diff --git "a/data_tmp/process_23/tokenized_finally.jsonl" "b/data_tmp/process_23/tokenized_finally.jsonl"
deleted file mode 100644--- "a/data_tmp/process_23/tokenized_finally.jsonl"
+++ /dev/null
@@ -1,10134 +0,0 @@
-{"id": "10104.png", "formula": "\\begin{align*} R ( X , Y ) \\xi = 0 . \\end{align*}"}
-{"id": "2761.png", "formula": "\\begin{align*} C _ f ^ * ( \\chi , s ) = C _ f ^ * ( T , s ) \\big | _ { T = \\chi ( 1 ) - 1 } , \\end{align*}"}
-{"id": "10107.png", "formula": "\\begin{align*} { J ( { \\boldsymbol \\omega _ k } ( i ) ) } = \\sum _ { k = 1 } ^ { N } { \\mathbb { E } \\big | { d _ k ( i ) } - { \\boldsymbol \\omega _ k } ^ H ( i ) { \\boldsymbol x _ k ( i ) } \\big | ^ 2 } , \\end{align*}"}
-{"id": "2357.png", "formula": "\\begin{align*} p ^ { - n } . z - x _ p = \\left ( c ^ 2 \\left ( n + \\frac { 1 } { c ( c z + d ) } \\right ) \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "8496.png", "formula": "\\begin{align*} g ( y ) = \\frac { b _ 2 } { y } \\varpi ^ { \\kappa - a _ 2 } + ( v ^ { - 1 } - b _ 1 ) \\varpi ^ { \\kappa - l } + b _ 1 \\sum _ { j \\geq 2 } ( - 1 ) ^ { j } y ^ { j - 1 } \\varpi ^ { ( j - 1 ) a _ 1 - j l + \\kappa } . \\end{align*}"}
-{"id": "8077.png", "formula": "\\begin{align*} \\alpha _ i ^ - : = \\left \\{ \\begin{array} { c c } \\alpha _ i & \\alpha _ i < 0 \\\\ 0 & \\alpha _ i \\ge 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "7228.png", "formula": "\\begin{align*} ( g * _ { U } f ) ( x ) = \\int _ { U } ^ { } g ( u ) f ( u ^ { - 1 } x ) d u . \\end{align*}"}
-{"id": "4137.png", "formula": "\\begin{align*} \\eta = K ^ { \\frac { 1 } { 4 } } \\xi + Z , \\end{align*}"}
-{"id": "7375.png", "formula": "\\begin{align*} [ a , b ] = - ( - 1 ) ^ { | a | | b | } [ b , a ] \\end{align*}"}
-{"id": "9940.png", "formula": "\\begin{align*} { \\cal C } _ { 1 } ( m , a , b ) : ~ ~ u _ { t } + ( u ^ m ) _ { x } + \\frac { 1 } { b } [ u ^ { a } ( u ^ { b } ) _ { x x } ] _ { x } = 0 , a \\geq 0 , m , b \\geq 2 , n : = a + b \\end{align*}"}
-{"id": "2117.png", "formula": "\\begin{align*} \\psi _ n ( \\Theta _ { i \\hat { T } _ n ^ i } ) _ { \\hat { T } _ n ^ i } = 0 , \\Theta _ { i \\hat { T } _ n ^ { i c } } = 0 . \\end{align*}"}
-{"id": "8465.png", "formula": "\\begin{align*} \\Delta = 1 + 4 v b _ 1 + 4 v ^ 2 b _ 2 ^ 2 \\varpi . \\end{align*}"}
-{"id": "894.png", "formula": "\\begin{align*} T _ n = \\sqrt { n } \\max _ { \\theta \\in \\mathcal { G } _ n } | U _ n ( \\theta ) | , \\end{align*}"}
-{"id": "7202.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ N \\xi _ { k , p } \\leq c . \\end{align*}"}
-{"id": "6689.png", "formula": "\\begin{align*} \\lim \\limits _ { \\Delta \\rightarrow 0 } \\frac { 1 } { \\Delta ^ { \\frac { n - 1 + p } { p } } } | H _ p ^ n ( \\Delta ) | _ n = & \\frac { 1 } { n } \\left ( \\frac { p } { p - 1 } \\right ) ^ { \\frac { n - 1 + p } { p } } | B _ p ^ { n - 1 } | _ { n - 1 } - \\frac { 1 } { n - 1 + p } \\left ( \\frac { p } { p - 1 } \\right ) ^ { \\frac { n - 1 + p } { p } } | B _ p ^ { n - 1 } | _ { n - 1 } \\\\ = & \\frac { p - 1 } { n ( n - 1 + p ) } \\left ( \\frac { p } { p - 1 } \\right ) ^ { \\frac { n - 1 + p } { p } } | B _ p ^ { n - 1 } | _ { n - 1 } . \\end{align*}"}
-{"id": "6436.png", "formula": "\\begin{align*} h _ { } = \\underset { \\left \\vert t \\right \\vert \\rightarrow \\infty } { \\lim } \\frac { I \\left ( t \\right ) } { \\left \\vert t \\right \\vert } . \\end{align*}"}
-{"id": "10053.png", "formula": "\\begin{align*} a _ F ( \\alpha , \\phi ) = \\Lambda ( 0 , \\chi _ E ) \\cdot E ' _ \\alpha ( \\vec { v } , 0 , \\phi ) . \\end{align*}"}
-{"id": "2366.png", "formula": "\\begin{align*} \\Gamma = \\Big \\langle a _ 1 , b _ 1 , \\ldots , a _ g , b _ g , e _ 1 , \\ldots , e _ r , & \\ , p _ 1 , \\ldots , p _ s , h _ 1 , \\ldots , h _ t \\ \\Big \\vert \\ e _ 1 ^ { m _ 1 } = \\ldots = e _ r ^ { m _ r } \\\\ & = \\big ( \\prod _ { j = 1 } ^ g [ a _ j , b _ j ] \\big ) e _ 1 \\cdots e _ r p _ 1 \\cdots p _ s h _ 1 \\cdots h _ t = 1 \\Big \\rangle , \\end{align*}"}
-{"id": "5300.png", "formula": "\\begin{align*} ( B ( u ) - B ( v ) ) u = B ( u ) u - B ( v ) v - B ( v ) ( u - v ) . \\end{align*}"}
-{"id": "2836.png", "formula": "\\begin{align*} A _ j ^ { ( n + 1 ) } = A _ { j _ 1 } ^ { ( n ) } \\ldots A _ { j _ { \\lambda } } ^ { ( n ) } , \\end{align*}"}
-{"id": "876.png", "formula": "\\begin{align*} \\max _ { 1 \\leq k \\leq d _ n } ( E [ F _ { n , k } ^ 4 ] - 3 E [ F _ { n , k } ^ 2 ] ^ 2 ) = O ( N _ n ^ { - 1 } ) . \\end{align*}"}
-{"id": "1380.png", "formula": "\\begin{align*} Z _ { t + 1 } = \\sum _ { x = 1 } ^ { Z _ t } u _ t ( x ) , \\end{align*}"}
-{"id": "5164.png", "formula": "\\begin{align*} U _ { 1 } ( x , y ) & = \\begin{cases} \\frac { f _ { 1 } ( x ) - g _ { 1 , [ y , 1 ] } ( x ) } { y - x } , & x < y , \\\\ - \\infty , & , \\end{cases} \\\\ U _ { 2 } ( x , y ) & = \\begin{cases} \\frac { f _ { 2 } ( y ) - g _ { 2 , [ 0 , x ] } ( y ) } { y - x } , & x < y , \\\\ - \\infty , & , \\end{cases} \\end{align*}"}
-{"id": "5603.png", "formula": "\\begin{align*} X _ t = x _ 0 + \\int _ 0 ^ t f ( X _ s ) d s + \\int _ 0 ^ t \\sigma ( X _ s ) d \\mathbf { B } ^ H _ s \\end{align*}"}
-{"id": "105.png", "formula": "\\begin{align*} \\Gamma _ 1 = \\bigcup _ { n \\in \\mathbb { Z } ^ k } ( n + \\Gamma ) , ( n + \\Gamma = \\{ n + \\gamma | \\ , \\gamma \\in \\Gamma \\} ) . \\end{align*}"}
-{"id": "3245.png", "formula": "\\begin{align*} ( \\eta , \\omega _ Z , \\theta ) _ V = ( i ^ \\ast ( \\eta ) , i ^ \\ast ( \\theta ) ) _ Z . \\end{align*}"}
-{"id": "2245.png", "formula": "\\begin{align*} \\sigma _ { \\gamma + I } = \\sum \\limits _ { j = 1 } ^ p \\frac { 1 } { z _ { j 1 } ^ { \\gamma _ 1 + 1 } \\cdots z _ { j n } ^ { \\gamma _ n + 1 } } , \\end{align*}"}
-{"id": "4736.png", "formula": "\\begin{align*} { \\varepsilon } = F ( x ^ 0 , Y , Z ) \\Delta \\sqrt { - \\det g ^ { i j } } . \\end{align*}"}
-{"id": "5747.png", "formula": "\\begin{align*} \\frac { f _ { x + 1 , y - 1 , z - 1 } ( \\textbf { a } ) f _ { x - 1 , y , z } ( \\textbf { a } ) } { f _ { x , y - 1 , z - 1 } ( \\textbf { a } ) f _ { x , y , z } ( \\textbf { a } ) } = 1 , \\end{align*}"}
-{"id": "1639.png", "formula": "\\begin{align*} \\lambda _ { i , j } = y ( n - m + \\vec { i } , \\vec { j } ) . \\end{align*}"}
-{"id": "2551.png", "formula": "\\begin{align*} E ^ { ( \\alpha ) } : = \\left \\{ x \\in \\R ^ n \\ , : \\ , \\theta ( E ) ( x ) = \\alpha \\right \\} \\ , . \\end{align*}"}
-{"id": "1498.png", "formula": "\\begin{align*} \\sigma ^ 2 = \\sum \\limits _ { k = - \\infty } ^ \\infty C _ k ( f , f ) . \\end{align*}"}
-{"id": "1505.png", "formula": "\\begin{align*} \\begin{gathered} \\textbf { i } ^ { 2 } = \\textbf { j } ^ { 2 } = \\textbf { k } ^ { 2 } = \\textbf { i } \\textbf { j } \\textbf { k } = - 1 , \\\\ \\textbf { i } \\textbf { j } = - \\textbf { j } \\textbf { i } = \\textbf { k } , \\ \\textbf { j } \\textbf { k } = - \\textbf { k } \\textbf { j } = \\textbf { i } , \\textbf { k } \\textbf { i } = - \\textbf { i } \\textbf { k } = \\textbf { j } . \\end{gathered} \\end{align*}"}
-{"id": "1081.png", "formula": "\\begin{align*} \\mathbf { Y } = | \\mathbf { H } \\boldsymbol { \\rho } | ^ 2 , \\end{align*}"}
-{"id": "6402.png", "formula": "\\begin{align*} \\mathcal { D } _ { \\theta } \\overset { } { = } { \\displaystyle \\bigotimes \\limits _ { k = 1 } ^ { n } } \\mathcal { I } _ { \\theta ^ { k } } = \\left ( \\mathcal { I } _ { \\theta ^ { 1 } } \\otimes \\mathcal { I } _ { \\theta ^ { 2 } } \\otimes \\mathcal { I } _ { \\theta ^ { n } } \\right ) \\subseteq \\mathbb { R } ^ { n } \\end{align*}"}
-{"id": "569.png", "formula": "\\begin{align*} \\begin{array} { l } \\beta _ 1 = - b \\sum _ { j = 2 } ^ m d _ j \\leq \\alpha _ 1 \\\\ \\beta _ j = b d _ j \\leq \\alpha _ j \\mbox { f o r } j = 2 , \\ldots , m . \\end{array} \\end{align*}"}
-{"id": "7595.png", "formula": "\\begin{align*} \\omega ( v _ 1 + u , v _ 2 - u ) = \\omega ( v _ 1 , v _ 2 ) + \\omega ( v _ 1 , - u ) + \\omega ( u , v _ 2 ) + \\omega ( u , - u ) = \\omega ( v _ 1 , v _ 2 ) \\end{align*}"}
-{"id": "3938.png", "formula": "\\begin{align*} u ( x _ j , t ^ { n + 1 } ) = \\sum \\limits _ { l = 0 } ^ { n } ( w _ { n - l } - w _ { n - l + 1 } ) u ( x _ j , t ^ { l } ) + w _ n u ( x _ j , t ^ { 0 } ) + \\nu \\alpha _ 0 ( u ( x _ j , t ^ { n + 1 } ) ) _ { x x } - \\\\ { } \\frac { \\alpha _ 0 } { 3 } \\left ( u ( x _ j , t ^ { n } ) ( u ( x _ j , t ^ { n + 1 } ) ) _ x + ( u ( x _ j , t ^ { n } ) u ( x _ j , t ^ { n + 1 } ) ) _ x \\right ) , \\end{align*}"}
-{"id": "8965.png", "formula": "\\begin{gather*} \\sum _ { \\alpha \\in \\Phi ^ - ( W _ I ) } T _ { w \\alpha } - \\sum _ { \\alpha \\in \\Phi ^ - ( W _ J ) } T _ \\alpha = \\ ! \\ ! \\sum _ { \\alpha \\in \\Phi ^ - ( W _ I ) \\setminus \\Phi ^ - ( W _ I \\cap w ^ { - 1 } W _ J w ) } \\ ! \\ ! \\ ! \\ ! T _ { w \\alpha } - \\ ! \\ ! \\sum _ { \\alpha \\in \\Phi ^ - ( W _ J ) \\setminus \\Phi ^ - ( W _ J \\cap w W _ I w ^ { - 1 } ) } \\ ! \\ ! \\ ! \\ ! T _ \\alpha . \\end{gather*}"}
-{"id": "8868.png", "formula": "\\begin{align*} F _ k ( T ) = 1 + ( \\Delta ^ + - k ) \\sum _ { i = 0 } ^ { r - 1 } ( \\Delta ^ + ) ^ i = 1 + ( \\Delta ^ + - k ) \\frac { ( \\Delta ^ + ) ^ r - 1 } { \\Delta ^ + - 1 } . \\end{align*}"}
-{"id": "8263.png", "formula": "\\begin{align*} r _ 1 + r _ 2 - 2 V _ 0 = 2 T _ 0 - 2 V _ 0 + O ( \\log ( L ) ^ { - 1 - \\delta } ) \\gg T _ 0 . \\end{align*}"}
-{"id": "7009.png", "formula": "\\begin{align*} \\epsilon _ 0 = \\epsilon _ 0 ( n , s , \\eta _ 0 , S , L C ) = \\sqrt { \\frac { n } { n - 1 } } C _ 4 ^ { 1 / s } ( \\frac { \\mu _ 0 } { \\mu _ 1 } ) ^ { 1 / s } . \\end{align*}"}
-{"id": "6785.png", "formula": "\\begin{align*} w _ { \\lambda , k } ( \\xi _ j ) = 8 \\pi G ( \\xi _ j , \\xi _ k ) - 4 \\lambda ^ 2 \\ln { \\lambda } + O ( \\lambda ^ 2 ) , \\end{align*}"}
-{"id": "3176.png", "formula": "\\begin{align*} \\alpha ( z , s ) : = \\frac { 2 b z } { \\sigma ^ { 2 } \\left ( e ^ { b s } - 1 \\right ) } \\quad \\quad \\beta ( z , s ) : = \\frac { 2 b e ^ { b s } } { \\sigma ^ { 2 } \\left ( e ^ { b s } - 1 \\right ) } , \\end{align*}"}
-{"id": "8889.png", "formula": "\\begin{align*} c _ 1 = \\omega + ( c - \\theta ) g . \\end{align*}"}
-{"id": "5062.png", "formula": "\\begin{align*} \\omega _ { i j } = \\sum _ k \\frac { B _ { i j , k } } { b _ i - b _ j } \\omega _ k = \\frac { B _ { i j , i } } { b _ i - b _ j } \\omega _ i + \\frac { B _ { i j , j } } { b _ i - b _ j } \\omega _ j . \\end{align*}"}
-{"id": "1744.png", "formula": "\\begin{align*} ( S _ \\lambda ^ { u n i v } ) ^ * ( f \\ , \\sqrt { d \\mu } ) : = ( f \\circ \\sigma _ \\lambda ) \\ , \\sqrt { d ( \\mu \\circ \\sigma _ \\lambda ) } . \\end{align*}"}
-{"id": "8379.png", "formula": "\\begin{align*} \\psi _ g ( f ) = \\frac { ( j ^ { [ 2 ] } ) ^ * \\tilde { \\psi } _ g ( f ^ { [ 2 ] } ) } { ( j ^ { [ 1 ] } ) ^ * \\tilde { \\psi } _ g ( f ^ { [ 1 ] } ) } \\end{align*}"}
-{"id": "77.png", "formula": "\\begin{align*} m ( T ( \\sigma ~ 0 1 ~ 1 0 ~ w ~ 1 1 ~ \\tau ) ) & = m ( T ( \\sigma ~ \\underline { 0 1 } ~ \\underline { 1 0 } ~ 1 0 ~ w ' ~ 1 1 ~ \\tau ) ) \\\\ & = m ( T ( \\sigma ~ 1 0 ~ 0 1 ~ 1 0 ~ w ' ~ 1 1 ~ \\tau ) ) . \\end{align*}"}
-{"id": "774.png", "formula": "\\begin{align*} B _ { k } = B _ { k } ( 0 ) , k = 0 , 1 , 2 , . . . \\end{align*}"}
-{"id": "4097.png", "formula": "\\begin{align*} K = \\frac { \\mathrm { d e t } ( h _ { i j } ) } { \\mathrm { d e t } ( b _ { i j } ) } , \\end{align*}"}
-{"id": "5838.png", "formula": "\\begin{align*} v = g e + w , w \\in V ' . \\end{align*}"}
-{"id": "8186.png", "formula": "\\begin{align*} J _ { \\nu } ( k _ { \\nu } ) = \\begin{cases} \\{ \\xi _ { \\nu } \\in F _ { \\nu } \\colon k _ { \\nu } \\abs { a } R _ { \\nu } - D < \\abs { \\xi _ { \\nu } } \\leq ( k _ { \\nu } + 1 ) \\abs { a } R _ { \\nu } + D \\} & k _ { \\nu } \\geq 1 , \\\\ \\{ \\xi _ { \\nu } \\in F _ { \\nu } \\colon - k _ { \\nu } - D \\leq \\abs { \\abs { \\xi _ { \\nu } } - \\frac { \\abs { a } T _ { \\nu } } { 2 \\pi \\abs { y _ { \\nu } } } } < - k _ { \\nu } + 1 + D \\} & \\end{cases} \\end{align*}"}
-{"id": "9458.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ n } { ( z q ^ n ; q ) _ { n + 1 } ( z q ^ { 2 n + 2 } ; q ^ 2 ) _ { \\infty } } & = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ { 2 n ^ 2 + 2 n + 1 } } { ( q ; q ^ 2 ) _ { n + 1 } ( z q ; q ^ 2 ) _ { n + 1 } } , \\\\ \\sum _ { n = 0 } ^ { \\infty } q ^ n ( - z q ^ { n + 1 } ; q ) _ { n } ( - z q ^ { 2 n + 2 } ; q ^ 2 ) _ { \\infty } & = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ { n ^ 2 + n } } { ( q ; q ^ 2 ) _ { n + 1 } } . \\end{align*}"}
-{"id": "9052.png", "formula": "\\begin{align*} A _ { { \\mathbf x } } ( x ) = ( a _ { 1 1 } ^ 1 { \\mathbf x } _ 1 x _ 1 , a _ { 2 2 } ^ 2 { \\mathbf x } _ 2 x _ 2 ) . \\end{align*}"}
-{"id": "1820.png", "formula": "\\begin{align*} \\langle \\sigma _ x ; \\sigma _ y \\rangle _ { a , h } = \\mathbb { P } _ h ^ a ( x \\longleftrightarrow y ) - \\mathbb { P } _ h ^ a ( x \\longleftrightarrow g ) \\mathbb { P } _ h ^ a ( y \\longleftrightarrow g ) . \\end{align*}"}
-{"id": "2300.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\| A u ^ N \\| _ { L ^ 2 } ^ 2 + \\nu \\| A ^ { 1 + s / 2 } u ^ N \\| _ { L ^ 2 } ^ 2 = - \\langle A ^ 2 u ^ N , u ^ N \\cdot \\nabla u ^ N \\rangle - \\langle A ^ 2 u ^ N , \\mathcal { U } ^ \\alpha ( u ^ N , u ^ N ) \\rangle . \\end{align*}"}
-{"id": "4169.png", "formula": "\\begin{align*} \\left ( - \\frac { 1 } { n } + \\frac { 1 } { n ^ 2 } \\right ) & e _ { i , g + 1 } ^ { n - 2 } + \\frac { 1 } { n } e _ { i , g + 2 } ^ { n - 2 } + \\frac { 1 } { n } \\sum _ { \\tau = 2 } ^ { n - s - 2 } \\left [ e ^ { n - \\tau - 1 } _ { i , g + \\tau + 1 } - e _ { i , g + \\tau } ^ { n - \\tau - 1 } \\right ] \\\\ & + \\left ( 1 - \\frac { 1 } { n } \\right ) \\left [ \\sum _ { \\tau = 0 } ^ { n - s - 3 } \\frac { 1 } { n ^ { n - \\tau - s - 1 } } e ^ { s } _ { i , g + \\tau + 1 } \\right ] - \\frac { 1 } { n ^ 2 } e _ { i , g + n - s - 1 } ^ s . \\end{align*}"}
-{"id": "9425.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } n \\| n \\alpha \\| \\| n \\beta \\| = 0 , \\end{align*}"}
-{"id": "1388.png", "formula": "\\begin{align*} ( u ( 1 ) , u ( 2 ) , \\ldots ) & = ( \\underbrace { 0 , \\ldots , 0 } _ { \\xi _ 1 } , \\eta _ { 1 } + 1 , \\underbrace { 0 , \\ldots , 0 } _ { \\xi _ 2 } , \\eta _ { 2 } + 1 , \\ldots ) , \\xi _ i , \\eta _ i \\ge 0 , i \\in \\mathbb N , \\end{align*}"}
-{"id": "2231.png", "formula": "\\begin{align*} \\frac { d F _ j \\left ( \\frac 1 { w _ 1 } , \\frac 1 { w _ 2 } , \\ldots , \\frac 1 { w _ n } , t \\right ) } { F _ j \\left ( \\frac 1 { w _ 1 } , \\frac 1 { w _ 2 } , \\ldots , \\frac 1 { w _ n } , t \\right ) } = \\frac { d { \\widetilde F _ j } ( w , t ) } { { \\widetilde F _ j } ( w , t ) } - \\sum _ { k = 1 } ^ n m _ { j k } \\cdot \\frac { d w _ k } { w _ k } . \\end{align*}"}
-{"id": "8921.png", "formula": "\\begin{gather*} \\zeta _ { g , h } \\colon \\ { \\cal L } _ { Q _ g , w _ g } \\otimes \\big ( g ^ { - 1 } \\big ) ^ * { \\cal L } _ { Q _ h , w _ h } \\cong { \\cal L } _ { Q _ g + g ^ { - t } Q _ h g ^ { - 1 } , w _ g + w _ h } = { \\cal L } _ { Q _ { g h } , w _ { g h } } . \\end{gather*}"}
-{"id": "2689.png", "formula": "\\begin{align*} L : = \\widehat { K _ v ^ { \\mathrm { u r } } } \\end{align*}"}
-{"id": "1201.png", "formula": "\\begin{align*} | \\nabla \\bar e ( x ) | = o \\left ( | x | ^ { \\frac { 1 - n } { p - 1 } } \\right ) \\ , \\ , \\mbox { a s } \\ , \\ , x \\to \\infty . \\end{align*}"}
-{"id": "153.png", "formula": "\\begin{align*} z = z ^ { \\sigma ^ 2 } = r ^ { \\sigma } r ^ { - 1 } z \\end{align*}"}
-{"id": "7231.png", "formula": "\\begin{align*} \\Theta _ { \\pi } ( h ) = T r [ \\hat { h } ( \\pi ) ] . \\end{align*}"}
-{"id": "6379.png", "formula": "\\begin{align*} \\sqrt { n } \\exp ( - ( n - ( 2 i - 1 ) ) K ) \\int _ 0 ^ \\infty \\exp \\big \\{ - ( 2 i - 1 ) ( V _ 1 ( x ) - V _ 1 ( z _ 1 ^ * ) + \\log ( z _ 1 ^ * ( 1 - z _ 1 ^ * ) ) ) \\big \\} \\ , d x = o ( 1 ) . \\end{align*}"}
-{"id": "9407.png", "formula": "\\begin{align*} \\tau _ { \\varepsilon , L } ( \\eta ) = \\{ t \\in \\mathbb Z ^ 2 : \\} \\end{align*}"}
-{"id": "1118.png", "formula": "\\begin{align*} & G _ i = ( V , F _ i ) , \\ ; i = 0 , \\dots , s , \\end{align*}"}
-{"id": "6048.png", "formula": "\\begin{align*} & \\Omega ^ { ( \\alpha , \\theta ) } ( Q _ { X Y U } ) \\\\ & = - \\log \\mathbb { E } _ { Q _ { X Y U } } \\Big [ \\exp \\big ( - \\theta \\omega _ { Q _ { X Y U } } ^ { ( \\alpha ) } ( X , Y | U ) \\big ) \\Big ] \\\\ & = - \\log \\sum _ { x , y , u } Q _ { X Y U } ^ { 1 - \\theta ( 1 + \\bar { \\alpha } ) } ( x , y , u ) \\left ( Q _ { U } ( u ) \\pi _ { X Y } ( x , y ) \\right ) ^ { \\theta } \\\\ & \\qquad \\times \\left ( Q _ { U | X Y } ( u | x , y ) Q _ { X | U } ( x | u ) Q _ { Y | U } ( y | u ) \\right ) ^ { \\theta \\bar { \\alpha } } \\end{align*}"}
-{"id": "5459.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { U ( z r ^ n ) } { ( z r ^ n ) ^ \\alpha \\ell ( r ^ n ) } = p ( z ) . \\end{align*}"}
-{"id": "937.png", "formula": "\\begin{align*} \\Delta _ { i , j } = - q \\sum _ { r = 1 } ^ { q - 1 } ( r - 1 ) ! \\binom { q - 1 } { r - 1 } ^ 2 I _ { 2 q - 2 r } ( f _ i \\widetilde { \\otimes } _ r f _ j ) . \\end{align*}"}
-{"id": "1791.png", "formula": "\\begin{align*} g ( t ) & : = 1 - \\frac { b ( q - 2 ) ( 6 - q ) } { a ( p - 2 ) ( 6 - p ) } t - \\frac { c ( r - 2 ) ( 6 - r ) } { a ( p - 2 ) ( 6 - p ) } t ^ { \\frac { r - p } { q - p } } \\\\ h ( t ) & : = 1 - \\frac { b ( q - 2 ) } { a ( p - 2 ) } t - \\frac { c ( r - 2 ) } { a ( p - 2 ) } t ^ { \\frac { r - p } { q - p } } . \\end{align*}"}
-{"id": "1495.png", "formula": "\\begin{align*} F ( x ) = T ^ { \\R ( x ) } ( x ) . \\end{align*}"}
-{"id": "4186.png", "formula": "\\begin{align*} g _ i ( m \\Lambda ) = m ^ { - 2 i } g _ i ( \\Lambda ) , \\ , i = 2 , 3 , m \\in \\mathbb { R } ^ * \\end{align*}"}
-{"id": "6407.png", "formula": "\\begin{align*} \\mathcal { S } \\left [ p | q \\right ] \\overset { } { = } - \\int d x p \\left ( x | \\theta \\right ) \\log \\left [ \\frac { p \\left ( x | \\theta \\right ) } { q \\left ( x \\right ) } \\right ] \\end{align*}"}
-{"id": "723.png", "formula": "\\begin{align*} \\Pi = \\frac { 1 } { 2 } \\left [ 1 + \\frac { \\theta _ - } { \\theta _ + } + \\frac { ( u _ - - u _ + ) ^ 2 } { a ^ 2 \\theta _ + } \\right ] \\pm \\sqrt { \\frac { 1 } { 4 } \\left [ 1 + \\frac { \\theta _ - } { \\theta _ + } + \\frac { ( u _ - - u _ + ) ^ 2 } { a ^ 2 \\theta _ + } \\right ] ^ { \\ ! 2 } - \\frac { \\theta _ - } { \\theta _ + } } . \\end{align*}"}
-{"id": "9968.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to 0 } \\bar { \\gamma } _ { k i } = \\rho _ { k i i } ^ { 2 } \\left ( \\frac { 1 } { \\tau } { \\displaystyle \\sum _ { j \\neq i } ^ { L } } { \\displaystyle \\sum _ { m = 1 } ^ { K _ { j } } \\rho _ { m j i } ^ { 2 } } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "4808.png", "formula": "\\begin{align*} \\mathsf { Z } ( x ) : = \\phi ^ { - 1 } ( x ) \\subseteq \\mathsf { Z } ( M ) \\end{align*}"}
-{"id": "6817.png", "formula": "\\begin{align*} - \\Delta \\tilde { g } _ j = \\frac { 2 } { | z _ { \\xi _ j } | ^ 3 } + 2 \\lambda ^ 2 \\textrm { i n } \\lambda ^ { - 1 } \\left ( \\mathbb { S } ^ 2 \\setminus \\Pi _ { \\xi _ j } ( B ( 0 , R _ 2 ) ) \\right ) , \\end{align*}"}
-{"id": "3036.png", "formula": "\\begin{align*} \\begin{cases} ( u _ { h , t } ( t ) , v _ h ) _ H + a ( u _ h ( t ) , v _ h ) + b ( v _ h , p _ h ( t ) ) = 0 , & \\forall v _ h \\in V _ h , \\\\ b ( u _ h ( t ) , q _ h ) = 0 , & \\forall q _ h \\in Q _ h , \\end{cases} \\end{align*}"}
-{"id": "258.png", "formula": "\\begin{align*} \\frac { 1 } { n ! } c _ { m , n } = \\frac { - 1 } { n ! ( m - n ) } c _ { m , n + 1 } = \\frac { d ! } { ( - 1 ) ^ { m - n } 2 ^ m n ! ( m - n ) ! ( d - m ) ! } = : \\widetilde { c } _ { m , n } \\end{align*}"}
-{"id": "5991.png", "formula": "\\begin{align*} f _ { \\mathrm { a } } ( x , y ) = \\cos \\left ( \\frac { ( x + y ) ^ 2 } { 2 . 8 8 } + \\frac { ( y - x ) ^ 2 } { 4 . 5 } \\right ) , x , y \\in [ - 1 , 1 ] \\end{align*}"}
-{"id": "1947.png", "formula": "\\begin{align*} | P _ { B } ( \\eta ) | \\le \\sum _ { j = 0 } ^ \\infty 2 ^ { j + 1 } \\chi _ { ( \\delta 2 ^ { j } , \\infty ) } ( < \\eta > ^ { s } ) \\int _ { | 1 - r | < 2 ^ { - j } } | { F _ r } ( \\eta ) | \\ , d r . \\end{align*}"}
-{"id": "947.png", "formula": "\\begin{align*} U _ { p , i } + Y _ i V _ { p , i } = Q _ p ( W ^ { ( i ) } ) , U _ { p , i } + G _ i V _ { p , i } = Q _ p ( W ^ { ( i - 1 ) } ) . \\end{align*}"}
-{"id": "3842.png", "formula": "\\begin{align*} T ' = \\lfloor T ^ \\epsilon \\rfloor , T '' = \\lfloor \\delta \\log T \\rfloor . \\end{align*}"}
-{"id": "4596.png", "formula": "\\begin{align*} \\Lambda _ { k , \\frac { \\pi } { 2 } } & = \\frac { \\pi } { 2 } \\Z \\ltimes _ \\varphi \\Gamma _ k , \\\\ \\Lambda _ { k , \\pi } & = \\pi \\Z \\ltimes _ \\varphi \\Gamma _ k , \\\\ \\Lambda _ { k , 2 \\pi } & = 2 \\pi \\Z \\ltimes _ \\varphi \\Gamma _ k . \\end{align*}"}
-{"id": "416.png", "formula": "\\begin{align*} \\Re [ ( \\lambda + i y ) \\coth ( \\lambda + i y ) ] = \\frac { \\lambda \\sinh ( 2 \\lambda ) + y \\sin ( 2 y ) } { 2 ( \\sinh ( \\lambda ) ^ 2 + \\sin ( y ) ^ 2 ) } > 0 . \\end{align*}"}
-{"id": "8748.png", "formula": "\\begin{align*} - \\frac { 1 } { M } \\Delta _ y - \\sum _ { i = 1 } ^ N \\Delta _ { x _ i } - g \\sum _ { i = 1 } ^ N \\delta ( x _ i - y ) , \\end{align*}"}
-{"id": "6146.png", "formula": "\\begin{align*} X : = \\{ ( \\vec { e } , a ) \\ , : \\ , a \\in A \\} \\cup \\{ ( \\vec { b } , e ' ) \\ , : \\ , b \\in B \\} . \\end{align*}"}
-{"id": "214.png", "formula": "\\begin{align*} \\chi ( A _ 1 ) = \\begin{bmatrix} 0 & 0 & \\dots & 0 \\\\ 0 & \\chi ( C _ 2 ) & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\dots & \\chi ( C _ N ) \\end{bmatrix} \\end{align*}"}
-{"id": "264.png", "formula": "\\begin{align*} \\ell ( N ) = N ^ { - k } \\cdot \\sqrt { c _ 0 | E | k N \\log \\frac { \\tau N } { c _ 1 k } } , \\end{align*}"}
-{"id": "4313.png", "formula": "\\begin{align*} \\begin{aligned} F _ { 0 } ( s , r ) & : = s \\log s + r \\log r + ( 1 - s - r ) \\log ( 1 - s - r ) , \\\\ F _ { 1 } ( s , r ) & : = \\frac { \\chi } { 2 } \\big ( r ( 1 - r ) + s ( 1 - s ) + ( 1 - r - s ) ( r + s ) \\big ) , \\end{aligned} \\end{align*}"}
-{"id": "5892.png", "formula": "\\begin{align*} \\begin{aligned} & P \\big ( \\cap _ { n = 1 } ^ { 2 M } \\big \\{ Z _ n ^ { N , i } \\in \\{ 0 , - 1 \\} \\big \\} \\big | \\cup _ { n = 1 } ^ M \\{ Z _ { 2 n } ^ { N , i } = - 1 \\} \\big ) \\ge ( 1 - b _ N ) ^ { 2 M } . \\end{aligned} \\end{align*}"}
-{"id": "8094.png", "formula": "\\begin{align*} c | | Y | | ^ 2 \\leq \\mathcal H \\rho ( z , w ) ( 0 , Y ) = \\tilde { \\mathcal H } \\rho ( z , w ) ( 0 , Y ) \\end{align*}"}
-{"id": "713.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } ( V _ + - V _ - ) ^ 2 = - ( p _ + - p _ - ) ( v _ + - v _ - ) , \\\\ H _ + - H _ - = \\frac { 1 } { 2 } ( p _ + - p _ - ) ( v _ + + v _ - ) . \\end{array} \\right . \\end{align*}"}
-{"id": "3976.png", "formula": "\\begin{align*} p ^ { \\alpha _ 2 } ( 2 , t ) = \\sum _ { k = 2 } ^ { \\infty } ( - \\lambda ) ^ k \\underset { \\Theta ^ { k } _ { 2 } } { \\sum } \\frac { t ^ { k _ 0 \\alpha _ 0 + k _ 1 \\alpha _ 1 + k _ 2 \\alpha _ 2 } } { \\Gamma \\left ( k _ 0 \\alpha _ 0 + k _ 1 \\alpha _ 1 + k _ 2 \\alpha _ 2 + 1 \\right ) } , \\end{align*}"}
-{"id": "1661.png", "formula": "\\begin{align*} \\tau ^ { e _ 2 } ( x ) = \\begin{cases} \\tau _ { d _ 0 } ^ { - 1 } ( x ) \\quad \\ ; \\ ; x \\in R _ { d _ 0 } \\\\ \\tau _ { d _ 1 } ^ { - 1 } ( x ) \\quad \\ ; \\ ; x \\in R _ { d _ 1 } \\\\ \\tau _ { b _ 0 } ^ { - 1 } ( x ) \\quad \\ ; \\ ; x \\in R _ { b _ 0 } \\\\ \\tau _ { b _ 1 } ^ { - 1 } ( x ) \\quad \\ ; \\ ; x \\in R _ { b _ 1 } \\end{cases} \\end{align*}"}
-{"id": "6227.png", "formula": "\\begin{align*} n \\cdot \\P ( \\tau ( k ) = n ) = k \\cdot \\P ( R ( n ) = k ) ~ , ~ n \\geq 1 ~ . \\end{align*}"}
-{"id": "2022.png", "formula": "\\begin{align*} \\sum _ { a = 1 } ^ l ( z _ 1 ^ { s - k _ a } g _ { a , s } \\Big ( \\frac { z _ 2 z _ 3 } { z _ 1 ^ 2 } \\Big ) f _ a ( z _ 2 , z _ 3 ) + z _ 2 ^ { s - k _ a } g _ { a , s } \\Big ( \\frac { z _ 3 z _ 1 } { z _ 2 ^ 2 } \\Big ) f _ a ( z _ 3 , z _ 1 ) + z _ 3 ^ { s - k _ a } g _ { a , s } \\Big ( \\frac { z _ 1 z _ 2 } { z _ 3 ^ 2 } \\Big ) f _ a ( z _ 1 , z _ 2 ) ) = 0 . \\end{align*}"}
-{"id": "2785.png", "formula": "\\begin{align*} { u } ( t _ { i } ) = g ( t _ { i } , { u } ( t _ { i } ) ) + \\int _ { 0 } ^ { t _ { i } } \\frac { 1 } { \\sqrt { t _ { i } - s } } k _ { 1 } ( t _ { i } , s , { u } ( t _ { i } ) , { u } ( s ) ) \\mathrm { d } s + \\int _ { 0 } ^ { t _ { i } } k _ { 2 } ( t _ { i } , s , { u } ( t _ { i } ) , { u } ( s ) ) \\mathrm { d } s , \\end{align*}"}
-{"id": "6267.png", "formula": "\\begin{align*} | I _ 5 | = & \\left | \\eta \\int _ { \\R ^ 3 } ( ( \\nabla \\times b ^ \\eta ) \\times b ^ \\eta ) \\cdot \\nabla \\times B \\ , d x \\right | \\\\ \\leq & C \\eta \\| b ^ \\eta \\| _ \\infty \\| \\nabla b ^ \\eta \\| _ 2 \\| \\nabla B \\| _ 2 \\\\ \\leq & C \\eta ^ 2 \\mu ^ { - 1 } \\| b ^ \\eta \\| _ \\infty ^ 2 \\| \\nabla b ^ \\eta \\| _ 2 ^ 2 + \\frac 1 { 4 } \\mu \\| \\nabla B \\| _ 2 ^ 2 \\end{align*}"}
-{"id": "7562.png", "formula": "\\begin{gather*} ( f ^ * _ { z ^ * } + i g ^ * _ { z ^ * } ) f _ { z z } + ( f _ z - i g _ z ) f ^ * _ { z ^ * z ^ * } = 0 , ( f ^ * _ { z ^ * } + i g ^ * _ { z ^ * } ) g _ { z z } + ( f _ z - i g _ z ) g ^ * _ { z ^ * z ^ * } = 0 . \\end{gather*}"}
-{"id": "10134.png", "formula": "\\begin{align*} \\boldsymbol S _ { D _ k } ( i ) \\in \\arg \\min _ { \\boldsymbol S _ { D _ k } ^ { \\rm o p t } \\in \\underline { \\boldsymbol S } _ { D _ k } ( i ) } { D } _ k \\bigg ( \\boldsymbol S _ { D _ k } ^ { \\rm o p t } , \\bar { \\boldsymbol \\omega } _ k ( i ) \\bigg ) \\ \\ \\ \\ \\textrm { f o r } \\ k = 1 , 2 , \\ldots , N , \\end{align*}"}
-{"id": "4903.png", "formula": "\\begin{align*} U = \\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix} \\end{align*}"}
-{"id": "3273.png", "formula": "\\begin{align*} r _ u ^ { } = \\frac { \\big | x _ u \\tan \\theta _ 0 - y _ u \\big | } { \\sqrt { 1 + \\tan ^ 2 \\theta _ 0 } } . \\end{align*}"}
-{"id": "5835.png", "formula": "\\begin{align*} \\phi ( x , v ) = x + v \\end{align*}"}
-{"id": "5602.png", "formula": "\\begin{align*} \\Psi ( Y , Y ' ) ( t ) : = \\biggl ( x _ 0 + \\int _ 0 ^ t f ( Y _ r ) d r + \\int _ 0 ^ t \\sigma ( Y _ r ) d \\mathbf { B } ^ H _ r , \\sigma ( Y _ t ) \\biggr ) \\end{align*}"}
-{"id": "1965.png", "formula": "\\begin{align*} \\gamma _ j & = - ( j - 1 ) + \\frac { ( n - 1 ) } { 2 } \\sum _ { \\ell = 1 } ^ { j - 1 } \\left ( \\frac { 1 } { t _ \\ell } - \\frac { 1 } { r _ \\ell } \\right ) \\\\ & = - ( j - 1 ) + \\frac { ( n - 1 ) } { 2 } \\sum _ { \\ell = 1 } ^ { j - 2 } \\left ( \\frac { 1 } { t _ \\ell } - \\frac { 1 } { r _ { \\ell + 1 } } \\right ) . \\end{align*}"}
-{"id": "9880.png", "formula": "\\begin{align*} \\lambda ^ { ( \\omega ) } _ { ( a , b , 0 ) } \\lambda ^ { ( \\omega ) } _ { ( c , d , 1 ) } & = \\exp \\left ( - \\frac { \\sqrt { - 1 } \\theta } { 2 } ( a d - b c ) \\right ) \\lambda ^ { ( \\omega ) } _ { ( a + c , b + d , 1 ) } , \\\\ \\lambda ^ { ( \\omega ) } _ { ( c , d , 1 ) } \\lambda ^ { ( \\omega ) } _ { ( a , b , 0 ) } & = \\exp \\left ( - \\frac { \\sqrt { - 1 } \\theta } { 2 } ( c a + a d + b d ) \\right ) \\lambda ^ { ( \\omega ) } _ { ( c - a - b , d + a , 1 ) } . \\end{align*}"}
-{"id": "9755.png", "formula": "\\begin{align*} Z _ A ^ { ( 0 ) } = Z _ { A , 0 } \\exp ( - \\frac { 1 } { \\rho _ { A , 0 } u _ { A , 0 } A ( 0 ) } \\int _ 0 ^ x A ( \\tau ) \\rho _ { A , 0 } \\phi ( T _ { A , 0 } ) d \\tau ) . \\end{align*}"}
-{"id": "7039.png", "formula": "\\begin{align*} \\eta _ 1 = \\frac { 1 } { 2 } ( Q + \\sigma _ n ) \\geq \\frac { 1 + Q ^ 2 / 2 } { 4 \\epsilon } . \\end{align*}"}
-{"id": "9405.png", "formula": "\\begin{align*} \\begin{aligned} & \\lambda > \\lambda _ 0 , \\\\ & \\Pr ( \\eta ^ { [ \\lambda ] } ( Q _ L , \\zeta ) \\in E ) \\leq 1 - p , \\end{aligned} \\end{align*}"}
-{"id": "8919.png", "formula": "\\begin{gather*} f ( \\vec { z } + \\vec { x } \\tau + \\vec { y } ) = \\prod _ { 1 \\le i \\le n } ( - 1 ) ^ { Q _ { i i } ( x _ i + y _ i ) } C _ i ^ { x _ i } \\prod _ { 1 \\le i , j \\le n } e ( - Q _ { i j } x _ i ( z _ j + x _ j \\tau / 2 ) ) f ( \\vec { z } ) , \\end{gather*}"}
-{"id": "1999.png", "formula": "\\begin{align*} \\sup _ { \\lambda / n \\leq s < t } \\frac { n ^ { \\eta } \\left \\vert \\widetilde { \\beta } _ { n } \\left ( s ; t \\right ) - \\widetilde { B } _ { n } \\left ( s \\right ) \\right \\vert } { s ^ { 1 / 2 - \\eta } } = O _ { \\mathbb { P } } \\left ( 1 \\right ) = \\sup _ { \\lambda / n \\leq s < t } \\frac { n ^ { \\nu } \\left \\vert \\widetilde { \\alpha } _ { n } \\left ( s ; t \\right ) - \\widetilde { B } _ { n } \\left ( s \\right ) \\right \\vert } { s ^ { 1 / 2 - \\nu } } . \\end{align*}"}
-{"id": "4818.png", "formula": "\\begin{align*} \\mathsf { n } ( r ) r ^ j - \\mathsf { n } ( r ) = \\mathsf { n } ( r ) ( r - 1 ) \\sum _ { i = 0 } ^ { j - 1 } r ^ i = r \\big ( \\mathsf { n } ( r ) - \\mathsf { d } ( r ) \\big ) \\sum _ { i = 0 } ^ { j - 1 } r ^ i = \\big ( \\mathsf { n } ( r ) - \\mathsf { d } ( r ) \\big ) \\sum _ { i = 0 } ^ { j - 1 } r ^ { i + 1 } \\in M _ r . \\end{align*}"}
-{"id": "2403.png", "formula": "\\begin{align*} p _ 3 ( x ) = a _ 3 + a _ 1 ^ 2 a _ 6 - ( a _ 5 + a _ 1 ( - a _ 3 + a _ 1 a _ 5 + 4 a _ 6 + 3 a _ 1 ^ 2 a _ 6 ) ) x . \\end{align*}"}
-{"id": "16.png", "formula": "\\begin{align*} \\hat { V } ^ { C } _ { \\sigma ' } ( C _ 1 , C _ 2 ) = \\int \\limits _ { - \\infty } ^ { \\infty } \\int \\limits _ { - \\infty } ^ { \\infty } \\ ! \\hat { f _ { \\sigma } } _ { X Y Z S } ( x , y , z , s ) \\ , \\mathrm { d } u _ { 1 } \\mathrm { d } u _ { 2 } \\Big | _ { x = y = u _ { 1 } , z = s = u _ { 2 } } \\end{align*}"}
-{"id": "7793.png", "formula": "\\begin{align*} \\lim _ { \\Vert \\xi _ h \\Vert _ X \\rightarrow \\infty } \\Vert F - B \\xi _ h \\Vert _ { Y _ h ^ * } = \\infty . \\end{align*}"}
-{"id": "3745.png", "formula": "\\begin{align*} g _ y = \\theta _ y g : = g \\circ \\theta _ y . \\end{align*}"}
-{"id": "3125.png", "formula": "\\begin{align*} \\tilde { K } : = \\frac { ( N - 1 ) ( 1 + \\mu _ f ) } { 2 \\mu _ f } \\log \\left ( \\frac { \\tilde { R } ^ 2 ( x ^ 0 ) } { \\rho \\epsilon } \\right ) . \\end{align*}"}
-{"id": "1993.png", "formula": "\\begin{align*} \\sup _ { \\lambda / n \\leq s < t } \\frac { n ^ { \\nu } \\left \\vert \\alpha _ { n } \\left ( s ; t \\right ) - B _ { n } \\left ( s \\right ) \\right \\vert } { s ^ { 1 / 2 - \\nu } } = O _ { \\mathbb { P } } \\left ( 1 \\right ) . \\bigskip \\end{align*}"}
-{"id": "7993.png", "formula": "\\begin{align*} | \\cup _ { u \\in B } X _ u | \\geq | A | ( | H _ 0 | - | P | ) + | A | ( | P | - 2 | A | ) = | A | | H _ 0 | - 2 | A | ^ 2 . \\end{align*}"}
-{"id": "9595.png", "formula": "\\begin{align*} \\big \\langle f , D _ { k ' + \\ell } D _ { k ' } ( R _ { N } ) ^ { m - 1 } D _ { k } ^ { N } D _ { k } ( g ) \\big \\rangle & = \\big \\langle D _ { k ' + \\ell } ( f ) , D _ { k ' } ( R _ { N } ) ^ { m - 1 } D _ { k } ^ { N } D _ { k } ( g ) \\big \\rangle \\\\ & = \\big \\langle D _ { k ' } D _ { k ' + \\ell } ( f ) , ( R _ { N } ) ^ { m - 1 } D _ { k } ^ { N } D _ { k } ( g ) \\big \\rangle . \\end{align*}"}
-{"id": "3883.png", "formula": "\\begin{align*} \\Psi _ { \\Gamma } ( x ) = x + O ( x ^ { 2 / 3 + \\epsilon } ) . \\end{align*}"}
-{"id": "5622.png", "formula": "\\begin{align*} \\ell = \\sum _ { i = 0 } ^ d ( a _ i - 2 ) + 1 = \\sum _ { i = 0 } ^ d a _ i - 2 d - 1 \\ , . \\end{align*}"}
-{"id": "697.png", "formula": "\\begin{align*} b _ n = | \\psi ( 0 , \\mu _ n ) | | \\dot { \\Psi } ( \\mu _ n ) | , \\end{align*}"}
-{"id": "2565.png", "formula": "\\begin{align*} \\cos \\gamma = \\frac { | p _ { 0 } - z | } { r } & \\le \\frac { 2 r ( 1 - d _ { 0 } ) + r d _ { 0 } - d _ { 0 } + 3 d _ { 0 } ^ { 2 } / 4 } { 2 r ( 1 - d _ { 0 } ) } \\\\ & = 1 - \\frac { d _ { 0 } ( 1 - r ) - 3 d _ { 0 } ^ { 2 } / 4 } { 2 r ( 1 - d _ { 0 } ) } < 1 - d _ { 0 } / 4 \\ , , \\end{align*}"}
-{"id": "7772.png", "formula": "\\begin{align*} \\sum _ { | \\ell | < R } \\sum _ { \\substack { \\rho \\in \\mathcal { N } ( \\ell ) - \\ell \\\\ | \\ell + \\rho | < R } } | D _ { \\rho } u _ { R } ( \\ell ) | ^ { 2 } = 0 . \\end{align*}"}
-{"id": "9717.png", "formula": "\\begin{align*} \\tilde { Z } = ( 1 - \\frac { \\phi ( T ) } { u } h ) Z . \\end{align*}"}
-{"id": "5312.png", "formula": "\\begin{align*} Z = \\partial _ t ( B ( \\theta ) ) \\quad \\mbox { i n } L ^ { \\ell ' } ( 0 , T ; ( V _ \\ell ( \\Omega ) ) ' ) . \\end{align*}"}
-{"id": "6510.png", "formula": "\\begin{align*} \\rho = \\frac { V } { \\mathcal { E } } = \\frac { 2 \\mu V } { \\hbar ^ { 2 } k _ { \\mathrm { o } } ^ { 2 } } \\approx \\frac { 3 \\sqrt { \\Sigma } } { 2 \\sqrt { \\pi } k _ { \\mathrm { o } } ^ { 2 } L ^ { 3 } } \\approx \\frac { 3 a _ { \\mathrm { s } } } { k _ { \\mathrm { o } } ^ { 2 } L ^ { 3 } } \\end{align*}"}
-{"id": "203.png", "formula": "\\begin{align*} F = \\{ \\emptyset , [ 0 ] , [ 1 ] , [ 0 , 1 ] \\} . \\end{align*}"}
-{"id": "2633.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\frac { \\log | r - \\tfrac 1 2 | } { \\phi _ 0 ( r ) } \\ , d r = - 2 \\pi \\log 2 . \\end{align*}"}
-{"id": "2882.png", "formula": "\\begin{align*} \\frac { \\cos ( a \\sin ( u ) + u v ) - e ^ { - a \\cos ( u ) } \\cos ( u v ) } { \\cosh ( a \\cos ( u ) ) - \\cos ( a \\sin ( u ) ) } = 2 \\textup { R e } \\left ( \\frac { e ^ { i u v } } { \\exp { \\left ( a e ^ { - i u } \\right ) } - 1 } \\right ) . \\end{align*}"}
-{"id": "6460.png", "formula": "\\begin{align*} d s ^ { 2 } = { \\displaystyle \\sum \\limits _ { j = 1 } ^ { 2 l } } \\frac { 1 } { \\sigma _ { j } ^ { 2 } } \\left ( d \\mu _ { j } ^ { 2 } + 2 d \\sigma _ { j } ^ { 2 } \\right ) , \\end{align*}"}
-{"id": "9460.png", "formula": "\\begin{align*} \\nu _ 1 ( z ; q ) = \\sum _ { n = 0 } ^ { \\infty } ( z q ; q ^ 2 ) _ n ( - q ) ^ n . \\end{align*}"}
-{"id": "2360.png", "formula": "\\begin{align*} \\delta _ { 0 , m } \\coloneqq \\begin{cases} 1 & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "7528.png", "formula": "\\begin{gather*} w ^ 1 _ 1 + w ^ 1 w ^ 1 _ 2 - w ^ 1 _ { 2 2 } - 2 w ^ 2 = 0 , \\\\ w ^ 2 _ 1 + w ^ 1 w ^ 2 _ 2 - w ^ 2 _ { 2 2 } + 2 w ^ 1 = 0 . \\end{gather*}"}
-{"id": "6582.png", "formula": "\\begin{align*} d ^ { * } ( f ( x ) , f ( y ) ) \\leq \\left ( \\max _ { j = 0 , 1 \\ldots , n } K _ j \\right ) \\cdot ( 1 + 2 \\kappa ^ \\alpha ) d ( x , y ) ^ \\alpha \\end{align*}"}
-{"id": "5799.png", "formula": "\\begin{align*} w _ 1 \\in L ^ { 1 + q } ( \\R ^ n , d \\sigma ) \\omega _ { 1 } : = { w _ { 1 } } ^ { q } d \\sigma + d \\mu \\in W ^ { - 1 , p ' } ( \\R ^ n ) . \\end{align*}"}
-{"id": "1294.png", "formula": "\\begin{align*} 2 \\sqrt { 6 } \\cos \\frac { 1 1 \\pi } { 3 6 } \\ + \\ 6 \\cos \\frac { 1 0 \\pi } { 3 6 } \\ = \\ \\left ( 3 \\sqrt { 2 } + \\sqrt { 6 } \\right ) \\cos \\frac { \\pi } { 3 6 } \\end{align*}"}
-{"id": "8330.png", "formula": "\\begin{align*} A = \\prod _ { \\substack { x \\in \\Z / N \\Z \\\\ x \\neq 0 } } \\left ( 1 - e ^ { 2 \\pi i x / N } \\right ) ^ { c ( 0 , x h ^ { - 1 } \\ell / N ) } . \\end{align*}"}
-{"id": "2114.png", "formula": "\\begin{align*} X _ t = X _ 0 + \\int _ 0 ^ t \\Theta ^ T \\phi ( X _ s ) d s + \\sigma W _ t , \\end{align*}"}
-{"id": "2884.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\sigma _ { k } ( n ) e ^ { - n y } = \\left ( \\frac { 2 \\pi } { y } \\right ) ^ { k + 1 } \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { \\frac { k + 1 } { 2 } } \\sigma _ { k } ( n ) e ^ { - \\frac { 4 \\pi ^ 2 n } { y } } + \\mathcal { P } ( y ) , \\end{align*}"}
-{"id": "5096.png", "formula": "\\begin{align*} g _ \\Gamma \\colon \\widehat { \\mathcal { L } } \\to [ 0 , + \\infty \\rangle , g _ \\Gamma ( \\tau ) : = \\frac { 1 } { \\mathop { \\textup { c a r d } } \\Gamma } \\Big | \\sum _ { k \\in \\Gamma } e ( k , \\tau ) \\Big | ^ 2 ; \\tau \\in \\widehat { \\mathcal { L } } \\end{align*}"}
-{"id": "3051.png", "formula": "\\begin{align*} a _ \\lambda ( u , v ) = \\lambda ( u , v ) _ H + a ( u , v ) \\end{align*}"}
-{"id": "4682.png", "formula": "\\begin{align*} d v _ { \\rm r a d } \\ = \\ ( V _ n ^ 2 ) ^ { \\frac { d - n } { 2 } } \\prod _ 1 ^ { \\frac { n ( n - 1 ) } { 2 } } \\ , d \\rho _ { i j } \\ , \\end{align*}"}
-{"id": "8410.png", "formula": "\\begin{align*} \\sum _ { x \\in \\mathbb { F } _ p ^ { \\times } } \\chi ( x ) \\exp \\left ( 2 \\pi i \\frac { a x + b x ^ { - 1 } } { p } \\right ) \\leftrightsquigarrow K _ { \\nu } ( x ) = \\frac { 1 } { 2 } \\int _ { 0 } ^ { \\infty } t ^ { - \\nu } \\exp ( - \\frac { x } { 2 } ( t + t ^ { - 1 } ) ) \\frac { d t } { t } . \\end{align*}"}
-{"id": "4385.png", "formula": "\\begin{align*} M _ { f \\circ \\tau _ { - z _ \\gamma } } ( A P _ \\alpha + Q _ \\alpha ) D _ \\gamma = M _ { f \\circ \\tau _ { - z _ \\gamma } } \\end{align*}"}
-{"id": "4606.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { C } _ { d , D } \\times [ 0 , 1 ) \\times \\mathbb { R } & \\to [ 0 , 1 ) \\times \\mathbb { R } \\\\ ( f , ( x , t ) ) & \\mapsto T _ f ( x , t ) \\end{aligned} \\end{align*}"}
-{"id": "2792.png", "formula": "\\begin{align*} \\beta _ { i } = ( - 1 ) ^ { i - d } \\sum _ { j \\in J _ { i } } \\binom { d } { i - j } , \\end{align*}"}
-{"id": "2126.png", "formula": "\\begin{align*} d ^ * ( f , g ) = s u p \\{ \\hat { d } ( f ( u ) , g ( v ) ) | ~ u , v \\in U \\} = s u p \\{ | f ( u ) - g ( v ) | ~ | u , v \\in U \\} . \\end{align*}"}
-{"id": "6989.png", "formula": "\\begin{align*} ( D f _ k ( \\Lambda ) ) _ { i i } = \\lim _ { \\epsilon \\to 0 } { \\frac { f _ k ( \\Lambda + \\epsilon E _ { i i } ) - f _ k ( \\Lambda ) } { \\epsilon } } > 0 . \\end{align*}"}
-{"id": "1029.png", "formula": "\\begin{align*} \\frac { ( n - 1 ) w } { w - n } - w + 1 = \\left ( \\frac { n - 1 } { w - n } \\right ) ^ { n } \\end{align*}"}
-{"id": "2573.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { k - 1 } ( b _ { m _ i n _ i } - b _ { m _ i n _ { i + 1 } } ) = 0 \\mod N , \\end{align*}"}
-{"id": "8691.png", "formula": "\\begin{align*} X _ { q ^ n } ( z ) : = \\frac { f ( 1 - q ^ n z ) } { \\sqrt { v ( 1 - q ^ n z ) } } , n \\in \\N , \\end{align*}"}
-{"id": "8472.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { t , l , v } ) = \\chi ( x _ 0 ) ^ 2 \\psi ( ( x _ 0 - b ) \\varpi ^ { - \\frac { k } { 2 } } ) \\end{align*}"}
-{"id": "7777.png", "formula": "\\begin{align*} e ( \\ell ) : = ( e _ \\rho ( \\ell ) ) _ { \\rho \\in \\L - \\ell } e _ \\rho ( \\ell ) : = \\left \\{ \\begin{array} { l l } S ^ * D _ \\rho S _ 0 u _ 0 ( \\ell ) , & \\ell \\in \\Omega _ { \\Gamma } , \\\\ D _ \\rho u _ 0 ( \\ell ) , & \\end{array} \\right . \\end{align*}"}
-{"id": "335.png", "formula": "\\begin{align*} \\phi ' ( x ) & = \\log \\prod _ { j \\in J } m _ j ( x ) ^ { \\tau _ j } \\prod _ { i \\in I } l _ i ( x ) ^ { - \\sigma _ i } , \\\\ \\phi '' ( x ) & = \\sum _ { j \\in J } \\dfrac { \\tau _ j ^ 2 } { m _ j ( x ) } - \\sum _ { i \\in I } \\dfrac { \\sigma _ i ^ 2 } { l _ i ( x ) } . \\end{align*}"}
-{"id": "7695.png", "formula": "\\begin{align*} & K ^ { \\top } \\Theta + \\Theta K = - V Z V ^ { \\top } \\\\ & = - \\left [ \\begin{array} { c | c | c } Z _ { 1 1 } & \\cdots & Z _ { 1 ( N - 1 ) } \\\\ \\hline \\vdots & \\ddots & \\vdots \\\\ \\hline Z _ { ( N - 1 ) 1 } & \\cdots & Z _ { ( N - 1 ) ( N - 1 ) } \\end{array} \\right ] \\ , , \\end{align*}"}
-{"id": "9580.png", "formula": "\\begin{align*} \\lvert u \\rvert _ { W ^ { s , p } ( \\R ^ n _ + ) } = \\biggl ( \\iint _ { \\R ^ n _ + \\times \\R ^ n _ + } \\frac { \\lvert u ( x ) - u ( y ) \\rvert ^ p } { \\lvert x - y \\rvert ^ { n + s p } } \\ , d y \\ , d x \\ , \\biggr ) ^ { 1 / p } \\ , . \\end{align*}"}
-{"id": "2708.png", "formula": "\\begin{align*} ( d _ { u } \\oplus d _ { v } ) ( x ) & = \\lambda _ { u } ( 2 e _ { \\emptyset } - 2 e _ { u } ) + 0 + \\lambda _ { \\emptyset } ( 2 e _ { u } - 2 e _ { \\emptyset } ) + \\mu _ { \\emptyset } ( f _ { v } - n f _ { \\emptyset } ) \\\\ & = ( 2 \\lambda _ { u } - 2 \\lambda _ { \\emptyset } ) e _ { \\emptyset } + ( 2 \\lambda _ { \\emptyset } - 2 \\lambda _ { u } ) e _ { u } + \\mu _ { \\emptyset } ( f _ { v } - n f _ { \\emptyset } ) , \\end{align*}"}
-{"id": "10079.png", "formula": "\\begin{align*} A _ { n } ( \\C ) = g \\mathfrak { a } _ n \\backslash \\mathfrak { a } _ { n \\C } / \\epsilon \\mathfrak { a } _ { n \\C } . \\end{align*}"}
-{"id": "9799.png", "formula": "\\begin{align*} \\lambda _ x [ A ] = \\theta \\circ \\kappa ( - A , x ^ { - 1 } ) [ \\pi ( x ^ { - t } A ) ] = \\tilde x [ A ] \\end{align*}"}
-{"id": "2369.png", "formula": "\\begin{align*} 1 = \\det ( \\chi ' _ 2 ( p _ 1 ) ) = \\det ( \\varrho ( p _ 1 ) ) = \\tau ^ 3 \\det ( \\chi ' _ 2 ( p _ 1 ) ) = \\tau ^ 3 , \\end{align*}"}
-{"id": "7203.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ N \\xi _ { k , p } \\geq c , \\end{align*}"}
-{"id": "409.png", "formula": "\\begin{align*} p _ { s , k _ 1 , k _ 2 } ( x , t ) = \\frac { 1 } { s ^ { n + m + k _ 1 + k _ 2 } } p _ { 1 , k _ 1 , k _ 2 } \\left ( \\frac { x } { \\sqrt { s } } , \\frac { t } { s } \\right ) \\end{align*}"}
-{"id": "8350.png", "formula": "\\begin{align*} \\mathrm { o b s t } _ x ^ { a n } | _ { \\widetilde { \\mathcal { S } } ( x ) } = \\mathrm { o b s t } _ x | _ { \\widetilde { \\mathcal { S } } ( x ) } , \\end{align*}"}
-{"id": "8754.png", "formula": "\\begin{align*} a _ p f ( P _ f ) = f ( P _ f + p ) a _ p \\textrm { a n d } a _ p g ( H _ f ) = g ( H _ f + p ^ 2 ) a _ p , \\end{align*}"}
-{"id": "9212.png", "formula": "\\begin{align*} [ z \\otimes \\alpha , u \\otimes b ] = z u \\otimes \\alpha b [ u ' \\otimes b ' , z \\otimes \\alpha ] = z ^ { t } u ' \\otimes b ' \\alpha . \\end{align*}"}
-{"id": "1499.png", "formula": "\\begin{align*} T _ { n } ( x ) = x ^ { 2 } T _ { n - 1 } ( x ) + x T _ { n - 2 } ( x ) + T _ { n - 3 } ( x ) \\end{align*}"}
-{"id": "2999.png", "formula": "\\begin{align*} u _ { n + 1 } ( t , x ) = & \\int _ F p ^ b _ t ( x , y ) u _ 0 ( y ) \\mu ( d y ) + \\int _ 0 ^ t \\int _ F p ^ b _ { t - s } ( x , y ) f ( s , u _ n ( s , y ) ) \\mu ( d y ) d s \\\\ & + \\int _ 0 ^ t \\int _ F p ^ b _ { t - s } ( x , y ) g ( s , u _ n ( s , y ) ) \\xi ( s , y ) \\mu ( d y ) d s . \\end{align*}"}
-{"id": "7324.png", "formula": "\\begin{align*} ( \\nu x u , \\nu y u ) ( \\nu y u , \\nu z u ) = ( \\nu x u , \\nu z u ) x , y = x y ' , z = y z ' \\in P , u \\in U . \\end{align*}"}
-{"id": "7717.png", "formula": "\\begin{align*} u _ { 0 } & = \\frac { 1 } { \\sqrt { N } } ( 1 , 1 , 1 , \\cdots , 1 , 1 , 1 ) ^ \\top \\ , \\\\ u _ { n } & = \\frac { 1 } { \\sqrt { n ( n + 1 ) } } ( 0 , \\underbrace { - 1 , \\cdots , - 1 } _ , n , 0 , 0 , \\cdots , 0 ) ^ \\top , \\\\ & n = 1 , 2 , \\cdots , N - 2 \\ , , \\\\ u _ { N - 1 } & = \\frac { 1 } { \\sqrt { N ( N - 1 ) } } ( 1 - N , 1 , \\cdots , 1 , 1 ) ^ \\top \\ , . \\end{align*}"}
-{"id": "1087.png", "formula": "\\begin{align*} m _ { s , i } = { \\mu \\rho _ i ^ 2 \\mid \\alpha _ { i i } \\mid ^ 2 } \\end{align*}"}
-{"id": "2959.png", "formula": "\\begin{align*} t = x + \\sum _ { e ' \\ni i , e ' \\neq e } \\left ( 1 - \\sum _ { j \\in e ' , j \\neq i } \\theta _ \\epsilon ^ { b _ x } ( e ' , j ) \\right ) . \\end{align*}"}
-{"id": "9062.png", "formula": "\\begin{align*} B _ { { \\mathbf x } } ( w ) = \\displaystyle \\frac { 1 } { 2 } \\left ( b _ { 1 1 } ^ 1 { \\mathbf x } _ 1 w _ 1 + b _ { 1 2 } ^ 1 { \\mathbf x } _ 1 w _ 2 + b _ { 1 3 } ^ 1 { \\mathbf x } _ 1 w _ 3 , 0 , 0 \\right ) . \\end{align*}"}
-{"id": "3163.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ e ^ { u X _ { t } ^ { x } } \\right ] = \\mathbb { E } \\left [ e ^ { u Y _ { t } ^ { x } } \\right ] \\mathbb { E } \\left [ e ^ { u Z _ { t } } \\right ] \\end{align*}"}
-{"id": "3445.png", "formula": "\\begin{align*} C : = h ( \\omega , \\cdot ) ^ { - 1 } ( B _ { \\varepsilon } ( 0 ) ) \\end{align*}"}
-{"id": "6103.png", "formula": "\\begin{align*} \\lambda _ { 1 , 2 } = 1 - n \\pm \\sqrt { n ^ 2 + c ^ 2 } . \\end{align*}"}
-{"id": "8003.png", "formula": "\\begin{align*} & \\sup _ { P \\in \\mathcal { D } _ { \\mu } } \\Big ( \\frac { 1 } { | P | } \\int _ P \\sum _ { k = \\mu } ^ { \\infty } { 2 ^ { s k q } \\big | \\Pi _ k \\mathfrak { S } _ { [ a ] } ^ { f a r } f ( x ) \\big | ^ q } d x \\Big ) ^ { 1 / q } \\\\ & \\lesssim \\sup _ { 0 \\leq k \\leq \\mu - 1 } { \\Vert 2 ^ { k ( s + m ) } \\Pi _ k f \\Vert _ { L ^ { \\infty } } } + \\sup _ { R \\in \\mathcal { D } _ { \\mu } } \\Big ( \\frac { 1 } { | R | } \\int _ R \\sum _ { k = \\mu } ^ { \\infty } { 2 ^ { ( s + m ) k q } \\big | \\Pi _ k f ( x ) \\big | ^ q } d x \\Big ) ^ { 1 / q } \\end{align*}"}
-{"id": "9336.png", "formula": "\\begin{align*} \\overline { p } = R _ { p } - I _ { p } = a _ { 0 } - \\sum _ { s = 1 } ^ { 7 } a _ { s } e _ { s } \\end{align*}"}
-{"id": "1670.png", "formula": "\\begin{align*} \\alpha _ i \\ ; = \\begin{cases} p _ i = \\frac { 1 } { 2 } + \\gamma _ i & \\ ; g _ i = f _ 1 , \\\\ q _ i = \\frac { 1 } { 2 } - \\gamma _ i \\ ; & \\ ; g _ i = f _ 2 , \\ ; 1 \\leq i \\leq n . \\end{cases} \\end{align*}"}
-{"id": "4147.png", "formula": "\\begin{align*} \\max _ { z \\geq 0 } \\biggl \\{ \\sum _ { i \\in N } \\sum _ { j \\in V } \\sum _ { t \\in [ n ] } w _ { i j } ^ t z _ { i j } ^ t : \\eqref { e q : p r o b _ b o u n d } , \\eqref { e q : i n e q _ J s i z e _ 1 } \\biggr \\} , \\end{align*}"}
-{"id": "4796.png", "formula": "\\begin{align*} x _ 1 = { \\frac { z _ { { 5 } } } { z _ { { 1 } } + 2 \\ , z _ { { 2 } } } } , x _ 2 = { \\frac { 2 \\ , z _ { { 2 } } z _ 5 } { \\left ( z _ { { 1 } } + 2 \\ , z _ { { 2 } } \\right ) z _ 3 } } , x _ 3 = { \\frac { z _ { { 2 } } z _ 5 } { \\left ( z _ { { 1 } } + 2 \\ , z _ { { 2 } } \\right ) z _ { { 4 } } } } . \\end{align*}"}
-{"id": "3599.png", "formula": "\\begin{align*} D _ { * 0 } ^ \\alpha y ( x ) = f ( x , y ( x ) ) , y ( 0 ) = y _ 0 , \\end{align*}"}
-{"id": "6219.png", "formula": "\\begin{align*} \\P ( R ( i ) > 0 ~ \\textrm { f o r e a c h $ 1 \\leq i \\leq n $ } ~ | ~ R ( n ) = k ) = \\frac { k } { n } ~ . \\end{align*}"}
-{"id": "9232.png", "formula": "\\begin{align*} [ x ^ { + } , y ^ { + } ] \\otimes a ^ { + } & = x ^ { + } \\circ y ^ { + } \\otimes \\frac { [ 1 ^ { - } , a ^ { - } ] _ { A ^ { - } } } { 2 } + [ x ^ { + } , y ^ { + } ] \\otimes \\frac { ( 1 ^ { - } \\circ a ^ { - } ) _ { A ^ { + } } } { 2 } + ( x ^ { + } \\mid y ^ { + } ) \\langle 1 ^ { - } , a ^ { - } \\rangle \\\\ { } [ x ^ { + } , y ^ { - } ] \\otimes a ^ { - } & = . x ^ { + } \\diamond y ^ { - } \\otimes \\frac { [ 1 ^ { - } , a ^ { + } ] _ { A ^ { + } } } { 2 } + [ x ^ { + } , y ^ { - } ] \\otimes \\frac { ( 1 ^ { - } \\circ a ^ { + } ) _ { A ^ { - } } } { 2 } \\end{align*}"}
-{"id": "9039.png", "formula": "\\begin{align*} G ( \\Omega _ 1 , v | w | ^ 2 ) = \\left \\{ \\left ( \\begin{array} { c c } \\rho & 0 \\\\ 0 & \\rho \\end{array} \\right ) , \\ , \\ , \\left ( \\begin{array} { c c } 0 & \\displaystyle \\eta \\frac { v _ 1 } { v _ 2 } \\\\ \\displaystyle \\eta \\frac { v _ 2 } { v _ 1 } & 0 \\end{array} \\right ) \\ , \\ , \\hbox { w i t h } \\ , \\ , \\rho , \\ , \\eta > 0 \\right \\} , \\end{align*}"}
-{"id": "4548.png", "formula": "\\begin{align*} & { \\rm p d e g } \\ , e _ i = - { \\rm p d e g } \\ , f _ i = 1 ( 1 \\le i \\le n - 1 ) , & { \\rm p d e g } \\ , q ^ h = 0 ( h \\in P ) , \\\\ & { \\rm p d e g } \\ , e _ 0 ^ { ( l ) } = - { \\rm p d e g } \\ , f _ 0 ^ { ( l ) } = 1 , & { \\rm p d e g } \\ , Z _ r = n r ( r \\neq 0 ) \\ , . \\end{align*}"}
-{"id": "9739.png", "formula": "\\begin{align*} U _ { h , 0 } ( y ) = \\frac { 1 } { 2 s } \\int _ { 2 n s } ^ { ( 2 n + 2 ) s } U _ 0 ( y ) d y . \\end{align*}"}
-{"id": "5818.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } \\vert \\nabla \\lambda _ { t } \\vert ^ { p } \\ ; d x & \\leq ( 1 - t ) \\int _ { \\R ^ n } \\vert \\nabla v \\vert ^ { p } \\ ; d x + t \\int _ { \\R ^ n } \\vert \\nabla u \\vert ^ { p } \\ ; d x \\\\ & = t \\left ( \\int _ { \\R ^ n } \\vert \\nabla u \\vert ^ { p } \\ ; d x - \\int _ { \\R ^ n } \\vert \\nabla v \\vert ^ { p } \\ ; d x \\right ) + \\int _ { \\R ^ n } \\vert \\nabla v \\vert ^ { p } \\ ; d x . \\end{align*}"}
-{"id": "4332.png", "formula": "\\begin{align*} ( z + { \\bf y } _ { \\leq \\max ( A ) } ^ { A } ) \\cdot t _ { A } ( { \\bf x } , { \\bf y } ; z ) = z \\cdot s _ { A } ( { \\bf x } , { \\bf y } ; z ) . \\end{align*}"}
-{"id": "7374.png", "formula": "\\begin{align*} L _ { \\omega } \\psi = - ( - 1 ) ^ p \\frac { p } { n } \\omega . \\displaystyle { \\not } D \\psi + \\frac { p } { 2 ( p + 1 ) } d \\omega . \\psi + \\frac { p } { 2 ( n - p + 1 ) } \\delta \\omega . \\psi . \\end{align*}"}
-{"id": "2428.png", "formula": "\\begin{align*} 2 a _ 1 a _ 2 + 4 a _ 3 + a _ 4 = 0 , \\\\ 1 2 a _ 2 + 3 a _ 5 - a _ 1 ( 6 a _ 3 + 2 a _ 4 ) = 0 , \\\\ 8 a _ 3 + 3 a _ 4 + 4 a _ 6 + a _ 1 ( 6 a _ 2 + 2 a _ 5 ) , \\\\ 4 a _ 2 + a _ 5 - 2 a _ 1 a _ 6 = 0 . \\end{align*}"}
-{"id": "931.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ d E \\left [ \\frac { \\partial \\varphi } { \\partial x _ j } ( \\sqrt { t } F + \\sqrt { 1 - t } Z ) \\frac { F _ j } { \\sqrt { t } } \\right ] = \\sum _ { i , j = 1 } ^ d E \\left [ \\frac { \\partial ^ 2 \\varphi } { \\partial x _ i \\partial x _ j } ( \\sqrt { t } F + \\sqrt { 1 - t } Z ) \\langle D F _ i , - D L ^ { - 1 } F _ j \\rangle _ H \\right ] . \\end{align*}"}
-{"id": "577.png", "formula": "\\begin{align*} H _ { F } ^ { \\infty } ( \\Omega ) = \\{ f \\in H ( \\Omega ) : f ^ { ( l ) } \\in H ^ { \\infty } ( \\Omega ) , \\ \\ l \\in F \\} . \\end{align*}"}
-{"id": "9720.png", "formula": "\\begin{align*} \\tilde { V } = V + \\widetilde { \\mathcal { V } } ( V , Z h ) Z h . \\end{align*}"}
-{"id": "8158.png", "formula": "\\begin{align*} a _ 1 a _ 2 ^ q a _ 5 ^ { q ^ 3 } + a _ 3 a _ 7 ^ q a _ 6 ^ { q ^ 3 } = 0 . \\end{align*}"}
-{"id": "7691.png", "formula": "\\begin{align*} \\mathcal { H } ^ 2 _ 2 = \\int _ { 0 } ^ { \\infty } B ^ { \\top } \\mathrm { e } ^ { - M ^ { \\top } t } Z \\mathrm { e } ^ { - M t } B \\mathrm { d } t \\ , , \\end{align*}"}
-{"id": "2260.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } \\frac { 1 } { ( k ^ 2 + a ^ 2 ) ^ 2 } = \\frac { - 1 } { 2 a ^ 4 } + \\frac { \\pi } { 4 a ^ 3 } \\coth ( \\pi a ) + \\frac { \\pi ^ 2 } { 4 a ^ 2 } \\cdot \\frac 1 { { \\sinh } \\ , ^ 2 ( \\pi a ) } \\end{align*}"}
-{"id": "9072.png", "formula": "\\begin{align*} z ^ 2 _ 1 = x _ 1 + b _ 1 z _ 2 , z _ 2 ^ 2 = x _ 2 + b _ 2 z _ 1 . \\end{align*}"}
-{"id": "3806.png", "formula": "\\begin{align*} X _ - : = \\left \\{ \\begin{array} { l l } \\hat { Z } - \\ell _ L & \\hat { Z } - \\ell _ L \\in 2 \\Z , \\\\ \\hat { Z } - \\ell _ L + 1 & \\end{array} \\right . \\end{align*}"}
-{"id": "3568.png", "formula": "\\begin{align*} \\left | \\psi _ { \\chi } ( x ) \\right | = \\left | \\underset { n \\leq x } \\sum \\chi ( n ) \\Lambda ( n ) \\right | \\ll x . \\end{align*}"}
-{"id": "911.png", "formula": "\\begin{align*} F = f ( W ( h _ 1 ) , \\dots , W ( h _ m ) ) , \\end{align*}"}
-{"id": "4407.png", "formula": "\\begin{align*} \\sigma _ { e s s } ( T _ f ) = f ( \\partial \\mathbb { C } ^ n ) . \\end{align*}"}
-{"id": "9136.png", "formula": "\\begin{align*} f ( x ) = D _ { 2 } ( x ) + f ( 1 ) x \\left ( x \\in R \\right ) . \\end{align*}"}
-{"id": "2236.png", "formula": "\\begin{align*} \\int _ { \\Gamma } \\frac { P ( z ) d z } { f _ 1 ( z ) \\ldots f _ n ( z ) } = 0 \\end{align*}"}
-{"id": "9547.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - L ^ { 1 / m } _ { + } \\otimes L ^ { 1 / m } _ { - } } & = \\frac { 1 } { 1 - L ^ { 1 / m } _ { - } - L ^ { 1 / m } _ { - } \\otimes ( L ^ { \\ 1 / m } _ { + } - 1 ) } \\\\ & = \\sum _ { k \\geq 0 } \\frac { L _ { - } ^ { k / m } } { ( 1 - L ^ { 1 / m } _ { - } ) ^ { k + 1 } } \\otimes ( L ^ { 1 / m } _ { + } - 1 ) ^ k . \\end{align*}"}
-{"id": "8436.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { - 2 , l , v } ) = q ^ { - 1 } G ( \\varpi ^ { - l } , \\chi ) \\chi ^ { - 1 } ( v ) + \\sum _ { \\substack { \\mu \\in \\mathfrak { X } _ l , \\\\ a ( \\mu \\chi ) = 1 } } \\epsilon ( \\frac { 1 } { 2 } , \\mu ^ { - 1 } \\chi ^ { - 1 } ) ^ 2 G ( \\varpi ^ { - l } , \\mu ^ { - 1 } ) \\mu ( v ) . \\end{align*}"}
-{"id": "29.png", "formula": "\\begin{align*} | C _ 1 | ^ 2 + | C _ 2 | ^ 2 - 2 | C _ 1 | | C _ 2 | ( c o s ( \\theta ) c o s ( \\phi ) + s i n ( \\theta ) s i n ( \\phi ) ) = | C _ 1 | ^ 2 + | C _ 2 | ^ 2 - 2 | C _ 1 | | C _ 2 | c o s ( \\theta - \\phi ) \\end{align*}"}
-{"id": "5231.png", "formula": "\\begin{align*} \\Omega ( Y _ 1 , Y _ 2 ) = \\lim \\limits _ { z \\rightarrow - \\infty } e ^ { c z } \\omega ( Y _ 1 , Y _ 2 ) = \\lim \\limits _ { z \\rightarrow - \\infty } \\omega ( e ^ { c z } Y _ 1 , Y _ 2 ) = 0 . \\end{align*}"}
-{"id": "5022.png", "formula": "\\begin{align*} G ^ { \\omega , \\lambda } _ { p , p , p } ( z ) & = G ^ \\omega _ { p , p } ( z ) ( I + \\lambda C _ p G ^ \\omega _ { p , p } ( z ) ) ^ { - 1 } , \\\\ G ^ { \\omega , \\lambda } _ { p , n , m } ( z ) & = G ^ \\omega _ { n , m } ( z ) - \\lambda G ^ \\omega _ { n , p } ( z ) ( I + \\lambda C _ p G ^ \\omega _ { p , p } ( z ) ) ^ { - 1 } C _ p G ^ \\omega _ { p , m } ( z ) . \\end{align*}"}
-{"id": "4572.png", "formula": "\\begin{align*} ( 1 - q _ 1 q _ 3 ^ { - 1 } ) F ^ { ( 1 ) } _ { 0 | 1 , l } & = \\sum _ { j \\ge 0 } q _ 1 ^ { - j - l } \\Bigl ( F _ { 1 , - j } F _ { 0 , l + j } - ( q _ 1 ^ { - 1 } + q _ 3 ^ { - 1 } ) F _ { 1 , - j - 1 } F _ { 0 , l + j + 1 } + q _ 1 ^ { - 1 } q _ 3 ^ { - 1 } F _ { 1 , - j - 2 } F _ { 0 , l + j + 2 } \\Bigr ) \\\\ & + \\sum _ { j \\ge 1 } q _ 1 ^ { j - l } \\Bigl ( q _ 1 ^ { - 1 } q _ 3 ^ { - 1 } F _ { 0 , l - j } F _ { 1 , j } - ( q _ 1 ^ { - 1 } + q _ 3 ^ { - 1 } ) F _ { 0 , l - j + 1 } F _ { 1 , j - 1 } + F _ { 0 , l - j + 2 } F _ { 1 , j - 2 } \\Bigr ) \\ , , \\end{align*}"}
-{"id": "5376.png", "formula": "\\begin{align*} \\frac { S ^ { ( \\ell ) } _ { a \\omega , r \\omega } } { S ^ { ( \\ell ) } _ { 0 , r \\omega } } = ( - 1 ) ^ { ( r + 1 ) b + 1 } . \\end{align*}"}
