } \\right ) , j = 1 , \\dots , k , \\end{align*}"}
-{"id": "3102.png", "formula": "\\begin{align*} \\lim _ { \\bar q \\rightarrow 0 } \\mathcal F ^ Z _ g ( q , \\bar q ) = F ^ { Y } _ g ( q ) . \\end{align*}"}
-{"id": "1321.png", "formula": "\\begin{align*} R \\big ( \\varPsi _ 1 \\big ) = \\rho [ Z _ 1 , P _ 0 ] \\le \\rho [ Z _ 2 , P _ 0 ] = R \\big ( \\varPsi _ 2 \\big ) , \\end{align*}"}
-{"id": "8982.png", "formula": "\\begin{align*} \\Omega ^ { \\omega } = \\{ \\omega ^ { \\prime } \\in \\Omega : \\omega _ { t } ^ { \\prime } = \\omega _ { t } \\ \\ [ 0 , \\tau _ { Q } ( \\omega ) ] \\} , \\ \\ \\ \\ \\omega \\in \\Omega . \\end{align*}"}
-{"id": "8709.png", "formula": "\\begin{align*} \\hat { \\nu } ( j ) = \\frac { 2 ^ { j - 1 } ( - 1 ) ^ { m + j } ( m - j + 1 ) { { N - 1 } \\choose { m } } { { m } \\choose { j - 1 } } } { ( N - j ) \\sum _ { k = 0 } ^ { m - 1 } { { N - 1 } \\choose { k } } } a n d p g f _ { \\hat { T } _ { j , N } } ( s ) = \\prod _ { k = j } ^ { N - 1 } \\left [ { ( 1 - \\frac { k - 1 } { N - 1 } ) s \\over 1 - \\frac { k - 1 } { N - 1 } s } \\right ] . \\end{align*}"}
-{"id": "8552.png", "formula": "\\begin{align*} \\zeta _ { \\alpha } - 1 = \\frac { D _ t ^ 2 \\zeta - i ( A - 1 ) } { i A } . \\end{align*}"}
-{"id": "4572.png", "formula": "\\begin{align*} I _ t ( \\mu ) = \\int \\int \\frac { d \\mu ( x ) d \\mu ( y ) } { | x - y | ^ t } . \\end{align*}"}
-{"id": "6856.png", "formula": "\\begin{align*} ( d \\varphi ) ^ T \\varphi '' + \\nabla p = 0 \\end{align*}"}
-{"id": "9373.png", "formula": "\\begin{align*} D _ i ( G _ k ) H _ k / D _ { i + 1 } ( G _ k ) H _ k = \\langle x ^ { 2 ^ l } \\rangle H _ k / \\langle x ^ { 2 ^ { l + 1 } } \\rangle H _ k \\cong C _ 2 . \\end{align*}"}
-{"id": "8469.png", "formula": "\\begin{align*} \\| q \\| _ { H ^ s } ^ 2 \\leq \\sum _ { n = 0 } ^ { s } \\int _ { - \\infty } ^ { \\infty } \\Big | \\partial _ { \\alpha } ^ n \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { 2 \\pi } \\frac { 1 } { \\zeta ( \\alpha , t ) - z _ j ( t ) } \\Big | ^ 2 d \\alpha \\end{align*}"}
-{"id": "3344.png", "formula": "\\begin{align*} [ u ^ n ] Z ( u ) = t ^ { n / 2 } [ u ^ n ] \\widetilde { Z } ( u ) . \\end{align*}"}
-{"id": "2474.png", "formula": "\\begin{align*} 2 ( g _ K - 1 ) = 2 ( g _ E - 1 ) | G | + | G | \\sum _ { i = 1 } ^ { r } \\frac { \\beta _ i } { e _ i } d _ i \\end{align*}"}
-{"id": "5685.png", "formula": "\\begin{align*} \\| x _ \\alpha ^ A & - x _ \\gamma ^ A \\| = \\| x _ \\beta ^ A - x _ \\gamma ^ A + h _ \\beta v _ \\beta ^ A \\| \\le \\| x _ \\beta ^ A - x _ \\gamma ^ A \\| + h _ \\beta \\| v _ \\beta ^ A \\| \\\\ & \\le M ( t _ \\beta - t _ \\gamma ) + M ( t _ \\alpha - t _ \\beta ) = M ( t _ \\alpha - t _ \\gamma ) \\end{align*}"}
-{"id": "2123.png", "formula": "\\begin{align*} d x ( t ) = A ( t ) x ( t ) d t + C ( t ) x ( t ) d Z ( t , \\omega ) , \\ x ( t _ 0 ) = x _ 0 \\in \\R ^ d , t \\geq t _ 0 , \\end{align*}"}
-{"id": "6299.png", "formula": "\\begin{align*} \\mathbb { P } _ { G \\sim G _ { n , p } } \\left ( \\bigcup _ { \\emptyset \\not = S \\subseteq [ n ] } A ^ c _ { \\rho , S , \\frac { C } { 2 } } ( p ) \\right ) = O ( \\frac { 1 } { n ^ { \\frac { C } { 1 6 } - 2 } } ) . \\end{align*}"}
-{"id": "2111.png", "formula": "\\begin{align*} E _ k ^ u ( \\omega ) : = \\big \\{ y \\in \\R ^ d \\bigm | \\limsup _ { t \\rightarrow \\infty } \\frac { 1 } { t } \\log | \\Phi _ \\omega ( u , t ) y | \\leq \\lambda _ k ( \\omega ) \\big \\} , k = 1 , \\ldots , d , \\ ; \\ ; \\end{align*}"}
-{"id": "5783.png", "formula": "\\begin{align*} R ( \\Gamma , q ^ { - 1 } , T ^ { - 1 } ) = \\epsilon ( \\Gamma ) + ( - 1 ) ^ { \\# E ( \\Gamma ) - 1 } q ^ { b _ 1 ( \\Gamma ) } R ( \\Gamma , q , T ) , \\end{align*}"}
-{"id": "1755.png", "formula": "\\begin{align*} X = \\frac { B _ { t } - B _ { s } } { ( t - s ) ^ { H } } , Y = \\frac { B _ { r } - B _ { u } } { ( r - u ) ^ { H } } , \\end{align*}"}
-{"id": "7341.png", "formula": "\\begin{align*} & \\theta \\in C ^ { \\infty } ( 0 , T ) , 0 \\leq \\theta ( t ) \\leq 1 , \\theta ( t ) = 1 \\ \\mbox { i n } \\ ( \\varepsilon , T - \\varepsilon ) . \\end{align*}"}
-{"id": "8251.png", "formula": "\\begin{align*} E _ 2 ( \\tau ) & = 1 - 2 4 \\sum _ { n = 1 } ^ \\infty \\sigma ( n ) q ^ n \\\\ & = \\left ( 1 - 2 4 \\sum _ { n = 1 } ^ \\infty \\sigma ( 2 n ) q ^ { 2 n } \\right ) - 2 4 \\sum _ { n = 0 } ^ \\infty \\sigma ( 2 n + 1 ) q ^ { 2 n + 1 } \\\\ & = 3 E _ 2 ( 2 \\tau ) - 2 E _ { 2 } ( 4 \\tau ) - 2 4 \\sum _ { n = 0 } ^ \\infty \\sigma ( 2 n + 1 ) q ^ { 2 n + 1 } . \\end{align*}"}
-{"id": "6058.png", "formula": "\\begin{align*} \\Pi _ s = \\{ \\gamma _ s , \\gamma ' _ s , \\gamma '' _ s \\} \\end{align*}"}
-{"id": "4073.png", "formula": "\\begin{align*} Y _ { + } ( x ) = Y _ { - } ( x ) \\begin{pmatrix} 1 & w ( x ) & x ^ { \\frac { 1 } { 2 } } w ( x ) \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} , x > 0 . \\end{align*}"}
-{"id": "589.png", "formula": "\\begin{align*} \\tilde { \\Lambda } ^ x _ t : = \\begin{cases} \\Lambda ^ x _ t \\qquad \\ , \\ ; \\ ; \\ ; t \\in [ 0 , \\tau ^ x \\wedge T ) , \\\\ - Z ^ x _ { \\tau ^ x } \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; t \\in ( \\tau ^ x \\wedge T , T ] . \\end{cases} \\end{align*}"}
-{"id": "3827.png", "formula": "\\begin{align*} v _ - ^ k = \\kappa ^ * w ^ k \\in \\mathcal { C } ^ \\infty _ c ( \\mathcal { M } ) , \\end{align*}"}
-{"id": "9614.png", "formula": "\\begin{align*} - \\sum _ { m = 1 } ^ { \\infty } \\frac { q ^ { N + m } } { 1 + q ^ { N + m } } - \\sum _ { m = 1 } ^ { \\infty } \\frac { q ^ { m } } { 1 - q ^ { m } } = - \\sum _ { m = 1 } ^ { N } \\frac { q ^ m } { 1 - q ^ m } - \\sum _ { m = N + 1 } ^ { \\infty } \\frac { 2 q ^ m } { 1 - q ^ { 2 m } } . \\end{align*}"}
-{"id": "3003.png", "formula": "\\begin{align*} \\Big ( \\sum _ { n = 0 } ^ \\infty p ' _ m ( n ) x ^ n \\Big ) \\Big ( \\sum _ { n = 0 } ^ \\infty n e _ { m + 2 , n } x ^ { n } \\Big ) & = - \\sum _ { n = 1 } ^ \\infty \\sigma ' _ m ( n ) x ^ n . \\end{align*}"}
-{"id": "7097.png", "formula": "\\begin{align*} u _ { n , \\theta } = u _ { n - 1 , \\theta } + F _ { n , \\omega } ^ \\omega + w _ { n , \\theta } ~ . \\end{align*}"}
-{"id": "7755.png", "formula": "\\begin{align*} \\Big | [ \\langle x , z \\rangle \\xi , \\xi ] \\Big | = \\Big | [ \\langle x , y \\rangle \\langle y , z \\rangle \\xi , \\xi ] \\Big | = \\left \\| x \\right \\| \\left \\| z \\right \\| . \\end{align*}"}
-{"id": "1423.png", "formula": "\\begin{align*} & \\| u \\| _ E = \\| u \\| _ { E _ 0 } = \\| \\partial u \\| _ { L _ T ^ \\infty L _ x ^ 2 } , \\ \\| u \\| _ { E _ m } = \\sum _ { | a | \\leq m } \\| Y ^ a u \\| _ E . \\end{align*}"}
-{"id": "888.png", "formula": "\\begin{align*} \\implies \\lambda ( \\phi _ u , \\phi _ v ) \\geq 0 . \\end{align*}"}
-{"id": "6807.png", "formula": "\\begin{align*} \\mathsf { d i m } \\big ( \\mathsf { V } ( I _ { 3 , 8 } ) \\big ) = 6 > 5 = \\mathsf { d i m } \\big ( \\mathsf { V } ( \\sqrt [ \\mathbb { R } ] { I _ { 3 , 8 } } ) \\big ) \\ , . \\end{align*}"}
-{"id": "7146.png", "formula": "\\begin{align*} \\chi _ { t t } = - \\frac { v _ t } { \\epsilon } ( \\chi ^ 3 - \\chi ) - \\frac { v } { \\epsilon } ( 3 \\chi ^ 2 - 1 ) \\chi _ t + \\epsilon v _ t \\Big ( \\frac { \\chi _ y } { v ^ 2 } \\Big ) _ y + \\epsilon v \\Big ( \\frac { \\chi _ y } { v ^ 2 } \\Big ) _ { y t } . \\end{align*}"}
-{"id": "9053.png", "formula": "\\begin{align*} & \\mathbb { E } \\left [ \\left ( \\frac { 1 } { \\pi R ^ 2 } \\int _ { B ( R ) } ( F _ { \\mu _ A } \\cdot F _ { \\mu _ B } ) d x \\right ) ^ 2 \\right ] = \\\\ = & \\frac { 1 } { \\pi ^ 2 R ^ 4 } \\int _ { B ( R ) } \\int _ { B ( R ) } \\mathbb { E } [ F _ { \\mu _ A } ( x ) \\cdot F _ { \\mu _ A } ( y ) ] \\mathbb { E } [ F _ { \\mu _ B } ( x ) F _ { \\mu _ B } ( y ) ] d x d y \\\\ = & \\frac { 1 } { \\pi ^ 2 R ^ 4 } \\int _ { B ( R ) } \\int _ { B ( R ) } r _ { \\mu _ { B } } ( x - y ) r _ { \\mu _ A } ( x - y ) d x d y \\end{align*}"}
-{"id": "1440.png", "formula": "\\begin{align*} w ^ { n + 1 } _ { t } + \\Delta w ^ { n + 1 } + \\varepsilon P \\Theta ( t , x , \\mu ^ { n } , D w ^ { n } ) = 0 , \\end{align*}"}
-{"id": "4208.png", "formula": "\\begin{align*} R _ - ^ { - 1 } ( z ) R _ + ( z ) & = P ( z ) S _ - ^ { - 1 } ( z ) S _ + ( z ) P ^ { - 1 } ( z ) \\\\ & = \\mathbb I + z ^ { - \\beta } e ^ { 2 n \\varphi ( z ) } P ( z ) \\begin{pmatrix} 0 & 0 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} P ^ { - 1 } ( z ) . \\end{align*}"}
-{"id": "4290.png", "formula": "\\begin{align*} \\binom { s - 2 } { r - 1 } + \\binom { s - 3 } { r - 2 } \\end{align*}"}
-{"id": "4708.png", "formula": "\\begin{align*} F _ { j + \\frac { 1 } { 2 } } ^ { \\mbox { \\small H y b r i d 6 } } = \\Phi F _ { j + \\frac { 1 } { 2 } } ^ { \\mbox { W E N O 5 } } + ( 1 - \\Phi ) F _ { j + \\frac { 1 } { 2 } } ^ { \\mbox { Q n B S Q I } } . \\end{align*}"}
-{"id": "5983.png", "formula": "\\begin{align*} \\sup _ { Q \\in { \\cal P } ^ * } \\left ( F ( Q ) - \\tilde { \\alpha } ^ g ( Q ) \\right ) & = \\sup _ { Q \\in \\Q } ( F ( Q ) - \\alpha ^ g ( Q ) ) . \\end{align*}"}
-{"id": "4066.png", "formula": "\\begin{align*} \\mathbb K ^ { ( \\alpha , \\theta ) } ( x , y ) = \\theta y ^ { \\alpha } \\int _ 0 ^ 1 J _ { \\frac { \\alpha + 1 } { \\theta } , \\frac { 1 } { \\theta } } ( u x ) J _ { \\alpha + 1 , \\theta } \\left ( ( u y ) ^ { \\theta } \\right ) u ^ { \\alpha } d u \\end{align*}"}
-{"id": "4502.png", "formula": "\\begin{align*} \\deg ( M _ 0 , \\mathcal { O } , 0 ) = \\deg ( M _ { 0 , z ^ 0 } , B _ 1 , 0 ) = \\pm 1 \\neq 0 , \\end{align*}"}
-{"id": "6534.png", "formula": "\\begin{align*} f ( t , x ( t ) ) = \\bigl [ g ( t , x ( t ) ) - g ( t , y ( t ) ) \\bigr ] + g ( t , y ( t ) ) + Q ( t , x ( t ) ) , t \\geq 0 . \\end{align*}"}
-{"id": "7470.png", "formula": "\\begin{gather*} \\deg a _ { i j } = 1 - d _ { i j } , \\end{gather*}"}
-{"id": "6075.png", "formula": "\\begin{align*} J '' = \\frac { 1 } { m } \\left ( \\phi ' + \\frac { \\phi ^ 2 } { m } \\right ) J \\leq - \\frac { R i c _ m ( \\dot \\gamma , \\mathcal H _ t ) } { m } J \\end{align*}"}
-{"id": "4916.png", "formula": "\\begin{align*} \\delta L & = \\int _ \\Omega g ( A u , \\delta u ) \\ , \\mu ( x ) = \\int _ \\Omega g ( m , \\delta u ) \\ , \\mu ( x ) \\ , , \\end{align*}"}
-{"id": "859.png", "formula": "\\begin{gather*} \\min _ { B \\in L ( \\R ^ n , \\R ^ m ) } \\norm { B } \\mbox { s u b j e c t t o } B ^ T v ^ * = u ^ \\ast , \\ ; B u = v , \\end{gather*}"}
-{"id": "2201.png", "formula": "\\begin{align*} \\dot { \\boldsymbol { g } } ( t ) + \\big [ A ( t ) - B ( t ) R ^ { - 1 } ( t ) B ^ T ( t ) P ( t ) \\big ] ^ { T } \\boldsymbol { g } ( t ) + C ^ T ( t ) Q ( t ) \\boldsymbol { z } ( t ) = \\boldsymbol { 0 } \\ \\forall t \\in [ t _ 0 , t _ f ] \\end{align*}"}
-{"id": "5464.png", "formula": "\\begin{align*} K _ t ^ n \\ = \\ n \\int _ 0 ^ t \\int _ \\Lambda \\big ( R _ s ^ n ( b ) \\big ) ^ + \\ , \\lambda _ 0 ( d b ) d s , 0 \\leq t \\leq T , \\ ; \\hat \\P \\end{align*}"}
-{"id": "9270.png", "formula": "\\begin{align*} \\begin{array} { r l } M _ 2 ^ * ( R ) = \\ ! \\ ! & \\ ! \\ ! \\displaystyle \\sup \\limits _ { B _ { \\frac { \\delta R } { 1 6 } } ( \\bar x ) } ( d _ x ^ 2 | D ^ 2 u ( x ) | ) \\le ( \\frac { 1 7 } { 1 6 } \\delta R ) ^ 2 \\sup \\limits _ { B _ { \\frac { \\delta R } { 1 6 } } ( \\bar x ) } | D ^ 2 u | \\\\ \\le \\ ! \\ ! & \\ ! \\ ! \\displaystyle ( \\frac { 1 7 } { 1 6 } \\delta R ) ^ 2 C ( 1 + \\frac { 1 } { ( \\delta R / 8 ) ^ 2 } ) , \\end{array} \\end{align*}"}
-{"id": "1031.png", "formula": "\\begin{align*} & \\limsup _ { n \\to + \\infty } \\sup _ { y _ 1 , \\cdots , y _ k \\in [ 0 , 2 \\pi ) } \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) \\leq \\prod _ { j = 1 } ^ k \\left ( e ^ { c _ 0 - x _ j } / 4 \\right ) ; \\end{align*}"}
-{"id": "5536.png", "formula": "\\begin{align*} L ^ * _ b u = - h . \\nabla \\mu _ { b + h } - d i v ( h ) \\mu _ { b + h } . \\end{align*}"}
-{"id": "5476.png", "formula": "\\begin{align*} \\hat v ^ n ( s , \\hat X _ s ^ { t , x , a } ) \\ = \\ Y _ s ^ { t , x , a } , \\hat \\P , \\ , t \\leq s \\leq T . \\end{align*}"}
-{"id": "4525.png", "formula": "\\begin{align*} a _ i = ( | x _ i - y _ i | , | x _ i - z _ i | , | y _ i - z _ i | ) . \\end{align*}"}
-{"id": "4295.png", "formula": "\\begin{align*} 6 | \\mathcal { I } _ { 3 } ' ( G ) | + 6 | \\mathcal { I } _ { 3 } '' ( G ) | = 6 | \\mathcal { I } _ 3 ( G ) | . \\end{align*}"}
-{"id": "1921.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\frac { A _ j ^ 2 } { Z _ j ^ p } \\leq \\frac { \\cos ^ 2 ( Z _ 1 ) } { Z _ 1 ^ { p - 1 } \\left [ Z _ 1 + \\sin ( Z _ 1 ) \\cos ( Z _ 1 ) \\right ] } + \\frac { 1 } { \\pi ^ p } \\sum _ { j = 2 } ^ \\infty \\frac { 1 } { ( j - 1 ) ^ p } . \\end{align*}"}
-{"id": "5547.png", "formula": "\\begin{align*} \\Upsilon _ { \\varepsilon } : = \\lambda \\| \\rho \\phi _ { \\varepsilon } \\| _ { L ^ { 4 , \\varepsilon } } ^ { 4 } + m ^ { 2 } \\| \\rho ^ { 2 } \\psi _ { \\varepsilon } \\| _ { L ^ { 2 , \\varepsilon } } ^ { 2 } + \\| \\rho ^ { 2 } \\nabla _ { \\varepsilon } \\psi _ { \\varepsilon } \\| _ { L ^ { 2 , \\varepsilon } } ^ { 2 } \\end{align*}"}
-{"id": "8381.png", "formula": "\\begin{align*} ( I + \\mathfrak { K } ^ { \\ast } ) a _ t | z _ { \\alpha } | = R e ( \\frac { i z _ { \\alpha } } { | z _ { \\alpha } | } ( g _ 1 + g _ 2 ) ) , \\end{align*}"}
-{"id": "97.png", "formula": "\\begin{align*} d ~ \\dot = ~ \\bar k \\ , C _ 1 \\ , , \\gamma ~ \\dot = ~ \\frac { d } { N } \\ , . \\end{align*}"}
-{"id": "5214.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { s } t ^ { - 1 } B ( t x ) d t = \\log \\left \\{ \\frac { h ( s x ) } { h ( x ) } \\right \\} \\rightarrow - ( 1 + \\rho ) \\log s x \\rightarrow \\infty , \\end{align*}"}
-{"id": "8972.png", "formula": "\\begin{align*} d _ E ( u , v ) ^ 2 = 1 - \\frac { | u \\cdot v | ^ 2 } { \\| u \\| ^ 2 \\cdot \\| v \\| ^ 2 } \\end{align*}"}
-{"id": "3756.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ \\Omega Y _ { l , m } \\ , \\overline { Y } _ { l ^ { \\prime } , m ^ { \\prime } } \\ , d \\hat { \\mathbf { r } } & = \\delta _ { l l ^ { \\prime } } \\delta _ { m m ^ { \\prime } } , \\\\ \\int _ \\Omega \\mathbf { Y } _ { l , m } \\cdot \\overline { \\mathbf { Y } } _ { l ^ { \\prime } , m ^ { \\prime } } \\ , d \\hat { \\mathbf { r } } & = l \\left ( l + 1 \\right ) \\delta _ { l l ^ { \\prime } } \\delta _ { m m ^ { \\prime } } , \\end{aligned} \\end{align*}"}
-{"id": "3405.png", "formula": "\\begin{align*} \\tau ^ 1 ( T ) = \\log ( 1 + \\deg ( T ) ) \\le \\deg ( T ) . \\end{align*}"}
-{"id": "7331.png", "formula": "\\begin{align*} \\begin{cases} \\rho ( x ) z _ { t t } - \\operatorname { d i v } ( K ( x ) \\nabla z ) = 0 & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ z = 0 & ~ \\hbox { i n } ~ \\omega \\times ( 0 , T ) , \\end{cases} \\end{align*}"}
-{"id": "3546.png", "formula": "\\begin{align*} \\mathtt { a } : = ( - a _ { 0 , } - b _ { 0 } , - c _ { 0 } , a _ { 1 } , a _ { 2 } , b _ { 1 } , b _ { 2 } , c _ { 1 } , c _ { 2 } ) . \\end{align*}"}
-{"id": "1761.png", "formula": "\\begin{align*} \\begin{aligned} \\langle D X _ { i } , D X _ { j } \\rangle _ { \\mathcal { H } } & = \\mathbb { E } [ X _ { j } X _ { k } ] \\\\ & = \\frac { 1 } { 2 } \\left [ \\left ( k - j + 1 \\right ) ^ { 2 H } + \\left ( k - j - 1 \\right ) ^ { 2 H } - 2 \\left ( k - j \\right ) ^ { 2 H } \\right ] . \\end{aligned} \\end{align*}"}
-{"id": "2865.png", "formula": "\\begin{align*} \\sigma _ i ( a ) = \\sigma _ i ( 1 ) a ^ d \\end{align*}"}
-{"id": "3064.png", "formula": "\\begin{align*} T _ j f ( x ) = K _ j * f ( x ) , K _ j ( x ) = \\frac { x _ j } { | x | ^ { n + 1 - \\beta } } , 0 < \\beta < n , j = 1 , 2 , \\cdots , n . \\end{align*}"}
-{"id": "4939.png", "formula": "\\begin{align*} f _ { i + 3 } = v _ i ^ n + v _ i z ^ { t - n } - ( v _ { i + 1 } - v _ { i + 2 } + \\cdots + ( - 1 ) ^ { d + i } ( v _ { d - 3 } ) + ( - 1 ) ^ { d + i + 1 } x ) ^ n + v _ i z ^ n - v _ i x ^ n \\end{align*}"}
-{"id": "3054.png", "formula": "\\begin{align*} \\mu ( x , y ) = \\begin{cases} 1 , x = y . \\\\ - \\sum _ { x \\preceq z \\prec y } \\mu ( x , z ) , x \\prec y . \\\\ 0 , \\quad \\end{cases} \\end{align*}"}
-{"id": "1088.png", "formula": "\\begin{align*} & \\lim _ { n \\to + \\infty } \\frac { D _ n ( ( 1 + z / \\ln n ) G _ n ( x ) / 2 ) } { D _ n ( G _ n ( x ) / 2 ) } \\frac { ( 1 + z / \\ln n ) a _ 1 } { \\sqrt { 4 - ( 1 + z / \\ln n ) ^ 2 a _ 1 ^ 2 } } \\\\ = & \\lim _ { n \\to + \\infty } \\frac { D _ n ( ( 1 + z / \\ln n ) G _ n ( x ) / 2 ) } { D _ n ( G _ n ( x ) / 2 ) } \\frac { a _ 1 } { \\sqrt { 4 - a _ 1 ^ 2 } } = e ^ { - 2 z } \\frac { a _ 1 } { \\sqrt { 4 - a _ 1 ^ 2 } } . \\end{align*}"}
-{"id": "8976.png", "formula": "\\begin{align*} d ( x , y _ \\theta ) ^ 2 = \\frac { | \\kappa ^ 2 - y _ 1 | } { 2 \\kappa ^ 2 } = \\frac { | w _ 2 | \\upsilon } { 2 \\kappa ^ 2 } | e ^ { i \\theta } - z | \\ , , \\end{align*}"}
-{"id": "9480.png", "formula": "\\begin{align*} \\psi ^ { + } ( x ) = \\psi ^ { + } ( y ) \\ \\ \\mbox { o r } \\ \\ Q _ { m } ( x , y ) = 0 , \\end{align*}"}
-{"id": "1027.png", "formula": "\\begin{align*} | \\Sigma _ k ( a _ 1 , \\cdots , a _ k ) | & = \\sum _ { i _ 1 , \\cdots , i _ { k } \\ } \\prod _ { j = 1 } ^ k ( \\theta _ { i _ j + 1 } - \\theta _ { i _ j } - a _ j ) _ + \\\\ & = \\sum _ { i _ 1 , \\cdots , i _ { k } \\ } \\prod _ { j = 1 } ^ k ( m _ { i _ j } - a _ j ) _ + , \\end{align*}"}
-{"id": "4972.png", "formula": "\\begin{align*} \\forall x , y \\in C \\ , \\forall \\gamma \\in [ 0 , 1 ] \\ , \\big ( U ( x ) = x \\wedge U ( y ) = y \\to U ( ( 1 - \\gamma ) x + \\gamma y ) = ( 1 - \\gamma ) x + \\gamma y \\big ) , \\end{align*}"}
-{"id": "1375.png", "formula": "\\begin{align*} \\kappa _ j ( i ) = \\begin{cases} i , & i < j \\\\ i + 1 , & j \\leq i \\leq n - 1 \\\\ j , & i = n \\end{cases} \\ ; . \\end{align*}"}
-{"id": "2759.png", "formula": "\\begin{align*} D S ^ { \\mathrm { s o f t } } ( \\kappa , \\pi ; u , v ) & = D S ^ { ( \\mathrm { s o f t } ) } ( \\kappa ; u , v ) \\\\ & + \\sum _ { p = 1 } ^ { m } \\Big ( \\phi _ { 1 , p - 1 } ( \\kappa , \\pi ; u ) \\phi _ { 1 , p } ( \\kappa , \\pi ; v ) - \\phi _ { 1 , p } ( \\kappa , \\pi ; u ) \\phi _ { 1 , p - 1 } ( \\kappa , \\pi ; v ) \\Big ) , \\end{align*}"}
-{"id": "3378.png", "formula": "\\begin{align*} N ( c , k ) = ( 1 + o ( 1 ) ) f ( c , k ) \\end{align*}"}
-{"id": "8437.png", "formula": "\\begin{align*} D _ t \\bar { \\zeta } = F \\circ \\zeta - \\frac { i } { 2 \\pi } \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j } { \\zeta ( \\alpha , t ) - z _ j ( t ) } , \\lambda _ 1 = - \\lambda _ 2 = \\lambda , \\end{align*}"}
-{"id": "9204.png", "formula": "\\begin{align*} L _ { s } : [ 0 , h ( s ) ] \\rightarrow M , L _ { s } ( t ) = H ( t , s ) , \\end{align*}"}
-{"id": "6575.png", "formula": "\\begin{gather*} e ^ i e _ { I , t } : = \\# ( i , I ) e _ { I \\backslash i , t } , e ^ 0 e _ { I , t + 1 } : = ( t + 1 ) e _ { I , t } , \\end{gather*}"}
-{"id": "2183.png", "formula": "\\begin{align*} \\int | \\tilde K _ \\ell ( \\mathbf x , \\mathbf y ) | ( 1 + d ( \\mathbf x , \\mathbf y ) ) ^ { \\delta ' } \\ , d w ( \\mathbf x ) & = \\int | \\tau _ { - \\mathbf y } ( \\mathcal { F } ^ { - 1 } m _ \\ell ) ( \\mathbf x ) | ( 1 + d ( \\mathbf x , \\mathbf y ) ) ^ { \\delta ' } \\ , d w ( \\mathbf x ) \\\\ & \\leq C _ { \\delta ' , s } \\| m _ { \\ell , 1 } \\| _ { W ^ s _ 2 } \\leq C M . \\end{align*}"}
-{"id": "447.png", "formula": "\\begin{align*} - \\Delta _ g \\int _ M G _ \\mu ( \\cdot , y ) f ( y ) { \\rm v } _ g ( \\dd y ) = f ( \\cdot ) - \\int _ { \\mathcal { C } } f ( y ) \\mu ( \\dd y ) \\end{align*}"}
-{"id": "6555.png", "formula": "\\begin{align*} m ^ P _ \\beta = m ^ { P ' } _ \\beta . \\end{align*}"}
-{"id": "132.png", "formula": "\\begin{align*} \\sum _ { \\substack { k = 1 \\\\ k \\ , } } ^ { 2 N - 1 } \\gamma ^ k S _ k & = \\gamma ~ \\sum _ { j = 0 } ^ { 2 N - 1 } { } _ 2 F _ 1 ( - j , - 2 N + j + 1 , 1 ; \\gamma ^ 2 ) B ( 0 ) ^ { 2 j } B _ 2 ( 0 ) \\ , . \\end{align*}"}
-{"id": "4491.png", "formula": "\\begin{align*} \\Phi ( x ) = \\mu ( x ) - \\nabla _ n ( u - \\underline { u } ) ( x ) , x \\in \\Omega _ \\delta \\end{align*}"}
-{"id": "2617.png", "formula": "\\begin{align*} F ( \\theta ) : = | G ( \\theta ) ^ { \\alpha - 1 } - G _ 0 ( \\theta ) ^ { \\alpha - 1 } | ^ { \\frac { 1 } { \\alpha - 1 } } , \\theta \\geq 0 . \\end{align*}"}
-{"id": "3578.png", "formula": "\\begin{align*} \\begin{alignedat} { 1 } Y _ { 0 } = Y _ { 0 } ( A ) & = [ 1 \\ , 2 \\ , 3 ] [ 4 \\ , 5 \\ , 6 ] , \\\\ Y _ { 1 } = Y _ { 1 } ( A ) & = [ 1 \\ , 2 \\ , 4 ] [ 3 \\ , 5 \\ , 6 ] , \\\\ Y _ { 2 } = Y _ { 2 } ( A ) & = [ 1 \\ , 2 \\ , 5 ] [ 3 \\ , 4 \\ , 6 ] , \\\\ Y _ { 3 } = Y _ { 3 } ( A ) & = [ 1 \\ , 3 \\ , 4 ] [ 2 \\ , 5 \\ , 6 ] , \\\\ Y _ { 4 } = Y _ { 4 } ( A ) & = [ 1 \\ , 3 \\ , 5 ] [ 2 \\ , 4 \\ , 6 ] , \\\\ Y _ { 5 } = Y _ { 5 } ( A ) & = [ 1 \\ , 2 \\ , 3 ] [ 1 \\ , 4 \\ , 5 ] [ 2 \\ , 4 \\ , 6 ] [ 3 \\ , 5 \\ , 6 ] - [ 1 \\ , 2 \\ , 4 ] [ 1 \\ , 3 \\ , 5 ] [ 2 \\ , 3 \\ , 6 ] [ 4 \\ , 5 \\ , 6 ] , \\end{alignedat} \\end{align*}"}
-{"id": "1634.png", "formula": "\\begin{align*} \\{ F , G \\} ( \\varphi ) : = d _ \\varphi F ( X _ G ) = - i \\int _ 0 ^ 1 \\big ( ( \\partial _ { \\varphi _ 1 } F ) ( \\partial _ { \\varphi _ 2 } G ) - ( \\partial _ { \\varphi _ 1 } G ) ( \\partial _ { \\varphi _ 2 } F ) \\big ) \\ , d x \\end{align*}"}
-{"id": "7630.png", "formula": "\\begin{align*} A \\varkappa _ 1 \\ = \\ h \\ = \\ f \\ - \\ b \\end{align*}"}
-{"id": "5384.png", "formula": "\\begin{align*} \\frac { k } { n } c _ { n + 1 \\ , m - 3 } | | v _ { n m } ^ k | | ^ 2 = - \\overline { b _ { n m } } \\frac { k } { n + 1 - k } | | v _ { n + 1 \\ , m - 3 } ^ { k } | | ^ 2 . \\end{align*}"}
-{"id": "6629.png", "formula": "\\begin{align*} y _ 1 = 1 , \\ \\ldots , \\ y _ { m - 2 } = 1 ( m = v - t - 1 ) . \\end{align*}"}
-{"id": "3630.png", "formula": "\\begin{align*} v _ { ( 1 , 2 ) } \\cdot ( a _ 3 \\cdot v _ { ( 1 , 2 ) } ) = \\frac { ( t - 1 ) t } { 2 ^ 2 } ( a _ 3 - a _ { - 3 } ) + \\frac { t } { 2 ^ 2 } v _ { ( 1 , 2 ) } + \\frac { 1 } { 2 } a _ 3 \\cdot v _ { ( 1 , 2 ) } \\end{align*}"}
-{"id": "2036.png", "formula": "\\begin{align*} \\left < \\frac { a } { d } \\right > ^ { \\pm } & = - \\pi \\int _ { \\infty } ^ 0 \\left ( f ( \\frac { a } { d } + i y ) \\pm f ( - \\frac { a } { d } + i y ) \\right ) \\ ; d y \\\\ & = \\pi \\left ( \\Lambda ( 1 , f , \\frac { a } { d } ) \\pm \\Lambda ( 1 , f , - \\frac { a } { d } ) \\right ) = \\frac { 1 } { 2 } \\bigg ( L ( 1 , f , \\frac { a } { d } ) \\pm L ( 1 , f , - \\frac { a } { d } ) \\bigg ) . \\end{align*}"}
-{"id": "2125.png", "formula": "\\begin{align*} \\sup \\limits _ { t _ 0 \\leq s \\leq t \\leq t _ 0 + 1 } \\log ^ + \\| \\Phi _ \\omega ( s , t ) ^ { - 1 } \\| = \\sup \\limits _ { t _ 0 \\leq s \\leq t \\leq t _ 0 + 1 } \\log ^ + \\| \\Psi _ \\omega ( s , t ) \\| \\end{align*}"}
-{"id": "7600.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { I _ 5 } _ { L ^ \\infty ( 0 , T ; C ^ { \\alpha , p } ) } \\le \\frac { C ( \\alpha ) } { \\nu } \\left ( 1 + t + \\left ( \\frac { t } { \\nu } \\right ) ^ 2 \\right ) \\left ( 1 + \\norm { X - \\mathrm { I d } } _ { L i p ( 0 , T ; C ^ { 1 + \\alpha } ) } \\right ) ^ 3 M _ X ^ { 1 + 2 \\alpha } \\\\ \\norm { X ' } _ { L ^ \\infty ( 0 , T ; C ^ { 1 + \\alpha } ) } \\norm { \\tau } _ { L i p ( 0 , T ; C ^ { \\alpha , p } ) } . \\end{gathered} \\end{align*}"}
-{"id": "3090.png", "formula": "\\begin{align*} m \\{ x : | T _ { 1 1 } g | > \\frac t 2 \\} \\le \\frac { C ( n ) } { t ^ 2 } \\| g \\| _ 2 ^ 2 \\le \\frac { C ( n ) } { t } \\| f \\| _ 1 . \\end{align*}"}
-{"id": "7365.png", "formula": "\\begin{align*} d ( b _ i , b _ j ) = d ( p , b _ j ) - d ( p , b _ i ) < r - ( r - \\frac { \\delta } { 2 ^ i } ) = \\frac { \\delta } { 2 ^ i } \\end{align*}"}
-{"id": "8900.png", "formula": "\\begin{align*} T _ { } & = \\frac { 1 } { 1 - \\eta } T ^ { \\hat x _ 1 } ( \\widetilde G ) - T ^ { \\hat x _ 1 } ( G ' ) \\shortintertext { a n d } T _ { } & = \\frac { 1 } { 1 - \\eta } T ^ { \\hat x _ 1 } ( \\widetilde G ) - T ^ { \\hat x _ 1 } ( H ) , \\end{align*}"}
-{"id": "1012.png", "formula": "\\begin{align*} \\tau _ k - x = ( F _ n ( \\tau _ k ) - F _ n ( x ) ) ( n / 4 ) ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } = ( m _ k - F _ n ( x ) ) ( n / 4 ) ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } , \\end{align*}"}
-{"id": "9913.png", "formula": "\\begin{align*} & \\lim _ { n \\to \\infty } \\sup _ { \\| g \\| \\leq 1 } \\\\ & \\left | \\int _ { \\mathcal { Y } } g ( y _ { n } ) P ^ { \\mu } ( d y _ { n } | y _ { [ 0 , n - 1 ] } ) - \\int _ { \\mathcal { Y } } g ( y _ { n } ) P ^ { \\nu } ( d y _ { n } | y _ { 0 , n - 1 ] } ) \\right | = 0 , \\end{align*}"}
-{"id": "3878.png", "formula": "\\begin{align*} Y ( I ) = ( C \\cap L ^ \\infty ) ( I , H ^ 1 ) \\cap L ^ 4 ( I , W ^ { 1 , \\infty } ) & d = 1 , \\\\ Y ( I ) = ( C \\cap L ^ \\infty ) ( I , H ^ 1 ) \\cap L ^ { q _ 0 } ( I , W ^ { 1 , { r _ 0 } } ) & d = 2 , \\end{align*}"}
-{"id": "8168.png", "formula": "\\begin{align*} \\begin{aligned} c _ { j + 1 } + b _ { j + 1 } + c _ { j } & \\leq k , \\\\ b _ { j + 1 } + a _ { j + 1 } + c _ { j } & \\leq k , \\\\ a _ { j + 1 } + c _ { j } + b _ { j } & \\leq k , \\\\ a _ { j + 1 } + b _ { j } + a _ { j } & \\leq k \\end{aligned} \\end{align*}"}
-{"id": "9372.png", "formula": "\\begin{align*} \\log _ 2 \\lvert D _ i ( G _ k ) \\rvert = \\begin{cases} \\log _ 2 \\lvert \\gamma _ i ( G _ k ) \\rvert + 1 & \\\\ \\log _ 2 \\lvert \\gamma _ i ( G _ k ) \\rvert + 2 & \\end{cases} \\end{align*}"}
-{"id": "7636.png", "formula": "\\begin{align*} G \\varphi ( \\xi ) \\ = \\ \\varphi ( \\xi ) \\ \\int \\limits _ { \\mathbb R ^ d } a ( \\xi - q ) \\mu ( \\xi , q ) \\ d q \\ = \\ \\varphi ( \\xi ) \\int \\limits _ { \\mathbb T ^ d } \\hat a ( \\xi - \\eta ) \\mu ( \\xi , \\eta ) \\ d \\eta , \\varphi \\in L ^ 2 ( \\mathbb T ^ d ) , \\end{align*}"}
-{"id": "8351.png", "formula": "\\begin{align*} A _ 1 ( \\alpha , t ) = & 1 + \\frac { \\lambda ^ 2 } { 4 \\pi ^ 2 } \\frac { 1 } { | \\alpha - z _ 1 | ^ 2 } \\frac { i } { - 2 i y } + \\frac { \\lambda ^ 2 } { 2 \\pi ^ 2 } \\frac { y } { | \\alpha - z _ 1 ( t ) | ^ 4 } . \\end{align*}"}
-{"id": "1244.png", "formula": "\\begin{align*} \\tilde S _ 1 ( \\eta , \\nu ) = S _ 1 \\left ( \\frac { \\eta + \\nu } { 2 } \\right ) S _ 1 \\left ( \\frac { \\eta - \\nu } { 2 } \\right ) , \\end{align*}"}
-{"id": "438.png", "formula": "\\begin{align*} \\dd \\langle Q ^ x , Q ^ y \\rangle _ t = A ^ x _ t A ^ y _ t Q _ t ( x , y ) \\dd t , \\end{align*}"}
-{"id": "9404.png", "formula": "\\begin{align*} \\theta ( f ) ( z ) : = \\frac { 1 } { 2 \\pi i } \\frac { d } { d z } f ( z ) . \\end{align*}"}
-{"id": "7691.png", "formula": "\\begin{align*} T \\psi _ { + } ^ \\mathsf { o } ( D ) = T \\bigl ( \\psi _ + - \\psi _ - \\bigr ) ( D ) = \\bigl ( \\psi _ - - \\psi _ + \\bigr ) ( D ) T = - \\psi _ { + } ^ \\mathsf { o } ( D ) T . \\end{align*}"}
-{"id": "4902.png", "formula": "\\begin{align*} ( b , c ) + ( a , 0 ) + ( a , 0 ) = ( b ' , c ' ) + ( a ' , 0 ) + ( a ' , 0 ) \\end{align*}"}
-{"id": "1518.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { q ^ { n } - 1 } \\min \\{ e , \\lfloor \\frac { \\log _ { q } k } { d } \\rfloor \\} = \\sum _ { k = 1 } ^ { q ^ { n } - 1 } \\min \\{ e , \\lfloor \\frac { k } { q ^ d } \\rfloor + \\lfloor \\frac { k } { q ^ { 2 d } } \\rfloor + \\cdots \\} . \\end{align*}"}
-{"id": "1488.png", "formula": "\\begin{align*} W ^ { j } _ { 1 } = \\int _ { \\mathbb { T } ^ { d } } \\left | \\nabla \\partial _ { x _ { j } } \\left ( w ^ { 1 } - w ^ { 2 } \\right ) \\right | ^ { 2 } \\ d x . \\end{align*}"}
-{"id": "7130.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l } \\displaystyle \\rho _ t + ( \\rho u ) _ x = 0 , \\ \\ & x \\in \\mathbb { R } , t > 0 , \\\\ \\displaystyle \\rho u _ t + \\rho u u _ x + p _ x = \\nu u _ { x x } - \\frac { \\epsilon } { 2 } \\big ( \\chi _ x ^ 2 \\big ) _ x , \\ \\ & x \\in \\mathbb { R } , t > 0 , \\\\ \\displaystyle \\rho \\chi _ t + \\rho u \\chi _ x = - \\frac { 1 } { \\epsilon } ( \\chi ^ 3 - \\chi ) + \\frac { \\epsilon } { \\rho } \\chi _ { x x } , \\ \\ & x \\in \\mathbb { R } , t > 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "1756.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } [ X Y ] & = ( t - s ) ^ { - H } ( r - u ) ^ { - H } \\left ( R ( t , r ) - R ( t , u ) - R ( s , r ) + R ( s , u ) \\right ) \\\\ & = \\frac { 1 } { 2 ( t - s ) ^ { H } ( u - r ) ^ { H } } \\left [ ( t - u ) ^ { 2 H } - ( t - r ) ^ { 2 H } - \\left ( ( s - u ) ^ { 2 H } - ( s - r ) ^ { 2 H } \\right ) \\right ] , \\end{aligned} \\end{align*}"}
-{"id": "5811.png", "formula": "\\begin{align*} 0 = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } f ( r e ^ { i \\theta } ) e ^ { i n \\theta } d \\theta \\end{align*}"}
-{"id": "7917.png", "formula": "\\begin{align*} R _ N = \\hat { R } _ n h ^ { n + 2 } M \\sum _ { i = 1 } ^ { M } \\frac { f ^ { ( n + 1 ) } ( \\xi _ i ) } { M } \\end{align*}"}
-{"id": "6045.png", "formula": "\\begin{align*} g _ k ( s ) \\leq f _ k ( s ) \\leq f _ k \\left ( k + \\tfrac { k + 2 } { 7 } \\right ) = \\tfrac { 9 6 } { 4 9 } k ^ 2 - \\tfrac { 3 6 } { 4 9 } k - \\tfrac { 1 5 } { 4 9 } , \\end{align*}"}
-{"id": "9032.png", "formula": "\\begin{align*} f _ { \\{ X \\} , \\{ x _ i \\} } ( A ) = \\begin{cases} \\{ x _ i \\} & A = X , \\\\ A & \\end{cases} \\end{align*}"}
-{"id": "6005.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u _ n ( t , x ) + \\frac { 1 } { 2 n } \\Delta u _ n ( t , x ) + g ^ * ( t , \\nabla u _ n ( t , x ) ) = 0 [ 0 , 1 ] \\times \\R ^ d \\\\ u _ n ( 1 , x ) = f ( x ) , x \\in \\R ^ d . \\end{cases} \\end{align*}"}
-{"id": "2457.png", "formula": "\\begin{align*} P ( R , t ) \\ = \\ \\frac { 1 } { 1 - t ^ d } \\left ( \\frac { 1 } { 1 - t ^ s } - p ( t ^ s ) \\right ) . \\end{align*}"}
-{"id": "528.png", "formula": "\\begin{align*} \\mathcal { Q } _ 2 ( x ' , F _ { 2 \\times 2 } ) = \\min \\Big \\{ \\mathcal { Q } _ 3 \\big ( \\bar A ( x ' ) ^ { - 1 } ( F _ { 2 \\times 2 } ^ * + c \\otimes e _ 3 ) \\bar A ( x ' ) ^ { - 1 } \\big ) ; ~ c \\in \\mathbb { R } ^ 3 \\Big \\} . \\end{align*}"}
-{"id": "2128.png", "formula": "\\begin{align*} \\Phi ( n r , \\omega ) = \\Phi ( r , \\theta _ { ( n - 1 ) r } \\omega ) \\circ \\Phi ( ( n - 1 ) r , \\omega ) , \\ \\forall n \\in \\Z . \\end{align*}"}
-{"id": "4626.png", "formula": "\\begin{align*} \\mathcal { R } _ { B D 2 } = \\begin{bmatrix} A _ 0 & & & A _ 2 & A _ 1 \\\\ A _ 1 & A _ 0 & & & A _ 2 \\\\ A _ 2 & A _ 1 & A _ 0 & & \\\\ & \\ddots & \\ddots & \\ddots & \\\\ & & A _ 2 & A _ 1 & A _ 0 \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "8431.png", "formula": "\\begin{align*} \\tilde { g } _ 1 = & ( I - \\mathfrak { H } ) \\partial _ t g _ c + ( I - \\mathfrak { H } ) \\partial _ t g _ d + ( I - \\mathfrak { H } ) i a _ t ( I - \\mathfrak { H } ) ( z - \\bar { z } ) _ { \\alpha } \\\\ : = & \\tilde { g } _ { 1 1 } + \\tilde { g } _ { 1 2 } + \\tilde { g } _ { 1 3 } . \\end{align*}"}
-{"id": "1268.png", "formula": "\\begin{align*} | f ( \\xi ) | = O ( | \\xi | ^ { - 1 / 2 } ) , | f ' ( \\xi ) | = O ( | \\xi | ^ { - 3 / 2 } ) . \\end{align*}"}
-{"id": "9985.png", "formula": "\\begin{align*} \\ell _ { E } ( B ( \\mathbf { p } ) ) = 0 , \\end{align*}"}
-{"id": "7743.png", "formula": "\\begin{align*} \\Big | [ \\langle x , y \\rangle \\xi , \\xi ] \\Big | = \\| y \\| . \\end{align*}"}
-{"id": "3602.png", "formula": "\\begin{align*} f _ { \\{ \\frac { 1 } { 3 2 } \\} } ( a d _ { \\hat { a } _ { \\pm i } } ) ( \\hat { a } _ j - \\hat { a } _ { - j } ) = f _ { \\{ 1 , 0 , \\frac { 1 } { 4 } \\} } ( a d _ { \\hat { a } _ { \\pm i } } ) ( \\hat { a } _ j + \\hat { a } _ { - j } ) = 0 \\end{align*}"}
-{"id": "5869.png", "formula": "\\begin{align*} q ( w _ i , w _ j ) = 0 , \\ \\ - n \\leq i \\neq - j \\leq n , \\ \\ q ( w _ i , w _ { - i } ) = 1 , \\ \\ 1 \\leq i \\leq n . \\end{align*}"}
-{"id": "9994.png", "formula": "\\begin{align*} b ^ i _ t = \\tilde { g } ^ { i k } \\partial _ x ^ { - 1 } \\frac { \\delta h } { \\delta b ^ k } = \\tilde { g } ^ { i k } \\left ( - \\frac { \\partial h } { \\partial b ^ k _ x } + \\partial _ x \\frac { \\partial h } { \\partial b ^ k _ { x x } } \\right ) = w ^ i _ j ( \\mathbf { b } _ x ) b ^ j _ { x x } . \\end{align*}"}
-{"id": "34.png", "formula": "\\begin{align*} \\lambda = \\sup _ { { \\pi } \\in \\bar { \\mathcal { A } } ^ { \\mathbb { G } } } \\limsup _ { T \\uparrow \\infty } \\frac { 1 } { T } \\ln \\mathbb { E } ^ { { \\mathbb { P } } ^ { { \\pi } } } \\left [ e ^ { \\int _ { 0 } ^ { T } L ^ { \\alpha _ { s - } } ( V _ { s } , { \\pi } _ { s } ) d s } \\right ] , \\end{align*}"}
-{"id": "2622.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { P } } _ { N } ( z ) & = \\frac { 1 } { ( \\pi \\sqrt { 1 - \\tau ^ 2 } ) ^ { N } \\prod _ { j = 1 } ^ { N } j ! } \\exp \\ ! \\Big \\{ - \\frac { 1 } { 1 - \\tau ^ 2 } \\sum _ { j = 1 } ^ { N } \\left ( | z _ { j } | ^ 2 - \\frac { \\tau } { 2 } \\big ( z _ { j } ^ { 2 } + \\overline { z } _ { j } ^ { 2 } \\big ) \\right ) \\Big \\} \\ , \\prod _ { j < k } | z _ k - z _ j | ^ 2 , \\end{align*}"}
-{"id": "6054.png", "formula": "\\begin{align*} \\alpha _ i = \\left \\{ \\begin{array} { l l } n , & \\mbox { i f } 1 \\leq i \\leq x - 1 \\\\ n - 2 , & \\mbox { i f } i = x \\\\ 3 , & \\mbox { i f } i = x + 1 \\\\ 0 , & \\mbox { i f } x + 2 \\leq i \\leq r . \\end{array} \\right . \\end{align*}"}
-{"id": "4797.png", "formula": "\\begin{align*} \\psi = \\eta \\in L ^ 2 ( | x | ^ { - 1 } ) ^ 4 , \\end{align*}"}
-{"id": "13.png", "formula": "\\begin{align*} \\delta Y _ t ^ i : = e ^ { - \\rho t } ( Y _ t ^ { i } - \\bar { Y } _ t ^ { i } ) \\ \\ \\ \\ \\delta Z _ t ^ i : = e ^ { - \\rho t } ( Z _ t ^ { i } - \\bar { Z } _ t ^ { i } ) . \\end{align*}"}
-{"id": "1519.png", "formula": "\\begin{align*} & U _ { 1 } ( n ) = \\{ g : \\ , \\textrm { $ g $ s q u a r e - f r e e , $ \\deg g = n $ } \\} , \\\\ & U _ { 2 } ( n ) = \\{ x ^ { 2 } g : \\ , \\textrm { $ g $ s q u a r e - f r e e , $ \\deg g = n - 2 , x \\nmid g $ } \\} , \\\\ & U _ { 3 } ( n ) = \\{ ( x + 1 ) ^ { 2 } g : \\ , \\textrm { $ g $ s q u a r e - f r e e , $ \\deg g = n - 2 , x + 1 \\nmid g $ } \\} , \\\\ & U _ { 4 } ( n ) = \\{ x ^ { 2 } ( x + 1 ) ^ { 2 } g : \\ , \\textrm { $ g $ s q u a r e - f r e e , $ \\deg g = n - 4 , x \\nmid g , x + 1 \\nmid g $ } \\} . \\end{align*}"}
-{"id": "4819.png", "formula": "\\begin{align*} \\int _ M R _ g d v = \\int _ { M \\setminus ( \\overline { D } \\times \\mathbb { S } ^ p ) } R _ g d v + \\int _ { ( \\overline { D } \\times \\mathbb { S } ^ p ) } R _ g d v . \\end{align*}"}
-{"id": "5650.png", "formula": "\\begin{align*} u ( t ) = E ( t ) u _ { 0 } + \\int _ { 0 } ^ { t } \\bar E ( t - s ) \\ , \\d W ( s ) , \\end{align*}"}
-{"id": "2096.png", "formula": "\\begin{align*} \\mathrm { T h r } = \\frac { { N p _ { \\rm o b s } ( t _ s ) } } { \\ln \\left ( 1 + \\frac { { N p _ { \\rm o b s } ( t _ s ) } } { \\sum _ { i = 1 } ^ { M } { N b _ i p _ { \\rm o b s } ( i T + t _ s ) } } \\right ) } . \\end{align*}"}
-{"id": "5305.png", "formula": "\\begin{align*} J = \\begin{bmatrix} I _ 4 & 0 \\\\ 0 & - I _ { 1 6 } \\end{bmatrix} . \\end{align*}"}
-{"id": "3014.png", "formula": "\\begin{align*} y _ n & = B _ n ( x _ 1 , x _ 2 , \\dots , x _ n ) , \\end{align*}"}
-{"id": "9650.png", "formula": "\\begin{align*} \\Bigl \\| \\sum _ { n = 1 } ^ N \\varphi _ n ( T ) x _ n \\Bigr \\| & = \\| A ^ \\ast \\bigl ( \\oplus _ { n = 1 } ^ N x _ n \\bigr ) \\| \\\\ & \\leq K C _ { { \\rm p o l } , T } \\mathop { } _ { \\zeta \\in \\mathbb T } \\Bigr ( \\sum _ { n = 1 } ^ N | \\varphi _ n ( \\zeta ) | ^ 2 \\Bigr ) ^ { 1 / 2 } \\Bigr ( \\sum _ { n = 1 } ^ N \\| x _ n \\| ^ 2 \\Bigr ) ^ { 1 / 2 } \\\\ & \\leq K C _ { { \\rm p o l } , T } a ^ { 1 / 2 } \\Bigr ( \\sum _ { n = 1 } ^ N \\| x _ n \\| ^ 2 \\Bigr ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "8486.png", "formula": "\\begin{align*} \\mathcal { T } : = \\Big \\{ T \\in [ 0 , T _ 0 ] | \\dot { y } ( t ) \\leq - \\frac { | \\lambda | } { 2 0 \\pi x ( 0 ) } , \\frac { 1 } { 2 } \\leq \\frac { x ( t ) } { x ( 0 ) } \\leq 2 , \\hat { d } _ I ( t ) \\geq 1 + \\frac { | \\lambda | } { 2 0 \\pi x ( 0 ) } t , \\forall ~ t \\in [ 0 , T ] \\Big \\} . \\end{align*}"}
-{"id": "6800.png", "formula": "\\begin{align*} \\mathcal { R } _ 2 \\quad \\equiv \\begin{cases} f _ 1 = y ^ 1 _ { 1 0 } y ^ 2 _ { 0 1 } - y ^ 1 _ { 0 1 } y ^ 2 _ { 1 0 } - 1 = 0 \\ , , \\\\ f _ 2 = y ^ 1 _ { 2 0 } y ^ 2 _ { 0 1 } + y ^ 1 _ { 1 0 } y ^ 2 _ { 1 1 } - y ^ 1 _ { 1 1 } y ^ 2 _ { 1 0 } - y ^ 1 _ { 0 1 } y ^ 2 _ { 2 0 } = 0 \\ , , \\\\ f _ 3 = y ^ 1 _ { 1 1 } y ^ 2 _ { 0 1 } + y ^ 1 _ { 1 0 } y ^ 2 _ { 0 2 } - y ^ 1 _ { 0 2 } y ^ 2 _ { 1 0 } - y ^ 1 _ { 0 1 } y ^ 2 _ { 1 1 } = 0 \\ , , \\\\ f _ 4 = y ^ 1 _ { 2 0 } + y ^ 1 _ { 0 2 } = 0 \\ , , \\\\ f _ 5 = y ^ 2 _ { 2 0 } + y ^ 2 _ { 0 2 } = 0 \\ , , \\end{cases} \\end{align*}"}
-{"id": "6027.png", "formula": "\\begin{align*} \\tilde Q _ { \\epsilon , \\delta } ^ 0 = \\mu \\tilde Q _ { \\epsilon , \\delta } ^ 1 = \\nu . \\end{align*}"}
-{"id": "7878.png", "formula": "\\begin{align*} \\frac { 1 } { m ! } \\sum _ { \\sigma \\in { \\rm S y m } ( I ) } \\prod _ { j \\in I } \\cos ( 2 \\pi y _ { \\sigma ^ { - 1 } ( j ) } \\xi _ j ) = \\sum _ { S \\subseteq I } a _ S ( \\xi ) \\Bbbk _ { l } ^ { ( m ) } ( | S | ) . \\end{align*}"}
-{"id": "5347.png", "formula": "\\begin{align*} H _ { \\alpha } = \\left [ \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & - 1 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right ] \\quad \\mbox { a n d } \\quad H _ { \\beta } = \\left [ \\begin{array} { c c c } 0 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & - 1 \\end{array} \\right ] . \\end{align*}"}
-{"id": "8787.png", "formula": "\\begin{align*} \\partial _ t u ( t , x ) = \\partial _ { x x } u ( t , x ) - u ( t , x ) + \\dot L ( t , x ) , ( t , x ) \\in [ 0 , T ] \\times [ 0 , \\pi ] , \\end{align*}"}
-{"id": "9054.png", "formula": "\\begin{align*} r _ { \\mu } ( x - y ) = \\int _ { \\mathbb { S } ^ 1 } e ( \\langle x - y , t \\rangle ) d \\mu ( t ) \\end{align*}"}
-{"id": "6193.png", "formula": "\\begin{align*} N ( x _ 0 , r ) : = \\frac { E ( x _ 0 , r ) } { H ( x _ 0 , r ) } \\end{align*}"}
-{"id": "4479.png", "formula": "\\begin{align*} \\nabla _ { i i } H = \\Delta ( v _ { i i } ) - n v _ { i i } + H . \\end{align*}"}
-{"id": "4075.png", "formula": "\\begin{align*} Y ( z ) = \\mathcal { O } \\begin{pmatrix} 1 & h _ { \\alpha } ( z ) & h _ { \\alpha + \\frac { 1 } { 2 } } ( z ) \\\\ 1 & h _ { \\alpha } ( z ) & h _ { \\alpha + \\frac { 1 } { 2 } } ( z ) \\\\ 1 & h _ { \\alpha } ( z ) & h _ { \\alpha + \\frac { 1 } { 2 } } ( z ) \\end{pmatrix} \\ , h _ { \\alpha } ( z ) = \\begin{cases} | z | ^ { \\alpha } , & \\alpha < 0 , \\\\ \\log { | z | } , & \\alpha = 0 , \\\\ 1 , & \\alpha > 0 . \\end{cases} \\end{align*}"}
-{"id": "600.png", "formula": "\\begin{align*} X _ n : = \\big \\{ g \\in B V \\cap L ^ { \\infty } \\ , \\big | \\ , \\| g \\| _ { L ^ { \\infty } } \\leq \\log n \\big \\} . \\end{align*}"}
-{"id": "7227.png", "formula": "\\begin{align*} ( \\mu - \\phi ) \\phi ^ { \\prime \\prime } ( 0 ) = L _ r \\phi ^ { \\prime \\prime } ( 0 ) . \\end{align*}"}
-{"id": "5551.png", "formula": "\\begin{align*} f = \\sum _ { - 1 \\leqslant j \\leqslant N - J } \\mathcal { F } ^ { - 1 } ( \\mathcal { F } _ { 2 ^ { - j - J } \\mathbb { Z } ^ d } ( \\lambda _ { j , \\cdot } ) ) , \\end{align*}"}
-{"id": "843.png", "formula": "\\begin{align*} f ( t ) = \\sum _ { n = 1 } ^ { \\infty } f ( t _ n ) \\frac { S _ { n } ( t ) } { a _ n } \\textrm { w i t h } S _ { n } ( t ) = \\langle y _ n , \\Phi ( t ) \\rangle t \\in \\Omega \\end{align*}"}
-{"id": "7065.png", "formula": "\\begin{align*} T ( b ) = \\oplus _ { a \\in \\L } T _ { a b } \\ , a , b \\in \\L . \\end{align*}"}
-{"id": "8962.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\P ( \\mbox { M L E e x i s t s } ) = 0 . \\end{align*}"}
-{"id": "2355.png", "formula": "\\begin{align*} \\sigma & = \\{ c _ s : s \\in \\mathfrak { p } _ + \\} , & S = & \\{ ( c _ { q _ i } , c _ { r _ i } ) : i \\in I \\} , & \\tau & = \\{ c _ t : t \\in \\mathfrak { p } _ - \\} \\end{align*}"}
-{"id": "5574.png", "formula": "\\begin{align*} & \\partial _ x ^ j u \\in \\mathcal { C } ( \\R _ x ^ + ; H ^ { ( 2 s + 1 - 2 j ) / 4 } ( 0 , T ) ) j = 0 , 1 , \\\\ & \\partial _ x ^ j v \\in \\mathcal { C } ( \\R _ x ^ + ; H ^ { ( \\kappa + 1 - j ) / 4 } ( 0 , T ) ) j = 0 , 1 , 2 . \\end{align*}"}
-{"id": "6241.png", "formula": "\\begin{align*} \\gamma : = \\sum _ { i = 1 } ^ n \\frac { 1 - \\beta } { \\alpha _ i } \\geq \\frac { 1 - \\beta } { \\alpha _ n } \\geq \\frac { \\beta _ m } { \\alpha _ n } \\geq 1 . \\end{align*}"}
-{"id": "3871.png", "formula": "\\begin{align*} g = \\left ( \\begin{matrix} a & b \\\\ c & d \\end{matrix} \\right ) \\end{align*}"}
-{"id": "79.png", "formula": "\\begin{align*} D = [ \\inf _ I { f _ 0 ^ - } , \\sup _ I { f _ 0 ^ - } ] \\times [ \\inf _ I { f _ 0 ^ + } , \\sup _ I { f _ 0 ^ + } ] \\ , , \\end{align*}"}
-{"id": "3540.png", "formula": "\\begin{align*} C _ { N E } = { \\rm C o n e } \\left \\{ \\begin{matrix} ( 1 , 0 , 0 , 0 ) , ( 0 , 1 , - 1 , 1 ) , ( 1 , - 1 , 1 , 0 ) , ( 0 , 0 , 0 , 1 ) \\\\ ( 0 , 0 , 1 , - 1 ) , ( - 1 , 1 , 0 , 0 ) \\end{matrix} \\right \\} , \\end{align*}"}
-{"id": "6874.png", "formula": "\\begin{align*} \\mathsf { d i m } \\big ( \\mathsf { V } ( \\sqrt [ \\mathbb { R } ] { I _ { 3 , 8 } } ) \\big ) = 5 \\ , . \\end{align*}"}
-{"id": "8432.png", "formula": "\\begin{align*} \\kappa _ t = b \\circ \\kappa . \\end{align*}"}
-{"id": "6210.png", "formula": "\\begin{align*} I _ V ' ( t ) = \\frac { \\sqrt { 2 } } { \\pi } \\int _ { 0 } ^ { \\infty } \\left [ L _ V ( u ) - e ^ { - t u } \\right ] e ^ { - t u } \\sqrt { u } d u . \\end{align*}"}
-{"id": "5308.png", "formula": "\\begin{align*} i = 1 : & - 2 ( H _ { q _ 1 } - E _ { 1 , 5 } ) - 2 ( H _ { p _ 1 } - E _ { 1 , 7 } ) - 3 E _ { 1 - 2 } - 2 E _ { 2 - 3 } - E _ { 3 - 4 } \\\\ & - E _ { 5 - 6 } - E _ { 7 - 8 } = - 2 H _ { q _ 1 } - 2 H _ { p _ 1 } + E _ { 1 , \\dots , 8 } , \\\\ i = 2 : & q _ 1 \\leftrightarrow q _ 2 , p _ 1 \\leftrightarrow p _ 2 , E _ j \\leftrightarrow E _ { j + 8 } \\ ( j = 1 , \\dots , 8 ) \\mbox { i n t h e a b o v e } , \\end{align*}"}
-{"id": "1691.png", "formula": "\\begin{align*} \\begin{aligned} s _ b ( N _ k ^ { n } ) & = \\frac { n + 1 } { 2 } ( b - 1 ) ( 2 ^ { k } - 1 ) + \\frac { n + 1 } { 2 } b - 2 ^ { n - 1 } + 2 ^ { n - 1 } - \\frac { n + 1 } { 2 } \\\\ & = \\frac { n + 1 } { 2 } ( b - 1 ) 2 ^ { k } . \\end{aligned} \\end{align*}"}
-{"id": "6274.png", "formula": "\\begin{align*} f = \\sum _ { j = 0 } ^ { N - 1 } f _ j \\mathbf { 1 } _ { [ t _ j , t _ { j + 1 } ) } , \\end{align*}"}
-{"id": "7549.png", "formula": "\\begin{align*} \\norm { f } _ { C ^ { 1 + \\alpha } ( \\mathbb { R } ^ d ) } = \\norm { f } _ { L ^ \\infty ( \\mathbb { R } ^ d ) } + \\norm { \\nabla f } _ { C ^ { \\alpha } ( \\mathbb { R } ^ d ) } , \\end{align*}"}
-{"id": "6382.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( A \\subseteq Y \\right ) = \\det ( K _ A ) , A \\subseteq V , A \\neq \\emptyset , \\end{align*}"}
-{"id": "6070.png", "formula": "\\begin{align*} \\mathcal A ' = h \\mathcal A . \\end{align*}"}
-{"id": "9466.png", "formula": "\\begin{align*} W ^ { s , m } ( \\Omega ) & = \\{ u \\in L ^ { m } ( \\Omega ) \\ ; \\ , \\ , [ u ] _ { s , m , \\Omega } < \\infty \\} , \\end{align*}"}
-{"id": "4385.png", "formula": "\\begin{align*} \\Upsilon ^ Z _ r : = d d ^ c \\lambda ^ T _ r = \\frac { n ! } { ( \\pi r ^ 2 ) ^ n } \\mathbf { 1 } _ { B _ r ( 0 ) } * [ Z ] , \\end{align*}"}
-{"id": "6273.png", "formula": "\\begin{align*} \\sigma ^ 2 _ H = ( 4 H - 1 ) + \\frac { 2 \\Gamma ( 2 - 4 H ) \\Gamma ( 4 H ) } { \\Gamma ( 2 H ) \\Gamma ( 1 - 2 H ) } . \\end{align*}"}
-{"id": "3093.png", "formula": "\\begin{align*} T _ 1 f = & T _ 1 g + T _ 1 b \\mathrm { I } _ { G ^ * } + T _ 1 b \\mathrm { I } _ { F ^ * } \\\\ = & ( T _ 1 g + T _ 1 b \\mathrm { I } _ { F ^ * } ) + T _ 1 b \\mathrm { I } _ { G ^ * } \\\\ \\equiv & T _ { 1 1 } f + T _ { 1 2 } f , \\end{align*}"}
-{"id": "6336.png", "formula": "\\begin{align*} \\sigma ^ 2 _ { R } = 1 . 2 3 C _ n ^ 2 \\kappa ^ { 7 / 6 } d ^ { 1 1 / 6 } , \\end{align*}"}
-{"id": "2691.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\phi } \\Re { f _ { \\tau , \\theta } \\Big ( \\tau + \\frac { 1 } { 2 \\cos \\theta } \\Big ( \\frac { 1 } { \\tau } - \\tau \\Big ) e ^ { i \\phi } \\Big ) } & = \\frac { 8 \\cos ^ { 2 } \\theta \\sin \\phi ( \\cos \\phi - \\cos \\theta ) } { 4 \\cos ^ { 2 } \\theta - 4 \\cos \\theta \\cos \\phi + 1 } . \\end{align*}"}
-{"id": "9012.png", "formula": "\\begin{align*} f _ { B , b } ( a _ 1 , . . . , a _ n ) = \\begin{cases} b & B = \\{ a _ 1 , . . . , a _ n \\} , \\\\ a _ 1 & \\end{cases} \\end{align*}"}
-{"id": "6236.png", "formula": "\\begin{align*} \\phi ( p _ 1 ) = \\alpha \\phi ( p _ 2 ) = 1 - \\alpha . \\end{align*}"}
-{"id": "8647.png", "formula": "\\begin{align*} \\frac { 2 } { \\varepsilon ^ 2 } \\hat { H } = & \\varepsilon ^ 2 w _ 2 ^ 2 + \\frac { 1 } { 4 } ( 1 - \\varepsilon ^ 2 w _ 1 ) ^ 2 \\varphi _ 2 ^ 2 \\\\ & - \\frac { \\varepsilon ^ 2 } { 2 } ( 2 w _ 1 - \\varepsilon ^ 2 w _ 1 ^ 2 ) ^ 2 - \\left [ \\frac { ( 1 - \\varepsilon ^ 2 w _ 1 ) ^ 2 } { 2 } \\sin \\varphi _ 1 - c _ 0 - \\tilde { \\omega } ( \\varepsilon ) \\right ] ^ 2 \\end{align*}"}
-{"id": "5042.png", "formula": "\\begin{align*} A _ n ( x , y ) = p _ n ( x ) ^ \\top w _ n ( y - x ) . \\end{align*}"}
-{"id": "4714.png", "formula": "\\begin{align*} E ( u , \\phi ) : = - \\int _ 0 ^ T \\int _ \\mathbb { R } \\{ u ( x , t ) \\phi _ t ( x , t ) + f ( u ) \\phi _ { x } ( x , t ) \\} d x d t + \\int _ \\mathbb { R } u ( x , 0 ) \\phi ( x , 0 ) d x = 0 , \\end{align*}"}
-{"id": "6295.png", "formula": "\\begin{align*} \\binom { N } { N p } & = \\Theta \\left ( \\frac { ( \\frac { N } { e } ) ^ { N } \\sqrt { 2 \\pi N } } { \\left [ ( \\frac { N p } { e } ) ^ { N p } \\sqrt { 2 \\pi N p } ( \\frac { N ( 1 - p ) } { e } ) ^ { N ( 1 - p ) } \\sqrt { 2 \\pi N ( 1 - p ) } \\right ] } \\right ) \\\\ & = \\Theta \\left ( \\frac { 1 } { 2 \\sqrt { N p ( 1 - p ) } } \\exp \\left ( - N \\left ( p \\log p + ( 1 - p ) \\log ( 1 - p ) \\right ) \\right ) \\right ) \\\\ & = \\Omega \\left ( \\frac { 1 } { \\sqrt { N } } \\right ) \\exp ( N H _ 2 ( p ) ) . \\end{align*}"}
-{"id": "4910.png", "formula": "\\begin{align*} u '' = - R ( u , w ) w + \\nabla _ { \\nabla _ w w } u - \\nabla _ w \\nabla _ w u \\ , . \\end{align*}"}
-{"id": "9742.png", "formula": "\\begin{align*} \\mathcal { N } _ \\ast : = \\mathcal { N } ( p _ \\ast ) \\end{align*}"}
-{"id": "5836.png", "formula": "\\begin{align*} { \\cal F } f ( x ) = \\int _ F f ( y ) \\psi ( x y ) d y \\end{align*}"}
-{"id": "8103.png", "formula": "\\begin{align*} 8 k \\ , d _ k = \\sum _ { j = 0 } ^ { k - 1 } \\alpha _ { j , k } \\ , d _ j . \\end{align*}"}
-{"id": "4014.png", "formula": "\\begin{align*} \\alpha _ { 1 , 1 } b _ 1 ( n ) = \\sum _ t \\beta _ { t } a _ H ( \\gamma _ { t } n ) . \\end{align*}"}
-{"id": "552.png", "formula": "\\begin{align*} \\Box \\phi = \\phi ( | \\nabla \\phi | ^ 2 - | \\partial _ t \\phi | ^ 2 ) \\end{align*}"}
-{"id": "1072.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to + \\infty } \\left ( \\ln D _ n ( \\alpha _ n ) + \\ln n + x + 2 z - \\frac { \\ln ( 2 \\ln n ) } { 2 } - c _ 0 \\right ) = 0 , \\end{align*}"}
-{"id": "3543.png", "formula": "\\begin{align*} \\begin{aligned} & \\sigma _ { i } ^ { ( 1 ) } = { \\rm C o n e } \\left \\{ \\rho _ { 1 } , \\rho _ { 2 } , \\rho _ { 3 } , \\rho _ { 6 - i } \\right \\} \\ ; ( i = 1 , 2 ) \\\\ & \\sigma _ { i } ^ { ( 2 ) } { \\rm = C o n e } \\left \\{ \\rho _ { j } , \\rho _ { k } , \\rho _ { 4 } , \\rho _ { 5 } \\right \\} \\ ; \\ ; ( \\{ i , j , k \\} = \\{ 1 , 2 , 3 \\} ) . \\end{aligned} \\end{align*}"}
-{"id": "2390.png", "formula": "\\begin{align*} m _ i \\in L ( \\mathfrak { C } ) : m _ i ( f ) & = z _ i f & d _ i ( f ) & = \\sum _ { i < l } H _ { i l } \\frac { \\partial f } { \\partial z _ l } . \\end{align*}"}
-{"id": "7902.png", "formula": "\\begin{align*} \\phi ^ n _ k ( x ) = \\prod _ { \\substack { i = - \\hat { n } \\\\ i \\neq k } } ^ { \\hat { n } } \\frac { x - x _ i } { x _ k - x _ i } = \\prod _ { \\substack { i = - \\hat { n } \\\\ i \\neq k } } ^ { \\hat { n } } \\frac { x - i h } { ( k - i ) h } \\end{align*}"}
-{"id": "6791.png", "formula": "\\begin{align*} \\mathsf { V } ( I ) = \\big \\{ a \\in \\mathbb { L } ^ { n } \\colon f ( a ) = 0 \\ { \\rm f o r \\ a l l \\ } ~ f \\in I \\big \\} \\subset \\mathbb { L } ^ { n } . \\end{align*}"}
-{"id": "9786.png", "formula": "\\begin{align*} f ( X ) : = \\begin{cases} N ( 0 ^ + , u ( X + \\ , \\cdot \\ , ) - p _ { * , X } ) & X \\in \\R ^ n \\times \\{ 0 \\} \\\\ 0 & \\textrm { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "2266.png", "formula": "\\begin{align*} M ( x + \\i y ) & = \\mu ( x ) \\delta ( y ) \\\\ \\mu ( x ) & = \\frac { 1 } { 2 \\pi \\i } ( R ( x - \\i 0 ) - R ( x + \\i 0 ) ) = | \\alpha _ x \\rangle \\langle \\alpha _ x | \\\\ | \\alpha _ x \\rangle & = ( 1 + \\pi ^ 2 ) ^ { - 1 / 2 } \\left ( \\frac { \\mathcal { P } } { x - \\Omega } | E \\rangle + | \\delta _ x \\rangle \\right ) \\end{align*}"}
-{"id": "7955.png", "formula": "\\begin{align*} \\sum _ { a \\in [ n ] } { x _ { a , 1 } \\otimes x _ { a , 2 } \\otimes x _ { a , 3 } } = 0 . \\end{align*}"}
-{"id": "4922.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { n \\rightarrow \\infty } \\dfrac { \\ell ( R / I _ n ) } { \\ell ( R / ( I _ n + P ) ) } = 1 . \\end{align*}"}
-{"id": "6846.png", "formula": "\\begin{align*} I _ 2 = I _ 1 + \\langle a _ { 2 1 } '' a _ { 2 2 } - a _ { 2 2 } '' a _ { 2 1 } + a _ { 1 1 } '' a _ { 1 2 } - a _ { 1 2 } '' a _ { 1 1 } \\rangle \\end{align*}"}
-{"id": "2966.png", "formula": "\\begin{align*} H ( z , w ) = \\left ( \\frac { 1 } { 2 \\pi i } \\right ) ^ { 2 } \\int _ { \\sigma _ { r } } \\int _ { \\sigma _ { s } } \\sum _ { m , n = 0 } ^ { \\infty } \\frac { z ^ { m } } { u ^ { m + 1 } } \\frac { w ^ { n } } { v ^ { n + 1 } } H ( u , v ) d u d v ( z \\in \\Delta _ { s } , w \\in \\Delta _ { r } , v \\in \\sigma _ { r } , u \\in \\sigma _ { s } ) \\end{align*}"}
-{"id": "7717.png", "formula": "\\begin{align*} \\gamma \\colon \\Gamma _ { c } ^ \\infty ( M ; S ) \\to \\Gamma _ { c } ^ \\infty ( M ; S ) , \\gamma u ( p ) : = \\gamma _ S \\bigl ( u ( p ) \\bigr ) , \\end{align*}"}
-{"id": "9248.png", "formula": "\\begin{align*} \\zeta = 0 \\ { \\rm o n } \\ \\partial B _ R ( x _ 0 ) \\cap S _ \\zeta , D _ \\beta \\zeta = 0 \\ { \\rm o n } \\ \\partial \\Omega \\cap S _ \\zeta , \\end{align*}"}
-{"id": "4480.png", "formula": "\\begin{align*} \\tilde F ^ { i i } H _ { i i } = \\tilde F ^ { i i } \\Delta ( v _ { i i } ) - n \\tilde F ^ { i i } v _ { i i } + H \\sum _ i \\tilde F ^ { i i } \\leq 0 . \\end{align*}"}
-{"id": "5604.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { - \\infty } ^ 0 | x | | u | ^ 2 d x & = 2 \\ , \\int _ { - \\infty } ^ 0 u \\bar { u } _ x d x \\\\ & = \\mathcal { Q } _ 0 ^ - + 2 \\int _ 0 ^ t | u _ x ( 0 , s ) | ^ 2 d s - \\frac { \\alpha } { \\gamma } \\int _ { - \\infty } ^ 0 v ^ 2 d x . \\end{align*}"}
-{"id": "6017.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\rho ^ { g _ n } \\left ( F \\left ( \\frac { W } { \\sqrt { n } } \\right ) \\right ) = \\lim _ { n \\rightarrow \\infty } \\ , \\rho ^ { G _ n } \\left ( F \\left ( \\frac 1 n \\sum _ { k = 1 } ^ n { W _ { ( n , k ) } } \\right ) \\right ) . \\end{align*}"}
-{"id": "353.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - a \\left ( \\int _ { \\Omega } u ^ { p } d x \\right ) \\Delta u = f ( u ) & \\mbox { i n $ \\Omega $ , } \\\\ u > 0 & \\mbox { i n $ \\Omega $ , } \\\\ u = 0 & \\mbox { o n $ \\partial \\Omega $ , } \\end{array} \\right . \\end{align*}"}
-{"id": "5068.png", "formula": "\\begin{align*} \\| I _ \\mathcal { H } - S _ { B , \\Psi } S _ { A , \\Psi } ^ { - 1 } \\| & = \\left \\| \\sum _ { j \\in \\mathbb { J } } A _ j ^ * \\Psi _ j S _ { A , \\Psi } ^ { - 1 } - \\sum _ { j \\in \\mathbb { J } } B _ j ^ * \\Psi _ j S _ { A , \\Psi } ^ { - 1 } \\right \\| = \\left \\| \\sum _ { j \\in \\mathbb { J } } ( A _ j ^ * - B _ j ^ * ) \\Psi _ j S _ { A , \\Psi } ^ { - 1 } \\right \\| \\\\ & \\leq \\sum _ { j \\in \\mathbb { J } } \\| A _ j - B _ j \\| \\| \\Psi _ j S _ { A , \\Psi } ^ { - 1 } \\| = \\beta < 1 . \\end{align*}"}
-{"id": "3852.png", "formula": "\\begin{align*} P _ 1 & = \\langle t _ { 1 3 } \\rangle , \\\\ P _ 2 & = \\langle t _ { 8 } - t _ { 1 2 } , t _ { 6 } - t _ { 1 0 } , t _ { 4 } + t _ { 5 } - t _ { 1 0 } - t _ { 1 1 } , t _ { 3 } - t _ { 1 1 } , t _ { 2 } + t _ { 7 } - t _ { 9 } - t _ { 1 2 } , t _ { 1 } - t _ { 9 } , \\\\ & t _ { 5 } t _ { 7 } - t _ { 5 } t _ { 9 } - t _ { 7 } t _ { 1 1 } + t _ { 9 } t _ { 1 0 } + t _ { 9 } t _ { 1 1 } - t _ { 9 } t _ { 1 2 } - t _ { 1 0 } t _ { 1 1 } + t _ { 1 1 } t _ { 1 2 } \\rangle . \\end{align*}"}
-{"id": "8531.png", "formula": "\\begin{align*} \\| G _ { d 4 } \\| _ { H ^ s } \\leq & 4 \\| D _ t \\zeta \\| _ { H ^ s } \\norm { \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j } { 2 \\pi ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } } _ { H ^ s } + 4 \\norm { \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j \\dot { z } _ j } { 2 \\pi ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } } _ { H ^ s } \\\\ \\leq & C \\epsilon ^ 2 d _ I ( t ) ^ { - 5 / 2 } + K _ s ^ { - 1 } \\epsilon \\frac { | \\lambda | } { x ( 0 ) } d _ I ( t ) ^ { - 5 / 2 } . \\end{align*}"}
-{"id": "641.png", "formula": "\\begin{align*} \\gamma \\left ( ( \\varphi , [ u ] , [ u ' x ] ) , ( \\varphi , [ u ' ] , [ x ] ) \\right ) : = ( \\varphi , [ u u ' ] , [ x ] ) \\ , \\iota ( \\varphi , [ x ] ) : = ( \\varphi , [ e ] , [ x ] ) \\ , \\end{align*}"}
-{"id": "5691.png", "formula": "\\begin{align*} h _ 0 \\le \\bar h : = \\min \\left \\{ \\bar \\delta , \\ , \\frac { \\delta } { 2 ( \\eta + \\| v _ 0 \\| ) } \\right \\} \\ , . \\end{align*}"}
-{"id": "2459.png", "formula": "\\begin{align*} P _ 1 = [ 1 : 0 : 1 ] , P _ 2 = [ \\theta : 0 : 1 ] , P _ 3 = [ \\theta ^ 2 : 0 : 1 ] , \\end{align*}"}
-{"id": "4159.png", "formula": "\\begin{align*} T _ { \\alpha } \\Phi _ { \\alpha } ( n ^ 3 f ( z ) ) D _ 0 ^ n ( z ) = \\frac { 2 \\pi } { \\sqrt { 3 } } \\left ( n ^ 3 f ( z ) \\right ) ^ { - \\frac { 2 \\beta } { 3 } } \\left ( \\mathbb { I } + \\frac { A _ { \\alpha } } { n ^ 3 f ( z ) } + O \\left ( \\frac { 1 } { n ^ 6 f ( z ) ^ 2 } \\right ) \\right ) L _ { \\alpha } ( n ^ 3 f ( z ) ) D _ 2 ^ n ( z ) \\end{align*}"}
-{"id": "1132.png", "formula": "\\begin{align*} \\det ( - B ) = \\prod _ { j = 1 } ^ k \\det ( - B _ j ) = \\prod _ { j = 1 } ^ k D _ n \\left ( a _ j \\sqrt { 4 - y _ j ^ 2 } / 2 \\right ) . \\end{align*}"}
-{"id": "1894.png", "formula": "\\begin{align*} u ^ \\mu ( \\tau ) = \\frac { 1 } { \\sqrt { 1 - v ^ 2 } } ( 1 , v ) \\end{align*}"}
-{"id": "8784.png", "formula": "\\begin{align*} f ( x , y ) = ( x - y ) ^ 2 + x = a _ 2 ( b _ 1 x + b _ 0 + c _ 1 y + c _ 0 ) ^ 2 + a _ 1 ( b _ 1 x + b _ 0 + c _ 1 y + c _ 0 ) + a _ 0 . \\end{align*}"}
-{"id": "4031.png", "formula": "\\begin{align*} \\dots \\rightarrow \\pi _ 1 ( \\mathbb { S } ^ 1 ) \\cong \\Z \\stackrel { \\alpha } { \\rightarrow } \\pi _ 1 ( S ) \\stackrel { \\beta } { \\rightarrow } \\pi _ 1 ( M ) \\rightarrow \\pi _ 0 ( \\mathbb { S } ^ 1 ) = \\{ 0 \\} \\rightarrow \\cdots \\end{align*}"}
-{"id": "6522.png", "formula": "\\begin{align*} ( \\pi _ \\Lambda ) _ * \\left ( \\widehat { T } ^ { k } \\wedge [ \\Gamma _ a ] \\right ) & = ( \\pi _ \\Lambda ) _ * \\left ( u \\cdot ( d d ^ c _ { \\lambda , z } v ( z ) ) ^ { k } \\wedge [ \\Gamma _ a ] \\right ) \\\\ & = u ( \\lambda , a ( \\lambda ) ) \\left ( d d ^ c _ { \\lambda } ( v \\circ a ( \\lambda ) ) \\right ) ^ { k } . \\end{align*}"}
-{"id": "5859.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i = 1 } ^ k \\sum _ { j = 1 } ^ { d _ i } f ( \\alpha _ { i j } ) \\frac { x ( x - 1 ) \\prod _ { ( k , l ) \\neq ( i , j ) } ( x - \\alpha _ { k l } ) } { \\alpha _ { i j } ( \\alpha _ { i j } - 1 ) \\prod _ { ( k , l ) \\neq ( i , j ) } ( \\alpha _ { i j } - \\alpha _ { k l } ) } + f ( 0 ) \\frac { ( x - 1 ) G ( x ) } { S ' ( 0 ) } + f ( 1 ) \\frac { x G ( x ) } { S ' ( 1 ) } . \\end{align*}"}
-{"id": "953.png", "formula": "\\begin{align*} \\| s _ j \\| \\ ; \\leq \\ ; \\frac { 2 \\| J _ { m _ j } ^ \\top r _ { m _ j } \\| } { \\gamma _ j } = \\frac { 2 } { \\mu _ j } , \\end{align*}"}
-{"id": "2464.png", "formula": "\\begin{align*} 3 + \\deg ( y - z ) > \\deg y > \\deg x > \\deg w = 3 . \\end{align*}"}
-{"id": "762.png", "formula": "\\begin{align*} r _ { } ( \\mu ) = \\sum _ { j = 1 } ^ J \\alpha _ j R ^ * q _ j ( \\mu ) \\end{align*}"}
-{"id": "7241.png", "formula": "\\begin{align*} ( \\mu - \\phi ( \\pi ) ) \\phi ^ \\prime ( x ) & \\geq ( \\mu - \\phi ( x ) ) \\phi ^ \\prime ( x ) \\\\ & = \\int _ { - \\pi } ^ { 0 } \\left ( K _ r ( x - y ) - K _ r ( x + y ) \\right ) \\phi ^ \\prime ( y ) \\ , d y \\\\ & \\geq \\int _ { - \\frac { 3 } { 4 } \\pi } ^ { - \\frac { 1 } { 4 } \\pi } \\left ( K _ r ( x - y ) - K _ r ( x + y ) \\right ) \\phi ^ \\prime ( y ) \\ , d y , \\end{align*}"}
-{"id": "4506.png", "formula": "\\begin{align*} \\varsigma _ X ( x ) = \\varsigma _ Q ( \\langle x , x \\rangle ) = \\varsigma _ Q ( \\langle x , 1 _ X \\rangle ) \\ ; . \\end{align*}"}
-{"id": "740.png", "formula": "\\begin{align*} T = 1 - \\sum _ { i = 1 } ^ { m } \\lambda _ { i } ^ { 2 } , \\end{align*}"}
-{"id": "140.png", "formula": "\\begin{align*} \\delta _ { \\pi } ( X ) = [ \\pi , X ] \\end{align*}"}
-{"id": "1910.png", "formula": "\\begin{align*} \\mbox { a ) } | x | - t = | x - t | \\mbox { a n d } | x | + t = | x + t | , \\end{align*}"}
-{"id": "6385.png", "formula": "\\begin{align*} q _ A : = \\mathbb { P } ( X _ A = 1 _ A ) = \\det ( K _ A ) , A \\subseteq V , A \\neq \\emptyset , \\end{align*}"}
-{"id": "2484.png", "formula": "\\begin{align*} \\| \\mu \\| _ 0 = \\max \\bigl \\{ \\| \\mu ^ + \\| _ 1 , \\| \\mu ^ - \\| _ 1 \\bigr \\} = \\frac { 1 } { 2 } \\Bigl ( \\| \\mu \\| _ 1 + \\bigl | \\mu ( K ) \\bigl | \\Bigr ) \\end{align*}"}
-{"id": "3527.png", "formula": "\\begin{align*} G = \\left ( \\begin{matrix} 0 & 2 \\\\ 2 & 0 \\end{matrix} \\right ) \\oplus \\left ( \\begin{matrix} 0 & 2 \\\\ 2 & 0 \\end{matrix} \\right ) \\oplus ( - 2 ) \\oplus ( - 2 ) . \\end{align*}"}
-{"id": "560.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ N \\| u \\| _ { \\widetilde { U } ^ p _ { \\mathfrak { B } } ( s _ k , s _ { k + 1 } ) } ^ p > 0 . \\end{align*}"}
-{"id": "9021.png", "formula": "\\begin{align*} f _ { \\{ b _ i , c _ i \\} , b _ { i + 1 } } ( x _ 1 , x _ 2 ) = \\begin{cases} b _ { i + 1 } & \\{ x _ 1 , x _ 2 \\} = \\{ b _ i , c _ i \\} , \\\\ x _ 1 & \\end{cases} \\end{align*}"}
-{"id": "2041.png", "formula": "\\begin{align*} f \\big ( \\frac { a } { M } + i y \\big ) = \\xi _ R ( - M ) \\overline { \\xi _ { q / R } ( a ) } i ^ { k } ( M R ^ { \\frac { 1 } { 2 } } y ) ^ { - k } \\tilde { f } _ R \\big ( - \\frac { \\overline { R a } } { M } + i \\frac { 1 } { M ^ 2 R y } \\big ) . \\end{align*}"}
-{"id": "525.png", "formula": "\\begin{align*} G ^ h ( x ) = \\bar { \\mathcal { G } } ( x ' ) + h \\mathcal { G } _ 1 ( x ' , \\frac { x _ 3 } { h } ) + \\frac { h ^ 2 } { 2 } \\mathcal { G } _ 2 ( x ' , \\frac { x _ 3 } { h } ) + o ( h ^ 2 ) \\mbox { f o r a l l } ~ x = ( x ' , x _ 3 ) \\in \\Omega ^ h , \\end{align*}"}
-{"id": "3821.png", "formula": "\\begin{align*} & \\langle a _ m , \\dots , a _ 1 ; \\langle b _ n , \\dots , b _ 1 ; c \\rangle \\rangle \\\\ = & \\sum \\langle V _ { 2 n } , \\langle V _ { 2 n - 1 } ; b _ n \\rangle , \\dots , V _ 4 , \\langle V _ 3 ; b _ 2 \\rangle , V _ 2 , \\langle V _ 1 ; b _ 1 \\rangle , V _ 0 ; c \\rangle , \\end{align*}"}
-{"id": "8997.png", "formula": "\\begin{align*} R ^ { \\prime } ( \\rho ) = C _ \\infty \\kappa ^ { \\gamma _ g } \\Big ( \\frac { ( 1 - 2 \\beta \\gamma _ g ) \\rho ^ 2 + 2 X ( 0 ) ( 1 - \\beta \\gamma _ g ) \\rho + 1 + X ( 0 ) ^ 2 } { [ 1 + ( X ( 0 ) + \\rho ) ^ 2 ] ^ { \\beta \\gamma _ g + 1 } } \\Big ) . \\end{align*}"}
-{"id": "9603.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\frac { q ^ n ( q ^ { n + 1 } ) _ { N - n } } { ( q ^ { 2 n } ; q ^ 2 ) _ { N - n + 1 } } = \\sum _ { n = 1 } ^ { \\infty } N _ { \\textup { S C } } ( n , N ) q ^ n . \\end{align*}"}
-{"id": "1376.png", "formula": "\\begin{align*} F ^ 0 _ j ( \\lambda ) = \\lambda ^ { \\kappa _ j } _ 1 \\ldots \\lambda ^ { \\kappa _ j } _ { n - 1 } , \\ S _ j ( \\lambda ) = \\lambda ^ { \\kappa _ j } _ { n } , \\ R _ j ( \\lambda ) = \\lambda ^ { \\kappa _ j \\kappa _ 1 ^ { - 1 } } _ { 1 } F ^ 1 _ j ( \\lambda ) = \\lambda ^ { \\kappa _ j \\kappa _ 1 ^ { - 1 } } _ { 2 } \\ldots \\lambda ^ { \\kappa _ j \\kappa _ 1 ^ { - 1 } } _ { n \\vphantom { 2 } } \\ ; . \\end{align*}"}
-{"id": "8318.png", "formula": "\\begin{align*} \\mathfrak { K } f ( \\alpha ) : = p . v . \\int _ { - \\infty } ^ { \\infty } R e \\{ \\frac { 1 } { \\pi i } \\frac { z _ { \\beta } } { z ( \\alpha , t ) - z ( \\beta , t ) } \\} f ( \\beta ) d \\beta . \\end{align*}"}
-{"id": "3407.png", "formula": "\\begin{align*} d _ * ( T ) & = \\prod _ i d _ 0 ( T _ i ) , \\\\ d _ 0 ( T ) & = \\prod _ i d ( T _ i ) - \\prod _ i d _ 0 ( T _ i ) , \\\\ d ( T ) & = d _ 0 ( T ) + \\prod _ i \\left ( d ( T _ i ) + d _ * ( T _ i ) \\right ) . \\end{align*}"}
-{"id": "2364.png", "formula": "\\begin{align*} [ a _ x , a ^ + _ y ] & = \\delta _ { x , y } & [ x , a _ y ] & = [ a ^ + _ x , a ^ + _ y ] = 0 . \\end{align*}"}
-{"id": "2320.png", "formula": "\\begin{align*} \\langle \\alpha _ x | \\alpha _ y \\rangle = \\delta ( x - y ) \\end{align*}"}
-{"id": "2074.png", "formula": "\\begin{align*} D R _ n ' ( \\rho | \\rho _ { \\rm t x } ) \\mid _ { \\rho = \\rho _ c } = - { k _ { f } } R _ n ( \\rho _ c | \\rho _ { \\rm t x } ) . \\end{align*}"}
-{"id": "9014.png", "formula": "\\begin{align*} f _ { \\{ a _ 1 \\} , 0 } ( x ) = \\begin{cases} 0 & a _ 1 = x , \\\\ x & \\end{cases} f _ { \\{ a _ 1 , a _ 2 \\} , 1 } ( x _ 1 , x _ 2 ) = \\begin{cases} 1 & \\{ a _ 1 , a _ 2 \\} = \\{ x _ 1 , x _ 2 \\} , \\\\ x _ 1 & \\end{cases} \\end{align*}"}
-{"id": "9097.png", "formula": "\\begin{align*} \\iint _ { \\mathbb { R } ^ N \\times \\mathbb { R } ^ M } | \\tau ( x , y ) | ^ p d x d y = \\left ( \\int _ { \\mathbb { R } ^ N } | \\varphi ( x ) | ^ p d x \\right ) \\left ( \\int _ { \\mathbb { R } ^ M } | v ( y ) | ^ p d y \\right ) . \\end{align*}"}
-{"id": "5918.png", "formula": "\\begin{align*} { \\rm c o e f f } _ { D _ i } ( \\Gamma _ i ) = { \\rm g l c t } ( K _ { X _ i } + B _ i + M _ i \\mid D _ i ) . \\end{align*}"}
-{"id": "3847.png", "formula": "\\begin{align*} 2 \\Phi ^ i _ l h _ { 0 0 } ^ { \\frac { m + 3 } { 2 } } W _ 0 ^ { m + 1 } & = & h _ { 0 0 } ^ { m + 2 } W _ 0 C ^ i _ { ( 0 ) l } + h _ { 0 0 } ^ { m + 2 } C ^ i _ { ( 1 1 ) l } + h _ { 0 0 } ^ { \\frac { m + 3 } { 2 } } W _ 0 ^ { m + 1 } C i _ { ( 1 2 ) l } + \\\\ & & ( h _ { 0 0 } ) ^ { m + 1 } W _ 0 C ^ i _ { ( 2 1 ) l } + h _ { 0 0 } W _ 0 ^ { 2 m + 1 } C ^ i _ { ( 2 2 ) l } + W _ 0 ^ { 2 m } h _ { 0 0 } C ^ i _ { ( 3 ) l } + \\\\ & & W _ 0 ^ { 2 m + 1 } C ^ i _ { ( 4 ) l } , \\end{align*}"}
-{"id": "1737.png", "formula": "\\begin{align*} C _ { r , p } \\left ( A \\right ) = \\inf \\left \\{ \\mathrm { C } _ { r , p } \\left ( O \\right ) : A \\subset O , \\ O \\ \\right \\} \\end{align*}"}
-{"id": "8587.png", "formula": "\\begin{align*} m ( 0 ; S ) = \\dim ( \\ker ( S ) ) . \\end{align*}"}
-{"id": "3224.png", "formula": "\\begin{align*} \\mu _ { k - 1 } ^ 2 = o ( \\mu _ k ) . \\end{align*}"}
-{"id": "7073.png", "formula": "\\begin{align*} & ( ( X _ 1 \\otimes X _ 2 ) . M ) . ( Y _ 2 \\otimes Y _ 1 ) \\to ( X _ 1 . ( ( X _ 2 . M ) . Y _ 2 ) . Y _ 1 \\\\ & l _ { X _ 1 \\boxtimes Y _ 1 , X _ 2 \\boxtimes Y _ 2 , M } = b _ { X _ 1 , X _ 2 . M , Y _ 2 } n _ { X _ 1 . ( X _ 2 . M ) , Y _ 2 , Y _ 1 } m _ { X _ 1 , X _ 2 , M } \\end{align*}"}
-{"id": "942.png", "formula": "\\begin{align*} J ^ { * } [ \\psi ^ { ( 4 ) } ] _ { T , t } = { \\int \\limits _ t ^ { * } } ^ T \\psi _ 4 ( t _ 4 ) { \\int \\limits _ t ^ { * } } ^ { t _ 4 } \\psi _ 3 ( t _ 3 ) { \\int \\limits _ t ^ { * } } ^ { t _ 3 } \\psi _ 2 ( t _ 2 ) { \\int \\limits _ t ^ { * } } ^ { t _ 2 } \\psi _ 1 ( t _ 1 ) d { \\bf w } _ { t _ 1 } ^ { ( i _ 1 ) } d { \\bf w } _ { t _ 2 } ^ { ( i _ 2 ) } d { \\bf w } _ { t _ 3 } ^ { ( i _ 3 ) } d { \\bf w } _ { t _ 4 } ^ { ( i _ 4 ) } \\end{align*}"}
-{"id": "7342.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\dot { x } = p _ \\xi ( x , \\xi ) = \\frac { \\partial p } { \\partial \\xi } ( x , \\xi ) , \\dot { \\xi } = - p _ x ( x , \\xi ) = - \\frac { \\partial p } { \\partial x } , \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "6334.png", "formula": "\\begin{align*} \\xi = \\frac { 3 . 9 1 } { \\mathcal { V } } \\left ( \\frac { \\lambda } { 5 5 0 } \\right ) ^ { - q } , \\end{align*}"}
-{"id": "3532.png", "formula": "\\begin{align*} \\begin{aligned} & \\Gamma : = \\left \\{ g \\in P G L ( 4 , \\mathbb { Z } [ i ] ) \\mid g ^ { \\dagger } J g = J \\right \\} , \\ ; g ^ { \\dagger } : = \\ , ^ { t } \\bar { g } , \\ , \\ , J : = \\left ( \\begin{smallmatrix} 0 & E _ { 2 } \\\\ - E _ { 2 } & 0 \\end{smallmatrix} \\right ) , \\\\ & \\Gamma _ { T } : = \\Gamma \\rtimes \\langle T \\rangle , T : W \\mapsto \\ , ^ { t } W \\ , \\ , ( W \\in \\mathbb { H } _ { 2 } ) , \\\\ & \\Gamma _ { M } : = \\left \\{ g T ^ { a } \\in \\Gamma _ { T } \\mid ( - 1 ) ^ { a } \\det ( g ) = 1 , a = 0 , 1 \\right \\} . \\end{aligned} \\end{align*}"}
-{"id": "7856.png", "formula": "\\begin{align*} \\int _ G ( \\xi \\cdot x ) ^ 2 { \\rm d } x = L ( G ) ^ 2 . \\end{align*}"}
-{"id": "6463.png", "formula": "\\begin{align*} \\mu ( \\sigma ) = 1 \\forall \\sigma \\in \\Lambda _ { \\Xi } . \\end{align*}"}
-{"id": "4113.png", "formula": "\\begin{align*} g _ { 1 } ( z ) & = \\int _ { 0 } ^ { q } \\log { ( z - s ) } \\ , d \\mu ^ * ( s ) , z \\in \\mathbb C \\setminus ( - \\infty , q ] , \\\\ g _ { 2 } ( z ) & = \\int _ { - \\infty } ^ { 0 } \\log { ( z - s ) } \\ , d \\nu ^ * ( s ) , z \\in \\mathbb C \\setminus ( - \\infty , 0 ] , \\end{align*}"}
-{"id": "1958.png", "formula": "\\begin{gather*} S = \\frac { R [ y ] } { ( y \\cdot F _ 1 ( y ) , \\dotsc , y \\cdot F _ n ( y ) ) } , \\\\ T = \\frac { R [ y ] } { ( y \\cdot F _ 1 ( y ) , \\dotsc , y \\cdot F _ n ( y ) , y \\cdot G _ 1 ( y ) , \\dotsc , y \\cdot G _ r ( y ) ) } . \\end{gather*}"}
-{"id": "5088.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n \\langle \\tau _ j , \\tau _ k \\rangle \\langle x _ k , x _ j \\rangle & = \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n \\langle \\tau _ j , x _ k \\rangle \\langle \\tau _ k , x _ j \\rangle = \\operatorname { T r a c e } ( S ^ 2 _ { x , \\tau } ) \\\\ & = \\operatorname { T r a c e } ( b ^ 2 I _ \\mathcal { H } ) = b ^ 2 m = \\left ( \\frac { 1 } { m } \\sum _ { j = 1 } ^ n \\langle x _ j , \\tau _ j \\rangle \\right ) ^ 2 m . \\end{align*}"}
-{"id": "223.png", "formula": "\\begin{align*} T u = 0 \\ \\ u ( e ^ { i \\theta } z , t ) = u ( z , t ) . \\end{align*}"}
-{"id": "8873.png", "formula": "\\begin{align*} e _ 7 & = 1 3 4 4 + ( 1 6 8 b + 9 0 7 2 ) p + ( - 1 2 b ^ 2 - 4 7 1 6 \\mu ^ 2 + 1 8 2 4 b - 4 4 3 4 0 \\mu - 3 0 2 4 ) p ^ 2 \\\\ & + ( 6 2 4 b \\mu ^ 2 - 2 7 6 b ^ 2 + 5 1 2 5 2 \\mu ^ 2 - 5 0 4 b + 1 9 7 6 4 \\mu - 1 4 4 ) p ^ 3 \\\\ & + ( - 2 4 b ^ 3 + 1 1 5 2 b ^ 2 \\mu - 8 7 6 0 b \\mu ^ 2 - 1 2 0 0 \\mu ^ 3 + 4 8 b ^ 2 - 2 6 5 6 8 \\mu ^ 2 - 1 5 8 4 \\mu + 4 8 ) p ^ 4 \\\\ & + ( - 9 6 b ^ 2 \\mu + 1 0 0 8 b \\mu ^ 2 + 7 0 7 2 \\mu ^ 3 + 3 6 0 0 \\mu ^ 2 - 4 8 \\mu ) p ^ 5 + ( 9 6 b \\mu ^ 2 - 1 5 3 6 \\mu ^ 3 - 9 6 \\mu ^ 2 ) p ^ 6 \\end{align*}"}
-{"id": "363.png", "formula": "\\begin{align*} x _ m \\sqrt { c } + z _ m \\sqrt { a } = ( x \\sqrt { c } + z \\sqrt { a } ) ( s + \\sqrt { a c } ) ^ m \\end{align*}"}
-{"id": "5692.png", "formula": "\\begin{align*} \\| x ^ A - x ^ B \\| = h _ 0 \\| v _ 0 ^ A - v _ 0 ^ B \\| = h _ 0 \\| v _ 0 ^ A - v _ 0 + v _ 0 - v _ 0 ^ B \\| \\le 2 \\eta h _ 0 \\end{align*}"}
-{"id": "792.png", "formula": "\\begin{align*} b _ { { s } } = \\mu b _ 1 \\mod q ^ { { s } } - 1 . \\end{align*}"}
-{"id": "6530.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow + \\infty } \\Biggl ( \\int ^ { t + 1 } _ { t } \\bigl \\| q ( s , x ( s ) ) \\bigr \\| ^ { p } \\ , d s \\Biggr ) ^ { 1 / p } = 0 . \\end{align*}"}
-{"id": "4648.png", "formula": "\\begin{align*} x _ 0 \\mapsto & \\omega x _ 0 & x _ 1 \\mapsto & \\omega ^ 2 x _ 1 & y \\mapsto & \\omega ^ 3 y = - y \\\\ \\end{align*}"}
-{"id": "884.png", "formula": "\\begin{align*} \\beta _ 1 & = \\alpha _ 3 & \\beta _ 2 & = \\alpha _ 2 + \\alpha _ 3 & \\beta _ 3 & = \\alpha _ 2 \\\\ \\beta _ 4 & = \\alpha _ 1 + \\alpha _ 2 + \\alpha _ 3 & \\beta _ 5 & = \\alpha _ 1 + \\alpha _ 2 & \\beta _ 6 & = \\alpha _ 1 \\end{align*}"}
-{"id": "3823.png", "formula": "\\begin{align*} w = & \\langle u _ n , \\dots , u _ 2 , u _ 1 ; \\langle v _ m , \\dots , v _ 2 , v _ 1 ; x \\rangle \\rangle \\\\ = & \\sum \\langle U _ { 2 m } , \\langle U _ { 2 m - 1 } ; v _ m \\rangle , \\dots , U _ 4 , \\langle U _ 3 ; v _ 2 \\rangle , U _ 2 , \\langle U _ 1 ; v _ 1 \\rangle , U _ 0 ; x \\rangle , \\end{align*}"}
-{"id": "9584.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\textup { s p t } ( n , N ) q ^ n = \\frac { 1 } { ( q ) _ N } \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - 1 ) ^ { n - 1 } n q ^ { n ( n + 1 ) / 2 } } { 1 - q ^ n } , \\end{align*}"}
-{"id": "3818.png", "formula": "\\begin{align*} \\| g \\| _ { T , \\rho } \\le \\sum _ { i = 1 } ^ n \\| s _ i \\| _ { T , \\rho } \\le c \\sum _ { i = 1 } ^ n \\| s _ i \\| _ { S , \\pi } \\le c \\ell . \\end{align*}"}
-{"id": "7979.png", "formula": "\\begin{align*} \\left | ^ 2 F _ 4 ( q ) \\right | = q ^ { 1 2 } ( q - 1 ) ^ 2 ( q + 1 ) ^ 2 ( q ^ 2 + 1 ) ^ 2 ( q ^ 2 - q + 1 ) ( q ^ 4 - q ^ 2 + 1 ) \\ , , \\end{align*}"}
-{"id": "365.png", "formula": "\\begin{align*} | z _ 0 | ^ 2 & \\leqslant \\frac { | c - a | } { | a | } + \\frac { | c | | c - a | } { | s | - 1 } \\leqslant \\frac { | c - a | } { | a | } + \\frac { 2 | c | \\cdot | c - a | } { | a | } \\\\ & = \\frac { | c - a | } { | a | } ( 2 | c | + 1 ) \\leqslant \\left ( \\Big | \\frac { c } { a } \\Big | + 1 \\right ) ( 2 | c | + 1 ) \\\\ & \\leqslant \\left ( \\frac { | c | } { 4 } + 1 \\right ) ( 2 | c | + 1 ) = \\frac { | c | ^ 2 } { 2 } + \\frac 9 4 | c | + 1 < | c | ^ 2 \\end{align*}"}
-{"id": "4374.png", "formula": "\\begin{align*} \\frac { ( n - 2 k - 2 ) } { n } \\ge \\prod _ { i = 2 } ^ { k } \\frac { n - k + 1 - i } { n - 1 - i } . \\end{align*}"}
-{"id": "4507.png", "formula": "\\begin{align*} \\pi ^ * _ 1 ( r ^ * ( a ) ) = \\pi ^ * _ 2 ( p ^ * ( a ) ) \\ ; , \\end{align*}"}
-{"id": "9555.png", "formula": "\\begin{align*} F ( a , b ; t ) = \\frac { ( a q ) _ { \\infty } } { ( b q ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { ( b / a ) _ n } { ( q ) _ n } \\frac { ( a q ) ^ n } { 1 - t q ^ n } . \\end{align*}"}
-{"id": "5903.png", "formula": "\\begin{align*} \\xi \\colon \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\mapsto \\frac { a } { c } \\mod \\pi \\end{align*}"}
-{"id": "4186.png", "formula": "\\begin{align*} \\left ( \\frac { A _ n ^ { ( 2 ) } ( 0 ) } { n ^ 6 z } \\right ) ^ 2 = \\frac { A _ n ^ { ( 1 ) } ( 0 ) ( A _ n ^ { ( 1 ) } ( z ) - A _ n ^ { ( 1 ) } ( 0 ) ) A _ n ^ { ( 1 ) } ( 0 ) ( A _ n ^ { ( 1 ) } ( z ) - A _ n ^ { ( 1 ) } ( 0 ) ) } { n ^ { 1 2 } z ^ 4 } = \\mathcal { O } \\left ( \\frac { 1 } { n } \\right ) \\end{align*}"}
-{"id": "7545.png", "formula": "\\begin{align*} [ \\eta _ \\epsilon \\cdot \\nabla _ x , \\mathbb { U } ] ( \\sigma _ \\epsilon ) = \\eta _ \\epsilon \\cdot \\nabla _ x \\left ( \\mathbb { U } ( \\sigma _ \\epsilon ) \\right ) - \\mathbb { U } \\left ( \\eta _ \\epsilon \\cdot \\nabla _ x \\sigma _ \\epsilon \\right ) \\end{align*}"}
-{"id": "4389.png", "formula": "\\begin{align*} \\psi ( z ^ 1 , . . . , z ^ { k + 1 } ) = \\frac { 1 } { \\pi } \\int _ { | \\zeta | < 2 r } \\chi \\left ( \\frac { \\zeta } { r } \\right ) \\omega _ { k + 1 \\overline { k + 1 } } ( z ^ 1 , z ^ 2 , . . . , z ^ k , \\zeta ) \\log | z ^ { k + 1 } - \\zeta | ^ 2 d A ( \\zeta ) , \\end{align*}"}
-{"id": "5796.png", "formula": "\\begin{align*} f \\circ \\ell _ \\gamma \\ : = \\ \\left ( \\left ( \\cdots \\left ( \\left ( f \\circ \\ell _ { \\omega ^ { \\beta _ k } } \\right ) \\circ \\ell _ { \\omega ^ { \\beta _ { k - 1 } } } \\right ) \\circ \\cdots \\right ) \\circ \\ell _ { \\omega ^ { \\beta _ 2 } } \\right ) \\circ \\ell _ { \\omega ^ { \\beta _ 1 } } . \\end{align*}"}
-{"id": "7254.png", "formula": "\\begin{align*} D _ { \\phi \\phi \\phi } ^ 3 \\Phi ( 0 , \\mu ^ * _ { k } ) [ \\phi ^ * _ { k } , \\phi ^ * _ { k } , \\phi ^ * _ { k } ] & = \\Pi D _ \\phi F ( 0 , \\mu ^ * _ { k } ) D _ { \\phi \\phi \\phi } ^ 3 \\psi ( 0 , \\mu ^ * _ { k } ) [ \\phi ^ * _ { k } , \\phi ^ * _ { k } , \\phi ^ * _ { k } ] \\\\ & \\quad + 3 \\Pi D _ { \\phi \\phi } ^ 2 F ( 0 , \\mu ^ * _ { k } ) [ \\phi ^ * _ { k } , D _ { \\phi \\phi } ^ 2 \\psi ( 0 , \\mu ^ * _ { k } ) [ \\phi ^ * _ { k } , \\phi ^ * _ { k } ] ] . \\end{align*}"}
-{"id": "2508.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } B _ { n , p } \\frac { t ^ { n } } { n ! } = \\ _ { 2 } F _ { 1 } \\left ( 1 , 1 ; p + 2 ; 1 - e ^ { t } \\right ) , \\end{align*}"}
-{"id": "6041.png", "formula": "\\begin{align*} g _ k ( s ) = ( 4 k + 1 - 2 s ) ( s - 1 ) - r ( s - r ) / 2 \\leq 2 k ^ 2 - 2 k + 1 = \\tfrac { n ^ 2 - 6 n + 1 3 } { 8 } , \\end{align*}"}
-{"id": "8205.png", "formula": "\\begin{align*} \\sum \\limits _ { n = 1 } ^ { \\infty } p _ { \\omega } ( n ) q ^ n = \\sum \\limits _ { n = 1 } ^ { \\infty } \\frac { q ^ n } { ( 1 - q ^ n ) ( q ^ { n + 1 } ; q ) _ { n } ( q ^ { 2 n + 2 } ; q ^ 2 ) _ { \\infty } } . \\end{align*}"}
-{"id": "3652.png", "formula": "\\begin{align*} b _ { i j , k l } = \\langle v _ i \\cdot v _ k , v _ j \\cdot v _ l \\rangle - \\langle v _ j \\cdot v _ k , v _ i \\cdot v _ l \\rangle . \\end{align*}"}
-{"id": "3109.png", "formula": "\\begin{align*} c _ 1 & = c _ 1 ( L _ 0 ) + \\ldots + c _ 1 ( L _ s ) \\ , , \\\\ | n | & = n _ 0 + \\ldots + n _ s \\ , , \\end{align*}"}
-{"id": "2133.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } \\frac { 1 } { n } \\sum _ { j = 0 } ^ { n - 1 } \\Big | \\mu ^ A \\Big ( S ^ A _ j ( Q _ 1 ) \\cap Q _ 2 \\Big ) - \\mu ^ A ( Q _ 1 ) \\mu ^ A ( Q _ 2 ) \\Big | = 0 , \\forall Q _ 1 \\ne Q _ 2 \\in \\mathcal { F } ^ A \\end{align*}"}
-{"id": "2000.png", "formula": "\\begin{align*} \\left \\langle \\phi _ { m _ 1 } \\psi ^ { a _ 1 } \\cdots \\phi _ { m _ n } \\psi ^ { a _ n } \\cdot \\phi _ 1 \\psi \\right \\rangle _ { 0 , n + 1 } ^ \\mathrm { F J R W } = ( n - 2 ) \\left \\langle \\phi _ { m _ 1 } \\psi ^ { a _ 1 } \\cdots \\phi _ { m _ n } \\psi ^ { a _ n } \\right \\rangle _ { 0 , n } ^ \\mathrm { F J R W } . \\end{align*}"}
-{"id": "7643.png", "formula": "\\begin{align*} \\psi _ 0 ( \\xi ) = \\big ( ( G ^ { - 1 } K ) ^ \\ast \\big ) ^ N \\psi _ 0 ( \\xi ) \\ge ( g _ 2 ^ { - 1 } \\alpha _ 1 ) ^ N \\gamma _ 0 \\int \\limits _ { \\mathbb { T } ^ d } \\psi _ 0 ( \\eta ) d \\eta = ( g _ 2 ^ { - 1 } \\alpha _ 1 ) ^ N \\gamma _ 0 \\| \\psi _ 0 \\| _ { L ^ 1 ( \\mathbb { T } ^ d ) } \\forall \\ ; \\xi \\in \\mathbb { T } ^ d , \\end{align*}"}
-{"id": "2361.png", "formula": "\\begin{align*} \\Phi _ n & = \\sqrt { n ! } | n \\rangle \\\\ \\langle \\Phi _ m | \\Phi _ m \\rangle & = \\delta _ { n , m } n ! \\\\ a \\Phi _ n & = n \\Phi _ { n - 1 } , \\\\ a ^ + \\Phi _ n & = \\Phi _ { n + 1 } . \\end{align*}"}
-{"id": "6066.png", "formula": "\\begin{align*} \\int _ { \\Sigma _ j } \\Omega \\wedge \\omega ^ { k + 1 } \\wedge \\cdots \\wedge \\omega ^ { j } \\ = \\ \\deg ( F ) \\ , . \\end{align*}"}
-{"id": "8105.png", "formula": "\\begin{align*} G _ { \\rm S } = \\{ \\ell _ 1 , \\dots , \\ell _ g \\} \\end{align*}"}
-{"id": "3635.png", "formula": "\\begin{align*} ( a _ 1 \\cdot v _ { ( 2 , 3 ) } ) \\cdot ( a _ 1 \\cdot v _ { ( 2 , 3 ) } ) & = \\frac { ( 1 0 t + 1 ) t } { 2 ^ 4 } a _ 1 - \\frac { ( 2 t - 1 ) t } { 2 ^ 4 } a _ { - 1 } + \\frac { t } { 2 ^ 4 } v _ { ( 2 , 3 ) } + \\frac { t } { 2 ^ 2 } a _ 1 \\cdot v _ { ( 2 , 3 ) } \\\\ ( a _ { - 1 } \\cdot v _ { ( 2 , 3 ) } ) \\cdot ( a _ 1 \\cdot v _ { ( 2 , 3 ) } ) & = - \\frac { ( 6 t - 1 ) t } { 2 ^ 4 } a _ 1 - \\frac { ( 2 t - 1 ) t } { 2 ^ 4 } a _ { - 1 } + \\frac { t } { 2 ^ 4 } v _ { ( 2 , 3 ) } + \\frac { t } { 2 ^ 2 } a _ 1 \\cdot v _ { ( 2 , 3 ) } \\end{align*}"}
-{"id": "6584.png", "formula": "\\begin{align*} \\bar { S } _ i ( r _ i , \\theta _ i ) = \\bar { \\varphi } ( r _ i ) r _ i ^ { 1 / 2 } \\sin ( \\theta _ i / 2 ) , \\end{align*}"}
-{"id": "618.png", "formula": "\\begin{align*} \\vec k = ( k _ { - \\infty } , k _ 1 , \\dots , k _ n , k _ \\infty ) \\end{align*}"}
-{"id": "5876.png", "formula": "\\begin{align*} ( \\phi \\circ j ) ^ { - 1 } ( t _ w ) = \\sum a _ { w , x } C ' _ x \\end{align*}"}
-{"id": "8934.png", "formula": "\\begin{align*} H _ 0 : d ( x , y ) = 0 \\textrm { ~ ~ ~ ~ ~ ~ ~ v s . ~ ~ ~ ~ ~ ~ ~ } H _ 1 : d ( x , y ) \\ge s , \\end{align*}"}
-{"id": "3431.png", "formula": "\\begin{gather*} c _ 2 \\left ( \\bigoplus n _ i V _ i \\right ) = 4 n _ 4 + 9 n _ 5 + n _ 6 \\pmod { 1 6 } , n _ 1 = n _ 2 , \\end{gather*}"}
-{"id": "8460.png", "formula": "\\begin{align*} \\mathcal { E } _ s : = \\sum _ { k = 0 } ^ s ( E _ k ^ { \\theta } + E _ k ^ { \\sigma } ) . \\end{align*}"}
-{"id": "9583.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { N } \\frac { ( c ) _ n q ^ n } { ( q ) _ n } = \\frac { ( c q ) _ N } { ( q ) _ N } . \\end{align*}"}
-{"id": "4684.png", "formula": "\\begin{align*} \\mu _ j ( u ) = \\frac { 4 7 } { 1 1 5 2 } ( u _ { j - 4 } + u _ j ) - \\frac { 1 0 7 } { 2 8 8 } ( u _ { j - 3 } + u _ { j - 1 } ) + \\frac { 3 1 9 } { 1 9 2 } u _ { j - 2 } ~ ~ ~ \\mbox { f o r } ~ ~ j = 5 , \\ldots , m , \\end{align*}"}
-{"id": "3296.png", "formula": "\\begin{align*} \\mu ^ 2 ( f ) = \\begin{cases} 1 & \\mbox { i f $ f $ i s s q u a r e f r e e , } \\\\ 0 & \\mbox { o t h e r w i s e } , \\end{cases} \\end{align*}"}
-{"id": "9361.png", "formula": "\\begin{align*} G _ k = \\gamma _ 1 ( G _ k ) = \\langle x , y \\rangle \\ ; \\gamma _ 2 ( G _ k ) G _ k / \\gamma _ 2 ( G _ k ) \\cong C _ { 2 ^ { k + 1 } } \\times C _ { 4 } \\end{align*}"}
-{"id": "2936.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { s } \\bigg ( \\frac { 1 } { \\sigma _ { i } } - \\frac { 1 } { \\widetilde { \\sigma } _ { i } } \\bigg ) ^ { 2 } + \\sum _ { i = s + 1 } ^ { r } \\frac { 1 } { \\sigma _ { i } ^ { 2 } } \\leq \\| B ^ { \\dagger } - A ^ { \\dagger } \\| _ { F } ^ { 2 } \\leq \\sum _ { i = 1 } ^ { s } \\bigg ( \\frac { 1 } { \\sigma _ { i } } + \\frac { 1 } { \\widetilde { \\sigma } _ { i } } \\bigg ) ^ { 2 } + \\sum _ { i = s + 1 } ^ { r } \\frac { 1 } { \\sigma _ { i } ^ { 2 } } . \\end{align*}"}
-{"id": "2210.png", "formula": "\\begin{align*} \\dot { P } + P A + A ^ T P - \\frac { 1 } { 2 } P B R ^ { - 1 } B ^ { T } P - \\frac { 1 } { 4 } P B R ^ { - 1 } B ^ { T } P ^ T - \\frac { 1 } { 4 } P ^ T B R ^ { - 1 } B ^ { T } P + C ^ { T } Q C = 0 \\end{align*}"}
-{"id": "3973.png", "formula": "\\begin{align*} f = \\Phi ( [ f ] ) = \\underset { n \\geq 1 } \\sum a _ { [ f ] } \\left ( \\begin{pmatrix} n & 0 \\\\ 0 & 0 \\end{pmatrix} \\right ) q ^ n . \\end{align*}"}
-{"id": "2690.png", "formula": "\\begin{align*} z _ { \\pm } = \\tau + \\frac { 1 } { 2 \\cos \\theta } \\Big ( \\frac { 1 } { \\tau } - \\tau \\Big ) e ^ { \\pm i \\theta } , \\end{align*}"}
-{"id": "7288.png", "formula": "\\begin{align*} D ( p , \\Phi _ p ) = \\min \\{ 3 \\lVert a \\rVert / 2 - \\lVert \\psi _ p ( a ) + \\Phi _ p ( a ) ( 1 - h _ p ^ + ) \\rVert : a \\in F _ p \\} . \\end{align*}"}
-{"id": "354.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - a \\left ( \\int _ { \\Omega } | \\nabla u | ^ { 2 } d x \\right ) \\Delta u = f ( u ) & \\mbox { i n $ \\Omega $ , } \\\\ u > 0 & \\mbox { i n $ \\Omega $ , } \\\\ u = 0 & \\mbox { o n $ \\partial \\Omega $ , } \\end{array} \\right . \\end{align*}"}
-{"id": "9339.png", "formula": "\\begin{align*} \\int _ { M } \\| n \\mathbf { H } + \\lambda _ { 1 } \\psi \\| ^ { 2 } \\ , d V = n ^ 2 \\int _ { M } \\| \\mathbf { H } \\| ^ { 2 } \\ , d V + \\lambda _ { 1 } ^ { 2 } \\int _ { M } \\| \\psi \\| ^ { 2 } \\ , d V - 2 \\lambda _ { 1 } n \\mathrm { V o l } ( M ) \\geq 0 . \\end{align*}"}
-{"id": "4356.png", "formula": "\\begin{align*} \\mathcal P _ a ' ( i ) : = \\ & \\Big \\{ P \\ : \\ P \\in { [ 2 , n ] \\choose k - 1 } , \\ P \\cap [ 2 , i ] = \\{ i \\} \\Big \\} , \\\\ \\mathcal P _ b ' ( i ) : = \\ & \\Big \\{ P \\ : \\ P \\in { [ 2 , n ] \\choose k } , \\ P \\cap [ 2 , i ] = [ 2 , i - 1 ] \\Big \\} , \\end{align*}"}
-{"id": "5304.png", "formula": "\\begin{align*} ( \\bar { v } _ 8 , \\bar { u } _ 8 , \\bar { q } _ 2 , \\bar { p } _ 2 ) = & \\left ( - v _ 4 , u _ 4 , a q _ 1 ^ { - 1 } + ( b _ 2 + v _ 2 - b _ 2 u _ 4 v _ 2 ) ( 1 - u _ 4 v _ 2 ) ^ { - 1 } , q _ 1 \\right ) \\end{align*}"}
-{"id": "1919.png", "formula": "\\begin{align*} \\sin Z _ j = ( - 1 ) ^ { j - 1 } \\frac { x } { \\sqrt { Z _ j ^ 2 + x ^ 2 } } \\qquad \\mbox { a n d } \\cos Z _ j = ( - 1 ) ^ { j - 1 } \\frac { Z _ j } { \\sqrt { Z _ j ^ 2 + x ^ 2 } } , \\end{align*}"}
-{"id": "7290.png", "formula": "\\begin{align*} \\phi ( a ) = \\Phi _ p ( a ) h _ p ^ - + R _ p \\Phi _ p ( a ) ( h _ p ^ + - h _ p ^ - ) . \\end{align*}"}
-{"id": "4900.png", "formula": "\\begin{align*} s _ X \\circ f = s ' _ X \\circ f ' & \\iff \\eta ( s \\otimes X ) \\circ f = \\eta ( s ' \\otimes X ) \\circ f ' \\\\ & \\iff \\eta ( ( s \\otimes X ) \\circ f ) ) = \\eta ( ( s ' \\otimes X ) \\circ f ' ) \\\\ & \\iff ( s \\otimes X ) \\circ f = ( s ' \\otimes X ) \\circ f ' \\end{align*}"}
-{"id": "218.png", "formula": "\\begin{align*} \\rho ( z , t ) = ( | z | ^ { 2 ( \\beta + 1 ) } + ( \\beta + 1 ) ^ 2 | t | ^ 2 ) ^ { \\frac { 1 } { 2 ( \\beta + 1 ) } } \\ \\ \\ \\ \\ \\ \\ \\psi ( z , t ) = \\frac { | z | ^ { 2 \\beta } } { \\rho ( z , t ) ^ { 2 \\beta } } , \\end{align*}"}
-{"id": "6729.png", "formula": "\\begin{align*} \\Box R _ { k l } ^ { ( 1 ) } - \\frac { 1 } { 2 } \\frac { \\partial ^ 2 R ^ { ( 1 ) } } { \\partial x ^ k \\partial x ^ l } = 0 , \\end{align*}"}
-{"id": "3019.png", "formula": "\\begin{align*} ( n + 1 ) ! \\cdot p ' _ m ( n + 1 ) & = \\sum _ { i = 0 } ^ n \\binom { n } { i } ( n - i ) ! \\cdot p ' _ m ( n - i ) \\cdot i ! \\cdot \\sigma ' _ m ( i + 1 ) \\\\ ( n + 1 ) p ' _ m ( n + 1 ) & = \\sum _ { i = 0 } ^ n p ' _ m ( n - i ) \\sigma ' _ m ( i + 1 ) . \\end{align*}"}
-{"id": "8519.png", "formula": "\\begin{align*} ( I - \\mathcal { H } ) \\bar { \\mathfrak { F } } = 2 \\bar { \\mathfrak { F } } - ( \\bar { \\mathcal { H } } + \\mathcal { H } ) \\bar { \\mathfrak { F } } , ( I - \\mathcal { H } ) q = 2 q , \\end{align*}"}
-{"id": "2165.png", "formula": "\\begin{align*} \\| \\tau _ { \\mathbf { x } } ( f - ( p g _ 1 ) ) \\| _ { L ^ 2 ( d w ) } = \\| \\tau _ { \\mathbf { x } } f - \\tau _ { \\mathbf { x } } ( p g _ 1 ) \\| _ { L ^ 2 ( d w ) } < \\varepsilon . \\end{align*}"}
-{"id": "932.png", "formula": "\\begin{align*} E _ 3 ^ q = I _ 3 - \\sum _ { j _ 3 , j _ 2 , j _ 1 = 0 } ^ q C _ { j _ 3 j _ 2 j _ 1 } ^ 2 - \\sum _ { j _ 3 , j _ 2 , j _ 1 = 0 } ^ q C _ { j _ 2 j _ 3 j _ 1 } C _ { j _ 3 j _ 2 j _ 1 } \\ \\ \\ ( i _ 1 \\ne i _ 2 = i _ 3 ) , \\end{align*}"}
-{"id": "7131.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l } ( \\rho , u , \\chi ) ( x , t ) = ( \\rho , u , \\chi ) ( x + L , t ) , \\ \\ & x \\in \\mathbb { R } , t > 0 , \\\\ ( \\rho , u , \\chi ) \\big | _ { t = 0 } = ( \\rho _ 0 , u _ 0 , \\chi _ 0 ) , \\ \\ & x \\in \\mathbb { R } . \\end{array} \\right . \\end{align*}"}
-{"id": "9698.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n _ s } I _ { \\mathbf { d } } ^ i = \\chi ( M ) . \\end{align*}"}
-{"id": "1370.png", "formula": "\\begin{align*} H \\big ( 1 , [ ( \\lambda , x ) ] \\big ) = \\left [ \\left ( \\alpha _ \\lambda , x + ( d ( \\alpha _ \\lambda ) - x ) \\right ) \\right ] = \\left [ \\left ( \\alpha _ \\lambda , d ( \\alpha _ \\lambda ) \\right ) \\right ] = [ ( \\alpha , 0 ) ] \\end{align*}"}
-{"id": "9349.png", "formula": "\\begin{align*} w = y _ { 2 ^ k - 1 } \\cdots y _ 1 y _ 0 [ w , x ] = [ w , y ] = [ y _ 0 , y _ { 2 ^ { k - 1 } } ] \\end{align*}"}
-{"id": "5206.png", "formula": "\\begin{align*} U ( x ) = c x ^ { - 1 } b ( x ) \\exp \\left ( - \\int _ { 1 } ^ { x } t ^ { - 1 } b ( t ) d t \\right ) \\end{align*}"}
-{"id": "6085.png", "formula": "\\begin{align*} \\begin{gathered} m = \\sum \\limits _ { \\substack { 1 \\leq i \\leq k _ j \\\\ 1 \\leq j \\leq r } } ( e ^ i _ j - 2 ) + \\sum \\limits _ { 1 \\leq j \\leq r } ( k _ j - 1 ) \\end{gathered} \\end{align*}"}
-{"id": "6845.png", "formula": "\\begin{align*} \\mathsf { N F } ( f _ 2 , I _ 1 ) = \\mathsf { N F } ( f _ 3 , I _ 1 ) = \\mathsf { N F } ( f _ 6 , I _ 1 ) = a _ { 2 1 } '' a _ { 2 2 } - a _ { 2 2 } '' a _ { 2 1 } + a _ { 1 1 } '' a _ { 1 2 } - a _ { 1 2 } '' a _ { 1 1 } \\ , . \\end{align*}"}
-{"id": "2918.png", "formula": "\\begin{align*} \\mu ( f _ \\Phi ) & = \\widehat { \\mu } ( f _ { \\widehat { \\Phi } } ) \\end{align*}"}
-{"id": "50.png", "formula": "\\begin{align*} \\delta \\mathcal { Y } ^ { v } _ 0 = { \\mathcal { Y } } ^ v _ 0 - \\bar { \\mathcal { Y } } ^ v _ 0 = ( \\mathbf { y } ^ { i } - \\bar { \\mathbf { y } } ^ { i } ) ( v ) = 0 . \\end{align*}"}
-{"id": "5524.png", "formula": "\\begin{align*} \\Lambda _ T ( u ) = \\frac { u } { \\sqrt { T } } \\int _ 0 ^ T \\gamma ( X _ t ) . d W _ t + \\frac { u ^ 2 } { 2 } \\| \\gamma \\| _ { \\mu _ 0 } ^ 2 + u r _ T + u ^ 2 \\tilde { r } _ T , ~ u \\in \\mathbb R , \\end{align*}"}
-{"id": "4080.png", "formula": "\\begin{align*} z ^ 2 \\zeta ^ 3 - z ^ 2 \\zeta ^ 2 + z \\zeta - \\frac { 1 } { 4 } = 0 \\end{align*}"}
-{"id": "7713.png", "formula": "\\begin{align*} t < 1 : \\ , f ( t ) < 1 , \\quad \\ ; \\ ; t \\geq 1 : \\ , f ( t ) = 1 . \\end{align*}"}
-{"id": "7134.png", "formula": "\\begin{align*} \\Phi ( \\rho ) = \\rho \\int _ { \\tilde { \\rho } } ^ { \\rho } \\frac { p ( s ) - p ( \\tilde { \\rho } ) } { s ^ 2 } d s . \\end{align*}"}
-{"id": "3331.png", "formula": "\\begin{align*} B ( u ) : = \\sum _ { \\lambda \\vdash n } \\omega ( B _ { \\lambda , \\mu } ) u ^ { \\ell ( \\lambda ) } . \\end{align*}"}
-{"id": "7427.png", "formula": "\\begin{align*} \\displaystyle d \\chi : = \\gamma ^ * ( 0 , \\mathbf { 1 } _ F , \\psi ' ) ^ { - \\dim ( A ) } d _ { \\psi ' } \\chi \\end{align*}"}
-{"id": "9284.png", "formula": "\\begin{align*} \\mathop { \\lim \\sup } \\limits _ { y \\rightarrow x _ 0 } \\bar \\lambda ( y ) \\le 0 , [ D v - D ( u - \\tau \\tilde \\eta ) ] ( x _ 0 ) = 0 , \\end{align*}"}
-{"id": "4054.png", "formula": "\\begin{align*} a ( x d y ) = ( a x ) d y , ~ ~ ~ ~ ( x d y ) a = x d ( y a ) - x y d a . \\end{align*}"}
-{"id": "5977.png", "formula": "\\begin{align*} \\overline { X } ^ { n , q } ( t ) : = \\frac { 1 } { \\sqrt { n } } X ^ { n , q } ( n t ) = \\frac { 1 } { \\sqrt { n } } \\int _ 0 ^ { n t } q ( s ) d s + \\frac { 1 } { \\sqrt { n } } B ^ n ( { n t } ) , t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "6965.png", "formula": "\\begin{align*} \\beta _ 2 = \\frac { x _ 2 } { h ' ( x _ 0 ) } + \\frac { h '' ( x _ 0 ) } { ( h ' ( x _ 0 ) ) ^ 2 } , \\beta _ 3 = \\frac { x _ 3 } { ( h ' ( x _ 0 ) ) ^ 2 } + \\frac { 3 h '' ( x _ 0 ) x _ 2 } { ( h ' ( x _ 0 ) ) ^ 3 } + \\frac { h ''' ( x _ 0 ) } { ( h ' ( x _ 0 ) ) ^ 3 } . \\end{align*}"}
-{"id": "5637.png", "formula": "\\begin{align*} \\frac { 4 } { 5 } ( 2 + \\frac { 2 A } { \\sqrt { A ^ 2 + \\frac { 3 2 } { 2 7 } a _ 3 } } ) & = 2 \\\\ \\frac { 3 2 } { 2 7 } \\frac { 1 } { \\sqrt { A ^ 2 + \\frac { 3 2 } { 2 7 } a _ 3 } } & = 2 . \\end{align*}"}
-{"id": "1454.png", "formula": "\\begin{align*} E _ { \\mu } ( t ) = \\frac { 1 } { 2 } \\int _ { \\mathbb { T } ^ { d } } \\left ( \\mu ^ { 1 } ( t , x ) - \\mu ^ { 2 } ( t , x ) \\right ) ^ { 2 } \\ d x , \\end{align*}"}
-{"id": "7248.png", "formula": "\\begin{align*} M : = \\mbox { s p a n } \\left \\{ \\cos \\left ( x l \\right ) \\mid l \\neq k \\right \\} , N : = \\ker D _ \\phi F ( 0 , \\mu ^ * _ { k } ) = \\mbox { s p a n } \\{ \\phi ^ * _ { k } \\} . \\end{align*}"}
-{"id": "7824.png", "formula": "\\begin{align*} \\lim _ { n } \\| T x _ n - \\lambda \\| T \\| x _ n \\| = 0 . \\end{align*}"}
-{"id": "3403.png", "formula": "\\begin{align*} \\rho ^ M _ { \\sup } ( T ) = \\frac { 1 } { 1 + \\sum _ { i } \\rho ^ { M - 1 } _ { \\inf } ( T _ i ) } \\ , \\ , \\rho ^ M _ { \\inf } ( T ) = \\frac { 1 } { 1 + \\sum _ { i } \\rho ^ { M - 1 } _ { \\sup } ( T _ i ) } . \\end{align*}"}
-{"id": "1801.png", "formula": "\\begin{align*} \\left [ c _ { 2 , 1 } \\left ( \\bigcap _ { n = l } ^ { \\infty } G _ { n } \\right ) \\right ] ^ { 2 } \\leq \\left [ c _ { 2 , 1 } \\left ( \\bigcap _ { n = l } ^ { N } G _ { n } \\right ) \\right ] ^ { 2 } \\leq C ' ( N - l ) ^ { 2 } C _ { H } e ^ { - C _ { 3 } ( N ^ { 1 - \\gamma } - l ^ { 1 - \\gamma } ) } , \\end{align*}"}
-{"id": "8000.png", "formula": "\\begin{align*} x _ i ^ * = \\begin{cases} 1 , y _ i \\geq 1 ; \\\\ y _ i , - 1 < y _ i < 1 ; \\\\ - 1 , y _ i \\leq - 1 . \\end{cases} \\end{align*}"}
-{"id": "3099.png", "formula": "\\begin{align*} \\textstyle { V ^ { 0 1 } } ( z , w ) : = \\sum _ { \\alpha = 1 } ^ N \\frac { R ^ { [ 1 ] } ( z ) - R ^ { [ 1 ] } ( - w ) } { z + w } \\mathbf 1 _ \\alpha \\otimes R ^ { [ 1 ] } ( w ) \\mathbf 1 ^ \\alpha . \\end{align*}"}
-{"id": "1707.png", "formula": "\\begin{align*} | f ( x ) - f ( y ) | & \\leq \\sum _ { n = 1 } ^ { M } | f \\left ( ( n \\delta + r ) \\cdot z + y \\right ) - f \\left ( ( ( n - 1 ) \\delta + r ) \\cdot z + y \\right ) | + | f ( r \\cdot z + y ) - f ( y ) | \\\\ & \\leq \\varepsilon \\cdot ( M + 1 ) \\leq \\varepsilon \\cdot ( \\| x - y \\| \\cdot \\delta ^ { - 1 } + 1 ) . \\end{align*}"}
-{"id": "1657.png", "formula": "\\begin{align*} \\frac { \\partial H ^ c } { \\partial p _ n } \\Big | _ { z \\bullet } = \\frac { \\partial H ^ c } { \\partial q _ n } \\Big | _ { z ^ \\bullet } = 0 \\end{align*}"}
-{"id": "2894.png", "formula": "\\begin{align*} \\{ \\Gamma _ { K \\cup L } , \\Gamma _ { L \\cup M } \\} = \\Gamma _ { K \\cup M } + 2 \\Gamma _ { L } \\Gamma _ { K \\cup L \\cup M } + 2 \\Gamma _ { K } \\Gamma _ { M } . \\end{align*}"}
-{"id": "6179.png", "formula": "\\begin{align*} U _ t h = H . \\end{align*}"}
-{"id": "7364.png", "formula": "\\begin{align*} r - d ( p , b _ 1 ) & = r - ( d ( p , a ) + d ( a , b _ 1 ) ) = ( r - d ( p , a ) ) - d ( a , b _ 1 ) \\\\ & = \\delta - d ( a , b _ 1 ) < \\delta - \\frac { \\delta } { 2 } = \\frac { \\delta } { 2 } \\end{align*}"}
-{"id": "5062.png", "formula": "\\begin{align*} \\theta _ A \\pi _ g & = \\sum \\limits _ { p \\in G } L _ p A _ p \\pi _ g = \\sum \\limits _ { p \\in G } L _ p A \\pi _ { p ^ { - 1 } } \\pi _ g = \\sum \\limits _ { p \\in G } L _ p A \\pi _ { { p ^ { - 1 } } g } \\\\ & = \\sum \\limits _ { q \\in G } L _ { g q } A \\pi _ { q ^ { - 1 } } = \\sum \\limits _ { q \\in G } ( \\lambda _ g \\otimes I _ { \\mathcal { H } _ 0 } ) L _ { q } A \\pi _ { q ^ { - 1 } } = ( \\lambda _ g \\otimes I _ { \\mathcal { H } _ 0 } ) \\theta _ A . \\end{align*}"}
-{"id": "5313.png", "formula": "\\begin{align*} \\varphi = \\sigma ^ { ( 1 2 ) } \\circ w _ { \\alpha _ 1 ^ { ( 2 ) } } \\circ \\sigma _ { 1 2 } ^ { ( 2 ) } \\circ \\sigma _ { 0 1 } ^ { ( 2 ) } \\circ w _ { \\alpha _ 1 ^ { ( 1 ) } } \\circ \\sigma _ { 1 2 } ^ { ( 1 ) } \\circ \\sigma _ { 0 1 } ^ { ( 1 ) } . \\end{align*}"}
-{"id": "5261.png", "formula": "\\begin{align*} H ( F , x ) = : H ( x ) = W ( F , x ) / R ( F , x ) ^ 2 , \\ x < u e p ( F ) . \\end{align*}"}
-{"id": "6099.png", "formula": "\\begin{align*} C ( p , n ) = 2 ^ { p ( n - 2 ) / 2 } \\pi ^ { p ( p - 1 ) / 4 } \\prod _ { i = 1 } ^ p \\Gamma ( \\{ n + 1 - i \\} / 2 ) \\end{align*}"}
-{"id": "6489.png", "formula": "\\begin{align*} \\sum _ { m \\geq 0 } X ^ m \\cdot \\frac { \\alpha ^ { m + 1 } - \\beta ^ { m + 1 } } { \\alpha - \\beta } & = \\frac { 1 } { \\alpha - \\beta } \\sum _ { m \\geq 0 } \\left ( \\alpha \\cdot ( X \\alpha ) ^ m - \\beta ( X \\beta ) ^ m \\right ) = \\frac { 1 } { \\alpha - \\beta } \\left ( \\frac { \\alpha } { 1 - \\alpha X } - \\frac { \\beta } { 1 - \\beta X } \\right ) \\\\ & = \\frac { 1 } { ( 1 - \\alpha X ) ( 1 - \\beta X ) } . \\end{align*}"}
-{"id": "2931.png", "formula": "\\begin{align*} A ^ N = \\{ a \\in A : ( \\forall n \\in N ) ~ n . a = a \\} , \\end{align*}"}
-{"id": "6004.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\mathbb L _ 1 \\cdots \\mathbb L _ { n - 1 } \\mathbb L _ n F ^ n & = \\rho ^ { G _ n } \\left ( \\widetilde { F } ^ n ( W _ { ( n , 1 ) } , \\ldots , W _ { ( n , n ) } ) \\right ) = \\rho ^ { G _ n } \\left ( F \\left ( \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\delta _ { W _ { ( n , k ) } ( 1 ) } \\right ) \\right ) . \\end{align*}"}
-{"id": "3715.png", "formula": "\\begin{align*} \\widehat { \\bar { Y } } _ S = W _ B \\bar { y } _ B + ( 1 - W _ B ) \\bar { y } _ w \\qquad \\bar { y } _ w = \\frac { \\sum _ { i \\in S } y _ i / \\pi _ i } { \\sum _ { i \\in S } 1 / \\pi _ i } \\end{align*}"}
-{"id": "6991.png", "formula": "\\begin{align*} \\displaystyle { \\pi ( \\underline { x } ) = \\begin{cases} \\frac { q } { q ^ 2 - 1 } & \\ \\ \\underline { x } \\neq \\underline { 0 } \\\\ \\frac { 1 } { q + 1 } & \\ \\ \\underline { x } = \\underline { 0 } . \\\\ \\end{cases} } \\end{align*}"}
-{"id": "6164.png", "formula": "\\begin{align*} 9 \\sum ^ { T - 1 } _ { t = 0 } \\left ( F _ t + \\frac { D _ y ^ 2 } { 2 \\lambda _ t } \\right ) & = \\sum ^ { T - 1 } _ { t = 0 } \\frac { 1 } { ( t + 2 ) ^ 2 } + 9 \\sum ^ { T - 1 } _ { t = 0 } \\frac { D _ y ^ 2 } { 2 ( t + 2 ) } \\leq \\frac { \\pi ^ 2 } { 6 } + \\ln ( T + 2 ) \\frac { 9 D _ y ^ 2 } { 2 } . \\end{align*}"}
-{"id": "3557.png", "formula": "\\begin{align*} \\mathcal { M } _ { 3 , 3 } = \\left \\{ X _ { 0 } X _ { 1 } X _ { 2 } = X _ { 3 } X _ { 4 } X _ { 5 } \\right \\} \\subset \\mathbb { P } ^ { 5 } . \\end{align*}"}
-{"id": "5394.png", "formula": "\\begin{align*} \\frac { \\overline { b _ { n \\ , m + 6 } } } { c _ { n + 1 \\ , m + 3 } } \\cdot \\frac { \\overline { d _ { n + 1 \\ , m + 3 } } } { a _ { n m } } = \\frac { \\overline { d _ { n \\ , m + 6 } } } { a _ { n - 1 \\ , m + 3 } } \\cdot \\frac { \\overline { b _ { n - 1 \\ , m + 3 } } } { c _ { n m } } . \\end{align*}"}
-{"id": "8169.png", "formula": "\\begin{align*} B = \\{ X _ { a b } \\mid b \\in \\{ 1 , 2 , \\cdots , \\ell , \\underline { \\ell } , \\cdots , \\underline { 2 } , \\underline { 1 } \\} , \\ a \\in \\{ 1 , \\cdots , b \\} \\} . \\end{align*}"}
-{"id": "9355.png", "formula": "\\begin{align*} c _ i = [ y , x , \\overset { i - 1 } { \\ldots } , x ] z _ i = [ c _ { i - 1 } , y ] = [ y , x , \\overset { i - 2 } { \\ldots } , x , y ] . \\end{align*}"}
-{"id": "2478.png", "formula": "\\begin{align*} P ( x ) = \\int _ K x ^ n \\ , d \\mu \\end{align*}"}
-{"id": "3456.png", "formula": "\\begin{align*} x _ N ( k ) = N ( \\psi ( 1 - k + N ) - \\log N ) , \\end{align*}"}
-{"id": "8749.png", "formula": "\\begin{align*} \\begin{aligned} \\omega _ n ^ { ( \\alpha , \\sigma ) } \\approx \\widehat { \\omega } _ n ^ { ( \\alpha , \\sigma ) } = & { \\tau ^ { 1 + \\alpha } e ^ { - n \\sigma \\tau } } \\Delta { x } \\sum _ { j = - \\infty } ^ { \\infty } ( 1 + e ^ { x _ j } \\tau ) ^ { - 1 - n } \\phi ( x _ j ) \\\\ = & { \\tau ^ { 1 + \\alpha } e ^ { - n \\sigma \\tau } } \\sum _ { j = - \\infty } ^ { \\infty } w _ j ( 1 + \\lambda _ j \\tau ) ^ { - 1 - n } , \\end{aligned} \\end{align*}"}
-{"id": "5883.png", "formula": "\\begin{align*} \\phi ( C _ w ) = - \\left ( q ^ { \\frac { 1 } { 2 } } + q ^ { - \\frac { 1 } { 2 } } \\right ) t _ { r s _ i } + t _ r + t _ { r s _ i s _ j } . \\end{align*}"}
-{"id": "6721.png", "formula": "\\begin{align*} Q _ { 1 2 } = \\bar { q } _ 2 , Q _ { 2 2 } = \\dot { \\bar { q } } _ 2 , P _ { 2 2 } = \\lambda m ( \\dot { \\bar { q } } _ 1 + \\lambda \\ddot { \\bar { q } } _ 2 ) , \\end{align*}"}
-{"id": "7006.png", "formula": "\\begin{align*} a _ j ' & = \\frac { \\lambda _ j ^ \\ell { \\textsl { \\footnotesize R } } _ j ( 0 ) } { d _ j } = \\frac { \\lambda _ j ^ \\ell \\sin ( \\theta _ j ) } { d _ j } , \\\\ a _ j & = \\frac { \\lambda _ j ^ \\ell { \\textsl { \\footnotesize R } } _ j ( 0 ) } { d _ j } \\left ( \\frac { \\ell } { \\lambda _ j } - \\frac { d _ j ' } { d _ j } \\right ) = \\frac { \\lambda _ j ^ \\ell \\sin ( \\theta _ j ) } { d _ j } \\left ( \\frac { \\ell } { \\lambda _ j } - \\frac { d _ j ' } { d _ j } \\right ) , \\end{align*}"}
-{"id": "3931.png", "formula": "\\begin{align*} \\eta \\bigl ( \\nu ( \\lambda ) \\bigr ) ( f ) = & \\ \\bigl ( S \\circ \\iota \\circ \\nu ( \\lambda ) _ { \\Bbbk [ H ] } \\bigr ) ( S f ) = ( S \\circ \\iota ) \\bigl ( \\sum \\lambda ( f _ 1 ) \\otimes S f _ 2 \\bigr ) = \\\\ & \\ S \\bigl ( \\sum S \\bigl ( \\lambda ( f _ 1 ) \\bigr ) f _ 2 ( 1 ) \\bigr ) = \\lambda ( f ) \\in k [ G ] \\ , . \\end{align*}"}
-{"id": "8239.png", "formula": "\\begin{align*} \\pi ( b ( 2 i , k ) ) & \\geq \\min \\limits _ { j \\geq 1 } \\pi ( b ( 2 i - 1 , j ) ) + \\pi ( x _ 1 ( j , k ) ) \\\\ & \\geq \\min \\limits _ { j \\geq 1 } 2 i - 1 + \\left \\lfloor \\frac { 5 j - 3 } { 3 } \\right \\rfloor + \\left \\lfloor \\frac { 5 k - j + 2 } { 3 } \\right \\rfloor \\\\ & = 2 i - 1 + \\left \\lfloor \\frac { 5 k + 1 } { 3 } \\right \\rfloor \\\\ & = 2 i + 1 + \\left \\lfloor \\frac { 5 k - 5 } { 3 } \\right \\rfloor . \\end{align*}"}
-{"id": "9256.png", "formula": "\\begin{align*} v \\le 0 \\ { \\rm o n } \\ B _ R \\cap \\partial \\Omega , { \\rm a n d } \\ v ( 0 ) = 0 . \\end{align*}"}
-{"id": "1970.png", "formula": "\\begin{align*} 2 d _ { 3 3 } = d _ { 1 1 } + d _ { 2 1 } + \\sum _ { \\substack { \\ell \\in \\mathcal { N } ( 3 ) \\\\ \\ell \\neq 1 , 2 } } d _ { \\ell 1 } , \\end{align*}"}
-{"id": "5116.png", "formula": "\\begin{align*} \\left \\| T \\left ( \\sum _ { j \\in \\mathbb { S } _ { \\mathcal { X } } } c _ j x _ j \\right ) \\right \\| ^ p = \\left \\| \\sum _ { j \\in \\mathbb { S } _ { \\mathcal { X } } } c _ j y _ j \\right \\| ^ p \\leq b \\sum _ { j \\in \\mathbb { S } _ { \\mathcal { X } } } | c _ j | ^ p \\leq \\frac { b } { a } \\left \\| \\sum _ { j \\in \\mathbb { S } _ \\mathcal { X } } c _ j x _ j \\right \\| ^ p . \\end{align*}"}
-{"id": "9164.png", "formula": "\\begin{align*} \\Delta = \\{ ( s , s ) : s \\in G \\} \\end{align*}"}
-{"id": "1408.png", "formula": "\\begin{align*} r _ j : = \\frac { p _ j ' ( x _ j ) } { p _ j ( x _ j ) } \\end{align*}"}
-{"id": "4391.png", "formula": "\\begin{align*} C _ { 1 \\pm } : = \\{ T _ { 1 \\pm } = 0 \\} C _ 1 = \\{ T _ 1 = 0 \\} . \\end{align*}"}
-{"id": "1790.png", "formula": "\\begin{align*} \\mathbb { E } [ X _ { j } X _ { k } ] = \\frac { 1 } { 2 } \\left [ ( k - j + 1 ) ^ { 2 H } + ( k - j - 1 ) ^ { 2 H } \\right ] - ( k - j ) ^ { 2 H } , \\end{align*}"}
-{"id": "8912.png", "formula": "\\begin{align*} s \\Lambda & = \\omega ( 1 ) \\\\ \\mu & = \\omega ( n / s ^ 2 ) , \\end{align*}"}
-{"id": "6563.png", "formula": "\\begin{align*} m _ \\beta ^ P = \\binom { e ( S _ r ) - \\eta } { 2 } + 5 . \\end{align*}"}
-{"id": "9405.png", "formula": "\\begin{align*} f _ \\theta ( z ) : = \\frac { \\theta f ( z ) } { f ( z ) } - \\frac { k } { 1 2 } E _ 2 ( z ) , \\end{align*}"}
-{"id": "3044.png", "formula": "\\begin{align*} \\kappa _ { i } & = \\min \\left \\{ \\kappa ( \\sigma _ { i } ) , \\kappa ( \\sigma _ { i } ^ { 0 } ) , \\kappa ( \\sigma _ { i } ^ { 1 } ) , \\kappa ( \\sigma _ { i } ^ { 2 } ) , \\kappa ( b _ { i } ) \\right \\} . \\end{align*}"}
-{"id": "2103.png", "formula": "\\begin{align*} K : = ( 1 - 2 ^ { 1 - \\theta } ) ^ { - 1 } , \\theta : = \\frac { 1 } { p } + \\frac { 1 } { q } > 1 . \\end{align*}"}
-{"id": "1940.png", "formula": "\\begin{align*} ( j - 1 ) \\pi + \\epsilon _ j = \\chi \\cot \\epsilon _ j \\end{align*}"}
-{"id": "480.png", "formula": "\\begin{align*} \\mathcal { C } _ { m } : = \\bigl \\{ L ^ { m } \\mid L \\in S _ 1 \\bigr \\} \\subset S _ { m } . \\end{align*}"}
-{"id": "2227.png", "formula": "\\begin{align*} u \\in { \\mathcal X } _ T = { \\mathcal C } _ { \\rm w } \\Big ( [ 0 , T ] , M ^ { d / 2 } ( \\R ^ d ) \\cap M ^ p ( \\R ^ d ) \\Big ) \\cap { \\mathcal Y } _ T , \\end{align*}"}
-{"id": "8758.png", "formula": "\\begin{align*} { } _ F D _ { \\tau } ^ { \\alpha , \\sigma , \\gamma , m , n } U = - U _ n + f ( U _ n , t _ n ) , U _ 0 = u _ 0 , \\end{align*}"}
-{"id": "8359.png", "formula": "\\begin{align*} A _ 1 ( 0 , t ) = & 1 + \\frac { \\lambda ^ 2 } { 4 \\pi ^ 2 } \\frac { x ^ 4 + 5 x ^ 2 y ^ 2 } { | y | ( x ^ 2 + y ^ 2 ) ^ 3 } + \\frac { \\lambda ^ 2 y ( 3 x ^ 2 - y ^ 2 ) } { \\pi ^ 2 ( x ^ 2 + y ^ 2 ) ^ 3 } \\\\ = & 1 + \\frac { \\lambda ^ 2 } { 4 \\pi ^ 2 } \\frac { x ^ 4 + 5 x ^ 2 y ^ 2 - 1 2 x ^ 2 y ^ 2 + 4 y ^ 4 } { | y | ( x ^ 2 + y ^ 2 ) ^ 3 } \\\\ = & 1 + \\frac { \\lambda ^ 2 } { 4 \\pi ^ 2 } \\frac { x ^ 4 + 4 y ^ 4 - 7 x ^ 2 y ^ 2 } { | y | ( x ^ 2 + y ^ 2 ) ^ 3 } \\end{align*}"}
-{"id": "8499.png", "formula": "\\begin{align*} I m \\{ \\zeta ( \\alpha , t ) - z _ j ( t ) \\} \\geq & \\inf _ { \\alpha \\in \\mathbb { R } } I m \\{ \\zeta ( \\alpha , 0 ) - z _ j ( 0 ) \\} + ( \\frac { | \\lambda | } { 1 0 \\pi x ( 0 ) } - 6 \\epsilon - ( C \\epsilon ^ 2 + K _ s ^ { - 1 } \\epsilon ) ) t \\\\ \\geq & 1 + \\frac { | \\lambda | } { 1 8 \\pi x ( 0 ) } t . \\end{align*}"}
-{"id": "3148.png", "formula": "\\begin{align*} f ( v , u ) \\ E _ i ( u ) E _ i ( v ) = f ( u , v ) \\ E _ i ( v ) E _ i ( u ) , \\end{align*}"}
-{"id": "5885.png", "formula": "\\begin{align*} f = T _ 1 + T _ { s _ 0 } + \\sum _ { n = 1 } ^ { \\infty } q ^ { - 2 n } \\left ( T _ { ( s _ 1 s _ 0 ) ^ n } + T _ { s _ 0 ( s _ 1 s _ 0 ) ^ n } - q \\left ( T _ { ( s _ 0 s _ 1 ) ^ n } + T _ { s _ 1 ( s _ 0 s _ 1 ) ^ n } \\right ) \\right ) \\end{align*}"}
-{"id": "6656.png", "formula": "\\begin{align*} [ R _ { ( p ) } : I _ { ( p ) } ] & = [ R _ { ( p ) } : R _ { I ( p ) } ] [ R _ { ( p ) } : x R _ { ( p ) } ] , \\\\ [ R _ { ( p ) } : J _ { ( p ) } ] & = [ R _ { ( p ) } : R _ { J ( p ) } ] [ R _ { ( p ) } : y R _ { ( p ) } ] , \\\\ [ R _ { ( p ) } : I _ { ( p ) } J _ { ( p ) } ] & = [ R _ { ( p ) } : x R _ { I ( p ) } y R _ { J ( p ) } ] = \\\\ & = [ R _ { ( p ) } : R _ { I ( p ) } R _ { J ( p ) } ] [ R _ { ( p ) } : x R _ { ( p ) } ] [ R _ { ( p ) } : y R _ { ( p ) } ] . \\end{align*}"}
-{"id": "8411.png", "formula": "\\begin{align*} \\Big | \\frac { d } { d t } | z _ j ( t ) - z _ k ( t ) | ^ { - 1 } \\Big | = & \\Big | \\frac { ( z _ j ( t ) - z _ k ( t ) ) \\cdot ( \\dot { z } _ j ( t ) - \\dot { z } _ k ( t ) ) } { | z _ j ( t ) - z _ k ( t ) | ^ 3 } \\Big | \\leq | \\dot { z } _ j ( t ) - \\dot { z } _ k ( t ) | d _ p ( t ) ^ { - 2 } , \\end{align*}"}
-{"id": "2244.png", "formula": "\\begin{align*} \\kappa ( x ) = \\begin{cases} p ( x ) + | \\alpha ( x ) \\rangle \\langle \\alpha ' ( x ) | & C _ 2 ( x ) \\ne 0 \\\\ \\delta ( x - \\Omega ) & C _ 2 ( x ) = 0 \\end{cases} \\end{align*}"}
-{"id": "7036.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\alpha _ { j } , \\beta _ { j } } ( z ) = \\Gamma ( \\beta _ { j } ) z E _ { \\alpha _ { j } , \\beta _ { j } } ( z ) . \\end{align*}"}
-{"id": "8376.png", "formula": "\\begin{align*} [ \\partial _ t ^ 2 + i a \\partial _ { \\alpha } , \\mathfrak { H } ] u = 2 [ z _ { t t } , \\mathfrak { H } ] \\frac { \\bar { z } _ { t \\alpha } } { z _ { \\alpha } } + 2 [ z _ t , \\mathfrak { H } ] \\frac { \\bar { z } _ { t t \\alpha } } { z _ { \\alpha } } - \\frac { 1 } { \\pi i } \\int \\Big ( \\frac { z _ t ( \\alpha , t ) - z _ t ( \\beta , t ) } { z ( \\alpha , t ) - z ( \\beta , t ) } \\Big ) ^ 2 \\bar { z } _ { t \\beta } d \\beta \\end{align*}"}
-{"id": "4757.png", "formula": "\\begin{align*} \\frac { \\partial f ( \\xi ) } { \\partial \\xi _ i } = \\frac { \\xi _ i } { r ^ 2 } \\end{align*}"}
-{"id": "1464.png", "formula": "\\begin{align*} V _ { 5 } = - \\varepsilon \\int _ { \\mathbb { T } ^ { d } } ( \\mu ^ { 1 } - \\mu ^ { 2 } ) ( \\mu ^ { 2 } ) \\sum _ { i = 1 } ^ { d } \\left [ \\Theta _ { p _ { i } x _ { i } } ( t , x , \\mu ^ { 1 } , D w ^ { 1 } ) - \\Theta _ { p _ { i } x _ { i } } ( t , x , \\mu ^ { 2 } , D w ^ { 2 } ) \\right ] \\ d x , \\end{align*}"}
-{"id": "8160.png", "formula": "\\begin{align*} S ( a _ j ) S ^ { - 1 } ( b _ k ) \\otimes S ^ { - 2 } ( b _ i ) b _ j \\otimes a _ k a _ i & = a _ j S ^ { - 1 } ( b _ k ) \\otimes S ^ { - 1 } \\ ! \\left ( b _ j S ^ { - 1 } ( b _ i ) \\right ) \\otimes a _ k a _ i = ( \\mathrm { i d } \\otimes S ^ { - 1 } \\otimes \\mathrm { i d } ) ( R _ { 1 2 } R _ { 3 1 } ^ { - 1 } R _ { 3 2 } ^ { - 1 } ) \\\\ & = ( \\mathrm { i d } \\otimes S ^ { - 1 } \\otimes \\mathrm { i d } ) ( R _ { 3 2 } ^ { - 1 } R _ { 3 1 } ^ { - 1 } R _ { 1 2 } ) = S ^ { - 1 } ( b _ k ) S ( a _ j ) \\otimes b _ j S ^ { - 2 } ( b _ i ) \\otimes a _ i a _ k \\end{align*}"}
-{"id": "5121.png", "formula": "\\begin{align*} \\mathring H _ { m i n } \\subseteq T = T ^ * \\subseteq H _ { m a x } , \\end{align*}"}
-{"id": "5126.png", "formula": "\\begin{align*} d : = \\sum _ { \\substack { j , m _ j , k _ j \\\\ ( k _ j + \\lambda ) ^ 2 + \\mu ^ 2 - \\nu ^ 2 < 1 / 4 } } 1 \\ , = \\sum _ { \\substack { k \\in \\Z \\setminus \\{ 0 \\} \\\\ ( k + \\lambda ) ^ 2 + \\mu ^ 2 - \\nu ^ 2 < 1 / 4 } } 2 \\abs { k } . \\end{align*}"}
-{"id": "6731.png", "formula": "\\begin{align*} P _ { 2 k l } = \\frac { 1 } { 4 c ^ 4 } \\frac { \\partial ^ 2 h _ { k l } } { \\partial t ^ 2 } , \\end{align*}"}
-{"id": "6189.png", "formula": "\\begin{align*} R - \\rho - x _ n = - \\frac { | x ' | ^ 2 } { R - x _ n + \\rho } . \\end{align*}"}
-{"id": "8740.png", "formula": "\\begin{align*} \\left [ D ^ { \\sigma , \\alpha } _ { 0 , t } ( u ( t ) - u ( 0 ) ) \\right ] _ { t = t _ n } = D _ { \\tau } ^ { \\alpha , \\sigma , \\gamma , m , n } u + O ( \\tau ^ { p } ) . \\end{align*}"}
-{"id": "6702.png", "formula": "\\begin{align*} \\left ( \\alpha ' _ { i k } + \\beta ' _ { i l k } \\ , \\dot { \\bar { q } } _ l - \\frac { d } { d t } \\beta _ { i k } \\right ) \\left ( m \\ddot { q } _ i + \\omega _ { i j } \\dot { q } _ j + \\frac { \\partial V } { \\partial q _ i } \\right ) = 0 . \\end{align*}"}
-{"id": "9118.png", "formula": "\\begin{align*} p _ \\gamma ( E ) : = \\alpha _ d ( F ) P ( E ) - \\alpha _ d ( E ) P ( F ) \\le 0 \\end{align*}"}
-{"id": "5369.png", "formula": "\\begin{align*} V ( c , 2 t ) | _ K = V _ { 1 \\ , 2 t } \\oplus \\bigoplus _ { n , m \\in \\Z , \\ , n > 1 } V _ { n m } \\end{align*}"}
-{"id": "8875.png", "formula": "\\begin{align*} G _ { f ; M } & = \\min _ { i _ 1 , i _ 2 , i _ 3 = 0 , \\dots , M - 1 } \\left [ f _ + \\left ( \\frac { i _ 1 } { M } , \\frac { i _ 2 } { M } , \\frac { i _ 3 } { M } \\right ) - f _ - \\left ( \\frac { i _ 1 + 1 } { M } , \\frac { i _ 2 + 1 } { M } , \\frac { i _ 3 + 1 } { M } \\right ) \\right ] . \\end{align*}"}
-{"id": "1897.png", "formula": "\\begin{align*} T _ { \\rm r i g h t } = \\sqrt { \\frac { 1 + v } { 1 - v } } L \\qquad \\mbox { a n d } T _ { \\rm l e f t } = \\sqrt { \\frac { 1 - v } { 1 + v } } L , \\end{align*}"}
-{"id": "307.png", "formula": "\\begin{align*} \\hat { p } ^ { - 1 } ( \\Theta ) \\ ; \\ ; = \\ ; \\bigcup _ { e \\in \\ker ( \\hat { p } ) } ( D + e ) , \\end{align*}"}
-{"id": "4065.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n ^ { 1 + 1 / \\theta } } K ^ { \\alpha , \\theta } _ { V , n } \\left ( \\frac { x } { n ^ { 1 + 1 / \\theta } } , \\frac { y } { n ^ { 1 + 1 / \\theta } } \\right ) = \\mathbb K ^ { ( \\alpha , \\theta ) } ( x , y ) \\end{align*}"}
-{"id": "4071.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } p _ n ( x ) x ^ { k \\theta } w ( x ) d x = 0 , k = 0 , 1 , \\ldots , n - 1 . \\end{align*}"}
-{"id": "3123.png", "formula": "\\begin{align*} S _ { 0 , K _ S } ^ { [ 1 , 1 , c _ 2 - 2 ] } & \\cong S _ { K _ S } ^ { [ 1 , c _ 2 - 2 ] } \\\\ & = \\{ p \\in S , I \\in S ^ { [ c _ 2 - 2 ] } , C \\in | K _ S | : Z _ I \\subset C \\cup p \\} \\end{align*}"}
-{"id": "9319.png", "formula": "\\begin{align*} x _ { i j } , ~ y _ i \\in \\{ 0 , 1 \\} , ~ i , j = 1 , \\dots , n . \\end{align*}"}
-{"id": "3468.png", "formula": "\\begin{align*} P _ 5 : = P _ 6 + P _ 7 + P _ 8 < \\frac { \\epsilon } { 4 } . \\end{align*}"}
-{"id": "7793.png", "formula": "\\begin{align*} Q ( G ) & = \\max \\big \\{ { \\rm V o l } _ { d - 1 } ( \\pi _ { \\xi } ( G ) ) : \\xi \\in \\mathbb S ^ { d - 1 } \\big \\} , \\end{align*}"}
-{"id": "1524.png", "formula": "\\begin{align*} a _ i Z _ i \\cdot \\Theta ^ { g - 1 } \\le H \\cdot \\Theta ^ { g - 1 } = n \\cdot g ! . \\end{align*}"}
-{"id": "6695.png", "formula": "\\begin{gather*} 0 \\to \\nu ( r ) \\to \\Omega _ { Y , d = 0 } ^ { r } \\overset { C - 1 } { \\longrightarrow } \\Omega _ { Y } ^ { r } \\to 0 , \\\\ 0 \\to \\nu ( r ) \\to \\Omega _ { Y } ^ { r } \\overset { C ^ { - 1 } - 1 } { \\longrightarrow } \\Omega _ { Y } ^ { r } / d \\Omega _ { Y } ^ { r - 1 } \\to 0 \\end{gather*}"}
-{"id": "6972.png", "formula": "\\begin{align*} q _ { t _ 0 } ( \\varphi ( y _ 0 ) , \\varphi ' ( y _ 0 ) y _ 1 ) ( \\varphi ' ( y _ 0 ) ) ^ 2 = q _ { y _ 0 } ( y _ 0 , y _ 1 ) - { \\rm S } ( \\varphi ) ( y _ 0 ) . \\end{align*}"}
-{"id": "9421.png", "formula": "\\begin{align*} d ( ( \\log _ { e } | F ( z ) | ^ 2 ) \\overline { g ( z ) } d \\overline { z } ) = \\frac { \\partial _ z F ( z ) \\overline { F ( z ) } } { F ( z ) \\overline { F ( z ) } } \\overline { g ( z ) } d z d \\overline { z } = \\frac { \\partial _ z F ( z ) } { F ( z ) } \\overline { g ( z ) } ( - 2 i ) d x d y . \\end{align*}"}
-{"id": "6208.png", "formula": "\\begin{align*} I _ { \\lambda V } ( t ' ) = \\frac { 1 } { \\sqrt { \\lambda } } I _ V ( t _ 0 ) . \\end{align*}"}
-{"id": "4262.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\delta ( a ( x _ { 0 } ) \\lvert d v _ { j } \\rvert ^ { p - 2 } d v _ { j } ) ) & = 0 & & \\frac { 1 } { 2 } B _ { j } , \\\\ \\delta v _ { j } & = 0 & & \\frac { 1 } { 2 } B _ { j } , \\\\ \\nu \\wedge v _ { j } & = \\nu \\wedge w _ { j } & & \\partial \\left ( \\frac { 1 } { 2 } B _ { j } \\right ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "5946.png", "formula": "\\begin{align*} \\textstyle d Y ( t ) = - g ^ * ( t , Z ( t ) ) \\ , d t + Z ( t ) \\ , d W ( t ) , Y ( 1 ) = X , \\end{align*}"}
-{"id": "2406.png", "formula": "\\begin{align*} M & : = \\tilde { R } _ { ( i , j ) ( f , b ) } = \\begin{array} { l l l l l l l } & i & p & a & j & f & b \\\\ i & 0 & 0 & 0 & 1 & 2 & 2 \\\\ p & & 0 & 0 & 1 & 1 & 2 \\\\ a & & & 0 & 0 & 1 & 1 \\\\ j & & & & 0 & 0 & 0 \\\\ f & & & & & 0 & 0 \\\\ b & & & & & & 0 \\end{array} \\end{align*}"}
-{"id": "5253.png", "formula": "\\begin{align*} r ( x ) = c + \\int _ { x _ 1 } ^ { x } b ( t ) d t \\end{align*}"}
-{"id": "8950.png", "formula": "\\begin{align*} \\mathbb { P } ( Y = 1 \\ , | \\ , X ) = \\left ( 1 + \\frac { 1 - p } { p } \\exp \\left \\{ \\langle X , m _ 0 - m _ 1 \\rangle _ { K } - \\mathbb { E } _ { m _ 1 } [ \\langle X , m _ 0 - m _ 1 \\rangle _ { K } ] - \\frac { 1 } { 2 } \\| m _ 0 - m _ 1 \\| ^ 2 _ { K } \\right \\} \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "7970.png", "formula": "\\begin{align*} J _ q ^ A \\subseteq \\bigsqcup _ { j = 2 } ^ m \\{ j _ 1 , \\dots , j _ { d _ j - 1 } \\} , \\end{align*}"}
-{"id": "8055.png", "formula": "\\begin{align*} \\Omega _ R / d R \\cong \\mathbb { C } \\omega _ 0 \\oplus \\bigoplus _ { i = 3 } ^ { 4 } U _ i ^ { \\frac { ( 1 - ( - 1 ) ^ k ) n } { 2 k } } \\oplus \\bigoplus _ { h = 1 } ^ { k - 1 } V _ { h } ^ { \\oplus \\frac { ( 1 - ( - 1 ) ^ h ) n } { k } } . \\end{align*}"}
-{"id": "6902.png", "formula": "\\begin{align*} q _ 1 = u ^ 3 _ { 1 0 } u ^ 2 _ { 0 1 } - u ^ 3 _ { 0 1 } u ^ 2 _ { 1 0 } - u ^ 4 _ { 1 0 } u ^ 1 _ { 0 1 } + u ^ 4 _ { 0 1 } u ^ 1 _ { 1 0 } \\quad { \\rm a n d } q _ 2 = u ^ 4 _ { 1 0 } u ^ 2 _ { 0 1 } - u ^ 4 _ { 0 1 } u ^ 2 _ { 1 0 } + u ^ 3 _ { 1 0 } u ^ 1 _ { 0 1 } - u ^ 3 _ { 0 1 } u ^ 1 _ { 1 0 } \\ , . \\end{align*}"}
-{"id": "3455.png", "formula": "\\begin{align*} t _ w = \\frac { 1 } { 2 } + w + \\frac { C _ { t _ w } } { N } , \\end{align*}"}
-{"id": "7111.png", "formula": "\\begin{align*} d \\Psi _ { m } = - m \\Psi _ { m } \\xi _ { m } = m \\Psi _ { m } \\begin{pmatrix} \\psi & - \\psi ^ { 2 } \\\\ 1 & - \\psi \\end{pmatrix} \\eta . \\end{align*}"}
-{"id": "3353.png", "formula": "\\begin{align*} L : = \\lfloor 2 \\log _ q ( \\ell + \\deg ( M ) + 1 ) \\rfloor . \\end{align*}"}
-{"id": "4841.png", "formula": "\\begin{align*} S ' = \\{ y \\in D \\mid x y = y x x \\in S \\} ; \\end{align*}"}
-{"id": "9084.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } g ( x ) | u _ n ( x ) | ^ p \\ , d x = 1 \\int _ { \\mathbb { R } ^ n } g ( x ) | v _ n ( x ) | ^ p \\ , d x = - 1 \\end{align*}"}
-{"id": "2679.png", "formula": "\\begin{align*} f ( x ; z ) = \\eta _ { - } x ( z + 1 ) + \\log ( \\tau z - 1 ) - \\log ( z - \\tau ) , \\end{align*}"}
-{"id": "3509.png", "formula": "\\begin{align*} \\sqrt { \\gamma } \\int ^ { \\frac { 1 } { 2 } + i \\infty } _ { \\frac { 1 } { 2 } - i \\infty } \\frac { d s } { 2 \\pi i } \\frac { e ^ { \\frac { 1 } { 2 } s ^ 2 - y s } } { s \\Gamma ( \\frac { s } { \\sqrt { \\gamma } } ) } = \\int ^ { \\frac { 1 } { 2 } + i \\infty } _ { \\frac { 1 } { 2 } - i \\infty } \\frac { d s } { 2 \\pi i } \\frac { e ^ { \\frac { 1 } { 2 } s ^ 2 - y s } } { \\Gamma ( \\frac { s } { \\sqrt { \\gamma } } + 1 ) } , \\end{align*}"}
-{"id": "508.png", "formula": "\\begin{align*} Q : = \\omega _ { 2 , 0 } \\odot \\omega _ { 2 , 2 } - \\omega _ { 2 , 1 } \\odot \\omega _ { 2 , 1 } . \\end{align*}"}
-{"id": "3766.png", "formula": "\\begin{align*} f _ { 1 } ^ { ( 1 ) } = 0 , \\ , \\forall l \\in \\mathbb { N } ^ { * } . \\end{align*}"}
-{"id": "862.png", "formula": "\\begin{align*} \\norm { \\overline { B } } _ F ^ 2 = \\norm { u ^ \\ast } _ 2 ^ 2 + \\norm { v } _ 2 ^ 2 - \\big ( { u ^ \\ast } ^ T u \\big ) ^ 2 . \\end{align*}"}
-{"id": "7117.png", "formula": "\\begin{align*} D \\lambda _ k ( q ) ( h ) = \\frac { 1 } { \\| \\phi _ k ( q ) \\| ^ 2 _ { L ^ 2 } } \\int ^ \\pi _ 0 \\phi _ k ^ 2 ( q ) h \\ , d x , ~ ~ \\forall \\ , q , h \\in L ^ 2 . \\end{align*}"}
-{"id": "3493.png", "formula": "\\begin{align*} \\frac { d } { d \\phi } f _ { M } ( q _ { M } ( \\phi ) ; \\theta ) = ( v _ { M } ( \\phi ) - v _ { M } ( \\theta ) ) \\frac { d } { d \\phi } q _ { M } ( \\phi ) . \\end{align*}"}
-{"id": "6816.png", "formula": "\\begin{align*} A ( t ) = \\cosh ( s ( t ) ) \\begin{pmatrix} \\cos ( \\mu ( t ) ) & - \\sin ( \\mu ( t ) ) \\\\ \\sin ( \\mu ( t ) ) & \\cos ( \\mu ( t ) ) \\end{pmatrix} + \\sinh ( s ( t ) ) \\begin{pmatrix} \\cos ( \\theta ( t ) ) & \\sin ( \\theta ( t ) ) \\\\ \\sin ( \\theta ( t ) ) & - \\cos ( \\theta ( t ) ) \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "3685.png", "formula": "\\begin{align*} E ( \\hat { y } _ i | z _ i , x _ i , i \\in S ) = E ( y _ i | x _ i , i \\in U ) \\neq E ( y _ i | z _ i , x _ i , i \\in U ) \\end{align*}"}
-{"id": "7475.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { \\aleph } \\rho _ s ( \\gamma _ j ) \\cdot \\prod _ { i = 1 } ^ b \\rho _ s ( \\delta _ i ) = 1 . \\end{align*}"}
-{"id": "6510.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\displaystyle d J _ t ^ \\varepsilon = A J _ t ^ \\varepsilon d t + \\left [ f ( X _ { t ( \\delta ) } ^ \\varepsilon , \\hat { Y } _ t ^ \\varepsilon ) - \\bar { f } ( X _ { t ( \\delta ) } ^ \\varepsilon ) \\right ] d t , \\\\ J _ 0 ^ \\varepsilon = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "4301.png", "formula": "\\begin{align*} \\textup { s a t } ( n , K _ 3 , C _ { 2 k + 1 } ) = 0 \\end{align*}"}
-{"id": "6121.png", "formula": "\\begin{align*} \\big ( f _ { 0 , d } , f _ { 1 , d } , f _ { 2 , d } \\big ) = \\alpha _ { d - \\mu _ 1 } \\big ( A _ { 0 , \\mu _ 1 } , A _ { 1 , \\mu _ 1 } , A _ { 2 , \\mu _ 1 } \\big ) + \\beta _ { d , - \\mu _ 2 } \\big ( B _ { 0 , \\mu _ 2 } , B _ { 1 , \\mu _ 2 } , B _ { 2 , \\mu _ 2 } \\big ) , \\end{align*}"}
-{"id": "8367.png", "formula": "\\begin{align*} \\bar { v } ( z , t ) = F ( z , t ) - \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i } { 2 \\pi } \\frac { 1 } { z - z _ j ( t ) } . \\end{align*}"}
-{"id": "3462.png", "formula": "\\begin{align*} \\frac { \\Gamma ( t ) } { \\Gamma ( s ) } = \\frac { \\sigma } { \\tau } \\Big ( 1 + C _ { \\sigma , \\tau } \\big ( | \\sigma | + | \\tau | \\big ) \\sqrt { \\frac { N } { M + 1 } } \\Big ) , \\frac { 1 } { s - t } = \\frac { 1 } { \\tau - \\sigma } \\sqrt { \\frac { M + 1 } { N } } , \\end{align*}"}
-{"id": "2445.png", "formula": "\\begin{gather*} F _ { r , s } ( x , y , p , q , z ) = q ^ 2 + r p ^ 2 + s y ^ 2 \\end{gather*}"}
-{"id": "144.png", "formula": "\\begin{align*} [ \\pi + \\lambda X , \\pi + \\lambda X ] = \\lambda ^ 2 [ X , X ] \\end{align*}"}
-{"id": "3897.png", "formula": "\\begin{align*} \\delta ( g ) = \\max \\{ \\ | \\lambda _ 1 - 1 | + | \\lambda _ i - 1 | \\ : \\ i = 2 , \\ldots , n + 1 \\} . \\end{align*}"}
-{"id": "5447.png", "formula": "\\begin{align*} V _ 0 \\ = \\ \\sup _ { \\alpha \\in \\mathcal A } J ( \\alpha ) . \\end{align*}"}
-{"id": "583.png", "formula": "\\begin{align*} \\mathcal { X } : = \\{ 0 \\} , Z ^ 0 _ t : = Z ^ 0 _ 0 + \\alpha t + \\sigma B _ t , \\ ; \\ ; t \\in [ 0 , T ] , \\kappa ( 0 , \\cdot ) : = \\delta _ 0 , \\end{align*}"}
-{"id": "4517.png", "formula": "\\begin{align*} D _ x ( F ) = \\{ | x - y | : y \\in F \\} . \\end{align*}"}
-{"id": "5407.png", "formula": "\\begin{align*} J ( x ) = { \\left . \\frac { d } { d s } \\right | } _ { s = 0 } \\gamma _ s ( 0 ) = \\dot \\beta ( t _ 0 ) . \\end{align*}"}
-{"id": "4160.png", "formula": "\\begin{align*} E _ { i n } ^ { ( 1 ) } ( z ) T _ { \\alpha } ^ { - 1 } \\frac { 2 \\pi } { \\sqrt { 3 } } \\left ( n ^ 3 f ( z ) \\right ) ^ { - \\frac { 2 \\beta } { 3 } } L _ { \\alpha } ( n ^ 3 f ( z ) ) D _ 2 ^ n ( z ) \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & z ^ { \\beta } & 0 \\\\ 0 & 0 & z ^ { \\beta } \\end{pmatrix} = E _ { o u t } ^ { ( 1 ) } ( z ) N ( z ) , | z | = r _ n , \\end{align*}"}
-{"id": "4212.png", "formula": "\\begin{align*} E _ { i n } ^ { - 1 } ( y _ n ) E _ { i n } ( x _ n ) = \\mathbb { I } + \\mathcal { O } \\left ( ( x - y ) n ^ { - \\frac { 1 } { 2 } } \\right ) \\end{align*}"}
-{"id": "5626.png", "formula": "\\begin{align*} \\int _ 0 ^ { + \\infty } u _ { x x } \\bar { u } _ t \\ d x = \\alpha \\ , \\int _ 0 ^ { + \\infty } u \\bar { u } _ t v \\ , d x + \\beta \\ , \\int _ 0 ^ { + \\infty } | u | ^ 2 u \\bar { u } _ t \\ , d x . \\end{align*}"}
-{"id": "3330.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\lambda \\vdash n \\\\ \\ell ( \\lambda ) = t } } w ( B _ { \\lambda , \\mu } ) = \\sum _ { \\substack { b _ 1 , \\ldots , b _ { r } \\ge 1 \\\\ b _ 1 + \\ldots + b _ r = t } } \\sum _ { \\substack { \\forall 1 \\le i \\le r : \\\\ a _ { i , 1 } , \\ldots , a _ { i , b _ i } \\ge 1 \\\\ \\sum _ { j = 1 } ^ { b _ i } a _ { i , j } = \\mu _ i } } \\prod _ { i = 1 } ^ { r } a _ { i , b _ i } . \\end{align*}"}
-{"id": "2097.png", "formula": "\\begin{align*} P _ { \\rm e r r o r } = \\left ( \\frac { 1 } { 2 } \\right ) ^ { M + 1 } \\sum _ { b _ 0 , \\cdots , b _ M } { { \\rm P r } ( E | b _ 0 , b _ 1 , \\cdots , b _ M ) } , \\end{align*}"}
-{"id": "6985.png", "formula": "\\begin{align*} \\begin{array} { l l } 1 - 2 \\cos \\left ( \\frac { 4 \\pi s } { p + 1 } \\right ) & = 1 - 2 \\left ( 1 - \\frac { 2 \\pi ^ 2 s ^ 2 } { ( p + 1 ) ^ 2 } + O \\left ( \\frac { r ^ 4 } { p ^ 4 } \\right ) \\right ) = - 1 + \\frac { 4 \\pi ^ 2 s ^ 2 } { ( p + 1 ) ^ 2 } + O \\left ( \\frac { r ^ 4 } { p ^ 4 } \\right ) \\\\ & = - 1 + \\frac { 4 \\pi ^ 2 s ^ 2 } { p ^ 2 } \\left ( 1 + O \\left ( \\frac { 1 } { p } \\right ) \\right ) + O \\big ( \\frac { r ^ 4 } { p ^ 4 } \\big ) \\\\ & = - 1 + \\frac { 4 \\pi ^ 2 s ^ 2 } { p ^ 2 } + O \\big ( \\frac { s ^ 2 } { p ^ 3 } \\big ) . \\end{array} \\end{align*}"}
-{"id": "60.png", "formula": "\\begin{align*} a = a ( x ) \\ , \\dot = \\int _ { 0 } ^ x k ( y ) \\ , d y \\ , , \\end{align*}"}
-{"id": "6131.png", "formula": "\\begin{align*} X ^ { a _ k } Y ^ { b _ k } = \\bigg ( \\sum _ { \\ell = 1 } ^ m \\Omega ( R _ \\ell ) G ^ * _ \\ell \\bigg ) . \\end{align*}"}
-{"id": "3770.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\nabla \\cdot ( \\gamma _ 0 \\nabla u _ 0 ) = 0 \\quad \\quad \\Omega , \\\\ & \\gamma _ 0 \\nabla u _ 0 \\cdot \\nu = h \\quad \\quad \\partial \\Omega , \\\\ & \\int _ { \\Omega } u _ 0 = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "7914.png", "formula": "\\begin{align*} R _ { 2 \\hat { n } + 1 } ( p ( x ) ) = R _ { 2 \\hat { n } + 1 } \\left ( \\sum _ { i = 0 } ^ { \\hat { n } } a _ { 2 i } x ^ { 2 i } \\right ) + R _ { 2 \\hat { n } + 1 } \\left ( \\sum _ { i = 0 } ^ { \\hat { n } } a _ { 2 i + 1 } x ^ { 2 i + 1 } \\right ) \\end{align*}"}
-{"id": "8071.png", "formula": "\\begin{align*} \\psi _ c ^ + ( \\overline { \\omega _ { \\frac { n + 3 } { 2 } } } ) = - \\overline { \\omega _ { \\frac { n + 3 } { 2 } } } \\mbox { a n d } \\phi _ { \\zeta } ^ + ( \\overline { \\omega _ { \\frac { n + 3 } { 2 } } } ) = \\zeta ^ { n / 2 } \\overline { \\omega _ { \\frac { n + 3 } { 2 } } } = - \\overline { \\omega _ { \\frac { n + 3 } { 2 } } } . \\end{align*}"}
-{"id": "5218.png", "formula": "\\begin{align*} 1 \\leq \\limsup _ { t \\rightarrow + \\infty } \\frac { U ( t - ) } { U ( t ) } \\leq \\lim _ { t \\rightarrow + \\infty } \\frac { U ( t x ) } { U ( t ) } = x ^ { \\rho } . \\end{align*}"}
-{"id": "1661.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 H ^ c } { \\partial p _ n \\partial q _ l } \\Big | _ { z ^ \\bullet } = \\frac { \\partial ^ 2 H ^ c } { \\partial p _ n \\partial p _ l } \\Big | _ { z ^ \\bullet } = 0 \\quad \\frac { \\partial ^ 2 H ^ c } { \\partial q _ n \\partial q _ l } \\Big | _ { z ^ \\bullet } = \\frac { \\partial ^ 2 H ^ c } { \\partial q _ n \\partial p _ l } \\Big | _ { z ^ \\bullet } = 0 . \\end{align*}"}
-{"id": "4461.png", "formula": "\\begin{align*} 0 < \\psi \\leq K _ 0 : = \\min _ { p \\in O } W _ k ( p ) . \\end{align*}"}
-{"id": "9151.png", "formula": "\\begin{align*} \\overline { \\Theta } _ { i } ^ { ( \\partial \\Omega ) } ( \\overline { \\mathbf { r } } ) \\equiv \\overline { \\Theta } _ { i } ^ { ( \\partial \\Omega ) } ( \\overline { \\mathbf { r } } _ { i } ) = \\overline { \\Theta } \\left ( \\left \\vert \\mathbf { r } _ { i } - \\mathbf { r } _ { W i } \\right \\vert - \\frac { \\sigma } { 2 } \\right ) , \\end{align*}"}
-{"id": "7150.png", "formula": "\\begin{align*} \\nu _ { R } \\left ( B \\right ) : = \\int 1 _ { B } \\left ( y / R \\right ) \\nu \\left ( d y \\right ) , B \\in \\mathcal { B } \\left ( \\mathbf { R } _ { 0 } ^ { d } \\right ) , ~ \\tilde { \\nu } _ { R } \\left ( d y \\right ) : = w \\left ( R \\right ) \\nu _ { R } \\left ( d y \\right ) . \\end{align*}"}
-{"id": "6828.png", "formula": "\\begin{align*} N = d y - d y ^ T = 0 \\ , . \\end{align*}"}
-{"id": "5452.png", "formula": "\\begin{align*} \\hat I _ t \\ = \\ \\sum _ { n \\geq 0 } \\hat \\eta _ n \\ , 1 _ { [ \\hat T _ n , \\hat T _ { n + 1 } ) } ( t ) , \\qquad t \\geq 0 , \\end{align*}"}
-{"id": "5056.png", "formula": "\\begin{align*} h = \\sum \\limits _ { j \\in \\mathbb { J } } \\Psi _ j ^ * A _ j h \\iff \\sum \\limits _ { j \\in \\mathbb { J } } ( 2 - c _ j ) \\langle A _ j h , \\Psi _ j h \\rangle = \\| h \\| ^ 2 \\iff \\sum \\limits _ { j \\in \\mathbb { J } } c _ j ^ 2 \\| A _ j h \\| ^ 2 = \\| h \\| ^ 2 . \\end{align*}"}
-{"id": "6470.png", "formula": "\\begin{align*} p _ { 0 } ' = \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} \\oplus 0 \\qquad p _ { 1 } ' = \\begin{pmatrix} 0 & 0 \\\\ 0 & 1 \\end{pmatrix} \\oplus 1 . \\end{align*}"}
-{"id": "8173.png", "formula": "\\begin{align*} \\begin{aligned} & X _ { { 1 2 } } ( - j - 1 ) \\sp { b _ { { 1 2 } } } X _ { { 1 1 } } ( - j - 1 ) \\sp { b _ { { 1 1 } } } X _ { { 2 2 } } ( - j ) \\sp { a _ { 2 2 } } \\ , , b _ { { 1 2 } } + b _ { { 1 1 } } + a _ { { 2 2 } } = k + 1 , \\\\ & X _ { { 2 2 } } ( - j - 1 ) \\sp { b _ { 2 2 } } X _ { { 1 2 } } ( - j - 1 ) \\sp { b _ { 1 2 } } X _ { { 2 2 } } ( - j ) \\sp { a _ { 2 2 } } \\ , , b _ { 2 2 } + b _ { 1 2 } + a _ { 2 2 } = k + 1 . \\end{aligned} \\end{align*}"}
-{"id": "9562.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - 1 ) ^ { n - 1 } q ^ { \\frac { n ( n + 1 ) } { 2 } } } { ( 1 - c q ^ n ) } = \\frac { 1 } { 1 - c } \\left ( 1 - \\frac { ( q ) _ N } { ( c q ) _ N } \\right ) . \\end{align*}"}
-{"id": "782.png", "formula": "\\begin{align*} Z _ { i _ 0 } : = \\left ( 1 - \\prod _ { \\substack { i = 1 \\\\ i \\neq i _ 0 } } ^ { s } ( \\rho _ { i _ 0 } - \\rho _ i ) \\right ) . \\end{align*}"}
-{"id": "6355.png", "formula": "\\begin{align*} \\mathcal { P } _ { \\rm o u t } ^ M & = { \\rm P r o b } \\left \\{ \\mathbb { P } _ { e , M | p , h } > \\mathbb { P } _ { e , t } \\right \\} \\\\ & \\simeq { \\rm P r o b } \\left \\{ g _ { 1 1 } h _ { 1 1 } - g _ { 2 1 } h _ { 2 1 } < \\mathcal { A } _ { t h 1 } \\right \\} , \\end{align*}"}
-{"id": "9471.png", "formula": "\\begin{align*} & \\vert u \\vert _ { C ^ { \\alpha } ( \\overline { \\Omega } ) } = \\displaystyle \\sup _ { x , y \\in \\overline { \\Omega } , x \\neq y } \\frac { \\vert u ( x ) - u ( y ) \\vert } { \\vert x - y \\vert ^ { \\alpha } } \\\\ & C ^ { 0 , \\alpha } ( \\overline { \\Omega } ) = \\{ u \\in C ( \\overline { \\Omega } ) : | u | _ { C ^ { \\alpha } ( \\overline { \\Omega } ) } < \\infty \\} . \\end{align*}"}
-{"id": "6335.png", "formula": "\\begin{align*} f _ { h _ a } ( h _ a ) = \\frac { 1 } { h _ a \\sigma _ { l , h _ a } \\sqrt { 2 \\pi } } \\exp \\left ( - \\frac { \\left ( \\ln h _ a - \\mu _ { l , h _ a } \\right ) ^ 2 } { 2 \\sigma ^ 2 _ { l , h _ a } } \\right ) . \\end{align*}"}
-{"id": "8524.png", "formula": "\\begin{align*} \\norm { \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { \\pi } \\frac { 2 D _ t \\zeta \\dot { z } _ j } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 3 } } _ { H ^ s } \\leq & 2 \\| D _ t \\zeta \\| _ { H ^ s } \\norm { \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { \\pi } \\frac { \\dot { z } _ j } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 3 } } _ { H ^ s } \\\\ \\leq & 1 2 \\epsilon K _ s ^ { - 1 } \\epsilon d _ I ( t ) ^ { - 5 / 2 } \\leq K _ s ^ { - 1 } \\epsilon ^ 2 d _ I ( t ) ^ { - 5 / 2 } . \\end{align*}"}
-{"id": "7532.png", "formula": "\\begin{align*} \\mathbb { L } _ \\nu ( u _ 0 ) ( x , t ) = g _ { \\nu t } * u _ 0 ( x ) = \\int _ { \\mathbb { R } ^ d } \\frac { 1 } { ( 4 \\pi \\nu t ) ^ { \\frac { d } { 2 } } } e ^ { - \\frac { | x - y | ^ 2 } { 4 \\nu t } } u _ 0 ( y ) d y . \\end{align*}"}
-{"id": "9648.png", "formula": "\\begin{align*} \\bigvee _ { n \\geq 0 } ( T ^ { ( a ) } ) ^ { - n } X \\mathcal H = \\mathcal H ^ { ( a ) } . \\end{align*}"}
-{"id": "2737.png", "formula": "\\begin{align*} \\Re { f ( z _ { 4 } ) } - \\Re { f ( z _ { 0 } ) } = - \\frac { 2 ( 1 - \\tau ) } { 1 + \\tau } + \\frac { 1 } { 2 } \\log \\Big ( 1 + \\frac { 4 ( 1 - \\tau ) } { 1 + \\tau } \\Big ) < 0 , \\end{align*}"}
-{"id": "6988.png", "formula": "\\begin{align*} \\begin{array} { l l } \\left ( \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\cos \\left ( \\frac { 2 \\pi } { p ^ 2 - 1 } \\right ) \\right ) ^ \\ell & = \\left ( 1 - \\frac { \\pi ^ 2 } { ( p ^ 2 - 1 ) ^ 2 } + O \\left ( \\frac { 1 } { p ^ 8 } \\right ) \\right ) ^ \\ell \\\\ & = \\mathsf { e } ^ { - \\frac { \\pi ^ 2 \\ell } { ( p ^ 2 - 1 ) ^ 2 } } \\left ( 1 + O \\left ( \\frac { \\ell } { p ^ 8 } \\right ) \\right ) . \\end{array} \\end{align*}"}
-{"id": "5112.png", "formula": "\\begin{align*} ( \\{ g _ j = f _ j \\widehat { S } _ { f , \\tau } ^ { - 1 } + h _ j U - f _ j \\widehat { S } _ { f , \\tau } ^ { - 1 } \\widehat { \\theta } _ \\tau U \\} _ { j \\in \\mathbb { J } } , \\{ \\omega _ j = \\widehat { S } _ { f , \\tau } ^ { - 1 } \\tau _ j + V e _ j - V \\theta _ f \\widehat { S } _ { f , \\tau } ^ { - 1 } \\tau _ j \\} _ { j \\in \\mathbb { J } } ) \\end{align*}"}
-{"id": "3472.png", "formula": "\\begin{align*} \\Sigma ^ { 3 , ' } _ - ( a ) = \\{ - t + \\frac { i } { 4 } \\mid t \\in [ \\frac { 1 } { 2 } - a , + \\infty ) \\} , \\Sigma ^ { 4 , ' } _ - ( a ) = \\{ t - \\frac { i } { 4 } \\mid t \\in ( - \\infty , a - \\frac { 1 } { 2 } ] \\} . \\end{align*}"}
-{"id": "3291.png", "formula": "\\begin{align*} \\frac { \\sum _ { x \\le n \\le x + x ^ { \\varepsilon } } \\alpha ( n ) } { x ^ { \\varepsilon } } = o ( 1 ) , x \\to \\infty \\end{align*}"}
-{"id": "310.png", "formula": "\\begin{align*} \\begin{cases} ( \\frac { 1 } { 2 } n ( n ^ 2 - 1 ) , k ) = ( \\frac { 1 } { 2 } n ( n ^ 2 - 1 ) , l ) , & ; \\\\ ( n ( n ^ 2 - 1 ) , k ) = ( n ( n ^ 2 - 1 ) , l ) , & . \\end{cases} \\end{align*}"}
-{"id": "7090.png", "formula": "\\begin{align*} \\begin{cases} - \\partial _ { t t } v _ n + \\Delta v _ n = | \\nabla v _ n | ^ 2 + 2 \\nabla u _ { n - 1 } \\cdot \\nabla v _ n \\\\ v _ n | _ { t = 0 } = Q _ { N } f _ 0 ^ \\omega ~ , ~ ~ \\partial _ t v _ n | _ { t = 0 } = Q _ { N } f _ 1 ^ \\omega ~ . \\end{cases} \\end{align*}"}
-{"id": "7822.png", "formula": "\\begin{align*} \\lim _ n [ \\langle x _ n , y \\rangle \\langle x , z \\rangle \\langle u , x _ n \\rangle \\xi _ n , \\xi _ n ] = \\lambda \\| \\theta _ { x , y } \\| \\| \\theta _ { z , u } \\| , \\end{align*}"}
-{"id": "7553.png", "formula": "\\begin{align*} \\norm { f } _ { L i p ( 0 , T ; Y ) } = \\sup _ { t \\ne s , t , s \\in [ 0 , T ] } \\frac { \\norm { f ( t ) - f ( s ) } _ Y } { | t - s | } + \\norm { f } _ { L ^ \\infty ( 0 , T ; Y ) } \\end{align*}"}
-{"id": "2156.png", "formula": "\\begin{align*} T _ { j } ( f g ) = f ( T _ j g ) + g ( T _ j f ) 1 \\leq j \\leq N . \\end{align*}"}
-{"id": "10013.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } a _ { n } n ^ { - s } = \\Big ( \\sum _ { n = 1 } ^ { N } \\frac { a _ { n } } { n ^ { \\sigma _ { 0 } + s } } \\Big ) N ^ { \\sigma _ { 0 } } + \\sum _ { n = 1 } ^ { N - 1 } \\Big ( \\sum _ { k = 1 } ^ { n } \\frac { a _ { k } } { k ^ { \\sigma _ { 0 } + s } } \\Big ) \\big ( n ^ { \\sigma _ { 0 } } - ( n + 1 ) ^ { \\sigma _ { 0 } } \\big ) \\end{align*}"}
-{"id": "8145.png", "formula": "\\begin{align*} \\overset { I } { C } = \\overset { I } { v } { ^ { 2 } } \\overset { I } { B } \\overset { I } { A } { ^ { - 1 } } \\overset { I } { B } { ^ { - 1 } } \\overset { I } { A } , \\ : \\ : \\ : \\ : \\ : \\overset { I } { C } { ^ { ( \\pm ) } } = \\Psi _ { 1 , 0 } ^ { - 1 } ( \\overset { I } { L } { ^ { ( \\pm ) } } \\overset { I } { \\widetilde { L } } { ^ { ( \\pm ) } } ) \\end{align*}"}
-{"id": "6161.png", "formula": "\\begin{align*} \\int _ { \\mathbb { D } } f ( z ) \\overline { g ( z ) } \\omega ( z ) d A ( z ) = \\int _ { \\mathbb { D } } R ^ { \\eta , \\eta _ { + N } } f ( z ) \\overline { R ^ { \\nu , \\nu _ { + M } } g ( z ) } \\omega _ { + N + M } ( z ) d A ( z ) . \\end{align*}"}
-{"id": "8808.png", "formula": "\\begin{align*} v ^ k ( t , x ) \\stackrel { d } { = } u ^ { \\varepsilon _ k } ( t , x ) { \\textrm { L e b } } _ { [ 0 , T ] \\times [ 0 , \\pi ] } \\textrm { - a l m o s t e v e r y w h e r e } , \\end{align*}"}
-{"id": "9528.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { n - 1 } z ^ n q ^ { \\frac { n ( n + 1 ) } { 2 } } } { ( 1 - c q ^ n ) ( z q ) _ n } = \\frac { z } { c } \\sum _ { n = 1 } ^ { \\infty } \\frac { ( z q / c ) _ { n - 1 } } { ( z q ) _ n } ( c q ) ^ n . \\end{align*}"}
-{"id": "8731.png", "formula": "\\begin{align*} \\tau ^ { - \\alpha } \\sum _ { j = 0 } ^ n \\omega ^ { ( \\alpha , \\sigma ) } _ { n - k } u _ k , 0 \\leq n \\leq n _ T , \\end{align*}"}
-{"id": "6681.png", "formula": "\\begin{align*} \\psi _ { e v e n , \\rho } ( x ) & = \\frac { 1 } { \\sqrt { 4 \\pi } } \\ , | x | ^ { - \\frac { 1 } { 2 } + i \\rho } , \\\\ \\psi _ { o d d , \\rho } ( x ) & = \\frac { { \\rm s g n } ( x ) } { \\sqrt { 4 \\pi } } \\ , | x | ^ { - \\frac { 1 } { 2 } + i \\rho } . \\end{align*}"}
-{"id": "3925.png", "formula": "\\begin{align*} Q _ { i j } ( s , \\phi ) = \\left ( P _ 1 ( s , \\phi ) , P _ { i j } ( P _ 2 ( s , \\phi ) ) \\right ) \\end{align*}"}
-{"id": "2675.png", "formula": "\\begin{align*} \\sum _ { k = l + 1 } ^ { \\infty } \\frac { 1 } { k ! } \\big ( v ( z + 1 ) \\big ) ^ k = ( z + 1 ) ^ { l + 1 } e ^ { v ( z + 1 ) } \\frac { 1 } { l ! } \\int _ { 0 } ^ { v } d s \\ , s ^ l e ^ { ( v - s ) ( z + 1 ) } , \\end{align*}"}
-{"id": "9462.png", "formula": "\\begin{align*} ( - \\Delta _ { p } ) u + ( - \\Delta _ { q } ) u = f ( x ) , x \\in \\mathbb { R } ^ N , \\end{align*}"}
-{"id": "3306.png", "formula": "\\begin{align*} \\frac { 1 } { X } \\int _ { X } ^ { 2 X } ( \\sum _ { n \\in [ x , x + H ] } \\mu ( n ) ) ^ 2 d x = o ( H ^ 2 ) \\end{align*}"}
-{"id": "804.png", "formula": "\\begin{align*} H ^ 1 = H _ { q ^ d , P ^ \\alpha } \\cap N _ 1 \\supset H ^ 2 = H _ { q ^ d , P ^ \\alpha } \\cap N _ 2 \\supset \\cdots \\supset H ^ i = H _ { q ^ d , P ^ \\alpha } \\cap N _ i \\supset \\cdots \\end{align*}"}
-{"id": "7809.png", "formula": "\\begin{align*} | Q \\cap ( B ^ 2 _ N - x ) | = | \\{ y \\in Q \\colon x + y \\in B ^ 2 _ N \\} | \\le 2 e ^ { - c t ^ 2 } , \\end{align*}"}
-{"id": "4467.png", "formula": "\\begin{align*} G ( \\nabla ^ 2 u , \\nabla u , u ) = \\Psi ( \\nabla u , u , x ) , x \\in \\Omega \\subset \\mathbb { S } ^ n , \\end{align*}"}
-{"id": "1117.png", "formula": "\\begin{align*} & \\mathbb { P } ^ { G U E ( n ) } ( \\lambda _ i \\not \\in [ x , x + \\delta _ n / \\rho _ { s c } ( x ) ] , 1 \\leq i \\leq n ) \\\\ \\leq & \\mathbb { P } ^ { G U E ( n ) } ( \\lambda _ i \\not \\in [ x , x + \\delta _ n ' / \\rho _ { s c } ( x ) ] , 1 \\leq i \\leq n ) \\\\ \\leq & \\mathbb { P } ^ { C U E ( n ) } ( \\theta _ i \\not \\in [ 0 , 2 \\pi \\delta _ n ' ] , 1 \\leq i \\leq n ) + O ( ( n \\ln n ) ^ { - 1 } ) = O ( ( n \\ln n ) ^ { - 1 } ) , \\end{align*}"}
-{"id": "5541.png", "formula": "\\begin{align*} \\| u \\| _ { H ^ t } & \\lesssim \\| L ^ * u \\| _ { H ^ { t - 2 } } + \\| b . \\nabla u + d i v ( b ) u \\| _ { H ^ { t - 2 } } \\\\ & \\lesssim \\| L ^ * u \\| _ { H ^ { t - 2 } } + \\| b \\| _ { B ^ { t - 2 } _ { \\infty \\infty } } \\| u \\| _ { H ^ { t - 1 } } + \\| b \\| _ { B ^ { t - 1 } _ { \\infty \\infty } } \\| u \\| _ { H ^ { t - 2 } } \\end{align*}"}
-{"id": "4960.png", "formula": "\\begin{align*} \\Delta _ K ( t ) = a _ 0 + \\sum _ { s > 0 } a _ s ( t ^ s + t ^ { - s } ) . \\end{align*}"}
-{"id": "8356.png", "formula": "\\begin{align*} \\dot { z } _ 1 = \\frac { - \\lambda i } { 2 \\pi } \\frac { 1 } { \\overline { z _ 1 - z _ 2 } } = \\frac { \\lambda i } { 4 \\pi } \\frac { 1 } { x } . \\end{align*}"}
-{"id": "10010.png", "formula": "\\begin{align*} \\sigma _ { \\mathfrak { X } ( X ) } ( D ) \\leq \\inf { \\{ \\sigma \\in \\mathbb { R } \\colon \\mbox { $ \\sup _ { N } \\big \\| \\sum _ { n = 1 } ^ { N } { \\frac { a _ { n } } { n ^ { \\sigma } } n ^ { - s } } \\big \\| _ { \\mathfrak { X } ( X ) } < \\infty $ } \\} } \\ , , \\end{align*}"}
-{"id": "3269.png", "formula": "\\begin{align*} p _ n - e _ 1 p _ { n - 1 } + e _ 2 p _ { n - 2 } - \\dots \\pm e _ { n - 1 } p _ 1 \\mp n e _ n = 0 . \\end{align*}"}
-{"id": "3226.png", "formula": "\\begin{align*} \\Delta _ 1 w ^ k = - Q _ { k + \\frac { 1 } { 2 } } ^ { - 1 } \\Phi _ { \\mu _ k } ( { w } ^ { k + \\frac { 1 } { 2 } } ) , \\end{align*}"}
-{"id": "1997.png", "formula": "\\begin{align*} H ^ 0 ( C , L ) - H ^ 1 ( C , L ) = \\deg ( L ) + 1 - g - \\sum \\frac { m _ i } { 5 } . \\end{align*}"}
-{"id": "8184.png", "formula": "\\begin{align*} \\Lambda _ { k + 1 } = \\sum _ { i \\in \\mathbb { N } } \\Lambda _ k \\Theta _ i ^ * \\Lambda _ { i + 1 } . \\end{align*}"}
-{"id": "4952.png", "formula": "\\begin{align*} 1 \\geq \\lim _ { n \\rightarrow \\infty } \\dfrac { \\ell ( H ^ 2 ( I _ n ; M ) ) } { \\ell ( R / I _ n ) } \\geq \\lim _ { n \\rightarrow \\infty } \\dfrac { t } { t + ( 3 n ^ { d - 2 } + n ^ { d - 3 } + \\cdots + n ^ 2 ) + ( 4 ( 2 n ) ^ { d - 2 } ) ^ d } = 1 . \\end{align*}"}
-{"id": "2581.png", "formula": "\\begin{align*} R ( r ) & = \\int _ r ^ \\infty \\big \\langle \\psi _ 0 \\big ( \\cdot , ( 1 + \\epsilon _ { R ( u ) } ) u \\phi \\big ) , \\phi ^ * \\big \\rangle _ m ^ { - 1 } d u = \\int _ r ^ \\infty u ^ { - \\gamma _ 0 } l ( u ) d u \\\\ & = - \\frac { 1 } { \\gamma _ 0 - 1 } \\int _ r ^ \\infty l ( u ) d u ^ { - ( \\gamma _ 0 - 1 ) } \\\\ & \\stackrel [ r \\to 0 ] { } { \\sim } C _ X ^ { - 1 } ( \\gamma _ 0 - 1 ) ^ { - 1 } r ^ { - ( \\gamma _ 0 - 1 ) } . \\end{align*}"}
-{"id": "146.png", "formula": "\\begin{align*} a \\left ( [ X _ { A } , Y _ { A } ] \\right ) = [ a \\left ( X _ { A } \\right ) , a \\left ( Y _ { A } \\right ) ] . \\end{align*}"}
-{"id": "6662.png", "formula": "\\begin{align*} H _ r ( T ) = ( - 1 ) ^ { | T | } | \\hat { K _ r } ( T ) | . \\end{align*}"}
-{"id": "2067.png", "formula": "\\begin{align*} D \\frac { { \\partial { C _ { \\rho \\varphi } } ( \\rho , \\varphi , t | { \\rho _ { \\rm t x } } , \\varphi _ { \\rm t x } , { t _ 0 } ) } } { { \\partial \\rho } } \\mid _ { \\rho = \\rho _ c } = - k _ { f } { C _ { \\rho \\varphi } } ( \\rho _ c , \\varphi , t | { \\rho _ { \\rm t x } } , \\varphi _ { \\rm t x } , { t _ 0 } ) . \\end{align*}"}
-{"id": "3382.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { k _ 0 } c ^ 2 & ( k - 1 ) ^ 2 ( G ( c k ) - G ( c ( k - 1 ) ) ) ( 1 + o ( 1 ) ) \\\\ & \\leq \\min _ { S \\subset [ n ] : | S | = \\lceil \\delta n \\rceil } \\| x _ S \\| _ 2 ^ 2 \\leq \\sum _ { k = 1 } { k _ 0 } c ^ 2 k ^ 2 ( G ( c k ) - G ( c ( k - 1 ) ) ) ( 1 + o ( 1 ) ) \\end{align*}"}
-{"id": "1908.png", "formula": "\\begin{align*} f ( x ) \\cos ( \\kappa _ j t ) \\stackrel { ? } { = } \\frac { 1 } { 2 } \\left [ f ( x - t ) + f ( x + t ) \\right ] . \\end{align*}"}
-{"id": "6960.png", "formula": "\\begin{align*} z _ p = \\frac { d ^ p f } { ( d t ) ^ p } ( 0 ) , p = 0 , 1 , 2 , \\dots \\end{align*}"}
-{"id": "6026.png", "formula": "\\begin{align*} \\limsup _ { \\epsilon \\downarrow 0 } \\ , \\ , \\ , \\inf \\left \\{ \\tilde { \\alpha } ^ \\mu _ \\epsilon ( Q ) \\ , : \\ , Q \\in \\mathcal { P } ( \\C ) , \\ Q ^ 1 = \\nu \\right \\} & \\leq \\inf \\left \\{ \\alpha ^ \\mu _ 0 ( Q ) \\ , : \\ , Q \\in \\mathcal { P } ( \\C ) , \\ Q ^ 1 = \\nu \\right \\} , \\end{align*}"}
-{"id": "4958.png", "formula": "\\begin{align*} \\ell \\left ( \\dfrac { V [ [ x _ 1 , x _ 2 ] ] } { I _ n + ( p ^ s x _ 1 ) } \\right ) = \\ell \\left ( \\dfrac { V [ [ x _ 1 , x _ 2 ] ] } { I _ n + ( p ^ s ) } \\right ) + \\ell \\left ( \\dfrac { V [ [ x _ 1 , x _ 2 ] ] } { I _ n + ( x _ 1 ) } \\right ) \\leq s \\cdot \\ell \\left ( \\dfrac { V [ [ x _ 1 , x _ 2 ] ] } { I _ n + ( p ) } \\right ) + \\ell \\left ( \\dfrac { V [ [ x _ 1 , x _ 2 ] ] } { I _ n + ( x _ 1 ) } \\right ) . \\end{align*}"}
-{"id": "881.png", "formula": "\\begin{align*} y _ { \\gamma _ 1 } y _ { \\gamma _ 2 } = q ^ { \\lambda ( \\gamma _ 1 , \\gamma _ 2 ) } y _ { \\gamma _ 2 } y _ { \\gamma _ 1 } . \\end{align*}"}
-{"id": "5871.png", "formula": "\\begin{align*} \\mathcal { L } : = \\Lambda ( 0 ) = \\mathfrak { p } _ F ^ { a _ { n } } w _ n \\oplus \\mathfrak { p } _ F ^ { a _ { n - 1 } } w _ { n - 1 } \\oplus \\cdots \\oplus \\mathfrak { p } _ F ^ { a _ { - n + 1 } } w _ { - n + 1 } \\oplus \\mathfrak { p } _ F ^ { a _ { - n } } w _ { - n } . \\end{align*}"}
-{"id": "61.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\rho + \\partial _ x J & = 0 \\ , , \\\\ \\partial _ t J + \\partial _ x \\rho + 2 g ( J ) \\partial _ x a & = 0 \\ , , \\\\ \\partial _ t a & = 0 \\ , , \\end{cases} \\end{align*}"}
-{"id": "86.png", "formula": "\\begin{align*} c ^ n _ j = \\frac { g ' ( s _ j ^ n ) \\delta _ j } { g ' ( s _ j ^ n ) \\delta _ j + 1 } \\ge 0 \\ , , s _ j ^ n \\in D _ J \\ , , j = 1 , \\ldots , N - 1 \\ , , n \\ge 1 , \\end{align*}"}
-{"id": "6178.png", "formula": "\\begin{align*} \\Delta _ { x _ n , z } ( U _ n g _ 1 ) = - U _ n \\Delta _ { x ' } g _ 1 , \\end{align*}"}
-{"id": "3986.png", "formula": "\\begin{align*} A _ F ( T ) = A _ F ^ 0 ( T ) + A _ F ( T ) ^ 1 + A _ G ( T ) \\end{align*}"}
-{"id": "4550.png", "formula": "\\begin{align*} x ' _ j : = \\begin{cases} \\frac { 1 } { x _ k } \\left ( \\prod \\limits _ { \\substack { b _ { l k } > 0 } } x _ l ^ { b _ { l k } } + \\prod \\limits _ { \\substack { b _ { l k } < 0 } } x _ l ^ { - b _ { l k } } \\right ) & \\\\ x _ j & \\end{cases} \\end{align*}"}
-{"id": "6703.png", "formula": "\\begin{align*} Q _ { 1 k } = \\bar { q } _ k , Q _ { 2 k } = \\dot { \\bar { q } } _ k , \\end{align*}"}
-{"id": "9759.png", "formula": "\\begin{align*} | D ^ \\alpha p _ * ( Z _ \\ell ) | = o ( 1 ) \\to 0 \\quad \\ell \\to \\infty \\end{align*}"}
-{"id": "2288.png", "formula": "\\begin{align*} & \\overline { \\partial } R ( z ) = M ( z ) \\\\ & M ( x + \\i y ) = \\mu ( x ) \\delta ( y ) \\\\ & \\mu ( x ) = 1 / ( 2 \\pi \\i ) ( R ( x - \\i 0 ) - R ( x + \\i 0 ) \\end{align*}"}
-{"id": "596.png", "formula": "\\begin{align*} d Y ( x ) = f ( x ) \\ , d x + \\frac { \\sigma } { \\sqrt { n } } \\ , d W ( x ) , \\ \\ x \\in [ 0 , 1 ) ^ d . \\end{align*}"}
-{"id": "3126.png", "formula": "\\begin{align*} W _ - ( q ^ { \\frac { 1 } { 2 } } , y ) & = \\frac { y ^ 2 + y + 1 } { y } \\ , q ^ \\frac { 1 } { 3 } \\left ( 1 + \\frac { 2 y ^ 2 + 3 y + 2 } { ( y + 1 ) ^ 2 } q + \\ldots \\right ) \\\\ W _ + ( q ^ { \\frac { 1 } { 2 } } , y ) & = \\frac { ( y + 1 ) ^ 2 \\ , y } { y ^ 2 + y + 1 } q ^ { - \\frac { 2 } { 3 } } \\left ( 1 + \\frac { y ^ 4 + 4 y ^ 3 + 3 y ^ 2 + 4 y + 1 } { ( y + 1 ) ^ 2 } q + \\ldots \\right ) \\ , . \\end{align*}"}
-{"id": "6306.png", "formula": "\\begin{align*} | E ( G ) - E ( G ' ) | = | E _ G ( S _ 0 , S _ 0 ) + E _ G ( S _ 0 , S ^ c _ 0 ) - E _ { G ' } ( S _ 0 , S _ 0 ) - E _ { G ' } ( S _ 0 , S ^ c _ 0 ) | . \\end{align*}"}
-{"id": "7139.png", "formula": "\\begin{align*} \\int _ 0 ^ L \\rho ( x , t ) d x = \\int _ 0 ^ L \\rho _ 0 ( x ) d x . \\end{align*}"}
-{"id": "8348.png", "formula": "\\begin{align*} A _ 1 = 1 + \\sum _ { 1 \\leq j , k \\leq N } \\frac { \\lambda _ j \\lambda _ k } { ( 2 \\pi ) ^ 2 } \\frac { 1 } { ( \\alpha - z _ j ) \\overline { ( \\alpha - z _ k ) } } \\frac { i } { \\overline { z _ k } - z _ j } - \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j } { \\pi } R e \\Big \\{ \\frac { D _ t Z - \\dot { z } _ j } { ( \\alpha - z _ j ) ^ 2 } \\Big \\} . \\end{align*}"}
-{"id": "3035.png", "formula": "\\begin{align*} ( L f ) ( x ) & = \\sum \\limits _ { i = 1 } ^ { d } c _ i x _ i \\frac { \\partial ^ 2 f ( x ) } { \\partial x _ i ^ 2 } + ( \\beta + B x ) \\cdot ( \\nabla f ) ( x ) + \\int \\limits _ { \\R _ + ^ d } ( f ( x + z ) - f ( x ) ) \\nu ( d z ) \\\\ & \\ \\ \\ + \\sum \\limits _ { i = 1 } ^ { d } x _ i \\int \\limits _ { \\R _ + ^ d } \\left ( f ( x + z ) - f ( x ) - \\frac { \\partial f ( x ) } { \\partial x _ i } ( 1 \\wedge z _ i ) \\right ) \\mu _ i ( d z ) , \\end{align*}"}
-{"id": "6257.png", "formula": "\\begin{align*} A _ 0 : = \\bigoplus _ { i = 1 } ^ N \\underset { \\alpha _ { i , 1 } , \\cdots , \\alpha _ { i , k _ { i } } } { \\overset { p _ { i , 1 } , \\cdots , p _ { i , k _ { i } } } { M _ { k _ { i } } ( \\C ) } } \\qquad B _ 0 : = \\bigoplus _ { j = 1 } ^ { M } \\underset { \\beta _ { j , 1 } , \\cdots , \\beta _ { j , \\ell _ { j } } } { \\overset { q _ { j , 1 } , \\cdots , q _ { j , \\ell _ { j } } } { M _ { \\ell _ { j } } ( \\C ) } } , \\end{align*}"}
-{"id": "3693.png", "formula": "\\begin{gather*} \\begin{cases} C o v _ N ( \\delta _ i , y _ i ) = \\frac { 1 } { N } \\sum \\limits _ { i \\in U } \\delta _ i y _ i - \\big ( \\frac { 1 } { N } \\sum \\limits _ { i \\in U } \\delta _ i \\big ) \\big ( \\frac { 1 } { N } \\sum \\limits _ { i \\in U } y _ i \\big ) = 0 \\\\ E _ N ( \\delta _ i ) = \\bar { p } _ N \\equiv \\sum \\limits _ { i \\in U } \\delta _ i / N > 0 \\end{cases} \\end{gather*}"}
-{"id": "8093.png", "formula": "\\begin{align*} n a _ n ' - ( 2 n - 1 ) a _ { n - 1 } ' + ( n - 2 ) a _ { n - 2 } ' = \\begin{cases} - 1 , & \\\\ \\phantom { - } 0 , & \\end{cases} \\end{align*}"}
-{"id": "10079.png", "formula": "\\begin{align*} \\varphi ( t ; z ) = \\gamma ( t ) + \\mathcal { O } ( \\| ( x , y ) \\| ^ N ) . \\end{align*}"}
-{"id": "1063.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to + \\infty } \\dfrac { n G _ n ( x ) } { ( 3 2 \\ln n ) ^ { \\frac { 1 } { 2 } } } = 1 , \\ \\lim \\limits _ { n \\to + \\infty } n G _ n ( x ) = + \\infty , \\ \\lim \\limits _ { n \\to + \\infty } n ^ { \\gamma } G _ n ( x ) = 0 , \\ \\forall \\ \\gamma < 1 . \\end{align*}"}
-{"id": "5708.png", "formula": "\\begin{align*} y ( 0 ) & = b _ 0 & & y ' ( 0 ) = b _ 1 \\end{align*}"}
-{"id": "2889.png", "formula": "\\begin{align*} \\gamma _ 1 = i \\sigma _ 1 , \\qquad \\gamma _ 2 = i \\sigma _ 2 , \\{ \\gamma _ \\mu , \\gamma _ \\nu \\} = - 2 \\delta _ { \\mu \\nu } . \\end{align*}"}
-{"id": "7442.png", "formula": "\\begin{align*} \\displaystyle R _ q ( Z _ 1 , \\ldots , Z _ q ) = \\frac { Q _ q ( Z _ 1 , \\ldots , Z _ q ) } { \\prod \\limits _ { 1 \\leqslant j \\leqslant q } 2 Z _ j } \\end{align*}"}
-{"id": "9430.png", "formula": "\\begin{align*} f _ C = \\left < f , u _ 0 \\right > = \\int _ { \\mathcal { F } _ N } f ( z ) \\ ; d \\mu ( z ) , \\end{align*}"}
-{"id": "3614.png", "formula": "\\begin{align*} a _ 1 \\cdot ( ( \\alpha _ 0 - \\beta _ 0 ) \\cdot ( \\alpha _ 1 - \\beta _ 1 ) ) = - \\frac { 1 } { 2 ^ 2 } ( \\alpha _ 0 \\cdot \\beta _ 1 + \\alpha _ 1 \\cdot \\beta _ 0 ) + \\frac { 1 } { 2 ^ 2 } \\langle \\beta _ 0 , \\beta _ 1 \\rangle a _ 1 . \\end{align*}"}
-{"id": "3841.png", "formula": "\\begin{align*} ^ { ( b ) } { { { R _ j } ^ i } _ { k l } } = A _ { ( k l ) } \\Bigl ( { \\frac { \\partial { { G _ j } ^ i } _ k } { \\partial x ^ l } } \\ + { { G _ j } ^ r } _ k { { G _ r } ^ i } _ l \\Bigr ) , \\end{align*}"}
-{"id": "9148.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } L _ { 1 \\left ( 1 \\right ) } \\rho _ { 1 } ^ { ( N ) } ( \\mathbf { x } _ { 1 } , t ) - \\mathcal { C } _ { 1 } \\left ( \\rho _ { 1 } ^ { ( N ) } | \\rho _ { 1 } ^ { ( N ) } \\right ) = 0 , \\\\ \\rho _ { 1 } ^ { ( N ) } ( \\mathbf { x } _ { 1 } , t _ { o } ) = \\rho _ { 1 o } ^ { ( N ) } ( \\mathbf { x } _ { 1 } ) , \\end{array} \\right . \\end{align*}"}
-{"id": "5741.png", "formula": "\\begin{align*} & 0 = u ( x , y , 0 ) = \\sum _ { m , n \\in \\N } \\Big [ b _ { m , n } E _ { \\alpha , 1 } ( 0 ) + 0 c _ { m , n } E _ { \\alpha , 2 } ( 0 ) \\Big ] J _ 0 ( \\gamma _ m x ) \\sin ( n \\pi y ) \\\\ & = \\sum _ { m , n \\in \\N } b _ { m , n } J _ 0 ( \\gamma _ m x ) \\sin ( n \\pi y ) . \\end{align*}"}
-{"id": "828.png", "formula": "\\begin{align*} \\mathrm { d i v } _ Y ( \\pi _ * ( R _ { X / Y } ) ) = \\prod _ { i = 1 } ^ s ( P _ i ) _ Y ^ { \\delta _ i \\frac { | H | } { e _ i } } , \\end{align*}"}
-{"id": "5234.png", "formula": "\\begin{align*} \\forall ( \\gamma > 0 ) , \\lim _ { x \\rightarrow + \\infty } \\frac { 1 - F ( \\gamma x ) } { 1 - F ( x ) } = \\gamma ^ { - \\alpha } . \\end{align*}"}
-{"id": "5518.png", "formula": "\\begin{align*} E _ { b _ 0 } \\Big [ \\int _ 0 ^ T \\nabla L ^ { - 1 } [ f _ h ] ( X _ t ) . d W _ t \\Big ] ^ 2 = E _ { b _ 0 } \\int _ 0 ^ T \\| \\nabla L ^ { - 1 } [ f _ h ] ( X _ t ) \\| ^ 2 d t \\lesssim T \\| L ^ { - 1 } [ f _ h ] \\| _ { C ^ 1 } ^ 2 \\end{align*}"}
-{"id": "673.png", "formula": "\\begin{align*} \\varpi : \\prod _ { i = 1 } ^ n \\mathcal C ( X _ i , Y _ i ) _ { \\nu _ i } \\to \\mathcal C ( X , Y ) _ { \\nu _ 1 \\diamond \\dots \\diamond \\nu _ n } \\end{align*}"}
-{"id": "8562.png", "formula": "\\begin{align*} \\norm { b _ { \\alpha } \\circ \\kappa } _ { \\infty } = \\norm { b _ { \\alpha } } _ { \\infty } \\leq \\norm { b _ { \\alpha } } _ { H ^ 1 } \\leq C \\epsilon ^ 2 + K _ s ^ { - 1 } d _ I ( t ) ^ { - 5 / 2 } \\epsilon . \\end{align*}"}
-{"id": "5937.png", "formula": "\\begin{align*} f ^ * ( m ' p M ) = m ' p M ' + m ' p E ' , \\end{align*}"}
-{"id": "4205.png", "formula": "\\begin{align*} S ( z ) \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & z ^ { - \\beta } & 0 \\\\ 0 & 0 & z ^ { - \\beta } \\end{pmatrix} D _ 0 ^ { - n } ( z ) = \\mathcal { O } \\begin{pmatrix} 1 & h _ { - \\alpha - \\frac { 1 } { 2 } } ( z ) & h _ { - \\alpha - \\frac { 1 } { 2 } } ( z ) \\\\ 1 & h _ { - \\alpha - \\frac { 1 } { 2 } } ( z ) & h _ { - \\alpha - \\frac { 1 } { 2 } } ( z ) \\\\ 1 & h _ { - \\alpha - \\frac { 1 } { 2 } } ( z ) & h _ { - \\alpha - \\frac { 1 } { 2 } } ( z ) \\end{pmatrix} , \\end{align*}"}
-{"id": "769.png", "formula": "\\begin{align*} M = \\prod _ { j = 1 } ^ r \\prod _ { i = 1 } ^ { s _ j } ( T - \\rho _ { i , j } ) ^ { \\alpha _ j } , \\rho _ { i , j } \\in \\mathbb { F } _ { q ^ d } . \\end{align*}"}
-{"id": "7681.png", "formula": "\\begin{align*} \\overline { \\mathrm { r a n g e } ( D \\pm \\mathbf { i } ) } = \\mathrm { r a n g e } ( D \\pm \\mathbf { i } ) ^ { \\perp \\perp } \\supset \\mathrm { k e r } ( D \\mp \\mathbf { i } ) ^ \\perp . \\end{align*}"}
-{"id": "7949.png", "formula": "\\begin{align*} \\sum _ { a \\in [ n + r ] \\setminus R } x _ a = 0 . \\end{align*}"}
-{"id": "7024.png", "formula": "\\begin{align*} \\Omega ( A , B ) = ( - 1 ) ^ { N } \\sum \\limits ^ { N - M } _ { l = 0 } \\sum \\limits ^ { l } _ { m = 0 } ( - 1 ) ^ { m } \\sigma _ { m } ( M , N ) S _ { l - m } T _ { N - M - l } \\end{align*}"}
-{"id": "131.png", "formula": "\\begin{align*} \\widetilde \\zeta _ { j , N } & = \\gamma ~ { } _ 2 F _ 1 ( - j , - 2 N + j + 1 , 1 ; \\gamma ^ 2 ) \\ , , \\gamma = \\frac d N \\end{align*}"}
-{"id": "8803.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow \\infty } \\sup _ { \\varepsilon > 0 } \\mathbb { P } \\left ( { \\Vert \\overline { u } _ t ^ { \\varepsilon } \\Vert } _ { q } > \\delta \\right ) = 0 \\end{align*}"}
-{"id": "4779.png", "formula": "\\begin{align*} \\ge ( 1 + o ( 1 ) ) 0 . 1 1 \\cdot \\log _ 2 n \\cdot \\log _ 2 \\log _ 2 n > 1 . 1 \\binom { h _ 1 } { 2 } . \\end{align*}"}
-{"id": "2238.png", "formula": "\\begin{align*} & R ( \\varphi ) = \\int d ^ 2 z \\varphi ( z ) / ( z - \\Omega ) , & & ( R ( \\varphi ) f ) ( y ) = \\int d ^ 2 z \\varphi ( z ) / ( z - P ( y ) ) f ( y ) \\end{align*}"}
-{"id": "4155.png", "formula": "\\begin{align*} \\begin{aligned} 2 \\varphi _ { 1 } ( z ) + \\varphi _ { 2 } ( z ) & = \\omega z ^ { 1 / 3 } f _ { 1 } ( z ) + \\omega ^ 2 z ^ { 2 / 3 } f _ { 2 } ( z ) , \\\\ \\varphi _ { 2 } ( z ) - \\varphi _ { 1 } ( z ) & = \\omega ^ 2 z ^ { 1 / 3 } f _ { 1 } ( z ) + \\omega z ^ { 2 / 3 } f _ { 2 } ( z ) , \\\\ - \\varphi _ { 1 } ( z ) - 2 \\varphi _ { 2 } ( z ) & = z ^ { 1 / 3 } f _ { 1 } ( z ) + z ^ { 2 / 3 } f _ { 2 } ( z ) , \\end{aligned} \\end{align*}"}
-{"id": "1616.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow \\infty } \\frac { X _ T ^ { Z ^ { b } } } { T } = 0 \\end{align*}"}
-{"id": "9997.png", "formula": "\\begin{align*} & c _ { i j k , l } + g ^ { p q } c _ { p j k } c _ { q i l } + \\\\ & \\hphantom { c _ { i j k , l } + } c ^ 1 w _ { 1 j k } w _ { 1 i l } + c ^ 2 ( w _ { 1 j k } w _ { 2 i l } + w _ { 1 i l } w _ { 2 j k } ) + c ^ 3 w _ { 2 j k } w _ { 2 i l } = 0 , \\\\ & w _ { a i j , s } \\frac { \\partial u ^ s } { \\partial q ^ l } - g ^ { p q } c _ { p i j } w _ { a q l } = 0 , a = 1 , 2 \\end{align*}"}
-{"id": "3467.png", "formula": "\\begin{align*} \\limsup _ { N \\to \\infty } P _ 3 = \\limsup _ { N \\to \\infty } P _ 2 \\leq \\frac { \\epsilon } { 4 } . \\end{align*}"}
-{"id": "8968.png", "formula": "\\begin{align*} \\pi \\mu ( B ( t , r ) ) & \\le C \\pi \\mu _ n ( B ( t , C 2 ^ { - n } ) ) + C 2 ^ { n ( \\theta - k ) } \\\\ & \\le C 2 ^ { n ( \\theta - k ) } + C \\int _ { t \\in B ( t , C 2 ^ { - n } ) } \\int \\mu _ n \\ , \\mathrm { d } \\eta _ t \\ , \\mathrm { d } t \\\\ & \\le C 2 ^ { n ( \\theta - k ) } \\ , , \\end{align*}"}
-{"id": "4894.png", "formula": "\\begin{align*} | f ( a ) | ( x ) = | f ( a ) \\cdot g | ( x ) \\leq \\| f ( a ) \\cdot g \\| = \\| f _ U ( a ) \\cdot g \\| \\leq \\| a \\| \\| g \\| \\leq \\| a \\| \\end{align*}"}
-{"id": "554.png", "formula": "\\begin{align*} w \\pm i | \\nabla | ^ { - 1 } \\partial _ t w = \\chi ( t | \\nabla | ) e ^ { \\mp i t | \\nabla | } ( f _ \\lambda \\pm i | \\nabla | ^ { - 1 } g _ \\lambda ) \\pm i \\chi ' ( t | \\nabla | ) v . \\end{align*}"}
-{"id": "3881.png", "formula": "\\begin{align*} M ( u ( t ) , v ( t ) ) & = \\| u ( t ) \\| ^ 2 _ { L ^ 2 } + 2 \\| v ( t ) \\| ^ 2 _ { L ^ 2 } , \\\\ E ( u ( t ) , v ( t ) ) & = \\frac { 1 } { 2 } ( \\| \\nabla u ( t ) \\| ^ 2 _ { L ^ 2 } + \\kappa \\| \\nabla v ( t ) \\| ^ 2 _ { L ^ 2 } ) - ( \\langle v ( t ) , u ^ 2 ( t ) \\rangle ) . \\end{align*}"}
-{"id": "125.png", "formula": "\\begin{align*} \\eta _ { j , N } ~ & \\dot = \\sum _ { \\substack { k = 2 \\\\ k \\ , } } ^ { \\min \\{ 2 j , 4 N - 2 j \\} } \\gamma ^ k \\begin{pmatrix} j \\\\ \\frac { k } { 2 } \\end{pmatrix} \\begin{pmatrix} 2 N - j - 1 \\\\ \\frac { k } { 2 } - 1 \\end{pmatrix} \\\\ & = \\sum _ { h = 1 } ^ { \\min \\{ j , 2 N - j \\} } \\gamma ^ { 2 h } \\begin{pmatrix} j \\\\ h \\end{pmatrix} \\begin{pmatrix} 2 N - j - 1 \\\\ h - 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "5054.png", "formula": "\\begin{align*} R ( \\mathbb V , \\mu ) = S t r ( \\mathbb V , \\mu ) \\mu + { \\bf k } \\mu . \\end{align*}"}
-{"id": "5620.png", "formula": "\\begin{align*} \\int _ 0 ^ { + \\infty } \\ ! \\ ! \\ ! i u _ t \\bar { u } _ x d x & = - \\int _ 0 ^ { + \\infty } \\ ! \\ ! \\ ! u _ t \\bar { u } _ x d x \\\\ & = - \\int _ 0 ^ { + \\infty } \\frac { d } { d t } ( u \\bar { u } _ x ) d x + \\int _ 0 ^ { + \\infty } \\ ! \\ ! \\ ! u \\bar { u } _ { x t } d x \\\\ & = - \\int _ 0 ^ { + \\infty } \\frac { d } { d t } ( u \\bar { u } _ x ) d x - \\int _ 0 ^ { + \\infty } \\ ! \\ ! \\ ! u _ x \\bar { u } _ t d x - ( u ( 0 , t ) \\bar { u } _ t ( 0 , t ) ) . \\end{align*}"}
-{"id": "4677.png", "formula": "\\begin{align*} B _ j ^ 0 = \\left \\{ \\begin{array} { l l } 1 ~ ~ ~ ~ x \\in [ x _ { j - 1 } , x _ { j } ] , \\\\ 0 ~ ~ ~ ~ ~ ~ . \\end{array} \\right . \\end{align*}"}
-{"id": "1311.png", "formula": "\\begin{align*} H ( U _ n ) & = \\{ a b c d \\in W _ 2 ( U _ n ) : ( a , b ^ { - 1 } ) , ( c , d ^ { - 1 } ) \\in U _ n \\} , \\\\ I ( U _ n ) & = \\{ a b c d \\in W _ 2 ( U _ n ) : ( a , d ^ { - 1 } ) , ( b , c ^ { - 1 } ) \\in U _ n , a \\neq d ^ { - 1 } \\} \\\\ J ( U _ n ) & = \\{ a b c d \\in W _ 2 ( U _ n ) : ( a , d ^ { - 1 } ) , ( b , c ^ { - 1 } ) \\in U _ n , a = d ^ { - 1 } \\} \\\\ & = \\{ a b c a ^ { - 1 } \\in W _ 2 ( U _ n ) : ( b , c ^ { - 1 } ) \\in U _ n \\} . \\end{align*}"}
-{"id": "5401.png", "formula": "\\begin{align*} c ( 2 k + 1 ) = - \\frac { k ^ 2 + k + 1 } { 2 } \\end{align*}"}
-{"id": "3073.png", "formula": "\\begin{align*} \\widehat { K _ 1 } ( y ) & = \\int _ { | x | < \\frac { 1 } { | y | } } e ^ { 2 \\pi i x \\cdot y } K _ 1 ( x ) \\ , d x + \\int _ { \\frac { 1 } { | y | } \\leq | x | \\leq \\frac { 2 } { \\beta } } e ^ { 2 \\pi i x \\cdot y } K _ 1 ( x ) \\ , d x \\\\ & = \\int _ { | x | < \\frac { 1 } { | y | } } ( e ^ { 2 \\pi i x \\cdot y } - 1 ) K _ 1 ( x ) \\ , d x + \\int _ { \\frac { 1 } { | y | } \\leq | x | \\leq \\frac { 2 } { \\beta } } e ^ { 2 \\pi i x \\cdot y } K _ 1 ( x ) \\ , d x . \\end{align*}"}
-{"id": "4556.png", "formula": "\\begin{align*} \\varphi _ { 1 } \\cdots \\varphi _ { m _ k } . ( u \\otimes ( v _ p ) _ z ) & = \\varphi _ { 1 } \\cdots \\varphi _ { m _ k } e ( h _ 1 , \\ldots , h _ { m _ k } , p ) . ( u \\otimes ( v _ p ) _ z ) \\\\ & = \\varphi _ { 1 } e ( h _ 1 , p , h _ 2 , \\ldots , h _ { m _ k } ) \\varphi _ 2 \\cdots \\varphi _ { m _ k } . ( u \\otimes ( v _ p ) _ z ) \\\\ & = ( \\tau _ 1 ( x _ 1 - x _ 2 ) + 1 ) \\varphi _ 2 \\cdots \\varphi _ { m _ k } . ( u \\otimes ( v _ p ) _ z ) . \\end{align*}"}
-{"id": "185.png", "formula": "\\begin{align*} c _ i \\circ q _ M \\textrm { a n d } l _ { d c _ i } = \\sum _ { s = 1 } ^ { N } \\dfrac { \\partial c _ i } { \\partial x _ s } ( { \\bf x } ) y _ s , i = 1 , \\dots r , \\end{align*}"}
-{"id": "1928.png", "formula": "\\begin{align*} \\frac { d Z _ j } { d x } = \\frac { A _ j ^ 2 } { Z _ j } . \\end{align*}"}
-{"id": "10015.png", "formula": "\\begin{align*} S ( \\ell _ { 1 } ( \\mathbb { C } ) , \\mathcal { D } _ { \\infty } ( \\mathbb { C } ) ) = \\sup { \\{ \\sigma _ { a } ( D ) - \\sigma _ { u } ( D ) \\colon D \\in \\mathfrak { D } ( \\mathbb { C } ) \\} } = \\frac { 1 } { 2 } \\ , . \\end{align*}"}
-{"id": "8806.png", "formula": "\\begin{align*} & ( i ) \\int _ { \\vert x \\vert \\leq h } x ^ 2 \\ , \\overline { \\nu } ^ k ( [ 0 , t ] \\times \\textrm { d } x ) \\longrightarrow \\overline { C } _ t , \\\\ & ( i i ) \\overline { B } _ t ^ k \\longrightarrow \\overline { B } _ t , \\\\ & ( i i i ) \\overline { \\nu } ^ k \\left ( [ 0 , t ] \\times \\{ \\vert x \\vert > 1 \\} \\right ) \\longrightarrow 0 \\end{align*}"}
-{"id": "2593.png", "formula": "\\begin{align*} J _ g ( t , r , \\xi ) : = ( \\gamma _ 0 - 1 ) t \\int _ 0 ^ 1 \\big ( \\kappa \\gamma \\cdot ( \\phi \\eta _ { u t } ) ^ { \\gamma - 1 } g ( u t , r u ^ { \\frac { 1 } { \\gamma _ 0 - 1 } } , \\cdot ) ^ { \\gamma - 1 } \\big ) ( \\xi _ { ( 1 - u ) t } ) d u . \\end{align*}"}
-{"id": "6445.png", "formula": "\\begin{align*} \\xi _ { k } = \\| y \\| ^ { 2 } = \\sum _ { j = 0 } ^ { k - 1 } \\frac { \\left ( \\sum _ { i = j } ^ { k - 1 } \\psi _ { i } \\right ) ^ { 2 } } { \\| r _ { j } \\| ^ { 2 } } . \\end{align*}"}
-{"id": "5722.png", "formula": "\\begin{align*} - \\int _ 0 ^ x f ( z ) \\frac { d } { d z } [ E _ { \\alpha , 1 } ( \\lambda ( x - z ) ^ { \\alpha } ) ] d z - \\lambda \\int _ 0 ^ x ( x - z ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } [ \\lambda ( x - z ) ^ { \\alpha } ] f ( z ) d z = 0 \\end{align*}"}
-{"id": "8553.png", "formula": "\\begin{align*} 4 \\sum _ { k = 0 } ^ s \\Big \\{ \\| \\partial _ { \\alpha } ^ k D _ t \\tilde { \\theta } \\| ^ 2 + \\| \\partial _ { \\alpha } ^ k D _ t \\tilde { \\sigma } \\| ^ 2 + \\| \\partial _ { \\alpha } ^ k | D | ^ { 1 / 2 } \\tilde { \\theta } \\| _ { L ^ 2 } ^ 2 + \\| \\partial _ { \\alpha } ^ k | D | ^ { 1 / 2 } \\tilde { \\sigma } \\| _ { L ^ 2 } ^ 2 \\Big \\} \\leq \\mathcal { E } _ s + C \\epsilon ^ 3 \\leq 2 0 \\epsilon ^ 2 . \\end{align*}"}
-{"id": "7573.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { \\int _ 0 ^ t \\Delta g _ { \\nu ( t - s ) } * \\Delta _ 1 \\tau ( s , t ) d s } _ { \\alpha , p } \\le \\frac { C t } { \\nu } \\norm { \\tau } _ { L i p ( 0 , T ; C ^ { \\alpha , p } ) } M _ X ^ { \\alpha } , \\\\ \\end{gathered} \\end{align*}"}
-{"id": "825.png", "formula": "\\begin{align*} \\mathrm { d i v } _ X ( \\pi ^ * ( d f ) ) = \\pi ^ * \\mathrm { d i v } _ Y ( d f ) \\cdot R _ { X / Y } , \\end{align*}"}
-{"id": "3171.png", "formula": "\\begin{align*} \\partial ^ * \\ker ( \\tau \\mp 1 ) = \\ker ( \\gamma \\mp 1 ) \\cap \\ker ( c - 1 ) . \\end{align*}"}
-{"id": "4085.png", "formula": "\\begin{align*} \\frac { \\Gamma ( s ) } { \\Gamma ( 1 + \\alpha - s ) \\Gamma ( \\frac { 3 } { 2 } + \\alpha - s ) } & = \\Gamma ( s ) \\Gamma ( s - \\alpha ) \\Gamma ( s - \\alpha - \\tfrac { 1 } { 2 } ) \\frac { \\sin \\pi ( s - \\alpha ) \\sin \\pi ( s - \\alpha - \\frac { 1 } { 2 } ) } { \\pi ^ 2 } \\\\ & = \\Gamma ( s ) \\Gamma ( s - \\alpha ) \\Gamma ( s - \\alpha - \\tfrac { 1 } { 2 } ) \\frac { 1 } { ( 2 \\pi i ) ^ 2 } \\left ( e ^ { 2 \\pi i ( s - \\beta ) } + e ^ { - 2 \\pi i ( s - \\beta ) } \\right ) . \\end{align*}"}
-{"id": "2075.png", "formula": "\\begin{align*} R _ n ( \\rho | \\rho _ { \\rm t x } ) = A _ n J _ n ( \\lambda _ { n } \\rho ) + B _ n Y _ n ( \\lambda _ { n } \\rho ) , \\end{align*}"}
-{"id": "2290.png", "formula": "\\begin{align*} a ^ + & = \\begin{pmatrix} 1 , & ( E | \\frac { \\mathcal { P } } { x \\Omega } \\end{pmatrix} & b ^ + & = \\begin{pmatrix} 0 , & ( \\delta _ x | \\end{pmatrix} . \\end{align*}"}
-{"id": "1332.png", "formula": "\\begin{align*} f \\ast g ( x ) = \\int _ { G ^ { d ( x ) } } f ( x t ) g ( t ^ { - 1 } ) \\ , d \\lambda ^ { d ( x ) } ( t ) \\end{align*}"}
-{"id": "6694.png", "formula": "\\begin{align*} \\psi _ { \\rho } ( x ) & = \\langle x | \\psi _ { \\rho } \\rangle = | x | ^ { - \\sigma + i \\rho } , \\\\ \\psi _ { \\rho ' } ( x ) & = \\langle x | \\psi _ { \\rho ' } \\rangle = | x | ^ { - \\sigma + i \\rho ' } . \\end{align*}"}
-{"id": "4118.png", "formula": "\\begin{align*} \\nu ^ * = \\int _ { 0 } ^ { q } \\nu _ t d \\mu ^ * ( t ) , \\end{align*}"}
-{"id": "2260.png", "formula": "\\begin{align*} D _ l & = \\mathfrak { H } R ( z ) = \\bigg \\{ \\xi \\in \\mathbb { C } \\oplus \\mathcal { L } : \\xi = c ( 1 , ( E | R ( z ) ) + ( 0 , ( f | ) \\bigg \\} \\\\ D _ r & = R ( z ) \\mathfrak { H } = \\bigg \\{ \\xi \\in \\mathbb { C } \\oplus \\mathcal { L } : \\xi = c \\binom { 1 } { - R ( z ) | E ) } + \\binom { 0 } { R ( z ) f } \\bigg \\} \\end{align*}"}
-{"id": "2980.png", "formula": "\\begin{align*} M & = \\begin{cases} ( - 1 ) ^ m , & n = 5 P _ { 5 , m } 5 Q _ { 5 , m } m ; \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "9587.png", "formula": "\\begin{align*} B ( q , N + 1 ) = \\sum _ { n = 1 } ^ { N } \\frac { ( - 1 ) ^ { n - 1 } n q ^ { n ( n + 1 ) / 2 } } { ( q ) _ n ( 1 - q ^ n ) ( q ) _ { N - n } } \\frac { 1 } { ( 1 - q ^ { N + 1 - n } ) } + \\frac { ( - 1 ) ^ N ( N + 1 ) q ^ { ( N + 1 ) ( N + 2 ) / 2 } } { ( q ) _ { N + 1 } ( 1 - q ^ { N + 1 } ) } . \\end{align*}"}
-{"id": "3355.png", "formula": "\\begin{align*} \\widetilde { H _ { \\chi } } ( u ) : = \\prod _ { d \\ge 1 } \\widetilde { H _ { d } } ( u ^ d ) , \\end{align*}"}
-{"id": "6490.png", "formula": "\\begin{align*} W _ f ( g _ 1 , g _ 2 ) & = \\mu _ 1 ( g _ 1 \\cdot f _ 1 ) \\cdot \\mu ( g _ 2 \\cdot f _ 2 ) = \\left ( \\int _ F f _ 1 ( w \\cdot m _ x \\cdot g _ 1 ) \\Psi _ 1 ( - x ) d x \\right ) \\cdot \\left ( \\int _ F f _ 2 ( w \\cdot m _ x \\cdot g _ 2 ) \\Psi _ 2 ( - x ) d x \\right ) \\\\ & = W _ { 1 , f _ 1 } ( g _ 1 ) \\cdot W _ { 2 , f _ 2 } ( g _ 2 ) , \\end{align*}"}
-{"id": "6231.png", "formula": "\\begin{align*} \\psi _ \\lambda ( e _ { i , j } ) : = \\begin{cases} \\lambda ^ i \\frac { ( 1 - \\lambda ) } { ( 1 - \\lambda ^ n ) } & i = j \\\\ 0 & \\end{cases} . \\end{align*}"}
-{"id": "5568.png", "formula": "\\begin{align*} \\begin{cases} i u _ t + u _ { x x } = \\alpha u v + \\beta u | u | ^ 2 , & ( x , t ) \\in \\R ^ - \\times ( 0 , T ) , \\\\ v _ t + v _ { x x x } + v v _ x = \\gamma ( | u | ^ 2 ) _ x , & ( x , t ) \\in \\R ^ - \\times ( 0 , T ) , \\\\ u ( x , 0 ) = u _ 0 ( x ) , \\ v ( x , 0 ) = v _ 0 ( x ) , & x \\in \\R ^ - , \\\\ u ( 0 , t ) = f ( t ) , \\ v ( 0 , t ) = g ( t ) , \\ v _ x ( 0 , t ) = h ( t ) , & t \\in ( 0 , T ) , \\end{cases} \\end{align*}"}
-{"id": "9677.png", "formula": "\\begin{align*} | \\psi _ { 1 n } | = \\begin{cases} | \\psi _ n | & \\sigma , \\\\ \\min ( | \\psi _ n | , C _ 1 | \\varphi _ n | ) & \\mathbb T \\setminus \\sigma . \\end{cases} \\end{align*}"}
-{"id": "6536.png", "formula": "\\begin{align*} \\det ( S _ \\beta ) = \\det ( S ) \\otimes \\det ( \\nu _ \\beta ) . \\end{align*}"}
-{"id": "8011.png", "formula": "\\begin{align*} 0 \\leq u ( t , x ) & \\leq \\int _ { \\mathbb { R } ^ d } t ^ { - \\frac { d } { 2 } } e ^ { c _ 3 ( t + 1 ) } e ^ { - \\frac { | x - y | ^ 2 } { c _ 4 t } } u _ 0 ( y ) \\ , \\mathrm d y \\\\ & = ( c _ 4 \\pi ) ^ { \\frac { d } { 2 } } e ^ { c _ 3 ( t + 1 ) } ( e ^ { \\frac { c _ 4 } { 4 } t \\Delta } u _ 0 ) ( x ) . \\end{align*}"}
-{"id": "5419.png", "formula": "\\begin{align*} \\hat { \\rho } ( \\mathcal M ) = \\lim _ { n \\to \\infty } \\sup \\{ \\lVert A _ n \\cdots A _ 1 \\rVert ^ { 1 / n } : A _ i \\in \\mathcal M \\} . \\end{align*}"}
-{"id": "2815.png", "formula": "\\begin{align*} \\hat { a } _ { i i } = \\lambda _ i + a _ { i i } , \\end{align*}"}
-{"id": "4884.png", "formula": "\\begin{align*} \\iota ( x + \\sqrt { - 1 } y ) = \\left ( x , \\ , \\dfrac { y } { \\sqrt { | x | ^ 2 + 1 } } \\right ) . \\end{align*}"}
-{"id": "9951.png", "formula": "\\begin{align*} U \\ , \\Delta \\ , U ^ { - 1 } = \\Delta _ 0 + R , \\end{align*}"}
-{"id": "5766.png", "formula": "\\begin{align*} \\sum _ { e \\in c , r ( e ) \\leq k } a _ { e , k } = 0 . \\end{align*}"}
-{"id": "8080.png", "formula": "\\begin{align*} ( b - a ) M ( a - 1 , b , z ) + ( 2 a - b + z ) M ( a , b , z ) - a M ( a + 1 , b , z ) = & \\ ; 0 , \\\\ U ( a - 1 , b , z ) + ( b - 2 a - z ) U ( a , b , z ) + a ( a - b + 1 ) U ( a + 1 , b , z ) = & \\ ; 0 . \\end{align*}"}
-{"id": "4135.png", "formula": "\\begin{align*} r _ n & = n ^ { - \\frac { 3 } { 2 } } \\end{align*}"}
-{"id": "6517.png", "formula": "\\begin{align*} \\pi \\circ F _ { r _ n ( x ) } ^ { n _ j + n q } ( \\tilde { a } _ j \\circ r _ n ( x ) ) & = f _ { r _ n ( x ) } ^ { n _ j + n q } \\circ \\pi ( \\tilde { a } _ j \\circ r _ n ( x ) ) = f _ { r _ n ( x ) } ^ { n _ j + n q } ( a _ j \\circ r _ n ( x ) ) \\\\ & = \\pi \\circ \\sigma _ j \\left ( a ^ { n , j } ( x ) \\right ) . \\end{align*}"}
-{"id": "7190.png", "formula": "\\begin{gather*} \\xi _ { \\ast } + \\zeta _ { \\ast } + h \\left ( \\xi _ { \\ast } , \\zeta _ { \\ast } \\right ) = \\xi _ { 0 } + \\zeta _ { 0 } . \\end{gather*}"}
-{"id": "326.png", "formula": "\\begin{align*} a ( \\tilde { \\alpha } _ { i , k } ) = k u ^ 2 \\otimes \\Sigma ^ { - 2 } c _ { n - 1 } ( \\alpha _ i ) \\oplus k u ^ 2 \\otimes \\Sigma ^ { - 2 } c _ { n } ( \\alpha _ i ) . \\end{align*}"}
-{"id": "5929.png", "formula": "\\begin{align*} \\pi ^ * ( K _ X + \\tilde { \\Delta } + \\beta M ) = K _ Y + \\sum _ i e _ i E _ i + \\beta M _ Y , \\end{align*}"}
-{"id": "5868.png", "formula": "\\begin{align*} \\Gamma _ \\C ( s ) : = \\Gamma _ \\C ( | \\cdot | _ { \\mathbb { C } } ^ s ) = \\frac { ( 2 \\pi ) ^ { 1 - s } \\Gamma ( s ) } { ( 2 \\pi ) ^ { s } \\Gamma ( 1 - s ) } = 2 ^ { 1 - 2 s } \\pi ^ { - 2 s } \\Gamma ( s ) ^ 2 \\sin ( \\pi s ) . \\end{align*}"}
-{"id": "4562.png", "formula": "\\begin{align*} \\left ( ( j + 1 \\ldots n - 1 ) ( j \\ldots n ) \\right ) \\odot \\hat { \\mu } _ { r _ { n - 2 } + j } \\odot \\mu & = \\left ( ( j + 1 \\ldots n ) ( j \\ldots n - 1 ) \\right ) \\odot \\mu \\\\ & > \\left ( ( j + 1 \\ldots n - 1 ) ( j \\ldots n ) \\right ) \\odot \\mu . \\end{align*}"}
-{"id": "9465.png", "formula": "\\begin{align*} [ u ] _ { s , m , \\Omega } = \\left ( \\displaystyle \\int _ { \\Omega } \\displaystyle \\int _ { \\Omega } \\frac { | u ( x ) - u ( y ) | ^ { m } } { | x - y | ^ { N + s m } } \\dd x \\dd y \\right ) ^ { 1 / m } \\end{align*}"}
-{"id": "1404.png", "formula": "\\begin{align*} f ( x _ 1 , \\dots , x _ d ) = \\tilde { H } ( \\exp ( \\sum _ j R _ j ( x _ j ) ) \\prod _ j \\tilde { R } _ j ( x _ j ) ) . \\end{align*}"}
-{"id": "2326.png", "formula": "\\begin{align*} [ a _ x , a ^ + _ y ] & = \\delta _ { x , y } & [ a _ x , a _ y ] & = [ a ^ + _ x , a ^ + _ y ] = 0 \\end{align*}"}
-{"id": "7932.png", "formula": "\\begin{align*} \\sum _ { a \\in [ n ] } \\alpha _ a v _ a = 0 , \\end{align*}"}
-{"id": "9239.png", "formula": "\\begin{align*} \\mathcal { G } [ u ] = G ( \\cdot , u , D u ) = \\beta \\cdot D u - \\varphi ( \\cdot , u ) = 0 , { \\rm o n } \\ \\partial \\Omega , \\end{align*}"}
-{"id": "6523.png", "formula": "\\begin{align*} D _ { S _ { l } } ^ { p } \\bigl [ f ( \\cdot ) , g ( \\cdot ) \\bigr ] : = \\sup _ { x \\in I } \\Biggl [ \\frac { 1 } { l } \\int _ { x } ^ { x + l } \\bigl \\| f ( t ) - g ( t ) \\bigr \\| ^ { p } \\ , d t \\Biggr ] ^ { 1 / p } . \\end{align*}"}
-{"id": "5866.png", "formula": "\\begin{align*} | \\ , \\Phi _ n ( z _ 1 , \\cdots , z _ n ) | _ { \\mathbb { C } } = | \\prod _ { i < j } ( z _ i - z _ j ) | _ { \\mathbb { C } } = | \\Delta ( f ) | _ { \\mathbb { C } } ^ { \\frac 1 2 } . \\end{align*}"}
-{"id": "8637.png", "formula": "\\begin{align*} F \\left ( \\begin{array} { c } p \\\\ q \\\\ u \\\\ z \\\\ \\varepsilon \\end{array} \\right ) = \\left ( \\begin{array} { c } p - 2 c _ 0 \\frac { u ^ 2 + z ^ 2 } { u ^ 4 + c _ 0 ^ 2 } + 2 c _ 0 - \\varepsilon h _ 1 ( u , z , \\varepsilon ) \\\\ q - \\varepsilon h _ 2 ( u , z , \\varepsilon ) \\\\ \\hat { H } ( p , q , u , z , \\varepsilon ) \\end{array} \\right ) , \\end{align*}"}
-{"id": "3544.png", "formula": "\\begin{align*} \\omega _ { C } ( a ) : = \\sqrt { a _ { 0 } b _ { 0 } c _ { 0 } } \\ ; \\bar { \\omega } _ { C } ( a ) . \\end{align*}"}
-{"id": "9002.png", "formula": "\\begin{align*} & \\langle X ( v _ { 1 } , \\dotsc , v _ { k } ) , v _ { i } \\rangle = 0 \\ , , 1 \\leq i \\leq k \\ , . \\\\ & \\langle X ( v _ { 1 } , \\dotsc , v _ { k } ) , X ( v _ { 1 } , \\dotsc , v _ { k } ) \\rangle = \\det ( \\langle v _ { i } , v _ { j } \\rangle ) \\ , . \\end{align*}"}
-{"id": "7221.png", "formula": "\\begin{align*} \\mu ( \\phi ( x _ M ) - \\phi ( x _ m ) ) & = K _ r * \\phi ( x _ M ) - K _ r * \\phi ( x _ m ) + \\frac { 1 } { 2 } \\left ( \\phi ^ 2 ( x _ M ) - \\phi ^ 2 ( x _ m ) \\right ) \\\\ & \\leq \\| K _ r \\| _ { L ^ 1 } ( \\phi ( x _ M ) - \\phi ( x _ m ) ) + \\frac { 1 } { 2 } \\left ( \\phi ( x _ M ) - \\phi ( x _ m ) \\right ) \\left ( \\phi ( x _ M ) + \\phi ( x _ m ) \\right ) , \\end{align*}"}
-{"id": "3118.png", "formula": "\\begin{align*} \\underline { \\chi ( \\O _ S ) } ( S , \\beta ) = \\chi ( \\O _ S ) \\ , , \\underline { K _ S ^ 2 } ( S , \\beta ) = K _ S ^ 2 \\ , , \\underline { K _ S \\beta ^ i } ( S , \\beta ) = K _ S \\beta ^ i \\ , , \\ldots \\ , . \\end{align*}"}
-{"id": "6160.png", "formula": "\\begin{align*} R ^ { \\omega , \\omega _ \\alpha } h ( z ) & = 4 \\int _ { \\mathbb { D } } h ' ( \\xi ) \\overline { ( B _ z ^ { \\omega _ \\alpha } ) ' ( \\xi ) } \\omega ^ * ( \\xi ) d A ( \\xi ) \\\\ & = 4 \\int _ { \\mathbb { D } } ( 1 - | \\xi | ) h ' ( \\xi ) \\overline { ( B _ z ^ { \\omega _ \\alpha } ) ' ( \\xi ) } \\frac { \\omega ^ * ( \\xi ) } { 1 - | \\xi | } d A ( \\xi ) . \\end{align*}"}
-{"id": "4543.png", "formula": "\\begin{align*} \\lambda ( u ) = \\bigcap _ { x , u \\in \\Delta ( x ) } \\Delta ( x ) ; \\lambda ^ \\circ ( u ) = \\bigcap _ { x , u \\in \\Delta ^ { \\ ! \\circ } ( x ) } \\Delta ^ \\circ ( x ) \\end{align*}"}
-{"id": "4018.png", "formula": "\\begin{align*} \\eta ^ { \\mathcal { N } } \\alpha ( r a ) = \\eta ^ { \\mathcal { N } } \\rho ( r ) \\alpha ( a ) \\end{align*}"}
-{"id": "8612.png", "formula": "\\begin{align*} \\begin{array} { l c r } ( u ^ 2 - v ^ 2 ) ( u ^ 2 + v ^ 2 - 1 ) & = & 0 , \\\\ & & \\\\ \\left ( u ^ 2 + v ^ 2 - \\frac { 1 } { 2 } \\right ) ^ 2 + 2 ( \\Lambda - 1 ) u ^ 2 v ^ 2 - 2 \\omega u v & = & \\frac { 1 } { 4 } . \\end{array} \\end{align*}"}
-{"id": "9779.png", "formula": "\\begin{align*} q ( X + Z _ \\infty ) = \\tau ^ { \\lambda _ * } q ( \\tau ^ { - 1 } X + Z _ \\infty ) = q ( X + \\tau Z _ \\infty ) \\quad X \\in \\R ^ n \\tau > 0 ; \\end{align*}"}
-{"id": "8464.png", "formula": "\\begin{align*} D _ t \\mathfrak { F } ( \\alpha , t ) = D _ t F ( \\zeta ( \\alpha , t ) , t ) = F _ t \\circ \\zeta + D _ t \\zeta F _ { \\zeta } \\circ \\zeta . \\end{align*}"}
-{"id": "2614.png", "formula": "\\begin{align*} G ( \\theta ) = \\int _ 0 ^ \\theta \\exp \\Big \\{ - \\frac { \\alpha } { \\alpha - 1 } \\int _ 0 ^ 1 G ( r u ^ { \\frac { 1 } { \\alpha - 1 } } ) ^ { \\alpha - 1 } \\frac { d u } { u } \\Big \\} d r , \\theta \\geq 0 , \\end{align*}"}
-{"id": "3423.png", "formula": "\\begin{align*} \\eta ( T ) = \\frac { 1 3 } { 7 } \\tau _ 0 ^ M ( T ) + \\tau _ * ^ M ( T ) \\le \\Sigma _ 0 + \\frac { 2 0 } { 2 1 } \\Sigma _ * \\le \\frac { 2 0 } { 2 1 } \\sum _ i \\eta ^ { M - 2 } ( T _ i ) . \\end{align*}"}
-{"id": "1417.png", "formula": "\\begin{align*} x _ { T + 1 } = \\frac { T + 1 } { T } x _ T - \\frac { 1 } { 2 } \\times \\frac { T + 1 } { T ^ 2 } x ^ 2 _ T . \\end{align*}"}
-{"id": "1199.png", "formula": "\\begin{align*} \\zeta _ 1 = \\zeta _ 1 ( \\rho , x , z ' ) = i \\sqrt { 1 + \\rho ^ 2 } \\sin \\theta + \\rho \\cos \\theta \\qquad ( \\rho \\in \\Bbb R , x \\in \\Bbb R ^ 3 _ - , z ' \\in \\Bbb R ^ 2 ) \\end{align*}"}
-{"id": "1659.png", "formula": "\\begin{align*} \\Big ( \\frac { \\partial ^ 2 H ^ c } { \\partial p _ n \\partial q _ l } \\ , p _ l - \\frac { \\partial ^ 2 H ^ c } { \\partial p _ n \\partial p _ l } \\ , q _ l \\Big ) \\Big | _ { z ^ \\bullet } = 0 \\quad \\Big ( \\frac { \\partial ^ 2 H ^ c } { \\partial q _ n \\partial q _ l } \\ , p _ l - \\frac { \\partial ^ 2 H ^ c } { \\partial q _ n \\partial p _ l } \\ , q _ l \\Big ) \\Big | _ { z ^ \\bullet } = 0 . \\end{align*}"}
-{"id": "708.png", "formula": "\\begin{align*} \\lim _ { x \\to 1 ^ - } \\left ( ( 1 - x ) ^ { k + 2 } f ^ { ( k + 1 ) } ( x ) \\right ) = \\frac { ( k + 2 ) ! } { 2 } - \\frac { k ( k + 1 ) ! } { 2 } = ( k + 1 ) ! , \\end{align*}"}
-{"id": "3129.png", "formula": "\\begin{align*} R ( u , v ) \\left ( T ( u ) \\otimes \\mathbf { I } \\right ) \\left ( \\mathbf { I } \\otimes T ( v ) \\right ) = \\left ( \\mathbf { I } \\otimes T ( v ) \\right ) \\left ( T ( u ) \\otimes \\mathbf { I } \\right ) R ( u , v ) . \\end{align*}"}
-{"id": "613.png", "formula": "\\begin{align*} \\gamma _ { \\mathcal G } ( x y ; x _ 1 y _ 1 , \\dots , x _ n y _ n ) = \\gamma _ { \\mathcal G } ( x ; x _ { y ^ { - 1 } ( 1 ) } , \\dots , x _ { y ^ { - 1 } ( n ) } ) \\gamma _ { \\mathcal G } ( y ; y _ 1 , \\dots , y _ n ) \\end{align*}"}
-{"id": "2633.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } & \\big ( 2 ^ { \\frac { 4 } { 3 } } N ^ { \\frac { 1 } { 3 } } \\big ) ^ k R _ { N } ^ { ( k ) } \\Big ( 4 N + 2 ^ { \\frac { 4 } { 3 } } N ^ { \\frac { 1 } { 3 } } u _ 1 , \\ldots , 4 N + 2 ^ { \\frac { 4 } { 3 } } N ^ { \\frac { 1 } { 3 } } u _ k \\Big ) = \\mathrm { P f } \\big [ { K ^ { ( \\mathrm { s o f t } ) } ( \\kappa , \\pi ; u _ i , u _ j ) } \\big ] _ { i , j = 1 } ^ { k } , \\end{align*}"}
-{"id": "4828.png", "formula": "\\begin{align*} J _ { s , t } & = I + \\sum _ { k = 1 } ^ n \\int _ s ^ t \\nabla A _ { k } ( Z _ r ) \\cdot J _ { s , r } \\mathrm { d } W _ { k , r } + \\int _ s ^ t \\nabla B ( Z _ r ) \\cdot J _ { s , r } \\mathrm { d } r , \\\\ & = I + \\sum _ { k = 1 } ^ n \\int _ s ^ t \\nabla A _ k ( Z _ r ) \\cdot J _ { s , r } \\circ \\mathrm { d } W _ { k , r } + \\int _ s ^ t \\nabla A _ 0 ( Z _ r ) \\cdot J _ { s , r } \\mathrm { d } r . \\end{align*}"}
-{"id": "924.png", "formula": "\\begin{align*} + \\Biggl . { \\bf 1 } _ { \\{ i _ 2 = i _ 5 \\ne 0 \\} } { \\bf 1 } _ { \\{ j _ 2 = j _ 5 \\} } { \\bf 1 } _ { \\{ i _ 3 = i _ 4 \\ne 0 \\} } { \\bf 1 } _ { \\{ j _ 3 = j _ 4 \\} } \\zeta _ { j _ 1 } ^ { ( i _ 1 ) } \\Biggr ) , \\end{align*}"}
-{"id": "389.png", "formula": "\\begin{align*} \\hat { \\mu } = ( - n , \\lambda - \\beta , \\kappa ' ) , \\end{align*}"}
-{"id": "513.png", "formula": "\\begin{align*} Q _ { i , j } = x t _ i \\odot y t _ j - x t _ j \\odot y t _ i \\in I _ 2 ( K _ C ) = I _ 2 ( K _ C ) ^ { \\sigma } . \\end{align*}"}
-{"id": "3321.png", "formula": "\\begin{align*} \\frac { S ( n , \\alpha \\cdot \\chi _ 0 ) } { \\phi ( M ) q ^ { \\ell } } = q ^ { n - \\ell - \\deg ( M ) } \\langle \\alpha \\rangle _ { \\mathcal { M } _ { n ; M } } . \\end{align*}"}
-{"id": "3025.png", "formula": "\\begin{align*} h ( z ) = z + \\sum \\limits _ { k = 2 } ^ \\infty a _ { k } { z ^ k } \\quad g ( z ) = \\sum \\limits _ { k = 1 } ^ \\infty b _ { k } { z ^ k } . \\end{align*}"}
-{"id": "4513.png", "formula": "\\begin{align*} \\mathcal { H } ^ s _ \\delta ( F ) = \\inf \\left \\{ \\sum _ { i = 1 } ^ { \\infty } ( \\mathrm { d i a m } ( U _ i ) ) ^ s : \\bigcup _ i U _ i \\supset F , \\mathrm { d i a m } ( U _ i ) < \\delta \\right \\} . \\end{align*}"}
-{"id": "8515.png", "formula": "\\begin{align*} \\partial _ { \\alpha } ^ k \\tilde { \\theta } = & \\partial _ { \\alpha } ^ k ( I - \\mathcal { H } ) ( \\zeta - \\bar { \\zeta } ) = \\partial _ { \\alpha } ^ k ( I - \\mathcal { H } ) ( \\zeta - \\alpha ) \\\\ = & \\partial _ { \\alpha } ^ k ( I + \\bar { \\mathcal { H } } - ( \\bar { \\mathcal { H } } + \\mathcal { H } ) ) ( \\zeta - \\alpha ) \\\\ = & 2 \\partial _ { \\alpha } ^ { k - 1 } ( \\zeta _ { \\alpha } - 1 ) - \\partial _ { \\alpha } ^ k ( \\bar { \\mathcal { H } } + \\mathcal { H } ) ) ( \\zeta - \\alpha ) . \\end{align*}"}
-{"id": "9745.png", "formula": "\\begin{align*} N ( 1 , q ) \\le \\liminf _ { \\ell \\to \\infty } N ( 1 , \\tilde v _ { r _ { j _ \\ell } } ) = \\liminf _ { \\ell \\to \\infty } N ( 1 , v _ { r _ { j _ \\ell } } ) = \\liminf _ { \\ell \\to \\infty } N ( r _ { j _ \\ell } , v _ { * } ) = \\lambda _ * . \\end{align*}"}
-{"id": "3320.png", "formula": "\\begin{align*} \\mathrm { V a r } _ M ( \\alpha ; n , n - \\ell - 1 ) = \\frac { \\sum _ { \\chi _ 0 \\neq \\chi \\in G ( R _ { \\ell , M } ) } \\left | S ( n , \\alpha \\cdot \\chi ) \\right | ^ 2 } { q ^ { 2 \\ell } \\phi ^ 2 ( M ) } . \\end{align*}"}
-{"id": "6243.png", "formula": "\\begin{align*} 1 > \\sum _ { i = 1 } ^ n \\frac { 1 - \\beta _ 1 } { \\alpha _ i } \\geq \\frac { 1 - \\beta _ 1 } { \\alpha _ 2 } \\geq \\frac { 1 - \\beta _ 1 } { 1 - \\alpha _ 1 } . \\end{align*}"}
-{"id": "1426.png", "formula": "\\begin{align*} m _ { t } - \\Delta m + \\mathrm { d i v } \\left ( m \\mathcal { H } _ { p } ( t , x , m , D u ) \\right ) = 0 , \\end{align*}"}
-{"id": "8366.png", "formula": "\\begin{align*} | \\dot { z } _ j ( t ) | = | \\bar { F } + \\sum _ { k \\neq j } \\frac { \\lambda _ k i } { 2 \\pi } \\frac { 1 } { \\overline { z _ j ( t ) - z _ k ( t ) } } | \\leq M _ 0 + \\frac { \\sum _ { j = 1 } ^ N | \\lambda _ j | } { 2 \\pi } \\tilde { d } _ P ( t ) ^ { - 1 } = M _ 0 + \\frac { \\tilde { \\lambda } } { 2 \\tilde { d } _ P ( t ) } . \\end{align*}"}
-{"id": "411.png", "formula": "\\begin{align*} S ^ { { \\rm c l } , 0 } _ { { \\rm L } } ( \\hat g , g ) : = & \\int _ M \\big ( | d \\omega | ^ 2 _ { g } + 2 K _ { g } \\omega \\big ) { \\rm d v } _ { g } , \\\\ S ^ { { \\rm c l } } _ { { \\rm A Y } } ( \\hat g , g ) : = & \\int _ M \\big ( \\tfrac { 1 } { 4 } \\phi \\Delta _ { g } \\phi + \\frac { \\phi } { V _ { g } } \\big ) { \\rm d v } _ { g } . \\end{align*}"}
-{"id": "2752.png", "formula": "\\begin{align*} \\Big ( \\frac { \\tau z - 1 } { z - \\tau } \\frac { w - \\tau } { \\tau w - 1 } \\Big ) ^ { N - n } & = \\Big ( 1 - \\frac { \\frac { 1 } { \\tau } - \\tau } { z - \\tau } \\Big ) ^ { N - n } \\Big ( 1 - \\frac { \\frac { 1 } { \\tau } - \\tau } { w - \\tau } \\Big ) ^ { - N + n } \\to e ^ { - \\frac { 2 \\kappa } { z - 1 } + \\frac { 2 \\kappa } { w - 1 } } . \\end{align*}"}
-{"id": "5900.png", "formula": "\\begin{align*} \\bar { \\varphi } _ n : = \\sum _ { k = n } ^ { \\infty } \\varphi _ k + \\psi _ k \\end{align*}"}
-{"id": "8216.png", "formula": "\\begin{align*} X ^ 5 + \\sum _ { j = 0 } ^ 4 a _ j ( t ) X ^ j = \\prod \\limits _ { \\lambda = 0 } ^ 4 ( X - t _ \\lambda ) , \\end{align*}"}
-{"id": "4246.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\left \\langle a ( x ) \\left \\lvert d u \\right \\rvert ^ { p - 2 } d u ; d \\phi \\right \\rangle = - \\int _ { \\Omega } \\left \\langle f ; \\phi \\right \\rangle \\phi \\in W ^ { 1 , p } _ { \\delta , T } \\left ( \\Omega ; \\Lambda ^ { k } \\mathbb { R } ^ { n } \\otimes \\mathbb { R } ^ { N } \\right ) . \\end{align*}"}
-{"id": "2543.png", "formula": "\\begin{align*} \\psi _ 0 ( x , z ) : = \\psi ( x , z ) + \\beta ( x ) z , x \\in E , z \\geq 0 . \\end{align*}"}
-{"id": "9840.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ r \\alpha _ j ( v ) \\sum _ { u \\in N ^ j _ G ( v ) } \\hat { w } _ { i + 1 } ( u ) & = \\alpha _ 0 ( v ) \\hat { w } _ { i + 1 } ( v ) + \\sum _ { j = 1 } ^ r \\alpha _ j ( v ) \\sum _ { u \\in N ^ j _ G ( v ) } \\hat { w } _ { i + 1 } ( u ) \\\\ & \\le \\alpha _ 0 ( v ) \\hat { w } _ { i + 1 } ( v ) + \\sum _ { j = 1 } ^ r \\alpha _ j ( v ) \\big | N ^ j _ G ( v ) \\big | \\end{align*}"}
-{"id": "9385.png", "formula": "\\begin{align*} K \\cap Z = \\langle \\{ y ^ 2 \\} \\cup \\{ [ y _ 0 , y _ j ] \\mid j \\in \\mathbb { N } j \\equiv _ { 2 ^ n } 0 , \\pm 1 , \\ldots , \\pm ( m - 1 ) \\} \\rangle . \\end{align*}"}
-{"id": "5479.png", "formula": "\\begin{align*} ( u - \\psi ) ( t , x ) = \\min _ { ( 0 , T ) \\times H } ( u - \\psi ) \\end{align*}"}
-{"id": "7571.png", "formula": "\\begin{align*} \\begin{gathered} \\Delta _ 1 \\tau ( x , s , t ) = \\tau ( X ^ { - 1 } ( x , s ) , s ) - \\tau ( X ^ { - 1 } ( x , s ) , t ) , \\\\ \\Delta _ 2 \\tau ( x , s , t ) = \\tau ( X ^ { - 1 } ( x , s ) , t ) - \\tau ( X ^ { - 1 } ( x , t ) , t ) . \\end{gathered} \\end{align*}"}
-{"id": "7599.png", "formula": "\\begin{align*} \\begin{gathered} \\Delta g _ { \\nu ( t - s ) } * \\left ( \\nabla \\cdot \\eta ( t ) \\Delta _ 2 \\tau ( s , t ) \\right ) ( x ) = \\int _ { \\mathbb { R } ^ d } ( K ( x , z , t , s ) \\left ( \\nabla \\cdot \\eta \\right ) ( X ( z , t ) , t ) \\\\ + \\Delta g _ { \\nu ( t - s ) } ( x - X ( z , t ) ) \\left ( \\left ( \\nabla \\cdot \\eta \\right ) ( X ( z , s ) , t ) - \\left ( \\nabla \\cdot \\eta \\right ) ( X ( z , t ) , t ) \\right ) ) d z , \\end{gathered} \\end{align*}"}
-{"id": "4049.png", "formula": "\\begin{align*} ( m _ n ( a _ 1 , \\ldots , a _ n ) , a _ { n + 1 } ) = ( - 1 ) ^ n ( - 1 ) ^ { | a _ 1 | ( | a _ 2 | + \\ldots + | a _ { n + 1 } | ) } ( m _ n ( a _ 2 , \\ldots , a _ { n + 1 } ) , a _ 1 ) \\end{align*}"}
-{"id": "1478.png", "formula": "\\begin{align*} V _ { 9 } + V _ { 1 4 } \\leq \\varepsilon ( \\mathcal { G } _ { 1 } ( K ) + \\mathcal { G } _ { 2 } ( K ) ) E _ { \\mu } + \\frac { 1 } { 4 } \\sum _ { j = 1 } ^ { d } \\int _ { \\mathbb { T } ^ { d } } ( \\partial _ { x _ { j } } ( D w ^ { 1 } - D w ^ { 2 } ) ) ^ { 2 } \\ d x . \\end{align*}"}
-{"id": "9485.png", "formula": "\\begin{align*} m _ { j } \\leq m _ { j + 1 } \\leq \\displaystyle \\inf _ { B _ { j + 1 } } u \\leq \\displaystyle \\sup _ { B _ { j + 1 } } u \\leq M _ { j + 1 } \\leq M _ { j } , \\ \\ M _ { j + 1 } - m _ { j + 1 } = \\lambda R _ { j + 1 } ^ { \\alpha } , \\end{align*}"}
-{"id": "5427.png", "formula": "\\begin{align*} \\mathcal B = E _ 1 ( x ) \\oplus \\cdots \\oplus E _ k ( x ) \\oplus \\ldots \\oplus E _ \\infty ( x ) \\end{align*}"}
-{"id": "4286.png", "formula": "\\begin{align*} \\binom { n } { 2 } - e ( G ) \\geq \\binom { n } { 2 } - n ^ { 5 / 3 + \\delta } \\end{align*}"}
-{"id": "3477.png", "formula": "\\begin{align*} C _ { M , N } = ( M + 1 ) ^ { 2 } \\Big ( 1 - \\big ( 1 - \\frac { 1 } { 2 N } \\big ) \\log N - \\frac { 1 } { 2 N } \\log ( 2 \\pi ) \\Big ) + \\frac { M + 1 } { 2 N } \\log \\frac { M + 1 } { N } . \\end{align*}"}
-{"id": "6780.png", "formula": "\\begin{align*} \\Phi ' ( t , \\beta ) = \\frac { \\partial } { \\partial t } \\Phi ( t , \\beta ) = u ( t , \\Phi ( t , \\beta ) ) \\ , . \\end{align*}"}
-{"id": "7218.png", "formula": "\\begin{align*} K _ 2 ( x ) = \\frac { 1 } { 2 } \\left ( | x | - \\pi \\right ) ^ 2 - \\frac { \\pi ^ 2 } { 6 } , K _ 4 ( x ) = \\frac { \\pi ^ 4 } { 4 5 } - \\frac { 1 } { 2 4 } \\left ( x ^ 2 - 2 \\pi | x | \\right ) ^ 2 , x \\in [ - \\pi , \\pi ] . \\end{align*}"}
-{"id": "4980.png", "formula": "\\begin{align*} B _ m \\| u _ { \\tilde { n } } - x \\| ^ 2 \\leq b ^ 2 \\exp \\left ( - \\sum _ { j = \\tilde { n } + 1 } ^ { m } \\lambda _ j \\right ) \\leq \\frac { 1 } { 3 ( k + 1 ) ^ 2 } . \\end{align*}"}
-{"id": "4247.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\delta \\left ( a ( x ) \\left \\lvert d u \\right \\rvert ^ { p - 2 } d u \\right ) & = f & & B _ { R } , \\\\ \\delta u & = \\delta u _ { 0 } & & B _ { R } , \\\\ \\nu \\wedge u & = \\nu \\wedge u _ { 0 } & & \\partial B _ { R } , \\end{aligned} \\right . \\end{align*}"}
-{"id": "9991.png", "formula": "\\begin{align*} B ^ i ( \\mathbf { p } ) = - g ^ { i j } p _ { j , x } - c ^ { i j } _ k b ^ k _ { x x } p _ j - c ^ \\alpha w ^ i _ { \\alpha k } b ^ k _ { x x } r _ \\alpha . \\end{align*}"}
-{"id": "1630.png", "formula": "\\begin{align*} \\{ F , G \\} ( \\varphi ) : = - i \\int _ 0 ^ 1 \\big ( ( \\partial _ { \\varphi _ 1 } F ) ( \\partial _ { \\varphi _ 2 } G ) - ( \\partial _ { \\varphi _ 1 } G ) ( \\partial _ { \\varphi _ 2 } F ) \\big ) \\ , d x , \\end{align*}"}
-{"id": "32.png", "formula": "\\begin{align*} & \\sum _ { j \\geq 1 } \\left [ F ^ { \\alpha _ { T _ j } } _ { T _ j } - F ^ { \\alpha _ { T _ { j } - } } _ { T _ { j } - } \\right ] \\chi _ { \\{ T _ { j } \\leq t \\} } \\\\ = & \\int _ { 0 } ^ { t } \\sum _ { k \\in I } q ^ { \\alpha _ { s - } k } [ F ^ k _ s - F ^ { \\alpha _ { s - } } _ s ] d s + \\int _ { 0 } ^ { t } \\sum _ { k , k ^ { \\prime } \\in I } [ F _ s ^ k - F _ s ^ { k ' } ] \\chi _ { \\{ \\alpha _ { s - } = k ^ { \\prime } \\} } d \\tilde { N } _ s ^ { k ^ { \\prime } k } , \\end{align*}"}
-{"id": "9225.png", "formula": "\\begin{align*} y _ { u _ 1 u _ 2 } ^ { } = \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } y _ { b _ { i 1 } b _ { i 2 } } . \\end{align*}"}
-{"id": "1544.png", "formula": "\\begin{align*} \\widetilde { T } \\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} z + a _ 2 z ^ 2 \\dots + a _ i z ^ i + a _ { i + 1 } ( x ) z ^ { i + 1 } + \\dots \\\\ \\delta x + b _ 1 x z + b _ 2 x z ^ 2 + \\dots + b _ j x z ^ j + b _ { j + 1 } ( x ) z ^ { j + 1 } + \\dots \\end{pmatrix} , \\end{align*}"}
-{"id": "1158.png", "formula": "\\begin{align*} & \\sum \\limits _ { j = k + 1 } ^ { + \\infty } y _ j ^ 2 \\geq \\sum \\limits _ { j = 1 } ^ { k } ( x _ j - \\epsilon ) ^ 2 \\geq \\sum \\limits _ { j = 1 } ^ { k } x _ j ^ 2 - 2 ( x _ 0 k ) ^ { \\frac { 1 } { 2 } } \\epsilon + k \\epsilon ^ 2 . \\end{align*}"}
-{"id": "4954.png", "formula": "\\begin{align*} f _ { i + 3 } = v _ i ^ n + v _ i z ^ { t - n } - ( v _ { i + 1 } - v _ { i + 2 } + \\cdots + ( - 1 ) ^ { d + i } v _ { d - 3 } + ( - 1 ) ^ { d + i + 1 } ( x - p ) ) ^ n + v _ i z ^ n - v _ i x ^ n \\end{align*}"}
-{"id": "6495.png", "formula": "\\begin{align*} \\langle M ( f _ { \\hat { \\phi _ t } , \\chi _ s , \\psi _ s } ) , z _ { \\varphi _ 0 } \\rangle & = L ( \\chi \\psi ^ { - 1 } , 1 - 2 s ) ^ { - 1 } \\cdot \\int _ { H ( \\Z _ { \\ell } ) } f _ { \\phi _ t , \\psi _ s , \\chi _ s } ( h ) z _ { \\varphi _ 0 } ( h ) d h \\\\ & = L ( \\chi \\psi ^ { - 1 } , 1 - 2 s ) ^ { - 1 } f _ { \\phi _ t , \\psi _ s , \\chi _ s } ( 1 ) \\cdot \\int _ { K _ 0 ( \\ell ^ t ) } z _ { s , \\varphi _ 0 } ( h ) d h . \\end{align*}"}
-{"id": "7289.png", "formula": "\\begin{align*} \\phi ( a ) = \\psi _ q ( a ) + ( 1 - h _ q ^ - ) \\Phi _ p ( a ) ( h _ p ^ - - h _ q ^ + ) + ( 1 - h _ q ^ - ) R _ p \\Phi _ p ( a ) ( h _ p ^ + - h _ p ^ - ) \\end{align*}"}
-{"id": "3385.png", "formula": "\\begin{align*} p _ k = \\Theta \\left ( \\frac { \\delta } { n } 2 ^ { k / 2 } \\right ) . \\end{align*}"}
-{"id": "8112.png", "formula": "\\begin{align*} \\widetilde { A } _ { n , r } ( t ) \\ = \\ \\sum _ { m = 0 } ^ n { n \\choose m } t ^ { n - m } A _ { m , r } ( t ) , \\end{align*}"}
-{"id": "1270.png", "formula": "\\begin{align*} S ( \\eta - \\xi ) = e ^ { i a \\ln | \\eta - \\xi | } \\left ( A + B \\frac { e ^ { i \\beta ( \\eta - \\xi ) ^ 3 } } { ( \\eta - \\xi ) ^ 3 } \\right ) . \\end{align*}"}
-{"id": "4276.png", "formula": "\\begin{align*} \\textup { s a t } ( n , H , F ) = \\Theta ( n ^ m ) \\end{align*}"}
-{"id": "2658.png", "formula": "\\begin{align*} g _ { j k } = \\frac { 4 \\pi } { \\eta _ { - } ^ { 2 } } \\frac { \\varrho _ { j } \\varrho _ { k } ( \\varrho _ { j } - \\varrho _ { k } ) } { ( 1 - \\varrho _ { j } - \\varrho _ { k } ) \\sqrt { 1 - 2 \\varrho _ { j } } \\sqrt { 1 - 2 \\varrho _ { k } } } , 1 \\leq j , k \\leq N . \\end{align*}"}
-{"id": "7775.png", "formula": "\\begin{align*} S _ n = P _ { 2 ^ { n - 1 } } - P _ { 2 ^ n } . \\end{align*}"}
-{"id": "196.png", "formula": "\\begin{align*} X _ { C } = \\sqrt { x ^ 3 } \\dfrac { \\partial } { \\partial x ^ 2 } + \\dfrac { y ^ 3 } { 2 \\sqrt { x ^ 3 } } \\dfrac { \\partial } { \\partial y ^ 2 } , & & X _ { V } = \\sqrt { x ^ 3 } \\dfrac { \\partial } { \\partial y ^ 2 } \\end{align*}"}
-{"id": "2613.png", "formula": "\\begin{align*} \\| f \\| _ { \\mu ; p } : = \\Big ( \\int _ { S } | f | ^ p d \\mu \\Big ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "1359.png", "formula": "\\begin{align*} \\delta _ { n } ( f ) ( \\lambda _ 0 , \\ldots , \\lambda _ { n } ) = & \\ f ( \\lambda _ 1 , \\ldots , \\lambda _ { n } ) + ( - 1 ) ^ { n + 1 } f ( \\lambda _ 0 , \\ldots , \\lambda _ { n - 1 } ) \\cdot \\lambda _ { n } \\\\ & + \\sum _ { i = 1 } ^ { n } ( - 1 ) ^ i f ( \\lambda _ 0 , \\ldots , \\lambda _ { i - 1 } \\lambda _ i , \\ldots , \\lambda _ { n } ) \\end{align*}"}
-{"id": "7446.png", "formula": "\\begin{align*} \\displaystyle \\lvert S _ { \\pi _ \\lambda } \\rvert = 2 ^ S P \\end{align*}"}
-{"id": "5154.png", "formula": "\\begin{align*} \\begin{array} { l l l } A _ { s s } : = k _ j + \\lambda + \\gamma _ { k _ j } , & B _ { s s } : = \\mu - \\nu , & s \\sim ( j , m _ j , k _ j ) \\in I _ 1 \\\\ A _ { s s } : = \\mu + \\nu , & B _ { s s } : = - ( k _ j + \\lambda - \\gamma _ { k _ j } ) , & s \\sim ( j , m _ j , k _ j ) \\in I _ 2 , \\\\ A _ { s t } = B _ { s t } = 0 , & & s \\sim ( j , m _ j , k _ j ) \\in I _ 1 \\cup I _ 2 , \\ , 1 \\leq t \\leq d , t \\neq s . \\end{array} \\end{align*}"}
-{"id": "6750.png", "formula": "\\begin{align*} | \\widehat { u } ( t _ 0 , \\xi ) | = \\int _ { r _ 0 - \\delta } ^ { r _ 0 + \\delta } e ^ { \\xi \\psi ( r ) } \\varphi ( t _ 0 - r ) d r \\geqslant \\int _ { r _ 0 - \\frac { \\delta } { 2 } } ^ { r _ 0 + \\frac { \\delta } { 2 } } e ^ { \\xi \\psi ( r ) } d r . \\end{align*}"}
-{"id": "4825.png", "formula": "\\begin{align*} \\mathcal B _ t ' \\psi ( t ) + \\mathcal B _ t \\psi ' ( t ) = \\lambda ' _ 1 ( \\mathcal B _ t , g ( t ) ) \\psi ( t ) + \\lambda _ 1 ( \\mathcal B _ t , g ( t ) ) \\psi ' ( t ) \\end{align*}"}
-{"id": "5087.png", "formula": "\\begin{align*} \\min \\{ \\lambda _ j \\} _ { j = 1 } ^ m \\| h \\| ^ 2 \\leq \\sum _ { j = 1 } ^ m \\lambda _ j | \\langle h , e _ j \\rangle | ^ 2 = \\langle S _ { x , \\tau } h , h \\rangle = \\sum _ { j = 1 } ^ n \\langle h , x _ j \\rangle \\langle \\tau _ j , h \\rangle \\leq \\max \\{ \\lambda _ j \\} _ { j = 1 } ^ m \\| h \\| ^ 2 , \\forall h \\in \\mathcal { H } . \\end{align*}"}
-{"id": "5745.png", "formula": "\\begin{align*} s ( x , y , t ) & = \\sum _ { m , n \\in \\N } \\Big ( \\int _ 0 ^ t ( t - \\xi ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } [ - \\mu _ { m , n } ( t - \\xi ) ^ { \\alpha } ] 0 d \\xi \\Big ) J _ 0 ( \\gamma _ m x ) \\sin ( n \\pi x ) \\\\ & = 0 . \\end{align*}"}
-{"id": "3730.png", "formula": "\\begin{align*} ( 2 n - 1 - q ) a \\| t _ * u \\| ^ { 2 n } & + ( n - 1 - q ) b \\| t _ * u \\| ^ n + q \\int _ { \\Omega } ( | x | ^ { - \\mu } * F ( t _ * u ) ) f ( t _ * u ) t _ * u ~ d x \\\\ & = \\int _ { \\Omega } ( | x | ^ { - \\mu } * F ( t _ * u ) ) f ^ { ' } ( t _ * u ) ( t _ * u ) ^ 2 ~ d x + \\int _ { \\Omega } ( | x | ^ { - \\mu } * f ( t _ * u ) t _ * u ) f ( t _ * u ) t _ * u ~ d x . \\end{align*}"}
-{"id": "5749.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 | \\sin ( n \\pi y ) | ^ 2 d y & \\geq \\int _ 0 ^ { \\frac { 1 } { n } } | \\sin ( n \\pi y ) | ^ 2 d y & & \\\\ & = \\int _ 0 ^ 1 \\frac { 1 } { n } | \\sin ( \\pi t ) | ^ 2 d t \\\\ & = \\frac { C _ 4 } { n } \\end{align*}"}
-{"id": "9001.png", "formula": "\\begin{align*} \\begin{aligned} & \\C : = \\left ( \\begin{array} { c c c c } 0 & 0 & 1 & 0 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\end{array} \\right ) , { \\mathcal N } _ 1 = \\{ 3 \\} , { \\mathcal N } _ 2 : = \\{ 1 \\} , \\\\ & { \\mathcal N } _ 3 : = \\{ 2 \\} , { \\mathcal N } _ 4 : = \\{ 3 \\} , n ^ { \\infty } = 1 , \\gamma _ g = 2 . \\end{aligned} \\end{align*}"}
-{"id": "4063.png", "formula": "\\begin{align*} K ^ { \\alpha , \\theta } _ { V , n } ( x , y ) = w ( y ) \\sum _ { j = 0 } ^ { n - 1 } p _ j ( x ) q _ j ( y ^ { \\theta } ) \\end{align*}"}
-{"id": "9930.png", "formula": "\\begin{align*} E ^ { \\nu } \\left [ \\left . \\frac { d \\mu } { d \\nu } ( X _ { 0 } ) \\right | \\bigcap _ { n \\geq 0 } \\mathcal { F } _ { 0 , \\infty } ^ { \\mathcal { Y } } \\vee \\mathcal { F } _ { n , \\infty } ^ { \\mathcal { X } } \\right ] & = E ^ { \\nu } \\left [ \\left . \\frac { d \\mu } { d \\nu } ( X _ { 0 } ) \\right | { F } _ { 0 , \\infty } ^ { \\mathcal { Y } } \\right | \\end{align*}"}
-{"id": "2203.png", "formula": "\\begin{align*} q ( \\boldsymbol { x } , t ) = \\boldsymbol { x } ^ T P \\boldsymbol { x } \\end{align*}"}
-{"id": "9876.png", "formula": "\\begin{align*} & \\pi _ { n + 1 } ^ { \\mu } ( d x _ { n + 1 } ) = F ( \\pi _ { n } ^ { \\mu } , y _ { n + 1 } ) ( d x _ { n + 1 } ) \\\\ & : = \\frac { g ( x _ { n + 1 } , y _ { n + 1 } ) \\int _ { \\mathcal { X } } T ( d x _ { n + 1 } | X _ { n } = x ) \\pi _ { n } ^ { \\mu } ( d x ) } { \\int _ { \\mathcal { X } } g ( x _ { n + 1 } , y _ { n + 1 } ) \\int _ { \\mathcal { X } } T ( d x _ { n + 1 } | X _ { n } = x ) \\pi _ { n } ^ { \\mu } ( d x ) } \\end{align*}"}
-{"id": "923.png", "formula": "\\begin{align*} + \\Biggl . { \\bf 1 } _ { \\{ i _ 1 = i _ 4 \\ne 0 \\} } { \\bf 1 } _ { \\{ j _ 1 = j _ 4 \\} } { \\bf 1 } _ { \\{ i _ 2 = i _ 3 \\ne 0 \\} } { \\bf 1 } _ { \\{ j _ 2 = j _ 3 \\} } \\Biggr ) , \\end{align*}"}
-{"id": "2172.png", "formula": "\\begin{align*} \\| \\tau _ { - \\mathbf y } ( f * \\phi ) ( \\mathbf x ) & ( 1 + d ( \\mathbf x , \\mathbf y ) ) ^ { \\delta ' } \\| _ { L ^ 1 d w ( \\mathbf x ) } \\\\ & \\leq C \\| f ( \\cdot ) ( 1 + \\| \\cdot \\| ) ^ { \\mathbf { N } / 2 + \\delta } \\| _ { L ^ 1 ( d w ) } \\| \\phi ( \\cdot ) ( 1 + \\| \\cdot \\| ) ^ { \\mathbf { N } + \\delta } \\| _ { L ^ { \\infty } } . \\end{align*}"}
-{"id": "7794.png", "formula": "\\begin{align*} - 2 \\pi | \\xi | m ^ G ( \\xi ) = \\sum _ { j = 1 } ^ d \\bigg ( - i \\frac { \\xi _ j } { | \\xi | } \\bigg ) ( - 2 \\pi i \\xi _ j m ^ G ( \\xi ) ) . \\end{align*}"}
-{"id": "9571.png", "formula": "\\begin{align*} S _ 1 = \\frac { 1 } { ( 1 - c ) ( q ) _ N } \\left ( 1 - \\frac { ( q ) _ N } { ( c q ) _ N } \\right ) . \\end{align*}"}
-{"id": "2917.png", "formula": "\\begin{align*} g _ 1 ( [ u ] \\cup [ v ] ) \\cap g _ 2 ( [ u ] \\cup [ v ] ) & = \\varnothing \\end{align*}"}
-{"id": "1389.png", "formula": "\\begin{align*} \\Box _ n \\circ \\triangledown _ n = \\operatorname { i d } _ { \\Z { \\widetilde { Q } } _ * ( \\widetilde { \\Lambda } ) } \\ ; . \\end{align*}"}
-{"id": "3145.png", "formula": "\\begin{align*} k ^ { \\pm } _ i ( u ) F _ i ( v ) k ^ { \\pm } _ i ( u ) ^ { - 1 } & = f ( v , u ) \\ F _ i ( v ) , \\\\ k ^ { \\pm } _ { i + 1 } ( u ) F _ i ( v ) k ^ { \\pm } _ { i + 1 } ( u ) ^ { - 1 } & = f ( u , v ) \\ F _ i ( v ) , \\end{align*}"}
-{"id": "9301.png", "formula": "\\begin{align*} p ^ v - p ^ u = - \\frac { \\tau K \\eta ( \\hat y ) } { 1 + \\tau K \\eta ( \\hat y ) } [ D \\underline u ( \\hat y ) + D _ y \\phi ( \\hat x , \\hat y ) ] - \\delta [ D d ( \\hat x ) + D d ( \\hat y ) - 2 ( \\hat x - z ) ] . \\\\ \\end{align*}"}
-{"id": "3503.png", "formula": "\\begin{align*} g ( \\xi ) = M \\log N ( 1 - u ) + \\log \\frac { 1 - u } { u } + \\frac { M + 1 } { N ( 1 - u ) } \\left ( N u - [ N u ] - \\frac { 1 } { 2 } \\right ) - \\xi . \\end{align*}"}
-{"id": "6573.png", "formula": "\\begin{gather*} e _ { I , t } : = e _ { i _ 1 } { \\circ } \\cdots { \\circ } e _ { i _ s } { \\circ } ( e _ 0 ) ^ { t } \\end{gather*}"}
-{"id": "3137.png", "formula": "\\begin{align*} f ( u , \\bar t ^ i ) = \\prod _ { t ^ i _ j \\in \\bar t ^ i } f ( u , t ^ i _ j ) , f ( \\bar t ^ s , \\bar x ^ p ) = \\prod _ { t ^ s _ j \\in \\bar t ^ s } \\prod _ { x ^ p _ k \\in \\bar x ^ p } f ( t ^ s _ j , x ^ p _ k ) . \\end{align*}"}
-{"id": "8248.png", "formula": "\\begin{align*} 2 4 S _ { C 5 } ( q ) = \\frac { 2 E _ 2 ( 2 \\tau ) - E _ 2 ( \\tau ) } { ( q ^ 2 ; q ^ 2 ) _ \\infty } - \\frac { E _ 2 ( 2 \\tau ) } { ( q ^ 2 ; q ^ 2 ) _ \\infty } . \\end{align*}"}
-{"id": "2887.png", "formula": "\\begin{align*} \\begin{aligned} G ^ { i } & = L _ { 2 i - 1 , 2 i } ^ { 2 } , \\\\ K ^ { i j } & = L _ { 2 i - 1 , 2 i } ^ 2 + L _ { 2 i - 1 , 2 j - 1 } ^ 2 + L _ { 2 i - 1 , 2 j } ^ 2 + L _ { 2 i , 2 j - 1 } ^ 2 + L _ { 2 i , 2 j } ^ 2 + L _ { 2 j - 1 , 2 j } ^ 2 , \\end{aligned} \\end{align*}"}
-{"id": "6147.png", "formula": "\\begin{align*} \\frac { b ' _ t } { a ' _ t } \\le \\frac { a ' _ 1 - 1 } { 4 e b ' _ 1 } = \\frac { ( 2 e b _ 1 ^ 2 + 1 ) - 1 } { 4 e a _ 1 b _ 1 } = \\frac { b _ 1 } { 2 a _ 1 } \\end{align*}"}
-{"id": "1480.png", "formula": "\\begin{align*} \\frac { d E ^ { j } _ { w } } { d t } = \\int _ { \\mathbb { T } ^ { d } } ( \\partial _ { x _ { j } } w ^ { 1 } - \\partial _ { x _ { j } } w ^ { 2 } ) \\partial _ { t } ( \\partial _ { x _ { j } } w ^ { 1 } - \\partial _ { x _ { j } } w ^ { 2 } ) \\ d x . \\end{align*}"}
-{"id": "6362.png", "formula": "\\begin{align*} \\chi \\mapsto \\int _ { X _ T } \\chi d t \\wedge ( \\omega _ t + d d ^ c \\varphi _ t ) ^ n : = \\int _ 0 ^ T d t \\left ( \\int _ { X } \\chi ( t , \\cdot ) ( \\omega _ t + d d ^ c \\varphi _ t ) ^ n \\right ) . \\end{align*}"}
-{"id": "1674.png", "formula": "\\begin{align*} \\rho _ A ( \\alpha _ A ( X ) ) \\varphi ^ * = \\varphi ^ * \\rho _ A ( X ) . \\end{align*}"}
-{"id": "8418.png", "formula": "\\begin{align*} \\| z _ { \\alpha } ( \\cdot , t ) - 1 \\| _ { H ^ { s - 1 } } \\leq \\| \\partial _ { \\alpha } \\xi _ 0 \\| _ { H ^ { s - 1 } } + T _ 0 M _ 2 : = M _ 3 , \\end{align*}"}
-{"id": "6870.png", "formula": "\\begin{align*} I _ 3 = I _ 2 + \\langle f _ 6 , \\dots , f _ { 1 2 } \\rangle \\subset \\mathbb { Q } [ y _ 1 , y _ 2 , y _ 3 ] \\ , . \\end{align*}"}
-{"id": "3310.png", "formula": "\\begin{align*} \\frac { 1 } { q ^ { n - \\deg ( M ) } \\phi ( M ) } \\sum _ { F \\in \\mathcal { M } _ { n } } \\chi ( F ) = \\begin{cases} 0 & \\chi \\neq \\chi _ 0 , \\\\ 1 & \\chi = \\chi _ 0 . \\end{cases} \\end{align*}"}
-{"id": "4493.png", "formula": "\\begin{align*} a _ { i j } = \\frac { e ^ v } { w } \\big ( \\sigma _ { i j } + \\gamma ^ { i k } \\nabla _ { k l } v \\gamma ^ { j l } \\big ) \\end{align*}"}
-{"id": "4767.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\| \\tilde { u } _ N \\| _ { \\mathcal { H } } + 2 \\nu \\int _ { \\mathcal { F } _ 0 } | D ( \\tilde { u } _ N ) | ^ 2 d x + 2 \\nu \\alpha \\int _ { \\partial \\mathcal { S } _ 0 } | \\tilde { u } _ N - \\tilde { u } _ { N , \\mathcal { S } } | ^ 2 d s = 2 \\nu a ( H , \\tilde { u } _ N ) - b ( \\tilde { u } _ N , \\tilde { u } _ N , H ) . \\end{align*}"}
-{"id": "4699.png", "formula": "\\begin{align*} u _ t + u _ x = 0 \\ ; \\ ; \\ ; ( 0 , 2 \\pi ) \\times ( 0 , T ] , \\end{align*}"}
-{"id": "7472.png", "formula": "\\begin{gather*} \\Pi ^ { - 1 } ( D _ p ^ * ) = \\coprod _ \\alpha \\widetilde { D ^ * _ p } ^ \\alpha \\end{gather*}"}
-{"id": "6730.png", "formula": "\\begin{align*} Q _ { 1 k l } = h _ { k l } , Q _ { 2 k l } = \\frac { \\partial h _ { k l } } { \\partial t } , \\end{align*}"}
-{"id": "5019.png", "formula": "\\begin{align*} \\lambda _ 1 ( t _ 0 , x _ 0 ) = \\displaystyle \\max _ { ( t , x ) \\in [ 0 , T ] \\times { \\bar \\Omega } } \\lambda _ 1 ( t , x ) . \\end{align*}"}
-{"id": "2082.png", "formula": "\\begin{align*} ( - D { \\beta ^ 2 } - { k _ { d } } - j \\beta v ) \\tilde C _ z ( \\beta , t \\mid z _ { \\rm t x } , { t _ 0 } ) + \\delta ( t - { t _ 0 } ) = \\\\ \\frac { { \\partial \\tilde C _ z ( \\beta , t \\mid z _ { \\rm t x } , { t _ 0 } ) } } { { \\partial t } } , \\end{align*}"}
-{"id": "377.png", "formula": "\\begin{align*} A B + C D \\bar { \\alpha } ^ 2 & = 0 , \\\\ A C + B D \\alpha ^ 2 & = 0 , \\\\ A D + B C & = 0 . \\end{align*}"}
-{"id": "767.png", "formula": "\\begin{align*} G _ { q , M } = ( G _ { q ^ d , M } ) _ { \\mathrm { G a l } ( \\mathbb { F } _ { q ^ d } / \\mathbb { F } _ q ) } = \\frac { G _ { q ^ d , M } } { ( \\sigma - 1 ) G _ { q ^ d , M } } . \\end{align*}"}
-{"id": "179.png", "formula": "\\begin{align*} & [ \\pi _ { X _ { C } , Y _ { V } } , \\pi _ { X _ { C } , Y _ { V } } ] = 2 [ X _ { C } , Y _ { V } ] \\wedge X _ { C } \\wedge Y _ { V } = 2 [ X , Y ] _ { V } \\wedge X _ { C } \\wedge Y _ { V } = 0 . \\end{align*}"}
-{"id": "10046.png", "formula": "\\begin{align*} \\dot x = - \\frac { 1 } { 4 } x ^ 3 y , \\dot y = - \\frac { M } { 4 } x ^ 4 + \\O ( x ^ 6 ) , \\dot \\theta = \\frac { 1 } { 2 } G x ^ 4 , \\dot G = \\O ( x ^ 6 ) . \\end{align*}"}
-{"id": "5737.png", "formula": "\\begin{align*} D _ { l , k } = \\frac { 1 } { | | J _ 0 ( \\gamma _ l x ) \\sin ( k \\pi y ) | | _ 2 ^ 2 } \\int _ 0 ^ 1 \\int _ 0 ^ 1 x \\psi ( x , y ) J _ 0 ( \\gamma _ l x ) \\sin ( k \\pi y ) d x d y . \\end{align*}"}
-{"id": "575.png", "formula": "\\begin{align*} \\xi ^ s ( 1 + p x _ s ) R _ n ^ * - P = \\xi ^ s \\left ( V \\setminus ( 1 + p x _ s ) R _ n ^ * \\right ) . \\end{align*}"}
-{"id": "6567.png", "formula": "\\begin{align*} \\begin{array} { r l l } P \\in E ( h ) , & M _ { h , P } & = \\left \\{ \\ , L _ P \\ , \\right \\} , \\end{array} \\end{align*}"}
-{"id": "6927.png", "formula": "\\begin{align*} \\sigma : = ( R ' < R _ 1 ' < \\cdots < R _ m ' ) \\mbox { a n d \\quad } x : = \\hat { \\beta } \\alpha ( z ) , \\end{align*}"}
-{"id": "8596.png", "formula": "\\begin{align*} | E ( H ) | \\geq \\left \\lceil \\frac { ( \\ell - 1 ) | B | } { k } \\right \\rceil + \\binom { | A | } k . \\end{align*}"}
-{"id": "5565.png", "formula": "\\begin{align*} \\omega = \\sum _ { \\substack { \\alpha \\in A ( r - 1 , n ) } } \\sum _ { \\substack { \\rho \\in \\Sigma _ { 0 } ( k , n ) \\\\ \\lfloor \\rho \\rfloor = 0 } } \\omega _ { \\alpha \\rho } \\lambda _ T ^ { \\alpha } \\phi ^ T _ { \\rho } . \\end{align*}"}
-{"id": "9243.png", "formula": "\\begin{align*} D _ { \\beta \\tau } u = \\tilde D _ { x _ \\tau } \\varphi - D _ \\tau \\beta \\cdot D u , { \\rm o n } \\ B _ R \\cap \\partial \\Omega , \\end{align*}"}
-{"id": "6744.png", "formula": "\\begin{align*} a _ { j 0 } = \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } a _ j ( s ) d s b _ { j 0 } = \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } b _ j ( s ) d s . \\end{align*}"}
-{"id": "1301.png", "formula": "\\begin{align*} p _ r ( h ) = \\inf \\Bigl \\{ \\sum _ { i = 1 } ^ n \\rho _ { g _ i } ( x _ i , y _ i ) : h = g _ 1 x _ 1 ^ { \\varepsilon _ 1 } y _ 1 ^ { - \\varepsilon _ 1 } g _ 1 ^ { - 1 } \\cdots g _ n x _ n ^ { \\varepsilon _ n } y _ n ^ { - \\varepsilon _ n } g _ n ^ { - 1 } , n \\in { \\mathbb N } \\Bigr \\} \\end{align*}"}
-{"id": "4077.png", "formula": "\\begin{align*} \\Psi ( z ) = \\begin{cases} \\Psi _ 0 ( z ) = 1 - F _ 1 ( z ) , & z \\in \\mathfrak { R } _ 0 , \\\\ \\Psi _ 1 ( z ) = F _ 1 ( z ) - F _ 2 ( z ) , & z \\in \\mathfrak { R } _ 1 , \\\\ \\Psi _ 2 ( z ) = F _ 2 ( z ) , & z \\in \\mathfrak { R } _ 2 , \\end{cases} \\end{align*}"}
-{"id": "6118.png", "formula": "\\begin{align*} \\deg ( T _ 0 ^ u T _ 1 ^ { s - u } X ^ j Y ^ k g ) = ( s - j \\mu _ 1 - k \\mu _ 2 + d - \\mu , j + k + 1 ) , \\end{align*}"}
-{"id": "68.png", "formula": "\\begin{align*} \\sigma _ { 1 } + \\sigma _ { - 1 } = ( J _ r - J _ * ) + ( J _ * - J _ \\ell ) = J _ r - J _ \\ell \\ , . \\end{align*}"}
-{"id": "8164.png", "formula": "\\begin{align*} \\prod _ { i = 1 } \\sp r x _ i ( n _ i ) = x _ 1 ( n _ 1 ) x _ 2 ( n _ 2 ) \\dots x _ r ( n _ r ) , x _ 1 ( n _ 1 ) \\preceq x _ 2 ( n _ 2 ) \\preceq \\dots \\preceq x _ r ( n _ r ) . \\end{align*}"}
-{"id": "6745.png", "formula": "\\begin{align*} L _ j = \\dfrac { \\partial } { \\partial t _ j } + ( a _ j + i b _ j ) ( t _ j ) \\dfrac { \\partial } { \\partial x } , \\ ; j = 1 , \\ldots , n , \\end{align*}"}
-{"id": "6658.png", "formula": "\\begin{align*} A & \\longrightarrow \\begin{bmatrix} k & V \\\\ 0 & k \\end{bmatrix} \\\\ a e + b f + v & \\longmapsto \\begin{pmatrix} b & v \\\\ 0 & a \\end{pmatrix} \\end{align*}"}
-{"id": "1391.png", "formula": "\\begin{align*} b = \\sum _ { i = 1 } ^ { n } \\sum _ { j = K _ { ( i ) } + 1 } ^ { k } d _ { j } ( \\lambda _ { i - 1 } ) e _ j \\quad c = \\sum _ { i = 1 } ^ { n } \\sum _ { j = K _ { ( i ) } } ^ { k } d _ { j } ( \\lambda _ { i - 1 } ) e _ j . \\end{align*}"}
-{"id": "6820.png", "formula": "\\begin{align*} \\varphi ( t , \\alpha ) = M _ 1 ( t ) v ( \\alpha ) + M _ 2 ( t ) w ( \\alpha ) \\ , , \\end{align*}"}
-{"id": "9632.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\infty } ( - 1 ) ^ { j - 1 } ( ( q ^ { j } ) _ { \\infty } - 1 ) = \\frac { 1 } { 2 } \\left ( \\frac { ( q ) _ { \\infty } } { ( - q ) _ { \\infty } } - 1 \\right ) = \\sum _ { j = 1 } ^ { \\infty } ( - 1 ) ^ j q ^ { j ^ 2 } , \\end{align*}"}
-{"id": "1234.png", "formula": "\\begin{align*} J _ j ( f , g ) ( \\xi ) = \\int e ^ { - 3 i \\Phi ( \\xi , \\eta ) } f ( \\eta ) g ( \\eta - \\xi ) \\varphi _ j ( \\eta / \\xi ) d \\eta \\end{align*}"}
-{"id": "1585.png", "formula": "\\begin{align*} \\Phi _ { \\widetilde { F } } ( \\widetilde { \\boldsymbol { F } } _ { n ^ 2 + k _ n , n ^ 2 + \\kappa _ 0 } ( z _ 0 , x _ 0 ) ) & = \\Phi _ { \\widetilde { F } } ( z _ 0 , x _ 0 ) + k _ n - \\kappa _ 0 + O \\left ( \\frac { k _ n ^ 3 } { n ^ 2 } \\right ) \\\\ & = \\Phi _ { \\widetilde { F } } ( z _ 0 , x _ 0 ) + k _ n - \\kappa _ 0 + o ( 1 ) , \\end{align*}"}
-{"id": "8966.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\frac { f ( r ) } { r ^ m } = 1 \\ , . \\end{align*}"}
-{"id": "3413.png", "formula": "\\begin{align*} \\rho ^ M _ { 0 , \\inf } ( T ) : = \\inf \\{ \\rho _ 0 ( S ) : S ^ { ( M ) } = T ^ { ( M ) } \\} , \\rho ^ M _ { 0 , \\sup } ( T ) : = \\sup \\{ \\rho _ 0 ( S ) : S ^ { ( M ) } = T ^ { ( M ) } \\} , \\end{align*}"}
-{"id": "6885.png", "formula": "\\begin{align*} \\sinh ^ 2 ( s ) \\theta '' - \\cosh ^ 2 ( s ) \\mu '' + 2 \\sinh ( s ) \\cosh ( s ) s ' ( \\theta ' - \\mu ' ) = 0 \\ , , \\end{align*}"}
-{"id": "943.png", "formula": "\\begin{align*} J ^ { * } [ \\psi ^ { ( 4 ) } ] _ { T , t } = \\hbox { \\vtop { \\offinterlineskip \\halign { \\hfil # \\hfil \\cr { \\rm l . i . m . } \\cr $ \\stackrel { } { { } _ { p \\to \\infty } } $ \\cr } } } \\sum \\limits _ { j _ 1 , j _ 2 , j _ 3 , j _ 4 = 0 } ^ { p } C _ { j _ 4 j _ 3 j _ 2 j _ 1 } \\zeta _ { j _ 1 } ^ { ( i _ 1 ) } \\zeta _ { j _ 2 } ^ { ( i _ 2 ) } \\zeta _ { j _ 3 } ^ { ( i _ 3 ) } \\zeta _ { j _ 4 } ^ { ( i _ 4 ) } , \\end{align*}"}
-{"id": "4017.png", "formula": "\\begin{align*} \\eta _ { \\pm } = \\lim _ { t \\rightarrow 0 _ { \\pm } } \\eta , \\ \\ \\eta ( \\nabla _ i ) = \\pm ( * \\nabla _ i - \\nabla _ i * ) , \\ \\ \\eta _ + = \\eta _ 0 + \\sum _ { i \\ge 1 } \\sigma _ i , \\ \\ \\eta _ - = \\eta _ 0 + \\sum _ { i \\ge 1 } ( - 1 ) ^ i \\sigma _ i \\end{align*}"}
-{"id": "3800.png", "formula": "\\begin{align*} \\delta ( n ) = \\frac { 1 } { Q } \\sum _ { 1 \\leq q \\leq Q } \\ ; \\frac { 1 } { q } \\ ; \\sideset { } { ^ \\star } \\sum _ { a \\bmod { q } } e \\left ( \\frac { n a } { q } \\right ) \\int _ \\mathbb { R } g ( q , x ) e \\left ( \\frac { n x } { q Q } \\right ) \\mathrm { d } x \\end{align*}"}
-{"id": "7120.png", "formula": "\\begin{align*} \\phi _ m ( q _ j ) = \\lambda _ m ( q _ j ) \\int _ 0 ^ 1 G _ 0 & ( x , \\xi ) ( \\phi _ m ( q _ j ) ( \\xi ) - \\phi ^ * _ m ( \\xi ) ) d \\xi \\\\ & - \\int _ 0 ^ 1 G _ 0 ( x , \\xi ) q _ j ( \\xi ) ( \\phi _ m ( q _ j ) ( \\xi ) - \\phi ^ * _ m ( \\xi ) ) d \\xi + \\\\ + & \\lambda _ m ( q _ j ) \\int _ 0 ^ 1 G _ 0 ( x , \\xi ) \\phi ^ * _ m ( \\xi ) d \\xi - \\int _ 0 ^ 1 G _ 0 ( x , \\xi ) q _ j ( \\xi ) \\phi ^ * _ m ( \\xi ) d \\xi . \\end{align*}"}
-{"id": "3096.png", "formula": "\\begin{align*} I ( q , z ) : = \\ , & z \\ , \\sum _ { d = 0 } ^ \\infty q ^ d \\frac { \\prod _ { m = 1 } ^ { 5 d } ( 5 H + m z ) } { \\prod _ { m = 1 } ^ d ( H + m z ) ^ 5 } = \\sum _ { i = 0 } ^ 3 I _ i ( q ) H ^ i z ^ { 1 - i } ; \\end{align*}"}
-{"id": "4586.png", "formula": "\\begin{align*} \\int 1 _ E ( x _ k , y _ k ) d \\mu ( x _ k ) d \\mu ( y _ k ) = \\nu _ g ( B _ { \\delta } ( x _ 1 - g y _ 1 ) ) . \\end{align*}"}
-{"id": "3273.png", "formula": "\\begin{align*} F _ n ( e _ 1 , e _ 2 ) = h _ { n - 1 } \\end{align*}"}
-{"id": "5256.png", "formula": "\\begin{align*} R _ p ( F , x ) = \\frac { 1 } { 1 - F ( x ) } \\int _ { x } ^ { u e p ( F ) } \\int _ { u _ 1 } ^ { u e p ( F ) } . . . \\int _ { u _ { p - 1 } } ^ { u e p ( F ) } 1 - F ( t ) \\ d u _ { 1 } \\ . . . \\ 1 - F ( t ) \\ d u _ { p - 1 } d t . \\end{align*}"}
-{"id": "9564.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { N } \\frac { ( 1 / c ) _ n q ^ n } { ( q ) _ n } \\frac { ( q ^ { - N } ) _ n } { \\left ( \\frac { q ^ { - N } } { c } \\right ) _ n } = \\frac { ( q ^ { - N } ) _ N } { \\left ( \\frac { q ^ { - N } } { c } \\right ) _ N c ^ N } = \\frac { ( q ) _ N } { ( c q ) _ { N } } . \\end{align*}"}
-{"id": "8360.png", "formula": "\\begin{align*} \\bar { F } ( z , t ) : = v ( z , t ) - \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i } { 2 \\pi } \\frac { 1 } { \\overline { z - z _ j ( t ) } } . \\end{align*}"}
-{"id": "1263.png", "formula": "\\begin{align*} I ( S , S , S ) ( \\xi ) = O ( 1 ) . \\end{align*}"}
-{"id": "4499.png", "formula": "\\begin{align*} H ( \\nabla ^ 2 v , \\nabla v , v ) & = \\Xi ^ s ( \\nabla v , v , x ) \\quad \\textrm { i n } \\ , \\Omega \\\\ v & = \\ln \\phi \\quad \\textrm { o n } \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "6011.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\ , \\rho ^ { G _ n } \\left ( F \\left ( \\frac 1 n \\sum _ { k = 1 } ^ n { W _ { ( n , k ) } } \\right ) \\right ) = \\sup _ { \\omega \\in \\C _ 0 } \\left ( F ( \\omega ) - \\int _ 0 ^ 1 g ( t , \\dot { \\omega } ( t ) ) d t \\right ) . \\end{align*}"}
-{"id": "6813.png", "formula": "\\begin{align*} p = - \\tfrac { 1 } { 2 } \\ , \\langle \\beta , A ^ T A '' \\beta \\rangle = - \\tfrac { 1 } { 2 } \\ , \\langle x , A '' A ^ { - 1 } x \\rangle \\ . \\end{align*}"}
-{"id": "433.png", "formula": "\\begin{align*} W ^ 0 _ t ( x ) \\geq W _ t ( x ) - \\sum _ { i = 1 } ^ { I _ t ^ x } \\big ( A + \\eta T ^ x _ i \\big ) e ^ { \\gamma A + ( \\gamma \\eta + \\frac { \\gamma ^ 2 } { 2 } ) T ^ x _ i } \\end{align*}"}
-{"id": "4608.png", "formula": "\\begin{align*} l _ { x _ 1 , x _ 3 } = T ( x _ 1 , x _ 3 , 0 ) , x _ 1 \\in F _ { 1 , \\delta } , x _ 3 \\in F _ { 2 , \\delta } \\end{align*}"}
-{"id": "3768.png", "formula": "\\begin{align*} { \\bf E } = { \\bf Q } _ L - i \\xi { \\bf Q } _ R , \\quad { \\bf H } = \\frac { 1 } { i \\xi } { \\bf Q } _ L + { \\bf Q } _ R , \\end{align*}"}
-{"id": "1336.png", "formula": "\\begin{align*} \\widetilde { \\tau } _ B \\left ( \\widetilde { \\theta } _ B ( y , t ) \\right ) & = \\widetilde { \\tau } _ B \\left ( \\theta _ { B , 1 } ( y ) , \\theta _ { B , 2 } ( y ) , \\theta _ { B , 2 } ( y ) - t \\right ) \\\\ & = \\left ( \\tau \\left ( \\theta ( y ) \\right ) , \\theta _ { B , 2 } ( y ) - \\left ( \\theta _ { B , 2 } ( y ) - t \\right ) \\right ) \\\\ & = ( y , t ) \\ ; . \\end{align*}"}
-{"id": "5202.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow + \\infty } U ( \\gamma x ) / U ( x ) = \\gamma ^ { \\rho } . \\end{align*}"}
-{"id": "3863.png", "formula": "\\begin{align*} c _ { t , x } ( { s , y } ) & = \\frac { 1 } { \\prod _ { ( l , w ) \\in \\mathcal { S } ( t , x ) } P ( Z _ l ^ { w } = w - l e _ d ) } \\\\ & \\qquad \\times \\sum _ { \\sigma \\in { \\rm S y m } ( \\mathcal { S } ( t , x ) ) } { \\rm s g n } ( \\sigma ) P ( Z _ { \\sigma _ 1 ( t , x ) } ^ { \\sigma _ 2 ( t , x ) } = y + s e _ d ) \\\\ & \\times \\prod _ { ( h , z ) \\in \\mathcal { S } ( t , x ) \\backslash \\{ ( t , x ) \\} } P ( Z _ { \\sigma _ 1 ( h , z ) } ^ { \\sigma _ 2 ( h , z ) } = z - h e _ d ) , \\end{align*}"}
-{"id": "314.png", "formula": "\\begin{align*} \\begin{array} { c c c } A _ i = \\sum ^ i _ { j = 1 } ( - 1 ) ^ { i + j } \\binom { i } { j } j ^ { n - 2 } & & B _ i = \\sum ^ i _ { j = 1 } ( - 1 ) ^ { i + j } \\binom { i } { j } j ^ { n - 1 } . \\end{array} \\end{align*}"}
-{"id": "9386.png", "formula": "\\begin{align*} L = \\langle y _ j \\mid j \\in \\mathbb { N } _ 0 j \\equiv _ { 2 ^ n } 0 , \\pm 1 , \\ldots , \\pm ( m - 1 ) \\rangle Z , \\end{align*}"}
-{"id": "6576.png", "formula": "\\begin{gather*} E ^ { p , q } _ 0 = G r ^ { p , q } . \\end{gather*}"}
-{"id": "915.png", "formula": "\\begin{align*} + \\sum _ { i _ { 1 } , i _ 2 , i _ 3 , i _ { 4 } = 1 } ^ { m } G _ 0 ^ { ( i _ { 4 } ) } G _ 0 ^ { ( i _ { 3 } ) } G _ 0 ^ { ( i _ { 2 } ) } \\Sigma _ { i _ { 1 } } \\hat I _ { 0 0 0 0 _ { \\tau _ { p + 1 } , \\tau _ p } } ^ { ( i _ { 4 } i _ { 3 } i _ { 2 } i _ { 1 } ) } + { \\bf u } _ { p + 1 , p } + { \\bf v } _ { p + 1 , p } , \\end{align*}"}
-{"id": "8533.png", "formula": "\\begin{align*} [ D _ t ^ 2 - i A \\partial _ { \\alpha } , \\mathcal { H } ] \\partial _ { \\alpha } ^ k \\tilde { \\theta } = 2 [ D _ t \\zeta , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } \\partial _ { \\alpha } ^ k \\tilde { \\theta } } { \\zeta _ { \\alpha } } - \\frac { 1 } { \\pi i } \\int \\Big ( \\frac { \\zeta ( \\alpha , t ) - \\zeta ( \\beta , t ) } { \\zeta ( \\alpha , t ) - \\zeta ( \\beta , t ) } \\Big ) ^ 2 \\partial _ { \\beta } \\partial _ { \\beta } ^ k \\tilde { \\theta } d \\beta \\end{align*}"}
-{"id": "7508.png", "formula": "\\begin{align*} u ( x , t ) = ( Z _ { t } \\ast u _ { 0 } ) ( x ) , u _ { 0 } ( x ) \\in \\mathcal { D } ( \\mathbb { Q } _ { p } ^ { n } ) , \\ x \\in \\mathbb { Q } _ { p } ^ { n } , t \\geq 0 , \\end{align*}"}
-{"id": "870.png", "formula": "\\begin{gather*} F ( x ) = \\begin{cases} x - K & \\mbox { i f } x \\in D , \\\\ \\emptyset & \\mbox { o t h e r w i s e } , \\end{cases} \\end{gather*}"}
-{"id": "8165.png", "formula": "\\begin{align*} \\pi = ( x _ 1 ( n _ 1 ) \\preceq x _ 2 ( n _ 2 ) \\preceq \\dots \\preceq x _ r ( n _ r ) ) \\end{align*}"}
-{"id": "4630.png", "formula": "\\begin{align*} \\mathcal { P } = ( U \\otimes \\mathbb { I } _ n ) [ \\mathbb { I } _ \\ell \\otimes A _ 0 + \\Lambda \\otimes A _ 1 ] ( U ^ { * } \\otimes \\mathbb { I } _ n ) . \\end{align*}"}
-{"id": "4415.png", "formula": "\\begin{align*} \\nu ( x , y , z ) = \\frac { 4 ! } { \\pi ^ 2 r ^ 4 } \\int _ { B _ r ( x , y ) } \\log ( 1 + 4 | z \\xi ^ 2 | + 4 | z \\eta ^ 2 | ) d V ( \\xi , \\eta ) . \\end{align*}"}
-{"id": "2561.png", "formula": "\\begin{align*} \\widehat \\psi ( z ) : = - ( \\| \\beta \\| _ \\infty + \\kappa _ 0 ) z + \\kappa _ 0 z ^ { \\gamma _ 0 } , z \\geq 0 . \\end{align*}"}
-{"id": "3906.png", "formula": "\\begin{align*} h _ n g h _ n ^ { - 1 } = \\begin{pmatrix} a _ n & b _ n & \\gamma _ n ^ * \\\\ c _ n & d _ n & \\delta _ n ^ * \\\\ \\alpha _ n & \\beta _ n & U _ n \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "6564.png", "formula": "\\begin{align*} g \\cdot ( v , \\varphi _ i ) = ( g v , g \\varphi _ i g ^ { - 1 } ) . \\end{align*}"}
-{"id": "6485.png", "formula": "\\begin{align*} \\ < \\frac { \\mu ' ( e _ i ) } { \\mu ' ( e _ j ) } \\colon i , j = 1 , \\ldots , n \\ > = H ' . \\end{align*}"}
-{"id": "701.png", "formula": "\\begin{align*} & 4 ( 3 f ( x ) e ^ { 3 F ( x ) } ) = 3 ( 2 p - 1 ) x f ( x ) ^ 2 f ^ { ( 1 ) } ( x ) + 1 2 p K f ( x ) f ^ { ( 1 ) } ( x ) - 3 f ( x ) ^ 3 , \\\\ & 4 Z ( x ) = 4 e ^ { 3 F ( x ) } = 3 ( 2 p - 1 ) \\int _ 0 ^ x t f ( t ) ^ 2 f ^ { ( 1 ) } ( t ) \\ ; d t + 6 p K f ( x ) ^ 2 - 3 \\int _ 0 ^ x f ( t ) ^ 3 \\ ; d t + 4 e ^ { 3 F ( 0 ) } . \\end{align*}"}
-{"id": "3594.png", "formula": "\\begin{align*} u _ { \\rho ( a _ 0 , a _ 1 ) } = u _ { \\rho ( a _ 0 , a _ { - 1 } ) } \\mbox { a n d } v _ { \\rho ( a _ 0 , a _ 1 ) } = v _ { \\rho ( a _ 0 , a _ { - 1 } ) } . \\end{align*}"}
-{"id": "5875.png", "formula": "\\begin{align*} \\phi \\left ( \\sum _ { x \\in \\tilde { W } } b _ x C _ x \\right ) = \\sum _ { \\substack { x , z \\in \\tilde { W } \\\\ d \\in \\mathcal { D } , ~ a ( d ) = a ( z ) } } b _ x h _ { x , d , z } t _ z , \\end{align*}"}
-{"id": "1983.png", "formula": "\\begin{align*} \\mathrm { T o t } ( \\O _ { \\P ^ n } ( 1 ) ) = \\frac { ( \\C ^ { n + 1 } \\setminus \\{ 0 \\} ) \\times \\C } { \\C ^ * } , \\end{align*}"}
-{"id": "8991.png", "formula": "\\begin{align*} n ^ { \\infty } : = \\max _ { 1 \\leq i \\leq N } | { \\mathcal N } _ i | , \\end{align*}"}
-{"id": "7072.png", "formula": "\\begin{align*} l _ { X _ 1 \\boxtimes Y _ 1 , X _ 2 \\boxtimes Y _ 2 , M } & : = ( ( X _ 1 \\otimes X _ 2 ) \\boxtimes ( Y _ 1 \\otimes _ { o p } Y _ 2 ) ) . M \\to ( X _ 1 \\boxtimes Y _ 1 ) . ( ( X _ 2 \\boxtimes Y _ 2 ) . M ) \\end{align*}"}
-{"id": "3432.png", "formula": "\\begin{gather*} c _ 2 ( V ) | _ { 2 D _ 8 } = 8 \\pmod { 1 6 } . \\end{gather*}"}
-{"id": "8925.png", "formula": "\\begin{align*} \\mathbb { E } [ T ^ { \\hat { x } } ( G ' ) - T ^ { \\hat { x } } ( H ) \\mid \\hat { x } ] = \\frac { ( a - b ) } { n } { \\Big ( } n ( & \\textrm { C o r r e c t , U n c h a n g e d } ) - n ( \\textrm { I n c o r r e c t , U n c h a n g e d } ) \\Big { ) } \\\\ & \\times \\Big { ( } n ( \\textrm { C o r r e c t , C h a n g e d } ) - n ( \\textrm { I n c o r r e c t , C h a n g e d } ) { \\Big ) } . \\end{align*}"}
-{"id": "1083.png", "formula": "\\begin{align*} & \\lim _ { n \\to + \\infty } n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } e ^ { 1 - ( S ( I _ 2 ) / S ( I ) - 1 ) \\ln n } D _ n ( G _ n ( x ) / 2 ) \\\\ = & \\lim _ { n \\to + \\infty } ( 2 \\ln n ) e ^ { 1 - ( S ( I _ 2 ) / S ( I ) - 1 ) \\ln n } \\lim _ { n \\to + \\infty } n ( 2 \\ln n ) ^ { - \\frac { 1 } { 2 } } D _ n ( G _ n ( x ) / 2 ) = 0 , \\end{align*}"}
-{"id": "2024.png", "formula": "\\begin{align*} \\widetilde { T } _ j ( - z ) = \\sum _ { j ' \\neq j \\atop d > 0 } \\frac { \\widetilde { T } _ j ^ { d , j ' } } { z - \\frac { \\alpha _ { j ' } - \\alpha _ { j } } { d } } . \\end{align*}"}
-{"id": "5226.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\frac { V ( t _ { n } x _ { 0 } ) } { V ( t _ { n } ) } = c \\neq x _ { 0 } ^ { \\rho } . \\end{align*}"}
-{"id": "9503.png", "formula": "\\begin{align*} I _ { \\lambda } ( u ) = \\frac { 1 } { p } \\Vert u \\Vert _ { s , p } ^ p + \\frac { 1 } { q } \\Vert u \\Vert _ { s , q } ^ { q } - \\frac { 1 } { p _ { s } ^ * } \\displaystyle \\int _ { \\mathbb { R } ^ N } ( u ^ + ) ^ { p _ { s } ^ * } \\dd x - \\frac { \\lambda } { r } \\displaystyle \\int _ { \\mathbb { R } ^ N } g ( u ^ + ) ^ r \\dd x . \\end{align*}"}
-{"id": "8619.png", "formula": "\\begin{align*} \\hat { H } = 0 \\ \\ \\textrm { o n } \\ \\ W ^ u ( 0 , 0 , \\bar { u } , 0 ) \\cup W ^ s ( 0 , 0 , \\bar { v } , 0 ) . \\end{align*}"}
-{"id": "7260.png", "formula": "\\begin{align*} \\left [ u _ t + u u _ x \\right ] _ x - u = 0 \\end{align*}"}
-{"id": "1696.png", "formula": "\\begin{align*} \\int \\varphi \\ , d { \\mathcal L } ^ * \\mu = \\int { \\mathcal L } \\varphi \\ , d \\mu \\ \\ \\ \\ \\forall \\ \\varphi \\in L _ { \\infty } \\ \\ \\ \\ \\ \\forall \\ \\mu \\in { \\mathcal M } . \\end{align*}"}
-{"id": "7718.png", "formula": "\\begin{align*} \\norm { \\tilde { F } _ { t } } \\leq \\mathsf { s u p } \\{ \\norm { \\phi _ { t } } _ { \\mathrm { L } ^ 1 } \\cdot \\norm { u } _ { 2 } \\ , : \\ , \\norm { u } _ { 2 } \\leq 1 \\} = \\norm { \\phi _ { t } } _ { \\mathrm { L } ^ 1 } = 1 , \\end{align*}"}
-{"id": "2284.png", "formula": "\\begin{align*} \\hat { H } : \\mathbb { C } \\oplus \\mathcal { L } & \\rightarrow \\mathbb { C } \\oplus \\mathcal { L } ^ * \\\\ \\hat { H } & = \\left ( \\begin{array} { c c } 0 & ( \\hat { E } | \\\\ | \\hat { E } ) & \\hat { \\Omega } \\end{array} \\right ) , \\end{align*}"}
-{"id": "2840.png", "formula": "\\begin{align*} S ' _ \\infty \\cup S ( A _ \\infty ) & \\subset \\bigcup _ { i = 1 } ^ M B _ { r _ i } ( x _ i ) = : U _ 1 \\ ; \\\\ \\sum _ { i = 1 } ^ M r _ i ^ { 2 n - 4 } & \\leq \\varepsilon \\cdot 2 ^ { 4 - 2 n } \\ . \\end{align*}"}
-{"id": "4534.png", "formula": "\\begin{align*} \\hat { \\phi } ^ { D } = \\hat { \\phi } ^ { D D } * \\hat { \\phi } ^ { D D } . \\end{align*}"}
-{"id": "7631.png", "formula": "\\begin{align*} ( A \\bar \\varphi ) ( \\xi ) \\ = \\ \\int \\limits _ { \\mathbb R ^ d } a ( \\xi - q ) \\mu ( \\xi , q ) ( \\bar \\varphi ( q ) - \\bar \\varphi ( \\xi ) ) \\ d q \\ = \\ \\int \\limits _ { \\mathbb T ^ d } \\hat a ( \\xi - \\eta ) \\mu ( \\xi , \\eta ) ( \\bar \\varphi ( \\eta ) - \\bar \\varphi ( \\xi ) ) \\ d \\eta , \\end{align*}"}
-{"id": "2755.png", "formula": "\\begin{align*} \\phi _ { 2 , p } ( \\kappa , \\pi ; v ) = & \\int _ { \\mathcal { C } _ { > } } \\frac { d z } { 2 \\pi i } e ^ { - \\frac { 1 } { 3 } ( z - \\kappa ) ^ 3 + v ( z - \\kappa ) } \\frac { 1 } { \\sqrt { 2 z } } \\prod _ { k = 1 } ^ { p } \\frac { z - \\pi _ { k } } { z + \\pi _ { k } } . \\end{align*}"}
-{"id": "7339.png", "formula": "\\begin{align*} \\begin{cases} \\rho ( x ) \\varphi _ { t t } - \\operatorname { d i v } ( K ( x ) \\nabla w ) + 2 \\alpha ^ 2 z w \\psi + \\alpha ^ 2 z ^ 2 \\varphi = 0 & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ \\rho ( x ) \\psi _ { t t } - \\operatorname { d i v } ( K ( x ) \\nabla \\psi ) + 2 \\alpha ^ 2 w z \\varphi + \\alpha ^ 2 w ^ 2 \\psi = 0 & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ \\varphi = \\psi = 0 & ~ \\hbox { i n } ~ \\omega \\times ( 0 , T ) . \\end{cases} \\end{align*}"}
-{"id": "4968.png", "formula": "\\begin{align*} s _ 0 & : = 1 \\\\ s _ { n + 1 } & : = \\prod _ { A \\subseteq \\{ 0 , \\ldots , n \\} , A \\neq \\varnothing } ~ s _ A . \\end{align*}"}
-{"id": "8139.png", "formula": "\\begin{align*} \\overset { I } { M } \\cdot h = \\overset { I } { h ' } \\overset { I } { M } \\overset { I } { S ( h '' ) } \\end{align*}"}
-{"id": "2844.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } [ \\mathcal { F } _ { i } | _ { S } ] = [ \\mathcal { F } _ { \\Vert S } ] \\in \\overline { M } _ { S } ^ { \\mu } \\ . \\end{align*}"}
-{"id": "142.png", "formula": "\\begin{align*} \\pi _ { \\lambda } = \\pi + \\lambda X , \\end{align*}"}
-{"id": "6268.png", "formula": "\\begin{align*} & ( \\alpha x ) ^ { [ p ] } = \\alpha ^ p x ^ { [ p ] } , \\\\ & ( \\ , x ^ { [ p ] } ) ( z ) = [ x ^ { [ p ] } , z ] = ( \\ , x ) ^ p ( z ) , \\\\ & ( x + y ) ^ { [ p ] } = x ^ { [ p ] } + y ^ { [ p ] } + \\sum _ i s _ i ( x , y ) , \\end{align*}"}
-{"id": "2667.png", "formula": "\\begin{align*} 0 & = \\frac { - 1 } { 2 \\pi i } \\int _ { \\mathcal { C } _ { \\{ 1 - \\varrho _ { 1 } , \\ldots , 1 - \\varrho _ { N } \\} } } d z \\frac { z - \\varrho _ { k } } { 1 - z - \\varrho _ { k } } \\frac { 1 } { \\sqrt { 2 z - 1 } } \\frac { 1 } { 1 - z } e ^ { - \\frac { \\eta _ { - } } { 1 - z } v } , k > N . \\end{align*}"}
-{"id": "7278.png", "formula": "\\begin{align*} \\Delta ^ { p , + } _ { a , b , \\lambda , \\mu } & : = \\psi _ p ( \\lambda a + \\mu b ) - \\lambda \\psi _ p ( a ) - \\mu \\psi _ p ( b ) , \\\\ \\Delta ^ { p , * } _ a & : = \\psi _ p ( a ^ * ) - \\psi _ p ( a ) ^ * , \\\\ \\Delta ^ { p , \\cdot } _ { a , b } & : = \\psi _ p ( a b ) - \\psi _ p ( a ) \\psi _ p ( b ) . \\end{align*}"}
-{"id": "8375.png", "formula": "\\begin{align*} & - i ( I - \\mathfrak { H } ) a _ t \\bar { z } _ { \\alpha } = ( I - \\mathfrak { H } ) ( u _ { t t } + i a u _ { \\alpha } ) \\\\ = & [ \\partial _ t ^ 2 + i a \\partial _ { \\alpha } , \\mathfrak { H } ] u + ( \\partial _ t ^ 2 + i a \\partial _ { \\alpha } ) ( I - \\mathfrak { H } ) u . \\end{align*}"}
-{"id": "9711.png", "formula": "\\begin{align*} I _ { \\textbf { d } } ( P _ i ) _ { c } = \\sum _ { j = 1 } ^ 2 I _ { \\textbf { d } } ^ j ( P _ { i } ) _ { c } . \\end{align*}"}
-{"id": "2121.png", "formula": "\\begin{align*} d _ p ( x , y ) : = \\sum _ { m \\geq 1 } \\frac { 1 } { 2 ^ m } ( \\| x - y \\| _ { p { \\rm - v a r } , [ - m , m ] } \\wedge 1 ) . \\end{align*}"}
-{"id": "6014.png", "formula": "\\begin{align*} I ( \\omega ) & : = \\inf \\left \\{ \\tilde \\alpha ^ g ( Q ) : Q \\in \\P ^ * \\cap \\P _ 1 ( \\C ) , \\ \\int _ { \\C } \\bar { \\omega } \\ , Q ( d \\bar { \\omega } ) = \\omega \\right \\} . \\end{align*}"}
-{"id": "3168.png", "formula": "\\begin{align*} { \\rm i n d } _ \\varGamma ( U ) = { \\rm i n d e x } ( \\alpha ) , \\end{align*}"}
-{"id": "8494.png", "formula": "\\begin{align*} | \\partial _ x F ( \\tilde { x } + i y ( t ) , t ) | \\leq & \\frac { 1 } { 2 \\pi } \\Big ( \\int \\frac { 1 } { | \\tilde { x } + i y ( t ) - \\zeta ( \\beta , t ) | ^ 4 } d \\beta \\Big ) ^ { 1 / 2 } \\| \\zeta _ { \\beta } \\| _ { L ^ { \\infty } } \\| \\mathfrak { F } \\| _ { L ^ 2 } \\\\ \\leq & C \\epsilon \\hat { d } _ I ( t ) ^ { - 3 / 2 } , \\forall ~ t \\in [ 0 , T _ 0 ] , \\end{align*}"}
-{"id": "3856.png", "formula": "\\begin{align*} \\mathbf { E } [ f ( Z ^ x _ t ) ] = E [ g ( Z ^ x _ t ) ] , \\end{align*}"}
-{"id": "4731.png", "formula": "\\begin{align*} \\mathcal { F } _ { A , B } ^ { \\mathcal { B } } = \\left \\lbrace ( T _ { p - q } ) _ { p , q = 0 } ^ { n - 1 } : T _ { j } \\in \\mathcal { B } , A T _ { j } = B T _ { j - n } , j = 1 , 2 , \\cdots n - 1 \\right \\rbrace . \\end{align*}"}
-{"id": "3570.png", "formula": "\\begin{align*} D _ { 0 } = \\left \\{ X _ { 0 } + X _ { 1 } + X _ { 2 } - X _ { 3 } - X _ { 4 } - X _ { 5 } = 0 \\right \\} . \\end{align*}"}
-{"id": "8804.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { \\pi } u ^ { \\varepsilon } ( t , x ) \\phi ( x ) \\ , \\textrm { d } x = \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { \\pi } u ^ { \\varepsilon } ( s , x ) \\phi '' ( x ) \\ , \\textrm { d } s \\ , \\textrm { d } x + \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { \\pi } \\frac { f ( u ^ { \\varepsilon } ( s , x ) ) } { \\sigma ( \\varepsilon ) } \\phi ( x ) \\ , L ^ { \\varepsilon } ( \\textrm { d } s , \\textrm { d } x ) \\end{align*}"}
-{"id": "2754.png", "formula": "\\begin{align*} \\phi _ { 1 , p } ( \\kappa , \\pi ; u ) = & \\int _ { \\mathcal { C } _ { < } } \\frac { d w } { 2 \\pi i } e ^ { \\frac { 1 } { 3 } ( w - \\kappa ) ^ 3 - u ( w - \\kappa ) } \\frac { 1 } { \\sqrt { 2 w } } \\prod _ { k = 1 } ^ { p } \\frac { w + \\pi _ { k } } { w - \\pi _ { k } } . \\end{align*}"}
-{"id": "7116.png", "formula": "\\begin{align*} \\phi _ k ( q _ j ) ( x ) = \\lambda _ k ( q _ j ) \\int _ 0 ^ 1 G _ 0 ( x , \\xi ) \\phi _ k ( q _ j ) ( \\xi ) d \\xi - \\int _ 0 ^ 1 G _ 0 ( x , \\xi ) q _ j ( \\xi ) \\phi _ k ( q _ j ) ( \\xi ) d \\xi , \\end{align*}"}
-{"id": "4631.png", "formula": "\\begin{align*} \\mathcal { P } ^ { - 1 } = ( U \\otimes \\mathbb { I } _ n ) [ \\mathbb { I } _ \\ell \\otimes A _ 0 + \\Lambda \\otimes A _ 1 ] ^ { - 1 } ( U ^ { * } \\otimes \\mathbb { I } _ n ) . \\end{align*}"}
-{"id": "9995.png", "formula": "\\begin{align*} \\Delta = \\sqrt { \\vert \\det ( g _ { i j } ) \\vert } = ( u ^ 1 - u ^ 2 ) ( u ^ 1 - u ^ 3 ) ( u ^ 2 - u ^ 3 ) u ^ 5 ; \\end{align*}"}
-{"id": "290.png", "formula": "\\begin{align*} p _ \\beta \\ ; = \\ ; \\prod _ { i = 1 } ^ \\ell \\Bigl ( \\sum _ j x _ j ^ { \\beta _ i } \\Bigr ) . \\end{align*}"}
-{"id": "3697.png", "formula": "\\begin{align*} \\widetilde { Y } = \\sum _ x \\sum _ { i \\in U _ x } \\delta _ i y _ i / p _ x \\end{align*}"}
-{"id": "761.png", "formula": "\\begin{align*} r ( \\mu ) = \\sum _ j \\alpha _ j R ^ * q _ j ( \\mu ) . \\end{align*}"}
-{"id": "1491.png", "formula": "\\begin{align*} E _ { \\mu } ( t ) = E _ { \\mu } ( 0 ) + \\int _ { 0 } ^ { t } \\sum _ { \\ell = 1 } ^ { 1 4 } V _ { \\ell } . \\end{align*}"}
-{"id": "8263.png", "formula": "\\begin{align*} \\bar { v } ( z ) + \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i } { 2 \\pi ( z - z _ j ( t ) ) } = \\frac { 1 } { 2 \\pi i } \\int \\frac { z _ { \\beta } } { z - z ( \\beta ) } \\Big ( \\bar { z } _ t ( \\beta ) + \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i } { 2 \\pi ( z ( \\beta ) - z _ j ( t ) } ) \\Big ) d \\beta . \\end{align*}"}
-{"id": "4957.png", "formula": "\\begin{align*} \\ell ( R / I _ n ) = \\ell ( R / ( x _ 1 , x _ 2 ^ { n ^ 3 } ) ) + \\ell ( R / ( x _ 1 ^ n - x _ 2 ^ n , x _ 2 ^ { n ^ 3 } + x _ 1 ^ n ) ) = n ^ 3 + n ^ 2 . \\end{align*}"}
-{"id": "7174.png", "formula": "\\begin{gather*} X ^ { t } ( x ) \\mathbb { L } _ { x } ^ { \\pm } = \\mathbb { L } _ { \\chi ^ { t } ( x ) } ^ { \\pm } \\quad \\forall ( t , x ) \\in \\mathbb { R } \\times \\mathcal { M } ; \\end{gather*}"}
-{"id": "175.png", "formula": "\\begin{align*} [ X \\wedge Y , X \\wedge Y ] = 2 [ X , Y ] \\wedge X \\wedge Y = \\end{align*}"}
-{"id": "723.png", "formula": "\\begin{align*} T = \\frac { 1 } { q - 1 } \\left ( 1 - \\sum _ { i = 1 } ^ { m } \\lambda _ { i } ^ { q } \\right ) , ~ ~ ~ ~ q \\in \\mathbb { R } \\backslash \\{ 0 \\} , \\end{align*}"}
-{"id": "5658.png", "formula": "\\begin{align*} { \\rm I } _ 1 & \\leq c h ^ { ( 2 - s - r ) p } \\int _ { 0 } ^ { t _ n } ( t _ n - t ) ^ { ( \\frac { r \\alpha } { 2 } + \\gamma - 1 ) p } \\d t \\\\ & \\leq c t _ n ^ { ( \\frac { r \\alpha } { 2 } + \\gamma - 1 ) p + 1 } h ^ { ( 2 - s - r ) p } . \\end{align*}"}
-{"id": "5161.png", "formula": "\\begin{align*} H _ { 2 } ( x ) = H _ { 1 } ( A x + B ) . \\end{align*}"}
-{"id": "758.png", "formula": "\\begin{align*} B = \\sum _ m \\sqrt { \\lambda _ m } ( h _ m \\otimes v _ m ) ; B ^ * = \\sum _ m \\sqrt { \\lambda _ m } ( v _ m \\otimes h _ m ) ; r = \\sum _ m \\sqrt { \\lambda _ m } \\ , h _ m v _ m = \\sum _ m B ^ * h _ m , \\end{align*}"}
-{"id": "2256.png", "formula": "\\begin{align*} \\mu ( x ) & = \\frac { 1 } { 2 \\pi \\i } ( R ( x - \\i 0 ) - R ( x + \\i 0 ) ) = | \\alpha _ x \\ > \\ < \\beta _ x | \\\\ | \\alpha _ x \\rangle & = ( x ^ 2 + \\pi ^ 2 ) ^ { - 1 / 2 } \\left ( \\binom { 1 } { - \\frac { \\mathcal { P } } { x - \\Omega } | E \\rangle } + x \\binom { 0 } { | \\delta _ x \\rangle } \\right ) \\\\ \\langle \\beta _ x | & = ( x ^ 2 + \\pi ^ 2 ) ^ { - 1 / 2 } \\bigg ( - ( 1 , \\langle E | \\frac { \\mathcal { P } } { x - \\Omega } ) + x ( 0 , \\langle \\delta _ x | ) \\bigg ) \\end{align*}"}
-{"id": "8327.png", "formula": "\\begin{align*} Z ( \\alpha , t ) : = z \\circ h ^ { - 1 } ( \\alpha , t ) , b = h _ t \\circ h ^ { - 1 } , D _ t : = \\partial _ t + b \\partial _ { \\alpha } , \\end{align*}"}
-{"id": "3034.png", "formula": "\\begin{align*} \\beta & = \\min \\{ \\gamma _ 2 , \\phi _ 2 \\} = \\min \\{ \\lambda _ 2 [ 2 ] ^ m _ { p , q } - \\alpha u _ 2 [ 2 ] ^ n _ { p , q } , \\mu _ 2 [ 2 ] ^ m _ { p , q } - ( - 1 ) ^ { n + j - ( m + i ) } \\alpha v _ 2 [ 2 ] ^ n _ { p , q } \\} . \\end{align*}"}
-{"id": "3381.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { k _ 0 } c ^ 2 & ( k - 1 ) ^ 2 N ( c , k ) - \\tau _ n c ^ 2 k _ 0 ^ 2 N ( c , k _ 0 ) \\\\ & \\leq \\min _ { S \\subset [ n ] : | S | = \\lceil \\delta n \\rceil } \\| x _ S \\| _ 2 ^ 2 \\leq \\sum _ { k = 1 } ^ { k _ 0 } c ^ 2 k ^ 2 N ( c , k ) + \\tau _ n c ^ 2 ( k _ 0 + 1 ) ^ 2 N ( c , k _ 0 + 1 ) . \\end{align*}"}
-{"id": "1564.png", "formula": "\\begin{align*} F _ w \\left ( \\begin{array} { c } z \\\\ x \\end{array} \\right ) = F \\left ( \\begin{array} { c } z \\\\ x \\end{array} \\right ) + \\left ( \\begin{array} { c } \\frac { \\pi ^ 2 } { 4 } w \\\\ 0 \\end{array} \\right ) = \\left ( \\begin{array} { c } z + q _ 1 ( z + \\delta x ) \\\\ \\delta x - q _ 1 ( z + \\delta x ) \\end{array} \\right ) + \\left ( \\begin{array} { c } \\frac { \\pi ^ 2 } { 4 } w \\\\ 0 \\end{array} \\right ) , \\end{align*}"}
-{"id": "2053.png", "formula": "\\begin{align*} M _ d ^ 2 R _ d ' = \\frac { d ^ 2 R _ d } { \\gcd ( R _ d , r _ d ^ 2 ) } = [ q , d ^ 2 ] . \\end{align*}"}
-{"id": "7515.png", "formula": "\\begin{align*} \\frac { 1 } { N ' } \\sum _ { l = 0 } ^ { N ' - 1 } [ J _ N ( T ^ l ( z _ 0 ) , r ) ] ^ { q - 1 } \\simeq \\Gamma _ \\mu ( r , q ) . \\end{align*}"}
-{"id": "8549.png", "formula": "\\begin{align*} ( I - \\mathbb { H } ) \\tilde { \\theta } = 2 \\tilde { \\theta } + ( \\mathcal { H } - \\mathbb { H } ) \\tilde { \\theta } , \\end{align*}"}
-{"id": "4448.png", "formula": "\\begin{align*} \\mathcal { T } = \\{ \\ , T _ 0 \\cup \\varepsilon ' \\cup T ' _ 1 \\mid T ' _ 1 \\in \\mathcal { T } _ 1 \\ , \\} . \\end{align*}"}
-{"id": "9651.png", "formula": "\\begin{align*} \\frac { \\psi } { \\psi _ t } ( 0 ) = \\exp \\Bigl ( \\int _ { \\sigma _ t } \\log \\frac { | \\psi | } { t } m \\Bigr ) \\to 1 \\ \\ t \\to 0 . \\end{align*}"}
-{"id": "4742.png", "formula": "\\begin{align*} S _ 1 & = \\{ x \\in \\mathbb { R } ^ 2 : \\dfrac { x _ 1 } { a _ 1 } + \\dfrac { x _ 2 } { a _ 2 } \\leq 1 \\} \\\\ S _ 2 & = \\{ x \\in \\mathbb { R } ^ 2 : \\dfrac { - x _ 1 } { a _ 1 } + \\dfrac { x _ 2 } { a _ 2 } \\leq 1 \\} \\\\ S _ 3 & = \\{ x \\in \\mathbb { R } ^ 2 : \\dfrac { x _ 1 } { a _ 1 } + \\dfrac { - x _ 2 } { a _ 2 } \\leq 1 \\} \\\\ S _ 4 & = \\{ x \\in \\mathbb { R } ^ 2 : \\dfrac { - x _ 1 } { a _ 1 } + \\dfrac { - x _ 2 } { a _ 2 } \\leq 1 \\} . \\end{align*}"}
-{"id": "5298.png", "formula": "\\begin{align*} B = J ( A ^ { - 1 } ) ^ T J , \\end{align*}"}
-{"id": "4933.png", "formula": "\\begin{align*} x ^ n y ^ n = x ^ n ( y ^ n - z ^ n x ) + z ^ n x ^ { n + 1 } \\end{align*}"}
-{"id": "2891.png", "formula": "\\begin{align*} \\begin{aligned} x _ { 2 j - 1 } = \\rho _ { j } \\cos \\theta _ j , \\\\ x _ { 2 j } = \\rho _ { j } \\sin \\theta _ j , \\end{aligned} j = 1 , 2 , 3 . \\end{align*}"}
-{"id": "3817.png", "formula": "\\begin{align*} \\ell \\ge \\| g \\| _ { S , \\pi } = | w | _ \\pi = \\sum _ { i = 1 } ^ n \\pi ( s _ i ) = \\sum _ { i = 1 } ^ n | s _ i | _ \\pi = \\sum _ { i = 1 } ^ n \\| s _ i \\| _ { S , \\pi } . \\end{align*}"}
-{"id": "2588.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\theta } ( - \\log \\mathbf P _ { \\delta _ x } [ e ^ { - \\theta X _ T ( f ) } ] ) = \\frac { \\mathbf P _ { \\delta _ x } [ X _ T ( f ) e ^ { - \\theta X _ T ( f ) } ] } { \\mathbf P _ { \\delta _ x } [ e ^ { - \\theta X _ T ( f ) } ] } = P ^ \\beta _ T f ( x ) \\dot { \\mathbf P } _ x ^ { ( T , f ) } [ e ^ { - \\theta Y _ T ( f ) } ] . \\end{align*}"}
-{"id": "6067.png", "formula": "\\begin{align*} E _ 0 \\ , : = \\ , P _ { S p i n _ 4 } ( S ^ 4 ) \\times _ \\lambda \\C l _ 4 \\ , , \\end{align*}"}
-{"id": "5663.png", "formula": "\\begin{align*} { \\rm I I } ^ 2 & \\leq c \\sum _ { j = 0 } ^ { n - 1 } \\int _ { t _ j } ^ { t _ { j + 1 } } \\| \\bar E ( t _ n - t ) - B _ { n - j } P _ h \\| _ { \\mathcal { L } _ 2 ^ 0 } ^ { 2 } \\d t \\\\ & \\leq c \\| A ^ { - \\frac { s } { 2 } } \\| _ { \\mathcal { L } _ 2 ^ 0 } ^ 2 \\sum _ { j = 0 } ^ { n - 1 } \\int _ { t _ j } ^ { t _ { j + 1 } } \\| A ^ \\frac { s } { 2 } ( \\bar E ( t _ { n } - t ) - B _ { n - j } P _ h ) \\| ^ { 2 } \\d t . \\end{align*}"}
-{"id": "1181.png", "formula": "\\begin{align*} | g ( \\lambda ) | & = | { \\det } _ 2 ( + B + \\lambda ( A - B ) ) | \\leq \\exp ( | B + \\lambda ( A - B ) | _ 2 ^ 2 / 2 ) \\\\ & \\leq \\exp ( ( | A | _ 2 + | B | _ 2 + 1 ) ^ 2 / 2 ) . \\end{align*}"}
-{"id": "5893.png", "formula": "\\begin{align*} g = \\left ( 1 + 2 \\frac { q ^ { - 1 } } { 1 - q ^ { - 1 } } \\right ) \\sum _ { w \\in \\tilde { W } } ( - 1 ) ^ { \\ell ( w ) } q ^ { - \\frac { \\ell ( w ) } { 2 } } C ' _ w . \\end{align*}"}
-{"id": "1503.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r @ { \\ , } c @ { \\ , } c @ { \\ } l @ { \\quad } l @ { \\quad } l @ { \\ , } c } n _ { t } & + & u \\cdot \\ ! \\nabla n & = \\Delta n ^ m - \\nabla \\ ! \\cdot ( n S ( x , n , c ) \\nabla c ) , \\ & x \\in \\Omega , & t > 0 , \\\\ c _ { t } & + & u \\cdot \\ ! \\nabla c & = \\Delta c - c + n , \\ & x \\in \\Omega , & t > 0 , \\\\ u _ { t } & + & ( u \\cdot \\nabla ) u & = \\Delta u + \\nabla P + n \\nabla \\phi , \\ & x \\in \\Omega , & t > 0 , \\\\ & & \\nabla \\cdot u & = 0 , \\ & x \\in \\Omega , & t > 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "9701.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n _ s = n _ i + n _ b } I _ { \\textbf { d } } ^ i = \\sum _ { i = 1 } ^ { n _ i } I _ { \\textbf { d } } ^ i + \\sum _ { i = 1 } ^ { n _ b } I _ { \\textbf { d } } ^ i . \\end{align*}"}
-{"id": "4382.png", "formula": "\\begin{align*} D _ p \\cap D _ q = \\emptyset . \\end{align*}"}
-{"id": "703.png", "formula": "\\begin{align*} 0 \\leq F ( y ) - F ( x ) = \\int _ x ^ y F ^ { ( 1 ) } ( t ) \\ ; d t \\leq ( y - x ) F ^ { ( 1 ) } ( y ) , \\end{align*}"}
-{"id": "4597.png", "formula": "\\begin{align*} \\int ( \\hat { f } _ { g , \\delta } \\overset { ( k - 1 ) - } { * \\dots * } \\hat { f } _ { g , \\delta } ) ( - \\omega ) \\hat { \\nu } _ { g } ( \\omega ) d \\omega = \\sum _ { j _ 1 , j _ 2 , \\dots , j _ k } \\int ( \\hat { f } _ { g , \\delta , j _ 1 } \\overset { ( k - 1 ) - } { * \\dots * } \\hat { f } _ { g , \\delta , j _ { k - 1 } } ) ( - \\omega ) \\hat { \\nu } _ { g , j _ k } ( \\omega ) d \\omega . \\end{align*}"}
-{"id": "2711.png", "formula": "\\begin{align*} K _ { \\mathrm { A i r y } } ( \\pi ; u , v ) = \\int _ { \\mathcal { C } _ { L } } \\frac { d z } { 2 i \\pi } \\int _ { \\mathcal { C } _ { R } } \\frac { d w } { 2 i \\pi } \\frac { 1 } { z - w } e ^ { \\frac { 1 } { 3 } w ^ 3 - \\frac { 1 } { 3 } z ^ 3 } e ^ { v z - u w } \\prod _ { k = 1 } ^ { m } \\frac { z - \\pi _ { k } } { w - \\pi _ { k } } \\end{align*}"}
-{"id": "394.png", "formula": "\\begin{align*} V = \\bigoplus _ { \\alpha \\in F } C _ { \\alpha } , F \\subseteq \\widehat { H } , \\end{align*}"}
-{"id": "2807.png", "formula": "\\begin{align*} P _ { t , t + s } ( \\mathbf { n } , \\mathbf { m } ) = \\sum _ { \\mathbf { n } _ 1 \\in \\mathbb { N } _ 0 } \\sum _ { \\mathbf { n } _ 2 \\in \\mathbb { N } _ 0 } \\dots \\sum _ { \\mathbf { n } _ { s - 1 } \\in \\mathbb { N } _ 0 } P ( \\mathbf { n } , \\mathbf { n } _ 1 , \\dots , \\mathbf { n } _ { s - 1 } , \\mathbf { m } ) . \\end{align*}"}
-{"id": "2275.png", "formula": "\\begin{align*} H _ 0 R ( z ) \\xi & = - \\xi + z R ( z ) \\xi \\\\ H _ 0 R ( \\overline { z } ) \\xi & = - \\xi + \\overline { z } R ( \\overline { z } ) \\xi \\end{align*}"}
-{"id": "7860.png", "formula": "\\begin{align*} | \\mathfrak m _ N ( \\xi ) - \\lambda _ N ^ 1 ( \\xi ) | \\le 1 7 \\min \\Big \\{ e ^ { - \\frac { c \\kappa ( d , N ) ^ 2 } { 4 0 0 } \\sum _ { i = 1 } ^ d \\sin ^ 2 ( \\pi \\xi _ i ) } , \\kappa ( d , N ) ^ 2 \\sum _ { i = 1 } ^ d \\sin ^ 2 ( \\pi \\xi _ i ) \\Big \\} , \\end{align*}"}
-{"id": "8314.png", "formula": "\\begin{align*} \\mathbb { H } f ( \\alpha ) : = \\frac { 1 } { \\pi i } p . v . \\int _ { - \\infty } ^ { \\infty } \\frac { 1 } { \\alpha - \\beta } f ( \\beta ) d \\beta . \\end{align*}"}
-{"id": "160.png", "formula": "\\begin{align*} X _ { C } \\left ( \\{ x ^ i , x ^ j \\} _ { T M } \\right ) & - \\{ X _ { C } ( x ^ i ) , x ^ j \\} _ { T M } - \\{ x ^ i , X _ { C } ( x ^ j ) \\} _ { T M } = \\\\ & = - \\{ v ^ i ( { \\bf x } ) , x ^ j \\} _ { T M } - \\{ x ^ i , v ^ j ( { \\bf x } ) \\} _ { T M } = 0 , \\end{align*}"}
-{"id": "6899.png", "formula": "\\begin{align*} I = \\langle c _ 1 ^ 2 + s _ 1 ^ 2 - 1 , c _ 2 ^ 2 + s _ 2 ^ 2 - 1 \\rangle \\subset \\mathbb { A } = \\mathbb { Q } ( \\mu , \\theta ) [ c _ 1 , s _ 1 , c _ 2 , s _ 2 ] \\ , , \\end{align*}"}
-{"id": "4705.png", "formula": "\\begin{align*} F _ { j + \\frac { 1 } { 2 } } ^ { \\mbox { \\small H y b r i d } } = \\Phi _ { j + \\frac { 1 } { 2 } } F _ { j + \\frac { 1 } { 2 } } ^ { \\mbox { W E N O } } + ( 1 - \\Phi _ { j + \\frac { 1 } { 2 } } ) F _ { j + \\frac { 1 } { 2 } } ^ { \\mbox { B S Q I } } , \\end{align*}"}
-{"id": "7685.png", "formula": "\\begin{align*} \\tilde { f } ( x ) : = f ^ \\mathrm { e } - f ^ \\mathrm { o } = f ( - x ) . \\end{align*}"}
-{"id": "6411.png", "formula": "\\begin{align*} \\widehat { z } = \\left [ \\begin{array} { c } s z \\\\ c \\end{array} \\right ] , \\end{align*}"}
-{"id": "4310.png", "formula": "\\begin{align*} \\begin{aligned} S _ K ( u , v ) = & \\ ; h _ K ^ { - 1 } \\ ; \\sum _ { F \\subset \\partial K } \\Bigl [ \\big ( Q _ K u - Q _ F u , Q _ K v - Q _ F v \\big ) _ { F } + \\\\ & \\epsilon _ F h _ F \\sum _ { e \\subset \\partial F } \\big ( u - Q _ F u , v - Q _ F v \\big ) _ e \\Bigr ] , \\end{aligned} \\end{align*}"}
-{"id": "4776.png", "formula": "\\begin{align*} L _ 1 : = & \\sum _ { i = 1 } ^ { h _ 0 - b } \\left | \\mathcal { A } _ i \\cap \\binom { [ b ] } { k } \\right | , \\\\ L _ 2 : = & \\sum _ { i = h _ 0 - b + 1 } ^ { \\chi } \\left | \\mathcal { A } _ i \\cap \\binom { [ b ] } { k } \\right | . \\end{align*}"}
-{"id": "1787.png", "formula": "\\begin{align*} C _ { i , n } = \\bigcap _ { j = 1 } ^ { N } \\left \\{ n ^ { H } \\left | B _ { \\frac { i + j } { n } } - B _ { \\frac { i + j - 1 } { n } } \\right | \\leq \\frac { ( 2 j + 1 ) M } { n ^ { 1 - H } } \\right \\} , \\quad 1 \\leq i \\leq n \\end{align*}"}
-{"id": "9183.png", "formula": "\\begin{align*} V ^ { \\prime } \\left ( t _ { i } ^ { - } \\right ) = \\lim \\nolimits _ { t \\rightarrow t _ { i } ^ { - } } V ^ { \\prime } \\left ( t \\right ) , \\quad V ^ { \\prime } \\left ( t _ { i } ^ { + } \\right ) = \\lim \\nolimits _ { t \\rightarrow t _ { i } ^ { + } } V ^ { \\prime } \\left ( t \\right ) . \\end{align*}"}
-{"id": "2138.png", "formula": "\\begin{align*} f ( x ) = \\sup _ { y \\in L ^ 0 ( L ^ q ( S , \\mathcal { S } , \\nu ) ) } \\{ \\langle x , y \\rangle - f ^ \\ast ( y ) \\} , \\end{align*}"}
-{"id": "3422.png", "formula": "\\begin{align*} \\tau _ 0 ^ M ( T ) \\le \\log \\left ( A e ^ { \\Sigma _ * / 3 } / A \\right ) = \\frac { 1 } { 3 } \\Sigma _ * . \\end{align*}"}
-{"id": "7469.png", "formula": "\\begin{gather*} \\deg d _ { i j } = \\sum _ { k = i } ^ { j - 1 } d _ { k k + 1 } \\end{gather*}"}
-{"id": "9180.png", "formula": "\\begin{align*} E : \\Omega _ { p , q } \\rightarrow \\lbrack 0 , \\infty ) , E ( \\alpha ) = \\frac { 1 } { 2 } \\int _ { 0 } ^ { 1 } \\ , | \\alpha ^ { \\prime } | ^ { 2 } \\ , d t \\end{align*}"}
-{"id": "3731.png", "formula": "\\begin{align*} \\psi _ u ( t _ * ) = \\frac { 1 } { ( 2 n - 1 - q ) t _ * ^ { q + 1 } } \\bigg [ B ( t _ * u ) - ( 2 n - 1 ) \\int _ { \\Omega } ( | x | ^ { - \\mu } * F ( t _ * u ) ) f ( t _ * u ) t _ * u ~ d x + n b \\| t _ * u \\| ^ n \\bigg ] . \\end{align*}"}
-{"id": "3033.png", "formula": "\\begin{align*} f ( z ) = z + \\sum \\limits _ { k = 2 } ^ \\infty \\frac { 1 - \\alpha } { \\lambda _ k [ k ] ^ m _ { p , q } - \\alpha u _ k [ k ] ^ n _ { p , q } } x _ k z ^ k + \\sum \\limits _ { k = 1 } ^ \\infty \\frac { 1 - \\alpha } { \\mu _ k [ k ] ^ m _ { p , q } - ( - 1 ) ^ { n + j - ( m + i ) } \\alpha v _ k [ k ] ^ n _ { p , q } } y _ k \\overline { z } ^ k , \\end{align*}"}
-{"id": "6705.png", "formula": "\\begin{align*} \\beta _ { i k } \\beta _ { i l } \\ , \\ddot { \\bar { q } } _ l = \\frac { 1 } { m } ( P _ { 2 k } - u _ i \\beta _ { i k } ) - ( \\alpha ' _ { i n } + \\beta ' _ { i l n } \\ , \\dot { \\bar { q } } _ l ) \\dot { \\bar { q } } _ n \\beta _ { i k } . \\end{align*}"}
-{"id": "9439.png", "formula": "\\begin{align*} \\dim _ { \\mathbb { C } } M ^ { E i s } _ 2 ( \\Gamma _ 0 ( N ) ) = D ( N ) - 1 . \\end{align*}"}
-{"id": "574.png", "formula": "\\begin{align*} p \\xi ^ { p ^ n - s + k ( p ^ n + 1 ) } - \\left ( p \\xi ^ { p ^ n - s + \\ell ( p ^ n + 1 ) } \\right ) = \\xi ^ { p ^ n - s } \\left ( p \\xi ^ { k ( p ^ n + 1 ) } - p \\xi ^ { \\ell ( p ^ n + 1 ) } \\right ) \\end{align*}"}
-{"id": "1294.png", "formula": "\\begin{align*} \\tilde \\Psi _ A ( z ) ' ( \\xi ) = \\frac { 3 i } { 4 \\pi ^ 2 } \\tilde I ( S _ A + z ) ( \\xi ) - S ' ( \\xi ) = ( \\| z \\| _ { Z ^ k } ^ 3 + | A | ^ 3 + | A | ) O ( | \\xi | ^ { - k - 1 } ) . \\end{align*}"}
-{"id": "7357.png", "formula": "\\begin{align*} d ( u , u ' ) & = d ( u , Y ) + d ( Y , Y ' ) + d ( Y ' , u ' ) \\\\ & = | d ( x , Y ) - d ( x , u ) | + d ( Y , Y ' ) + | d ( x ' , Y ' ) - d ( x ' , u ' ) | \\end{align*}"}
-{"id": "9785.png", "formula": "\\begin{align*} L _ \\ast = \\{ ( 0 , 0 ) \\} \\quad \\quad \\mathcal { N } _ \\ast = \\{ x _ 1 = 0 \\} \\cup \\{ x _ 2 = 0 \\} . \\end{align*}"}
-{"id": "9453.png", "formula": "\\begin{align*} b ( m ) = - 2 4 \\left ( \\sum _ { 1 \\leq j \\leq D ( N ) - 1 } \\frac { \\det ( A _ { f , j } ) } { \\det ( A _ N ) } \\right ) \\sigma _ 1 ( m ) . \\end{align*}"}
-{"id": "1723.png", "formula": "\\begin{align*} \\limsup _ { \\varepsilon \\to 0 ^ { + } } \\frac { B _ { t + \\varepsilon } - B _ { t } } { \\sqrt { 2 \\varepsilon ^ { 2 H } \\log \\log ( 1 / \\varepsilon ) } } = 1 , \\quad \\end{align*}"}
-{"id": "6655.png", "formula": "\\begin{align*} a = \\lambda _ 1 a _ 1 + \\cdots + \\lambda _ l a _ l \\in I , b = \\lambda _ 1 b _ 1 + \\cdots + \\lambda _ l b _ l \\in J \\end{align*}"}
-{"id": "5441.png", "formula": "\\begin{align*} \\log \\hat { \\rho } _ s ( A ) \\le \\max _ { f ^ k ( p ) = p , k \\in \\N } \\left \\lbrace \\frac { s - \\lfloor s \\rfloor } { k } r _ { \\lfloor s \\rfloor + 1 } ( A ^ k ( p ) ) + \\frac { 1 - s + \\lfloor s \\rfloor } { k } r _ { \\lfloor s \\rfloor } ( A ^ k ( p ) ) \\right \\rbrace , \\end{align*}"}
-{"id": "8993.png", "formula": "\\begin{align*} \\rho _ + : = \\frac { X [ 0 ] ( 1 - \\beta \\gamma _ g ) + \\sqrt { \\beta ^ 2 \\gamma _ g ^ 2 X [ 0 ] ^ 2 + 2 \\beta \\gamma _ g - 1 } } { 2 \\beta \\gamma _ g - 1 } . \\end{align*}"}
-{"id": "2616.png", "formula": "\\begin{align*} \\frac { d } { d \\theta } G ( \\theta ) & = - \\frac { 1 } { \\alpha - 1 } \\big ( 1 + \\theta ^ { - ( \\alpha - 1 ) } \\big ) ^ { - \\frac { 1 } { \\alpha - 1 } - 1 } \\frac { d } { d \\theta } \\theta ^ { - ( \\alpha - 1 ) } \\\\ & = \\big ( 1 + \\theta ^ { - ( \\alpha - 1 ) } \\big ) ^ { - \\frac { \\alpha } { \\alpha - 1 } } \\theta ^ { - \\alpha } . \\end{align*}"}
-{"id": "8534.png", "formula": "\\begin{align*} ( I + \\mathcal { H } ) \\partial _ { \\alpha } ^ k \\tilde { \\theta } = & ( I + \\mathcal { H } ) \\partial _ { \\alpha } ^ k ( I - \\mathcal { H } ) ( \\zeta - \\bar { \\zeta } ) = - [ \\partial _ { \\alpha } ^ k , \\mathcal { H } ] \\tilde { \\theta } \\\\ = & - \\sum _ { m = 0 } ^ { k - 1 } \\partial _ { \\alpha } ^ m [ \\zeta _ { \\alpha } - 1 , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } \\partial _ { \\alpha } ^ { k - m - 1 } \\tilde { \\theta } } { \\zeta _ { \\alpha } } . \\end{align*}"}
-{"id": "4849.png", "formula": "\\begin{align*} r _ s = i _ 1 \\vert x _ { 1 } \\vert + \\dots + i _ n \\vert x _ { n } \\vert - 1 , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , i _ 1 \\geq 0 , \\dots , i _ n \\geq 0 . \\end{align*}"}
-{"id": "1879.png", "formula": "\\begin{align*} \\mathcal { C } \\equiv \\frac { ( \\xi L ) ^ 2 } { 2 \\pi } \\sum _ { j = 1 } ^ { \\infty } \\frac { A _ j ^ 2 } { Z _ j ^ 2 } = \\frac { ( \\xi L ) ^ 2 } { 2 \\pi } \\sum _ { j = 1 } ^ { \\infty } \\frac { A _ j ^ 2 } { Z _ j ^ 2 - ( \\pi m ) ^ 2 } . \\end{align*}"}
-{"id": "8882.png", "formula": "\\begin{align*} \\tilde { \\tilde { g } } _ { \\alpha , \\beta } = \\frac { \\gamma _ { \\alpha , \\beta } } { g _ { \\alpha , \\beta } + \\epsilon g _ { \\alpha , \\beta } \\phi _ { \\alpha , \\beta } } = \\frac { \\gamma _ { \\alpha , \\beta } } { g _ { \\alpha , \\beta } } [ 1 + \\epsilon ( - \\phi _ { \\alpha , \\beta } ) ] . \\end{align*}"}
-{"id": "3252.png", "formula": "\\begin{align*} \\Pi _ n ^ { } ( z ) = \\prod _ { j = 1 } ^ { n - 1 } \\left ( z - ( \\zeta ^ { j } a + \\zeta ^ { - j } b ) \\right ) , \\end{align*}"}
-{"id": "5167.png", "formula": "\\begin{align*} \\{ \\log ( \\frac { U _ { j + 1 , n } } { U _ { j , n } } ) ^ { j } , j = 1 , . . . , n \\} = _ { d } \\{ E _ { 1 , n } , . . . , E _ { n , n } \\} , \\end{align*}"}
-{"id": "6878.png", "formula": "\\begin{align*} x = \\Phi ^ t ( \\beta ) = \\varphi ^ t \\circ v ^ { - 1 } = A ( t ) \\beta \\ , , \\end{align*}"}
-{"id": "971.png", "formula": "\\begin{align*} S _ q ( n , k ) = \\begin{cases} q ^ { k - 1 } S _ q ( n - 1 , k - 1 ) + [ k ] _ q S _ q ( n - 1 , k ) & \\ ; \\\\ \\delta _ { n , k } & \\ \\end{cases} \\end{align*}"}
-{"id": "9641.png", "formula": "\\begin{align*} S _ 1 ( z , q , N ) = \\frac { z q ( q ^ 2 ; q ^ 2 ) _ N } { ( z q ^ 2 ; q ^ 2 ) _ N } \\sum _ { n = 0 } ^ { N - 1 } \\frac { ( z q ^ 2 ; q ^ 2 ) _ n q ^ { 2 n } } { ( q ^ 2 ; q ^ 2 ) _ n ( 1 - z q ^ { 2 n + 1 } ) } , \\end{align*}"}
-{"id": "7032.png", "formula": "\\begin{align*} \\frac { z \\mathbb { F } ^ { \\prime } ( z ) } { \\mathbb { F } ( z ) } : = \\delta + ( 1 - \\delta ) p ( z ) , \\end{align*}"}
-{"id": "6630.png", "formula": "\\begin{align*} T _ 1 & : = \\left \\langle \\{ 1 \\} , \\ldots , \\{ t + 1 \\} \\right \\rangle , \\\\ T _ 2 & : = \\langle \\{ 1 \\} , \\ldots , \\{ t - 1 \\} , \\{ t + 2 \\} , \\{ t + 3 \\} \\rangle . \\end{align*}"}
-{"id": "1922.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\frac { A _ j ^ 2 } { Z _ j ^ p } \\leq \\frac { \\cos ^ 2 ( Z _ 1 ) } { Z _ 1 ^ { p - 1 } \\left [ Z _ 1 + \\sin ( Z _ 1 ) \\cos ( Z _ 1 ) \\right ] } + \\frac { \\zeta ( p ) } { \\pi ^ p } . \\end{align*}"}
-{"id": "129.png", "formula": "\\begin{align*} ( a ) _ n = \\begin{cases} 1 & n = 0 \\\\ a ( a + 1 ) \\cdots ( a + n - 1 ) & n \\ge 1 \\ , . \\end{cases} \\end{align*}"}
-{"id": "2186.png", "formula": "\\begin{align*} \\| f \\| _ { H ^ 1 _ { { \\rm m a x } , H } } = \\| \\mathcal M f \\| _ { L ^ 1 ( d w ) } . \\end{align*}"}
-{"id": "4516.png", "formula": "\\begin{align*} D ( F ) = \\{ | x - y | : ( x , y ) \\in F \\times F \\} . \\end{align*}"}
-{"id": "3051.png", "formula": "\\begin{align*} X _ { i } ( t ) & = X _ { i } ( 0 ) + \\int \\limits _ { 0 } ^ { t } \\left ( \\beta _ { i } + \\sum \\limits _ { j = 1 } ^ { d } b _ { i j } X _ { j } ( s ) \\right ) d s + \\int \\limits _ { 0 } ^ { t } X _ { i } ( s - ) ^ { 1 / \\alpha _ { i } } d Z _ { i } ( s ) , \\end{align*}"}
-{"id": "9909.png", "formula": "\\begin{align*} & E ^ { \\mu } [ | \\int _ { \\mathcal { X } } f ( x ) \\pi _ { n } ^ { \\mu } ( d x ) - \\int _ { \\mathcal { X } } f ( x ) \\pi _ { n } ^ { \\nu } ( d x ) | ] \\\\ = & E ^ { \\mu } [ E ^ { \\mu } [ | \\int _ { \\mathcal { X } } f ( x ) F ( \\pi _ { n - 1 } ^ { \\mu } , y _ { n } ) ( d x ) \\\\ & - \\int _ { X } f ( x ) F ( \\pi _ { n - 1 } ^ { \\nu } , y _ { n } ) ( d x ) | | Y _ { [ 0 , n - 1 ] } ] ] \\end{align*}"}
-{"id": "9333.png", "formula": "\\begin{align*} = \\sum _ { i , k = 1 } ^ { n } \\Big [ \\langle e _ { i } , \\nabla _ { e _ { k } } W ^ { T } \\rangle - \\langle e _ { i } , A _ { W ^ { \\perp } } ( e _ { k } ) \\rangle \\Big ] ^ { 2 } \\end{align*}"}
-{"id": "6625.png", "formula": "\\begin{align*} \\sum _ { X \\supseteq S } \\tau _ X = 0 , \\mbox { f o r e v e r y $ S \\in 2 ^ V $ s u c h t h a t $ | S | \\le t $ . } \\end{align*}"}
-{"id": "6533.png", "formula": "\\begin{align*} \\frac { 1 } { l } & \\sum _ { i = 1 } ^ { k } \\Biggl ( \\int ^ { t + l } _ { t } \\bigl \\| f ( s + \\tau , x _ { i } ) - f ( s , x _ { i } ) \\bigr \\| ^ { q } \\ , d s \\Biggr ) ^ { 1 / q } \\\\ \\leq & \\frac { 1 } { l } l ^ { ( 1 / q ) - ( 1 / p ) } \\sum _ { i = 1 } ^ { k } \\Biggl ( \\int ^ { t + l } _ { t } \\bigl \\| f ( s + \\tau , x _ { i } ) - f ( s , x _ { i } ) \\bigr \\| ^ { p } \\ , d s \\Biggr ) ^ { 1 / p } \\leq \\epsilon l ^ { ( 1 / q ) - 1 } , t \\in I . \\end{align*}"}
-{"id": "5025.png", "formula": "\\begin{align*} r s _ d ( m , n , I ) = \\Omega _ d \\left ( \\left ( \\frac { I } { m n } \\right ) ^ { d - 1 } m n \\right ) . \\end{align*}"}
-{"id": "505.png", "formula": "\\begin{align*} 2 g + 6 = 3 r _ { 4 } + 2 r _ { 2 } . \\end{align*}"}
-{"id": "5856.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i = 1 } ^ k \\sum _ { j = 1 } ^ { d _ i } f ( \\alpha _ { i j } ) \\frac { \\prod _ { ( k , l ) \\neq ( i , j ) } ( x - \\alpha _ { k l } ) } { \\prod _ { ( k , l ) \\neq ( i , j ) } ( \\alpha _ { i j } - \\alpha _ { k l } ) } \\end{align*}"}
-{"id": "3113.png", "formula": "\\begin{align*} \\chi ( \\beta ^ { j } + \\ldots + \\beta ^ { i + j } ) & = \\frac { ( \\beta ^ { j } + \\ldots + \\beta ^ { i + j } ) \\cdot ( \\beta ^ { j } + \\ldots + \\beta ^ { i + j } - K _ S ) } { 2 } + \\chi ( \\O _ S ) \\\\ & = \\sum _ { j \\leq k < l \\leq i + j } \\beta ^ k \\beta ^ l + \\chi ( \\O _ S ) \\ , . \\end{align*}"}
-{"id": "699.png", "formula": "\\begin{align*} D e t \\left [ g _ { i \\bar { j } } \\right ] & = e ^ { 3 g } , \\\\ & = e ^ { 3 F \\circ X - \\frac { 3 K } { p } L o g \\left ( 1 - 4 p \\pi _ 1 ^ \\mathbb { R } \\right ) } , \\\\ & = \\frac { e ^ { 3 F \\circ X } } { \\left ( 1 - 4 p \\pi _ 1 ^ \\mathbb { R } \\right ) ^ { 2 + \\frac { 1 } { p } } } , \\end{align*}"}
-{"id": "5325.png", "formula": "\\begin{align*} X = \\frac 1 2 \\left ( \\left [ \\begin{array} { c c } 1 & 0 \\\\ 0 & - 1 \\end{array} \\right ] + i \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & 0 \\end{array} \\right ] \\right ) = \\frac 1 2 \\left ( A + i B \\right ) \\end{align*}"}
-{"id": "2898.png", "formula": "\\begin{align*} S _ { \\mu \\nu } & = \\left ( L _ { \\mu \\nu } \\gamma _ \\mu \\gamma _ \\nu + \\frac { 1 } { 2 } \\right ) \\\\ S _ { \\mu \\nu \\rho \\sigma } & = \\left ( L _ { \\mu \\nu } \\gamma _ \\mu \\gamma _ \\nu + L _ { \\mu \\rho } \\gamma _ \\mu \\gamma _ \\rho + L _ { \\mu \\sigma } \\gamma _ \\mu \\gamma _ \\sigma + L _ { \\nu \\rho } \\gamma _ \\nu \\gamma _ \\rho + L _ { \\nu \\sigma } \\gamma _ \\nu \\gamma _ \\sigma + L _ { \\rho \\sigma } \\gamma _ \\rho \\gamma _ \\sigma + \\frac { 3 } { 2 } \\right ) \\end{align*}"}
-{"id": "5200.png", "formula": "\\begin{align*} H _ { 1 } ^ { t } ( x _ { 0 } ) = H _ { 1 } ( x _ { 0 } t ^ { \\rho } ) , t > 0 , t \\neq 1 . \\end{align*}"}
-{"id": "6064.png", "formula": "\\begin{align*} { \\bf i } = { \\bf j } _ 0 , \\ \\ { \\bf i } ' = { \\bf i } _ 0 , \\end{align*}"}
-{"id": "5758.png", "formula": "\\begin{align*} \\mu _ \\R \\left ( \\rho _ \\R ( g ) ( v ) , \\rho ^ \\vee ( g ) ( f ) \\right ) ) & = \\left \\langle \\varrho _ \\R ( x ) ( \\rho _ \\R ( g ) ( v ) ) , \\rho ^ \\vee _ \\R ( g ) ( f ) \\right \\rangle \\\\ & = \\left \\langle \\rho _ \\R ( g ^ { - 1 } ) \\circ \\varrho _ \\R ( x ) \\circ \\rho _ \\R ( g ) ( v ) , f \\right \\rangle \\\\ & = \\left \\langle \\varrho _ \\R ( { \\rm A d } ( g ^ { - 1 } ) ( x ) ) ( v ) , f \\right \\rangle \\\\ & = \\mu _ \\R ( v , f ) ( { \\rm A d } ( g ^ { - 1 } ) ( x ) ) \\end{align*}"}
-{"id": "7434.png", "formula": "\\begin{align*} \\displaystyle d _ { \\psi ' _ E } \\pi = \\gamma ^ * ( 0 , \\mathbf { 1 } _ E , \\psi ' _ E ) ^ k d ^ E \\pi \\end{align*}"}
-{"id": "8712.png", "formula": "\\begin{align*} \\rho ( ( i _ 1 , \\ldots , i _ d ) ) = { \\displaystyle \\prod _ { j = 1 } ^ d \\left ( \\sum _ { n _ j = 1 } ^ { i _ j } \\prod _ { r = 1 } ^ { n _ j } \\left ( { q _ j ( r ) \\over p _ j ( r ) } \\right ) \\right ) \\over \\displaystyle \\prod _ { j = 1 } ^ d \\left ( \\sum _ { n _ j = 1 } ^ { N _ j } \\prod _ { r = 1 } ^ { n _ j } \\left ( { q _ j ( r ) \\over p _ j ( r ) } \\right ) \\right ) } . \\end{align*}"}
-{"id": "6607.png", "formula": "\\begin{align*} p _ j : = - \\varepsilon _ j ( - \\varphi ) ^ { - n _ j } = \\sum _ { s = j + 1 } ^ m ( - \\varphi ) ^ { n _ s - n _ j } , \\end{align*}"}
-{"id": "7704.png", "formula": "\\begin{align*} \\zeta ( x ) : = \\int \\limits _ { - \\infty } ^ { \\infty } \\frac { \\sin ( t x ) } { \\mathbf { i } t } h \\left ( \\frac { t } { 2 \\pi } \\right ) \\ , \\mathrm { d } t . \\end{align*}"}
-{"id": "520.png", "formula": "\\begin{align*} \\mu _ { 1 , | K | } ( v _ i \\wedge w _ j ) = \\frac { ( \\phi ' ( y ) ) ^ 2 } { y ^ 6 } ( ( i - j ) \\phi ' \\tilde { f } \\tilde { g } + \\phi ( \\tilde { f } ' \\tilde { g } - \\tilde { f } \\tilde { g } ' ) ) \\phi ^ { i + j - 1 } ( d y ) ^ 3 . \\end{align*}"}
-{"id": "1647.png", "formula": "\\begin{align*} \\omega \\big ( F _ k , G _ k \\big ) = \\omega \\big ( F _ { - k } , G _ { - k } \\big ) = - 2 i \\quad \\omega \\big ( F _ k , G _ { - k } \\big ) = \\omega \\big ( F _ { - k } , G _ k \\big ) = 0 \\end{align*}"}
-{"id": "6785.png", "formula": "\\begin{align*} \\varphi ( t , \\alpha ) = \\varphi ^ t ( \\alpha ) = A ( t ) v ( \\alpha ) \\ , , \\end{align*}"}
-{"id": "3728.png", "formula": "\\begin{align*} \\psi ^ { ' } _ u ( t ) & = ( n - 1 - q ) t ^ { n - 2 - q } m ( \\| t u \\| ^ n ) \\| u \\| ^ n + n t ^ { 2 n - 2 - q } m { ' } ( \\| t u \\| ^ n ) \\| u \\| ^ { 2 n } \\\\ & + \\frac { q } { t ^ { q + 1 } } \\int _ { \\Omega } ( | x | ^ { - \\mu } * F ( t u ) ) f ( t u ) . u ~ d x - t ^ { - q } \\bigg [ \\int _ { \\Omega } ( | x | ^ { - \\mu } * f ( t u ) . u ) f ( t u ) . u ~ d x \\\\ & + \\int _ { \\Omega } ( | x | ^ { - \\mu } * F ( t u ) ) f ^ { ' } ( t u ) . u ^ 2 ~ d x \\bigg ] . \\end{align*}"}
-{"id": "2094.png", "formula": "\\begin{align*} & \\mathbb { E } ( \\textbf { y } _ R | b _ 0 , { b } _ 1 , \\cdots , { b } _ M ) = \\sum _ { i = 0 } ^ { M } { { b } _ i N \\delta ( i T + t ) * p _ { \\rm o b s } ( i T + t ) } = \\sum _ { i = 0 } ^ { M } { { b } _ i N p _ { \\rm o b s } ( i T + t ) } , \\end{align*}"}
-{"id": "4176.png", "formula": "\\begin{align*} \\left ( A _ n ^ { ( 1 ) } ( z ) \\right ) ^ k = \\frac { 1 } { f ' ( 0 ) ^ k } E _ n ( z ) T _ \\alpha ^ { - 1 } A _ \\alpha ^ k T _ \\alpha E _ n ^ { - 1 } ( z ) . \\end{align*}"}
-{"id": "8406.png", "formula": "\\begin{align*} e r r o r _ k = & 2 R e \\int i \\partial _ { \\alpha } D ^ k u \\overline { \\partial _ t D ^ k u } - 2 R e \\int \\bold { n } \\frac { 1 } { | z _ { \\alpha } | ^ { 2 k - 1 } } \\partial _ { \\alpha } ^ k D u \\overline { \\partial _ t \\partial _ { \\alpha } ^ k u } \\\\ \\leq & C ( \\| \\partial _ { \\alpha } \\xi _ 0 \\| _ { H ^ { s - 1 } } , \\| v _ 0 \\| _ { H ^ s } , \\| w _ 0 \\| _ { H ^ s } , \\mathcal { E } ( t ) , d _ I ( t ) ^ { - 1 } , d _ P ( t ) ^ { - 1 } , N \\lambda _ { m a x } , C _ 1 , C _ 2 ) . \\end{align*}"}
-{"id": "6865.png", "formula": "\\begin{align*} u _ \\nu ^ j = \\partial ^ \\nu u ^ j = \\frac { \\partial ^ { | \\nu | } u ^ j } { \\partial x _ 1 ^ { \\nu _ 1 } \\cdots \\partial x _ n ^ { \\nu _ n } } \\end{align*}"}
-{"id": "2318.png", "formula": "\\begin{align*} \\xi = \\left ( \\begin{array} { c } \\xi _ 0 \\\\ \\xi _ 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "4644.png", "formula": "\\begin{align*} f ^ n ( x ) - \\beta = \\prod _ { i = 1 } ^ { q ^ { n - 1 } } \\left ( f ( x ) - z _ i \\right ) \\end{align*}"}
-{"id": "3377.png", "formula": "\\begin{align*} - \\int _ { 1 - \\delta } ^ 1 H ( u ) d u & = \\int _ 0 ^ \\delta F ^ { - 1 } ( u ) d u \\\\ & = \\int _ 0 ^ { \\sqrt { F ^ { - 1 } ( \\delta ) } } 2 x ^ 3 F ' ( x ) d x \\\\ & = \\int _ 0 ^ { G ^ { - 1 } ( \\delta ) } x ^ 2 G ' ( x ) d x \\\\ & = - \\int _ 0 ^ \\delta \\log ( 1 - u ) d u \\end{align*}"}
-{"id": "6722.png", "formula": "\\begin{align*} \\dot { Q } _ { 1 1 } = Q _ { 2 1 } , \\dot { Q } _ { 1 2 } = Q _ { 2 2 } , , \\end{align*}"}
-{"id": "8971.png", "formula": "\\begin{align*} d _ E ( u , v ) = \\frac { \\| u \\wedge v \\| } { \\| u \\| \\cdot \\| v \\| } \\end{align*}"}
-{"id": "4097.png", "formula": "\\begin{align*} \\Psi _ { \\alpha } ( z ) = \\mathcal O \\begin{pmatrix} 1 & z ^ { \\alpha } & z ^ { \\alpha } \\\\ 1 & z ^ { \\alpha } & z ^ { \\alpha } \\\\ 1 & z ^ { \\alpha } & z ^ { \\alpha } \\end{pmatrix} \\end{align*}"}
-{"id": "9266.png", "formula": "\\begin{align*} \\bar v \\le \\frac { 1 } { 2 } a C [ 1 + ( M ^ \\prime _ 2 ( R ) ) ^ 2 + M _ 2 ( R ) ] ( | x - x _ 0 | ^ 2 + d ) \\le \\bar \\phi , { \\rm o n } \\ ( B _ R \\cap \\Omega ) \\cap \\{ G ( \\cdot , u , D u ) = 0 \\} , \\end{align*}"}
-{"id": "5312.png", "formula": "\\begin{align*} w _ { \\alpha _ i ^ { ( j ) } } ( D ) = D + \\langle D , \\check { \\alpha } _ i ^ { ( j ) } \\rangle \\alpha _ i ^ { ( j ) } , w _ { \\alpha _ i ^ { ( j ) } } ( d ) = d + \\langle \\alpha _ i ^ { ( j ) } , d \\rangle \\check { \\alpha } _ i ^ { ( j ) } \\end{align*}"}
-{"id": "547.png", "formula": "\\begin{align*} \\begin{aligned} | J _ 1 ^ h | & \\leq C \\big ( 1 + | \\nabla v ^ h | \\big ) , \\\\ | J _ 2 ^ h | & \\leq C \\big ( 1 + | \\nabla w ^ h | + | \\nabla v ^ h | ^ 2 + | \\nabla ^ 2 v ^ h | + | \\nabla \\vec r ^ { \\ , h } | \\big ) , \\\\ | J _ 3 ^ h | & \\leq C h ^ 3 \\big ( 1 + | \\nabla w ^ h | + | \\nabla ^ 2 w ^ h | + | \\nabla v ^ h | ^ 2 + | \\nabla ^ 2 v ^ h | + | \\nabla v ^ h | \\cdot | \\nabla ^ 2 v ^ h | + | \\nabla \\vec r ^ { \\ , h } | \\big ) + o ( h ^ 2 ) . \\end{aligned} \\end{align*}"}
-{"id": "9039.png", "formula": "\\begin{align*} \\mathbb { E } [ F ( x ) \\cdot F ( y ) ] = \\mathbb { E } [ F ( x - y ) \\cdot F ( 0 ) ] \\end{align*}"}
-{"id": "4343.png", "formula": "\\begin{align*} \\Phi _ A ( a ) Z _ b ( X ) = \\omega ( a , b ) Z _ b ( X ) \\quad \\Phi _ B ( b ) Z _ { b ' } ( X ) = Z _ { b + b ' } ( X ) . \\end{align*}"}
-{"id": "9182.png", "formula": "\\begin{align*} E ^ { \\prime \\prime } \\left ( 0 \\right ) = - \\int _ { 0 } ^ { b } \\left \\langle V ^ { \\prime \\prime } + R \\left ( V , \\dot { \\gamma } \\right ) \\dot { \\gamma } , V \\right \\rangle d t + \\left . \\left \\langle V ^ { \\prime } - S _ { \\gamma ^ { \\prime } } V , V \\right \\rangle \\right \\vert _ { 0 } ^ { b } + \\sum _ { i } \\left \\langle V ^ { \\prime } \\left ( t _ { i } ^ { - } \\right ) - V ^ { \\prime } \\left ( t _ { i } ^ { + } \\right ) , V \\left ( t _ { i } \\right ) \\right \\rangle , \\end{align*}"}
-{"id": "10032.png", "formula": "\\begin{align*} w ( z q ^ { - n } ) = \\frac { \\theta ( - q z ; q ) } { \\theta ( - z a ^ 2 ; q ) } a ^ { - 2 n } q ^ n \\Big ( 1 + \\mathcal O ( q ^ n ) \\Big ) , n \\rightarrow \\infty . \\end{align*}"}
-{"id": "3775.png", "formula": "\\begin{align*} \\tau = \\frac { ( 2 r _ 1 / \\lambda ) ^ { - s } - r _ 2 ^ { - s } } { r _ 0 ^ { - s } - r _ 2 ^ { - s } } . \\end{align*}"}
-{"id": "3789.png", "formula": "\\begin{align*} \\alpha & = 1 ^ { c + d } a 1 ^ c & & \\beta = 1 ^ { c + d + 1 } a 1 ^ { c - 1 } \\\\ \\alpha & = ( a - 1 ) 1 ^ { c - 1 } 2 1 ^ { c + d } & & \\beta = ( a - 1 ) 1 ^ { c + d } 2 1 ^ { c - 1 } \\\\ \\alpha & = 1 ^ { c - 1 } 2 1 ^ { c + d - 1 } a & & \\beta = 1 ^ { c + d } 2 1 ^ { c - 2 } a \\\\ \\alpha & = 1 ^ { c - 1 } a 1 ^ { c + d - 1 } 2 & & \\beta = 1 ^ { c + d } a 1 ^ { c - 2 } 2 \\\\ \\alpha & = 1 ^ d 2 1 ^ { c - 1 } a 1 ^ { c - 1 } & & \\beta = 1 ^ d 2 1 ^ { c - 2 } a 1 ^ c \\end{align*}"}
-{"id": "7348.png", "formula": "\\begin{align*} B ( 0 , r ) : = \\{ v \\in T _ p M : \\| v \\| \\leq r \\} \\end{align*}"}
-{"id": "9511.png", "formula": "\\begin{align*} u _ { \\varepsilon , R } ( r ) = \\varepsilon ^ { - \\frac { N - s p } { p ^ 2 } } \\overline { u } _ { \\delta , R } ( r ) , \\ \\mbox { w i t h } \\ \\delta = \\varepsilon ^ { \\frac { p - 1 } { p } } . \\end{align*}"}
-{"id": "2451.png", "formula": "\\begin{align*} \\lim _ { t \\to 1 } \\ , ( 1 - t ) ^ { \\dim R } P ( R , t ) \\ = \\ ( \\deg D ) ^ { \\dim R - 1 } , \\end{align*}"}
-{"id": "1874.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { \\infty } \\frac { 1 } { ( Z _ j + \\pi m ) ( Z _ j + \\pi ( n + m ) ) } = \\frac { 1 } { \\pi n } \\sum _ { m = 1 } ^ { n } \\frac { 1 } { Z _ j + \\pi m } \\end{align*}"}
-{"id": "5124.png", "formula": "\\begin{align*} & { D } ( h _ { m _ j , k _ j } ) = C ^ \\infty _ c ( 0 , + \\infty ) ^ 2 , \\\\ & h _ { m _ j , k _ j } ( f ^ + , f ^ - ) : = \\left ( \\begin{array} { c c } m + \\frac { \\nu + \\mu } { r } & - \\partial _ r + \\frac { k _ j + \\lambda } { r } \\\\ \\partial _ r + \\frac { k _ j + \\lambda } { r } & - m + \\frac { \\nu - \\mu } { r } \\end{array} \\right ) \\begin{pmatrix} f ^ + \\\\ f ^ - \\end{pmatrix} ; \\end{align*}"}
-{"id": "7547.png", "formula": "\\begin{align*} C ^ { \\alpha , p } = C ^ { \\alpha } ( \\mathbb { R } ^ d ) \\cap L ^ p ( \\mathbb { R } ^ d ) \\end{align*}"}
-{"id": "1255.png", "formula": "\\begin{align*} T _ { 4 , 4 } & = B \\int e ^ { i \\Theta ( \\eta , \\nu ) } \\frac { \\eta - 2 \\nu } { \\eta + \\nu } \\Bigg ( e ^ { 2 i a \\ln | ( \\eta + \\nu ) / 2 | } \\left ( 1 + \\frac { 2 i a } { ( \\eta + \\nu ) ^ 3 } \\right ) \\chi \\left ( \\frac { \\eta + \\nu } { 2 } \\right ) \\\\ & \\qquad + e ^ { - 2 i a \\ln | ( \\eta + \\nu ) / 2 | } \\left ( 1 - \\frac { 2 i a } { ( \\eta + \\nu ) ^ 3 } \\right ) \\chi \\left ( - \\frac { \\eta + \\nu } { 2 } \\right ) \\Bigg ) S \\left ( \\frac { \\eta - \\nu } { 2 } \\right ) d \\nu \\end{align*}"}
-{"id": "8260.png", "formula": "\\begin{align*} ( I - \\mathfrak { H } ) \\Big ( \\bar { z } _ t + \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i } { 2 \\pi ( z ( \\alpha , t ) - z _ j ( t ) ) } \\Big ) = 0 , \\end{align*}"}
-{"id": "9810.png", "formula": "\\begin{align*} \\mathcal { I } _ a = c _ { n , a } ( - \\Delta ) ^ { - \\frac { a } { 2 } } \\equiv c _ { n , a } ( - \\Delta ) ^ { s - \\frac { 1 } { 2 } } , \\end{align*}"}
-{"id": "9852.png", "formula": "\\begin{align*} a ^ r ( b ^ { r + 1 } - b ^ r ) = a ^ r b ^ r ( b - 1 ) = - w ( 1 + w ^ 2 ) ^ r ( 1 - w ) ^ r . \\end{align*}"}
-{"id": "7059.png", "formula": "\\begin{align*} E ( t ) : = E ( u ( t ) ) \\end{align*}"}
-{"id": "4373.png", "formula": "\\begin{align*} \\frac { k - t } { n - t } \\Big ( 1 - \\prod _ { i = 1 } ^ { k } \\frac { n - k + 1 - i } { n - t - i } \\Big ) \\ge \\frac { k ( k - 1 ) ( 3 n - 2 k - 2 ) } { n ( n - 1 ) ( n - 2 ) } . \\end{align*}"}
-{"id": "2853.png", "formula": "\\begin{align*} f ( x ) = x ^ 4 - 4 t x ^ 3 + ( 6 t + 2 ) x ^ 2 + 4 t x + 1 \\in \\Z _ M [ x ] . \\end{align*}"}
-{"id": "858.png", "formula": "\\begin{align*} & \\hat { v } _ { k } : = v _ { k } + h ( \\bar { x } + t _ { k } u _ { k } ) / t _ { k } \\\\ & \\hat { u } ^ { * } _ { k } : = \\nabla h ( \\bar { x } + t _ { k } u _ { k } ) ^ { T } v ^ { * } _ { k } - u ^ { * } _ { k } \\end{align*}"}
-{"id": "9848.png", "formula": "\\begin{align*} \\frac { 1 } { ( 1 - t ^ 2 ) ^ { r + 1 } } \\begin{bmatrix} \\frac { t ^ 3 } { 1 + t } [ t ( ( 1 + t ) ^ r - 1 ) + ( ( 1 - t ^ 2 ) ^ r - 1 ) ] \\\\ - \\frac { t ^ 3 } { 1 + t ^ 3 } [ t ^ 3 ( ( 1 + t ) ^ r - 1 ) + ( ( 1 - t ^ 4 ) ^ r - 1 ) ] \\end{bmatrix} . \\end{align*}"}
-{"id": "5430.png", "formula": "\\begin{align*} \\lVert A \\rVert _ \\alpha : = \\sup _ { x , y \\in M } \\lVert A ( x ) - A ( y ) \\rVert + \\sup _ { x \\neq y } \\bigg { ( } \\frac { \\lVert A ( x ) - A ( y ) \\rVert } { d ( x , y ) ^ \\alpha } \\bigg { ) } . \\end{align*}"}
-{"id": "9717.png", "formula": "\\begin{align*} \\tilde { u } : = u - \\varphi , \\end{align*}"}
-{"id": "2105.png", "formula": "\\begin{align*} M ^ * = M ^ * ( t _ 0 , T ) : = \\max \\{ \\| A \\| _ { \\infty , [ t _ 0 , t _ 0 + T ] } , 2 K \\| C \\| _ { q { \\rm - v a r } , [ t _ 0 , t _ 0 + T ] } \\} < \\infty , \\end{align*}"}
-{"id": "3.png", "formula": "\\begin{align*} p ( y ) : = \\max \\{ - K _ y , \\min \\{ y , K _ y \\} \\} \\ \\ \\ \\ q ( z ) : = \\frac { \\min \\left \\{ | z | , K _ z \\right \\} } { | z | } z \\mathbf { 1 } _ { \\{ z \\neq 0 \\} } . \\end{align*}"}
-{"id": "8204.png", "formula": "\\begin{align*} \\omega ( q ) : = & \\sum \\limits _ { n = 0 } ^ { \\infty } \\frac { q ^ { 2 n ^ 2 + 2 n } } { ( q ; q ^ 2 ) _ { n + 1 } ^ { 2 } } , \\end{align*}"}
-{"id": "5138.png", "formula": "\\begin{align*} \\left ( \\Gamma ^ { \\pm } ( \\psi ) \\right ) _ { m _ j , k _ j } = \\Gamma _ { m _ j , k _ j } ^ { \\pm } ( f _ { m _ j , k _ j } ) \\in \\C . \\end{align*}"}
-{"id": "124.png", "formula": "\\begin{align*} \\sum _ { \\substack { k = 2 \\\\ k \\ , } } ^ { 2 N } \\gamma ^ k S _ k ( B ( 0 ) , B _ 1 ) = \\sum _ { j = 1 } ^ { 2 N - 1 } \\eta _ { j , N } B ( 0 ) ^ { 2 j } , \\end{align*}"}
-{"id": "1351.png", "formula": "\\begin{align*} ( n \\lambda ) \\cdot ( m \\mu ) = \\begin{cases} ( n m ) ( \\lambda \\mu ) \\ , , & s ( \\lambda ) = r ( \\mu ) \\\\ 0 \\ , , & s ( \\lambda ) \\not = r ( \\mu ) \\end{cases} \\end{align*}"}
-{"id": "5558.png", "formula": "\\begin{align*} Q _ R = \\bigcup _ { p \\nmid n } \\bigcup _ i \\{ q _ i ( p ) ^ a : a \\in \\N \\} . \\end{align*}"}
-{"id": "2513.png", "formula": "\\begin{align*} H _ { p } = - { \\displaystyle \\sum \\limits _ { s = 1 } ^ { p } } \\left ( - 1 \\right ) ^ { s } \\frac { 1 } { s } \\dbinom { p } { s } . \\end{align*}"}
-{"id": "9338.png", "formula": "\\begin{align*} \\mathrm { t r a c e } ( A _ { Q _ { 1 } } ) = n - \\| a ^ { T } \\| ^ 2 + 2 \\| a ^ { T } \\| ^ 2 = n + \\| a ^ { T } \\| ^ 2 . \\end{align*}"}
-{"id": "7725.png", "formula": "\\begin{align*} F _ { t } \\colon \\mathrm { L } ^ 2 ( K ; S ) \\to \\mathrm { L } ^ 2 ( M ; S ) , F _ { t } u : = \\sum _ { i = 1 } ^ l F _ { t } ^ i ( \\rho _ { i } \\cdot u ) . \\end{align*}"}
-{"id": "5556.png", "formula": "\\begin{align*} k = k S ^ 0 \\subseteq k S ^ 1 \\subseteq \\cdots k S ^ i \\subseteq k S ^ { i + 1 } \\subseteq \\cdots . \\end{align*}"}
-{"id": "2407.png", "formula": "\\begin{align*} N & : = \\tilde { R } _ { ( i , j ) ( f , b ) } = \\begin{array} { l l l l l l l } & i & p & a & j & f & b \\\\ i & 0 & 0 & 0 & 1 & 1 & 2 \\\\ p & & 0 & 0 & 1 & 1 & 2 \\\\ a & & & 0 & 0 & 0 & 1 \\\\ j & & & & 0 & 0 & 1 \\\\ f & & & & & 0 & 1 \\\\ b & & & & & & 0 \\end{array} \\end{align*}"}
-{"id": "3528.png", "formula": "\\begin{align*} \\mathcal { G } : = \\left \\{ g \\in P G L ( 6 , \\mathbb { Z } ) \\mid \\ , ^ { t } g G g = G , H ( g ) > 0 \\right \\} \\end{align*}"}
-{"id": "3139.png", "formula": "\\begin{align*} \\hat \\lambda _ { i } ( u ) = \\frac { 1 } { \\lambda _ { N - i + 1 } ( u - ( N - i ) c ) ) } \\prod _ { \\ell = 1 } ^ { N - i } \\frac { \\lambda _ { \\ell } ( u - \\ell c ) } { \\lambda _ { \\ell } ( u - ( \\ell - 1 ) c ) } . \\end{align*}"}
-{"id": "5453.png", "formula": "\\begin{align*} \\hat X _ t & = e ^ { t A } \\ , \\hat x _ 0 + \\int _ 0 ^ t e ^ { ( t - s ) A } \\ , b ( \\hat X , \\hat I _ s ) \\ , d s + \\int _ 0 ^ t e ^ { ( t - s ) A } \\ , \\sigma ( \\hat X , \\hat I _ s ) \\ , d \\hat W _ s \\\\ & \\ + \\ , \\int _ 0 ^ t \\int _ { U \\setminus \\{ 0 \\} } e ^ { ( t - s ) A } \\ , \\gamma ( \\hat X , \\hat I _ { s - } , z ) \\ , ( \\hat \\pi ( d s \\ , d z ) - \\lambda _ \\pi ( d z ) \\ , d s ) , 0 \\leq t \\leq T . \\end{align*}"}
-{"id": "5727.png", "formula": "\\begin{align*} y ' ( 0 ) & = \\frac { d } { d x } ( b _ 1 x E _ { \\alpha , 2 } ( \\lambda x ^ { \\alpha } ) ) | _ { x = 0 } + \\frac { d } { d x } \\left ( \\int _ 0 ^ x ( x - t ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } [ \\lambda ( x - t ) ^ { \\alpha } ] f ( t ) d t \\right ) \\Big | _ { x = 0 } \\end{align*}"}
-{"id": "9158.png", "formula": "\\begin{align*} G ( H ) = \\{ l \\in H \\mid \\mbox { $ l $ i s i n v e r t i b l e w i t h $ l ^ { - 1 } = S ( l ) $ a n d $ \\Delta ( l ) = ( l \\otimes l ) ( S \\otimes S ) ( f _ { 2 1 } ^ { - 1 } ) f $ } \\} . \\end{align*}"}
-{"id": "1792.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\left [ P \\left ( \\bigcap _ { j = 1 } ^ { N } \\{ - \\alpha _ { j } - c \\leq X _ { j } \\leq \\alpha _ { j } + c \\} \\right ) \\right ] ^ { 1 / p } \\leq O ( n \\cdot n ^ { - N ( 1 - H ) / p } ) \\to 0 \\end{align*}"}
-{"id": "4302.png", "formula": "\\begin{align*} \\textup { s a t } ( n , K _ 3 , C _ 4 ) = 0 ~ ~ \\mbox { f o r } ~ ~ n \\in \\{ 8 , 9 , \\dots , 2 4 \\} . \\end{align*}"}
-{"id": "635.png", "formula": "\\begin{align*} N ( L _ \\varphi ) = \\left \\{ x \\in G _ m \\ ; \\middle | \\ ; x L _ \\varphi x ^ { - 1 } = L _ \\varphi \\right \\} \\ ; \\end{align*}"}
-{"id": "5642.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\frac { \\sin ( n x ) } { n ^ { 2 m - 1 } } = \\frac { ( - 1 ) ^ { m } ( 2 \\pi ) ^ { 2 m - 1 } } { 2 ( 2 m - 1 ) ! } \\ , B _ { 2 m - 1 } \\left ( \\frac { x } { 2 \\pi } \\right ) \\end{align*}"}
-{"id": "1623.png", "formula": "\\begin{align*} \\frac { \\psi _ r ( x ) - x \\psi _ r ' ( x ) } { S ' ( x ) } = \\int _ 0 ^ x \\psi _ r ( t ) \\theta _ r ( t ) m ' ( t ) d t \\end{align*}"}
-{"id": "2104.png", "formula": "\\begin{align*} x ( t ) = x _ 0 + \\int _ { t _ 0 } ^ t A ( s ) x ( s ) d s + \\int _ { t _ 0 } ^ t C ( s ) x ( s ) d \\omega ( s ) , \\end{align*}"}
-{"id": "3663.png", "formula": "\\begin{align*} \\mu _ i = \\mu \\end{align*}"}
-{"id": "7950.png", "formula": "\\begin{align*} v = \\sum _ { a \\in [ n ] } x _ a = \\sum _ { a \\in [ n ] } y _ a , \\end{align*}"}
-{"id": "1407.png", "formula": "\\begin{align*} r _ j : = p _ j ' ( x _ j ) , \\end{align*}"}
-{"id": "5522.png", "formula": "\\begin{align*} R _ T : = 2 ^ { J [ d + ( d / 2 + \\kappa - 1 ) _ + ] } M _ T \\varepsilon _ T | \\Gamma _ T | _ 2 \\left ( 1 + \\sqrt { \\log ( 1 / ( M _ T \\varepsilon _ T ) ) } + \\sqrt { \\log ( 1 / | \\Gamma _ T | _ 2 ) } \\right ) \\end{align*}"}
-{"id": "5940.png", "formula": "\\begin{align*} c _ \\infty = 1 - \\frac { 1 } { l } + \\sum _ { i = 1 } ^ n \\frac { \\alpha _ i \\bar { \\lambda } _ i } { l } + \\frac { \\beta } { l p } , \\end{align*}"}
-{"id": "8272.png", "formula": "\\begin{align*} A _ 1 = 1 + \\frac { 1 } { 2 \\pi } \\int \\frac { | D _ t Z ( \\alpha , t ) - D _ t Z ( \\beta , t ) | ^ 2 } { ( \\alpha - \\beta ) ^ 2 } d \\beta - \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j } { \\pi } R e \\Big \\{ \\frac { D _ t Z - \\dot { z } _ j } { c _ 0 ^ j ( \\alpha - w _ 0 ^ j ) ^ 2 } \\Big \\} , \\end{align*}"}
-{"id": "9851.png", "formula": "\\begin{align*} & - \\left [ [ ( 1 + t ^ 2 ) ^ r - 1 ] ( 1 - t ) ^ r + t ( 1 - t ) [ ( 1 - t ) ^ r - 1 ] \\right ] \\\\ & = - [ ( 1 + t ^ 2 ) ^ r ( 1 - t ) ^ r - ( 1 - t ) ^ r + t ( 1 - t ) [ ( 1 - t ) ^ r - 1 ] ] \\\\ & = - ( 1 + t ^ 2 ) ^ r ( 1 - t ) ^ r + ( 1 - t ) ^ r - t ( 1 - t ) ^ { r + 1 } + t ( 1 - t ) . \\end{align*}"}
-{"id": "5735.png", "formula": "\\begin{align*} C _ { l , k } = \\frac { \\langle \\varphi ( x , y ) , J _ 0 ( \\gamma _ l x ) \\sin ( k \\pi y ) \\rangle } { | | J _ 0 ( \\gamma _ l x ) \\sin ( k \\pi y ) | | _ 2 ^ 2 } = \\frac { 1 } { | | J _ 0 ( \\gamma _ l x ) \\sin ( k \\pi y ) | | _ 2 ^ 2 } \\int _ 0 ^ 1 \\int _ 0 ^ 1 x \\varphi ( x , y ) J _ 0 ( \\gamma _ l x ) \\sin ( k \\pi y ) d x d y . \\end{align*}"}
-{"id": "127.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { 2 N - 1 } \\eta _ { j , N } & \\le \\sum _ { h = 1 } ^ { \\infty } \\frac { d ^ { 2 h } } { h ! ( h - 1 ) ! } \\left \\{ \\frac 1 N \\sum _ { j = 1 } ^ { 2 N - 1 } ( 1 - x _ j ^ 2 ) ^ { h - 1 } \\left ( 1 + x _ j \\right ) \\right \\} \\ , . \\end{align*}"}
-{"id": "5815.png", "formula": "\\begin{align*} | a _ n | r ^ n + 2 \\Re ( a _ 0 ) \\leq \\frac { 1 } { \\pi } \\int _ 0 ^ { 2 \\pi } | \\Re ( f ( r e ^ { i \\theta } ) ) | + | \\Re ( f ( r e ^ { i \\theta } ) | d \\theta \\leq 4 \\ , \\ , \\underset { | z | = r } { m a x } ( \\Re ( f ( z ) ) , 0 ) . \\end{align*}"}
-{"id": "1387.png", "formula": "\\begin{align*} \\Box _ { n - 1 } ( \\lambda _ 0 , \\ldots , \\lambda _ { m ( J , q ) - 1 } \\lambda _ { m ( J , q ) } , \\ldots , \\lambda _ n ; J ) = \\sum _ { \\varepsilon = 0 } ^ 1 \\Xi ( \\lambda _ 0 , \\ldots , \\lambda _ n ; J , q - \\varepsilon ) \\ ; . \\end{align*}"}
-{"id": "395.png", "formula": "\\begin{align*} S ^ { { \\rm c l } , \\mu } _ { { \\rm L } } ( \\hat g , g ) : = \\int _ M \\big ( | d \\omega | ^ 2 _ { g } + 2 K _ { g } \\omega + 4 \\pi \\mu e ^ { \\omega } \\big ) { \\rm d v } _ { g } , \\end{align*}"}
-{"id": "947.png", "formula": "\\begin{align*} { \\sf M } \\left \\{ \\left ( J ^ { * } [ \\psi ^ { ( 5 ) } ] _ { T , t } - \\sum \\limits _ { j _ 1 , \\ldots , j _ 5 = 0 } ^ { p } C _ { j _ 5 \\ldots j _ 1 } \\zeta _ { j _ 1 } ^ { ( i _ 1 ) } \\ldots \\zeta _ { j _ 5 } ^ { ( i _ 5 ) } \\right ) ^ 2 \\right \\} \\le \\frac { C } { p ^ { 1 - \\varepsilon } } \\end{align*}"}
-{"id": "400.png", "formula": "\\begin{align*} \\tfrac { 1 } { 4 \\pi \\gamma ^ 2 } S ^ { { \\rm c l } , \\mu \\gamma ^ 2 } _ { { \\rm L } } ( \\hat g , g ) : = \\frac { 1 } { 4 \\pi } \\int _ M \\big ( | d \\omega | ^ 2 _ { g } + \\frac { 2 } { \\gamma } K _ { g } \\omega + 4 \\pi \\mu e ^ { \\gamma \\omega } \\big ) { \\rm d v } _ { g } \\end{align*}"}
-{"id": "9468.png", "formula": "\\begin{align*} W ^ { s , m } _ { 0 } ( \\Omega ) & = \\{ u \\in W ^ { s , m } ( \\mathbb { R } ^ N ) ; \\ , \\ , u = 0 \\ \\ \\mathbb { R } ^ N \\backslash \\Omega \\} , \\\\ W ^ { - s , m ' } ( \\Omega ) & = \\left ( W ^ { s , m } ( \\Omega ) \\right ) ^ * , \\ m ' = \\frac { m } { m - 1 } \\ ( \\mbox { d u a l s p a c e } ) . \\end{align*}"}
-{"id": "6215.png", "formula": "\\begin{align*} f ( x ) = \\frac { C } { ( 1 + | | x | | ^ 2 ) ^ p } , \\end{align*}"}
-{"id": "2833.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\displaystyle { x _ 1 ' ( t ) = x _ 1 ( t ) \\biggl ( b _ 1 - \\mu _ 1 x _ 1 ( t ) - a _ { 1 1 } \\int _ 0 ^ { \\infty } K _ { 1 1 } ( s ) x _ 1 ( t - s ) \\ , d s - c _ 1 \\int _ 0 ^ \\infty G _ 1 ( s ) u _ 1 ( t - s ) \\ , d s \\biggr ) } \\\\ \\displaystyle { u _ 1 ' ( t ) = - e _ l u _ 1 ( t ) + d _ 1 x _ 1 ( t ) } \\end{array} \\right . \\end{align*}"}
-{"id": "7972.png", "formula": "\\begin{align*} p ^ { s ( s + 1 ) / 2 } = ( g ( | A | ) ) _ p \\ , \\ , \\big | \\ , \\ , g ( | A | ) \\ , \\ , \\big | \\ , \\ , g ( m ) \\ , , \\end{align*}"}
-{"id": "4383.png", "formula": "\\begin{align*} \\Phi ( s , t ) : = ( s , t , s t ^ 2 ) \\Psi ( s , t ) = ( s , t , s ^ 2 t ^ 2 ) \\end{align*}"}
-{"id": "6351.png", "formula": "\\begin{align*} \\mathbb { P } _ { e , M } = \\int _ 0 ^ \\infty \\int _ 0 ^ \\infty \\mathbb { P } _ { e , M | p } f _ { x _ p } ( x _ p ) f _ { y _ p } ( y _ p ) d x _ p d y _ p , \\end{align*}"}
-{"id": "6266.png", "formula": "\\begin{align*} \\mbox { } x = \\sum _ { g \\in S } \\pi _ g ( x ) , \\pi _ h ( x ) = 0 , . \\end{align*}"}
-{"id": "5971.png", "formula": "\\begin{align*} \\widetilde { W } ( t ) : = X ( t ) - \\int _ 0 ^ t \\widetilde { q } ( s , X ) d s \\end{align*}"}
-{"id": "2540.png", "formula": "\\begin{align*} \\frac { \\int _ E u ^ { \\alpha ( x ) } t ^ { \\alpha ( x ) } \\rho ( d x ) } { \\int _ E t ^ { \\alpha ( x ) } \\rho ( d x ) } \\leq \\frac { \\int _ E u ^ { \\alpha _ 0 } t ^ { \\alpha ( x ) } \\rho ( d x ) } { \\int _ E t ^ { \\alpha ( x ) } \\rho ( d x ) } = u ^ { \\alpha _ 0 } , t \\in ( 0 , \\infty ) . \\end{align*}"}
-{"id": "4055.png", "formula": "\\begin{align*} w = x ^ { n + 1 } + . \\end{align*}"}
-{"id": "6714.png", "formula": "\\begin{align*} \\varphi ^ { ( 2 ) } _ k = \\dot { \\varphi } ^ { ( 1 ) } _ k = 0 . \\end{align*}"}
-{"id": "1907.png", "formula": "\\begin{align*} f ( x ) \\cos ( \\kappa _ j t ) \\stackrel { ? } { = } \\frac { 1 } { 2 } \\left ( \\frac { \\xi L } { 2 } \\left \\{ \\frac { 1 } { Z _ j ^ 2 } + 2 \\sum _ { n = 1 } ^ \\infty \\frac { \\cos [ k _ n ( x - t ) ] } { Z _ j ^ 2 - ( \\pi n ) ^ 2 } \\right \\} + \\frac { \\xi L } { 2 } \\left \\{ \\frac { 1 } { Z _ j ^ 2 } + 2 \\sum _ { n = 1 } ^ \\infty \\frac { \\cos [ k _ n ( x + t ) ] } { Z _ j ^ 2 - ( \\pi n ) ^ 2 } \\right \\} \\right ) , \\end{align*}"}
-{"id": "6048.png", "formula": "\\begin{align*} d e g _ { H ( w ) } ( x ' + \\lambda m , y ' ) = \\sum _ { \\gamma \\in \\Gamma } d e g _ { H ^ * + \\gamma } ( x ' + \\lambda m , y ' ) . \\end{align*}"}
-{"id": "9190.png", "formula": "\\begin{align*} \\mathrm { i n d e x } \\left ( \\gamma \\right ) & \\geq \\ell - k + 1 - \\left ( n - 1 - \\ell \\right ) \\\\ & = 2 \\ell - n - k + 2 , \\end{align*}"}
-{"id": "2383.png", "formula": "\\begin{align*} & h ^ { \\otimes k } : \\mathfrak { A } ^ { \\otimes ^ g k } \\to \\mathfrak { B } ^ { \\otimes ^ g k } \\\\ & h ^ { \\otimes k } ( a _ 1 \\otimes \\cdots \\otimes a _ k ) = h ( a _ 1 ) \\otimes \\cdots \\otimes h ( a _ k ) \\end{align*}"}
-{"id": "347.png", "formula": "\\begin{align*} ( \\frac { 1 } { 2 } n ( n - 1 ) , k ) \\cdot ( n ( n + 1 ) , k ) = ( \\frac { 1 } { 2 } n ( n - 1 ) , l ) \\cdot ( n ( n + 1 ) , l ) . \\end{align*}"}
-{"id": "9956.png", "formula": "\\begin{align*} h = ( a x + i y ) ^ 2 + \\frac 1 3 ( 1 - a ^ 2 ) ( x ^ 2 + y ^ 2 + z ^ 2 ) \\end{align*}"}
-{"id": "731.png", "formula": "\\begin{align*} X _ { N } \\left ( x _ { 1 } , \\dots , x _ { N } \\right ) = \\det \\left ( K \\left ( x _ { i } , x _ { j } \\right ) \\right ) _ { i , j = 1 } ^ { N } \\end{align*}"}
-{"id": "8942.png", "formula": "\\begin{align*} q p _ { b l } ( z , w ) = q p _ { b l } ( T z , T w ) & \\geq a _ 1 [ q p _ { b l } ( z , w ) + q p _ { b l } ( w , z ) ] + a _ 2 [ q p _ { b l } ( z , T z ) + \\\\ & q p _ { b l } ( T z , z ) ] + a _ 3 [ q p _ { b l } ( w , T w ) + q p _ { b l } ( T w , w ) ] \\\\ & \\quad + a _ 4 [ q p _ { b l } ( z , T w ) + q p _ { b l } ( T w , z ) ] \\\\ & = ( a _ 1 + a _ 4 ) [ q p _ { b l } ( z , w ) + q p _ { b l } ( w , z ) ] + 2 a _ 2 q p _ { b l } ( z , z ) \\end{align*}"}
-{"id": "4632.png", "formula": "\\begin{align*} ( U \\otimes \\mathbb { I } _ n ) = \\left [ \\begin{array} { c | c | c } U _ { 1 1 } \\mathbb { I } _ n & \\dots & U _ { 1 \\ell } \\mathbb { I } _ n \\\\ \\hline \\vdots & \\ddots & \\vdots \\\\ \\hline U _ { \\ell 1 } \\mathbb { I } _ n & \\dots & U _ { \\ell \\ell } \\mathbb { I } _ n \\end{array} \\right ] . \\end{align*}"}
-{"id": "6916.png", "formula": "\\begin{align*} \\hat A = \\hat A ( t ) = \\begin{pmatrix} a _ { 1 1 } & a _ { 1 2 } \\\\ a _ { 2 1 } & a _ { 2 2 } \\end{pmatrix} \\in \\mathbb { S L } ( 2 ) \\end{align*}"}
-{"id": "2831.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l } \\displaystyle { U _ 2 ( t ) = \\sum _ { i = 1 } ^ p \\eta _ i \\biggl ( \\sum _ { j = 1 } ^ p \\frac { | { a } _ { i j } | } { 2 } \\int _ 0 ^ { \\infty } K _ { i j } ( s ) \\int _ { t - s } ^ t ( x _ j ( u ) - x _ j ^ * ) ^ 2 d u d s } \\\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\displaystyle { + \\frac { c _ i } { 2 \\varepsilon } \\int _ 0 ^ { \\infty } G _ { i } ( s ) \\int _ { t - s } ^ t ( u _ i ( u ) - u _ i ( t ) ) ^ 2 d u d s \\biggr ) } . \\end{array} \\right . \\end{align*}"}
-{"id": "7788.png", "formula": "\\begin{align*} ( c L ( G ) ) ^ d | B ^ 2 | \\le | G | = 1 , \\end{align*}"}
-{"id": "6202.png", "formula": "\\begin{align*} f ( x ) = \\frac { a } { 2 } e ^ { - a | x | } \\end{align*}"}
-{"id": "4340.png", "formula": "\\begin{align*} \\alpha ( t ) = e ^ { 2 \\pi i d \\alpha ( x ) } = e ^ { \\pi i n } \\in \\{ \\pm 1 \\} . \\end{align*}"}
-{"id": "6089.png", "formula": "\\begin{align*} f ( x ) = ( x - a ^ 1 _ j ) ^ { e ^ 1 _ j - 1 } \\ldots ( x - a ^ { k _ j } _ j ) ^ { e ^ { k _ j } _ j - 1 } \\Big ( ( x - a ^ 1 _ j ) \\ldots ( x - a ^ { k _ j } _ j ) h ' _ j ( x ) + \\\\ e ^ 1 _ j ( \\widehat { x - a ^ 1 _ j } ) ( x - a ^ 2 _ j ) \\ldots ( x - a ^ { k _ j } _ j ) h _ j ( x ) + \\\\ e ^ 2 _ j ( x - a ^ 1 _ j ) ( \\widehat { x - a ^ 2 _ j } ) \\ldots ( x - a ^ { k _ j } _ j ) h _ j ( x ) + \\\\ e ^ { k _ j } _ j ( x - a ^ 1 _ j ) \\ldots ( x - a ^ { k _ j - 1 } _ j ) ( \\widehat { x - a ^ { k _ j } _ j } ) \\Big ) \\\\ \\ , \\ , 1 \\leq j \\leq r \\end{align*}"}
-{"id": "8265.png", "formula": "\\begin{align*} A : = ( a h _ { \\alpha } ) \\circ h ^ { - 1 } . \\end{align*}"}
-{"id": "3533.png", "formula": "\\begin{align*} \\begin{aligned} & \\Gamma ( 1 + i ) : = \\left \\{ g \\mid g \\equiv E _ { 4 } \\right \\} , \\\\ & \\Gamma _ { T } ( 1 + i ) : = \\Gamma ( 1 + i ) \\rtimes \\left \\langle T \\right \\rangle . \\end{aligned} \\end{align*}"}
-{"id": "6586.png", "formula": "\\begin{align*} 0 = \\bar { \\Phi } ( \\rho , \\theta _ i ) - \\bar { \\Phi } ( 0 , \\theta _ i ) = \\int _ 0 ^ { \\rho } \\partial _ { r _ i } \\bar { \\Phi } ( r _ i , \\theta _ i ) \\ , d r _ i = \\int _ 0 ^ { \\rho } \\left ( \\partial _ { r _ i } \\bar { S } _ i \\bar { \\psi } ^ { ( i ) } + r _ i \\partial _ { r _ i } ^ 2 \\bar { S } _ i \\bar { \\psi } ^ { ( i ) } + r _ i \\partial _ { r _ i } \\bar { S } _ i \\partial _ { r _ i } \\bar { \\psi } ^ { ( i ) } \\right ) \\ , d r _ i . \\end{align*}"}
-{"id": "4824.png", "formula": "\\begin{align*} \\mathcal L _ t ' \\varphi ( t ) + \\mathcal L _ t \\varphi ' ( t ) = \\lambda _ 1 ' ( \\mathcal L _ t , g ( t ) ) \\varphi ( t ) + \\lambda _ 1 ( \\mathcal L _ t , g ( t ) ) \\varphi ' ( t ) \\end{align*}"}
-{"id": "5828.png", "formula": "\\begin{align*} \\phi _ { \\bar X _ n } ( t _ 1 ) = \\prod _ { i = 1 } ^ n \\int e ^ { i t _ 1 x _ i / n } f ( x _ i ) d x _ i = [ \\phi _ { X } ( t _ 1 / n ) ] ^ n \\end{align*}"}
-{"id": "4236.png", "formula": "\\begin{align*} \\operatorname * { c u r l } ( a ( x ) \\lvert \\operatorname * { c u r l } u \\rvert ^ { p - 2 } \\operatorname * { c u r l } u ) = f & & \\Omega . \\end{align*}"}
-{"id": "3972.png", "formula": "\\begin{align*} c _ { \\phi _ m } ( \\frac { r ^ 2 } { 4 m } , r ) = a _ { [ f ] } \\left ( \\begin{pmatrix} a s ^ 2 & a b s \\\\ a b s & a b ^ 2 \\end{pmatrix} \\right ) . \\end{align*}"}
-{"id": "8588.png", "formula": "\\begin{align*} & \\int ^ x d t \\ , \\sinh ( a t + b ) \\sinh ( a t + c ) \\\\ & = - ( t / 2 ) \\cosh ( b - c ) + ( 4 a ) ^ { - 1 } \\sinh ( 2 a t + b + c ) + C . \\end{align*}"}
-{"id": "1037.png", "formula": "\\begin{align*} & \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) = ( n / 4 ) ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } \\times \\\\ & \\mathbb { P } ( \\xi ^ { ( n ) } ( I _ { n , k } ) = 0 , \\ \\xi ^ { ( n ) } ( J _ { n , k , j } ) > 0 , \\ \\forall \\ 1 \\leq j \\leq k ) . \\end{align*}"}
-{"id": "9391.png", "formula": "\\begin{align*} \\varphi _ { 1 } ( t w _ { 0 } ) & \\leq \\Lambda ^ { 2 } t ^ { p } [ w _ { 0 } ] _ { s , p } ^ { p } - t ^ { p _ { s } ^ { \\ast } } \\int _ \\Omega | w _ { 0 } | ^ { p ^ { \\ast } _ { s } } d x - \\lambda c _ { 3 } \\int _ { \\Omega } | t w _ { 0 } | ^ { q } d x \\\\ & \\leq A _ { 2 } t ^ { p } - B t ^ { p _ { s } ^ { \\ast } } - E _ { 2 } t ^ { q } \\end{align*}"}
-{"id": "7680.png", "formula": "\\begin{align*} \\label [ p r o p e r t y ] { C l i f f o r d M u l t } \\varepsilon _ { i } \\varepsilon _ { j } + \\varepsilon _ { j } \\varepsilon _ { i } = 0 i \\neq j , \\varepsilon _ { i } ^ * = - \\varepsilon _ { i } , \\ ; \\ ; \\varepsilon _ { i } ^ 2 = - 1 . \\end{align*}"}
-{"id": "3575.png", "formula": "\\begin{align*} \\begin{matrix} \\begin{aligned} & C _ { 0 } = \\mathrm { C o n e } \\left \\{ ( 0 , 1 , 0 , 0 ) , ( 1 , 0 , 0 , 1 ) , ( 0 , 0 , 1 , 0 ) , ( 1 , 0 , 1 , 0 ) , ( 0 , 1 , 0 , 1 ) \\right \\} , \\\\ & C _ { N E } = { \\rm C o n e } \\left \\{ \\begin{matrix} ( 1 , 0 , 0 , 0 ) , ( 0 , 1 , - 1 , 1 ) , ( 1 , - 1 , 1 , 0 ) , ( 0 , 0 , 0 , 1 ) \\\\ ( 0 , 0 , 1 , - 1 ) , ( - 1 , 1 , 0 , 0 ) \\end{matrix} \\right \\} . \\end{aligned} \\end{matrix} \\end{align*}"}
-{"id": "2801.png", "formula": "\\begin{align*} \\mathcal { R } _ { \\mathrm { s t a t } } ( \\alpha ) = \\mathcal { G } _ { \\mathrm { s t a t } } \\left ( C ( \\alpha ) \\right ) = \\prod _ { r = 0 } ^ { \\infty } \\left ( 1 - p q ^ r + p q ^ r e ^ { C ( \\alpha ) } \\right ) . \\end{align*}"}
-{"id": "6076.png", "formula": "\\begin{align*} Y '' = \\frac { 1 } { n - m - 1 } \\left ( \\psi ' + \\frac { \\psi ^ 2 } { n - m - 1 } \\right ) Y \\leq - \\frac { R i c _ { n - m - 1 } ( \\dot \\gamma , \\mathcal V _ t ) } { n - m - 1 } Y . \\end{align*}"}
-{"id": "8280.png", "formula": "\\begin{align*} d _ I ( t ) : = \\min _ { 1 \\leq j \\leq N } \\{ d ( z _ j ( t ) , \\Sigma ( t ) ) \\} , d _ P ( t ) : = \\min _ { \\substack { 1 \\leq i , j \\leq N \\\\ i \\neq j } } \\{ d ( z _ i ( t ) , z _ j ( t ) ) \\} . \\end{align*}"}
-{"id": "2115.png", "formula": "\\begin{align*} | \\lambda _ k ( \\omega ) | \\leq \\eta \\Big [ 2 + \\Big ( \\frac { 2 M _ 0 } { \\mu } \\Big ) ^ { p } ( 1 + \\Gamma _ p ( \\omega ) ) \\Big ] , \\forall k = 1 , \\dots , d , \\end{align*}"}
-{"id": "8073.png", "formula": "\\begin{align*} n _ 1 = n _ 2 = n _ 3 = 0 , n _ 4 = 2 , m _ 1 = 2 . \\end{align*}"}
-{"id": "958.png", "formula": "\\begin{align*} \\| g _ { m _ j } \\| \\ge \\| \\nabla f ( x _ j ) \\| - \\frac { \\kappa _ { e g } } { \\mu _ j ^ 2 } \\ge \\epsilon ' - \\frac { \\epsilon ' } { 2 } = \\frac { \\epsilon ' } { 2 } . \\end{align*}"}
-{"id": "9402.png", "formula": "\\begin{align*} \\gamma z : = \\frac { a z + b } { c z + d } . \\end{align*}"}
-{"id": "992.png", "formula": "\\begin{align*} & = 1 + ( d - 1 ) ( 1 - t ) ^ { d - 1 } t + ( 1 - t ) ^ { d - 1 } - 1 - d t ^ { r } ( 1 - t ) ^ { d - 1 } \\\\ & = ( 1 - t ) ^ { d - 1 } [ 1 + ( d - 1 ) t - d t ^ r ] \\\\ & = ( 1 - t ) ^ { d - 1 } [ ( 1 - t ^ r ) + t ( d - 1 ) ( 1 - t ^ { r - 1 } ) ] \\\\ & = ( 1 - t ) ^ { d - 1 } [ ( 1 - t ) ( 1 + \\cdots + t ^ { r - 1 } ) + t ( d - 1 ) ( 1 - t ) ( 1 + \\cdots + t ^ { r - 2 } ) ] \\\\ & = ( 1 - t ) ^ d [ 1 + d t + d t ^ 2 + \\cdots + d t ^ { r - 1 } ] . \\end{align*}"}
-{"id": "6269.png", "formula": "\\begin{align*} h \\ast ( h _ 1 \\otimes \\cdots \\otimes h _ n ) = \\sum ( h _ { ( 1 ) } h _ 1 ) \\otimes \\cdots \\otimes ( h _ { ( n ) } h _ n ) . \\end{align*}"}
-{"id": "8292.png", "formula": "\\begin{align*} g _ 2 : = \\frac { i } { \\pi } \\sum _ { j = 1 } ^ N \\lambda _ j \\Big ( \\frac { 2 z _ { t t } + i - \\partial _ t ^ 2 z _ j } { ( z ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } - 2 \\frac { ( z _ t - \\dot { z } _ j ( t ) ) ^ 2 } { ( z ( \\alpha , t ) - z _ j ( t ) ) ^ 3 } \\Big ) . \\end{align*}"}
-{"id": "1632.png", "formula": "\\begin{align*} \\omega ( f , g ) : = - i \\int _ 0 ^ 1 \\det \\begin{pmatrix} f _ 1 & g _ 1 \\\\ f _ 2 & g _ 2 \\end{pmatrix} \\ , d x \\end{align*}"}
-{"id": "9812.png", "formula": "\\begin{align*} \\bar { v } ( x ' , x _ n , y ) = \\int _ { \\R ^ { n - 1 } } v ( z ' ) P _ a ( x ' - z ' , x _ n , y ) \\ , \\d z ' . \\end{align*}"}
-{"id": "2263.png", "formula": "\\begin{align*} a & = \\begin{pmatrix} 1 \\\\ - \\frac { \\mathcal { P } } { x - \\Omega } | E \\rangle \\end{pmatrix} & b & = \\begin{pmatrix} 0 \\\\ | \\delta _ x \\rangle \\end{pmatrix} \\end{align*}"}
-{"id": "9244.png", "formula": "\\begin{align*} M _ 2 ( R ) = \\sup _ { B _ R \\cap \\Omega } | D ^ 2 u | , \\end{align*}"}
-{"id": "7774.png", "formula": "\\begin{align*} g ( f ) ( x ) = \\left ( \\int _ { 0 } ^ { \\infty } t \\Big | \\frac { \\rm d } { { \\rm d } t } P _ t f ( x ) \\Big | ^ 2 { \\rm d } t \\right ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "4311.png", "formula": "\\begin{align*} V _ k ( F ) : = \\big \\{ v \\in H ^ 1 ( F ) : \\Delta _ { F } v \\in \\mathbb { P } _ { k - 2 } ( F ) , v | _ { \\partial F } \\in B _ k ( \\partial F ) \\big \\} , \\end{align*}"}
-{"id": "170.png", "formula": "\\begin{align*} X = \\sum _ { i = 1 } ^ N v ^ i ( { \\bf x } ) \\frac { \\partial } { \\partial x ^ i } , Y = \\sum _ { i = 1 } ^ N w ^ i ( { \\bf x } ) \\frac { \\partial } { \\partial x ^ i } , \\end{align*}"}
-{"id": "8753.png", "formula": "\\begin{align*} \\begin{aligned} { } _ H ^ F { u } _ { n , n _ 0 } ^ { ( \\alpha , \\sigma ) } = e ^ { - n _ 0 \\tau \\sigma } ( 1 + \\lambda _ j \\tau ) ^ { - ( n _ 0 + 1 ) \\tau } \\sum _ { j = 0 } ^ { Q - 1 } w _ j y ^ { ( j ) } _ { n - n _ 0 } , \\end{aligned} \\end{align*}"}
-{"id": "9211.png", "formula": "\\begin{align*} e ( G ' ) & = e ( S ) + e ( \\bar { S } ) + e ( \\bar { S } , S ) \\\\ & = e ( S ) + \\frac { 1 } { 2 } \\sum _ { x \\in \\bar { S } } \\left ( d _ S ( x ) + d _ { G ' } ( x ) \\right ) \\\\ & \\leq \\binom { s } { 2 } + \\frac { 1 } { 2 } \\left ( s - 1 + \\left \\lceil \\frac { k } { 2 } \\right \\rceil - 1 \\right ) ( n - s ) . \\end{align*}"}
-{"id": "203.png", "formula": "\\begin{align*} \\pi _ { T M , X _ { 1 , V } , Y _ { 1 , V } , X _ { 2 , V } , Y _ { 2 , V } , c } ( { \\bf x } , { \\bf y } ) = \\left ( \\begin{array} { c c c | c c c } 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & x ^ 1 \\\\ 0 & 0 & 0 & 0 & - x ^ 1 & 0 \\\\ \\hline 0 & 0 & 0 & 0 & 0 & - x ^ 2 \\\\ 0 & 0 & - x ^ 1 & 0 & 0 & y ^ 1 + x ^ 1 \\\\ 0 & - x ^ 1 & 0 & x ^ 2 & - y ^ 1 - x ^ 1 & 0 \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "4103.png", "formula": "\\begin{align*} \\mathbb K ^ { ( \\alpha , \\frac { 1 } { 2 } ) } ( x , y ) = \\frac { - 1 } { x - y } \\sum _ { j = 0 } ^ 2 \\sum _ { i = 0 } ^ { 2 - j } ( - 1 ) ^ j a _ { i + j } \\left ( \\vartheta _ x ^ j f ( x ) \\right ) \\left ( \\vartheta _ y ^ i g ( y ) \\right ) \\end{align*}"}
-{"id": "6825.png", "formula": "\\begin{align*} \\det ( \\sigma L ) = \\big ( w ^ 1 _ { 0 1 } \\xi _ 1 - w ^ 1 _ { 1 0 } \\xi _ 2 ) ^ 2 + \\big ( w ^ 2 _ { 0 1 } \\xi _ 1 - w ^ 2 _ { 1 0 } \\xi _ 2 ) ^ 2 \\ne 0 \\end{align*}"}
-{"id": "1347.png", "formula": "\\begin{align*} \\pi ( x ) \\cdot \\pi _ * ^ { d ( x ) } ( f ) ( y ) & = \\pi _ * ^ { d ( x ) } ( f ) \\left ( \\left ( \\pi ( x ) \\right ) ^ { - 1 } y \\right ) \\\\ & = \\sum _ { z \\in \\pi ^ { - 1 } \\left ( \\left ( \\pi ( x ) \\right ) ^ { - 1 } y \\right ) } f ( z ) \\\\ & = \\sum _ { z \\in x ^ { - 1 } \\cdot \\pi ^ { - 1 } \\left ( y \\right ) } f ( z ) \\\\ & = \\sum _ { z ' \\in \\pi ^ { - 1 } \\left ( y \\right ) } f ( x ^ { - 1 } z ' ) \\\\ & = \\pi _ * ^ { r ( x ) } ( x \\cdot f ) ( y ) \\end{align*}"}
-{"id": "4266.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\delta ( a ( x ) \\lvert d u \\rvert ^ { p - 2 } d u ) & = \\delta F & & \\Omega , \\\\ \\delta u & = 0 & & \\Omega , \\end{aligned} \\right . \\end{align*}"}
-{"id": "5340.png", "formula": "\\begin{align*} ( \\lambda + k + 1 ) ( \\lambda - ( k + 1 ) ) = \\lambda ^ 2 - ( k + 1 ) ^ 2 \\in \\left ( - \\infty , 0 \\right ) , \\quad \\forall k . \\end{align*}"}
-{"id": "6897.png", "formula": "\\begin{align*} N _ { i j } = & y _ { i ; j } - y _ { j ; i } = u ^ \\ell _ { ; i } B _ { \\ell k } u ^ k _ { ; j } - u ^ \\ell _ { ; j } B _ { \\ell k } u ^ k _ { ; i } \\\\ = & B _ { \\ell k } \\big ( u ^ \\ell _ { ; i } u ^ k _ { ; j } - u ^ \\ell _ { ; j } u ^ k _ { ; i } \\big ) = u ^ \\ell _ { ; i } \\big ( B _ { \\ell k } - B _ { k \\ell } \\big ) u ^ k _ { ; j } \\ , . \\end{align*}"}
-{"id": "4874.png", "formula": "\\begin{align*} G _ r ( \\Gamma ) = \\left \\{ z \\in \\mathbb { C } ^ n \\ , ; \\ , | z | > r , \\frac { z } { | z | } \\in \\Gamma \\right \\} \\cup \\Gamma \\infty , \\end{align*}"}
-{"id": "3517.png", "formula": "\\begin{align*} \\hat \\theta _ \\varnothing & = \\d u - p _ a \\d x ^ a \\\\ \\hat \\theta _ a & = \\d p _ a - p _ { a b } \\d x ^ b , \\end{align*}"}
-{"id": "2192.png", "formula": "\\begin{align*} \\dot { P } ( t ) + P ( t ) A ( t ) + A ^ { T } ( t ) P ( t ) + Q ( t ) - P ( t ) B ( t ) R ^ { - 1 } ( t ) B ^ { T } ( t ) P ( t ) = 0 \\end{align*}"}
-{"id": "8547.png", "formula": "\\begin{align*} \\tilde { \\theta } = \\frac { 1 } { 2 } ( I + \\mathbb { H } ) \\tilde { \\theta } + \\frac { 1 } { 2 } ( I - \\mathbb { H } ) \\tilde { \\theta } \\end{align*}"}
-{"id": "71.png", "formula": "\\begin{align*} \\sigma ^ + _ 1 + \\sigma ^ + _ { - 1 } = \\sigma _ 1 ^ - + \\sigma ^ - _ { - 1 } \\ , . \\end{align*}"}
-{"id": "7154.png", "formula": "\\begin{align*} u \\left ( t , x \\right ) = \\int _ { 0 } ^ { t } \\left [ L ^ { \\nu } u \\left ( r , x \\right ) - \\lambda u \\left ( r , x \\right ) \\right ] d r , ~ \\left ( t , x \\right ) \\in \\left [ 0 , T \\right ] \\times \\mathbf { R } ^ { d } . \\end{align*}"}
-{"id": "2248.png", "formula": "\\begin{align*} & | \\alpha _ \\pm \\rangle = \\frac { 1 } { \\sqrt { \\langle h | ( \\pm z _ 0 - \\Omega ) ^ { - 2 } | g \\rangle } } \\frac { 1 } { \\pm z _ 0 - \\Omega } | g \\rangle & & \\langle \\alpha _ \\pm ' | = \\frac { 1 } { \\sqrt { \\langle h | ( \\pm z _ 0 - \\Omega ) ^ { - 2 } | g \\rangle } } \\langle h | \\frac { 1 } { \\pm z _ 0 - \\Omega } \\end{align*}"}
-{"id": "1899.png", "formula": "\\begin{align*} - \\frac { \\pi } { 6 L ^ 2 } + \\frac { \\mathcal { B - C } } { L ^ 2 } + \\frac { \\mathcal { C } } { L ^ 2 } \\sum _ { n = - \\infty } ^ \\infty \\delta \\left ( \\frac { t \\pm x } { L } - n \\right ) > 0 . \\end{align*}"}
-{"id": "1572.png", "formula": "\\begin{align*} \\psi _ w ( x \\pm i ) & = \\frac { \\zeta ^ - ( w ) \\cdot e ^ { 2 \\pi i x } e ^ { \\mp 2 \\pi } - \\zeta ^ + ( w ) } { e ^ { 2 \\pi i x } e ^ { \\mp 2 \\pi } - 1 } \\\\ & = \\mp \\frac { e ^ { 2 \\pi i x } e ^ { \\mp 2 \\pi } + 1 } { e ^ { 2 \\pi i x } e ^ { \\mp 2 \\pi } - 1 } \\zeta ^ \\pm ( w ) + o ( w ) \\\\ \\end{align*}"}
-{"id": "9150.png", "formula": "\\begin{align*} \\overline { \\Theta } ^ { ( N ) } ( \\overline { \\mathbf { r } } ) \\equiv \\prod \\limits _ { i = 1 , N } \\overline { \\Theta } _ { i } ( \\overline { \\mathbf { r } } ) \\overline { \\Theta } _ { i } ^ { ( \\partial \\Omega ) } ( \\overline { \\mathbf { r } } ) . \\end{align*}"}
-{"id": "9811.png", "formula": "\\begin{align*} P _ a ( x ' , x _ n , y ) = C _ { n , a } \\frac { ( x _ n ^ 2 + y ^ 2 ) ^ { - \\frac { a } { 2 } } } { ( | x ' | ^ 2 + x _ n ^ 2 + y ^ 2 ) ^ { \\frac { n - 1 - a } { 2 } } } . \\end{align*}"}
-{"id": "760.png", "formula": "\\begin{align*} C _ { \\C { H } } = B B ^ * = W M _ { \\gamma } W ^ * \\end{align*}"}
-{"id": "157.png", "formula": "\\begin{align*} & [ X _ { C } , Y _ { C } ] = [ X , Y ] _ { C } , \\\\ & [ X _ { C } , Y _ { V } ] = [ X , Y ] _ { V } , \\\\ & [ X _ { V } , Y _ { V } ] = 0 . \\end{align*}"}
-{"id": "9530.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( \\frac { b } { a } ) _ { n } ( q ) _ { n } ( a ) _ { N - n } a ^ { n } } { ( 1 - c q ^ { n } ) ( b ) _ n ( a ) _ N } = \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( \\frac { b } { c } ) _ { n - 1 } ( q ) _ n ( c q ) _ { N - n } c ^ { n - 1 } } { ( b ) _ { n - 1 } ( c q ) _ N } \\left ( \\frac { a q ^ { n - 1 } } { 1 - a q ^ { n - 1 } } - \\frac { b q ^ { n - 1 } } { 1 - b q ^ { n - 1 } } \\right ) . \\end{align*}"}
-{"id": "8820.png", "formula": "\\begin{align*} \\min _ { y \\in \\mathbb R ^ N } f ^ * ( - A ^ \\top y ) + \\sum _ { i = 1 } ^ n g ^ * _ i ( y ^ { ( i ) } ) \\ ; . \\end{align*}"}
-{"id": "2694.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y } \\Re { f _ { \\tau , \\theta } ( r + i y ) } & = \\frac { 2 y \\big ( r - \\frac { 1 } { 2 } ( \\tau + \\frac { 1 } { \\tau } ) \\big ) \\big ( \\frac { 1 } { \\tau } - \\tau \\big ) } { \\big ( ( r - \\frac { 1 } { \\tau } ) ^ { 2 } + y ^ { 2 } \\big ) \\big ( ( r - \\tau ) ^ { 2 } + y ^ { 2 } \\big ) } , \\end{align*}"}
-{"id": "5589.png", "formula": "\\begin{align*} \\begin{cases} ( \\partial _ t + \\partial _ x ^ 3 ) \\mathcal { K } w ( x , t ) = w ( x , t ) & ( x , t ) \\in \\mathbb { R } \\times \\mathbb { R } , \\\\ \\mathcal { K } w ( x , t ) = 0 & x \\in \\mathbb { R } . \\end{cases} \\end{align*}"}
-{"id": "9102.png", "formula": "\\begin{align*} \\begin{aligned} | I _ 2 | & \\le \\left | \\Upsilon \\left ( \\dfrac { x } 2 \\right ) \\right | ^ { p - 1 } \\int _ { \\left \\{ \\frac { | x | } { 2 } \\le | y | \\le \\frac { 3 } { 2 } | x | \\right \\} \\setminus B _ { \\frac { | x | } 2 } ( x ) } \\dfrac { 1 } { | x - y | ^ { N + p s } } d y \\le \\dfrac { C ( N , s , p , C _ 1 , \\alpha ) } { | x | ^ { \\alpha ( p - 1 ) + p s } } . \\end{aligned} \\end{align*}"}
-{"id": "6970.png", "formula": "\\begin{align*} \\sigma ^ * \\theta _ 1 = \\eta . \\end{align*}"}
-{"id": "7811.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | [ \\langle x , y \\rangle \\xi _ { n } , \\xi _ { n } ] | = \\left \\| x \\right \\| \\left \\| y \\right \\| . \\end{align*}"}
-{"id": "9664.png", "formula": "\\begin{align*} \\operatorname { R e } \\theta _ 1 \\overline \\theta _ 2 \\overline \\theta _ 0 = 0 \\ \\tau . \\end{align*}"}
-{"id": "9658.png", "formula": "\\begin{align*} ( Y _ 0 \\oplus I _ { H ^ 2 _ - } ) U _ { \\mathbb T } = T ( Y _ 0 \\oplus I _ { H ^ 2 _ - } ) , \\end{align*}"}
-{"id": "1178.png", "formula": "\\begin{align*} | g '' ( t ) | = \\frac { 2 } { 2 \\pi } \\left | \\int _ { | \\zeta | = \\rho } \\frac { g ( \\zeta + t ) } { \\zeta ^ 3 } d t \\right | \\leq \\frac { 2 } { \\rho ^ 2 } \\sup _ { | \\zeta | = \\rho } | g ( \\zeta + t ) | \\leq \\frac { 2 } { \\rho ^ 2 } \\sup _ { | \\lambda - 1 / 2 | \\leq \\rho + 1 / 2 } | g ( \\lambda ) | . \\end{align*}"}
-{"id": "3995.png", "formula": "\\begin{align*} h _ { \\mu } ( \\tau ) = \\sum _ n a _ { \\mu } ( n - T ^ { - 1 } [ \\mu / 2 ] ) e ^ { 2 \\pi i ( n - T ^ { - 1 } [ \\mu / 2 ] ) \\cdot \\tau } \\end{align*}"}
-{"id": "1705.png", "formula": "\\begin{align*} \\mbox { F o r } \\mu \\mbox { - a . e . } x \\in A _ 0 , \\ \\ \\ \\varphi ( x ) = \\int \\varphi ^ + ( y ) \\ P ( x , d y ) \\geq \\int \\varphi ( y ) \\ P ( x , d y ) , \\end{align*}"}
-{"id": "4288.png", "formula": "\\begin{align*} \\binom { s - 2 } { r } + \\binom { s - 3 } { r - 1 } + ( n - s + 1 ) \\left ( \\binom { s - 2 } { r - 1 } + \\binom { s - 3 } { r - 2 } \\right ) \\end{align*}"}
-{"id": "1584.png", "formula": "\\begin{align*} \\Phi _ { \\widetilde { F } } ( z _ 0 , x _ 0 ) & = \\Phi _ { \\widetilde { F } } \\circ U \\circ \\boldsymbol { F } _ { n ^ 2 + \\kappa _ 0 , n ^ 2 } ( z , x ) \\\\ & = \\Phi _ F \\circ \\boldsymbol { F } _ { n ^ 2 + \\kappa _ 0 , n ^ 2 } ( z , x ) \\\\ & = \\Phi _ F \\circ F ^ { \\kappa _ 0 } ( z , x ) + o ( 1 ) \\\\ & = \\Phi _ F ( z , x ) + \\kappa _ 0 + o ( 1 ) \\end{align*}"}
-{"id": "245.png", "formula": "\\begin{align*} \\Delta _ { \\mathbb { H } } u = V u \\end{align*}"}
-{"id": "9000.png", "formula": "\\begin{align*} \\begin{aligned} & i = i _ 1 , j = i _ 2 , t _ 1 = T \\gamma _ g ( 2 \\tau + 1 ) , t _ 2 = T \\gamma _ g ( 2 \\tau + 1 ) + 2 \\tau + 1 , \\\\ & \\alpha _ d = \\frac { h \\psi ( X [ 0 ] + \\rho ) ( 1 - h n ^ { \\infty } \\kappa ) ^ { 3 \\tau + 1 } } { 1 + n ^ { \\infty } \\kappa } \\mbox { a n d } \\underline { \\psi } : = \\psi ( X [ 0 ] + \\rho ) \\end{aligned} \\end{align*}"}
-{"id": "8958.png", "formula": "\\begin{align*} \\Psi _ X ( U ) ( s ) = \\mathbb { E } \\left [ \\int _ 0 ^ 1 \\alpha ( t ) ( X ( t ) - m ( t ) ) d t \\cdot ( X ( s ) - m ( s ) ) \\right ] = \\int _ 0 ^ 1 K ( s , t ) \\alpha ( t ) d t = \\beta ( s ) , \\end{align*}"}
-{"id": "989.png", "formula": "\\begin{align*} \\beta _ { i , i + 2 } ( { \\rm i n } ( I _ { G _ { r , d } } ) ) & = \\sum _ { p = 1 } ^ { \\frac { d ( d - 1 ) } { 2 } } \\binom { n _ { p } } { i } = \\binom { 0 } { i } + 2 \\binom { 1 } { i } + 3 \\binom { 2 } { i } + \\ldots + ( d - 1 ) \\binom { d - 2 } { i } \\\\ & = \\sum _ { q = 1 } ^ { d - 1 } q \\binom { q - 1 } { i } = ( i + 1 ) \\sum _ { q = 1 } ^ { d - 1 } \\binom { q } { i + 1 } = ( i + 1 ) \\binom { d } { i + 2 } . \\end{align*}"}
-{"id": "1811.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\omega _ { t _ { i } } \\left | \\omega _ { t _ { 0 } } , \\omega _ { t _ { 1 } } , \\cdots \\omega _ { t _ { i - 1 } } , \\omega _ { t _ { i + 1 } } , \\omega _ { t _ { n } } \\vphantom { \\frac { t _ { i + 1 } } { t _ { i + 1 } } } \\right . \\right ] = \\frac { t _ { i + 1 } - t _ { i } } { t _ { i + 1 } - t _ { i - 1 } } \\omega _ { t _ { i - 1 } } + \\frac { t _ { i } - t _ { i - 1 } } { t _ { i + 1 } - t _ { i - 1 } } \\omega _ { t _ { i + 1 } } . \\end{align*}"}
-{"id": "1733.png", "formula": "\\begin{align*} L u = \\lim _ { t \\downarrow 0 } \\frac { T _ { t } u - u } { t } \\textrm { f o r } u \\in \\mathcal { D } ( L ) . \\end{align*}"}
-{"id": "7484.png", "formula": "\\begin{gather*} n = 2 g - 2 + b + \\sum _ { i = 1 } ^ b k _ i . \\end{gather*}"}
-{"id": "2881.png", "formula": "\\begin{align*} \\{ \\gamma _ \\mu , \\gamma _ \\nu \\} = - 2 \\delta _ { \\mu \\nu } . \\end{align*}"}
-{"id": "610.png", "formula": "\\begin{align*} \\varphi _ \\ast ( x ) ^ v = ( \\varphi ^ v ) _ \\ast ( x ^ { \\varphi ^ \\ast ( v ) } ) \\ . \\end{align*}"}
-{"id": "2449.png", "formula": "\\begin{gather*} \\boxed { \\Lambda : = \\frac { 6 4 \\sigma _ 4 } { 1 0 0 \\sigma _ 4 - 9 \\sigma _ 2 ^ 2 } } \\end{gather*}"}
-{"id": "752.png", "formula": "\\begin{align*} \\mathbb { E } _ { r } \\ ! \\left [ r ^ { 2 } \\ln ^ { 2 } r \\right ] = m n ( m n + 1 ) \\left ( \\psi _ { 1 } ( m n + 2 ) + \\psi ^ { 2 } ( m n + 2 ) \\right ) \\end{align*}"}
-{"id": "1290.png", "formula": "\\begin{align*} \\left | \\int _ { | \\nu | > 2 0 } \\right | \\lesssim \\int _ { | \\nu | > 2 0 } | \\nu | | \\nu | ^ { - 2 k - 1 } d \\nu = O ( 1 ) . \\end{align*}"}
-{"id": "1269.png", "formula": "\\begin{align*} I ( S , S , S ) ( \\xi ) = O ( 1 ) , \\end{align*}"}
-{"id": "976.png", "formula": "\\begin{align*} H _ n \\left ( q , \\frac { 1 } { 1 - q } \\right ) = \\frac { 1 } { ( 1 - q ) ^ n } , \\end{align*}"}
-{"id": "3579.png", "formula": "\\begin{align*} \\begin{matrix} Y _ { 6 } = [ 1 \\ , 2 \\ , 6 ] [ 3 \\ , 4 \\ , 5 ] , & Y _ { 7 } = [ 1 \\ , 3 \\ , 6 ] [ 2 \\ , 4 \\ , 5 ] , & Y _ { 8 } = [ 1 \\ , 4 \\ , 6 ] [ 2 \\ , 3 \\ , 5 ] , \\\\ Y _ { 9 } = [ 1 \\ , 5 \\ , 6 ] [ 2 \\ , 3 \\ , 4 ] , & Y _ { 1 0 } = [ 1 \\ , 4 \\ , 5 ] [ 2 \\ , 3 \\ , 6 ] . \\end{matrix} \\end{align*}"}
-{"id": "4657.png", "formula": "\\begin{align*} \\begin{cases} ( I ) : \\hphantom { ( I I ) } m _ 1 + 4 + 2 b + n _ 1 + b n _ 2 \\equiv _ n 0 \\\\ ( I I ) : \\hphantom { ( I ) } m _ 2 - 2 b - b n _ 1 - n _ 2 \\equiv _ n 0 \\\\ 0 \\leq m _ 1 , n _ 1 , m _ 1 + m _ 2 , n _ 1 + n _ 2 \\leq 2 n - 6 \\\\ 0 \\leq m _ 2 , n _ 2 \\leq n - 1 \\\\ m _ 0 + m _ 1 + m _ 2 = n _ 0 + n _ 1 + n _ 2 = 2 b - 6 \\end{cases} \\end{align*}"}
-{"id": "4692.png", "formula": "\\begin{align*} u _ t + \\displaystyle { \\sum _ { k = 1 } ^ { m + d } \\mu _ k ( f ) ( B ^ d _ k ) ' } \\approx 0 , \\ ; \\ ; \\ ; x \\in [ a , b ] , ~ t \\in ( 0 , T ] . \\end{align*}"}
-{"id": "6228.png", "formula": "\\begin{align*} d = \\lim _ { \\delta \\to 0 } \\max _ { \\theta \\in \\bar { B } _ r ( \\theta ^ \\star ) \\setminus B _ \\delta ( \\theta ^ \\star ) } \\frac { \\mathcal { V } ( \\theta ) } { \\| \\theta - \\theta ^ \\star \\| } , \\end{align*}"}
-{"id": "1082.png", "formula": "\\begin{align*} & n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } \\int _ { I _ 2 } D _ n ( \\sqrt { 4 - y ^ 2 } / S ( I ) \\cdot G _ n ( x ) / 2 ) d y \\\\ \\leq & n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } \\int _ { I _ 2 } e ^ { 1 - ( \\sqrt { 4 - y ^ 2 } / S ( I ) - 1 ) \\ln n } D _ n ( G _ n ( x ) / 2 ) d y \\\\ \\leq & n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } \\int _ { I _ 2 } e ^ { 1 - ( S ( I _ 2 ) / S ( I ) - 1 ) \\ln n } D _ n ( G _ n ( x ) / 2 ) d y \\\\ = & n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } ( b - b _ 0 ) e ^ { 1 - ( S ( I _ 2 ) / S ( I ) - 1 ) \\ln n } D _ n ( G _ n ( x ) / 2 ) , \\end{align*}"}
-{"id": "2794.png", "formula": "\\begin{align*} E _ { i , j } : = e _ i \\delta _ { i , j } , \\end{align*}"}
-{"id": "25.png", "formula": "\\begin{align*} d S _ { t } & = S _ { t } \\sigma ( V _ t ) ( \\theta ^ { \\alpha _ { t - } } ( V _ t ) d t + d W _ { t } ) , \\\\ d V _ { t } & = \\eta \\left ( V _ { t } \\right ) d t + \\kappa \\ , d W _ { t } . \\end{align*}"}
-{"id": "4624.png", "formula": "\\begin{align*} \\mathcal { R } _ { C D } = \\begin{bmatrix} A _ 0 & M & & & M \\\\ M & A _ 0 & M & & \\\\ & M & A _ 0 & \\ddots & \\\\ & & \\ddots & \\ddots & M \\\\ M & & & M & A _ 0 \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "4152.png", "formula": "\\begin{align*} z ^ { 1 / 3 } f _ { 1 } ( z ) + z ^ { 2 / 3 } f _ { 2 } ( z ) & = - \\varphi _ { 1 } ( z ) - 2 \\varphi _ { 2 } ( z ) \\\\ \\omega ^ 2 z ^ { 1 / 3 } f _ { 1 } ( z ) + \\omega z ^ { 2 / 3 } f _ { 2 } ( z ) & = \\varphi _ { 2 } ( z ) - \\varphi _ { 1 } ( z ) . \\end{align*}"}
-{"id": "9705.png", "formula": "\\begin{align*} T _ { \\mathbf { d } } ( \\partial M ) = - \\chi ( M ) . \\end{align*}"}
-{"id": "2370.png", "formula": "\\begin{align*} C ( \\mathfrak { p } ) = \\prod _ { S \\in \\mathfrak { p } } \\omega ( a _ { A | S } ) \\end{align*}"}
-{"id": "5959.png", "formula": "\\begin{align*} \\inf _ { \\substack { Q \\in \\mathcal { P } ^ * _ \\epsilon ( \\C ) \\\\ Q ^ 0 = \\mu , \\ , Q \\circ H ^ { - 1 } = \\nu } } E ^ Q \\left [ \\int _ 0 ^ 1 g \\left ( t , q ^ Q ( t ) \\right ) d t \\right ] , \\end{align*}"}
-{"id": "268.png", "formula": "\\begin{align*} \\widetilde { b } _ f ( - \\partial _ t t - 1 ) \\partial _ t \\delta = R ( - \\partial _ t t ) \\delta , \\quad R ( s ) = - P ( s - 1 ) . \\end{align*}"}
-{"id": "5166.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sup _ { x \\in \\mathbb { R } } \\left \\vert F ^ { n } ( a _ { n } x + b _ { n } ) - H ( x ) \\right \\vert = 0 . \\end{align*}"}
-{"id": "6873.png", "formula": "\\begin{align*} \\sqrt [ \\mathbb { R } ] { I _ { 3 , 8 } } = \\langle y ^ 1 _ { 1 0 } y ^ 2 _ { 0 1 } - y ^ 1 _ { 0 1 } y ^ 2 _ { 1 0 } - 1 \\rangle + \\langle y ^ k _ \\nu , | \\nu | = 2 \\ , , k = 1 , 2 \\rangle \\ , . \\end{align*}"}
-{"id": "5445.png", "formula": "\\begin{align*} V _ 0 \\ = \\ \\sup _ \\alpha J ( \\alpha ) . \\end{align*}"}
-{"id": "672.png", "formula": "\\begin{gather*} ( \\nu ' _ 1 \\nu _ 1 \\diamond \\dots \\diamond \\nu ' _ n \\nu _ n ) = ( \\nu ' _ 1 \\diamond \\dots \\diamond \\nu ' _ n ) \\circ ( \\nu _ 1 \\diamond \\dots \\diamond \\nu _ n ) \\ , \\\\ \\rho ^ { ( \\vec l ) } _ i \\circ ( \\nu _ 1 \\diamond \\dots \\diamond \\nu _ n ) = \\nu _ i \\circ \\rho ^ { ( \\vec k ) } _ i \\ , \\end{gather*}"}
-{"id": "2857.png", "formula": "\\begin{align*} y ^ 2 - x z - z ^ 2 t ^ 2 = 0 . \\end{align*}"}
-{"id": "5477.png", "formula": "\\begin{align*} v ( s , \\hat X _ s ^ { t , x , a } ) \\ = \\ Y _ s ^ { t , x , a } , t \\leq s \\leq T , \\ , \\hat \\P \\end{align*}"}
-{"id": "3398.png", "formula": "\\begin{align*} \\tau ^ { M } ( T ) \\leq 2 ^ { - M } \\sum _ { i = 1 } ^ { w _ { M } ( T ) } \\tau ^ 0 ( T _ { v _ i } ) = \\frac { \\log 2 } { 2 ^ M } w _ { M } ( T ) . \\end{align*}"}
-{"id": "5911.png", "formula": "\\begin{align*} T _ { s _ 1 s _ 0 } T _ { ( s _ 1 s _ 0 ) ^ n } \\star \\varphi _ m & = T _ { s _ 1 s _ 0 } \\star q ^ { 2 n } \\varphi _ { m + n } + ( q - 1 ) \\sum _ { k = 1 } ^ { 2 n } q ^ { 2 n - k } \\psi _ { m + n - k } \\\\ & = q ^ { 2 n + 2 } \\varphi _ { m + n + 1 } + ( q - 1 ) q ^ { 2 n } \\left ( q \\psi _ { m + n } + \\psi _ { m + n - 1 } \\right ) + ( q - 1 ) \\sum _ { k = 1 } ^ { 2 n } q ^ { 2 n - k } \\psi _ { m + n - 1 - k } \\\\ & = q ^ { 2 n + 2 } \\varphi _ { m + n + 1 } + ( q - 1 ) \\left ( q ^ { 2 n + 1 } \\psi _ { m + n } + q ^ { 2 n } \\psi _ { m + n - 1 } + \\sum _ { k = 3 } ^ { 2 n + 2 } q ^ { 2 n + 2 - k } \\psi _ { m + n + 1 - k } \\right ) , \\end{align*}"}
-{"id": "1226.png", "formula": "\\begin{align*} \\MoveEqLeft \\int _ { 0 } ^ { b } ( g ' ( \\nu ) - g ' ( - \\nu ) ) \\left ( \\int _ { \\sqrt { \\xi } \\nu } ^ \\infty e ^ { i \\mu ^ 2 } d \\mu \\right ) d \\nu = \\int _ 0 ^ b ( g ' ( \\nu ) - g ' ( - \\nu ) ) \\left ( \\frac { \\sqrt \\pi } { 2 } e ^ { i \\pi / 4 } + O ( \\sqrt { \\xi } \\nu ) \\right ) d \\nu \\\\ & = \\frac { \\sqrt \\pi } { 2 } e ^ { i \\pi / 4 } ( g ( b ) - g ( - b ) ) + O \\left ( \\sqrt { \\xi } \\int _ { | \\nu | \\le b } | \\nu g ' ( \\nu ) | d \\nu \\right ) \\end{align*}"}
-{"id": "373.png", "formula": "\\begin{align*} v _ { 2 m + 3 } & = 2 s v _ { 2 m + 2 } - v _ { 2 m + 1 } \\\\ & \\equiv 2 s z _ 0 - ( s z _ 0 + c x _ 0 ) \\\\ & \\equiv s z _ 0 - c x _ 0 \\\\ & \\equiv s z _ 0 + c x _ 0 \\pmod { 2 c } , \\end{align*}"}
-{"id": "9509.png", "formula": "\\begin{align*} U _ { \\delta } ( x ) = \\delta ^ { - \\frac { N - s p } { p } } U ( | x | / \\delta ) \\end{align*}"}
-{"id": "719.png", "formula": "\\begin{align*} \\lim _ { x \\to 1 ^ - } \\sup _ { v , w \\in S ( 0 , 1 ) ^ 2 } \\left ( B i s _ { ( 0 , x ) } ( v , w ) + 1 + \\frac { \\left \\lvert \\displaystyle \\sum _ { 1 \\leq i , j \\leq 2 } g _ { i \\bar { j } } ( 0 , x ) v _ i \\overline { w _ j } \\right \\rvert ^ 2 } { \\left ( \\displaystyle \\sum _ { 1 \\leq i , j \\leq 2 } g _ { i \\bar { j } } ( 0 , x ) v _ i \\overline { v _ j } \\right ) \\left ( \\displaystyle \\sum _ { 1 \\leq i , j \\leq 2 } g _ { i \\bar { j } } ( 0 , x ) w _ i \\overline { w _ j } \\right ) } \\right ) = 0 . \\end{align*}"}
-{"id": "2535.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { f ( u t ) } { f ( t ) } = u ^ \\alpha \\Big ( \\lim _ { t \\to 0 } \\frac { f ( u t ) } { f ( t ) } = u ^ \\alpha \\Big ) . \\end{align*}"}
-{"id": "8610.png", "formula": "\\begin{align*} \\varphi _ 0 ' = 2 c _ 0 - \\sin \\varphi _ 0 , \\ \\ x \\in \\mathbb { R } , \\end{align*}"}
-{"id": "6458.png", "formula": "\\begin{align*} \\lambda : = \\frac { i + \\ < v , w \\ > _ U } { i - \\ < v , w \\ > _ U } \\end{align*}"}
-{"id": "9507.png", "formula": "\\begin{align*} 0 & = \\frac { d } { d t } \\left [ I _ { \\lambda } ( t u _ 0 ) \\right ] _ { t = t _ { \\lambda } } \\\\ & = t _ { \\lambda } ^ { p - 1 } \\Vert u _ 0 \\Vert _ { s , p } ^ p + t _ { \\lambda } ^ { q - 1 } \\Vert u _ 0 \\Vert _ { s , q } ^ q - \\lambda t _ { \\lambda } ^ { r - 1 } \\displaystyle \\int _ { \\mathbb { R } ^ N } g u _ 0 ^ r \\dd x - t _ { \\lambda } ^ { p _ { s } ^ * - 1 } \\end{align*}"}
-{"id": "6380.png", "formula": "\\begin{align*} ( \\theta + d d ^ c \\rho ) ^ n = \\left ( a + \\frac { ( g - f ) _ + } { \\Vert ( g - f ) _ + \\Vert _ p } \\right ) d V , \\end{align*}"}
-{"id": "788.png", "formula": "\\begin{align*} L = \\prod _ { i = 1 } ^ { { s } } \\lambda _ i ^ { \\frac { q ^ d - 1 } { q ^ s - 1 } b _ i } , \\mbox { w h i c h s a t i s f i e s } L ^ { q ^ s - 1 } = ( - 1 ) ^ s \\prod _ { i = 1 } ^ { { s } } ( T - \\rho _ i ) ^ { b _ i } . \\end{align*}"}
-{"id": "8651.png", "formula": "\\begin{align*} \\sin \\bar { \\varphi } _ 0 = 2 c _ 0 , \\ \\ \\bar { \\varphi } _ 0 \\in ( 0 , \\frac { \\pi } { 2 } ) , \\end{align*}"}
-{"id": "5743.png", "formula": "\\begin{align*} u ( x , y , t ) & = \\sum _ { m , n \\in \\N } T _ { m , n } ( t ) J _ 0 ( \\gamma _ m x ) \\sin ( n \\pi y ) , \\\\ T _ { m , n } ( t ) & = C _ { m , n } E _ { \\alpha , 1 } ( - \\mu _ { m , n } t ^ \\alpha ) + D _ { m , n } t E _ { \\alpha , 2 } ( - \\mu _ { m , n } t ^ \\alpha ) + \\int _ 0 ^ t ( t - \\xi ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } [ - \\mu _ { m , n } ( t - \\xi ) ^ { \\alpha } ] F _ { m , n } ( \\xi ) d \\xi , \\end{align*}"}
-{"id": "3677.png", "formula": "\\begin{align*} \\sum _ { i \\in S } H ( \\delta _ i ; \\eta ) = 0 \\end{align*}"}
-{"id": "786.png", "formula": "\\begin{align*} \\sigma _ { \\lambda _ i } ^ * ( \\sigma _ { f _ j } ) = \\lambda _ { i } ^ { \\sigma _ { f _ j } } \\lambda _ { i } ^ { - 1 } = \\begin{cases} Z _ 1 ^ { q ^ { j - 1 } } & \\mbox { i f } i = j \\\\ 1 & \\mbox { i f } i \\neq j . \\end{cases} \\end{align*}"}
-{"id": "3100.png", "formula": "\\begin{align*} & \\sum _ v ( 3 g _ v - 3 + n _ v - \\sum _ { e \\in E _ v } b _ { ( e , v ) } - \\sum _ { i \\in L _ v } b _ i ) + \\sum _ { e = ( v _ 1 , v _ 2 ) } ( b _ { ( e , v _ 1 ) } + b _ { ( e , v _ 2 ) } + 1 ) \\\\ = & ( \\sum _ v 3 g _ v ) - 3 | V | + 3 | E | + m - \\sum _ i b _ i = 3 h - 3 + m - \\sum _ i b _ i ; \\end{align*}"}
-{"id": "2312.png", "formula": "\\begin{align*} \\hat { \\Omega } \\delta _ x & = x \\delta _ x & \\hat { \\Omega } \\frac { \\mathcal { P } } { x - \\Omega } 1 & = - \\hat { 1 } + x \\frac { \\mathcal { P } } { x - \\Omega } 1 \\\\ \\langle \\hat { 1 } | \\delta _ x ) & = x & ( \\hat { 1 } | \\frac { \\mathcal { P } } { x - \\Omega } | 1 ) & = 0 \\end{align*}"}
-{"id": "5894.png", "formula": "\\begin{align*} \\tilde { f } = f - T _ 1 - T _ { s _ 0 } = \\sum _ { m = 1 } ^ { \\infty } q ^ { - 2 m } \\left ( \\underbrace { T _ { s _ 0 ( s _ 1 s _ 0 ) ^ m } } _ { A } + \\underbrace { T _ { ( s _ 1 s _ 0 ) ^ m } } _ { B } - q \\left ( \\underbrace { T _ { ( s _ 0 s _ 1 ) ^ m } } _ { C } + \\underbrace { T _ { s _ 1 ( s _ 0 s _ 1 ) ^ m } } _ { D } \\right ) \\right ) . \\end{align*}"}
-{"id": "764.png", "formula": "\\begin{align*} H _ { q ^ d , M } = ( \\sigma - 1 ) \\mathrm { G a l } ( K _ { q ^ d , M } / \\mathbb { F } _ { q ^ d } ( T ) ) . \\end{align*}"}
-{"id": "5484.png", "formula": "\\begin{align*} v ( t , x ) - \\psi ( t , x ) \\ = \\ 0 , \\end{align*}"}
-{"id": "3943.png", "formula": "\\begin{align*} A _ { i } \\varphi ( x ) = - \\sum _ { \\alpha , \\beta = 1 } ^ { n } a _ { i } ^ { \\alpha \\beta } ( x ) \\dfrac { \\partial ^ { 2 } \\varphi ( x ) } { \\partial x _ { \\alpha } \\partial x _ { \\beta } } \\ , . \\end{align*}"}
-{"id": "383.png", "formula": "\\begin{align*} p _ 2 ( x ) = & \\ , x ^ 8 - 4 \\left ( ( 1 6 ( \\mu ^ 2 + \\nu ^ 2 ) - 1 6 \\mu - 4 ) ( \\mu ^ 2 + \\nu ^ 2 ) + 1 6 \\nu ^ 2 + 4 \\mu + 1 \\right ) x ^ 6 + \\\\ & \\ , + \\left ( ( 1 2 8 \\mu ^ 4 - 1 2 8 \\mu ^ 3 - 3 2 \\mu ^ 2 + 3 2 \\mu + 6 ) + ( - 7 6 8 \\mu ^ 2 + 3 8 4 \\mu + 3 2 ) \\nu ^ 2 + 1 2 8 \\nu ^ 4 \\right ) x ^ 4 \\\\ & \\ , - 4 \\left ( ( 1 6 ( \\mu ^ 2 + \\nu ^ 2 ) - 1 6 \\mu - 4 ) ( \\mu ^ 2 + \\nu ^ 2 ) + 1 6 \\nu ^ 2 + 4 \\nu + 1 ) \\right ) x ^ 2 + 1 . \\end{align*}"}
-{"id": "3056.png", "formula": "\\begin{align*} Z _ { r } ( s ) = \\sum _ { k = 1 } ^ { \\infty } a _ { k } t _ { k } ^ { - 2 s } . \\end{align*}"}
-{"id": "4587.png", "formula": "\\begin{align*} \\hat { \\phi } ^ { D } = \\hat { \\phi } ^ { D D } * \\hat { \\phi } ^ { D D } . \\end{align*}"}
-{"id": "4.png", "formula": "\\begin{align*} Y _ t ^ i ( m ) = & \\int _ t ^ m \\left [ f ^ i ( V _ s , q ( Z _ s ^ i ( m ) ) ) + \\sum _ { k \\in I } q ^ { i k } ( e ^ { p ( Y _ s ^ k ( m ) ) - p ( Y _ s ^ { i } ( m ) ) } - 1 ) - \\rho Y _ s ^ i ( m ) \\right ] d s \\\\ & - \\int _ t ^ m ( Z _ s ^ i ( m ) ) ^ { t r } d W _ s . \\end{align*}"}
-{"id": "6657.png", "formula": "\\begin{align*} ( \\# k ) ^ { \\dim _ k V } = \\# V & = \\# ( V _ 1 \\cup \\cdots \\cup V _ n ) \\leq \\\\ & \\leq \\left ( \\sum _ { i = 1 } ^ n \\# V _ i \\right ) - ( n - 1 ) < \\sum _ { i = 1 } ^ n \\# V _ i \\leq n ( \\# k ) ^ { \\dim _ k V - 1 } . \\end{align*}"}
-{"id": "6764.png", "formula": "\\begin{align*} \\phi ( s x ) = \\phi ( x ) \\tau ( s ) ^ { - 1 } \\end{align*}"}
-{"id": "3984.png", "formula": "\\begin{align*} \\{ \\ ( \\sum _ { ( \\lambda , \\mu ) } \\phi \\mid _ { k , m } [ \\lambda , \\mu ] ) \\ \\mid U _ l , \\psi \\} & = \\{ \\phi , \\sum _ { ( \\lambda , \\mu ) } \\psi \\mid A _ l \\mid _ { k , m } [ - \\lambda , - \\mu ] \\} \\end{align*}"}
-{"id": "7968.png", "formula": "\\begin{align*} \\abs { J _ 1 \\cap J _ 2 } = \\abs { J _ 1 } + \\abs { J _ 2 } - \\abs { J _ 1 + J _ 2 } , \\end{align*}"}
-{"id": "7617.png", "formula": "\\begin{align*} X _ \\epsilon ' = X _ 2 - X _ 1 , v _ \\epsilon ' = v _ 2 - v _ 1 , u _ { \\epsilon , 0 } ' = 0 , \\sigma _ { \\epsilon , 0 } ' = 0 . \\end{align*}"}
-{"id": "972.png", "formula": "\\begin{align*} \\left \\{ ( b , c ) \\in B \\times C \\ , : \\ , \\{ \\min ( b , c ) + 1 , \\dots , \\max ( b , c ) - 1 \\} \\cap \\left ( B \\cup C \\right ) = \\emptyset \\right \\} , \\end{align*}"}
-{"id": "1435.png", "formula": "\\begin{align*} \\mu _ { t } - \\Delta \\mu + \\varepsilon \\mathrm { d i v } ( \\mu \\Theta _ { p } ( t , x , \\mu , D w ) ) + \\varepsilon \\bar { m } \\mathrm { d i v } ( \\Theta _ { p } ( t , x , \\mu , D w ) ) = 0 . \\end{align*}"}
-{"id": "4674.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\frac { n - 1 } 2 } \\binom n j = \\frac 1 2 \\sum _ { j = 1 } ^ { n - 1 } \\binom n j & = \\frac 1 2 ( 2 ^ n - 2 ) = 2 ^ { n - 1 } - 1 \\\\ \\sum _ { j = 1 } ^ { h - 1 } \\binom { 2 h } j + \\frac 1 2 \\binom { 2 h } h & = \\frac 1 2 2 ^ { 2 h } - 1 = 2 ^ { n - 1 } - 1 \\end{align*}"}
-{"id": "7496.png", "formula": "\\begin{align*} ( X * Y , Z ) = ( X , Y * Z ) \\end{align*}"}
-{"id": "1374.png", "formula": "\\begin{align*} \\left ( \\triangledown _ n \\right ) _ { v } ( \\lambda , \\mu ) : = \\sum _ { \\sigma \\in \\Sigma _ { n } } \\operatorname { s g n } ( \\sigma ) ( \\lambda _ 1 ^ \\sigma , \\ldots , \\lambda _ n ^ \\sigma , \\mu ) \\end{align*}"}
-{"id": "5870.png", "formula": "\\begin{align*} \\Lambda ( i ) = \\mathfrak { p } _ F ^ { a _ { n } + i } w _ n \\oplus \\mathfrak { p } _ F ^ { a _ { ( n - 1 ) } + i } w _ { n - 1 } \\oplus \\cdots \\oplus \\mathfrak { p } _ F ^ { a _ { ( - n + 1 ) } + i } w _ { - n + 1 } \\oplus \\mathfrak { p } _ F ^ { a _ { ( - n ) } + i } w _ { - n } . \\end{align*}"}
-{"id": "1743.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t \\wedge u } K ( t , s ) K ( u , s ) d s = R ( t , u ) . \\end{align*}"}
-{"id": "8684.png", "formula": "\\begin{align*} \\mathcal { F } ( \\abs { X } ^ 2 \\partial _ S ) & = \\langle S , X \\rangle \\Delta _ { \\mathfrak { v } } + 2 \\partial _ S , \\\\ \\mathcal { F } ( \\abs { X } ^ 2 \\partial _ { [ S , X ] } ) & = - \\partial _ { J _ Z S } \\Delta _ { \\mathfrak { v } } , \\\\ \\mathcal { F } ( \\partial _ { J _ Z S } ) & = - \\partial _ { [ S , X ] } . \\end{align*}"}
-{"id": "337.png", "formula": "\\begin{align*} | I m ( \\partial ' _ k ) _ * | = \\frac { 2 m ( 4 m + 2 ) ( 4 m + 1 ) } { ( 2 m ( 4 m + 2 ) ( 4 m + 1 ) , k ) } \\cdot \\frac { 4 m + 1 } { ( 4 m + 1 , k ) } . \\end{align*}"}
-{"id": "9697.png", "formula": "\\begin{align*} \\tau = \\frac { e _ { n } \\cdot b _ { n } } { | | e _ { n } | | ^ { 2 } } . \\end{align*}"}
-{"id": "6544.png", "formula": "\\begin{align*} n _ \\beta = | E ( \\beta ) | \\cdot w \\cdot \\frac { ( - 1 ) ^ { w - 1 } } { w ^ 2 } \\ , m _ \\beta ^ { t o t } , \\end{align*}"}
-{"id": "5470.png", "formula": "\\begin{align*} v ( t , x ) \\ = \\ \\sup _ { \\alpha \\in \\mathcal A _ t } J ( t , x , \\alpha ) , ( t , x ) \\in [ 0 , T ] \\times H . \\end{align*}"}
-{"id": "2566.png", "formula": "\\begin{align*} \\langle v _ t , \\phi ^ * \\rangle _ m + \\int _ s ^ t \\langle \\psi _ 0 ( \\cdot , v _ t ) , \\phi ^ * \\rangle _ m d r = \\langle v _ s , \\phi ^ * \\rangle _ m \\in [ 0 , \\infty ) , s , t > 0 . \\end{align*}"}
-{"id": "4975.png", "formula": "\\begin{align*} F = F i x ( T _ { \\ell - 1 } \\cdots T _ 1 T _ 0 ) = F i x ( T _ 0 T _ { \\ell - 1 } \\cdots T _ 1 ) = \\cdots = F i x ( T _ { \\ell - 2 } \\cdots T _ 0 T _ { \\ell - 1 } ) . \\end{align*}"}
-{"id": "1061.png", "formula": "\\begin{align*} G _ n ( x ) = \\frac { 8 x - 5 \\ln ( 2 \\ln n ) } { 2 n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } } + \\frac { ( 3 2 \\ln n ) ^ { \\frac { 1 } { 2 } } } { n } . \\end{align*}"}
-{"id": "8233.png", "formula": "\\begin{align*} L _ { 2 i - 1 } & = ( q ^ { 1 0 } ; q ^ { 1 0 } ) _ \\infty \\sum _ { n = 0 } ^ \\infty c \\left ( 5 ^ { 2 i - 1 } n + \\frac { 7 \\cdot 5 ^ { 2 i - 1 } + 1 } { 1 2 } \\right ) q ^ { n + 1 } , \\\\ L _ { 2 i } & = ( q ^ 2 ; q ^ 2 ) _ \\infty \\sum _ { n = 0 } ^ \\infty c \\left ( 5 ^ { 2 i } n + \\frac { 1 1 \\cdot 5 ^ { 2 i } + 1 } { 1 2 } \\right ) q ^ { n + 1 } . \\end{align*}"}
-{"id": "6132.png", "formula": "\\begin{align*} ( i , j ) = ( d - \\mu - a \\mu _ 1 - b \\mu _ 2 , a + b + 1 ) \\end{align*}"}
-{"id": "2836.png", "formula": "\\begin{align*} ( \\widehat { Q } _ { i } ) ^ { \\vee \\vee } \\cong ( Q _ { i } ) ^ { \\vee \\vee } = ( \\mathcal { E } _ { i } / \\mathcal { E } _ { i - 1 } ) ^ { \\vee \\vee } , \\end{align*}"}
-{"id": "3617.png", "formula": "\\begin{align*} ( \\alpha _ 0 - \\beta _ 0 ) \\cdot \\alpha _ 1 = - \\frac { 1 } { 2 ^ 3 } ( a _ 1 - a _ { - 1 } ) + \\frac { 7 } { 2 ^ 2 \\cdot 3 } ( a _ 2 + a _ { - 2 } ) + \\frac { 2 } { 3 } ( a _ 3 + a _ { - 3 } ) + \\frac { 1 } { 2 ^ 3 } v _ { ( 1 , 2 ) } - v _ { ( 2 , 3 ) } + \\frac { 8 } { 3 } ( a _ 2 + a _ { - 2 } ) \\cdot v _ { ( 1 , 3 ) } \\end{align*}"}
-{"id": "4866.png", "formula": "\\begin{align*} \\mathcal { W } = \\{ W \\ , ; \\ , \\mbox { E a c h $ W $ i s e i t h e r $ X $ i t s e l f o r t h e f i n i t e i n t e r s e c t i o n o f $ H ^ * _ i $ w i t h $ * = + $ o r $ - $ . } \\} . \\end{align*}"}
-{"id": "2832.png", "formula": "\\begin{align*} \\dot { U } ( t ) \\leq - \\sum _ { i = 1 } ^ p \\biggl ( \\gamma _ i ( x _ i ( t ) - x _ i ^ * ) ^ 2 + \\eta _ i \\frac { c _ i e _ i } { d _ i } ( u _ i ( t ) - u _ i ^ * ) ^ 2 \\biggr ) + h ( t ) \\end{align*}"}
-{"id": "1881.png", "formula": "\\begin{align*} \\mathcal { B - C } = \\frac { ( \\xi L ) ^ 2 } { 2 \\pi ^ 2 } \\sum _ { j = 1 } ^ { \\infty } \\frac { A _ j ^ 2 } { Z _ j } \\left [ \\psi ^ { ( 1 ) } \\left ( 1 + \\frac { Z _ j } { \\pi } \\right ) + \\frac { \\pi ^ 2 } { 2 Z _ j ^ 2 } - \\frac { \\pi } { Z _ j } \\right ] . \\end{align*}"}
-{"id": "840.png", "formula": "\\begin{align*} H ^ 0 ( Y , \\Omega _ Y ) \\subset L _ Y ( \\Omega _ Y ( \\mathcal { D } ) ) = W _ { \\bar { \\mu } } ^ { H _ { q ^ d , P ^ \\alpha } } . \\end{align*}"}
-{"id": "3721.png", "formula": "\\begin{align*} & \\max _ { t \\in [ 0 , \\infty ) } E ( t w _ k ) = E ( t _ k w _ k ) \\geq \\frac { 1 } { n } M \\left ( \\left ( \\frac { 2 n - \\mu } { 2 n } \\alpha _ n \\right ) ^ { n - 1 } \\right ) \\\\ & \\ ; \\frac { d } { d t } ( E ( t w _ k ) ) | _ { t = t _ k } = 0 . \\end{align*}"}
-{"id": "35.png", "formula": "\\begin{align*} \\lambda = \\sup _ { \\pi \\in \\bar { \\mathcal { A } } ^ { \\mathbb { G } } } \\limsup _ { T \\uparrow \\infty } \\frac { 1 } { T } \\ln \\mathbb { E } \\left [ \\frac { ( X _ { T } ( \\pi ) ) ^ { \\delta } } { \\delta } \\right ] . \\end{align*}"}
-{"id": "8978.png", "formula": "\\begin{align*} M _ { U R \\ 1 , 2 } & = ( - t ^ { - 1 } ) M _ { U L \\ 1 , 2 } ; \\ \\ M _ { U L \\ 1 , 2 } = M _ { L R \\ 1 , 2 } = ( - t ^ { - 1 } ) M _ { L L \\ 1 , 2 } \\\\ M _ { U R \\ 2 , 1 } & = ( - t ^ { - 1 } ) M _ { U L \\ 2 , 1 } ; \\ \\ M _ { U L \\ 2 , 1 } = M _ { L R \\ 2 , 1 } = ( - t ^ { - 1 } ) M _ { L L \\ 2 , 1 } \\\\ M _ { U R \\ 2 , 3 } & = q ^ 2 M _ { U L \\ 2 , 3 } ; \\ \\ \\ \\ \\ \\ \\ \\ M _ { U L \\ 2 , 3 } = M _ { L R \\ 2 , 3 } = q ^ 2 M _ { L L \\ 2 , 3 } \\end{align*}"}
-{"id": "8888.png", "formula": "\\begin{align*} \\emptyset \\sqcup w = w \\sqcup \\emptyset & = w , \\\\ ( \\omega _ 1 \\cdots \\omega _ k ) \\sqcup ( \\omega _ 1 ' \\cdots \\omega _ n ' ) & = ( \\omega _ 1 \\cdots \\omega _ k \\omega _ 1 ' \\cdots \\omega _ n ' ) \\end{align*}"}
-{"id": "9955.png", "formula": "\\begin{align*} = ( x _ z y _ \\theta - x _ \\theta y _ z ) \\ , ( F _ x ^ 2 + F _ y ^ 2 ) . \\end{align*}"}
-{"id": "6389.png", "formula": "\\begin{align*} \\frac { \\det ( K _ { A B C } ) } { \\det ( K _ C ) } = \\frac { \\det ( K _ { A C } ) } { \\det ( K _ C ) } \\frac { \\det ( K _ { B C } ) } { \\det ( K _ C ) } , \\end{align*}"}
-{"id": "4928.png", "formula": "\\begin{align*} I _ n = ( z ^ { n ^ 4 } - z ^ n x ^ n , y ^ n - z ^ n x , x ^ { n + 1 } - x z ^ { n ^ 4 - n } + y z ^ n ) . \\end{align*}"}
-{"id": "1253.png", "formula": "\\begin{align*} T _ { 4 , 1 } = i a A ^ 2 \\sqrt { \\frac { 4 \\pi } { 3 | \\eta | } } e ^ { i \\pi / 4 } e ^ { i a \\ln ( \\eta ^ 2 / 4 ) } + O ( | \\eta | ^ { - 2 } ) . \\end{align*}"}
-{"id": "9952.png", "formula": "\\begin{align*} \\gamma = S ^ 2 \\cap \\pi _ \\gamma . \\end{align*}"}
-{"id": "6420.png", "formula": "\\begin{align*} \\rho _ { k } = \\| B _ { k } z _ { k } \\| ^ { 2 } , \\qquad \\sigma _ { k } = \\alpha _ { k } \\beta _ { k } e _ { k } ^ { T } z _ { k } , \\qquad \\tau _ { k } = \\beta _ { k } ^ { 2 } + \\alpha _ { k + 1 } ^ { 2 } . \\end{align*}"}
-{"id": "6196.png", "formula": "\\begin{align*} \\gamma ( \\omega ) = \\sqrt { \\left ( \\frac { n - 1 } { 2 } \\right ) ^ 2 + \\lambda _ 1 ( \\omega ) } - \\frac { n - 1 } { 2 } \\ ; . \\end{align*}"}
-{"id": "3057.png", "formula": "\\begin{align*} \\int _ { n } ^ { \\infty } a ( n ) x ^ { s - 1 } d x = \\lim \\limits _ { b \\to \\infty } [ a ( n ) x ^ { s } / s ] _ { x = n } ^ { x = b } \\end{align*}"}
-{"id": "4996.png", "formula": "\\begin{align*} \\Phi ( x ) = \\frac { x } { | x | ^ 2 } , \\Phi ^ { - 1 } ( y ) = \\frac { y } { | y | ^ 2 } . \\end{align*}"}
-{"id": "5720.png", "formula": "\\begin{align*} _ c D _ { 0 t } ^ { \\alpha } \\left ( \\int _ 0 ^ x ( x - t ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } [ \\lambda ( x - t ) ^ { \\alpha } ] f ( t ) d t \\right ) & = f ( 0 ) E _ { \\alpha , 1 } ( \\lambda x ^ { \\alpha } ) + \\int _ 0 ^ x f ' ( z ) E _ { \\alpha , 1 } ( \\lambda ( x - z ) ^ { \\alpha } ) d z . \\end{align*}"}
-{"id": "3243.png", "formula": "\\begin{align*} \\theta ( x ) & = \\lim _ { r \\rightarrow 0 ^ + } \\frac { \\| T \\| ( B _ { r } ( x ) ) } { \\mathcal { H } ^ k ( S \\bigcap B _ r ( x ) ) } = \\lim _ { r \\rightarrow 0 ^ + } \\frac { \\| T \\| ( B _ { r } ( x ) ) } { \\Omega ( k ) r ^ k } \\frac { \\Omega ( k ) r ^ k } { \\mathcal { H } ^ k ( S \\bigcap B _ r ( x ) ) } \\\\ & \\leq 2 ^ k \\lim _ { r \\rightarrow 0 ^ + } \\frac { \\| T \\| ( B _ { r } ( x ) ) } { \\Omega ( k ) r ^ k } = 2 ^ k \\Theta ^ k ( | | T | | , x ) \\end{align*}"}
-{"id": "8102.png", "formula": "\\begin{align*} A ( h ) + B ( h ) + C ( h ) + D ( h ) & = - 4 h ^ 2 + O ( h ^ 3 ) , \\\\ B ( h ) + 2 C ( h ) + 3 D ( h ) & = 8 h + O ( h ^ 2 ) , \\ ; \\\\ B ( h ) + 2 ^ 2 C ( h ) + 3 ^ 2 D ( h ) & = O ( h ) . \\end{align*}"}
-{"id": "668.png", "formula": "\\begin{align*} \\widehat \\rho _ { \\vec a } : = \\widehat { [ \\rho , e _ m ] } _ { \\vec a } \\ , \\widehat x _ { \\vec a } : = \\widehat { [ \\mathrm { i d } , x ] } _ { \\vec a } \\ . \\end{align*}"}
-{"id": "5585.png", "formula": "\\begin{align*} \\begin{cases} v _ t + v _ { x x x } = 0 , & x , t \\in \\R ^ { + } , \\\\ v ( x , 0 ) = 0 & x \\in \\R ^ + , \\\\ v ( 0 , t ) = g ( t ) \\in H ^ { ( \\kappa + 1 ) / 3 } ( \\R ^ + ) , & \\end{cases} \\end{align*}"}
-{"id": "8466.png", "formula": "\\begin{align*} F _ { \\zeta \\zeta } ( \\zeta ( \\alpha , t ) , t ) = ( \\frac { \\partial _ { \\alpha } } { \\zeta _ { \\alpha } } ) ^ 2 F ( \\zeta ( \\alpha , t ) , t ) = \\frac { 1 } { \\zeta _ { \\alpha } ^ 2 } \\mathfrak { F } _ { \\alpha \\alpha } - \\frac { \\zeta _ { \\alpha \\alpha } } { \\zeta _ { \\alpha } ^ 3 } \\mathfrak { F } _ { \\alpha } . \\end{align*}"}
-{"id": "1335.png", "formula": "\\begin{align*} & \\left \\langle \\varphi _ v ( \\operatorname { I n d } ^ G v ( f ) ) \\cdot \\xi \\ ; , \\ ; \\eta \\right \\rangle _ { L ^ 2 ( H _ v , ( \\lambda _ H ) _ v ) } \\\\ = & \\int _ { x \\in H _ v } \\int _ { t \\in H _ v } f ( x t ^ { - 1 } ) \\cdot \\xi ( t ) \\cdot \\overline { \\eta ( x ) } \\ , d \\lambda _ v ( t ) \\ , d \\lambda _ v ( x ) \\\\ = & 0 \\ ; \\end{align*}"}
-{"id": "3580.png", "formula": "\\begin{align*} \\begin{array} { c c } Y _ { 0 } - Y _ { 1 } + Y _ { 2 } - Y _ { 6 } = 0 , & Y _ { 0 } - Y _ { 6 } + Y _ { 7 } - Y _ { 1 0 } = 0 , \\\\ Y _ { 2 } - Y _ { 3 } - Y _ { 7 } + Y _ { 8 } = 0 , & Y _ { 2 } - Y _ { 3 } - Y _ { 6 } + Y _ { 9 } = 0 , \\\\ Y _ { 3 } - Y _ { 4 } + Y _ { 6 } + Y _ { 1 0 } = 0 . \\end{array} \\end{align*}"}
-{"id": "4025.png", "formula": "\\begin{align*} \\phi _ { a } : = \\phi , \\xi _ { a } : = \\frac { 1 } { a } \\xi , \\eta _ { a } : = a \\eta , g _ { a } : = a g + a ( a - 1 ) \\eta \\otimes \\eta \\end{align*}"}
-{"id": "4917.png", "formula": "\\begin{align*} \\int _ { t _ 1 } ^ { t _ 2 } \\int _ \\Omega g ( \\dot m + \\nabla _ u m + ( \\nabla u ) ^ T m , w ) \\ , \\mu ( x ) \\ , \\d t = 0 \\ , . \\end{align*}"}
-{"id": "8008.png", "formula": "\\begin{align*} u ( t , x ) = \\frac { 1 } { ( 4 \\pi t ) ^ { n / 2 } } \\int _ { \\mathbb { R } ^ d } e ^ { - \\frac { | x - y | ^ 2 } { 4 t } } u _ 0 ( y ) \\ , \\mathrm d y , x \\in \\mathbb { R } ^ d . \\end{align*}"}
-{"id": "9033.png", "formula": "\\begin{align*} \\Delta _ g f _ { E _ i } + E _ i f _ { E _ i } = 0 \\end{align*}"}
-{"id": "2496.png", "formula": "\\begin{align*} \\aligned ( { \\rm I m } \\ , { \\bf w } _ 2 ) ^ t & = A _ 2 \\cdot \\big ( \\ , { \\sf t } ^ 2 _ 1 \\ , \\ldots \\ , { \\sf t } ^ 2 _ { k _ 2 } \\ , \\big ) ^ t , \\\\ & \\ \\ \\vdots \\\\ ( { \\rm I m } \\ , { \\bf w } _ { \\rho } ) ^ t & = A _ { \\rho } \\cdot \\big ( \\ , { \\sf t } ^ \\rho _ 1 \\ , \\ldots \\ , { \\sf t } ^ { \\rho } _ { k _ { \\rho } } \\ , \\big ) ^ t . \\endaligned \\end{align*}"}
-{"id": "7792.png", "formula": "\\begin{align*} \\sigma ( G ) ^ { - 1 } = \\max \\big \\{ \\varphi ^ G _ { \\xi } ( 0 ) : \\xi \\in \\mathbb S ^ { d - 1 } \\big \\} , \\end{align*}"}
-{"id": "6466.png", "formula": "\\begin{align*} v _ i v _ i ^ * = 1 \\otimes ( e _ { i k , i k } + e _ { i k + 1 , i k + 1 } + \\cdots + e _ { ( i + 1 ) k - 1 , ( i + 1 ) k - 1 } ) \\otimes 1 . \\end{align*}"}
-{"id": "8833.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathbb R ^ { n \\times q } } \\sum _ { i = 1 } ^ m \\log \\Big ( \\sum _ { j = 1 } ^ q \\exp \\big ( \\sum _ { l = 1 } ^ n A _ { i , l } x _ { l , j } \\big ) \\Big ) + \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ q x _ { i , j } b _ { i , j } + \\sum _ { l = 1 } ^ n \\sqrt { \\sum _ { j = 1 } ^ q x _ { l , j } ^ 2 } \\end{align*}"}
-{"id": "4250.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} d \\psi & = d \\phi & & B _ { R } \\\\ \\psi & = 0 & & \\partial B _ { R } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "1104.png", "formula": "\\begin{align*} A _ n : = & \\{ ( y _ 1 , \\cdots , y _ k ) \\in ( a , b ) ^ k | [ y _ i , y _ i + G _ n ( x _ i ) / S ( I ) ] \\\\ & \\cap [ y _ j , y _ j + G _ n ( x _ j ) / S ( I ) ] = \\emptyset , \\forall \\ 1 \\leq i < j \\leq k \\} , \\end{align*}"}
-{"id": "1960.png", "formula": "\\begin{align*} \\pi \\cdot T ' = 0 . \\end{align*}"}
-{"id": "9921.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\| \\pi _ { m _ { i } - } ^ { \\mu } - \\pi _ { m _ { i } - } ^ { \\nu } \\| _ { T V } = 0 \\end{align*}"}
-{"id": "4067.png", "formula": "\\begin{align*} I _ { V , \\theta } ( \\mu ) = \\frac { 1 } { 2 } I ( \\mu ) + \\frac { 1 } { 2 } I _ { \\theta } ( \\mu ) + \\int V ( x ) d \\mu ( x ) \\end{align*}"}
-{"id": "7253.png", "formula": "\\begin{align*} \\ddot \\mu _ { k } ( 0 ) = - \\frac { 1 } { 3 } \\frac { \\left < D _ { \\phi \\phi \\phi } ^ 3 \\Phi ( 0 , \\mu ^ * _ { k } ) [ \\phi ^ * _ { k } , \\phi ^ * _ { k } , \\phi ^ * _ { k } ] , \\phi ^ * _ { k } \\right > _ { L _ 2 } } { \\left < D _ { \\phi \\mu } ^ 2 F ( 0 , \\mu ^ * _ { k } ) \\phi ^ * _ { k } , \\phi ^ * _ { k } \\right > _ { L ^ 2 } } , \\end{align*}"}
-{"id": "9492.png", "formula": "\\begin{align*} \\beta _ { n + 1 } = p _ { s } ^ * \\frac { \\beta _ n + p - 1 } { p \\theta ' } , \\ \\beta _ 0 = \\frac { p _ { s } ^ * } { \\theta ' } > 1 \\ \\mbox { a n d } \\ \\sigma _ { n } = \\frac { \\beta _ n } { \\beta _ n + p - 1 } < 1 . \\end{align*}"}
-{"id": "188.png", "formula": "\\begin{align*} & \\pi _ { X _ { 1 , C } , X _ { 2 , C } , X _ { 3 , V } , X _ { 4 , V } } = X _ { 1 , C } \\wedge X _ { 2 , C } + X _ { 3 , V } \\wedge X _ { 4 , V } \\end{align*}"}
-{"id": "2944.png", "formula": "\\begin{align*} \\| E \\| _ { F } ^ { 2 } = \\| \\widetilde { \\Sigma } _ { 1 } \\widetilde { V } _ { 1 } ^ { \\ast } V _ { 1 } - \\widetilde { U } _ { 1 } ^ { \\ast } U _ { 1 } \\Sigma _ { 1 } \\| _ { F } ^ { 2 } + \\| \\widetilde { \\Sigma } _ { 1 } \\widetilde { V } _ { 1 } ^ { \\ast } V _ { 2 } \\| _ { F } ^ { 2 } + \\| \\widetilde { U } _ { 2 } ^ { \\ast } U _ { 1 } \\Sigma _ { 1 } \\| _ { F } ^ { 2 } . \\end{align*}"}
-{"id": "6713.png", "formula": "\\begin{align*} \\dot { \\varphi } ^ { ( 1 ) } _ k = - \\left ( \\alpha ' _ { i k } + \\beta ' _ { i l k } \\ , \\dot { \\bar { q } } _ l \\right ) \\left ( m \\ddot { q } _ i + \\omega _ { i j } \\dot { q } _ j + \\frac { \\partial V } { \\partial q _ i } \\right ) , \\end{align*}"}
-{"id": "6552.png", "formula": "\\begin{align*} m ^ P _ \\beta = m ^ { P ' } _ \\beta . \\end{align*}"}
-{"id": "3000.png", "formula": "\\begin{align*} - \\frac { x G ' _ m ( x ) } { G _ m ( x ) } = \\sum _ { k = 1 } ^ \\infty \\Big ( \\frac { k m x ^ { k m } } { 1 - x ^ { k m } } + \\frac { ( k m - ( m - 1 ) ) x ^ { k m - ( m - 1 ) } } { 1 - x ^ { k m - ( m - 1 ) } } + \\frac { ( k m - 1 ) x ^ { k m - 1 } } { 1 - x ^ { k m - 1 } } \\Big ) . \\end{align*}"}
-{"id": "3740.png", "formula": "\\begin{align*} a _ { l , m } ^ { ( j ) } = ( \\mathcal { B } _ { l , m } ^ { ( j ) } , \\mathbf { J } ) . \\end{align*}"}
-{"id": "5785.png", "formula": "\\begin{align*} R _ d ( q ) = \\psi ( q ^ { d - 1 } ) * \\cdots * \\psi ( 1 ) , d \\geq 1 . \\end{align*}"}
-{"id": "2539.png", "formula": "\\begin{align*} \\int _ E t ^ { \\alpha ( x ) } \\rho ( d x ) \\stackrel [ t \\to 0 ] { } { \\sim } \\rho \\{ x : \\alpha ( x ) = \\alpha _ 0 \\} t ^ { \\alpha _ 0 } . \\end{align*}"}