-{"id": "3000.png", "formula": "\\begin{align*} \\kappa ( \\alpha ) = \\frac { \\alpha } { 1 + \\alpha ^ \\frac { d _ s } { 2 } \\Gamma ( 1 - d _ s / 2 ) } \\end{align*}"}
-{"id": "3243.png", "formula": "\\begin{align*} \\mathcal A _ k ( X ) = \\sum _ { p = - \\infty } ^ { + \\infty } A ^ { p } H ^ { 2 p + k } ( X ) . \\end{align*}"}
-{"id": "8650.png", "formula": "\\begin{align*} \\pi [ A _ k ] \\leq C \\frac { k ^ { \\alpha d } } { k ^ { { d } / { 2 } } } = \\frac { C } { k ^ { ( \\frac 1 2 - \\alpha ) d } } \\end{align*}"}
-{"id": "7758.png", "formula": "\\begin{align*} \\big | \\partial _ { y ( n _ 1 ) } \\cdots \\partial _ { y ( n _ j ) } \\Phi _ \\ell ( y ) \\big | \\leq C \\exp \\big ( - \\gamma \\sum _ { i = 1 } ^ j | y _ { n _ i } - y _ \\ell | \\big ) , \\end{align*}"}
-{"id": "2703.png", "formula": "\\begin{align*} \\phi _ { 1 } ( x ) + \\phi _ 2 ( x ) = \\phi _ 1 ( x + 1 ) \\phi _ 2 ( x + 1 ) - \\phi _ 1 ( x ) \\phi _ 2 ( x ) - 1 \\in E ^ { * } _ 0 , \\end{align*}"}
-{"id": "2322.png", "formula": "\\begin{align*} p _ k g . z = g . z + k , \\end{align*}"}
-{"id": "1107.png", "formula": "\\begin{align*} \\zeta _ i = \\frac { \\mu P _ t \\left ( \\sum _ { j \\in \\mathcal { M } _ i } \\beta _ j ^ { ( i ) } | \\alpha _ { i j } | ^ 2 \\right ) ^ 2 } { N \\sum _ { j \\in \\mathcal { M } _ i } \\left ( \\beta _ j ^ { ( i ) } \\right ) ^ 2 \\left [ | \\alpha _ { i j } | ^ 2 \\sigma _ { Z _ { s , j } ^ { ( i ) } } ^ 2 + \\frac { N } { \\mu P _ t } \\sigma _ { Z _ { 0 , j } ^ { ( i ) } } ^ 2 \\right ] } , \\end{align*}"}
-{"id": "2414.png", "formula": "\\begin{align*} a _ 3 = \\frac { - a _ 2 ^ 2 } { a _ 6 } , a _ 5 = \\frac { a _ 2 ( a _ 4 a _ 6 + 3 a _ 2 - 3 a _ 2 ^ 2 ) } { a _ 6 ^ 2 } , a _ 4 = \\frac { 3 a _ 2 ^ 2 - a _ 6 ^ 2 } { a _ 6 } . \\end{align*}"}
-{"id": "366.png", "formula": "\\begin{align*} \\sum _ { r , s \\in \\mathbb { Z } } b _ { n , r , s } ^ 2 = 1 \\ \\ \\rho _ n ^ 2 = \\max _ { r , s \\in \\mathbb { Z } } b _ { n , r , s } ^ { 2 } \\rightarrow 0 \\ \\ n \\rightarrow \\infty . \\end{align*}"}
-{"id": "10093.png", "formula": "\\begin{align*} ( \\tilde { \\nabla } _ { X } \\pi ) ( Y ) = - \\frac { n - 1 } { n + 1 } \\pi ( X ) \\pi ( Y ) \\end{align*}"}
-{"id": "5066.png", "formula": "\\begin{align*} & R _ { i j i j } = \\rho ^ 2 \\tilde { R } _ { i j i j } + \\rho \\rho _ { i i } + \\rho \\rho _ { j j } - | \\nabla \\rho | ^ 2 , i \\neq j \\\\ & R _ { i j i k } = \\rho ^ 2 \\tilde { R } _ { i j i k } + \\rho \\rho _ { j k } , ~ ~ w h e n ~ \\{ i , j , k \\} ~ a r e ~ d i s t i n c t , \\\\ \\end{align*}"}
-{"id": "607.png", "formula": "\\begin{align*} \\dim L _ q ( m ) = \\sum _ r b _ r c _ { r , m } . \\end{align*}"}
-{"id": "2641.png", "formula": "\\begin{align*} w ( b ) : = b - \\sum _ { i = 1 } ^ n \\left \\lfloor \\frac { q _ i b } { 1 + q _ 1 + \\cdots + q _ n } \\right \\rfloor \\ , . \\end{align*}"}
-{"id": "6012.png", "formula": "\\begin{align*} T _ { 1 + s } ( \\pi _ { X Y } ) & = \\widetilde { T } _ { 1 + s } ( \\pi _ { X Y } ) \\\\ & = \\begin{cases} C _ { \\mathsf { W y n e r } } ( X ; Y ) & s \\in ( - 1 , 1 ] \\\\ 0 & s = - 1 \\end{cases} . \\end{align*}"}
-{"id": "6511.png", "formula": "\\begin{align*} \\frac { V } { L ^ { 3 } } = \\frac { 4 \\hbar ^ { 2 } k _ { \\mathrm { o } } ^ { 4 } \\left ( 2 k _ { \\mathrm { o } } ^ { 2 } + \\sigma _ { k \\mathrm { o } } ^ { 2 } \\right ) R _ { \\mathrm { o } } } { 3 \\mu } , \\end{align*}"}
-{"id": "4285.png", "formula": "\\begin{align*} H ^ { \\ast } ( X ) = H ^ { \\ast } _ { + } \\oplus H ^ { \\ast } _ - \\oplus H ^ { \\ast } _ { \\perp } \\end{align*}"}
-{"id": "6697.png", "formula": "\\begin{align*} a _ { \\delta } \\geq \\frac { 1 - c _ 4 \\delta ^ { \\alpha \\frac { 2 } { n + 1 } } ( 1 + s _ 4 ( \\delta ' ) ) } { 1 - c _ 2 \\delta ^ { \\frac { 2 } { n + 1 } } ( 1 + s _ 2 ( \\delta ) ) } = 1 + c _ 2 \\delta ^ { \\frac { 2 } { n + 1 } } ( 1 + o _ { \\delta } ( 1 ) + o _ { \\delta ' } ( 1 ) ) \\quad . \\end{align*}"}
-{"id": "2033.png", "formula": "\\begin{align*} f ( z _ 1 , . . . , z _ k ) = \\sum _ { \\substack { \\alpha _ 1 , . . . , \\alpha _ k \\in \\Z , \\\\ \\alpha _ 1 \\leq . . . \\leq \\alpha _ k , \\\\ \\alpha _ 1 + . . . + \\alpha _ k = \\frac { k ( k - 1 ) } { 2 } \\\\ \\sigma \\in S _ k } } a _ { \\alpha _ 1 , . . . , \\alpha _ k } z _ { \\sigma ( 1 ) } ^ { \\alpha _ 1 + 1 } z _ { \\sigma ( 2 ) } ^ { \\alpha _ 2 + 1 } . . . z _ { \\sigma ( k ) } ^ { \\alpha _ k + 1 } \\end{align*}"}
-{"id": "1001.png", "formula": "\\begin{align*} \\widehat { z } _ n ( t ) = \\frac { \\widehat { \\sigma } ^ 2 _ n ( t ) - \\sigma ^ 2 ( t ) } { \\widehat { \\mathfrak { s } } _ n ( t ) } , t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "6873.png", "formula": "\\begin{align*} \\Phi _ 1 ( \\lambda _ { m } ) & = 2 - \\sqrt { 2 + 2 \\sqrt { 1 - p \\lambda _ { m } + p ^ 2 \\lambda _ { m } } } \\\\ \\Phi _ 2 ( \\lambda _ { m } ) & = 2 - \\sqrt { 2 - 2 \\sqrt { 1 - p \\lambda _ { m } + p ^ 2 \\lambda _ { m } } } \\\\ \\Phi _ 3 ( \\lambda _ { m } ) & = 2 + \\sqrt { 2 - 2 \\sqrt { 1 - p \\lambda _ { m } + p ^ 2 \\lambda _ { m } } } \\\\ \\Phi _ 4 ( \\lambda _ { m } ) & = 2 + \\sqrt { 2 + 2 \\sqrt { 1 - p \\lambda _ { m } + p ^ 2 \\lambda _ { m } } } \\end{align*}"}
-{"id": "3515.png", "formula": "\\begin{align*} J ^ { n } ( u ^ { n } ( \\cdot ) ) \\leq J ^ { n } ( \\bar { u } ( \\cdot ) ) = \\frac { 1 } { n } , \\ \\tilde { d } ( u ^ { n } ( \\cdot ) , \\bar { u } ( \\cdot ) ) \\leq \\sqrt { \\frac { 1 } { n } } , \\end{align*}"}
-{"id": "9134.png", "formula": "\\begin{align*} f _ { n + 1 - i } = ( - 1 ) ^ { i } \\sum _ { k = 0 } ^ { i } \\binom { n + 1 - i + k } { k } D _ { n - i + k } \\left ( i = 0 , \\ldots , n \\right ) , \\end{align*}"}
-{"id": "8884.png", "formula": "\\begin{align*} T = S _ \\omega + ( \\cdot , \\overline \\chi \\omega ) u . \\end{align*}"}
-{"id": "7459.png", "formula": "\\begin{align*} \\Phi : = a \\beta a ^ * + a \\gamma a ^ * - \\beta ^ 4 - \\gamma ^ 4 - ( - \\beta - \\gamma ) ^ 3 . \\end{align*}"}
-{"id": "5537.png", "formula": "\\begin{align*} z \\left ( t \\right ) = \\Phi \\left ( t _ { 1 } , 0 \\right ) M ^ { k } z \\left ( 0 \\right ) \\end{align*}"}
-{"id": "6824.png", "formula": "\\begin{align*} \\frac { 4 } { ( 1 + | x _ { \\xi _ j } | ) ^ 2 } L ( \\eta _ { R _ 3 , \\xi _ j } \\varphi _ { 0 , j } ) = O ( \\lambda ^ 2 ) + \\lambda \\left [ O \\left ( \\frac { r } { ( 1 + r ^ 2 ) ^ 2 } \\right ) + O \\left ( \\frac { 1 } { ( 1 + r ^ 2 ) ^ 2 } \\right ) \\right ] . \\end{align*}"}
-{"id": "5397.png", "formula": "\\begin{align*} | x ^ K \\cup y ^ K \\cup z ^ K \\cup ( x y ) ^ K \\cup ( x z ) ^ K \\cup ( ( y z ) ^ 2 ) ^ K | = 1 0 \\end{align*}"}
-{"id": "9884.png", "formula": "\\begin{align*} \\rho ( K _ t ) = \\lim _ { \\delta \\to 0 } f _ t ( \\delta ) \\ , . \\end{align*}"}
-{"id": "9017.png", "formula": "\\begin{align*} \\left \\{ \\aligned & \\partial _ { t } \\theta ^ { N } + \\mathcal { J } _ { N } ( u ^ { N } \\cdot \\nabla \\theta ^ { N } ) + \\Lambda _ { x _ { 1 } } ^ { 2 \\alpha } \\theta ^ { N } + \\Lambda _ { x _ { 2 } } ^ { 2 \\beta } \\theta ^ { N } = 0 , \\\\ & u ^ { N } = \\mathcal { R } ^ { \\perp } \\theta ^ { N } , \\\\ & \\theta ^ { N } ( x , 0 ) = \\mathcal { J } _ { N } \\theta _ { 0 } ( x ) . \\endaligned \\right . \\end{align*}"}
-{"id": "6501.png", "formula": "\\begin{align*} k _ { r } \\overset { } { = } \\sqrt { 1 - \\rho } k _ { \\mathrm { o } } 0 < x < L \\end{align*}"}
-{"id": "608.png", "formula": "\\begin{align*} \\dim C _ q ^ n ( m ) = \\sum _ s \\sum _ { j = 0 } ^ { \\min \\{ s , m \\} } \\dim C _ q ^ { n - 1 } ( s ) a _ { m - s + 2 j } . \\end{align*}"}
-{"id": "6208.png", "formula": "\\begin{align*} \\begin{pmatrix} - 2 & 0 & a \\\\ 0 & - 2 & 0 \\\\ a & 0 & c ' \\end{pmatrix} . \\end{align*}"}
-{"id": "7311.png", "formula": "\\begin{align*} \\widehat { \\Gamma } = \\Gamma \\circ \\widehat { } \\ : \\widehat { A } \\overset { \\widehat { } } { \\longrightarrow } \\gamma ( A ) \\overset { \\Gamma } { \\longrightarrow } \\overline { A } , \\ \\ f ( \\widehat { q } ) \\mapsto f ( 0 ) . \\end{align*}"}
-{"id": "1160.png", "formula": "\\begin{align*} t _ k = \\begin{cases} x ^ \\frac { k } { 2 } , & \\cr y x ^ \\frac { k - 3 } { 2 } , & \\end{cases} \\end{align*}"}
-{"id": "7970.png", "formula": "\\begin{align*} \\nu : = \\frac { 1 } { 2 } ( 1 + r ^ 2 ) < 1 . \\end{align*}"}
-{"id": "2744.png", "formula": "\\begin{align*} E ^ { K } ( t ) = E _ { s , q } ^ { K } ( t ) = \\sum _ { k = 1 } ^ { K } \\ , | | k ^ q h _ k | | _ { H _ { x , v } ^ s } ^ 2 , \\end{align*}"}
-{"id": "2384.png", "formula": "\\begin{align*} \\rho ' = A ( \\theta ) \\rho ^ 3 + B ( \\theta ) \\rho ^ 2 + a _ 1 \\rho , \\end{align*}"}
-{"id": "4108.png", "formula": "\\begin{align*} \\mathrm { h e s s } _ b g ( X , Y ) | _ p = X ( Y g ) | _ p = - \\frac { \\langle D _ X Y , \\xi \\rangle } { \\langle \\eta , \\xi \\rangle } = - h ( X , Y ) . \\end{align*}"}
-{"id": "1611.png", "formula": "\\begin{align*} \\sigma ^ m ( x ) ( p , q ) = x ( p + m , q + m ) \\end{align*}"}
-{"id": "7608.png", "formula": "\\begin{align*} ( h , k ) = \\left ( h _ 1 + h _ 2 , A h _ 1 + B h _ 2 \\right ) , h _ 1 = - U ^ { - 1 } k + U ^ { - 1 } B h , h _ 2 = U ^ { - 1 } k - U ^ { - 1 } A h . \\end{align*}"}
-{"id": "6272.png", "formula": "\\begin{align*} [ \\Delta _ q , u _ { \\leq { p - 2 } } \\cdot \\nabla ] v _ p = \\Delta _ q ( u _ { \\leq { p - 2 } } \\cdot \\nabla v _ p ) - u _ { \\leq { p - 2 } } \\cdot \\nabla \\Delta _ q v _ p . \\end{align*}"}
-{"id": "5554.png", "formula": "\\begin{align*} z = - \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { t _ { 1 } } g \\left ( t _ { 2 } \\right ) x \\left ( t _ { 2 } \\right ) d t _ { 2 } d t _ { 1 } + b t + a \\end{align*}"}
-{"id": "6864.png", "formula": "\\begin{align*} c ( I _ { k } ^ { ( m ) } ) = \\frac { 1 } { R ( I _ { k } ^ { ( m ) } ) } . \\end{align*}"}
-{"id": "2808.png", "formula": "\\begin{align*} p _ { t } & = p _ { x x } + ( \\rho - 1 ) p _ { x } , x > b ( t ) , t > 0 , \\\\ b ( 0 ) & = 0 , \\\\ p ( 0 , x ) & = e ^ { x } - 1 , x \\geq 0 , \\\\ p ( t , b ( t ) ) & = e ^ { \\rho t } - 1 , p _ { x } ( t , b ( t ) ) = 0 , t > 0 . \\end{align*}"}
-{"id": "7103.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\frac { \\partial } { \\partial t } X ( x , t ) = & ~ F ( \\mathcal { W } ( x , t ) ) ^ { - \\alpha } \\nu ( x , t ) , \\\\ X ( \\cdot , 0 ) = & ~ X _ 0 ( \\cdot ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "4084.png", "formula": "\\begin{align*} \\lambda \\in \\mathbb { R } ^ { n _ y } , ( \\alpha , 0 , 0 ) \\in \\partial \\langle \\lambda , h \\rangle ( \\bar x , \\bar y , \\bar u ) + \\{ ( 0 , 0 ) \\} \\times N _ U ( \\bar u ) \\Longrightarrow \\alpha = 0 \\end{align*}"}
-{"id": "1270.png", "formula": "\\begin{align*} \\hat \\nu _ l ( \\hat E _ l ) = \\mbox { C a p } _ { \\mathcal { A } } ( \\hat { E _ l } ) \\mbox { f o r } l = 1 , 2 , \\dots . \\end{align*}"}
-{"id": "9685.png", "formula": "\\begin{align*} U | _ { x = x _ 0 } = \\begin{cases} U _ { b } = ( u _ b , v _ b , p _ b , \\rho _ b , Z _ b ) ^ \\top , & y > y _ 0 , \\\\ U _ { a } = ( u _ a , v _ a , p _ a , \\rho _ a , Z _ a ) ^ \\top , & y < y _ 0 , \\\\ \\end{cases} \\end{align*}"}
-{"id": "5550.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t } w _ { 0 } \\left ( t _ { 1 } \\right ) d t _ { 1 } = \\left [ \\begin{array} { c c c c } \\frac { 1 } { 2 } & - \\frac { 1 } { 4 } & - \\frac { 1 } { 8 } & 0 \\end{array} \\right ] \\bar { w } _ { 4 } \\left ( t \\right ) \\end{align*}"}
-{"id": "2781.png", "formula": "\\begin{align*} p ^ { E } ( t , S ) = K e ^ { - r t } \\aleph ( - d _ { 2 } ( S , t , K ) ) - \\mathcal { B } ( t ) e ^ { - \\delta t } \\aleph ( - d _ { 1 } ( S , t , K ) ) , \\end{align*}"}
-{"id": "8049.png", "formula": "\\begin{align*} \\alpha _ { + } ( s ) : = \\angle \\gamma _ { - t _ * } ( s ) \\gamma _ { t _ * } ( s ) p , \\ , \\ , \\ , \\ , \\ , \\alpha _ { - } ( s ) : = \\angle \\gamma _ { t _ * } ( s ) \\gamma _ { - t _ * } ( s ) p , \\\\ \\tilde \\alpha _ { + } ( s ) : = \\tilde \\angle \\gamma _ { - t _ * } ( s ) \\gamma _ { t _ * } ( s ) p , \\ , \\ , \\ , \\ , \\ , \\tilde \\alpha _ { - } ( s ) : = \\tilde \\angle \\gamma _ { t _ * } ( s ) \\gamma _ { - t _ * } ( s ) p . \\end{align*}"}
-{"id": "5741.png", "formula": "\\begin{align*} V ( x , n , m ) = \\prod _ { i = 0 } ^ { m - 1 } [ x + 2 i ] _ { n - 2 i } . \\end{align*}"}
-{"id": "8611.png", "formula": "\\begin{align*} V ( t , x ) = \\int _ { \\R ^ { d + 1 } } \\phi ( t - s ) \\psi ( x - y ) d W ( s , y ) , \\end{align*}"}
-{"id": "3476.png", "formula": "\\begin{align*} \\begin{cases} u _ { t } ( t , x ) - u _ { x x } ( t , x ) + b ( u ( t , x ) ) = f ( t , x ) , & ( t , x ) \\in ( 0 , 1 ) ^ 2 , \\\\ u ( t , 0 ) = u ( t , 1 ) = 0 , & t \\in ( 0 , 1 ) , \\\\ u ( 0 , x ) = u _ 0 ( x ) , & x \\in ( 0 , 1 ) , \\end{cases} \\end{align*}"}
-{"id": "3526.png", "formula": "\\begin{align*} \\begin{array} [ c ] { r l } & \\hat { J } ( u ^ { \\varepsilon } ( \\cdot ) ) - \\hat { J } ( \\bar { u } ( \\cdot ) ) \\\\ = & \\psi ( X ^ { \\varepsilon } ( t _ 1 ) , X ^ { \\varepsilon } ( t _ 2 ) , \\cdots , X ^ { \\varepsilon } ( t _ n ) ) - \\psi ( \\bar { X } ( t _ 1 ) , \\bar { X } ( t _ 2 ) , \\cdots , \\bar { X } ( t _ n ) ) \\\\ & + { \\displaystyle \\int \\limits _ { 0 } ^ { T } } \\{ f ( X { } ^ { \\varepsilon } ( t ) , u ^ { \\varepsilon } ( t ) ) - f ( \\bar { X } { ( t ) } , \\bar { u } ( t ) ) \\} d t . \\\\ \\end{array} \\end{align*}"}
-{"id": "4740.png", "formula": "\\begin{align*} { \\varepsilon } = F ( x ^ 0 , x ^ 1 , Z ) \\Delta \\sqrt { - \\det g ^ { i j } } . \\end{align*}"}
-{"id": "2906.png", "formula": "\\begin{align*} H _ a ^ X = \\{ \\langle y , 0 \\rangle : y \\in X \\} \\cup \\{ \\langle y , b + 1 \\rangle : b < _ { \\mathcal { O } } ^ X a \\wedge y \\in T J ( H _ b ^ X ) \\} \\end{align*}"}
-{"id": "2107.png", "formula": "\\begin{align*} \\sum _ a \\big ( | | v ^ a _ s | | _ { C ^ 0 } + | | \\nabla v ^ a _ s | | _ { C ^ 0 } \\big ) \\big | _ { t = \\frac { T _ 3 } { 2 } } \\leq C _ 9 ' | | \\phi _ { s + s _ 0 } - \\phi _ { s _ 0 } | | _ { C ^ 1 } = O ( s ) . \\end{align*}"}
-{"id": "8133.png", "formula": "\\begin{align*} U _ i = \\{ x \\in X : d ( x , X \\setminus V _ i ) > \\eta ' \\} \\end{align*}"}
-{"id": "9970.png", "formula": "\\begin{align*} \\bar { \\gamma } _ { k i } ^ { p } = \\bar { \\gamma } _ { i } ^ { p } = \\frac { \\bar { \\upsilon } _ { i } ^ { p } } { { \\displaystyle \\sum _ { j = 1 } ^ { L } \\alpha _ { j } \\mathbb { E } \\left [ \\mathsf { \\bar { L } } _ { j i } \\right ] + \\bar { \\upsilon } _ { i } ^ { p } \\sum _ { j \\neq i } ^ { L } \\kappa _ { j } \\mathbb { E } \\left [ \\mathsf { \\bar { L } } _ { j i } ^ { 2 } \\right ] } } \\end{align*}"}
-{"id": "702.png", "formula": "\\begin{align*} \\langle v _ g p _ t ( \\xi ) , v _ g p _ t ( \\eta ) \\rangle & = \\langle T u _ g p _ t ( \\xi ) , v _ g p _ t ( \\eta ) \\rangle \\\\ & = \\langle T p _ t u _ g p _ t ( \\xi ) , v _ g p _ t ( \\eta ) \\rangle \\\\ & = \\langle u _ g p _ t ( \\xi ) , T v _ g p _ t ( \\eta ) \\rangle \\\\ & = \\langle p _ t ( \\xi ) , T u _ g ^ * v _ g p _ t ( \\eta ) \\rangle \\\\ & = \\langle p _ t ( \\xi ) , v _ { g ^ { - 1 } } v _ g p _ t ( \\eta ) \\rangle \\\\ & = \\langle p _ t ( \\xi ) , p _ t ( \\eta ) \\rangle , \\end{align*}"}
-{"id": "8624.png", "formula": "\\begin{align*} R _ \\psi ( x ) = \\int _ { \\R ^ d } \\psi ( x - y ) \\psi ( y ) d y , \\ \\ R _ \\phi ( t _ 1 , t _ 2 ) = \\int _ 0 ^ \\infty \\phi ( s - t _ 1 ) \\phi ( s - t _ 2 ) d s . \\end{align*}"}
-{"id": "7721.png", "formula": "\\begin{align*} D ^ 2 _ B ( 0 ) & = ( N - 1 ) \\cdot \\frac { N - 1 } { N } = \\frac { ( N - 1 ) ^ 2 } { N } \\ , , \\\\ C _ B ( 0 ) & = \\frac { N ^ 2 } { ( N - 1 ) ^ 2 } \\ , , \\end{align*}"}
-{"id": "5380.png", "formula": "\\begin{align*} G ^ { ( m , n , p ) } : = \\langle a , b , c \\mid a ^ 2 , b ^ 2 , c ^ 2 , ( a b ) ^ 2 , ( a c ) ^ m , ( b c ) ^ n , ( a b c ) ^ p \\rangle \\end{align*}"}
-{"id": "1443.png", "formula": "\\begin{align*} D & = \\mathbb { Z } _ { ( 2 ) } [ x _ 2 ^ 2 , x _ 3 ^ 2 , x _ { 4 1 } ^ 2 , \\dots , x _ { 4 n } ^ 2 , v _ 1 , \\dots , v _ m ] / ( v _ 0 x _ 3 ^ 2 ) , \\\\ C & = \\mathbb { Z } _ { ( 2 ) } [ x _ 2 ^ 2 , x _ { 4 1 } ^ 2 , \\dots , x _ { 4 n } ^ 2 , v _ 1 , \\dots , v _ m ] . \\end{align*}"}
-{"id": "4765.png", "formula": "\\begin{align*} { L } _ i = \\Big ( \\alpha , { L } _ 1 ( x ^ 1 ) , { L } _ 2 ( x ^ 2 ) , { L } _ 3 ( x ^ 3 ) \\Big ) , \\alpha - c o n s t . \\end{align*}"}
-{"id": "8176.png", "formula": "\\begin{align*} \\phi _ { \\circ } = \\frac { \\sigma ( v ^ { \\circ } ) } { \\Vert \\sigma ( v ^ { \\circ } ) \\Vert _ 2 } \\end{align*}"}
-{"id": "4654.png", "formula": "\\begin{align*} \\left | \\sum _ { n = 0 } ^ { N - 1 } 1 _ { I _ k } ( T ^ n x ) - 1 _ { I _ k } ( T ^ n y ) \\right | \\leq E _ 2 \\max \\{ 1 , ( | I _ k | N ) ^ { \\zeta _ 2 } \\} \\end{align*}"}
-{"id": "7091.png", "formula": "\\begin{align*} k ( p ) = p ^ m , \\end{align*}"}
-{"id": "2665.png", "formula": "\\begin{align*} a _ 0 ( a _ 1 + e ' ) = a _ 1 + a _ 0 ( e ' ) . \\end{align*}"}
-{"id": "1535.png", "formula": "\\begin{gather*} A ^ * \\ : = \\ \\left [ b _ 1 - \\frac { 1 } { 4 } , c _ 1 + \\frac { 1 } { 4 } \\right ] \\cup \\dots \\cup \\left [ b _ k - \\frac { 1 } { 4 } , c _ k + \\frac { 1 } { 4 } \\right ] . \\end{gather*}"}
-{"id": "1783.png", "formula": "\\begin{align*} V ( R _ 0 ( 0 ) ) = \\omega _ 0 , V ( s ) : = - \\frac { 2 G ( s ) } { s ^ 2 } \\end{align*}"}
-{"id": "8143.png", "formula": "\\begin{align*} h _ { k , l , c , i , j } ^ * f h _ { k , l , c , i , j } & = \\sum _ { q = 1 } ^ Q \\sum _ { t \\in B _ { k , l , c , q } } \\frac { q ^ 2 } { Q ^ 2 } \\alpha _ { t c \\Lambda _ { k , i , j } ( c ) ^ { - 1 } t ^ { - 1 } } ( f ) \\alpha _ { t c } ( h _ k ^ 2 ) \\end{align*}"}
-{"id": "3029.png", "formula": "\\begin{align*} W _ B \\Sigma W _ B ^ * & = \\frac { 1 } { 4 } ( 2 I - 2 T T ^ * ) = \\frac { 1 } { 4 } I . \\end{align*}"}
-{"id": "9080.png", "formula": "\\begin{align*} p ( x ) = \\sum _ { k = 0 } ^ { n } a _ { k } x ^ { k } , \\end{align*}"}
-{"id": "112.png", "formula": "\\begin{align*} \\mathrm { I n t } _ { E + ( 2 L + 1 ) \\sqrt { k } } W _ 0 ( x , n ) \\subset \\bigcup _ { m \\in \\mathbb { G } _ x } W ( x , m ) , \\\\ \\bigcup _ { m \\in \\mathbb { B } _ x } W ( x , m ) \\subset \\bigcup _ { n \\in \\mathbb { Z } ^ k } \\partial _ { E + ( 2 L + 1 ) \\sqrt { k } } W _ 0 ( x , n ) . \\end{align*}"}
-{"id": "7652.png", "formula": "\\begin{align*} c = \\lambda , \\Lambda = \\frac { 1 } { 2 } \\ , \\lambda . \\end{align*}"}
-{"id": "9485.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ n } { ( - q ; q ^ 2 ) _ { n + 1 } } = \\sum _ { j = 0 } ^ { \\infty } ( - 1 ) ^ j q ^ { 6 j ^ 2 + 4 j } ( 1 + q ^ { 4 j + 2 } ) . \\end{align*}"}
-{"id": "9599.png", "formula": "\\begin{align*} \\langle f , D _ k ( g ) \\rangle = & \\lim _ { M \\to \\infty } \\lim _ { J \\to \\infty } \\langle f , g ^ 2 _ { k , M , J } \\rangle \\\\ = & \\lim _ { M \\to \\infty } \\lim _ { J \\to \\infty } \\sum _ { j = 1 } ^ { N _ J } D _ k ( f ) ( y _ j ) \\int _ { Q _ j } g ( y ) d \\mu ( y ) , \\end{align*}"}
-{"id": "6895.png", "formula": "\\begin{align*} \\mathcal { E } ( u , v ) = - \\int f v d \\mu \\end{align*}"}
-{"id": "2160.png", "formula": "\\begin{align*} \\frac { d } { d s } \\int _ { I ( s ) } | \\tilde { w } ( \\xi , s ) | ^ 2 \\rho _ d \\ , d \\xi & \\le - 2 \\int _ { I ( s ) } | \\tilde { w } ( \\xi , s ) | ^ 2 \\rho _ d \\ , d \\xi + O ( e ^ { - d \\theta _ * s } ) + O ( e ^ { - \\frac { \\theta } { 4 } s } ) \\\\ & = - 2 \\int _ { I ( s ) } | \\tilde { w } ( \\xi , s ) | ^ 2 \\rho _ d \\ , d \\xi + O ( e ^ { - 2 \\theta ' s } ) \\end{align*}"}
-{"id": "2057.png", "formula": "\\begin{align*} \\frac { \\partial f } { \\partial t } = \\tau ( f ) \\end{align*}"}
-{"id": "6138.png", "formula": "\\begin{align*} \\mathbb { P } _ { c o u p l e } \\left ( \\sum _ { i = 1 } ^ { M _ { 1 } \\left ( n \\right ) } n ^ { - \\frac { 1 } { \\alpha } } W _ { i } \\leq T \\right ) & \\leq C n ^ { - c ' } \\\\ \\mathbb { P } _ { c o u p l e } \\left ( \\sum _ { i = 1 } ^ { M _ { 2 } \\left ( n \\right ) } n ^ { - \\frac { 1 } { \\alpha } } W _ { i } \\leq n ^ { - c ' } \\right ) & \\leq C n ^ { - c ' } . \\end{align*}"}
-{"id": "9710.png", "formula": "\\begin{align*} & V _ m = \\tilde { \\Phi } ( \\alpha _ 5 , \\alpha _ 3 , \\alpha _ 2 , \\alpha _ 1 ; V _ a ) ) ) , Z _ m = Z _ a + \\alpha _ 4 , \\\\ & V _ b = \\tilde { \\Phi } _ 5 ( \\beta _ 5 ; \\mathcal { F } ( \\sigma _ 3 , \\sigma _ 2 ; \\tilde { \\Phi } _ 1 ( \\beta _ 1 ; V _ m ) ) ) , Z _ b = Z _ m + \\beta _ 4 , \\\\ & V _ b = \\tilde { \\Phi } _ 5 ( \\gamma _ 5 ; \\mathcal { F } ( \\sigma ' _ 3 , \\sigma ' _ 2 ; \\tilde { \\Phi } _ 1 ( \\gamma _ 1 ; V _ a ) ) ) , Z _ b = Z _ a + \\gamma _ 4 , \\end{align*}"}
-{"id": "10073.png", "formula": "\\begin{align*} \\| x \\| ^ 2 = \\frac { | [ x , \\overline { x } ] | } { 4 \\pi e ^ \\gamma } \\end{align*}"}
-{"id": "2583.png", "formula": "\\begin{align*} H ^ { - s } ( \\R ) : = \\big \\{ \\zeta \\in \\mathcal S ' ( \\R ) : { \\textstyle \\int _ \\R } ( 1 + \\omega ^ 2 ) ^ { - s } \\big | \\widehat \\zeta ( \\omega ) \\big | ^ 2 \\ , d \\omega < \\infty \\big \\} , \\end{align*}"}
-{"id": "3466.png", "formula": "\\begin{align*} \\mathrm { d i s t } _ H ( v , V _ h ) & : = \\inf _ { v _ h \\in V _ h } \\| v - v _ h \\| _ H . \\end{align*}"}
-{"id": "8093.png", "formula": "\\begin{align*} \\frac { \\partial \\rho } { \\partial w } = 0 . \\end{align*}"}
-{"id": "5771.png", "formula": "\\begin{align*} \\| u \\| _ { \\dot { W } ^ { 1 , p } _ { 0 } ( \\Omega ) } = \\| \\nabla u \\| _ { L ^ { p } ( \\Omega ) } . \\end{align*}"}
-{"id": "8809.png", "formula": "\\begin{align*} C _ { I } & : = \\max \\{ C _ { m _ { i } } \\mid 1 \\leq i \\leq r \\} ; \\\\ c _ { j , p } & : = \\sum _ { i = 1 } ^ { r } a _ { i , p } m _ { j } ^ { \\beta _ { i } } p ^ { ( \\sigma - \\lambda _ { i } ) m _ { j } } , 1 \\leq j \\leq r ; \\\\ D _ { p } & : = \\det ( B _ { p } ) ; \\\\ D _ { k , l , p } & : = ( - 1 ) ^ { k + l } \\det \\big ( ( m _ { j } ^ { \\beta _ { i } } p ^ { ( \\sigma - \\lambda _ { i } ) m _ { j } } ) _ { j \\neq k , i \\neq l } \\big ) , 1 \\leq k , l \\leq r . \\end{align*}"}
-{"id": "8127.png", "formula": "\\begin{align*} B _ - & = \\{ x \\in X : d ( x , X \\setminus B ) > \\eta \\} , \\\\ A _ + & = \\{ x \\in X : d ( x , A ) \\leq \\eta \\} \\end{align*}"}
-{"id": "5661.png", "formula": "\\begin{align*} \\inf \\{ \\mathfrak { E } _ { W } ( { v } ) \\ ; : \\ ; { v } \\in \\mathcal { S } ( a ^ - , a ^ + ) \\} = d _ K ( a ^ - , a ^ + ) , \\end{align*}"}
-{"id": "5341.png", "formula": "\\begin{align*} \\langle \\xi \\cdot \\lambda , w \\rangle = - \\langle \\lambda , \\xi \\cdot w \\rangle \\end{align*}"}
-{"id": "245.png", "formula": "\\begin{align*} & \\chi _ { [ - L , 0 ] ^ d } \\{ h ( g ( H ^ { R _ { \\sigma } } ) _ { \\Lambda _ L } ) - f ^ { R _ { \\sigma } } _ 0 \\} \\\\ & \\quad = U ^ * _ L \\chi _ { [ 0 , L ] ^ d } \\{ h ( g ( H ^ { T _ L R _ \\sigma } ) _ { [ 0 , 2 L ] ^ d } ) - f _ { 0 } ^ { T _ L R _ \\sigma } \\} U _ L , \\end{align*}"}
-{"id": "886.png", "formula": "\\begin{align*} \\bar { Q } _ n ( Y ) = \\sum _ { i , j = 1 } ^ { N _ n } \\bar { \\gamma } _ n ( i , j ) Y _ i Y _ j . \\end{align*}"}
-{"id": "745.png", "formula": "\\begin{align*} \\| w _ n \\| _ { \\dot H } ^ 2 & = \\| w _ n - u \\| _ { \\dot H } ^ 2 + \\| u \\| _ { \\dot H } ^ 2 + o _ n ( 1 ) , \\\\ \\| w _ n \\| _ { q } ^ q & = \\| w _ n - u \\| _ { q } ^ q + \\| u \\| _ { q } ^ q + o _ n ( 1 ) \\ \\ \\ \\ 1 \\leq q < \\infty . \\end{align*}"}
-{"id": "4433.png", "formula": "\\begin{align*} \\partial _ 1 ^ j \\partial _ 2 ^ l u _ T ( x ) & = \\int _ { \\mathbb { R } ^ 2 } \\partial _ 1 ^ j \\partial _ 2 ^ l \\psi _ T ( x - y ) ( u ( y ) - u ( x ) ) d y , \\\\ \\partial _ 1 ^ j \\partial _ 2 ^ l u _ T ( x ) & = \\int _ { \\mathbb { R } ^ 2 } \\partial _ 1 ^ j \\partial _ 2 ^ l \\psi _ T ( x - y ) ( u ( y ) - u ( x ) - ( y - x ) _ 1 \\partial _ 1 u ( x ) ) d y , \\end{align*}"}
-{"id": "6316.png", "formula": "\\begin{align*} \\Tilde { A } _ m ( z _ m ) = ( z _ m - 2 \\alpha _ { m } ) p _ m ^ 2 ( z _ m ) \\mbox { w i t h } p _ m ( z _ m ) = z _ m ^ { \\ell _ m } + \\beta _ { m , 1 } z ^ { \\ell _ m - 1 } + \\ ! \\ldots \\ ! + \\beta _ { m , \\ell _ m } \\mbox { a n d } \\beta _ { m , 1 } = \\alpha _ { m } . \\end{align*}"}
-{"id": "9871.png", "formula": "\\begin{align*} t _ i : = \\left \\{ \\begin{array} { l l } \\ s _ i & \\mbox { i f $ i = \\iota ( i ) $ i n $ \\Delta $ } , \\\\ \\ s _ i s _ { \\iota ( i ) } s _ i & \\mbox { i f t h e r e i s a n e d g e $ i \\stackrel { } { \\mbox { - - - } } \\iota ( i ) $ i n $ \\Delta $ } , \\\\ \\ s _ i s _ { \\iota ( i ) } & \\mbox { i f n o e d g e b e t w e e n $ i $ a n d $ \\iota ( i ) $ i n $ \\Delta $ } , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "5736.png", "formula": "\\begin{align*} \\lim \\limits _ { x _ 2 \\to \\pm \\infty } \\| { u } ( \\cdot , x _ 2 ) - z ^ \\pm ( \\cdot - m ( x _ 2 ) ) \\| _ { L ^ 2 ( \\R ) } = 0 . \\end{align*}"}
-{"id": "4212.png", "formula": "\\begin{align*} \\mathbf { B } : = \\bigcup _ { j \\in J } \\bigcup _ { w \\in X _ j } \\{ \\mathrm { e v } _ { \\mathtt { v } } \\circ \\partial _ w \\circ \\xi _ j \\} \\qquad \\underline { \\mathbf { B } } : = \\bigcup _ { j \\in J } \\bigcup _ { w \\in G _ { \\mathtt { v } } } \\{ \\mathrm { e v } _ { \\mathtt { v } } \\circ \\partial _ w \\circ \\xi _ j \\} . \\end{align*}"}
-{"id": "1015.png", "formula": "\\begin{align*} g _ { p s } & = \\hat { g } _ { p s } + \\Delta g _ { p s } \\\\ g _ { s p } & = \\hat { g } _ { s p } + \\Delta g _ { s p } \\end{align*}"}
-{"id": "5865.png", "formula": "\\begin{align*} a _ i = 0 ~ i \\notin \\{ 0 , 8 , 1 2 , 1 6 , 2 4 \\} . \\end{align*}"}
-{"id": "370.png", "formula": "\\begin{align*} X _ { n , r , s } ^ { \\prime } = b _ { n , r , s } \\xi _ { - r , - s } I ( b _ { n , r , s } \\xi _ { - r , - s } \\leq \\varepsilon x ) . \\end{align*}"}
-{"id": "8448.png", "formula": "\\begin{align*} Y = \\begin{cases} 0 & \\delta _ 0 \\geq n , \\\\ Y _ 0 & \\end{cases} \\delta = \\begin{cases} \\lfloor \\frac { n } { 2 } \\rfloor & \\\\ \\delta ' & \\delta _ 0 = 2 \\delta ' < n . \\end{cases} \\end{align*}"}
-{"id": "2650.png", "formula": "\\begin{align*} w ( b ) = b - \\alpha \\left \\lfloor \\frac { b } { 2 ( k + 1 ) } \\right \\rfloor - \\beta \\left \\lfloor \\frac { b } { 3 ( k + 1 ) } \\right \\rfloor , \\end{align*}"}
-{"id": "7418.png", "formula": "\\begin{align*} \\mathbb { H } _ { \\Gamma } : = \\frac { \\mathbb { C } [ t _ 0 , t _ 1 , t _ 2 , t _ 3 , t _ 4 , t _ 5 , t _ 6 , t _ 7 ] } { ( t _ 0 + 2 t _ 1 + 4 t _ 2 + 3 t _ 3 + 2 t _ 4 + 3 t _ 5 + 2 t _ 6 + t _ 7 ) } . \\end{align*}"}
-{"id": "2826.png", "formula": "\\begin{align*} F _ n = \\left ( \\begin{array} { c c c c c c c } F ^ { ( n ) } _ { B _ 1 } & * & \\cdots & * & * \\\\ * & F ^ { ( n ) } _ { B _ 2 } & \\cdots & * & * \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ * & * & \\cdots & F ^ { ( n ) } _ { B _ l } & * \\\\ * & * & \\cdots & * & F ^ { ( n ) } _ { B _ 0 } \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "1307.png", "formula": "\\begin{align*} f ( x , y ) = f ( y ^ { - 1 } w _ 0 , w _ 0 x ^ { - 1 } ) . \\end{align*}"}
-{"id": "7200.png", "formula": "\\begin{align*} \\sum _ j \\xi _ j ( u _ j ) _ \\nu ^ 2 ( x ) = 1 . \\end{align*}"}
-{"id": "3444.png", "formula": "\\begin{align*} U _ { \\varepsilon } ^ { - 1 } ( A ) = U _ { \\varepsilon } ^ { - 1 } ( A \\cap \\Q ^ d ) . \\end{align*}"}
-{"id": "9618.png", "formula": "\\begin{align*} C = W \\log _ 2 \\left ( 1 + \\frac { P _ { \\rm { t r } , \\rm { u s e r } , \\xi } \\big ( r , h _ \\beta \\big ) } { L _ \\xi \\big ( r , h _ \\beta \\big ) N _ 0 W } \\right ) , \\end{align*}"}
-{"id": "5763.png", "formula": "\\begin{align*} v _ { l , 1 , 1 } ^ 2 + v _ { l , 2 , 1 } ^ 2 = q _ l ( v _ { l , 1 , 2 } ^ 2 + v _ { l , 2 , 2 } ^ 2 ) \\end{align*}"}
-{"id": "6591.png", "formula": "\\begin{align*} g _ n = \\psi _ { n } \\circ g \\circ \\varphi _ { n } ^ { - 1 } \\colon \\varphi _ { n } ( U _ n \\cap g ^ { - 1 } ( V _ n ) ) \\longrightarrow \\psi _ { n } ( g ( U _ n ) \\cap V _ n ) \\ . \\end{align*}"}
-{"id": "4769.png", "formula": "\\begin{align*} { \\varepsilon } = F ( X , Y , Z ) \\ , { \\Delta \\sqrt { - \\det g ^ { i j } } } / { ( { L } _ 2 \\ , { L } _ 3 ) } , \\end{align*}"}
-{"id": "2857.png", "formula": "\\begin{align*} \\overline { a } _ { n 0 } & = 1 , \\ n = 0 , 1 , . . . , \\\\ a _ { n - 1 , v } & \\geq a _ { n v } , \\ \\textnormal { f o r } ~ ~ n \\geq v + 1 , \\\\ a _ { n n } & = O ( \\frac { p _ { n } } { P _ { n } } ) , \\\\ \\sum _ { v = 1 } ^ { n - 1 } a _ { v v } \\hat { a } _ { n , v + 1 } & = O ( a _ { n n } ) \\end{align*}"}
-{"id": "1536.png", "formula": "\\begin{gather*} | A + B | \\ = \\ \\mu ( A ^ * + B ^ * ) \\end{gather*}"}
-{"id": "3008.png", "formula": "\\begin{align*} Q = ( Q _ { i j } ) _ { \\substack { 1 \\leq i \\leq N \\\\ 1 \\leq j \\leq N } } & = \\begin{cases} ( - 1 ) ^ { i - 1 } P _ { i + j - 1 } & i + j \\leq N + 1 \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "4879.png", "formula": "\\begin{align*} \\lambda _ 1 ^ 2 ( t ) = Z ( 1 , t ) = \\sum _ { k = 0 } p ( k ) t ^ k = \\prod _ { n = 1 } ^ { \\infty } \\frac { 1 } { 1 - t ^ n } \\end{align*}"}
-{"id": "7421.png", "formula": "\\begin{align*} \\mathbb { H } _ { \\Gamma } : = \\frac { \\mathbb { C } [ t _ 0 , t _ 1 , t _ 2 , t _ 3 , t _ 4 , t _ 5 , t _ 6 , t _ 7 , t _ 8 ] } { ( t _ 0 + 2 t _ 1 + 3 t _ 2 + 4 t _ 3 + 5 t _ 4 + 3 t _ 5 + 2 t _ 6 + 4 t _ 7 + 6 t _ 8 ) } . \\end{align*}"}
-{"id": "7912.png", "formula": "\\begin{align*} x _ 1 = [ x _ 1 ] \\leq \\left [ x _ 1 + y _ 1 \\over 2 \\right ] \\end{align*}"}
-{"id": "3985.png", "formula": "\\begin{align*} p ^ { \\beta _ 1 } ( 1 , t ) = - \\sum _ { k = 1 } ^ { \\infty } ( - \\lambda ) ^ k \\underset { \\Omega ^ { k } _ { 1 } } { \\sum } \\frac { t ^ { k _ 0 \\beta _ 0 + k _ 1 \\beta _ 1 } } { \\Gamma \\left ( k _ 0 \\beta _ 0 + k _ 1 \\beta _ 1 + 1 \\right ) } , \\end{align*}"}
-{"id": "4347.png", "formula": "\\begin{align*} \\widetilde { \\varphi } _ \\sigma ( x _ 1 , \\dots , x _ { 2 n } ) = \\widetilde { \\varphi } _ 0 ( x _ 1 - \\sigma _ 1 , \\dots , x _ { 2 n } - \\sigma _ { 2 n } ) . \\end{align*}"}
-{"id": "530.png", "formula": "\\begin{align*} \\varphi ^ * \\omega _ 2 = \\omega _ 1 \\end{align*}"}
-{"id": "2006.png", "formula": "\\begin{align*} x _ i y _ { i + 1 } - x _ { i + 2 } y _ i = 0 , ~ ~ ~ i \\in \\Z . \\end{align*}"}
-{"id": "1826.png", "formula": "\\begin{align*} \\tilde { \\mathbb { P } } ^ a _ { Q , \\rho , h } ( E ) & = \\frac { \\sum _ { \\omega \\in E } \\mathbb { P } ^ a _ { Q , \\rho , 0 } ( \\omega ) \\prod _ { i } \\cosh ( h A _ i ( \\omega ) ) } { \\mathbb { E } ^ a _ { Q , \\rho , 0 } \\left ( \\prod _ { i } \\cosh ( A _ i ) \\right ) } \\geq \\frac { \\mathbb { P } ^ a _ { Q , \\rho , 0 } ( E ) } { E ^ a _ { Q , + , 0 } ( e ^ { h m ^ a _ Q } ) } \\\\ & \\geq e ^ { - C _ 5 ( h + h ^ 2 ) } \\mathbb { P } ^ a _ { Q , \\rho , 0 } ( E ) , \\end{align*}"}
-{"id": "10116.png", "formula": "\\begin{align*} \\bar { \\boldsymbol R } _ k ( i ) = \\mathbb { E } [ \\boldsymbol S _ { D _ k } ^ H ( i ) \\boldsymbol x _ k ( i ) \\boldsymbol x _ k ^ H ( i ) \\boldsymbol S _ { D _ k } ( i ) ] = \\mathbb { E } [ \\bar { \\boldsymbol x } _ k ( i ) \\bar { \\boldsymbol x } _ k ^ H ( i ) ] \\end{align*}"}
-{"id": "5612.png", "formula": "\\begin{align*} \\phi ( s , t , x ) - \\phi ( s , t , y ) = ( x - y ) \\exp \\biggl [ 2 \\int _ 0 ^ 1 \\{ I _ t ^ u - J _ t ^ u \\} d u \\biggr ] \\end{align*}"}
-{"id": "6131.png", "formula": "\\begin{align*} p _ { X , Y } \\left ( \\left | X - Y \\right | > \\epsilon \\right ) & = \\\\ p _ { X , Y } \\left ( \\left | X - Y \\right | 1 _ { \\left \\{ X \\in A \\right \\} \\cup \\left \\{ Y \\in B \\right \\} } + \\left | X - Y \\right | 1 _ { \\left \\{ X \\in A ^ { c } \\right \\} \\cap \\left \\{ Y \\in B ^ { c } \\right \\} } > \\epsilon \\right ) & < \\epsilon . \\end{align*}"}
-{"id": "8064.png", "formula": "\\begin{align*} \\sum _ { i \\in I } c _ i m _ i v _ i = 0 \\end{align*}"}
-{"id": "3836.png", "formula": "\\begin{align*} \\begin{aligned} & \\xi ( z , i , 0 ) = 1 z \\ge 0 , z \\in 2 \\Z i \\leq N ( z , 0 ) , \\\\ & \\xi ( z , i , 0 ) = 0 \\end{aligned} \\end{align*}"}
-{"id": "5414.png", "formula": "\\begin{align*} V ^ { \\partial H } : = \\bigcup _ { H \\subset K } V ^ K \\end{align*}"}
-{"id": "1352.png", "formula": "\\begin{align*} \\theta ' ( X ; H ) = \\displaystyle \\sum _ { i = 1 } ^ { s - 1 } { \\rm T r } ( \\widehat H _ { a _ i a _ i } ) - \\displaystyle \\sum _ { i = s + 1 } ^ r { \\rm T r } ( \\widehat H _ { a _ i a _ i } ) + \\| \\widehat H _ { a _ s a _ s } \\| _ * . \\end{align*}"}
-{"id": "7420.png", "formula": "\\begin{align*} \\delta : = d - ( t _ 1 + 2 t _ 2 ) / 3 , \\beta : = b - ( t _ 3 ) / 2 , \\gamma : = c - ( t _ 4 + 2 t _ 5 + 3 t _ 6 ) / 4 . \\end{align*}"}
-{"id": "2280.png", "formula": "\\begin{align*} a _ { 1 2 } ( p ) : = & f _ { x y } ( p ) ^ 2 f _ { x x x } ( p ) - 2 f _ { x x } ( p ) f _ { x y } ( p ) f _ { x x y } ( p ) + f _ { x x } ( p ) ^ 2 f _ { x y y } ( p ) , \\\\ a _ { 0 4 } ( p ) : = & f _ { x y } ( p ) ^ 4 f _ { x x x x } ( p ) - 4 f _ { x x } ( p ) f _ { x y } ( p ) ^ 3 f _ { x x x y } ( p ) \\\\ & + 6 f _ { x x } ( p ) ^ 2 f _ { x y } ( p ) ^ 2 f _ { x x y y } ( p ) - 4 f _ { x x } ( p ) ^ 3 f _ { x y } ( p ) f _ { x y y y } ( p ) + f _ { x x } ( p ) ^ 4 f _ { y y y y } ( p ) . \\end{align*}"}
-{"id": "7138.png", "formula": "\\begin{align*} q ( m + 1 ) = \\max \\{ q ( m ) + E ( m , a _ m ^ * ) - \\alpha B / M , 0 \\} , \\end{align*}"}
-{"id": "6675.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\frac { 1 } { \\delta ^ { \\frac { 2 } { n + 1 } } } \\frac { \\| y ^ { \\delta } \\| - \\| y \\| } { \\| y \\| } = n ^ { \\frac { 2 } { n + 1 } } c ( S ^ { \\circ } , n ) \\frac { \\kappa ( y ) ^ { \\frac { 1 } { n + 1 } } } { \\langle y , u ( y ) \\rangle } = n ^ { \\frac { 2 } { n + 1 } } c ( S ^ { \\circ } , n ) \\frac { \\langle x , u ( x ) \\rangle } { \\kappa ( x ) ^ { \\frac { 1 } { n + 1 } } } = \\frac { \\tilde { c } ( S , n ) } { G ( x ) } \\quad . \\end{align*}"}
-{"id": "8823.png", "formula": "\\begin{align*} _ { t ^ { m - 1 } } \\prod _ { i \\in I } \\frac { t ^ { N _ { i } } p ^ { - \\nu _ { i } } } { 1 - t ^ { N _ { i } } p ^ { - \\nu _ { i } } } = \\sum _ { ( a _ { i } ) _ { i \\in I } \\in J _ { I , m } } p ^ { - \\sum _ { i \\in I } \\nu _ { i } ( a _ { i } + 1 ) } , \\end{align*}"}
-{"id": "1128.png", "formula": "\\begin{align*} n _ { - } ( - 1 ) \\{ w B , B , n _ { - } ( 1 ) B \\} = \\{ w B , n _ { - } ( - 1 ) B , B \\} . \\end{align*}"}
-{"id": "8479.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { t , l , v } ) = q ^ { \\frac { n } { 1 2 } } \\gamma _ F ( - 2 v , \\rho ) \\chi ^ { - 1 } ( 4 v ^ 2 ) \\psi ( \\frac { 3 } { 4 v } \\varpi ^ { - \\frac { n } { 2 } } ) _ { \\psi } ( - 1 6 b v ^ 3 \\varpi ^ { r + 2 \\rho - 3 \\delta } ; \\Delta \\varpi ^ { - r - \\delta } ) . \\end{align*}"}
-{"id": "2487.png", "formula": "\\begin{align*} \\left ( - i g ^ { - 1 } \\xi \\cdot \\eta + L \\right ) \\left ( g \\cdot e _ { j } \\right ) & = \\left ( - i \\xi \\cdot g \\eta + L \\right ) \\left ( g \\cdot e _ { j } \\right ) \\\\ = \\left ( - i \\xi _ { 1 } \\left | \\eta \\right | + L \\right ) \\left ( g \\cdot e _ { j } \\right ) & = \\varrho _ { j } \\left ( \\eta \\right ) \\left ( g \\cdot e _ { j } \\right ) . \\end{align*}"}
-{"id": "2412.png", "formula": "\\begin{align*} \\lambda = \\frac { a _ 6 } { - a _ 2 + a _ 1 a _ 6 } . \\end{align*}"}
-{"id": "9742.png", "formula": "\\begin{align*} N ( \\theta _ { k + 1 } , n ) = \\left \\{ \\begin{array} { c c c } P _ { k + 1 , n } & \\mbox { $ \\theta _ { k + 1 } \\le 0 $ } , \\\\ P _ { k + 1 , n - 1 } & \\mbox { $ \\theta _ { k + 1 } > 0 $ } , \\end{array} \\right . S ( \\theta _ { k } , n ) = \\left \\{ \\begin{array} { c c c } P _ { k - 1 , n - 1 } & \\mbox { $ \\theta _ { k } \\le 0 $ } , \\\\ P _ { k + 1 , n } & \\mbox { $ \\theta _ { k } > 0 $ } . \\end{array} \\right . \\end{align*}"}
-{"id": "1726.png", "formula": "\\begin{align*} t _ \\lambda t _ \\lambda ^ * 1 & = t _ \\lambda ( ( \\overline { f _ \\lambda } \\circ \\sigma _ \\lambda ) \\cdot ( 1 \\circ \\sigma _ \\lambda ) \\cdot \\Phi _ \\lambda ) \\\\ & = f _ \\lambda \\cdot ( \\overline { f _ \\lambda } \\circ \\sigma _ \\lambda \\circ \\sigma ^ n ) \\cdot ( 1 \\circ \\sigma _ \\lambda \\circ \\sigma ^ n ) \\cdot ( \\Phi _ \\lambda \\circ \\sigma ^ n ) , \\end{align*}"}
-{"id": "8098.png", "formula": "\\begin{align*} B _ - & = \\{ x \\in X : d ( x , X \\setminus B ) > \\eta \\} , \\\\ A _ + & = \\{ x \\in X : d ( x , A ) \\leq \\eta \\} \\end{align*}"}
-{"id": "110.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ k } \\chi _ 1 ( z - t ) \\ , d t _ 1 \\dots d t _ k = 1 , ( z \\in \\mathbb { C } ^ k ) . \\end{align*}"}
-{"id": "10004.png", "formula": "\\begin{align*} A _ 0 ( \\C ) = \\mathfrak { a } _ { 0 \\R } / g \\mathfrak { a } _ 0 , A _ 1 ( \\C ) = \\mathfrak { a } _ { 1 \\R } / g \\mathfrak { a } _ 1 , B ( \\C ) = \\mathfrak { b } _ { \\R } / g \\mathfrak { b } , \\end{align*}"}
-{"id": "4127.png", "formula": "\\begin{align*} h ( E _ 1 , E _ 1 ) = - \\frac { \\langle E _ 1 , d \\xi _ q E _ 1 \\rangle } { \\langle \\eta , \\xi \\rangle } = - \\frac { \\lambda _ 1 } { \\langle \\eta , \\xi \\rangle } \\ \\ \\mathrm { a n d } \\\\ h ( E _ 2 , E _ 2 ) = - \\frac { \\langle E _ 2 , d \\xi _ q E _ 2 \\rangle } { \\langle \\eta , \\xi \\rangle } = - \\frac { \\lambda _ 2 } { \\langle \\eta , \\xi \\rangle } , \\end{align*}"}
-{"id": "4830.png", "formula": "\\begin{align*} \\frac { \\mathsf { v } _ { g _ n } ( A _ i ) } { \\mathsf { v } _ { g _ n } ( A _ j ) } = \\frac { c _ j } { c _ i } = \\frac { \\mathsf { v } _ { g _ 1 } ( A _ i ) } { \\mathsf { v } _ { g _ 1 } ( A _ j ) } \\end{align*}"}
-{"id": "543.png", "formula": "\\begin{align*} d \\alpha ( R _ { \\alpha } , v ) = 0 \\forall v \\in T Y , \\alpha ( R _ { \\alpha } ) = 1 , \\end{align*}"}
-{"id": "7741.png", "formula": "\\begin{align*} T _ N ( l ) = \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N - 1 } \\frac { 1 - e ^ { i l \\phi _ n } } { ( 1 - e ^ { i \\phi } ) ^ 2 } . \\end{align*}"}
-{"id": "9979.png", "formula": "\\begin{align*} \\lambda _ { 0 } v + \\lambda _ { 1 } T _ 1 { v } + \\dots + \\lambda _ { n - 1 } T _ { n - 1 } \\cdots T _ { 1 } v + \\lambda _ { n } T _ { n } \\cdots T _ { 1 } v = 0 , \\end{align*}"}
-{"id": "1989.png", "formula": "\\begin{align*} \\sup _ { c n ^ { - 1 } \\log n \\leq s < t } \\frac { \\left \\vert \\alpha _ { n } \\left ( s ; t \\right ) - B _ { n } \\left ( s ; t \\right ) \\right \\vert } { s ^ { \\nu } } = o _ { \\mathbb { P } } \\left ( 1 \\right ) , \\end{align*}"}
-{"id": "9625.png", "formula": "\\begin{align*} P _ { \\rm { t r } , 1 } \\left ( { h _ \\beta } / { R _ { \\beta } } \\right ) = \\int _ { 0 } ^ { 1 } 2 \\pi r \\bar { L } ( r , h _ \\beta / R _ { \\beta } ) N _ 0 W { \\rm { d } } r . \\end{align*}"}
-{"id": "6148.png", "formula": "\\begin{align*} g ( u , v ) \\dd _ g = \\frac { 1 } { 6 } \\epsilon _ { i j k } \\iota _ u \\omega _ i \\wedge \\iota _ v \\omega _ j \\wedge \\omega _ k , \\ ; \\ ; \\forall u , v \\in T X \\end{align*}"}
-{"id": "657.png", "formula": "\\begin{align*} | P \\widetilde A - A | ^ 2 = g ^ { i k } g ^ { j l } h ( P \\widetilde A _ { i j } - A _ { i j } , P \\widetilde A _ { k l } - A _ { k l } ) . \\end{align*}"}
-{"id": "9558.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in B _ n } { ( - 1 ) ^ { \\ell ( \\sigma ) } x ^ { L _ { e o e } ( \\sigma ) } } = ( 1 - x ^ { \\left \\lfloor \\frac { n } { 2 } \\right \\rfloor } ) \\prod _ { i = i } ^ { n - 1 } ( 1 - x ^ i ) , \\end{align*}"}
-{"id": "6181.png", "formula": "\\begin{align*} g ( t ) = V ^ 2 ( x _ 0 , t ) \\dd x _ 0 ^ 2 + \\sum _ { i = 1 } ^ 3 f _ i ( x _ 0 , t ) \\dd x _ i ^ 2 = \\dd y ^ 2 + \\sum _ { i = 1 } ^ 3 \\tilde f _ i ( y , t ) \\dd x _ i ^ 2 \\end{align*}"}
-{"id": "8775.png", "formula": "\\begin{align*} \\begin{array} { l } P = ( x - 1 - c _ 2 ^ 2 \\chi _ { G _ 1 } ( B ) ) I _ n + \\dfrac { A ( G ) } { 2 r + n _ 1 } - \\frac { c _ 1 ^ 2 } { x - 1 - c _ 3 ^ 2 \\chi _ { G _ 2 } ( C ) } M M ^ T \\\\ \\ ; \\ ; \\ ; \\ ; = ( x - 1 - c _ 2 ^ 2 \\chi _ { G _ 1 } ( B ) ) I _ n + \\dfrac { A ( G ) } { 2 r + n _ 1 } - \\frac { c _ 1 ^ 2 } { x - 1 - c _ 3 ^ 2 \\chi _ { G _ 2 } ( C ) } ( r I _ n + A ( G ) ) . \\end{array} \\end{align*}"}
-{"id": "6377.png", "formula": "\\begin{align*} W _ 1 ( z _ 1 ^ * + x / \\sqrt { n } ) = W _ 1 ( z _ 1 ^ * ) + \\frac { x ^ 2 } { 2 n } W _ 1 '' ( z _ 1 ^ * + \\lambda x / \\sqrt { n } ) \\end{align*}"}
-{"id": "2285.png", "formula": "\\begin{align*} \\sharp \\Delta _ { \\Z } - 4 & = \\sharp V ( S ) - 1 - \\sum _ { k = 1 } ^ N ( \\sharp V ( \\Delta _ k ) - 3 ) \\\\ & = \\sharp V ( S ) - 1 - N _ 4 ' . \\end{align*}"}
-{"id": "1668.png", "formula": "\\begin{align*} \\tau _ { \\alpha _ { 1 1 } } \\circ \\tau _ { \\beta _ { 2 1 } } = \\tau _ { \\beta _ { 1 1 } } \\circ \\tau _ { \\alpha _ { 2 1 } } , \\tau _ { \\alpha _ { 2 1 } } \\circ \\tau _ { \\beta _ { 1 2 } } = \\tau _ { \\beta _ { 2 1 } } \\circ \\tau _ { \\alpha _ { 1 2 } } , \\tau _ { \\alpha _ { 1 2 } } \\circ \\tau _ { \\beta _ { 2 1 } } = \\tau _ { \\beta _ { 1 2 } } \\circ \\tau _ { \\alpha _ { 2 1 } } . \\end{align*}"}
-{"id": "27.png", "formula": "\\begin{align*} ( x - y ) ^ 2 = ( | C _ 1 | c o s ( \\theta ) - | C _ 2 | c o s ( \\phi ) ) ^ 2 = \\\\ = | C _ 1 | ^ 2 c o s ( \\theta ) ^ 2 - 2 | C _ 1 | | C _ 2 | c o s ( \\theta ) c o s ( \\phi ) + | C _ 2 | ^ 2 c o s ( \\phi ) ^ 2 \\end{align*}"}
-{"id": "7487.png", "formula": "\\begin{align*} \\Psi & = \\dfrac { 1 } { p ! q ! } \\psi _ { A _ p \\bar { B } _ q } \\mathcal { Z } ^ { A _ p } \\wedge \\mathcal { Z } ^ { \\bar { B } _ q } , \\\\ \\Phi & = \\dfrac { 1 } { p ! q ! } \\phi _ { A _ p \\bar { B } _ q } \\mathcal { Z } ^ { A _ p } \\wedge \\mathcal { Z } ^ { \\bar { B } _ q } , \\end{align*}"}
-{"id": "8686.png", "formula": "\\begin{align*} ( x + \\lambda ) ^ { [ 2 p ] } & = \\Big ( \\frac { K ( \\lambda x , \\lambda x ) } { 2 } \\Big ) ^ { \\frac { p - 1 } { 2 } } \\lambda x \\\\ & = \\lambda ^ p \\Big ( \\frac { K ( x , x ) } { 2 } \\Big ) ^ { \\frac { p - 1 } { 2 } } x . \\end{align*}"}
-{"id": "9213.png", "formula": "\\begin{align*} z _ { 1 } = E _ { 1 , 2 } + \\varepsilon _ { 1 } E _ { 2 , 1 } , \\ z _ { 2 } = E _ { 2 , 3 } + \\varepsilon _ { 2 } E _ { 3 , 2 } u = u ' = e _ { 3 } \\varepsilon _ { i } = \\pm 1 . \\end{align*}"}
-{"id": "6041.png", "formula": "\\begin{align*} D ( \\widetilde { Q } _ { X Y } \\| \\pi _ { X Y } ) & = 0 , \\\\ D ( \\widetilde { Q } _ { X Y | U } \\| \\widetilde { Q } _ { X | U } \\widetilde { Q } _ { Y | U } | \\widetilde { Q } _ { U } ) & = 0 . \\end{align*}"}
-{"id": "7112.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Phi = ~ \\Phi ' \\dot { F } ^ { i j } \\nabla _ i \\nabla _ j \\Phi + \\Phi \\Phi ' \\left ( \\dot { F } ^ { i j } h _ i ^ k h _ { k j } + K _ N \\dot { F } ^ { i j } g _ { i j } \\right ) , \\end{align*}"}
-{"id": "7967.png", "formula": "\\begin{align*} a _ 1 ( s ) : = L ( \\sigma _ s ) . \\end{align*}"}
-{"id": "8550.png", "formula": "\\begin{align*} | x \\cdot \\xi | \\le H _ 0 ( x ) H ( \\xi ) , H ( \\xi ) = \\sup _ { x \\in { \\bf R } ^ N \\setminus \\{ 0 \\} } \\frac { x \\cdot \\xi } { H _ 0 ( x ) } . \\end{align*}"}
-{"id": "5711.png", "formula": "\\begin{align*} \\begin{dcases} \\int _ \\R ( { u } ( x _ 1 , x _ 2 ) - z ^ + ( x _ 1 ) ) ^ 2 \\d x _ 1 < + \\infty \\mathrm { f o r \\ a . e . \\ } x _ 2 \\in \\R ; \\\\ \\inf \\left \\{ \\int _ \\R ( { u } ( x _ 1 , x _ 2 ) - z ^ \\pm ( x _ 1 - c ) ) ^ 2 \\d x _ 1 \\ ; : \\ ; c \\in \\R \\right \\} \\to 0 \\quad \\mathrm { w h e n \\ } x _ 2 \\to \\pm \\infty . \\end{dcases} \\end{align*}"}
-{"id": "2025.png", "formula": "\\begin{align*} z _ 1 ^ 2 g _ { 1 , 0 } \\Big ( \\frac { z _ 2 z _ 3 } { z _ 1 ^ 2 } \\Big ) f _ 1 ( z _ 2 , z _ 3 ) + z _ 2 ^ 2 g _ { 1 , 0 } \\Big ( \\frac { z _ 3 z _ 1 } { z _ 2 ^ 2 } \\Big ) f _ 1 ( z _ 3 , z _ 1 ) + z _ 3 ^ 2 g _ { 1 , 0 } \\Big ( \\frac { z _ 1 z _ 2 } { z _ 3 ^ 2 } \\Big ) f _ 1 ( z _ 1 , z _ 2 ) = 0 \\end{align*}"}
-{"id": "7749.png", "formula": "\\begin{align*} \\phi ( \\vec { x } ) = ( x _ 1 \\neq 1 ) \\wedge \\bigwedge _ { j = 1 } ^ b ( r _ j ( \\vec { x } ) = 1 ) . \\end{align*}"}
-{"id": "7306.png", "formula": "\\begin{align*} \\omega _ i : = ( \\nabla _ { \\frac { \\partial } { \\partial t _ 1 } } ) ^ { i - 1 } ( \\omega _ 1 ) , \\end{align*}"}
-{"id": "6809.png", "formula": "\\begin{align*} \\int _ { \\mathbb { S } ^ 2 _ { \\lambda } } \\chi _ { R _ 1 , j } \\varphi _ { i , j } \\phi = 0 \\textrm { f o r a l l } i = 0 , 1 , 2 , j = 1 , 2 , 3 , 4 . \\end{align*}"}
-{"id": "5334.png", "formula": "\\begin{align*} { \\mathcal { Y } } ( L _ { - 1 } w _ { ( 1 ) } , x ) = \\frac d { d x } { \\mathcal { Y } } ( w _ { ( 1 ) } , x ) . \\end{align*}"}
-{"id": "2874.png", "formula": "\\begin{align*} S ( x ) = S ( x ; N , h ) = \\frac { ( - 1 ) ^ { h + 1 } } { N } \\left ( \\frac { 2 \\pi } { x } \\right ) ^ { \\frac { N - 2 h + 1 } { N } } \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ { \\frac { 2 h - 1 } { N } } } \\bigg \\{ f _ { 0 } ( x ; n , N ) + \\sum _ { j = 1 } ^ { \\frac { N - 1 } { 2 } } f _ { 2 j } ( x ; n , N , h ) \\bigg \\} \\end{align*}"}
-{"id": "4174.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { k \\in I } e _ { k , j } ^ { n - 1 } + \\frac { 1 } { n } \\sum _ { \\tau = 1 } ^ { n - 2 } e _ { i , j + \\tau } ^ { n - \\tau - 1 } . \\end{align*}"}
-{"id": "640.png", "formula": "\\begin{align*} ( D _ t - \\Delta ) \\nabla ^ k F = \\sum _ { l = 0 } ^ { k - 1 } \\nabla ^ l [ h ( A , A ) * g ^ { - 2 } + \\bar R * ( \\nabla F ) ^ a * g ^ { - b } ] * \\nabla ^ { k - l } F , \\end{align*}"}
-{"id": "10077.png", "formula": "\\begin{align*} g \\mathfrak { a } = \\mathfrak { a } _ 1 \\oplus \\cdots \\oplus \\mathfrak { a } _ n \\end{align*}"}
-{"id": "9104.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } \\dfrac { a _ { n + 1 - i } } { \\binom { n + 1 } { i } } \\sum _ { \\mathrm { c a r d } ( I ) = i } \\left ( \\prod _ { j \\in I } x _ { j } \\right ) \\cdot A \\left ( \\prod _ { k \\in \\left \\{ 1 , \\ldots , n + 1 \\right \\} \\setminus I } x _ { k } \\right ) = 0 , \\end{align*}"}
-{"id": "7207.png", "formula": "\\begin{align*} w _ T ( z ) = \\begin{cases} \\frac { ( | z - x _ T | ^ { 2 - n } - R _ T ^ { 2 - n } ) ^ + } { ( n - 2 ) R _ T ^ { 1 - n } } & T < \\ 8 \\\\ ( e \\cdot ( z - q ) ) ^ + & T = \\ 8 \\end{cases} \\end{align*}"}
-{"id": "2515.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { F } ( t , h ) & \\equiv \\int h ^ 2 \\varrho \\ , m _ 1 + \\alpha _ 1 t \\int \\left | \\nabla _ \\xi h \\right | ^ 2 \\varrho \\ , m _ 0 + \\alpha _ 2 t ^ 2 \\int \\nabla _ x h \\cdot \\nabla _ { \\xi } h \\ , \\varrho \\ , m _ 0 \\\\ & + \\alpha _ 3 t ^ 3 \\int \\left | \\nabla _ x h \\right | ^ 2 \\varrho \\ , m _ 0 , \\end{aligned} \\end{align*}"}
-{"id": "9337.png", "formula": "\\begin{align*} N r ^ { 2 } ( p ) = \\overline { p } \\cdot p = p \\cdot \\overline { p } = \\sum _ { s = 0 } ^ { 7 } a _ { s } ^ { 2 } . \\end{align*}"}
-{"id": "2174.png", "formula": "\\begin{align*} F _ d ^ 0 ( r , t ) & = \\int _ 0 ^ r s ^ { 1 - d } [ \\nu _ d ( s ) ] ^ { - 1 } \\left ( \\int _ 0 ^ s \\tau ^ { d - 1 } \\nu _ d ( \\tau ) ( \\partial _ t u _ * ) ( \\tau , t ) \\ , d \\tau \\right ) \\ , d s \\\\ & = \\int _ 0 ^ r s ^ { 1 - d } [ \\nu _ d ( s ) ] ^ { - 1 } \\left ( \\int _ 0 ^ s \\tau ^ { d - 1 } \\nu _ d ( \\tau ) \\biggr [ ( \\partial _ t u _ * ) ( 0 , t ) + F _ d ^ 1 ( \\tau , t ) \\biggr ] \\ , d \\tau \\right ) \\ , d s \\\\ & = ( \\partial _ t u _ * ) ( 0 , t ) F _ d ( r ) + G _ d ( r , t ) \\end{align*}"}
-{"id": "5390.png", "formula": "\\begin{align*} a & : = ( 1 , 2 ) ( 3 , 4 ) ( 5 , 6 ) ( 7 , 8 ) ( 9 , 1 0 ) ( 1 1 , 1 2 ) \\\\ b & : = ( 1 , 3 ) ( 2 , 4 ) ( 5 , 6 ) ( 7 , 8 ) ( 1 3 , 1 4 ) ( 1 5 , 1 6 ) \\\\ c & : = ( 1 , 1 2 ) ( 3 , 1 4 ) ( 4 , 6 ) ( 5 , 1 6 ) ( 7 , 1 1 ) ( 9 , 1 3 ) . \\end{align*}"}
-{"id": "2723.png", "formula": "\\begin{align*} - \\frac { d } { d t } \\int _ { \\mathbb R ^ d } \\ , f \\log f \\ , d v = - \\int _ { \\mathbb R ^ d } \\ , \\mathcal Q ( f , f ) \\log ( f ) \\ , d v \\geq 0 . \\end{align*}"}
-{"id": "449.png", "formula": "\\begin{align*} \\Upsilon ( x , t ) = c _ { k _ 1 , k _ 2 + 1 , ( k _ 2 + 1 ) / 2 } \\frac { | t | } { | x | ^ { k _ 2 + 1 } } + O \\left ( \\frac { | t | } { | x | ^ { k _ { 2 } + 3 } } \\right ) ; \\end{align*}"}
-{"id": "3542.png", "formula": "\\begin{align*} 0 \\leq \\sum _ { i = 1 } ^ n P ( A _ i ) - P ( \\bigcup _ { i = 1 } ^ n A _ i ) \\leq \\sum _ { 1 \\leq i < j \\leq n } P ( A _ i \\cap A _ j ) , \\end{align*}"}
-{"id": "451.png", "formula": "\\begin{align*} p _ { 1 , k _ 1 , k _ 2 } ( x , t ) = \\frac { 1 } { \\abs { x } ^ m } e ^ { - \\frac { 1 } { 4 } d ( x , t ) ^ 2 } \\Psi ( \\omega ) \\left [ \\sum _ { j = 0 } ^ { k } \\frac { 4 ^ j L _ { j , \\psi _ \\omega } a _ { k _ 1 , k _ 2 , \\omega } } { | x | ^ { 2 j } } + O \\left ( \\frac { 1 } { \\abs { x } ^ { 2 k + 2 } } \\right ) \\right ] \\end{align*}"}
-{"id": "3449.png", "formula": "\\begin{align*} C _ 1 = 2 \\big ( 1 + \\| u \\| _ { C ( [ 0 , T ] ; \\R ^ d ) } \\big ) ^ 2 \\Big ( 1 + T ^ { \\frac { 1 } { 2 } } \\big \\| L _ { K _ u } + g \\big \\| _ { L ^ 2 ( 0 , T ; \\R ) } \\Big ) ^ 2 . \\end{align*}"}
-{"id": "738.png", "formula": "\\begin{align*} ( u \\ast v ) ( x ) = ( u \\ast v ) ( x ' , x _ 3 ) : = \\int _ { 0 } ^ { \\infty } \\chi _ { \\{ | u ( x ' , y ) | > t \\} \\ast \\{ | v ( x ' , y ) | > t \\} } ( x _ 3 ) \\ , d t , \\end{align*}"}
-{"id": "9659.png", "formula": "\\begin{align*} B : = \\Sigma ^ \\downarrow _ \\pi ( A ) , \\end{align*}"}
-{"id": "590.png", "formula": "\\begin{align*} \\{ z \\in \\overline { \\Omega } : | z | \\leqslant m \\} = \\overline { \\Omega } \\cap \\overline { D ( 0 , m ) } & \\subseteq \\overline { \\Omega \\cap D ( 0 , m + 1 ) } = \\overline { \\Omega _ { m + 1 } } \\\\ & \\subseteq \\overline { \\Omega } \\cap \\overline { D ( 0 , m + 1 ) } = \\{ z \\in \\overline { \\Omega } : | z | \\leqslant m + 1 \\} , \\end{align*}"}
-{"id": "2153.png", "formula": "\\begin{align*} w ( \\xi , s ) = ( 1 + t ) ^ { \\frac { d } { 2 } } r ^ { - A } u ( r , t ) = ( 1 + t ) ^ { \\frac { d } { 2 } } r ^ { - A } U ( r ) u _ * ( r , t ) = ( 1 + t ) ^ { \\frac { d } { 2 } } U _ d ( r ) u _ * ( r , t ) & \\\\ \\mbox { w i t h $ \\xi = ( 1 + t ) ^ { - \\frac { 1 } { 2 } } r $ a n d $ s = \\log ( 1 + t ) $ } , & \\end{align*}"}
-{"id": "8374.png", "formula": "\\begin{align*} g a ( \\mathbb { H } \\otimes \\widehat { \\Z } ) = \\alpha ( \\mathbb { H } \\otimes \\widehat { \\Z } ) . \\end{align*}"}
-{"id": "743.png", "formula": "\\begin{align*} \\lambda _ 0 : = \\inf _ { \\int _ { \\R ^ 2 } | v | ^ 2 \\ , d x ' = 1 } \\int _ { \\R ^ 2 } | \\partial _ { x _ 1 } v | ^ 2 + | \\partial _ { x _ 2 } v | ^ 2 + ( x _ 1 ^ 2 + x _ 2 ^ 2 ) | v | ^ 2 \\ , d x ' . \\end{align*}"}
-{"id": "8408.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } H = & \\mathcal { L } H + \\ddot { F } ^ { k l , r s } \\nabla ^ i h _ { k l } \\nabla _ i h _ { r s } + \\dot { F } ^ { k l } h _ { k m } h ^ m _ l H + ( 1 - \\alpha ) F | h | ^ 2 \\\\ [ 5 p t ] \\frac { \\partial } { \\partial t } G = & \\mathcal { L } G + [ \\dot { G } ^ { i j } \\ddot { F } ^ { k l , r s } - \\dot { F } ^ { i j } \\ddot { G } ^ { k l , r s } ] \\nabla _ i h _ { k l } \\nabla _ j h _ { r s } + \\dot { F } ^ { k l } h _ { k m } h _ l ^ m \\dot { G } ^ { i j } h _ { i j } \\\\ & + ( 1 - \\alpha ) F \\dot { G } ^ { i j } h _ { i m } h ^ m _ j \\end{align*}"}
-{"id": "6957.png", "formula": "\\begin{align*} F _ s [ u ] ( x ) = \\inf _ { M \\in \\mathcal { M } } \\{ - C _ { n , s } ^ { - 1 } ( - \\Delta ) ^ s ( u \\circ \\sqrt { M } ) ( x ) \\} . \\end{align*}"}
-{"id": "2908.png", "formula": "\\begin{align*} f ( \\langle n _ \\sigma , k \\rangle ) = 1 \\Leftrightarrow y \\not \\in T J ( H _ c ^ X ) \\Leftrightarrow k \\not \\in H _ b ^ X \\Leftrightarrow \\sigma ( k ) = H _ b ^ X ( k ) . \\end{align*}"}
-{"id": "8177.png", "formula": "\\begin{align*} T _ { \\nu } = \\max \\left ( \\tfrac { 1 } { 2 } , \\abs { t _ { \\nu } } \\right ) . \\end{align*}"}
-{"id": "1058.png", "formula": "\\begin{align*} \\vert P _ { k } ( \\zeta ) \\vert = \\vert V _ { k , 0 } ( \\zeta ) \\vert = \\max _ { 0 \\leq j \\leq n - 2 } \\vert V _ { k , j } ( \\zeta ) \\vert , k \\geq 1 . \\end{align*}"}
-{"id": "2222.png", "formula": "\\begin{align*} \\left | q _ i \\left ( \\dfrac 1 { w _ 1 } , \\ldots , \\dfrac 1 { w _ n } \\right ) \\right | > \\left | t \\cdot Q _ i \\left ( \\dfrac 1 { w _ 1 } , \\ldots , \\dfrac 1 { w _ n } \\right ) \\right | , i = 1 , \\ldots , n . \\end{align*}"}
-{"id": "5260.png", "formula": "\\begin{align*} \\dim ( E ^ u ( \\lambda ^ * , z ) \\cap E ^ s ( \\lambda ^ * , z ) ) = 2 . \\end{align*}"}
-{"id": "10101.png", "formula": "\\begin{align*} \\tilde { R } ( Y , Z ) X = \\lambda \\pi ( X ) \\{ \\pi ( Y ) Z - \\pi ( Z ) Y \\} , \\end{align*}"}
-{"id": "1033.png", "formula": "\\begin{align*} \\Phi _ { n } ( \\underline { x } ) = \\det \\Lambda _ { n } ( \\underline { x } ) , \\end{align*}"}
-{"id": "8219.png", "formula": "\\begin{align*} 1 + U _ i = ( 1 + \\hat { H _ i } ) ( 1 + K _ i ) \\\\ \\end{align*}"}
-{"id": "6002.png", "formula": "\\begin{gather*} E _ 3 = \\frac { e _ 0 } { 4 } ( 1 + x - y - x y ) \\end{gather*}"}
-{"id": "4206.png", "formula": "\\begin{align*} a ^ 2 = \\frac { \\Omega _ 2 \\Omega _ 3 } { \\Omega _ 1 } , b ^ 2 = \\frac { \\Omega _ 3 \\Omega _ 1 } { \\Omega _ 2 } , c ^ 2 = \\frac { \\Omega _ 1 \\Omega _ 2 } { \\Omega _ 3 } . \\end{align*}"}
-{"id": "83.png", "formula": "\\begin{align*} \\Gamma _ W ( a ) = W ^ * ( 1 \\otimes a ) W \\ \\mbox { a n d } \\ \\Gamma ' _ V ( b ) = V ( b \\otimes 1 ) V ^ * . \\end{align*}"}
-{"id": "10068.png", "formula": "\\begin{align*} \\| \\eta \\| _ s ^ 2 = \\Big | \\int _ { A _ s ( \\C ) } \\eta \\wedge \\overline { \\eta } \\ \\Big | . \\end{align*}"}