-{"id": "3318.png", "formula": "\\begin{align*} \\left | \\sum _ { d \\mid n , \\ , d \\neq 1 } \\frac { n } { d } \\sum _ { f \\in \\mathcal { P } _ { \\frac { n } { d } } } \\chi ^ d ( f ) \\right | & \\le \\sum _ { d \\mid n , \\ , d \\neq 1 } \\frac { n } { d } | \\mathcal { P } _ { n / d } | \\le \\sum _ { d \\mid n , \\ , d \\neq 1 } q ^ { n / d } \\le 2 q ^ { \\frac { n } { 2 } } , \\end{align*}"}
-{"id": "8455.png", "formula": "\\begin{align*} G _ k ^ { \\sigma } = ( I - \\mathcal { H } ) ( \\partial _ { \\alpha } ^ k \\tilde { G } + [ D _ t ^ 2 - i A \\partial _ { \\alpha } , \\partial _ { \\alpha } ^ k ] \\tilde { \\sigma } ) - [ D _ t ^ 2 - i A \\partial _ { \\alpha } , \\mathcal { H } ] \\partial _ { \\alpha } ^ k \\tilde { \\sigma } . \\end{align*}"}
-{"id": "6696.png", "formula": "\\begin{align*} L ( q , \\dot { q } ) = \\frac { 1 } { 2 } m \\dot { q } _ i \\dot { q } _ i + \\dot { q } _ i \\ , u _ i ( q ) - V ( q ) , \\end{align*}"}
-{"id": "4995.png", "formula": "\\begin{align*} B ( x _ j , \\theta d ( x _ j , a ) ) \\cap G = \\emptyset j = 1 , 2 , \\ldots . \\end{align*}"}
-{"id": "3278.png", "formula": "\\begin{align*} \\Omega _ n ' ( z ) = \\frac { d } { d z } L _ n ( z , a b ) = \\frac { d } { d z } V _ n ( z , - a b ) . \\end{align*}"}
-{"id": "8797.png", "formula": "\\begin{align*} & \\overline { B } _ t = \\int _ { 0 } ^ { t } \\langle v ( s , \\cdot ) , \\phi '' \\rangle \\ , \\textrm { d } s , \\overline { C } _ t = \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { \\pi } f ^ 2 ( v ( s , x ) ) \\phi ^ 2 ( x ) \\ , \\textrm { d } s \\ , \\textrm { d } x , \\overline { A } _ t = i \\xi \\overline { B } _ t - \\frac { 1 } { 2 } \\xi ^ 2 \\overline { C } _ t \\end{align*}"}
-{"id": "1722.png", "formula": "\\begin{align*} C ( H ) = \\int _ { - \\infty } ^ { 0 } \\left [ ( 1 - s ) ^ { H - \\frac { 1 } { 2 } } - ( - s ) ^ { H - \\frac { 1 } { 2 } } \\right ] ^ { 2 } d s + \\frac { 1 } { 2 H } . \\end{align*}"}
-{"id": "6414.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } s \\\\ c \\end{array} \\right ] = \\frac { u } { \\| u \\| } , u = \\left [ \\begin{array} { c } \\rho - \\tau + \\chi \\\\ 2 \\sigma \\end{array} \\right ] , \\end{align*}"}
-{"id": "5697.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow 0 } x u _ x & = 0 & & u ( 1 , y , t ) = 0 \\\\ u ( x , 0 , t ) & = 0 & & u ( x , 1 , t ) = 0 \\end{align*}"}
-{"id": "8176.png", "formula": "\\begin{align*} T _ \\Lambda \\big ( \\{ g _ i \\} _ { i \\in I } \\big ) = \\sum _ { i \\in I } \\Lambda _ i ^ * g _ i . \\end{align*}"}
-{"id": "222.png", "formula": "\\begin{align*} \\Delta _ { \\mathbb { H } } u = \\mathcal { B } _ 1 u , \\ . \\end{align*}"}
-{"id": "1859.png", "formula": "\\begin{align*} a = \\frac { 1 } { 2 \\sqrt { \\kappa _ j \\ell } } ( 1 + \\kappa _ j \\ell ) Y _ { j , 0 } \\qquad \\mbox { a n d } b = \\frac { 1 } { 2 \\sqrt { \\kappa _ j \\ell } } ( 1 - \\kappa _ j \\ell ) Y _ { j , 0 } , \\end{align*}"}
-{"id": "3505.png", "formula": "\\begin{align*} f _ { N } ( \\xi , t ) = - t g ( \\xi ) + \\log \\frac { \\Gamma ( 1 + [ N u ] - t ) } { \\Gamma ( 1 + [ N u ] ) } + ( M + 1 ) \\log \\frac { \\Gamma ( N - [ N u ] + t ) } { \\Gamma ( N - [ N u ] ) } , \\end{align*}"}
-{"id": "5954.png", "formula": "\\begin{align*} \\omega \\oplus _ t \\overline \\omega ( s ) : = { \\omega ( s \\wedge t ) } + \\overline \\omega ( s ) 1 _ { [ t , 1 ] } ( s ) . \\end{align*}"}
-{"id": "6105.png", "formula": "\\begin{align*} k \\pi _ k ( \\eta ^ k y ) = \\frac { 1 } { \\eta ^ k y } \\frac { 1 } { \\sqrt { 2 \\pi } } \\exp \\left ( - \\frac { ( \\log \\eta ^ k + \\log y ) ^ 2 } { 2 k ^ 2 } \\right ) . \\end{align*}"}
-{"id": "5588.png", "formula": "\\begin{align*} \\begin{cases} ( i \\partial _ t + \\partial _ x ^ 2 ) \\mathcal { S } w ( x , t ) = w ( x , t ) & ( x , t ) \\in \\mathbb { R } \\times \\mathbb { R } , \\\\ \\mathcal { S } w ( x , t ) = 0 & x \\in \\mathbb { R } \\end{cases} \\end{align*}"}
-{"id": "5978.png", "formula": "\\begin{align*} \\overline { X } ^ { n , q } ( t ) = \\int _ 0 ^ t \\overline { q } ( s ) d s + \\overline { B } ^ n ( t ) , \\end{align*}"}
-{"id": "10065.png", "formula": "\\begin{align*} \\begin{aligned} \\dot q & = - \\frac { 1 } { 4 } ( q + p ) ^ 3 \\left ( q + \\O ( ( q + p ) ^ 3 ) \\right ) , & \\dot p & = \\frac { 1 } { 4 } ( q + p ) ^ 3 \\left ( p + \\O ( ( q + p ) ^ 3 ) \\right ) , \\\\ \\dot z & = \\O ( ( q + p ) ^ 6 , \\rho ^ 6 ) , & \\dot \\rho & = \\O ( ( q + p ) ^ { 1 2 } , \\rho ^ { 1 2 } ) , \\\\ \\dot \\phi & = \\omega ^ 0 + \\O ( ( q + p ) ^ 6 , \\tilde \\rho ) . \\end{aligned} \\end{align*}"}
-{"id": "5595.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { - \\infty } ^ 0 | x | ^ 2 | u | ^ 2 d x = 4 \\emph { I m } \\int _ { - \\infty } ^ 0 x \\bar { u } u _ x d x \\end{align*}"}
-{"id": "8133.png", "formula": "\\begin{align*} & \\forall \\ , h \\in H , \\ : \\ : \\ : \\mu ^ r ( h ? ) \\varphi = \\mu ^ r \\ ! \\left ( h ' ? \\right ) \\varphi \\ ! \\left ( S ^ { - 1 } ( h '' ) \\right ) , \\\\ & \\mu ^ l = \\mu ^ r ( g ^ 2 ? ) , \\\\ & \\forall \\ , x , y \\in H , \\ : \\ : \\ : \\mu ^ r ( x y ) = \\mu ^ r ( S ^ 2 ( y ) x ) , \\ : \\ : \\ : \\mu ^ l ( x y ) = \\mu ^ l ( S ^ { - 2 } ( y ) x ) , \\\\ & \\forall \\ , x , y \\in H , \\ : \\ : \\ : \\mu ^ r ( g x y ) = \\mu ^ r ( g y x ) , \\ : \\ : \\ : \\mu ^ l ( g ^ { - 1 } x y ) = \\mu ^ l ( g ^ { - 1 } y x ) . \\end{align*}"}
-{"id": "6203.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } e ^ { - t \\theta } g ( t ) d t = e ^ { - A \\theta ^ { \\alpha } } . \\end{align*}"}
-{"id": "3974.png", "formula": "\\begin{align*} \\begin{pmatrix} a s ^ 2 & a b s \\\\ a b s & a b ^ 2 \\end{pmatrix} = U ' \\begin{pmatrix} \\alpha & 0 \\\\ 0 & 0 \\end{pmatrix} U , \\end{align*}"}
-{"id": "4833.png", "formula": "\\begin{align*} ( 1 - q ^ { \\alpha _ { i _ 1 } - \\alpha _ { i _ 2 } } ) = \\prod _ { \\substack { \\pi \\in \\mathbb F _ q [ T ] \\\\ \\textrm { m o n i c } \\\\ \\textrm { i r r e d u c i b l e } } } ( 1 - | \\pi | ^ { \\alpha _ { i _ 1 } - \\alpha _ { i _ 2 } - 1 } ) . \\end{align*}"}
-{"id": "4040.png", "formula": "\\begin{align*} I _ M ( P , V ) = \\sum _ { x \\in P } i ( x ) . \\end{align*}"}
-{"id": "6784.png", "formula": "\\begin{align*} ( d \\varphi ) ^ T \\varphi '' + \\nabla p = 0 \\end{align*}"}
-{"id": "5915.png", "formula": "\\begin{align*} K _ Y + B _ Y + M _ Y = \\pi ^ * ( K _ X + B + M ) , \\end{align*}"}
-{"id": "7977.png", "formula": "\\begin{align*} m ( G _ 2 ( q ) ) \\geq q ^ 3 > 2 \\left ( \\frac { q ^ 3 - 1 } { q - 1 } \\right ) = 2 ( q ^ 2 + q + 1 ) \\ , , \\end{align*}"}
-{"id": "6946.png", "formula": "\\begin{align*} \\hbox { b o u n d e d , c o n n e c t e d o p e n s e t } \\Omega \\subset \\R ^ N \\hbox { w i t h L i p s c h i t z b o u n d a r y , } \\ N = 2 , 3 . \\end{align*}"}
-{"id": "44.png", "formula": "\\begin{align*} ( \\delta Y _ t ^ { i + } ) ^ 2 = & \\ ( \\delta \\xi ^ { i + } ) ^ 2 + \\int _ t ^ T 2 \\delta Y _ s ^ { i + } [ F _ s ^ { i } ( Z _ s ^ i ) - \\bar { F } _ s ^ { i } ( \\bar { Z } _ s ^ i ) ] d s \\\\ & + \\int _ t ^ T 2 \\delta Y _ s ^ { i + } [ G _ s ^ { i } ( Y _ s ^ i , Y _ s ^ { - i } ) - \\bar { G } _ s ^ { i } ( \\bar { Y } _ s ^ i , \\bar { Y } _ s ^ { - i } ) ] d s \\\\ & - \\int _ t ^ T \\chi _ { \\{ \\delta Y _ s ^ i > 0 \\} } | \\delta Z _ s ^ i | ^ 2 d s - \\int _ t ^ T 2 \\delta Y _ s ^ { i + } ( \\delta Z _ s ^ i ) ^ { t r } d W _ s . \\end{align*}"}
-{"id": "854.png", "formula": "\\begin{gather*} u _ k \\to u , \\ ; v ^ * _ k \\to v ^ * , \\ ; u _ { h , k } ^ * \\to u _ h ^ * , \\ ; \\tilde v _ k \\to \\tilde v , \\ ; \\| u \\| = \\| v ^ * \\| _ * = 1 . \\end{gather*}"}
-{"id": "9476.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { m = p , q } \\displaystyle \\int _ { \\mathbb { R } ^ N } \\displaystyle \\int _ { \\mathbb { R } ^ N } & \\frac { J _ m ( u ( x ) - u ( y ) ) ( \\varphi ( x ) - \\varphi ( y ) ) } { \\vert x - y \\vert ^ { N + s m } } \\dd x \\dd y \\\\ & \\leq \\displaystyle \\sum _ { m = p , q } \\displaystyle \\int _ { \\mathbb { R } ^ N } \\displaystyle \\int _ { \\mathbb { R } ^ N } \\frac { J _ m ( v ( x ) - v ( y ) ) ( \\varphi ( x ) - \\varphi ( y ) ) } { \\vert x - y \\vert ^ { N + s m } } \\dd x \\dd y . \\end{align*}"}
-{"id": "9577.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { N - k } \\frac { ( 1 / c ) _ j q ^ j \\left ( q ^ { - ( N - k ) } \\right ) _ j } { ( q ) _ j \\left ( q ^ { - ( N - k ) } / c \\right ) _ j } = \\frac { ( q ) _ { N - k } } { ( c q ) _ { N - k } } , \\end{align*}"}
-{"id": "59.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t f ^ - - \\partial _ x f ^ - = { k ( x ) } \\ , g ( f ^ + - f ^ - ) , & \\\\ \\partial _ t f ^ + + \\partial _ x f ^ + = - { k ( x ) } \\ , g ( f ^ + - f ^ - ) \\ , . & \\end{cases} \\end{align*}"}
-{"id": "1656.png", "formula": "\\begin{align*} ( \\Phi ^ { - 1 } ) ^ * \\omega = \\sum _ { k \\in \\Z } d p _ k \\wedge d q _ k \\end{align*}"}
-{"id": "7169.png", "formula": "\\begin{gather*} \\frac { \\mathrm { d } } { \\mathrm { d } t } \\frac { \\partial \\chi ^ { t } ( x ) } { \\partial x } = v ^ { \\prime } \\left ( \\chi ^ { t } ( x ) \\right ) \\frac { \\partial \\chi ^ { t } ( x ) } { \\partial x } \\end{gather*}"}
-{"id": "2579.png", "formula": "\\begin{align*} R ( r ) = \\int _ r ^ \\infty \\big \\langle \\psi _ 0 \\big ( \\cdot , ( 1 + \\epsilon _ { R ( u ) } ) u \\phi \\big ) , \\phi ^ * \\big \\rangle _ m ^ { - 1 } d u , r \\in ( 0 , \\infty ) . \\end{align*}"}
-{"id": "534.png", "formula": "\\begin{align*} \\partial _ { i j } y _ 0 = \\sum _ { m = 1 } ^ 2 \\gamma _ { i j } ^ m \\partial _ m y _ 0 - ( \\Pi _ { y _ 0 } ) _ { i j } \\vec \\nu _ 0 \\mbox { f o r } i , j = 1 , 2 , \\end{align*}"}
-{"id": "7263.png", "formula": "\\begin{align*} - \\mu \\phi + L \\phi + \\frac { 1 } { 2 } \\left ( \\phi ^ { 2 } - \\widehat { \\phi ^ 2 } ( 0 ) \\right ) = 0 . \\end{align*}"}
-{"id": "7047.png", "formula": "\\begin{align*} E ( z ) : = E ( u ( z ) ) \\end{align*}"}
-{"id": "1138.png", "formula": "\\begin{align*} & a _ j ^ 2 \\leq ( G _ n ( C _ 0 ) / S ( I ) ) ^ 2 = O \\left ( \\frac { \\ln n } { n ^ 2 } \\right ) , \\end{align*}"}
-{"id": "5754.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 } { \\partial x ^ 2 } J _ 0 ( \\gamma _ m x ) & = \\frac { \\partial } { \\partial x } \\left ( \\gamma _ m J _ 1 ( \\gamma _ m x ) \\right ) , \\\\ & = \\frac { - \\gamma _ m ^ 2 } { 2 } \\left ( J _ 0 ( \\gamma _ m x ) - J _ 2 ( \\gamma _ m x ) \\right ) . \\end{align*}"}
-{"id": "3098.png", "formula": "\\begin{align*} R ( z ) = R _ 0 + R _ 1 z + R _ 2 z ^ 2 + \\cdots . \\end{align*}"}
-{"id": "8470.png", "formula": "\\begin{align*} \\partial _ { \\alpha } ^ n f _ j ( g ) = \\sum _ { k = 1 } ^ n \\sum \\frac { n ! } { ( k _ 1 ) ! . . . ( k _ n ) ! } \\partial _ { \\alpha } ^ k f _ j ( \\cdot , t ) \\circ g \\prod _ { l = 1 } ^ n \\Big ( \\frac { \\partial _ { \\alpha } ^ l g } { l ! } \\Big ) ^ { k _ l } , \\end{align*}"}
-{"id": "3217.png", "formula": "\\begin{align*} \\tilde { \\epsilon } _ { k } : = \\gamma _ 1 \\mu _ { k } ^ { 1 + \\tilde { c } \\alpha } \\end{align*}"}
-{"id": "8497.png", "formula": "\\begin{align*} \\zeta ( \\alpha , t ) - z _ j ( t ) = & \\zeta ( \\alpha , 0 ) - z _ j ( 0 ) + \\int _ 0 ^ t \\partial _ { \\tau } ( \\zeta ( \\alpha , \\tau ) - z _ j ( \\tau ) ) d \\tau \\\\ = & \\zeta ( \\alpha , 0 ) - z _ j ( 0 ) + \\int _ 0 ^ t D _ { \\tau } \\zeta ( \\alpha , \\tau ) d \\tau - \\int _ 0 ^ t b ( \\alpha , \\tau ) \\partial _ { \\alpha } \\zeta ( \\alpha , \\tau ) d \\tau - \\int _ 0 ^ t \\dot { z } _ j ( \\tau ) d \\tau \\end{align*}"}
-{"id": "5420.png", "formula": "\\begin{align*} \\bar { \\rho } ( \\mathcal M ) = \\limsup _ { n \\to \\infty } \\sup \\{ \\rho ( A _ n \\cdots A _ 1 ) ^ { 1 / n } : A _ i \\in \\mathcal M \\} , \\end{align*}"}
-{"id": "1551.png", "formula": "\\begin{align*} U _ 2 ^ \\delta \\circ F _ 1 ^ \\delta \\left ( \\begin{array} { c } z \\\\ x \\end{array} \\right ) & = \\left ( \\begin{array} { c } P ( b _ 0 ^ \\delta ( x ) + \\dots ) ( a _ 1 ^ \\delta ( x ) z + \\dots ) \\\\ b _ 0 ^ \\delta ( x ) + \\dots \\end{array} \\right ) \\\\ & = \\left ( \\begin{array} { c } \\left [ a _ 1 ^ \\delta ( x ) \\prod _ { n = 0 } ^ \\infty a _ 1 ^ \\delta ( ( b _ 0 ^ \\delta ) ^ { n + 1 } ( x ) ) \\right ] z + \\dots \\\\ b _ 0 ^ \\delta ( x ) + \\dots \\end{array} \\right ) \\end{align*}"}
-{"id": "4505.png", "formula": "\\begin{align*} x = \\langle x , x \\rangle \\cdot 1 _ X \\wedge x \\ ; . \\end{align*}"}
-{"id": "1356.png", "formula": "\\begin{align*} \\left ( h _ { n } \\right ) _ w ( \\lambda _ 0 , \\ldots , \\lambda _ n ) = ( - 1 ) ^ { n + 1 } ( \\lambda _ 0 , \\ldots , \\lambda _ n , w ) \\end{align*}"}
-{"id": "4809.png", "formula": "\\begin{align*} E \\left [ e ^ { 4 8 c _ { d , \\sigma } k ^ 2 T \\sup _ { 0 \\leq t \\leq T } | \\sigma _ i \\cdot B ^ i _ t | ^ 2 } \\right ] & = 1 + \\sum _ { p = 1 } ^ \\infty \\frac { ( 4 8 c _ { d , \\sigma } \\sigma _ i ^ 2 k ^ 2 T ) ^ p } { p ! } E \\left [ \\sup _ { 0 \\leq t \\leq T } | B _ t | ^ { 2 p } \\right ] \\\\ & \\leq 1 + \\sum _ { p = 1 } ^ \\infty \\frac { ( 4 8 k ^ 2 d T ) ^ p } { p ! } \\left ( \\frac { 2 p } { 2 p - 1 } \\right ) ^ 2 \\frac { ( 2 p ) ! } { 2 ^ p \\cdot p ! } T ^ p . \\end{align*}"}
-{"id": "9867.png", "formula": "\\begin{align*} r _ { Q - Q } ( z ) \\le | A | ^ { - 2 } | \\{ q _ 1 a ^ { - 1 } _ 1 - q _ 2 a ^ { - 1 } _ 2 = z ~ : ~ a _ 1 , a _ 2 \\in A , \\ , q _ 1 , q _ 2 \\in Q A \\} | = | A | ^ { - 2 } { I } ( Q A \\times Q A , A \\times A ) \\ , , \\end{align*}"}
-{"id": "2953.png", "formula": "\\begin{align*} G ( f ) = \\bigcup _ { n \\in \\mathbb N } \\{ G ( f _ { | 0 \\ldots n } ) : n \\in \\mathbb N \\} , ~ H ( f ) = \\bigcup _ { n \\in \\mathbb N } \\{ H ( f _ { | 0 \\ldots n } ) : n \\in \\mathbb N \\} \\end{align*}"}
-{"id": "9121.png", "formula": "\\begin{align*} \\rho _ { 1 M } ^ { ( N ) } ( \\mathbf { v } _ { 1 } ) = \\frac { n _ { o } } { \\pi ^ { 3 / 2 } \\left ( 2 T _ { o } / m \\right ) ^ { 3 / 2 } } \\exp \\left \\{ - \\frac { m \\left ( \\mathbf { v } _ { 1 } - \\mathbf { V } _ { o } \\right ) ^ { 2 } } { 2 T _ { o } } \\right \\} , \\end{align*}"}
-{"id": "4614.png", "formula": "\\begin{align*} M \\frac { \\mathbf { u } } { t } = - K \\mathbf { u } , \\end{align*}"}
-{"id": "1439.png", "formula": "\\begin{align*} \\mu ^ { n + 1 } ( 0 , x ) = \\mathbb { P } _ { \\delta } \\mu _ { 0 } ( x ) . \\end{align*}"}
-{"id": "4577.png", "formula": "\\begin{align*} E ( \\mu , g , 0 ) = \\mu ^ { 4 } \\{ ( x _ 1 , x _ 2 , y _ 1 , y _ 2 ) \\in A ^ 4 : x _ 1 - x _ 2 = g ( y _ 1 - y _ 2 ) \\} , \\end{align*}"}
-{"id": "5285.png", "formula": "\\begin{align*} & P E ^ d = Q E ^ d = E ^ d = A \\ ; , \\\\ & P E ^ { d p ^ r } = { ^ { ( r ) } } A \\ ; \\\\ & Q E ^ { d p ^ r } = { ^ { ( r ) } } A \\ ; \\end{align*}"}
-{"id": "6094.png", "formula": "\\begin{gather*} \\tau _ 1 = ( 1 \\ldots e _ 1 ) , \\\\ \\tau _ 2 = ( e _ 1 \\ldots ( e _ 1 + e _ 2 - 1 ) ) , \\\\ \\tau _ 3 = ( ( e _ 1 + e _ 2 - 1 ) \\ldots ( e _ 1 + e _ 2 + e _ 3 - 2 ) ) , \\\\ \\vdots \\\\ \\tau _ k = \\Big ( \\big ( \\sum _ { i = 1 } ^ { k - 1 } e _ i - ( k - 2 ) \\big ) \\ldots \\big ( \\sum _ { i = 1 } ^ { k } e _ i - ( k - 1 ) \\big ) \\Big ) , \\\\ \\vdots \\end{gather*}"}
-{"id": "6063.png", "formula": "\\begin{align*} \\min _ { \\pi _ 1 } \\left \\{ \\sum _ { k = 1 } ^ { M } d _ k ( \\pi _ 1 ) c _ k \\right \\} - \\min _ { \\pi _ 2 } \\left \\{ \\sum _ { k = 1 } ^ { M } d _ k ( \\pi _ 2 ) c _ k \\right \\} , \\end{align*}"}
-{"id": "4279.png", "formula": "\\begin{align*} \\textup { s a t } ( n , K _ 3 , C _ k ) = 0 \\end{align*}"}
-{"id": "4695.png", "formula": "\\begin{align*} \\frac { d u _ j } { d t } = - \\frac { 1 } { \\Delta x } \\left ( F _ { j + \\frac { 1 } { 2 } } - F _ { j - \\frac { 1 } { 2 } } \\right ) = : L ( u ) , \\end{align*}"}
-{"id": "8707.png", "formula": "\\begin{align*} p g f _ { T ^ * _ { 2 , 3 } } ( s ) = { p ( q + p + \\sqrt { q p } ) ( - q - p + \\sqrt { q p } ) u ( 1 - u ( 1 - q - p ) ) \\over ( p ^ 2 + q p + q ^ 2 ) ( 1 - u ( 1 - q - p - \\sqrt { q p } ) ( - 1 + u ( 1 - q - p + \\sqrt { q p } ) } \\end{align*}"}
-{"id": "8386.png", "formula": "\\begin{align*} g : = \\frac { \\bar { z } _ { t t } - i } { | z _ { t t } + i | } ( I + \\mathfrak { K } ^ { \\ast } ) ^ { - 1 } \\Big \\{ R e ( \\frac { i z _ { \\alpha } } { | z _ { \\alpha } | } ( g _ 1 + g _ 2 ) ) \\Big \\} \\end{align*}"}
-{"id": "7228.png", "formula": "\\begin{align*} \\phi = \\mu - \\sqrt { \\mu ^ { 2 } - 2 L _ r \\phi + \\hat { \\phi } ^ { 2 } ( 0 ) } . \\end{align*}"}
-{"id": "3227.png", "formula": "\\begin{align*} \\| \\Delta _ 1 w ^ k \\| = O ( \\| \\Phi _ { \\mu _ k } ( { w } ^ { k + \\frac { 1 } { 2 } } ) \\| ) = O ( \\mu _ k ^ { 1 + \\tilde { c } \\alpha } ) . \\end{align*}"}
-{"id": "774.png", "formula": "\\begin{align*} C _ { q ^ d } [ T - \\rho _ i ] : = \\{ z \\in \\overline { \\mathbb { F } _ q ( T ) } : z ^ { q ^ d } + ( T - \\rho _ i ) z = 0 \\} . \\end{align*}"}
-{"id": "9407.png", "formula": "\\begin{align*} f ( z ) = q ^ { h _ \\infty } \\prod _ { n = 1 } ^ \\infty ( 1 - q ^ n ) ^ { c ( n ) } \\end{align*}"}
-{"id": "1186.png", "formula": "\\begin{align*} V - V _ { x x } + c - \\sqrt { c ^ 2 + 2 V } = 0 , \\end{align*}"}
-{"id": "9344.png", "formula": "\\begin{align*} G = F / N , N = [ R , F ] R ^ 2 \\trianglelefteq _ \\mathrm { c } F . \\end{align*}"}
-{"id": "9692.png", "formula": "\\begin{align*} y = \\underset { x } { \\underbrace { z \\cdot G } } + \\eta , \\end{align*}"}
-{"id": "8342.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } \\frac { 1 } { ( \\beta - w _ 1 ) ( \\beta - \\overline { w _ 2 } ) } d \\beta = \\frac { 2 \\pi i } { \\overline { w _ 2 } - w _ 1 } \\end{align*}"}
-{"id": "9539.png", "formula": "\\begin{align*} M _ { S _ 2 } ( m , n ) : = \\sum _ { \\vec { \\pi } \\in S _ 2 , | \\vec { \\pi } | = n \\atop \\mathrm { c r a n k } ( \\vec { \\pi } ) = m } w _ c ( \\vec { \\pi } ) , \\end{align*}"}
-{"id": "7621.png", "formula": "\\begin{align*} \\frac { \\partial u ^ { \\varepsilon } } { \\partial t } = L ^ { \\varepsilon } u ^ { \\varepsilon } , u ^ { \\varepsilon } ( x , 0 ) = \\varphi ( x ) , \\varphi \\in L ^ 2 ( \\mathbb R ^ d ) , \\end{align*}"}
-{"id": "4094.png", "formula": "\\begin{align*} \\vartheta ( \\vartheta - \\alpha ) ( \\vartheta - \\alpha - \\tfrac { 1 } { 2 } ) \\psi - z \\psi = 0 . \\end{align*}"}
-{"id": "8878.png", "formula": "\\begin{align*} e _ { 9 b } & = e _ 9 | _ { b = 1 } = 1 6 p ^ 5 s ^ 3 - 2 4 p ^ 4 s ^ 3 - 7 2 p ^ 4 s ^ 2 + 1 2 p ^ 3 s ^ 3 + 4 6 8 p ^ 3 s ^ 2 - 2 s ^ 3 p ^ 2 - 2 4 p ^ 3 s - 4 2 6 p ^ 2 s ^ 2 \\\\ & + 2 4 s p ^ 2 + 1 1 1 s ^ 2 p + 2 p ^ 2 - 1 8 p s - 3 s ^ 2 + 6 s \\end{align*}"}
-{"id": "4904.png", "formula": "\\begin{align*} \\partial _ t \\eta ( t , x ) = u ( t , \\eta ( t , x ) ) \\end{align*}"}
-{"id": "2903.png", "formula": "\\begin{align*} v _ i = ( 0 , \\dots , 0 , s ^ \\prime , 0 , \\dots 0 ) , \\end{align*}"}
-{"id": "1105.png", "formula": "\\begin{align*} I _ { k , n } = \\cup _ { j = 1 } ^ k [ y _ j , y _ j + G _ n ( x _ j ) / S ( I ) ] , \\ J _ { k , n , j } = [ z _ j , z _ { j + 1 } ] , \\ 0 \\leq j \\leq k , \\end{align*}"}
-{"id": "1154.png", "formula": "\\begin{align*} 0 \\leq & \\mathbb { P } ( \\xi ^ { ( n ) } ( I _ { n , k } ) = \\xi ^ { ( n ) } ( J _ { n , k , j } ) = 0 ) = \\mathbb { P } ( \\xi ^ { ( n ) } ( I _ { n , k } \\cup J _ { n , k , j } ) = 0 ) \\\\ \\leq & \\mathbb { P } ( [ z _ j ' , z _ j ' + d _ 0 ] \\cup \\left ( \\cup _ { i = 1 } ^ k [ z _ i , z _ i + d _ 0 ] \\right ) ) = 0 ) \\\\ \\leq & \\mathbb { P } ( \\xi ^ { ( n ) } ( [ z _ j ' , z _ j ' + d _ 0 ] ) = 0 ) \\prod _ { i = 1 } ^ k \\mathbb { P } ( \\xi ^ { ( n ) } ( [ z _ i , z _ i + d _ 0 ] ) = 0 ) \\leq p _ { n , k } ^ { k + 1 } , \\end{align*}"}
-{"id": "2724.png", "formula": "\\begin{align*} I _ { N } ^ { 1 } & = - \\int _ { z _ { - } } ^ { z _ { + } } \\frac { d w } { 2 \\pi i } \\frac { \\eta _ { - } } { \\varphi ( x ) } e ^ { \\frac { \\eta _ { - } } { \\varphi ( x ) } ( v - u ) ( w - \\Re { z _ { - } } ) } = - \\frac { \\sin \\pi ( u - v ) } { \\pi ( u - v ) } . \\end{align*}"}
-{"id": "6698.png", "formula": "\\begin{align*} \\dot { q } _ i = \\alpha ' _ { i k } ( \\bar { q } ) \\ , \\dot { \\bar { q } } _ k + \\beta ' _ { i k l } ( \\bar { q } ) \\ , \\dot { \\bar { q } } _ k \\dot { \\bar { q } } _ l + \\beta _ { i k } ( \\bar { q } ) \\ , \\ddot { \\bar { q } } _ k , \\end{align*}"}
-{"id": "3147.png", "formula": "\\begin{align*} f ( u , v ) \\ F _ i ( u ) F _ i ( v ) = f ( v , u ) \\ F _ i ( v ) F _ i ( u ) , \\end{align*}"}
-{"id": "1953.png", "formula": "\\begin{align*} \\mathfrak { Z } & = \\textrm { t h e r a y c l a s s g r o u p o f $ K $ m o d u l o $ \\mathfrak { C } p ^ \\infty $ , } \\\\ Z ^ - & = \\textrm { t h e m a x i m a l q u o t i e n t o f $ \\mathfrak { Z } $ w h e r e c o m p l e x c o n j u g a t i o n a c t s a s $ - 1 $ } , \\\\ Z _ p ^ - & = \\textrm { t h e m a x i m a l $ p $ - p r o f i n i t e q u o t i e n t o f $ Z ^ - $ } . \\end{align*}"}
-{"id": "7929.png", "formula": "\\begin{align*} M _ { N - 2 } ( f ) \\triangleq h \\sum _ { i = 0 } ^ { N - 3 } f ( a + ( i + \\tfrac { 1 } { 2 } ) h ) \\end{align*}"}
-{"id": "9436.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq j \\leq D ( N ) - 1 } a _ j E _ 2 ( z ) = \\sum _ { 1 \\leq j \\leq D ( N ) - 1 } a _ j d _ j E _ 2 ( d _ j z ) . \\end{align*}"}
-{"id": "1592.png", "formula": "\\begin{align*} \\Phi _ { v ^ o _ n } ( z ^ o _ n , x ^ o _ n ) = \\Phi _ { v ^ \\iota _ n } ( z ^ \\iota _ n , x ^ \\iota _ n ) + \\sum _ { j = n ^ 2 + k _ n } ^ { n ^ 2 + 2 n - k _ n } \\left ( \\frac { \\sqrt { w _ j } } { 2 } + o \\left ( \\frac { 1 } { n ^ 2 } \\right ) \\right ) . \\end{align*}"}
-{"id": "1173.png", "formula": "\\begin{align*} & \\lim _ { n \\to + \\infty } n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } D _ n ( F _ n ( x _ 0 ) / 2 ) / 4 = e ^ { c _ 0 - x _ 0 } / 4 , \\end{align*}"}
-{"id": "9686.png", "formula": "\\begin{align*} C _ 2 = & K ^ 2 C _ { { \\rm p o l } , T } ^ 2 ( 1 + \\varepsilon _ 0 ) \\frac { ( 2 ( 1 + c _ 1 ^ 2 ) ) ^ { 1 / 2 } } { 1 - c _ 1 } \\mathop { } _ { \\mathbb T } \\sum _ { n \\in \\mathbb N } | \\eta _ n | ^ 2 \\\\ & \\ \\ + \\| X _ * ^ * \\| \\bigl ( 1 + \\sum _ { n \\geq 2 } \\xi _ n ^ 2 \\bigr ) \\mathop { } _ { \\mathbb T \\setminus \\sigma } | h _ 0 | . \\end{align*}"}
-{"id": "5701.png", "formula": "\\begin{align*} \\frac { d } { d z } z ^ { k - 1 } E _ { \\alpha , k } ( \\lambda z ^ \\alpha ) = z ^ { k - 2 } E _ { \\alpha , k - 1 } ( \\lambda z ^ \\alpha ) . \\end{align*}"}
-{"id": "5064.png", "formula": "\\begin{align*} \\mathcal { U } & = \\{ \\sigma ( U V ) : U \\in \\mathcal { U } \\} = \\{ \\sigma ( V U ^ { - 1 } ) : U \\in \\mathcal { U } \\} \\\\ & = \\{ \\sigma ( V U ^ { - 1 } W ) : U \\in \\mathcal { U } \\} = \\{ \\sigma ( V ^ { - 1 } U ) : U \\in \\mathcal { U } \\} . \\end{align*}"}
-{"id": "4401.png", "formula": "\\begin{align*} S _ { \\delta } : = S \\cap \\Omega _ { \\delta } \\end{align*}"}
-{"id": "2727.png", "formula": "\\begin{align*} f ( z ) : = f ( 4 ; z ) = 4 \\eta _ { - } ( z + 1 ) + \\log ( \\tau z - 1 ) - \\log ( z - \\tau ) , \\end{align*}"}
-{"id": "6782.png", "formula": "\\begin{align*} ( d \\Phi ^ t ) ^ T \\Phi '' + \\nabla _ { \\ ! \\beta } p = 0 \\ , . \\end{align*}"}
-{"id": "3383.png", "formula": "\\begin{align*} \\P \\Big ( \\| v _ I \\| _ 2 \\leq \\frac { C } { \\log ^ c n } \\frac { m } { n } I \\subset [ n ] , | I | = m \\Big ) & \\leq \\P \\Big ( \\| v _ I \\| _ 2 \\leq \\frac { C } { \\log ^ c n } \\frac { m } { n } I \\mathcal { E } \\Big ) + \\P ( \\mathcal { E } ^ c ) \\\\ & \\leq \\P \\left ( \\sum _ { i = 1 } ^ m Y _ i ^ 2 \\leq \\frac { C } { \\log ^ c n } \\frac { m ^ 2 } { n } \\right ) + \\exp ( - c n ) . \\end{align*}"}
-{"id": "7067.png", "formula": "\\begin{align*} \\sum _ { { { \\beta } } } \\eta _ { c , { { \\beta } } } \\ H ^ { a b } _ { c ; ( e { \\gamma } ; f { \\theta } ) , ( d { \\alpha } ; c { { \\beta } } ) } & = \\delta _ { f , d } \\delta _ { e , a } \\delta _ { { \\theta } , { \\alpha } } \\eta _ { a , { \\gamma } } . \\end{align*}"}
-{"id": "5215.png", "formula": "\\begin{align*} U ( x ) = C ( 1 + p ( x ) ) x ^ { \\rho } \\exp ( \\int _ { 1 } ^ { x } t ^ { - 1 } \\ell ( t ) d \\lambda ( t ) ) . \\end{align*}"}
-{"id": "1812.png", "formula": "\\begin{align*} X _ { i } : = \\omega _ { t _ { i } } - \\frac { t _ { i + 1 } - t _ { i } } { t _ { i + 1 } - t _ { i - 1 } } \\omega _ { t _ { i - 1 } } - \\frac { t _ { i } - t _ { i - 1 } } { t _ { i + 1 } - t _ { i - 1 } } \\omega _ { t _ { i + 1 } } \\end{align*}"}
-{"id": "1605.png", "formula": "\\begin{align*} 0 \\leq v ( X _ { T _ n } ^ Z ) = v ( x ) & + \\int _ 0 ^ { T _ n } ( \\mathcal { A } v ) ( X _ s ^ Z ) d s + \\int _ 0 ^ { T _ n } \\sigma ( X _ s ^ Z ) v ' ( X _ s ^ Z ) d B _ s \\\\ & - \\int _ 0 ^ { T _ n } v ' ( X _ s ^ Z ) d Z _ s ^ c + \\sum _ { s \\leq T _ n } ( v ( X _ s ^ Z ) - v ( X _ { s - } ^ Z ) ) , \\end{align*}"}
-{"id": "10003.png", "formula": "\\begin{align*} \\sup \\{ \\sigma _ { a } ( D ) - \\sigma _ { u } ( D ) \\colon D \\in \\mathfrak { D } ( \\mathbb { C } ) \\} = \\frac { 1 } { 2 } \\ , . \\end{align*}"}
-{"id": "8551.png", "formula": "\\begin{align*} \\mathcal { T } : = \\Big \\{ T \\in [ 0 , \\delta \\epsilon ^ { - 2 } ] : \\| \\zeta _ { \\alpha } - 1 \\| _ { H ^ s } \\leq 5 \\epsilon , ~ ~ \\| \\mathfrak { F } \\| _ { H ^ { s + 1 / 2 } } \\leq 5 \\epsilon , ~ ~ \\| D _ t \\mathfrak { F } \\| { H ^ s } \\leq 5 \\epsilon , \\forall ~ t \\in [ 0 , T ] \\Big \\} \\end{align*}"}
-{"id": "9547.png", "formula": "\\begin{align*} \\mathrm { s p t } ( n , N ) = \\frac { 1 } { 2 } \\left ( M _ { 2 , N } ( n ) - N _ { 2 , N } ( n ) \\right ) . \\end{align*}"}
-{"id": "6388.png", "formula": "\\begin{align*} z _ B = z _ { C _ 1 } \\cdots z _ { C _ k } , \\end{align*}"}
-{"id": "561.png", "formula": "\\begin{align*} \\| u \\| _ { U ^ p _ \\Phi } = \\| t \\mapsto e ^ { i t \\Phi ( - i \\nabla ) } u ( t ) \\| _ { U ^ p } , \\| v \\| _ { V ^ p _ \\Phi } = \\| t \\mapsto e ^ { i t \\Phi ( - i \\nabla ) } v ( t ) \\| _ { V ^ p } \\end{align*}"}
-{"id": "7986.png", "formula": "\\begin{align*} \\log g ( n ) = \\sum _ { p \\leq n } { \\frac { \\xi _ { p } ( 1 + \\xi _ { p } ) } { 2 } } \\log p \\ , , \\end{align*}"}
-{"id": "2045.png", "formula": "\\begin{align*} F _ { \\chi } ( z ) = \\sum _ { n = 1 } ^ \\infty a _ { \\chi } ( n ) e ^ { 2 \\pi i n z } \\end{align*}"}
-{"id": "5776.png", "formula": "\\begin{align*} R ( \\bullet , q , T ) = \\frac 1 { 1 - T } . \\end{align*}"}
-{"id": "139.png", "formula": "\\begin{align*} [ X , [ Y , Z ] ] = [ [ X , Y ] , Z ] + ( - 1 ) ^ { ( k - 1 ) ( l - 1 ) } [ Y , [ X , Z ] ] \\end{align*}"}
-{"id": "590.png", "formula": "\\begin{align*} \\tilde { \\Lambda } ^ x _ t = \\Lambda ^ x _ t \\ge - \\eta ^ x _ n , \\ ; \\ ; t \\in [ 0 , \\delta ^ x _ n ) \\quad \\quad \\{ \\tau ^ x \\ge \\delta ^ x _ n \\} \\end{align*}"}
-{"id": "3895.png", "formula": "\\begin{align*} g = \\mathrm { d i a g } ( \\lambda _ 1 , \\ldots , \\lambda _ { n + 1 } ) , \\end{align*}"}
-{"id": "9331.png", "formula": "\\begin{align*} \\int _ { v \\in \\S _ { a } ^ { m - 2 } } F _ { v } ^ 2 ( x ) \\ , d V _ { a } = \\frac { \\mathrm { V o l } ( \\S ^ { m - 2 } ) } { m - 1 } \\Big [ m F _ { a } ^ 2 ( x ) + \\mathrm { t r a c e } ( A _ { Q _ { 2 } ^ { x } } ) \\Big ] . \\end{align*}"}
-{"id": "4871.png", "formula": "\\begin{align*} { \\rm E x t } ^ 1 ( \\mathbb { C } _ { \\overline { \\sigma } } , { \\rm K e r } \\ , d ^ k _ { \\chi ' } ) = 0 . \\end{align*}"}
-{"id": "6956.png", "formula": "\\begin{align*} \\left \\{ \\textbf { S } _ k ( \\textbf { x } ) \\right \\} _ { i i } & = \\textbf { S } _ 1 ( \\textbf { x } _ { i } ) \\\\ \\left \\{ \\textbf { S } _ k ( \\textbf { x } ) \\right \\} _ { i j } & = \\left \\{ \\textbf { S } _ 2 ( \\textbf { x } _ { i j } ) \\right \\} _ { 1 2 } , \\end{align*}"}
-{"id": "6727.png", "formula": "\\begin{align*} \\bar { q } _ { k l } = h _ { k l } , \\end{align*}"}
-{"id": "8326.png", "formula": "\\begin{align*} \\dot { z } _ j ( t ) = ( v - \\frac { \\lambda _ j i } { \\overline { 2 \\pi ( z - z _ j ( t ) } ) } ) \\Big | _ { z = z _ j ( t ) } = \\bar { F } ( z _ j ( t ) , t ) - \\sum _ { k : k \\neq j } \\frac { \\lambda _ k i } { 2 \\pi \\overline { z _ k ( t ) - z _ j ( t ) } } \\end{align*}"}
-{"id": "221.png", "formula": "\\begin{align*} \\Delta _ { \\mathbb { H } } = \\Delta _ z + \\frac { | z | ^ 2 } { 4 } \\partial _ { t } ^ 2 + \\sum _ { i = 1 } ^ n \\partial _ t ( y _ j \\partial _ { x _ j } - x _ j \\partial _ { y _ j } ) \\end{align*}"}
-{"id": "4253.png", "formula": "\\begin{align*} \\delta ( a ( x ) \\lvert d w \\rvert ^ { p - 2 } d w ) = 0 \\delta w = 0 \\Omega . \\end{align*}"}
-{"id": "7463.png", "formula": "\\begin{align*} [ a x + b y , z ] = a [ x , z ] + b [ y , z ] , [ x , a y + b z ] = a [ x , y ] + b [ x , z ] , \\quad , \\\\ [ x , x ] = 0 , \\quad , \\\\ \\left [ \\alpha ( x ) , [ y , z ] \\right ] + \\left [ \\alpha ( z ) , [ x , y ] \\right ] + \\left [ \\alpha ( y ) , [ z , x ] \\right ] = 0 , \\quad . \\end{align*}"}
-{"id": "6034.png", "formula": "\\begin{align*} \\overline { \\lambda } ( G ) = \\sum _ { \\{ u , v \\} \\subseteq V ( G ) } \\lambda _ G ( u , v ) / \\tbinom { n } { 2 } . \\end{align*}"}
-{"id": "3114.png", "formula": "\\begin{align*} N ^ { [ n ] } _ { L + L ' , 0 } = \\sum _ { i + j = n } p ^ * N ^ { [ i ] } _ { L , 0 } + q ^ * N ^ { [ j ] } _ { L ' , 0 } \\ , \\ , . \\end{align*}"}
-{"id": "955.png", "formula": "\\begin{align*} \\mu _ j \\ ; \\ge \\ ; \\max \\left \\{ \\frac { \\alpha + \\sqrt { \\alpha ^ 2 + 4 \\alpha \\kappa _ { J _ m } ^ 2 ( 1 - \\eta _ 1 ) } } { 2 ( 1 - \\eta _ 1 ) } , \\eta _ 2 \\right \\} \\frac { 1 } { \\| J _ { m _ j } ^ \\top r _ { m _ j } \\| } : = \\frac { \\kappa _ { \\mu g } } { \\| J _ { m _ j } ^ \\top r _ { m _ j } \\| } , \\end{align*}"}
-{"id": "4935.png", "formula": "\\begin{align*} x ^ n y z ^ n = - y ( z ^ { n ^ 4 } - z ^ n x ^ n ) + y z ^ { n ^ 4 } \\end{align*}"}
-{"id": "9144.png", "formula": "\\begin{align*} \\rho _ { 1 } ^ { \\left ( N \\right ) ( + ) } ( \\mathbf { x } _ { 1 } ^ { ( + ) } ( t _ { 1 } ) , t _ { i } ) = \\rho ^ { \\left ( N \\right ) ( - ) } ( \\mathbf { x } _ { 1 } ^ { ( + ) } ( t _ { i } ) , t _ { i } ) , \\end{align*}"}
-{"id": "339.png", "formula": "\\begin{align*} \\begin{array} { l l } c _ { n - 1 } ( \\alpha _ 1 ) = ( n - 2 ) ! \\Sigma ^ 2 v _ { 2 n - 4 } & c _ n ( \\alpha _ 1 ) = 0 \\\\ c _ { n - 1 } ( \\alpha _ 2 ) = 0 & c _ n ( \\alpha _ 2 ) = ( n - 1 ) ! \\Sigma ^ 2 v _ { 2 n - 2 } . \\end{array} \\end{align*}"}
-{"id": "1230.png", "formula": "\\begin{align*} \\int _ { | \\eta | \\ge 1 0 } e ^ { - 3 i \\Phi ( \\zeta , \\eta ) } f ( \\eta ) i ( \\partial _ \\zeta \\Phi ) g ( \\eta - \\zeta ) d \\eta = \\int _ { | \\eta | \\ge 1 0 } e ^ { - 3 i \\Phi ( \\zeta , \\eta ) } \\partial _ \\eta \\left ( \\frac { \\partial _ \\zeta \\Phi } { \\partial _ \\eta \\Phi } f ( \\eta ) g ( \\eta - \\zeta ) \\right ) d \\eta . \\end{align*}"}
-{"id": "7673.png", "formula": "\\begin{align*} \\tau ( \\alpha ( x ) ) \\beta ( y ) = \\tau ( \\beta ( x ) ) \\alpha ( y ) & \\implies \\tau ( x ) \\beta ( y ) = \\tau ( x ) \\alpha ( y ) \\\\ & \\implies \\beta ( y ) = \\alpha ( y ) . \\end{align*}"}
-{"id": "5534.png", "formula": "\\begin{align*} & \\sqrt { T } E ^ { \\Pi ^ { D _ T } } [ \\max _ { k , j } | Z _ { \\lambda , k , j } | | X ^ T ] \\\\ & \\lesssim \\sqrt { 2 \\log 2 ( V _ \\lambda ^ { \\otimes d } ) } ( C _ T ' + 1 ) \\max _ { k , j } ( \\| \\tilde { \\Phi } _ { \\lambda , k , j } \\| _ { \\mu _ 0 } ^ 2 + c _ T ' ) = O _ { P _ { b _ 0 } } ( \\sqrt { \\lambda } ) . \\end{align*}"}
-{"id": "1092.png", "formula": "\\begin{align*} & n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } \\int _ I D _ n ( \\sqrt { 4 - y ^ 2 } / S ( I ) \\cdot G _ n ( x ) / 2 ) d y \\\\ = & n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } \\int _ { - I } D _ n ( \\sqrt { 4 - y ^ 2 } / S ( - I ) \\cdot G _ n ( x ) / 2 ) d y \\\\ \\to & M ( - I ) e ^ { c _ 0 - x } = M ( I ) e ^ { c _ 0 - x } , \\end{align*}"}
-{"id": "6981.png", "formula": "\\begin{align*} K _ \\rho Z _ \\rho ^ { * n } 1 = & S ^ { * n } K _ \\rho 1 = S ^ { * n } P ( \\rho ) = S ^ { * n } P ( 1 - | b | ^ 2 ) \\\\ = & - S ^ { * n } P ( | b | ^ 2 ) = - S ^ { * n } T _ { \\bar b } b = - T _ { \\bar b } S ^ { * n } b . \\end{align*}"}
-{"id": "9139.png", "formula": "\\begin{align*} W _ { M } ( \\rho _ { 1 \\infty } ^ { ( N ) } ( \\mathbf { x } _ { 1 } ) , \\mathbf { b } ) = 0 \\end{align*}"}
-{"id": "4061.png", "formula": "\\begin{align*} \\frac { 1 } { Z _ { n } } \\prod _ { j < k } ( x _ { k } - x _ { j } ) ( x _ { k } ^ { \\theta } - x _ { j } ^ { \\theta } ) \\prod _ { j = 1 } ^ { n } w ( x _ { j } ) . \\end{align*}"}
-{"id": "1250.png", "formula": "\\begin{align*} \\int e ^ { 3 i \\eta \\nu ^ 2 / 4 } \\left ( \\eta - \\frac { \\nu } { 2 } \\right ) S ' \\left ( \\frac { \\eta + \\nu } { 2 } \\right ) S \\left ( \\frac { \\eta - \\nu } { 2 } \\right ) d \\nu = O ( 1 ) . \\end{align*}"}
-{"id": "7945.png", "formula": "\\begin{align*} \\sum _ { a \\in [ n + r ] } x _ a = 0 . \\end{align*}"}
-{"id": "7267.png", "formula": "\\begin{align*} \\int _ { - \\pi } ^ \\pi \\left [ \\frac { 1 } { 2 } \\phi ^ { 2 } ( x ) - \\mu \\phi ( x ) \\right ] \\psi _ { x x } ( x ) \\ , d x = \\int _ { - \\pi } ^ \\pi \\phi ( x ) \\psi ( x ) \\ , d x \\end{align*}"}
-{"id": "9160.png", "formula": "\\begin{align*} s ( \\varphi \\# h ) = [ \\varepsilon \\# S ( h ) ] [ \\sigma ^ { - 1 } ( h , S ( h ) ) S ( \\varphi \\nu _ 1 ^ { - 1 } ( h ) ) \\nu _ 2 ^ { - 1 } ( h ) \\# 1 _ H ] , \\end{align*}"}
-{"id": "7426.png", "formula": "\\begin{align*} \\displaystyle \\ell ( I ) = \\int _ X \\ell ( f ( x ) ) d x \\end{align*}"}
-{"id": "6550.png", "formula": "\\begin{align*} m ^ P _ \\beta : = \\sum _ { \\left \\{ k | \\beta \\ , : \\ , P \\in E ( \\beta / k ) \\right \\} } \\frac { ( - 1 ) ^ { ( k - 1 ) w / k } } { k ^ 2 } \\ , \\mu ( k ) \\ , \\mathcal N ^ P _ { \\beta / k } ( S , E ) . \\end{align*}"}
-{"id": "2180.png", "formula": "\\begin{align*} K _ \\ell ( \\mathbf x , \\mathbf y ) = 2 ^ { \\mathbf N \\ell } \\tilde K _ \\ell ( 2 ^ \\ell \\mathbf x , 2 ^ { \\ell } \\mathbf y ) . \\end{align*}"}
-{"id": "6645.png", "formula": "\\begin{align*} M _ { 1 , Q } f = \\sum _ { Q \\leq q < 2 Q } \\sum _ { a \\in \\mathbb Z _ q ^ { \\times } } e _ q ( - \\lambda ^ 2 a ) C ^ { a / q , 1 } _ { \\tau } f , 1 \\leq Q \\leq N / 2 , \\end{align*}"}
-{"id": "7252.png", "formula": "\\begin{align*} D _ { \\phi \\phi } ^ 2 F [ 0 , \\mu ^ * _ { k } ] ( \\phi ^ * _ { k } , \\phi ^ * _ { k } ) & = ( \\phi ^ * _ { k } ) ^ 2 , \\\\ D _ { \\phi , \\mu } ^ 2 F [ 0 , \\mu ^ * _ { k } ] \\phi ^ * _ { k } & = - \\phi ^ * _ { k } . \\end{align*}"}
-{"id": "2896.png", "formula": "\\begin{align*} [ P ^ { i j } , P ^ { j k } ] & = 2 F ^ { i j k } , \\\\ [ P ^ { j k } , F ^ { i j k } ] & = P ^ { i k } P ^ { j k } - P ^ { j k } P ^ { i j } + 2 P ^ { i k } C ^ j - 2 P ^ { i j } C ^ k , \\\\ [ P ^ { k l } , F ^ { i j k } ] & = P ^ { i k } P ^ { j l } - P ^ { i l } P ^ { j k } , \\\\ [ F ^ { i j k } , F ^ { j k l } ] & = F ^ { j k l } P ^ { i j } - F ^ { i k l } \\big ( P ^ { j k } + 2 C ^ j \\big ) - F ^ { i j k } P ^ { j l } , \\\\ [ F ^ { i j k } , F ^ { k l m } ] & = F ^ { i l m } P ^ { j k } - P ^ { i k } F ^ { j l m } , \\end{align*}"}
-{"id": "4729.png", "formula": "\\begin{align*} \\mathcal { A } = \\Big \\{ T = \\begin{pmatrix} t _ { 0 } & b t _ { 1 } & \\ldots & b t _ { n - 1 } \\\\ a t _ { n - 1 } & t _ { 0 } & \\ldots & b t _ { n - 2 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a t _ { 1 } & a t _ { 2 } & \\ldots & t _ { 0 } \\end{pmatrix} : t _ 0 , \\dots , t _ { n - 1 } \\in \\mathbb { C } \\Big \\} . \\end{align*}"}
-{"id": "6964.png", "formula": "\\begin{align*} \\begin{aligned} \\beta _ 0 & = h ( x _ 0 ) , \\beta _ 1 = x _ 1 + \\ln | h ' ( x _ 0 ) | , \\\\ \\beta _ n & = n ! \\sum _ { k = 1 } ^ { n - 1 } \\frac { 1 } { k ! } \\frac { h ^ { ( k ) } ( x _ 0 ) } { ( h ' ( x _ 0 ) ) ^ n } \\sum _ { i _ 1 + \\dots + i _ k = n } \\frac { x _ { i _ 1 } } { i _ 1 ! } \\dots \\frac { x _ { i _ k } } { i _ k ! } + \\frac { h ^ { ( n ) } ( x _ 0 ) } { ( h ' ( x _ 0 ) ) ^ n } , n \\geq 2 , \\end{aligned} \\end{align*}"}
-{"id": "5185.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } F _ { n } ( a _ { n } x + b _ { n } ) = H _ { 1 } ( x ) , x \\in C ( H _ { 1 } ) \\end{align*}"}
-{"id": "165.png", "formula": "\\begin{align*} & X _ C ( f \\circ q _ M ) = 0 , & X _ C ( l _ { d f } ) = 0 , \\\\ & X _ V ( f \\circ q _ M ) = 0 , & X _ V ( l _ { d f } ) = 0 . \\end{align*}"}
-{"id": "6009.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } \\tilde { \\rho } ^ g _ n ( n F \\circ L _ n ) = \\sup _ { Q \\in \\P _ 1 ( \\C ) } ( F ( Q ) - \\tilde { \\alpha } ^ g ( Q ) ) . \\end{align*}"}
-{"id": "8194.png", "formula": "\\begin{align*} J _ { n } ( F , G ) = \\frac { { \\rm t r } ( F - G ) ( F - G ) ^ { T } } { n ( n - 1 ) } = \\frac { \\sum _ { i , j \\in [ n ] } ( F ^ { i j } - G ^ { i j } ) ^ { 2 } } { n ( n - 1 ) } , \\end{align*}"}
-{"id": "8488.png", "formula": "\\begin{align*} \\dot { y } ( t ) = - F _ 2 ( z _ 2 ( t ) , t ) - \\frac { | \\lambda | } { 4 \\pi x ( t ) } . \\end{align*}"}
-{"id": "6875.png", "formula": "\\begin{align*} \\mathsf { d i m } \\big ( \\mathsf { V } ( I _ { 3 , 8 } ) \\big ) = 4 > 3 = \\mathsf { d i m } \\big ( \\mathsf { V } ( \\sqrt [ \\mathbb { R } ] { I _ { 3 , 8 } } ) \\big ) \\ , . \\end{align*}"}
-{"id": "7771.png", "formula": "\\begin{align*} \\| T \\| _ { B _ 1 \\to B _ 2 } = \\sup _ { \\| f \\| _ { B _ 1 } \\le 1 } \\| T ( f ) \\| _ { B _ 2 } . \\end{align*}"}
-{"id": "1450.png", "formula": "\\begin{align*} \\mu ^ { n + 1 } ( t , \\cdot ) = \\mathbb { P } _ { \\delta } \\mu _ { 0 } + \\int _ { 0 } ^ { t } \\left [ \\Delta \\mu ^ { n + 1 } ( \\tau , \\cdot ) - \\varepsilon \\mathrm { d i v } \\left ( \\left ( \\bar { m } + \\mu ^ { n } ( \\tau , \\cdot ) \\right ) \\Theta _ { p } ( \\tau , \\cdot , \\mu ^ { n } , D w ^ { n } ) \\right ) \\right ] \\ d \\tau , \\end{align*}"}
-{"id": "3092.png", "formula": "\\begin{align*} | T b _ l | = & | \\int _ { K _ l } ( \\frac { x _ j - y _ j } { | x - y | ^ { n + 1 - \\beta } } - \\frac { x _ j - \\bar y _ j } { | x - \\bar y | ^ { n + 1 - \\beta } } ) b _ l d y | \\\\ \\le & C \\delta _ l \\int _ { K _ l } \\frac { 1 } { | x - y | ^ { n + 1 - \\beta } } | b _ l ( y ) | d y \\\\ = & C \\delta _ l \\int _ { R ^ n } \\frac { | b _ l ( y ) | \\mathrm { I } _ { K _ l } ( y ) } { | x - y | ^ { n + 1 - \\beta } } d y , \\end{align*}"}
-{"id": "7462.png", "formula": "\\begin{align*} \\displaystyle \\left \\lvert D \\left ( \\lambda \\mapsto ( \\pi _ \\lambda ( g ) e , e ) \\right ) _ { \\lambda = 0 } \\right \\rvert \\ll \\sigma ( g ) ^ { \\deg ( D ) } \\int _ K \\delta _ P ( m _ P ( k g ) ) ^ { 1 / 2 } \\lvert ( \\tau ( m _ P ( k g ) ) e ( k _ P ( k g ) ) , e ( k ) ) \\rvert d k \\end{align*}"}
-{"id": "8086.png", "formula": "\\begin{align*} \\mu ( h ) = - \\sum _ { k = 1 } ^ \\infty \\frac { B _ { 2 k } } { ( 2 k ) ! } h ^ { 2 k - 1 } = - \\frac { h } { 1 2 } + \\frac { h ^ 3 } { 7 2 0 } - O ( h ^ 5 ) . \\end{align*}"}
-{"id": "1162.png", "formula": "\\begin{align*} & A _ { 1 } = \\frac { 1 } { 2 4 } , \\ A _ 2 = \\frac { 1 } { 2 4 \\pi } , \\ A _ 4 = \\frac { 1 } { 2 7 0 \\pi } . \\end{align*}"}
-{"id": "4218.png", "formula": "\\begin{align*} E _ n ^ { - 1 } ( z ) A _ n ^ { ( 2 ) } ( s ) E _ n ( z ) & = n ^ 3 \\left ( E _ n ^ { - 1 } ( z ) A _ n ^ { ( 1 ) } ( 0 ) E _ n ( z ) \\right ) \\left ( E _ n ^ { - 1 } ( z ) B _ n ^ { ( 1 ) } ( s ) E _ n ( z ) \\right ) \\\\ & = n ^ 3 \\cdot \\mathcal { O } ( 1 ) \\cdot \\mathcal { O } \\left ( n ^ { - \\frac { 1 } { 2 } } \\right ) = \\mathcal { O } \\left ( n ^ { \\frac { 5 } { 2 } } \\right ) \\end{align*}"}
-{"id": "2253.png", "formula": "\\begin{align*} ( \\delta _ x | \\delta _ y ) & = \\delta ( x - y ) & \\int \\d x | \\delta _ x ) ( \\delta _ x | & = 1 . \\end{align*}"}
-{"id": "1834.png", "formula": "\\begin{align*} T _ { \\mu \\nu } = ( \\partial _ \\mu \\phi ) ( \\partial _ \\nu \\phi ) - \\frac { 1 } { 2 } g _ { \\mu \\nu } \\left [ g ^ { \\alpha \\beta } ( \\partial _ \\alpha \\phi ) ( \\partial _ \\beta \\phi ) - V \\phi ^ 2 \\right ] . \\end{align*}"}
-{"id": "8311.png", "formula": "\\begin{align*} d _ I ( t ) ^ { - 1 } = \\Big ( \\min _ { j = 1 , 2 } \\inf _ { \\alpha \\in \\mathbb { R } } | \\zeta ( \\alpha , t ) - z _ j ( t ) | \\Big ) ^ { - 1 } \\leq ( 1 + \\frac { | \\lambda | } { 2 0 \\pi x ( 0 ) } t ) ^ { - 1 } , \\end{align*}"}
-{"id": "3141.png", "formula": "\\begin{align*} \\hat { \\widetilde { \\mathbb { B } } } ( \\bar t ) = \\frac { \\widehat { T } _ { N - 1 , N } ( \\bar t ^ { N - 1 } ) \\widehat { T } _ { N - 2 , N - 1 } ( \\bar t ^ { N - 2 } ) \\cdots \\widehat { T } _ { 2 3 } ( \\bar t ^ 2 ) \\widehat { T } _ { 1 2 } ( \\bar t ^ 1 ) | 0 \\rangle } { \\prod _ { i = 1 } ^ { N - 1 } \\hat \\lambda _ { i + 1 } ( \\bar t ^ { i } ) \\prod _ { i = 1 } ^ { N - 2 } f ( \\bar t ^ { i + 1 } , \\bar t ^ i ) } . \\end{align*}"}
-{"id": "2516.png", "formula": "\\begin{align*} P ( Z _ n > 0 ) = n ^ { - 1 / \\alpha } L ( n ) , \\end{align*}"}
-{"id": "836.png", "formula": "\\begin{align*} ( \\omega _ i \\wp ^ { - 1 } ) ^ \\sigma = ( \\omega _ i ^ \\sigma ) ( \\wp ^ { - 1 } ) ^ \\sigma = \\omega _ i \\rho ( \\sigma ) ( \\wp ^ { - 1 } ) ^ \\sigma = \\omega _ i \\wp ^ { - 1 } \\wp ^ \\sigma ( \\wp ^ { - 1 } ) ^ \\sigma = \\omega _ i \\wp ^ { - 1 } . \\end{align*}"}
-{"id": "770.png", "formula": "\\begin{align*} \\mathrm { G a l } ( K _ { q ^ d , M } / \\mathbb { F } _ { q ^ d } ( T ) ) \\cong { \\bigtimes } _ { j = 1 } ^ r \\mathrm { G a l } ( K _ { q ^ d , M _ j ^ { a _ j } } / \\mathbb { F } _ { q ^ d } ( T ) ) \\end{align*}"}
-{"id": "9126.png", "formula": "\\begin{align*} \\rho _ { 1 } ^ { ( N ) } ( t _ { o } ) \\equiv \\rho _ { 1 } ^ { ( N ) } ( \\mathbf { x } _ { 1 } , t _ { o } ) = \\rho _ { 1 ( o ) } ^ { ( N ) } ( \\mathbf { x } _ { 1 } ) , \\end{align*}"}
-{"id": "8242.png", "formula": "\\begin{align*} F ( z ) = ( - q ; q ) _ \\infty ( z q , z ^ { - 1 } q , q ^ 2 ; q ^ 2 ) _ \\infty . \\end{align*}"}
-{"id": "1385.png", "formula": "\\begin{align*} \\Box _ { n - 1 } ( \\lambda _ 0 , \\ldots , \\lambda _ { l - 1 } \\lambda _ { l } , \\ldots , \\lambda _ n ; J ) = \\sum _ { \\varepsilon = 0 } ^ 1 \\Xi ( \\lambda _ 0 , \\ldots , \\lambda _ n ; J , J _ { ( l ) } - \\varepsilon ) . \\end{align*}"}
-{"id": "9916.png", "formula": "\\begin{align*} \\pi _ { n - } ^ { \\mu } ( A ) & = \\int _ { \\mathcal { X } } T ( A | x ) \\pi _ { n - 1 } ^ { \\mu } ( d x ) = \\int _ { \\mathcal { X } } \\int _ { A } \\frac { d T ( \\cdot | x ) } { d \\bar { \\mu } } ( a ) \\bar { \\mu } ( d a ) \\pi _ { n - 1 } ^ { \\mu } ( d x ) \\\\ & = \\int _ { A } \\left ( \\int _ { \\mathcal { X } } \\frac { d T ( \\cdot | x ) } { d \\bar { \\mu } } ( a ) \\pi _ { n - 1 } ^ { \\mu } ( d x ) \\right ) \\bar { \\mu } ( d a ) \\end{align*}"}
-{"id": "6763.png", "formula": "\\begin{align*} \\phi ( t x ) = \\phi ( x ) \\tau _ i ( t ) ^ { - 1 } . \\end{align*}"}
-{"id": "3216.png", "formula": "\\begin{align*} & \\| \\Delta X _ r \\circ \\Delta Y _ r \\| _ F = O ( \\mu _ r ^ 2 ) , \\\\ & \\| X _ r \\circ Y _ r - \\mu _ r I \\| _ F = O ( \\mu _ r ^ { 1 + \\zeta } ) . \\end{align*}"}
-{"id": "7033.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n } \\frac { 1 } { \\lambda _ { j } } \\left ( \\frac { z \\mathbb { E } _ { \\alpha _ { j } , \\beta _ { j } } ^ { \\prime } ( z ) } { \\mathbb { E } _ { \\alpha _ { j } , \\beta _ { j } } ( z ) } \\right ) = \\zeta ( 1 - \\delta ) p ( z ) + \\frac { ( 1 - \\delta ) z p ^ { \\prime } ( z ) } { \\delta + ( 1 - \\delta ) p ( z ) } + \\sum _ { j = 1 } ^ { n } \\frac { 1 } { \\lambda _ { j } } - \\zeta ( 1 - \\delta ) . \\end{align*}"}
-{"id": "5972.png", "formula": "\\begin{align*} \\rho _ n ^ g ( f ) : = \\rho _ { n - 1 } ^ g \\bigl ( \\ , ( \\omega _ 1 , \\ldots , \\omega _ { n - 1 } ) \\mapsto \\rho ( f ( \\omega _ 1 , \\ldots , \\omega _ { n - 1 } , \\cdot ) ) \\ , \\bigr ) . \\end{align*}"}
-{"id": "5039.png", "formula": "\\begin{align*} E [ u _ { n x } ( V _ 1 , v _ 2 , \\dots , v _ m ) ] = \\dots = E [ u _ { n x } ( v _ 1 , v _ 2 , \\dots , V _ m ) ] = 0 . \\end{align*}"}
-{"id": "2526.png", "formula": "\\begin{align*} \\psi ( x , z ) & = - \\beta ( x ) z + \\kappa ( x ) \\int _ 0 ^ \\infty ( e ^ { - z y } - 1 + z y ) \\frac { d y } { \\Gamma ( - \\gamma ( x ) ) y ^ { 1 + \\gamma ( x ) } } \\\\ & = - \\beta ( x ) z + \\kappa ( x ) z ^ { \\gamma ( x ) } , x \\in E , z \\geq 0 , \\end{align*}"}
-{"id": "5359.png", "formula": "\\begin{align*} \\frac { n - k } { n - 1 } \\eqref { b 2 0 } + \\frac { k - 1 } { n - 1 } \\eqref { b 2 5 } = \\eqref { b 5 0 } . \\end{align*}"}
-{"id": "9212.png", "formula": "\\begin{align*} e ( G ' ) & \\leq \\binom { \\frac { k } { 2 } } { 2 } + \\left ( \\frac { k } { 2 } - 1 \\right ) \\left ( n - \\frac { k } { 2 } \\right ) = \\binom { \\frac { k } { 2 } - 1 } { 2 } + \\left ( \\frac { k } { 2 } - 1 \\right ) \\left ( n - \\frac { k } { 2 } + 1 \\right ) . \\end{align*}"}
-{"id": "7080.png", "formula": "\\begin{align*} \\xi ( x , y ) = \\xi ( x , y ) \\xi ( y , y ) = \\xi ( x + y , y ) \\implies \\xi ( x , y ) = \\xi ( x , x + y ) = \\xi ( x + y , y ) \\end{align*}"}
-{"id": "3662.png", "formula": "\\begin{align*} \\mu _ i = E ( y _ i | \\delta _ i , i \\in U ) \\end{align*}"}
-{"id": "9297.png", "formula": "\\begin{align*} \\tilde u : = u - \\tau \\tilde \\eta , \\end{align*}"}
-{"id": "1213.png", "formula": "\\begin{align*} V ( t , \\xi ) : = \\inf _ { u \\in \\mathcal { U } } J ( t , \\xi , u ) , \\textnormal { w h e r e } J ( t , \\xi , u ) : = \\mathbb { E } ^ { \\gamma _ 0 } \\left [ \\int _ { 0 } ^ { t } \\Vert \\mathbf { Z } _ { r } ^ { u , t , \\xi } ( \\theta ) \\Vert ^ 2 \\hat { q } ( d r , d \\theta ) \\right ] . \\end{align*}"}
-{"id": "3807.png", "formula": "\\begin{align*} \\Omega = \\sum _ { n _ 2 \\ll N _ 0 / n _ 1 ^ 2 } \\Bigl | \\sum _ { q _ 2 \\sim C / q _ 1 } \\ ; \\sum _ { m \\sim M _ 1 } \\frac { \\lambda _ f ( m ) } { m ^ { 1 / 4 } } \\ ; \\mathcal { C } ( \\dots ) \\ : I ( m , n _ 1 ^ 2 n _ 2 , q ) \\Bigr | ^ 2 , \\end{align*}"}
-{"id": "2008.png", "formula": "\\begin{align*} e \\left ( H ^ 1 ( C , L ^ { \\oplus 5 } ) ^ \\vee \\right ) = \\prod _ { 0 < k < \\frac { d + 1 } { 5 } \\atop \\langle k \\rangle = \\left \\langle \\frac { d + 1 } { 5 } \\right \\rangle } ( k z ) ^ 5 . \\end{align*}"}
-{"id": "7020.png", "formula": "\\begin{align*} \\alpha _ R \\circ T _ { g , R } = T _ { g , R } \\circ \\alpha _ R , g \\in G \\end{align*}"}
-{"id": "7512.png", "formula": "\\begin{align*} f ( \\alpha ) = \\min _ q \\{ \\alpha \\ q - \\tau ( q ) \\} , \\end{align*}"}
-{"id": "1121.png", "formula": "\\begin{align*} & \\gamma _ n ( u ) = - \\sqrt { 4 - ( 1 + u / \\ln n ) ^ 2 S ( I ) ^ 2 } , \\ \\ \\beta _ n ( u ) = \\frac { ( 1 + u / \\ln n ) S ( I ) } { \\sqrt { 4 - ( 1 + u / \\ln n ) ^ 2 S ( I ) ^ 2 } } , \\end{align*}"}
-{"id": "8618.png", "formula": "\\begin{align*} \\hat { H } = & \\frac { z ^ 2 } { 2 } + \\frac { \\left ( \\varepsilon q u - \\left ( \\varepsilon ^ 2 p + c _ 0 + \\tilde { \\omega } ( \\varepsilon ) \\right ) z \\right ) ^ 2 } { 2 u ^ 4 } \\\\ & - \\frac { \\left ( 1 - u ^ 2 - \\frac { 1 } { u ^ 2 } \\left ( \\varepsilon ^ 2 p + c _ 0 + \\tilde { \\omega } ( \\varepsilon ) \\right ) ^ 2 \\right ) ^ 2 } { 4 } - \\frac { \\varepsilon ^ 2 } { 2 } p ^ 2 \\end{align*}"}
-{"id": "7008.png", "formula": "\\begin{align*} & \\cos ( z ) \\le \\mathsf { e } ^ { \\frac { - z ^ 2 } { 2 } } \\ ; \\ ; ( 0 \\le z \\le \\frac { \\pi } { 2 } ) , | \\sin ( z ) | \\leq | z | , \\\\ & \\frac { n ^ 2 } { 2 } n \\sum _ { j = 1 } ^ { \\lfloor m / 2 \\rfloor } \\frac { \\mathsf { e } ^ { - \\theta _ j ^ 2 \\frac { \\ell } { 2 } } } { n ^ 2 } 1 6 \\ , \\theta _ j ^ 4 = 8 \\frac { ( 2 \\pi ) ^ 4 } { n ^ 6 } n ^ 3 \\sum _ { j = 1 } ^ { \\lfloor m / 2 \\rfloor } j ^ 4 \\mathsf { e } ^ { - \\theta _ j ^ 2 \\frac { \\ell } { 2 } } . \\end{align*}"}
-{"id": "8858.png", "formula": "\\begin{align*} c _ 1 & = 1 6 ( 1 - \\nu ) ^ 2 c _ { 1 a } + 3 ( 1 - \\mu ) c _ { 1 b } , \\quad \\\\ c _ { 1 a } & = - 2 7 \\nu ^ 6 + 1 4 4 \\nu ^ 5 - 1 0 2 \\nu ^ 4 + 1 6 2 0 \\nu ^ 3 - 9 8 8 3 \\nu ^ 2 + 1 2 4 8 4 \\nu + 1 7 3 2 \\\\ c _ { 1 b } & = 8 1 \\mu \\nu ^ 8 - 1 0 8 \\mu \\nu ^ 7 + 1 3 5 \\nu ^ 8 - 1 2 6 0 \\mu \\nu ^ 6 - 8 2 8 \\nu ^ 7 - 1 2 2 7 6 \\mu \\nu ^ 5 + 2 7 6 \\nu ^ 6 + 8 4 7 7 4 \\mu \\nu ^ 4 \\\\ & - 4 4 0 4 \\nu ^ 5 - 1 5 7 1 4 0 \\mu \\nu ^ 3 + 6 9 1 7 0 \\nu ^ 4 + 1 5 2 6 2 8 \\mu \\nu ^ 2 - 1 9 8 3 7 2 \\nu ^ 3 - 1 5 6 1 0 8 \\mu \\nu + 1 8 2 0 8 4 \\nu ^ 2 \\\\ & + 2 7 9 6 9 \\mu - 6 0 5 8 8 \\nu + 7 3 9 6 7 \\end{align*}"}
-{"id": "1283.png", "formula": "\\begin{align*} \\MoveEqLeft O ( | \\eta | ^ { - 3 / 2 } ) + \\int _ { | \\mu | > | \\eta | ^ { 3 / 2 } / 2 } \\frac { 1 } { \\mu \\sqrt { \\eta } } | \\mu / \\sqrt { \\eta } | ^ { - k - 1 } + \\frac { 1 } { \\mu \\sqrt { \\eta } } | \\mu / \\sqrt { \\eta } | ^ { - k } d \\mu + \\left ( \\frac { 1 } { \\mu } \\left | z \\left ( \\eta + \\frac { \\mu } { \\sqrt { \\eta } } \\right ) \\right | \\right ) \\Bigg | _ { \\mu = | \\eta | ^ { 3 / 2 } } \\\\ & = O ( | \\eta | ^ { - 3 / 2 } ) + O ( | \\eta | ^ { - 3 / 2 - k } ) + O ( | \\eta | ^ { - 1 / 2 - k } ) \\end{align*}"}
-{"id": "78.png", "formula": "\\begin{align*} \\int _ I v _ 0 \\ , d x = 0 \\ , . \\end{align*}"}
-{"id": "279.png", "formula": "\\begin{align*} T ( P ) ( x ) = \\sum _ { j = l } ^ m a _ j S _ \\lambda ^ j ( P ) ( x ) . \\end{align*}"}
-{"id": "4652.png", "formula": "\\begin{align*} \\deg x _ { 0 1 } = & ( 1 , 0 , 0 ) & \\deg x _ { 1 1 } = & ( 1 , 0 , 0 ) & \\deg y _ { 1 } = & ( 2 , 0 , 0 ) \\\\ \\deg x _ { 0 2 } = & ( 0 , 1 , 0 ) & \\deg x _ { 1 2 } = & ( 0 , 1 , 0 ) & \\deg y _ { 2 } = & ( 0 , 2 , 0 ) \\\\ \\deg x _ { 0 3 } = & ( 0 , 0 , 1 ) & \\deg x _ { 1 3 } = & ( 0 , 0 , 1 ) & \\deg y _ { 3 } = & ( 0 , 0 , 2 ) \\\\ \\end{align*}"}
-{"id": "6221.png", "formula": "\\begin{align*} \\mathcal { L } ( \\theta ) = p _ \\theta ( y ) \\end{align*}"}
-{"id": "444.png", "formula": "\\begin{align*} e ^ { - \\frac { \\gamma ^ 2 } { 2 } t _ 0 } = s | \\log s | ^ { 2 \\kappa } . \\end{align*}"}
-{"id": "23.png", "formula": "\\begin{align*} \\bar { \\mathbf { y } } ( v ) & = \\chi _ { \\{ v \\geq 0 \\} } \\int _ 0 ^ { v } ( \\frac 1 2 { e ^ { \\frac { - y ^ 2 } { 2 } } } - e ^ { \\frac { y ^ 2 } { 2 } } ) d y - \\chi _ { \\{ v < 0 \\} } \\int _ 0 ^ { v } \\frac 1 2 { e ^ { - \\frac { y ^ 2 } { 2 } } } d y ; \\\\ \\bar { \\mathbf { z } } ( v ) & = \\chi _ { \\{ v \\geq 0 \\} } ( \\frac 1 2 { e ^ { \\frac { - v ^ 2 } { 2 } } } - e ^ { \\frac { v ^ 2 } { 2 } } ) - \\chi _ { \\{ v < 0 \\} } \\frac 1 2 { e ^ { - \\frac { v ^ 2 } { 2 } } } , \\end{align*}"}
-{"id": "587.png", "formula": "\\begin{align*} Z ^ x _ t : = Y ^ x _ 0 + \\sigma \\ , W _ t + \\int _ 0 ^ t L \\left [ x , \\ , P ( x , C ^ b _ s , C ^ l _ s ) - C ^ b _ s r _ s ( x ) + \\int _ { \\mathcal { X } } C ^ l _ s r _ s ( x ' ) \\nu _ s ( x ' ) \\mu ( \\mathrm { d } x ' ) \\right ] \\ , \\mathrm { d } s , \\end{align*}"}
-{"id": "816.png", "formula": "\\begin{align*} U ^ { q ^ d } - U = - \\frac { 1 } { T - \\rho } . \\end{align*}"}
-{"id": "7106.png", "formula": "\\begin{align*} g = ( 1 + \\phi \\bar { \\phi } ) e _ { 0 } + ( \\phi + \\bar { \\phi } ) e _ { 1 } - i ( \\phi - \\bar { \\phi } ) e _ { 2 } + ( - 1 + \\phi \\bar { \\phi } ) e _ { 3 } \\in \\Gamma G , \\end{align*}"}
-{"id": "1306.png", "formula": "\\begin{align*} Y & = Z \\times ( D \\oplus D ^ { - 1 } ) \\subseteq Z \\times F ( D ) \\subseteq A \\bigl ( Z \\times F ( D ) \\bigr ) \\\\ f & = ( \\pi \\circ \\widetilde { \\psi } ) | _ { ( \\pi \\circ \\widetilde { \\psi } ) ^ { - 1 } ( A ( Y ) ) } : ( \\pi \\circ \\widetilde { \\psi } ) ^ { - 1 } \\bigl ( A ( Y ) \\bigr ) \\to A ( Y ) , \\end{align*}"}
-{"id": "211.png", "formula": "\\begin{align*} \\Delta _ { \\mathbb { H } } u = V u \\end{align*}"}
-{"id": "2258.png", "formula": "\\begin{align*} \\langle \\alpha _ x | \\beta _ y \\rangle & = \\delta ( x - y ) \\\\ \\langle \\alpha _ x | \\beta _ { \\pm \\i \\pi } \\rangle & = 0 & ( \\alpha _ { \\pm \\i \\pi } | \\beta _ x \\rangle & = 0 \\\\ \\langle \\alpha _ { \\pm \\ i \\pi } | \\beta _ { \\pm \\i \\pi } \\rangle & = 1 & \\alpha _ { \\pm \\ i \\pi } | \\beta _ { \\mp \\i \\pi } \\rangle & = 0 \\end{align*}"}
-{"id": "9552.png", "formula": "\\begin{align*} ( z ) _ N & = \\sum _ { n = 0 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] ( - 1 ) ^ { n } z ^ n q ^ { n ( n - 1 ) / 2 } , \\\\ \\frac { 1 } { ( z ) _ { N } } & = \\sum _ { j = 0 } ^ { \\infty } \\left [ \\begin{matrix} N + j - 1 \\\\ j \\end{matrix} \\right ] z ^ j . \\end{align*}"}
-{"id": "2214.png", "formula": "\\begin{align*} s = - 2 \\dot { g } x + & 2 a p x ^ 2 + 2 p b x u + - 2 g a x - 2 g b u \\\\ & + m z ^ 2 + m c ^ 2 x ^ 2 - 2 m c x z + n u ^ 2 \\end{align*}"}
-{"id": "5309.png", "formula": "\\begin{align*} i = 1 : & - 2 ( H _ { q _ 1 } - E _ { 9 , 1 3 } ) - 2 ( H _ { p _ 1 } - E _ { 1 , 7 } ) - E _ { 1 - 2 } - E _ { 7 - 8 } - 2 E _ { 9 - 1 0 } - 2 E _ { 1 0 - 1 1 } \\\\ & - E _ { 1 1 - 1 2 } - E _ { 1 3 - 1 4 } = - 2 H _ { q _ 1 } - 2 H _ { p _ 1 } + E _ { 1 , 2 , 7 , 8 , 1 1 , 1 2 , 1 3 , 1 4 } ( i = 1 ) , \\\\ i = 2 : & q _ 1 \\leftrightarrow q _ 2 , p _ 1 \\leftrightarrow p _ 2 , E _ j \\leftrightarrow E _ { j + 8 } \\ ( j = 1 , \\dots , 8 ) \\mbox { i n t h e a b o v e } . \\end{align*}"}
-{"id": "171.png", "formula": "\\begin{align*} \\pi _ { X _ { V } , Y _ { V } } = X _ { V } \\wedge Y _ { V } = \\left ( X \\wedge Y \\right ) _ { V } \\end{align*}"}
-{"id": "180.png", "formula": "\\begin{align*} & [ \\pi _ { X _ { C } , Y _ { C } } , \\pi _ { X _ { C } , Y _ { C } } ] = 2 [ X _ { C } , Y _ { C } ] \\wedge X _ { C } \\wedge Y _ { C } = 2 [ X , Y ] _ { C } \\wedge X _ { C } \\wedge Y _ { C } = 0 . \\end{align*}"}
-{"id": "6834.png", "formula": "\\begin{align*} q _ 1 = u ^ 3 _ { 1 0 } u ^ 2 _ { 0 1 } - u ^ 3 _ { 0 1 } u ^ 2 _ { 1 0 } - u ^ 4 _ { 1 0 } u ^ 1 _ { 0 1 } + u ^ 4 _ { 0 1 } u ^ 1 _ { 1 0 } \\quad { \\rm a n d } q _ 2 = u ^ 4 _ { 1 0 } u ^ 2 _ { 0 1 } - u ^ 4 _ { 0 1 } u ^ 2 _ { 1 0 } + u ^ 3 _ { 1 0 } u ^ 1 _ { 0 1 } - u ^ 3 _ { 0 1 } u ^ 1 _ { 1 0 } \\ , . \\end{align*}"}
-{"id": "9531.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - 1 ) ^ { n - 1 } z ^ n q ^ { \\frac { n ( n + 1 ) } { 2 } } ( q ) _ { n } } { ( 1 - c q ^ { n } ) ( z q ) _ n } = \\frac { z } { c } \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( z q / c ) _ { n - 1 } ( q ) _ { n } ( c q ) _ { N - n } ( c q ) ^ { n } } { ( z q ) _ { n } ( c q ) _ { N } } . \\end{align*}"}
-{"id": "6310.png", "formula": "\\begin{align*} \\left ( \\binom { n } { 2 } - \\left ( k ( n - k ) + \\binom { k } { 2 } \\right ) \\right ) | e ( G ) - e ( G ' ) | \\leq 2 C \\max \\{ \\sqrt { \\rho } , \\sqrt { \\frac { \\log n } { n } } \\} k \\sqrt { n \\log n } . \\end{align*}"}
-{"id": "7430.png", "formula": "\\begin{align*} \\displaystyle \\gamma ( s , \\varphi \\oplus \\varphi ' , \\psi ' ) = \\gamma ( s , \\varphi , \\psi ' ) \\gamma ( s , \\varphi ' , \\psi ' ) \\end{align*}"}
-{"id": "4162.png", "formula": "\\begin{align*} D _ { 2 , + } ( x ) & = \\begin{pmatrix} 0 & 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} D _ { 2 , - } ( x ) \\begin{pmatrix} 0 & 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} , x > 0 , \\\\ D _ { 2 , + } ( x ) & = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 1 & 0 \\end{pmatrix} D _ { 2 , - } ( x ) \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 1 & 0 \\end{pmatrix} , x < 0 . \\end{align*}"}
-{"id": "8335.png", "formula": "\\begin{align*} \\Phi ^ { - 1 } ( z , t ) - z _ 0 = c _ 1 ( z - w _ 0 ) + \\sum _ { n = 2 } ^ { \\infty } c _ n ( z - w _ 0 ) ^ n , w h e r e c _ 1 = ( \\Phi ^ { - 1 } ) _ z ( w _ 0 ) \\neq 0 . \\end{align*}"}
-{"id": "9581.png", "formula": "\\begin{align*} S : = \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { q ^ { n ( n + 1 ) } } { ( q ) _ n ( 1 - q ^ n ) } F ( q ^ N , q ^ n ; q ^ n ) . \\end{align*}"}
-{"id": "500.png", "formula": "\\begin{align*} \\rho ( Q ) ( ( \\xi _ { p _ 1 } + \\xi _ { p _ 2 } ) \\odot ( \\xi _ { p _ 1 } + \\xi _ { p _ 2 } ) ) = 2 \\rho ( Q ) ( \\xi _ { p _ 1 } \\odot \\xi _ { p _ 1 } ) + 2 \\rho ( Q ) ( \\xi _ { p _ 1 } \\odot \\xi _ { p _ 2 } ) . \\end{align*}"}
-{"id": "8558.png", "formula": "\\begin{align*} \\partial _ t \\kappa _ { \\alpha } = b _ { \\alpha } \\circ \\kappa \\kappa _ { \\alpha } . \\end{align*}"}
-{"id": "5799.png", "formula": "\\begin{align*} T _ g ( \\phi ) \\ = \\ T _ g ( \\phi ' ) \\log _ 3 ( g ) ' . \\end{align*}"}
-{"id": "2993.png", "formula": "\\begin{align*} ( q ; q ^ { g - 2 } ) _ \\infty ( q ^ { g - 3 } ; q ^ { g - 2 } ) _ \\infty ( q ^ { g - 2 } ; q ^ { g - 2 } ) _ \\infty & = \\sum _ { n = 0 } ^ \\infty e _ { g , n } q ^ n . \\end{align*}"}
-{"id": "6410.png", "formula": "\\begin{align*} \\widehat { R } = \\left [ \\begin{array} { c c } R & v \\\\ & \\eta \\end{array} \\right ] , v \\in \\mathbb { R } ^ { k } , \\quad \\eta \\in \\mathbb { R } , \\end{align*}"}
-{"id": "3428.png", "formula": "\\begin{gather*} \\big ( z ^ i t ^ u \\big ) \\big ( z ^ j t ^ v \\big ) : = z ^ { i + j + \\omega ( u , v ) } t ^ { u + v } . \\end{gather*}"}
-{"id": "1455.png", "formula": "\\begin{align*} E _ { w } ( t ) = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { d } \\int _ { \\mathbb { T } ^ { d } } \\left ( \\partial _ { x _ { i } } w ^ { 1 } ( t , x ) - \\partial _ { x _ { i } } w ^ { 2 } ( t , x ) \\right ) ^ { 2 } \\ d x . \\end{align*}"}
-{"id": "4736.png", "formula": "\\begin{align*} T _ { p } U _ { q - n } = T _ { p - n } U _ { q } ( q = 1 , 2 , \\cdots n - 1 ) . \\end{align*}"}
-{"id": "4585.png", "formula": "\\begin{align*} E = \\{ ( x _ k , y _ k ) : | ( x _ k - g y _ k ) - ( x _ 1 - g y _ 1 ) | \\leq \\delta \\} . \\end{align*}"}
-{"id": "9804.png", "formula": "\\begin{align*} P _ { X _ \\circ } ( X ) : = p _ { \\ast , X _ \\circ } ( X - X _ \\circ ) + q _ { \\ast , X _ \\circ } ( X - X _ \\circ ) . \\end{align*}"}
-{"id": "1447.png", "formula": "\\begin{align*} V I _ { A } = \\varepsilon \\int _ { t } ^ { T } \\int _ { \\mathbb { T } ^ { d } } ( \\partial ^ { \\alpha } \\partial _ { x _ { j } } w ^ { n + 1 } ) ( \\Theta _ { q } ( \\tau , x , \\mu ^ { n } , D w ^ { n } ) ) ( \\partial ^ { \\alpha } \\partial _ { x _ { j } } \\mu ^ { n } ) \\ d x d \\tau , \\end{align*}"}
-{"id": "5555.png", "formula": "\\begin{align*} P _ { 2 , 3 } ( T ) \\to \\left \\{ \\begin{array} { l } 0 , \\ ; T = \\ , ^ 2 \\ ! B _ 2 ( 2 ^ a ) , \\ , P S p _ 4 ( 2 ^ a ) , \\ , P S p _ 4 ( 3 ^ a ) \\\\ \\frac { 1 } { 2 } , \\ , T = P S p _ 4 ( p ^ a ) , \\ , p \\ne 2 , 3 , p \\hbox { p r i m e } \\\\ 1 , \\ , \\hbox { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "5622.png", "formula": "\\begin{align*} \\int _ 0 ^ { + \\infty } i u _ t \\bar { u } _ x d x = - \\frac 1 2 \\frac { d } { d t } \\int _ 0 ^ { + \\infty } u \\bar { u } _ x d x - \\frac 1 2 ( u ( 0 , t ) \\bar { u } _ t ( 0 , t ) ) . \\end{align*}"}
-{"id": "1405.png", "formula": "\\begin{align*} f ( x _ 1 , \\dots , x _ d ) = \\tilde { H } ( \\prod _ { j = 1 } ^ d \\tilde { R _ j } ( x _ j ) ) , \\end{align*}"}
-{"id": "7628.png", "formula": "\\begin{align*} \\int \\limits _ { \\mathbb R ^ d } \\Big ( - z ^ i + \\varkappa ^ i _ 1 ( \\xi - z ) - \\varkappa ^ i _ 1 ( \\xi ) \\Big ) \\ a ( z ) \\mu ( \\xi , \\xi - z ) \\ d z + b ^ i \\ = \\ 0 \\forall \\ i = 1 , \\ldots , d . \\end{align*}"}
-{"id": "238.png", "formula": "\\begin{align*} ( x , y , t ) \\circ ( x ' , y ' , t ' ) = ( x + x ' , y + y ' , t + t ' + 2 ( x ' . y - x . y ' ) ) \\end{align*}"}
-{"id": "7490.png", "formula": "\\begin{align*} \\beta _ 0 = \\beta - \\epsilon , \\theta _ 0 = \\theta + \\epsilon = \\pi + \\epsilon \\end{align*}"}
-{"id": "5632.png", "formula": "\\begin{align*} I I ( u , v ) = - \\frac { \\alpha } { 2 \\gamma } \\int _ 0 ^ { + \\infty } \\Big [ \\int _ 0 ^ x \\Big ( v _ t + v _ { x x x } + \\tfrac 1 2 ( v ^ 2 ) _ x \\Big ) ( x ' , t ) d x ' \\Big ] v _ t d x . \\end{align*}"}
-{"id": "4038.png", "formula": "\\begin{align*} I _ M ( P , V ) = O ( M | P | + | P | ^ { 3 / 2 } + | P | ^ { 1 / 2 } | V | ) . \\end{align*}"}
-{"id": "4754.png", "formula": "\\begin{align*} f ( \\xi ) : = \\log \\left ( \\norm { \\xi - { \\bf e } } ^ 2 - n + \\frac 3 2 \\right ) , \\qquad \\norm { \\xi } \\geq C _ 1 , \\end{align*}"}
-{"id": "4732.png", "formula": "\\begin{align*} K e r A \\cap K e r B = \\{ 0 \\} . \\end{align*}"}
-{"id": "7825.png", "formula": "\\begin{align*} \\lim _ n [ \\langle T x _ n , x _ n \\rangle \\xi _ n , \\xi _ n ] = \\lambda \\| T \\| \\lim _ n [ \\langle T x _ n , T x _ n \\rangle \\xi _ n , \\xi _ n ] = \\| T \\| ^ 2 , \\end{align*}"}
-{"id": "4308.png", "formula": "\\begin{align*} a _ h ( u , v ) = \\ ; \\sum _ { K \\in \\mathcal { T } _ h } \\Bigl [ \\big ( \\nabla \\Pi _ K u , \\nabla \\Pi _ K v \\big ) _ K + S _ K ( u , v ) \\Bigr ] , \\end{align*}"}
-{"id": "5889.png", "formula": "\\begin{align*} q ^ { \\frac { \\ell ( w ) } { 2 } } C ' _ w - q ^ { \\frac { \\ell ( s _ i r ) } { 2 } } C ' _ { s _ i r } = T _ w + T _ { r s _ j } . \\end{align*}"}
-{"id": "8575.png", "formula": "\\begin{align*} [ E _ 1 ^ + , E _ 2 ^ + ] = \\sum _ j x _ j \\otimes x _ j = - E _ 3 ^ + \\qquad \\textup { a n d } [ E _ 1 ^ + , E _ 3 ^ - ] = - \\sum _ j x _ j \\otimes y _ j = - E _ 2 ^ - . \\end{align*}"}
-{"id": "62.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t f ^ - - \\partial _ x f ^ - - g ( f ^ + - f ^ - ) \\partial _ x a & = 0 \\ , , \\\\ \\partial _ t f ^ + + \\partial _ x f ^ + + g ( f ^ + - f ^ - ) \\partial _ x a & = 0 \\ , , \\\\ \\partial _ t a & = 0 \\ , . \\end{cases} \\end{align*}"}
-{"id": "1487.png", "formula": "\\begin{align*} W ^ { j } _ { 6 } = - \\varepsilon \\sum _ { i = 1 } ^ { d } \\int _ { \\mathbb { T } ^ { d } } ( \\partial _ { x _ { j } } w ^ { 1 } - \\partial _ { x _ { j } } w ^ { 2 } ) \\left ( \\Theta _ { p _ { i } } ( t , x , \\mu ^ { 2 } , D w ^ { 2 } ) \\partial ^ { 2 } _ { x _ { i } x _ { j } } w ^ { 1 } - \\Theta _ { p _ { i } } ( t , x , \\mu ^ { 2 } , D w ^ { 2 } ) \\partial ^ { 2 } _ { x _ { i } x _ { j } } w ^ { 2 } \\right ) \\ d x . \\end{align*}"}
-{"id": "6471.png", "formula": "\\begin{align*} \\alpha : = 1 + \\mu ( e _ 1 ) \\left [ 1 - \\sum _ { i = 1 } ^ { n - 1 } \\frac { 1 } { \\mu ( e _ i ) } \\right ] = \\mu ( e _ { 1 } ) \\left [ 1 - \\sum _ { i = 2 } ^ { n - 1 } \\frac { 1 } { \\mu ( e _ i ) } \\right ] . \\end{align*}"}
-{"id": "8572.png", "formula": "\\begin{align*} e _ 1 ^ - v _ 4 = e _ 2 ^ + [ e _ 1 ^ - , e _ 1 ^ + ] v _ \\lambda - \\lambda ( h ) [ e _ 1 ^ - , e _ 3 ^ + ] v _ \\lambda = - \\lambda ( h ) e _ 2 ^ + v _ \\lambda + \\lambda ( h ) e _ 2 ^ + v _ \\lambda = 0 , \\end{align*}"}
-{"id": "5809.png", "formula": "\\begin{align*} \\mathbb { E } [ V | \\mathcal { F } _ s ] = \\mathbb { E } [ V ] . \\end{align*}"}
-{"id": "5191.png", "formula": "\\begin{align*} \\psi _ { \\gamma } ( x ) = \\exp ( - ( x ) ^ { - \\gamma } ) \\mathbb { I } _ { \\left ] - \\infty , 0 \\right ] } ( x ) + ( 1 - 1 _ { \\left ] - \\infty , 0 \\right ] } ( x ) ) , \\ x \\in \\mathbb { R } , \\ \\end{align*}"}
-{"id": "1390.png", "formula": "\\begin{align*} \\triangledown ^ n _ { \\mathcal { M } } ( f ) ( \\eta ) = \\sum _ { \\sigma \\in \\Sigma _ { n } } \\operatorname { s g n } ( \\sigma ) \\ ; f \\left ( \\eta _ 1 ^ \\sigma , \\ldots , \\eta _ n ^ \\sigma \\right ) \\end{align*}"}
-{"id": "6477.png", "formula": "\\begin{align*} t _ { B } = \\frac { 1 } { b ^ { 2 } } [ t ( a + b ) ^ { 2 } - 4 a b ] . \\end{align*}"}
-{"id": "263.png", "formula": "\\begin{align*} \\sum _ { j = i } ^ { p + 1 } { j \\choose i } Q _ { j - i } ( \\alpha ) f ^ { p + 1 - j } v _ j \\in I '' _ { p + 1 - i } ( D ) \\quad 0 \\leq i \\leq p + 1 . \\end{align*}"}
-{"id": "5176.png", "formula": "\\begin{align*} u e p ( F ) - F ^ { - 1 } ( 1 - u ) = c ( 1 + a ( u ) ) u ^ { - \\gamma } \\exp ( \\int _ { u } ^ { 1 } \\frac { \\ell ( t ) } { t } d t ) . \\end{align*}"}
-{"id": "1022.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to + \\infty } \\left ( \\ln D _ n ( \\alpha _ n ) + \\ln n + x + \\frac { \\ln ( 2 \\ln n ) } { 2 } - c _ 0 \\right ) = 0 , \\end{align*}"}
-{"id": "1686.png", "formula": "\\begin{align*} \\begin{aligned} s _ b ( N _ k ^ { 2 n } ) & = n ( b - 1 ) ( 2 ^ { k } - 1 ) \\\\ & + ( b - C _ { 2 n } ^ 1 ) + ( b - C _ { 2 n } ^ 3 ) + \\cdots + ( b - C _ { 2 n } ^ { 2 n - 1 } ) \\\\ & + ( C _ { 2 n } ^ 2 - 1 ) + ( C _ { 2 n } ^ 4 - 1 ) + \\cdots + ( C _ { 2 n } ^ { 2 n - 2 } - 1 ) \\\\ & + 1 , \\end{aligned} \\end{align*}"}
-{"id": "1607.png", "formula": "\\begin{align*} \\begin{aligned} u ' ( x ) & = \\begin{cases} \\ell ^ \\ast S ' ( x ) m ( ( 0 , x ) ) , & \\mbox { i f $ x \\in ( 0 , b ^ * ) $ } \\\\ 1 & \\mbox { i f $ x \\geq b ^ * $ } \\end{cases} \\end{aligned} \\end{align*}"}
-{"id": "4812.png", "formula": "\\begin{align*} \\widehat { B } _ t : = B _ { T - t } , \\ , \\ , 0 \\leq t \\leq T . \\end{align*}"}
-{"id": "2555.png", "formula": "\\begin{align*} \\mathbf m ^ \\xi ( d s , d w ) : = 2 \\alpha ( \\xi _ s ) d s \\cdot \\mathbb N _ { \\xi _ s } ( d w ) + d s \\cdot \\int _ { ( 0 , \\infty ) } y \\mathbf P _ { y \\delta _ { \\xi _ s } } ( X \\in d w ) \\pi ( \\xi _ s , d y ) ; \\end{align*}"}
-{"id": "9346.png", "formula": "\\begin{align*} R _ k = \\langle \\{ x ^ { 2 ^ k } , y ^ 2 \\} \\cup \\{ [ y _ 0 , y _ i ] \\mid 1 \\le i \\le 2 ^ { k - 1 } \\} \\rangle ^ F \\trianglelefteq _ \\mathrm { o } F \\end{align*}"}
-{"id": "3110.png", "formula": "\\begin{align*} c _ 1 ( E _ { 0 } ) = \\ldots = c _ 1 ( E _ s ) c _ 2 ( E _ { 0 } ) \\leq \\ldots \\leq c _ 2 ( E _ s ) \\ , . \\end{align*}"}
-{"id": "6561.png", "formula": "\\begin{align*} n ( d ; ( a _ i ) _ { i = 1 } ^ r ; ) = n ( d ; ( a _ i ) _ { i = 1 } ^ { r - 1 } ; a _ r ) + \\mu \\cdot n ( d ; ( a _ i ) _ { i = 1 } ^ { r - 1 } ; a _ r + 1 ) , \\end{align*}"}
-{"id": "3177.png", "formula": "\\begin{align*} { \\rm i n d } _ \\varGamma ( U ) = ( M _ - - m _ - ) - ( M _ + - m _ + ) . \\end{align*}"}
-{"id": "2220.png", "formula": "\\begin{align*} P A + A ^ T P - \\frac { 1 } { 2 } P B R ^ { - 1 } B ^ { T } P - \\frac { 1 } { 4 } P B R ^ { - 1 } B ^ { T } P ^ T - \\frac { 1 } { 4 } P ^ T B R ^ { - 1 } B ^ { T } P = 0 \\end{align*}"}
-{"id": "2907.png", "formula": "\\begin{align*} \\psi ( \\mu ) & = \\lim _ { n \\to \\infty } \\frac { \\Psi _ \\mu ( F _ n \\ , | \\ , F _ n ^ \\complement ) } { \\abs { F _ n } } \\end{align*}"}
-{"id": "4715.png", "formula": "\\begin{align*} \\phi _ { j } ^ { n } ( x , t ) = B _ { j } ( x ) B ^ { n } ( t ) , \\end{align*}"}
-{"id": "7168.png", "formula": "\\begin{gather*} \\dot { x } = v ( x ) . \\end{gather*}"}
-{"id": "2680.png", "formula": "\\begin{align*} \\frac { d } { d z } f ( x ; z ) = \\eta _ { - } x + \\frac { \\tau } { \\tau z - 1 } - \\frac { 1 } { z - \\tau } = 0 , \\end{align*}"}
-{"id": "5467.png", "formula": "\\begin{align*} \\hat \\tau \\ = \\ \\inf \\Big \\{ t \\geq 0 \\colon K _ t \\ , < \\ , \\sup _ { 0 \\leq s \\leq t } K _ s \\Big \\} , \\hat \\tau ' \\ = \\ \\inf \\Big \\{ t \\geq \\hat \\tau \\colon K _ t \\ , = \\ , \\sup _ { 0 \\leq s \\leq t } K _ s \\Big \\} , \\end{align*}"}
-{"id": "1998.png", "formula": "\\begin{align*} L ^ { \\otimes 5 } \\cong \\omega _ { C } \\left ( \\sum _ { i = 1 } ^ { n + 1 } 5 [ q _ i ] \\right ) \\Longrightarrow \\left ( L \\otimes \\O \\left ( - [ q _ { n + 1 } ] \\right ) \\right ) ^ { \\otimes 5 } \\cong \\omega _ { C } \\left ( \\sum _ { i = 1 } ^ { n } 5 [ q _ i ] \\right ) . \\end{align*}"}
-{"id": "2570.png", "formula": "\\begin{align*} Y ^ G _ t : = \\int _ { G \\times \\mathbb W } w _ { t - s } \\mathbf n ( d s , d w ) . \\end{align*}"}
-{"id": "2804.png", "formula": "\\begin{align*} P _ { t , t + s } ( \\mathbf { n } , \\mathbf { m } ) : = \\mathbb { P } \\left ( \\mathbf { N } ^ { t } = \\mathbf { n } ; \\mathbf { N } ^ { t + s } = \\mathbf { m } \\right ) , \\end{align*}"}
-{"id": "720.png", "formula": "\\begin{align*} \\forall z \\in D , \\forall v , w \\in \\mathbb { C } ^ 2 \\setminus \\{ 0 \\} , B i s _ { z ^ { ( \\nu ) } } ^ D \\left ( \\left ( \\partial \\Lambda ^ { ( \\nu ) } _ { z ^ { ( \\nu ) } } \\right ) ^ { - 1 } ( v ) , \\left ( \\partial \\Lambda ^ { ( \\nu ) } _ { z ^ { ( \\nu ) } } \\right ) ^ { - 1 } ( w ) \\right ) = B i s _ { 0 } ^ { \\Omega _ \\nu } \\left ( v , w \\right ) , \\end{align*}"}
-{"id": "181.png", "formula": "\\begin{align*} & \\pi _ { T M , X _ { C } , Y _ { V } , c } = \\pi _ { T M } + \\lambda c ( { \\bf x } ) X _ { C } \\wedge Y _ { V } , \\\\ & \\pi _ { T M , X _ { C } , Y _ { C } , c } = \\pi _ { T M } + \\lambda c ( { \\bf x } ) X _ { C } \\wedge Y _ { C } \\end{align*}"}
-{"id": "2295.png", "formula": "\\begin{align*} H _ 0 ' & = \\sum _ { \\lambda \\in \\Lambda } \\omega _ \\lambda a _ \\lambda ^ + a _ \\lambda + \\sum _ { \\lambda \\in \\Lambda } ( g _ \\lambda a _ \\lambda b + \\overline { g } _ \\lambda a _ \\lambda ^ + b ^ + ) \\\\ H _ 0 '' & = \\omega _ 0 ( - b ^ + b + \\sum _ { \\lambda \\in \\Lambda } a _ \\lambda ^ + a _ \\lambda ) \\end{align*}"}
-{"id": "113.png", "formula": "\\begin{align*} 0 \\le \\sum _ { j = 0 } ^ { 2 N - 1 } \\zeta _ { j , N } & \\le \\sinh ( 2 d ) - 2 d + { \\frac 1 N } f _ 0 ( d ) \\\\ 0 \\le \\sum _ { j = 1 } ^ { 2 N - 1 } \\eta _ { j , N } & \\le \\cosh ( 2 d ) - 1 ~ + ~ { \\frac 1 N } f _ 1 ( d ) \\ , , \\end{align*}"}
-{"id": "8594.png", "formula": "\\begin{align*} \\phi ( x ) = \\frac { x ! } { ( k ! ) ^ { x / k } ( x / k ) ! } . \\end{align*}"}
-{"id": "6436.png", "formula": "\\begin{align*} \\pmb { u } _ k ^ { \\scriptscriptstyle ( \\mu _ k ) } = \\frac { \\| r _ { k } \\| } { \\sqrt { \\mu _ k } } \\frac { \\| r _ { k } \\| } { \\| p _ { k } \\| } \\end{align*}"}
-{"id": "8399.png", "formula": "\\begin{align*} \\| z _ t ( t ) \\| _ { H ^ s } \\leq & \\| v _ 0 \\| _ { H ^ s } + \\norm { \\int _ 0 ^ t z _ { t t } ( \\cdot , \\tau ) d \\tau } _ { H ^ s } \\leq \\| v _ 0 \\| _ { H ^ s } + t \\sup _ { \\tau \\in [ 0 , t ] } \\| z _ { t t } ( t = \\tau ) \\| _ { H ^ s } \\\\ \\leq & \\| v _ 0 \\| _ { H ^ s } + \\frac { ( 2 \\| w _ 0 \\| _ { H ^ s } + 1 ) ^ { 1 / 2 } } { ( 2 C _ 2 ) ^ { - s + 1 / 2 } } \\mathcal { E } ( t ) ^ { 1 / 2 } \\\\ \\leq & C ( \\| v _ 0 \\| _ { H ^ s } , \\| w _ 0 \\| _ { H ^ s } , C _ 2 , \\mathcal { E } ( t ) ) . \\end{align*}"}
-{"id": "4529.png", "formula": "\\begin{align*} \\Delta _ k ( F ) = \\{ ( r _ { i j } , 1 \\leq i < j \\leq k ) \\in \\mathbb { R } ^ { k ( k - 1 ) / 2 } : x _ 1 , \\dots , x _ k \\in F , | x _ i - x _ j | = r _ { i j } , 1 \\leq i < j \\leq k \\} . \\end{align*}"}
-{"id": "371.png", "formula": "\\begin{align*} z _ { m + 2 } & = s z _ { m + 1 } + c x _ { m + 1 } \\\\ & = s z _ { m + 1 } + c ( s x _ m + a z _ m ) \\\\ & = s z _ { m + 1 } + a c z _ m + s c x _ m \\\\ & = s z _ { m + 1 } + a c z _ m + s ( z _ { m + 1 } - s z _ m ) \\\\ & = 2 s z _ { m + 1 } + ( a c - s ^ 2 ) z _ m \\\\ & = 2 s z _ { m + 1 } - z _ m . \\end{align*}"}
-{"id": "6898.png", "formula": "\\begin{align*} f = \\sum _ { i = 1 } ^ 6 f _ i g _ i = B _ { \\ell k } \\big ( u ^ \\ell _ { ; 1 } u ^ k _ { ; 2 } - u ^ \\ell _ { ; 2 } u ^ k _ { ; 1 } \\big ) \\ , , \\end{align*}"}
-{"id": "9443.png", "formula": "\\begin{align*} \\mathcal { E } _ 2 ( z ) = \\sum _ { n = 0 } ^ { \\infty } b ( n ) q ^ n . \\end{align*}"}
-{"id": "1814.png", "formula": "\\begin{align*} D B _ { t } ( s ) = \\int _ { 0 } ^ { s \\wedge t } K ( t , u ) d u , \\end{align*}"}
-{"id": "9300.png", "formula": "\\begin{align*} D _ x \\phi ( \\hat x , \\hat y ) = & \\frac { 1 } { \\epsilon } a ( z ) ( \\hat x - \\hat y ) + \\varphi ( z , v ( z ) ) \\beta ( z ) - \\delta D d ( \\hat x ) + 2 \\delta ( \\hat x - z ) , \\\\ - D _ y \\phi ( \\hat x , \\hat y ) = & \\frac { 1 } { \\epsilon } a ( z ) ( \\hat x - \\hat y ) + \\varphi ( z , v ( z ) ) \\beta ( z ) + \\delta D d ( \\hat y ) . \\end{align*}"}
-{"id": "2091.png", "formula": "\\begin{align*} \\textbf { I } = \\sum _ { i = 1 } ^ { M } \\textbf { I } _ i , \\end{align*}"}
-{"id": "8747.png", "formula": "\\begin{align*} { } _ F u ^ { ( \\alpha , \\sigma ) } _ n = u _ n ^ { ( 0 ) } + \\widehat { u } _ n ^ { ( 1 ) } + \\cdot \\cdot \\cdot + \\widehat { u } _ n ^ { ( L ) } . \\end{align*}"}
-{"id": "7758.png", "formula": "\\begin{align*} \\| A \\| ^ D = \\max \\{ | \\langle A , B \\rangle | : \\| B \\| \\ \\leq 1 \\} . \\end{align*}"}
-{"id": "5750.png", "formula": "\\begin{align*} \\left | \\int \\limits _ 0 ^ 1 x \\varphi _ { y y y } ( x , y ) J _ 0 ( \\gamma _ m x ) d x \\right | = \\left | \\int \\limits _ 0 ^ 1 \\sqrt { x } ( \\sqrt x \\ , \\varphi _ { y y y } ( x , y ) ) J _ 0 ( \\gamma _ m x ) d x \\right | \\leq \\frac { C _ 8 } { ( \\gamma _ m ) ^ { 5 / 2 } } \\max _ { x , y \\in [ 0 , 1 ] } \\left ( \\sqrt x \\ , \\varphi _ { y y y x x } ( x , y ) \\right ) \\leq \\frac { C _ 9 } { m ^ { 5 / 2 } } \\end{align*}"}
-{"id": "2951.png", "formula": "\\begin{align*} \\exists z _ 1 z _ 2 ( = \\ ! \\ ! ( z _ 1 ) \\wedge = \\ ! \\ ! ( z _ 2 ) \\wedge ( ( z _ 1 = z _ 2 \\wedge \\phi ) \\vee ( z _ 1 \\not = z _ 2 \\wedge \\psi ) ) ) \\end{align*}"}
-{"id": "4095.png", "formula": "\\begin{align*} \\Phi _ { \\alpha } \\Psi _ { \\alpha } ^ T = - 4 \\pi ^ 2 \\begin{pmatrix} 1 & 0 & 0 \\\\ - 2 \\alpha - \\tfrac { 1 } { 2 } & - 1 & 0 \\\\ \\alpha ( \\alpha + \\tfrac { 1 } { 2 } ) & 2 \\alpha + \\tfrac { 1 } { 2 } & 1 \\end{pmatrix} ^ { - 1 } \\end{align*}"}
-{"id": "1725.png", "formula": "\\begin{align*} \\int _ { \\boldsymbol { W } } e ^ { i \\gamma ( \\omega ) } P ( d \\omega ) = \\exp \\left [ - \\frac { 1 } { 2 } \\Vert \\gamma \\Vert _ { \\mathcal { H } } ^ { 2 } \\right ] \\end{align*}"}
-{"id": "7138.png", "formula": "\\begin{align*} \\sup _ { t \\in [ 0 , T ] } \\int _ 0 ^ L \\Big ( \\frac { \\rho u ^ 2 } { 2 } + \\frac { \\epsilon } { 2 } \\chi _ x ^ 2 + \\Phi ( \\rho ) + \\frac { \\rho ( \\chi ^ 2 - 1 ) ^ 2 } { 4 \\epsilon } \\Big ) d x + \\int _ 0 ^ T \\int _ 0 ^ L \\Big ( \\mu ^ 2 + \\nu u ^ 2 _ x \\Big ) d x d \\tau = E _ 0 , \\end{align*}"}
-{"id": "6499.png", "formula": "\\begin{align*} B ( m , l ) = \\frac { \\Gamma ( m ) \\Gamma ( l ) } { \\Gamma ( m + l ) } \\end{align*}"}
-{"id": "6344.png", "formula": "\\begin{align*} \\sigma ^ 2 _ { t h } = \\frac { 2 K _ B T _ R T _ s } { R _ L e ^ 2 } , \\end{align*}"}
-{"id": "1331.png", "formula": "\\begin{align*} f \\ast g ( x ) = \\int _ { G ^ { r ( x ) } } f ( y ) g ( y ^ { - 1 } x ) \\ , d \\lambda ^ { r ( x ) } ( y ) \\end{align*}"}
-{"id": "637.png", "formula": "\\begin{align*} ( \\psi , [ v ] , [ y ] ) \\circ ( \\varphi , [ u ] , [ x ] ) = ( \\psi \\varphi ^ y , [ \\varphi ^ \\ast ( v y ) u \\varphi ^ \\ast ( y ) ^ { - 1 } ] , [ \\varphi ^ \\ast ( y ) x ] ) \\ . \\end{align*}"}
-{"id": "6742.png", "formula": "\\begin{align*} L ( p _ n , { ( m _ i + 1 ) } ^ 2 ) & = p _ n ( 2 ( m _ i + 1 ) - p _ n ) . \\end{align*}"}
-{"id": "7646.png", "formula": "\\begin{align*} - A \\varkappa _ 2 ( \\xi ) \\ = \\ F ( \\xi ) - \\Theta \\end{align*}"}
-{"id": "255.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ R ( e _ i - 2 ) a _ i + \\sum _ { q \\in U _ { T _ 0 , f } } \\bigl ( e ( q | t _ 0 ) - 1 \\bigr ) \\leq 1 . \\end{align*}"}
-{"id": "7891.png", "formula": "\\begin{align*} U _ { \\sigma } \\left ( \\Xi _ { j } ^ { * } \\Xi _ { j } \\{ g _ { n } \\} _ { n = 1 } ^ { \\infty } \\right ) = \\left \\{ \\begin{array} { l } ( C C ' ) ^ { \\frac { 1 } { 2 } } \\Lambda _ { j } ^ { * } ( g _ { j } ) \\ , \\ j \\in \\sigma \\\\ \\\\ ( C C ' ) ^ { \\frac { 1 } { 2 } } \\Omega _ { j } ^ { * } ( g _ { j } ) \\ , \\ j \\in \\sigma ^ { c } \\end{array} \\right . \\end{align*}"}
-{"id": "7889.png", "formula": "\\begin{align*} U _ { \\sigma } ^ { * } ( f ) = & \\left \\{ \\Lambda _ { j } ( C C ' ) ^ { \\frac { 1 } { 2 } } f \\right \\} _ { j \\in \\sigma } \\cup \\left \\{ \\Omega _ { j } ( C C ' ) ^ { \\frac { 1 } { 2 } } f \\right \\} _ { j \\in \\sigma ^ { c } } \\end{align*}"}
-{"id": "4944.png", "formula": "\\begin{align*} - v _ 1 z ^ { 3 n ^ 2 + n } = v _ 1 z ^ { 3 n ^ 2 - n } ( z ^ t - z ^ n x ^ n ) - z ^ { 3 n ^ 2 } f _ 4 + ( v _ 1 z ^ { 3 n } ) ^ n - \\left ( \\left ( \\sum _ { j = 2 } ^ { d - 3 } ( - 1 ) ^ { j } v _ i z ^ { 3 n } \\right ) + ( - 1 ) ^ { d } ( x z ^ { 3 n } ) \\right ) ^ n , \\end{align*}"}
-{"id": "9638.png", "formula": "\\begin{align*} S _ 1 ( z , q , N ) = \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] _ { q ^ 2 } \\left ( \\frac { ( q ; q ) _ { 2 n - 2 } z ^ { 2 n - 1 } q ^ { n ( 2 n - 1 ) } } { ( z q ; q ) _ { 2 n - 1 } } + \\frac { ( q ; q ) _ { 2 n - 1 } z ^ { 2 n } q ^ { n ( 2 n + 1 ) } } { ( z q ; q ) _ { 2 n } } \\right ) \\frac { ( q ^ { 2 } ; q ^ 2 ) _ n } { ( z q ^ { 2 N + 1 } ; q ^ 2 ) _ { n } } . \\end{align*}"}
-{"id": "3536.png", "formula": "\\begin{align*} \\mathcal { M } _ { 3 , 3 } \\simeq \\left \\{ X _ { 1 } X _ { 2 } X _ { 3 } = X _ { 4 } X _ { 5 } X _ { 6 } \\right \\} \\subset \\mathbb { P } ^ { 5 } . \\end{align*}"}
-{"id": "6450.png", "formula": "\\begin{align*} \\Vert \\widetilde x \\Vert ^ 2 = \\Vert x _ k \\Vert _ M ^ 2 , \\Vert \\widetilde r \\Vert ^ 2 = z _ k ^ T r _ k = \\Vert r _ k \\Vert _ { M ^ { - 1 } } ^ 2 , \\Vert \\widehat b \\Vert ^ 2 = \\Vert b \\Vert _ { M ^ { - 1 } } ^ 2 . \\end{align*}"}
-{"id": "4476.png", "formula": "\\begin{align*} \\tilde { F } ( \\nabla ^ 2 u + u \\sigma ) = \\tilde \\psi \\textrm { i n } \\ , \\ , \\eta ( M ) \\subset \\mathbb { S } ^ n \\end{align*}"}
-{"id": "8659.png", "formula": "\\begin{align*} \\o ( \\l _ * , d \\xi ) = | \\Psi _ { \\l _ * } ( \\xi ) | \\frac { d \\xi } { \\sqrt { \\xi } } , \\ \\xi \\in E , \\end{align*}"}
-{"id": "2514.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } B _ { n , p } ( x ) \\frac { t ^ { n } } { n ! } = \\frac { ( p + 1 ) ( t - H _ { p } ) e ^ { ( x + p ) t } } { ( e ^ { t } - 1 ) ^ { p + 1 } } + ( p + 1 ) e ^ { x t } \\sum _ { k = 1 } ^ { p } \\binom { p } { k } \\frac { H _ { k } } { ( e ^ { t } - 1 ) ^ { k + 1 } } . \\end{align*}"}
-{"id": "3983.png", "formula": "\\begin{align*} \\langle \\phi \\mid \\eta , \\psi \\rangle & = \\int _ { F } \\phi ( \\tau , l z ) \\overline { \\psi ( \\tau , z ) } e ^ { - 4 \\pi m l ^ 2 y ^ 2 / v } v ^ k d V \\\\ & = l ^ { - 2 } \\int _ { F ' } \\phi ( \\tau , z ) \\overline { \\psi ( \\tau , z / l ) } e ^ { - 4 \\pi m y ^ 2 / v } v ^ k d V \\\\ & = l ^ { - 2 } \\int _ { F ' } \\phi ( \\tau , z ) \\overline { \\psi ( \\tau , z ) \\mid \\eta ^ { - 1 } } e ^ { - 4 \\pi m y ^ 2 / v } v ^ k d V , \\end{align*}"}
-{"id": "9963.png", "formula": "\\begin{align*} \\widehat { R } = \\exp \\left ( \\frac { \\hbar } { 2 } \\sum _ { \\mu _ 1 , \\mu _ 2 } \\sum _ { k _ 1 , k _ 2 } \\Delta ^ { \\mu _ 1 \\mu _ 2 } _ { k _ 1 k _ 2 } \\frac { \\partial } { \\partial t ^ { \\mu _ 1 } _ { k _ 1 } } \\frac { \\partial } { \\partial t ^ { \\mu _ 2 } _ { k _ 2 } } \\right ) , \\end{align*}"}
-{"id": "9437.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq j \\leq D ( N ) - 1 } a _ j E _ 2 ( z ) = \\sum _ { 1 \\leq j \\leq D ( N ) - 1 } a _ j d _ j E _ 2 ( d _ j z ) = 0 . \\end{align*}"}
-{"id": "702.png", "formula": "\\begin{align*} \\int _ 0 ^ x t f ( t ) ^ 2 f ^ { ( 1 ) } ( t ) \\ ; d t & = \\left [ \\frac { t f ( t ) ^ 3 } { 3 } \\right ] _ 0 ^ x - \\frac { 1 } { 3 } \\int _ 0 ^ x f ( t ) ^ 3 \\ ; d t = \\frac { x f ( x ) ^ 3 } { 3 } - \\frac { 1 } { 3 } \\int _ 0 ^ x f ( t ) ^ 3 \\ ; d t . \\end{align*}"}
-{"id": "682.png", "formula": "\\begin{align*} H ( [ \\mu _ m ; f ; e _ m ] ) = [ \\mu _ m ; H ^ \\circ ( f ) ; e _ m ] \\ . \\end{align*}"}
-{"id": "3994.png", "formula": "\\begin{align*} \\Phi ( F ) ( Z ) : = \\underset { t \\rightarrow \\infty } \\lim F \\left ( \\begin{smallmatrix} Z & 0 \\\\ 0 & i t \\end{smallmatrix} \\right ) = \\sum _ { T \\in \\Lambda _ { n - 1 } } a _ F ( \\begin{psmallmatrix} T & 0 \\\\ 0 & 0 \\end{psmallmatrix} ) e ( T Z ) . \\end{align*}"}
-{"id": "7198.png", "formula": "\\begin{align*} u _ t + L _ r u _ x + u u _ x = 0 , \\end{align*}"}
-{"id": "5423.png", "formula": "\\begin{align*} A ( x ) E _ i ( x ) = E _ i ( f ( x ) ) , i = 1 , \\ldots , k , \\end{align*}"}
-{"id": "2160.png", "formula": "\\begin{align*} T _ { j } ( p g _ { L + 1 } ) ( \\mathbf { x } ) & = p ( \\mathbf { x } ) T _ { j } g _ { L + 1 } ( \\mathbf { x } ) + g _ { L + 1 } ( \\mathbf { x } ) T _ { j } p ( \\mathbf { x } ) \\\\ & = 2 ( L + 1 ) x _ j p ( \\mathbf { x } ) g _ { L } ( \\mathbf { x } ) + g _ { L + 1 } ( \\mathbf { x } ) T _ { j } p ( \\mathbf { x } ) . \\end{align*}"}
-{"id": "4647.png", "formula": "\\begin{align*} \\epsilon ^ { \\ast } d K _ { \\widehat { X } } = & \\epsilon ^ { \\ast } ( \\psi ^ { \\ast } d K _ X + d E ) \\\\ = & \\epsilon ^ { \\ast } \\psi ^ { \\ast } d K _ X + \\epsilon ^ { \\ast } d E \\\\ = & \\epsilon ^ { \\ast } \\psi ^ { \\ast } d K _ X + \\epsilon ^ { \\ast } d E \\\\ = & \\phi ^ { \\ast } \\pi ^ { \\ast } d K _ X + \\epsilon ^ { \\ast } d E \\\\ = & \\phi ^ { \\ast } ( d K _ Y - d R ) + \\epsilon ^ { \\ast } d E . \\end{align*}"}
-{"id": "5502.png", "formula": "\\begin{align*} \\| b - \\bar { b } \\| _ { \\mu _ 0 } ^ 2 = \\sum _ { j = 1 } ^ d ( \\theta _ { j , \\cdot } - \\bar { \\theta } _ { j , \\cdot } ) ^ T \\Gamma ( \\theta _ { j , \\cdot } - \\bar { \\theta } _ { j , \\cdot } ) , \\end{align*}"}
-{"id": "1433.png", "formula": "\\begin{align*} \\Theta ( t , x , \\mu , D w ) = \\mathcal { H } ( t , x , m , D u ) . \\end{align*}"}
-{"id": "8573.png", "formula": "\\begin{align*} \\sigma ( \\tau _ \\alpha ) ( x _ { i _ 1 } \\wedge \\cdots \\wedge x _ { i _ l } ) = s _ \\alpha ( x _ { i _ 1 } ) \\wedge \\cdots \\wedge s _ \\alpha ( x _ { i _ l } ) , \\end{align*}"}
-{"id": "2664.png", "formula": "\\begin{align*} f _ { j } ( \\varrho _ { k } ) = \\sum _ { l = 1 } ^ { N + n } \\tilde { g } _ { k l } \\tilde { c } _ { l j } = \\delta _ { k j } \\end{align*}"}
-{"id": "875.png", "formula": "\\begin{align*} f ( x ) & > | x | - \\sum _ { k = n } ^ \\infty ( b _ k - a _ { k + 1 } ) \\ge | x | - \\left ( \\max _ { k \\ge n } \\frac { b _ k - a _ { k + 1 } } { a _ k - b _ k } \\right ) \\sum _ { k = n } ^ \\infty ( a _ k - b _ k ) \\\\ & > | x | - \\left ( \\max _ { k \\ge n } \\frac { b _ k - a _ { k + 1 } } { a _ k - b _ k } \\right ) \\sum _ { k = n } ^ \\infty ( a _ k - a _ { k + 1 } ) \\ge | x | \\left ( 1 - \\max _ { k \\ge n } \\frac { b _ k - a _ { k + 1 } } { a _ k - b _ k } \\right ) . \\end{align*}"}
-{"id": "4057.png", "formula": "\\begin{align*} x _ 0 ^ { - 1 } \\delta \\left ( \\frac { x _ 1 - z } { x _ 0 } \\right ) Y _ { W _ 3 } ( v , x _ 1 ) I ( w _ 1 \\otimes w _ 2 ) & - x _ 0 ^ { - 1 } \\delta \\left ( \\frac { - z + x _ 1 } { x _ 0 } \\right ) I ( w _ 1 \\otimes Y _ { W _ 2 } ( v , x _ 1 ) w _ 2 ) \\\\ & = z ^ { - 1 } \\delta \\left ( \\frac { x _ 1 - x _ 0 } { z } \\right ) I ( Y _ { W _ 1 } ( v , x _ 0 ) w _ 1 \\otimes w _ 2 ) \\end{align*}"}
-{"id": "7742.png", "formula": "\\begin{align*} | [ \\langle x , x \\rangle \\langle x , y \\rangle \\xi , \\xi ] | = \\| x \\| ^ 3 \\| y \\| . \\end{align*}"}
-{"id": "7639.png", "formula": "\\begin{align*} b \\ = \\ \\int \\limits _ { \\mathbb R ^ d } \\int \\limits _ { \\mathbb T ^ d } a ( \\xi - q ) \\mu ( \\xi , q ) ( \\xi - q ) \\ d q \\ , v _ 0 ( \\xi ) d \\xi , \\end{align*}"}
-{"id": "6446.png", "formula": "\\begin{align*} \\vartheta _ { k } \\equiv \\sum _ { j = 0 } ^ { k - 1 } \\frac { \\sum _ { i = j } ^ { k - 1 } \\psi _ { i } } { \\| r _ { j } \\| ^ { 2 } } . \\end{align*}"}
-{"id": "619.png", "formula": "\\begin{align*} k ^ { ( \\varphi ) } _ j : = \\begin{cases} \\# \\varphi ^ { - 1 } \\{ j \\} & 1 \\le j \\le n \\ , \\\\ \\# \\varphi ^ { - 1 } \\{ j \\} - 1 & j = \\pm \\infty \\ . \\end{cases} \\end{align*}"}
-{"id": "6491.png", "formula": "\\begin{align*} | \\alpha _ i | _ { \\C } = | \\beta _ i | _ { \\C } \\ \\ \\ \\ \\ | \\alpha | _ { \\C } = | \\beta | _ { \\C } \\ \\ \\end{align*}"}
-{"id": "9449.png", "formula": "\\begin{align*} ( f _ { \\theta } - \\mathcal { E } _ 2 , \\xi _ 0 ( j _ { N , m } ) ) _ { r e g } = 0 . \\end{align*}"}
-{"id": "7534.png", "formula": "\\begin{align*} ( \\nabla u ) ( x , t ) = \\mathbb { L } _ \\nu ( \\nabla u _ 0 ) ( x , t ) + \\int _ 0 ^ t \\left ( g _ { \\nu ( t - s ) } * \\left ( \\mathbb { H } \\nabla \\mathrm { d i v } \\ , \\ , ( \\sigma - u \\otimes u ) \\right ) \\right ) ( x , s ) d s . \\end{align*}"}
-{"id": "5851.png", "formula": "\\begin{align*} { \\cal F } _ D 1 _ { R ^ n } ( x ) = C q ^ { - \\frac { n \\delta } 2 } 1 _ { \\pi ^ { - \\delta } D ^ { - 1 } R ^ n } ( x ) \\end{align*}"}
-{"id": "2235.png", "formula": "\\begin{align*} \\overline { \\partial } ( 1 / z ) = \\pi \\delta ( z ) . \\end{align*}"}
-{"id": "2619.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta _ p u ( z ) = \\hat { \\lambda } | u ( z ) | ^ { p - 2 } u ( z ) & \\mbox { i n } \\ \\Omega , \\\\ \\frac { \\partial u } { \\partial n _ p } + \\beta ( z ) | u | ^ { p - 2 } u = 0 & \\mbox { o n } \\ \\partial \\Omega . \\end{array} \\right \\} \\end{align*}"}
-{"id": "6505.png", "formula": "\\begin{align*} C _ t ( \\alpha ) = \\{ v \\in \\R ^ n : \\ \\langle \\bar v , \\bar \\alpha \\rangle > t \\} \\ , . \\end{align*}"}
-{"id": "4346.png", "formula": "\\begin{align*} \\widetilde { G } _ \\tau : = \\frac { \\widetilde G \\times T } { Z ( \\widetilde { G } ) } , \\end{align*}"}
-{"id": "2216.png", "formula": "\\begin{align*} u ^ * = - \\frac { b p x ^ * } { n } + \\frac { b } { n } \\beta \\left [ - ( a n - { p b ^ 2 } ) s i n ( \\omega t ) + n \\omega c o s ( \\omega t ) \\right ] . \\end{align*}"}
-{"id": "3589.png", "formula": "\\begin{align*} v = \\bigoplus _ { \\lambda \\in \\mathcal { F } } v _ \\lambda \\end{align*}"}
-{"id": "7257.png", "formula": "\\begin{align*} \\ddot \\phi _ k ( 0 ) = & D ^ 2 _ { \\phi \\phi } \\psi ( 0 , \\mu ^ * _ { k } ) [ \\phi ^ * _ k , \\phi ^ * _ k ] + 2 D ^ 2 _ { \\phi \\mu } \\psi ( 0 , \\mu ^ * _ { k } ) [ \\phi ^ * _ k , \\dot \\mu _ { k } ( 0 ) ] + D ^ 2 _ { \\mu \\mu } \\psi ( 0 , \\mu ^ * _ { k } ) [ \\dot \\mu _ k ( 0 ) , \\dot \\mu _ { k } ( 0 ) ] \\\\ & + D _ \\mu \\psi ( 0 , \\mu ^ * _ { k } ) \\dot \\mu _ { k } ( 0 ) . \\end{align*}"}
-{"id": "6797.png", "formula": "\\begin{align*} \\begin{cases} \\partial ^ 2 _ { j k } u ^ i + \\partial ^ 2 _ { i k } u ^ j = 0 \\ , , \\\\ \\partial ^ 2 _ { k j } u ^ i + \\partial ^ 2 _ { i j } u ^ k = 0 \\ , , \\\\ \\partial ^ 2 _ { j i } u ^ k + \\partial ^ 2 _ { k i } u ^ j = 0 \\ , , \\end{cases} \\end{align*}"}
-{"id": "4469.png", "formula": "\\begin{align*} G ( \\nabla ^ 2 u , \\nabla u , u ) & = \\Psi ( \\nabla u , u , x ) \\quad \\textrm { i n } \\ , \\Omega \\\\ u & = \\varphi \\quad \\textrm { o n } \\ , \\partial \\Omega \\end{align*}"}
-{"id": "2469.png", "formula": "\\begin{align*} g ^ { \\ast } x & = \\frac { x } { 1 + b x } = x - b x ^ { 2 } + b ^ { 2 } x ^ { 3 } - \\cdots + h _ { 1 } ( x ) , \\\\ g ^ { \\ast } x ^ { 2 } & = \\left ( \\frac { x } { 1 + b x } \\right ) ^ { 2 } = x ^ { 2 } - 2 b x ^ { 3 } + 3 b ^ { 2 } x ^ { 4 } - \\cdots + h _ { 2 } ( x ) , \\\\ g ^ { \\ast } x ^ { 3 } & = \\left ( \\frac { x } { 1 + b x } \\right ) ^ { 3 } = x ^ { 3 } - 3 b x ^ { 4 } + 6 b ^ { 2 } x ^ { 5 } - \\cdots + h _ { 3 } ( x ) , \\ \\\\ & \\cdots , \\end{align*}"}
-{"id": "8285.png", "formula": "\\begin{align*} \\hat { d } _ I ( t ) : = \\min _ { j = 1 , 2 } \\inf _ { \\alpha \\in \\mathbb { R } } I m \\{ z ( \\alpha , t ) - z _ j ( t ) \\} . \\end{align*}"}
-{"id": "9288.png", "formula": "\\begin{align*} \\phi ( x , y ) : = \\frac { 1 } { 2 \\epsilon } a ^ { i j } ( z ) ( x _ i - y _ i ) ( x _ j - y _ j ) + \\varphi ( z , v ( z ) ) \\beta ( z ) \\cdot ( x - y ) - \\delta ( d ( x ) + d ( y ) ) + \\delta | x - z | ^ 2 , \\end{align*}"}
-{"id": "4485.png", "formula": "\\begin{align*} v = u - \\underline { u } + t d - N d ^ 2 \\end{align*}"}
-{"id": "4599.png", "formula": "\\begin{align*} \\int \\left | \\tilde { \\nu } _ { g , j } ( z ) \\right | ^ 2 d z = \\int | \\hat { \\tilde { \\nu } } _ { g , j } ( \\omega ) | ^ 2 d \\omega . \\end{align*}"}
-{"id": "2545.png", "formula": "\\begin{align*} \\mathbb N _ \\mu ( \\| W _ t \\| \\neq 0 ) & = - \\log \\mathbf P _ { \\mu } ( \\| X _ t \\| = 0 ) = \\lim _ { \\lambda \\to \\infty } ( - \\log \\mathbf P _ \\mu [ e ^ { - \\lambda X _ t ( \\mathbf 1 _ E ) } ] ) \\\\ & = \\lim _ { \\lambda \\to \\infty } \\langle \\mu , V _ t ( \\lambda \\mathbf 1 _ E ) \\rangle = \\mu ( v _ t ) . \\end{align*}"}
-{"id": "3459.png", "formula": "\\begin{align*} \\Re ( M + 1 ) F ( s ; x _ { N } ( k ) ) \\geq \\Re ( M + 1 ) F ( s _ { \\pm } ; x _ { N } ( k ) ) + \\epsilon N ^ { 1 / 8 } ( M + 1 ) ^ { 1 / 8 } ( | y | - N ^ { 1 / 8 } ( M + 1 ) ^ { 1 / 8 } ) , \\end{align*}"}
-{"id": "1348.png", "formula": "\\begin{align*} & \\ \\int _ { H ^ { \\pi ( d ( x ) ) } } \\pi _ * ^ { d ( x ) } \\left ( x ^ { - 1 } \\cdot f \\middle | _ { G ^ { r ( x ) } } \\right ) \\ , d \\lambda _ H ^ { \\pi ( d ( x ) ) } \\\\ & = \\int _ { H ^ { \\pi ( d ( x ) ) } } \\pi ( x ^ { - 1 } ) \\cdot \\pi _ * ^ { r ( x ) } \\left ( f \\middle | _ { G ^ { r ( x ) } } \\right ) \\ , d \\lambda _ H ^ { \\pi ( d ( x ) ) } \\\\ & = \\int _ { H ^ { \\pi ( r ( x ) ) } } \\pi _ * ^ { r ( x ) } \\left ( f \\middle | _ { G ^ { r ( x ) } } \\right ) \\ , d \\lambda _ H ^ { \\pi ( r ( x ) ) } \\ ; . \\end{align*}"}
-{"id": "5687.png", "formula": "\\begin{align*} \\| x ^ A & - x ^ B + t _ m ( w ^ A _ m - w ^ B _ m ) \\| = \\Big \\| x ^ A - x ^ B + t _ m \\alpha \\frac { x ^ B - x ^ A } { \\| x ^ B - x ^ A \\| } \\Big \\| \\\\ & = \\| x ^ A - x ^ B \\| \\left | 1 - \\frac { t _ m \\alpha } { \\| x ^ B - x ^ A \\| } \\right | = \\| x ^ A - x ^ B \\| - t _ m \\eta \\ , . \\end{align*}"}
-{"id": "8001.png", "formula": "\\begin{align*} \\begin{cases} | y _ { i } | \\geq 1 , j + 1 \\leq i \\leq d \\\\ - 1 < y _ i < 1 , i \\leq j . \\end{cases} \\end{align*}"}
-{"id": "254.png", "formula": "\\begin{align*} 2 n - 2 & \\geq R n - \\sum _ { i = 1 } ^ R ( a _ i + b _ i ) \\\\ & \\geq R n + \\sum _ { i = 1 } ^ R ( e _ { i } - 2 ) a _ i + \\sum _ { q \\in U _ { T _ 0 , f } } \\bigl ( e ( q | t _ 0 ) - 1 \\bigr ) - ( 2 n - 2 ) - \\sum _ { i = 1 } ^ R b _ i . \\end{align*}"}
-{"id": "6806.png", "formula": "\\begin{align*} \\mathsf { d i m } \\big ( \\mathsf { V } ( \\sqrt [ \\mathbb { R } ] { I _ { 3 , 8 } } ) \\big ) = 5 \\ , . \\end{align*}"}
-{"id": "16.png", "formula": "\\begin{align*} G _ s ^ { i j } ( y ^ { i j } , y ^ { - i j } ) = & \\ q ^ { i j } e ^ { - y ^ { i j } } - q ^ { j i } e ^ { y ^ { i j } } - \\rho y ^ { i j } + \\sum _ { k \\neq j } q ^ { i k } e ^ { y ^ { k i } } - \\sum _ { k \\neq i } q ^ { j k } e ^ { - y ^ { j k } } , \\end{align*}"}
-{"id": "9326.png", "formula": "\\begin{align*} \\| \\mathbf { H } _ { \\psi _ { \\alpha } } \\| ^ 2 ( t , y ) = 1 + \\frac { ( 1 - t ^ 2 ) ^ 2 } { n ^ 2 } \\| \\alpha '' ( t ) \\| ^ 2 , \\end{align*}"}
-{"id": "2287.png", "formula": "\\begin{align*} \\hat { \\Omega } | \\delta _ x ) & = | \\delta _ x ) & ( \\delta _ x | \\hat { \\Omega } & = ( \\delta _ x | \\\\ \\hat { \\Omega } \\frac { \\mathcal { P } } { x - \\Omega } | E ) & = - | \\hat { E } ) + x \\frac { \\mathcal { P } } { x \\Omega } | E ) & ( E | \\frac { \\mathcal { P } } { x - \\Omega } \\hat { \\Omega } & = - ( \\hat { E } | + x ( E | \\frac { \\mathcal { P } } { x - \\Omega } \\\\ \\end{align*}"}
-{"id": "4012.png", "formula": "\\begin{align*} F ( \\tau ) = \\sum _ { n = 1 } ^ { \\infty } A ( n ) q ^ { n } : = \\prod _ { 2 \\le j \\le s } ( b _ { 1 } ( q _ { 1 , j } ) - b _ { j } ( q _ { 1 , j } ) ) \\sum _ { \\delta | N } \\alpha _ { 1 , \\delta } f _ { 1 } ( \\delta \\tau ) . \\end{align*}"}
-{"id": "6486.png", "formula": "\\begin{align*} \\sum _ { \\substack { e \\in E \\\\ s ( e ) = v } } \\mu ( e ) \\geq 1 \\end{align*}"}
-{"id": "4173.png", "formula": "\\begin{align*} A _ n ^ { ( 1 ) } ( z _ 1 ) \\cdot \\cdots \\cdot A _ n ^ { ( 1 ) } ( z _ k ) = \\mathcal { O } ( n ^ { k + 1 } ) . \\end{align*}"}
-{"id": "780.png", "formula": "\\begin{align*} f _ { i _ 0 } ( x ) : = \\left ( 1 - \\prod _ { \\substack { i = 1 \\\\ i \\neq i _ 0 } } ^ { s } ( x - \\rho _ i ) \\right ) = 1 - \\frac { P ( x ) } { x - \\rho _ { i _ 0 } } \\in \\mathbb { F } _ { q ^ d } [ x ] . \\end{align*}"}
-{"id": "9828.png", "formula": "\\begin{align*} N ( \\rho , \\tilde u _ { X _ \\circ , 0 } ) = \\lim _ { r _ { j _ \\ell } \\downarrow 0 } N ( \\rho , \\tilde { u } _ { X _ \\circ , r _ { j _ \\ell } } ) , \\end{align*}"}
-{"id": "1360.png", "formula": "\\begin{align*} \\partial _ { n } \\big ( { ( \\lambda _ 0 , \\ldots , \\lambda _ { n - 1 } ) , a } \\big ) = & \\ \\big ( { ( \\lambda _ 1 , \\ldots , \\lambda _ { n - 1 } ) } , a \\big ) + ( - 1 ) ^ n \\big ( { ( \\lambda _ 0 , \\ldots , \\lambda _ { n - 2 } ) } , \\lambda _ { n - 1 } \\cdot a \\big ) \\\\ & + \\sum _ { i = 1 } ^ { n - 1 } ( - 1 ) ^ i \\big ( { ( \\lambda _ 0 , \\ldots , \\lambda _ { i - 1 } \\lambda _ i , \\ldots , \\lambda _ { n - 1 } ) , a } \\big ) \\ ; , \\end{align*}"}
-{"id": "4503.png", "formula": "\\begin{align*} f _ ! ( a \\wedge f ^ * ( b ) ) = f _ ! ( a ) \\wedge b \\ , , \\end{align*}"}
-{"id": "3327.png", "formula": "\\begin{align*} w _ T ( b ) = \\begin{cases} 1 & \\mbox { i f $ b $ i s n o t a t t h e e n d o f a r o w i n $ T $ , } \\\\ | b | & \\mbox { i f $ b $ i s a t t h e e n d o f a r o w i n $ T $ . } \\end{cases} \\end{align*}"}
-{"id": "5899.png", "formula": "\\begin{align*} K \\pi ^ n \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} = \\begin{pmatrix} \\pi ^ n e \\\\ \\pi ^ n g \\end{pmatrix} . \\end{align*}"}
-{"id": "553.png", "formula": "\\begin{align*} v \\pm i | \\nabla | ^ { - 1 } \\partial _ t v = e ^ { \\mp i t | \\nabla | } ( f _ \\lambda \\pm i | \\nabla | ^ { - 1 } g _ \\lambda ) , \\end{align*}"}
-{"id": "1543.png", "formula": "\\begin{align*} T \\begin{pmatrix} z \\\\ x \\end{pmatrix} = \\begin{pmatrix} z + \\sum _ { i + j \\geq 2 } a _ { i , j } z ^ i x ^ j \\\\ \\delta x + \\sum _ { i + j \\geq 2 } b _ { i , j } z ^ i x ^ j \\end{pmatrix} \\end{align*}"}
-{"id": "9080.png", "formula": "\\begin{align*} \\mu _ K ( A ) = \\mu _ { f } ( A ) + O \\left ( \\frac { 1 } { K } \\right ) \\end{align*}"}
-{"id": "6855.png", "formula": "\\begin{align*} ( d \\Phi ) ^ T \\Phi '' + \\nabla p = 0 \\ , . \\end{align*}"}
-{"id": "5192.png", "formula": "\\begin{align*} F ^ { [ n t ] } ( a _ { n } x + b _ { n } ( t ) ) = ( F ^ { n } ( a _ { n } x + b _ { n } ( t ) ) ) ^ { ( [ n t ] / n ) } \\rightarrow H ^ { t } ( x ) . \\end{align*}"}
-{"id": "6213.png", "formula": "\\begin{align*} \\nu _ { \\mu } ( d \\lambda ) = \\int _ { \\mathcal { D } } \\alpha ( d b ) \\rho _ b ( d \\lambda ) . \\end{align*}"}
-{"id": "5833.png", "formula": "\\begin{align*} \\frac { - \\phi ( u ) \\frac { \\partial ^ 2 \\phi ( u ) } { \\partial u ^ 2 } + \\left ( \\frac { \\partial \\phi ( u ) } { \\partial u } \\right ) ^ 2 } { \\phi ( u ) ^ 2 } = \\sigma ^ 2 \\end{align*}"}
-{"id": "7016.png", "formula": "\\begin{align*} e ^ { i \\theta _ 2 - i r } U _ 2 W v _ { 1 \\mu } W ^ * U _ 2 ^ * = ( - 1 ) ^ { S + \\mu } c _ 2 v _ { 2 , - \\mu } c _ 2 = e ^ { i r + i \\theta _ 1 } c _ 2 W c _ 1 U _ 1 v _ { 1 \\mu } U _ 1 ^ * c _ 1 W ^ * c _ 2 . \\end{align*}"}
-{"id": "1127.png", "formula": "\\begin{align*} & \\mathbb { P } ( \\xi ^ { ( n ) } ( \\cup _ { j = 1 } ^ k [ y _ j , y _ j + a _ j ] ) = 0 ) \\geq ( 1 - ( \\ln n ) ^ { - 1 } ) \\prod _ { j = 1 } ^ k D _ n \\left ( a _ j \\sqrt { 4 - y _ j ^ 2 } / 2 \\right ) . \\end{align*}"}
-{"id": "1217.png", "formula": "\\begin{align*} 3 C ^ { \\varepsilon } = 2 C ^ { k } \\qquad \\qquad \\qquad ( a s \\ , \\ , n \\geq 4 ) . \\end{align*}"}
-{"id": "6448.png", "formula": "\\begin{align*} \\underbrace { L ^ { - 1 } A L ^ { - T } } _ { \\widehat { A } } \\underbrace { L ^ T x } _ { \\widehat { x } } = \\underbrace { L ^ { - 1 } b } _ { \\widehat { b } } . \\end{align*}"}
-{"id": "6280.png", "formula": "\\begin{align*} & R _ k ( \\rho , \\epsilon , n ) = \\min _ { \\substack { \\mathcal { A } \\\\ \\epsilon - } } \\max _ { W k - } \\operatorname * { \\mathbb { E } } _ { G \\sim G _ n ( \\rho W ) } [ \\delta _ 2 ( \\mathcal { A } ( G ) , W ) ^ 2 ] . \\end{align*}"}
-{"id": "9858.png", "formula": "\\begin{align*} B = \\frac { 1 } { 2 } \\int _ { \\partial M } { \\left \\langle { \\nabla ( { { \\left | { \\nabla u } \\right | } ^ 2 } ) , A \\overrightarrow { n } } \\right \\rangle d v o { l _ { \\bar g } } } - \\int _ { \\partial M } { ( { L _ A } u ) \\left \\langle { \\nabla u , \\overrightarrow { n } } \\right \\rangle d v o { l _ { \\bar g } } } , \\end{align*}"}
-{"id": "8436.png", "formula": "\\begin{align*} \\mathcal { H } f ( \\alpha ) : = \\frac { 1 } { \\pi i } p . v . \\int _ { - \\infty } ^ { \\infty } \\frac { \\zeta _ { \\beta } ( \\beta , t ) } { \\zeta ( \\alpha , t ) - \\zeta ( \\beta , t ) } f ( \\beta ) d \\beta . \\end{align*}"}
-{"id": "1798.png", "formula": "\\begin{align*} G _ { n } = \\left \\{ X _ { n } < \\frac { 1 - \\theta ^ { H } } { ( 1 - \\theta ) ^ { H } } \\sqrt { 2 \\log \\log ( \\theta ^ { - n } ) } \\right \\} . \\end{align*}"}
-{"id": "6588.png", "formula": "\\begin{align*} \\bar { h } ( r , \\theta ) = \\bar { h } ( r , 0 ) + \\int _ 0 ^ { \\theta } \\partial _ { \\theta } \\bar { h } ( r , \\alpha ) \\ , d \\alpha , \\end{align*}"}
-{"id": "8824.png", "formula": "\\begin{align*} \\begin{cases} \\theta ^ 3 + \\theta ^ 2 + \\theta _ k ^ 2 \\theta - \\theta _ k ^ 2 = 0 & h \\neq 0 \\\\ \\phantom { \\theta ^ 3 ~ ~ ~ ~ } \\theta ^ 2 + \\theta _ k ^ 2 \\theta - \\theta _ k ^ 2 = 0 & h = 0 \\\\ \\end{cases} \\end{align*}"}
-{"id": "4730.png", "formula": "\\begin{align*} T _ { p } U _ { q - n } = T _ { p - n } U _ { q } \\quad p , q = 1 , 2 , \\cdots n - 1 . \\end{align*}"}
-{"id": "1818.png", "formula": "\\begin{align*} \\phi ^ { \\rm O U T } ( x , t ) = \\frac { 1 } { 2 } \\left [ \\phi ^ { \\rm I N } ( x + t , 0 ) + \\phi ^ { \\rm I N } ( x - t , 0 ) \\right ] + \\frac { 1 } { 2 } \\int _ { x - t } ^ { x + t } \\partial _ t \\phi ^ { \\rm I N } ( z , 0 ) \\ , d z . \\end{align*}"}
-{"id": "2657.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } I _ { 0 } ( \\beta \\sqrt { t } ) e ^ { - \\alpha t } d t = \\frac { 1 } { \\alpha } e ^ { \\frac { \\beta ^ { 2 } } { 4 \\alpha } } , \\Re \\{ \\alpha \\} > 0 , \\end{align*}"}
-{"id": "2672.png", "formula": "\\begin{align*} g ( z , v ) & = \\int _ { \\mathcal { C } _ { \\{ 0 \\} } } \\frac { d s } { 2 \\pi i s } e ^ { \\frac { v } { s } } \\frac { ( z + 1 ) s - z } { 1 - ( z + 1 ) s } \\frac { 1 } { \\sqrt { 1 - 2 s } } . \\end{align*}"}
-{"id": "7985.png", "formula": "\\begin{align*} f ( n ) = e ^ { \\frac { n } { 2 } + O \\left ( \\frac { n } { \\log n } \\right ) } \\ , . \\end{align*}"}
-{"id": "8995.png", "formula": "\\begin{align*} \\begin{aligned} & \\mbox { C a s e ( i ) } : ~ v ^ k _ { r } ( T \\gamma _ g ( 2 \\tau + 1 ) ) \\leq \\frac { \\bar { v } ^ k ( T \\gamma _ g ( 2 \\tau + 1 ) ) + \\underline { v } ^ k ( T \\gamma _ g ( 2 \\tau + 1 ) ) } { 2 } . \\\\ & \\mbox { C a s e ( i i ) } : ~ v ^ k _ { r } ( T \\gamma _ g ( 2 \\tau + 1 ) ) \\geq \\frac { \\bar { v } ^ k ( T \\gamma _ g ( 2 \\tau + 1 ) ) + \\underline { v } ^ k ( T \\gamma _ g ( 2 \\tau + 1 ) ) } { 2 } . \\end{aligned} \\end{align*}"}
-{"id": "3796.png", "formula": "\\begin{align*} | \\{ x _ 1 x _ 2 x _ 3 : \\ \\sum _ { i = 1 } ^ 3 x _ i = M , \\ 0 \\leq x _ 1 \\leq 1 , \\ 0 \\leq x _ 2 \\leq 3 , \\ 0 \\leq x _ 3 \\leq 4 \\} | = | E _ { \\alpha , M } | . \\end{align*}"}
-{"id": "7668.png", "formula": "\\begin{align*} \\tilde \\varphi _ 0 ^ { \\boldsymbol { \\ell } } = 2 \\varkappa _ { \\rm s y m } + r _ { \\boldsymbol { \\ell } } , \\tilde \\varphi _ 0 ^ { \\boldsymbol { \\ell } } \\in ( L ^ 2 ( \\mathbb T ^ d ) ) ^ d , \\end{align*}"}
-{"id": "906.png", "formula": "\\begin{align*} y _ { e _ { t a } } y _ { e _ { h a } } = q ^ { \\lambda ( e _ { t a } , e _ { h a } ) } y _ { e _ { h a } } y _ { e _ { t a } } = q \\ , y _ { e _ { h a } } y _ { e _ { t a } } . \\end{align*}"}
-{"id": "10069.png", "formula": "\\begin{align*} \\Psi _ t \\circ K ^ s ( u , \\varphi , \\theta _ n ^ 0 , G _ n ^ 0 ) = K ^ s ( \\widetilde \\Psi _ t ( u , \\theta _ n ^ 0 , G _ n ^ 0 ) , \\varphi + \\omega ^ 0 t , \\theta _ n ^ 0 , G _ n ^ 0 ) , \\end{align*}"}
-{"id": "7612.png", "formula": "\\begin{align*} \\left \\{ \\begin{gathered} X ^ { n e w } ( t ) = \\mathrm { I d } + \\int _ 0 ^ t \\mathcal { V } ( X ( s ) , \\tau ( s ) , v ( s ) ) d s , \\\\ \\tau ^ { n e w } ( t ) = \\sigma _ 0 + \\int _ 0 ^ t \\mathcal { T } ( X ( s ) , \\tau ( s ) , v ( s ) ) d s , \\\\ v ^ { n e w } ( t ) = \\mathcal { V } ( X , \\tau , v ) . \\end{gathered} \\right . \\end{align*}"}
-{"id": "3354.png", "formula": "\\begin{align*} [ u ^ k ] \\log H _ { \\chi } ( u ) \\le \\frac { 1 } { k } \\sum _ { d i = k } 2 ^ i q ^ d \\le \\frac { 8 q ^ k } { k } . \\end{align*}"}
-{"id": "7815.png", "formula": "\\begin{align*} \\| e \\langle x + \\lambda y , x + \\lambda y \\rangle e \\| = \\| e ( x + \\lambda y ) \\| ^ 2 = \\| x + \\lambda y \\| ^ 2 . \\end{align*}"}
-{"id": "4364.png", "formula": "\\begin{align*} T = \\frac { 1 } { 2 K \\eta } \\log \\left ( \\frac { \\alpha _ 1 - 1 } { \\alpha _ 0 - 1 } \\right ) , \\end{align*}"}
-{"id": "5964.png", "formula": "\\begin{align*} F \\left ( \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\delta _ { W _ { ( n , k ) } ( 1 ) } \\right ) = F \\left ( \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\delta _ { \\sqrt { n } ( W ( k / n ) - W ( ( k - 1 ) / n ) ) } \\right ) . \\end{align*}"}
-{"id": "3169.png", "formula": "\\begin{align*} & m _ \\pm = \\dim \\ker ( \\gamma \\mp 1 ) \\cap \\ker ( c - 1 ) , \\\\ & M _ \\pm = \\dim \\ker ( \\gamma \\pm 1 ) \\cap \\ker ( c + 1 ) , \\end{align*}"}
-{"id": "5318.png", "formula": "\\begin{align*} I _ 1 ^ { { \\rm N A } } = & H _ { { \\rm I V } } ( - q _ 1 , p _ 1 ; a _ 1 ^ { ( 1 ) } , a _ 2 ^ { ( 1 ) } ; b ^ { ( 1 ) } ) = q _ 1 p _ 1 ( q _ 1 + p _ 1 - b ^ { ( 1 ) } ) - a _ 1 ^ { ( 1 ) } q _ 1 + a _ 2 ^ { ( 1 ) } p _ 1 \\\\ I _ 2 ^ { { \\rm N A } } = & H _ { { \\rm I V } } ( - q _ 2 , p _ 2 ; a _ 1 ^ { ( 2 ) } , a _ 2 ^ { ( 2 ) } ; b ^ { ( 2 ) } ) = p _ 2 q _ 2 ( p _ 2 + q _ 2 - b ^ { ( 2 ) } ) - a _ 1 ^ { ( 2 ) } q _ 2 + a _ 2 ^ { ( 2 ) } p _ 2 , \\end{align*}"}
-{"id": "886.png", "formula": "\\begin{align*} \\beta ^ 1 _ 1 & = \\alpha _ 1 & \\beta ^ 2 _ 1 & = \\alpha _ 3 & \\beta ^ 2 _ 2 & = \\alpha _ 2 + \\alpha _ 3 & \\beta ^ 2 _ 3 & = \\alpha _ 2 . \\end{align*}"}
-{"id": "4941.png", "formula": "\\begin{align*} \\left ( \\sum _ { j = i + 1 } ^ { d - 3 } ( - 1 ) ^ { j } v _ j z ^ { 3 n ^ { i + 2 } + n ^ { i + 1 } + \\cdots + n ^ 2 + n } \\right ) + ( - 1 ) ^ { d } ( x z ^ { 3 n ^ { i + 2 } + n ^ { i + 1 } + \\cdots + n ^ 2 + n } ) \\end{align*}"}
-{"id": "3149.png", "formula": "\\begin{align*} ( u - v - c ) \\ F _ i ( u ) F _ { i + 1 } ( v ) = ( u - v ) \\ F _ { i + 1 } ( v ) F _ i ( u ) , \\end{align*}"}
-{"id": "8671.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t u + \\div ( u \\otimes u ) + \\nabla p = 0 \\\\ \\\\ \\div u = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "8199.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\frac { 1 } { m ! } \\sum _ { \\sigma \\in [ m ] } | d ( F ^ \\sigma , G ^ \\sigma ) | ^ \\alpha & = \\lim _ { m \\to \\infty } \\frac { 1 } { m ! } \\sum _ { \\sigma \\in [ m ] } | d ( ( F - G ) ^ \\sigma , 0 ) | ^ \\alpha \\\\ & = \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { n ^ \\alpha } ( 1 - p ) ^ { n \\choose 2 } ( 1 - ( 1 - p ) ^ n ) \\\\ & \\leq p \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { n ^ { \\alpha - 1 } } ( 1 - p ) ^ { n \\choose 2 } \\\\ & \\leq p \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { n ^ { \\alpha - 1 } } \\end{align*}"}
-{"id": "4907.png", "formula": "\\begin{gather*} L ( \\dot \\eta , \\eta ) = \\tfrac { 1 } { 2 } \\int _ \\Omega g ( u , u ) \\ , \\mu ( x ) \\equiv \\tfrac { 1 } { 2 } \\int _ \\Omega \\lvert u \\rvert ^ 2 \\ , \\mu ( x ) \\ , , \\end{gather*}"}
-{"id": "4297.png", "formula": "\\begin{align*} 6 \\left ( \\frac { n ^ 3 } { 6 } - o ( n ^ 3 ) \\right ) = ( 1 - o ( 1 ) ) n ^ 3 . \\end{align*}"}
-{"id": "8400.png", "formula": "\\begin{align*} & R e \\frac { d } { d t } \\int i \\partial _ { \\alpha } D ^ k u \\overline { D ^ k u } = R e ~ i \\int \\partial _ { \\alpha } \\partial _ t D ^ k u \\overline { D ^ k u } + \\partial _ { \\alpha } D ^ k u \\overline { \\partial _ t D ^ k u } \\\\ = & 2 R e \\int i \\partial _ { \\alpha } D ^ k u \\overline { \\partial _ t D ^ k u } . \\end{align*}"}
-{"id": "4033.png", "formula": "\\begin{align*} I ( P , V ) = O ( | P | + M ^ { 2 / 3 } | P | ^ { 2 / 3 } | V | ^ { 2 / 3 } + | P | ^ { 2 / 3 } | V | ) . \\end{align*}"}
-{"id": "5947.png", "formula": "\\begin{align*} \\textstyle \\begin{cases} Y ( s ) - \\int _ { s } ^ { t } g ^ * ( u , Z ( u ) ) d u + \\int _ { s } ^ { t } Z ( u ) d W ( u ) \\geq Y ( t ) , 0 \\leq s \\leq t \\leq 1 \\\\ \\displaystyle Y ( 1 ) \\geq X , \\end{cases} \\end{align*}"}
-{"id": "8294.png", "formula": "\\begin{align*} u _ { t t } + | D | u = u _ t ^ p + \\frac { C } { ( \\alpha + i t ) ^ m } , p \\geq 2 , m \\geq 2 . \\end{align*}"}
-{"id": "9281.png", "formula": "\\begin{align*} \\eta = u - u _ 0 + K ( u - u _ 0 ) ^ 2 \\end{align*}"}
-{"id": "4721.png", "formula": "\\begin{align*} f ( u ) = \\left \\{ \\begin{array} { l l } \\dfrac { u ( 1 - u ) } { 4 } ~ ~ ~ ~ & u < \\dfrac { 1 } { 2 } , \\medskip \\\\ \\dfrac { 1 } { 2 } u ^ 2 - \\dfrac { 1 } { 4 } u + \\dfrac { 3 } { 1 6 } ~ ~ ~ ~ & u \\geq \\dfrac { 1 } { 2 } , \\end{array} \\right . \\end{align*}"}
-{"id": "1760.png", "formula": "\\begin{align*} f _ { i } ( x ) = \\begin{cases} 1 , & x \\in ( a _ { i } , b _ { i } ) , \\\\ 0 , & x \\in ( - \\infty , a _ { i } - c _ { i } ) \\cup ( b _ { i } + c _ { i } , \\infty ) , \\end{cases} \\end{align*}"}
-{"id": "6227.png", "formula": "\\begin{align*} \\mu & = \\lim _ { k \\to \\infty } \\frac { \\| \\theta _ { k + 1 } - \\theta ^ \\star \\| } { \\| \\theta _ k - \\theta ^ \\star \\| } \\\\ & \\leq \\lim _ { k \\to \\infty } \\frac { c \\cdot e ^ { - \\gamma \\times 0 } \\| \\theta _ k - \\theta ^ \\star \\| } { \\| \\theta _ k - \\theta ^ \\star \\| } \\\\ & = c , \\end{align*}"}
-{"id": "1286.png", "formula": "\\begin{align*} \\left | \\int _ { | \\nu | > 2 0 } \\right | \\lesssim \\int _ { | \\nu | > 2 0 } | \\nu | ^ { - 2 k } d \\nu = O ( 1 ) . \\end{align*}"}
-{"id": "1520.png", "formula": "\\begin{align*} \\rho & = \\lim _ { m \\to \\infty } \\frac { ( q - 1 ) ( S ( 1 ) + \\cdots + S ( m ) ) } { q ^ { m + 1 } - q } \\\\ & = \\lim _ { m \\to \\infty } \\frac { ( q - 1 ) q ^ { m } } { q ^ { m + 1 } - q } \\\\ & = \\frac { q - 1 } { q } . \\end{align*}"}
-{"id": "1770.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } \\left [ Y _ { i } Y _ { j } \\right ] & = \\mathbb { E } \\left [ ( \\omega _ { t _ { i } } - \\omega _ { s } + \\xi _ { s , t } ) ( \\omega _ { t _ { j } } - \\omega _ { s } + \\xi _ { s , t } ) \\right ] \\\\ & = \\mathbb { E } \\left [ ( \\omega _ { t _ { i } } - \\omega _ { s } ) ( \\omega _ { t _ { j } } - \\omega _ { s } ) \\right ] + \\mathbb { E } \\left [ \\xi _ { s , t } ^ { 2 } \\right ] \\\\ & = t _ { i } - s + \\gamma _ { H } ( t - s ) ^ { 2 H } \\\\ & \\geq \\gamma _ { H } ( t - s ) ^ { 2 H } , \\end{aligned} \\end{align*}"}
-{"id": "4951.png", "formula": "\\begin{align*} & v _ { d - 3 - k } ^ { N _ k } \\equiv \\left ( - v _ { d - 3 - k } z ^ { t - n } + \\left ( \\left ( \\sum _ { j = d - k - 2 } ^ { d - 3 } ( - 1 ) ^ { j + d + k } v _ j \\right ) + ( - 1 ) ^ k x \\right ) ^ n - v _ { d - k - 3 } z ^ n + v _ { d - k - 3 } x ^ n \\right ) ^ { N _ k } \\equiv 0 , \\end{align*}"}
-{"id": "5380.png", "formula": "\\begin{align*} a _ { n - 1 \\ , m - 3 } | | v _ { n m } ^ 1 | | ^ 2 = - \\overline { d _ { n m } } | | v _ { n - 1 \\ , m - 3 } ^ 1 | | ^ 2 . \\end{align*}"}
-{"id": "1574.png", "formula": "\\begin{align*} \\frac { \\partial \\chi _ w } { \\partial w } ( \\mathcal { Z } ) & = - \\frac { ( 1 - a ) } { 4 \\sqrt { w } } \\log \\left ( \\frac { 2 \\sin ( \\pi \\mathcal { Z } ) } { \\pi \\sqrt { w } } \\right ) + \\frac { \\sqrt { w } ( 1 - a ) } { 2 } \\frac { \\pi \\sqrt { w } } { 2 \\sin ( \\pi \\mathcal { Z } ) } \\frac { 2 \\sin ( \\pi \\mathcal { Z } ) } { 2 \\pi w ^ { 3 / 2 } } \\\\ & = O \\left ( \\frac { \\log ( w ) } { \\sqrt { w } } \\right ) + O \\left ( \\frac { 1 } { \\sqrt { w } } \\right ) = O \\left ( \\frac { 1 } { w } \\right ) . \\end{align*}"}
-{"id": "9055.png", "formula": "\\begin{align*} & \\frac { 1 } { \\pi ^ 2 R ^ 4 } \\int _ { B ( R ) } \\int _ { B ( R ) } \\int _ { \\mathbb { S } ^ 1 } \\int _ { \\mathbb { S } ^ 1 } e ( \\langle x - y , t - w \\rangle ) d x d y d \\mu _ { B } ( t ) d \\mu _ { A } ( w ) \\\\ = & \\int _ { \\mathbb { S } ^ 1 } \\int _ { \\mathbb { S } ^ 1 } \\left ( \\frac { J _ 1 ( R | | t - w | | ) } { R | | t - w | | } \\right ) ^ 2 d \\mu _ { B } ( t ) d \\mu _ { A } ( w ) \\end{align*}"}
-{"id": "7384.png", "formula": "\\begin{align*} \\alpha = \\langle p , D _ { \\lambda _ 1 } f v _ 2 - D _ { \\lambda _ 2 } f v _ 1 \\rangle \\end{align*}"}
-{"id": "10070.png", "formula": "\\begin{align*} \\dot u = - \\frac { 1 } { 4 } u ^ 4 + b ( \\theta _ n ^ 0 , G _ n ^ 0 ) u ^ 7 , \\end{align*}"}
-{"id": "5375.png", "formula": "\\begin{align*} \\langle v _ { n m } ^ k , v _ { r s } ^ l \\rangle = 0 , v _ { n m } ^ k \\in V _ { n m } , \\ ; v _ { r s } ^ l \\in V _ { r s } \\end{align*}"}
-{"id": "5474.png", "formula": "\\begin{align*} v ( t , x ) \\ = \\ Y _ t ^ { t , x , a } , \\hat \\P \\end{align*}"}
-{"id": "9433.png", "formula": "\\begin{align*} \\mathcal { E } _ 2 ( z ) = \\sum _ { 1 \\leq j \\leq D ( N ) - 1 } a _ j ( E _ 2 ( z ) - d _ j E _ 2 ( d _ j z ) ) . \\end{align*}"}
-{"id": "232.png", "formula": "\\begin{align*} \\int _ { B _ r } u ^ { q } \\psi d z d t = O ( r ^ { l } ) , \\ \\end{align*}"}
-{"id": "2358.png", "formula": "\\begin{align*} & [ a , a ^ + ] = 1 \\\\ & H = \\omega ( a ^ + a + 1 / 2 ) \\end{align*}"}
-{"id": "8700.png", "formula": "\\begin{align*} u ^ C _ { \\lambda , \\nu } = \\sum _ { h + i + 2 j = k } \\frac { 2 ^ { - 2 i - 2 h } \\Gamma ( \\frac { 2 \\nu + p ' + 2 } { 4 } ) \\Gamma ( \\frac { \\lambda + \\rho + \\nu - \\rho ' } { 2 } + i ) } { h ! i ! j ! \\Gamma ( \\frac { 2 \\nu + p ' + 2 } { 4 } - j ) \\Gamma ( \\frac { p '' } { 2 } + i ) \\Gamma ( \\frac { \\lambda + \\rho + \\nu - \\rho ' } { 2 } ) } \\Delta _ { \\mathfrak { v } ' } ^ h \\Delta _ { \\mathfrak { v } '' } ^ i \\square ^ j \\delta , \\end{align*}"}
-{"id": "3813.png", "formula": "\\begin{align*} | w | _ \\pi : = \\sum _ { i = 1 } ^ n \\pi ( s _ i ) . \\end{align*}"}
-{"id": "9166.png", "formula": "\\begin{align*} \\phi \\bigl ( ( b , s ) \\otimes ( c , s ) \\bigr ) = ( b \\otimes c , s ) \\end{align*}"}
-{"id": "8175.png", "formula": "\\begin{align*} f _ { j + 1 } = \\sum _ { i \\in \\mathbb { N } } \\langle f _ j , g _ i \\rangle f _ { i + 1 } , \\forall j \\in \\mathbb { N } . \\end{align*}"}
-{"id": "628.png", "formula": "\\begin{align*} \\begin{multlined} q ( \\varphi ^ y , \\varphi ^ \\ast ( y ) x \\varphi ^ \\ast ( y ) ^ { - 1 } ) \\\\ = q ( \\mathrm { i d } , y ) q ( \\varphi , x ) q ( \\mathrm { i d } , \\varphi ^ \\ast ( y ) ^ { - 1 } ) = q ( \\mathrm { i d } , y ) q ( \\varphi , \\varphi ^ \\ast ( y ) ^ { - 1 } ) = \\varphi ^ y \\ , \\end{multlined} \\end{align*}"}
-{"id": "8142.png", "formula": "\\begin{align*} \\overset { J } { V } ( j ) _ 2 \\overset { I } { U } ( i ) _ 1 = ( \\overset { I } { a ' _ k } ) _ 1 \\overset { I } { U } ( i ) _ 1 \\overset { I } { S ( a '' _ k ) } _ 1 ( \\overset { J } { b ' _ k } ) _ 2 \\overset { J } { V } ( j ) _ 2 \\overset { J } { S ( b '' _ k ) } _ 2 = ( \\overset { I } { a _ l } ) _ 1 \\overset { I J } { R } _ { 1 2 } \\overset { I } { U } ( i ) _ 1 \\overset { I J } { R } { ^ { - 1 } _ { 1 2 } } \\overset { J } { V } ( j ) _ 2 \\overset { I J } { R } _ { 1 2 } \\overset { J } { S ( b _ l ) } _ 2 \\end{align*}"}
-{"id": "4643.png", "formula": "\\begin{align*} \\bar { f } ^ n ( x ) - \\bar { \\beta } = x ^ q h ( x ) \\end{align*}"}
-{"id": "8447.png", "formula": "\\begin{align*} \\tilde { \\theta } : = ( I - \\mathcal { H } ) ( \\zeta - \\bar { \\zeta } ) , \\tilde { \\sigma } : = ( I - \\mathcal { H } ) D _ t \\tilde { \\theta } . \\end{align*}"}
-{"id": "7081.png", "formula": "\\begin{align*} \\xi ( x , y ) = \\xi ( x + y , y ) = \\xi ( x + y , x + 2 y = x ) = \\xi ( x + y , x ) = \\xi ( y , x ) \\implies \\xi \\end{align*}"}
-{"id": "1069.png", "formula": "\\begin{align*} \\frac { n ^ 2 G _ n ^ 2 ( x ) } { 3 2 } & = \\frac { 3 2 \\ln n } { 3 2 } + \\frac { 8 x - 5 \\ln ( 2 \\ln n ) } { ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } } \\frac { ( 3 2 \\ln n ) ^ { \\frac { 1 } { 2 } } } { 3 2 } + \\frac { ( 8 x - 5 \\ln ( 2 \\ln n ) ) ^ 2 } { 3 2 \\cdot 4 \\cdot ( 2 \\ln n ) } \\\\ & = \\ln n + \\frac { 8 x - 5 \\ln ( 2 \\ln n ) } { 8 } + o ( 1 ) , \\end{align*}"}
-{"id": "9700.png", "formula": "\\begin{align*} I _ { \\textbf { d } } ( P _ i ) = \\frac { 1 } { 2 \\pi } \\int _ { \\gamma ( s ) = \\partial \\Omega ( P _ i ) } \\kappa _ { \\textbf { d } } d s . \\end{align*}"}
-{"id": "6153.png", "formula": "\\begin{align*} R ^ { \\omega , \\nu } f _ k ( r z ) & = \\int _ { \\mathbb { D } } f _ k ( \\sqrt r \\xi ) \\overline { B _ { \\sqrt r z } ^ \\nu ( \\xi ) } \\omega ( \\xi ) d A ( \\xi ) \\\\ & \\to \\int _ { \\mathbb { D } } f ( \\sqrt r \\xi ) \\overline { B _ { \\sqrt r z } ^ \\nu ( \\xi ) } \\omega ( \\xi ) d A ( \\xi ) \\\\ & = R ^ { \\omega , \\nu } f ( r z ) \\end{align*}"}
-{"id": "5077.png", "formula": "\\begin{align*} A _ { g p } A _ { g q } ^ * = A _ p A _ q ^ * , ~ A _ { g p } \\Psi _ { g q } ^ * = A _ p \\Psi _ q ^ * , ~ \\Psi _ { g p } \\Psi _ { g q } ^ * = \\Psi _ p \\Psi _ q ^ * , ~ \\forall g , p , q \\in G . \\end{align*}"}
-{"id": "2886.png", "formula": "\\begin{align*} L _ { a b } L _ { c d } + L _ { a c } L _ { d b } + L _ { a d } L _ { b c } = 0 \\end{align*}"}
-{"id": "7849.png", "formula": "\\begin{align*} \\sum _ { j \\in \\sigma ( I ) \\cap J } u _ j \\ge \\sum _ { k = 0 } ^ { k _ 0 - 1 } \\sum _ { j \\in \\sigma ( I ) \\cap J \\cap U _ k } u _ j \\ge \\sum _ { k = 0 } ^ { k _ 0 - 1 } u _ { d - k m } | \\sigma ( I ) \\cap J \\cap U _ k | . \\end{align*}"}
-{"id": "3180.png", "formula": "\\begin{align*} \\mathcal { B } _ \\pm ^ \\prime & = \\{ \\psi \\mid d ^ \\prime \\psi = 0 , \\ ( \\varGamma ^ \\prime \\pm 1 ) \\psi = 0 \\} \\\\ & = \\{ \\psi \\mid d ( V ^ { - 1 } \\psi ) = 0 , \\ ( \\varGamma \\pm 1 ) ( V ^ { - 1 } \\psi ) = 0 \\} \\\\ & = V \\mathcal { B } _ \\pm , \\end{align*}"}
-{"id": "1600.png", "formula": "\\begin{align*} d X _ { t } = X _ { t } \\mu ( X _ { t } ) \\ , d t + \\sigma ( X _ { t } ) \\ , d B _ { t } , \\end{align*}"}
-{"id": "6314.png", "formula": "\\begin{align*} \\frac { \\mathbb { P } _ 2 [ G = G _ 0 ] } { \\mathbb { P } _ 1 [ G = G _ 0 ] } = \\frac { 1 } { n } \\sum _ { v \\in V ( G _ 0 ) } \\frac { \\binom { n - d ^ { G _ 0 } ( v ) - 1 } { k } } { \\binom { d ^ { G _ 0 } ( v ) + k } { k } } . \\end{align*}"}
-{"id": "8809.png", "formula": "\\begin{align*} ( J _ { \\alpha } v ^ k ) ( t , x ) \\stackrel { d } { = } ( J _ { \\alpha } u ^ { \\varepsilon _ k } ) ( t , x ) \\end{align*}"}
-{"id": "5301.png", "formula": "\\begin{align*} A = \\begin{bmatrix} 3 & 1 & 1 & 1 & 1 \\\\ - 2 & 0 & - 1 & - 1 & - 1 \\\\ - 2 & - 1 & 0 & - 1 & - 1 \\\\ - 2 & - 1 & - 1 & 0 & - 1 \\\\ - 2 & - 1 & - 1 & - 1 & 0 \\end{bmatrix} , B = \\begin{bmatrix} 3 & 2 & 2 & 2 & 2 \\\\ - 1 & 0 & - 1 & - 1 & - 1 \\\\ - 1 & - 1 & 0 & - 1 & - 1 \\\\ - 1 & - 1 & - 1 & 0 & - 1 \\\\ - 1 & - 1 & - 1 & - 1 & 0 \\end{bmatrix} \\end{align*}"}
-{"id": "4451.png", "formula": "\\begin{align*} q \\circ \\bar { h } = h = h ' \\circ \\bar { h } ' = ( q \\circ b ' ) \\circ \\bar { h } ' = q \\circ ( b ' \\circ \\bar { h } ' ) . \\end{align*}"}
-{"id": "621.png", "formula": "\\begin{align*} ( \\gamma , \\delta ) \\circ ( \\alpha , \\beta ) = ( \\gamma \\alpha , \\beta \\delta ) \\ . \\end{align*}"}
-{"id": "3659.png", "formula": "\\begin{align*} \\sum _ { j \\in [ b ] } N _ j = \\Big | \\big ( \\bigcup _ { i \\in [ b ] } X _ i \\big ) \\cap \\big ( \\bigcup _ { j \\in [ b ] } Y _ j \\big ) \\Big | \\ge \\frac { b } { 2 } \\cdot \\frac { \\log a } { 3 \\log \\log n } \\ge \\frac { b } { 1 8 } \\cdot \\frac { \\log n } { \\log \\log n } = \\frac { a } { 1 8 } \\end{align*}"}
-{"id": "2173.png", "formula": "\\begin{align*} \\int _ { \\mathcal { O } ( B ( \\mathbf { y } , r ) ) ^ { c } } & | \\tau _ { - \\mathbf y } ( f * \\phi ) ( \\mathbf x ) | \\ , d w ( \\mathbf x ) \\\\ & \\leq C r ^ { - \\delta ' } \\| f ( \\cdot ) ( 1 + \\| \\cdot \\| ) ^ { \\mathbf { N } / 2 + \\delta } \\| _ { L ^ 1 ( d w ) } \\| \\phi ( \\cdot ) ( 1 + \\| \\cdot \\| ) ^ { \\mathbf { N } + \\delta } \\| _ { L ^ { \\infty } } . \\end{align*}"}
-{"id": "5366.png", "formula": "\\begin{align*} \\left ( \\frac { a _ { n m } d _ { n + 1 \\ , m + 3 } } { a _ { n - 1 \\ , m + 3 } d _ { n \\ , m + 6 } } \\right ) ^ 2 = \\frac { n ^ 2 } { ( n + 1 ) ^ 2 } \\end{align*}"}
-{"id": "5896.png", "formula": "\\begin{align*} j ( \\tilde { f } ) = ( 1 - q ^ { - 1 } ) \\sum _ { n = 1 } ^ { \\infty } q ^ n C _ { ( s _ 0 s _ 1 ) ^ n } + q ^ { n + \\frac { 1 } { 2 } } C _ { s _ 0 ( s _ 1 s _ 0 ) ^ n } , \\end{align*}"}
-{"id": "5827.png", "formula": "\\begin{align*} \\frac { 1 } { \\phi _ { \\bar X _ n } ( t _ 1 ) } \\left . \\frac { \\phi _ { ( \\bar X _ n , s _ n ^ 2 ) } ( t _ 1 , t _ 2 ) } { \\partial t _ 2 } \\right | _ { t _ 2 = 0 } = \\left . \\frac { \\partial \\phi _ { s _ n ^ 2 } ( t _ 2 ) } { \\partial t _ 2 } \\right | _ { t _ 2 = 0 } . \\end{align*}"}
-{"id": "2270.png", "formula": "\\begin{align*} a R ( z ) & = - 1 + z R ( z ) \\ ; \\quad & R ( z ) a & = - 1 + z R ( z ) . \\end{align*}"}
-{"id": "9933.png", "formula": "\\begin{align*} & \\| \\pi _ { n - } ^ { \\mu } - \\pi _ { n - } ^ { \\gamma } \\| _ { T V } = \\\\ & \\frac { E ^ { \\gamma } \\left [ \\left . | E ^ { \\gamma } [ \\frac { d \\mu } { d \\gamma } ( X _ { 0 } ) | Y _ { [ 0 , n - 1 ] } , X _ { n } ] - E ^ { \\gamma } [ \\frac { d \\mu } { d \\gamma } ( X _ { 0 } ) | Y _ { [ 0 , n - 1 ] } ] | \\right | Y _ { [ 0 , n - 1 ] } \\right ] } { E ^ { \\gamma } \\left [ \\left . \\frac { d \\mu } { d \\gamma } ( X _ { 0 } ) \\right | Y _ { [ 0 , n - 1 ] } \\right ] } \\end{align*}"}
-{"id": "6256.png", "formula": "\\begin{align*} \\tilde { \\gamma } ( x ) = \\sum _ { j , k = 1 } ^ n u _ j \\gamma ( u _ j ^ * x u _ k ) u _ k ^ * . \\end{align*}"}
-{"id": "5052.png", "formula": "\\begin{align*} [ L _ b ^ { \\mathcal P } , [ L _ a ^ { \\mathcal P } , { \\mathcal P } ] ] = - [ L _ { 2 / 3 { \\mathcal P } ( a , b ) + 1 / 3 { \\mathcal P } ( b , a ) } ^ { \\mathcal P } , { \\mathcal P } ] . \\end{align*}"}
-{"id": "7962.png", "formula": "\\begin{align*} \\bigsqcup _ { q \\in Q } S _ q = \\bigsqcup _ { r \\in R } T _ r = N . \\end{align*}"}
-{"id": "1421.png", "formula": "\\begin{align*} \\overline { \\textrm { O P T } } _ { \\eta } ( T ) \\leq 3 + 3 \\times T \\times \\bigg ( \\sum _ { k = t _ { \\eta } ( T ) } ^ T P ( \\tau = k ) \\times c ^ 2 _ k \\bigg ) ^ { \\frac { 1 } { 2 } } + 1 5 \\times T ^ 2 \\times \\sum _ { k = t _ { \\eta } ( T ) } ^ T P ( \\tau = k ) \\times c ^ 2 _ k . \\end{align*}"}
-{"id": "6496.png", "formula": "\\begin{align*} \\bold { c } ( K ^ \\circ ) = \\int _ { K ^ \\circ } z d z = 0 \\ , . \\end{align*}"}
-{"id": "2628.png", "formula": "\\begin{align*} D S ^ { ( \\mathrm { s o f t } ) } ( \\kappa , \\pi ; u , v ) = & \\int _ { \\mathcal { C } _ { < } } \\frac { d z } { 2 \\pi i } \\int _ { \\mathcal { C } _ { < } } \\frac { d w } { 2 \\pi i } e ^ { \\frac { 1 } { 3 } ( w - \\kappa ) ^ 3 - u ( w - \\kappa ) + \\frac { 1 } { 3 } ( z - \\kappa ) ^ 3 - v ( z - \\kappa ) } \\\\ & \\times \\frac { 1 } { \\sqrt { 4 z w } } \\frac { w - z } { z + w } \\prod _ { k = 1 } ^ { m } \\frac { z + \\pi _ { k } } { z - \\pi _ { k } } \\frac { w + \\pi _ { k } } { w - \\pi _ { k } } , \\end{align*}"}
-{"id": "6024.png", "formula": "\\begin{align*} \\liminf _ { \\epsilon \\downarrow 0 } \\ , \\ , \\ , \\inf \\left \\{ \\tilde { \\alpha } ^ \\mu _ \\epsilon ( Q ) \\ , : \\ , Q \\in \\mathcal { P } ( \\C ) , \\ Q \\circ H ^ { - 1 } = \\nu _ \\epsilon \\right \\} & \\geq \\inf \\left \\{ \\alpha ^ \\mu _ 0 ( Q ) \\ , : \\ , Q \\in \\mathcal { P } ( \\C ) , \\ Q \\circ H ^ { - 1 } = \\nu \\right \\} . \\end{align*}"}
-{"id": "9353.png", "formula": "\\begin{align*} [ [ y , x , \\overset { i - 1 } \\ldots , x ] , x ] ^ 2 = ( [ y , x , \\overset { i - 1 } \\ldots , x ] ^ { - 1 } ) ^ 2 ( [ y , x , \\overset { i - 1 } \\ldots , x ] ^ x ) ^ 2 [ [ y , x , \\overset { i - 1 } \\ldots , x ] ^ x , [ y , x , \\overset { i - 1 } \\ldots , x ] ^ { - 1 } ] . \\end{align*}"}
-{"id": "8631.png", "formula": "\\begin{align*} | u _ \\varepsilon - \\bar { u } | \\cdot | u _ \\varepsilon - \\bar { v } | + | z _ \\varepsilon | = O ( 1 ) e ^ { - \\left ( \\sqrt { 2 } \\sqrt { 1 - 4 c _ 0 ^ 2 } + O ( \\varepsilon ) \\right ) | x | } , \\end{align*}"}
-{"id": "4549.png", "formula": "\\begin{align*} | s _ 1 | = \\max \\{ | s _ 1 | , | s _ 1 ^ { - 1 } | , \\ldots , | s _ k | , | s _ k ^ { - 1 } | \\} . \\end{align*}"}
-{"id": "550.png", "formula": "\\begin{align*} R _ { 1 2 1 2 } = - \\phi ' ( x _ 3 ) ^ 2 e ^ { 2 \\phi ( x _ 3 ) } , R _ { 1 3 1 3 } = R _ { 2 3 2 3 } = - \\phi '' ( x _ 3 ) e ^ { 2 \\phi ( x _ 3 ) } . \\end{align*}"}
-{"id": "4249.png", "formula": "\\begin{align*} \\int _ { B _ { R } } \\left \\langle a ( x ) \\left \\lvert d u \\right \\rvert ^ { p - 2 } d u ; d \\phi \\right \\rangle = \\int _ { B _ { R } } \\left \\langle f ; \\phi \\right \\rangle \\phi \\in W _ { \\delta , T } ^ { 1 , p } \\left ( B _ { R } ; \\Lambda ^ { k } \\mathbb { R } ^ { n } \\otimes \\mathbb { R } ^ { N } \\right ) . \\end{align*}"}
-{"id": "3725.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\int _ { | u | \\geq M } \\frac { F ( y , u _ k ) } { | x - y | ^ { \\mu } } F ( x , u ) d y ~ d x = o ( M ) , \\end{align*}"}
-{"id": "7773.png", "formula": "\\begin{align*} P _ { * } f ( x ) = \\sup _ { t > 0 } | P _ t f ( x ) | , \\end{align*}"}
-{"id": "7352.png", "formula": "\\begin{align*} d ( a , b ) = d ( a , e _ a ) + d ( e _ a , e _ b ) + d ( b , e _ b ) \\end{align*}"}
-{"id": "3914.png", "formula": "\\begin{align*} r _ 0 = r _ 1 , \\ r _ 1 = r _ 2 , \\ r _ 2 = r _ 3 , \\end{align*}"}
-{"id": "9788.png", "formula": "\\begin{align*} \\Sigma _ { \\kappa } = \\bigcup _ { h = 1 } ^ { \\infty } E _ h \\end{align*}"}
-{"id": "1038.png", "formula": "\\begin{align*} \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) & \\leq ( n / 4 ) ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } \\mathbb { P } ( \\xi ^ { ( n ) } ( I _ { n , k } ) = 0 ) \\\\ & \\leq ( n / 4 ) ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } \\prod _ { j = 1 } ^ k \\mathbb { P } ( \\xi ^ { ( n ) } ( I ( y _ j , F _ n ( x _ j ) ) ) = 0 ) \\\\ & = ( n / 4 ) ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } \\prod _ { j = 1 } ^ k D _ n ( F _ n ( x _ j ) / 2 ) . \\end{align*}"}
-{"id": "2892.png", "formula": "\\begin{align*} e ^ { B } = \\prod _ { j = 1 } ^ { 3 } \\frac { 1 } { \\sqrt { \\rho _ j } } . \\end{align*}"}
-{"id": "4402.png", "formula": "\\begin{align*} \\tilde F _ { \\delta } - F _ { \\delta } = T \\cdot h \\end{align*}"}
-{"id": "9004.png", "formula": "\\begin{align*} Z = \\frac { \\partial \\sigma } { \\partial t } \\times \\frac { \\partial \\sigma } { \\partial u ^ { 1 } } \\times \\dotsb \\times \\frac { \\partial \\sigma } { \\partial u ^ { m - 1 } } \\ , , \\end{align*}"}
-{"id": "5335.png", "formula": "\\begin{align*} \\langle w ^ k , w ^ l \\rangle = 0 \\quad \\mbox { f o r a l l } \\ k \\neq l . \\end{align*}"}
-{"id": "1303.png", "formula": "\\begin{align*} j _ n \\bigl ( ( x _ 1 , x _ 2 , \\ldots , x _ n ) , ( y _ 1 , y _ 2 , \\ldots , y _ n ) \\bigr ) = x _ 1 + x _ 2 + \\cdots x _ n - ( y _ 1 + y _ 2 + \\cdots + y _ n ) \\end{align*}"}
-{"id": "1606.png", "formula": "\\begin{align*} \\lim _ { x \\downarrow 0 } \\frac { u ' ( x ) } { S ' ( x ) } & = 0 \\\\ ( \\mathcal { A } u ) ( x ) & = \\ell ^ \\ast , x \\in ( 0 , b ^ \\ast ) , \\\\ u ' ( x ) & = 1 , x \\geq b ^ \\ast . \\end{align*}"}
-{"id": "4022.png", "formula": "\\begin{align*} F ^ { \\ast } \\eta _ { 2 } = \\eta _ { 1 } , F ^ { \\ast } g _ { 2 } = g _ { 1 } . \\end{align*}"}
-{"id": "1676.png", "formula": "\\begin{align*} \\alpha _ A ( [ X , Y ] ) = [ \\alpha _ A ( X ) , \\alpha _ A ( Y ) ] . \\end{align*}"}
-{"id": "3264.png", "formula": "\\begin{align*} ( a ^ n + b ^ n ) = ( a + b ) ( a ^ { n - 1 } + b ^ { n - 1 } ) - a b ( a ^ { n - 2 } + b ^ { n - 2 } ) , \\end{align*}"}
-{"id": "2650.png", "formula": "\\begin{align*} D S _ { N } ( x , y ) & = \\sum _ { j , k = 1 } ^ { N } \\phi _ { j } ( x ) c _ { k j } \\phi _ { k } ( y ) , \\\\ S _ { N } ( x , y ) & = \\sum _ { j = 1 } ^ { N } \\phi _ { j } ( x ) \\sum _ { k = 1 } ^ { N + n } c _ { k j } \\epsilon \\phi _ { k } ( y ) , \\\\ I S _ { N } ( x , y ) & = - \\epsilon ( x , y ) + \\sum _ { j , k = 1 } ^ { N + n } \\epsilon \\phi _ { j } ( x ) c _ { k j } \\epsilon \\phi _ { k } ( y ) . \\end{align*}"}
-{"id": "536.png", "formula": "\\begin{align*} P _ 0 ( x ' , x _ 3 ) = \\left [ x _ 3 \\partial _ 1 \\vec b _ 0 ( x ' ) , ~ x _ 3 \\partial _ 2 \\vec b _ 0 ( x ' ) , ~ \\partial _ 3 \\vec d _ 0 ( x ' , x _ 3 ) \\right ] . \\end{align*}"}
-{"id": "9232.png", "formula": "\\begin{align*} \\alpha ^ * ( x _ 0 ) = \\pm \\frac 1 { \\norm { z } } \\sqrt { ( p _ { 1 0 } - x _ 0 ) ^ 2 - \\norm { M ( u + x _ 0 v ) } ^ 2 } . \\end{align*}"}
-{"id": "5774.png", "formula": "\\begin{align*} \\epsilon _ 1 ( \\Gamma ) : = \\begin{cases} 1 & \\Gamma = S _ m , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "516.png", "formula": "\\begin{align*} \\wedge ^ 2 H ^ 0 ( K - L ) = \\wedge ^ 2 W _ 1 \\oplus \\wedge ^ 2 W _ 3 \\oplus \\Big ( W _ 1 \\otimes W _ 3 \\Big ) . \\end{align*}"}
-{"id": "385.png", "formula": "\\begin{align*} 4 D ( v , w ) & = - 8 ( w ^ 3 - 3 2 w ^ 2 + 6 5 5 3 6 v ^ 2 w - 2 0 9 7 1 5 2 v ^ 2 + 6 4 0 v w ^ 2 - 2 0 4 8 0 v w ) \\\\ & = - 4 \\cdot 2 ( ( w ^ 2 ( w - 3 2 ) + 6 5 5 3 6 v ^ 2 ( w - 3 2 ) + 6 4 0 w v ( w - 3 2 ) ) , \\end{align*}"}
-{"id": "3174.png", "formula": "\\begin{align*} \\mathcal { T } _ \\pm = \\ker ( \\varGamma \\mp 1 ) \\cap \\ker ( C - 1 ) . \\end{align*}"}
-{"id": "2997.png", "formula": "\\begin{align*} p ' _ { e , m } ( n ) - p ' _ { o , m } ( n ) & = \\begin{cases} ( - 1 ) ^ k , & n = P _ { m + 2 , k } Q _ { m + 2 , k } k \\in \\mathbb { N } ; \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "8379.png", "formula": "\\begin{align*} g _ 1 : = 2 [ z _ { t t } , \\mathfrak { H } ] \\frac { \\bar { z } _ { t \\alpha } } { z _ { \\alpha } } + 2 [ z _ t , \\mathfrak { H } ] \\frac { \\bar { z } _ { t t \\alpha } } { z _ { \\alpha } } - \\frac { 1 } { \\pi i } \\int \\Big ( \\frac { z _ t ( \\alpha , t ) - z _ t ( \\beta , t ) } { z ( \\alpha , t ) - z ( \\beta , t ) } \\Big ) ^ 2 \\bar { z } _ { t \\beta } d \\beta . \\end{align*}"}
-{"id": "5676.png", "formula": "\\begin{align*} C ^ \\circ \\subset ( \\rho \\bar { \\mbox { \\bf B } } ) ^ \\circ = \\frac 1 \\rho \\bar { \\mbox { \\bf B } } _ { X ^ * } \\ , . \\end{align*}"}
-{"id": "1002.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\mathbb { E } \\dfrac { ( { \\chi } ^ { ( n ) } ( A ) ) ! } { ( { \\chi } ^ { ( n ) } ( A ) - k ) ! } = \\lim _ { n \\to + \\infty } \\mathbb { E } \\rho ^ { ( n , k ) } ( A ^ k ) = ( - f ' ( x ) ) ^ k , \\end{align*}"}
-{"id": "9630.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\infty } ( - 1 ) ^ { j - 1 } ( ( q ^ { j } ) _ { m } - 1 ) = \\frac { 1 } { 2 } \\left ( \\frac { ( q ) _ { m } } { ( - q ) _ { m } } - 1 \\right ) . \\end{align*}"}
-{"id": "9794.png", "formula": "\\begin{align*} | D ^ \\alpha P _ { Z _ \\circ } ( r X ) - D ^ { \\alpha } P _ { X _ \\circ } ( r X ) | \\leq C _ h r ^ { \\ell + \\beta - | \\alpha | } = C _ h | X _ \\circ - Z _ \\circ | ^ { \\ell + \\beta - | \\alpha | } X \\in B _ { 1 / 2 } ( r ^ { - 1 } X _ \\circ ) . \\end{align*}"}
-{"id": "115.png", "formula": "\\begin{align*} \\left [ B ( 0 ) + \\gamma B _ 1 \\right ] ^ { 2 N } = & I _ { 2 N } + \\left [ \\sum _ { \\substack { k = 1 \\\\ k \\ , } } ^ { 2 N - 1 } + \\sum _ { \\substack { k = 2 \\\\ k \\ , } } ^ { 2 N } \\right ] \\gamma ^ k S _ k ( B ( 0 ) , B _ 1 ) \\end{align*}"}
-{"id": "4148.png", "formula": "\\begin{align*} \\varphi _ { 2 , - } ( x ) = - \\varphi _ { 2 , + } ( x ) , x \\in \\Delta _ 2 , \\end{align*}"}
-{"id": "6126.png", "formula": "\\begin{align*} D _ B \\big ( F _ { i , j } \\big ) : = \\det \\left ( \\begin{array} { c c c } F ^ { ( 0 ) } _ { i - \\mu , j } & F ^ { ( 1 ) } _ { i - \\mu , j } & F ^ { ( 2 ) } _ { i - \\mu , j } \\\\ Z _ 0 & Z _ 1 & Z _ 2 \\\\ A _ { 0 , \\mu _ 1 } & A _ { 1 , \\mu _ 1 } & A _ { 2 , \\mu _ 1 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "1944.png", "formula": "\\begin{align*} \\mathcal { A } = \\lim _ { p \\rightarrow \\infty } \\sum _ { j = 1 } ^ p ( \\epsilon _ j - \\tan \\epsilon _ j ) + ( \\chi + 1 ) F _ 3 ( \\chi ) . \\end{align*}"}
-{"id": "9778.png", "formula": "\\begin{align*} ( 1 - 2 ^ { \\lambda _ * - \\kappa - 1 } ) \\kappa ! c _ \\infty = 0 . \\end{align*}"}
-{"id": "7957.png", "formula": "\\begin{align*} g _ 2 ( { x _ { a , 2 } } ) = \\alpha _ { a , b } g _ 2 ( { x _ { b , 2 } } ) \\neq 0 , \\end{align*}"}
-{"id": "6551.png", "formula": "\\begin{align*} \\sum _ { P \\in E ( \\beta ) } m _ \\beta ^ P = ( - 1 ) ^ { w - 1 } w \\ , n _ \\beta . \\end{align*}"}
-{"id": "6646.png", "formula": "\\begin{align*} M _ { 1 , Q } f & \\leq \\overline M _ { 1 , Q } f + N ^ { - \\epsilon } A _ { \\tau } f , \\\\ \\lvert E \\rvert ^ { - 1 } \\langle \\overline M _ { 1 , Q } f , g \\rangle & \\lesssim N ^ { 1 + \\epsilon / 2 } \\langle f \\rangle _ { E } ^ { \\frac 1 { \\bar p + \\epsilon } } \\langle g \\rangle _ { E } ^ { \\frac 1 { \\bar q + \\epsilon } } . \\end{align*}"}
-{"id": "1307.png", "formula": "\\begin{align*} ( d , 0 ) , \\bigl ( e , ( x , e ) \\bigr ) , \\bigl ( e , ( y , e ) \\bigr ) \\in G , ( d , 0 ) ^ { - 1 } = ( d ^ { - 1 } , 0 ) , \\bigl ( e , ( y , e ) \\bigr ) ^ { - 1 } = \\bigl ( e , - ( y , e ) \\bigr ) , \\end{align*}"}
-{"id": "3654.png", "formula": "\\begin{align*} A : = L D L ^ T \\end{align*}"}
-{"id": "5196.png", "formula": "\\begin{align*} \\forall ( s , t ) \\in ( \\mathbb { R } _ { + } \\setminus { 0 } ) ^ { 2 } , \\alpha ( s t ) = \\alpha ( s ) \\alpha ( t ) \\end{align*}"}
-{"id": "372.png", "formula": "\\begin{align*} v _ { 2 m + 2 } & = 2 s v _ { 2 m + 1 } - v _ { 2 m } \\\\ & \\equiv 2 s ( s z _ 0 + c x _ 0 ) - z _ 0 \\\\ & \\equiv 2 s ^ 2 z _ 0 + 2 c x _ 0 - z _ 0 \\\\ & \\equiv 2 ( a c + 1 ) z _ 0 - z _ 0 \\\\ & \\equiv z _ 0 \\pmod { 2 c } . \\end{align*}"}
-{"id": "2265.png", "formula": "\\begin{align*} \\langle \\alpha _ x | \\beta _ y \\rangle & = \\delta ( x - y ) \\\\ \\langle \\alpha _ x | \\beta _ { \\pm \\i \\pi } \\rangle & = 0 & ( \\alpha _ { \\pm \\i \\pi } | \\beta _ x \\rangle & = 0 \\\\ \\langle \\alpha _ { \\pm \\ i \\pi } | \\beta _ { \\pm \\i \\pi } \\rangle & = 1 & \\alpha _ { \\pm \\ i \\pi } | \\beta _ { \\mp \\i \\pi } \\rangle & = 0 \\end{align*}"}
-{"id": "1561.png", "formula": "\\begin{align*} \\widetilde { F } \\left ( \\begin{array} { c } z \\\\ x \\end{array} \\right ) : = U \\circ F \\circ U ^ { - 1 } \\left ( \\begin{array} { c } z \\\\ x \\end{array} \\right ) = \\left ( \\begin{array} { c } z + z ^ 2 + a _ 3 z ^ 3 + O ( z ^ 4 ) + \\dots \\\\ b _ 0 ( x ) + b _ 1 ( x ) z + b _ 2 ( x ) z ^ 2 + \\dots \\end{array} \\right ) , \\end{align*}"}
-{"id": "2682.png", "formula": "\\begin{align*} z _ 0 = \\frac { 1 } { 2 } ( \\frac { 1 } { \\tau } + \\tau ) ; \\end{align*}"}
-{"id": "7655.png", "formula": "\\begin{align*} \\frac { \\partial v ^ { \\varepsilon } ( x , t ) } { \\partial t } - L ^ { \\varepsilon } v ^ { \\varepsilon } ( x , t ) = \\phi ^ \\varepsilon ( x ^ \\varepsilon , t ) , v ^ { \\varepsilon } ( x , 0 ) = \\psi ^ \\varepsilon ( x ) . \\end{align*}"}
-{"id": "5012.png", "formula": "\\begin{align*} A _ { m _ 1 \\ldots m _ l } : = \\psi _ { m _ 1 } \\circ \\ldots \\circ \\psi _ { m _ l } ( A ) , 1 \\leq m _ i \\leq M , l \\geq 1 . \\end{align*}"}
-{"id": "9250.png", "formula": "\\begin{align*} v _ \\tau = \\zeta ^ 2 w _ { \\tau } \\end{align*}"}
-{"id": "2767.png", "formula": "\\begin{align*} \\int _ { \\mathcal { C } _ { > } } \\frac { d z } { 2 \\pi i } ( 2 z ) ^ { - \\frac { 3 } { 2 } } e ^ { - \\frac { 1 } { 3 } ( z - \\kappa ) ^ 3 + v ( z - \\kappa ) } = - \\phi _ { 1 } '' ( \\kappa ; v ) + v \\phi _ { 1 } ( \\kappa ; v ) . \\end{align*}"}
-{"id": "5686.png", "formula": "\\begin{align*} \\max & \\left \\{ \\| x ^ A - x _ 0 \\| , \\| x ^ B - x _ 0 \\| \\right \\} + \\frac { M } { \\eta } \\| x ^ A - x ^ B \\| \\\\ & \\le \\zeta + \\frac { M } { \\eta } \\left ( \\| x ^ A - x _ 0 \\| + \\| x _ 0 - x ^ B \\| \\right ) \\le \\zeta + \\frac { M } { \\eta } 2 \\zeta = \\delta \\ , . \\end{align*}"}
-{"id": "288.png", "formula": "\\begin{align*} \\delta _ Z \\ ; \\hookrightarrow \\ ; \\delta _ X * \\delta _ Y \\ ; = \\ ; a _ \\dag ( \\delta _ X \\boxtimes \\delta _ Y ) \\end{align*}"}
-{"id": "4429.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ T \\Bigl { \\langle } \\dfrac { \\dd u _ p ( \\tau ) } { \\dd t } , w ( \\tau ) \\Bigr { \\rangle } \\dd \\tau \\underset { p \\rightarrow \\infty } { \\longrightarrow } & \\int _ { \\mathcal { Q } _ T } \\big ( \\nabla u + u _ { \\alpha } \\otimes \\nabla \\mathcal { V } \\big ) : \\nabla w \\dd x \\dd \\tau \\\\ & + \\int _ { \\Gamma _ T } \\sigma ( \\tau , x , u ( \\tau , x ) , \\mathcal { V } ( \\tau , x ) ) \\cdot w ( x ) \\dd \\mu \\dd \\tau . \\end{aligned} \\end{align*}"}
-{"id": "1195.png", "formula": "\\begin{align*} \\nabla _ x ^ k \\Phi _ \\tau ( x , y ) = \\frac { { e } ^ { - { \\tau } l ( x , y ) } } { 8 \\pi \\gamma _ + \\gamma _ - \\sqrt { { \\rm d e t } H ( x , y ) } } \\Big ( \\frac { - \\tau } { \\sqrt { \\gamma _ - } } \\Big ) ^ k \\Big ( \\sum _ { j = 0 } ^ { N } { \\tau ^ { - j } } \\Phi _ { j } ^ { ( k ) } ( x , y ) + Q _ { N , \\tau } ^ { ( k ) } ( x , y ) \\Big ) , \\end{align*}"}
-{"id": "5248.png", "formula": "\\begin{align*} 2 \\log ( 1 / s ) / x ^ { 2 } = 1 + ( 2 \\log C ) / x ^ { 2 } + ( 2 \\log x ) / x ^ { 2 } + O ( 1 / x ^ { 3 } ) . \\end{align*}"}
-{"id": "2940.png", "formula": "\\begin{align*} & \\alpha _ { 1 } = \\| B ^ { \\dagger } \\| _ { 2 } ^ { 2 } \\big ( \\| E B ^ { \\dagger } \\| _ { F } ^ { 2 } - \\| A A ^ { \\dagger } E B ^ { \\dagger } \\| _ { F } ^ { 2 } \\big ) + ( n - s ) \\| A ^ { \\dagger } \\| _ { 2 } ^ { 2 } , \\\\ & \\alpha _ { 2 } = \\| A ^ { \\dagger } \\| _ { 2 } ^ { 2 } \\big ( \\| E A ^ { \\dagger } \\| _ { F } ^ { 2 } - \\| B B ^ { \\dagger } E A ^ { \\dagger } \\| _ { F } ^ { 2 } \\big ) . \\end{align*}"}
-{"id": "8220.png", "formula": "\\begin{align*} a _ j ( t ) = ( - 1 ) ^ { j + 1 } t s _ j ( t _ 0 ^ { - 1 } , \\dots , t _ 4 ^ { - 1 } ) , 0 \\leq j \\leq 4 , \\end{align*}"}
-{"id": "5457.png", "formula": "\\begin{align*} V _ 0 \\ = \\ \\hat V _ 0 ^ { \\mathcal R } . \\end{align*}"}
-{"id": "1481.png", "formula": "\\begin{align*} \\frac { d E ^ { j } _ { w } } { d t } = \\sum _ { \\ell = 1 } ^ { 6 } W ^ { j } _ { \\ell } , \\end{align*}"}
-{"id": "8835.png", "formula": "\\begin{align*} \\mu _ N ( x _ 1 , \\dots , x _ N ) & : = N ^ { - 1 } \\sum _ { i = 1 } ^ N x _ i . \\end{align*}"}
-{"id": "7000.png", "formula": "\\begin{align*} { \\textsl { \\footnotesize Y } } _ j = [ \\xi ^ { j } + \\xi ^ { - j } , \\xi ^ { 2 j } + \\xi ^ { - 2 j } , \\ldots , \\xi ^ { ( n - 1 ) j } + \\xi ^ { - ( n - 1 ) j } , 1 ] , \\end{align*}"}
-{"id": "5324.png", "formula": "\\begin{align*} H = - i \\left [ \\begin{array} { c c } 0 & 1 \\\\ - 1 & 0 \\end{array} \\right ] , \\end{align*}"}
-{"id": "5961.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u ( t , x ) + g ^ * ( t , \\nabla u ( t , x ) ) = 0 [ 0 , 1 ] \\times \\R ^ d \\\\ u ( 1 , x ) = f ( x ) , x \\in \\R ^ d . \\end{cases} \\end{align*}"}
-{"id": "1581.png", "formula": "\\begin{align*} \\Phi _ G ( w _ { n ^ 2 + k } , y _ { n ^ 2 + k } ) & = \\frac { 1 } { w _ { n ^ 2 + k } + O ( w _ { n ^ 2 + k } ^ 2 ) } + c \\log \\left ( \\frac { 1 } { w _ { n ^ 2 + k } + O ( w _ { n ^ 2 + k } ^ 2 ) } \\right ) + o ( 1 ) \\\\ & = \\frac { 1 } { w _ { n ^ 2 + k } } + c \\log \\left ( \\frac { 1 } { w _ { n ^ 2 + k } } \\right ) + O ( 1 ) . \\end{align*}"}
-{"id": "6610.png", "formula": "\\begin{align*} \\overline { C } _ n ( \\varepsilon ) = \\prod _ { t = 1 } ^ { ( F _ { n } - 1 ) / 2 } \\left ( 1 - \\frac { v _ { n } ^ 2 } { s _ { n t } ^ 2 } \\right ) , \\end{align*}"}
-{"id": "4472.png", "formula": "\\begin{align*} \\Delta _ s \\overline u + n L & = 0 \\quad \\textrm { i n } \\ , \\Omega \\\\ \\overline u & = \\varphi \\quad \\textrm { o n } \\ , \\partial \\Omega \\end{align*}"}
-{"id": "1845.png", "formula": "\\begin{align*} v ^ { \\rm o d d } ( n , x ) = u ^ { \\rm o d d } ( n , x ) , \\end{align*}"}
-{"id": "7373.png", "formula": "\\begin{align*} d ( p , x _ { \\sigma ( n ) } ) = d ( p , x _ { \\sigma ( 1 ) } ) + \\sum _ { i = 1 } ^ { n - 1 } d ( x _ { \\sigma ( i ) } , x _ { \\sigma ( i + 1 ) } ) \\end{align*}"}
-{"id": "8143.png", "formula": "\\begin{align*} \\Psi _ { g , n } = \\left ( \\Psi _ { 1 , 0 } ^ { \\otimes g } \\otimes \\Psi _ { 0 , 1 } ^ { \\otimes n } \\right ) \\circ \\alpha _ { g , n } \\ : : \\ : \\mathcal { L } _ { g , n } ( H ) \\overset { \\sim } { \\rightarrow } \\mathcal { H } ( \\mathcal { O } ( H ) ) ^ { \\otimes g } \\otimes H ^ { \\otimes n } . \\end{align*}"}
-{"id": "9885.png", "formula": "\\begin{align*} M = \\begin{pmatrix} A & T A & \\cdots & T ^ { n - 1 } A \\end{pmatrix} \\end{align*}"}
-{"id": "1292.png", "formula": "\\begin{align*} E = \\{ z \\in Z ^ k : \\| z \\| _ { Z ^ k } \\le M \\} \\end{align*}"}
-{"id": "7311.png", "formula": "\\begin{align*} \\begin{cases} \\rho ( x ) u _ { t t } - \\operatorname { d i v } ( K ( x ) \\nabla u ) + V _ 1 ( x , t ) u = V _ 3 ( x , t ) v & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ \\rho ( x ) v _ { t t } - \\operatorname { d i v } [ K ( x ) \\nabla v ] + V _ 2 ( x , t ) v = V _ 3 ( x , t ) u & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ u = v = 0 & ~ \\hbox { o n } ~ \\omega , \\\\ \\end{cases} \\end{align*}"}
-{"id": "3769.png", "formula": "\\begin{align*} { \\bf H } = - \\frac { i } { \\xi } \\overline { \\bf E } . \\end{align*}"}
-{"id": "1507.png", "formula": "\\begin{align*} n ( x , 0 ) = n _ 0 ( x ) , c ( x , 0 ) = c _ 0 ( x ) , u ( x , 0 ) = u _ 0 ( x ) , x \\in \\Omega , \\end{align*}"}
-{"id": "6212.png", "formula": "\\begin{align*} \\rho _ b = \\nu _ { \\mu _ b } . \\end{align*}"}
-{"id": "5137.png", "formula": "\\begin{align*} \\Gamma ^ { \\pm } ( \\psi ) = \\left ( \\Gamma _ { m _ j , k _ j } ^ { \\pm } ( f _ { m _ j , k _ j } ) \\right ) _ { ( j , m _ j , k _ j ) \\in I } \\in \\C ^ d . \\end{align*}"}
-{"id": "3988.png", "formula": "\\begin{align*} R S = ( s - k ) ( s + k - 2 ) \\frac { \\Gamma ( s + k - 2 ) } { ( 4 \\pi ) ^ { s + k - 2 } } \\sum _ N \\{ \\phi _ N , \\psi _ N \\} N ^ { - s - k + 2 } . \\end{align*}"}
-{"id": "7850.png", "formula": "\\begin{align*} \\delta _ 0 \\le \\sum _ { k = 0 } ^ { k _ 0 - 1 } u _ { d - k m } \\le 2 . \\end{align*}"}
-{"id": "9371.png", "formula": "\\begin{align*} D _ i ( G _ k ) = G _ k ^ { \\ , 2 ^ k } \\gamma _ i ( G _ k ) = \\langle x ^ { 2 ^ k } , [ y , x , \\overset { 2 ^ k - 3 } { \\ldots } , x , y ] \\rangle \\ ; \\gamma _ i ( G _ k ) , \\end{align*}"}
-{"id": "820.png", "formula": "\\begin{align*} K _ { q ^ d , P ^ \\alpha } ^ { \\mathbb { F } _ { q ^ d } ^ * } = \\prod _ { i = 1 } ^ { { s } } K _ { q ^ d , ( T - \\rho _ i ) ^ \\alpha } ^ { \\mathbb { F } _ { q ^ d } ^ * } . \\end{align*}"}
-{"id": "5169.png", "formula": "\\begin{align*} \\lim _ { u \\rightarrow 0 } \\frac { U ( \\mu u ) - U ( u ) } { s ( u ) } = \\log \\mu \\end{align*}"}
-{"id": "517.png", "formula": "\\begin{align*} y ^ 4 = x \\prod _ { i = 2 } ^ { g + 1 } ( x - t _ i ) ^ { a _ i } , \\end{align*}"}
-{"id": "7393.png", "formula": "\\begin{align*} D ( I , J ) = \\frac { 2 N _ I N _ J } { N _ I + N _ J } | | \\overline { \\mathbf { x } } _ I - \\overline { \\mathbf { x } } _ J | | ^ 2 \\end{align*}"}
-{"id": "9383.png", "formula": "\\begin{align*} K & \\varrho _ k \\cap Z \\varrho _ k \\\\ & = \\langle \\{ \\overline { y } ^ 2 \\} \\cup \\{ [ \\overline { y _ 0 } , \\overline { y _ j } ] \\mid 0 \\le j \\le 2 ^ { k - 1 } , \\ ; j \\equiv _ { 2 ^ n } 0 , \\pm 1 , \\ldots , \\pm ( m - 1 ) \\} \\rangle \\\\ & \\cong C _ 2 ^ { \\ , ( 2 m - 1 ) 2 ^ { k - n - 1 } + 1 } . \\end{align*}"}
-{"id": "8743.png", "formula": "\\begin{align*} b _ \\ell ^ { ( n ) } = q _ { \\ell } B ^ l n - n _ 0 + 1 - b _ \\ell ^ { ( n ) } \\in [ B ^ { \\ell - 1 } , 2 B ^ \\ell - 1 ] . \\end{align*}"}
-{"id": "3608.png", "formula": "\\begin{align*} B : = \\{ a _ 1 , a _ { - 1 } , a _ 2 , a _ { - 2 } , a _ 3 , a _ { - 3 } , a _ { \\rho } \\} \\end{align*}"}
-{"id": "2863.png", "formula": "\\begin{align*} x \\odot ( x \\vee y ) = x . \\end{align*}"}
-{"id": "2729.png", "formula": "\\begin{align*} z _ { 2 } & = z _ { 1 } + \\eta _ { - } ^ { - 1 } \\delta e ^ { i \\frac { 2 } { 3 } \\pi } , w _ { 2 } = w _ { 1 } + \\eta _ { - } ^ { - 1 } \\delta e ^ { i \\frac { 1 } { 3 } \\pi } \\end{align*}"}
-{"id": "10036.png", "formula": "\\begin{align*} L f = \\mu ( \\lambda ) f , \\mu ( \\lambda ) = \\lambda + \\lambda ^ { - 1 } , \\end{align*}"}
-{"id": "375.png", "formula": "\\begin{align*} | \\alpha | ^ 2 = A + B \\alpha + C \\bar { \\alpha } . \\end{align*}"}
-{"id": "7217.png", "formula": "\\begin{align*} K _ { 2 n } ( x ) = \\frac { ( - 1 ) ^ { n - 1 } } { ( 2 n ) ! } ( 2 \\pi ) ^ { 2 n } B _ { 2 n } \\left ( \\frac { x } { 2 \\pi } \\right ) , x \\in [ 0 , \\pi ] , \\end{align*}"}
-{"id": "2851.png", "formula": "\\begin{align*} x = a ^ 2 , \\ ; \\ ; y = \\pm a b \\ ; \\ ; z = b ^ 2 \\end{align*}"}
-{"id": "6636.png", "formula": "\\begin{align*} \\mathcal A f ( x ) = \\sup _ { \\lambda > 0 } \\mathcal A _ { \\lambda } f , \\end{align*}"}
-{"id": "8308.png", "formula": "\\begin{align*} ( D _ t ^ 2 - i A \\partial _ { \\alpha } ) \\tilde { \\sigma } = \\tilde { G } , \\end{align*}"}
-{"id": "7181.png", "formula": "\\begin{gather*} x = \\chi ^ { t } ( \\xi ) + y \\end{gather*}"}
-{"id": "7649.png", "formula": "\\begin{align*} I ^ { i j } _ 1 \\ = \\ \\int \\limits _ { \\mathbb T ^ d } \\ ! \\int \\limits _ { \\mathbb R ^ d } \\ ! a ( \\xi \\ ! - \\ ! q ) \\mu ( \\xi , q ) ( \\xi - q ) ^ i ( \\xi - q ) ^ j v _ 0 ( \\xi ) d q d \\xi , \\end{align*}"}
-{"id": "5239.png", "formula": "\\begin{align*} \\frac { b _ { [ n s ] } - b _ { n } } { a _ { n } } \\rightarrow 0 \\frac { a _ { [ n s ] } } { a _ { n } } \\rightarrow ( 1 / s ) ^ { - 1 / \\alpha } = \\rho > 1 n \\rightarrow + \\infty . \\end{align*}"}
-{"id": "2823.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } R _ i ^ p > 0 , \\ & \\mbox { f o r a n y } \\ i = 1 , \\ldots , p , \\\\ R _ q ^ { p + 1 , q } \\leq 0 , \\ & \\mbox { f o r a n y } \\ q = p + 1 , \\dots , n . \\end{array} \\right . \\end{align*}"}
-{"id": "5779.png", "formula": "\\begin{align*} \\sum _ { k \\geq 0 } ( k + 1 ) ^ n T ^ k = \\frac { \\sum _ { j = 0 } ^ { n - 1 } A ( n , j ) T ^ j } { ( 1 - T ) ^ { n + 1 } } . \\end{align*}"}
-{"id": "665.png", "formula": "\\begin{align*} [ \\varphi \\rho ; f _ 1 , \\dots , f _ l ; \\rho ^ \\ast ( y ) ] = [ \\varphi ; f _ 1 , \\dots , f _ l ; y ] \\circ \\widehat \\rho _ { \\vec a } \\ , \\end{align*}"}
-{"id": "8604.png", "formula": "\\begin{align*} u _ \\Lambda ( x ) - u _ 0 ( x ) = O \\left ( \\frac { 1 } { \\sqrt { \\Lambda } } \\right ) , \\end{align*}"}
-{"id": "3534.png", "formula": "\\begin{align*} \\begin{aligned} \\Gamma _ { M } ( 1 + i ) : = \\Gamma _ { M } \\cap \\Gamma _ { T } ( 1 + i ) , \\end{aligned} \\end{align*}"}
-{"id": "9227.png", "formula": "\\begin{align*} C _ { i + 1 } = \\left \\lfloor \\frac { C _ i - f _ i } { s _ i } \\right \\rfloor + 1 . \\end{align*}"}
-{"id": "8652.png", "formula": "\\begin{align*} H _ { r e d } = \\frac { 1 } { 2 } \\varphi _ 2 ^ 2 - \\frac { 1 } { 2 } ( \\sin \\varphi _ 1 - 2 c _ 0 ) ^ 2 \\end{align*}"}
-{"id": "5593.png", "formula": "\\begin{align*} \\int _ 0 ^ { + \\infty } \\ ! \\ ! \\ ! x \\bar { u } u _ t d x + \\int _ 0 ^ { + \\infty } \\ ! \\ ! \\ ! x \\bar { u } u _ { x x } d x = \\alpha \\int _ 0 ^ { + \\infty } \\ ! \\ ! \\ ! x | u | ^ 2 v d x + \\beta \\int _ 0 ^ { + \\infty } \\ ! \\ ! \\ ! x | u | ^ 4 d x . \\end{align*}"}
-{"id": "9626.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\frac { q ^ n } { 1 - q ^ n } + \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { q ^ { n ( n + 1 ) } } { ( q ) _ n ( 1 - q ^ n ) } = \\ \\sum _ { n = 1 } ^ { N } \\frac { q ^ n } { ( 1 - q ^ n ) ( q ) _ n } . \\end{align*}"}
-{"id": "7048.png", "formula": "\\begin{align*} \\int _ M ( \\alpha ) d \\mu _ g = 0 . \\end{align*}"}
-{"id": "6631.png", "formula": "\\begin{align*} T ^ + _ 5 : = ( T ^ + _ 3 \\cup T ^ + _ 4 ) \\setminus ( T ^ - _ 3 \\cup T ^ - _ 4 ) , ~ ~ T ^ - _ 5 : = ( T ^ - _ 3 \\cup T ^ - _ 4 ) \\setminus ( T ^ + _ 3 \\cup T ^ + _ 4 ) \\end{align*}"}
-{"id": "8870.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 e _ 5 } { \\partial a ^ 2 } & = 4 9 + 4 1 \\nu - 2 \\nu ^ 2 > 0 \\end{align*}"}
-{"id": "6409.png", "formula": "\\begin{align*} \\chi ^ { 2 } = ( \\rho - \\tau ) ^ { 2 } + 4 \\sigma { } ^ { 2 } . \\end{align*}"}
-{"id": "929.png", "formula": "\\begin{align*} E _ k ^ q = I _ k - \\sum _ { j _ 1 , \\ldots , j _ k = 0 } ^ { q } C _ { j _ k \\ldots j _ 1 } ^ 2 , \\end{align*}"}
-{"id": "7852.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { k _ 0 - 1 } u _ { d - k m } \\le \\frac { 1 } { m } \\sum _ { k = 0 } ^ { k _ 0 - 1 } \\sum _ { j \\in U _ { k } } u _ j \\le \\frac { \\bar { m } } { m } \\le 2 . \\end{align*}"}
-{"id": "8482.png", "formula": "\\begin{align*} | \\dot { z } _ 1 ( t ) - \\dot { z } _ 2 ( t ) | = | F ( z _ 1 , t ) - F ( z _ 2 , t ) | \\leq 2 \\| F \\| _ { L ^ { \\infty } ( \\partial \\Omega ( t ) ) } \\leq 1 0 \\epsilon , \\end{align*}"}
-{"id": "9736.png", "formula": "\\begin{align*} \\lambda _ \\ast : = N ( 0 ^ + , v _ \\ast ) \\geq \\kappa _ \\ast = N ( 0 ^ + , u ) , \\end{align*}"}
-{"id": "603.png", "formula": "\\begin{align*} & \\Theta _ n ^ { W } : = \\big \\{ ( j , k , e ) \\in \\Theta ^ W \\ , \\big | \\ , j = 0 , \\ldots , J - 1 \\big \\} , \\\\ & \\Theta _ n : = \\big \\{ ( j , \\tilde { \\theta } ) \\in \\Theta \\ , \\big | \\ , j = 0 , \\ldots , \\widetilde { J } - 1 \\big \\} , \\end{align*}"}
-{"id": "6777.png", "formula": "\\begin{align*} \\nabla \\cdot u = 0 \\end{align*}"}
-{"id": "419.png", "formula": "\\begin{align*} s : = \\frac { 2 Q } { \\gamma } ( \\mathbf { h } - 1 ) - \\frac { 2 \\beta } { 1 - \\frac { \\gamma ^ 2 } { 4 } } . \\end{align*}"}
-{"id": "3258.png", "formula": "\\begin{align*} \\mathcal { P } _ n ( z ) = a ^ n + b ^ n . \\end{align*}"}
-{"id": "6968.png", "formula": "\\begin{align*} d \\omega = \\eta \\wedge \\omega , \\end{align*}"}
-{"id": "9048.png", "formula": "\\begin{align*} F _ { \\mu } = \\sqrt { \\alpha } F _ { \\mu _ { A } } + \\sqrt { \\beta } F _ { \\mu _ B } \\end{align*}"}
-{"id": "4656.png", "formula": "\\begin{align*} K _ { \\widehat { X } _ { a , b } } ^ 2 = K _ { X _ { a , b } } ^ 2 - b ^ 2 \\frac { ( b - 2 ) ^ 2 } { b ^ 2 } = 2 ( n - 3 ) ^ 2 - ( b - 2 ) ^ 2 . \\end{align*}"}
-{"id": "5252.png", "formula": "\\begin{align*} \\Gamma ( F , x , \\alpha ( t ) , \\beta ( t ) ) = \\frac { 1 - F ( \\alpha ( t ) x + \\beta ( t ) ) } { 1 - F ( t ) } , \\ t < u e p ( F ) . \\end{align*}"}
-{"id": "8785.png", "formula": "\\begin{align*} 2 a _ 2 ( b _ 0 + c _ 0 ) + a _ 1 = 0 . \\end{align*}"}
-{"id": "2430.png", "formula": "\\begin{align*} \\varphi _ { \\ell } ^ { H _ 1 } ( x _ 0 , x _ 1 ) = \\frac { 1 } { | H _ 1 | } \\sum _ { \\sigma \\in H _ 1 } ( \\sigma x _ 0 ) ^ { \\ell } . \\end{align*}"}
-{"id": "2100.png", "formula": "\\begin{align*} d y ( t ) = f ( y ( t ) ) d t + g ( y ( t ) ) d \\omega ( t ) \\end{align*}"}
-{"id": "1980.png", "formula": "\\begin{align*} e ^ { - 1 } ( V \\otimes \\O _ \\alpha ) = \\frac { 1 } { \\alpha ^ r } \\sum _ { 0 \\leq k \\leq \\dim ( X ) } \\left ( - \\frac { c _ 1 ( V ) } { \\alpha } - \\cdots - \\frac { c _ { r - 1 } ( V ) } { \\alpha ^ { r - 1 } } - \\frac { c _ r ( V ) } { \\alpha ^ r } \\right ) ^ k . \\end{align*}"}
-{"id": "611.png", "formula": "\\begin{align*} ( \\mu _ n ) _ \\ast \\left ( ( x _ 1 , \\dots , x _ n ) ^ \\sigma \\right ) = ( \\mu _ n ) _ \\ast ( x _ { \\sigma ^ { - 1 } ( 1 ) } , \\dots , x _ { \\sigma ^ { - 1 } ( n ) } ) = x _ { \\sigma ^ { - 1 } ( 1 ) } \\dots x _ { \\sigma ^ { - 1 } ( n ) } \\ . \\end{align*}"}
-{"id": "5328.png", "formula": "\\begin{align*} X . w ^ k = \\frac 1 2 \\left ( \\lambda + ( k + 1 ) \\right ) w ^ { k + 2 } = a _ k w ^ { k + 2 } \\end{align*}"}
-{"id": "2035.png", "formula": "\\begin{align*} \\Lambda ( s , f , a / d ) = \\int _ 0 ^ \\infty f \\left ( \\frac { a } { d } + i y \\right ) y ^ s \\ ; \\frac { d y } { y } = \\left ( 2 \\pi \\right ) ^ { - s } \\Gamma \\left ( s \\right ) L ( s , f , a / d ) . \\end{align*}"}
-{"id": "2150.png", "formula": "\\begin{align*} g ( \\mathbf x , \\mathbf y ) = \\tau _ { \\mathbf x } g ( - \\mathbf y ) = \\tau _ { - \\mathbf y } g ( \\mathbf x ) \\end{align*}"}
-{"id": "9553.png", "formula": "\\begin{align*} { } _ { 2 } \\phi _ { 1 } \\left [ \\begin{matrix} a , q ^ { - M } \\\\ d \\end{matrix} \\ , ; q , q \\right ] = \\frac { ( d / a ) _ M a ^ M } { ( d ) _ M } , \\end{align*}"}
-{"id": "5436.png", "formula": "\\begin{align*} \\log \\hat { \\rho } _ s ( A ) & = \\max _ { \\mu \\in \\mathcal { M } _ f } \\left \\lbrace \\gamma _ 1 ( A , \\mu ) + \\gamma _ 2 ( A , \\mu ) + \\cdots + \\gamma _ s ( A , \\mu ) \\right \\rbrace \\\\ & = \\max _ { \\mu \\in \\mathcal { M } _ f ( P e r ) } \\left \\lbrace \\gamma _ 1 ( A , \\mu ) + \\gamma _ 2 ( A , \\mu ) + \\cdots + \\gamma _ s ( A , \\mu ) \\right \\rbrace , \\end{align*}"}
-{"id": "8247.png", "formula": "\\begin{align*} \\frac { x } { ( 1 - x ) ^ 2 } = \\sum _ { n = 1 } ^ \\infty n x ^ n , \\end{align*}"}
-{"id": "7659.png", "formula": "\\begin{align*} a ( z ) \\ = \\ a _ { \\rm s y m } ( z ) \\ + \\ \\boldsymbol { \\ell } \\cdot c ( z ) , \\end{align*}"}
-{"id": "4150.png", "formula": "\\begin{align*} P _ { - } ^ { - 1 } ( x ) P _ { + } ( x ) = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 0 & i e ^ { - 2 \\pi i \\alpha } x _ - ^ { - \\beta } x _ + ^ { \\beta } \\\\ 0 & - i e ^ { 2 \\pi i \\alpha } x _ - ^ { - \\beta } x _ + ^ { \\beta } & 0 \\end{pmatrix} , x \\in \\Delta _ 2 \\cap O _ V . \\end{align*}"}
-{"id": "9770.png", "formula": "\\begin{align*} Z _ \\infty = ( z _ \\infty , 0 ) \\in L _ * \\end{align*}"}
-{"id": "5209.png", "formula": "\\begin{align*} U ( x ) = c x ^ { - 1 } b ( x ) \\exp \\left ( \\int _ { 1 } ^ { x } t ^ { - 1 } b ( t ) d t \\right ) , a . e . , \\end{align*}"}
-{"id": "9563.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - 1 ) ^ { n - 1 } q ^ { \\frac { n ( n + 1 ) } { 2 } } } { ( 1 - c q ^ n ) } & = \\frac { 1 } { c - 1 } \\sum _ { n = 1 } ^ { N } \\frac { ( 1 / c ) _ n ( c q ) ^ n } { ( q ) _ n } \\frac { ( q ^ { N + 1 - n } ) _ n } { ( c q ^ { N + 1 - n } ) _ n } \\\\ & = \\frac { 1 } { c - 1 } \\sum _ { n = 1 } ^ { N } \\frac { ( 1 / c ) _ n q ^ n } { ( q ) _ n } \\frac { ( q ^ { - N } ) _ n } { \\left ( \\frac { q ^ { - N } } { c } \\right ) _ n } , \\end{align*}"}
-{"id": "3104.png", "formula": "\\begin{align*} A ^ { ( 2 ) } ( y ) & = \\frac { y ^ { \\frac { 1 } { 2 } } - y ^ { - \\frac { 1 } { 2 } } } { \\phi _ { - 2 , 1 } ( q ^ 2 , y ^ 2 ) ^ \\frac { 1 } { 2 } \\ , \\tilde \\Delta ( q ^ 2 ) ^ \\frac { 1 } { 2 } } \\ , , \\\\ B ^ { ( 2 ) } ( y ) & = \\frac { \\tilde \\eta ( q ) ^ 2 } { \\theta _ 3 ( q , y ) } \\ , , \\\\ C ^ { ( 2 ) } _ { 1 1 } ( y ) & = \\frac { - \\theta _ 3 ( q , y ) } { \\theta _ 2 ( q , y ) } \\ , . \\end{align*}"}
-{"id": "2915.png", "formula": "\\begin{align*} \\mu ( [ u ] \\cap Q ) & = \\mu \\big ( z ^ { - 1 } ( [ u ] \\cap Q ) \\big ) = \\mu ( [ w ^ { - 1 } u ] \\cap Q ) = \\mu ( [ v ] \\cap Q ) \\ ; . \\end{align*}"}
-{"id": "6407.png", "formula": "\\begin{align*} \\lambda _ { \\max } ( T _ { k } ) = \\left \\Vert L _ { k } \\right \\Vert ^ { 2 } , \\qquad \\lambda _ { \\min } ( T _ { k } ) = \\left \\Vert L _ { k } ^ { - 1 } \\right \\Vert ^ { - 2 } . \\end{align*}"}
-{"id": "9578.png", "formula": "\\begin{align*} S _ 2 = \\frac { - 1 } { ( 1 - c ) ^ 2 ( q ) _ N } \\left ( 1 - \\frac { ( q ) _ N } { ( c q ) _ N } \\right ) + \\frac { 1 } { ( 1 - c ) ( c q ) _ N } \\sum _ { k = 1 } ^ N \\frac { ( c q ) _ k q ^ k } { ( q ) _ k ( 1 - q ^ k ) } , \\end{align*}"}
-{"id": "5639.png", "formula": "\\begin{align*} - \\frac { 1 2 0 } { 2 1 } + 2 + \\frac { 2 C } { \\sqrt { C ^ 2 + \\frac { 3 2 } { 2 7 } a _ 1 } } = 0 \\end{align*}"}
-{"id": "1649.png", "formula": "\\begin{align*} F _ { - k } : = \\sigma \\big ( F _ k \\big ) = - \\frac { 1 } { \\overline { \\varkappa _ k } } \\begin{pmatrix} a _ k + i \\sqrt [ + ] { b ^ 2 - a _ k ^ 2 } \\\\ - b \\end{pmatrix} e ^ { - 2 \\pi i k x } \\end{align*}"}
-{"id": "250.png", "formula": "\\begin{align*} q _ j \\leq \\frac { N - p _ j } { 2 } \\leq \\frac { N } { 2 } j = 2 , \\ldots , r v _ 1 \\leq \\frac { N - p _ 1 - 2 u _ 1 } { 3 } \\leq \\frac { N } { 3 } . \\end{align*}"}
-{"id": "1672.png", "formula": "\\begin{align*} \\rho _ A ( g X ) = \\varphi ^ * ( g ) \\rho _ A ( X ) . \\end{align*}"}
-{"id": "1745.png", "formula": "\\begin{align*} c _ { p , r } ( | B _ { t } - B _ { s } | > \\eta ) \\leq \\sqrt [ p ] { 2 \\left [ \\sum _ { l = 0 } ^ { r } \\left ( \\frac { \\eta } { p ( t - s ) ^ { H } } \\right ) ^ { l p } \\right ] } e ^ { - \\frac { \\eta ^ { 2 } } { 2 p ( t - s ) ^ { 2 H } } } \\end{align*}"}
-{"id": "8385.png", "formula": "\\begin{align*} \\bold { n } = \\frac { i z _ { \\alpha } } { | z _ { \\alpha } | } , a | z _ { \\alpha } | = | z _ { t t } + i | = | \\bar { u } _ t + i | , \\end{align*}"}
-{"id": "4362.png", "formula": "\\begin{align*} \\rho _ { \\sigma } : = \\Gamma _ { \\sigma } ^ { \\frac { 1 - \\alpha } { \\alpha } } ( \\rho ) \\equiv \\sigma ^ { \\frac { 1 - \\alpha } { 2 \\alpha } } \\rho \\sigma ^ { \\frac { 1 - \\alpha } { 2 \\alpha } } . \\end{align*}"}
-{"id": "1160.png", "formula": "\\begin{align*} & \\frac { C _ { \\beta , n - 2 , 1 } } { C _ { \\beta , n } } = ( 2 \\pi ) ^ { - 1 } \\frac { \\Gamma ( \\beta ( n + 1 ) / 2 + 1 ) ( \\Gamma ( \\beta / 2 + 1 ) ) ^ { 3 } } { \\Gamma ( 3 \\beta / 2 + 1 ) \\Gamma ( \\beta + 1 ) \\Gamma ( \\beta ( n - 1 ) / 2 + 1 ) } . \\end{align*}"}
-{"id": "502.png", "formula": "\\begin{align*} \\rho ( Q ) ( ( \\xi _ { p _ 1 } + \\xi _ { p _ 2 } ) \\odot ( \\xi _ { p _ 1 } + \\xi _ { p _ 2 } ) ) = - 4 \\pi i \\mu _ 2 ( Q ) ( p _ 1 ) \\neq 0 . \\end{align*}"}
-{"id": "459.png", "formula": "\\begin{align*} f \\Big ( \\frac x y , \\ , y \\Big ) & = \\sum _ { i = 0 } ^ { \\infty } \\sum _ { j = 0 } ^ { i } [ b + c ( i + j ) ] v _ { i , i + j } x ^ i y ^ j \\\\ & = b \\ , G ( x , y ) + c \\big ( x \\ , \\partial _ x G ( x , y ) + y \\ , \\partial _ y G ( x , y ) \\big ) . \\end{align*}"}
-{"id": "8945.png", "formula": "\\begin{align*} { 3 ^ k } ( x _ i ( t ) - x _ i ( t ( k ) ) ) = ( - 1 ) ^ { \\sigma ( k ) } x _ i ( 3 ^ { k n } ( t - t ( k ) ) ) , \\end{align*}"}
-{"id": "9013.png", "formula": "\\begin{align*} f _ { \\emptyset , 0 } = 0 f _ { \\{ a _ 1 , a _ 2 \\} , 0 } ( x _ 1 , x _ 2 ) = \\begin{cases} 0 & \\{ a _ 1 , a _ 2 \\} = \\{ x _ 1 , x _ 2 \\} , \\\\ x _ 1 & \\end{cases} \\end{align*}"}
-{"id": "213.png", "formula": "\\begin{align*} \\Delta u = V u \\end{align*}"}
-{"id": "2564.png", "formula": "\\begin{align*} \\mathbf P _ { \\delta _ x } ( \\| X _ t \\| = 0 ) < 1 , t > 0 , x \\in E . \\end{align*}"}
-{"id": "6171.png", "formula": "\\begin{align*} U ( t , z ) = r ^ { 1 / 2 } \\cos \\frac \\theta 2 , \\end{align*}"}
-{"id": "7705.png", "formula": "\\begin{align*} \\int \\limits _ { - \\infty } ^ { - \\epsilon } \\frac { h ( t ) } { t } \\varphi ( 0 ) \\ , \\mathrm { d } t = - \\int \\limits _ { \\epsilon } ^ { \\infty } \\frac { h ( t ) } { t } \\varphi ( 0 ) \\ , \\mathrm { d } t . \\end{align*}"}
-{"id": "7530.png", "formula": "\\begin{align*} \\left \\{ \\begin{gathered} \\partial _ t u - \\nu \\Delta u = \\mathbb { H } \\left ( \\mathrm { d i v } \\ , \\ , ( b \\otimes b - u \\otimes u ) \\right ) , \\\\ \\nabla \\cdot u = 0 , \\\\ \\nabla \\cdot b = 0 , \\\\ \\partial _ t b + u \\cdot \\nabla b = ( \\nabla u ) b , \\\\ u ( x , 0 ) = u _ 0 ( x ) , b ( x , 0 ) = b _ 0 ( x ) . \\end{gathered} \\right . \\end{align*}"}
-{"id": "2519.png", "formula": "\\begin{align*} \\mathbf 1 - \\mathbf F ( \\mathbf s ) = ( M - G ( \\mathbf s ) ) ( \\mathbf 1 - \\mathbf s ) , \\mathbf s \\in \\mathbb R _ + ^ d . \\end{align*}"}
-{"id": "8369.png", "formula": "\\begin{align*} \\dot { z } _ j ( t ) = & ( v - \\frac { \\lambda _ j i } { \\overline { z - z _ j ( t ) } } ) \\Big | _ { z = z _ j ( t ) } = \\sum _ { 1 \\leq k \\leq N , k \\neq j } \\frac { \\lambda _ k i } { 2 \\pi ( \\overline { z _ j ( t ) - z _ k ( t ) } ) } + \\bar { F } ( z _ j ( t ) ) . \\end{align*}"}
-{"id": "1784.png", "formula": "\\begin{align*} A = \\bigcup _ { t \\in [ 0 , 1 ] } \\bigcup _ { M = 1 } ^ { \\infty } \\bigcup _ { k = 1 } ^ { \\infty } A _ { k , M } ^ { t } . \\end{align*}"}
-{"id": "8545.png", "formula": "\\begin{align*} \\theta _ k = \\partial _ { \\alpha } ^ k ( I - \\mathcal { H } ) \\tilde { \\theta } + [ \\partial _ { \\alpha } ^ k , \\mathcal { H } ] \\tilde { \\theta } = 2 \\partial _ { \\alpha } ^ k \\tilde { \\theta } + [ \\partial _ { \\alpha } ^ k , \\mathcal { H } ] \\tilde { \\theta } . \\end{align*}"}
-{"id": "2786.png", "formula": "\\begin{align*} \\mathbb { P } ( J _ 1 ^ t = j _ 1 ; J _ 2 ^ t = j _ 2 ; J _ 3 ^ t = j _ 3 ) = : f _ { 1 , 2 } ( j _ 1 , j _ 2 ) f _ 3 ( j _ 3 ) . \\end{align*}"}
-{"id": "7730.png", "formula": "\\begin{align*} e ( V \\otimes L ) = \\sum _ { i = 0 } ^ n c _ i ( V ) c _ 1 ( L ) ^ { n - i } . \\end{align*}"}
-{"id": "108.png", "formula": "\\begin{align*} \\begin{pmatrix} \\alpha + \\frac { k - 1 } 2 \\\\ \\frac { k - 1 } 2 \\end{pmatrix} \\begin{pmatrix} 2 N - \\alpha - \\frac { k - 1 } 2 - 1 \\\\ \\frac { k - 1 } 2 \\end{pmatrix} \\ , , \\alpha = 0 , \\ldots , 2 N - k \\ , . \\end{align*}"}
-{"id": "5945.png", "formula": "\\begin{align*} g ^ * ( t , z ) : = \\sup _ { q \\in \\R ^ d } \\left ( q \\cdot z - g ( t , q ) \\right ) . \\end{align*}"}
-{"id": "6728.png", "formula": "\\begin{align*} \\Box \\Box h _ { k l } - \\Box \\frac { \\partial ^ 2 h _ { n l } } { \\partial x ^ k \\partial x _ n } - \\Box \\frac { \\partial ^ 2 h _ { k n } } { \\partial x _ n \\partial x ^ l } + \\frac { \\partial ^ 4 h _ { m n } } { \\partial x _ m \\partial x _ n \\partial x ^ k \\partial x ^ l } = 0 , \\end{align*}"}
-{"id": "8305.png", "formula": "\\begin{align*} G _ d : = - 2 [ \\bar { q } , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } \\bar { \\mathfrak { F } } } { \\zeta _ { \\alpha } } - 2 [ \\bar { \\mathfrak { F } } , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } \\bar { q } } { \\zeta _ { \\alpha } } - 2 [ \\bar { q } , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } \\bar { q } } { \\zeta _ { \\alpha } } - 4 D _ t q . \\end{align*}"}
-{"id": "2490.png", "formula": "\\begin{align*} \\varphi ( x ) = 1 \\varphi ( y ) < 1 y \\in \\overline { B } _ E \\backslash \\{ x \\} . \\end{align*}"}
-{"id": "6768.png", "formula": "\\begin{align*} \\sqrt { A + E } = \\sqrt { A } + U ( [ \\sqrt { \\cdot } , \\alpha ] \\circ \\hat E ) U ^ * + O ( \\| E \\| ^ 2 ) , \\end{align*}"}
-{"id": "6772.png", "formula": "\\begin{align*} \\begin{aligned} & \\| Z ^ { 1 / p } - { A } ^ { 1 / p } \\| _ 2 \\leq \\| { Z } ^ { 1 / p } - { ( Z + c I ) } ^ { 1 / p } \\| _ 2 + \\\\ & \\| ( { Z + c I } ) ^ { 1 / p } - { ( A + c I ) } ^ { 1 / p } \\| _ 2 + \\| ( { A + c I } ) ^ { 1 / p } - { A } ^ { 1 / p } \\| _ 2 . \\end{aligned} \\end{align*}"}
-{"id": "2122.png", "formula": "\\begin{align*} \\int _ a ^ b x ( u ) d \\omega ( u ) = \\int _ { a - r } ^ { b - r } x ( u + r ) d \\theta _ r \\omega ( u ) . \\end{align*}"}
-{"id": "1341.png", "formula": "\\begin{align*} \\psi _ { > R } ^ { ( l ) } & = \\bigoplus _ { i \\in I ^ { ( l ) } } \\psi _ i \\colon C ^ * _ r ( Y \\rtimes \\mathbb { R } ) \\to A _ { > R } ^ { ( l ) } \\\\ \\varphi _ { > R } ^ { ( l ) } & = \\sum _ { i \\in I ^ { ( l ) } } \\varphi _ i \\colon A _ { > R } ^ { ( l ) } \\to C ^ * _ r ( Y \\rtimes \\mathbb { R } ) \\ ; . \\end{align*}"}
-{"id": "1005.png", "formula": "\\begin{align*} \\varphi _ k \\left ( \\lambda \\right ) = 1 - \\sum _ { j = 0 } ^ { k - 1 } \\frac { \\lambda ^ j } { j ! } e ^ { - \\lambda } , \\end{align*}"}
-{"id": "7728.png", "formula": "\\begin{align*} \\pi ( h , \\ell _ 1 , \\ldots , \\ell _ l , \\ell , A _ 1 , \\ldots , A _ l , C _ 1 , \\ldots , C _ l ) = ( A _ 1 , \\ldots , A _ l , C _ 1 J , \\ldots , C _ l J ) . \\end{align*}"}
-{"id": "8480.png", "formula": "\\begin{align*} \\dot { z } _ 1 = \\frac { \\lambda _ 2 i } { 2 \\pi ( z _ 1 - z _ 2 ) } + \\bar { F } ( z _ 1 , t ) = \\frac { \\lambda i } { 4 \\pi x ( t ) } + \\bar { F } ( z _ 1 , t ) . \\end{align*}"}
-{"id": "8217.png", "formula": "\\begin{align*} & a _ 0 ( t ) = - t , a _ 1 ( t ) = - t ( 2 \\cdot 5 + 5 ^ 2 t ) , a _ 2 ( t ) = - t ( 1 1 \\cdot 5 + 2 \\cdot 5 ^ 3 t + 5 ^ 4 t ^ 2 ) , \\\\ & a _ 3 ( t ) = - t ( 2 8 \\cdot 5 + 1 1 \\cdot 5 ^ 3 t + 2 \\cdot 5 ^ 5 t ^ 2 + 5 ^ 6 t ^ 3 ) , \\\\ & a _ 4 ( t ) = - t ( 7 \\cdot 5 ^ 2 + 2 8 \\cdot 5 ^ 3 t + 1 1 \\cdot 5 ^ 5 t ^ 2 + 2 \\cdot 5 ^ 7 t ^ 3 + 5 ^ 8 t ^ 4 ) . \\end{align*}"}
-{"id": "1521.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k - 1 } v _ { P } ( b _ { i } ^ { ' } ) \\ge \\sum _ { i = 1 } ^ { k } v _ { P } ( b _ { i } ) - v _ P ( b _ { k } ^ { ' } ) = \\sum _ { j \\ge 1 } \\lfloor \\frac { k } { q ^ { d j } } \\rfloor - \\mu ( k ) , \\end{align*}"}
-{"id": "1873.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\left ( 2 \\kappa _ j - k _ n - \\frac { 1 } { \\ell } \\right ) Y _ { j , 0 } Y _ { j , n } = 2 \\sum _ { j = 1 } ^ \\infty \\kappa _ j Y _ { j , 0 } Y _ { j , n } - \\left ( k _ n + \\frac { 1 } { \\ell } \\right ) \\sum _ { j = 1 } ^ \\infty Y _ { j , 0 } Y _ { j , n } = 2 \\sum _ { j = 1 } ^ \\infty \\kappa _ j Y _ { j , 0 } Y _ { j , n } . \\end{align*}"}
-{"id": "1419.png", "formula": "\\begin{align*} \\textrm { O P T } _ { \\eta } ( T ) = \\textrm { O P T } \\big ( t _ { \\eta } ( T ) \\big ) , \\end{align*}"}
-{"id": "110.png", "formula": "\\begin{align*} \\widehat P ~ \\dot = ~ \\frac 1 2 \\left ( e e ^ t + v _ - v _ - ^ t \\right ) \\ , , \\end{align*}"}
-{"id": "8151.png", "formula": "\\begin{align*} X _ i \\otimes Y _ i ' \\otimes Y _ i '' = a _ j X _ i b _ k \\otimes Y _ i \\otimes b _ j a _ k . \\end{align*}"}
-{"id": "7597.png", "formula": "\\begin{align*} I _ 5 = \\int _ 0 ^ t \\Delta g _ { \\nu ( t - s ) } * \\left ( \\nabla \\cdot \\eta ( t ) \\left ( \\Delta _ 1 \\tau ( s , t ) + \\Delta _ 2 \\tau ( s , t ) \\right ) \\right ) d s \\end{align*}"}
-{"id": "4745.png", "formula": "\\begin{align*} e ^ { \\alpha \\xi _ i + \\beta ( A \\xi ) _ i } = e ^ { \\alpha \\xi _ i + \\beta \\sum _ { j : j \\sim i } \\xi _ j } , \\end{align*}"}
-{"id": "4853.png", "formula": "\\begin{align*} \\alpha ( w ^ * _ { ( j , r _ 1 ) } ) = a w ^ * _ { ( j , r _ 1 ) } . \\end{align*}"}
-{"id": "4563.png", "formula": "\\begin{align*} \\Delta _ k ( F ) = \\{ ( r _ { i j } , 1 \\leq i < j \\leq k ) \\in \\mathbb { R } ^ { k ( k - 1 ) / 2 } : x _ 1 , \\dots , x _ k \\in F , | x _ i - x _ j | = r _ { i j } , 1 \\leq i < j \\leq k \\} . \\end{align*}"}
-{"id": "2193.png", "formula": "\\begin{align*} \\frac { \\partial J ^ { \\ast } } { \\partial t } + \\mathcal { H } \\left [ \\boldsymbol { x } ^ \\ast ( t ) , \\frac { \\partial J ^ * } { \\partial \\boldsymbol { x } ^ * } , \\boldsymbol { u } ^ * ( t ) , t \\right ] = 0 \\ \\ \\forall t \\in \\ [ t _ 0 , t _ f ] \\end{align*}"}
-{"id": "2806.png", "formula": "\\begin{align*} f ( \\mathbf { j } _ 0 , \\mathbf { j } _ 1 , \\dots , \\mathbf { j } _ { s - 1 } ) : = \\mathbb { P } \\left ( \\mathbf { J } ^ { t } = \\mathbf { j } _ 0 ; \\mathbf { J } ^ { t + 1 } = \\mathbf { j } _ 1 ; \\dots , \\mathbf { J } ^ { t + s - 1 } = \\mathbf { j } _ { s - 1 } \\right ) . \\end{align*}"}
-{"id": "8939.png", "formula": "\\begin{align*} B _ { { q p } _ { b l } } ( 1 ; \\epsilon ) & = \\{ y \\in X : q p _ { b l } ( 1 , y ) < q p _ { b l } ( 1 , 1 ) + \\epsilon \\thinspace \\mbox { a n d } \\thinspace q p _ { b l } ( y , 1 ) < q p _ { b l } ( 1 , 1 ) + \\epsilon \\} \\\\ & = \\{ 0 , 1 , 2 \\} . \\end{align*}"}
-{"id": "2462.png", "formula": "\\begin{align*} r + \\deg x _ 0 > \\deg x _ 1 > \\cdots > \\deg x _ r = r . \\end{align*}"}
-{"id": "5732.png", "formula": "\\begin{align*} \\frac { Y '' ( y ) } { Y ( y ) } = \\frac { X '' ( x ) + \\frac { 1 } { x } X ' ( x ) + \\mu X ( x ) } { - X ( x ) } = - \\lambda \\end{align*}"}
-{"id": "7363.png", "formula": "\\begin{align*} d ( c ' , b _ 1 ) & = ( 1 - \\frac { d ( a , c ) } { \\delta } ) ( r - d ( p , c ) ) \\geq ( 1 - \\frac { d ( a , c ) } { \\delta } ) \\frac { \\delta } { 2 } = \\frac { \\delta } { 2 } - \\frac { d ( a , c ) } { 2 } \\\\ \\tag * { T h e n } d ( a , b _ 1 ) & = d ( a , c ' ) + d ( c ' , b _ 1 ) > ( d ( a , c ) - \\epsilon ) + d ( c ' , b _ 1 ) \\\\ & > d ( a , c ) - \\frac { d ( a , c ) } { 2 } + \\frac { \\delta } { 2 } - \\frac { d ( a , c ) } { 2 } = \\frac { \\delta } { 2 } \\end{align*}"}
-{"id": "4363.png", "formula": "\\begin{align*} \\sigma = \\frac { 1 } { 7 } \\begin{bmatrix} 2 & 3 \\\\ 3 & 5 \\end{bmatrix} . \\end{align*}"}
-{"id": "9877.png", "formula": "\\begin{align*} & D ( P ( X | Y ) \\| Q ( X | Y ) ) = \\int \\log \\left ( \\frac { d P _ { X | Y } } { d Q _ { X | Y } } ( x , y ) \\right ) P ( d ( x , y ) ) \\\\ & = \\int \\left ( \\int \\log \\left ( \\frac { d P _ { X | Y } } { d Q _ { X | Y } } ( x , y ) \\right ) P ( d x | Y = y ) \\right ) P ( d y ) \\end{align*}"}
-{"id": "1242.png", "formula": "\\begin{align*} \\int e ^ { 3 i \\mu ^ 2 / 4 } e ^ { i a \\ln | \\eta ^ 2 - \\mu ^ 2 / \\eta | } d \\mu & = e ^ { i a \\ln | \\eta | ^ 2 } \\int _ { | \\mu | \\le | \\eta | ^ { 3 / 2 } / 2 } e ^ { 3 i \\mu ^ 2 / 4 } \\left ( 1 + \\frac { \\mu ^ 2 } { | \\eta | ^ 3 } \\phi \\left ( \\frac { \\mu ^ 2 } { | \\eta | ^ 3 } \\right ) \\right ) d \\mu \\\\ & = e ^ { i a \\ln | \\eta | ^ 2 } | \\eta | ^ { 3 / 2 } \\int _ { | z | \\le 1 / 2 } e ^ { 3 i | \\eta | ^ 3 z ^ 2 / 4 } \\left ( 1 + z ^ 2 \\phi ( z ) \\right ) d z \\end{align*}"}
-{"id": "811.png", "formula": "\\begin{align*} B = ( q ^ d - 2 ) - ( q - 2 ) \\frac { q ^ d - 1 } { q - 1 } = q \\left ( \\frac { q ^ { d - 1 } - 1 } { q - 1 } \\right ) . \\end{align*}"}
-{"id": "2615.png", "formula": "\\begin{align*} G ( \\theta ) = \\Big ( \\frac { 1 } { 1 + \\theta ^ { - ( \\alpha - 1 ) } } \\Big ) ^ { \\frac { 1 } { \\alpha - 1 } } , \\theta \\geq 0 . \\end{align*}"}
-{"id": "6592.png", "formula": "\\begin{align*} F _ n \\varphi = F _ { n - 1 } - ( - \\varphi ) ^ { n } , \\end{align*}"}