-{"id": "2705.png", "formula": "\\begin{align*} T ( A ) : = N S ( A ) ^ { \\perp } \\subset H ^ { 2 } ( A , \\Z ( 1 ) ) . \\end{align*}"}
-{"id": "9835.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { [ \\frac { 2 n + 1 - \\nu } { 2 } ] } { { 2 n + 1 - 2 i } \\choose { \\nu } } A _ i = - q ^ { n - \\nu + 1 / 2 } \\sum _ { i = 0 } ^ { [ \\frac { \\nu } { 2 } ] } { { 2 n + 1 - 2 i } \\choose { 2 n + 1 - \\nu } } A _ i \\qquad ( \\nu = 0 , 1 , \\cdots , 2 n + 1 ) , \\end{align*}"}
-{"id": "910.png", "formula": "\\begin{align*} I _ p ( f ) I _ q ( g ) = \\sum _ { r = 0 } ^ { p \\wedge q } r ! \\binom { p } { r } \\binom { q } { r } I _ { p + q - 2 r } ( f \\widetilde { \\otimes } _ r g ) . \\end{align*}"}
-{"id": "5156.png", "formula": "\\begin{align*} \\tau _ 1 = \\inf \\{ t \\geq 0 : X _ t \\leq \\ell \\} \\tau _ 2 = \\inf \\{ t \\geq 0 : X _ t \\geq r \\} , \\end{align*}"}
-{"id": "3579.png", "formula": "\\begin{align*} \\| \\bar \\xi _ { x _ 1 } - \\bar \\xi _ { x _ 2 } \\| _ 1 = \\| \\eta ( x _ 1 ) \\cdot \\xi _ { u ( x _ 1 ) } - \\eta ( x _ 2 ) \\cdot \\xi _ { u ( x _ 2 ) } \\| _ 1 \\end{align*}"}
-{"id": "7791.png", "formula": "\\begin{align*} \\Vert F - B ( x _ h ) \\Vert _ { Y _ h ^ * } = \\min _ { \\xi _ h \\in X _ h } \\Vert F - B ( \\xi _ h ) \\Vert _ { Y _ h ^ * } . \\end{align*}"}
-{"id": "7533.png", "formula": "\\begin{gather*} \\tilde w ^ j _ 1 = \\tilde w ^ j _ { 2 2 } , j = 1 , 2 . \\end{gather*}"}
-{"id": "7276.png", "formula": "\\begin{align*} \\sum _ { k , \\ell = 1 } ^ M c _ k c _ { \\ell } \\frac { ( n _ k , n _ \\ell ) ^ 2 } { n _ k n _ \\ell } \\ll ( \\log \\log M ) ^ 2 \\sum _ k c _ k ^ 2 . \\end{align*}"}
-{"id": "4896.png", "formula": "\\begin{align*} \\mathrm { t r } ( ( X ^ * X ) ^ 2 ) = x _ 1 ^ 2 + x _ 2 ^ 2 = y _ 1 ^ 2 + y _ 2 ^ 2 = \\mathrm { t r } ( ( Y ^ * Y ) ^ { 2 } ) . \\end{align*}"}
-{"id": "9345.png", "formula": "\\begin{align*} N r ^ { 2 } ( J O _ { n } ^ { ( 3 ) } ) = \\sum _ { s = 0 } ^ { 7 } ( J _ { n + s } ^ { ( 3 ) } ) ^ { 2 } \\ \\textrm { a n d } \\ N r ^ { 2 } ( j O _ { n } ^ { ( 3 ) } ) = \\sum _ { s = 0 } ^ { 7 } ( j _ { n + s } ^ { ( 3 ) } ) ^ { 2 } . \\end{align*}"}
-{"id": "6583.png", "formula": "\\begin{align*} d _ M ( z _ 0 , z _ { m _ { N } } ) \\leq \\sum _ { n = 0 } ^ { N } d _ M ( z _ { m _ { n + 1 } } , z _ { m _ n } ) \\leq \\sum _ { n = 0 } ^ { N } \\left ( \\frac { 3 } { 4 } \\right ) ^ n d _ M ( z _ { m _ 0 } , z _ { m _ 1 } ) = \\frac { 4 } { 5 } \\eta \\ . \\end{align*}"}
-{"id": "4115.png", "formula": "\\begin{align*} \\mathrm { d i v } _ { \\nabla } X | _ p = \\mathrm { t r } \\{ Y \\mapsto \\nabla _ Y X : Y \\in T _ p M \\} \\end{align*}"}
-{"id": "8971.png", "formula": "\\begin{gather*} c _ w + { } ^ w h _ { w ^ { - 1 } r w } \\zeta _ { w , w ^ { - 1 } r w } ^ { - 1 } c _ { r w } = 0 \\in { \\cal Z } _ w ( [ X ^ r ] ) | _ { [ X ^ r ] } , \\end{gather*}"}
-{"id": "4235.png", "formula": "\\begin{align*} \\delta ( a \\star g ) & = \\delta \\circ \\left ( \\sum _ { i = 0 } ^ { f - 1 } \\sigma ^ { i } ( a ) g _ i \\right ) \\circ \\delta ^ { - 1 } \\\\ & = \\sum _ { i = 0 } ^ { f - 1 } \\sigma ^ { i } ( \\delta ( a ) ) \\delta \\circ g _ i \\circ \\delta ^ { - 1 } \\\\ & = \\delta ( a ) \\star \\delta ( g ) \\end{align*}"}
-{"id": "9406.png", "formula": "\\begin{align*} \\lambda _ 0 = \\frac { 2 | Q _ { L + 1 } | } { \\beta ^ 2 \\varepsilon ^ d p q } \\end{align*}"}
-{"id": "421.png", "formula": "\\begin{align*} \\Im \\ , \\psi ( \\lambda , y ) = \\frac { \\lambda \\cosh ( \\lambda ) \\sinh ( \\lambda ) - y \\cot ( y ) \\sinh ( \\lambda ) ^ 2 } { \\sinh ( \\lambda ) ^ 2 + \\sin ( y ) ^ 2 } \\end{align*}"}
-{"id": "7005.png", "formula": "\\begin{align*} P ^ T \\sqrt { M } ^ { - 1 } P = J = d i a g \\{ \\lambda _ 1 , . . . , \\lambda _ n \\} . \\end{align*}"}
-{"id": "7285.png", "formula": "\\begin{align*} ( \\tt ) \\cdot \\phi = ( x \\ , y \\ , z ) \\cdot B . \\end{align*}"}
-{"id": "8229.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ k \\binom { k } { \\ell } ( - 1 ) ^ { \\ell + k } \\ell ^ d = \\Delta ^ k i ^ d \\end{align*}"}
-{"id": "2612.png", "formula": "\\begin{align*} \\mathcal C _ a ( A + B t ) = \\frac { \\cos \\tfrac { a \\pi } 2 } { \\pi \\phi _ a ( t ) } \\Big ( \\frac B a t + \\Big [ A - \\frac { 1 - a } { 2 a } B \\Big ] \\Big ) . \\end{align*}"}
-{"id": "3400.png", "formula": "\\begin{align*} \\epsilon _ 1 = \\min \\{ \\epsilon ( t ) : - 1 \\le t \\le 1 \\} > 0 . \\end{align*}"}
-{"id": "97.png", "formula": "\\begin{align*} p _ { \\mathbf { S } | \\mathbf { W } } ( S | W ) = \\left \\{ \\begin{array} { l l } \\frac { 1 } { \\binom { K - 1 } { M } } , & \\textrm { i f } \\ : \\ : W \\not \\in S \\ : \\ : \\textrm { a n d } \\ : \\ : | S | = M , \\\\ 0 , & \\textrm { o t h e r w i s e } . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "9644.png", "formula": "\\begin{align*} \\begin{cases} \\bar { \\Phi } _ { \\rm { d n } } | ^ { \\tau ^ { - } _ { i } } _ { \\tau _ { i - 1 } } > \\bar { \\Phi } _ { \\rm { d n } } | ^ { \\tau _ { i } } _ { \\tau _ { i - 1 } } \\\\ \\bar { \\Phi } _ { \\rm { d n } } | ^ { \\tau ^ { + } _ { i } } _ { \\tau _ { i - 1 } } > \\bar { \\Phi } _ { \\rm { d n } } | ^ { \\tau _ { i } } _ { \\tau _ { i - 1 } } . \\end{cases} \\end{align*}"}
-{"id": "804.png", "formula": "\\begin{align*} \\begin{aligned} S _ { k } ( 2 m ) = m ^ { k } + \\sum _ { j = 1 } ^ { m } ( ( m - j ) ^ k + ( m + j ) ^ k ) = & m ^ { k } + 2 \\sum _ { j = 1 } ^ { m } \\sum _ { i = 0 } ^ { [ \\frac { k } { 2 } ] } { \\binom { k } { 2 i } } m ^ { k - 2 i } j ^ { 2 i } \\\\ = & m ^ { k } + 2 \\sum _ { i = 0 } ^ { [ \\frac { k } { 2 } ] } { \\binom { k } { 2 i } } m ^ { k - 2 i } S _ { 2 i } ( m ) . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "6315.png", "formula": "\\begin{align*} \\Tilde { \\xi } _ m ^ 2 = \\Tilde { A } _ m ( \\Tilde { \\lambda } ) = \\Tilde { A } _ m \\circ \\phi _ m ^ { - 1 } ( \\Tilde { \\lambda } ) = \\Tilde { A } _ m ( z _ m ) \\quad \\hbox { f o r } m = 1 , \\ldots M \\end{align*}"}
-{"id": "2356.png", "formula": "\\begin{align*} H _ { s , m } = H _ { s , m } \\circ Q _ M + H _ { s , m } \\circ P _ M . \\end{align*}"}
-{"id": "8669.png", "formula": "\\begin{align*} 1 = \\sum _ j ( P ) _ { k j } = \\sum _ { j \\in A } ( P ) _ { k j } + ( P ) _ { k \\ell } \\end{align*}"}
-{"id": "3412.png", "formula": "\\begin{align*} \\tau _ x \\circ \\theta _ n \\circ \\tau _ x = \\theta _ n . \\end{align*}"}
-{"id": "621.png", "formula": "\\begin{align*} B ( z , w ) = \\sum _ { \\alpha _ 1 \\ge 0 , \\alpha _ 2 \\in \\Z } c _ { \\alpha } ^ 2 z _ 1 ^ { \\alpha _ 1 } z _ 2 ^ { \\alpha _ 2 } \\bar w _ 1 ^ { \\alpha _ 1 } \\bar w _ 2 ^ { \\alpha _ 2 } . \\end{align*}"}
-{"id": "8526.png", "formula": "\\begin{align*} T _ + = \\sum _ { ( j , l ) \\in { \\mathbb Z } ^ 3 \\times { \\mathbb Z } ^ 3 } T _ { ( j , l ) } \\end{align*}"}
-{"id": "4757.png", "formula": "\\begin{align*} \\alpha \\Omega P _ { , 0 } + V ^ \\mu ( { L } _ \\mu P _ { , \\mu } + 2 { L } _ \\mu ' ) = 0 . \\end{align*}"}
-{"id": "6439.png", "formula": "\\begin{align*} d s ^ { 2 } = { \\displaystyle \\sum \\limits _ { k = 1 } ^ { l } } \\left ( \\frac { 1 } { \\sigma _ { k } ^ { 2 } } d \\mu _ { k } ^ { 2 } + \\frac { 2 } { \\sigma _ { k } ^ { 2 } } d \\sigma _ { k } ^ { 2 } \\right ) \\end{align*}"}
-{"id": "3601.png", "formula": "\\begin{align*} w \\Big ( ( n , k ) \\rightarrow ( n + 1 , m ) \\Big ) : = \\langle x P _ { k } ^ N , Q _ { m } ^ N \\rangle . \\end{align*}"}
-{"id": "5448.png", "formula": "\\begin{align*} \\mathcal { Q } _ { r , \\rho } = \\Big \\{ q : ( 0 , \\infty ) \\to ( 0 , \\infty ) \\ , : \\ , s ^ { - \\rho } q ( s ) q ( s r ) = q ( s ) , \\ \\forall s > 0 \\Big \\} . \\end{align*}"}
-{"id": "2244.png", "formula": "\\begin{gather*} \\sum _ { j = 1 } ^ p \\frac { 1 } { z _ { j 1 } ( t ) ^ { \\gamma _ 1 + 1 } \\cdots z _ { j n } ( t ) ^ { \\gamma _ n + 1 } } \\\\ = \\sum _ { K \\in \\Re } ( - t ) ^ { | | K | | + n } \\sum _ J \\frac { ( - 1 ) ^ { s ( J ) } } { \\beta ( K , J ) ! } \\cdot \\frac { \\partial ^ { | | \\beta ( K , J ) | | } } { \\partial w ^ \\beta ( K , J ) } \\left [ \\widetilde \\Delta ( t ) \\cdot w _ 1 ^ { \\gamma _ 1 + 1 } \\cdots w _ n ^ { \\gamma _ n + 1 } \\cdot \\frac { \\widetilde Q ^ K } { \\widetilde q ^ { K + I } ( J ) } \\right ] _ { w = a _ J } . \\end{gather*}"}
-{"id": "1509.png", "formula": "\\begin{align*} t _ { n } ( x ) = \\underline { \\alpha } \\alpha ^ { n } ( x ) + \\underline { \\omega _ { 1 } } \\omega _ { 1 } ^ { n } ( x ) + \\underline { \\omega _ { 2 } } \\omega _ { 2 } ^ { n } ( x ) , \\end{align*}"}
-{"id": "765.png", "formula": "\\begin{align*} P _ { g } = \\{ ( y , \\omega ) \\in Y \\times P Z \\mid \\omega ( 0 ) = g ( y ) \\} , \\end{align*}"}
-{"id": "800.png", "formula": "\\begin{align*} a _ 1 = 1 . 0 2 \\cdot ( \\rho + 3 ) \\log ( 2 x ) , \\end{align*}"}
-{"id": "1632.png", "formula": "\\begin{align*} T _ \\lambda [ \\xi ^ i _ \\mu ] = \\left \\{ \\begin{array} { c l } [ \\xi ^ i _ { \\lambda \\mu } ] , & s ( \\lambda ) = r ( \\mu ) \\\\ 0 , & \\end{array} \\right . \\end{align*}"}
-{"id": "9808.png", "formula": "\\begin{align*} w ' ( t ) = \\frac { \\l f ( t ) - \\mu ( 1 - p _ 0 ( t ) ) } { \\mu } , 0 \\leq t \\leq T _ { e _ 2 } . \\end{align*}"}
-{"id": "675.png", "formula": "\\begin{gather*} \\mu _ n ( q , \\alpha , \\beta ) = \\mu _ n ( q _ 0 , \\alpha _ 0 , \\beta _ 0 ) , \\\\ a _ n ( q , \\alpha , \\beta ) = a _ n ( q _ 0 , \\alpha _ 0 , \\beta _ 0 ) , \\end{gather*}"}
-{"id": "6444.png", "formula": "\\begin{align*} \\rho _ { i j } = \\rho \\left ( x _ { i } , x _ { j } \\right ) \\overset { } { = } \\frac { \\left \\langle x _ { i } x _ { j } \\right \\rangle - \\left \\langle x _ { i } \\right \\rangle \\left \\langle x _ { j } \\right \\rangle } { \\sigma _ { i } \\sigma _ { j } } , \\end{align*}"}
-{"id": "6886.png", "formula": "\\begin{align*} - \\Delta f _ i & = \\lambda _ i f _ i \\\\ - \\Delta f _ j & = \\lambda _ j f _ j \\\\ \\implies ( \\lambda _ j - \\lambda _ i ) f _ i f _ j & = f _ i ( - \\Delta f _ j ) - f _ j ( - \\Delta f _ i ) & & \\end{align*}"}
-{"id": "1170.png", "formula": "\\begin{align*} C = \\frac { 1 } { d _ { k - 1 } } + \\frac { 1 + D ^ 2 _ { k - 1 } + D _ { k - 2 } ^ 4 + \\cdots + D _ 2 ^ { 2 ^ { k - 2 } } } { d _ k } , \\end{align*}"}
-{"id": "3891.png", "formula": "\\begin{align*} \\widetilde { \\omega } ( 1 ) \\frac { \\zeta ( 2 s ) } { \\zeta ( s ) } \\sum _ { q = 1 } ^ { \\infty } \\frac { 1 } { q ^ { 1 + s } } \\sum _ { a , b = 1 } ^ { q } \\delta _ q ( a b - l ^ 2 ) = \\widetilde { \\omega } ( 1 ) \\sigma _ { - s } ( l ^ 2 ) \\frac { \\zeta ( 2 s ) } { \\zeta ( 1 + s ) } . \\end{align*}"}
-{"id": "542.png", "formula": "\\begin{align*} \\omega ( X _ { \\lambda } , \\cdot ) = \\lambda \\end{align*}"}
-{"id": "6380.png", "formula": "\\begin{align*} g _ { i , j } : = \\begin{cases} 1 & , i = 0 , \\\\ 0 & , i \\ne 0 , i > j , \\\\ g _ { i , j - 1 } + z _ { j - i + 1 } g _ { i - 1 , j } & , i \\ne 0 , i \\le j . \\end{cases} \\end{align*}"}
-{"id": "980.png", "formula": "\\begin{align*} M ^ 0 _ n ( t ) = 2 \\sum _ { i = 1 } ^ n K _ h ( t _ { i - 1 } - t ) \\int _ { t _ { i - 1 } } ^ { t _ i } \\int _ { t _ { i - 1 } } ^ s d B _ r d B _ s , t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "9805.png", "formula": "\\begin{align*} & \\mathbb { E } \\left ( \\left [ \\sum \\limits _ { i = 1 } ^ { N ( t ) } X _ i + t \\right ] ^ + \\right ) = \\sum _ { j = 0 } ^ { \\infty } p _ j ( t ) \\int _ { s = 0 } ^ { \\infty } \\sum _ { k = 0 } ^ { j - 1 } \\frac { 1 } { k ! } e ^ { - \\mu ( s - t ) } ( \\mu ( s - t ) ) ^ { k } d s , \\end{align*}"}
-{"id": "9568.png", "formula": "\\begin{align*} u _ E = \\frac { 1 } { \\lvert E \\rvert } \\int _ { E } u ( y ) \\ , d y \\ , . \\end{align*}"}
-{"id": "1372.png", "formula": "\\begin{align*} A ( Q , P ) = U [ \\Sigma ( Q , P ) \\ , \\ , \\ , \\ , \\ , 0 ] R ^ T \\ , . \\end{align*}"}
-{"id": "3070.png", "formula": "\\begin{align*} \\epsilon \\dot { u } = - \\nabla _ { { x } } F ( t , u ) \\end{align*}"}
-{"id": "1155.png", "formula": "\\begin{align*} \\rho _ { a , 0 } & = a & & \\\\ \\rho _ { a , 1 } & = a ^ 2 + a & & \\end{align*}"}
-{"id": "4809.png", "formula": "\\begin{align*} \\mathsf { L } ( x ) : = \\{ | z | : z \\in \\mathsf { Z } ( x ) \\} . \\end{align*}"}
-{"id": "5433.png", "formula": "\\begin{align*} \\hat { f } ( P ) : = ( 1 , f ( P ) ) = ( f _ { - 1 } , f _ 0 , f _ 1 , \\dots , f _ { d - 1 } ) \\in \\Z ^ { d + 1 } . \\end{align*}"}
-{"id": "4424.png", "formula": "\\begin{align*} [ f ] _ \\beta : = \\inf \\{ | c | + [ g ] _ { \\beta + 2 } + [ h ] _ { \\beta + \\frac { 3 } { 2 } } \\ , : \\ , f = c + \\partial _ 1 ^ 2 g + \\partial _ 2 h \\} . \\end{align*}"}
-{"id": "4198.png", "formula": "\\begin{align*} E _ T : \\ \\ y ^ 2 - 4 ( x - T _ 1 ) ( x - T _ 2 ) ( x - T _ 3 ) = 0 , \\\\ \\textnormal { a n d m o d u l i } \\mathtt { T _ H } = \\{ ( T _ 1 , T _ 2 , T _ 3 ) \\in \\C ^ 3 | \\ , T _ 1 \\neq T _ 2 \\neq T _ 3 \\} . \\end{align*}"}
-{"id": "7710.png", "formula": "\\begin{align*} C _ R ( j ) = \\left ( \\frac { 1 } { N } D _ R ( j ) \\right ) ^ { - 1 } \\ , . \\end{align*}"}
-{"id": "5513.png", "formula": "\\begin{align*} \\mu ( ( x , \\infty ) ) = x ^ { - \\alpha } p _ + ( x ) , \\ \\mu ( ( - \\infty , x ) ) = x ^ { - \\alpha } p _ - ( x ) , \\ x > 0 , \\end{align*}"}
-{"id": "6161.png", "formula": "\\begin{align*} f _ 1 f _ 2 f _ 3 \\equiv 1 \\end{align*}"}
-{"id": "3331.png", "formula": "\\begin{align*} & I \\left ( W _ { k : K } ; \\mathbb { H } , Q _ { 1 : N } ^ { [ k - 1 ] } , A _ { 1 : N } ^ { [ k - 1 ] } | W _ { 1 : k - 1 } , Z \\right ) \\\\ & = I \\left ( W _ { k : K } ; \\mathbb { H } , Q _ { 1 : N } ^ { [ k - 1 ] } , A _ { 1 : N } ^ { [ k - 1 ] } , W _ { k - 1 } | W _ { 1 : k - 2 } , Z \\right ) - I \\left ( W _ { k : K } ; W _ { k - 1 } | W _ { 1 : k - 2 } , Z \\right ) \\end{align*}"}
-{"id": "4550.png", "formula": "\\begin{align*} K _ 0 = \\prod _ { i = 1 } ^ { n - 1 } K _ i ^ { - 1 } . \\end{align*}"}
-{"id": "8681.png", "formula": "\\begin{align*} & H ( e _ i ) = ( \\alpha - 2 [ i ] ) e _ i ~ ; \\\\ & E ( e _ 0 ) = 0 ~ ; & E ( e _ i ) = & [ i ] ( \\alpha - ( [ i ] - 1 ) ) e _ { i - 1 } , 1 \\leq i \\leq m ~ ; \\\\ & F ( e _ m ) = \\beta e _ 0 ~ ; & F ( e _ i ) = & e _ { i + 1 } , 0 \\leq i \\leq m - 1 . \\end{align*}"}
-{"id": "9953.png", "formula": "\\begin{align*} { \\cal N } ( w ) - V ( J _ 1 ) = { \\cal N } ( \\psi ( w ) ) - V ( J _ 2 ) , \\end{align*}"}
-{"id": "8535.png", "formula": "\\begin{align*} & \\mathcal { T } ^ * _ { ( j , - l ) } \\mathcal { T } _ { ( k , - m ) } g ( \\xi ) = \\int \\mathcal { K } _ { ( j , - l ) , ( k , - m ) } ( \\xi , \\eta ) g ( \\eta ) d \\eta \\\\ & = \\int \\mathcal { K } _ { ( 0 , j - l ) , ( 1 , j - m ) } ( \\tilde { \\xi } , \\tilde { \\eta } ) g _ { - j } ( \\tilde { \\eta } ) d \\tilde { \\eta } = \\mathcal { T } ^ * _ { ( 0 , j - l ) } \\mathcal { T } _ { ( 1 , j - m ) } g _ { - j } ( \\tilde { \\xi } ) . \\end{align*}"}
-{"id": "5408.png", "formula": "\\begin{align*} \\mathcal { I } _ p : = \\{ u \\in L ^ p ( E , m ) : \\alpha U _ \\alpha u = u \\ ; m \\mbox { - a . e . } , \\ ; \\alpha > 0 \\} . \\end{align*}"}
-{"id": "9153.png", "formula": "\\begin{align*} f _ { 3 } ( x ^ { 3 } ) + x f _ { 2 } ( x ^ { 2 } ) + x ^ { 2 } f _ { 1 } ( x ) = 0 \\left ( x \\in R \\right ) \\end{align*}"}
-{"id": "7444.png", "formula": "\\begin{align*} ( a ) = \\begin{pmatrix} 1 , 0 \\end{pmatrix} , \\ ; ( a ^ * ) = \\begin{pmatrix} a _ 0 ^ * \\\\ a _ 1 ^ * \\end{pmatrix} , \\ ; ( \\beta ) = \\begin{pmatrix} 0 & 1 \\\\ b _ { 1 0 } & b _ { 1 1 } \\\\ \\end{pmatrix} , \\ ; ( \\gamma ) = \\begin{pmatrix} c _ { 0 0 } & c _ { 0 1 } \\\\ c _ { 1 0 } & c _ { 1 1 } \\\\ \\end{pmatrix} , \\ ; ( \\delta ) = \\begin{pmatrix} d _ { 0 0 } & d _ { 0 1 } \\\\ d _ { 1 0 } & d _ { 1 1 } \\\\ \\end{pmatrix} \\end{align*}"}
-{"id": "6062.png", "formula": "\\begin{align*} \\Phi ^ { '' } ( r ) + ( n - 1 ) \\tfrac { f ' ( r ) } { f ( r ) } \\Phi ' ( r ) - 2 \\mathfrak { e } \\tfrac { \\varphi ( \\Phi ( r ) ) \\varphi ' ( \\Phi ( r ) ) } { f ( r ) ^ 2 } = 0 . \\end{align*}"}
-{"id": "779.png", "formula": "\\begin{align*} S _ { k } ( x ) = \\frac { 1 } { k + 1 } ( B _ { k + 1 } ( x + 1 ) - B _ { k + 1 } ( 0 ) ) . \\end{align*}"}
-{"id": "5192.png", "formula": "\\begin{align*} \\hat { U } _ { i } ( s ^ { * } ) = \\sup \\limits _ { ( s _ { i } , s _ { - i } ^ { * } ) \\in \\hat { \\mathcal { C } } } \\hat { U } _ { i } ( s _ { i } , s _ { - i } ^ { * } ) , \\enskip i \\in \\{ 1 , 2 , 3 \\} , \\end{align*}"}
-{"id": "1943.png", "formula": "\\begin{align*} \\delta _ \\eta : = { \\delta } { < \\eta > ^ { - s } } . \\end{align*}"}
-{"id": "1568.png", "formula": "\\begin{align*} - \\nu ( C _ { j _ k } ( \\alpha ( k ) ) ) = \\begin{cases} - c ( j _ k , - \\alpha ( k ) - 1 ) , & \\alpha ( k ) < 0 ; \\\\ - \\infty , & 0 \\leq \\alpha ( k ) \\leq j _ k ; \\\\ c ( j _ k , \\alpha ( k ) - j _ k ) , & \\alpha ( k ) > j _ k . \\end{cases} \\end{align*}"}
-{"id": "1482.png", "formula": "\\begin{align*} f ( z ) = \\frac { w _ 1 ( z ) } { w _ 2 ( z ) } = \\cfrac { 1 } { z } + b _ 0 + b _ 1 z + \\cdots . \\end{align*}"}
-{"id": "1195.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { u ( x ) } { G ( x ) } = \\mbox { C a p } _ { \\mathcal { A } } ( E ) ^ { \\frac { 1 } { p - 1 } } . \\end{align*}"}
-{"id": "7484.png", "formula": "\\begin{align*} \\Delta ^ h f & = h ^ { \\bar { \\gamma } \\alpha } \\left [ \\nabla _ { \\mathcal { X } _ \\alpha } \\nabla _ { \\mathcal { X } _ { \\bar { \\gamma } } } f - \\mathcal { C } _ \\alpha \\left ( \\nabla _ { \\mathcal { X } _ { \\bar { \\gamma } } } f \\right ) \\right ] \\\\ \\Delta ^ v f & = h ^ { \\bar { \\gamma } \\alpha } \\left [ \\nabla _ { \\mathcal { V } _ \\alpha } \\nabla _ { \\mathcal { V } _ { \\bar { \\gamma } } } f + C _ \\alpha \\left ( \\nabla _ { \\mathcal { V } _ { \\bar { \\gamma } } } f \\right ) \\right ] \\end{align*}"}
-{"id": "923.png", "formula": "\\begin{align*} E \\left [ \\exp \\left ( \\max _ { 1 \\leq k \\leq d } \\left ( \\frac { | Y _ k | } { M _ k } \\right ) ^ { 2 / q } \\right ) \\right ] & = E \\left [ \\max _ { 1 \\leq k \\leq d } \\exp \\left ( \\left ( \\frac { | Y _ k | } { M _ k } \\right ) ^ { 2 / q } \\right ) \\right ] \\leq \\sum _ { k = 1 } ^ d E \\left [ \\exp \\left ( \\left ( \\frac { | Y _ k | } { M _ k } \\right ) ^ { 2 / q } \\right ) \\right ] \\leq 2 d , \\end{align*}"}
-{"id": "8901.png", "formula": "\\begin{align*} \\lim _ k ( g \\psi _ { - n _ k } , g p \\kappa _ { - n _ k } ) = ( g J _ { \\theta , 1 } ^ { - 1 } ( p \\overline \\gamma ) , g p J _ { \\theta , 1 } ^ { - 1 } \\overline \\gamma ) - \\| g p J _ { \\theta , 1 } ^ { - 1 } \\overline \\gamma \\| ^ 2 . \\end{align*}"}
-{"id": "2772.png", "formula": "\\begin{align*} \\gamma ( Q _ 1 ) : = \\pi ^ { m _ \\ell } \\prod _ { i = 1 } ^ { \\ell } \\Big ( \\zeta _ { \\sum \\limits _ { j = i } ^ { \\ell } \\# S _ { j } ( Q _ 1 ) } \\Big ) ^ { m _ i - m _ { i - 1 } } . \\end{align*}"}
-{"id": "2706.png", "formula": "\\begin{align*} \\beta ^ 2 ( \\phi _ 1 , \\phi _ 2 ) = 2 - \\sqrt { 4 - \\gamma ^ 2 ( \\phi _ 1 , \\phi _ 2 ) } . \\end{align*}"}
-{"id": "5440.png", "formula": "\\begin{align*} f ( P ' _ \\vartriangle \\# P ' _ \\vartriangle ) = f ( P ' _ \\vartriangle ) + f ( P ' _ \\vartriangle ) - ( 1 , 0 , 0 , 1 ) . \\end{align*}"}
-{"id": "5289.png", "formula": "\\begin{align*} \\frac { } { t } \\| u \\| ^ 2 & - \\mu u _ { x } ( t , 1 ) u ( t , 1 ) + \\mu u _ { x } ( t , 0 ) u ( t , 0 ) + \\mu \\| u _ x \\| ^ 2 \\\\ = & \\int _ { 0 } ^ 1 f ( t , x , u , u _ x ) u x \\\\ \\leq & \\int _ { 0 } ^ 1 \\big ( | d _ 2 ( t ) | | u | + M _ 1 u ^ 2 + M _ 2 u _ x u \\big ) x : = I _ 2 . \\end{align*}"}
-{"id": "9930.png", "formula": "\\begin{gather*} u ^ { ( n ) } ( t , x ) = \\int _ t ^ T \\left ( A _ x u ^ { ( n ) } ( s , x ) + b ( s , x ) \\cdot \\nabla _ x u ^ { ( n ) } ( s , x ) + g ( s , x ) \\right ) \\ , d s \\intertext { w e g e t } u ( t , x ) = \\int _ t ^ T \\left ( A _ x u ( s , x ) + b ( s , x ) \\cdot \\nabla _ x u ( s , x ) + g ( s , x ) \\right ) \\ , d s . \\end{gather*}"}
-{"id": "5145.png", "formula": "\\begin{align*} z _ 3 = - i \\int _ 0 ^ t S ( t - t ' ) | z _ 1 | ^ 2 z _ 1 ( t ' ) d t ' , \\end{align*}"}
-{"id": "7423.png", "formula": "\\begin{align*} \\begin{aligned} R _ 1 & : = \\tau _ 1 + \\tau _ 2 + \\tau _ 3 , \\\\ R _ 2 & : = \\tau _ 1 \\tau _ 2 + \\tau _ 2 \\tau _ 3 + \\tau _ 3 \\tau _ 1 , \\\\ R _ 3 & : = \\tau _ 1 \\tau _ 2 \\tau _ 3 , \\end{aligned} \\begin{aligned} S _ 1 & : = \\tau _ 4 + \\tau _ 5 , \\\\ S _ 2 & : = \\tau _ 4 \\tau _ 5 , \\end{aligned} \\end{align*}"}
-{"id": "6841.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ T \\theta _ t \\| u _ t - u ^ * \\| & \\leq \\frac { 2 } { T ( T + 1 ) } \\ O \\ ! \\left ( \\sum _ { t = 1 } ^ T t \\beta ^ t \\right ) \\\\ & \\leq \\frac { 2 } { T ( T + 1 ) } \\ O \\ ! \\left ( \\frac { \\beta ( 1 - ( T + 1 ) \\beta ^ T + T \\beta ^ { T + 1 } ) } { ( 1 - \\beta ) ^ 2 } \\right ) = O \\ ! \\left ( \\frac { 1 } { T ^ 2 } \\right ) = o \\ ! \\left ( \\frac { 1 } { T } \\right ) . \\end{align*}"}
-{"id": "4876.png", "formula": "\\begin{align*} \\Lambda _ x ( \\mathcal { I } ) = \\bigcap _ k \\bigcup _ { m \\ge k } \\mathrm { L } _ { x \\upharpoonright P _ m } . \\end{align*}"}
-{"id": "1007.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { x \\in \\mathbb { R } } P \\left ( \\left | \\sup _ { t \\in [ a _ n , T - a _ n ] } | Z _ n ( t ) | - x \\right | \\leq M ( v _ n + 7 \\varepsilon ) \\right ) = \\lim _ { n \\to \\infty } P ( w _ n ( Z _ n ; n ^ { - 1 } ) > M \\varepsilon ) = 0 . \\end{align*}"}
-{"id": "6875.png", "formula": "\\begin{align*} \\Phi _ { 1 } ( x ) = \\frac { 4 } { p q } x + O ( x ^ { 2 } ) \\end{align*}"}
-{"id": "7812.png", "formula": "\\begin{align*} \\frac { \\partial F ( m , r , t , s ) } { \\partial r } = \\frac { \\partial { \\rm I } } { \\partial r } + \\frac { \\partial { \\rm I I } } { \\partial r } + \\frac { \\partial { \\rm I I I } } { \\partial r } \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\end{align*}"}
-{"id": "5281.png", "formula": "\\begin{align*} \\frac { 1 } { \\mu } \\langle \\mathcal { A } u , u \\rangle = \\int _ { 0 } ^ 1 u _ { x x } u x = - \\frac { a _ 1 } { a _ 2 } u ^ 2 ( t , 1 ) - \\| u _ x \\| ^ 2 . \\end{align*}"}
-{"id": "3741.png", "formula": "\\begin{align*} \\omega : = \\sum _ { z \\in \\Z ^ d } \\sum _ { i \\le N ( z , 0 ) } \\delta _ { S ^ { z , i } } , \\end{align*}"}
-{"id": "4049.png", "formula": "\\begin{align*} F ( u ( x ) , d u _ x v ) = | | v | | , \\end{align*}"}
-{"id": "5207.png", "formula": "\\begin{align*} \\partial _ { x } \\Gamma _ { 1 } ( x , y ) & = f _ { 1 } ' ( x ) + f _ { 1 } '' ( x ) ( y - x ) - f _ { 1 } ' ( x ) \\\\ & = f _ { 1 } '' ( x ) ( y - x ) \\end{align*}"}
-{"id": "261.png", "formula": "\\begin{align*} M ( N ) \\le \\begin{cases} C \\sqrt { m _ { - 2 k } } N ^ k \\ell , & k > 0 \\\\ C ( s ) \\sqrt { m _ { - s } } N \\ell , & k = 0 . \\end{cases} \\end{align*}"}
-{"id": "528.png", "formula": "\\begin{align*} \\ddot q ( t ) = - \\nabla V _ t ( q ( t ) ) . \\end{align*}"}
-{"id": "8793.png", "formula": "\\begin{align*} E _ { I } : = \\cap _ { i \\in I } E _ { i } \\overset { \\circ } { E _ { I } } : = E _ { I } \\backslash \\cup _ { j \\in T \\backslash I } E _ { j } . \\end{align*}"}
-{"id": "2643.png", "formula": "\\begin{align*} L ( K _ n ) = \\begin{bmatrix} n - 1 & - 1 & \\dots & - 1 \\\\ - 1 & n - 1 & \\dots & - 1 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ - 1 & - 1 & \\dots & n - 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "7372.png", "formula": "\\begin{align*} L _ { \\omega } \\psi = ( i _ { X ^ a } \\omega ) . \\nabla _ { X _ a } \\psi + \\frac { p } { 2 ( p + 1 ) } d \\omega . \\psi - \\frac { n - p } { 2 ( n - p + 1 ) } \\delta \\omega . \\psi . \\end{align*}"}
-{"id": "4930.png", "formula": "\\begin{align*} H ( \\bf { k } ) = \\bf { d } ( \\bf { k } ) . \\bf { \\sigma } \\end{align*}"}
-{"id": "7315.png", "formula": "\\begin{align*} ( \\widehat { q } + 1 ) ^ { k ( x _ i ) } ( \\psi ^ { - 1 } \\sigma \\psi ) ( x _ i ) = \\sum _ { j \\in I } a _ j ( \\widehat { q } + 1 ) ^ { \\ell _ j } \\xi _ j , \\end{align*}"}
-{"id": "6830.png", "formula": "\\begin{align*} N ( \\phi ) = \\left [ \\frac { \\rho } { \\int _ { \\mathbb { S } ^ 2 } e ^ { w _ { \\lambda } + \\phi } } e ^ { \\phi } - \\frac { \\rho } { \\int _ { \\mathbb { S } ^ 2 } e ^ { w _ { \\lambda } } } - \\left ( \\phi - \\frac { \\int _ { \\mathbb { S } ^ 2 } e ^ { w _ { \\lambda } } \\phi } { \\int _ { \\mathbb { S } ^ 2 } e ^ { w _ { \\lambda } } } \\right ) \\right ] e ^ { w _ { \\lambda } } . \\end{align*}"}
-{"id": "2675.png", "formula": "\\begin{align*} \\tilde { M } _ { f ^ { ( 3 ) } , 2 } = \\begin{pmatrix} a & b \\\\ b & c \\end{pmatrix} \\end{align*}"}
-{"id": "4823.png", "formula": "\\begin{align*} \\mathcal { B } ( G ) : = \\bigg \\{ X \\in \\mathcal { F } ( G ) \\ \\bigg { | } \\ \\sum _ { g \\in G } \\mathsf { v } _ g ( X ) g = 0 \\bigg \\} \\end{align*}"}
-{"id": "7257.png", "formula": "\\begin{align*} \\sum _ { n \\le X } \\Bigl ( 1 - \\frac { \\Phi ( n ) } n \\Bigr ) ^ 2 & \\le \\sum _ { n \\le X } \\sum _ { \\substack { d , e \\mid n \\\\ d , e > T } } \\frac 1 { d e } = \\sum _ { T < d , e \\le X } \\frac 1 { d e } \\sum _ { \\substack { n \\le X : \\\\ [ d , e ] \\mid n } } 1 \\\\ & \\le X \\sum _ { T < d , e \\le X } \\frac { ( d , e ) } { d ^ 2 e ^ 2 } . \\end{align*}"}
-{"id": "3816.png", "formula": "\\begin{align*} \\mathcal { D } _ L : = \\{ H ^ { ( t ) } _ - > H ^ { ( t ) } _ + \\wedge ( L ^ { \\alpha } - t ) \\ ; \\forall \\ ; t \\in [ 0 , L ^ \\alpha ] \\} . \\end{align*}"}
-{"id": "7489.png", "formula": "\\begin{align*} ( \\Psi , \\Phi ) = \\int _ E < \\Psi , \\Phi > d \\mathcal { V } , | | \\Psi | | ^ 2 = \\int _ E < \\Psi , \\Psi > d \\mathcal { V } . \\end{align*}"}
-{"id": "6822.png", "formula": "\\begin{align*} \\langle L ( \\phi ) , \\eta _ { R _ 3 , \\xi _ j } \\varphi _ { 0 , j } \\rangle = \\langle h , \\eta _ { R _ 3 , \\xi _ j } \\varphi _ { 0 , j } \\rangle + c _ j \\int _ { \\mathbb { S } ^ 2 _ { \\lambda } } \\chi _ { R _ 1 , j } | \\varphi _ { 0 , j } | ^ 2 . \\end{align*}"}
-{"id": "9266.png", "formula": "\\begin{align*} f ( \\xi | \\nu ) : = | V _ \\xi ( \\nu ) | ^ 2 , \\end{align*}"}
-{"id": "2915.png", "formula": "\\begin{align*} E ^ \\ast [ s , s ' ] = \\{ w \\in E ^ \\ast \\colon \\iota ( w ) = ( s , s ' ) \\} , \\end{align*}"}
-{"id": "8192.png", "formula": "\\begin{align*} I ( \\mathfrak { m } ) = \\{ q \\in \\imath ^ { - 1 } \\vert \\abs { q } _ { \\nu } \\leq \\abs { R _ { \\nu } } _ { \\nu } , ( q ) = \\mathfrak { m } \\} . \\end{align*}"}
-{"id": "1515.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } T _ { n + 2 } ( x ) \\\\ T _ { n + 1 } ( x ) \\\\ T _ { n } ( x ) \\end{array} \\right ] = \\left [ \\begin{array} { c c c } x ^ { 2 } & x & 1 \\\\ 1 & 0 & 0 \\\\ 0 & 1 & 0 \\end{array} \\right ] ^ { n } \\left [ \\begin{array} { c } T _ { 2 } ( x ) \\\\ T _ { 1 } ( x ) \\\\ T _ { 0 } ( x ) \\end{array} \\right ] \\end{align*}"}
-{"id": "2715.png", "formula": "\\begin{align*} \\partial _ t h + v \\cdot \\nabla _ x h = \\frac { 1 } { \\epsilon } \\mathcal L ( h ) + \\mathcal F ( h , h ) \\ , . \\end{align*}"}
-{"id": "165.png", "formula": "\\begin{align*} \\vert \\mathfrak { X } \\vert = \\{ a \\in A : \\mathfrak { X } ( a ) > 0 \\} . \\end{align*}"}
-{"id": "8951.png", "formula": "\\begin{gather*} c _ w w + c _ { r w } r w = c _ w w + c _ { w _ 1 ^ { - 1 } s _ i w _ 1 w } w _ 1 ^ { - 1 } s _ i w _ 1 w = w _ 1 ^ { - 1 } \\big ( { } ^ { w _ 1 } c _ w + { } ^ { w _ 1 } c _ { w _ 1 ^ { - 1 } s _ i w _ 1 w } s _ i \\big ) w _ 1 w , \\end{gather*}"}
-{"id": "4878.png", "formula": "\\begin{align*} \\bigcup _ { \\underline { k } : \\sum i k _ i = k } \\left ( \\prod _ i M ^ { k _ i } \\setminus \\Delta \\right ) / \\prod _ i S _ { k _ i } = \\bigcup _ { \\underline { k } : \\sum i k _ i = k } S ^ k _ { \\underline { k } } M = S ^ k M . \\end{align*}"}
-{"id": "3647.png", "formula": "\\begin{align*} S ( \\tilde P ) ^ { ( G , b ) } = \\mathbb C [ C ( \\tilde P ) \\cap \\tilde M _ { \\mathbb R , b } \\cap ( M \\times \\mathbb Z ) ] , \\end{align*}"}
-{"id": "6755.png", "formula": "\\begin{align*} h ( \\mu ) = [ l : k ] ^ { - 1 } h ( \\beta ) . \\end{align*}"}
-{"id": "4031.png", "formula": "\\begin{align*} - b ( Y , Z ) = K ( p ) \\cdot b ( d \\eta _ p Y , d \\eta _ p Z ) . \\end{align*}"}
-{"id": "7145.png", "formula": "\\begin{align*} q ^ 2 ( m + 1 ) & \\leq ( q ( m ) + y ( m ) ) ^ 2 \\\\ & = q ^ 2 ( m ) + y ^ 2 ( m ) + 2 q ( m ) y ( m ) . \\end{align*}"}
-{"id": "3726.png", "formula": "\\begin{align*} ( p _ 3 , p ' _ 3 ) = \\omega \\cdot ( p _ 1 , p ' _ 1 ) , ( p _ 5 , p ' _ 5 ) = \\omega ^ 2 \\cdot ( p _ 1 , p ' _ 1 ) . \\end{align*}"}
-{"id": "636.png", "formula": "\\begin{align*} \\omega \\lVert h _ { G _ 0 } \\rVert _ { 2 , p } ^ p + ( 1 - \\omega ) \\lVert h _ { H _ 0 } \\rVert _ { 2 , p } ^ p & \\leq \\omega \\frac { a ^ { p / 2 - 1 } ( 1 + \\delta _ { a k } ) } { 1 - \\delta _ { ( a + 1 ) k } } \\lVert h _ { G _ 0 ^ c } \\rVert _ { 2 , p } ^ p + ( 1 - \\omega ) \\frac { ( a k / s ) ^ { p / 2 - 1 } ( 1 + \\delta _ { a k } ) } { 1 - \\delta _ { ( a + 1 ) k } } \\lVert h _ { G _ 0 ^ c } \\rVert _ { 2 , p } ^ p \\\\ & = \\frac { a ^ { p / 2 - 1 } \\gamma ( 1 + \\delta _ { a k } ) } { 1 - \\delta _ { ( a + 1 ) k } } \\lVert h _ { G _ 0 ^ c } \\rVert _ { 2 , p } ^ p . \\end{align*}"}
-{"id": "9863.png", "formula": "\\begin{align*} \\theta ( x ) \\triangleq \\max _ i \\{ g _ i ( x ) _ + \\} - \\kappa ( x ) = \\lambda \\left ( \\max _ i \\{ g _ i ( x ) _ + \\} - \\min _ d \\left \\{ \\max _ i \\left \\{ \\tilde g _ i ( d ; x ) _ + \\right \\} \\ , | \\ , \\| d \\| _ \\infty \\le \\rho , \\ , d \\in K - x \\right \\} \\right ) \\end{align*}"}
-{"id": "5471.png", "formula": "\\begin{align*} \\int _ 0 ^ { r ^ m } s ^ { \\rho - 1 } p ( s ) \\dd s = \\frac { r ^ { m \\rho } } { r ^ \\rho - 1 } \\int _ 1 ^ r s ^ { \\rho - 1 } p ( s ) \\dd s , \\end{align*}"}
-{"id": "6025.png", "formula": "\\begin{align*} & \\inf \\left \\{ R : D _ { 1 + s } ( P _ { X ^ { n } Y ^ { n } | U _ { n } } \\| \\pi _ { X ^ { n } Y ^ { n } } | P _ { U _ { n } } ) \\rightarrow 0 \\right \\} \\\\ & \\leq \\min _ { Q _ { X Y W } : \\ , Q _ { X Y } = \\pi _ { X Y } , \\ , X - W - Y } I _ { Q } ( X Y ; W ) \\\\ & = C _ { \\mathsf { W y n e r } } ( X ; Y ) . \\end{align*}"}
-{"id": "4847.png", "formula": "\\begin{align*} y z = a t ^ l + \\cdots , x z = t ^ m , x y = a t ^ { m + l } + \\cdots . \\end{align*}"}
-{"id": "2235.png", "formula": "\\begin{align*} \\Gamma = \\{ z \\in \\mathbb C ^ n : \\ | f _ 1 ( z ) | = r _ 1 , \\ldots , | f _ n ( z ) | = r _ n \\} \\end{align*}"}
-{"id": "8525.png", "formula": "\\begin{align*} T h ( x ) = ( 2 \\pi ) ^ { - 3 } \\int _ { \\mathbb { R } ^ 3 } e ^ { i x \\cdot \\xi } a ( x , \\xi ) \\widehat { h } ( \\xi ) d \\xi \\end{align*}"}
-{"id": "8783.png", "formula": "\\begin{align*} \\mathcal { E } ( t ) : = \\iint _ { \\R ^ { 2 } \\times \\R ^ { 2 } } \\frac { 1 } { 2 } | v | ^ { 2 } f ( t , x , v ) \\ , d x d v \\leq C , \\forall \\ , t \\geq 0 , \\end{align*}"}
-{"id": "4173.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { k ' \\in N } e _ { k ' , j + n - 2 } ^ n + \\frac { 1 } { n } e ^ { n - 1 } _ { k , j } + \\frac { 1 } { n } \\left ( 1 - \\frac { 1 } { n } \\right ) e _ { i , j } ^ { n - 2 } + \\frac { 1 } { n } \\sum _ { \\tau = 1 } ^ { n - 3 } e _ { i , j + \\tau } ^ { n - \\tau - 2 } . \\end{align*}"}
-{"id": "9205.png", "formula": "\\begin{align*} [ \\lambda \\otimes e , \\lambda ' \\otimes e ' ] = \\lambda \\circ \\lambda ' \\otimes \\frac { [ e , e ' ] _ { A ^ { - } } } { 2 } + [ \\lambda , \\lambda ' ] \\otimes \\frac { ( e \\circ e ' ) _ { A ^ { + } } } { 2 } + ( \\lambda \\mid \\lambda ' ) \\langle e , e ' \\rangle . \\end{align*}"}
-{"id": "799.png", "formula": "\\begin{align*} \\rho = 6 . 2 . \\end{align*}"}
-{"id": "8051.png", "formula": "\\begin{align*} \\cos \\zeta ( 0 ) = \\frac { m ( r ( \\delta / 2 ) ) } { m ( r ( 0 ) ) } , \\end{align*}"}
-{"id": "4050.png", "formula": "\\begin{align*} K _ e ( p ) = \\mathrm { d e t } \\left ( d \\xi _ p \\right ) = \\mathrm { d e t } \\left ( d u ^ { - 1 } _ { \\eta ( p ) } \\right ) \\cdot \\mathrm { d e t } \\left ( d \\eta _ p \\right ) = K _ { \\partial B } ( \\eta ( p ) ) \\cdot K ( p ) . \\end{align*}"}
-{"id": "3031.png", "formula": "\\begin{align*} ( A { \\mathcal H } x ) ( \\zeta ) & = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix} \\frac { \\partial } { \\partial \\zeta } \\left ( \\begin{bmatrix} \\frac { 1 } { \\rho ( \\zeta ) } & 0 \\\\ 0 & T ( \\zeta ) \\end{bmatrix} x ( \\zeta ) \\right ) ; \\\\ D ( A { \\mathcal H } ) & = \\left \\{ x \\in { \\cal W } ^ { 1 , 2 } ( ( 0 , 1 ) ; \\mathbb F ^ 2 ) \\mid \\hat W _ B ( { \\mathcal H } x ) ( 0 , t ) = 0 \\right \\} , \\end{align*}"}
-{"id": "4926.png", "formula": "\\begin{align*} \\mathrm { r a n k } \\ , C = \\mathrm { r a n k } \\ , C ^ * C & = \\mathrm { r a n k } \\ , ( I - A A ^ * ) = d _ { A ^ * } < \\infty , \\mbox { w h i l e } \\\\ \\mathrm { r a n k } \\ , B = \\mathrm { r a n k } \\ , B B ^ * & = \\mathrm { r a n k } \\ , ( I - A ^ * A ) = d _ { A } . \\end{align*}"}
-{"id": "1172.png", "formula": "\\begin{align*} E + F = \\{ x + y : x \\in E , y \\in F \\} \\end{align*}"}
-{"id": "2593.png", "formula": "\\begin{align*} \\phi _ a ( t ) : = \\varphi ^ { \\tfrac { 1 - a } 2 } ( t ) = [ t ( 1 - t ) ] ^ { \\tfrac { 1 - a } 2 } . \\end{align*}"}
-{"id": "5385.png", "formula": "\\begin{align*} a \\cdot ( a b c ) ^ 3 & = ( ( b c ) ^ 3 ) ^ { a c } \\in T \\\\ b \\cdot ( a b c ) ^ 3 & = ( ( a c ) ^ 3 ) ^ { b c } \\in T \\\\ a b \\cdot ( a b c ) ^ 3 & = c ^ { a b c } \\in T \\end{align*}"}
-{"id": "4095.png", "formula": "\\begin{align*} ( \\lambda _ 2 - \\lambda _ 1 ) ( \\sin ^ 2 \\theta _ 0 + \\sin ^ 2 \\theta _ 1 ) = \\lambda _ 2 - \\lambda _ 1 . \\end{align*}"}
-{"id": "1556.png", "formula": "\\begin{align*} A = \\pi \\left [ 1 - \\sum _ { m = 1 } ^ { \\infty } { m | b _ { m } | ^ 2 } \\right ] . \\end{align*}"}
-{"id": "6395.png", "formula": "\\begin{align*} P _ { } \\left ( \\theta \\right ) \\overset { } { = } \\int d x P _ { } \\left ( x \\theta \\right ) = \\frac { \\exp \\left [ \\beta f \\left ( \\theta \\right ) \\right ] P _ { } \\left ( x ^ { \\prime } \\theta \\right ) } { \\int d \\theta \\exp \\left [ \\beta f \\left ( \\theta \\right ) \\right ] P _ { } \\left ( x ^ { \\prime } \\theta \\right ) } \\end{align*}"}
-{"id": "5507.png", "formula": "\\begin{align*} \\widehat F ( s ) = \\sum _ { n = 1 } ^ \\infty e ^ { - 2 ^ { n / \\alpha } s } 2 ^ { - n } \\end{align*}"}
-{"id": "8830.png", "formula": "\\begin{align*} Q '''' + y Q + 5 \\alpha ( 2 Q ^ 2 Q '' + 3 Q ( Q ' ) ^ 2 ) + 5 \\beta Q ^ 5 = 0 , \\end{align*}"}
-{"id": "5342.png", "formula": "\\begin{align*} Z ^ N = \\{ \\lambda \\in ( W _ 1 \\otimes \\cdots \\otimes W _ m ) ^ * | G _ N \\cdot \\lambda = 0 \\} . \\end{align*}"}
-{"id": "6111.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\frac { \\mathbb { P } \\left ( Y > t \\right ) } { \\mathbb { P } \\left ( \\Phi _ { \\psi _ { n } } Y _ { n } > t \\right ) } = 1 . \\end{align*}"}
-{"id": "6693.png", "formula": "\\begin{align*} c _ 1 = \\frac { ( n - 1 + p ) ^ { \\frac { p } { n - 1 + p } } } { p } \\left ( \\frac { | B _ p ^ n | _ n } { | B _ p ^ { n - 1 } | _ { n - 1 } } \\right ) ^ { \\frac { p } { n - 1 + p } } , & c _ 2 = G _ { B _ p ^ n } ( ( n ^ { - 1 / p } , \\dots , n ^ { - 1 / p } ) ) \\quad . \\end{align*}"}
-{"id": "2646.png", "formula": "\\begin{align*} \\frac { 1 } { c _ j } + \\sum _ { i \\neq j } \\left \\{ \\frac { c _ i } { c _ j } \\right \\} = 1 . \\end{align*}"}
-{"id": "50.png", "formula": "\\begin{align*} ( u _ 1 - x _ { i } ) ^ { 2 } + ( u _ 1 - y _ { i } ) ^ { 2 } + ( u _ 2 - z _ { i } ) ^ { 2 } + ( u _ 2 - s _ { i } ) ^ { 2 } = a ( u _ 1 - b ) ^ 2 + a ' ( u _ 2 - b ' ) ^ 2 + c \\end{align*}"}
-{"id": "4400.png", "formula": "\\begin{align*} \\| A P _ \\alpha | _ F \\| : = \\sup \\{ \\| A P _ \\alpha f \\| ; f \\in L _ \\alpha ^ p , \\| f \\| = 1 , \\operatorname { s u p p } f \\subseteq F \\} \\end{align*}"}
-{"id": "6251.png", "formula": "\\begin{align*} \\sigma = v ( \\omega _ 0 , \\mu _ 0 ) , \\omega = \\omega _ 0 + u ( \\omega _ 0 , \\mu _ 0 ) , \\mu = \\mu _ 0 + w ( \\omega _ 0 , \\mu _ 0 ) \\end{align*}"}
-{"id": "5329.png", "formula": "\\begin{align*} Y ( a ( - 1 ) 1 , x ) = \\sum _ { n \\in \\Z } a ( n ) x ^ { - n - 1 } . \\end{align*}"}
-{"id": "9922.png", "formula": "\\begin{align*} q = \\left ( \\begin{array} { c c c c c c c c c } z _ 1 ^ 2 & z _ 1 z _ 2 & z _ 2 ^ 2 & z _ 1 w _ 1 & \\frac { 1 } { 2 } ( z _ 1 w _ 2 + z _ 2 w _ 1 ) & z _ 2 w _ 2 & w _ 1 ^ 2 & w _ 1 w _ 2 & w _ 2 ^ 2 \\end{array} \\right ) ; \\end{align*}"}
-{"id": "1731.png", "formula": "\\begin{align*} f ^ T _ \\lambda \\frac { h \\circ \\sigma ^ n } { h } = f ^ S _ \\lambda \\end{align*}"}
-{"id": "1129.png", "formula": "\\begin{align*} \\Bigg \\{ \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} , \\begin{bmatrix} 0 & 1 \\\\ - 1 & 1 \\end{bmatrix} , \\begin{bmatrix} - 1 & 1 \\\\ - 1 & 0 \\end{bmatrix} , \\begin{bmatrix} 0 & \\sqrt { - 1 } \\\\ \\sqrt { - 1 } & 0 \\end{bmatrix} , \\begin{bmatrix} \\sqrt { - 1 } & 0 \\\\ \\sqrt { - 1 } & - \\sqrt { - 1 } \\end{bmatrix} , \\begin{bmatrix} \\sqrt { - 1 } & - \\sqrt { - 1 } \\\\ 0 & - \\sqrt { - 1 } \\end{bmatrix} \\Bigg \\} . \\end{align*}"}
-{"id": "8907.png", "formula": "\\begin{align*} T \\cong \\begin{pmatrix} V _ 1 & \\ast & \\ast \\\\ \\mathbb O & T _ 1 & \\ast \\\\ \\mathbb O & \\mathbb O & V _ 2 \\end{pmatrix} , \\ \\ \\ T _ 1 = U _ \\nu + ( \\cdot , \\psi ) \\varphi . \\end{align*}"}
-{"id": "5544.png", "formula": "\\begin{align*} w _ { n } \\left ( t \\right ) = s i g n \\left [ \\left ( \\sin 2 \\pi t \\right ) ^ { b _ { 0 } } \\prod \\nolimits _ { k = 1 } ^ { m } \\left ( \\cos 2 ^ { k } \\pi t \\right ) ^ { b _ { k } } \\right ] \\end{align*}"}
-{"id": "6635.png", "formula": "\\begin{align*} \\int _ { D } | D f | ^ 2 \\ ; & = \\ ; \\int _ { D \\setminus \\bigcup _ { n = 1 } ^ { \\infty } Q _ n } | D f | ^ 2 + \\int _ { \\bigcup _ { n = 1 } ^ { \\infty } Q _ n } | D f | ^ 2 \\\\ & \\leq \\ ; { \\mathrm { A r e a } ( D ) } + C \\sum _ { n = 1 } ^ { \\infty } n ^ { 2 p } { \\mathrm { A r e a } ( Q _ n ) } \\\\ & = \\ ; { \\mathrm { A r e a } ( D ) } + C \\sum _ { n = 1 } ^ { \\infty } n ^ { 2 p } 2 ^ { - 2 n } < \\infty \\ . \\\\ \\end{align*}"}
-{"id": "3679.png", "formula": "\\begin{align*} \\begin{aligned} a _ 1 + \\dots + a _ k & \\geqslant - \\frac { k ^ 2 ( n + 2 ) - k ( 2 n ^ 2 + 3 n ) } { 2 ( n + 1 ) } \\mbox { f o r } k = 1 , \\dots , n - 1 , \\mbox { a n d } \\\\ a _ 1 + \\dots + a _ n & = - \\frac { n ^ 2 } 2 , \\end{aligned} \\end{align*}"}
-{"id": "824.png", "formula": "\\begin{align*} \\vec { u } ( \\vec { x } , t ) = \\left ( \\begin{array} { c } \\sin ( \\pi x ) \\cos ( \\pi y ) \\\\ - \\cos ( \\pi x ) \\sin ( \\pi y ) \\\\ \\end{array} \\right ) e ^ { - 2 \\pi ^ 2 t } , \\\\ p ( \\vec { x } , t ) = 0 . \\end{align*}"}
-{"id": "3014.png", "formula": "\\begin{align*} 0 \\geq \\Re \\ < A x , x \\ > _ { L ^ 2 } & = \\Re \\ < P _ 0 x , x \\ > _ { L ^ 2 } = \\Re \\ < P _ 0 \\Psi z , \\Psi z \\ > _ { L ^ 2 } \\\\ & = \\Re \\int _ 0 ^ 1 \\abs { \\Psi ( \\zeta ) } ^ 2 \\ < P _ 0 z , z \\ > _ { H } d \\zeta \\\\ & = \\norm { \\Psi } ^ 2 _ { L ^ 2 } \\Re \\ < P _ 0 z , z \\ > _ { H } . \\end{align*}"}
-{"id": "1519.png", "formula": "\\begin{align*} Q _ { T , n + 2 } ( x ) = Q _ { T , 2 } ( x ) T _ { n + 1 } ( x ) + ( x Q _ { T , 1 } ( x ) + Q _ { T , 0 } ( x ) ) T _ { n } ( x ) + Q _ { T , 1 } ( x ) T _ { n - 1 } ( x ) , \\end{align*}"}
-{"id": "4782.png", "formula": "\\begin{align*} { L } _ { 0 } = 0 , { L } _ i = \\sqrt { \\alpha a _ i + \\beta b _ i + \\gamma } , i \\neq 0 . \\end{align*}"}
-{"id": "7658.png", "formula": "\\begin{align*} & F _ { \\mu \\nu } = X _ { \\mu \\nu } - X _ { \\nu \\mu } , F _ { \\mu \\nu } = - F _ { \\nu \\mu } \\\\ [ 1 e x ] & \\widetilde { F } _ { \\mu \\nu } = X _ { \\mu \\nu } + X _ { \\nu \\mu } , \\widetilde { F } _ { \\mu \\nu } = + \\widetilde { F } _ { \\nu \\mu } , \\end{align*}"}
-{"id": "5441.png", "formula": "\\begin{align*} f ( F ) = ( n + 2 , 3 n + 1 , 3 n + 1 , n + 2 ) , \\end{align*}"}
-{"id": "7187.png", "formula": "\\begin{align*} \\langle z _ \\infty , \\zeta \\rangle = 0 \\qquad \\zeta \\in T _ h \\mathfrak { H } _ { 1 + s } . \\end{align*}"}
-{"id": "4252.png", "formula": "\\begin{align*} \\mathcal { F } _ { 1 2 } = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & \\frac { t } { t - 1 } \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "3532.png", "formula": "\\begin{align*} J ^ { \\theta } ( u ^ { \\theta } ( \\cdot ) ) \\leq J ^ { \\theta } ( \\bar { u } ( \\cdot ) ) = \\theta , \\ \\tilde { d } ( u ^ { \\theta } ( \\cdot ) , \\bar { u } ( \\cdot ) ) \\leq \\sqrt { \\theta } , \\end{align*}"}
-{"id": "4910.png", "formula": "\\begin{align*} T = - \\alpha \\beta ^ { - 1 } ( I - \\alpha ^ { - 1 } F ) + 0 I \\end{align*}"}
-{"id": "7516.png", "formula": "\\begin{gather*} \\psi _ t + \\frac 1 2 ( \\psi _ x ) ^ 2 + \\frac 1 2 ( \\psi _ y ) ^ 2 - \\psi _ { x x } - \\psi _ { y y } = 0 . \\end{gather*}"}
-{"id": "3170.png", "formula": "\\begin{align*} \\lim _ { u \\to - \\infty } e ^ { - u \\delta / 2 } e ^ { \\Delta _ t ( u ) - u \\delta / 2 } = \\infty . \\end{align*}"}
-{"id": "8440.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { - 2 , 1 , v } ) = q ^ { 1 - \\frac { a ( \\chi ) } { 2 } } \\zeta _ F ( 1 ) ^ { - 1 } \\epsilon ( \\frac { 1 } { 2 } , \\chi ^ { - 1 } ) \\int _ { - v ^ { - 1 } + \\varpi ^ { 1 - a ( \\chi ) } \\mathcal { O } ^ { \\times } } \\chi ( y _ 1 ) \\chi ^ { - 1 } ( y _ 1 v + \\varpi ^ { a ( \\chi ) - 1 } ) \\psi ( y _ 1 \\varpi ^ { - 1 } ) d ^ { \\times } y _ 1 . \\end{align*}"}
-{"id": "3230.png", "formula": "\\begin{align*} N ^ i H ^ { 2 i + k } ( X ) = \\left \\{ \\begin{array} { c c c } N ^ i H ^ { 2 i + k } ( X ) & \\mbox { f o r } & i \\in [ 0 , 2 d i m ( X ) - k ] \\\\ 0 & \\mbox { f o r } & 2 i + k \\not \\in [ 0 , 2 d i m ( X ) ] \\\\ H ^ { 2 i + k } ( X ; \\mathbb Q ) & \\mbox { f o r } & 2 i + k \\in [ 0 , k ] \\cup [ 2 d i m ( X ) - k , 2 d i m ( X ) ] \\end{array} \\right . \\end{align*}"}
-{"id": "7808.png", "formula": "\\begin{align*} V _ 0 = \\rho ^ { \\prime } \\Delta r + \\rho ^ { \\prime \\prime } - \\rho ^ { \\prime 2 } . \\end{align*}"}
-{"id": "4086.png", "formula": "\\begin{align*} h ( X , Y ) = \\frac { \\langle D _ X Y , \\xi \\rangle } { \\langle \\eta , \\xi \\rangle } = - \\frac { \\langle Y , d \\xi _ p X \\rangle } { \\langle \\eta , \\xi \\rangle } = - \\frac { \\langle d u ^ { - 1 } _ { \\eta ( p ) } Y , d \\eta _ p X \\rangle } { \\langle \\eta , \\xi \\rangle } , \\end{align*}"}
-{"id": "4931.png", "formula": "\\begin{align*} H ( { \\bf { k } } ) u _ n ( { \\bf { k } } ) = E _ n ( { \\bf { k } } ) u _ n ( { \\bf { k } } ) \\end{align*}"}
-{"id": "6052.png", "formula": "\\begin{align*} \\frac { \\partial ^ { 2 } \\Omega ^ { ( \\alpha , \\theta ) } ( Q _ { X Y U } ) } { \\partial \\theta ^ { 2 } } & = - \\mathrm { V a r } _ { Q _ { X Y U } ^ { ( \\alpha , \\theta ) } } \\Big [ \\omega _ { Q _ { X Y U } } ^ { ( \\alpha ) } ( X , Y | U ) \\Big ] . \\end{align*}"}
-{"id": "4430.png", "formula": "\\begin{align*} \\lim _ { T \\uparrow \\infty } \\| \\partial _ 1 ^ j \\partial _ 2 ^ l f _ T \\| = 0 \\quad \\mbox { p r o v i d e d } \\ ; j + l > 0 , \\end{align*}"}
-{"id": "6747.png", "formula": "\\begin{align*} \\bar { \\mathbb { P } } \\Big ( | \\mathbb { P } ( \\Theta > N ^ { \\gamma } t ) - e ^ { - t } | > \\epsilon \\Big ) = 0 . \\end{align*}"}
-{"id": "5457.png", "formula": "\\begin{align*} p ( x ) = p ( x / r ) , x \\in C _ p . \\end{align*}"}
-{"id": "8616.png", "formula": "\\begin{align*} \\partial _ t \\bar { u } = \\frac 1 2 \\Delta \\bar { u } , \\ \\ \\bar { u } ( 0 , x ) = u _ 0 ( x ) , \\end{align*}"}
-{"id": "1512.png", "formula": "\\begin{align*} g _ { t } ( y ) & = \\frac { Q _ { t , 0 } ( x ) + ( Q _ { t , 1 } ( x ) - x ^ { 2 } Q _ { t , 0 } ( x ) ) y + ( Q _ { t , 2 } ( x ) - x ^ { 2 } Q _ { t , 1 } ( x ) - x Q _ { t , 0 } ( x ) ) y ^ { 2 } } { 1 - x ^ { 2 } y - x y ^ { 2 } - y ^ { 3 } } \\\\ & = \\frac { \\left ( \\begin{array} { c } 3 - 2 x ^ { 2 } y - x y ^ { 2 } + ( x ^ { 2 } + 2 x y + 3 y ^ { 2 } ) \\textbf { i } + ( x ^ { 4 } + 2 x + x ^ { 3 } y + 3 y + x ^ { 2 } y ^ { 2 } ) \\textbf { j } \\\\ + ( x ^ { 6 } + 3 x ^ { 3 } + 3 + x ^ { 5 } y + 3 x ^ { 2 } y + x ^ { 4 } y ^ { 2 } + 2 x y ^ { 2 } ) \\textbf { k } \\end{array} \\right ) } { 1 - x ^ { 2 } y - x y ^ { 2 } - y ^ { 3 } } . \\end{align*}"}
-{"id": "3498.png", "formula": "\\begin{align*} X ( s ) = x _ 0 + \\int _ { 0 } ^ { s } b ( X ( t ) , u ( t ) ) d t , \\end{align*}"}
-{"id": "3612.png", "formula": "\\begin{align*} a = \\inf _ { x \\in \\Omega } x _ 1 . \\end{align*}"}
-{"id": "1483.png", "formula": "\\begin{align*} { \\rm R e } \\Big ( 1 + \\frac { z f '' ( z ) } { f ' ( z ) } \\Big ) = 1 - 2 { \\rm R e } \\Big ( \\frac { z w _ 2 ' ( z ) } { w _ 2 ( z ) } \\Big ) . \\end{align*}"}
-{"id": "7756.png", "formula": "\\begin{align*} P _ { 0 } ( \\ell _ { 1 } , \\ell _ { 2 } ) = ( \\ell _ { 1 } , \\ell _ { 2 } , P _ { 0 } ( \\ell _ { 1 } , \\ell _ { 2 } ) _ { 3 } ) , \\end{align*}"}
-{"id": "6983.png", "formula": "\\begin{align*} D f _ 2 ( B ) E _ { i j } = - \\frac { 1 } { 2 f _ 2 ( B ) } b _ { j i } , \\end{align*}"}
-{"id": "6507.png", "formula": "\\begin{align*} \\left \\vert f \\left ( k _ { \\mathrm { o } } \\right ) \\right \\vert ^ { 2 } \\approx \\frac { \\theta _ { \\mathrm { o } } ^ { 2 } } { k _ { \\mathrm { o } } ^ { 2 } } \\approx \\frac { \\rho ^ { 2 } k _ { \\mathrm { o } } ^ { 4 } L ^ { 6 } } { 9 } = \\frac { 4 \\mu ^ { 2 } V ^ { 2 } L ^ { 6 } } { 9 \\hbar ^ { 4 } } \\approx a _ { \\mathrm { s } } ^ { 2 } . \\end{align*}"}
-{"id": "3407.png", "formula": "\\begin{align*} \\tau _ x ( x + \\imath y ) = - x + \\imath y , \\tau _ y ( x + \\imath y ) = x - \\imath y . \\end{align*}"}
-{"id": "4310.png", "formula": "\\begin{align*} [ D _ 1 , D _ 2 ] \\Omega = r _ { [ M _ 1 , M _ 2 ] + D _ 1 ( M _ 2 ) - D _ 2 ( M _ 1 ) } \\Omega . \\end{align*}"}
-{"id": "5065.png", "formula": "\\begin{align*} g = \\rho ^ 2 I = \\left [ 4 H _ u ^ 2 - 3 ( K _ u + 1 ) \\right ] ( I _ { \\mathbb { S } ^ 1 } + I _ u ) , \\end{align*}"}
-{"id": "2871.png", "formula": "\\begin{align*} f _ { 0 } ( x ; n , N ) = \\frac { e ^ { - A _ { N } \\left ( \\frac { n } { x } \\right ) } } { 2 \\sinh \\left ( A _ { N } \\left ( \\frac { n } { x } \\right ) \\right ) } \\end{align*}"}
-{"id": "9708.png", "formula": "\\begin{align*} & \\delta _ i = \\beta _ i + O ( 1 ) \\Delta ' ( \\alpha _ { 5 } , \\boldsymbol { \\beta } ^ { * } ) , i = 1 , 2 , 3 , \\\\ & \\delta _ 5 = \\alpha _ 5 + \\beta _ 5 + O ( 1 ) \\Delta ' ( \\alpha _ { 5 } , \\boldsymbol { \\beta } ^ { * } ) , \\end{align*}"}
-{"id": "8191.png", "formula": "\\begin{align*} S _ 1 ^ 4 \\ll \\abs { y } _ { \\infty } ^ 2 \\abs { T } _ { \\infty } ^ { - 2 } \\prod _ { \\nu } \\sum _ { k _ { \\nu } = 0 } ^ { \\lfloor \\abs { a } R _ { \\nu } \\rfloor } g _ { \\nu } ( - k _ { \\nu } ) ^ 4 f _ { \\nu } ( - k _ { \\nu } ) . \\end{align*}"}
-{"id": "7218.png", "formula": "\\begin{align*} E _ { 1 } ^ { p , q } = \\prod _ { i _ { 1 } \\le \\dots \\le i _ { p } } { } ^ { P _ { i _ { 1 } , \\dots , i _ { p } } } K _ { G } ^ { q } ( U _ { i _ { 1 } , \\dots , i _ { p } } ) . \\end{align*}"}
-{"id": "3363.png", "formula": "\\begin{align*} p = p _ 1 \\wedge p _ 2 \\wedge \\dots \\wedge p _ n \\in \\tau _ 1 ( q _ 1 ) \\vee \\tau _ 2 ( q _ 2 ) \\vee \\dots \\vee \\tau _ n ( q _ n ) \\subseteq \\tau _ 1 ( q ) \\vee \\tau _ 2 ( q ) \\vee \\dots \\vee \\tau _ n ( q ) \\subseteq \\bigl ( \\bigvee T \\bigr ) ( q ) \\end{align*}"}
-{"id": "1674.png", "formula": "\\begin{align*} \\mu ( \\sigma _ { f _ 1 } ( Z ( \\lambda _ { N } ) ) = ( 1 / 2 - \\gamma _ 1 ) \\prod _ { i = 2 } ^ { N + 1 } [ \\frac { 1 } { 2 } + ( - 1 ) ^ { m _ i } \\gamma _ i ] \\end{align*}"}
-{"id": "5976.png", "formula": "\\begin{align*} \\phi _ n ( x ) = \\frac { \\sqrt { \\beta } } { \\sqrt { 2 ^ n n ! } } \\exp ( - \\delta ^ 2 x ^ 2 ) h _ { n - 1 } ( \\alpha \\beta x ) , \\end{align*}"}
-{"id": "5461.png", "formula": "\\begin{align*} \\widehat U ( x ^ { - 1 } ) & = \\int _ 0 ^ \\infty e ^ { - y / x } \\dd U ( y ) \\\\ & \\leq U ( x ) + \\sum _ { n = 1 } ^ { \\infty } e ^ { - 2 ^ { n - 1 } } U ( 2 ^ n x ) \\\\ & \\leq 2 K x ^ { \\rho } \\ell ( x ) \\left [ 1 + \\sum _ { n = 1 } ^ \\infty e ^ { - 2 ^ { n - 1 } } 2 ^ { n ( \\rho + 1 ) } \\right ] . \\end{align*}"}
-{"id": "3108.png", "formula": "\\begin{align*} P _ t ^ { \\mu } = 1 - \\prod _ { j \\in \\mathcal { J } } ( 1 - \\Theta _ j ^ { \\mu } ) . \\end{align*}"}
-{"id": "3081.png", "formula": "\\begin{align*} \\nabla _ x F ( t , u ( t ) ) = 0 \\iff u ( t ) = 0 . \\end{align*}"}
-{"id": "7124.png", "formula": "\\begin{align*} \\partial _ t Y = ~ \\sqrt { ( 1 - | Y | ^ 2 ) ( 1 - \\langle N , Y \\rangle ^ 2 ) } F ^ { - \\alpha } ( \\mathcal { W } ^ X ) N \\end{align*}"}
-{"id": "2014.png", "formula": "\\begin{align*} z _ i ( q ) = \\sum _ { \\substack { \\alpha _ 1 , . . . , \\alpha _ { k + 1 } \\in \\Z , \\\\ \\alpha _ 1 \\leq . . . \\leq \\alpha _ { k + 1 } , \\\\ \\alpha _ 1 + . . . + \\alpha _ { k + 1 } = \\frac { k ( k + 1 ) } { 2 } } } q ^ { \\alpha _ 1 ^ 2 + . . . + \\alpha _ { k + 1 } ^ 2 - 1 ^ 2 - . . . k ^ 2 } a _ { \\alpha _ 1 , . . . , \\alpha _ { k + 1 } } x _ { i + \\alpha _ 1 } x _ { i + \\alpha _ 2 } . . . x _ { i + \\alpha _ { k + 1 } } , ~ ~ ~ i \\in \\Z \\end{align*}"}
-{"id": "8474.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { t , a ( \\chi ) , v } ) = q ^ { - \\frac { t } { 4 } - \\frac { n } { 4 } } \\sum _ { \\pm } \\gamma _ F \\left ( - \\frac { b } { 2 } , \\rho \\right ) \\gamma _ F \\left ( \\pm Y , 2 \\{ - \\frac { t } { 4 } \\} \\right ) \\chi \\left ( - \\frac { b } { v } \\right ) \\psi ( ( \\pm Y \\varpi ^ { l + \\frac { t } { 2 } } - b ) \\varpi ^ { - l } ) \\end{align*}"}
-{"id": "8719.png", "formula": "\\begin{align*} & u ^ \\varepsilon ( 0 , x _ { \\sigma , \\varepsilon } ) = u ( 0 , x ' _ { \\sigma , \\varepsilon } ) - \\varepsilon ^ { - 1 } | x _ { \\sigma , \\varepsilon } - x ' _ { \\sigma , \\varepsilon } | ^ 2 \\quad \\\\ & v _ \\varepsilon ( 0 , y _ { \\sigma , \\varepsilon } ) = v ( 0 , y ' _ { \\sigma , \\varepsilon } ) + \\varepsilon ^ { - 1 } | y _ { \\sigma , \\varepsilon } - y ' _ { \\sigma , \\varepsilon } | ^ 2 . \\end{align*}"}
-{"id": "7155.png", "formula": "\\begin{align*} & = \\mathbb { E } \\left [ \\sum _ { k = 1 } ^ { K } z ( a ^ k ) - Z ( a ^ * ) \\right ] \\\\ & = \\mathbb { E } \\left [ \\sum _ { n \\in \\mathcal { A } ( L ^ m ) } \\theta _ { n , K } \\frac { Z ( n ) } { K } - \\theta _ { a ^ * , K } \\frac { Z ( a ^ * ) } { K } \\right ] \\\\ & = \\sum _ { n \\neq a ^ * } \\beta \\delta _ n \\mathbb { E } [ \\theta _ { n , K } ] . \\end{align*}"}
-{"id": "6610.png", "formula": "\\begin{align*} \\lim _ { J \\ni n \\to \\infty } \\bigl \\| w _ { i j } ^ { ( n ) } - u _ { i j } \\bigr \\| _ { L ^ { p } } = 0 \\ . \\end{align*}"}
-{"id": "8989.png", "formula": "\\begin{gather*} { \\cal D } ^ { ( n ) } _ { q , t } ( c ) D ^ { ( n ) } _ q ( c \\pm u ; t ) | _ { u = z _ i } = 0 \\end{gather*}"}
-{"id": "9698.png", "formula": "\\begin{align*} U _ { k } = \\Phi _ 1 ( \\gamma _ 1 ; U _ a ) , ( u _ { k } , v _ { k } ) \\cdot \\textbf { n } _ { k } = 0 , \\end{align*}"}
-{"id": "7107.png", "formula": "\\begin{align*} \\ddot { F } ^ { i j , k l } B _ { i j } B _ { k l } = \\sum _ { i , k } \\ddot { f } ^ { i k } B _ { i i } B _ { k k } + 2 \\sum _ { i > k } \\frac { \\dot { f } ^ i - \\dot { f } ^ k } { \\kappa _ i - \\kappa _ k } B _ { i k } ^ 2 . \\end{align*}"}
-{"id": "8029.png", "formula": "\\begin{align*} g _ k ( x ) : = 2 ^ { k { d } / { p } } g ( 2 ^ k x ) . \\end{align*}"}
-{"id": "3873.png", "formula": "\\begin{align*} 0 & = 2 ( x _ 2 - 1 ) + \\gamma _ 2 y _ 2 \\Longrightarrow \\gamma _ 2 y _ 2 \\approx 2 , \\\\ 0 & = 2 ( x _ 3 - 2 ) + \\gamma _ 3 y _ 3 \\Longrightarrow \\gamma _ 3 y _ 3 \\approx 4 , \\\\ 0 & = \\nu + \\gamma \\circ x , \\nu \\geq 0 , \\gamma \\circ x \\geq 0 \\end{align*}"}
-{"id": "4745.png", "formula": "\\begin{align*} { \\varepsilon } = F ( x ^ 0 , Y , Z ) \\ , { \\Delta \\sqrt { - \\det g ^ { i j } } } / { { L } _ 1 } , \\end{align*}"}
-{"id": "8684.png", "formula": "\\begin{align*} x ^ { [ 2 p ] } = \\Big ( \\frac { 1 } { 2 } P ( x , x ) \\Big ) ^ { [ p ] } \\forall x \\in \\gg _ 1 . \\end{align*}"}
-{"id": "2526.png", "formula": "\\begin{align*} T _ 1 = - \\int ( \\partial _ { k } \\sigma ^ { i j } ) ( \\partial _ { i } \\partial _ { x _ { k } } h ) ( \\partial _ j h ) \\varrho \\ , m _ 0 - \\int ( \\partial _ { k } \\sigma ^ { i j } ) ( \\partial _ { x _ { k } } h ) ( \\partial _ { j } h ) \\partial _ { i } ( \\varrho \\ , m _ 0 ) = : T _ { 1 1 } + T _ { 1 2 } . \\end{align*}"}
-{"id": "8453.png", "formula": "\\begin{align*} \\abs { S _ { \\chi } ( 1 , a \\varpi ^ l , m ) } \\leq \\begin{cases} 2 \\zeta _ F ( 1 ) q ^ { - \\frac { m } { 2 } } & l = 0 a \\in \\mathcal { O } ^ { \\times 2 } , \\\\ \\zeta _ F ( 1 ) q ^ { - 1 } & m = 1 l \\geq 1 , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "315.png", "formula": "\\begin{align*} L _ { \\tau _ { i + 1 } - 1 } - L _ { \\eta _ i - 1 } & = 0 , \\\\ L _ { \\eta _ i - 1 } - L _ { \\tau _ i - 1 } & = Z _ { \\tau _ i - 1 } ( 1 + \\xi _ { \\tau _ i } + \\xi _ { \\tau _ i } \\xi _ { \\tau _ i + 1 } + \\dots + \\xi _ { \\tau _ i } \\xi _ { \\tau _ i + 1 } \\cdots \\xi _ { \\eta _ i - 2 } ) \\\\ & \\le 1 + \\xi _ { \\tau _ i } + \\xi _ { \\tau _ i } \\xi _ { \\tau _ i + 1 } + \\dots + \\xi _ { \\tau _ i } \\xi _ { \\tau _ i + 1 } \\cdots \\xi _ { \\eta _ i - 2 } \\le \\tilde V _ i , \\end{align*}"}
-{"id": "2282.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "3584.png", "formula": "\\begin{align*} A _ + ( z ) : = \\sum _ { i > 0 } \\alpha _ i [ p ] z ^ i , A _ - ( z ) : = \\sum _ { i < 0 } \\alpha _ i [ p ] z ^ i \\end{align*}"}
-{"id": "8714.png", "formula": "\\begin{align*} & J [ u ^ * ] ( \\hat { t } , \\hat { x } ) + K _ { ( 0 , \\rho ) } [ \\varphi ] ( \\hat { t } , \\hat { x } ) + K _ { ( \\rho , \\hat { t } ) } [ u ^ * ] ( \\hat { t } , \\hat { x } ) \\\\ & + F _ * ( \\hat { t } , \\hat { x } , u ^ * ( \\hat { t } , \\hat { x } ) , \\nabla \\varphi ( \\hat { t } , \\hat { x } ) , \\nabla ^ 2 \\varphi ( \\hat { t } , \\hat { x } ) ) \\le 0 . \\end{align*}"}