-{"id": "4438.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Q } _ t } u _ p \\nabla \\mathcal { V } _ p \\cdot \\nabla u _ p \\dd x \\dd \\tau = - \\frac 1 2 \\int _ { \\mathcal { Q } _ t } u _ p ^ 2 \\Delta \\mathcal { V } _ p \\dd x \\dd \\tau + \\frac 1 2 \\int _ { \\mathcal { Q } _ t } u _ p ^ 2 \\frac { \\partial \\mathcal { V } _ p } { \\partial \\nu } \\dd \\mu \\dd \\tau . \\end{align*}"}
-{"id": "2334.png", "formula": "\\begin{align*} \\mathcal { V } ( f ) & = \\mathcal { V } ( \\tilde { f } ) = \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { n ! } \\sum _ { x \\in X ^ n } f ( x ) | x \\rangle \\\\ \\langle f | g \\rangle & = \\langle \\tilde { f } | \\tilde { g } \\rangle = \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { n ! } \\sum _ { x \\in X ^ n } f ( x ) g ( x ) \\end{align*}"}
-{"id": "1110.png", "formula": "\\begin{align*} & \\limsup _ { n \\to + \\infty } n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } \\sup _ { y \\in I } \\Big ( \\mathbb { P } ( \\xi ^ { ( n ) } ( [ y , y + G _ n ( x ) / S ( I ) ] ) = 0 ) \\\\ & - D _ n ( \\sqrt { 4 - y ^ 2 } / S ( I ) \\cdot G _ n ( x ) / 2 ) \\Big ) \\leq 0 . \\end{align*}"}
-{"id": "3831.png", "formula": "\\begin{align*} ^ { h } { { \\gamma _ { j } } ^ { i } } _ k = ^ { \\alpha } { { \\gamma _ { j } } ^ { i } } _ k + \\frac { 1 } { 2 } k _ j \\delta ^ i _ k + \\frac { 1 } { 2 } k _ k \\delta ^ i _ j - \\frac { 1 } { 2 } k ^ i a _ { j k } , \\end{align*}"}
-{"id": "48.png", "formula": "\\begin{align*} g ( x ) : = e ^ { x } - 1 - x , \\ \\ | x | \\leq \\frac { 1 } { q ^ { \\min } } \\left ( K _ f + \\frac { C _ v C _ { \\eta } C _ z } { ( C _ { \\eta } - C _ v ) ^ 2 } \\right ) . \\end{align*}"}
-{"id": "8906.png", "formula": "\\begin{align*} \\Psi : = \\mathbb { E } _ { \\textrm { $ e \\sim $ B e r n $ ( p ) $ } } [ f _ { e } ( q , p ) ^ 2 ] \\mathbb { E } _ { \\textrm { $ e \\sim $ B e r n $ ( q ) $ } } [ f _ { e } ( p , q ) ^ 2 ] \\end{align*}"}
-{"id": "6258.png", "formula": "\\begin{align*} A = \\bigoplus _ { i = 1 } ^ \\infty A _ { 0 , i } \\oplus \\bigoplus _ { i = 1 } ^ \\infty A _ { 1 , i } \\oplus \\bigoplus _ { i = 1 } ^ \\infty A _ { 2 , i } B = \\bigoplus _ { j = 1 } ^ \\infty B _ { 0 , j } \\oplus \\bigoplus _ { j = 1 } ^ \\infty B _ { 1 , j } \\oplus \\bigoplus _ { j = 1 } ^ \\infty B _ { 2 , j } \\end{align*}"}
-{"id": "3058.png", "formula": "\\begin{align*} g = \\begin{pmatrix} u & 0 \\\\ 0 & u \\\\ \\end{pmatrix} \\begin{pmatrix} y ^ { 1 / 2 } & x y ^ { - 1 / 2 } \\\\ 0 & y ^ { - 1 / 2 } \\\\ \\end{pmatrix} \\kappa _ { \\theta } , \\ \\kappa _ { \\theta } = \\begin{pmatrix} \\cos ( \\theta ) & \\sin ( \\theta ) \\\\ - \\sin ( \\theta ) & \\cos ( \\theta ) \\\\ \\end{pmatrix} \\end{align*}"}
-{"id": "3106.png", "formula": "\\begin{align*} A ^ { ( r ) } ( y ) = \\frac { y ^ \\frac { 1 } { 2 } - y ^ { - \\frac { 1 } { 2 } } } { \\phi _ { - 2 , 1 } ( q ^ r , y ^ r ) ^ \\frac { 1 } { 2 } \\ , \\tilde \\Delta ( q ^ r ) ^ \\frac { 1 } { 2 } } \\end{align*}"}
-{"id": "429.png", "formula": "\\begin{align*} \\mathcal { D } _ t ( \\dd x ) : = ( X _ t ( x ) - \\gamma t ) e ^ { \\gamma X _ t ( x ) - \\frac { \\gamma ^ 2 } { 2 } K _ t ( x , x ) } \\ , \\mu ( \\dd x ) . \\end{align*}"}
-{"id": "4808.png", "formula": "\\begin{align*} & \\leq C e ^ { 4 8 c _ { d , \\sigma } k ^ 2 T ( 1 + | x | ) ^ 2 } E \\left [ e ^ { 4 8 c _ { d , \\sigma } k ^ 2 T \\sum _ { i = 1 } ^ d \\sup _ { 0 \\leq t \\leq T } | \\sigma _ i \\cdot B ^ i _ t | ^ 2 } \\right ] ^ { \\frac { 1 } { 2 } } \\\\ & \\leq C _ { T , d , \\sigma , M _ 2 } e ^ { 4 8 c _ { d , \\sigma } k ^ 2 T ( 1 + | x | ) ^ 2 } \\prod _ { i = 1 } ^ { d } E \\left [ e ^ { 4 8 c _ { d , \\sigma } k ^ 2 T \\sup _ { 0 \\leq t \\leq T } | \\sigma _ i \\cdot B ^ i _ t | ^ 2 } \\right ] ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "8465.png", "formula": "\\begin{align*} | R e F ( x + i y , t ) | = | R e F ( x + i y , t ) - R e F ( 0 + i y , t ) | \\leq \\| F _ { \\zeta } \\| _ { \\infty } | x | \\leq 6 \\epsilon | x | . \\end{align*}"}
-{"id": "8050.png", "formula": "\\begin{align*} \\sigma _ k ( a ) = \\begin{pmatrix} e ^ { \\pi \\imath k / n } & 0 \\\\ 0 & e ^ { - \\pi \\imath k / n } \\end{pmatrix} , \\sigma _ k ( x ) = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "5579.png", "formula": "\\begin{align*} t \\longmapsto \\eta ( t ) : = \\int _ { \\R ^ - } | x | ^ 2 | u ( x , t ) | ^ 2 d x + \\int _ { 0 } ^ t \\int _ { \\R ^ - } x | u | ^ 2 ( x , t ) d x - \\frac { 2 \\alpha } { \\gamma } \\int _ { 0 } ^ t \\int _ { \\R ^ - } x v ^ 2 ( x , t ) d x . \\end{align*}"}
-{"id": "2472.png", "formula": "\\begin{align*} w ( P ) & = \\sum _ { i = 1 } ^ g n _ i - \\sum _ { i = 1 } ^ g i \\\\ & = \\sum _ { i = 1 } ^ { 2 g } i - \\sum _ { i = 1 } ^ g \\alpha _ i - \\sum _ { i = 1 } ^ g i \\\\ & = \\sum _ { i = g + 1 } ^ { 2 g - 1 } i - \\sum _ { i = 1 } ^ { g - 1 } \\alpha _ i . \\end{align*}"}
-{"id": "4403.png", "formula": "\\begin{align*} T ( x , y , z ) : = z - 1 . \\end{align*}"}
-{"id": "6488.png", "formula": "\\begin{align*} \\chi \\psi \\cdot \\chi _ { \\sigma } = 1 \\end{align*}"}
-{"id": "5163.png", "formula": "\\begin{align*} \\Lambda ( x ) = \\exp ( - \\exp ( - x ) ) , x \\in R \\end{align*}"}
-{"id": "578.png", "formula": "\\begin{align*} \\frac { p ^ r ( p ^ r - 1 ) ( p ^ r - 2 ) } { p ^ r ( p ^ r - 1 ) } = p ^ r - 2 \\end{align*}"}
-{"id": "8414.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { E } ( t ) + d _ P ( t ) ^ { - 1 } + d _ I ( t ) ^ { - 1 } \\leq 2 ( \\mathcal { E } ( 0 ) + d _ P ( 0 ) ^ { - 1 } + d _ I ( 0 ) ^ { - 1 } ) , \\\\ \\| z _ { t t } ( \\cdot , t ) \\| _ { H ^ s } \\leq 2 \\| w _ 0 \\| _ { H ^ s } , \\\\ \\frac { C _ 1 } { 2 } | \\alpha - \\beta | \\leq | z ( \\alpha , t ) - z ( \\beta , t ) | \\leq 2 C _ 2 | \\alpha - \\beta | \\\\ \\inf _ { \\alpha \\in \\mathbb { R } } a ( \\alpha , t ) | z _ { \\alpha } | \\geq \\frac { 1 } { 2 } \\alpha _ 0 . \\end{cases} \\end{align*}"}
-{"id": "1386.png", "formula": "\\begin{align*} \\widehat { F } ^ 1 _ { J , l } \\left ( \\widehat { \\Xi } \\left ( \\vec { \\lambda } ; J , J _ { ( l ) } \\right ) \\right ) & = \\widehat { F } ^ 0 _ { J , l } \\left ( \\widehat { \\Xi } \\left ( \\vec { \\lambda } ; J , J _ { ( l ) } - 1 \\right ) \\right ) \\ ; . \\end{align*}"}
-{"id": "705.png", "formula": "\\begin{align*} \\displaystyle ( 2 p - 1 ) \\frac { x } { f ( x ) } + ( 2 p - 1 ) \\int _ x ^ 1 \\frac { d t } { f ( t ) } + \\frac { 2 p K } { f ( x ) ^ 2 } & = \\displaystyle \\int _ x ^ 1 \\left ( ( 2 p - 1 ) t + \\frac { 4 p K } { f ( t ) } \\right ) \\frac { f ^ { ( 1 ) } } { f ^ 2 } ( t ) \\ ; d t , \\\\ & = \\frac { 2 p - 1 } { 2 } ( 1 - x ^ 2 ) + B ( x ) , \\end{align*}"}
-{"id": "755.png", "formula": "\\begin{align*} r ( \\mu ) \\approx r _ { } ( \\mu ) = \\sum _ { m = 1 } ^ n \\sqrt { \\lambda _ m } \\ , s _ m ( \\mu ) v _ m = \\sum _ { m = 1 } ^ n ( R ^ * s _ m ) ( \\mu ) , \\end{align*}"}
-{"id": "8542.png", "formula": "\\begin{align*} \\mathcal { E } _ s = \\sum _ { k = 0 } ^ s \\Big \\{ \\int \\frac { 1 } { A } | D _ t \\theta _ k | ^ 2 + i \\theta _ k \\overline { \\partial _ { \\alpha } \\theta _ k } d \\alpha + \\int \\frac { 1 } { A } | D _ t \\sigma _ k | ^ 2 + i \\sigma _ k \\overline { \\partial _ { \\alpha } \\sigma _ k } d \\alpha \\Big \\} , \\end{align*}"}
-{"id": "541.png", "formula": "\\begin{align*} f ^ { s , h } \\to S \\vec b _ 0 = \\vec p \\mbox { s t r o n g l y i n } \\ ; L ^ 2 ( \\Omega , \\mathbb { R } ^ 3 ) , \\mbox { a s } h \\to 0 . \\end{align*}"}
-{"id": "9455.png", "formula": "\\begin{align*} \\frac { 1 } { \\sigma _ 1 ( m ) } T _ m F _ { n , \\epsilon } ( z _ 0 ) = \\frac { 1 } { \\sigma _ 1 ( m ) } \\left \\{ \\sum _ { \\substack { a d = m , \\\\ b \\pmod { d } } } J _ { N , n } \\left ( \\frac { a z _ 0 + b } { d } \\right ) - \\sum _ { \\substack { a d = m , \\\\ b \\pmod { d } } } P _ { n , \\epsilon } \\left ( \\frac { a z _ 0 + b } { d } \\right ) \\right \\} . \\end{align*}"}
-{"id": "6381.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( A \\subseteq Y \\right ) = \\det ( K _ A ) , \\end{align*}"}
-{"id": "648.png", "formula": "\\begin{align*} \\vec a ^ { \\psi \\varphi } _ s = \\vec a ^ \\varphi _ { j ^ s _ 1 } \\dots \\vec a ^ \\varphi _ { j ^ s _ r } \\ . \\end{align*}"}
-{"id": "4008.png", "formula": "\\begin{align*} g = \\sum _ \\chi c _ \\chi g _ \\chi , \\end{align*}"}
-{"id": "5817.png", "formula": "\\begin{align*} g ( \\zeta ) = a _ 0 + i a _ 1 \\zeta + \\frac { a _ 2 } { 2 } \\zeta ^ 2 \\end{align*}"}
-{"id": "6225.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\theta } \\Big \\{ \\log \\mathcal { L } ( \\theta ) - \\mathcal { D } _ \\textnormal { K L } ( \\theta ^ \\star \\| \\theta ) \\Big \\} \\Big \\vert _ { \\theta = \\theta ^ \\star } = 0 , \\end{align*}"}
-{"id": "5933.png", "formula": "\\begin{align*} \\pi _ X ^ * ( K _ X + B + M ) = K _ W + B _ W + M _ W \\pi _ Y ^ * ( K _ Y + D _ Y + M _ Y ) = K _ W + D _ W + M _ W , \\end{align*}"}
-{"id": "3969.png", "formula": "\\begin{align*} H ( Z ) : = f ( \\tau , z ) \\cdot \\det ( Y ) ^ k \\cdot e ^ { - 4 \\pi N v ' } \\end{align*}"}
-{"id": "8284.png", "formula": "\\begin{align*} \\sup _ { \\substack { \\alpha \\neq \\beta \\\\ 0 \\leq t < T _ 0 ^ * } } \\Big | \\frac { \\alpha - \\beta } { z ( \\alpha , t ) - z ( \\beta , t ) } \\Big | + \\sup _ { \\substack { \\alpha \\neq \\beta \\\\ 0 \\leq t < T _ 0 ^ * } } \\Big | \\frac { z ( \\alpha , t ) - z ( \\beta , t ) } { \\alpha - \\beta } \\Big | = \\infty . \\end{align*}"}
-{"id": "91.png", "formula": "\\begin{align*} \\bar J _ 0 ( x ) = \\begin{cases} J _ 0 ( x ) & 0 < x < 1 \\\\ 0 & x = 0 \\mbox { o r } 1 \\ , . \\end{cases} \\end{align*}"}
-{"id": "9792.png", "formula": "\\begin{align*} { \\rm I } = \\| u ( X _ \\circ + r \\ , \\cdot \\ , ) - p _ { * , X _ \\circ } ( r \\ , \\cdot \\ , ) \\| _ { L ^ 2 ( B _ { 1 / 2 } , | y | ^ a ) } \\leq \\| u ( X _ \\circ + r \\ , \\cdot \\ , ) - p _ { * , X _ \\circ } ( r \\ , \\cdot \\ , ) \\| _ { L ^ 2 ( B _ { 1 } , | y | ^ a ) } \\leq C _ h r ^ \\lambda \\end{align*}"}
-{"id": "1463.png", "formula": "\\begin{align*} V _ { 4 } = - \\varepsilon \\int _ { \\mathbb { T } ^ { d } } ( \\mu ^ { 1 } - \\mu ^ { 2 } ) ( \\nabla \\mu ^ { 2 } ) \\cdot \\left ( \\Theta _ { p } ( t , x , \\mu ^ { 1 } , D w ^ { 1 } ) - \\Theta _ { p } ( t , x , \\mu ^ { 2 } , D w ^ { 2 } ) \\right ) \\ d x , \\end{align*}"}
-{"id": "8503.png", "formula": "\\begin{align*} | F _ t ( z _ 1 , t ) - F _ t ( z _ 2 , t ) | = & 2 | R e F _ t ( z _ 2 , t ) - R e F _ t ( 0 , y , t ) | x = 2 | R e F _ { t x } ( \\tilde { x } , y , t ) | x ( t ) \\\\ \\leq & 2 \\| F _ { t \\zeta } \\| _ { L ^ { \\infty } ( \\Omega ( t ) ) } x ( t ) \\leq 2 0 \\epsilon x ( t ) . \\end{align*}"}
-{"id": "413.png", "formula": "\\begin{align*} G _ { \\hat g } ( x , y ) = G _ { g } ( x , y ) + \\frac { 1 } { 2 } ( \\phi ( x ) + \\phi ( y ) ) - S ^ { { \\rm c l } } _ { { \\rm A Y } } ( \\hat g , g ) . \\end{align*}"}
-{"id": "1299.png", "formula": "\\begin{align*} | A | ^ 2 = 2 \\ln \\left ( \\frac { 1 } { 1 - \\kappa ^ 2 } \\right ) , a = \\frac { 3 | A | ^ 2 } { 4 \\pi } = \\frac { 3 } { 2 \\pi } \\ln \\left ( \\frac { 1 } { 1 - \\kappa ^ 2 } \\right ) . \\end{align*}"}
-{"id": "1232.png", "formula": "\\begin{align*} P ( 2 / 3 ) = 8 / 2 7 , P ' ( 2 / 3 ) = 0 , P '' ( 2 / 3 ) = - 3 / 2 . \\end{align*}"}
-{"id": "10011.png", "formula": "\\begin{align*} \\sum _ { n = M + 1 } ^ { N } { \\frac { a _ { n } } { n ^ { L + \\varepsilon + s } } } = \\Big ( \\sum _ { k = 1 } ^ { N } { \\frac { a _ { k } } { k ^ { s } } } \\Big ) \\frac { 1 } { N ^ { L + \\varepsilon } } - \\Big ( \\sum _ { k = 1 } ^ { M } { \\frac { a _ { k } } { k ^ { s } } } \\Big ) \\frac { 1 } { M ^ { L + \\varepsilon } } + \\sum _ { n = M } ^ { N - 1 } \\Big ( \\sum _ { k = 1 } ^ { n } { \\frac { a _ { k } } { k ^ { s } } } \\Big ) \\Big [ \\frac { 1 } { n ^ { L + \\varepsilon } } - \\frac { 1 } { ( n + 1 ) ^ { L + \\varepsilon } } \\Big ] \\ , \\end{align*}"}
-{"id": "4328.png", "formula": "\\begin{align*} a _ h ( u _ h , v ) = & - \\sum _ { K \\in \\mathcal { T } _ h } ( \\Delta u , Q _ K v ) _ K \\\\ = & \\sum _ { K \\in \\mathcal { T } _ h } ( \\nabla \\Pi _ K u , \\nabla \\Pi _ K v ) _ K + \\sum _ { K \\in \\mathcal { T } _ h } \\langle \\nabla ( u - \\Pi _ K u ) \\cdot n , Q _ K v - Q _ F v \\rangle _ { \\partial K } . \\end{align*}"}
-{"id": "9939.png", "formula": "\\begin{align*} E _ k : = A \\ k , \\end{align*}"}
-{"id": "9482.png", "formula": "\\begin{align*} Q ( u ; x _ 0 ; R ) = \\vert \\vert u \\vert \\vert _ { L ^ { \\infty } ( B _ { R } ( x _ 0 ) } + T a i l _ { p } ( u ; x _ 0 ; R ) , \\ \\ \\ Q ( u ; R ) = Q ( u ; 0 ; R ) . \\end{align*}"}
-{"id": "3710.png", "formula": "\\begin{align*} \\sum _ x \\frac { \\partial p ( x ; \\lambda ) } { \\partial \\lambda } \\left [ \\frac { n _ { x B } } { p ( x ; \\lambda ) } - \\frac { N _ x - n _ { x B } } { 1 - p ( x ; \\lambda ) } \\right ] = 0 \\end{align*}"}
-{"id": "6591.png", "formula": "\\begin{align*} \\liminf _ { N \\to \\infty } P _ N ( \\varphi ) = \\liminf _ { N \\to \\infty } \\prod _ { r = 1 } ^ N \\left | 2 \\sin ( \\pi r \\varphi ) \\right | > 0 . \\end{align*}"}
-{"id": "6124.png", "formula": "\\begin{align*} f _ { 1 d , } = p _ { 2 , \\mu } q _ { 0 , d - \\mu } - p _ { 0 , \\mu } q _ { 2 , d - \\mu } & = \\alpha _ { d - \\mu _ 1 } ' A _ { 1 , \\mu _ 1 } + \\beta _ { d - \\mu _ 2 } ' B _ { 1 , \\mu _ 2 } , \\\\ f _ { 2 , d } = p _ { 0 , \\mu } q _ { 1 , d - \\mu } - p _ { 1 , \\mu } q _ { 0 , d - \\mu } & = \\alpha _ { d - \\mu _ 1 } ' A _ { 2 , \\mu _ 1 } + \\beta _ { d - \\mu _ 2 } ' B _ { 2 , \\mu _ 2 } . \\end{align*}"}
-{"id": "8293.png", "formula": "\\begin{align*} \\begin{cases} ( \\partial _ t ^ 2 + i a \\partial _ { \\alpha } ) \\bar { z } _ t = - i a _ t \\bar { z } _ { \\alpha } \\\\ \\dot { z } _ j ( t ) = ( v - \\frac { \\lambda _ j i } { 2 \\pi } \\frac { 1 } { \\overline { z ( \\alpha , t ) - z _ j ( t ) } } ) \\Big | _ { z = z _ j ( t ) } \\\\ ( I - \\mathfrak { H } ) ( \\bar { z } _ t + \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i } { 2 \\pi } \\frac { 1 } { z ( \\alpha , t ) - z _ j ( t ) } ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "5804.png", "formula": "\\begin{align*} \\ell _ { \\rho } * \\ell _ { \\omega ^ \\beta } \\ & = \\ \\log ( \\ell _ { \\mu } * \\ell _ { \\omega ^ \\beta } ) \\ = \\ \\log ( \\ell _ { \\mu } - 1 ) \\\\ & = \\ \\log \\big ( \\ell _ { \\mu } ( 1 - \\ell _ { \\mu } ^ { - 1 } ) \\big ) \\ = \\ \\log ( \\ell _ { \\mu } ) + \\log ( 1 - \\ell _ { \\mu } ^ { - 1 } ) \\\\ & = \\ \\ell _ { \\rho } + \\epsilon _ { \\rho } , \\epsilon _ { \\rho } \\asymp \\ell _ { \\mu } ^ { - 1 } \\prec 1 . \\end{align*}"}
-{"id": "4810.png", "formula": "\\begin{align*} T \\le T _ 1 : = \\frac { 1 } { 4 \\sqrt { 3 } d k ^ 2 _ 1 } . \\end{align*}"}
-{"id": "573.png", "formula": "\\begin{align*} \\xi ^ { s + a _ 1 } p ( a _ 2 - b _ 2 ) , \\end{align*}"}
-{"id": "9087.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { R } ^ N } \\int _ { \\mathbb { R } ^ N } \\dfrac { | \\varphi _ n ( x ) - \\varphi _ n ( y ) | ^ p | u ( x ) | ^ p } { | x - y | ^ { N + s p } } \\ , d x d y = \\\\ & \\int _ { B _ n } \\int _ { B _ n ^ c } \\dfrac { | \\varphi _ n ( x ) - 1 | ^ p | u ( x ) | ^ p } { | x - y | ^ { N + s p } } \\ , d x d y + \\int _ { B _ n ^ c } \\int _ { \\mathbb { R } ^ N } \\dfrac { | \\varphi _ n ( x ) - \\varphi _ n ( y ) | ^ p | u ( x ) | ^ p } { | x - y | ^ { N + s p } } \\ , d x d y \\\\ & = I _ 1 + I _ 2 . \\end{align*}"}
-{"id": "4244.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\left ( a ( x ) \\left \\lvert d u \\right \\rvert ^ { p } + \\langle f ; u \\rangle \\right ) & = \\int _ { \\Omega } \\left ( a ( x ) \\left \\lvert d u \\right \\rvert ^ { p } - \\langle F ; d u \\rangle \\right ) + \\int _ { \\partial \\Omega } \\langle F ; \\nu \\wedge u _ { 0 } \\rangle . \\end{align*}"}
-{"id": "364.png", "formula": "\\begin{align*} a z _ 1 ^ 2 - c x _ 1 ^ 2 & = a ( s z + c x ) ^ 2 - c ( s x + a z ) ^ 2 = s ^ 2 ( a z ^ 2 - c x ^ 2 ) + a c ( c x ^ 2 - a z ^ 2 ) \\\\ & = s ^ 2 ( a - c ) + a c ( c - a ) = ( s ^ 2 - a c ) ( a - c ) = a - c . \\end{align*}"}
-{"id": "6894.png", "formula": "\\begin{align*} \\varphi ( t , \\alpha ) = A ( t ) u ( \\alpha ) \\ , , \\end{align*}"}
-{"id": "5224.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow + \\infty } \\frac { U ( a _ { n } ) } { U ( b _ { n } ) } = c ^ { \\rho } . \\end{align*}"}
-{"id": "9984.png", "formula": "\\begin{align*} & w _ { \\alpha i j } + w _ { \\alpha j i } = 0 , \\\\ & w _ { \\alpha i j , l } - c ^ s _ { i j } w _ { \\alpha s l } = 0 , \\\\ & c _ { n m l , k } + c ^ s _ { m l } c _ { s n k } + c ^ \\alpha w _ { \\alpha m l } w _ { \\alpha n k } = 0 , \\end{align*}"}
-{"id": "5182.png", "formula": "\\begin{align*} \\underset { 1 / n \\leq s \\leq 1 - 1 / n } { \\sup } \\frac { \\left \\vert B _ { n } ( s ) - \\sqrt { n } ( s - V _ { n } ( s ) ) \\right \\vert } { ( s ( 1 - s ) ) ^ { 1 / 2 - \\nu } } = O _ { p } ( n ^ { - \\nu } ) , \\end{align*}"}
-{"id": "7858.png", "formula": "\\begin{align*} \\mathbb E \\bigg [ \\exp \\Big ( - 1 6 \\varepsilon ^ 2 \\kappa ( d , N ) ^ 2 \\sum _ { j \\in \\sigma ( I _ x ) \\cap J _ r } \\| \\xi _ j \\| ^ 2 \\Big ) \\bigg ] \\le C \\exp \\Big ( - c ' \\kappa ( d , N ) ^ 2 \\sum _ { j = r + 1 } ^ d \\| \\xi _ j \\| ^ 2 \\Big ) , \\end{align*}"}
-{"id": "7653.png", "formula": "\\begin{align*} \\tilde D ^ { i j } _ 2 = \\int \\limits _ { \\mathbb T ^ d } \\varkappa ^ j _ 1 ( \\xi ) v _ 0 ( \\xi ) A \\varkappa ^ i _ 1 ( \\xi ) d \\xi \\ + \\ \\int \\limits _ { \\mathbb T ^ d } \\varkappa ^ i _ 1 ( \\xi ) v _ 0 ( \\xi ) A \\varkappa ^ j _ 1 ( \\xi ) d \\xi . \\end{align*}"}
-{"id": "6925.png", "formula": "\\begin{align*} \\eta ( \\sigma ) = \\begin{cases} Q _ 0 < \\cdots < Q _ m < Q _ m R & \\mbox { i f $ i = m $ } , \\\\ Q _ 0 < \\cdots < Q _ i < Q _ { i + 2 } < \\cdots < Q _ m & \\mbox { i f $ Q _ i R = Q _ { i + 1 } $ , a n d } \\\\ Q _ 0 < \\cdots Q _ i < Q _ i R < Q _ { i + 1 } < \\cdots < Q _ m & \\mbox { i f $ Q _ i R < Q _ { i + 1 } $ } . \\end{cases} \\end{align*}"}
-{"id": "9062.png", "formula": "\\begin{align*} \\int _ { \\mathbb { T } ^ 2 } ( M _ f ( x , r ) - 1 ) ^ 2 \\rightarrow V a r \\left ( \\alpha + \\beta W ( \\mu _ { A } ) \\right ) = \\beta ^ 2 V a r ( W ( \\mu _ A ) ) \\end{align*}"}
-{"id": "3934.png", "formula": "\\begin{align*} E _ N \\nu \\bigl ( \\eta ( \\sigma ) \\bigr ) _ N ( n ) = & \\ \\sum E _ N \\bigl ( \\eta ( \\sigma ) ( S n _ 1 ) \\otimes n _ 0 \\bigr ) = \\\\ & \\ \\sum \\eta ( \\sigma ) ( S n _ 1 ) ( 1 ) n _ 0 = \\\\ & \\ \\sum S n _ 1 ( 1 ) n _ 0 = n \\ , . \\end{align*}"}
-{"id": "4392.png", "formula": "\\begin{align*} f _ { \\delta } ( z ) = \\left \\{ \\begin{array} { l @ { , \\quad } l } g _ { \\delta } ( z ) & z \\in C _ { 1 + } \\\\ 0 & z \\in C _ { 1 - } \\end{array} \\right . , \\end{align*}"}
-{"id": "2666.png", "formula": "\\begin{align*} \\sqrt { 1 - 2 \\varrho _ { k } } \\frac { 1 } { \\varrho _ { k } } e ^ { - \\frac { \\eta _ { - } } { \\varrho _ { k } } v } & = \\frac { - 1 } { 2 \\pi i } \\int _ { \\mathcal { C } _ { 1 - \\varrho _ { k } } } d z \\frac { z - \\varrho _ { k } } { 1 - z - \\varrho _ { k } } \\frac { 1 } { \\sqrt { 2 z - 1 } } \\frac { 1 } { 1 - z } e ^ { - \\frac { \\eta _ { - } } { 1 - z } v } , \\end{align*}"}
-{"id": "9426.png", "formula": "\\begin{align*} \\partial ^ * \\mathcal { F } _ N ( f , \\varepsilon ) = \\bigcup _ { \\tau \\in \\mathrm { S i n g } ( f ) \\cup \\mathcal { C } _ N } \\gamma _ { \\tau } ( \\varepsilon ) , \\end{align*}"}
-{"id": "9760.png", "formula": "\\begin{align*} \\lambda _ * q ( Z _ \\infty ) = Z _ \\infty \\cdot \\nabla _ { x } q ( Z _ \\infty ) = - Z _ \\infty \\cdot \\nabla _ { x } q _ \\infty ( 0 ) = - \\lim _ { \\ell \\to \\infty } \\frac { r _ \\ell } { h _ { r _ \\ell } } ( Z _ \\infty \\cdot \\nabla _ { x } p _ * ( X _ \\ell ) ) = 0 . \\end{align*}"}
-{"id": "1951.png", "formula": "\\begin{align*} X ^ - _ \\infty ( \\psi ^ - ) = 0 X ( \\psi ^ - ) = 0 . \\end{align*}"}
-{"id": "7298.png", "formula": "\\begin{align*} \\phi ( a ) = \\psi _ p ( a ) + ( 1 - h ^ - ) \\Phi ' _ p ( a ) ( h _ s ^ - - h ^ + ) + ( 1 - h ^ - ) R _ s \\Phi ' _ p ( a ) ( h _ s ^ + - h _ s ^ - ) \\end{align*}"}
-{"id": "187.png", "formula": "\\begin{align*} & \\pi _ { X _ { 1 , C } , X _ { 2 , C } , X _ { 3 , C } , X _ { 4 , V } } = X _ { 1 , C } \\wedge X _ { 2 , C } + X _ { 3 , C } \\wedge X _ { 4 , V } , \\\\ & \\pi _ { X _ { 1 , C } , X _ { 2 , C } , X _ { 3 , C } , X _ { 4 , C } } = X _ { 1 , C } \\wedge X _ { 2 , C } + X _ { 3 , C } \\wedge X _ { 4 , C } \\end{align*}"}
-{"id": "6904.png", "formula": "\\begin{align*} \\begin{cases} a _ { 1 1 } a _ { 2 2 } - a _ { 1 2 } a _ { 2 1 } - 1 = 0 \\ , , \\\\ a _ { 1 3 } a _ { 2 4 } - a _ { 1 4 } a _ { 2 3 } - 1 = 0 \\ , , \\\\ a _ { 1 4 } - a _ { 1 2 } + a _ { 1 1 } = 0 \\ , , \\\\ a _ { 2 4 } - a _ { 2 2 } + a _ { 2 1 } = 0 \\ , . \\end{cases} \\end{align*}"}
-{"id": "1793.png", "formula": "\\begin{align*} c _ { 2 , 1 } \\left ( \\limsup _ { t \\downarrow 0 } \\frac { B _ { t } } { \\sqrt { 2 t ^ { 2 H } \\log \\log ( 1 / t ) } } > 1 \\right ) = 0 . \\end{align*}"}
-{"id": "3989.png", "formula": "\\begin{align*} D ( F , G , s ) : = \\zeta ( 2 s ) \\sum _ { N } \\{ \\phi _ N , \\psi _ N \\} N ^ { - s - k + 2 } \\end{align*}"}
-{"id": "9973.png", "formula": "\\begin{align*} \\begin{dcases*} \\frac { \\partial } { \\partial t } u ( t , x ) - \\sum _ { j = 0 } ^ k a _ j D ^ j u ( t , x ) = U ( t ) u ( t , \\cdot ) ( x ) + h ( t , x ) & i n $ I \\times \\R { } $ \\\\ u ( 0 , x ) = \\mathring { u } ( x ) & o n $ \\R { } $ \\\\ | | h ( t , \\cdot ) | | _ 2 \\ < r ( t , u ( t , \\cdot ) ) & o n $ I $ . \\end{dcases*} \\end{align*}"}
-{"id": "5515.png", "formula": "\\begin{align*} \\ell _ T ( b _ 0 + h / \\sqrt { T } ) - \\ell _ T ( b _ 0 ) = W _ T ( h ) - \\frac { 1 } { 2 T } \\int _ 0 ^ T \\| h ( X _ t ) \\| ^ 2 d t , \\end{align*}"}
-{"id": "4044.png", "formula": "\\begin{align*} I _ M ( P , V ) = I _ M ( P _ 1 , V ) + I _ M ( P _ 2 , V ) \\leq C M m + c ^ { - 1 / 2 } m ^ { 1 / 2 } ( n + C ' m ) = O ( M m + m ^ { 3 / 2 } + m ^ { 1 / 2 } n ) . \\end{align*}"}
-{"id": "909.png", "formula": "\\begin{align*} \\nu _ m = \\left \\{ ( x _ a ) _ { a \\in Q _ 1 } \\in \\eta _ m : ( x _ a ) _ { a \\in Q ^ j _ 1 } = \\nu _ { m ^ j } \\right \\} . \\end{align*}"}
-{"id": "4728.png", "formula": "\\begin{align*} ( \\rho , u , p ) ( x , 0 ) = \\left \\{ \\begin{array} { l l } ( 0 . 4 4 5 , 0 . 6 9 8 0 , 3 . 5 2 8 ) ~ ~ ~ ~ x < 0 , \\\\ ( 0 . 5 , 0 . 0 , 0 . 5 7 1 ) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ x > 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "227.png", "formula": "\\begin{align*} \\nabla _ H u = ( X _ 1 u , \\dots , X _ N u , Y _ 1 u , \\dots Y _ N u ) . \\end{align*}"}
-{"id": "5516.png", "formula": "\\begin{align*} \\ell _ T ( b _ 0 + h / \\sqrt { T } ) - \\ell _ T ( b _ 0 ) & = \\frac { 1 } { \\sqrt { T } } \\int _ 0 ^ T h ( X _ t ) . d X _ t - \\frac { 1 } { \\sqrt { T } } \\int _ 0 ^ T b _ 0 ( X _ t ) . h ( X _ t ) d t \\\\ & - \\frac { 1 } { 2 T } \\int _ 0 ^ T \\| h ( X _ t ) \\| ^ 2 d t \\\\ & = \\frac { 1 } { \\sqrt { T } } \\int _ 0 ^ T h ( X _ t ) . d W _ t - \\frac { 1 } { 2 T } \\int _ 0 ^ T \\| h ( X _ t ) \\| ^ 2 d t . \\end{align*}"}
-{"id": "392.png", "formula": "\\begin{align*} \\Phi _ \\lambda ( F ) : = j _ * ( p _ Z ^ * F \\otimes L _ \\lambda ( U ) ) ; \\end{align*}"}
-{"id": "753.png", "formula": "\\begin{align*} \\psi _ { 0 } ( l + n ) = \\psi _ { 0 } ( l ) + \\sum _ { k = 0 } ^ { n - 1 } \\frac { 1 } { l + k } , ~ ~ ~ ~ \\psi _ { 1 } ( l + n ) = \\psi _ { 1 } ( l ) - \\sum _ { k = 0 } ^ { n - 1 } \\frac { 1 } { ( l + k ) ^ 2 } . \\end{align*}"}
-{"id": "9608.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\infty } ( - 1 ) ^ j \\frac { ( q ) _ { m + j - 1 } - ( q ) _ j } { ( q ) _ j } = \\frac { 1 } { 2 } + \\frac { ( q ) _ { m - 1 } } { ( - 1 ) _ m } - ( q ) _ { m - 1 } , \\end{align*}"}
-{"id": "831.png", "formula": "\\begin{align*} D \\mod P _ i ^ \\alpha = a _ { i , 0 } + a _ { i , 1 } P _ i + a _ { i , 2 } P _ i ^ 2 + \\cdots a _ { i , \\alpha - 1 } P _ i ^ { \\alpha - 1 } \\mod P _ i ^ \\alpha , \\end{align*}"}
-{"id": "9287.png", "formula": "\\begin{align*} \\Phi ( x , y ) : = v ( x ) - u ( y ) - \\phi ( x , y ) , \\end{align*}"}
-{"id": "5058.png", "formula": "\\begin{align*} 0 & \\leq \\left \\| h - \\sum _ { j \\in \\mathbb { J } } \\Psi _ j ^ * A _ j h \\right \\| ^ 2 = \\left \\langle h - \\sum _ { j \\in \\mathbb { J } } c _ j A _ j ^ * A _ j h , h - \\sum _ { k \\in \\mathbb { J } } c _ k A _ k ^ * A _ k h \\right \\rangle \\\\ & = \\| h \\| ^ 2 - 2 \\sum _ { j \\in \\mathbb { J } } c _ j \\| A _ j h \\| ^ 2 + \\sum _ { j \\in \\mathbb { J } } c ^ 2 _ j \\| A _ j h \\| ^ 2 = \\| h \\| ^ 2 - \\sum _ { j \\in \\mathbb { J } } ( 2 c _ j - c _ j ^ 2 ) \\| A _ j h \\| ^ 2 , \\end{align*}"}
-{"id": "9965.png", "formula": "\\begin{align*} \\begin{dcases*} \\dot { u } ( t ) \\in A u ( t ) + F ( t , u ( t ) ) & o n $ I $ , \\\\ u ( 0 ) = x _ 0 , \\end{dcases*} \\end{align*}"}
-{"id": "2524.png", "formula": "\\begin{align*} \\psi ( z ) = c z ^ \\gamma l ( z ) , z \\geq 0 , \\end{align*}"}
-{"id": "5704.png", "formula": "\\begin{align*} J _ p ( x ) = \\sum _ { m = 0 } ^ \\infty \\frac { ( - 1 ) ^ m } { \\Gamma ( m + 1 ) \\Gamma ( p + m + 1 ) } \\left ( \\frac { x } { 2 } \\right ) ^ { p + 2 m } . \\end{align*}"}
-{"id": "3241.png", "formula": "\\begin{align*} R _ { 0 } ( w ^ k ) = 0 , \\ { F ( x ^ k ) \\in S ^ m _ + , \\ V _ k \\in S ^ m _ + , \\ y ^ k \\in \\mathcal { M } _ + ( T ) . } \\end{align*}"}
-{"id": "5886.png", "formula": "\\begin{align*} g = \\sum _ { w \\in \\tilde { W } } ( - 1 ) ^ { \\ell ( w ) } q ^ { - \\ell ( w ) } T _ w \\end{align*}"}
-{"id": "1576.png", "formula": "\\begin{align*} & \\sup \\left | \\frac { \\partial \\varphi _ w } { \\partial w } ( z ) \\right | | w _ { m + 1 } - g ( w _ m ) | \\\\ = & \\sup \\left | \\frac { \\partial \\chi _ w } { \\partial w } ( \\psi _ w ^ { - 1 } ( z ) ) + \\frac { \\partial \\chi _ w } { \\partial \\mathcal { Z } } ( \\psi _ w ^ { - 1 } ( z ) ) \\frac { \\partial \\psi _ w ^ { - 1 } } { \\partial w } ( z ) \\right | | w _ { m + 1 } - g ( w _ m ) | \\\\ = & o ( w _ m ) \\end{align*}"}
-{"id": "7830.png", "formula": "\\begin{align*} \\lim _ n \\| \\langle ( T + \\lambda S ) ( x _ { n _ k } ) , y \\rangle - \\langle ( T + \\lambda S ) ( x ) , y \\rangle \\| = 0 . \\end{align*}"}
-{"id": "9494.png", "formula": "\\begin{align*} \\beta _ n = \\gamma ^ n \\beta _ 0 + ( p - 1 ) \\frac { \\gamma ^ { n + 1 } - \\gamma } { \\gamma - 1 } . \\end{align*}"}
-{"id": "1400.png", "formula": "\\begin{align*} \\frac { \\tfrac { \\partial f } { \\partial x _ 1 } ( x _ 1 , \\ldots , x _ d ) } { r _ 1 ( x _ 1 ) } = \\cdots = \\frac { \\tfrac { \\partial f } { \\partial x _ d } ( x _ 1 , \\ldots , x _ d ) } { r _ d ( x _ d ) } \\end{align*}"}
-{"id": "3352.png", "formula": "\\begin{align*} H _ { \\chi } ( u ) : = \\prod _ { d \\ge 1 } H _ d ( u ^ d ) . \\end{align*}"}
-{"id": "9163.png", "formula": "\\begin{align*} A ^ D = Q _ A ( A ) . \\end{align*}"}
-{"id": "4557.png", "formula": "\\begin{align*} \\varphi _ { m _ k } \\cdots \\varphi _ 1 . ( v _ m \\otimes u _ z ) & = \\tau _ { m _ k } \\cdots \\tau _ { i + 1 } ( \\tau _ i ( x _ i - x _ { i + 1 } ) + 1 ) \\varphi _ { i - 1 } \\cdots \\varphi _ 1 . ( v _ m \\otimes u _ z ) \\\\ & = \\tau _ { m _ k } \\cdots \\tau _ { i + 1 } \\varphi _ { i - 1 } \\cdots \\varphi _ 1 . ( v _ m \\otimes u _ z ) \\pm z . \\tau _ { m _ k } \\cdots \\tau _ { i + 1 } \\tau _ i \\varphi _ { i - 1 } \\cdots \\varphi _ 1 . ( v _ m \\otimes u _ z ) . \\end{align*}"}
-{"id": "3842.png", "formula": "\\begin{align*} { { { R _ 0 } ^ i } _ { 0 l } } = K F ^ 2 ( \\delta ^ i _ l - l ^ i l _ l ) , \\end{align*}"}
-{"id": "9452.png", "formula": "\\begin{align*} f _ { \\theta } ( z ) = h _ \\infty + \\sum _ { n = 1 } ^ { \\infty } \\sum _ { d \\mid n } d c ( d ) q ^ n - \\frac { k } { 1 2 } + 2 k \\sum _ { n = 1 } ^ { \\infty } \\sigma _ 1 ( n ) q ^ n . \\end{align*}"}
-{"id": "3732.png", "formula": "\\begin{align*} B ( u _ k ) & = \\int _ { \\Omega } ( | x | ^ { - \\mu } * F ( u _ k ) ) f ^ { ' } ( u _ k ) u _ k ^ 2 ~ d x + \\int _ { \\Omega } ( | x | ^ { - \\mu } * f ( u _ k ) u _ k ) f ( u _ k ) u _ k ~ d x \\\\ & \\leq C ( n , \\mu ) \\left ( \\| f ( u _ k ) u _ k \\| ^ 2 _ { L ^ { 2 n / ( 2 n - \\mu ) } ( \\Omega ) } + \\| F ( u _ k ) \\| _ { L ^ { 2 n / ( 2 n - \\mu ) } ( \\Omega ) } \\| f ^ { ' } ( u _ k ) ( u _ k ) ^ 2 \\| _ { L ^ { 2 n / ( 2 n - \\mu ) } ( \\Omega ) } \\right ) . \\end{align*}"}
-{"id": "9741.png", "formula": "\\begin{align*} v _ \\ast : = u - p _ \\ast , \\quad \\kappa _ \\ast : = \\kappa _ 0 , L _ \\ast : = L ( p _ \\ast ) , m _ \\ast : = m _ 0 , \\quad \\lambda _ \\ast : = N ( 0 ^ + , v _ \\ast ) . \\end{align*}"}
-{"id": "1413.png", "formula": "\\begin{align*} Z ^ 1 _ t - \\textrm { M A R } _ t = Z ^ j _ t - E \\big [ \\sum _ { k = j } ^ { \\infty } \\min _ { i \\in [ 1 , T ] } Z ^ k _ i | { \\mathcal F } _ t \\big ] . \\end{align*}"}
-{"id": "7896.png", "formula": "\\begin{align*} \\dim \\mathcal { M } ( L ) & \\leq \\dim \\mathcal { M } ( L / I ) + \\dim ( ( L ) ^ { a b } \\otimes I ) - \\dim ( L ^ { 2 } \\cap I ) \\cr & \\leq \\frac { 1 } { 2 } ( n - m - 1 ) ( n + m - 2 ) - ( \\sum \\limits _ { i = 2 } ^ { m i n \\lbrace n - m , c \\rbrace } n - m - i ) + ( n - m ) - 1 \\cr & = \\frac { 1 } { 2 } ( n - m - 1 ) ( n + m ) - \\sum \\limits _ { i = 2 } ^ { m i n \\lbrace n - m , c \\rbrace } n - m - i = \\dim \\mathcal { M } ( L ) . \\end{align*}"}
-{"id": "4667.png", "formula": "\\begin{align*} f ( t _ { 1 } w _ { 1 } + \\cdots + t _ { k } w _ { k } ) = \\sum _ { \\substack { \\alpha = ( \\alpha _ { 1 } , \\ldots , \\alpha _ { k } ) \\\\ \\alpha _ { 1 } + \\cdots + \\alpha _ { k } = k } } t ^ \\alpha f _ \\alpha ( w _ 1 , \\dots , w _ k ) , t ^ \\alpha : = t _ { 1 } ^ { \\alpha _ { 1 } } \\cdots t _ { k } ^ { \\alpha _ { k } } . \\end{align*}"}
-{"id": "7493.png", "formula": "\\begin{gather*} f = z ^ { k _ 1 } + O ( z ^ { k _ 1 + 1 } ) . \\end{gather*}"}
-{"id": "1857.png", "formula": "\\begin{align*} \\partial _ t \\phi ( x , 0 ) = - \\frac { i } { \\sqrt { 2 L } } v ^ { \\rm t o p } ( x ) ( a - b ) - i \\sum _ { n = 1 } ^ \\infty \\sqrt { \\frac { k _ n } { 2 } } \\left [ v ^ { \\rm o d d } ( n , x ) ( a _ n - b _ n ) + v ^ { \\rm e v e n } ( n , x ) ( c _ n - d _ n ) \\right ] . \\end{align*}"}
-{"id": "1293.png", "formula": "\\begin{gather*} \\tilde \\Psi _ A ( z ) ( \\xi ) = ( \\| z \\| _ { Z ^ k } ^ 3 + | A | ^ 3 + | A | ) O ( | \\xi | ^ { - k } ) . \\end{gather*}"}
-{"id": "7432.png", "formula": "\\begin{align*} \\displaystyle d \\mu _ { G _ n ( E ) } ( \\pi ) = \\mu _ { G _ n ( E ) } ^ E ( \\pi ) d _ { \\psi ' _ E } \\pi \\end{align*}"}
-{"id": "1146.png", "formula": "\\begin{align*} \\sqrt { 4 - y _ j ^ 2 } = ( 1 + | u _ j | / \\ln n ) S ( I ) , \\ \\sqrt { 4 - y _ j ^ 2 } a _ j = ( 1 + | u _ j | / \\ln n ) G _ n ( x _ j ) . \\end{align*}"}
-{"id": "8668.png", "formula": "\\begin{align*} \\tilde \\mu ( A ) = \\int _ X \\left | \\pi ^ { - 1 } ( x ) \\cap A \\right | d \\mu ( x ) ; \\end{align*}"}
-{"id": "1716.png", "formula": "\\begin{align*} c _ n & : = { s _ n } ^ { - 1 } \\cdot \\sup _ { ( t , x ) \\in [ 0 , 1 ] \\times [ j _ n , j _ { n + 1 } ] } | f ( \\varphi ( s _ n + t , x ) ) - f ( \\varphi ( s _ n , x ) ) | , \\\\ d _ n & : = \\max \\{ M _ n , 1 \\} \\cdot \\max _ { n - 1 \\leq i \\leq n + 1 } \\left \\{ | a _ { i } | , | a _ { i } - a _ n | , c _ i \\right \\} \\end{align*}"}
-{"id": "8729.png", "formula": "\\begin{align*} \\hat { \\nu } ( j ) = \\frac { 2 ^ { j - 1 } ( - 1 ) ^ { m + j } ( m - j + 1 ) { { N - 1 } \\choose { m } } { { m } \\choose { j - 1 } } } { ( N - j ) \\sum _ { k = 0 } ^ { m - 1 } { { N - 1 } \\choose { k } } } a n d p g f _ { \\hat { T } _ { j , N } } ( s ) = \\prod _ { k = j } ^ { N - 1 } \\left [ { ( 1 - \\frac { k - 1 } { N - 1 } ) s \\over 1 - \\frac { k - 1 } { N - 1 } s } \\right ] . \\end{align*}"}
-{"id": "7045.png", "formula": "\\begin{align*} | d u | ^ 2 : = ( d u , d u ) = 2 g ^ { i j } u ^ v _ i \\o { u ^ v _ j } \\phi _ { v \\b { v } } , \\end{align*}"}
-{"id": "2199.png", "formula": "\\begin{align*} & \\boldsymbol { u } ^ * ( t ) = - R ^ { - 1 } ( t ) B ^ T ( t ) P ( t ) \\boldsymbol { x } ^ * ( t ) + R ^ { - 1 } ( t ) B ^ T ( t ) \\boldsymbol { g } ( t ) \\end{align*}"}
-{"id": "3360.png", "formula": "\\begin{align*} h \\cdot v = S ^ 2 ( h ) v , \\ h \\in H , \\ v \\in V . \\end{align*}"}
-{"id": "559.png", "formula": "\\begin{align*} & C _ d P _ { \\lambda _ 0 } ( C _ { \\ll d } u _ { \\lambda _ 1 } C _ { \\ll d } v _ { \\lambda _ 2 } ) \\\\ = { } & C _ d P _ { \\lambda _ 0 } ( C ^ + _ { \\ll d } u _ { \\lambda _ 1 } C ^ + _ { \\ll d } v _ { \\lambda _ 2 } ) + C _ d P _ { \\lambda _ 0 } ( C ^ - _ { \\ll d } u _ { \\lambda _ 1 } C ^ - _ { \\ll d } v _ { \\lambda _ 2 } ) . \\end{align*}"}
-{"id": "1068.png", "formula": "\\begin{align*} n ^ 2 \\ln \\cos \\frac { \\alpha _ n } { 2 } + \\frac { n ^ 2 \\alpha _ n ^ 2 } { 8 } = n ^ 2 O ( \\alpha _ n ^ 4 ) = n ^ 2 O ( G _ n ^ 4 ( x ) ) \\to 0 , \\end{align*}"}
-{"id": "1193.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\displaystyle ( \\partial _ t ^ 2 - \\nabla \\cdot \\gamma \\nabla ) u = 0 & \\ , ( 0 , T ) \\times \\Bbb R ^ 3 , \\\\ \\displaystyle u ( 0 , x ) = 0 , \\partial _ t u ( 0 , x ) = f ( x ) & \\ , \\Bbb R ^ 3 , \\end{array} \\right . \\end{align*}"}
-{"id": "4985.png", "formula": "\\begin{align*} Z ( C / \\mathbb { F } _ q , T ) = \\displaystyle \\sum _ { i = 0 } ^ { \\infty } \\frac { ( N _ 1 T + \\frac { N _ 2 } { 2 } T ^ 2 + \\ldots + \\frac { N _ { 2 g } } { 2 g } T ^ { 2 g } ) ^ i } { i ! } . \\end{align*}"}
-{"id": "2605.png", "formula": "\\begin{align*} M ( r , x ) : = \\limsup _ { t \\to \\infty } M ( t , r , x ) ; M ( r ) : = \\sup _ { x \\in E } M ( r , x ) , r \\geq 0 , x \\in E . \\end{align*}"}
-{"id": "281.png", "formula": "\\begin{align*} T ( L _ n ) ( x ) = \\sum _ { j = l } ^ m a _ j ( x - j \\lambda ) ^ n = x ^ n \\sum _ { j = l } ^ m a _ j \\left ( 1 - \\frac { j \\lambda } { x } \\right ) ^ n = : x ^ n S _ n ( x ) \\in { \\mathcal H P } . \\end{align*}"}
-{"id": "7776.png", "formula": "\\begin{align*} S _ n f ( x ) = - \\int _ { 2 ^ { n - 1 } } ^ { 2 ^ n } \\frac { \\rm d } { { \\rm d } t } P _ t f ( x ) { \\rm d } t . \\end{align*}"}
-{"id": "8148.png", "formula": "\\begin{align*} \\left ( \\tau _ { b _ g } \\tau _ { d _ g } \\ldots \\tau _ { b _ 1 } \\tau _ { d _ 1 } \\tau _ { d _ 1 } \\tau _ { b _ 1 } \\ldots \\tau _ { d _ g } \\tau _ { b _ g } \\right ) \\omega = \\omega \\left ( \\tau _ { b _ g } \\tau _ { d _ g } \\ldots \\tau _ { b _ 1 } \\tau _ { d _ 1 } \\tau _ { d _ 1 } \\tau _ { b _ 1 } \\ldots \\tau _ { d _ g } \\tau _ { b _ g } \\right ) \\end{align*}"}
-{"id": "3979.png", "formula": "\\begin{align*} \\mathcal { E } _ { k , m } = \\underset { 1 \\leq s \\leq b } \\sum a _ f ( a \\cdot ( s , b ) ^ 2 ) E _ { k , m , s } . \\end{align*}"}
-{"id": "6749.png", "formula": "\\begin{align*} \\widehat { u } ( t '' , \\eta , \\xi ) = - i ( \\xi a _ { M 0 } + \\eta _ M ) ^ { - 1 } \\widehat { f } _ M ( t '' , \\eta , \\xi ) . \\end{align*}"}
-{"id": "524.png", "formula": "\\begin{align*} \\omega _ { n , \\nu } = \\frac { m } { g ' ( \\phi ( y ) ) } \\ , y ^ { n - 1 } \\phi ( y ) ^ \\nu \\prod _ { i = 1 } ^ N ( \\phi ( y ) - t _ i ) ^ { l ( i , n ) + a _ i } \\ , d y . \\end{align*}"}
-{"id": "6701.png", "formula": "\\begin{align*} \\omega _ { i j } ( q ) = \\frac { \\partial u _ i ( q ) } { \\partial q _ j } - \\frac { \\partial u _ j ( q ) } { \\partial q _ i } . \\end{align*}"}
-{"id": "6168.png", "formula": "\\begin{align*} J _ \\beta ( U ) = \\sum _ { i = 1 } ^ k \\int _ { B ^ + _ 1 } \\frac 1 2 \\nabla u _ i ( x , z ) | ^ 2 d x \\ , d z + \\beta \\sum _ { 1 \\le i < j \\le k } \\int _ { \\mathcal { B } _ 1 } u _ i ^ 2 ( x , 0 ) u _ j ^ 2 ( x , 0 ) \\ , d x \\end{align*}"}
-{"id": "7034.png", "formula": "\\begin{align*} \\Phi ( u , v ) = \\zeta ( 1 - \\delta ) u + \\frac { ( 1 - \\delta ) v } { \\delta + ( 1 - \\delta ) u } + \\sum _ { j = 1 } ^ { n } \\frac { 1 - \\eta _ { j } } { \\lambda _ { j } } - \\zeta ( 1 - \\delta ) \\end{align*}"}
-{"id": "7286.png", "formula": "\\begin{align*} M ( p , F ) = 3 \\max \\{ 3 M _ p + 1 , L ( F ) , 2 J ( F ) + 1 \\} . \\end{align*}"}
-{"id": "4087.png", "formula": "\\begin{align*} \\gamma & = \\frac { 2 } { 3 } { \\alpha } + \\frac { 1 } { 2 } = \\frac { 2 } { 3 } \\beta + \\frac { 1 } { 3 } , \\\\ M _ 1 & = \\frac { 1 } { 3 } \\alpha ^ 2 + \\frac { 1 } { 6 } \\alpha - \\frac { 1 } { 3 6 } , \\\\ M _ 2 & = \\frac { 1 } { 1 8 } \\alpha ^ 4 + \\frac { 1 } { 5 4 } \\alpha ^ 3 - \\frac { 1 7 } { 2 1 6 } \\alpha ^ 2 - \\frac { 1 } { 5 4 } \\alpha + \\frac { 2 5 } { 2 5 9 2 } , \\end{align*}"}
-{"id": "4559.png", "formula": "\\begin{align*} \\varphi _ 1 \\cdots \\varphi _ { m _ k } . ( u \\otimes ( v _ q ) _ z ) = \\tau _ 1 \\cdots \\tau _ { b - 1 } ( \\tau _ b ( x _ b - x _ { b + 1 } ) + 1 ) \\tau _ { b + 1 } \\cdots \\tau _ { c - 1 } ( \\tau _ c ( x _ c - x _ { c + 1 } ) + 1 ) Q ( z ) \\end{align*}"}
-{"id": "3293.png", "formula": "\\begin{align*} \\frac { \\sum _ { n \\le x , \\ , n \\equiv a \\bmod q } \\alpha ( n ) } { x / \\phi ( q ) } = o ( 1 ) , x \\to \\infty \\end{align*}"}
-{"id": "2413.png", "formula": "\\begin{align*} T ' = ( X _ 0 , \\dots , X _ s , W _ k , \\dots , W _ { k + t } , X _ r , \\dots , X _ n ) , \\end{align*}"}
-{"id": "8238.png", "formula": "\\begin{align*} L _ { 2 i } = F \\left ( \\sum _ { j = 1 } ^ \\infty a ( 2 i , j ) t ^ j + \\rho \\sum _ { j = 1 } ^ \\infty b ( 2 i , j ) t ^ j \\right ) . \\end{align*}"}
-{"id": "2388.png", "formula": "\\begin{align*} u _ 1 : f _ 1 \\in \\mathfrak { A } _ 1 & \\mapsto u _ 1 ( f _ 1 ) = f _ 1 \\otimes 1 \\otimes \\cdots \\otimes 1 \\in \\textstyle \\bigotimes \\mathfrak { A } _ i . \\\\ & \\vdots \\\\ u _ n : f _ n \\in \\mathfrak { A } _ n & \\mapsto u _ n ( f _ n ) = 1 \\otimes \\cdots \\otimes 1 \\otimes f _ n \\in \\textstyle \\bigotimes \\mathfrak { A } _ i . \\end{align*}"}
-{"id": "4743.png", "formula": "\\begin{align*} S _ 1 = \\{ x \\in \\mathbb { R } ^ 2 : \\dfrac { x _ 1 } { a _ 1 } + \\dfrac { x _ 2 } { a _ 2 } \\leq t _ 1 \\} \\end{align*}"}
-{"id": "7947.png", "formula": "\\begin{align*} S & = R \\cap [ n ] , \\\\ T & = R \\cap \\{ n + 1 , \\dots , n + r \\} . \\end{align*}"}
-{"id": "3738.png", "formula": "\\begin{align*} \\mathbf { E } ( \\mathbf { r } ) = \\sum _ { j , l , m } a _ { l , m } ^ { ( j ) } \\mathbf { \\Lambda } _ { l , m } ^ { ( j ) } ( \\mathbf { r } ) , \\quad \\mathbf { r } \\notin V , \\end{align*}"}
-{"id": "45.png", "formula": "\\begin{align*} F _ s ^ { i } ( Z _ s ^ i ) - \\bar { F } _ s ^ { i } ( \\bar { Z } _ s ^ i ) & = F _ s ^ { i } ( Z _ s ^ i ) - { F } _ s ^ { i } ( \\bar { Z } _ s ^ i ) + F _ s ^ { i } ( \\bar { Z } _ s ^ i ) - \\bar { F } _ s ^ { i } ( \\bar { Z } _ s ^ i ) \\leq C _ f | \\delta Z _ s ^ i | . \\end{align*}"}
-{"id": "2200.png", "formula": "\\begin{align*} \\dot { P } ( t ) + P ( t ) A ( t ) + A ^ T ( t ) P ( t ) + C ^ T ( t ) Q ( t ) C ( t ) - P ( t ) B ( t ) R ^ { - 1 } ( t ) B ^ T ( t ) P ( t ) = 0 \\ \\forall t \\in [ t _ 0 , t _ f ] \\end{align*}"}
-{"id": "3901.png", "formula": "\\begin{align*} T _ { s , \\zeta } = \\begin{pmatrix} 1 & 0 & 0 \\\\ s & 1 & { \\zeta } ^ { \\ast } \\\\ \\zeta & 0 & I \\end{pmatrix} , \\end{align*}"}
-{"id": "7482.png", "formula": "\\begin{align*} n = 2 g - 2 + b + \\sum _ { i = 1 } ^ b k _ i . \\end{align*}"}
-{"id": "6958.png", "formula": "\\begin{align*} \\theta _ c ( V _ 1 , \\dots , V _ k ) = c ( \\theta ( V _ 1 ) , \\dots , \\theta ( V _ k ) ) , \\end{align*}"}
-{"id": "4082.png", "formula": "\\begin{align*} \\phi _ 0 ( z ) & : = { { } _ { 0 } F _ { 2 } } \\left ( { - \\atop 1 + \\alpha , \\frac { 3 } { 2 } + \\alpha } ; - z \\right ) \\end{align*}"}
-{"id": "8666.png", "formula": "\\begin{align*} R _ 0 ( \\l ) = R _ 0 ( \\l _ * ) \\sqrt { \\frac { \\l _ * } { \\l } } \\prod _ { j \\ge 1 } \\frac { ( \\l - \\l _ j ) \\sqrt { ( \\l _ * - a _ j ) ( \\l _ * - b _ j ) } } { ( \\l _ * - \\l _ j ) \\sqrt { ( \\l - a _ j ) ( \\l - b _ j ) } } . \\end{align*}"}
-{"id": "10001.png", "formula": "\\begin{align*} ( F _ { 2 m } A ) ( 2 n ) = ( O _ { 2 n } ) _ { + } \\wedge _ { O _ { 2 n - 2 m } } ( A \\wedge T ^ { 2 n - 2 m } ) . \\end{align*}"}
-{"id": "2753.png", "formula": "\\begin{align*} & \\lim _ { N \\to \\infty } \\frac { 4 } { N ^ { 2 } } \\Big ( I _ { 2 } \\big ( \\frac { 2 u } { N } , \\frac { 2 v } { N } \\big ) + I _ { 3 } \\big ( \\frac { 2 u } { N } , \\frac { 2 v } { N } \\big ) \\Big ) = \\frac { 1 } { 4 \\pi \\kappa ^ { 2 } } \\int _ { 0 } ^ { \\delta } \\frac { d t } { \\pi } \\Im \\left \\{ \\frac { 1 } { 1 + i t } e ^ { - \\frac { u } { \\kappa } ( 2 + i t ) - \\frac { v } { \\kappa } \\frac { 2 + i t } { 1 + i t } } \\right \\} . \\end{align*}"}
-{"id": "876.png", "formula": "\\begin{align*} \\chi ( \\gamma _ 1 , \\gamma _ 2 ) = \\sum _ { i \\in Q _ 0 } \\gamma _ 1 ( i ) \\gamma _ 2 ( i ) - \\sum _ { a \\in Q _ 1 } \\gamma _ 1 ( t a ) \\gamma _ 2 ( h a ) . \\end{align*}"}
-{"id": "887.png", "formula": "\\begin{gather*} \\implies \\lambda _ { Q ^ j _ 1 } ( \\phi _ u , \\phi _ v ) \\geq 0 \\lambda _ { Q _ 1 \\setminus Q ^ j _ 1 } ( \\phi _ u , \\phi _ v ) \\leq 0 ; \\\\ \\implies \\lambda ( \\phi _ u , \\phi _ v ) \\leq 0 . \\end{gather*}"}
-{"id": "5310.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\alpha _ 0 ^ { ( 1 ) } = H _ { q _ 1 } + H _ { p _ 1 } - E _ { 1 , 2 , 3 , 4 } , & \\alpha _ 1 ^ { ( 1 ) } = H _ { p _ 1 } - E _ { 5 , 6 } , & \\alpha _ 2 ^ { ( 1 ) } = H _ { q _ 1 } - E _ { 7 , 8 } , \\\\ \\alpha _ 0 ^ { ( 2 ) } = H _ { p _ 2 } + H _ { q _ 2 } - E _ { 9 , 1 0 , 1 1 , 1 2 } , & \\alpha _ 1 ^ { ( 2 ) } = H _ { p _ 2 } - E _ { 1 3 , 1 4 } , & \\alpha _ 2 ^ { ( 2 ) } = H _ { q _ 2 } - E _ { 1 5 , 1 6 } \\end{array} \\end{align*}"}
-{"id": "9257.png", "formula": "\\begin{align*} v = - \\zeta ^ 2 ( 0 ) w _ 1 ( 0 ) f \\le 0 , { \\rm o n } \\ \\partial B _ R \\cap \\Omega . \\end{align*}"}
-{"id": "797.png", "formula": "\\begin{align*} H ' _ { q ^ d , P ^ \\alpha } : = \\mathrm { G a l } \\left ( K _ { q ^ d , P ^ \\alpha } ^ { G _ { q ^ d , P } } / \\mathbb { F } _ { q ^ d } K _ { q , P ^ \\alpha } ^ { G _ { q , P } } \\right ) . \\end{align*}"}
-{"id": "9129.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } I _ { M } ( \\rho _ { 1 } ^ { ( N ) } ( t ) ) = 0 \\\\ \\frac { \\partial } { \\partial t } I _ { M } ( \\rho _ { 1 } ^ { ( N ) } ( t ) ) = 0 \\end{array} \\Leftrightarrow \\rho _ { 1 } ^ { ( N ) } ( \\mathbf { x } _ { 1 } , t ) \\equiv \\rho _ { 1 M } ^ { ( N ) } ( \\mathbf { v } _ { 1 } ) , \\right . \\end{align*}"}
-{"id": "106.png", "formula": "\\begin{align*} \\boxed { S _ k ( B ( 0 ) , B _ 1 ) = \\sum _ { j = \\frac k 2 } ^ { 2 N - \\frac k 2 } \\begin{pmatrix} j \\\\ \\frac k 2 \\end{pmatrix} \\begin{pmatrix} 2 N - j - 1 \\\\ \\frac k 2 - 1 \\end{pmatrix} B ( 0 ) ^ { 2 j } \\ , , k = 2 , 4 , \\dots , 2 N \\ , . } \\end{align*}"}
-{"id": "9953.png", "formula": "\\begin{align*} V ( x , y , z ) = f ( \\zeta ) , f ( \\zeta ) = \\sum _ { l = 1 } ^ \\infty a _ l \\zeta ^ { 2 l } \\end{align*}"}
-{"id": "2416.png", "formula": "\\begin{align*} ( h \\otimes i d ) \\circ \\Delta ( a ) = ( i d \\otimes h ) \\circ \\Delta ( a ) = h ( a ) 1 . \\end{align*}"}
-{"id": "6824.png", "formula": "\\begin{align*} \\begin{cases} v ^ 1 _ { 1 0 } \\big ( w ^ 1 _ { 1 0 } - w ^ 2 _ { 0 1 } \\big ) + v ^ 1 _ { 0 1 } \\big ( w ^ 1 _ { 0 1 } + w ^ 2 _ { 1 0 } \\big ) = 0 \\ , , \\\\ v ^ 1 _ { 1 0 } \\big ( w ^ 1 _ { 0 1 } + w ^ 2 _ { 1 0 } \\big ) + v ^ 1 _ { 0 1 } \\big ( w ^ 1 _ { 1 0 } - w ^ 2 _ { 0 1 } \\big ) = 0 \\ , . \\end{cases} \\end{align*}"}
-{"id": "4454.png", "formula": "\\begin{align*} \\bar \\iota _ 1 = ( \\phi _ { 1 2 } ) _ W \\circ \\bar \\iota _ 2 = ( \\phi _ { 1 2 } ) _ W \\circ ( \\phi _ { 2 3 } ) _ W \\circ \\bar \\iota _ 3 \\bar \\iota _ 1 = ( \\phi _ { 1 3 } ) _ W \\circ \\bar \\iota _ 3 . \\end{align*}"}
-{"id": "6952.png", "formula": "\\begin{align*} \\textbf { S } \\left ( \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ n \\textbf { x } _ i \\right ) = \\frac { 1 } { n } \\textbf { S } ( \\textbf { x } _ 1 ) + \\ldots + \\frac { 1 } { n } \\textbf { S } ( \\textbf { x } _ n ) = \\textbf { S } ( \\textbf { x } ) , \\end{align*}"}
-{"id": "1469.png", "formula": "\\begin{align*} V _ { 1 1 } = - \\varepsilon \\bar { m } \\int _ { \\mathbb { T } ^ { d } } ( \\mu ^ { 1 } - \\mu ^ { 2 } ) \\sum _ { i = 1 } ^ { d } \\left [ \\Theta _ { p _ { i } q } ( t , x , \\mu ^ { 1 } , D w ^ { 1 } ) \\frac { \\partial \\mu ^ { 1 } } { \\partial x _ { i } } - \\Theta _ { p _ { i } q } ( t , x , \\mu ^ { 2 } , D w ^ { 1 } ) \\frac { \\partial \\mu ^ { 1 } } { \\partial x _ { i } } \\right ] \\ d x , \\end{align*}"}
-{"id": "4930.png", "formula": "\\begin{align*} x z ^ { 3 n } = - z ^ { 2 n } ( y ^ n - z ^ n x ) + y ^ n z ^ { 2 n } \\end{align*}"}
-{"id": "1773.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ e ^ { \\alpha M _ { s , t } ^ { ( n ) } } \\right ] = \\mathbb { E } \\left [ \\exp \\left ( \\alpha M _ { s ; t _ { 1 } , \\cdots , t _ { n } } \\right ) \\right ] \\leq 2 \\exp \\left ( \\frac { \\alpha ^ { 2 } } { 2 } \\left [ \\gamma _ { H } ( t - s ) ^ { 2 H } + ( t - s ) \\right ] \\right ) . \\end{align*}"}
-{"id": "3945.png", "formula": "\\begin{align*} H _ { i } ( x , m , q _ { i } ) = \\inf _ { v _ { i } } \\big [ f _ { i } ( x , m , v _ { i } ) + q _ { i } . g _ { i } ( x , m , v _ { i } ) \\big ] \\ , , \\end{align*}"}
-{"id": "2507.png", "formula": "\\begin{align*} h _ { r } ^ { \\langle r \\rangle } = { r + 2 \\choose r + 1 } = r + 2 > h _ { r + 1 } . \\end{align*}"}
-{"id": "576.png", "formula": "\\begin{align*} u - u ' = \\alpha ( 1 + p \\alpha _ 1 ) - \\alpha ( 1 + p \\alpha _ 1 ' ) = p \\alpha ( \\alpha _ 1 - \\alpha _ 1 ' ) , \\end{align*}"}
-{"id": "10048.png", "formula": "\\begin{align*} \\begin{aligned} \\dot u & = \\frac { 1 } { 4 } ( u + v ) ^ 3 \\left ( u + \\O ( ( u + v ) ^ 3 ) \\right ) , \\\\ \\dot v & = - \\frac { 1 } { 4 } ( u + v ) ^ 3 \\left ( v + \\O ( ( u + v ) ^ 3 ) \\right ) , \\\\ \\dot { \\tilde { z } } & = \\frac { 1 } { 4 } ( u + v ) ^ 2 ( u - v ) \\tilde z + \\O ( ( u + v ) ^ 5 ) , \\\\ \\dot \\phi & = \\omega , \\end{aligned} \\end{align*}"}
-{"id": "8730.png", "formula": "\\begin{align*} E ( T ^ * _ { m , N } ) = ( N - 1 ) \\sum _ { j \\leq m } \\hat { \\nu } ( j ) \\sum _ { k = j } ^ { N - 1 } { 1 \\over N - k } . \\end{align*}"}
-{"id": "2858.png", "formula": "\\begin{align*} y ^ 2 - x z + 4 t y z + ( 6 t + 2 ) z ^ 2 = 0 . \\end{align*}"}
-{"id": "2239.png", "formula": "\\begin{align*} H = \\Omega + | g \\rangle \\langle h | \\end{align*}"}
-{"id": "6277.png", "formula": "\\begin{align*} \\sqrt { \\frac { T } { \\theta \\sigma ^ 2 _ H } } ( \\hat { \\theta } _ T - \\theta ) = \\frac { I _ 2 ( f _ T ) } { I _ 2 ( g _ T ) + b _ T } , \\end{align*}"}
-{"id": "8760.png", "formula": "\\begin{align*} \\nabla _ k \\overline { g } _ { i j } = 2 \\psi _ k \\overline { g } _ { i j } + \\psi _ i \\overline { g } _ { j k } + \\psi _ j \\overline { g } _ { i k } \\ , . \\end{align*}"}
-{"id": "128.png", "formula": "\\begin{align*} \\frac 1 N \\sum _ { j = 0 } ^ { 2 N - 1 } { ( 1 - x _ j ^ 2 ) } ^ { h - 1 } & \\le \\int _ { - 1 } ^ 1 ( 1 - x ^ 2 ) ^ { h - 1 } \\ , d x ~ + ~ \\frac 1 N \\\\ & = \\frac { 2 ^ { 2 h - 1 } \\cdot ( ( h - 1 ) ! ) ^ 2 } { ( 2 h - 1 ) ! } ~ + ~ \\frac 1 N \\ , . \\end{align*}"}
-{"id": "9780.png", "formula": "\\begin{align*} D ^ \\alpha q ( Z _ \\infty ) = D ^ \\alpha q _ \\infty ( 0 ) , \\end{align*}"}
-{"id": "1131.png", "formula": "\\begin{align*} \\det ( - B _ j ) & = \\mathbb { P } ^ { C U E ( n ) } ( \\theta _ i \\not \\in [ 0 , 2 \\pi \\rho _ { s c } ( y _ j ) a _ j ] , 1 \\leq i \\leq n ) \\\\ & = D _ n ( \\pi \\rho _ { s c } ( y _ j ) a _ j ) = D _ n \\left ( a _ j \\sqrt { 4 - y _ j ^ 2 } / 2 \\right ) , \\end{align*}"}
-{"id": "7070.png", "formula": "\\begin{align*} S ( x ) = x , S ( y ) = - y . \\end{align*}"}
-{"id": "1315.png", "formula": "\\begin{align*} \\rho \\big [ Z , \\delta _ { x } \\big ] = \\rho \\big [ \\ 1 _ { x } Z , \\delta _ { x } \\big ] = \\rho \\big [ Z ( x ) \\ 1 , \\delta _ { x } \\big ] = Z ( x ) + \\rho [ 0 , \\delta _ { x } ] = Z ( x ) . \\end{align*}"}
-{"id": "5321.png", "formula": "\\begin{align*} \\frac { d b _ j } { d t } = & \\frac { 1 } { 2 } \\lambda b _ j = \\frac { 1 } { 2 } ( a _ 0 + a _ 1 + \\cdots + a _ 5 ) b _ j , ( j = 1 , 2 ) \\end{align*}"}
-{"id": "333.png", "formula": "\\begin{align*} \\begin{array} { c c c } A = \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ - 2 & 0 & 1 & 0 \\\\ 4 m + 2 & 1 & - ( 2 m + 1 ) & 0 \\\\ 4 & - 2 & - 2 & 1 \\end{pmatrix} & & B = \\begin{pmatrix} 1 & - m & - ( 2 m + 1 ) \\\\ 0 & 0 & 1 \\\\ 0 & 1 & 2 \\end{pmatrix} . \\end{array} \\end{align*}"}
-{"id": "4837.png", "formula": "\\begin{align*} r _ 0 = r _ 1 , \\ r _ 1 = r _ 2 , \\ r _ 2 = r _ 3 , \\end{align*}"}
-{"id": "5839.png", "formula": "\\begin{align*} \\eta : M _ n \\rightarrow F ^ { n } , f ( x ) = x ^ { n } + b _ { n - 1 } x ^ { n - 1 } + \\cdots + b _ 0 \\mapsto ( b _ { n - 1 } , \\cdots , b _ 0 ) . \\end{align*}"}
-{"id": "8949.png", "formula": "\\begin{align*} \\mathbb { P } ( Y = 1 \\ , | \\ , X ) \\ = \\ \\frac { p \\frac { \\dd P _ { m _ 1 } } { \\dd P _ { m _ 0 } } ( X ) } { p \\frac { \\dd P _ { m _ 1 } } { \\dd P _ { m _ 0 } } ( X ) + ( 1 - p ) } \\ = \\ \\left ( 1 + \\frac { 1 - p } { p } \\frac { \\dd P _ { m _ 0 } } { \\dd P _ { m _ 1 } } ( X ) \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "6776.png", "formula": "\\begin{align*} ( X + Z ) ^ * ( X + Z ) = V ( \\Lambda _ { \\sigma } ^ 2 + \\Lambda _ { \\sigma } \\check Z + \\check Z ^ * \\Lambda _ { \\sigma } + \\check Z ^ * \\check Z ) V ^ * . \\end{align*}"}
-{"id": "6328.png", "formula": "\\begin{align*} q _ t = q ^ { 2 } ( q _ { x x } - \\nu ^ { - \\frac { 1 } { 2 } } f ) Q _ T , \\end{align*}"}
-{"id": "2721.png", "formula": "\\begin{align*} l _ { \\min } : = \\min \\Big \\{ \\frac { 1 } { 2 } \\big ( \\frac { 1 } { \\tau } + \\tau \\big ) , \\frac { \\sigma _ { 1 } } { \\tau } , \\ldots , \\frac { \\sigma _ { n } } { \\tau } \\Big \\} , \\end{align*}"}
-{"id": "8769.png", "formula": "\\begin{align*} \\frac { \\widetilde { \\kappa } } { ( n - 3 ) ( n - 2 ) } = B ^ 2 _ 2 - C B ^ 2 \\ , . \\end{align*}"}
-{"id": "5669.png", "formula": "\\begin{align*} { \\rm I I } ^ { p ' } & \\leq c \\sum _ { j = 0 } ^ { n - 1 } \\int _ { t _ j } ^ { t _ { j + 1 } } \\| \\bar E ( t _ n - t ) - B _ { n - j } P _ h \\| _ { \\mathcal { L } _ 2 ^ 0 } ^ { p ' } \\d t \\\\ & \\leq c \\| A ^ { - \\frac { s } { 2 } } \\| _ { \\mathcal { L } _ 2 ^ 0 } ^ { p ' } \\sum _ { j = 0 } ^ { n - 1 } \\int _ { t _ j } ^ { t _ { j + 1 } } \\| A ^ { \\frac { s } { 2 } } ( \\bar E ( t _ { n } - t ) - B _ { n - j } P _ { h } ) \\| ^ { p ' } \\ , \\d s . \\end{align*}"}
-{"id": "7727.png", "formula": "\\begin{align*} \\norm { F _ { t } u - u } _ { 2 } = \\norm { \\sum _ { i = 1 } ^ l ( F _ { t } ^ i ( \\rho _ { i } u ) - \\rho _ { i } u ) } _ { 2 } \\leq \\sum _ { i = 1 } ^ l \\norm { F _ { t } ^ i ( \\rho _ { i } u ) - \\rho _ { i } u } _ { 2 } \\longrightarrow 0 . \\end{align*}"}
-{"id": "1340.png", "formula": "\\begin{align*} A _ { > R } ^ { ( l ) } & = \\bigoplus _ { i \\in I ^ { ( l ) } } \\overline { f _ i ^ { \\frac { 1 } { 2 } } C ^ * _ r ( \\mathcal { H } _ { \\widehat { B } _ i } ) f _ i ^ { \\frac { 1 } { 2 } } } \\\\ A _ { > R } & = \\bigoplus _ { l = 0 } ^ { 5 ( d + 1 ) - 1 } A _ { > R } ^ { ( l ) } = \\bigoplus _ { i \\in I } \\overline { f _ i ^ { \\frac { 1 } { 2 } } C ^ * _ r ( \\mathcal { H } _ { \\widehat { B } _ i } ) f _ i ^ { \\frac { 1 } { 2 } } } \\end{align*}"}
-{"id": "1221.png", "formula": "\\begin{gather*} J ( f , g ) ( \\xi ) = \\int _ \\eta e ^ { - 3 i \\Phi ( \\xi , \\eta ) } \\bar f ( \\eta - \\xi ) g ( \\eta ) d \\eta , \\end{gather*}"}
-{"id": "6461.png", "formula": "\\begin{align*} \\ell ( e ) & p _ { v } = \\delta _ { v , t ( e ) } e \\\\ \\ell ( e ) & e _ { 1 } \\otimes \\cdots \\otimes e _ { n } = e \\otimes e _ { 1 } \\otimes \\cdots \\otimes e _ { n } . \\end{align*}"}
-{"id": "5765.png", "formula": "\\begin{align*} X _ { \\mathcal { R } ^ \\varphi / K } ( T ) & = X / _ K ( \\mathcal { R } ^ \\varphi \\otimes _ K T ) \\\\ & = X / _ K ( \\mathcal { R } \\otimes _ A T ) \\\\ & = X / _ A ( \\mathcal { R } \\otimes _ A T ) . \\end{align*}"}
-{"id": "6648.png", "formula": "\\begin{align*} M _ { \\zeta , q } ( x ) = e _ q ( a \\lvert x \\rvert ^ 2 ) \\zeta ( x ) . \\end{align*}"}
-{"id": "8621.png", "formula": "\\begin{align*} \\mathcal { M } _ 0 = \\left \\{ p = 2 c _ 0 \\frac { u ^ 2 + z ^ 2 } { u ^ 4 + c _ 0 ^ 2 } - 2 c _ 0 , \\ \\ q = 0 , \\ \\ ( u , z ) \\in \\mathbb { R } ^ 2 , \\ u \\neq 0 \\right \\} . \\end{align*}"}
-{"id": "9125.png", "formula": "\\begin{align*} \\rho ^ { ( + ) ( N ) } ( \\mathbf { x } ^ { ( + ) } ( t _ { i } ) , t _ { i } ) = \\rho ^ { ( N ) } ( \\mathbf { x } ^ { ( + ) } ( t _ { i } ) , t _ { i } ) \\end{align*}"}
-{"id": "4191.png", "formula": "\\begin{align*} \\left ( \\frac { A _ n ^ { ( 2 ) } ( z ) - A _ n ^ { ( 2 ) } ( 0 ) } { n ^ 6 z } \\right ) ^ 2 & = \\mathcal { O } \\left ( n ^ { - 1 } \\right ) \\\\ \\frac { A _ n ^ { ( 2 ) } ( z ) - A _ n ^ { ( 2 ) } ( 0 ) } { n ^ 6 z } \\frac { A _ n ^ { ( 1 ) } ( 0 ) A _ n ^ { ( 1 ) } ( z ) A _ n ^ { ( 1 ) } ( 0 ) } { n ^ 9 z ^ 3 } & = \\mathcal { O } \\left ( n ^ { - \\frac { 3 } { 2 } } \\right ) \\end{align*}"}
-{"id": "4497.png", "formula": "\\begin{align*} H ( \\nabla ^ 2 v , \\nabla v , v ) & = e ^ { 2 ( v - \\underline v ) } \\big ( t \\Psi ( \\nabla v , v , x ) + ( 1 - t ) \\underline { \\psi } ( x ) \\big ) \\quad \\textrm { i n } \\ , \\Omega \\\\ v & = \\ln \\varphi \\quad \\textrm { o n } \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "3196.png", "formula": "\\begin{align*} Q _ 1 - Q _ 2 & = \\frac { 1 } { i } [ \\varGamma , P _ 1 - P _ 2 ] \\end{align*}"}
-{"id": "8089.png", "formula": "\\begin{align*} n a _ n - ( 2 n - 1 ) a _ { n - 1 } + ( n - 2 ) a _ { n - 2 } = 0 \\ ; \\ ; n \\ge 2 . \\end{align*}"}
-{"id": "8899.png", "formula": "\\begin{align*} x _ i & = \\begin{cases} 0 , 0 \\leq i \\leq n / 2 \\\\ 1 , n / 2 < i \\leq n \\end{cases} \\intertext { f o r t h e n u l l d i s t r i b u t i o n , a n d t h e s h i f t e d b a l a n c e d p a r t i t i o n $ y = [ y _ i ] _ { i = 1 } ^ n $ w i t h } y _ i & = \\begin{cases} 0 , s / 2 < i \\leq n / 2 + s / 2 \\\\ 1 , i \\in ( n / 2 , n ] \\cup [ 0 , s / 2 ] \\end{cases} \\end{align*}"}
-{"id": "9753.png", "formula": "\\begin{align*} \\lambda _ * = N ( \\rho , q ) \\quad \\rho \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "4306.png", "formula": "\\begin{align*} \\textup { s a t } ( n , H , C _ k ) = 0 \\end{align*}"}
-{"id": "3372.png", "formula": "\\begin{align*} H ( s ) : = - Q ( 1 - s ) , 0 \\le s < 1 . \\end{align*}"}
-{"id": "7264.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { k \\in \\Z } \\hat f \\left ( k \\right ) e ^ { i x k } . \\end{align*}"}
-{"id": "9837.png", "formula": "\\begin{align*} & \\qquad \\left \\{ \\begin{array} { r c l l } w _ 2 ( x ' , 0 ) & \\geq & \\psi ( x ' ) & \\R ^ { n - 1 } \\\\ ( - \\Delta ) ^ s w _ 2 & = & 0 & \\R ^ { n } \\setminus \\{ ( x ' , 0 ) : w _ 2 ( x ' , 0 ) = \\psi ( x ' ) \\} \\\\ ( - \\Delta ) ^ s w _ 2 & \\leq & 0 & \\R ^ { n } \\\\ \\lim _ { | x | \\to \\infty } w _ 2 ( x ) & = & 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "6117.png", "formula": "\\begin{align*} \\Psi ^ t _ { { i _ \\ell , j _ \\ell , k _ \\ell + 1 } } = A ^ t _ { { i _ \\ell , j _ \\ell , k _ \\ell + 1 } } - B ^ t _ { i _ \\ell , j _ \\ell + 1 , k _ \\ell } , \\end{align*}"}
-{"id": "7754.png", "formula": "\\begin{align*} \\sum _ { \\alpha \\in \\Lambda } \\| x \\| ^ 2 = [ \\langle x , x \\rangle \\xi , \\xi ] = \\Big | [ \\langle x , x \\rangle \\xi , \\xi ] \\Big | \\leq \\| [ \\langle x , x \\rangle \\xi , \\xi ] \\| \\leq \\| x \\| ^ 2 , \\end{align*}"}
-{"id": "5449.png", "formula": "\\begin{align*} V _ 0 \\ = \\ \\hat V _ 0 . \\end{align*}"}
-{"id": "7240.png", "formula": "\\begin{align*} \\frac { \\phi ( 0 ) - \\phi ( x ) } { | x | } = \\phi ' ( \\xi ) | \\xi | \\ll 1 . \\end{align*}"}
-{"id": "6936.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow T ^ { * } } \\left ( \\| \\rho \\| _ { L ^ \\infty ( 0 , T ; L ^ \\infty ) } + \\| \\theta \\| _ { L ^ \\infty ( 0 , T ; L ^ \\infty ) } \\right ) = \\infty , \\end{align*}"}
-{"id": "457.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } e ^ { - \\alpha t } Z ^ { \\phi } ( t ) = Y _ \\infty m ^ { \\phi } _ { \\infty } \\end{align*}"}
-{"id": "7458.png", "formula": "\\begin{align*} \\displaystyle \\left \\lvert D ( \\lambda \\mapsto f _ { \\pi _ \\lambda } ( g ) ) _ { \\lambda = 0 } \\right \\rvert \\ll \\Xi ^ G ( g ) \\sigma ( g ) ^ { \\deg ( D ) } , \\ ; \\ ; \\ ; \\tau \\in \\Pi _ 2 ( M ) , g \\in G ( F ) . \\end{align*}"}
-{"id": "7409.png", "formula": "\\begin{align*} \\mathcal { Z } _ U = \\bigcup _ { i = 1 } ^ \\infty \\mathcal { Z } _ i ^ { U } . \\end{align*}"}
-{"id": "73.png", "formula": "\\begin{align*} \\sigma ^ + _ { 1 } - \\sigma ^ + _ { - 1 } & = \\sigma _ 1 ^ - - \\sigma _ { - 1 } ^ - + g ' ( s ) \\left ( \\sigma _ { - 1 } ^ + - \\sigma _ { 1 } ^ + + \\sigma _ { - 1 } ^ - - \\sigma _ { 1 } ^ - \\right ) \\delta \\ , , \\end{align*}"}
-{"id": "159.png", "formula": "\\begin{align*} X _ { V } = \\sum _ { i = 1 } ^ N v ^ i ( { \\bf x } ) \\frac { \\partial } { \\partial y ^ i } \\end{align*}"}
-{"id": "3069.png", "formula": "\\begin{align*} \\chi _ \\lambda ( s ) = \\chi ( \\lambda s ) , \\end{align*}"}
-{"id": "852.png", "formula": "\\begin{align*} \\overline { N } _ { A } ( \\bar { x } ) = \\bigcup _ { \\| u \\| = 1 } \\overline { N } _ { A } ( \\bar { x } ; u ) \\cup N _ { A } ( \\bar { x } ) . \\end{align*}"}
-{"id": "9904.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Y } } g ( y _ { n } ) P ^ { \\mu } ( d y _ { n } | X _ { n } = x ) = \\int _ { \\mathcal { Z } } g ( h _ { x } ( z ) ) Q ( d z ) \\end{align*}"}
-{"id": "2426.png", "formula": "\\begin{align*} f ( M Z ) = \\det ( C Z + D ) ^ k f ( Z ) \\mbox { f o r } M = \\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix} \\in \\Gamma _ g \\end{align*}"}
-{"id": "4658.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } t _ * ( a _ { n _ k } , b _ { n _ k } , c _ { n _ k } ) = t _ * ( 1 , \\beta , 1 ) = \\beta , \\end{align*}"}
-{"id": "1890.png", "formula": "\\begin{align*} \\mathcal { B } - \\mathcal { C } \\leq \\frac { \\pi } { 3 } \\chi ^ 2 \\sum _ { j = 1 } ^ \\infty \\frac { A _ j ^ 2 } { Z _ j ^ 4 } = \\frac { \\pi } { 3 } F _ 4 ( \\chi ) . \\end{align*}"}
-{"id": "9088.png", "formula": "\\begin{align*} u _ n ( x ) - u _ n ( y ) = \\begin{cases} \\varphi _ n ( x ) u _ n ( x ) & x \\in B _ n ^ c \\cap B _ { 2 n } , y \\in B _ { 2 n } ^ c , \\\\ - \\varphi _ n ( y ) u _ n ( y ) & y \\in B _ n ^ c \\cap B _ { 2 n } , x \\in B _ { 2 n } ^ c , \\\\ \\varphi _ n ( x ) u _ n ( x ) - \\varphi _ n ( x ) u _ n ( y ) & x , y \\in B _ n ^ c \\cap B _ { 2 n } , \\\\ 0 & x , y \\in B _ { 2 n } ^ c . \\end{cases} \\end{align*}"}
-{"id": "2981.png", "formula": "\\begin{align*} ( p _ { 1 , 5 } + p _ { 4 , 5 } ) ( 9 ) & = 4 , 9 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 4 + 1 + 1 + 1 + 1 + 1 = 4 + 4 + 1 , \\\\ ( p _ { 1 , 5 } + p _ { 4 , 5 } ) ( 8 ) & = 3 , 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 4 + 1 + 1 + 1 + 1 = 4 + 4 , \\\\ ( p _ { 1 , 5 } + p _ { 4 , 5 } ) ( 5 ) & = 2 , 1 + 1 + 1 + 1 + 1 = 4 + 1 , \\\\ ( p _ { 1 , 5 } + p _ { 4 , 5 } ) ( 2 ) & = 1 , 1 + 1 . \\end{align*}"}
-{"id": "1785.png", "formula": "\\begin{align*} c _ { p , r } \\Big ( \\bigcup _ { t \\in [ 0 , 1 ] } A _ { k , M } ^ { t } \\Big ) = 0 \\end{align*}"}
-{"id": "4613.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } & = \\frac { \\partial ^ 2 u } { \\partial x ^ 2 } \\Omega \\times ( 0 , T ] , \\\\ u & = 0 \\mathbf { x } \\in \\partial \\Omega , \\\\ u ( 0 , \\mathbf { x } ) & = s ( \\mathbf { x } ) , \\end{align*}"}
-{"id": "8653.png", "formula": "\\begin{align*} \\varphi _ { 1 , 0 } ( 0 ) = \\frac { \\pi } { 2 } . \\end{align*}"}
-{"id": "5587.png", "formula": "\\begin{align*} & \\mathcal { L } _ { \\lambda } q ( t ) = \\frac { 1 } { \\pi } \\int _ 0 ^ { + \\infty } e ^ { \\lambda ( i z ^ 2 t - i z x ) } z \\ , \\widehat { q } ( \\lambda z ^ 2 ) d z , \\lambda = \\pm 1 , \\\\ \\intertext { a n d } & \\widehat { q } ( \\tau ) = \\mathcal { F } ( \\chi _ { ( 0 , + \\infty ) } f ) ( \\tau ) = \\int _ { 0 } ^ { + \\infty } e ^ { - i \\tau t } f ( t ) d t , \\end{align*}"}
-{"id": "2930.png", "formula": "\\begin{align*} \\theta ^ f = \\theta + \\{ \\textrm { c o b o u n d a r y } \\} . \\end{align*}"}
-{"id": "3079.png", "formula": "\\begin{align*} | I | & = | \\int _ { \\frac { 1 } { | y | } \\leq | x | < \\frac { 1 } { \\beta } , ~ | x - z | \\leq \\frac { 1 } { \\beta } } ( \\frac { x } { | x | ^ { n + 1 - \\beta } } - \\frac { x - z } { | x - z | ^ { n + 1 - \\beta } } ) e ^ { 2 \\pi i x \\cdot y } \\ , d x | \\\\ & \\le C ( \\frac { 2 ^ \\beta - 1 } { \\beta } \\beta ^ { - \\beta } + \\frac { 1 - 2 ^ { - \\beta } } { \\beta } \\beta ^ { - \\beta } + \\frac { \\beta ^ { - \\beta } } { 1 - \\beta } ) \\end{align*}"}
-{"id": "9569.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( q ) _ { n - 1 } ( q ) _ n ( - q ) _ { N - n } } { ( - q ) _ { n - 1 } ( - q ) _ { N } } \\frac { ( - q ) ^ n } { 1 - q ^ { 2 n } } = \\sum _ { n = 1 } ^ { \\infty } \\frac { n ( - q ) ^ n ( 1 - q ^ { N n } ) } { 1 - q ^ { n } } . \\end{align*}"}
-{"id": "7199.png", "formula": "\\begin{align*} - \\mu \\phi + L _ r \\phi + \\frac { 1 } { 2 } \\phi ^ 2 = B , \\end{align*}"}
-{"id": "5236.png", "formula": "\\begin{align*} F ^ { - 1 } ( 1 - s ) = c ( 1 + p ( x ) ) x ^ { - 1 / \\alpha } \\exp ( \\int _ { s } ^ { 1 } s ^ { - 1 } \\ell ( s ) d \\lambda ( s ) ) , \\end{align*}"}
-{"id": "174.png", "formula": "\\begin{align*} \\begingroup \\makeatletter \\def \\f @ s i z e { 8 } \\check @ m a t h f o n t s \\pi _ { T , X _ { V } , Y _ { V } , c } ( { \\bf x } , { \\bf y } ) = \\left ( \\begin{array} { c | c } 0 & \\pi ( { \\bf x } ) \\\\ \\hline \\pi ( { \\bf x } ) & \\sum _ { s = 1 } ^ { N } \\dfrac { \\partial \\pi } { \\partial x ^ s } ( { \\bf x } ) y ^ s + \\lambda c ( { \\bf x } ) \\left ( v ( { \\bf x } ) w ^ { \\top } ( { \\bf x } ) - w ( { \\bf x } ) v ^ { \\top } ( { \\bf x } ) \\right ) \\end{array} \\right ) , \\endgroup \\end{align*}"}
-{"id": "4890.png", "formula": "\\begin{align*} f \\circ \\lambda _ A \\circ ( s \\otimes A ) & = \\lambda _ { S B } \\circ ( s \\otimes S \\otimes B ) \\circ ( S \\otimes f ) \\\\ & = \\lambda _ { S B } \\circ ( s \\otimes S \\otimes B ) \\circ ( S \\otimes g ) = g \\circ \\lambda _ A \\circ ( s \\otimes A ) \\text , \\end{align*}"}
-{"id": "7101.png", "formula": "\\begin{align*} \\begin{aligned} \\sigma ^ \\prime - s + 1 - \\gamma ( \\sigma - 1 ) - \\frac { 1 } { 2 } ( 1 - \\gamma ) & < 0 ~ , \\\\ \\nu - s - \\gamma ( \\sigma ^ \\prime - 1 ) & < 0 ~ , \\\\ ( 1 - \\gamma ) ( \\nu - 1 ) + 1 - \\sigma ^ \\prime & < 0 ~ . \\end{aligned} \\end{align*}"}
-{"id": "8695.png", "formula": "\\begin{align*} j c _ { h , i , j } = 8 ( h + 1 ) ( h + 2 ) ( 2 h + p ' + 2 ) c _ { h + 2 , i , j - 2 } + 8 ( h + 1 ) ( i + 1 ) ( 2 i + p '' ) c _ { h + 1 , i + 1 , j - 2 } \\end{align*}"}
-{"id": "7640.png", "formula": "\\begin{align*} ( G ^ { - 1 } K ) ^ \\ast \\psi _ 0 \\ = \\ \\psi _ 0 \\end{align*}"}
-{"id": "3037.png", "formula": "\\begin{align*} 0 < a _ 1 , \\dots , a _ d < \\infty a _ 1 + \\dots + a _ d = d . \\end{align*}"}
-{"id": "7100.png", "formula": "\\begin{align*} \\nu > 2 > s > 1 ~ , \\sigma = \\nu - 1 - , \\sigma ^ \\prime > \\sigma , ~ \\gamma \\in ( 0 , 1 ) ~ . \\end{align*}"}
-{"id": "5563.png", "formula": "\\begin{align*} \\lambda _ T ^ { \\alpha } & : = \\prod _ { i = 0 } ^ { n } ( \\lambda ^ T _ i ) ^ { \\alpha ( i ) } . \\end{align*}"}
-{"id": "9617.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { \\infty } \\frac { q ^ { m } } { 1 - q ^ { m } } \\frac { ( - 1 ) _ m } { ( - q ^ N ) _ m } - \\sum _ { m = N } ^ { \\infty } \\frac { 2 q ^ m } { 1 - q ^ { 2 m } } = \\sum _ { m = 1 } ^ { N - 1 } \\frac { q ^ m } { 1 - q ^ m } ( - 1 ) _ m , \\end{align*}"}
-{"id": "2436.png", "formula": "\\begin{align*} N _ { \\widetilde { \\varphi _ \\ell ^ { H _ 1 } } } ( t ) & = \\sum _ { 0 < j < \\ell , j \\equiv 0 \\pmod { 4 } } \\frac { ( - 1 ) ^ { \\frac { \\ell } { 4 } } } { ( - 1 ) ^ { \\ell / 4 } + 2 ^ { ( \\ell - 4 ) / 2 } } t ^ { j - 4 } + t ^ { \\ell - 4 } \\\\ & \\equiv \\frac { ( - 1 ) ^ { \\ell / 4 } } { ( - 1 ) ^ { \\ell / 4 } + 2 ^ { ( \\ell - 4 ) / 2 } } \\frac { 1 } { 1 - t ^ 4 } + \\left ( 1 - \\frac { ( - 1 ) ^ { \\ell / 4 } } { ( - 1 ) ^ { \\ell / 4 } + 2 ^ { ( \\ell - 4 ) / 2 } } \\right ) t ^ { \\ell - 4 } \\pmod { t ^ { \\ell - 3 } } . \\end{align*}"}
-{"id": "7513.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\frac { \\log \\mu ( B ( z , r ) ) } { \\log r } = \\alpha , \\end{align*}"}
-{"id": "9214.png", "formula": "\\begin{align*} e ( G ' ) & \\leq \\max \\left \\{ f ( k ) , f \\left ( \\left \\lceil \\frac { k + 1 } { 2 } \\right \\rceil \\right ) \\right \\} \\\\ & = \\max \\left \\{ \\binom { k } { 2 } , \\binom { \\left \\lceil \\frac { k + 1 } { 2 } \\right \\rceil } { 2 } + \\left \\lfloor \\frac { k - 1 } { 2 } \\right \\rfloor \\left ( n - \\left \\lceil \\frac { k + 1 } { 2 } \\right \\rceil \\right ) \\right \\} . \\end{align*}"}
-{"id": "2945.png", "formula": "\\begin{align*} \\mathcal { A } _ p ( X ) = \\{ [ \\Omega ] _ A / \\Omega \\mbox { s t r i c t l y w e a k l y p o s i t i v e s u c h t h a t } \\partial \\bar { \\partial } \\Omega = 0 \\} \\subset H _ A ^ { p , p } ( X , \\mathbb { R } ) \\subset H _ A ^ { p , p } ( X , \\mathbb { C } ) , \\end{align*}"}
-{"id": "359.png", "formula": "\\begin{align*} A : = \\min _ { k \\in \\{ 1 , \\ldots , K \\} } \\max _ { t \\in [ t _ { k - 1 } , t _ k ] } a ( t ) t > 0 \\quad \\textrm { a n d } M : = \\lambda _ { 1 } ^ { 1 / 2 } A / C _ { 1 } | \\Omega | ^ { 1 / 2 } t _ { \\ast } ^ { p - 1 } . \\end{align*}"}
-{"id": "1889.png", "formula": "\\begin{align*} f ( y ) \\leq \\frac { 1 } { 1 2 } \\int _ 0 ^ \\infty e ^ { - y t } t ^ 2 d t = \\frac { 1 } { 6 y ^ 3 } . \\end{align*}"}
-{"id": "5552.png", "formula": "\\begin{align*} C _ { \\varepsilon } ( f , g , h ) = h \\circ ( f \\prec g ) - f ( h \\circ g ) . \\end{align*}"}
-{"id": "5132.png", "formula": "\\begin{align*} M _ { k _ j } : = \\begin{pmatrix} - ( k _ j + \\lambda ) & - \\nu + \\mu \\\\ \\nu + \\mu & k _ j + \\lambda \\end{pmatrix} ; \\end{align*}"}
-{"id": "9107.png", "formula": "\\begin{align*} \\phi ( x ) = K \\Upsilon ( R x ) \\end{align*}"}
-{"id": "3990.png", "formula": "\\begin{align*} \\{ e _ { k , N } , e _ { k , N } \\} = \\{ V _ N ^ * V _ N E _ { k , 1 } , E _ { k , 1 } \\} \\end{align*}"}
-{"id": "4783.png", "formula": "\\begin{align*} M _ 4 = \\Big \\{ z _ 2 ( T ) \\in Y \\ \\Big | \\ z _ 2 \\mbox { i s t h e s o l u t i o n t o } ( \\ref { 6 2 1 } ) \\mbox { w i t h s o m e } v ( \\cdot ) \\in \\mathcal { U } [ 0 , T ] \\Big \\} . \\end{align*}"}
-{"id": "9175.png", "formula": "\\begin{align*} S _ { t } : \\dot { \\gamma } \\left ( t \\right ) ^ { \\perp } \\longrightarrow \\dot { \\gamma } \\left ( t \\right ) ^ { \\perp } , S _ { t } ( v ) = J _ { v } ^ { \\prime } \\left ( t \\right ) , \\end{align*}"}
-{"id": "4146.png", "formula": "\\begin{align*} \\varphi _ { 2 , - } ( x ) = \\varphi _ { 1 , + } ( x ) + \\varphi _ { 2 , + } ( x ) , x \\in \\Delta _ 1 . \\end{align*}"}
-{"id": "738.png", "formula": "\\begin{align*} \\mathbb { A } _ { s , t } = A _ { s , t } ^ { ( n - m , n - m ) } ( n - m + q ) , \\end{align*}"}
-{"id": "8571.png", "formula": "\\begin{align*} \\begin{array} { r c l c r c l c r c l } \\{ e _ 2 ^ + , e _ 2 ^ - \\} & = & h + z , & & \\{ e _ 3 ^ + , e _ 3 ^ - \\} & = & h - z , & & \\{ e _ 2 ^ \\mp , e _ 3 ^ \\pm \\} & = & \\pm 2 e _ 1 ^ \\pm , \\\\ \\{ e _ 2 ^ \\pm , e _ 2 ^ \\pm \\} & = & 0 , & & \\{ e _ 3 ^ \\pm , e _ 3 ^ \\pm \\} & = & 0 , & & \\{ e _ 2 ^ \\pm , e _ 3 ^ \\pm \\} & = & 0 . \\end{array} \\end{align*}"}
-{"id": "10058.png", "formula": "\\begin{align*} \\dot u = - \\frac { 1 } { 4 } u ^ 4 + b ( \\theta _ n ^ 0 , G _ n ^ 0 ) u ^ 7 , \\end{align*}"}
-{"id": "5189.png", "formula": "\\begin{align*} \\Lambda ( x ) = \\exp ( - \\exp ( - x ) ) , x \\in \\mathbb { R } , \\end{align*}"}
-{"id": "9460.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\frac { 1 } { \\sigma _ 1 ( m ) } \\left ( J _ { N , 1 } ( T _ m . D _ f ) - e \\left ( \\left ( T _ m . D _ f \\right ) _ { \\ ; > 1 } \\right ) \\right ) = h _ f \\lim _ { \\epsilon '' \\to 0 } \\int _ { \\mathcal { F } _ { N } ( \\epsilon '' ) } J _ { N , 1 } ( z ) d \\mu ( z ) . \\end{align*}"}
-{"id": "9708.png", "formula": "\\begin{align*} T _ { \\textbf { d } } ( \\partial D ) = - \\chi ( D ) = - \\chi ( D _ c ) + b = - 1 . \\end{align*}"}
-{"id": "9395.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } \\int _ { \\mathbb { R } ^ N } \\Phi ( u _ { \\pm } ( x ) - u _ { \\pm } ( y ) ) ( u _ { \\pm } ( x ) - u _ { \\pm } ( y ) ) K ( x , y ) d x d y = \\int _ \\Omega | u _ { \\pm } | ^ { p ^ { \\ast } _ { s } } d x + \\lambda \\int _ { \\Omega } f ( x , u ) u _ { \\pm } d x \\end{align*}"}
-{"id": "1319.png", "formula": "\\begin{align*} P _ 0 \\big \\{ x : Z ( x ) \\le z \\big \\} = P _ 0 \\big \\{ x : \\varPsi ( Z _ 0 ( x ) ) \\le z \\big \\} = P _ 0 \\big \\{ x : Z _ 0 ( x ) \\le \\varPsi ^ { - 1 } ( z ) \\big \\} = \\varPsi ^ { - 1 } ( z ) . \\end{align*}"}
-{"id": "7710.png", "formula": "\\begin{align*} \\pi \\int \\limits _ { - \\frac { x } { 2 \\pi } } ^ { \\frac { x } { 2 \\pi } } \\hat { g } ( t ) ^ 2 \\ , \\mathrm { d } t \\leq \\pi \\int \\limits _ { - \\infty } ^ { \\infty } \\hat { g } ( t ) ^ 2 \\ , \\mathrm { d } t = \\pi \\norm { \\hat { g } } _ { 2 } ^ 2 \\stackrel { ( \\ast ) } { = } \\pi \\norm { { g } } _ { 2 } ^ 2 = \\pi f ( 0 ) = 1 , \\end{align*}"}
-{"id": "557.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\| u ( t ) - V ( t ) ( f _ \\infty , g _ \\infty ) \\| _ { L ^ \\infty _ t \\dot { B } ^ { \\frac { n } { 2 } } _ { 2 , 1 } } + \\lim _ { t \\to \\infty } \\| \\partial _ t u ( t ) - \\partial _ t V ( t ) ( f _ \\infty , g _ \\infty ) \\| _ { L ^ \\infty _ t \\dot { B } ^ { \\frac { n } { 2 } - 1 } _ { 2 , 1 } } = 0 . \\end{align*}"}
-{"id": "2684.png", "formula": "\\begin{align*} I _ { N } & = \\frac { \\eta _ { - } } { \\varphi ( x ) } \\int _ { \\{ z \\in \\mathcal { C } : \\ , \\Re { z } > 1 \\} } \\frac { d z } { 2 \\pi i } \\int _ { \\Sigma } \\frac { d w } { 2 \\pi i } e ^ { N ( f ( x ; z ) - f ( x ; w ) ) } e ^ { \\frac { \\eta _ { - } } { \\varphi ( x ) } ( v ( z - \\Re { z _ { - } } ) - u ( w - \\Re { z _ { - } } ) ) } H ( z , w ) \\end{align*}"}
-{"id": "3764.png", "formula": "\\begin{align*} \\begin{aligned} f _ { 0 } ^ { ( 2 ) } \\coloneqq \\frac { 2 } { a ^ 3 \\left [ j _ l ^ 2 ( k a ) - j _ { l - 1 } ( k a ) j _ { l + 1 } ( k a ) \\right ] } \\end{aligned} \\end{align*}"}
-{"id": "5908.png", "formula": "\\begin{align*} T _ { ( s _ 0 s _ 1 ) ^ n } \\star \\psi _ m = q ^ { 2 n } \\psi _ { m + n } + ( q - 1 ) \\sum _ { k = 1 } ^ { 2 n } q ^ { 2 n - k } \\varphi _ { m + n + 1 - k } ; \\end{align*}"}
-{"id": "1633.png", "formula": "\\begin{align*} X _ F ( \\varphi ) = i \\big ( - \\partial _ { \\varphi _ 2 } F , \\partial _ { \\varphi _ 1 } F \\big ) . \\end{align*}"}
-{"id": "5070.png", "formula": "\\begin{align*} - \\frac { b - a } { b + a } \\| h \\| ^ 2 = \\| h \\| ^ 2 - \\frac { 2 b } { b + a } \\| h \\| ^ 2 & \\leq \\| h \\| ^ 2 - \\frac { 2 } { b + a } \\langle S _ { x , \\tau } h , h \\rangle = \\left \\langle \\left ( I _ \\mathcal { H } - \\frac { 2 } { b + a } S _ { x , \\tau } \\right ) h , h \\right \\rangle \\\\ & \\leq \\| h \\| ^ 2 - \\frac { 2 a } { b + a } \\| h \\| ^ 2 = \\frac { b - a } { b + a } \\| h \\| ^ 2 . \\end{align*}"}
-{"id": "4655.png", "formula": "\\begin{align*} g _ 2 \\cdot ( x _ 0 : x _ 1 : x _ 2 ) : = & g _ 1 \\cdot ( x _ 0 : x _ 1 : x _ 2 ) & h _ 2 \\cdot ( x _ 0 : x _ 1 : x _ 2 ) : = & h _ 1 \\cdot ( x _ 0 : x _ 1 : x _ 2 ) \\end{align*}"}
-{"id": "2447.png", "formula": "\\begin{gather*} x ^ 1 = \\lambda , x ^ 2 = s \\cos \\lambda + t \\sin \\lambda + \\mu , x ^ 3 = - s \\sin \\lambda + t \\cos \\lambda , \\\\ x ^ 4 = r , x ^ 5 = \\tan \\mu , \\end{gather*}"}
-{"id": "3985.png", "formula": "\\begin{align*} \\psi \\mid U _ l ^ * ( \\tau , z ) = l ^ { - 1 } \\underset { ( D , r ) } \\sum \\left ( \\underset { \\substack { r ' \\bmod { 2 m l } \\\\ r ' \\equiv r \\bmod { 2 m } } } \\sum c _ \\psi ( l ^ 2 D , l r ' ) \\right ) e \\left ( \\frac { r ^ 2 - D } { 4 m } \\tau + r z \\right ) . \\end{align*}"}
-{"id": "316.png", "formula": "\\begin{align*} ( \\tilde { \\Gamma } \\circ T ) ^ * ( \\bar { x } _ { 2 i + 1 } ) = T ^ * \\circ \\tilde { \\Gamma } ^ * ( \\bar { x } _ { 2 i + 1 } ) = T ^ * ( u ^ 2 \\otimes \\alpha ^ * ( x _ { 2 i - 3 } ) ) = \\Sigma u ^ 2 \\otimes \\Sigma ^ { - 1 } \\alpha ^ * ( x _ { 2 i - 3 } ) , \\end{align*}"}
-{"id": "4938.png", "formula": "\\begin{align*} f _ 3 = y ^ n + v _ 1 z ^ n - v _ 2 z ^ n + \\cdots + ( - 1 ) ^ { i + 1 } v _ i z ^ n + \\cdots + ( - 1 ) ^ { d - 2 } v _ { d - 3 } z ^ n + ( - 1 ) ^ { d - 3 } x z ^ n , \\end{align*}"}
-{"id": "6640.png", "formula": "\\begin{align*} Z _ 2 = \\bigl ( \\tfrac { d ^ { 3 } - 4 d ^ 2 + 4 d + 1 } { d ^ { 3 } - 2 d ^ 2 + d - 2 } , \\tfrac { d ^ { 3 } - 4 d ^ 2 + 6 d - 7 } { d ^ { 3 } - 2 d ^ 2 + d - 2 } \\bigr ) . \\end{align*}"}
-{"id": "7946.png", "formula": "\\begin{align*} \\sum _ { a \\in R } x _ a = 0 . \\end{align*}"}
-{"id": "4704.png", "formula": "\\begin{align*} \\frac { d u _ j } { d t } = - \\frac { 1 } { \\Delta x } \\Big ( F _ { j + \\frac { 1 } { 2 } } ^ { \\mbox { H y b r i d } } - F _ { j - \\frac { 1 } { 2 } } ^ { \\mbox { H y b r i d } } \\Big ) , \\end{align*}"}
-{"id": "8024.png", "formula": "\\begin{align*} \\Xi _ 1 = \\frac { 1 } { 2 } - \\frac { 1 } { 2 } \\sum _ { i = n + 3 } ^ { 2 n } c ^ { \\frac { n + 3 - 2 i } { 2 } } P _ { i - n - 3 , - i } \\mbox { a n d } \\Xi _ 2 = \\frac { 3 } { 2 } + \\frac { 1 } { 2 } \\sum _ { i = n + 3 } ^ { 2 n } c ^ { \\frac { n + 3 - 2 i } { 2 } } P _ { i - n - 3 , - i } , \\end{align*}"}
-{"id": "9128.png", "formula": "\\begin{align*} I _ { M } \\left ( \\rho _ { 1 o } ^ { ( N ) } ( \\mathbf { x } _ { 1 } ) \\right ) = 1 . \\end{align*}"}
-{"id": "3230.png", "formula": "\\begin{align*} \\| { w } ^ { k + \\frac { 1 } { 2 } } + \\Delta _ { 1 } { w } ^ k - { w } ^ { \\ast } \\| & \\le \\| { w } ^ { k + \\frac { 1 } { 2 } } - w ^ { \\ast } \\| + \\| \\Delta _ { 1 } { w } ^ k \\| \\\\ & = \\| { w } ^ { k + \\frac { 1 } { 2 } } - w ^ { \\ast } \\| + O ( \\| { w } ^ { k + \\frac { 1 } { 2 } } - { w } ^ { \\ast } \\| ^ { 1 + \\tilde { c } \\alpha } ) \\\\ & = O ( \\| { w } ^ { k + \\frac { 1 } { 2 } } - { w } ^ { \\ast } \\| ) , \\end{align*}"}
-{"id": "6033.png", "formula": "\\begin{align*} \\overline { \\kappa } ( G ) = \\sum _ { \\{ u , v \\} \\subseteq V ( G ) } \\kappa ( u , v ) / \\tbinom { n } { 2 } . \\end{align*}"}
-{"id": "1071.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to + \\infty } \\left ( n ^ 2 \\ln \\cos \\frac { \\alpha _ n } { 2 } + \\ln n + x - \\frac { 5 \\ln ( 2 \\ln n ) } { 8 } + 2 z \\right ) = 0 . \\end{align*}"}
-{"id": "4619.png", "formula": "\\begin{align*} \\mathcal { Q } = \\mathcal { P } + \\sigma \\otimes K , \\end{align*}"}
-{"id": "2629.png", "formula": "\\begin{align*} I S ^ { ( \\mathrm { s o f t } ) } ( \\kappa , \\pi ; u , v ) = & \\int _ { \\mathcal { C } _ { > } } \\frac { d z } { 2 \\pi i } \\int _ { \\mathcal { C } _ { > } } \\frac { d w } { 2 \\pi i } e ^ { - \\frac { 1 } { 3 } ( w - \\kappa ) ^ 3 + u ( w - \\kappa ) - \\frac { 1 } { 3 } ( z - \\kappa ) ^ 3 + v ( z - \\kappa ) } \\\\ & \\times \\frac { 1 } { \\sqrt { 4 z w } } \\frac { w - z } { z + w } \\prod _ { k = 1 } ^ { m } \\frac { z - \\pi _ { k } } { z + \\pi _ { k } } \\frac { w - \\pi _ { k } } { w + \\pi _ { k } } , \\end{align*}"}
-{"id": "1952.png", "formula": "\\begin{align*} ( \\phi _ k \\cdot \\chi ' ) \\vert _ { ( \\Z / p N \\Z ) ^ \\times } = \\chi \\end{align*}"}
-{"id": "5508.png", "formula": "\\begin{align*} E ^ { \\Pi ^ { D _ T } } \\big [ \\max _ { \\lambda \\le J , k , j } \\sqrt T | \\langle b _ j - b _ { 0 , j } , a _ \\lambda \\Phi _ { \\lambda , k } \\rangle _ { L ^ 2 } | | X ^ T \\big ] = O _ { P _ { b _ 0 } } ( \\sqrt J ) . \\end{align*}"}
-{"id": "3334.png", "formula": "\\begin{align*} f ( \\pi ) = \\prod _ { C \\in \\pi , \\ , m \\nmid \\left | C \\right | } z _ 1 \\prod _ { C \\in \\pi , \\ , m \\mid \\left | C \\right | } z _ 2 \\end{align*}"}
-{"id": "6137.png", "formula": "\\begin{align*} \\Phi ' \\big ( \\Gamma ( D _ A ^ a D _ B ^ b ( q ) ) \\big ) = ( - 1 ) ^ { b + 1 } X ^ a Y ^ b \\big ( \\alpha _ { d - \\mu _ 1 } Y - \\beta _ { d - \\mu _ 2 } X \\big ) . \\end{align*}"}
-{"id": "6641.png", "formula": "\\begin{align*} M _ { 2 , 2 } = \\Bigl \\lvert \\sum _ { N ^ { 1 + \\epsilon } \\leq q \\leq \\tau } e _ q ( - \\lambda ^ 2 a ) C ^ { a / q } _ { \\tau } f \\Bigr \\rvert . \\end{align*}"}
-{"id": "7344.png", "formula": "\\begin{align*} \\tilde \\Delta u = \\frac { 1 } { \\sqrt { \\det G _ { i j } } } \\sum _ { i , j = 1 } ^ n \\frac { \\partial } { \\partial x _ i } \\left ( \\sqrt { \\det G _ { i j } } G ^ { i j } \\frac { \\partial u } { \\partial x _ j } \\right ) = \\frac { 1 } { \\rho ( x ) } ( K ( x ) \\nabla u ) \\end{align*}"}
-{"id": "9757.png", "formula": "\\begin{align*} q _ \\ell ( X ) : = \\frac { p _ { * } ( X _ \\ell + r _ \\ell X ) - p _ { * } ( r _ \\ell X ) } { h _ { r _ \\ell } } , \\end{align*}"}
-{"id": "8660.png", "formula": "\\begin{align*} m _ - ( \\l ) = - \\overline { m _ + ( \\l ) } \\quad \\end{align*}"}
-{"id": "9763.png", "formula": "\\begin{align*} q _ \\infty = q - q ( Z _ \\infty + \\ , \\cdot \\ , ) , \\end{align*}"}
-{"id": "6993.png", "formula": "\\begin{align*} \\pi ( a , a ) \\cdot ( p ^ 3 - 1 ) ( p ^ 2 - 1 ) = \\begin{cases} 7 & \\ a = 0 , \\\\ 1 2 ( a + 1 ) ^ 3 & \\ 1 \\le a \\le \\frac { p - 3 } { 2 } , \\\\ 6 \\left ( ( a + 1 ) ^ 3 - ( p - a - 1 ) ^ 3 \\right ) & \\ \\frac { p - 1 } { 2 } \\le a < p - 1 , \\\\ p ^ 3 & \\ a = p - 1 . \\end{cases} \\end{align*}"}
-{"id": "6109.png", "formula": "\\begin{align*} k _ { i i } = k , k _ { i j } = k ^ { ( i - j ) k } i > j \\end{align*}"}
-{"id": "4578.png", "formula": "\\begin{align*} A ( \\mu , R ) = \\int _ { B _ R ( 0 ) } | \\hat { \\mu } ( \\omega ) | ^ 2 d \\omega . \\end{align*}"}
-{"id": "3187.png", "formula": "\\begin{align*} \\dim \\ker H & = \\dim \\ker ( 1 - U ) \\oplus \\ker ( 1 + U ) \\\\ & = m _ + + M _ + + m _ - + M _ - \\\\ & = \\dim \\ker ( 1 - T ^ 2 ) + M _ + + M _ - . \\end{align*}"}
-{"id": "3761.png", "formula": "\\begin{align*} \\left | \\chi _ { 0 } \\right | = \\inf \\limits _ { \\chi \\in \\Xi } \\left \\{ \\left | \\chi \\right | \\right \\} . \\end{align*}"}
-{"id": "5404.png", "formula": "\\begin{align*} a _ { r s } d _ { r + 1 \\ , s + 3 } = - \\frac { r ( s + r + 1 ) } { 2 ( r + 1 ) } \\quad \\mbox { a n d } b _ { r s } c _ { r + 1 \\ , s - 3 } = \\frac { r ( s - r - 1 ) } { 2 ( r + 1 ) } . \\end{align*}"}
-{"id": "6245.png", "formula": "\\begin{align*} v ^ { * } v & = i _ { 1 } ( e _ { 1 1 } ) v v ^ { * } = i _ { 2 } ( e _ { 1 1 } ) \\\\ w ^ { * } w & = i _ { 1 } ( p ) \\qquad \\ w w ^ { * } = i _ { 2 } ( p ) \\end{align*}"}
-{"id": "5143.png", "formula": "\\begin{align*} ( \\mu + \\nu ) \\left ( \\Gamma ^ + ( \\psi ) \\right ) _ { m _ j , k _ j } = - ( k _ j + \\lambda ) \\left ( \\Gamma ^ - ( \\psi ) \\right ) _ { m _ j , k _ j } . \\end{align*}"}
-{"id": "2895.png", "formula": "\\begin{align*} \\{ \\Gamma _ { A } , \\Gamma _ { B } \\} = \\Gamma _ { ( A \\cup B ) \\setminus ( A \\cap B ) } + 2 \\Gamma _ { A \\cap B } \\Gamma _ { A \\cup B } + 2 \\Gamma _ { A \\setminus ( A \\cap B ) } \\Gamma _ { B \\setminus ( A \\cap B ) } , \\end{align*}"}
-{"id": "3240.png", "formula": "\\begin{align*} R _ { \\mu } ( x , y , V ) : = \\sqrt { \\theta ( x ) ^ 2 + \\| \\phi _ 1 ( x , { y } , { V } ) \\| ^ 2 + \\phi _ 2 ( x , y ) ^ 2 + \\| \\phi _ 3 ( x , V , \\mu ) \\| ^ 2 } , \\end{align*}"}
-{"id": "6906.png", "formula": "\\begin{align*} \\begin{cases} - u ^ 3 _ { 1 0 } u ^ 2 _ { 0 1 } + u ^ 3 _ { 0 1 } u ^ 2 _ { 1 0 } - u ^ 3 _ { 1 0 } u ^ 1 _ { 0 1 } + u ^ 3 _ { 0 1 } u ^ 1 _ { 1 0 } = 0 \\ , , \\\\ - u ^ 4 _ { 1 0 } u ^ 3 _ { 0 1 } + u ^ 4 _ { 0 1 } u ^ 3 _ { 1 0 } - u ^ 3 _ { 1 0 } u ^ 1 _ { 0 1 } + u ^ 3 _ { 0 1 } u ^ 1 _ { 1 0 } = 0 \\ , . \\end{cases} \\end{align*}"}
-{"id": "4974.png", "formula": "\\begin{align*} U _ n : = T _ { n \\ , \\ell } . \\end{align*}"}
-{"id": "5299.png", "formula": "\\begin{align*} & f _ * : H \\mapsto 3 H - 2 E _ { 0 , 1 , 2 , 3 } , E _ i \\mapsto H - E _ { i + 1 , i + 2 , i + 3 } ( i = 0 , 1 , 2 , 3 \\mod 4 ) , \\end{align*}"}
-{"id": "5668.png", "formula": "\\begin{align*} \\| u ( t _ n ) - U ^ n \\| _ { M ^ { 1 , p } ( H ) ^ { * } } \\leq & \\| ( E ( t _ n ) - B _ n P _ h ) u _ 0 \\| _ { M ^ { 1 , p } ( H ) ^ { * } } \\\\ & + \\| \\int _ { 0 } ^ { t _ n } \\bar { E } ( t _ n - t ) \\d W ( t ) - \\sum _ { j = 0 } ^ { n - 1 } \\int _ { t _ j } ^ { t _ { j + 1 } } B _ { n - j } P _ h \\d W ( t ) \\| _ { M ^ { 1 , p } ( H ) ^ { * } } : = { \\rm I + I I } . \\end{align*}"}
-{"id": "8026.png", "formula": "\\begin{align*} \\mathbb U _ n = \\langle x _ 1 , y _ 1 : y _ 1 ^ { 2 n } = 1 , x _ 1 ^ { 2 } = y _ 1 ^ n , y _ 1 x _ 1 y _ 1 = x _ 1 ^ { - 1 } \\rangle . \\end{align*}"}
-{"id": "9639.png", "formula": "\\begin{align*} & ( 1 - z q ^ { 2 N + 3 } ) \\sum _ { n = 0 } ^ { N } \\frac { ( q ; q ^ 2 ) _ { n } ( q ^ 2 ; q ^ 2 ) _ N ( z q ^ 2 ; q ^ 2 ) _ { N - n } z ^ n q ^ { 2 n } } { ( z q ; q ^ 2 ) _ { n + 1 } ( q ^ 2 ; q ^ 2 ) _ { N - n } ( z q ^ 2 ; q ^ 2 ) _ { N + 1 } } \\\\ & = \\sum _ { n = 0 } ^ { N } \\left ( \\frac { ( q ; q ) _ { 2 n } z ^ { 2 n } q ^ { 2 n ^ 2 + 3 n } } { ( z q ; q ) _ { 2 n + 1 } } + \\frac { ( q ; q ) _ { 2 n + 1 } z ^ { 2 n + 1 } q ^ { ( 2 n + 1 ) ( n + 2 ) } } { ( z q ; q ) _ { 2 n + 2 } } \\right ) \\frac { ( q ^ 2 ; q ^ 2 ) _ N ( z q ^ 5 ; q ^ 2 ) _ N } { ( q ^ 2 ; q ^ 2 ) _ { N - n } ( z q ^ 5 ; q ^ 2 ) _ { N + n } } . \\end{align*}"}
-{"id": "7842.png", "formula": "\\begin{align*} \\frac { \\pi ^ { d / 2 } } { \\Gamma ( d / 2 + 1 ) } = \\frac { \\pi ^ m } { m ! } \\le \\frac { \\pi ^ m e ^ m } { ( 2 \\pi ) ^ { 1 / 2 } m ^ { m + 1 / 2 } } \\le \\frac { ( 2 \\pi e ) ^ { d / 2 } } { 2 d ^ { d / 2 } } . \\end{align*}"}
-{"id": "3027.png", "formula": "\\begin{align*} \\Phi _ i ( z ) = z + \\sum \\limits _ { k = 2 } ^ \\infty \\lambda _ { k } { z ^ k } + ( - 1 ) ^ { i } \\sum \\limits _ { k = 1 } ^ \\infty \\mu _ { k } { \\overline { z } ^ k } , \\end{align*}"}
-{"id": "2481.png", "formula": "\\begin{align*} | P | ( x ) \\le 2 \\ , \\sup \\bigl \\{ | P ( y ) | : 0 \\le y \\le x \\bigr \\} = 2 \\ , \\sup \\bigl \\{ | P ( y ) | : | y | \\le x \\bigr \\} \\ , , \\end{align*}"}
-{"id": "6059.png", "formula": "\\begin{align*} \\sigma _ i ( { \\bf c } ) = ( \\underbrace { - \\cdots - } _ { c _ { \\gamma ' _ 1 } } \\ , \\underbrace { + \\cdots + } _ { c _ { \\gamma _ 1 } } \\ \\cdots \\ \\underbrace { - \\cdots - } _ { c _ { \\gamma ' _ t } } \\ , \\underbrace { + \\cdots + } _ { c _ { \\gamma _ t } } ) . \\end{align*}"}
-{"id": "294.png", "formula": "\\begin{align*} V _ i \\ ; : = \\ ; \\Psi ^ e ( U _ i ) \\ ; \\in \\ ; R ( G ) . \\end{align*}"}
-{"id": "7795.png", "formula": "\\begin{align*} \\Big \\| \\big ( \\sum _ { j = 1 } ^ d | R _ j f | ^ 2 \\big ) ^ { 1 / 2 } \\Big \\| _ { L ^ p } \\leq C _ p \\| f \\| _ { L ^ p } , \\end{align*}"}
-{"id": "6294.png", "formula": "\\begin{align*} Z _ { D _ 1 } & = \\int _ { \\Omega } \\inf _ { D ' \\in \\mathcal { H } } \\left [ \\exp \\left ( \\epsilon d ( D _ 1 , D ' ) \\right ) f _ { \\hat { \\mathcal { A } } ( D ' ) } \\right ] d \\mu \\\\ & \\leq \\exp \\left ( \\epsilon d ( D _ 1 , D _ 2 ) \\right ) \\int _ { \\Omega } \\inf _ { D ' \\in \\mathcal { H } } \\left [ \\exp \\left ( \\epsilon d ( D _ 2 , D ' ) \\right ) f _ { \\hat { \\mathcal { A } } ( D ' ) } ( \\omega ) \\right ] d \\mu \\\\ & = \\exp \\left ( \\epsilon d ( D _ 1 , D _ 2 ) \\right ) Z _ { D _ 2 } . \\end{align*}"}
-{"id": "4709.png", "formula": "\\begin{align*} f ( u ) = f ( u ) ^ { + } + f ( u ) ^ { - } , \\end{align*}"}
-{"id": "5323.png", "formula": "\\begin{align*} \\frac { d f _ i } { d t } = & f _ i ( - f _ { i + 1 } f _ { i + 2 } - f _ { i + 1 } f _ { i + 4 } - f _ { i + 3 } f _ { i + 4 } + f _ { i + 2 } f _ { i + 3 } + f _ { i + 2 } f _ { i + 5 } + f _ { i + 4 } f _ { i + 5 } ) \\\\ & - \\frac { 1 } { 2 } ( a _ { i } + a _ { i + 2 } + a _ { i + 4 } - a _ { i + 1 } - a _ { i + 3 } - a _ { i + 5 } ) f _ i + a _ i ( f _ i + f _ { i + 2 } + f _ { i + 4 } ) \\end{align*}"}
-{"id": "4170.png", "formula": "\\begin{align*} \\mathcal L _ n ( z _ 2 ) - \\mathcal L _ n ( z _ 1 ) = \\frac { z _ 1 - z _ 2 } { 2 \\pi i } \\oint _ { | z | = 2 r _ n } \\frac { \\mathcal L _ n ( z ) } { ( z - z _ 1 ) ( z - z _ 2 ) } d z = \\mathcal O \\left ( n ^ { \\frac { 3 } { 2 } } ( z _ 1 - z _ 2 ) \\right ) \\end{align*}"}
-{"id": "9990.png", "formula": "\\begin{align*} r _ { \\alpha t } = V ^ i _ j w ^ j _ { \\alpha k } b ^ k _ { x x } p _ i , r _ { \\alpha x } = w ^ i _ { \\alpha k } b ^ k _ { x x } p _ i . \\end{align*}"}
-{"id": "4620.png", "formula": "\\begin{align*} \\mathcal { Q } _ i ^ { - 1 } = \\mathcal { P } ^ { - 1 } - \\mathcal { P } ^ { - 1 } ( [ \\sigma \\otimes K ] \\mathcal { P } ^ { - 1 } ) + \\mathcal { P } ^ { - 1 } ( [ \\sigma \\otimes K ] \\mathcal { P } ^ { - 1 } ) ^ 2 + . . . ( - 1 ) ^ { i - 1 } \\mathcal { P } ^ { - 1 } ( [ \\sigma \\otimes K ] \\mathcal { P } ^ { - 1 } ) ^ { i - 1 } , \\end{align*}"}
-{"id": "5923.png", "formula": "\\begin{align*} B ^ + = B _ m + \\psi _ * ( E + \\phi ^ * ( B ^ + _ m - B '' _ m ) ) . \\end{align*}"}
-{"id": "10067.png", "formula": "\\begin{align*} \\begin{aligned} \\dot q & = - \\frac { 1 } { 4 } ( q + p ) ^ 3 \\left ( q + \\O ( ( q + p ) ^ 3 ) \\right ) , & \\dot p & = \\frac { 1 } { 4 } ( q + p ) ^ 3 \\left ( p + \\O ( ( q + p ) ^ 3 ) \\right ) , \\\\ \\dot w & = \\frac { 1 } { 4 } ( q + p ) ^ 2 ( p - q ) \\tilde w + \\O ( ( q + p ) ^ 5 ) , & \\dot \\phi & = \\omega ^ 0 + \\O ( ( q + p ) ^ 6 ) . \\end{aligned} \\end{align*}"}
-{"id": "8905.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\frac { P _ { T _ 1 } P _ { T _ 2 } } { P _ 0 ^ 2 } \\right ] = \\left ( \\Psi \\right ) ^ { | 1 F 2 F ^ + | \\cdot | 1 S 2 S ^ + | + | 1 F 2 F ^ - | \\cdot | 1 S 2 S ^ - | + | 1 S 2 F ^ + | \\cdot | 1 F 1 S ^ - | + | 1 F 2 S ^ + | \\cdot | 1 S 2 F ^ - | } , \\end{align*}"}
-{"id": "7438.png", "formula": "\\begin{align*} \\displaystyle P _ { m , n } = \\frac { 1 } { \\lvert W ' \\rvert } \\sum _ { w \\in W } w \\cdot S _ { m , n } . \\end{align*}"}
-{"id": "7808.png", "formula": "\\begin{align*} | B ^ 2 _ { N _ 1 } | = \\frac { \\pi ^ { d / 2 } N _ 1 ^ d } { \\Gamma ( d / 2 + 1 ) } = | B ^ 2 _ N | \\bigg ( 1 + \\frac { d } { 4 N ^ 2 } \\bigg ) ^ { d / 2 } \\le | B ^ 2 _ N | \\bigg ( 1 + \\frac { 1 } { 4 C N } \\bigg ) ^ { d / 2 } \\le e ^ { { 1 } / { ( 8 C ^ 2 ) } } | B ^ 2 _ N | , \\end{align*}"}
-{"id": "229.png", "formula": "\\begin{align*} R _ s = \\sum _ { \\beta = 0 } ^ m R _ s ^ \\beta , \\end{align*}"}
-{"id": "3620.png", "formula": "\\begin{align*} \\alpha _ { - 1 } & : = \\frac { 3 } { 2 ^ 4 } ( \\alpha _ 0 \\cdot \\alpha _ 1 + \\beta _ 0 \\cdot \\beta _ 1 - \\frac { 1 } { 2 ^ 2 } ( \\beta _ 0 , \\beta _ 0 ) a _ 1 + \\frac { 1 } { 3 } ( \\alpha _ 0 + \\alpha _ 1 ) ) \\in M _ 0 ^ { ( a _ 1 ) } \\\\ \\beta _ { - 1 } & : = \\frac { 3 } { 2 ^ 4 } ( \\alpha _ 0 \\cdot \\beta _ 1 + \\alpha _ 1 \\cdot \\beta _ 0 - \\frac { 3 } { 2 ^ 3 } ( \\beta _ 0 + \\beta _ 0 ) ) \\in M _ { \\frac { 1 } { 2 ^ 2 } } ^ { ( a _ 1 ) } . \\end{align*}"}
-{"id": "4072.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } p _ n ( x ) x ^ k w _ i ( x ) d x = 0 , i = 1 , \\ldots , r , k = 0 , \\ldots , \\lfloor \\tfrac { n - i } { r } \\rfloor , \\end{align*}"}
-{"id": "8814.png", "formula": "\\begin{align*} ( v , \\overline { v } ) \\stackrel { d } { = } ( u , \\overline { u } ) . \\end{align*}"}
-{"id": "1849.png", "formula": "\\begin{align*} \\delta ( x - x ' ) = v ^ { \\rm t o p . } ( x ) v ^ { \\rm t o p . } ( x ' ) + \\sum _ { n = 1 } ^ \\infty \\left ( v ^ { \\rm o d d } ( n , x ) v ^ { \\rm o d d } ( n , x ' ) + v ^ { \\rm e v e n } ( n , x ) v ^ { \\rm e v e n } ( n , x ' ) \\right ) . \\end{align*}"}
-{"id": "4649.png", "formula": "\\begin{align*} x _ 0 \\mapsto & \\frac { d u } { \\frac { \\partial f } { \\partial v } } = \\frac { d u } { 2 v } & x _ 1 \\mapsto & u \\frac { d u } { 2 v } & y \\mapsto & v \\left ( \\frac { d u } { 2 v } \\right ) ^ { \\otimes 3 } & \\end{align*}"}
-{"id": "9903.png", "formula": "\\begin{align*} & \\int _ { \\mathcal { Y } ^ { k + 1 } } g ( y _ { [ n , n + k ] } ) P ^ { \\mu } ( d y _ { [ n , n + k ] } | Y _ { [ 0 , n - 1 ] } ) \\\\ & = \\int _ { \\mathcal { Y } ^ { k + 1 } \\times \\mathcal { X } } g ( y _ { [ n , n + k ] } ) P ^ { \\mu } ( d ( y _ { [ n , n + k ] } , x _ { n } ) | Y _ { [ 0 , n - 1 ] } ) \\end{align*}"}
-{"id": "5242.png", "formula": "\\begin{align*} F ^ { n } ( a _ { [ n s ] } x + b _ { [ n s ] } ) \\rightarrow \\psi _ { \\beta } ( x ) ^ { 1 / s } = \\psi _ { \\beta } ( ( 1 / s ) ^ { 1 / \\beta } x ) . \\end{align*}"}
-{"id": "9285.png", "formula": "\\begin{align*} 0 & \\le \\mathcal { G } [ \\psi ] ( z ) - \\mathcal { G } [ \\phi ] ( z ) \\\\ & = \\beta ( z ) \\cdot D ( \\psi - \\phi ) ( z ) + [ \\varphi ( z , \\phi ( z ) ) - \\varphi ( z , \\psi ( z ) ) ] \\\\ & < 0 , \\end{align*}"}
-{"id": "6396.png", "formula": "\\begin{align*} \\| x - x _ { 0 } \\| _ { A } ^ { 2 } = \\sum _ { j = 0 } ^ { k - 1 } \\gamma _ { j } \\| r _ { j } \\| ^ { 2 } + \\| x - x _ { k } \\| _ { A } ^ { 2 } . \\end{align*}"}
-{"id": "4041.png", "formula": "\\begin{align*} P _ 1 = \\{ x \\in P : i ( x ) \\leq C M \\} . \\end{align*}"}
-{"id": "2839.png", "formula": "\\begin{align*} \\deg ( g _ { 1 } g _ { 2 } ) = \\deg g _ { 1 } + \\deg g _ { 2 } \\ . \\end{align*}"}
-{"id": "8329.png", "formula": "\\begin{align*} F _ z \\circ Z ( \\alpha , t ) D _ t Z + & F _ t \\circ Z ( \\alpha , t ) + \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i ( D _ t Z ( \\alpha , t ) - \\dot { z } _ j ( t ) ) } { 2 \\pi ( Z ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } + i A \\bar { Z } _ { \\alpha } = i . \\end{align*}"}
-{"id": "7334.png", "formula": "\\begin{align*} \\begin{cases} \\rho ( x ) w _ { t t } - \\operatorname { d i v } ( K ( x ) \\nabla w ) + V _ 1 ( x , t ) w = V _ 3 ( x , t ) z & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ \\rho ( x ) z _ { t t } - \\operatorname { d i v } ( K ( x ) \\nabla z ) + V _ 2 ( x , t ) z = V _ 3 ( x , t ) w & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ w = z = 0 & ~ \\hbox { i n } ~ \\omega \\times ( 0 , T ) , \\end{cases} \\end{align*}"}
-{"id": "1933.png", "formula": "\\begin{align*} F _ 2 ( x ) \\equiv x ^ 2 \\sum _ { j = 1 } ^ \\infty \\frac { A _ j ^ 2 } { Z _ j ^ 2 } = \\frac { 1 } { 2 } x \\end{align*}"}
-{"id": "5216.png", "formula": "\\begin{align*} \\lim _ { y \\rightarrow + \\infty } V ( y ) = + \\infty . \\end{align*}"}
-{"id": "8120.png", "formula": "\\begin{align*} d _ { n , r } ( t ) \\ = \\ d ^ + _ { n , r } ( t ) \\ , + \\ , d ^ - _ { n , r } ( t ) , \\end{align*}"}
-{"id": "6650.png", "formula": "\\begin{align*} \\mathsf c _ { q } ( m ) = \\sum _ { a \\in \\mathbb Z ^ { \\times } _ { q } } e _ q ( a m ) , m \\in \\mathbb Z , \\end{align*}"}
-{"id": "5279.png", "formula": "\\begin{align*} ( 1 - F ( x ) ) / r ( x ) = k ( x _ 0 ) \\exp \\left ( \\int _ { x _ 0 } ^ { x } - \\frac { 1 } { r ( t ) } d t \\right ) , \\ x _ 0 \\leq x < u e p ( F ) . \\end{align*}"}
-{"id": "9320.png", "formula": "\\begin{align*} B = \\left [ I _ { n ( n - 1 ) } C \\right ] , \\end{align*}"}
-{"id": "1726.png", "formula": "\\begin{align*} F = f ( [ h _ { 1 } ] , , \\cdots , [ h _ { n } ] ) , h _ { i } \\in \\mathcal { H } , \\end{align*}"}
-{"id": "2979.png", "formula": "\\begin{align*} ( p _ { 1 , 5 } + p _ { 4 , 5 } ) ( n ) & = M + \\sum _ { k = 1 } ^ \\infty ( - 1 ) ^ { k + 1 } \\big ( ( p _ { 1 , 5 } + p _ { 4 , 5 } ) ( n - P _ { 7 , k } ) + ( p _ { 1 , 5 } + p _ { 4 , 5 } ) ( n - Q _ { 7 , k } ) ) \\big ) . \\end{align*}"}
-{"id": "1245.png", "formula": "\\begin{align*} \\tilde S ( \\eta , \\nu ) = e ^ { i a \\ln \\left | \\frac { \\eta + \\nu } { \\eta - \\nu } \\right | } \\left ( A + 8 B e ^ { 2 i a \\ln | ( \\eta + \\nu / 2 | } \\frac { e ^ { i \\beta ( \\eta + \\nu ) ^ 3 / 8 } } { ( \\eta + \\nu ) ^ 3 } \\right ) \\left ( \\overline { A } - 8 \\overline { B } e ^ { - 2 i a \\ln | ( \\eta - \\nu ) / 2 | } \\frac { e ^ { i \\beta ( \\eta - \\nu ) ^ 3 / 8 } } { ( \\eta - \\nu ) ^ 3 } \\right ) \\end{align*}"}
-{"id": "511.png", "formula": "\\begin{align*} & ( a ) a = ( [ 1 ] ^ 5 , [ 3 ] , [ 2 ] ^ { g - 6 } ) \\mbox { o r } \\\\ & ( b ) a = ( [ 1 ] ^ 3 , [ 3 ] ^ 3 , [ 2 ] ^ { g - 6 } ) , \\end{align*}"}
-{"id": "6933.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow T ^ { * } } \\left ( \\| \\nabla \\mathbf { u } \\| _ { L ^ 1 ( 0 , T ; L ^ \\infty ) } + \\| \\theta \\| _ { L ^ \\infty ( 0 , T ; L ^ \\infty ) } \\right ) = \\infty . \\end{align*}"}
-{"id": "2073.png", "formula": "\\begin{align*} \\rho ^ 2 R _ n '' ( \\rho | \\rho _ { \\rm t x } ) + \\rho R _ n ' ( \\rho | \\rho _ { \\rm t x } ) + ( \\lambda _ { n } ^ 2 \\rho ^ 2 - { n ^ 2 } ) R _ n ( \\rho | \\rho _ { \\rm t x } ) = 0 , \\end{align*}"}
-{"id": "6811.png", "formula": "\\begin{align*} A ' ( t ) \\beta = x ' ( t ) = u ( t , x ( t ) ) = u ( t , A ( t ) \\beta ) \\ , , \\end{align*}"}
-{"id": "6983.png", "formula": "\\begin{align*} \\mathsf { r } _ c ( \\c ) = \\frac { \\c ( c ) } { \\c ( 1 ) } . \\end{align*}"}
-{"id": "3861.png", "formula": "\\begin{align*} W ^ { - } _ { t , x } \\mathcal { N } _ { t , x } f ( t , x ) = W ^ { - } _ { x } \\mathcal { N } _ { t , x } f ( t ) = \\mathbf { E } [ \\mathcal { N } _ { t , x } f ( Z _ t ^ x ) ] . \\end{align*}"}
-{"id": "4653.png", "formula": "\\begin{align*} g _ 1 \\cdot ( x _ 0 : x _ 1 : x _ 2 ) = & ( \\lambda x _ 0 : \\lambda \\omega x _ 1 : \\lambda x _ 2 ) & h _ 1 \\cdot ( x _ 0 : x _ 1 : x _ 2 ) = & ( \\lambda x _ 0 : \\lambda x _ 1 : \\lambda \\omega x _ 2 ) . \\end{align*}"}
-{"id": "6666.png", "formula": "\\begin{align*} \\zeta ( s ) = 2 ( 2 \\pi ) ^ { s - 1 } \\sin \\frac { \\pi s } { 2 } \\ , \\Gamma ( 1 - s ) \\zeta ( 1 - s ) , { \\rm R e } \\ , s < 1 . \\end{align*}"}
-{"id": "5272.png", "formula": "\\begin{align*} \\forall ( x , y ) \\in \\mathbb { R } ^ { 2 } , h ( x + y ) = h ( x ) + h ( y ) . \\end{align*}"}
-{"id": "2242.png", "formula": "\\begin{align*} C _ 1 ( x ) = & 1 - \\langle h | \\frac { \\mathcal { P } } { x - \\Omega } | g \\rangle & C _ 2 ( x ) = & \\langle h | \\delta ( x - \\Omega ) | g \\rangle \\end{align*}"}
-{"id": "3788.png", "formula": "\\begin{align*} \\alpha & = b 1 ^ d a & & \\beta = ( b - 1 ) 1 ^ d ( a + 1 ) \\\\ \\alpha & = a b 1 ^ d & & \\beta = ( b - 1 ) ( a + 1 ) 1 ^ d \\\\ \\alpha & = 1 ^ { d + 1 } a ( b - 1 ) & & \\beta = 1 ^ { d + 1 } ( b - 1 ) a \\\\ \\alpha & = 1 a 1 ^ d ( b - 1 ) & & \\beta = 1 ( b - 1 ) 1 ^ d a \\\\ \\alpha & = ( b - 1 ) 1 ^ d b ( a - b + 1 ) & & \\beta = b 1 ^ d ( b - 1 ) ( a - b + 1 ) \\end{align*}"}
-{"id": "4686.png", "formula": "\\begin{align*} u ^ { ( l ) } ( x ) \\approx ( Q _ d u ) ^ { ( l ) } ( x ) = \\displaystyle { \\sum _ { k = 1 } ^ { m + d } \\mu _ k ( u ) \\Big ( B _ k ^ d ( x ) \\Big ) ^ { ( l ) } } , ~ ~ ~ ~ ~ l \\in \\{ 1 , 2 , \\ldots , d - 1 \\} . \\end{align*}"}
-{"id": "730.png", "formula": "\\begin{align*} \\mathbb { E } _ { f } \\ ! \\left [ T ^ { 2 } \\right ] = \\frac { 1 } { ( q - 1 ) ^ { 2 } } \\left ( 1 - \\frac { 2 \\Gamma ( m n ) } { \\Gamma ( m n + q ) } \\mathbb { E } _ { g } \\ ! \\left [ L \\right ] + \\frac { \\Gamma ( m n ) } { \\Gamma ( m n + 2 q ) } \\mathbb { E } _ { g } \\ ! \\left [ L ^ { 2 } \\right ] \\right ) , \\end{align*}"}
-{"id": "2999.png", "formula": "\\begin{align*} G _ m ( x ) & = \\prod _ { k = 1 } ^ \\infty ( 1 - x ^ { k m } ) ( 1 - x ^ { k m - ( m - 1 ) } ) ( 1 - x ^ { k m - 1 } ) . \\end{align*}"}
-{"id": "8744.png", "formula": "\\begin{align*} u ^ { ( \\alpha , \\sigma ) } _ n = \\tau ^ { - \\alpha } \\sum _ { k = 0 } ^ { n } \\omega ^ { ( \\alpha , \\sigma ) } _ { n - k } u _ { k } = \\sum _ { \\ell = 0 } ^ { n } u _ n ^ { ( \\ell ) } , \\end{align*}"}
-{"id": "7602.png", "formula": "\\begin{align*} \\begin{gathered} \\Delta _ 1 \\nabla _ a X ' ( x , s , t ) = \\nabla _ a X ' ( X ^ { - 1 } ( x , s ) , s ) - \\nabla _ a X ' ( X ^ { - 1 } ( x , s ) , t ) , \\\\ \\Delta _ 2 \\nabla _ a X ' ( x , s , t ) = \\nabla _ a X ' ( X ^ { - 1 } ( x , s ) , t ) - \\nabla _ a X ' ( X ^ { - 1 } ( x , t ) , t ) , \\end{gathered} \\end{align*}"}
-{"id": "2292.png", "formula": "\\begin{align*} a _ { \\mathbf { m } , j } = & \\sum _ { j = 1 , 2 } ( \\mathbf { e } _ { \\mathbf { m } , j } ) _ i b _ { \\mathbf { m } , i } & a _ { \\mathbf { m } , j } ^ + & = \\sum _ { j = 1 , 2 } ( \\mathbf { e } _ { \\mathbf { m } , j } ) _ i b _ { \\mathbf { m } , i } ^ + . \\end{align*}"}
-{"id": "6886.png", "formula": "\\begin{align*} \\cosh ^ 2 ( s ) \\mu ' - \\sinh ^ 2 ( s ) \\theta ' = \\mathrm { c o n s t a n t } \\ , . \\end{align*}"}
-{"id": "737.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\ ! \\ ! x ^ { q } K ( x , x ) \\dd { x } = \\mathbb { E } _ { g } \\ ! \\left [ L \\right ] . \\end{align*}"}
-{"id": "9441.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c } d _ j & 0 \\\\ 0 & 1 \\\\ \\end{array} \\right ) \\left ( \\begin{array} { c c } 1 & 0 \\\\ d _ i & 1 \\\\ \\end{array} \\right ) = \\gamma \\left ( \\begin{array} { c c } 1 & \\mu _ j \\\\ 0 & d _ j / \\gcd ( d _ j , d _ i ) \\\\ \\end{array} \\right ) \\left ( \\begin{array} { c c } \\gcd ( d _ j , d _ i ) & 0 \\\\ 0 & 1 \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "1368.png", "formula": "\\begin{align*} d ( \\alpha _ \\lambda ) - x = d ( \\alpha _ \\lambda ) - \\lfloor x \\rfloor + ( \\lfloor x \\rfloor - x ) = d ( \\alpha _ \\eta ) - z \\ ; . \\end{align*}"}
-{"id": "9672.png", "formula": "\\begin{align*} \\mathcal L = \\bigvee _ { n \\geq 0 } ( U _ \\tau ^ * \\oplus U _ { \\mathbb T } ^ * ) ^ { - n } X _ * \\mathcal M _ 0 ^ \\perp . \\end{align*}"}
-{"id": "6543.png", "formula": "\\begin{align*} n _ \\beta = ( - 1 ) ^ { w - 1 } \\ , w \\ , m ^ P _ \\beta . \\end{align*}"}
-{"id": "9442.png", "formula": "\\begin{align*} c _ { d _ i } = \\sum _ { 1 \\leq j \\leq D ( N ) - 1 } \\left ( 1 - \\frac { \\gcd ( d _ j , d _ i ) ^ 2 } { d _ j } \\right ) a _ j \\end{align*}"}
-{"id": "2924.png", "formula": "\\begin{align*} \\left < z ^ k f , f \\right > _ H = \\delta _ k ( 0 ) . \\end{align*}"}
-{"id": "6553.png", "formula": "\\begin{align*} n _ \\beta = ( - 1 ) ^ { w - 1 } \\ , w \\ , m ^ P _ \\beta . \\end{align*}"}
-{"id": "9606.png", "formula": "\\begin{align*} ( - q ) _ { m - 1 } = 1 + \\sum _ { n = 1 } ^ { m - 1 } ( - q ) _ { n - 1 } q ^ n , \\end{align*}"}
-{"id": "6879.png", "formula": "\\begin{align*} A ' ( t ) \\beta = x ' ( t ) = u ( t , x ( t ) ) = u ( t , A ( t ) \\beta ) \\ , , \\end{align*}"}
-{"id": "7806.png", "formula": "\\begin{align*} \\Big \\| \\big ( \\sum _ { n \\in \\mathbb Z } \\sum _ { m = 0 } ^ { 2 ^ { l } - 1 } \\abs { ( M ^ G _ { 2 ^ n + { 2 ^ { n - l } ( m + 1 ) } } - M ^ G _ { 2 ^ n + { 2 ^ { n - l } m } } ) S _ { j + n } f } ^ 2 \\big ) ^ { 1 / 2 } \\Big \\| _ { L ^ 2 } \\lesssim 2 ^ { - l / 2 } \\min \\big \\{ 1 , 2 ^ { l } 2 ^ { - | j | / 2 } \\big \\} \\| f \\| _ { L ^ 2 } , \\end{align*}"}
-{"id": "6107.png", "formula": "\\begin{align*} L ( \\delta / \\sigma ^ c ) = \\delta / \\sigma ^ c - \\log ( \\delta / \\sigma ^ c ) - 1 . \\end{align*}"}
-{"id": "4635.png", "formula": "\\begin{align*} P _ { e f f } ^ p = \\frac { } { p \\times p } . \\end{align*}"}
-{"id": "9463.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } ( - \\Delta _ { p } ) ^ { s } u + ( - \\Delta _ { q } ) ^ { s } u = \\vert u \\vert ^ { p ^ { * } _ { s } - 2 } u + \\lambda g \\vert u \\vert ^ { r - 2 } u \\ , \\mbox { i n } \\mathbb { R } ^ { N } \\\\ u ( x ) \\geq 0 , x \\in \\mathbb { R } ^ { N } \\end{array} \\right . \\end{align*}"}
-{"id": "5965.png", "formula": "\\begin{align*} F ^ n ( x _ 1 , \\dots , x _ n ) : = F \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\delta _ { x _ i } \\right ) \\end{align*}"}
-{"id": "1461.png", "formula": "\\begin{align*} V _ { 2 } = - \\varepsilon \\int _ { \\mathbb { T } ^ { d } } ( \\mu ^ { 1 } - \\mu ^ { 2 } ) \\left ( \\nabla \\left ( \\mu ^ { 1 } - \\mu ^ { 2 } \\right ) \\right ) \\cdot \\Theta _ { p } ( t , x , \\mu ^ { 1 } , D w ^ { 1 } ) \\ d x , \\end{align*}"}
-{"id": "2007.png", "formula": "\\begin{align*} H ^ 1 ( C , L ) = \\C \\left \\{ x _ 0 ^ { - k } x _ 1 ^ { k - \\frac { d + 1 } { 5 } } : 0 < k < \\frac { d + 1 } { 5 } , \\ ; \\langle k \\rangle = \\left \\langle \\frac { d + 1 } { 5 } \\right \\rangle \\right \\} . \\end{align*}"}
-{"id": "969.png", "formula": "\\begin{align*} \\Box _ g u + F ( t , x , u ) = 0 , ( t , x ) \\in ( 0 , T ) \\times M , \\end{align*}"}
-{"id": "6460.png", "formula": "\\begin{align*} \\frac { s ( v ) + i s ( w ) } { 2 } = \\frac { 1 } { \\sqrt { 1 + \\lambda } } ( \\ell ( g ) + \\sqrt { \\lambda } \\ell ( h ) ^ * ) . \\end{align*}"}
-{"id": "10059.png", "formula": "\\begin{align*} \\Phi _ t ( \\widetilde K _ { G _ n ^ 0 } ( u , \\theta _ n ^ 0 , \\varphi ) ) = \\widetilde K _ { G _ n ^ 0 } ( \\tilde \\Phi _ t ( u ; \\theta _ n ^ 0 , G _ n ^ 0 ) , \\theta _ n ^ 0 , \\varphi + \\omega ^ 0 t ) , t \\ge 0 . \\end{align*}"}
-{"id": "7780.png", "formula": "\\begin{align*} L ( G ) ^ 2 = \\frac { 1 } { d } \\int _ { G } | x | ^ 2 { \\rm d } x . \\end{align*}"}
-{"id": "2385.png", "formula": "\\begin{align*} & \\varphi : \\mathfrak { F } \\to \\mathfrak { F } ' : \\\\ & \\varphi ( X _ A ) = X _ A A \\ne \\emptyset , \\\\ & \\varphi ( 1 ) = 0 . \\end{align*}"}
-{"id": "4911.png", "formula": "\\begin{align*} w = w ^ { i } e _ i \\ , . \\end{align*}"}
-{"id": "588.png", "formula": "\\begin{align*} r : = A ^ * \\otimes \\alpha , \\end{align*}"}
-{"id": "5725.png", "formula": "\\begin{align*} y ( 0 ) & = b _ 0 E _ { \\alpha , 1 } ( \\lambda \\cdot 0 ^ { \\alpha } ) + b _ 1 \\cdot 0 \\cdot E _ { \\alpha , 2 } ( \\lambda x ^ { \\alpha } ) + \\int _ 0 ^ 0 ( 0 - t ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } [ \\lambda ( 0 - t ) ^ { \\alpha } ] f ( t ) d t = b _ 0 . \\end{align*}"}
-{"id": "9980.png", "formula": "\\begin{align*} & q ^ { 1 } = \\frac { 1 } { 4 } ( u ^ { 1 } + u ^ { 2 } + u ^ { 3 } ) ^ { 2 } - ( u ^ { 1 } u ^ { 2 } + u ^ { 1 } u ^ { 3 } + u ^ { 2 } u ^ { 3 } ) - \\frac { 1 } { 4 } ( u ^ { 6 } ) ^ { 2 } - u ^ { 4 } u ^ { 5 } , \\\\ & q ^ { 2 } = \\frac { 2 u ^ { 1 } u ^ { 2 } u ^ { 3 } + ( u ^ { 1 } + u ^ { 2 } + u ^ { 3 } - u ^ { 6 } ) q ^ { 1 } } { 2 u ^ { 5 } } , \\\\ & q ^ { 3 } = \\frac { 1 } { 2 } ( u ^ { 1 } + u ^ { 2 } + u ^ { 3 } + u ^ { 6 } ) , \\\\ & q ^ { 4 } = u ^ { 4 } , \\\\ & q ^ { 5 } = u ^ { 5 } , \\\\ & q ^ { 6 } = \\frac { 1 } { 2 } ( u ^ { 1 } + u ^ { 2 } + u ^ { 3 } - u ^ { 6 } ) . \\end{align*}"}
-{"id": "2890.png", "formula": "\\begin{align*} \\begin{aligned} \\hat { \\gamma } _ 1 & = \\gamma _ 1 \\otimes ( i \\gamma _ 1 \\gamma _ 2 ) & = & ( i \\sigma _ 1 ) \\otimes \\sigma _ 3 \\\\ \\hat { \\gamma } _ 2 & = \\gamma _ 2 \\otimes ( i \\gamma _ 1 \\gamma _ 2 ) & = & ( i \\sigma _ 2 ) \\otimes \\sigma _ 3 \\\\ \\hat { \\gamma } _ 3 & = 1 \\otimes \\gamma _ 1 & = & ~ ~ \\ , 1 \\ , ~ \\otimes ( i \\sigma _ 1 ) \\\\ \\hat { \\gamma } _ 4 & = 1 \\otimes \\gamma _ 2 & = & ~ ~ \\ , 1 \\ , ~ \\otimes ( i \\sigma _ 2 ) . \\end{aligned} \\end{align*}"}
-{"id": "9688.png", "formula": "\\begin{align*} \\alpha _ t = \\sum _ { n \\in \\mathbb N } \\eta _ n ^ 2 \\frac { \\omega _ n \\varphi _ n g h _ { 0 1 } } { \\psi _ { n t } } \\ \\ \\ \\ \\alpha = \\sum _ { n \\in \\mathbb N } \\eta _ n ^ 2 \\frac { \\omega _ n \\varphi _ n g h _ { 0 1 } } { \\psi _ { 1 n } } \\ \\ \\mathbb T . \\end{align*}"}
-{"id": "5747.png", "formula": "\\begin{align*} F ' _ { m , n } ( t ) & = \\frac { d } { d t } \\left ( \\frac { 1 } { | | J _ 0 ( \\gamma _ m x ) \\sin ( n \\pi y ) | | _ 2 ^ 2 } \\int \\limits _ 0 ^ 1 \\int \\limits _ 0 ^ 1 x f ( x , y , t ) J _ 0 ( \\gamma _ m x ) \\sin ( n \\pi y ) d x d y \\right ) \\\\ & = \\frac { 1 } { | | J _ 0 ( \\gamma _ m x ) \\sin ( n \\pi y ) | | _ 2 ^ 2 } \\int \\limits _ 0 ^ 1 \\int \\limits _ 0 ^ 1 x f _ t ( x , y , t ) J _ 0 ( \\gamma _ m x ) \\sin ( n \\pi y ) d x d y , \\end{align*}"}
-{"id": "2058.png", "formula": "\\begin{align*} \\sum _ { n \\in \\mathbb Z } \\hat h ( n ) \\alpha _ { \\pm n , M } ( 1 ) = \\sum _ { n \\in \\mathbb Z } \\hat h ( n ) \\sum _ { m \\equiv \\pm n \\mod M } \\frac { a ( m ) } { m } V \\left ( \\frac { X } { 2 \\pi m } \\right ) + \\mathcal { O } \\bigg ( X ^ { - \\frac { 1 } { 2 } - \\epsilon } M ^ { \\frac { 1 } { 2 } + \\epsilon } q ^ { \\frac { 1 } { 2 } + \\epsilon } \\prod _ { \\substack { p \\mid \\gcd ( q , M ) \\\\ p ^ 2 \\mid q } } p \\bigg ) , \\end{align*}"}
-{"id": "6156.png", "formula": "\\begin{align*} R ^ { \\omega , \\nu } R ^ { \\eta , \\sigma } & = R ^ { \\eta , \\sigma } R ^ { \\omega , \\nu } \\\\ R ^ { \\omega , \\nu } R ^ { \\eta , \\omega } & = R ^ { \\eta , \\nu } \\\\ R ^ { \\omega , \\nu } R ^ { \\nu , \\omega } & = R ^ { \\nu , \\omega } R ^ { \\omega , \\nu } = I \\end{align*}"}
-{"id": "7997.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u = \\Delta u , ( t , x ) \\in \\mathbb { R } ^ + \\times \\mathbb { R } ^ d ; \\\\ u ( 0 , x ) = u _ 0 ( x ) \\in L ^ 2 ( \\mathbb { R } ^ d ) . \\end{cases} \\end{align*}"}
-{"id": "4219.png", "formula": "\\begin{align*} E _ n ^ { - 1 } ( z ) A _ n ^ { ( 3 ) } ( s ) E _ n ( z ) & = - n ^ 6 \\left ( E _ n ^ { - 1 } ( z ) B _ n ^ { ( 1 ) } ( s ) E _ n ( z ) \\right ) \\left ( E _ n ^ { - 1 } ( z ) A _ n ^ { ( 1 ) } ( s ) E _ n ( z ) \\right ) \\left ( E _ n ^ { - 1 } ( z ) B _ n ^ { ( 1 ) } ( s ) E _ n ( z ) \\right ) \\\\ & = n ^ 6 \\cdot \\mathcal { O } \\left ( n ^ { - \\frac { 1 } { 2 } } \\right ) \\cdot \\mathcal { O } ( 1 ) \\cdot \\mathcal { O } \\left ( n ^ { - \\frac { 1 } { 2 } } \\right ) = \\mathcal { O } \\left ( n ^ { 5 } \\right ) \\end{align*}"}
-{"id": "518.png", "formula": "\\begin{align*} \\omega _ { 1 , 0 } = f ( y ) d y , \\omega _ { 3 , 0 } = g ( y ) d y . \\end{align*}"}
-{"id": "3782.png", "formula": "\\begin{align*} ( z _ { R - k } - 2 ) + \\cdots + ( z _ 1 - 2 ) & + \\chi ( p ' _ 1 = - 1 ) + \\chi ( p ' _ { R - k + 1 } = - 1 ) \\\\ & = ( z _ 1 + \\cdots + z _ { R - k } ) - 2 ( R - k ) + \\chi ( \\alpha _ R \\neq 1 ) + \\chi ( \\alpha _ 1 \\neq 1 ) \\\\ & = ( N - k ) - 2 ( R - k ) + ( 2 - \\delta _ \\alpha ) = N - 2 R + k + 2 - \\delta _ \\alpha \\end{align*}"}
-{"id": "2145.png", "formula": "\\begin{align*} \\sigma _ \\alpha \\mathbf x = \\mathbf x - 2 \\frac { \\langle \\mathbf x , \\alpha \\rangle } { \\| \\alpha \\| ^ 2 } \\alpha . \\end{align*}"}
-{"id": "6928.png", "formula": "\\begin{align*} \\sigma ' = ( Q _ { - 1 } < Q _ 0 < \\cdots Q _ m ) . \\end{align*}"}
-{"id": "1124.png", "formula": "\\begin{align*} & \\int _ { I ^ k } \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) d y _ 1 \\cdots d y _ k \\geq \\int _ { I _ 1 ^ k } \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) d y _ 1 \\cdots d y _ k \\\\ = & S ( I ) ^ k ( \\ln n ) ^ { - k } \\int _ { ( - l _ n , 0 ) ^ k } \\phi _ { k , n } \\big ( \\gamma _ n ( u _ 1 ) , \\cdots , \\gamma _ n ( u _ k ) \\big ) \\prod _ { j = 1 } ^ k \\beta _ n ( u _ j ) d u _ 1 \\cdots d u _ k . \\end{align*}"}
-{"id": "6229.png", "formula": "\\begin{align*} \\overset { p } { \\underset { t } { ( A , \\phi ) } } : = ( A , t \\phi ) . \\end{align*}"}
-{"id": "4027.png", "formula": "\\begin{gather*} \\varphi ^ { \\ast } g _ { 2 } = g _ { 1 } , \\varphi ^ { \\ast } \\eta _ { 2 } = \\eta _ { 1 } , \\\\ \\varphi _ { { \\ast } } \\xi _ { { 1 } } = \\xi _ { { 2 } } , \\varphi _ { { \\ast } } \\circ \\phi _ { 1 } = \\phi _ { 2 } \\circ \\varphi _ { { \\ast } } . \\end{gather*}"}
-{"id": "3443.png", "formula": "\\begin{align*} K _ { N } ( x , y ) = \\int _ { c - i \\infty } ^ { c + i \\infty } \\frac { d s } { 2 \\pi i } \\oint _ { \\Sigma } \\frac { d t } { 2 \\pi i } \\frac { x ^ { t } y ^ { - s - 1 } } { s - t } \\frac { \\Gamma ( t ) } { \\Gamma ( s ) } \\left ( \\frac { \\Gamma ( s + N ) } { \\Gamma ( t + N ) } \\right ) ^ { M + 1 } , \\end{align*}"}
-{"id": "8062.png", "formula": "\\begin{align*} \\phi _ { \\zeta } ^ + ( \\omega _ i ) = \\zeta ^ { \\frac { 2 n + 3 - 2 i } { 2 } } \\omega _ i \\mbox { f o r } n + 3 \\leq i \\leq 2 n \\end{align*}"}
-{"id": "4453.png", "formula": "\\begin{align*} ( \\phi _ { 1 3 } ) _ Z = ( \\phi _ { 1 2 } ) _ Z \\circ ( \\phi _ { 2 3 } ) _ Z \\end{align*}"}
-{"id": "6633.png", "formula": "\\begin{align*} \\boldsymbol Y & : = \\{ \\underline { Y _ 1 } , \\underline { Y _ 2 } , \\underline { Y _ 3 } , Y _ 1 Y _ 2 , Y _ 1 Y _ 3 , Y _ 2 Y _ 3 \\} , \\\\ \\boldsymbol Z & : = \\{ Z _ 1 , \\underline { Z _ 1 Z _ 2 } , \\underline { Z _ 1 Y _ 1 Y _ 2 Y _ 3 } , Z _ 1 Z _ 2 Y _ 1 Y _ 2 Y _ 3 \\} , \\\\ \\boldsymbol Z ' & : = \\{ Z _ 2 , \\underline { Z _ 1 Z _ 2 } , \\underline { Z _ 2 Y _ 1 Y _ 2 Y _ 3 } , Z _ 1 Z _ 2 Y _ 1 Y _ 2 Y _ 3 \\} . \\end{align*}"}
-{"id": "7722.png", "formula": "\\begin{align*} \\left ( \\bigcap _ { k = 1 } ^ { \\infty } V ^ 1 _ { k } \\right ) \\cup \\ldots \\cup \\left ( \\bigcap _ { k = 1 } ^ { \\infty } V ^ l _ { k } \\right ) \\supset \\bigcap _ { k = 1 } ^ { \\infty } \\left ( V ^ 1 _ { k } \\cup \\ldots \\cup V ^ l _ { k } \\right ) , \\end{align*}"}
-{"id": "9127.png", "formula": "\\begin{align*} L _ { \\rho } ( t ) = \\inf _ { \\mathbf { x } _ { 1 } \\in \\Gamma _ { 1 } } \\left \\{ \\left \\vert \\frac { \\partial \\ln \\rho _ { 1 } ^ { ( N ) } ( \\mathbf { x } _ { 1 } , t ) } { \\partial \\mathbf { r } _ { 1 } } \\right \\vert ^ { - 1 } \\right \\} , \\end{align*}"}
-{"id": "515.png", "formula": "\\begin{align*} \\omega _ { n , 0 } = y ^ n \\prod _ { i = 1 , . . . , g + 1 } ( x - t _ i ) ^ { l ( i , n ) } d x , \\ \\ l ( i , n ) = \\left \\lfloor { \\frac { - n a _ i } { 4 } } \\right \\rfloor , \\ \\ i = 1 , 2 , 3 . \\end{align*}"}
-{"id": "2775.png", "formula": "\\begin{align*} \\mathbf { Q } = \\left [ \\begin{array} { c c c } 0 & 1 / 2 & 1 / 4 \\\\ 1 / 4 & 1 / 4 & 0 \\\\ 1 / 4 & 1 / 2 & 1 / 4 \\end{array} \\right ] \\end{align*}"}
-{"id": "8193.png", "formula": "\\begin{align*} G ^ { i j } = \\left \\{ \\begin{array} { l l } 1 & \\\\ 0 & \\end{array} \\right . . \\end{align*}"}
-{"id": "2369.png", "formula": "\\begin{align*} u _ i : \\mathfrak { A } & \\to \\mathfrak { A } ^ { \\otimes N } & u _ 1 ( b ) = & b \\otimes 1 \\otimes \\cdots \\otimes 1 , & \\cdots & , & u _ N ( b ) & = 1 \\otimes \\cdots \\otimes 1 \\otimes b , \\end{align*}"}
-{"id": "1884.png", "formula": "\\begin{align*} f ( y ) = \\int _ 0 ^ \\infty e ^ { - y t } \\left ( \\frac { t } { 1 - e ^ { - t } } - \\frac { 1 } { 2 } t - 1 \\right ) d t = \\int _ 0 ^ \\infty \\frac { e ^ { - y t } } { 1 - e ^ { - t } } \\left [ \\left ( \\frac { 1 } { 2 } t - 1 \\right ) + \\left ( \\frac { 1 } { 2 } t + 1 \\right ) e ^ { - t } \\right ] d t . \\end{align*}"}
-{"id": "3154.png", "formula": "\\begin{align*} Z \\bigl ( \\mu ( \\bar x ) | \\mu ( \\bar t ) \\bigr ) = Z ( \\bar x | \\bar t ) \\prod _ { k = 1 } ^ { N - 2 } f ( \\bar x ^ { k + 1 } , \\bar x ^ { k } ) f ( \\bar t ^ { k + 1 } , \\bar t ^ { k } ) . \\end{align*}"}
-{"id": "907.png", "formula": "\\begin{align*} \\prod _ { 1 \\leq j \\leq \\ell } ^ { \\to } \\left ( \\prod _ { i \\in Q _ 0 ^ j } ^ { \\to } y _ { e _ i } ^ { \\gamma ( i ) } \\right ) = q ^ { \\sum \\gamma ( t a ) \\gamma ( h a ) } y _ { e _ 1 } ^ { \\gamma ( 1 ) } \\cdots y _ { e _ n } ^ { \\gamma ( n ) } = \\Gamma ^ { - 1 } \\cdot y _ { e _ 1 } ^ { \\gamma ( 1 ) } \\cdots y _ { e _ n } ^ { \\gamma ( n ) } \\end{align*}"}
-{"id": "7799.png", "formula": "\\begin{align*} A _ { \\zeta } ( u ) = \\{ x \\in G \\colon x \\cdot \\zeta = u \\} , \\end{align*}"}
-{"id": "6580.png", "formula": "\\begin{gather*} E ^ { 1 , n } _ 1 = \\ker ( \\delta ) . \\end{gather*}"}
-{"id": "6805.png", "formula": "\\begin{align*} \\sqrt [ \\mathbb { R } ] { I _ { 3 , 8 } } = \\langle y ^ 1 _ { 1 0 } y ^ 2 _ { 0 1 } - y ^ 1 _ { 0 1 } y ^ 2 _ { 1 0 } - 1 \\rangle + \\langle y ^ k _ \\nu , | \\nu | = 2 \\ , , k = 1 , 2 \\rangle \\ , . \\end{align*}"}
-{"id": "2427.png", "formula": "\\begin{align*} \\Delta _ { g , 0 } : = \\left \\{ \\begin{pmatrix} \\ast & \\ast \\\\ { \\bf 0 _ n } & \\ast \\end{pmatrix} \\in \\Gamma _ g \\right \\} , \\end{align*}"}
-{"id": "8658.png", "formula": "\\begin{align*} \\begin{array} { c } w _ { 1 , \\varepsilon } = \\frac { 1 } { 8 } { \\varphi _ { 2 , \\varepsilon } ^ 2 + \\frac { 1 } { 4 } \\sin ^ 2 \\varphi _ { 1 , \\varepsilon } - \\frac { c _ 0 } { 2 } \\sin \\varphi _ { 1 , \\varepsilon } } + O ( \\varepsilon ) e ^ { - \\left ( \\sqrt { 1 - 4 c _ 0 ^ 2 } + O ( \\varepsilon ) \\right ) | x | } , \\\\ \\\\ w _ { 2 , \\varepsilon } = O ( \\varepsilon ) e ^ { - \\left ( \\sqrt { 1 - 4 c _ 0 ^ 2 } + O ( \\varepsilon ) \\right ) | x | } , \\end{array} \\end{align*}"}
-{"id": "39.png", "formula": "\\begin{align*} \\mathcal { Y } _ t ^ { i , v } ( T ) = & \\ h ^ { i } ( V _ T ^ { v } ) + \\int _ t ^ T \\left [ f ^ i ( V _ s ^ v , \\mathcal { Z } _ s ^ { i , v } ( T ) ) + \\sum _ { k \\in I } q ^ { i k } ( e ^ { \\mathcal { Y } _ s ^ { k , v } ( T ) - \\mathcal { Y } _ s ^ { i , v } ( T ) } - 1 ) \\right ] d s \\\\ & - \\int _ t ^ T ( \\mathcal { Z } _ s ^ { i , v } ( T ) ) ^ { t r } d W _ s . \\end{align*}"}
-{"id": "1714.png", "formula": "\\begin{align*} g _ { f , s } ( y _ n ) = | f ( \\varphi ( t _ n , y _ n ) ) | g _ { f , s } ( x ) = | f ( \\varphi ( t , x ) ) | . \\end{align*}"}
-{"id": "795.png", "formula": "\\begin{align*} \\sum _ { \\nu = 1 } ^ { { s } } b _ i = 1 + q + \\cdots + q ^ { { { s } } - 1 } . \\end{align*}"}
-{"id": "4043.png", "formula": "\\begin{align*} I _ M ( P _ 2 , V ) = \\sum _ { x \\in P _ 2 } i ( x ) \\leq | P _ 2 | ^ { 1 / 2 } \\left ( \\sum _ { x \\in P _ 2 } i ( x ) ^ 2 \\right ) ^ { 1 / 2 } \\leq m ^ { 1 / 2 } \\left ( \\sum _ { x \\in P _ 2 } i ( x ) ^ 2 \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "2140.png", "formula": "\\begin{align*} r ( x _ 1 , x _ 2 , \\dots , x _ n ) = ( \\psi ( x _ 1 ) , \\psi ( x _ 2 ) , \\dots , \\psi ( x _ n ) ) , \\end{align*}"}
-{"id": "1346.png", "formula": "\\begin{align*} \\pi _ * ^ { r ( x ) } ( x \\cdot f ) = \\pi ( x ) \\cdot \\pi _ * ^ { d ( x ) } ( f ) \\ ; , \\end{align*}"}
-{"id": "6439.png", "formula": "\\begin{align*} \\eta = \\frac { \\| \\widetilde r \\| } { \\| A \\| \\| \\widetilde x \\| + \\| b \\| } \\ , . \\end{align*}"}
-{"id": "8066.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 2 n } \\zeta ^ { \\frac { ( 2 n + 3 - 2 i ) k } { 2 } } & = \\zeta ^ { \\frac { 3 } { 2 } k } \\sum _ { i = 1 } ^ { 2 n } \\zeta ^ { ( n - i ) k } = \\zeta ^ { \\frac { 3 } { 2 } k } \\left ( \\zeta ^ { ( n - 1 ) k } + \\zeta ^ { ( n - 2 ) k } + \\ldots + \\zeta ^ k + 1 + \\zeta ^ { - k } + \\ldots + \\zeta ^ { - 2 n k } \\right ) = 0 . \\end{align*}"}
-{"id": "3046.png", "formula": "\\begin{align*} m ( d \\xi ) = \\sum \\limits _ { j = 1 } ^ { d } \\mu _ { j } ( d z ) d r \\delta _ { j } ( d k ) + \\nu ( d z ) \\delta _ { 0 } ( d r ) \\delta _ { d + 1 } ( d k ) . \\end{align*}"}
-{"id": "9592.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { n q ^ n ( 1 - q ^ { N n } ) } { 1 - q ^ n } = \\sum _ { k = 1 } ^ { \\infty } \\sigma ( k , N ) q ^ k , \\end{align*}"}
-{"id": "3779.png", "formula": "\\begin{align*} \\begin{aligned} \\Re \\delta W & = \\int _ \\Omega B _ 0 ( v _ 0 - v _ 1 ) \\cdot ( v _ 0 - v _ 1 ) + \\int _ D ( B _ 1 - B _ 0 ) v _ 1 \\cdot v _ 1 \\\\ & = - \\int _ \\Omega B _ 1 ( v _ 0 - v _ 1 ) \\cdot ( v _ 0 - v _ 1 ) + \\int _ D ( B _ 1 - B _ 0 ) v _ 0 \\cdot v _ 0 , \\end{aligned} \\end{align*}"}
-{"id": "1126.png", "formula": "\\begin{align*} & \\liminf _ { n \\to + \\infty } ( \\ln n ) ^ { - k } \\phi _ { k , n } \\big ( \\gamma _ n ( u _ 1 ) , \\cdots , \\gamma _ n ( u _ k ) \\big ) \\geq 2 ^ k e ^ { \\sum _ { j = 1 } ^ k ( c _ 0 - x _ j - 2 | u _ j | ) } . \\end{align*}"}
-{"id": "7185.png", "formula": "\\begin{gather*} \\varkappa : = 1 1 K C R / ( 6 \\alpha ) < 1 . \\end{gather*}"}
-{"id": "9666.png", "formula": "\\begin{align*} | h _ { 0 1 } | = | h _ 0 | \\ \\mathbb T . \\end{align*}"}
-{"id": "4261.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\delta ( a ( x ) \\lvert d w _ { j } \\rvert ^ { p - 2 } d w _ { j } ) ) & = 0 & & B _ { j } , \\\\ \\delta w _ { j } & = 0 & & B _ { j } , \\\\ \\nu \\wedge w _ { j } & = \\nu \\wedge u & & \\partial B _ { j } , \\end{aligned} \\right . \\end{align*}"}
-{"id": "7598.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { \\int _ 0 ^ t \\Delta g _ { \\nu ( t - s ) } * \\left ( \\nabla \\cdot \\eta ( t ) \\Delta _ 1 \\tau ( s , t ) \\right ) d s } _ { \\alpha , p } \\\\ \\le \\frac { C t } { \\nu } \\norm { \\eta } _ { L ^ \\infty ( 0 , T ; C ^ { 1 + \\alpha } ) } \\norm { \\tau } _ { L i p ( 0 , T ; C ^ { \\alpha , p } ) } M _ X ^ { \\alpha } . \\end{gathered} \\end{align*}"}
-{"id": "9203.png", "formula": "\\begin{align*} H : [ 0 , b ] \\times \\lbrack 0 , 1 ] \\rightarrow M , H = H ( t , s ) , \\end{align*}"}
-{"id": "5448.png", "formula": "\\begin{align*} \\hat V _ 0 \\ = \\ \\sup _ { \\hat \\alpha \\in \\hat { \\mathcal A } } \\hat J ( \\hat \\alpha ) . \\end{align*}"}
-{"id": "5418.png", "formula": "\\begin{align*} Q _ \\mu ( u , v ) & = c _ { N , \\alpha } \\iint _ { \\R ^ N \\times \\R ^ N } \\frac { | ( u ( x ) - u ( y ) ) ( v ( x ) - v ( y ) ) } { | x - y | ^ { N + \\alpha } } \\ , d x \\ , d y + \\int _ { \\R ^ N } V ( x ) u v \\ , d x \\\\ & - \\frac { \\mu } { 2 } \\int _ { \\R ^ N } \\frac { u v } { | x | ^ \\alpha } \\ , d x \\end{align*}"}
-{"id": "9787.png", "formula": "\\begin{align*} | Y _ 1 ^ \\ell \\wedge \\dots \\wedge Y _ { n - 1 } ^ \\ell | \\ge \\varepsilon _ \\circ ^ { n - 1 } \\quad Y ^ \\ell _ i : = r _ \\ell ^ { - 1 } X _ i ^ \\ell \\in B _ 1 \\setminus B _ { \\varepsilon _ \\circ } , \\end{align*}"}
-{"id": "2281.png", "formula": "\\begin{align*} & ( \\delta ' + L U _ K \\check { Y } G ) * Z \\check { Y } = \\delta \\\\ & Z ' = \\int _ t ^ 0 d t _ 1 L U _ K ( t - t _ 1 ) G Z ( t _ 1 ) , & Z ( 0 ) & = 1 , & Z ( t ) & = 0 t > 0 \\end{align*}"}
-{"id": "7575.png", "formula": "\\begin{align*} K ( x , z , t , s ) = \\Delta g _ { \\nu ( t - s ) } ( x - X ( z , s ) ) - \\Delta g _ { \\nu ( t - s ) } ( x - X ( z , t ) ) . \\end{align*}"}
-{"id": "4492.png", "formula": "\\begin{align*} L ( \\nabla _ n u ) & = - G _ u \\nabla _ n u - g ^ { i n } \\sigma \\big ( \\nabla \\psi ( N ) , \\frac { 1 } { u w } e _ i - \\frac { \\nabla _ i u } { u ^ 2 w } x \\big ) \\\\ & - G ^ { i n } \\nabla _ i u + \\nabla _ n u \\sum G ^ { i i } . \\end{align*}"}
-{"id": "1014.png", "formula": "\\begin{align*} & \\lim _ { n \\to + \\infty } ( n / 4 ) ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } D _ n ( \\alpha ) = 0 . \\end{align*}"}
-{"id": "8742.png", "formula": "\\begin{align*} \\omega _ n ^ { ( \\alpha , \\sigma ) } \\approx \\widetilde { \\omega } _ n ^ { ( \\alpha , \\sigma ) } = 2 \\tau ^ { 1 + \\alpha } e ^ { - n \\sigma \\tau } \\mathbf { I m } \\left ( \\sum _ { j = 0 } ^ { N - 1 } w _ j ^ { ( \\ell ) } ( \\lambda _ j ^ { ( \\ell ) } ) ^ { \\alpha } ( 1 - \\lambda ^ { ( \\ell ) } _ j \\tau ) ^ { - 1 - n } F _ { \\omega } ( \\lambda ^ { ( \\ell ) } _ j ) \\right ) , \\end{align*}"}
-{"id": "8229.png", "formula": "\\begin{align*} U _ 5 ( F Z \\rho t ^ i ) = F \\Big ( ( \\sum _ { j = \\left \\lceil \\frac { i + 2 } { 5 } \\right \\rceil } ^ \\infty z _ 0 ( i , j ) t ^ j ) + \\rho ( \\sum _ { j = \\left \\lceil \\frac { i + 2 } { 5 } \\right \\rceil } ^ \\infty z _ 1 ( i , j ) t ^ j ) \\Big ) , \\end{align*}"}
-{"id": "6077.png", "formula": "\\begin{align*} v ( t ) = \\int _ { \\hat \\nu } J ^ m ( t , \\xi ) Y ^ { n - m - 1 } ( t , \\xi ) d \\xi . \\end{align*}"}
-{"id": "1013.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to + \\infty } \\dfrac { n F _ n ( x ) } { ( 3 2 \\ln n ) ^ { \\frac { 1 } { 2 } } } = 1 , \\ \\lim \\limits _ { n \\to + \\infty } n F _ n ( x ) = + \\infty , \\ \\lim \\limits _ { n \\to + \\infty } n ^ { \\gamma } F _ n ( x ) = 0 , \\ \\forall \\ \\gamma < 1 . \\end{align*}"}
-{"id": "3437.png", "formula": "\\begin{align*} \\lambda ( v _ 0 ) : = \\limsup _ { t \\rightarrow \\infty } \\frac { 1 } { t } \\log \\| v ( t ) \\| . \\end{align*}"}
-{"id": "4434.png", "formula": "\\begin{align*} \\forall i = 1 , 2 , 3 : \\ ; u ^ i ( 0 , x ) = u _ 0 ^ i , x \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "9659.png", "formula": "\\begin{align*} P _ { \\mathcal N } p ( T ) | _ { H ^ 2 _ - } = Y _ 0 P _ { H ^ 2 } p ( U _ { \\mathbb T } ) | _ { H ^ 2 _ - } . \\end{align*}"}
-{"id": "5526.png", "formula": "\\begin{align*} \\frac { \\int _ { D _ T } e ^ { \\ell _ T ( b _ u ) } d \\Pi ( b ) } { \\int _ { D _ T } e ^ { \\ell _ T ( b ) } d \\Pi ( b ) } = 1 + \\zeta _ T ( u ) \\leq C _ T e ^ { r _ T u ^ 2 } , \\end{align*}"}
-{"id": "2783.png", "formula": "\\begin{align*} H ^ { ( r ) } ( \\alpha ) = \\log \\left ( 1 - q ^ r + q ^ r e ^ { \\alpha } \\right ) . \\end{align*}"}
-{"id": "7389.png", "formula": "\\begin{align*} \\dot { x } & = a + y + \\mu _ 1 x ^ 2 \\\\ \\dot { y } & = b y + \\mu _ 2 x y . \\end{align*}"}
-{"id": "3323.png", "formula": "\\begin{align*} E : = \\{ e _ i : 1 \\le i \\le k \\} \\cup \\{ 0 : 1 \\le i \\le | \\mathcal { P } _ d | - k \\} . \\end{align*}"}
-{"id": "9093.png", "formula": "\\begin{align*} \\int _ { \\Omega } g ( x ) v _ R ( x ) ^ p d x = 1 . \\end{align*}"}
-{"id": "7005.png", "formula": "\\begin{align*} \\sum _ { a = 1 } ^ m ( m + 1 - a ) \\xi ^ { 2 a - 1 } = \\frac { \\xi } { \\left ( \\xi ^ 2 - 1 \\right ) ^ 2 } \\Bigg ( \\big ( \\xi ^ { 2 ( m + 1 ) } - 1 \\big ) - ( m + 1 ) \\big ( \\xi ^ 2 - 1 \\big ) \\Bigg ) . \\end{align*}"}
-{"id": "9343.png", "formula": "\\begin{align*} A = \\begin{bmatrix} A _ { 1 , 1 } & 0 \\\\ 0 & 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "138.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { N } \\left ( \\dfrac { \\partial \\pi ^ { i j } } { \\partial x ^ s } v ^ s - \\pi ^ { s j } \\dfrac { \\partial v ^ i } { \\partial x ^ s } - \\pi ^ { i s } \\dfrac { \\partial v ^ j } { \\partial x ^ s } \\right ) = 0 . \\end{align*}"}
-{"id": "10054.png", "formula": "\\begin{align*} \\frac { 1 } { | 1 - z | } = \\frac { 1 } { ( 1 - z ) ^ { 1 / 2 } ( 1 - \\bar { z } ) ^ { 1 / 2 } } = \\sum _ { \\ell \\ge 0 } c _ { \\ell } z ^ { \\ell } \\sum _ { k \\ge 0 } c _ { k } \\bar { z } ^ { k } = \\sum _ { \\ell , k \\ge 0 } c _ { \\ell } c _ k z ^ { \\ell } \\bar { z } ^ k , \\end{align*}"}
-{"id": "7424.png", "formula": "\\begin{align*} \\xi ( A _ { 1 , i } ) = A _ { 1 , - i } z \\in \\Z . \\end{align*}"}
-{"id": "1152.png", "formula": "\\begin{align*} & ( \\ln n ) ^ { - k } \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) \\geq ( 2 n ) ^ k ( 2 \\ln n ) ^ { - \\frac { k } { 2 } } \\mathbb { P } ( \\xi ^ { ( n ) } ( I _ { n , k } ) = 0 ) \\\\ & - ( 2 n ) ^ k ( 2 \\ln n ) ^ { - \\frac { k } { 2 } } \\sum _ { j = 0 } ^ k \\mathbb { P } ( \\xi ^ { ( n ) } ( I _ { n , k } ) = \\xi ^ { ( n ) } ( J _ { n , k , j } ) = 0 ) . \\end{align*}"}
-{"id": "2635.png", "formula": "\\begin{align*} \\Lambda = \\begin{bmatrix} \\Lambda _ { 0 } \\\\ 0 _ { n \\times N } \\end{bmatrix} , \\Lambda _ { 0 } = \\mathrm { d i a g } \\Big ( \\sqrt { \\lambda _ { 1 } } , \\ldots , \\sqrt { \\lambda _ { N } } \\Big ) . \\end{align*}"}
-{"id": "1976.png", "formula": "\\begin{align*} \\alpha _ i = c _ 1 ( \\O _ { \\alpha _ i } ) \\end{align*}"}
-{"id": "9474.png", "formula": "\\begin{align*} G ( t ) = \\displaystyle \\int _ { 0 } ^ { t } ( g ' ( \\tau ) ) ^ { \\frac { 1 } { m } } \\dd \\tau , \\ t \\in \\mathbb { R } , \\end{align*}"}
-{"id": "8959.png", "formula": "\\begin{align*} \\langle X , \\beta \\rangle _ K = \\sum _ { j = 1 } ^ p \\beta _ j \\langle u _ j , X - m \\rangle _ 2 . \\end{align*}"}
-{"id": "2805.png", "formula": "\\begin{align*} P ( \\mathbf { n } _ 0 , \\mathbf { n } _ 1 , \\dots , \\mathbf { n } _ s ) : = \\mathbb { P } \\left ( \\mathbf { N } ^ t = \\mathbf { n } _ 0 ; \\mathbf { N } ^ { t + 1 } = \\mathbf { n } _ 1 ; \\dots ; \\mathbf { N } ^ { t + s } = \\mathbf { n } _ s \\right ) . \\end{align*}"}
-{"id": "8135.png", "formula": "\\begin{align*} \\overset { I } { L } \\ , \\ ! ^ { ( \\pm ) } _ 1 \\overset { J } { T } _ 2 = \\overset { J } { T } _ 2 \\overset { I } { L } \\ , \\ ! ^ { ( \\pm ) } _ 1 \\overset { I J } { R } \\ , \\ ! ^ { ( \\pm ) } _ { 1 2 } . \\end{align*}"}
-{"id": "6766.png", "formula": "\\begin{align*} \\phi ( \\gamma x t ^ { - 1 } ) = \\rho ( \\gamma ) \\phi ( x ) \\tau _ i ( t ) ^ { - 1 } . \\end{align*}"}
-{"id": "8462.png", "formula": "\\begin{align*} \\| F \\| _ { L ^ { \\infty } ( \\Omega ( t ) ) } = \\| F \\| _ { L ^ { \\infty } ( \\Sigma ( t ) ) } = \\| \\mathfrak { F } ( t ) \\| _ { \\infty } \\leq 5 \\epsilon . \\end{align*}"}
-{"id": "7734.png", "formula": "\\begin{align*} | [ \\langle x , y \\rangle \\xi , \\xi ] | = \\left \\| y \\right \\| . \\end{align*}"}
-{"id": "6340.png", "formula": "\\begin{align*} f _ { n , r _ 2 } ( n ) = L a g \\Big \\{ & n , G _ 1 G _ 2 m _ s h _ { s r } h _ { r r } , G _ 1 G _ 2 ( m _ { b r _ 1 } + n _ { s p } ) h _ { r r } \\\\ & + G _ 2 ( m _ { b r _ 2 } + n _ { s p } ) , D \\Big \\} , \\end{align*}"}
-{"id": "910.png", "formula": "\\begin{align*} F _ 0 \\subseteq F _ 1 \\subseteq \\cdots \\subseteq F _ { \\dim _ \\R ( X ) } = X . \\end{align*}"}
-{"id": "9649.png", "formula": "\\begin{align*} & W S _ { | \\psi | ^ 2 m } = T W , \\ \\ \\ \\| W \\| \\leq K C _ { { \\rm p o l } , T } \\| x \\| \\\\ \\ \\ \\ & W \\varphi = \\varphi ( T ) x \\ \\ \\varphi \\in H ^ \\infty . \\end{align*}"}
-{"id": "10030.png", "formula": "\\begin{align*} \\lim _ { k , l , m \\rightarrow \\infty } \\langle f , g \\rangle _ { k , l , m } = \\langle f , g \\rangle _ z . \\end{align*}"}
-{"id": "542.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 } \\frac { 1 } { h ^ 4 } \\mathcal E ^ h ( u ^ h ) = \\mathcal { I } _ 4 ^ O ( V , \\mathbb { S } ) . \\end{align*}"}
-{"id": "5365.png", "formula": "\\begin{align*} \\frac { a _ { n m } d _ { n + 1 \\ , m + 3 } } { a _ { n - 1 \\ , m + 3 } d _ { n \\ , m + 6 } } \\frac { b _ { n \\ , m + 6 } c _ { n + 1 \\ , m + 3 } } { b _ { n - 1 \\ , m + 3 } c _ { n m } } = \\frac { n ^ 2 } { ( n + 1 ) ^ 2 } . \\end{align*}"}
-{"id": "9684.png", "formula": "\\begin{align*} W _ t \\varphi = \\sum _ { n \\in \\mathbb N } ( \\eta _ n \\varphi ) ( T ) x _ { n t } + X _ * ^ * ( \\beta _ t \\varphi F _ { \\tau , \\mathbb T \\setminus \\sigma } \\oplus \\beta _ t \\varphi F _ { \\mathbb T , \\mathbb T \\setminus \\sigma } ) . \\end{align*}"}
-{"id": "1597.png", "formula": "\\begin{align*} \\begin{aligned} | \\pi _ x \\circ \\widetilde { F } ( z ^ o _ { n , k } , u ^ o _ { n , k } ) - \\pi _ x \\circ \\widetilde { F } _ { w _ { n , k } } ( z ^ o _ { n , k } , x ^ o _ { n , k } ) | & = | \\pi _ x \\circ \\widetilde { F } ( z ^ o _ { n , k } , u ^ o _ { n , k } ) - \\pi _ x \\circ \\widetilde { F } ( z ^ o _ { n , k } , x ^ o _ { n , k } ) | \\\\ & < \\sup | \\frac { \\partial \\pi _ x \\circ \\widetilde { F } } { \\partial x } | | u ^ o _ { n , k } - x ^ o _ { n , k } | . \\end{aligned} \\end{align*}"}
-{"id": "8739.png", "formula": "\\begin{align*} \\omega ^ { ( \\alpha , \\sigma ) } ( z ) = ( \\omega ( z e ^ { - \\sigma \\tau } ) ) ^ { \\alpha } = \\sum _ { k = 0 } ^ { \\infty } \\omega ^ { ( \\alpha , \\sigma ) } _ k z ^ k . \\end{align*}"}
-{"id": "1780.png", "formula": "\\begin{align*} X _ { j } = 2 ^ { n H } \\left ( B _ { \\frac { j } { 2 ^ { n } } } - B _ { \\frac { j - 1 } { 2 ^ { n } } } \\right ) \\sim N ( 0 , 1 ) , \\end{align*}"}
-{"id": "116.png", "formula": "\\begin{align*} \\sum _ { \\substack { k = 1 \\\\ k \\ , } } ^ { 2 N - 1 } \\gamma ^ k S _ k ( B ( 0 ) , B _ 1 ) & = \\sum _ { j = 0 } ^ { 2 N - 1 } \\widetilde \\zeta _ { j , N } B ( 0 ) ^ { 2 j } B _ 2 ( 0 ) , \\end{align*}"}
-{"id": "991.png", "formula": "\\begin{align*} \\sum _ { i } ( - 1 ) ^ i \\left ( \\sum _ j \\beta _ { i , j } t ^ { j } \\right ) & = 1 + \\sum ^ { d - 1 } _ { i = 1 } ( - 1 ) ^ { i } \\cdot i \\cdot \\binom { d } { i + 1 } t ^ { i + 1 } + \\sum ^ { d } _ { i = 1 } ( - 1 ) ^ { i } \\cdot d \\cdot \\binom { d - 1 } { i - 1 } t ^ { i + r - 1 } \\\\ & = 1 + \\sum _ { i = 1 } ^ { d } ( - 1 ) ^ { i + 1 } ( i - 1 ) \\binom { d } { i } t ^ { i } - d t ^ { r } ( 1 - t ) ^ { d - 1 } \\end{align*}"}
-{"id": "7990.png", "formula": "\\begin{align*} \\sum _ { \\sqrt { n } < p \\leq \\frac { n } { 2 } } \\log p = \\sum _ { p \\leq \\frac { n } { 2 } } \\log p - \\sum _ { p \\leq \\sqrt { n } } \\log p & = \\frac { n } { 2 } + O \\left ( \\frac { n } { \\log n } \\right ) + O ( \\log n \\sqrt { n } ) \\\\ & = \\frac { n } { 2 } + O \\left ( \\frac { n } { \\log n } \\right ) \\ , . \\end{align*}"}
-{"id": "11.png", "formula": "\\begin{align*} \\delta Y _ t ^ i ( m , n ) : = Y _ t ^ { i } ( m ) - Y _ t ^ { i } ( n ) \\ \\ \\ \\ \\delta Z _ t ^ i ( m , n ) : = Z _ t ^ { i } ( m ) - Z _ t ^ { i } ( n ) . \\end{align*}"}
-{"id": "391.png", "formula": "\\begin{align*} S ^ m U \\in \\langle \\ , S ^ i U \\ , \\rangle _ { 0 \\leq i \\leq m } = \\langle \\ , \\Lambda ^ i Q \\ , \\rangle _ { 0 \\leq i \\leq m } = \\langle \\ , \\Lambda ^ i Q \\ , \\rangle _ { 0 \\leq i \\leq n - k } = \\langle \\ , S ^ i U \\ , \\rangle _ { 0 \\leq i \\leq n - k } . \\end{align*}"}
-{"id": "2841.png", "formula": "\\begin{align*} s = \\sum _ { j = 1 } ^ { \\tau ( m + k ) } f ^ j e ^ { ( k ) } _ { \\infty , j } \\otimes \\sigma ^ { - m } \\ , \\end{align*}"}
-{"id": "2647.png", "formula": "\\begin{align*} \\epsilon \\phi _ { k } ( x ) = \\begin{cases} \\int _ { \\Omega } \\epsilon ( x , z ) \\phi _ { k } ( z ) d \\nu ( z ) , & k = 1 , \\ldots , N , \\\\ h _ { k - N } ( x ) , & k > N , \\end{cases} \\end{align*}"}
-{"id": "7675.png", "formula": "\\begin{align*} [ e _ 1 , e _ 2 , e _ 3 ] & = t _ 1 c _ 2 e _ 3 ; \\ [ e _ 1 , e _ 3 , e _ 1 ] = ( t _ 1 c _ 1 + t _ 1 c _ 3 ) e _ 3 ; \\ [ e _ 1 , e _ 3 , e _ 2 ] = t _ 1 c _ 4 e _ 3 ; \\ [ e _ 1 , e _ 3 , e _ 3 ] = t _ 1 c _ 5 e _ 3 ; \\\\ [ e _ 2 , e _ 1 , e _ 3 ] & = - t _ 1 c _ 2 e _ 3 ; \\ [ e _ 2 , e _ 3 , e _ 1 ] = t _ 1 c _ 2 e _ 3 ; \\ [ e _ 3 , e _ 1 , e _ 2 ] = - t _ 1 c _ 4 e _ 3 ; \\ [ e _ 3 , e _ 1 , e _ 3 ] = - t _ 1 c _ 5 e _ 3 ; \\\\ [ e _ 3 , e _ 2 , e _ 1 ] & = t _ 1 c _ 4 e _ 3 ; \\ [ e _ 3 , e _ 3 , e _ 3 ] = t _ 1 c _ 5 e _ 3 . \\end{align*}"}
-{"id": "5424.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac 1 n \\log \\lVert A ^ n ( x ) v \\rVert = \\lambda _ i ( A , \\mu ) , \\end{align*}"}
-{"id": "2268.png", "formula": "\\begin{align*} \\mu ( x ) & = \\frac { 1 } { 2 \\pi \\i } ( R ( x - \\i 0 ) - R ( x + \\i 0 ) ) = | \\alpha _ x \\rangle \\langle \\alpha _ x | \\\\ | \\alpha _ x \\rangle & = ( 1 + \\pi ^ 2 ) ^ { - 1 / 2 } \\left ( \\frac { \\mathcal { P } } { x - \\Omega } | E \\rangle + | \\delta _ x \\rangle \\right ) \\end{align*}"}
-{"id": "3417.png", "formula": "\\begin{align*} \\tau _ 0 ^ 1 ( T ) \\ll \\deg ( T ) = w _ 1 ( T ) . \\end{align*}"}
-{"id": "3072.png", "formula": "\\begin{align*} | \\widehat { K _ 1 } ( y ) | & \\leq C | y | \\int _ { | x | \\leq \\frac 2 \\beta } | x | \\frac { 1 } { | x | ^ { n - \\beta } } \\ , d x \\\\ & \\leq \\frac { 2 ^ \\beta } { \\beta + 1 } \\beta ^ { - \\beta } . \\end{align*}"}
-{"id": "9869.png", "formula": "\\begin{align*} \\begin{aligned} \\tau | S _ \\tau | | A | \\le { I } ( Q A \\times S _ \\tau , A ^ { - 1 } \\times R ) & \\ll | Q A | ^ { 5 / 8 } | S _ \\tau | ^ { 1 / 2 } | A | ^ { 2 7 / 1 6 } + | S _ \\tau | ^ { 1 / 2 } | A | ^ 2 \\cdot \\sqrt { \\max \\{ 1 , | Q A | ^ 2 / p \\} } \\\\ & \\ll | Q A | ^ { 5 / 8 } | S _ \\tau | ^ { 1 / 2 } | A | ^ { 2 7 / 1 6 } \\ , . \\end{aligned} \\end{align*}"}
-{"id": "5258.png", "formula": "\\begin{align*} 1 - F ( x ) = c ( x ) \\exp \\left ( \\int _ { x _ 1 } ^ { u e p ( F ) } - \\frac { a ( t ) } { r ( t ) } d t \\right ) , \\ x _ 1 \\geq x < u e p ( F ) , \\end{align*}"}
-{"id": "29.png", "formula": "\\begin{align*} f ^ i ( v , z ) = \\frac 1 2 \\delta ( \\delta - 1 ) ^ 2 \\left ( \\Pi , \\frac { z + \\theta ^ i ( v ) } { 1 - \\delta } \\right ) + \\frac { \\delta } { 2 ( 1 - \\delta ) } | z + \\theta ^ i ( v ) | ^ 2 + \\frac { | z | ^ 2 } { 2 } . \\end{align*}"}
-{"id": "3791.png", "formula": "\\begin{align*} m _ { \\mathcal { M } ( \\alpha ) } ( \\mu ) = \\begin{cases} m _ { \\mathcal { M } ( \\beta ) } ( \\mu ) + ( - 1 ) ^ { R - d - 1 } & \\mu = a 1 ^ d \\\\ m _ { \\mathcal { M } ( \\beta ) } ( \\mu ) & \\mu \\neq a 1 ^ d . \\end{cases} \\end{align*}"}
-{"id": "3075.png", "formula": "\\begin{align*} | I _ 2 | & \\leq \\int _ { \\frac { 1 } { \\beta } \\leq | x | \\leq \\frac { 2 } { \\beta } } \\frac { 1 } { | x | ^ { n - \\beta } } \\ , d x + \\int _ { \\frac { 1 } { 2 \\beta } \\leq | x - z | \\leq \\frac { 1 } { \\beta } } \\frac { 1 } { | x - z | ^ { n - \\beta } } \\ , d x \\\\ & \\leq C \\frac { 2 ^ \\beta - 1 } { \\beta } \\beta ^ { - \\beta } + C \\frac { 1 - 2 ^ { - \\beta } } { \\beta } \\beta ^ { - \\beta } . \\end{align*}"}
-{"id": "2644.png", "formula": "\\begin{align*} B = \\big [ \\beta _ { r , s } \\big ] _ { r , s = 0 } ^ { \\infty } = : \\begin{bmatrix} B _ { 1 1 } & B _ { 1 2 } \\\\ B _ { 2 1 } & B _ { 2 2 } \\end{bmatrix} \\end{align*}"}
-{"id": "6735.png", "formula": "\\begin{align*} \\Box \\Box h _ { k l } = 0 , \\end{align*}"}
-{"id": "9027.png", "formula": "\\begin{align*} f _ { \\{ c _ i \\} , b _ { i + 1 } } ( x ) = \\begin{cases} b _ { i + 1 } & x = c _ i , \\\\ x & \\end{cases} \\end{align*}"}
-{"id": "8162.png", "formula": "\\begin{align*} \\begin{aligned} & Z ( - j ) Z ( - j ) + 2 \\sum _ { i > 0 } Z ( - j - i ) Z ( - j + i ) \\approx 0 , \\\\ & Z ( - j - 1 ) Z ( - j ) + \\sum _ { i > 0 } Z ( j - 1 - i ) Z ( - j + i ) \\approx 0 , \\end{aligned} \\end{align*}"}
-{"id": "6529.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow + \\infty } \\Biggl ( \\int ^ { t + 1 } _ { t } \\bigl \\| g ( s , x ( s ) ) - g ( s , y ( s ) ) \\bigr \\| ^ { p } \\ , d s \\Biggr ) ^ { 1 / p } = 0 \\end{align*}"}
-{"id": "1613.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow \\infty } \\frac { 1 } { T } \\mathbb { E } _ x \\int _ 0 ^ T \\mu \\left ( X _ t ^ { Z ^ { b ^ \\ast } } \\right ) X _ t ^ { Z ^ { b ^ \\ast } } \\ , d t = \\int _ 0 ^ { b ^ \\ast } \\mu ( x ) x \\frac { m ' ( x ) } { m ( ( 0 , b ^ \\ast ) ) } \\ , d x = \\mu ( b ^ \\ast ) b ^ \\ast = \\ell ^ \\ast , \\end{align*}"}
-{"id": "1497.png", "formula": "\\begin{align*} \\widetilde { I V } _ { A , i i } = \\varepsilon \\int _ { 0 } ^ { t } \\int _ { \\mathbb { T } ^ { d } } ( \\partial ^ { \\alpha } \\mu ^ { n + 1 } ) \\sum _ { i = 1 } ^ { d } \\sum _ { j = 1 } ^ { d } \\left ( \\partial _ { x _ { i } } \\left ( ( \\mu ^ { n } + \\bar { m } ) \\Theta _ { p _ { i } p _ { j } } \\right ) \\right ) ( \\partial ^ { \\alpha } \\partial _ { x _ { j } } w ^ { n } ) \\ d x d \\tau . \\end{align*}"}
-{"id": "5787.png", "formula": "\\begin{align*} \\tilde R _ { d - 1 } ( q ) = \\psi ( q ^ { - d } ) * \\tilde R _ d ( q ) , d > 1 , \\end{align*}"}
-{"id": "2709.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\Big ( \\frac { 1 } { \\varphi ( x ) } \\Big ) ^ k R _ { N } ^ { ( k ) } \\Big ( N x + \\frac { u _ 1 } { \\varphi ( x ) } , \\ldots , N x + \\frac { u _ k } { \\varphi ( x ) } \\Big ) = \\det \\Big [ \\frac { \\sin \\pi ( u _ { i } - u _ { j } ) } { \\pi ( u _ { i } - u _ { j } ) } \\Big ] _ { i , j = 1 } ^ { k } \\end{align*}"}
-{"id": "6394.png", "formula": "\\begin{align*} v _ { j + 1 } = ( - 1 ) ^ { j } \\frac { r _ { j } } { \\| r _ { j } \\| } \\ , , j = 0 , \\dots , k . \\end{align*}"}
-{"id": "6990.png", "formula": "\\begin{align*} \\begin{array} { l } 4 \\cos ^ 2 \\left ( \\frac { 2 \\pi r } { p + 1 } \\right ) - 1 = 3 - \\frac { 1 6 \\pi ^ 2 r ^ 2 } { p ^ 2 } + O \\left ( \\frac { r ^ 2 } { p ^ 3 } \\right ) , \\\\ 1 - 4 \\cos \\left ( \\frac { 2 ( p - 1 ) \\pi s } { p ^ 2 + 1 } \\right ) \\cos \\left ( \\frac { 2 ( p + 1 ) \\pi s } { p ^ 2 + 1 } \\right ) = - 3 + \\frac { 1 6 \\pi ^ 2 s ^ 2 } { p ^ 2 } + O \\left ( \\frac { s ^ 2 } { p ^ 3 } \\right ) . \\end{array} \\end{align*}"}
-{"id": "4540.png", "formula": "\\begin{align*} \\int \\int | \\nu _ { g , j } ( x ) | ^ 2 d x d g = \\int \\int | \\hat { \\nu } _ { g , j } ( \\omega ) | ^ 2 d \\omega d g = \\int \\int | \\hat { \\nu } _ g ( \\omega ) | ^ 2 | \\psi ( 2 ^ { - j } \\omega ) | ^ 2 d \\omega d g . \\end{align*}"}
-{"id": "7076.png", "formula": "\\begin{align*} c ( \\vec { e } _ i , \\vec { e } _ j ) = \\begin{cases} 0 , & i = j , \\\\ \\frac { 1 } { 2 ^ k } , & i \\neq j . \\end{cases} \\end{align*}"}
-{"id": "3239.png", "formula": "\\begin{align*} 2 \\Delta x ^ { \\top } H _ P ( x , V ) \\Delta x & = \\Delta F \\bullet \\left ( \\mathcal { L } _ { F _ P ( x ) } ^ { - 1 } \\mathcal { L } _ { { V } _ P } + \\mathcal { L } _ { V _ P } \\mathcal { L } _ { F _ P ( x ) } ^ { - 1 } \\right ) \\Delta F \\\\ & = \\mathcal { L } _ { F _ P ( x ) } ^ { - 1 } ( \\Delta F ) \\bullet \\left ( \\mathcal { L } _ { V _ P } \\mathcal { L } _ { F _ P ( x ) } + \\mathcal { L } _ { F _ P ( x ) } \\mathcal { L } _ { V _ P } \\right ) \\mathcal { L } _ { F _ P ( x ) } ^ { - 1 } ( \\Delta F ) > 0 \\end{align*}"}
-{"id": "2920.png", "formula": "\\begin{align*} \\Pi _ 0 ^ \\bullet ( \\psi ) u : = \\psi | _ M \\ , u \\ , , \\quad \\forall \\ , \\psi \\in C ( X ) \\ , , \\ , u \\in L ^ 2 ( M ) \\ , . \\end{align*}"}
-{"id": "2726.png", "formula": "\\begin{align*} \\mathcal { C } ^ { \\mathrm { l o c a l } } = \\left \\{ z \\in \\mathcal { C } ^ { 2 } : \\abs { z - z _ { + } } < \\delta _ { 2 } \\ , \\ , \\abs { z - z _ { - } } < \\delta _ { 2 } \\right \\} . \\end{align*}"}
-{"id": "4183.png", "formula": "\\begin{align*} \\frac { A _ n ^ { ( 2 ) } ( z ) } { n ^ 6 z } & = \\mathcal { O } ( 1 ) , \\frac { A _ n ^ { ( 1 ) } ( 0 ) A _ n ^ { ( 1 ) } ( z ) A _ n ^ { ( 1 ) } ( 0 ) } { n ^ 9 z ^ 3 } = \\mathcal { O } ( n ^ { - \\frac { 1 } { 2 } } ) . \\end{align*}"}
-{"id": "9215.png", "formula": "\\begin{align*} e ( G ' ) & \\leq \\max \\left \\{ \\binom { k } { 2 } , \\binom { \\frac { k + 1 } { 2 } } { 2 } + \\frac { k - 1 } { 2 } \\left ( n - \\frac { k + 1 } { 2 } \\right ) \\right \\} \\\\ & = \\max \\left \\{ \\binom { k } { 2 } , \\binom { \\frac { k - 1 } { 2 } } { 2 } + \\frac { k - 1 } { 2 } \\left ( n - \\frac { k - 1 } { 2 } \\right ) \\right \\} . \\end{align*}"}
-{"id": "8589.png", "formula": "\\begin{align*} \\zeta ( z ) & = \\sin ( \\pi z / 2 ) \\pi ^ { z - 1 } \\int \\limits _ 1 ^ \\infty d s \\ , s ^ { - z - 1 } ( s - 1 ) + E ( z ) \\\\ & = \\sin ( \\pi z / 2 ) \\pi ^ { z - 1 } \\left [ \\frac 1 { z - 1 } - \\frac 1 z \\right ] + E ( z ) , \\end{align*}"}
-{"id": "8110.png", "formula": "\\begin{align*} \\widetilde { A } _ n ( t ) \\ = \\ 1 \\ , + \\ , t \\ , \\sum _ { m = 1 } ^ n { n \\choose m } A _ m ( t ) , \\end{align*}"}
-{"id": "6982.png", "formula": "\\begin{align*} \\mathsf { B D } _ n = \\langle a , x \\ ; | \\ ; a ^ { 2 n } = 1 , x ^ 2 = a ^ n , x ^ { - 1 } a x = a ^ { - 1 } \\rangle \\end{align*}"}
-{"id": "7921.png", "formula": "\\begin{align*} R _ N = \\hat { R } _ n \\frac { ( b - a ) ^ { n + 2 } f ^ { ( n + 1 ) } ( \\xi ) } { M ^ { n + 1 } } \\end{align*}"}
-{"id": "7322.png", "formula": "\\begin{align*} \\begin{cases} u - w = p , \\\\ v - z = q , \\\\ w - \\frac { 1 } { \\rho } \\operatorname { d i v } ( K ( x ) \\nabla u + a ( x ) \\nabla w ) = r , \\\\ z - \\frac { 1 } { \\rho } \\operatorname { d i v } ( K ( x ) \\nabla v + b ( x ) \\nabla z ) = s , \\end{cases} \\end{align*}"}
-{"id": "3390.png", "formula": "\\begin{align*} f _ { \\mathcal { S } } ( T ) = \\begin{cases} f ( T ) \\ , & \\ , | T | \\in \\mathcal { S } , \\\\ 0 \\ , & \\end{cases} \\end{align*}"}
-{"id": "9671.png", "formula": "\\begin{align*} X x _ 0 = g h _ { 0 1 } | _ \\sigma . \\end{align*}"}
-{"id": "3081.png", "formula": "\\begin{align*} | K | & \\leq \\int _ { 2 | z | \\leq | x | \\leq 3 | z | } \\frac { 1 } { | x | ^ { n - \\beta } } \\ , d x \\\\ & \\leq \\frac { ( \\frac 3 2 ) ^ { \\beta } - 1 } { \\beta } \\beta ^ { - \\beta } . \\end{align*}"}
-{"id": "683.png", "formula": "\\begin{align*} H _ { \\mathbb G } ( [ \\mu _ m ; f ; x , e _ m ] ) = \\left [ \\mu _ m ; \\gamma _ { \\mathcal N } \\left ( \\Theta ( H ) ( f ) ; \\lambda _ { \\vec a , x ^ { - 1 } ( 1 ) } ^ { - 1 } , \\dots , \\lambda _ { \\vec a , x ^ { - 1 } ( m ) } ^ { - 1 } \\right ) ; x , e _ m \\right ] \\ . \\end{align*}"}
-{"id": "9490.png", "formula": "\\begin{align*} u _ { M } = g _ { \\beta , M } ( u ) \\in \\mathcal { W } \\cap L ^ { \\infty } ( \\mathbb { R } ^ N ) . \\end{align*}"}
-{"id": "8791.png", "formula": "\\begin{align*} \\left [ \\begin{array} { l l } \\partial _ t u ( t , x ) = \\partial _ { x x } u ( t , x ) + f ( u ( t , x ) ) \\dot { W } ( t , x ) , & ( t , x ) \\in [ 0 , T ] \\times [ 0 , \\pi ] , \\\\ u ( t , 0 ) = u ( t , \\pi ) = 0 , & \\textrm { f o r a l l } \\ , \\ , t \\in [ 0 , T ] , \\\\ u ( 0 , x ) = 0 , & \\textrm { f o r a l l } \\ , \\ , x \\in [ 0 , \\pi ] . \\end{array} \\right . \\end{align*}"}
-{"id": "4923.png", "formula": "\\begin{align*} \\ell ( H ^ i ( f _ 1 , \\ldots , f _ d ; M ) ) \\leq \\sum _ { j = 0 } ^ { i } \\ell ( H ^ { d - i + j } ( f _ 1 , \\ldots , f _ d ; H ^ j _ m ( M ) ^ \\vee ) ) . \\end{align*}"}
-{"id": "5967.png", "formula": "\\begin{align*} X _ k ( t ) = \\int _ 0 ^ t q _ k ( s ) d s + W _ k ( t ) , \\end{align*}"}
-{"id": "4792.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } \\int _ { \\{ \\abs { x } > \\epsilon \\} } \\left [ \\abs { x } \\abs * { \\left ( - i \\alpha \\cdot \\nabla + m \\beta \\right ) \\psi _ C ^ 0 } ^ 2 - \\frac { | \\psi _ C ^ 0 | ^ 2 } { | x | } \\right ] \\ , d x = 0 . \\end{align*}"}
-{"id": "1610.png", "formula": "\\begin{align*} \\begin{aligned} Z _ t ^ { b ^ \\ast } & = \\begin{cases} \\left ( x - b ^ \\ast \\right ) ^ + & \\mbox { i f $ t = 0 $ , } \\\\ L ( t , b ^ \\ast ) & \\mbox { i f $ t > 0 $ } \\end{cases} \\end{aligned} \\end{align*}"}
-{"id": "9119.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\tau \\equiv t - t _ { o } \\rightarrow \\tau ^ { \\prime } \\equiv t ^ { \\prime } - t _ { o } = - t + t _ { o } \\equiv - \\tau , \\\\ \\mathbf { r } _ { 1 } \\rightarrow \\mathbf { r } _ { 1 } ^ { \\prime } = \\mathbf { r } _ { 1 } , \\\\ \\mathbf { v } _ { 1 } \\rightarrow \\mathbf { v } _ { 1 } ^ { \\prime } = - \\mathbf { v } _ { 1 } . \\end{array} \\right . \\end{align*}"}
-{"id": "8917.png", "formula": "\\begin{align*} \\mathbb { E } [ T ^ { \\hat { x } } ( G ' ) - T ^ { \\hat { x } } ( H ) \\mid \\hat { x } ] = \\frac { ( a - b ) } { n } \\Big { ( } & n ( \\textrm { U n c h a n g e d } ) - 2 n ( \\textrm { I n c o r r e c t , U n c h a n g e d } ) \\Big { ) } \\\\ & \\times \\Big { ( } n ( \\textrm { C h a n g e d } ) - 2 n ( \\textrm { I n c o r r e c t , C h a n g e d } ) \\Big { ) } , \\end{align*}"}
-{"id": "9704.png", "formula": "\\begin{align*} \\mathbf { d } _ 1 \\equiv _ t \\mathbf { d } _ 2 \\Leftrightarrow \\forall \\gamma \\in H ( M ) = H _ { g } ( M ) \\cup \\partial M , T _ { \\mathbf { d } _ 1 } ( \\gamma ) = T _ { \\mathbf { d } _ 2 } ( \\gamma ) . \\end{align*}"}
-{"id": "4512.png", "formula": "\\begin{align*} | \\langle x - y , y - z \\rangle | = | x - y | | y - z | | \\sin \\theta ( x - y , y - z ) | > r . \\end{align*}"}
-{"id": "4256.png", "formula": "\\begin{align*} H _ { 1 } : = 1 0 ^ { 5 n } \\sigma ^ { - 2 n } . \\end{align*}"}
-{"id": "2957.png", "formula": "\\begin{align*} g ( n ) = 3 ( f ( n - 1 ) + 1 ) \\end{align*}"}
-{"id": "8943.png", "formula": "\\begin{align*} q p _ { b l } ( T x , T y ) & = \\Big ( 3 x \\sqrt { 1 + x ^ 2 } + 3 y \\sqrt { 1 + y ^ 2 } ) ^ 2 \\\\ & \\geq ( 3 x + 3 y ) ^ 2 \\\\ & = \\frac { 9 } { 2 } [ ( x + y ) ^ 2 + ( y + x ) ^ 2 ] . \\end{align*}"}
-{"id": "3156.png", "formula": "\\begin{align*} Z ^ q \\bigl ( \\mu ( \\bar x ) | \\mu ( \\bar t ) \\bigr ) = Z ^ q ( \\bar x | \\bar t ) \\prod _ { k = 1 } ^ { n - 2 } f ^ q ( \\bar x ^ { k + 1 } , \\bar x ^ { k } ) f ^ q ( \\bar t ^ { k + 1 } , \\bar t ^ { k } ) , \\end{align*}"}
-{"id": "8884.png", "formula": "\\begin{align*} \\tilde { s } _ { \\alpha } = s _ { \\alpha } + \\epsilon \\acute { s } _ { \\alpha } . \\end{align*}"}
-{"id": "9789.png", "formula": "\\begin{align*} S _ { \\kappa , \\lambda } ^ h : = \\{ X _ \\circ \\in E _ h : N ( 0 ^ + , u ( X _ \\circ + \\ , \\cdot \\ , ) - p _ { * , X _ \\circ } ) \\geq \\lambda \\} \\end{align*}"}
-{"id": "1669.png", "formula": "\\begin{align*} & B _ { t } ( m , \\ell ^ { 2 n + 1 } d ) - \\ell ^ { \\lambda - 1 } \\left ( \\frac { ( - 1 ) ^ { \\lambda } m } { \\ell } \\right ) B _ { t } ( m , \\ell ^ { 2 n - 1 } d ) \\\\ = & B _ { t } ( m , \\ell ^ { 2 n } ( \\ell d ) ) - \\ell ^ { \\lambda - 1 } \\left ( \\frac { ( - 1 ) ^ { \\lambda } m } { \\ell } \\right ) B _ { t } ( m , \\ell ^ { 2 n - 2 } ( \\ell d ) ) \\\\ = & \\ell ^ { ( k - 2 ) n } \\ , B _ { t } ( \\ell ^ { 2 n } m , \\ell d ) \\equiv 0 \\pmod { \\ell ^ { ( k - 2 ) n } } . \\end{align*}"}
-{"id": "8543.png", "formula": "\\begin{align*} \\theta _ k = ( I - \\mathcal { H } ) \\partial _ { \\alpha } ^ k \\tilde { \\theta } , \\sigma _ k = ( I - \\mathcal { H } ) \\partial _ { \\alpha } ^ k \\tilde { \\sigma } , \\tilde { \\theta } : = ( I - \\mathcal { H } ) ( \\zeta - \\bar { \\zeta } ) , \\tilde { \\sigma } : = ( I - \\mathcal { H } ) D _ t \\tilde { \\theta } . \\end{align*}"}
-{"id": "5544.png", "formula": "\\begin{align*} ( \\partial _ t + m ^ 2 - \\Delta ) \\varphi + \\lambda \\varphi ^ 3 - \\infty \\varphi = \\xi , ( t , x ) \\in \\mathbb { R } _ + \\times \\mathbb { R } ^ 3 , \\end{align*}"}
-{"id": "3686.png", "formula": "\\begin{align*} E ( \\bar { y } _ S | x _ S , S ) = E ( \\bar { y } _ S | S ) \\neq E ( \\bar { Y } ) = E ( \\bar { Y } | x _ U ) \\end{align*}"}
-{"id": "963.png", "formula": "\\begin{align*} \\begin{array} { r l } \\displaystyle \\min _ { z _ 0 , \\ldots , z _ T \\in \\real ^ n } & \\frac { 1 } { 2 } \\left ( \\| z _ 0 - b \\| ^ 2 _ { ( B ^ { \\infty } ) ^ { - 1 } } + \\sum _ { t = 0 } ^ { T } \\| y _ t - \\mathcal { H } _ t ( z _ { t } ) \\| ^ 2 _ { R _ t ^ { - 1 } } \\right ) , \\\\ \\mbox { s . t . } & z _ t = \\mathcal { M } _ t ( z _ { t - 1 } ) , ~ t = 1 , \\ldots , T . \\end{array} \\end{align*}"}
-{"id": "2016.png", "formula": "\\begin{align*} F ( j ) = \\left ( \\frac { 5 \\alpha _ j } { \\prod \\limits _ { j ' \\neq j } ( \\alpha _ j - \\alpha _ { j ' } ) } \\right ) ^ { - 1 } , \\end{align*}"}
-{"id": "3883.png", "formula": "\\begin{align*} d ( \\omega ) : = \\inf \\{ S _ \\omega ( u , v ) \\ : \\ ( u , v ) \\in H ^ 1 \\times H ^ 1 \\backslash \\{ ( 0 , 0 ) \\} , K _ \\omega ( u , v ) = 0 \\} . \\end{align*}"}
-{"id": "2147.png", "formula": "\\begin{align*} T _ { j , \\mathbf x } E ( \\mathbf x , \\mathbf y ) = y _ j E ( \\mathbf x , \\mathbf y ) , \\end{align*}"}
-{"id": "3625.png", "formula": "\\begin{align*} \\beta _ { - 1 } \\cdot \\alpha _ 0 = v _ { ( 1 , 2 ) } \\cdot ( a _ 1 \\cdot v _ { ( 2 , 3 ) } ) & - \\frac { 3 t } { 2 ^ 3 } a _ 1 + \\frac { t } { 2 ^ 2 } ( a _ 2 + a _ { - 2 } ) + \\frac { 3 } { 2 ^ 4 } ( a _ 1 + a _ { - 1 } ) \\cdot v _ { ( 2 , 3 ) } \\\\ & - \\frac { 1 } { 2 ^ 3 } ( a _ 2 + a _ { - 2 } ) \\cdot v _ { ( 1 , 3 ) } + \\frac { 1 } { 2 ^ 2 } ( a _ 3 + a _ { - 3 } ) \\cdot v _ { ( 1 , 2 ) } . \\end{align*}"}
-{"id": "5995.png", "formula": "\\begin{align*} L _ n ( \\omega ) : = \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\delta _ { \\omega _ { ( n , k ) } } . \\end{align*}"}
-{"id": "9863.png", "formula": "\\begin{align*} ( a - d ) \\gamma + \\tau ( 1 - a ) + c t = 0 \\ , . \\end{align*}"}
-{"id": "645.png", "formula": "\\begin{align*} x _ \\ast \\vec a : = a _ { x ^ { - 1 } ( 1 ) } \\dots a _ { x ^ { - 1 } ( n ) } \\ . \\end{align*}"}
-{"id": "9652.png", "formula": "\\begin{align*} \\Bigl \\| \\frac { \\psi } { \\psi _ t } - \\frac { \\psi } { \\psi _ t } ( 0 ) \\Bigr \\| _ { H ^ 2 } ^ 2 = \\Bigl \\| \\frac { \\psi } { \\psi _ t } \\Bigr \\| _ { H ^ 2 } ^ 2 - \\Bigl | \\frac { \\psi } { \\psi _ t } ( 0 ) \\Bigr | ^ 2 \\to 0 \\ \\ t \\to 0 . \\end{align*}"}
-{"id": "2502.png", "formula": "\\begin{align*} h _ { S / I } ( \\lambda ) = h _ 0 + h _ 1 \\lambda + h _ 2 \\lambda ^ 2 + \\cdots + h _ s \\lambda ^ s \\end{align*}"}
-{"id": "8516.png", "formula": "\\begin{align*} \\norm { \\partial _ { \\alpha } ^ k \\tilde { \\theta } - 2 \\partial _ { \\alpha } ^ { k - 1 } ( \\zeta _ { \\alpha } - 1 ) } _ { L ^ 2 } \\leq C \\epsilon ^ 2 . \\end{align*}"}
-{"id": "4398.png", "formula": "\\begin{align*} | d T | ^ 2 _ { \\omega _ o } = 1 + 4 | x y | ^ 2 + | y ^ 2 | ^ 2 . \\end{align*}"}
-{"id": "8481.png", "formula": "\\begin{align*} \\dot { z } _ 2 = \\frac { \\lambda i } { 4 \\pi x ( t ) } + \\bar { F } ( z _ 2 , t ) . \\end{align*}"}
-{"id": "9502.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { c l l l } ( - \\Delta _ { p } ) ^ { s } u + ( - \\Delta _ { q } ) ^ { s } u = \\vert u \\vert ^ { p _ { s } ^ * - 2 } u + \\lambda g ( x ) \\vert u \\vert ^ { r - 2 } u \\ , \\ , \\ , \\ , \\ , \\ , \\mathbb { R } ^ { N } \\\\ u ( x ) \\geq 0 \\ , \\ , \\ , \\ , x \\in \\mathbb { R } ^ { N } . \\end{array} \\right . \\end{align*}"}
-{"id": "8410.png", "formula": "\\begin{align*} d _ P ( t ) ^ { - 1 } = \\max _ { 1 \\leq j \\neq k \\leq N } \\frac { 1 } { | z _ j ( t ) - z _ k ( t ) | } . \\end{align*}"}
-{"id": "2534.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\eta _ t ^ { - 1 } \\mathbf P _ { \\mu } ( \\| X _ t \\| \\neq 0 ) = \\mu ( \\phi ) . \\end{align*}"}
-{"id": "8337.png", "formula": "\\begin{align*} ( I - \\mathbb { H } ) \\frac { 1 } { Z ( \\alpha , t ) - z _ 0 } = ( I - \\mathbb { H } ) \\frac { 1 } { c _ 1 ( \\alpha - w _ 0 ) } = \\frac { 2 } { c _ 1 ( \\alpha - w _ 0 ) } . \\end{align*}"}
-{"id": "8877.png", "formula": "\\begin{align*} e _ { 9 a } & = e _ 9 | _ { b = 0 } = 2 s ^ 3 p ^ 2 - 1 8 p ^ 2 s ^ 2 + 3 0 s p ^ 2 + 9 9 s ^ 2 p - 1 2 p ^ 2 - 1 9 2 p s - 3 s ^ 2 + 9 3 p + 3 \\\\ & = 2 p ^ 2 + ( 1 - s ) [ 6 ( 1 - p ) + ( 1 - s ) ( 9 9 p + 2 p ^ 2 s - 1 4 p ^ 2 - 3 ) ] \\ge 0 \\end{align*}"}
-{"id": "1453.png", "formula": "\\begin{align*} w ( t , \\cdot ) = w _ { T } + \\int _ { t } ^ { T } \\left [ \\Delta w ( \\tau , \\cdot ) + \\varepsilon P \\Theta ( \\tau , \\cdot , \\mu , D w ) \\right ] \\ d \\tau . \\end{align*}"}
-{"id": "397.png", "formula": "\\begin{align*} S ^ { { \\rm c l } } _ { \\rm M } ( \\hat g , g ) = \\int _ M \\Big ( 2 \\pi ( 1 - \\textbf { h } ) \\phi \\Delta _ g \\phi + ( \\frac { 8 \\pi ( 1 - \\textbf { h } ) } { V _ { g } } - K _ { g } ) \\phi + \\frac { 2 } { V _ { \\hat g } } \\omega e ^ { \\omega } \\Big ) d { \\rm v } _ { g } \\end{align*}"}
-{"id": "8928.png", "formula": "\\begin{align*} P ( \\eta _ 1 = k - j , \\eta _ 2 = j \\mid d ( \\hat { x } , x ) = k ) = \\frac { \\binom { n - s } { k - j } \\binom { s } { j } } { \\binom { n } { k } } . \\end{align*}"}
-{"id": "1638.png", "formula": "\\begin{align*} W _ k ^ \\R \\otimes \\C = V _ k ^ \\C \\oplus _ \\C V _ { - k } ^ \\C \\quad W _ 0 ^ \\R \\otimes \\C = V _ 0 ^ \\C . \\end{align*}"}
-{"id": "217.png", "formula": "\\begin{align*} \\mathcal { B } _ \\beta u = V u \\end{align*}"}
-{"id": "9920.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } f _ { m _ { i } } ( x ) = \\tilde { f } ( x ) ~ ~ \\forall x \\in \\mathcal { X } \\end{align*}"}
-{"id": "9138.png", "formula": "\\begin{align*} \\lim _ { \\tau \\rightarrow + \\infty } \\frac { \\partial } { \\partial t } I _ { M } ( \\rho _ { 1 } ^ { ( N ) } ( \\mathbf { x } _ { 1 } , t ) , \\mathbf { b } ) = W _ { M } ( \\rho _ { 1 \\infty } ^ { ( N ) } ( \\mathbf { x } _ { 1 } ) , \\mathbf { b } ) \\end{align*}"}
-{"id": "442.png", "formula": "\\begin{align*} u ( y - x ) = \\frac { 1 - e ^ { - 2 \\gamma ( x - y ) } } { 1 - e ^ { - 2 \\gamma ( A + \\eta t ) } } \\le C | x - y | \\end{align*}"}
-{"id": "1111.png", "formula": "\\begin{align*} & K ^ { G U E ( n ) } ( x , y ) = \\sqrt { n } \\frac { \\psi _ { n } ( x \\sqrt { n } ) \\psi _ { n - 1 } ( y \\sqrt { n } ) - \\psi _ { n - 1 } ( x \\sqrt { n } ) \\psi _ { n } ( y \\sqrt { n } ) } { x - y } . \\end{align*}"}
-{"id": "7721.png", "formula": "\\begin{align*} K = \\overline { K \\cap U _ { 1 } } \\cup \\ldots \\cup \\overline { K \\cap U _ l } = \\left ( \\bigcap _ { k = 1 } ^ { \\infty } V ^ 1 _ { k } \\right ) \\cup \\ldots \\cup \\left ( \\bigcap _ { k = 1 } ^ { \\infty } V ^ l _ { k } \\right ) \\subset \\bigcap _ { k = 1 } ^ { \\infty } \\left ( V ^ 1 _ { k } \\cup \\ldots \\cup V ^ l _ { k } \\right ) . \\end{align*}"}
-{"id": "1465.png", "formula": "\\begin{align*} V _ { 6 } = - \\varepsilon \\int _ { \\mathbb { T } ^ { d } } ( \\mu ^ { 1 } - \\mu ^ { 2 } ) ( \\mu ^ { 2 } ) \\sum _ { i = 1 } ^ { d } \\left [ \\Theta _ { p _ { i } q } ( t , x , \\mu ^ { 1 } , D w ^ { 1 } ) \\frac { \\partial \\mu ^ { 1 } } { \\partial x _ { i } } - \\Theta _ { p _ { i } q } ( t , x , \\mu ^ { 2 } , D w ^ { 2 } ) \\frac { \\partial \\mu ^ { 1 } } { \\partial x _ { i } } \\right ] \\ d x , \\end{align*}"}
-{"id": "8438.png", "formula": "\\begin{align*} D _ t \\bar { \\zeta } = & D _ t ( \\bar { \\zeta } - \\alpha ) + b = D _ t \\zeta \\Psi _ { \\zeta } \\circ \\zeta + \\Psi _ t \\circ \\zeta + b . \\end{align*}"}
-{"id": "6162.png", "formula": "\\begin{align*} & \\sum ^ { T - 1 } _ { t = 0 } E _ t = \\sum ^ { T - 1 } _ { t = 0 } \\bigg [ \\frac { 7 M _ x D _ x + 7 M _ y D _ y } { 2 ( t + 3 ) } + \\frac { Q _ g + Q _ r } { ( t + 3 ) ^ 2 } \\bigg ] \\\\ \\leq & \\ln ( T + 1 ) \\left ( 3 . 5 M _ x D _ x + 3 . 5 M _ y D _ y \\right ) + 2 Q _ g + 2 Q _ r . \\end{align*}"}
-{"id": "2848.png", "formula": "\\begin{align*} F ( u , v ) = \\pm \\frac { d ^ { 6 } m } { n } \\end{align*}"}
-{"id": "3556.png", "formula": "\\begin{align*} \\left \\{ [ x , y , z , u , v , 1 ] \\in \\mathbb { P } ^ { 5 } \\mid x y z = 1 \\ , u v \\right \\} \\subset \\mathcal { M } _ { 3 , 3 } . \\end{align*}"}
-{"id": "5932.png", "formula": "\\begin{align*} t _ { i } ( s ) = \\frac { 1 - e _ i - ( 1 - s ) f _ i } { g _ i + s f _ i } . \\end{align*}"}
-{"id": "1393.png", "formula": "\\begin{align*} C ^ n _ { } ( \\varphi , \\psi ) ( f ) ( \\lambda _ 0 , \\ldots , \\lambda _ { n - 1 } ) & = \\psi \\left ( f \\left ( \\varphi ( \\lambda _ 0 ) , \\ldots , \\varphi ( \\lambda _ { n - 1 } ) \\right ) \\right ) \\\\ C ^ n _ { \\operatorname { c u b } } ( \\varphi , \\psi ) ( g ) ( \\lambda ) & = \\psi \\left ( g \\left ( \\varphi ( \\lambda ) \\right ) \\right ) \\end{align*}"}
-{"id": "7454.png", "formula": "\\begin{align*} \\displaystyle \\lvert \\tau \\rvert _ E ^ { - \\dim ( N ' _ S ) / 2 } O _ s ( \\xi _ { + } , \\varphi ) = \\int _ { ( F ^ \\times ) ^ n } f _ \\varphi ( a _ 1 , \\ldots , a _ n ) \\prod _ { k = 1 } ^ n \\eta _ { E / F } ( a _ k ) ^ k \\lvert a _ k \\rvert ^ { k s - ( k - 1 ) } d ^ \\times a _ 1 \\ldots d ^ \\times a _ n \\end{align*}"}
-{"id": "1395.png", "formula": "\\begin{align*} f ( x _ 1 , \\ldots , x _ d ) & = h ( p _ 1 ( x _ 1 ) + \\cdots + p _ d ( x _ d ) ) ~ ~ \\\\ f ( x _ 1 , \\ldots , x _ d ) & = h ( p _ 1 ( x _ 1 ) \\cdot \\ldots \\cdot p _ d ( x _ d ) ) , \\end{align*}"}
-{"id": "6327.png", "formula": "\\begin{align*} \\tfrac { d } { d t } \\mathcal { E } ( q | q _ { \\min } ) = - \\int _ 0 ^ 1 { w _ t w _ { x x } \\ , d x } = - \\int _ 0 ^ 1 { q ^ 2 w ^ 2 _ { x x } \\ , d x } \\leqslant 0 . \\end{align*}"}
-{"id": "4104.png", "formula": "\\begin{align*} \\mathbb K ^ { ( \\alpha , \\frac { 1 } { 2 } ) } ( x , y ) = \\frac { - 1 } { x - y } \\begin{pmatrix} \\vartheta _ y ^ 2 g ( y ) & \\vartheta _ y g ( y ) & g ( y ) \\end{pmatrix} \\begin{pmatrix} 1 & 0 & 0 \\\\ - 2 \\alpha - \\frac { 1 } { 2 } & 1 & 0 \\\\ \\alpha ( \\alpha + \\frac { 1 } { 2 } ) & - 2 \\alpha - \\frac { 1 } { 2 } & 1 \\end{pmatrix} \\begin{pmatrix} f ( x ) \\\\ - \\vartheta _ x f ( x ) \\\\ \\vartheta _ x ^ 2 f ( x ) \\end{pmatrix} \\end{align*}"}
-{"id": "4533.png", "formula": "\\begin{align*} \\int 1 _ E ( x _ k , y _ k ) d \\mu ( x _ k ) d \\mu ( y _ k ) = \\nu _ g ( B _ { \\delta } ( x _ 1 - g y _ 1 ) ) . \\end{align*}"}
-{"id": "3976.png", "formula": "\\begin{align*} g _ k ( m ) = \\sum _ { y ^ 2 \\mid m } \\mu ( y ) \\sigma _ { k - 1 } ( m / y ^ 2 ) . \\end{align*}"}
-{"id": "1466.png", "formula": "\\begin{align*} V _ { 7 } = - \\varepsilon \\int _ { \\mathbb { T } ^ { d } } ( \\mu ^ { 1 } - \\mu ^ { 2 } ) ( \\mu ^ { 2 } ) \\sum _ { i = 1 } ^ { d } \\left [ \\Theta _ { p _ { i } q } ( t , x , \\mu ^ { 2 } , D w ^ { 2 } ) \\left ( \\frac { \\partial ( \\mu ^ { 1 } - \\mu ^ { 2 } ) } { \\partial x _ { i } } \\right ) \\right ] \\ d x , \\end{align*}"}
-{"id": "8563.png", "formula": "\\begin{align*} \\kappa _ { \\alpha } ( \\alpha , t ) - \\kappa _ { \\alpha } ( \\alpha , 0 ) = & \\int _ 0 ^ t \\kappa _ { \\alpha \\tau } ( \\alpha , \\tau ) d \\tau \\\\ = & \\int _ 0 ^ t b _ { \\alpha } \\circ \\kappa ( \\alpha , \\tau ) \\kappa _ { \\alpha } ( \\alpha , \\tau ) d \\tau . \\end{align*}"}
-{"id": "271.png", "formula": "\\begin{align*} \\eta ( n ) = \\prod _ { i = 1 } ^ n a _ { X _ i } \\eta ( 0 ) \\end{align*}"}
-{"id": "2473.png", "formula": "\\begin{align*} f ( x , y ) = y ^ m + u _ 1 ( x ) y ^ { m - 1 } + \\dots + u _ { m - 1 } ( x ) y + u _ m ( x ) , \\end{align*}"}
-{"id": "9229.png", "formula": "\\begin{align*} \\alpha _ i = \\frac { { y _ { i 0 } } / { p _ { i 0 } - x _ 0 ^ * } } { \\sum _ j { y _ { i 0 } } / { p _ { i 0 } - x _ 0 ^ * } } , \\end{align*}"}
-{"id": "8788.png", "formula": "\\begin{align*} \\partial _ t u ( t , x ) = \\partial _ { x x } u ( t , x ) - u ( t , x ) + \\dot W ( t , x ) , ( t , x ) \\in [ 0 , T ] \\times [ 0 , \\pi ] , \\end{align*}"}
-{"id": "3667.png", "formula": "\\begin{align*} E ( \\sum _ { i \\in B } y _ i / p ) = \\sum _ { i \\in U } E ( \\delta _ i ; y _ i , i \\in U ) y _ i / p = \\sum _ { i \\in U } p y _ i / p = Y \\end{align*}"}
-{"id": "6949.png", "formula": "\\begin{align*} \\overline { \\mathbb K } = { \\mathbb K } \\cup { M ^ { N \\times N } _ { s k e w } } . \\end{align*}"}
-{"id": "5537.png", "formula": "\\begin{align*} L _ b ^ * v _ h = - h . \\nabla \\mu _ b - d i v ( h ) \\mu _ b = - \\sum _ { j = 1 } ^ d \\frac { \\partial } { \\partial x _ j } ( h _ j \\mu _ b ) \\equiv f _ h \\end{align*}"}
-{"id": "6514.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\left ( \\| [ S _ n ] \\| ^ { - 1 } \\cdot I _ n \\right ) & \\geq \\liminf _ { n \\to \\infty } \\int \\bigwedge _ { j = 1 } ^ m ( \\pi _ j ) ^ * \\left ( \\widehat { T } ^ k \\right ) \\wedge \\frac { [ S _ n ] } { \\| [ S _ n ] \\| } \\\\ & \\geq \\int \\bigwedge _ { j = 1 } ^ m ( \\pi _ j ) ^ * \\left ( \\widehat { T } ^ k \\right ) \\wedge S . \\end{align*}"}
-{"id": "8686.png", "formula": "\\begin{align*} \\widehat { u } ^ C _ { \\lambda , \\nu } : = \\sum _ { h + i + 2 j = k } c _ { h , i , j } ( \\lambda , \\nu ) \\abs { X ' } ^ { 2 h } \\abs { X '' } ^ { 2 i } \\abs { Z } ^ { 2 j } , \\end{align*}"}
-{"id": "2877.png", "formula": "\\begin{align*} \\{ \\Gamma _ { i j } , \\Gamma _ { j k } \\} = \\Gamma _ { i k } + 2 \\Gamma _ { j } \\Gamma _ { i j k } + 2 \\Gamma _ { i } \\Gamma _ { k } , \\end{align*}"}
-{"id": "2061.png", "formula": "\\begin{align*} \\langle \\sqrt { - 1 } \\partial \\bar { \\partial } \\log W , \\frac { 1 } { \\sqrt { - 1 } } v \\wedge \\bar { v } \\rangle = R ^ M _ { 1 \\bar { 1 } v \\bar { v } } - R ^ N _ { 1 \\bar { 1 } \\partial f ( v ) \\overline { \\partial f ( v ) } } + \\frac { \\sum _ { i \\ne 1 } | f ^ i _ { 1 v } | ^ 2 } { W } . \\end{align*}"}
-{"id": "4489.png", "formula": "\\begin{align*} \\mu = \\frac { \\underline u + \\nabla _ { 1 1 } \\underline u - M } { \\Pi _ { 1 1 } } \\end{align*}"}
-{"id": "4112.png", "formula": "\\begin{align*} \\begin{aligned} & 2 U ^ { \\mu ^ * } - U ^ { \\nu ^ * } + V \\begin{cases} = - \\ell & [ 0 , q ] , \\\\ > - \\ell & ( q , \\infty ) , \\end{cases} \\\\ & 2 U ^ { \\nu ^ * } = U ^ { \\mu ^ * } ( - \\infty , 0 ] . \\end{aligned} \\end{align*}"}
-{"id": "1546.png", "formula": "\\begin{align*} \\widetilde { F } ^ \\delta \\begin{pmatrix} z \\\\ x \\end{pmatrix} = \\begin{pmatrix} z + z ^ 2 + a _ 3 ^ \\delta z ^ 2 + \\dots + a _ l ^ \\delta z ^ l + a _ { l + 1 } ^ \\delta ( x ) z ^ { l + 1 } + \\dots \\\\ b _ 0 ( x ) + b _ 1 ^ \\delta ( x ) z + b _ 2 ^ \\delta ( x ) z ^ 2 + \\dots \\end{pmatrix} , \\end{align*}"}
-{"id": "8246.png", "formula": "\\begin{align*} S _ { C 5 } ( q ) & = - \\lim _ { z \\rightarrow 1 } \\frac { F ( z ) - F ( 1 ) } { ( z - 1 ) ^ 2 } \\cdot \\frac { 1 } { ( - q ; q ) _ \\infty ( q ; q ) _ \\infty ^ 2 } \\\\ & = - \\frac { F '' ( 1 ) } { 2 ( - q , q , q ; q ) _ \\infty } \\\\ & = \\frac { 1 } { ( q ^ 2 ; q ^ 2 ) _ \\infty } \\sum _ { n = 0 } ^ \\infty \\frac { q ^ { 2 n + 1 } } { ( 1 - q ^ { 2 n + 1 } ) ^ 2 } . \\end{align*}"}
-{"id": "7557.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { \\tau \\circ X ^ { - 1 } } _ { L ^ \\infty ( 0 , T ; C ^ { \\alpha , p } ) } \\le \\norm { \\tau } _ { L ^ \\infty ( 0 , T ; C ^ { \\alpha , p } ) } M _ X ^ \\alpha . \\end{gathered} \\end{align*}"}
-{"id": "4885.png", "formula": "\\begin{align*} \\rho _ S \\circ ( S \\otimes t ) \\circ g = \\rho _ S \\circ ( S \\otimes s ) \\circ ( S \\otimes s ) ^ { - 1 } \\circ { \\rho _ S } ^ { - 1 } = S \\text . \\end{align*}"}
-{"id": "6135.png", "formula": "\\begin{align*} F _ { i , j } ^ A = h _ 1 \\rho ^ A _ 1 + h _ 2 \\rho ^ A _ 2 . \\end{align*}"}
-{"id": "4970.png", "formula": "\\begin{align*} \\forall k \\exists x \\in C \\left ( U ( x ) = x \\wedge \\forall y \\in C \\left ( U ( y ) = y \\to \\Vert x - v _ 0 \\Vert < \\Vert y - v _ 0 \\Vert + \\frac { 1 } { k + 1 } \\right ) \\right ) . \\end{align*}"}
-{"id": "5282.png", "formula": "\\begin{align*} 1 - F ( x ) = 1 - \\int _ { x } ^ { u e p ( G ) } 1 - G ( t ) \\ d t , \\ \\ x _ 0 \\leq x < u e p ( F ) \\end{align*}"}
-{"id": "1694.png", "formula": "\\begin{align*} A \\in { \\mathcal A } \\mbox { i s } \\mu \\mbox { - a . e . } { \\mathcal L } \\mbox { - i n v a r i a n t } \\ \\ \\ \\Leftrightarrow \\ \\ \\ \\chi _ A ( x ) = ( { \\mathcal L } \\varphi ) ( x ) = P ( x , A ) \\ \\ \\mbox { f o r } \\mu \\mbox { - a . e . } x \\in X . \\end{align*}"}
-{"id": "4367.png", "formula": "\\begin{align*} \\frac { { n - u - 1 \\choose n - k - 1 } } { { n - u - 1 \\choose k - 1 } } \\le \\prod _ { i = k } ^ { n - k - 1 } \\frac { n - 3 - i } { n - 1 - i } \\le \\frac { ( k - 1 ) ( k - 2 ) } { ( n - k - 1 ) ( n - k - 2 ) } \\ \\ \\ \\ \\ \\ 3 \\le u \\le k . \\end{align*}"}
-{"id": "2225.png", "formula": "\\begin{align*} u _ t - \\Delta u + \\nabla \\cdot ( u \\nabla E _ d \\ast u ) & = 0 , & & x \\in \\R ^ d , \\ , \\\\ u ( x , 0 ) & = u _ 0 ( x ) , & & x \\in \\R ^ d . \\end{align*}"}
-{"id": "4414.png", "formula": "\\begin{align*} { \\rm R i c c i } ( \\omega _ o | _ { \\Sigma } ) = - \\Pi ^ * d d ^ c \\log ( 1 + | z ^ 1 | ^ 2 + | z ^ 2 | ^ 2 ) , \\end{align*}"}
-{"id": "539.png", "formula": "\\begin{align*} { Z } ^ h ( x ) = \\frac { 1 } { h ^ 2 } \\Big ( \\nabla u ^ h ( x ' , h x _ 3 ) - R ^ h ( x ' ) \\big ( Q _ 0 ( x ' ) + h P _ 0 ( x ' , x _ 3 ) \\big ) \\Big ) \\rightharpoonup Z \\qquad \\mbox { w e a k l y i n } L ^ 2 ( \\Omega , \\mathbb { R } ^ { 3 \\times 3 } ) . \\end{align*}"}
-{"id": "7223.png", "formula": "\\begin{align*} \\frac { d } { d x } ( \\mu - \\phi ) ^ 2 ( x ) = - 2 K _ r ^ \\prime * \\phi ( x ) . \\end{align*}"}
-{"id": "9817.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } L _ a ( \\hat { u } - \\hat { w } ) & = & 0 & \\R ^ { n + 1 } \\setminus \\{ x _ n = y = 0 \\} \\\\ \\hat { u } - \\hat { w } & \\geq & 0 & \\R ^ { n - 1 } \\times \\{ 0 \\} \\times \\{ 0 \\} . \\end{array} \\right . \\end{align*}"}
-{"id": "3195.png", "formula": "\\begin{align*} \\varphi = d ^ * f + \\varphi _ 0 , \\end{align*}"}
-{"id": "3793.png", "formula": "\\begin{align*} \\sum _ { \\mu : \\ \\mu _ 1 = a + d + 1 } m _ { \\mathcal { M } ( \\alpha ) } ( \\mu ) & = \\sum _ { \\substack { ( i , j ) \\in S _ \\alpha : \\\\ i \\geq 2 j \\geq 2 } } 2 ^ { \\max \\{ i - 1 , 0 \\} + \\max \\{ j - 1 , 0 \\} } \\\\ & + \\chi ( ( 0 , 0 ) \\in S _ \\alpha ) + \\chi ( ( 1 , 0 ) \\in S _ \\alpha ) + \\chi ( ( 0 , 1 ) \\in S _ \\alpha ) + \\chi ( ( 1 , 1 ) \\in S _ \\alpha ) \\end{align*}"}
-{"id": "1629.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot { \\varphi } _ 1 = i \\varphi _ { 1 x x } - 2 i \\varphi _ 1 ^ 2 \\varphi _ 2 , \\\\ \\dot { \\varphi } _ 2 = - i \\varphi _ { 2 x x } + 2 i \\varphi _ 1 \\varphi _ 2 ^ 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "9364.png", "formula": "\\begin{align*} & G _ k = P _ 1 ( G _ k ) = \\langle x , y \\rangle \\ ; P _ 2 ( G _ k ) & & G _ k / P _ 2 ( G _ k ) \\cong C _ 2 \\times C _ 2 , \\\\ & P _ 2 ( G _ k ) = \\langle x ^ 2 , y ^ 2 , [ y , x ] \\rangle \\ ; P _ 3 ( G _ k ) & & P _ 2 ( G _ k ) / P _ 3 ( G _ k ) \\cong C _ 2 \\times C _ 2 \\times C _ 2 , \\end{align*}"}
-{"id": "7266.png", "formula": "\\begin{align*} L \\phi ( x ) : = \\sum _ { k \\neq 0 } k ^ { - 2 } \\hat \\phi ( k ) e ^ { i x k } . \\end{align*}"}
-{"id": "9290.png", "formula": "\\begin{align*} D _ x \\Phi = D _ y \\Phi = 0 , { \\rm a n d } \\ \\ D ^ 2 _ { x , y } \\Phi \\le 0 , \\end{align*}"}
-{"id": "5225.png", "formula": "\\begin{align*} \\forall ( x > 0 ) , \\lim _ { t \\rightarrow \\infty } \\frac { V ( t x ) } { V ( t ) } = x ^ { \\rho } . \\end{align*}"}
-{"id": "1496.png", "formula": "\\begin{align*} \\widetilde { I V } _ { A , i } = \\varepsilon \\int _ { 0 } ^ { t } \\int _ { \\mathbb { T } ^ { d } } \\sum _ { i = 1 } ^ { d } \\sum _ { j = 1 } ^ { d } ( \\partial ^ { \\alpha } \\partial _ { x _ { i } } \\mu ^ { n + 1 } ) ( \\mu ^ { n } + \\bar { m } ) ( \\Theta _ { p _ { i } p _ { j } } ) ( \\partial ^ { \\alpha } \\partial _ { x _ { j } } w ^ { n } ) \\ d x d \\tau , \\end{align*}"}
-{"id": "9252.png", "formula": "\\begin{align*} \\begin{array} { r l } \\tilde v _ 1 : = \\ ! \\ ! & \\ ! \\ ! \\displaystyle v _ 1 - \\zeta ^ 2 [ g ( 0 ) \\nu _ 1 + C _ 1 ( 1 + M _ 2 ( R ) ) | x | ^ 2 ] \\\\ = \\ ! \\ ! & \\ ! \\ ! \\displaystyle \\zeta ^ 2 [ w _ 1 - g ( 0 ) \\nu _ 1 - C _ 1 ( 1 + M _ 2 ( R ) ) | x | ^ 2 ] \\end{array} \\end{align*}"}
-{"id": "577.png", "formula": "\\begin{align*} u - u ' = \\alpha _ 0 ( 1 + p \\alpha _ 1 ) - \\alpha _ 0 ' ( 1 + p \\alpha _ 1 ' ) = \\alpha _ 0 - \\alpha _ 0 ' + p ( \\alpha _ 1 - \\alpha _ 1 ' ) , \\end{align*}"}
-{"id": "1748.png", "formula": "\\begin{align*} D ^ { l } F _ { N } = f _ { N } ^ { ( l ) } ( M _ { s , t } ) D M _ { s , t } \\otimes \\cdots \\otimes D M _ { s , t } \\end{align*}"}
-{"id": "6594.png", "formula": "\\begin{align*} P _ N ( \\varphi ) = \\prod _ { j = 1 } ^ { m } \\prod _ { r = 1 } ^ { F _ { n _ j } } \\left | 2 \\sin \\pi ( r \\varphi + k _ j \\varphi ) \\right | , \\end{align*}"}
-{"id": "76.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t v + \\partial _ x w = 0 & \\\\ \\partial _ t w + \\partial _ x v = - 2 k ( x ) \\widetilde g ( w ; J _ b ) & ~ ~ ~ \\widetilde g ( w ; J _ b ) = g ( J _ b + w ) - g ( J _ b ) \\end{cases} \\end{align*}"}