-{"id": "8623.png", "formula": "\\begin{align*} \\int _ { \\R ^ { d + 1 } } | \\Phi _ { t , x , B } ( s ' , y ' ) | ^ 2 d s ' d y ' = \\int _ { [ 0 , t ] ^ 2 } R ( s - u , B _ s - B _ u ) d s d u . \\end{align*}"}
-{"id": "538.png", "formula": "\\begin{align*} u ' ( s ) - \\nabla H ( u ( s ) ) = 0 . \\end{align*}"}
-{"id": "1375.png", "formula": "\\begin{align*} ( x _ c ) ' ( \\overline y ; \\Delta y ) = { \\cal A } _ c ( \\overline y , W _ 1 , W _ 2 ) ^ { - 1 } \\left ( - { \\rm D } F ( \\overline x ) ^ * W _ 1 ( \\Delta Y / c ) - { \\cal J } h ( \\overline x ) ^ T ( \\Delta \\mu ) + { \\rm D } g ( \\overline x ) ^ * W _ 2 ( \\Delta \\Gamma ) \\right ) . \\end{align*}"}
-{"id": "2639.png", "formula": "\\begin{align*} 1 + \\sum _ { n \\ge 1 } | n P \\cap \\Z ^ d | z ^ n = \\frac { h _ d ^ * z ^ d + \\cdots + h _ 1 ^ * z + 1 } { ( 1 - z ) ^ { d + 1 } } , \\end{align*}"}
-{"id": "7739.png", "formula": "\\begin{align*} G _ N ( l ) = Z _ N ( l ) + \\bigg ( \\frac { l ^ 2 } { 2 } - \\frac { l } { 2 } \\bigg ) G _ N ( 2 ) - ( l ^ 2 - 2 l ) G _ N ( 1 ) , \\end{align*}"}
-{"id": "2670.png", "formula": "\\begin{align*} f ^ { ( 1 ) } ( i , j ) & = i \\pmod { 2 } , \\\\ f ^ { ( 2 ) } ( i , j ) & = ( i - j ) \\cdot i \\pmod { 2 } , \\\\ f ^ { ( 3 ) } ( i , j ) & = ( i - j + 1 ) \\cdot i \\pmod { 2 } , \\end{align*}"}
-{"id": "7498.png", "formula": "\\begin{align*} ( \\Box ^ h \\Phi ) _ { A _ p \\overline { B } _ q } = - h ^ { \\bar { \\varepsilon } \\gamma } \\nabla _ { \\mathcal { X } _ \\gamma } \\circ \\nabla _ { \\mathcal { X } _ { \\bar { \\varepsilon } } } ( \\phi _ { A _ p \\overline { B } _ q } ) + \\sum _ i h ^ { \\bar { \\varepsilon } \\gamma } [ \\nabla _ { \\mathcal { X } _ \\gamma } , \\nabla _ { \\mathcal { X } _ { \\bar { \\beta } _ i } } ] \\phi _ { A _ p \\bar { \\varepsilon } \\bar { \\beta } _ 1 \\dots \\hat { \\bar { \\beta } } _ i \\dots \\bar { \\beta } _ q } . \\end{align*}"}
-{"id": "8407.png", "formula": "\\begin{align*} f _ \\sigma = \\frac { | \\mathring { h } | ^ 2 } { H ^ { 2 - \\sigma } } \\end{align*}"}
-{"id": "1192.png", "formula": "\\begin{align*} E _ l ( t ) = \\{ x : u ' _ l ( x ) > t \\} \\ , \\ , \\ , \\mbox { i s c o n v e x f o r } \\ , \\ , l = 1 , 2 , \\dots , \\ , \\ , \\mbox { a n d } \\ , \\ , t \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "4856.png", "formula": "\\begin{align*} s = 1 5 d ^ 2 - 3 3 d - 3 0 \\sum \\delta _ p - \\sum _ I ( 4 m _ p + 4 l _ p - 1 5 ) - \\sum _ J ( 1 0 m _ p + c _ p - 1 5 ) , \\end{align*}"}
-{"id": "6445.png", "formula": "\\begin{align*} p ( x _ { 1 } , x _ { 2 } | \\mu , \\sigma ) = \\frac { \\exp \\left \\{ - \\frac { 1 } { 2 \\sigma ^ { 2 } ( 1 - \\rho ^ { 2 } ) } \\left [ ( x _ { 1 } - \\mu ) ^ { 2 } - 2 \\rho ( x _ { 1 } - \\mu ) ( x _ { 2 } - \\mu ) + ( x _ { 2 } - \\mu ) ^ { 2 } \\right ] \\right \\} } { 2 \\pi \\sigma ^ { 2 } \\sqrt { 1 - \\rho ^ { 2 } } } \\end{align*}"}
-{"id": "1379.png", "formula": "\\begin{align*} \\underline \\kappa _ { 1 4 } ( c ) = c ^ { - 1 } - { \\rho _ 0 \\overline { \\nu } _ 0 \\sqrt { \\overline { \\nu } } } c ^ { - 2 } - \\frac { \\overline { \\sigma } \\overline { \\eta } } { c ( \\sqrt { c } + \\sqrt { c _ 0 } ) ^ 2 } \\ , . \\end{align*} % \\end{align*}"}
-{"id": "9642.png", "formula": "\\begin{align*} \\Phi _ { \\rm { s t } , \\beta | ( \\tau _ 0 , \\cdots , \\tau _ i ) } ( t ) = \\Phi _ { \\rm { s t } , \\beta | \\tau _ i } ( t ) , \\end{align*}"}
-{"id": "7344.png", "formula": "\\begin{align*} M \\ ; > \\ ; 2 C \\ , \\Big ( \\Vert f \\Vert _ { H ^ 1 ( \\R ^ 3 ) } + \\sum _ { i = 1 } ^ N \\| F _ i \\| _ { L ^ { s _ i } ( [ 0 , T ] , L ^ { p _ i } ( \\R ^ 3 ) } + \\sum _ { i = 1 } ^ N \\| G _ i \\| _ { L ^ { \\widetilde { s } _ i } ( [ 0 , T ] , L ^ { \\widetilde { p } _ i } ( \\R ^ 3 ) } \\Big ) \\end{align*}"}
-{"id": "7607.png", "formula": "\\begin{align*} U ^ { - 1 } ( p _ 2 ^ A - p _ 1 ^ A ) ^ 2 \\le ( p _ 2 ^ A - p _ 1 ^ A ) \\cdot ( x _ 2 - x _ 1 ) = U ^ { - 1 } ( p _ 2 ^ A - p _ 1 ^ A ) \\cdot U ( x _ 2 - x _ 1 ) . \\end{align*}"}
-{"id": "4074.png", "formula": "\\begin{align*} \\Xi = \\Phi ( \\langle \\nu , \\lambda _ F \\rangle ) = \\Phi ( F ) \\ , . \\end{align*}"}
-{"id": "9188.png", "formula": "\\begin{align*} g ( T ) & = 2 V ( - T ) + T - \\Re V \\ ! \\left ( \\frac { i } { 2 } - T \\right ) - \\Re V \\ ! \\left ( \\frac { i } { 2 } + T \\right ) \\\\ & = 2 V ( - T ) - 2 \\Re V \\ ! \\left ( \\frac { i } { 2 } - T \\right ) , \\end{align*}"}
-{"id": "2844.png", "formula": "\\begin{align*} w _ { n } = \\frac { 1 } { P _ { n } } \\sum _ { v = 0 } ^ { n } p _ { v } s _ { v } \\end{align*}"}
-{"id": "2717.png", "formula": "\\begin{align*} | | h | | _ { H _ { x , v } ^ { s , r } L _ z ^ { \\infty } } = \\sup _ { z \\in I _ z } \\ , | | h | | _ { H _ { x , v } ^ { s , r } } \\ , , | | h | | _ { H _ { x , v } ^ s L _ z ^ { \\infty } } = \\sup _ { z \\in I _ z } \\ , | | h | | _ { H _ { x , v } ^ s } \\ , . \\end{align*}"}
-{"id": "4314.png", "formula": "\\begin{align*} \\begin{aligned} & \\big ( \\nabla F _ { 0 } ( s , r ) - \\nabla F _ { 0 } ( S , R ) \\big ) \\cdot ( s - S , r - R ) ^ { \\top } \\\\ & = \\nabla F _ { 0 } ( s , r ) \\cdot ( s - S , r - R ) ^ { \\top } + \\nabla F _ { 0 } ( S , R ) \\cdot ( S - s , R - r ) ^ { \\top } \\\\ & \\geq c _ { 3 } | \\nabla F _ 0 ( s , r ) | + c _ { 3 } | \\nabla F _ 0 ( S , R ) | - 2 c _ 4 \\\\ & \\geq c _ 3 | \\nabla F _ 0 ( s , r ) - \\nabla F _ 0 ( S , R ) | - 2 c _ 4 . \\end{aligned} \\end{align*}"}
-{"id": "8005.png", "formula": "\\begin{align*} ( \\ref { d d d } ) & \\lesssim \\sup _ { R \\in \\mathcal { D } _ { \\mu } } \\Big ( \\frac { 1 } { | R | } \\int _ R \\sum _ { k = \\max { ( 3 , \\mu - 2 ) } } ^ { \\infty } { 2 ^ { ( s + m ) k q } \\big | \\Pi ^ * _ k f ( x ) \\big | ^ q } d x \\Big ) ^ { 1 / q } \\\\ & \\lesssim \\sup _ { 0 \\leq k \\leq \\mu - 1 } { \\big \\Vert 2 ^ { k ( s + m ) } \\Pi _ k f \\big \\Vert _ { L ^ { \\infty } } } + \\sup _ { R \\in \\mathcal { D } _ { \\mu } } \\Big ( \\frac { 1 } { | R | } \\int _ R \\sum _ { k = \\mu } ^ { \\infty } { 2 ^ { ( s + m ) k q } \\big | \\Pi _ k f ( x ) \\big | ^ q } d x \\Big ) ^ { 1 / q } \\end{align*}"}
-{"id": "8908.png", "formula": "\\begin{align*} \\begin{pmatrix} V _ 1 & \\ast \\\\ \\mathbb O & T _ 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "5855.png", "formula": "\\begin{align*} \\Phi _ { F S } = \\sum H \\left [ Y _ i | X _ i \\right ] - H [ Y | X ] , \\end{align*}"}
-{"id": "806.png", "formula": "\\begin{align*} S _ { k } ( 2 ^ t q ) = 2 ^ { k ( t - 1 ) } q ^ k + 2 ^ { t - 1 } \\frac { S _ { k } ( 2 ^ { t - 1 } q ) } { 2 ^ { t - 2 } } & & \\\\ + 2 ^ { 2 t - 1 } \\sum _ { i = 0 } ^ { \\frac { k - 2 } { 2 } } { \\binom { k } { 2 i } } 2 ^ { ( t - 1 ) ( k - 2 i - 2 ) } q ^ { k - 2 i } S _ { 2 i } ( 2 ^ { t - 1 } q ) . \\end{align*}"}
-{"id": "596.png", "formula": "\\begin{align*} E _ { \\gamma } ( A , \\mathcal { V } _ { N } \\setminus A ) = \\frac { \\Vert \\gamma . F - F \\Vert ^ { 2 } } { \\vert \\mathcal { V } _ { N } \\vert ^ { 2 } } \\geq \\frac { c _ { 0 } ^ { 2 } \\Vert F \\Vert ^ { 2 } } { \\vert \\mathcal { V } _ { N } \\vert ^ { 2 } } = c _ { 0 } ^ { 2 } \\left ( 1 - \\frac { \\vert A \\vert } { \\vert \\mathcal { V } _ { N } \\vert } \\right ) \\vert A \\vert . \\end{align*}"}
-{"id": "2662.png", "formula": "\\begin{align*} g = \\ell ( z + \\kappa ) g _ C + \\frac { d z ^ 2 } { \\Theta ( z ) } + \\Theta ( z ) \\theta ^ 2 , \\ , \\omega = \\ell ( z + \\kappa ) \\omega _ C + d z \\wedge \\theta , \\end{align*}"}
-{"id": "1093.png", "formula": "\\begin{align*} u _ { o , i } & = m _ { s , i } + m _ { c , i } , \\\\ \\sigma _ { o , i } ^ 2 & = u _ { o , i } + m _ { c , i } ^ 2 + 2 m _ { s , i } m _ { c , i } + \\sigma _ { t h } ^ 2 . \\end{align*}"}
-{"id": "6663.png", "formula": "\\begin{align*} | B _ x ( \\Delta ) | _ { n - 1 } \\geq \\frac { ( 2 \\Delta ) ^ { \\frac { n - 1 } { 2 } } | \\mathcal { E } _ x | _ { n - 1 } } { ( 1 + \\varepsilon ) ^ { \\frac { n - 1 } { 2 } } } = \\frac { ( 2 \\Delta ) ^ { \\frac { n - 1 } { 2 } } | B _ 2 ^ { n - 1 } | _ { n - 1 } } { \\kappa ( x ) ^ { 1 / 2 } } ( 1 + \\varepsilon ) ^ { - \\frac { n - 1 } { 2 } } , \\end{align*}"}
-{"id": "5770.png", "formula": "\\begin{align*} - \\Delta _ { p } u = \\sigma u ^ { q } \\ ; \\ ; \\mathbb { R } ^ n , \\end{align*}"}
-{"id": "2623.png", "formula": "\\begin{align*} \\big ( S ( \\phi _ a g ) \\big ) ( t ) = ( S \\psi ) ( t ) + \\int _ 0 ^ { t _ 1 } \\frac { ( \\phi _ a g - \\psi ) ( s ) } { t - s } \\ , d s \\\\ + \\int _ { t _ 2 } ^ 1 \\frac { ( \\phi _ a g - \\psi ) ( s ) } { t - s } \\ , d s , t _ 1 < t < t _ 2 . \\end{align*}"}
-{"id": "9531.png", "formula": "\\begin{align*} | x | ^ 2 \\ , v _ { \\lambda _ 1 } ^ { 1 - m } ( x ) = ( n - 1 ) ( n - 2 ) - B \\ , | x | ^ { - \\gamma } + o ( | x | ^ { - \\gamma } ) \\end{align*}"}
-{"id": "5561.png", "formula": "\\begin{align*} x = - \\tau ^ { 2 } \\left ( \\alpha \\bar { \\nu } ^ { T } + \\beta \\bar { \\nu } ^ { T } \\Lambda _ { \\bar { r } } \\right ) P ^ { 2 } \\bar { w } _ { 2 ^ { k } } \\left ( t \\right ) + \\tau \\dot { x } _ { 0 } e _ { 1 } ^ { T } P \\bar { w } _ { 2 ^ { k } } \\left ( t \\right ) + x _ { 0 } \\end{align*}"}
-{"id": "308.png", "formula": "\\begin{align*} \\P \\left ( Z _ { d + 1 } \\in B \\ | \\ Z _ 0 = z \\right ) & \\ge \\P \\left ( Z _ { d + 1 } \\in B , Z _ 1 \\in V | Z _ 0 = z \\right ) \\\\ & = \\P \\left ( Z _ { d + 1 } \\in B \\ | \\ Z _ 1 \\in V , Z _ 0 = z \\right ) \\cdot \\P ( Z _ 1 \\in V \\ | \\ Z _ 0 = z ) \\\\ & \\ge \\eta ~ \\P \\left ( Z _ { d + 1 } \\in B \\ | \\ Z _ 1 \\in V \\right ) \\ge ~ \\lambda _ d \\left ( B \\cap K \\right ) . \\end{align*}"}
-{"id": "8841.png", "formula": "\\begin{align*} \\int _ { \\R } \\chi _ 1 ( \\zeta , a ) d \\zeta = 1 + O \\left ( \\frac { 1 } { a } \\right ) \\end{align*}"}
-{"id": "7433.png", "formula": "\\begin{align*} \\delta : = d - ( t _ 1 + 2 t _ 2 + 3 t _ 3 + 4 t _ 4 ) / 5 , \\beta : = b - ( t _ 5 ) / 2 , \\gamma : = c - ( 2 t _ 7 + t _ 6 ) / 3 \\end{align*}"}
-{"id": "5265.png", "formula": "\\begin{align*} \\beta _ 2 ( ( z _ n ) _ { n \\in \\N } ) = ( z ^ 2 _ n ) _ { n \\in \\N } \\hbox { a n d } \\beta _ 3 ( ( z _ n ) _ { n \\in \\N } ) = ( z ^ 3 _ n ) _ { n \\in \\N } . \\end{align*}"}
-{"id": "4748.png", "formula": "\\begin{align*} { \\varepsilon } = F ( x ^ 0 , x ^ 1 , Z ) \\ , \\Delta \\sqrt { - \\det g ^ { i j } } , \\end{align*}"}
-{"id": "8417.png", "formula": "\\begin{align*} L ( s , \\pi ) = L ( s , \\chi _ 1 ) L ( s , \\chi _ 2 ) \\epsilon ( \\frac { 1 } { 2 } , \\pi ) = \\epsilon ( \\frac { 1 } { 2 } , \\chi _ 1 ) \\epsilon ( \\frac { 1 } { 2 } , \\chi _ 2 ) . \\end{align*}"}
-{"id": "4199.png", "formula": "\\begin{align*} \\nabla \\begin{pmatrix} \\frac { d x } { y } \\\\ \\frac { x d x } { y } \\end{pmatrix} = A \\begin{pmatrix} \\frac { d x } { y } \\\\ \\frac { x d x } { y } \\end{pmatrix} \\end{align*}"}
-{"id": "6270.png", "formula": "\\begin{align*} \\| u \\| _ { \\dot H ^ s } \\sim \\left ( \\sum _ { q = - 1 } ^ \\infty \\lambda _ q ^ { 2 s } \\| u _ q \\| _ 2 ^ 2 \\right ) ^ { 1 / 2 } , \\ \\ \\ s \\in \\R . \\end{align*}"}
-{"id": "3556.png", "formula": "\\begin{align*} \\chi ( n ) = \\left \\{ \\begin{array} { l l } \\omega ^ { \\log n } & \\gcd ( n , q ) = 1 , \\\\ 0 & \\gcd ( n , q ) \\neq 1 , \\end{array} \\right . \\end{align*}"}
-{"id": "9040.png", "formula": "\\begin{align*} s = n ^ 2 - 4 n + 4 - 2 m , \\end{align*}"}
-{"id": "4841.png", "formula": "\\begin{align*} p _ 1 & = ( 0 : 0 : 1 ) , \\\\ p _ 2 & = \\left ( \\tfrac { 6 4 } { 3 } : \\tfrac { 2 5 6 } { 3 } : 1 \\right ) , \\\\ p _ 3 & = \\left ( \\tfrac { 4 9 } { 2 4 } + i \\tfrac { 7 7 \\sqrt { 7 } } { 2 4 } : \\tfrac { - 6 3 7 } { 4 8 } + i \\tfrac { 3 4 3 \\sqrt { 7 } } { 4 8 } : 1 \\right ) , \\\\ p _ 4 & = \\left ( \\tfrac { 4 9 } { 2 4 } - i \\tfrac { 7 7 \\sqrt { 7 } } { 2 4 } : \\tfrac { - 6 3 7 } { 4 8 } - i \\tfrac { 3 4 3 \\sqrt { 7 } } { 4 8 } : 1 \\right ) . \\end{align*}"}
-{"id": "552.png", "formula": "\\begin{align*} \\frac { d } { d t } K ( \\phi ^ t ( z ) ) \\Big | _ { t = 0 } \\leqslant \\frac { d } { d t } a ( t ) \\Big | _ { t = 0 } = a ' ( 0 ) < 0 . \\end{align*}"}
-{"id": "1114.png", "formula": "\\begin{align*} \\tilde { m } _ { s , i } & = \\mu \\rho _ i ^ 2 \\sum _ { j \\in \\mathcal { M } _ i } \\beta _ j ^ { ( i ) } | \\alpha _ { i j } | ^ 2 , \\ , \\ , \\mathcal { Z } _ 0 ^ { ( i ) } = \\sum _ { j \\in \\mathcal { M } _ i } \\beta _ j ^ { ( i ) } Z _ { 0 , j } ^ { ( i ) } , \\\\ \\mathcal { Z } _ s ^ { ( i ) } & = \\frac { 1 } { \\sqrt { \\sum _ { j \\in \\mathcal { M } _ i } \\beta _ j ^ { ( i ) } | \\alpha _ { i j } | ^ 2 } } \\sum _ { j \\in \\mathcal { M } _ i } \\beta _ j ^ { ( i ) } | \\alpha _ { i j } | Z _ { s , j } ^ { ( i ) } . \\end{align*}"}
-{"id": "2634.png", "formula": "\\begin{align*} & \\int _ Q f - P _ Q f \\ d x = 0 , \\\\ & \\int _ Q ( f - P _ Q f ) x _ i \\ d x = 0 , i = 1 , \\dots , n . \\end{align*}"}
-{"id": "6604.png", "formula": "\\begin{align*} & \\lim _ { J _ 2 \\ni m \\to \\infty } \\int _ { \\varphi _ n ( \\mathsf { U } _ n ) } \\Bigl | \\sum _ { 1 \\leq k \\leq d } u _ { i k } ^ { ( n ) } ( y ) v _ { k j } ^ { ( m , n ) } \\left ( \\phi _ { m , n } ^ { - 1 } ( y ) \\right ) \\ , \\Bigr | ^ p J _ { \\phi _ { m , n } ^ { - 1 } } ( y ) \\ , d \\mu ( y ) \\\\ & = \\int _ { \\varphi _ n ( \\mathsf { U } _ n ) } \\left | u _ { i j } ( y ) \\right | ^ p \\ , d \\mu ( y ) \\ . \\end{align*}"}
-{"id": "714.png", "formula": "\\begin{align*} \\frac { d ^ 2 \\mathcal { T } ( \\alpha ) } { d \\alpha ^ 2 } = - \\left . \\frac { \\eta _ { \\alpha \\alpha } \\eta _ T ^ 2 - 2 \\eta _ { \\alpha T } \\eta _ { \\alpha } \\eta _ T + \\eta _ { T T } \\eta _ { \\alpha } ^ 2 } { \\eta _ T ^ 3 } \\right | { } _ { \\left ( \\alpha , \\mathcal { T } ( \\alpha ) \\right ) } , \\end{align*}"}
-{"id": "1878.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ d \\big ( f _ j ( x _ j ) - g _ j ( x _ j ) ) + \\sum _ { i \\in I } a ^ i 1 _ { A ^ i } ( x ) \\geq f ( x ) \\mbox { f o r a l l } x \\in \\mathbb { R } ^ d , \\end{align*}"}
-{"id": "3939.png", "formula": "\\begin{align*} L _ j ^ n c _ { j - 1 } ^ { n + 1 } + M _ j ^ n c _ { j } ^ { n + 1 } + N _ j ^ n c _ { j + 1 } ^ { n + 1 } = \\sum \\limits _ { l = 1 } ^ { n } ( b _ { n - l } - b _ { n + 1 - l } ) ( a _ 1 c _ { j - 1 } ^ { l } + a _ 2 c _ { j } ^ { l } + a _ 1 c _ { j + 1 } ^ { l } ) \\\\ { } + b _ n ( a _ 1 c _ { j - 1 } ^ { 0 } + a _ 2 c _ { j } ^ { 0 } + a _ 1 c _ { j + 1 } ^ { 0 } ) , \\end{align*}"}
-{"id": "460.png", "formula": "\\begin{align*} \\partial ^ \\beta [ ( \\psi _ \\omega - P _ { 2 , 0 } \\psi _ \\omega ) ^ \\mu ] ( 0 ) \\partial ^ { 2 \\alpha - \\beta } a _ { k _ 1 , k _ 2 } ( i y _ \\omega u _ 1 ) = O \\left ( y _ \\omega ^ { k _ 2 - \\abs { 2 \\alpha - \\beta } } \\right ) = O \\left ( y _ \\omega ^ { k _ 2 - 2 j + 1 } \\right ) \\end{align*}"}
-{"id": "5908.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\liminf _ { n \\to \\infty } \\frac { X ^ { N ; I } _ n } n = \\mu _ 1 . \\end{align*}"}
-{"id": "9700.png", "formula": "\\begin{align*} K _ { b 5 } = 1 , K _ { b i } = 0 , . \\end{align*}"}
-{"id": "3247.png", "formula": "\\begin{align*} \\mathcal H _ k = L _ 0 \\oplus L _ 1 \\cdots \\oplus \\cdots \\end{align*}"}
-{"id": "3834.png", "formula": "\\begin{align*} \\widehat { N } _ L : = \\sum _ { z \\in B } \\sum _ { i \\le N ( z , 0 ) } \\mathbf { 1 } _ { \\{ \\exists s \\in [ 0 , L ^ \\alpha ] \\colon \\ , S ^ { z , i } _ s \\in [ - c _ 1 L ^ \\beta , c _ 1 L ^ { \\beta } ] \\} } \\end{align*}"}
-{"id": "6814.png", "formula": "\\begin{align*} - \\Delta \\tilde { V } \\geq 2 \\sum \\limits _ { j = 1 } ^ 4 \\frac { a ^ 2 } { ( 1 + a ^ 2 | z _ { \\xi _ j } | ) ^ 2 } \\cdot \\frac { ( 1 + | x _ { \\xi _ j } | ^ 2 ) ^ 2 } { 4 } \\geq \\sum \\limits _ { j = 1 } ^ 4 \\frac { a ^ { - 2 } } { | z _ { \\xi _ j } | ^ 4 } \\cdot \\frac { ( 1 + | x _ { \\xi _ j } | ^ 2 ) ^ 2 } { 4 } . \\end{align*}"}
-{"id": "5262.png", "formula": "\\begin{align*} D ( \\lambda ) = \\Omega ( u _ 1 ( \\lambda , z ) , u _ 4 ( \\lambda , z ) ) \\Omega ( u _ 2 ( \\lambda , z ) , u _ 3 ( \\lambda , z ) ) - \\Omega ( u _ 1 ( \\lambda , z ) , u _ 3 ( \\lambda , z ) ) \\Omega ( u _ 2 ( \\lambda , z ) , u _ 4 ( \\lambda , z ) ) . \\end{align*}"}
-{"id": "3419.png", "formula": "\\begin{align*} \\Delta = \\bigl \\{ ( z , w ) \\in \\overline D \\times \\overline D : \\sigma \\circ F ( z ) = \\sigma \\circ F ( w ) \\bigr \\} = \\Delta _ 0 \\cup \\Delta ' , \\end{align*}"}
-{"id": "9461.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ n } { ( - q ^ n ; q ) _ { n + 1 } ( - q ^ { 2 n + 2 } ; q ^ 2 ) _ { \\infty } } & = \\sum _ { j = 0 } ^ { \\infty } ( - 1 ) ^ j q ^ { 6 j ^ 2 + 4 j + 1 } ( 1 + q ^ { 4 j + 2 } ) , \\\\ \\sum _ { n = 0 } ^ { \\infty } q ^ n ( q ^ { n + 1 } ; q ) _ n ( q ^ { 2 n + 2 } ; q ^ 2 ) _ { \\infty } & = \\sum _ { j = 0 } ^ { \\infty } ( - 1 ) ^ j q ^ { 3 j ^ 2 + 2 j } ( 1 + q ^ { 2 j + 1 } ) . \\end{align*}"}
-{"id": "8831.png", "formula": "\\begin{align*} \\| u ( t ) \\| _ { X ^ s } : = \\| u ( t ) \\| _ { H ^ s } + \\| \\Lambda u ( t ) \\| _ { L ^ 2 } \\end{align*}"}
-{"id": "6261.png", "formula": "\\begin{align*} s \\geq & 1 + \\frac n 2 - 2 \\alpha + ( \\alpha - 1 ) \\delta _ 2 + 2 \\alpha \\left ( \\frac 1 { \\theta _ 2 } + \\frac 1 { \\theta _ 4 } + \\frac 1 { \\theta _ 6 } \\right ) \\\\ = & 1 + \\frac n 2 - 2 \\alpha + ( \\alpha - 1 ) \\delta _ 2 + \\epsilon \\end{align*}"}
-{"id": "10017.png", "formula": "\\begin{align*} f _ 0 ^ + ( \\tau ) = \\sum _ { m \\in \\Q } c _ 0 ^ + ( m ) q ^ m . \\end{align*}"}
-{"id": "1202.png", "formula": "\\begin{align*} | \\bar a _ { i j } ( \\hat x ) - \\bar a _ { i j } ( \\hat y ) | & \\leq \\breve c \\ , | \\hat x - \\hat y | \\max _ { B ( x , | x | / 2 ) } \\left \\{ \\left ( | \\nabla \\bar u ( z ) | + | \\nabla G ( z ) | \\right ) ^ { p - 3 } \\sum _ { i , j = 1 } ^ n \\ , ( | u _ { z _ i z _ j } ( z ) | + | G _ { z _ i z _ j } ( z ) | ) \\right \\} \\\\ & \\leq \\breve c ^ 2 | \\hat x - \\hat y | \\ , | x | ^ { \\frac { ( 2 - p ) n - 1 } { p - 1 } } \\end{align*}"}
-{"id": "1022.png", "formula": "\\begin{align*} \\partial _ t \\vec u = \\Delta \\vec u - \\vec u \\cdot \\vec \\nabla \\vec u - \\vec \\nabla p + \\vec f \\mathcal { D } ' ( Q ) . \\end{align*}"}
-{"id": "4308.png", "formula": "\\begin{align*} M - M _ + = m _ v z ^ { - 1 } , \\end{align*}"}
-{"id": "3398.png", "formula": "\\begin{align*} \\alpha _ 0 = d z + \\sum _ { i = 1 } ^ n x _ i d y _ i = d z + x d y . \\end{align*}"}
-{"id": "671.png", "formula": "\\begin{align*} & | | U ( t ) A ( t ) U ^ * ( t ) - B ( t ) | | \\\\ & \\leq | | U ( t ) P _ { m ( n ) } A ( t ) P _ { m ( n ) } U ^ * ( t ) - P _ { m ( n ) } B ( t ) P _ { m ( n ) } | | + \\frac { 1 8 } { n } \\\\ & = | | U _ { 2 m ( n ) } '' ( t ) P _ { m ( n ) } A ( t ) P _ { m ( n ) } U _ { 2 m ( n ) } ''^ * ( t ) - P _ { m ( n ) } B ( t ) P _ { m ( n ) } | | + \\frac { 1 8 } { n } \\leq \\frac { 3 7 } { n } \\end{align*}"}
-{"id": "5887.png", "formula": "\\begin{align*} \\begin{aligned} & P ( \\frac { S ^ { ( i ) } _ { \\tau ^ { N , i } , N + \\tau ^ { N , i } } } N < r _ i ) \\ge P ( \\cup _ { n = 1 } ^ M \\{ Z _ { 2 n } ^ { N , i } = - 1 \\} ) \\times \\\\ & P \\big ( \\cap _ { n = 1 } ^ { 2 M } \\big \\{ Z _ n ^ { N , i } \\in \\{ 0 , - 1 \\} \\big \\} \\big | \\cup _ { n = 1 } ^ M \\{ Z _ { 2 n } ^ { N , i } = - 1 \\} \\big ) . \\end{aligned} \\end{align*}"}
-{"id": "5159.png", "formula": "\\begin{align*} L ^ { + } _ { \\alpha } = \\left \\{ x \\in \\mathcal { D } \\colon F ( x ) \\ge \\alpha \\right \\} \\end{align*}"}
-{"id": "7042.png", "formula": "\\begin{align*} \\eta _ j = \\frac { 1 } { 2 } ( \\sum _ { i \\neq j } { \\sigma _ i } ) . \\end{align*}"}
-{"id": "6298.png", "formula": "\\begin{align*} \\{ 0 ; ( \\varepsilon , g , 0 , t ) ; \\{ ( \\alpha _ i , \\beta _ i ) \\} _ { i = 1 } ^ n \\} . \\end{align*}"}
-{"id": "2852.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { m } \\frac { | s _ { n } | ^ { k } } { n X _ { n } ^ { k - 1 } } = O ( X _ { m } ) m \\rightarrow \\infty , \\end{align*}"}
-{"id": "3718.png", "formula": "\\begin{align*} i _ l = m _ { k _ l } < m _ { k _ l + 1 } < \\dots < m _ { k _ l + \\beta _ l - 1 } < m _ { k _ l + \\beta _ l } = m _ { k _ { l + 1 } } = i _ { l + 1 } . \\end{align*}"}
-{"id": "1678.png", "formula": "\\begin{align*} S _ \\lambda ^ \\mu ( \\chi _ { Z ( \\eta ) } ) ( x ) = \\left ( \\frac { d \\mu } { d ( \\mu \\circ \\sigma ^ n ) } ( x ) \\right ) ^ { - 1 / 2 } \\chi _ { Z ( \\lambda \\eta ) } ( x ) . \\end{align*}"}
-{"id": "8136.png", "formula": "\\begin{align*} C _ { k , l } = C _ { k , l , 1 } \\sqcup C _ { k , l , 2 } \\sqcup \\cdots \\sqcup C _ { k , l , M } \\end{align*}"}
-{"id": "725.png", "formula": "\\begin{align*} 1 < \\frac { \\theta _ - } { \\theta _ + } < 1 + \\frac { D + 2 \\sqrt { 4 D ^ 2 + 1 5 D } } { 1 5 } = 3 . 5 7 0 8 \\times 1 0 . \\end{align*}"}
-{"id": "304.png", "formula": "\\begin{align*} ( \\zeta _ { 1 } , \\dots , \\zeta _ { d } ) & : = T ( \\xi _ 0 , . . . , \\xi _ { d - 1 } ) , \\\\ \\zeta _ 0 & : = \\displaystyle \\prod _ { j = 0 } ^ { d - 1 } ( 1 - \\xi _ j ) = 1 - \\sum _ { j = 1 } ^ d \\zeta _ j . \\end{align*}"}
-{"id": "8301.png", "formula": "\\begin{align*} \\mathrm { w t } _ { - 3 } H = 0 , \\mathrm { w t } _ { - 2 } H = \\mathrm { w t } _ { - 1 } H = I H , \\mathrm { w t } _ 0 H = H , \\end{align*}"}
-{"id": "3641.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - \\Delta u & = \\lambda u & & \\mbox { i n } \\ \\Omega \\ , , \\\\ \\frac { \\partial u } { \\partial \\nu } & = 0 & & \\mbox { o n } \\ \\partial \\Omega \\ , , \\end{aligned} \\right . \\end{align*}"}
-{"id": "2177.png", "formula": "\\begin{align*} w ( \\xi , s ) = ( 1 + t ) ^ { \\frac { d } { 2 } } U _ d ( r ) u _ * ( r , t ) = ( 1 + t ) ^ { \\frac { d } { 2 } } U _ d ( r ) \\left [ u _ * ( 0 , t ) + F _ d ^ 0 ( r , t ) \\right ] \\end{align*}"}
-{"id": "1921.png", "formula": "\\begin{align*} \\zeta _ { i } ^ { s } ( p ) = \\pi ^ { s } ( p ) \\zeta _ { i } ( p ) \\quad \\zeta _ { i } ^ { u } ( p ) = \\pi ^ { u } ( p ) \\zeta _ { i } ( p ) , \\quad i \\in \\mathbb { Z } , p \\in M _ { i } , \\end{align*}"}
-{"id": "3082.png", "formula": "\\begin{align*} u ^ 1 _ \\epsilon ( t ) : = \\langle u _ \\epsilon ( t ) , e _ 1 \\rangle \\quad \\mbox { a n d } u ^ \\perp _ \\epsilon ( t ) : = u _ \\epsilon ( t ) - u ^ 1 _ \\epsilon ( t ) e _ 1 . \\end{align*}"}
-{"id": "721.png", "formula": "\\begin{align*} B = \\sqrt { \\frac { T _ { + } T _ { \\rm i } ^ { 3 / 2 } } { \\kappa p _ { + } } } , g ( x ) = x ^ \\frac 3 4 e ^ { - \\frac { 1 } { 2 x } } . \\end{align*}"}
-{"id": "8978.png", "formula": "\\begin{gather*} \\phi _ { d + 1 } ( x ) f _ { d + 1 } = \\phi _ { d + 1 } ( x ) \\phi _ d ( y ) f _ d = \\phi _ { d + 1 } ( y ) \\phi _ d ( x ) f _ d = \\lambda _ d ( x ) \\phi _ { d + 1 } ( y ) f _ { d + 1 } \\end{gather*}"}
-{"id": "2446.png", "formula": "\\begin{align*} ( \\alpha \\beta ) ^ { \\alpha } = 1 - 3 \\epsilon . \\end{align*}"}
-{"id": "2324.png", "formula": "\\begin{align*} \\binom { u - a } { k } = \\sum _ { p = 0 } ^ k u ^ p b _ p ( a , k ) . \\end{align*}"}
-{"id": "6929.png", "formula": "\\begin{align*} \\langle \\Gamma _ f g , h \\rangle _ { L ^ 2 ( \\mathbb { R } _ + ^ d ) } = \\langle \\check f J \\check { g } , \\check h \\rangle _ { H ^ 2 _ d } . \\end{align*}"}
-{"id": "6685.png", "formula": "\\begin{align*} | C _ p ^ n ( \\Delta ) | _ n = \\frac { 1 } { n - 1 + p } | B _ p ^ { n - 1 } | _ { n - 1 } ( p \\Delta ) ^ { \\frac { n - 1 + p } { p } } ( 1 - \\phi _ p ^ n ( \\Delta ) ) , \\end{align*}"}
-{"id": "8652.png", "formula": "\\begin{align*} B _ k : = A _ k ^ c \\cap \\{ \\min _ { s \\in [ T _ k , T _ { k + 1 } ] } | x + \\omega _ { X _ 0 } ( s ) - y - \\omega _ { Y _ 0 } ( s ) | \\leq 1 \\} , \\end{align*}"}
-{"id": "6577.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\rightarrow 0 } 2 \\ \\sqrt { 2 } \\cdot \\frac { ( 1 + \\varepsilon ) ^ { 1 / 2 } ( 1 - \\varepsilon ) ^ { 3 / 2 } } { 3 - 2 \\varepsilon } = \\frac { 2 \\ \\sqrt { 2 } } { 3 } \\neq \\frac { \\sqrt { 2 } } { 2 } = G ( B _ 1 ^ 2 ) . \\end{align*}"}
-{"id": "6289.png", "formula": "\\begin{align*} \\| u \\| _ \\infty \\leq C \\left ( 1 + \\| u \\| _ { B M O } \\ln ^ { \\frac 1 2 } ( e + \\| u \\| _ { H ^ { m - 1 } } ) \\right ) \\\\ \\| \\nabla b \\| _ \\infty \\leq C \\left ( 1 + \\| \\nabla b \\| _ { B M O } \\ln ^ { \\frac 1 2 } ( e + \\| b \\| _ { H ^ m } ) \\right ) \\end{align*}"}
-{"id": "7752.png", "formula": "\\begin{align*} ( x u x ^ { - 1 } ) w ( x ^ w b x ^ { - 1 } ) = x u x ^ { - 1 } x w b x ^ { - 1 } = x ( u w b ) x ^ { - 1 } \\in [ \\gamma ^ \\Gamma \\cup ( \\gamma ^ { - 1 } ) ^ \\Gamma ] ^ { C _ 1 } \\end{align*}"}
-{"id": "6242.png", "formula": "\\begin{align*} { \\rm r a n k } \\left ( { \\bf Q } _ { \\rm s } \\right ) = { \\rm r a n k } \\left ( \\boldsymbol \\Sigma { \\bf Q } _ { \\rm s } \\right ) \\le { \\rm r a n k } \\left ( \\bar { \\bf H } _ { \\rm b } { \\boldsymbol \\Psi } _ { \\rm b } \\bar { \\bf H } _ { \\rm b } ^ H { \\bf Q } _ { \\rm s } \\right ) \\end{align*}"}
-{"id": "8071.png", "formula": "\\begin{align*} T _ h ^ D = \\frac { 1 } { h ^ 2 } \\ \\operatorname { d i a g } ( - 1 , 2 , - 1 ) \\in \\mathbb { R } ^ { ( n - 1 ) \\times ( n - 1 ) } . \\end{align*}"}
-{"id": "378.png", "formula": "\\begin{align*} | b _ { n , r , s } | & \\le C \\sum _ { j , k = 1 } ^ n ( j + r + | k + s | ) ^ { - \\beta } L ( j + r + | k + s | ) \\\\ & \\propto n \\sum _ { k = 1 } ^ n ( r + | k + s | ) ^ { - \\beta } L ( r + | k + s | ) \\\\ & \\propto n ^ 2 ( r + | s | ) ^ { - \\beta } L ( r + | s | ) . \\end{align*}"}
-{"id": "2133.png", "formula": "\\begin{align*} \\mbox { $ A : = A ^ + ( \\lambda _ 2 ) $ i f $ L $ i s s u b c r i t i c a l } , \\mbox { $ A : = A ^ - ( \\lambda _ 2 ) $ i f $ L $ i s c r i t i c a l , } \\end{align*}"}
-{"id": "3188.png", "formula": "\\begin{align*} V ' ( x ) : = \\frac { \\partial } { \\partial x } V ( x ) = ( 1 + x ) ^ { - 1 } \\quad V '' ( x ) : = \\frac { \\partial ^ { 2 } } { \\partial x ^ { 2 } } V ( x ) = - ( 1 + x ) ^ { - 2 } . \\end{align*}"}
-{"id": "2159.png", "formula": "\\begin{align*} & ( \\partial _ t u _ * ) ( 0 , t ( s ) ) = O ( e ^ { - \\frac { d } { 2 } s - s } ) , \\\\ & ( \\partial _ r F _ N ^ 0 ) ( r ( s ) , t ( s ) ) = O ( e ^ { - \\frac { d } { 2 } s - s } r ( s ) ) , \\\\ & ( \\partial _ t F _ N ^ 0 ) ( r ( s ) , t ( s ) ) = F _ N ^ 1 ( r ( s ) , t ( s ) ) = O ( e ^ { - \\frac { d } { 2 } s - 2 s } r ( s ) ^ 2 ) , \\end{align*}"}
-{"id": "7792.png", "formula": "\\begin{align*} a ( v _ h , \\eta _ h ) + b ( u _ h , t _ h ; \\eta _ h ) & = F ( \\eta _ h ) \\quad \\eta _ h \\in Y _ h , \\\\ b ' ( u _ h , t _ h ; w _ h , s _ h , v _ h ) & = 0 \\quad ( w _ h , s _ h ) \\in X _ h . \\end{align*}"}
-{"id": "605.png", "formula": "\\begin{align*} r ( t + 2 ^ s ) = \\begin{cases} { r ( t ) s > r ( t ) } \\\\ { s s < r ( t ) } \\end{cases} , \\end{align*}"}
-{"id": "3166.png", "formula": "\\begin{align*} \\mathbb { P } ( Z _ { t } = 0 ) = 0 . \\end{align*}"}
-{"id": "5703.png", "formula": "\\begin{align*} P \\varphi = \\left ( \\frac { \\varphi _ 1 ( \\cdot , \\cdot ) - \\varphi _ 1 ( - \\cdot , \\cdot ) } { 2 } , \\varphi _ 2 , \\dots , \\varphi _ n \\right ) . \\end{align*}"}
-{"id": "7724.png", "formula": "\\begin{align*} d _ B ( j , k ) = \\sqrt { \\frac { l ^ 4 } { 1 2 N } - \\frac { l ^ 3 } { 6 } + \\frac { l ^ 2 N } { 1 2 } - \\frac { l ^ 2 } { 6 N } + \\frac { l } { 6 } } \\ , . \\end{align*}"}
-{"id": "2288.png", "formula": "\\begin{align*} \\left | \\bigcup _ { i = 1 } ^ { r } T _ { i } \\right | & = \\sum _ { \\emptyset \\subsetneqq I \\subseteq [ r ] } ( - 1 ) ^ { | I | - 1 } \\cdot | T _ { I } | = \\sum _ { \\emptyset \\subsetneqq I \\subseteq [ r ] } ( - 1 ) ^ { | I | - 1 } \\cdot | U _ { I } | = \\left | \\bigcup _ { i = 1 } ^ { r } U _ { i } \\right | . \\end{align*}"}
-{"id": "2855.png", "formula": "\\begin{align*} A _ { n } ( s ) & = \\sum _ { v = 0 } ^ { n } a _ { n v } s _ { v } = \\sum _ { v = 0 } ^ { n } \\bar { a } _ { n v } a _ { v } \\end{align*}"}
-{"id": "3347.png", "formula": "\\begin{align*} \\bar { D } ( r _ s ) & = \\frac { \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K } { s + 1 + i } ( N - 1 ) ^ i N } { \\binom { K - 2 } { s - 1 } + \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K - 1 } { s + i } ( N - 1 ) ^ i N } \\\\ & = \\frac { \\frac { \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K } { s + 1 + i } ( N - 1 ) ^ i } { \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K - 1 } { s + i } ( N - 1 ) ^ i } } { \\frac { \\binom { K - 2 } { s - 1 } } { \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K - 1 } { s + i } ( N - 1 ) ^ i N } + 1 } \\\\ & = \\frac { \\psi _ 1 ( N , K , s ) } { \\psi _ 2 ( N , K , s ) + 1 } . \\end{align*}"}
-{"id": "4943.png", "formula": "\\begin{align*} k ^ 2 = 1 + p ^ { y - x } = 1 + p ^ { y - 2 e } \\end{align*}"}
-{"id": "2469.png", "formula": "\\begin{align*} \\Vert g \\Vert _ { L ^ { 2 } } ^ { 2 } = \\int _ { { \\mathbb { R } ^ { 3 } } } | g | _ { L _ { \\xi } ^ { 2 } } ^ { 2 } d x \\ , , \\quad \\Vert g \\Vert _ { L ^ { 2 } ( m ) } ^ { 2 } = \\int _ { { \\mathbb { R } ^ { 3 } } } | g | _ { L _ { \\xi } ^ { 2 } ( m ) } ^ { 2 } d x \\ , , \\end{align*}"}
-{"id": "4542.png", "formula": "\\begin{align*} S _ { { t } ^ * } ( f ( { t } ) ) : = f ( { t } - { t } ^ \\star ) \\end{align*}"}
-{"id": "3840.png", "formula": "\\begin{align*} \\mathcal { F } _ k = \\Big \\{ B \\in \\sigma ( \\omega , U ) \\colon \\ , \\ , \\forall \\ , y \\in \\Z ^ 2 , \\ , \\exists \\ , B _ { y } \\in \\mathcal { F } _ { y } \\ , B \\cap \\{ Y _ { R _ k } = y \\} = B _ { y } \\cap \\{ Y _ { R _ k } = y \\} \\Big \\} , \\end{align*}"}
-{"id": "5777.png", "formula": "\\begin{align*} \\| \\mu \\| _ { W ^ { - 1 , p ' } ( \\Omega ) } = \\sup \\frac { | \\langle \\mu , u \\rangle | } { \\| u \\| _ { \\dot { W } _ { 0 } ^ { 1 , p } ( \\Omega ) } } < + \\infty , \\end{align*}"}
-{"id": "4462.png", "formula": "\\begin{align*} \\lefteqn { \\langle \\xi ( k ' ) \\xi ( k '' ) \\overline { \\xi ( l ' ) } \\ , \\overline { \\xi ( l '' ) } \\rangle } \\\\ & = \\left \\{ \\begin{array} { c l } \\langle | \\xi ( k ' ) | ^ 2 | \\xi ( k '' ) | ^ 2 \\rangle & \\mbox { f o r } \\ ; \\{ k ' , k '' \\} = \\{ l ' , l '' \\} \\\\ \\langle | \\xi ( k ' ) | ^ 2 | \\xi ( l ' ) | ^ 2 \\rangle & \\mbox { f o r } \\ ; k ' + k '' = l ' + l '' = 0 \\\\ 0 & \\mbox { e l s e } \\end{array} \\right \\} . \\end{align*}"}
-{"id": "365.png", "formula": "\\begin{align*} \\mathbb { P } \\big ( S _ { n } \\geq x \\big ) & = \\big ( 1 + o ( 1 ) \\big ) \\sum _ { r , s \\in \\mathbb { Z } } \\mathbb { P } \\big ( b _ { n , r , s } \\xi _ { - r , - s } \\geq x \\big ) \\\\ & = \\big ( 1 + o ( 1 ) \\big ) x ^ { - t } \\sum _ { r , s \\in \\mathbb { Z } } b _ { n , r , s } ^ t h \\Big ( \\frac { x } { b _ { n , r , s } } \\Big ) , \\ \\ \\ \\ n \\rightarrow \\infty . \\end{align*}"}
-{"id": "711.png", "formula": "\\begin{align*} G _ + ( \\alpha _ - , T _ - ) = 0 \\hbox { a n d } H _ + ( \\alpha _ - , T _ - ) = 0 \\end{align*}"}
-{"id": "2967.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { K } R ^ { } _ k \\leq \\left [ \\sum _ { k = 1 } ^ { K } R _ k - R _ { _ k } \\right ] ^ + . \\end{align*}"}
-{"id": "10081.png", "formula": "\\begin{align*} \\tilde { \\nabla } _ { X } Y = { \\nabla } _ { X } Y + \\psi ( Y ) X + \\psi ( X ) Y + \\phi ( Y ) X - \\phi ( X ) Y , \\end{align*}"}
-{"id": "8533.png", "formula": "\\begin{align*} \\overline { a } _ { ( j , - l ) } ( x , \\xi ) = \\overline { a } _ { ( 0 , j - l ) } ( \\tilde { x } , \\tilde { \\xi } ) \\ ; , \\ ; a _ { ( k , - m ) } ( x , \\eta ) = a _ { ( 1 , j - m ) } ( \\tilde { x } , \\tilde { \\eta } ) , \\end{align*}"}
-{"id": "2748.png", "formula": "\\begin{align*} I \\leq 2 \\sum _ { m = 1 } ^ K \\ , | | m ^ q h _ m | | _ { H _ { \\Lambda } ^ s } ^ 2 \\cdot \\sum _ { n = 1 } ^ K \\ , | | n ^ q h _ n | | _ { H _ { x , v } ^ s } ^ 2 \\ , , I I \\leq c \\ , \\sum _ { k = 1 } ^ K \\ , | | k ^ q f _ k | | _ { \\Lambda } ^ 2 \\ , , \\end{align*}"}
-{"id": "4264.png", "formula": "\\begin{align*} \\theta ' ( e _ { 1 2 } ' ) = \\frac 1 d \\left ( \\frac { f _ o \\mathrm { d } h _ o - h _ o \\mathrm { d } f _ o } { \\mathrm { d } \\log \\frac { t - \\lambda } { t - 1 } } + f _ o ^ 2 \\left ( \\frac { t - a } { \\lambda - 1 } \\right ) ^ p \\right ) \\cdot \\left ( e ' _ { 1 1 } \\otimes \\mathrm { d } \\log \\frac { t - \\lambda } { t - 1 } \\right ) \\end{align*}"}
-{"id": "7640.png", "formula": "\\begin{align*} \\omega _ { 1 , 1 } & = ( j k x _ 1 ) ( \\overline { j k } x _ 1 ) + ( j x _ 2 ) ( \\bar { j } x _ 2 ) + ( k x _ 3 ) ( \\bar { k } x _ 3 ) + ( x _ 3 ) ( x _ 3 ) \\\\ & = j k \\overline { j k } x _ 1 x _ 1 + j \\bar { j } x _ 2 x _ 2 + k \\bar { k } x _ 3 x _ 3 + x _ 3 x _ 3 \\\\ & = x _ 1 ^ 2 + x _ 2 ^ 2 + 2 x _ 3 ^ 2 \\end{align*}"}
-{"id": "1142.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ \\infty \\frac { ( t - t ^ { q ^ i } ) \\cdot z ^ { q ^ i } } { \\ell _ i } = \\sum _ { i = 0 } ^ \\infty \\frac { z ^ { q ^ { i + 1 } } } { \\ell _ i } \\end{align*}"}
-{"id": "5719.png", "formula": "\\begin{align*} | \\dot { \\tilde { \\gamma } } | ( t ) = \\| \\partial _ t \\gamma ( t , s - m ( t ) ) - \\partial _ s \\gamma ( t , s - m ( t ) ) m ' ( t ) \\| _ { L ^ 2 } , \\end{align*}"}
-{"id": "222.png", "formula": "\\begin{align*} f _ { n } : = h ( g ( H ) _ { \\mathbb { R } _ { \\geq 0 } ^ n \\times \\R ^ { d - n } } ) , \\end{align*}"}
-{"id": "1113.png", "formula": "\\begin{align*} \\tilde { n } _ { o , i } = \\tilde { m } _ { s , i } + \\sqrt { \\tilde { m } _ { s , i } } \\mathcal { Z } _ s ^ { ( i ) } + \\mathcal { Z } _ 0 ^ { ( i ) } , \\end{align*}"}
-{"id": "3662.png", "formula": "\\begin{align*} \\sigma ^ { ( n ) \\vee } = \\begin{pmatrix} 1 & 1 & 1 & \\cdots & 1 & 0 & 0 & 0 & \\cdots & 0 \\\\ 0 & - 1 & 0 & \\cdots & 0 & 0 & 1 & 0 & \\cdots & 0 \\\\ 0 & 0 & - 1 & \\cdots & 0 & 0 & 0 & 1 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & 0 & \\dots & - 1 & 0 & 0 & 0 & \\cdots & 1 \\\\ 0 & 0 & 0 & \\cdots & 0 & 1 & 1 & 1 & \\cdots & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "6966.png", "formula": "\\begin{align*} D f ( B ) A = \\lim _ { \\epsilon \\to 0 } { \\frac { f ( B + \\epsilon A ) - f ( B ) } { \\epsilon } } . \\end{align*}"}
-{"id": "5111.png", "formula": "\\begin{align*} \\rho ^ v _ M ( v _ j ) : = \\sum _ { i = 1 } ^ d M _ { i j } v _ i \\end{align*}"}
-{"id": "8553.png", "formula": "\\begin{align*} \\partial _ t u = \\Delta _ H \\ , u , x \\in { \\bf R } ^ N , \\ , \\ , t > 0 , u ( \\cdot , 0 ) = \\mu \\quad \\mbox { i n } \\quad { \\bf R } ^ N , \\end{align*}"}
-{"id": "2163.png", "formula": "\\begin{align*} | a ( s ) - a _ \\infty | = O ( e ^ { - 2 \\theta ' s } ) \\quad \\mbox { a s } s \\to \\infty . \\end{align*}"}
-{"id": "4351.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty M _ { a _ j } A M _ { b _ j } \\end{align*}"}
-{"id": "3660.png", "formula": "\\begin{align*} B _ e = \\begin{pmatrix} 1 & 0 & \\cdots & \\cdots & 0 & 0 \\\\ - 1 & 1 & \\cdots & \\cdots & 0 & 0 \\\\ 0 & - 1 & \\ddots & & 0 & 0 \\\\ \\vdots & \\vdots & \\ddots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & & \\ddots & 1 & 0 \\\\ 0 & 0 & \\cdots & \\cdots & - 1 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "3287.png", "formula": "\\begin{align*} w _ { \\mu } ( y ) : = \\log \\frac { 8 \\mu ^ 2 } { ( \\mu ^ 2 + | y | ^ 2 ) ^ 2 } , \\mu > 0 . \\end{align*}"}
-{"id": "7227.png", "formula": "\\begin{align*} \\phi \\left ( \\int _ { \\R ^ { d } } f F d \\| T \\| \\right ) - \\phi \\left ( \\int _ { \\R ^ { d } } f F d \\| S \\| \\right ) = \\int _ { \\R ^ { d } } \\langle L , f \\rangle F \\ , d \\left ( \\| T \\| - \\| S \\| \\right ) \\ , . \\end{align*}"}
-{"id": "5520.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } G ( r ^ { - n } z ) ( r ^ { - n } z ) ^ { - \\alpha } = p ( z ) , \\end{align*}"}
-{"id": "9241.png", "formula": "\\begin{align*} \\epsilon ( [ \\alpha _ { 1 } , \\alpha _ { 2 } ] , \\alpha _ { 3 } ) + \\epsilon ( [ \\alpha _ { 2 } , \\alpha _ { 3 } ] , \\alpha _ { 1 } ) + \\epsilon ( [ \\alpha _ { 3 } , \\alpha _ { 1 } ] , \\alpha _ { 2 } ) = 0 \\end{align*}"}
-{"id": "1137.png", "formula": "\\begin{align*} M _ p \\sim \\begin{bmatrix} 1 & - ( i - j ) \\\\ 0 & 1 \\end{bmatrix} , M _ m \\sim \\begin{bmatrix} 1 & \\frac { 4 } { i - j } \\\\ 0 & 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "3263.png", "formula": "\\begin{align*} ( \\alpha _ 0 ( 1 ) + \\alpha _ 0 ( 2 ) , \\omega ) _ X = ( \\alpha _ 0 , \\omega ) _ X . \\end{align*}"}
-{"id": "664.png", "formula": "\\begin{align*} \\max _ { x \\in [ \\alpha ( n ) , 1 ] } \\{ | \\lambda _ i ' ( x ) | \\} = \\max _ { x \\in [ \\alpha ( n ) , 1 ] } \\{ | f _ i ( x ) | \\} \\leq \\max \\{ | \\lambda _ i ( \\alpha ( n ) ) | , | \\lambda _ { \\sigma '^ { - 1 } ( i ) } ( 0 ) | \\} \\leq \\frac { 2 } { n } \\end{align*}"}
-{"id": "3903.png", "formula": "\\begin{align*} R ( r ) = e ^ { - a r ^ 2 } r ^ \\ell . \\end{align*}"}
-{"id": "7523.png", "formula": "\\begin{gather*} w ^ i w ^ 1 _ i - w ^ 1 _ { i i } - 2 \\kappa w ^ 2 + ( 1 - \\kappa ^ 2 ) z _ 1 = 0 , \\\\ w ^ i w ^ 2 _ i - w ^ 2 _ { i i } + 2 \\kappa w ^ 1 + ( 1 - \\kappa ^ 2 ) z _ 2 = 0 . \\end{gather*}"}
-{"id": "2989.png", "formula": "\\begin{align*} - \\int _ \\Omega z \\Delta \\varphi & = - \\int _ \\Omega w \\Delta \\varphi - \\int _ \\Omega v \\Delta \\varphi \\\\ & = \\int _ { \\Omega } h ( v ) f \\varphi + \\int _ { \\Omega } \\varphi d \\mu , ~ \\forall \\varphi \\in C _ 0 ^ 2 ( \\bar { \\Omega } ) . \\end{align*}"}
-{"id": "2451.png", "formula": "\\begin{align*} \\binom { m _ s - 1 } { q _ s - 1 } \\cdot \\binom { m _ r } { k - q _ s } > \\binom { m _ r - 1 } { q _ r - 1 } \\cdot \\binom { m _ s } { k - q _ r } \\end{align*}"}
-{"id": "7601.png", "formula": "\\begin{align*} W _ 2 ( y _ 1 + y _ 2 ) = y _ 1 , W _ 1 ( y _ 1 + y _ 2 ) = y _ 2 . \\end{align*}"}
-{"id": "8999.png", "formula": "\\begin{gather*} { \\cal D } ^ { ( n ) } _ { q , t : * * } ( q ^ { - 1 / 2 } ) = \\sum _ { I \\subset \\{ 1 , \\dots , n \\} } ( - 1 ) ^ { | I | } t ^ { | I | ( | I | - 1 ) / 2 } \\prod _ { 1 \\le i \\le n } z _ i ^ { - 1 } \\prod _ { i \\in I , j \\notin I } \\frac { z _ j - t z _ i } { z _ j - z _ i } \\prod _ { i \\in I } T _ i ^ { 1 / 2 } \\prod _ { i \\notin I } T _ i ^ { - 1 / 2 } , \\end{gather*}"}
-{"id": "4476.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| \\Phi _ n ( f ) - f \\| = 0 \\end{align*}"}
-{"id": "10031.png", "formula": "\\begin{align*} \\langle x , y \\rangle _ { h } = \\frac { 1 } { \\mathrm { r a t } ( \\nu ( h ) ) ^ { n - 1 } } \\cdot \\langle x , y \\rangle . \\end{align*}"}
-{"id": "8585.png", "formula": "\\begin{align*} \\delta < \\| \\pi _ { t _ \\mu } ( j _ { \\lambda _ \\mu } ) b _ { t _ \\mu } \\| = \\| U _ { t _ \\mu } \\pi ( j _ { \\lambda _ \\mu } ) U ^ * _ { t _ \\mu } b _ { t _ \\mu } \\| \\leq \\| \\pi ( j _ { \\lambda _ \\mu } ) U ^ * _ { t _ \\mu } b _ { t _ \\mu } \\| . \\end{align*}"}
-{"id": "9574.png", "formula": "\\begin{align*} \\lvert u ( x ) \\rvert & = \\lim _ { i \\to \\infty } \\lvert u ( x ) - u _ { B _ i } \\rvert \\le C ( M , n , c _ 1 , s , p ) \\limsup _ { i \\to \\infty } \\int _ { D _ i } \\frac { g _ i ( y ) } { \\lvert x - y \\rvert ^ { n - s } } \\ , d y \\\\ & \\le C ( M , n , c _ 1 , s , p ) \\int _ D \\frac { g ( y ) } { \\lvert x - y \\rvert ^ { n - s } } \\ , d y \\ , . \\end{align*}"}
-{"id": "4029.png", "formula": "\\begin{align*} h ( X , Y ) = - \\frac { \\langle d u ^ { - 1 } _ { \\eta ( p ) } Y , d \\eta _ p X \\rangle } { \\langle \\eta , \\xi \\rangle } = - b ( Y , d \\eta _ p X ) . \\end{align*}"}
-{"id": "2818.png", "formula": "\\begin{align*} V ( t , x ) = e ^ { - r ( T - t ) } E _ { t , x } ( G ( X _ { T } ) ) + r K \\int _ { 0 } ^ { T - t } e ^ { - r u } P _ { t , x } ( X \\leq \\mathcal { B } ( t + u ) ) \\mathrm { d } u , \\end{align*}"}
-{"id": "7252.png", "formula": "\\begin{align*} \\begin{cases} 1 , & x \\le 2 0 , \\\\ ( \\log x ) ^ { - 1 } ( \\log \\log x ) ^ { - C } , & x > 2 0 . \\end{cases} \\end{align*}"}
-{"id": "8715.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to 0 } ( K _ { ( 0 , \\rho ) } [ \\varphi ] ( \\hat { t } , \\hat { x } ) + K _ { ( \\rho , \\hat { t } ) } [ u ^ * ] ( \\hat { t } , \\hat { x } ) ) = K _ { ( 0 , \\hat { t } ) } [ u ^ * ] ( \\hat { t } , \\hat { x } ) . \\end{align*}"}
-{"id": "4003.png", "formula": "\\begin{align*} \\mathrm { P r } \\{ \\mathcal { X } > t \\} = \\mathrm { P r } \\{ \\mathcal { N } ( t , \\lambda ) = 1 \\} = E _ { \\nu _ 1 } ( - \\lambda _ 1 t ^ { \\nu _ 1 } ) . \\end{align*}"}
-{"id": "8886.png", "formula": "\\begin{align*} X ^ \\ast ( \\overline \\chi \\omega ) = g _ 1 ( 0 ) \\overline \\chi \\theta . \\end{align*}"}
-{"id": "3090.png", "formula": "\\begin{align*} \\epsilon \\dot { v } _ \\epsilon ( t ) = \\epsilon ( \\dot { u } _ \\epsilon ( t ) - \\dot { \\varphi } ( t ) ) = - \\nabla _ x F ( t , u _ \\epsilon ( t ) ) + \\nabla _ x F ( t , \\varphi ( t ) ) - \\epsilon \\dot { \\varphi } ( t ) \\ , . \\end{align*}"}
-{"id": "917.png", "formula": "\\begin{align*} L F = - \\sum _ { q = 1 } ^ \\infty q J _ q F , F \\in \\mathrm { D o m } ( L ) , \\end{align*}"}
-{"id": "4195.png", "formula": "\\begin{align*} \\nabla _ R ( \\alpha ) = - \\omega , \\nabla _ R ( \\omega ) = 0 . \\end{align*}"}
-{"id": "183.png", "formula": "\\begin{align*} { \\vdash _ { \\delta } } = \\{ \\langle a , b \\rangle : b \\in \\delta ( a ) \\} \\quad \\delta _ { \\vdash } ( a ) = \\mathrm { T h } _ { \\vdash } ( a ) . \\end{align*}"}
-{"id": "3738.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { X _ t } { t } = v \\end{align*}"}
-{"id": "2164.png", "formula": "\\begin{align*} \\left | \\int _ { I ( s ) ^ c } \\hat { w } \\xi ^ { d - 1 } \\ , d \\xi \\right | \\le \\left ( \\int _ { I ( s ) ^ c } \\hat { w } ^ 2 \\rho _ d \\ , d \\xi \\right ) ^ { 1 / 2 } \\left ( \\int _ { I ( s ) ^ c } \\xi ^ { d - 1 } e ^ { - \\frac { \\xi ^ 2 } { 4 } } \\ , d \\xi \\right ) ^ { 1 / 2 } = O ( e ^ { - d \\theta _ * s } ) \\end{align*}"}
-{"id": "1771.png", "formula": "\\begin{align*} \\tilde { t _ \\lambda } X \\ ; = \\ ; X t _ { \\lambda } \\ ; \\ ; \\ ; \\ ; ( \\tilde t _ \\lambda ) ^ * X = X t _ { \\lambda } ^ * \\quad \\ ; \\ ; \\lambda \\in \\Lambda . \\end{align*}"}
-{"id": "9240.png", "formula": "\\begin{align*} \\epsilon ( \\alpha _ { 1 } \\circ \\alpha _ { 2 } , \\alpha _ { 3 } ) + \\epsilon ( \\alpha _ { 2 } \\circ \\alpha _ { 3 } , \\alpha _ { 1 } ) + \\epsilon ( \\alpha _ { 3 } \\circ \\alpha _ { 1 } , \\alpha _ { 2 } ) = 0 \\end{align*}"}
-{"id": "1348.png", "formula": "\\begin{align*} { \\cal C } _ { { \\cal S } _ + ^ p } ( \\overline { M } ) \\ , = \\ , \\left \\{ B \\in { \\cal S } ^ p \\ , \\Big { | } \\ , P _ { \\beta } ^ T B P _ { \\beta } \\succeq 0 , \\ P _ { \\beta } ^ T B P _ { \\gamma } = 0 , \\ P _ { \\gamma } ^ T B P _ { \\gamma } = 0 \\right \\} . \\end{align*} % \\end{align*}"}
-{"id": "8058.png", "formula": "\\begin{align*} \\lambda ( G ( \\Lambda ( \\mu _ { 1 } , \\dots , \\mu _ { m } ) ) ) \\underset { \\log } { \\prec } G ( \\lambda _ { * } \\Lambda ( \\mu _ { 1 } , \\dots , \\mu _ { m } ) ) = G ( ( \\lambda \\circ G ) _ { * } ( \\mu _ { 1 } \\times \\cdots \\times \\mu _ { m } ) ) . \\end{align*}"}
-{"id": "2815.png", "formula": "\\begin{align*} p ( \\tau , x ) = \\int _ { - \\infty } ^ { 0 } ( 1 - e ^ { y } ) \\Gamma ( x - y , \\tau ) \\mathrm { d } y + \\rho \\int _ { 0 } ^ { \\tau } \\int _ { - \\infty } ^ { b ( \\tau - s ) } \\Gamma ( x - y , s ) \\mathrm { d } y \\mathrm { d } s , \\end{align*}"}
-{"id": "7637.png", "formula": "\\begin{align*} X X ^ { \\rm T } = \\Big ( \\sum _ { j = 1 } ^ k c _ j x _ j ^ 2 \\Big ) I _ n , \\end{align*}"}
-{"id": "1005.png", "formula": "\\begin{align*} E \\left [ \\widetilde { Z } _ n ^ * ( t ) ^ 2 | \\mathcal { F } ^ X \\right ] & = \\frac { 2 } { 3 \\mathfrak { s } ^ 2 _ n ( t ) } \\sum _ { i = 1 } ^ n K _ h ( t _ { i - 1 } - t ) ^ 2 ( X _ { t _ { i } } - X _ { t _ { i - 1 } } ) ^ 4 \\leq c _ 0 r _ n ^ 2 n h \\cdot \\frac { n h } { h ^ 2 } = c _ 0 n ^ 2 r _ n ^ 2 \\end{align*}"}
-{"id": "9933.png", "formula": "\\begin{align*} \\frac { d } { d r } p _ t ^ { ( d ) } ( r ) = - 2 \\pi r p _ t ^ { ( d + 2 ) } ( r ) , r > 0 , \\ ; d \\geq 1 . \\end{align*}"}
-{"id": "5018.png", "formula": "\\begin{align*} \\frac { | S | } { r _ 1 } \\le \\frac { | \\Gamma _ 2 ( S ) | } { r _ 2 } = \\frac { | \\Gamma _ 2 ( S ) | } { k _ 1 r _ 1 + 1 } . \\end{align*}"}
-{"id": "8570.png", "formula": "\\begin{align*} { } ^ \\# \\left \\{ y _ { j , \\ell } \\ , : \\ , B _ { H _ 0 } ( y _ { j , \\ell } , 4 \\sqrt { t } ) \\ , \\cap \\ , B _ { H _ 0 } ( y _ { i , k } , 4 \\sqrt { t } ) \\not = \\emptyset \\right \\} \\le m \\end{align*}"}
-{"id": "7442.png", "formula": "\\begin{align*} ( x + t v , 0 , 0 , 0 ) = ( 0 , y , - z , t ) \\end{align*}"}
-{"id": "3372.png", "formula": "\\begin{align*} \\nabla _ { X } ^ { \\Sigma ^ { \\mathbb { C } } Q } \\varphi = \\nabla _ { X } ^ { \\Sigma ^ { a d \\mathbb { C } } } \\varphi + \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { n } e _ i \\cdot B ( e _ i , X ) \\cdot \\varphi , \\end{align*}"}
-{"id": "7744.png", "formula": "\\begin{align*} F _ N ( l ) = & \\sum ^ { l } _ { i = 2 } \\sum ^ { i } _ { j = 2 } 4 \\mathrm { R e } ( T _ N ( l ) ) + \\bigg ( \\frac { l ^ 2 } { 2 } - \\frac { l } { 2 } \\bigg ) F _ N ( 2 ) \\\\ & - ( l ^ 2 - 2 l ) F _ N ( 1 ) \\ , . \\end{align*}"}
-{"id": "4471.png", "formula": "\\begin{align*} \\sum _ { k \\not = 0 } \\exp ( - 2 T d ^ 3 ( k , 0 ) ) & \\min \\{ d ^ { - 1 } ( k , 0 ) , \\ell ^ 2 d ( k , 0 ) \\} \\lesssim \\min \\{ ( T ^ \\frac { 1 } { 3 } ) ^ { - \\frac { 3 } { 2 } } , \\ell ^ 2 ( T ^ \\frac { 1 } { 3 } ) ^ { - \\frac { 7 } { 2 } } \\} . \\end{align*}"}
-{"id": "8175.png", "formula": "\\begin{align*} \\lambda _ { \\nu } = \\begin{cases} \\frac { 1 } { 4 } + t _ { \\nu } ^ 2 & \\\\ 1 + 4 t _ { \\nu } ^ 2 & \\end{cases} \\end{align*}"}
-{"id": "4408.png", "formula": "\\begin{align*} \\bigcup _ { x \\in \\mathcal { M } \\setminus \\mathbb { C } ^ n } \\sigma ( { ( T _ f ) _ x } ) = f ( \\partial \\mathbb { C } ^ n ) . \\end{align*}"}
-{"id": "9870.png", "formula": "\\begin{align*} x ^ { \\nu + 1 } = x ^ \\nu + \\bar \\gamma d ( x ^ \\nu ) . \\end{align*}"}
-{"id": "3807.png", "formula": "\\begin{align*} \\sigma _ { k + 1 } : = \\inf \\{ s > \\sigma _ k \\colon \\ , X _ s - X _ { \\sigma _ k } \\ge L ^ \\beta \\} , \\ ; \\ ; k \\ge 0 . \\end{align*}"}
-{"id": "3746.png", "formula": "\\begin{align*} L _ 0 = 1 0 ^ { 5 0 } L _ { k + 1 } = \\lfloor L _ k ^ { 1 / 2 } \\rfloor L _ k , . \\end{align*}"}
-{"id": "1257.png", "formula": "\\begin{align*} C = \\{ ( x ' , x _ n ) \\in \\mathbb { R } ^ n : | x ' | \\leq r _ j , - r _ j < x _ n < r _ j \\} . \\end{align*}"}
-{"id": "7875.png", "formula": "\\begin{align*} \\begin{aligned} \\overline { \\mathcal { C } } & : = \\{ u \\in \\mathcal { C } \\mid \\} , \\\\ \\underline { \\mathcal { C } } & : = \\{ u \\in \\mathcal { C } \\mid \\} , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "7962.png", "formula": "\\begin{align*} \\mu = \\sum _ { I \\in \\mathcal S } \\lambda _ { I } ^ s ( g _ I ) _ * ( \\mu ) = \\sum _ { I \\in \\mathcal S , \\alpha \\in A _ { I } } \\lambda _ { I J _ { \\alpha } } ^ s ( g _ { I } ) _ { * } ( \\mu ) . \\end{align*}"}
-{"id": "7835.png", "formula": "\\begin{align*} F _ { X _ { 1 } , X _ { 2 } } ( x _ { 1 } , x _ { 2 } ) = \\left ( 1 - e ^ { - \\eta ^ { \\alpha } ( x _ { 1 } ) } \\right ) ^ { \\theta _ { 1 } } \\left ( 1 - e ^ { - \\eta ^ { \\alpha } ( x _ { 2 } ) } \\right ) ^ { \\theta _ { 2 } } \\left ( 1 - e ^ { - \\eta ^ { \\alpha } ( z ) } \\right ) ^ { \\theta _ { 3 } } , \\ \\ z = \\min ( x _ { 1 } , x _ { 2 } ) . \\ \\end{align*}"}
-{"id": "9025.png", "formula": "\\begin{align*} \\| g ^ { \\natural } _ { s t } \\| _ { E _ { - 3 } } \\lesssim _ { A , I } \\| g \\| _ { L ^ \\infty ( s , t ; E _ { - 0 } ) } \\ , \\omega _ A ( s , t ) ^ { \\frac { 3 } { p } } + \\omega _ \\mu ( s , t ) \\omega _ A ( s , t ) ^ { \\frac { 3 - p } { p } } . \\end{align*}"}
-{"id": "4201.png", "formula": "\\begin{gather*} \\rm \\left \\{ \\begin{aligned} \\frac { \\partial T _ 1 } { \\partial \\tau } = T _ 1 ( T _ 2 + T _ 3 ) - T _ 2 T _ 3 , \\\\ \\frac { \\partial T _ 2 } { \\partial \\tau } = T _ 2 ( T _ 1 + T _ 3 ) - T _ 1 T _ 3 , \\\\ \\frac { \\partial T _ 3 } { \\partial \\tau } = T _ 3 ( T _ 1 + T _ 2 ) - T _ 1 T _ 2 . \\end{aligned} \\right . \\end{gather*}"}
-{"id": "1208.png", "formula": "\\begin{align*} \\xi = \\frac { \\nabla u _ 1 ( y ) } { | \\nabla u _ 1 ( y ) | } = \\frac { \\nabla u _ 2 ( z ) } { | \\nabla u _ 2 ( z ) | } = \\frac { \\nabla u ( x ) } { | \\nabla u ( x ) | } \\end{align*}"}
-{"id": "8634.png", "formula": "\\begin{align*} \\begin{aligned} & I _ { 0 , 1 } ( x _ 0 , x _ 1 ) = \\lambda ^ 2 \\int _ 0 ^ \\tau d u \\int _ { 0 } ^ { 1 } d s \\ R ( s + \\tau - u , x _ 1 ( s ) + x _ 0 ( \\tau ) - x _ 0 ( u ) ) , \\\\ & I _ { N , N + 1 } ( x _ N , x _ { N + 1 } ) = \\lambda ^ 2 \\int _ 0 ^ 1 d u \\int _ 0 ^ { T - \\tau - N } d s \\ R ( s + 1 - u , x _ { N + 1 } ( s ) + x _ N ( 1 ) - x _ N ( u ) ) . \\end{aligned} \\end{align*}"}
-{"id": "406.png", "formula": "\\begin{align*} p _ s ( x , t ) = \\frac { 1 } { ( 4 \\pi ) ^ n ( 2 \\pi ) ^ m s ^ { n + m } } \\int _ { \\R ^ m } e ^ { \\frac { i } { s } ( \\lambda , t ) - \\frac { \\abs * { x } ^ 2 } { 4 s } | \\lambda | \\coth ( | \\lambda | ) } \\left ( \\frac { | \\lambda | } { \\sinh | \\lambda | } \\right ) ^ n \\ , \\dd \\lambda , \\end{align*}"}
-{"id": "2104.png", "formula": "\\begin{align*} \\Phi _ s ^ a ( p , t ) = \\int _ M H ( p , q , t ) \\big ( \\phi ^ a _ { s + s _ 0 } ( q ) - \\phi ^ a _ { s _ 0 } ( q ) \\big ) d V _ q , \\end{align*}"}
-{"id": "3943.png", "formula": "\\begin{align*} u ( x , t ) = \\frac { \\mu + \\sigma + ( \\sigma - \\mu ) \\exp [ \\frac { \\mu } { \\nu } ( x - \\sigma t - \\lambda ) ] } { 1 + \\exp [ \\frac { \\mu } { \\nu } ( x - \\sigma t - \\lambda ) ] } \\end{align*}"}
-{"id": "10098.png", "formula": "\\begin{align*} ( \\tilde { \\nabla } _ { X } { \\tilde { S } } ) ( Y , Z ) = ( \\nabla _ { X } S ) ( Y , Z ) . \\end{align*}"}
-{"id": "968.png", "formula": "\\begin{align*} P \\left ( T _ n < q _ n ^ * ( 1 - \\alpha ) \\right ) & \\leq P \\left ( T _ n < q _ n ^ Z ( 1 - \\alpha + \\varepsilon _ n ) \\right ) + P ( \\mathcal { E } _ n ^ c ) \\\\ & \\leq P \\left ( \\max _ { \\theta \\in \\mathcal { G } _ n } | Z _ n ( \\theta ) | < q _ n ^ Z ( 1 - \\alpha + \\varepsilon _ n ) \\right ) + 2 \\varepsilon _ n = 1 - \\alpha + 3 \\varepsilon _ n . \\end{align*}"}
-{"id": "9298.png", "formula": "\\begin{align*} d ( p , C ) : = \\inf \\{ \\| p - q \\| : q \\in C \\} . \\end{align*}"}
-{"id": "8779.png", "formula": "\\begin{align*} \\small \\det P = \\det \\left [ \\left ( x - 1 - \\dfrac { c _ 2 ^ 2 n _ 1 } { x - \\dfrac { 1 } { r _ 1 + 1 } } + \\dfrac { r } { 2 r + n _ 1 } - \\dfrac { c _ 1 ^ 2 2 r } { x - 1 - c _ 3 ^ 2 \\dfrac { n _ 2 } { x - \\frac { 1 } { r _ 2 + 1 } } } \\right ) I _ n + \\dfrac { c _ 1 ^ 2 r \\mathcal { L } ( G ) } { x - 1 - c _ 3 ^ 2 \\dfrac { n _ 2 } { x - \\frac { 1 } { r _ 2 + 1 } } } - \\dfrac { r \\mathcal { L } ( G ) } { 2 r + n _ 1 } \\right ] . \\end{align*}"}
-{"id": "8478.png", "formula": "\\begin{align*} \\abs { W _ { \\pi } ( g _ { t , l , v } ) } \\leq 2 q ^ { \\frac { r } { 3 } + \\frac { \\rho } { 6 } } = 2 q ^ { \\frac { n } { 1 2 } } . \\end{align*}"}
-{"id": "3967.png", "formula": "\\begin{align*} u = f + L ( u ) + H ( u ) , \\end{align*}"}
-{"id": "5841.png", "formula": "\\begin{align*} B = \\sqrt { 1 + \\frac { A ^ 2 + 1 } { 2 C _ 0 } } . \\end{align*}"}
-{"id": "2148.png", "formula": "\\begin{align*} W _ * ( x , t ) : = 2 \\zeta ( t ) \\left [ 1 - \\kappa t ^ { - 1 } F ( x ) \\right ] . \\end{align*}"}
-{"id": "763.png", "formula": "\\begin{align*} K _ \\omega = \\{ \\sigma \\in K \\mid \\sigma \\subseteq \\omega \\} . \\end{align*}"}
-{"id": "5565.png", "formula": "\\begin{align*} \\Gamma = \\left ( I _ { 2 ^ { k } } + \\tau ^ { 2 } \\left ( \\alpha I _ { 2 ^ { k } } + \\beta \\Lambda _ { \\bar { r } } \\right ) P ^ { 2 } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "1110.png", "formula": "\\begin{align*} \\zeta _ i ^ \\mathrm { \\infty } = \\sum _ { j \\in \\mathcal { M } _ i } \\frac { | \\alpha _ { i j } | ^ 4 } { 2 | \\alpha _ { i j } | ^ 2 \\sum _ { k \\in \\mathcal { N } , k \\neq i } E [ | \\alpha _ { k j } | ^ 2 ] + \\left ( \\sum _ { k \\in \\mathcal { N } , k \\neq i } E [ | \\alpha _ { k j } | ^ 2 ] \\right ) ^ 2 } , \\end{align*}"}
-{"id": "2391.png", "formula": "\\begin{align*} \\begin{array} { r c l } f _ 1 ( x _ 1 , \\ldots , x _ n ) & = & 0 , \\\\ & \\vdots & \\\\ f _ s ( x _ 1 , \\ldots , x _ n ) & = & 0 . \\end{array} \\end{align*}"}
-{"id": "4204.png", "formula": "\\begin{align*} d \\sigma _ i = - \\sigma _ j \\wedge \\sigma _ k , \\end{align*}"}
-{"id": "9414.png", "formula": "\\begin{align*} a _ { i , j } = a _ { i ' , j } b _ { i , k } = b _ { i ' , k } i + i ' = \\frac { d } { \\ell } + 1 , \\ , k + k ' = 2 b + 1 \\end{align*}"}
-{"id": "9398.png", "formula": "\\begin{align*} \\frac { d } { d t } T _ { j , t } h ( x ) = \\frac { d } { d t } [ \\nu _ { j , t } \\ast h ( x ) ] & = \\frac { d } { d t } \\bigg [ \\int _ { 2 ^ j t < | y | \\le 2 ^ { j + 1 } } \\frac { \\Omega ( y ' ) } { | y | ^ n } h ( x - y ) d y \\bigg ] \\\\ & = \\frac { d } { d t } \\bigg [ \\int _ { \\mathbf S ^ { n - 1 } } \\Omega ( y ' ) \\int _ { 2 ^ j t } ^ { 2 ^ { j + 1 } } \\frac { 1 } { r } h ( x - r y ' ) d r d \\sigma ( y ' ) \\bigg ] \\\\ & = - \\frac { 1 } { t } \\int _ { \\mathbf S ^ { n - 1 } } \\Omega ( y ' ) h ( x - 2 ^ j t y ' ) d \\sigma ( y ' ) . \\end{align*}"}
-{"id": "2099.png", "formula": "\\begin{align*} f ( p , 0 ) = \\phi ( p ) \\end{align*}"}
-{"id": "7995.png", "formula": "\\begin{align*} T _ { [ a ] } f : = \\lim _ { N \\to \\infty } { \\sum _ { k = 0 } ^ { N } { \\sum _ { j = 0 } ^ { N } { T _ { [ a _ { j , k } ] } f } } } , f \\in S ' \\end{align*}"}