-{"id": "8344.png", "formula": "\\begin{align*} D _ t Z ( \\alpha , t ) - D _ t ( \\beta , t ) = \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i } { 2 \\pi } \\frac { \\beta - \\alpha } { \\overline { ( \\alpha - z _ j ) ( \\beta - z _ j ) } } . \\end{align*}"}
-{"id": "3389.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } a _ n = 0 , \\ , \\ , \\sum _ { n = 1 } ^ { \\infty } \\frac { a _ n } { n } < \\infty , \\end{align*}"}
-{"id": "957.png", "formula": "\\begin{align*} \\mu _ j > b : = \\max \\left \\{ \\frac { 2 \\kappa _ { \\mu g } } { \\epsilon ' } , \\frac { 1 6 ( \\kappa _ { e f } + \\kappa _ { e f s } ) } { \\eta _ 1 \\theta _ { f c d } \\epsilon ' } , \\frac { 2 \\kappa _ { J _ m } ^ 2 } { \\epsilon ' } , \\frac { 2 \\eta _ 2 } { \\epsilon ' } , \\lambda \\ , \\mu _ { \\min } \\right \\} . \\end{align*}"}
-{"id": "4024.png", "formula": "\\begin{align*} \\lambda + \\nu = 2 n \\end{align*}"}
-{"id": "4320.png", "formula": "\\begin{align*} \\begin{aligned} \\Tilde { W } _ k ( D ) : = \\big \\{ v \\in H ^ 1 ( D ) : \\ ; \\ ; & \\Delta _ D v \\in \\mathbb { P } _ { k } ( D ) , v | _ { \\partial D } \\in C ^ 0 ( \\partial D ) , \\\\ & v | _ { F } \\in \\Tilde { W } _ k ( F ) \\ ; \\forall F \\subset \\partial D , \\Pi _ D ^ k v _ h - Q _ D ^ k v _ h \\in \\mathbb { P } _ { k - 2 } ( D ) \\big \\} . \\end{aligned} \\end{align*}"}
-{"id": "1128.png", "formula": "\\begin{align*} & \\mathbb { P } ( \\xi ^ { ( n ) } ( \\cup _ { j = 1 } ^ k [ y _ j , y _ j + a _ j ] ) = 0 ) = \\mathbb { P } ( \\xi ^ { ( n ) } ( J ) = 0 ) = \\det ( - A ) . \\end{align*}"}
-{"id": "1977.png", "formula": "\\begin{align*} H _ T ^ * ( X ) = H ^ * ( X ) \\otimes \\Q [ \\alpha _ 1 , \\dots , \\alpha _ n ] . \\end{align*}"}
-{"id": "6365.png", "formula": "\\begin{align*} \\lambda _ s : = \\frac { | 1 - s | } { s } , \\ \\alpha _ s : = s ( 1 - \\lambda _ s ) ( 1 - A _ 1 | s - 1 | ) \\in ] 0 , 1 [ . \\end{align*}"}
-{"id": "3066.png", "formula": "\\begin{align*} \\| T _ j f \\| _ q \\le \\| \\int _ { \\R ^ n } \\frac { | f ( y ) | } { | x - y | ^ { n - \\beta } } \\ , d y \\| _ q \\le C ( \\beta ) \\| f \\| _ p , \\frac 1 q = \\frac 1 p - \\frac \\beta n , 0 < \\beta < n . \\end{align*}"}
-{"id": "2876.png", "formula": "\\begin{align*} \\{ \\Gamma _ { A } , \\Gamma _ { B } \\} = \\Gamma _ { ( A \\cup B ) \\setminus ( A \\cap B ) } + 2 \\Gamma _ { A \\cap B } \\Gamma _ { A \\cup B } + 2 \\Gamma _ { A \\setminus ( A \\cap B ) } \\Gamma _ { B \\setminus ( A \\cap B ) } , \\end{align*}"}
-{"id": "7693.png", "formula": "\\begin{align*} \\int \\limits _ { U } h \\circ \\varphi \\ , \\mathrm { d } \\mu = \\int \\limits _ { \\varphi ( U ) } h \\cdot f \\ , \\mathrm { d } \\lambda . \\end{align*}"}
-{"id": "4552.png", "formula": "\\begin{align*} k , \\ldots k + d _ 1 , l \\ldots & = k ' , \\ldots , k ' + d _ 1 , k ' \\\\ & < k ' , \\ldots , k ' + d _ 1 , k ' + d _ 1 + 1 \\\\ & \\leq k ' , \\ldots k ' + d ' _ 1 \\\\ & = { \\bf i ' } ^ { ( 1 ) } \\end{align*}"}
-{"id": "3650.png", "formula": "\\begin{align*} a = \\langle a , b \\rangle b + a _ 0 + a _ { \\frac { 1 } { 4 } } + a _ { \\frac { 1 } { 3 2 } } \\end{align*}"}
-{"id": "285.png", "formula": "\\begin{align*} P \\boxplus Q \\ ( x _ 0 ) = 0 \\ \\Leftrightarrow \\ P ( - x ) \\ \\mbox { a n d } \\ Q ( x + x _ 0 ) \\ \\mbox { a r e a p o l a r . } \\end{align*}"}
-{"id": "8014.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\| B ^ * \\widetilde \\varphi ^ n ( \\tau ) \\| _ { \\ell ^ 2 } = 0 . \\end{align*}"}
-{"id": "4089.png", "formula": "\\begin{align*} t _ 1 = - \\gamma - M _ 1 , t _ 2 = \\gamma ( \\gamma - \\tfrac { 1 } { 3 } ) + M _ 1 ( M _ 1 + \\gamma - \\tfrac { 2 } { 3 } ) - M _ 2 , t _ 3 = 2 \\gamma - \\tfrac { 1 } { 3 } + M _ 1 \\end{align*}"}
-{"id": "951.png", "formula": "\\begin{align*} \\min _ { x \\in \\real ^ n } \\ ; f ( x ) : = \\frac { 1 } { 2 } \\| r ( x ) \\| ^ 2 , \\end{align*}"}
-{"id": "7828.png", "formula": "\\begin{align*} \\| T + \\lambda S \\| = \\lim _ n \\| ( T + \\lambda S ) ( x _ n ) \\| . \\end{align*}"}
-{"id": "4668.png", "formula": "\\begin{align*} m _ { h _ 1 , \\dots , h _ n } = \\sum _ { k _ 1 , \\dots , k _ n } \\alpha _ { h _ 1 , \\dots , h _ n , k _ 1 , \\dots , k _ n } e _ 1 ^ { k _ 1 } \\dots e _ n ^ { k _ n } . \\end{align*}"}
-{"id": "4676.png", "formula": "\\begin{align*} S ( n ) = \\mathbb C [ c _ 2 , \\dots , c _ n ] [ a _ 0 , a _ 0 ^ { - 1 } ] \\cap \\mathbb C [ a _ 0 , \\dots , a _ n ] . \\end{align*}"}
-{"id": "1531.png", "formula": "\\begin{align*} u _ 2 ( x , t ) = 2 { \\int _ { 0 } ^ { t } \\left [ \\sum _ { n = 1 } ^ { \\infty } \\left ( \\int _ { 0 } ^ { 1 } { F ( y , s ) \\sin { ( n \\pi y ) } d y } \\right ) e ^ { - n ^ 2 \\pi ^ 2 ( t - s ) } \\sin { ( n \\pi x ) } \\right ] d s } . \\end{align*}"}
-{"id": "9417.png", "formula": "\\begin{align*} B _ { \\tau } ( \\varepsilon ) : = \\left \\{ \\begin{array} { l l } \\{ z \\in \\mathbb { H } \\ ; : \\ ; | z - \\tau | < \\varepsilon \\} , & \\tau \\in \\mathbb { H } , \\\\ \\{ z \\in \\mathcal { F } _ N \\ ; : \\ ; \\mathrm { I m } ( \\sigma _ { \\tau } z ) > 1 / \\varepsilon \\} , & \\tau \\in \\{ i \\infty \\} \\cup \\mathbb { Q } . \\end{array} \\right . \\end{align*}"}
-{"id": "5734.png", "formula": "\\begin{align*} E _ { \\alpha , 1 } ( 0 ) = \\sum _ { k = 1 } ^ \\infty \\frac { ( 0 ^ \\alpha ) ^ k } { \\Gamma ( k \\alpha + 1 ) } = \\frac { 1 } { \\Gamma ( 1 ) } = 1 . \\end{align*}"}
-{"id": "8963.png", "formula": "\\begin{align*} \\mathbb { P } ( Y = 1 \\ , | \\ , X ) = \\frac { 1 } { 1 + \\exp \\left \\{ - \\beta _ 0 ^ * - \\alpha _ { T _ n } ' X ( T _ n ) \\right \\} } , \\end{align*}"}
-{"id": "1734.png", "formula": "\\begin{align*} ( I - L ) ^ { - \\frac { r } { 2 } } = \\frac { 1 } { \\Gamma ( r / 2 ) } \\int _ { 0 } ^ { \\infty } t ^ { \\frac { r } { 2 } - 1 } e ^ { - t } T _ { t } d t \\end{align*}"}
-{"id": "6565.png", "formula": "\\begin{align*} m ^ P _ \\beta = \\begin{cases} 1 & l = 1 , \\\\ 0 & l \\geq 2 . \\end{cases} \\end{align*}"}
-{"id": "1880.png", "formula": "\\begin{align*} \\mathcal { C } = \\frac { 2 } { \\pi } F _ 2 ( \\chi ) = \\frac { \\xi L } { 2 \\pi } . \\end{align*}"}
-{"id": "6086.png", "formula": "\\begin{align*} \\begin{gathered} \\Phi : X ( \\lambda ) / \\mathfrak { S } _ { \\lvert N ( \\lambda ) \\rvert } \\xrightarrow { \\cong } \\Z ( \\lambda ) \\end{gathered} \\end{align*}"}
-{"id": "3275.png", "formula": "\\begin{align*} \\frac { a ^ n - b ^ n } { a - b } - ( a + b ) \\frac { a ^ { n - 1 } - b ^ { n - 1 } } { a - b } + ( a b ) \\frac { a ^ { n - 2 } - b ^ { n - 2 } } { a - b } = 0 . \\end{align*}"}
-{"id": "4206.png", "formula": "\\begin{align*} \\Psi _ \\alpha ^ T ( z ) = \\mathcal { O } \\begin{pmatrix} h _ { \\alpha } ( z ) & h _ { \\alpha } ( z ) & h _ { \\alpha } ( z ) \\\\ z ^ \\alpha & z ^ \\alpha & z ^ \\alpha \\\\ z ^ \\alpha & z ^ \\alpha & z ^ \\alpha \\end{pmatrix} \\end{align*}"}
-{"id": "4662.png", "formula": "\\begin{align*} x ^ { [ i ] } : = \\frac { x ^ i } { i ! } , p ( x ) : = \\sum _ { j = 0 } ^ n a _ j x ^ { [ n - j ] } = \\sum _ { j = 0 } ^ n b _ j x ^ { n - j } , b _ j : = \\frac { a _ j } { ( n - j ) ! } . \\end{align*}"}
-{"id": "3946.png", "formula": "\\begin{align*} H _ { i } ( x , m ^ { \\hat v } _ { t } , D u ^ { \\hat v } _ { i } ( x , t ) ) & = f _ { i } ( x , m ^ { \\hat v } _ { t } , \\hat { v } _ { i } ( x , m ^ { \\hat v } _ { t } ) ) + D u ^ { \\hat v } _ { i } ( x , t ) . g _ { i } ( x , m ^ { \\hat v } _ { t } , \\hat { v } _ { i } ( x , m ^ { \\hat v } _ { t } ) ) \\ , , \\end{align*}"}
-{"id": "10055.png", "formula": "\\begin{align*} r _ n = \\frac { 2 } { x _ n ^ 2 } . \\end{align*}"}
-{"id": "3573.png", "formula": "\\begin{align*} \\mathcal { V } : = \\left \\{ \\frac { 1 } { 4 } ( \\psi _ { T _ { i } } - \\psi _ { T _ { 1 } } ) \\mid i = 1 , . . . , 6 \\right \\} . \\end{align*}"}
-{"id": "7483.png", "formula": "\\begin{align*} B ^ { ( m ) } = S ^ 1 \\vee \\cdots \\vee S ^ 1 . \\end{align*}"}
-{"id": "8154.png", "formula": "\\begin{align*} \\mu ^ l ( v ) ^ { - 1 } \\mu ^ l ( v X _ i ) Y _ i = v ^ { - 1 } . \\end{align*}"}
-{"id": "2914.png", "formula": "\\begin{align*} \\mu ( [ u ] \\cap Q ) & = \\mu ( [ w ^ { - 1 } u ] \\cap Q ) = \\mu \\big ( z ^ { - 1 } ( [ u ] \\cap Q ) \\big ) \\end{align*}"}
-{"id": "3999.png", "formula": "\\begin{align*} \\left ( e ( \\frac { \\mu \\nu } { d ' } ) \\right ) _ { \\begin{array} { c } \\mu \\bmod d ' , ( \\mu , d ' ) = 1 \\\\ \\nu \\bmod d ' , \\nu ^ 2 \\equiv \\nu _ 0 ^ 2 \\bmod d ' \\end{array} } . \\end{align*}"}
-{"id": "8312.png", "formula": "\\begin{align*} E ( t ) = \\sum _ { 0 \\leq k \\leq s } \\Big \\{ \\int \\frac { 1 } { A } | D _ t \\theta _ k | ^ 2 + \\int \\frac { 1 } { A } | D _ t \\sigma _ k | ^ 2 + i \\int \\theta _ k \\overline { \\partial _ { \\alpha } \\theta _ k } + i \\int \\sigma _ k \\overline { \\partial _ { \\alpha } \\sigma _ k } \\Big \\} . \\end{align*}"}
-{"id": "8317.png", "formula": "\\begin{align*} ( I - \\mathfrak { H } ) \\frac { 1 } { z ( \\alpha , t ) - z _ j ( t ) } = \\frac { 2 } { z ( \\alpha , t ) - z _ j ( t ) } . \\end{align*}"}
-{"id": "8059.png", "formula": "\\begin{align*} \\Omega _ R / d R \\cong \\mathbb { C } \\omega _ 0 \\oplus U _ 1 ^ { \\Xi _ 1 } \\oplus U _ 2 ^ { \\Xi _ 2 } \\oplus \\bigoplus _ { j = 1 } ^ { \\frac { n - 1 } { 2 } } V _ { j } , \\end{align*}"}
-{"id": "4977.png", "formula": "\\begin{align*} \\| u _ { n + m + \\ell } - u _ { n + m } \\| \\leq b \\cdot \\sum ^ { n + m } _ { j = n } | \\lambda _ { j + \\ell } - \\lambda _ j | + \\| u _ { n + \\ell - 1 } - u _ { n - 1 } \\| \\cdot \\prod ^ { n + m } _ { j = n } ( 1 - \\lambda _ { j + \\ell } ) . \\end{align*}"}
-{"id": "5267.png", "formula": "\\begin{align*} \\lambda ( A _ { k , n } ) \\downarrow \\lambda ( A _ { k } ) = 0 . \\end{align*}"}
-{"id": "3271.png", "formula": "\\begin{align*} U _ n ( a + b , - a b ) = \\frac { a ^ n - b ^ n } { a - b } . \\end{align*}"}
-{"id": "7414.png", "formula": "\\begin{align*} \\varepsilon _ k ^ { - 1 } ( ( 0 , \\beta ) ) = & \\{ x : \\varepsilon _ { k } ( x ) \\in ( 0 , \\beta ) \\} \\\\ = & \\bigcap _ { \\beta \\leq r \\leq 1 } w ( \\cdot , r ) ^ { - 1 } ( [ 0 , k ) ) \\\\ = & \\bigcap _ { \\beta \\leq r \\leq 1 , r \\in \\mathbb Q } w ( \\cdot , r ) ^ { - 1 } ( [ 0 , k ) ) . \\end{align*}"}
-{"id": "2261.png", "formula": "\\begin{align*} \\mu ( x ) & = \\frac { 1 } { 2 \\pi \\i } ( R ( x - \\i 0 ) - R ( x + \\i 0 ) ) = | \\alpha _ x ) ( \\beta _ x | \\\\ | \\alpha _ x \\rangle & = ( x ^ 2 + \\pi ^ 2 ) ^ { - 1 / 2 } \\left ( \\binom { 1 } { - \\frac { \\mathcal { P } } { x - \\Omega } | E \\rangle } + x \\binom { 0 } { | \\delta _ x \\rangle } \\right ) \\langle \\beta _ x | & = ( x ^ 2 + \\pi ^ 2 ) ^ { - 1 / 2 } \\bigg ( - ( 1 , \\langle E | \\frac { \\mathcal { P } } { x - \\Omega } ) + x ( 0 , \\langle \\delta _ x | ) \\bigg ) \\end{align*}"}
-{"id": "5891.png", "formula": "\\begin{align*} b _ w = ( - 1 ) ^ { \\ell ( w ) } q ^ { - \\ell ( w ) } q ^ { \\frac { \\ell ( w ) } { 2 } } + 2 \\left ( \\sum _ { n = \\ell ( w ) + 1 } ^ { \\infty } ( - 1 ) ^ n ( - 1 ) ^ { n - \\ell ( w ) } q ^ { \\frac { \\ell ( w ) } { 2 } } q ^ { - n } \\right ) . \\end{align*}"}
-{"id": "1338.png", "formula": "\\begin{align*} \\eta = \\frac { \\varepsilon ^ 2 } { 4 L ^ 3 \\left ( \\sup _ { b \\in F } \\left \\| b \\right \\| _ 1 \\right ) ^ 2 } \\ , . \\end{align*}"}
-{"id": "1350.png", "formula": "\\begin{align*} \\int _ { G ^ u } f \\ , d \\lambda ^ u = \\int _ { H ^ { \\pi ( u ) } } \\pi _ * ( f ) \\ , d \\lambda ^ { \\pi ( u ) } \\ ; , \\end{align*}"}
-{"id": "9919.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\sup _ { x \\in K _ { j } } | f _ { n _ { k } ^ { j } } ( x ) - f ^ { j } ( x ) | = 0 \\end{align*}"}
-{"id": "4548.png", "formula": "\\begin{align*} \\mu ( z _ 1 : \\ldots : z _ n ) = \\frac { 1 } { \\sum _ { i = 1 } ^ n | z _ i | ^ 2 } \\sum _ { j = 1 } ^ { k } ( | z _ { 2 j - 1 } | ^ 2 - | z _ { 2 j } | ^ 2 ) e _ j . \\end{align*}"}
-{"id": "1754.png", "formula": "\\begin{align*} \\left [ c _ { p , r } \\left ( | M _ { s , t } | > \\eta \\right ) \\right ] ^ { p } \\leq 2 \\left [ \\sum _ { l = 0 } ^ { r } \\left ( \\frac { \\eta } { p ( t - s ) ^ { H } } \\right ) ^ { l p } \\right ] e ^ { - \\frac { \\eta ^ { 2 } } { 2 ( t - s ) ^ { 2 H } } } . \\end{align*}"}
-{"id": "9999.png", "formula": "\\begin{align*} ( F _ { 2 m } A ) ( 2 n ) = ( S L _ { 2 n } ) _ { + } \\wedge _ { S L _ { 2 n - 2 m } } ( A \\wedge T ^ { 2 n - 2 m } ) . \\end{align*}"}
-{"id": "4252.png", "formula": "\\begin{align*} \\delta ( a ( x _ { 0 } ) \\lvert d v \\rvert ^ { p - 2 } d v ) = 0 \\Omega . \\end{align*}"}
-{"id": "6371.png", "formula": "\\begin{align*} ( \\theta / 2 + d d ^ c \\phi _ 2 ) ^ n = e ^ { c _ 1 } g d V , \\end{align*}"}
-{"id": "6386.png", "formula": "\\begin{align*} \\mathbb { P } ( X _ A = 0 _ A ) = \\det ( I - K _ A ) . \\end{align*}"}
-{"id": "2071.png", "formula": "\\begin{align*} \\Phi '' ( \\varphi | \\varphi _ { \\rm t x } ) + \\alpha \\Phi ( \\varphi | \\varphi _ { \\rm t x } ) = 0 . \\end{align*}"}
-{"id": "2591.png", "formula": "\\begin{align*} J _ G ( r ) : = \\gamma _ 0 \\int _ 0 ^ 1 G ( r u ^ { \\frac { 1 } { \\gamma _ 0 - 1 } } ) ^ { \\gamma _ 0 - 1 } \\frac { d u } { u } , r \\geq 0 . \\end{align*}"}
-{"id": "2453.png", "formula": "\\begin{align*} R / x R \\ = \\ ( R / x R ) ^ { ( s ) } \\ = \\ R ^ { ( s ) } / x ^ s R ^ { ( s ) } , \\end{align*}"}
-{"id": "3030.png", "formula": "\\begin{align*} ( L ^ n _ { p , q } f \\ast \\Psi _ j ) ( z ) = z + \\sum \\limits _ { k = 2 } ^ \\infty u _ k [ k ] _ { p , q } ^ n a _ { k } { z ^ k } + ( - 1 ) ^ { n + j } \\sum \\limits _ { k = 1 } ^ \\infty v _ k [ k ] _ { p , q } ^ n b _ { k } { \\overline { z } ^ k } . \\end{align*}"}
-{"id": "7012.png", "formula": "\\begin{align*} x \\to \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "6861.png", "formula": "\\begin{align*} f ^ { 2 r } + \\sum _ { i = 1 } ^ m g _ i ^ 2 \\in I \\end{align*}"}
-{"id": "3357.png", "formula": "\\begin{align*} Y _ i = X _ i + Z _ i \\end{align*}"}
-{"id": "8708.png", "formula": "\\begin{align*} p g f _ { T _ { m , d } ^ * } ( s ) = \\sum _ { j \\leq m } \\hat { \\nu } ( j ) p g f _ { \\hat { T } _ { j , d } } ( s ) , \\end{align*}"}
-{"id": "9670.png", "formula": "\\begin{align*} x _ 0 = X _ * ^ * ( F _ { \\tau , \\sigma } \\oplus F _ { \\mathbb T , \\sigma } ) . \\end{align*}"}
-{"id": "304.png", "formula": "\\begin{align*} [ Y ] \\ ; = \\ ; c _ { M , g - 1 } ( \\Lambda _ 1 \\circ \\Lambda _ 2 ) \\ ; \\geq \\ ; \\deg ( \\Lambda _ 2 ) \\cdot [ Y ] \\ ; > \\ ; [ Y ] \\end{align*}"}
-{"id": "3363.png", "formula": "\\begin{align*} \\Gamma ( H ) : = \\left \\{ \\left ( \\dim V , \\dim P ( V ) \\right ) \\in \\mathbb N ^ 2 \\ , | \\ , V \\in \\mathrm { I r r ( H ) } \\right \\} . \\end{align*}"}
-{"id": "2435.png", "formula": "\\begin{align*} P _ { \\widetilde { \\varphi _ { \\ell } ^ { H _ 1 } } } ( T ) = \\frac { 1 } { ( - 1 ) ^ { \\ell / 4 } + 2 ^ { ( \\ell - 4 ) / 2 } } \\frac { ( - 1 ) ^ { \\ell / 4 } + 2 ^ { ( \\ell - 4 ) / 2 } T ^ { \\ell - 4 } } { 1 - 2 T + 2 T ^ 2 } . \\end{align*}"}
-{"id": "1161.png", "formula": "\\begin{align*} & \\lim _ { n \\to + \\infty } \\frac { C _ { \\beta , n - 2 , 1 } } { C _ { \\beta , n } n ^ { \\beta } } = ( 2 \\pi ) ^ { - 1 } \\frac { ( \\beta / 2 ) ^ { \\beta } ( \\Gamma ( \\beta / 2 + 1 ) ) ^ { 3 } } { \\Gamma ( 3 \\beta / 2 + 1 ) \\Gamma ( \\beta + 1 ) } : = A _ { \\beta } . \\end{align*}"}
-{"id": "7220.png", "formula": "\\begin{align*} - \\mu \\phi + L _ r \\phi + \\frac { 1 } { 2 } \\phi ^ 2 - \\frac { 1 } { 2 } \\widehat { \\phi ^ 2 } ( 0 ) = 0 . \\end{align*}"}
-{"id": "9818.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } \\mathcal { F } _ a ( u \\xi ) ( x ' ) & = & 0 & u ( x ' , 0 , 0 ) > 0 \\\\ \\mathcal { F } _ a ( u \\xi ) ( x ' ) & \\leq & 0 & u ( x ' , 0 , 0 ) = 0 ; \\end{array} \\right . \\end{align*}"}
-{"id": "1614.png", "formula": "\\begin{align*} \\sup _ { Z \\in \\Lambda } \\liminf _ { T \\rightarrow \\infty } \\frac { 1 } { T } \\int _ 0 ^ T \\ , d Z _ t = \\sup _ { Z \\in \\Lambda } \\liminf _ { T \\rightarrow \\infty } \\frac { Z _ T } { T } . \\end{align*}"}
-{"id": "7973.png", "formula": "\\begin{align*} \\lvert O _ { 2 n + 1 } ( q ) \\rvert = \\frac { 1 } { \\gcd ( 2 , q - 1 ) } q ^ { n ^ 2 } \\prod _ { i = 1 } ^ { n } ( q ^ { 2 i } - 1 ) \\ , , \\end{align*}"}
-{"id": "9236.png", "formula": "\\begin{align*} - \\frac { \\rho _ k } { \\sigma _ k } : = \\min _ { j = 1 , \\dots , s } \\left \\{ - \\frac { \\rho _ j } { \\sigma _ j } : \\sigma _ j < 0 \\right \\} . \\end{align*}"}
-{"id": "3482.png", "formula": "\\begin{align*} \\frac { d } { d \\phi } \\Re f _ M ( q _ M ( \\phi ) ; \\theta ) \\begin{cases} < 0 , & \\phi \\in ( 0 , \\theta ) , \\\\ > 0 , & \\phi \\in ( \\theta , \\pi ) , \\end{cases} \\frac { d } { d \\phi } \\Re f _ M ( \\overline { q _ M ( \\phi ) } ; \\theta ) \\begin{cases} < 0 , & \\phi \\in ( 0 , \\theta ) , \\\\ > 0 , & \\phi \\in ( \\theta , \\pi ) . \\end{cases} \\end{align*}"}
-{"id": "437.png", "formula": "\\begin{align*} \\dd Q ^ x _ t : = A ^ x _ t \\dd X _ t ( x ) . \\end{align*}"}
-{"id": "2606.png", "formula": "\\begin{align*} M ( t , r , x ) \\leq | r ^ { \\gamma _ 0 - 1 } + c _ f ^ { \\gamma _ 0 - 1 } r ^ { \\gamma _ 0 - 1 } | ^ { \\frac { 1 } { \\gamma _ 0 - 1 } } = : c _ 6 r , \\end{align*}"}
-{"id": "9843.png", "formula": "\\begin{align*} ( \\ell ( u _ 1 ) \\ell ( u _ 2 ) ) ^ { - 1 } = \\Pr ( B _ { c _ 1 c _ 2 } ) \\le x _ { c _ 1 c _ 2 } \\prod _ { c _ 1 ' c _ 2 ' \\in \\Gamma ( c _ 1 c _ 2 ) } ( 1 - x _ { c _ 1 ' c _ 2 ' } ) . \\end{align*}"}
-{"id": "9082.png", "formula": "\\begin{align*} ( - \\Delta _ p ) ^ s u ( x ) = \\lambda g ( x ) | u ( x ) | ^ { p - 2 } u ( x ) \\mathbb { R } ^ N , \\end{align*}"}
-{"id": "1867.png", "formula": "\\begin{align*} G ^ + _ { \\rm I N } ( x , t ; x ' , t ' ) = \\frac { 1 } { 2 } \\left [ \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { k _ n } u ^ { \\rm o d d } ( n , x ) \\overline { u ^ { \\rm o d d } ( n , x ' ) } e ^ { - i k _ n ( t - t ' ) } + \\sum _ { j = 1 } ^ \\infty \\frac { 1 } { \\kappa _ j } u ^ { \\rm e v e n } ( j , x ) \\overline { u ^ { \\rm e v e n } ( j , x ' ) } e ^ { - i \\kappa _ j ( t - t ' ) } \\right ] . \\end{align*}"}
-{"id": "5179.png", "formula": "\\begin{align*} F ^ { - 1 } ( 1 - u ) = d + \\int _ { u } ^ { u _ { 0 } } \\frac { r ( t ) } { t } d t , 0 < u < u _ { 0 } < 1 , \\end{align*}"}
-{"id": "532.png", "formula": "\\begin{align*} u ^ h ( x ' , x _ 3 ) = y ^ h ( x ' ) + x _ 3 b ^ h ( x ' ) + h ^ 2 d ^ h \\big ( x ' , \\frac { x _ 3 } { h } \\big ) , \\end{align*}"}
-{"id": "4420.png", "formula": "\\begin{align*} \\forall i \\in \\{ 1 , \\ldots , n \\} : \\ , \\Bigl { ( } \\frac { \\partial u ^ i } { \\partial \\nu } + \\alpha ^ i u ^ i \\frac { \\partial \\mathcal { V } } { \\partial \\nu } \\Bigr { ) } ( t , x ) = \\sigma ^ i ( t , x , u ^ i _ { | _ { \\partial \\Omega } } , \\mathcal { V } _ { | _ { \\partial \\Omega } } ) , ( t , x ) \\in [ 0 , T ] \\times \\partial \\Omega , \\end{align*}"}
-{"id": "87.png", "formula": "\\begin{align*} \\frac { \\inf g ' } 2 \\delta _ j \\le c ^ n _ j \\le \\min \\{ C _ 1 \\delta _ j , 1 / 2 \\} \\ , , j = 1 , \\ldots , N - 1 \\ , . \\end{align*}"}
-{"id": "8920.png", "formula": "\\begin{align*} \\mathbb { E } [ N _ w ^ { \\hat { x } } ( G ' [ + ] ) \\mid \\hat { x } ] = \\binom { n ^ { + + } + n ^ { + - } } { 2 } \\frac { a } { n } - \\frac { ( a - b ) } { n } n ^ { + + } n ^ { + - } . \\end{align*}"}
-{"id": "5629.png", "formula": "\\begin{align*} \\alpha \\ , \\int _ 0 ^ { + \\infty } u \\bar { u } _ t v d x & = \\frac { \\alpha } { 2 } \\int _ 0 ^ { + \\infty } v \\frac { d } { d t } | u | ^ 2 d x \\\\ & = \\frac { \\alpha } { 2 } \\frac { d } { d t } \\int _ 0 ^ { + \\infty } | u | ^ 2 v d x - \\frac { \\alpha } { 2 } \\int _ 0 ^ { + \\infty } | u | ^ 2 v _ t d x \\\\ \\end{align*}"}
-{"id": "8098.png", "formula": "\\begin{align*} n ^ 2 ( n - 1 ) ( 2 n - 3 ) \\rho _ n = & \\ ; \\ ; ( n - 1 ) ( 2 n - 1 ) ( 3 n ^ 2 - 5 n + 1 ) \\rho _ { n - 1 } \\\\ & - ( n - 2 ) ( 2 n - 3 ) ( 3 n ^ 2 - 5 n + 1 ) \\rho _ { n - 2 } \\\\ & + ( n - 2 ) ( n - 3 ) ^ 2 ( 2 n - 1 ) \\rho _ { n - 3 } \\end{align*}"}
-{"id": "207.png", "formula": "\\begin{align*} & X _ 1 = \\dfrac { \\partial } { \\partial x ^ 3 } , Y _ 1 = x ^ 1 \\dfrac { \\partial } { \\partial x ^ 2 } , \\\\ & X _ 3 = \\dfrac { \\partial } { \\partial x ^ 2 } , Y _ 2 = x ^ 3 \\dfrac { \\partial } { \\partial x ^ 2 } . \\end{align*}"}
-{"id": "698.png", "formula": "\\begin{align*} Z = e ^ { 3 F } . \\end{align*}"}
-{"id": "112.png", "formula": "\\begin{align*} \\left [ B ( 0 ) + { \\frac d N } B _ 1 \\right ] ^ { 2 N } & = ~ I _ { 2 N } + { \\frac { 2 d } N } \\widehat P + \\sum _ { j = 0 } ^ { 2 N - 1 } \\zeta _ { j , N } B ( 0 ) ^ { 2 j } B _ 2 ( 0 ) + \\sum _ { j = 1 } ^ { 2 N - 1 } \\eta _ { j , N } B ( 0 ) ^ { 2 j } \\ , , \\end{align*}"}
-{"id": "4446.png", "formula": "\\begin{align*} x y ^ + _ { G - \\varepsilon } = \\left \\{ \\begin{array} { c l } x y _ G ^ + & x y _ G ^ + \\neq \\varepsilon \\\\ ( x y _ G ^ + ) _ G ^ + & x y _ G ^ + = \\varepsilon \\end{array} \\right . \\end{align*}"}
-{"id": "8459.png", "formula": "\\begin{align*} \\frac { d } { d t } E _ k ^ { \\sigma } = \\int \\frac { 2 } { A } R e D _ t \\sigma _ k \\overline { \\tilde { G } } _ k - \\int \\frac { 1 } { A } \\frac { a _ t } { a } \\circ \\kappa ^ { - 1 } | D _ t \\sigma _ k | ^ 2 \\end{align*}"}
-{"id": "4883.png", "formula": "\\begin{align*} U _ 2 = \\bigcap _ { x ^ * \\in \\mathbb { R } ^ n , \\ , | x ^ * | \\le 1 } U \\left ( x ^ * , \\ , r ( x ^ * ) / 2 \\right ) . \\end{align*}"}
-{"id": "3937.png", "formula": "\\begin{align*} \\sum ( x \\cdot f _ { 1 } ) \\bigl ( x \\cdot \\sigma ( S f _ { 2 } | _ H ) \\bigr ) = \\sum f _ 1 \\sigma ( S f _ { 2 } | _ H ) \\ , , \\end{align*}"}
-{"id": "5067.png", "formula": "\\begin{align*} ( \\lambda _ U \\otimes I _ { \\mathcal { H } _ 0 } ) \\theta _ A \\theta _ A ^ * = \\theta _ A \\theta _ A ^ * ( \\lambda _ U \\otimes I _ { \\mathcal { H } _ 0 } ) , ~ ( \\lambda _ U \\otimes I _ { \\mathcal { H } _ 0 } ) \\theta _ A \\theta _ \\Psi ^ * = \\theta _ A \\theta _ \\Psi ^ * ( \\lambda _ U \\otimes I _ { \\mathcal { H } _ 0 } ) , \\\\ ( \\lambda _ U \\otimes I _ { \\mathcal { H } _ 0 } ) \\theta _ \\Psi \\theta _ \\Psi ^ * = \\theta _ \\Psi \\theta _ \\Psi ^ * ( \\lambda _ U \\otimes I _ { \\mathcal { H } _ 0 } ) . \\end{align*}"}
-{"id": "8083.png", "formula": "\\begin{align*} \\widetilde { a _ 2 } = e ^ { - 1 } M ( 3 , 2 , 1 ) = e ^ { - 1 } \\sum _ { k \\ge 0 } \\frac { ( 3 ) _ k } { ( 2 ) _ k \\ , k ! } = e ^ { - 1 } \\sum _ { k \\ge 0 } \\frac { k + 2 } { 2 \\ , k ! } = 3 / 2 = a _ 2 , \\end{align*}"}
-{"id": "1052.png", "formula": "\\begin{align*} & A - B = \\sum _ { i \\neq j } \\chi _ { I ( y _ i , F _ n ( x _ i ) ) } P _ n \\chi _ { I ( y _ j , F _ n ( x _ j ) ) } : = \\sum _ { i \\neq j } \\chi _ { i } P _ n \\chi _ { j } , \\end{align*}"}
-{"id": "945.png", "formula": "\\begin{align*} J ^ { * } [ \\psi ^ { ( 5 ) } ] _ { T , t } = { \\int \\limits _ t ^ { * } } ^ T \\psi _ 5 ( t _ 5 ) \\ldots { \\int \\limits _ t ^ { * } } ^ { t _ 2 } \\psi _ 1 ( t _ 1 ) d { \\bf w } _ { t _ 1 } ^ { ( i _ 1 ) } \\ldots d { \\bf w } _ { t _ 5 } ^ { ( i _ 5 ) } \\end{align*}"}
-{"id": "4908.png", "formula": "\\begin{align*} \\dot \\eta = u \\circ \\eta \\end{align*}"}
-{"id": "6739.png", "formula": "\\begin{align*} L ( p _ n , { ( p _ n + 1 ) } ^ 2 ) & = p _ n ( p _ n + 2 ) . \\\\ L ( p _ n , { ( p _ n + 1 ) } ^ 2 ) & = p _ n ( 2 ( p _ { n } + 1 ) - p _ n ) . \\end{align*}"}
-{"id": "9103.png", "formula": "\\begin{align*} | I _ 4 | & \\le \\dfrac { 2 ^ { N + p s } } { | x | ^ { N + p s } } \\int _ { B _ { \\frac { | x | } 2 } ( 0 ) } | \\Upsilon ( y ) | ^ { p - 1 } d y \\\\ & \\le \\dfrac { C ( N , s , p ) } { | x | ^ { N + p s } } \\int _ { B _ { \\frac { | x | } 2 } ( 0 ) } | \\Upsilon ( y ) | ^ { p - 1 } d y = \\dfrac { C ( N , s , p ) } { | x | ^ { N + p s } } I _ 4 ' . \\end{align*}"}
-{"id": "6248.png", "formula": "\\begin{align*} U _ j : = e _ { ( j - 1 ) a + 1 , 1 } + e _ { ( j - 1 ) a + 2 , 2 } + \\cdots + e _ { j a , a } . \\end{align*}"}
-{"id": "3648.png", "formula": "\\begin{align*} ( a _ 1 - a _ { - 1 } ) \\cdot v _ { ( 2 , 3 ) } - t ( a _ 1 - a _ { - 1 } ) = 0 \\\\ ( a _ 2 - a _ { - 2 } ) \\cdot v _ { ( 1 , 3 ) } - t ( a _ 2 - a _ { - 2 } ) = 0 \\\\ ( a _ 3 - a _ { - 3 } ) \\cdot v _ { ( 1 , 2 ) } - t ( a _ 3 - a _ { - 3 } ) = 0 . \\end{align*}"}
-{"id": "9275.png", "formula": "\\begin{align*} F ^ \\epsilon ( r ) = F ( r + \\epsilon F _ 1 ( r ) I ) , \\Gamma ^ \\epsilon = \\{ r + \\epsilon F _ 1 ( r ) I | \\ r \\in \\Gamma \\} . \\end{align*}"}
-{"id": "441.png", "formula": "\\begin{align*} g _ t ( u ) : = \\big ( A + \\eta ( u + t ) \\big ) e ^ { ( \\gamma \\eta + \\frac { \\gamma ^ 2 } { 2 } ) ( u + t ) } \\end{align*}"}
-{"id": "3524.png", "formula": "\\begin{align*} P ( 3 , 6 ) : = G L ( 3 , \\mathbb { C } ) \\diagdown M _ { 3 , 6 } ^ { o } \\diagup ( \\mathbb { C } ^ { * } ) ^ { 6 } , \\end{align*}"}
-{"id": "9640.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] _ { q ^ 2 } \\frac { ( - 1 ) ^ { n - 1 } z ^ n q ^ { n ^ 2 } ( q ^ 2 ; q ^ 2 ) _ n } { ( z q ; q ^ 2 ) _ n ( 1 - z q ^ { 2 n } ) } = \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] _ { q ^ 2 } \\frac { ( q ; q ^ 2 ) _ { n - 1 } ( q ^ 2 ; q ^ 2 ) _ { n } ( z q ^ 2 ; q ^ 2 ) _ { N - n } z ^ { n } q ^ { 2 n - 1 } } { ( z q ; q ^ 2 ) _ { n } ( z q ^ 2 ; q ^ 2 ) _ N } . \\end{align*}"}
-{"id": "4251.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\delta d \\theta & = - \\delta \\psi & & B _ { R } \\\\ \\delta \\theta & = 0 & & B _ { R } \\\\ \\nu \\wedge \\theta & = 0 & & \\partial B _ { R } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "7386.png", "formula": "\\begin{align*} \\dot { x } = a x + b x ^ 2 + x ^ 3 . \\end{align*}"}
-{"id": "6006.png", "formula": "\\begin{align*} u _ n ( t , x ) = Y _ n ( t ) & \\to \\sup _ { \\omega \\in \\C _ 0 [ t , 1 ] } \\left ( F ( 0 \\oplus _ t \\omega ) - \\int _ t ^ 1 g ( s , \\dot \\omega ( s ) ) d s \\right ) \\\\ & = \\sup _ { \\omega \\in \\C _ 0 [ t , 1 ] } \\left ( f ( x + \\omega ( 1 ) ) - \\int _ t ^ 1 g ( s , \\dot \\omega ( s ) ) d s \\right ) = u ( t , x ) . \\end{align*}"}
-{"id": "1948.png", "formula": "\\begin{align*} f = \\sum _ { n \\geq 1 } a _ n ( f ) q ^ n \\end{align*}"}
-{"id": "9271.png", "formula": "\\begin{align*} u _ { \\beta \\beta } ( y ) \\le \\epsilon \\sup \\limits _ { B _ { \\frac { \\delta R } { 4 } } ( y ) \\cap \\Omega } | D ^ 2 u | + C _ \\epsilon ( 1 + \\sup \\limits _ { B _ { \\frac { \\delta R } { 4 } } ( y ) \\cap \\partial \\Omega } \\sup \\limits _ { | \\tau | = 1 , \\tau \\cdot \\nu = 0 } | u _ { \\tau \\tau } | ) + \\frac { C } { ( \\delta R / 4 ) ^ 2 } , \\end{align*}"}
-{"id": "3213.png", "formula": "\\begin{align*} y ^ k & = ( y ^ k _ 1 , y ^ k _ 2 , \\ldots , y ^ k _ { p ( x ^ { \\ast } ) } ) ^ { \\top } \\in \\R ^ { p ( x ^ { \\ast } ) } , \\ y ^ k _ i : = y ^ k ( \\tau _ { x ^ { \\ast } } ^ i ( x ^ k ) ) \\ ( i = 1 , 2 , \\ldots , p ( x ^ { \\ast } ) ) , \\\\ y ^ { \\ast } & = ( y ^ { \\ast } _ 1 , y ^ { \\ast } _ 2 , \\ldots , y ^ { \\ast } _ { p ( x ^ { \\ast } ) } ) ^ { \\top } \\in \\R ^ { p ( x ^ { \\ast } ) } , \\ y ^ { \\ast } _ i : = y ^ { \\ast } ( \\tau _ { x ^ { \\ast } } ^ i ( x ^ { \\ast } ) ) \\ ( i = 1 , 2 , \\ldots , p ( x ^ { \\ast } ) ) . \\end{align*}"}
-{"id": "4663.png", "formula": "\\begin{align*} ( a + b ) ^ { [ i ] } = \\frac { ( a + b ) ^ i } { i ! } = \\sum _ { j = 0 } ^ i \\frac 1 { i ! } \\binom i j a ^ { i - j } b ^ j = \\sum _ { j = 0 } ^ i a ^ { [ i - j ] } b ^ { [ j ] } . \\end{align*}"}
-{"id": "1896.png", "formula": "\\begin{align*} \\langle \\rho _ { \\rm c l o u d } \\rangle _ { \\rm R e n . } = \\frac { 1 + v ^ 2 } { 1 - v ^ 2 } \\left ( - \\frac { \\pi } { 6 L ^ 2 } + \\frac { \\mathcal { B - C } } { L ^ 2 } \\right ) \\end{align*}"}
-{"id": "9976.png", "formula": "\\begin{align*} \\frac { \\partial ^ { 2 } h } { \\partial a ^ { i } \\partial a ^ { j } } = \\lambda \\frac { \\partial ^ { 2 } c ^ { m } } { \\partial a ^ { i } \\partial a ^ { j } } \\frac { \\partial h } { \\partial a ^ { m } } \\end{align*}"}
-{"id": "5275.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow + \\infty } \\frac { x F ^ { \\prime } ( x ) } { 1 - F ( x ) } = \\alpha . . \\end{align*}"}
-{"id": "6827.png", "formula": "\\begin{align*} \\det ( d \\varphi ) = \\sum _ { i = 1 } ^ 6 p _ i g _ i \\ , , \\end{align*}"}
-{"id": "8195.png", "formula": "\\begin{align*} J ( F , G ) : = \\lim _ { n \\to \\infty } J _ { n } ( F | _ n , G | _ n ) \\end{align*}"}
-{"id": "7536.png", "formula": "\\begin{align*} \\begin{gathered} v = \\frac { \\partial X } { \\partial t } = u \\circ X , \\\\ \\tau = \\sigma \\circ X . \\end{gathered} \\end{align*}"}
-{"id": "6201.png", "formula": "\\begin{align*} L _ V ( u ) = \\int _ 0 ^ { \\infty } e ^ { - u v } \\mu ( d v ) . \\end{align*}"}
-{"id": "1460.png", "formula": "\\begin{align*} V _ { 1 } = \\int _ { \\mathbb { T } ^ { d } } ( \\mu ^ { 1 } - \\mu ^ { 2 } ) \\Delta ( \\mu ^ { 1 } - \\mu ^ { 2 } ) \\ d x , \\end{align*}"}
-{"id": "807.png", "formula": "\\begin{align*} \\mathcal { M } ^ { H _ { q ^ d , P ^ \\alpha } } = \\left ( \\left ( \\mathcal { M } ^ { H _ { q ^ d , P ^ \\alpha } ^ \\alpha } \\right ) ^ { H _ { q ^ d , P ^ \\alpha } ^ { \\alpha - 1 } / H _ { q ^ d , P ^ \\alpha } ^ \\alpha } \\cdots \\right ) ^ { H _ { q ^ d , P ^ \\alpha } ^ 1 / H _ { q ^ d , P ^ \\alpha } ^ 2 } . \\end{align*}"}
-{"id": "4388.png", "formula": "\\begin{align*} \\lambda ( z ) = \\frac { 1 } { { \\rm V o l } B _ r ( 0 ) } \\int _ { B _ r ( 0 ) } \\log | T ( z - \\zeta ) | ^ 2 d V ( \\zeta ) . \\end{align*}"}
-{"id": "3445.png", "formula": "\\begin{align*} g ( k ; \\xi ) = N \\log N + x _ N ( k ) + \\xi \\sqrt { \\frac { N } { M + 1 } } . \\end{align*}"}
-{"id": "10049.png", "formula": "\\begin{align*} m _ i \\ddot { q _ i } = \\sum _ { j = 0 , i \\neq j } ^ n m _ i m _ j \\frac { q _ j - q _ i } { \\| q _ j - q _ i \\| ^ 3 } = \\frac { \\partial U } { \\partial q _ i } ( q _ 0 , \\dots , q _ n ) , i = 0 , \\dots , n , \\end{align*}"}
-{"id": "2568.png", "formula": "\\begin{align*} \\mathbf P _ \\mu ( \\| X _ t \\| \\neq 0 ) \\stackrel [ t \\to \\infty ] { } { \\sim } - \\log \\mathbf P _ \\mu ( \\| X _ t \\| = 0 ) = \\mu ( \\phi \\phi ^ { - 1 } v _ t ) \\stackrel [ t \\to \\infty ] { } { \\sim } \\mu ( \\phi ) \\langle v _ t , \\phi ^ * \\rangle _ m . \\end{align*}"}
-{"id": "7657.png", "formula": "\\begin{align*} \\frac { \\partial v _ \\phi } { \\partial t } - L ^ { \\varepsilon } v _ \\phi = \\phi , v _ \\phi ( x , 0 ) = 0 , \\end{align*}"}
-{"id": "9016.png", "formula": "\\begin{align*} f _ { \\{ a _ 1 , a _ 2 \\} , a _ 2 } ( x _ 1 , x _ 2 ) = \\begin{cases} a _ 2 & \\{ a _ 1 , a _ 2 \\} = \\{ x _ 1 , x _ 2 \\} , \\\\ x _ 1 & \\end{cases} \\end{align*}"}
-{"id": "6111.png", "formula": "\\begin{align*} k _ { i i } = k , k _ { i j } = k ^ { ( i - j ) k } . \\end{align*}"}
-{"id": "3919.png", "formula": "\\begin{align*} S ( g _ i , g _ j ) = \\sum _ { l = 1 } ^ s b _ l g _ l + h \\end{align*}"}
-{"id": "6441.png", "formula": "\\begin{align*} \\| x _ { k } \\| ^ { 2 } = \\| r _ { 0 } \\| ^ { 2 } e _ { 1 } ^ { T } T _ { k } ^ { - 2 } e _ { 1 } . \\end{align*}"}
-{"id": "5442.png", "formula": "\\begin{align*} \\log \\hat { \\rho } _ s ( A ) = \\max _ { f ^ k ( p ) = p , k \\in \\N } \\left \\lbrace \\frac { s - \\lfloor s \\rfloor } { k } r _ { \\lfloor s \\rfloor + 1 } ( A ^ k ( p ) ) + \\frac { 1 - s + \\lfloor s \\rfloor } { k } r _ { \\lfloor s \\rfloor } ( A ^ k ( p ) ) \\right \\rbrace . \\end{align*}"}
-{"id": "3495.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } } { d y ^ { 2 } } \\Re { f _ { M } ( x + i y ; \\theta ) } = { } & - \\left . \\Re { \\frac { d ^ { 2 } } { d z ^ { 2 } } f _ { M } ( z ; \\theta ) } \\right | _ { z = x + i y } \\\\ = { } & - \\frac { ( M + 1 + x ) x ( M x - ( M + 1 ) ) + y ^ { 2 } ( ( M + 1 ) ( M + 1 + x ) - x ) } { ( ( M + 1 + x ) ^ { 2 } + y ^ { 2 } ) ( x ^ { 2 } + y ^ { 2 } ) } . \\end{align*}"}
-{"id": "9245.png", "formula": "\\begin{align*} \\sup _ { | \\tau | = 1 , \\tau \\cdot \\nu = 0 } u _ { \\tau \\tau } ( x _ 0 ) \\le \\epsilon M _ 2 ( R ) + C _ \\epsilon ( 1 + \\frac { 1 } { R ^ 2 } ) , \\end{align*}"}
-{"id": "9826.png", "formula": "\\begin{align*} 0 = \\int _ { B _ \\rho } \\tilde { u } _ { X _ \\circ , r _ { j _ \\ell } } L _ a \\tilde { u } _ { X _ \\circ , r _ { j _ \\ell } } | y | ^ a \\to \\int _ { B _ \\rho } \\tilde u _ { X _ \\circ , 0 } L _ a \\tilde u _ { X _ \\circ , 0 } | y | ^ a \\quad \\ell \\to \\infty , \\end{align*}"}
-{"id": "5036.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\Big ( Y _ j - \\beta _ 0 - \\beta _ 1 ( Z _ j - x ) - \\beta _ 2 ( Z _ j - x ) ^ 2 \\Big ) ^ 2 \\frac { 1 } { c _ n } w \\Big ( \\frac { Z _ j - x } { c _ n } \\Big ) . \\end{align*}"}
-{"id": "2750.png", "formula": "\\begin{align*} D S _ { N } ( u , v ) & = \\int _ { \\mathcal { C } _ { \\{ \\tau / \\sigma _ { l } \\} } } \\frac { d z } { 2 \\pi i } \\int _ { \\mathcal { C } _ { \\{ \\sigma _ { l } / \\tau \\} } } \\frac { d w } { 2 \\pi i } \\frac { \\eta _ { - } ^ 2 } { 4 \\pi } \\frac { e ^ { - \\eta _ { - } v ( 1 + \\frac { 1 } { z } ) } } { z \\sqrt { 1 - z ^ { 2 } } } \\frac { e ^ { - \\eta _ { - } u ( w + 1 ) } } { \\sqrt { w ^ { 2 } - 1 } } \\psi _ { N , \\alpha , \\tau } ( z , w ) , \\end{align*}"}
-{"id": "8674.png", "formula": "\\begin{align*} \\| f ( t ) \\| _ { C ^ \\theta } : = \\| f ( t ) \\| _ 0 + [ f ( t ) ] _ \\theta \\ , . \\end{align*}"}
-{"id": "6608.png", "formula": "\\begin{align*} \\left | \\frac { \\overline { A } _ { n _ j } ( \\varepsilon _ j ) } { A _ { n _ j } } \\right | & = \\frac { \\pi \\varphi ^ { n _ j } ( 1 + p _ j ) \\left ( 1 + \\mathcal { O } ( \\varphi ^ { 2 n _ j } ) \\right ) } { \\pi \\varphi ^ { n _ j } \\left ( 1 + \\mathcal { O } ( \\varphi ^ { 2 n _ j } ) \\right ) } \\\\ & = 1 + p _ j + \\mathcal { O } ( \\varphi ^ { 2 n _ j } ) . \\end{align*}"}
-{"id": "3572.png", "formula": "\\begin{align*} \\psi _ { T } = \\bigg ( \\sum _ { v _ { 1 } \\prec \\sigma } v o l ( \\sigma ) , \\sum _ { v _ { 2 } \\prec \\sigma } v o l ( \\sigma ) , \\cdots , \\sum _ { v _ { 9 } \\prec \\sigma } v o l ( \\sigma ) \\bigg ) \\in \\mathbb { Z } ^ { 9 } . \\end{align*}"}
-{"id": "7243.png", "formula": "\\begin{align*} \\widehat { \\psi } ( k ) \\left ( \\mu - | k | ^ { - r } \\right ) = 0 , \\ \\ \\ k \\neq 0 . \\end{align*}"}
-{"id": "7015.png", "formula": "\\begin{align*} \\tilde \\Psi ( X ) : = \\left \\{ \\begin{gathered} \\Psi ( X ) , X \\subset [ 0 , \\infty ) \\quad X \\subset ( - \\infty , - 1 ] \\\\ 0 , \\end{gathered} \\right . . \\end{align*}"}
-{"id": "3720.png", "formula": "\\begin{align*} l ^ * = \\inf _ { \\gamma \\in \\Gamma } \\max _ { t \\in [ 0 , 1 ] } E ( \\gamma ( t ) ) . \\end{align*}"}
-{"id": "7444.png", "formula": "\\begin{align*} \\displaystyle \\lim \\limits _ { s \\to 0 ^ + } s \\widetilde { D } _ s ( \\varphi ) = \\left \\{ \\begin{array} { l l } D ' ( \\varphi ) & \\mbox { i f } m _ i = n _ i \\mbox { f o r a l l } 1 \\leqslant i \\leqslant r \\mbox { a n d } p _ j \\mbox { i s e v e n f o r a l l } 1 \\leqslant j \\leqslant s \\\\ 0 & \\mbox { o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "9573.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } \\frac { q ^ k } { 1 - q ^ k } - \\sum _ { k = 1 } ^ { n - 1 } \\frac { x q ^ k } { 1 - x q ^ k } = \\frac { x } { 1 - x } - \\frac { 1 } { ( x ) _ n } \\sum _ { k = 1 } ^ { n } \\left [ \\begin{matrix} n \\\\ k \\end{matrix} \\right ] \\frac { ( q / x ) _ k ( x ) _ { n - k } x ^ k } { 1 - q ^ k } . \\end{align*}"}
-{"id": "6881.png", "formula": "\\begin{align*} p = - \\tfrac { 1 } { 2 } \\ , \\langle \\beta , A ^ T A '' \\beta \\rangle = - \\tfrac { 1 } { 2 } \\ , \\langle x , A '' A ^ { - 1 } x \\rangle \\ . \\end{align*}"}
-{"id": "235.png", "formula": "\\begin{align*} \\mathcal { L } u = V u \\ B _ { r _ 0 } , \\end{align*}"}
-{"id": "8409.png", "formula": "\\begin{align*} \\mathcal { E } ( t & ) \\leq \\mathcal { E } ( 0 ) \\\\ & + \\int _ 0 ^ t C ( \\| \\partial _ { \\alpha } \\xi _ 0 \\| _ { H ^ { s - 1 } } , \\| v _ 0 \\| _ { H ^ s } , \\| w _ 0 \\| _ { H ^ s } , \\mathcal { E } ( \\tau ) , d _ I ( \\tau ) ^ { - 1 } , d _ P ( \\tau ) ^ { - 1 } , N \\lambda _ { m a x } , C _ 1 , C _ 2 , \\alpha _ 0 ) d \\tau . \\end{align*}"}
-{"id": "3301.png", "formula": "\\begin{align*} H ( x , y , 2 y ) - H ( x - \\Delta , y , 2 y ) = \\frac { \\Delta } { x } H ( x , y , 2 y ) ( 1 + o ( 1 ) ) , y _ 0 \\le y \\le \\sqrt { x } , \\ \\frac { x } { \\log ^ { 1 0 } x } \\le \\Delta \\le x , \\end{align*}"}
-{"id": "3647.png", "formula": "\\begin{align*} \\bar { B } : = B \\cup \\{ a _ { - 1 } \\cdot v _ { ( 2 , 3 ) } , a _ { - 2 } \\cdot v _ { ( 1 , 3 ) } , a _ { - 3 } \\cdot v _ { ( 1 , 2 ) } \\} . \\end{align*}"}
-{"id": "8121.png", "formula": "\\begin{align*} A _ { n , r } ( t ) \\ = \\ A ^ + _ { n , r } ( t ) \\ , + \\ , A ^ - _ { n , r } ( t ) . \\end{align*}"}
-{"id": "5784.png", "formula": "\\begin{align*} R _ { d + 1 } = \\psi ( q ^ d ) * R _ d , R _ 0 = \\epsilon , d \\geq 0 , \\end{align*}"}
-{"id": "4259.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\delta d \\theta & = \\delta u & & B _ { R } \\\\ \\delta \\theta & = 0 & & B _ { R } \\\\ \\nu \\wedge \\theta & = 0 & & \\partial B _ { R } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "4697.png", "formula": "\\begin{align*} u _ j ( t ) = \\hat { u } ( t ) e ^ { i j \\theta } , ~ j = d - 1 , \\ldots , m - d + 1 , \\end{align*}"}
-{"id": "300.png", "formula": "\\begin{align*} \\dim ( S _ i ) \\ ; \\leq \\ ; \\dim ( S ) \\textnormal { f o r } i \\ ; = \\ ; 1 , 2 . \\end{align*}"}
-{"id": "544.png", "formula": "\\begin{align*} & v ^ h \\to V \\quad \\mbox { s t r o n g l y i n } \\ , W ^ { 2 , 2 } ( \\omega , \\R ^ 3 ) ~ ~ \\mbox { a s } \\ , h \\to 0 , \\\\ & h \\| v ^ h \\| _ { W ^ { 2 , \\infty } ( \\omega , \\R ^ 3 ) } \\leq \\varepsilon \\quad \\mbox { a n d } \\quad \\lim _ { h \\to 0 } \\frac { 1 } { h ^ 2 } \\ , \\big | \\{ x ' \\in \\omega ; ~ v ^ h ( x ' ) \\neq V ( x ' ) \\} \\big | = 0 . \\end{align*}"}
-{"id": "8433.png", "formula": "\\begin{align*} \\mathfrak { F } = f \\circ \\kappa ^ { - 1 } , q = p \\circ \\kappa ^ { - 1 } . \\end{align*}"}
-{"id": "6883.png", "formula": "\\begin{align*} R _ 1 = \\begin{pmatrix} \\cos ( \\mu ) & - \\sin ( \\mu ) \\\\ \\sin ( \\mu ) & \\cos ( \\mu ) \\end{pmatrix} \\quad { \\rm a n d } R _ 2 = \\begin{pmatrix} \\cos ( \\theta ) & \\sin ( \\theta ) \\\\ \\sin ( \\theta ) & - \\cos ( \\theta ) \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "4851.png", "formula": "\\begin{align*} \\delta ( \\mu ) = \\frac { 1 } { 2 } \\sum _ { r _ s , t } ^ { } [ w _ { ( t , r _ s ) } , w ^ * _ { ( t , r _ s ) } ] , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , 1 \\leq s \\leq m , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , 1 \\leq t \\leq n _ { r _ s } . \\end{align*}"}
-{"id": "7588.png", "formula": "\\begin{align*} T f ( x ) = \\int F ( z ) f ( x , z ) d z \\end{align*}"}
-{"id": "4574.png", "formula": "\\begin{align*} S ( \\mu , R ) = \\int _ { S ^ { n - 1 } } | \\hat { \\mu } ( R \\sigma ) | ^ 2 d \\sigma , \\end{align*}"}
-{"id": "3664.png", "formula": "\\begin{align*} E ( \\bar { y } _ B - Y / N | B ) = \\sum _ { i \\in B } \\mu / n _ B - \\mu = 0 \\end{align*}"}
-{"id": "92.png", "formula": "\\begin{align*} \\rho ( x , t ^ n + ) & = \\rho ( 0 + , t ^ n + ) + \\sum _ { y _ \\ell < x } \\Delta \\rho ( y _ \\ell , t ^ n + ) + \\sum _ { x _ j < x } \\Delta \\rho ( x _ j , t ^ n + ) \\end{align*}"}
-{"id": "7924.png", "formula": "\\begin{align*} p _ 2 ( u ) = \\frac { u ( u - 1 ) } { ( - \\tfrac { 1 } { 2 } ) ( - \\tfrac { 3 } { 2 } ) } f ( a ) + \\frac { ( u + \\tfrac { 1 } { 2 } ) ( u - 1 ) } { ( \\tfrac { 1 } { 2 } ) ( - 1 ) } f ( a + \\tfrac { 1 } { 2 } h ) + \\frac { ( u + \\tfrac { 1 } { 2 } ) u } { ( \\tfrac { 3 } { 2 } ) ( 1 ) } f ( a + \\tfrac { 3 } { 2 } h ) \\end{align*}"}
-{"id": "3916.png", "formula": "\\begin{align*} E ( Z ) = E ^ { \\mathrm { T F } } ( Z ) + \\left ( \\frac q 4 - s ( \\gamma ) \\right ) Z ^ 2 + O ( Z ^ { 4 7 / 2 4 } ) \\end{align*}"}
-{"id": "2033.png", "formula": "\\begin{align*} \\widehat { h _ { \\delta } } ( n ) = \\int _ { - 1 / 2 } ^ { 1 / 2 } h _ { \\delta } ( x ) e ^ { - 2 \\pi i n x } \\ ; d x = \\widehat { 1 _ { [ - \\delta , x + \\delta ] } } ( n ) \\cdot \\widehat { \\phi _ { \\delta } } ( n ) , \\end{align*}"}
-{"id": "9412.png", "formula": "\\begin{align*} ( f | _ 2 { \\sigma _ t ) } ( z ) = \\sum a _ t ( n ) q ^ { n / \\alpha _ t } , \\end{align*}"}
-{"id": "8289.png", "formula": "\\begin{align*} ( \\partial _ t ^ 2 + i a \\partial _ { \\alpha } ) \\bar { z } _ t = - i a _ t \\bar { z } _ { \\alpha } . \\end{align*}"}
-{"id": "474.png", "formula": "\\begin{align*} f = f _ 1 ( x _ 1 , \\dots , x _ a ) + f _ 2 ( x _ { a + 1 } , \\dots , x _ n ) . \\end{align*}"}
-{"id": "7687.png", "formula": "\\begin{align*} \\psi _ \\pm ( x ) : = \\frac { 1 } { \\mathbf { i } \\pm x } = \\overline { \\psi _ \\mp ( x ) } \\end{align*}"}
-{"id": "2050.png", "formula": "\\begin{align*} \\widetilde { f ^ \\chi } _ { R ' } ( z ) = \\sum _ { m = 1 } ^ \\infty b _ { \\chi , R ' } ( m ) e ^ { 2 \\pi i m z } . \\end{align*}"}
-{"id": "7958.png", "formula": "\\begin{align*} S = \\bigcup _ { a \\in [ d _ 1 ] } ( \\{ a \\} \\cup S _ a ) . \\end{align*}"}
-{"id": "9067.png", "formula": "\\begin{align*} I _ k = \\left ( \\frac { k - 1 } { 2 K } , \\frac { k } { 2 K } \\right ] \\end{align*}"}
-{"id": "2367.png", "formula": "\\begin{align*} f ( X _ i \\mapsto ( a _ i \\otimes 1 \\otimes \\cdots \\otimes 1 + \\cdots + 1 \\otimes \\cdots \\otimes 1 \\otimes a _ i ) ) = \\sum _ { S _ 1 + \\cdots S _ N = [ 1 , k ] } a _ { A | S _ 1 } \\otimes \\cdots \\otimes a _ { A | S _ N } \\end{align*}"}
-{"id": "8003.png", "formula": "\\begin{align*} \\sup _ { x \\in Q _ 2 ( 0 ) } e ^ { - a | x - y | ^ 2 } = e ^ { - a | x ^ * - y | ^ 2 } = \\prod _ { j + 1 \\leq i \\leq d } e ^ { - a | x _ i ^ * - y _ i | ^ 2 } . \\end{align*}"}
-{"id": "6374.png", "formula": "\\begin{align*} v _ s ( t , x ) : = \\psi ( t + s , x ) + C s t + C s - C s \\log s , \\end{align*}"}
-{"id": "4807.png", "formula": "\\begin{align*} u _ { i , n } = \\frac { \\sigma _ i } { \\sigma _ 1 ^ 2 + \\cdots + \\sigma _ d ^ 2 } b _ n . \\end{align*}"}
-{"id": "5106.png", "formula": "\\begin{align*} \\langle x _ j , x _ k \\rangle = \\frac { \\| x _ j + x _ k \\| ^ 2 - \\| x _ j - x _ k \\| ^ 2 } { 4 } = \\frac { ( \\| x _ j \\| ^ 2 + \\| x _ k \\| ^ 2 ) - ( \\| x _ j \\| ^ 2 + \\| - x _ k \\| ^ 2 ) } { 4 } = 0 . \\end{align*}"}
-{"id": "3822.png", "formula": "\\begin{align*} \\langle a , b ; \\langle c ; d \\rangle \\rangle = & \\langle a , b , c ; d \\rangle + \\langle a , \\langle b ; c \\rangle ; d \\rangle + \\langle \\langle a , b ; c \\rangle ; d \\rangle \\\\ & + \\langle a , c , b ; d \\rangle + \\langle \\langle a ; c \\rangle , b ; d \\rangle + \\langle c , a , b ; d \\rangle . \\end{align*}"}
-{"id": "1966.png", "formula": "\\begin{align*} 2 d _ { i i } = \\sum _ { k \\in \\mathcal { N } ( i ) } d _ { k j } = d _ { j j } , \\end{align*}"}
-{"id": "5656.png", "formula": "\\begin{align*} \\tau \\sum _ { j = 1 } ^ { n } \\| B _ { n - j } P _ { h } \\| _ { \\mathcal { L } _ 2 ^ 0 } ^ p & \\leq \\tau \\sum _ { j = 1 } ^ { n } \\| A _ h ^ { \\frac { s } { 2 } } B _ { n - j } \\| ^ p \\| A _ h ^ { - \\frac { s } { 2 } } P _ h A ^ { \\frac { s } { 2 } } \\| ^ p \\| A ^ { - \\frac { s } { 2 } } \\| _ { \\mathcal { L } _ 2 ^ 0 } ^ { p } \\\\ & \\leq c \\tau \\sum _ { j = 1 } ^ n t _ { n - j + 1 } ^ { ( ( 1 - \\frac { s } { 2 } ) \\alpha + \\gamma - 1 ) p } \\| A ^ { - \\frac { s } { 2 } } \\| _ { \\mathcal { L } _ 2 ^ 0 } ^ p < \\infty . \\end{align*}"}
-{"id": "5089.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j \\in \\mathbb { S } } A _ j ^ * A _ j x \\right \\| ^ 2 & = \\left \\| \\left \\langle \\sum _ { j \\in \\mathbb { S } } A _ j ^ * A _ j x , \\sum _ { k \\in \\mathbb { S } } A _ k ^ * A _ k x \\right \\rangle \\right \\| = \\left \\| \\sum _ { j \\in \\mathbb { S } } \\langle A _ j x , A _ j x \\rangle \\right \\| , \\end{align*}"}
-{"id": "5276.png", "formula": "\\begin{align*} 1 - F ( x ) = \\int _ { x } ^ { + \\infty } F ^ { \\prime } ( t ) d t . \\end{align*}"}
-{"id": "6642.png", "formula": "\\begin{align*} \\widehat { C _ { \\lambda } ^ { a / q , 1 } } ( \\xi ) = \\sum _ { \\ell \\in \\mathbb Z ^ { d } } G ( a , \\ell , q ) \\widetilde \\psi _ { q } ( \\xi - \\ell / q ) \\widetilde \\psi _ { \\lambda Q / N } ( \\xi - \\ell / q ) \\widetilde { d \\sigma _ { \\lambda } } ( \\xi - \\ell / q ) . \\end{align*}"}
-{"id": "9550.png", "formula": "\\begin{align*} \\textup { s p t } ( n , N ) = \\frac { n ^ N } { ( N ! ) ^ 2 } + O ( n ^ { N - 1 } ) . \\end{align*}"}
-{"id": "2257.png", "formula": "\\begin{align*} p _ { \\pm \\i \\pi } & = | \\alpha _ { \\pm \\i \\pi } \\rangle \\langle \\beta _ { \\pm \\i \\pi } | & | \\alpha _ { \\pm \\i \\pi } \\rangle & = \\binom { 1 } { - \\displaystyle \\frac { 1 } { \\pm \\i \\pi - \\Omega } | E \\rangle } \\\\ \\langle \\beta _ { \\pm \\i \\pi } | & = ( 1 , \\langle E | \\frac { 1 } { \\pm \\i \\pi - \\Omega } ) \\end{align*}"}
-{"id": "7751.png", "formula": "\\begin{align*} ( \\langle x _ { \\alpha } , x \\rangle E ) ^ * ( \\langle x _ { \\alpha } , x \\rangle E ) = \\lambda _ \\alpha \\overline { \\lambda _ \\alpha } \\| x \\| ^ 2 E ^ 2 = \\| x \\| ^ 2 E \\end{align*}"}
-{"id": "4561.png", "formula": "\\begin{align*} \\tau _ { a } . ( \\tau _ { a } \\cdots \\tau _ { b - 1 } \\tau _ { b + 1 } \\cdots \\tau _ { c } Q ( z ) ) & = ( x _ a - x _ { a + 1 } ) \\tau _ { a + 1 } \\cdots \\tau _ { b - 1 } \\tau _ { b + 1 } \\cdots \\tau _ { c } Q ( z ) \\\\ & = \\tau _ { a + 1 } \\cdots \\tau _ { b - 1 } x _ { b } \\tau _ { b + 1 } \\cdots \\tau _ { c } Q ( z ) \\\\ \\end{align*}"}
-{"id": "9522.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\frac { q ^ n } { 1 - q ^ n } = \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - 1 ) ^ { n - 1 } q ^ { n ( n + 1 ) / 2 } } { ( 1 - q ^ n ) } , \\end{align*}"}
-{"id": "273.png", "formula": "\\begin{align*} R ^ j f _ * C ^ { \\bullet } = 0 \\quad j > n - p . \\end{align*}"}
-{"id": "4139.png", "formula": "\\begin{align*} \\varphi _ { 1 } ( z ) = \\varphi ( z ) + \\begin{cases} \\pi i , & z \\in O _ V , \\ , \\Im ( z ) > 0 , \\\\ - \\pi i , & z \\in O _ V , \\ , \\Im ( z ) < 0 . \\end{cases} \\end{align*}"}
-{"id": "2402.png", "formula": "\\begin{align*} \\sigma _ T = ( i _ 1 , b _ 1 ) , \\ldots , ( i _ k , b _ k ) , \\end{align*}"}
-{"id": "2137.png", "formula": "\\begin{align*} \\inf _ { x \\in X } \\Phi ( x , 0 ) = \\max _ { z \\in L ^ 0 ( Z ) } \\{ - \\Phi ^ \\ast ( 0 , z ) \\} . \\end{align*}"}
-{"id": "646.png", "formula": "\\begin{align*} \\vec a ^ \\varphi _ j = a _ { i _ 1 } \\dots a _ { i _ { k ^ { ( \\varphi ) } _ j } } \\ . \\end{align*}"}
-{"id": "7121.png", "formula": "\\begin{align*} \\phi ^ * _ m ( x ) = \\lambda ^ * _ m \\int _ 0 ^ 1 G _ 0 ( x , \\xi ) \\phi ^ * _ m ( \\xi ) d \\xi - \\int _ 0 ^ 1 G _ 0 ( x , \\xi ) \\hat { q } ( \\xi ) \\phi ^ * _ m ( \\xi ) d \\xi , ~ ~ ~ m = 1 , 2 , \\end{align*}"}
-{"id": "1470.png", "formula": "\\begin{align*} V _ { 1 2 } = - \\varepsilon \\bar { m } \\int _ { \\mathbb { T } ^ { d } } ( \\mu ^ { 1 } - \\mu ^ { 2 } ) \\sum _ { i = 1 } ^ { d } \\Theta _ { p _ { i } q } ( t , x , \\mu ^ { 2 } , D w ^ { 2 } ) \\left ( \\frac { \\partial ( \\mu ^ { 1 } - \\mu ^ { 2 } ) } { \\partial x _ { i } } \\right ) \\ d x , \\end{align*}"}
-{"id": "386.png", "formula": "\\begin{align*} j _ { - \\check { \\lambda } , ! } \\star \\mathbb { D } _ I A = \\mathbb { D } _ I ( j _ { - \\check { \\lambda } , * } \\star A ) , \\end{align*}"}
-{"id": "7492.png", "formula": "\\begin{align*} f ( z ) \\colon = g ( z ) ^ { 1 \\slash ( s - 2 ) } = e ^ { \\frac { 1 } { s - 2 } \\log g ( z ) } \\end{align*}"}
-{"id": "2101.png", "formula": "\\begin{align*} x ( t ) = x _ 0 + \\int _ { t _ 0 } ^ t A ( s ) x ( s ) d s + \\int _ { t _ 0 } ^ t C ( s ) x ( s ) d \\omega ( s ) , \\ t \\geq t _ 0 , \\end{align*}"}
-{"id": "7481.png", "formula": "\\begin{align*} \\int _ { q _ * \\gamma } \\psi _ s = \\int _ { \\gamma } q ^ * \\psi _ s = q _ s \\int _ { \\gamma } \\psi _ s \\end{align*}"}
-{"id": "5844.png", "formula": "\\begin{align*} \\int _ { F } c _ 1 ( x ) | x | ^ { - 1 } c _ 2 ( 1 - x ) | 1 - x | ^ { - 1 } d x = \\frac { \\Gamma ( c _ 1 ) \\Gamma ( c _ 2 ) } { \\Gamma ( c _ 1 c _ 2 ) } . \\end{align*}"}
-{"id": "5355.png", "formula": "\\begin{align*} X _ \\alpha X _ \\beta . v _ { n m } ^ 1 = - \\lambda v _ { n + 1 \\ , m + 3 } ^ 1 . \\end{align*}"}
-{"id": "7216.png", "formula": "\\begin{align*} K _ r ( x ) = \\frac { 2 } { \\Gamma ( r ) } \\int _ 0 ^ \\infty \\frac { t ^ { r - 1 } ( e ^ t \\cos \\left ( x \\right ) - 1 ) } { 1 - 2 e ^ t \\cos \\left ( x \\right ) + e ^ { 2 t } } \\ , d t , x \\in [ 0 , \\pi ] , r > 1 . \\end{align*}"}
-{"id": "1911.png", "formula": "\\begin{align*} \\mbox { b ) } | x | - t = | x + t | \\mbox { a n d } | x | + t = | x - t | . \\end{align*}"}
-{"id": "9896.png", "formula": "\\begin{align*} g ' ( x ) & = 1 _ { \\mathcal { X } } ( x ) \\frac { f ' ( x ) } { Q ( Z \\leq x ) } \\\\ g ( x ) & = c + \\int _ { - \\infty } ^ { x } 1 _ { \\mathcal { X } } ( u ) \\frac { f ' ( u ) } { Q ( Z \\leq u ) } d u \\end{align*}"}
-{"id": "4228.png", "formula": "\\begin{align*} \\phi _ { i } ( s ) = e ^ { \\tan ( s ) X _ { \\beta } } w b _ { 0 } \\ , , \\ , s \\in \\left ( - \\pi / 2 , \\pi / 2 \\right ) . \\end{align*}"}
-{"id": "2919.png", "formula": "\\begin{align*} \\widetilde { E } _ { A | A ^ \\complement } ( \\theta , ( x , y ) ) & = E _ { A | A ^ \\complement } ( \\theta , x ) \\ ; . \\end{align*}"}
-{"id": "5029.png", "formula": "\\begin{align*} r _ 0 : = \\frac { c _ 5 I ^ { 3 / 2 } } { m ^ { 1 / 2 } n ^ { 3 / 2 } ( \\log m n ) ^ 3 } \\end{align*}"}
-{"id": "4839.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big ( t ^ 2 \\frac { d u } { d t } \\Big ) + t ^ 2 u ^ p = 0 , \\end{align*}"}
-{"id": "3388.png", "formula": "\\begin{align*} f ( T ) = O ( \\deg ( T ) ^ { \\alpha } ) . \\end{align*}"}
-{"id": "9184.png", "formula": "\\begin{align*} \\mathcal { K } & \\equiv \\mathrm { s p a n } \\left \\{ \\left . J \\in \\Lambda _ { N } \\right \\vert J \\left ( t \\right ) = 0 t \\in \\left ( 0 , b \\right ] \\right \\} \\\\ \\mathcal { K } _ { b } & \\equiv \\left \\{ \\left . J \\in \\Lambda _ { N } \\right \\vert J \\left ( b \\right ) = 0 \\right \\} . \\end{align*}"}
-{"id": "6724.png", "formula": "\\begin{align*} \\dot { P } _ { 1 1 } = - h ( Q _ { 1 1 } + \\lambda Q _ { 2 2 } ) , \\dot { P } _ { 1 2 } = - h Q _ { 1 2 } , \\end{align*}"}
-{"id": "1620.png", "formula": "\\begin{align*} V _ r ( x ) = \\sup _ { Z \\in \\Lambda } \\mathbb { E } _ x \\int _ 0 ^ \\infty e ^ { - r s } d Z _ s \\end{align*}"}
-{"id": "4153.png", "formula": "\\begin{align*} i \\sqrt { 3 } z ^ { 1 / 3 } f _ { 1 } ( z ) - i \\sqrt { 3 } z ^ { 2 / 3 } f _ { 2 } ( z ) & = 3 \\varphi _ { 1 } ( z ) , \\Im z > 0 . \\end{align*}"}
-{"id": "8693.png", "formula": "\\begin{align*} \\langle J _ Z X ' , J _ T X ' \\rangle & = \\frac { 1 } { 2 } \\left ( \\langle J _ { Z + T } X ' , J _ { Z + T } X ' \\rangle - \\langle J _ { Z } X ' , J _ { Z } X ' \\rangle - \\langle J _ { T } X ' , J _ { T } X ' \\rangle \\right ) \\\\ & = 1 6 \\abs { X ' } ^ 2 \\langle Z , T \\rangle \\end{align*}"}
-{"id": "9169.png", "formula": "\\begin{align*} \\psi ( c ) ( s ) = \\gamma _ s ( c ) . \\end{align*}"}
-{"id": "431.png", "formula": "\\begin{align*} W _ t ( x ) : = \\int _ { [ 0 , t ] \\cap \\mathcal R ^ x } \\dd W _ t ( x ) = \\int _ 0 ^ t \\Big ( \\gamma ( X _ s ( x ) - \\gamma s ) + 1 \\Big ) e ^ { \\gamma X _ s ( x ) - \\frac { \\gamma ^ 2 } { 2 } s } \\mathbf { 1 } _ { \\{ s \\in \\mathcal { R } ^ x \\} } d X _ s ( x ) \\end{align*}"}
-{"id": "3279.png", "formula": "\\begin{align*} \\Omega _ n ' ( a + b ) = n U _ n ( a + b , - a b ) . \\end{align*}"}
-{"id": "2174.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\Phi } f ( \\mathbf { x } ) = \\sup _ { t > 0 } | \\Phi _ t * f ( \\mathbf { x } ) | = \\sup _ { t > 0 } \\left | \\int \\Phi _ t ( \\mathbf { x } , \\mathbf { y } ) f ( \\mathbf { y } ) \\ , d w ( \\mathbf { y } ) \\right | , \\end{align*}"}
-{"id": "6754.png", "formula": "\\begin{align*} x ^ 2 y \\approx x y x \\approx y x ^ 2 \\approx 0 . \\end{align*}"}
-{"id": "5229.png", "formula": "\\begin{align*} \\lim \\sup _ { h \\rightarrow + \\infty } \\lim _ { \\lambda \\rightarrow + \\infty } \\inf \\mathbb { P } ( B ^ { \\ast } ( h ) / A ( h ) > \\lambda ) = 0 . \\end{align*}"}
-{"id": "1345.png", "formula": "\\begin{align*} \\pi _ * ^ u ( f ) ( y ) = \\sum _ { x \\in \\pi ^ { - 1 } ( y ) } f ( x ) f \\in C _ c ( G ^ u ) y \\in H ^ { \\pi ( u ) } \\end{align*}"}
-{"id": "7546.png", "formula": "\\begin{align*} \\nabla _ a \\mathcal { V } = \\left ( \\nabla _ a X \\right ) g . \\end{align*}"}
-{"id": "6138.png", "formula": "\\begin{align*} \\chi ( X , M ) = \\binom { \\frac 1 2 q _ X ( M ) + m + 1 } { m } ; \\end{align*}"}
-{"id": "7390.png", "formula": "\\begin{align*} \\dot { x } & = y \\\\ \\dot { y } & = \\lambda _ 1 + \\lambda _ 2 x + a _ 2 x ^ 2 + b _ 2 x y . \\end{align*}"}
-{"id": "4691.png", "formula": "\\begin{align*} { u } _ t + { f } ( { u } ) _ x = 0 , \\ ; \\ ; \\ ; ( x , t ) \\in ( a , b ) \\times ( 0 , T ] . \\end{align*}"}
-{"id": "8019.png", "formula": "\\begin{align*} ( 2 m - 3 ) a _ 1 Q _ { m , i } = \\sum _ { j = 2 } ^ { r + 1 } ( 3 j - 2 m ) a _ j Q _ { m - j + 1 , i } \\mbox { f o r a l l } m > 0 \\end{align*}"}
-{"id": "2359.png", "formula": "\\begin{align*} a | n > = & \\sqrt { n } | n - 1 > \\\\ a ^ + | n > & = \\sqrt { n + 1 } | n + 1 > \\\\ a ^ + a | n > & = n | n > \\end{align*}"}
-{"id": "5341.png", "formula": "\\begin{align*} - \\frac 1 2 i H , \\quad \\frac 1 2 ( X - Y ) = \\frac 1 2 i B \\quad \\mbox { a n d } - \\frac 1 2 i ( X + Y ) = - \\frac 1 2 i A . \\end{align*}"}
-{"id": "4880.png", "formula": "\\begin{align*} r ( x ) = \\operatorname { d i s t } ( x , \\ , ( \\mathbb { C } ^ n \\setminus V ) \\cap ( \\{ x \\} \\times \\sqrt { - 1 } \\mathbb { R } ^ n ) ) ( x \\in \\mathbb { R } ^ n ) . \\end{align*}"}
-{"id": "6053.png", "formula": "\\begin{align*} \\alpha _ i = \\left \\{ \\begin{array} { l l } n , & \\mbox { i f } 1 \\leq i \\leq x \\\\ y , & \\mbox { i f } i = x + 1 \\\\ 0 , & \\mbox { i f } x + 2 \\leq i \\leq r . \\end{array} \\right . \\end{align*}"}
-{"id": "5450.png", "formula": "\\begin{align*} \\bar V _ 0 \\ = \\ \\sup _ { \\bar \\alpha \\in \\bar { \\mathcal A } ^ { \\bar \\theta } } \\bar J ( \\bar \\alpha ) . \\end{align*}"}
-{"id": "4712.png", "formula": "\\begin{align*} \\omega _ k = \\frac { \\alpha _ k } { \\displaystyle { \\sum _ { l = 0 } ^ 2 \\alpha _ l } } , ~ ~ \\alpha _ k = \\frac { C ^ 2 _ k } { ( \\epsilon + I S _ k ) ^ 2 } , \\ ; \\ ; \\ ; k = 0 , 1 , 2 . \\end{align*}"}
-{"id": "153.png", "formula": "\\begin{align*} H _ i \\circ q _ M \\textrm { a n d } l _ { d H _ i } = \\sum _ { s = 1 } ^ { N } \\dfrac { \\partial H _ i } { \\partial x ^ s } ( { \\bf x } ) y ^ s , i = 1 , \\dots k , \\end{align*}"}
-{"id": "7474.png", "formula": "\\begin{gather*} \\aleph = \\sum _ { i = 1 } ^ b k _ i . \\end{gather*}"}
-{"id": "7603.png", "formula": "\\begin{align*} \\nabla _ x \\left ( X ^ { - 1 } ( x , s ) - X ^ { - 1 } ( x , t ) \\right ) = \\left ( \\nabla _ a X \\circ X ^ { - 1 } \\right ) ( x , t ) \\left ( \\nabla _ a \\left ( X - \\mathrm { I d } \\right ) \\right ) \\left ( X ^ { - 1 } ( x , t ) , t - s \\right ) \\end{align*}"}
-{"id": "1490.png", "formula": "\\begin{align*} \\sum _ { \\ell = 2 } ^ { 6 } \\sum _ { j = 1 } ^ { d } W _ { \\ell } ^ { j } \\leq \\varepsilon ( \\mathcal { G } _ { 1 } ( K ) + \\mathcal { G } _ { 2 } ( K ) ) ( E _ { w } + E _ { \\mu } ) + \\frac { 1 } { 4 } \\int _ { \\mathbb { T } ^ { d } } | D \\mu ^ { 1 } - D \\mu ^ { 2 } | ^ { 2 } \\ d x . \\end{align*}"}
-{"id": "14.png", "formula": "\\begin{align*} | \\Delta Y _ t ^ { i j } ( m ) | : = | Y _ t ^ { i } ( m ) - Y _ t ^ { j } ( m ) | \\leq \\frac { 1 } { q ^ { \\min } } \\left ( K _ f + \\frac { C _ v C _ { \\eta } C _ z } { ( C _ { \\eta } - C _ v ) ^ 2 } \\right ) . \\end{align*}"}
-{"id": "7841.png", "formula": "\\begin{align*} \\Big | \\prod _ { j = 1 } ^ d a _ j - \\prod _ { j = 1 } ^ d b _ j \\Big | \\le \\sum _ { j = 1 } ^ d | a _ j - b _ j | . \\end{align*}"}
-{"id": "7275.png", "formula": "\\begin{align*} \\lVert \\xi _ i - \\eta _ i \\rVert < \\frac { \\epsilon } { n } , \\ i = 1 , \\dots , n . \\end{align*}"}
-{"id": "5594.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ 0 ^ { + \\infty } \\ ! \\ ! \\ ! x | u | ^ 2 d x & = - 2 \\ , \\int _ 0 ^ { + \\infty } \\ ! \\ ! \\ ! u \\bar { u } _ x d x \\\\ & = \\frac { \\alpha } { \\gamma } \\int _ 0 ^ { + \\infty } \\ ! \\ ! \\ ! v ^ 2 - \\mathcal { Q } ^ + _ 0 + \\int _ 0 ^ t \\Big ( 2 | u _ x ( 0 , s ) | ^ 2 + \\frac { \\alpha } { \\gamma } v _ x ^ 2 ( 0 , s ) \\Big ) d s \\\\ & > - \\mathcal { Q } ^ + _ 0 , \\end{align*}"}
-{"id": "3192.png", "formula": "\\begin{align*} \\varGamma d ^ * f = \\varGamma C _ + \\varphi = \\varGamma C C _ + \\varphi = U C _ + \\varphi = C _ + \\varphi = d ^ * f , \\end{align*}"}
-{"id": "9390.png", "formula": "\\begin{align*} \\varphi _ { 1 } ( t w _ { 0 } ) & \\geq \\Lambda ^ { - 2 } t ^ { p } [ w _ { 0 } ] _ { s , p } ^ { p } - t ^ { p _ { s } ^ { \\ast } } \\int _ \\Omega | w _ { 0 } | ^ { p ^ { \\ast } _ { s } } d x - \\lambda c _ { 4 } \\int _ { \\Omega } | t w _ { 0 } | ^ { q } d x \\\\ & \\geq A _ { 1 } t ^ { p } - B t ^ { p _ { s } ^ { \\ast } } - E _ { 1 } t ^ { q } \\end{align*}"}
-{"id": "5153.png", "formula": "\\begin{align*} { \\S } = \\frac { 1 } { 2 } \\begin{pmatrix} \\sigma & 0 \\\\ 0 & \\sigma \\end{pmatrix} . \\end{align*}"}
-{"id": "6427.png", "formula": "\\begin{align*} B _ { k } ^ { - T } w _ { k } = \\left [ \\begin{array} { c c } B _ { k - 1 } ^ { - T } \\\\ - w _ { k - 1 } ^ { T } \\frac { \\beta _ { k - 1 } } { \\alpha _ { k } } & \\frac { 1 } { \\alpha _ { k } } \\end{array} \\right ] \\left [ \\begin{array} { c } - w _ { k - 1 } \\frac { \\beta _ { k - 1 } } { \\alpha _ { k } } \\\\ \\frac { 1 } { \\alpha _ { k } } \\end{array} \\right ] = \\left [ \\begin{array} { c } - \\frac { \\beta _ { k - 1 } } { \\alpha _ { k } } \\left ( B _ { k - 1 } ^ { - T } w _ { k - 1 } \\right ) \\\\ \\| w _ { k } \\| ^ { 2 } \\end{array} \\right ] . \\end{align*}"}
-{"id": "3626.png", "formula": "\\begin{align*} ( a _ 1 - a _ { - 1 } ) \\cdot v _ { ( 2 , 3 ) } - t ( a _ 1 - a _ { - 1 } ) = 0 \\\\ ( a _ 2 - a _ { - 2 } ) \\cdot v _ { ( 1 , 3 ) } - t ( a _ 2 - a _ { - 2 } ) = 0 \\\\ ( a _ 3 - a _ { - 3 } ) \\cdot v _ { ( 1 , 2 ) } - t ( a _ 3 - a _ { - 3 } ) = 0 . \\end{align*}"}
-{"id": "2465.png", "formula": "\\begin{align*} \\ell ( D ) : = \\dim _ k ( \\L ( D ) ) . \\end{align*}"}
-{"id": "968.png", "formula": "\\begin{align*} S S ^ { ( - 1 ) } = ( k - \\lambda ) 1 _ H + \\lambda w H . \\end{align*}"}
-{"id": "5303.png", "formula": "\\begin{align*} ( x _ 1 , \\dots , x _ N ) = & ( u _ 1 ^ { ( i ) } u _ i ^ { ( i ) } + h _ 1 , \\dots , u _ { i - 1 } ^ { ( i ) } u _ i ^ { ( i ) } + h _ { i - 1 } , u _ i ^ { ( i ) } + h _ i , \\\\ & u _ { i + 1 } ^ { ( i ) } u _ i ^ { ( i ) } + h _ { i + 1 } \\dots , u _ k ^ { ( i ) } u _ i ^ { ( i ) } + h _ k , x _ { k + 1 } , \\dots , x _ N ) . \\end{align*}"}
-{"id": "5159.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } F _ { n } ( a _ { n } x + b _ { n } ) = H _ { 1 } ( x ) , x \\in C ( H _ { 1 } ) \\end{align*}"}
-{"id": "8750.png", "formula": "\\begin{align*} \\begin{aligned} \\widehat { \\omega } _ n ^ { ( \\alpha , \\sigma ) } = & { \\tau ^ { 1 + \\alpha } e ^ { - n \\sigma \\tau } } \\sum _ { j = 0 } ^ { Q - 1 } w _ j ( 1 + \\lambda _ j \\tau ) ^ { - 1 - n } , \\end{aligned} \\end{align*}"}
-{"id": "6426.png", "formula": "\\begin{align*} \\| w _ { k + 1 } \\| ^ { 2 } = \\frac { 1 } { \\alpha _ { k + 1 } ^ { 2 } } \\left ( \\beta _ { k } ^ { 2 } \\| w _ { k } \\| ^ { 2 } + 1 \\right ) , \\end{align*}"}
-{"id": "8349.png", "formula": "\\begin{align*} A _ 1 ( \\alpha , t ) > 0 , \\forall ~ \\alpha \\neq x ( t ) ; a n d ~ ~ A _ 1 ( x ( t ) , t ) = 0 . \\end{align*}"}
-{"id": "8704.png", "formula": "\\begin{align*} \\frac { d ^ n } { d x ^ n } \\Big [ ( f \\circ g ) ( x ) \\Big ] = \\sum _ { k _ 1 + 2 k _ 2 + \\cdots + n k _ n = n } \\frac { n ! } { k _ 1 ! \\cdots k _ n ! } \\Bigg [ \\frac { d ^ { k _ 1 + \\cdots + k _ n } f } { d x } \\circ g \\Bigg ] ( x ) \\prod _ { m = 1 } ^ n \\Bigg ( \\frac { 1 } { m ! } \\frac { d ^ m g } { d x ^ m } \\Bigg ) ^ { k _ m } . \\end{align*}"}
-{"id": "6455.png", "formula": "\\begin{align*} \\phi ^ p ( \\ \\cdot \\ ) : = \\frac { 1 } { \\phi ( p ) } \\phi ( p \\ \\cdot \\ p ) . \\end{align*}"}
-{"id": "773.png", "formula": "\\begin{align*} C _ { q ^ d } [ P ] = \\bigoplus _ { i = 1 } ^ { s } C _ { q ^ d } [ T - \\rho _ i ] . \\end{align*}"}
-{"id": "9200.png", "formula": "\\begin{align*} N = \\left \\{ \\left . \\left ( z _ { 1 } , z _ { 2 } \\right ) \\in \\mathbb { C } ^ { 2 } \\right \\vert \\left \\vert z _ { 1 } \\right \\vert = \\left \\vert z _ { 2 } \\right \\vert = \\frac { 1 } { \\sqrt { 2 } } \\right \\} \\end{align*}"}
-{"id": "5072.png", "formula": "\\begin{align*} \\langle V U ^ * h , h \\rangle & = \\langle U ^ * h , V ^ * h \\rangle = \\sum \\limits _ { j \\in \\mathbb { J } } \\langle U ^ * h , f _ j \\rangle \\langle f _ j , V ^ * h \\rangle \\\\ & = \\sum \\limits _ { j \\in \\mathbb { J } } \\langle h , U f _ j \\rangle \\langle V f _ j , h \\rangle = \\sum \\limits _ { j \\in \\mathbb { J } } \\langle h , x _ j \\rangle \\langle \\tau _ j , h \\rangle , ~ \\forall h \\in \\mathcal { H } , \\end{align*}"}
-{"id": "7581.png", "formula": "\\begin{align*} \\delta _ h \\left ( \\int _ 0 ^ t \\Delta g _ { \\nu ( t - s ) } * \\Delta _ 2 \\tau ( s , t ) d s \\right ) = \\int _ 0 ^ t \\delta _ h ( \\Delta g _ { \\nu ( t - s ) } ) * \\Delta _ 2 \\tau ( s , t ) d s . \\end{align*}"}
-{"id": "4416.png", "formula": "\\begin{align*} g | _ { C _ 1 } = f . \\end{align*}"}
-{"id": "6200.png", "formula": "\\begin{align*} f ( x ) = \\int _ { 0 } ^ { \\infty } e ^ { - \\frac { x ^ 2 } { 2 v } } \\frac { \\mu ( d v ) } { \\sqrt { 2 \\pi v } } \\end{align*}"}
-{"id": "2757.png", "formula": "\\begin{align*} \\phi _ { 1 } ( \\kappa ; u ) = & - \\int _ { \\mathcal { C } _ { < } } \\frac { d w } { 2 \\pi i } \\big ( ( w - \\kappa ) ^ 2 - u \\big ) \\sqrt { 2 w } e ^ { \\frac { 1 } { 3 } ( w - \\kappa ) ^ 3 - u ( w - \\kappa ) } \\end{align*}"}
-{"id": "4702.png", "formula": "\\begin{align*} u _ t + \\Big ( \\frac { u ^ 2 } { 2 } \\Big ) _ x = 0 . \\end{align*}"}
-{"id": "9152.png", "formula": "\\begin{align*} \\overline { \\Theta } _ { i } ( \\overline { \\mathbf { r } } ) \\equiv \\prod \\limits _ { \\substack { j = 1 , N ; \\\\ i < j } } \\overline { \\Theta } \\left ( \\left \\vert \\mathbf { r } _ { i } - \\mathbf { r } _ { j } \\right \\vert - \\sigma \\right ) , \\end{align*}"}
-{"id": "2461.png", "formula": "\\begin{align*} D \\ = \\ \\frac { 1 } { 2 } Q _ 1 + \\frac { 1 } { 2 } Q _ 2 - \\frac { 1 } { 2 } O . \\end{align*}"}
-{"id": "10034.png", "formula": "\\begin{align*} u & ( z q ^ n ) D ( f , g ) ( z q ^ { n } ) - u ( - q ^ n ) D ( f , g ) ( - q ^ n ) = \\\\ & \\left [ f ( z q ^ n ) - f ( - q ^ n ) \\right ] g ( z q ^ { n + 1 } ) - f ( z q ^ { n + 1 } ) \\left [ g ( z q ^ n ) - g ( - q ^ n ) \\right ] \\\\ & + f ( - q ^ n ) \\left [ g ( z q ^ { n + 1 } ) - g ( - q ^ { n + 1 } ) \\right ] - \\left [ f ( z q ^ { n + 1 } ) - f ( - q ^ { n + 1 } ) \\right ] g ( - q ^ n ) . \\end{align*}"}
-{"id": "3774.png", "formula": "\\begin{align*} \\tau = \\frac { ( 2 \\rho _ 1 ) ^ { - s } - \\rho _ 2 ^ { - s } } { \\rho _ 0 ^ { - s } - \\rho _ 2 ^ { - s } } \\end{align*}"}
-{"id": "626.png", "formula": "\\begin{align*} K ^ { \\mathcal Q } _ \\varphi : = \\{ x \\in G _ m \\mid q ( \\varphi , x ) = \\varphi \\} \\ . \\end{align*}"}
-{"id": "2410.png", "formula": "\\begin{align*} q = q _ { Q L } + \\rho , \\end{align*}"}
-{"id": "664.png", "formula": "\\begin{align*} \\widehat \\rho _ { \\vec a } = [ \\rho ; \\mathrm { i d } _ { a _ { \\delta ( 1 ) } } , \\dots , \\mathrm { i d } _ { a _ { \\delta ( n ) } } ; e _ m ] : \\vec a \\to a _ { \\delta ( 1 ) } \\dots a _ { \\delta ( n ) } \\end{align*}"}
-{"id": "6761.png", "formula": "\\begin{align*} \\begin{array} { r l } = \\ { \\strut \\displaystyle - c _ { 1 } \\over \\displaystyle c _ { 1 } + c _ { 0 } ( s - 1 ) - { \\strut \\displaystyle 2 c _ { 0 } c _ { 2 } ( s - 1 ) \\over \\displaystyle 2 c _ { 2 } + c _ { 1 } ( s + 1 ) - { \\strut \\displaystyle 4 c _ { 1 } c _ { 3 } ( s + 1 ) \\over \\displaystyle 4 c _ { 3 } + c _ { 2 } ( s + 3 ) } } } & \\\\ & \\ddots \\end{array} \\end{align*}"}
-{"id": "3438.png", "formula": "\\begin{align*} v ( t + 1 ) = X _ { t + 1 } v ( t ) , t = 0 , 1 , 2 , \\ldots , \\end{align*}"}
-{"id": "4145.png", "formula": "\\begin{align*} \\varphi _ { 2 , + } ( x ) - \\varphi _ { 2 , - } ( x ) = \\frac { 1 } { 2 } ( g _ { 1 , + } ( x ) - g _ { 1 , - } ( x ) ) - \\pi i = \\pi i \\mu ^ * ( [ x , \\infty ) ) - \\pi i \\end{align*}"}
-{"id": "8417.png", "formula": "\\begin{align*} \\| z _ t ( \\cdot , t ) \\| _ { H ^ s } \\leq \\| z _ t ( \\cdot , 0 ) \\| _ { H ^ s } + T _ 0 M _ 1 : = M _ 2 . \\end{align*}"}
-{"id": "2825.png", "formula": "\\begin{align*} U _ 1 ( t ) = \\sum _ { i = 1 } ^ n \\eta _ i \\biggl ( G ( x _ i ( t ) , x _ i ^ * ) + \\frac { c _ i } { d _ i } \\frac { ( u _ i ( t ) - u _ i ^ * ) ^ 2 } { 2 } \\biggr ) , \\end{align*}"}
-{"id": "7582.png", "formula": "\\begin{align*} \\begin{gathered} \\frac { 1 } { | h | ^ { \\alpha } } \\norm { \\delta _ h \\left ( \\int _ 0 ^ t \\Delta g _ { \\nu ( t - s ) } * \\Delta _ 2 \\tau ( s , t ) d s \\right ) } _ { L ^ \\infty } \\le \\frac { C ( \\alpha ) } { \\nu } \\norm { X - \\mathrm { I d } } _ { L i p ( 0 , T ; C ^ { 1 + \\alpha } ) } ^ \\alpha \\norm { \\tau } _ { L ^ \\infty ( 0 , T ; C ^ { \\alpha , p } ) } . \\end{gathered} \\end{align*}"}
-{"id": "4019.png", "formula": "\\begin{align*} K _ { { \\mathcal F } } = \\sum _ n e ^ { i \\pi s n ^ 2 } P _ N , \\end{align*}"}
-{"id": "4504.png", "formula": "\\begin{align*} \\theta \\circ \\langle \\boldsymbol x , \\boldsymbol x ' \\rangle & = \\theta \\circ \\langle \\pi _ 2 , \\pi _ 1 \\rangle \\circ \\langle \\boldsymbol x ' , \\boldsymbol x \\rangle \\\\ & = i \\circ \\theta \\circ \\langle \\boldsymbol x ' , \\boldsymbol x \\rangle \\ ; . \\end{align*}"}
-{"id": "3159.png", "formula": "\\begin{align*} \\mu ( \\bar t ) = \\{ \\bar t ^ { m + n - 1 } + ( n - 1 ) c , \\bar t ^ { n + m - 2 } + ( n - 2 ) c , \\dots , \\bar t ^ { m + 1 } + c , \\bar t ^ { m } , \\bar t ^ { m - 1 } + c , \\dots , \\bar t ^ { 1 } + ( m - 1 ) c \\} . \\end{align*}"}
-{"id": "2822.png", "formula": "\\begin{align*} x _ i ^ { * , p } = \\frac { R _ i ^ p } { R _ 0 ^ p } > 0 , u _ i ^ { * , p } = \\frac { d _ i } { e _ i } x _ i ^ { * , p } > 0 , i = 1 , 2 , \\ldots , p . \\end{align*}"}
-{"id": "1001.png", "formula": "\\begin{align*} & ( a _ { - 1 } - a _ 1 ) ^ k \\rho ^ { ( n , k ) } ( [ a _ { - 1 } , + \\infty ) ^ k ) \\\\ \\leq & \\sum _ { i _ 1 , \\cdots , i _ { k } \\ } \\sum _ { \\varepsilon _ 1 , \\cdots , \\varepsilon _ { k } \\in \\{ \\pm 1 \\} } \\prod _ { j = 1 } ^ k \\varepsilon _ { j } ( \\tau ^ { ( n ) } _ { i _ j } - a _ { \\varepsilon _ { j } } ) _ + \\\\ \\leq & ( a _ { - 1 } - a _ 1 ) ^ k \\rho ^ { ( n , k ) } ( ( a _ { 1 } , + \\infty ) ^ k ) . \\end{align*}"}
-{"id": "6596.png", "formula": "\\begin{align*} P _ { F _ n } ( \\varphi ) = \\prod _ { r = 1 } ^ { F _ n } \\left | 2 \\sin \\pi r \\varphi \\right | = A _ n B _ n C _ n , \\end{align*}"}
-{"id": "3611.png", "formula": "\\begin{align*} \\bar { B } : = B \\cup \\{ a _ { - 1 } \\cdot v _ { ( 2 , 3 ) } , a _ { - 2 } \\cdot v _ { ( 1 , 3 ) } , a _ { - 3 } \\cdot v _ { ( 1 , 2 ) } \\} . \\end{align*}"}
-{"id": "1073.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to + \\infty } \\ln \\left ( n ( 2 \\ln n ) ^ { - \\frac { 1 } { 2 } } D _ n ( \\alpha _ n ) \\right ) = c _ 0 - x - 2 z . \\end{align*}"}
-{"id": "6872.png", "formula": "\\begin{align*} \\mathcal { R } _ 2 ^ { ( 1 ) } \\quad \\equiv \\begin{cases} y ^ 1 _ { 1 0 } y ^ 2 _ { 0 1 } - y ^ 1 _ { 0 1 } y ^ 2 _ { 1 0 } - 1 = 0 \\ , , \\\\ y ^ k _ \\nu = 0 \\ , , & \\ | \\nu | = 2 \\ , , k = 1 , 2 \\ , . \\end{cases} \\end{align*}"}
-{"id": "6583.png", "formula": "\\begin{align*} \\partial \\Omega = \\overline { \\Gamma _ D } \\cup \\overline { \\Gamma _ N } . \\end{align*}"}
-{"id": "9246.png", "formula": "\\begin{align*} \\sup _ { | \\tau | = 1 , \\tau \\cdot \\nu = 0 } | u _ { \\tau \\tau } ( x _ 0 ) | \\le \\epsilon M _ 2 ( R ) + C _ \\epsilon ( 1 + \\frac { 1 } { R ^ 2 } ) , \\end{align*}"}
-{"id": "8889.png", "formula": "\\begin{align*} P ( a ) P ( b ) = P ( a P ( b ) ) + P ( P ( a ) b ) + \\lambda P ( a b ) \\end{align*}"}
-{"id": "1800.png", "formula": "\\begin{align*} \\sum _ { n = l } ^ { N } P \\left ( Y _ { n } \\geq \\frac { 1 - \\theta ^ { H } } { ( 1 - \\theta ) ^ { H } } \\sqrt { 2 \\log \\log ( \\theta ^ { - n } ) } + c _ { n } \\right ) \\geq \\sum _ { n = l } ^ { N } \\frac { C _ { 3 } } { n ^ { \\gamma } } \\geq C _ { 3 } ( N ^ { 1 - \\gamma } - l ^ { 1 - \\gamma } ) , \\end{align*}"}
-{"id": "3924.png", "formula": "\\begin{align*} \\rho _ \\infty ( x , y ) = \\sup _ i \\left | x _ i - y _ i \\right | \\end{align*}"}
-{"id": "5702.png", "formula": "\\begin{align*} E _ { \\alpha , \\beta } ( z ) - z E _ { \\alpha , \\alpha + \\beta } ( z ) = \\frac { 1 } { \\Gamma ( \\beta ) } , \\end{align*}"}
-{"id": "8309.png", "formula": "\\begin{align*} \\dot { z } _ j ( t ) = ( v - \\frac { \\lambda _ j i } { 2 \\pi } \\frac { 1 } { \\overline { z - z _ j ( t ) } } ) \\Big | _ { z = z _ j ( t ) } \\end{align*}"}
-{"id": "3696.png", "formula": "\\begin{align*} \\begin{cases} E \\big ( C o v _ N ( \\delta _ i , y _ i ) ; y _ U \\big ) = \\frac { 1 } { N } \\sum \\limits _ { i \\in U } p _ i y _ i - \\big ( \\frac { 1 } { N } \\sum \\limits _ { i \\in U } p _ i \\big ) \\big ( \\frac { 1 } { N } \\sum \\limits _ { i \\in U } y _ i \\big ) \\rightarrow 0 \\\\ E \\big ( E _ N ( \\delta _ i ) \\big ) = \\sum _ { i \\in U } p _ i / N \\rightarrow p > 0 \\end{cases} \\end{align*}"}
-{"id": "8916.png", "formula": "\\begin{align*} \\textrm { I n c o r r e c t } & : = \\{ u \\in [ 1 : n ] : x ( u ) \\neq \\hat { x } ( u ) \\} \\\\ \\textrm { U n c h a n g e d } & : = \\{ u \\in [ 1 : n ] : x ( u ) = y ( u ) \\} . \\end{align*}"}
-{"id": "6324.png", "formula": "\\begin{align*} q ( x , 0 ) = q _ 0 ( x ) : = \\nu ^ { \\frac { 1 } { 2 } } u ^ { - 1 } _ 0 ( x ) \\quad x \\in ( 0 , \\ , 1 ) . \\end{align*}"}
-{"id": "8868.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 e _ 4 } { \\partial a ^ 2 } & = 2 0 9 \\nu + 3 1 7 - 5 \\nu ^ 3 - 1 7 \\nu ^ 2 > 0 \\end{align*}"}
-{"id": "7887.png", "formula": "\\begin{align*} U _ { \\sigma } \\left ( \\Xi _ { j } ^ { * } \\Xi _ { j } \\{ g _ { n } \\} _ { n = 1 } ^ { \\infty } \\right ) = \\left \\{ \\begin{array} { l } ( C C ' ) ^ { \\frac { 1 } { 2 } } \\Lambda _ { j } ^ { * } ( g _ { j } ) \\ , \\ j \\in \\sigma \\\\ \\\\ ( C C ' ) ^ { \\frac { 1 } { 2 } } \\Omega _ { j } ^ { * } ( g _ { j } ) \\ , \\ j \\in \\sigma ^ { c } \\end{array} \\right . \\end{align*}"}
-{"id": "7984.png", "formula": "\\begin{align*} h ( n ) = \\prod _ { \\frac { n } { 2 } < p \\leq n } p \\ , . \\end{align*}"}
-{"id": "1147.png", "formula": "\\begin{align*} & 4 | y _ i - y _ j | \\geq | y _ i + y _ j | \\cdot | y _ i - y _ j | = \\left | y _ i ^ 2 - y _ j ^ 2 \\right | = \\left | ( \\gamma _ n ( u _ i ) ) ^ 2 - ( \\gamma _ n ( u _ j ) ) ^ 2 \\right | \\\\ = & \\left | ( 1 + | u _ j | / \\ln n ) ^ 2 - ( 1 + | u _ i | / \\ln n ) ^ 2 \\right | S ( I ) ^ 2 \\\\ = & \\big | | u _ j | - | u _ i | \\big | / \\ln n \\cdot ( 2 + | u _ j | / \\ln n + | u _ i | / \\ln n ) S ( I ) ^ 2 \\\\ \\geq & \\big | | u _ j | - | u _ i | \\big | / \\ln n \\cdot 2 S ( I ) ^ 2 \\geq \\varepsilon _ 1 / \\ln n \\cdot 2 S ( I ) ^ 2 , \\end{align*}"}
-{"id": "2645.png", "formula": "\\begin{align*} V = \\big [ ( \\eta _ { - } \\lambda _ { r } ) ^ { s - 1 } \\big ] _ { r = 1 , \\ldots , N ; s = 1 , \\ldots , \\infty } = : [ V _ { 1 } , V _ { 2 } ] \\end{align*}"}
-{"id": "3864.png", "formula": "\\begin{align*} \\{ e \\} = H _ 0 \\triangleleft H _ 1 \\triangleleft H _ 2 \\triangleleft \\dotsb \\triangleleft H _ k = G \\end{align*}"}
-{"id": "1930.png", "formula": "\\begin{align*} x F _ 2 ' ( x ) - 2 F _ 2 ( x ) = - \\frac { 1 } { 2 } x , \\end{align*}"}
-{"id": "644.png", "formula": "\\begin{align*} [ ( \\psi ^ { u ^ { - 1 } x } ) ^ \\ast ( v u ) ] & = [ ( \\psi ^ x ) ^ \\ast ( v ) ( \\psi ^ { u ^ { - 1 } x } ) ^ \\ast ( u ) ] \\\\ & = [ ( \\psi ^ x ) ^ \\ast ( v ' ) ( \\psi ^ { u '^ { - 1 } x ' } ) ^ \\ast ( u ' ) ] \\\\ & = [ ( \\psi ^ { u '^ { - 1 } x ' } ) ^ \\ast ( v ' u ' ) ] \\ . \\end{align*}"}
-{"id": "6700.png", "formula": "\\begin{align*} m \\ddot { q } _ i + \\omega _ { i j } \\dot { q } _ j + \\frac { \\partial V } { \\partial q _ i } = 0 , \\end{align*}"}
-{"id": "5269.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\lambda ( A _ { n } ) < \\lambda ( E ) + \\varepsilon / 2 . \\end{align*}"}
-{"id": "744.png", "formula": "\\begin{align*} \\mathbb { E } _ { f } \\ ! \\left [ T ^ { 2 } \\right ] = \\frac { 1 } { ( q - 1 ) ^ { 2 } } \\left ( 1 - \\frac { 2 \\Gamma ( m n ) } { \\Gamma ( m n + q ) } \\mathbb { E } _ { g } \\ ! \\left [ L \\right ] + \\frac { \\Gamma ( m n ) } { \\Gamma ( m n + 2 q ) } \\mathbb { E } _ { g } \\ ! \\left [ L ^ { 2 } \\right ] \\right ) . \\end{align*}"}
-{"id": "4765.png", "formula": "\\begin{align*} ( \\tilde { u } _ R ( t ) , \\varphi ( t ) ) _ { \\mathcal { H } } - ( \\tilde { u } _ { R , 0 } , \\varphi ( 0 ) ) _ { \\mathcal { H } } = \\int _ 0 ^ t \\Big [ & ( \\tilde { u } _ R , \\partial _ t \\varphi ) _ { \\mathcal { H } } + 2 \\nu a ( \\tilde { u } _ R , \\varphi ) + 2 \\nu a ( H , \\varphi ) \\\\ & - b _ R ( \\tilde { u } _ R , \\varphi , \\tilde { u } _ R ) - b ( H , \\varphi , \\tilde { u } _ R ) - b ( \\tilde { u } _ R , \\varphi , H ) \\Big ] d t . \\end{align*}"}
-{"id": "4366.png", "formula": "\\begin{align*} \\rho _ { \\sigma , t } : = \\Gamma _ { \\sigma } ^ { \\frac { 1 - \\beta _ t } { \\beta _ t } } ( \\rho _ t ) . \\end{align*}"}
-{"id": "2280.png", "formula": "\\begin{align*} & ( \\delta ' + L U _ K Y G ) * Z Y = \\delta \\\\ & Z ' = - \\int _ 0 ^ t d t _ 1 L U _ K ( t - t _ 1 ) G Z ( t _ 1 ) , & Z ( 0 ) & = 1 , & Z ( t ) & = 0 t < 0 \\end{align*}"}
-{"id": "5913.png", "formula": "\\begin{align*} - q ^ { - 1 } ( 1 + q ) ( t _ { s _ 0 } \\star ( \\varphi _ 0 + \\psi _ 0 ) ) = - ( 1 + q ^ { - 1 } ) ( \\varphi _ 0 + \\psi _ 0 ) \\end{align*}"}
-{"id": "8989.png", "formula": "\\begin{align*} F ( x ) = \\prod _ { \\gamma \\in \\mathbb { A } ( \\mathbb { F } _ q ) \\setminus \\overline { \\mathcal { R } } } ( x - \\gamma ) \\in \\mathbb { F } _ q [ x ] _ { \\le q - r - 1 } . \\end{align*}"}
-{"id": "1211.png", "formula": "\\begin{align*} z ' ( x , y ) \\in \\overline { { \\mathcal U } _ { \\sin \\theta _ { m a x } } ( x ) } = \\overline { { \\mathcal U } _ { \\delta _ 0 } ( x ) } ( x \\in \\overline { D } , y \\in \\overline { B } ) , \\end{align*}"}
-{"id": "9947.png", "formula": "\\begin{align*} B ( \\psi ) = \\sum _ { k = 1 } ^ \\infty B ( \\psi _ k ) \\end{align*}"}
-{"id": "2537.png", "formula": "\\begin{align*} - \\int _ { s } ^ { s _ 0 } l ( u ) d u ^ { \\alpha } = \\int _ { s _ 0 ^ { - 1 } } ^ { s ^ { - 1 } } L ( u ) d u ^ { - \\alpha } \\stackrel [ s \\to 0 ] { } { \\sim } ( s ^ { - 1 } ) ^ { - \\alpha } L ( s ^ { - 1 } ) = s ^ \\alpha l ( s ) , \\end{align*}"}
-{"id": "368.png", "formula": "\\begin{align*} v _ 0 ^ { ( i ) } = z _ 0 ^ { ( i ) } , v _ 1 ^ { ( i ) } = s z _ 0 ^ { ( i ) } + c x _ 0 ^ { ( i ) } , v _ { m + 2 } ^ { ( i , ) } = 2 s v _ { m + 1 } ^ { ( i ) } - v _ m ^ { ( i ) } i = 1 , \\dotsc , i _ 0 . \\end{align*}"}
-{"id": "4016.png", "formula": "\\begin{align*} f ( z ) = \\sum _ \\chi \\sum _ M \\sum _ \\ell c _ { \\chi , M , \\ell } \\ , f _ { M } ( \\ell z ) , \\end{align*}"}
-{"id": "6556.png", "formula": "\\begin{align*} U _ d = \\left \\{ ( P , E ) \\mid E \\ { \\rm i s \\ a \\ s m o o t h \\ p l a n e \\ c u b i c \\ a n d \\ } P { \\rm \\ i s \\ a \\ } d { \\rm p r i m i t i v e \\ p o i n t \\ o f \\ } E \\right \\} \\end{align*}"}
-{"id": "6545.png", "formula": "\\begin{align*} \\tau _ \\eta + [ ( u _ { \\eta , x } ) ] = 0 \\end{align*}"}
-{"id": "6081.png", "formula": "\\begin{align*} \\int _ 0 ^ t ( V ' ) ^ { 1 - \\delta _ 1 } d s & \\leq t ^ { \\delta _ 1 } \\left ( \\int _ 0 ^ t V ' d s \\right ) ^ { 1 - \\delta _ 1 } = t ^ { \\delta _ 1 } V ( t ) ^ { 1 - \\delta _ 1 } \\end{align*}"}
-{"id": "9147.png", "formula": "\\begin{align*} L _ { 1 \\left ( 1 \\right ) } \\rho _ { 1 } ^ { ( N ) } ( \\mathbf { x } _ { 1 } , t ) = \\widehat { \\rho } _ { 1 } ^ { ( N ) } ( \\mathbf { x } _ { 1 } , t ) L _ { 1 \\left ( 1 \\right ) } k _ { 1 } ^ { ( N ) } ( \\mathbf { r } _ { 1 } , t ) , \\end{align*}"}
-{"id": "41.png", "formula": "\\begin{align*} \\delta \\mathcal { Y } _ 0 ^ { v } ( T ) = \\mathbb { E } ^ { \\mathbb { Q } } [ h ^ { \\alpha _ T } ( V _ T ^ v ) - \\mathbf { y } ^ { \\alpha _ T } ( V _ T ^ { v } ) ] . \\end{align*}"}
-{"id": "8825.png", "formula": "\\begin{align*} \\sigma _ C ( x ) = \\sup _ { y \\in C } \\langle y , x \\rangle \\ ; . \\end{align*}"}
-{"id": "2381.png", "formula": "\\begin{align*} ( T ^ { N , l } _ + ) _ { k k ' } & = \\delta _ { k + 1 , k ' } \\frac { \\sqrt { ( 2 l ( k + 1 ) - k k ^ 2 } } { \\sqrt { N } } \\approx \\delta _ { k + 1 , k ' } \\sqrt { 2 z ( k + 1 ) } \\\\ ( T ^ { N , l } _ - ) _ { k k ' } & = \\delta _ { k - 1 , k ' } \\frac { \\sqrt { ( 2 l k + k - k ^ 2 } } { \\sqrt { N } } \\approx \\delta _ { k - 1 , k ' } \\sqrt { 2 z k } \\end{align*}"}
-{"id": "4795.png", "formula": "\\begin{align*} \\left ( H _ 0 + \\frac { \\nu } { | x | } + a _ 1 \\right ) \\psi ^ { - a _ 1 } _ C = 0 , \\end{align*}"}
-{"id": "251.png", "formula": "\\begin{align*} 2 N - 2 \\geq \\sum _ { \\ell = 1 } ^ { r _ 1 } ( e _ { 1 , \\ell } - 1 ) + \\sum _ { j = 2 } ^ r \\sum _ { \\ell = 1 } ^ { r _ j } ( e _ { j , \\ell } - 1 ) = N - p _ 1 - u _ 1 - v _ 1 + \\sum _ { j = 2 } ^ r ( N - p _ j - q _ j ) . \\end{align*}"}
-{"id": "1009.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\mathbb { P } ( \\tau _ k ^ { ( n ) } \\in I ) = \\lim _ { n \\to + \\infty } \\mathbb { P } ( \\tau _ k ^ { ( n ) } \\in A _ 1 ) - \\lim _ { n \\to + \\infty } \\mathbb { P } ( \\tau _ k ^ { ( n ) } \\in A _ 2 ) \\\\ = \\varphi _ k \\left ( - f ' ( a ) \\right ) - \\varphi _ k \\left ( - f ' ( b ) \\right ) = \\int _ a ^ { b } \\frac { f '' ( x ) ( - f ' ( x ) ) ^ { k - 1 } } { ( k - 1 ) ! } e ^ { f ' ( x ) } d x . \\end{align*}"}
-{"id": "1941.png", "formula": "\\begin{align*} \\mathcal { A } = \\sum _ { j = 1 } ^ \\infty \\left [ ( j - 1 ) \\pi + \\epsilon _ j - \\tan \\epsilon _ j + ( \\chi + 1 ) \\chi ^ 2 \\frac { A _ j ^ 2 } { Z _ j ^ 3 } \\right ] - \\sum _ { j = 1 } ^ \\infty ( j - 1 ) \\pi . \\end{align*}"}
-{"id": "2867.png", "formula": "\\begin{align*} K _ p = \\begin{cases} 0 & p = 0 \\\\ ( n - 1 ) ^ { - 1 } ( \\textrm { l o w e r b o u n d o f R i c c i c u r v a t u r e } ) & p = 1 \\\\ \\textrm { l o w e r b o u n d o f t h e c u r v a t u r e o p e r a t o r } & p \\geq 2 \\end{cases} \\end{align*}"}
-{"id": "9273.png", "formula": "\\begin{align*} \\sup \\limits _ { | \\tau | = 1 , \\tau \\cdot \\nu = 0 } | u _ { \\tau \\tau } ( \\tilde y ) | \\le \\epsilon \\sup \\limits _ { B _ { \\frac { \\delta R } { 1 6 } } ( \\tilde y ) \\cap \\Omega } | D ^ 2 u | + C _ \\epsilon ( 1 + \\frac { 1 } { ( \\delta R / 1 6 ) ^ 2 } ) , \\end{align*}"}
-{"id": "842.png", "formula": "\\begin{align*} f ( t ) = \\langle x , \\Phi ( t ) \\rangle t \\in \\Omega , \\end{align*}"}
-{"id": "7441.png", "formula": "\\begin{align*} \\displaystyle R _ q = \\frac { 1 } { \\lvert W ' \\rvert } \\sum _ { w \\in W } w \\cdot U _ q . \\end{align*}"}
-{"id": "151.png", "formula": "\\begin{align*} \\pi _ { T M } ( { \\bf x } , { \\bf y } ) = \\left ( \\begin{array} { c | c } 0 & \\pi ( { \\bf x } ) \\\\ \\hline \\pi ( { \\bf x } ) & \\sum _ { s = 1 } ^ { N } \\dfrac { \\partial \\pi } { \\partial x ^ s } ( { \\bf x } ) y ^ s \\end{array} \\right ) , \\end{align*}"}
-{"id": "10007.png", "formula": "\\begin{gather*} \\sigma _ { \\mathfrak { X } ( X ) } ( D ) = \\inf { \\{ \\sigma \\in \\mathbb { R } \\colon \\mbox { $ \\sup _ { N } \\big \\| \\sum _ { n = 1 } ^ { N } { \\frac { a _ { n } } { n ^ { \\sigma } } n ^ { - s } } \\big \\| _ { \\mathfrak { X } ( X ) } < \\infty $ } \\} } \\\\ [ 2 m m ] \\sigma _ { \\mathfrak { X } ( X ) } ( D ) \\leq \\limsup _ { N } { \\frac { \\log { \\left \\| \\sum _ { n = 1 } ^ { N } { a _ { n } n ^ { - s } } \\right \\| _ { \\mathfrak { X } ( X ) } } } { \\log { N } } } \\end{gather*}"}
-{"id": "9037.png", "formula": "\\begin{align*} | | f | | ^ 2 _ { L ^ 2 ( \\mathbb { T } ^ 2 ) } = \\sum _ { | \\xi | ^ 2 = E } | a _ { \\xi } | ^ 2 = 1 . \\end{align*}"}
-{"id": "7061.png", "formula": "\\begin{align*} E ( 0 ) = \\frac { 1 } { 2 } \\int _ M g ^ { i j } u ^ v _ i \\o { u ^ v _ j } \\phi _ 0 d \\mu _ g . \\end{align*}"}
-{"id": "1424.png", "formula": "\\begin{align*} & \\| u \\| _ { L E } = \\| u \\| _ E + \\| \\partial u \\| _ { l _ \\infty ^ { - 1 / 2 } ( L _ t ^ 2 L _ x ^ 2 ) } + \\| r ^ { - 1 } u \\| _ { l _ \\infty ^ { - 1 / 2 } ( L _ t ^ 2 L _ x ^ 2 ) } , \\ \\| u \\| _ { L E _ m } = \\sum _ { | a | \\leq m } \\| Y ^ a u \\| _ { L E } , \\end{align*}"}
-{"id": "7213.png", "formula": "\\begin{align*} | k | ^ { - r } = \\int _ 0 ^ 1 t ^ { | k | - 1 } \\ , d \\sigma _ r ( t ) \\qquad \\mbox { f o r a n y } k \\neq 0 . \\end{align*}"}
-{"id": "9764.png", "formula": "\\begin{align*} q _ \\infty ( x + \\xi , 0 ) = q _ \\infty ( x , 0 ) \\quad ( \\xi , x ) \\in ( L _ \\ast + L ( q ) ) \\times \\R ^ n . \\end{align*}"}
-{"id": "3757.png", "formula": "\\begin{align*} \\mbox { \\boldmath $ { \\nabla } $ } \\times \\left [ f _ { l } ( r ) \\mathbf { Y } _ { l , m } \\right ] = \\frac { i \\ , l ( l + 1 ) } { r } f _ { l } ( r ) { Y } _ { l , m } \\ , \\hat { \\mathbf { r } } + \\frac { 1 } { r } \\frac { d } { d r } [ r f _ { l } ( r ) ] \\ , \\hat { \\mathbf { r } } \\times \\mathbf { Y } _ { l , m } , \\end{align*}"}
-{"id": "3285.png", "formula": "\\begin{align*} X = 2 \\sqrt { - Y } \\cos \\frac { ( 1 + 2 j ) \\pi } { 2 n } \\end{align*}"}
-{"id": "7230.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } [ ( \\mu - \\phi ) ^ 2 ] ^ { \\prime \\prime } ( \\xi ) = - K _ r ^ { \\prime } * \\phi ^ \\prime ( \\xi ) . \\end{align*}"}
-{"id": "9838.png", "formula": "\\begin{align*} & \\left \\{ \\begin{array} { r c l l } w _ 3 ( x ' ) & \\geq & \\psi ( x ' ) & \\R ^ { n - 1 } \\\\ ( - \\Delta ) ^ { s - \\frac { 1 } { 2 } } w _ 3 & = & 0 & \\R ^ { n - 1 } \\setminus \\{ x ' : w _ 3 ( x ' ) = \\psi ( x ' ) \\} \\\\ ( - \\Delta ) ^ { s - \\frac { 1 } { 2 } } w _ 3 & \\leq & 0 & \\R ^ { n - 1 } \\\\ \\lim _ { | x ' | \\to \\infty } w _ 3 ( x ' ) & = & 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "3960.png", "formula": "\\begin{align*} u _ i ( x , t ) = \\frac { 1 } { 2 } x ^ * P ^ i _ t x + x ^ * \\nu ^ i _ t + \\tau ^ i _ t . \\end{align*}"}
-{"id": "5640.png", "formula": "\\begin{align*} 2 ( A + 3 C - \\frac { 8 } { 3 } ) - 4 A + \\frac { 2 2 4 } { 8 1 } & = 0 \\\\ 6 ( A + 3 C - \\frac { 8 } { 3 } ) - 4 C + \\frac { 6 4 0 } { 8 1 } & = 0 , \\end{align*}"}
-{"id": "1.png", "formula": "\\begin{align*} d V _ { t } = \\eta ( V _ { t } ) d t + \\kappa d W _ { t } \\end{align*}"}
-{"id": "7746.png", "formula": "\\begin{align*} \\langle x , x \\rangle \\langle x , y \\rangle \\langle x , x \\rangle = - \\bar { \\mu } \\| y \\| \\langle x , x \\rangle . \\end{align*}"}
-{"id": "9703.png", "formula": "\\begin{align*} T _ { \\textbf { d } } ( \\partial \\Omega ( P _ i ) ) = I _ { \\textbf { d } } ( P _ i ) - 1 . \\end{align*}"}
-{"id": "3339.png", "formula": "\\begin{align*} f ' ( t ) = f ( t ) ( \\log x - \\frac { n \\log 2 } { t ^ 2 } ) , \\end{align*}"}
-{"id": "6494.png", "formula": "\\begin{align*} & \\int _ { \\Q _ { \\ell } ^ { \\times } } | y | ^ { s } \\gamma ^ { - 1 } ( y ) ( f ( y ) \\gamma ( y ) + g ( y ) \\gamma ( y ) v ( y ) ) d ^ { \\times } y = P _ 1 ( s ) L ( \\mathbf { 1 } , s ) + P _ 2 ( s ) L ( \\mathbf { 1 } , s ) ^ 2 . \\end{align*}"}
-{"id": "3448.png", "formula": "\\begin{align*} g ( \\xi ) = M \\log N + \\log ( M + 1 ) + v _ M ( \\theta ) + \\frac { \\xi } { \\rho _ { M , N } } , \\end{align*}"}
-{"id": "839.png", "formula": "\\begin{align*} W _ { \\bar { \\mu } } : = \\{ f \\in \\mathbb { F } _ { q ^ d } [ \\lambda _ 0 , \\ldots , \\lambda _ { \\alpha - 1 } ] : \\mathrm { m d e g } ( f ) \\leq ( \\mu _ 0 , \\ldots , \\mu _ { \\alpha - 1 } ) \\} ( \\bar { \\mu } = ( \\mu _ 0 , \\ldots , \\mu _ { \\alpha - 1 } ) ) . \\end{align*}"}
-{"id": "4734.png", "formula": "\\begin{align*} ( \\mathbf { T } \\mathbf { U } ) _ { i , j } = \\sum _ { k = 0 } ^ { n - 1 } T _ { i - k } U _ { k - j } , i , j = 0 , 1 , \\cdots n - 1 , \\end{align*}"}
-{"id": "1779.png", "formula": "\\begin{align*} C _ { s , t , \\eta , H } = \\sqrt { \\frac { \\eta ^ { 2 } ( t - s ) ^ { 2 H } } { 2 \\left [ \\gamma _ { H } ( t - s ) ^ { 2 H } + ( t - s ) \\right ] ^ { 2 } } + 2 } . \\end{align*}"}
-{"id": "8225.png", "formula": "\\begin{align*} t _ \\lambda ^ 5 + \\sum _ { l = 0 } ^ 4 a _ l ( t ) t _ { \\lambda } ^ l = 0 . \\end{align*}"}
-{"id": "8537.png", "formula": "\\begin{align*} D _ t G = ( \\partial _ t g ) \\circ \\kappa ^ { - 1 } . \\end{align*}"}
-{"id": "9545.png", "formula": "\\begin{align*} n p ( n ) = \\frac { 1 } { 2 } M _ { 2 } ( n ) , \\end{align*}"}
-{"id": "759.png", "formula": "\\begin{align*} B = W M _ { \\sqrt { \\mu } } V ^ * , R ^ * = V M _ { \\sqrt { \\mu } } W ^ * . \\end{align*}"}
-{"id": "5473.png", "formula": "\\begin{align*} v ( t , x ) \\ = \\ \\hat v ^ { \\mathcal R } ( t , x , a ) , \\forall \\ , ( t , x , a ) \\in [ 0 , T ] \\times H \\times \\Lambda . \\end{align*}"}
-{"id": "2933.png", "formula": "\\begin{align*} & A = U \\begin{pmatrix} \\Sigma _ { 1 } & 0 \\\\ 0 & 0 \\end{pmatrix} V ^ { \\ast } = U _ { 1 } \\Sigma _ { 1 } V _ { 1 } ^ { \\ast } , \\\\ & B = \\widetilde { U } \\begin{pmatrix} \\widetilde { \\Sigma } _ { 1 } & 0 \\\\ 0 & 0 \\end{pmatrix} \\widetilde { V } ^ { \\ast } = \\widetilde { U } _ { 1 } \\widetilde { \\Sigma } _ { 1 } \\widetilde { V } _ { 1 } ^ { \\ast } , \\end{align*}"}
-{"id": "1192.png", "formula": "\\begin{align*} \\| F ( w ) ( t ) \\| & \\le \\| u _ 0 \\| + \\int _ 0 ^ t \\| e ^ { - ( t - s ) A } \\| \\left [ \\| w ( s ) \\| _ { L ^ { \\infty } } \\| w _ x ( s ) \\| + \\| ( 1 - \\partial _ x ^ 2 ) ^ { - 1 } w _ x ( s ) \\| \\right ] d s \\\\ & \\le \\| u _ 0 \\| + \\int _ 0 ^ t \\left [ c R ^ 2 s ^ { - 3 / 4 } + c R \\right ] d s = \\| u _ 0 \\| + 4 c R ^ 2 t ^ { 1 / 4 } + c R t . \\end{align*}"}
-{"id": "4860.png", "formula": "\\begin{align*} \\delta ( \\alpha ( \\mu ) ) = a \\delta ( \\mu ) = \\frac { a } { 2 } [ w _ { ( j , r _ s ) } , w ^ * _ { ( j , r _ s ) } ] + \\frac { a } { 2 } \\ , \\ , \\sum _ { t \\neq j } [ w _ { ( t , r _ s ) } , w ^ * _ { ( t , r _ s ) } ] . \\end{align*}"}
-{"id": "5790.png", "formula": "\\begin{align*} R ( \\Gamma , q , T ) - R ( \\Gamma \\backslash e , q , T ) - R ( \\Gamma / e , q , T ) & = \\frac { ( ( q + 2 ) T ^ 2 - 1 ) T } { ( 1 - T ) ^ 3 } \\\\ & = - T + ( 2 q + 1 ) T ^ 3 + ( 2 q ^ 2 + 4 q + 2 ) T ^ 4 + O ( T ^ 5 ) . \\end{align*}"}
-{"id": "8378.png", "formula": "\\begin{align*} - i ( I - \\mathfrak { H } ) a _ t \\bar { z } _ { \\alpha } = g _ 1 + g _ 2 , \\end{align*}"}
-{"id": "9024.png", "formula": "\\begin{align*} f _ { \\{ b _ k , c _ k \\} , 1 } ( x _ 1 , x _ 2 ) = \\begin{cases} 1 & \\{ x _ 1 , x _ 2 \\} = \\{ b _ k , c _ k \\} , \\\\ x _ 1 & \\end{cases} \\end{align*}"}
-{"id": "3963.png", "formula": "\\begin{align*} M _ k ^ 2 = \\bigoplus \\limits _ { r = 0 } ^ 2 \\ , M _ k ^ { n , r } . \\end{align*}"}
-{"id": "1310.png", "formula": "\\begin{align*} \\bigl \\{ \\bigl ( \\varepsilon ( x , a ) , \\varepsilon ( y , a ) \\bigr ) \\in \\widetilde { Y } \\times \\widetilde { Y } : \\varepsilon ( x , a ) - \\varepsilon ( y , a ) \\in f ( B ) \\bigl \\} \\ , \\cap \\ , O = \\emptyset . \\end{align*}"}
-{"id": "2904.png", "formula": "\\begin{align*} [ a ] \\eta = [ \\overline { ( a ) \\bullet m } ] \\stackrel { } { = } [ \\overline { T \\circ ( m \\bullet ( a ) ) } ] \\stackrel { p r o p \\ref { p r o p : c o m p o s a } ( 2 ) } { = } [ \\overline { ( m \\bullet ( a ) ) \\circ T } ] \\stackrel { } { = } [ \\overline { ( m \\circ T ) \\bullet ( a ) } ] \\stackrel { r e m \\ref { r m : T m } } { = } [ \\overline { m \\bullet ( a ) } ] = \\eta [ a ] \\end{align*}"}
-{"id": "5825.png", "formula": "\\begin{align*} \\ln ( \\phi _ { ( \\bar X _ n , s _ n ^ 2 ) } ( t _ 1 , t _ 2 ) ) = \\ln ( \\phi _ { \\bar X _ n } ( t _ 1 ) ) + \\ln ( \\phi _ { s _ n ^ 2 } ( t _ 2 ) ) . \\end{align*}"}
-{"id": "5091.png", "formula": "\\begin{align*} x = \\sum \\limits _ { j \\in \\mathbb { J } } \\Psi _ j ^ * A _ j x \\iff \\sum \\limits _ { j \\in \\mathbb { J } } ( 2 e - c _ j ) \\langle A _ j x , \\Psi _ j x \\rangle = \\langle x , x \\rangle \\iff \\sum \\limits _ { j \\in \\mathbb { J } } c _ j ^ 2 \\langle A _ j x , A _ j x \\rangle = \\langle x , x \\rangle . \\end{align*}"}
-{"id": "5898.png", "formula": "\\begin{align*} \\log ( 1 + | p ( g ) | ) & \\leq \\log ( q ^ { n M _ p } + q ^ { n M _ p } C _ p ) \\\\ & = \\log ( q ^ { n M _ p } ( 1 + C _ P ) ) \\\\ & = n M _ p \\log ( q ( 1 + C _ p ) ^ { 1 / n M _ p } ) \\\\ & \\leq n M _ p \\log ( q ( 1 + C _ p ) ) \\\\ & = n M _ p D _ p \\end{align*}"}
-{"id": "7107.png", "formula": "\\begin{align*} \\zeta = g \\wedge d g \\circ Q & = \\frac { 1 } { 2 } \\{ ( e _ { 0 } + \\phi e _ { 1 } - i \\phi e _ { 2 } - e _ { 3 } ) \\wedge ( \\phi e _ { 0 } + e _ { 1 } + i e _ { 2 } + \\phi e _ { 3 } ) \\omega \\\\ & + ( e _ { 0 } + \\bar { \\phi } e _ { 1 } + i \\bar { \\phi } e _ { 2 } - e _ { 3 } ) \\wedge ( \\bar { \\phi } e _ { 0 } + e _ { 1 } - i e _ { 2 } + \\bar { \\phi } e _ { 3 } ) \\bar { \\omega } \\} . \\end{align*}"}
-{"id": "492.png", "formula": "\\begin{align*} \\bar \\AA ( \\pi _ 1 ( U ) ) = \\pi _ 2 \\left ( z _ 1 ^ { n ( d - 1 ) } + \\cdots + z _ k ^ { n ( d - 1 ) - 1 } + G ( z _ { k + 1 } , \\dots , z _ n ) \\right ) \\end{align*}"}
-{"id": "6060.png", "formula": "\\begin{align*} a _ k = \\varepsilon _ { i _ k } ( \\tilde { e } _ { i _ { k - 1 } } ^ { a _ { k - 1 } } \\cdots \\tilde { e } _ { i _ 1 } ^ { a _ 1 } b ^ * ) , \\end{align*}"}
-{"id": "4369.png", "formula": "\\begin{align*} \\delta _ t ( T ) \\le \\frac { \\gamma { k \\choose t } + \\Delta { k - 1 \\choose t } } { { n - 1 \\choose t } } = \\frac { { k - 1 \\choose t } } { { n - 1 \\choose t } } \\Big ( \\frac k { k - t } \\gamma + \\Delta \\Big ) \\overset { \\eqref { e q 0 2 } } { \\le } \\frac { { k - 1 \\choose t } { n - 1 \\choose k - 1 } } { { n - 1 \\choose t } } = { n - t - 1 \\choose k - t - 1 } . \\end{align*}"}
-{"id": "6073.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l r } \\Delta u = n u \\\\ u \\vert _ \\Sigma = c > 0 \\end{array} \\right . \\end{align*}"}
-{"id": "8617.png", "formula": "\\begin{align*} h = \\varepsilon ^ 2 p , \\ \\ q = \\varepsilon p ' , \\ \\ z = u ' . \\end{align*}"}
-{"id": "8252.png", "formula": "\\begin{align*} & \\sum _ { n = 0 } ^ \\infty c ( 2 n ) q ^ n = \\frac { 2 E _ 2 ( 2 \\tau ) - E _ 2 ( \\tau ) } { ( q ; q ) _ \\infty } , \\\\ & \\sum _ { n = 0 } ^ \\infty c ( 2 n + 1 ) q ^ n = \\frac { 2 4 } { ( q ; q ) _ \\infty } \\sum _ { n = 0 } ^ \\infty \\sigma ( 2 n + 1 ) q ^ n . \\end{align*}"}
-{"id": "8298.png", "formula": "\\begin{align*} ( D _ t ^ 2 - i A \\partial _ { \\alpha } ) \\theta \\circ \\kappa ^ { - 1 } = - 2 [ D _ t \\zeta , \\mathcal { H } \\frac { 1 } { \\zeta _ { \\alpha } } + \\bar { \\mathcal { H } } \\frac { 1 } { \\bar { \\zeta } _ { \\alpha } } ] \\partial _ { \\alpha } D _ t \\zeta + \\frac { 1 } { \\pi i } \\int _ { - \\infty } ^ { \\infty } \\Big ( \\frac { D _ t \\zeta ( \\alpha , t ) - D _ t \\zeta ( \\beta , t ) } { \\zeta ( \\alpha , t ) - \\zeta ( \\beta , t ) } \\Big ) ^ 2 ( \\zeta - \\bar { \\zeta } ) _ { \\beta } d \\beta , \\end{align*}"}
-{"id": "4413.png", "formula": "\\begin{align*} ( \\omega _ o | _ { \\Sigma } ) ^ 2 = ( 1 + | 2 s t ^ 2 | ^ 2 + | 2 t s ^ 2 | ^ 2 ) d d ^ c | s | ^ 2 \\wedge d d ^ c | t | ^ 2 . \\end{align*}"}
-{"id": "497.png", "formula": "\\begin{align*} \\eta _ p : T _ p C & \\longrightarrow H ^ 0 ( C , K _ C ( 2 p ) ) , \\\\ u = \\lambda \\frac { \\partial } { \\partial z } ( p ) & \\longmapsto \\eta _ p ( u ) = \\lambda \\phi . \\end{align*}"}
-{"id": "5109.png", "formula": "\\begin{align*} \\| x _ n - x _ m \\| ^ 4 & = \\| ( x _ n - x ) - ( x _ m - x ) \\| ^ 4 \\\\ & \\leq 2 \\| x _ n - x \\| ^ 4 + 2 \\| x _ m - x \\| ^ 4 + 1 2 \\| x _ n - x \\| ^ 2 \\| x _ m - x \\| ^ 2 - \\| ( x _ n - x ) + ( x _ m - x ) \\| ^ 4 \\\\ & \\leq 2 \\| x _ n - x \\| ^ 4 + 2 \\| x _ m - x \\| ^ 4 + 1 2 \\| x _ n - x \\| ^ 2 \\| x _ m - x \\| ^ 2 - 1 6 d ^ 4 , ~ \\forall n , m \\in \\mathbb { N } . \\end{align*}"}
-{"id": "6020.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\downarrow 0 } \\ , \\rho ^ \\mu _ \\epsilon \\left ( F \\right ) = \\int _ { \\R ^ d } \\ , \\ , \\sup _ { \\omega \\in \\C _ x } \\left ( F ( \\omega ) - \\int _ 0 ^ 1 g ( t , \\dot { \\omega } ( t ) ) d t \\right ) \\mu ( d x ) . \\end{align*}"}
-{"id": "3716.png", "formula": "\\begin{align*} \\mbox { R E } = \\big [ ( 1 - W _ B ) ^ 2 V ( \\bar { y } _ w ) \\big ] / V ( \\widehat { \\bar { Y } ' } ) \\end{align*}"}
-{"id": "8641.png", "formula": "\\begin{align*} y = \\varepsilon x . \\end{align*}"}
-{"id": "2860.png", "formula": "\\begin{align*} p ^ 4 - i t p ^ 2 q ^ 2 + q ^ 4 = k ^ 2 u . \\end{align*}"}
-{"id": "7525.png", "formula": "\\begin{align*} \\nabla _ x u = \\mathbb { G } ( \\sigma ) , \\end{align*}"}
-{"id": "3343.png", "formula": "\\begin{align*} \\widetilde { Z } ( u ) : = Z ( u / \\sqrt { t } ) , \\end{align*}"}
-{"id": "5217.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow + \\infty } \\frac { U ( t - ) } { U ( t ) } = \\lim _ { t \\rightarrow + \\infty } \\frac { U ( t + ) } { U ( t ) } = 1 . \\end{align*}"}
-{"id": "1008.png", "formula": "\\begin{align*} \\varphi _ k \\left ( - f ' ( a ) \\right ) = \\int _ 0 ^ { - f ' ( a ) } \\frac { s ^ { k - 1 } } { ( k - 1 ) ! } e ^ { - s } d s = \\int _ a ^ { + \\infty } \\frac { f '' ( x ) ( - f ' ( x ) ) ^ { k - 1 } } { ( k - 1 ) ! } e ^ { f ' ( x ) } d x \\end{align*}"}
-{"id": "4185.png", "formula": "\\begin{align*} E _ { i n } ^ { ( 3 ) } ( z ) & = \\left ( \\mathbb { I } - \\frac { A _ n ^ { ( 2 ) } ( z ) - A _ n ^ { ( 2 ) } ( 0 ) } { n ^ 6 z } \\right ) E _ { i n } ^ { ( 2 ) } ( z ) , & & z \\in D ( 0 , r _ n ) \\\\ E _ { o u t } ^ { ( 3 ) } ( z ) & = \\left ( \\mathbb { I } - \\frac { A _ n ^ { ( 2 ) } ( 0 ) } { n ^ 6 z } \\right ) ^ { - 1 } E _ { o u t } ^ { ( 2 ) } ( z ) , & & z \\in A ( 0 ; r _ n , R ) , \\end{align*}"}
-{"id": "9772.png", "formula": "\\begin{align*} | D ^ \\alpha p _ * ( Z _ \\ell ) | = o ( 1 ) \\to 0 \\quad \\ell \\to \\infty \\end{align*}"}
-{"id": "3221.png", "formula": "\\begin{align*} \\max \\left ( \\| \\Delta _ { \\frac { 1 } { 2 } } x ^ k \\| , \\| \\Delta _ { \\frac { 1 } { 2 } } V _ k \\| \\right ) \\le \\| \\Delta _ { \\frac { 1 } { 2 } } { w } ^ k \\| = O ( \\mu _ { k - 1 } ) . \\end{align*}"}
-{"id": "6758.png", "formula": "\\begin{align*} x ^ m x ^ \\omega \\approx x ^ p \\ , \\overline x \\ , ^ q \\end{align*}"}
-{"id": "5618.png", "formula": "\\begin{align*} \\alpha \\int _ { 0 } ^ { + \\infty } ( | u | ^ 2 ) _ x v d x = 2 \\alpha \\ , \\int _ 0 ^ { + \\infty } u v \\bar { u } _ x d x \\end{align*}"}
-{"id": "605.png", "formula": "\\begin{align*} \\delta : = \\inf _ { \\epsilon \\neq \\epsilon ' } \\| g ^ { \\epsilon } - g ^ { \\epsilon ' } \\| _ { L ^ q } & = \\| \\gamma \\psi _ { j , k , e } \\| _ { L ^ q } = \\gamma \\ , 2 ^ { j d ( \\frac { 1 } { 2 } - \\frac { 1 } { q } ) } \\ , \\| \\psi \\| _ { L ^ q } , \\end{align*}"}
-{"id": "3860.png", "formula": "\\begin{align*} \\mathbf { E } [ f ( Z ^ x _ t ) ] & = W ^ { + } _ { t , x } f ( t , x ) \\\\ & = W ^ { - } _ { t , x } ( W ^ { - } _ { t , x } ) ^ { - 1 } W ^ { + } _ { t , x } f ( t , x ) \\\\ & = W ^ { - } _ { t , x } \\mathcal { N } _ { t , x } f ( t , x ) . \\end{align*}"}
-{"id": "5699.png", "formula": "\\begin{align*} _ { c } D _ { a x } ^ { \\alpha } y ( x ) = _ { R L } D _ { a x } ^ { \\alpha } y ( x ) - \\sum _ { k = 0 } ^ { n - 1 } \\frac { y ^ { ( k ) } ( a ) } { \\Gamma ( k - \\alpha + 1 ) } ( x - a ) ^ { k - \\alpha } \\end{align*}"}
-{"id": "415.png", "formula": "\\begin{align*} \\log \\frac { { \\det } ' ( \\Delta _ { \\hat { g } } ) } { { \\rm V } _ { \\hat { g } } } = \\log \\frac { { \\det } ' ( \\Delta _ g ) } { { \\rm V } _ { g } } - \\frac { 1 } { 4 8 \\pi } S ^ { { \\rm c l } , 0 } _ { { \\rm L } } ( \\hat g , g ) . \\end{align*}"}
-{"id": "7506.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\frac { \\partial u } { \\partial t } ( x , t ) = - ( f ( \\mathcal { \\partial } , \\beta ) u ) ( x , t ) , & t \\in \\lbrack 0 , \\infty ) , x \\in \\mathbb { Q } _ { p } ^ { n } \\\\ & \\\\ u ( x , 0 ) = u _ { 0 } ( x ) \\in \\mathcal { D } ( \\mathcal { \\mathbf { \\mathbb { Q } } } _ { p } ^ { n } ) & \\end{array} \\right . \\end{align*}"}
-{"id": "8341.png", "formula": "\\begin{align*} c _ 0 ^ j = ( \\Phi ^ { - 1 } ) _ z ( \\omega _ 0 ^ j ) , \\omega _ 0 ^ j = \\Phi ( z _ j ) . \\end{align*}"}
-{"id": "9622.png", "formula": "\\begin{align*} ( - q ) _ N = \\frac { 1 - q ^ { N + 1 } } { 2 } \\sum _ { n = 0 } ^ { \\infty } q ^ n \\left \\{ ( - q ^ { n + 1 } ) _ N + ( q ^ { n + 1 } ) _ N \\right \\} . \\end{align*}"}
-{"id": "2135.png", "formula": "\\begin{align*} ( \\inf _ { z \\in L ^ 0 ( Z ) } \\Phi ^ \\ast ( \\cdot , z ) ) ^ \\ast ( x ) & = \\sup _ { y \\in L ^ 0 ( Y ) } \\{ \\langle x , y \\rangle - \\inf _ { z \\in L ^ 0 ( Z ) } \\Phi ^ \\ast ( y , z ) \\} \\\\ & = \\sup _ { y \\in L ^ 0 ( Y ) , \\ , z \\in L ^ 0 ( Z ) } \\{ \\langle x , y \\rangle - \\Phi ^ \\ast ( y , z ) \\} \\\\ & = \\sup _ { y \\in L ^ 0 ( Y ) , \\ , z \\in L ^ 0 ( Z ) } \\{ \\langle x , y \\rangle + \\langle 0 , z \\rangle - \\Phi ^ \\ast ( y , z ) \\} \\\\ & = \\Phi ( x , 0 ) . \\end{align*}"}
-{"id": "5778.png", "formula": "\\begin{align*} A ( C _ n , q , T ) = & ( q - 1 ) \\frac { T A _ n ( T ) } { ( 1 - T ) ^ n ( 1 - q T ) } + n \\frac { T A _ { n - 1 } ( T ) } { ( 1 - T ) ^ n } \\\\ = & \\frac { T \\sum _ { j = 0 } ^ { n - 2 } A ( n - 1 , j ) \\left [ q ( j + 1 ) + n - 1 - j \\right ] T ^ j } { ( 1 - T ) ^ { n - 1 } ( 1 - q T ) } , \\end{align*}"}
-{"id": "2499.png", "formula": "\\begin{align*} \\aligned 2 \\ , a _ 3 + b _ 4 + ( \\rho - 3 ) \\ , a _ 2 = 0 , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 2 \\ , b _ 2 + \\frac { \\rho - 1 } { 2 } a _ 1 = 0 , \\\\ 2 \\ , b _ 2 + a _ 1 + ( \\rho - 3 ) \\ , b _ 3 = 0 , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 2 \\ , a _ 3 + \\frac { \\rho - 1 } { 2 } b _ 4 = 0 . \\endaligned \\end{align*}"}
-{"id": "8611.png", "formula": "\\begin{align*} \\begin{array} { l c r } u ^ 3 - u + \\Lambda v ^ 2 u - \\omega v & = & 0 , \\\\ v ^ 3 - v + \\Lambda u ^ 2 v - \\omega u & = & 0 . \\end{array} \\end{align*}"}
-{"id": "6337.png", "formula": "\\begin{align*} f _ { n , r _ 1 } ( n ) = L a g \\left \\{ n , G _ 1 m _ s h _ { s r } , G _ 1 ( m _ { b r _ 1 } + n _ { s p } ) , D \\right \\} , \\end{align*}"}
-{"id": "2348.png", "formula": "\\begin{align*} \\Phi _ \\alpha : \\mathfrak { X } & \\to X ^ \\alpha \\\\ \\Phi _ \\alpha & = a _ \\alpha ^ + \\Phi \\end{align*}"}
-{"id": "7465.png", "formula": "\\begin{align*} m _ { \\sigma } ( E ) : = \\sum _ { i = 1 } ^ m | Z ( A _ i ) | . \\end{align*}"}
-{"id": "1653.png", "formula": "\\begin{align*} F _ { - k } = \\alpha _ k - i \\alpha _ { - k } \\quad G _ { - k } = \\beta _ k - i \\beta _ { - k } . \\end{align*}"}
-{"id": "2313.png", "formula": "\\begin{align*} \\mathbf { k } _ \\lambda & = \\mathbf { k } _ { \\mathbf { m } , j } = \\frac { 2 \\pi } { L } \\mathbf { m } , & \\omega _ \\lambda = | \\mathbf { k } _ \\lambda | & = \\frac { 2 \\pi } { L } | \\mathbf { m } | & \\mathbf { e } _ \\lambda & = \\mathbf { e } _ j ( \\mathbf { k } _ \\lambda ) . \\end{align*}"}
-{"id": "2423.png", "formula": "\\begin{align*} \\begin{cases} u _ { i j } = u ^ 2 _ { i j } = u _ { i j } ^ * \\mbox { f o r } 1 \\le i , j \\le n , \\\\ \\sum _ { i = 1 } ^ n u _ { i j } = 1 = \\sum _ { i = 1 } ^ n u _ { j i } \\quad \\mbox { f o r } 1 \\le j \\le n , \\end{cases} \\end{align*}"}
-{"id": "2299.png", "formula": "\\begin{align*} D _ l & = \\mathfrak { H } R ( z ) = \\bigg \\{ \\xi \\in \\mathbb { C } \\oplus \\mathcal { L } : \\xi = c ( 1 , ( E | R ( z ) ) + ( 0 , ( f | ) \\bigg \\} \\\\ D _ r & = R ( z ) \\mathfrak { H } = \\bigg \\{ \\xi \\in \\mathbb { C } \\oplus \\mathcal { L } : \\xi = c \\binom { 1 } { - R ( z ) | E ) } + \\binom { 0 } { R ( z ) f } \\bigg \\} \\end{align*}"}
-{"id": "4021.png", "formula": "\\begin{align*} [ \\phi X , \\phi Y ] = - d \\eta ( X , Y ) \\xi , \\end{align*}"}
-{"id": "8871.png", "formula": "\\begin{align*} c _ { 5 a } & = 1 5 - \\nu ^ 2 + 1 5 \\nu > 0 \\\\ c _ { 5 b } & = 2 ( 1 - \\nu ) ( 1 4 \\nu ^ 2 + 2 8 \\nu + 1 9 - \\nu ^ 3 ) + ( 1 - \\mu ) ( 1 3 \\nu ^ 3 + 4 0 \\nu ^ 2 + 4 9 \\nu + 1 1 - \\nu ^ 4 ) \\ge 0 . \\end{align*}"}
-{"id": "8388.png", "formula": "\\begin{align*} ( \\partial _ { \\alpha } ^ k u ) _ { t t } + A \\bold { n } \\partial _ { \\alpha } ^ k ( \\frac { \\partial _ { \\alpha } } { z _ { \\alpha } } u ) = \\partial _ { \\alpha } ^ k g - [ \\partial _ { \\alpha } ^ k , A \\bold { n } ] D u . \\end{align*}"}
-{"id": "902.png", "formula": "\\begin{align*} Y ^ j _ m = y _ { \\beta ^ j _ 1 } ^ { m ^ j _ 1 } \\cdots y _ { \\beta ^ j _ { r _ j } } ^ { m ^ j _ { r _ j } } ; \\end{align*}"}
-{"id": "7543.png", "formula": "\\begin{align*} \\begin{gathered} \\mathcal { X } _ \\epsilon ' = \\int _ 0 ^ t \\frac { d } { d \\epsilon } \\mathcal { V } _ \\epsilon d s , \\ , \\ , \\pi _ \\epsilon = \\int _ 0 ^ t \\frac { d } { d \\epsilon } \\mathcal { T } _ \\epsilon d s + \\sigma _ { \\epsilon , 0 } ' , \\\\ \\mathcal { V } _ \\epsilon = \\mathcal { V } ( X _ \\epsilon , \\tau _ \\epsilon ) , \\ , \\mathcal { T } _ \\epsilon = \\mathcal { T } ( X _ \\epsilon , \\tau _ \\epsilon ) . \\end{gathered} \\end{align*}"}
-{"id": "3464.png", "formula": "\\begin{align*} x = \\frac { M + 1 } { N } g ( k + 1 ; \\xi ) = \\frac { M + 1 } { N } g ( k ; \\xi ' ) , \\end{align*}"}
-{"id": "6823.png", "formula": "\\begin{align*} \\begin{cases} w ^ 1 _ { 1 0 } v ^ 2 _ { 0 1 } - w ^ 1 _ { 0 1 } v ^ 2 _ { 1 0 } - w ^ 2 _ { 1 0 } v ^ 1 _ { 0 1 } + w ^ 2 _ { 0 1 } v ^ 1 _ { 1 0 } = 0 \\ , , \\\\ w ^ 2 _ { 1 0 } v ^ 2 _ { 0 1 } - w ^ 2 _ { 0 1 } v ^ 2 _ { 1 0 } + w ^ 1 _ { 1 0 } v ^ 1 _ { 1 0 } - w ^ 1 _ { 0 1 } v ^ 1 _ { 1 0 } = 0 \\ , , \\end{cases} \\end{align*}"}
-{"id": "7892.png", "formula": "\\begin{align*} \\{ g _ { n } \\} _ { n = 1 } ^ { \\infty } \\in \\left ( \\sum _ { n = 1 } ^ { \\infty } \\oplus \\mathcal { Z } _ { n } ^ { \\sigma } \\right ) _ { \\ell ^ { 2 } } , \\end{align*}"}
-{"id": "9379.png", "formula": "\\begin{align*} c _ { \\nu ( i ) + 1 } = [ y , x , \\overset { \\nu ( i ) } { \\ldots } , x ] \\equiv [ y , x , x ^ 2 , \\ldots , x ^ { 2 ^ { i - 1 } } ] C _ i . \\end{align*}"}
-{"id": "964.png", "formula": "\\begin{align*} \\frac { d x } { d t } \\ ; = \\ ; - \\sigma ( x - y ) , \\frac { d y } { d t } \\ ; = \\ ; \\rho x - y - x z , \\frac { d z } { d t } \\ ; = \\ ; x y - \\beta z , \\end{align*}"}
-{"id": "8087.png", "formula": "\\begin{align*} \\exp ( \\mu ( h ) ) = 1 - \\frac { h } { 1 2 } + \\frac { h ^ 2 } { 2 8 8 } + \\frac { 6 7 h ^ 3 } { 5 1 8 4 0 } + O ( h ^ 4 ) . \\end{align*}"}
-{"id": "54.png", "formula": "\\begin{align*} \\partial _ { t t } u - \\partial _ { x x } u + 2 k ( x ) g ( \\partial _ t u ) = 0 \\ , . \\end{align*}"}
-{"id": "9575.png", "formula": "\\begin{align*} S _ 2 ^ { * } = \\frac { 1 } { ( 1 - c ) ( c q ) _ N } \\sum _ { n = 1 } ^ N \\frac { ( c q ) _ { N - n } ( c q ) ^ n } { ( q ) _ n ( q ) _ { N - n } } \\sum _ { k = 1 } ^ n \\left [ \\begin{matrix} n \\\\ k \\end{matrix} \\right ] \\frac { ( q c ) _ k ( 1 / c ) _ { n - k } c ^ { - k } } { 1 - q ^ k } . \\end{align*}"}
-{"id": "8226.png", "formula": "\\begin{align*} U _ 5 ( F t ^ i ) = F \\sum _ { j = \\left \\lceil \\frac { i } { 5 } \\right \\rceil } ^ \\infty m ( i , j ) t ^ j , \\end{align*}"}
-{"id": "2144.png", "formula": "\\begin{align*} \\tau _ { \\mathbf x } f ( \\mathbf y ) = c _ k ^ { - 1 } \\int E ( i \\xi , \\mathbf x ) E ( i \\xi , \\mathbf y ) \\mathcal F f ( \\xi ) \\ , d w ( \\xi ) \\end{align*}"}
-{"id": "9008.png", "formula": "\\begin{align*} E _ { 1 } \\cdot ( \\overline { D } _ { t } X _ { j } ^ { i } E _ { i } + X _ { j } ^ { i } \\overline { D } _ { t } E _ { i } ) \\times X _ { 1 } ^ { i } E _ { i } \\times \\dotsb \\times X _ { m - 1 } ^ { i } E _ { i } = 0 \\ , , \\end{align*}"}
-{"id": "1640.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\alpha _ k = \\xi _ k ' + O ( 1 / k ^ 2 ) , \\quad \\beta _ k = \\eta _ k ' + O ( 1 / k ^ 2 ) , \\\\ \\alpha _ { - k } = \\xi _ { - k } ' + O ( 1 / k ^ 2 ) , \\quad \\beta _ { - k } = \\eta _ { - k } ' + O ( 1 / k ^ 2 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "643.png", "formula": "\\begin{align*} \\varphi \\psi ^ { u ^ { - 1 } x } = \\varphi \\psi ^ { u '^ { - 1 } x ' } \\ , [ \\psi ^ \\ast ( u ^ { - 1 } x ) ] = [ \\psi ^ \\ast ( u '^ { - 1 } x ' ) ] \\end{align*}"}
-{"id": "9776.png", "formula": "\\begin{align*} q ( X + Z _ \\infty ) = \\frac { q ( X ) + q ( X + 2 Z _ \\infty ) } { 2 } . \\end{align*}"}
-{"id": "9878.png", "formula": "\\begin{align*} \\| f \\| _ { L } = \\sup \\left \\{ \\left . \\frac { | f ( x ) - f ( y ) | } { d ( x , y ) } \\right | d ( x , y ) \\neq 0 \\right \\} \\end{align*}"}
-{"id": "3469.png", "formula": "\\begin{align*} \\frac { d } { d x } \\Re N F ( x \\pm i / 4 ; x _ N ( k ) ) \\big \\rvert _ { x = - k } = 1 + o ( 1 ) > 0 . \\end{align*}"}
-{"id": "9117.png", "formula": "\\begin{align*} \\begin{bmatrix} A & 0 \\\\ C & D \\end{bmatrix} \\mapsto ( A , D ) . \\end{align*}"}
-{"id": "9946.png", "formula": "\\begin{align*} f _ 1 : = - \\frac { H _ 0 ^ { - 1 } } { 2 T } \\int _ 0 ^ { T } d t \\int _ 0 ^ t \\ , \\phi _ s ^ * ( V + q _ 0 ) \\ , d s \\end{align*}"}
-{"id": "2987.png", "formula": "\\begin{align*} \\Big ( \\frac { 1 } { ( q ^ 2 ; q ^ 4 ) _ \\infty } \\Big ) & = \\prod _ { k = 1 } ^ \\infty ( 1 + q ^ { 2 k } ) = \\sum _ { n = 0 } ^ \\infty q ( n ) q ^ { 2 n } . \\end{align*}"}
-{"id": "8898.png", "formula": "\\begin{align*} L = - ( e _ 3 e _ 2 ^ { ( 2 ) } ) \\otimes ( f _ 1 \\wedge f _ 2 \\wedge f _ 3 ) \\otimes ( e _ 1 \\otimes f _ 3 ) & - ( e _ 3 e _ 2 ^ { ( 2 ) } ) \\otimes ( e _ 1 \\otimes f _ 1 \\wedge f _ 3 ) \\otimes ( f _ 2 \\wedge f _ 3 ) \\\\ & + ( e _ 3 e _ 2 ^ { ( 2 ) } ) \\otimes ( e _ 1 \\otimes f _ 2 \\wedge f _ 3 ) \\otimes ( f _ 1 \\wedge f _ 3 ) , \\\\ \\end{align*}"}
-{"id": "5047.png", "formula": "\\begin{align*} \\sup _ { c _ n < x , y < 1 - c _ n , | x - y | \\leq c _ n } \\Big \\| \\overline p _ n ( y ) - \\dfrac { 1 } { g ( x ) } v \\Big \\| = o ( 1 ) \\end{align*}"}
-{"id": "4903.png", "formula": "\\begin{align*} \\chi ^ N ( g ) = \\left \\{ \\begin{array} { @ { } l l } | G / N | & \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "332.png", "formula": "\\begin{align*} A \\cdot M _ { \\Phi } \\cdot B = ( 4 m + 3 ) L \\begin{pmatrix} ( 2 m + 1 ) & & \\\\ & ( 2 m + 1 ) & \\\\ & & ( 2 m + 1 ) ( 4 m + 4 ) \\\\ & & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "872.png", "formula": "\\begin{gather*} y ^ 2 + ( y \\mp 1 ) ^ 2 + \\left ( x - \\frac { 1 } { \\sqrt { 2 } } \\right ) ^ 2 + \\frac { 1 } { 2 } - \\frac { 1 } { 2 } ( - y + y \\mp 1 ) ^ 2 = x ^ 2 - \\sqrt { 2 } x + 2 y ^ 2 \\mp 2 y + \\frac { 3 } { 2 } , \\end{gather*}"}
-{"id": "239.png", "formula": "\\begin{align*} X _ i & = \\partial _ { x _ i } + 2 y _ i \\partial _ t , \\ i = 1 , . . , n \\\\ X _ { n + j } & = \\partial _ { y _ j } - 2 x _ j \\partial _ t , \\ j = 1 , . . , n \\end{align*}"}
-{"id": "5631.png", "formula": "\\begin{align*} I ( u , v ) = - \\frac { \\alpha } { 2 } | u ( 0 , t ) | ^ 2 \\Big ( v _ { x x } ( 0 , t ) + \\frac 1 2 v ^ 2 ( 0 , t ) - \\gamma | u ( 0 , t ) | ^ 2 \\Big ) \\end{align*}"}
-{"id": "4838.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } t ^ 2 u '' - u _ { x x } + \\int _ { 1 } ^ { t } \\mu \\left ( s \\right ) u _ { x x } \\left ( t - s \\right ) d s = u ^ p \\hbox { i n } [ 1 , T ) \\times ( r _ 1 , r _ 2 ) , \\\\ u ( 1 , x ) = u _ 0 ( x ) \\in H ^ 2 ( r _ 1 , r _ 2 ) \\cap H _ 0 ^ { 1 } ( r _ 1 , r _ 2 ) , \\\\ u ' ( 1 , x ) = u _ 1 ( x ) \\in H _ 0 ^ { 1 } ( r _ 1 , r _ 2 ) \\end{array} \\right . \\end{align*}"}
-{"id": "2736.png", "formula": "\\begin{align*} \\max _ { z \\in \\mathcal { C } _ { + } ^ { 2 } } & \\Re { f ( z ) } - \\Re { f ( z _ { 0 } ) } = \\Re { f ( z _ 3 ) } - \\Re { f ( z _ { 0 } ) } \\\\ & \\leq \\Re { f ( z _ 2 ) } - \\Re { f ( z _ { 0 } ) } \\leq - 2 N ^ { - \\frac { 1 } { 6 } } . \\end{align*}"}
-{"id": "9223.png", "formula": "\\begin{align*} \\hat { \\psi } _ { v _ { i 1 } v _ { i 2 } } + \\hat { \\zeta } _ { v _ { i 1 } v _ { i 2 } } = y _ { v _ { i 1 } v _ { i 2 } } - { G } _ 0 ^ \\star + 1 0 { \\eta } ^ \\star \\log _ { 1 0 } L _ { v _ { i 1 } v _ { i 2 } } . \\end{align*}"}
-{"id": "564.png", "formula": "\\begin{align*} ( 0 , 0 ) _ e & = \\eta ^ 2 - ( e - 3 ) \\eta - 1 , & & \\\\ ( 0 , i ) _ e = ( i , 0 ) _ e = ( i , i ) _ e & = \\eta ^ 2 + \\eta & & \\textrm { f o r } i \\ne 0 , \\\\ ( i , j ) _ e & = \\eta ^ 2 & & \\textrm { f o r } i \\ne j \\textrm { a n d } i , j \\ne 0 , \\end{align*}"}
-{"id": "4305.png", "formula": "\\begin{align*} P _ 1 = a _ 1 u _ 2 v u _ 1 x _ 1 y _ 1 x _ 2 y _ 2 \\dots x _ { k - 1 } y _ { k - 1 } x _ k a _ 2 \\end{align*}"}
-{"id": "5797.png", "formula": "\\begin{align*} f ^ { \\uparrow _ 3 } \\circ \\ell _ 3 \\ = \\ ( f \\circ \\ell _ 3 ) ^ { \\uparrow _ 3 } \\ = \\ f . \\end{align*}"}
-{"id": "4435.png", "formula": "\\begin{align*} \\partial _ t u = \\nabla \\cdot ( \\nabla u + u \\nabla \\mathcal { V } ) , & ( t , x ) \\in ( 0 , T ) \\times \\Omega , \\\\ \\Delta \\mathcal { V } = u , & ( t , x ) \\in ( 0 , T ) \\times \\Omega , \\end{align*}"}
-{"id": "4495.png", "formula": "\\begin{align*} \\Theta ^ t ( p , z , x ) = e ^ { 2 ( z - \\underline v ( x ) ) } \\big ( t \\Psi ( p , z , x ) + ( 1 - t ) \\underline { \\psi } ( x ) \\big ) \\end{align*}"}
-{"id": "6158.png", "formula": "\\begin{align*} \\int _ { \\mathbb { D } } f ( \\xi ) \\overline { g ( \\xi ) } \\omega ( \\xi ) d A ( \\xi ) = 4 \\int _ { \\mathbb { D } } f ' ( \\xi ) \\overline { g ' ( \\xi ) } \\omega ^ * ( \\xi ) d A ( \\xi ) + \\omega ( \\mathbb { D } ) f ( 0 ) \\overline { g ( 0 ) } . \\end{align*}"}
-{"id": "5905.png", "formula": "\\begin{align*} \\xi ' \\colon \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\mapsto \\frac { d } { \\pi b } \\mod \\pi , \\end{align*}"}
-{"id": "911.png", "formula": "\\begin{align*} B _ G F _ 0 \\subseteq B _ G F _ 1 \\subseteq \\cdots \\subseteq B _ G F _ { \\dim _ \\R ( X ) } = B _ G X . \\end{align*}"}
-{"id": "8295.png", "formula": "\\begin{align*} E _ s ( t ) = \\sum _ { k \\leq s } \\int | \\partial _ { \\alpha } ^ k u _ t ( \\alpha , t ) | ^ 2 + | | D | ^ { 1 / 2 } \\partial _ { \\alpha } ^ k u | ^ 2 d \\alpha . \\end{align*}"}
-{"id": "9667.png", "formula": "\\begin{align*} F _ { \\tau , \\sigma } \\oplus F _ { \\mathbb T , \\sigma } = J h _ { 0 1 } | _ \\sigma + \\theta _ 1 h _ { 0 2 } \\oplus ( - \\theta _ 2 h _ { 0 2 } ) , \\ \\ \\ \\ \\theta _ 1 h _ { 0 2 } \\oplus ( - \\theta _ 2 h _ { 0 2 } ) \\in \\mathcal E _ 0 , \\end{align*}"}
-{"id": "7583.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { \\tau ( t ) } _ { \\alpha , p } \\le \\norm { \\tau ( 0 ) } _ { \\alpha , p } + t \\norm { \\tau } _ { L i p ( 0 , T ; C ^ { \\alpha , p } ) } . \\\\ \\end{gathered} \\end{align*}"}
-{"id": "8429.png", "formula": "\\begin{align*} g _ d : = - 2 [ \\bar { p } , \\mathfrak { H } ] \\frac { \\partial _ { \\alpha } \\bar { f } } { z _ { \\alpha } } - 2 [ \\bar { f } , \\mathfrak { H } ] \\frac { \\partial _ { \\alpha } \\bar { p } } { z _ { \\alpha } } - 2 [ \\bar { p } , \\mathfrak { H } ] \\frac { \\partial _ { \\alpha } \\bar { p } } { z _ { \\alpha } } - 4 p _ t . \\end{align*}"}
-{"id": "10061.png", "formula": "\\begin{align*} \\sigma = \\phi \\ , d \\rho + R \\ , d \\Phi + \\frac { 2 } { x _ n ^ 2 } \\ , d y _ n - \\frac { 4 } { X _ n ^ 3 } Y _ n \\ , d X _ n + \\theta _ n \\ , d G _ n + \\tilde G _ n \\ , d \\Theta _ n , \\end{align*}"}
-{"id": "4617.png", "formula": "\\begin{align*} ( K + \\tau _ i M ) \\mathbf { u } _ k = M \\mathbf { u } _ { k - 1 } , k \\in [ 1 , \\ell ] . \\end{align*}"}
-{"id": "6842.png", "formula": "\\begin{align*} g _ 4 + g _ 5 = - u ^ 3 _ { 1 0 } u ^ 2 _ { 0 1 } + u ^ 3 _ { 0 1 } u ^ 2 _ { 1 0 } - u ^ 3 _ { 1 0 } u ^ 1 _ { 0 1 } + u ^ 3 _ { 0 1 } u ^ 1 _ { 1 0 } \\ , . \\end{align*}"}
-{"id": "5950.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 g ( t , \\dot \\omega ( t ) ) \\ , d t = + \\infty \\end{align*}"}
-{"id": "1860.png", "formula": "\\begin{align*} Y _ { j , 0 } = \\left ( u ^ { \\rm e v e n } ( j , x ) , v ^ { \\rm t o p . } ( x ) \\right ) _ { L ^ 2 } = \\frac { \\xi L A _ j } { \\sqrt { 2 } Z _ j ^ 2 } . \\end{align*}"}
-{"id": "4581.png", "formula": "\\begin{align*} \\dim _ { L ^ 2 } \\mu = n - \\limsup _ { R \\to \\infty } \\frac { \\log A ( \\mu , R ) } { \\log R } . \\end{align*}"}
-{"id": "8643.png", "formula": "\\begin{align*} u = R \\cos \\frac { \\varphi } { 2 } , \\ \\ v = R \\sin \\frac { \\varphi } { 2 } , \\end{align*}"}
-{"id": "6919.png", "formula": "\\begin{align*} \\dim F = m - ( n - 2 k ) = : d \\in ( k , 2 k ) . \\end{align*}"}
-{"id": "5046.png", "formula": "\\begin{align*} \\sup _ { c _ n < x , y < 1 - c _ n , | x - y | \\leq c _ n } \\| \\bar Q _ n ( y ) - g ( x ) \\Sigma \\| = o ( 1 ) . \\end{align*}"}
-{"id": "5963.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } v _ n ( 0 , 0 ) = \\lim _ { n \\rightarrow \\infty } u _ n ( 0 , 0 ) = u ( 0 , 0 ) = \\sup _ { x \\in \\R ^ d } ( f ( x ) - g ( x ) ) , \\end{align*}"}
-{"id": "1830.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty f ( x ) g ( y - x ) d x = \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } \\hat { f } ( \\alpha ) \\hat { g } ( \\alpha ) \\ , e ^ { - i \\alpha y } d \\alpha , \\end{align*}"}
-{"id": "9359.png", "formula": "\\begin{align*} \\log _ 2 \\lvert Z G ^ { 2 ^ k } : G ^ { 2 ^ k } \\rvert = 2 ^ k + 2 ^ { k - 1 } + k - 1 - ( 2 ^ k + k - 1 ) = 2 ^ { k - 1 } . \\end{align*}"}
-{"id": "7416.png", "formula": "\\begin{align*} \\mu ^ U _ { \\bar { x } } ( \\Pi ^ { \\infty } _ U \\cap \\mathcal F | U ( \\bar { x } ) ) = 1 . \\end{align*}"}
-{"id": "2585.png", "formula": "\\begin{align*} \\mathbf P _ \\mu [ 1 - e ^ { - \\theta \\eta _ t X _ t ( f ) } | \\| X _ t \\| \\neq 0 ] = \\frac { 1 - \\exp \\{ - \\mu \\big ( V _ t ( \\theta \\eta _ t f ) \\big ) \\} } { \\mathbf P _ \\mu ( \\| X _ t \\| \\neq 0 ) } \\xrightarrow [ t \\to \\infty ] { } \\int _ { [ 0 , \\infty ) } ( 1 - e ^ { - \\theta u } ) F _ f ( d u ) . \\end{align*}"}
-{"id": "6065.png", "formula": "\\begin{align*} & s _ { j _ 1 } \\cdots s _ { j _ { { k _ l } - 1 } } ( \\varpi _ { j _ { k _ l } } ) = s _ { j _ 1 } \\cdots s _ { j _ { { k _ l } - 1 } } ( \\varpi _ n ) = { \\rm w t } ( \\underbrace { - , \\cdots , - } _ { 2 l - 2 } , \\underbrace { + , \\cdots , + } _ { n - 2 l + 2 } ) , \\\\ & s _ { j _ 1 } \\cdots s _ { j _ { k _ l } } ( \\varpi _ { j _ { k _ l } } ) = { \\rm w t } ( \\underbrace { - , \\cdots , - } _ { 2 l } , \\underbrace { + , \\cdots , + } _ { n - 2 l } ) . \\end{align*}"}
-{"id": "1287.png", "formula": "\\begin{align*} \\partial _ \\eta K ( z , w ) & = - \\frac { 1 } { 2 \\eta } K ( z , w ) \\\\ & + \\frac { 1 } { 2 \\eta } \\int e ^ { 3 i \\eta \\nu ^ 2 / 4 } \\left ( \\eta - \\frac { \\nu } { 2 } \\right ) \\left ( z ' \\left ( \\frac { \\eta + \\nu } { 2 } \\right ) w \\left ( \\frac { \\eta - \\nu } { 2 } \\right ) + z \\left ( \\frac { \\eta + \\nu } { 2 } \\right ) w ' \\left ( \\frac { \\eta - \\nu } { 2 } \\right ) \\right ) d \\nu \\end{align*}"}
-{"id": "2869.png", "formula": "\\begin{align*} C _ S = \\left ( \\frac { 2 ( n - 1 ) } { n - 2 } \\right ) ^ { \\frac 2 n } C _ 0 ^ { \\frac 2 n } . \\end{align*}"}
-{"id": "2428.png", "formula": "\\begin{align*} \\psi _ k ^ { \\Gamma _ g } ( Z ) = \\sum _ { \\begin{pmatrix} A & B \\\\ C & D \\end{pmatrix} : \\Delta _ g \\backslash \\Gamma _ g } \\det ( C Z + D ) ^ { - k } , \\end{align*}"}
-{"id": "9381.png", "formula": "\\begin{align*} \\frac { \\log _ 2 \\lvert K P _ i ( G ) \\cap Z : P _ i ( G ) \\cap Z \\rvert } { \\log _ 2 \\lvert Z : P _ i ( G ) \\cap Z \\rvert } = \\frac { \\log _ 2 \\lvert K \\varrho _ k \\cap Z \\varrho _ k \\rvert } { \\log _ 2 \\lvert Z \\varrho _ k \\rvert } . \\end{align*}"}
-{"id": "2763.png", "formula": "\\begin{align*} \\mathrm { L H S \\ o f \\ } \\eqref { i d e } & = \\frac { z + w } { z - w } + \\frac { z + w } { z - w } \\sum _ { p = 1 } ^ { m } \\Big ( \\frac { z - \\pi _ { p } } { z + \\pi _ { p } } \\frac { w + \\pi _ { p } } { w - \\pi _ { p } } - 1 \\Big ) \\prod _ { k = 1 } ^ { p - 1 } \\frac { z - \\pi _ { k } } { z + \\pi _ { k } } \\frac { w + \\pi _ { k } } { w - \\pi _ { k } } \\end{align*}"}
-{"id": "30.png", "formula": "\\begin{align*} U ^ i ( x , t ) = \\frac { x ^ { \\delta } } { \\delta } e ^ { \\mathcal { Y } _ t ^ { i } - \\lambda t } = \\frac { x ^ { \\delta } } { \\delta } e ^ { \\mathbf { y } ^ i ( V _ t ^ v ) - \\lambda t } , \\ i \\in I , \\end{align*}"}
-{"id": "1525.png", "formula": "\\begin{align*} g ( x t \\otimes t ^ { - 1 } y ) & = ( x t ) * g ^ { - 1 } \\otimes g t ^ { - 1 } y = ( x * g ^ { - 1 } ) \\cdot ( g t g ^ { - 1 } ) \\otimes g t ^ { - 1 } y \\\\ & = x * g ^ { - 1 } \\otimes ( g t g ^ { - 1 } ) g t ^ { - 1 } y = x * g ^ { - 1 } \\otimes g y = g ( x \\otimes y ) \\end{align*}"}
-{"id": "4281.png", "formula": "\\begin{align*} V ( G ) = \\{ v \\} \\cup N ( v ) \\cup N _ 2 ( v ) . \\end{align*}"}
-{"id": "6429.png", "formula": "\\begin{align*} \\rho _ { 1 } = \\frac { 1 } { \\alpha _ { 1 } ^ { 2 } } , \\quad \\tau _ { 0 } = \\frac { 1 } { \\alpha _ { 1 } ^ { 2 } } , \\quad \\sigma _ { 0 } = 0 , s _ { 0 } = 0 , c _ { 0 } = 1 , \\end{align*}"}
-{"id": "9440.png", "formula": "\\begin{align*} \\mathcal { E } _ 2 ( z ) = \\sum _ { 1 \\leq j \\leq D ( N ) - 1 } a _ j ( E _ 2 ( z ) - d _ j E _ 2 ( d _ j z ) ) . \\end{align*}"}
-{"id": "3125.png", "formula": "\\begin{align*} C ^ { ( 3 ) } _ { 1 2 } = { } & q ^ { - \\frac { 1 } { 3 } } \\frac { ( 1 + y ) ^ 2 } { 1 + y + y ^ 2 } A ^ { ( 3 ) } _ { \\underline { \\beta ^ 1 \\beta ^ 2 } } ( y ) \\\\ = { } & q ^ { - \\frac { 1 } { 3 } } \\frac { ( 1 + y ) ^ 2 } { 1 + y + y ^ 2 } \\left ( 1 + \\frac { y ^ 4 + 6 y ^ 3 + 6 y ^ 2 + 6 y + 1 } { ( y + 1 ) ^ 2 y } \\ , q + \\ldots \\right ) \\ , . \\end{align*}"}
-{"id": "2916.png", "formula": "\\begin{align*} \\widehat { K } _ A \\big ( ( \\theta , x ) , [ u ] \\big ) & = K _ { A F } ( y , [ u ] ) \\ ; . \\end{align*}"}
-{"id": "1019.png", "formula": "\\begin{align*} n ^ 2 \\ln \\cos \\frac { \\alpha _ n } { 2 } + \\frac { n ^ 2 \\alpha _ n ^ 2 } { 8 } = n ^ 2 O ( \\alpha _ n ^ 4 ) = n ^ 2 O ( F _ n ^ 4 ( x ) ) = O \\left ( ( n ^ { 1 / 2 } F _ n ( x ) ) ^ 4 \\right ) \\to 0 , \\end{align*}"}
-{"id": "3049.png", "formula": "\\begin{align*} L _ { 1 } ^ { x } ( t ) & = \\sum \\limits _ { j \\in I _ { 1 } } e _ { j } B _ { j } ( 2 c _ { j } x _ { j } t ) + \\sum \\limits _ { j \\in I _ { 2 } } e _ { j } Z _ { j } ( x _ { j } t ) \\end{align*}"}
-{"id": "496.png", "formula": "\\begin{align*} \\phi : = \\bigg ( \\frac { 1 } { z ^ 2 } + h ( z ) \\bigg ) d z , \\end{align*}"}
-{"id": "9286.png", "formula": "\\begin{align*} 0 \\le & \\{ \\mathcal { F } [ v ] ( x _ 0 ) - B ( x _ 0 , v ( x _ 0 ) , D v ( x _ 0 ) ) \\} - \\{ \\mathcal { F } [ u ] ( x _ 0 ) - B ( x _ 0 , u ( x _ 0 ) , D u ( x _ 0 ) ) \\} \\\\ = & F ( D ^ 2 v ( x _ 0 ) - A ( x _ 0 , v ( x _ 0 ) , D v ( x _ 0 ) ) - F ( D ^ 2 u ( x _ 0 ) - A ( x _ 0 , u ( x _ 0 ) , D u ( x _ 0 ) ) \\\\ & + B ( x _ 0 , u ( x _ 0 ) , D u ( x _ 0 ) ) - B ( x _ 0 , v ( x _ 0 ) , D v ( x _ 0 ) ) \\\\ < & 0 , \\end{align*}"}
-{"id": "5607.png", "formula": "\\begin{align*} \\mathcal { M } ( t ) = \\int _ 0 ^ { + \\infty } | u _ { \\delta , \\epsilon } ( x , t ) | ^ 2 d x = \\mathcal { M } ( 0 ) + 2 \\int _ 0 ^ t ( \\partial _ x u _ { \\delta , \\epsilon } ( 0 , s ) \\bar { u } _ { \\delta , \\epsilon } ( 0 , s ) ) d s . \\end{align*}"}
-{"id": "1963.png", "formula": "\\begin{align*} d _ { i i } = \\frac { 1 } { 2 \\ , \\deg ( i ) } \\sum _ { k \\in \\mathcal { N } ( i ) } d _ { k k } . \\end{align*}"}
-{"id": "9957.png", "formula": "\\begin{align*} V | _ { S ^ 2 } = h | _ { S ^ 2 } - \\frac 1 3 ( 1 - a ^ 2 ) . \\end{align*}"}
-{"id": "3977.png", "formula": "\\begin{align*} E _ { k , m } = g _ k ( m ) ^ { - 1 } \\underset { t ^ 2 \\mid m } \\sum \\mu ( t ) E _ { k , 1 } \\mid U _ { t } \\circ V _ { \\frac { m } { t ^ 2 } } . \\end{align*}"}
-{"id": "3260.png", "formula": "\\begin{align*} L _ n ( a + b , a b ) = a ^ n + b ^ n \\end{align*}"}
-{"id": "5415.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } \\frac { 1 } { | x | ^ \\alpha } | \\varphi _ m ( \\cdot - x _ n ) | ^ 2 \\ , d x & = \\int _ { \\R ^ N } \\frac { 1 } { | x + x _ n | ^ \\alpha } | \\varphi _ m | ^ 2 \\ , d x = \\int _ { B ( 0 , R _ m ) } \\frac { 1 } { | x + x _ n | ^ \\alpha } | \\varphi _ m | ^ 2 \\ , d x \\\\ & \\leq \\frac { 1 } { ( | x _ n | - R _ m ) ^ \\alpha } \\int _ { B ( 0 , R _ m ) } | \\varphi _ m | ^ 2 \\ , d x \\leq \\frac { 1 } { m ^ \\alpha } \\int _ { \\R ^ N } | \\varphi _ m | ^ 2 \\ , d x \\to 0 \\end{align*}"}
-{"id": "7391.png", "formula": "\\begin{align*} V _ { \\pi } ( \\mathfrak { F } ) = T _ { \\pi } ( \\mathfrak { F } ) \\otimes _ { A _ \\pi } K _ { \\pi } \\\\ \\Lambda _ { \\pi } ( \\mathfrak { F } ) = \\bigcup _ { i \\geq 1 } \\mathfrak { F } [ \\pi ^ i ] . \\end{align*}"}
-{"id": "1462.png", "formula": "\\begin{align*} V _ { 3 } = - \\varepsilon \\int _ { \\mathbb { T } ^ { d } } \\left ( \\mu ^ { 1 } - \\mu ^ { 2 } \\right ) ^ { 2 } \\mathrm { d i v } \\left ( \\Theta _ { p } ( t , x , \\mu ^ { 1 } , D w ^ { 1 } ) \\right ) \\ d x , \\end{align*}"}
-{"id": "920.png", "formula": "\\begin{align*} J [ \\psi ^ { ( 1 ) } ] _ { T , t } = \\hbox { \\vtop { \\offinterlineskip \\halign { \\hfil # \\hfil \\cr { \\rm l . i . m . } \\cr $ \\stackrel { } { { } _ { p _ 1 \\to \\infty } } $ \\cr } } } \\sum _ { j _ 1 = 0 } ^ { p _ 1 } C _ { j _ 1 } \\zeta _ { j _ 1 } ^ { ( i _ 1 ) } , \\end{align*}"}
-{"id": "4357.png", "formula": "\\begin{align*} \\mathcal P _ a ( t ' , I ) : = \\ & \\Big \\{ P \\ : \\ P \\in { [ 2 , n ] \\choose k - 1 } , \\ I \\subset P , \\ [ 2 , t ' ] \\cap P = \\emptyset \\Big \\} , \\\\ \\mathcal P _ b ( t ' , I ) : = \\ & \\Big \\{ P \\ : \\ P \\in { [ 2 , n ] \\choose k } , \\ [ 2 , t ' ] \\subset P , \\ I \\cap P = \\emptyset \\Big \\} , \\end{align*}"}
-{"id": "3294.png", "formula": "\\begin{align*} \\Lambda ( f ) = \\begin{cases} \\deg ( P ) & \\mbox { i f $ f = P ^ k $ f o r $ P \\in \\mathcal { P } $ a n d $ k \\ge 1 $ } , \\\\ 0 & \\mbox { o t h e r w i s e } , \\end{cases} \\end{align*}"}
-{"id": "7548.png", "formula": "\\begin{align*} \\norm { f } _ { \\alpha , p } = \\norm { f } _ { C ^ { \\alpha } ( \\mathbb { R } ^ d ) } + \\norm { f } _ { L ^ p ( \\mathbb { R } ^ d ) } \\end{align*}"}
-{"id": "3483.png", "formula": "\\begin{align*} \\frac { d } { d y } \\Re f _ M ( \\Re q _ M ( \\pm \\theta ) + i y ; \\theta ) \\begin{cases} < 0 , & y > \\Im q _ M ( \\theta ) , \\\\ > 0 , & y \\in ( 0 , \\Im q _ M ( \\theta ) ) , \\\\ < 0 , & y \\in ( \\Im q _ M ( - \\theta ) , 0 ) , \\\\ > 0 , & y < \\Im q _ M ( - \\theta ) . \\end{cases} \\end{align*}"}
-{"id": "6346.png", "formula": "\\begin{align*} \\gamma = \\left ( \\eta G _ 1 G _ 2 m _ s h _ { s r } h _ { r r } h _ { r d } \\right ) ^ 2 / \\sigma _ d ^ 2 . \\end{align*}"}
-{"id": "5080.png", "formula": "\\begin{align*} \\langle x _ { \\sigma ( U V ) } , x _ { \\sigma ( U W ) } \\rangle & = \\langle \\pi ( { \\sigma ( U V ) } ) x _ I , \\pi ( { \\sigma ( U W ) } ) x _ I \\rangle = \\langle \\overline { f ( U V ) } \\pi ( U ) \\pi ( V ) x _ I , \\overline { f ( U W ) } \\pi ( U ) \\pi ( W ) x _ I \\rangle \\\\ & = \\overline { f ( U V ) } f ( U W ) \\langle \\pi ( V ) x _ I , \\pi ( W ) x _ I \\rangle = \\overline { f ( U V ) } f ( U W ) \\langle x _ V , x _ W \\rangle , \\forall U , V , W \\in \\mathcal { U } . \\end{align*}"}
-{"id": "5802.png", "formula": "\\begin{align*} ( \\ell _ \\gamma \\circ g ) \\circ h \\ & = \\ \\ ( \\ell _ { \\lambda _ g + \\gamma } - \\lambda _ { g ; \\gamma } + \\epsilon ) \\circ h \\ = \\ ( \\ell _ { \\lambda _ g + \\gamma } \\circ h ) - \\lambda _ { g ; \\gamma } + ( \\epsilon \\circ h ) \\\\ & = \\ \\big ( \\ell _ { \\lambda _ h + \\lambda _ g + \\gamma } - \\lambda _ { h ; \\lambda _ g + \\gamma } + \\epsilon ^ * \\big ) - \\lambda _ { g ; \\gamma } + ( \\epsilon \\circ h ) . \\end{align*}"}
-{"id": "261.png", "formula": "\\begin{align*} \\sum _ { j = i } ^ p { j \\choose i } Q _ { j - i } ( \\alpha ) f ^ { p - j } v _ j \\in I '' _ { p - i } ( D ) \\quad 0 \\leq i \\leq p . \\end{align*}"}
-{"id": "9924.png", "formula": "\\begin{align*} & 0 = \\lim _ { n \\to \\infty } \\sqrt { \\frac { 2 } { \\log ( e ) } E ^ { \\mu } [ D ( \\pi ^ { \\mu } _ { n } \\| \\pi _ { n } ^ { \\nu } ) ] } \\\\ & \\geq \\lim _ { n \\to \\infty } E ^ { \\mu } \\left [ \\sqrt { \\frac { 2 } { \\log ( e ) } D ( \\pi ^ { \\mu } _ { n } \\| \\pi _ { n } ^ { \\nu } ) } \\right ] \\end{align*}"}
-{"id": "8099.png", "formula": "\\begin{align*} n ( n - 1 ) & ( 2 n - 3 ) \\sigma _ n = ( 2 n - 1 ) ( 3 n ^ 2 - 5 n + 1 ) \\sigma _ { n - 1 } \\\\ & - ( 2 n - 3 ) ( 3 n ^ 2 - 5 n + 1 ) \\sigma _ { n - 2 } + ( n - 2 ) ( n - 3 ) ( 2 n - 1 ) \\sigma _ { n - 3 } \\end{align*}"}
-{"id": "4967.png", "formula": "\\begin{align*} s _ A : = \\sum _ { n \\in A } ~ s _ n . \\end{align*}"}
-{"id": "9932.png", "formula": "\\begin{align*} & \\| \\pi _ { n - } ^ { \\mu } - \\pi _ { n - } ^ { \\gamma } \\| _ { T V } = \\\\ & \\frac { E ^ { \\gamma } \\left [ \\left . \\left | E ^ { \\gamma } [ \\frac { d \\mu } { d \\gamma } ( X _ { 0 } ) | Y _ { [ 0 , \\infty ) } , X _ { [ n , \\infty ) } ] - E ^ { \\gamma } [ \\frac { d \\mu } { d \\gamma } ( X _ { 0 } ) | Y _ { [ 0 , n - 1 ] } ] \\right | \\right | Y _ { [ 0 , n - 1 ] } \\right ] } { E ^ { \\gamma } \\left [ \\left . \\frac { d \\mu } { d \\gamma } ( X _ { 0 } ) \\right | Y _ { [ 0 , n - 1 ] } \\right ] } \\end{align*}"}
-{"id": "8662.png", "formula": "\\begin{align*} O ( \\l , D ) = e ^ { \\int \\left ( \\frac 1 { \\xi - \\l } - \\frac 1 { \\xi - \\l _ * } \\right ) \\chi _ D ( \\xi ) d \\xi } , \\chi _ D ( \\xi ) = \\begin{cases} 1 / 2 , & \\xi \\in ( a _ j , \\l _ j ) \\\\ - 1 / 2 , & \\xi \\in ( a _ j , \\l _ j ) \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "1055.png", "formula": "\\begin{align*} & \\lambda _ 1 ( B ) f = B f = \\sum \\limits _ { j = 1 } ^ k B _ j f . \\end{align*}"}
-{"id": "6104.png", "formula": "\\begin{align*} \\pi _ k ( \\sigma ) = \\frac { 1 } { k } \\phi ( \\log \\sigma / k ) \\frac { 1 } { \\sigma } , \\end{align*}"}
-{"id": "1828.png", "formula": "\\begin{align*} \\hat { f } ( \\alpha ) = \\int _ { - \\infty } ^ \\infty f ( x ) \\ , e ^ { i \\alpha x } d x . \\end{align*}"}
-{"id": "5435.png", "formula": "\\begin{align*} \\xi _ 2 ( T ) & = \\lim _ { n \\to + \\infty } \\frac { 1 } { n } \\log c _ 2 ( T ^ n ) \\ge - \\lim _ { n \\to + \\infty } \\frac { 1 } { n } \\log \\lVert ( ( T _ { \\mid \\mathcal { N } _ { \\lambda _ 1 } } ) ^ { - 1 } ) ^ n \\rVert = \\log | \\lambda _ 1 | , \\end{align*}"}
-{"id": "4141.png", "formula": "\\begin{align*} D _ 0 ( z ) & = \\begin{pmatrix} e ^ { \\frac { 2 } { 3 } ( 2 \\varphi _ { 1 } ( z ) + \\varphi _ { 2 } ( z ) ) } & 0 & 0 \\\\ 0 & e ^ { \\frac { 2 } { 3 } ( \\varphi _ { 2 } ( z ) - \\varphi _ { 1 } ( z ) ) } & 0 \\\\ 0 & 0 & e ^ { - \\frac { 2 } { 3 } ( \\varphi _ { 1 } ( z ) + 2 \\varphi _ { 2 } ( z ) ) } \\end{pmatrix} . \\end{align*}"}
-{"id": "3771.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\nabla \\cdot ( \\gamma _ 1 ( \\nabla u ) ) = 0 \\quad \\quad \\Omega , \\\\ & \\gamma _ 1 ( \\nabla u ) \\cdot \\nu = \\gamma _ 0 \\nabla u \\cdot \\nu = h \\quad \\quad \\partial \\Omega , \\\\ & \\int _ { \\Omega } u = 0 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "9839.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ r \\alpha _ j ( v ) \\big | N ^ j _ G ( v ) \\big | \\ge \\sum _ { j = 0 } ^ r \\alpha _ j ( v ) \\sum _ { u \\in N ^ j _ G ( v ) } \\hat { w } _ { i + 1 } ( u ) & \\geq \\hat { w } _ { i + 1 } ( G ) = \\gamma ( v ) , \\end{align*}"}
-{"id": "5043.png", "formula": "\\begin{align*} \\overline q _ { n m } ( x ) = E [ q _ { n m } ( x ) ] = \\int g ( x + c _ n t ) t ^ m w ( t ) \\ , d t . \\end{align*}"}
-{"id": "10027.png", "formula": "\\begin{align*} ( A B x - C ) \\varphi ( q x ) + [ C + q - ( A + B ) x ] \\varphi ( x ) + ( x - q ) \\varphi ( x / q ) = 0 , \\end{align*}"}
-{"id": "3911.png", "formula": "\\begin{align*} \\mathbf { T } ( r _ 1 ) \\cdots \\mathbf { T } ( r _ k ) = \\begin{pmatrix} p _ k ( r _ 1 , \\dots , r _ k ) & p _ { k - 1 } ( r _ 1 , \\dots , r _ { k - 1 } ) \\\\ p _ { k - 1 } ( r _ 2 , \\dots , r _ k ) & p _ { k - 2 } ( r _ 2 , \\dots , r _ { k - 1 } ) \\end{pmatrix} . \\end{align*}"}
-{"id": "3276.png", "formula": "\\begin{align*} h _ n - h _ { n - 1 } e _ 1 + h _ { n - 2 } e _ 2 - \\dots \\pm h e _ { n - 1 } \\mp e _ n = 0 \\end{align*}"}
-{"id": "3183.png", "formula": "\\begin{align*} { \\rm i n d } _ \\varGamma ( - U ) & = ( \\dim \\mathcal { T } _ + ( U , \\varGamma ) - \\dim \\mathcal { B } _ + ( U , \\varGamma ) ) \\\\ & - ( \\dim \\mathcal { T } _ - ( U , \\varGamma ) - \\dim \\mathcal { B } _ - ( U , \\varGamma ) ) \\\\ & = { \\rm i n d } _ \\varGamma ( U ) . \\end{align*}"}
-{"id": "2341.png", "formula": "\\begin{align*} \\Phi ( \\emptyset ) & = 1 & \\Phi ( x _ \\alpha ) & = 0 \\mbox { f o r } \\alpha \\neq \\emptyset \\end{align*}"}
-{"id": "827.png", "formula": "\\begin{align*} \\pi _ * ( \\mathrm { d i v } _ X ( \\omega ) ) = \\mathrm { d i v } _ Y ( h ) ^ { | H | } \\cdot \\mathrm { d i v } _ Y ( d f ) ^ { | H | } \\cdot \\mathrm { d i v } _ Y ( \\pi _ * ( R _ { X / Y } ) ) , \\end{align*}"}
-{"id": "5219.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow + \\infty } \\frac { U ( t - ) } { U ( t ) } = 1 . \\end{align*}"}
-{"id": "7854.png", "formula": "\\begin{align*} | B _ { R - r ^ { 1 / 2 } } | = | B _ R | \\bigg ( 1 - \\frac { r ^ { 1 / 2 } } { R } \\bigg ) ^ r \\ge | B _ R | \\big ( 1 - r ^ { 3 / 2 } R ^ { - 1 } \\big ) . \\end{align*}"}
-{"id": "7214.png", "formula": "\\begin{align*} f ( t , x ) = \\sum _ { k \\neq 0 } t ^ { | k | - 1 } e ^ { i x k } + a _ 0 ( t ) \\end{align*}"}
-{"id": "2847.png", "formula": "\\begin{align*} \\vartheta = \\frac { 1 } { d } \\left ( a + x \\xi + y \\xi ^ 2 + z \\xi ^ 3 \\right ) \\end{align*}"}
-{"id": "9228.png", "formula": "\\begin{align*} e = \\sqrt { \\frac { 1 } { \\vert \\mathcal { T } \\vert } \\sum _ { i \\in \\mathcal { T } } ( g _ i - y _ i ) ^ 2 } , \\end{align*}"}
-{"id": "3050.png", "formula": "\\begin{align*} \\Psi _ { x } ^ { 1 } ( \\lambda ) & = \\int \\limits _ { \\R _ { + } ^ { d } } \\left ( 1 - e ^ { i \\lambda \\cdot z } \\right ) \\nu ( d z ) , \\\\ \\Psi _ { x } ^ { 2 } ( \\lambda ) & = \\sum \\limits _ { j = 1 } ^ { d } 2 c _ { j } x _ { j } \\ 1 _ { \\R _ { + } } ( x _ { j } ) \\lambda _ { j } ^ { 2 } + \\sum \\limits _ { j = 1 } ^ { d } x _ { j } \\ 1 _ { \\R _ { + } } ( x _ { j } ) \\int \\limits _ { | z | \\leq 1 } \\left ( 1 + i \\lambda \\cdot z - e ^ { i \\lambda \\cdot z } \\right ) \\mu _ { j } ( d z ) . \\end{align*}"}
-{"id": "832.png", "formula": "\\begin{align*} I : = \\langle T _ { i , i + \\mu } - T _ { j , j + \\mu } 1 \\leq i \\leq \\alpha 0 \\leq j \\leq n - 1 \\rangle , \\end{align*}"}
-{"id": "6934.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow T ^ { * } } \\left ( \\| \\rho \\| _ { L ^ \\infty ( 0 , T ; L ^ \\infty ) } + \\| \\theta \\| _ { L ^ \\infty ( 0 , T ; L ^ \\infty ) } \\right ) = \\infty \\end{align*}"}
-{"id": "6715.png", "formula": "\\begin{align*} \\varphi ^ { ( 3 ) } _ k = \\dot { \\varphi } ^ { ( 3 ) } _ k = 0 . \\end{align*}"}
-{"id": "9384.png", "formula": "\\begin{align*} \\varliminf _ { i \\to \\infty } \\frac { \\log _ 2 \\lvert K P _ i ( G ) \\cap Z : P _ i ( G ) \\cap Z \\rvert } { \\log _ 2 \\lvert Z : P _ i ( G ) \\cap Z \\rvert } & \\le \\varliminf _ { k \\to \\infty } \\frac { \\log _ 2 \\lvert K \\varrho _ k \\cap Z \\varrho _ k \\rvert } { \\log _ 2 \\lvert Z \\varrho _ k \\rvert } \\\\ & = \\lim _ { k \\to \\infty } \\frac { ( 2 m - 1 ) 2 ^ { k - n - 1 } + 1 } { 2 ^ { k - 1 } + 1 } = ( 2 m - 1 ) / 2 ^ n . \\end{align*}"}
-{"id": "4142.png", "formula": "\\begin{align*} P ( z ) & = E _ { i n } ( z ) \\Phi _ { \\alpha } ( n ^ 3 f ( z ) ) D _ 0 ^ n ( z ) \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & z ^ { \\beta } & 0 \\\\ 0 & 0 & z ^ { \\beta } \\end{pmatrix} , & & z \\in D ( 0 , r _ n ) , \\\\ P ( z ) & = E _ { o u t } ( z ) N ( z ) , & & z \\in A ( 0 ; r _ n , R ) , \\end{align*}"}
-{"id": "2510.png", "formula": "\\begin{align*} { \\displaystyle \\sum \\limits _ { n \\geq 0 } } B _ { n , p } \\frac { t ^ { n } } { n ! } = \\left ( p + 1 \\right ) { \\displaystyle \\int \\limits _ { 0 } ^ { 1 } } \\frac { \\left ( 1 - x \\right ) ^ { p } } { 1 - \\left ( 1 - e ^ { t } \\right ) x } d x . \\end{align*}"}
-{"id": "8210.png", "formula": "\\begin{align*} \\rho = \\frac { \\eta ^ 2 ( 2 \\tau ) \\eta ^ 4 ( 5 \\tau ) } { \\eta ^ 4 ( \\tau ) \\eta ^ 2 ( 1 0 \\tau ) } , t = \\frac { \\eta ^ 2 ( 5 \\tau ) \\eta ^ 2 ( 1 0 \\tau ) } { \\eta ^ 2 ( \\tau ) \\eta ^ 2 ( 2 \\tau ) } . \\end{align*}"}
-{"id": "4634.png", "formula": "\\begin{align*} \\mathbf { y } _ i = U \\mathbf { x } _ i , \\end{align*}"}
-{"id": "937.png", "formula": "\\begin{align*} \\Biggl . \\Biggl . + \\frac { ( i + 1 ) ^ 2 \\left ( \\zeta _ { i + 1 } ^ { ( i _ 2 ) } \\zeta _ { i } ^ { ( i _ 1 ) } - \\zeta _ { i } ^ { ( i _ 2 ) } \\zeta _ { i + 1 } ^ { ( i _ 1 ) } \\right ) } { \\sqrt { ( 2 i + 1 ) ( 2 i + 3 ) } ( 2 i - 1 ) ( 2 i + 5 ) } \\Biggr ) \\Biggr ] - \\frac { 1 } { 2 4 } { \\bf 1 } _ { \\{ i _ 1 = i _ 2 \\} } { \\Delta ^ 3 } , \\end{align*}"}
-{"id": "6995.png", "formula": "\\begin{align*} \\textstyle { \\lambda _ j = \\mathsf { c o s } \\left ( \\frac { 2 \\pi j } { n } \\right ) , 1 \\leq j \\leq ( n - 1 ) / 2 ; } \\end{align*}"}
-{"id": "3465.png", "formula": "\\begin{align*} I _ { k } [ a , b ] = \\left [ T g ( k ; a ) , T g ( k ; b ) \\right ] . \\end{align*}"}
-{"id": "1489.png", "formula": "\\begin{align*} W _ { 6 } ^ { j } = \\frac { \\varepsilon } { 2 } \\sum _ { i = 1 } ^ { d } \\int _ { \\mathbb { T } ^ { d } } ( \\partial _ { x _ { j } } w ^ { 1 } - \\partial _ { x _ { j } } w ^ { 2 } ) ^ { 2 } \\partial _ { x _ { i } } \\left ( \\Theta _ { p _ { i } } ( t , x , \\mu ^ { 2 } , D w ^ { 2 } ) \\right ) \\ d x . \\end{align*}"}
-{"id": "3369.png", "formula": "\\begin{align*} p \\ , = \\ , \\sum _ { j = 0 } ^ 2 \\ , \\left ( d _ j + \\delta _ { j k } d _ i - N _ { i j } ^ k \\right ) \\ , D _ j 0 \\le i , k \\le 2 . \\end{align*}"}
-{"id": "292.png", "formula": "\\begin{align*} s _ \\alpha ( \\Lambda ) \\ ; = \\ ; \\sum _ { \\beta } m _ { \\alpha \\beta } \\cdot \\Lambda _ { [ \\beta ] } \\textnormal { w h e r e } \\Lambda _ { [ \\beta ] } \\ ; = \\ ; [ \\beta _ 1 ] _ * ( \\Lambda ) \\circ \\cdots \\circ [ \\beta _ \\ell ] _ * ( \\Lambda ) \\end{align*}"}
-{"id": "3274.png", "formula": "\\begin{align*} F _ n ( X , Y ) & = h _ { n - 1 } \\\\ & = U _ n ( a + b , - a b ) \\\\ & = U _ n ( X , - Y ) . \\end{align*}"}
-{"id": "4983.png", "formula": "\\begin{align*} L ( C / \\mathbb { F } _ q , T ) = 1 + C _ 1 T + . . . + C _ { 2 g } T ^ { 2 g } . \\end{align*}"}
-{"id": "8493.png", "formula": "\\begin{align*} \\partial _ x F ( z , t ) = - \\frac { 1 } { 2 \\pi i } \\int \\frac { \\zeta _ { \\beta } } { ( z - \\zeta ( \\beta , t ) ) ^ 2 } \\mathfrak { F } ( \\beta , t ) d \\beta . \\end{align*}"}
-{"id": "9219.png", "formula": "\\begin{align*} e ( G ' ) \\leq \\max \\left \\{ f ( 2 k + 1 ) , f ( k + 1 ) \\right \\} = \\max \\left \\{ \\binom { 2 k + 1 } { 2 } , \\binom { k } { 2 } + k ( n - k ) \\right \\} . \\end{align*}"}
-{"id": "10071.png", "formula": "\\begin{align*} \\omega ( \\partial _ u K , \\partial _ { \\theta _ n ^ 0 } K ) = \\omega ( \\partial _ u K , \\partial _ { \\varphi _ i } K ) = \\omega ( \\partial _ { \\theta _ n ^ 0 } K , \\partial _ { \\varphi _ i } K ) = \\omega ( \\partial _ { \\varphi _ j } K , \\partial _ { \\varphi _ i } K ) = 0 , \\end{align*}"}
-{"id": "8590.png", "formula": "\\begin{align*} \\zeta ( m ; - \\Delta _ D ) = \\frac { 2 ^ { 2 m - 1 } | B _ { 2 m } | } { ( 2 m ) ! } , \\end{align*}"}
-{"id": "9367.png", "formula": "\\begin{align*} [ x ^ { 4 } , y ] \\equiv [ x , y ] ^ { 4 } [ x , y , x ] ^ { \\binom { 4 } { 2 } } \\equiv 1 \\gamma _ { 4 } ( G _ k ) , \\end{align*}"}
-{"id": "3061.png", "formula": "\\begin{align*} \\widetilde { ( g _ { e } . 1 ) ^ { 2 s } } ( 4 q ) = ( a b ) ^ { 2 s } c ^ { - s } \\sum _ { k > 2 q } { - s \\choose k } { 2 k \\choose k - 2 q } \\frac { c ^ { - k } d ^ { k } } { 2 ^ { 2 k - 1 } } = c ^ { - s } \\sum _ { j = 0 } ^ { \\infty } { 2 j + 4 q \\choose j } \\frac { d ^ { j + 2 q } \\Gamma ( j + 2 q - s ) } { 2 ^ { 2 q - 1 } c ^ { j + 2 q } \\Gamma ( - s ) } ( 1 / 2 ) ^ { j } . \\end{align*}"}
-{"id": "8777.png", "formula": "\\begin{align*} \\overline { S } _ { a b } = - \\frac { n - 1 } { 4 p } ( D + 4 q C ) \\overline { g } _ { a b } \\ , . \\end{align*}"}
-{"id": "2277.png", "formula": "\\begin{align*} K R _ K & = - 1 + z R _ K & L R _ K G Z & = - 1 + z Z \\end{align*}"}
-{"id": "4125.png", "formula": "\\begin{align*} \\varphi _ { \\pm } ( x ) & = \\mp \\pi i \\mu ^ * ( [ x , \\infty ) ) = \\pm \\pi i \\mu ^ * ( [ 0 , x ] ) \\mp \\pi i , x \\in [ 0 , q ] . \\end{align*}"}
-{"id": "4456.png", "formula": "\\begin{align*} ( \\eta _ i ) _ { U _ { i j } } \\circ \\phi _ { i j } = ( \\eta _ j ) _ { U _ { i j } } . \\end{align*}"}
-{"id": "1509.png", "formula": "\\begin{align*} \\frac { 2 q ( m + \\alpha - \\frac 7 6 ) } { q - 1 } = \\frac { 1 2 r ( m + \\alpha - \\frac 7 6 ) } { 7 r - 6 } > \\frac { 2 r } { 2 - r } . \\end{align*}"}
-{"id": "5003.png", "formula": "\\begin{align*} I _ K ( A ) : = \\inf _ \\mu \\int K ( x , y ) \\textup { d } \\mu ( x ) \\textup { d } \\mu ( y ) , \\end{align*}"}
-{"id": "2027.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n = 1 } ^ { \\infty } a ( n ) e ^ { 2 \\pi i n z } = \\sum _ { n = 1 } ^ { \\infty } A ( n ) n ^ { 1 / 2 } e ^ { 2 \\pi i n z } \\end{align*}"}
-{"id": "4606.png", "formula": "\\begin{align*} T ( x _ 1 , x _ 3 , \\delta ) = \\{ g \\in U ( 2 ) : | x _ 1 - g ( x _ 3 ) | \\leq \\delta \\} . \\end{align*}"}
-{"id": "8237.png", "formula": "\\begin{align*} L _ { 2 i - 1 } = F \\left ( \\sum _ { j = 1 } ^ \\infty a ( 2 i - 1 , j ) t ^ j + \\rho \\sum _ { j = 1 } ^ \\infty b ( 2 i - 1 , j ) t ^ j \\right ) . \\end{align*}"}
-{"id": "845.png", "formula": "\\begin{align*} f ( t ) = \\langle f , K ( t , \\cdot ) \\rangle t \\in \\Omega . \\end{align*}"}
-{"id": "1883.png", "formula": "\\begin{align*} \\frac { 1 } { y ^ n } = \\frac { 1 } { \\Gamma ( n ) } \\int _ 0 ^ \\infty t ^ { n - 1 } e ^ { - y t } d t . \\end{align*}"}
-{"id": "1187.png", "formula": "\\begin{align*} d ( x ) = \\cos ( 2 k \\pi x ) ( k = 2 , 3 , 4 ) d ( x ) = \\left \\{ \\begin{array} { l l } 1 - \\cos ( 4 \\pi x ) & ( 0 \\le x < 1 / 2 ) , \\\\ 0 & ( \\hbox { o t h e r w i s e } ) . \\end{array} \\right . \\end{align*}"}
-{"id": "3270.png", "formula": "\\begin{align*} V _ n ( X , Y ) = a ^ n + b ^ n , \\end{align*}"}
-{"id": "5028.png", "formula": "\\begin{align*} I ' : = I ( \\S , \\P _ i ) > 4 C _ { \\beta , 3 } ( | \\P _ i | | \\S | ) ^ { 3 / 4 } + | \\P _ i | | \\S | ^ { 1 / 2 } \\end{align*}"}
-{"id": "198.png", "formula": "\\begin{align*} X = x ^ 1 \\dfrac { \\partial } { \\partial x ^ 1 } + x ^ 2 \\dfrac { \\partial } { \\partial x ^ 2 } , Y = \\dfrac { \\partial } { \\partial x ^ 3 } . \\end{align*}"}
-{"id": "3121.png", "formula": "\\begin{align*} \\Gamma ( U _ { \\sigma \\tau } , \\mathcal { E } x t ^ i ( I , J \\otimes M ) ) & = \\Gamma ( U _ \\sigma \\cap U _ \\tau , \\mathcal { E } x t ^ i ( I | _ { U _ { \\sigma \\tau } } , ( J \\otimes M ) | _ { U _ { \\sigma \\tau } } ) ) \\\\ & = \\Gamma ( U _ { \\sigma \\tau } , \\mathcal { E } x t ^ i ( \\O , M ) ) \\end{align*}"}
-{"id": "2493.png", "formula": "\\begin{align*} [ J \\ , { \\sf x } , J \\ , { \\sf y } ] = [ { \\sf x } , { \\sf y } ] , \\ \\ \\ \\ \\ \\textrm { f o r e a c h } \\ \\ \\ { \\sf x , y } \\in \\frak g _ { - 1 } . \\end{align*}"}
-{"id": "1112.png", "formula": "\\begin{align*} & A ( u , v ) = - \\frac { 1 } { n \\rho _ { s c } ( x ) } K _ { ( 0 , n \\delta _ n ) } ^ { G U E ( n ) } \\left ( x + \\frac { u } { n \\rho _ { s c } ( x ) } , x + \\frac { v } { n \\rho _ { s c } ( x ) } \\right ) \\end{align*}"}
-{"id": "4260.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\delta ( a ( x ) \\lvert d u \\rvert ^ { p - 2 } d u ) ) & = f & & B _ { 2 r } , \\\\ \\delta u & = 0 & & B _ { 2 r } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "8522.png", "formula": "\\begin{align*} \\norm { \\frac { 2 } { \\pi } \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } } _ { H ^ s } \\leq K _ s ^ { - 1 } \\epsilon d _ I ( t ) ^ { - 5 / 2 } \\end{align*}"}
-{"id": "1473.png", "formula": "\\begin{align*} V _ { 2 } = \\frac { \\varepsilon } { 2 } \\int _ { \\mathbb { T } ^ { d } } ( \\mu ^ { 1 } - \\mu ^ { 2 } ) ^ { 2 } \\mathrm { d i v } \\left ( \\Theta _ { p } ( t , x , \\mu ^ { 1 } , D w ^ { 1 } ) \\right ) \\ d x , \\end{align*}"}
-{"id": "3053.png", "formula": "\\begin{align*} h ( \\Delta ' , t ) = h ( \\Delta , t ) + t ^ { l + 1 } h ( \\Sigma , t ) . \\end{align*}"}
-{"id": "3138.png", "formula": "\\begin{align*} \\widetilde { \\mathbb { B } } ( \\bar t ) = \\frac { T _ { N - 1 , N } ( \\bar t ^ { N - 1 } ) T _ { N - 2 , N - 1 } ( \\bar t ^ { N - 2 } ) \\cdots T _ { 2 3 } ( \\bar t ^ 2 ) T _ { 1 2 } ( \\bar t ^ 1 ) | 0 \\rangle } { \\prod _ { i = 1 } ^ { N - 1 } \\lambda _ { i + 1 } ( \\bar t ^ { i } ) \\prod _ { i = 1 } ^ { N - 2 } f ( \\bar t ^ { i + 1 } , \\bar t ^ i ) } . \\end{align*}"}
-{"id": "4163.png", "formula": "\\begin{align*} P ^ { ( 1 ) } ( z ) = \\begin{cases} E _ { i n } ^ { ( 1 ) } ( z ) \\Phi _ { \\alpha } ( n ^ 3 f ( z ) ) D _ 0 ^ n ( z ) \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & z ^ { \\beta } & 0 \\\\ 0 & 0 & z ^ { \\beta } \\end{pmatrix} , & z \\in D ( 0 , r _ n ) \\\\ E _ { o u t } ^ { ( 1 ) } ( z ) N ( z ) = N ( z ) , & z \\in A ( 0 ; r _ n , R ) , \\end{cases} \\end{align*}"}
-{"id": "3324.png", "formula": "\\begin{align*} p _ { \\lambda } ( X _ i : 1 \\le i \\le n ) : = \\prod _ { i = 1 } ^ { k } p _ { \\lambda _ i } ( X _ i : 1 \\le i \\le n ) . \\end{align*}"}
-{"id": "8585.png", "formula": "\\begin{align*} \\limsup _ { y \\in A \\to x } \\varphi ( y ) = \\varphi ( x ) . \\end{align*}"}
-{"id": "6801.png", "formula": "\\begin{align*} \\mathcal { R } _ 3 \\quad \\equiv \\begin{cases} \\mathcal { R } _ 2 \\ , , \\\\ \\partial ^ \\nu f _ 1 = 0 \\ , , & \\ | \\nu | = 2 \\ , , \\\\ \\partial ^ \\nu f _ 4 = 0 \\ , , & \\ | \\nu | = 1 \\ , , \\\\ \\partial ^ \\nu f _ 5 = 0 \\ , , & \\ | \\nu | = 1 \\ , . \\end{cases} \\end{align*}"}
-{"id": "8946.png", "formula": "\\begin{align*} \\P ( Y = 1 | X = x ) = \\frac { 1 } { 1 + \\exp \\{ - \\beta _ 0 - \\langle \\beta , x \\rangle _ 2 \\} } , \\end{align*}"}
-{"id": "8525.png", "formula": "\\begin{align*} & \\norm { \\frac { 1 } { \\pi } \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j \\ddot { z } _ j ( t ) } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } } _ { H ^ s } + \\norm { \\frac { 1 } { \\pi } \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j 2 ( \\dot { z } _ j ( t ) ) ^ 2 } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 3 } } _ { H ^ s } \\\\ \\leq & K _ s ^ { - 1 } \\epsilon \\frac { | \\lambda | } { x ( 0 ) } d _ I ( t ) ^ { - 5 / 2 } + K _ s ^ { - 1 } \\epsilon ^ 2 d _ I ( t ) ^ { - 5 / 2 } . \\end{align*}"}
-{"id": "8630.png", "formula": "\\begin{align*} u _ \\varepsilon = u _ 0 + O ( \\varepsilon ) , \\ \\ \\ z _ \\varepsilon = z _ 0 + O ( \\varepsilon ) , \\end{align*}"}
-{"id": "1225.png", "formula": "\\begin{align*} \\int _ \\lambda ^ \\infty e ^ { i \\eta ^ 2 } d \\eta & = \\int _ \\lambda ^ \\infty \\frac { 2 i \\eta } { 2 i \\eta } e ^ { i \\eta ^ 2 } d \\eta = \\frac { 1 } { 2 i \\lambda } + \\int _ { \\lambda } ^ \\infty \\frac { e ^ { i \\eta ^ 2 } } { 2 i \\eta ^ 2 } d \\eta \\\\ & = \\frac { 1 } { 2 i \\lambda } - \\frac { 1 } { 4 \\lambda ^ 3 } - \\int _ { \\lambda } ^ \\infty \\frac { e ^ { i \\eta ^ 2 } } { 1 2 \\eta ^ 4 } d \\eta . \\end{align*}"}
-{"id": "1688.png", "formula": "\\begin{align*} \\begin{gathered} N _ k ^ { n } = ( b ^ { 2 ^ { k } } - 1 ) ^ { n } = \\sum _ { i = 1 } ^ { \\frac { n + 1 } { 2 } } \\Big ( C _ { n } ^ { n + 2 - 2 i } b ^ { ( n + 2 - 2 i ) \\cdot 2 ^ { k } } - C _ { n } ^ { n + 1 - 2 i } b ^ { ( n + 1 - 2 i ) \\cdot 2 ^ { k } } \\Big ) . \\end{gathered} \\end{align*}"}
-{"id": "940.png", "formula": "\\begin{align*} J ^ { * } [ \\psi ^ { ( 3 ) } ] _ { T , t } = \\hbox { \\vtop { \\offinterlineskip \\halign { \\hfil # \\hfil \\cr { \\rm l . i . m . } \\cr $ \\stackrel { } { { } _ { p \\to \\infty } } $ \\cr } } } \\sum \\limits _ { j _ 1 , j _ 2 , j _ 3 = 0 } ^ { p } C _ { j _ 3 j _ 2 j _ 1 } \\zeta _ { j _ 1 } ^ { ( i _ 1 ) } \\zeta _ { j _ 2 } ^ { ( i _ 2 ) } \\zeta _ { j _ 3 } ^ { ( i _ 3 ) } , \\end{align*}"}
-{"id": "1964.png", "formula": "\\begin{align*} 2 d _ { i i } = \\sum _ { k \\in \\mathcal { N } ( i ) } d _ { k j } . \\end{align*}"}
-{"id": "2809.png", "formula": "\\begin{align*} R ^ t _ j = \\sum _ { i = 1 } ^ S B _ { i , j } ^ t , \\end{align*}"}
-{"id": "1817.png", "formula": "\\begin{align*} \\phi ^ { \\rm I N } ( x , 0 ) = \\phi ^ { \\rm O U T } ( x , 0 ) \\qquad \\mbox { a n d } \\partial _ t \\phi ^ { \\rm I N } ( x , 0 ) = \\partial _ t \\phi ^ { \\rm O U T } ( x , 0 ) . \\end{align*}"}
-{"id": "5342.png", "formula": "\\begin{align*} \\frac 1 2 ( X - Y ) . v ^ k = \\frac 1 2 \\left ( - ( k - 1 ) v ^ { k - 1 } + ( n - k ) v ^ { k + 1 } \\right ) \\end{align*}"}
-{"id": "1247.png", "formula": "\\begin{align*} \\int _ { | \\nu | > 2 0 } e ^ { 3 i \\eta \\nu ^ 2 / 4 } e ^ { i a \\ln | \\frac { \\eta + \\nu } { \\eta - \\nu } | } e ^ { 2 i a \\ln | ( \\eta + \\nu ) / 2 | } \\frac { e ^ { i \\beta ( \\eta + \\nu ) ^ 3 / 8 } } { ( \\eta + \\nu ) ^ 3 } d \\nu = \\int _ { | \\nu | > 2 0 } e ^ { i \\Theta ( \\eta , \\nu ) } e ^ { 2 i a \\ln | ( \\eta + \\nu ) / 2 | } \\frac { e ^ { i a \\ln | \\frac { \\eta + \\nu } { \\eta - \\nu } | } } { ( \\eta + \\nu ) ^ 3 } d \\nu \\end{align*}"}
-{"id": "2482.png", "formula": "\\begin{align*} | P | ( x ) & \\ge \\sup \\Bigl \\{ \\ , \\sum _ { i } | A ( u ^ 1 _ { i } , x , \\dots , x ) | : u ^ 1 \\in \\Pi ( x ) \\Bigr \\} \\\\ & = \\sup \\Bigl \\{ \\ , \\sum _ { i } \\Bigl | \\int _ K u ^ 1 _ i x ^ { n - 1 } \\ , d \\mu \\Bigr | : u ^ 1 \\in \\Pi ( x ) \\Bigr \\} = \\int _ K x ^ n \\ , d | \\mu | \\ , , \\end{align*}"}
-{"id": "8937.png", "formula": "\\begin{align*} A = \\Big \\{ n \\in \\mathbb { N } : q p _ { b l } ( x , x ) > \\frac { \\epsilon } { 2 s ^ { n + 1 } ( 2 s - 1 ) } \\Big \\} . \\end{align*}"}
-{"id": "2271.png", "formula": "\\begin{align*} ( z - a ) R ( z ) f & = f f \\in V \\\\ R ( z ) ( z - a ) f & = f f \\in D . \\end{align*}"}
-{"id": "2728.png", "formula": "\\begin{align*} z _ { 1 } & = z _ { 0 } + 2 ^ { - \\frac { 4 } { 3 } } N ^ { - \\frac { 1 } { 3 } } \\eta _ { - } ^ { - 1 } ( \\pi _ { * } - 2 ) , w _ { 1 } = z _ { 0 } + 2 ^ { - \\frac { 4 } { 3 } } N ^ { - \\frac { 1 } { 3 } } \\eta _ { - } ^ { - 1 } ( \\pi _ { * } - 1 ) , \\end{align*}"}
-{"id": "8924.png", "formula": "\\begin{align*} \\mathbb { E } [ N _ a ^ { \\hat { x } } ( G ' ) - N _ a ^ { \\hat { x } } ( H ) \\mid \\hat { x } ] = & \\frac { ( a - b ) } { n } { \\Big ( } \\left ( ( n ^ { + + } - n _ C ^ { + + } ) - ( n ^ { + - } - n _ C ^ { + - } ) \\right ) ( n _ C ^ { - - } - n _ C ^ { - + } ) \\\\ & \\quad + \\left ( ( n ^ { - - } - n _ C ^ { - - } ) - ( n ^ { - + } - n _ C ^ { - + } ) \\right ) ( n _ C ^ { + + } - n _ C ^ { + - } ) { \\Big ) } . \\end{align*}"}
-{"id": "7804.png", "formula": "\\begin{align*} \\Big \\| \\big ( \\sum _ { n \\in \\mathbb Z } \\sum _ { m = 0 } ^ { 2 ^ { l } - 1 } | ( M ^ G _ { 2 ^ n + { 2 ^ { n - l } ( m + 1 ) } } & - M ^ G _ { 2 ^ n + { 2 ^ { n - l } m } } ) S _ { j + n } f | ^ 2 \\big ) ^ { 1 / 2 } \\Big \\| _ { L ^ p } \\\\ & \\lesssim 2 ^ { - \\theta l / 2 + ( 1 - \\theta ) l } \\min \\big \\{ 1 , 2 ^ { l } 2 ^ { - | j | / 2 } \\big \\} ^ { \\theta } \\| f \\| _ { L ^ p } , \\end{align*}"}
-{"id": "7338.png", "formula": "\\begin{align*} \\begin{cases} \\rho ( x ) w _ { t t } - \\operatorname { d i v } ( K ( x ) \\nabla w ) + \\alpha ^ 2 z ^ 2 w = 0 & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ \\rho ( x ) z _ { t t } - \\operatorname { d i v } ( K ( x ) \\nabla z ) + \\alpha ^ 2 w ^ 2 z = 0 & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ w _ t = z _ t = 0 & ~ \\hbox { i n } ~ \\omega \\times ( 0 , T ) . \\end{cases} \\end{align*}"}
-{"id": "3507.png", "formula": "\\begin{align*} \\frac { \\sin \\pi ( s - t ) } { s - t } = \\frac { \\pi } { 2 } \\int _ { - 1 } ^ { 1 } d w \\ , e ^ { - i \\pi ( s - t ) w } , \\end{align*}"}
-{"id": "3759.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\left | r \\ , u _ { l } ( \\alpha r ) \\right | ^ { 2 } d r = 0 < \\infty ; \\alpha \\in \\mathbb { R } ^ { * } \\end{align*}"}
-{"id": "7805.png", "formula": "\\begin{align*} \\Big \\| \\big ( \\sum _ { n \\in \\mathbb Z } \\sum _ { m = 0 } ^ { 2 ^ { l } - 1 } \\abs { ( M ^ G _ { 2 ^ n + { 2 ^ { n - l } ( m + 1 ) } } - M ^ G _ { 2 ^ n + { 2 ^ { n - l } m } } ) S _ { j + n } f } ^ 2 \\big ) ^ { 1 / 2 } \\Big \\| _ { L ^ 2 } \\lesssim 2 ^ { l / 2 } 2 ^ { - | j | / 2 } \\| f \\| _ { L ^ 2 } . \\end{align*}"}
-{"id": "5065.png", "formula": "\\begin{align*} A _ { \\sigma ( U V ) } A _ { \\sigma ( U W ) } ^ * & = f ( U V ) \\overline { f ( U W ) } A _ V A _ W ^ * , \\\\ A _ { \\sigma ( U V ) } \\Psi _ { \\sigma ( U W ) } ^ * & = f ( U V ) \\overline { f ( U W ) } A _ V \\Psi _ W ^ * , \\\\ \\Psi _ { \\sigma ( U V ) } \\Psi _ { \\sigma ( U W ) } ^ * & = f ( U V ) \\overline { f ( U W ) } \\Psi _ V \\Psi _ W ^ * \\end{align*}"}
-{"id": "1799.png", "formula": "\\begin{align*} \\alpha = \\frac { c _ { n } } { \\sqrt { 2 \\log \\log ( \\theta ^ { - n } ) } } + \\frac { 1 - \\theta ^ { H } } { ( 1 - \\theta ) ^ { H } } \\end{align*}"}
-{"id": "7055.png", "formula": "\\begin{align*} _ { \\Phi } h = 0 \\delta _ { \\Phi } h = 0 . \\end{align*}"}
-{"id": "3309.png", "formula": "\\begin{align*} \\frac { 1 } { q ^ { \\ell } \\phi ( M ) } \\sum _ { F } \\chi _ 1 ( F ) \\overline { \\chi _ 2 } ( F ) = \\begin{cases} 0 & \\chi _ 1 \\neq \\chi _ 2 , \\\\ 1 & \\chi _ 1 = \\chi _ 2 , \\end{cases} \\end{align*}"}
-{"id": "9450.png", "formula": "\\begin{align*} a _ \\theta ( m ) - b ( m ) = - \\sum _ { \\tau \\in \\mathcal { F } _ N } \\nu _ \\tau ^ { ( N ) } ( f ) J _ { N , m } ( \\tau ) \\end{align*}"}
-{"id": "5584.png", "formula": "\\begin{align*} \\begin{cases} i u _ t + u _ { x x } = 0 , & x , t \\in \\R ^ { + } , \\\\ u ( x , 0 ) = 0 , & x \\in \\R ^ + , \\\\ u ( 0 , t ) = f ( t ) \\in H ^ { ( 2 s + 1 ) / 4 } ( \\R ^ + ) & \\end{cases} \\end{align*}"}
-{"id": "6321.png", "formula": "\\begin{align*} v _ t + v v _ y & = \\frac { \\nu } { h } [ h v _ y ] _ y , \\\\ h _ t & = - \\left ( h v \\right ) _ y , \\end{align*}"}
-{"id": "2486.png", "formula": "\\begin{align*} \\bigl ( T x \\bigr ) ( s ) = \\alpha ( s ) x ( \\varphi ( s ) ) \\end{align*}"}
-{"id": "846.png", "formula": "\\begin{align*} \\overline { s p a n } \\{ \\phi ( t ) : t \\in \\Omega \\} = X _ d \\textrm { a n d } \\overline { s p a n } \\{ \\phi ^ * ( t ) : t \\in \\Omega \\} = X _ d ^ * \\end{align*}"}
-{"id": "4856.png", "formula": "\\begin{align*} \\alpha ( w _ { ( j , r _ 1 ) } ) = w _ { ( j , r _ 1 ) } . \\end{align*}"}
-{"id": "8350.png", "formula": "\\begin{align*} - \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j } { \\pi } R e \\Big \\{ \\frac { D _ t Z - \\dot { z } _ j } { ( \\alpha - z _ j ) ^ 2 } \\Big \\} = & - \\frac { \\lambda } { \\pi } R e \\Big \\{ \\frac { D _ t Z ( \\alpha , t ) } { ( \\alpha - z _ 1 ( t ) ) ^ 2 } \\Big \\} \\\\ = & - \\frac { \\lambda } { \\pi } R e \\Big \\{ \\frac { 1 } { ( \\alpha - z _ 1 ( t ) ) ^ 2 } \\frac { \\lambda i } { 2 \\pi \\overline { \\alpha - z _ 1 ( t ) } } \\Big \\} \\\\ = & \\frac { \\lambda ^ 2 } { 2 \\pi ^ 2 } \\frac { y } { | \\alpha - z _ 1 ( t ) | ^ 4 } . \\end{align*}"}
-{"id": "3981.png", "formula": "\\begin{align*} \\langle \\phi , \\psi \\rangle : = \\frac { 1 } { [ \\Gamma ^ J \\colon G ] } \\int _ { F _ G } \\phi ( \\tau , z ) \\overline { \\psi ( \\tau , z ) } e ^ { - 4 \\pi m y ^ 2 / v } v ^ k d V , \\end{align*}"}
-{"id": "9187.png", "formula": "\\begin{align*} m _ { T } & = \\dim \\mathcal { K } _ { b } \\left ( b \\right ) - \\dim ( \\mathcal { K } _ { b } \\left ( b \\right ) \\cap T \\tilde { N } ^ { \\perp } ) \\\\ & = b N - \\dim ( \\mathcal { K } _ { b } \\left ( b \\right ) \\cap T \\tilde { N } ^ { \\perp } ) , \\end{align*}"}
-{"id": "1438.png", "formula": "\\begin{align*} \\mu ^ { n + 1 } _ { t } - \\Delta \\mu ^ { n + 1 } + \\varepsilon \\mathrm { d i v } ( \\mu ^ { n } \\Theta _ { p } ( t , x , \\mu ^ { n } , D w ^ { n } ) ) + \\varepsilon \\bar { m } \\mathrm { d i v } ( \\Theta _ { p } ( t , x , \\mu ^ { n } , D w ^ { n } ) ) = 0 , \\end{align*}"}
-{"id": "4450.png", "formula": "\\begin{align*} \\iota = \\phi _ 1 \\circ \\iota '' \\iota = \\phi _ 2 \\circ \\iota '' , \\end{align*}"}
-{"id": "6756.png", "formula": "\\begin{align*} x ^ m \\approx x ^ p \\ , \\overline x \\ , ^ q . \\end{align*}"}
-{"id": "3478.png", "formula": "\\begin{align*} f _ M ( t ; \\theta ) = ( M + 1 ) ^ { 2 } ( 1 + \\frac { t } { M + 1 } ) \\log ( 1 + \\frac { t } { M + 1 } ) - t \\log { t } - M t - v _ { M } ( \\theta ) t . \\end{align*}"}
-{"id": "8435.png", "formula": "\\begin{align*} \\begin{cases} D _ t ^ 2 \\zeta - i A \\zeta _ { \\alpha } = - i \\\\ \\frac { d } { d t } z _ j ( t ) = ( v - \\frac { \\lambda _ j i } { 2 \\pi ( \\overline { z - z _ j } ) } ) \\Big | _ { z = z _ j } \\\\ ( I - \\mathcal { H } ) ( D _ t \\bar { \\zeta } + \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i } { 2 \\pi ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) } ) = 0 \\\\ ( I - \\mathcal { H } ) ( \\bar { \\zeta } - \\alpha ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "9870.png", "formula": "\\begin{align*} c / a = c ' / a ' = c '' / a '' , \\ , ( d - b ) / a = ( d ' - b ' ) / a ' = ( d '' - b '' ) / a '' , \\end{align*}"}
-{"id": "3401.png", "formula": "\\begin{align*} \\rho ( T ) = \\frac { 1 } { 1 + \\sum _ { i } \\rho ( T _ i ) } . \\end{align*}"}
-{"id": "5151.png", "formula": "\\begin{align*} H _ { m a x } \\psi = \\sum _ { j , m _ j , k _ j } h ^ * _ { m _ j , k _ j } f _ { m _ j , k _ j } , H _ { m a x } \\widetilde \\psi = \\sum _ { j , m _ j , k _ j } h ^ * _ { m _ j , k _ j } \\widetilde f _ { m _ j , k _ j } , \\end{align*}"}
-{"id": "1056.png", "formula": "\\begin{align*} & \\lambda _ 1 ( B ) \\chi _ { I ( y _ i , F _ n ( x _ i ) ) } f = \\chi _ { I ( y _ i , F _ n ( x _ i ) ) } B f \\\\ & = \\chi _ { I ( y _ i , F _ n ( x _ i ) ) } B _ i f = B _ i f = B _ i \\chi _ { I ( y _ i , F _ n ( x _ i ) ) } f , \\end{align*}"}
-{"id": "6457.png", "formula": "\\begin{align*} \\ell ( v ) ^ * & \\Omega = 0 \\\\ \\ell ( v ) ^ * & w _ 1 \\otimes \\cdots \\otimes w _ d = \\ < v , w _ 1 \\ > w _ 2 \\otimes \\cdots \\otimes w _ d . \\end{align*}"}
-{"id": "8661.png", "formula": "\\begin{align*} R _ 0 ( \\l ) = - \\frac { 1 } { m _ + ( \\l ) + m _ - ( \\l ) } , R _ 1 ( \\l ) = \\frac { m _ + ( \\l ) m _ - ( \\l ) } { m _ + ( \\l ) + m _ - ( \\l ) } \\end{align*}"}
-{"id": "1504.png", "formula": "\\begin{align*} n _ t = \\nabla \\cdot \\big ( D ( n ) \\nabla n - S ( n , c ) \\nabla c \\big ) & & c _ t = \\Delta c - c + n , \\end{align*}"}
-{"id": "9958.png", "formula": "\\begin{align*} = \\frac a 2 \\left ( 2 a ^ 2 x ^ 2 z - 2 y ^ 2 ( - z ) \\right ) = a z \\left ( a ^ 2 x ^ 2 + y ^ 2 \\right ) . \\end{align*}"}
-{"id": "8796.png", "formula": "\\begin{align*} \\overline { \\mathcal { F } } _ t = \\bigcap _ { u > t } \\sigma \\left ( v ^ k ( s , x ) , \\overline { v } ^ k _ s \\mid 0 \\leq x \\leq \\pi , \\ , k \\in \\mathbb { N } , \\ , s \\leq u \\right ) , t \\leq T . \\end{align*}"}
-{"id": "5463.png", "formula": "\\begin{align*} Y _ t ^ n \\ & = \\ g ( \\hat X ) + \\int _ t ^ T f ( \\hat X , \\hat I _ s ) \\ , d s + K _ T ^ n - K _ t ^ n - \\int _ t ^ T \\int _ \\Lambda R _ s ^ n ( b ) \\ , \\hat \\theta ( d s \\ , d b ) \\\\ & \\ - \\int _ t ^ T Z _ s ^ n \\ , d \\hat W _ s - \\int _ t ^ T \\int _ U L _ s ^ n ( z ) \\ , ( \\hat \\pi ( d s \\ , d z ) - \\lambda _ \\pi ( d z ) d s ) , 0 \\leq t \\leq T , \\ ; \\hat \\P \\end{align*}"}
-{"id": "24.png", "formula": "\\begin{align*} d S _ t = ( S _ t ) \\sigma ( V _ { t } ) ( \\theta ^ { \\alpha _ { t - } } ( V _ t ) d t + d W _ t ) \\end{align*}"}
-{"id": "4978.png", "formula": "\\begin{align*} b \\cdot \\exp \\left ( - \\sum _ { i = N } ^ { N + m } \\lambda _ { i + \\ell } \\right ) \\leq \\frac { 1 } { 2 ( k + 1 ) } . \\end{align*}"}
-{"id": "2495.png", "formula": "\\begin{align*} \\frak { a u t } _ { C R } ( M ) : = \\underbrace { \\frak g _ { - \\rho } \\oplus \\cdots \\oplus \\frak g _ { - 1 } } _ { \\frak g _ - } \\oplus \\ , \\frak g _ { 0 } \\oplus \\underbrace { \\frak g _ { 1 } \\oplus \\cdots \\oplus \\frak g _ { \\mu } } _ { \\frak g _ + } , \\ \\ \\ \\ \\ \\ \\mu \\ , \\in \\ , \\mathbb N . \\end{align*}"}
-{"id": "3404.png", "formula": "\\begin{align*} \\tau ^ M ( T ) = - \\log \\left ( \\frac { 1 + \\sum _ { i } \\rho ^ { M - 1 } _ { \\inf } ( T _ i ) } { 1 + \\sum _ { i } \\rho ^ { M - 1 } _ { \\sup } ( T _ i ) } \\right ) = - \\log \\left ( \\frac { 1 + \\sum _ { i } \\rho _ i \\exp ( - \\tau ^ { M - 1 } ( T _ i ) ) } { 1 + \\sum _ { i } \\rho _ i } \\right ) . \\end{align*}"}
-{"id": "2030.png", "formula": "\\begin{align*} \\phi _ { \\delta } ( t ) = \\delta ^ { - 1 } \\phi ( t / \\delta ) \\end{align*}"}
-{"id": "8085.png", "formula": "\\begin{align*} c _ m = ( - 1 ) ^ m \\sum _ { j = 0 } ^ m \\ , [ h ^ { m - j } ] \\exp ( \\mu ( h ) ) \\ ; \\frac { ( m - 2 j + 3 / 2 ) _ { 2 j } } { 4 ^ j j ! } \\end{align*}"}
-{"id": "6262.png", "formula": "\\begin{align*} u ( z e _ { i , j } x _ { i } \\varepsilon e _ { j , } ) = u ( z x _ { i } \\varepsilon e _ { i , } ) = a z x _ { i } \\varepsilon e _ { i , } ; \\end{align*}"}
-{"id": "3684.png", "formula": "\\begin{align*} E ( \\hat { y } _ i | x _ i , i \\in S ) = E ( y _ i | x _ i , i \\in U ) \\neq E ( y _ i | x _ i , i \\in S ) \\end{align*}"}
-{"id": "5274.png", "formula": "\\begin{align*} \\left \\vert h ( x ) \\right \\vert = \\left \\vert \\frac { h ( n x ) } { n } \\right \\vert \\leq \\frac { A ( \\delta ) } { n } . \\end{align*}"}
-{"id": "7313.png", "formula": "\\begin{align*} \\begin{aligned} \\rho ( x ) \\partial _ t ^ 2 w _ t ^ n - \\operatorname { d i v } ( K ( x ) \\nabla w _ t ^ n ) \\rightarrow 0 ~ \\hbox { i n } ~ H ^ { - 1 } _ { l o c } ( \\Omega \\times ( 0 , T ) ) , \\\\ \\rho ( x ) \\partial _ t ^ 2 z _ t ^ n - \\operatorname { d i v } ( K ( x ) \\nabla z _ t ^ n ) \\rightarrow 0 ~ \\hbox { i n } ~ H ^ { - 1 } _ { l o c } ( \\Omega \\times ( 0 , T ) ) , \\end{aligned} \\end{align*}"}
-{"id": "4070.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { ( c _ V n ) ^ { 1 + 1 / \\theta } } K ^ { \\alpha , \\theta } _ { V , n } \\left ( \\frac { x } { ( c _ V n ) ^ { 1 + 1 / \\theta } } , \\frac { y } { ( c _ V n ) ^ { 1 + 1 / \\theta } } \\right ) = \\mathbb K ^ { ( \\alpha , \\theta ) } ( x , y ) . \\end{align*}"}
-{"id": "5333.png", "formula": "\\begin{align*} Y . v ^ k = - ( n - k ) v ^ { k + 1 } . \\end{align*}"}
-{"id": "2317.png", "formula": "\\begin{align*} \\mathfrak { H } = \\mathbb { C } \\oplus L ^ 2 ( \\mathbb R ) , \\end{align*}"}
-{"id": "9434.png", "formula": "\\begin{align*} E _ 2 ( z ) = 1 - 2 4 \\sum _ { n = 1 } ^ \\infty \\sigma _ 1 ( n ) q ^ n . \\end{align*}"}
-{"id": "3867.png", "formula": "\\begin{align*} A _ 0 = \\left \\{ \\left ( \\begin{array} { c c } 1 & 3 \\\\ 0 & 1 \\end{array} \\right ) , \\left ( \\begin{array} { c c } 1 & 0 \\\\ 3 & 1 \\end{array} \\right ) \\right \\} , \\end{align*}"}
-{"id": "1355.png", "formula": "\\begin{align*} \\left ( h _ { - 1 } \\right ) _ w ( 1 ) = w \\in \\Z \\Lambda ^ { * 1 } w \\end{align*}"}
-{"id": "493.png", "formula": "\\begin{align*} \\bar A ( \\pi _ 0 ( f ) ) = \\pi _ 2 ( z _ 1 ^ { n ( d - 1 ) } + \\cdots + z _ n ^ { n ( d - 1 ) } ) . \\end{align*}"}
-{"id": "1240.png", "formula": "\\begin{align*} & \\left | \\int _ { | \\mu | \\le | \\eta | ^ { 3 / 2 } / 2 } e ^ { 3 i \\mu ^ 2 / 4 } e ^ { i a \\ln ( \\eta ^ 2 - \\mu ^ 2 / \\eta ) } e ^ { 2 i a \\ln | ( \\eta + \\mu / \\sqrt { | \\eta | } ) | } \\frac { e ^ { i \\beta ( \\eta + \\mu / \\sqrt { \\eta } ) ^ 3 } } { ( \\eta + \\mu / \\sqrt { \\eta } ) ^ 3 } d \\mu \\right | \\\\ \\le & \\int _ { - | \\eta | ^ { 3 / 2 } / 2 } ^ { | \\eta | ^ { 3 / 2 } / 2 } \\frac { 1 } { ( \\eta + \\mu / \\sqrt { \\eta } ) ^ 3 } d \\mu = O ( | \\eta | ^ { - 3 / 2 } ) . \\end{align*}"}
-{"id": "1832.png", "formula": "\\begin{align*} S _ { \\rm m a t t e r } ( \\phi , g ^ { \\mu \\nu } ) = \\frac { 1 } { 2 } \\int \\left [ g ^ { \\mu \\nu } \\left ( \\partial _ \\mu \\phi \\right ) \\left ( \\partial _ \\nu \\phi \\right ) - V ( x ) \\phi ^ 2 \\right ] \\sqrt { - g } \\ , d ^ n x , \\end{align*}"}
-{"id": "7995.png", "formula": "\\begin{align*} A ^ { p , q } ( \\mathcal { U } , U _ { 0 } ) : = \\{ \\ , \\xi \\in A ^ { p , q } ( \\mathcal { U } ) \\mid \\xi _ { 0 } = 0 \\ , \\} = A ^ { p , q } ( U _ { 1 } ) \\oplus A ^ { p , q - 1 } ( U _ { 0 1 } ) . \\end{align*}"}
-{"id": "1291.png", "formula": "\\begin{gather*} I ( z , S _ A , S _ A ) = J ( K ( S _ A , S _ A ) , z ) , I ( S _ A , z , w ) = J ( K ( z , w ) , S _ A ) , \\\\ I ( z , w , u ) = J ( K ( w , u ) , z ) , \\end{gather*}"}
-{"id": "204.png", "formula": "\\begin{align*} X = x ^ 3 \\dfrac { \\partial } { \\partial x ^ 2 } + x ^ 1 \\dfrac { \\partial } { \\partial x ^ 3 } , Y = \\dfrac { \\partial } { \\partial x ^ 2 } \\end{align*}"}
-{"id": "5557.png", "formula": "\\begin{align*} W _ r ' ( k ) = \\{ x \\in G ( k ) ^ r \\ ; | \\ ; { \\langle x \\rangle } \\hbox { a c t s r e d u c i b l y o n s o m e m o d u l e i n } { \\mathcal S } \\} . \\end{align*}"}
-{"id": "1243.png", "formula": "\\begin{align*} S ( \\xi ) = S _ 1 ( \\xi ) + S _ 2 ( \\xi ) , S _ 1 ( \\xi ) = A e ^ { i a \\ln | \\xi | } \\chi ( \\xi ) + \\overline { A } e ^ { - i a \\ln | \\xi | } \\chi ( - \\xi ) . \\end{align*}"}
-{"id": "985.png", "formula": "\\begin{align*} I ( p ) = \\left \\langle \\frac { L C M ( m _ 1 , m _ p ) } { m _ p } , \\frac { L C M ( m _ 2 , m _ p ) } { m _ p } , \\ldots , \\frac { L C M ( m _ { p - 1 } , m _ p ) } { m _ p } \\right \\rangle . \\end{align*}"}
-{"id": "8332.png", "formula": "\\begin{align*} ( I - \\mathbb { H } ) F _ z \\circ Z Z _ { \\alpha } D _ t Z = & [ D _ t Z , \\mathbb { H } ] ( \\partial _ { \\alpha } D _ t \\bar { Z } - \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i Z _ { \\alpha } } { 2 \\pi ( Z ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } ) \\\\ = & [ D _ t Z , \\mathbb { H } ] \\partial _ { \\alpha } D _ t \\bar { Z } - \\sum _ { j = 1 } ^ N [ D _ t Z , \\mathbb { H } ] \\frac { \\lambda _ j i Z _ { \\alpha } } { 2 \\pi ( Z ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } ) \\end{align*}"}
-{"id": "2558.png", "formula": "\\begin{align*} q _ t ( x , y ) : = \\frac { \\phi ( y ) } { \\phi ( x ) } p ^ \\beta _ t ( x , y ) , x , y \\in E , t > 0 . \\end{align*}"}
-{"id": "1377.png", "formula": "\\begin{align*} b = \\sum _ { i = 1 } ^ { n } \\sum _ { j = K _ { ( i ) } + 1 } ^ { k } d _ { j } ( \\lambda _ { i - 1 } ) e _ j \\quad c = \\sum _ { i = 1 } ^ { n } \\sum _ { j = K _ { ( i ) } } ^ { k } d _ { j } ( \\lambda _ { i - 1 } ) e _ j . \\end{align*}"}
-{"id": "5456.png", "formula": "\\begin{align*} \\bar I _ t \\ = \\ \\sum _ { n \\geq 0 } \\bar \\eta _ n \\ , 1 _ { [ \\bar T _ n , \\bar T _ { n + 1 } ) } ( t ) , \\qquad t \\geq 0 , \\end{align*}"}
-{"id": "4623.png", "formula": "\\begin{align*} M \\mathbf { u } _ { n - 1 } + ( \\tau ^ 2 K - 2 M ) \\mathbf { u } _ { n } + M \\mathbf { u } _ { n + 1 } = \\mathbf { 0 } . \\end{align*}"}
-{"id": "7137.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ 0 ^ L \\Big ( \\frac { \\rho u ^ 2 } { 2 } + \\frac { \\epsilon } { 2 } \\chi _ x ^ 2 + \\Phi ( \\rho ) + \\frac { \\rho ( \\chi ^ 2 - 1 ) ^ 2 } { 4 \\epsilon } \\Big ) d x + \\int _ 0 ^ L \\Big ( \\mu ^ 2 + \\nu u ^ 2 _ x \\Big ) d x d \\tau = 0 . \\end{align*}"}
-{"id": "3786.png", "formula": "\\begin{align*} c _ { \\alpha , \\mu } = \\begin{cases} c _ { \\beta , \\mu } & \\mu \\neq \\nu \\\\ c _ { \\beta , \\mu } + 1 & \\mu = \\nu \\end{cases} \\end{align*}"}
-{"id": "7115.png", "formula": "\\begin{align*} \\mathcal { L } [ q ] u : = - u '' + q ( x ) u \\ , \\ , \\ , \\ , \\end{align*}"}
-{"id": "6613.png", "formula": "\\begin{align*} \\prod _ { t = \\eta + 1 } ^ { ( F _ { n } - 1 ) / 2 } \\left ( 1 - \\frac { v _ n ^ 2 } { s _ { n t } ^ 2 } \\right ) = 1 - \\mathcal { O } ( \\varphi ^ { n / 5 } ) . \\end{align*}"}
-{"id": "5542.png", "formula": "\\begin{align*} \\| ( L ^ * _ b ) ^ { - 1 } [ f ] \\| _ { L ^ 2 } & = \\sup _ { \\| \\phi \\| _ { L ^ 2 } \\le 1 } \\left | \\int ( L ^ * _ b ) ^ { - 1 } [ f ] L _ b L _ b ^ { - 1 } [ \\bar \\phi ] \\right | = \\sup _ { \\| \\phi \\| _ { L ^ 2 } \\le 1 } \\left | \\int f L _ b ^ { - 1 } [ \\bar \\phi ] \\right | \\\\ & \\leq \\| f \\| _ { H ^ { - 2 } } \\sup _ { \\| \\phi \\| _ { L ^ 2 } \\le 1 } \\| L _ b ^ { - 1 } [ \\bar \\phi ] \\| _ { H ^ 2 } \\lesssim \\| f \\| _ { H ^ { - 2 } } , \\end{align*}"}
-{"id": "6826.png", "formula": "\\begin{align*} \\varphi ( t , \\alpha ) = A ( t ) u ( \\alpha ) \\ , , \\end{align*}"}
-{"id": "8189.png", "formula": "\\begin{align*} f ( x ) = f ( x ^ i ) + f ( x ^ j ) - f ( x ^ { i j } ) . \\end{align*}"}
-{"id": "659.png", "formula": "\\begin{align*} \\delta ^ { ( \\psi \\varphi ^ y ) \\ast } _ j ( \\varphi ^ \\ast ( v y ) u \\varphi ^ \\ast ( y ) ^ { - 1 } ) = \\gamma ( \\delta ^ { ( \\psi ) \\ast } _ j ( v ) ; ( y _ \\ast \\vec u ) ^ \\psi _ j ) \\end{align*}"}
-{"id": "4298.png", "formula": "\\begin{align*} \\textup { s a t } ( n , H , C _ { 2 k + 1 } ) = 0 . \\end{align*}"}
-{"id": "8591.png", "formula": "\\begin{align*} \\zeta ( z ) & = \\sin ( \\pi z / 2 ) \\pi ^ { z - 1 } \\int _ a ^ \\infty d s \\ , s ^ { - z - 1 } [ s \\coth ( s ) - 1 ] \\\\ & + \\sin ( \\pi z / 2 ) \\pi ^ { z - 1 } \\int _ 0 ^ a d s \\ , s ^ { - z - 1 } [ s \\coth ( s ) - 1 ] , \\end{align*}"}
-{"id": "6239.png", "formula": "\\begin{align*} H = \\ < \\{ \\alpha _ { i } / \\alpha _ { j } \\ , : \\ , 1 \\leq i , j \\leq n \\} \\ > , \\end{align*}"}
-{"id": "5009.png", "formula": "\\begin{align*} \\lim N ( \\delta ) ^ { 1 / d } \\delta = C _ \\theta , \\end{align*}"}
-{"id": "4509.png", "formula": "\\begin{align*} \\left \\langle V \\phi _ i \\phi _ j , \\phi _ i \\phi _ j \\right \\rangle & = \\int _ { \\Omega } { V \\phi _ i ^ 2 \\phi _ j ^ 2 d x } \\\\ & \\leq \\| V \\| _ { L ^ { \\infty } } \\max _ { 1 \\leq i \\leq n } { \\| \\phi _ i \\| ^ 2 _ { L ^ { \\infty } } } \\lesssim _ { V } \\max _ { 1 \\leq i \\leq n } { \\| \\phi _ i \\| ^ 2 _ { L ^ { \\infty } } } \\end{align*}"}
-{"id": "9307.png", "formula": "\\begin{align*} x _ { i j } \\geq 0 , ~ i , j = 1 , \\dots , n . \\end{align*}"}
-{"id": "1457.png", "formula": "\\begin{align*} m ^ { 1 } ( 0 , \\cdot ) = m ^ { 2 } ( 0 , \\cdot ) , u ^ { 1 } ( T , \\cdot ) = u ^ { 2 } ( T , \\cdot ) . \\end{align*}"}
-{"id": "3830.png", "formula": "\\begin{align*} { { \\gamma _ { j } } ^ { * i } } _ k = { { \\gamma _ { j } } ^ { i } } _ k + \\rho _ j \\delta ^ i _ k + \\rho _ k \\delta ^ i _ j - \\rho ^ i g _ { j k } , \\end{align*}"}
-{"id": "9975.png", "formula": "\\begin{align*} \\frac { \\partial ^ { 2 } c ^ { i } } { \\partial a ^ { j } \\partial a ^ { m } } \\frac { \\partial ^ { 2 } c ^ { m } } { \\partial a ^ { k } \\partial a ^ { n } } = \\frac { \\partial ^ { 2 } c ^ { i } } { \\partial a ^ { k } \\partial a ^ { m } } \\frac { \\partial ^ { 2 } c ^ { m } } { \\partial a ^ { j } \\partial a ^ { n } } \\end{align*}"}
-{"id": "8864.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 c _ 2 } { \\partial \\mu ^ 2 } & = \\nu ^ 4 ( 2 1 6 \\nu ^ 3 - 1 4 4 \\nu ^ 2 - 9 6 9 6 \\nu - 7 1 8 3 2 ) + \\nu ^ 3 ( 4 9 5 0 9 6 \\nu - 5 0 5 4 4 0 ) + ( 2 8 8 9 1 2 \\nu - 1 2 9 5 3 5 2 ) < 0 \\end{align*}"}
-{"id": "6442.png", "formula": "\\begin{align*} \\xi _ k = \\sum _ { j = 0 } ^ { k - 1 } \\| r _ { j } \\| ^ { - 2 } \\left ( \\sum _ { i = j } ^ { k - 1 } \\psi _ { i } \\right ) ^ { 2 } , \\psi _ { i } = \\gamma _ { i } \\| r _ { i } \\| ^ { 2 } , \\end{align*}"}
-{"id": "5079.png", "formula": "\\begin{align*} \\langle x _ { \\sigma ( U V ) } , x _ { \\sigma ( U W ) } \\rangle & = \\overline { f ( U V ) } f ( U W ) \\langle x _ V , x _ W \\rangle , \\\\ \\langle x _ { \\sigma ( U V ) } , \\tau _ { \\sigma ( U W ) } \\rangle & = \\overline { f ( U V ) } f ( U W ) \\langle x _ V , \\tau _ W \\rangle , \\\\ \\langle \\tau _ { \\sigma ( U V ) } , \\tau _ { \\sigma ( U W ) } \\rangle & = \\overline { f ( U V ) } f ( U W ) \\langle \\tau _ V , \\tau _ W \\rangle \\end{align*}"}
-{"id": "2828.png", "formula": "\\begin{align*} \\mu _ i - | a _ { i i } | > 0 , \\ i = 1 , 2 , \\mbox { a n d } ( \\mu _ 1 - | a _ { 1 1 } | ) ( \\mu _ 2 - | a _ { 2 2 } | ) > | a _ { 1 2 } a _ { 2 1 } | , \\end{align*}"}
-{"id": "9945.png", "formula": "\\begin{align*} e ^ F \\bigl ( \\Delta + M _ V \\bigr ) e ^ { - F } = e ^ { F } \\bigl ( \\frac { 4 \\pi ^ 2 } { T ^ 2 } ( A + \\frac { \\beta } 4 ) ^ 2 + Q _ 0 + M _ V \\bigr ) e ^ { - F } = \\frac { 4 \\pi ^ 2 } { T ^ 2 } ( A + \\frac { \\beta } { 4 } ) ^ 2 + B + S , \\end{align*}"}
-{"id": "3760.png", "formula": "\\begin{align*} \\Xi \\coloneqq \\left \\{ \\chi \\in \\mathbb { R } : \\Im \\left [ \\mathcal { P } \\right ] = 0 \\right \\} , \\end{align*}"}
-{"id": "6678.png", "formula": "\\begin{align*} H _ { - } & = A ^ { \\dagger } ( \\omega ) A ( \\omega ) = | x | ^ { \\frac { i \\omega } { 2 } } O ^ { \\dagger } O \\ , | x | ^ { - \\frac { i \\omega } { 2 } } , \\\\ H _ { + } & = A ( \\omega ) A ^ { \\dagger } ( \\omega ) = | x | ^ { - \\frac { i \\omega } { 2 } } O O ^ { \\dagger } \\ , | x | ^ { \\frac { i \\omega } { 2 } } . \\end{align*}"}
-{"id": "6922.png", "formula": "\\begin{align*} | \\langle Z _ { m } , W ' _ { i } \\rangle - { \\bf 1 } _ { \\{ i = q \\} } | \\leq \\sqrt { 2 } m \\delta ' + { \\bf 1 } _ { \\{ m - 1 \\geq q \\} } \\sqrt { 2 } q \\delta ' + | \\zeta - 1 | | \\langle Z '' _ { m - 1 } , W ' _ { m } \\rangle | , \\forall i \\geq m . \\end{align*}"}
-{"id": "6746.png", "formula": "\\begin{align*} \\Delta ( m ) = \\{ ( k _ 1 , \\ldots , k _ m ) \\in \\mathbb { N } ^ m _ 0 : k _ 1 + 2 k _ 2 + \\cdots + m k _ m = m \\} . \\end{align*}"}
-{"id": "2685.png", "formula": "\\begin{align*} f _ 1 ( x ; z ) = 2 \\eta _ { - } x + \\log ( \\tau z - 1 ) - \\log ( z - \\tau ) , \\end{align*}"}
-{"id": "9884.png", "formula": "\\begin{align*} f ( x ) = \\int _ { \\mathcal { Y } ^ { N } } g ( y _ { [ 1 , N ] } ) d P ^ { \\mu } ( y _ { [ 1 , N ] } | x _ { 1 } = x ) \\end{align*}"}
-{"id": "6129.png", "formula": "\\begin{align*} F _ { { i _ k , j _ k } } = \\sum _ { \\ell = 1 } ^ m R _ \\ell G _ \\ell \\end{align*}"}
-{"id": "5561.png", "formula": "\\begin{align*} \\lambda ^ T _ { i } ( v _ j ) = \\delta _ { i j } , i , j \\in [ 0 : n ] . \\end{align*}"}
-{"id": "3514.png", "formula": "\\begin{align*} g ^ { ( \\nu ) } ( \\xi ) = \\sum ^ M _ { j = 0 } \\left ( \\log ( \\nu _ j + N ) - \\frac { 1 } { 2 ( \\nu _ j + N ) } \\right ) + \\xi , \\end{align*}"}
-{"id": "1514.png", "formula": "\\begin{align*} \\sigma = \\sum _ { k = 0 } ^ { q ^ { n } - 1 } c _ { k } B _ { k } , \\end{align*}"}
-{"id": "3095.png", "formula": "\\begin{align*} \\| T b \\mathrm { I } _ { G ^ * } \\| _ q \\le & \\sum _ l \\| T b _ l \\| _ { q ; \\R ^ n \\backslash B _ l } \\\\ \\le & C \\sum _ l \\| b _ l \\| _ { 1 ; K _ l } \\\\ = & C \\| b \\| _ { 1 ; F } \\le C \\| f \\| _ 1 \\end{align*}"}
-{"id": "2357.png", "formula": "\\begin{align*} M & = a ^ { \\vartheta _ n } _ { c _ n } \\cdots a ^ { \\vartheta _ 1 } _ { c _ 1 } \\\\ M _ 2 & = a ^ { \\vartheta _ n } _ { c _ n } \\cdots a ^ { \\vartheta _ k } _ { c _ k } \\\\ M _ 1 & = a ^ { \\vartheta _ { k 1 } } _ { c _ { k - 1 } } \\cdots a ^ { \\vartheta _ 1 } _ { c _ 1 } \\end{align*}"}
-{"id": "5265.png", "formula": "\\begin{align*} \\Gamma ( F , x , r ( t ) , t ) = ( 1 + o ( 1 ) ) \\left ( - \\int _ { t } ^ { x r ( t ) + t } \\frac { a ( s ) } { r ( s ) } d s \\right ) . \\end{align*}"}
-{"id": "9609.png", "formula": "\\begin{align*} { } Q ( q , m + 1 ) - Q ( q , m ) & = \\frac { 1 } { 1 - q ^ N } \\left \\{ \\frac { ( - 1 ) _ { m + 1 } } { ( - q ^ N ) _ { m + 1 } } - \\frac { ( - 1 ) _ { m } } { ( - q ^ N ) _ { m } } \\right \\} \\\\ { } & = \\frac { 1 } { ( 1 - q ^ N ) } \\frac { ( - 1 ) _ { m } } { ( - q ^ N ) _ { m + 1 } } \\left \\{ ( 1 + q ^ m ) - ( 1 + q ^ { N + m } ) \\right \\} \\\\ { } & = \\frac { ( - 1 ) _ m q ^ m } { ( - q ^ N ) _ { m + 1 } } = P ( q , m + 1 ) - P ( q , m ) . \\end{align*}"}
-{"id": "5922.png", "formula": "\\begin{align*} \\psi ^ * ( K _ { X } + B _ m + M ) + E = \\phi ^ * ( K _ { X '' } + B '' _ m + M '' ) , \\end{align*}"}
-{"id": "4670.png", "formula": "\\begin{align*} \\bar \\pi _ { n , g } ( \\bar \\lambda _ 1 , \\ldots , \\bar \\lambda _ n ; a _ 0 , a _ 1 , \\ldots , a _ g ) = \\sum _ { \\substack { k _ 2 , \\ldots , k _ n \\geq 0 \\\\ \\sum i k _ i = g } } \\bar e _ 2 ^ { k _ 2 } \\dots \\bar e _ n ^ { k _ n } U _ { k _ 2 , \\ldots , k _ n } . \\end{align*}"}
-{"id": "9538.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ N \\left [ \\begin{matrix} N \\\\ j \\end{matrix} \\right ] \\frac { q ^ { j ^ 2 } ( q ) _ j } { ( z q ) _ j ( z ^ { - 1 } q ) _ j } = \\sum _ { n = 1 } ^ { \\infty } \\sum _ { m = - \\infty } ^ { \\infty } N _ { S _ 1 } ( m , n ) z ^ m q ^ n . \\end{align*}"}
-{"id": "9833.png", "formula": "\\begin{align*} \\limsup _ { \\ell \\to \\infty } N ( 0 ^ + , \\tilde u _ { X _ \\circ , 0 , j _ \\ell } ( X _ \\ell + \\ , \\cdot \\ , ) ) & \\leq \\inf _ { \\rho > 0 } \\limsup _ { \\ell \\to \\infty } N ( \\rho , \\tilde u _ { X _ \\circ , 0 , j _ \\ell } ( X _ \\ell + \\ , \\cdot \\ , ) ) \\\\ & = \\inf _ { \\rho > 0 } N ( \\rho , \\tilde u _ { X _ \\circ , 0 } ^ { ( \\infty ) } ( X _ \\infty + \\ , \\cdot \\ , ) ) \\\\ & = N ( 0 ^ + , \\tilde u _ { X _ \\circ , 0 } ^ { ( \\infty ) } ( X _ \\infty + \\ , \\cdot \\ , ) ) . \\end{align*}"}
-{"id": "2575.png", "formula": "\\begin{align*} \\epsilon _ { t } ( x ) : = \\frac { v _ t ( x ) } { \\langle v _ t , \\phi ^ * \\rangle _ m \\phi ( x ) } - 1 , t > 0 , x \\in E , \\end{align*}"}
-{"id": "3568.png", "formula": "\\begin{align*} \\begin{matrix} \\begin{aligned} W _ { 1 } W _ { 2 } & = V z , & & & W _ { 1 } W _ { 3 } & = V y , & & & W _ { 2 } W _ { 3 } & = V x , \\\\ W _ { 1 } x & = V u , & & & W _ { 2 } y & = V u , & & & W _ { 3 } z & = V u \\end{aligned} \\end{matrix} \\end{align*}"}
-{"id": "640.png", "formula": "\\begin{align*} s ( \\varphi , [ u ] , [ x ] ) = ( \\varphi , [ x ] ) \\ , t ( \\varphi , [ u ] , [ x ] ) = ( \\varphi , [ u x ] ) \\ . \\end{align*}"}
-{"id": "4478.png", "formula": "\\begin{align*} v _ { i j } = \\nabla _ { i j } u + u \\delta _ { i j } \\textrm { a n d } \\tilde { F } ^ { i j } = \\frac { \\partial \\tilde F } { \\partial v _ { i j } } \\big ( \\{ v _ { i j } \\} \\big ) . \\end{align*}"}
-{"id": "122.png", "formula": "\\begin{align*} 0 \\le \\sum _ { j = 0 } ^ { 2 N - 1 } \\zeta _ { j , N } & \\le { \\frac 1 N } \\sum _ { j = 0 } ^ { 2 N - 1 } \\sum _ { { \\ell = 1 } } ^ { \\infty } \\frac { d ^ { 2 \\ell + 1 } } { ( \\ell ! ) ^ 2 } { ( 1 - x _ j ^ 2 ) } ^ \\ell \\\\ & = \\sum _ { \\ell = 1 } ^ { \\infty } \\frac { d ^ { 2 \\ell + 1 } } { ( \\ell ! ) ^ 2 } \\left \\{ \\frac 1 N \\sum _ { j = 0 } ^ { 2 N - 1 } { ( 1 - x _ j ^ 2 ) } ^ \\ell \\right \\} \\end{align*}"}
-{"id": "5782.png", "formula": "\\begin{align*} A ( \\Gamma , q ^ { - 1 } , T ^ { - 1 } ) = \\epsilon _ 1 ( \\Gamma ) + ( - 1 ) ^ { \\# V ( \\Gamma ) } A ( \\Gamma , q , T ) , \\end{align*}"}
-{"id": "863.png", "formula": "\\begin{gather*} \\overline { B } ^ T v ^ \\ast = \\left ( u ^ \\ast { v ^ \\ast } ^ T + u v ^ T - ( { u ^ \\ast } ^ T u ) u { v ^ \\ast } ^ T \\right ) v ^ \\ast = \\norm { v ^ \\ast } _ 2 ^ 2 u ^ \\ast + \\left ( v ^ T v ^ \\ast - ( { u ^ \\ast } ^ T u ) \\norm { v ^ \\ast } _ 2 ^ 2 \\right ) u = u ^ \\ast , \\\\ \\overline { B } u = ( v ^ \\ast { u ^ \\ast } ^ T + v u ^ T - ( { u ^ \\ast } ^ T u ) v ^ \\ast u ^ T ) u = ( { u ^ \\ast } ^ T u ) v ^ \\ast + \\norm { u } _ 2 ^ 2 v - ( { u ^ \\ast } ^ T u ) \\norm { u } _ 2 ^ 2 v ^ \\ast = v . \\end{gather*}"}
-{"id": "3618.png", "formula": "\\begin{align*} a _ 1 \\cdot ( ( \\alpha _ 0 - \\beta _ 0 ) \\cdot \\alpha _ 1 ) = \\frac { 8 } { 3 } a _ 1 \\cdot ( ( a _ 2 & + a _ { - 2 } ) \\cdot v _ { ( 1 , 3 ) } ) + \\frac { 1 } { 2 ^ 5 } ( a _ 1 + a _ { - 1 } ) + \\frac { 1 } { 2 ^ 4 \\cdot 3 } ( a _ 2 + a _ { - 2 } ) \\\\ & + \\frac { 1 } { 2 ^ 3 \\cdot 3 } ( a _ 3 + a _ { - 3 } ) - \\frac { 1 } { 2 ^ 5 } ( v _ { ( 1 , 2 ) } + 2 v _ { ( 1 , 3 ) } ) + a _ 1 \\cdot v _ { ( 2 , 3 ) } . \\end{align*}"}
-{"id": "7781.png", "formula": "\\begin{align*} M ( \\xi ) = \\int _ { G } \\langle x , \\xi \\rangle x { \\rm d } x = \\bigg ( \\int _ { G } \\langle x , \\xi \\rangle x _ 1 { \\rm d } x , \\ldots , \\int _ { G } \\langle x , \\xi \\rangle x _ d { \\rm d } x \\bigg ) \\end{align*}"}
-{"id": "3166.png", "formula": "\\begin{align*} U = \\varGamma C . \\end{align*}"}
-{"id": "4193.png", "formula": "\\begin{align*} - \\frac { A _ n ^ { ( 1 ) } ( 0 ) A _ n ^ { ( 1 ) } ( z ) A _ n ^ { ( 1 ) } ( 0 ) } { n ^ 9 z ^ 3 } = \\frac { A _ n ^ { ( 3 ) } ( z ) } { n ^ 9 z ^ 3 } + \\mathcal { O } \\left ( n ^ { - \\frac { 3 } { 2 } } \\right ) \\end{align*}"}
-{"id": "1494.png", "formula": "\\begin{align*} \\widetilde { M } _ { n } = \\sup _ { t \\in [ 0 , T ] } \\left ( \\| D w ^ { n } \\| _ { s - 1 } ^ { 2 } + \\| \\mu ^ { n } \\| _ { s } ^ { 2 } \\right ) , \\end{align*}"}
-{"id": "7883.png", "formula": "\\begin{align*} | \\mathfrak m _ N ( \\xi ) | \\le 1 6 e ^ { - \\frac { c \\kappa ( d , N ) ^ 2 } { 4 0 0 } \\sum _ { i = 1 } ^ d \\sin ^ 2 ( \\pi \\xi _ i ) } , \\end{align*}"}
-{"id": "5805.png", "formula": "\\begin{align*} f \\circ g \\ & = \\ ( f ^ { \\uparrow 3 } \\circ \\ell _ 3 ) \\circ g \\ = \\ f ^ { \\uparrow 3 } \\circ ( \\ell _ 3 \\circ g ) \\ = \\ f ^ { \\uparrow 3 } \\circ ( \\ell _ \\gamma + \\epsilon ) , \\\\ f \\ast g \\ & = \\ ( f ^ { \\uparrow 3 } \\ast \\ell _ 3 ) \\ast g \\ = \\ f ^ { \\uparrow 3 } \\ast ( \\ell _ 3 \\ast g ) \\ = \\ f ^ { \\uparrow 3 } \\ast ( \\ell _ { \\gamma } + \\epsilon ) , \\end{align*}"}
-{"id": "8773.png", "formula": "\\begin{align*} \\phi = \\frac { 1 } { ( n - 3 ) ( \\mu _ 2 - \\mu _ 1 ) } \\ , , \\ \\ \\mu = - \\frac { \\mu _ 1 } { ( n - 3 ) ( \\mu _ 2 - \\mu _ 1 ) } \\ , , \\ \\ \\eta = \\rho _ 1 + \\frac { \\mu _ 1 ^ 2 } { ( n - 3 ) ( \\mu _ 2 - \\mu _ 1 ) } \\ , . \\end{align*}"}
-{"id": "88.png", "formula": "\\begin{align*} \\det ( B ) = - \\left ( 1 - 2 c _ 1 \\right ) \\cdots \\left ( 1 - 2 c _ { N - 1 } \\right ) \\ , . \\end{align*}"}
-{"id": "1837.png", "formula": "\\begin{align*} - u _ { x x } ( x ) + 2 \\xi \\ , \\delta ( x ) u ( x ) = \\lambda u ( x ) . \\end{align*}"}
-{"id": "2717.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\mathbb { P } \\Big ( 2 ^ { - \\frac { 4 } { 3 } } N ^ { - \\frac { 1 } { 3 } } \\big ( \\lambda _ { \\mathrm { m a x } } - 4 N \\big ) \\leq x \\Big ) = F _ m ( \\pi ; x ) . \\end{align*}"}
-{"id": "4703.png", "formula": "\\begin{align*} u ( x , 0 ) = \\sin ( x ) , \\ ; \\ ; \\ ; x \\in [ 0 , 2 \\pi ] \\end{align*}"}
-{"id": "2054.png", "formula": "\\begin{align*} V ( y ) : = \\frac { 1 } { 2 \\pi i } \\int _ { ( 2 ) } ( 2 \\pi y ) ^ { u } G ( u ) \\Gamma ( u ) d u . \\end{align*}"}
-{"id": "3358.png", "formula": "\\begin{align*} \\mathsf { Q } ( l ) = \\begin{cases} \\left \\lfloor \\textsf { c l i p } ( l ) / \\Delta + \\frac { 1 } { 2 } \\right \\rfloor , & l > \\frac { \\Delta } { 2 } \\\\ \\left \\lceil \\textsf { c l i p } ( l ) / \\Delta - \\frac { 1 } { 2 } \\right \\rceil , & l < - \\frac { \\Delta } { 2 } \\\\ 0 , & . \\end{cases} \\end{align*}"}
-{"id": "5757.png", "formula": "\\begin{align*} ( u + \\epsilon v ) ( x + \\epsilon y ) = u ( x ) + \\epsilon ( u ( y ) + v ( x ) ) . \\end{align*}"}
-{"id": "2001.png", "formula": "\\begin{align*} I ( q , z ) : = z \\sum _ { d \\geq 0 } \\frac { q ^ d } { z ^ d d ! } \\left ( \\prod _ { 0 \\leq k < \\frac { d + 1 } { d } \\atop \\langle k \\rangle = \\langle \\frac { d + 1 } { 5 } \\rangle } ( k z ) ^ 5 \\right ) \\phi _ { d + 1 } . \\end{align*}"}
-{"id": "7555.png", "formula": "\\begin{align*} T f ( x ) = \\int _ { \\mathbb { R } ^ d } K ( x , y ) f ( y ) d y \\end{align*}"}
-{"id": "6627.png", "formula": "\\begin{align*} | T ' | & = 2 ^ { v - m - 2 \\nu + 2 } \\sum _ { j \\ , { \\rm o d d } } \\binom { \\nu } { j } 3 ^ { \\nu - j } \\\\ & = 2 ^ { v - m - 2 \\nu + 2 } \\cdot \\frac { 1 } { 2 } \\big ( ( 3 + 1 ) ^ \\nu - ( 3 - 1 ) ^ \\nu \\big ) \\\\ & = 2 ^ { t + 2 } - 2 ^ { t + 2 - \\nu } ~ ~ ~ ~ \\hbox { ( a s $ t = v - m - 1 $ ) . } \\end{align*}"}
-{"id": "9065.png", "formula": "\\begin{align*} X ( \\mu _ { f } ) = \\frac { 1 } { \\pi } \\int _ { B ( 1 ) } | F ^ R _ { \\mu _ { f } } | ^ 2 d x \\end{align*}"}
-{"id": "3085.png", "formula": "\\begin{align*} b ( x ) = 0 , x \\in G , \\int _ { K _ l } b = 0 , l = 1 , 2 , \\cdots . \\end{align*}"}
-{"id": "2397.png", "formula": "\\begin{align*} ( X _ A ) ^ * & = X _ { A ^ * } \\\\ \\Delta ( X _ A ) & = \\sum _ { M _ 1 + M _ 2 = [ 1 , k ] } X _ { A | M _ 1 } \\otimes X _ { A | M _ 2 } \\\\ \\Delta ( X _ A ) ^ * & = \\sum _ { M _ 1 + M _ 2 = [ 1 , k ] } X _ { A ^ * | M _ 1 } \\otimes X _ { A ^ * | M _ 2 } = \\sum _ { M _ 1 + M _ 2 = [ 1 , k ] } ( X _ { A | M _ 1 } ) ^ * \\otimes ( X _ { A | M _ 2 } ) ^ * . \\end{align*}"}
-{"id": "2998.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\sigma ' _ m ( n ) q ^ n & = \\sum _ { \\{ n | n \\equiv 0 m , n \\equiv 1 m , n \\equiv m - 1 m \\} } \\frac { n q ^ n } { 1 - q ^ n } . \\end{align*}"}
-{"id": "7114.png", "formula": "\\begin{align*} x = x ^ { M } + \\frac { \\mu + 1 } { 2 } \\tilde { g } & = \\Psi _ { m } \\left ( \\begin{pmatrix} 1 & 0 \\\\ 0 & - \\mu \\end{pmatrix} + \\frac { \\mu + 1 } { 1 - \\mu | \\psi | ^ { 2 } } \\begin{pmatrix} | \\psi | ^ { 2 } & \\psi \\\\ \\bar { \\psi } & 1 \\end{pmatrix} \\right ) \\Psi ^ { * } _ { m } \\\\ & = \\frac { 1 } { 1 - \\mu | \\psi | ^ { 2 } } \\Psi _ { m } \\begin{pmatrix} 1 + | \\psi | ^ { 2 } & ( \\mu + 1 ) \\psi \\\\ ( \\mu + 1 ) \\bar { \\psi } & 1 + \\mu ^ { 2 } | \\psi | ^ { 2 } \\end{pmatrix} \\Psi _ { m } ^ { * } . \\end{align*}"}
-{"id": "9718.png", "formula": "\\begin{align*} \\Lambda ( \\tilde { u } ) : = \\{ ( x , 0 ) : \\tilde { u } ( x , 0 ) = 0 \\} = \\Lambda ( u ) . \\end{align*}"}
-{"id": "8851.png", "formula": "\\begin{align*} s _ 3 & \\ge s _ 3 | _ { a = 1 } = - 2 M ^ 3 \\mu w ^ 2 + M ^ 3 \\mu w + 4 { M } ^ 2 \\mu { w } ^ 2 - 3 { M } ^ 3 \\mu + { M } ^ 3 w - { M } ^ 2 \\mu w + { M } ^ 2 { w } ^ 2 \\\\ & - 2 M \\mu w ^ 2 + 3 M ^ 3 - M ^ 2 \\mu - 5 M ^ 2 w - 2 M w ^ 2 + 2 M ^ 2 + 2 M \\mu + 4 M w + w ^ 2 - 2 M - 1 = : s _ 4 \\end{align*}"}
-{"id": "1333.png", "formula": "\\begin{align*} f ^ * ( x ) = \\overline { f ( x ^ { - 1 } ) } \\end{align*}"}
-{"id": "228.png", "formula": "\\begin{align*} \\begin{cases} \\sum _ { \\beta } \\Phi _ \\beta ( r ) = 1 , r > 0 \\\\ \\operatorname { s u p p } \\Phi _ \\beta \\subset \\{ r : 2 ^ { \\beta - 2 } \\leq r \\leq 2 ^ \\beta \\} , \\ \\beta = 1 , 2 , \\dots , m - 1 \\\\ \\operatorname { s u p p } \\Phi _ 0 \\subset \\{ r : 0 < r \\leq 1 \\} \\\\ \\operatorname { s u p p } \\Phi _ m \\subset \\{ r : r \\geq \\frac { s } { 4 0 } \\} \\end{cases} \\end{align*}"}
-{"id": "2777.png", "formula": "\\begin{align*} K ( \\mathbf { k } , \\mathbf { n } ) : = \\mathbb { P } ( \\mathbf { N } ^ { t + 1 } = \\mathbf { n } | \\mathbf { N } ^ { t } = \\mathbf { k } ) , \\end{align*}"}
-{"id": "3181.png", "formula": "\\begin{align*} { \\rm i n d } _ { \\varGamma ^ \\prime } ( U ^ \\prime ) & = ( M _ - ^ \\prime - m _ - ^ \\prime ) - ( M _ + ^ \\prime - m _ + ^ \\prime ) \\\\ & = ( M _ - - m _ - ) - ( M _ + - m _ + ) = { \\rm i n d } _ \\varGamma ( U ) . \\end{align*}"}
-{"id": "4158.png", "formula": "\\begin{align*} \\overline { f ( x ) } & = \\frac { 8 } { 7 2 9 } \\left ( \\omega \\overline { \\varphi _ { 1 + } ( x ) } - \\overline { \\varphi _ { 2 + } ( x ) } \\right ) ^ 3 = \\frac { 8 } { 7 2 9 } \\left ( \\omega \\varphi _ { 1 - } ( x ) - \\varphi _ { 2 - } ( x ) \\right ) ^ 3 = f ( x ) . \\end{align*}"}
-{"id": "9120.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } S ( \\rho _ { 1 } ^ { ( N ) } ( t ) ) = 0 \\end{align*}"}
-{"id": "1074.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } n ( 2 \\ln n ) ^ { - \\frac { 1 } { 2 } } D _ n ( ( 1 + z / \\ln n ) G _ n ( x ) / 2 ) = e ^ { c _ 0 - x - 2 z } , \\end{align*}"}
-{"id": "4088.png", "formula": "\\begin{align*} T _ { \\alpha } = \\begin{pmatrix} 1 & 0 & 0 \\\\ t _ 1 & - 1 & 0 \\\\ t _ 2 & t _ 3 & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "4157.png", "formula": "\\begin{align*} f ( z ) & = \\frac { 8 } { 7 2 9 } z f _ 1 ^ 3 ( z ) \\\\ & = \\frac { 8 } { 7 2 9 } \\times \\begin{cases} \\left ( \\omega ^ 2 \\varphi _ 1 ( z ) - \\varphi _ 2 ( z ) \\right ) ^ 3 , & \\Im z > 0 , \\\\ [ 5 p t ] \\left ( \\omega \\varphi _ 1 ( z ) - \\varphi _ 2 ( z ) \\right ) ^ 3 , & \\Im z < 0 . \\end{cases} \\end{align*}"}
-{"id": "729.png", "formula": "\\begin{align*} \\mathbb { E } _ { f } \\ ! \\left [ T \\right ] = \\frac { 1 } { q - 1 } \\left ( 1 - \\frac { \\Gamma ( m n ) } { \\Gamma ( m n + q ) } \\mathbb { E } _ { g } \\ ! \\left [ L \\right ] \\right ) \\end{align*}"}
-{"id": "7509.png", "formula": "\\begin{align*} T _ { t } u ( x ) : = \\left \\{ \\begin{array} { l l } u ( x ) & t = 0 \\\\ & \\\\ \\int _ { \\mathbf { \\mathbb { Q } } _ { p } ^ { n } } Z _ { t } ( x - y ) u ( y ) d ^ { n } y = ( Z _ { t } \\ast u ) ( x ) & t > 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "1184.png", "formula": "\\begin{align*} u _ t + u u _ x = 0 ( x \\in \\mathbb { T } , t > 0 ) . \\end{align*}"}
-{"id": "9621.png", "formula": "\\begin{align*} V ( q , N + 1 ) - V ( q , N ) = \\frac { q ^ { N + 1 } } { 2 } \\sum _ { n = 0 } ^ { \\infty } q ^ n \\left \\{ ( - q ^ { n + 1 } ) _ N + ( q ^ { n + 1 } ) _ N \\right \\} . \\end{align*}"}
-{"id": "1277.png", "formula": "\\begin{align*} \\mathcal { I } = \\int _ 0 ^ 1 \\tilde I ( S ) ( \\eta ) d \\eta + \\int _ 1 ^ \\infty \\left ( \\tilde I ( S ) ( \\eta ) - E \\frac { e ^ { i a \\ln | \\eta | } } { | \\eta | } \\right ) d \\eta . \\end{align*}"}
-{"id": "2191.png", "formula": "\\begin{align*} \\boldsymbol { { \\lambda } } ^ \\ast ( t ) = P ( t ) \\boldsymbol { x } ^ \\ast ( t ) \\end{align*}"}
-{"id": "421.png", "formula": "\\begin{align*} A _ 3 = \\frac { 2 } { 1 - \\frac { \\gamma ^ 2 } { 4 } } \\frac { \\mathcal { D } _ { g } ^ \\gamma ( M ) } { \\mathcal { G } _ { g } ^ \\gamma ( M ) } + \\frac { \\gamma ^ 2 - \\gamma Q } { ( 1 - \\frac { \\gamma ^ 2 } { 4 } ) } \\frac { 1 } { \\mathcal { G } _ { g } ^ \\gamma ( M ) } \\int _ M \\omega ( x ) \\mathcal { G } _ { g } ^ \\gamma ( \\dd x ) . \\end{align*}"}
-{"id": "9142.png", "formula": "\\begin{align*} \\mathbf { v } _ { 1 } ^ { ( + ) } = \\mathbf { v } _ { 1 } ^ { ( - ) } - 2 \\mathbf { n } _ { 1 } \\mathbf { n } _ { 1 } \\cdot \\mathbf { v } _ { 1 } ^ { ( - ) } . \\end{align*}"}
-{"id": "7356.png", "formula": "\\begin{align*} 2 d ( Y , Y ' ) = d ( x , x ' ) + d ( y , y ' ) - d ( x , y ) - d ( x ' , y ' ) \\end{align*}"}
-{"id": "7176.png", "formula": "\\begin{gather*} T _ { x } \\mathbb { R } ^ { n } = T _ { x } \\mathcal { M } \\oplus \\mathbb { J } _ { x } ^ { - } , T _ { x } \\mathcal { M } = \\mathbb { K } _ { x } ^ { - } \\oplus \\mathbb { L } _ { x } ^ { + } \\oplus \\mathbb { L } _ { x } ^ { 0 } . \\end{gather*}"}
-{"id": "9557.png", "formula": "\\begin{align*} { } _ { 2 } \\phi _ { 1 } \\left [ \\begin{matrix} a , b \\\\ c \\end{matrix} \\ , ; q , z \\right ] = \\frac { ( b , a z ; q ) _ { \\infty } } { ( c , z ; q ) _ { \\infty } } { } _ { 2 } \\phi _ { 1 } \\left [ \\begin{matrix} \\frac { c } { b } , z \\\\ a z \\end{matrix} \\ , ; q , b \\right ] , \\end{align*}"}
-{"id": "5888.png", "formula": "\\begin{align*} C ' _ w = q ^ { - \\frac { \\ell ( w ) } { 2 } } \\left ( T _ w + T _ { r s _ j } + T _ { s _ i r } + \\cdots \\right ) = q ^ { - \\frac { \\ell ( w ) } { 2 } } \\left ( T _ w + T _ { r s _ j } + q ^ { \\frac { \\ell ( s _ i r ) } { 2 } } C ' _ { s _ i r } \\right ) \\end{align*}"}
-{"id": "6097.png", "formula": "\\begin{gather*} \\Big ( \\widetilde { H } _ { p - 2 } ( \\widehat { 0 } , \\lambda ) \\Big ) ^ { \\mathfrak { S } _ n } = 0 \\lambda \\in \\P _ n , l ( \\lambda ) \\geq 2 . \\end{gather*}"}
-{"id": "6512.png", "formula": "\\begin{align*} \\| \\Psi ^ { n } ( d d ^ { c } \\varphi ) \\| _ U & = \\int _ U \\left ( \\Psi ^ { n } ( d d ^ { c } \\varphi ) \\right ) \\wedge \\omega ^ { k - 1 } = \\int _ U \\frac { 1 } { N ^ { n } } \\left ( ( g ^ { n } ) _ * ( d d ^ { c } \\varphi ) \\right ) \\wedge \\omega ^ { k - 1 } \\\\ & = \\frac { 1 } { N ^ { n } } \\int _ U d d ^ { c } \\varphi \\wedge ( g ^ { n } ) ^ { * } \\omega ^ { k - 1 } \\\\ & = \\frac { 1 } { N ^ { n } } \\int _ U d d ^ { c } \\varphi \\wedge ( d ^ { m n } \\omega + d d ^ { c } u _ { n m } ) ^ { k - 1 } \\end{align*}"}
-{"id": "1776.png", "formula": "\\begin{align*} D F = \\frac { \\alpha } { 2 } e ^ { \\frac { \\alpha } { 2 } M _ { s , t } ^ { ( n ) } } D M _ { s , t } ^ { ( n ) } \\end{align*}"}
-{"id": "5257.png", "formula": "\\begin{align*} c = 1 - \\int _ { x _ 0 } ^ { u e p ( F ) } 1 - F ( t ) d t > 0 . \\end{align*}"}
-{"id": "3781.png", "formula": "\\begin{align*} & \\sum _ { j \\geq 0 } q _ j = | \\{ i : \\ p _ i \\geq 0 \\} | = R - k + 1 \\hbox { a n d } \\\\ & \\sum _ { j \\geq 0 } q ' _ j = R - k + 1 - \\chi ( p ' _ 1 = - 1 ) - \\chi ( p ' _ { R - k + 1 } = - 1 ) = R - k - 1 + \\delta _ \\alpha . \\end{align*}"}
-{"id": "2906.png", "formula": "\\begin{align*} \\Psi _ \\mu ( A \\cup B \\ , | \\ , C ) & = \\Psi _ \\mu ( B \\ , | \\ , C ) + \\Psi _ \\mu ( A \\ , | \\ , B \\cup C ) \\ ; . \\end{align*}"}
-{"id": "7022.png", "formula": "\\begin{align*} y ^ { 2 } = x ^ { 3 } \\end{align*}"}
-{"id": "7086.png", "formula": "\\begin{align*} \\widehat { f } ( \\xi ) = \\sum _ { k \\in \\mathbb { Z } ^ d } \\varphi ( \\xi - k ) \\widehat { f } ( \\xi ) ~ . \\end{align*}"}
-{"id": "4950.png", "formula": "\\begin{align*} & f _ { d - k } = v _ { d - 3 - k } ^ n + v _ { d - 3 - k } z ^ { t - n } \\\\ & - ( v _ { d - 3 - k + 1 } - v _ { d - 3 - k + 2 } + \\cdots + ( - 1 ) ^ { d + d - 3 - k } ( v _ { d - 3 } ) + ( - 1 ) ^ { d + d - 3 - k + 1 } x ) ^ n + v _ { d - 3 - k } z ^ n - v _ { d - 3 - k } x ^ n . \\end{align*}"}
-{"id": "8517.png", "formula": "\\begin{align*} ( I + \\bar { \\mathcal { H } } ) ) \\bar { \\mathfrak { F } } = 2 \\bar { \\mathfrak { F } } , ( I + \\bar { \\mathcal { H } } ) \\bar { q } = 0 . \\end{align*}"}
-{"id": "9104.png", "formula": "\\begin{align*} ( - \\Delta _ p ) ^ s \\Upsilon ( x ) = I _ 1 + I _ 2 + I _ 3 + I _ 4 \\le I _ 4 + \\dfrac { C } { | x | ^ { \\alpha ( p - 1 ) + p s } } . \\end{align*}"}
-{"id": "9388.png", "formula": "\\begin{align*} \\varphi _ { 1 } ( t w _ { 0 } ) & = \\int _ { \\mathbb { R } ^ N } \\int _ { \\mathbb { R } ^ N } \\Phi ( t w _ { 0 } ( x ) - t w _ { 0 } ( y ) ) ( t w _ { 0 } ( x ) - t w _ { 0 } ( y ) ) K ( x , y ) d x d y - \\int _ \\Omega | t w _ { 0 } | ^ { p ^ { \\ast } _ { s } } d x \\\\ & - \\lambda \\int _ { \\Omega } f ( x , t w _ { 0 } ) t w _ { 0 } d x \\end{align*}"}
-{"id": "2351.png", "formula": "\\begin{align*} \\varrho & = S _ - \\setminus ( S _ + + \\sigma ) , & \\pi & = S _ + + \\sigma , & n & = \\# \\sigma . \\end{align*}"}
-{"id": "8243.png", "formula": "\\begin{align*} \\frac { F ' ( z ) } { F ( z ) } = \\sum _ { n = 0 } ^ \\infty \\left ( \\frac { - q ^ { 2 n + 1 } } { 1 - z q ^ { 2 n + 1 } } + \\frac { z ^ { - 2 } q ^ { 2 n + 1 } } { 1 - z ^ { - 1 } q ^ { 2 n + 1 } } \\right ) . \\end{align*}"}
-{"id": "6921.png", "formula": "\\begin{align*} | \\langle Z _ m , W ' _ { i } \\rangle - { \\bf 1 } _ { \\{ i = q \\} } | \\leq \\sqrt { 2 } ( m + 1 ) \\delta ' + \\sqrt { 2 } q \\delta ' \\quad \\ , i \\ge m \\end{align*}"}
-{"id": "1046.png", "formula": "\\begin{align*} & \\prod _ { j = 1 } ^ k \\mathbb { P } ( \\xi ^ { ( n ) } ( I ( y _ j , F _ n ( x _ j ) ) ) = 0 ) = \\prod _ { j = 1 } ^ k \\det ( - B _ j ) = \\det ( - B ) . \\end{align*}"}
-{"id": "2019.png", "formula": "\\begin{align*} T _ j ( - z ) : = F ( j ) \\cdot \\left ( \\right ) . \\end{align*}"}
-{"id": "7343.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\dot { t } = \\frac { \\partial \\tilde p } { \\partial \\tau } = - \\tau , \\\\ & \\dot { x } = \\frac { \\partial \\tilde p } { \\partial \\xi } = \\frac { K ( x ) } { \\rho ( x ) } \\xi , \\\\ & \\dot { \\tau } = - \\frac { \\partial \\tilde p } { \\partial t } = 0 , \\\\ & \\dot { \\xi } = - \\frac { \\partial \\tilde p } { \\partial x } = - \\frac 1 2 \\nabla \\left ( \\frac { K ( x ) } { \\rho ( x ) } \\right ) ( \\xi \\cdot \\xi ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "3326.png", "formula": "\\begin{align*} w ( T ) = \\prod _ { b \\in T } w _ T ( b ) , \\end{align*}"}
-{"id": "5821.png", "formula": "\\begin{align*} \\psi ( t ) = \\psi ( \\frac { t } { 2 ^ n } ) ^ { 2 ^ n } \\end{align*}"}
-{"id": "6519.png", "formula": "\\begin{align*} d ^ { n q + n _ j } ( r _ n ) ^ * T _ { f , a _ j } & = d d ^ c G _ { r _ n ( x ) } \\left ( \\sigma _ j \\circ a ^ { n , j } ( x ) \\right ) , \\end{align*}"}
-{"id": "2052.png", "formula": "\\begin{align*} \\widetilde { f ^ \\chi } _ { R ' } ( z ) = \\lambda _ { R _ * ' \\frac { R } { R _ * } } ( F _ \\chi ) \\sum _ { \\ell \\mid r _ { * 0 } } \\beta _ { F _ \\chi } ( \\ell ) \\ell ^ { - \\frac { k } { 2 } } \\left ( Q _ * \\frac { r _ { * 0 } } { \\ell } \\right ) ^ { \\frac { k } { 2 } } \\sum _ { n = 1 } ^ \\infty \\tilde { a } _ { \\chi , R _ * ' \\frac { R } { R _ * } } ( n ) e ^ { 2 \\pi i n Q _ * \\frac { r _ { * 0 } } { \\ell } z } . \\end{align*}"}
-{"id": "6323.png", "formula": "\\begin{align*} q _ x ( 0 , t ) = q _ x ( 1 , t ) = 0 \\quad t > 0 , \\end{align*}"}
-{"id": "1358.png", "formula": "\\begin{align*} C _ n ( \\Lambda , \\mathcal { N } ) : = \\bigoplus _ { ( \\lambda _ 0 , \\ldots , \\lambda _ { n - 1 } ) \\in \\Lambda ^ { * n } } \\mathcal { N } ( s ( \\lambda _ { n - 1 } ) ) \\ ; , \\end{align*}"}
-{"id": "9572.png", "formula": "\\begin{align*} & \\sum _ { k = 0 \\atop k \\neq m } ^ { n } \\left [ \\begin{matrix} n \\\\ k \\end{matrix} \\right ] \\frac { ( q / x ) _ k ( x ) _ { n - k } } { 1 - q ^ { k - m } } x ^ k \\\\ & = ( - 1 ) ^ m q ^ { \\frac { m ( m + 1 ) } { 2 } } \\left [ \\begin{matrix} n \\\\ m \\end{matrix} \\right ] ( x q ^ { - m } ) _ n \\left ( \\sum _ { k = 0 } ^ { n - 1 } \\frac { x q ^ { k - m } } { 1 - x q ^ { k - m } } - \\sum _ { k = 0 \\atop k \\neq m } ^ { n } \\frac { q ^ { k - m } } { 1 - q ^ { k - m } } \\right ) . \\end{align*}"}
-{"id": "6195.png", "formula": "\\begin{align*} H ^ 2 ( r ) N ' ( r ) = E ' ( r ) H ( r ) - E ( r ) H ' ( r ) r \\in ( r _ 1 , r _ 2 ) . \\end{align*}"}
-{"id": "3067.png", "formula": "\\begin{align*} T f ( x ) = K * f ( x ) , K ( x ) = \\frac { x _ j } { | x | ^ { n + 1 - \\beta } } , 0 < \\beta < n \\end{align*}"}
-{"id": "6537.png", "formula": "\\begin{align*} E ( L ) : = \\left \\{ P \\in E \\ ; \\big { | } \\ ; L | _ E \\sim w P \\right \\} \\end{align*}"}
-{"id": "6316.png", "formula": "\\begin{align*} \\lim _ { n } \\frac { k } { n } \\mathbb { E } _ { G _ 0 \\sim \\mathbb { P } _ 1 } [ | n - 2 d ^ { G _ 0 } ( v ) | \\bigg { | } \\mathcal { E } ] = 0 . \\end{align*}"}
-{"id": "995.png", "formula": "\\begin{align*} T : = \\lceil \\gamma \\cdot \\zeta ^ { \\frac { 2 } { d - 2 } } \\rceil , \\end{align*}"}
-{"id": "9003.png", "formula": "\\begin{align*} \\overline { D } _ { t } E _ { i } = D _ { t } E _ { i } + \\tau _ { i } W _ { m + 1 } \\ , , \\end{align*}"}
-{"id": "1081.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } \\int _ I D _ n ( \\sqrt { 4 - y ^ 2 } / S ( I ) \\cdot G _ n ( x ) / 2 ) d y = M ( I ) e ^ { c _ 0 - x } , \\end{align*}"}
-{"id": "6568.png", "formula": "\\begin{gather*} \\hat \\theta _ \\varnothing : = \\d u - p _ a \\d x ^ a , \\hat \\theta _ a : = \\d p _ a - p _ { a b } \\d x ^ b , \\d x ^ a , \\qquad \\d p _ { a b } . \\end{gather*}"}
-{"id": "3723.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\int _ { \\Omega } \\left ( \\int _ { \\Omega } \\frac { | F ( y , u _ k ) - F ( y , u ) | } { | x - y | ^ { \\mu } } d y \\right ) | F ( x , u _ k ) - F ( x , u ) | ~ d x = 0 . \\end{align*}"}
-{"id": "3349.png", "formula": "\\begin{align*} F _ { \\chi } ( u ) : = \\prod _ { d \\ge 1 } F _ d ( u ^ d ) \\end{align*}"}
-{"id": "6353.png", "formula": "\\begin{align*} \\mathbb { P } _ { e , M | p , h } = & \\frac { 1 } { 2 } Q \\left ( \\frac { R P _ t \\sqrt { T _ s } \\left ( g _ { 1 1 } h _ { 1 1 } + g _ { 2 1 } h _ { 2 1 } \\right ) } { \\sqrt { N _ 0 } } \\right ) \\\\ + & \\frac { 1 } { 2 } Q \\left ( \\frac { R P _ t \\sqrt { T _ s } \\left ( g _ { 1 1 } h _ { 1 1 } - g _ { 2 1 } h _ { 2 1 } \\right ) } { \\sqrt { N _ 0 } } \\right ) . \\end{align*}"}
-{"id": "8123.png", "formula": "\\begin{align*} \\ell _ V ( \\Gamma , t ) \\ = \\sum _ { F \\subseteq V } \\ , ( - 1 ) ^ { n - | F | } \\ , h ( \\Gamma _ F , t ) , \\end{align*}"}
-{"id": "2362.png", "formula": "\\begin{align*} f = & \\sum \\frac { 1 } { n ! } f _ n \\Phi _ n , & g & = \\sum \\frac { 1 } { n ! } g _ n \\Phi _ n \\end{align*}"}
-{"id": "2986.png", "formula": "\\begin{align*} ( q ; q ^ 4 ) _ \\infty ( q ^ 3 ; q ^ 4 ) _ \\infty ( q ^ 4 ; q ^ 4 ) _ \\infty & = \\Big ( ( q ; q ) _ \\infty \\Big ) \\Big ( \\frac { 1 } { ( q ^ 2 ; q ^ 4 ) _ \\infty } \\Big ) . \\end{align*}"}
-{"id": "9554.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( - 1 / a ) _ n ( a c ) ^ n q ^ { n ( n + 1 ) / 2 } } { ( c q ) _ n } = \\frac { ( - a c q ) _ { N } } { ( c q ) _ N } . \\end{align*}"}
-{"id": "7297.png", "formula": "\\begin{align*} \\psi _ q ( a ) \\xi _ j = \\psi _ q ( a ) h _ { p _ n } ^ + \\xi _ j = \\psi _ { p _ n } ( a ) \\xi _ j \\end{align*}"}
-{"id": "7786.png", "formula": "\\begin{align*} L ( r _ d B ^ 2 ) ^ 2 = \\frac { 1 } { d } \\int _ { r _ d B ^ 2 } | x | ^ 2 { \\rm d } x = \\frac { | B ^ 2 | r _ d ^ { d + 2 } } { d + 2 } = \\frac { r _ d ^ { 2 } } { d + 2 } \\simeq 1 . \\end{align*}"}
-{"id": "9938.png", "formula": "\\begin{align*} \\Delta = \\frac { 4 \\pi ^ 2 } { T ^ 2 } \\left ( A + \\frac { \\beta } { 4 } \\right ) ^ 2 + Q _ 0 . \\end{align*}"}
-{"id": "694.png", "formula": "\\begin{align*} g = F \\circ X + \\frac { K } { p } L o g \\left ( \\frac { 1 } { 1 - 4 p \\pi _ 1 ^ \\mathbb { R } } \\right ) . \\end{align*}"}
-{"id": "3687.png", "formula": "\\begin{align*} \\hat { S } _ 1 = \\{ i ; \\min _ { j \\in B } \\| x _ i - x _ j \\| < \\epsilon \\} \\end{align*}"}
-{"id": "784.png", "formula": "\\begin{align*} Z _ { i _ 0 } & = \\left ( 1 - \\prod _ { \\substack { i = 1 \\\\ i \\neq i _ 0 } } ^ { { { s } } } ( \\zeta _ 1 ^ { q ^ { i _ 0 - 1 } } - \\zeta _ 1 ^ { q ^ { i - 1 } } ) \\right ) \\\\ & = \\left ( 1 - \\prod _ { i = 2 } ^ { { { s } } } ( \\zeta _ 1 ^ { q ^ { i _ 0 - 1 } } - \\zeta _ 1 ^ { q ^ { i + i _ 0 - 2 } } ) \\right ) \\\\ & = \\left ( 1 - \\prod _ { i = 2 } ^ { { { s } } } ( \\zeta _ 1 - \\zeta _ 1 ^ { q ^ { i - 1 } } ) \\right ) ^ { q ^ { i _ 0 - 1 } } = Z _ 1 ^ { q ^ { i _ 0 - 1 } } , \\end{align*}"}
-{"id": "7001.png", "formula": "\\begin{align*} \\textsl { \\footnotesize X } _ j ' ( a ) \\ = \\ a \\eta _ a + ( a - 2 ) \\eta _ { a - 2 } + \\cdots + \\left ( a - 2 \\lfloor \\textstyle { \\frac { a } { 2 } } \\rfloor \\right ) \\eta _ { a - 2 \\lfloor \\frac { a } { 2 } \\rfloor } . \\end{align*}"}
-{"id": "5002.png", "formula": "\\begin{align*} \\lefteqn { \\log L ( { \\rm S y m } _ f ^ { \\mu } , \\sigma ) - \\log L ( { \\rm S y m } _ f ^ { \\nu } , \\sigma ) } \\\\ & = - \\sum _ { p \\neq q } \\bigl ( \\log ( 1 - \\alpha _ f ^ { \\mu } ( p ) p ^ { - \\sigma } ) + \\log ( 1 - \\beta _ f ^ { \\mu } ( p ) p ^ { - \\sigma } ) \\bigr ) . \\end{align*}"}
-{"id": "4565.png", "formula": "\\begin{align*} \\int f ( t ) d \\nu ( t ) = \\int f ( | x _ 1 - x _ 2 | , \\dots , | x _ i - x _ j | , \\dots , | x _ { k - 1 } - x _ k | ) d \\mu ( x _ 1 ) \\dots d \\mu ( x _ k ) , \\forall f \\in C _ 0 ( \\mathbb { R } ^ { k ( k - 1 ) / 2 } ) , \\end{align*}"}
-{"id": "2241.png", "formula": "\\begin{align*} M ( x + \\i y ) = & \\mu ( x ) \\delta ( y ) , & \\mu ( x ) = & \\frac { 1 } { 2 \\pi \\i } ( R ( x - \\i 0 ) - R ( x + \\i 0 ) . \\end{align*}"}
-{"id": "8907.png", "formula": "\\begin{align*} \\mathbb { E } _ { P _ 0 } \\frac { P _ { T _ 1 } P _ { T _ 2 } } { P _ 0 ^ 2 } & = \\exp { ( \\log \\Psi ) ( 2 k ( m + k - 2 t ) + 2 ( t - k ) ^ 2 ) } \\\\ & = \\exp { ( \\log \\Psi ) ( 2 m k + 2 k ^ 2 - 4 k t + 2 k ^ 2 + 2 t ^ 2 - 4 k t ) } \\\\ & = \\exp { ( \\log \\Psi ) ( 2 m k + 4 k ^ 2 - 8 k t + 2 t ^ 2 ) } \\\\ & \\le \\exp { ( \\log \\Psi ) ( ( 2 m - 4 t ) k + 2 t ^ 2 ) } , \\end{align*}"}
-{"id": "7542.png", "formula": "\\begin{align*} \\left \\{ \\begin{gathered} X _ 2 ( a , t ) - X _ 1 ( a , t ) = \\int _ 1 ^ 2 \\mathcal { X } _ \\epsilon ' d \\epsilon , \\\\ \\tau _ 2 ( a , t ) - \\tau _ 1 ( a , t ) = \\int _ 1 ^ 2 \\pi _ \\epsilon d \\epsilon , \\\\ v _ 2 ( a , t ) - v _ 1 ( a , t ) = \\int _ 1 ^ 2 \\frac { d } { d \\epsilon } \\mathcal { V } _ \\epsilon d \\epsilon , \\end{gathered} \\right . \\end{align*}"}
-{"id": "5826.png", "formula": "\\begin{align*} \\frac { 1 } { \\phi _ { ( \\bar X _ n , s _ n ^ 2 ) } ( t _ 1 , t 2 ) } \\left . \\frac { \\partial \\phi _ { ( \\bar X _ n , s _ n ^ 2 ) } ( t _ 1 , t _ 2 ) } { \\partial t _ 2 } \\right | _ { t _ 2 = 0 } = \\frac { 1 } { \\phi _ { s _ n ^ 2 } ( t 2 ) } \\left . \\frac { \\partial \\phi _ { s _ n ^ 2 } ( t _ 2 ) } { \\partial t _ 2 } \\right | _ { t _ 2 = 0 } . \\end{align*}"}
-{"id": "3688.png", "formula": "\\begin{align*} \\widehat { Y } = \\sum _ { i \\in \\hat { S } _ 1 } d _ i w _ { 2 i } \\hat { y } _ i \\end{align*}"}
-{"id": "4605.png", "formula": "\\begin{align*} G _ j = \\{ g \\in U ( 2 ) : 2 ^ j \\leq \\mu _ 1 \\times \\mu _ 2 \\{ x _ 1 , x _ 3 : | x _ 1 - g ( x _ 3 ) | \\leq 2 \\delta \\} \\leq 2 ^ { j + 1 } \\} . \\end{align*}"}
-{"id": "7367.png", "formula": "\\begin{align*} d ( b _ i , b _ j ) & = d ( b _ i , e _ u ) + d ( e _ u , e _ v ) + d ( e _ v , b _ j ) \\\\ & = d ( f _ u ( b _ i ) , f _ u ( e _ u ) ) + d ( e _ u , e _ v ) + d ( f _ v ( e _ v ) , f _ v ( b _ j ) ) \\\\ & = d ( f ( b _ i ) , e _ u ) + d ( e _ u , e _ v ) + d ( e _ v , f ( b _ j ) ) \\\\ & = d ( f ( b _ i ) , f ( b _ j ) ) . \\end{align*}"}
-{"id": "6023.png", "formula": "\\begin{align*} \\limsup _ { \\epsilon \\downarrow 0 } \\ , \\ , \\ , \\inf \\left \\{ \\tilde { \\alpha } ^ \\mu _ \\epsilon ( Q ' ) \\ , : \\ , Q ' \\in \\mathcal { P } ( \\C ) , \\ Q ' \\circ H ^ { - 1 } = \\nu _ \\epsilon \\right \\} \\leq \\limsup _ { \\epsilon \\downarrow 0 } \\tilde { \\alpha } ^ \\mu _ \\epsilon ( \\tilde Q _ \\epsilon ) \\leq \\alpha ^ \\mu _ 0 ( Q ) . \\end{align*}"}
-{"id": "2129.png", "formula": "\\begin{align*} d ( x , y ) : = \\sum _ { m \\geq 1 } \\frac { 1 } { 2 ^ m } ( \\| x - y \\| _ { \\alpha , [ - m , m ] } \\wedge 1 ) . \\end{align*}"}
-{"id": "7037.png", "formula": "\\begin{align*} \\mathbb { F } _ { \\alpha _ { j } , \\beta _ { j } , \\lambda _ { j } } ( z ) = \\int \\limits _ { 0 } ^ { z } \\prod _ { j = 1 } ^ { n } \\left ( \\frac { \\mathbb { E } _ { \\alpha _ { j } , \\beta _ { j } } ( t ) } { t } \\right ) ^ { 1 / \\lambda _ { j } } d t , \\end{align*}"}
-{"id": "8282.png", "formula": "\\begin{align*} \\inf _ { \\alpha \\in \\mathbb { R } } a ( \\alpha , 0 ) | z _ { \\alpha } ( \\alpha , t = 0 ) | \\geq \\alpha _ 0 > 0 . \\end{align*}"}
-{"id": "7064.png", "formula": "\\begin{align*} u ( x , i ) = v _ i ( x ) + \\sum _ { j = 1 \\atop { j \\not = i } } ^ m G _ D ^ i ( q _ { i j } u ( \\cdot , j ) ) ( x ) . \\end{align*}"}
-{"id": "3639.png", "formula": "\\begin{align*} \\beta _ { - 1 } \\cdot \\alpha _ { - 1 } = ( a _ 1 \\cdot v _ { ( 2 , 3 ) } ) \\cdot ( a _ 1 \\cdot v _ { ( 2 , 3 ) } ) - \\frac { ( 1 1 t - 1 ) t } { 2 ^ 4 } a _ 1 - \\frac { ( 2 t - 1 ) t } { 2 ^ 4 } a _ { - 1 } + \\frac { t } { 2 ^ 4 } v _ { ( 2 , 3 ) } + \\frac { 3 t + 2 } { 2 ^ 4 } a _ 1 \\cdot v _ { ( 2 , 3 ) } . \\end{align*}"}
-{"id": "4627.png", "formula": "\\begin{align*} ( 2 M + \\tau ^ 2 K ) \\mathbf { u } _ n - 5 M \\mathbf { u } _ { n - 1 } + 4 M \\mathbf { u } _ { n - 2 } - M \\mathbf { u } _ { n - 3 } = \\mathbf { 0 } . \\end{align*}"}
-{"id": "3744.png", "formula": "\\begin{align*} \\inf \\limits _ { \\mathbf { J } \\in X } \\mathcal { E } \\left ( \\mathbf { J } \\right ) = \\inf \\left \\{ \\mathcal { E } \\left ( \\mathbf { J } \\right ) : \\mathbf { J } \\in X \\cap \\overline { B _ { \\Gamma } \\left ( \\mathbf { J } _ { 0 } \\right ) } \\right \\} . \\end{align*}"}
-{"id": "6071.png", "formula": "\\begin{align*} S ' + S ^ 2 = - R _ { \\partial _ r } \\end{align*}"}
-{"id": "6157.png", "formula": "\\begin{align*} \\omega _ \\alpha ( r ) & = ( 1 - r ) ^ \\alpha \\omega ( r ) ; \\\\ \\widetilde { \\omega } ( r ) & = \\frac { \\widehat { \\omega } ( r ) } { 1 - r } ; \\\\ \\omega ^ * ( r ) & = \\int _ { r } ^ 1 \\omega ( s ) s \\log ( s / r ) d s . \\end{align*}"}
-{"id": "5525.png", "formula": "\\begin{align*} E ^ { \\Pi ^ { D _ T } } [ e ^ { u \\sqrt { T } G ( b ) } | X ^ { T } ] & = E ^ { \\Pi ^ { D _ T } } [ e ^ { S _ T ( b ) + u \\sqrt { T } \\langle b - b _ 0 , \\gamma \\rangle _ { \\mu _ 0 } } | X ^ { T } ] \\\\ & = Z _ T ^ { - 1 } \\int _ { D _ T } e ^ { S _ T ( b ) + u \\sqrt { T } \\langle b - b _ { 0 } , \\gamma \\rangle _ { \\mu _ 0 } + \\ell _ T ( b _ u ) + \\ell _ T ( b ) - \\ell _ T ( b _ u ) } d \\Pi ( b ) , \\end{align*}"}
-{"id": "8401.png", "formula": "\\begin{align*} \\partial _ { \\alpha } D ^ k u = \\frac { 1 } { z _ { \\alpha } ^ { k - 1 } } \\partial _ { \\alpha } ^ k D u + \\frac { 1 } { z _ { \\alpha } ^ { k - 2 } } \\partial _ { \\alpha } ^ { k - 1 } ( \\frac { 1 } { z _ { \\alpha } } ) \\partial _ { \\alpha } D u + F _ k , \\end{align*}"}
-{"id": "7368.png", "formula": "\\begin{align*} d ( a _ i , b _ j ) & = d ( a _ i , e _ v ) + d ( e _ v , b _ j ) \\\\ & = d ( a _ i , e _ v ) + d ( f ( e _ v ) , f ( b _ i ) ) \\\\ & = d ( a _ i , e _ v ) + d ( e _ v , c _ j ) = d ( a _ i , c _ j ) . \\end{align*}"}
-{"id": "3283.png", "formula": "\\begin{align*} a , b = \\sqrt [ n ] { \\pm \\sqrt { - ( - Y ) ^ n } } = \\sqrt [ n ] { \\pm i \\sqrt { - Y } ^ n } = \\xi \\sqrt { - Y } , \\xi ^ { - 1 } \\sqrt { - Y } , \\end{align*}"}
-{"id": "7803.png", "formula": "\\begin{align*} B _ p ( N ) = \\sup _ { \\| f \\| _ { L ^ p } \\le 1 } \\big \\| \\sup _ { | n | \\le N } | M ^ G _ { 2 ^ n } f | \\big \\| _ { L ^ p } . \\end{align*}"}
-{"id": "6473.png", "formula": "\\begin{align*} p _ 0 = 1 \\oplus \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & 0 \\end{array} \\right ) p _ { n - 1 } = 0 \\oplus \\left ( \\begin{array} { c c } 0 & 0 \\\\ 0 & 1 \\end{array} \\right ) , \\end{align*}"}
-{"id": "6953.png", "formula": "\\begin{align*} \\| \\textbf { S } \\left ( \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ n \\textbf { x } _ i \\right ) - \\textbf { S } \\left \\{ \\mathcal { N } ( \\textbf { 0 } , \\boldsymbol { \\Sigma } ) \\right \\} \\| < \\epsilon , \\forall n > N . \\end{align*}"}
-{"id": "2409.png", "formula": "\\begin{align*} q ( x + y ) = q ( x ) + q ( y ) + b ( x , y ) \\end{align*}"}
-{"id": "4846.png", "formula": "\\begin{align*} \\delta ( \\beta ( w ) ) + \\delta ( y ) = \\delta \\alpha ( w ) = \\alpha _ { } ( \\delta ( w ) ) = F \\circ e ^ { \\theta } ( \\delta ( w ) ) = F \\circ e ^ { \\theta } ( D ( w ) ) . \\end{align*}"}
-{"id": "425.png", "formula": "\\begin{align*} G _ B ( x , x ' ) = 2 \\int _ { 0 } ^ { \\infty } p _ B ( e ^ { - 2 u } , x , x ' ) e ^ { - 2 u } \\dd u + \\bar G ( x , x ' ) . \\end{align*}"}
-{"id": "1136.png", "formula": "\\begin{align*} & \\frac { 1 } { n \\rho _ { s c } ( x ) } K ^ { G U E ( n ) } ( x , y ) - \\frac { \\sin ( n \\pi \\rho _ { s c } ( x ) ( x - y ) ) } { n \\pi \\rho _ { s c } ( x ) ( x - y ) } = O \\left ( \\frac { 1 } { n } \\right ) + O \\left ( a _ j \\right ) + O \\left ( n a _ j ^ 2 \\right ) , \\\\ & \\frac { 2 \\pi } { n } K ^ { C U E ( n ) } ( { 2 \\pi } \\rho _ { s c } ( y _ j ) x , { 2 \\pi } \\rho _ { s c } ( y _ j ) y ) - \\frac { \\sin ( n \\pi \\rho _ { s c } ( y _ j ) ( x - y ) ) } { n \\pi \\rho _ { s c } ( y _ j ) ( x - y ) } = O \\left ( \\frac { a _ j } { n } \\right ) , \\end{align*}"}
-{"id": "7716.png", "formula": "\\begin{align*} \\mathsf { s u p p } \\left ( \\mathsf { e } ^ { \\mathbf { i } s D } u \\right ) = \\mathsf { s u p p } ( u _ { s } ) \\subseteq L _ { s } \\stackrel { } { \\subseteq } W \\end{align*}"}
-{"id": "8461.png", "formula": "\\begin{align*} | \\zeta ( \\alpha , t ) - \\zeta ( \\beta , t ) | = | \\alpha - \\beta + ( \\zeta ( \\alpha , t ) - \\alpha ) - ( \\zeta ( \\beta , t ) - \\beta ) | . \\end{align*}"}
-{"id": "5529.png", "formula": "\\begin{align*} ( I ) & = a _ \\lambda \\left | \\sum _ { l > J } \\sum _ r \\langle \\mu _ 0 ( b _ j - P _ { V _ J } b _ { 0 , j } ) , \\Phi _ { l , r } \\rangle _ { L ^ 2 } \\langle \\Phi _ { \\lambda , k } / \\mu _ 0 , \\Phi _ { l , r } \\rangle _ { L ^ 2 } \\right | \\\\ & \\leq a _ \\lambda \\sum _ { l > J } \\max _ r | \\langle \\mu _ 0 ( b _ j - P _ { V _ J } b _ { 0 , j } ) , \\Phi _ { l , r } \\rangle _ { L ^ 2 } | \\sum _ r | \\langle \\Phi _ { \\lambda , k } / \\mu _ 0 , \\Phi _ { l , r } \\rangle _ { L ^ 2 } | . \\end{align*}"}
-{"id": "4245.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\left \\langle a ( x ) \\left \\lvert d u \\right \\rvert ^ { p - 2 } d u - F ; d \\phi \\right \\rangle = 0 \\phi \\in W ^ { 1 , p } _ { \\delta , T } \\left ( \\Omega ; \\Lambda ^ { k } \\mathbb { R } ^ { n } \\otimes \\mathbb { R } ^ { N } \\right ) , \\end{align*}"}
-{"id": "3231.png", "formula": "\\begin{align*} \\| w ^ { k + 1 } - w ^ { \\ast } \\| = O \\left ( \\| w ^ k - w ^ { \\ast } \\| ^ { 1 + c \\alpha } \\right ) . \\end{align*}"}
-{"id": "7410.png", "formula": "\\begin{align*} \\mu _ x ( F ) = \\mu ^ U _ x ( F | B _ { d _ x } ( x , \\mathfrak r ) ) . \\end{align*}"}
-{"id": "7251.png", "formula": "\\begin{align*} \\dot \\mu _ { k } ( 0 ) = - \\frac { 1 } { 2 } \\frac { \\left < D _ { \\phi \\phi } ^ 2 F ( 0 , \\mu ^ * _ { k } ) [ \\phi ^ * _ { k } , \\phi ^ * _ { k } ] , \\phi ^ * _ { k } \\right > _ { L ^ 2 } } { \\left < D _ { \\phi \\mu } ^ 2 F ( 0 , \\mu ^ * _ { k } ) \\phi ^ * _ { k } , \\phi ^ * _ { k } \\right > _ { L ^ 2 } } , \\end{align*}"}
-{"id": "7330.png", "formula": "\\begin{align*} \\begin{cases} \\rho ( x ) v _ { t t } - \\operatorname { d i v } ( K ( x ) \\nabla v ) = 0 & ~ \\hbox { i n } ~ \\Omega \\times ( 0 , T ) , \\\\ v _ t = 0 & ~ \\hbox { i n } ~ \\omega \\times ( 0 , T ) , \\end{cases} \\end{align*}"}
-{"id": "9233.png", "formula": "\\begin{align*} & \\alpha ^ { * + } ( x _ 0 ) = \\frac 1 { \\norm { z } } \\sqrt { ( p _ { 1 0 } - x _ 0 ) ^ 2 - \\norm { M ( u + x _ 0 v ) } ^ 2 } , \\\\ & \\alpha _ 1 ( x _ 0 ) = 1 - { 1 } _ { s - 1 } ^ T ( u + x _ 0 v ) - \\alpha ^ { * + } ( x _ 0 ) \\left ( { 1 } _ { s - 1 } ^ T w + 1 \\right ) , \\\\ & \\alpha _ { 2 : s } ( x _ 0 ) = u + x _ 0 v + \\alpha ^ { * + } ( x _ 0 ) \\frac w { \\norm { z } } , \\end{align*}"}
-{"id": "1713.png", "formula": "\\begin{align*} \\varphi ( 0 , x ) = x \\varphi ( t + s , x ) = \\varphi ( t , \\varphi ( s , x ) ) \\end{align*}"}
-{"id": "2764.png", "formula": "\\begin{align*} \\frac { z + w } { z - w } \\Big ( \\frac { z - \\pi _ { p } } { z + \\pi _ { p } } \\frac { w + \\pi _ { p } } { w - \\pi _ { p } } - 1 \\Big ) = \\frac { w + \\pi _ { p } } { w - \\pi _ { p } } - \\frac { z - \\pi _ { p } } { z + \\pi _ { p } } . \\end{align*}"}
-{"id": "3184.png", "formula": "\\begin{align*} \\mathcal { B } _ \\pm ( U , - \\varGamma ) & = \\ker ( - \\varGamma \\pm 1 ) \\cap \\ker ( - C + 1 ) \\\\ & = \\ker ( \\varGamma \\mp 1 ) \\cap \\ker ( C - 1 ) = \\mathcal { T } _ \\pm ( U , \\varGamma ) , \\\\ \\mathcal { T } _ \\pm ( U , - \\varGamma ) & = \\ker ( - \\varGamma \\mp 1 ) \\cap \\ker ( - C - 1 ) \\\\ & = \\ker ( \\varGamma \\pm 1 ) \\cap \\ker ( C + 1 ) = \\mathcal { B } _ \\pm ( U , \\varGamma ) , \\end{align*}"}
-{"id": "2487.png", "formula": "\\begin{align*} T ^ t ( \\delta _ { \\alpha } ) = & \\delta _ a , \\\\ T ^ t ( \\delta _ { \\beta } ) = & \\delta _ a - \\delta _ b . \\end{align*}"}
-{"id": "1566.png", "formula": "\\begin{align*} \\Phi _ { \\widetilde { F } } ( z , x ) - \\phi _ f ( z ) = o ( 1 ) , ~ w \\rightarrow 0 . \\end{align*}"}
-{"id": "5818.png", "formula": "\\begin{align*} f ( \\eta ) = 1 + \\int _ { 0 } ^ { + \\infty } 2 x \\eta ^ 2 e ^ { \\eta ^ 2 x ^ 2 } \\mathbb { P } ( | X | \\geq x ) d x \\end{align*}"}
-{"id": "9325.png", "formula": "\\begin{align*} \\| \\mathbf { H } \\| ^ 2 ( t , y ) = 1 - \\frac { ( 1 - t ^ 2 ) ^ 2 } { n ^ 2 } . \\end{align*}"}
-{"id": "5824.png", "formula": "\\begin{align*} \\ln ( \\phi _ { s _ n ^ 2 } ( t _ 2 ) ) = \\ln \\left ( \\iiint e ^ { i t _ 2 s _ n ^ 2 } \\prod _ { i = 1 } ^ n f ( x _ i ) d x _ i \\right ) . \\end{align*}"}
-{"id": "7732.png", "formula": "\\begin{align*} m ( i _ * \\underline { \\Q } _ p [ l - 2 ] , \\P _ t \\otimes \\Q _ p ) = 1 \\end{align*}"}
-{"id": "2741.png", "formula": "\\begin{align*} a = \\frac { \\eta _ { - } } { \\varphi ( x ) } ( z _ { - } - z _ { 1 } ) = \\pi _ { * } + 1 , b = \\frac { \\eta _ { - } } { \\varphi ( x ) } \\frac { \\pi _ { * } h ( \\theta ) } { \\sqrt { N } } = \\pi _ { * } . \\end{align*}"}
-{"id": "4771.png", "formula": "\\begin{align*} W = \\sum _ { i = 1 } ^ M \\alpha _ i \\beta _ i + \\sum _ { i = M + 1 } ^ N ( \\alpha _ i \\beta _ i + \\alpha _ i p _ i ) + W _ 1 \\end{align*}"}
-{"id": "1961.png", "formula": "\\begin{align*} d ( e _ i ) = \\sum _ { k = 1 } ^ { n } d _ { i k } e _ k , \\end{align*}"}
-{"id": "4304.png", "formula": "\\begin{align*} P = x _ 1 y _ 1 x _ 2 y _ 2 \\dots x _ { k - 1 } y _ { k - 1 } x _ k \\end{align*}"}
-{"id": "2289.png", "formula": "\\begin{align*} a & = \\begin{pmatrix} 1 \\\\ \\frac { \\mathcal { P } } { x \\Omega } | E ) \\end{pmatrix} & b & = \\begin{pmatrix} 0 \\\\ | \\delta _ x ) \\end{pmatrix} \\end{align*}"}
-{"id": "1402.png", "formula": "\\begin{align*} \\tilde R _ j = \\prod _ k \\left ( \\frac { x - a _ k } { a _ k } \\right ) ^ { \\alpha _ k } \\end{align*}"}
-{"id": "6920.png", "formula": "\\begin{align*} & Z _ { m - 1 } = Z _ { m - 1 } ' + Z _ { m - 1 } '' , \\end{align*}"}
-{"id": "7738.png", "formula": "\\begin{align*} \\| x \\| ^ 3 \\| y \\| = | [ \\langle x , x \\rangle \\langle x , y \\rangle \\xi , \\xi ] | & \\leq \\| x \\| ^ 2 \\| \\langle x , y \\rangle \\| \\leq \\| x \\| ^ 2 \\| x \\| \\| y \\| . \\end{align*}"}
-{"id": "8155.png", "formula": "\\begin{align*} \\alpha _ X = \\mathcal { T } \\circ i _ X , \\ : \\ : \\ : \\ : \\ : \\beta _ { X , Y } = \\mathcal { O } \\circ ( i _ X \\otimes i _ Y ) , \\ : \\ : \\ : \\ : \\ : \\gamma _ X ^ Y = \\mathcal { Q } _ Y \\circ ( i _ X \\otimes \\mathrm { i d } _ Y ) . \\end{align*}"}
-{"id": "8427.png", "formula": "\\begin{align*} - 2 [ \\bar { f } , \\mathfrak { H } ] \\frac { \\partial _ { \\alpha } \\bar { f } } { z _ { \\alpha } } = - 2 [ \\bar { f } , \\mathfrak { H } \\frac { 1 } { z _ { \\alpha } } + \\bar { \\mathfrak { H } } \\frac { 1 } { \\bar { z } _ { \\alpha } } ] \\bar { f } _ { \\alpha } , \\end{align*}"}
-{"id": "9262.png", "formula": "\\begin{align*} \\bar v : = [ D _ \\beta u - \\varphi ( \\cdot , u ) ] + \\frac { a } { 2 } | D u - D u ( x _ 0 ) | ^ 2 , { \\rm i n } \\ B _ R ( x _ 0 ) \\cap \\Omega , \\end{align*}"}
-{"id": "8434.png", "formula": "\\begin{align*} \\mathfrak { F } ( \\alpha , t ) = F ( \\zeta ( \\alpha , t ) , t ) , \\end{align*}"}
-{"id": "8084.png", "formula": "\\begin{align*} b _ n = - \\Gamma ( n ) \\ , U ( n , 0 , 1 ) . \\end{align*}"}
-{"id": "2947.png", "formula": "\\begin{align*} ( H _ k ) : \\forall \\ , \\Gamma \\in C _ { p + k , p - k + 1 } ^ \\infty ( X , \\mathbb { C } ) \\mbox { s u c h t h a t } d \\Gamma = 0 , \\ , \\Gamma \\in I m \\ , \\partial \\Rightarrow \\Gamma \\in I m \\ , \\bar { \\partial } . \\end{align*}"}
-{"id": "5356.png", "formula": "\\begin{align*} X _ { \\alpha + \\beta } . v _ { n m } ^ 1 = [ X _ \\alpha , X _ \\beta ] . v _ { n m } ^ 1 = - \\lambda v _ { n + 1 \\ , m + 3 } ^ 1 . \\end{align*}"}
-{"id": "2336.png", "formula": "\\begin{align*} a _ x \\mathcal { V } ( f ) & = \\mathcal { V } ( a _ x f ) , & a _ x ^ + \\mathcal { V } ( f ) & = \\mathcal { V } ( a _ x ^ + f ) \\end{align*}"}
-{"id": "6905.png", "formula": "\\begin{align*} \\begin{cases} a _ { 1 1 } a _ { 2 2 } - a _ { 1 2 } a _ { 2 1 } - 1 = 0 \\ , , \\\\ a _ { 1 3 } a _ { 2 4 } - a _ { 1 4 } a _ { 2 3 } - 1 = 0 \\ , , \\\\ a _ { 1 4 } - a _ { 1 2 } + a _ { 1 1 } = 0 \\ , , \\\\ a _ { 2 4 } - a _ { 2 2 } + a _ { 2 1 } = 0 \\ , , \\\\ a _ { 2 1 } '' a _ { 2 2 } - a _ { 2 2 } '' a _ { 2 1 } + a _ { 1 1 } '' a _ { 1 2 } - a _ { 1 2 } '' a _ { 1 1 } = 0 \\ , , \\end{cases} \\end{align*}"}
-{"id": "8206.png", "formula": "\\begin{align*} \\sum \\limits _ { n = 1 } ^ { \\infty } p _ { \\omega } ( n ) q ^ n = q \\omega ( q ) . \\end{align*}"}
-{"id": "3621.png", "formula": "\\begin{align*} \\alpha _ { - 1 } & = - \\frac { 3 t } { 2 ^ 2 } a _ 1 - \\frac { 1 } { 2 ^ 2 } v _ { ( 2 , 3 ) } + a _ 1 \\cdot v _ { ( 2 , 3 ) } \\\\ \\beta _ { - 1 } & = - t a _ 1 + a _ 1 \\cdot v _ { ( 2 , 3 ) } . \\end{align*}"}
-{"id": "335.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 \\\\ 4 m + 2 & 1 \\end{pmatrix} \\cdot \\begin{pmatrix} k L & 0 & - k L \\\\ 0 & ( 2 m + 1 ) k L & ( 4 m + 2 ) k L \\end{pmatrix} = \\begin{pmatrix} k L & 0 & - k L \\\\ ( 4 m + 2 ) k L & ( 2 m + 1 ) k L & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "7647.png", "formula": "\\begin{align*} I ^ { i j } = \\int \\limits _ { \\mathbb T ^ d } \\ ! \\int \\limits _ { \\mathbb R ^ d } \\ ! a ( \\xi \\ ! - \\ ! q ) \\mu ( \\xi , q ) \\big ( ( \\xi - q ) + ( \\varkappa _ 1 ( \\xi ) - \\varkappa _ 1 ( q ) ) \\big ) ^ i \\big ( ( \\xi - q ) + ( \\varkappa _ 1 ( \\xi ) - \\varkappa _ 1 ( q ) ) \\big ) ^ j v _ 0 ( \\xi ) d q d \\xi . \\end{align*}"}
-{"id": "5363.png", "formula": "\\begin{align*} Y _ \\alpha . v _ { n + 1 \\ , m + 3 } ^ 1 = \\frac 1 { a _ { n m } } \\left ( X _ { \\alpha + \\beta } Y _ \\alpha + X _ \\beta \\right ) . v _ { n m } ^ 1 . \\end{align*}"}
-{"id": "939.png", "formula": "\\begin{align*} J ^ { * } [ \\psi ^ { ( 3 ) } ] _ { T , t } = \\hbox { \\vtop { \\offinterlineskip \\halign { \\hfil # \\hfil \\cr { \\rm l . i . m . } \\cr $ \\stackrel { } { { } _ { p \\to \\infty } } $ \\cr } } } \\sum \\limits _ { j _ 1 , j _ 2 , j _ 3 = 0 } ^ { p } C _ { j _ 3 j _ 2 j _ 1 } \\zeta _ { j _ 1 } ^ { ( i _ 1 ) } \\zeta _ { j _ 2 } ^ { ( i _ 2 ) } \\zeta _ { j _ 3 } ^ { ( i _ 3 ) } \\end{align*}"}
-{"id": "4121.png", "formula": "\\begin{align*} g _ 2 ( z ) & = \\log z ^ { \\frac { 1 } { 2 } } + \\int _ 0 ^ q \\log \\left ( 1 + \\frac { \\sqrt { t } } { z ^ { \\frac { 1 } { 2 } } } \\right ) d \\mu ^ * ( t ) \\\\ & = \\frac { 1 } { 2 } \\log z + \\int _ 0 ^ q \\left ( \\frac { \\sqrt { t } } { z ^ { \\frac { 1 } { 2 } } } - \\frac { t } { 2 z } + \\mathcal { O } \\left ( z ^ { - \\frac { 3 } { 2 } } \\right ) \\right ) d \\mu ^ * ( t ) , \\end{align*}"}
-{"id": "1109.png", "formula": "\\begin{align*} & \\limsup _ { n \\to + \\infty } n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } \\int _ { I } \\mathbb { P } ( \\xi ^ { ( n ) } ( [ y , y + G _ n ( x ) / S ( I ) ] ) = 0 ) d y \\leq M ( I ) e ^ { c _ 0 - x } , \\end{align*}"}
-{"id": "4128.png", "formula": "\\begin{align*} S ( z ) & = \\left ( \\mathbb { I } + \\mathcal { O } \\left ( \\frac { 1 } { z } \\right ) \\right ) \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & z ^ { \\frac { 1 } { 4 } } & 0 \\\\ 0 & 0 & z ^ { - \\frac { 1 } { 4 } } \\end{pmatrix} \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & \\frac { 1 } { \\sqrt { 2 } } & \\frac { i } { \\sqrt { 2 } } \\\\ 0 & \\frac { i } { \\sqrt { 2 } } & \\frac { 1 } { \\sqrt { 2 } } \\end{pmatrix} . \\end{align*}"}
-{"id": "9784.png", "formula": "\\begin{align*} \\bar q ( x ) = q _ 1 ( x ^ l ) + \\sum _ { j = l + 1 } ^ { n } q _ j ( x ^ l ) x _ j = : \\bar q _ 1 ( x ^ l ) + \\bar q _ 2 ( x ) , \\end{align*}"}
-{"id": "1865.png", "formula": "\\begin{align*} \\phi ^ { \\rm e v e n } ( j , x , t ) = \\phi ^ { \\rm e v e n } _ { \\rm O U T } ( j , x , t ) \\end{align*}"}
-{"id": "4265.png", "formula": "\\begin{align*} q \\geq \\left ( p ^ { \\ast } \\right ) ^ { ' } p < n q ' = \\left ( p ' \\right ) ^ { \\ast } . \\end{align*}"}
-{"id": "7351.png", "formula": "\\begin{align*} d ( x , z ) = \\frac { d ( x , y ) } { 2 } , d ( y , z ) = \\frac { d ( x , y ) } { 2 } \\end{align*}"}
-{"id": "7345.png", "formula": "\\begin{align*} \\tilde \\nabla ^ 2 \\phi ( X , Y ) = X Y ( \\phi ) - \\tilde \\nabla _ X Y ( \\phi ) , \\end{align*}"}
-{"id": "2582.png", "formula": "\\begin{align*} \\eta _ t : = ( C _ X ( \\gamma _ 0 - 1 ) t ) ^ { - \\frac { 1 } { \\gamma _ 0 - 1 } } , t \\geq 0 , \\end{align*}"}
-{"id": "4915.png", "formula": "\\begin{align*} \\bar L = \\tfrac { 1 } { 2 } \\int _ \\Omega g ( A u , u ) \\ , \\mu ( x ) \\ , , \\end{align*}"}
-{"id": "8182.png", "formula": "\\begin{align*} T _ \\Lambda \\mathcal { T } ^ * \\{ g _ i \\} _ { i \\in \\mathbb { Z } } = \\sum _ { i \\in \\mathbb { Z } } \\Lambda _ { i - 1 } ^ * g _ i = \\sum _ { i \\in \\mathbb { Z } } ( T ^ { i - 1 } ) ^ * \\Lambda _ 0 ^ * g _ i & = ( T ^ { - 1 } ) ^ * \\Big ( \\sum _ { i \\in \\mathbb { Z } } ( T ^ { i } ) ^ * \\Lambda _ 0 ^ * g _ i \\Big ) \\\\ & = ( T ^ { - 1 } ) ^ * \\Big ( \\sum _ { i \\in \\mathbb { Z } } \\Lambda _ i ^ * g _ i \\Big ) \\\\ & = ( T ^ { - 1 } ) ^ * T _ \\Lambda \\{ g _ i \\} _ { i \\in \\mathbb { Z } } = 0 . \\end{align*}"}
-{"id": "5934.png", "formula": "\\begin{align*} { \\rm g l c t } ( K _ X + B + M \\mid f _ X ^ * P ) = { \\rm g l c t } ( K _ Y + D + M _ Y \\mid f _ Y ^ * P ) - { \\rm c o e f f } _ P N . \\end{align*}"}
-{"id": "2941.png", "formula": "\\begin{align*} & \\alpha _ { 1 } = \\| B ^ { \\dagger } \\| _ { 2 } ^ { 2 } \\big ( \\| E B ^ { \\dagger } \\| _ { F } ^ { 2 } - \\| A A ^ { \\dagger } E B ^ { \\dagger } \\| _ { F } ^ { 2 } \\big ) , \\\\ & \\alpha _ { 2 } = \\| A ^ { \\dagger } \\| _ { 2 } ^ { 2 } \\big ( \\| E A ^ { \\dagger } \\| _ { F } ^ { 2 } - \\| B B ^ { \\dagger } E A ^ { \\dagger } \\| _ { F } ^ { 2 } \\big ) . \\end{align*}"}
-{"id": "9964.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ n \\deg ( \\alpha _ i ) < ( 1 - g ) + k + n . \\end{align*}"}
-{"id": "7091.png", "formula": "\\begin{align*} \\begin{cases} - \\partial _ { t t } F _ { n } ^ \\omega + \\Delta F _ { n } ^ \\omega = 2 \\nabla P _ { \\leq N ^ \\gamma } u _ { n - 1 } \\cdot \\nabla F _ { n } ^ \\omega \\\\ F _ { n } ^ \\omega | _ { t = 0 } = Q _ N f _ 0 ^ \\omega ~ , ~ ~ \\partial _ t F _ { n } ^ \\omega | _ { t = 0 } = Q _ N f _ 1 ^ \\omega ~ . \\end{cases} \\end{align*}"}
-{"id": "8483.png", "formula": "\\begin{align*} | \\dot { z } _ j ( t ) - \\frac { \\lambda i } { 4 \\pi x ( t ) } | = | \\bar { F } ( z _ 1 ( t ) , t ) | \\leq 5 \\epsilon . \\end{align*}"}
-{"id": "2386.png", "formula": "\\begin{align*} Y _ j ^ 1 & = Y _ j \\otimes 1 \\cdots \\otimes 1 \\otimes 1 , & & \\cdots , & Y ^ { p + 1 } _ j & = 1 \\otimes \\cdots \\otimes 1 \\otimes Y _ j \\end{align*}"}
-{"id": "6393.png", "formula": "\\begin{align*} \\| x - x _ { k } \\| _ { A } = \\min _ { y \\in x _ { 0 } + \\mathcal { K } _ { k } ( A , r _ { 0 } ) } \\| x - y \\| _ { A } , \\end{align*}"}
-{"id": "9045.png", "formula": "\\begin{align*} G ( A ) = N ( 0 , \\mu ( A ) ) . \\end{align*}"}
-{"id": "669.png", "formula": "\\begin{align*} ( \\widehat \\rho _ j ) _ { H ( x _ \\ast \\vec a ) } \\circ H ( \\widehat x _ { \\vec a } ) \\circ \\lambda _ { \\vec a } & = \\lambda _ { x _ \\ast \\vec a , j } \\circ H ( ( \\widehat \\rho _ j ) _ { x _ \\ast \\vec a } \\circ \\widehat x _ { \\vec a } ) \\circ \\lambda _ { \\vec a } \\\\ & = \\lambda _ { x _ \\ast \\vec a , j } \\circ H ( \\widehat { [ \\rho _ j , x ] } _ { \\vec a } ) \\circ \\lambda _ { \\vec a } \\ . \\end{align*}"}
-{"id": "4269.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\delta ( a ( x _ { 0 } ) \\lvert d v \\rvert ^ { p - 2 } d v ) ) & = 0 & & B _ { R } , \\\\ \\delta v & = 0 & & B _ { R } , \\\\ \\nu \\wedge v & = \\nu \\wedge w _ { j } & & \\partial B _ { R } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "7374.png", "formula": "\\begin{align*} \\mathcal { M } _ { p e r } : = \\{ \\phi \\in \\mathcal { M } : \\phi ( x , y ) = \\phi ( x + 2 \\pi , y ) = \\phi ( x , y + 2 \\pi ) = \\phi ( x + 2 \\pi , y + 2 \\pi ) \\ \\} . \\end{align*}"}
-{"id": "4831.png", "formula": "\\begin{align*} \\sum _ { \\chi \\in S _ { n , q } } \\left ( q ^ { - \\frac { n + 1 } { 2 } } \\sum _ { a _ 1 , \\dots , a _ n \\in \\mathbb F _ q } \\chi ( 1 + \\sum _ i a _ i x ^ i ) \\psi ( a _ n ) \\right ) ^ m = \\sum _ { \\substack { \\chi \\in S _ { n , q } \\\\ \\psi _ \\chi = \\psi } } \\epsilon _ \\chi ^ m = \\sum _ { \\substack { \\chi \\in S _ { n , q } \\\\ \\psi _ \\chi = \\psi } } \\mu = q ^ { n - 1 } \\mu \\end{align*}"}
-{"id": "2673.png", "formula": "\\begin{align*} I _ 1 = \\int _ { \\mathcal { C } _ { \\{ 0 \\} } } \\frac { d s } { 2 \\pi i s } e ^ { \\frac { v } { s } } \\frac { 1 } { \\sqrt { 1 - 2 s } } , I _ 2 = \\int _ { \\mathcal { C } _ { \\{ 0 \\} } } \\frac { d s } { 2 \\pi i s } e ^ { \\frac { v } { s } } \\frac { 1 } { 1 - ( z + 1 ) s } \\frac { 1 } { \\sqrt { 1 - 2 s } } . \\end{align*}"}
-{"id": "7321.png", "formula": "\\begin{align*} \\mathcal { F } ( u , v , w , z ) = \\left ( 0 , 0 , - \\frac { 1 } { \\rho } v ^ 2 u , - \\frac { 1 } { \\rho } u ^ 2 v \\right ) . \\end{align*}"}
-{"id": "9800.png", "formula": "\\begin{align*} w \\ , L _ a w = - ( p _ { \\ast } + q _ \\circ ) \\ , L _ a u . \\end{align*}"}
-{"id": "8812.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { \\pi } f ( v ( s , x ) ) \\phi ( x ) \\ , \\widetilde { W } ( \\textrm { d } s , \\textrm { d } x ) = M _ t ( \\phi ) = \\langle \\overline { v } _ t , \\phi \\rangle - \\int _ { 0 } ^ { t } \\langle v ( s , \\cdot ) , \\phi '' \\rangle \\ , \\textrm { d } s , t \\leq T , \\end{align*}"}
-{"id": "556.png", "formula": "\\begin{align*} \\Box u = u Q _ 0 ( u , u ) . \\end{align*}"}
-{"id": "2424.png", "formula": "\\begin{align*} A _ s : = \\begin{bmatrix} a ^ d _ s & n ^ d _ s \\\\ 0 & ( a ^ d _ s ) ^ { - 1 } \\end{bmatrix} \\end{align*}"}
-{"id": "6708.png", "formula": "\\begin{align*} \\frac { \\partial H } { \\partial P _ { 1 k } } = \\dot { \\bar { q } } _ k , \\end{align*}"}
-{"id": "2090.png", "formula": "\\begin{align*} I _ i ( t ) & = \\int _ { - i T } ^ { ( 1 - i ) T } s _ i ( \\tau + i T ) p _ { \\rm o b s } ( t - \\tau ) d \\tau \\\\ & = \\int _ { 0 } ^ { T } s _ i ( \\tau ) p _ { \\rm o b s } ( i T + t - \\tau ) d \\tau \\\\ & = s _ i ( i T + t ) * p _ { \\rm o b s } ( i T + t ) . \\end{align*}"}
-{"id": "1783.png", "formula": "\\begin{align*} A = \\left \\{ \\limsup _ { h \\downarrow 0 } \\frac { \\left | B _ { t + h } - B _ { t } \\right | } { h } < \\infty \\ t \\in [ 0 , 1 ] \\right \\} . \\end{align*}"}
-{"id": "5205.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow + \\infty } \\frac { U ^ { \\ast } ( \\gamma x ) } { U ^ { \\ast } ( x ) } = \\gamma \\lim _ { x \\rightarrow + \\infty } \\frac { U ( \\gamma x ) } { U ( x ) } = \\gamma ^ { \\rho + 1 } . \\end{align*}"}
-{"id": "6095.png", "formula": "\\begin{gather*} \\phi ' = ( x - a _ 1 ) ^ { e _ 1 - 1 } \\ldots ( x - a _ r ) ^ { e _ r - 1 } , \\phi ( a _ i ) \\neq \\phi ( a _ j ) i < j , \\end{gather*}"}
-{"id": "2776.png", "formula": "\\begin{align*} p _ t ( \\mathbf { n } ) : = \\mathbb { P } ( \\mathbf { N } ^ t = \\mathbf { n } ) . \\end{align*}"}
-{"id": "1693.png", "formula": "\\begin{align*} \\int \\varphi \\ , d { \\mathcal L } ^ * \\mu = \\int { \\mathcal L } \\varphi \\ , d \\mu \\ \\ \\ \\ \\forall \\ \\varphi \\in C ^ 0 ( X , \\mathbb { C } ) \\ \\ \\ \\ \\ \\forall \\ \\mu \\in { \\mathcal M } . \\end{align*}"}
-{"id": "7959.png", "formula": "\\begin{align*} T & = \\bigcup _ { a \\in T ' } ( \\{ a \\} \\cup S _ a ) , \\\\ Q & = \\bigcup _ { a \\in Q ' } ( \\{ a \\} \\cup S _ a ) \\end{align*}"}
-{"id": "9264.png", "formula": "\\begin{align*} \\bar \\phi = \\phi + \\frac { b } { 2 } | x - x _ 0 | ^ 2 , \\end{align*}"}
-{"id": "6252.png", "formula": "\\begin{align*} u _ { a + 1 } ^ * y _ i u _ k & = q _ 1 ( u _ { a + 1 } ^ { ( t ) } ) ^ * y _ i ^ { ( t ) } u _ k \\\\ u _ j ^ * y _ i u _ { a + 1 } & = u _ j ^ * y _ i ^ { ( t ) } u _ { a + 1 } ^ { ( t ) } q _ 1 \\\\ u _ { a + 1 } ^ * y _ i u _ { a + 1 } & = q _ 1 ( u _ { a + 1 } ^ { ( t ) } ) ^ * y _ i ^ { ( t ) } u _ { a + 1 } ^ { ( t ) } q _ 1 \\end{align*}"}
-{"id": "358.png", "formula": "\\begin{align*} \\mathcal { P } _ { k } ( \\alpha ) = \\frac { 1 } { a ( \\alpha ) } \\int _ { \\Omega } f _ { \\ast } ( u _ { \\alpha } ) w _ { \\alpha } d x . \\end{align*}"}
-{"id": "2020.png", "formula": "\\begin{align*} T _ j ( - z ) = \\sum _ { j ' \\neq j \\atop d > 0 } \\frac { T _ j ^ { d , j ' } } { z - \\frac { \\alpha _ { j ' } - \\alpha _ { j } } { d } } . \\end{align*}"}
-{"id": "3899.png", "formula": "\\begin{align*} \\P ( 0 , 1 , \\ , \\ldots , \\ , 0 , 0 ) ^ t = 0 . \\end{align*}"}
-{"id": "5123.png", "formula": "\\begin{align*} \\mathring H _ { m i n } & \\cong \\bigoplus _ { j = \\frac { 1 } { 2 } , \\frac { 3 } { 2 } , \\dots } ^ \\infty \\ , \\bigoplus _ { m _ j = - j } ^ j \\ , \\bigoplus _ { k _ j = \\pm ( j + 1 / 2 ) } \\ , h _ { m _ j , k _ j } , \\\\ H _ { m a x } & \\cong \\bigoplus _ { j = \\frac { 1 } { 2 } , \\frac { 3 } { 2 } , \\dots } ^ \\infty \\ , \\bigoplus _ { m _ j = - j } ^ j \\ , \\bigoplus _ { k _ j = \\pm ( j + 1 / 2 ) } \\ , { h } ^ * _ { m _ j , k _ j } , \\end{align*}"}
-{"id": "1033.png", "formula": "\\begin{align*} & \\mathbb { P } ( \\xi ^ { ( n ) } ( I _ 1 \\cup I _ 2 ) = 0 ) \\leq \\mathbb { P } ( \\xi ^ { ( n ) } ( I _ 1 ) = 0 ) \\mathbb { P } ( \\xi ^ { ( n ) } ( I _ 2 ) = 0 ) . \\end{align*}"}
-{"id": "6320.png", "formula": "\\begin{align*} u ( x , 0 ) = u _ 0 ( x ) \\quad x \\in ( 0 , \\ , 1 ) , \\end{align*}"}
-{"id": "1710.png", "formula": "\\begin{align*} T ( s + h ) y - T ( s ) y = T ( s ) ( T ( h ) y - y ) \\in V _ 1 \\subset V _ 0 \\end{align*}"}
-{"id": "9083.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } g ( x ) | u _ n ( x ) | ^ p \\ , d x = 1 \\quad \\forall n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "6072.png", "formula": "\\begin{align*} S ( t ) = \\frac { 1 } { t } P _ { \\xi } + S _ \\xi + O ( t ) \\end{align*}"}
-{"id": "9358.png", "formula": "\\begin{align*} \\log _ 2 \\lvert G : Z G ^ { 2 ^ k } \\rvert = \\log _ 2 \\lvert W _ k : W _ k ^ { \\ , 2 ^ k } \\rvert = 2 ^ k + k - 1 , \\end{align*}"}
-{"id": "4997.png", "formula": "\\begin{align*} { \\rm A v g } _ { \\chi } \\Phi \\left ( \\frac { L ' } { L } ( s , \\chi ) \\right ) = \\int _ { \\mathbb { C } } M _ { \\sigma } ( w ) \\Phi ( w ) | d w | \\end{align*}"}
-{"id": "7634.png", "formula": "\\begin{align*} h ( \\xi ) \\ = \\ \\int \\limits _ { \\mathbb R ^ d } a ( \\xi - q ) \\mu ( \\xi , q ) ( \\xi - q ) \\ d q \\ - \\ b \\ \\in \\ ( L ^ 2 ( \\mathbb T ^ d ) ) ^ d , \\end{align*}"}
-{"id": "3646.png", "formula": "\\begin{align*} ( \\alpha _ { - 1 } - \\beta _ { - 1 } ) \\cdot ( \\alpha _ { - 2 } - \\beta _ { - 2 } ) & = \\frac { ( 3 t + 2 2 ) t } { 2 ^ 2 \\cdot 3 ^ 2 } a _ 2 - \\frac { ( 3 t - 1 0 ) t } { 2 ^ 2 \\cdot 3 ^ 2 } a _ { - 2 } - \\frac { ( 3 t - 1 4 ) t } { 2 ^ 2 \\cdot 3 ^ 2 } a _ 3 + \\frac { ( 3 t + 2 ) t } { 2 ^ 2 \\cdot 3 ^ 2 } a _ { - 3 } \\\\ & - \\frac { 2 t - 1 } { 2 ^ 2 \\cdot 3 } v _ { ( 2 , 3 ) } + \\frac { t } { 2 \\cdot 3 } a _ 1 \\cdot v _ { ( 2 , 3 ) } + \\frac { t } { 2 } ( a _ 2 \\cdot v _ { ( 1 , 3 ) } - a _ 3 \\cdot v _ { ( 1 , 2 ) } ) . \\end{align*}"}
-{"id": "5354.png", "formula": "\\begin{align*} X _ \\beta . v _ { n m } ^ 1 = \\lambda v _ { n + 1 \\ , m + 3 } ^ 2 + c _ { n m } v _ { n - 1 \\ , m + 3 } ^ 1 . \\end{align*}"}
-{"id": "6649.png", "formula": "\\begin{align*} M _ { 1 , Q } ( n ) & = \\psi _ { \\tau Q / N } \\ast d \\sigma _ { \\tau } ( n ) \\sum _ { Q \\leq q \\leq 2 Q } \\sum _ { a \\in \\mathbb Z ^ { \\times } _ { q } } e _ q ( a ( \\lvert n \\rvert ^ 2 - \\lambda ^ 2 ) ) \\\\ & = P _ { Q , \\tau } ( n ) \\cdot \\mathsf C _ { Q } ( \\lvert n \\rvert ^ 2 - \\lambda ^ 2 ) . \\end{align*}"}
-{"id": "662.png", "formula": "\\begin{align*} \\widetilde \\Phi ( [ \\varphi ; f _ 1 , \\dots , f _ n ; u , x ] ) = \\left [ \\varphi ; ( f _ 1 , \\delta ^ { ( \\varphi ) \\ast } _ 1 ( u ) ) , \\dots , ( f _ n , \\delta ^ { ( \\varphi ) \\ast } _ n ( u ) ) ; x \\right ] \\ . \\end{align*}"}
-{"id": "9409.png", "formula": "\\begin{align*} - J _ { N , 1 } ( T _ m . D _ f ) = \\sum _ { d \\mid m } d c ( d ) + 2 4 \\left ( \\sum _ { 1 \\leq j \\leq D ( N ) - 1 } \\frac { \\det ( A _ { f , j } ) } { \\det ( A _ N ) } + \\frac { k } { 1 2 } \\right ) \\sigma _ 1 ( m ) . \\end{align*}"}
-{"id": "2923.png", "formula": "\\begin{align*} \\left < z ^ k f , f \\right > = \\delta _ k ( 0 ) , \\end{align*}"}
-{"id": "1850.png", "formula": "\\begin{align*} \\psi ^ { \\rm o d d } ( n , x , t ) = ( 2 k _ n ) ^ { - 1 / 2 } v ^ { \\rm o d d } ( n , x ) e ^ { - i k _ n t } \\end{align*}"}
-{"id": "2217.png", "formula": "\\begin{align*} & s = \\dot { p } ( t ) x ^ 2 - \\left ( \\frac { 2 p ( t ) x ^ 2 } { t + 1 } \\right ) + 2 p ( t ) x u + x ^ 2 + u ^ 2 \\\\ & s _ f = x ^ 2 ( t _ f ) - p ( t _ f ) x ^ 2 ( t _ f ) \\end{align*}"}
-{"id": "149.png", "formula": "\\begin{align*} l _ { d f } ( X ) = \\big < X , d f ( q _ M ( X ) ) \\big > , \\forall X \\in T M . \\end{align*}"}
-{"id": "6163.png", "formula": "\\begin{align*} 9 \\sum ^ { T - 1 } _ { t = 0 } \\left ( F _ t + \\frac { D _ y ^ 2 } { 2 \\lambda _ t } \\right ) & = \\sum ^ { T - 1 } _ { t = 0 } \\frac { 1 } { ( t + 2 ) ^ 2 } \\leq \\frac { \\pi ^ 2 } { 6 } . \\end{align*}"}
-{"id": "5374.png", "formula": "\\begin{align*} W ( r , s ) | _ K = V _ { r s } \\oplus \\bigoplus _ { n , m \\in \\Z , \\ , n > r } V _ { n m } \\end{align*}"}
-{"id": "726.png", "formula": "\\begin{align*} T ^ { k } = \\frac { 1 } { ( q - 1 ) ^ { k } } \\left ( 1 - \\frac { L } { r ^ { q } } \\right ) ^ { k } = \\frac { 1 } { ( q - 1 ) ^ { k } } \\sum _ { i = 0 } ^ { k } ( - 1 ) ^ { i } \\binom { k } { i } \\frac { L ^ { i } } { r ^ { q i } } \\end{align*}"}
-{"id": "7520.png", "formula": "\\begin{align*} \\mathfrak { S } _ { ( x , y ) } ( \\xi , \\eta ) = ( \\Psi _ { i j p } \\Psi _ { r s q } \\Psi ^ { p q } - \\Psi _ { i j r s } ) \\Psi ^ { r k } \\Psi ^ { s l } \\xi ^ { i } \\xi ^ { j } \\eta ^ { k } \\eta ^ { l } . \\end{align*}"}
-{"id": "8904.png", "formula": "\\begin{align*} f _ { e } ( q , p ) : = ( q / p ) ^ { e } ( ( 1 - q ) / ( 1 - p ) ) ^ { 1 - e } . \\end{align*}"}
-{"id": "5388.png", "formula": "\\begin{align*} \\langle A _ { \\alpha + \\beta } . v _ { n + 1 \\ , m + 3 } ^ { k } , v _ { n m } ^ k \\rangle = \\langle v _ { n + 1 \\ , m + 3 } ^ { k } , A _ { \\alpha + \\beta } ^ * . v _ { n m } ^ k \\rangle . \\end{align*}"}
-{"id": "4755.png", "formula": "\\begin{align*} { \\sf L } f ( \\xi ) & = \\log \\left ( x ^ 2 + ( y - 1 ) ^ 2 - \\frac 1 2 \\right ) + \\log \\left ( ( x - 1 ) ^ 2 + y ^ 2 - \\frac 1 2 \\right ) \\\\ & + \\log \\left ( ( x - 2 ) ^ 2 + ( y - 1 ) ^ 2 - \\frac 1 2 \\right ) + \\log \\left ( ( x - 1 ) ^ 2 + ( y - 2 ) ^ 2 - \\frac 1 2 \\right ) \\\\ & - 4 \\log \\left ( ( x - 1 ) ^ 2 + ( y - 1 ) ^ 2 - \\frac 1 2 \\right ) \\\\ & = \\log \\left ( 1 - \\frac { 6 4 ( x - y ) ^ 2 ( x + y - 2 ) ^ 2 } { ( 2 x ^ 2 + 2 y ^ 2 - 4 x - 4 y + 3 ) ^ 4 } \\right ) \\le 0 . \\end{align*}"}
-{"id": "1757.png", "formula": "\\begin{align*} \\begin{aligned} \\left [ c _ { p , r } \\left ( \\bigcap _ { i = 1 } ^ { N } \\left \\{ a _ { i } < X _ { i } < b _ { i } \\right \\} \\right ) \\right ] ^ { p } \\leq & \\left [ \\sum _ { l = 0 } ^ { r } N ^ { l p } C _ { H } ^ { l p / 2 } \\left ( \\frac { M _ { r } } { c } \\right ) ^ { l p } \\right ] P \\left ( \\bigcap _ { i = 1 } ^ { N } \\left \\{ a _ { i } - c _ { i } < X _ { i } < b _ { i } + c _ { i } \\right \\} \\right ) \\end{aligned} \\end{align*}"}
-{"id": "9588.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\textup { s p t } ( n , N ) q ^ n = \\frac { 1 } { 2 } \\frac { d ^ 2 } { d z ^ 2 } \\left ( \\frac { ( q ) _ N } { ( z q ) _ N ( z ^ { - 1 } q ) _ N } \\right ) _ { z = 1 } - \\frac { 1 } { 2 } \\frac { d ^ 2 } { d z ^ 2 } \\left ( \\sum _ { j = 1 } ^ N \\left [ \\begin{matrix} N \\\\ j \\end{matrix} \\right ] \\frac { q ^ { j ^ 2 } ( q ) _ j } { ( z q ) _ j ( z ^ { - 1 } q ) _ j } \\right ) _ { z = 1 } . \\end{align*}"}
-{"id": "868.png", "formula": "\\begin{gather*} S ( y ) : = \\{ x \\in D \\mid g ( x ) + y \\in K \\} , y \\in \\R ^ m \\end{gather*}"}
-{"id": "649.png", "formula": "\\begin{align*} ( \\varphi ^ \\ast ( y ) _ \\ast \\vec a ) ^ { \\varphi ^ y } _ j = \\beta ^ { \\varepsilon ^ y _ { y ^ { - 1 } ( j ) } } _ \\ast \\vec a ^ \\varphi _ { y ^ { - 1 } ( j ) } \\ . \\end{align*}"}
-{"id": "6853.png", "formula": "\\begin{align*} \\Phi ' ( t , \\beta ) = \\frac { \\partial } { \\partial t } \\Phi ( t , \\beta ) \\ , . \\end{align*}"}
-{"id": "4035.png", "formula": "\\begin{align*} I ( P , V ) = \\sum _ { x \\in P } i ( x ) \\leq m ^ { 2 / 3 } \\left ( \\sum _ { x \\in P } i ( x ) ^ 3 \\right ) ^ { 1 / 3 } . \\end{align*}"}
-{"id": "1098.png", "formula": "\\begin{align*} | \\Sigma _ k ( a _ 1 , \\cdots , a _ k ) | & = \\sum _ { i _ 1 , \\cdots , i _ { k } \\in \\Lambda ( I ) \\ } \\prod _ { j = 1 } ^ k ( \\lambda _ { i _ j + 1 } - \\lambda _ { i _ j } - a _ j ) _ + \\\\ & = \\sum _ { i _ 1 , \\cdots , i _ { k } \\in \\Lambda ( I ) \\ } \\prod _ { j = 1 } ^ k ( { m } ^ * _ { i _ j } - a _ j ) _ + . \\end{align*}"}
-{"id": "6007.png", "formula": "\\begin{align*} \\P _ 1 ( \\C ) : = \\left \\{ Q \\in \\P ( \\C ) : \\int _ \\C \\| \\omega \\| _ \\infty \\ , Q ( d \\omega ) < \\infty \\right \\} , \\end{align*}"}
-{"id": "9505.png", "formula": "\\begin{align*} I _ { \\lambda } ( t u _ 0 ) = \\frac { t ^ p } { p } \\Vert u _ 0 \\Vert _ { s , p } ^ p + \\frac { t ^ q } { q } \\Vert u _ 0 \\Vert _ { s , q } ^ q - \\frac { \\lambda t ^ r } { r } \\displaystyle \\int _ { \\mathbb { R } ^ N } g u _ 0 ^ r \\dd x - \\frac { t ^ { p _ { s } ^ * } } { p _ { s } ^ * } , \\ t > 0 . \\end{align*}"}
-{"id": "3117.png", "formula": "\\begin{align*} F ( S , \\beta , y ) = \\left ( F ^ { ( r ) } _ 0 ( y ) \\right ) ^ { \\chi ( \\O _ S ) } \\prod _ { k \\leq l } \\left ( F ^ { ( r ) } _ { k l } ( y ) \\right ) ^ { \\beta ^ k \\beta ^ l } \\ , , \\end{align*}"}
-{"id": "262.png", "formula": "\\begin{align*} w _ i = \\sum _ { j = i } ^ p { j \\choose i } Q _ { j - i } ( \\alpha ) \\frac { v _ j } { f ^ { j - i + 1 } } , \\end{align*}"}
-{"id": "4607.png", "formula": "\\begin{align*} S ( x ) + s = S ( x - x _ 0 ) + S ( x _ 0 ) + s = S ( x - x _ 0 ) + x _ 0 - s + s = S ( x - x _ 0 ) + x _ 0 . \\end{align*}"}
-{"id": "7580.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { \\int _ 0 ^ t \\Delta g _ { \\nu ( t - s ) } * \\Delta _ 2 \\tau ( s , t ) d s } _ { L ^ p \\cap L ^ \\infty } \\le \\frac { C } { \\nu } \\left ( \\left ( \\frac { t } { \\nu } \\right ) ^ { \\frac { 1 } { 2 } } \\norm { X - \\mathrm { I d } } _ { L i p ( 0 , T ; C ^ { 1 + \\alpha } ) } \\right ) \\norm { \\tau } _ { L ^ \\infty ( 0 , T ; L ^ p \\cap L ^ \\infty ) } . \\end{gathered} \\end{align*}"}
-{"id": "3932.png", "formula": "\\begin{align*} \\sum n _ i \\otimes S ( g _ i ) = \\sum { n _ i } _ 0 \\otimes S g _ i { n _ i } _ 1 ( 1 ) = \\sum n _ 0 \\otimes \\iota \\bigl ( \\sigma _ { \\Bbbk [ H ] } ( n _ 1 ) \\bigr ) \\ , . \\end{align*}"}
-{"id": "2124.png", "formula": "\\begin{align*} \\Phi _ \\omega ( s , t ) E _ k ^ s ( \\omega ) = E _ k ^ t ( \\omega ) , \\qquad \\hbox { f o r a l l } \\ ; s , t \\in [ t _ 0 , \\infty ) , \\omega \\in \\Omega , k = 1 , \\ldots , d . \\end{align*}"}
-{"id": "8996.png", "formula": "\\begin{align*} \\begin{aligned} & i = i _ 1 , j = i _ 2 , t _ 1 = T \\gamma _ g ( 2 \\tau + 1 ) , t _ 2 = T \\gamma _ g ( 2 \\tau + 1 ) + 2 \\tau + 1 , \\\\ & \\alpha _ c = \\frac { \\psi ( X ( 0 ) + \\rho ) e ^ { - n ^ { \\infty } \\kappa ( 3 \\tau + 2 ) } } { 1 + n ^ { \\infty } \\kappa } \\mbox { a n d } \\underline { \\psi } = \\psi ( X ( 0 ) + \\rho ) \\end{aligned} \\end{align*}"}
-{"id": "4324.png", "formula": "\\begin{align*} a _ h ( u _ h , v _ h ) = \\sum _ { K \\in \\mathcal { T } _ h } ( f , Q _ K v _ h ) \\forall v _ h \\in V _ h . \\end{align*}"}
-{"id": "6699.png", "formula": "\\begin{align*} \\alpha ' _ { i k } ( \\bar { q } ) = \\frac { \\partial \\alpha _ i ( \\bar { q } ) } { \\partial \\bar { q } _ k } , \\beta ' _ { i k l } ( \\bar { q } ) = \\frac { \\partial \\beta _ { i k } ( \\bar { q } ) } { \\partial \\bar { q } _ l } . \\end{align*}"}
-{"id": "7318.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\frac { \\partial W } { \\partial t } ( t ) = { } & \\mathcal { A } W ( t ) + \\mathcal { F } ( W ( t ) ) \\\\ W ( 0 ) = { } & ( u _ { 0 } , v _ { 0 } , u _ { 1 } , v _ { 1 } ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "7943.png", "formula": "\\begin{align*} 2 r - 1 \\leq \\sum _ { j = 1 } ^ m ( k _ j ^ S - 1 ) , \\end{align*}"}
-{"id": "7563.png", "formula": "\\begin{align*} \\beta _ \\tau = \\tau t + ( 1 - \\tau ) s . \\end{align*}"}
-{"id": "4345.png", "formula": "\\begin{align*} H _ { \\tilde { C } _ \\sigma } ( C ) & = \\{ h Z \\in ( \\widetilde { H } / Z ) ^ W \\ , | \\ , s _ \\alpha ( h ' ) = h ' , \\ , \\forall h ' \\in h Z \\} = \\widetilde { H } ^ W / Z = \\widetilde { H } _ { \\tilde { C } _ \\sigma } ( C ) / Z . \\end{align*}"}
-{"id": "7480.png", "formula": "\\begin{gather*} q ^ * \\psi _ s = q _ s \\psi _ s \\end{gather*}"}
-{"id": "4681.png", "formula": "\\begin{align*} \\mathcal T _ j ^ d = \\{ \\xi _ l : \\ ; \\ ; \\xi _ l = { ( x _ { l - 1 } + x _ { l } ) } / { 2 } , \\ ; \\ ; x _ l \\in \\mathcal { N } _ j ^ d \\cup \\{ x _ { j - d - 1 } , x _ j \\} \\} , \\ ; \\ ; j = 1 , \\ldots , m + d . \\end{align*}"}
-{"id": "10075.png", "formula": "\\begin{align*} \\varphi ( \\theta + \\omega , \\lambda ) - \\varphi ( \\theta , \\lambda ) = h ( \\theta , \\lambda ) , \\end{align*}"}
-{"id": "8775.png", "formula": "\\begin{align*} \\overline { T } _ { a b } = \\frac { D + 4 q C } { 2 p } \\ , \\overline { F } \\ , \\overline { g } _ { a b } \\ , . \\end{align*}"}
-{"id": "1807.png", "formula": "\\begin{align*} \\begin{aligned} c _ { 2 , 1 } \\left ( \\bigcap _ { i = 1 } ^ { d } \\left \\{ - 2 \\eta < B _ { s _ { 1 } } ^ { i } - B _ { t _ { 0 } } ^ { i } < 2 \\eta \\right \\} \\right ) \\leq & \\left ( 1 + d ^ { 2 } C _ { H } \\left ( \\frac { M } { \\eta } \\right ) ^ { 2 } \\right ) \\\\ & \\cdot P \\left ( \\bigcap _ { i = 1 } ^ { d } \\left \\{ \\left | B _ { s _ { 1 } } ^ { i } - B _ { t _ { 0 } } ^ { i } \\right | < 3 \\eta \\right \\} \\right ) , \\end{aligned} \\end{align*}"}
-{"id": "9291.png", "formula": "\\begin{align*} D v ( \\hat x ) = & \\frac { 1 } { \\epsilon } a ( z ) ( \\hat x - \\hat y ) + \\varphi ( z , v ( z ) ) \\beta ( z ) - \\delta D d ( \\hat x ) + 2 \\delta ( \\hat x - z ) , \\\\ D u ( \\hat y ) = & \\frac { 1 } { \\epsilon } a ( z ) ( \\hat x - \\hat y ) + \\varphi ( z , v ( z ) ) \\beta ( z ) + \\delta D d ( \\hat y ) , \\end{align*}"}
-{"id": "9019.png", "formula": "\\begin{align*} f _ { \\{ b _ i , a \\} , 1 } ( x _ 1 , x _ 2 ) = \\begin{cases} 1 & \\{ x _ 1 , x _ 2 \\} = \\{ b _ i , a \\} , \\\\ x _ 1 & \\end{cases} \\end{align*}"}
-{"id": "4823.png", "formula": "\\begin{align*} \\left \\{ \\begin{matrix} \\mathcal L \\varphi = 0 & \\textrm { i n $ M $ } \\\\ \\mathcal B \\varphi = \\lambda _ 1 ( \\mathcal B ) \\varphi & \\textrm { o n $ \\partial M $ . } \\end{matrix} \\right . \\end{align*}"}
-{"id": "4424.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Gamma _ T } ( \\phi - z ) h \\partial _ j \\big ( \\chi ^ { \\pm } _ { \\varepsilon } ( \\phi - z ) \\big ) \\dd x \\dd \\tau \\underset { \\varepsilon \\rightarrow 0 } { \\longrightarrow } 0 . \\end{aligned} \\end{align*}"}
-{"id": "5497.png", "formula": "\\begin{align*} Z _ T ( f ) & = \\int _ 0 ^ T \\nabla L ^ { - 1 } [ f ] ( X _ s ) . d W _ s \\\\ & = L ^ { - 1 } [ f ] ( X _ T ) - L ^ { - 1 } [ f ] ( X _ 0 ) - \\int _ 0 ^ T L L ^ { - 1 } [ f ] ( X _ s ) d s , \\end{align*}"}
-{"id": "5535.png", "formula": "\\begin{align*} A _ T = \\left \\{ \\max _ { \\lambda , k , j } \\left | \\frac { 1 } { T } \\int _ 0 ^ T \\| \\tilde { \\Phi } _ { \\lambda , k , j } ( X _ t ) \\| ^ 2 d t - \\| \\tilde { \\Phi } _ { \\lambda , k , j } \\| _ { \\mu _ 0 } ^ 2 \\right | \\leq \\epsilon \\right \\} . \\end{align*}"}
-{"id": "4375.png", "formula": "\\begin{align*} \\prod _ { i = 0 } ^ { t - 1 } \\frac { n - i } { n - s - i } - 1 \\le \\frac { 2 t s } { n - s - t } . \\end{align*}"}
-{"id": "7310.png", "formula": "\\begin{align*} * ^ { - 1 } \\frac { \\partial } { \\partial \\xi _ i } * = \\eta ^ { i \\bar { j } } \\bar { \\xi } _ j , & & * ^ { - 1 } \\frac { \\partial } { \\partial \\bar { \\xi } _ i } * = \\eta ^ { \\bar { i } j } \\xi _ j , & & * ^ { - 1 } \\xi _ i * = \\eta _ { \\bar { j } i } \\frac { \\partial } { \\partial \\bar { \\xi } _ j } , & & * ^ { - 1 } \\bar { \\xi } _ i * = \\eta _ { j \\bar { i } } \\frac { \\partial } { \\partial \\xi _ j } , \\end{align*}"}
-{"id": "5608.png", "formula": "\\begin{align*} \\partial _ x u _ { \\delta , \\epsilon } ( x _ 1 , t ) = \\frac { 1 } { L } \\left [ u _ { \\delta , \\epsilon } ( L , t ) - u _ { \\delta , \\epsilon } ( 0 , t ) \\right ] . \\end{align*}"}
-{"id": "1135.png", "formula": "\\begin{align*} | A _ { i , j } | _ 2 ^ 2 = & \\int _ { [ y _ i , y _ i + a _ i ] } d x \\int _ { [ y _ j , y _ j + a _ j ] } | K ^ { G U E ( n ) } ( x , y ) | ^ 2 d y \\\\ = & \\int _ { [ y _ i , y _ i + a _ i ] } d x \\int _ { [ y _ j , y _ j + a _ j ] } O ( ( \\ln n ) ^ 2 ) d y = a _ i a _ j O ( ( \\ln n ) ^ 2 ) \\\\ \\leq & ( G _ n ( C _ 0 ) / S ( I ) ) ^ 2 O ( ( \\ln n ) ^ 2 ) = O \\left ( \\frac { \\ln n } { n ^ 2 } \\right ) O ( ( \\ln n ) ^ 2 ) = O \\left ( \\frac { ( \\ln n ) ^ 3 } { n ^ 2 } \\right ) . \\end{align*}"}
-{"id": "10005.png", "formula": "\\begin{align*} \\sigma _ { u } ( D ) = \\inf \\big \\{ \\sigma \\ , \\colon \\ , \\sum \\frac { a _ { n } } { n ^ { \\sigma } } n ^ { - s } \\in \\mathcal { D } _ { \\infty } ( X ) \\big \\} \\ , . \\end{align*}"}
-{"id": "5994.png", "formula": "\\begin{align*} G _ n ^ { ( t ) } ( s , q ) & : = ( G _ n ) ^ { ( t ) } ( s , q ) = ( 1 - t ) G _ n \\left ( t + s ( 1 - t ) , \\frac { q } { \\sqrt { 1 - t } } \\right ) \\\\ & = ( 1 - t ) g \\left ( n t + n s ( 1 - t ) - \\lfloor n t + n s ( 1 - t ) \\rfloor , \\frac { q } { \\sqrt { n ( 1 - t ) } } \\right ) . \\end{align*}"}
-{"id": "2842.png", "formula": "\\begin{align*} e ^ { ( k ) } _ { i , j } \\otimes \\sigma = \\sum _ { q = 1 } ^ { \\tau ( m + k + 1 ) } \\langle e ^ { ( k ) } _ { i , j } \\otimes \\sigma , e ^ { ( k + 1 ) } _ { i , q } \\rangle e ^ { ( k + 1 ) } _ { i , q } \\ , \\end{align*}"}
-{"id": "5050.png", "formula": "\\begin{align*} b ( a ( x y ) - ( a x ) y - x ( a y ) ) - a ( ( b x ) y ) + ( a ( b x ) ) y + ( b x ) ( a y ) - a ( x ( b y ) ) + ( a x ) ( b y ) + x ( a ( b y ) ) = \\\\ = - \\mathcal F ( a , b ) ( x y ) + ( \\mathcal F ( a , b ) x ) y + x ( \\mathcal F ( a , b ) y ) . \\end{align*}"}
-{"id": "6609.png", "formula": "\\begin{align*} u _ t : = 2 \\left ( \\sqrt { 5 } t - \\{ t \\varphi \\} + \\frac { 1 } { 2 } \\right ) . \\end{align*}"}
-{"id": "7642.png", "formula": "\\begin{align*} \\lim _ { { r \\to 0 ^ { + } } } { \\frac { 1 } { | B _ r ( x ) | } } \\int \\limits _ { { B _ r ( x ) } } \\ ! | f ( y ) - f ( x ) | \\ , d y = 0 , \\end{align*}"}
-{"id": "8072.png", "formula": "\\begin{align*} \\psi _ { c } ^ { + } = \\begin{pmatrix} 0 & 0 & 0 & 0 & - \\frac { 1 } { c ^ 2 } & 0 \\\\ 0 & 0 & 0 & - \\frac { 1 } { c } & 0 & 0 \\\\ 0 & 0 & - 1 & 0 & 0 & 0 \\\\ 0 & - c & 0 & 0 & 0 & 0 \\\\ - c ^ 2 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & - \\frac { \\alpha _ 1 ^ 3 \\alpha _ 2 ^ 3 } { c ^ 6 } \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "4861.png", "formula": "\\begin{align*} \\alpha ( w ^ * _ { ( j , \\xi ) } ) = \\sum _ { i \\geq 1 } ^ { } \\mu _ { i } w ^ * _ { ( i , \\xi ) } + \\sum _ { \\tau \\geq 1 } ^ { } \\gamma _ { \\tau } w _ { ( \\tau , \\xi ) } , \\end{align*}"}
-{"id": "3087.png", "formula": "\\begin{align*} \\| T _ { 1 1 } f \\| _ 2 \\le & \\| T _ { 1 1 } g \\| _ 2 + \\| T _ { 1 1 } b \\mathrm { I } _ { F ^ * } \\| _ 2 \\\\ \\le & C \\| g \\| _ 2 + \\| b \\| _ 2 . \\end{align*}"}
-{"id": "7376.png", "formula": "\\begin{align*} c _ { T } = c _ { W } + c _ R R . \\end{align*}"}
-{"id": "8467.png", "formula": "\\begin{align*} \\norm { F _ { t \\zeta } \\circ \\zeta } _ { L ^ { \\infty } } = & \\norm { \\frac { \\partial _ { \\alpha } F _ t ( \\zeta ( \\alpha , t ) , t ) } { \\zeta _ { \\alpha } } } _ { L ^ { \\infty } } \\leq \\norm { \\partial _ { \\alpha } F _ t ( \\zeta ( \\cdot , t ) , t ) } _ { \\infty } \\norm { \\frac { 1 } { \\zeta _ { \\alpha } } } _ { \\infty } \\\\ = & \\norm { D _ t \\mathfrak { F } ( \\alpha , t ) - D _ t \\zeta F _ { \\zeta } \\circ \\zeta } _ { H ^ s } \\norm { \\frac { 1 } { \\zeta _ { \\alpha } } } _ { \\infty } \\leq 1 0 \\epsilon , \\\\ \\end{align*}"}
-{"id": "882.png", "formula": "\\begin{align*} \\beta _ v = \\sum _ { i \\in Q _ 0 } d ^ i _ v \\alpha _ i \\end{align*}"}
-{"id": "9131.png", "formula": "\\begin{align*} I _ { M } ( \\rho _ { 1 o } ^ { ( N ) } , \\mathbf { b } ) = \\left \\{ \\begin{array} { c } 1 , \\\\ K _ { M } ( \\rho _ { 1 o } ^ { ( N ) } ( \\mathbf { x } _ { 1 } ) , \\mathbf { b } ) . \\end{array} \\right . \\end{align*}"}
-{"id": "8223.png", "formula": "\\begin{align*} & s _ 1 = p _ 1 = - 2 \\cdot 5 - 5 ^ 2 t , \\\\ & s _ 2 = ( s _ 1 p _ 1 - p _ 2 ) / 2 = 1 1 \\cdot 5 + 2 \\cdot 5 ^ 3 t + 5 ^ 4 t ^ 2 , \\\\ & s _ 3 = ( s _ 2 p _ 1 - s _ 1 p _ 2 + p _ 3 ) / 3 = - ( 2 8 \\cdot 5 + 1 1 \\cdot 5 ^ 3 t + 2 \\cdot 5 ^ 5 t ^ 2 + 5 ^ 6 t ^ 3 ) , \\\\ & s _ 4 = ( s _ 3 p _ 1 - s _ 2 p _ 2 + s _ 1 p _ 3 - p _ 4 ) / 4 = 7 \\cdot 5 ^ 2 + 2 8 \\cdot 5 ^ 3 t + 1 1 \\cdot 5 ^ 5 t ^ 2 + 2 \\cdot 5 ^ 7 t ^ 3 + 5 ^ 8 t ^ 4 . \\end{align*}"}
-{"id": "6092.png", "formula": "\\begin{align*} \\begin{gathered} \\Big ( \\bigcap \\limits _ { j } V _ j \\big ( ( a ^ 1 _ { 1 } , \\ldots , a ^ { k _ 1 } _ 1 ) , \\ldots , ( a ^ 1 _ j , \\ldots , a ^ { k _ j } _ j ) , \\ldots , ( a ^ 1 _ r , \\ldots , a ^ { k _ r } _ r ) \\big ) \\Big ) \\\\ \\geq \\Big ( \\varphi ^ { - 1 } ( \\underbrace { a , \\ldots \\ldots \\ldots , a } _ \\mathrm { \\sum \\limits _ { i , j } e ^ i _ j - r } ) \\Big ) \\\\ = \\sum \\limits _ { i , j } e ^ i _ j - r \\end{gathered} . \\end{align*}"}
-{"id": "4336.png", "formula": "\\begin{align*} I m ( \\Gamma ^ { \\boxtimes m - k } \\boxtimes ( \\Delta _ 2 - \\Gamma ) ^ { \\boxtimes k } ) ^ { S _ { m - k } \\times S _ { k } } & = \\gamma ^ { \\boxtimes m - k } \\boxtimes I m ( ( \\Delta _ 2 - \\Gamma ) ^ { \\boxtimes k } ) ^ { S _ { k } } \\\\ I m ( \\Gamma ^ { \\boxtimes \\{ 1 , \\ldots , n \\} \\setminus B } \\boxtimes ( \\Delta _ 2 - \\Gamma ) ^ { \\boxtimes B } ) & = \\gamma ^ { \\boxtimes \\{ 1 , \\ldots , n \\} \\setminus B } \\boxtimes I m ( ( \\Delta _ 2 - \\Gamma ) ^ { \\boxtimes B } ) . \\end{align*}"}
-{"id": "5751.png", "formula": "\\begin{align*} z ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } ( - \\mu _ { m , n } z ^ \\alpha ) = \\frac { - 1 } { z \\mu _ { m , n } } ( - \\mu _ { m , n } z ^ \\alpha E _ { \\alpha , \\alpha } ( - \\mu _ { m , n } z ^ \\alpha ) ) = \\frac { - 1 } { \\mu _ { m , n } } z ^ { - 1 } E _ { \\alpha , 0 } ( - \\mu _ { m , n } z ^ { \\alpha } ) . \\end{align*}"}
-{"id": "3815.png", "formula": "\\begin{align*} B _ { G , S , \\pi } ( x , \\ell ) : = \\{ y \\in G : d _ { S , \\pi } ( x , y ) \\le \\ell \\} . \\end{align*}"}
-{"id": "6103.png", "formula": "\\begin{align*} L ( \\delta / \\sigma ) = \\delta / \\sigma - \\log ( \\delta / \\sigma ) - 1 . \\end{align*}"}
-{"id": "7632.png", "formula": "\\begin{align*} \\hat a ( \\eta ) \\ = \\ \\sum _ { k \\in \\mathbb Z ^ d } a ( \\eta + k ) , \\eta \\in \\mathbb T ^ d , \\end{align*}"}
-{"id": "6016.png", "formula": "\\begin{align*} I ( \\omega ) = \\int _ 0 ^ 1 g ( t , \\dot { \\omega } ( t ) ) d t . \\end{align*}"}
-{"id": "8240.png", "formula": "\\begin{align*} S _ { C 5 } ( z , q ) = \\frac { ( - q ; q ) _ \\infty ( z q , z ^ { - 1 } q , q ^ 2 ; q ^ 2 ) _ { \\infty } - ( q ; q ) _ \\infty } { ( - q , z , z ^ { - 1 } ; q ) _ \\infty } , \\end{align*}"}
-{"id": "3851.png", "formula": "\\begin{align*} h _ { 0 0 } ^ { 2 m + 2 } { { \\gamma _ 1 } ^ i } _ { ( 2 m + 3 ) l } + h _ { 0 0 } ^ { m + 1 } { { \\gamma _ 2 } ^ i } _ { ( 2 m + 7 ) l } + W _ 0 ^ { 2 m + 2 } { { \\gamma _ 3 } ^ i } _ { ( 2 m + 7 ) l } = 0 , \\end{align*}"}
-{"id": "8144.png", "formula": "\\begin{align*} x \\triangleright \\varphi _ 1 \\otimes \\ldots \\otimes \\varphi _ g \\otimes v _ 1 \\otimes \\ldots \\otimes v _ n = \\Psi _ { g , n } ( x ) \\cdot \\varphi _ 1 \\otimes \\ldots \\otimes \\varphi _ g \\otimes v _ 1 \\otimes \\ldots \\otimes v _ n \\end{align*}"}
-{"id": "4781.png", "formula": "\\begin{align*} \\langle \\phi _ T , y _ T ^ 1 \\rangle _ { Y ' , Y } = 0 . \\end{align*}"}
-{"id": "1314.png", "formula": "\\begin{align*} \\rho \\big [ Z , \\delta _ { x } \\big ] = Z ( x ) . \\end{align*}"}
-{"id": "8984.png", "formula": "\\begin{align*} { \\tau } _ { Q } = { \\tau } _ { \\overline { Q } } , \\ \\ \\ \\end{align*}"}
-{"id": "6110.png", "formula": "\\begin{align*} y _ { i j } \\exp ( k _ { j j } \\omega _ { j j } ) + \\sum _ { l = j + 1 } ^ { i - 1 } y _ { i l } \\exp ( k _ { l l } \\omega _ { l l } ) k _ { l j } \\omega _ { l j } + y _ { i i } k _ { i j } \\omega _ { i j } \\exp ( k _ { i i } \\omega _ { i i } ) . \\end{align*}"}
-{"id": "4878.png", "formula": "\\begin{align*} X = \\{ z = x + \\sqrt { - 1 } y \\in \\mathbb { C } ^ n ; \\ , x \\in \\Omega , \\ , | y | ^ 2 < | x - a | ^ 2 + 1 \\} . \\end{align*}"}
-{"id": "1312.png", "formula": "\\begin{align*} J _ 1 ( U _ n ) = \\{ a b c a ^ { - 1 } \\in J ( U _ n ) : a \\in C \\cup C ^ { - 1 } \\} J _ 2 ( U _ n ) = \\{ a b c a ^ { - 1 } \\in J ( U _ n ) : a \\not \\in C \\cup C ^ { - 1 } \\} \\end{align*}"}
-{"id": "177.png", "formula": "\\begin{align*} \\pi _ { T M , X _ { 1 , V } , X _ { 2 , V } , X _ { 3 , V } , X _ { 4 , V } , c } = \\pi _ { T M } + \\lambda c ( { \\bf x } ) \\left ( X _ { 1 , V } \\wedge X _ { 2 , V } + X _ { 3 , V } \\wedge X _ { 4 , V } \\right ) \\end{align*}"}
-{"id": "8183.png", "formula": "\\begin{align*} A _ \\Lambda \\| f \\| ^ 2 \\leq \\sum _ { i \\in \\mathbb { Z } } \\| \\Lambda _ i f \\| ^ 2 = \\sum _ { i \\in \\mathbb { Z } } \\| \\Lambda _ 0 T ^ i f \\| ^ 2 & = \\sum _ { i \\in \\mathbb { Z } } \\| \\Lambda _ 0 T ^ i T ^ { - n } T ^ n f \\| ^ 2 \\\\ & = \\sum _ { i \\in \\mathbb { Z } } \\| \\Lambda _ 0 T ^ { i - n } T ^ n f \\| ^ 2 \\\\ & = \\sum _ { i \\in \\mathbb { Z } } \\| \\Lambda _ i T ^ n f \\| ^ 2 \\\\ & \\leq B _ \\Lambda \\| T ^ n f \\| ^ 2 \\leq B _ \\Lambda \\| T \\| ^ { 2 n } \\| f \\| ^ 2 , \\end{align*}"}
-{"id": "1327.png", "formula": "\\begin{align*} \\rho [ Z , P ] & = \\min _ { \\eta ( \\cdot ) } \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\Big [ \\eta ( x ) + \\frac { 1 } { \\alpha } \\big ( x - \\eta ( x ) \\big ) _ + \\Big ] \\ ; d y \\ ; d x \\\\ & = \\int _ 0 ^ 1 \\min _ { \\eta } \\Big [ \\eta + \\frac { 1 } { \\alpha } \\big ( x - \\eta \\big ) _ + \\Big ] \\ ; d x = \\int _ 0 ^ 1 x \\ ; d x = \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "4598.png", "formula": "\\begin{align*} \\int f _ { g , \\delta , j _ 1 } ( z ) \\dots f _ { g , \\delta , j _ { k - 1 } } ( z ) \\nu _ { g , j _ k } ( z ) d z = \\int ( \\hat { f } _ { g , \\delta , j _ 1 } \\overset { ( k - 1 ) - } { * \\dots * } \\hat { f } _ { g , \\delta , j _ { k - 1 } } ) ( - \\omega ) \\hat { \\nu } _ { g , j _ k } ( \\omega ) d \\omega = 0 . \\end{align*}"}
-{"id": "9005.png", "formula": "\\begin{align*} \\overline { \\nabla } _ { X } N = 0 \\ , . \\end{align*}"}
-{"id": "5277.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow u e p ( F ) ^ { - } } \\frac { ( u e p ( F ) - x ) F ^ { \\prime } ( x ) } { 1 - F ( x ) } = \\alpha . . \\end{align*}"}
-{"id": "8805.png", "formula": "\\begin{align*} \\overline { M } _ t ^ k & = e ^ { i \\xi \\langle \\overline { v } ^ k _ t , \\phi \\rangle } - \\int _ { 0 } ^ { t } e ^ { i \\xi \\langle \\overline { v } ^ k _ s , \\phi \\rangle } \\ , \\overline { A } ^ k ( \\textrm { d } s ) , t \\leq T . \\end{align*}"}
-{"id": "7177.png", "formula": "\\begin{gather*} \\mathcal { U } _ { r } ( \\mathcal { M } ) : = \\left \\{ ( \\xi , \\zeta ) : \\xi \\in \\mathcal { M } , \\ ; \\zeta \\in \\mathbb { J } _ { \\xi } ^ { - } , \\ ; \\left \\Vert \\zeta \\right \\Vert < r \\right \\} \\end{gather*}"}
-{"id": "7982.png", "formula": "\\begin{align*} g ( n ) = \\prod _ { p \\leq n } p ^ { \\frac { \\xi _ { p } ( 1 + \\xi _ { p } ) } { 2 } } \\ , , \\end{align*}"}
-{"id": "5996.png", "formula": "\\begin{align*} Y _ n ( t _ n , \\omega ) = \\rho ^ { G ^ { ( t _ n ) } _ n } \\left ( F \\circ L _ n \\left ( \\omega \\otimes _ { t _ n } W \\right ) \\right ) . \\end{align*}"}
-{"id": "1189.png", "formula": "\\begin{align*} Y Y ' & = Z , \\\\ Z ' & = \\frac { Y ^ 2 } { 2 } + Y - \\beta = \\frac { 1 } { 2 } \\left ( ( Y + 1 ) ^ 2 - ( 2 \\beta + 1 ) \\right ) , \\end{align*}"}
-{"id": "8056.png", "formula": "\\begin{align*} \\Omega _ R / d R \\cong \\mathbb { C } \\omega _ 0 \\oplus \\bigoplus _ { i = 3 } ^ { 4 } U _ i ^ { \\oplus \\Upsilon _ i ( \\epsilon _ i , \\nu _ i ) } \\oplus \\bigoplus _ { j = 1 } ^ { k - 1 } V _ j ^ { \\oplus \\frac { ( 1 - ( - 1 ) ^ j ) n } { k } } , \\end{align*}"}
-{"id": "9680.png", "formula": "\\begin{align*} | \\psi _ { n t } | = \\max ( | \\psi _ { 1 n } | , t ) \\ \\ \\mathbb T . \\end{align*}"}
-{"id": "8430.png", "formula": "\\begin{align*} ( \\partial _ t ^ 2 - i a \\partial _ { \\alpha } ) \\partial _ t ( I - \\mathfrak { H } ) ( z - \\bar { z } ) = & \\partial _ t ( \\partial _ t ^ 2 - i a \\partial _ { \\alpha } ) ( I - \\mathfrak { H } ) ( z - \\bar { z } ) + i a _ t ( ( I - \\mathfrak { H } ) ( z - \\bar { z } ) ) _ { \\alpha } \\\\ = & \\partial _ t g + i a _ t ( ( I - \\mathfrak { H } ) ( z - \\bar { z } ) ) _ { \\alpha } . \\end{align*}"}
-{"id": "1742.png", "formula": "\\begin{align*} \\begin{aligned} K ( t , s ) = & \\sqrt { \\frac { 2 H } { ( 1 - 2 H ) \\beta ( 1 - 2 H , H + \\frac { 1 } { 2 } ) } } \\\\ & \\cdot \\left [ \\left ( \\frac { t } { s } \\right ) ^ { H - \\frac { 1 } { 2 } } ( t - s ) ^ { H - \\frac { 1 } { 2 } } - \\left ( H - \\frac { 1 } { 2 } \\right ) s ^ { \\frac { 1 } { 2 } - H } \\int _ { s } ^ { t } u ^ { H - \\frac { 3 } { 2 } } ( u - s ) ^ { H - \\frac { 1 } { 2 } } d u \\right ] , \\end{aligned} \\end{align*}"}
-{"id": "6893.png", "formula": "\\begin{align*} \\det ( \\sigma L ) = \\big ( w ^ 1 _ { 0 1 } \\xi _ 1 - w ^ 1 _ { 1 0 } \\xi _ 2 ) ^ 2 + \\big ( w ^ 2 _ { 0 1 } \\xi _ 1 - w ^ 2 _ { 1 0 } \\xi _ 2 ) ^ 2 \\ne 0 \\end{align*}"}
-{"id": "7739.png", "formula": "\\begin{align*} | [ \\langle x , x \\rangle \\langle x , y \\rangle \\xi , \\xi ] | = \\| x \\| ^ 2 \\| \\langle x , y \\rangle \\| = \\| \\langle x , x \\rangle \\| \\| \\langle x , y \\rangle \\| . \\end{align*}"}
-{"id": "5102.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j \\in \\mathbb { S } } \\widehat { A } _ j U _ j y \\right \\| ^ p = \\left \\| \\sum _ { j \\in \\mathbb { S } } \\widehat { A } _ j \\widehat { L } _ j \\left ( \\sum _ { k \\in \\mathbb { S } } L _ k U _ k y \\right ) \\right \\| ^ p = \\sum _ { j \\in \\mathbb { S } } \\left \\| \\widehat { L } _ j \\left ( \\sum _ { k \\in \\mathbb { S } } L _ k U _ k y \\right ) \\right \\| ^ p = \\sum _ { j \\in \\mathbb { S } } \\| U _ j y \\| ^ p . \\end{align*}"}
-{"id": "5500.png", "formula": "\\begin{align*} \\| \\gamma \\| ^ 2 _ { H ^ p } = \\sum _ { l \\le J } \\sum _ r 2 ^ { 2 l p } | \\langle \\gamma , \\Phi _ { l , r } \\rangle | ^ 2 \\le 2 ^ { 2 J p } \\| \\gamma \\| ^ 2 _ { L ^ 2 } . \\end{align*}"}
-{"id": "7236.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } ( \\mu - \\phi ) ^ 2 ( x ) = K _ r * \\phi ( 0 ) - K _ r * \\phi ( x ) \\end{align*}"}
-{"id": "9934.png", "formula": "\\begin{align*} E ^ { \\gamma } [ \\frac { d \\mu } { d \\nu } ( X _ { 0 } ) | Y _ { [ 0 , n - 1 ] } , X _ { n } ] & = E ^ { \\gamma } [ \\frac { d \\mu } { d \\nu } ( X _ { 0 } ) | Y _ { [ 0 , \\infty ) } , X _ { [ n , \\infty ) } ] \\end{align*}"}
-{"id": "7993.png", "formula": "\\begin{align*} \\log f ( n ) = \\log n + \\log g ( n ) - \\log h ( n ) = \\frac { n } { 2 } + O \\left ( \\frac { n } { \\log n } \\right ) , \\end{align*}"}
-{"id": "2068.png", "formula": "\\begin{align*} C _ { \\rho \\varphi } ( \\rho , \\varphi , t = t _ 0 | { { \\bar r } _ { \\rm t x } } , { t _ 0 } ) = \\frac { { \\delta ( \\rho - { \\rho _ { \\rm t x } } ) } } { \\rho } \\delta ( \\varphi - { \\varphi _ { \\rm t x } } ) . \\end{align*}"}
-{"id": "6789.png", "formula": "\\begin{align*} f ^ { 2 r } + \\sum _ { i = 1 } ^ m g _ i ^ 2 \\in I \\end{align*}"}
-{"id": "5266.png", "formula": "\\begin{align*} z _ { N _ { 1 } } = \\underset { p \\geq n } { \\sup } x _ { p } > \\ell - 1 , \\end{align*}"}
-{"id": "8291.png", "formula": "\\begin{align*} g _ 1 : = 2 [ z _ { t t } , \\mathfrak { H } ] \\frac { \\bar { z } _ { t \\alpha } } { z _ { \\alpha } } + 2 [ z _ t , \\mathfrak { H } ] \\frac { \\bar { z } _ { t t \\alpha } } { z _ { \\alpha } } - \\frac { 1 } { \\pi i } \\int \\Big ( \\frac { z _ t ( \\alpha , t ) - z _ t ( \\beta , t ) } { z ( \\alpha , t ) - z ( \\beta , t ) } \\Big ) ^ 2 \\bar { z } _ { t \\beta } d \\beta . \\end{align*}"}
-{"id": "7897.png", "formula": "\\begin{align*} & \\dim \\mathcal { M } ( L ) = \\dim \\mathcal { M } ( L / I ) + ( \\dim L / \\gamma _ { 2 } ( L ) - 1 ) \\dim I - \\dim \\ker ( \\lambda _ { c } ) \\cr & \\leq \\frac { 1 } { 2 } ( n - m - 1 ) ( n + m - 2 ) - ( \\sum \\limits _ { i = 2 } ^ { c - 1 } n - m - i ) + ( n - m - 1 ) - ( n - m - c ) \\cr & = \\dim \\mathcal { M } ( L ) , \\end{align*}"}
-{"id": "8764.png", "formula": "\\begin{align*} F = F ( x ^ 1 , x ^ 2 ) = f ^ 2 ( x ^ 1 , x ^ 2 ) \\end{align*}"}
-{"id": "9473.png", "formula": "\\begin{align*} J ^ { m } = \\vert t \\vert ^ { m - 2 } t . \\end{align*}"}
-{"id": "4093.png", "formula": "\\begin{align*} R _ { \\alpha , - } ^ { - 1 } ( z ) R _ { \\alpha , + } ( z ) & = L _ { \\alpha } ( z ) \\begin{pmatrix} 1 & 0 & 0 \\\\ e ^ { \\pm 3 ( \\omega - \\omega ^ 2 ) z ^ { 1 / 3 } } & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} L _ { \\alpha } ^ { - 1 } ( z ) , z \\in i \\mathbb R ^ { \\pm } , \\\\ & = L _ { \\alpha } ( z ) \\begin{pmatrix} 1 & 0 & 0 \\\\ e ^ { - \\frac { 3 \\sqrt { 3 } } { 2 } | z | ^ { 1 / 3 } } & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} L _ { \\alpha } ^ { - 1 } ( z ) . \\end{align*}"}
-{"id": "5596.png", "formula": "\\begin{align*} A & = \\left ( - i \\int _ { - \\infty } ^ 0 x \\bar { u } _ { x x } u _ x d x + i \\alpha \\int _ { - \\infty } ^ 0 x \\bar { u } v u _ x d x + i \\beta \\int _ { - \\infty } ^ 0 x | u | ^ 2 \\bar { u } u _ x d x \\right ) \\\\ & = - \\int _ { - \\infty } ^ 0 x \\bar { u } _ { x x } u _ x d x + \\alpha \\ , \\int _ { - \\infty } ^ 0 x \\bar { u } v u _ x d x + \\beta \\ , \\int _ { - \\infty } ^ 0 x | u | ^ 2 \\bar { u } u _ x d x \\\\ & : = A _ 1 + A _ 2 + A _ 3 . \\end{align*}"}
-{"id": "135.png", "formula": "\\begin{align*} \\left ( \\mathcal { L } _ { X } \\pi \\right ) ( d f , d g ) = \\mathcal { L } _ { X } \\left ( \\pi ( d f , d g ) \\right ) - \\pi ( \\mathcal { L } _ { X } ( d f ) , d g ) - \\pi ( d f , \\mathcal { L } _ { X } ( d g ) ) , \\end{align*}"}
-{"id": "3234.png", "formula": "\\begin{align*} \\mu _ { k + 1 } : = \\min \\left ( \\beta \\mu _ k , \\gamma _ 2 \\mu _ k ^ { 1 + c \\alpha } \\right ) , \\varepsilon _ { k + 1 } : = \\gamma _ 1 \\mu _ { k + 1 } ^ { 1 + \\alpha } . \\end{align*}"}
-{"id": "5302.png", "formula": "\\begin{align*} U _ i = \\{ ( u _ 1 ^ { ( i ) } , \\dots , u _ k ^ { ( i ) } , x _ { k + 1 } , \\dots x _ N ) \\in \\C ^ N \\} ( i = 1 , \\dots , k ) \\end{align*}"}
-{"id": "9461.png", "formula": "\\begin{align*} ( - \\Delta _ { m } ) ^ { s } \\phi ( x ) = 2 \\displaystyle \\lim _ { \\varepsilon \\rightarrow 0 } \\displaystyle \\int _ { \\mathbb { R } ^ { N } \\setminus B _ { \\varepsilon } ( x ) } \\frac { \\vert \\phi ( x ) - \\phi ( y ) \\vert ^ { m - 2 } ( \\phi ( x ) - \\phi ( y ) ) } { \\vert x - y \\vert ^ { N + s m } } \\dd y , \\ \\ \\ \\forall x \\in \\mathbb { R } ^ { N } , \\end{align*}"}
-{"id": "3535.png", "formula": "\\begin{align*} \\Theta ( W ) ^ { 2 } = \\frac { 3 \\cdot 5 ^ { 2 } } { 2 ^ { 6 } } \\bigg \\{ \\bigg ( \\sum _ { i = 1 } ^ { 1 0 } \\Theta _ { i } ( W ) ^ { 4 } \\bigg ) ^ { 2 } - 4 \\ , \\sum _ { i = 1 } ^ { 1 0 } \\Theta _ { i } ( W ) ^ { 8 } \\bigg \\} \\end{align*}"}
-{"id": "1523.png", "formula": "\\begin{align*} \\gamma = \\underset { 1 \\le i \\le m } { \\rm m i n } \\frac { b _ i + 1 } { a _ i } . \\end{align*}"}
-{"id": "3729.png", "formula": "\\begin{align*} ( 2 n - 1 - q ) t _ * ^ { 2 n - 2 - q } a \\| u \\| ^ { 2 n } & + ( n - 1 - q ) t _ * ^ { n - 2 - q } b \\| u \\| ^ n + \\dfrac { q } { t _ * ^ { q + 1 } } \\int _ { \\Omega } ( | x | ^ { - \\mu } * F ( t _ * u ) ) f ( t _ * u ) u ~ d x \\\\ & = t _ * ^ { - q } \\bigg [ \\int _ { \\Omega } ( | x | ^ { - \\mu } * F ( t _ * u ) ) f ^ { ' } ( t _ * u ) u ^ 2 ~ d x + \\int _ { \\Omega } ( | x | ^ { - \\mu } * f ( t _ * u ) u ) f ( t _ * u ) . u ~ d x \\bigg ] . \\end{align*}"}
-{"id": "8798.png", "formula": "\\begin{align*} \\overline { M } _ t = e ^ { i \\xi \\langle \\overline { v } _ t , \\phi \\rangle } - \\int _ { 0 } ^ { t } e ^ { i \\xi \\langle \\overline { v } _ s , \\phi \\rangle } \\ , \\overline { A } ( \\textrm { d } s ) . \\end{align*}"}
-{"id": "889.png", "formula": "\\begin{align*} & \\implies \\lambda ( \\phi _ u , \\phi _ v ) \\geq 0 ; \\\\ & \\implies \\lambda ( \\phi _ u , \\phi _ v ) \\leq 0 . \\end{align*}"}
-{"id": "1516.png", "formula": "\\begin{align*} | P | ^ { e q ^ { n } - ( q - 1 ) \\sum _ { k = 1 } ^ { n - 1 } q ^ { k } \\min \\{ e , \\lfloor \\frac { k } { d } \\rfloor \\} } = | P | ^ { e q ^ { n } - \\sum _ { k = 1 } ^ { q ^ { n } - 1 } \\min \\{ e , \\lfloor \\frac { k } { q ^ d } \\rfloor + \\lfloor \\frac { k } { q ^ { 2 d } } \\rfloor + \\cdots \\} } . \\end{align*}"}
-{"id": "507.png", "formula": "\\begin{align*} r _ 2 = g - 3 g ' \\geq 0 ; r _ { 4 } = 2 g ' + 2 . \\end{align*}"}
-{"id": "9464.png", "formula": "\\begin{align*} ( - \\Delta _ { p } ) u + ( - \\Delta _ { q } ) u = \\vert u \\vert ^ { p ^ * - 2 } u + \\lambda g ( x ) \\vert u \\vert ^ { r - 2 } u , \\ \\ \\ x \\in \\mathbb { R } ^ N . \\end{align*}"}
-{"id": "8472.png", "formula": "\\begin{align*} \\partial _ { \\alpha } ^ n q = \\sum _ { k = 1 } ^ n \\sum \\frac { n ! } { ( k _ 1 ) ! . . . ( k _ n ) ! } \\Big ( \\sum _ { j = 1 } ^ 2 \\partial _ { \\alpha } ^ k f _ j ( \\cdot , t ) \\circ g \\Big ) \\prod _ { l = 1 } ^ n \\Big ( \\frac { \\partial _ { \\alpha } ^ l g } { l ! } \\Big ) ^ { k _ l } \\end{align*}"}
-{"id": "7004.png", "formula": "\\begin{align*} \\gamma _ 1 ^ 2 = \\gamma _ 2 + 2 , \\gamma _ 1 \\gamma _ a = \\gamma _ { a + 1 } + \\gamma _ { a - 1 } \\ \\ . \\end{align*}"}
-{"id": "1575.png", "formula": "\\begin{align*} | w _ { m + 1 } - g ( w _ m ) | & = | \\pi _ w \\circ G ( w _ m , y _ m ) - g ( w _ m ) | \\\\ & = O ( w _ m ^ 3 , w _ m y _ m , y _ m ^ 2 ) \\\\ & = O ( w _ m ^ 3 ) \\end{align*}"}
-{"id": "6638.png", "formula": "\\begin{align*} A _ { \\lambda } f ( x ) = \\lambda ^ { 2 - d } \\sum _ { n \\in \\mathbb Z ^ { d } \\ ; : \\ ; \\lvert n \\rvert = \\lambda } f ( x - n ) \\end{align*}"}
-{"id": "7395.png", "formula": "\\begin{align*} \\zeta _ { A } ( G ) = \\{ g \\in G \\ , | \\ , g \\cdot A \\cap A \\neq \\emptyset \\} . \\end{align*}"}
-{"id": "7439.png", "formula": "\\begin{align*} \\displaystyle \\lim \\limits _ { s \\to 0 ^ + } s ^ n P _ { m , n , s } ( \\underline { x } ) = \\lim \\limits _ { s \\to 0 ^ + } s ^ n S _ { m , n , s } ( \\underline { x } ) = P _ { m , n } ^ * ( \\underline { x } ) = S _ { m , n } ^ * ( \\underline { x } ) \\end{align*}"}
-{"id": "4312.png", "formula": "\\begin{align*} B _ k ( \\partial F ) : = \\{ v \\in C ^ 0 ( \\partial F ) : v | _ e \\in \\mathbb { P } _ k ( e ) e \\subset \\partial F \\} . \\end{align*}"}
-{"id": "8269.png", "formula": "\\begin{align*} c _ 0 ^ j : = ( \\Phi ^ { - 1 } ) _ z ( \\omega _ 0 ^ j , t ) , \\omega _ 0 ^ j : = \\Phi ( z _ j ( t ) , t ) . \\end{align*}"}
-{"id": "2233.png", "formula": "\\begin{align*} & R ( z _ 1 ) - R ( z _ 2 ) = ( z _ 2 - z _ 1 ) R ( z _ 1 ) R ( z _ 2 ) \\\\ & R ( z _ 1 ) R ( z _ 2 ) = 1 / ( z _ 2 - z _ 1 ) ( R ( z _ 1 ) - R ( z _ 2 ) . \\end{align*}"}
-{"id": "69.png", "formula": "\\begin{align*} \\begin{pmatrix} \\sigma ^ + _ { - 1 } \\\\ \\sigma ^ + _ 1 \\end{pmatrix} = \\begin{pmatrix} 1 - c & c \\\\ c & 1 - c \\end{pmatrix} \\begin{pmatrix} \\sigma ^ - _ { - 1 } \\\\ \\sigma ^ - _ 1 \\end{pmatrix} , c = \\frac { g ' ( s ) \\delta } { g ' ( s ) \\delta + 1 } \\ , , \\end{align*}"}
-{"id": "6403.png", "formula": "\\begin{align*} \\phi _ { k } = \\frac { \\phi _ { k - 1 } } { \\phi _ { k - 1 } + \\delta _ { k } } , \\qquad \\phi _ { 0 } = 1 . \\end{align*}"}
-{"id": "790.png", "formula": "\\begin{align*} \\lambda _ 1 ^ { b _ s } \\prod _ { i = 2 } ^ { { { s } } } \\lambda _ i ^ { b _ { i - 1 } } = \\prod _ { i = 1 } ^ { { s } } \\lambda _ i ^ { b _ i \\mu } a ^ { q ^ { { s } } - 1 } , \\end{align*}"}
-{"id": "9303.png", "formula": "\\begin{align*} \\tilde \\Theta ( \\hat x , \\hat y ) = - \\tau \\mathcal { L } \\tilde \\eta ( \\hat y ) , \\end{align*}"}
-{"id": "5831.png", "formula": "\\begin{align*} \\frac { \\phi _ { X } ( \\frac { t _ 1 } { n } ) \\int x ^ 2 e ^ { \\frac { i t _ 1 x } { n } } f ( x ) d x - ( \\int x ^ 2 e ^ { \\frac { i t _ 1 x } { n } } f ( x ) d x ) ^ 2 } { [ \\phi _ { X } ( \\frac { t _ 1 } { n } ) ] ^ 2 } = \\sigma ^ 2 \\end{align*}"}
-{"id": "3673.png", "formula": "\\begin{align*} p _ i = p ( x _ i ; \\eta ) > 0 \\end{align*}"}
-{"id": "8005.png", "formula": "\\begin{align*} \\sum _ { \\substack { n _ i \\in \\{ - 1 , 0 , 1 \\} , \\\\ i = 1 , 2 , \\cdots , j } } \\prod _ { i \\leq j } e ^ { - a | n _ i - y _ i | ^ 2 } = \\prod _ { i \\leq j } A _ i , \\end{align*}"}
-{"id": "877.png", "formula": "\\begin{align*} \\lambda ( \\gamma _ 1 , \\gamma _ 2 ) = \\chi ( \\gamma _ 2 , \\gamma _ 1 ) - \\chi ( \\gamma _ 1 , \\gamma _ 2 ) . \\end{align*}"}
-{"id": "2219.png", "formula": "\\begin{align*} & s = \\dot { p } ( t ) x ^ 2 - 2 \\dot { g } ( t ) x - \\frac { 2 p ( t ) x ^ 2 } { t + 1 } + 2 p ( t ) x u + \\frac { 2 g ( t ) x } { t + 1 } - 2 g ( t ) u + 1 0 0 0 t ^ 2 + 1 0 0 0 x ^ 2 - 2 0 0 0 z x \\\\ & s _ f = 1 0 x ^ 2 ( t _ f ) + 1 0 z ^ 2 ( t _ f ) - 2 x ( t _ f ) z ( t _ f ) - p ( t _ f ) x ^ 2 ( t _ f ) + 2 g ( t _ f ) x ( t _ f ) \\end{align*}"}
-{"id": "9727.png", "formula": "\\begin{align*} \\Sigma _ { \\kappa } ^ { m , { \\rm g } } ( u ) : = \\Sigma _ { \\kappa } ^ { m } ( u ) \\setminus \\Sigma _ { \\kappa } ^ { m , { \\rm a } } ( u ) \\end{align*}"}
-{"id": "9230.png", "formula": "\\begin{align*} ( \\alpha ^ * ) ^ 2 \\norm { z } ^ 2 + \\norm { M ( u + x _ 0 v ) } ^ 2 - ( p _ { 1 0 } - x _ 0 ) ^ 2 = 0 . \\end{align*}"}
-{"id": "1305.png", "formula": "\\begin{align*} \\psi ( t ) = \\begin{cases} \\bigl ( e , ( t , e ) \\bigr ) & , \\\\ ( t , 0 ) & , \\end{cases} \\end{align*}"}
-{"id": "9646.png", "formula": "\\begin{align*} & \\sum _ { n = 0 } ^ { N } \\frac { ( q ; q ^ 2 ) _ { n } ( q ^ 2 ; q ^ 2 ) _ N ( z q ^ 2 ; q ^ 2 ) _ { N - n } z ^ n q ^ { 2 n } } { ( z q ; q ^ 2 ) _ { n + 1 } ( q ^ 2 ; q ^ 2 ) _ { N - n } ( z q ^ 2 ; q ^ 2 ) _ { N + 1 } } \\\\ & = \\frac { 1 } { ( 1 - z q ^ { 2 N + 3 } ) } \\sum _ { n = 0 } ^ { N } \\left ( \\frac { ( q ; q ) _ { 2 n } z ^ { 2 n } q ^ { 2 n ^ 2 + 3 n } } { ( z q ; q ) _ { 2 n + 1 } } + \\frac { ( q ; q ) _ { 2 n + 1 } z ^ { 2 n + 1 } q ^ { ( 2 n + 1 ) ( n + 2 ) } } { ( z q ; q ) _ { 2 n + 2 } } \\right ) \\frac { ( q ^ 2 ; q ^ 2 ) _ N ( z q ^ 5 ; q ^ 2 ) _ N } { ( q ^ 2 ; q ^ 2 ) _ { N - n } ( z q ^ 5 ; q ^ 2 ) _ { N + n } } . \\end{align*}"}
-{"id": "681.png", "formula": "\\begin{align*} \\{ \\lambda _ { \\vec a } = [ \\mathrm { i d } ; \\lambda _ { \\vec a , 1 } , \\dots , \\lambda _ { \\vec a , m } ; e _ m ] \\} _ { \\vec a = a _ 1 \\dots a _ m } \\end{align*}"}
-{"id": "486.png", "formula": "\\begin{align*} \\sigma _ i ( A ( f _ i ) ) = A ( f _ i ) . \\end{align*}"}
-{"id": "6712.png", "formula": "\\begin{align*} \\varphi ^ { ( 1 ) } _ k = P _ { 1 k } - ( m \\dot { q } _ i + u _ i ) ( \\alpha ' _ { i k } + \\beta ' _ { i l k } \\ , \\dot { \\bar { q } } _ l ) = 0 . \\end{align*}"}
-{"id": "7222.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } ( \\mu - \\phi ) ^ { 2 } = \\frac { 1 } { 2 } \\mu ^ { 2 } - L _ r \\phi + \\frac { 1 } { 2 } \\widehat { \\phi ^ 2 } ( 0 ) . \\end{align*}"}
-{"id": "3065.png", "formula": "\\begin{align*} \\widehat { T _ j f } ( \\xi ) = \\gamma _ \\beta \\frac { \\xi _ j } { | \\xi | ^ { \\beta + 1 } } , 0 < \\beta < n , \\gamma _ \\beta = i \\pi ^ { n / 2 - \\beta } \\frac { \\Gamma ( \\frac { \\beta + 1 } { 2 } ) } { \\Gamma ( \\frac { n + 1 - \\beta } { 2 } ) } , \\end{align*}"}
-{"id": "7829.png", "formula": "\\begin{align*} \\lim _ n \\| \\langle x _ { n _ k } , y \\rangle - \\langle x , y \\rangle \\| = 0 , \\end{align*}"}
-{"id": "1011.png", "formula": "\\begin{align*} F _ n ( x ) = \\frac { 8 x + 3 \\ln ( 2 \\ln n ) } { 2 n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } } + \\frac { ( 3 2 \\ln n ) ^ { \\frac { 1 } { 2 } } } { n } , \\end{align*}"}
-{"id": "2169.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty 2 ^ { n ( \\mathbf N + \\delta ) } \\| \\phi ( \\cdot ) \\Psi _ n ( \\| \\cdot \\| ) \\| _ { L ^ \\infty } = A < \\infty \\end{align*}"}
-{"id": "2471.png", "formula": "\\begin{align*} y ^ { n } = ( x - a _ { 1 } ) ^ { n _ { 1 } } \\cdots ( x - a _ { t } ) ^ { n _ { t } } , \\end{align*}"}
-{"id": "4361.png", "formula": "\\begin{align*} \\Gamma _ { \\sigma } ^ { \\gamma } ( A ) = \\sigma ^ { \\gamma / 2 } A \\sigma ^ { \\gamma / 2 } . \\end{align*}"}
-{"id": "3488.png", "formula": "\\begin{align*} h _ M ( \\varphi ) = q _ M ( 0 ) - ( 2 ( M + 1 ) / N ) ^ { 1 / 3 } - \\frac { 1 } { 2 } ( 2 ( M + 1 ) / N ) ^ { 3 / 1 0 } , \\end{align*}"}
-{"id": "2599.png", "formula": "\\begin{align*} g ( t , \\theta , x ) ^ { \\gamma _ 0 - 1 } = \\Big \\| \\| ( \\phi ^ { - 1 } f ) ( \\xi _ t ) ^ { \\gamma _ 0 - 1 } e ^ { - J _ g ( t , r , \\xi ) } \\| _ { \\Pi _ x ^ { ( \\phi ) } ; \\frac { 1 } { \\gamma _ 0 - 1 } } \\Big \\| _ { \\mathbf 1 _ { 0 \\leq r \\leq \\theta } d r ; \\frac { 1 } { \\gamma _ 0 - 1 } } , t \\geq 0 , \\theta \\geq 0 , x \\in E . \\end{align*}"}
-{"id": "695.png", "formula": "\\begin{align*} \\forall \\psi \\in A u t ( T _ p ) , \\left [ g _ { i \\bar { j } } \\right ] = J a c _ \\mathbb { C } \\left ( \\psi \\right ) ^ T \\left [ g _ { i \\bar { j } } \\circ \\psi \\right ] \\overline { J a c _ \\mathbb { C } \\left ( \\psi \\right ) } . \\end{align*}"}
-{"id": "6052.png", "formula": "\\begin{align*} \\epsilon ( C ) = \\bigg | \\left | \\{ i \\in \\Z _ \\ell \\mid ( x _ i , x _ { i + 1 } ) = ( m - 1 , 0 ) \\} \\big | - \\right | \\{ i \\in \\Z _ \\ell \\mid ( x _ i , x _ { i + 1 } ) = ( 0 , m - 1 ) \\} \\big | \\bigg | . \\end{align*}"}
-{"id": "7262.png", "formula": "\\begin{align*} \\phi _ p ( x ) = \\frac { 3 x ^ 2 - \\pi ^ 2 } { 1 8 } , \\end{align*}"}
-{"id": "8697.png", "formula": "\\begin{align*} J _ { T _ 1 } X & = 4 ( X _ 2 S _ 1 - X _ 1 S _ 2 + X _ 4 S _ 3 - X _ 3 S _ 4 ) \\\\ J _ { T _ 2 } X & = 4 ( X _ 3 S _ 1 - X _ 4 S _ 2 - X _ 1 S _ 3 + X _ 2 S _ 4 ) \\\\ J _ { T _ 3 } X & = 4 ( X _ 4 S _ 1 + X _ 3 S _ 2 - X _ 2 S _ 3 - X _ 1 S _ 4 ) . \\end{align*}"}
-{"id": "4537.png", "formula": "\\begin{align*} \\int f ^ { k - 1 } _ { g , \\delta } ( z ) d \\nu _ g ( z ) = \\int \\hat { \\nu } _ g ( \\omega ) \\hat { f } _ { g , \\delta } * \\dots * \\hat { f } _ { g , \\delta } ( - \\omega ) d \\omega . \\end{align*}"}
-{"id": "5576.png", "formula": "\\begin{align*} \\mathcal { M } ^ { \\pm } ( t ) = \\mathcal { M } ^ { \\pm } ( 0 ) , \\end{align*}"}
-{"id": "8582.png", "formula": "\\begin{align*} ( \\lambda _ + ( H ) , \\lambda _ + ( Z ) ) = ( \\tfrac { 1 } { 2 } - c , - \\tfrac { 1 } { 2 } + c ) \\qquad \\textup { a n d } ( \\lambda _ - ( H ) , \\lambda _ - ( Z ) ) = ( \\tfrac { 3 } { 2 } - c , - \\tfrac { 1 } { 2 } - c ) . \\end{align*}"}
-{"id": "6604.png", "formula": "\\begin{align*} P _ N ( \\varphi ) = \\prod _ { j = 1 } ^ m P _ { F _ { n _ j } } ( \\varphi , \\varepsilon _ j ) , \\varepsilon _ j = - \\sum _ { s = j + 1 } ^ m ( - \\varphi ) ^ { n _ s } . \\end{align*}"}
-{"id": "3362.png", "formula": "\\begin{align*} \\dim H = \\sum _ { 1 \\le i \\le \\ell } | \\mathcal O _ i | \\cdot \\dim V _ i \\cdot \\dim P ( V _ i ) , \\end{align*}"}
-{"id": "4886.png", "formula": "\\begin{align*} \\big ( s \\colon S \\rightarrowtail I \\big ) \\wedge \\big ( t \\colon T \\rightarrowtail I \\big ) = \\big ( \\lambda _ I \\circ ( s \\otimes t ) \\colon S \\otimes T \\rightarrowtail I \\big ) \\text , \\end{align*}"}
-{"id": "2538.png", "formula": "\\begin{align*} \\alpha _ 0 : = \\operatorname * { e s s \\ , i n f } _ { \\rho ( d x ) } \\alpha ( x ) : = \\sup \\{ r : \\rho ( x : \\alpha ( x ) < r ) = 0 \\} \\in \\mathbb R . \\end{align*}"}
-{"id": "5617.png", "formula": "\\begin{align*} \\frac { \\alpha } { \\gamma } \\frac { d } { d t } \\int _ { 0 } ^ { + \\infty } v ^ 2 d x + \\mathcal { Q } ^ { v } ( t ) = 2 \\alpha \\int _ { 0 } ^ { + \\infty } ( | u | ^ 2 ) _ x v d x , \\end{align*}"}
-{"id": "5195.png", "formula": "\\begin{align*} H ^ { s t } ( x ) = H ( \\alpha ( t ) \\left \\{ \\alpha ( s ) x + \\beta ( s ) \\right \\} + \\beta ( t ) ) . \\end{align*}"}
-{"id": "5830.png", "formula": "\\begin{align*} \\left . \\frac { \\partial \\phi _ { s _ n ^ 2 } ( t _ 1 , t _ 2 ) } { \\partial t _ 2 } \\right | _ { t _ 2 = 0 } & = \\left . \\frac { \\partial \\phi _ { ( \\bar X _ n , s _ n ^ 2 ) } ( 0 , t _ 2 ) } { \\partial t _ 2 } \\right | _ { t _ 2 = 0 } = i \\sigma ^ 2 \\end{align*}"}
-{"id": "2531.png", "formula": "\\begin{align*} \\mathbf P _ \\mu [ X _ t ( \\phi ) ] = e ^ { \\lambda t } \\mu ( \\phi ) , t \\geq 0 , \\end{align*}"}
-{"id": "1631.png", "formula": "\\begin{align*} \\langle f , g \\rangle _ { L ^ 2 } : = \\int _ 0 ^ 1 ( f _ 1 g _ 1 + f _ 2 g _ 2 ) \\ , d x \\ , . \\end{align*}"}
-{"id": "8188.png", "formula": "\\begin{align*} f = \\sum _ { i \\in \\mathbb { Z } } \\Lambda _ i ^ * \\Theta _ i f = \\sum _ { i \\in \\mathbb { Z } } ( T ^ * ) ^ i \\Lambda _ 0 ^ * \\Theta _ 0 S ^ i f & = T ^ * \\Big ( \\sum _ { i \\in \\mathbb { Z } } ( T ^ * ) ^ { i - 1 } \\Lambda _ 0 ^ * \\Theta _ 0 S ^ { i - 1 } \\Big ) S f \\\\ & = T ^ * \\Big ( \\sum _ { i \\in \\mathbb { Z } } \\Lambda _ i ^ * \\Theta _ i \\Big ) S f = T ^ * S f . \\end{align*}"}
-{"id": "9719.png", "formula": "\\begin{align*} \\tilde { u } \\ , L _ a \\tilde { u } = 0 \\quad B _ 1 . \\end{align*}"}
-{"id": "2692.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial r } \\Re { f _ { \\tau , \\theta } ( r + i y ) } & = \\frac { 4 \\tau } { 1 - \\tau ^ { 2 } } \\cos ^ { 2 } \\theta + \\frac { 1 - \\tau ^ { 2 } } { \\tau } \\frac { r ^ { 2 } - r ( \\tau + \\frac { 1 } { \\tau } ) + 1 - y ^ { 2 } } { \\big ( ( r - \\frac { 1 } { \\tau } ) ^ { 2 } + y ^ { 2 } \\big ) \\big ( ( r - \\tau ) ^ { 2 } + y ^ { 2 } \\big ) } . \\end{align*}"}
-{"id": "4013.png", "formula": "\\begin{align*} A ( n ) = \\sum _ { \\delta \\mid N } \\alpha _ { 1 , \\delta } b _ { 1 } ( n / \\delta ) \\equiv \\sum _ { t } \\beta _ { t } a _ H ( \\gamma _ { t } n ) \\end{align*}"}
-{"id": "8313.png", "formula": "\\begin{align*} \\mathfrak { H } f ( \\alpha ) : = \\frac { 1 } { \\pi i } p . v . \\int _ { - \\infty } ^ { \\infty } \\frac { z _ { \\beta } ( \\beta , t ) } { z ( \\alpha , t ) - z ( \\beta , t ) } f ( \\beta , t ) d \\beta . \\end{align*}"}
-{"id": "5048.png", "formula": "\\begin{align*} \\sup _ { 0 \\leq x \\leq 1 } | \\hat r ( x ) - r ( x ) | = O _ p ( q _ n ) . \\end{align*}"}
-{"id": "5114.png", "formula": "\\begin{align*} m = \\sum _ { j = 1 } ^ m p _ j ( e _ j ) = \\sum _ { j = 1 } ^ m p _ j \\left ( \\sum _ { k = 1 } ^ n f _ k ( e _ j ) \\tau _ k \\right ) = \\sum _ { k = 1 } ^ n \\sum _ { j = 1 } ^ m f _ k ( e _ j ) p _ j ( \\tau _ k ) = \\sum _ { k = 1 } ^ n f _ k \\left ( \\sum _ { j = 1 } ^ m p _ j ( \\tau _ k ) e _ j \\right ) = \\sum _ { k = 1 } ^ n f _ k ( \\tau _ k ) . \\end{align*}"}
-{"id": "7094.png", "formula": "\\begin{align*} c _ { N , D } ( M ) : = \\max \\left ( \\frac { N } { M } , \\frac { M } { N } \\right ) ^ { D } ~ . \\end{align*}"}
-{"id": "5739.png", "formula": "\\begin{align*} F _ { m , n } ( t ) = \\frac { 1 } { | | J _ 0 ( \\gamma m x ) \\sin ( n \\pi y ) | | _ 2 ^ 2 } \\int \\limits _ 0 ^ 1 \\int \\limits _ 0 ^ 1 x f ( x , y , t ) J _ 0 ( \\gamma _ m x ) \\sin ( n \\pi y ) d x d y . \\end{align*}"}
-{"id": "4134.png", "formula": "\\begin{align*} Q ( z ) N ^ { - 1 } ( z ) = \\mathbb { I } + \\mathcal { O } ( n ^ { - 1 } ) n \\to \\infty , \\end{align*}"}
-{"id": "8034.png", "formula": "\\begin{align*} \\tau ( f ( t ) \\partial ) = \\phi ( f ) ( \\phi \\circ \\partial \\circ \\phi ^ { - 1 } ) \\enspace \\mbox { f o r a l l } f \\in R _ 2 ( p ) . \\end{align*}"}
-{"id": "8741.png", "formula": "\\begin{align*} F _ { \\omega } ( \\lambda ) = ( \\tau \\lambda ) ^ { - \\alpha } \\omega ^ { ( \\alpha , 0 ) } ( 1 - \\tau \\lambda ) . \\end{align*}"}
-{"id": "1824.png", "formula": "\\begin{align*} Z = \\frac { \\xi L } { 2 } \\cot ( Z ) , \\end{align*}"}
-{"id": "5803.png", "formula": "\\begin{align*} \\big ( \\ell _ \\gamma \\circ ( g \\circ h ) \\big ) ' \\ & = \\ \\big ( \\ell _ \\gamma ' \\circ ( g \\circ h ) \\big ) ( g ' \\circ h ) h ' \\ = \\ \\big ( ( \\ell _ \\gamma ' \\circ g ) \\circ h ) \\big ) ( g ' \\circ h ) h ' \\\\ & = \\ \\big ( \\big ( ( \\ell _ \\gamma ' \\circ g ) g ' \\big ) \\circ h \\big ) h ' \\ = \\ \\big ( ( \\ell _ \\gamma \\circ g ) ' \\circ h \\big ) h ' \\ = \\ \\big ( ( \\ell _ \\gamma \\circ g ) \\circ h \\big ) ' \\end{align*}"}
-{"id": "1428.png", "formula": "\\begin{align*} m ( 0 , x ) = m _ { 0 } ( x ) , u ( T , x ) = G ( x , m ( T , x ) ) , \\end{align*}"}
-{"id": "9673.png", "formula": "\\begin{align*} X P _ { \\mathcal M _ 0 } X _ * ^ * | _ { \\mathcal L ^ \\perp } = X X _ * ^ * | _ { \\mathcal L ^ \\perp } \\ \\ \\ \\ X X _ * ^ * \\mathcal L ^ \\perp \\subset L ^ 2 ( \\sigma ) . \\end{align*}"}
-{"id": "7684.png", "formula": "\\begin{align*} f ^ { \\mathrm { e } } ( x ) = \\frac { f ( x ) + f ( - x ) } { 2 } \\quad \\ ; \\ ; f ^ { \\mathrm { o } } ( x ) = \\frac { f ( x ) - f ( - x ) } { 2 } , f = f ^ \\mathrm { e } + f ^ \\mathrm { o } . \\end{align*}"}
-{"id": "9068.png", "formula": "\\begin{align*} \\mathcal { E } ^ { ( k ) } = \\{ \\xi \\in \\mathcal { E } : \\xi / \\sqrt { E } \\in I _ k \\} . \\end{align*}"}
-{"id": "7862.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 2 \\Big \\| \\Big ( \\sum _ { N \\in \\mathbb D _ { C _ 0 } } \\big | \\mathcal F ^ { - 1 } \\big ( ( \\mathfrak m _ N - \\lambda _ N ^ i ) \\hat { f } _ i \\big ) \\big | ^ 2 \\Big ) \\Big \\| _ { \\ell ^ 2 } \\lesssim \\| f \\| _ { \\ell ^ 2 } . \\end{align*}"}
-{"id": "6002.png", "formula": "\\begin{align*} \\mathbb L _ 1 \\cdots \\mathbb L _ { n - 1 } \\mathbb L _ n f = \\rho ^ g _ n ( \\widetilde { f } ) = n \\rho ^ { G _ n } \\left ( \\frac { 1 } { n } \\widetilde { f } ( W _ { ( n , 1 ) } , \\ldots , W _ { ( n , n ) } ) \\right ) . \\end{align*}"}
-{"id": "4869.png", "formula": "\\begin{align*} { \\rm S t } _ \\chi ( | \\sigma | ) = \\left ( \\underset { \\sigma \\subset H _ i \\in H _ \\chi } { \\cap } H _ i \\right ) \\cap S ^ { m - 1 } . \\end{align*}"}
-{"id": "7605.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { I _ 2 ( t ) } _ { \\alpha , p } \\le \\frac { C ( \\alpha ) } { \\nu } \\left ( 1 + t + \\left ( \\frac { t } { \\nu } \\right ) ^ 2 \\right ) \\left ( 1 + \\norm { X - \\mathrm { I d } } _ { L i p ( 0 , T ; C ^ { 1 + \\alpha } ) } \\right ) ^ { 4 } M _ X ^ { 1 + 3 \\alpha } \\\\ \\norm { X ' } _ { L ^ \\infty ( 0 , T ; C ^ { 1 + \\alpha } ) } \\norm { \\tau } _ { L i p ( 0 , T ; C ^ { \\alpha , p } ) } . \\end{gathered} \\end{align*}"}
-{"id": "775.png", "formula": "\\begin{align*} \\lambda _ i ^ { q ^ d - 1 } = - ( T - \\rho _ i ) . \\end{align*}"}
-{"id": "4003.png", "formula": "\\begin{align*} \\left \\langle U ( \\beta ) f _ j , U ( \\beta ) f _ j \\right \\rangle _ { d ^ 2 \\beta } = Q _ \\beta \\left \\langle f _ j , f _ j \\right \\rangle _ { d ^ 2 \\beta } , \\end{align*}"}
-{"id": "2961.png", "formula": "\\begin{align*} \\theta _ 1 & = \\alpha _ 1 \\nabla _ 1 + \\alpha _ 2 \\nabla _ 2 + \\alpha _ 3 \\nabla _ 3 , \\\\ \\theta _ 2 & = \\beta _ 1 \\nabla _ 1 + \\beta _ 2 \\nabla _ 2 . \\end{align*}"}
-{"id": "240.png", "formula": "\\begin{align*} [ X _ i , X _ { n + j } ] = - 4 \\delta _ { i j } \\partial _ t , \\ \\forall i , j \\in \\{ 1 , \\dots , n \\} \\end{align*}"}
-{"id": "4318.png", "formula": "\\begin{align*} \\begin{aligned} W _ k ( F ) : = \\big \\{ v \\in H ^ 1 ( F ) : \\ ; \\ ; & \\Delta _ F v \\in \\mathbb { P } _ { k } ( F ) , v | _ { \\partial F } \\in B _ k ( \\partial F ) , \\\\ & ( v , q ) _ { F } = ( \\Pi _ F ^ k v , q ) _ { F } , \\ ; \\forall q \\in \\widetilde { \\mathbb { P } } _ k ( F ) \\cup \\widetilde { \\mathbb { P } } _ { k - 1 } ( F ) \\big \\} . \\end{aligned} \\end{align*}"}
-{"id": "3632.png", "formula": "\\begin{align*} a _ 1 \\cdot ( ( \\alpha _ 0 - \\beta _ 0 ) \\cdot \\alpha _ { - 2 } ) = - \\frac { 1 } { 2 ^ 2 } \\beta _ 0 \\cdot \\alpha _ { - 2 } . \\end{align*}"}
-{"id": "7797.png", "formula": "\\begin{align*} \\kappa _ q ( d , N ) = N d ^ { - 1 / q } , \\end{align*}"}
-{"id": "6836.png", "formula": "\\begin{align*} \\begin{cases} a _ { 1 1 } a _ { 2 2 } - a _ { 1 2 } a _ { 2 1 } - 1 = 0 \\ , , \\\\ a _ { 1 3 } a _ { 2 4 } - a _ { 1 4 } a _ { 2 3 } - 1 = 0 \\ , , \\\\ a _ { 1 4 } - a _ { 1 2 } + a _ { 1 1 } = 0 \\ , , \\\\ a _ { 2 4 } - a _ { 2 2 } + a _ { 2 1 } = 0 \\ , . \\end{cases} \\end{align*}"}
-{"id": "5768.png", "formula": "\\begin{align*} R _ d ( \\Gamma , q ) : = \\sum _ r q ^ { \\delta ( \\Gamma , r ) } \\end{align*}"}
-{"id": "6955.png", "formula": "\\begin{align*} \\{ \\textbf { S } _ p ( \\textbf { y } ) \\} _ { 1 1 } = \\mathrm { V a r } ( y _ 1 + y _ 2 ) = \\mathrm { V a r } ( x _ 1 + x _ 2 ) = ( \\boldsymbol { \\Sigma } ) _ { 1 1 } + 2 ( \\boldsymbol { \\Sigma } ) _ { 1 2 } + ( \\boldsymbol { \\Sigma } ) _ { 2 2 } . \\end{align*}"}
-{"id": "3820.png", "formula": "\\begin{align*} B _ { G , S , \\pi } ( \\ell ) \\hookrightarrow \\bigcup _ { \\ell _ 1 + \\dots + \\ell _ { d ^ 2 } \\le \\eta ^ 2 \\ell + c _ 1 } K ^ { d + 1 } \\times \\prod _ { i = 1 } ^ { d ^ 2 } B _ { G , S , \\pi } ( \\ell _ i ) , \\end{align*}"}
-{"id": "8819.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathbb R ^ N } f ( x ) + \\sum _ { i = 1 } ^ n g _ i ( x ^ { ( i ) } ) \\end{align*}"}
-{"id": "1641.png", "formula": "\\begin{align*} F _ k : = - \\frac { 1 } { \\varkappa _ k } \\begin{pmatrix} - b \\\\ a _ k + \\sqrt [ + ] { a _ k ^ 2 - b ^ 2 } \\end{pmatrix} e ^ { - 2 \\pi i k x } \\end{align*}"}
-{"id": "6361.png", "formula": "\\begin{align*} [ E ] = [ F ] \\textup { i n C H } _ 0 ( M _ \\sigma ( v ) ) \\iff c _ 2 ( E ) = c _ 2 ( F ) \\textup { i n C H } _ 0 ( X ) . \\end{align*}"}
-{"id": "7043.png", "formula": "\\begin{align*} \\Phi = \\phi _ { v \\b { v } } ( d v \\otimes d \\b { v } + d \\b { v } \\otimes d v ) \\end{align*}"}
-{"id": "3387.png", "formula": "\\begin{align*} F ( T ) = \\sum _ { i = 1 } ^ k F ( T _ i ) + f ( T ) , \\end{align*}"}
-{"id": "8914.png", "formula": "\\begin{align*} P ( ( e , v ) \\in G _ 1 | x ) = \\begin{cases} \\frac { a } { 2 n } & x _ u = x _ v \\\\ \\frac { b } { 2 n } & x _ u \\neq x _ v , \\end{cases} \\end{align*}"}
-{"id": "3446.png", "formula": "\\begin{align*} g ( \\xi ) = ( M + 1 ) \\left ( \\log N - \\frac { 1 } { 2 N } \\right ) + \\xi . \\end{align*}"}
-{"id": "5939.png", "formula": "\\begin{align*} c _ i = 1 - \\frac { 1 } { l } + \\sum _ { i = 1 } ^ n \\frac { \\alpha _ i \\lambda _ i } { l } + \\frac { \\beta } { l p } \\end{align*}"}
-{"id": "1140.png", "formula": "\\begin{align*} & | B _ j | _ 2 ^ 2 = \\int _ { y _ j } ^ { y _ j + a _ j } \\int _ { y _ j } ^ { y _ j + a _ j } | B _ j ( u , v ) | ^ 2 d u d v \\\\ = & \\int _ { y _ j } ^ { y _ j + a _ j } \\int _ { y _ j } ^ { y _ j + a _ j } O \\left ( \\dfrac { n ^ 2 } { ( 1 + n | u - v | ) ^ 2 } \\right ) d u d v \\\\ = & \\int _ { y _ j } ^ { y _ j + a _ j } O \\left ( \\int _ { \\mathbb { R } } \\dfrac { n ^ 2 } { ( 1 + n | u - v | ) ^ 2 } d u \\right ) d v \\\\ = & \\int _ { y _ j } ^ { y _ j + a _ j } O \\left ( n \\right ) d v = O ( n a _ j ) = O \\left ( ( \\ln n ) ^ { 1 / 2 } \\right ) , \\end{align*}"}
-{"id": "9173.png", "formula": "\\begin{align*} \\omega \\left ( J _ { 1 } , J _ { 2 } \\right ) = \\left \\langle J _ { 1 } ^ { \\prime } , J _ { 2 } \\right \\rangle - \\left \\langle J _ { 1 } , J _ { 2 } ^ { \\prime } \\right \\rangle , \\end{align*}"}
-{"id": "9751.png", "formula": "\\begin{align*} \\bar w ( x , y ) : = \\Gamma _ a ( \\ , \\cdot \\ , , y ) * _ x ( \\zeta n _ w ) , \\end{align*}"}
-{"id": "10018.png", "formula": "\\begin{align*} \\mathcal { H } _ { p } ( X ) = \\mathfrak { B } _ { X } ( H _ { p } ( \\mathbb { T } ^ { \\infty } , X ) ) \\end{align*}"}
-{"id": "427.png", "formula": "\\begin{align*} | K _ t ( x , x ) - t | \\le C \\lim _ { t \\to \\infty } \\sup _ { x \\in D } | \\partial _ t K _ t ( x , x ) - 1 | = 0 , \\end{align*}"}
-{"id": "5679.png", "formula": "\\begin{align*} N _ { C } & ( x _ 0 , f _ 1 ( x _ 0 ) , f _ 2 ( x _ 0 ) ) = \\hat T _ { C } ^ \\circ ( x _ 0 , f _ 1 ( x _ 0 ) , f _ 2 ( x _ 0 ) ) \\\\ & = \\{ ( x ^ * , s _ 1 , s _ 2 ) \\in X ^ * \\times \\mathbb { R } \\times \\mathbb { R } \\ | \\ \\langle x ^ * v \\rangle + s _ 1 r _ 1 + s _ 2 r _ 2 \\le 0 \\\\ & \\mbox { f o r a l l } ( v , r _ 1 + r _ 2 ) \\in \\hat T _ { e p i ( f _ 1 + f _ 2 ) } ( x _ 0 , f _ 1 ( x _ 0 ) + f _ 2 ( x _ 0 ) ) \\} \\ , . \\end{align*}"}
-{"id": "6341.png", "formula": "\\begin{align*} f _ { n , d } ( n ) = & L a g \\Big \\{ n , G _ 1 G _ 2 m _ s h _ { s r } h _ { r r } h _ { r d } , G _ 1 G _ 2 ( m _ { b r _ 1 } + n _ { s p } ) \\\\ & \\times h _ { r r } h _ { r d } + G _ 2 ( m _ { b r _ 2 } + n _ { s p } ) h _ { r d } + m _ { b d } , D \\Big \\} , \\end{align*}"}
-{"id": "3059.png", "formula": "\\begin{align*} \\pi ^ { - s } \\Gamma ( s ) ( E ^ { \\sharp } . f _ { q } ) ( g , s ) = \\pi ^ { - ( 1 - s ) } \\Gamma ( 1 - s ) \\frac { \\Gamma ( s ) ^ { 2 } ( - i ) ^ { q } } { \\Gamma ( s + q / 2 ) \\Gamma ( s - q / 2 ) } ( E ^ { \\sharp } . f _ { q } ) ( g , 1 - s ) \\end{align*}"}
-{"id": "6839.png", "formula": "\\begin{align*} I = & \\langle a _ { 1 1 } a _ { 2 2 } - a _ { 1 2 } a _ { 2 1 } - 1 , a _ { 1 3 } a _ { 2 4 } - a _ { 1 4 } a _ { 2 3 } - 1 , a _ { 1 4 } - a _ { 1 2 } + a _ { 1 1 } , a _ { 2 4 } - a _ { 2 2 } + a _ { 2 1 } \\rangle \\\\ \\subset & \\ \\mathbb { A } = \\mathbb { Q } [ a _ { 1 1 } , a _ { 1 2 } , a _ { 1 3 } , a _ { 1 4 } , a _ { 2 1 } , a _ { 2 2 } , a _ { 2 3 } , a _ { 2 4 } ] \\ , . \\end{align*}"}
-{"id": "9868.png", "formula": "\\begin{align*} S _ \\tau = \\{ y ~ : ~ r _ { Q - R } ( y ) \\ge \\tau \\} \\ , . \\end{align*}"}
-{"id": "8077.png", "formula": "\\begin{align*} B _ i ( s + t ) - B _ i ( s ) \\le \\frac { - B _ i ( s ) t } { 1 - s } + C ^ { k - i + 1 } _ 1 ( s + t ) . \\end{align*}"}
-{"id": "336.png", "formula": "\\begin{align*} | I m ( \\partial ' _ k ) _ * | = \\frac { ( 2 m + 2 ) ( 4 m + 3 ) ( 4 m + 2 ) } { ( ( 2 m + 2 ) ( 4 m + 3 ) ( 4 m + 2 ) , k ) } \\cdot \\frac { 4 m + 3 } { ( 4 m + 3 , k ) } . \\end{align*}"}
-{"id": "8715.png", "formula": "\\begin{align*} \\rho ( ( i _ 1 , \\ldots , i _ d ) ) = { \\displaystyle \\prod _ { j = 1 } ^ d \\left ( \\sum _ { n _ j = 1 } ^ { i _ j } \\prod _ { r = 1 } ^ { n _ j } \\left ( { q _ j ( r ) \\over p _ j ( r ) } \\right ) \\right ) \\over \\displaystyle \\prod _ { j = 1 } ^ d \\left ( \\sum _ { n _ j = 1 } ^ { N _ j } \\prod _ { r = 1 } ^ { n _ j } \\left ( { q _ j ( r ) \\over p _ j ( r ) } \\right ) \\right ) } . \\end{align*}"}
-{"id": "3201.png", "formula": "\\begin{align*} 2 N = \\dim \\mathcal { H } = M _ + + M _ - + 2 . \\end{align*}"}
-{"id": "5246.png", "formula": "\\begin{align*} F ^ { - 1 } ( 1 - u ) = d + s ( u ) + \\int _ { u } ^ { 1 } t ^ { - 1 } s ( t ) d t . \\end{align*}"}
-{"id": "7606.png", "formula": "\\begin{align*} M _ \\epsilon = 1 + \\norm { X _ \\epsilon - \\mathrm { I d } } _ { L ^ \\infty ( 0 , T ; C ^ { 1 + \\alpha } ) } . \\end{align*}"}
-{"id": "2824.png", "formula": "\\begin{align*} \\tilde { a } _ { i j } : = a _ { i j } ^ - = \\max \\{ 0 , - a _ { i j } \\} \\ \\mbox { f o r } \\ i , j = p + 1 , \\dots , n , \\ \\mbox { a n d } \\ \\tilde { a } _ { i j } = a _ { i j } \\ \\ \\mbox { o t h e r w i s e } , \\end{align*}"}
-{"id": "6113.png", "formula": "\\begin{align*} \\deg ( T _ 0 ) = \\deg ( T _ 1 ) = ( 1 , 0 ) , \\ \\deg ( X ) = ( - \\mu _ 1 , 1 ) , \\ \\deg ( Y ) = ( - \\mu _ 2 , 1 ) , \\end{align*}"}
-{"id": "8530.png", "formula": "\\begin{align*} D _ t \\bar { q } = \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { 2 \\pi } \\frac { D _ t \\zeta - \\dot { z } _ j } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } , \\end{align*}"}
-{"id": "2376.png", "formula": "\\begin{align*} S ^ 2 & = S _ 1 ^ 2 + S _ 2 ^ 2 + S _ 3 ^ 2 & S _ \\pm & = S _ 1 \\pm S _ 2 . \\end{align*}"}
-{"id": "9283.png", "formula": "\\begin{align*} \\{ F ^ { i j } ( r _ 0 ) \\} _ { 1 \\le i , j \\le n } : = \\partial ^ + F ( r _ 0 ) , \\end{align*}"}
-{"id": "1863.png", "formula": "\\begin{align*} u ^ { \\rm e v e n } ( j , x ) = Y _ { j , 0 } v ^ { \\rm t o p . } ( x ) + \\sum _ { n = 1 } ^ { \\infty } Y _ { j , n } v ^ { \\rm e v e n } ( n , x ) . \\end{align*}"}
-{"id": "1412.png", "formula": "\\begin{align*} \\textrm { O P T } = \\inf _ { \\tau \\in { \\mathcal T } } E [ Z ^ 1 _ { \\tau } ] = E [ \\min _ { t \\in [ 1 , T ] } Z ^ 1 _ t ] + \\inf _ { \\tau \\in { \\mathcal T } } E [ Z ^ 2 _ { \\tau } ] . \\end{align*}"}
-{"id": "126.png", "formula": "\\begin{align*} \\eta _ { j , N } & \\le \\frac 1 N \\sum _ { h = 1 } ^ { \\infty } \\frac { d ^ { 2 h } } { h ! ( h - 1 ) ! } ( 1 + x _ j ) ^ h \\left ( 1 - x _ j - \\frac 1 N \\right ) ^ { h - 1 } \\\\ & \\le \\frac 1 N \\sum _ { h = 1 } ^ { \\infty } \\frac { d ^ { 2 h } } { h ! ( h - 1 ) ! } ( 1 - x _ j ^ 2 ) ^ { h - 1 } \\left ( 1 + x _ j \\right ) \\ , . \\end{align*}"}
-{"id": "6661.png", "formula": "\\begin{align*} H _ r ( T ) = & \\sum _ { S \\in U P ( T ) } ( - 1 ) ^ { | S | } | K _ r ( S ) | = \\sum _ { S \\in U P ( T ) } ( - 1 ) ^ { | S | } | i _ { S } ( K _ r ( S ) ) | \\\\ = & \\sum _ { S \\in U P ( T ) } ( - 1 ) ^ { | S | } ( \\sum _ { P \\in K _ r ( \\{ \\{ t _ 1 \\} , \\{ t _ 2 \\} , \\cdots , \\{ t _ m \\} \\} ) } \\chi _ S ( P ) ) \\\\ = & \\sum _ { P \\in K _ r ( \\{ \\{ t _ 1 \\} , \\{ t _ 2 \\} , \\cdots , \\{ t _ m \\} \\} ) } ( \\sum _ { S \\in U P ( T ) } ( - 1 ) ^ { | S | } \\chi _ S ( P ) ) . \\end{align*}"}
-{"id": "7366.png", "formula": "\\begin{align*} d ( b , b _ j ) & = d ( b , q ) + d ( q , b _ j ) = d ( b , q ) + d ( q , b _ i ) + d ( b _ i , b _ j ) \\\\ & \\geq d ( b , q ) + d ( q , b _ i ) = d ( b , b _ i ) \\end{align*}"}
-{"id": "4233.png", "formula": "\\begin{align*} \\delta ( a ( x ) \\lvert d u \\rvert ^ { p - 2 } d u ) ) = f & & \\Omega . \\end{align*}"}
-{"id": "9500.png", "formula": "\\begin{align*} \\Vert u _ n \\Vert _ { \\mathcal { W } } = \\Vert u _ n \\Vert _ { s , p } + \\Vert u _ n \\Vert _ { s , q } \\leq C , \\ \\ \\forall n \\in \\mathbb { N } . \\end{align*}"}
-{"id": "2181.png", "formula": "\\begin{align*} \\mathcal { T } _ \\ell f ( \\mathbf x ) = \\int K _ \\ell ( \\mathbf x , \\mathbf y ) f ( \\mathbf y ) \\ , d w ( \\mathbf y ) . \\end{align*}"}
-{"id": "3978.png", "formula": "\\begin{align*} \\mathcal { E } _ { k , m } = \\underset { \\substack { 1 \\leq \\lambda \\leq b \\\\ \\lambda \\mid b } } \\sum a _ f ( a \\lambda ^ 2 ) \\underset { \\substack { 1 \\leq d \\leq b \\\\ d \\lambda \\mid b } } \\sum \\mu ( d ) E _ { k , a \\lambda ^ 2 d ^ 2 } \\mid U _ { \\frac { b } { \\lambda d } } . \\end{align*}"}
-{"id": "4317.png", "formula": "\\begin{align*} \\big ( Q _ D ^ k v - v , q \\big ) _ { D } = 0 , \\forall q \\in \\mathbb { P } _ k ( D ) . \\end{align*}"}
-{"id": "990.png", "formula": "\\begin{align*} \\beta _ { i , i + r } ( { \\rm i n } ( I _ { G _ { r , d } } ) ) = \\sum _ { p = { \\frac { d ( d - 1 ) } { 2 } } + 1 } ^ { { \\frac { d ( d + 1 ) } { 2 } } } \\binom { n _ { p } } { i } = d \\binom { d - 1 } { i } . \\end{align*}"}
-{"id": "3167.png", "formula": "\\begin{align*} Q = \\begin{pmatrix} 0 & \\alpha ^ * \\\\ \\alpha & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "6237.png", "formula": "\\begin{align*} p _ { 0 } = \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} \\oplus 0 \\qquad p _ { 1 } = \\begin{pmatrix} 0 & 0 \\\\ 0 & 1 \\end{pmatrix} \\oplus 1 . \\end{align*}"}
-{"id": "6377.png", "formula": "\\begin{align*} \\varphi _ { \\varepsilon , j } ( t , x ) : = ( 1 - \\delta _ j ) \\varphi ( t + \\varepsilon , x ) + \\delta _ j \\rho _ j + n \\log ( 1 - \\delta _ j ) - ( B \\delta _ j + C \\varepsilon + \\eta _ j ) t , \\end{align*}"}
-{"id": "6219.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } \\frac { \\lambda ^ 0 _ j } { \\lambda ^ 0 _ j + \\lambda } \\times \\frac { \\nu ( d \\lambda ) } { \\prod _ { i = 1 } ^ n ( \\lambda ^ 0 _ i + \\lambda ) ^ { 1 / 2 } } = \\frac { \\pi ^ { n / 2 } } { 2 \\sqrt { \\lambda ^ 0 _ 1 \\ldots \\lambda ^ 0 _ n } } . \\end{align*}"}
-{"id": "5909.png", "formula": "\\begin{align*} T _ { s _ 1 ( s _ 0 s _ 1 ) ^ n } \\star \\psi _ m = q ^ { 2 n + 1 } \\varphi _ { m + n + 1 } + ( q - 1 ) \\sum _ { k = 0 } ^ { 2 n } q ^ { 2 n - k } \\psi _ { m + n - k } . \\end{align*}"}
-{"id": "3695.png", "formula": "\\begin{align*} E \\big ( C o v _ N ( \\delta _ i , y _ i ) | \\delta _ U \\big ) = \\frac { 1 } { N } \\sum \\limits _ { i \\in U } \\delta _ i \\mu _ i - \\big ( \\frac { 1 } { N } \\sum \\limits _ { i \\in U } \\delta _ i \\big ) \\big ( \\frac { 1 } { N } \\sum \\limits _ { i \\in U } \\mu _ i \\big ) \\rightarrow 0 \\end{align*}"}
-{"id": "6428.png", "formula": "\\begin{align*} \\tau _ { k } = \\| w _ { k + 1 } \\| ^ { 2 } = \\frac { 1 } { \\alpha _ { k + 1 } ^ { 2 } } \\left ( \\beta _ { k } ^ { 2 } \\| w _ { k } \\| ^ { 2 } + 1 \\right ) = \\frac { 1 } { \\alpha _ { k + 1 } ^ { 2 } } \\left ( \\beta _ { k } ^ { 2 } \\tau _ { k - 1 } + 1 \\right ) , \\end{align*}"}
-{"id": "3539.png", "formula": "\\begin{align*} C _ { N E } = \\mathrm { C o n e } \\left \\{ \\begin{aligned} ( - 1 , \\ ; \\ ; \\ , 0 , \\ ; \\ ; \\ , 0 , 1 , 0 , 0 , 0 , 1 , - 1 ) , \\\\ ( \\ ; \\ ; \\ , 0 , - 1 , \\ , \\ ; \\ ; 0 , 1 , - 1 , 1 , 0 , 0 , 0 ) , \\\\ ( \\ ; \\ ; \\ , 0 , \\ ; \\ ; \\ , 0 , - 1 , 0 , 0 , 1 , - 1 , 1 , 0 ) , \\end{aligned} \\ ; \\begin{aligned} ( - 1 , \\ ; \\ ; \\ , 0 , \\ ; \\ ; \\ , 0 , 0 , 1 , - 1 , 1 , 0 , 0 ) \\\\ ( \\ ; \\ ; \\ , 0 , - 1 , \\ , \\ ; \\ ; 0 , 0 , 0 , 0 , 1 , - 1 , 1 ) \\\\ ( \\ ; \\ ; \\ , 0 , \\ ; \\ ; \\ , 0 , - 1 , - 1 , 1 , 0 , 0 , 0 , 1 ) \\end{aligned} \\right \\} . \\end{align*}"}
-{"id": "439.png", "formula": "\\begin{align*} \\hat R ^ t _ k : = & \\inf \\{ s \\geq \\hat T ^ t _ { k - 1 } \\ : \\ B _ s \\leq \\gamma ( t + s ) \\} , \\\\ \\hat T ^ t _ k : = & \\inf \\{ s \\geq \\hat R ^ t _ { k } \\ : \\ B _ s = A + ( \\eta + \\gamma ) ( t + s ) \\} \\end{align*}"}
-{"id": "8714.png", "formula": "\\begin{align*} \\rho ( ( i _ 1 , \\ldots , i _ d ) ) = P ( \\tau _ { ( N _ 1 , \\ldots , N _ d ) } < \\tau _ { - \\infty } | Z _ 0 = ( i _ 1 , \\ldots , i _ d ) ) , \\end{align*}"}
-{"id": "6347.png", "formula": "\\begin{align*} \\mathcal { C } _ { \\rm e r g } = & \\int _ 0 ^ \\infty \\ ! \\ ! \\int _ 0 ^ \\infty \\ ! \\ ! \\int _ 0 ^ \\infty \\log \\Big ( 1 + \\gamma ( h _ { s r } , h _ { r r } , h _ { r d } ) \\Big ) d h _ { s r } d h _ { r r } d h _ { r d } . \\end{align*}"}
-{"id": "741.png", "formula": "\\begin{align*} \\mathbb { V } \\ ! _ { f } \\ ! \\left [ T \\right ] = \\frac { 2 \\left ( m ^ { 2 } - 1 \\right ) \\left ( n ^ { 2 } - 1 \\right ) } { \\left ( m n + 1 \\right ) ^ 2 \\left ( m n + 2 \\right ) \\left ( m n + 3 \\right ) } . \\end{align*}"}
-{"id": "3491.png", "formula": "\\begin{align*} G _ N ( x , r ) = { } \\oint _ { \\Sigma } g _ N ( t ; x , r ) \\frac { d t } { 2 \\pi i } , \\ g _ N ( t ; x , r ) = e ^ { \\frac { N } { M + 1 } \\big ( f _ { M , N } ( q _ { M } ( 0 ) ) - f _ { M , N } ( t ) + \\frac { ( x + r - C ) ( t - q _ M ( 0 ) ) } { \\rho _ { M , N } } \\big ) } , \\end{align*}"}
-{"id": "2206.png", "formula": "\\begin{align*} J ( \\boldsymbol { x } ( t ) , \\boldsymbol { u } ( t ) , t ) = 0 . 5 \\Big [ \\int _ { t _ 0 } ^ { \\infty } \\boldsymbol { x } ^ T ( t ) Q \\boldsymbol { x } ( t ) + \\boldsymbol { u } ^ T ( t ) R \\boldsymbol { u } ( t ) d t \\Big ] \\end{align*}"}
-{"id": "5396.png", "formula": "\\begin{align*} c ( 0 ) = 0 . \\end{align*}"}
-{"id": "4832.png", "formula": "\\begin{align*} \\sum _ { \\substack { a _ { i , j } \\in \\mathbb F _ q , i = 1 , \\dots , n , j = 1 , \\dots m \\\\ \\prod _ { j = 1 } ^ m ( 1 + \\sum _ { i = 1 } ^ n a _ { i , j } x ^ i ) \\equiv 0 \\mod x ^ { n + 1 } } } \\psi \\left ( \\sum _ { j = 1 } ^ m a _ { n , j } \\right ) = q ^ { m \\frac { n + 1 } { 2 } - 1 } \\mu \\end{align*}"}
-{"id": "9403.png", "formula": "\\begin{align*} T _ m . D = \\sum n _ z \\sum _ { \\gamma \\in T ( m ) } Q _ { \\gamma z } . \\end{align*}"}
-{"id": "1353.png", "formula": "\\begin{align*} \\mathcal F : = \\bigoplus _ { v \\in \\operatorname { O b j } \\Lambda } \\bigoplus _ { b \\in B _ v } \\Z v \\Lambda \\ast . \\end{align*}"}
-{"id": "4144.png", "formula": "\\begin{align*} \\varphi _ { 1 , - } ( x ) = - \\varphi _ { 1 , + } ( x ) , x \\in \\Delta _ 1 . \\end{align*}"}
-{"id": "6430.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } \\rho _ { 1 } & \\sigma _ { 1 } \\\\ \\sigma _ { 1 } & \\tau _ { 1 } \\end{array} \\right ] = \\left [ \\begin{array} { c c } \\frac { 1 } { \\alpha _ { 1 } ^ { 2 } } & - \\frac { \\beta _ { 1 } } { \\alpha _ { 2 } \\alpha _ { 1 } ^ { 2 } } \\\\ - \\frac { \\beta _ { 1 } } { \\alpha _ { 2 } \\alpha _ { 1 } ^ { 2 } } & \\frac { 1 } { \\alpha _ { 2 } ^ { 2 } } + \\left ( \\frac { \\beta _ { 1 } } { \\alpha _ { 2 } \\alpha _ { 1 } } \\right ) ^ { 2 } \\end{array} \\right ] = B _ { 2 } ^ { - T } B _ { 2 } ^ { - 1 } , \\end{align*}"}
-{"id": "8492.png", "formula": "\\begin{align*} F ( z , t ) = \\frac { 1 } { 2 \\pi i } \\int \\frac { \\zeta _ { \\beta } } { z - \\zeta ( \\beta , t ) } \\mathfrak { F } ( \\beta , t ) d \\beta . \\end{align*}"}
-{"id": "7087.png", "formula": "\\begin{align*} \\widehat { f ^ \\omega } ( \\xi ) = \\sum _ { k \\in \\mathbb { Z } ^ d } g _ k ( \\omega ) \\varphi ( \\xi - k ) \\widehat { f } ( \\xi ) ~ . \\end{align*}"}
-{"id": "1875.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sum _ { m = 1 } ^ { n - 1 } \\frac { 1 } { ( Z _ j - \\pi m ) ( Z _ j + \\pi ( n - m ) ) } = \\frac { 1 } { \\pi n } \\sum _ { m = 1 } ^ { n - 1 } \\frac { \\pi m } { Z _ j ^ 2 - ( \\pi m ) ^ 2 } . \\end{align*}"}
-{"id": "7152.png", "formula": "\\begin{align*} \\int \\left \\vert H ^ { j } \\left ( t , x \\right ) \\right \\vert d x = \\int \\left \\vert G ^ { j } \\left ( t , x \\right ) \\right \\vert d x , t > 0 , j \\geq 0 , \\end{align*}"}
-{"id": "2438.png", "formula": "\\begin{align*} P _ { \\widetilde { \\varphi _ { \\ell } ^ { H _ 1 } } } ( T ) & = \\frac { 1 } { ( - 1 ) ^ { \\ell / 4 } + 2 ^ { ( \\ell - 4 ) / 2 } } \\\\ & \\sum _ { i = 1 } ^ { \\ell / 4 - 1 } ( - 1 ) ^ { i - 1 } ( 4 ^ { i - 1 } T ^ { 4 ( i - 1 ) } + 2 \\times 4 ^ { i - 1 } T ^ { 4 ( i - 1 ) + 1 } + 2 \\times 4 ^ { i - 1 } T ^ { 4 ( i - 1 ) + 2 } ) . \\end{align*}"}
-{"id": "5210.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { s } \\frac { b ( t x ) - b ( x ) } { t } d \\lambda ( t ) = \\int _ { 1 } ^ { s } t ^ { - 1 } b ( t x ) d \\lambda ( t ) - b ( x ) \\log s \\rightarrow 0 . \\end{align*}"}
-{"id": "3188.png", "formula": "\\begin{align*} & \\dim \\ker \\alpha = \\dim \\ker ( 1 - T ) + \\dim \\mathcal { B } _ - , \\\\ & \\dim \\ker \\alpha ^ * = \\dim \\ker ( 1 + T ) + \\dim \\mathcal { B } _ + . \\end{align*}"}
-{"id": "6943.png", "formula": "\\begin{align*} [ f ] _ { B M O } = \\sup _ { x \\in \\Omega , r \\in ( 0 , d ) } \\frac { 1 } { | \\Omega _ r ( x ) | } \\int _ { \\Omega _ r ( x ) } | f ( y ) - f _ { \\Omega _ r } ( x ) | d y , \\end{align*}"}
-{"id": "2868.png", "formula": "\\begin{align*} Q = C ( n ) \\left ( \\frac { - p ( n - p ) K _ p } { C _ S } \\right ) ^ { n / 2 } \\exp \\left [ \\frac { C ( n ) C _ S } { - p ( n - p ) K _ p V ^ { 2 / n } } \\right ] \\quad \\quad Q = V \\ ; K _ p = 0 . \\end{align*}"}
-{"id": "9832.png", "formula": "\\begin{align*} g _ \\infty ( z ' ) : = \\begin{cases} N ( 0 ^ + , \\tilde u _ { X _ \\circ , 0 } ^ { ( \\infty ) } ( Z + \\ , \\cdot \\ , ) ) & X _ \\circ \\in \\mathcal { N } ( u ) \\\\ 0 & X _ \\circ \\notin \\mathcal { N } ( u ) . \\end{cases} \\end{align*}"}
-{"id": "7664.png", "formula": "\\begin{align*} \\mathrm { s p a n } ( \\mathbf { 1 } ) = \\mathrm { K e r } \\ , ( G _ { \\rm s y m } - K _ { \\rm s y m } ) ^ \\ast = \\mathrm { K e r } \\ , ( G _ { \\rm s y m } - K _ { \\rm s y m } ) , \\end{align*}"}
-{"id": "522.png", "formula": "\\begin{align*} y ^ m = g ( x ) : = \\prod _ { i = 1 } ^ N ( x - t _ i ) ^ { a _ i } . \\end{align*}"}
-{"id": "8890.png", "formula": "\\begin{align*} \\sigma _ { ( N ) } ( x ) : = \\sigma ( x ) - \\sum _ { j = 0 } ^ { N - 1 } a _ { \\alpha - j } x ^ { \\alpha - j } \\end{align*}"}
-{"id": "5099.png", "formula": "\\begin{align*} ( \\{ y _ j = S _ { x , \\tau } ^ { - 1 } x _ j + V e _ j - V \\theta _ \\tau S _ { x , \\tau } ^ { - 1 } x _ j \\} _ { j \\in \\mathbb { J } } , \\{ \\omega _ j = S _ { x , \\tau } ^ { - 1 } \\tau _ j + U e _ j - U \\theta _ x S _ { x , \\tau } ^ { - 1 } \\tau _ j \\} _ { j \\in \\mathbb { J } } ) \\end{align*}"}
-{"id": "7847.png", "formula": "\\begin{align*} \\mathbb P [ \\{ \\sigma \\in { \\rm S y m } ( d ) \\colon | \\sigma ( I ) \\cap J | = k \\} ] = \\binom { r } { k } \\binom { d - r } { | I | - k } \\binom { d } { | I | } ^ { - 1 } , \\end{align*}"}
-{"id": "9888.png", "formula": "\\begin{align*} T = \\begin{pmatrix} 0 & \\frac { 1 } { 4 } & \\frac { 1 } { 4 } & \\frac { 1 } { 2 } \\\\ \\frac { 1 } { 2 } & 0 & 0 & \\frac { 1 } { 2 } \\\\ 0 & \\frac { 1 } { 4 } & \\frac { 1 } { 4 } & \\frac { 1 } { 2 } \\\\ \\frac { 1 } { 2 } & 0 & 0 & \\frac { 1 } { 2 } \\end{pmatrix} \\end{align*}"}
-{"id": "311.png", "formula": "\\begin{align*} h ^ * \\alpha \\boxtimes h ^ * \\beta = h ^ * ( \\alpha + \\beta ) . \\end{align*}"}
-{"id": "526.png", "formula": "\\begin{align*} \\vec b = ( \\nabla y ) ( \\bar { \\mathcal { G } } _ { 2 \\times 2 } ) ^ { - 1 } \\left [ \\begin{matrix} \\bar { \\mathcal { G } } _ { 1 3 } \\\\ \\bar { \\mathcal { G } } _ { 2 3 } \\end{matrix} \\right ] + \\frac { \\sqrt { \\det \\bar { \\mathcal { G } } } } { \\sqrt { \\det \\bar { \\mathcal { G } } _ { 2 \\times 2 } } } \\ , \\vec \\nu . \\end{align*}"}
-{"id": "2669.png", "formula": "\\begin{align*} g ( z , v ) & = - z M ( \\frac { 1 } { 2 } , 1 , 2 v ) + ( 1 - z ^ 2 ) \\int _ { 0 } ^ { v } d s \\ , e ^ { ( v - s ) ( z + 1 ) } M ( \\frac { 1 } { 2 } , 1 , 2 s ) , \\end{align*}"}
-{"id": "3665.png", "formula": "\\begin{align*} p _ i = \\mbox { P r } ( \\delta _ i = 1 ; y _ i , i \\in U ) \\end{align*}"}
-{"id": "1644.png", "formula": "\\begin{align*} F _ k = \\alpha _ k + i \\beta _ k \\quad F _ { - k } = \\alpha _ { - k } + i \\beta _ { - k } \\end{align*}"}
-{"id": "9231.png", "formula": "\\begin{align*} \\alpha _ { 2 : s } = u + x _ 0 v + \\alpha ^ * w , \\end{align*}"}
-{"id": "7541.png", "formula": "\\begin{align*} u _ { \\epsilon , 0 } ' = \\frac { d } { d \\epsilon } u _ \\epsilon ( 0 ) , \\sigma _ { \\epsilon , 0 } ' = \\frac { d } { d \\epsilon } \\sigma _ { \\epsilon } ( 0 ) . \\end{align*}"}
-{"id": "1000.png", "formula": "\\begin{align*} ( a _ { - 1 } - a _ 1 ) ^ k \\prod _ { j = 1 } ^ k \\chi _ { \\{ \\tau ^ { ( n ) } _ { i _ j } \\geq a _ { - 1 } \\} } \\leq \\prod _ { j = 1 } ^ k ( ( \\tau ^ { ( n ) } _ { i _ j } - a _ { 1 } ) _ + - ( \\tau ^ { ( n ) } _ { i _ j } - a _ { - 1 } ) _ + ) \\\\ = \\sum _ { \\varepsilon _ 1 , \\cdots , \\varepsilon _ { k } \\in \\{ \\pm 1 \\} } \\prod _ { j = 1 } ^ k \\varepsilon _ { j } ( \\tau ^ { ( n ) } _ { i _ j } - a _ { \\varepsilon _ { j } } ) _ + \\leq ( a _ { - 1 } - a _ 1 ) ^ k \\prod _ { j = 1 } ^ k \\chi _ { \\{ \\tau ^ { ( n ) } _ { i _ j } > a _ { 1 } \\} } . \\end{align*}"}
-{"id": "4759.png", "formula": "\\begin{align*} g _ { V \\cup W } ( s ) = \\frac { f _ { V \\cup W } ( s ) } { 1 - f _ { V \\cup W } ( s ) } \\approx | \\log s | . \\end{align*}"}
-{"id": "6039.png", "formula": "\\begin{align*} K ( G ) & = 2 \\left [ \\tbinom { n } { 2 } - \\tbinom { s } { 2 } \\right ] + K ( H ) \\\\ & \\leq 2 \\left [ \\tbinom { n } { 2 } - \\tbinom { s } { 2 } \\right ] + P ( H ) , \\end{align*}"}
-{"id": "33.png", "formula": "\\begin{align*} f ^ { i } ( v , z ; \\pi ) : = \\frac 1 2 \\delta ( \\delta - 1 ) | \\pi | ^ 2 + \\delta \\pi ^ { t r } \\theta ^ i ( v ) + \\delta \\pi ^ { t r } z + \\frac 1 2 | z | ^ 2 , \\end{align*}"}
-{"id": "3041.png", "formula": "\\begin{align*} \\frac { 1 } { \\overline { \\alpha } } = \\frac { 1 } { d } \\left ( \\frac { 1 } { \\alpha _ { 1 } } + \\cdots + \\frac { 1 } { \\alpha _ { d } } \\right ) , a _ { i } = \\frac { \\overline { \\alpha } } { \\alpha _ { i } } , \\ \\ i \\in \\{ 1 , \\dots , d \\} . \\end{align*}"}
-{"id": "5173.png", "formula": "\\begin{align*} \\lim \\sup _ { h \\rightarrow + \\infty } \\lim _ { \\lambda \\rightarrow + \\infty } \\inf \\mathbb { P } ( B ^ { \\ast } ( h ) / A ( h ) > \\lambda ) = 0 . \\end{align*}"}
-{"id": "956.png", "formula": "\\begin{align*} \\Pr \\left ( \\liminf _ { j \\rightarrow \\infty } \\| J ( X _ j ) ^ \\top r ( X _ j ) \\| = 0 \\right ) = 1 . \\end{align*}"}
-{"id": "107.png", "formula": "\\begin{align*} B ( 0 ) ^ { \\ell _ 1 } \\cdot B _ 1 \\cdot B ( 0 ) ^ { \\ell _ 2 } \\cdot B _ 1 \\cdots B ( 0 ) ^ { \\ell _ k } \\cdot B _ 1 \\cdot B ( 0 ) ^ { \\ell _ { k + 1 } } & = B ( 0 ) ^ { \\alpha - \\beta } B _ 1 \\\\ & = B ( 0 ) ^ { 2 \\alpha + k } B _ 1 \\\\ & = B ( 0 ) ^ { 2 \\alpha + k - 1 } B _ 2 ( 0 ) \\ , , \\end{align*}"}
-{"id": "2246.png", "formula": "\\begin{align*} & \\kappa ( x ) \\kappa ( x ' ) = \\delta ( x - x ' ) \\kappa ( x ) & & p ( x ) p ( x ' ) = \\delta ( x - x ' ) p ( x ) \\\\ & p ( x ) | \\alpha ( x ' ) \\rangle = 0 , \\ ; \\langle \\alpha ' ( x ) | p ( x ' ) = 0 & & \\langle \\alpha ' ( x ) | \\alpha ( x ' ) \\rangle = \\delta ( x - x ' ) \\end{align*}"}
-{"id": "259.png", "formula": "\\begin{align*} F _ p M = \\sum _ { i \\geq 0 } \\partial _ t ^ i ( V ^ 0 M \\cap j _ * j ^ * F _ { p - i } M ) \\quad p , \\end{align*}"}
-{"id": "684.png", "formula": "\\begin{align*} H ( [ \\mu _ m ; f ^ x ; e _ m ] ) & = \\left [ \\mu _ m ; \\gamma _ { \\mathcal N } \\left ( \\Theta ( H ) ( f ) ; \\lambda _ { \\vec a , x ^ { - 1 } ( 1 ) } ^ { - 1 } , \\dots , \\lambda _ { \\vec a , x ^ { - 1 } ( m ) } ^ { - 1 } \\right ) ^ x ; e _ m \\right ] \\\\ & = \\left [ \\mu _ m ; \\gamma _ { \\mathcal N } \\left ( \\Theta ( H ) ( f ) ^ x ; \\lambda _ { \\vec a , 1 } ^ { - 1 } , \\dots , \\lambda _ { \\vec a , m } ^ { - 1 } \\right ) ; e _ m \\right ] \\\\ & = [ \\mu _ m ; \\Theta ( H ) ( f ) ^ x ; e _ m ] \\circ \\lambda _ { \\vec a } ^ { - 1 } \\end{align*}"}
-{"id": "9783.png", "formula": "\\begin{align*} \\mathcal { A } _ E : = \\{ Z \\in \\overline { B _ { 1 / 2 } } : \\exists \\{ Z _ \\ell \\} _ { \\ell \\in \\N } , \\{ j _ \\ell \\} _ { \\ell \\in \\N } Z _ \\ell \\in r ^ { - 1 } _ { j _ \\ell } E \\cap B _ { 1 / 2 } Z _ \\ell \\to Z \\} . \\end{align*}"}
-{"id": "8908.png", "formula": "\\begin{align*} \\mathbb { E } _ { P _ 0 } [ L ^ 2 ( G ) ] \\le \\frac { e ^ { \\frac { 4 t ^ 2 } { n } \\nu } } { \\binom { m } { t } } \\sum _ { k = 0 } ^ t \\binom { t } { k } \\binom { m - t } { t - k } \\exp { 2 k \\nu ( 1 - 4 t / n ) } , \\end{align*}"}
-{"id": "8191.png", "formula": "\\begin{align*} t ^ { T } \\frac { 1 } { \\sqrt { \\mathbb { V } ( D _ n ) } } \\begin{pmatrix} D _ n - \\mathbb { E } ( D _ n ) \\\\ D ' _ n - \\mathbb { E } ( D ' _ n ) \\\\ \\end{pmatrix} \\stackrel { D } { \\rightarrow } N ( 0 , 1 ) \\end{align*}"}
-{"id": "8439.png", "formula": "\\begin{align*} F \\circ \\zeta - \\frac { i } { 2 \\pi } \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j } { \\zeta ( \\alpha , t ) - z _ j ( t ) } = D _ t \\zeta \\Psi _ { \\zeta } \\circ \\zeta + \\Psi _ t \\circ \\zeta + b . \\end{align*}"}
-{"id": "1004.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\mathbb { P } ( \\tau _ k ^ { ( n ) } \\in A ) = \\lim _ { n \\to + \\infty } \\mathbb { P } ( \\chi ^ { ( n ) } ( A ) \\geq k ) = \\mathbb { P } ( \\chi ( A ) \\geq k ) = \\varphi _ k \\left ( - f ' ( x ) \\right ) , \\end{align*}"}
-{"id": "4400.png", "formula": "\\begin{align*} L _ o : = \\{ ( x , y , z ) \\in S \\ ; \\ y = 0 \\} \\end{align*}"}
-{"id": "3219.png", "formula": "\\begin{align*} Q _ k : = Q ( { w } ^ k ) , \\ \\mathcal { J } _ k : = \\mathcal { J } \\Phi _ 0 ( { w } ^ k ) \\end{align*}"}
-{"id": "7633.png", "formula": "\\begin{align*} f \\ = \\ \\int \\limits _ { \\mathbb R ^ d } a ( \\xi - q ) \\mu ( \\xi , q ) ( \\xi - q ) \\ d q . \\end{align*}"}
-{"id": "4010.png", "formula": "\\begin{align*} \\underset { n \\le X ^ { 1 + \\epsilon } , \\ , ( n , M ) = 1 } { \\sum } | a ' _ f ( n ) | ^ { 2 } e ^ { - n / X } + \\underset { n > X ^ { 1 + \\epsilon } , \\ , ( n , M ) = 1 } { \\sum } | a ' _ f ( n ) | ^ { 2 } e ^ { - n / X } \\gg _ f X . \\end{align*}"}
-{"id": "5296.png", "formula": "\\begin{align*} \\langle H _ i , h _ j \\rangle = \\delta _ { i j } , \\langle E _ l , e _ l \\rangle = - \\delta _ { k l } , \\langle H _ i , e _ k \\rangle = 0 . \\end{align*}"}
-{"id": "1843.png", "formula": "\\begin{align*} \\phi ^ { \\rm e v e n } ( j , x , t ) = ( 2 \\kappa _ j ) ^ { - 1 / 2 } u ^ { \\rm e v e n } ( j , x ) \\ , e ^ { - i \\kappa _ j t } . \\end{align*}"}
-{"id": "1786.png", "formula": "\\begin{align*} \\begin{aligned} \\left | B _ { \\frac { i + j } { n } } ( \\omega ) - B _ { \\frac { i + j - 1 } { n } } ( \\omega ) \\right | & \\leq \\left | B _ { t + ( \\frac { i + j } { n } - t ) } ( \\omega ) - B _ { t } ( \\omega ) \\right | + \\left | B _ { t } ( \\omega ) - B _ { t + ( \\frac { i + j - 1 } { n } - t ) } ( \\omega ) \\right | \\\\ & \\leq \\left [ \\left ( \\frac { i + j } { n } - t \\right ) + \\left ( \\frac { i + j - 1 } { n } - t \\right ) \\right ] M \\\\ & \\leq \\frac { ( 2 j + 1 ) M } { n } . \\end{aligned} \\end{align*}"}
-{"id": "8999.png", "formula": "\\begin{align*} R ^ { \\prime } ( \\rho ) = 0 , ~ ~ \\rho > 0 \\Longleftrightarrow h ( \\rho ) = 0 , ~ ~ \\rho > 0 \\Longleftrightarrow \\rho = \\rho _ + . \\end{align*}"}
-{"id": "7127.png", "formula": "\\begin{align*} - \\phi _ 2 '' ( \\hat { q } ) + q _ 0 \\phi _ 2 ( \\hat { q } ) = \\lambda ^ * \\phi _ 2 ( \\hat { q } ) + \\nu \\phi _ 2 ^ 3 ( \\hat { q } ) . \\end{align*}"}
-{"id": "3858.png", "formula": "\\begin{align*} \\mathbf { E } [ f ( Z ^ x _ t ) ] = E [ \\mathcal { N } f ( Z ^ x _ t ) ] . \\end{align*}"}
-{"id": "3077.png", "formula": "\\begin{align*} | I _ 4 | & \\leq \\int _ { \\frac { 1 } { \\beta } \\leq | x | \\leq \\frac { 2 } { \\beta } } \\frac { 1 } { | x | ^ { n - \\beta } } \\ , d x + \\int _ { \\frac { 1 } { \\beta } \\leq | x - z | \\leq \\frac { 2 } { \\beta } } \\frac { 1 } { | x - z | ^ { n - \\beta } } \\ , d x \\\\ & \\leq C \\frac { 2 ^ \\beta - 1 } { \\beta } \\beta ^ { - \\beta } . \\end{align*}"}
-{"id": "878.png", "formula": "\\begin{align*} \\lambda ( e _ i , e _ j ) = \\left | \\{ a \\in Q _ 1 : t a = i , \\ , h a = j \\} \\right | - \\left | \\{ a \\in Q _ 1 : t a = j , \\ , h a = i \\} \\right | . \\end{align*}"}
-{"id": "8923.png", "formula": "\\begin{align*} \\mathbb { E } [ N _ w ^ { \\hat { x } } ( G ' ) - N _ w ^ { \\hat { x } } ( H ) \\mid \\hat { x } ] = & \\frac { ( a - b ) } { n } { \\Big ( } \\left ( ( n ^ { + + } - n _ C ^ { + + } ) - ( n ^ { + - } - n _ C ^ { + - } ) \\right ) ( n _ C ^ { + + } - n _ C ^ { + - } ) \\\\ & \\quad + \\left ( ( n ^ { - - } - n _ C ^ { - - } ) - ( n ^ { - + } - n _ C ^ { - + } ) \\right ) ( n _ C ^ { - - } - n _ C ^ { - + } ) { \\Big ) } . \\end{align*}"}
-{"id": "10008.png", "formula": "\\begin{align*} \\sum _ { n = M + 1 } ^ { N } { \\frac { a _ { n } } { n ^ { \\sigma + \\varepsilon + s } } } = \\Big ( \\sum _ { k = 1 } ^ { N } { \\frac { a _ { k } } { k ^ { \\sigma + s } } } \\Big ) \\frac { 1 } { N ^ { \\varepsilon } } - \\Big ( \\sum _ { k = 1 } ^ { M } { \\frac { a _ { k } } { k ^ { \\sigma + s } } } \\Big ) \\frac { 1 } { M ^ { \\varepsilon } } + \\sum _ { n = M } ^ { N - 1 } \\Big ( \\sum _ { k = 1 } ^ { n } { \\frac { a _ { k } } { k ^ { \\sigma + s } } } \\Big ) \\Big [ \\frac { 1 } { n ^ { \\varepsilon } } - \\frac { 1 } { ( n + 1 ) ^ { \\varepsilon } } \\Big ] \\ , . \\end{align*}"}
-{"id": "2527.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\eta ^ { - 1 } _ t \\mathbf P _ { \\mu } ( \\| X _ t \\| \\neq 0 ) = \\mu ( \\phi ) , \\end{align*}"}
-{"id": "8528.png", "formula": "\\begin{align*} G _ d : = - 2 [ \\bar { q } , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } \\bar { \\mathfrak { F } } } { \\zeta _ { \\alpha } } - 2 [ \\bar { \\mathfrak { F } } , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } \\bar { q } } { \\zeta _ { \\alpha } } - 2 [ \\bar { q } , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } \\bar { q } } { \\zeta _ { \\alpha } } - 4 D _ t q : = G _ { d 1 } + G _ { d 2 } + G _ { d 3 } + G _ { d 4 } . \\end{align*}"}
-{"id": "9337.png", "formula": "\\begin{align*} \\psi _ { a } : = \\psi + \\langle \\psi , a \\rangle a \\in \\mathfrak { X } _ { \\psi } ( M ) , \\end{align*}"}
-{"id": "9749.png", "formula": "\\begin{align*} L _ a ^ { x _ n , y } w = L _ a w - y ^ a \\Delta _ { x ' } w = - y ^ a \\Delta _ { x ' } w \\quad D _ 1 \\cap \\{ y > 0 \\} \\end{align*}"}
-{"id": "8022.png", "formula": "\\begin{align*} \\Upsilon _ i ( \\epsilon _ i , \\nu _ i ) & = \\frac { ( 1 - ( - 1 ) ^ k ) n } { 2 k } ( \\delta _ { i , 3 } + \\delta _ { i , 4 } ) + ( - 1 ) ^ i \\frac { 1 - ( - 1 ) ^ n } { 4 } + \\frac { 1 } { 2 } ( - 1 ) ^ i \\displaystyle { \\sum _ { i = n + 3 } ^ { 2 n } } c ^ { n + 3 - 2 i } P _ { i - n - 3 , - i } . \\end{align*}"}
-{"id": "8952.png", "formula": "\\begin{align*} \\P ( Y = 1 | X ) = \\bigg ( 1 + \\exp \\Big \\{ - \\beta _ 0 - \\sum _ { j = 1 } ^ p \\beta _ j ( X ( t _ j ) - m ( t _ j ) ) \\Big \\} \\bigg ) ^ { - 1 } . \\end{align*}"}
-{"id": "6305.png", "formula": "\\begin{align*} \\mathbb { P } _ { G _ 1 \\sim G ( n , m ) } \\left [ \\mathcal { H } ^ c _ { \\rho , C } \\right ] \\Omega ( \\frac { 1 } { n } ) \\leq \\mathbb { P } _ { G _ 1 \\sim G ( n , m ) } \\left [ \\mathcal { H } ^ c _ { \\rho , C } \\right ] \\mathbb { P } _ { G \\sim G _ { n , p } } \\left [ E ( G ) = m \\right ] \\leq O \\left ( \\frac { 1 } { n ^ { \\frac { C } { 1 6 } - 2 } } \\right ) . \\end{align*}"}
-{"id": "7359.png", "formula": "\\begin{align*} d ( x , p ) \\leq d ( p , c ) + d ( c , x ) \\leq s + d ( c , x ) < s + d ( x , p ) - s = d ( x , p ) \\end{align*}"}
-{"id": "4379.png", "formula": "\\begin{align*} \\left [ x _ { 1 , 2 } ^ n , \\left [ x _ { 1 , 3 } ^ n , \\dots , \\left [ x _ { 1 , r - 1 } ^ n , y _ { 1 , r } ^ n \\right ] \\dots \\right ] \\right ] = \\left ( w _ { n k } ( a _ 1 , b _ 1 ) , 1 , \\dots , 1 \\right ) = h _ { k n } \\in H _ r \\end{align*}"}
-{"id": "6547.png", "formula": "\\begin{align*} f ^ * \\omega = \\omega '' \\wedge p ^ * \\omega ' , \\end{align*}"}
-{"id": "10077.png", "formula": "\\begin{align*} \\mathcal { E } ( \\Delta ) = F \\circ ( K ^ { \\leq } + \\Delta ) - F \\circ K ^ \\leq - M \\Delta \\end{align*}"}
-{"id": "1730.png", "formula": "\\begin{align*} c _ { p , r } \\left ( A \\right ) = \\inf \\left \\{ c _ { p , r } \\left ( O \\right ) : A \\subset O , \\ O \\ \\right \\} . \\end{align*}"}
-{"id": "7942.png", "formula": "\\begin{align*} ( \\Pi _ 1 x _ { a , 1 } ) \\otimes x _ { a , 2 } \\otimes x _ { a , 3 } = ( \\alpha _ a \\Pi _ 1 x _ { a , 1 } ) \\otimes y _ { \\sigma ( a ) , 2 } \\otimes y _ { \\sigma ( a ) , 3 } a \\in [ n ] \\setminus \\{ 1 \\} . \\end{align*}"}
-{"id": "6056.png", "formula": "\\begin{align*} ( \\alpha _ 0 , \\beta _ 0 ) = \\left \\{ \\begin{array} { l l } ( 0 , \\gamma ) , & \\mbox { i f $ \\alpha \\leq \\gamma + 1 $ } , \\\\ ( \\gamma , 0 ) , & \\mbox { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "1431.png", "formula": "\\begin{align*} u _ { t } + \\Delta u + \\varepsilon \\mathcal { H } ( t , x , m , D u ) = 0 , \\end{align*}"}
-{"id": "4098.png", "formula": "\\begin{align*} \\tilde { \\gamma } & = - \\frac { 2 } { 3 } \\alpha + \\frac { 1 } { 6 } = - \\frac { 2 } { 3 } \\beta + \\frac { 1 } { 3 } , \\\\ \\tilde { M } _ 1 & = \\frac { 1 } { 3 } \\alpha ^ 2 + \\frac { 1 } { 6 } \\alpha - \\frac { 1 } { 3 } , \\\\ \\tilde { M } _ 2 & = \\frac { 1 } { 1 8 } \\alpha ^ 4 + \\frac { 5 } { 5 4 } \\alpha ^ 3 - \\frac { 5 } { 2 1 6 } \\alpha ^ 2 - \\frac { 5 } { 1 0 8 } \\alpha + \\frac { 1 } { 2 5 9 2 } . \\end{align*}"}
-{"id": "3699.png", "formula": "\\begin{align*} \\hat { \\tau } _ 1 = \\sum _ x \\sum _ { i \\in U _ x } \\delta _ i y _ i ^ 2 / p _ x ^ 2 = \\sum _ x p _ x ^ { - 2 } \\sum _ { i \\in B _ x } y _ i ^ 2 \\end{align*}"}
-{"id": "4178.png", "formula": "\\begin{align*} A _ n ^ { ( 1 ) } ( z _ 1 ) \\cdot \\cdots \\cdot A _ n ^ { ( 1 ) } ( z _ k ) & = \\frac { 1 } { f ' ( 0 ) ^ k } E _ n ( z _ 1 ) \\left ( \\prod _ { j = 1 } ^ { k - 1 } T _ \\alpha ^ { - 1 } A _ \\alpha T _ \\alpha E _ n ^ { - 1 } ( z _ j ) E _ n ( z _ { j + 1 } ) \\right ) T _ \\alpha ^ { - 1 } A _ \\alpha T _ \\alpha E _ n ^ { - 1 } ( z _ k ) \\\\ & = E _ n ( z _ 1 ) \\mathcal { O } ( n ^ { k - 1 } ) E _ n ^ { - 1 } ( z _ k ) . \\end{align*}"}
-{"id": "190.png", "formula": "\\begin{align*} [ X \\wedge Y , Z \\wedge W ] = [ X , Z ] \\wedge Y \\wedge W + [ Z , Y ] \\wedge X \\wedge W + [ X , W ] \\wedge Z \\wedge Y + [ W , Y ] \\wedge Z \\wedge X \\end{align*}"}
-{"id": "3980.png", "formula": "\\begin{align*} \\mathcal { E } _ { k , m } & = \\underset { \\lambda \\mid b } \\sum a _ f ( a \\lambda ^ 2 ) \\underset { ( s , b ) = \\lambda } \\sum E _ { k , m , s } \\\\ & = \\underset { \\lambda \\mid b } \\sum a _ f ( a \\lambda ^ 2 ) \\underset { ( s ' , b ' ) = 1 } \\sum E _ { k , m , \\lambda s ' } . \\end{align*}"}
-{"id": "8250.png", "formula": "\\begin{align*} \\sigma ( 2 n ) = 3 \\sigma ( n ) - 2 \\sigma ( n / 2 ) . \\end{align*}"}
-{"id": "7204.png", "formula": "\\begin{align*} \\sum _ { j \\geq 0 } \\varphi _ j ( \\xi ) = 1 \\qquad \\mbox { f o r a l l } \\xi \\in \\R , \\end{align*}"}
-{"id": "1308.png", "formula": "\\begin{align*} \\widetilde { \\psi } ( g ) = \\widetilde { \\psi } ( a ) \\widetilde { \\psi } ( x ) ^ { \\varepsilon } \\widetilde { \\psi } ( y ) ^ { - \\varepsilon } \\widetilde { \\psi } ( a ) ^ { - 1 } = \\bigl ( e , \\varepsilon ( x , a ) - \\varepsilon ( y , a ) \\bigr ) . \\end{align*}"}
-{"id": "7315.png", "formula": "\\begin{align*} P ( x , D ) u + V ( x ) u = 0 , ~ x \\in \\Omega \\subset \\mathbb { R } ^ { d } \\end{align*}"}
-{"id": "4875.png", "formula": "\\begin{align*} N ^ 1 _ \\infty ( U ) = S ^ { 2 n - 1 } \\infty \\setminus { \\rm c l o s } ^ 1 _ \\infty ( \\mathbb { C } ^ n \\setminus U ) . \\end{align*}"}
-{"id": "5767.png", "formula": "\\begin{align*} \\delta ( \\Gamma , r ) = \\sum _ { k = 1 } ^ { d - 1 } [ b _ 1 ( \\Gamma ) - b _ 1 ( \\Gamma _ k ) ] . \\end{align*}"}
-{"id": "7888.png", "formula": "\\begin{align*} \\{ g _ { n } \\} _ { n = 1 } ^ { \\infty } \\in \\left ( \\sum _ { n = 1 } ^ { \\infty } \\oplus \\mathcal { Z } _ { n } ^ { \\sigma } \\right ) _ { \\ell ^ { 2 } } , \\end{align*}"}
-{"id": "465.png", "formula": "\\begin{align*} g _ k = & \\sum _ { j = 0 } ^ { a _ i - k } { \\lambda _ j u ^ { j } v ^ { a _ i - k + j } } = v ^ { d - a _ i } \\sum _ { j = 0 } ^ { a _ i - k } \\lambda _ j u ^ j v ^ { a _ i - k + j } & \\lambda _ j \\in \\mathbb { K } . \\end{align*}"}
-{"id": "9847.png", "formula": "\\begin{align*} \\sum _ { i = 2 } ^ { r } ( - t ) ^ { 4 - i } x ^ { 1 - i } ( 1 - x ^ 2 ) ^ r _ { [ i ] } . \\end{align*}"}
-{"id": "2963.png", "formula": "\\begin{align*} f ( z ) = \\frac { 1 } { 2 \\pi i } \\int _ { \\partial \\Omega } \\frac { \\widetilde { f } ( \\zeta ) } { \\zeta - z } \\Omega \\zeta ( z \\in \\Omega ) . \\end{align*}"}
-{"id": "8395.png", "formula": "\\begin{align*} p = - \\frac { i } { 2 \\pi } \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j } { z ( \\alpha , t ) - z _ j ( t ) } . \\end{align*}"}
-{"id": "3191.png", "formula": "\\begin{align*} & C _ + \\varphi = d ^ * f , \\\\ & C _ - \\varphi = \\varphi _ 0 . \\end{align*}"}
-{"id": "6959.png", "formula": "\\begin{align*} d c _ r = \\sum _ { k = 0 } ^ r \\left ( \\begin{smallmatrix} r \\\\ k \\end{smallmatrix} \\right ) c _ { r - k + 1 } \\wedge c _ k = c _ { r + 1 } \\wedge c _ 0 + ( r - 1 ) c _ r \\wedge c _ 1 + \\sum _ { k = 2 } ^ { r - 1 } \\left ( \\begin{smallmatrix} r \\\\ k \\end{smallmatrix} \\right ) c _ { r - k + 1 } \\wedge c _ k . \\end{align*}"}
-{"id": "4182.png", "formula": "\\begin{align*} A _ n ^ { ( 2 ) } ( z ) = \\frac { A _ n ^ { ( 1 ) } ( 0 ) \\left ( A _ n ^ { ( 1 ) } ( z ) - A _ n ^ { ( 1 ) } ( 0 ) \\right ) } { z } \\end{align*}"}
-{"id": "4986.png", "formula": "\\begin{align*} Z ( C / \\mathbb { F } _ q , T ) = 1 + ( N _ 1 ) T + \\bigg ( \\frac { N _ 2 } { 2 } + \\frac { N _ 1 ^ 2 } { 2 } \\bigg ) T ^ 2 + \\bigg ( \\frac { N _ 3 } { 3 } + \\frac { N _ 1 N _ 2 } { 2 } + \\frac { N _ 1 ^ 3 } { 6 } \\bigg ) T ^ 3 + \\ldots . \\end{align*}"}
-{"id": "319.png", "formula": "\\begin{align*} \\begin{array} { c c c } m = \\frac { 1 } { 2 } t n ( n ^ 2 - 1 ) , & s = - 2 r , & s = t ( n - 1 ) . \\end{array} \\end{align*}"}
-{"id": "1026.png", "formula": "\\begin{align*} & \\Sigma _ k ( a _ 1 , \\cdots , a _ k ) : = \\bigcup _ { i _ 1 , \\cdots , i _ { k } \\ } \\prod _ { j = 1 } ^ k J _ { i _ j } ( a _ j ) \\subset [ 0 , 2 \\pi ) ^ k , \\end{align*}"}
-{"id": "7907.png", "formula": "\\begin{align*} \\Delta _ 3 \\approx \\Delta _ 3 ' = \\frac { h ^ 2 } { 2 4 } ( f ' ( b ) - f ' ( a ) ) \\end{align*}"}
-{"id": "9558.png", "formula": "\\begin{align*} _ 3 \\phi _ 2 \\left [ \\begin{matrix} q ^ { - N } , & \\alpha , & \\beta \\\\ \\gamma , & \\frac { q ^ { 1 - N } } { \\tau } \\end{matrix} ; q , q \\right ] = \\frac { ( \\beta , \\alpha \\tau ; q ) _ N } { ( \\gamma , \\tau ; q ) _ N } { } _ 3 \\phi _ 2 \\left [ \\begin{matrix} q ^ { - N } , & \\frac { \\gamma } { \\beta } , & \\tau \\\\ \\alpha \\tau , & \\frac { q ^ { 1 - N } } { \\beta } \\end{matrix} ; q , q \\right ] . \\end{align*}"}
-{"id": "1587.png", "formula": "\\begin{align*} \\frac { 2 } { \\sqrt { v ^ \\iota _ n } } \\Phi _ { v ^ \\iota _ n } ( z ^ \\iota _ n , x ^ \\iota _ n ) = \\Phi _ { \\widetilde { F } } ( z ^ \\iota _ n , x ^ \\iota _ n ) + o ( 1 ) . \\end{align*}"}
-{"id": "1089.png", "formula": "\\begin{align*} & \\lim _ { n \\to + \\infty } 2 S ( I ) \\int _ { 0 } ^ { ( b _ 1 / a _ 1 - 1 ) \\ln n } \\frac { D _ n ( ( 1 + z / \\ln n ) G _ n ( x ) / 2 ) } { D _ n ( G _ n ( x ) / 2 ) } \\frac { ( 1 + z / \\ln n ) a _ 1 } { \\sqrt { 4 - ( 1 + z / \\ln n ) ^ 2 a _ 1 ^ 2 } } d z \\\\ = & 2 S ( I ) \\int _ { 0 } ^ { + \\infty } e ^ { - 2 z } \\frac { a _ 1 } { \\sqrt { 4 - a _ 1 ^ 2 } } d z = \\frac { S ( I ) a _ 1 } { \\sqrt { 4 - a _ 1 ^ 2 } } = \\frac { S ( I ) \\sqrt { 4 - a ^ 2 } } { | a | } , \\end{align*}"}
-{"id": "5902.png", "formula": "\\begin{align*} g x = \\begin{pmatrix} a x + b y \\\\ c x + d y \\end{pmatrix} \\end{align*}"}
-{"id": "1815.png", "formula": "\\begin{align*} \\phi ^ { \\rm I N } ( x , t ) = \\frac { 1 } { \\sqrt { 2 \\omega _ { \\xi , j } } } \\ , \\chi _ { \\xi , j } ( x ) \\ , e ^ { - i \\omega _ { \\xi , j } t } , \\end{align*}"}
-{"id": "4116.png", "formula": "\\begin{align*} g _ { 1 } ( z ) & = \\log { z } - \\frac { m _ { 1 } } { z } + \\mathcal { O } \\left ( z ^ { - 2 } \\right ) \\\\ g _ { 2 } ( z ) & = \\frac { 1 } { 2 } \\log { z } + \\frac { m _ { \\frac { 1 } { 2 } } } { \\sqrt { z } } - \\frac { m _ { 1 } } { 2 z } + \\mathcal { O } \\left ( z ^ { - \\frac { 3 } { 2 } } \\right ) , \\end{align*}"}
-{"id": "3569.png", "formula": "\\begin{align*} \\begin{aligned} W _ { 1 } u = y z , & & W _ { 2 } u = z x , & & W _ { 3 } u = x y \\end{aligned} \\end{align*}"}
-{"id": "7877.png", "formula": "\\begin{align*} \\sum _ { \\substack { J \\subseteq I \\\\ | J | = l } } w _ S ( \\varepsilon ( J ) ) = \\sum _ { j = 0 } ^ { l } \\sum _ { \\substack { J \\subseteq I \\\\ | J | = l , \\ , | J \\cap S | = j } } ( - 1 ) ^ { j } = \\sum _ { j = 0 } ^ { l } ( - 1 ) ^ { j } \\binom { | S | } { j } \\binom { m - | S | } { l - j } . \\end{align*}"}
-{"id": "5399.png", "formula": "\\begin{align*} 2 c + ( q + 1 ) t - q ( q + 2 ) = - ( q - l ) ( q - ( 2 ( k - 1 ) - l ) ) \\end{align*}"}
-{"id": "8682.png", "formula": "\\begin{align*} \\pi _ \\lambda ( g ) f ( X , Z ) = a ( g ^ { - 1 } \\bar { n } _ { ( X , Z ) } ) ^ { - ( \\lambda + \\rho ) } f ( \\log \\bar { n } ( g ^ { - 1 } \\bar { n } _ { ( X , Z ) } ) ) , \\end{align*}"}
-{"id": "7113.png", "formula": "\\begin{align*} \\tilde { g } = - \\frac { T _ { m } ^ { - 1 } g _ { m } } { ( T _ { m } ^ { - 1 } g _ { m } , x ) } = - \\frac { \\Psi _ { m } g _ { m } \\Psi _ { m } ^ { * } } { \\left ( \\Psi _ { m } g _ { m } \\Psi _ { m } ^ { * } , \\Psi _ { m } \\begin{pmatrix} 1 & 0 \\\\ 0 & - \\mu \\end{pmatrix} \\Psi ^ { * } _ { m } \\right ) } = \\frac { \\Psi _ { m } g _ { m } \\Psi _ { m } ^ { * } } { 1 - \\mu | \\psi | ^ { 2 } } . \\end{align*}"}
-{"id": "6207.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } \\frac { \\mu ( d v ) } { ( 1 + v y ) ^ { 3 / 2 } } = \\frac { 1 } { 2 ^ { 3 / 2 } } . \\end{align*}"}
-{"id": "746.png", "formula": "\\begin{align*} R _ { a } = \\sum _ { i = 1 } ^ { m } \\theta _ { i } \\ln ^ { a } \\theta _ { i } \\end{align*}"}
-{"id": "4130.png", "formula": "\\begin{align*} N _ { + } ( x ) & = N _ { - } ( x ) \\begin{pmatrix} 0 & x ^ \\beta & 0 \\\\ - x ^ { - \\beta } & 0 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} , & & x \\in ( 0 , q ) , \\\\ N _ { + } ( x ) & = N _ { - } ( x ) \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 0 & - 1 \\\\ 0 & 1 & 0 \\end{pmatrix} , & & x < 0 . \\end{align*}"}
-{"id": "5956.png", "formula": "\\begin{align*} u ( t , \\omega ) : = \\sup _ { \\overline \\omega \\in \\C _ 0 [ t , 1 ] } \\left ( F ( \\omega \\oplus _ t \\overline \\omega ) - \\int _ t ^ 1 g ( s , \\dot { \\overline \\omega } ( s ) ) d s \\right ) . \\end{align*}"}
-{"id": "6472.png", "formula": "\\begin{align*} \\beta & : = \\left ( \\alpha + 1 + \\frac { \\mu ( e _ 1 ) } { \\mu ( e _ n ) } \\right ) - 1 - \\mu ( e _ 1 ) = \\alpha + \\frac { \\mu ( e _ 1 ) } { \\mu ( e _ n ) } - \\mu ( e _ 1 ) \\\\ \\gamma & : = \\left ( \\alpha + \\mu ( e _ 1 ) - \\frac { \\mu ( e _ 1 ) } { \\mu ( e _ n ) } \\right ) - 1 - \\mu ( e _ 1 ) = \\mu ( e _ 1 ) \\left ( 1 - \\sum _ { i = 1 } ^ n \\frac { 1 } { \\mu ( e _ i ) } \\right ) . \\end{align*}"}
-{"id": "9351.png", "formula": "\\begin{align*} w ^ 2 & = ( y _ { 2 ^ k - 1 } \\cdots y _ 1 y _ 0 ) ^ 2 \\\\ & = ( y _ { 2 ^ k - 1 } \\cdots y _ 1 ) ^ 2 y _ 0 ^ 2 [ y _ 0 , y _ { 2 ^ { k } - 1 } ] \\ldots [ y _ 0 , y _ { 2 ^ { k - 1 } + 1 } ] [ y _ 0 , y _ { 2 ^ { k - 1 } } ] [ y _ 0 , y _ { 2 ^ { k - 1 } - 1 } ] \\ldots [ y _ 0 , y _ 1 ] \\\\ & \\ , \\ , \\vdots \\\\ & = y _ { 2 ^ k - 1 } ^ { \\ , 2 } \\cdots y _ 1 ^ { \\ , 2 } y _ 0 ^ { \\ , 2 } \\\\ & = y ^ { 2 ^ { k + 1 } } \\\\ & = 1 . \\end{align*}"}
-{"id": "9317.png", "formula": "\\begin{align*} x _ { i j } \\leq y _ i , ~ i , j = 1 , \\dots , n ; \\end{align*}"}
-{"id": "8856.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 e _ 1 } { \\partial a ^ 2 } & = 4 [ 9 \\nu ^ 4 - 6 6 \\nu ^ 3 + 7 6 \\nu ^ 2 + 2 \\nu + 2 3 5 ] > 0 , \\end{align*}"}
-{"id": "2845.png", "formula": "\\begin{align*} W ( a , b ) : = \\log ( S ( R , R ) ) = a ^ 2 \\log ( S ( P , P ) ) + 2 a b \\log ( S ( P , Q ) ) + b ^ 2 \\log ( S ( Q , Q ) ) \\end{align*}"}
-{"id": "4568.png", "formula": "\\begin{align*} \\hat { f } ( \\omega ) = \\int f ( x ) e ^ { - 2 \\pi i ( \\omega , x ) } d x , \\end{align*}"}
-{"id": "2722.png", "formula": "\\begin{align*} w _ { 4 } & = 1 + \\max \\Big \\{ \\frac { 1 } { \\tau } + \\frac { 1 } { 2 \\cos \\theta } \\big ( \\frac { 1 } { \\tau } - \\tau \\big ) , \\frac { \\sigma _ { 1 } } { \\tau } , \\ldots , \\frac { \\sigma _ { n } } { \\tau } \\Big \\} . \\end{align*}"}