-{"id": "6680.png", "formula": "\\begin{align*} C _ p ^ n ( \\Delta ) = \\{ x \\in B _ p ^ n : x _ 1 \\geq 1 - \\Delta \\} . \\end{align*}"}
-{"id": "5637.png", "formula": "\\begin{align*} A C _ { p l o c } ( I , X ) : = \\big \\{ \\gamma \\in \\mathcal { C } ( I , X ) \\ ; : \\ ; \\exists t _ 0 = \\inf I < t _ 1 < \\dots < t _ n = \\sup I , \\ , \\forall i , \\ , \\gamma \\in A C _ { l o c } ( ( t _ i , t _ { i + 1 } ) , X ) \\big \\} . \\end{align*}"}
-{"id": "2301.png", "formula": "\\begin{align*} & \\ ; | \\langle A ^ 2 u ^ N , u ^ N \\cdot \\nabla u ^ N \\rangle | \\\\ \\leq & \\ ; C \\| A u ^ N \\| _ { L ^ { \\frac { 2 n } { n - 2 s } } } \\| \\nabla u ^ N \\| _ { L ^ { \\frac { n } { s } } } ( \\| A u ^ N \\| _ { L ^ 2 } + \\| \\nabla ^ 2 u ^ N \\| _ { L ^ 2 } ) \\\\ \\leq & \\ ; C \\| u ^ N \\| _ { D ( A ^ { 1 + s / 2 } ) } \\| u ^ N \\| _ { D ( A ^ { ( 1 + \\frac { n } { 2 } - s ) / 2 } ) } \\| u ^ N \\| _ { D ( A ) } \\\\ \\leq & \\ ; C \\| u ^ N \\| _ { D ( A ^ { 1 + s / 2 } ) } \\| u ^ N \\| _ { D ( A ) } ^ 2 . \\end{align*}"}
-{"id": "833.png", "formula": "\\begin{align*} \\Delta _ n = \\{ ( x , y ) : x \\geq 0 , \\ , y \\geq 0 , \\ , x + y \\leq n \\} \\end{align*}"}
-{"id": "9702.png", "formula": "\\begin{align*} \\frac { \\partial \\gamma _ 1 } { \\partial \\omega _ k } = \\frac { { u } _ 2 ^ { ( 0 ) } } { \\kappa _ 1 ( { U } _ 2 ^ { ( 0 ) } ) } , \\frac { \\partial \\gamma _ 1 } { \\partial \\alpha _ i } = \\frac { r _ { i } ^ { ( 2 ) } ( { U } _ 2 ^ { ( 0 ) } ) } { \\kappa _ 1 ( { U } _ 2 ^ { ( 0 ) } ) } , i = 2 , 3 , 5 , \\end{align*}"}
-{"id": "1717.png", "formula": "\\begin{align*} \\mu _ \\pi ( Z ( \\lambda ) ) = \\langle \\xi , P ( Z ( \\lambda ) ) \\xi \\rangle = \\langle \\xi , t _ \\lambda t ^ * _ \\lambda \\xi \\rangle . \\end{align*}"}
-{"id": "505.png", "formula": "\\begin{align*} N _ 0 ( J , t ) ^ \\sigma = \\\\ \\{ ( 0 , 0 , 0 ) , ( 0 , 0 , 2 ) , ( 0 , 0 , 1 ) , ( 0 , 2 , 0 ) , ( 0 , 1 , 0 ) , ( 2 , 0 , 0 ) , ( 4 , 0 , 0 ) , \\end{align*}"}
-{"id": "8188.png", "formula": "\\begin{align*} R _ { \\nu } = \\frac { T _ { \\nu } + T _ { \\nu } ^ { \\frac { 1 } { 3 } + \\epsilon } } { 2 \\pi \\abs { y _ { \\nu } } } \\asymp \\frac { T _ { \\nu } } { y _ { \\nu } } . \\end{align*}"}
-{"id": "3520.png", "formula": "\\begin{align*} \\hat { J } ( u ( \\cdot ) ) = \\displaystyle \\int _ 0 ^ T f ( X ^ u ( t ) , u ( t ) ) d t + \\psi ( X ^ u ( t _ 1 ) , X ^ u ( t _ 2 ) , \\cdots , X ^ u ( t _ n ) ) , \\end{align*}"}
-{"id": "2431.png", "formula": "\\begin{align*} v ^ i ( x _ 1 ^ \\ast , y _ 1 ^ \\ast ) & = \\sum \\limits _ { x _ 2 \\in X } p ( x _ 2 | x _ 1 ^ \\ast , \\mu _ 1 ^ \\ast ) v ^ i ( x _ 2 , y _ 2 ^ \\ast ) . \\end{align*}"}
-{"id": "9689.png", "formula": "\\begin{align*} \\begin{cases} & \\gamma _ i = \\alpha _ i + \\beta _ i + O ( 1 ) \\Delta ( \\boldsymbol { \\alpha } ^ { * } , \\boldsymbol { \\beta } ^ { * } ) , \\ , \\ , i = 1 , 2 , 3 , 5 , \\\\ & \\gamma _ 4 = \\alpha _ 4 + \\beta _ 4 , \\end{cases} \\end{align*}"}
-{"id": "2845.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { P _ { n } } { p _ { n } } \\right ) ^ { k - 1 } \\left | w _ { n } - w _ { n - 1 } \\right | ^ { k } < \\infty . \\end{align*}"}
-{"id": "6678.png", "formula": "\\begin{align*} \\left | \\mathrm { c o n v } [ K , \\zeta - \\Delta e _ n ] \\backslash K \\right | _ n = \\delta | K | _ n \\end{align*}"}
-{"id": "874.png", "formula": "\\begin{align*} B _ n = \\sqrt { N _ n } \\max _ { 1 \\leq k \\leq d _ n } \\max _ { 1 \\leq i \\leq N _ n } | \\lambda _ { n , k } ( i ) | . \\end{align*}"}
-{"id": "7712.png", "formula": "\\begin{align*} \\lambda _ 0 & = 0 \\\\ \\lambda _ n & = N , \\ \\ \\ \\ \\ n = 1 , 2 , \\cdots , N - 1 \\\\ u _ { n m } & = \\frac { 1 } { \\sqrt { N } } e ^ { i 2 \\pi n m / N } , \\ \\ \\ \\ \\ n , m = 0 , 1 , \\cdots , N - 1 \\ , . \\end{align*}"}
-{"id": "8097.png", "formula": "\\begin{align*} f _ 0 = X _ 1 ^ 2 X _ 2 ^ 4 - X _ 1 ^ 4 X _ 2 ^ 2 , & f _ 2 = X _ 0 ^ 2 X _ 1 ^ 2 X _ 2 ^ 2 - X _ 0 ^ 2 X _ 1 ^ 4 , \\\\ f _ 1 = X _ 0 ^ 4 X _ 2 ^ 2 - X _ 2 ^ 6 , \\quad \\ ; & f _ 3 = X _ 0 ^ 4 X _ 1 ^ 2 - X _ 1 ^ 2 X _ 2 ^ 4 . \\end{align*}"}
-{"id": "6021.png", "formula": "\\begin{align*} & \\liminf _ { n \\to \\infty } \\frac { 1 } { n } D _ { 1 + s } ( P _ { X ^ { n } Y ^ { n } } \\| \\pi _ { X ^ { n } Y ^ { n } } ) \\\\ & \\geq \\liminf _ { n \\to \\infty } \\left \\{ \\min \\left \\{ 1 , \\frac { 1 - s } { s } \\right \\} \\delta _ { n } - \\frac { 1 } { n s } \\log 2 \\right \\} \\\\ & = \\min \\left \\{ 1 , \\frac { 1 - s } { s } \\right \\} \\liminf _ { n \\to \\infty } \\delta _ { n } \\\\ & > 0 . \\end{align*}"}
-{"id": "1300.png", "formula": "\\begin{align*} \\sqrt [ 3 ] { 2 + 2 \\sqrt { 3 } \\cos \\frac { \\pi } { 1 8 } } + \\sqrt [ 3 ] { 2 + 2 \\sqrt { 3 } \\cos \\frac { 1 1 \\pi } { 1 8 } } + \\sqrt [ 3 ] { 2 + 2 \\sqrt { 3 } \\cos \\frac { 1 3 \\pi } { 1 8 } } = \\sqrt [ 3 ] { 9 } . \\end{align*}"}
-{"id": "4508.png", "formula": "\\begin{align*} \\begin{bmatrix} \\phi ( \\Delta _ n ) ^ * \\phi ( \\Delta _ n ) & \\phi ( \\Delta _ n ) ^ * \\phi ( a ) ^ * \\\\ \\phi ( a ) \\phi ( \\Delta _ n ) & \\phi ( a ) \\phi ( a ) ^ * \\end{bmatrix} \\leq \\begin{bmatrix} \\phi ( \\Delta _ n ^ * \\Delta _ n ) & \\phi ( \\Delta _ n ^ * a ^ * ) \\\\ \\phi ( a \\Delta _ n ) & \\phi ( a a ^ * ) \\end{bmatrix} , \\end{align*}"}
-{"id": "2354.png", "formula": "\\begin{align*} Q _ M ( f ) ( z ) \\coloneqq \\sum _ { k = 0 } ^ M \\frac { f ^ { ( k ) } ( x _ p ) } { k ! } ( z - x _ p ) ^ k \\end{align*}"}
-{"id": "898.png", "formula": "\\begin{align*} a _ { i j } = \\left \\{ \\begin{array} { l l } 1 & j - i = \\pm 1 , \\\\ 0 & . \\end{array} \\right . \\end{align*}"}
-{"id": "309.png", "formula": "\\begin{align*} Z \\overset { d } { = } ( 1 - \\xi ) Z + \\xi \\Theta , \\end{align*}"}
-{"id": "1156.png", "formula": "\\begin{align*} \\rho _ a ( z ) & = \\sum _ { k = 0 } ^ \\infty \\left ( \\sum _ { j = 0 } ^ k a _ j b _ { k - j } ^ { 2 ^ j } a ^ { 2 ^ j } \\right ) z ^ { 2 ^ k } \\\\ & = \\sum _ { k = 0 } ^ \\infty \\left ( \\sum _ { j = 0 } ^ k \\frac { a ^ { 2 ^ j } } { d _ j \\ell _ { k - j } ^ { 2 ^ j } } \\right ) z ^ { 2 ^ k } . \\end{align*}"}
-{"id": "6682.png", "formula": "\\begin{align*} | C _ p ^ n ( \\Delta ) | _ n = \\frac { 1 } { n - 1 + p } | B _ p ^ { n - 1 } | _ { n - 1 } ( p \\Delta ) ^ { \\frac { n - 1 + p } { p } } ( 1 - \\phi _ p ^ n ( \\Delta ) ) \\quad , \\end{align*}"}
-{"id": "8203.png", "formula": "\\begin{align*} E _ { \\Psi } ( s , g ) = \\sum _ { \\gamma \\in B ( F ) \\setminus G ( F ) } \\Psi ( \\gamma g ) H ( \\gamma g ) ^ s . \\end{align*}"}
-{"id": "9797.png", "formula": "\\begin{align*} \\tilde x _ { i j } ( \\alpha ) . [ A ] = [ \\pi ( \\tilde x _ { j i } ( - \\alpha ) A ) ] = [ B ] \\end{align*}"}
-{"id": "781.png", "formula": "\\begin{align*} T _ { k } ( x ) = \\frac { B _ { k + 1 } ( 2 x + 1 ) - B _ { k + 1 } ( x + 1 ) } { k + 1 } . \\end{align*}"}
-{"id": "7985.png", "formula": "\\begin{align*} h _ i : = f _ { I _ 0 i } : W \\to W _ { i } = h _ i ( W ) \\subset W , \\end{align*}"}
-{"id": "2226.png", "formula": "\\begin{align*} ( 1 - \\varepsilon ^ 2 ) \\left | w _ j - \\frac { a _ { j i _ { j } } } { 1 - \\varepsilon ^ 2 } \\right | ^ 2 = \\frac { \\varepsilon ^ 2 \\cdot | a _ { j i _ { j } } | ^ 2 } { ( 1 - \\varepsilon ^ 2 ) } , \\end{align*}"}
-{"id": "1951.png", "formula": "\\begin{align*} \\begin{cases} 0 < \\lambda < 1 \\\\ 0 < \\lambda \\le \\frac { n - 1 } { 2 } \\end{cases} \\Longleftrightarrow \\begin{cases} \\beta < \\alpha < \\beta + 1 \\\\ \\beta + 1 - \\frac { n - 1 } { 2 } \\le \\alpha < \\beta + 1 . \\end{cases} \\end{align*}"}
-{"id": "6404.png", "formula": "\\begin{align*} \\mathbb { R } ^ { l } \\subseteq \\mathcal { X } \\ni x \\mapsto y \\overset { } { = } f \\left ( x \\right ) \\in \\mathcal { Y } \\subseteq \\mathbb { R } ^ { l } \\Longrightarrow p \\left ( x | \\theta \\right ) \\mapsto p ^ { \\prime } \\left ( y | \\theta \\right ) = \\left [ \\frac { 1 } { \\left \\vert \\frac { \\partial f } { \\partial x } \\right \\vert } p \\left ( x | \\theta \\right ) \\right ] _ { x = f ^ { - 1 } \\left ( y \\right ) } \\end{align*}"}
-{"id": "3594.png", "formula": "\\begin{align*} \\frac { f ( b ) - f ( a ) } { b - a } = f ' ( \\xi ) \\end{align*}"}
-{"id": "975.png", "formula": "\\begin{align*} n h \\cdot \\mathfrak { s } _ n ^ 2 ( t ) = \\frac { 2 } { n h } \\sum _ { i = 1 } ^ n K \\left ( \\frac { t _ { i - 1 } - t } { h } \\right ) ^ 2 = 2 \\int _ { - \\infty } ^ \\infty K ( s ) ^ 2 d s + O ( ( n h ) ^ { - 1 } ) \\end{align*}"}
-{"id": "9347.png", "formula": "\\begin{align*} J O _ { n + 2 } ^ { ( 3 ) } + J O _ { n + 1 } ^ { ( 3 ) } + J O _ { n } ^ { ( 3 ) } = 2 ^ { n + 1 } \\underline { \\alpha } , \\end{align*}"}
-{"id": "5169.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 2 } U _ { i } ( x , y ) = x \\left ( \\frac { a - x } { y - x } \\right ) \\left ( 1 + \\frac { 1 - y } { 1 - x } \\right ) - \\left ( \\frac { ( 1 - y ) ( b - y ) } { y - x } \\right ) \\left ( \\frac { y + x } { y } \\right ) , \\end{align*}"}
-{"id": "977.png", "formula": "\\begin{align*} M _ n ( t ) = 2 \\sum _ { i = 1 } ^ n K _ h ( t _ { i - 1 } - t ) \\int _ { t _ { i - 1 } } ^ { t _ i } \\int _ { t _ { i - 1 } } ^ s \\sigma ( r ) d B _ r \\sigma ( s ) d B _ s \\end{align*}"}
-{"id": "5839.png", "formula": "\\begin{align*} B _ 1 v \\in \\mathrm { k e r } P _ { Q ^ \\perp } = \\mathrm { s p a n } ( Q ) . \\end{align*}"}
-{"id": "7504.png", "formula": "\\begin{align*} \\| \\widetilde { \\omega } _ { N , t } \\| _ { \\mathrm { H S } } = \\sqrt { N } \\ , \\| \\widetilde { W } _ { N , t } \\| _ 2 = \\sqrt { N } \\ , \\| W _ N \\| _ 2 \\end{align*}"}
-{"id": "8635.png", "formula": "\\begin{align*} \\int _ { \\Omega } e ^ { I ( x , y ) } \\Psi ( y ) \\pi ( d y ) = \\rho \\Psi ( x ) , \\end{align*}"}
-{"id": "9539.png", "formula": "\\begin{align*} d ( \\phi ) = \\max _ { z \\in B ( x , r ) } d ( z , \\phi ( z ) ) . \\end{align*}"}
-{"id": "9518.png", "formula": "\\begin{align*} f ( d \\cdot n , m ) = \\tilde { a } _ { v , 0 } ^ { ( m , d ) } - \\sum \\limits _ { l \\mid m } \\sum \\limits _ { k = 0 } ^ { \\tilde { d } - 2 } \\dfrac { l ^ k } { ( \\tilde { d } - 2 - k ) ! } \\cdot \\tilde { p } _ { k } ^ { \\ , ( \\tilde { d } , l ) } \\cdot \\tilde { a } _ { v - \\tilde { d } , k } ^ { ( m , d ) } , \\end{align*}"}
-{"id": "3452.png", "formula": "\\begin{align*} f ( t , x ) : = \\lambda ( x - g ( t ) ) + \\dot { g } ( t ) , t \\in [ 0 , T ] , \\ ; x \\in \\R , \\end{align*}"}
-{"id": "1524.png", "formula": "\\begin{align*} \\begin{cases} \\max \\left ( F ( \\lfloor \\alpha _ { m , t } ( 3 ) \\rfloor ) , G ( \\lfloor \\displaystyle \\frac { t - 1 } { 2 } \\rfloor ) \\right ) , & \\ \\ \\{ \\alpha _ { m , t } ( 3 ) \\} \\leq 1 / 2 ; \\\\ \\max \\left ( F ( \\lfloor \\alpha _ { m , t } ( 3 ) \\rfloor + 1 ) , G ( \\lfloor \\displaystyle \\frac { t - 1 } { 2 } \\rfloor ) \\right ) , & \\ \\ \\{ \\alpha _ { m , t } ( 3 ) \\} > 1 / 2 . \\end{cases} \\end{align*}"}
-{"id": "4113.png", "formula": "\\begin{align*} p - a = \\rho ( p ) \\eta ( p ) + V ( p ) , \\end{align*}"}
-{"id": "7470.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { Z } } _ \\beta & = W ^ \\alpha _ \\beta \\left ( \\mathcal { Z } _ \\alpha - \\rho ^ h _ \\alpha \\dfrac { \\partial M ^ \\gamma _ \\varepsilon } { \\partial z ^ h } W ^ \\tau _ \\gamma u ^ \\varepsilon \\mathcal { V } _ \\tau \\right ) , \\\\ \\widetilde { \\mathcal { V } } _ \\beta & = W ^ \\alpha _ \\beta \\mathcal { V } _ \\alpha , \\end{align*}"}
-{"id": "4041.png", "formula": "\\begin{align*} W _ 1 = ( 1 - c \\mu _ 1 ) d g _ p W _ 1 \\ \\ \\mathrm { a n d } \\\\ W _ 2 = ( 1 - c \\mu _ 2 ) d g _ p W _ 2 . \\end{align*}"}
-{"id": "2382.png", "formula": "\\begin{align*} \\frac { d r } { d \\theta } = \\frac { a _ 1 r + f ( \\theta ) r ^ 2 } { 1 + g ( \\theta ) r } , \\end{align*}"}
-{"id": "6504.png", "formula": "\\begin{align*} V = \\rho \\mathcal { E } = \\rho \\frac { \\hbar ^ { 2 } k _ { \\mathrm { o } } ^ { 2 } } { 2 \\mu } = \\rho \\frac { p _ { \\mathrm { o } } ^ { 2 } } { 2 \\mu } . \\end{align*}"}
-{"id": "4500.png", "formula": "\\begin{align*} s ^ * a s = \\langle a \\xi , \\xi \\rangle s = \\omega ( a ) s \\end{align*}"}
-{"id": "6617.png", "formula": "\\begin{align*} & \\mathcal { E } _ { 1 } ( \\phi ^ { - 1 } ) \\ ; = \\ ; \\int _ { D ^ * } | D \\phi ^ { - 1 } | \\ , d \\mu \\\\ & \\ ; = \\ ; \\int _ { D ^ * } | U _ x | \\ , d \\mu + \\int _ { D ^ * } | U _ y | \\ , d \\mu + \\int _ { D ^ * } | V _ x | \\ , d \\mu + \\int _ { D ^ * } | V _ y | \\ , d \\mu \\end{align*}"}
-{"id": "5234.png", "formula": "\\begin{align*} \\mathrm { s i g n } ( Q ) = n _ + ( Q ) - n _ - ( Q ) . \\end{align*}"}
-{"id": "3818.png", "formula": "\\begin{align*} | \\eta | : = \\sum _ { z \\in \\Z } \\eta ( z ) \\ ; \\in [ 0 , \\infty ] \\end{align*}"}
-{"id": "6619.png", "formula": "\\begin{align*} \\| f - g \\| _ { C ^ 0 ( \\Omega ) } \\leq \\mathrm { d i a m } ( \\mathcal { O } ^ * ) \\ ; = \\ ; 2 \\delta \\ ; < \\ ; \\epsilon _ 0 \\ . \\end{align*}"}
-{"id": "1712.png", "formula": "\\begin{align*} R H S = t _ { \\lambda \\eta } t _ { \\lambda \\eta } ^ * = t _ { \\lambda } t _ { \\eta } t _ { \\eta } ^ * t _ { \\lambda } ^ * . \\end{align*}"}
-{"id": "8549.png", "formula": "\\begin{align*} c _ { 2 } ^ 2 = p ^ { 3 } x ^ 2 y ^ 2 \\frac { 2 - x - y \\pm \\sqrt { ( 2 - x - y ) ^ 2 - ( x - y ) ^ 2 } } { ( x - y ) ^ 2 } . \\end{align*}"}
-{"id": "1990.png", "formula": "\\begin{align*} \\sup _ { c n ^ { - 1 } \\log n \\leq s < t } \\frac { \\left \\vert \\beta _ { n } \\left ( s ; t \\right ) - B _ { n } \\left ( s ; t \\right ) \\right \\vert } { s ^ { \\nu } } = o _ { \\mathbb { P } } \\left ( 1 \\right ) . \\end{align*}"}
-{"id": "857.png", "formula": "\\begin{align*} U _ n ( \\theta ) = \\sum _ { i , j } ( X ^ 1 _ { t ^ 1 _ i } - X ^ 1 _ { t ^ 1 _ { i - 1 } } ) ( X ^ 2 _ { t ^ 2 _ j } - X ^ 2 _ { t ^ 2 _ { j - 1 } } ) 1 _ { \\{ ( t ^ 1 _ { i - 1 } , t ^ 1 _ i ] \\cap ( t ^ 2 _ { j - 1 } - \\theta , t ^ 2 _ j - \\theta ] \\neq \\emptyset \\} } . \\end{align*}"}
-{"id": "6178.png", "formula": "\\begin{align*} R _ { 0 0 } = - \\frac { 1 } { 4 } \\sum _ { i = 1 } ^ 3 \\frac { f _ i '^ 2 } { f _ i ^ 2 } \\ ; \\ ; , \\ ; \\ ; R _ { i i } = - \\frac { f _ i '' } { 2 V ^ 2 } + \\frac { V ' } { 2 V ^ 3 } f _ i ' + \\frac { f _ i '^ 2 } { 4 V ^ 2 f _ i } \\ ; \\ ; \\ ; ; \\ ; \\ ; i = 1 , 2 , 3 \\end{align*}"}
-{"id": "2975.png", "formula": "\\begin{align*} \\int _ \\Omega | \\nabla w | ^ 2 & = \\int _ \\Omega | \\nabla G ( v ) | ^ 2 \\\\ & \\leq C ' . C ( n , \\gamma ) , \\end{align*}"}
-{"id": "1175.png", "formula": "\\begin{align*} \\| h \\| _ { 1 , q } = \\| h \\| _ q + \\| \\ , | \\nabla h | \\ , \\| _ { q } \\end{align*}"}
-{"id": "3589.png", "formula": "\\begin{align*} D _ a ^ \\alpha f : = D ^ { \\lceil \\alpha \\rceil } J _ a ^ { \\lceil \\alpha \\rceil - \\alpha } f , \\end{align*}"}
-{"id": "4062.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } f ( t ) g _ k ( t ) \\ , \\d t & = \\int _ { \\Omega _ 1 } f ( t ) g _ k ( t ) \\ , \\d t + \\int _ { \\Omega _ 2 } f ( t ) g _ k ( t ) \\ , \\d t \\\\ & \\le ( i _ { \\pi ( k ) } + \\varepsilon ) \\int _ { \\Omega _ 1 } f ( t ) \\ , \\d t + ( i _ { \\pi ( k ) } - \\varepsilon ) \\int _ { \\Omega _ 2 } f ( t ) \\ , \\d t \\\\ & \\le \\frac 1 2 ( i _ { \\pi ( k ) } + \\varepsilon ) - \\frac 1 2 ( i _ { \\pi ( k ) } - \\varepsilon ) \\le \\varepsilon . \\end{align*}"}
-{"id": "8721.png", "formula": "\\begin{align*} & u ^ { \\varepsilon , \\delta } ( t , x ) = \\sup _ { t ' \\in [ 0 , T ] } \\{ u ^ \\varepsilon ( t ' , x ) - \\delta ^ { - 1 } | t - t ' | ^ 2 \\} \\quad \\\\ & v _ { \\varepsilon , \\delta } ( t , x ) = \\inf _ { t ' \\in [ 0 , T ] } \\{ v _ \\varepsilon ( t ' , x ) + \\delta ^ { - 1 } | t - t ' | ^ 2 \\} \\end{align*}"}
-{"id": "5330.png", "formula": "\\begin{align*} \\omega = \\frac { 1 } { 2 ( k + h ^ { \\vee } ) } \\sum _ { i = 1 } ^ { \\mbox { d i m } \\ ; \\mathfrak { g } } g ^ i ( - 1 ) ^ 2 { \\bf 1 } , \\end{align*}"}
-{"id": "7963.png", "formula": "\\begin{align*} \\mu = \\sum _ { I _ i \\in \\mathcal S , \\alpha _ i \\in A _ { I _ i } } \\lambda _ { I _ 1 J _ { \\alpha _ 1 } } ^ s \\cdots \\lambda _ { I _ { \\ell } J _ { \\alpha _ { \\ell } } } ^ s ( g _ { I _ { 1 } } \\circ \\cdots \\circ g _ { I _ { \\ell } } ) _ { * } ( \\mu ) . \\end{align*}"}
-{"id": "6752.png", "formula": "\\begin{align*} \\mathbb { P } \\Big \\{ | \\bar { Y } _ { N } - E ( \\bar { Y } _ { N } ) | \\geq \\epsilon \\Big \\} \\leq 2 \\exp \\Bigg \\{ \\frac { - 2 N ^ { 2 } \\epsilon ^ { 2 } } { \\sum _ { n = 1 } ^ { N } ( b _ { n } - a _ { n } ) ^ { 2 } } \\Bigg \\} . \\end{align*}"}
-{"id": "8240.png", "formula": "\\begin{align*} E ( s , g ) = E _ { v ^ { \\circ } } ( s , g ) . \\end{align*}"}
-{"id": "8217.png", "formula": "\\begin{align*} \\alpha _ 0 = \\bigl ( ( 1 + \\hat { H _ { i + 1 } } ) ( 1 + K _ { i + 1 } ) - ( 1 + \\hat { H _ { i } ) } ( 1 + K _ { i } ) \\bigr ) \\end{align*}"}
-{"id": "7403.png", "formula": "\\begin{align*} \\mathbb { H } _ { \\Gamma } : = \\frac { \\mathbb { C } [ t _ 0 , t _ 1 , t _ 2 , t _ 3 , t _ 4 ] } { ( t _ 0 + t _ 1 + t _ 2 + t _ 3 + 2 t _ 4 ) } \\end{align*}"}
-{"id": "8688.png", "formula": "\\begin{align*} ( a , b ) ^ { [ 2 p ] } & = \\Big ( - 4 \\cdot d e t \\begin{pmatrix} - a b & a ^ 2 \\\\ - b ^ 2 & a b \\end{pmatrix} \\Big ) ^ { \\frac { p - 1 } { 2 } } \\begin{pmatrix} - a b & a ^ 2 \\\\ - b ^ 2 & a b \\end{pmatrix} \\\\ & = 0 . \\end{align*}"}
-{"id": "5895.png", "formula": "\\begin{align*} P ( \\cup _ { n = 1 } ^ { [ \\frac 1 { b _ N } ] } \\{ Z _ { 2 n } ^ { N , i } = - 1 \\} ) \\ge 1 - e ^ { - a _ N [ \\frac 1 { b _ N } ] } \\ge \\min ( c , \\frac { a _ N } { 2 b _ N } ) , \\ \\ c > 0 . \\end{align*}"}
-{"id": "6701.png", "formula": "\\begin{align*} \\sum _ { \\tau \\in \\Theta } \\sum _ { j = 0 } ^ n { \\tau \\theta \\choose \\theta } \\tau ( y _ j ) \\cdot \\frac { \\partial f } { \\partial \\tau \\theta ( y _ j ) } = \\left \\{ \\begin{array} { l l l } r f , & \\quad & \\theta = 1 \\\\ 0 , & \\quad & \\theta \\in \\Theta , \\ , \\theta \\neq 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "1617.png", "formula": "\\begin{align*} t _ v = \\sum _ { \\lambda \\in v \\Lambda ^ n } t _ \\lambda t ^ * _ \\lambda . \\end{align*}"}
-{"id": "1130.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & 1 \\\\ - 1 & 0 \\end{bmatrix} \\begin{bmatrix} \\delta & 0 \\\\ c & \\delta \\end{bmatrix} \\begin{bmatrix} 0 & - 1 \\\\ 1 & 0 \\end{bmatrix} = \\begin{bmatrix} \\delta & - c \\\\ 0 & \\delta \\end{bmatrix} . \\end{align*}"}
-{"id": "3232.png", "formula": "\\begin{align*} ( Z _ 2 \\circ Z _ 1 ) _ \\ast = \\langle Z _ 2 \\rangle _ \\ast \\circ \\langle Z _ 1 \\rangle _ \\ast \\end{align*}"}
-{"id": "5922.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { s \\in A - A _ { } } L ( s ) Q ^ N ( s ) \\le \\sum _ { s \\in A _ { } } L ( s ) Q ^ N ( \\mathcal { M } ^ { - 1 } ( \\{ s \\} ) ) \\le \\\\ & \\sum _ { r = 2 } ^ \\infty \\sum _ { s \\in A _ { } : r _ s = r } L ( s ) \\big ( ( 1 + 2 ^ { l - 1 } e ^ { - c _ 0 N } ) ^ r - 1 \\big ) Q ^ N ( s ) , \\end{aligned} \\end{align*}"}
-{"id": "1829.png", "formula": "\\begin{align*} \\frac { d \\tilde { \\mathbb { P } } _ { \\mathcal { G } , \\vec { H } } } { d \\tilde { \\mathbb { P } } _ { \\mathcal { G } , 0 } } = \\frac { \\prod _ { \\mathcal { C } } \\cosh ( H ( \\mathcal { C } ) ) } { \\tilde { \\mathbb { E } } _ { \\mathcal { G } , 0 } \\left ( \\prod _ { \\mathcal { C } } \\cosh \\left ( H ( \\mathcal { C } ) \\right ) \\right ) } , \\end{align*}"}
-{"id": "8083.png", "formula": "\\begin{align*} f _ { F , m } ( \\hat { x } _ G ^ \\ell ) = \\left \\{ \\begin{array} { c } m , G = F , \\\\ 0 , G \\not = F , \\end{array} \\right . \\end{align*}"}
-{"id": "2807.png", "formula": "\\begin{align*} F ( t ) = \\int _ { 0 } ^ { t } \\exp \\left ( - \\frac { 2 r } { \\sigma ^ { 2 } } ( t - s ) - \\frac { ( \\sqrt { t } \\eta ( t ) - \\sqrt { s } \\eta ( s ) ) ^ { 2 } } { t - s } \\right ) \\Big \\{ 1 + \\frac { \\sqrt { t } \\eta ( t ) - \\sqrt { s } \\eta ( s ) } { t - s } \\Big \\} \\frac { \\mathrm { d } s } { \\sqrt { t - s } } . \\end{align*}"}
-{"id": "9168.png", "formula": "\\begin{align*} g _ 4 ( x ^ 4 ) + x g _ 3 ( x ^ 3 ) + x ^ 2 g _ 2 ( x ^ 2 ) + x ^ 3 g _ 1 ( x ) = 0 \\left ( x \\in K \\right ) \\end{align*}"}
-{"id": "8372.png", "formula": "\\begin{align*} c ( m , \\mu ) & = \\sum _ { k \\ge 0 } r ^ { [ 2 ] } ( k ) \\cdot c ^ { [ 2 ] } ( m - k , \\mu ) \\\\ & - \\sum _ { k \\ge 0 } r ^ { [ 1 ] } ( k ) \\cdot c ^ { [ 1 ] } ( m - k , \\mu ) . \\end{align*}"}
-{"id": "2456.png", "formula": "\\begin{align*} \\mu ^ * ( \\pi \\rtimes \\sigma ) = M ^ * ( \\pi ) \\rtimes \\mu ^ * ( \\sigma ) \\end{align*}"}
-{"id": "6459.png", "formula": "\\begin{align*} p \\left ( x | \\theta \\right ) = { \\displaystyle \\prod \\limits _ { k = 1 } ^ { 2 l } } p \\left ( x _ { k } | \\mu _ { k } \\sigma _ { k } \\right ) p \\left ( x _ { k } | \\mu _ { k } \\sigma _ { k } \\right ) \\overset { } { = } \\frac { 1 } { \\sqrt { 2 \\pi \\sigma _ { k } ^ { 2 } } } \\exp \\left [ - \\frac { \\left ( x _ { k } - \\mu _ { k } \\right ) ^ { 2 } } { 2 \\sigma _ { k } ^ { 2 } } \\right ] \\end{align*}"}
-{"id": "428.png", "formula": "\\begin{align*} \\abs { t } = \\frac { \\kappa } { 2 \\pi \\delta } , R = \\frac { \\kappa \\delta } { 2 } , \\end{align*}"}
-{"id": "500.png", "formula": "\\begin{align*} ( 2 I _ { \\nu - 1 } I _ { \\nu + 1 } - I _ \\nu ^ 2 ) ' ( \\zeta ) = I _ { \\nu - 2 } ( \\zeta ) I _ { \\nu + 1 } ( \\zeta ) + I _ { \\nu - 1 } ( \\zeta ) I _ { \\nu + 2 } ( \\zeta ) \\end{align*}"}
-{"id": "3227.png", "formula": "\\begin{align*} j ^ \\ast ( \\omega _ { \\Delta _ X \\subset ( X \\times X ) } ) = \\omega _ { \\Delta _ { V , X } \\subset ( X \\cap V ) \\times X } . \\end{align*}"}
-{"id": "2314.png", "formula": "\\begin{align*} \\| f ( w _ 1 , w _ 2 ) ( t ) \\| _ { D ( A ^ { 1 - s / 2 } ) } \\leq C \\| w _ 1 \\| _ { D ( A ) } \\| A ^ { s / 2 } w _ 2 \\| _ { D ( A ) } \\leq C R _ 1 R _ 2 t ^ { - 1 / 2 } , \\end{align*}"}
-{"id": "6586.png", "formula": "\\begin{align*} \\phi ( x ^ { k _ 0 } _ 0 ) = x ^ 0 _ 0 \\ . \\end{align*}"}
-{"id": "4527.png", "formula": "\\begin{align*} \\pi ( \\Delta _ n ) ^ * \\Pi ( a ^ * a ) \\pi ( \\Delta _ n ) \\leq \\pi ( \\Delta _ n ) ^ * \\pi ( s ) \\pi ( \\Delta _ n ) = \\pi ( \\Delta _ n ^ * s \\Delta _ n ) \\end{align*}"}
-{"id": "3420.png", "formula": "\\begin{align*} \\delta : = \\inf \\bigl \\{ | F ( z ) - F ( w ) | : ( z , w ) \\in \\Delta ' \\bigr \\} > 0 . \\end{align*}"}
-{"id": "2667.png", "formula": "\\begin{align*} a _ { i , j } & = a _ { i + k \\pmod * { N } , j + k \\pmod * { N } } \\\\ a _ { i , j } & = a _ { j , i } , \\end{align*}"}
-{"id": "2912.png", "formula": "\\begin{align*} P = \\{ \\rho _ e \\colon e \\in E \\} , \\end{align*}"}
-{"id": "8497.png", "formula": "\\begin{align*} 1 - v b _ 1 = \\beta \\varpi ^ { l - a _ 2 } . \\end{align*}"}
-{"id": "8001.png", "formula": "\\begin{align*} \\mathfrak { S } _ { [ a ] } ^ { n e a r } f : = \\sum _ { k , j : | j - k | \\leq 2 } { T _ { [ a _ { j , k } ] } f } \\end{align*}"}
-{"id": "1373.png", "formula": "\\begin{align*} \\overline { c } : = \\max \\left \\{ ( 2 + \\sqrt { 2 } ) c _ 0 , \\ , ( \\overline { \\sigma } \\overline { \\eta } - c _ 0 ) ^ 2 / c _ 0 , \\ ( \\underline { \\sigma } \\underline { \\eta } / 2 - c _ 0 ) ^ 2 / c _ 0 \\right \\} \\ , \\end{align*}"}
-{"id": "432.png", "formula": "\\begin{align*} I _ \\nu ( \\zeta ) = \\sum _ { k \\in \\N } \\frac { \\zeta ^ { 2 k + \\nu } } { 2 ^ { 2 k + \\nu } k ! \\ , \\Gamma ( k + \\nu + 1 ) } . \\end{align*}"}
-{"id": "1486.png", "formula": "\\begin{align*} h ( x \\sqrt { c } ) = 2 x \\sqrt { c } - ( 1 + \\alpha ) \\tan ( x \\sqrt { c } ) < 0 \\mbox { f o r $ \\sqrt { c _ \\alpha } < x \\sqrt { c } < \\pi / 2 $ } . \\end{align*}"}
-{"id": "5267.png", "formula": "\\begin{align*} B = \\left ( \\begin{array} { c c } - 1 & 0 \\\\ 0 & - 1 \\end{array} \\right ) \\end{align*}"}
-{"id": "3895.png", "formula": "\\begin{align*} g ( \\omega ; 1 ; k ) = \\pi ^ { 1 / 2 } \\int _ { 2 } ^ { \\infty } \\frac { \\omega ( x ) } { \\sqrt { x ^ 2 - 4 } } \\frac { 2 ^ { 2 k - 1 } } { ( x + \\sqrt { x ^ 2 - 4 } ) ^ { 2 k - 1 } } d x . \\end{align*}"}
-{"id": "3160.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ e ^ { u Y _ { t } ^ { x } } \\right ] = \\left ( 1 - \\tfrac { \\sigma ^ { 2 } u } { 2 b } \\left ( 1 - e ^ { - b t } \\right ) \\right ) ^ { - \\tfrac { 2 a } { \\sigma ^ { 2 } } } \\exp \\left \\lbrace \\tfrac { x u e ^ { - b t } } { 1 - \\tfrac { \\sigma ^ { 2 } u } { 2 b } \\left ( 1 - e ^ { - b t } \\right ) } \\right \\rbrace \\end{align*}"}
-{"id": "7604.png", "formula": "\\begin{align*} \\begin{aligned} v & = ( x _ 1 + x _ 2 , S _ 1 x _ 1 + S _ 2 x _ 2 ) = ( x _ 1 + x _ 2 , S _ 1 ( x _ 1 + x _ 2 ) + U x _ 2 ) \\\\ & = ( x , S _ 1 x + U _ 1 x ) = ( x , ( S _ 1 + U _ 1 ) x ) . \\end{aligned} \\end{align*}"}
-{"id": "8124.png", "formula": "\\begin{align*} \\psi ( a _ 0 , \\dots , a _ d ) = \\psi _ 0 ( a _ 0 ) + \\dots + \\psi _ d ( a _ d ) . \\end{align*}"}
-{"id": "998.png", "formula": "\\begin{align*} \\tau _ k = \\inf \\left \\{ t \\in [ 0 , T ] : | \\sigma ( t ) | > k \\frac { 1 } { | \\sigma ( t ) | } > k \\sup _ { \\eta \\in ( 0 , 1 ) } \\eta ^ { - \\gamma } w ( \\sigma ; \\eta , t ) > k \\right \\} , \\end{align*}"}
-{"id": "8457.png", "formula": "\\begin{align*} \\abs { W _ { \\pi } ( g _ { t , \\frac { n } { 2 } , v } ) } \\leq \\begin{cases} 2 q ^ { \\frac { \\rho } { 2 } } & t = - n , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "10054.png", "formula": "\\begin{align*} a _ F ( 0 , \\phi ) = \\Lambda ( 0 , \\chi ) \\cdot E ' _ 0 ( \\vec { v } , 0 , \\phi ) - \\Lambda ( 0 , \\chi _ E ) \\cdot \\phi ( 0 ) \\log \\mathrm { N m } ( \\vec { v } ) . \\end{align*}"}
-{"id": "3765.png", "formula": "\\begin{align*} I _ { y , L } = y + ( [ 0 , \\mathfrak { R } L ) ^ d \\times \\lbrace 0 \\rbrace ) \\cap ( \\mathbb { Z } ^ { d } \\times \\mathbb { Z } ) . \\end{align*}"}
-{"id": "4358.png", "formula": "\\begin{align*} \\Big | \\sum _ { j = 1 } ^ \\infty ( M _ { a _ j } P _ \\alpha M _ { b _ j } f ) ( z ) \\Big | e ^ { - \\frac { \\alpha } { 2 } | z | ^ 2 } \\leq 2 ^ n \\| f \\| _ { \\infty , \\alpha } \\end{align*}"}
-{"id": "2867.png", "formula": "\\begin{align*} P _ P = \\{ x \\in \\R _ + ^ { n } \\ , | \\ , x _ i + x _ j \\le 1 \\ \\forall i , j \\in [ n ] , \\ i < j \\} . \\end{align*}"}
-{"id": "7158.png", "formula": "\\begin{align*} & = \\sum _ { n \\neq a ^ { * } } \\beta \\delta ( n ) \\mathbb { E } [ \\theta _ { n , K } ] \\\\ & \\leq \\beta \\left [ 8 \\sum _ { n \\neq a ^ { * } } \\frac { \\ln K } { \\delta _ n } + \\left ( 1 + \\frac { \\pi ^ 2 } { 3 } \\right ) \\sum _ { n \\neq a ^ { * } } \\delta _ n \\right ] . \\end{align*}"}
-{"id": "5700.png", "formula": "\\begin{align*} \\mathcal { C } : = \\{ { v } \\in X \\ ; : \\ ; \\pm ( s - s ^ \\pm ) \\geq 0 , \\ , | { v } ( s ) - a ^ \\pm | \\leq E ( s ) \\} . \\end{align*}"}
-{"id": "8928.png", "formula": "\\begin{gather*} \\sum _ { x \\in \\phi _ { N , 1 } ^ { - 1 } ( \\sigma _ N - \\phi _ { N , 1 } ( \\sigma _ 1 ) ) } x = 0 . \\end{gather*}"}
-{"id": "8326.png", "formula": "\\begin{align*} f _ 0 ( \\tau ) = \\sum _ { \\substack { m \\in \\Q \\\\ m \\gg - \\infty } } \\sum _ { \\lambda \\in V _ { 0 \\Z } ^ \\vee / V _ { 0 \\Z } } c _ 0 ( m , \\lambda ) \\cdot q ^ m \\in M ^ ! _ { 1 - \\frac { n } { 2 } } ( \\overline { \\rho } _ { V _ { 0 \\Z } } ) \\end{align*}"}
-{"id": "2310.png", "formula": "\\begin{align*} & \\ ; \\| A ^ { ( r _ { k + 1 } - s ) / 2 } w _ k ( t _ { k + 1 } ) \\| _ { L ^ 2 } ^ 2 + \\int _ { t _ k } ^ { \\infty } \\| A ^ { r _ { k + 1 } / 2 } w _ k \\| _ { L ^ 2 } ^ 2 \\\\ \\leq & \\ ; C \\| u _ 0 \\| _ { D ( A ) } ^ 2 ( \\nu \\delta ) ^ { - \\frac { 2 s - 1 } { s } } \\| A ^ { ( r _ { k - 1 } - s ) / 2 } u _ { k - 1 } \\| _ { L ^ 2 } ^ 2 \\\\ & \\ ; + C \\| u _ 0 \\| _ { D ( A ) } ^ 2 \\left [ 1 + \\int _ { t _ { k - 1 } } ^ \\infty \\| A ^ { r _ k / 2 } w _ { k - 1 } \\| _ { L ^ 2 } ^ 2 \\ , d \\tau \\right ] . \\end{align*}"}
-{"id": "3507.png", "formula": "\\begin{align*} X ^ { \\hat { u } ^ 1 } ( t _ 2 ) = 0 . \\end{align*}"}
-{"id": "8660.png", "formula": "\\begin{align*} \\begin{aligned} & \\pi \\Big [ \\int _ { \\tau _ 1 } ^ \\infty 1 _ { \\{ | x + \\omega _ { X _ 0 } ( s ) - y - \\omega _ { Y _ 0 } ( s ) | \\leq 1 \\} } d s > K | X _ { \\tau _ 1 } , Y _ { \\tau _ 1 } \\Big ] \\\\ & = \\pi \\Big [ \\int _ 0 ^ \\infty 1 _ { \\{ x + \\omega _ { X _ 0 } ( \\tau _ 1 ) + \\tilde { \\omega } _ { X _ { \\tau _ 1 } } ( s ) - y - \\omega _ { Y _ 0 } ( \\tau _ 1 ) - \\tilde { \\omega } _ { Y _ { \\tau _ 1 } } ( s ) \\} } d s > K | X _ { \\tau _ 1 } , Y _ { \\tau _ 1 } \\Big ] < \\frac 1 2 \\end{aligned} \\end{align*}"}
-{"id": "5620.png", "formula": "\\begin{align*} R ^ { \\sigma ( Y ) } _ { s , t } = & \\int _ 0 ^ 1 \\int _ 0 ^ 1 r \\sigma '' ( u r Y _ t + u ( 1 - r ) Y _ s + ( 1 - u ) Y _ s ) ( Y _ s ' B ^ H _ { s , t } + R ^ Y _ { s , t } ) ( Y _ s ' B ^ H _ { s , t } ) d r d u \\\\ + & \\int _ 0 ^ 1 \\sigma ' ( r Y _ t + ( 1 - r ) Y _ s ) R ^ Y _ { s , t } d r . \\end{align*}"}
-{"id": "4499.png", "formula": "\\begin{align*} \\| \\rho ( s ) \\| = | \\chi ( \\rho ( s ) ) | < \\psi ( s ) \\leq \\| \\sigma _ \\psi ( s ) \\| \\end{align*}"}
-{"id": "9997.png", "formula": "\\begin{align*} \\forall g \\in G \\ ; \\exists f \\in F : \\ ; g x \\in f V \\cap Y = f ( V \\cap Y ) ; \\end{align*}"}
-{"id": "5430.png", "formula": "\\begin{align*} P _ { k , \\ell } ^ m : = \\Delta _ { m - 1 } * ( \\Delta _ k \\oplus \\Delta _ \\ell ) \\qquad \\textrm { f o r } k , \\ell , m \\ge 0 . \\end{align*}"}
-{"id": "469.png", "formula": "\\begin{align*} ( \\psi _ { \\pi / 2 } '' ( 0 ) ^ { - 1 } \\partial , \\partial ) ^ { ( k _ 1 + 1 ) / 2 } a _ { k _ 1 , k _ 2 , \\pi / 2 } ( 0 ) & = i ^ { - ( k _ 1 + 1 ) / 2 } \\sum _ { \\abs { \\alpha } = ( k _ 1 + 1 ) / 2 } \\frac { [ ( k _ 1 + 1 ) / 2 ] ! } { 2 ^ { \\alpha _ 1 } \\alpha ! } \\partial ^ { 2 \\alpha } a _ { k _ 1 , k _ 2 , \\pi / 2 } ( 0 ) . \\end{align*}"}
-{"id": "5671.png", "formula": "\\begin{align*} \\mathfrak { E } _ { W } ( f \\sigma , I ) = \\int _ I \\frac 1 2 ( f ' ) ^ 2 + W ( f \\sigma ) + \\frac 1 2 | \\sigma ' | ^ 2 f ^ 2 , \\end{align*}"}
-{"id": "9813.png", "formula": "\\begin{align*} p _ { N _ { \\max } } ' ( t ) = - p _ { N _ { \\max } } ( t ) \\mu + p _ { N _ { \\max } - 1 } ( t ) \\l f ( t ) , - T _ { e _ 1 } \\leq t \\leq T _ { e _ 2 } . \\end{align*}"}
-{"id": "8499.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { t , l , v } ) = \\chi _ 2 ( v ^ { - 1 } ) \\epsilon ( \\frac { 1 } { 2 } , \\chi _ 1 ^ { - 1 } ) q ^ { \\frac { l } { 2 } - 1 } \\chi _ 2 ( y _ 0 ) \\chi _ 1 ^ { - 1 } ( 1 + \\varpi ^ { \\frac { a _ 1 - a _ 2 } { 2 } } y _ 0 ) \\psi ( \\frac { y _ 0 } { v } \\varpi ^ { - \\frac { a _ 1 + a _ 2 } { 2 } } ) S ( f _ { y _ 0 } , a _ 2 - 1 ) . \\end{align*}"}
-{"id": "1933.png", "formula": "\\begin{align*} { \\chi } ( \\xi ) = 1 \\ ; \\ ; \\ ; \\ ; | \\xi | > 2 C _ 0 \\ ; \\ ; \\ ; \\ ; \\chi ( \\xi ) = 0 \\ ; \\ ; \\ ; \\ ; | \\xi | < C _ 0 . \\end{align*}"}
-{"id": "680.png", "formula": "\\begin{align*} \\cot \\alpha = \\cot \\alpha _ 0 + \\sum _ { n = 0 } ^ { \\infty } c _ n . \\end{align*}"}
-{"id": "6723.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\mathbb { P } \\Bigg ( \\Gamma _ { 1 } < N ^ { 1 + \\delta } < \\min \\limits _ { 2 \\leq l \\leq \\frac { N ^ { 1 + \\delta } } { 2 } } \\Gamma _ { l } \\Bigg ) = 1 \\end{align*}"}
-{"id": "7639.png", "formula": "\\begin{align*} X X ^ { * } = \\Bigg ( \\sum _ { i = 1 } ^ k u _ i x _ i ^ 2 \\Bigg ) I _ n , \\end{align*}"}
-{"id": "5989.png", "formula": "\\begin{align*} f _ { \\mathrm { u } } ( x ) = \\sin ( 2 x ) + \\cos ( 4 x ) + \\frac { 1 } { 2 + x } , x \\in [ - 1 , 1 ] . \\end{align*}"}
-{"id": "10111.png", "formula": "\\begin{align*} \\boldsymbol S _ { D _ k } ( i ) = \\boldsymbol R _ k ^ { - 1 } ( i ) \\boldsymbol P _ { D _ k } ( i ) \\bar { \\boldsymbol R } _ { \\bar { \\boldsymbol \\omega } _ k } ^ { - 1 } ( i ) , \\end{align*}"}
-{"id": "9030.png", "formula": "\\begin{align*} J _ { [ a b } \\nu _ { c d e f \\cdots g ] } = 0 \\iff J ^ { c d } \\nu _ { c d e f \\cdots g } = 0 . \\end{align*}"}
-{"id": "4690.png", "formula": "\\begin{align*} u _ { x _ 0 , r } : = \\frac { u ( x _ 0 + r x ) } { r ^ 2 } . \\end{align*}"}
-{"id": "5607.png", "formula": "\\begin{align*} \\int _ 0 ^ T D G ( B ^ H _ r ) d \\mathbf { B } ^ H _ r & = \\lim \\limits _ { | \\mathcal { P } | \\to 0 } \\sum _ { [ s , t ] \\in \\mathcal { P } } D G ( B ^ H _ s ) B ^ H _ { s , t } + D ^ 2 G ( B ^ H _ { s } ) \\mathbb { B } ^ H _ { s , t } \\\\ & = \\lim \\limits _ { | \\mathcal { P } | \\to 0 } \\sum _ { [ s , t ] \\in \\mathcal { P } } D G ( B ^ H _ s ) B ^ H _ { s , t } + \\frac { 1 } { 2 } D ^ 2 G ( B ^ H _ { s } ) { B ^ H _ { s , t } } ^ { \\otimes 2 } + \\frac { 1 } { 2 } D ^ 2 G ( B ^ H _ { s } ) A n t i ( \\mathbb { B } ^ H _ { s , t } ) . \\end{align*}"}
-{"id": "3148.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } _ { \\geqslant 0 } } x ^ { \\kappa } m _ { \\alpha , \\beta } ( \\mathrm { d } x ) \\leqslant \\left ( \\int _ { \\mathbb { R } _ { \\geqslant 0 } } x m _ { \\alpha , \\beta } ( \\mathrm { d } x ) \\right ) ^ { \\kappa } = \\left ( \\frac { \\alpha } { \\beta } \\right ) ^ { \\kappa } , \\end{align*}"}
-{"id": "318.png", "formula": "\\begin{align*} \\frac { v _ j } { \\displaystyle 1 - \\sum _ { l = j + 1 } ^ d v _ l } & \\ge v _ j = u _ j - z _ j \\frac { u _ k } { z _ k } \\ge u _ j - z _ j \\ge s ( 1 - t ) ^ { d - 1 } - \\delta . \\end{align*}"}
-{"id": "1667.png", "formula": "\\begin{align*} \\tau _ { \\beta _ { 1 1 } } : = \\tau _ { \\alpha _ { 1 1 } } = \\tau _ e , \\tau _ { \\beta _ { 1 2 } } : = \\tau _ { \\alpha _ { 1 2 } } = \\tau _ f , \\tau _ { \\beta _ { 2 1 } } : = \\tau _ { \\alpha _ { 2 1 } } = \\tau _ g . \\end{align*}"}
-{"id": "5731.png", "formula": "\\begin{align*} | m - m ( t _ 0 ) | < \\theta / 2 \\quad G ( t , m ) = 0 , \\end{align*}"}
-{"id": "2745.png", "formula": "\\begin{align*} \\left | \\ , \\sum _ { k = 1 } ^ K \\ , k ^ { 2 q } \\ , \\langle \\partial _ l ^ j \\mathcal F _ { k } ( h ^ K , h ^ K ) , \\ , f _ { k } \\rangle _ { L ^ 2 _ { x , v } } \\ , \\right | \\leq C ( \\delta ) \\ , \\sum _ { m = 1 } ^ K \\ , | | m ^ q h _ m | | _ { H _ { \\Lambda } ^ s } ^ 2 \\ , \\sum _ { n = 1 } ^ K \\ , | | n ^ q h _ n | | _ { H _ { x , v } ^ s } ^ 2 + \\delta \\ , \\sum _ { k = 1 } ^ K \\ , | | k ^ q f _ k | | _ { \\Lambda } ^ 2 \\ , . \\end{align*}"}
-{"id": "10082.png", "formula": "\\begin{align*} ( \\tilde { \\nabla } _ { X } g ) ( Y , Z ) = \\frac { 1 } { n + 1 } [ 2 \\pi ( X ) g ( Y , Z ) - n \\pi ( Y ) g ( X , Z ) - n \\pi ( Z ) g ( X , Y ) ] , \\end{align*}"}
-{"id": "776.png", "formula": "\\begin{align*} B _ { k } ( 1 - x ) = ( - 1 ) ^ { k } B _ { k } ( x ) \\end{align*}"}
-{"id": "1631.png", "formula": "\\begin{align*} \\pi _ S ( b ) \\xi ( x ) & = S _ \\mu S _ \\mu ^ * ( \\xi ) ( x ) - \\rho ( \\Lambda ) ^ { ( m - n ) / 2 } S _ \\nu S _ \\mu ^ * ( \\xi ) ( x ) \\\\ & = \\chi _ { Z ( \\mu ) } ( x ) \\rho ( \\Lambda ) ^ { m / 2 } S _ \\mu ^ * \\xi ( \\sigma ^ m ( x ) ) - \\chi _ { Z ( \\nu ) } ( x ) \\rho ( \\Lambda ) ^ { m / 2 } S _ \\mu ^ * \\xi ( \\sigma ^ n ( x ) ) . \\end{align*}"}
-{"id": "7856.png", "formula": "\\begin{align*} h _ { 1 2 } ( x _ { 1 } | x _ { 2 } ) = \\frac { f _ { X _ { 1 } } ( x _ { 1 } ) \\left ( 1 - e ^ { - \\eta ^ { \\alpha } ( x _ { 1 } ) } \\right ) ^ { \\theta _ { 3 } } } { 1 - \\left ( 1 - e ^ { - \\eta ^ { \\alpha } ( x _ { 1 } ) } \\right ) ^ { \\theta _ { 1 } } } \\end{align*}"}
-{"id": "7583.png", "formula": "\\begin{align*} r _ \\Omega u _ 1 ^ k v _ 1 ^ k - r _ \\Omega u _ 2 ^ k v _ 2 ^ k = r _ \\Omega ( u _ 1 ^ k - u _ 2 ^ k ) v _ 1 ^ k + r _ \\Omega u _ 2 ^ k ( v _ 1 ^ k - v _ 2 ^ k ) \\end{align*}"}
-{"id": "4703.png", "formula": "\\begin{align*} & \\psi _ { ( \\infty , s , t ) } : = L _ t D _ { S _ s } \\Phi = \\Phi ( S _ s ^ { - 1 } ( \\cdot - t ) ) \\mbox { a n d } \\\\ & \\psi _ { ( a , s , t ) } : = L _ t D _ { S _ s } D _ { A _ a } \\Psi = \\abs { \\det A _ a } ^ { - \\frac { 1 } { 2 } } \\Psi ( A _ a ^ { - 1 } S _ s ^ { - 1 } ( \\cdot - t ) ) . \\end{align*}"}
-{"id": "5183.png", "formula": "\\begin{align*} u _ r ( x ) = V _ { 1 } ^ { [ r , 1 ] } ( x ) - g _ { 1 , [ r , 1 ] } ( x ) , \\forall x \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "2185.png", "formula": "\\begin{align*} & \\lim _ { t \\to \\infty } \\ , t ^ { \\frac { N + A } { 2 } } \\left [ e ^ { - t L _ k } \\phi ^ { k , i } \\right ] \\left ( t ^ { \\frac { 1 } { 2 } } y , t \\right ) = 0 \\quad \\mbox { i n } L ^ 2 ( { \\bf R } ^ N , e ^ { | y | ^ 2 / 4 } \\ , d y ) \\ , \\cap \\ , L ^ \\infty ( K ) , \\\\ & \\lim _ { t \\to \\infty } t ^ { \\frac { N + 2 A } { 2 } } \\frac { \\left [ e ^ { - t L _ k } \\phi ^ { k , i } \\right ] ( x ) } { U _ k ( | x | ) } = 0 \\quad \\ , \\ , \\qquad \\mbox { i n } L ^ \\infty ( B ( 0 , R ) ) , \\end{align*}"}
-{"id": "5276.png", "formula": "\\begin{align*} \\| u \\| ^ 2 _ { L ^ 2 ( a , b ) } & = \\| u \\| ^ 2 _ { L ^ 2 ( a , c ) } + \\| u \\| ^ 2 _ { L ^ 2 ( c , b ) } \\\\ & \\leq ( b - a ) u ^ 2 ( c ) + \\frac { ( b - a ) ^ 2 } { 2 } \\| u \\| ^ 2 _ { L ^ 2 ( a , b ) } . \\end{align*}"}
-{"id": "6515.png", "formula": "\\begin{align*} h _ { \\mathcal { M } } ^ { } \\approx 2 A _ { \\mathrm { o } } = \\lambda _ { \\mathcal { M } } . \\end{align*}"}
-{"id": "7570.png", "formula": "\\begin{align*} \\mu ( f , g ) ( x ) = f \\cdot g ( x ) = f ( x ) g ( x ) , \\end{align*}"}
-{"id": "9542.png", "formula": "\\begin{align*} \\sum _ v w ( Z _ v ) & \\le \\sum _ v w ( Z _ v ) \\sum _ { A \\ni v } x _ A = \\sum _ A x _ A \\sum _ { v \\in A } w ( Z _ v ) \\\\ & \\le \\sum _ A x _ A \\sum _ { v \\in A } w ( A ) = \\sum _ A x _ A \\ , | A | \\ , w ( A ) \\le k \\sum _ A x _ A \\ , w ( A ) = k \\sigma ^ * . \\end{align*}"}
-{"id": "5240.png", "formula": "\\begin{align*} \\mathrm { M a s l o v } ( \\varphi ) : = \\sum _ { z ^ * \\in ( - \\infty , \\tau ) } \\mathrm { s i g n } \\ , \\Gamma ( E ^ u , E ^ s ( 0 , \\tau ) , z ^ * ) + n _ + ( \\Gamma ( E ^ u , E ^ s ( 0 , \\tau ) , \\tau ) ) , \\end{align*}"}
-{"id": "3772.png", "formula": "\\begin{align*} & \\eta ( z , i , 0 ) = 1 z \\leq 0 i \\leq N ( z , 0 ) , \\\\ & \\eta ( z , i , 0 ) = 0 \\end{align*}"}
-{"id": "1215.png", "formula": "\\begin{align*} \\bar h _ { X _ k X _ n } = 0 \\mbox { f o r } \\ , \\ , k = 1 , \\ldots , n . \\end{align*}"}
-{"id": "9844.png", "formula": "\\begin{align*} ( f \\ast g ) \\ , ( x ) = \\int _ { - \\infty } ^ { \\infty } f ( x - t ) g ( t ) d t , \\end{align*}"}
-{"id": "3812.png", "formula": "\\begin{align*} \\mathcal { E } _ L : = \\{ N ( y ) = 0 \\ ; \\forall \\ ; y \\in [ \\hat { Z } - \\ell _ L , \\hat { Z } + \\ell _ L ] \\times [ 0 , L ^ \\alpha ] \\} . \\end{align*}"}
-{"id": "9453.png", "formula": "\\begin{align*} ( a ; q ) _ 0 & : = 1 , \\\\ ( a ; q ) _ n & : = ( 1 - a ) ( 1 - a q ) \\cdots ( 1 - a q ^ { n - 1 } ) , \\\\ ( a ; q ) _ { \\infty } & : = \\lim _ { n \\to \\infty } ( a ; q ) _ n , | q | < 1 . \\end{align*}"}
-{"id": "9883.png", "formula": "\\begin{align*} f _ t ( \\delta ) = \\begin{cases} \\frac { 3 t - 1 0 } { 3 t - 4 } + \\delta - \\delta ^ 2 & \\cr \\frac { t - 3 } { t - 1 } + \\delta & \\end{cases} \\end{align*}"}
-{"id": "4365.png", "formula": "\\begin{align*} B _ { j , t } A M _ { \\psi _ { j , t } } = M _ { \\psi _ { j , t } } = M _ { \\psi _ { j , t } } A C _ { j , t } . \\end{align*}"}
-{"id": "6297.png", "formula": "\\begin{align*} \\{ b ; \\varepsilon , g , ( f , k _ 1 ) , ( t , k _ 2 ) , ( s , k _ 3 ) ; \\{ ( \\alpha _ i , \\beta _ i ) \\} _ { i = 1 } ^ { n } ; ( r _ 1 , \\ldots , r _ { s - k _ 3 } ) ; ( q _ 1 , q _ 2 , \\ldots , q _ { k _ 3 } ) \\} . \\end{align*}"}
-{"id": "5374.png", "formula": "\\begin{align*} & \\left ( \\mathcal F \\left ( L ( \\ell + 1 , 0 ) \\otimes M _ { 1 , r + 1 } \\right ) , \\mathcal F \\left ( L ( \\ell + 1 , 0 ) \\otimes M _ { 1 , r ' + 1 } \\right ) \\right ) = \\\\ & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad = \\left ( L ( \\ell , r \\omega ) \\otimes L ( 1 , \\overline { r } \\omega ) , L ( \\ell , r ' \\omega ) \\otimes L ( 1 , \\overline { r } ' \\omega ) \\right ) \\end{align*}"}
-{"id": "9980.png", "formula": "\\begin{align*} R _ { d , N } = \\bigoplus _ { \\lambda \\in { \\N } ^ { N } } { R _ { d , N , \\lambda } } , \\end{align*}"}
-{"id": "1906.png", "formula": "\\begin{align*} \\Vert \\psi _ { n + 1 } ( x ) - \\psi _ { n + 1 } ( y ) \\Vert \\leq \\alpha \\Vert r _ { n } ( w ) - r _ { n } ( z ) \\Vert = \\alpha \\Vert x - y \\Vert , \\end{align*}"}
-{"id": "6166.png", "formula": "\\begin{align*} \\partial _ t A _ i = \\Big ( ( \\log \\frac { A _ i } { V } ) ' \\frac { A _ i } { V ^ 2 } \\Big ) ' , \\ ; \\ ; i = 1 , 2 , 3 \\end{align*}"}
-{"id": "1894.png", "formula": "\\begin{align*} G ( x _ 1 , \\dots , x _ d ) \\le \\min _ { i = 1 , \\dots , d } F ^ \\ast _ i ( x _ i ) \\wedge \\min \\Big \\{ \\pi _ s + \\sum _ { i = 1 } ^ d \\big ( F _ i ^ \\ast ( x _ i ) - F _ i ^ \\ast ( s _ i ) \\big ) ^ + : s \\in S \\Big \\} , \\end{align*}"}
-{"id": "2931.png", "formula": "\\begin{align*} \\left | \\int f d \\mu _ n - \\int f d \\mu \\right | & \\leq \\norm { f } _ \\infty \\left ( 1 - \\sum _ { j = 1 } ^ m \\mu _ n ( A _ { ( T _ j , i _ j ) } ) \\right ) + \\norm { f } _ \\infty \\left | \\sum _ { j = 1 } ^ m \\mu _ n ( A _ { ( T _ j , i _ j ) } ) - \\mu ( A _ { ( T _ j , i _ j ) } ) \\right | \\\\ & \\qquad + \\epsilon ( 2 + \\norm { f } _ \\infty ) . \\end{align*}"}
-{"id": "4811.png", "formula": "\\begin{align*} \\varphi ( r ) = \\frac 1 { \\mathsf { d } ( r ) } \\varphi ( \\mathsf { n } ( r ) ) = \\frac 1 { \\mathsf { d } ( r ) } \\sum _ { i = 1 } ^ k c _ i \\varphi ( n _ i ) = \\frac 1 { \\mathsf { d } ( r ) } \\sum _ { i = 1 } ^ k c _ i n _ i \\frac { \\varphi ( n _ j ) } { n _ j } = r q . \\end{align*}"}
-{"id": "4801.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c c c c } - z _ 1 & z _ 2 & 0 & 0 & 0 \\\\ 1 & 1 & 0 & 0 & 0 \\\\ 0 & z _ 2 & - z _ 3 - z _ 4 & z _ 5 & 0 \\\\ 0 & 0 & z _ 3 & - z _ 5 & 0 \\\\ 0 & 0 & z _ 3 & z _ 5 & - z _ 6 \\end{array} \\right ) \\left ( \\begin{array} { c } x _ 1 \\\\ x _ 2 \\\\ x _ 3 \\\\ x _ 4 \\\\ x _ 5 \\end{array} \\right ) + \\left ( \\begin{array} { c } 0 \\\\ - z _ 7 \\\\ z _ 8 \\\\ 0 \\\\ z _ 9 \\end{array} \\right ) = \\left ( \\begin{array} { c } 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "2328.png", "formula": "\\begin{align*} \\Phi ( r , a , y , s ) = \\cfrac { 2 \\pi ^ s } { i ^ r \\Gamma ( s ) } y ^ { \\frac 1 2 - s } \\cfrac { \\partial ^ r } { \\partial a ^ r } \\left ( | a | ^ { s - \\frac 1 2 } K _ { s - \\frac 1 2 } \\big ( 2 \\pi | a | y \\big ) \\right ) . \\end{align*}"}
-{"id": "9499.png", "formula": "\\begin{align*} T _ 2 ( t ) = z _ a ( t ) \\cdot D ( z _ a ( t ) , t ) , \\end{align*}"}
-{"id": "4112.png", "formula": "\\begin{align*} \\left . \\frac { d ^ 2 } { d s ^ 2 } \\left ( D _ a \\circ \\gamma \\right ) \\right | _ { s = 0 } = 0 \\ \\Leftrightarrow \\ k _ { M , p } ( V ) = \\frac { 1 } { D _ a ( p ) } . \\end{align*}"}
-{"id": "6214.png", "formula": "\\begin{align*} X ( t ) = c t - C ( t ) + Z ( t ) ~ , ~ ~ t \\geq 0 ~ , \\end{align*}"}
-{"id": "5125.png", "formula": "\\begin{align*} { \\mathcal A } _ p ( ( X _ \\lambda ) _ { \\lambda \\in \\Lambda } ) = { \\mathcal A } _ { V _ p } ( ( X _ \\lambda ) _ { \\lambda \\in \\Lambda _ V } ) { \\mathcal A } _ { W _ p } ( ( X _ \\lambda ) _ { \\lambda \\in \\Lambda _ W } ) . \\end{align*}"}
-{"id": "9111.png", "formula": "\\begin{align*} ( n + 1 - i ) a _ { n + 1 - i } + ( i + 1 ) a _ { n + 1 - ( i + 1 ) } = 0 \\left ( 0 \\leq i \\leq n - 1 \\right ) . \\end{align*}"}
-{"id": "2250.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ \\infty \\frac { d \\ , f _ { j , s } ( z ) } { f _ { j , s } ( z ) } = \\frac { d \\prod \\limits _ { s = 1 } ^ \\infty f _ { j , s } ( z ) } { \\prod \\limits _ { s = 1 } ^ \\infty f _ { j , s } ( z ) } = \\lim _ { m \\to \\infty } \\frac { d \\prod \\limits _ { s = 1 } ^ m f _ { j , s } } { \\prod \\limits _ { s = 1 } ^ m f _ { j , s } } \\end{align*}"}
-{"id": "1477.png", "formula": "\\begin{align*} d _ u ( E _ u ) \\subset ( D _ r \\langle x _ 3 ^ 2 y _ r , x _ 3 z _ r \\rangle ) ^ { * , - u + 1 } = \\{ 0 \\} . \\end{align*}"}
-{"id": "9130.png", "formula": "\\begin{align*} \\widetilde { f _ { k } } ( x ) = f _ { k } ( x ) - f _ { k } ( 1 ) x \\left ( x \\in R \\right ) . \\end{align*}"}
-{"id": "6339.png", "formula": "\\begin{align*} W _ 1 ( p ) : = V _ 1 ( p ) - \\log ( p ( 1 - p ) ) W _ 2 ( p ) : = V _ 2 ( p ) - \\log ( p ( 1 - p ) ) \\end{align*}"}
-{"id": "1069.png", "formula": "\\begin{align*} \\Omega _ { k } ^ { \\prime } : = \\left \\{ P _ { k - 1 } ^ { n , \\zeta } , \\ ; T P _ { k - 1 } ^ { n , \\zeta } , \\ldots , \\ ; T ^ { d _ { k } - 1 } P _ { k - 1 } ^ { n , \\zeta } , \\ ; P _ { k } ^ { n , \\zeta } , \\ ; T P _ { k } ^ { n , \\zeta } , \\ldots , \\ ; T ^ { m - d _ { k } } P _ { k } ^ { n , \\zeta } \\right \\} \\end{align*}"}
-{"id": "8811.png", "formula": "\\begin{align*} | a _ { i , p } | \\leq \\dfrac { \\sum _ { j = 1 } ^ { r } | c _ { j , p } D _ { j , i , p } | } { | D _ { p } | } \\leq C _ { I } \\dfrac { \\sum _ { j = 1 } ^ { r } | D _ { j , i , p } | } { | D _ { p } | } , \\end{align*}"}
-{"id": "2121.png", "formula": "\\begin{align*} C _ { \\Gamma } ( f ) ( x ) : = { \\rm p . v . } \\frac { 1 } { \\pi i } \\int _ { \\R } \\frac { f ( y ) } { y - x + i ( A ( y ) - A ( x ) ) } \\ , d y . \\end{align*}"}
-{"id": "3357.png", "formula": "\\begin{align*} \\tilde { D } ( \\tilde { r } _ i ) & = \\left ( 1 + \\frac { 1 } { N } + \\cdots + \\frac { 1 } { N ^ i } \\right ) - \\frac { 1 } { 1 + N + \\cdots + N ^ i } \\left ( ( i + 1 ) + \\frac { i } { N } + \\cdots + \\frac { 1 } { N ^ i } \\right ) \\\\ & = \\frac { 1 + N + \\cdots + N ^ i } { N ^ i } - \\frac { ( i + 1 ) N ^ i + i N ^ { i - 1 } + \\cdots + 1 } { N ^ i ( 1 + N + \\cdots + N ^ i ) } \\\\ & = \\frac { ( 1 + N + \\cdots + N ^ i ) ^ 2 - ( 1 + 2 N + 3 N ^ 2 + \\cdots + ( i + 1 ) N ^ i ) } { N ^ i ( 1 + N + \\cdots + N ^ i ) } \\end{align*}"}
-{"id": "737.png", "formula": "\\begin{align*} \\mathcal { L } ^ 1 ( \\{ u ^ * ( x ' , y ) > t \\} ) = \\mathcal { L } ^ 1 ( \\{ | u ( x ' , y ) | > t \\} ) . \\end{align*}"}
-{"id": "7858.png", "formula": "\\begin{align*} V ( x ^ { \\ast } ) = \\left ( v _ { X } ( x ) , v _ { 1 2 } ( x _ { 1 } | x _ { 2 } ) , v _ { 2 1 } ( x _ { 2 } | x _ { 1 } ) \\right ) , \\end{align*}"}
-{"id": "7935.png", "formula": "\\begin{align*} \\mathop { \\Pi } \\limits _ { i = 0 } ^ { \\infty } ( 1 + \\varphi ( \\nu ^ i | V | ) < \\infty , \\ , \\ , \\ , \\ , \\mathop { \\Pi } \\limits _ { i = 0 } ^ { \\infty } ( 1 - \\varphi ( \\nu ^ i | V | ) > 0 . \\end{align*}"}
-{"id": "9598.png", "formula": "\\begin{align*} \\| g ^ 1 _ { k , M , J } \\| _ { \\dot { \\mathcal B } ^ { \\alpha , q } _ { p , \\mathcal F } } & \\le C 2 ^ { - J \\varepsilon } \\bigg \\{ \\sum _ { \\ell \\in \\Bbb Z } 2 ^ { \\ell \\alpha q - | \\ell - k | \\varepsilon q } \\bigg \\} ^ { 1 / q } \\| g \\| _ { L ^ p _ \\mu } \\\\ & \\le C 2 ^ { - J \\varepsilon } 2 ^ { k \\alpha } \\| g \\| _ { L ^ p _ \\mu } \\\\ & \\to 0 \\qquad \\ J \\to \\infty . \\end{align*}"}
-{"id": "1173.png", "formula": "\\begin{align*} B ( z , r ) = \\{ y \\in \\mathbb R ^ { n } : | z - y | < r \\} \\mbox { w h e n e v e r } \\ , \\ , z \\in \\mathbb R ^ { n } , \\ , r > 0 , \\end{align*}"}
-{"id": "7101.png", "formula": "\\begin{align*} \\begin{aligned} k _ t & = k _ p ( \\nabla \\cdot ( k ( p ) \\nabla p ) ) \\\\ & = ( k _ p | \\nabla p | ) ^ 2 + k k _ p \\Delta p \\\\ & = | \\nabla k | ^ 2 + k k _ p \\Delta p . \\end{aligned} \\end{align*}"}
-{"id": "7206.png", "formula": "\\begin{align*} v _ k ^ s ( y ) = \\frac { u _ { k , p } ( x + s y ) } { r } , \\end{align*}"}
-{"id": "7932.png", "formula": "\\begin{align*} \\mu _ 0 = \\sum _ { i = 1 } ^ m a _ i ( f _ i ) _ * ( \\mu _ 0 ) , \\end{align*}"}
-{"id": "4857.png", "formula": "\\begin{align*} 0 = 3 d ^ 2 - 6 d - 6 \\sum \\delta _ p - \\sum _ { I \\cup J } ( m _ p + l _ p - 3 ) . \\end{align*}"}
-{"id": "3523.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c l } d { y } ( t ) = & \\big { [ } b _ x ( \\bar { X } { ( t ) } , \\bar { u } ( t ) ) y ( t ) + b ( \\bar { X } { ( t ) } , u ^ { \\varepsilon } ( t ) ) - b ( \\bar { X } { ( t ) } , \\bar { u } ( t ) ) \\big { ] } d t , \\\\ y ( 0 ) = & 0 , t \\in ( 0 , T ] . \\end{array} \\end{align*}"}
-{"id": "1059.png", "formula": "\\begin{align*} L _ { P _ { k - 1 } } ^ { \\ast } ( q _ { k } ) = L _ { P _ { k } } ^ { \\ast } ( q _ { k } ) . \\end{align*}"}
-{"id": "4342.png", "formula": "\\begin{align*} P _ \\alpha M _ { \\chi _ D } ( f ) ( z ) & = \\Big ( \\frac { \\alpha } { \\pi } \\Big ) ^ n \\int _ { \\mathbb { C } ^ n } e ^ { \\alpha \\langle z , w \\rangle } \\chi _ D ( w ) e ^ { - \\alpha | w | ^ 2 } f ( w ) d w \\\\ & = \\Big ( \\frac { 2 } { p } \\Big ) ^ n \\int _ { \\mathbb { C } ^ n } e ^ { \\alpha \\langle z , w \\rangle - \\frac { 2 \\alpha - p \\alpha } { 2 } | w | ^ 2 } \\chi _ D ( w ) f ( w ) d \\mu _ { p \\alpha / 2 } ( w ) . \\end{align*}"}
-{"id": "7780.png", "formula": "\\begin{align*} \\Gamma _ { \\varepsilon } : = \\left \\{ \\ , ( x _ { 1 } , x _ { 2 } ) \\ , | \\ , x _ { 1 } \\geq \\hat { x } _ { 1 } , | x _ { 2 } - \\hat { x } _ { 2 } | \\leq \\varepsilon \\ , \\right \\} . \\end{align*}"}
-{"id": "8843.png", "formula": "\\begin{align*} \\chi _ 1 ( \\zeta , a ) : = \\sqrt { 2 } ( 1 + i ) e ^ { - \\frac { i } { 5 } ( \\zeta ^ 5 a ^ { - 3 } + 5 \\zeta ^ 4 a ^ { - 2 } + 1 0 \\zeta ^ 3 a ^ { - 1 } + 1 0 \\zeta ^ 2 ) } \\mathcal { F } [ e ^ { - \\frac { i } { 8 } z ^ 2 + i R ( z , a ) } \\chi ] ( \\zeta ) . \\end{align*}"}
-{"id": "3244.png", "formula": "\\begin{align*} i _ ! i ^ \\ast ( \\theta ) = \\theta \\cdot \\omega _ Z . \\end{align*}"}
-{"id": "399.png", "formula": "\\begin{align*} { \\mathbb E } \\left [ \\bigg ( \\sum _ { ( r , s ) \\in \\Upsilon _ { n } \\backslash \\Upsilon _ m } a _ { r , s } \\xi _ { j - r , k - s } \\bigg ) ^ 2 \\right ] = \\sum _ { ( r , s ) \\in \\Upsilon _ { n } \\backslash \\Upsilon _ m } a _ { r , s } ^ 2 , \\end{align*}"}
-{"id": "4803.png", "formula": "\\begin{align*} g _ \\zeta ( j _ i ) & = \\begin{cases} \\beta _ i & \\textrm { i f } k \\neq i , \\\\ \\ell & \\textrm { i f } k = i , \\end{cases} & g _ \\zeta ( m + 1 ) & = \\begin{cases} m + 1 & \\textrm { i f } k \\neq d + 1 , \\\\ \\ell & \\textrm { i f } k = d + 1 , \\end{cases} \\end{align*}"}
-{"id": "60.png", "formula": "\\begin{align*} \\hat { V } ^ { C } _ { \\sigma } ( C _ { 1 } , C _ { 2 } ) = \\frac { 1 } { 2 \\pi \\sigma ^ { 2 } } \\frac { 1 } { N } \\sum \\limits _ { n = 1 } ^ N e x p \\left ( - \\frac { ( x _ { n } - y _ { n } ) ^ { 2 } + ( z _ { n } - s _ { n } ) ^ { 2 } } { 2 \\sigma ^ 2 } \\right ) \\end{align*}"}
-{"id": "9826.png", "formula": "\\begin{align*} W _ { 2 , q } ( x , y ) = x ^ 2 + ( q - 1 ) y ^ 2 \\end{align*}"}
-{"id": "5822.png", "formula": "\\begin{align*} \\tilde { w } = w d \\sigma , \\end{align*}"}
-{"id": "9712.png", "formula": "\\begin{align*} W ( \\tilde { U } ) = W ( U ) + G ( U ) h , \\end{align*}"}
-{"id": "1209.png", "formula": "\\begin{align*} & u _ 1 ( y + \\rho \\eta ) = u _ 1 ( y ) + A _ 1 \\rho + A _ 2 \\rho ^ { 2 } + o ( \\rho ^ { 2 } ) , \\\\ & u _ 2 ( z + \\rho \\eta ) = u _ 2 ( z ) + B _ 1 \\rho + B _ 2 \\rho ^ { 2 } + o ( \\rho ^ { 2 } ) , \\\\ & u ( x + \\rho \\eta ) = u ( x ) + C _ 1 \\rho + C _ 2 \\rho ^ { 2 } + o ( \\rho ^ { 2 } ) \\end{align*}"}
-{"id": "1910.png", "formula": "\\begin{align*} ( \\chi _ { \\psi } \\xi _ i ) \\rtimes \\omega _ 0 = ( \\chi _ { \\psi } \\xi _ i ^ { - 1 } ) \\rtimes \\omega _ 0 . \\end{align*}"}
-{"id": "8810.png", "formula": "\\begin{align*} a _ { i , p } = \\dfrac { \\sum _ { j = 1 } ^ { r } c _ { j , p } D _ { j , i , p } } { D _ { p } } , \\end{align*}"}
-{"id": "4927.png", "formula": "\\begin{align*} P T P ^ \\perp & = P _ N T P _ N ^ \\perp \\\\ & = P _ { N - 1 } T P _ { N - 1 } ^ \\perp P _ N ^ \\perp + Q _ N T P _ N ^ \\perp , \\end{align*}"}
-{"id": "2154.png", "formula": "\\begin{align*} u _ * ( | x | , t ) = u _ * ( 0 , t ) + F _ N ^ 0 ( | x | , t ) \\quad \\mbox { i n } \\quad { \\bf R } ^ N \\times ( 0 , \\infty ) . \\end{align*}"}
-{"id": "303.png", "formula": "\\begin{align*} Z _ d = \\zeta _ 0 Z _ 0 & + \\zeta _ 1 \\Theta _ 0 + \\zeta _ 2 \\Theta _ 1 + \\dots + \\zeta _ d \\Theta _ { d - 1 } , \\end{align*}"}
-{"id": "7334.png", "formula": "\\begin{align*} \\mathcal N ( u ) = \\lambda _ 1 | u | ^ { \\gamma - 1 } u + \\lambda _ 2 ( | \\cdot | ^ { - \\alpha } \\ast | u | ^ 2 ) u , \\begin{array} { l } \\gamma \\in ( 1 , 5 ] \\ , , \\\\ \\alpha \\in ( 0 , 3 ) \\ , , \\\\ \\lambda _ 1 , \\lambda _ 2 \\geqslant 0 \\end{array} \\end{align*}"}
-{"id": "4889.png", "formula": "\\begin{align*} G P ( G ) & = d ( 1 , \\alpha _ 1 ( 1 ) ) + d ( 6 , \\alpha _ 1 ( 6 ) ) + d ( 4 , \\alpha _ 2 ( 4 ) ) + d ( 7 , \\alpha _ 2 ( 7 ) ) + d ( 5 , \\alpha _ 2 ( 5 ) ) + d ( 8 , \\alpha _ 2 ( 8 ) ) \\\\ & + d ( 1 , \\alpha _ 3 ( 1 ) ) + d ( 6 , \\alpha _ 3 ( 6 ) ) + d ( 4 , \\alpha _ 3 ( 4 ) ) + d ( 7 , \\alpha _ 3 ( 7 ) ) + d ( 5 , \\alpha _ 3 ( 5 ) ) + d ( 8 , \\alpha _ 3 ( 8 ) ) \\\\ & = 2 + 2 + 2 + 2 + 4 + 4 + 2 + 2 + 2 + 2 + 4 + 4 \\\\ & = 3 2 . \\end{align*}"}
-{"id": "2473.png", "formula": "\\begin{align*} \\displaystyle \\mathbb { G } ^ { t } f _ { 0 } = f ( t , x , \\xi ) = \\int _ { \\R ^ { 3 } } e ^ { i \\eta x + ( - i \\xi \\cdot \\eta + L ) t } \\widehat { f } _ { 0 } ( \\eta , \\xi ) d \\eta \\ , , \\end{align*}"}
-{"id": "8656.png", "formula": "\\begin{align*} \\pi [ \\ell ( x , y , X _ 0 , Y _ 0 ) > t ] \\leq \\pi [ \\tau _ n < \\infty ] = \\pi [ \\tau _ n < \\infty | \\tau _ 1 < \\infty ] \\pi [ \\tau _ 1 < \\infty ] \\leq \\frac 1 2 \\pi [ \\tau _ n < \\infty | \\tau _ 1 < \\infty ] . \\end{align*}"}
-{"id": "7958.png", "formula": "\\begin{align*} \\mu _ m = \\sum _ { | J | = m } \\lambda _ J ^ s ( g _ J ) _ * ( \\mu _ m ) . \\end{align*}"}
-{"id": "7283.png", "formula": "\\begin{align*} u ( \\phi ) = \\min \\{ u _ { p } \\ , | \\ , p \\in { \\rm A s s } ( R / I _ { n - 1 } ( \\varphi ) ) \\} \\end{align*}"}