-{"id": "2788.png", "formula": "\\begin{align*} \\mathbf { Q } ^ k = \\frac { q ^ n } { 3 } \\left [ \\begin{array} { c c c } 2 ^ k + 2 ( - 1 ) ^ k & 2 ^ n - ( - 1 ) ^ n & 2 ^ k - ( - 1 ) ^ k \\\\ 2 ^ k - ( - 1 ) ^ n & 2 ^ k + 2 ( - 1 ) ^ n & 2 ^ k - ( - 1 ) ^ k \\\\ 2 ^ k - ( - 1 ) ^ n & 2 ^ k - ( - 1 ) ^ n & 2 ^ k + 2 ( - 1 ) ^ k \\end{array} \\right ] . \\end{align*}"}
-{"id": "7637.png", "formula": "\\begin{align*} \\mathrm { K e r } \\ , ( G - K ) ^ \\ast = G ^ { - 1 } ( \\xi ) \\psi _ 0 ( \\xi ) = : v _ 0 ( \\xi ) . \\end{align*}"}
-{"id": "2060.png", "formula": "\\begin{align*} \\limsup _ { r ( x ) \\to \\infty } \\frac { \\| d f \\| ( x ) } { r ^ \\epsilon ( x ) } = 0 \\end{align*}"}
-{"id": "2732.png", "formula": "\\begin{align*} \\min _ { w \\in \\Sigma _ { + } ^ { 1 } } \\Re { f ( w ) } - \\Re { f ( z _ { 0 } ) } = \\Re { f ( w _ { 2 } ) } - \\Re { f ( z _ { 0 } ) } \\geq 2 N ^ { - \\frac { 1 } { 6 } } . \\end{align*}"}
-{"id": "8275.png", "formula": "\\begin{align*} \\xi _ 0 ( \\alpha ) : = z ( \\alpha , t = 0 ) - \\alpha , v _ 0 : = z _ t ( t = 0 ) , w _ 0 : = z _ { t t } ( t = 0 ) , \\end{align*}"}
-{"id": "4229.png", "formula": "\\begin{align*} \\phi ( w ) - \\phi ( w ^ { \\prime } ) = \\beta + \\sum _ { j = i } ^ { n - 1 } d _ { j } ( r _ { 1 } \\cdots r _ { i - 1 } ) \\left ( r _ { i } ( \\gamma _ { j } ) - \\gamma _ { j } \\right ) \\end{align*}"}
-{"id": "9269.png", "formula": "\\begin{align*} M _ 2 ^ * ( R ) = d _ { \\bar x } ^ 2 | D ^ 2 u ( \\bar x ) | = ( \\delta R ) ^ 2 | D ^ 2 u ( \\bar x ) | , \\end{align*}"}
-{"id": "282.png", "formula": "\\begin{align*} T = S _ { i \\beta } ^ l \\prod _ { k = 1 } ^ { m - l } ( S _ { i \\beta } - e ^ { i \\theta _ k } I ) . \\end{align*}"}
-{"id": "4611.png", "formula": "\\begin{align*} \\sum _ { k \\geq 2 } k ^ { 1 . 5 } \\# _ k = \\sum _ { j \\geq 1 } \\sum _ { k \\in [ 2 ^ j , 2 ^ { j + 1 } ] } k ^ { 1 . 5 } \\# _ k \\leq \\sum _ { j \\geq 1 } 2 ^ { 1 . 5 j + 1 . 5 } \\delta ^ { - 3 } 2 ^ { - 1 . 5 j } \\leq C ' \\delta ^ { - 3 } ( - \\log \\delta ) . \\end{align*}"}
-{"id": "6291.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\mathcal { A } ' ( D ) \\in S \\right ) & = \\int _ { S } f _ { \\mathcal { A } ' ( D ) } d \\mu \\\\ & \\leq \\exp \\left ( \\epsilon d ( D , D ' ) \\right ) \\int _ { S } f _ { \\mathcal { A } ' ( D ' ) } d \\mu \\\\ & = \\exp \\left ( \\epsilon d ( D , D ' ) \\right ) \\mathbb { P } \\left ( \\mathcal { A } ' ( D ) \\in S \\right ) . \\end{align*}"}
-{"id": "9309.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { m } x _ { i j } \\geq d _ j , ~ j = 1 , \\dots , n ; \\end{align*}"}
-{"id": "6593.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } P _ { F _ n } ( \\varphi ) = \\lim _ { n \\to \\infty } \\prod _ { r = 1 } ^ { F _ n } | 2 \\sin ( \\pi r \\varphi ) | = 2 . 4 0 7 \\ldots \\ , . \\end{align*}"}
-{"id": "9499.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { n \\rightarrow \\infty } \\Vert u _ n - u \\Vert _ { s , m } ^ { m } = \\displaystyle \\lim _ { n \\rightarrow \\infty } \\left ( \\Vert u _ n \\Vert _ { s , m } ^ { m } - \\Vert u \\Vert _ { s , m } ^ { m } \\right ) . \\end{align*}"}
-{"id": "716.png", "formula": "\\begin{align*} \\\\ & \\left [ g _ { i \\bar { j } } \\right ] = \\left [ \\begin{matrix} \\displaystyle & 4 p K & 0 \\\\ \\\\ & 0 & \\frac { f ^ { ( 1 ) } ( 0 ) } { 4 } \\end{matrix} \\right ] , \\left [ g ^ { \\bar { \\alpha } \\beta } \\right ] = \\left [ \\begin{matrix} \\displaystyle & \\frac { 1 } { 4 p K } & 0 \\\\ \\\\ & 0 & \\frac { 4 } { f ^ { ( 1 ) } ( 0 ) } \\end{matrix} \\right ] , \\end{align*}"}
-{"id": "4314.png", "formula": "\\begin{align*} \\int _ { D } ( \\Pi _ D ^ k v - v ) = 0 , k \\geq 2 , \\end{align*}"}
-{"id": "4323.png", "formula": "\\begin{align*} a _ h ( u , v ) = \\ ; \\sum _ { K \\in \\mathcal { T } _ h } \\Bigl [ \\big ( \\nabla \\Pi _ K u , \\nabla \\Pi _ K v \\big ) _ K + S _ K ( u , v ) \\Bigr ] . \\end{align*}"}
-{"id": "2862.png", "formula": "\\begin{align*} x \\rightarrow y = ( y \\wedge x ) \\vee x ' . \\end{align*}"}
-{"id": "8690.png", "formula": "\\begin{align*} j c _ { h , i , j } = 4 ( h + 1 ) ( h + 2 ) ( 2 h + p ' + 2 ) c _ { h + 2 , i , j - 1 } + 4 ( h + 1 ) ( i + 1 ) ( 2 i + p '' ) c _ { h + 1 , i + 1 , j - 1 } , \\end{align*}"}
-{"id": "3268.png", "formula": "\\begin{align*} \\sum x _ j a _ j ^ n - e _ 1 \\sum x _ j a _ j ^ { n - 1 } + e _ 2 \\sum x _ j a _ j ^ { n - 2 } - \\dots \\pm e _ m \\sum x _ j a _ j ^ { n - m } = 0 , \\end{align*}"}
-{"id": "5334.png", "formula": "\\begin{align*} \\langle i H . w ^ k , w ^ l \\rangle = \\langle w ^ k , ( i H ) ^ * . w ^ l \\rangle \\end{align*}"}
-{"id": "3855.png", "formula": "\\begin{align*} \\tau _ { \\partial B } : = \\inf \\{ s > 0 : \\ Z _ s ^ x = \\partial B \\} . \\end{align*}"}
-{"id": "2414.png", "formula": "\\begin{align*} \\Pi = ( X _ s , W _ k , \\dots , W _ { k + t } , X _ r ) \\end{align*}"}
-{"id": "8267.png", "formula": "\\begin{align*} \\mathbb { H } f ( \\alpha ) : = \\frac { 1 } { \\pi i } p . v . \\int _ { - \\infty } ^ { \\infty } \\frac { 1 } { \\alpha - \\beta } f ( \\beta ) d \\beta . \\end{align*}"}
-{"id": "7920.png", "formula": "\\begin{align*} R _ N = \\hat { R } _ n ( b - a ) h ^ { n + 1 } f ^ { ( n + 1 ) } ( \\xi ) \\end{align*}"}
-{"id": "3392.png", "formula": "\\begin{align*} I _ 0 ( T ) & = \\prod _ { i } I ( T _ i ) , \\\\ I ( T ) & = I _ 0 ( T ) + \\prod _ { i } I _ 0 ( T _ i ) . \\end{align*}"}
-{"id": "1937.png", "formula": "\\begin{align*} F _ { p + 2 } ( x ) = - \\frac { x ^ 2 } { p } \\frac { d } { d x } \\sum _ { j = 1 } ^ \\infty \\frac { 1 } { Z _ j ^ p } . \\end{align*}"}
-{"id": "1795.png", "formula": "\\begin{align*} G _ { n } = \\left \\{ B _ { \\theta ^ { n } } - B _ { \\theta ^ { n + 1 } } < ( 1 - \\theta ^ { H } ) h ( \\theta ^ { n } ) \\right \\} . \\end{align*}"}
-{"id": "9354.png", "formula": "\\begin{align*} [ [ y , x , \\overset { i - 1 } \\ldots , x ] , x ] ^ 2 & = [ [ y , x , \\overset { i - 1 } \\ldots , x ] ^ x , [ y , x , \\overset { i - 1 } \\ldots , x ] ^ { - 1 } ] \\\\ & = [ [ y , x , \\overset { i - 1 } \\ldots , x ] [ y , x , \\overset { i } \\ldots , x ] , [ y , x , \\overset { i - 1 } \\ldots , x ] ^ { - 1 } ] \\\\ & = [ [ y , x , \\overset { i } \\ldots , x ] , [ y , x , \\overset { i - 1 } \\ldots , x ] ^ { - 1 } ] \\in [ H _ k , H _ k ] \\cap \\gamma _ { 2 i + 1 } ( G _ k ) , \\end{align*}"}
-{"id": "1080.png", "formula": "\\begin{align*} & D _ n ( w G _ n ( x ) / 2 ) = D _ n ( w \\alpha _ n ) = \\exp ( \\ln ( D _ n ( w \\alpha _ n ) / D _ n ( \\alpha _ n ) ) ) D _ n ( \\alpha _ n ) \\\\ \\leq & e ^ { - ( w - 1 ) \\ln n + 1 } D _ n ( \\alpha _ n ) = e ^ { 1 - ( w - 1 ) \\ln n } D _ n ( G _ n ( x ) / 2 ) , \\end{align*}"}
-{"id": "925.png", "formula": "\\begin{align*} \\Biggl . - { \\bf 1 } _ { \\{ i _ 6 = i _ 5 \\ne 0 \\} } { \\bf 1 } _ { \\{ j _ 6 = j _ 5 \\} } { \\bf 1 } _ { \\{ i _ 1 = i _ 3 \\ne 0 \\} } { \\bf 1 } _ { \\{ j _ 1 = j _ 3 \\} } { \\bf 1 } _ { \\{ i _ 2 = i _ 4 \\ne 0 \\} } { \\bf 1 } _ { \\{ j _ 2 = j _ 4 \\} } \\Biggr ) , \\end{align*}"}
-{"id": "224.png", "formula": "\\begin{align*} \\Delta _ z u + | z | ^ 2 \\Delta _ { t } u = V u , \\ ( z , t ) \\in \\mathbb { R } ^ N \\times \\mathbb { R } ^ m \\end{align*}"}
-{"id": "4984.png", "formula": "\\begin{align*} C _ k = \\sum _ { \\gamma \\in { \\rm p a r } ( k ) } \\frac { \\displaystyle \\prod _ { j \\in \\gamma } \\frac { N _ j } { j } } { \\rm l e n ( \\gamma ) ! } - \\sum _ { i = 0 } ^ { k - 1 } ( C _ i \\sum _ { \\mu = 0 } ^ { k - i } q ^ \\mu ) . \\end{align*}"}
-{"id": "2569.png", "formula": "\\begin{align*} \\langle \\mu , \\phi \\rangle \\dot { \\mathbf P } ^ { ( \\phi ) } _ \\mu [ Y _ t ( \\phi ) ^ { - 1 } ] = \\mathbb N _ \\mu [ W _ t ( \\phi ) ] \\mathbb N ^ { W _ t ( \\phi ) } _ \\mu [ W _ t ( \\phi ) ^ { - 1 } ] = \\mathbb N _ \\mu ( W _ t ( \\phi ) > 0 ) = \\mu ( v _ t ) . \\end{align*}"}
-{"id": "1403.png", "formula": "\\begin{align*} f ( x _ 1 , \\dots , x _ d ) = H ( \\sum _ { j = 1 } ^ d R _ j ( x _ j ) + \\log ( \\prod _ { j = 1 } ^ d \\tilde { R } _ j ( x _ j ) ) ) , \\end{align*}"}
-{"id": "265.png", "formula": "\\begin{align*} ( f h - f ^ { p + 1 } v _ 0 ) - \\alpha f g & = \\sum _ { j = 1 } ^ p f ^ { p - j + 1 } v _ j \\big ( Q _ j ( \\alpha ) - \\alpha j Q _ { j - 1 } ( \\alpha ) \\big ) \\\\ & = ( 1 - \\alpha ) \\cdot \\sum _ { j = 2 } ^ p ( j - 1 ) Q _ { j - 1 } ( \\alpha ) f ^ { p - j + 1 } v _ j . \\end{align*}"}
-{"id": "8346.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } \\frac { 1 } { ( \\alpha - z _ j ) ( \\beta - z _ j ) \\overline { ( \\alpha - z _ k ) ( \\beta - z _ k ) } } d \\beta = \\frac { 1 } { ( \\alpha - z _ j ) \\overline { ( \\alpha - z _ k ) } } \\frac { 2 \\pi i } { \\overline { z _ k } - z _ j } . \\end{align*}"}
-{"id": "3610.png", "formula": "\\begin{align*} v _ { ( i , j ) } = a _ i + a _ j + \\frac { 1 } { 3 } a _ { - i } + \\frac { 1 } { 3 } a _ { - j } - \\frac { 2 ^ 6 } { 3 } a _ i \\cdot a _ j \\end{align*}"}
-{"id": "6282.png", "formula": "\\begin{align*} \\max _ { p \\in [ 0 , \\rho ] } \\mathbb { E } _ { G \\sim G _ { n , p } } [ ( A ( G ) - p ) ^ 2 ] = O \\left ( \\frac { \\rho } { n ^ 2 } + \\frac { \\log n } { n ^ 3 \\epsilon ^ 2 } \\right ) . \\end{align*}"}
-{"id": "2439.png", "formula": "\\begin{align*} P _ { \\widetilde { \\varphi _ { \\ell } ^ { H _ 1 } } } ( T ) & = \\frac { 1 } { ( - 1 ) ^ { \\ell / 4 } + 2 ^ { ( \\ell - 4 ) / 2 } } \\\\ & \\sum _ { i = 1 } ^ { \\ell / 4 - 1 } ( - 1 ) ^ { i } ( 4 ^ { i - 1 } T ^ { 4 ( i - 1 ) } + 2 \\times 4 ^ { i - 1 } T ^ { 4 ( i - 1 ) + 1 } + 2 \\times 4 ^ { i - 1 } T ^ { 4 ( i - 1 ) + 2 } ) . \\end{align*}"}
-{"id": "1588.png", "formula": "\\begin{align*} ( z _ m , x _ m ) : = \\widetilde { \\boldsymbol { F } } _ { m , n ^ 2 + k _ n } ( z ^ \\iota _ n , x ^ \\iota _ n ) \\in ( V _ { w _ m } \\times \\mathbb { C } ) \\cap \\Omega _ { w _ m } \\end{align*}"}
-{"id": "10014.png", "formula": "\\begin{align*} \\Big \\Vert \\sum _ { n = 1 } ^ { N } a _ { n } n ^ { - s } \\Big \\Vert _ { \\mathfrak { X } ( X ) } \\leq 2 B N ^ { \\sigma _ { 0 } } \\ , . \\end{align*}"}
-{"id": "8555.png", "formula": "\\begin{align*} \\norm { | D | ^ { 1 / 2 } ( \\Big ( \\tilde { \\theta } - 2 ( \\zeta - \\alpha ) \\Big ) } _ { H ^ s } = \\| | D | ^ { 1 / 2 } ( \\mathcal { H } + \\bar { \\mathcal { H } } ) ( \\zeta - \\alpha ) \\| _ { H ^ s } \\leq C \\epsilon ^ 2 . \\end{align*}"}
-{"id": "4675.png", "formula": "\\begin{align*} \\pi _ { n , g } ( \\bar \\lambda _ 1 , \\ldots , \\bar \\lambda _ n ; a _ 0 , a _ 1 , \\ldots , a _ g ) = \\sum _ { \\substack { k _ 2 , \\ldots , k _ n , \\\\ \\sum i k _ i = g } } e _ 2 ^ { k _ 2 } \\dots e _ n ^ { k _ n } U _ { k _ 2 , \\ldots , k _ n } \\end{align*}"}
-{"id": "8826.png", "formula": "\\begin{align*} \\sigma _ C ^ \\circ ( \\bar x ) = \\sup _ { x : \\sigma _ C ( x ) \\leq 1 } \\langle \\bar x , x \\rangle \\ ; . \\end{align*}"}
-{"id": "382.png", "formula": "\\begin{align*} \\alpha _ 1 & = | k + \\sqrt { k ^ 2 - 1 } | , \\\\ \\alpha _ 2 & = | 4 k ^ 2 - 2 k - 1 + \\sqrt { ( 1 6 k ^ 3 - 4 k ) ( k - 1 ) } | \\\\ \\alpha _ 3 & = \\left | \\frac { \\sqrt { 1 6 k ^ 3 - 4 k } ( \\sqrt { k - 1 } + \\sqrt { k + 1 } ) } { \\sqrt { k + 1 } ( x _ 1 \\sqrt { 1 6 k ^ 3 - 4 k } + z _ 1 \\sqrt { k + 1 } ) } \\right | . \\end{align*}"}
-{"id": "823.png", "formula": "\\begin{align*} \\sigma \\left ( \\sum _ { i = 1 } ^ { { s } } \\frac { b _ i } { U _ 1 ^ { ( i ) } } \\right ) = \\sum _ { i = 1 } ^ { { s } } \\frac { b _ i } { U _ 1 ^ { ( i + 1 ) } } + c _ i ^ p - c _ i . \\end{align*}"}
-{"id": "9761.png", "formula": "\\begin{align*} ( \\lambda _ * - | \\alpha | ) D ^ \\alpha q ( Z _ \\infty ) = - \\lim _ { \\ell \\to \\infty } \\frac { r _ \\ell ^ \\kappa } { h _ { r _ \\ell } } ( Z _ \\infty \\cdot \\nabla _ { x } D ^ \\alpha p _ * ( Z _ \\ell ) ) = 0 . \\end{align*}"}
-{"id": "2229.png", "formula": "\\begin{align*} \\# \\left ( I _ { n } \\cap J _ { k } \\right ) = 1 \\forall k \\in \\left [ n \\right ] . \\end{align*}"}
-{"id": "954.png", "formula": "\\begin{align*} | s _ j ^ { \\top } \\ , ( \\gamma _ j \\ , s _ j + J _ { m _ j } ^ \\top r _ { m _ j } ) | \\ ; \\leq \\ ; \\frac { 4 \\ , \\| J _ { m _ j } \\| ^ 2 \\ , \\| J _ { m _ j } ^ \\top r _ { m _ j } \\| ^ 2 + 2 \\theta _ { i n } \\ , \\| J _ { m _ j } ^ \\top r _ { m _ j } \\| ^ 2 } { \\gamma _ j ^ 2 } \\ ; = \\ ; \\frac { 4 \\| J _ { m _ j } \\| ^ 2 + 2 \\theta _ { i n } } { \\mu _ j ^ 2 } . \\end{align*}"}
-{"id": "9314.png", "formula": "\\begin{align*} B = \\left [ \\begin{matrix} I _ n & I _ n & I _ n & \\dots & I _ n \\\\ \\end{matrix} \\right ] \\end{align*}"}
-{"id": "6144.png", "formula": "\\begin{align*} \\frac { a } { b } < \\nu _ e \\qquad { a n d } \\gcd ( a , b ) = 1 . \\end{align*}"}
-{"id": "183.png", "formula": "\\begin{align*} \\pi _ { T M , X _ { C } , Y _ { V } , c } ( { \\bf x } , { \\bf y } ) = \\end{align*}"}
-{"id": "2942.png", "formula": "\\begin{align*} & \\beta _ { 1 } : = \\| A ^ { \\dagger } \\| _ { 2 } ^ { 4 } \\big ( \\| E \\| _ { F } ^ { 2 } - \\| E B ^ { \\dagger } B \\| _ { F } ^ { 2 } \\big ) + \\| B ^ { \\dagger } \\| _ { 2 } ^ { 4 } \\big ( \\| E \\| _ { F } ^ { 2 } - \\| A A ^ { \\dagger } E \\| _ { F } ^ { 2 } \\big ) , \\\\ & \\beta _ { 2 } : = \\| A ^ { \\dagger } \\| _ { 2 } ^ { 4 } \\big ( \\| E \\| _ { F } ^ { 2 } - \\| B B ^ { \\dagger } E \\| _ { F } ^ { 2 } \\big ) + \\| B ^ { \\dagger } \\| _ { 2 } ^ { 4 } \\big ( \\| E \\| _ { F } ^ { 2 } - \\| E A ^ { \\dagger } A \\| _ { F } ^ { 2 } \\big ) . \\end{align*}"}
-{"id": "3613.png", "formula": "\\begin{align*} ( \\alpha _ 0 - \\beta _ 0 ) \\cdot ( \\alpha _ 1 - \\beta _ 1 ) = \\frac { 1 } { 2 ^ 2 } a _ { - 1 } + \\frac { 1 1 } { 2 ^ 2 \\cdot 3 } ( a _ 2 + a _ { - 2 } + a _ 3 ) - \\frac { 1 } { 2 ^ 3 \\cdot 3 } ( v _ { ( 1 , 2 ) } + v _ { ( 1 , 3 ) } ) - \\frac { 4 } { 3 } v _ { ( 2 , 3 ) } \\end{align*}"}
-{"id": "1789.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { i = 1 } ^ { n } c _ { p , r } ( C _ { i , n } ) \\leq & \\left [ \\sum _ { l = 0 } ^ { r } \\left ( N ^ { l p } C _ { H } ^ { l p / 2 } \\left ( \\frac { M _ { r } } { c } \\right ) ^ { l p } \\right ) \\right ] ^ { 1 / p } \\\\ & \\cdot \\begin{aligned} \\sum _ { i = 1 } ^ { n } \\left [ P \\left ( \\bigcap _ { j = 1 } ^ { N } \\{ - \\alpha _ { j } - c \\leq X _ { j } \\leq \\alpha _ { j } + c \\} \\right ) \\right ] ^ { 1 / p } , \\end{aligned} \\end{aligned} \\end{align*}"}
-{"id": "6042.png", "formula": "\\begin{align*} g _ k ( k - i ) = 3 4 i ^ 2 + ( 1 4 - 4 k ) i + 2 k ^ 2 - 2 k + 1 \\end{align*}"}
-{"id": "2305.png", "formula": "\\begin{align*} \\mu ( x ) & = \\frac { 1 } { 2 \\pi \\i } ( R ( x - \\i 0 ) - R ( x + \\i 0 ) ) = | \\alpha _ x ) ( \\alpha _ x | \\\\ | \\alpha _ x ) & = ( 1 + \\pi ^ 2 ) ^ { 1 / 2 } \\left ( \\frac { \\mathcal { P } } { x - \\Omega } | E ) + | \\delta _ x ) \\right ) \\end{align*}"}
-{"id": "1608.png", "formula": "\\begin{align*} \\ell ^ \\ast = \\mu ( b ^ \\ast ) b ^ \\ast = \\frac { 1 } { S ' ( b ^ \\ast ) m ( ( 0 , b ^ \\ast ) ) } . \\end{align*}"}
-{"id": "6664.png", "formula": "\\begin{align*} \\zeta ( s ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ { s } } , \\ \\ s = \\sigma + i \\lambda , \\ ( \\sigma , \\lambda \\ , { \\rm r e a l } ) , { \\rm R e } \\ , s > 1 . \\end{align*}"}
-{"id": "2483.png", "formula": "\\begin{align*} \\| \\mu \\| _ 0 & = \\frac { 1 } { 2 } \\bigl ( \\bigl | \\| \\mu ^ + \\| _ 1 + \\| \\mu ^ - \\| _ 1 \\bigr | + \\bigl | \\| \\mu ^ + \\| _ 1 - \\| \\mu ^ - \\| _ 1 \\bigr | \\bigr ) \\\\ & = \\frac { 1 } { 2 } \\bigl ( \\| \\mu \\| _ 1 + \\bigl | \\mu ^ + ( K ) - \\mu ^ - ( K ) \\bigr | \\bigr ) = \\frac { 1 } { 2 } \\bigl ( \\| \\mu \\| _ 1 + \\bigl | \\mu ( K ) \\bigl | \\bigr ) \\end{align*}"}
-{"id": "494.png", "formula": "\\begin{align*} \\mu _ { 1 , L } ( \\sigma _ 1 \\wedge \\sigma _ 2 ) = ( f _ i ' ( z ) f _ j ( z ) - f _ i ( z ) \\ , f _ j ' ( z ) ) d z \\otimes l ^ 2 . \\end{align*}"}
-{"id": "6557.png", "formula": "\\begin{align*} \\iota : E _ { \\omega _ 1 , \\omega _ 2 } \\to \\mathbb { P } ^ 2 , \\iota ( z ) = ( \\wp ( z ) , \\wp ' ( z ) , 1 ) \\end{align*}"}
-{"id": "5511.png", "formula": "\\begin{align*} P _ { ( \\kappa ) } ( Z _ i ) \\equiv ( Z _ i ( P _ { V _ \\kappa ^ { \\otimes d } } [ \\eta ] ) : \\eta \\in \\mathcal B ^ \\rho ( 1 ) ) , i = 1 , 2 , \\end{align*}"}
-{"id": "3835.png", "formula": "\\begin{align*} 2 G ^ i = ^ { h } { \\gamma _ { 0 } ^ { i } } _ 0 + 2 \\Phi ^ i , \\end{align*}"}
-{"id": "5120.png", "formula": "\\begin{align*} \\quad { \\sigma } _ 1 = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} , { \\sigma } _ 2 = \\begin{pmatrix} 0 & - i \\\\ i & 0 \\end{pmatrix} , \\quad { \\sigma } _ 3 = \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "8512.png", "formula": "\\begin{align*} & \\norm { \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { 2 \\pi } \\frac { \\ddot { z } _ j } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } } _ { H ^ s } \\\\ \\leq & \\norm { \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i \\ddot { z } _ 1 ( t ) } { 2 \\pi } \\frac { 1 } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } } _ { H ^ s } + \\norm { \\frac { \\lambda i ( \\ddot { z } _ 1 ( t ) - \\ddot { z } _ 2 ( t ) ) } { 2 \\pi } \\frac { 1 } { ( \\zeta ( \\alpha , t ) - z _ 2 ( t ) ) ^ 2 } } _ { H ^ s } : = I + \\it { I I } . \\end{align*}"}
-{"id": "10037.png", "formula": "\\begin{align*} \\Psi _ \\lambda ( z q ^ { - n } ) = ( a \\lambda ) ^ n \\Big ( 1 + \\mathcal O ( q ^ n ) \\Big ) , \\end{align*}"}
-{"id": "9133.png", "formula": "\\begin{align*} \\Delta M ( \\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } , \\mathbf { b } ) = \\mathbf { b \\cdot n } _ { 1 2 } \\left \\vert \\mathbf { n } _ { 1 2 } \\cdot \\mathbf { v } _ { 1 2 } ^ { ( + ) } \\right \\vert \\mathbf { v } _ { 1 2 } ^ { ( + ) } \\cdot \\mathbf { b } - \\left ( \\mathbf { b \\cdot n } _ { 1 2 } \\right ) ^ { 2 } \\left ( \\mathbf { n } _ { 1 2 } \\cdot \\mathbf { v } _ { 1 2 } ^ { ( + ) } \\right ) ^ { 2 } , \\end{align*}"}
-{"id": "1829.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ \\infty \\hat { f } ( \\alpha ) \\ , e ^ { - i \\alpha x } d \\alpha . \\end{align*}"}
-{"id": "3516.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\sum ^ M _ { j = 0 } \\Big ( \\log ( \\nu _ j + N ) - \\psi ( \\nu _ j + N ) + \\frac { w } { \\nu _ j + N } \\Big ) = \\gamma ( w + \\frac { 1 } { 2 } ) , \\lim _ { N \\to \\infty } \\sum ^ M _ { j = 0 } \\psi ' ( \\nu _ j + N ) = \\gamma , \\end{align*}"}
-{"id": "8222.png", "formula": "\\begin{align*} & U _ 5 ( t ^ { - 1 } ) = - 2 - 5 t , U _ 5 ( t ^ { - 2 } ) = - 2 - 5 ^ 3 t ^ 2 , \\\\ & U _ 5 ( t ^ { - 3 } ) = 4 6 - 5 ^ 5 t ^ 3 , U _ 5 ( t ^ { - 4 } ) = - 4 2 \\cdot 5 - 5 ^ 7 t ^ 4 . \\end{align*}"}
-{"id": "8396.png", "formula": "\\begin{align*} ( p ) _ t = \\frac { i } { 2 \\pi } \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j ( z _ t - \\dot { z } _ j ( t ) ) } { ( z ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } . \\end{align*}"}
-{"id": "173.png", "formula": "\\begin{align*} \\pi _ { T M , X _ { V } , Y _ { V } , c } = \\pi _ { T M } + \\lambda c ( { \\bf x } ) X _ { V } \\wedge Y _ { V } \\end{align*}"}
-{"id": "7144.png", "formula": "\\begin{align*} \\tilde { L } : = \\bar \\rho L , \\end{align*}"}
-{"id": "5213.png", "formula": "\\begin{align*} U ( x ) = c x ^ { - 1 } B ( x ) \\exp \\left ( - \\int _ { 1 } ^ { x } t ^ { - 1 } B ( t ) d t \\right ) . \\end{align*}"}
-{"id": "2937.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { r } \\bigg ( \\frac { 1 } { \\sigma _ { i } } - \\frac { 1 } { \\widetilde { \\sigma } _ { i } } \\bigg ) ^ { 2 } + \\sum _ { i = r + 1 } ^ { s } \\frac { 1 } { \\widetilde { \\sigma } _ { i } ^ { 2 } } \\leq \\| B ^ { \\dagger } - A ^ { \\dagger } \\| _ { F } ^ { 2 } \\leq \\sum _ { i = 1 } ^ { r } \\bigg ( \\frac { 1 } { \\sigma _ { i } } + \\frac { 1 } { \\widetilde { \\sigma } _ { i } } \\bigg ) ^ { 2 } + \\sum _ { i = r + 1 } ^ { s } \\frac { 1 } { \\widetilde { \\sigma } _ { i } ^ { 2 } } . \\end{align*}"}
-{"id": "8615.png", "formula": "\\begin{align*} \\tilde { H } = & \\frac { ( u ' ) ^ 2 } { 2 } + \\frac { \\left ( h ' u - \\left ( h + \\frac { \\omega } { \\Lambda - 1 } \\right ) u ' \\right ) ^ 2 } { 2 u ^ 4 } \\\\ & - \\frac { \\left ( 1 - u ^ 2 - \\frac { 1 } { u ^ 2 } \\left ( h + \\frac { \\omega } { \\Lambda - 1 } \\right ) ^ 2 \\right ) ^ 2 } { 4 } - \\frac { \\Lambda - 1 } { 2 } h ^ 2 \\end{align*}"}
-{"id": "1279.png", "formula": "\\begin{align*} | z | _ { k , c + \\frac { 3 i } { 2 \\pi } \\alpha } : = \\| ( z - c - \\frac { 3 i } { 2 \\pi } \\alpha ) | \\eta | ^ { - 1 } \\| _ { L ^ { \\infty } ( 0 , 1 ) } + \\| z | \\eta | ^ k \\| _ { L ^ { \\infty } ( 1 , \\infty ) } + \\| ( 1 + | \\eta | ^ { k + 1 } ) z ' \\| _ { L ^ { \\infty } ( \\real ^ + ) } . \\end{align*}"}
-{"id": "6074.png", "formula": "\\begin{align*} \\mathcal A '' = ( h ' + h ^ 2 ) \\mathcal A . \\end{align*}"}
-{"id": "7798.png", "formula": "\\begin{align*} | \\mathfrak m _ N ^ { B ^ { q } } ( \\xi ) - 1 | & \\le C _ q \\kappa _ q ( d , N ) | \\xi | , \\\\ | \\mathfrak m _ N ^ { B ^ { q } } ( \\xi ) | & \\le C _ q ( \\kappa _ q ( d , N ) | \\xi | ) ^ { - 1 } , \\\\ | \\mathfrak m _ { N + 1 } ^ { B ^ { q } } ( \\xi ) - \\mathfrak m _ { N } ^ { B ^ { q } } ( \\xi ) | & \\le C _ q N ^ { - 1 } , \\end{align*}"}
-{"id": "2787.png", "formula": "\\begin{align*} ( \\mathbf { 1 } - \\mathbf { Q } ) ^ { - 1 } = \\frac { 1 } { 1 - 3 q ^ 2 - 2 q ^ 3 } \\left [ \\begin{array} { c c c } 1 - q ^ 2 & q + q ^ 2 & q + q ^ 2 \\\\ q + q ^ 2 & 1 - q ^ 2 & q + q ^ 2 \\\\ q + q ^ 2 & q + q ^ 2 & 1 - q ^ 2 \\end{array} \\right ] . \\end{align*}"}
-{"id": "3634.png", "formula": "\\begin{align*} v _ { ( 1 , 2 ) } \\cdot ( ( a _ 3 - a _ { - 3 } ) \\cdot v _ { ( 1 , 2 ) } ) = t ( a _ 3 - a _ { - 3 } ) \\cdot v _ { ( 1 , 2 ) } . \\end{align*}"}
-{"id": "3263.png", "formula": "\\begin{align*} ( a + b ) ^ 3 - 3 a b ( a + b ) - ( a ^ 3 + b ^ 3 ) = 0 \\end{align*}"}
-{"id": "5966.png", "formula": "\\begin{align*} X ^ q ( t ) = \\int _ 0 ^ t q ( s ) d s + W ( t ) . \\end{align*}"}
-{"id": "5238.png", "formula": "\\begin{align*} F ^ { n } ( a _ { [ n s ] } x + b _ { [ n s ] } ) \\rightarrow \\varphi _ { \\alpha } ( x ) ^ { 1 / s } = \\varphi _ { \\alpha } ( ( 1 / s ) ^ { - 1 / \\alpha } x ) n \\rightarrow + \\infty . \\end{align*}"}
-{"id": "7361.png", "formula": "\\begin{align*} \\hat d ( x , x ' ) = \\hat d ( x ' , x ) = d _ i ( x , x _ i ) + d ( x _ i , x _ j ) + d _ j ( x _ j , x ' ) \\end{align*}"}
-{"id": "4518.png", "formula": "\\begin{align*} A = \\{ ( a _ 1 , a _ 2 , \\dots , a _ n ) \\in \\mathbb { N } ^ n : \\forall i \\in [ 1 , n ] , a _ i \\in [ 1 , K ] \\} . \\end{align*}"}
-{"id": "8393.png", "formula": "\\begin{align*} E ( t ) \\geq & \\sum _ { k = 0 } ^ s \\int \\frac { \\inf _ { \\alpha \\in \\mathbb { R } } | z _ { \\alpha } | ^ { - 2 k + 1 } } { \\sup _ { \\alpha \\in \\mathbb { R } } a | z _ { \\alpha } | } | \\partial _ { \\alpha } ^ k u _ t ( \\alpha , t ) | ^ 2 d \\alpha \\\\ \\geq & \\sum _ { k = 0 } ^ s \\frac { ( 2 C _ 2 ) ^ { - 2 k + 1 } } { 2 \\| w _ 0 \\| _ { H ^ s } + 1 } \\int | \\partial _ { \\alpha } ^ k u _ t ( \\alpha , t ) | ^ 2 d \\alpha \\\\ \\geq & \\frac { ( 2 C _ 2 ) ^ { - 2 s + 1 } } { 2 \\| w _ 0 \\| _ { H ^ s } + 1 } \\| z _ { t t } ( \\cdot , t ) \\| _ { H ^ s } ^ 2 . \\end{align*}"}
-{"id": "9525.png", "formula": "\\begin{align*} t ( n , N ) : = \\sum _ { \\pi \\in \\mathcal { D } ( n , N ) } ( - 1 ) ^ { \\# ( \\pi ) - 1 } s ( \\pi ) , \\end{align*}"}
-{"id": "1468.png", "formula": "\\begin{align*} V _ { 1 0 } = - \\varepsilon \\bar { m } \\int _ { \\mathbb { T } ^ { d } } ( \\mu ^ { 1 } - \\mu ^ { 2 } ) \\sum _ { i = 1 } ^ { d } \\left [ \\Theta _ { p _ { i } x _ { i } } ( t , x , \\mu ^ { 1 } , D w ^ { 1 } ) - \\Theta _ { p _ { i } x _ { i } } ( t , x , \\mu ^ { 2 } , D w ^ { 2 } ) \\right ] \\ d x , \\end{align*}"}
-{"id": "7140.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\mathrm { m e s } \\big \\{ ( x , t ) \\in [ 0 , L ] \\times [ 0 , T ] \\big | \\rho ( x , t ) \\geq 3 - \\delta \\big \\} = 0 , \\end{align*}"}
-{"id": "2662.png", "formula": "\\begin{align*} \\widetilde { G } & = [ \\tilde { g } _ { j k } ] : = \\left [ \\frac { \\varrho _ { j } - \\varrho _ { k } } { 1 - \\varrho _ { j } - \\varrho _ { k } } \\right ] _ { 1 \\leq j , k \\leq N + n } . \\end{align*}"}
-{"id": "4270.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} d F & = 0 & & B _ { R } \\\\ \\delta F & = f & & B _ { R } \\\\ \\nu \\wedge F & = 0 & & \\partial B _ { R } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "2328.png", "formula": "\\begin{align*} ( a ^ + ) ^ \\mathfrak { m } & = ( a ^ + _ { x _ 1 } ) ^ { m _ 1 } \\cdots ( a ^ + _ { x _ k } ) ^ { m _ k } & a ^ \\mathfrak { m } & = ( a _ { x _ 1 } ) ^ { m _ 1 } \\cdots ( a _ { x _ k } ) ^ { m _ k } . \\end{align*}"}
-{"id": "6146.png", "formula": "\\begin{align*} \\frac { b _ t } { a _ t } \\le \\frac { a _ 1 - 1 } { e b _ 1 } = \\frac { a _ 1 ^ 2 - a _ 1 } { e a _ 1 b _ 1 } = \\frac { b _ 1 } { a _ 1 } - \\frac { a _ 1 - 1 } { e a _ 1 b _ 1 } < \\frac { b _ 1 } { a _ 1 } . \\end{align*}"}
-{"id": "8951.png", "formula": "\\begin{align*} \\beta ( \\cdot ) = \\sum _ { j = 1 } ^ p \\beta _ j K ( t _ j , \\cdot ) , \\end{align*}"}
-{"id": "10060.png", "formula": "\\begin{align*} \\omega = \\sum _ { j = 1 } ^ { 2 ( n - 1 ) } d \\phi _ j \\wedge d \\rho _ j - \\frac { 4 } { x _ n ^ 3 } \\ , d x _ n \\wedge d y _ n + d \\theta _ n \\wedge d G _ n . \\end{align*}"}
-{"id": "5575.png", "formula": "\\begin{align*} & \\partial _ x ^ j u \\in \\mathcal { C } ( \\R _ x ^ + ; H ^ { ( 2 s + 1 - 2 j ) / 4 } ( 0 , T ) ) j = 0 , 1 , \\\\ & \\partial _ x ^ j v \\in \\mathcal { C } ( \\R _ x ^ + ; H ^ { ( \\kappa + 1 - j ) / 4 } ( 0 , T ) ) j = 0 , 1 , 2 . \\end{align*}"}
-{"id": "306.png", "formula": "\\begin{align*} c _ { M , g - 1 } \\bigl ( \\Lambda _ { m Z } \\circ \\Lambda _ 2 ' + \\Lambda _ 1 ' \\circ \\Lambda _ { m Z } + \\Lambda _ 1 ' \\circ \\Lambda _ 2 ' \\bigr ) \\ ; = \\ ; 0 . \\end{align*}"}
-{"id": "5241.png", "formula": "\\begin{align*} \\forall ( \\gamma > 0 ) , \\lim _ { x \\rightarrow + \\infty } \\frac { 1 - F ^ { \\ast } ( \\gamma x ) } { 1 - F ^ { \\ast } ( x ) } = \\gamma ^ { - \\beta } . \\end{align*}"}
-{"id": "3777.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ \\Omega B _ j v _ j \\cdot v _ k & = \\int _ \\Omega \\Re \\nabla u _ j \\cdot \\Re \\gamma _ k ( \\nabla u _ k ) + \\Im \\gamma _ j ( \\nabla u _ j ) \\cdot \\Im \\nabla u _ k \\\\ & = \\Re \\int _ \\Omega \\div ( \\gamma _ k ( \\nabla u _ k ) \\Re u _ j ) + \\Im \\int _ \\Omega \\div ( \\gamma _ j ( \\nabla u _ j ) \\Im u _ k ) \\\\ & = \\int _ { \\partial \\Omega } ( \\Re u _ j \\Re h + \\Im u _ k \\Im h ) . \\end{aligned} \\end{align*}"}
-{"id": "19.png", "formula": "\\begin{align*} | \\bar { \\mathbf { y } } ^ { i , \\rho } ( v ) | & = | \\mathbf { y } ^ { i , \\rho } ( v ) - \\mathbf { y } ^ { i , \\rho } ( v _ 0 ) + \\mathbf { y } ^ { i , \\rho } ( v _ 0 ) - \\mathbf { y } ^ { m ^ 0 , \\rho } ( v _ 0 ) | \\\\ & \\leq \\frac { C _ { v } } { C _ { \\eta } - C _ v } | v - v _ 0 | + \\frac { 1 } { q ^ { \\min } } \\left ( K _ f + \\frac { C _ v C _ { \\eta } C _ z } { ( C _ { \\eta } - C _ v ) ^ 2 } \\right ) . \\end{align*}"}
-{"id": "1833.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { - g } } \\partial _ \\mu \\sqrt { - g } g ^ { \\mu \\nu } \\partial _ \\nu \\phi + V \\phi = 0 , \\end{align*}"}
-{"id": "6278.png", "formula": "\\begin{align*} \\frac { 2 \\sigma ^ 2 _ H } { H ^ 2 } I _ 2 ( T ) = \\frac { 1 } { T } I _ { 1 2 } ( T ) . \\end{align*}"}
-{"id": "7554.png", "formula": "\\begin{align*} ( \\mathbb { K } \\sigma ) ( x ) = P . V . \\int _ { \\mathbb { R } ^ d } k ( x - y ) \\sigma ( y ) d y \\end{align*}"}
-{"id": "3479.png", "formula": "\\begin{align*} f _ { M , N } ( t ; \\theta ) = f _ { M } ( t ; \\theta ) + \\frac { M + 1 } { N } \\varphi _ { M , N } ( t ; \\theta ) , \\end{align*}"}
-{"id": "9805.png", "formula": "\\begin{align*} \\frac { 1 } { r ^ { \\kappa + 1 } } \\| [ P _ { Z _ \\circ } - P _ { X _ \\circ } ] ( r \\ , \\cdot \\ , ) \\| _ { L ^ 2 ( B _ { 1 / 2 } ( r ^ { - 1 } X _ \\circ ) , | y | ^ a ) } \\leq 2 \\omega _ { K _ j } ( r ) . \\end{align*}"}
-{"id": "7724.png", "formula": "\\begin{align*} \\int \\limits _ E h \\circ \\varphi _ { i } \\ , \\mathrm { d } \\mu = \\int \\limits _ { \\varphi _ { i } ( E ) } h \\cdot f _ { i } \\ , \\mathrm { d } \\lambda \\quad \\ ; \\ ; \\int \\limits _ E ( h \\cdot g _ { i } ) \\circ \\varphi _ { i } \\ , \\mathrm { d } \\mu = \\int \\limits _ { \\varphi _ { i } ( E ) } h \\ , \\mathrm { d } \\lambda . \\end{align*}"}
-{"id": "9078.png", "formula": "\\begin{align*} & \\sum _ { k \\in \\mathcal { K } } \\mu _ { f } ( I _ k ) = 1 - \\sum _ { k \\not \\in \\mathcal { K } } \\mu _ { f } ( I _ k ) = 1 + O ( \\delta K ) \\end{align*}"}
-{"id": "1598.png", "formula": "\\begin{align*} | u ^ o _ { n , k + 1 } - x ^ o _ { n , k + 1 } | & \\leq \\frac { 1 } { 2 } | u ^ o _ { n , k } - x ^ o _ { n , k } | + \\frac { D } { n ^ 2 } \\\\ & \\leq \\frac { 1 } { 2 } \\left ( \\frac { \\left ( \\frac { 1 } { 2 } \\right ) ^ k C } { k _ n ^ 2 } + \\sum _ { j = 1 } ^ { k } \\frac { \\left ( \\frac { 1 } { 2 } \\right ) ^ { j - 1 } D } { n ^ 2 } \\right ) + \\frac { D } { n ^ 2 } \\\\ & = \\frac { \\left ( \\frac { 1 } { 2 } \\right ) ^ { k + 1 } C } { k _ n ^ 2 } + \\sum _ { j = 1 } ^ { k + 1 } \\frac { \\left ( \\frac { 1 } { 2 } \\right ) ^ { j - 1 } D } { n ^ 2 } , \\end{align*}"}
-{"id": "2185.png", "formula": "\\begin{align*} \\mathcal M f ( \\mathbf x ) = \\sup _ { \\| \\mathbf x - \\mathbf y \\| < t } | \\exp ( t ^ 2 \\Delta ) f ( \\mathbf x ) | \\end{align*}"}
-{"id": "638.png", "formula": "\\begin{align*} \\varphi ^ \\ast ( v y ) u \\varphi ^ \\ast ( y ) ^ { - 1 } = ( \\varphi ^ y ) ^ \\ast ( v ) \\cdot \\varphi ^ \\ast ( y ) u \\varphi ^ \\ast ( y ) ^ { - 1 } \\ , \\end{align*}"}
-{"id": "8554.png", "formula": "\\begin{align*} \\tilde { \\theta } = ( I - \\mathcal { H } ) ( \\bar { \\zeta } - \\zeta ) = ( I - \\mathcal { H } ) ( \\zeta - \\alpha ) = 2 ( \\zeta - \\alpha ) - ( \\mathcal { H } + \\bar { \\mathcal { H } } ) ( \\zeta - \\alpha ) . \\end{align*}"}
-{"id": "8793.png", "formula": "\\begin{align*} u ^ { \\varepsilon } _ t \\colon H _ { r } ( [ 0 , \\pi ] ) \\longrightarrow \\mathbb { R } , \\phi \\mapsto \\langle u ^ { \\varepsilon } ( t , \\cdot ) , \\phi \\rangle = \\int _ { 0 } ^ { \\pi } u ^ { \\varepsilon } ( t , y ) \\phi ( y ) \\ , \\textrm { d } y \\end{align*}"}
-{"id": "7193.png", "formula": "\\begin{gather*} \\left ( q , p \\right ) ^ { \\mathrm { T } } = A ^ { \\mathrm { - 1 } } H ( q , p ) . \\end{gather*}"}
-{"id": "8556.png", "formula": "\\begin{align*} \\zeta _ { \\alpha } - 1 = \\frac { D _ t ^ 2 \\zeta - i ( A - 1 ) } { i A } . \\end{align*}"}
-{"id": "256.png", "formula": "\\begin{align*} t ( m \\otimes \\partial _ t ^ j \\delta ) = f m \\otimes \\partial _ t ^ j \\delta - j m \\otimes \\partial _ t ^ { j - 1 } \\delta \\end{align*}"}
-{"id": "4221.png", "formula": "\\begin{align*} C _ n ( z ) = \\mathcal { O } ( n ^ { - \\frac { 1 } { 2 } } ) , z = \\mathcal { O } ( n ^ { - 3 } ) . \\end{align*}"}
-{"id": "3565.png", "formula": "\\begin{align*} E _ { x } | _ { \\pi _ { 1 } ^ { - 1 } ( o ) } = \\mathbb { P } _ { 2 } ^ { 2 } \\cup \\mathbb { P } _ { 3 } ^ { 2 } , \\ ; \\ ; E _ { y } | _ { \\pi _ { 1 } ^ { - 1 } ( o ) } = \\mathbb { P } _ { 3 } ^ { 2 } \\cup \\mathbb { P } _ { 1 } ^ { 2 } , \\ ; \\ ; E _ { z } | _ { \\pi _ { 1 } ^ { - 1 } ( o ) } = \\mathbb { P } _ { 1 } ^ { 2 } \\cup \\mathbb { P } _ { 2 } ^ { 2 } . \\end{align*}"}
-{"id": "3739.png", "formula": "\\begin{align*} \\mathbf { \\Lambda } _ { l , m } ^ { ( j ) } ( \\mathbf { r } ) = \\left \\{ \\begin{array} [ c ] { l } \\mbox { \\boldmath $ { \\nabla } $ } \\times \\lbrack h _ { l } ^ { ( + ) } ( k _ { 0 } r ) \\mathbf { Y } _ { l , m } ] \\quad ; j = 1 \\\\ i k _ { 0 } h _ { l } ^ { ( + ) } ( k _ { 0 } r ) \\mathbf { Y } _ { l , m } \\quad ; j = 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "10042.png", "formula": "\\begin{align*} H ( r , \\theta , y , G , t ) = \\frac { 1 } { 2 } \\left ( y ^ 2 + \\frac { G ^ 2 } { r ^ 2 } \\right ) - V ( r , \\theta , t ) , \\end{align*}"}
-{"id": "2674.png", "formula": "\\begin{align*} I _ 2 & = \\sum _ { k = 0 } ^ { \\infty } \\sum _ { l = 0 } ^ { \\infty } \\frac { ( \\frac { 1 } { 2 } ) _ { l } } { l ! } \\int _ { \\mathcal { C } _ { \\{ 0 \\} } } \\frac { d s } { 2 \\pi i s } e ^ { \\frac { v } { s } } \\big ( s ( z + 1 ) \\big ) ^ k ( 2 s ) ^ l \\\\ & = \\sum _ { l = 0 } ^ { \\infty } \\big ( \\frac { 2 } { z + 1 } \\big ) ^ l \\frac { ( \\frac { 1 } { 2 } ) _ { l } } { l ! } \\sum _ { k = l } ^ { \\infty } \\frac { 1 } { k ! } \\big ( v ( z + 1 ) \\big ) ^ k , \\end{align*}"}
-{"id": "9064.png", "formula": "\\begin{align*} \\mathbb { E } [ F ^ R _ { \\mu _ { f } } ( x ) F ^ R _ { \\mu _ { f } } ( y ) ] = \\int _ { \\mathbb { S } ^ 1 } e ( \\langle \\lambda , R ( x - y ) \\rangle ) d \\mu _ { f } ( \\lambda ) . \\end{align*}"}
-{"id": "2017.png", "formula": "\\begin{align*} i _ * J ( \\tau , z ) = 5 H z + 5 H \\tau ( - z ) + \\sum _ { n , d \\atop 0 \\leq i \\leq 4 } \\frac { q ^ d } { n ! } \\left \\langle \\tau ( \\psi ) ^ n \\frac { H ^ i } { z - \\psi } \\right \\rangle _ { 0 , n + 1 , d } ^ \\mathrm { G W } H ^ { 4 - i } , \\end{align*}"}
-{"id": "6437.png", "formula": "\\begin{align*} ( A + \\Delta A ) \\ , \\widetilde x = b + \\Delta b \\ , . \\end{align*}"}
-{"id": "7255.png", "formula": "\\begin{align*} D _ { \\phi \\phi \\phi } ^ 3 \\Phi ( 0 , \\mu ^ * _ { k } ) [ \\phi ^ * _ { k } , \\phi ^ * _ { k } , \\phi ^ * _ { k } ] = 3 \\Pi \\phi ^ * _ { k } D _ { \\phi \\phi } ^ 2 \\psi ( 0 , \\mu ^ * _ { k } ) [ \\phi ^ * _ { k } , \\phi ^ * _ { k } ] . \\end{align*}"}
-{"id": "5283.png", "formula": "\\begin{align*} & S _ { r , i } = S ( F _ { r , i } ) , \\\\ & \\Lambda _ { r , i } = \\Lambda ( F _ { r , i } ) , \\\\ & T _ { r , i } = \\mathrm { c o k e r } \\ , \\left [ \\ , { ^ { ( 1 ) } } S ( F _ { r , i } ) \\xrightarrow [ ] { \\mathbf { F } } S ( F _ { r , i } ) \\right ] \\ ; . \\end{align*}"}
-{"id": "7360.png", "formula": "\\begin{align*} \\phi ( x _ 1 , x _ 2 , x _ 3 , x ) = \\max \\left \\{ ( x _ 1 \\cdot x _ 2 ) _ x , ( x _ 1 \\cdot x _ 3 ) _ x , ( x _ 2 \\cdot x _ 3 ) _ x \\right \\} \\end{align*}"}
-{"id": "199.png", "formula": "\\begin{align*} X _ { V } = x ^ 1 \\dfrac { \\partial } { \\partial y ^ 1 } + x ^ 2 \\dfrac { \\partial } { \\partial y ^ 2 } , Y _ { V } = \\dfrac { \\partial } { \\partial y ^ 3 } \\end{align*}"}
-{"id": "4532.png", "formula": "\\begin{align*} E = \\{ ( x _ k , y _ k ) : | ( x _ k - g y _ k ) - ( x _ 1 - g y _ 1 ) | \\leq \\delta \\} . \\end{align*}"}
-{"id": "1925.png", "formula": "\\begin{align*} ( x ^ 2 + x ) F _ { p + 2 } ( x ) = x ^ 2 \\sum _ { j = 1 } ^ \\infty \\left ( \\frac { 1 } { Z _ j ^ p } - \\frac { A _ j ^ 2 } { Z _ j ^ p } \\right ) . \\end{align*}"}
-{"id": "7622.png", "formula": "\\begin{align*} \\frac { \\partial u ^ { 0 } } { \\partial t } = \\Theta \\cdot \\nabla \\nabla u ^ { 0 } , u ^ { 0 } ( x , 0 ) = \\varphi ( x ) , \\varphi \\in L ^ 2 ( \\mathbb R ^ d ) . \\end{align*}"}
-{"id": "4818.png", "formula": "\\begin{align*} \\left \\{ \\begin{matrix} \\mathcal L \\varphi = 0 & \\textrm { i n $ M $ } \\\\ \\mathcal B \\varphi = \\lambda _ 1 ( \\mathcal B ) \\varphi & \\textrm { o n $ \\partial M $ . } \\end{matrix} \\right . \\end{align*}"}
-{"id": "3336.png", "formula": "\\begin{align*} G ( \\pi ) = \\prod _ { C \\in \\pi } g ( | C | ) , \\end{align*}"}
-{"id": "8862.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 e _ 2 } { \\partial a ^ 2 } & = 2 1 \\nu ^ 4 - 2 0 6 \\nu ^ 3 + 1 3 6 \\nu ^ 2 + 3 9 8 \\nu + 1 5 7 1 > 0 \\end{align*}"}
-{"id": "4672.png", "formula": "\\begin{align*} \\bar \\pi _ { n , g , h } = \\sum _ { j = 0 } ^ g \\pi _ { h , j } ( \\tilde \\lambda _ 1 , \\ldots , \\tilde \\lambda _ h ; a ) \\cdot \\pi _ { n - h , g - j } ( \\tilde \\lambda _ { h + 1 } , \\ldots , \\tilde \\lambda _ n ; a ) \\end{align*}"}
-{"id": "1482.png", "formula": "\\begin{align*} W ^ { j } _ { 1 } = - \\int _ { \\mathbb { T } ^ { d } } ( \\partial _ { x _ { j } } w ^ { 1 } - \\partial _ { x _ { j } } w ^ { 2 } ) \\Delta ( \\partial _ { x _ { j } } w ^ { 1 } - \\partial _ { x _ { j } } w ^ { 2 } ) \\ d x , \\end{align*}"}
-{"id": "867.png", "formula": "\\begin{gather*} \\widetilde { F } ( x ) : = \\begin{bmatrix} g ( x ) - K \\\\ x - D \\end{bmatrix} , x \\in \\R ^ n . \\end{gather*}"}
-{"id": "2661.png", "formula": "\\begin{align*} D _ { 1 } & = \\mathrm { d i a g } \\big ( \\varrho _ 1 / \\eta _ { - } , \\ldots , \\varrho _ N / \\eta _ { - } , 1 , \\eta _ { - } , \\ldots , \\eta _ { - } ^ { n - 1 } \\big ) , \\\\ D _ { 2 } & = \\mathrm { d i a g } \\big ( ( 1 - 2 \\varrho _ { 1 } ) ^ { - 1 / 2 } , \\ldots , ( 1 - 2 \\varrho _ { N + n } ) ^ { - 1 / 2 } \\big ) , \\end{align*}"}
-{"id": "7558.png", "formula": "\\begin{align*} \\begin{gathered} \\norm { X ' \\circ X ^ { - 1 } } _ { L i p ( 0 , T ; C ^ { \\alpha } ) } \\le \\norm { X ' } _ { L i p ( 0 , T ; C ^ { 1 + \\alpha } ) } \\norm { X - \\mathrm { I d } } _ { L i p ( 0 , T ; C ^ { 1 + \\alpha } ) } M _ X ^ { 1 + 3 \\alpha } . \\end{gathered} \\end{align*}"}
-{"id": "2329.png", "formula": "\\begin{align*} & f \\in \\mathfrak { W } ( X ) \\mapsto T _ l ( f ) : \\mathfrak { W } ( X ) / \\mathfrak { I } _ l \\to \\mathfrak { W } ( X ) / \\mathfrak { I } _ l \\\\ & T _ l ( f ) ( g + \\mathfrak { I } _ l ) = f g + \\mathfrak { I } _ l \\end{align*}"}
-{"id": "2018.png", "formula": "\\begin{align*} J _ j ( \\tau , z ) : = \\frac { i _ j ^ * i _ * J ( \\tau , z ) } { e ( N _ { p _ j / \\P ^ 4 } ) } = \\frac { 5 \\alpha _ j } { \\prod \\limits _ { j ' \\neq j } ( \\alpha _ j - \\alpha _ { j ' } ) } \\left ( z + \\tau _ j ( - z ) \\right ) + \\sum _ { n , d } \\frac { q ^ d } { n ! } \\left \\langle \\tau ( \\psi ) ^ n \\frac { \\rho _ j } { z - \\psi } \\right \\rangle _ { 0 , n + 1 , d } ^ \\mathrm { G W } , \\end{align*}"}
-{"id": "2949.png", "formula": "\\begin{align*} \\begin{array} { l l } ( * _ k ) : & \\mbox { t h e i m p l i c a t i o n } u \\in I m \\ , \\partial \\Rightarrow u \\in I m \\ , \\partial \\bar { \\partial } \\mbox { h o l d s f o r a l l } d - c l o s e d \\mbox { f o r m s o f t y p e s } \\\\ & ( p , q ) , ( q , p ) , ( p + 1 , q ) , \\mbox { a n d } ( q + 1 , p ) \\\\ & \\mbox { f o r a l l } p , q \\mbox { s u c h t h a t } p + q = k \\mbox { o r } p + q = 2 n - k \\end{array} \\end{align*}"}
-{"id": "8931.png", "formula": "\\begin{align*} \\eta _ 1 & : = n ( \\textrm { U n c h a n g e d , I n c o r r e c t } ) \\\\ \\eta _ 2 & : = n ( \\textrm { C h a n g e d , I n c o r r e c t } ) \\end{align*}"}
-{"id": "5892.png", "formula": "\\begin{align*} b _ w = ( - 1 ) ^ { \\ell ( w ) } q ^ { - \\frac { \\ell ( w ) } { 2 } } \\left ( 1 + 2 \\frac { q ^ { - 1 } } { 1 - q ^ { - 1 } } \\right ) , \\end{align*}"}
-{"id": "3348.png", "formula": "\\begin{align*} \\omega _ { \\lambda , d } : = \\{ ( d , \\lambda _ 1 ) , \\ldots , ( d , \\lambda _ k ) \\} \\in \\Omega _ d . \\end{align*}"}
-{"id": "7413.png", "formula": "\\begin{align*} w ( x , \\varepsilon ) = \\frac { \\mu _ x ( B _ { d _ x } ( x , \\varepsilon ) ) } { 2 \\varepsilon } . \\end{align*}"}
-{"id": "7672.png", "formula": "\\begin{align*} \\Big ( K _ { \\rm s y m } \\ - \\ G _ { \\rm s y m } \\Big ) r _ { \\boldsymbol { \\ell } } \\ = \\ 2 g _ { \\boldsymbol { \\ell } } , \\end{align*}"}
-{"id": "6793.png", "formula": "\\begin{align*} u _ \\nu ^ j = \\partial ^ \\nu u ^ j = \\frac { \\partial ^ { | \\nu | } u ^ j } { \\partial x _ 1 ^ { \\nu _ 1 } \\cdots \\partial x _ n ^ { \\nu _ n } } \\end{align*}"}
-{"id": "3718.png", "formula": "\\begin{align*} \\gamma = \\frac { V ( \\bar { y } _ w ) + W _ B ( \\bar { y } _ B - \\bar { Y } _ B ^ c ) ^ 2 } { V ( \\bar { y } _ w ) + ( \\bar { y } _ B - \\bar { Y } _ B ^ c ) ^ 2 } \\end{align*}"}
-{"id": "4780.png", "formula": "\\begin{align*} M = \\Big \\{ y ( T ) \\in Y \\ \\Big | \\ y \\mbox { i s t h e m i l d s o l u t i o n t o } ( \\ref { l l z 5 1 } ) \\mbox { w i t h s o m e } u ( \\cdot ) \\in \\widetilde { \\mathcal { U } } \\Big \\} . \\end{align*}"}
-{"id": "2415.png", "formula": "\\begin{align*} B _ { W , w } ( y ; \\delta ) = \\{ y \\} + B _ { W , w } ( 0 ; \\delta ) & \\subseteq \\{ y \\} + T ( B _ { V , w } ( 0 ; r ) ) \\\\ & = T ( \\{ x \\} + B _ { V , w } ( 0 ; r ) ) \\\\ & \\subseteq T ( U ) . \\Box \\end{align*}"}
-{"id": "1727.png", "formula": "\\begin{align*} D F = \\sum _ { i = 1 } ^ { n } \\partial _ { i } f ( [ h _ { 1 } ] , , \\cdots , [ h _ { n } ] ) h _ { i } , \\end{align*}"}
-{"id": "6711.png", "formula": "\\begin{align*} \\frac { \\partial V ( q ) } { \\partial q _ i } \\beta _ { i k } = \\frac { \\partial \\bar { V } ( Q _ 1 , Q _ 2 ) } { \\partial Q _ 2 } . \\end{align*}"}
-{"id": "9221.png", "formula": "\\begin{align*} \\mathbb { E } \\{ \\psi _ { i j } \\psi _ { l m } \\} = \\sigma _ \\psi ^ 2 \\exp \\left ( - \\frac { d ( i j , l m ) } { d _ 0 } \\right ) , \\end{align*}"}
-{"id": "810.png", "formula": "\\begin{align*} A & = ( { \\alpha q ^ { d \\alpha } - ( \\alpha + 1 ) q ^ { d ( \\alpha - 1 ) } } ) - ( { \\alpha q ^ { s \\alpha } - ( \\alpha + 1 ) q ^ { s ( \\alpha - 1 ) } } ) \\frac { q ^ { d \\alpha } - q ^ { d ( \\alpha - 1 ) } } { q ^ { \\alpha s } - q ^ { ( \\alpha - 1 ) s } } \\\\ & = \\frac { q ^ { d ( \\alpha - 1 ) } \\left ( q ^ d - q ^ s \\right ) } { q ^ s - 1 } , \\end{align*}"}
-{"id": "4473.png", "formula": "\\begin{align*} w \\nabla _ i w ( x _ 0 ) = ( u \\sigma _ { i j } + \\nabla _ { i j } u ) \\sigma ^ { j k } \\nabla _ k u = 0 \\end{align*}"}
-{"id": "2900.png", "formula": "\\begin{align*} \\gamma _ 1 = i \\sigma _ 1 , \\qquad \\gamma _ 2 = i \\sigma _ 2 , \\qquad \\Sigma _ { 1 2 } = \\sigma _ 3 , \\end{align*}"}
-{"id": "5590.png", "formula": "\\begin{align*} & \\tilde { h } _ 1 ( t ) = \\psi ( t ) \\tilde { f } ( t ) - \\psi ( t ) e ^ { i t \\partial _ x ^ 2 } \\tilde { u } _ 0 ( x ) - \\psi ( t ) \\mathcal { S } \\big ( \\alpha \\psi _ T u \\psi _ T v + \\beta | \\psi _ T u | ^ 2 \\psi _ T u \\big ) ( 0 , t ) , \\\\ & \\tilde { h } _ 2 ( t ) = \\psi ( t ) \\tilde { g } ( t ) - \\psi ( t ) e ^ { - t \\partial _ x ^ 3 } \\tilde { v } _ 0 ( x ) - \\psi ( t ) \\mathcal { K } \\big ( \\gamma \\partial _ x ( | \\psi _ T ^ 2 u | ^ 2 ) - \\tfrac { 1 } { 2 } \\partial _ x ( ( \\psi _ T ^ 2 v ) ^ 2 ) \\big ) ( x , t ) + \\big ) ( 0 , t ) . \\end{align*}"}
-{"id": "4443.png", "formula": "\\begin{align*} \\langle w _ 1 ( \\det ( D _ \\bullet \\# 0 ) ) , v \\rangle _ { C ^ { \\infty } _ { X , L } } = \\langle w _ 2 ( T L ) , v _ * [ S ^ 1 \\times \\partial D ] \\rangle _ L + I _ { \\mu _ L } ( A ) \\langle w _ 1 ( T L ) , v _ * [ S ^ 1 \\times \\{ 1 \\} ] \\rangle _ L . \\end{align*}"}
-{"id": "8771.png", "formula": "\\begin{align*} \\mu _ 1 = - \\frac { n - 1 } { 4 } \\ , D , \\ \\ \\mu _ 2 = \\frac 1 F \\left ( \\frac { \\widetilde { \\kappa } } { n - 2 } - \\frac { n - 1 } { 4 } \\ , F D - ( n - 3 ) ( B ^ 2 _ 2 - C B ^ 2 ) \\right ) \\ , . \\end{align*}"}
-{"id": "5251.png", "formula": "\\begin{align*} \\phi ^ { - 1 } ( 1 - 1 / n ) = \\left \\{ ( 2 \\log n ) ^ { 1 / 2 } - \\frac { \\log 4 \\pi + \\log \\log n } { 2 ( 2 \\log n ) ^ { 1 / 2 } } + O ( \\varepsilon _ { n } ) \\right \\} \\end{align*}"}
-{"id": "623.png", "formula": "\\begin{align*} \\varphi \\psi ^ { \\varphi ^ \\ast ( y ) x \\varphi ^ \\ast ( y ) ^ { - 1 } } = ( \\varphi \\psi ^ { x \\varphi ^ \\ast ( y ) ^ { - 1 } } ) ^ y = ( \\varphi \\psi ^ { \\varphi ^ \\ast ( y ) ^ { - 1 } } ) ^ y = \\varphi \\psi \\ . \\end{align*}"}
-{"id": "3941.png", "formula": "\\begin{align*} x \\sim y \\Leftrightarrow \\textrm { f o r a n y c o n t i n o u s m a p } C \\xrightarrow { g } D \\textrm { w h e r e } D \\textrm { H a u s d o r f f , } g ( x ) = g ( y ) , \\end{align*}"}
-{"id": "5391.png", "formula": "\\begin{align*} \\frac { c _ { n + 2 \\ , m } } { \\overline { b _ { n + 1 \\ , m + 3 } } } \\cdot \\frac { \\overline { d _ { n + 1 \\ , m + 3 } } } { a _ { n m } } = \\frac { \\overline { d _ { n + 2 \\ , m } } } { a _ { n + 1 \\ , m - 3 } } \\cdot \\frac { c _ { n + 1 \\ , m - 3 } } { \\overline { b _ { n m } } } . \\end{align*}"}
-{"id": "1522.png", "formula": "\\begin{align*} \\prod _ { k = 0 } ^ { q ^ { n } - 1 } \\# \\{ c _ { k } \\} & = | P | ^ e \\prod _ { k = 1 } ^ { q ^ { n } - 1 } | P | ^ { e - \\min \\{ e , \\mu ( k ) \\} } \\\\ & = | P | ^ { e + \\sum _ { k = 1 } ^ { q ^ { n } - 1 } ( e - \\min \\{ e , \\lfloor \\frac { \\log _ { q } k } { d } \\rfloor \\} ) } \\\\ & = | P | ^ { e q ^ { n } - \\sum _ { k = 1 } ^ { q ^ { n } - 1 } \\min \\{ e , \\lfloor \\frac { \\log _ { q } k } { d } \\rfloor \\} } \\\\ & = | P | ^ { e q ^ { n } - ( q - 1 ) \\Sigma _ { k = 1 } ^ { n - 1 } q ^ { k } \\min \\{ e , \\lfloor \\frac { k } { d } \\rfloor \\} } , \\end{align*}"}
-{"id": "9559.png", "formula": "\\begin{align*} _ 3 \\phi _ 2 \\left [ \\begin{matrix} q ^ { - N } , & \\alpha , & \\beta \\\\ \\gamma , & \\frac { q ^ { 1 - N } } { \\tau } \\end{matrix} ; q , q \\right ] = \\frac { ( \\frac { \\gamma } { \\beta } , \\beta \\tau ; q ) _ N } { ( \\gamma , \\tau ; q ) _ N } { } _ 3 \\phi _ 2 \\left [ \\begin{matrix} q ^ { - N } , & \\frac { \\alpha \\beta \\tau } { \\gamma } , & \\beta \\\\ \\beta \\tau , & \\frac { \\beta q ^ { 1 - N } } { \\gamma } \\end{matrix} ; q , q \\right ] \\end{align*}"}
-{"id": "1451.png", "formula": "\\begin{align*} w ^ { n + 1 } ( t , \\cdot ) = \\mathbb { P } _ { \\delta } w _ { T } + \\int _ { t } ^ { T } \\left [ \\Delta w ^ { n + 1 } ( \\tau , \\cdot ) + \\varepsilon \\Theta ( \\tau , \\cdot , \\mu ^ { n } , D w ^ { n } ) \\right ] \\ d \\tau . \\end{align*}"}
-{"id": "2518.png", "formula": "\\begin{align*} \\mathbf v G ( \\mathbf 1 - x \\mathbf u ) \\mathbf u = x ^ \\alpha l ( x ) , x > 0 , \\end{align*}"}
-{"id": "7638.png", "formula": "\\begin{align*} \\int \\limits _ { \\mathbb T ^ d } h ( \\xi ) v _ 0 ( \\xi ) \\ d \\xi \\ = \\ \\int \\limits _ { \\mathbb R ^ d } \\int \\limits _ { \\mathbb T ^ d } a ( \\xi - q ) \\mu ( \\xi , q ) ( \\xi - q ) \\ d q \\ , v _ 0 ( \\xi ) d \\xi \\ - \\ b \\int \\limits _ { \\mathbb T ^ d } \\ , v _ 0 ( \\xi ) d \\xi \\ = \\ 0 . \\end{align*}"}
-{"id": "952.png", "formula": "\\begin{align*} p _ j ^ * \\ ; : = \\ ; P ( U _ j | \\mathcal { F } ^ { M \\cdot R } _ { j - 1 } ) \\ ; \\geq \\ ; p , \\end{align*}"}
-{"id": "5882.png", "formula": "\\begin{align*} \\phi ( C _ { s _ i } ) = - \\left ( q ^ { \\frac { 1 } { 2 } } + q ^ { - \\frac { 1 } { 2 } } \\right ) t _ { s _ i } + t _ { s _ i s _ j } . \\end{align*}"}
-{"id": "4179.png", "formula": "\\begin{align*} A _ n ^ { ( 1 ) } ( z _ 1 ) E _ n ( z _ 2 ) = \\frac { 1 } { f ' ( 0 ) } E _ n ( z _ 1 ) T _ \\alpha ^ { - 1 } A _ \\alpha T _ \\alpha \\left ( E _ n ^ { - 1 } ( z _ 1 ) E _ n ( z _ 2 ) \\right ) \\end{align*}"}
-{"id": "2706.png", "formula": "\\begin{align*} & \\mathrm { P } \\Big ( 2 ^ { - \\frac { 4 } { 3 } } N ^ { - \\frac { 1 } { 3 } } \\big ( \\lambda _ { \\mathrm { m a x } } - 4 N \\big ) \\leq x \\Big ) = 1 + \\\\ & \\sum _ { k = 1 } ^ { N } \\frac { ( - 1 ) ^ k } { k ! } \\frac { 1 } { ( \\varphi ( 4 ) ) ^ k } \\int _ { x } ^ { \\infty } \\cdots \\int _ { x } ^ { \\infty } \\mathrm { P f } \\Big [ K _ { N } \\Big ( 4 N + \\frac { u _ i } { \\varphi ( 4 ) } , 4 N + \\frac { u _ j } { \\varphi ( 4 ) } \\Big ) \\Big ] _ { i , j = 1 } ^ { k } d u _ 1 \\cdots d u _ k , \\end{align*}"}
-{"id": "6244.png", "formula": "\\begin{align*} ( A , \\phi ) = \\bigoplus _ { i = 1 } ^ { n } \\underset { \\alpha _ { i , 1 } , \\cdots , \\alpha _ { i , k _ { i } } } { \\overset { p _ { i , 1 } , \\cdots , p _ { i , k _ { i } } } { M _ { k _ { i } } ( \\C ) } } \\qquad ( B , \\psi ) = \\bigoplus _ { j = 1 } ^ { m } \\underset { \\beta _ { j , 1 } , \\cdots , \\beta _ { j , \\ell _ { j } } } { \\overset { q _ { j , 1 } , \\cdots , q _ { j , \\ell _ { j } } } { M _ { \\ell _ { j } } ( \\C ) } } . \\end{align*}"}
-{"id": "4758.png", "formula": "\\begin{align*} f _ W ( s ) = \\frac { g _ W ( s ) } { 1 + g _ W ( s ) } \\approx \\frac { | \\log ( s ) | } { 1 + | \\log ( s ) | } \\approx 1 - 1 / | \\log s | . \\end{align*}"}
-{"id": "8088.png", "formula": "\\begin{align*} ( 1 - z ) ^ 2 f _ 0 ' ( z ) - f _ 0 ( z ) = 0 , \\end{align*}"}
-{"id": "4137.png", "formula": "\\begin{align*} P _ { + } ( x ) & = P _ { - } ( x ) \\begin{pmatrix} 0 & x ^ \\beta & 0 \\\\ - x ^ { - \\beta } & 0 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} , & & x \\in \\Delta _ { 1 } \\cap A ( 0 ; r _ n , R ) , \\\\ P _ { + } ( x ) & = P _ { - } ( x ) \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 0 & - 1 \\\\ 0 & 1 & 0 \\end{pmatrix} , & & x \\in \\Delta _ { 2 } \\cap A ( 0 ; r _ n , R ) , \\\\ P _ + ( z ) & = P _ - ( z ) , & & z \\in \\Delta _ 1 ^ { \\pm } \\cap A ( 0 ; r _ n , R ) . \\end{align*}"}
-{"id": "6643.png", "formula": "\\begin{align*} \\widehat { C ^ { a / q , 2 } _ { \\lambda } } ( \\xi ) & = \\widehat { M _ { \\psi _ { Q / 2 } , q } } ( \\xi ) \\cdot \\sum _ { \\ell \\in \\mathbb Z ^ { d } } \\widetilde \\psi _ { q } ( \\xi - \\ell / q ) ( 1 - \\widetilde \\psi _ { \\lambda Q / N } ( \\xi - \\ell / q ) ) \\widetilde { d \\sigma _ { \\lambda } } ( \\xi - \\ell / q ) \\\\ & : = \\widehat { M _ { \\psi _ { q / 2 } , q } } ( \\xi ) \\cdot \\widehat { C ^ { a / q , 3 } _ { \\lambda } } ( \\xi ) . \\end{align*}"}
-{"id": "7018.png", "formula": "\\begin{align*} S _ X ^ { ( k ) } : = \\frac { a } { 2 v _ a } d ( \\Lambda _ { n _ k } ^ c , X ) . \\end{align*}"}
-{"id": "3844.png", "formula": "\\begin{align*} 2 h _ { 0 0 } ^ { \\frac { m + 1 } { 2 } } W _ 0 ^ m \\Phi ^ i _ { \\parallel l } & = & h _ { 0 0 } ^ { m + 1 } A ^ i _ { ( 1 ) \\parallel l } - m h _ { 0 0 } ^ { m + 1 } W _ { 0 \\parallel l } A ^ i _ { ( 1 ) } + ( h _ { 0 0 } ) ^ { \\frac { m + 1 } { 2 } } W _ 0 ^ m A ^ i _ { ( 2 ) \\parallel l } + \\\\ & & W _ 0 ^ { 2 m } A ^ i _ { ( 3 ) \\parallel l } + m W _ 0 ^ { 2 m - 1 } W _ { 0 \\parallel l } A ^ i _ { ( 3 ) } . \\end{align*}"}
-{"id": "1827.png", "formula": "\\begin{align*} \\mathcal { E } = - \\frac { \\pi } { 6 L } + \\frac { \\mathcal { B } } { L } . \\end{align*}"}
-{"id": "8783.png", "formula": "\\begin{align*} T = \\left \\{ ( k , k + \\ell , k + \\ell ^ 2 ) : k , \\ell \\in \\Z , ~ 0 \\leq k \\leq n / 2 , ~ 0 \\leq \\ell \\le \\sqrt { n } / 2 \\right \\} \\subset A \\times A \\times A . \\end{align*}"}
-{"id": "1228.png", "formula": "\\begin{align*} \\int _ { | \\eta | \\ge 1 0 } G ( \\xi , \\eta ) d \\eta = \\int _ { | \\eta | \\ge 1 0 } G ( 0 , \\eta ) d \\eta + \\int _ 0 ^ \\xi \\int _ { | \\eta | \\ge 1 0 } \\partial _ \\zeta G ( \\zeta , \\eta ) d \\eta d \\zeta . \\end{align*}"}
-{"id": "3199.png", "formula": "\\begin{align*} T f & = d \\varGamma d ^ * f = \\langle \\chi _ 0 , ( D _ 0 \\otimes 1 ) \\chi _ 0 \\rangle f \\\\ & = \\left ( 2 | \\langle x _ 0 , \\phi _ 0 \\rangle | ^ 2 - 1 \\right ) f \\\\ & = ( 2 / N - 1 ) f . \\end{align*}"}
-{"id": "6976.png", "formula": "\\begin{align*} v ^ { ( n ) } ( x ) ( v ( x ) ) ^ { n - 1 } = - \\frac { f ^ { ( n + 1 ) } ( x ) } { ( f ^ \\prime ( x ) ) ^ { n + 1 } } + Q _ n \\left ( \\frac { f ^ { \\prime \\prime } ( x ) } { ( f ^ \\prime ( x ) ) ^ { 2 } } , \\ldots , \\frac { f ^ { ( n ) } ( x ) } { ( f ^ \\prime ( x ) ) ^ { n } } \\right ) , n \\geq 1 , \\end{align*}"}
-{"id": "4241.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} d u & = f & & B _ { R } , \\\\ \\delta u & = g & & B _ { R } \\\\ \\nu \\wedge u & = 0 & & \\partial B _ { R } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "2444.png", "formula": "\\begin{gather*} \\Lambda = \\frac { 6 4 s } { 1 0 0 s - 9 r ^ 2 } = \\frac { 1 6 } { 2 5 } \\big ( I ^ 2 + 1 \\big ) , \\end{gather*}"}
-{"id": "2014.png", "formula": "\\begin{align*} \\eqref { e q : g w i n v a r i a n t s } = \\sum _ { \\Gamma \\in G _ { n , d } } \\frac { 1 } { | \\mathrm { A u t } ( \\Gamma ) | } \\int _ { F _ \\Gamma } \\frac { i _ \\Gamma ^ * \\left ( \\mathrm { e v } _ 1 ^ * ( \\varphi _ 1 ) \\psi _ 1 ^ { a _ 1 } \\cdots \\mathrm { e v } _ n ^ * ( \\varphi _ n ) \\psi ^ { a _ n } e ( R ^ 0 \\pi _ * f ^ * \\O _ { \\P ^ 4 } ( 5 ) ) \\right ) } { e ( N _ { F _ \\Gamma } ) } . \\end{align*}"}
-{"id": "7488.png", "formula": "\\begin{gather*} q ^ * \\xi = q _ s ^ 2 \\xi . \\end{gather*}"}
-{"id": "933.png", "formula": "\\begin{align*} E _ 3 ^ q = I _ 3 - \\sum _ { j _ 3 , j _ 2 , j _ 1 = 0 } ^ q C _ { j _ 3 j _ 2 j _ 1 } ^ 2 - \\sum _ { j _ 3 , j _ 2 , j _ 1 = 0 } ^ q C _ { j _ 3 j _ 2 j _ 1 } C _ { j _ 1 j _ 2 j _ 3 } \\ \\ \\ ( i _ 1 = i _ 3 \\ne i _ 2 ) , \\end{align*}"}
-{"id": "8681.png", "formula": "\\begin{align*} T \\varphi ( h ) = \\langle u ^ T , \\varphi ( h \\exp ( \\ ; \\cdot \\ ; ) ) \\rangle . \\end{align*}"}
-{"id": "4719.png", "formula": "\\begin{align*} u ( x , 0 ) = \\left \\{ \\begin{array} { l l } 1 ~ ~ ~ ~ ~ ~ | x | \\leq 1 / 3 , \\\\ 0 ~ ~ ~ ~ , \\end{array} \\right . \\end{align*}"}
-{"id": "9927.png", "formula": "\\begin{align*} \\frac { d P ^ { \\mu } | _ { \\mathcal { G } } } { d P ^ { \\nu } | _ { \\mathcal { G } } } & = E ^ { \\nu } \\left [ \\left . \\frac { d \\mu } { d \\nu } ( X _ { 0 } ) \\right | \\mathcal { G } \\right ] ~ ~ ~ P ^ { \\mu } ~ a . s . \\end{align*}"}
-{"id": "1618.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow \\infty } \\frac { 1 } { T } \\int _ 0 ^ T \\sigma \\left ( X _ t ^ { Z ^ { b } } \\right ) \\ , d B _ { t } = 0 . \\end{align*}"}
-{"id": "6224.png", "formula": "\\begin{align*} \\log \\mathcal { L } ( F ^ \\textnormal { E M } ( \\theta ) ) - & \\mathcal { D } _ \\textnormal { K L } ( \\theta \\| F ^ \\textnormal { E M } ( \\theta ) ) \\\\ & \\geq \\log \\mathcal { L } ( \\theta ) - \\underbrace { \\mathcal { D } _ \\textnormal { K L } ( \\theta \\| \\theta ) } _ { = 0 } , \\end{align*}"}
-{"id": "7417.png", "formula": "\\begin{align*} \\displaystyle J _ \\lambda ( \\pi _ { \\lambda } ( t ) e ) & = J _ \\lambda ( \\pi _ \\lambda ( \\varphi _ { \\overline { B } } ^ 1 ) \\pi _ \\lambda ( \\Delta _ { \\overline { B } } ^ m ) \\pi _ \\lambda ( t ) e ) + J _ \\lambda ( \\pi _ \\lambda ( \\varphi _ { \\overline { B } } ^ 2 ) \\pi _ \\lambda ( t ) e ) \\\\ & = J _ \\lambda ( \\pi _ \\lambda ( \\varphi ^ 1 ) \\pi _ \\lambda ( \\Delta _ { \\overline { B } } ^ m ) \\pi _ \\lambda ( t ) e ) + J _ \\lambda ( \\pi _ \\lambda ( \\varphi ^ 2 ) \\pi _ \\lambda ( t ) e ) . \\end{align*}"}
-{"id": "5948.png", "formula": "\\begin{align*} \\rho ^ g ( X ) = \\bar { Y } ( 0 ) , \\end{align*}"}
-{"id": "8415.png", "formula": "\\begin{align*} \\mathcal { T } : = \\{ T \\in [ 0 , T _ 0 ] : ~ ~ ( \\ref { a p r i o r i } ) ~ a n d ~ ( \\ref { a p r i o r i 2 } ) 0 \\leq t \\leq T \\} \\end{align*}"}
-{"id": "2790.png", "formula": "\\begin{align*} I _ t = \\sum _ { i = 1 } ^ s J ^ t _ i . \\end{align*}"}
-{"id": "3605.png", "formula": "\\begin{align*} u ^ { \\psi _ { \\sigma } } \\cdot v ^ { \\psi _ { \\sigma } } = ( u \\cdot v ) ^ { \\psi _ { \\sigma } } \\textrm { a n d } \\langle u ^ { \\psi _ { \\sigma } } , v ^ { \\psi _ { \\sigma } } \\rangle = \\langle u , v \\rangle ^ { \\phi _ { \\sigma } } \\end{align*}"}
-{"id": "5008.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } N ^ { 1 / d } \\delta ( A , N ) = \\lim _ { \\delta \\to 0 } N ( \\delta ) ^ { 1 / d } \\delta = C . \\end{align*}"}
-{"id": "1251.png", "formula": "\\begin{gather*} \\int _ { | \\nu | > 2 0 } e ^ { 3 i \\eta \\nu ^ 2 / 4 } \\left ( \\eta - \\frac { \\nu } { 2 } \\right ) S ' \\left ( \\frac { \\eta + \\nu } { 2 } \\right ) S \\left ( \\frac { \\eta - \\nu } { 2 } \\right ) d \\nu = \\sum _ { i , j = 1 , 2 } T _ { 3 , i j } \\\\ T _ { 3 , i j } : = \\int _ { | \\nu | > 2 0 } e ^ { 3 i \\eta \\nu ^ 2 / 4 } \\left ( \\eta - \\frac { \\nu } { 2 } \\right ) S _ i ' \\left ( \\frac { \\eta + \\nu } { 2 } \\right ) S _ j \\left ( \\frac { \\eta - \\nu } { 2 } \\right ) d \\nu . \\end{gather*}"}
-{"id": "2846.png", "formula": "\\begin{align*} I ( x _ 2 , \\ldots , x _ n ) = \\pm 1 \\ ; \\ ; \\ ; { \\rm w i t h } \\ ; \\ ; \\ ; x _ 2 , \\ldots , x _ n \\in \\Z . \\end{align*}"}
-{"id": "7294.png", "formula": "\\begin{align*} \\Phi _ p ( a ^ * ) ( h _ p ^ - - h _ q ^ - ) = h _ p ^ + \\Phi _ p ( a ^ * ) ( h _ p ^ - - h _ q ^ - ) \\end{align*}"}
-{"id": "9908.png", "formula": "\\begin{align*} & | \\int _ { \\mathcal { Y } ^ { N ' + 1 } } g ( y _ { [ n , n + N ' ] } ) P ^ { \\mu } ( d y _ { [ n , n + N ' ] } | Y _ { [ 0 , n - 1 ] } ) \\\\ & - \\int _ { \\mathcal { Y } ^ { N ' + 1 } } g ( y _ { [ n , n + N ' ] } ) P ^ { \\nu } ( d y _ { [ n , n + N ' ] } | Y _ { [ 0 , n - 1 ] } ) | + \\frac { 2 } { 3 } \\epsilon \\\\ & \\leq \\| g \\| _ { \\infty } \\frac { \\epsilon } { 3 \\| g \\| _ { \\infty } } + \\frac { 2 } { 3 } \\epsilon = \\epsilon \\end{align*}"}
-{"id": "3598.png", "formula": "\\begin{align*} \\hat { R } = \\mathbf { k } \\left [ \\{ \\lambda _ { [ \\hat { x } , \\hat { y } ] } \\} _ { [ \\hat { x } , \\hat { y } ] \\in [ \\hat { X } \\times \\hat { X } ] } \\right ] \\end{align*}"}
-{"id": "3272.png", "formula": "\\begin{align*} h _ { n - 1 } = a ^ { n - 1 } + a ^ { n - 2 } b + \\dots + a b ^ { n - 2 } + b ^ { n - 1 } , \\end{align*}"}
-{"id": "3223.png", "formula": "\\begin{align*} \\Gamma _ k : = \\frac { 1 } { 2 } { { \\displaystyle \\sum _ { i = 1 } ^ n } \\Delta _ { \\frac { 1 } { 2 } } x _ i ^ k \\left ( \\mathcal { L } _ { { F } ( x ^ k ) } \\mathcal { L } _ { { V } _ k } \\mathcal { L } _ { { F } ( x ^ k ) } ^ { - 1 } - \\mathcal { L } _ { V _ k } \\right ) F _ i } . \\end{align*}"}
-{"id": "798.png", "formula": "\\begin{align*} { e = \\frac { q ^ { d \\alpha } - q ^ { d ( \\alpha - 1 ) } } { q ^ { s \\alpha } - q ^ { s ( \\alpha - 1 ) } } = q ^ { ( d - s ) ( \\alpha - 1 ) } \\frac { q ^ { d } - 1 } { q ^ s - 1 } . } \\end{align*}"}
-{"id": "6559.png", "formula": "\\begin{align*} m ^ P _ \\beta = 1 + \\# \\left \\{ \\right \\} + 2 \\cdot \\# \\left \\{ \\right \\} = e ( S ' ) = e ( S ) - \\eta \\end{align*}"}
-{"id": "3386.png", "formula": "\\begin{align*} \\P \\left ( \\sum _ { i = 1 } ^ m Y _ i ^ 2 \\geq \\sum _ { k = \\log \\log n } ^ { L } \\delta 2 ^ { k / 2 } \\frac { \\delta ^ 2 2 ^ k } { n ^ 2 } \\right ) & \\geq \\P \\left ( \\sum _ { i = 1 } ^ m Y _ i ^ 2 \\geq \\frac { C \\delta ^ 3 2 ^ { 3 L / 2 } } { n ^ 2 } \\right ) \\\\ & \\geq \\P \\left ( \\sum _ { i = 1 } ^ m Y _ i ^ 2 \\geq \\frac { C m ^ { 3 } } { n ^ 2 \\log ^ c n } \\right ) \\\\ & = 1 - O ( \\delta ) . \\end{align*}"}
-{"id": "3237.png", "formula": "\\begin{align*} B _ P ( x , y , V ) : = \\nabla _ { x x } ^ 2 { L } ( x , y ) + { H } _ { P } ( x , V ) \\end{align*}"}
-{"id": "1298.png", "formula": "\\begin{align*} V ( y ) & = \\frac { 1 } { \\pi } \\Re A e ^ { i R ( \\xi _ 0 ) } \\sqrt { \\frac { 2 \\pi } { R '' ( \\xi _ 0 ) } } e ^ { i a \\ln \\xi _ 0 } \\ + \\ L ^ 2 \\mbox { - r e m a i n d e r } \\\\ & = \\frac { | A | } { \\sqrt { \\pi } | 3 y | ^ { 1 / 4 } } \\cos \\left ( - \\frac { 2 } { 3 \\sqrt 3 } | y | ^ { 3 / 2 } + \\frac { a } { 2 } \\ln | y | + \\theta _ 0 \\right ) \\ + \\ L ^ 2 , \\end{align*}"}
-{"id": "9446.png", "formula": "\\begin{align*} \\xi _ 0 ( f ) ( z ) : = 2 i \\overline { \\frac { \\partial } { \\partial \\bar { z } } f ( z ) } . \\end{align*}"}
-{"id": "2852.png", "formula": "\\begin{align*} f ( x ) = x ^ 4 - t ^ 2 x ^ 2 + 1 \\in \\Z _ M [ x ] . \\end{align*}"}
-{"id": "7105.png", "formula": "\\begin{align*} T ^ { t } _ { s } T _ { t } = T _ { t + s } , T ^ { t } _ { - t } = ( T _ { t } ) ^ { - 1 } . \\end{align*}"}
-{"id": "2136.png", "formula": "\\begin{align*} ( \\Phi ( \\cdot , 0 ) ) ^ \\ast ( y ) = \\sup _ { x \\in X } \\{ \\langle x , y \\rangle - \\Phi ( x , 0 ) \\} = \\min _ { z \\in L ^ 0 ( Z ) } \\Phi ^ \\ast ( y , z ) \\end{align*}"}
-{"id": "2625.png", "formula": "\\begin{align*} R _ { N } ^ { ( k ) } ( \\lambda _ 1 , \\ldots , \\lambda _ k ) = \\mathrm { P f } \\left [ K _ { N } ( \\lambda _ { i } , \\lambda _ { j } ) \\right ] _ { i , j = 1 } ^ { k } . \\end{align*}"}
-{"id": "5820.png", "formula": "\\begin{align*} \\phi _ { X _ 1 } ( 2 t ) \\overline { \\phi _ { X _ 1 } } ( 2 t ) = \\phi _ { X _ 1 } ( t ) ^ 4 \\overline { \\phi _ { X _ 1 } } ( t ) ^ 4 \\end{align*}"}
-{"id": "950.png", "formula": "\\begin{align*} \\le \\frac { \\Delta ^ 2 } { 2 } \\int \\limits _ { q } ^ { \\infty } \\frac { 1 } { 4 x ^ 2 - 1 } d x = - \\frac { \\Delta ^ 2 } { 8 } { \\rm l n } \\left | 1 - \\frac { 2 } { 2 q + 1 } \\right | \\le C _ 1 \\frac { \\Delta ^ 2 } { q } , \\end{align*}"}
-{"id": "3435.png", "formula": "\\begin{gather*} c ( V ) = 1 ^ { 1 9 3 8 } ( 1 - t ) ^ { 1 8 9 0 } ( 1 - 2 t ) ^ { 1 9 3 2 } ( 1 + t ) ^ { 1 8 9 0 } = 1 - 2 t ^ 2 + \\cdots \\pmod { 4 t } . \\end{gather*}"}
-{"id": "7882.png", "formula": "\\begin{align*} \\sin ^ 2 ( \\pi \\xi _ i ' ) = \\cos ^ 2 ( \\pi \\xi _ i ) i \\in V _ { \\xi } . \\end{align*}"}
-{"id": "9607.png", "formula": "\\begin{align*} ( - 1 ) _ { m } - 1 = 1 + \\sum _ { n = 1 } ^ { m - 1 } ( - 1 ) _ n q ^ n = \\sum _ { n = 0 } ^ { m - 1 } ( - 1 ) _ n q ^ n . \\end{align*}"}
-{"id": "1401.png", "formula": "\\begin{align*} Q ( x _ 1 , \\dots , x _ d ) & = Q ( y _ 1 - \\sum _ { i = 2 } ^ d x _ i , x _ 2 , \\dots , x _ d ) \\\\ & = Q \\circ S \\circ r ( x _ 2 , \\dots , x _ d ) \\\\ & = Q \\circ S \\circ r ( x _ 2 ' , \\dots , x _ d ' ) \\\\ & = Q ( x _ 1 ' , \\dots , x _ d ' ) . \\end{align*}"}
-{"id": "4300.png", "formula": "\\begin{align*} \\textup { s a t } ( m ( r - 1 ) + 1 , C _ t , C _ k ) = 0 \\end{align*}"}
-{"id": "3460.png", "formula": "\\begin{align*} s _ { \\pm } = 1 - k + ( 2 \\pm i N ^ { 1 / 8 } ( M + 1 ) ^ { 1 / 8 } ) \\sqrt { N / ( M + 1 ) } . \\end{align*}"}
-{"id": "3904.png", "formula": "\\begin{align*} | a _ { 1 , 1 } ^ { ( k + 1 ) } | ^ 2 \\leq | a _ { 1 , 1 } ^ { ( k ) } | ^ 4 + | \\beta ^ { ( k ) } | ^ 4 - \\sum _ { i = 2 } ^ { n + 1 } | a _ { 1 , 1 } ^ { ( k ) } | ^ 2 | a _ { 1 , i } ^ { ( k ) } | ^ 2 ( 2 - | u _ 1 - u _ { i } | ^ 2 ) , \\end{align*}"}
-{"id": "9081.png", "formula": "\\begin{align*} a _ { n } : = \\left \\Vert u _ { n } - p \\right \\Vert , b _ { n } : = \\left \\Vert v _ { n } - p \\right \\Vert \\quad ( n \\geq 0 ) . \\end{align*}"}
-{"id": "7669.png", "formula": "\\begin{align*} \\int \\limits _ { \\mathbb { R } ^ d } a _ { \\rm s y m } ( \\xi - q ) \\ , \\mu ( \\xi , q ) \\ , \\big ( \\varkappa _ { \\rm s y m } ( q ) - \\varkappa _ { \\rm s y m } ( \\xi ) \\big ) \\ , d q \\ = \\ \\int \\limits _ { \\mathbb { R } ^ d } a _ { \\rm s y m } ( \\xi - q ) \\ , ( \\xi - q ) \\ , \\mu ( \\xi , q ) \\ , d q , \\end{align*}"}
-{"id": "1903.png", "formula": "\\begin{align*} \\left [ \\cos ( \\kappa _ j x ) + \\frac { \\xi } { \\kappa _ j } \\sin ( \\kappa _ j | x | ) \\right ] e ^ { - i \\kappa _ j t } = \\frac { \\xi L } { 2 } \\left \\{ \\frac { 1 - i \\kappa _ j t } { Z _ j ^ 2 } + 2 \\sum _ { n = 1 } ^ \\infty \\frac { \\cos ( k _ n x ) } { Z _ j ^ 2 - ( \\pi n ) ^ 2 } \\left [ \\cos ( k _ n t ) - i \\frac { \\kappa _ j } { k _ n } \\sin ( k _ n t ) \\right ] \\right \\} \\end{align*}"}
-{"id": "5095.png", "formula": "\\begin{align*} y _ n \\coloneqq x _ n - \\sum _ { k = 1 } ^ { n - 1 } \\langle x _ n , z _ k \\rangle z _ k , ~ z _ n \\coloneqq \\langle y _ n , y _ n \\rangle ^ { - \\frac { 1 } { 2 } } y _ n , ~ \\forall n \\geq 2 . \\end{align*}"}
-{"id": "4973.png", "formula": "\\begin{align*} \\phi _ b ( k , f ) = 1 2 b ^ 2 ( k + 1 ) ^ 2 ( L _ { k , f , b } + 1 ) . \\end{align*}"}
-{"id": "3442.png", "formula": "\\begin{align*} \\lambda _ { k } = \\frac { 1 } { 2 } \\left ( \\log \\frac { 2 } { \\beta } + \\psi \\big ( \\frac { \\beta } { 2 } ( N - k + 1 ) \\big ) \\right ) , k = 1 , \\dotsc , N , \\end{align*}"}
-{"id": "8669.png", "formula": "\\begin{align*} ( \\partial _ i f ) ( v _ 0 , \\dots , v _ { i - 1 } ) : = \\sum _ { v _ i : ( v _ 0 , \\dots , v _ i ) \\in \\Sigma ^ { ( i ) } } f ( v _ 0 , \\dots , v _ i ) , \\end{align*}"}
-{"id": "1553.png", "formula": "\\begin{align*} F _ { 3 , j } ^ \\delta \\left ( \\begin{array} { c } z \\\\ x \\end{array} \\right ) = \\left ( \\begin{array} { c } z + \\dots + a ^ \\delta _ j z ^ j + a ^ \\delta _ { j + 1 } ( x ) z ^ { j + 1 } + \\dots \\\\ b ^ \\delta _ 0 ( x ) + b ^ \\delta _ 1 ( x ) z + b ^ \\delta _ 2 ( x ) z ^ 2 + \\dots \\end{array} \\right ) , \\end{align*}"}
-{"id": "5877.png", "formula": "\\begin{align*} ( \\phi \\circ j ) ^ { - 1 } ( t _ w ) = \\sum b _ { w , x } T _ x \\end{align*}"}
-{"id": "6333.png", "formula": "\\begin{align*} h _ l = \\frac { A _ a } { \\left ( \\theta _ { \\rm d i v } d / 2 \\right ) ^ 2 } h _ l ' ~ , \\end{align*}"}
-{"id": "1655.png", "formula": "\\begin{align*} I _ k : = p _ k q _ k + p _ { - k } q _ { - k } \\quad I _ { - k } : = p _ k q _ { - k } - p _ { - k } q _ k , \\end{align*}"}
-{"id": "7952.png", "formula": "\\begin{align*} x _ 1 & = e _ 1 \\otimes e _ 1 \\otimes e _ 1 \\\\ x _ 2 & = e _ 2 \\otimes e _ 2 \\otimes e _ 1 \\\\ x _ 3 & = e _ 3 \\otimes e _ 3 \\otimes e _ 2 , \\end{align*}"}
-{"id": "3717.png", "formula": "\\begin{align*} \\widehat { \\bar { Y } } _ C = \\gamma \\bar { y } _ B + ( 1 - \\gamma ) \\bar { y } _ w \\end{align*}"}
-{"id": "7873.png", "formula": "\\begin{align*} \\prod _ { j \\in J } \\cos ( 2 \\pi \\xi _ j ) & = \\prod _ { j \\in I } \\bigg ( \\frac { 1 + \\cos ( 2 \\pi \\xi _ j ) } { 2 } + \\varepsilon ( J ) _ j \\frac { 1 - \\cos ( 2 \\pi \\xi _ j ) } { 2 } \\bigg ) \\\\ & = \\prod _ { j \\in I } \\big ( \\cos ^ 2 ( \\pi \\xi _ j ) + \\varepsilon ( J ) _ j \\sin ^ 2 ( \\pi \\xi _ j ) \\big ) \\\\ & = \\sum _ { S \\subseteq I } a _ S ( \\xi ) w _ S ( \\varepsilon ( J ) ) , \\end{align*}"}
-{"id": "3915.png", "formula": "\\begin{align*} C _ { c , Z } : = \\sum _ { \\nu = 1 } ^ N \\left ( \\sqrt { - c ^ 2 \\Delta _ \\nu + c ^ 4 } - c ^ 2 - \\frac { Z } { | x _ \\nu | } \\right ) + \\sum _ { 1 \\leq \\nu < \\mu \\leq N } \\frac { 1 } { | x _ \\nu - x _ \\mu | } \\end{align*}"}
-{"id": "4462.png", "formula": "\\begin{align*} K _ O = \\min _ { p \\in O } W _ k ( p ) , O \\in X , \\end{align*}"}
-{"id": "4683.png", "formula": "\\begin{align*} \\mu _ j ( u ) = \\frac { 1 } { 6 } ( - u _ { j - 3 } + 8 u _ { j - 2 } - u _ { j - 1 } ) ~ ~ ~ \\mbox { f o r } ~ ~ j = 3 , \\ldots , m + 1 , \\end{align*}"}
-{"id": "3604.png", "formula": "\\begin{align*} \\sigma = ( a _ 1 , \\ , a _ 2 , \\ , a _ 3 ) ( a _ { - 1 } , \\ , a _ { - 2 } , \\ , a _ { - 3 } ) \\end{align*}"}
-{"id": "6871.png", "formula": "\\begin{align*} y ^ 1 _ { 2 0 } = y ^ 1 _ { 1 1 } = y ^ 2 _ { 2 0 } = y ^ 2 _ { 1 1 } = 0 \\ , . \\end{align*}"}
-{"id": "7284.png", "formula": "\\begin{align*} L ( F ) = \\max \\{ | \\lambda | : \\lambda \\in \\C \\ \\ \\exists \\mu \\in \\C , & \\exists a , b \\in F \\\\ & a \\neq 0 \\lambda a + \\mu b \\in F \\} \\end{align*}"}
-{"id": "5975.png", "formula": "\\begin{align*} x _ { ( c , n , k ) } ( t ) = x ( k - 1 + t ) - x ( k - 1 ) , t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "6398.png", "formula": "\\begin{align*} \\int _ { \\zeta } ^ { \\xi } f ( \\lambda ) \\ , d \\omega ( \\lambda ) = \\sum _ { i = 1 } ^ { N } \\omega _ { i } f ( \\lambda _ { i } ) . \\end{align*}"}
-{"id": "10041.png", "formula": "\\begin{align*} U ( q , t ) = \\sum _ { j = 1 } ^ n \\frac { m _ j } { \\| q - q _ j ( \\omega t ) \\| } . \\end{align*}"}
-{"id": "7452.png", "formula": "\\begin{align*} \\displaystyle \\prod _ { l = n , n + 1 } \\int _ { Y _ l } \\varphi _ { 1 , l } ' ( x ) \\overline { \\varphi _ { 2 , l } ' ( x ) } d x \\end{align*}"}
-{"id": "327.png", "formula": "\\begin{align*} \\begin{array} { c c c } \\tilde { \\alpha } _ { 1 , k } = ( n - 2 ) ! k u ^ 2 v _ { 2 n - 4 } + \\frac { ( n - 1 ) ! } { 2 } k u ^ 2 v _ { 2 n - 2 } & & \\tilde { \\alpha } _ { 2 , k } = ( n - 1 ) ! k u ^ 2 v _ { 2 n - 2 } . \\end{array} \\end{align*}"}
-{"id": "6270.png", "formula": "\\begin{align*} \\mathrm { d } X _ t = - \\theta X _ t \\mathrm { d } t + \\mathrm { d } B ^ { H } _ t , X _ 0 = 0 , 0 \\le t \\le T , \\end{align*}"}
-{"id": "5724.png", "formula": "\\begin{align*} y ( 0 ) & = b _ 0 , & y ' ( 0 ) & = b _ 1 . \\end{align*}"}
-{"id": "2849.png", "formula": "\\begin{align*} a ^ 4 - g b ^ 4 = \\pm 1 \\ ; \\ ; ( { \\rm i n } \\ ; a , b \\in \\Z ) . \\end{align*}"}
-{"id": "824.png", "formula": "\\begin{align*} H ^ 0 ( Y , \\Omega _ Y ) \\subset L _ Y ( \\Omega ( D ) ) = H ^ 0 ( X , \\Omega _ X ) ^ H , \\end{align*}"}
-{"id": "8473.png", "formula": "\\begin{align*} ( \\sum _ { j = 1 } ^ 2 \\partial _ { \\alpha } ^ k f _ j ( \\cdot , t ) ) \\circ g = & \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { 2 \\pi } \\frac { ( - 1 ) ^ k k ! } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ { k + 1 } } \\\\ = & \\frac { \\lambda i ( - 1 ) ^ k k ! } { 2 \\pi } \\sum _ { m = 0 } ^ k \\frac { z _ 1 - z _ 2 } { ( \\zeta ( \\alpha , t ) - z _ 1 ( t ) ) ^ { k + 1 - m } ( \\zeta ( \\alpha , t ) - z _ 2 ( t ) ) ^ { m + 1 } } \\end{align*}"}
-{"id": "5063.png", "formula": "\\begin{align*} ( \\lambda _ g \\otimes I _ { \\mathcal { H } _ 0 } ) \\theta _ A \\theta _ A ^ * = \\theta _ A \\theta _ A ^ * ( \\lambda _ g \\otimes I _ { \\mathcal { H } _ 0 } ) , ~ ( \\lambda _ g \\otimes I _ { \\mathcal { H } _ 0 } ) \\theta _ A \\theta _ \\Psi ^ * = \\theta _ A \\theta _ \\Psi ^ * ( \\lambda _ g \\otimes I _ { \\mathcal { H } _ 0 } ) , \\\\ ( \\lambda _ g \\otimes I _ { \\mathcal { H } _ 0 } ) \\theta _ \\Psi \\theta _ \\Psi ^ * = \\theta _ \\Psi \\theta _ \\Psi ^ * ( \\lambda _ g \\otimes I _ { \\mathcal { H } _ 0 } ) . \\end{align*}"}
-{"id": "258.png", "formula": "\\begin{align*} g = \\sum _ { j = 0 } ^ p Q _ j ( \\alpha ) f ^ { p - j } v _ j . \\end{align*}"}
-{"id": "7056.png", "formula": "\\begin{align*} \\Phi _ t = ( d v , d \\b { v } ) \\Phi \\otimes \\left ( \\begin{matrix} { } d v \\\\ d \\b { v } \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "7054.png", "formula": "\\begin{align*} \\ell ( \\gamma ( z ) ) = E ( z ) ^ { 1 / 2 } \\ell ^ { 1 / 2 } _ 0 . \\end{align*}"}
-{"id": "1557.png", "formula": "\\begin{align*} \\Phi _ { T } ( z , x ) : = \\lim _ { n \\rightarrow \\infty } ( \\omega ^ \\iota \\circ \\pi _ z \\circ T ^ n ( z , x ) - n ) \\end{align*}"}
-{"id": "197.png", "formula": "\\begin{align*} \\pi _ { T M , X _ { C } , X _ { V } , c } ( { \\bf x } , { \\bf y } ) = \\left ( \\begin{array} { c c c | c c c } 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & x ^ 3 & x ^ 1 \\\\ 0 & 0 & 0 & 0 & - x ^ 1 & 0 \\\\ \\hline 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & - x ^ 3 & x ^ 1 & 0 & 0 & y ^ 1 \\\\ 0 & - x ^ 1 & 0 & 0 & - y ^ 1 & 0 \\end{array} \\right ) , \\end{align*}"}
-{"id": "9931.png", "formula": "\\begin{align*} \\frac { d \\pi _ { n - } ^ { \\mu } } { d \\pi _ { n - } ^ { \\nu } } ( x ) & = \\frac { E ^ { \\nu } [ \\frac { d \\mu } { d \\nu } ( X _ { 0 } ) | Y _ { [ 0 , n - 1 ] } , X _ { n } = x ] } { E ^ { \\nu } [ \\frac { d \\mu } { d \\nu } ( X _ { 0 } ) | Y _ { [ 0 , n - 1 ] } ] } ~ ~ ~ ~ ~ P ^ { \\mu } ~ a . s . \\end{align*}"}
-{"id": "2870.png", "formula": "\\begin{align*} S ^ N _ 1 f = P _ N ( z _ 1 f ) , \\ \\ S ^ N _ 2 f = P _ N z _ 2 f , \\ \\ \\ f \\in N . \\end{align*}"}
-{"id": "6317.png", "formula": "\\begin{align*} \\lim _ { n } \\frac { k } { n } \\mathbb { E } _ { G _ 0 \\sim \\mathbb { P } _ 1 } [ | n - 2 d ^ { G _ 0 } ( v ) ] = 0 . \\end{align*}"}
-{"id": "5392.png", "formula": "\\begin{align*} | | v _ { n \\ , m + 6 } ^ 1 | | ^ 2 = - \\frac { \\overline { b _ { n \\ , m + 6 } } } { c _ { n + 1 \\ , m + 3 } } | | v _ { n + 1 \\ , m + 3 } ^ 1 | | ^ 2 = \\frac { \\overline { b _ { n \\ , m + 6 } } } { c _ { n + 1 \\ , m + 3 } } \\cdot \\frac { \\overline { d _ { n + 1 \\ , m + 3 } } } { a _ { n m } } | | v _ { n m } ^ 1 | | ^ 2 . \\end{align*}"}
-{"id": "2456.png", "formula": "\\begin{align*} P ( R / x R , t ) \\ = \\ \\frac { 1 } { 1 - t ^ s } - p ( t ^ s ) , \\end{align*}"}
-{"id": "4268.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\delta ( a ( x ) \\lvert d w \\rvert ^ { p - 2 } d w ) ) & = 0 & & B _ { R } , \\\\ \\delta w & = 0 & & B _ { R } , \\\\ \\nu \\wedge w & = \\nu \\wedge u & & \\partial B _ { R } , \\end{aligned} \\right . \\end{align*}"}
-{"id": "1796.png", "formula": "\\begin{align*} c _ { 2 , 1 } \\left ( \\liminf _ { n \\to \\infty } G _ { n } \\right ) = 0 , \\end{align*}"}
-{"id": "5788.png", "formula": "\\begin{align*} \\tilde R _ d ( q ) = ( - 1 ) ^ \\delta \\psi ( q ^ { - d } ) * \\tilde R _ { d - 1 } ( q ) , d > 1 . \\end{align*}"}
-{"id": "2697.png", "formula": "\\begin{align*} \\mathcal { C } _ { \\pm } ^ { \\mathrm { l o c a l } } & = \\left \\{ z _ { 1 } + r e ^ { \\pm i \\frac { 2 } { 3 } \\pi } : r \\in [ 0 , 2 ^ { \\frac { 5 } { 3 } } N ^ { - \\frac { 1 } { 3 } } \\kappa ] \\right \\} , \\\\ \\mathcal { C } _ { \\pm } ^ { 1 } & = \\left \\{ t \\pm i 2 ^ { \\frac { 2 } { 3 } } \\sqrt { 3 } N ^ { - \\frac { 1 } { 3 } } \\kappa : t \\in [ z _ { 1 } - 2 ^ { \\frac { 2 } { 3 } } N ^ { - \\frac { 1 } { 3 } } \\kappa , 1 ] \\right \\} , \\end{align*}"}
-{"id": "1210.png", "formula": "\\begin{align*} \\frac { \\sin \\theta _ - } { \\sqrt { \\gamma _ - } } = \\frac { \\sin \\theta _ + } { \\sqrt { \\gamma _ + } } , \\end{align*}"}
-{"id": "3481.png", "formula": "\\begin{align*} \\Sigma _ + = \\{ t = q _ { M } ( \\phi ) \\mid \\phi \\in [ 0 , \\pi ) \\} , \\Sigma _ - = \\{ t = q _ { M } ( - \\phi ) = \\overline { q _ { M } ( \\phi ) } \\mid \\phi \\in [ 0 , \\pi ) \\} , \\end{align*}"}
-{"id": "7014.png", "formula": "\\begin{align*} \\varphi _ 2 = \\varphi _ 1 \\circ \\alpha \\quad \\alpha \\circ \\Xi = \\Xi \\circ \\alpha . \\end{align*}"}
-{"id": "8897.png", "formula": "\\begin{align*} \\check { H } _ { \\bullet } \\left ( X ; G \\right ) & : = \\varprojlim _ { \\lambda \\in \\mathrm { C o v } _ { X } } H _ { \\bullet } \\left ( V \\left ( \\lambda \\right ) ; G \\right ) . \\\\ \\check { H } ^ { \\bullet } \\left ( X ; G \\right ) & : = \\varinjlim _ { \\lambda \\in \\mathrm { C o v } _ { X } } H ^ { \\bullet } \\left ( V \\left ( \\lambda \\right ) ; G \\right ) . \\end{align*}"}
-{"id": "4056.png", "formula": "\\begin{align*} d ( \\theta ^ { 2 k } ) = 0 , ~ ~ ~ ~ ~ d ( \\theta ^ { 2 k + 1 } ) = x ^ n \\theta ^ { 2 k } . \\end{align*}"}
-{"id": "4316.png", "formula": "\\begin{align*} ( \\nabla v _ h , \\nabla q ) _ D = - ( v _ h , \\Delta q ) _ D + ( v _ h , \\nabla q \\cdot n ) _ { \\partial D } , \\end{align*}"}
-{"id": "7550.png", "formula": "\\begin{align*} C ^ { 1 + \\alpha , p } = C ^ { 1 + \\alpha } ( \\mathbb { R } ^ d ) \\cap W ^ { 1 , p } ( \\mathbb { R } ^ d ) \\end{align*}"}
-{"id": "5912.png", "formula": "\\begin{align*} t _ { s _ 0 } \\star \\bar { \\varphi } _ 0 = \\sum _ { n = 0 } ^ { \\infty } t _ { s _ 0 } \\star ( \\varphi _ n + \\psi _ n ) . \\end{align*}"}
-{"id": "3954.png", "formula": "\\begin{align*} g _ { 1 } ( x , m ^ { \\hat v } _ t , \\hat { v } _ { 1 } ( x , m ^ { \\hat v } _ t ) ) = \\dfrac { \\partial H _ { 1 } } { \\partial q _ { 1 } } ( x , m ^ { \\hat v } _ t , D _ { x } U _ { 1 } ( x , m ^ { \\hat v } _ t , t ) ) \\ , . \\end{align*}"}
-{"id": "7196.png", "formula": "\\begin{gather*} \\left \\Vert \\chi ^ { t + s } \\left ( \\xi + \\eta + h ( \\xi , \\eta ) \\right ) - \\chi ^ { t + s } ( \\xi ) \\right \\Vert = O \\left ( \\mathrm { e } ^ { - \\alpha ( t + s ) } \\right ) , t \\to \\infty , \\end{gather*}"}
-{"id": "3520.png", "formula": "\\begin{align*} F ( x ^ a , u , p _ a , p _ { a b } ) = \\sum _ { \\substack { I , J \\subseteq \\{ 0 , \\ldots , n \\} \\\\ | I | = | J | } } A _ { I , J } ( x ^ a , u , p _ a ) H _ { I , J } = 0 , \\end{align*}"}
-{"id": "1690.png", "formula": "\\begin{align*} \\begin{gathered} C _ { n } ^ 0 + C _ { n } ^ 2 + \\cdots + C _ { n } ^ { n - 1 } = 2 ^ { n - 1 } - 1 \\\\ C _ { n } ^ 1 + C _ { n } ^ 3 + \\cdots + C _ { n } ^ { n } = 2 ^ { n - 1 } - 1 , \\end{gathered} \\end{align*}"}
-{"id": "8042.png", "formula": "\\begin{align*} x ( x y ^ k ) x = y ^ { k } x = x y ^ { k ( n - 1 ) } = x y ^ { k ( 4 s - 1 ) } = x y ^ { 4 k s - k } = y ^ k ( x y ^ k ) y ^ { - k } , \\end{align*}"}
-{"id": "119.png", "formula": "\\begin{align*} \\zeta _ { j , N } & \\le \\sum _ { \\substack { \\ell = 1 } } ^ { \\infty } \\frac { d ^ { 2 \\ell + 1 } } { ( \\ell ! ) ^ 2 } \\frac { j ^ \\ell } { N ^ \\ell } \\frac { ( 2 N - j - 1 ) ^ \\ell } { N ^ \\ell } \\ , . \\end{align*}"}
-{"id": "104.png", "formula": "\\begin{align*} \\alpha = \\sum _ { j = 1 , \\ j \\mbox { { \\small o d d } } } ^ { k + 1 } \\ell _ j \\ , , \\beta = \\sum _ { j = 2 , \\ j \\mbox { { \\small e v e n } } } ^ { k + 1 } \\ell _ j = 2 N - k - \\alpha \\ , . \\end{align*}"}
-{"id": "9272.png", "formula": "\\begin{align*} \\sup \\limits _ { B _ { \\frac { \\delta R } { 2 } } ( \\bar x ^ \\prime ) \\cap \\partial \\Omega } u _ { \\beta \\beta } \\le \\epsilon \\sup \\limits _ { B _ { \\frac { 3 \\delta R } { 4 } } ( \\bar x ^ \\prime ) \\cap \\Omega } | D ^ 2 u | + C _ \\epsilon ( 1 + \\sup \\limits _ { B _ { \\frac { 3 \\delta R } { 4 } } ( \\bar x ^ \\prime ) \\cap \\partial \\Omega } \\sup \\limits _ { | \\tau | = 1 , \\tau \\cdot \\nu = 0 } | u _ { \\tau \\tau } | ) + \\frac { C } { ( \\delta R / 4 ) ^ 2 } , \\end{align*}"}
-{"id": "7763.png", "formula": "\\begin{align*} \\nu ^ D ( x ) = \\max \\{ | \\langle x , y \\rangle | : y \\in \\mathcal { M } _ { n \\times 1 } , \\nu ( y ) \\leq 1 \\} \\end{align*}"}
-{"id": "3040.png", "formula": "\\begin{align*} \\rho _ { I } ( x ) : = \\min \\{ x _ { j } ^ { 1 / \\alpha _ { j } } \\ | \\ j \\in I \\} \\ 1 _ { \\R _ { + } ^ { d } } ( x ) , \\qquad \\rho _ { \\emptyset } = \\ 1 _ { \\R _ { + } ^ { d } } ( x ) , \\qquad \\Gamma ( I ) = \\{ x \\in \\R ^ { d } \\ | \\ \\rho _ { I } ( x ) > 0 \\} . \\end{align*}"}
-{"id": "3105.png", "formula": "\\begin{align*} P & = c q ^ { \\frac { z } { 2 r } } ( 1 + p _ 1 q + p _ 2 q ^ 2 + \\ldots ) \\\\ P ' & = c ' q ^ { \\frac { z ' } { 2 r } } ( 1 + p ' _ 1 q + p ' _ 2 q ^ 2 + \\ldots ) \\end{align*}"}
-{"id": "9218.png", "formula": "\\begin{align*} e ( G ' ) \\leq e ( S ) + \\sum _ { x \\in \\bar { S } } d _ { G ' } ( x ) \\leq \\binom { s } { 2 } + ( 2 k - s + 1 ) ( n - s ) . \\end{align*}"}
-{"id": "5432.png", "formula": "\\begin{align*} r _ s ( T ) = \\sum _ { j = 1 } ^ s \\xi _ j ( T ) . \\end{align*}"}
-{"id": "1172.png", "formula": "\\begin{align*} 0 \\leq & ( n / 4 ) ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } \\mathbb { P } ( \\xi ^ { ( n ) } ( I _ { n , k } ) = \\xi ^ { ( n ) } ( J _ { n , k , j } ) = 0 ) \\\\ = & ( n / 4 ) ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } D _ n ( ( z _ { j + 1 } - z _ j ) / 2 ) \\cdot \\left ( n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } D _ n ( F _ n ( x _ 0 ) / 2 ) / 4 \\right ) ^ { n - 1 } , \\end{align*}"}
-{"id": "7880.png", "formula": "\\begin{align*} \\frac { 1 } { m ! } \\Big | \\sum _ { \\sigma \\in { \\rm S y m } ( I ) } \\prod _ { j \\in I } \\cos ( 2 \\pi y _ { \\sigma ^ { - 1 } ( j ) } \\xi _ j ) \\Big | & \\le \\sum _ { \\substack { S \\subseteq I \\\\ \\emptyset \\not = S \\not = I } } a _ S ( \\xi ) e ^ { - \\frac { c l | S | } { m } } + \\sum _ { \\substack { S \\subseteq I \\\\ \\emptyset \\not = S \\not = I } } a _ S ( \\xi ) e ^ { - \\frac { c l ( m - | S | ) } { m } } \\\\ & + e ^ { - \\sum _ { i \\in I } \\sin ^ 2 ( \\pi \\xi _ i ) } + e ^ { - \\sum _ { i \\in I } \\cos ^ 2 ( \\pi \\xi _ i ) } , \\end{align*}"}
-{"id": "8485.png", "formula": "\\begin{align*} | \\dot { z } _ 1 ^ 2 - \\dot { z } _ 2 ^ 2 | = & | \\frac { 2 \\lambda i } { 4 \\pi x ( t ) } ( F ( z _ 1 ( t ) , t ) - F ( z _ 2 ( t ) , t ) ) + F ( z _ 1 ( t ) , t ) ^ 2 - F ( z _ 2 ( t ) , t ) ^ 2 | \\\\ = & | \\frac { \\lambda i } { \\pi x ( t ) } R e F ( z _ 1 ( t ) , t ) + F ( z _ 1 ( t ) , t ) ^ 2 - F ( z _ 2 ( t ) , t ) ^ 2 | \\\\ \\leq & \\frac { | \\lambda | } { \\pi x ( t ) } \\| F _ { \\zeta } \\| _ { \\infty } x ( t ) + | F ( z _ 1 ( t ) , t ) ^ 2 - F ( z _ 2 ( t ) , t ) ^ 2 | \\\\ \\leq & 6 | \\lambda | \\epsilon + 1 2 0 \\epsilon ^ 2 x ( t ) . \\end{align*}"}
-{"id": "7644.png", "formula": "\\begin{align*} \\int \\limits _ { \\mathbb R ^ d } a ( q - \\xi ) \\mu ( q , \\xi ) v _ 0 ( q ) \\ d q \\ = \\ v _ 0 ( \\xi ) \\ , \\int \\limits _ { \\mathbb R ^ d } a ( \\xi - q ) \\mu ( \\xi , q ) d q , \\end{align*}"}
-{"id": "5075.png", "formula": "\\begin{align*} & ( \\left \\{ y _ j = S _ { x , \\tau } ^ { - 1 } \\theta _ x ^ * e _ j + V e _ j - V \\theta _ \\tau S _ { x , \\tau } ^ { - 1 } \\theta _ x ^ * e _ j = S _ { x , \\tau } ^ { - 1 } x _ j + V e _ j - V \\theta _ \\tau S _ { x , \\tau } ^ { - 1 } x _ j \\right \\} _ { j \\in \\mathbb { J } } , \\\\ & \\left \\{ \\omega _ j = S _ { x , \\tau } ^ { - 1 } \\theta _ \\tau ^ * e _ j + U e _ j - U \\theta _ x S _ { x , \\tau } ^ { - 1 } \\theta _ \\tau ^ * e _ j = S _ { x , \\tau } ^ { - 1 } \\tau _ j + U e _ j - U \\theta _ x S _ { x , \\tau } ^ { - 1 } \\tau _ j \\right \\} _ { j \\in \\mathbb { J } } ) \\end{align*}"}
-{"id": "5434.png", "formula": "\\begin{align*} c _ 2 ( T ^ n ) = \\inf _ V \\lVert ( T ^ n ) _ { \\rvert _ V } \\rVert \\ge \\inf _ V \\lVert T ^ n v _ V \\rVert \\ge \\lVert ( ( T _ { \\mid \\mathcal { N } _ { \\lambda _ 1 } } ) ^ { - 1 } ) ^ n \\rVert ^ { - 1 } . \\end{align*}"}
-{"id": "2899.png", "formula": "\\begin{align*} C ^ { i } & = \\frac { 1 } { 4 } \\left ( S _ { 2 i - 1 , 2 i } ^ { 2 } - S _ { 2 i - 1 , 2 i } - \\frac { 3 } { 4 } \\right ) , \\\\ C ^ { i j } & = \\frac { 1 } { 4 } \\left ( S _ { 2 i - 1 , 2 i , 2 j - 1 , 2 j } ^ { 2 } - S _ { 2 i - 1 , 2 i , 2 j - 1 , 2 j } - \\frac { 3 } { 4 } \\right ) . \\end{align*}"}
-{"id": "8514.png", "formula": "\\begin{align*} \\norm { \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { 2 \\pi } \\frac { \\ddot { z } _ j } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } } _ { H ^ s } \\leq K _ s ^ { - 1 } \\epsilon ^ 2 d _ I ( t ) ^ { - 5 / 2 } . \\end{align*}"}
-{"id": "4553.png", "formula": "\\begin{align*} k , k + 1 , \\ldots , k + p , k ' & = k , k + 1 , \\ldots , k + p , k \\\\ & < k , k + 1 , \\ldots , k + p , k + p + 1 \\\\ & \\leq k , k + 1 , \\ldots , k + d _ 1 \\\\ & < k , k + 1 , \\ldots , k + p , k + d ' _ 1 \\\\ & = { \\bf i ' } ^ { ( 1 ) } \\end{align*}"}
-{"id": "837.png", "formula": "\\begin{align*} \\omega = \\sum _ { i = 1 } ^ g a _ i \\omega _ i \\end{align*}"}
-{"id": "8738.png", "formula": "\\begin{align*} \\left [ D ^ { \\sigma , \\alpha } _ { 0 , t } u ( t ) \\right ] _ { t = t _ n } = D _ { \\tau } ^ { \\alpha , \\sigma , \\gamma , 0 , n } u + O ( t _ n ^ { \\alpha + \\delta - p } \\tau ^ { p } ) + O ( t _ n ^ { \\alpha - 1 } \\tau ^ { \\delta + 1 } ) . \\end{align*}"}
-{"id": "2874.png", "formula": "\\begin{align*} C = \\Delta _ { R ^ * } = I - R _ 1 R _ 1 ^ * - R _ 2 R _ 2 ^ * + R _ 1 R _ 2 R _ 1 ^ * R _ 2 ^ * . \\end{align*}"}
-{"id": "3071.png", "formula": "\\begin{align*} \\widehat { K _ 1 } ( y ) = \\int _ { \\R ^ n } e ^ { 2 \\pi i x \\cdot y } K _ 1 ( x ) \\ , d x = \\int _ { | x | \\leq \\frac { 2 } { \\beta } } ( e ^ { 2 \\pi i x \\cdot y } - 1 ) K _ 1 ( x ) \\ , d x \\end{align*}"}
-{"id": "9544.png", "formula": "\\begin{align*} \\mathrm { s p t } ( n ) = n p ( n ) - \\frac { 1 } { 2 } N _ { 2 } ( n ) . \\end{align*}"}
-{"id": "7277.png", "formula": "\\begin{align*} p = ( F _ p , n _ p , \\epsilon _ p , \\psi _ p ) \\end{align*}"}
-{"id": "7559.png", "formula": "\\begin{align*} \\norm { \\tau \\circ X ^ { - 1 } } _ { L ^ p \\cap L ^ \\infty } = \\norm { \\tau } _ { L ^ p \\cap L ^ \\infty } , \\end{align*}"}
-{"id": "935.png", "formula": "\\begin{align*} \\Biggl . \\Biggl . + \\frac { ( i ^ 2 + i - 3 ) \\zeta _ { i + 1 } ^ { ( i _ 2 ) } \\zeta _ { i } ^ { ( i _ 1 ) } - ( i ^ 2 + 3 i - 1 ) \\zeta _ { i } ^ { ( i _ 2 ) } \\zeta _ { i + 1 } ^ { ( i _ 1 ) } } { \\sqrt { ( 2 i + 1 ) ( 2 i + 3 ) } ( 2 i - 1 ) ( 2 i + 5 ) } \\Biggr ) \\Biggr ] - \\frac { 1 } { 2 4 } { \\bf 1 } _ { \\{ i _ 1 = i _ 2 \\} } { \\Delta ^ 3 } , \\end{align*}"}
-{"id": "3903.png", "formula": "\\begin{align*} t = { \\rm S u p } \\{ | b | , | \\beta | , | \\gamma | , | U - I | \\} , \\ M = | s | + 2 | \\zeta | . \\end{align*}"}
-{"id": "632.png", "formula": "\\begin{align*} \\psi ^ x = ( \\delta \\psi ' ) ^ x = \\delta ^ x \\psi '^ { \\delta ^ \\ast ( x ) } = \\delta \\psi ' = \\psi \\ , \\end{align*}"}
-{"id": "2875.png", "formula": "\\begin{align*} \\left ( \\widehat { H F K } ( K ) \\right ) ^ 2 & = \\widehat { H F K } ( S ^ 3 , K \\# K ^ r ) \\\\ & \\le \\widetilde { K h } _ 2 ( K ) \\\\ & \\le 2 4 . \\end{align*}"}
-{"id": "8879.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 } { \\partial s ^ 2 } e _ { 9 b } & = 6 ( 2 p - 1 ) ^ 2 ( 1 4 + ( 2 p - 1 ) ( 2 p ^ 2 s - 3 p + 1 5 ) ) \\ge 0 \\end{align*}"}
-{"id": "6456.png", "formula": "\\begin{align*} \\ell ( v ) & \\Omega = v \\\\ \\ell ( v ) & w _ 1 \\otimes \\cdots \\otimes w _ d = v \\otimes w _ 1 \\otimes \\cdots \\otimes w _ d . \\end{align*}"}
-{"id": "9105.png", "formula": "\\begin{align*} I _ 4 & = - \\int _ { B _ { \\frac { | x | } { 2 } } ( 0 ) } \\dfrac { ( \\Upsilon ( y ) - \\Upsilon ( x ) ) ^ { p - 1 } } { | x - y | ^ { N + p s } } d y \\\\ & \\le - \\left ( \\dfrac { 2 } 3 \\right ) ^ { N + p s } \\dfrac 1 { | x | ^ { N + p s } } \\int _ { B _ { \\frac { | x | } { 2 } } ( 0 ) } ( \\Upsilon ( y ) - \\Upsilon ( x ) ) ^ { p - 1 } d y . \\end{align*}"}
-{"id": "1102.png", "formula": "\\begin{align*} & \\limsup _ { n \\to + \\infty } \\int _ { I ^ k } \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) d y _ 1 \\cdots d y _ k \\leq ( M ( I ) ) ^ k \\prod _ { j = 1 } ^ k \\left ( e ^ { c _ 0 - x _ j } \\right ) , \\end{align*}"}
-{"id": "8218.png", "formula": "\\begin{align*} a _ 0 ( t ) = - \\prod \\limits _ { \\lambda = 0 } ^ 4 t _ \\lambda = - t . \\end{align*}"}
-{"id": "9330.png", "formula": "\\begin{align*} \\int _ { v \\in \\S _ { a } ^ { m - 2 } } \\| \\nabla F _ { v } \\| ^ 2 ( x ) \\ , d V _ { a } = \\frac { \\mathrm { V o l } ( \\S ^ { m - 2 } ) } { m - 1 } \\Big [ m \\| \\nabla F _ { a } \\| ^ 2 ( x ) + \\mathrm { t r a c e } ( A _ { Q _ { 1 } ^ { x } } ) \\Big ] \\end{align*}"}
-{"id": "966.png", "formula": "\\begin{align*} G = \\sum _ { g \\in G } g , D = \\sum _ { d \\in D } d , D ^ { ( - 1 ) } = \\sum _ { d \\in D } d ^ { - 1 } , g D = \\sum _ { d \\in D } g d , D ^ \\phi = \\sum _ { d \\in D } \\phi ( d ) \\end{align*}"}
-{"id": "4174.png", "formula": "\\begin{align*} A _ n ^ { ( 1 ) } ( z _ 1 ) \\cdot \\cdots \\cdot A _ n ^ { ( 1 ) } ( z _ k ) = \\mathcal { O } ( n ^ { k + 1 - \\ell } ) . \\end{align*}"}
-{"id": "3512.png", "formula": "\\begin{align*} \\lambda ^ { ( \\nu ) } _ { k } = \\lim _ { M \\to \\infty } \\frac { 1 } { 2 ( M + 1 ) } \\sum _ { j = 0 } ^ { M } \\psi \\big ( \\nu _ j + N - k + 1 \\big ) , k = 1 , \\ldots , N , \\end{align*}"}
-{"id": "803.png", "formula": "\\begin{align*} N _ t : = \\{ D \\mod P ^ \\alpha : D \\equiv 1 \\mod P ^ t \\} . \\end{align*}"}
-{"id": "4427.png", "formula": "\\begin{align*} \\begin{aligned} \\dfrac { 2 } { p } - \\eta - d \\Bigl ( \\dfrac { 1 } { 2 } - \\dfrac { 1 } { p } \\Bigr ) = \\dfrac { 1 } { p } + \\eta . \\end{aligned} \\end{align*}"}
-{"id": "6774.png", "formula": "\\begin{align*} P _ i ( \\tilde E ) = \\int _ { \\Gamma _ i } ( z I - ( \\Lambda _ { \\alpha } + \\tilde E ) ) ^ { - 1 } \\frac { d z } { 2 \\pi i } , i = 0 , 1 \\end{align*}"}
-{"id": "8224.png", "formula": "\\begin{align*} U _ 5 ( u t ^ j ) = - \\sum _ { l = 0 } ^ 4 a _ l ( t ) U _ 5 ( u t ^ { j + l - 5 } ) . \\end{align*}"}
-{"id": "3489.png", "formula": "\\begin{align*} f ' _ M ( q _ M ( 0 ) ) = 0 . \\end{align*}"}
-{"id": "6635.png", "formula": "\\begin{align*} T ' = ( 1 - X ) \\bar \\sigma ' , \\end{align*}"}
-{"id": "2982.png", "formula": "\\begin{align*} K & = \\begin{cases} 1 , & n = \\Delta _ m m ; \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "5999.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t v ( t , x ) + \\frac 1 2 \\Delta v ( t , x ) + g ( t , \\nabla v ( t , x ) ) = 0 [ 0 , 1 ] \\times \\R ^ d \\\\ v ( 1 , x ) = f ( x ) , x \\in \\R ^ d \\end{cases} \\end{align*}"}
-{"id": "3023.png", "formula": "\\begin{align*} L ^ m _ { p , q } g ( z ) = \\sum _ { { k = 1 } } ^ { \\infty } [ k ] ^ m _ { p , q } b _ { k } z ^ { k } . \\end{align*}"}
-{"id": "3292.png", "formula": "\\begin{align*} \\frac { \\sum _ { n \\le x , \\ , n \\equiv a \\bmod q } \\alpha ( n ) } { x / \\phi ( q ) } = \\frac { \\sum _ { n \\le x , \\ , ( n , q ) = 1 } \\alpha ( n ) } { x ( \\phi ( q ) / q ) } ( 1 + o ( 1 ) ) , x \\to \\infty \\end{align*}"}
-{"id": "9305.png", "formula": "\\begin{align*} \\min \\sum _ { i = 1 } ^ m \\sum _ { j = 1 } ^ n c _ { i j } x _ { i j } \\end{align*}"}
-{"id": "2187.png", "formula": "\\begin{align*} J ( \\boldsymbol { x } ( t ) , \\boldsymbol { u } ( t ) , t ) = l _ f ( \\boldsymbol { x } ( t _ f ) ) + \\int _ { t _ 0 } ^ { t _ f } l ( \\boldsymbol { x } ( t ) , \\boldsymbol { u } ( t ) , t ) d t \\end{align*}"}
-{"id": "9616.png", "formula": "\\begin{align*} ( 1 - q ^ { N } ) \\sum _ { n = 1 } ^ { \\infty } \\textup { s s p t d } _ { o } ( n , N ) q ^ n = \\frac { 1 } { 2 } \\left \\{ \\sum _ { m = 1 } ^ { \\infty } \\frac { q ^ { m } } { 1 - q ^ { m } } \\frac { ( - 1 ) _ m } { ( - q ^ N ) _ m } - \\sum _ { m = N } ^ { \\infty } \\frac { 2 q ^ m } { 1 - q ^ { 2 m } } + \\frac { q ^ N } { 1 - q ^ N } ( - 1 ) _ N \\right \\} . \\end{align*}"}
-{"id": "3259.png", "formula": "\\begin{align*} \\Omega _ n ' ( a + b ) & = P _ n ' ( a + b ) \\\\ & = \\lim _ { \\theta \\to 0 } \\frac { e ^ { i n \\theta } a ^ n + e ^ { - i n \\theta } b ^ n - ( a ^ n + b ^ n ) } { e ^ { i \\theta } a + e ^ { - i \\theta } b - ( a + b ) } \\\\ & = \\lim _ { \\theta \\to 0 } \\frac { i n e ^ { i n \\theta } a ^ n - i n e ^ { i n \\theta } b ^ n } { i e ^ { i \\theta } a - i e ^ { i \\theta } b } \\\\ & = n \\frac { a ^ n - b ^ n } { a - b } . \\end{align*}"}
-{"id": "10078.png", "formula": "\\begin{align*} { E } ^ { ( j ) } = & { E } ^ { ( j - 1 ) } + \\big [ F \\circ { K } ^ { ( j ) } - F \\circ { K } ^ { ( j - 1 ) } - ( D F \\circ { K } ^ { ( j - 1 ) } ) { \\mathcal K } ^ { ( j ) } \\big ] \\\\ & + ( D F \\circ { K } ^ { ( j - 1 ) } ) { \\mathcal K } ^ { ( j ) } - { \\mathcal K } ^ { ( j ) } \\circ { R } ^ { ( j - 1 ) } \\\\ & - \\big [ { K } ^ { ( j ) } \\circ { R } ^ { ( j ) } - { K } ^ { ( j ) } \\circ { R } ^ { ( j - 1 ) } \\big ] . \\end{align*}"}
-{"id": "6986.png", "formula": "\\begin{align*} \\textstyle { ( p + 1 ) \\cos \\left ( \\frac { 2 a \\pi r } { p - 1 } \\right ) - ( p - 1 ) \\cos \\left ( \\frac { ( 2 a + 4 ) \\pi r } { p + 1 } \\right ) = O ( r ) } \\end{align*}"}
-{"id": "318.png", "formula": "\\begin{align*} m ( 1 , 0 , 1 ) = s ( \\frac { 1 } { 2 } n ( n - 1 ) , n , \\frac { 1 } { 2 } n ( n + 1 ) ) + t ( n ( n - 1 ) , 0 , 0 ) + r ( 0 , 2 n , 0 ) \\end{align*}"}
-{"id": "6675.png", "formula": "\\begin{align*} x \\frac { d x ^ { - s } } { d x } = - s x ^ { - s } , ( f ( x \\frac { d } { d x } ) x ^ { - s } ) = f ( - s ) x ^ { - s } . \\end{align*}"}
-{"id": "9874.png", "formula": "\\begin{align*} \\tau | S _ \\tau | \\le | \\{ p - p ' = s ~ : ~ p , p ' \\in P \\ , , s \\in S _ \\tau \\} | \\le \\end{align*}"}
-{"id": "3307.png", "formula": "\\begin{align*} \\left | \\sum _ { f \\in \\mathcal { M } _ n } \\alpha ( f ) \\chi ( f ) \\right | \\le 7 \\max _ { f \\in \\mathcal { M } _ { n ; M } } \\left | \\alpha ( f ) \\right | q ^ { \\frac { n } { 2 } } \\binom { 2 0 ( \\ell + \\deg ( M ) + 1 ) + n - 1 } { n } . \\end{align*}"}
-{"id": "5320.png", "formula": "\\begin{align*} H _ { { \\rm V } } ( q , p ; \\alpha , \\beta , \\gamma ; s , t ) = & p q ( p + t ) ( q - s ) + \\alpha t q + \\beta s p + \\gamma p q \\end{align*}"}
-{"id": "7998.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u = ( \\Delta + V ( x ) ) u , ( t , x ) \\in \\mathbb { R } ^ + \\times \\mathbb { R } ^ d ; \\\\ u ( 0 , x ) = u _ 0 ( x ) \\in L ^ 2 ( \\mathbb { R } ^ d ) . \\end{cases} \\end{align*}"}
-{"id": "7089.png", "formula": "\\begin{align*} \\begin{cases} - \\partial _ { t t } u _ n + \\Delta u _ n = | \\nabla u _ n | ^ 2 \\\\ u _ n | _ { t = 0 } = Q _ { \\leq N } f _ 0 ^ \\omega ~ , ~ ~ \\partial _ t u _ n | _ { t = 0 } = Q _ { \\leq N } f _ 1 ^ \\omega ~ . \\end{cases} \\end{align*}"}
-{"id": "6562.png", "formula": "\\begin{align*} n ( d ; ( a _ i ) _ { i = 1 } ^ { r - 1 } ; 1 ) = n ( d ; ( a _ i ) _ { i = 1 } ^ { r - 1 } ; ) . \\end{align*}"}
-{"id": "8602.png", "formula": "\\begin{align*} \\frac { \\omega } { \\Lambda - 1 } = c _ 0 + \\tilde { \\omega } \\left ( \\frac { 1 } { \\sqrt { \\Lambda - 1 } } \\right ) \\end{align*}"}
-{"id": "2604.png", "formula": "\\begin{align*} M ( t , r , x ) : = | G ( r ) ^ { \\gamma _ 0 - 1 } - g ( t , r , x ) ^ { \\gamma _ 0 - 1 } | ^ { \\frac { 1 } { \\gamma _ 0 - 1 } } , t \\geq 0 , r \\geq 0 , x \\in E , \\end{align*}"}
-{"id": "9299.png", "formula": "\\begin{align*} & ( p ^ v , X ^ v ) : = ( D _ x \\phi ( \\hat x , \\hat y ) , X - \\delta D ^ 2 d ( \\hat x ) + 2 \\delta I ) \\in \\bar J ^ { 2 , + } _ { \\bar \\Omega } v ( \\hat x ) , \\\\ & ( p ^ { \\tilde u } , Y ^ { \\tilde u } ) : = ( - D _ y \\phi ( \\hat x , \\hat y ) , Y + \\delta D ^ 2 d ( \\hat y ) ) \\in \\bar J ^ { 2 , - } _ { \\bar \\Omega } \\tilde u ( \\hat y ) , \\end{align*}"}
-{"id": "6639.png", "formula": "\\begin{align*} Z _ j = \\tfrac { d - 4 } { d - 2 } R _ j + \\tfrac 2 { d - 2 } ( \\tfrac 1 2 , \\tfrac 1 2 ) , j = 0 , 1 , 2 , \\end{align*}"}
-{"id": "3703.png", "formula": "\\begin{align*} \\widehat { Y } = \\sum _ { i \\in B } w _ i y _ i \\end{align*}"}
-{"id": "7436.png", "formula": "\\begin{align*} \\displaystyle d ^ E \\pi = 2 ^ { - k } ( \\frac { \\log ( q _ E ) } { \\log ( q _ F ) } ) ^ k d \\pi \\end{align*}"}
-{"id": "8529.png", "formula": "\\begin{align*} G _ { c 1 } = - \\frac { 4 } { \\pi } \\int \\frac { ( D _ t \\bar { \\mathfrak { F } } ( \\alpha , t ) - D _ t \\bar { \\mathfrak { F } } ( \\beta , t ) ) I m \\{ \\zeta ( \\alpha , t ) - \\zeta ( \\beta , t ) \\} } { | \\zeta ( \\alpha , t ) - \\zeta ( \\beta , t ) | ^ 2 } \\partial _ { \\beta } \\bar { \\mathfrak { F } } ( \\beta , t ) d \\beta . \\end{align*}"}
-{"id": "133.png", "formula": "\\begin{align*} \\{ f , g \\} ( { \\bf x } ) = \\sum _ { i , j = 1 } ^ { N } \\pi ^ { i j } ( { \\bf x } ) \\dfrac { \\partial f } { \\partial x ^ i } \\dfrac { \\partial g } { \\partial x ^ j } , \\end{align*}"}
-{"id": "5993.png", "formula": "\\begin{align*} | Y _ n ( t , \\omega ) - Y _ n ( t , 0 ) | & = | u _ n ( t , \\omega / \\sqrt { n } ) - u _ n ( t , 0 ) | \\\\ & \\le \\left \\| F \\left ( \\frac { 1 } { \\sqrt { n } } ( \\omega \\otimes _ t \\cdot ) \\right ) - F \\left ( \\frac { 1 } { \\sqrt { n } } ( 0 \\otimes _ t \\cdot ) \\right ) \\right \\| _ \\infty , \\end{align*}"}
-{"id": "4594.png", "formula": "\\begin{align*} \\int \\int \\hat { f } _ { g , \\delta } ( - \\omega ) \\hat { \\nu } _ { g } ( \\omega ) \\psi ( 2 ^ { - j } \\omega ) d \\omega d g = \\int \\int | \\hat { \\mu } ( \\omega ) | ^ 2 | \\hat { \\mu } ( g \\omega ) | ^ 2 \\hat { \\phi ^ D _ \\delta } ( \\omega ) \\psi ( 2 ^ { - j } \\omega ) d \\omega d g . \\end{align*}"}
-{"id": "3971.png", "formula": "\\begin{align*} \\mathcal { E } _ { k , m } = \\underset { 1 \\leq s \\leq b } \\sum a _ f ( a \\cdot ( s , b ) ^ 2 ) E _ { k , m , s } . \\end{align*}"}
-{"id": "4452.png", "formula": "\\begin{align*} \\tilde q \\circ i = g = g ' \\circ i ' = ( \\tilde q \\circ a ) \\circ i ' = \\tilde q \\circ ( a \\circ i ' ) . \\end{align*}"}
-{"id": "1446.png", "formula": "\\begin{align*} I V _ { A } = - \\varepsilon \\int _ { 0 } ^ { t } \\int _ { \\mathbb { T } ^ { d } } \\left ( \\partial ^ { \\alpha } \\mu ^ { n + 1 } \\right ) ( \\mu ^ { n } + \\bar { m } ) \\sum _ { i = 1 } ^ { d } \\sum _ { j = 1 } ^ { d } \\left [ \\left ( \\Theta _ { p _ { i } p _ { j } } ( \\tau , x , \\mu ^ { n } , D w ^ { n } ) \\right ) \\left ( \\partial ^ { \\alpha } \\partial _ { x _ { i } x _ { j } } ^ { 2 } w ^ { n } \\right ) \\right ] \\ d x d \\tau , \\end{align*}"}
-{"id": "1323.png", "formula": "\\begin{align*} \\varPsi _ { ( a , b ] } ( p ) = \\begin{cases} \\displaystyle { \\frac { 1 } { b - a } \\int _ a ^ b \\varPsi ( s ) \\ ; d s } & p \\in ( a , b ] , \\\\ \\varPsi ( p ) & . \\end{cases} \\end{align*}"}
-{"id": "1846.png", "formula": "\\begin{align*} v ^ { \\rm e v e n } ( n , x ) = \\sqrt { \\frac { 2 } { L } } \\cos ( k _ n x ) , \\end{align*}"}
-{"id": "4447.png", "formula": "\\begin{align*} \\varepsilon _ { G } ( k ) = a _ 0 \\binom { d + k } { d } + a _ 1 \\binom { d + k - 1 } { d } + \\cdots + a _ i \\binom { d + k - i } { d } + \\cdots . \\end{align*}"}
-{"id": "8607.png", "formula": "\\begin{align*} u _ \\Lambda ( x ) v _ \\Lambda ( x ) - \\frac { \\omega } { \\Lambda - 1 } = O \\left ( \\frac { 1 } { \\Lambda } \\right ) e ^ { - \\left ( \\sqrt { 2 } \\sqrt { 1 - 4 c _ 0 ^ 2 } + O \\left ( \\frac { 1 } { \\sqrt { \\Lambda } } \\right ) \\right ) | x | } , \\end{align*}"}
-{"id": "1750.png", "formula": "\\begin{align*} \\begin{aligned} \\langle D ^ { l } e ^ { \\frac { \\alpha } { p } M _ { s , t } } , D ^ { l } e ^ { \\frac { \\alpha } { p } M _ { s , t } } \\rangle _ { \\mathcal { H } ^ { \\otimes l } } & = \\left ( \\frac { \\alpha } { p } \\right ) ^ { 2 l } e ^ { \\frac { 2 \\alpha } { p } M _ { s , t } } ( \\langle D M _ { s , t } , D M _ { s , t } \\rangle _ { \\mathcal { H } } ) ^ { l } \\\\ & = \\left ( \\frac { \\alpha } { p } \\right ) ^ { 2 l } e ^ { \\frac { 2 \\alpha } { p } M _ { s , t } } ( t - s ) ^ { 2 l H } , \\end{aligned} \\end{align*}"}
-{"id": "8131.png", "formula": "\\begin{align*} v \\ : \\ : \\ : \\ : \\ : v ^ 2 = u S ( u ) , \\ : \\ : \\ : \\ : \\ : \\Delta ( v ) = ( R ' R ) ^ { - 1 } v \\otimes v , \\ : \\ : \\ : \\ : \\ : S ( v ) = v . \\end{align*}"}
-{"id": "7273.png", "formula": "\\begin{align*} - \\mu \\phi + L \\phi + \\frac { 1 } { 2 } \\left ( \\phi ^ { 2 } - \\widehat { \\phi ^ 2 } ( 0 ) \\right ) = 0 , \\end{align*}"}
-{"id": "6765.png", "formula": "\\begin{align*} \\phi ( x s ^ { - 1 } ) = \\phi ( x ) \\tau ( s ) ^ { - 1 } . \\end{align*}"}
-{"id": "491.png", "formula": "\\begin{align*} \\bar \\AA ( \\pi _ 1 ( U ) ) = \\pi _ 2 \\left ( z _ 1 ^ { n ( d - 1 ) } + \\cdots + z _ n ^ { n ( d - 1 ) } \\right ) . \\end{align*}"}
-{"id": "7155.png", "formula": "\\begin{align*} p _ { 1 } = \\lim _ { \\varepsilon \\rightarrow 0 } \\frac { \\log r _ { 1 } \\left ( \\varepsilon \\right ) } { \\log \\varepsilon } \\leq q _ { 1 } = \\lim _ { \\varepsilon \\rightarrow \\infty } \\frac { \\log r _ { 1 } \\left ( \\varepsilon \\right ) } { \\log \\varepsilon } . \\end{align*}"}
-{"id": "2356.png", "formula": "\\begin{align*} \\varphi _ + \\mathfrak { p } & = \\{ \\mathfrak { p } _ + \\setminus \\{ n + 1 \\} , \\ ; ( \\mathfrak { p } _ i ) _ { i \\in I } , \\ ; \\mathfrak { p } _ - \\} \\\\ \\varphi _ + ^ { - 1 } \\mathfrak { q } & = \\{ \\mathfrak { q } _ + + \\{ n + 1 \\} , ( \\mathfrak { q } _ j ) _ { j \\in J } , \\mathfrak { q } _ - + \\} \\\\ a _ { c _ { n + 1 } } ^ + \\lfloor V \\rfloor _ \\mathfrak { q } & = \\lfloor W \\rfloor _ { \\varphi _ + ^ { - 1 } \\mathfrak { q } } . \\end{align*}"}
-{"id": "3554.png", "formula": "\\begin{align*} \\omega _ { 0 } ( x , y , z , u ) : = \\sum _ { l = - \\infty } ^ { \\infty } \\sum _ { \\ ; \\ ; \\ ; m , n , k \\geq \\max \\left \\{ 0 , - l \\right \\} } c ( n , m , k , l ) x ^ { n } y ^ { m } z ^ { k } u ^ { l } , \\end{align*}"}
-{"id": "5680.png", "formula": "\\begin{align*} \\{ \\partial _ C ^ \\infty f _ 1 ( x _ 0 ) \\} \\cap \\{ - \\partial _ C ^ \\infty f _ 2 ( x _ 0 ) \\} = \\{ \\mathbf { 0 } \\} \\ , , \\end{align*}"}
-{"id": "3198.png", "formula": "\\begin{align*} \\sigma ( U ) = \\{ e ^ { i \\arccos ( 1 - 2 / N ) } , e ^ { - i \\arccos ( 1 - 2 / N ) } , 1 , - 1 \\} . \\end{align*}"}
-{"id": "1678.png", "formula": "\\begin{align*} [ [ X , Y ] , \\alpha _ A ( Z ) ] + [ [ Y , Z ] , \\alpha _ A ( X ) ] + [ [ Z , X ] , \\alpha _ A ( Y ) ] = 0 . \\end{align*}"}
-{"id": "3656.png", "formula": "\\begin{align*} \\mathrm { g r } ^ { ( t ) } _ { 4 A } : \\langle a , b \\mid a ^ 2 = b ^ 2 = ( a b ) ^ 2 = 1 \\rangle \\rightarrow P ( \\mathcal { F } ^ { ( t ) } _ { 4 A } ) \\end{align*}"}
-{"id": "7667.png", "formula": "\\begin{align*} \\int \\limits _ { \\mathbb R ^ d } \\int \\limits _ { \\mathbb T ^ d } \\beta ( \\xi - q ) \\mu ( \\xi , q ) \\ , d \\xi d q = - \\int \\limits _ { \\mathbb R ^ d } \\int \\limits _ { \\mathbb T ^ d } \\beta ( \\xi - q ) \\mu ( q , \\xi ) \\ d \\xi \\ d q . \\end{align*}"}
-{"id": "7326.png", "formula": "\\begin{align*} \\begin{aligned} a ( ( u _ 1 , v _ 1 ) , ( u _ 2 , v _ 2 ) ) = { } & \\int _ { \\Omega } \\rho u _ 1 u _ 2 + ( \\nabla u _ 1 ) ^ { \\top } K ( x ) \\nabla u _ 2 + ( \\nabla u _ 1 ) ^ { \\top } a ( x ) \\nabla u _ 2 \\ , d x \\\\ & { } + { } \\int _ { \\Omega } \\rho v _ 1 v _ 2 + ( \\nabla v _ 1 ) ^ { \\top } K ( x ) \\nabla v _ 2 + ( \\nabla v _ 1 ) ^ { \\top } b ( x ) \\nabla v _ 2 \\ , d x . \\end{aligned} \\end{align*}"}
-{"id": "1667.png", "formula": "\\begin{align*} C _ { n } ( d ) = \\ell ^ { ( k - 2 ) n } B _ { t } ( \\ell ^ { 2 n } m , d ) . \\end{align*}"}
-{"id": "9892.png", "formula": "\\begin{align*} E [ a ( z ) ^ { n } ] & = \\frac { 1 } { 2 } \\int _ { - 1 } ^ { 1 } z ^ { 2 n } d z = \\frac { 1 } { 2 n + 1 } \\neq 0 \\end{align*}"}
-{"id": "5635.png", "formula": "\\begin{align*} 2 & > \\frac { 8 } { 2 7 } ( \\frac { 1 } { b _ { 2 3 } + x _ 2 + x _ 3 } + \\frac { 1 } { b _ { 1 3 } + x _ 1 + x _ 3 } + \\frac { 1 } { b _ { 1 2 } + x _ 1 + x _ 2 } ) \\\\ & \\ge \\frac { 8 } { 2 7 } \\frac { ( 1 + 1 + 1 ) ^ 2 } { b _ { 2 3 } + x _ 2 + x _ 3 + b _ { 1 3 } + x _ 1 + x _ 3 + b _ { 1 2 } + x _ 1 + x _ 2 } \\\\ & = \\frac { 8 } { 3 ( 1 + x _ 1 + x _ 2 + x _ 3 ) } \\ge \\frac { 8 } { 3 ( 1 + \\frac { 1 } { 3 } ) } = 2 , \\end{align*}"}
-{"id": "1724.png", "formula": "\\begin{align*} \\lVert \\omega \\rVert = \\sum _ { n = 1 } ^ { \\infty } 2 ^ { - n } \\max _ { 0 \\leq t \\leq n } \\left | \\omega ( t ) \\right | , \\end{align*}"}
-{"id": "8081.png", "formula": "\\begin{align*} a _ n = e ^ { - 1 } M ( n + 1 , 2 , 1 ) . \\end{align*}"}
-{"id": "8236.png", "formula": "\\begin{align*} L _ 1 = F \\left ( ( 4 9 \\cdot 5 t + 6 \\cdot 5 ^ 4 t ^ 2 + 5 ^ 6 t ^ 3 ) + \\rho ( - 5 ^ 3 t - 5 ^ 5 t ^ 2 ) \\right ) . \\end{align*}"}
-{"id": "8688.png", "formula": "\\begin{align*} c _ { h , i , j } ( \\lambda , \\nu ) = \\frac { 2 ^ { - i } \\Gamma ( \\frac { \\nu } { 2 } - j ) } { i ! j ! \\Gamma ( \\frac { p } { 2 } + i ) \\Gamma ( \\frac { \\nu } { 2 } - \\lfloor \\frac { k } { 2 } \\rfloor ) } . \\end{align*}"}
-{"id": "7974.png", "formula": "\\begin{align*} \\lvert O _ { 2 n } ^ { + } ( q ) \\rvert = \\frac { 1 } { \\gcd ( 4 , q ^ n - 1 ) } q ^ { n ( n - 1 ) } ( q ^ n - 1 ) \\prod _ { i = 1 } ^ { n - 1 } ( q ^ { 2 i } - 1 ) \\ , , \\end{align*}"}
-{"id": "8628.png", "formula": "\\begin{align*} p = 2 c _ 0 \\frac { u ^ 2 + z ^ 2 } { u ^ 4 + c _ 0 ^ 2 } - 2 c _ 0 + \\varepsilon h _ 1 ( u , z , \\varepsilon ) , \\ \\ q = \\varepsilon h _ 2 ( u , z , \\varepsilon ) , \\ \\ ( u , z ) \\in \\mathcal { K } , \\end{align*}"}
-{"id": "6218.png", "formula": "\\begin{align*} \\int _ { \\mathcal { P } } \\frac { ( v + t ) ^ { - 1 } } { \\sqrt { \\det ( v + t ) } } \\mu ( d v ) = \\frac { 1 } { 2 ^ { 1 + \\d n } } \\frac { t ^ { - 1 } } { \\sqrt { \\det t } } . \\end{align*}"}
-{"id": "9375.png", "formula": "\\begin{align*} & \\Phi _ { k } ( G _ k ) = \\langle x ^ { 2 ^ k } , [ y , x , \\overset { 2 ^ k - 1 } { \\ldots } , x ] , [ y , x , \\overset { 2 ^ k } { \\ldots } , x ] , [ y , x , \\overset { 2 ^ k - 3 } { \\ldots } , x , y ] \\rangle \\Phi _ { k + 1 } ( G _ k ) \\\\ & \\Phi _ k ( G _ k ) / \\Phi _ { k + 1 } ( G _ k ) \\cong C _ 2 ^ { \\ , 4 } . \\end{align*}"}
-{"id": "3668.png", "formula": "\\begin{align*} & \\widehat { Y } = \\sum _ { i \\in B } w _ i y _ i \\qquad \\begin{cases} \\sum _ { i \\in B } w _ i x _ i = X & X \\\\ \\sum _ { i \\in B } w _ i x _ i = \\widehat { X } ( S ) & X \\end{cases} \\end{align*}"}
-{"id": "7153.png", "formula": "\\begin{align*} u _ { n } \\left ( t , x \\right ) = \\int _ { 0 } ^ { t } \\left [ L ^ { \\nu } u _ { n } \\left ( r , x \\right ) - \\lambda u _ { n } \\left ( r , x \\right ) + f _ { n } \\left ( r , x \\right ) \\right ] d r , \\left ( t , x \\right ) \\in \\left [ 0 , T \\right ] \\times \\mathbf { R } ^ { d } . \\end{align*}"}
-{"id": "796.png", "formula": "\\begin{align*} P _ { q ^ d , P ^ \\alpha } = \\{ D \\in \\mathbb { F } _ { q ^ d } [ T ] \\mod P ^ \\alpha , D \\equiv 1 \\mod P \\} . \\end{align*}"}
-{"id": "8532.png", "formula": "\\begin{align*} \\| \\partial _ { \\alpha } ^ m ( D _ t b _ { \\alpha } ) \\partial _ { \\alpha } ^ { k - m } \\tilde { \\theta } \\| _ { L ^ 2 } \\leq \\| D _ t b _ { \\alpha } \\| _ { H ^ m } \\| \\partial _ { \\alpha } ^ { k - m } \\tilde { \\theta } \\| _ { H ^ m } \\end{align*}"}
-{"id": "9669.png", "formula": "\\begin{align*} F _ { \\tau , \\sigma } \\oplus F _ { \\mathbb T , \\sigma } + F _ { \\tau , \\mathbb T \\setminus \\sigma } \\oplus F _ { \\mathbb T , \\mathbb T \\setminus \\sigma } = \\theta _ 0 h _ 0 | _ \\tau \\oplus h _ 0 \\in L ^ 2 ( \\tau ) \\oplus H ^ 2 . \\end{align*}"}
-{"id": "2182.png", "formula": "\\begin{align*} \\mathcal F ^ { - 1 } m _ \\ell ( \\mathbf x ) = ( \\mathcal F ^ { - 1 } m _ { \\ell , 1 } ) * h _ 1 ( \\mathbf x ) , \\end{align*}"}
-{"id": "430.png", "formula": "\\begin{align*} T ^ x _ k : = & \\inf \\{ t \\geq R _ { k - 1 } ^ x \\ : \\ X _ t ( x ) = ( \\gamma + \\eta ) t + A \\} \\\\ R ^ x _ k : = & \\inf \\{ t \\geq T _ { k } ^ x \\ : \\ X _ t ( x ) = \\gamma t \\} . \\end{align*}"}
-{"id": "8770.png", "formula": "\\begin{align*} S _ { a b } = - \\frac { n - 1 } { 4 } \\ , D g _ { a b } \\ , . \\end{align*}"}
-{"id": "5178.png", "formula": "\\begin{align*} s ( u ) = c ( 1 + a ( u ) ) \\exp ( \\int _ { u } ^ { 1 } \\frac { \\ell ( t ) } { t } d t ) . \\end{align*}"}
-{"id": "3176.png", "formula": "\\begin{align*} \\ker ( U \\mp 1 ) = \\mathcal { T } _ \\pm \\oplus \\mathcal { B } _ \\pm . \\end{align*}"}
-{"id": "4202.png", "formula": "\\begin{align*} \\Sigma _ R = \\partial D ( 0 , r _ n ) \\cup \\partial D ( 0 , R ) \\cup \\partial D ( q , r _ q ) \\cup \\left ( \\Delta _ 1 ^ { \\pm } \\setminus \\left ( D ( 0 , r _ n ) \\cup D ( q , r _ q ) \\right ) \\right ) \\cup [ q + r _ q , \\infty ) \\end{align*}"}
-{"id": "4947.png", "formula": "\\begin{align*} & \\left ( \\sum _ { j = i + 2 } ^ { d - 3 } ( - 1 ) ^ { j } v _ j z ^ { 3 n ^ { i + 2 } + n ^ { i + 1 } + \\cdots + n ^ 2 + n } \\right ) + ( - 1 ) ^ { d } ( x z ^ { 3 n ^ { i + 2 } + n ^ { i + 1 } + \\cdots + n ^ 2 + n } ) \\\\ & = - z ^ { 3 n ^ { i + 2 } + n ^ { i + 1 } + \\cdots + n ^ 2 } f _ 3 + \\left ( \\sum _ { j = 1 } ^ { i + 1 } ( - 1 ) ^ { j + 1 } v _ j z ^ { 3 n ^ { i + 2 } + n ^ { i + 1 } + \\cdots + n ^ 2 + n } \\right ) + y ^ n z ^ { 3 n ^ { i + 2 } + n ^ { i + 1 } + \\cdots + n ^ 2 } \\in I _ n . \\end{align*}"}
-{"id": "7478.png", "formula": "\\begin{gather*} \\psi _ s : = \\sqrt { \\xi } \\end{gather*}"}
-{"id": "7901.png", "formula": "\\begin{align*} p _ { n - 1 } ( x ) = \\sum _ { k = - \\hat { n } } ^ { \\hat { n } } \\phi ^ n _ k ( x ) f ( x _ k ) \\end{align*}"}
-{"id": "158.png", "formula": "\\begin{align*} X _ { C } = \\sum _ { i = 1 } ^ N v ^ i ( { \\bf x } ) \\frac { \\partial } { \\partial x ^ i } + \\sum _ { i , s = 1 } ^ N \\frac { \\partial v ^ i } { \\partial x ^ s } ( { \\bf x } ) y ^ s \\frac { \\partial } { \\partial y ^ i } , \\end{align*}"}
-{"id": "6116.png", "formula": "\\begin{align*} s _ \\ell = i _ \\ell + ( j _ \\ell + 1 ) \\mu _ 1 + ( k _ \\ell + 1 ) \\mu _ 2 - d \\geq 0 . \\end{align*}"}
-{"id": "7053.png", "formula": "\\begin{align*} D _ 2 ^ { - 1 } - D _ 1 ^ { - 1 } = D _ 2 ^ { - 1 } ( D _ 1 - D _ 2 ) D _ 1 ^ { - 1 } , \\end{align*}"}
-{"id": "7764.png", "formula": "\\begin{align*} \\| A \\| _ { \\nu } ^ D = \\max \\{ | \\langle A , B \\rangle | : B \\in \\mathcal { M } _ { n \\times n } , \\| B \\| _ { \\nu } \\leq 1 \\} . \\end{align*}"}
-{"id": "5227.png", "formula": "\\begin{align*} x _ { 0 } = \\frac { t _ { n } x _ { 0 } } { t _ { n } } \\leq \\frac { U ( V ( t _ { n } x _ { 0 } ) + ) } { U ( V ( t _ { n } ) - ) } \\leq \\frac { U ( V ( t _ { n } x _ { 0 } ) + 1 ) } { U ( V ( t _ { n } ) - 1 ) } . \\end{align*}"}
-{"id": "3803.png", "formula": "\\begin{align*} \\frac { N ^ { 3 / 4 - i ( t - v ) } } { q ^ { 1 / 2 } } \\sum _ { m = 1 } ^ \\infty \\frac { \\lambda _ f ( m ) } { m ^ { 1 / 4 } } e \\left ( \\frac { \\bar { a } m } { q } \\right ) \\int _ 0 ^ \\infty U ( y ) y ^ { - i ( t + v ) } e \\left ( - \\frac { N x y } { q Q } \\pm \\frac { 2 \\sqrt { m N y } } { q } \\right ) \\mathrm { d } y . \\end{align*}"}
-{"id": "6948.png", "formula": "\\begin{align*} \\mathbb K : = \\{ \\tau ( \\mathbf R - \\mathbf I ) : \\tau > 0 , \\ \\mathbf R \\in S O ( N ) \\} \\end{align*}"}
-{"id": "7309.png", "formula": "\\begin{gather*} [ E _ i , F _ i ] = H _ i , [ H _ i , E _ i ] = 2 E _ i , [ H _ i , F _ i ] = - 2 F _ i , \\\\ [ E _ i , [ E _ i , E _ { i \\pm 1 } ] ] = 0 , \\ [ F _ i , [ F _ i , F _ { i \\pm 1 } ] ] = 0 , \\\\ [ E _ i , F _ j ] = 0 \\mbox { f o r } i \\neq j , \\\\ [ E _ i , E _ { i + 2 } ] = 0 \\end{gather*}"}
-{"id": "6775.png", "formula": "\\begin{align*} ( \\Xi _ { \\sigma ^ + } ) _ { i , j } = \\frac { 1 } { \\sigma _ i + \\sigma _ j } \\end{align*}"}
-{"id": "7440.png", "formula": "\\begin{align*} \\displaystyle Q _ p = \\frac { 1 } { \\lvert W ' \\rvert } \\sum _ { w \\in W } w \\cdot T _ p . \\end{align*}"}
-{"id": "4992.png", "formula": "\\begin{align*} q \\circ \\varphi ( x , y ) & = x ^ 2 - x ( x - y ) + ( x - y ) ^ 2 \\\\ & = x ^ 2 - x ^ 2 + x y + x ^ 2 - 2 x y + y ^ 2 \\\\ & = x ^ 2 - x y + y ^ 2 \\\\ & = q ( x , y ) . \\end{align*}"}
-{"id": "9660.png", "formula": "\\begin{align*} ( ( T ^ * ) ^ n & ( x _ 1 \\oplus h _ 1 ) , ( T ^ * ) ^ n ( x _ 2 \\oplus h _ 2 ) ) \\\\ & = ( h _ 1 , h _ 2 ) + ( P _ { H ^ 2 _ - } \\overline \\chi ^ n Y _ 0 ^ * x _ 1 , P _ { H ^ 2 _ - } \\overline \\chi ^ n Y _ 0 ^ * x _ 2 ) + ( ( T _ 0 ^ * ) ^ n x _ 1 , ( T _ 0 ^ * ) ^ n x _ 2 ) . \\end{align*}"}
-{"id": "2927.png", "formula": "\\begin{align*} k _ a ( z ) = \\sum _ { k = 0 } ^ \\infty \\frac { \\overline { a } ^ k z ^ k } { \\omega _ k } . \\end{align*}"}
-{"id": "5060.png", "formula": "\\begin{align*} \\Psi _ j h & = \\sum _ { k \\in \\mathbb { L } _ j } \\langle \\Psi _ j h , e _ { j , k } \\rangle e _ { j , k } = \\sum _ { k \\in \\mathbb { L } _ j } \\langle h , v _ { j , k } \\rangle e _ { j , k } = \\sum _ { k \\in \\mathbb { L } _ j } \\langle h , c _ j u _ { j , k } \\rangle e _ { j , k } \\\\ & = c _ j \\sum _ { k \\in \\mathbb { L } _ j } \\langle h , u _ { j , k } \\rangle e _ { j , k } = c _ j \\sum _ { k \\in \\mathbb { L } _ j } \\langle h , A _ j ^ * e _ { j , k } \\rangle e _ { j , k } = c _ j A _ j h , ~ \\forall j \\in \\mathbb { J } . \\end{align*}"}
-{"id": "2249.png", "formula": "\\begin{align*} | \\alpha _ x \\rangle = \\frac { 1 } { \\sqrt { C _ 1 ^ 2 + \\pi ^ 2 C _ 2 ^ 2 } } ( C _ 1 ( x ) | \\delta _ x \\rangle + h ( x ) A ( x ) ) & & \\langle \\alpha _ x ' | = \\frac { 1 } { \\sqrt { C _ 1 ^ 2 + \\pi ^ 2 C _ 2 ^ 2 } } ( C _ 1 ( x ) \\langle \\delta _ x | + g ( x ) A ' ( x ) ) \\end{align*}"}
-{"id": "102.png", "formula": "\\begin{align*} B ( 0 ) ^ { \\ell } B _ 1 & = B ( 0 ) ^ { 2 N - ( 2 N - \\ell ) } B _ 1 = B ( 0 ) ^ { 2 N } B ( 0 ) ^ { - ( 2 N - \\ell ) } B _ 1 \\\\ & = B ( 0 ) ^ { - ( 2 N - \\ell ) } B _ 1 , \\end{align*}"}
-{"id": "5633.png", "formula": "\\begin{align*} - \\frac { \\alpha } { 2 } \\int _ 0 ^ { + \\infty } | u | ^ 2 v _ t d x = \\frac 1 2 \\mathcal { E } _ 2 ( t ) + \\frac { d } { d t } \\int _ 0 ^ { + \\infty } \\Big ( \\frac { \\alpha } { 4 \\gamma } v _ x ^ 2 - \\frac { \\alpha } { 1 2 \\gamma } v ^ 3 \\Big ) d x + \\frac { \\alpha } { 2 \\gamma } v _ x ( 0 , t ) v _ t ( 0 , t ) \\end{align*}"}
-{"id": "4876.png", "formula": "\\begin{align*} N _ \\infty ( U ) = S ^ { 2 n - 1 } \\infty \\setminus \\overline { \\left ( \\mathbb { C } ^ n \\setminus U \\right ) } , \\end{align*}"}
-{"id": "2079.png", "formula": "\\begin{align*} \\frac { { \\delta ( \\rho - { \\rho _ { \\rm t x } } ) } } { \\rho } = \\sum \\limits _ { m = 1 } ^ \\infty { \\frac { { { J _ n } ( { \\lambda _ { n m } } { \\rho _ { \\rm t x } } ) } } { { { N _ { n m } } } } } { J _ n } ( { \\lambda _ { n m } } \\rho ) , \\end{align*}"}
-{"id": "8384.png", "formula": "\\begin{align*} u _ { t t } + a | z _ { \\alpha } | \\bold { n } \\frac { u _ { \\alpha } } { z _ { \\alpha } } = g , \\end{align*}"}
-{"id": "6044.png", "formula": "\\begin{align*} g _ k ( s ) \\leq f _ k ( s ) \\leq f _ k \\left ( k - \\tfrac { k - 2 } { 9 } \\right ) = \\tfrac { 1 6 0 } { 8 1 } k ^ 2 - \\tfrac { 1 0 0 } { 8 1 } k - \\tfrac { 3 5 } { 8 1 } , \\end{align*}"}
-{"id": "6500.png", "formula": "\\begin{align*} g ( K , y ) : = \\inf \\{ \\lambda > 0 : \\ y \\in \\lambda K \\} \\ , , y \\in \\R ^ n \\setminus \\{ 0 \\} \\ , , \\end{align*}"}
-{"id": "6996.png", "formula": "\\begin{align*} { \\textsl { \\footnotesize R } } _ j = \\textstyle { \\left [ \\sin \\left ( \\frac { 2 \\pi j } { n } \\right ) , \\frac { 1 } { 2 } \\sin \\left ( \\frac { 4 \\pi j } { n } \\right ) , \\ldots , \\frac { 1 } { n - 1 } \\sin \\left ( \\frac { 2 ( n - 1 ) \\pi j } { n } \\right ) , 0 \\right ] ^ { \\tt T } , } \\end{align*}"}
-{"id": "2441.png", "formula": "\\begin{align*} \\widetilde { \\varphi _ { \\ell } ^ { H _ 1 } } ( x _ 0 , x _ 1 ) = ( x _ 0 ^ \\ell + x _ 1 ^ \\ell ) + \\sum _ { 0 < j < \\ell , j \\equiv 0 \\pmod { 4 } } \\frac { ( - 1 ) ^ { \\ell / 4 } \\binom { \\ell } { j } } { ( ( - 1 ) ^ { \\ell / 4 } + 2 ^ { \\ell / 2 - 2 } ) } x _ 0 ^ { \\ell - j } x _ 1 ^ j \\end{align*}"}
-{"id": "291.png", "formula": "\\begin{align*} s _ \\alpha ( x ) = \\sum _ { \\beta } m _ { \\alpha \\beta } \\cdot \\Psi ^ { \\beta _ 1 } ( x ) \\cdot \\Psi ^ { \\beta _ 2 } ( x ) \\cdots \\Psi ^ { \\beta _ \\ell } ( x ) . \\end{align*}"}
-{"id": "2905.png", "formula": "\\begin{align*} \\mu ( f _ \\Phi ) & = \\lim _ { n \\to \\infty } \\frac { \\mu \\big ( E _ { F _ n } \\big ) } { \\abs { F _ n } } \\end{align*}"}
-{"id": "2304.png", "formula": "\\begin{align*} ( \\alpha _ x | \\beta _ y ) & = \\delta ( x - y ) \\\\ ( \\alpha _ x | \\beta _ { \\pm \\i \\pi } ) & = 0 & ( \\alpha _ { \\pm \\i \\pi } | \\beta _ x ) & = 0 \\\\ ( \\alpha _ { \\pm \\ i \\pi } | \\beta _ { \\pm \\i \\pi } ) & = 1 & ( \\alpha _ { \\pm \\ i \\pi } | \\beta _ { \\mp \\i \\pi } ) & = 0 \\end{align*}"}
-{"id": "7933.png", "formula": "\\begin{align*} x _ a = x _ { a , 1 } \\otimes \\dots \\otimes x _ { a , m } a \\in [ n ] . \\end{align*}"}
-{"id": "4161.png", "formula": "\\begin{align*} F _ n ( z ) & = L _ { \\alpha } ( n ^ 3 f ( z ) ) D _ 2 ^ n ( z ) , \\\\ E _ n ( z ) & = N ( z ) \\begin{pmatrix} z ^ { \\frac { 2 \\beta } { 3 } } & 0 & 0 \\\\ 0 & z ^ { - \\frac { \\beta } { 3 } } & 0 \\\\ 0 & 0 & z ^ { - \\frac { \\beta } { 3 } } \\end{pmatrix} F _ n ^ { - 1 } ( z ) T _ { \\alpha } , \\\\ E _ { i n } ^ { ( 1 ) } ( z ) & = \\frac { \\sqrt { 3 } } { 2 \\pi } n ^ { 2 \\beta } \\left ( \\frac { f ( z ) } { z } \\right ) ^ { \\frac { 2 \\beta } { 3 } } E _ n ( z ) , \\\\ E _ { o u t } ^ { ( 1 ) } ( z ) & = \\mathbb { I } . \\end{align*}"}
-{"id": "908.png", "formula": "\\begin{align*} Y _ m = ( - 1 ) ^ { s _ m } \\cdot q ^ { w _ m } \\cdot y _ { e _ 1 } ^ { \\gamma ( 1 ) } \\cdots y _ { e _ n } ^ { \\gamma ( n ) } \\end{align*}"}
-{"id": "5108.png", "formula": "\\begin{align*} \\| x + y \\| ^ 4 + \\| x - y \\| ^ 4 & \\leq 2 ( \\| x \\| ^ 4 + \\| y \\| ^ 4 ) + 1 2 \\| x \\| ^ 2 y \\| ^ 2 \\\\ & = 2 ( ( \\| x \\| ^ 2 + \\| y \\| ^ 2 ) ^ 2 + 4 \\| x \\| ^ 2 \\| y \\| ^ 2 ) , ~ \\forall x , y \\in \\mathcal { X } . \\end{align*}"}
-{"id": "7516.png", "formula": "\\begin{align*} D _ q = \\frac { \\log _ 2 ( p _ 1 ^ q + p _ 2 ^ q + p _ 3 ^ q ) } { 1 - q } . \\end{align*}"}
-{"id": "3083.png", "formula": "\\begin{align*} | f | \\le t , a . e . \\ { \\rm o n } \\ G = \\R ^ n \\setminus \\cup K _ l . \\end{align*}"}
-{"id": "10062.png", "formula": "\\begin{align*} S ( \\Phi , r , y _ n , X _ n ) = A ( \\Phi ) X _ n ^ i y _ n ^ j \\rho ^ k , \\end{align*}"}
-{"id": "661.png", "formula": "\\begin{align*} ( \\mathcal M \\wr \\mathbb E _ { \\mathcal G } ) \\times _ { \\mathbb E _ { \\mathcal G } } \\mathbb G _ { \\mathcal G } = \\mathcal M \\wr \\mathbb G _ { \\mathcal G } \\xrightarrow [ \\cong ] { \\Phi } ( \\mathcal M \\rtimes \\mathcal G ) \\wr \\mathbb E _ { \\mathcal G } \\to \\mathcal M \\wr \\mathbb E _ { \\mathcal G } \\ . \\end{align*}"}
-{"id": "3112.png", "formula": "\\begin{align*} \\chi ( \\beta ^ { 1 + j } + \\ldots + \\beta ^ { i + j } + K _ S ) = { } & \\frac { ( \\beta ^ { 1 + j } + \\ldots + \\beta ^ { i + j } + K _ S ) \\cdot ( \\beta ^ { 1 + j } + \\ldots + \\beta ^ { i + j } ) } { 2 } \\\\ & + \\chi ( \\O _ S ) \\\\ = { } & \\sum _ { j < k \\leq l \\leq i + j } \\beta ^ k \\beta ^ l + \\chi ( \\O _ S ) \\ , . \\end{align*}"}
-{"id": "9438.png", "formula": "\\begin{align*} - a _ { j _ 0 } d _ { j _ 0 } E _ 2 ( d _ { j _ 0 } z ) = \\sum _ { 1 \\leq j \\leq D ( N ) - 1 } a _ j d _ j E _ 2 ( d _ j z ) - a _ { j _ 0 } d _ { j _ 0 } E _ 2 ( d _ { j _ 0 } z ) . \\end{align*}"}
-{"id": "594.png", "formula": "\\begin{align*} \\min _ { t _ 1 \\le s \\le t _ 2 } ( Z ^ x _ s - Z ^ x _ t ) = Z ^ x _ { t _ 1 } - Z ^ x _ t + \\min _ { t _ 1 \\le s \\le t _ 2 } ( Z ^ x _ s - Z ^ x _ { t _ 1 } ) , \\end{align*}"}
-{"id": "5097.png", "formula": "\\begin{align*} \\langle U x , U x \\rangle = \\left \\langle \\sum _ { j \\in \\mathbb { J } } c _ j x _ j , \\sum _ { k \\in \\mathbb { J } } c _ k x _ k \\right \\rangle = \\sum _ { j \\in \\mathbb { J } } c _ j c _ j ^ * = \\langle x , x \\rangle , \\end{align*}"}
-{"id": "7820.png", "formula": "\\begin{align*} \\lim _ n [ \\langle T x _ n , S x _ n \\rangle \\xi _ n , \\xi _ n ] = \\lambda \\| T \\| \\| S \\| \\end{align*}"}
-{"id": "9312.png", "formula": "\\begin{align*} x _ { i j } ~ { \\rm a r e ~ i n t e g e r s } , ~ i = 1 , \\dots , m , ~ j = 1 , \\dots , n , \\end{align*}"}
-{"id": "4788.png", "formula": "\\begin{align*} \\rho = \\psi \\psi ^ \\dagger + [ W , W ^ \\dagger ] \\ , . \\end{align*}"}
-{"id": "4735.png", "formula": "\\begin{align*} ( \\mathbf { J } \\mathbf { T } ) _ { i , j } = \\begin{cases} A T _ { n - 1 - j } & i = 0 , j = 0 , \\dots , n - 1 , \\\\ B T _ { i - 1 - j } & i \\not = 0 , j = 0 , \\dots , n - 1 . \\end{cases} \\end{align*}"}
-{"id": "5490.png", "formula": "\\begin{align*} L _ b ^ * \\mu _ b \\equiv \\frac { 1 } { 2 } \\Delta \\mu _ b - b . \\nabla \\mu _ b - d i v ( b ) \\mu _ b = 0 . \\end{align*}"}
-{"id": "7747.png", "formula": "\\begin{align*} \\Big | [ \\langle x , y \\rangle \\xi , \\xi ] \\Big | \\| \\xi \\otimes \\xi \\| = \\| y \\| \\| \\xi \\otimes \\xi \\| , \\end{align*}"}
-{"id": "979.png", "formula": "\\begin{align*} W = ( \\{ x _ { 1 , 1 } , x _ { 1 , 2 } \\} , \\{ x _ { 1 , 2 } , x _ { 1 , 3 } \\} , \\ldots , \\{ x _ { 1 , n } , x _ { 1 , 1 } \\} , \\{ x _ { 2 , 1 } , x _ { 2 , 2 } \\} , \\{ x _ { 2 , 2 } , x _ { 2 , 3 } \\} , \\ldots , \\{ x _ { 2 , m } , x _ { 2 , 1 } \\} ) \\end{align*}"}
-{"id": "8070.png", "formula": "\\begin{align*} \\begin{aligned} \\phi _ { \\zeta } ^ + ( \\omega _ 1 ) = \\zeta ^ { \\frac { 5 } { 2 } } \\omega _ 1 , \\phi _ { \\zeta } ^ + ( \\omega _ 2 ) = \\zeta ^ { \\frac { 3 } { 2 } } \\omega _ 2 , \\phi _ { \\zeta } ^ + ( \\omega _ 3 ) = \\zeta ^ { \\frac { 1 } { 2 } } \\omega _ 3 , \\phi _ { \\zeta } ^ + ( \\omega _ 4 ) = \\zeta ^ { - \\frac { 1 } { 2 } } \\omega _ 4 . \\end{aligned} \\end{align*}"}
-{"id": "6969.png", "formula": "\\begin{align*} \\sigma ^ * \\theta _ 0 = - \\omega , \\end{align*}"}
-{"id": "748.png", "formula": "\\begin{align*} \\mathbb { E } _ { r } \\ ! \\left [ r ^ { k } \\right ] = \\frac { \\Gamma ( m n + k ) } { \\Gamma ( m n ) } . \\end{align*}"}
-{"id": "2864.png", "formula": "\\begin{align*} { \\sigma _ 1 ( a ) } { \\sigma _ i ( b ) } = { \\sigma _ 1 ( 1 ) } { \\sigma _ i ( a b ) } . \\end{align*}"}
-{"id": "3670.png", "formula": "\\begin{align*} E ( y _ i | x _ i , i \\in U ) = \\mu ( x _ i ) = x _ i ^ { \\top } \\beta \\end{align*}"}
-{"id": "4274.png", "formula": "\\begin{align*} \\binom { n } { 2 } - n ^ { 5 / 3 + \\delta } \\leq \\textup { s a t } ( n , C _ 4 , K _ 4 ) \\leq \\binom { n - 2 } { 2 } . \\end{align*}"}
-{"id": "3913.png", "formula": "\\begin{align*} r _ 1 = r _ 2 , \\ r _ 2 = r _ { 1 2 } , \\ r _ { 1 2 } = r _ { 1 3 } , \\ r _ { 1 3 } = r _ { 2 3 } . \\end{align*}"}
-{"id": "8054.png", "formula": "\\begin{align*} \\sum _ { \\{ g \\} \\in \\Xi } \\frac { | \\{ g \\} | } { | D _ n | } \\chi _ \\pi ( g ) \\overline { \\chi _ \\rho ( g ) } = \\begin{cases} 1 & \\pi \\cong \\rho , \\\\ 0 & , \\end{cases} \\end{align*}"}
-{"id": "1595.png", "formula": "\\begin{align*} | u ^ o _ { n , k } - x ^ o _ { n , k } | \\leq \\frac { \\left ( \\frac { 1 } { 2 } \\right ) ^ k C } { k _ n ^ 2 } + \\frac { 2 D } { n ^ 2 } = o ( 1 ) . \\end{align*}"}
-{"id": "2888.png", "formula": "\\begin{align*} J _ - ^ { A } = - i \\sum _ { \\mu \\in A } \\gamma _ \\mu \\partial _ \\mu ~ , J _ + ^ { A } = - i \\sum _ { \\mu \\in A } \\gamma _ \\mu x _ \\mu ~ , J _ 0 ^ { A } = \\tfrac { | A | } { 2 } + \\sum _ { \\mu \\in A } x _ \\mu \\partial _ \\mu , \\end{align*}"}
-{"id": "4843.png", "formula": "\\begin{align*} D ( w ) = \\partial ( w ) , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , D ( s w ) = w ' , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , D ( w ' ) = 0 . \\end{align*}"}
-{"id": "5017.png", "formula": "\\begin{align*} \\begin{aligned} a - 2 \\epsilon & \\leq \\{ \\log _ M n _ 1 \\} \\leq a - \\epsilon \\\\ a + \\epsilon & \\leq \\{ \\log _ M n _ 2 \\} \\leq a + 2 \\epsilon . \\end{aligned} \\end{align*}"}
-{"id": "9134.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } K _ { M } ( \\rho _ { 1 } ^ { ( N ) } ( t ) , \\mathbf { b } ) = W _ { M } ( \\rho _ { 1 } ^ { ( N ) } ( t ) , \\mathbf { b } ) , \\end{align*}"}
-{"id": "5171.png", "formula": "\\begin{align*} \\frac { s ( \\lambda u ) - s ( u ) } { s ( \\kappa u ) - s ( u ) } = \\frac { \\log \\mu } { \\log \\kappa } . \\end{align*}"}
-{"id": "6425.png", "formula": "\\begin{align*} R _ { k } = B _ { k } ^ { - 1 } , v _ { k } = - w _ { k } \\frac { \\beta _ { k } } { \\alpha _ { k + 1 } } , \\quad \\eta _ { k } = \\frac { 1 } { \\alpha _ { k + 1 } } . \\end{align*}"}
-{"id": "56.png", "formula": "\\begin{align*} C = \\int _ 0 ^ 1 \\rho _ 0 ( x ) \\ , d x + 2 g ( J _ b ) \\int _ 0 ^ 1 k ( y ) ( 1 - y ) \\ , d y \\ , . \\end{align*}"}
-{"id": "4514.png", "formula": "\\begin{align*} \\mathcal { H } ^ s ( F ) = \\lim _ { \\delta \\to 0 } \\mathcal { H } ^ s _ { \\delta } ( F ) . \\end{align*}"}
-{"id": "2360.png", "formula": "\\begin{align*} \\Phi & = | 0 > , & \\Phi _ n & = a ^ { + n } \\Phi = a ^ { + n } | 0 > . \\end{align*}"}
-{"id": "8964.png", "formula": "\\begin{align*} E _ n = \\{ \\mbox { T h e r e e x i s t s } \\ \\alpha \\in \\mathbb { R } ^ { p _ n } : \\ , \\alpha ' x _ i ( T _ n ) \\geq 0 , \\ \\mbox { i f } \\ y _ i = 1 ; \\ \\ \\alpha ' x _ i ( T _ n ) \\leq 0 , \\ \\mbox { i f } \\ y _ i = 0 \\} . \\end{align*}"}
-{"id": "2066.png", "formula": "\\begin{align*} & D \\frac { { { \\partial ^ 2 } { C _ z } ( z , t | { z _ { \\rm t x } } , { t _ 0 } ) } } { { \\partial { z ^ 2 } } } - v \\frac { { \\partial { C _ z } ( z , t | { z _ { \\rm t x } } , { t _ 0 } ) } } { { \\partial z } } - { k _ { d } } { C _ z } ( z , t | { z _ { \\rm t x } } , { t _ 0 } ) + \\delta ( z - z _ { \\rm t x } ) \\delta ( t - { t _ 0 } ) \\\\ & = \\frac { { \\partial { C _ z } ( z , t | { z _ { \\rm t x } } , { t _ 0 } ) } } { { \\partial t } } . \\end{align*}"}
-{"id": "9155.png", "formula": "\\begin{align*} \\rho ^ { ( + ) \\left ( N \\right ) } ( \\mathbf { x } ^ { \\left ( + \\right ) } ( t _ { i } ) , t _ { i } ) = \\rho ^ { ( - ) \\left ( N \\right ) } ( \\mathbf { x } ^ { ( + ) } ( t _ { i } ) , t _ { i } ) , \\end{align*}"}
-{"id": "9428.png", "formula": "\\begin{align*} | ( \\log _ { e } | F ( z ) | ^ 2 ) | & \\leq | ( \\log _ { e } | ( z - z _ 0 ) | ^ { 2 4 \\nu _ \\tau ^ { ( N ) } ( f ) } ) | + | ( \\log _ { e } | F _ 0 ( z ) | ^ 2 ) | \\\\ & \\leq | ( \\log _ { e } | ( z - z _ 0 ) | ^ { 2 4 \\nu _ \\tau ^ { ( N ) } ( f ) } ) | + M _ 2 \\ ; \\ ; ( M _ 2 ) \\\\ & = | 2 4 \\nu _ \\tau ^ { ( N ) } ( f ) \\log _ { e } \\varepsilon | + M _ 2 . \\end{align*}"}
-{"id": "865.png", "formula": "\\begin{gather*} F ( x ) : = \\begin{cases} K - g ( x ) & \\mbox { i f } x \\in D , \\\\ \\emptyset & \\mbox { o t h e r w i s e } . \\end{cases} \\end{gather*}"}
-{"id": "7976.png", "formula": "\\begin{align*} | G _ 2 ( q ) | = q ^ 6 ( q ^ 6 - 1 ) ( q ^ 2 - 1 ) = q ^ 6 ( q - 1 ) ^ 2 ( q + 1 ) ^ 2 ( q ^ 2 - q + 1 ) ( q ^ 2 + q + 1 ) \\ , , \\end{align*}"}
-{"id": "9431.png", "formula": "\\begin{align*} b ( n ) = c \\sigma _ 1 ( n ) . \\end{align*}"}
-{"id": "6654.png", "formula": "\\begin{align*} M _ { 2 , 2 } = \\sum _ { N / 1 0 0 \\leq q \\leq \\lambda } \\sum _ { a \\in \\mathbb Z _ q ^ { \\times } } \\lvert C ^ { a / q } _ { \\tau } f \\rvert . \\end{align*}"}
-{"id": "3801.png", "formula": "\\begin{align*} g ( q , x ) & = 1 + h ( q , x ) , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; h ( q , x ) = O \\left ( \\frac { 1 } { q Q } \\left ( \\frac { q } { Q } + | x | \\right ) ^ A \\right ) , \\\\ g ( q , x ) & \\ll | x | ^ { - A } \\end{align*}"}
-{"id": "9397.png", "formula": "\\begin{align*} T _ { u } W _ { 0 } ^ { s , p } ( \\Omega ) = T _ { u } M _ { i } \\oplus s p a n \\lbrace u _ { + } , u _ { - } \\rbrace \\end{align*}"}
-{"id": "3593.png", "formula": "\\begin{align*} \\langle u , v \\rangle & = \\langle u _ 1 + u _ 0 + u _ { \\frac { 1 } { 4 } } + u _ { \\frac { 1 } { 3 2 } } , v _ 1 + v _ 0 + v _ { \\frac { 1 } { 4 } } + v _ { \\frac { 1 } { 3 2 } } \\rangle \\\\ & = \\langle u _ 1 , v _ 1 \\rangle + \\langle u _ 0 , v _ 0 \\rangle + \\langle u _ { \\frac { 1 } { 4 } } , v _ { \\frac { 1 } { 4 } } \\rangle + \\langle u _ { \\frac { 1 } { 3 2 } } , v _ { \\frac { 1 } { 3 2 } } \\rangle . \\end{align*}"}
-{"id": "2391.png", "formula": "\\begin{align*} \\mathfrak { P } & = \\{ \\mathfrak { p } _ 1 , \\cdots , \\mathfrak { p } _ m \\} & \\mathfrak { p } _ i & = ( r _ i , s _ i ) , & r _ i & > s _ i . \\end{align*}"}
-{"id": "8502.png", "formula": "\\begin{align*} \\ddot { z } _ j ( t ) = - \\frac { \\lambda i x ' ( t ) } { 4 \\pi x ( t ) ^ 2 } + \\bar { F } _ \\zeta ( z _ j ( t ) , t ) \\dot { z } _ j ( t ) + \\bar { F } _ t ( z _ j , t ) . \\end{align*}"}
-{"id": "2551.png", "formula": "\\begin{align*} \\{ ( Y _ t ) _ { 0 \\leq t \\leq T } ; \\dot { \\mathbf P } ^ { ( g , T ) } _ \\mu \\} \\overset { f . d . d . } { = } \\{ ( W _ t ) _ { 0 \\leq t \\leq T } ; \\mathbb N _ \\mu ^ { W _ T ( g ) } \\} . \\end{align*}"}
-{"id": "8711.png", "formula": "\\begin{align*} \\rho ( ( i _ 1 , \\ldots , i _ d ) ) = P ( \\tau _ { ( N _ 1 , \\ldots , N _ d ) } < \\tau _ { - \\infty } | Z _ 0 = ( i _ 1 , \\ldots , i _ d ) ) , \\end{align*}"}
-{"id": "4386.png", "formula": "\\begin{align*} \\int _ { B _ r ( z ) } [ Z ] \\wedge \\frac { \\omega _ o ^ { n - 1 } } { ( n - 1 ) ! } = O ( r ^ { 2 n - 2 } ) \\end{align*}"}
-{"id": "4847.png", "formula": "\\begin{align*} \\Gamma _ { n } = \\ker \\big ( H _ { n } ( \\L ( W _ { \\leq n } ) ) \\overset { j _ n } { \\longrightarrow } W _ n \\big ) , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\forall n \\geq 2 . \\end{align*}"}
-{"id": "8049.png", "formula": "\\begin{align*} \\omega _ 1 ( y ) & = 1 , \\omega _ 1 ( x ) = - 1 , \\\\ \\omega _ 2 ( y ) & = - 1 , \\omega _ 2 ( x ) = 1 , \\\\ \\omega _ 3 ( y ) & = - 1 , \\omega _ 3 ( x ) = - 1 . \\end{align*}"}
-{"id": "9469.png", "formula": "\\begin{align*} D ^ { s , m } ( \\mathbb { R } ^ { N } ) : = \\{ u \\in L ^ { m ^ { * } _ { s } } ( \\mathbb { R } ^ { N } ) ; [ u ] _ { s , m } < \\infty \\} \\end{align*}"}
-{"id": "4836.png", "formula": "\\begin{align*} \\chi = \\underbrace { 1 \\oplus \\cdots \\oplus 1 } _ { { k } \\choose { 2 } } \\oplus \\underbrace { \\lambda _ { n - k } \\oplus \\cdots \\oplus \\lambda _ { n - k } } _ { k } \\end{align*}"}
-{"id": "94.png", "formula": "\\begin{gather*} \\bigl ( \\textbf { x } ^ { ( 1 ) } \\bigr ) _ { k _ 2 } \\cdot \\bigl ( \\textbf { x } ^ { ( 1 ) } \\bigr ) _ { h _ 2 } < 0 \\ , , 0 < \\bigl | \\bigl ( \\textbf { x } ^ { ( 1 ) } \\bigr ) _ { k _ 2 } \\bigr | = \\min _ { k = 1 , \\dots , N \\ , , \\bigl ( \\textbf { x } ^ { ( 1 ) } \\bigr ) _ { k } \\ne 0 } | \\bigl ( \\textbf { x } ^ { ( 1 ) } \\bigr ) _ { k } | \\ , . \\end{gather*}"}
-{"id": "6932.png", "formula": "\\begin{align*} \\mathbf { u } \\in L ^ s ( 0 , T ; L ^ r ) , \\ \\ \\frac 2 s + \\frac 3 r = 1 , \\ 3 < r \\leq \\infty , \\end{align*}"}
-{"id": "9352.png", "formula": "\\begin{align*} [ y , x ] ^ 2 & = [ y _ 1 , y _ 0 ] = [ x ^ { - 1 } y x , y ] \\equiv [ x ^ { - 1 } , y ] [ x , y ] \\gamma _ 3 ( G _ k ) \\\\ & \\equiv [ x , y ] ^ { 2 ^ { k + 1 } - 1 } [ x , y ] \\gamma _ 3 ( G _ k ) \\\\ & = 1 \\gamma _ 3 ( G _ k ) , \\end{align*}"}
-{"id": "8565.png", "formula": "\\begin{align*} T _ \\eta ( p ) = \\partial _ \\eta ( p ) - \\sum _ { \\alpha > 0 } c _ \\alpha ( \\alpha , \\eta ) \\Delta _ \\alpha ( p ) , \\end{align*}"}
-{"id": "7013.png", "formula": "\\begin{align*} \\c ( g ) = \\xi _ 1 + \\cdots + \\xi _ n . \\end{align*}"}
-{"id": "5263.png", "formula": "\\begin{align*} r ( x ) = \\int _ { x _ 1 } ^ { x } b ( t ) d t + r ( x _ 1 ) . \\end{align*}"}
-{"id": "9523.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\frac { q ^ n } { 1 - q ^ n } = \\sum _ { n = 1 } ^ \\infty n q ^ n \\left ( q ^ { n + 1 } \\right ) _ { N - 1 } - \\sum _ { n = 1 } ^ \\infty n q ^ { n + N } \\left ( q ^ { n + 1 } \\right ) _ { N - 1 } , \\end{align*}"}
-{"id": "7179.png", "formula": "\\begin{gather*} P _ { x } ^ { \\pm , 0 } = \\left [ X ^ { t } ( x ) \\right ] ^ { - 1 } P _ { \\chi ^ { t } ( x ) } ^ { \\pm , 0 } X ^ { t } ( x ) . \\end{gather*}"}
-{"id": "7723.png", "formula": "\\begin{align*} \\Psi _ { i } \\colon S _ { | { U } _ { i } } \\stackrel { \\approx } { \\longrightarrow } { U } _ { i } \\times \\mathbb { C } ^ k , \\psi _ { i } : = \\mathrm { p r } _ { 2 } \\circ \\Psi _ { i } . \\end{align*}"}
-{"id": "1229.png", "formula": "\\begin{align*} \\int _ { | \\eta | \\ge 1 0 } \\partial _ \\zeta G ( \\zeta , \\eta ) d \\eta = & \\int _ { | \\eta | \\ge 1 0 } e ^ { - 3 i \\Phi ( \\zeta , \\eta ) } f ( \\eta ) i ( \\partial _ \\zeta \\Phi ) g ( \\eta - \\zeta ) d \\eta \\\\ & - \\int _ { | \\eta | \\ge 1 0 } e ^ { - 3 i \\Phi ( \\zeta , \\eta ) } f ( \\eta ) g ' ( \\eta - \\zeta ) d \\eta \\end{align*}"}
-{"id": "844.png", "formula": "\\begin{align*} J ( f ) = \\{ f ^ * \\in E ^ * : f ^ * ( f ) = \\Vert f \\Vert \\Vert f ^ * \\Vert _ { * } , \\ \\Vert f \\Vert = \\Vert f ^ * \\Vert _ { * } \\} f \\in E \\end{align*}"}
-{"id": "4271.png", "formula": "\\begin{align*} \\textup { s a t } ( n , K _ 2 , F ) = \\textup { s a t } ( n , F ) . \\end{align*}"}
-{"id": "6242.png", "formula": "\\begin{align*} ( C , \\psi ) : = \\underset { \\beta _ { 1 } , \\cdots , \\beta _ { m } } { M _ { m } ( \\C ) } \\oplus \\underset { \\beta } { \\overset { q } { ( B , \\phi ) } } . \\end{align*}"}
-{"id": "1609.png", "formula": "\\begin{align*} f ( y ) - f ( x ) & = \\frac { 1 } { S ' ( y ) } - \\mu ( y ) y m ( ( 0 , y ) ) - \\left ( \\frac { 1 } { S ' ( x ) } - \\mu ( x ) x m ( ( 0 , x ) ) \\right ) \\\\ & = \\int _ x ^ y \\mu ( t ) t m ' ( t ) d t - \\mu ( y ) y m ( ( 0 , y ) ) + \\mu ( x ) x m ( ( 0 , x ) ) \\\\ & = \\int _ x ^ y ( \\mu ( t ) t - \\mu ( y ) y ) m ' ( t ) d t + ( \\mu ( x ) x - \\mu ( y ) y ) m ( ( 0 , x ) ) . \\end{align*}"}
-{"id": "1415.png", "formula": "\\begin{align*} P \\bigg ( \\bigg | A ^ 3 - \\big ( N ( \\frac { \\epsilon } { 4 } , \\frac { \\delta } { 4 } ) \\big ) ^ { - 1 } \\sum _ { i = 1 } ^ { N ( \\frac { \\epsilon } { 4 } , \\frac { \\delta } { 4 } ) } A ^ 0 _ i - Z ^ { k + 1 } _ t ( \\gamma ) \\bigg | \\geq \\epsilon \\bigg ) \\leq \\delta \\ \\ \\ \\textrm { a s d e s i r e d . } \\end{align*}"}
-{"id": "1615.png", "formula": "\\begin{align*} \\frac { Z _ T ^ { b } } { T } = \\frac { x } { T } + \\frac { 1 } { T } \\int _ 0 ^ T \\mu \\left ( X _ t ^ { Z ^ { b } } \\right ) X _ t ^ { Z ^ { b } } \\ , d t + \\frac { 1 } { T } \\int _ 0 ^ T \\sigma \\left ( X _ t ^ { Z ^ { b } } \\right ) \\ , d B _ { t } - \\frac { X _ T ^ { Z ^ { b } } } { T } . \\end{align*}"}
-{"id": "5808.png", "formula": "\\begin{align*} S ^ 2 \\backslash \\Omega = S ^ 2 \\backslash \\bigcup _ { r \\in ( 1 , 2 ) } \\gamma _ r = \\bigcap _ { r \\in ( 1 , 2 ) } ( S ^ 2 \\backslash \\gamma _ r ) = \\bigcap _ { r \\in ( 1 , 2 ) } ( A _ r \\cup B _ r ) = \\Bigl ( \\bigcap _ { r \\in ( 1 , 2 ) } A _ r \\Bigr ) \\cup \\Bigl ( \\bigcap _ { r \\in ( 1 , 2 ) } B _ r \\Bigr ) . \\end{align*}"}
-{"id": "8956.png", "formula": "\\begin{align*} \\Psi _ X ( X ( t _ j ) - m ( t _ j ) ) ( s ) = \\mathbb { E } [ ( X ( s ) - m ( s ) ) ( X ( t _ j ) - m ( t _ j ) ) ] = K ( s , t _ j ) . \\end{align*}"}
-{"id": "7834.png", "formula": "\\begin{align*} \\| x \\| ^ 2 \\| y \\| ^ 2 = \\lim _ n | [ \\langle x _ n , x \\rangle \\langle x , y \\rangle \\langle y , x _ n \\rangle \\xi _ n , \\xi _ n ] | \\leq \\| x \\| \\| \\langle x , y \\rangle \\| \\| y \\| \\leq \\| x \\| ^ 2 \\| y \\| ^ 2 , \\end{align*}"}
-{"id": "6284.png", "formula": "\\begin{align*} & R _ k ( \\rho , \\epsilon , \\Lambda , n ) = \\Omega \\left ( \\min \\{ \\Lambda ^ 2 \\sqrt { \\frac { k } { n } } + \\Lambda \\left ( \\frac { \\log k } { \\rho n } + \\frac { k ^ 2 } { \\rho n ^ 2 } \\right ) , \\Lambda ^ 2 \\} \\right ) \\end{align*}"}
-{"id": "9274.png", "formula": "\\begin{align*} \\sup \\limits _ { B _ { \\frac { 3 \\delta R } { 4 } } ( \\bar x ^ \\prime ) \\cap \\partial \\Omega } \\sup \\limits _ { | \\tau | = 1 , \\tau \\cdot \\nu = 0 } | u _ { \\tau \\tau } | \\le \\epsilon \\sup \\limits _ { B _ { \\frac { 1 3 \\delta R } { 1 6 } } ( \\bar x ^ \\prime ) \\cap \\Omega } | D ^ 2 u | + C _ \\epsilon ( 1 + \\frac { 1 } { ( \\delta R / 1 6 ) ^ 2 } ) , \\end{align*}"}
-{"id": "1579.png", "formula": "\\begin{align*} \\Psi _ { \\widetilde { F } } ^ { - 1 } ( z ^ o _ n , u ^ o _ n ) = \\Phi _ F ( z , x ) - k _ n + o ( 1 ) . \\end{align*}"}
-{"id": "1571.png", "formula": "\\begin{align*} \\left | \\frac { \\partial \\psi _ w ^ { - 1 } } { \\partial z } ( z ) \\right | = O ( \\sqrt { { w _ m } } / z ^ 2 ) . \\end{align*}"}
-{"id": "5390.png", "formula": "\\begin{align*} | | v _ { n + 2 \\ , m } ^ 1 | | ^ 2 = - \\frac { \\overline { d _ { n + 2 \\ , m } } } { a _ { n + 1 \\ , m - 3 } } | | v _ { n + 1 \\ , m - 3 } ^ 1 | | ^ 2 = \\frac { \\overline { d _ { n + 2 \\ , m } } } { a _ { n + 1 \\ , m - 3 } } \\cdot \\frac { c _ { n + 1 \\ , m - 3 } } { \\overline { b _ { n m } } } | | v _ { n m } ^ 1 | | ^ 2 . \\end{align*}"}
-{"id": "9498.png", "formula": "\\begin{align*} D ^ { s , m } ( \\mathbb { R } ^ { N } ) = \\{ u \\in L ^ { m _ { s } ^ * } ( \\mathbb { R } ^ N ) ; \\Vert u \\Vert _ { s , m } < \\infty \\} \\ \\mbox { w i t h } \\ m _ { s } ^ * = \\frac { N m } { N - s m } . \\end{align*}"}
-{"id": "278.png", "formula": "\\begin{align*} T ( P ) ( x ) = \\sum _ { j = l } ^ m a _ j P ( x - j \\lambda ) , \\end{align*}"}
-{"id": "1036.png", "formula": "\\begin{align*} I _ { k , n } = \\cup _ { j = 1 } ^ k I ( y _ j , F _ n ( x _ j ) ) , \\ J _ { k , n , j } = ( z _ j , z _ { j + 1 } ) ( \\ 2 \\pi ) , \\ 1 \\leq j \\leq k , \\end{align*}"}
-{"id": "7455.png", "formula": "\\begin{align*} \\displaystyle \\int _ X \\sum _ { \\Pi \\in p ^ { - 1 } ( \\chi ) } L _ \\Pi ( f ) \\mu _ \\chi ( \\Pi ) \\overline { \\mu } ( \\chi ) = 0 \\end{align*}"}
-{"id": "7756.png", "formula": "\\begin{align*} \\Big | [ \\langle x , z \\rangle y , y ] \\Big | = \\| x \\| \\| z \\| . \\end{align*}"}
-{"id": "4327.png", "formula": "\\begin{align*} - ( \\Delta u , Q _ K v ) _ K = \\ ; & ( \\nabla \\Pi _ K u , \\nabla v ) _ K \\\\ & + \\langle ( \\nabla u - \\nabla \\Pi _ K u ) \\cdot n , Q _ K v \\rangle _ { \\partial K } \\\\ & + \\langle \\nabla \\Pi _ K u \\cdot n , Q _ F v \\rangle _ { \\partial K } . \\end{align*}"}
-{"id": "8119.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } d _ { n , r } ( t ) \\ , \\frac { z ^ n } { n ! } \\ = \\ \\frac { ( 1 - t ) e ^ { ( r - 1 ) t z } } { e ^ { r t z } - t e ^ { r z } } . \\end{align*}"}
-{"id": "8627.png", "formula": "\\begin{align*} z _ 0 = u _ 0 ' > 0 , \\end{align*}"}
-{"id": "9556.png", "formula": "\\begin{align*} F ( a , b ; t ) = \\frac { 1 - b } { 1 - t } F \\left ( \\frac { a t } { b } , t ; b \\right ) . \\end{align*}"}
-{"id": "6605.png", "formula": "\\begin{align*} P _ { F _ { n _ j } } ( \\varphi , \\varepsilon _ j ) = P _ { F _ { n _ j } } ( \\varphi ) \\cdot \\frac { P _ { F _ { n _ j } } ( \\varphi , \\varepsilon _ j ) } { P _ { F _ { n _ j } } ( \\varphi ) } = P _ { F _ { n _ j } } ( \\varphi ) \\cdot \\frac { \\overline { A } _ { n _ j } ( \\varepsilon _ j ) \\overline { C } _ { n _ j } ( \\varepsilon _ j ) } { A _ { n _ j } C _ { n _ j } } , \\end{align*}"}
-{"id": "2343.png", "formula": "\\begin{align*} \\kappa : X _ 1 \\times X _ 2 & \\to \\mathcal { M } _ + ( Y _ 1 \\times Y _ 2 ) \\\\ \\kappa ( x _ 1 , x _ 2 ; d y _ 1 , d y _ 2 ) & = \\kappa _ 1 ( x _ 1 , d y _ 1 ) \\kappa _ 2 ( x _ 2 , d y _ 2 ) \\end{align*}"}
-{"id": "9972.png", "formula": "\\begin{align*} \\begin{cases} \\dot { u } ( t ) = A u ( t ) , \\\\ u ( 0 ) = x , \\end{cases} \\end{align*}"}
-{"id": "231.png", "formula": "\\begin{align*} u \\in L ^ { 2 ^ * } _ { l o c } ( B _ 1 ) , \\ \\ 2 ^ * = \\frac { 2 ( N + 2 ) } { N } \\end{align*}"}
-{"id": "2594.png", "formula": "\\begin{align*} J ' _ G ( t , r , \\xi ) : = \\gamma _ 0 ( \\gamma _ 0 - 1 ) t \\int _ 0 ^ 1 \\big ( \\mathbf 1 _ { \\gamma ( \\cdot ) = \\gamma _ 0 } \\kappa \\cdot ( \\phi \\eta _ { u t } ) ^ { \\gamma _ 0 - 1 } G \\big ( r u ^ { \\frac { 1 } { \\gamma _ 0 - 1 } } \\big ) ^ { \\gamma _ 0 - 1 } \\big ) ( \\xi _ { ( 1 - u ) t } ) d u \\end{align*}"}
-{"id": "4778.png", "formula": "\\begin{align*} 2 ^ { m ^ { r ( k - 1 ) + 1 } } = o ( 2 ^ { \\frac { m } 3 \\log _ 2 n } ) = o ( n ^ { m / 3 } ) . \\end{align*}"}
-{"id": "8122.png", "formula": "\\begin{align*} h ( \\Delta , t ) \\ = \\ \\sum _ { i = 0 } ^ n \\ , f _ { i - 1 } ( \\Delta ) \\ , t ^ i ( 1 - t ) ^ { n - i } , \\end{align*}"}
-{"id": "9094.png", "formula": "\\begin{align*} w _ m ( x ) = \\dfrac { v _ R ( x ) ^ p } { u _ m ( x ) ^ { p - 1 } } \\in \\widetilde { W } \\ ! \\prescript { s , p } { } ( B _ R ) . \\end{align*}"}
-{"id": "4294.png", "formula": "\\begin{align*} \\frac { 1 } { r - 1 } \\binom { s - 2 } { r - 1 } n - 2 \\binom { s - 2 } { r - 1 } . \\end{align*}"}
-{"id": "1935.png", "formula": "\\begin{align*} F _ 6 ( x ) \\equiv x ^ 2 \\sum _ { j = 1 } ^ \\infty \\frac { A _ j ^ 2 } { Z _ j ^ 6 } = \\frac { 1 } { 2 } \\left ( \\frac { 1 } { x } + \\frac { 1 } { 3 } \\right ) . \\end{align*}"}
-{"id": "8679.png", "formula": "\\begin{align*} L & = \\{ ( - \\rho + q - 1 - 2 i , - \\rho ' + q - 1 - 2 j ) : i , j \\in \\Z , 0 \\leq j \\leq i \\} \\intertext { a n d f o r $ m = 0 $ } L & = \\{ ( - \\rho + q - 1 - 2 i , \\pm ( 1 + 2 j ) ) : i , j \\in \\Z , 0 \\leq j \\leq i \\} , \\intertext { a s w e l l a s } S _ 1 & = \\{ ( - \\rho + q - 1 - 2 i , \\rho ' + 2 j ) : i , j \\in \\Z , 0 \\leq j \\leq i \\} , \\\\ S _ 2 & = \\{ ( - \\rho + q - 1 - 2 i , \\rho ' + 2 i ) : i \\in \\Z _ { \\geq 0 } \\} , \\\\ S _ 3 & = \\{ ( - \\rho + q - 1 , \\rho ' ) \\} . \\end{align*}"}
-{"id": "5385.png", "formula": "\\begin{align*} c _ { n + 1 \\ , m - 3 } | | v _ { n m } ^ k | | ^ 2 = - \\overline { b _ { n m } } \\frac { n } { n + 1 - k } | | v _ { n + 1 \\ , m - 3 } ^ { k } | | ^ 2 . \\end{align*}"}
-{"id": "1627.png", "formula": "\\begin{align*} d X _ t = \\mu X _ t ( 1 - \\gamma X _ { t } ) d t + \\sigma X _ t d W _ t , X _ 0 = x \\in \\mathbb { R } _ + , \\end{align*}"}
-{"id": "6400.png", "formula": "\\begin{align*} \\| x - x _ { k } \\| _ { A } ^ { 2 } = \\sum _ { j = k } ^ { k + d - 1 } \\gamma _ { j } \\| r _ { j } \\| ^ { 2 } + \\| x - x _ { k + d } \\| _ { A } ^ { 2 } , \\end{align*}"}
-{"id": "4277.png", "formula": "\\begin{align*} ( 1 - o ( 1 ) ) n ^ 3 \\leq \\textup { s a t } ( n , C _ 6 , K _ 5 ) \\leq 6 \\binom { n - 3 } { 3 } . \\end{align*}"}
-{"id": "9046.png", "formula": "\\begin{align*} & F _ { \\mu _ n } ( x ) = \\int _ { \\mathbb { S } ^ 1 } e ( \\langle x \\cdot \\lambda \\rangle ) G _ n ( d \\lambda ) & F _ { \\mu } ( x ) = \\int _ { \\mathbb { S } ^ 1 } e ( \\langle x \\cdot \\lambda \\rangle ) G ( d \\lambda ) . \\end{align*}"}
-{"id": "1821.png", "formula": "\\begin{align*} V ( x , t ) = 2 \\xi \\ , \\delta ( x ) \\ , \\theta ( - t ) , \\end{align*}"}
-{"id": "6821.png", "formula": "\\begin{align*} M _ 1 ( t ) = \\begin{pmatrix} \\cos ( \\mu t ) & - \\sin ( \\mu t ) \\\\ \\sin ( \\mu t ) & \\cos ( \\mu t ) \\end{pmatrix} \\quad { \\rm a n d } M _ 2 ( t ) = \\begin{pmatrix} \\cos ( \\theta t ) & - \\sin ( \\theta t ) \\\\ \\sin ( \\theta t ) & \\cos ( \\theta t ) \\end{pmatrix} \\end{align*}"}
-{"id": "4815.png", "formula": "\\begin{align*} Q ^ { a , b } _ g ( u ) = \\frac { E _ g ( u ) } { a ( \\int _ { M } u ^ { \\frac { 2 n } { n - 2 } } d v ) ^ { \\frac { n - 2 } { n } } + b ( \\int _ { \\partial M } u ^ { \\frac { 2 ( n - 1 ) } { n - 2 } } d \\sigma ) ^ { \\frac { n - 2 } { n - 1 } } } . \\end{align*}"}
-{"id": "10022.png", "formula": "\\begin{align*} D _ { x ^ { \\ast } } = \\sum x ^ { * } ( a _ { n } ) n ^ { - s } \\ , . \\end{align*}"}
-{"id": "6032.png", "formula": "\\begin{align*} \\overline { \\lambda } ( G ) = \\displaystyle \\sum _ { \\{ u , v \\} \\subseteq V ( G ) } \\lambda _ G ( u , v ) / \\tbinom { n } { 2 } . \\end{align*}"}
-{"id": "6406.png", "formula": "\\begin{align*} \\| x - x _ { k } \\| _ { A } ^ { 2 } < \\sum _ { j = k } ^ { k + d - 1 } \\gamma _ { j } \\| r _ { j } \\| ^ { 2 } + \\frac { \\| r _ { k + d } \\| ^ { 2 } } { \\mu } \\frac { \\| r _ { k + d } \\| ^ { 2 } } { \\| p _ { k + d } \\| ^ { 2 } } . \\end{align*}"}
-{"id": "4426.png", "formula": "\\begin{align*} & \\frac { 1 } p = \\frac { 1 - \\theta } s + \\frac { \\theta } 2 , \\\\ & q = \\theta r , \\end{align*}"}
-{"id": "1328.png", "formula": "\\begin{align*} \\underline { f } ( x _ J , x _ { J ^ c } ) = \\underline { \\bar { f } } ( x _ L ) = f ( x _ J ) . \\end{align*}"}
-{"id": "9072.png", "formula": "\\begin{align*} \\phi _ x ( y ) = \\sum _ { k \\in \\mathcal { K } } \\mu _ { f } ( I _ k ) ^ { 1 / 2 } b _ k ( x ) e ( \\langle R \\zeta ^ { k } , y \\rangle ) . \\end{align*}"}
-{"id": "571.png", "formula": "\\begin{align*} \\xi ^ s ( 1 + p x _ s ) \\xi ^ { a _ 1 } ( 1 + p a _ 2 ) - \\xi ^ s ( 1 + p x _ s ) \\xi ^ { a _ 1 } ( 1 + p b _ 2 ) . \\end{align*}"}
-{"id": "4858.png", "formula": "\\begin{align*} \\alpha ( w ^ * _ { ( j , r _ s ) } ) = \\sum _ { \\tau \\geq 1 } ^ { } \\lambda _ { \\tau } w ^ * _ { ( \\tau , r _ s ) } , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\lambda _ { \\tau } \\in \\Q . \\end{align*}"}
-{"id": "9859.png", "formula": "\\begin{align*} \\nabla v . { \\Delta _ A } v = - \\lambda \\frac { a } { { a + u } } \\left | { \\nabla v } \\right | _ { } ^ 2 - \\left ( { A \\nabla v } \\right ) . { \\left | { \\nabla v } \\right | ^ 2 } . \\end{align*}"}
-{"id": "0.png", "formula": "\\begin{align*} d Y _ t = - ( f ( V _ t , Z _ t ) - \\lambda ) d t + ( Z _ t ) ^ { t r } d W _ t , t \\geq 0 . \\end{align*}"}
-{"id": "2893.png", "formula": "\\begin{align*} J _ { 1 2 } \\mapsto \\widetilde { J } _ { 1 2 } & = e ^ { - B } S ^ { - 1 } \\left ( - i \\frac { \\partial } { \\partial \\theta _ 1 } + \\frac { 1 } { 2 } \\Sigma _ { 1 2 } \\right ) S e ^ { B } \\\\ & = - i \\frac { \\partial } { \\partial \\theta _ 1 } + \\frac { 1 } { 2 } \\Sigma _ { 1 2 } + ( - i ) \\left ( \\frac { - i } { 2 } \\Sigma _ { 1 2 } \\right ) = - i \\frac { \\partial } { \\partial \\theta _ 1 } \\end{align*}"}
-{"id": "252.png", "formula": "\\begin{align*} N \\leq 6 ( p _ 1 + u _ 1 + \\sum _ { i = 2 } ^ 4 p _ { j _ i } ) . \\end{align*}"}
-{"id": "2084.png", "formula": "\\begin{align*} \\tilde C _ z ( \\beta , t \\mid z _ { \\rm t x } , { t _ 0 } ) = \\frac { e ^ { ( - D { \\beta ^ 2 } - { k _ { d } } - j \\beta v ) ( t - { t _ 0 } ) } } { 2 \\pi } . \\end{align*}"}
-{"id": "3882.png", "formula": "\\begin{align*} M ( u ( t ) , v ( t ) ) & : = \\| u ( t ) \\| ^ 2 _ { L ^ 2 } + 2 \\| v ( t ) \\| ^ 2 _ { L ^ 2 } = M ( u _ 0 , v _ 0 ) , \\\\ E ( u ( t ) , v ( t ) ) & : = \\frac { 1 } { 2 } ( \\| \\nabla u ( t ) \\| ^ 2 _ { L ^ 2 } + \\kappa \\| \\nabla v ( t ) \\| ^ 2 _ { L ^ 2 } ) - \\emph { R e } ( \\langle v ( t ) , u ^ 2 ( t ) \\rangle ) = E ( u _ 0 , v _ 0 ) . \\end{align*}"}
-{"id": "498.png", "formula": "\\begin{align*} & \\rho ( Q ) ( \\xi _ u \\odot \\xi _ v ) = - 4 \\pi i \\eta _ { p } ( u ) ( v ) Q ( u , v ) , \\\\ & \\rho ( Q ) ( \\xi _ u \\otimes \\xi _ u ) = - 2 \\pi i \\mu _ 2 ( Q ) ( u ^ { \\otimes 4 } ) . \\end{align*}"}
-{"id": "2720.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\mathbb { P } \\Big ( \\frac { \\tau h ( \\theta ) } { 1 - \\tau ^ 2 } N ^ { - \\frac { 1 } { 2 } } \\big ( \\lambda _ { \\mathrm { m a x } } - N x ( \\theta ) \\big ) \\leq x \\Big ) = G _ m ( \\pi ; x ) . \\end{align*}"}
-{"id": "3513.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\sum _ { j = 0 } ^ M \\frac { 1 } { \\nu _ j + N } = \\gamma \\in ( 0 , \\infty ) . \\end{align*}"}
-{"id": "9238.png", "formula": "\\begin{align*} \\mathcal { G } [ u ] : = G ( \\cdot , u , D u ) = 0 , \\mbox { o n } \\ \\partial \\Omega , \\end{align*}"}
-{"id": "1740.png", "formula": "\\begin{align*} B _ { t } = \\int _ { 0 } ^ { t } K ( t , s ) d \\omega ( s ) , \\end{align*}"}
-{"id": "5413.png", "formula": "\\begin{align*} \\frac { \\tilde { c } } { 2 } & \\leq \\int _ { | x | \\geq R } | u _ n | ^ 2 \\ , d x = \\int _ { | x | \\geq R } \\frac { V ( x ) | u _ n | ^ 2 } { V ( x ) } \\ , d x \\leq \\frac { 1 } { \\inf _ { | x | \\geq R } V ( x ) } \\int _ { | x | \\geq R } V ( x ) | u _ n | ^ 2 \\ , d x \\\\ & \\leq \\frac { \\| u _ n \\| ^ 2 } { \\inf _ { | x | \\geq R } V ( x ) } \\leq \\frac { \\sup \\| u _ n \\| ^ 2 } { \\inf _ { | x | \\geq R } V ( x ) } . \\end{align*}"}
-{"id": "7694.png", "formula": "\\begin{align*} \\int \\limits _ U ( h \\cdot g ) \\circ \\varphi \\ , \\mathrm { d } \\mu = \\int \\limits _ { \\varphi ( U ) } h \\ , \\mathrm { d } \\lambda . \\end{align*}"}
-{"id": "2332.png", "formula": "\\begin{align*} \\Psi a ^ + _ x & = \\langle 0 | a ^ + _ x = 0 \\\\ \\langle \\mathfrak { m } | a _ x & = \\langle \\mathfrak { m } + \\ 1 _ x | \\\\ \\langle \\mathfrak { m } | a ^ + _ x & = \\sum _ { y \\in X } \\delta _ { x , y } \\langle \\mathfrak { m } - \\ 1 _ x | \\\\ \\langle \\mathfrak { m } | ( a ^ + ) ^ \\mathfrak { l } & = \\frac { \\mathfrak { m } ! } { ( \\mathfrak { m } \\mathfrak { l } ) ! } \\langle \\mathfrak { m } - \\mathfrak { l } | . \\end{align*}"}
-{"id": "9015.png", "formula": "\\begin{align*} f _ { \\{ a _ 1 \\} , a _ 1 } ( x ) = \\begin{cases} a _ 1 & a _ 1 = x , \\\\ x & \\end{cases} f _ { \\{ a _ 1 , a _ 2 \\} , a _ 1 } ( x _ 1 , x _ 2 ) = \\begin{cases} a _ 1 & \\{ a _ 1 , a _ 2 \\} = \\{ x _ 1 , x _ 2 \\} , \\\\ x _ 1 & \\end{cases} \\end{align*}"}
-{"id": "8675.png", "formula": "\\begin{align*} \\| f ( t ) \\| _ { W ^ { \\alpha , p } } : = \\| f ( t ) \\| _ { L ^ p } + [ f ( t ) ] _ { W ^ { \\alpha , p } } \\ , . \\end{align*}"}
-{"id": "7855.png", "formula": "\\begin{align*} | B _ { R + | z | } | - | B _ { R } | & = \\big ( ( 1 + | z | R ^ { - 1 } ) ^ r - 1 \\big ) | B _ R | \\\\ & \\le r | z | R ^ { - 1 } ( 1 + | z | R ^ { - 1 } ) ^ r | B _ R | \\\\ & \\le r | z | R ^ { - 1 } e ^ { r | z | R ^ { - 1 } } | B _ R | . \\end{align*}"}
-{"id": "9937.png", "formula": "\\begin{align*} \\Lambda _ k = \\frac { 4 \\pi ^ 2 } { T ^ 2 } \\left ( k + \\frac { \\beta } { 4 } \\right ) ^ 2 , k = 1 , 2 , \\ldots , \\end{align*}"}
-{"id": "5332.png", "formula": "\\begin{align*} X . v ^ k = - ( k - 1 ) v ^ { k - 1 } \\end{align*}"}
-{"id": "2834.png", "formula": "\\begin{align*} \\left ( \\Lambda \\Omega \\right ) \\omega ^ { n } = n ( \\Omega \\wedge \\omega ^ { n - 1 } ) \\ . \\end{align*}"}
-{"id": "4564.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } f ( z ) d \\nu _ g ( z ) = \\int _ F \\int _ F f ( u - g v ) d \\mu ( u ) d \\mu ( v ) , \\forall f \\in C _ 0 ( \\mathbb { R } ^ n ) , \\end{align*}"}
-{"id": "7354.png", "formula": "\\begin{align*} \\tag * { ( D ) } d ( v , v ' ) = d ( v , Y ) + d ( Y , Y ' ) + d ( Y ' , v ' ) \\end{align*}"}
-{"id": "3153.png", "formula": "\\begin{align*} Z ( \\bar x | \\bar t ) = W _ { } ( \\bar x , \\emptyset | \\bar t , \\emptyset ) . \\end{align*}"}
-{"id": "2113.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ d \\lambda _ i \\geq \\displaystyle \\limsup _ { t \\to \\infty } \\frac { 1 } { t } \\log | \\det \\Phi _ { \\omega } ( t _ 0 , t ) | . \\end{align*}"}
-{"id": "1388.png", "formula": "\\begin{align*} \\left ( \\Box _ n \\circ \\triangledown _ n \\right ) _ v ( \\lambda , \\mu ) = \\left ( \\Box _ n \\right ) _ v \\big ( \\lambda _ 1 ^ e , \\ldots , \\lambda _ n ^ e , \\mu ; C ( \\lambda ) \\big ) = ( \\lambda , \\mu ) \\ ; , \\end{align*}"}
-{"id": "6191.png", "formula": "\\begin{align*} \\tilde \\Omega = : \\Omega \\cap \\R ^ n \\times \\{ 0 \\} = \\bigcup _ { i = 1 } ^ k \\mathcal N ( u _ i ) , \\mathcal N ^ o ( u _ i ) \\cap \\mathcal N ^ o ( \\sum _ { j \\neq i } u _ j ) = \\emptyset \\ ; \\forall i = 1 , \\dots , k \\ ; , \\end{align*}"}
-{"id": "7911.png", "formula": "\\begin{align*} h = \\sqrt { \\left | \\frac { 2 4 \\bar { R } } { f ' ( b ) - f ' ( a ) } \\right | } \\end{align*}"}
-{"id": "2766.png", "formula": "\\begin{align*} \\int _ { \\mathcal { C } _ { < } } \\frac { d w } { 2 \\pi i } ( 2 w ) ^ { - \\frac { 3 } { 2 } } e ^ { \\frac { 1 } { 3 } ( w - \\kappa ) ^ 3 - u ( w - \\kappa ) } = \\phi _ { 1 } '' ( \\kappa ; u ) - u \\phi _ { 1 } ( \\kappa ; u ) , \\end{align*}"}
-{"id": "5835.png", "formula": "\\begin{align*} \\Psi ( u ) = i \\mu u - \\frac { \\sigma ^ 2 u ^ 2 } { 2 } \\end{align*}"}
-{"id": "593.png", "formula": "\\begin{align*} \\inf \\{ s \\in [ 0 , t ] : \\ , Z ^ x _ s + \\tilde { \\Lambda } ^ { x , \\infty } _ s \\le 0 \\} = t \\end{align*}"}
-{"id": "5237.png", "formula": "\\begin{align*} F ^ { [ n s ] } ( a _ { [ n s ] } x + b _ { [ n s ] } ) = \\left \\{ F ^ { n } ( a _ { [ n s ] } x + b _ { [ n s ] } ) \\right \\} ^ { [ n s ] / n } \\rightarrow \\varphi _ { \\alpha } ( x ) , \\end{align*}"}
-{"id": "3550.png", "formula": "\\begin{align*} c ( n ) : = \\frac { 1 } { \\Gamma ( \\frac { 1 } { 2 } ) ^ { 3 } } \\frac { \\Gamma ( n _ { 1 } + \\frac { 1 } { 2 } ) \\Gamma ( n _ { 2 } + \\frac { 1 } { 2 } ) \\Gamma ( n _ { 3 } + \\frac { 1 } { 2 } ) } { \\Pi _ { _ { i = 1 } } ^ { 3 } \\Gamma ( n _ { 4 } - n _ { i } + 1 ) \\cdot \\Pi _ { 1 \\leq j < k \\leq 3 } \\Gamma ( n _ { j } + n _ { k } - n _ { 4 } + 1 ) } . \\end{align*}"}
-{"id": "5652.png", "formula": "\\begin{align*} E _ h ( t ) = \\frac { 1 } { 2 \\pi \\rm i } \\int _ { \\Gamma _ { \\theta , \\delta } } e ^ { z t } z ^ { \\alpha - 1 } ( z ^ \\alpha + A _ h ) ^ { - 1 } \\d z \\quad \\mbox { a n d } \\bar E _ h ( t ) = \\frac { 1 } { 2 \\pi \\rm i } \\int _ { \\Gamma _ { \\theta , \\delta } } e ^ { z t } z ^ { - \\gamma } ( z ^ \\alpha + A _ h ) ^ { - 1 } \\d z . \\end{align*}"}
-{"id": "2085.png", "formula": "\\begin{align*} { C _ z } ( z , t | { z _ { \\rm t x } } , { t _ 0 } ) = \\\\ \\frac { 1 } { { { \\sqrt { ( 4 \\pi D ( t - { t _ 0 } ) ) } } } } { e ^ { \\frac { { - { { ( z - { z _ { \\rm t x } } - v ( t - { t _ 0 } ) ) } ^ 2 } } } { { 4 D ( t - { t _ 0 } ) } } - { k _ d } ( t - { t _ 0 } ) } } u ( t - t _ 0 ) . \\end{align*}"}
-{"id": "4092.png", "formula": "\\begin{align*} R _ { \\alpha } = T _ \\alpha \\Phi _ \\alpha \\widehat { L } _ \\alpha ^ { - 1 } = \\mathbb { I } + \\mathcal { O } ( z ^ { - 1 } ) \\end{align*}"}
-{"id": "4047.png", "formula": "\\begin{align*} \\sum _ r m _ r ^ { 3 / 2 } \\leq \\left ( \\sum _ r m _ r \\right ) ^ { 1 / 2 } \\left ( \\sum _ r m _ r ^ 2 \\right ) ^ { 1 / 2 } = O ( N ^ { 5 / 2 } \\log ^ { 1 / 2 } ( N ) ) . \\end{align*}"}
-{"id": "9663.png", "formula": "\\begin{align*} X X _ * ^ * J h = g h \\ h \\in L ^ 2 ( \\mathbb T ) \\ \\ L ^ 2 ( \\tau ) \\oplus L ^ 2 ( \\mathbb T ) = J L ^ 2 ( \\mathbb T ) \\oplus \\mathcal E _ 0 . \\end{align*}"}
-{"id": "3929.png", "formula": "\\begin{align*} ( \\iota \\circ \\sigma _ { \\Bbbk [ H ] } ) ( f \\cdot y ) = \\sum f _ { i } ( y ) S ( g _ { i } ) = \\bigl ( \\sum f _ { i } ( 1 ) S ( g _ { i } ) \\bigr ) \\cdot y = \\bigl ( \\iota \\circ \\sigma _ { \\Bbbk [ H ] } ( f ) \\bigr ) \\cdot y \\ , . \\end{align*}"}
-{"id": "7300.png", "formula": "\\begin{align*} s = ( F _ p \\cup F _ q , \\gamma / 6 , h _ s , R _ s , \\psi _ s ) \\end{align*}"}
-{"id": "629.png", "formula": "\\begin{align*} \\varphi ^ \\ast ( y ) \\cdot K ^ { \\mathcal Q } _ \\varphi \\cdot \\varphi ^ \\ast ( y ) ^ { - 1 } = K ^ { \\mathcal Q } _ { \\varphi ^ y } \\ . \\end{align*}"}
-{"id": "6392.png", "formula": "\\begin{align*} T _ { k } = \\left [ \\begin{array} { c c c c } \\widetilde { \\alpha } _ { 1 } & \\widetilde { \\beta } _ { 1 } \\\\ \\widetilde { \\beta } _ { 1 } & \\ddots & \\ddots \\\\ & \\ddots & \\ddots & \\widetilde { \\beta } _ { k - 1 } \\\\ & & \\widetilde { \\beta } _ { k - 1 } & \\widetilde { \\alpha } _ { k } \\end{array} \\right ] \\end{align*}"}
-{"id": "4171.png", "formula": "\\begin{align*} \\mathcal L _ n ^ { - 1 } ( z _ 1 ) \\mathcal L _ n ( z _ 2 ) = \\mathbb { I } + \\mathcal L _ n ^ { - 1 } ( z _ 1 ) ( \\mathcal L _ n ( z _ 2 ) - \\mathcal L _ n ( z _ 1 ) ) = \\mathbb { I } + \\mathcal { O } \\left ( n ^ \\frac { 3 } { 2 } ( z _ 1 - z _ 2 ) \\right ) . \\end{align*}"}
-{"id": "9561.png", "formula": "\\begin{align*} \\left ( \\frac { q ^ { - N } } { c } \\right ) _ n & = \\frac { ( - 1 ) ^ n ( c q ^ { N - n + 1 } ) _ n q ^ { \\frac { n ( n - 1 ) } { 2 } } } { c ^ n q ^ { N n } } , \\\\ \\frac { \\left ( q ^ { - N } \\right ) _ n } { \\left ( { \\frac { q ^ { - N } } { c } } \\right ) _ n } & = \\frac { \\left ( q ^ { N - n + 1 } \\right ) _ n c ^ n } { \\left ( c q ^ { N - n + 1 } \\right ) _ n } = \\frac { ( q ) _ N ( c q ) _ { N - n } c ^ n } { ( q ) _ { N - n } ( c q ) _ N } \\end{align*}"}
-{"id": "2929.png", "formula": "\\begin{align*} [ f ] _ \\omega = [ h _ { Z _ 0 } ] _ \\omega . \\end{align*}"}
-{"id": "2422.png", "formula": "\\begin{align*} | | v ^ { ( g ) } | | _ \\infty = | | V | | ^ n _ \\infty \\cdot | | V ^ { - 1 } | | ^ m _ \\infty , \\end{align*}"}
-{"id": "6732.png", "formula": "\\begin{align*} P _ { 1 k l } = \\frac { 1 } { 4 c ^ 2 } \\frac { \\partial } { \\partial t } \\left ( \\Box h _ { k l } + \\frac { \\partial ^ 2 h _ { k l } } { \\partial x _ n \\partial x ^ n } - \\frac { \\partial ^ 2 h _ { n l } } { \\partial x ^ k \\partial x _ n } - \\frac { \\partial ^ 2 h _ { k n } } { \\partial x _ n \\partial x ^ l } \\right ) , \\end{align*}"}
-{"id": "452.png", "formula": "\\begin{align*} \\mathcal { Z } _ { q , m } ( g ) : = \\int _ { \\Sigma } e ^ { - \\frac { 1 } { 4 \\pi } \\int _ M \\big ( | d X | _ { g } ^ 2 + i q K _ { g } X + m ^ 2 X ^ 2 \\big ) \\ , { \\rm d v } _ { g } } D X \\end{align*}"}
-{"id": "9316.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } x _ { i j } = 1 , ~ j = 1 , \\dots , n ; \\end{align*}"}
-{"id": "8475.png", "formula": "\\begin{align*} \\prod _ { l = 1 } ^ n \\| \\partial _ { \\alpha } ^ l g \\| _ { \\infty } ^ { k _ l } \\leq \\prod _ { j = 1 } ^ n ( 1 + 5 \\epsilon ) ^ { k _ l } \\leq ( 1 + 5 \\epsilon ) ^ s , \\end{align*}"}
-{"id": "5670.png", "formula": "\\begin{align*} \\kappa = \\left \\{ \\begin{array} { l l } 2 , & \\mbox { i f } 1 / 2 < \\gamma \\leq 1 , \\\\ 2 - \\epsilon , & \\mbox { i f } \\gamma = 1 / 2 , \\\\ 2 - \\frac { 1 - 2 \\gamma } { \\alpha } , & \\mbox { i f } 0 \\leq \\gamma < 1 / 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "8786.png", "formula": "\\begin{align*} ( x - y ) ^ 2 + x = f ( x , y ) = a _ 1 ( ( b _ 1 x + b _ 0 ) ( c _ 1 y + c _ 0 ) ) + a _ 0 . \\end{align*}"}
-{"id": "4642.png", "formula": "\\begin{align*} f ( x ) = a _ d x ^ d + a _ { d - 1 } x ^ { d - 1 } + \\dots + a _ 1 x + a _ 0 , \\end{align*}"}
-{"id": "6620.png", "formula": "\\begin{align*} \\frac { \\overline { C } _ { n _ j } ( \\varepsilon _ j ) } { C _ { n _ j } } & \\geq 1 - \\sum _ { t = 1 } ^ { \\infty } \\frac { ( 1 + 2 p _ j ) ^ 2 } { u _ t ^ 2 } - \\mathcal { O } ( \\varphi ^ { n _ j / 5 } ) \\\\ & > 1 - \\frac { 1 } { 7 } ( 1 + 2 p _ j ) ^ 2 - \\mathcal { O } ( \\varphi ^ { n _ j / 5 } ) . \\end{align*}"}
-{"id": "1904.png", "formula": "\\begin{align*} \\left [ \\cos ( \\kappa _ j x ) + \\frac { \\xi } { \\kappa _ j } \\sin ( \\kappa _ j | x | ) \\right ] \\cos ( \\kappa _ j t ) = \\frac { \\xi L } { 2 } \\left [ \\frac { 1 } { Z _ j ^ 2 } + 2 \\sum _ { n = 1 } ^ \\infty \\frac { \\cos ( k _ n x ) \\cos ( k _ n t ) } { Z _ j ^ 2 - ( \\pi n ) ^ 2 } \\right ] \\end{align*}"}
-{"id": "8261.png", "formula": "\\begin{align*} \\mathfrak { H } f ( \\alpha ) : = \\frac { 1 } { \\pi i } p . v . \\int _ { - \\infty } ^ { \\infty } \\frac { z _ { \\beta } } { z ( \\alpha , t ) - z ( \\beta , t ) } f ( \\beta ) d \\beta . \\end{align*}"}
-{"id": "8759.png", "formula": "\\begin{align*} \\psi _ { i j } \\ = \\ \\nabla _ j \\psi _ i - \\psi _ i \\psi _ j \\ , , \\ \\ \\psi _ i \\ = \\ \\frac { 1 } { 2 ( n + 1 ) } \\frac { \\partial } { \\partial x ^ i } \\left ( \\log \\left | \\frac { \\det \\overline { g } } { \\det g } \\right | \\right ) \\ , . \\end{align*}"}
-{"id": "4319.png", "formula": "\\begin{align*} V _ k ( K ) : = \\big \\{ v \\in H ^ 1 ( K ) : \\Delta v \\in \\mathbb { P } _ { k - 2 } ( K ) , v | _ { \\partial K } \\in B _ k ( \\partial K ) \\big \\} , \\end{align*}"}
-{"id": "4026.png", "formula": "\\begin{align*} \\lambda _ { a } = \\frac { \\lambda + 2 - 2 a } { a } , \\nu _ { a } = 2 n - \\frac { \\lambda + 2 - 2 a } { a } . \\end{align*}"}
-{"id": "1120.png", "formula": "\\begin{align*} & n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } \\sup _ { y \\in I } \\Big ( \\mathbb { P } ( \\xi ^ { ( n ) } ( [ y , y + G _ n ( x ) / S ( I ) ] ) = 0 ) - D _ n ( \\sqrt { 4 - y ^ 2 } / S ( I ) \\cdot G _ n ( x ) / 2 ) \\Big ) \\\\ & \\leq n ( 2 \\ln n ) ^ { \\frac { 1 } { 2 } } O ( ( n \\ln n ) ^ { - 1 } ) = O ( ( \\ln n ) ^ { - 1 / 2 } ) \\to 0 , \\ , \\ , \\ , n \\to + \\infty , \\end{align*}"}
-{"id": "8300.png", "formula": "\\begin{align*} ( I - \\mathcal { H } ) b = - [ D _ t \\zeta , \\mathcal { H } ] \\frac { \\bar { \\zeta } _ { \\alpha } - 1 } { \\zeta _ { \\alpha } } , \\end{align*}"}
-{"id": "2765.png", "formula": "\\begin{align*} S ^ { ( \\mathrm { s o f t } ) } ( \\kappa ; u , v ) & = \\frac { 1 } { u - v } \\Big ( - 2 \\phi _ { 1 } '' ( \\kappa ; u ) \\phi _ { 2 } ( \\kappa ; v ) - 2 \\phi _ { 1 } ( \\kappa ; u ) \\phi _ { 2 } '' ( \\kappa ; v ) + 2 \\phi _ { 1 } ' ( \\kappa ; u ) \\phi _ { 2 } ' ( \\kappa ; v ) \\\\ & + 2 \\kappa \\big ( \\phi _ { 1 } ' ( \\kappa ; u ) \\phi _ { 2 } ( \\kappa ; v ) - \\phi _ { 1 } ( \\kappa ; u ) \\phi _ { 2 } ' ( \\kappa ; v ) \\big ) + ( u + v ) \\phi _ { 1 } ( \\kappa ; u ) \\phi _ { 2 } ( \\kappa ; v ) \\Big ) . \\end{align*}"}
-{"id": "9167.png", "formula": "\\begin{align*} C ^ * ( \\phi ) \\circ Q ( b , s ) & = C ^ * ( \\phi ) \\bigl ( ( b , s ) \\otimes ( 1 , s ) \\bigr ) \\\\ & = ( b \\otimes 1 , s ) \\\\ & = ( \\psi \\rtimes G ) ( b , s ) . \\end{align*}"}
-{"id": "3997.png", "formula": "\\begin{align*} F ^ o ( \\tau , Z ) : = \\sum _ { \\lambda : \\nu ( \\lambda ) = \\nu _ 0 } F _ { \\lambda } ( \\tau , Z ) X _ 2 ^ { \\lambda _ 2 } , \\dots X _ n ^ { \\lambda _ n } . \\end{align*}"}
-{"id": "4961.png", "formula": "\\begin{align*} \\delta _ i = \\begin{cases} 0 & \\mbox { i f } i = k \\\\ \\delta _ { i + 1 } - 2 ( n _ { i + 1 } - n _ i ) + 1 & \\mbox { i f } k - i \\mbox { i s o d d } \\\\ \\delta _ { i + 1 } - 1 & \\mbox { i f } k - i > 0 \\mbox { i s e v e n } , \\end{cases} \\end{align*}"}
-{"id": "2714.png", "formula": "\\begin{align*} G _ m & ( \\pi ; x ) = 1 + \\sum _ { k = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ k } { k ! } \\int _ { x } ^ { \\infty } \\cdots \\int _ { x } ^ { \\infty } \\det [ { K _ { \\mathrm { G U E } } ( \\pi ; u _ i , u _ j ) } ] _ { i , j = 1 } ^ { k } d u _ 1 \\cdots d u _ k . \\end{align*}"}
-{"id": "9829.png", "formula": "\\begin{align*} \\int _ { E } \\nabla \\tilde { u } _ { X _ \\circ , r _ { j _ \\ell } } ( X ) \\cdot ( X _ \\circ + r _ { j _ \\ell } X ) | y | ^ a = \\int _ { E } \\lambda r _ { j _ \\ell } \\tilde { u } _ { X _ \\circ , r _ { j _ \\ell } } ( X ) | y | ^ a \\end{align*}"}
-{"id": "9674.png", "formula": "\\begin{align*} \\| X X _ * ^ * u \\| & \\leq \\mathop { } _ { \\sigma } | g | \\| h \\| _ { L ^ 2 ( \\sigma ) } \\leq \\mathop { } _ { \\sigma } | g | ( \\| h \\| _ { L ^ 2 ( \\mathbb T ) } ^ 2 + \\| v \\| ^ 2 ) ^ { 1 / 2 } \\\\ & = \\mathop { } _ { \\sigma } | g | \\| u \\| \\end{align*}"}
-{"id": "1658.png", "formula": "\\begin{align*} 0 = ( d H ^ c ) ( X _ { I _ l } ) = p _ l \\frac { \\partial H ^ c } { \\partial q _ l } - q _ l \\frac { \\partial H ^ c } { \\partial p _ l } \\end{align*}"}
-{"id": "8271.png", "formula": "\\begin{align*} \\tilde { \\lambda } = : \\frac { \\sum _ { j = 1 } ^ N | \\lambda _ j | } { 2 \\pi } , \\tilde { d } _ I ( t ) : = \\min _ { 1 \\leq j \\leq N } \\inf _ { \\alpha \\in \\mathbb { R } } | \\alpha - \\Phi ( z _ j ( t ) , t ) | , \\tilde { d } _ P ( t ) = \\min _ { j \\neq k } | z _ j ( t ) - z _ k ( t ) | . \\end{align*}"}
-{"id": "4622.png", "formula": "\\begin{align*} M \\frac { ^ 2 \\mathbf { u } } { t ^ 2 } = - K \\mathbf { u } . \\end{align*}"}
-{"id": "3492.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\iint _ { ( C , \\infty ) ^ 2 } d x d r \\left \\lvert G _ N ( x , r ) - G _ { \\infty , \\gamma } ( x , r ) \\right \\rvert ^ 2 = { } & 0 , \\\\ \\lim _ { N \\to \\infty } \\iint _ { ( C , \\infty ) ^ 2 } d r d y \\left \\lvert H _ N ( r , y ) - H _ { \\infty , \\gamma } ( r , y ) \\right \\rvert ^ 2 = { } & 0 . \\end{align*}"}
-{"id": "1143.png", "formula": "\\begin{align*} b _ 3 : = & | B - A | _ 2 ^ 2 \\| ( - B ) ^ { - 1 } \\| ^ 2 \\leq O \\left ( \\frac { ( \\ln n ) ^ { 3 } } { n ^ 2 } \\right ) O \\left ( n ^ 2 ( \\ln n ) ^ { - 1 } e ^ { - ( \\ln n ) ^ { \\frac { 1 } { 2 } } } \\right ) \\\\ = & O \\left ( { ( \\ln n ) ^ { 2 } } e ^ { - ( \\ln n ) ^ { \\frac { 1 } { 2 } } } \\right ) = O \\left ( { ( \\ln n ) ^ { - 2 } } \\right ) . \\end{align*}"}
-{"id": "9902.png", "formula": "\\begin{align*} & \\int _ { \\mathcal { Y } ^ { k + 1 } } g ( y _ { [ n , n + k ] } ) P ^ { \\mu } ( d y _ { [ n , n + k ] } | Y _ { [ 0 , n - 1 ] } ) \\\\ & = \\int _ { \\mathcal { X } } \\int _ { \\mathcal { Y } ^ { k + 1 } } g ( y _ { [ n , n + k ] } ) P ( d y _ { [ n , n + k ] } | X _ { n } = x _ { n } ) \\pi _ { n - } ^ { \\mu } ( d x _ { n } ) \\end{align*}"}
-{"id": "7195.png", "formula": "\\begin{gather*} \\mathcal { L } _ { \\xi } ^ { - } : = \\bigcup _ { \\eta \\in \\mathbb { L } _ { \\xi } ^ { - } \\cap \\mathcal { N } _ { 2 \\varepsilon } ( \\mathcal { M } ) } \\left \\{ \\xi + \\eta + h ( \\eta , \\xi ) \\right \\} \\end{gather*}"}
-{"id": "4254.png", "formula": "\\begin{align*} B _ { i } \\equiv B ( x , R _ { i } ) , R _ { i } : = \\sigma ^ { i } R , \\sigma \\in ( 0 , \\frac { 1 } { 2 } ) . \\end{align*}"}
-{"id": "1430.png", "formula": "\\begin{align*} \\| f \\| _ { s } ^ { 2 } = \\sum _ { 0 \\leq | \\alpha | \\leq s } \\| \\partial ^ { \\alpha } f \\| _ { 0 } ^ { 2 } . \\end{align*}"}
-{"id": "9993.png", "formula": "\\begin{align*} h _ { 0 1 } + h _ { 0 2 } + h _ { 0 3 } = 0 . \\end{align*}"}
-{"id": "456.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } e ^ { - \\alpha t } \\ , \\mu ( d t ) = 1 , \\end{align*}"}
-{"id": "6535.png", "formula": "\\begin{align*} \\| \\nabla s \\| ^ 2 & = ( s , \\nabla ^ \\ast \\nabla s ) _ { W _ R } - ( \\sigma _ \\nabla ( - \\nu ) s , \\nabla s ) _ { N _ R } \\\\ & = ( s , \\nabla ^ \\ast \\nabla s ) _ { W _ R } + ( \\nu \\otimes s , \\nabla s ) _ { N _ R } \\\\ & = ( s , \\nabla ^ \\ast \\nabla s ) _ { W _ R } + ( s , \\iota ( \\nu ) \\nabla s ) _ { N _ R } \\end{align*}"}
-{"id": "700.png", "formula": "\\begin{align*} \\left ( ( 2 p - 1 ) x f ( x ) + 4 p K \\right ) f ^ { ( 1 ) } ( x ) & = ( 2 p - 1 ) x f ( x ) ^ 3 + \\left ( 6 p K + 1 \\right ) f ( x ) ^ 2 - 2 ( p + 1 ) \\int _ 0 ^ x f ( t ) ^ 3 \\ ; d t + 4 e ^ { 3 F ( 0 ) } . \\end{align*}"}
-{"id": "1698.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } ( { \\mathcal L } \\varphi _ n ) ( x ) = ( { \\mathcal L } \\chi _ V ) ( x ) \\ \\ \\forall \\ x \\in X . \\end{align*}"}
-{"id": "1888.png", "formula": "\\begin{align*} \\ln h ( 0 ) = 0 \\qquad \\mbox { a n d } \\left [ \\ln h ( t ) \\right ] ' \\leq 1 . \\end{align*}"}
-{"id": "1166.png", "formula": "\\begin{align*} \\mathbb { E } \\sum _ { k = 1 } ^ n ( m _ k - 2 \\alpha ) _ + = 2 \\pi D _ n ( \\alpha ) . \\end{align*}"}
-{"id": "5020.png", "formula": "\\begin{align*} f ( t _ 0 , x _ 0 ) = \\displaystyle \\max _ { ( t , x ) \\in [ 0 , T ] \\times { \\bar \\Omega } } f ( t , x ) \\end{align*}"}
-{"id": "4538.png", "formula": "\\begin{align*} \\int | f _ { g , \\delta , j } ( x ) | ^ 2 d x = \\int | \\hat { f } _ { g , \\delta , j } ( \\omega ) | ^ 2 d \\omega = \\int | \\hat { \\nu } _ { g , j } ( \\omega ) | ^ 2 | \\hat { \\phi } ^ D _ \\delta ( \\omega ) | ^ 2 d \\omega \\leq ( c '''' ) ^ 2 \\int | \\hat { \\nu } _ { g , j } ( \\omega ) | ^ 2 d \\omega . \\end{align*}"}
-{"id": "8365.png", "formula": "\\begin{align*} | D _ t Z | \\leq | F | + \\sum _ { j = 1 } ^ N \\frac { | \\lambda _ j | } { 2 \\pi \\beta _ 0 } \\frac { 1 } { | \\alpha - \\omega _ 0 ^ j | } = M _ 0 + \\frac { \\tilde { \\lambda } } { 2 \\tilde { d } _ I ( t ) \\beta _ 0 } . \\end{align*}"}
-{"id": "1196.png", "formula": "\\begin{align*} R ( \\rho ) = \\frac { 4 \\sqrt { \\gamma _ - } \\sqrt { a _ 0 ^ 2 + \\rho ^ 2 } \\sqrt { 1 + \\rho ^ 2 } } { \\sqrt { a _ 0 ^ 2 + \\rho ^ 2 } + a _ 0 ^ 2 \\sqrt { 1 + \\rho ^ 2 } } \\quad ( \\rho \\geq 0 ) \\quad a _ 0 = \\sqrt { \\frac { \\gamma _ - } { \\gamma _ + } } . \\end{align*}"}
-{"id": "8009.png", "formula": "\\begin{align*} u _ M ( t , x _ 0 ) & = ( 4 \\pi ( t + 1 ) ) ^ { - \\frac { d - 1 } { 2 } } ( 4 \\pi t ) ^ { - \\frac { 1 } { 2 } } \\int _ M ^ { M + 1 } e ^ { - \\frac { | 1 - y _ 1 | ^ 2 } { 4 t } } \\ , \\mathrm d y _ 1 \\\\ & \\geq \\frac { 1 } { 2 } ( 4 \\pi ( t + 1 ) ) ^ { - \\frac { d - 1 } { 2 } } ( 4 \\pi t ) ^ { - \\frac { 1 } { 2 } } e ^ { - \\frac { | \\frac { 1 } { 2 } - M | ^ 2 } { 4 t } } , \\end{align*}"}
-{"id": "6571.png", "formula": "\\begin{gather*} W ' = \\R \\{ e _ 1 , \\dots , e _ n \\} \\subset W = \\R \\{ e _ 0 , \\dots , e _ n \\} . \\end{gather*}"}
-{"id": "2703.png", "formula": "\\begin{align*} \\frac { \\eta _ { - } } { \\varphi ( 4 ) } \\big ( 1 - z _ { 0 } \\big ) & = - \\kappa \\leq - a , \\frac { \\eta _ { - } } { \\varphi ( 4 ) } \\big ( z _ { 0 } - \\Re w \\big ) \\leq b , \\end{align*}"}
-{"id": "6716.png", "formula": "\\begin{align*} L = \\frac { 1 } { 2 } m \\left ( \\dot { q } _ 1 ^ 2 + \\dot { q } _ 2 ^ 2 + \\dot { q } _ 3 ^ 2 \\right ) - \\frac { 1 } { 2 } \\left ( h _ 1 q _ 1 ^ 2 + h _ 2 q _ 2 ^ 2 + h _ 3 q _ 3 ^ 2 \\right ) , \\end{align*}"}
-{"id": "777.png", "formula": "\\begin{align*} ( \\sigma \\lambda _ i ) ^ { q ^ d - 1 } = - \\sigma ( T - \\rho _ i ) = - ( T - \\sigma ( \\rho _ i ) ) = - ( T - \\sigma _ { i + 1 } ) = \\lambda _ { i + 1 } ^ { q ^ d - 1 } . \\end{align*}"}
-{"id": "462.png", "formula": "\\begin{align*} \\ell ^ c \\circ f = \\sum ^ { d - c } _ { k = 0 } \\sum ^ { c } _ { i = 0 } \\dfrac { ( k + c - i ) ! } { k ! } \\binom { c } { i } y ^ k _ j { \\ell ^ \\prime } ^ i \\circ g _ { k + c - i } . \\end{align*}"}
-{"id": "3418.png", "formula": "\\begin{align*} \\tau _ * ^ 2 ( T ) = \\log \\frac { \\rho _ { * , \\sup } ^ 2 ( T ) } { \\rho _ { * , \\inf } ^ 2 ( T ) } \\le \\log \\frac { 1 + 2 ^ { \\deg ( T ) } } { \\prod _ i 1 / ( 1 + 2 ^ { \\deg ( T _ i ) + 1 } ) } \\ll \\deg ( T ) + \\sum _ i \\deg ( T _ i ) = w _ 1 ( T ) + w _ 2 ( T ) . \\end{align*}"}
-{"id": "3466.png", "formula": "\\begin{align*} \\frac { 1 } { 4 \\pi } \\iint _ { [ - C , C ] ^ 2 } \\begin{vmatrix} e ^ { - \\frac { 1 } { 2 } \\xi ^ 2 } & e ^ { - \\frac { 1 } { 2 } \\eta ^ 2 } \\\\ e ^ { - \\frac { 1 } { 2 } \\xi ^ 2 } & e ^ { - \\frac { 1 } { 2 } \\eta ^ 2 } \\end{vmatrix} d \\xi d \\eta = 0 . \\end{align*}"}
-{"id": "5609.png", "formula": "\\begin{align*} \\partial _ x u _ { \\delta , \\epsilon } ( x _ 1 , t ) - \\partial _ x u _ { \\delta , \\epsilon } ( 0 , t ) = x _ 1 \\partial _ x ^ 2 u _ { \\delta , \\epsilon } ( x _ 2 , t ) . \\end{align*}"}
-{"id": "2153.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ N } h _ t ( \\mathbf x , \\mathbf y ) \\ , d w ( \\mathbf y ) = 1 . \\end{align*}"}
-{"id": "3552.png", "formula": "\\begin{align*} \\mathcal { D } _ { \\ell } : = ( a _ { 0 } b _ { 0 } c _ { 0 } ) ^ { \\frac { 1 } { 2 } } \\left ( \\mathtt { a } ^ { \\ell _ { + } } \\square _ { l } \\right ) ( a _ { 0 } b _ { 0 } c _ { 0 } ) ^ { - \\frac { 1 } { 2 } } \\end{align*}"}
-{"id": "1059.png", "formula": "\\begin{align*} 0 \\leq & \\mathbb { P } ( \\xi ^ { ( n ) } ( I _ { n , k } ) = \\xi ^ { ( n ) } ( J _ { n , k , j } ) = 0 ) = \\mathbb { P } ( \\xi ^ { ( n ) } ( I _ { n , k } \\cup J _ { n , k , j } ) = 0 ) \\\\ \\leq & \\mathbb { P } ( \\xi ^ { ( n ) } ( J _ { n , k , j } \\cup \\left ( \\cup _ { i \\neq j } I ( z _ i , F _ n ( x _ 0 ) ) \\right ) ) = 0 ) \\\\ \\leq & \\mathbb { P } ( \\xi ^ { ( n ) } ( J _ { n , k , j } ) = 0 ) \\prod _ { i \\neq j } \\mathbb { P } ( \\xi ^ { ( n ) } ( I ( z _ i , F _ n ( x _ 0 ) ) ) = 0 ) \\\\ = & D _ n ( ( z _ { j + 1 } - z _ j ) / 2 ) ( D _ n ( F _ n ( x _ 0 ) / 2 ) ) ^ { k - 1 } . \\end{align*}"}
-{"id": "8186.png", "formula": "\\begin{align*} T ^ * T _ \\Lambda \\{ g _ i \\} _ { i \\in \\mathbb { Z } } = \\sum _ { i \\in \\mathbb { Z } } T ^ * \\Lambda _ i ^ * g _ i & = \\sum _ { i \\in \\mathbb { Z } } ( \\Lambda _ i T ) ^ * g _ i \\\\ & = \\sum _ { i \\in \\mathbb { Z } } \\Lambda _ { i + 1 } ^ * g _ i = T _ \\Lambda \\mathcal { T } \\{ g _ i \\} _ { i \\in \\mathbb { Z } } , \\end{align*}"}
-{"id": "8101.png", "formula": "\\begin{align*} B ( h ) & = - 6 + 1 3 h - 7 h ^ 2 + h ^ 3 \\ ; \\ ; \\ ; \\ , = - ( 2 - h ) ( 3 - 5 h + h ^ 2 ) , \\\\ C ( h ) & = + 6 - 1 9 h + 1 7 h ^ 2 - 3 h ^ 3 = ( 2 - 3 h ) ( 3 - 5 h + h ^ 2 ) , \\ ; \\\\ D ( h ) & = - 2 + 1 1 h - 1 7 h ^ 2 + 6 h ^ 3 = - ( 1 - 2 h ) ( 1 - 3 h ) ( 2 - h ) . \\end{align*}"}
-{"id": "4707.png", "formula": "\\begin{align*} F _ { j + \\frac { 1 } { 2 } } ^ { \\mbox { \\small H y b r i d 4 } } = \\Phi F _ { j + \\frac { 1 } { 2 } } ^ { \\mbox { W E N O 3 } } + ( 1 - \\Phi ) F _ { j + \\frac { 1 } { 2 } } ^ { \\mbox { C B S Q I } } , \\end{align*}"}
-{"id": "8665.png", "formula": "\\begin{align*} \\lim _ { \\l \\to - 0 } R _ 0 ( \\l ) = \\infty , \\lim _ { \\l \\to - \\infty } R _ 0 ( \\l ) = 0 . \\end{align*}"}
-{"id": "5562.png", "formula": "\\begin{align*} 1 = \\lambda ^ T _ 0 + \\lambda ^ T _ 1 + \\dots + \\lambda ^ T _ n . \\end{align*}"}
-{"id": "7867.png", "formula": "\\begin{align*} Z _ j ( x ) = \\begin{cases} x _ j , & \\textrm { i f } | x _ j | \\ge 2 \\\\ 0 , & \\textrm { i f } | x _ j | \\le 1 , \\end{cases} \\end{align*}"}
-{"id": "7909.png", "formula": "\\begin{align*} \\int _ { - h / 2 } ^ { h / 2 } f ( x ) \\ , d x \\approx \\int _ { - h / 2 } ^ { h / 2 } p _ 2 ( x ) \\ , d x = \\frac { h ^ 2 } { 2 4 } ( f ' ( h / 2 ) - f ' ( - h / 2 ) ) + h f ( 0 ) \\end{align*}"}
-{"id": "879.png", "formula": "\\begin{align*} \\lambda _ A ( e _ i , e _ j ) = \\left | \\{ a \\in A : t a = i , h a = j \\} \\right | - \\left | \\{ a \\in A : t a = j , h a = i \\} \\right | \\end{align*}"}
-{"id": "289.png", "formula": "\\begin{align*} \\gamma _ \\Lambda : \\Lambda \\ ; \\hookrightarrow \\ ; T ^ * A = A \\times V \\ ; \\twoheadrightarrow \\ ; V \\end{align*}"}
-{"id": "2935.png", "formula": "\\begin{align*} & \\| B ^ { \\dagger } E A ^ { \\dagger } \\| _ { F } ^ { 2 } = \\| \\widetilde { V } _ { 1 } ^ { \\ast } V _ { 1 } \\Sigma _ { 1 } ^ { - 1 } - \\widetilde { \\Sigma } _ { 1 } ^ { - 1 } \\widetilde { U } _ { 1 } ^ { \\ast } U _ { 1 } \\| _ { F } ^ { 2 } , \\\\ & \\| A ^ { \\dagger } E B ^ { \\dagger } \\| _ { F } ^ { 2 } = \\| \\Sigma _ { 1 } ^ { - 1 } U _ { 1 } ^ { \\ast } \\widetilde { U } _ { 1 } - V _ { 1 } ^ { \\ast } \\widetilde { V } _ { 1 } \\widetilde { \\Sigma } _ { 1 } ^ { - 1 } \\| _ { F } ^ { 2 } . \\end{align*}"}
-{"id": "1114.png", "formula": "\\begin{align*} & \\det ( + A ) = \\mathbb { P } ^ { G U E ( n ) } ( \\lambda _ i \\not \\in [ x , x + \\delta _ n / \\rho _ { s c } ( x ) ] , 1 \\leq i \\leq n ) \\end{align*}"}
-{"id": "2973.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\frac { q ^ { n ^ 2 + n } } { ( q ; q ) _ n } & = \\frac { 1 } { ( q ^ 2 ; q ^ 5 ) _ \\infty ( q ^ 3 ; q ^ 5 ) _ \\infty } = \\frac { ( q ; q ^ 5 ) _ \\infty ( q ^ 4 ; q ^ 5 ) _ \\infty ( q ^ 5 ; q ^ 5 ) _ \\infty } { ( q ; q ^ 5 ) _ \\infty ( q ^ 2 ; q ^ 5 ) _ \\infty ( q ^ 3 ; q ^ 5 ) _ \\infty ( q ^ 4 ; q ^ 5 ) _ \\infty ( q ^ 5 ; q ^ 5 ) _ \\infty } \\\\ & = \\Big ( \\frac { 1 } { ( q ; q ) _ \\infty } \\Big ) \\Big ( ( q ; q ^ 5 ) _ \\infty ( q ^ 4 ; q ^ 5 ) _ \\infty ( q ^ 5 ; q ^ 5 ) _ \\infty \\Big ) . \\end{align*}"}
-{"id": "7324.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} u - \\frac { 1 } { \\rho } \\operatorname { d i v } ( K ( x ) \\nabla u + a ( x ) \\nabla u ) = { } & - \\frac { 1 } { \\rho } \\operatorname { d i v } ( a ( x ) \\nabla p ) + p + r , \\\\ v - \\frac { 1 } { \\rho } \\operatorname { d i v } ( K ( x ) \\nabla v + a ( x ) \\nabla v ) = { } & - \\frac { 1 } { \\rho } \\operatorname { d i v } ( b ( x ) \\nabla q ) + q + s . \\end{aligned} \\right . \\end{align*}"}
-{"id": "9114.png", "formula": "\\begin{align*} f _ R ^ { \\# } ( a _ i ) = \\epsilon _ i ( \\pi ^ { n _ i - \\frac { d _ i m } { d } } ) ( \\pi ^ { \\frac { m } { d } } t ^ { - 1 } ) ^ { d _ i } = \\epsilon _ i ( \\pi ^ { n _ i - \\frac { d _ i m } { d } } ) s ^ { d _ i } . \\end{align*}"}
-{"id": "5286.png", "formula": "\\begin{align*} & P E ^ { 2 p ^ s } = { ^ { ( s ) } } ( \\mathrm { K e r } \\ , V _ { s - 1 } ) \\ ; , & Q E ^ { 2 p ^ s } = { ^ { ( s ) } } ( \\mathrm { C o k e r } \\ , F _ { s - 1 } ) \\ ; . \\end{align*}"}
-{"id": "4510.png", "formula": "\\begin{align*} \\left \\langle ( - \\mbox { d i v } ( a ( x ) \\nabla ( \\phi _ i \\phi _ j ) ) , \\phi _ i \\phi _ j \\right \\rangle & = \\int _ { \\Omega } { a ( x ) ( \\nabla ( \\phi _ i \\phi _ j ) \\cdot \\nabla ( \\phi _ i \\phi _ j ) ) d x } \\\\ & \\leq \\left ( \\max _ { x \\in \\Omega } { a ( x ) } \\right ) \\| \\nabla ( \\phi _ i \\phi _ j ) \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "4187.png", "formula": "\\begin{align*} P ^ { ( 3 ) } _ + ( z ) \\left ( P ^ { ( 3 ) } _ - ( z ) \\right ) ^ { - 1 } = \\mathbb { I } + \\frac { A _ n ^ { ( 3 ) } ( z ) } { n ^ 9 z } + \\mathcal { O } ( n ^ { - 1 } ) \\end{align*}"}
-{"id": "9029.png", "formula": "\\begin{align*} f _ { \\{ b _ 1 , c _ 1 \\} , 1 } ( x _ 1 , x _ 2 ) = \\begin{cases} 1 & \\{ x _ 1 , x _ 2 \\} = \\{ b _ 1 , c _ 1 \\} , \\\\ x _ 1 & \\end{cases} \\end{align*}"}
-{"id": "9762.png", "formula": "\\begin{align*} D ^ \\alpha q ( Z _ \\infty ) = 0 \\quad \\alpha = ( \\alpha ' , 0 ) | \\alpha | \\leq \\kappa - 2 . \\end{align*}"}
-{"id": "5400.png", "formula": "\\begin{align*} c ( l , 2 k ) = \\frac { l ^ 2 - 2 ( k - 1 ) l - 2 k } { 2 } . \\end{align*}"}
-{"id": "7679.png", "formula": "\\begin{align*} T \\iff T = \\left ( \\begin{smallmatrix} * & 0 \\\\ 0 & * \\end{smallmatrix} \\right ) , \\ ; \\ ; \\ ; \\ ; T \\iff T = \\left ( \\begin{smallmatrix} 0 & * \\\\ * & 0 \\end{smallmatrix} \\right ) . \\end{align*}"}
-{"id": "6569.png", "formula": "\\begin{gather*} \\hat \\theta _ I = \\d p _ I - p _ { I a } \\d x ^ a , | I | < r . \\end{gather*}"}
-{"id": "4214.png", "formula": "\\begin{align*} B _ n ^ { ( k ) } ( z ) = \\frac { A _ n ^ { ( k ) } ( z ) - A _ n ^ { ( k ) } ( 0 ) } { n ^ { 3 k } z } , k = 1 , 2 , 3 . \\end{align*}"}
-{"id": "814.png", "formula": "\\begin{align*} u ^ { q ^ d } + ( T - \\rho ) u = \\lambda . \\end{align*}"}
-{"id": "2057.png", "formula": "\\begin{align*} \\alpha _ { n , M } ( 1 ) = \\sum _ { \\ell \\equiv n \\bmod M } \\frac { a ( \\ell ) } { \\ell } V \\left ( \\frac { X } { 2 \\pi \\ell } \\right ) \\\\ + \\mathcal { O } \\bigg ( M ^ { \\frac { 1 } { 2 } + 2 \\epsilon + \\epsilon ' } \\sigma _ 0 ( M ) ^ 2 X ^ { - \\frac { 1 } { 2 } - \\epsilon } q ^ { \\frac { 1 } { 2 } + \\epsilon } \\prod _ { \\substack { p \\mid \\gcd ( q , M ) \\\\ p ^ 2 \\mid q } } p \\bigg ) . \\end{align*}"}
-{"id": "1285.png", "formula": "\\begin{align*} \\int _ { | \\nu | < 2 0 } e ^ { 3 i \\eta \\nu ^ 2 / 4 } z \\left ( \\frac { \\eta + \\nu } { 2 } \\right ) w \\left ( \\frac { \\eta - \\nu } { 2 } \\right ) d \\nu = O ( 1 ) . \\end{align*}"}
-{"id": "8286.png", "formula": "\\begin{align*} \\bar { v } _ 0 = \\frac { 1 } { 2 } ( I + \\mathfrak { H } _ 0 ) f - \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j i } { 2 \\pi } \\frac { 1 } { z _ 0 ( \\alpha ) - z _ j ( 0 ) } , \\end{align*}"}
-{"id": "926.png", "formula": "\\begin{align*} \\sum _ { \\stackrel { ( \\{ \\{ g _ 1 , g _ 2 \\} , \\ldots , \\{ g _ { 2 r - 1 } , g _ { 2 r } \\} \\} , \\{ q _ 1 , \\ldots , q _ { k - 2 r } \\} ) } { { } _ { \\{ g _ 1 , g _ 2 , \\ldots , g _ { 2 r - 1 } , g _ { 2 r } , q _ 1 , \\ldots , q _ { k - 2 r } \\} = \\{ 1 , 2 , \\ldots , k \\} } } } a _ { g _ 1 g _ 2 , \\ldots , g _ { 2 r - 1 } g _ { 2 r } , q _ 1 \\ldots q _ { k - 2 r } } . \\end{align*}"}
-{"id": "1032.png", "formula": "\\begin{align*} & \\liminf _ { n \\to + \\infty } \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) \\geq \\prod _ { j = 1 } ^ k \\left ( e ^ { c _ 0 - x _ j } / 4 \\right ) . \\end{align*}"}
-{"id": "20.png", "formula": "\\begin{align*} d \\mathcal { Y } _ t ^ { i , v } = - f ^ i ( V _ t ^ { v } , \\mathcal { Z } _ t ^ { i , v } ) d t - \\sum _ { k \\in I } q ^ { i k } ( e ^ { \\mathcal { Y } _ t ^ { k , v } - \\mathcal { Y } _ t ^ { i , v } } - 1 ) d t + \\lambda d t + ( \\mathcal { Z } _ t ^ { i , v } ) ^ { t r } d W _ t , \\end{align*}"}
-{"id": "5292.png", "formula": "\\begin{align*} I _ 1 = & ( q _ 1 p _ 1 - q _ 2 p _ 2 ) ^ 2 + b _ 1 b _ 2 ( q _ 1 p _ 1 + q _ 2 p _ 2 ) \\\\ & + b _ 1 \\left ( a ( p _ 1 + q _ 2 ) - q _ 1 p _ 1 ^ 2 - q _ 2 ^ 2 p _ 2 \\right ) + b _ 2 \\left ( a ( q _ 1 + p _ 2 ) - q _ 1 ^ 2 p _ 1 - q _ 2 p _ 2 ^ 2 \\right ) \\\\ I _ 2 = & ( a ( q _ 1 + p _ 2 ) + q _ 1 p _ 2 ( b _ 2 - q _ 2 - p _ 1 ) ) ( a ( q _ 2 + p _ 1 ) + q _ 2 p _ 1 ( b _ 1 - q _ 1 - p _ 2 ) ) . \\end{align*}"}
-{"id": "6154.png", "formula": "\\begin{align*} R ^ { \\omega , \\nu } f ( z ) = \\int _ { \\mathbb { D } } f ( \\xi ) \\overline { B _ z ^ \\nu ( \\xi ) } \\omega ( \\xi ) d A ( \\xi ) , \\end{align*}"}
-{"id": "2746.png", "formula": "\\begin{align*} \\left | \\frac { 1 } { ( \\varphi ( x ) ) ^ k } \\mathrm { P f } \\Big [ K _ { N } \\Big ( x N + \\frac { u _ i } { \\varphi ( x ) } , x N + \\frac { u _ j } { \\varphi ( x ) } \\Big ) \\Big ] _ { i , j = 1 } ^ { k } \\right | \\leq ( 2 k ) ^ { \\frac { k } { 2 } } C ^ { k } \\prod _ { j = 1 } ^ { k } e ^ { - ( a - v ) u _ { j } } . \\end{align*}"}
-{"id": "7412.png", "formula": "\\begin{align*} \\Delta ( x ) = \\Delta . \\end{align*}"}
-{"id": "9797.png", "formula": "\\begin{align*} f ( X ) : = N ( 0 ^ + , u ( X + \\ , \\cdot \\ , ) - p _ { * , X } ) \\quad X \\in \\R ^ n \\times \\{ 0 \\} , \\end{align*}"}
-{"id": "9201.png", "formula": "\\begin{align*} N = \\left \\{ \\left . \\left ( z _ { 1 } , z _ { 2 } \\right ) \\in \\mathbb { C } ^ { 2 } \\right \\vert \\left \\vert z _ { 1 } \\right \\vert = \\left \\vert z _ { 2 } \\right \\vert = \\frac { 1 } { \\sqrt { 2 } } \\right \\} , \\end{align*}"}
-{"id": "8276.png", "formula": "\\begin{align*} a _ 0 : = a ( t = 0 ) . \\end{align*}"}
-{"id": "9635.png", "formula": "\\begin{align*} S _ 1 ( z , q , N ) = \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] _ { q ^ 2 } \\frac { ( q ; q ^ 2 ) _ { n - 1 } ( q ^ 2 ; q ^ 2 ) _ { n } ( z q ^ 2 ; q ^ 2 ) _ { N - n } z ^ n q ^ { 2 n - 1 } } { ( z q ; q ^ 2 ) _ { n } ( z q ^ 2 ; q ^ 2 ) _ { N } } . \\end{align*}"}
-{"id": "7103.png", "formula": "\\begin{align*} ( d x \\wedge d x ^ { * } ) = 0 d x \\curlywedge d x ^ { * } = 0 , \\end{align*}"}
-{"id": "2448.png", "formula": "\\begin{gather*} x = x ^ 1 , y = x ^ 2 , p = x ^ 3 , q = \\partial _ { x ^ 5 } f , z = x ^ 4 , \\end{gather*}"}
-{"id": "4217.png", "formula": "\\begin{align*} E _ n ^ { - 1 } ( z ) \\frac { A _ n ^ { ( 1 ) } ( s ) - A _ n ^ { ( 1 ) } ( 0 ) } { s } E _ n ( z ) = \\frac { \\mathcal { O } \\left ( 1 \\right ) } { n ^ { - \\frac { 5 } { 2 } } } = \\mathcal { O } \\left ( n ^ { \\frac { 5 } { 2 } } \\right ) \\end{align*}"}
-{"id": "6330.png", "formula": "\\begin{align*} q ( x , 0 ) = q _ 0 ( x ) : = \\nu ^ { \\frac { 1 } { 2 } } u ^ { - 1 } _ 0 ( x ) \\quad x \\in ( 0 , \\ , 1 ) , \\end{align*}"}
-{"id": "83.png", "formula": "\\begin{align*} \\sigma _ { 1 } ^ + = \\Delta J = J _ r = \\sigma _ { - 1 } ^ - \\ , . \\end{align*}"}
-{"id": "4873.png", "formula": "\\begin{align*} B _ \\varepsilon ( z _ 0 ) = \\{ z \\in \\mathbb { C } ^ n \\ , ; \\ , | z - z _ 0 | < \\varepsilon \\} \\end{align*}"}
-{"id": "475.png", "formula": "\\begin{align*} f = x _ 1 ^ { m } + \\cdots + x _ k ^ { m } + g ( x _ { k + 1 } , \\dots , x _ n ) . \\end{align*}"}
-{"id": "8256.png", "formula": "\\begin{align*} \\textrm { t h e s e n s i t i v i t y } : = \\frac { \\textrm { T P } } { \\textrm { T P } + \\textrm { F N } } , \\textrm { t h e s p e c i f i c i t y } : = \\frac { \\textrm { T N } } { \\textrm { T N } + \\textrm { F P } } , \\end{align*}"}
-{"id": "8202.png", "formula": "\\begin{align*} & ( a ; q ) _ \\infty = \\prod \\limits _ { n = 0 } ^ \\infty ( 1 - a q ^ n ) , \\\\ & ( a _ 1 , a _ 2 , \\dots , a _ m ; q ) _ \\infty = \\prod \\limits _ { k = 1 } ^ { m } ( a _ k ; q ) _ \\infty , \\end{align*}"}
-{"id": "8541.png", "formula": "\\begin{align*} [ D _ t \\zeta , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } D _ t \\partial _ { \\alpha } ^ k \\tilde { \\sigma } } { \\zeta _ { \\alpha } } = [ D _ t \\zeta , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } \\partial _ { \\alpha } ^ k D _ t \\tilde { \\sigma } } { \\zeta _ { \\alpha } } + [ D _ t \\zeta , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } [ D _ t , \\partial _ { \\alpha } ^ k ] \\tilde { \\sigma } } { \\zeta _ { \\alpha } } \\end{align*}"}
-{"id": "10035.png", "formula": "\\begin{align*} V _ \\mu ^ - & = \\{ f \\in F ( I ^ - _ * ) \\mid L f = \\mu f \\} , \\\\ V _ \\mu ^ + & = \\{ f \\in F ( I ^ + ) \\mid L f = \\mu f \\} , \\\\ V _ \\mu & = \\Big \\{ f \\in F ( I _ * ) \\ | \\ L f = \\mu f , f ( z q ^ n ) = o ( q ^ { - n } ) , f ( - q ^ n ) = o ( q ^ { - n } ) \\\\ & f ( z q ^ n ) - f ( - q ^ n ) = o ( 1 ) \\ , n \\to \\infty \\Big \\} \\end{align*}"}
-{"id": "5847.png", "formula": "\\begin{align*} \\int _ { S } c _ 1 ( x _ 1 ) | x _ 1 | _ { E _ 1 } ^ { - 1 } d s = \\int _ { \\phi ^ { - 1 } ( 0 ) } | y + v | _ { E _ 1 } ^ { r _ 1 - 1 } d y \\end{align*}"}
-{"id": "3350.png", "formula": "\\begin{align*} z _ 1 = z _ { 1 , d } : = \\min \\{ \\frac { q ^ { \\frac { d } { 2 } } } { d } ( \\ell + \\deg ( M ) + 1 ) , \\frac { q ^ { d } } { d } \\} , z _ 2 = z _ { 2 , d } : = \\frac { q ^ d } { d } , \\end{align*}"}
-{"id": "598.png", "formula": "\\begin{align*} \\Omega & : = \\big \\{ ( j , k , e ) \\ , \\big | \\ , j \\geq 0 , \\ k \\in P _ j ^ d , e \\in E _ j \\big \\} , \\ \\ \\textup { w i t h } \\\\ P _ j ^ d & : = \\big \\{ k = ( k _ 1 , \\ldots , k _ d ) \\ , \\big | \\ , k _ i = 0 , \\ldots , 2 ^ j - 1 , \\ i = 1 , \\ldots , d \\big \\} , \\\\ E _ j & : = \\begin{cases} \\{ 0 , 1 \\} ^ d & \\textup { i f } j = 0 , \\\\ \\{ 0 , 1 \\} ^ d \\backslash ( 0 , \\ldots , 0 ) & \\textup { e l s e . } \\end{cases} \\end{align*}"}
-{"id": "6632.png", "formula": "\\begin{align*} T ' = ( 1 - X Y _ 1 ) ( 1 - X Y _ 2 ) ( x _ s - X Y _ 3 ) - ( 1 - X Z _ 1 ) ( 1 - X Z _ 2 ) ( x _ s - X Z _ 3 ) . \\end{align*}"}
-{"id": "1804.png", "formula": "\\begin{align*} c _ { 2 , 1 } \\left ( B _ { s } = B _ { t } , s \\in I t \\in J \\right ) = 0 . \\end{align*}"}
-{"id": "4531.png", "formula": "\\begin{align*} \\int f ( t ) d \\nu ( t ) = \\int f ( | x _ 1 - x _ 2 | , \\dots , | x _ i - x _ j | , \\dots , | x _ { k - 1 } - x _ k | ) d \\mu ( x _ 1 ) \\dots d \\mu ( x _ k ) , f \\in C _ 0 ( \\mathbb { R } ^ { k ( k - 1 ) / 2 } ) , \\end{align*}"}
-{"id": "2286.png", "formula": "\\begin{align*} H R ( z ) | f ) & = - | f ) + z R ( z ) | f ) \\\\ ( f | R ( z ) H f & = - ( f | + z ( f | R ( z ) f . \\end{align*}"}
-{"id": "8037.png", "formula": "\\begin{align*} \\{ \\underline { 1 } , y , \\widehat { y ^ 2 } , \\dots , & \\boxed { y ^ { 2 j + 1 } } , \\dots , \\underline { y ^ l } , \\dots , \\boxed { y ^ { 2 l - 2 j - 1 } } , \\dots , \\underline { y ^ { 2 l } } , \\dots , \\\\ & \\overline { y ^ { n + 2 r + 1 } } , \\dots , \\underline { y ^ { 3 l } } , \\dots , \\overline { y ^ { 2 n - 2 r - 1 } } , \\dots , \\widehat { y ^ { 2 n - 2 } } , y ^ { 2 n - 1 } \\} \\\\ \\end{align*}"}
-{"id": "3987.png", "formula": "\\begin{align*} a _ F ( T ) = c ^ { - 1 } _ k \\frac { g _ { \\Phi F } ( m ) } { g _ k ( m ) } a _ 2 ^ k ( T ) + c _ { \\phi ^ \\circ _ m } ( n , r ) + a _ G ( T ) , \\end{align*}"}
-{"id": "9872.png", "formula": "\\begin{align*} \\sum _ { x \\in A } \\sum _ { s \\in S } A ( s x ) = | S | | A | / K \\ , , \\end{align*}"}
-{"id": "3203.png", "formula": "\\begin{align*} \\| \\mathcal { L } _ X \\mathcal { L } _ Y - \\mathcal { L } _ Y \\mathcal { L } _ X \\| _ 2 & = \\max _ { \\| Z \\| _ F = 1 } \\| \\mathcal { L } _ X \\mathcal { L } _ Y Z - \\mathcal { L } _ Y \\mathcal { L } _ X Z \\| _ F \\\\ & = \\max _ { \\| Z \\| _ F = 1 } \\frac { \\| ( X Y - Y X ) Z - Z ( X Y - Y X ) \\| _ F } { 4 } \\\\ & \\le \\frac { \\| X Y - Y X \\| _ F } { 2 } \\\\ & \\le \\| X \\circ Y - \\mu I \\| _ F , \\end{align*}"}
-{"id": "7173.png", "formula": "\\begin{gather*} T _ { x } \\mathbb { R } ^ { n } = \\mathbb { L } _ { x } ^ { - } \\oplus \\mathbb { L } _ { x } ^ { + } \\oplus \\mathbb { L } _ { x } ^ { 0 } , \\quad \\dim \\mathbb { L } _ { x } ^ { \\pm , 0 } = n _ { \\pm , 0 } , \\end{gather*}"}
-{"id": "8397.png", "formula": "\\begin{align*} D ^ m \\frac { 1 } { z ( \\alpha , t ) - z _ j ( t ) } = \\frac { ( - 1 ) ^ m m ! } { ( z ( \\alpha , t ) - z _ j ( t ) ) ^ { m + 1 } } , \\end{align*}"}
-{"id": "3798.png", "formula": "\\begin{align*} \\lambda ( \\alpha ) = x y ( b - 1 ) 1 ^ d \\end{align*}"}
-{"id": "657.png", "formula": "\\begin{align*} \\varphi ^ { - 1 } \\{ j \\} = \\left \\{ i _ 1 < \\dots < i _ { k ^ { ( \\varphi ) } _ j } \\right \\} \\ . \\end{align*}"}
-{"id": "3164.png", "formula": "\\begin{align*} { \\rm i n d } _ \\varGamma ( U ) = { \\rm i n d e x } ( \\varGamma _ + , C _ + ) + { \\rm i n d e x } ( \\varGamma _ + , C _ - ) . \\end{align*}"}
-{"id": "9529.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { ( b / a ) _ n a ^ n } { ( 1 - c q ^ n ) ( b ) _ n } = \\sum _ { m = 0 } ^ { \\infty } \\frac { ( b / c ) _ m c ^ m } { ( b ) _ m } \\left ( \\frac { a q ^ m } { 1 - a q ^ m } - \\frac { b q ^ m } { 1 - b q ^ m } \\right ) \\end{align*}"}
-{"id": "8634.png", "formula": "\\begin{align*} z _ \\varepsilon ^ - = z _ \\varepsilon ^ + . \\end{align*}"}
-{"id": "2654.png", "formula": "\\begin{align*} \\mathrm { P f } \\begin{bmatrix} K _ { N } ( x _ { j } , x _ { k } ) \\end{bmatrix} = \\sqrt { \\det \\begin{bmatrix} K _ { N } ( x _ { j } , x _ { k } ) \\end{bmatrix} } \\propto \\det ( \\Phi ) \\ , \\mathrm { P f } \\ ! \\begin{bmatrix} E & H \\\\ - H ^ { t } & \\alpha \\end{bmatrix} . \\end{align*}"}
-{"id": "7212.png", "formula": "\\begin{align*} ( k + 1 ) ^ { - r } = \\int _ 0 ^ 1 t ^ { k } \\ , d \\sigma _ r ( t ) \\qquad \\mbox { f o r a n y } k \\geq 0 . \\end{align*}"}
-{"id": "3364.png", "formula": "\\begin{align*} p = D _ 0 + n d _ 1 D _ 1 \\ , . \\end{align*}"}
-{"id": "9601.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { q ^ { n ( n + 1 ) } } { ( q ) _ n ( 1 - q ^ n ) } F \\left ( q ^ N , q ^ n ; - q ^ n \\right ) = \\frac { 1 } { 2 ( - q ) _ N } \\sum _ { k = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ k \\end{matrix} \\right ] \\frac { q ^ { k ( k + 1 ) / 2 } } { ( 1 - q ^ k ) } \\left ( \\frac { ( - q ) _ k } { ( q ) _ k } - 1 \\right ) . \\end{align*}"}
-{"id": "1239.png", "formula": "\\begin{align*} \\tilde S ( \\eta , \\mu / \\sqrt { \\eta } ) & = e ^ { - i a \\ln 4 } e ^ { i a \\ln | \\eta ^ 2 - \\mu ^ 2 / \\eta | } \\left ( A + B e ^ { 2 i a \\ln | ( \\eta + \\mu / \\sqrt { | \\eta | } ) / 2 | } \\frac { e ^ { i \\beta ( \\eta + \\mu / \\sqrt { \\eta } ) ^ 3 } } { ( \\eta + \\mu / \\sqrt { \\eta } ) ^ 3 } \\right ) \\\\ & \\times \\left ( A + B e ^ { 2 i a \\ln | ( \\eta - \\mu / \\sqrt { | \\eta | } ) / 2 | } \\frac { e ^ { i \\beta ( \\eta - \\mu / \\sqrt { \\eta } ) ^ 3 } } { ( \\eta - \\mu / \\sqrt { \\eta } ) ^ 3 } \\right ) \\end{align*}"}
-{"id": "9198.png", "formula": "\\begin{align*} \\mathrm { c o n j } _ { p } = \\mathrm { i n j } _ { p } = \\pi \\sqrt { \\frac { 1 } { 3 } } > \\frac { \\pi } { 2 } . \\end{align*}"}
-{"id": "2638.png", "formula": "\\begin{align*} I _ { 1 } = \\prod _ { k = 0 } ^ { N - 1 } k ! \\ , \\frac { \\det [ e ^ { - \\eta _ { + } \\sigma _ { i } \\lambda _ { j } } ] _ { i , j = 1 } ^ N } { \\Delta _ { N } ( \\sigma ) \\Delta _ { N } ( \\lambda ) } . \\end{align*}"}
-{"id": "8424.png", "formula": "\\begin{align*} \\sup _ { \\substack { \\alpha \\neq \\beta \\\\ 0 \\leq t < T _ 0 ^ * } } \\Big | \\frac { z ( \\alpha , t ) - z ( \\beta , t ) } { \\alpha - \\beta } \\Big | + \\sup _ { \\substack { \\alpha \\neq \\beta \\\\ 0 \\leq t < T _ 0 ^ * } } \\Big | \\frac { \\alpha - \\beta } { z ( \\alpha , t ) - z ( \\beta , t ) } \\Big | = \\infty . \\end{align*}"}
-{"id": "3706.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { i \\in B } w _ i \\epsilon _ i = \\sum _ t \\frac { w _ t } { N } \\sum _ { i \\in U _ t } \\delta _ i \\epsilon _ i = \\sum _ t w _ t \\frac { N _ t } { N } \\big ( C o v _ { N _ t } ( \\delta _ i , \\epsilon _ i ) + ( \\frac { 1 } { N _ t } \\sum _ { i \\in U _ t } \\delta _ i ) ( \\frac { 1 } { N _ t } \\sum _ { i \\in U _ t } \\epsilon _ i ) \\big ) \\rightarrow 0 \\end{align*}"}
-{"id": "5665.png", "formula": "\\begin{align*} \\| U ^ { n } \\| _ { L ^ { p } ( \\Omega ; H ) } & \\leq \\| U _ { h } ^ { n } \\| _ { L ^ { p } ( \\Omega ; H ) } + \\Big \\| \\int _ { 0 } ^ { T } \\sum _ { j = 0 } ^ { n - 1 } \\chi _ { [ t _ { j } , t _ { j + 1 } ) } ( s ) B _ { n - j } P _ { h } \\d W ( s ) \\Big \\| _ { L ^ { p } ( \\Omega ; H ) } : = { \\rm I } _ { 1 } + { \\rm I } _ { 2 } . \\end{align*}"}
-{"id": "7143.png", "formula": "\\begin{align*} \\frac { 1 } { L } \\int _ 0 ^ L \\rho d x = \\frac 1 L \\int _ 0 ^ L \\rho _ 0 d x : = \\bar \\rho . \\end{align*}"}
-{"id": "4575.png", "formula": "\\begin{align*} \\gamma _ s = \\begin{cases} s & s \\in ( 0 , ( n - 1 ) / 2 ] ; \\\\ ( n - 1 ) / 2 & s \\in [ ( n - 1 ) / 2 , n / 2 ] ; \\\\ ( n + 2 s - 2 ) / 4 & s \\in [ n / 2 , ( n + 2 ) / 2 ] ; \\\\ s - 1 & s \\in [ ( n + 2 ) / 2 , n ) . \\end{cases} \\end{align*}"}
-{"id": "9451.png", "formula": "\\begin{align*} f _ { \\theta } ( z ) - \\mathcal { E } _ 2 ( z ) = - \\sum _ { \\tau \\in \\mathcal { F } _ N } \\nu _ \\tau ^ { ( N ) } ( f ) - \\sum _ { m = 1 } ^ \\infty \\biggl ( \\sum _ { \\tau \\in \\mathcal { F } _ N } \\nu _ \\tau ^ { ( N ) } ( f ) J _ { N , m } ( \\tau ) \\biggr ) q ^ m . \\end{align*}"}
-{"id": "5085.png", "formula": "\\begin{align*} & \\operatorname { T r a c e } ( S _ { x , \\tau } ) = \\sum _ { j = 1 } ^ n \\langle x _ j , \\tau _ j \\rangle = \\sum _ { j = 1 } ^ n \\langle \\tau _ j , x _ j \\rangle ; \\\\ & \\operatorname { T r a c e } ( S ^ 2 _ { x , \\tau } ) = \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n \\langle \\tau _ j , x _ k \\rangle \\langle \\tau _ k , x _ j \\rangle = \\sum _ { j = 1 } ^ n \\sum _ { k = 1 } ^ n \\langle \\tau _ j , \\tau _ k \\rangle \\langle x _ k , x _ j \\rangle . \\end{align*}"}
-{"id": "4879.png", "formula": "\\begin{align*} \\mathcal { W } = \\underset { \\Gamma } { \\bigcup } \\ , \\ , \\mathcal { W } ( \\sqrt { - 1 } \\Gamma ) . \\end{align*}"}
-{"id": "7279.png", "formula": "\\begin{align*} R _ { n _ q } \\psi _ p ( a ) ( 1 - R _ { n _ q } ) = ( 1 - R _ { n _ q } ) \\psi _ p ( a ) R _ { n _ q } = 0 , \\end{align*}"}
-{"id": "9625.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } d ( n , N ) q ^ n + \\sum _ { n = 1 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { q ^ { n ( n + 1 ) / 2 } } { 1 - q ^ n } \\frac { ( - q ) _ n } { ( q ) _ n } = \\frac { 1 } { 2 } \\left \\{ \\frac { ( - q ) _ N } { ( q ) _ N } - 1 \\right \\} + \\sum _ { n = 1 } ^ { N } \\frac { ( - q ) _ n } { ( q ) _ n } \\frac { q ^ n } { 1 - q ^ n } \\end{align*}"}
-{"id": "9043.png", "formula": "\\begin{align*} W ( \\mu ) : = \\sum _ { \\xi \\in ( \\mu _ A ) } | X _ { \\xi } | ^ 2 \\end{align*}"}
-{"id": "477.png", "formula": "\\begin{align*} \\mathfrak { D } _ { \\omega } ( a _ 1 , \\dots , a _ n ) = \\omega \\bigl ( ( a _ 1 x _ 1 + \\cdots + a _ n x _ n ) ^ m / m ! \\bigr ) . \\end{align*}"}
-{"id": "10043.png", "formula": "\\begin{align*} V ( r , \\theta , t ) = U ( r e ^ { i \\theta } , t ) = \\sum _ { j = 1 } ^ n \\frac { m _ j } { | r e ^ { i \\theta } - q _ j ( \\omega t ) | } . \\end{align*}"}
-{"id": "8362.png", "formula": "\\begin{align*} \\Big | \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j } { \\pi } R e \\Big \\{ \\frac { D _ t Z - \\dot { z } _ j } { c _ 0 ^ j ( \\alpha - \\omega _ 0 ^ j ) ^ 2 } \\Big \\} \\Big | \\leq & \\frac { \\sum _ { j = 1 } ^ N | \\lambda _ j | } { \\pi } \\max _ { 1 \\leq j \\leq N } | c _ 0 ^ j | ^ { - 1 } ( \\| D _ t Z \\| _ { \\infty } + | \\dot { z } _ j | ) ( \\inf _ { \\alpha \\in \\mathbb { R } } | \\alpha - \\Phi ( z _ j ) | ) ^ { - 2 } \\end{align*}"}
-{"id": "5004.png", "formula": "\\begin{align*} A = \\bigcup _ { m = 1 } ^ M \\psi _ m ( A ) , \\end{align*}"}
-{"id": "7028.png", "formula": "\\begin{align*} \\Omega ( A _ { n } , B _ { k n } ) = ( - 1 ) ^ { N _ { k n } } \\sum \\limits ^ { N _ { k n } - M _ { k n } } _ { l = 0 } \\sum \\limits ^ { l } _ { m = 0 } ( - 1 ) ^ { m } \\sigma _ { m } ( M _ { k n } , N _ { k n } ) S _ { l - m } ^ { n } T _ { N - M - l } ^ { k n } \\end{align*}"}
-{"id": "1601.png", "formula": "\\begin{align*} S ' ( x ) = \\exp \\left ( - \\int _ c ^ x \\frac { 2 \\mu ( y ) y } { \\sigma ^ 2 ( y ) } \\ , d y \\right ) , \\end{align*}"}
-{"id": "5027.png", "formula": "\\begin{align*} s _ 0 : = \\frac { c _ 1 I ^ { 3 / 2 } } { m ^ { 3 / 2 } n ^ { 1 / 2 } ( \\log m n ) ^ 4 } \\end{align*}"}
-{"id": "8509.png", "formula": "\\begin{align*} \\lambda ^ 2 + | \\lambda x ( 0 ) | \\leq c _ 0 \\epsilon , c _ 0 = \\frac { 1 } { ( ( s + 1 2 ) ! ) ^ 2 } . \\end{align*}"}
-{"id": "1991.png", "formula": "\\begin{align*} \\chi ( \\P ^ n ) = \\int _ { \\P ^ n } e ( T \\P ^ n ) . \\end{align*}"}
-{"id": "8296.png", "formula": "\\begin{align*} ( \\partial _ t ^ 2 - i a \\partial _ { \\alpha } ) \\theta = - 2 [ z _ t , \\mathfrak { H } \\frac { 1 } { z _ { \\alpha } } + \\bar { \\mathfrak { H } } \\frac { 1 } { \\bar { z } _ { \\alpha } } ] z _ { t \\alpha } + \\frac { 1 } { \\pi i } \\int _ { - \\infty } ^ { \\infty } \\Big ( \\frac { z _ t ( \\alpha , t ) - z _ t ( \\beta , t ) } { z ( \\alpha , t ) - z ( \\beta , t ) } \\Big ) ^ 2 ( z - \\bar { z } ) _ { \\beta } d \\beta : = g . \\end{align*}"}
-{"id": "5952.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\ , \\rho ^ { g _ n } \\left ( h \\left ( \\frac { W ( 1 ) } { \\sqrt { n } } \\right ) \\right ) = \\sup _ { x \\in \\R ^ d } ( h ( x ) - g ( x ) ) . \\end{align*}"}
-{"id": "3810.png", "formula": "\\begin{align*} \\mathfrak { C } _ 2 = \\mathop { \\sum \\sum } _ { \\substack { d _ 2 | q _ 2 \\\\ d _ 2 ' | q _ 2 ' } } d _ 2 d _ 2 ' \\mathop { \\mathop { \\sideset { } { ^ \\star } \\sum } _ { \\substack { \\alpha \\bmod { q _ 2 } \\\\ n _ 1 \\alpha \\equiv - m \\bmod { d _ 2 } } } \\ ; \\mathop { \\sideset { } { ^ \\star } \\sum } _ { \\substack { \\alpha ' \\bmod { q _ 2 ' } \\\\ n _ 1 \\alpha ' \\equiv - m ' \\bmod { d _ 2 ' } } } } _ { \\bar { \\alpha } q _ 2 ' - \\bar { \\alpha } ' q _ 2 \\equiv n _ 2 \\bmod { q _ 2 q _ 2 ' } } \\ ; 1 . \\end{align*}"}
-{"id": "8843.png", "formula": "\\begin{align*} 0 & \\le x _ 1 \\le \\d ; \\\\ x _ 1 & \\le x _ i \\le x _ N i = 2 , \\dots , N - 1 ; \\\\ x _ k , x _ { k + 1 } , \\dots , x _ N & \\in [ a ( 1 - \\Delta ) , a ) \\end{align*}"}
-{"id": "6859.png", "formula": "\\begin{align*} ( a '' ) ^ k + \\Gamma _ { i j } ^ k ( a ' ) ^ i ( a ' ) ^ j = 0 \\ , . \\end{align*}"}
-{"id": "1895.png", "formula": "\\begin{align*} r ^ \\mu = \\frac { 1 } { \\sqrt { 1 - v ^ 2 } } ( v , 1 ) . \\end{align*}"}
-{"id": "927.png", "formula": "\\begin{align*} E _ k ^ { q } \\le k ! \\left ( I _ k - \\sum _ { j _ 1 , \\ldots , j _ k = 0 } ^ { q } C ^ 2 _ { j _ k \\ldots j _ 1 } \\right ) , \\end{align*}"}
-{"id": "5331.png", "formula": "\\begin{align*} B . w ^ k = i \\left ( - X + Y \\right ) . w ^ k = i \\left ( - a _ k w ^ { k + 2 } + b _ k w ^ { k - 2 } \\right ) . \\end{align*}"}
-{"id": "2141.png", "formula": "\\begin{align*} \\mathcal F f ( \\xi ) = c _ k ^ { - 1 } \\int _ { \\mathbb R ^ N } E ( - i \\xi , \\mathbf x ) f ( \\mathbf x ) \\ , d w ( \\mathbf x ) , \\end{align*}"}
-{"id": "2148.png", "formula": "\\begin{align*} \\tau _ { \\mathbf { x } } f ( - \\mathbf { y } ) = \\int _ { \\mathbb { R } ^ N } { ( \\tilde { f } \\circ A ) } ( \\mathbf { x } , \\mathbf { y } , \\eta ) \\ , d \\mu _ { \\mathbf { x } } ( \\eta ) \\mathbf { x } , \\mathbf { y } \\in \\mathbb { R } ^ N . \\end{align*}"}
-{"id": "8126.png", "formula": "\\begin{align*} \\sum _ { k \\ge 0 } \\chi _ { k P } \\ , t ^ k \\ = \\ \\frac { \\varphi ^ \\ast _ P ( t ) } { ( 1 - t ) \\det ( I - \\rho t ) } , \\end{align*}"}
-{"id": "1866.png", "formula": "\\begin{align*} \\Phi ( x , t ) = \\sum _ { n = 1 } ^ \\infty \\left [ \\alpha _ n \\Phi ^ { \\rm o d d } ( n , x , t ) + \\alpha _ n ^ * \\overline { \\Phi ^ { \\rm o d d } ( n , x , t ) } \\right ] + \\sum _ { j = 1 } ^ \\infty \\left [ \\beta _ j \\Phi ^ { \\rm e v e n } ( j , x , t ) + \\beta _ j ^ * \\overline { \\Phi ^ { \\rm e v e n } ( j , x , t ) } \\right ] \\end{align*}"}
-{"id": "8491.png", "formula": "\\begin{align*} \\dot { x } ( t ) = R e F ( z _ 2 ( t ) , t ) - R e F ( 0 + i y ( t ) , t ) = R e ~ F _ x ( \\tilde { x } + i y ( t ) , t ) x ( t ) , \\end{align*}"}
-{"id": "1495.png", "formula": "\\begin{align*} \\widetilde { N } _ { n } = \\sum _ { 1 \\leq | \\alpha | \\leq s } \\int _ { 0 } ^ { T } \\left [ \\| \\partial ^ { \\alpha } D w ^ { n } \\| _ { 0 } ^ { 2 } + \\| \\partial ^ { \\alpha } D \\mu ^ { n } \\| _ { 0 } ^ { 2 } \\right ] \\ d \\tau , \\end{align*}"}
-{"id": "3486.png", "formula": "\\begin{align*} f ' _ M ( q _ M ( \\pm \\theta ) ) = 0 . \\end{align*}"}
-{"id": "6736.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Box h _ { k l } = 0 . \\end{align*}"}
-{"id": "9318.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } y _ { i } = k ; \\end{align*}"}
-{"id": "2796.png", "formula": "\\begin{align*} T ( \\mathbf { n } ; \\mathbf { m } ) = \\prod _ { i = 1 } ^ S \\frac { n _ i ! } { ( n _ i - m _ i ) ! m _ i ! } e _ i ^ { m _ i } ( 1 - e _ i ) ^ { n _ i - m _ i } . \\end{align*}"}
-{"id": "6770.png", "formula": "\\begin{align*} \\| \\sqrt { A + E } - \\sqrt { A } \\| _ 2 = O ( \\sqrt { \\| E \\| _ 2 } ) , \\end{align*}"}
-{"id": "2725.png", "formula": "\\begin{align*} \\Sigma ^ { \\mathrm { l o c a l } } = \\left \\{ w \\in \\Sigma _ { + } ^ { 2 } \\cup \\overline { \\Sigma _ { + } ^ { 2 } } : \\abs { w - z _ { + } } < \\delta _ { 1 } \\ , \\ , \\abs { w - z _ { - } } < \\delta _ { 1 } \\right \\} . \\end{align*}"}
-{"id": "4583.png", "formula": "\\begin{align*} \\dim _ { L ^ 2 , t } \\mu = n - t - \\limsup \\frac { \\log A ( \\mu , R , | . | ^ { - 1 } ) } { \\log R } . \\end{align*}"}
-{"id": "8638.png", "formula": "\\begin{align*} F \\left ( p _ \\varepsilon ^ \\pm , q _ \\varepsilon ^ \\pm , u _ 0 ( 0 ) , z _ \\varepsilon ^ \\pm , \\varepsilon \\right ) = 0 \\ \\ \\textrm { f o r s m a l l } \\ \\varepsilon > 0 . \\end{align*}"}
-{"id": "195.png", "formula": "\\begin{align*} X = \\sqrt { x ^ 3 } \\dfrac { \\partial } { \\partial x ^ 2 } \\end{align*}"}
-{"id": "2371.png", "formula": "\\begin{align*} \\gamma _ Q ( 1 ) = & 1 \\\\ \\gamma _ Q ( X _ { i _ 1 } \\cdots X _ { i _ s } ) = & Q _ { i _ 1 , \\cdots , i _ s } \\\\ \\gamma _ Q ( X _ { i _ 1 } \\cdots X _ { i _ k } ) = & 0 \\\\ \\gamma _ Q ( X _ { A ( 1 ) } \\cdots X _ { A ( p s ) } ) = & \\sum _ { \\{ S _ 1 , \\cdots , S _ p \\} \\in \\mathfrak { P } ( [ 1 , p s ] ) , \\# S _ i = s } Q _ { A | S _ 1 } \\cdots Q _ { A | S _ p } \\\\ & A : [ 1 , p s ] \\to [ 1 , n ] . \\end{align*}"}
-{"id": "9537.png", "formula": "\\begin{align*} N _ { S _ 1 } ( m , n ) : = \\sum _ { j = 1 } ^ N N _ { S _ 1 } \\left ( m , n ; \\boxed { j } \\right ) , \\end{align*}"}
-{"id": "3433.png", "formula": "\\begin{gather*} c _ t ( S ^ n ( V ) ) : = \\big ( 1 + c _ 2 ( S ^ n ( V ) ) t ^ 2 + c _ 4 ( S ^ n ) t ^ 4 + \\cdots \\big ) = ( 1 + n b t ) ( 1 + ( n - 2 ) b t ) \\cdots ( 1 - n b t ) . \\end{gather*}"}
-{"id": "2114.png", "formula": "\\begin{align*} \\lambda _ k ( \\omega ) : = \\inf _ { V \\in \\mathcal { G } _ { d - k + 1 } } \\sup _ { y \\in V } \\limsup \\limits _ { \\mathbb { N } \\ni t \\to \\infty } \\frac { 1 } { t } \\log | \\Phi _ { \\omega } ( t _ 0 , t ) y | , \\ k = 1 , \\ldots , d . \\end{align*}"}
-{"id": "3458.png", "formula": "\\begin{align*} s = ( 1 - k ) + \\sigma \\sqrt { \\frac { N } { M + 1 } } , t = ( 1 - k ) + \\tau \\sqrt { \\frac { N } { M + 1 } } . \\end{align*}"}
-{"id": "8166.png", "formula": "\\begin{align*} u ( \\pi ) = x _ 1 ( n _ 1 ) x _ 2 ( n _ 2 ) \\dots x _ r ( n _ r ) \\in U ( \\hat { \\mathfrak { g } } ) . \\end{align*}"}
-{"id": "2365.png", "formula": "\\begin{align*} a ^ + _ x \\Phi ( \\mathfrak { m } ) & = \\Phi ( \\mathfrak { m } + \\ 1 _ x ) \\\\ a _ x \\Phi ( \\mathfrak { m } ) & = \\mathfrak { m } ( x ) \\Phi ( \\mathfrak { m } - \\ 1 _ x ) \\mathfrak { m } ( x ) > 0 \\\\ a _ x \\Phi ( \\emptyset ) & = 0 x . \\end{align*}"}
-{"id": "813.png", "formula": "\\begin{align*} ( T - \\rho ) * _ d X = X ^ { q ^ d } + ( T - \\rho ) X = 0 \\end{align*}"}
-{"id": "833.png", "formula": "\\begin{align*} \\boxed { \\sigma _ D \\circ \\sigma ( x ) = \\sigma \\circ \\sigma _ D ( x ) , D \\in \\mathbb { F } _ { q ^ d } [ T ] / M . } \\end{align*}"}
-{"id": "5740.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial ^ 2 } { \\partial x ^ 2 } + \\frac { 1 } { x } \\frac { \\partial } { \\partial x } + \\frac { \\partial ^ 2 } { \\partial y ^ 2 } \\right ) \\left [ \\sum _ { m , n \\in \\N } T _ { m , n } ( t ) J _ 0 ( \\gamma _ m x ) \\sin ( n \\pi y ) \\right ] = - \\sum _ { m , n \\in \\N } T _ { m , n } ( t ) ( ( n \\pi ) ^ 2 + \\gamma _ m ^ 2 ) J _ 0 ( \\gamma _ m x ) \\sin ( n \\pi y ) . \\end{align*}"}
-{"id": "10072.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { u } { ( 1 + 3 a _ + u ^ 3 t ) ^ { 1 / 3 } } & \\leq \\widetilde \\Psi _ t ( u , \\theta _ n ^ 0 , G _ n ^ 0 ) \\leq \\frac { u } { ( 1 + 3 a _ - u ^ 3 t ) ^ { 1 / 3 } } , \\\\ \\frac { 1 } { ( 1 + 3 a _ + u ^ 3 t ) ^ { 1 / ( 3 \\alpha ) } } & \\leq \\partial _ u \\widetilde \\Psi _ t ( u , \\theta _ n ^ 0 , G _ n ^ 0 ) \\leq \\frac { 1 } { ( 1 + 3 a _ - u ^ 3 t ) ^ { \\alpha / 3 } } . \\end{aligned} \\end{align*}"}
-{"id": "7853.png", "formula": "\\begin{align*} | B _ { R + r ^ { 1 / 2 } } | = | B _ R | \\bigg ( 1 + \\frac { r ^ { 1 / 2 } } { R } \\bigg ) ^ r \\le e ^ { r ^ { 3 / 2 } / R } | B _ R | \\le | B _ R | \\big ( 1 + r ^ { 3 / 2 } R ^ { - 1 } e ^ { r ^ { 3 / 2 } / R } \\big ) , \\end{align*}"}
-{"id": "1764.png", "formula": "\\begin{align*} C _ { s , t , \\eta , H } = \\sqrt { \\frac { \\eta ^ { 2 } ( t - s ) ^ { 2 H } } { 2 \\left [ \\gamma _ { H } ( t - s ) ^ { 2 H } + ( t - s ) \\right ] ^ { 2 } } + 2 } . \\end{align*}"}
-{"id": "2897.png", "formula": "\\begin{align*} \\begin{aligned} C ^ { i } & = - \\frac { 1 } { 4 } G ^ { i } - \\frac { 1 } { 4 } , \\\\ C ^ { i j } & = - \\frac { 1 } { 4 } K ^ { i j } , \\\\ P ^ { i j } & = C ^ { i j } - C ^ { i } - C ^ { j } = - \\frac { 1 } { 4 } K ^ { i j } + \\frac { 1 } { 4 } \\left ( G ^ { i } + G ^ { j } \\right ) + \\frac { 1 } { 2 } , \\end{aligned} \\end{align*}"}
-{"id": "3574.png", "formula": "\\begin{align*} C _ { N E } = \\sum _ { i = 1 } ^ { 1 0 8 } \\mathbb { R } _ { \\geq 0 } ( \\psi _ { T _ { i } } - \\psi _ { T _ { 1 } } ) . \\end{align*}"}
-{"id": "8586.png", "formula": "\\begin{align*} 0 \\le T _ \\infty : = ( \\lim _ { N \\to \\infty } T _ N ) ^ * \\le T ^ * _ { V \\cap U _ 1 } \\end{align*}"}
-{"id": "8718.png", "formula": "\\begin{align*} { \\Lambda } ( k , \\cdot ) = \\mathbf { Q } _ k ( 1 , \\cdot ) , k = 1 , \\ldots , M . \\end{align*}"}
-{"id": "2379.png", "formula": "\\begin{align*} m = & 0 , \\pm 1 , \\pm 2 , \\cdots , \\pm N / 2 \\\\ m = & \\pm 1 / 2 , \\pm 3 / 2 , \\cdots , \\pm N / 2 \\end{align*}"}
-{"id": "7249.png", "formula": "\\begin{align*} \\Pi \\phi = \\left < \\phi , \\phi ^ * _ { k } \\right > _ { L _ 2 } \\phi ^ * _ { k } \\end{align*}"}
-{"id": "2051.png", "formula": "\\begin{align*} \\tilde { a } _ { \\chi , R _ * ' \\frac { R } { R _ * } } ( p ) = \\begin{cases} \\overline { \\chi } ^ 2 ( p ) a _ \\chi ( p ) & p \\nmid R _ * ' \\frac { R } { R _ * } \\\\ \\overline { a _ { \\chi } ( p ) } & p \\mid R _ * ' \\frac { R } { R _ * } . \\end{cases} \\end{align*}"}
-{"id": "3642.png", "formula": "\\begin{align*} \\alpha _ { - 2 } : = - \\frac { 2 t } { 3 } a _ 1 - \\frac { 1 } { 3 } v _ { ( 2 , 3 ) } + \\frac { 4 } { 3 } ( a _ 2 + a _ { - 2 } ) \\cdot v _ { ( 1 , 3 ) } + \\frac { 4 } { 3 } ( a _ 3 + a _ { - 3 } ) \\cdot v _ { ( 1 , 2 ) } \\in M _ 0 ^ { ( a _ 1 ) } . \\end{align*}"}
-{"id": "901.png", "formula": "\\begin{align*} \\Gamma = q ^ { - \\sum _ a \\gamma ( t a ) \\gamma ( h a ) } \\end{align*}"}
-{"id": "4192.png", "formula": "\\begin{align*} P ^ { ( 3 ) } _ + ( z ) \\left ( P ^ { ( 3 ) } _ - ( z ) \\right ) ^ { - 1 } = \\mathbb { I } - \\frac { A _ n ^ { ( 1 ) } ( 0 ) A _ n ^ { ( 1 ) } ( z ) A _ n ^ { ( 1 ) } ( 0 ) } { n ^ 9 z ^ 3 } + \\mathcal { O } ( n ^ { - 1 } ) . \\end{align*}"}
-{"id": "9576.png", "formula": "\\begin{align*} S _ 2 ^ { * } & = \\frac { 1 } { ( 1 - c ) ( c q ) _ N } \\sum _ { k = 1 } ^ N \\frac { ( c q ) _ k q ^ k } { ( q ) _ k ( 1 - q ^ k ) } \\sum _ { j = 0 } ^ { N - k } \\frac { ( 1 / c ) _ j ( c q ) ^ j ( c q ) _ { N - j - k } } { ( q ) _ j ( q ) _ { N - j - k } } \\\\ & = \\frac { 1 } { ( 1 - c ) ( c q ) _ N } \\sum _ { k = 1 } ^ N \\frac { ( c q ) _ k ( c q ) _ { N - k } q ^ k } { ( q ) _ k ( 1 - q ^ k ) ( q ) _ { N - k } } \\sum _ { j = 0 } ^ { N - k } \\frac { ( 1 / c ) _ j q ^ j \\left ( q ^ { - ( N - k ) } \\right ) _ j } { ( q ) _ j \\left ( q ^ { - ( N - k ) } / c \\right ) _ j } , \\end{align*}"}
-{"id": "5717.png", "formula": "\\begin{align*} & \\frac { 1 } { \\Gamma ( 2 - \\alpha ) } \\int _ 0 ^ x { ( x - t ) ^ { 1 - \\alpha } } \\int _ 0 ^ t ( t - z ) ^ { \\alpha - 2 } E _ { \\alpha , \\alpha - 1 } ( \\lambda ( t - z ) ^ { \\alpha } ) f ' ( z ) d z d t \\\\ & = \\frac { 1 } { \\Gamma ( 2 - \\alpha ) } \\int _ 0 ^ x f ' ( z ) \\int _ z ^ x { ( x - t ) ^ { 1 - \\alpha } } ( t - z ) ^ { \\alpha - 2 } E _ { \\alpha , \\alpha - 1 } ( \\lambda ( t - z ) ^ { \\alpha } ) d t d z \\end{align*}"}
-{"id": "9489.png", "formula": "\\begin{align*} g _ { \\beta , M } ( t ) = ( m i n \\{ t , M \\} ) ^ { \\beta } = \\left \\{ \\begin{array} { r c } t ^ { \\beta } , & \\mbox { s e } \\ t \\leq M , \\\\ M ^ { \\beta } , & \\mbox { s e } \\ t > M , \\end{array} \\right . \\end{align*}"}
-{"id": "2154.png", "formula": "\\begin{align*} \\frac { C } { w ( B ( \\mathbf { x } , t ) ) ^ { 1 / 2 } } \\geq \\| h _ { t ^ 2 } ( \\mathbf { x } , \\cdot ) \\| _ { L ^ 2 ( d w ) } = \\| E ( i \\cdot , \\mathbf { x } ) e ^ { - t ^ 2 \\| \\cdot \\| ^ 2 } \\| _ { L ^ 2 ( d w ) } \\geq e ^ { - 1 } \\| E ( i \\cdot , \\mathbf { x } ) \\| _ { L ^ 2 ( B ( 0 , t ^ { - 1 } ) , d w ) } . \\end{align*}"}
-{"id": "7750.png", "formula": "\\begin{align*} \\langle x _ { \\alpha } , x \\rangle E = E \\langle x _ { \\alpha } , x \\rangle E = - \\overline { \\lambda _ \\alpha } \\| x \\| E \\ , \\ , \\ , \\ , \\ , ( \\alpha \\in \\Lambda ) . \\end{align*}"}
-{"id": "4582.png", "formula": "\\begin{align*} A ( \\mu , R , h ) = \\int _ { B ( 0 , R ) } | \\hat { \\mu } ( \\omega ) | ^ 2 h ( \\omega ) d \\omega \\end{align*}"}
-{"id": "379.png", "formula": "\\begin{align*} v _ n & = \\frac { \\sqrt { k + 1 } + \\sqrt { k - 1 } } { 2 \\sqrt { k + 1 } } ( k + \\sqrt { k ^ 2 - 1 } ) ^ n + \\frac { \\sqrt { k + 1 } - \\sqrt { k - 1 } } { 2 \\sqrt { k + 1 } } ( k - \\sqrt { k ^ 2 - 1 } ) ^ n , \\\\ w _ m & = \\frac { 1 } { 2 \\sqrt { 1 6 k ^ 3 - 4 k } } \\big ( ( x _ 1 \\sqrt { 1 6 k ^ 3 - 4 k } + z _ 1 \\sqrt { k - 1 } ) ( 4 k ^ 2 - 2 k - 1 + \\sqrt { ( 1 6 k ^ 3 - 4 k ) ( k - 1 ) } ) ^ m + \\\\ & \\ , + ( x _ 1 \\sqrt { 1 6 k ^ 3 - 4 k } - z _ 1 \\sqrt { k - 1 } ) ( 4 k ^ 2 - 2 k - 1 - \\sqrt { ( 1 6 k ^ 3 - 4 k ) ( k - 1 ) } ) ^ m \\big ) . \\end{align*}"}
-{"id": "4842.png", "formula": "\\begin{align*} \\delta _ { \\infty } \\colon m \\mapsto \\sum _ { n = 1 } ^ { \\infty } v _ n ^ { } m v _ n ^ { * } \\end{align*}"}
-{"id": "8215.png", "formula": "\\begin{align*} t _ { \\lambda } ( \\tau ) : = t \\left ( \\frac { \\tau + \\lambda } { 5 } \\right ) . \\end{align*}"}
-{"id": "676.png", "formula": "\\begin{align*} \\rho _ j \\circ [ \\rho , x ] = [ \\rho _ { \\delta ( j ) } , x ] = \\rho _ { x ^ { - 1 } ( \\delta ( j ) ) } \\end{align*}"}
-{"id": "2338.png", "formula": "\\begin{align*} \\beta _ 1 + \\cdots \\beta _ k & = [ 1 , n ] , & \\# \\beta _ i & = n _ i . \\end{align*}"}
-{"id": "6413.png", "formula": "\\begin{align*} \\rho = \\| R z \\| ^ { 2 } , \\qquad \\sigma = v ^ { T } R z , \\qquad \\tau = v ^ { T } v + \\eta ^ { 2 } . \\end{align*}"}
-{"id": "2791.png", "formula": "\\begin{align*} \\mathbb { E } [ I _ t ] = \\mathbb { E } [ O _ t ] , \\end{align*}"}
-{"id": "666.png", "formula": "\\begin{align*} \\mathbf { O p e r } ^ { \\mathsf { a l g } } _ { \\mathcal G } : = \\mathbf { P S h } ( \\widetilde { \\mathbb G } _ { \\mathcal G } \\rightrightarrows \\widetilde { \\mathbb E } _ { \\mathcal G } ) \\times _ { \\mathbf { C a t } ^ { / \\widetilde { \\mathbb E } _ { \\mathcal G } } } \\mathbf { O p e r } ' _ { \\mathcal G } \\ , \\end{align*}"}
-{"id": "9188.png", "formula": "\\begin{align*} 0 = \\pi _ { 1 } ( \\Omega _ { p } , E ^ { - 1 } [ 0 , b ^ { 2 } / 2 ] ) \\longrightarrow \\pi _ { 0 } ( E ^ { - 1 } [ 0 , b ^ { 2 } / 2 ] ) \\longrightarrow \\pi _ { 0 } ( \\Omega _ { p } ) = 0 . \\end{align*}"}
-{"id": "2700.png", "formula": "\\begin{align*} a = \\kappa - \\frac { 4 } { 9 } \\pi _ { * } , b = \\kappa - \\frac { 5 } { 9 } \\pi _ { * } , \\end{align*}"}
-{"id": "8947.png", "formula": "\\begin{align*} \\mathbb { P } ( Y = 1 \\ , | \\ , X = x ) \\ = \\ \\frac { 1 } { 1 + \\exp \\left \\{ - \\beta _ 0 - \\langle \\beta , x \\rangle _ K \\right \\} } , \\end{align*}"}
-{"id": "1813.png", "formula": "\\begin{align*} D B _ { t } ^ { ( n ) } ( s ) = \\int _ { 0 } ^ { s } u _ { t } ^ { ( n ) } ( v ) d v , \\end{align*}"}
-{"id": "7349.png", "formula": "\\begin{align*} P _ { \\scriptscriptstyle { T , k } } \\cdot Q _ { \\scriptscriptstyle { T , k } } + P _ { \\scriptscriptstyle { S , k } } \\cdot Q _ { \\scriptscriptstyle { S , k } } = 1 . \\end{align*}"}
-{"id": "4496.png", "formula": "\\begin{align*} \\Xi ^ t ( p , z , x ) = \\big ( t + ( 1 - t ) e ^ { 2 ( z - \\underline { v } ( x ) ) } \\big ) \\Psi ( p , z , x ) \\end{align*}"}
-{"id": "1399.png", "formula": "\\begin{align*} \\frac { \\tfrac { \\partial f } { \\partial x _ 1 } } { r _ 1 ( x _ 1 ) } & = \\frac { \\tfrac { \\partial f } { \\partial x _ 2 } } { r _ 2 ( x _ 2 ) } = \\frac { \\tfrac { \\partial f } { \\partial x _ 3 } } { r _ 3 ( x _ 3 ) } \\\\ \\frac { \\tfrac { \\partial f } { \\partial x _ 1 } } { r _ { 1 , j } ( x _ 1 ) } & = \\frac { \\tfrac { \\partial f } { \\partial x _ 2 } } { r _ { 2 , j } ( x _ 2 ) } = \\frac { \\tfrac { \\partial f } { \\partial x _ i } } { \\tilde r _ j ( x _ j ) } , ~ ~ j = 4 , \\ldots , d , \\end{align*}"}
-{"id": "1565.png", "formula": "\\begin{align*} \\widetilde { F } _ w : = U \\circ F _ w \\circ U ^ { - 1 } \\begin{pmatrix} z \\\\ x \\end{pmatrix} = \\widetilde { F } \\begin{pmatrix} z \\\\ x \\end{pmatrix} + \\begin{pmatrix} \\frac { \\pi ^ 2 } { 4 } w + \\mathcal { O } ( w x , w z ^ 2 ) \\\\ 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "7609.png", "formula": "\\begin{align*} \\begin{gathered} \\mathcal { P } _ 1 = L i p ( 0 , T ; C ^ { 1 + \\alpha , p } ) \\times L i p ( 0 , T ; C ^ { \\alpha , p } ) \\times L ^ \\infty ( 0 , T ; C ^ { 1 + \\alpha , p } ) \\\\ \\mathcal { I } = \\{ ( X , \\tau , v ) : \\norm { ( X - \\mathrm { I d } , \\tau , v ) } _ { \\mathcal { P } _ 1 } \\le \\Gamma , v = \\frac { d X } { d t } \\} , \\end{gathered} \\end{align*}"}
-{"id": "3080.png", "formula": "\\begin{align*} | J | & \\leq \\int _ { | z | \\leq | x | \\leq 2 | z | } \\frac { 1 } { | x | ^ { n - \\beta } } \\ , d x \\\\ & \\leq \\frac { 1 - 2 ^ { - \\beta } } { \\beta } \\beta ^ { - \\beta } . \\end{align*}"}
-{"id": "380.png", "formula": "\\begin{align*} | Q ' | & = \\frac { 1 } { \\sqrt { | b | } } | \\sqrt { a } + \\sqrt { b } | \\cdot | r + \\sqrt { a b } | ^ n \\geqslant \\frac { | b - a | } { \\sqrt { | b | } } \\cdot \\frac { 1 } { | \\sqrt { b } - \\sqrt { a } | } | \\sqrt { a b } | ^ 3 \\\\ & \\geqslant \\frac { 2 } { \\sqrt { | b | } } \\cdot \\frac { 1 } { | \\sqrt { b } | + | \\sqrt { a } | } | a b | ^ { 3 / 2 } \\geqslant 1 2 \\frac { | b | } { | a | } , \\end{align*}"}
-{"id": "6838.png", "formula": "\\begin{align*} \\begin{cases} - u ^ 3 _ { 1 0 } u ^ 2 _ { 0 1 } + u ^ 3 _ { 0 1 } u ^ 2 _ { 1 0 } - u ^ 3 _ { 1 0 } u ^ 1 _ { 0 1 } + u ^ 3 _ { 0 1 } u ^ 1 _ { 1 0 } = 0 \\ , , \\\\ - u ^ 4 _ { 1 0 } u ^ 3 _ { 0 1 } + u ^ 4 _ { 0 1 } u ^ 3 _ { 1 0 } - u ^ 3 _ { 1 0 } u ^ 1 _ { 0 1 } + u ^ 3 _ { 0 1 } u ^ 1 _ { 1 0 } = 0 \\ , . \\end{cases} \\end{align*}"}
-{"id": "6275.png", "formula": "\\begin{align*} \\nu _ { f } = \\sum _ { j = 1 } ^ { N - 1 } ( f _ j - f _ { j - 1 } ) \\delta _ { t _ j } + f ( 0 + ) \\delta _ 0 - f ( T - ) \\delta _ T . \\end{align*}"}
-{"id": "3671.png", "formula": "\\begin{align*} E ( y _ i | x _ i , i \\in U ) = E ( y _ i | \\delta _ i , x _ i , i \\in U ) = E ( y _ i | x _ i , i \\in B ) \\end{align*}"}
-{"id": "7088.png", "formula": "\\begin{align*} \\begin{cases} - \\partial _ { t t } w + \\Delta w = | \\nabla w | ^ 2 + 2 \\nabla w \\cdot \\nabla F ^ \\omega + | \\nabla F ^ \\omega | ^ 2 \\qquad ~ ( t , x ) \\in \\mathbb { R } ^ { 1 + d } \\\\ w | _ { t = 0 } = 0 , ~ \\partial _ t w | _ { t = 0 } = 0 \\end{cases} ~ . \\end{align*}"}
-{"id": "206.png", "formula": "\\begin{align*} \\pi _ { T M , X _ { C } , Y _ { V } , c } ( { \\bf x } , { \\bf y } ) = \\left ( \\begin{array} { c c c | c c c } 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & x ^ 3 & x ^ 1 \\\\ 0 & 0 & 0 & 0 & 0 & 0 \\\\ \\hline 0 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & - x ^ 3 & 0 & 0 & 0 & 0 \\\\ 0 & - x ^ 1 & 0 & 0 & 0 & 0 \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "8936.png", "formula": "\\begin{align*} T ^ { \\hat { x } } ( G ) : = N _ w ^ { \\hat { x } } ( G ) - N _ a ^ { \\hat { x } } ( G ) . \\end{align*}"}
-{"id": "2983.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty q ^ { \\Delta _ n } & = \\frac { ( q ^ 2 ; q ^ 2 ) _ \\infty } { ( q ; q ^ 2 ) _ \\infty } = \\Big ( ( q ^ 2 ; q ^ 2 ) _ \\infty ) \\Big ) \\cdot \\Big ( \\frac { 1 } { ( q ; q ^ 2 ) _ \\infty } \\Big ) = \\Big ( ( q ^ 2 ; q ^ 2 ) _ \\infty ) \\Big ) \\cdot \\Big ( \\sum _ { n = 1 } ^ \\infty q ( n ) q ^ n \\Big ) \\end{align*}"}
-{"id": "2023.png", "formula": "\\begin{align*} I _ j ( q , z ) = \\frac { 5 \\alpha _ j } { \\prod \\limits _ { j ' \\neq j } ( \\alpha _ j - \\alpha _ { j ' } ) } \\left ( z + \\tau _ j ( - z ) + \\widetilde { T } _ j ( - z ) + \\O ( z ^ { - 1 } ) \\right ) \\end{align*}"}
-{"id": "7700.png", "formula": "\\begin{align*} \\norm { u } _ { 1 } \\geq \\sum _ { i = 1 } ^ l \\norm { \\Psi _ { i } ^ * ( \\rho _ { i } \\cdot u ) } _ { 2 } . \\end{align*}"}
-{"id": "9998.png", "formula": "\\begin{align*} ( F _ m A ) ( n ) = ( G L _ { n } ) _ + \\wedge _ { G L _ { n - m } } ( A \\wedge T ^ { n - m } ) . \\end{align*}"}
-{"id": "1537.png", "formula": "\\begin{align*} { \\cal L } ( u _ 1 ) ( x , s ) = \\int _ { 0 } ^ { 1 } { g ( \\sqrt { s } , x , y ) f ( y ) d y } . \\end{align*}"}
-{"id": "2401.png", "formula": "\\begin{align*} S ( t ) \\approx \\sum _ { j = 1 } ^ m U _ j c _ j ( t ) . \\end{align*}"}
-{"id": "4920.png", "formula": "\\begin{align*} e _ I ( R ) = \\displaystyle \\sum _ { i = 0 } ^ d ( - 1 ) ^ { i } \\ell ( H ^ { d - i } ( x _ 1 , \\ldots , x _ d ; R ) ) . \\end{align*}"}
-{"id": "8419.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\int | u _ t | ^ 2 d \\alpha = R e \\Big \\{ - i \\int a u _ { \\alpha } \\bar { u } _ t d \\alpha - i \\int a _ t \\bar { z } _ { \\alpha } \\bar { u } _ t d \\alpha \\Big \\} . \\end{align*}"}
-{"id": "3450.png", "formula": "\\begin{align*} \\rho _ { M , N } = \\rho _ { M , N } ( \\theta ) = \\begin{cases} \\frac { N \\sin \\theta \\sin \\frac { \\theta } { M + 1 } } { \\pi \\sin ( 1 - \\frac { 1 } { M + 1 } ) \\theta } , & \\theta \\in ( 0 , \\pi ) , \\\\ 2 ^ { \\frac { 1 } { 3 } } \\Big ( \\frac { N } { M + 1 } \\Big ) ^ { \\frac { 2 } { 3 } } , & \\theta = 0 . \\end{cases} \\end{align*}"}
-{"id": "4198.png", "formula": "\\begin{align*} \\left ( \\frac { A _ n ^ { ( 3 ) } ( z ) - A _ n ^ { ( 3 ) } ( 0 ) } { n ^ 9 z } \\right ) ^ 2 & = \\mathcal { O } \\left ( n ^ { - 2 } \\right ) \\end{align*}"}
-{"id": "579.png", "formula": "\\begin{align*} ( 0 , 0 ) _ { p ^ r + 1 } & = p ^ r - 2 , \\\\ ( 0 , i ) _ { p ^ r + 1 } = ( i , 0 ) _ { p ^ r + 1 } = ( i , i ) _ { p ^ r + 1 } & = 0 & & \\textrm { f o r } i \\ne 0 , \\\\ ( i , j ) _ { p ^ r + 1 } & = 1 & & \\textrm { f o r } i \\ne j \\textrm { a n d } i , j \\ne 0 , \\end{align*}"}
-{"id": "5372.png", "formula": "\\begin{align*} b _ { 1 + p + q \\ , 2 t + 3 p - 3 q } c _ { 2 + p + q \\ , 2 t + 3 p - 3 q - 3 } = \\frac { q + 1 } { p + q + 2 } \\left ( 2 c + ( q + 1 ) t - q ( q + 2 ) \\right ) . \\end{align*}"}
-{"id": "1378.png", "formula": "\\begin{align*} m ( J , q ) = \\max \\{ i \\in \\{ 0 , \\ldots , n - 1 \\} \\colon J _ { ( i ) } \\leq q \\} \\ ; . \\end{align*}"}
-{"id": "606.png", "formula": "\\begin{align*} \\delta \\ , 2 ^ { j \\Delta / q } = c \\ , \\gamma \\ , 2 ^ { j d ( \\frac { 1 } { 2 } - \\frac { 1 } { q } ) } \\ , 2 ^ { j \\Delta / q } & = c \\ , \\bigg ( \\frac { 1 } { n } \\bigg ) ^ { \\frac { 1 } { 2 } - \\big ( \\frac { 1 } { 2 } - \\frac { 1 } { q } \\big ) - \\frac { \\Delta } { d q } } = c \\ , n ^ { - \\frac { 1 } { d q } } . \\end{align*}"}
-{"id": "9567.png", "formula": "\\begin{align*} \\sigma ( q , N ) : = \\sum _ { n = 0 } ^ { N } \\left [ \\begin{matrix} N \\\\ n \\end{matrix} \\right ] \\frac { ( q ) _ { n } q ^ { \\frac { n ( n + 1 ) } { 2 } } } { ( - q ) _ n } . \\end{align*}"}
-{"id": "1715.png", "formula": "\\begin{align*} | g _ { f , s } ( x ) | & = | f ( \\varphi ( t , x ) ) | \\leq | f ( \\varphi ( t , x ) ) - f ( \\varphi ( t , y _ { n _ K } ) ) | + | f ( \\varphi ( t , y _ { n _ K } ) ) | \\leq \\frac { \\varepsilon } { 2 } + g _ { f , s } ( y _ { n _ K } ) \\\\ & \\leq \\frac { \\varepsilon } { 2 } + | g _ { f , s } ( y _ { n _ K } ) - f ( \\varphi ( t ^ * , x ) ) | + | f ( \\varphi ( t ^ * , x ) ) | \\leq \\varepsilon + | f ( \\varphi ( t ^ * , x ) ) | . \\end{align*}"}
-{"id": "7757.png", "formula": "\\begin{align*} \\| A + \\lambda B \\| = \\| A \\| + \\| B \\| \\end{align*}"}
-{"id": "2179.png", "formula": "\\begin{align*} K _ \\ell ( \\mathbf { x } , \\mathbf { y } ) = \\tau _ { - \\mathbf { y } } \\mathcal { F } ^ { - 1 } ( m ( \\cdot ) \\phi ( 2 ^ { - \\ell } \\cdot ) ) ( \\mathbf { x } ) . \\end{align*}"}
-{"id": "687.png", "formula": "\\begin{align*} \\varpi : \\prod _ { i = 1 } ^ n \\mathcal C ( W _ i , V _ i ) _ { \\nu _ i } \\to \\mathcal C ( \\varpi ( W _ 1 \\dots W _ n ) , \\varpi ( V _ 1 \\dots V _ n ) ) _ { \\nu _ 1 \\diamond \\dots \\diamond \\nu _ n } \\end{align*}"}
-{"id": "9944.png", "formula": "\\begin{align*} \\Pi _ k = \\frac { 1 } { 2 \\pi \\imath } \\oint _ { { \\mathcal C } _ k } \\left ( z - \\Delta \\right ) ^ { - 1 } d z . \\end{align*}"}
-{"id": "6408.png", "formula": "\\begin{align*} \\lambda _ { + } = \\frac { 1 } { 2 } \\left ( \\rho + \\tau + \\chi \\right ) , \\qquad \\lambda _ { - } = \\frac { 1 } { 2 } \\left ( \\rho + \\tau - \\chi \\right ) \\end{align*}"}
-{"id": "918.png", "formula": "\\begin{align*} t = \\tau _ 0 < \\ldots < \\tau _ N = T , \\ \\ \\ \\Delta _ N = \\hbox { \\vtop { \\offinterlineskip \\halign { \\hfil # \\hfil \\cr { \\rm m a x } \\cr $ \\stackrel { } { { } _ { 0 \\le j \\le N - 1 } } $ \\cr } } } \\Delta \\tau _ j \\to 0 \\ \\ \\hbox { i f } \\ \\ N \\to \\infty , \\ \\ \\ \\Delta \\tau _ j = \\tau _ { j + 1 } - \\tau _ j . \\end{align*}"}
-{"id": "3889.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } ( \\| x u ( t ) \\| ^ 2 _ { L ^ 2 } + 2 \\| x v ( t ) \\| ^ 2 _ { L ^ 2 } ) = 8 \\left ( \\| \\nabla u ( t ) \\| ^ 2 _ { L ^ 2 } + \\frac { 1 } { 2 } \\| \\nabla v ( t ) \\| ^ 2 _ { L ^ 2 } \\right ) - 2 0 \\emph { R e } \\int \\overline { v } ( t ) u ^ 2 ( t ) d x . \\end{align*}"}
-{"id": "4488.png", "formula": "\\begin{align*} M & = u ( x _ 0 ) + \\nabla _ { 1 1 } u ( x _ 0 ) \\\\ & = \\underline { u } ( x _ 0 ) + \\nabla _ { 1 1 } \\underline { u } ( x _ 0 ) - \\nabla _ { n } ( u - \\underline { u } ) ( x _ 0 ) \\Pi _ { 1 1 } ( x _ 0 ) \\end{align*}"}
-{"id": "9085.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } g ( x ) | u _ n ( x ) | ^ p \\ , d x = 1 \\quad \\forall n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "897.png", "formula": "\\begin{align*} Y _ m = y _ { \\beta ^ { j ( 1 ) } _ { k ( 1 ) } } ^ { m _ 1 } \\cdots y _ { \\beta ^ { j ( r ) } _ { k ( r ) } } ^ { m _ r } = \\Gamma \\cdot \\left ( y _ { \\beta ^ 1 _ 1 } ^ { m ^ 1 _ 1 } \\cdots y _ { \\beta ^ 1 _ { r _ 1 } } ^ { m ^ 1 _ { r _ 1 } } \\right ) \\cdots \\left ( y _ { \\beta ^ \\ell _ 1 } ^ { m ^ \\ell _ 1 } \\cdots y _ { \\beta ^ \\ell _ { r _ \\ell } } ^ { m ^ \\ell _ { r _ \\ell } } \\right ) . \\end{align*}"}
-{"id": "8392.png", "formula": "\\begin{align*} R e \\int \\bold { n } | z _ { \\alpha } | D ^ { k + 1 } f \\overline { D ^ k f } d \\alpha = R e \\int i \\partial _ { \\alpha } D ^ k f \\overline { D ^ k f } = \\int _ { \\Omega ( t ) } | \\nabla D ^ k f | ^ 2 d x d y \\geq 0 . \\end{align*}"}
-{"id": "1006.png", "formula": "\\begin{align*} \\ \\varphi _ k ( 0 ) = 0 , \\ \\varphi _ k ' \\left ( \\lambda \\right ) = - \\sum _ { j = 1 } ^ { k - 1 } \\frac { \\lambda ^ { j - 1 } } { ( j - 1 ) ! } e ^ { - \\lambda } + \\sum _ { j = 0 } ^ { k - 1 } \\frac { \\lambda ^ j } { j ! } e ^ { - \\lambda } = \\frac { \\lambda ^ { k - 1 } } { ( k - 1 ) ! } e ^ { - \\lambda } \\end{align*}"}
-{"id": "3337.png", "formula": "\\begin{align*} 1 + \\sum _ { i \\ge 1 } \\left ( \\mathrm { E } _ { \\pi \\in S _ i } G ( \\pi ) \\right ) u ^ i = \\exp ( \\sum _ { j \\ge 1 } \\frac { g ( j ) } { j } u ^ j ) . \\end{align*}"}
-{"id": "1853.png", "formula": "\\begin{align*} ( \\psi _ j , \\psi _ { j ' } ) _ { \\rm Q F T } = \\delta _ { j j ' } , ( \\overline { \\psi _ j } , \\overline { \\psi _ { j ' } } ) _ { \\rm Q F T } = - \\delta _ { j j ' } , \\qquad \\mbox { a n d } ( \\psi _ j , \\overline { \\psi _ { j ' } } ) _ { \\rm Q F T } = 0 . \\end{align*}"}
-{"id": "6424.png", "formula": "\\begin{align*} B _ { k + 1 } ^ { - 1 } = \\left [ \\begin{array} { c c } B _ { k } ^ { - 1 } & - w _ { k } \\frac { \\beta _ { k } } { \\alpha _ { k + 1 } } \\\\ & \\frac { 1 } { \\alpha _ { k + 1 } } \\end{array} \\right ] , w _ { k + 1 } = \\frac { 1 } { \\alpha _ { k + 1 } } \\left [ \\begin{array} { c } - w _ { k } \\beta _ { k } \\\\ 1 \\end{array} \\right ] , \\end{align*}"}
-{"id": "4166.png", "formula": "\\begin{align*} T _ { \\alpha } \\Phi _ { \\alpha } ( n ^ 3 f ( z ) ) D _ 0 ^ n ( z ) & = \\frac { 2 \\pi } { \\sqrt { 3 } } \\left ( n ^ 3 f ( z ) \\right ) ^ { - \\frac { 2 \\beta } { 3 } } \\left ( \\mathbb { I } + \\frac { A _ { \\alpha } } { n ^ 3 f ' ( 0 ) z } + O \\left ( n ^ { - 3 } \\right ) \\right ) F _ n ( z ) , \\end{align*}"}
-{"id": "5402.png", "formula": "\\begin{align*} 2 c + ( q + 1 ) t - q ( q + 2 ) = - ( q - l ) ( q - ( 2 k - 1 - l ) ) \\end{align*}"}
-{"id": "8683.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { p } J _ { [ X , S _ i ] } S _ i & = - 1 6 \\sum _ { i = 1 } ^ { p } \\Im ( X S _ i ^ * ) S _ i \\\\ & = - 1 6 \\sum _ { i = 1 } ^ { p } ( X S _ i ^ * ) S _ i + 1 6 \\sum _ { i = 1 } ^ { p } \\langle X , S _ i \\rangle S _ i \\\\ & = - 1 6 X \\sum _ { i = 1 } ^ { p } S _ i ^ * S _ i + 1 6 X \\\\ & = - 1 6 ( q + 1 ) X + 1 6 X \\\\ & = - 1 6 q X , \\end{align*}"}
-{"id": "3047.png", "formula": "\\begin{align*} \\sigma _ { i } ^ { 0 } ( x , \\xi ) = z _ { i } \\ 1 _ { \\{ r \\leq x _ { k } \\} } \\ 1 _ { \\R _ { + } } ( x _ { k } ) \\ 1 _ { \\{ 1 , \\dots , d \\} } ( k ) , \\qquad \\sigma _ { i } ^ { 2 } ( x , \\xi ) = z _ { i } \\ 1 _ { \\{ r \\leq x _ { k } \\} } \\ 1 _ { \\R _ { + } } ( x _ { k } ) \\ 1 _ { \\{ 1 , \\dots , d \\} } ( k ) . \\end{align*}"}
-{"id": "3329.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\lambda \\vdash n \\\\ \\ell ( \\lambda ) \\le k } } w ( B _ { \\lambda , \\mu } ) \\le \\sum _ { i = 0 } ^ { k } \\binom { n } { i } . \\end{align*}"}
-{"id": "8710.png", "formula": "\\begin{align*} E ( T ^ * _ { m , N } ) = ( N - 1 ) \\sum _ { j \\leq m } \\hat { \\nu } ( j ) \\sum _ { k = j } ^ { N - 1 } { 1 \\over N - k } . \\end{align*}"}
-{"id": "5193.png", "formula": "\\begin{align*} H ( \\alpha ( t ) x + \\beta ( t ) ) = H ^ { t } ( x ) . \\end{align*}"}
-{"id": "6997.png", "formula": "\\begin{align*} { \\textsl { \\footnotesize L } } _ j = \\textstyle { \\left [ \\cos \\left ( \\frac { 2 \\pi j } { n } \\right ) , 2 \\cos \\left ( \\frac { 4 \\pi j } { n } \\right ) , \\ldots , ( n - 1 ) \\cos \\left ( \\frac { 2 ( n - 1 ) \\pi j } { n } \\right ) , \\frac { n } { 2 } \\right ] } ; \\end{align*}"}
-{"id": "5030.png", "formula": "\\begin{align*} I '' : = I ( \\L _ j , \\P _ i ) \\geq C _ { \\beta , 2 } \\left ( | \\P _ i | ^ { 2 / 3 } | \\L _ j | ^ { 2 / 3 } + | \\P _ i | + | \\L _ j | \\right ) \\end{align*}"}
-{"id": "6345.png", "formula": "\\begin{align*} \\sigma _ d ^ 2 = \\sigma '^ 2 _ d + \\sigma _ { t h } ^ 2 . \\end{align*}"}
-{"id": "4753.png", "formula": "\\begin{align*} p _ { \\xi , \\eta } : = \\frac { q _ { \\xi , \\eta } } { q _ \\xi } = \\frac { C _ { \\xi , \\eta } } { C _ \\xi } . \\end{align*}"}
-{"id": "2884.png", "formula": "\\begin{align*} L _ { \\mu \\nu } & = x _ \\mu \\frac { \\partial } { \\partial x _ \\nu } - x _ \\nu \\frac { \\partial } { \\partial x _ \\mu } , \\end{align*}"}
-{"id": "4371.png", "formula": "\\begin{align*} \\Delta + \\frac { k } { k - t } \\gamma \\le \\Big ( 1 - \\prod _ { i = 1 } ^ { k } \\frac { n - k + 1 - i } { n - t - i } \\Big ) { n - 1 \\choose k - 1 } , \\end{align*}"}
-{"id": "2077.png", "formula": "\\begin{align*} { C _ { \\rho \\varphi } } ( \\rho , \\varphi , t | { { \\bar r } _ { \\rm t x } } , { t _ 0 } ) = \\sum \\limits _ { n = 0 } ^ \\infty \\sum \\limits _ { m = 1 } ^ \\infty { H _ { n m } J _ n ( \\lambda _ { n m } \\rho ) \\cos ( n ( \\varphi - \\varphi _ { \\rm t x } ) ) e ^ { - D \\lambda _ { n m } ^ 2 ( t - t _ 0 ) } u ( t - t _ 0 ) } , \\end{align*}"}
-{"id": "5451.png", "formula": "\\begin{align*} \\alpha _ t ^ { \\omega ' } ( \\omega ) \\ = \\ \\bar \\alpha _ t ( \\omega , \\omega ' ) , ( \\omega , \\omega ' ) \\in \\bar \\Omega = \\Omega \\times \\Omega ' , \\ ; t \\geq 0 , \\end{align*}"}
-{"id": "7507.png", "formula": "\\begin{align*} Z ( x , t ) : = \\mathcal { F } _ { \\xi \\rightarrow x } ^ { - 1 } ( e ^ { - t | f ( \\xi ) | _ { p } ^ { \\beta } } ) = \\int \\nolimits _ { \\mathbf { \\mathbb { Q } } _ { p } ^ { n } } \\chi _ { p } ( - x , \\xi ) e ^ { - t | f ( \\xi ) | _ { p } ^ { \\beta } } d ^ { n } \\xi , t > 0 , x \\in \\mathbf { \\mathbb { Q } } _ { p } ^ { n } . \\end{align*}"}
-{"id": "2525.png", "formula": "\\begin{align*} \\{ \\lambda _ t Y _ t ; P ( \\cdot | Y _ t > 0 , Y _ 0 = x ) \\} \\xrightarrow [ t \\to \\infty ] { \\operatorname { l a w } } \\mathbf z ^ { ( \\gamma - 1 ) } \\end{align*}"}
-{"id": "5164.png", "formula": "\\begin{align*} \\varphi _ { \\alpha } ( x ) = \\exp ( - x ^ { - \\alpha } ) 1 _ { ( x \\geq 0 ) } . \\end{align*}"}
-{"id": "9313.png", "formula": "\\begin{align*} x _ { i j } \\leq s _ j , ~ j = 1 , \\dots , n , ~ i = 1 , \\dots , m . \\end{align*}"}
-{"id": "7531.png", "formula": "\\begin{align*} \\begin{gathered} u ( x , t ) = \\mathbb { L } _ \\nu ( u _ 0 ) ( x , t ) + \\int _ 0 ^ t g _ { \\nu ( t - s ) } * \\left ( \\mathbb { H } \\left ( \\mathrm { d i v } \\ , \\ , ( \\sigma - u \\otimes u ) \\right ) \\right ) ( x , s ) d s . \\end{gathered} \\end{align*}"}
-{"id": "5997.png", "formula": "\\begin{align*} L _ n ( \\omega \\otimes _ { t _ n } W ) & = \\frac { 1 } { n } \\sum _ { k = 1 } ^ { \\lfloor n t \\rfloor } \\delta _ { \\omega _ { ( n , k ) } } + \\frac { 1 } { n } \\sum _ { k = \\lfloor n t \\rfloor + 1 } ^ n \\delta _ { W _ { ( n - \\lfloor n t \\rfloor , k - \\lfloor n t \\rfloor ) } } \\\\ & = t _ n \\frac { 1 } { \\lfloor n t \\rfloor } \\sum _ { k = 1 } ^ { \\lfloor n t \\rfloor } \\delta _ { \\omega _ { ( n , k ) } } + ( 1 - t _ n ) L _ { n - \\lfloor n t \\rfloor } ( W ) . \\end{align*}"}
-{"id": "3958.png", "formula": "\\begin{align*} & f _ i ( x , m , v _ i ) = \\chi ( v _ i ) + \\Phi _ { i , \\Lambda } ( x , m ) \\ , , \\\\ & \\chi ( v ) = \\frac { | v | ^ 2 } { 2 } , \\Phi _ { i , \\Lambda } ( x , m ) = \\sum _ { k = 1 } ^ 2 \\Lambda _ { i , k } \\ , \\phi * \\mu _ k ( x ) \\ , , \\end{align*}"}
-{"id": "8288.png", "formula": "\\begin{align*} ( I - \\mathbb { H } ) ( Z _ { \\alpha } - 1 ) = 0 , ( I - \\mathbb { H } ) ( D _ t \\bar { Z } + \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i } { 2 \\pi } \\frac { 1 } { Z ( \\alpha , t ) - z _ j ( t ) } ) = 0 , \\end{align*}"}
-{"id": "1100.png", "formula": "\\begin{align*} & \\phi _ { k , n } ( y _ 1 , \\cdots , y _ k ) = n ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } \\times \\\\ & \\mathbb { P } ( ( y _ 1 , \\cdots , y _ k ) \\in \\Sigma _ k ( G _ n ( x _ 1 ) / S ( I ) , \\cdots , G _ n ( x _ k ) / S ( I ) ) ) , \\end{align*}"}
-{"id": "8180.png", "formula": "\\begin{align*} T _ \\Lambda \\mathcal { T } \\{ g _ i \\} _ { i \\in \\mathbb { N } } = \\sum _ { i \\in \\mathbb { N } } \\Lambda _ { i + 1 } ^ * g _ i = \\sum _ { i \\in \\mathbb { N } } T ^ * \\Lambda _ i ^ * g _ i = T ^ * \\Big ( \\sum _ { i \\in \\mathbb { N } } \\Lambda _ i ^ * g _ i \\Big ) = 0 . \\end{align*}"}
-{"id": "3010.png", "formula": "\\begin{align*} j _ 1 + j _ 2 + \\dots + j _ { n - k + 1 } & = k , \\\\ j _ 1 + 2 j _ 2 + \\dots + ( n - k + 1 ) j _ { n - k + 1 } & = n . \\end{align*}"}
-{"id": "6890.png", "formula": "\\begin{align*} M _ 2 = \\begin{pmatrix} \\cos ( k t ) & - \\sin ( k t ) \\\\ \\sin ( k t ) & \\cos ( k t ) \\end{pmatrix} \\ , , { \\rm a n d } w ( \\alpha ) = \\frac { e ^ { k \\alpha _ 2 } } { k } \\begin{pmatrix} \\sin ( k \\alpha _ 1 ) \\\\ - \\cos ( k \\alpha _ 1 ) \\end{pmatrix} \\ , , \\end{align*}"}
-{"id": "7961.png", "formula": "\\begin{align*} \\bigsqcup _ { q \\in Q } S _ q = \\bigsqcup _ { r \\in R } T _ r = N \\end{align*}"}
-{"id": "6482.png", "formula": "\\begin{align*} t = 1 + \\frac { 2 \\mu _ { 1 } \\cdots \\mu _ { n - 1 } - 1 } { ( 1 + \\mu _ { 1 } \\cdots \\mu _ { n - 1 } ) ^ { 2 } } = 1 + 2 a b - a ^ { 2 } . \\end{align*}"}
-{"id": "5370.png", "formula": "\\begin{align*} b _ { 1 \\ , 2 t } c _ { 2 \\ , 2 t - 3 } = c + \\frac 1 2 t . \\end{align*}"}
-{"id": "7835.png", "formula": "\\begin{align*} \\lim _ n [ \\langle x _ n , x \\rangle \\langle x , y \\rangle \\langle y , x _ n \\rangle \\xi _ n , \\xi _ n ] = \\lambda \\| x \\| ^ 2 \\| y \\| ^ 2 \\end{align*}"}
-{"id": "7408.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\mu ^ U _ x ( B _ n ) & \\leq \\mu ^ U _ x \\left ( \\bigcap _ { m = 1 } ^ { \\infty } \\bigcup _ { n \\geq m } B _ n \\right ) \\\\ & \\leq \\mu ^ U _ x ( \\partial B _ { d _ x } ( y , r ) ) = 0 . \\end{align*}"}
-{"id": "7736.png", "formula": "\\begin{align*} | [ \\langle x , x \\rangle \\langle x , y \\rangle \\xi , \\xi ] | = \\| x \\| ^ 3 \\| y \\| . \\end{align*}"}
-{"id": "2689.png", "formula": "\\begin{align*} \\mathcal { C } _ { 1 / \\tau , \\theta } = \\left \\{ \\frac { 1 } { \\tau } + \\frac { 1 } { 2 \\cos \\theta } \\Big ( \\frac { 1 } { \\tau } - \\tau \\Big ) e ^ { i \\phi } : \\phi \\in ( - \\pi , \\pi ] \\right \\} , \\end{align*}"}
-{"id": "8830.png", "formula": "\\begin{align*} \\mathrm { G a p } ( x , \\bar \\zeta , \\bar \\omega ) = \\frac 1 2 x ^ \\top Q x + f ( A ^ f x ) + G ( x ) + G ^ * ( - ( A ^ f ) ^ \\top \\bar \\zeta - \\bar \\omega ) + \\frac 1 2 \\bar \\omega ^ \\top Q ^ { \\dagger } \\bar \\omega + f ^ * ( \\bar \\zeta ) \\ ; . \\end{align*}"}
-{"id": "5475.png", "formula": "\\begin{align*} Y _ s ^ n \\ & = \\ g ( \\hat X _ T ^ { t , x , a } ) + \\int _ s ^ T f ( r , \\hat X _ r ^ { t , x , a } , \\hat I _ r ^ { t , a } ) d r + n \\int _ s ^ T \\int _ \\Lambda \\big ( R _ r ^ n ( b ) \\big ) _ + \\lambda _ \\theta ( d b ) d r \\\\ & \\ - \\int _ s ^ T Z _ r ^ n d \\hat W _ r - \\int _ s ^ T \\int _ \\Lambda R _ r ^ n ( b ) \\hat \\theta ( d r , d b ) - \\int _ s ^ T \\int _ { U \\setminus \\{ 0 \\} } L _ r ^ n ( z ) \\ , ( \\hat \\pi ( d r \\ , d z ) - \\lambda _ \\pi ( d z ) \\ , d r ) , \\end{align*}"}
-{"id": "8854.png", "formula": "\\begin{align*} s _ 5 = 3 ( 3 - 2 p ) & \\left [ ( 1 - a ) ^ 2 ( 8 p - 5 ) + ( 3 2 ( 1 - a ) ^ 2 + 1 4 4 a ) ( 1 - p ) ^ 4 \\right . \\\\ & \\left . + 1 2 ( 1 - p ) ^ 2 ( 4 p + a ( 4 a p + 1 0 p - 3 ) ) \\right ] \\ge 0 . \\end{align*}"}
-{"id": "5158.png", "formula": "\\begin{align*} G ^ { - 1 } ( u ) = \\log F ^ { - 1 } ( u ) , 0 \\leq u \\leq 1 \\end{align*}"}
-{"id": "5311.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\check { \\alpha } _ 0 ^ { ( 1 ) } = h _ { q _ 1 } + h _ { p _ 1 } - e _ { 1 , 2 , 3 , 4 } , & \\check { \\alpha } _ 1 ^ { ( 1 ) } = h _ { q _ 1 } - e _ { 5 , 6 } , & \\check { \\alpha } _ 2 ^ { ( 1 ) } = h _ { p _ 1 } - e _ { 7 , 8 } , \\\\ \\check { \\alpha } _ 0 ^ { ( 2 ) } = h _ { q _ 2 } + h _ { p _ 2 } - e _ { 9 , 1 0 , 1 1 , 1 2 } , & \\check { \\alpha } _ 1 ^ { ( 2 ) } = h _ { q _ 2 } - e _ { 1 3 , 1 4 } , & \\check { \\alpha } _ 2 ^ { ( 2 ) } = h _ { p _ 2 } - e _ { 1 5 , 1 6 } \\end{array} . \\end{align*}"}
-{"id": "962.png", "formula": "\\begin{align*} \\displaystyle \\min _ { z _ 0 , \\dots , z _ T } \\frac { 1 } { 2 } \\left ( \\| z _ 0 - b \\| ^ 2 _ { ( B ^ { \\infty } ) ^ { - 1 } } + \\sum _ { t = 1 } ^ { T } \\| z _ t - \\mathcal { M } _ t ( z _ { t - 1 } ) \\| ^ 2 _ { Q _ t ^ { - 1 } } + \\sum _ { t = 0 } ^ { T } \\| y _ t - \\mathcal { H } _ t ( z _ { t } ) \\| ^ 2 _ { R _ t ^ { - 1 } } \\right ) . \\end{align*}"}
-{"id": "977.png", "formula": "\\begin{align*} H _ n \\left ( q , \\frac { 1 } { 1 - q } \\right ) = q ^ { { n \\choose 2 } } \\frac { q ^ { - { n \\choose 2 } } } { ( 1 - q ) ^ n } = \\frac { 1 } { ( 1 - q ) ^ n } . \\end{align*}"}
-{"id": "9025.png", "formula": "\\begin{align*} f _ { \\{ 1 \\} , c _ k } ( x ) = \\begin{cases} c _ k & x = 1 , \\\\ x & \\end{cases} \\end{align*}"}
-{"id": "4584.png", "formula": "\\begin{align*} A ( \\mu , \\delta ^ { - 1 } ) \\lesssim \\| \\mu _ \\delta \\| ^ 2 _ 2 = \\int \\mu ^ 2 _ \\delta ( x ) d x , \\end{align*}"}
-{"id": "340.png", "formula": "\\begin{align*} \\begin{array} { c c c } \\tilde { \\alpha } _ { 1 , k } = ( n - 2 ) ! k u ^ 2 v _ { 2 n - 4 } & & \\tilde { \\alpha } _ { 2 , k } = ( n - 1 ) ! k u ^ 2 v _ { 2 n - 2 } . \\end{array} \\end{align*}"}
-{"id": "3150.png", "formula": "\\begin{align*} ( u - v ) \\ E _ i ( u ) E _ { i + 1 } ( v ) = ( u - v - c ) \\ E _ { i + 1 } ( v ) E _ i ( u ) , \\end{align*}"}
-{"id": "9875.png", "formula": "\\begin{align*} S _ \\tau = \\{ y ~ : ~ r _ { Q / B } ( y ) \\ge \\tau \\} \\ , . \\end{align*}"}
-{"id": "1622.png", "formula": "\\begin{align*} \\int _ 0 ^ { x _ r ^ \\ast } \\psi _ r ( z ) ( \\theta _ r ( t ) - \\theta _ r ( x _ r ^ \\ast ) ) m ' ( z ) d z = 0 . \\end{align*}"}
-{"id": "4679.png", "formula": "\\begin{align*} Q _ d u ( x ) = \\displaystyle \\sum \\limits _ { j = 1 } ^ { m + d } \\mu _ j ( u ) { B _ j ^ d } ( x ) , \\end{align*}"}
-{"id": "8649.png", "formula": "\\begin{align*} \\mathcal { M } _ 0 = \\left \\{ w _ 1 = \\frac { 1 } { 8 } { \\varphi _ 2 ^ 2 + \\frac { 1 } { 4 } \\sin ^ 2 \\varphi _ 1 - \\frac { c _ 0 } { 2 } \\sin \\varphi _ 1 } , \\ w _ 2 = 0 , \\ ( \\varphi _ 1 , \\varphi _ 2 ) \\in \\mathbb { R } ^ 2 \\right \\} . \\end{align*}"}
-{"id": "4887.png", "formula": "\\begin{align*} ( \\mu \\otimes \\mu ) \\circ ( M \\otimes \\sigma \\otimes M ) \\circ ( \\delta \\otimes \\delta ) = \\delta \\circ \\mu \\end{align*}"}
-{"id": "3089.png", "formula": "\\begin{align*} m \\{ x : | T _ { 1 1 } f | > t \\} & \\le m \\{ x : | T _ { 1 1 } g | > \\frac t 2 \\} + m \\{ x : | T _ { 1 1 } b \\mathrm { I } _ { F ^ * } | > \\frac t 2 \\} . \\end{align*}"}
-{"id": "8140.png", "formula": "\\begin{align*} & m _ { A \\widetilde { \\otimes } B } = ( m _ A \\otimes m _ B ) \\circ ( \\mathrm { i d } _ A \\otimes c _ { B , A } \\otimes \\mathrm { i d } _ B ) \\ : : \\ : A \\otimes B \\to A \\otimes B , \\\\ & 1 _ { A \\widetilde { \\otimes } B } = 1 _ A \\otimes 1 _ B \\ : : \\ : \\mathbb { C } \\to A \\otimes B . \\end{align*}"}
-{"id": "9822.png", "formula": "\\begin{align*} \\frac { H _ \\lambda ' } { H _ \\lambda } ( r , u ) = \\frac { 2 } { r } ( N ( r , u ) - \\lambda ) . \\end{align*}"}
-{"id": "6773.png", "formula": "\\begin{align*} ( [ ( \\cdot ) ^ { s } , \\alpha ] _ 1 ) _ { i , j } = \\left \\{ \\begin{array} { c c } \\frac { \\alpha _ { i } ^ { s } - \\alpha _ { j } ^ { s } } { \\alpha _ { i } - \\alpha _ { j } } & \\alpha _ { i } \\neq \\alpha _ { j } \\\\ s \\alpha _ { i } ^ { s - 1 } & \\alpha _ { i } = \\alpha _ { j } > 0 \\\\ 1 & \\alpha _ { i } = \\alpha _ { j } = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "7890.png", "formula": "\\begin{align*} A \\Vert K ^ { * } f \\Vert ^ { 2 } & = A \\langle K K ^ { * } f , \\ f \\rangle \\leq \\langle U _ { \\sigma } U _ { \\sigma } ^ { * } f , \\ f \\rangle = \\Vert U _ { \\sigma } ^ { * } f \\Vert ^ { 2 } \\\\ & = \\sum _ { j \\in \\sigma } \\Vert \\Lambda _ { j } ( C C ' ) ^ { \\frac { 1 } { 2 } } f \\Vert ^ { 2 } + \\sum _ { j \\in \\sigma ^ { c } } \\Vert \\Omega _ { j } ( C C ' ) ^ { \\frac { 1 } { 2 } } f \\Vert ^ { 2 } \\end{align*}"}
-{"id": "1337.png", "formula": "\\begin{align*} \\widetilde { \\theta } _ B \\left ( \\widetilde { \\tau } _ B ( y , t , s ) \\right ) & = \\widetilde { \\theta } _ B \\left ( \\tau ( y , t ) , t - s \\right ) \\\\ & = \\left ( \\theta _ { B , 1 } ( \\tau ( y , t ) ) , \\theta _ { B , 2 } ( \\tau ( y , t ) ) , \\theta _ { B , 2 } ( \\tau ( y , t ) ) - ( t - s ) \\right ) \\\\ & = ( y , t , s ) \\ ; . \\end{align*}"}
-{"id": "6255.png", "formula": "\\begin{align*} z _ { s , t } ^ \\epsilon : = \\begin{cases} y ^ \\ell _ { s , t } q _ 1 + y ^ \\ell _ { s , t + 1 } ( p _ 1 - q _ 1 ) & \\epsilon = \\ell \\\\ q _ 1 y _ { s , r } ^ r + ( p _ 1 - q _ 1 ) y _ { s , t + 1 } ^ r & \\epsilon = r \\\\ q _ 1 y ^ c _ { s , t } q _ 1 + \\left [ y _ { s , t + 1 } ^ c - q _ 1 y _ { s , t + 1 } ^ c q _ 1 \\right ] & \\epsilon = c \\end{cases} . \\end{align*}"}
-{"id": "2631.png", "formula": "\\begin{align*} F ^ { ( \\mathrm { s o f t } ) } & ( \\kappa , \\pi ; x ) : = \\mathrm { P f } [ J - K ^ { ( \\mathrm { s o f t } ) } ( \\kappa , \\pi ) ] \\\\ & = 1 + \\sum _ { k = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ k } { k ! } \\int _ { x } ^ { \\infty } \\cdots \\int _ { x } ^ { \\infty } \\mathrm { P f } [ { K ^ { ( \\mathrm { s o f t } ) } ( \\kappa , \\pi ; u _ i , u _ j ) } ] _ { i , j = 1 } ^ { k } d u _ 1 \\cdots d u _ k , \\end{align*}"}
-{"id": "5494.png", "formula": "\\begin{align*} b _ j = \\sum _ { l \\le J } \\sum _ { r = 0 } ^ { 2 ^ { l d } - 1 } \\sigma _ l g _ { l , r , j } \\Phi _ { l , r } , ~ ~ g _ { l , r , j } \\sim ^ { i i d } \\mathcal N ( 0 , 1 ) , j = 1 , \\dots , d , \\end{align*}"}
-{"id": "8273.png", "formula": "\\begin{align*} \\norm { f } _ { H ^ s } ^ 2 \\leq \\sum _ { k = 0 } ^ s \\norm { \\partial _ { \\alpha } ^ k f } _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "9011.png", "formula": "\\begin{align*} X _ { j } ( Z ) = Z \\times E _ { 2 } \\times \\dotsb \\times \\widetilde { E } _ { m - j + 1 } \\times \\dotsb \\times E _ { m } \\ , , \\end{align*}"}
-{"id": "1280.png", "formula": "\\begin{gather*} I ( f , g , h ) ( \\xi ) = \\frac { 1 } { 2 } \\int _ \\eta e ^ { - 3 i \\Phi ( \\xi , \\eta ) } \\bar h ( \\eta - \\xi ) \\left ( \\int _ \\nu e ^ { \\frac { 3 i } 4 \\eta \\nu ^ 2 } f \\left ( \\frac { \\eta + \\nu } { 2 } \\right ) g \\left ( \\frac { \\eta - \\nu } { 2 } \\right ) d \\nu \\right ) d \\eta \\\\ \\Phi ( \\xi , \\eta ) = \\eta \\xi ^ 2 - \\xi \\eta ^ 2 + \\frac 1 4 \\eta ^ 3 , \\end{gather*}"}
-{"id": "6100.png", "formula": "\\begin{align*} L ( \\delta - \\mu ) = ( \\delta - \\mu ) ^ 2 , L ( \\delta / \\sigma ) = \\delta / \\sigma - \\log ( \\delta / \\sigma ) - 1 \\end{align*}"}
-{"id": "10023.png", "formula": "\\begin{align*} \\sup \\{ \\sigma _ { a } ( D ) - \\sigma _ { u } ( D ) \\} = \\frac { m - 1 } { 2 m } \\ , . \\end{align*}"}
-{"id": "4234.png", "formula": "\\begin{align*} \\delta ( a ( x ) \\lvert d u \\rvert ^ { p - 2 } d u ) ) = f & & \\Omega . \\end{align*}"}
-{"id": "8495.png", "formula": "\\begin{align*} x ( t ) \\leq & x ( 0 ) e x p \\Big \\{ 2 \\epsilon \\int _ 0 ^ t ( 1 + \\frac { | \\lambda | } { 2 0 \\pi x ( 0 ) } \\tau ) ^ { - 3 / 2 } d \\tau \\Big \\} \\\\ \\leq & x ( 0 ) e x p \\Big \\{ 4 \\epsilon \\frac { 2 0 \\pi x ( 0 ) } { | \\lambda | } \\Big \\} \\\\ \\leq & x ( 0 ) e ^ { 2 \\epsilon \\frac { 4 0 \\pi } { 2 0 0 \\pi \\epsilon } } = e ^ { \\frac { 2 } { 5 } } x ( 0 ) \\leq \\frac { 3 } { 2 } x ( 0 ) . \\end{align*}"}
-{"id": "9656.png", "formula": "\\begin{align*} \\mathbb N _ { \\varepsilon _ 0 , \\delta _ 0 } = \\{ n \\in \\mathbb N \\ : \\ & \\| X \\varphi _ n ( T ) x \\| \\geq \\delta _ 0 \\| \\varphi _ n ( T ) x \\| \\\\ & \\ \\ 1 - \\varepsilon _ 0 \\leq \\| \\varphi _ n ( T ) x \\| \\leq 1 + \\varepsilon _ 0 \\} \\end{align*}"}
-{"id": "2523.png", "formula": "\\begin{align*} \\{ c _ t ^ { - 1 } Y _ t ; P ( \\cdot | Y _ t > 0 , Y _ 0 = x ) \\} \\xrightarrow [ t \\to \\infty ] { \\operatorname { l a w } } \\mathbf z ^ { ( \\gamma - 1 ) } , \\end{align*}"}
-{"id": "10017.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { n = 1 } ^ { N } a _ { n } n ^ { - s } \\Big \\| _ { \\mathfrak { X } _ { 1 } ( X ) } \\leq C _ { \\sigma } \\ , N ^ { \\sigma } \\ , \\Big \\| \\sum _ { n = 1 } ^ { N } a _ { n } n ^ { - s } \\Big \\| _ { \\mathfrak { X } _ { 2 } ( X ) } \\ , . \\end{align*}"}
-{"id": "237.png", "formula": "\\begin{align*} \\mathcal { L } u = V u \\ B _ { r _ 0 } , \\end{align*}"}
-{"id": "3616.png", "formula": "\\begin{align*} a _ 1 \\cdot ( ( \\alpha _ 0 - \\beta _ 0 ) \\cdot \\alpha _ 1 ) = - \\frac { 1 } { 2 ^ 2 } \\alpha _ 1 \\cdot \\beta _ 0 \\end{align*}"}
-{"id": "1034.png", "formula": "\\begin{align*} & \\mathbb { P } ( \\xi ^ { ( n ) } ( \\cup _ { j = 1 } ^ { k } I _ j ) = 0 ) \\leq \\prod _ { j = 1 } ^ { k } \\mathbb { P } ( \\xi ^ { ( n ) } ( I _ j ) = 0 ) . \\end{align*}"}
-{"id": "2546.png", "formula": "\\begin{align*} \\mathbb N _ { \\mu } [ W _ t ( f ) ] = \\mathbf P _ { \\mu } [ X _ t ( f ) ] = \\mu ( P ^ \\beta _ t f ) , t \\geq 0 . \\end{align*}"}
-{"id": "10050.png", "formula": "\\begin{align*} \\dot q _ i = \\frac { \\partial H } { \\partial p _ i } , \\dot p _ i = - \\frac { \\partial H } { \\partial q _ i } , i = 0 , \\dots , n . \\end{align*}"}
-{"id": "7449.png", "formula": "\\begin{align*} \\displaystyle \\int _ { H _ 1 ( F ) } ( L ( h _ 1 ) f _ 1 , f _ 2 ) _ { X _ 2 , \\Pi } d h _ 1 = \\lvert \\tau \\rvert _ E ^ { - n ( n - 1 ) / 2 } I _ \\Pi ( f _ 1 ) \\overline { I _ \\Pi ( f _ 2 ) } \\end{align*}"}
-{"id": "4919.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { m i n _ i \\{ n _ i \\} \\rightarrow \\infty } \\dfrac { e _ { ( x _ 1 ^ { n _ 1 } , \\ldots , x _ d ^ { n _ d } ) } ( M ) } { \\ell ( M / ( x _ 1 ^ { n _ 1 } , \\ldots , x _ d ^ { n _ d } ) M ) } = 1 . \\end{align*}"}
-{"id": "9493.png", "formula": "\\begin{align*} \\Vert u _ M \\Vert _ { L ^ { \\theta ' \\beta _ { n + 1 } } ( \\mathbb { R } ^ N ) } \\leq C ^ { \\displaystyle \\sum _ { j = 1 } ^ { n + 1 } \\frac { 1 } { \\beta _ j } } \\left ( \\displaystyle \\prod _ { j = 1 } ^ { n + 1 } \\beta _ j ^ { \\frac { 1 } { \\beta _ j } } \\right ) ^ { \\beta _ 0 } \\Vert u _ M \\Vert _ { L ^ { p _ { s } ^ * } ( \\mathbb { R } ^ N ) } ^ { \\displaystyle \\prod _ { j = 0 } ^ { n } \\sigma _ j } \\end{align*}"}
-{"id": "7226.png", "formula": "\\begin{align*} \\widehat { \\phi ^ 2 } ( 0 ) & = \\frac { 1 } { 2 \\pi } \\int _ { - \\pi } ^ \\pi \\phi ^ 2 ( x ) \\ , d x = \\frac { 1 } { 2 \\pi } \\int _ { - \\pi } ^ \\pi \\left [ ( \\mu - \\phi ) ^ 2 \\right ] ^ \\prime ( \\xi ) ( x - x _ 0 ) \\ , d x , \\end{align*}"}
-{"id": "2850.png", "formula": "\\begin{align*} a ^ 4 - m b ^ 4 = \\pm 1 \\ ; \\ ; ( { \\rm i n } \\ ; a , b \\in \\Z ) . \\end{align*}"}
-{"id": "8266.png", "formula": "\\begin{align*} \\begin{cases} ( D _ t ^ 2 - i A \\partial _ { \\alpha } ) Z = - i \\\\ \\frac { d } { d t } z _ j ( t ) = ( v - \\frac { \\lambda _ j i } { 2 \\pi ( \\overline { z - z _ j } ) } ) \\Big | _ { z = z _ j } \\\\ ( I - \\mathbb { H } ) ( D _ t \\bar { Z } + \\sum _ { j = 1 } ^ N \\frac { \\lambda _ j i } { 2 \\pi ( Z ( \\alpha , t ) - z _ j ( t ) ) } ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "5762.png", "formula": "\\begin{align*} A _ n ( q ) & = \\frac { 1 } { ( q - 1 ) q ^ { 2 n + 1 } } \\left ( ( q - 2 ) q ^ { 2 n + 1 } + q ^ { 3 n } ( q ^ n - 1 ) + q ^ { 2 n + 1 } ( q ^ n - 1 ) + q ^ { 2 n + 2 } + ( q - 1 ) q ^ { 3 n + 1 } \\right ) \\\\ & = 2 + q ^ n + \\sum _ { i = 0 } ^ { n - 1 } q ^ { i + n - 1 } + \\sum _ { i = 0 } ^ { n - 1 } q ^ i . \\end{align*}"}
-{"id": "8390.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t ^ 2 \\partial _ { \\alpha } ^ k u + A \\bold { n } \\partial _ { \\alpha } ^ k \\frac { \\partial _ { \\alpha } } { z _ { \\alpha } } u = g _ k , \\\\ A = a | z _ { \\alpha } | \\\\ \\bold { n } = \\frac { i z _ { \\alpha } } { | z _ { \\alpha } | } = \\frac { \\bar { u } _ t + i } { | \\bar { u } _ t + i | } \\\\ g _ k = \\partial _ { \\alpha } ^ k g - \\sum _ { m = 1 } ^ k c _ { m , k } \\partial _ { \\alpha } ^ m ( A \\bold { n } ) \\partial _ { \\alpha } ^ { k - m } D u . \\end{cases} \\end{align*}"}
-{"id": "8780.png", "formula": "\\begin{align*} \\frac { \\overline { \\widehat { \\kappa } } } { 2 } \\ , \\overline { F } + \\frac { \\widetilde { \\kappa } } { ( n - 3 ) ( n - 2 ) } + \\frac { { \\rm t r } ( \\overline { T } ) } { 2 } = \\frac { \\overline { \\Delta } _ 1 \\overline { F } } { 4 \\overline { F } } \\ , . \\end{align*}"}
-{"id": "3107.png", "formula": "\\begin{align*} \\Z _ { S , r , K _ S } ( q , y ) = \\left ( - A ^ { ( r ) } ( y ) \\right ) ^ { \\chi ( \\O _ S ) } \\left ( B ^ { ( r ) } ( y ) C _ { 1 1 } ^ { ( r ) } ( y ) \\right ) ^ { K _ S ^ 2 } \\end{align*}"}
-{"id": "2117.png", "formula": "\\begin{align*} z ( t ) = z _ 0 e ^ { \\int _ { 0 } ^ t a ( s ) d s + \\int _ { 0 } ^ t c ( s ) d \\omega ( s ) } . \\end{align*}"}
-{"id": "8025.png", "formula": "\\begin{align*} \\mathbb V _ n & : = \\langle x , y \\ , | \\ , x ^ 4 , y ^ n , ( x y ) ^ 2 , ( x ^ { - 1 } y ) ^ 2 \\rangle , \\\\ \\mathbb H _ n & : = \\langle x , y \\ , | \\ , x ^ 4 , y ^ 2 x ^ 2 , ( x y ) ^ n \\rangle , \\\\ \\mathbb G _ n & : = \\langle x , y \\ , | \\ , x ^ 2 y ^ n , y ^ { 2 n } , x ^ { - 1 } y x y \\rangle , \\\\ \\mathbb U _ n & : = \\langle x , y \\ , | \\ , x ^ 2 , y ^ { 2 n } , x y x y ^ { n + 1 } \\rangle , \\\\ W _ 2 & : = \\langle x , y | x ^ 4 , y ^ 3 , y x ^ 2 y ^ { - 1 } x ^ 2 , ( x y ) ^ 4 \\rangle , \\\\ W _ 3 & : = \\langle x , y | x ^ 4 , y ^ 3 , x ^ 2 ( x y ) ^ 4 , ( x y ) ^ 8 \\rangle . \\end{align*}"}
-{"id": "3281.png", "formula": "\\begin{align*} X ^ 2 - q X + p ^ n = 0 . \\end{align*}"}
-{"id": "8230.png", "formula": "\\begin{align*} L _ 0 = L _ 0 ( \\tau ) : = 2 E _ 2 ( 2 \\tau ) - E _ 2 ( \\tau ) . \\end{align*}"}
-{"id": "6962.png", "formula": "\\begin{align*} y _ 0 = z _ 0 , y _ 1 = z _ 1 , y _ 2 = \\frac { z _ 2 } { z _ 1 ^ 2 } , y _ 3 = \\frac { z _ 3 } { z _ 1 ^ 3 } , \\dots \\end{align*}"}
-{"id": "4872.png", "formula": "\\begin{align*} \\mathbb { D } _ { \\mathbb { R } ^ n } = \\mathbb { R } ^ n \\sqcup S ^ { n - 1 } \\infty . \\end{align*}"}
-{"id": "5343.png", "formula": "\\begin{align*} - \\frac 1 2 i ( X + Y ) . v ^ k = \\frac 1 2 i \\left ( ( k - 1 ) v ^ { k - 1 } + ( n - k ) v ^ { k + 1 } \\right ) . \\end{align*}"}
-{"id": "9501.png", "formula": "\\begin{align*} \\mathcal { U } _ { n } ( x , y ) = \\frac { u _ n ( x ) - u _ n ( y ) } { \\vert x - y \\vert ^ { \\frac { N } { p } + s } } \\in L ^ { m } \\left ( \\mathbb { R } ^ N \\times \\mathbb { R } ^ N \\right ) \\end{align*}"}
-{"id": "7699.png", "formula": "\\begin{align*} \\rho _ { 1 } , \\ldots , \\rho _ l \\colon M \\to [ 0 , 1 ] \\mathsf { s u p p } ( \\rho _ { i } ) \\subseteq U _ { i } \\ ; \\ ; \\sum _ { i = 1 } ^ l \\rho _ { i } ( p ) = 1 p \\in K . \\end{align*}"}
-{"id": "5863.png", "formula": "\\begin{align*} T _ n : = \\int _ { P \\in M _ { n - 1 } } \\int _ { Q \\in M _ { n } } \\alpha \\chi _ 0 ^ { - 1 } ( Q ( 0 ) ) \\beta \\chi _ 0 ^ { - 1 } ( Q ( 1 ) ) \\gamma \\chi _ 0 ^ { - 1 } ( R ( P , Q ) ) \\dd Q \\dd P . \\end{align*}"}
-{"id": "6150.png", "formula": "\\begin{align*} r ^ 2 - k ^ 2 = 2 p ( w ^ 2 - 2 a ) . \\end{align*}"}
-{"id": "5610.png", "formula": "\\begin{align*} \\sup _ { \\delta \\leq y \\leq - 2 \\delta + L } \\| u _ { \\delta , \\epsilon } | u _ { \\delta , \\epsilon } | ^ 2 \\| _ { L ^ 2 ( 0 , T ) } & \\leq \\sup _ { \\delta \\leq y \\leq - 2 \\delta + L } \\| u \\| ^ 2 _ { L ^ { \\infty } ( 0 , T ) } \\| u _ { \\delta , \\epsilon } \\| _ { L ^ { 2 } ( 0 , T ) } \\\\ & \\leq \\sup _ { \\delta \\leq y \\leq - 2 \\delta + L } \\| u _ { \\delta , \\epsilon } \\| _ { H ^ 1 ( 0 , T ) } ^ 2 \\| u _ { \\delta , \\epsilon } \\| _ { L ^ { 2 } ( 0 , T ) } \\\\ & \\leq 1 + \\delta ^ { - 2 } \\end{align*}"}
-{"id": "6139.png", "formula": "\\begin{align*} n a + b \\sqrt { 4 n e ' } = \\pm ( n a _ { - 5 } \\pm b _ { - 5 } \\sqrt { 4 n e ' } ) x _ 1 ^ m \\ , m \\in \\Z , \\end{align*}"}
-{"id": "4138.png", "formula": "\\begin{align*} P _ + ( z ) P _ - ^ { - 1 } ( z ) & = \\mathbb { I } + \\mathcal { O } ( n ^ { - 1 } ) n \\to \\infty , & & z \\in \\partial D \\left ( 0 , r _ n \\right ) , \\\\ P ( z ) N ^ { - 1 } ( z ) & = \\mathbb { I } + \\mathcal { O } ( n ^ { - 1 } ) n \\to \\infty , & & z \\in \\partial D \\left ( 0 , R \\right ) . \\end{align*}"}
-{"id": "5134.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\abs * { \\begin{pmatrix} f _ { m _ j , k _ j } ^ + ( r ) \\\\ f _ { m _ j , k _ j } ^ - ( r ) \\end{pmatrix} - E _ { k _ j } \\begin{pmatrix} A ^ + r ^ { i \\gamma _ { k _ j } } \\\\ A ^ - r ^ { - i \\gamma _ { k _ j } } \\end{pmatrix} } r ^ { - 1 / 2 } = 0 , \\end{align*}"}
-{"id": "6096.png", "formula": "\\begin{gather*} x ^ n + s _ 1 x ^ { n - 1 } + \\ldots s _ { n - 1 } x + s _ n = ( x ^ { e - 1 } + t _ 1 x ^ { e - 1 } + \\ldots + t _ { e - 1 } ) ( x - a _ 2 ) ^ { e _ 2 - 1 } \\ldots ( x - a _ r ) ^ { e _ r - 1 } \\end{gather*}"}
-{"id": "4387.png", "formula": "\\begin{align*} F | _ Z = f \\int _ X | F | ^ 2 e ^ { - \\psi } d V _ { \\omega } \\le \\frac { 2 4 \\pi } { s } \\int _ Z \\frac { | f | ^ 2 e ^ { - \\psi } } { | d T | _ { \\omega } ^ 2 e ^ { - \\lambda } } d A _ { \\omega } . \\end{align*}"}
-{"id": "1547.png", "formula": "\\begin{align*} U _ 1 ^ \\delta \\left ( \\begin{array} { c } z \\\\ x \\end{array} \\right ) : = \\left ( \\begin{array} { c } z - S ^ \\delta ( x ) \\\\ x \\end{array} \\right ) = \\left ( \\begin{array} { c } z + O ( x ^ 2 ) \\\\ x \\end{array} \\right ) . \\end{align*}"}
-{"id": "1794.png", "formula": "\\begin{align*} c _ { 2 , 1 } \\left ( \\limsup _ { t \\downarrow 0 } \\frac { B _ { t } } { \\sqrt { 2 t ^ { 2 H } \\log \\log ( 1 / t ) } } < 1 \\right ) = 0 . \\end{align*}"}
-{"id": "8027.png", "formula": "\\begin{align*} \\phi ( t ) = \\xi ^ 2 t , \\phi ( u ) = \\xi u . \\end{align*}"}
-{"id": "5988.png", "formula": "\\begin{align*} \\omega ^ { ( t ) } ( s ) : = \\frac { 1 } { \\sqrt { 1 - t } } \\left ( \\omega ( t + s ( 1 - t ) ) - \\omega ( t ) \\right ) . \\end{align*}"}
-{"id": "2688.png", "formula": "\\begin{align*} \\mathcal { C } _ { \\tau , \\theta } = \\left \\{ \\tau + \\frac { 1 } { 2 \\cos \\theta } \\Big ( \\frac { 1 } { \\tau } - \\tau \\Big ) e ^ { i \\phi } : \\phi \\in ( - \\pi , \\pi ] \\right \\} , \\end{align*}"}
-{"id": "3229.png", "formula": "\\begin{align*} \\| \\Delta _ { 1 } F _ k \\circ \\Delta _ { 1 } V _ k \\| _ F = O ( \\| \\Delta _ 1 w ^ k \\| ^ 2 ) = O ( \\mu _ k ^ 2 ) . \\end{align*}"}
-{"id": "248.png", "formula": "\\begin{align*} \\Delta _ { \\mathbb { H } } u = \\Delta _ z u + \\frac { | z | ^ 2 } { 4 } \\partial _ { t } ^ 2 u , z = ( x , y ) \\in \\mathbb { R } ^ { 2 n } \\end{align*}"}
-{"id": "8232.png", "formula": "\\begin{align*} L _ 1 = ( q ^ { 1 0 } ; q ^ { 1 0 } ) _ \\infty \\sum _ { n = 0 } ^ \\infty c ( 5 n + 3 ) q ^ { n + 1 } . \\end{align*}"}
-{"id": "9895.png", "formula": "\\begin{align*} f ( x ) & = \\int _ { - \\infty } ^ { x } g ( x ) q ( z ) d z + \\int _ { x } ^ { \\infty } g ( z ) q ( z ) d z \\\\ f ' ( x ) & = g ( x ) q ( x ) + \\int _ { - \\infty } ^ { x } g ' ( x ) q ( z ) d z - g ( x ) q ( x ) \\\\ & = g ' ( x ) Q ( Z \\leq x ) \\end{align*}"}
-{"id": "2816.png", "formula": "\\begin{align*} R _ 0 ^ p = \\left | \\begin{array} { c c c c } \\hat { a } _ { 1 1 } & a _ { 1 2 } & \\cdots & a _ { 1 p } \\\\ a _ { 2 1 } & \\hat { a } _ { 2 2 } & \\cdots & a _ { 2 p } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a _ { p 1 } & a _ { p 2 } & \\cdots & \\hat { a } _ { p p } \\\\ \\end{array} \\right | , \\end{align*}"}
-{"id": "6211.png", "formula": "\\begin{align*} I _ V ' ( t ) = \\frac { \\sqrt { 2 } } { \\pi } \\frac { \\Gamma ( 3 / 2 ) } { t ^ { 3 / 2 } } \\left [ F \\left ( \\frac { 1 } { t } \\right ) - \\frac { 1 } { 2 ^ { 3 / 2 } } \\right ] . \\end{align*}"}
-{"id": "7133.png", "formula": "\\begin{align*} \\mu = \\frac { 1 } { \\epsilon } ( \\chi ^ 3 - \\chi ) - \\frac { \\epsilon } { \\rho } \\chi _ { x x } . \\end{align*}"}
-{"id": "4850.png", "formula": "\\begin{align*} \\delta ( w _ { ( j , r _ s ) } ) = \\sum _ { } ^ { } \\lambda _ { ( i , t ) } [ w _ { ( i , r _ p ) } , w _ { ( t , r _ q ) } ] , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , r _ p \\leq r _ q \\ , \\ , , \\ , \\ , r _ p + r _ q = r _ s - 1 , \\end{align*}"}
-{"id": "2785.png", "formula": "\\begin{align*} \\mathbf { Q } = \\left [ \\begin{array} { c c c } 0 & q & q \\\\ q & 0 & q \\\\ q & q & 0 \\end{array} \\right ] \\end{align*}"}
-{"id": "37.png", "formula": "\\begin{align*} ( I I ) = & \\int _ 0 ^ T e ^ { \\mathcal { Y } _ s ^ { \\alpha _ { s - } } } \\sum _ { k , k ^ { \\prime } \\in I } \\left ( e ^ { \\mathcal { Y } _ s ^ k - \\mathcal { Y } _ s ^ { k ^ { \\prime } } } - 1 \\right ) \\chi _ { \\{ \\alpha _ { s - } = k ^ { \\prime } \\} } d \\tilde { N } _ s ^ { k ^ { \\prime } k } \\\\ & + \\int _ 0 ^ T e ^ { \\mathcal { Y } _ s ^ { \\alpha _ { s - } } } \\sum _ { k \\in I } q ^ { \\alpha _ { s - } k } \\left ( e ^ { \\mathcal { Y } _ s ^ k - \\mathcal { Y } _ s ^ { \\alpha _ { s - } } } - 1 \\right ) d s . \\end{align*}"}
-{"id": "5133.png", "formula": "\\begin{align*} \\begin{pmatrix} \\Gamma _ { m _ j , k _ j } ^ + ( f _ { m _ j , k _ j } ) \\\\ \\Gamma _ { m _ j , k _ j } ^ - ( f _ { m _ j , k _ j } ) \\end{pmatrix} : = \\begin{pmatrix} A ^ + \\\\ A ^ - \\end{pmatrix} ; \\end{align*}"}
-{"id": "1041.png", "formula": "\\begin{align*} & \\lim _ { n \\to + \\infty } \\mathbb { P } ( \\xi ^ { ( n ) } ( I _ { n , k } ) = 0 ) / \\prod _ { j = 1 } ^ k \\mathbb { P } ( \\xi ^ { ( n ) } ( I ( y _ j , F _ n ( x _ j ) ) ) = 0 ) = 1 . \\end{align*}"}
-{"id": "9807.png", "formula": "\\begin{align*} \\Lambda ( u ) : = \\{ ( x ' , 0 , 0 ) : u ( x ' , 0 , 0 ) = 0 \\} \\end{align*}"}
-{"id": "1877.png", "formula": "\\begin{align*} \\frac { 1 } { ( \\pi m ) ^ 2 } \\sum _ { j = 1 } ^ \\infty \\left [ \\frac { A _ j ^ 2 } { Z _ j ^ 2 - ( \\pi m ) ^ 2 } - \\frac { A _ j ^ 2 } { Z _ j ^ 2 } \\right ] = 0 . \\end{align*}"}
-{"id": "4397.png", "formula": "\\begin{align*} \\lambda ( x , y , z ) : = \\frac { 1 } { { \\rm V o l } ( B _ R ( 0 ) ) } \\int _ { B _ R ( x , y , z ) } \\log | T ( \\xi , \\eta , \\zeta ) | ^ 2 d V ( \\xi , \\eta , \\zeta ) . \\end{align*}"}
-{"id": "1709.png", "formula": "\\begin{align*} T ( s + t ) x - T ( s ) x & = [ T ( s + t ) ( x - x _ { \\alpha _ 0 } ) - T ( s ) ( x - x _ { \\alpha _ 0 } ) ] \\\\ & + [ T ( s + t ) x _ { \\alpha _ 0 } - T ( s ) x _ { \\alpha _ 0 } ] \\in V _ 1 + V _ 1 \\subset V _ 0 \\end{align*}"}
-{"id": "2977.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty ( p _ { 2 , 5 } + p _ { 3 , 5 } ) ( n ) q ^ n & = \\frac { 1 } { ( q ^ 2 ; q ^ 5 ) _ \\infty ( q ^ 3 ; q ^ 5 ) _ \\infty } \\\\ & = \\Big ( \\sum _ { n = 0 } ^ \\infty p ( n ) q ^ n \\Big ) \\Big ( 1 + \\sum _ { n = 1 } ^ \\infty ( - 1 ) ^ n ( q ^ { P _ { 7 , n } } + q ^ { Q _ { 7 , n } } ) \\Big ) . \\end{align*}"}
-{"id": "164.png", "formula": "\\begin{align*} \\mathcal { L } _ { X } c _ i = X ( c _ i ) = c _ j . \\end{align*}"}
-{"id": "1365.png", "formula": "\\begin{align*} C ^ n _ { \\operatorname { c u b } } ( \\Lambda , \\mathcal { M } ) = \\prod _ { \\lambda \\in Q _ n ( \\Lambda ) } \\mathcal { M } ( { s ( \\lambda ) } ) \\end{align*}"}
-{"id": "9448.png", "formula": "\\begin{align*} a _ \\theta ( 0 ) - b ( 0 ) = \\frac { 1 } { \\beta _ { N , m } } \\left ( \\sum _ { \\tau \\in \\mathcal { F } _ N } \\nu _ \\tau ^ { ( N ) } ( f ) J _ { N , m } ( \\tau ) - \\sum _ { \\tau \\in \\mathcal { F } _ N } \\nu _ \\tau ^ { ( N ) } ( f ) j _ { N , m } ( \\tau ) \\right ) = - \\sum _ { \\tau \\in \\mathcal { F } _ N } \\nu _ \\tau ^ { ( N ) } ( f ) . \\end{align*}"}
-{"id": "1788.png", "formula": "\\begin{align*} \\bigcup _ { t \\in [ 0 , 1 ] } A _ { k , M } ^ { t } \\subset \\bigcup _ { i = 1 } ^ { n } C _ { i , n } . \\end{align*}"}
-{"id": "6963.png", "formula": "\\begin{align*} x _ 0 = y _ 0 , x _ 1 = \\ln | y _ 1 | , x _ 2 = y _ 2 , x _ 3 = y _ 3 , \\dots \\end{align*}"}
-{"id": "2349.png", "formula": "\\begin{align*} \\Gamma & = \\Gamma ( \\sigma , S ) = \\Gamma _ 1 + \\Gamma _ 2 \\\\ \\Gamma _ 1 & = \\{ \\{ s , ( s , c _ { s , 1 } ) , ( c _ { s , 1 } , c _ { s , 2 } ) , \\cdots , ( c _ { s , k _ s 1 } , c _ { s , k _ s } ) \\} ; s \\in \\sigma \\} \\\\ \\Gamma _ 2 & = \\{ \\{ ( t , c _ { t , 1 } ) , ( c _ { t , 1 } , c _ { t , 2 } ) , \\cdots , ( c _ { t , k _ t - 1 } , c _ { t , k _ t } ) \\} ; t \\in S _ \\setminus ( S _ + + \\sigma ) \\} . \\end{align*}"}
-{"id": "8807.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { \\pi } \\left ( f ( v ^ k ( s , x ) ) + f ( v ( s , x ) ) \\right ) ^ 2 \\ , \\textrm { d } s \\ , \\textrm { d } x \\\\ & \\leq C \\left ( 1 + \\sup _ { k \\in \\mathbb { N } } \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { \\pi } v ^ k ( s , x ) ^ 2 \\ , \\textrm { d } s \\ , \\textrm { d } x + \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { \\pi } v ( s , x ) ^ 2 \\ , \\textrm { d } s \\ , \\textrm { d } x \\right ) < \\infty , \\end{align*}"}
-{"id": "2396.png", "formula": "\\begin{align*} ( c f ) ^ * & = \\overline { c } f ^ * , & ( f _ 1 f _ 2 ) ^ * & = f _ 2 ^ * f _ 1 ^ * , & 1 ^ * & = 1 . \\end{align*}"}
-{"id": "4775.png", "formula": "\\begin{align*} | \\mathcal A _ i \\setminus \\mathcal S _ { i } | < \\frac { 1 } { 2 ^ { 3 8 k } h } \\binom { h } { k } . \\end{align*}"}
-{"id": "2793.png", "formula": "\\begin{align*} O _ t : = \\sum _ { i = 1 } ^ S U _ { i } ^ t . \\end{align*}"}
-{"id": "7787.png", "formula": "\\begin{align*} d L ( G ) ^ 2 & = \\int _ { G } | x | ^ 2 { \\rm d } x = \\int _ { G \\cap r _ d B ^ 2 } | x | ^ 2 { \\rm d } x + \\int _ { G \\setminus r _ d B ^ 2 } | x | ^ 2 { \\rm d } x \\ge \\int _ { G \\cap r _ d B ^ 2 } | x | ^ 2 { \\rm d } x + r _ d ^ 2 | G \\setminus r _ d B ^ 2 | \\\\ & \\ge \\int _ { G \\cap r _ d B ^ 2 } | x | ^ 2 { \\rm d } x + \\int _ { r _ d B ^ 2 \\setminus G } | x | ^ 2 { \\rm d } x = d L ( r _ d B ^ 2 ) ^ 2 . \\end{align*}"}
-{"id": "3991.png", "formula": "\\begin{align*} \\{ e _ { k , N } , e _ { k , N } \\} = \\sum _ { t \\mid N } \\psi ( t ) t ^ { k - 2 } \\sigma _ { 2 k - 3 } ( N / t ) \\cdot \\{ E _ { k , 1 } , E _ { k , 1 } \\} . \\end{align*}"}
-{"id": "7938.png", "formula": "\\begin{align*} Y = \\left [ \\begin{array} { @ { } c | c @ { } } \\begin{matrix} 1 & & \\\\ & \\ddots & \\\\ & & 1 \\end{matrix} & \\begin{matrix} & & \\\\ & { \\normalfont \\Huge Z } & \\\\ & & \\end{matrix} \\end{array} \\right ] , \\end{align*}"}
-{"id": "3021.png", "formula": "\\begin{align*} D _ { p , q } f ( z ) = \\frac { f ( p z ) - f ( q z ) } { ( p - q ) z } , ( p \\neq q , z \\neq 0 ) , \\end{align*}"}
-{"id": "6812.png", "formula": "\\begin{align*} u _ t + u \\nabla u = A '' A ^ { - 1 } x = - \\nabla _ x p \\ , . \\end{align*}"}
-{"id": "900.png", "formula": "\\begin{align*} \\textstyle \\sum _ v m _ u d ^ { h a } _ u m _ v d ^ { t a } _ v = m _ u d ^ { h a } _ u \\gamma ( t a ) . \\end{align*}"}
-{"id": "7698.png", "formula": "\\begin{align*} A ^ j ( p ) \\psi ( \\eta ) = H ( p ) E ^ j ( p ) \\omega ( \\eta ) \\end{align*}"}
-{"id": "1499.png", "formula": "\\begin{align*} \\bar { H } ( t , x , m , D u ) = \\phi ( \\psi ( m ) ) H _ { 1 } ( t , x , m , D u ) + H _ { 2 } ( t , x , m , D u ) . \\end{align*}"}
-{"id": "1169.png", "formula": "\\begin{align*} ( b - a ) \\limsup _ { n \\to + \\infty } \\mathbb { E } \\chi ^ { ( n ) } ( ( b , + \\infty ) ) \\leq \\lim _ { n \\to + \\infty } \\mathbb { E } \\sum _ { k = 1 } ^ n ( ( \\tau _ k - a ) _ + - ( \\tau _ k - b ) _ + ) \\\\ = ( \\pi / 2 ) e ^ { c _ 0 - a } - ( \\pi / 2 ) e ^ { c _ 0 - b } \\leq ( b - a ) \\liminf _ { n \\to + \\infty } \\mathbb { E } \\chi ^ { ( n ) } ( ( a , + \\infty ) ) . \\end{align*}"}
-{"id": "8045.png", "formula": "\\begin{gather*} \\rho _ 2 ( x ) = 1 , \\enspace \\rho _ 2 ( y ) = - 1 , \\rho _ 3 ( x ) = - 1 , \\enspace \\rho _ 3 ( y ) = - 1 , \\rho _ 4 ( x ) = - 1 , \\enspace \\rho _ 4 ( y ) = 1 , \\end{gather*}"}
-{"id": "4433.png", "formula": "\\begin{align*} \\mathcal { V } ( x ) = \\int _ 0 ^ 1 G ( x , y ) \\varphi ( y ) \\dd y + \\frac { x - \\mathcal { A } _ 0 } { 1 + \\mathcal { A } _ 1 - \\mathcal { A } _ 0 } ( \\mathcal { V } _ 1 - \\mathcal { V } _ 0 ) + \\mathcal { V } _ 0 , \\end{align*}"}
-{"id": "5683.png", "formula": "\\begin{align*} \\| x _ \\alpha ^ A - x _ 0 \\| \\le & \\| x _ \\beta ^ A - x _ 0 \\| + h _ \\beta \\| w ^ A _ \\beta \\| \\\\ \\leq & \\| x _ 1 ^ A - x _ 0 \\| + t _ \\beta M + h _ \\beta M = \\| x _ 1 ^ A - x _ 0 \\| + t _ \\alpha M \\ . \\end{align*}"}
-{"id": "7702.png", "formula": "\\begin{align*} \\norm { u } _ { 1 } \\geq \\frac { 1 } { \\sqrt { L } } \\sum _ { i = 1 } ^ l \\norm { \\rho _ { i } \\cdot u } _ { 2 } . \\end{align*}"}
-{"id": "3637.png", "formula": "\\begin{align*} a _ 1 \\cdot ( ( \\alpha _ { - 1 } - \\beta _ { - 1 } ) \\cdot \\alpha _ { - 1 } ) = - \\frac { 1 } { 2 ^ 2 } \\beta _ { - 1 } \\cdot \\alpha _ { - 1 } . \\end{align*}"}
-{"id": "2972.png", "formula": "\\begin{align*} ( q ; q ^ { g - 2 } ) _ \\infty ( q ^ { g - 3 } ; q ^ { g - 2 } ) _ \\infty ( q ^ { g - 2 } ; q ^ { g - 2 } ) _ \\infty & = 1 + \\sum _ { n = 1 } ^ \\infty ( - 1 ) ^ n ( q ^ { P _ { g , n } } + q ^ { Q _ { g , n } } ) . \\end{align*}"}
-{"id": "1190.png", "formula": "\\begin{align*} Y ( 0 ) \\neq Y ( 1 ) \\quad Y ( 0 ) + Y ( 1 ) = 0 , \\end{align*}"}
-{"id": "538.png", "formula": "\\begin{align*} \\nabla u ^ h ( x ) A ^ h ( x ) ^ { - 1 } = Q _ 0 ( x ' ) \\bar A ( x ' ) ^ { - 1 } \\big ( I d _ 3 + h S ^ h ( x ) + \\mathcal { O } ( h ^ 2 ) \\big ) , \\end{align*}"}
-{"id": "3742.png", "formula": "\\begin{align*} X \\coloneqq \\left \\{ \\mathbf { J } \\in L ^ { 2 } \\left ( V ; \\mathbb { C } ^ { 3 } \\right ) : \\left ( \\mathcal { B } _ { l , m } ^ { ( j ) } , \\mathbf { J } \\right ) = a _ { l , m } ^ { ( j ) } , \\ ; \\Im \\left [ \\mathcal { P } \\right ] = 0 \\right \\} . \\end{align*}"}
-{"id": "3819.png", "formula": "\\begin{align*} B _ { G , S , \\pi } ( \\ell ) \\hookrightarrow \\bigcup _ { \\ell _ 1 + \\dots + \\ell _ d \\le \\eta \\ell + c _ 0 } K \\times \\prod _ { i = 1 } ^ d B _ { G , S , \\pi } ( \\ell _ i ) \\end{align*}"}
-{"id": "7242.png", "formula": "\\begin{align*} u _ t + L _ r u _ x + u u _ x = 0 \\qquad \\mbox { f o r } r > 1 . \\end{align*}"}
-{"id": "2437.png", "formula": "\\begin{align*} P _ { \\widetilde { \\varphi _ { \\ell } ^ { H _ 1 } } } ( T ) = \\frac { 1 } { ( - 1 ) ^ { \\ell / 4 } + 2 ^ { ( \\ell - 4 ) / 2 } } \\frac { ( - 1 ) ^ { \\ell / 4 } + 2 ^ { ( \\ell - 4 ) / 2 } T ^ { \\ell - 4 } } { 1 - 2 T + 2 T ^ 2 } . \\end{align*}"}
-{"id": "1441.png", "formula": "\\begin{align*} w ^ { n + 1 } ( T , x ) = \\mathbb { P } _ { \\delta } w _ { T } ( x ) , \\end{align*}"}
-{"id": "9983.png", "formula": "\\begin{align*} B ^ { i j } = \\partial _ x \\circ F ^ { i j } \\circ \\partial _ x = \\partial _ x ^ { } ( g ^ { i j } \\partial _ x ^ { } + c ^ { i j } _ k u ^ k _ x + c ^ \\alpha w ^ i _ { \\alpha k } u ^ k _ x \\partial _ x ^ { - 1 } w ^ j _ { \\alpha h } u ^ h _ x ) \\partial _ x ^ { } \\end{align*}"}
-{"id": "8106.png", "formula": "\\begin{align*} W _ { \\rm K } : = \\sum _ { i = 1 } ^ { g - 1 } ( \\ell _ i - i ) + g - 1 . \\end{align*}"}
-{"id": "6971.png", "formula": "\\begin{align*} q _ t ( \\varphi ( w ) ) \\cdot ( \\varphi ' ( w ) ) ^ 2 = q _ w ( w ) - { \\rm S } ( \\varphi ) ( w ) , \\end{align*}"}
-{"id": "8673.png", "formula": "\\begin{align*} \\partial _ t \\frac { | u | ^ 2 } { 2 } + u \\cdot \\div ( u \\otimes u ) + u \\cdot \\nabla p = \\partial _ t \\frac { | u | ^ 2 } { 2 } + \\div \\bigg ( \\frac { | u | ^ 2 } { 2 } u + p u \\bigg ) = 0 \\ , . \\end{align*}"}
-{"id": "5884.png", "formula": "\\begin{align*} q ^ 2 t _ { s _ 0 s _ 1 s _ 0 } - q ^ { \\frac { 3 } { 2 } } t _ { s _ 0 s _ 1 s _ 0 s _ 1 } + q ^ { \\frac { 5 } { 2 } } t _ { s _ 0 s _ 1 } - q ^ 2 t _ { s _ 0 s _ 1 s _ 0 } + q ^ 3 t _ { s _ 0 } - q ^ \\frac { 5 } { 2 } t _ { s _ 0 s _ 1 } - q ^ 3 t _ { s _ 0 } = - q ^ { \\frac { 3 } { 2 } } t _ { s _ 0 s _ 1 s _ 0 s _ 1 } . \\end{align*}"}
-{"id": "2501.png", "formula": "\\begin{align*} \\Gamma : M \\longrightarrow \\frac { { \\sf A u t } _ { C R } ( M ) } { { \\sf G _ 0 } \\cdot \\sf G _ + } \\cong { \\sf G _ - } \\ \\ \\ \\ \\ \\ \\ { \\rm w i t h } \\ \\ \\ \\ \\Gamma ( 0 ) = { \\tt e } , \\end{align*}"}
-{"id": "633.png", "formula": "\\begin{align*} \\psi ^ \\ast ( x u x ^ { - 1 } ) = ( \\psi ^ { u x ^ { - 1 } } ) ^ \\ast ( x ) ( \\psi ^ { x ^ { - 1 } } ) ^ \\ast ( u ) \\psi ^ \\ast ( x ^ { - 1 } ) = ( \\psi ^ { u x ^ { - 1 } } ) ^ \\ast ( x ) \\end{align*}"}
-{"id": "2541.png", "formula": "\\begin{align*} \\lim _ { ( 0 , \\infty ) \\ni t \\to 0 } \\frac { \\int _ E u ^ { \\alpha ( x ) } t ^ { \\alpha ( x ) } \\rho ( d x ) } { \\int _ E t ^ { \\alpha ( x ) } \\rho ( d x ) } = u ^ { \\alpha _ 0 } , u \\in ( 0 , 1 ] . \\end{align*}"}
-{"id": "3712.png", "formula": "\\begin{align*} \\begin{cases} z _ B \\equiv \\sum _ x n _ { x B } \\bar { z } _ { x B } = \\sum _ x \\hat { p } _ x N _ x \\bar { Z } _ x \\\\ Z = \\sum _ x n _ { x B } \\bar { z } _ { x B } / \\hat { p } _ x \\end{cases} \\end{align*}"}
-{"id": "2479.png", "formula": "\\begin{align*} | P | ( x ) = \\sup \\Bigl \\{ \\ , \\sum _ { i _ 1 , \\dots , i _ n } | A ( u ^ 1 _ { i _ 1 } , \\dots , u ^ n _ { i _ n } ) | : u ^ 1 , \\dots , u ^ n \\in \\Pi ( x ) \\Bigr \\} \\ , , \\end{align*}"}
-{"id": "7098.png", "formula": "\\begin{align*} N ^ { \\nu - s - \\gamma ( \\sigma ^ \\prime - 1 ) } = \\underbrace { N ^ { \\nu - 1 } } _ { } \\cdot \\underbrace { N ^ { - \\gamma ( \\sigma ^ \\prime - 1 ) } } _ { F _ { \\leq n - 1 } ^ \\omega } ~ \\cdot \\underbrace { N ^ { 1 - s } } _ { F _ n ^ \\omega } . \\end{align*}"}
-{"id": "4326.png", "formula": "\\begin{align*} \\begin{aligned} a _ h ( u _ h , v ) = \\sum _ { K \\in \\mathcal { T } _ h } ( \\nabla \\Pi _ K u , \\nabla \\Pi _ K v ) _ K + \\sum _ { K \\in \\mathcal { T } _ h } \\langle \\nabla ( u - \\Pi _ K u ) \\cdot n , Q _ K v - Q _ F v \\rangle _ { \\partial K } . \\end{aligned} \\end{align*}"}
-{"id": "1719.png", "formula": "\\begin{align*} | ( T _ \\varphi ( h ) f ) ( x ) - f ( x ) | & = | f ( \\varphi ( h , x ) ) - f ( x ) | \\leq \\varepsilon \\cdot ( | \\varphi ( h , x ) - x | \\cdot \\delta ^ { - 1 } + 1 ) \\leq \\varepsilon \\cdot ( 2 + | x | ) \\end{align*}"}
-{"id": "8082.png", "formula": "\\begin{align*} \\widetilde { a _ 1 } = e ^ { - 1 } M ( 2 , 2 , 1 ) = e ^ { - 1 } \\sum _ { k \\ge 0 } \\frac { ( 2 ) _ k } { ( 2 ) _ k \\ , k ! } = 1 = a _ 1 , \\end{align*}"}
-{"id": "983.png", "formula": "\\begin{align*} F _ { d } = \\{ a _ { i } b _ { j } \\mid 1 \\leq j < i \\leq d \\} H _ { r , d } = \\{ a _ { i } e _ { 2 } e _ { 4 } \\cdots e _ { 2 r - 2 } \\mid 1 \\leq i \\leq d \\} . \\end{align*}"}
-{"id": "9356.png", "formula": "\\begin{align*} \\gamma _ m ( G _ k ) = \\langle [ y , x , \\overset { m - 1 } { \\ldots } , x ] , [ y , x , \\overset { m - 2 } { \\ldots } , x , y ] \\rangle \\ ; \\gamma _ { m + 1 } ( G _ k ) , \\end{align*}"}
-{"id": "7832.png", "formula": "\\begin{align*} \\| T \\| \\| S \\| = | [ \\langle T x , S x \\rangle \\xi , \\xi ] | \\leq \\| T x \\| \\| S x \\| \\leq \\| T \\| \\| S \\| . \\end{align*}"}
-{"id": "8580.png", "formula": "\\begin{align*} H \\theta = ( m + \\tfrac { r } { 2 } - N _ c ( \\tau ) ) \\theta \\qquad \\textup { a n d } Z \\theta = ( l - \\tfrac { r } { 2 } + N _ c ( \\sigma ) ) \\theta . \\end{align*}"}
-{"id": "6022.png", "formula": "\\begin{align*} \\limsup _ { \\epsilon \\downarrow 0 } \\ , \\ , \\ , \\inf \\left \\{ \\tilde { \\alpha } ^ \\mu _ \\epsilon ( Q ) \\ , : \\ , Q \\in \\mathcal { P } ( \\C ) , \\ Q \\circ H ^ { - 1 } = \\nu _ \\epsilon \\right \\} & \\leq \\inf \\left \\{ \\alpha ^ \\mu _ 0 ( Q ) \\ , : \\ , Q \\in \\mathcal { P } ( \\C ) , \\ Q \\circ H ^ { - 1 } = \\nu \\right \\} . \\end{align*}"}
-{"id": "7729.png", "formula": "\\begin{align*} H ^ * ( Z ) \\cong \\Z [ w ] / ( w ^ q ) \\otimes ( \\bigotimes _ { i = 1 } ^ l \\Z [ a _ i ] / ( a _ i ^ q ) ) \\otimes \\Z [ z ] / ( z ^ q ) \\end{align*}"}
-{"id": "9925.png", "formula": "\\begin{align*} & D ( P ^ { \\mu } | _ { \\mathcal { F } ^ { \\mathcal { X } } _ { n } \\vee \\mathcal { F } ^ { \\mathcal { Y } } _ { 0 , n } } \\| P ^ { \\nu } | _ { \\mathcal { F } ^ { \\mathcal { X } } _ { n } \\vee \\mathcal { F } ^ { \\mathcal { Y } } _ { 0 , n } } ) \\\\ & = D ( P ^ { \\mu } | _ { \\mathcal { F } ^ { \\mathcal { X } } _ { n , \\infty } \\vee \\mathcal { F } ^ { \\mathcal { Y } } _ { 0 , \\infty } } \\| P ^ { \\nu } | _ { \\mathcal { F } ^ { \\mathcal { X } } _ { n , \\infty } \\vee \\mathcal { F } ^ { \\mathcal { Y } } _ { 0 , \\infty } } ) \\end{align*}"}
-{"id": "7224.png", "formula": "\\begin{align*} ( \\mu - \\phi ) ^ 2 ( x _ 0 ) = \\mu ^ 2 . \\end{align*}"}
-{"id": "9196.png", "formula": "\\begin{align*} \\dim H _ { b } & = \\dim W _ { 0 } \\\\ & = \\dim ( T _ { \\gamma \\left ( 0 \\right ) } N \\cap \\mathcal { K } ( 0 ) ^ { \\perp } ) \\\\ & \\geq n - \\dim \\mathcal { K } . \\end{align*}"}
-{"id": "986.png", "formula": "\\begin{align*} I ( p ) = \\langle a _ { d } b _ { d - 1 } , \\dots , a _ { d } b _ { j + 1 } , \\dots , a _ { j } b _ { j + 1 } \\rangle : \\langle a _ { d } b _ { j } \\rangle = \\langle b _ { d - 1 } , \\dots , b _ { j + 1 } \\rangle . \\end{align*}"}
-{"id": "1330.png", "formula": "\\begin{align*} \\begin{aligned} \\min _ { u _ 1 , \\pi } & \\ \\rho \\big [ Z ^ { u _ 1 , \\pi } , M ( u _ 1 ) \\big ] , \\\\ & \\ u _ 1 \\in U _ 1 , \\\\ & \\ \\pi ( \\cdot ) \\lessdot U _ 2 ( \\cdot , u _ 1 ) . \\end{aligned} \\end{align*}"}
-{"id": "5289.png", "formula": "\\begin{align*} A _ 2 ^ { ( 1 ) } + A _ 2 ^ { ( 1 ) } : & \\left \\{ \\begin{array} { r c l } \\bar { q } _ 1 & = & - p _ 2 - q _ 2 + a q _ 2 ^ { - 1 } + b \\\\ \\bar { p } _ 1 & = & q _ 2 \\\\ \\bar { q } _ 2 & = & - q _ 1 - p _ 1 + a q _ 1 ^ { - 1 } + b \\\\ \\bar { p } _ 2 & = & q _ 1 \\end{array} \\right . \\end{align*}"}
-{"id": "3886.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty K _ \\omega ( U ^ j , V ^ j ) \\leq 0 \\sum _ { j = 1 } ^ \\infty H _ \\omega ( U ^ j , V ^ j ) \\leq 6 d ( \\omega ) . \\end{align*}"}
-{"id": "4612.png", "formula": "\\begin{align*} P _ r = \\{ x \\in \\mathbb { R } ^ 2 : x r l ^ { \\delta } _ i , i \\leq L \\} . \\end{align*}"}
-{"id": "9324.png", "formula": "\\begin{align*} \\mathbf { H } ( t , y ) = \\frac { 1 - t ^ 2 } { n } \\mathbf { N _ { 1 } } ( t , y ) - \\mathbf { N _ { 2 } } ( t , y ) \\end{align*}"}
-{"id": "219.png", "formula": "\\begin{align*} u = f ( \\rho ( z , t ) ) \\end{align*}"}
-{"id": "2177.png", "formula": "\\begin{align*} \\| \\mathcal F m _ { \\ell } ( \\mathbf x ) ( 1 + \\| \\mathbf x \\| ) ^ \\beta \\| _ { L ^ 2 ( d w ( \\mathbf x ) ) } \\leq C \\| m _ { \\ell } \\| _ { W ^ \\alpha _ 2 } = C \\| \\widehat m _ { \\ell } ( \\mathbf x ) ( 1 + \\| \\mathbf x \\| ) ^ { \\alpha } \\| _ { L ^ 2 ( d \\mathbf x ) } , \\end{align*}"}
-{"id": "5174.png", "formula": "\\begin{align*} \\lim \\sup _ { h \\rightarrow + \\infty } \\lim _ { \\lambda \\rightarrow + \\infty } \\inf \\mathbb { P } ( A ^ { \\ast } ( h ) / B ^ { \\ast } ( h ) < 1 / \\lambda ) = 0 . \\end{align*}"}
-{"id": "7316.png", "formula": "\\begin{align*} \\begin{cases} P _ 1 ( x , D ) u + V _ 1 u = & V _ 3 v , \\\\ P _ 2 ( x , D ) v + V _ 2 v = V _ 3 u , \\end{cases} \\end{align*}"}
-{"id": "8954.png", "formula": "\\begin{align*} \\mathbb { P } ( Y = 1 \\ , | \\ , X = x ) = \\frac { 1 } { 1 + \\exp \\left \\{ - \\beta _ 0 - \\int _ 0 ^ 1 \\alpha ( t ) ( x ( t ) - m ( t ) ) d t \\right \\} } , \\end{align*}"}
-{"id": "5630.png", "formula": "\\begin{align*} - \\frac { \\alpha } { 2 } \\int _ 0 ^ { + \\infty } | u | ^ 2 v _ t d x & = - \\frac { \\alpha } { 2 } \\int _ 0 ^ { + \\infty } \\Big ( \\int _ 0 ^ x ( | u | ^ 2 ) _ x ( x ' , t ) d x ' + | u ( 0 , t ) | ^ 2 \\Big ) v _ t d x \\\\ & = - \\frac { \\alpha } { 2 } | u ( 0 , t ) | ^ 2 \\int _ { 0 } ^ { + \\infty } v _ t d x - \\frac { \\alpha } { 2 } \\int _ 0 ^ { + \\infty } \\Big [ \\int _ 0 ^ x ( | u | ^ 2 ) _ x ( x ' , t ) d x ' \\Big ] v _ t d x \\\\ & : = I ( u , v ) + I I ( u , v ) . \\end{align*}"}
-{"id": "6329.png", "formula": "\\begin{align*} q _ x ( 0 , t ) = q _ x ( 1 , t ) = 0 \\quad t > 0 , \\end{align*}"}
-{"id": "9771.png", "formula": "\\begin{align*} p _ { * , X _ \\ell } ( X ) : = \\lim _ { r \\downarrow 0 } \\frac { u ( X _ \\ell + r X ) } { r ^ \\kappa } . \\end{align*}"}
-{"id": "1699.png", "formula": "\\begin{align*} ( { \\mathcal L } ^ * \\mu ) ( V ) = \\int ( { \\mathcal L } \\chi _ V ) \\ , d \\mu . \\end{align*}"}
-{"id": "1077.png", "formula": "\\begin{align*} & \\ln \\frac { D _ n ( w \\alpha _ n ) } { D _ n ( \\alpha _ n ) } \\leq n ^ 2 \\ln \\cos \\frac { w \\alpha _ n } { 2 } - n ^ 2 \\ln \\cos \\frac { \\alpha _ n } { 2 } + 1 = n ^ 2 ( F ( w \\alpha _ n ) - F ( \\alpha _ n ) ) + 1 . \\end{align*}"}
-{"id": "1227.png", "formula": "\\begin{align*} J ( f , g ) ( \\xi ) = \\int e ^ { - 3 i \\Phi ( \\xi , \\eta ) } f ( \\eta ) g ( \\eta - \\xi ) d \\eta . \\end{align*}"}
-{"id": "2494.png", "formula": "\\begin{align*} \\aligned \\begin{array} { l } w _ j = \\Phi _ j ( { \\bf z } , \\overline { \\bf z } , \\overline { \\bf w } ) + { \\rm O } ( [ w _ j ] ) , \\ \\ \\ \\ \\ \\ \\ \\ \\ j = 1 , \\ldots , k \\end{array} \\endaligned \\end{align*}"}
-{"id": "8717.png", "formula": "\\begin{align*} { \\nu } ^ * = \\hat { \\nu } \\Lambda \\end{align*}"}
-{"id": "9726.png", "formula": "\\begin{align*} \\Sigma _ { \\kappa } ^ { m , { \\rm a } } ( u ) : = \\{ X _ \\circ \\in \\Sigma _ { \\kappa } ^ { m } ( u ) : \\lambda _ { \\ast , X _ \\circ } \\in [ \\kappa , \\kappa + 1 ) \\} , \\end{align*}"}
-{"id": "997.png", "formula": "\\begin{align*} \\frac { 1 } { Z _ n } e ^ { - n \\sum \\limits _ { i = 1 } ^ n \\lambda _ i ^ 2 / 2 } \\prod _ { 1 \\leq i < j \\leq n } | \\lambda _ i - \\lambda _ j | ^ 2 \\end{align*}"}
-{"id": "2276.png", "formula": "\\begin{align*} ( \\delta _ x | \\delta _ y ) & = \\delta ( x - y ) & \\int \\d x | \\delta _ x ) ( \\delta _ x | & = 1 . \\end{align*}"}
-{"id": "4237.png", "formula": "\\begin{align*} W _ { T } ^ { 1 , p } \\left ( \\Omega ; \\Lambda ^ { k } \\right ) = \\left \\{ \\omega \\in W ^ { 1 , p } \\left ( \\Omega ; \\Lambda ^ { k } \\right ) : \\nu \\wedge \\omega = 0 \\partial \\Omega \\right \\} , \\end{align*}"}
-{"id": "8006.png", "formula": "\\begin{align*} A _ i \\geq \\begin{cases} \\sum _ { k \\in \\{ 0 , 1 \\} } e ^ { - a | k - y _ i | ^ 2 } , 0 \\leq y _ i < 1 ; \\\\ \\sum _ { k \\in \\{ - 1 , 0 \\} } e ^ { - a | k - y _ i | ^ 2 } , - 1 < y _ i < 0 . \\end{cases} \\end{align*}"}
-{"id": "8053.png", "formula": "\\begin{align*} \\phi ( \\overline { r d s } ) = \\overline { \\phi ( r ) d \\phi ( s ) } . \\end{align*}"}
-{"id": "984.png", "formula": "\\begin{align*} ( n _ 1 , \\dots , n _ { \\frac { d ( d + 1 ) } { 2 } } ) = ( 0 , \\underbrace { 1 , 1 } _ { 2 } , \\underbrace { 2 , 2 , 2 } _ { 3 } , \\ldots , \\underbrace { d - 2 , d - 2 , \\dots , d - 2 } _ { d - 1 } , \\underbrace { d - 1 , d - 1 , \\ldots , d - 1 } _ { d } ) \\end{align*}"}
-{"id": "10039.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n m _ j q _ j ( \\omega t ) = 0 , t \\in \\R . \\end{align*}"}
-{"id": "7868.png", "formula": "\\begin{align*} B ( N ) = \\bigcup _ { z \\in Z ( B ( N ) ) } z + Y ( N , z ) , \\end{align*}"}
-{"id": "9063.png", "formula": "\\begin{align*} F _ x ( y ) = f \\left ( x + \\frac { R } { \\sqrt { E } } y \\right ) \\end{align*}"}
-{"id": "8974.png", "formula": "\\begin{align*} d ( x , L _ w ) \\asymp d _ E ( x , L _ w ) \\asymp d _ E ( x , w ^ \\perp ) = \\frac { | \\langle x , w \\rangle | } { \\| x \\| \\cdot \\| w \\| } \\end{align*}"}
-{"id": "6159.png", "formula": "\\begin{align*} \\partial _ { \\overline { z } } ^ n B _ z ^ \\omega ( \\xi ) = \\left ( \\frac { \\xi } { \\overline { z } } \\right ) ^ n ( B _ z ^ \\omega ) ^ { ( n ) } ( \\xi ) . \\end{align*}"}
-{"id": "10047.png", "formula": "\\begin{align*} \\dot x = - \\frac { 1 } { 4 } x ^ 3 y , \\dot y = - \\frac { 1 } { 4 } x ^ 4 + \\O ( x ^ 6 ) , \\dot \\alpha = \\O ( x ^ 6 ) , \\dot G = \\O ( x ^ 6 ) . \\end{align*}"}
-{"id": "8921.png", "formula": "\\begin{align*} \\mathbb { E } [ N _ w ^ { \\hat { x } } ( H [ + ] ) \\mid \\hat { x } ] = \\binom { n ^ { + + } + n ^ { + - } } { 2 } \\frac { a } { n } - \\frac { ( a - b ) } { n } ( n ^ { + + } - n _ C ^ { + + } + n _ C ^ { + - } ) ( n ^ { + - } + n _ C ^ { + + } - n _ C { ^ + - } ) , \\end{align*}"}
-{"id": "8767.png", "formula": "\\begin{align*} { \\rm t r } ( T ) = D F \\ , . \\end{align*}"}
-{"id": "4802.png", "formula": "\\begin{align*} \\Phi ^ + _ { m _ j , \\pm ( j + 1 / 2 ) } : = \\left ( \\begin{array} { c } i \\ , \\psi ^ { m _ j } _ { j \\pm 1 / 2 } \\\\ 0 \\end{array} \\right ) , \\Phi ^ - _ { m _ j , \\pm ( j + 1 / 2 ) } : = \\left ( \\begin{array} { c } 0 \\\\ \\psi ^ { m _ j } _ { j \\mp 1 / 2 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "40.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow \\infty } ( \\mathbf { y } ^ i ( T , v ) - \\lambda T - \\mathbf { y } ^ i ( v ) ) = L , \\end{align*}"}
-{"id": "5520.png", "formula": "\\begin{align*} E ^ { \\Pi ^ { D _ T } } [ e ^ { u \\sqrt { T } G ( b ) } | X ^ { T } ] = e ^ { \\Lambda _ T ( u ) } \\frac { \\int _ { D _ T } e ^ { S _ T ( b ) + \\ell _ T ( b _ u ) } d \\Pi ( b ) } { \\int _ { D _ T } e ^ { \\ell _ T ( b ) } d \\Pi ( b ) } , \\end{align*}"}
-{"id": "3582.png", "formula": "\\begin{align*} P _ { i j k } = \\left \\{ X _ { 0 } = X _ { i } , \\ , X _ { 1 } = X _ { j } , \\ , X _ { 2 } = X _ { k } \\right \\} \\subset \\mathcal { M } _ { 3 , 3 } \\end{align*}"}
-{"id": "622.png", "formula": "\\begin{align*} ( \\alpha , \\beta ) ^ \\ast : K _ { \\varphi _ 2 } \\xrightarrow { \\alpha ^ \\ast } K _ { \\varphi _ 2 \\alpha } \\hookrightarrow K _ { \\beta \\varphi _ 2 \\alpha } = K _ { \\varphi _ 1 } \\ . \\end{align*}"}
-{"id": "4431.png", "formula": "\\begin{align*} \\forall i \\in \\{ 1 , \\ldots , n \\} : \\ ; \\partial _ t u ^ i = \\partial _ x ( \\partial _ x u ^ i + \\alpha _ i ( u ^ i \\partial _ x \\mathcal { V } ) ) , \\quad ( t , x ) \\in ( 0 , T ) \\times ( 0 , 1 ) , \\\\ \\partial _ { x x } \\mathcal { V } = \\sum _ { i = 1 } ^ n \\beta ^ i u ^ i + \\zeta , \\quad ( t , x ) \\in ( 0 , T ) \\times ( 0 , 1 ) , \\end{align*}"}
-{"id": "3804.png", "formula": "\\begin{align*} & \\sum _ { n = 1 } ^ \\infty \\lambda _ \\pi ( n , r ) e \\left ( \\frac { a n } { q } \\right ) g ( n ) \\\\ = & q \\sum _ \\pm \\sum _ { n _ 1 | q r } \\sum _ { n _ 2 = 1 } ^ \\infty \\frac { \\lambda _ \\pi ( n _ 1 , n _ 2 ) } { n _ 1 n _ 2 } S ( r \\bar a , \\pm n _ 2 ; q r / n _ 1 ) G _ \\pm \\left ( \\frac { n _ 1 ^ 2 n _ 2 } { q ^ 3 r } \\right ) . \\end{align*}"}
-{"id": "5600.png", "formula": "\\begin{align*} A _ 3 & = \\frac { \\beta } { 2 } \\int _ { - \\infty } ^ 0 x | u | ^ 2 ( \\bar { u } u _ x + u \\bar { u } _ x ) d x \\\\ & = \\frac { \\beta } { 2 } \\int _ { - \\infty } ^ 0 x \\frac { 1 } { 2 } ( | u | ^ 4 ) _ x d x \\\\ & = - \\frac { \\beta } { 4 } \\int _ { - \\infty } ^ 0 | u | ^ 4 d x . \\end{align*}"}
-{"id": "785.png", "formula": "\\begin{align*} \\sigma _ { \\lambda _ i } ^ * ( \\sigma ) = \\lambda _ i ^ { \\sigma } \\lambda _ i ^ { - 1 } , \\end{align*}"}
-{"id": "2978.png", "formula": "\\begin{align*} p ( 8 ) - p ( 8 - P _ { 7 , 1 } ) - p ( 8 - Q _ { 7 , 1 } ) + p ( 8 - P _ { 7 , 2 } ) & = p ( 8 ) - p ( 7 ) - p ( 4 ) + p ( 1 ) \\\\ & = 2 2 - 1 5 - 5 + 1 = 3 \\\\ & = ( p _ { 2 , 5 } + p _ { 3 , 5 } ) ( 8 ) . \\end{align*}"}
-{"id": "1820.png", "formula": "\\begin{align*} \\eta = \\eta ( \\xi , a ) = \\frac { \\xi } { 2 \\pi a } \\left ( \\int _ 0 ^ \\infty \\frac { y e ^ { - y } } { y e ^ { y } + \\frac { \\xi a } { 2 } \\sinh y } d y - \\int _ 0 ^ \\infty \\frac { y e ^ { - y } } { y e ^ { y } + \\frac { \\xi a } { 2 } \\cosh y } d y \\right ) . \\end{align*}"}
-{"id": "8581.png", "formula": "\\begin{align*} \\Lambda ( m , l ) ' _ 1 + \\Lambda ( m , l ) ' _ 2 = \\left \\{ \\begin{array} { l l } l + 1 \\leq r , & \\textup { i f } m = 1 \\\\ l + 2 \\leq r , & \\textup { i f } m > 1 , \\end{array} \\right . \\end{align*}"}
-{"id": "225.png", "formula": "\\begin{align*} \\operatorname { s i n } \\phi = \\psi \\end{align*}"}
-{"id": "3332.png", "formula": "\\begin{align*} B ( u ) = \\sum _ { \\substack { \\forall 1 \\le i \\le r : \\\\ b _ i , y _ i \\ge 1 } } \\prod _ { i = 1 } ^ { r } c ( b _ i , y _ i , \\mu _ i ) y _ i u ^ { b _ i } = \\prod _ { i = 1 } ^ { r } ( \\sum _ { b _ i , y _ i \\ge 1 } c ( b _ i , y _ i , \\mu _ i ) y _ i u ^ { b _ i } ) . \\end{align*}"}
-{"id": "8118.png", "formula": "\\begin{align*} A _ { n , r } ( t ) \\ = \\ \\sum _ { k = 0 } ^ n { n \\choose k } d _ { k , r } ( t ) \\end{align*}"}
-{"id": "4006.png", "formula": "\\begin{align*} A _ { U ( \\beta ) f } = \\sum _ j | c _ j | ^ 2 \\beta ^ { 1 - \\kappa } Q _ \\beta ( f _ j ) \\cdot A _ { f _ j } . \\end{align*}"}
-{"id": "482.png", "formula": "\\begin{align*} ( a _ 1 x _ 1 + \\cdots + a _ n x _ n ) ^ { \\nu - \\ell } \\in \\ker H _ { \\nu - \\ell } = I _ { \\nu - \\ell } . \\end{align*}"}
-{"id": "3938.png", "formula": "\\begin{align*} ( \\operatorname { \\mathcal R } \\sigma ) ( g ) = \\sum \\sigma ( g _ { 1 } ) \\sigma ( S g _ { 2 } ) = \\sum \\sigma ( g _ { 1 } S g _ { 2 } ) = \\varepsilon ( g ) 1 \\ , . \\end{align*}"}
-{"id": "1704.png", "formula": "\\begin{align*} \\chi _ { A _ 0 } ( x ) \\cdot \\varphi ( x ) = \\varphi ^ + ( x ) = ( { \\mathcal L } ( \\varphi ^ + ) ) ( x ) = \\int \\varphi ^ + ( y ) \\ P ( x , d y ) \\ \\ \\ \\ \\mu \\mbox { - a . e . } x \\in X . \\end{align*}"}
-{"id": "7859.png", "formula": "\\begin{align*} \\exp \\Big ( - c ' \\kappa ( d , N ) ^ 2 \\sum _ { j = r + 1 } ^ d \\| \\xi _ j \\| ^ 2 \\Big ) \\le \\exp \\Big ( - \\frac { c ' \\kappa ( d , N ) ^ 2 } { 4 } \\sum _ { j = 1 } ^ d \\| \\xi _ j \\| ^ 2 \\Big ) . \\end{align*}"}
-{"id": "2969.png", "formula": "\\begin{align*} P ( \\beta ) = h _ { \\max } ( \\beta ) + \\beta \\partial _ - P ( \\beta ) = h _ { \\min } ( \\beta ) + \\beta \\partial _ + P ( \\beta ) . \\end{align*}"}
-{"id": "3936.png", "formula": "\\begin{align*} x \\cdot g = & \\ \\sum x \\cdot \\bigl [ \\sigma \\bigl ( S ( \\pi ( f _ { 2 } ) ) \\gamma \\bigr ) \\bigr ] ( x \\cdot f _ { 1 } ) = \\gamma ( x ) \\sum \\sigma \\bigl ( x \\cdot S \\bigl ( \\pi ( f _ { 2 } ) \\bigr ) \\gamma \\bigr ) ( x \\cdot f _ { 1 } ) = \\\\ & \\ \\gamma ( x ) \\sum \\sigma \\bigl ( S \\bigl ( \\pi ( f _ { 2 } \\cdot x ^ { - 1 } ) \\bigr ) \\gamma \\bigr ) x \\cdot f _ { 1 } = \\gamma ( x ) \\sum \\sigma \\bigl ( S \\bigl ( \\pi ( f _ { 2 } ) \\bigr ) \\gamma \\bigr ) f _ { 1 } = \\\\ & \\ \\gamma ( x ) g . \\end{align*}"}
-{"id": "6554.png", "formula": "\\begin{align*} a _ i = \\frac { w } { w ' } a ' _ i . \\end{align*}"}
-{"id": "873.png", "formula": "\\begin{gather*} u ^ * { } ^ T u = \\frac { 1 } { \\sqrt { 2 } } ( - y + y \\mp 1 ) = \\mp \\frac { 1 } { \\sqrt { 2 } } , v ^ * { } ^ T v = ( \\pm 1 ) \\left ( - \\frac { 1 } { \\sqrt { 2 } } \\right ) = \\mp \\frac { 1 } { \\sqrt { 2 } } . \\end{gather*}"}
-{"id": "9444.png", "formula": "\\begin{align*} \\frac { 2 \\pi i } { e _ \\tau } \\mathrm { R e s } _ \\tau f _ \\theta = \\nu _ \\tau ^ { ( N ) } ( f ) \\end{align*}"}
-{"id": "9595.png", "formula": "\\begin{align*} \\frac { 1 } { ( q ) _ N } \\sum _ { n = 1 } ^ N \\frac { ( q ) _ { n - 1 } q ^ n } { ( 1 - q ^ n ) } & = \\sum _ { n = 1 } ^ { \\infty } \\textup { s p t } ( n , N ) q ^ n , \\\\ \\frac { 1 } { ( q ) _ N } \\sum _ { n = 1 } ^ N \\frac { q ^ n } { ( 1 - q ^ n ) ^ 2 } & = \\frac { 1 } { 2 } \\sum _ { n = 1 } ^ { \\infty } M _ { 2 , N } ( n ) q ^ n . \\end{align*}"}
-{"id": "793.png", "formula": "\\begin{align*} L ^ { q ^ { { s } } - 1 } = ( - 1 ) ^ s \\prod _ { i = 1 } ^ { { s } } \\left ( T - \\zeta _ 1 ^ { q ^ { i - 1 } } \\right ) ^ { q ^ i } . \\end{align*}"}
-{"id": "8560.png", "formula": "\\begin{align*} ( I - \\mathcal { H } ) b _ { \\alpha } = [ \\zeta _ { \\alpha } , \\mathcal { H } ] b - \\partial _ { \\alpha } [ D _ t \\zeta , \\mathcal { H } ] \\frac { \\bar { \\zeta } _ { \\alpha } - 1 } { \\zeta _ { \\alpha } } + \\frac { i } { \\pi } \\sum _ { j = 1 } ^ 2 \\frac { \\lambda _ j \\zeta _ { \\alpha } } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } . \\end{align*}"}
-{"id": "7046.png", "formula": "\\begin{align*} E ( u ) : = \\frac { 1 } { 2 } \\| d u \\| ^ 2 : = \\frac { 1 } { 2 } \\int _ M | d u | ^ 2 d \\mu _ g = \\int _ M ( g ^ { i j } u ^ v _ i \\o { u ^ v _ j } \\phi _ { v \\b { v } } ) d \\mu _ g , \\end{align*}"}
-{"id": "3694.png", "formula": "\\begin{align*} \\begin{cases} \\lim \\limits _ { N \\rightarrow \\infty } C o v _ N ( \\delta _ i , y _ i ) = 0 & \\\\ \\lim \\limits _ { N \\rightarrow \\infty } E _ N ( \\delta _ i ) = p > 0 & \\end{cases} \\end{align*}"}
-{"id": "6526.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\ , \\lim _ { l \\rightarrow \\infty } \\ \\ \\sup _ { \\xi \\geq 0 , u \\in K } \\Biggl [ \\frac { 1 } { l } \\int _ { \\xi } ^ { \\xi + l } \\bigl \\| q ( t + s , u ) \\bigr \\| ^ { p } \\ , d s \\Biggr ] ^ { 1 / p } = 0 . \\end{align*}"}
-{"id": "8117.png", "formula": "\\begin{align*} h _ n ( t ) \\ = \\ \\sum _ { i = 0 } ^ { \\lfloor n / 2 \\rfloor } \\left ( \\ , \\sum _ { k = 2 i } ^ n { n \\choose k } \\xi _ { k , i } \\right ) t ^ i ( 1 + t ) ^ { n - 2 i } \\end{align*}"}
-{"id": "3890.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } ( \\| x u ( t ) \\| ^ 2 _ { L ^ 2 } + 2 \\| x v ( t ) \\| ^ 2 _ { L ^ 2 } ) = 8 Q ( u ( t ) , v ( t ) ) . \\end{align*}"}
-{"id": "1604.png", "formula": "\\begin{align*} d X ^ { Z } _ { t } = X ^ { Z } _ { t } \\mu ( X ^ { Z } _ { t } ) \\ , d t + \\sigma ( X ^ { Z } _ { t } ) \\ , d B _ { t } - d Z _ t , ~ X ^ Z _ 0 = x > 0 . \\end{align*}"}
-{"id": "4969.png", "formula": "\\begin{align*} F S ( a _ 1 , a _ 2 , \\ldots ) = F S ( 1 , a _ 2 , a _ 3 , \\ldots ) \\subseteq A ^ { \\ast } . \\end{align*}"}
-{"id": "3266.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ m c _ j g _ { n - j } = 0 , \\ ; n \\geq m , \\end{align*}"}
-{"id": "1406.png", "formula": "\\begin{align*} f ( x _ 1 , \\ldots , x _ d ) & = h ( p _ 1 ( x _ 1 ) + \\cdots + p _ d ( x _ d ) ) ~ ~ \\\\ f ( x _ 1 , \\ldots , x _ d ) & = h ( p _ 1 ( x _ 1 ) \\cdot \\ldots \\cdot p _ d ( x _ d ) ) , \\end{align*}"}
-{"id": "9850.png", "formula": "\\begin{align*} \\left ( \\frac { u ^ 4 } { 1 - u } - \\frac { u ^ 6 } { 1 - u ^ 3 } \\right ) [ ( 1 - u ) ^ r - 1 ] = \\frac { u ^ 4 ( 1 + u ) } { 1 - u ^ 3 } [ ( 1 - u ) ^ r - 1 ] . \\end{align*}"}
-{"id": "4239.png", "formula": "\\begin{align*} V ( z ) : = \\left \\lvert z \\right \\rvert ^ { \\frac { p - 2 } { 2 } } z , \\end{align*}"}
-{"id": "9336.png", "formula": "\\begin{align*} \\mathrm { t r a c e } ( A _ { Q _ { 1 } ^ { x } } ) = n \\end{align*}"}
-{"id": "9894.png", "formula": "\\begin{align*} \\int _ { \\mathcal { Z } } g ( h ( x , z ) ) Q ( d z ) & = \\int _ { - \\infty } ^ { x } g ( x ) q ( z ) d z + \\int _ { x } ^ { \\infty } g ( z ) q ( z ) d z \\end{align*}"}
-{"id": "6300.png", "formula": "\\begin{align*} \\mathbb { P } \\left [ A ^ c _ { \\rho , S , \\frac { C } { 2 } } ( p ) \\right ] \\leq 2 \\exp \\left ( - \\frac { C ^ 2 } { 4 } \\frac { c _ 0 ^ 2 k ^ 2 n \\log n / 2 } { \\left ( k ( n - k ) + \\binom { k } { 2 } \\right ) p + C c _ 0 k \\sqrt { n \\log n } / 6 } \\right ) . \\end{align*}"}
-{"id": "3157.png", "formula": "\\begin{align*} \\mu ( \\bar t ) = \\{ q ^ { - 2 } \\ , \\bar t ^ { n - 1 } , q ^ { - 4 } \\ , \\bar t ^ { n - 2 } , \\dots , q ^ { - 2 ( n - 1 ) } \\ , \\bar t ^ { 1 } \\} \\end{align*}"}
-{"id": "9739.png", "formula": "\\begin{align*} \\xi \\cdot \\nabla _ { x } p ( x ) = \\lim _ { h \\downarrow 0 } \\frac { p ( x + h \\xi , 0 ) - p ( x ) } { h } = 0 , \\end{align*}"}
-{"id": "6937.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow T ^ { * } } \\left ( \\| \\rho \\| _ { L ^ { \\infty } ( 0 , T ; L ^ \\infty ) } + \\| P \\| _ { L ^ { \\infty } ( 0 , T ; L ^ \\infty ) } \\right ) = \\infty . \\end{align*}"}
-{"id": "706.png", "formula": "\\begin{align*} ( 2 p - 1 ) \\left ( 1 + \\frac { 2 p K } { f ( x ) } \\right ) \\frac { x } { f ( x ) ( 1 - x ) } + ( 2 p - 1 ) \\frac { \\displaystyle \\int _ x ^ 1 \\frac { d t } { f ( t ) } } { 1 - x } & = \\frac { 2 p - 1 } { 2 } ( 1 + x ) + \\frac { B ( x ) } { 1 - x } . \\end{align*}"}
-{"id": "9406.png", "formula": "\\begin{align*} F _ { N , n } ( z , s ) : = \\sum _ { \\gamma \\in \\Gamma _ 0 ( N ) _ { \\infty } \\backslash \\Gamma _ 0 ( N ) } \\pi \\left | n \\mathrm { I m } ( \\gamma z ) \\right | ^ { 1 / 2 } I _ { s - \\frac 1 2 } ( | 2 \\pi n \\mathrm { I m } ( \\gamma z ) | ) e ( - n \\mathrm { R e } ( \\gamma z ) ) , \\end{align*}"}
-{"id": "3515.png", "formula": "\\begin{align*} F ^ { ( \\nu ) } ( t ; w ) = \\frac { 1 } { M + 1 } \\sum ^ M _ { j = 0 } \\left ( \\Big ( \\log ( \\nu _ j + N ) + \\frac { w } { \\nu _ j + N } \\Big ) t - \\log \\Gamma ( t + \\nu _ j + N ) + \\log \\Gamma ( \\nu _ j + N ) \\right ) . \\end{align*}"}
-{"id": "2988.png", "formula": "\\begin{align*} 1 - \\sum _ { k = 0 } ^ \\infty \\big ( q ^ { \\Delta _ { 4 k + 1 } } + q ^ { \\Delta _ { 4 k + 2 } } \\big ) + \\sum _ { k = 0 } ^ \\infty \\big ( q ^ { \\Delta _ { 4 k + 3 } } + q ^ { \\Delta _ { 4 k + 4 } } \\big ) & = \\Big ( 1 + \\sum _ { n = 1 } ^ \\infty ( - 1 ) ^ n ( q ^ { P _ { 5 , n } } + q ^ { Q _ { 5 , n } } ) \\Big ) \\cdot \\Big ( \\sum _ { n = 1 } ^ \\infty q ( n ) q ^ { 2 n } \\Big ) . \\end{align*}"}
-{"id": "8051.png", "formula": "\\begin{align*} \\gamma _ k ( a ) = \\begin{pmatrix} e ^ { \\pi \\imath k / n } & 0 \\\\ 0 & e ^ { - \\pi \\imath k / n } \\end{pmatrix} , \\gamma _ k ( x ) = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "5403.png", "formula": "\\begin{align*} c ( l , 2 k + 1 ) = \\frac { l ^ 2 - ( 2 k - 1 ) l - 2 k - 1 } { 2 } . \\end{align*}"}
-{"id": "9889.png", "formula": "\\begin{align*} & \\begin{pmatrix} A & \\cdots & T ^ { 3 } A \\end{pmatrix} = \\\\ & \\begin{pmatrix} 0 & 1 & 0 . 7 5 & 0 . 2 5 & 0 . 5 6 2 5 & 0 . 4 3 7 5 & 0 . 6 0 9 3 7 5 & 0 . 3 9 0 6 2 5 \\\\ 0 & 1 & 0 . 5 0 & 0 . 5 0 & 0 . 6 2 5 0 & 0 . 3 7 5 0 & 0 . 5 9 3 7 5 0 & 0 . 4 0 6 2 5 0 \\\\ 1 & 0 & 0 . 7 5 & 0 . 2 5 & 0 . 5 6 2 5 & 0 . 4 3 7 5 & 0 . 6 0 9 3 7 5 & 0 . 3 9 0 6 2 5 \\\\ 1 & 0 & 0 . 5 0 & 0 . 5 0 & 0 . 6 2 5 0 & 0 . 3 7 5 0 & 0 . 5 9 3 7 5 0 & 0 . 4 0 6 2 5 0 \\end{pmatrix} \\end{align*}"}
-{"id": "899.png", "formula": "\\begin{align*} \\Gamma = q ^ { - \\sum m _ u d ^ { h a } _ u m _ v d ^ { t a } _ v } \\end{align*}"}
-{"id": "7099.png", "formula": "\\begin{align*} N ^ { ( 1 - \\gamma ) ( \\nu - 1 ) + 1 - \\sigma ^ \\prime } = \\underbrace { N ^ { \\nu - 1 } } _ { } \\cdot \\underbrace { N ^ { - \\gamma ( \\nu - 1 ) } } _ { w _ { \\leq n - 1 } } \\cdot \\underbrace { N ^ { 1 - \\sigma ^ \\prime } } _ { F _ n ^ \\omega } ~ . \\end{align*}"}
-{"id": "9690.png", "formula": "\\begin{align*} X W h = \\alpha h \\ \\ \\ \\ h \\in H ^ 2 . \\end{align*}"}
-{"id": "912.png", "formula": "\\begin{align*} { \\bf x } _ t = { \\bf x } _ 0 + \\int \\limits _ 0 ^ t { \\bf a } ( { \\bf x } _ { \\tau } , \\tau ) d \\tau + \\int \\limits _ 0 ^ t \\Sigma ( { \\bf x } _ { \\tau } , \\tau ) d { \\bf f } _ { \\tau } , \\ { \\bf x } _ 0 = { \\bf x } ( 0 , \\omega ) . \\end{align*}"}
-{"id": "9899.png", "formula": "\\begin{align*} \\int g ( h ( x , z ) ) Q ( d z ) & = \\frac { 1 } { 2 } ( g ( x + 1 ) + g ( x - 1 ) ) \\end{align*}"}
-{"id": "6069.png", "formula": "\\begin{align*} 1 \\ , = \\ , \\| \\epsilon _ i \\wedge \\epsilon _ 4 \\| _ x \\ , = \\ , \\lambda _ i \\lambda _ 4 \\| f _ \\ast ( e _ i ) \\wedge f _ \\ast ( e _ 4 ) \\| _ x \\ , \\leq \\ , \\lambda _ i \\lambda _ 4 \\frac { c } { R } \\ , , \\end{align*}"}
-{"id": "2533.png", "formula": "\\begin{align*} \\Gamma ( x ) : = \\int _ 0 ^ \\infty t ^ { x - 1 } \\Big ( e ^ { - t } - \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( - t ) ^ k } { k ! } \\Big ) d t , - n < x < - n + 1 , n \\in \\mathbb N . \\end{align*}"}
-{"id": "1479.png", "formula": "\\begin{align*} \\sum _ { \\ell = 2 } ^ { 1 4 } V _ { \\ell } \\leq \\varepsilon ( \\mathcal { G } _ { 1 } ( K ) + \\mathcal { G } _ { 2 } ( K ) ) ( E _ { w } + E _ { \\mu } ) + \\frac { 1 } { 4 } \\sum _ { j = 1 } ^ { d } \\int _ { \\mathbb { T } ^ { d } } ( \\partial _ { x _ { j } } ^ { 2 } ( D w ^ { 1 } - D w ^ { 2 } ) ) ^ { 2 } \\ d x . \\end{align*}"}
-{"id": "8948.png", "formula": "\\begin{align*} \\frac { \\dd P _ { m _ 0 } } { \\dd P _ { m _ 1 } } ( X ) = \\frac { \\dd P _ { m _ 0 - m _ 1 } } { \\dd P _ { 0 } } ( X - m _ 1 ) = \\exp \\left \\{ \\langle X - m _ 1 , m _ 0 - m _ 1 \\rangle _ { K } - \\frac { 1 } { 2 } \\| m _ 0 - m _ 1 \\| ^ 2 _ { K } \\right \\} . \\end{align*}"}
-{"id": "2010.png", "formula": "\\begin{align*} I ( q , z ) : = z \\sum _ { d \\geq 0 } q ^ d \\frac { \\prod _ { k = 1 } ^ { 5 d } ( 5 H + k z ) } { \\prod _ { k = 1 } ^ d ( H + k z ) ^ 5 } , \\end{align*}"}
-{"id": "5730.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow 0 } x v _ x & = 0 & & v ( 1 , y , t ) = 0 \\\\ v ( x , 0 , t ) & = 0 & & v ( x , 1 , t ) = 0 \\\\ v ( x , y , 0 ) & = \\varphi ( x , y ) & & v _ t ( x , y , 0 ) = \\psi ( x , y ) , \\end{align*}"}
-{"id": "5071.png", "formula": "\\begin{align*} h = \\sum \\limits _ { j \\in \\mathbb { J } } \\langle h , x _ j \\rangle \\tau _ j \\iff \\sum \\limits _ { j \\in \\mathbb { J } } ( 2 - c _ j ) \\langle h , x _ j \\rangle \\langle \\tau _ j , h \\rangle = \\| h \\| ^ 2 \\iff \\sum \\limits _ { j \\in \\mathbb { J } } c _ j ^ 2 | \\langle h , x _ j \\rangle | ^ 2 = \\| h \\| ^ 2 . \\end{align*}"}
-{"id": "2026.png", "formula": "\\begin{align*} \\widetilde { \\tau } _ j ( z ) : = I _ j ( q , - z ) + z = \\tau _ j ( z ) + \\widetilde { T } _ j ( z ) . \\end{align*}"}
-{"id": "2039.png", "formula": "\\begin{align*} f ^ \\chi ( z ) : = \\sum _ { n = 1 } ^ \\infty a ( n ) \\chi ( n ) e ^ { 2 \\pi i n z } = \\frac { 1 } { \\tau ( \\overline { \\chi } ) } ( f \\mid R _ \\chi ) ( z ) . \\end{align*}"}
-{"id": "3202.png", "formula": "\\begin{align*} { \\rm R a n } \\varGamma _ + = { \\rm s p a n } \\{ | \\phi _ 0 \\rangle \\otimes | + \\rangle , \\ | \\phi _ 0 \\rangle \\otimes | - \\rangle \\} \\end{align*}"}
-{"id": "8540.png", "formula": "\\begin{align*} [ D _ t ^ 2 - i A \\partial _ { \\alpha } , \\mathcal { H } ] \\partial _ { \\alpha } ^ k \\tilde { \\sigma } = & 2 [ D _ t \\zeta , \\mathcal { H } ] \\frac { \\partial _ { \\alpha } D _ t \\partial _ { \\alpha } ^ k \\tilde { \\sigma } } { \\zeta _ { \\alpha } } - \\frac { 1 } { \\pi i } \\int \\Big ( \\frac { D _ t \\zeta ( \\alpha , t ) - D _ t \\zeta ( \\beta , t ) } { \\zeta ( \\alpha , t ) - \\zeta ( \\beta , t ) } \\Big ) ^ 2 \\partial _ { \\beta } ^ { k + 1 } \\tilde { \\sigma } ( \\beta , t ) d \\beta \\\\ : = & I _ 1 + I _ 2 . \\end{align*}"}
-{"id": "9782.png", "formula": "\\begin{align*} \\dim _ { \\mathcal { H } } E : = \\inf \\{ \\beta > 0 : \\mathcal { H } ^ \\beta _ \\infty ( E ) = 0 \\} . \\end{align*}"}
-{"id": "1840.png", "formula": "\\begin{align*} Z = \\frac { \\xi L } { 2 } \\cot ( Z ) . \\end{align*}"}
-{"id": "7658.png", "formula": "\\begin{align*} \\frac 1 2 \\int \\limits _ { \\mathbb { R } ^ d } v _ \\phi ^ 2 ( x , s ) \\ , v _ 0 ^ \\varepsilon ( x ) \\ , d x \\ - \\ \\int \\limits _ 0 ^ s \\int \\limits _ { \\mathbb { R } ^ d } \\phi ( x , t ) \\ , v _ \\phi ( x , t ) \\ , v _ 0 ^ \\varepsilon ( x ) \\ , d x \\ , d t \\ = \\ \\int \\limits _ 0 ^ s ( L ^ \\varepsilon v _ \\phi , v _ \\phi ) _ { v _ 0 } \\ , d t \\ \\le \\ 0 . \\end{align*}"}
-{"id": "9693.png", "formula": "\\begin{align*} \\hat { z } = \\underset { z \\in \\mathcal { M } } { } \\ | | y - \\underset { x } { \\underbrace { z . G } } | | . \\end{align*}"}
-{"id": "5960.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t v _ n ( t , x ) + \\frac 1 2 \\Delta v _ n ( t , x ) + g ^ * ( t , \\frac { 1 } { \\sqrt { n } } \\nabla v _ n ( t , x ) ) = 0 [ 0 , 1 ] \\times \\R ^ d \\\\ v _ n ( 1 , x ) = f ( \\frac { x } { \\sqrt { n } } ) , x \\in \\R ^ d , \\end{cases} \\end{align*}"}
-{"id": "297.png", "formula": "\\begin{align*} \\gamma : \\Lambda _ X \\ ; \\subset \\ ; T ^ * A \\ ; = \\ ; A \\times V \\ ; \\twoheadrightarrow \\ ; V \\end{align*}"}
-{"id": "1570.png", "formula": "\\begin{align*} | \\pi _ z \\circ \\widetilde { F } _ { w _ m } ( z , x ) - f _ { w _ m } ( z ) | = O ( z ^ 4 , { w _ m } x , w _ m z ^ 2 ) = O ( z ^ 4 , { w _ m } z ^ 2 , { w _ m } ^ 3 ) . \\end{align*}"}
-{"id": "3487.png", "formula": "\\begin{align*} \\Re \\Big ( \\Big ( \\frac { - \\theta } { \\sin ^ 2 \\theta } + \\frac { e ^ { i \\theta } } { \\sin \\theta } \\Big ) ^ 2 ( 1 - \\frac { \\sin \\theta } { \\theta } e ^ { - i \\theta } ) \\Big ) = \\frac { 1 } { \\sin ^ 2 \\theta } \\Big ( \\Big ( \\frac { \\theta } { \\sin \\theta } - \\cos \\theta \\Big ) ^ 2 + \\sin ^ 2 \\theta \\Big ) > 0 , \\end{align*}"}
-{"id": "5752.png", "formula": "\\begin{align*} u _ { y y } ( x , y , t ) = \\sum _ { m , n \\in \\N } ( n \\pi ) ^ 2 \\Bigg ( & C _ { m , n } E _ { \\alpha , 1 } ( - \\mu _ { m , n } t ^ \\alpha ) + D _ { m , n } t E _ { \\alpha , 2 } ( - \\mu _ { m , n } t ^ \\alpha ) \\\\ & + \\int _ 0 ^ t ( t - \\xi ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } [ - \\mu _ { m , n } ( t - \\xi ) ^ { \\alpha } ] F _ { m , n } ( \\xi ) d \\xi \\Bigg ) J _ { 0 } ( \\gamma _ m x ) \\sin ( n \\pi y ) . \\end{align*}"}
-{"id": "2955.png", "formula": "\\begin{align*} c ( g h , x ) = c ( g , h \\cdot x ) c ( h , x ) \\end{align*}"}
-{"id": "7510.png", "formula": "\\begin{align*} | T _ { t } u ( x ) | = \\left \\vert \\int _ { \\mathbf { \\mathbb { Q } } _ { p } ^ { n } } Z _ { t } ( x - y ) u ( y ) d ^ { n } y \\right \\vert \\leq | | u | | _ { L ^ { \\infty } } \\int _ { \\mathbf { \\mathbb { Q } } _ { p } ^ { n } } Z _ { t } ( x - y ) d ^ { n } y = | | u | | _ { L ^ { \\infty } } . \\end{align*}"}
-{"id": "7569.png", "formula": "\\begin{align*} \\begin{gathered} \\int _ 0 ^ t \\Delta g _ { \\nu ( t - s ) } * ( \\tau \\circ X ^ { - 1 } ) ( t ) d s = \\tau \\circ X ^ { - 1 } ( t ) - g _ { \\nu t } * ( \\tau \\circ X ^ { - 1 } ) ( t ) \\end{gathered} \\end{align*}"}
-{"id": "2395.png", "formula": "\\begin{align*} \\delta _ Q : \\mathfrak { F } & \\to \\mathbb { C } \\\\ \\delta _ Q ( X _ i X _ j ) & = Q _ { i j } \\\\ \\delta _ Q ( W ) & = 0 \\end{align*}"}
-{"id": "1637.png", "formula": "\\begin{align*} \\xi _ k ' : = \\frac { 1 } { \\sqrt { 2 } } \\big ( \\xi _ k - \\eta _ { - k } \\big ) = \\frac { 1 } { \\sqrt { 2 } } \\begin{pmatrix} e ^ { 2 \\pi k i x } \\\\ - e ^ { - 2 \\pi k i x } \\end{pmatrix} , \\eta _ k ' : = \\frac { i } { \\sqrt { 2 } } \\big ( \\xi _ k + \\eta _ { - k } \\big ) = \\frac { i } { \\sqrt { 2 } } \\begin{pmatrix} e ^ { 2 \\pi k i x } \\\\ e ^ { - 2 \\pi k i x } \\end{pmatrix} . \\end{align*}"}
-{"id": "3902.png", "formula": "\\begin{align*} A = \\begin{pmatrix} a & b & \\gamma ^ { \\ast } \\\\ c & d & \\delta ^ { \\ast } \\\\ \\alpha & \\beta & U \\end{pmatrix} , \\end{align*}"}
-{"id": "9028.png", "formula": "\\begin{align*} f _ { \\{ c _ i \\} , c _ { i + 1 } } ( x ) = \\begin{cases} c _ i & x = c _ { i + 1 } , \\\\ x & \\end{cases} \\end{align*}"}
-{"id": "6475.png", "formula": "\\begin{align*} a = \\frac { 1 } { 1 + \\mu _ { 1 } \\cdots \\mu _ { n - 1 } } b = \\frac { \\mu _ { 1 } \\cdots \\mu _ { n - 2 } } { 1 + \\mu _ { 1 } \\cdots \\mu _ { n - 1 } } c = \\frac { \\mu _ { 1 } \\cdots \\mu _ { n - 2 } ( \\mu _ { n - 1 } - 1 ) } { 1 + \\mu _ { 1 } \\cdots \\mu _ { n - 1 } } \\end{align*}"}
-{"id": "4845.png", "formula": "\\begin{align*} y ' = y - F \\Big ( \\underset { n \\geq 1 } { \\sum } \\frac { 1 } { n ! } ( S \\circ D ) ^ { n } ( w ) \\Big ) . \\end{align*}"}
-{"id": "4602.png", "formula": "\\begin{align*} D ( F _ 1 , F _ 2 ) = \\{ | x _ 1 - x _ 2 | : x _ 1 \\in F _ 1 , x _ 2 \\in F _ 2 \\} . \\end{align*}"}
-{"id": "4999.png", "formula": "\\begin{align*} L _ f ( s ) & = \\prod _ { p | N } ( 1 - \\lambda _ f ( p ) p ^ { - s } ) ^ { - 1 } \\prod _ { p \\nmid N } ( 1 - \\lambda _ f ( p ) p ^ { - s } + p ^ { - 2 s } ) ^ { - 1 } \\\\ & = \\prod _ { p | N } ( 1 - \\lambda _ f ( p ) p ^ { - s } ) ^ { - 1 } \\prod _ { p \\nmid N } ( 1 - \\alpha _ f ( p ) p ^ { - s } ) ^ { - 1 } ( 1 - \\beta _ f ( p ) p ^ { - s } ) ^ { - 1 } . \\end{align*}"}
-{"id": "1955.png", "formula": "\\begin{align*} \\begin{pmatrix} \\rho _ 1 & \\rho _ 1 \\cdot a \\\\ & \\rho _ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "8650.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\varphi _ 1 ' = \\varphi _ 2 , \\\\ \\varphi _ 2 ' = \\cos \\varphi _ 1 ( \\sin \\varphi _ 1 - 2 c _ 0 ) ; \\\\ \\\\ ( \\varphi _ 1 , \\varphi _ 2 ) \\to ( \\pi - \\bar { \\varphi } _ 0 , 0 ) \\ \\textrm { a s } \\ y \\to - \\infty , \\ \\ ( \\varphi _ 1 , \\varphi _ 2 ) \\to ( \\bar { \\varphi } _ 0 , 0 ) \\ \\textrm { a s } \\ y \\to + \\infty , \\end{array} \\right . \\end{align*}"}
-{"id": "4339.png", "formula": "\\begin{align*} s _ \\alpha ( x ) = x - d \\alpha ( x ) H _ \\alpha . \\end{align*}"}
-{"id": "9010.png", "formula": "\\begin{align*} E _ { 1 } \\cdot X _ { j } ^ { i } \\tau _ { i } E _ { m + 1 } \\times ( X _ { 1 } ^ { 2 } E _ { 2 } + \\dotsb + X _ { 1 } ^ { m } E _ { m } ) \\times \\dotsb \\times ( X _ { m - 1 } ^ { 2 } E _ { 2 } + \\dotsb + X _ { m - 1 } ^ { m } E _ { m } ) = 0 \\ , . \\end{align*}"}
-{"id": "8821.png", "formula": "\\begin{align*} \\nabla _ i F ( x _ k ) = \\sum _ { j = 1 } ^ J c ^ f _ j ( A ^ f ) ^ \\top _ { j , i } \\nabla f _ j ( A ^ f _ j x _ k - b ^ f _ j ) + Q x _ k = \\sum _ { j \\in \\mathcal J ^ f ( i ) } c ^ f _ j ( A ^ f ) ^ \\top _ { j , i } \\nabla f _ j ( ( r ^ { f , x } _ k ) _ j ) + \\sum _ { j \\in \\mathcal J ^ Q ( i ) } U _ i r ^ { Q , x } _ k \\end{align*}"}
-{"id": "7291.png", "formula": "\\begin{align*} \\lVert \\psi _ q ( a ) + \\Phi _ p ( a ) ( 1 - h ^ + _ q ) \\rVert & = \\lVert \\psi _ q ( a ) + \\Phi ( a ) ( 1 - h ^ + _ q ) \\rVert \\\\ & \\approx _ { \\frac { \\gamma } { 3 6 M } } \\lVert \\psi _ q ( a ) + \\Phi _ q ( a ) ( 1 - h ^ + _ q ) \\rVert < \\frac { 3 } { 2 } \\lVert a \\rVert , \\end{align*}"}
-{"id": "9400.png", "formula": "\\begin{align*} \\langle I ' ( \\tilde { u } _ n ) , \\tilde { u } _ n \\rangle & = \\int _ { \\mathbb { R } ^ N } \\int _ { \\mathbb { R } ^ N } \\Phi ( \\tilde { u } _ n ( x ) - \\tilde { u } _ n ( y ) ) ( \\tilde { u } _ n ( x ) - \\tilde { u } _ n ( y ) ) K ( x , y ) d x d y \\\\ & - \\lambda \\int _ \\Omega | \\tilde { u } _ n | ^ { p ^ { \\ast } _ { s } } d x - \\int _ \\Omega f ( x , \\tilde { u } _ n ) \\tilde { u } _ n d x \\end{align*}"}
-{"id": "8592.png", "formula": "\\begin{align*} \\zeta ( z ) = ( z / 2 ) \\pi ^ { z - 1 } \\sin ( \\pi z / 2 ) \\ , \\int _ 0 ^ \\infty \\ , d t \\ , t ^ { - z / 2 - 1 } \\ , [ F ( t ) - F _ n ( t ) ] , & \\\\ \\Re ( z ) \\in ( \\max ( 2 n , 1 ) , 2 n + 2 ) . & \\end{align*}"}
-{"id": "1366.png", "formula": "\\begin{align*} \\delta _ { n - 1 } ( f ) ( \\lambda ) = \\sum _ { i = 1 } ^ n ( - 1 ) ^ { i } \\left ( f ( F _ i ^ 0 ( \\lambda ) ) \\cdot S _ i ( \\lambda ) - f ( F _ i ^ 1 ( \\lambda ) ) \\right ) \\end{align*}"}
-{"id": "103.png", "formula": "\\begin{align*} B ( 0 ) ^ { \\ell _ 1 } \\cdot B _ 1 \\cdot B ( 0 ) ^ { \\ell _ 2 } \\cdot B _ 1 \\cdots B ( 0 ) ^ { \\ell _ k } \\cdot B _ 1 \\cdot B ( 0 ) ^ { \\ell _ { k + 1 } } = B ( 0 ) ^ { \\alpha - \\beta } \\ , , \\end{align*}"}
-{"id": "3920.png", "formula": "\\begin{align*} d _ r ( \\phi , \\psi ) = \\left ( \\sum _ { i , j } \\left | \\phi _ { i j } - \\psi _ { i j } \\right | ^ r \\right ) ^ { 1 / r } \\end{align*}"}
-{"id": "1206.png", "formula": "\\begin{align*} P ( \\zeta _ 1 , \\zeta _ 2 ) & = \\sqrt { \\zeta _ 1 ^ 2 + b _ 0 ( \\zeta _ 2 ) ^ 2 } \\to \\sqrt { w ^ 2 - b _ 0 ( \\zeta _ 2 ) ^ 2 } e ^ { \\pm \\pi { i } / 2 } \\end{align*}"}
-{"id": "275.png", "formula": "\\begin{align*} E ^ { k , i } _ r = E ^ { k , i } _ { r + 1 } , \\ , \\ , \\ , \\ , \\ , { \\rm f o r ~ a l l } \\ , \\ , \\ , \\ , \\ , \\ , r \\ge 1 . \\end{align*}"}
-{"id": "8064.png", "formula": "\\begin{align*} \\psi _ { c } ^ + ( \\omega _ i ) = - c ^ { n + 3 - 2 i } \\sum _ { k = 1 } ^ { 2 n } P _ { i - n - 3 , - k } \\omega _ k . \\end{align*}"}
-{"id": "727.png", "formula": "\\begin{align*} \\mathbb { E } _ { f } \\ ! \\left [ T ^ { k } \\right ] = \\frac { 1 } { ( q - 1 ) ^ { k } } \\sum _ { i = 0 } ^ { k } ( - 1 ) ^ { i } \\binom { k } { i } \\mathbb { E } _ { f } \\ ! \\left [ \\frac { L ^ { i } } { r ^ { q i } } \\right ] . \\end{align*}"}
-{"id": "70.png", "formula": "\\begin{align*} \\begin{aligned} \\sigma ^ + _ { - 1 } & = ( 1 - c ) \\sigma ^ - _ { - 1 } + c \\sigma ^ - _ 1 \\ , , \\\\ \\sigma ^ + _ { 1 } & = ( 1 - c ) \\sigma ^ - _ { 1 } + c \\sigma ^ - _ { - 1 } \\ , . \\end{aligned} \\end{align*}"}
-{"id": "2032.png", "formula": "\\begin{align*} h _ { \\delta } ( t ) = 1 , \\quad \\end{align*}"}
-{"id": "5395.png", "formula": "\\begin{align*} \\langle C . w , v _ { n m } ^ 1 \\rangle = \\langle w , ( - C ) . v _ { n m } ^ 1 \\rangle \\end{align*}"}
-{"id": "6267.png", "formula": "\\begin{align*} \\left ( \\otimes ^ m A \\right ) _ g = \\sum _ { g _ 1 g _ 2 \\cdots g _ m = g } A _ { g _ 1 } \\otimes \\cdots \\otimes A _ { g _ m } , \\end{align*}"}
-{"id": "4341.png", "formula": "\\begin{align*} \\Upsilon ( ( a , b ) , ( c , d ) ) = \\Upsilon _ 2 ( a , c ) \\Upsilon _ 2 ( b , d ) . \\end{align*}"}
-{"id": "2670.png", "formula": "\\begin{align*} h ( z , w ) & = \\frac { \\eta _ { - } ^ { 2 } } { 4 \\pi } \\frac { e ^ { - \\eta _ { - } u ( w + 1 ) } } { \\sqrt { w ^ { 2 } - 1 } \\sqrt { z ^ { 2 } - 1 } } \\frac { z - w } { 1 - z w } \\big ( z w \\big ) ^ { M - N } \\prod _ { k = 1 } ^ { N } \\frac { \\sigma _ { k } z - \\tau } { \\tau z - \\sigma _ { k } } \\frac { \\sigma _ { k } w - \\tau } { \\tau w - \\sigma _ { k } } . \\end{align*}"}
-{"id": "64.png", "formula": "\\begin{align*} U ( x , t ) = \\begin{cases} U _ \\ell & x / t < - 1 \\\\ U _ * = ( \\rho _ { * , \\ell } , J _ * , a _ \\ell ) & - 1 < x / t < 0 \\\\ U _ { * * } = ( \\rho _ { * , r } , J _ * , a _ r ) & 0 < x / t < 1 \\\\ U _ r & x / t > 1 \\end{cases} \\end{align*}"}
-{"id": "3564.png", "formula": "\\begin{align*} \\begin{matrix} \\begin{aligned} W _ { 1 } u & = y z U , & & & W _ { 2 } u & = z x U , & & & W _ { 3 } u & = x y U , \\\\ W _ { 1 } v & = y z V , & & & W _ { 2 } v & = z x V , & & & W _ { 3 } v & = x y V . \\end{aligned} \\end{matrix} \\end{align*}"}
-{"id": "9306.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n x _ { i j } \\leq s _ i , ~ i = 1 , \\dots , m ; \\\\ \\sum _ { i = 1 } ^ m x _ { i j } \\geq d _ j , ~ j = 1 , \\dots , n ; \\\\ \\end{align*}"}
-{"id": "3581.png", "formula": "\\begin{align*} \\begin{alignedat} { 1 } Y _ { 0 } = & X _ { 0 } + X _ { 1 } + X _ { 2 } - X _ { 3 } - X _ { 4 } - X _ { 5 } , \\\\ Y _ { 1 } = & X _ { 1 } - X _ { 5 } , \\\\ Y _ { 2 } = & X _ { 3 } - X _ { 2 } , \\\\ Y _ { 3 } = & X _ { 4 } - X _ { 2 } , \\\\ Y _ { 4 } = & X _ { 0 } - X _ { 5 } , \\\\ Y _ { 5 } = & X _ { 0 } X _ { 1 } + X _ { 0 } X _ { 2 } + X _ { 1 } X _ { 2 } - X _ { 3 } X _ { 4 } - X _ { 3 } X _ { 5 } - X _ { 4 } X _ { 5 } , \\end{alignedat} \\end{align*}"}
-{"id": "8800.png", "formula": "\\begin{align*} u ^ { \\varepsilon } ( t , x ) = \\frac { \\sin ( \\delta \\pi ) } { \\pi } \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { \\pi } \\frac { G _ { t - s } ( x , y ) } { ( t - s ) ^ { 1 - \\delta } } Y _ { \\delta } ^ { \\varepsilon } ( s , y ) \\ , \\mathrm { d } s \\ , \\mathrm { d } y \\end{align*}"}
-{"id": "7992.png", "formula": "\\begin{align*} \\log g ( n ) - \\log h ( n ) = \\frac { n } { 2 } + O \\left ( \\frac { n } { \\log n } \\right ) , \\end{align*}"}
-{"id": "7287.png", "formula": "\\begin{align*} N ( p , \\Phi _ p ) = \\max \\{ \\lVert ( \\psi _ p ( a ) - \\Phi _ p ( a ) ) ( h _ p ^ + - k _ p ) \\rVert : a \\in F _ p \\} \\end{align*}"}
-{"id": "1975.png", "formula": "\\begin{align*} H _ T ^ * ( \\mathrm { p t } ) = \\Q [ \\alpha _ 1 , \\dots , \\alpha _ n ] , \\end{align*}"}
-{"id": "8523.png", "formula": "\\begin{align*} \\norm { \\frac { 1 } { \\pi } \\sum _ { j = 1 } ^ 2 \\frac { 2 \\lambda _ j D _ t ^ 2 \\zeta } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } } _ { H ^ s } \\leq \\| D _ t ^ 2 \\zeta \\| _ { H ^ s } \\norm { \\frac { 1 } { \\pi } \\sum _ { j = 1 } ^ 2 \\frac { 2 \\lambda _ j } { ( \\zeta ( \\alpha , t ) - z _ j ( t ) ) ^ 2 } } _ { H ^ s } \\leq K _ s ^ { - 1 } \\epsilon ^ 2 . \\end{align*}"}
-{"id": "1577.png", "formula": "\\begin{align*} ( z ^ \\iota _ n , x ^ \\iota _ n ) : = \\widetilde { \\boldsymbol { F } } _ { n ^ 2 + k _ n , n ^ 2 + \\kappa _ 0 } \\circ U \\circ \\boldsymbol { F } _ { n ^ 2 + \\kappa _ 0 , n ^ 2 } ( z , x ) \\end{align*}"}
-{"id": "8450.png", "formula": "\\begin{align*} G _ c : = - 2 [ \\bar { \\mathfrak { F } } , \\mathcal { H } \\frac { 1 } { \\zeta _ { \\alpha } } + \\bar { \\mathcal { H } } \\frac { 1 } { \\bar { \\zeta } _ { \\alpha } } ] \\bar { \\mathfrak { F } } _ { \\alpha } + \\frac { 1 } { \\pi i } \\int \\Big ( \\frac { D _ t \\zeta ( \\alpha , t ) - D _ t \\zeta ( \\beta , t ) } { \\zeta ( \\alpha , t ) - \\zeta ( \\beta , t ) } \\Big ) ^ 2 ( \\zeta - \\bar { \\zeta } ) _ { \\beta } d \\beta . \\end{align*}"}
-{"id": "9108.png", "formula": "\\begin{align*} ( - \\Delta _ p ) ^ s \\phi ( x ) = \\dfrac { K } { R ^ { s p } } ( - \\Delta _ p ) ^ s \\Upsilon ( R x ) \\le - \\dfrac { c _ 1 } { R ^ { s p } } | \\phi ( x ) | ^ { p - 2 } \\phi ( x ) \\forall | x | > \\dfrac { k } { R } . \\end{align*}"}
-{"id": "7923.png", "formula": "\\begin{align*} R _ N = \\hat { R } _ n \\frac { ( b - a ) ^ { n + 2 } f ^ { ( n + 1 ) } ( \\xi ) } { [ N - ( n - 1 ) ] ^ { n + 1 } } \\xi \\in [ a - ( n - 2 ) h / 2 , b + ( n - 2 ) h / 2 ] \\end{align*}"}
-{"id": "7183.png", "formula": "\\begin{align*} \\left \\Vert y _ { \\ast } ( t , \\xi , \\eta ) \\right \\Vert \\le R \\mathrm { e } ^ { - \\alpha t } \\quad \\forall t \\ge 0 \\quad \\textrm { a n d } P _ { \\xi } ^ { - } y ( 0 ) = \\eta ; \\end{align*}"}
-{"id": "8849.png", "formula": "\\begin{align*} s _ 2 & = M ^ 3 \\mu p - 4 M ^ 2 \\mu p ^ 2 - M ^ 3 a + 2 M ^ 2 a p + 5 M ^ 2 \\mu p + 2 M a p ^ 2 - 3 M ^ 2 \\mu - 6 M a * p + 4 M \\mu p \\\\ & + 3 M a - 3 M \\mu - 2 a p + 2 a , \\end{align*}"}
-{"id": "9835.png", "formula": "\\begin{align*} g _ \\infty ( z ' ) : = \\begin{cases} N ( 0 ^ + , \\tilde u _ { X _ \\circ , 0 } ( Z + \\ , \\cdot \\ , ) ) & X _ \\circ \\in \\mathcal { N } ( u ) \\\\ 0 & X _ \\circ \\notin \\mathcal { N } ( u ) \\end{cases} \\end{align*}"}
-{"id": "8403.png", "formula": "\\begin{align*} \\partial _ t D ^ k u = \\frac { 1 } { z _ { \\alpha } ^ k } \\partial _ t \\partial _ { \\alpha } ^ k u + \\frac { 1 } { z _ { \\alpha } ^ { k - 1 } } \\partial _ { \\alpha } ^ { k - 1 } \\partial _ t ( \\frac { 1 } { z _ { \\alpha } } ) u _ { \\alpha } + G _ k , \\end{align*}"}
-{"id": "3865.png", "formula": "\\begin{align*} C ( g ) = \\{ h \\in G : h g = g h \\} . \\end{align*}"}
-{"id": "5249.png", "formula": "\\begin{align*} \\phi ^ { - 1 } ( 1 - s ) = \\left \\{ ( 2 \\log ( 1 / s ) ) ^ { 1 / 2 } - \\frac { \\log 4 \\pi + \\log \\log ( 1 / s ) } { 2 ( 2 \\log ( 1 / s ) ) ^ { 1 / 2 } } + O ( ( \\log \\log ( 1 / s ) ^ { 2 } ( \\log 1 / s ) ^ { - 1 / 2 } ) ) \\right \\} . \\end{align*}"}
-{"id": "1118.png", "formula": "\\begin{align*} & \\mathbb { P } ^ { C U E ( n ) } ( \\theta _ i \\not \\in [ 0 , 2 \\pi \\delta _ n ' ] , 1 \\leq i \\leq n ) = D _ n ( \\pi \\delta _ n ' ) = D _ n ( \\alpha _ n ) = ( n \\ln n ) ^ { - 1 } . \\end{align*}"}
-{"id": "8794.png", "formula": "\\begin{align*} \\Omega ^ * = L ^ 2 ( [ 0 , T ] \\times [ 0 , \\pi ] ) \\times D ( [ 0 , T ] , H _ { - r } ( [ 0 , \\pi ] ) ) , \\end{align*}"}
-{"id": "472.png", "formula": "\\begin{align*} G & = \\prod ^ { 2 d - 1 } _ { j = 0 } ( \\xi ^ { j } x + \\xi ^ { - j } y + z ) \\\\ & = \\prod ^ { d - 1 } _ { j = 0 } ( \\xi ^ { j } x + \\xi ^ { - j } y + z ) ( \\xi ^ { j + d } x + \\xi ^ { - j + d } y + z ) \\\\ & = \\prod ^ { d - 1 } _ { j = 0 } ( \\xi ^ { j } x + \\xi ^ { - j } y + z ) ( - \\xi ^ { j } x - \\xi ^ { - j } y + z ) \\\\ & = ( - 1 ) ^ d \\prod ^ { d - 1 } _ { j = 0 } ( \\xi ^ { j } x + \\xi ^ { - j } y + z ) ( \\xi ^ { j } x + \\xi ^ { - j } y - z ) = F \\\\ \\end{align*}"}
-{"id": "3421.png", "formula": "\\begin{align*} \\eta ^ M ( T ) \\le \\left ( \\frac { 1 3 } { 7 \\cdot 3 } + \\frac { 8 } { 7 } \\right ) \\Sigma _ 0 + \\left ( \\frac { 1 3 } { 7 \\cdot 3 } + \\frac { 1 } { 3 } \\right ) \\Sigma _ * \\le \\frac { 2 0 \\cdot 1 3 } { 2 1 \\cdot 7 } \\Sigma _ 0 + \\frac { 2 0 } { 2 1 } \\Sigma _ * = \\frac { 2 0 } { 2 1 } \\sum _ i \\eta ^ { M - 1 } ( T _ i ) . \\end{align*}"}
-{"id": "7504.png", "formula": "\\begin{align*} ( f ( \\mathcal { \\partial } , \\beta ) \\varphi ) ( x ) : = \\mathcal { F } _ { \\xi \\rightarrow x } ^ { - 1 } ( | f ( \\xi ) | _ { p } ^ { \\beta } \\mathcal { F } _ { x \\rightarrow \\xi } \\varphi ) , \\varphi \\in \\mathcal { D } ( \\mathcal { \\mathbf { \\mathbb { Q } } } _ { p } ^ { n } ) \\beta > 0 , \\end{align*}"}
-{"id": "2164.png", "formula": "\\begin{align*} \\| \\Phi _ t * ( f \\chi _ { B ( 0 , 1 - \\delta ) } ) - p g _ 1 \\| _ { L ^ 2 ( d w ) } & = \\| h g _ 1 - p g _ 1 \\| _ { L ^ 2 ( d w ) } \\\\ & \\leq w ( B ( 0 , 1 ) ) ^ { 1 \\slash 2 } \\| h g _ 1 - p g _ 1 \\| _ { L ^ \\infty ( B ( 0 , 1 ) ) } \\\\ & \\leq w ( B ( 0 , 1 ) ) ^ { 1 \\slash 2 } \\| g _ 1 \\| _ { L ^ { \\infty } ( B ( 0 , 1 ) ) } \\| h - p \\| _ { L ^ \\infty ( B ( 0 , 1 ) ) } \\\\ & \\leq w ( B ( 0 , 1 ) ) ^ { 1 \\slash 2 } \\varepsilon . \\end{align*}"}
-{"id": "8013.png", "formula": "\\begin{align*} \\| e ^ { t ( \\Delta + V ) } \\| _ { L ^ 2 ( \\mathbb { R } ^ d ) , L ^ 2 ( \\mathbb { R } ^ d ) } & \\leq \\| t ^ { - \\frac { d } { 2 } } e ^ { c _ 3 ( t + 1 ) } e ^ { - \\frac { | x | ^ 2 } { c _ 4 t } } \\| _ { L ^ 1 ( \\mathbb { R } ^ d ) } \\\\ & = ( c _ 4 \\pi ) ^ { d / 2 } e ^ { c _ 3 ( t + 1 ) } . \\end{align*}"}
-{"id": "2811.png", "formula": "\\begin{align*} P _ { t , t + s } ( \\mathbf { n } , \\mathbf { m } ) = \\sum _ { \\mathbf { n } _ 1 \\in \\mathbb { N } _ 0 } \\sum _ { \\mathbf { n } _ 2 \\in \\mathbb { N } _ 0 } \\dots \\sum _ { \\mathbf { n } _ { s - 1 } \\in \\mathbb { N } _ 0 } P ( \\mathbf { n } , \\mathbf { n } _ 1 , \\dots , \\mathbf { n } _ { s - 1 } , \\mathbf { m } ) . \\end{align*}"}
-{"id": "8910.png", "formula": "\\begin{align*} \\mathrm { B C } ( P _ { y } , P _ { x _ 0 } ) & = \\left ( \\sqrt { \\frac { a b } { n ^ 2 } } + \\sqrt { \\left ( 1 - \\frac { a } { n } \\right ) \\left ( 1 - \\frac { b } { n } \\right ) } \\right ) ^ { s ( n - s ) } \\end{align*}"}
-{"id": "8015.png", "formula": "\\begin{align*} \\langle y ( T ; v ) , \\varphi _ T \\rangle - \\langle y _ 0 , \\varphi ( 0 ) \\rangle = \\langle v , B ^ * \\varphi ( \\tau ) \\rangle _ { \\ell ^ 2 } . \\end{align*}"}
-{"id": "3850.png", "formula": "\\begin{align*} & 2 ^ { 2 m } K h _ { 0 0 } ^ { 2 m + 4 } W _ 0 ^ { m + 1 } h ^ i _ l = 2 ^ { 2 m + 2 } h _ { 0 0 } ^ { 2 m + 2 } W _ 0 ^ { 2 m + 2 } + \\\\ & ^ { h } { { { R _ 0 } ^ i } _ { 0 l } } 2 ^ { 2 m + 1 } h _ { 0 0 } ^ { 2 m + 1 } W _ 0 ^ { m + 1 } . 2 ^ { 2 m } W _ 0 ^ { m + 1 } \\Phi ^ i _ { \\parallel l } - 2 ^ { 2 m } h _ { 0 0 } ^ { m + 1 } W _ 0 . 2 ^ { 2 m } h _ { 0 0 } ^ { 2 m } W _ 0 ^ { 2 m + 1 } \\Phi ^ i _ { l \\parallel 0 } + \\\\ & 2 ^ { 2 m + 3 } h _ { 0 0 } ^ { 2 m + 2 } W _ 0 ^ { 2 m + 2 } \\Phi ^ r { { { \\Phi _ r } ^ i } _ l } , \\end{align*}"}
-{"id": "5462.png", "formula": "\\begin{align*} K _ t - \\tilde K _ t + \\int _ 0 ^ t \\int _ U \\big ( L _ s ( z ) - \\tilde L _ s ( z ) \\big ) \\ , \\lambda _ \\pi ( d z ) d s & = \\int _ 0 ^ t \\big ( Z _ s - \\tilde Z _ s \\big ) \\ , d \\hat W _ s , \\\\ \\int _ 0 ^ t \\int _ U \\big ( L _ s ( z ) - \\tilde L _ s ( z ) \\big ) \\ , \\hat \\pi ( d s \\ , d z ) & = \\int _ 0 ^ t \\int _ \\Lambda \\big ( R _ s ( b ) - \\tilde R _ s ( b ) \\big ) \\ , \\hat \\theta ( d s \\ , d b ) . \\end{align*}"}
-{"id": "2116.png", "formula": "\\begin{align*} d z ( t ) = a ( t ) z ( t ) d t + c ( t ) z ( t ) d \\omega ( t ) , \\ ; z ( 0 ) = z _ 0 . \\end{align*}"}
-{"id": "510.png", "formula": "\\begin{align*} \\omega _ { 1 , 0 } = y \\prod _ { i = 1 } ^ g ( x - t _ i ) ^ { l ( i , 1 ) } d x , \\ \\ \\ l ( i , 1 ) = \\left \\lfloor { \\frac { - a _ i } { 4 } } \\right \\rfloor . \\end{align*}"}
-{"id": "3280.png", "formula": "\\begin{align*} \\Omega _ 3 ( z ) = z ^ 3 - 3 a b z - ( a ^ 3 + b ^ 3 ) \\end{align*}"}
-{"id": "739.png", "formula": "\\begin{align*} I _ { 2 } = \\sum _ { k = 0 } ^ { m - 1 } \\frac { k ! ^ { 2 } \\mathbb { A } _ { k , k } ^ { 2 } } { ( n - m + k ) ! ^ 2 } + 2 \\sum _ { j = 1 } ^ { m - 1 } \\sum _ { i = 0 } ^ { j - 1 } \\frac { i ! j ! \\mathbb { A } _ { i , j } ^ { 2 } } { ( n - m + i ) ! ( n - m + j ) ! } , \\end{align*}"}
-{"id": "3074.png", "formula": "\\begin{align*} | \\int _ { | x | < \\frac { 1 } { | y | } } ( e ^ { 2 \\pi i x \\cdot y } - 1 ) K _ 1 ( x ) \\ , d x | & \\leq C | y | \\int _ { | x | < \\frac { 1 } { | y | } } | x | \\frac { 1 } { | x | ^ { n - \\beta } } \\ , d x \\\\ & \\leq \\frac { 1 } { \\beta + 1 } \\beta ^ { - \\beta } . \\end{align*}"}
-{"id": "1448.png", "formula": "\\begin{align*} V I I _ { A } = \\varepsilon \\int _ { t } ^ { T } \\int _ { \\mathbb { T } ^ { d } } ( \\partial ^ { \\alpha } \\partial _ { x _ { j } } w ^ { n + 1 } ) \\sum _ { i = 1 } ^ { d } \\left [ \\left ( \\Theta _ { p _ { i } } ( \\tau , x , \\mu ^ { n } , D w ^ { n } ) \\right ) \\left ( \\partial ^ { \\alpha } \\partial _ { x _ { i } x _ { j } } ^ { 2 } w ^ { n } \\right ) \\right ] \\ d x d \\tau , \\end{align*}"}
-{"id": "2458.png", "formula": "\\begin{align*} \\frac { 1 } { d s } \\ = \\ \\lim _ { t \\to 1 } \\ , ( 1 - t ) ^ 2 P ( R , t ) \\ = \\ \\deg D \\ = \\ \\frac { 1 } { d } , \\end{align*}"}
-{"id": "8423.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow T _ 0 ^ * - } \\| ( z _ t , z _ { t t } ) \\| _ { C ( [ 0 , T ] ; H ^ s \\times H ^ s ) } + \\sup _ { t \\rightarrow T _ 0 ^ * } ( d _ I ( t ) ^ { - 1 } + d _ P ( t ) ^ { - 1 } ) = \\infty . \\end{align*}"}
-{"id": "4740.png", "formula": "\\begin{align*} | t C \\cap \\mathbb { Z } ^ d | = p ( t ) + O \\left ( t ^ { \\frac { ( d - 1 ) ( d - 2 ) } { 2 d - 3 } + \\epsilon } \\right ) \\end{align*}"}
-{"id": "5243.png", "formula": "\\begin{align*} \\frac { b _ { [ n s ] } - b _ { n } } { a _ { n } } \\rightarrow 0 \\frac { a _ { [ n s ] } } { a _ { n } } \\rightarrow ( 1 / s ) ^ { 1 / \\beta } = \\rho < 1 . \\end{align*}"}
-{"id": "5855.png", "formula": "\\begin{align*} \\int _ { f \\in M _ { n - 1 } } \\chi \\chi _ 0 ^ { - 1 } ( R ( G , f ) ) \\dd f = \\chi _ 0 ( \\Delta ( G ) ) ^ { - \\frac 1 { 2 } } \\chi ( R ( G , G ' ) ) \\frac { \\prod _ { i = 1 } ^ { k } \\Gamma _ { g _ i } ( \\chi ) } { \\Gamma ( \\chi ^ n ) } . \\end{align*}"}
-{"id": "9900.png", "formula": "\\begin{align*} g ( x + 1 ) = 2 f ( x + 1 - 1 ) - g ( x + 1 - 2 ) = 2 f ( x ) - g ( x - 1 ) \\end{align*}"}
-{"id": "5503.png", "formula": "\\begin{align*} \\Pi _ T \\left ( \\| b - b _ 0 \\| _ { \\mu _ 0 } \\leq 4 \\sqrt { d } \\| \\mu _ 0 \\| _ \\infty ^ { 1 / 2 } \\varepsilon _ T \\right ) & \\geq \\Pi _ T \\left ( \\sup _ { j = 1 , \\dots , d } \\| b _ j - b _ { 0 , j } \\| _ { L ^ 2 } \\leq 4 \\varepsilon _ T \\right ) \\\\ & \\geq \\prod _ { j = 1 } ^ d \\Pi _ T \\left ( \\| b _ j - b _ { 0 , j } \\| _ \\infty \\leq 4 \\varepsilon _ T \\right ) \\geq e ^ { - d T \\varepsilon _ T ^ 2 } , \\end{align*}"}
-{"id": "5129.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\abs * { \\begin{pmatrix} f _ { m _ j , k _ j } ^ + ( r ) \\\\ f _ { m _ j , k _ j } ^ - ( r ) \\end{pmatrix} - D _ { k _ j } \\begin{pmatrix} A ^ + r ^ { \\gamma _ { k _ j } } \\\\ A ^ - r ^ { - \\gamma _ { k _ j } } \\end{pmatrix} } r ^ { - 1 / 2 } = 0 , \\end{align*}"}
-{"id": "1822.png", "formula": "\\begin{align*} \\Phi ^ { \\rm o d d } ( n , x , t ) = ( k _ n L ) ^ { - 1 / 2 } \\ , \\sin ( k _ n x ) \\ , e ^ { - i k _ n t } , \\end{align*}"}
-{"id": "3172.png", "formula": "\\begin{align*} \\ker ( \\gamma \\mp 1 ) \\cap \\ker ( c - 1 ) = \\{ \\partial ^ * f \\mid \\gamma \\partial ^ * f = \\pm \\partial ^ * f \\} . \\end{align*}"}
-{"id": "8320.png", "formula": "\\begin{align*} S _ 1 ( A , f ) = p . v . \\int \\prod _ { j = 1 } ^ m \\frac { A _ j ( \\alpha ) - A _ j ( \\beta ) } { \\gamma _ j ( \\alpha ) - \\gamma _ j ( \\beta ) } \\frac { f ( \\beta ) } { \\gamma _ 0 ( \\alpha ) - \\gamma _ 0 ( \\beta ) } d \\beta . \\end{align*}"}
-{"id": "1028.png", "formula": "\\begin{align*} & \\sum _ { i _ 1 , \\cdots , i _ { k } \\ } \\prod _ { j = 1 } ^ k ( \\tau _ { i _ j } - x _ j ) _ + \\\\ = & ( n / 4 ) ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } \\sum _ { i _ 1 , \\cdots , i _ { k } \\ } \\prod _ { j = 1 } ^ k ( m _ { i _ j } - F _ n ( x _ j ) ) _ + \\\\ = & ( n / 4 ) ^ k ( 2 \\ln n ) ^ { \\frac { k } { 2 } } | \\Sigma _ k ( F _ n ( x _ 1 ) , \\cdots , F _ n ( x _ k ) ) | . \\end{align*}"}
-{"id": "1149.png", "formula": "\\begin{align*} & | y _ 0 - y _ { k + 1 } | > | y _ i - y _ { k + 1 } | > b - b _ 0 = b _ 0 - a > | y _ 0 - y _ i | > \\left | y _ 0 ^ 2 - y _ i ^ 2 \\right | / 4 \\\\ = & \\left | ( 1 + | u _ i | / \\ln n ) ^ 2 - 1 \\right | S ( I ) ^ 2 / 4 \\geq | u _ i | / \\ln n \\cdot S ( I ) ^ 2 / 2 \\\\ \\geq & \\varepsilon _ 1 S ( I ) ^ 2 / ( 2 \\ln n ) \\geq \\varepsilon _ 0 ( \\ln n ) ^ { - 1 } . \\end{align*}"}
-{"id": "2240.png", "formula": "\\begin{align*} & R ( z ) = R _ \\Omega ( z ) + R _ \\Omega ( z ) | g \\rangle \\langle h | R _ \\Omega ( z ) / C ( z ) , & & C ( z ) = 1 - \\langle h | R _ \\Omega ( z ) | g \\rangle , \\end{align*}"}
-{"id": "1782.png", "formula": "\\begin{align*} \\limsup _ { h \\downarrow 0 } \\frac { \\lvert B _ { t + h } - B _ { t } \\rvert } { h } = \\infty \\quad t \\in [ 0 , 1 ] \\quad \\end{align*}"}
-{"id": "1914.png", "formula": "\\begin{align*} E ( x , t ; x ' , t ' ) = - \\frac { ( t - t ' ) } { L } - \\frac { 1 } { 2 } + \\frac { 1 } { 2 L } \\left \\{ \\left [ ( t - t ' ) - ( x - x ' ) \\right ] \\bmod L + \\left [ ( t - t ' ) + ( x - x ' ) \\right ] \\bmod L \\right \\} . \\end{align*}"}
-{"id": "9432.png", "formula": "\\begin{align*} \\mathcal { E } _ 2 ( z ) = \\sum _ { 1 \\leq j \\leq D ( N ) - 1 } \\frac { \\det ( A _ { j } ) } { \\det ( A _ N ) } ( E _ 2 ( z ) - d _ j E _ 2 ( d _ j z ) ) . \\end{align*}"}
-{"id": "5006.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ M r _ m ^ d = 1 . \\end{align*}"}
-{"id": "2155.png", "formula": "\\begin{align*} \\Big | ( \\mathcal { F } ^ { - 1 } m ) ( \\mathbf x , \\mathbf y ) \\Big | & = c _ k ^ { - 1 } \\Big | \\int _ { B ( 0 , 1 / t ) } m ( \\xi ) E ( - i \\xi , \\mathbf { x } ) E ( i \\xi , \\mathbf { y } ) \\ , d w ( \\xi ) \\Big | \\\\ & \\leq c _ k ^ { - 1 } \\| m \\| _ { L ^ { \\infty } } \\| E ( i \\cdot , \\mathbf { x } ) \\| _ { L ^ 2 ( B ( 0 , 1 / t ) , d w ) } \\| E ( i \\cdot , \\mathbf { y } ) \\| _ { L ^ 2 ( B ( 0 , 1 / t ) , d w ) } \\\\ & \\leq C \\| m \\| _ { L ^ { \\infty } } w ( B ( \\mathbf { x } , t ) ) ^ { - 1 / 2 } w ( B ( \\mathbf { y } , t ) ) ^ { - 1 / 2 } . \\end{align*}"}
-{"id": "4744.png", "formula": "\\begin{align*} | \\textbf { t } C \\cap \\mathbb { Z } ^ 2 | & = \\sum _ { i = 1 } ^ { 4 } | t _ i S _ i \\cap \\mathbb { Z } ^ 2 | \\\\ & = \\sum _ { i = 1 } ^ { 4 } q ( t _ i ) + O \\left ( t _ { } ^ { \\frac { ( d - 1 ) ( d - 2 ) } { 2 d - 3 } } \\right ) \\end{align*}"}
-{"id": "5090.png", "formula": "\\begin{align*} 0 \\leq \\left \\langle x - \\sum _ { j \\in \\mathbb { J } } A _ j ^ * A _ j x , x - \\sum _ { k \\in \\mathbb { J } } A _ k ^ * A _ k x \\right \\rangle & = \\langle x , x \\rangle - 2 \\sum _ { j \\in \\mathbb { J } } \\langle A _ j x , A _ j x \\rangle + \\sum _ { j \\in \\mathbb { J } } \\langle A _ j x , A _ j x \\rangle \\\\ & = \\langle x , x \\rangle - \\sum _ { j \\in \\mathbb { J } } \\langle A _ j x , A _ j x \\rangle , \\end{align*}"}
-{"id": "5577.png", "formula": "\\begin{align*} \\mathcal { Q } ^ { \\pm } ( t ) = \\mathcal { Q } ^ { \\pm } ( 0 ) \\mp \\int _ 0 ^ t \\Big ( 2 | u _ x ( 0 , s ) | ^ 2 + \\frac { \\alpha } { \\gamma } v _ x ^ 2 ( 0 , s ) \\Big ) d s \\end{align*}"}
-{"id": "7021.png", "formula": "\\begin{align*} y ^ { 2 } = x ^ { 3 } + a x + b \\end{align*}"}
-{"id": "2812.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\displaystyle { x _ i ^ { \\prime } ( t ) = x _ i ( t ) \\biggl ( b _ i - \\mu _ i x _ i ( t ) - \\sum _ { j = 1 } ^ n a _ { i j } \\int _ 0 ^ { \\infty } K _ { i j } ( s ) x _ j ( t - s ) \\ , d s } \\\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\displaystyle { - c _ i \\int _ 0 ^ { \\infty } G _ i ( s ) u _ i ( t - s ) \\ , d s \\biggr ) } , \\\\ \\displaystyle { u _ i ^ { \\prime } ( t ) = - e _ i u _ i ( t ) + d _ i x _ i ( t ) , i = 1 , \\ldots , n } , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "7285.png", "formula": "\\begin{align*} M _ p = \\max \\{ 3 \\lVert a \\rVert , 3 \\lVert \\psi _ p ( a ) \\rVert , L ( F _ p ) : a \\in F _ p \\} . \\end{align*}"}
-{"id": "52.png", "formula": "\\begin{align*} ( \\rho , J ) ( \\cdot , 0 ) = ( \\rho _ 0 , J _ 0 ) ( \\cdot ) \\ , , J ( 0 , t ) = J ( 1 , t ) = J _ b \\end{align*}"}
-{"id": "930.png", "formula": "\\begin{align*} E _ 2 ^ q = I _ 2 - \\sum _ { j _ 1 , j _ 2 = 0 } ^ q C _ { j _ 2 j _ 1 } ^ 2 - \\sum _ { j _ 1 , j _ 2 = 0 } ^ q C _ { j _ 2 j _ 1 } C _ { j _ 1 j _ 2 } \\ \\ \\ ( i _ 1 = i _ 2 ) , \\end{align*}"}
-{"id": "1163.png", "formula": "\\begin{align*} & \\lim _ { n \\to + \\infty } \\frac { C _ { \\beta , n - 2 k , k } } { C _ { \\beta , n } n ^ { k \\beta } } = A _ { \\beta } ^ k . \\end{align*}"}
-{"id": "7125.png", "formula": "\\begin{align*} \\int _ \\Omega ( - 2 \\mu _ 1 ( q _ 0 - \\hat { q } ) + \\mu _ 2 \\phi _ 2 ^ 2 ( \\hat { q } ) ) h \\ , d x = 0 , \\ , \\ , \\forall h \\in L ^ 2 , \\end{align*}"}
-{"id": "2610.png", "formula": "\\begin{align*} \\lim _ { | x | \\to \\infty } \\frac { V ( x ) } { \\log | x | } = \\infty . \\end{align*}"}
-{"id": "4231.png", "formula": "\\begin{align*} \\delta ( a ( x ) \\lvert d u \\rvert ^ { p - 2 } d u ) ) = f & & \\Omega , \\end{align*}"}
-{"id": "9018.png", "formula": "\\begin{align*} f _ { \\{ 1 \\} , a } ( x ) = \\begin{cases} a & x = 1 , \\\\ x & \\end{cases} \\end{align*}"}
-{"id": "2571.png", "formula": "\\begin{align*} Y _ t = Y ^ { ( 0 , t _ 0 ] } _ t + Y ^ { ( t _ 0 , t ] } _ t . \\end{align*}"}
-{"id": "2885.png", "formula": "\\begin{align*} [ J _ { \\mu \\nu } , L _ { \\mu \\rho } \\Sigma _ { \\mu \\rho } + L _ { \\nu \\rho } \\Sigma _ { \\nu \\rho } ] = 0 . \\end{align*}"}
-{"id": "8255.png", "formula": "\\begin{align*} m = N + 1 = s ( k + 2 ) + ( r - s ) ( k + 1 ) + r - 1 \\ , . \\end{align*}"}
-{"id": "5498.png", "formula": "\\begin{align*} [ Z _ \\cdot ( f ) ] _ T = \\int _ 0 ^ T \\| \\nabla L ^ { - 1 } [ f ] ( X _ s ) \\| ^ 2 d s \\leq T \\sum _ { i = 1 } ^ d \\| \\partial _ { x _ i } L ^ { - 1 } [ f ] \\| _ \\infty ^ 2 = T d _ L ^ 2 ( f , 0 ) . \\end{align*}"}
-{"id": "2810.png", "formula": "\\begin{align*} D \\mathbf { H } ( \\mathbf { 0 } ) = \\mathbf { Q } . \\end{align*}"}
-{"id": "9.png", "formula": "\\begin{align*} \\kappa ^ { t r } \\nabla _ v \\mathbf { y } ^ i ( t , V _ t ^ { r , v } ; m ) = Z _ t ^ { i , r , v } ( m ) . \\end{align*}"}
-{"id": "3208.png", "formula": "\\begin{align*} d ^ { \\top } \\Phi _ r ( \\bar { s } ) d & = \\frac { d ^ { \\top } ( X _ r + \\bar { s } \\Delta X _ r ) ( Y _ r + \\bar { s } \\Delta Y _ r ) d + d ^ { \\top } ( Y _ r + \\bar { s } \\Delta Y _ r ) ( X _ r + \\bar { s } \\Delta X _ r ) d } { 2 } = 0 , \\end{align*}"}
-{"id": "7870.png", "formula": "\\begin{align*} \\frac { 1 } { m ! } \\sum _ { \\sigma \\in { \\rm S y m } ( I ) } \\prod _ { j \\in I } \\cos ( 2 \\pi y _ { \\sigma ^ { - 1 } ( j ) } \\xi _ j ) = \\frac { 1 } { { m \\choose l } } \\sum _ { \\substack { J \\subseteq I \\\\ | J | = l } } \\prod _ { j \\in J } \\cos ( 2 \\pi \\xi _ j ) . \\end{align*}"}
-{"id": "8570.png", "formula": "\\begin{align*} \\begin{array} { r c l c r c l c r c l c r c l } \\ ; \\ ! [ h , e _ 2 ^ \\pm ] & = & \\mp e _ 2 ^ \\pm , & & [ z , e _ 2 ^ \\pm ] & = & \\pm e _ 2 ^ \\pm , & & \\ ; \\ ! [ h , e _ 3 ^ \\pm ] & = & \\pm e _ 3 ^ \\pm , & & [ z , e _ 3 ^ \\pm ] & = & \\pm e _ 3 ^ \\pm , \\\\ \\ ; \\ ! [ e _ 1 ^ \\pm , e _ 2 ^ \\mp ] & = & 0 , & & [ e _ 1 ^ \\pm , e _ 2 ^ \\pm ] & = & - e _ 3 ^ \\pm , & & \\ ; \\ ! [ e _ 1 ^ \\pm , e _ 3 ^ \\pm ] & = & 0 , & & [ e _ 1 ^ \\pm , e _ 3 ^ \\mp ] & = & - e _ 2 ^ \\mp . \\end{array} \\end{align*}"}
-{"id": "1087.png", "formula": "\\begin{align*} & \\sup _ { z \\in [ 0 , ( b _ 1 / a _ 1 - 1 ) \\ln n ] } e ^ z \\frac { D _ n ( ( 1 + z / \\ln n ) G _ n ( x ) / 2 ) } { D _ n ( G _ n ( x ) / 2 ) } \\frac { ( 1 + z / \\ln n ) a _ 1 } { \\sqrt { 4 - ( 1 + z / \\ln n ) ^ 2 a _ 1 ^ 2 } } \\\\ = & \\sup _ { w \\in [ 1 , b _ 1 / a _ 1 ] } e ^ { ( w - 1 ) \\ln n } \\frac { D _ n ( w G _ n ( x ) / 2 ) } { D _ n ( G _ n ( x ) / 2 ) } \\frac { w a _ 1 } { \\sqrt { 4 - w ^ 2 a _ 1 ^ 2 } } \\\\ \\leq & \\sup _ { w \\in [ 1 , b _ 1 / a _ 1 ] } e ^ { ( w - 1 ) \\ln n } e ^ { 1 - ( w - 1 ) \\ln n } \\frac { b _ 1 } { \\sqrt { 4 - b _ 1 ^ 2 } } = \\frac { e b _ 1 } { \\sqrt { 4 - b _ 1 ^ 2 } } \\end{align*}"}
-{"id": "9006.png", "formula": "\\begin{align*} \\partial _ { 1 } \\widehat { N } = \\dotsb = \\partial _ { m - 1 } \\widehat { N } = 0 \\ , , \\end{align*}"}
-{"id": "4466.png", "formula": "\\begin{align*} F ( A ) = W _ k ^ { 1 / k } \\big ( \\lambda ( A ) \\big ) , A \\in \\mathcal { K } . \\end{align*}"}
-{"id": "4439.png", "formula": "\\begin{align*} j ^ * ( \\mu _ L ) = 2 c _ 1 ( X ) . \\end{align*}"}
-{"id": "5927.png", "formula": "\\begin{align*} G _ { S ' } \\sim ( n \\Delta ' - \\lfloor ( n + 1 ) \\Delta ' \\rfloor + n N ' + P ' ) | _ { S ' } = ( L ' + P ' ) | _ { S ' } = L _ { S ' } + P _ { S ' } . \\end{align*}"}
-{"id": "1334.png", "formula": "\\begin{align*} \\rho _ H ( f _ 1 g f _ 2 ) = \\rho _ H \\big ( f _ 1 \\rho _ H ( g ) f _ 2 \\big ) + \\rho _ H \\big ( f _ 1 \\rho _ { G \\setminus H } ( g ) f _ 2 \\big ) = f _ 1 \\rho _ H ( g ) f _ 2 \\ ; . \\end{align*}"}
-{"id": "487.png", "formula": "\\begin{align*} \\lambda _ 1 \\leq \\cdots \\leq \\lambda _ { a } < \\lambda _ { a + 1 } = \\cdots = \\lambda _ n \\end{align*}"}
-{"id": "1666.png", "formula": "\\begin{align*} G _ { n } ^ { ( 1 ) } = G _ { n - 1 } ^ { ( 1 ) } \\mid T _ { k / 2 , 4 N } ( \\ell ^ { 2 } ) - \\ell ^ { k - 2 } \\cdot G _ { n - 2 } ^ { ( 1 ) } \\quad \\textnormal { f o r } n \\geq 2 . \\end{align*}"}
-{"id": "9401.png", "formula": "\\begin{align*} f ( D ) : = \\sum n _ P f ( P ) . \\end{align*}"}
-{"id": "8623.png", "formula": "\\begin{align*} \\bar { u } _ 0 = \\sqrt { \\frac { 1 - \\sqrt { 1 - 4 c _ 0 ^ 2 } } { 2 } } \\ \\ \\textrm { a n d } \\ \\ \\bar { v } _ 0 = \\sqrt { \\frac { 1 + \\sqrt { 1 - 4 c _ 0 ^ 2 } } { 2 } } . \\end{align*}"}
-{"id": "1869.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\left ( \\alpha _ { m j } \\ , \\overline { \\alpha _ { n j } } - \\beta _ { m j } \\ , \\overline { \\beta _ { n j } } \\right ) = \\delta _ { n m } \\qquad \\mbox { a n d } \\sum _ { j = 1 } ^ \\infty \\left ( \\alpha _ { m j } \\ , \\beta _ { n j } - \\beta _ { m j } \\ , \\alpha _ { n j } \\right ) = 0 , \\end{align*}"}
-{"id": "3628.png", "formula": "\\begin{align*} \\alpha _ { - 1 } \\cdot \\alpha _ 2 = - \\frac { t } { 2 ^ 3 } ( 2 a _ { - 1 } + a _ 3 + a _ { - 3 } ) + \\frac { 1 } { 2 ^ 3 } ( a _ 3 - a _ { - 3 } ) \\cdot v _ { ( 1 , 2 ) } \\in M _ 0 ^ { ( a _ 1 ) } . \\end{align*}"}
-{"id": "9942.png", "formula": "\\begin{align*} \\bigcup _ { k = 0 } ^ \\infty D _ k , D _ k : = \\Lambda _ k + D . \\end{align*}"}