-{"id": "9217.png", "formula": "\\begin{align*} ( z _ { 2 } z _ { 1 } ) ^ { t } u ' \\otimes ( b ' \\alpha _ { 2 } ) \\alpha _ { 1 } = - ( z _ { 1 } \\circ z _ { 2 } ) ^ { t } u ' \\otimes b ' \\frac { [ \\alpha _ { 1 } , \\alpha _ { 2 } ] } { 2 } - [ z _ { 1 } , z _ { 2 } ] ^ { t } u ' \\otimes b ' \\frac { \\alpha _ { 1 } \\circ \\alpha _ { 2 } } { 2 } . \\end{align*}"}
-{"id": "6506.png", "formula": "\\begin{align*} f \\left ( k _ { \\mathrm { o } } \\right ) = \\frac { e ^ { i \\theta _ { \\mathrm { o } } } \\sin \\theta _ { \\mathrm { o } } } { k _ { \\mathrm { o } } } \\approx \\frac { \\theta _ { \\mathrm { o } } } { k _ { \\mathrm { o } } } \\approx - a _ { \\mathrm { s } } \\end{align*}"}
-{"id": "3130.png", "formula": "\\begin{align*} | T ^ { \\prime } | & = t + ( q + 1 ) ( t - 2 ) [ ( m + 1 ) ( k - 3 ) + m ] + ( 2 - q ) ( t - 2 ) [ ( ( m - 1 ) + 1 ) ( k - 3 ) + ( m - 1 ) ] \\\\ & = t + ( t - 2 ) [ ( q + 1 ) ( m k - 2 m + k - 3 ) + ( 2 - q ) ( m k - 2 m - 1 ) ] \\\\ & = t + ( t - 2 ) [ ( q + 1 + 2 - q ) ( m k - 2 m ) + ( q + 1 ) ( k - 3 ) - ( 2 - q ) ] \\\\ & = t + ( k ^ 2 - k - 5 ) ( t - 2 ) \\end{align*}"}
-{"id": "7167.png", "formula": "\\begin{align*} W _ { 1 + s } ^ { x _ 0 } ( r , u ) : = \\frac { 1 } { r ^ { n + 1 } } \\int _ { B _ r ( x _ 0 ) } | \\nabla u _ r | ^ 2 \\ , d \\mu _ a - \\frac { 1 + s } { r ^ { n + 1 } } \\int _ { \\partial B _ r ( x _ 0 ) } | u _ r | ^ 2 \\ , | x _ n | ^ a \\ , d \\mathcal { H } ^ { n - 1 } , \\end{align*}"}
-{"id": "395.png", "formula": "\\begin{align*} S & = \\sum _ { n = 1 6 } ^ \\infty \\frac { 1 } { n \\ln n } \\mathbb P \\left ( | S _ n | > ( 1 + \\varepsilon ) \\sigma _ n \\sqrt { 2 \\ln \\ln \\ln n } \\right ) \\\\ & \\propto \\sum _ { n = 1 6 } ^ \\infty \\frac { 1 } { n \\ln n ( \\ln \\ln n ) ^ { ( 1 + \\varepsilon ) ^ 2 } \\sqrt { \\ln \\ln \\ln n } } . \\end{align*}"}
-{"id": "2598.png", "formula": "\\begin{align*} ( I ^ a f ) ( t ) & : = \\frac 1 { \\Gamma ( a ) } \\int _ 0 ^ t \\frac { f ( s ) } { ( t - s ) ^ { { 1 - a } } } \\ , d s & t > 0 , \\\\ ( D ^ a f ) ( t ) & : = \\frac 1 { \\Gamma ( 1 - a ) } \\frac d { d t } \\int _ 0 ^ t \\frac { f ( s ) } { ( t - s ) ^ a } \\ , d s = \\frac d { d t } ( I ^ { 1 - a } f ) ( t ) & t > 0 , \\end{align*}"}
-{"id": "3758.png", "formula": "\\begin{align*} h _ \\sigma ( t ) : = H ( \\theta _ { ( \\sigma ( t ) , t ) } ( \\omega , U ) , \\sigma ( t + 1 ) - \\sigma ( t ) ) . \\end{align*}"}
-{"id": "9290.png", "formula": "\\begin{align*} \\Delta _ { h } ^ k f ( x ) : = \\Delta ^ { k - 1 } _ { ( h _ 1 , \\dots , h _ { k - 1 } ) } f ( x + h _ k ) - \\Delta ^ { k - 1 } _ { ( h _ 1 , \\dots , h _ { k - 1 } ) } f ( x ) . \\end{align*}"}
-{"id": "6550.png", "formula": "\\begin{align*} \\beta _ { \\xi } = \\frac { n | P ^ { \\circ } | _ n \\| \\xi \\| } { | F _ { \\xi } | _ { n - 1 } } \\ \\ \\ \\ \\beta = \\max _ { \\xi \\in \\mathrm { e x t } ( P ) } \\beta _ { \\xi } . \\end{align*}"}
-{"id": "7850.png", "formula": "\\begin{align*} r _ { X _ { 1 } , X _ { 2 } } ( x _ { 1 } , x _ { 2 } ) = \\left \\{ \\begin{array} { l } r _ { 1 } ( x _ { 1 } , x _ { 2 } ) \\ \\ \\ \\ \\ \\ 0 < x _ { 1 } < x _ { 2 } \\\\ r _ { 2 } ( x _ { 1 } , x _ { 2 } ) \\ \\ \\ \\ \\ \\ 0 < x _ { 2 } < x _ { 1 } \\\\ r _ { 3 } ( x , x ) \\ \\ \\ \\ \\ \\ \\ \\ x _ { 1 } = x _ { 2 } = x , \\end{array} \\right . \\end{align*}"}
-{"id": "9555.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in B _ { n } } { ( - 1 ) ^ { \\ell ( \\sigma ) } x ^ { e ( \\sigma ) } y ^ { o ( \\sigma ) } z ^ { o ( \\sigma ) } } = \\left \\{ \\begin{array} { l l } \\displaystyle ( 1 - x ) ( z ^ \\frac { n } { 2 } - y ^ \\frac { n } { 2 } ) \\prod _ { i = 1 } ^ { \\lfloor \\frac { n - 1 } { 2 } \\rfloor } ( 1 - x z ^ { 2 i } ) ( 1 - y ^ { 2 i } ) , & \\mbox { i f $ n \\equiv 0 \\pmod 2 $ , } \\\\ \\displaystyle 0 , & \\mbox { i f $ n \\equiv 1 \\pmod 2 $ . } \\end{array} \\right . \\end{align*}"}
-{"id": "1463.png", "formula": "\\begin{align*} d _ { u } ( E ^ { e v e n , 0 } ) \\subset ( D _ r \\langle x _ 3 z _ r \\rangle ) ^ { * , - u + 1 } = \\{ 0 \\} . \\end{align*}"}
-{"id": "6922.png", "formula": "\\begin{align*} \\partial _ n u ( x , t ) | _ { \\partial \\Omega } = 0 \\end{align*}"}
-{"id": "2202.png", "formula": "\\begin{align*} F _ i ( z , t ) = q _ i ( z ) + t \\cdot Q _ i ( z ) , i = 1 , \\ldots , n \\end{align*}"}
-{"id": "3592.png", "formula": "\\begin{align*} \\int _ a ^ b f ( t ) g ( t ) \\dd t = f ( \\xi ) \\int _ a ^ b g ( t ) d t \\end{align*}"}
-{"id": "2478.png", "formula": "\\begin{align*} w = e ^ { \\alpha \\vartheta ( t , x , \\xi ) } . \\end{align*}"}
-{"id": "3790.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { X _ n } { n } = v \\ ; \\ ; \\ ; \\P ^ \\rho \\end{align*}"}
-{"id": "9335.png", "formula": "\\begin{align*} p \\pm q = \\sum _ { s = 0 } ^ { 7 } ( a _ { s } \\pm b _ { s } ) e _ { s } , \\end{align*}"}
-{"id": "7182.png", "formula": "\\begin{align*} I _ k ^ { ( 1 ) } \\leq \\int _ { B _ { 1 - 2 \\varepsilon } \\setminus I _ { 3 \\varepsilon } } | \\nabla v _ { \\varepsilon } | ^ 2 \\ , d \\mu _ a + O ( \\varepsilon ) . \\end{align*}"}
-{"id": "7555.png", "formula": "\\begin{gather*} u = - 2 \\frac { \\theta ^ 1 _ x } { \\theta ^ 1 } , v = \\left ( - 2 \\frac { \\theta ^ 1 _ x } { \\theta ^ 1 } y + \\frac { \\theta ^ 0 } { \\theta ^ 1 } \\right ) _ x \\quad \\mbox { w i t h } \\theta ^ i = \\theta ^ i ( t , x ) \\colon \\ \\theta ^ i _ t = \\theta ^ i _ { x x } , i = 0 , 1 . \\end{gather*}"}
-{"id": "4944.png", "formula": "\\begin{align*} k ^ 2 - 2 ^ { y - 2 e } = 1 \\end{align*}"}
-{"id": "10078.png", "formula": "\\begin{align*} A _ i ( \\C ) = g \\mathfrak { a } _ i \\backslash \\mathfrak { a } _ { i \\C } / \\overline { \\epsilon } \\mathfrak { a } _ { i \\C } . \\end{align*}"}
-{"id": "5780.png", "formula": "\\begin{align*} \\Delta _ { p } u = \\nabla \\cdot \\left ( \\vert \\nabla u \\vert ^ { p - 2 } \\nabla u \\right ) , u \\in W ^ { 1 , p } _ { l o c } ( \\Omega ) , \\end{align*}"}
-{"id": "9519.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 0 } ^ { \\tilde { d } - 2 } \\dfrac { m ^ k } { ( d - 2 - k ) ! } \\cdot \\tilde { p } _ { d - 2 - k } ^ { \\ , ( d , m ) } \\cdot \\tilde { a } _ { v - d , k } ^ { ( l , d ) } . \\end{align*}"}
-{"id": "7694.png", "formula": "\\begin{align*} & V M ^ { \\top } \\Sigma V ^ { \\top } + V \\Sigma M V ^ { \\top } = - V Z V ^ { \\top } \\\\ & ( V M ^ { \\top } V ^ { \\top } ) ( V \\Sigma V ^ { \\top } ) + ( V \\Sigma V ^ { \\top } ) ( V M V ^ { \\top } ) = - V Z V ^ { \\top } \\end{align*}"}
-{"id": "10096.png", "formula": "\\begin{align*} P ( X , Y ) Z = R ( X , Y ) Z - \\frac { 1 } { n - 1 } \\{ S ( Y , Z ) X - S ( X , Z ) Y \\} , \\end{align*}"}
-{"id": "3173.png", "formula": "\\begin{align*} \\mu _ { Z _ { t } } = \\mu _ { Z _ { t } ^ { 1 } } \\ast \\mu _ { Z _ { t } ^ { 2 } } . \\end{align*}"}
-{"id": "328.png", "formula": "\\begin{align*} \\sum _ { i \\in I } \\sigma _ i = \\sum _ { j \\in J } \\tau _ j , \\sum _ { i \\in I } \\lambda _ i = \\sum _ { j \\in J } \\mu _ j . \\end{align*}"}
-{"id": "7588.png", "formula": "\\begin{align*} L ( w ) = \\frac { 1 } { 2 } | \\{ ( i , j ) \\in [ - n , n ] ^ 2 \\mid i < j , \\ , w ( i ) > w ( j ) , \\ , i \\not \\equiv j \\pmod 2 \\} | , \\end{align*}"}
-{"id": "2045.png", "formula": "\\begin{align*} R e s ( q z _ 1 , . . . , z _ n ) = q ^ { - k ( k + 1 ) } z _ 1 ^ { - 2 k ^ 2 - 2 k } z _ 2 ^ { - k ^ 2 - k } . . . z _ n ^ { - k ^ 2 - k } R e s ( z _ 1 , . . . , z _ n ) . \\end{align*}"}
-{"id": "7000.png", "formula": "\\begin{align*} y _ { n - 1 } = z _ { n - 1 } \\cos \\theta , \\end{align*}"}
-{"id": "9356.png", "formula": "\\begin{align*} j O _ { n } ^ { ( 3 ) } - J O _ { n + 2 } ^ { ( 3 ) } & = ( j _ { n } ^ { ( 3 ) } - J _ { n + 2 } ^ { ( 3 ) } ) e _ { 0 } + ( j _ { n + 1 } ^ { ( 3 ) } - J O _ { n + 3 } ^ { ( 3 ) } ) e _ { 1 } + \\cdots + ( j _ { n + 7 } ^ { ( 3 ) } - J _ { n + 9 } ^ { ( 3 ) } ) e _ { 7 } \\\\ & = 1 - e _ { 1 } + e _ { 3 } - e _ { 4 } + e _ { 6 } - e _ { 7 } \\end{align*}"}
-{"id": "7626.png", "formula": "\\begin{align*} \\int _ { K _ s } m _ t \\varphi ^ p m \\ , d z = \\int _ { K _ s } \\Big [ \\big ( \\varphi ^ p \\frac { m ^ 2 } { 2 } \\big ) _ t - \\frac { p } { 2 } \\varphi ^ { p - 1 } \\varphi _ t m ^ 2 \\Big ] \\ , d z = \\frac 1 2 \\int _ { B _ \\frac 5 2 } ( \\varphi ^ p m ^ 2 ) ( x , s ) \\ , d x - \\frac { p } { 2 } \\int _ { K _ s } \\varphi ^ { p - 1 } \\varphi _ t m ^ 2 \\ , d z , \\end{align*}"}
-{"id": "3598.png", "formula": "\\begin{align*} y ' ( x ) = f ( x , y ( x ) ) , y ( 0 ) = y _ 0 , \\end{align*}"}
-{"id": "3235.png", "formula": "\\begin{align*} ( P _ { W } ^ { X W } \\circ P _ { X W } ^ { X Y W } ) _ \\ast ( \\beta ) = ( P _ { W } ^ { X Y W } ) _ \\ast ( \\beta ) , \\end{align*}"}
-{"id": "1761.png", "formula": "\\begin{align*} f \\sqrt { d \\mu / d \\lambda } = g \\sqrt { d \\nu / d \\lambda } , \\ \\ ( \\lambda - a . e . ) . \\end{align*}"}
-{"id": "115.png", "formula": "\\begin{align*} c _ 1 = \\frac { \\rho _ 1 \\dots \\rho _ k } { 2 } - c _ 0 \\frac { \\dim C P } { N ^ k } > 0 . \\end{align*}"}
-{"id": "6842.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ T \\theta _ t q _ t ( x _ t ) - \\inf _ { x \\in X } \\sum _ { t = 1 } ^ T \\theta _ t q _ t ( x ) \\leq \\frac { 2 G _ { f , X } ^ 2 } { \\alpha _ { f , X } ( T + 1 ) } = O \\left ( \\frac { 1 } { T } \\right ) . \\end{align*}"}
-{"id": "9559.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in B _ n } { ( - 1 ) ^ { \\ell ( \\sigma ) } x ^ { L _ { e o o } ( \\sigma ) } } = 0 . \\end{align*}"}
-{"id": "1309.png", "formula": "\\begin{align*} \\dim \\mathrm { H o m } ( \\theta _ { \\mathtt { 1 3 5 } } ^ { \\mathrm { o n } } \\theta _ { \\mathtt { 3 4 2 } } L ( \\mathtt { 2 3 2 4 3 2 } ) , \\theta _ { \\mathtt { 1 3 5 } } ^ { \\mathrm { o n } } L ( \\mathtt { 2 3 2 4 3 2 1 5 } ) ) = 1 . \\end{align*}"}
-{"id": "5611.png", "formula": "\\begin{align*} X _ t = x + \\int _ 0 ^ t f ( X _ r ) d r + B _ t . \\end{align*}"}
-{"id": "9504.png", "formula": "\\begin{align*} A ^ 2 ( x , t ) = B ( x , t ) \\end{align*}"}
-{"id": "9957.png", "formula": "\\begin{align*} \\overline { { \\cal N } } ( \\varphi ( u ) ) - V _ 2 = V ( { \\cal G } ) - ( { \\cal N } ( \\varphi ( u ) ) \\cup \\{ \\varphi ( u ) \\} \\cup V _ 2 ) = V ( { \\cal G } ) - ( { \\cal N } ( \\varphi ( u ) ) \\cup V _ 2 ) . \\end{align*}"}
-{"id": "8319.png", "formula": "\\begin{align*} \\Psi _ g ( f ) = \\Psi _ \\mathrm { c l a s s i c a l } ( g f ) . \\end{align*}"}
-{"id": "1982.png", "formula": "\\begin{align*} ( \\tilde { F } _ { p } v , \\tilde { F } _ { p } w ) & = ( \\tau _ { f ( p ) } D _ { p } f \\tau _ { p } ^ { - 1 } ( v ) , \\tau _ { f ( p ) } D _ { p } f \\tau _ { p } ^ { - 1 } ( w ) ) = \\tau _ { f ( p ) } D _ { p } f \\tau _ { p } ^ { - 1 } ( v , w ) \\\\ & = D _ { 0 } ( \\tilde { f } _ { p } ) ( v , w ) = ( D _ { 0 } ( \\tilde { f } _ { p } ) _ { 1 } ( v , w ) , D _ { 0 } ( \\tilde { f } _ { p } ) _ { 2 } ( v , w ) ) . \\end{align*}"}
-{"id": "4262.png", "formula": "\\begin{align*} ( v _ { 1 1 } , v _ { 1 2 } ) = ( e _ { 1 1 } \\otimes _ \\Phi 1 , e _ { 1 2 } \\otimes _ \\Phi 1 ) \\left ( \\begin{array} { c c } h _ 1 & h _ o \\\\ f _ 1 & f _ o \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "8645.png", "formula": "\\begin{align*} T _ 0 = 0 , \\ \\ T _ i = \\inf \\{ j > T _ { i - 1 } : \\eta _ j = 1 \\} , \\ \\ i \\geq 1 , \\end{align*}"}
-{"id": "4899.png", "formula": "\\begin{align*} \\norm P _ e T P _ e \\norm = \\norm P _ e T ^ * P _ e \\norm , \\end{align*}"}
-{"id": "6499.png", "formula": "\\begin{align*} \\rho _ { \\mathrm { Q M } } \\overset { } { = } \\sqrt { 8 \\left ( 2 k _ { \\mathrm { o } } ^ { 2 } + \\sigma _ { k \\mathrm { o } } ^ { 2 } \\right ) R _ { \\mathrm { o } } a _ { \\mathrm { s } } } \\ll 1 , \\end{align*}"}
-{"id": "6353.png", "formula": "\\begin{align*} I ( m ) : = { } & \\sum \\limits _ { j = 1 } ^ { \\lfloor \\frac { k + 1 } { 2 } \\rfloor } \\big \\{ W _ 1 ( y _ { 2 j - 1 } ) - W _ 1 ( y _ 1 ^ * ) \\big \\} + \\sum \\limits _ { j = 1 } ^ { \\lfloor \\frac { k } { 2 } \\rfloor } \\big \\{ W _ 2 ( y _ { 2 j } ) - W _ 2 ( y _ 2 ^ * ) \\big \\} . \\end{align*}"}
-{"id": "2922.png", "formula": "\\begin{align*} U _ s : = \\bigl \\{ u [ g ] \\colon g \\in G [ \\ast , s ] \\bigr \\} \\mbox { w h e r e } u [ g ] = [ A _ { \\iota _ 2 ( g ) } \\times \\{ g \\} ] = \\bigl \\{ [ ( a , g ) ] \\colon a \\in A _ { \\iota _ 2 ( g ) } \\bigr \\} \\end{align*}"}
-{"id": "8598.png", "formula": "\\begin{align*} ( t _ { g } ( b ) \\cdot \\Psi ) _ { i } ( \\delta ) : = g _ { i } ^ { - 1 } ( b ) \\Psi _ { i } ( \\delta ) , ( t _ { g } ( u _ { \\gamma } ) \\Psi ) _ { i } ( \\delta ) : = t _ { i } ( \\gamma ^ { - 1 } ) ^ { - 1 } ( \\Psi _ { \\gamma ^ { - 1 } ( i ) } ( t _ { i } ( \\gamma ^ { - 1 } ) \\delta ) ) . \\end{align*}"}
-{"id": "9809.png", "formula": "\\begin{align*} f ( t ) = \\frac { ( 1 - p _ 0 ( t ) ) \\mu } { \\l } - \\frac { \\beta _ 2 \\mu } { ( \\alpha + \\beta _ 2 ) \\l } , 0 \\leq t \\leq T _ { e _ 2 } . \\end{align*}"}
-{"id": "2437.png", "formula": "\\begin{align*} \\frac { n - d ( c _ 1 + c _ 2 ) } { d c _ 1 c _ 2 } = \\frac { l ( n - q - 1 ) } { q } . \\end{align*}"}
-{"id": "7953.png", "formula": "\\begin{align*} \\mathcal S _ m : = \\mathcal S \\cap \\mathcal I ^ m , \\ , \\ , \\ , \\mathcal S ' : = ( \\mathcal S \\setminus \\mathcal S _ m ) \\cup \\{ I _ - \\ , | \\ , I \\in \\mathcal S _ m \\} . \\end{align*}"}
-{"id": "7807.png", "formula": "\\begin{align*} - \\Delta v + 2 \\rho ^ { \\prime } \\frac { \\partial v } { \\partial r } + ( V + V _ 0 ) v = \\lambda v , \\end{align*}"}
-{"id": "4451.png", "formula": "\\begin{align*} \\lefteqn { ( - \\partial _ 1 ^ 2 - | \\partial _ 1 | ^ { - 1 } \\partial _ 2 ^ 2 ) u ^ \\ell } \\\\ & + P \\Big ( u ^ \\ell \\partial _ 2 R u ^ \\ell + \\partial _ 2 \\frac { 1 } { 2 } R ( u ^ \\ell ) ^ 2 - u ^ \\ell \\partial _ 1 \\frac { 1 } { 2 } R ( u ^ \\ell ) ^ 2 - \\sigma \\xi _ \\ell \\Big ) = 0 . \\end{align*}"}
-{"id": "5932.png", "formula": "\\begin{align*} U ^ { \\perp } = \\{ v \\in V \\colon ( v , U ) = 0 \\} \\end{align*}"}
-{"id": "8630.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { [ 0 , T ] ^ 2 } R ( s - u , \\omega ( s ) - \\omega ( u ) ) d s d u = \\sum _ { k = 0 } ^ { N + 1 } Q _ { k k } + 2 \\sum _ { k = 0 } ^ N Q _ { k , k + 1 } . \\end{aligned} \\end{align*}"}
-{"id": "7874.png", "formula": "\\begin{align*} d ( y , y ' ) = \\sqrt { ( \\theta _ { y ' } - \\theta _ y ) ^ 2 + ( y ' _ 3 - y ' _ 3 ) ^ 2 } , d ( z , z ' ) = \\sqrt { \\left ( { \\theta _ { y ' } - \\theta _ y \\over 2 } \\right ) ^ 2 + \\left ( { y ' _ 3 - y ' _ 3 \\over 2 } \\right ) ^ 2 } , \\end{align*}"}
-{"id": "4951.png", "formula": "\\begin{align*} f \\left ( \\sum _ { i = 1 } ^ { n } w _ i A _ i \\right ) \\leq \\sum _ { i = 1 } ^ { n } w _ i f ( A _ i ) , \\ , \\ , \\ , \\ , w _ i > 0 , \\ , \\ , \\ , \\sum _ { i = 1 } ^ { n } w _ i = 1 . \\end{align*}"}
-{"id": "7019.png", "formula": "\\begin{align*} C _ 2 = C _ 2 ( n , s ) = \\int _ { \\mathbb { R } ^ { n - 1 } } { \\frac { 1 } { ( 1 + | \\bar { z } | ^ 2 ) ^ { \\frac { n + 2 s } { 2 } } } d \\bar { z } } . \\end{align*}"}
-{"id": "8431.png", "formula": "\\begin{align*} S _ { \\chi } ( A , B , m ) = \\int _ { \\mathcal { O } ^ { \\times } } \\chi ( x ) \\psi \\left ( \\frac { A x + B x ^ { - 1 } } { \\varpi ^ { m } } \\right ) d ^ { \\times } x A , B \\in \\mathcal { O } m \\in \\N _ 0 . \\end{align*}"}
-{"id": "2417.png", "formula": "\\begin{align*} p _ 3 ( x ) = \\frac { ( a _ 2 + ( 2 a _ 3 + a _ 4 ) x - a _ 2 x ^ 2 ) ( a _ 3 - a _ 2 x + a _ 1 ( a _ 2 + a _ 3 x ) ) } { a _ 2 } . \\end{align*}"}
-{"id": "6808.png", "formula": "\\begin{align*} L ( \\phi ) = h \\textrm { i n } \\mathbb { S } ^ 2 _ { \\lambda } , \\end{align*}"}
-{"id": "4226.png", "formula": "\\begin{align*} g \\circ f = r \\circ s ' \\circ r ' \\circ s = r \\circ e \\circ s = r \\circ s \\circ r \\circ s = 1 _ N \\circ 1 _ N = 1 _ N , \\end{align*}"}
-{"id": "2577.png", "formula": "\\begin{align*} D D = \\left [ \\begin{array} { c c c c } ( \\infty ) & ( \\infty ) & ( 0 , 9 ) & ( \\infty ) \\\\ ( \\infty ) & ( \\infty ) & ( \\infty ) & ( \\infty ) \\\\ ( \\infty ) & ( 3 , 7 ) & ( \\infty ) & ( \\infty ) \\end{array} \\right ] , \\end{align*}"}
-{"id": "5106.png", "formula": "\\begin{align*} \\int _ { \\mathcal { M } ^ \\perp } \\ ! e ( k , \\tau ) d \\tau & = \\int _ { \\mathcal { M } ^ \\perp } \\ ! e ( k , \\tau _ 0 + \\tau - \\tau _ 0 ) d \\tau = e ( k , \\tau _ 0 ) \\int _ { \\mathcal { M } ^ \\perp } \\ ! e ( k , \\tau - \\tau _ 0 ) d \\tau \\\\ & = e ( k , \\tau _ 0 ) \\int _ { - \\tau _ 0 + \\mathcal { M } ^ \\perp } \\ ! e ( k , \\tau ) d \\tau = \\underbrace { e ( k , \\tau _ 0 ) } _ { \\neq 1 } \\int _ { \\mathcal { M } ^ \\perp } \\ ! e ( k , \\tau ) d \\tau , \\end{align*}"}
-{"id": "1893.png", "formula": "\\begin{align*} \\hat F \\in \\mathcal { F } ^ { S , \\pi } ( F _ 1 ^ \\ast , F _ 2 ^ \\ast ) : = \\big \\{ F \\in \\mathcal { F } ( F _ 1 ^ \\ast , F _ 2 ^ \\ast ) \\colon F ( s ) = \\pi _ s s \\in S \\big \\} . \\end{align*}"}
-{"id": "7730.png", "formula": "\\begin{align*} G _ N ( 2 ) & = \\frac { 1 } { N } \\sum ^ { N - 1 } _ { n = 1 } \\frac { 1 - \\cos 4 \\phi _ n } { ( 1 - \\cos 2 \\phi _ n ) ^ 2 } \\\\ & = \\frac { 1 } { 2 N } \\sum ^ { N - 1 } _ { n = 1 } \\frac { \\sin ^ 2 2 \\phi _ n } { \\sin ^ 4 \\phi _ n } = \\frac { 2 } { N } \\sum ^ { N - 1 } _ { n = 1 } \\frac { \\cos ^ 2 \\phi _ n } { \\sin ^ 2 \\phi _ n } \\\\ & = \\frac { 2 N } { 3 } - 2 + \\frac { 4 } { 3 N } \\end{align*}"}
-{"id": "2223.png", "formula": "\\begin{align*} \\begin{cases} \\left | 1 - a _ { 1 i _ { 1 } } \\frac 1 { w _ 1 } \\right | = \\varepsilon _ 1 , \\\\ \\ldots \\\\ \\left | 1 - a _ { n i _ { n } } \\frac 1 { w _ n } \\right | = \\varepsilon _ n , \\end{cases} \\end{align*}"}
-{"id": "7011.png", "formula": "\\begin{align*} B = d i a g \\{ \\frac { 2 } { n - 1 } \\epsilon , \\frac { 2 } { n - 1 } \\epsilon , . . . , \\frac { 2 } { n - 1 } \\epsilon , h ( \\epsilon ) \\} . \\end{align*}"}
-{"id": "8031.png", "formula": "\\begin{align*} { \\Gamma } _ { { t _ j } } \\ast g _ { t _ { n + j } } ( x ) & = 2 ^ { { d } { t _ n } / p } 2 ^ { { d } { t _ j } / p } \\int _ { | y | \\leq 2 ^ { - M } } { { \\Gamma } ( 2 ^ { { t _ j } } x - y ) g ( 2 ^ { { t _ n } } y ) } d y \\\\ & \\geq 2 ^ { { d } t _ n / { p } } 2 ^ { { d } t _ j / { p } } 2 ^ { - { t _ n } d } \\Vert g \\Vert _ { L ^ 1 } \\end{align*}"}
-{"id": "7333.png", "formula": "\\begin{align*} & \\int e ^ { - i x \\cdot \\xi } e ^ { i ( x - y ) \\cdot \\eta } e ^ { i y \\cdot \\zeta } a ( x - h ( x - y ) , \\eta ) \\widehat { u } ( \\zeta ) \\ , d \\zeta d y d \\eta d x \\\\ & = \\int e ^ { i \\widetilde { y } \\cdot \\varphi _ 1 ( \\xi , \\zeta , \\eta ) } \\ , e ^ { i \\widetilde { x } \\cdot \\varphi _ 2 ( \\xi , \\zeta , \\eta ) } \\ , a ( \\widetilde { x } , \\eta ) \\widehat { u } ( \\zeta ) \\ , d \\zeta d \\widetilde { y } d \\eta d \\widetilde { x } \\end{align*}"}
-{"id": "7486.png", "formula": "\\begin{align*} \\int _ E \\nabla _ { \\mathcal { X } _ \\alpha } Z ^ \\alpha d \\mathcal { V } = 0 , \\int _ E \\nabla _ { \\mathcal { X } _ { \\bar { \\alpha } } } \\overline { Z ^ \\alpha } d \\mathcal { V } = 0 . \\end{align*}"}
-{"id": "9492.png", "formula": "\\begin{align*} S ( x , t ) - 1 = \\dfrac { t } { x ^ { r - 2 } } \\left [ S ( x , t ) - 1 - \\sum \\limits _ { i = 1 } ^ { r - 2 } x ^ i \\ , S _ i ( t ) \\right ] + t \\ , x ^ 2 \\ , S ^ 2 ( x , t ) , S _ i ( t ) : = \\sum \\limits _ { j = 0 } ^ { \\infty } s _ { n , j } \\ , t ^ j . \\end{align*}"}
-{"id": "1295.png", "formula": "\\begin{align*} \\textstyle \\sqrt [ 3 ] { r _ 1 } \\ + \\ \\sqrt [ 3 ] { r _ 2 } \\ + \\ \\sqrt [ 3 ] { r _ 3 } \\ = \\ \\sqrt [ 3 ] { \\left ( \\frac { 3 + B } { 2 } \\right ) \\ - \\ 6 \\ + \\ 3 \\sqrt [ 3 ] { \\frac { 2 7 + B ^ 2 } { 4 } } } \\end{align*}"}
-{"id": "502.png", "formula": "\\begin{align*} V _ 1 ( x , t ) & \\sim - \\frac { R } { 4 } \\frac { \\frac { \\pi ^ 2 } { ( n ! ) ^ 2 } \\abs * { t } ^ { 2 } + \\frac { \\pi ^ 2 } { [ ( n - 1 ) ! ] ^ 2 } } { \\frac { 1 } { [ ( n - 1 ) ! ] ^ 2 } } + \\frac { R } { 2 } \\frac { \\frac { \\pi ^ 2 } { ( n + 1 ) ! } \\abs * { t } ^ { 2 } + \\frac { \\pi ^ 2 } { ( n - 1 ) ! } + \\frac { n \\pi } { R \\ , n ! } \\abs * { t } + \\frac { ( m - 1 ) \\pi } { ( n - 1 ) ! \\abs { t } } } { \\frac { 1 } { ( n - 1 ) ! } } \\\\ & \\sim \\frac { \\pi } { 2 } \\abs * { t } \\asymp d ( x , t ) ^ 2 , \\end{align*}"}
-{"id": "4339.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\| M _ { f _ j } g \\| _ { p , \\alpha } ^ p \\leq N \\| g \\| _ { p , \\alpha } ^ p . \\end{align*}"}
-{"id": "7216.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ N \\xi _ k | ( u _ k ) _ \\nu | ^ 2 = 1 \\end{align*}"}
-{"id": "9034.png", "formula": "\\begin{gather*} \\begin{pmatrix} 0 & 2 a _ 1 ^ 2 y _ 1 ( \\neq 0 ) & 0 & 2 a _ 3 y _ 3 \\\\ k y _ 1 ^ 3 ( \\neq 0 ) & 0 & 0 & 0 \\end{pmatrix} \\end{gather*}"}
-{"id": "709.png", "formula": "\\begin{align*} p _ \\eta ( \\rho , \\eta ) = \\frac { 2 p \\left [ 1 + \\frac { 1 } { 2 } \\alpha ( 1 - \\alpha ) \\left ( \\frac { 5 } { 2 } + \\frac { T _ i } { T } \\right ) \\right ] } { 3 ( 1 + \\alpha ) \\left [ 1 + \\alpha ( 1 - \\alpha ) \\left ( \\frac { 5 } { 4 } + \\frac { T _ i } { T } + \\frac { T _ i ^ 2 } { 3 T ^ 2 } \\right ) \\right ] } > 0 . \\end{align*}"}
-{"id": "9239.png", "formula": "\\begin{align*} \\epsilon ( \\alpha _ { 1 } \\circ \\alpha _ { 2 } , \\alpha _ { 3 } ) + \\epsilon ( \\alpha _ { 2 } \\circ \\alpha _ { 3 } , \\alpha _ { 1 } ) + \\epsilon ( \\alpha _ { 3 } \\circ \\alpha _ { 1 } , \\alpha _ { 2 } ) = 0 \\end{align*}"}
-{"id": "9872.png", "formula": "\\begin{align*} m _ \\Delta ( i , j ) : = \\left \\{ \\begin{array} { l l } 1 \\ \\ \\ \\ \\ & \\mbox { i f $ i = j $ , } \\\\ 2 \\ \\ \\ \\ \\ & \\mbox { i f n o e d g e b e t w e e n $ i $ a n d $ j $ i n $ \\Delta $ , } \\\\ 3 \\ \\ \\ \\ \\ & \\mbox { i f t h e r e i s a n e d g e $ i \\stackrel { } { \\mbox { - - - } } j $ i n $ \\Delta $ , } \\\\ 4 \\ \\ \\ \\ & \\mbox { i f t h e r e i s a n e d g e $ i \\stackrel { 4 } { \\mbox { - - - } } j $ i n $ \\Delta $ . } \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "3733.png", "formula": "\\begin{align*} ( p _ { m _ { k _ l + \\beta _ { l , i } + \\kappa r _ l + j } } , p ' _ { m _ { k _ l + \\beta _ { l , i } + \\kappa r _ l + j } } ) = \\tau _ l ^ j ( p _ { m _ { k _ l + \\beta _ { l , i } + \\kappa r _ l } } , p ' _ { m _ { k _ l + \\beta _ { l , i } + \\kappa r _ l } } ) . \\end{align*}"}
-{"id": "2454.png", "formula": "\\begin{align*} \\binom { m _ r } { k - q _ s } / \\binom { m _ r - 1 } { q _ r - 1 } > \\binom { m _ s } { k - q _ r } / \\binom { m _ s - 1 } { q _ s - 1 } . \\end{align*}"}
-{"id": "651.png", "formula": "\\begin{align*} | P | = | P ^ { - 1 } | & = 1 , \\\\ | \\nabla P | , | \\nabla P ^ { - 1 } | & \\le C ^ { l o c } _ 0 d \\end{align*}"}
-{"id": "9140.png", "formula": "\\begin{align*} f ( x ) = D _ { 4 } ( x ) + f ( 1 ) x \\left ( x \\in R \\right ) . \\end{align*}"}
-{"id": "959.png", "formula": "\\begin{align*} & E [ U _ n ( \\theta ) U _ n ( \\theta ' ) ] - E [ U ^ * _ n ( \\theta ) U ^ * _ n ( \\theta ' ) | \\mathcal { F } ^ X ] \\\\ & = \\sum _ { I , J } \\{ E [ X ^ 1 ( I ) ^ 2 X ^ 2 ( J ) ^ 2 ] - X ^ 1 ( I ) ^ 2 X ^ 2 ( J ) ^ 2 \\} K ( I , J _ { - \\theta } ) K ( I , J _ { - \\theta ' } ) \\in \\overline { \\mathcal { P } } _ 4 ( \\mathbb { H } ) \\end{align*}"}
-{"id": "3648.png", "formula": "\\begin{align*} \\Phi ' [ n ] : T [ n ] = ( \\mathbb C ^ * ) ^ { n + 2 } \\to ( \\mathbb A ^ 2 \\times \\mathbb A ^ { n + 1 } ) \\times ( \\mathbb P ^ 1 ) ^ { n } \\end{align*}"}
-{"id": "2486.png", "formula": "\\begin{align*} \\left ( g \\cdot f \\right ) \\left ( \\xi \\right ) = f \\left ( g ^ { - 1 } \\xi \\right ) . \\end{align*}"}
-{"id": "8789.png", "formula": "\\begin{align*} c ( f ) = \\inf _ { m \\geq 1 } \\frac { \\textnormal { c o d i m } \\textnormal { C o n t } ^ { \\geq m } ( f ) } { m } , \\end{align*}"}
-{"id": "10072.png", "formula": "\\begin{align*} 2 \\pi i \\cdot \\psi ( a , b ) = \\psi ( \\alpha , a ) \\psi ( \\beta , b ) - \\psi ( \\alpha , b ) \\psi ( \\beta , a ) \\end{align*}"}
-{"id": "4877.png", "formula": "\\begin{align*} ( A ( t ) ) ^ { [ M ] } = 1 + \\sum _ { k = 1 } ^ { \\infty } \\left \\{ \\sum _ { \\underline { k } : \\sum i k _ i = k } \\left [ \\left ( \\prod _ i M ^ { k _ i } \\setminus \\Delta \\right ) \\times \\prod _ i A _ i ^ { k _ i } \\middle / \\prod _ i S _ { k _ i } \\right ] \\right \\} \\cdot t ^ k . \\end{align*}"}
-{"id": "3256.png", "formula": "\\begin{align*} ( \\alpha _ { i , Y } ^ a , \\omega _ Y ) _ Y = ( \\alpha _ { i , Y } , \\omega _ Y ) _ Y . \\end{align*}"}
-{"id": "3674.png", "formula": "\\begin{align*} \\alpha _ W [ n ] L [ n ] = \\left ( \\begin{array} { c c c | c c c | c c c | c c c } - 1 & \\cdots & - 1 & 0 & \\cdots & 0 & & & & 0 & \\cdots & 0 \\\\ 0 & \\cdots & 0 & - 1 & \\cdots & - 1 & & & & 0 & \\cdots & 0 \\\\ & & & & & & & \\ddots & \\\\ 0 & \\cdots & 0 & 0 & \\cdots & 0 & & & & - 1 & \\cdots & - 1 \\end{array} \\right ) , \\end{align*}"}
-{"id": "2230.png", "formula": "\\begin{align*} F _ j \\left ( \\frac 1 { w _ 1 } , \\ldots , \\frac 1 { w _ n } , t \\right ) = \\left ( \\frac 1 { w _ 1 } \\right ) ^ { m _ { i 1 } } \\cdot \\ldots \\cdot \\left ( \\frac 1 { w _ n } \\right ) ^ { m _ { i n } } \\cdot \\widetilde { F _ j } ( w , t ) , j = 1 , \\ldots , n . \\end{align*}"}
-{"id": "10045.png", "formula": "\\begin{align*} ( \\mathfrak { B } , \\beta ) = \\alpha ^ { - 1 } ( \\mathfrak { B } , \\beta ) = ( \\mathfrak { A } , 1 ) \\end{align*}"}
-{"id": "4501.png", "formula": "\\begin{align*} \\| s ^ * a s \\| = | \\omega ( a ) | \\end{align*}"}
-{"id": "4132.png", "formula": "\\begin{align*} h ( Z , X ) = X \\left ( K ^ { \\frac { 1 } { 4 } } \\right ) \\end{align*}"}
-{"id": "1871.png", "formula": "\\begin{align*} \\phi ( f ^ n ) \\leq \\pi ( 0 , \\dots , 0 , a ) = \\underline { \\pi } ^ i . \\end{align*}"}
-{"id": "6080.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow - \\infty } \\tfrac { 1 } { 2 } e ^ { - t } P ( t ) = \\Phi ' ( 0 ) = : c \\end{align*}"}
-{"id": "8798.png", "formula": "\\begin{align*} \\theta _ { F } ( x ) : = \\# ( \\pi ^ { - 1 } ( x ) ) . \\end{align*}"}
-{"id": "6009.png", "formula": "\\begin{align*} D ( P _ { Y | X } \\| Q _ { Y | X } | P _ { X } ) & : = D ( P _ { X } P _ { Y | X } \\| P _ { X } Q _ { Y | X } ) \\\\ * D _ { 1 + s } ( P _ { Y | X } \\| Q _ { Y | X } | P _ { X } ) & : = D _ { 1 + s } ( P _ { X } P _ { Y | X } \\| P _ { X } Q _ { Y | X } ) , \\end{align*}"}
-{"id": "3187.png", "formula": "\\begin{align*} \\mathbb { E } _ { x } \\left [ V ( X _ { t } ) \\right ] = V ( x ) + \\mathbb { E } _ { x } \\left [ \\int _ { 0 } ^ { t } \\mathcal { A } V \\left ( X _ { s } \\right ) \\mathrm { d } s \\right ] , \\end{align*}"}
-{"id": "5346.png", "formula": "\\begin{align*} W = \\coprod _ { n \\in \\C } W _ { [ n ] } \\end{align*}"}
-{"id": "5913.png", "formula": "\\begin{align*} \\begin{aligned} & i _ 0 = 0 , i _ 1 = 1 , i _ 2 = 2 , \\cdots , i _ j = j , i _ { j + 1 } = j - 1 , \\cdots , i _ { j + k } = j - k , i _ { j + k + 1 } = j - k + 1 , \\\\ & \\ 2 \\le j \\le l \\ \\ 2 \\le k \\le j - 1 , \\end{aligned} \\end{align*}"}
-{"id": "2733.png", "formula": "\\begin{align*} \\partial _ t g + \\frac { 1 } { \\epsilon } v \\cdot \\nabla _ x g = \\frac { 1 } { \\epsilon ^ 2 } \\mathcal L ( g ) , \\end{align*}"}
-{"id": "9511.png", "formula": "\\begin{align*} c _ { n , k , t } ^ { ( d ) } = \\sum \\limits _ { i = 0 } ^ { d - 1 } { { d - 1 } \\choose i } { t \\choose t + i - k } { { d ( n - 1 ) - t } \\choose d - 2 i - t + k - 1 } , \\end{align*}"}
-{"id": "7361.png", "formula": "\\begin{align*} C _ { a b } . \\psi = 0 \\end{align*}"}
-{"id": "1729.png", "formula": "\\begin{align*} \\int _ { \\Lambda ^ \\infty } | W ( f ) | ^ 2 d \\mu _ T = \\int _ { \\Lambda ^ \\infty } | f | ^ 2 | h | ^ 2 d \\mu _ T = \\int _ { \\Lambda ^ \\infty } | f | ^ 2 d \\mu _ S \\quad , \\end{align*}"}
-{"id": "9864.png", "formula": "\\begin{align*} \\lim \\inf _ { \\nu \\to \\infty } \\| d ( x ^ \\nu ) \\| = 0 . \\end{align*}"}
-{"id": "6947.png", "formula": "\\begin{align*} [ a ] H ( j ) ( [ z ] ) & = [ a ] H ( i _ n ) ( z ) = [ a \\oplus e _ { k - m } ] H ( i _ { k } ) ( z \\oplus e _ { k - n } ) \\\\ & = \\alpha _ k ( a \\oplus e _ { k - m } ) \\bar { \\alpha } _ k \\alpha _ k ( z \\oplus e _ { k - n } ) \\ , , \\end{align*}"}
-{"id": "2794.png", "formula": "\\begin{align*} \\omega _ { i , n } = \\int _ { 0 } ^ { T } \\mathcal { L } _ { i } ( t ) \\mathrm { d } t , i = 0 , \\ldots , n . \\end{align*}"}
-{"id": "3762.png", "formula": "\\begin{align*} A _ m = \\Big \\{ \\chi ^ g _ \\sigma < v _ k \\Big \\} , m \\in M _ k . \\end{align*}"}
-{"id": "5894.png", "formula": "\\begin{align*} \\liminf _ { N \\to \\infty } P \\big ( \\cap _ { n = 1 } ^ { 2 [ \\frac 1 { b _ N } ] } \\big \\{ Z _ n ^ { N , i } \\in \\{ 0 , - 1 \\} \\big \\} \\big | \\cup _ { n = 1 } ^ { [ \\frac 1 { b _ N } ] } \\{ Z _ { 2 n } ^ { N , i } = - 1 \\} \\big ) \\ge e ^ { - 2 } . \\end{align*}"}
-{"id": "5585.png", "formula": "\\begin{align*} M _ { 2 ^ { k } } \\left ( t \\right ) = \\left [ \\bar { w } _ { 2 ^ { k } } \\left ( t \\right ) , \\Lambda _ { 1 } ^ { \\left ( 2 ^ { k } \\right ) } \\bar { w } _ { 2 ^ { k } } \\left ( t \\right ) , . . . , \\Lambda _ { 2 ^ { k } - 1 } ^ { \\left ( 2 ^ { k } \\right ) } \\bar { w } _ { 2 ^ { k } } \\left ( t \\right ) \\right ] \\end{align*}"}
-{"id": "913.png", "formula": "\\begin{align*} \\| F \\| _ { k , p } = \\left ( E [ | F | ^ p ] + \\sum _ { j = 1 } ^ k E \\left [ \\| D ^ j F \\| ^ p _ { H ^ { \\otimes j } } \\right ] \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "7140.png", "formula": "\\begin{align*} e ^ * _ G \\leq \\alpha B + \\sum _ { r = 0 } ^ { R - 1 } \\sqrt { 2 U J ^ 2 + 2 V _ r J g ^ * _ r } . \\end{align*}"}
-{"id": "2340.png", "formula": "\\begin{align*} \\max _ { j = 1 } ^ r \\max _ { \\ell = 0 } ^ { k _ j - 1 } \\left \\{ | \\lambda _ j | ^ { m - \\ell } { m \\choose \\ell } \\right \\} \\ll m ^ { d _ p - 1 } . \\end{align*}"}
-{"id": "4797.png", "formula": "\\begin{align*} L = \\left ( \\begin{array} { c c | c } - 2 & \\phantom { - } 1 & \\phantom { - } 2 \\\\ \\phantom { - } 1 & - 2 & \\phantom { - } 2 \\\\ \\hline \\phantom { - } 1 & \\phantom { - } 1 & - 4 \\end{array} \\right ) , \\widehat L = \\left ( \\begin{array} { c c | c } - 1 & \\phantom { - } 2 & - 2 \\\\ \\phantom { - } 2 & - 1 & - 2 \\\\ \\hline - 1 & - 1 & \\phantom { - } 4 \\end{array} \\right ) . \\end{align*}"}
-{"id": "2408.png", "formula": "\\begin{align*} 4 a _ 4 - 9 a _ 6 = 4 a _ 3 + 5 a _ 6 = 9 a _ 2 + a _ 5 = 9 a _ 1 a _ 6 + a _ 5 = 8 a _ 1 ^ 2 - 1 = 0 . \\end{align*}"}
-{"id": "3543.png", "formula": "\\begin{align*} S _ { \\Lambda } = \\sum _ { \\boldsymbol { t } \\in \\Lambda } Z _ { \\boldsymbol { t } } , \\ \\ S _ { \\emptyset } = 0 . \\end{align*}"}
-{"id": "2259.png", "formula": "\\begin{align*} \\sigma _ { ( 1 , 1 ) } = \\sum _ { j = 1 } ^ \\infty \\frac { 1 } { z _ { j 1 } } \\cdot \\frac { 1 } { z _ { j 2 } } = \\sum _ { k , m = 1 } ^ { \\infty } \\frac { 4 a _ 2 b _ 1 } { \\pi ^ 4 k ^ 2 m ^ 2 } - \\sum _ { k , m = 1 } ^ { \\infty } \\frac { a _ 3 b _ 2 } { \\pi ^ 2 ( a _ 2 m ^ 2 - b _ 2 k ^ 2 ) ^ 2 } . \\end{align*}"}
-{"id": "3121.png", "formula": "\\begin{align*} f ^ \\ast ( \\gamma ) = \\overline { \\sigma ( \\gamma , \\gamma ^ { - 1 } ) } \\ ; \\ ; \\overline { f ( \\gamma ^ { - 1 } ) } \\ , . \\end{align*}"}
-{"id": "4949.png", "formula": "\\begin{align*} f \\left ( \\sum \\limits _ { i = 1 } ^ { n } { { { w } _ { i } } { { x } _ { i } } } \\right ) \\le \\sum \\limits _ { i = 1 } ^ { n } { { { w } _ { i } } f \\left ( { { x } _ { i } } \\right ) } , \\end{align*}"}
-{"id": "6573.png", "formula": "\\begin{align*} \\beta _ 2 : = \\beta _ { \\xi } = \\frac { 2 \\cdot \\frac { 4 - 2 \\varepsilon } { \\sqrt { 1 - \\varepsilon ^ 2 } } } { \\frac { 1 } { \\sqrt { 1 - \\varepsilon ^ 2 } } } = 8 - 4 \\varepsilon . \\end{align*}"}
-{"id": "7618.png", "formula": "\\begin{align*} \\frac { 1 } { M ^ l } \\int _ { Q _ R } f ^ l \\ , d \\mu & \\leq ( c _ 0 \\lambda _ 0 ) ^ l \\mu ( Q _ R ) + \\Bigg [ ( c _ 0 \\lambda _ 0 ) ^ l \\mu ( Q _ { 2 R } ) + \\int _ { Q _ { 2 R } } g ^ l d \\nu + \\frac { M ^ l - 1 } { ( M ^ { \\frac { l } { \\hat p } } - 1 ) ^ { \\hat p } } \\Big ( \\int _ { Q _ { 2 R } } \\hat g ^ { \\frac { l } { \\hat p } } d \\hat \\nu \\Big ) ^ { \\hat p } \\Bigg ] \\sum _ { j = 1 } ^ \\infty ( \\alpha M ^ l ) ^ j , \\end{align*}"}
-{"id": "4043.png", "formula": "\\begin{align*} \\frac { 1 } { \\mu _ 1 } + \\frac { 1 } { \\lambda _ 2 } = c , \\end{align*}"}
-{"id": "2747.png", "formula": "\\begin{align*} & \\displaystyle \\chi _ { m n k } = \\begin{cases} 0 , S _ { m n k } = 0 , \\\\ [ 2 p t ] 1 , S _ { m n k } \\neq 0 , \\end{cases} \\end{align*}"}
-{"id": "5191.png", "formula": "\\begin{align*} \\hat { \\mathcal { C } } = \\{ ( x , y , z ) \\in [ a _ { 1 } , a _ { 2 } ] \\times [ a _ { 1 } , a _ { 2 } ] \\times [ b , 1 ] \\colon x \\le y \\} . \\end{align*}"}
-{"id": "523.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ d c _ k ( f ) \\ , z ^ k \\ , = \\ , \\sum _ { k = 0 } ^ d b _ k ( M ) \\ , z ^ k + ( 1 + z ) \\ , Q ( z ) . \\end{align*}"}
-{"id": "9464.png", "formula": "\\begin{align*} \\nu ( z ; q ) = \\sum _ { n = 0 } ^ { \\infty } ( - z q ; q ^ 2 ) _ n q ^ . \\end{align*}"}
-{"id": "9938.png", "formula": "\\begin{align*} u _ t + ( u ^ m ) _ x + ( u ^ n ) _ { x x x } = 0 , m , n \\geq 2 \\end{align*}"}
-{"id": "9624.png", "formula": "\\begin{align*} P _ { \\rm { t r } } \\big ( R _ { \\beta } , \\lambda _ { \\beta } , h _ \\beta \\big ) = \\gamma _ { \\rm { t r } } P _ { \\rm { t r } , 1 } \\big ( { h _ \\beta } / { R _ { \\beta } } \\big ) , \\end{align*}"}
-{"id": "734.png", "formula": "\\begin{align*} \\| u \\| _ p \\le C ( N , p ) \\| \\nabla u \\| _ 2 ^ \\alpha \\| u \\| _ 2 ^ { 1 - \\alpha } , \\ , \\ , \\alpha : = \\frac { N ( p - 2 ) } { 2 p } , \\end{align*}"}
-{"id": "4586.png", "formula": "\\begin{align*} q q _ 1 ^ 2 \\tilde \\theta ^ { - 1 } \\bigl ( \\tilde H _ { 0 , 1 } \\bigr ) & \\equiv q _ 1 \\bigl ( [ F _ { 1 , 1 } , F _ { 0 , - 1 } ] _ { q ^ { 2 } } + q _ 1 [ F _ { 0 , 0 } , F _ { 1 , 0 } ] _ { q ^ { 2 } } \\bigr ) \\\\ & = q q _ 1 ^ 2 \\bigl ( \\theta ^ { - 1 } ( H _ { 0 , 1 } ) + q _ 1 \\theta ^ { - 1 } ( H _ { 1 , 1 } ) \\bigr ) \\ , . \\end{align*}"}
-{"id": "6852.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { T } \\theta _ t \\left ( \\frac { 1 } { \\gamma _ t } V _ { x _ t } ( x ) - \\frac { 1 } { \\gamma _ t } V _ { x _ { t + 1 } } ( x ) - \\alpha V _ { ( x _ t ) } ( x ) + \\frac { \\gamma _ t G ^ 2 } { 2 } \\right ) \\leq \\frac { 2 G ^ 2 } { \\alpha \\ , ( T + 1 ) } , \\end{align*}"}
-{"id": "1936.png", "formula": "\\begin{align*} \\widehat { R _ k ( f ) } ( \\xi ) = ( - | \\xi | ^ 2 + k ^ 2 + i 0 ) ^ { - 1 } \\widehat { f } ( \\xi ) , \\end{align*}"}
-{"id": "5084.png", "formula": "\\begin{align*} | R i c | ^ 2 = \\frac { 2 } { 9 } + 5 t r ( A ) ^ 2 + | A | ^ 2 - \\frac { 4 } { 3 } t r ( A ) - 2 [ b _ 1 ^ 2 a _ 1 + b _ 2 ^ 2 a _ 2 + b _ 3 ^ 2 a _ 3 ] , \\end{align*}"}
-{"id": "8792.png", "formula": "\\begin{align*} E _ { \\Phi } ( z , \\mathfrak { p } , f ) : = \\int _ { K _ { \\mathfrak { p } } ^ { n } } \\Phi ( x ) \\Psi ( z f ( x ) ) | d x | . \\end{align*}"}
-{"id": "2933.png", "formula": "\\begin{align*} \\Theta _ \\epsilon ( H , e , i ) : = \\frac { \\exp ( - F _ \\epsilon ( H , i ) / \\epsilon ) } { \\sum _ { j \\in e } \\exp ( - F _ \\epsilon ( H , j ) / \\epsilon ) } . \\end{align*}"}
-{"id": "3024.png", "formula": "\\begin{align*} x - \\Theta x ' = y , x ( 0 ) = x _ 0 , \\end{align*}"}
-{"id": "1980.png", "formula": "\\begin{align*} \\sigma _ { 1 , p } & = \\sup \\{ \\Vert D _ { ( v , w ) } ( \\tilde { a } _ { p } , \\tilde { b } _ { p } ) \\Vert : ( v , w ) \\in B ^ { k } ( 0 , \\beta _ { i } ) \\times B ^ { d - k } ( 0 , \\beta _ { i } ) \\} \\\\ \\sigma _ { 2 , p } & = \\sup \\{ \\Vert D _ { ( v , w ) } ( \\tilde { c } _ { p } , \\tilde { d } _ { p } ) \\Vert : ( v , w ) \\in B ^ { k } ( 0 , \\beta _ { i + 1 } ) \\times B ^ { d - k } ( 0 , \\beta _ { i + 1 } ) \\} . \\end{align*}"}
-{"id": "6877.png", "formula": "\\begin{align*} \\Phi _ i ( \\lambda _ { n , p } ^ { ( m - 1 ) } ) & = \\lambda _ { - n + i ( s + 1 ) , p } ^ { ( m ) } \\\\ & = \\lambda _ { - n + i ( s + 1 ) , q } ^ { ( m ) } \\end{align*}"}
-{"id": "6230.png", "formula": "\\begin{align*} X ( n ) = n - C ( n ) ~ , ~ ~ n \\geq 0 ~ , \\end{align*}"}
-{"id": "6163.png", "formula": "\\begin{align*} \\tau _ i = ( \\log \\frac { A _ i } { V } ) ' \\frac { A _ i } { V ^ 2 } \\dd x _ i \\end{align*}"}
-{"id": "1468.png", "formula": "\\begin{align*} d _ { 2 ^ { r + 2 } - 1 } ( \\phi _ r ( x _ I ) ) & = v _ { r + 1 } \\phi _ { r + 1 } ( x _ I ) x _ 3 , \\\\ d _ { 2 ^ { r + 2 } - 1 } ( \\phi _ r ( x _ I ) x _ 3 ) & = v _ { r + 1 } \\phi _ { r + 1 } ( x _ I ) x _ 3 ^ 2 + \\alpha _ I , \\end{align*}"}
-{"id": "3731.png", "formula": "\\begin{align*} 0 = \\beta _ { l , 0 } < \\beta _ { l , 1 } < \\cdots < \\beta _ { l , c _ l - 1 } < \\beta _ l \\end{align*}"}
-{"id": "4170.png", "formula": "\\begin{align*} \\frac { 1 } { n } e ^ { n - 1 } _ { i , g + 1 } + \\left ( - 1 + \\frac { 1 } { n } \\right ) e _ { i , g } ^ { n - 1 } & + \\left ( \\frac { 1 } { n } - \\frac { 1 } { n ^ 2 } - 1 \\right ) e _ { i , g + 1 } ^ { n - 2 } \\\\ & + \\left ( - 1 + \\frac { 1 } { n } \\right ) \\sum _ { \\tau = 2 } ^ { n - 2 } e _ { i , g + \\tau } ^ { n - 1 - \\tau } - \\frac { 1 - n } { n } \\sum _ { \\tau = 0 } ^ { n - 2 } e _ { i , g + \\tau } ^ { n - \\tau - 1 } , \\end{align*}"}
-{"id": "4390.png", "formula": "\\begin{align*} \\| \\hat { A } _ x - ( A _ m ) _ x \\| \\leq \\sup _ { \\gamma } \\| C _ { z _ \\gamma } ( \\hat { A } - A _ m ) C _ { - z _ \\gamma } \\| = \\| \\hat { A } - A _ m \\| < \\frac { \\varepsilon } { 4 } . \\end{align*}"}
-{"id": "8321.png", "formula": "\\begin{align*} \\gamma \\cdot \\Psi _ g ( f ) ( z ) = \\xi _ g ( \\gamma ) \\cdot \\Psi _ g ( f ) ( \\gamma z ) \\end{align*}"}
-{"id": "7247.png", "formula": "\\begin{align*} ( \\mathcal { W _ { \\psi } } f ) ( 1 ) = \\int _ { \\hat { G } } ^ { } \\Phi _ { \\pi } ^ { \\psi } ( f ) d \\mu _ { \\pi } ( f \\in \\mathcal { C } _ { c } ( G ) ) , \\end{align*}"}
-{"id": "7711.png", "formula": "\\begin{align*} d _ B ^ 2 ( j , k ) = L ^ { 2 \\dagger } _ { j j } + L ^ { 2 \\dagger } _ { k k } - 2 L ^ { 2 \\dagger } _ { j k } = \\sum _ { n = 1 } ^ { N - 1 } \\frac { 1 } { \\lambda _ n ^ 2 } | u _ { n j } - u _ { n k } | ^ 2 \\ , , \\end{align*}"}
-{"id": "9430.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\mathcal { M } ( n ) \\psi ( n ) = \\infty . \\end{align*}"}
-{"id": "8532.png", "formula": "\\begin{align*} \\mathcal { K } _ { ( j , - l ) , ( k , - m ) } ( \\xi , \\eta ) = \\int e ^ { i x \\cdot ( \\eta - \\xi ) } \\overline { a } _ { ( j , - l ) } ( x , \\xi ) a _ { ( k , - m ) } ( x , \\eta ) d x . \\end{align*}"}
-{"id": "5356.png", "formula": "\\begin{align*} Y _ { P ( z ) } ' ( \\omega , x ) = \\sum _ { n \\in \\Z } L _ { P ( z ) } ' ( n ) x ^ { - n - 2 } . \\end{align*}"}
-{"id": "1074.png", "formula": "\\begin{align*} x y = \\omega , \\ \\ \\omega \\in R . \\end{align*}"}
-{"id": "9214.png", "formula": "\\begin{align*} [ z _ { 1 } \\otimes \\alpha _ { 1 } , [ z _ { 2 } \\otimes \\alpha _ { 2 } , u \\otimes b ] ] = [ [ z _ { 1 } \\otimes \\alpha _ { 1 } , z _ { 2 } \\otimes \\alpha _ { 2 } ] , u \\otimes b ] + [ z _ { 2 } \\otimes \\alpha _ { 2 } , [ z _ { 1 } \\otimes \\alpha _ { 1 } , u \\otimes b ] ] . \\end{align*}"}
-{"id": "5175.png", "formula": "\\begin{align*} V _ { 1 } ^ { [ r , 1 ] } ( x ) \\coloneqq \\sup \\limits _ { \\tau _ { 1 } \\in \\mathcal { T } } M ^ { x } _ { 1 } ( \\tau _ { 1 } , D _ { [ r , 1 ] } ) = M ^ { x } _ { 1 } ( D _ { [ 0 , \\ell _ { r } ] } , D _ { [ r , 1 ] } ) , \\forall x \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "7974.png", "formula": "\\begin{align*} J _ s ( t ) = Y _ 1 ( t ) ) + Y _ 2 ( 1 - t ) . \\end{align*}"}
-{"id": "4929.png", "formula": "\\begin{align*} \\mathrm { r a n k } \\ , ( Q _ k + Q _ N ) T P ^ \\perp = \\mathrm { r a n k } \\ , ( Q _ k + Q _ N ) [ R T R ^ \\perp ] P ^ \\perp \\le \\mathrm { r a n k } \\ , R T R ^ \\perp = 1 , \\end{align*}"}
-{"id": "1949.png", "formula": "\\begin{align*} \\widehat { \\widetilde { Q } _ { 2 } ( q ) } ( \\eta ) = ( i \\pi d + P ) \\widetilde { S } _ { r } ( q ) ( \\eta ) . \\end{align*}"}
-{"id": "8288.png", "formula": "\\begin{align*} C _ \\Phi ^ * = \\bigcup _ { \\Phi \\to \\Phi _ 1 } \\gamma ^ { - 1 } C _ { \\Phi _ 1 } \\gamma \\subset U _ \\Phi ( \\R ) ( - 1 ) , \\end{align*}"}
-{"id": "9078.png", "formula": "\\begin{align*} f ( p + q ) = f ( p ) + f ( q ) \\end{align*}"}
-{"id": "6569.png", "formula": "\\begin{align*} \\beta _ 1 : = \\beta _ { \\xi } = \\frac { 2 \\cdot \\frac { 4 - 2 \\varepsilon } { \\sqrt { 1 - \\varepsilon ^ 2 } } \\cdot 1 } { 2 \\cdot \\frac { 1 - \\varepsilon } { \\sqrt { 1 - \\varepsilon ^ 2 } } } = \\frac { 4 - 2 \\varepsilon } { 1 - \\varepsilon } . \\end{align*}"}
-{"id": "2040.png", "formula": "\\begin{align*} f _ { n + 1 , k } ( z _ 1 , . . . , z _ { n + 1 } ) = - \\sum _ { i = 1 } ^ { n + 1 } \\frac { g ( z _ 1 ^ { - 1 } . . . z _ i ^ k . . . z _ { n + 1 } ^ { - 1 } ) } { g ( z _ 1 ^ { - 1 } . . . z _ { n + 1 } ^ { - 1 } ) } f _ { n , k } ( z _ 1 , . . . , \\widehat { z _ i } , . . . , z _ { n + 1 } ) . \\end{align*}"}
-{"id": "7194.png", "formula": "\\begin{align*} ( \\widehat { \\mathcal { O } } ^ \\times _ F K ^ \\times / K ^ \\times , K ^ { \\mathrm { a b } } / K ) = \\bigcap _ i ( \\widehat { \\mathcal { O } } _ i ^ \\times K ^ \\times / K ^ \\times , K ^ { \\mathrm { a b } } / K ) . \\end{align*}"}
-{"id": "4428.png", "formula": "\\begin{align*} \\psi _ T ( x _ 1 , x _ 2 ) = \\frac { 1 } { ( T ^ { 1 / 3 } ) ^ { 1 + \\frac { 3 } { 2 } } } \\psi ( \\frac { x _ 1 } { T ^ { 1 / 3 } } , \\frac { x _ 2 } { ( T ^ { 1 / 3 } ) ^ \\frac { 3 } { 2 } } ) , \\forall x \\in \\R ^ 2 , \\end{align*}"}
-{"id": "2171.png", "formula": "\\begin{align*} 1 = \\| \\varphi \\| _ { L ^ 2 ( { \\bf R } ^ N , e ^ { | x | ^ 2 / 4 } \\ , d x ) } = \\| \\varphi _ * \\| _ { L ^ 2 ( { \\bf R } ^ N , \\ , \\nu e ^ { | x | ^ 2 / 4 } \\ , d x ) } = | { \\bf S } ^ { N - 1 } | ^ { 1 / 2 } \\| w ( 0 ) \\| _ { L ^ 2 ( { \\bf R } _ + , \\ , \\rho _ d \\ , d \\xi ) } . \\end{align*}"}
-{"id": "1869.png", "formula": "\\begin{align*} f _ j ( x ) : = h ( x ) \\ , \\ , f _ k ( x ) : = 0 x \\in \\R , \\ , a = 0 \\end{align*}"}
-{"id": "107.png", "formula": "\\begin{align*} \\Lambda _ 1 \\cap B _ { r ' } = \\Lambda _ 2 \\cap B _ { r ' } , | u _ 1 ( \\lambda ) - u _ 2 ( \\lambda ) | < \\varepsilon ' ( \\forall \\lambda \\in \\Lambda _ 1 \\cap B _ { r ' } ) \\end{align*}"}
-{"id": "34.png", "formula": "\\begin{align*} \\frac { \\partial J _ { M C C C } } { \\partial w ^ { * } } = \\frac { \\partial E _ { D Y } [ G ^ { C } _ { \\sigma \\ , \\sqrt { 2 } } ( e ) ] } { \\partial w ^ { * } } = E _ { D Y } \\left [ G ^ { C } _ { \\sigma \\ , \\sqrt { 2 } } ( e ) \\frac { \\partial ( e e ^ * ) } { \\partial w ^ { * } } \\right ] = \\textbf { 0 } \\end{align*}"}
-{"id": "8853.png", "formula": "\\begin{align*} \\limsup _ { N \\to \\infty } V ( X _ N , t N ^ { - \\frac 1 d } ) = \\mathcal { O } ( t ^ { d - 1 } ) \\quad t \\to \\infty . \\end{align*}"}
-{"id": "6861.png", "formula": "\\begin{align*} \\mu ( A ) = \\left ( \\frac { p } { 2 } \\right ) ^ { i ( A ) } \\left ( \\frac { 1 - p } { 2 } \\right ) ^ { m - i ( A ) } = \\frac { p ^ { i ( A ) } ( 1 - p ) ^ { m - i ( A ) } } { 2 ^ { m } } . \\end{align*}"}
-{"id": "6309.png", "formula": "\\begin{align*} \\xi _ U - \\xi _ O = \\Tilde { \\xi } _ U - \\Tilde { \\xi } _ O . \\end{align*}"}
-{"id": "4220.png", "formula": "\\begin{align*} a _ j = \\sum _ { r = j } ^ { k + 1 } ( - 1 ) ^ { r - j } e _ { r - j } ( 1 , \\dots , k ) \\mathcal { F } ( 2 k , j ) . \\end{align*}"}
-{"id": "9967.png", "formula": "\\begin{align*} \\bar { \\upsilon } _ { k i } ^ { s } = \\dfrac { \\rho _ { k i i } } { \\rho _ { k i i } + { \\displaystyle \\sum _ { j \\neq i } ^ { L } } \\rho _ { k j i } + \\sigma ^ { 2 } } = \\dfrac { \\rho _ { k i i } } { { \\displaystyle \\sum _ { j = 1 } ^ { L } } \\rho _ { k j i } + \\sigma ^ { 2 } } . \\end{align*}"}
-{"id": "228.png", "formula": "\\begin{align*} \\chi _ a | h | ^ 2 ( B _ G ) \\chi _ a & = \\chi _ a \\vert B _ G \\vert ^ { \\gamma _ h } \\widetilde { h } ^ 2 ( B _ G ) \\vert B _ G \\vert ^ { \\gamma _ h } \\chi _ a \\\\ & \\leq C _ h ^ 2 \\chi _ a \\vert B _ G \\vert ^ { 2 \\gamma _ h } \\chi _ a \\end{align*}"}
-{"id": "3956.png", "formula": "\\begin{align*} \\frac { \\partial Z _ j } { \\partial m _ { i } } = \\sum _ { S \\neq S _ 1 , \\ldots , S _ k } \\frac { \\partial Z _ { j } } { \\partial x _ S } \\frac { \\partial x _ S } { \\partial m _ { i } } \\end{align*}"}
-{"id": "9067.png", "formula": "\\begin{gather*} \\begin{pmatrix} \\frac { \\partial f } { \\partial t } & a _ 5 y _ 3 & a _ 6 y _ 3 & 0 \\\\ 0 & 0 & 0 & 0 \\end{pmatrix} \\end{gather*}"}
-{"id": "4754.png", "formula": "\\begin{align*} { L } _ i = \\left ( 0 , { L } _ 1 ( x ^ 1 ) , 0 , 0 \\right ) , \\alpha = \\beta = \\gamma = 0 . \\end{align*}"}
-{"id": "6996.png", "formula": "\\begin{align*} \\tilde { e } _ j = ( 0 , 0 , . . . , 0 , 1 , 0 , . . . , 0 ) \\ \\ j = 1 , 2 , . . . , n - 2 , \\end{align*}"}
-{"id": "4705.png", "formula": "\\begin{align*} m _ { v _ { r } } ( \\infty , \\infty ) & = 1 , \\\\ m _ { v _ { r } } ( a , \\infty ) & = m _ { v _ r } ( \\infty , a ) = | a | ^ { - r } , \\\\ m _ { v _ { r } } ( a , a ' ) & = \\max \\left \\{ \\frac { \\abs { a } } { \\abs { a ' } } , \\frac { \\abs { a ' } } { \\abs { a } } \\right \\} ^ { - r } . \\end{align*}"}
-{"id": "3783.png", "formula": "\\begin{align*} | C _ i | = L ^ d \\ ; \\ ; \\forall \\ ; i \\in I , \\ ; \\ ; \\bigcup _ { i \\in I } C _ i = \\Z ^ d C _ i \\cap C _ j = \\emptyset \\ ; \\ ; \\forall \\ ; i \\neq j \\in I . \\end{align*}"}
-{"id": "44.png", "formula": "\\begin{align*} \\hat { V } ^ { C } _ { \\sigma } ( C _ { 1 } , C _ { 2 } ) = \\frac { 1 } { 2 \\pi \\sigma ^ { 2 } } \\frac { 1 } { N } \\sum \\limits _ { n = 1 } ^ N e x p \\left ( - \\frac { ( x _ { n } - y _ { n } ) ^ { 2 } + ( z _ { n } - s _ { n } ) ^ { 2 } } { 2 \\sigma ^ 2 } \\right ) \\end{align*}"}
-{"id": "8627.png", "formula": "\\begin{align*} \\omega _ s = \\left \\{ \\begin{array} { l l } x _ 0 ( s ) & s \\in [ 0 , \\tau ] , \\\\ \\omega _ { \\tau + k - 1 } + x _ k ( s - \\tau - k + 1 ) & s \\in [ \\tau + k - 1 , \\tau + k ] , k = 1 , \\ldots , N , \\\\ \\omega _ { \\tau + N } + x _ { N + 1 } ( s - \\tau - N ) & s \\in [ \\tau + N , T ] , \\end{array} \\right . \\end{align*}"}
-{"id": "5721.png", "formula": "\\begin{align*} d _ X ( { v } , \\Sigma ) = d _ { L ^ 2 } ( { v } , \\mathcal { C } ( z ^ - ) \\cup \\mathcal { C } ( z ^ + ) ) . \\end{align*}"}
-{"id": "6959.png", "formula": "\\begin{align*} F _ { 2 , s } [ u ] ( x ) = F _ { 2 , s } ^ { \\epsilon _ 0 } [ u ] ( x ) \\end{align*}"}
-{"id": "3946.png", "formula": "\\begin{align*} & G _ k \\subset \\bigcup _ i Q ( x _ { i } , t _ { i } ; \\ , { 5 r } ) , \\\\ & ( x _ { i } , t _ { i } ) \\in G _ k , \\\\ & Q ( x _ { m } , t _ { m } ; \\ , { r } ) \\cap Q ( x _ { n } , t _ { n } ; \\ , { r } ) = \\varnothing , \\ m \\neq n , \\\\ & r \\leq \\varepsilon ^ { - 1 } \\iint _ { Q ( x _ i , t _ i ; \\ , { r } ) } | \\nabla u | ^ 2 d x d s . \\end{align*}"}
-{"id": "1489.png", "formula": "\\begin{align*} \\sqrt { c _ \\alpha } r \\cot ( \\sqrt { c _ \\alpha } r ) = { \\rm R e } ( \\sqrt { c _ \\alpha } r \\cot ( \\sqrt { c _ \\alpha } r ) ) > \\cfrac { \\alpha + 1 } { 2 } . \\end{align*}"}
-{"id": "6668.png", "formula": "\\begin{align*} S _ { \\delta } = \\bigcap _ { x \\in \\partial S } \\{ y \\in \\mathbb { R } ^ n : \\langle y - x , u ( x ) \\rangle \\leq - \\Delta _ x ( \\delta ) \\} , \\end{align*}"}
-{"id": "5455.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty e ^ { - s x } \\dd g ( x ) = \\int _ 0 ^ \\infty e ^ { - s x } \\dd [ r ^ \\rho g ( x / r ) ] . \\end{align*}"}
-{"id": "3720.png", "formula": "\\begin{align*} \\begin{aligned} p _ 2 & = \\omega p _ 1 , p _ 3 = \\omega ^ 2 p _ 1 , & p ' _ 1 & = p ' _ 2 = p ' _ 3 \\\\ p _ 5 & = \\omega p _ 4 , p _ 6 = \\omega ^ 2 p _ 4 , & p ' _ 4 & = p ' _ 5 = p ' _ 6 \\\\ p _ 8 & = \\omega p _ 7 , p _ 9 = \\omega ^ 2 p _ 7 , & p ' _ 7 & = p ' _ 8 = p ' _ 9 \\end{aligned} \\end{align*}"}
-{"id": "8826.png", "formula": "\\begin{align*} Z _ { \\Phi _ { y , p } } ( p , \\chi _ { } , 0 , f _ j ) & + _ { t ^ { m - 1 } } \\Big ( \\frac { ( t - p ) Z _ { \\Phi _ { y , p } } ( p , \\chi _ { } , s , f _ j ) } { ( p - 1 ) ( 1 - t ) } \\Big ) \\\\ & + \\sum _ { \\substack { \\chi \\neq \\chi _ { } , \\\\ c ( \\chi ) = 1 } } g _ { \\chi ^ { - 1 } } \\chi ( u ) _ { t ^ { m - 1 } } ( Z _ { \\Phi _ { y , p } } ( p , \\chi , s , f _ j ) ) . \\end{align*}"}
-{"id": "2944.png", "formula": "\\begin{align*} \\varrho ( H ) : = \\max _ { v \\in V ( H ) } \\partial \\theta ( v ) , \\end{align*}"}
-{"id": "8042.png", "formula": "\\begin{align*} M - B ( p , R _ M / 2 + \\delta ) & \\subset A \\subset \\mathrm { g e x p } _ p ^ { - D ^ 2 } ( A _ T ) . \\end{align*}"}
-{"id": "8832.png", "formula": "\\begin{align*} \\| u \\| _ { Y ^ { s , b } _ T } : = \\inf \\left \\{ \\| v \\| _ { Y ^ { s , b } } \\colon \\ v | _ { [ 0 , T ] } = u \\right \\} , \\end{align*}"}
-{"id": "3471.png", "formula": "\\begin{align*} \\| E ^ n \\| ^ 2 _ H = \\| \\Theta ^ n \\| ^ 2 _ H + \\| \\Xi ^ n \\| _ H ^ 2 \\end{align*}"}
-{"id": "6116.png", "formula": "\\begin{align*} A _ { \\delta , T } ^ { n } = \\left \\{ \\omega : \\sup _ { 1 \\leq i \\leq i _ { f } ^ { n } } \\left | T _ { i } ^ { n } - T _ { i - 1 } ^ { n } \\right | < \\delta \\right \\} . \\end{align*}"}
-{"id": "9766.png", "formula": "\\begin{align*} E _ { i } ( h , \\theta ) = \\tilde { E } _ { i } ( h , \\theta ) - \\int \\limits _ { y _ { i , s } } ^ { y _ { i } } G ( U _ { h , \\theta } ( i h - , y _ { i , n } ) ) h d y . \\end{align*}"}
-{"id": "3534.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l l } \\beta ^ { 0 , \\theta } = \\displaystyle \\frac { \\big { [ } J ( u ^ { \\theta } ( \\cdot ) ) + \\theta \\big { ] } ^ + } { J ^ { \\theta } ( u ^ { \\theta } ( \\cdot ) ) } , \\\\ \\beta ^ { j , \\theta } = \\displaystyle \\frac { - \\big { [ } - X ^ { \\theta } ( t _ j ) \\big { ] } ^ + } { J ^ { \\theta } ( u ^ { \\theta } ( \\cdot ) ) } , \\ j = 1 , 2 , \\cdots , n . \\\\ \\end{array} \\end{align*}"}
-{"id": "3805.png", "formula": "\\begin{align*} \\hat { Z } : = \\max \\left \\{ z < - 2 \\ell _ L \\colon \\ , N ( x , 0 ) = 0 \\ ; \\forall \\ ; x \\in \\Z , | x - z | \\le 2 \\ell _ L \\right \\} \\end{align*}"}
-{"id": "3105.png", "formula": "\\begin{align*} h _ { 0 , 0 } & = 2 \\left [ W \\left ( \\frac { 1 - ( 2 p ) ^ m } { 1 - 2 p } \\right ) \\right . \\\\ & + \\left . \\frac { 1 - p ^ m } { 1 - p } + \\frac { ( 2 ^ m W + 1 + 2 \\beta p ) p ^ m } { 1 - p + \\alpha \\beta p } \\right ] ^ { - 1 } . \\end{align*}"}
-{"id": "8367.png", "formula": "\\begin{align*} \\vartheta ^ { [ 2 ] } - \\vartheta ^ { [ 1 ] } = 2 4 \\Delta . \\end{align*}"}
-{"id": "1998.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { t _ { n } - 1 } \\mathbb { P } \\left \\{ \\sup _ { \\mathbf { s } \\in \\left [ \\mathbf { a } _ { k } - h ^ { \\ast } , \\mathbf { a } _ { k } + h ^ { \\ast } \\right ] \\cap \\left [ 0 , 1 \\right ] } \\left \\vert \\widetilde { B } _ { n } \\left ( \\mathbf { a } _ { k } \\right ) - \\widetilde { B } _ { n } \\left ( \\mathbf { s } _ { k } \\right ) \\right \\vert \\geq d k ^ { 1 / 4 - \\nu - \\delta } h _ { \\ast } ^ { 1 / 2 } \\right \\} . \\end{align*}"}
-{"id": "1859.png", "formula": "\\begin{align*} \\mathcal { F } ^ { S , \\pi } ( F _ 1 ^ \\ast , \\dots , F _ d ^ \\ast ) : = \\big \\{ F \\in \\mathcal { F } ( F _ 1 ^ \\ast , \\dots , F _ d ^ \\ast ) \\colon F ( s ) = \\pi _ s s \\in S \\big \\} , \\end{align*}"}
-{"id": "5254.png", "formula": "\\begin{align*} \\begin{aligned} E ^ s ( \\lambda , z ) = \\mathrm { s p } \\{ u _ 1 ( \\lambda , z ) , u _ 2 ( \\lambda , z ) \\} \\\\ E ^ u ( \\lambda , z ) = \\mathrm { s p } \\{ u _ 3 ( \\lambda , z ) , u _ 4 ( \\lambda , z ) \\} \\end{aligned} . \\end{align*}"}
-{"id": "2176.png", "formula": "\\begin{align*} | G _ d ( r , t ) | & \\le C t ^ { - \\frac { d } { 2 } - 2 } \\int _ 0 ^ r s ^ { 1 - d } [ \\nu _ d ( s ) ] ^ { - 1 } \\left ( \\int _ 0 ^ s \\tau ^ { d + 1 } \\nu _ d ( \\tau ) \\ , d \\tau \\right ) \\ , d s \\\\ & \\le C t ^ { - \\frac { d } { 2 } - 2 } \\int _ 0 ^ r s ^ { 1 - d } [ \\nu _ d ( s ) ] ^ { - 1 } \\cdot s ^ { d + 2 } \\nu _ d ( s ) \\ , d s \\le C t ^ { - \\frac { d } { 2 } - 2 } r ^ 4 \\end{align*}"}
-{"id": "4019.png", "formula": "\\begin{align*} X ^ t = ( 1 + t g \\lambda _ 1 ) X + t X ( g ) \\eta \\ \\ \\mathrm { a n d } \\\\ Y ^ t = ( 1 + t g \\lambda _ 2 ) Y + t Y ( g ) \\eta , \\end{align*}"}
-{"id": "5362.png", "formula": "\\begin{align*} M _ { r , s } \\boxtimes _ { P ( 1 ) } M _ { r ' , s ' } \\cong \\coprod _ { r '' = 1 } ^ { a - 1 } \\coprod _ { s '' = 1 } ^ { b - 1 } N ^ a _ { r , r ' , r '' } N ^ b _ { s , s ' , s '' } M _ { r '' , s '' } . \\end{align*}"}
-{"id": "1581.png", "formula": "\\begin{align*} V ( \\mathbf { j } ) & = \\sum _ { k = 1 } ^ { n - 1 } [ ( n - k + 1 ) j _ k - s ( j _ k ) ] - c ( j _ n , - \\alpha ( n ) - 1 ) \\\\ & \\leq \\sum _ { k = 1 } ^ { n - 1 } [ ( n - k + 1 ) j _ k - s ( j _ k ) ] ] = v ( \\mathbf { j } ) \\\\ & < v ( \\mathbf { j } ' ) = V ( \\mathbf { j } ' ) \\end{align*}"}
-{"id": "7413.png", "formula": "\\begin{align*} \\mathbb { H } _ { \\Gamma } : = \\frac { \\mathbb { C } [ t _ 0 , t _ 1 , t _ 2 , t _ 3 , t _ 4 , t _ 5 , t _ 6 ] } { ( t _ 0 + 2 t _ 1 + t _ 2 + 2 t _ 3 + t _ 4 + 2 t _ 5 + 3 t _ 6 ) } . \\end{align*}"}
-{"id": "5704.png", "formula": "\\begin{align*} 0 = \\int _ { \\R ^ 2 } - { u } \\cdot \\Delta P \\varphi + \\nabla W ( { u } ) \\cdot P \\varphi = \\int _ { \\R ^ 2 } - P { u } \\cdot \\Delta \\varphi + P \\nabla W ( { u } ) \\cdot \\varphi , \\end{align*}"}
-{"id": "1863.png", "formula": "\\begin{align*} A ^ i : = ( - \\infty , A ^ i _ 1 ] \\times \\cdots \\times ( - \\infty , A ^ i _ d ] \\subset \\mathbb { R } ^ d \\end{align*}"}
-{"id": "2194.png", "formula": "\\begin{align*} q _ i ( z _ 1 , \\ldots , z _ n ) = ( 1 - a _ { i 1 } z _ 1 ) ^ { m _ { i 1 } } \\cdot \\ldots \\cdot ( 1 - a _ { i n } z _ n ) ^ { m _ { i n } } \\end{align*}"}
-{"id": "10083.png", "formula": "\\begin{align*} \\tilde { R } ( X , Y ) Z = R ( X , Y ) Z + \\beta ( X , Y ) Z + \\theta ( X , Z ) Y - \\theta ( Y , Z ) X , \\end{align*}"}
-{"id": "2268.png", "formula": "\\begin{align*} \\psi ( x ) = \\int _ { \\mathbb { R } ^ + } \\Omega _ { p ^ * } ( x r ^ 2 ) \\textnormal { d } F ( r ) . \\end{align*}"}
-{"id": "2309.png", "formula": "\\begin{align*} & \\ ; \\| A ^ { ( r _ { k + 1 } - s ) / 2 } w _ k ( t ) \\| _ { L ^ 2 } ^ 2 + \\int _ { t _ k } ^ { t } \\| A ^ { r _ { k + 1 } / 2 } w _ k \\| _ { L ^ 2 } ^ 2 \\\\ \\leq & \\ ; C \\| u _ 0 \\| _ { D ( A ) } ^ 2 \\left ( 1 + \\int _ { t _ k } ^ t \\| A ^ { r _ k / 2 } u _ * \\| _ { L ^ 2 } ^ 2 \\ , d \\tau \\right ) , \\end{align*}"}
-{"id": "6505.png", "formula": "\\begin{align*} \\tan \\theta _ { \\mathrm { o } } \\overset { k _ { \\mathrm { o } } L = p _ { \\mathrm { o } } L / \\hbar \\ll 1 } { \\approx } \\theta _ { \\mathrm { o } } \\approx - \\frac { \\rho \\left ( k _ { \\mathrm { o } } L \\right ) ^ { 3 } } { 3 } = - \\frac { 2 \\mu V k _ { \\mathrm { o } } L ^ { 3 } } { 3 \\hbar ^ { 2 } } = - \\frac { 2 \\mu V p _ { \\mathrm { o } } L ^ { 3 } } { 3 \\hbar ^ { 3 } } , \\end{align*}"}
-{"id": "8437.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { - 2 , l , v } ) = q \\zeta _ F ( 1 ) ^ { - 2 } \\int _ { ( \\mathcal { O } ^ { \\times } ) ^ 3 } \\chi ( y _ 1 y _ 2 ) \\psi ( y _ 1 \\varpi ^ { - 1 } + y _ 2 \\varpi ^ { - 1 } + y _ 3 v \\varpi ^ { - l } ) \\sum _ { \\mu \\in \\mathfrak { X } _ l } \\mu ( y _ 1 y _ 2 y _ 3 ^ { - 1 } ) d ^ { \\times } y _ 3 d ^ { \\times } y _ 2 d ^ { \\times } y _ 1 . \\end{align*}"}
-{"id": "2810.png", "formula": "\\begin{align*} \\frac { 1 } { s - 1 } = \\frac { 2 r } { \\sigma ^ { 2 } } \\int _ { 0 } ^ { \\infty } \\exp \\Big ( \\frac { 2 r } { \\sigma ^ { 2 } } t - s { b } ( t ) - s ( s + \\frac { 2 r } { \\sigma ^ { 2 } } - 1 ) \\Big ) \\mathrm { d } t , \\Re ( s ) > 1 , \\end{align*}"}
-{"id": "5860.png", "formula": "\\begin{align*} A ' = - \\theta ^ { - 1 } _ { Y Y } \\theta _ { X Y } . \\end{align*}"}
-{"id": "1801.png", "formula": "\\begin{align*} d ( G ) : = \\lim _ { t \\to \\infty } \\frac { \\log \\deg ( R ( t ) ) } { t } . \\end{align*}"}
-{"id": "7822.png", "formula": "\\begin{align*} \\bar { F } ( r ) = \\int _ { S _ r } [ \\frac { 1 } { 2 } \\bar { q } \\bar { v } ^ 2 + | \\frac { \\partial \\bar { v } } { \\partial r } | ^ 2 - \\frac { 1 } { 2 } | \\nabla \\bar { v } | ^ 2 ] e ^ { - 2 \\rho } d x , \\end{align*}"}
-{"id": "5836.png", "formula": "\\begin{align*} \\psi : M _ x \\times \\R _ \\alpha ^ { n - d } \\to \\R ^ n , \\psi ( x , \\alpha ) = x + \\sum _ { j = 1 } ^ { n - d } \\alpha _ j w _ j ( x ) , \\end{align*}"}
-{"id": "7493.png", "formula": "\\begin{align*} ( \\bar { \\partial } ^ { * h } \\Phi ) _ { A _ p \\bar { \\beta } _ 2 \\dots \\bar { \\beta } _ q } = ( - 1 ) ^ { p + 1 } h ^ { \\bar { \\varepsilon } \\gamma } \\delta _ \\gamma ( \\phi _ { A _ p \\bar { \\varepsilon } \\bar { \\beta } _ 2 \\dots \\bar { \\beta } _ q } ) . \\end{align*}"}
-{"id": "2440.png", "formula": "\\begin{align*} \\ell \\frac { n - 2 q } { q ^ 2 } = \\frac { n / d - c _ 1 - c _ 2 } { c _ 1 c _ 2 } . \\end{align*}"}
-{"id": "2432.png", "formula": "\\begin{align*} v ^ i ( x _ 1 ^ \\ast , y _ 1 ^ \\ast ) & = \\sum \\limits _ { x _ 2 , \\dots , x _ { k - 1 } \\in X } p ( x _ { k - 1 } | x _ { k - 2 } , \\mu _ { k - 2 } ^ \\ast ) \\dots p ( x _ 2 | x _ 1 ^ \\ast , \\mu _ 1 ^ \\ast ) v ^ i ( x _ { k - 1 } , y _ { k - 1 } ^ \\ast ) . \\end{align*}"}
-{"id": "3510.png", "formula": "\\begin{align*} \\begin{array} [ c ] { r l } & X ^ { u ^ 0 } ( t _ i ) - X ^ u ( t _ i ) - X ^ { u ^ 0 } ( t _ { i - 1 } ) + X ^ u ( t _ { i - 1 } ) \\\\ = & \\displaystyle \\int _ { t _ { i - 1 } } ^ { t _ i } \\left ( b ( X ^ { u ^ 0 } ( t ) , u ^ 0 ( t ) ) - b ( X ^ { u } ( t ) , u ^ 0 ( t ) ) \\right ) d t \\\\ & + \\displaystyle \\int _ { t _ { i - 1 } } ^ { t _ i } \\left ( b ( X ^ { u } ( t ) , u ^ 0 ( t ) ) - b ( X ^ { u } ( t ) , u ( t ) ) \\right ) d t , \\end{array} \\end{align*}"}
-{"id": "952.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { N _ n } E \\left [ \\max _ { 1 \\leq k \\leq d _ n } \\left | \\sum _ { j = 1 } ^ { N _ n } \\gamma _ { n , k } ( i , j ) W ^ { ( i ) } _ j \\right | ^ 3 \\right ] \\leq d _ n ^ { 3 / p } \\eta ^ { - 3 } \\sum _ { i = 1 } ^ { N _ n } \\max _ { 1 \\leq k \\leq d _ n } \\left ( \\sum _ { j = 1 } ^ { N _ n } \\gamma _ { n , k } ( i , j ) ^ 2 \\right ) ^ { 3 / 2 } . \\end{align*}"}
-{"id": "7353.png", "formula": "\\begin{align*} \\begin{aligned} \\Vert & \\partial _ t u _ n | _ \\Omega \\Vert _ { L ^ 1 ( [ 0 , T ] , H ^ { - 1 } ( \\Omega ) ) } \\ ; \\lesssim _ { A , T } \\ ; 1 \\ , . \\end{aligned} \\end{align*}"}
-{"id": "7033.png", "formula": "\\begin{align*} B = \\int _ { 0 } ^ { \\infty } { \\int _ { \\{ x \\in \\partial B _ 1 ^ { n - 1 } ( 0 ) , u ( r x ) - u ( 0 ) \\leq 0 \\} } { \\frac { u ( r x ) - u ( 0 ) } { r ^ { 1 + 2 s } } d x } d r } \\leq 0 . \\end{align*}"}
-{"id": "7947.png", "formula": "\\begin{align*} o ( n _ 0 ) = \\lambda _ { \\max } \\varphi ( \\nu ^ { n _ 0 } | V | ) , \\end{align*}"}
-{"id": "2627.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\tilde h _ { \\rho } ( \\tilde t ) = C ( t _ 1 , t _ 2 ) , \\\\ & \\tilde h _ { \\rho } ' ( t _ 1 - ) \\leq 0 \\leq \\tilde h _ { \\rho } ' ( t _ 2 + ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "4045.png", "formula": "\\begin{align*} c = \\frac { 1 } { \\lambda _ 1 } + \\frac { 1 } { \\mu _ 2 } \\geq \\frac { c } { 2 } + \\frac { 1 } { \\mu _ 2 } , \\end{align*}"}
-{"id": "9747.png", "formula": "\\begin{align*} L ^ { 2 } ( J ) - L ^ { 2 } ( I ) \\leq ( | K _ { b 0 } | - K ^ { * } _ { 2 0 } ) | \\omega _ { k } | + \\sum \\limits _ { i = 2 , 3 , 5 } ( | K _ { b i } | - K ^ { * } _ { 2 i } ) | \\alpha _ i | . \\end{align*}"}
-{"id": "391.png", "formula": "\\begin{align*} S & = \\sum _ { n = 3 } ^ \\infty \\frac { 1 } { n ( \\ln \\ln n ) ^ b } \\mathbb P \\left ( | S _ n | > ( 1 + \\varepsilon ) \\sigma _ n \\sqrt { 2 \\ln \\ln n } \\right ) \\\\ & \\propto \\sum _ { n = 3 } ^ \\infty \\frac { 1 } { n ( \\ln \\ln n ) ^ { b + 1 / 2 } } ( \\ln n ) ^ { - ( 1 + \\varepsilon ) ^ 2 } . \\end{align*}"}
-{"id": "4472.png", "formula": "\\begin{align*} \\sum _ { k \\not = 0 } \\exp ( - 2 T d ^ 3 ( k , 0 ) ) d ^ { - 1 } ( k , 0 ) \\lesssim ( T ^ \\frac { 1 } { 3 } ) ^ { - \\frac { 3 } { 2 } } , \\\\ \\sum _ { k \\not = 0 } \\exp ( - 2 T d ^ 3 ( k , 0 ) ) d ( k , 0 ) \\lesssim ( T ^ \\frac { 1 } { 3 } ) ^ { - \\frac { 7 } { 2 } } , \\end{align*}"}
-{"id": "9681.png", "formula": "\\begin{align*} S _ i ( U _ a ) : \\ , \\ , [ p ] = \\frac { \\hat { c } _ a ^ 2 } { \\hat { \\gamma } } [ \\rho ] , \\ , \\ , [ u ] = - s _ i [ v ] , \\ , \\ , \\rho _ a ( s _ i u _ a - v _ a ) [ v ] = [ p ] , \\ , \\ , [ Z ] = 0 . \\end{align*}"}
-{"id": "2588.png", "formula": "\\begin{align*} { \\overline \\rho } ( t ) = C _ i | t - t _ i | ^ { \\tfrac { 1 + a } 2 } + R _ i ( | t - t _ i | ) \\end{align*}"}
-{"id": "2345.png", "formula": "\\begin{align*} z \\mapsto \\alpha _ s ( g ) f _ b ( z ) = \\big ( ( g ^ { - 1 } ) ' ( z ) \\big ) ^ s \\chi ( g ) f _ b ( g ^ { - 1 } . z ) \\end{align*}"}
-{"id": "3837.png", "formula": "\\begin{align*} \\xi ( z , i , n + 1 ) = \\left \\{ \\begin{array} { l l } 1 & \\begin{array} { l } i \\le N ( z , 0 ) \\\\ \\exists \\ , z ' \\in \\Z , i ' \\in \\N \\eta ( z ' , i ' , n ) = 1 , S ^ { z ' , i ' } _ n = S ^ { z , i } _ n , \\end{array} \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "8475.png", "formula": "\\begin{align*} \\sum _ { j \\geq 2 } - v \\left ( - \\frac { v } { b } x \\right ) ^ { j - 1 } \\frac { ( - 1 ) ^ j A _ { \\pm } ^ j - A _ { \\mp } ^ j } { \\varpi ^ { l + \\frac { t } { 2 } } } \\varpi ^ { - r j - ( j - 1 ) \\frac { t } { 2 } } = 0 . \\end{align*}"}
-{"id": "9041.png", "formula": "\\begin{align*} { \\mathcal H } ( w , w ' ) : = ( \\alpha \\bar w _ 1 w _ 1 ' + \\beta \\bar w _ 2 w _ 2 ' , \\gamma \\bar w _ 1 w _ 1 ' + \\delta \\bar w _ 2 w _ 2 ' ) . \\end{align*}"}
-{"id": "127.png", "formula": "\\begin{align*} [ B , A ] ( t , x , y ) & = \\left ( \\frac { \\partial b _ 1 ( 1 - t , x , y ) } { \\partial x } + \\frac { \\partial b _ 1 ( 1 - t , x , y ) } { \\partial y } \\right ) \\frac { \\partial } { \\partial x } \\\\ & + \\left ( \\frac { \\partial b _ 2 ( 1 - t , x , y ) } { \\partial x } + \\frac { \\partial b _ 2 ( 1 - t , x , y ) } { \\partial y } \\right ) \\frac { \\partial } { \\partial y } . \\end{align*}"}
-{"id": "2492.png", "formula": "\\begin{align*} \\left ( - i \\xi _ { 1 } \\left ( - \\left | \\eta \\right | \\right ) + L \\right ) \\overline { \\psi _ { j } \\left ( \\left | \\eta \\right | \\right ) } = \\overline { \\varrho _ { j } \\left ( \\left | \\eta \\right | \\right ) } \\ , \\overline { \\psi _ { j } \\left ( \\left | \\eta \\right | \\right ) } . \\end{align*}"}
-{"id": "6946.png", "formula": "\\begin{align*} H ( j ) ( [ a ] [ z ] ) & = H ( j ) ( [ ( a \\oplus e _ { k - m } ) ( z \\oplus e _ { k - n } ) ] ) \\\\ & = H ( i _ k ) ( ( a \\oplus e _ { k - m } ) ( z \\oplus e _ { k - n } ) ) \\\\ & = \\alpha _ k ( a \\oplus e _ { k - m } ) ( z \\oplus e _ { k - n } ) \\ , , \\end{align*}"}
-{"id": "4900.png", "formula": "\\begin{align*} \\langle T T ^ * e , e \\rangle = \\langle T ^ * T e , e \\rangle . \\end{align*}"}
-{"id": "1056.png", "formula": "\\begin{align*} V _ { k , j } ( T ) = T ^ { j } P _ { k } ( T ) , 0 \\leq j \\leq n - 2 , \\end{align*}"}
-{"id": "1655.png", "formula": "\\begin{align*} \\tau _ { f _ 1 } \\circ \\tau _ { e } = \\tau _ { e } \\circ \\tau _ { f _ 2 } , \\tau _ { e } \\circ \\tau _ { f _ 1 } = \\tau _ { f _ 2 } \\circ \\tau _ { e } . \\end{align*}"}
-{"id": "1330.png", "formula": "\\begin{align*} I _ G ( p , m ) : = \\int _ { \\sigma _ G } \\omega _ G ( p , m ) , \\end{align*}"}
-{"id": "9800.png", "formula": "\\begin{align*} C _ { d a } = B _ { d a } - \\beta B _ { d s } \\quad C _ { d \\bar s } = B _ { d \\bar s } + \\varepsilon \\beta B _ { d \\bar a } s \\neq n + 1 , \\end{align*}"}
-{"id": "2132.png", "formula": "\\begin{align*} A ^ \\pm ( \\lambda ) : = \\frac { - ( N - 2 ) \\pm \\sqrt { ( N - 2 ) ^ 2 + 4 \\lambda } } { 2 } \\quad \\mbox { f o r } \\quad \\lambda \\ge \\lambda _ * . \\end{align*}"}
-{"id": "1807.png", "formula": "\\begin{align*} ( X ' , Q ' ) : = \\mu _ { k _ m } \\circ \\mu _ { k _ { m - 1 } } \\circ \\dots \\circ \\mu _ { k _ 1 } ( X , Q ) . \\end{align*}"}
-{"id": "6580.png", "formula": "\\begin{align*} K _ 0 = [ f ] _ { \\alpha , \\Omega _ 0 } , K _ k = [ f ] _ { \\alpha , \\mathrm { c l } ( \\Omega _ k ) } , k = 1 , 2 , \\ldots , m \\end{align*}"}
-{"id": "2042.png", "formula": "\\begin{align*} f _ { n + 1 , k } ( z _ 1 , . . . , z _ n , 1 ) = ( k - n ) f _ { n , k } ( z _ 1 , . . . , z _ n ) \\end{align*}"}
-{"id": "9113.png", "formula": "\\begin{align*} a _ { n + 1 } \\cdot \\sum _ { i = 0 } ^ { n } ( - 1 ) ^ { i } \\sum _ { \\mathrm { c a r d } ( I ) = i } \\left ( \\prod _ { j \\in I } x _ { j } \\right ) \\cdot A \\left ( \\prod _ { k \\in \\left \\{ 1 , \\ldots , n + 1 \\right \\} \\setminus I } x _ { k } \\right ) = 0 \\left ( x \\in R \\right ) . \\end{align*}"}
-{"id": "1376.png", "formula": "\\begin{align*} A = U [ \\Sigma \\ , \\ , \\ , \\ , \\ , 0 ] R ^ T \\ , , \\end{align*}"}
-{"id": "5108.png", "formula": "\\begin{align*} & k _ 1 \\phi _ { 1 , s } \\ln ^ { \\frac 1 2 } \\left ( \\frac { 2 } { \\phi _ { 1 , s } } \\right ) \\leq w \\leq k _ 2 \\phi _ { 1 , s } \\ln ^ { \\frac 1 2 } \\left ( \\frac { 2 } { \\phi _ { 1 , s } } \\right ) , \\ ; \\ ; q = 1 \\\\ & k _ 1 \\phi _ { 1 , s } ^ { \\frac { 2 } { q + 1 } } \\leq w \\leq k _ 2 \\phi _ { 1 , s } ^ { \\frac { 2 } { q + 1 } } , \\ ; \\ ; q > 1 \\end{align*}"}
-{"id": "3907.png", "formula": "\\begin{align*} \\cal { E } = - 4 K = 4 e ^ 2 . \\end{align*}"}
-{"id": "302.png", "formula": "\\begin{align*} K & : = \\left \\{ ( u _ 1 , \\dots , u _ d ) \\in \\mathcal { S } _ d : s \\le \\frac { u _ j } { 1 - \\sum _ { l = j + 1 } ^ d u _ l } \\le t , j = 1 , 2 , \\dots , d \\right \\} = T \\left ( [ s , t ] ^ d \\right ) . \\end{align*}"}
-{"id": "559.png", "formula": "\\begin{align*} \\imath ^ * ( \\lambda ) = c \\lambda ' + d f \\end{align*}"}
-{"id": "8584.png", "formula": "\\begin{align*} \\int _ 0 ^ \\epsilon \\int _ { B _ { H _ 0 } ( 0 , R ) } \\left [ | u _ { m , n _ m } | + H ( \\nabla u _ { m , n _ m } ) | \\right ] \\ , d y \\ , d s \\le C [ \\epsilon + \\epsilon ^ { \\sigma ' } ] \\end{align*}"}
-{"id": "63.png", "formula": "\\begin{align*} \\frac { \\partial ( e e ^ * ) } { \\partial w ^ * } = \\frac { \\partial ( D - \\textbf { w } ^ { H } \\textbf { X } ) ( D - \\textbf { w } ^ { H } \\textbf { X } ) ^ * } { \\partial w ^ * } \\end{align*}"}
-{"id": "2086.png", "formula": "\\begin{align*} | u ^ a _ 0 ( p , t ) - \\phi ^ a ( p ) | = & \\left | \\int _ M H ( p , q , t ) \\big ( \\phi ^ a ( q ) - \\phi ^ a ( p ) \\big ) d V _ q \\right | \\\\ \\leq & \\tilde { C } _ \\beta t ^ \\beta ( | | \\nabla _ H \\phi ^ a | | _ { C ^ 0 } + | | \\phi ^ a | | _ { C ^ 0 } ) \\end{align*}"}
-{"id": "2644.png", "formula": "\\begin{align*} \\lambda _ i = \\begin{cases} \\frac { a + 1 } { n } , & \\mbox { i f } i = 1 \\\\ 0 , & \\mbox { i f } 2 \\leq i \\leq a + 1 \\\\ \\frac { 1 } { n } , & \\mbox { i f } a + 2 \\leq i \\leq n \\end{cases} . \\end{align*}"}
-{"id": "2560.png", "formula": "\\begin{align*} | B _ { 3 / 4 } \\setminus E | \\le | B _ { 1 } | - | E | \\le \\pi - \\frac { \\pi } { ( 1 + \\delta ) ^ { 2 } } = \\frac { 2 + \\delta } { ( 1 + \\delta ) ^ { 2 } } \\pi \\delta \\le \\frac { 2 + \\delta } { 1 + \\delta } \\pi \\delta \\le 2 \\pi \\delta \\le \\frac { \\pi ( 3 / 4 ) ^ { 2 } } { 3 6 } \\ , , \\end{align*}"}
-{"id": "8129.png", "formula": "\\begin{align*} \\mu ( \\tilde { B } ) = \\mu ( B ) - \\mu ( C ) \\geq \\mu ( A ) + \\frac { \\eta } { 2 } > \\mu ( A ) \\end{align*}"}
-{"id": "4401.png", "formula": "\\begin{align*} \\| A _ x \\| \\leq \\sup _ { \\gamma } \\| \\widetilde { C } _ { z _ \\gamma } ( A + K ) \\widetilde { C } _ { - z _ \\gamma } \\| = \\| A + K \\| . \\end{align*}"}
-{"id": "6820.png", "formula": "\\begin{align*} \\Delta \\phi + \\frac { 8 } { ( 1 + | z | ^ 2 ) ^ 2 } \\phi = 0 . \\end{align*}"}
-{"id": "5573.png", "formula": "\\begin{align*} \\Delta \\left ( \\alpha , \\beta \\right ) = \\sum _ { n = 0 } ^ { 2 ^ { k } - 1 } \\left ( b _ { n } + c _ { n } \\right ) \\end{align*}"}
-{"id": "1895.png", "formula": "\\begin{align*} H ( x _ 1 , \\dots , x _ d ) \\le \\min _ { i = 1 , \\dots , d } F ^ \\ast _ i ( x _ i ) \\wedge \\min \\{ \\pi _ s : s \\in S x \\le s \\} , \\end{align*}"}
-{"id": "9000.png", "formula": "\\begin{gather*} \\sum _ { \\pi \\in S _ n } \\pi \\frac { 1 } { \\prod _ { 1 \\le i < j \\le n } \\vartheta ( z _ i \\pm z _ j ) } D ' _ 1 ( z _ 1 ) \\cdots D ' _ n ( z _ n ) = \\frac { 1 } { \\prod _ { 1 \\le i < j \\le n } \\vartheta ( z _ i \\pm z _ j ) } \\det _ { 1 \\le i , j \\le n } D ' _ i ( z _ j ) \\end{gather*}"}
-{"id": "8469.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { t , l , v } ) = \\begin{cases} \\chi ^ 2 ( x _ 0 ) \\psi ( ( x _ 0 - b ) \\varpi ^ { - \\frac { k } { 2 } } ) & t = - k , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "7773.png", "formula": "\\begin{align*} | \\ell - m + u _ { R } ( \\ell ) - u _ { R } ( m ) | = | \\ell - m | \\geq \\frak { m } | \\ell - m | . \\end{align*}"}
-{"id": "4805.png", "formula": "\\begin{align*} \\det ( A _ { \\ell \\rightarrow - b } ) = \\sum _ { k = 1 } ^ d ( - b _ { j _ k } ) ( - 1 ) ^ { j _ k + \\ell } A _ { ( \\{ j _ k \\} , \\{ \\ell \\} ) } + \\ ! \\ ! \\ ! \\sum _ { k = m - m _ 0 + 1 } ^ m ( - b _ k ) ( - 1 ) ^ { k + \\ell } A _ { ( \\{ k \\} , \\{ \\ell \\} ) } . \\end{align*}"}
-{"id": "9463.png", "formula": "\\begin{align*} \\omega ( z ; q ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ { 2 n ^ 2 + 2 n } } { ( q ; q ^ 2 ) _ { n + 1 } ( z q ; q ^ 2 ) _ { n + 1 } } . \\end{align*}"}
-{"id": "6240.png", "formula": "\\begin{align*} \\boldsymbol \\Sigma & \\triangleq { \\bf I } _ { N _ { \\rm T } } + \\gamma _ { \\rm d } \\bar { \\bf H } _ { \\rm d } \\bar { \\bf H } _ { \\rm d } ^ H + \\gamma _ { \\rm r } \\bar { \\bf H } _ { \\rm r } \\bar { \\bf H } _ { \\rm r } ^ H + \\sum _ { k = 1 } ^ K \\bar { \\bf H } _ { { \\rm e } , k } { \\boldsymbol \\Psi } _ { { \\rm e } , k } \\bar { \\bf H } ^ H _ { { \\rm e } , k } . \\end{align*}"}
-{"id": "7031.png", "formula": "\\begin{align*} \\mu _ 1 = \\mu _ 1 ( n , s , L , S C ) = ( 1 - s ) \\int _ { \\mathbb { R } ^ { n - 1 } } { \\frac { \\max \\{ 2 L | \\bar { z } | , S C | \\bar { z } | ^ 2 \\} } { | \\bar { z } | ^ { n + 2 s - 1 } } d \\bar { z } } . \\end{align*}"}
-{"id": "9071.png", "formula": "\\begin{align*} z ^ 2 _ 1 = x _ 1 , z _ 2 ^ 2 = x _ 2 . \\end{align*}"}
-{"id": "7534.png", "formula": "\\begin{gather*} ( \\varphi ^ 2 - \\kappa \\varphi ^ 1 - 2 \\kappa ) \\varphi ^ 1 _ \\omega - ( \\kappa ^ 2 + 1 ) \\varphi ^ 1 _ { \\omega \\omega } + 2 \\varphi ^ 2 _ \\omega - ( \\varphi ^ 1 ) ^ 2 - ( \\varphi ^ 2 ) ^ 2 = 0 , \\\\ ( \\varphi ^ 2 - \\kappa \\varphi ^ 1 - 2 \\kappa ) \\varphi ^ 2 _ \\omega - ( \\kappa ^ 2 + 1 ) \\varphi ^ 2 _ { \\omega \\omega } - 2 \\varphi ^ 1 _ \\omega = 0 . \\end{gather*}"}
-{"id": "6078.png", "formula": "\\begin{align*} V ' = - 2 ( n - 2 ) \\Psi '^ 2 , \\end{align*}"}
-{"id": "3844.png", "formula": "\\begin{align*} ( t ) _ 3 \\cdot ( t ^ 2 - t + 2 ) = t ^ 5 - 4 t ^ 4 + 7 t ^ 3 - 8 t ^ 3 + 4 t . \\end{align*}"}
-{"id": "3635.png", "formula": "\\begin{align*} \\| w \\| _ \\varrho \\ , : = \\ , \\| | \\nabla w | \\| _ { L ^ 2 ( \\Omega ' , \\varrho ) } : = \\left ( \\int _ { \\Omega ' } \\rho | \\nabla w | ^ 2 \\ , d x \\right ) ^ { \\frac 1 2 } \\ , \\end{align*}"}
-{"id": "8339.png", "formula": "\\begin{align*} H _ \\Z = C ( V _ \\Z ) \\end{align*}"}
-{"id": "1328.png", "formula": "\\begin{align*} \\begin{bmatrix} \\pi ( P _ { s ( e ) } ) & 0 \\\\ 0 & * \\end{bmatrix} = \\rho ( S _ e ^ * S _ e ) = \\rho ( S _ e ) ^ * \\rho ( S _ e ) = \\begin{bmatrix} \\pi ( S _ e ) ^ * \\pi ( S _ e ) + Y _ e ^ * Y _ e & * \\\\ * & * \\end{bmatrix} , \\end{align*}"}
-{"id": "837.png", "formula": "\\begin{align*} x ( t ) = \\sum _ { i = 1 } ^ { n ( t ) } \\xi _ i \\end{align*}"}
-{"id": "3366.png", "formula": "\\begin{align*} m \\leq n \\Leftrightarrow ( \\exists p ) ( m = n \\circ p ) . \\end{align*}"}
-{"id": "525.png", "formula": "\\begin{align*} \\frac { d } { d s } f ( u ( s ) ) \\ , = \\ , - \\| \\nabla f ( u ( s ) ) \\| _ { u ( s ) } ^ 2 , \\end{align*}"}
-{"id": "1845.png", "formula": "\\begin{align*} \\mathfrak { P } = ( \\langle \\mathcal { P } ( R ) \\rangle , \\langle \\mathcal { I } ( R ^ { \\rm o p } ) \\rangle ) = ( \\langle R \\rangle , \\langle R ^ + \\rangle ) . \\end{align*}"}
-{"id": "1902.png", "formula": "\\begin{align*} \\theta _ { i } = \\min _ { p \\in M _ { i } } \\{ \\theta _ { p } : \\theta _ { p } E _ { p } ^ { s } E _ { p } ^ { u } \\} . \\end{align*}"}
-{"id": "3472.png", "formula": "\\begin{align*} \\| \\Xi ^ n \\| _ H = \\mathrm { d i s t } _ H ( u ( t _ n ) , V _ h ) \\end{align*}"}
-{"id": "8466.png", "formula": "\\begin{align*} \\abs { W _ { \\pi } ( g _ { - n , \\frac { n } { 2 } , v } ) } \\leq 2 q ^ { \\frac { r } { 6 } + \\frac { 1 } { 2 } } = 2 q ^ { \\frac { n } { 1 2 } + \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "2801.png", "formula": "\\begin{align*} \\Vert \\mathcal { B } - \\mathcal { B } _ { n } \\Vert _ { \\infty } & = \\Vert \\mathcal { B } - \\mathcal { P } _ { n } \\mathcal { B } + \\mathcal { P } _ { n } \\mathcal { B } - \\mathcal { B } _ { n } \\Vert _ { \\infty } \\\\ & \\leq \\Vert \\mathcal { B } - \\mathcal { P } _ { n } \\mathcal { B } \\Vert _ { \\infty } + \\Vert \\mathcal { P } _ { n } \\mathcal { B } - \\mathcal { B } _ { n } \\Vert _ { \\infty } , \\end{align*}"}
-{"id": "3665.png", "formula": "\\begin{align*} \\tau _ i = \\prod _ { j = 1 } ^ i t _ j ( i = 1 , \\dots , n ) . \\end{align*}"}
-{"id": "1823.png", "formula": "\\begin{align*} E ^ a _ { Q _ { 1 / 2 } , + , h } \\left ( m ^ a _ { Q _ { 1 / 2 } , h } \\right ) = \\frac { E ^ a _ { Q _ { 1 / 2 } , + , 0 } \\left ( m ^ a _ { Q _ { 1 / 2 } } e ^ { h m ^ a _ { Q _ { 1 / 2 } } } \\right ) } { E ^ a _ { Q _ { 1 / 2 } , + , 0 } \\left ( e ^ { h m ^ a _ { Q _ { 1 / 2 } } } \\right ) } . \\end{align*}"}
-{"id": "9521.png", "formula": "\\begin{align*} \\tilde { b } _ n = \\dfrac { 1 } { 2 d n } \\sum \\limits _ { m \\ , | \\ , d n } \\phi ( m ) \\ , f ( d n , m ) + \\dfrac { h ^ { ( 0 ) } ( n ) + 2 h ^ { ( 1 ) } ( n ) + h ^ { ( 2 ) } ( n ) } { 2 } , \\end{align*}"}
-{"id": "752.png", "formula": "\\begin{align*} J ( u _ 1 , u _ 2 ) = m ( b _ 1 , b _ 2 ) . \\end{align*}"}
-{"id": "7089.png", "formula": "\\begin{align*} ( m , M , N , t , r = ( r _ 1 , r _ 2 , r _ { 7 } , r _ { 1 4 } ) ) = ( 4 9 , 1 4 , 1 4 , 4 7 , ( 4 , 1 , - 1 , 0 ) ) \\in \\Delta ^ * \\end{align*}"}
-{"id": "3735.png", "formula": "\\begin{align*} & C _ { k , k _ 0 } = \\big ( 1 + 1 7 \\sqrt { 1 + \\frac { k _ 0 } { p } } \\big ) \\sigma _ { k _ 0 + 1 } + \\frac { 8 \\sqrt { k } } { p + 1 } ( \\sum \\limits _ { j > k _ 0 } \\sigma _ j ^ 2 ) ^ { \\frac { 1 } { 2 } } \\end{align*}"}
-{"id": "6972.png", "formula": "\\begin{align*} \\det { M } = n ^ { - n } . \\end{align*}"}
-{"id": "9201.png", "formula": "\\begin{align*} [ x \\otimes a _ { 1 } , y \\otimes a _ { 2 } ] & = x \\circ y \\otimes \\frac { [ a _ { 1 } , a _ { 2 } ] } { 2 } + [ x , y ] \\otimes \\frac { a _ { 1 } \\circ a _ { 2 } } { 2 } + ( x \\mid y ) \\langle a _ { 1 } , a _ { 2 } \\rangle \\end{align*}"}
-{"id": "8993.png", "formula": "\\begin{gather*} { \\cal D } ^ { ( n ) } _ { q , t } ( c ) D ^ { ( n ) } _ q ( c \\pm ( t _ 0 + d ) ; t ) = \\prod _ { 1 \\le i \\le n } \\vartheta ( z _ i \\pm ( t _ 0 + d ) ) { \\cal D } ^ { ( n ) } _ { q , t } ( c - q / 2 ) , \\end{gather*}"}
-{"id": "3414.png", "formula": "\\begin{align*} \\Omega _ n = \\Theta _ n \\cup R _ n \\end{align*}"}
-{"id": "890.png", "formula": "\\begin{align*} X ^ \\nu _ t = X ^ \\nu _ 0 + \\int _ 0 ^ t \\sigma _ \\nu ( s ) d B ^ \\nu _ s , t \\geq 0 , \\end{align*}"}
-{"id": "8340.png", "formula": "\\begin{align*} N \\Z = [ V _ \\Z , \\ell ] . \\end{align*}"}
-{"id": "5792.png", "formula": "\\begin{align*} - \\Delta _ { p } u = \\mu \\ ; \\ ; \\mathbb { R } ^ n - \\Delta _ { p } v = \\omega \\ ; \\ ; \\mathbb { R } ^ n , \\end{align*}"}
-{"id": "3629.png", "formula": "\\begin{align*} E _ 1 : = \\int _ { \\mathcal { M } } d \\sigma \\int _ 0 ^ { \\varepsilon } \\frac { 1 } { r ^ { ( 2 - p ) t - ( k - 1 ) } } \\ , d r < + \\infty , \\end{align*}"}
-{"id": "7806.png", "formula": "\\begin{align*} - \\Delta u + V u = \\lambda u \\end{align*}"}
-{"id": "4028.png", "formula": "\\begin{align*} b ( X , Y ) : = \\frac { \\langle d u ^ { - 1 } _ { \\eta ( p ) } X , Y \\rangle } { \\langle \\eta ( p ) , \\xi ( p ) \\rangle } , \\end{align*}"}
-{"id": "2434.png", "formula": "\\begin{align*} v ^ i ( x _ 1 ^ \\ast , y _ 1 ^ \\ast ) = \\sum \\limits _ { x _ 2 , \\dots , x _ { k - 1 } , x _ k \\in X } p ( x _ k | x _ { k - 1 } , \\mu _ { k - 1 } ^ \\ast ) & p ( x _ { k - 1 } | x _ { k - 2 } , \\mu _ { k - 2 } ^ \\ast ) \\dots \\\\ & p ( x _ 2 | x _ 1 ^ \\ast , \\mu _ 1 ^ \\ast ) v ^ i ( x _ k , y _ k ^ \\ast ) . \\end{align*}"}
-{"id": "4054.png", "formula": "\\begin{align*} \\mathbf { I } _ \\alpha ( \\eta ) = \\mathbf { E } _ \\alpha ( \\eta ) . \\end{align*}"}
-{"id": "7546.png", "formula": "\\begin{gather*} u = \\frac { 2 C _ 0 } { r ^ 2 } \\left ( x - y \\tan \\left ( C _ 0 \\arctan \\frac y x \\right ) \\right ) , v = \\frac { 2 C _ 0 } { r ^ 2 } \\left ( y + x \\tan \\left ( C _ 0 \\arctan \\frac y x \\right ) \\right ) , \\\\ u = \\frac y { r ^ 2 \\arctan ( y / x ) } , v = - \\frac x { r ^ 2 \\arctan ( y / x ) } . \\end{gather*}"}
-{"id": "5519.png", "formula": "\\begin{align*} K ( s ) = s ^ \\alpha Q ( f ( s ) ) , \\end{align*}"}
-{"id": "1096.png", "formula": "\\begin{align*} f _ { \\rho } ( \\rho ) = \\sqrt { \\frac { 2 N } { \\pi P _ t } } \\ , \\mathrm { e x p } \\left ( - \\frac { N \\rho ^ 2 } { 2 P _ t } \\right ) . \\end{align*}"}
-{"id": "9268.png", "formula": "\\begin{align*} \\mathcal { P } X : = \\sum _ { \\nu \\in \\sigma ( \\mathcal { N } ) } P _ { \\nu } X P _ { \\nu } , \\mbox { f o r } X \\in B ( \\mathcal { H } ) , \\end{align*}"}
-{"id": "198.png", "formula": "\\begin{align*} \\hat { f } ^ { - 1 } \\circ i \\circ \\hat { g } \\circ g ^ { \\ast } = f ^ { \\ast } \\circ h . \\end{align*}"}
-{"id": "9007.png", "formula": "\\begin{align*} \\left \\{ \\aligned & \\partial _ { t } \\theta + ( u \\cdot \\nabla ) \\theta + \\mu \\Lambda _ { x _ { 1 } } ^ { 2 \\alpha } \\theta + \\nu \\Lambda _ { x _ { 2 } } ^ { 2 \\beta } \\theta = 0 , x = ( x _ { 1 } , x _ { 2 } ) \\in \\mathbb { R } ^ 2 , \\ , \\ , t > 0 , \\\\ & \\theta ( x , 0 ) = \\theta _ { 0 } ( x ) , \\endaligned \\right . \\end{align*}"}
-{"id": "8815.png", "formula": "\\begin{align*} | E _ { p , m } ^ { 0 } | = \\Big | \\sum _ { i = 1 } ^ { s } a _ { i , p } m ^ { \\beta _ { i } } p ^ { - \\lambda _ { i } m } \\ 1 1 _ { A _ { i } } ( m ) \\Big | \\leq C m ^ { n - 1 } p ^ { - m \\sigma } , \\end{align*}"}
-{"id": "3590.png", "formula": "\\begin{align*} D _ { * a } ^ \\alpha f : = J _ a ^ { \\lceil \\alpha \\rceil - \\alpha } D ^ { \\lceil \\alpha \\rceil } f . \\end{align*}"}
-{"id": "6110.png", "formula": "\\begin{align*} \\psi _ { n } \\left ( s \\right ) = s + \\mu _ { n } ^ { - 1 } \\int _ { 0 } ^ { \\infty } \\left ( 1 - e ^ { - s y } \\right ) f \\left ( d y \\right ) , \\end{align*}"}
-{"id": "6819.png", "formula": "\\begin{align*} \\tilde { \\phi } = 2 \\norm { \\phi } _ i \\tilde { V } + \\norm { h } _ { \\ast } \\sum \\limits _ { j = 1 } ^ 2 \\tilde { g } _ j . \\end{align*}"}
-{"id": "6029.png", "formula": "\\begin{align*} & P _ { M _ { n } X ^ { n } Y ^ { n } } ( m , x ^ { n } , y ^ { n } ) \\\\ & = \\frac { 1 } { | { \\cal M } _ { n } | } P _ { X ^ { n } | M _ { n } } ( x ^ { n } | m ) P _ { Y ^ { n } | M _ { n } } ( y ^ { n } | m ) . \\end{align*}"}
-{"id": "924.png", "formula": "\\begin{align*} \\Psi ' ( t ) = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ d E \\left [ \\frac { \\partial \\varphi } { \\partial x _ i } ( \\sqrt { 1 - t } F + \\sqrt { t } Z ) \\left ( \\frac { Z _ i } { \\sqrt { t } } - \\frac { F _ i } { \\sqrt { 1 - t } } \\right ) \\right ] \\end{align*}"}
-{"id": "2283.png", "formula": "\\begin{align*} \\hat { \\Delta } _ 1 & = \\mathrm { C o n v } \\{ ( 2 , 0 ) , ( 0 , 1 ) , ( 0 , - 1 ) \\} , \\\\ \\hat { \\Delta } _ 2 & = \\mathrm { C o n v } \\{ ( 2 , 0 ) , ( 0 , 2 ) , ( 0 , - 1 ) \\} , \\\\ \\hat { \\Delta } _ 3 & = \\mathrm { C o n v } \\{ ( 3 , 0 ) , ( 0 , 1 ) , ( 0 , - 1 ) \\} , \\end{align*}"}
-{"id": "1070.png", "formula": "\\begin{align*} g = \\delta - m + 1 . \\end{align*}"}
-{"id": "3059.png", "formula": "\\begin{align*} \\tilde { \\Lambda } = \\Lambda \\ , \\bigcup \\ , \\{ \\rho _ n ( \\xi ) + \\Lambda \\} \\end{align*}"}
-{"id": "5226.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d } { d z } \\omega ( Y _ 1 , Y _ 2 ) & = \\omega ( Y _ 1 , A ( \\lambda , z ) Y _ 2 ) + \\omega ( A ( \\lambda , z ) Y _ 1 , Y _ 2 ) \\\\ & = \\langle Y _ 1 , J A ( \\lambda , z ) Y _ 2 \\rangle + \\langle A ( \\lambda , z ) Y _ 1 , J Y _ 2 \\rangle \\\\ & = \\langle Y _ 1 , \\left [ J A + A ^ T J \\right ] Y _ 2 \\rangle . \\end{aligned} \\end{align*}"}
-{"id": "5864.png", "formula": "\\begin{align*} \\min \\left \\{ \\langle F _ { \\emptyset } , X \\rangle \\ , \\ , \\big { | } \\ , \\ , \\langle F _ { \\omega } , X \\rangle = b _ { \\omega } , \\ , ~ X \\succeq 0 \\right \\} . \\end{align*}"}
-{"id": "7666.png", "formula": "\\begin{align*} \\zeta _ s = \\zeta _ s ( B ) = \\begin{cases} \\xi _ { B ^ { ( n _ s ^ \\ast ) } } & \\\\ \\eta _ { B ^ { ( n _ s ^ \\ast ) } } & \\end{cases} \\end{align*}"}
-{"id": "4151.png", "formula": "\\begin{align*} & r \\sum _ { \\tau = t + h } ^ { n } Z _ { N , J } ^ { \\tau } + n Z _ { I , J } ^ { t + h - 1 } + \\sum _ { \\tau = t } ^ { t + h - 2 } n ^ { t + h - \\tau } Z _ { i _ { \\tau } , J } ^ { \\tau } \\leq r + \\sum _ { \\tau = 1 } ^ { h - 1 } n ^ { \\tau } , \\\\ & \\forall \\ J \\subseteq V , \\lvert J \\rvert = h \\in [ n - 1 ] , \\ I \\subseteq N , \\lvert I \\rvert = r \\in [ n - 1 ] , t \\in [ n - h ] , \\ i _ t , \\dotsc , i _ { t + h - 2 } \\in N . \\end{align*}"}
-{"id": "7326.png", "formula": "\\begin{align*} V _ { ( n _ 1 , n _ 2 ) } = \\nu ( e _ 1 ^ { n _ 1 } e _ 2 ^ { n _ 2 } ) U = \\left ( ( p ^ { - n _ 1 } \\mathbb { Z } _ p / \\mathbb { Z } _ p ) \\times ( p ^ { - n _ 1 - n _ 2 } \\mathbb { Z } _ p / \\mathbb { Z } _ p ) \\times ( p ^ { - n _ 2 } \\mathbb { Z } _ p / \\mathbb { Z } _ p ) \\right ) ( e _ 1 ^ { n _ 1 } e _ 2 ^ { n _ 2 } ) \\end{align*}"}
-{"id": "6713.png", "formula": "\\begin{align*} \\Gamma _ { l } = \\min \\big ( t \\geq 2 l : \\big ( I ( t - ( 2 l - 1 ) ) , \\dots , I ( t ) \\big ) \\in J _ { l } \\big ) . \\end{align*}"}
-{"id": "7386.png", "formula": "\\begin{align*} \\det { ( 1 - \\mathcal { G } ) } _ { [ n , n + 1 , . . . ] } \\coloneqq \\sum _ { m = 0 } ^ { \\infty } ( - 1 ) ^ m \\sum _ { n \\leq z _ 1 < . . . < z _ m } ^ { \\infty } \\det { \\mathcal { G } \\big ( ( z _ i , z _ j ) \\big ) _ { i , j = 1 } ^ { m } } . \\end{align*}"}
-{"id": "5649.png", "formula": "\\begin{align*} \\mathfrak { L } _ { K } ( \\overline { \\gamma } ) = A _ K ( \\overline { \\gamma } ) \\leq A _ K ( \\gamma ) \\leq \\liminf \\limits _ { n \\to \\infty } A _ K ( \\gamma _ n ) = \\liminf \\limits _ { n \\to \\infty } \\mathfrak { L } _ { K } ( \\gamma _ n ) . \\end{align*}"}
-{"id": "7909.png", "formula": "\\begin{align*} d ( y , z ) = { | x _ 1 - y _ 1 | \\over 2 } + { | x _ 2 - y _ 2 | \\over 2 } = { 1 \\over 2 } d ( x , y ) . \\end{align*}"}
-{"id": "6317.png", "formula": "\\begin{align*} \\Tilde { A } ( z ) = ( z - 2 \\alpha ) p ^ 2 ( z ) = \\alpha ^ { 2 \\ell + 1 } ( w - 2 ) q ^ 2 ( w ) \\quad \\mbox { w i t h } w = \\tfrac { z } { \\alpha } . \\end{align*}"}
-{"id": "3303.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\frac { \\pi } { 2 } } \\sin ^ { \\mu - 1 } \\theta \\cos ^ { \\nu - 1 } \\theta d \\theta = \\frac { 1 } { 2 } B \\left ( \\frac { \\mu } { 2 } , \\frac { \\nu } { 2 } \\right ) . \\end{align*}"}
-{"id": "136.png", "formula": "\\begin{align*} \\hat h ( z ) = \\hat f ( z ) = \\hat g ( z ) = \\Phi ( \\hat V ( z ) ) \\end{align*}"}
-{"id": "6432.png", "formula": "\\begin{align*} \\mathcal { V } _ { \\mathcal { M } } \\left ( s \\right ) \\mathbf { = } \\int d \\theta ^ { 1 } \\int d \\theta ^ { 2 } \\int \\sqrt { g \\left ( \\theta ^ { 1 } \\theta ^ { n } \\right ) } d \\theta ^ { n } = { \\displaystyle \\prod \\limits _ { \\kappa = 1 } ^ { n } } \\int _ { s _ { 0 } } ^ { s _ { 0 } + s } \\sqrt { g _ { \\kappa } \\left ( \\theta ^ { \\kappa } \\left ( \\alpha \\right ) \\right ) } \\frac { d \\theta ^ { \\kappa } } { d \\alpha } d \\alpha \\end{align*}"}
-{"id": "4467.png", "formula": "\\begin{align*} \\sum _ { \\stackrel { k ' + k '' = k } { k ' , k '' \\not = 0 } } { \\rm m i n } ^ 2 \\{ 1 , \\ell ( d ( k ' , 0 ) + d ( k '' , 0 ) ) \\} d ^ { - 4 } ( k ' , 0 ) d ^ { - 1 } ( k '' , 0 ) \\lesssim \\min \\{ d ^ { - 1 } ( k , 0 ) , \\ell ^ 2 d ( k , 0 ) \\} . \\end{align*}"}
-{"id": "2011.png", "formula": "\\begin{align*} x _ i z ( q ) _ { i + 1 } = x _ { i + 2 } z ( q ) _ i + O ( q ) \\end{align*}"}
-{"id": "4350.png", "formula": "\\begin{align*} \\| A - A _ m \\| & = \\Big \\| \\sum _ { j = 1 } ^ \\infty ( M _ { \\varphi _ { j , \\frac { 1 } { m } } } A - M _ { \\varphi _ { j , \\frac { 1 } { m } } } A M _ { \\psi _ { j , \\frac { 1 } { m } } } ) \\Big \\| \\\\ & = \\Big \\| \\sum _ { j = 1 } ^ \\infty M _ { \\varphi _ { j , \\frac { 1 } { m } } } A M _ { 1 - \\psi _ { j , \\frac { 1 } { m } } } \\Big \\| \\\\ & \\to 0 \\end{align*}"}
-{"id": "1452.png", "formula": "\\begin{align*} E _ { 2 ^ { r + 1 } } & = C \\langle 1 , y _ r , v _ s y _ s \\rangle \\oplus D _ 1 / ( v _ t e _ t ) \\{ x _ 3 ^ 2 \\} \\oplus D _ r \\langle x _ 3 ^ 2 y _ r , x _ 3 z _ r \\rangle \\end{align*}"}
-{"id": "3945.png", "formula": "\\begin{align*} u ( x , 0 ) = \\frac { \\mu + \\sigma + ( \\sigma - \\mu ) \\exp [ \\frac { \\mu } { \\nu } ( x - \\lambda ) ] } { 1 + \\exp [ \\frac { \\mu } { \\nu } ( x - \\lambda ) ] } \\end{align*}"}
-{"id": "9358.png", "formula": "\\begin{align*} 4 9 ( J _ { n + r } ^ { ( 3 ) } j _ { n + s } ^ { ( 3 ) } ) & = 2 ^ { 2 n + 4 + r + s } + 3 \\cdot 2 ^ { n + r + 1 } ( a \\omega _ { 1 } ^ { n + s } + b \\omega _ { 2 } ^ { n + s } ) - 2 ^ { n + s + 3 } ( a \\omega _ { 1 } ^ { n + r } + b \\omega _ { 2 } ^ { n + r } ) \\\\ & \\ \\ - 3 ( a ^ { 2 } \\omega _ { 1 } ^ { 2 n + r + s } + b ^ { 2 } \\omega _ { 2 } ^ { 2 n + r + s } ) - 7 ( \\omega _ { 1 } ^ { r } \\omega _ { 2 } ^ { s } + \\omega _ { 1 } ^ { s } \\omega _ { 2 } ^ { r } ) , \\end{align*}"}
-{"id": "8579.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } \\int _ \\Omega e ^ { - 2 g } u ( y , s ) ^ 2 \\ , d y \\biggr | _ { s = 0 } ^ { s = t } + ( 1 - \\mu ) \\int _ 0 ^ t \\int _ \\Omega e ^ { - 2 g } H ( \\nabla u ) ^ 2 \\ , d y \\ , d s \\\\ & \\qquad \\le \\int _ 0 ^ t \\int _ \\Omega e ^ { - 2 g } u ^ 2 \\left [ \\mu ^ { - 1 } H ( \\nabla g ) ^ 2 - \\partial _ t g \\right ] \\ , d y \\ , d s \\end{align*}"}
-{"id": "5915.png", "formula": "\\begin{align*} \\begin{aligned} & i _ 1 = 0 , \\cdots , i _ l = l , i _ { l + 1 } = l - 1 , i _ { l + 2 } = l , \\cdots , i _ { l + 2 j - 1 } = l - 1 , l _ { l + 2 j } = l , \\\\ & i _ { l + 2 j + 1 } = l - 1 , i _ { l + 2 j + 2 } = l - 2 , i _ { l + 2 j + k } = l - k , i _ { l + 2 j + k + 1 } = l - k + 1 , \\\\ & \\ j \\ge 1 \\ \\ 2 \\le k \\le l - 1 . \\end{aligned} \\end{align*}"}
-{"id": "2007.png", "formula": "\\begin{align*} z _ i = x _ i x _ { i + 1 } + a _ 1 x _ { i - 1 } x _ { i + 2 } + a _ 2 x _ { i - 2 } x _ { i + 3 } + . . . \\end{align*}"}
-{"id": "9514.png", "formula": "\\begin{align*} a _ { n , k , l > 1 } ^ { ( d ) } = \\sum \\limits _ { R } \\dfrac { \\alpha _ R } { \\beta _ 1 ! \\cdot \\ldots \\cdot \\beta _ { d - 1 } ! } \\sum \\limits _ { m = 0 } ^ { l d - k } p _ { n , R , m } \\cdot a _ { n - l , m } ^ { ( d ) } . \\end{align*}"}
-{"id": "6729.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\mathbb { P } ( R _ { N } \\leq N ^ { 1 + \\delta } ) = \\lim \\limits _ { N \\rightarrow \\infty } \\mathbb { P } ( \\sigma ^ { + } ( N ^ { \\gamma } + 1 ) = \\sigma ^ { + } ( N ^ { \\gamma } ) ) \\leq \\lim \\limits _ { N \\rightarrow \\infty } \\frac { N ^ { \\gamma } } { N } = 0 \\end{align*}"}
-{"id": "1810.png", "formula": "\\begin{align*} Y _ { t } Y _ { t + 1 0 } & = Y _ { t + 3 } Y _ { t + 7 } Z _ { t } Z _ { t + 1 } Z _ { t + 2 } + Y _ { t + 2 } ^ 2 Y _ { t + 5 } ^ 2 Y _ { t + 8 } ^ 2 \\quad \\\\ Z _ { t } Z _ { t + 1 } Z _ { t + 2 } Z _ { t + 3 } & = Y _ { t + 2 } Y _ { t + 3 } Y _ { t + 5 } Y _ { t + 6 } Y _ { t + 8 } Y _ { t + 9 } + Y _ { t + 1 } Y _ { t + 4 } Y _ { t + 7 } Y _ { t + 1 0 } \\end{align*}"}
-{"id": "2474.png", "formula": "\\begin{align*} B _ { N } ( | x | , t ) = \\left ( 1 + \\frac { | x | ^ { 2 } } { 1 + t } \\right ) ^ { - N } . \\end{align*}"}
-{"id": "9019.png", "formula": "\\begin{align*} \\| \\theta ( t _ { 1 } ) - \\theta ( t _ { 2 } ) \\| _ { H ^ { s } } = \\Big \\{ ( \\sum _ { k < N } + \\sum _ { k \\geq N } ) ( 2 ^ { k s } \\| \\Delta _ { k } \\theta ( t _ { 1 } ) - \\Delta _ { k } \\theta ( t _ { 2 } ) \\| _ { L ^ { 2 } } ) ^ { 2 } \\Big \\} ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "3957.png", "formula": "\\begin{align*} \\langle D v , w \\rangle = \\sum _ { i , j } w _ j \\frac { \\partial Z _ j } { \\partial m _ { i } } v _ i = \\sum _ { i , j } \\sum _ { S \\neq S _ 1 , \\ldots , S _ k } w _ j \\left ( \\frac { \\partial Z _ j } { \\partial x _ S } \\right ) \\left ( \\frac { \\partial x _ S } { \\partial m _ i } \\right ) v _ i \\end{align*}"}
-{"id": "3757.png", "formula": "\\begin{align*} \\mathfrak { S } : = \\left \\{ \\sigma : \\Z _ + \\to \\Z ^ d \\colon \\ , \\sigma ( 0 ) = 0 , \\left | \\sigma ( i + 1 ) - \\sigma ( i ) \\right | \\le \\mathfrak { R } \\ ; \\forall \\ ; i \\in \\Z _ + \\right \\} , \\end{align*}"}
-{"id": "2162.png", "formula": "\\begin{align*} \\| \\hat { w } ( s ) \\| _ { L ^ 2 ( { \\bf R } _ + , \\rho _ d \\ , d \\xi ) } = O ( e ^ { - s } ) , \\| w ( s ) \\| _ { L ^ 2 ( { \\bf R } _ + , \\rho _ d \\ , d \\xi ) } = O ( e ^ { - s } ) , \\end{align*}"}
-{"id": "8292.png", "formula": "\\begin{align*} \\mathrm { w t } _ \\bullet ( g N _ \\Z ) = g N _ { \\widehat { \\Z } } \\cap \\mathrm { w t } _ \\bullet N . \\end{align*}"}
-{"id": "9216.png", "formula": "\\begin{align*} [ z _ { 1 } \\otimes \\alpha _ { 1 } , [ z _ { 2 } \\otimes \\alpha _ { 2 } , u ' \\otimes b ' ] ] = [ [ z _ { 1 } \\otimes \\alpha _ { 1 } , z _ { 2 } \\otimes \\alpha _ { 2 } ] , u ' \\otimes b ' ] + [ z _ { 2 } \\otimes \\alpha _ { 2 } , [ z _ { 1 } \\otimes \\alpha _ { 1 } , u ' \\otimes b ' ] ] . \\end{align*}"}
-{"id": "2503.png", "formula": "\\begin{align*} \\phi \\left ( s \\right ) & = \\frac { \\phi \\left ( s \\right ) + \\phi \\left ( - s \\right ) } { 2 } + \\frac { \\phi \\left ( s \\right ) - \\phi \\left ( - s \\right ) } { 2 } \\\\ & \\equiv \\phi ^ { e } \\left ( s \\right ) + \\phi ^ { o } \\left ( s \\right ) . \\end{align*}"}
-{"id": "2120.png", "formula": "\\begin{align*} \\widetilde { C } _ { \\Gamma } ( f ) ( x ) : = { \\rm p . v . } \\frac { 1 } { \\pi i } \\int _ { \\R } \\frac { ( 1 + i A ' ( y ) ) f ( y ) } { y - x + i ( A ( y ) - A ( x ) ) } \\ , d y , \\end{align*}"}
-{"id": "2714.png", "formula": "\\begin{align*} \\partial _ t h + \\frac { 1 } { \\epsilon } v \\cdot \\nabla _ x h = \\frac { 1 } { \\epsilon ^ 2 } \\mathcal L ( h ) + \\frac { 1 } { \\epsilon } \\mathcal F ( h , h ) \\ , . \\end{align*}"}
-{"id": "8490.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { t , l , v } ) = \\zeta _ F ( 1 ) ^ { - 1 } q ^ { - \\frac { a _ 1 + t } { 2 } } q ^ { s ( l _ 1 - l _ 2 ) } \\epsilon ( \\frac { 1 } { 2 } , \\chi _ 1 ^ { - 1 } ) \\sum _ { \\mu \\in \\mathfrak { X } _ l } G ( \\varpi ^ { - l _ 2 } , \\mu \\chi _ 2 ) G ( - v b _ 1 \\varpi ^ { - l } , \\mu ^ { - 1 } ) . \\end{align*}"}
-{"id": "786.png", "formula": "\\begin{align*} 1 - \\frac { ( 2 x ) ^ k } { y ^ n } = \\frac { s } { y ^ n } , \\end{align*}"}
-{"id": "9476.png", "formula": "\\begin{align*} \\sum _ { s = 0 } ^ N \\frac { q ^ { 2 s } } { ( q ^ 2 ; q ^ 2 ) _ s } \\bigg ( \\frac { 1 } { ( q ^ { 1 + N + s } ; q ) _ { N - s + 1 } } - 1 \\bigg ) = \\frac { q ^ { N + 1 } } { ( q ; q ^ 2 ) _ { N + 1 } } . \\end{align*}"}
-{"id": "9794.png", "formula": "\\begin{align*} p _ { r s } = - t _ { \\bar s \\ , \\bar r } - \\sum _ { r < l < s } t _ { \\bar s \\ , \\bar l } p _ { r l } \\end{align*}"}
-{"id": "5560.png", "formula": "\\begin{align*} M _ { 2 ^ { k } } \\left ( t \\right ) = \\left [ \\bar { w } _ { 2 ^ { k } } \\left ( t \\right ) , \\Lambda _ { 1 } ^ { \\left ( 2 ^ { k } \\right ) } \\bar { w } _ { 2 ^ { k } } \\left ( t \\right ) , . . . , \\Lambda _ { 2 ^ { k } - 1 } ^ { \\left ( 2 ^ { k } \\right ) } \\bar { w } _ { 2 ^ { k } } \\left ( t \\right ) \\right ] \\end{align*}"}
-{"id": "2247.png", "formula": "\\begin{align*} f _ j ( z ) = \\prod _ { s = 1 } ^ \\infty f _ { j , s } ( z ) , j = 1 , \\ldots , n . \\end{align*}"}
-{"id": "1386.png", "formula": "\\begin{align*} q ( t _ 0 , t ) & = \\Big ( 1 + \\int _ { t _ 0 } ^ t e ^ { \\rho ( t _ 0 , u ) } \\mu ( u ) d u \\Big ) ^ { - 1 } , p ( t _ 0 , t ) = e ^ { \\rho ( t _ 0 , t ) } q ( t _ 0 , t ) , \\rho ( t _ 0 , t ) = \\int _ { t _ 0 } ^ t ( \\mu ( u ) - \\lambda ( u ) ) d u . \\end{align*}"}
-{"id": "7146.png", "formula": "\\begin{align*} \\Delta _ 1 ( m ) = L ( q ( m + 1 ) ) - L ( q ( m ) ) \\leq \\frac { 1 } { 2 } y ^ 2 ( m ) + q ( m ) y ( m ) . \\end{align*}"}
-{"id": "5291.png", "formula": "\\begin{align*} u _ t = { \\mu u _ { x x } + \\alpha u } - \\beta | u | ^ 2 u , \\end{align*}"}
-{"id": "4076.png", "formula": "\\begin{align*} \\frac { | F _ \\alpha ( u ) | } { \\eta ( u ) } = o ( 1 ) | u | \\to \\infty , \\quad \\alpha = 1 , \\dots , d \\ , , \\end{align*}"}
-{"id": "3410.png", "formula": "\\begin{align*} \\phi ( 0 ) = 0 \\ \\ \\ \\ \\phi ' ( 0 ) > 0 . \\end{align*}"}
-{"id": "1928.png", "formula": "\\begin{align*} K _ { \\alpha , \\mathcal { F } , z } ^ { s } & = \\{ ( v , w ) \\in \\tilde { E } _ { z } ^ { s } \\oplus \\tilde { E } _ { z } ^ { u } : \\Vert w \\Vert _ { \\star } < \\alpha \\Vert v \\Vert _ { \\star } \\} \\cup \\{ ( 0 , 0 ) \\} , \\\\ K _ { \\alpha , \\mathcal { F } , z } ^ { u } & = \\{ ( v , w ) \\in \\tilde { E } _ { z } ^ { s } \\oplus \\tilde { E } _ { z } ^ { u } : \\Vert v \\Vert _ { \\star } < \\alpha \\Vert w \\Vert _ { \\star } \\} \\cup \\{ ( 0 , 0 ) \\} . \\end{align*}"}
-{"id": "742.png", "formula": "\\begin{align*} \\Lambda _ 0 : = \\inf _ { \\int _ { \\R ^ 3 } | u | ^ 2 \\ , d x = 1 } \\int _ { \\R ^ 3 } | \\nabla u | ^ 2 + ( x _ 1 ^ 2 + x _ 2 ^ 2 ) | u | ^ 2 \\ , d x , \\end{align*}"}
-{"id": "3453.png", "formula": "\\begin{align*} g ( i p ) = \\begin{cases} p , & i \\in \\{ 0 , \\dots , 2 ^ K \\} , \\\\ 0 , & i \\in \\{ 0 , \\dots , 2 ^ K \\} , \\end{cases} \\end{align*}"}
-{"id": "3140.png", "formula": "\\begin{align*} J _ { t } = \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { \\infty } z N ( \\mathrm { d } s , \\mathrm { d } z ) , t \\geqslant 0 , \\end{align*}"}
-{"id": "5328.png", "formula": "\\begin{align*} M ( \\ell , 0 ) = U ( \\hat { \\mathfrak { g } } ) \\otimes _ { U ( \\mathfrak { g } \\oplus \\C { \\bf k } \\oplus \\hat { \\mathfrak { g } } _ { + } ) } \\C \\simeq U ( \\hat { \\mathfrak { g } } _ { - } ) \\otimes \\C = U ( \\hat { \\mathfrak { g } } _ { - } ) . \\end{align*}"}
-{"id": "4560.png", "formula": "\\begin{align*} & \\theta ^ { - 1 } ( H _ { 0 , 1 } ) = - ( - d ) ^ { - n + 1 } [ [ \\cdots [ F _ { 1 , 1 } , F _ { 2 , 0 } ] _ { q } , \\cdots , F _ { n - 1 , 0 } ] _ { q } , F _ { 0 , - 1 } ] _ { q ^ 2 } \\ , , \\\\ & \\theta ^ { - 1 } ( H _ { 0 , - 1 } ) = - ( - d ) ^ { n - 1 } [ E _ { 0 , 1 } , [ E _ { n - 1 , 0 } , \\cdots , [ E _ { 2 , 0 } , E _ { 1 , - 1 } ] _ { q ^ { - 1 } } \\cdots ] _ { q ^ { - 1 } } ] _ { q ^ { - 2 } } \\ , . \\end{align*}"}
-{"id": "9676.png", "formula": "\\begin{align*} \\kappa _ 1 ( { U } _ k ^ { ( 0 ) } ) = \\kappa _ 5 ( { U } _ k ^ { ( 0 ) } ) = 1 / ( \\nabla _ { U } \\lambda _ i \\cdot ( - \\lambda _ i , 1 , \\rho u \\lambda _ i , { \\rho u \\lambda _ i } / { c ^ 2 } , 0 ) | _ { U = { U } _ k ^ { ( 0 ) } } ) > 0 , i = 1 , 5 . \\end{align*}"}
-{"id": "536.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial s } + J ( s , t , u ) \\left ( \\frac { \\partial u } { \\partial t } - X _ { H _ t } ( u ) \\right ) = 0 . \\end{align*}"}
-{"id": "8224.png", "formula": "\\begin{align*} \\biggl [ \\frac { 1 } { n ^ d } \\biggr ] ( t _ + ^ 2 - t _ - ^ 2 ) = \\sum _ { s = 1 } ^ { d - 1 } \\Biggl [ \\biggl ( \\biggl [ \\frac { 1 } { n ^ s } \\biggr ] t _ + \\biggr ) \\biggl ( \\biggl [ \\frac { 1 } { n ^ { d - s } } \\biggr ] t _ + \\biggr ) - \\biggl ( \\biggl [ \\frac { 1 } { n ^ s } \\biggr ] t _ - \\biggr ) \\biggl ( \\biggl [ \\frac { 1 } { n ^ { d - s } } \\biggr ] t _ - \\biggr ) \\Biggr ] \\end{align*}"}
-{"id": "6853.png", "formula": "\\begin{align*} S G = \\bigcup _ { j = 0 } ^ 2 F _ j ( S G ) \\end{align*}"}
-{"id": "7390.png", "formula": "\\begin{align*} v _ N ( z ) : = \\begin{cases} - \\frac { z ^ N } { 1 - z ^ N } & z \\in S ^ 1 _ { i n } \\\\ \\frac { z ^ { - N } } { 1 - z ^ { - N } } & z \\in S ^ 1 _ { o u t } \\\\ \\end{cases} , \\end{align*}"}
-{"id": "8251.png", "formula": "\\begin{align*} - \\omega n ( x ) = \\left ( \\begin{matrix} x ^ { - 1 } & - 1 \\\\ 0 & x \\end{matrix} \\right ) \\left ( \\begin{matrix} 1 & 0 \\\\ x ^ { - 1 } & 1 \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "3485.png", "formula": "\\begin{align*} 2 \\tau ( \\Delta f ) ^ { \\prime \\prime } + f ^ { \\prime \\prime } = \\tau \\Big \\{ ( \\big | \\nabla f | ^ 2 \\big ) ^ { \\prime \\prime } - R ^ { \\prime \\prime } \\Big \\} - R \\tau ^ { \\prime \\prime } . \\end{align*}"}
-{"id": "3055.png", "formula": "\\begin{align*} b ( v _ h , p _ h ( t ) - q _ h ) = b ( v _ h , p ( t ) - q _ h ) + ( u _ t ( t ) - u _ { h , t } ( t ) , v _ h ) _ H + a ( u ( t ) - u _ h ( t ) , v _ h ) , \\end{align*}"}
-{"id": "5251.png", "formula": "\\begin{align*} ( L _ q - a - b i ) ^ { - 1 } = \\left ( ( L _ q - a ) ^ 2 + b ^ 2 \\right ) ^ { - 1 } ( L _ q - a ) + i b \\left ( ( L _ q - a ) ^ 2 + b ^ 2 \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "6147.png", "formula": "\\begin{align*} C _ { G _ V , X _ V } \\subseteq \\bigcup _ { U \\subsetneq V } C _ { G _ U , X _ U } \\bigcup C _ { G _ V , = X _ V } . \\end{align*}"}
-{"id": "5668.png", "formula": "\\begin{align*} E ( s ) = \\begin{dcases} \\varepsilon _ 0 \\exp { \\left ( - \\sqrt { c } \\ , ( s - s _ 0 ) \\right ) } & p _ 0 = 2 , \\\\ \\frac { c ( s - s _ \\ast ) ^ { - \\alpha } } { \\alpha ( \\alpha + 1 ) } & p _ 0 \\in ( 2 , 6 ) , \\end{dcases} \\end{align*}"}
-{"id": "2668.png", "formula": "\\begin{align*} \\begin{pmatrix} B _ 0 & B _ 1 & \\cdots & B _ { N / k - 1 } \\\\ B _ { - 1 } & B _ 0 & \\cdots & B _ { N / k - 2 } \\\\ B _ { - 2 } & B _ { - 1 } & \\cdots & B _ { N / k - 3 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ B _ { 1 - N / k } & B _ { 2 - N / k } & \\cdots & B _ { 0 } \\end{pmatrix} , \\end{align*}"}
-{"id": "10119.png", "formula": "\\begin{align*} { \\boldsymbol { \\omega } } _ k ( i ) = \\boldsymbol S _ { D _ k } ( i ) { \\boldsymbol { \\bar { \\omega } } } _ k ( i ) , \\end{align*}"}
-{"id": "5946.png", "formula": "\\begin{align*} A & : = | \\{ i : f _ { l _ i } \\equiv \\phi \\} | + | \\{ j : f _ { r _ j } \\equiv \\psi \\} | \\\\ B & : = | \\{ i : f _ { l _ i } \\equiv \\psi \\} | + | \\{ j : f _ { r _ j } \\equiv \\phi \\} | . \\end{align*}"}
-{"id": "8698.png", "formula": "\\begin{align*} 1 = F _ { n , 0 } \\geq \\dots \\geq F _ { n , k _ 0 ( n ) - 1 } \\geq F _ { n , k _ 0 ( n ) } \\geq q ^ { n / 4 ( 1 \\wedge \\gamma ) } > F _ { n , k _ 0 ( n ) + 1 } \\geq \\ldots , \\\\ 1 = \\tilde { F } _ { n , 0 } \\geq \\dots \\geq \\tilde { F } _ { n , d _ 0 ( n ) - 1 } \\geq \\tilde { F } _ { n , d _ 0 ( n ) } \\geq q ^ { n / 4 ( 1 \\wedge \\gamma ) } > \\tilde { F } _ { n , d _ 0 ( n ) + 1 } \\geq \\ldots . \\end{align*}"}
-{"id": "4991.png", "formula": "\\begin{align*} \\partial _ { } ^ \\gamma V _ m = \\sum _ { m ' < m } \\left ( \\partial _ { } ^ { \\gamma } h _ { m ' } - \\sum _ { 0 < \\alpha \\leq \\gamma } c _ { \\alpha , \\gamma } \\partial _ { } ^ { \\alpha } U _ { m ' } \\partial _ { } ^ { \\gamma - \\alpha } V _ { m ' } \\right ) \\prod _ { m ' < m '' < m } ( 1 - U _ { m '' } ) \\end{align*}"}
-{"id": "9475.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ n } { ( z q ^ n ; q ) _ { n + 1 } ( z q ^ { 2 n + 2 } ; q ^ 2 ) _ { \\infty } } = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ { n + 1 } } { ( q ; q ^ 2 ) _ { n + 1 } } . \\end{align*}"}
-{"id": "2228.png", "formula": "\\begin{align*} \\begin{cases} \\left | w _ 1 - a _ { 1 j _ { 1 } } \\right | = \\varepsilon _ 1 , \\\\ \\ldots \\\\ \\left | w _ n - a _ { n j _ { n } } \\right | = \\varepsilon _ n . \\end{cases} \\end{align*}"}
-{"id": "7775.png", "formula": "\\begin{align*} | \\ell + m + u _ { R } ( \\ell ) - u _ { R } ( m ) | & \\geq | \\ell - m | - | u _ { R } ( \\ell ) - u _ { R } ( m ) | \\geq | \\ell - m | - C _ { 1 } \\lambda | \\ell - m | ^ { 1 / 2 } \\\\ & = | \\ell - m | ^ { 1 / 2 } \\ , ( | \\ell - m | ^ { 1 / 2 } - C _ { 1 } \\lambda ) \\geq \\frak { m } | \\ell - m | . \\end{align*}"}
-{"id": "8273.png", "formula": "\\begin{align*} F ( i T _ 0 , g ) ^ 2 = \\frac { 1 } { 2 \\pi i } \\int _ { \\mathcal { C } } F ( i T _ 0 + u , g ) ^ 2 \\frac { \\exp ( u ^ 2 ) } { u } d u , \\end{align*}"}
-{"id": "1043.png", "formula": "\\begin{align*} L _ { P _ { k } } ^ { \\ast } ( q ) = \\max \\left \\{ \\log H ( P _ { k } ) - \\frac { q } { m } , \\log \\vert P _ { k } ( \\zeta ) \\vert + q \\right \\} = L _ { m , 1 } ^ { \\ast } ( q ) , k \\geq 1 , \\ ; q \\in I _ { k } . \\end{align*}"}
-{"id": "3404.png", "formula": "\\begin{align*} h ( z ) = T [ u ] ( z ) = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } \\frac { e ^ { \\imath \\theta } + z } { e ^ { \\imath \\theta } - z } \\ , u ( e ^ { \\imath \\theta } ) \\ , d \\theta , z \\in \\d . \\end{align*}"}
-{"id": "5367.png", "formula": "\\begin{align*} | ( X \\boxtimes Y ) | = | S _ { X , Y } | . \\end{align*}"}
-{"id": "7270.png", "formula": "\\begin{align*} \\rho = \\frac { \\sqrt { \\log T } } T ( \\widetilde { E } \\delta ^ { - 1 } ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "1630.png", "formula": "\\begin{align*} \\pi _ T ( t _ \\lambda ) x = \\begin{cases} \\xi _ { \\lambda x } , & r ( x ) = s ( \\lambda ) \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "2146.png", "formula": "\\begin{align*} \\tilde { v } '' ( r ) + \\frac { N - 1 } { r } \\tilde { v } ' ( r ) - \\lambda _ 2 r ^ { - 2 } \\tilde { v } ( r ) = 0 \\quad \\mbox { i n } \\quad ( 0 , \\infty ) . \\end{align*}"}
-{"id": "9302.png", "formula": "\\begin{align*} A ' : = \\{ ( p , q ) \\in V \\times \\R ^ n : ( g ( p ) , q ) \\in A \\} . \\end{align*}"}
-{"id": "6995.png", "formula": "\\begin{align*} s p a n \\{ \\tilde { e } ( \\theta ) \\} ^ { \\perp } = s p a n \\{ \\tilde { e } _ 1 , \\tilde { e } _ 2 , . . . , \\tilde { e } _ { n - 1 } \\} \\end{align*}"}
-{"id": "1272.png", "formula": "\\begin{align*} \\mbox { e i t h e r } \\ , \\ , \\ , \\mu ( \\{ \\xi \\} ) \\ , \\ , \\ , \\mbox { o r } \\ , \\ , \\ , \\mu ( \\{ - \\xi \\} ) = 0 \\mbox { w h e n e v e r } \\xi \\in \\mathbb { S } ^ { n - 1 } . \\end{align*}"}
-{"id": "9783.png", "formula": "\\begin{align*} F _ k V _ \\infty : = \\mathrm { I m } \\left ( \\Gamma ( \\P ^ 1 _ \\lambda , U _ 0 H \\otimes \\mathcal { O } _ { \\P ^ 1 _ \\lambda } ( k \\{ 0 \\} ) ) \\to V _ \\infty \\right ) \\end{align*}"}
-{"id": "9131.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } x ^ { p _ { k } } \\widetilde { f _ { k } } ( x ^ { q _ { k } } ) = \\sum _ { k = 1 } ^ { n } x ^ { p _ { k } } \\left [ f _ { k } ( x ^ { q _ { k } } ) - f _ { k } ( 1 ) x ^ { q _ { k } } \\right ] = \\sum _ { k = 1 } ^ { n } x ^ { p _ { k } } f _ { k } ( x ^ { q _ { k } } ) - x ^ { l } \\sum _ { k = 1 } ^ { n } f _ { k } ( 1 ) = 0 \\end{align*}"}
-{"id": "7644.png", "formula": "\\begin{align*} S = \\big < s _ 1 , \\ldots , s _ { 2 ^ n - 1 } \\colon \\ , s _ { \\alpha } ^ 2 = - 1 , s _ { \\alpha } s _ { \\beta } = - s _ { \\beta } s _ { \\alpha } , 1 \\le \\alpha \\neq \\beta \\le 2 ^ n - 1 \\big > , \\end{align*}"}
-{"id": "3554.png", "formula": "\\begin{align*} \\chi ( u ) = e ^ { i 2 \\pi k \\log ( u ) / d } , \\end{align*}"}
-{"id": "3509.png", "formula": "\\begin{align*} X ^ { u } ( s ) = x _ 0 + \\displaystyle \\int _ 0 ^ { s } b ( X ^ { u } ( t ) , u ( t ) ) d t . \\end{align*}"}
-{"id": "8198.png", "formula": "\\begin{align*} f _ { } = \\sum _ { \\alpha \\in \\mathcal { O } _ F } y _ \\alpha \\frac { \\kappa _ { \\alpha } } { \\sqrt { \\mathcal { N } ( \\alpha ) } } . \\end{align*}"}
-{"id": "5243.png", "formula": "\\begin{align*} E ^ u ( \\lambda _ \\mathrm { m a x } , z ) \\cap S ( \\lambda _ \\mathrm { m a x } ) = \\{ 0 \\} . \\end{align*}"}
-{"id": "9687.png", "formula": "\\begin{align*} \\begin{cases} & V _ b = \\tilde { \\Phi } ( \\alpha _ 5 , \\alpha _ 3 , \\alpha _ 2 , \\alpha _ 1 ; V _ a ) , \\\\ & Z _ b = Z _ a + \\alpha _ 4 , \\end{cases} \\end{align*}"}
-{"id": "9352.png", "formula": "\\begin{align*} j O _ { n + 2 } ^ { ( 3 ) } + j O _ { n + 1 } ^ { ( 3 ) } + j O _ { n } ^ { ( 3 ) } = 2 ^ { n + 3 } \\underline { \\alpha } , \\end{align*}"}
-{"id": "7836.png", "formula": "\\begin{align*} f _ { X _ { 1 } , X _ { 2 } } ( x _ { 1 } , x _ { 2 } ) = \\left \\{ \\begin{array} { l } f _ { 1 } ( x _ { 1 } , x _ { 2 } ) \\ \\ \\ \\ \\ \\ 0 < x _ { 1 } < x _ { 2 } \\\\ f _ { 2 } ( x _ { 1 } , x _ { 2 } ) \\ \\ \\ \\ \\ \\ 0 < x _ { 2 } < x _ { 1 } \\\\ f _ { 3 } ( x , x ) \\ \\ \\ \\ \\ \\ \\ \\ \\ x _ { 1 } = x _ { 2 } = x , \\end{array} \\right . \\end{align*}"}
-{"id": "9660.png", "formula": "\\begin{align*} \\begin{cases} ( \\rho u ) _ x + ( \\rho v ) _ y = 0 , \\\\ ( \\rho u ^ 2 + p ) _ x + ( \\rho u v ) _ y = 0 , \\\\ ( \\rho u v ) _ x + ( \\rho v ^ 2 + p ) _ y = 0 , \\\\ \\big ( ( \\rho E + p ) u \\big ) _ x + \\big ( ( \\rho E + p ) v \\big ) _ y = 0 , \\\\ ( \\rho u Z ) _ x + ( \\rho v Z ) _ y = - \\rho \\phi ( T ) Z . \\end{cases} \\end{align*}"}
-{"id": "4335.png", "formula": "\\begin{align*} L _ \\alpha ^ p = \\{ f : \\mathbb { C } ^ n \\to \\mathbb { C } ; f \\| f \\| _ { p , \\alpha } < \\infty \\} = L ^ p ( \\mathbb { C } ^ n , d \\mu _ { p \\alpha / 2 } ) \\end{align*}"}
-{"id": "5854.png", "formula": "\\begin{align*} \\theta ^ { X Y } _ { 1 2 } = \\theta ^ { X Y } _ { 2 1 } = \\theta ^ { Y Y } _ { 1 2 } = 0 \\end{align*}"}
-{"id": "3788.png", "formula": "\\begin{align*} v _ \\circ : = 2 p _ \\circ - 1 , v _ \\bullet : = 2 p _ \\bullet - 1 . \\end{align*}"}
-{"id": "7669.png", "formula": "\\begin{align*} \\zeta _ s = \\zeta _ t \\zeta _ { t + 1 } ^ { m _ { t + 1 } + m _ { t + 2 } } \\zeta _ { t + 2 } ^ { m _ { t + 2 } + m _ { t + 3 } } \\cdots \\zeta _ { s - 1 } ^ { m _ { s - 1 } + m _ s } . \\end{align*}"}
-{"id": "5144.png", "formula": "\\begin{align*} \\| F \\| _ { N ^ s ( I ) } = \\bigg \\| \\int _ { t _ 0 } ^ t S ( t - t ' ) F ( t ' ) d t ' \\bigg \\| _ { X ^ s ( I ) } . \\end{align*}"}
-{"id": "6778.png", "formula": "\\begin{align*} m _ 0 = 2 + O ( \\lambda ^ 2 ) . \\end{align*}"}
-{"id": "4411.png", "formula": "\\begin{align*} \\sigma _ { e s s } ( T _ f ) = \\tilde { f } ( \\partial \\mathbb { C } ^ n ) , \\end{align*}"}
-{"id": "7839.png", "formula": "\\begin{align*} M _ { X _ { 1 } , X _ { 2 } } = 4 F _ { X _ { 1 } , X _ { 2 } } ( M _ { X _ { 1 } } , M _ { X _ { 2 } } ) - 1 , \\end{align*}"}
-{"id": "4353.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { j = 1 } ^ \\infty M _ { a _ j } P _ \\alpha M _ { 1 - b _ j } \\Big \\| \\leq N ^ 2 \\beta _ { p , \\alpha } ( \\sigma ) . \\end{align*}"}
-{"id": "986.png", "formula": "\\begin{align*} \\sup _ { t \\in [ 0 , T ] } \\left | \\mathbf { A } _ n ( t ) \\right | \\leq \\max _ { j = 0 , 1 , \\dots , \\lfloor T / h \\rfloor } \\left ( \\left | \\mathbf { A } _ n ( u _ j ) \\right | + \\sup _ { t \\in [ u _ j , u _ { j + 1 } ] } \\left | \\mathbf { A } _ n ( t ) - \\mathbf { A } _ n ( u _ j ) \\right | \\right ) . \\end{align*}"}
-{"id": "1615.png", "formula": "\\begin{align*} M ( Z ( \\lambda ) ) = ( \\rho ( \\Lambda ) ) ^ { - d ( \\lambda ) } \\kappa ^ { \\Lambda } _ { s ( \\lambda ) } \\quad \\ ; \\ ; \\lambda \\in \\Lambda , \\end{align*}"}
-{"id": "3057.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ z u = 0 , & \\Gamma _ u : = G \\times \\{ 0 \\} , \\\\ u = 0 , & \\Gamma _ b : = G \\times \\{ - D \\} , \\\\ & \\Gamma _ l : = \\partial G \\times ( - D , 0 ) , \\end{cases} \\end{align*}"}
-{"id": "632.png", "formula": "\\begin{align*} x = ( \\underbrace { x _ 1 \\cdots x _ { d _ 1 } } _ { x ^ { T } [ 1 ] } \\underbrace { x _ { d _ 1 + 1 } \\cdots x _ { d _ 1 + d _ 2 } } _ { x ^ { T } [ 2 ] } \\cdots \\underbrace { x _ { N - d _ n + 1 } \\cdots x _ N } _ { x ^ { T } [ n ] } ) ^ T . \\end{align*}"}
-{"id": "2191.png", "formula": "\\begin{align*} J _ \\beta = \\frac 1 { ( 2 \\pi i ) ^ n } \\int \\limits _ { \\gamma ( r ) } \\frac 1 { z ^ { \\beta + U } } \\cdot \\frac { d f } { f } \\end{align*}"}
-{"id": "2392.png", "formula": "\\begin{align*} A ( \\theta ) = A ( \\theta + \\pi ) , B ( \\theta ) = - B ( \\theta + \\pi ) . \\end{align*}"}
-{"id": "3492.png", "formula": "\\begin{align*} \\int _ { \\gamma _ 1 \\cdots \\gamma _ r } \\omega _ 1 \\cdots \\omega _ k = \\sum _ { \\substack { g \\in \\Delta ( k , r ) \\\\ g = ( n _ 1 , \\ldots , n _ r ) } } \\Big ( \\int _ { \\gamma _ 1 } \\omega _ 1 \\cdots \\omega _ { n _ 1 } \\Big ) \\Big ( \\int _ { \\gamma _ 2 } \\omega _ { n _ 1 + 1 } \\cdots \\omega _ { n _ 1 + n _ 2 } \\Big ) \\cdots \\Big ( \\int _ { \\gamma _ r } \\omega _ { n _ 1 + \\cdots + n _ { r - 1 } + 1 } \\cdots \\omega _ k \\Big ) . \\end{align*}"}
-{"id": "4619.png", "formula": "\\begin{align*} | U _ i ' | \\ge \\beta ( 1 - 2 \\eta ) \\sum _ { n = 0 } ^ { 2 ^ { k _ i } - 1 } 1 _ { [ - B - b , B + b ] } \\left ( t + \\sum _ { j = 0 } ^ { n - 1 } f ( T ^ j x ) \\right ) \\end{align*}"}
-{"id": "447.png", "formula": "\\begin{align*} \\Upsilon ( x , t ) = \\sum _ { j = 0 } ^ { k _ 2 / 2 } c _ { k _ 1 , k _ 2 , j } \\frac { \\omega ^ { k _ 2 - 2 j } } { | x | ^ { 2 j } } + O \\left ( \\sum _ { j = 0 } ^ { k _ 2 / 2 } \\frac { \\omega ^ { k _ 2 - 2 j + 1 } } { | x | ^ { 2 j } } + \\frac { 1 } { | x | ^ { k _ 2 + 2 } } \\right ) ; \\end{align*}"}
-{"id": "5028.png", "formula": "\\begin{align*} ( I + \\lambda C _ n G ^ \\omega _ { n , n } ( E + \\iota 0 ) ) \\left [ C _ n \\lim _ { \\epsilon \\downarrow 0 } \\frac { 1 } { t r ( G ^ { \\omega , \\lambda } _ { n , n , n } ( E + \\iota \\epsilon ) ) } G ^ { \\omega , \\lambda } _ { n , n , n } ( E + \\iota \\epsilon ) \\right ] = 0 , \\end{align*}"}
-{"id": "7291.png", "formula": "\\begin{align*} B = \\left ( \\begin{array} { c c c c c c c } t _ 1 & 0 & \\ldots & 0 \\\\ \\ell _ { 2 , 1 } & \\ell _ { 2 , 2 } & \\ldots & \\ell _ { 2 , n - 1 } \\\\ \\ell _ { 3 , 1 } & \\ell _ { 3 , 2 } & \\ldots & \\ell _ { 3 , n - 1 } \\end{array} \\right ) , \\end{align*}"}
-{"id": "427.png", "formula": "\\begin{align*} q ( \\delta , \\zeta ) = q _ \\delta ( \\zeta ) \\coloneqq \\cosh ( \\zeta ) + \\frac { \\delta } { 2 } r ( - \\delta e ^ { - \\zeta } ) . \\end{align*}"}
-{"id": "8458.png", "formula": "\\begin{align*} A _ { \\pm } = - \\frac { 1 } { 2 v } + b _ 2 \\sqrt { \\zeta } \\pm \\frac { Y } { 2 v } \\varpi ^ { \\delta } \\in \\mathfrak { O } ^ { \\times } . \\end{align*}"}
-{"id": "6302.png", "formula": "\\begin{align*} & \\exists \\ \\gamma _ 1 > 0 , \\ , \\gamma _ 2 > 0 : \\quad | f _ 1 ' ( r ) | \\le \\gamma _ 1 | f _ { \\Gamma 1 } ' ( r ) | + \\gamma _ 2 \\quad \\forall \\ , r \\in ( - 1 , 1 ) , \\\\ [ 1 m m ] & \\lim _ { r \\searrow - 1 } f _ 1 ' ( r ) = \\lim _ { r \\searrow - 1 } f _ { \\Gamma 1 } ' ( r ) = - \\infty , \\lim _ { r \\nearrow + 1 } f _ 1 ' ( r ) = \\lim _ { r \\nearrow + 1 } f _ { \\Gamma 1 } ' ( r ) = + \\infty . \\end{align*}"}
-{"id": "6140.png", "formula": "\\begin{align*} - c & = \\left ( c ' - \\frac { 1 } { \\alpha } \\right ) \\xi + 1 , \\\\ & > \\left ( c ' - \\frac { 1 } { \\alpha } \\right ) \\xi _ { 0 } + 1 , \\end{align*}"}
-{"id": "7538.png", "formula": "\\begin{gather*} \\varphi ^ 1 \\varphi ^ 1 _ \\omega - \\varphi ^ 1 _ { \\omega \\omega } - 2 \\varphi ^ 2 = 0 , \\\\ \\varphi ^ 1 \\varphi ^ 2 _ \\omega - \\varphi ^ 2 _ { \\omega \\omega } + 2 \\varphi ^ 1 + 2 \\mu = 0 . \\end{gather*}"}
-{"id": "3634.png", "formula": "\\begin{align*} A _ { \\lambda _ 0 + \\tau } & = \\{ x \\in \\Omega _ { \\lambda _ 0 + \\tau } \\setminus K \\ : \\ | \\nabla u _ { \\lambda _ 0 + \\tau } ( x ) | < \\dot C | \\nabla u ( x ) | \\} \\\\ \\\\ \\\\ \\\\ B _ { \\lambda _ 0 + \\tau } & = \\{ x \\in \\Omega _ { \\lambda _ 0 + \\tau } \\setminus K \\ : \\ | \\nabla u _ { \\lambda _ 0 + \\tau } ( x ) | \\geq \\dot C | \\nabla u ( x ) | \\} . \\end{align*}"}
-{"id": "7351.png", "formula": "\\begin{align*} \\Vert | u _ n | ^ { \\gamma - 1 } u \\Vert _ { L ^ { \\infty } ( [ 0 , T ] , L ^ { M / \\gamma } ( \\mathbb { R } ^ 3 ) ) } \\ ; & \\leqslant \\ ; \\Vert u _ n \\Vert _ { L ^ { \\infty } ( [ 0 , T ] , L ^ M ( \\mathbb { R } ^ 3 ) ) } ^ { \\gamma } \\\\ & \\lesssim \\ ; \\Vert u _ n \\Vert _ { L ^ { \\infty } ( [ 0 , T ] , H ^ 1 ( \\mathbb { R } ^ 3 ) ) } ^ { \\gamma } \\ ; \\lesssim _ { A , T } \\ ; 1 \\ , , \\end{align*}"}
-{"id": "9762.png", "formula": "\\begin{align*} E _ { i } ( h , \\theta ) = \\int \\limits _ { y _ { i - 1 , s } + s ^ { ( i - 1 ) } h } ^ { y _ { i } } ( W ( U _ { h , \\theta } ) ) ( i h - , y ) d y - \\int \\limits _ { y _ { i , s } } ^ { y _ { i } } ( W ( U _ { h , \\theta } ) ) ( i h + , y ) d y . \\end{align*}"}
-{"id": "9382.png", "formula": "\\begin{align*} z = h _ { < , 2 m } + h _ { = } + h _ { > } \\qquad h = h _ { < , \\alpha } + h _ { = } + h _ { > } . \\end{align*}"}
-{"id": "4293.png", "formula": "\\begin{align*} \\underline { \\mu } = \\{ ( \\mu _ i , \\Delta _ { D , \\ell _ i } ) _ { i = 1 } ^ { s } \\} , \\underline { \\mu } ^ { \\vee } = \\{ ( \\mu _ i , \\Delta ^ { \\vee } _ { D , \\ell _ i } ) _ { i = 1 } ^ { s } \\} . \\end{align*}"}
-{"id": "10124.png", "formula": "\\begin{align*} \\bar { { \\boldsymbol { \\omega } } } _ k ( i ) = \\sum \\limits _ { l \\in \\mathcal { N } _ k } c _ { k l } \\boldsymbol { \\bar { \\psi } } _ l ( i ) , \\end{align*}"}
-{"id": "2173.png", "formula": "\\begin{align*} ( \\partial _ t ^ j u _ * ) ( | x | , t ) = ( \\partial _ t ^ j u _ * ) ( 0 , t ) + F _ d ^ j ( | x | , t ) \\quad \\mbox { i n } \\quad { \\bf R } ^ N \\times ( 0 , \\infty ) , \\end{align*}"}
-{"id": "2977.png", "formula": "\\begin{align*} - \\int _ { \\Omega } | \\nabla ( u _ n - v _ n ) ^ - | ^ 2 & = \\int _ { \\Omega } \\nabla ( u _ n - v _ n ) . \\nabla ( u _ n - v _ n ) ^ - \\\\ & = \\int _ { \\Omega } \\bigg ( h _ n \\left ( u _ n + \\frac { 1 } { n } \\right ) - h _ n \\left ( v _ n + \\frac { 1 } { n } \\right ) \\bigg ) f _ n \\cdot ( u _ n - v _ n ) ^ - \\\\ & + \\int _ { \\Omega } \\mu _ n \\cdot ( u _ n - v _ n ) ^ - \\\\ & \\geq 0 . \\end{align*}"}
-{"id": "9582.png", "formula": "\\begin{align*} [ \\Gamma : \\Pi ] \\beta _ { v _ 1 } \\wedge \\dots \\wedge \\beta _ { v _ d } = & \\frac { [ \\Gamma : \\Pi ] } { n _ 1 \\dots n _ d } \\beta _ { n _ 1 v _ 1 } \\wedge \\dots \\wedge \\beta _ { n _ d v _ d } \\\\ = & \\frac { [ \\Gamma : \\Pi ] } { n _ 1 \\dots n _ d } \\iota _ * ( \\beta _ { \\Sigma \\cap \\Pi } ) \\end{align*}"}
-{"id": "10001.png", "formula": "\\begin{align*} [ \\widehat { \\theta } ( g ) : \\mathcal { Y } _ \\mathrm { b i g } ] = \\frac { - 1 } { n } \\cdot \\deg _ \\C ( \\mathcal { Y } _ \\mathrm { b i g } ) \\cdot \\frac { d } { d s } \\langle E ( s ) , \\tilde { g } \\rangle _ \\mathrm { P e t } \\big | _ { s = 0 } . \\end{align*}"}
-{"id": "9251.png", "formula": "\\begin{align*} \\mathcal { L } ( \\mathfrak { b } ) : = ( \\mathfrak { g ^ { + } } \\otimes A ^ { - } ) \\oplus ( \\mathfrak { g } ^ { - } \\otimes A ^ { + } ) \\oplus ( V \\otimes B ) \\oplus \\cdots \\oplus ( \\Lambda ' \\otimes E ' ) \\oplus D _ { \\mathfrak { b } , \\mathfrak { b } } \\end{align*}"}
-{"id": "1866.png", "formula": "\\begin{align*} \\pi ( \\lambda f _ 1 , \\dots , \\lambda f _ d , \\lambda a ) = \\lambda \\pi ( f _ 1 , \\dots , f _ d , a ) , \\end{align*}"}
-{"id": "8140.png", "formula": "\\begin{align*} \\varphi ( e _ { i j } ) = \\sum _ { k = 1 } ^ K \\sum _ { l = 1 } ^ L \\sum _ { m = 1 } ^ M \\sum _ { c \\in C _ { k , l , m } ^ { ( j ) } } h _ { k , l , c , i , j } \\end{align*}"}
-{"id": "9467.png", "formula": "\\begin{align*} S _ { n - 2 } ( 3 ) & = S _ { n - 2 } ( 1 ) - q S _ { n - 1 } ( 1 ) + q ^ { n - 1 } ( 1 + q ) ( q ; q ^ 2 ) _ { n - 1 } . \\end{align*}"}
-{"id": "3017.png", "formula": "\\begin{align*} 2 \\Re \\ < f _ { \\partial , x } , e _ { \\partial , x } \\ > _ { H ^ N } & = \\ < f _ { \\partial , x } , e _ { \\partial , x } \\ > _ { H ^ N } + \\ < e _ { \\partial , x } , f _ { \\partial , x } \\ > _ { H ^ N } \\\\ & = \\ < ( I - V ) l , ( - I - V ) l \\ > _ { H ^ N } + \\ < ( - I - V ) l , ( I - V ) l \\ > _ { H ^ N } \\\\ & = 2 \\ < l , ( - I + V ^ * V ) l \\ > _ { H ^ N } \\leq 0 \\end{align*}"}
-{"id": "9381.png", "formula": "\\begin{align*} M ^ 2 \\le \\Big ( \\sum _ { \\{ j \\ , : \\ , \\alpha _ j < 2 m \\} } | c _ j | \\Big ) ^ 2 \\le C _ 1 \\sum _ { \\{ j \\ , : \\ , \\alpha _ j < 2 m \\} } c _ j ^ 2 = C _ 1 \\| Q \\| ^ 2 _ { L ^ 2 ( \\partial B _ 1 ) } . \\end{align*}"}
-{"id": "1263.png", "formula": "\\begin{align*} \\left . \\frac { d } { d t } \\mbox { C a p } _ { \\mathcal { A } } ( E _ 1 + t E _ 2 ) \\ , \\right | _ { t = t _ 2 } = ( p - 1 ) \\int _ { \\mathbb { S } ^ { n - 1 } } h _ 2 ( \\mathbf { g } ( z , E _ 1 + t _ 2 E _ 2 ) ) \\ , d \\mu _ { E _ 1 + t _ 2 E _ 2 } ( z ) . \\end{align*}"}
-{"id": "2167.png", "formula": "\\begin{align*} \\int _ 0 ^ { ( 1 + t ) ^ { \\frac { 1 } { 2 } - \\theta _ * } } u _ * ( r , t ) \\nu _ d ( r ) r ^ { d - 1 } \\ , d r & = O ( t ^ { - \\frac { d } { 2 } } ) \\int _ 0 ^ { ( 1 + t ) ^ { \\frac { 1 } { 2 } - \\theta _ * } } \\nu _ d ( r ) r ^ { d - 1 } \\ , d r \\\\ & = O ( t ^ { - \\frac { d } { 2 } } ) O ( t ^ { \\frac { d } { 2 } - d \\theta _ * } ) = O ( t ^ { - d \\theta _ * } ) \\end{align*}"}
-{"id": "7582.png", "formula": "\\begin{align*} \\cal F ^ { - 1 } \\Phi _ 0 \\cal F ( \\theta _ { N , + } ^ { ( s _ 0 ) } \\theta _ { N , - } ^ { ( s _ 1 ) } ) = \\cal F ^ { - 1 } \\Phi _ 0 \\cal F \\big ( \\sum _ { k , l = 1 } ^ { N } 2 ^ { - k s _ 0 - l s _ 1 } e ^ { ( i ( 2 ^ { k } - 2 ^ l ) x _ { j _ 0 } ) } \\theta ^ 2 \\big ) = N \\theta ^ 2 , \\end{align*}"}
-{"id": "254.png", "formula": "\\begin{align*} c _ { m , n } : = \\frac { ( - 1 ) ^ { m - n } 2 ^ { m } d ! } { ( m - n ) ! ( d - m ) ! } , \\end{align*}"}
-{"id": "133.png", "formula": "\\begin{align*} \\Phi ^ { - 1 } \\bigg ( \\int Q _ { 1 - t } h ( \\sqrt { t } x ) \\ , \\gamma _ 1 ( d x ) \\bigg ) & = \\Phi ^ { - 1 } \\bigg ( \\int h \\ , d \\gamma _ 1 \\bigg ) \\\\ & = \\lambda \\Phi ^ { - 1 } \\bigg ( \\int f \\ , d \\gamma _ 1 \\bigg ) + \\mu \\Phi ^ { - 1 } \\bigg ( \\int g \\ , d \\gamma _ 1 \\bigg ) \\\\ & = \\Phi ^ { - 1 } \\bigg ( \\int Q _ { 1 - t } f ( \\sqrt { t } x ) \\ , \\gamma _ 1 ( d x ) \\bigg ) , \\end{align*}"}
-{"id": "9895.png", "formula": "\\begin{align*} X ( i ) = \\bigl \\{ v \\in V _ a \\colon | N ( v ) \\cap V _ i | \\le \\tfrac \\theta 2 | V _ i | \\bigr \\} \\end{align*}"}
-{"id": "5874.png", "formula": "\\begin{align*} I _ i ( r ) = \\sup _ { \\lambda \\in \\mathbb { R } } \\big ( \\lambda r - \\log M _ { { P ^ { ( ) } _ i } } ( \\lambda ) \\big ) , \\ r \\in \\mathbb { R } . \\end{align*}"}
-{"id": "364.png", "formula": "\\begin{align*} \\mathbb { P } ( \\xi _ { 0 } \\geq x ) = \\frac { h ( x ) } { x ^ { t } } , \\ ; \\ ; \\ \\hbox { a s } \\ , x \\rightarrow \\infty . \\end{align*}"}
-{"id": "4699.png", "formula": "\\begin{align*} \\Psi _ { w _ r } ( 1 ) - \\Phi ( r ) & \\geq \\frac { 1 } { 1 - k } \\left ( \\Phi ( r ) - \\theta - C \\left ( r ^ { 1 - \\frac { n } { p ^ * } } + \\omega ( r ) \\right ) \\right ) + \\theta - \\Phi ( r ) \\\\ & = \\frac { k } { 1 - k } ( \\Phi ( r ) - \\theta ) - C \\left ( r ^ { 1 - \\frac { n } { p ^ * } } + \\omega ( r ) \\right ) . \\end{align*}"}
-{"id": "1090.png", "formula": "\\begin{align*} \\Psi ( j \\omega ) = \\frac { 1 } { 1 + m _ { c , i } ( 1 - e ^ { j \\omega } ) } \\mathrm { e x p } \\left [ \\frac { - m _ { s , i } ( 1 - e ^ { j \\omega } ) } { 1 + m _ { c , i } ( 1 - e ^ { j \\omega } ) } \\right ] . \\end{align*}"}
-{"id": "5528.png", "formula": "\\begin{align*} A = A _ { 0 } - A _ { 1 } + A _ { 2 } - A _ { 3 } + \\ldots + \\left ( - 1 \\right ) ^ { n } A _ { n } + \\ldots \\end{align*}"}
-{"id": "5990.png", "formula": "\\begin{align*} f _ 3 ( x ) = \\cos ( \\Vert x \\Vert ^ 2 ) \\end{align*}"}
-{"id": "4676.png", "formula": "\\begin{align*} \\Gamma \\ = \\ F _ 1 ^ { \\frac { 2 - d } { 4 } } \\ , F _ 2 ^ { - \\frac { 1 } { 4 } } \\ = \\ ( S ^ 2 _ { \\triangle } ) ^ { \\frac { 2 - d } { 4 } } \\ , ( P _ m ) ^ { - \\frac { 1 } { 4 } } \\ , \\end{align*}"}
-{"id": "5815.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } \\vert \\nabla u \\vert ^ { p } \\ ; d x = \\int _ { \\R ^ n } u ^ { 1 + q } \\ ; d \\sigma + \\int _ { \\R ^ n } u \\ ; d \\mu , \\ ; \\ ; \\end{align*}"}
-{"id": "8425.png", "formula": "\\begin{align*} \\sum _ { t = - \\infty } ^ { \\infty } q ^ { ( t + 1 ) ( \\frac { 1 } { 2 } - s ) } c _ { t , 0 } ( 1 ) = - L ( s , \\abs { \\cdot } ^ { \\frac { 1 } { 2 } } ) = - \\sum _ { a = 0 } ^ { \\infty } q ^ { - a ( \\frac { 1 } { 2 } + s ) } . \\end{align*}"}
-{"id": "9122.png", "formula": "\\begin{align*} D _ { n } = f _ { n + 1 } D _ { n - ( k + 1 ) } = - \\dfrac { \\widetilde { D } _ { n - ( k + 1 ) } } { k + 1 } \\left ( k = 0 , \\ldots , n - 1 \\right ) . \\end{align*}"}
-{"id": "9958.png", "formula": "\\begin{align*} { \\cal N } _ s ( u ) - V _ { 1 s } = { \\cal N } _ s ( \\varphi ( u ) ) - V _ { 2 s } \\end{align*}"}
-{"id": "7754.png", "formula": "\\begin{align*} A = \\left \\{ ( u , b , x ) \\in U ' \\times P \\times ( P \\cap P ^ w ) \\mid u w b \\in \\Gamma , [ x , u ] \\in \\Gamma , x \\in \\Gamma \\right \\} \\end{align*}"}
-{"id": "8067.png", "formula": "\\begin{align*} & m _ U ( \\emptyset ) = 1 \\\\ & m _ U ( e ) = 1 \\textnormal { f o r } e = 1 , 2 , 3 \\\\ & m _ U ( 1 , 2 ) = 2 \\\\ & m _ U ( 1 , 3 ) = 3 \\\\ & m _ U ( 2 , 3 ) = 5 \\\\ & m _ U ( 1 , 2 , 3 ) = 1 \\end{align*}"}
-{"id": "8462.png", "formula": "\\begin{align*} \\abs { W _ { \\pi } ( g _ { - n , l , v } ) } = 1 . \\end{align*}"}
-{"id": "1000.png", "formula": "\\begin{align*} \\mathbb { H } = \\left \\{ \\int _ { 0 } ^ T f ( t ) d B _ t : f \\in L ^ 2 ( 0 , T ) \\right \\} . \\end{align*}"}
-{"id": "8736.png", "formula": "\\begin{align*} \\mathcal { F } ( \\delta ) & = F ( t _ \\delta , x _ \\delta , u ( t _ \\delta , x _ \\delta ) , \\nabla _ x \\varphi ( x _ \\delta , y _ \\delta ) , \\nabla _ x ^ 2 \\varphi ( x _ \\delta , y _ \\delta ) ) \\\\ & \\quad - F ( s _ \\delta , y _ \\delta , v ( s _ \\delta , y _ \\delta ) , - \\nabla _ y \\varphi ( x _ \\delta , y _ \\delta ) , - \\nabla _ y ^ 2 \\varphi ( x _ \\delta , y _ \\delta ) ) . \\end{align*}"}
-{"id": "8774.png", "formula": "\\begin{align*} \\det S = \\det [ ( x - 1 - c _ 3 ^ 2 \\chi _ { G _ 2 } ( C ) ) I _ m ] \\cdot \\det P , \\end{align*}"}
-{"id": "548.png", "formula": "\\begin{align*} \\widehat { M } \\ , = \\ , M \\cup _ { \\partial M } ( [ 1 + \\infty ) \\times \\partial M ) . \\end{align*}"}
-{"id": "8870.png", "formula": "\\begin{align*} \\Bigg \\lfloor \\frac { \\Big \\lceil \\frac { \\lceil d \\rceil } { 2 } \\Big \\rceil } { 2 } \\Bigg \\rfloor = \\Bigg \\lfloor \\frac { \\lceil d \\rceil } { 4 } \\Bigg \\rfloor = \\Bigg \\lfloor \\frac { \\lceil d \\rceil + 1 } { 4 } \\Bigg \\rfloor , \\end{align*}"}
-{"id": "3984.png", "formula": "\\begin{align*} p ^ { \\beta _ 1 } _ { k - 1 } ( 1 , t ) = - ( - \\lambda ) ^ { k - 1 } \\underset { \\Omega ^ { k - 1 } _ { 1 } } { \\sum } \\frac { t ^ { k _ 0 \\beta _ 0 + k _ 1 \\beta _ 1 } } { \\Gamma \\left ( k _ 0 \\beta _ 0 + k _ 1 \\beta _ 1 + 1 \\right ) } . \\end{align*}"}
-{"id": "9869.png", "formula": "\\begin{align*} x ^ \\nu + d ( x ^ \\nu ) = P _ K \\left ( x ^ \\nu + d ( x ^ \\nu ) - \\frac { \\nabla _ 1 \\tilde f ( d ( x ^ \\nu ) ; x ^ \\nu ) + \\nabla _ 1 \\tilde g ( d ( x ^ \\nu ) ; x ^ \\nu ) \\xi ^ \\nu } { 1 + \\| \\xi ^ \\nu \\| } \\right ) . \\end{align*}"}
-{"id": "9534.png", "formula": "\\begin{align*} { \\phi } ^ * _ { 1 } ( z _ 1 ) = x , { \\phi } ^ * _ { 1 } ( z _ i ) = x _ i \\textrm { f o r } \\ i > 1 \\quad \\mbox { a n d } { \\phi } ^ * _ { 2 } ( z _ 1 ) = y , { \\phi } ^ * _ { 2 } ( x _ i ) = x _ i \\quad \\mbox { f o r } \\ i > 1 . \\end{align*}"}
-{"id": "9949.png", "formula": "\\begin{align*} { \\cal N } ( u ) - V _ 1 = { \\cal N } ( \\varphi ( u ) ) - V _ 2 \\end{align*}"}
-{"id": "6850.png", "formula": "\\begin{align*} \\sup _ { y \\in Y } \\sum _ { t = 1 } ^ { T } \\theta _ t \\phi _ t ( x _ t , y ) - \\inf _ { x \\in X } \\sum _ { t = 1 } ^ { T } \\theta _ t \\phi _ t ( x , y _ t ) \\leq r ( T ) , \\lim _ { T \\to \\infty } r ( T ) = 0 . \\end{align*}"}
-{"id": "2010.png", "formula": "\\begin{align*} x _ i x _ { i + 1 } = z _ i ( q ) + O ( q ) \\end{align*}"}
-{"id": "3593.png", "formula": "\\begin{align*} \\int _ a ^ b f ( t ) \\dd t = ( b - a ) f ( \\xi ) , \\end{align*}"}
-{"id": "2663.png", "formula": "\\begin{align*} \\Theta ( \\pm 1 ) = 0 , \\ \\Theta ' ( \\pm 1 ) = \\mp 2 , \\end{align*}"}
-{"id": "7082.png", "formula": "\\begin{align*} 0 \\to C F ^ - \\to C F ^ \\infty \\to C F ^ + : = C F ^ \\infty / C F ^ - \\to 0 . \\end{align*}"}
-{"id": "7934.png", "formula": "\\begin{align*} g _ I : = f _ I \\circ \\cdots \\circ f _ { i _ 1 i _ 2 } \\circ f _ { i _ 1 } : \\bar V \\to \\bar V , \\ , \\ , \\ , \\bar V _ I : = g _ I ( \\bar V ) \\subset \\bar V . \\end{align*}"}
-{"id": "5177.png", "formula": "\\begin{align*} u _ { r } ( x ) \\coloneqq M ^ { x } _ { 1 } ( D _ { [ 0 , \\ell _ { r } ] } , D _ { [ r , 1 ] } ) - g _ { 1 , [ r , 1 ] } ( x ) = \\begin{cases} f _ { 1 } ( x ) - g _ { 1 , [ r , 1 ] } ( x ) , & x \\in [ 0 , \\ell _ { r } ) , \\\\ \\left ( f _ { 1 } ( \\ell _ { r } ) - g _ { 1 , [ r , 1 ] } ( \\ell _ { r } ) \\right ) \\frac { r - x } { r - \\ell _ { r } } , & x \\in [ \\ell _ { r } , r ) , \\\\ 0 , & x \\in [ r , 1 ] , \\end{cases} \\end{align*}"}
-{"id": "7889.png", "formula": "\\begin{align*} \\varphi ( t , r , s ) = ( t - r ) ^ 2 + ( t - s ) ^ 2 + ( r - s ) ^ 2 \\ge 0 , \\end{align*}"}
-{"id": "8784.png", "formula": "\\begin{align*} F ^ N ( x , v ) = \\begin{cases} \\displaystyle V ' ( | x | , v ) \\frac { x } { | x | } \\mathcal { H } ( x , v ) , & | x | \\ge N ^ { - \\theta } , \\\\ [ 5 m m ] \\displaystyle N ^ { \\theta } V ' ( | x | , v ) x \\mathcal { H } ( x , v ) , & | x | < N ^ { - \\theta } . \\end{cases} \\end{align*}"}
-{"id": "4357.png", "formula": "\\begin{align*} & \\Big | \\sum _ { j = 1 } ^ \\infty ( M _ { a _ j } P _ \\alpha M _ { 1 - b _ j } f ) ( z ) \\Big | e ^ { - \\frac { \\alpha } { 2 } | z | ^ 2 } \\\\ & \\leq \\Big ( \\frac { \\alpha } { \\pi } \\Big ) ^ n \\sum _ { j = 1 } ^ \\infty | a _ j ( z ) | \\int _ { \\mathbb { C } ^ n } | 1 - b _ j ( w ) | | f ( w ) | e ^ { - \\frac { \\alpha } { 2 } | w | ^ 2 } e ^ { - \\frac { \\alpha } { 2 } | w - z | ^ 2 } d w \\\\ & \\leq 2 ^ n \\| f \\| _ { \\infty , \\alpha } \\end{align*}"}
-{"id": "2592.png", "formula": "\\begin{align*} C ^ \\alpha _ 0 ( \\varphi ^ \\beta ) : = \\{ f : ( 0 , 1 ) \\to \\R \\ | \\ \\varphi ^ \\beta f \\in C ^ \\alpha _ 0 ( [ 0 , 1 ] ) \\} , \\end{align*}"}
-{"id": "1781.png", "formula": "\\begin{gather*} \\exists s _ 0 > 0 G ( s _ 0 ) < 0 \\\\ - C | s | ^ { p ^ * } \\leq G ( s ) , s \\geq s _ * , 2 < p _ * < 6 \\\\ G ( 0 ) = G ' ( 0 ) , | G '' ( s ) | \\leq C ( | s | ^ { p - 2 } + | s | ^ { q - 2 } ) , 2 < p < q \\end{gather*}"}
-{"id": "1870.png", "formula": "\\begin{align*} C : = ( - \\infty , A _ 1 ^ i + \\delta ) \\times \\cdots \\times ( - \\infty , A _ d ^ i + \\delta ) . \\end{align*}"}
-{"id": "6097.png", "formula": "\\begin{align*} \\int _ { B ^ n } \\cos ( g ( r ) ) { f _ { \\frak { e } _ k } ( x / r ) _ j } \\tfrac { \\partial \\eta } { \\partial x _ i } d x = - \\int _ { B ^ n } \\tfrac { \\partial h } { \\partial x _ i } \\eta d x , \\end{align*}"}
-{"id": "6832.png", "formula": "\\begin{align*} \\int _ { \\mathbb { S } ^ 2 _ { \\lambda } } e ^ { w _ { \\lambda } } \\phi = 0 . \\end{align*}"}
-{"id": "4932.png", "formula": "\\begin{align*} x . y + y . x = - 2 q ( x , y ) \\end{align*}"}
-{"id": "3763.png", "formula": "\\begin{align*} v _ k - v _ { k + 1 } = v _ k L _ k ^ { - 1 / 1 6 } \\geq \\frac { 1 } { L ^ { 1 / 8 } _ k } \\geq \\frac { 4 } { \\lfloor L _ k ^ { 1 / 2 } \\rfloor } = 4 \\ , \\frac { L _ k } { L _ { k + 1 } } , \\end{align*}"}
-{"id": "1284.png", "formula": "\\begin{align*} \\mbox { C a p } _ { \\mathcal { A } } ( E ) \\not = 0 . \\end{align*}"}
-{"id": "2769.png", "formula": "\\begin{align*} C ^ * _ f ( \\chi _ 0 , s ) = \\Big ( \\prod \\limits _ { j = 0 } ^ { \\infty } L ^ * _ f ( \\chi _ 0 , q ^ j s ) ^ { \\tbinom { n + j - 1 } { n - 1 } } \\Big ) ^ { ( - 1 ) ^ { n - 1 } } . \\end{align*}"}
-{"id": "7511.png", "formula": "\\begin{gather*} u _ y = v _ x , \\end{gather*}"}
-{"id": "2333.png", "formula": "\\begin{align*} \\tilde { \\pi } _ { 0 } = \\pi _ { 0 } | _ { ^ { r } ( \\textbf { M } ) } : ( ^ { r } ( \\textbf { M } ) , \\tilde { \\tau } ) \\rightarrow ( ^ { r } ( M _ { 0 } , M _ { 1 } ) , d ^ { r } ) \\end{align*}"}
-{"id": "7393.png", "formula": "\\begin{align*} \\Phi _ 2 = \\begin{pmatrix} x - v t & y & z & t \\\\ - u y - 2 z v & x + v t & - u t & z \\\\ - w z & w t & x - v t & - y \\\\ - u w t & - w z & u y + 2 v z & x + v t \\end{pmatrix} \\end{align*}"}
-{"id": "4689.png", "formula": "\\begin{align*} & \\Phi ( r ) - \\Phi ( 0 ^ + ) \\\\ & \\phantom { A } \\geq \\Phi ( r ) \\ , e ^ { \\bar { C _ 3 } r ^ { 1 - \\frac { n } { \\Theta } } } + C _ 4 \\ , \\int _ 0 ^ r \\left ( t ^ { - \\frac { n } { \\Theta } } + \\frac { \\omega ( t ) } { t } \\right ) e ^ { \\bar { C _ 3 } t ^ { 1 - \\frac { n } { \\Theta } } } \\ , d t - \\Phi ( 0 ^ + ) - c \\ , \\left ( r ^ { 1 - \\frac { n } { \\Theta } } + \\int _ 0 ^ r \\frac { \\omega ( t ) } { t } \\ , d t \\right ) . \\end{align*}"}
-{"id": "7401.png", "formula": "\\begin{align*} a = a _ 0 a _ 1 a _ 2 a _ 3 a _ 4 , a ^ * = a _ 4 ^ * a _ 3 ^ * a _ 2 ^ * a _ 1 ^ * a _ 0 ^ * , b = a _ 5 ^ * a _ 5 , c = a _ 7 ^ * a _ 7 . \\end{align*}"}
-{"id": "4835.png", "formula": "\\begin{align*} D F _ p ( x , y , z ) & = x F _ x ( p ) + y F _ y ( p ) + z F _ z ( p ) , \\\\ D ^ 2 F _ p ( x , y , z ) & = x ^ 2 F _ { x x } ( p ) + y ^ 2 F _ { y y } ( p ) + z ^ 2 F _ { z z } ( p ) \\\\ & + 2 x y F _ { x y } ( p ) + 2 x z F _ { x z } ( p ) + 2 y z F _ { y z } ( p ) . \\end{align*}"}
-{"id": "3067.png", "formula": "\\begin{align*} \\nabla _ x F ( t , u ( t ) ) = 0 \\end{align*}"}
-{"id": "6072.png", "formula": "\\begin{align*} E ( \\phi ) = c \\int _ 0 ^ { \\pi / 2 } ( r ' ( t ) ^ 2 + \\lambda _ 1 \\tfrac { \\sin ^ 2 r } { \\sin ^ 2 t } + \\lambda _ 2 \\tfrac { \\cos ^ 2 r } { \\cos ^ 2 t } ) \\sin ^ { p _ 1 } t \\cos ^ { p _ 2 } t d t , \\end{align*}"}
-{"id": "3311.png", "formula": "\\begin{align*} D ^ * ( r ) = \\inf \\left \\{ \\frac { D ( r ) } { L } : \\left ( D ( r ) , L \\right ) \\right \\} , \\end{align*}"}
-{"id": "4507.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( \\phi ( \\Delta _ n ^ * \\Delta _ n ) - \\phi ( \\Delta _ n ) ^ * \\phi ( \\Delta _ n ) ) = 0 . \\end{align*}"}
-{"id": "8248.png", "formula": "\\begin{align*} E _ { \\tilde { v } } ( s , g ) = [ \\omega _ { \\pi } ^ { - 1 } \\omega _ { \\pi } ^ { \\mathfrak { L } } ] ( \\det ( g ) ) E _ { v } ( s , g ) . \\end{align*}"}
-{"id": "2553.png", "formula": "\\begin{align*} f _ \\delta ( x ) = \\begin{cases} 1 - \\sqrt { 1 - \\left ( | x | - \\delta \\right ) ^ 2 } & x \\in ( - \\delta , \\delta ) , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "6803.png", "formula": "\\begin{align*} \\mathcal { L } ( u ) = \\Delta _ g u + \\frac { \\rho } { \\int _ { \\mathbb { S } ^ 2 } e ^ { w _ { \\lambda } } } e ^ { w _ { \\lambda } } u . \\end{align*}"}
-{"id": "6115.png", "formula": "\\begin{align*} Y _ { t } & = t - T _ { N _ { t } } \\\\ Z _ { t } & = T _ { N _ { t } + 1 } - t . \\end{align*}"}
-{"id": "8117.png", "formula": "\\begin{align*} \\dim ( A _ 1 ) \\leq \\dim ( F A _ 0 ' ) = \\dim ( A _ 0 ' ) < \\dim ( A _ 0 ) . \\end{align*}"}
-{"id": "3482.png", "formula": "\\begin{align*} v : = - 2 f ^ \\prime + H - 2 \\frac { \\tau ^ \\prime } { \\tau } ( f - \\nu ) \\end{align*}"}
-{"id": "682.png", "formula": "\\begin{align*} \\cot \\beta = \\cot \\beta _ 0 + \\sum _ { n = 0 } ^ { \\infty } \\cfrac { c _ n \\varphi _ 0 ^ 2 ( \\pi , \\mu _ n ^ 0 ) } { 1 + c _ n a _ n ^ 0 } . \\end{align*}"}
-{"id": "9845.png", "formula": "\\begin{align*} ( f \\ast g ) \\hat { \\ , } ( y ) = \\hat { f } ( y ) \\cdot \\hat { g } ( y ) . \\end{align*}"}
-{"id": "4484.png", "formula": "\\begin{align*} \\psi ( s ) = 1 > | \\phi ( s ) | \\end{align*}"}
-{"id": "885.png", "formula": "\\begin{align*} \\bar { Q } _ n ( Y ) = \\sum _ { i , j = 1 } ^ { N _ n } \\bar { \\gamma } _ n ( i , j ) Y _ i Y _ j , \\end{align*}"}
-{"id": "3268.png", "formula": "\\begin{align*} M ^ c ( u , k ) = \\min \\Big \\lbrace \\bigg \\lfloor \\frac { \\bar { R } ^ c ( u , k ) t _ u ^ c } { B } \\bigg \\rfloor , \\frac { \\Psi _ u } { B } \\Big \\rbrace , \\end{align*}"}
-{"id": "2777.png", "formula": "\\begin{gather*} \\frac { \\partial V } { \\partial t } + \\frac { 1 } { 2 } \\sigma ^ { 2 } S ^ { 2 } \\frac { \\partial ^ { 2 } V } { \\partial S ^ { 2 } } + ( r - \\delta ) S \\frac { \\partial V } { \\partial S } - r V = 0 , \\end{gather*}"}
-{"id": "6300.png", "formula": "\\begin{align*} \\{ \\tilde { b } ; \\tilde { \\varepsilon } , \\tilde { g } , ( \\tilde { f } , \\tilde { k } _ 1 ) , ( \\tilde { t } , \\tilde { k } _ 2 ) ; \\{ ( \\tilde { \\alpha } _ i , \\tilde { \\beta } _ i ) \\} _ { i = 1 } ^ { n } \\} . \\end{align*}"}
-{"id": "1752.png", "formula": "\\begin{align*} \\nu _ { f \\sqrt { d \\mu } } = | f | ^ 2 \\mu . \\end{align*}"}
-{"id": "7726.png", "formula": "\\begin{align*} \\lambda _ n = & 2 ( 1 - \\cos \\phi _ n ) , \\ \\ \\ \\ \\ n = 0 , 1 , 2 , \\cdots , N - 1 \\\\ u _ { 0 m } = & \\frac { 1 } { \\sqrt { N } } , \\ \\ \\ \\ \\ m = 0 , 1 , \\cdots N - 1 \\\\ u _ { n m } = & \\sqrt { \\frac { 2 } { N } } \\cos ( m + 1 / 2 \\phi _ n ) , \\ \\ \\ \\ \\\\ & n = 1 , 2 , \\cdots , N - 1 , m = 0 , 1 , \\cdots N - 1 \\end{align*}"}
-{"id": "3439.png", "formula": "\\begin{align*} \\begin{cases} \\dot { u } ( t ) = f ( t , u ( t ) ) , t \\in ( 0 , T ] , \\\\ u ( 0 ) = u _ 0 , \\end{cases} \\end{align*}"}
-{"id": "4741.png", "formula": "\\begin{align*} { \\varepsilon } = F ( X , Y , Z ) \\ , { \\Delta \\sqrt { - \\det g ^ { i j } } } / ( { { L } _ 0 \\ , { L } _ 1 } ) , \\end{align*}"}
-{"id": "4969.png", "formula": "\\begin{align*} f = \\sum _ { j = - \\infty } ^ { \\infty } \\Delta _ j f \\end{align*}"}
-{"id": "4785.png", "formula": "\\begin{align*} { L } _ 2 = \\sqrt { \\alpha a _ 2 + \\beta b _ 2 + \\gamma } , { L } _ 3 = \\sqrt { \\alpha a _ 3 + \\beta b _ 3 + \\gamma } . \\end{align*}"}
-{"id": "7387.png", "formula": "\\begin{align*} \\Z _ { N _ f } ( f ) = D _ { N _ { f } } ( f ) = \\det ( I - \\mathcal { G } ) , \\end{align*}"}
-{"id": "7825.png", "formula": "\\begin{align*} \\int _ { r \\leq r ( x ) \\leq r _ j } | \\nabla v | ^ 2 e ^ { - 2 \\rho } d x + \\int _ { S _ { r } } \\frac { \\partial v } { \\partial r } v e ^ { - 2 \\rho } d x - \\int _ { S _ { r _ j } } \\frac { \\partial v } { \\partial r } v e ^ { - 2 \\rho } d x = \\int _ { r \\leq r ( x ) \\leq r _ j } [ \\lambda - V _ 0 - V _ 1 - V _ 2 ] v ^ 2 e ^ { - 2 \\rho } d x . \\end{align*}"}
-{"id": "4953.png", "formula": "\\begin{align*} f \\left ( t \\right ) = f \\left ( \\frac { M - t } { M - m } m + \\frac { t - m } { M - m } M \\right ) \\le L _ { f } \\left ( t \\right ) , \\end{align*}"}
-{"id": "2416.png", "formula": "\\begin{align*} p _ 1 ( x ) = \\frac { ( a _ 3 x + a _ 2 ) ( a _ 2 + ( 2 a _ 3 + a _ 4 ) x - a _ 2 x ^ 2 ) } { a _ 2 } , \\end{align*}"}
-{"id": "341.png", "formula": "\\begin{align*} \\vert N _ { \\mathbf { S } } ( Q ) \\vert _ { p ' } = \\vert N _ { S _ 1 } ( P _ 1 ) \\vert _ { p ' } , \\end{align*}"}
-{"id": "4908.png", "formula": "\\begin{align*} T = - \\alpha \\beta ^ { - 1 } I - \\beta ^ { - 1 } F . \\end{align*}"}
-{"id": "3222.png", "formula": "\\begin{align*} \\begin{array} { c } \\mathcal H _ { k , p r i m } ( X ) = \\mathcal H _ k ( X ) \\cap \\biggl ( \\sum _ { p \\leq n } H ^ p _ { p r i m } ( X ; \\mathbb Q ) + \\sum _ { p > n } u ^ { 2 p - n } H ^ p _ { p r i m } ( X ; \\mathbb Q ) \\biggr ) . \\end{array} \\end{align*}"}
-{"id": "132.png", "formula": "\\begin{align*} u _ h ' ( 1 - t , \\lambda x + \\mu y ) & = u _ f ' ( 1 - t , x ) = u _ g ' ( 1 - t , y ) , \\\\ u _ h '' ( 1 - t , \\lambda x + \\mu y ) & = u _ f '' ( 1 - t , x ) = u _ g '' ( 1 - t , y ) \\end{align*}"}
-{"id": "5339.png", "formula": "\\begin{align*} [ a \\otimes f _ 1 , b \\otimes f _ 2 ] = [ a , b ] \\otimes f _ 1 f _ 2 + \\{ f _ 1 , f _ 2 \\} ( a , b ) { \\bf k } , \\end{align*}"}
-{"id": "2419.png", "formula": "\\begin{align*} & R _ 2 = 0 , \\ D _ 1 < 0 , \\ D _ 3 < 0 , \\ \\\\ & a _ 6 \\neq 0 , \\ a _ 2 - a _ 1 a _ 6 \\neq 0 , \\ a _ 3 \\neq a _ 6 , \\ a _ 2 ^ 2 + ( a _ 4 + 2 a _ 3 ) ^ 2 \\neq 0 . \\end{align*}"}
-{"id": "7133.png", "formula": "\\begin{align*} ( \\mathcal { W } _ X ^ { - 1 } ) _ { i j } = & ~ \\left ( \\bar { g } ^ { j q } - \\frac { \\langle \\bar { g } ^ { j a } \\bar { \\nabla } _ a s , \\bar { g } ^ { q b } \\bar { \\nabla } _ b s \\rangle } { 1 + s ^ 2 + | \\bar { \\nabla } s | ^ 2 } \\right ) \\tau _ { q i } \\sqrt { \\frac { 1 + s ^ 2 } { 1 + s ^ 2 + | \\bar { \\nabla } s | ^ 2 } } \\end{align*}"}
-{"id": "2635.png", "formula": "\\begin{align*} W _ { r , \\alpha } ^ x = \\left \\{ ( w _ 1 , w _ 2 , \\dots , w _ { n } ) \\in ( \\textbf { B } ( 0 , r ) ) ^ n \\subset ( \\R ^ n ) ^ n : x + w _ i \\in U , | w _ 1 \\wedge \\dots \\wedge w _ n | \\geq \\alpha r ^ n \\right \\} . \\end{align*}"}
-{"id": "9946.png", "formula": "\\begin{align*} u _ t + ( u ^ m ) _ x - ( u ^ n ) _ { x x } = 0 , m , n \\geq 2 , \\end{align*}"}
-{"id": "3955.png", "formula": "\\begin{align*} g _ { S ' } ( x _ { S ' } ) = \\prod _ { T \\in \\kappa _ \\phi ^ { - 1 } ( S ' ) } g _ T ( x _ T ) = g _ { S } ( x _ { S } ) \\end{align*}"}
-{"id": "4576.png", "formula": "\\begin{align*} [ \\cdots [ F _ { 0 | 1 , 0 } , \\tilde F _ { n - 2 , 0 } ] _ q \\cdots \\tilde F _ { i + 1 , 0 } ] _ q & \\equiv - q ^ { - 1 } [ \\cdots [ [ F _ { 0 , 0 } , F _ { 1 , 0 } ] _ q , F _ { n - 1 , 0 } ] _ q \\cdots F _ { i + 2 , 0 } ] _ q \\\\ & = - q ^ { - 1 } [ \\cdots [ [ F _ { 0 , 0 } , F _ { n - 1 , 0 } ] _ q \\cdots F _ { i + 2 , 0 } ] _ q F _ { 1 , 0 } ] _ q \\end{align*}"}
-{"id": "6226.png", "formula": "\\begin{align*} \\sum _ { u = 1 } ^ { t } \\delta ( u ) = \\left \\{ \\begin{array} { l l } t - \\varphi ( t ) , & \\hbox { $ 0 \\leq \\varphi ( t ) \\leq t $ ; } \\\\ 0 , & \\hbox { $ \\varphi ( t ) \\geq t $ . } \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "3062.png", "formula": "\\begin{align*} \\langle f , K _ z \\rangle = f ( z ) . \\end{align*}"}
-{"id": "6519.png", "formula": "\\begin{align*} \\rho = \\frac { V } { \\mathcal { E } } = \\frac { \\Delta \\mathcal { C } ^ { 2 } } { \\mathcal { C } _ { } ^ { 2 } } \\end{align*}"}
-{"id": "9906.png", "formula": "\\begin{align*} A ^ * _ s = \\bigcup _ { i \\in W _ s } V _ i \\ , . \\end{align*}"}
-{"id": "514.png", "formula": "\\begin{align*} & \\frac { m _ 1 } { n } = H _ b ( q ) - \\delta \\\\ & \\frac { m _ 1 + m _ 2 } { n } = H _ b ( q * p _ A ) + \\delta . \\end{align*}"}
-{"id": "3215.png", "formula": "\\begin{align*} \\partial _ t ^ 2 \\phi - \\Delta \\phi + h ( { \\tilde \\phi } ) \\Delta \\phi = F ( \\partial { \\tilde \\phi } ) \\ , \\ , ( t , x ) \\in S _ T \\ . \\end{align*}"}
-{"id": "7540.png", "formula": "\\begin{gather*} 2 . 1 . \\ \\varphi ^ 1 _ \\omega + \\kappa \\varphi ^ 2 _ \\omega = 0 . \\mbox { 2 . 2 . - - 2 . 4 . } \\ \\omega \\varphi ^ 2 _ \\omega + \\varphi ^ 2 = 0 . 2 . 5 . \\ \\varphi ^ 2 _ \\omega = - 2 . 2 . 6 . \\ 1 = 0 . \\end{gather*}"}
-{"id": "6284.png", "formula": "\\begin{align*} Q = C \\log \\Lambda _ r \\leq C ( r , \\nu ) \\left ( 1 + \\log \\| u _ Q \\| _ { H ^ s } \\right ) . \\end{align*}"}
-{"id": "1342.png", "formula": "\\begin{align*} \\nabla _ x L ( x _ c ( y ) , Y _ c ( y ) , \\mu _ c ( y ) , \\lambda _ c ( y ) ) = \\nabla _ x L _ c ( x _ c ( y ) , y ) = 0 \\ , , y \\in \\mathbb { B } _ { \\delta _ 0 } ( \\overline { y } ) \\ , . \\end{align*}"}
-{"id": "8411.png", "formula": "\\begin{align*} L ( s , \\chi ) = \\begin{cases} \\frac { 1 } { 1 - \\chi ( \\varpi ) q ^ { - s } } & \\\\ 1 & \\end{cases} \\end{align*}"}
-{"id": "5786.png", "formula": "\\begin{align*} T _ { k } ( u ) : = ( u , k ) \\end{align*}"}
-{"id": "6732.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\mathbb { P } ( Z > 0 ) = \\lim \\limits _ { N \\rightarrow \\infty } \\mathbb { P } ( R ^ { \\eta } _ { N } \\leq t ( N ) ) = 1 , \\\\ \\end{align*}"}
-{"id": "5972.png", "formula": "\\begin{align*} \\Phi ^ { \\mathrm { c o l } } \\alpha = f , \\Phi _ { i j } ^ { \\mathrm { c o l } } = \\phi _ j ( x _ i ^ { \\mathrm { c o l } } ) . \\end{align*}"}
-{"id": "9404.png", "formula": "\\begin{align*} \\begin{array} { l l l } R ' _ { i * } & = & [ ( 2 0 i - 5 ) K , ( 2 0 i + 1 5 ) K ] \\times [ - 2 0 n K , 2 0 n K ] , \\\\ R _ { i * } & = & [ ( 2 0 i - 1 ) K , ( 2 0 i + 1 ) K ] \\times [ - 2 0 n K , 2 0 n K ] , \\\\ R ' _ { * i } & = & [ - 2 0 n K , 2 0 n K ] \\times [ ( 2 0 i - 5 ) K , ( 2 0 i + 5 ) K ] , \\\\ R _ { * i } & = & [ - 2 0 n K , 2 0 n K ] \\times [ ( 2 0 i - 1 ) K , ( 2 0 i + 1 ) K ] , \\end{array} \\end{align*}"}
-{"id": "7160.png", "formula": "\\begin{align*} \\Delta _ J ( r J ) + V _ r \\sum _ { m = r J + 1 } ^ { ( r + 1 ) J } d _ L ^ * ( m ) \\leq U J ^ 2 + V _ r J g ^ * _ r + W J . \\end{align*}"}
-{"id": "8728.png", "formula": "\\begin{align*} u ^ \\varepsilon ( t ' , \\hat { x } ) - \\bar { u } ^ \\varepsilon ( t ' - \\tau , \\hat { x } ) & = u ^ { \\varepsilon , \\delta } ( \\hat { t } , \\hat { x } ) - ( - \\delta ^ { - 1 } | \\hat { t } - t ' | ^ 2 + u ^ \\varepsilon ( t ' - \\tau , \\hat { x } ) ) \\\\ & \\ge u ^ { \\varepsilon , \\delta } ( \\hat { t } , \\hat { x } ) - u ^ { \\varepsilon , \\delta } ( \\hat { t } - \\tau , \\hat { x } ) . \\end{align*}"}
-{"id": "7147.png", "formula": "\\begin{align*} & \\Delta _ J ( r J ) + V \\sum _ { m = r J + 1 } ^ { ( r + 1 ) J } D ( m , a _ m ^ * ) \\\\ \\leq & U J + V \\sum _ { m = r J + 1 } ^ { ( r + 1 ) J } D ( m , a _ m ^ * ) + \\sum _ { m = r J + 1 } ^ { ( r + 1 ) J } q ( m ) y ( m ) \\\\ = & U J + V \\sum _ { m = r J + 1 } ^ { ( r + 1 ) J } D ( m , a _ m ^ * ) + \\sum _ { m = r J + 1 } ^ { ( r + 1 ) J } q ( r J + 1 ) y ( m ) \\\\ & + \\sum _ { m = r J + 1 } ^ { ( r + 1 ) J } ( q ( m ) - q ( r J + 1 ) ) y ( m ) . \\end{align*}"}
-{"id": "8503.png", "formula": "\\begin{align*} x _ 2 = - \\frac { b _ 2 } { v x _ 1 } = \\frac { b _ 2 } { v } \\sum _ { j \\geq 0 } \\frac { \\alpha ^ j } { b _ { \\chi _ 1 \\chi _ 2 ^ { - 1 } } ^ { j + 1 } } \\varpi ^ { j a ( \\chi _ 1 \\chi _ 2 ^ { - 1 } ) - j r } . \\end{align*}"}
-{"id": "3931.png", "formula": "\\begin{align*} u ( x _ { j } , t _ { n } ) = u _ j ^ n = \\sum \\limits _ { j = i - 1 } ^ { i + 1 } c _ j ^ n ( t ) T B ^ 4 _ j ( x ) . \\end{align*}"}
-{"id": "7610.png", "formula": "\\begin{align*} G _ { T } ( \\gamma ( t ) ) = \\partial ^ 2 _ { 2 2 } A ^ T ( \\gamma ( s ) , \\gamma ( t ) ) , G _ { - T } ( \\gamma ( s ) ) = - \\partial ^ 2 _ { 1 1 } A ^ T ( \\gamma ( s ) , \\gamma ( t ) ) \\end{align*}"}
-{"id": "3284.png", "formula": "\\begin{align*} \\psi _ k ( \\xi , m ) = & \\ , ( \\log 1 6 - 2 ) \\sum _ { j = 1 } ^ k m _ j ^ 2 + \\sum _ { j = 1 } ^ k m _ j ^ 2 \\log m _ j ^ 2 - 4 \\pi \\sum _ { j = 1 } ^ k m _ j ^ 2 H ( \\xi _ j , \\xi _ j ) \\\\ & \\ , - 4 \\pi \\sum _ { i = 1 } ^ k \\sum _ { j = 1 , j \\ne i } ^ k m _ i m _ j G ( \\xi _ i , \\xi _ j ) , \\end{align*}"}
-{"id": "491.png", "formula": "\\begin{align*} \\abs { g ( \\abs { x } , \\abs { t } \\sigma ( s ) ) } \\leq \\begin{cases} C '' \\left ( \\delta ( s ) + \\frac { 1 } { \\kappa ( s ) } \\right ) & \\\\ C '' \\delta ( s ) & \\\\ C '' \\left ( \\frac { 1 } { \\abs { t } \\sigma ( s ) } + \\kappa ( s ) \\right ) & \\end{cases} \\end{align*}"}
-{"id": "3072.png", "formula": "\\begin{align*} \\nabla _ x F ( t , u ( t ) ) = 0 \\end{align*}"}
-{"id": "1131.png", "formula": "\\begin{align*} \\begin{bmatrix} 1 & 0 \\\\ \\frac { a - d } { 2 b } & 1 \\end{bmatrix} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\begin{bmatrix} 1 & 0 \\\\ - \\frac { a - d } { 2 b } & 1 \\end{bmatrix} = \\begin{bmatrix} \\delta & b \\\\ 0 & \\delta \\end{bmatrix} . \\end{align*}"}
-{"id": "683.png", "formula": "\\begin{align*} F ( x , y ) = \\sum _ { n = 0 } ^ { \\infty } \\left ( \\cfrac { 1 } { a _ n } - \\cfrac { 1 } { a _ n ^ 0 } \\right ) \\varphi _ 0 ( x , \\mu _ n ^ 0 ) \\varphi _ 0 ( y , \\mu _ n ^ 0 ) . \\end{align*}"}
-{"id": "9605.png", "formula": "\\begin{align*} f = \\sum _ { k \\in \\Bbb Z } D _ k D ^ N _ k T _ N ^ { - 1 } ( f ) , \\end{align*}"}
-{"id": "8255.png", "formula": "\\begin{align*} F ( s , g ) = - [ M ( s ) v ^ { \\circ } ( s ) ] ( g ) + \\sum _ { 1 \\neq \\gamma \\in B ( F ) \\setminus G ( F ) } v ^ { \\circ } ( s ) ( \\gamma g ) . \\end{align*}"}
-{"id": "3829.png", "formula": "\\begin{align*} \\tau : = \\inf \\{ n \\in \\N \\colon \\ , N ( \\bar { X } _ n , n ) \\ge 1 \\} \\in [ 1 , \\infty ] . \\end{align*}"}
-{"id": "2477.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\| u \\| ^ { 2 } _ { L ^ { 2 } } + \\frac { 1 } { D } \\int _ { \\R ^ { 6 } } \\Big ( M - \\frac { x \\cdot \\xi } { \\left < x \\right > } \\Big ) u ^ { 2 } d \\xi d x + \\int _ { \\R ^ { 6 } } ( - L u ) u d \\xi d x = 0 \\ , . \\end{align*}"}
-{"id": "2677.png", "formula": "\\begin{align*} \\nu _ A ^ { ( m ) } : = \\int _ { - \\infty } ^ \\infty x ^ m d \\nu _ A . \\end{align*}"}
-{"id": "6598.png", "formula": "\\begin{align*} \\liminf _ { J \\ni m \\to \\infty } \\int _ { \\Omega _ n } | D f _ n - D g _ { m , n } | ^ p \\ , d \\mu = 0 \\ . \\end{align*}"}
-{"id": "6351.png", "formula": "\\begin{align*} ( D \\varphi _ k ^ { \\mathbb { R } _ + } ( \\vec { y } ^ * ) ) _ { i , j } = ( z _ 1 ^ * ) ^ { i - 1 } \\left ( \\binom { 2 i } { i - j } - \\binom { 2 i } { i - j - 1 } \\right ) . \\end{align*}"}
-{"id": "3923.png", "formula": "\\begin{align*} \\begin{aligned} 3 ^ { l _ 1 + 1 } x ^ { r + 1 8 } & + a ( r ) x ^ { 1 5 } + a ( r ) x ^ { 3 3 } + 3 ^ { 2 l _ 1 + 1 } x ^ { 2 r + 9 } \\\\ & + 2 \\cdot 3 ^ { l _ 1 + 1 } a ( r ) x ^ { 2 4 + r } + 2 \\cdot 3 ^ { l _ 1 + 1 } b ( r ) x ^ { 4 2 + r } \\in \\langle F \\rangle . \\end{aligned} \\end{align*}"}
-{"id": "7696.png", "formula": "\\begin{align*} & Z _ { i m } = \\left [ \\begin{array} { c c } ( Q _ { i j } Q _ { m j } - Q _ { i j } Q _ { m k } - Q _ { i k } Q _ { m j } + Q _ { i k } Q _ { m k } ) & 0 \\\\ 0 & 0 \\end{array} \\right ] \\\\ & = \\left [ \\begin{array} { c c } ( u _ { i j } u _ { m j } - u _ { i j } u _ { m k } - u _ { i k } u _ { m j } + u _ { i k } u _ { m k } ) & 0 \\\\ 0 & 0 \\end{array} \\right ] \\end{align*}"}
-{"id": "4261.png", "formula": "\\begin{align*} f _ o = \\frac { \\alpha _ { m + 1 } t ^ { m + 1 } + \\alpha _ { m + 2 } t ^ { m + 2 } + \\cdots + \\alpha _ p t ^ p } { t ^ p } h _ o = - [ f _ o g ] . \\end{align*}"}
-{"id": "7746.png", "formula": "\\begin{align*} & \\sum ^ { 2 N - 1 } _ { n = 1 } \\frac { 1 } { \\sin ^ 2 \\frac { n \\pi } { 2 N } } = \\frac { 4 N ^ 2 } { 3 } - \\frac { 1 } { 3 } \\\\ & = \\frac { 1 } { \\sin ^ 2 \\frac { \\pi } { 2 N } } + \\frac { 1 } { \\sin ^ 2 \\frac { 2 \\pi } { 2 N } } + \\cdots + \\frac { 1 } { \\sin ^ 2 \\frac { ( N - 1 ) \\pi } { 2 N } } + \\frac { 1 } { \\sin ^ 2 \\frac { N \\pi } { 2 N } } \\\\ & \\ \\ \\ + \\frac { 1 } { \\sin ^ 2 \\frac { ( N + 1 ) \\pi } { 2 N } } + \\cdots + \\frac { 1 } { \\sin ^ 2 \\frac { ( 2 N - 2 ) \\pi } { 2 N } } + \\frac { 1 } { \\sin ^ 2 \\frac { ( 2 N - 1 ) \\pi } { 2 N } } \\ , . \\end{align*}"}
-{"id": "4631.png", "formula": "\\begin{align*} \\epsilon > \\left | \\sum _ { n = 0 } ^ { L _ j - 1 } f ( T ^ n x ) - \\sum _ { n = 0 } ^ { L _ i - 1 } f ( T ^ n x ) \\right | = \\left | \\sum _ { n = 0 } ^ { L _ j - L _ i - 1 } f ( T ^ n T ^ { L _ i } x ) \\right | \\end{align*}"}
-{"id": "5619.png", "formula": "\\begin{align*} \\sigma ( Y _ t ) - \\sigma ( Y _ s ) = & \\sigma ' ( Y _ s ) Y _ s ' ( B ^ H _ { s , t } ) + R ^ { \\sigma ( Y ) } _ { s , t } \\\\ = & \\sigma ' ( Y _ s ) Y _ { 0 , s } ' ( B ^ H _ { s , t } ) + \\sigma ' ( Y _ s ) \\sigma ( x _ 0 ) ( B ^ H _ { s , t } ) + R ^ { \\sigma ( Y ) } _ { s , t } . \\end{align*}"}
-{"id": "849.png", "formula": "\\begin{align*} v ( t , x ) = \\inf _ { u \\in \\mathcal { U } } \\mathbb { E } _ { x , t } \\left [ \\int _ t ^ \\theta L ( s , X _ s , u _ s ) d E _ s + v ( \\theta , X _ \\theta ) \\right ] . \\end{align*}"}
-{"id": "9798.png", "formula": "\\begin{align*} g [ A ] = \\chi _ { - A } ( \\pi ( g ^ { - 1 } ) ) [ \\pi ( g ^ { - t } A ) ] = \\theta \\circ \\kappa ( - A , g ^ { - 1 } ) [ \\pi ( g ^ { - t } A ) ] . \\end{align*}"}
-{"id": "3209.png", "formula": "\\begin{align*} \\phi _ { t t } - \\Delta \\phi = a | \\partial _ { t } \\phi | ^ { p } + b | \\nabla \\phi | ^ { p } , \\end{align*}"}
-{"id": "4009.png", "formula": "\\begin{align*} K _ { p } ^ { \\ast } \\left ( w \\right ) = \\sup _ { u \\in L ^ { 1 } ( \\mu ) } \\left ( \\int w u \\ d \\mu - K _ { p } \\left ( u \\right ) \\right ) , w \\in L ^ { \\infty } ( \\mu ) \\end{align*}"}
-{"id": "7325.png", "formula": "\\begin{align*} s ( x ) = p ^ { m ( x ) } . \\end{align*}"}
-{"id": "441.png", "formula": "\\begin{align*} e ^ { - \\frac { 3 \\pi \\abs { t } } { 2 } } = o \\left ( e ^ { - \\pi \\abs { t } - R } \\abs { t } ^ { n + k _ 1 - 2 } \\right ) \\end{align*}"}
-{"id": "415.png", "formula": "\\begin{align*} p _ { 1 , k _ 1 , k _ 2 } ( x , t ) = \\frac { 1 } { | x | } e ^ { - \\frac { 1 } { 4 } d ( x , t ) ^ 2 } \\Psi ( \\omega ) \\left [ ( - 1 ) ^ { k _ 1 + k _ 2 } \\frac { y _ \\omega ^ { n + k _ 1 + k _ 2 } \\cos ( y _ \\omega ) ^ { k _ 1 } } { \\sin ( y _ \\omega ) ^ { n + k _ 1 } } + O \\left ( \\frac { 1 } { | x | ^ 2 } \\right ) \\right ] \\end{align*}"}
-{"id": "8578.png", "formula": "\\begin{align*} & \\sup _ { x \\in \\Omega } \\left ( \\int _ { \\Omega \\ , \\cap \\ , B _ { H _ 0 } ( x , 1 ) } | e ^ { - h ( y , t ) } u ( y , t ) | ^ 2 \\ , d y \\right ) ^ { \\frac { 1 } { 2 } } \\\\ & \\qquad \\le C _ 2 t ^ { - \\frac { N } { 4 } } \\sup _ { x \\in \\Omega } \\int _ { \\Omega \\ , \\cap \\ , B _ { H _ 0 } ( x , 1 ) } e ^ { - H _ 0 ( y ) ^ 2 } | \\phi ( y ) | \\ , d y \\end{align*}"}
-{"id": "9029.png", "formula": "\\begin{align*} \\textstyle J \\wedge \\nabla _ \\perp \\psi = \\frac 2 n \\nabla \\psi \\end{align*}"}
-{"id": "6091.png", "formula": "\\begin{align*} \\Phi ^ { '' } ( t ) + ( n - 2 ) \\Phi ' ( t ) - \\frak { e } _ k \\sin ( 2 \\Phi ( t ) ) - \\sin ( 2 \\Phi ( t ) ) g ' ( t ) ^ 2 = 0 . \\end{align*}"}
-{"id": "2631.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\frac { \\log | t - s | } { \\phi _ 0 ( t ) } \\ , u ( s ) \\ , d s \\ , d t = - \\int _ 0 ^ 1 \\frac { f ( s ) } { \\phi _ 0 ( s ) } \\ , d s \\end{align*}"}
-{"id": "1570.png", "formula": "\\begin{align*} - \\nu ( C _ { j _ k } ( \\alpha ) ) & = - \\nu \\left ( \\frac { \\alpha ! } { ( \\alpha - j _ k ) ! j _ k ! } \\right ) \\\\ & = \\alpha - s ( \\alpha ) - ( \\alpha - j _ k ) + s ( \\alpha - j _ k ) - j _ k + s ( j _ k ) \\\\ & = s ( j _ k ) + s ( \\alpha - j _ k ) - s ( \\alpha ) \\\\ & = c ( j _ k , \\alpha ( k ) - j _ k ) \\end{align*}"}
-{"id": "4164.png", "formula": "\\begin{align*} \\frac { 1 } { n } e ^ { n - 1 } _ { i , g + 1 } + \\left ( - 1 + \\frac { 1 } { n } \\right ) e _ { i , g } ^ { n - 1 } + \\left ( \\frac { 1 } { n } - \\frac { 1 } { n ^ 2 } - 1 \\right ) e _ { i , g + 1 } ^ { n - 2 } + \\left ( - 1 + \\frac { 1 } { n } \\right ) \\sum _ { \\tau = 2 } ^ { n - 2 } e _ { i , g + \\tau } ^ { n - 1 - \\tau } . \\end{align*}"}
-{"id": "6978.png", "formula": "\\begin{align*} M _ { i j } = - \\frac { 1 } { 2 f _ 2 ( B ) } b _ { j i } , \\end{align*}"}
-{"id": "1475.png", "formula": "\\begin{align*} ( D _ r \\{ x _ 3 ^ 2 \\} ) ^ { * , - u + 1 } = \\{ 0 \\} . \\end{align*}"}
-{"id": "9465.png", "formula": "\\begin{align*} S _ n ( i ) - S _ { n - 1 } ( i ) & = \\sum _ { j = 0 } ^ { n - 1 } \\frac { q ^ { i j } ( q ; q ) _ { n - 1 + j } ( 1 - q ^ { n + j } - 1 ) } { ( q ^ 2 ; q ^ 2 ) _ j } + \\frac { q ^ { i n } ( q ; q ) _ { 2 n } } { ( q ^ 2 ; q ^ 2 ) _ n } \\\\ & = - q ^ n S _ { n - 1 } ( i + 1 ) + q ^ { i n } ( q ; q ^ 2 ) _ n . \\end{align*}"}
-{"id": "6500.png", "formula": "\\begin{align*} f \\left ( k _ { \\mathrm { o } } \\right ) = \\frac { e ^ { i \\theta _ { \\mathrm { o } } } \\sin \\theta _ { \\mathrm { o } } } { k _ { \\mathrm { o } } } \\overset { \\theta \\left ( k _ { \\mathrm { o } } \\right ) \\ll 1 } { \\approx } \\frac { \\theta \\left ( k _ { \\mathrm { o } } \\right ) } { k _ { \\mathrm { o } } } + \\mathcal { O } \\left ( \\theta ^ { 2 } \\right ) , \\end{align*}"}
-{"id": "3683.png", "formula": "\\begin{align*} \\psi ^ { ( n ) , G H H } = \\varepsilon ^ { ( n ) } \\circ \\Psi : I ^ n _ { S / C } \\to Z ^ { ( n ) } . \\end{align*}"}
-{"id": "795.png", "formula": "\\begin{align*} n = B k + r B \\ge 1 , \\ 0 \\le | r | \\le \\left \\lfloor \\frac { k } { 2 } \\right \\rfloor . \\end{align*}"}
-{"id": "7220.png", "formula": "\\begin{align*} \\theta _ { \\alpha _ { j } } ^ { n d _ { j } } ( \\exp ( t ) ) & = e ^ { n d _ { j } \\alpha _ { j } \\left ( \\frac { 1 } { n } \\left ( ( m _ { 1 } + 1 ) \\omega _ { 1 } ^ { * } + \\cdots + ( m _ { r } + 1 ) \\omega _ { r } ^ { * } \\right ) \\right ) } = e ^ { d _ { j } \\alpha _ { j } \\left ( ( m _ { 1 } + 1 ) \\omega _ { 1 } ^ { * } + \\cdots + ( m _ { r } + 1 ) \\omega _ { r } ^ { * } \\right ) } \\\\ & = e ^ { d _ { j } \\alpha _ { j } \\left ( ( m _ { j } + 1 ) \\omega _ { j } ^ { * } \\right ) } = e ^ { 2 \\pi i ( m _ { j } + 1 ) } = 1 . \\end{align*}"}
-{"id": "3906.png", "formula": "\\begin{align*} \\lim _ { \\ell \\to \\infty } \\frac { \\langle H \\rangle ^ { \\ell } _ { m i n } } { E ^ { \\ell } _ { 0 } } = \\lim _ { \\ell \\to \\infty } \\frac { ( \\ell + 1 ) ^ 2 } { ( \\ell + \\frac { 3 } { 2 } ) } \\Big ( \\frac { \\Gamma ( \\ell + 1 ) } { \\Gamma ( \\ell + \\frac { 3 } { 2 } ) } \\Big ) ^ 2 = 1 . \\end{align*}"}
-{"id": "8354.png", "formula": "\\begin{align*} m = \\begin{cases} \\frac { n } { 2 } & \\mbox { i f $ V _ { \\Z _ p } $ i s a n o r t h o g o n a l s u m o f h y p e r b o l i c p l a n e s } \\\\ \\lfloor \\frac { n } { 2 } \\rfloor & \\mbox { i f $ n $ i s o d d } \\\\ \\frac { n } { 2 } - 1 & \\mbox { o t h e r w i s e , } \\end{cases} \\end{align*}"}
-{"id": "8735.png", "formula": "\\begin{align*} & ( x , y ) \\mapsto u ( T , x ) - v ( T , y ) - \\varphi ( x , y ) \\quad \\\\ & \\varphi ( x , y ) : = \\frac { | x - \\hat { x } | ^ 4 + | y - \\hat { y } | ^ 4 } { \\eta } - | x - \\tilde { x } | ^ 4 - | y - \\tilde { y } | ^ 4 \\end{align*}"}
-{"id": "4798.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c c c c } A _ 1 & 0 & \\cdots & 0 & 0 \\\\ 0 & A _ 2 & \\cdots & 0 & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & \\cdots & A _ d & 0 \\\\ \\hline \\multicolumn { 5 } { c } { A _ 0 } \\end{array} \\right ) \\in R ^ { m \\times m } , b = \\left ( \\begin{array} { c } b ^ 1 \\\\ b ^ 2 \\\\ \\vdots \\\\ b ^ d \\\\ \\hline b ^ 0 \\end{array} \\right ) \\in R ^ m , \\end{align*}"}
-{"id": "3145.png", "formula": "\\begin{align*} m _ { \\alpha , \\beta } ( \\mathrm { d } x ) : = e ^ { - \\alpha } \\delta _ { 0 } ( \\mathrm { d } x ) + \\beta e ^ { - \\alpha - \\beta x } \\sqrt { \\alpha ( \\beta x ) ^ { - 1 } } I _ { 1 } \\left ( 2 \\sqrt { \\alpha \\beta x } \\right ) \\mathrm { d } x , x \\in \\mathbb { R } _ { \\geqslant 0 } , \\end{align*}"}
-{"id": "386.png", "formula": "\\begin{align*} D _ { n p } & = \\sum _ { r , s \\in \\mathbb { Z } } | b _ { n , r , s } | ^ p = \\sum _ { r , s \\in \\mathbb { Z } , | b _ { n , r , s } | \\ge 1 } | b _ { n , r , s } | ^ p + \\sum _ { r , s \\in \\mathbb { Z } , | b _ { n , r , s } | < 1 } | b _ { n , r , s } | ^ p \\\\ & \\le \\sum _ { r , s \\in \\mathbb { Z } , | b _ { n , r , s } | \\ge 1 } A ^ p + \\sum _ { r , s \\in \\mathbb { Z } , | b _ { n , r , s } | < 1 } b _ { n , r , s } ^ 2 \\\\ & \\le \\lceil a ^ 2 n ^ 2 \\rceil A ^ p + \\sigma _ n ^ 2 \\end{align*}"}
-{"id": "4218.png", "formula": "\\begin{align*} \\tau ^ 0 [ G ( U _ N , U _ N ^ * ) ] = \\frac { 1 } { N } \\sum _ { \\substack { \\kappa : V \\to [ N ] \\\\ \\kappa } } \\mathbb { E } \\left [ \\prod _ { ( u , v ) \\in E _ 1 } U _ N ( \\kappa ( v ) , \\kappa ( u ) ) \\prod _ { ( r , s ) \\in E _ 2 } \\overline { U _ N ( \\kappa ( r ) , \\kappa ( s ) ) } \\right ] . \\end{align*}"}
-{"id": "1134.png", "formula": "\\begin{align*} M _ p = \\begin{bmatrix} a - b _ p j & b _ p \\\\ a i - b _ p i j - a ^ { - 1 } j & b _ p i + a ^ { - 1 } \\end{bmatrix} , M _ m = \\begin{bmatrix} a - b _ m j & b _ m \\\\ a i - b _ m i j - a ^ { - 1 } j & b _ m i + a ^ { - 1 } \\end{bmatrix} . \\end{align*}"}
-{"id": "9585.png", "formula": "\\begin{align*} T _ k \\bigg ( \\frac 1 { T _ k ( 1 ) } \\bigg ) ( x ) & = \\int _ { \\Bbb R ^ n } \\psi ( 2 ^ k { \\bar \\rho } ( x , y ) ) \\frac 1 { T _ k ( 1 ) ( y ) } d \\mu ( y ) \\\\ & \\approx \\int _ { \\Bbb R ^ n } \\psi ( 2 ^ k { \\bar \\rho } ( x , y ) ) \\frac 1 { V _ k ( y ) } d \\mu ( y ) \\\\ & \\approx \\frac 1 { V _ k ( x ) } \\int _ { \\Bbb R ^ n } \\psi ( 2 ^ k { \\bar \\rho } ( x , y ) ) d \\mu ( y ) \\\\ & = \\frac 1 { V _ k ( x ) } T _ k ( 1 ) ( x ) \\approx 1 . \\end{align*}"}
-{"id": "8548.png", "formula": "\\begin{align*} \\lambda _ 1 = \\frac { G _ 1 ( T ) - G _ 3 ( T ) + ( \\alpha _ 1 - \\alpha _ 3 ) \\tau } { 2 \\pi } , \\lambda _ 2 = \\frac { G _ 2 ( T ) - G _ 3 ( T ) + ( \\alpha _ 2 - \\alpha _ 3 ) \\tau } { 2 \\pi } \\in \\mathbb { Q } . \\end{align*}"}
-{"id": "7871.png", "formula": "\\begin{align*} \\{ ( x _ 1 , x _ 2 , x _ 3 ) \\in \\R ^ 3 \\mid x _ 1 ^ 2 + x _ 2 ^ 2 = \\varphi ^ 2 ( x _ 3 ) , x _ 3 \\in I \\} \\end{align*}"}
-{"id": "7512.png", "formula": "\\begin{gather*} \\phi _ t - \\phi _ { x x } - \\phi _ { y y } = 0 . \\end{gather*}"}
-{"id": "8769.png", "formula": "\\begin{align*} \\Delta _ 1 ( X _ 1 ) & = 0 \\ , , \\\\ \\Delta _ 1 ( [ X _ 1 , X _ 2 ] ) = \\Delta _ 1 ( [ X _ 1 , X _ 2 ] ) & = 1 \\ , , \\\\ \\Delta _ 1 ( [ [ X _ 1 , X _ 2 ] , X _ 3 ] ) & = 2 \\ , , \\\\ \\Delta _ 1 ( [ X _ 1 , [ X _ 2 , X _ 3 ] ] ) & = 1 \\ , , \\\\ \\Delta _ 2 ( [ X _ 1 , [ X _ 2 , X _ 3 ] ] ) = \\Delta _ 3 ( [ X _ 1 , [ X _ 2 , X _ 3 ] ] ) & = 2 \\ , . \\end{align*}"}
-{"id": "1106.png", "formula": "\\begin{align*} \\tilde { n } _ { o , i } = \\mu \\rho _ i ^ 2 \\sum _ { j \\in \\mathcal { M } _ i } \\beta _ j ^ { ( i ) } | \\alpha _ { i j } | ^ 2 + \\sqrt { \\mu \\rho _ i ^ 2 } \\sum _ { j \\in \\mathcal { M } _ i } \\beta _ j ^ { ( i ) } | \\alpha _ { i j } | Z _ { s , j } ^ { ( i ) } + \\sum _ { j \\in \\mathcal { M } _ i } \\beta _ j ^ { ( i ) } Z _ { 0 , j } ^ { ( i ) } , \\end{align*}"}
-{"id": "10120.png", "formula": "\\begin{align*} { \\rm M S E } = \\sigma _ { d _ k } ^ 2 - \\bar { \\boldsymbol p } _ k ^ H ( i ) \\bar { \\boldsymbol R } _ k ^ { - 1 } ( i ) \\bar { \\boldsymbol p } _ k ( i ) \\end{align*}"}
-{"id": "1889.png", "formula": "\\begin{align*} \\ f _ 1 & = \\ 1 _ { ( \\infty , B _ 1 ] } , f _ 2 = 0 a = 0 , \\\\ \\ f _ 1 & = 0 , f _ 2 = \\ 1 _ { ( - \\infty , B _ 2 ] } a = 0 , \\\\ \\ f _ 1 & = \\ 1 _ { ( A ^ i _ 1 , B _ 1 ] } , f _ 2 = \\ 1 _ { ( A ^ i _ 2 , B _ 2 ] } a ^ j = 1 j = i a ^ j = 0 ; \\end{align*}"}
-{"id": "7776.png", "formula": "\\begin{align*} \\mathcal { P } ( 3 ) = \\bigg \\{ \\Big \\{ \\{ 1 \\} , \\{ 2 \\} , \\{ 3 \\} \\Big \\} , \\Big \\{ \\{ 1 , 2 \\} , \\{ 3 \\} \\Big \\} , \\Big \\{ \\{ 1 , 3 \\} , \\{ 2 \\} \\Big \\} , \\Big \\{ \\{ 1 \\} , \\{ 2 , 3 \\} \\Big \\} , \\Big \\{ 1 , 2 , 3 \\Big \\} \\bigg \\} , \\end{align*}"}
-{"id": "4832.png", "formula": "\\begin{align*} A _ { m + 1 } = \\prod _ { j = 1 } ^ { | G _ 0 | } g _ j ^ { \\mathsf { v } _ { g _ j } ( A _ { m + 1 } ) } = \\prod _ { j = 1 } ^ { | G _ 0 | } \\prod _ { i = 1 } ^ m g _ j ^ { \\alpha _ i \\mathsf { v } _ { g _ j } ( A _ i ) } = \\prod _ { i = 1 } ^ m \\bigg ( \\prod _ { j = 1 } ^ { | G _ 0 | } g _ j ^ { \\mathsf { v } _ { g _ j } ( A _ i ) } \\bigg ) ^ { \\alpha _ i } = \\prod _ { i = 1 } ^ m A _ i ^ { \\alpha _ i } . \\end{align*}"}
-{"id": "9280.png", "formula": "\\begin{align*} \\| \\mathcal { X } \\| _ { 1 , } : = \\underset { \\| X \\| _ { 1 } = 1 } { \\sup } \\| \\mathcal { X } X \\| _ { 1 } , \\| \\mathcal { X } \\| _ { \\infty , } : = \\underset { \\| X \\| = 1 } { \\sup } \\| \\mathcal { X } X \\| , \\end{align*}"}
-{"id": "1671.png", "formula": "\\begin{align*} \\mu ( Z ( e g _ 1 e g _ 2 e g _ 3 \\cdots e g _ n ) ) = \\mu ( Z ( e g _ 1 e g _ 2 e g _ 3 \\cdots e g _ { n - 1 } ) ) \\ , \\alpha _ n . \\end{align*}"}
-{"id": "362.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( S _ { n } \\leq - x _ { n } \\sigma _ { n } \\right ) = ( 1 - \\Phi ( x _ { n } ) ) ( 1 + o ( 1 ) ) n \\rightarrow \\infty . \\end{align*}"}
-{"id": "7956.png", "formula": "\\begin{align*} \\begin{aligned} \\mu ( B _ i ) & = \\sum _ { I \\in \\mathcal S } \\lambda _ I ^ s ( g _ I ) _ * ( \\mu ) ( B _ i ) = \\sum _ { I \\in \\mathcal S } \\lambda _ I ^ s ( g _ I ) _ * ( \\mu ) ( B _ i \\cap \\bar W _ I ) \\\\ & \\le C ( \\delta ) \\max _ { I \\in \\mathcal S , \\bar W _ I \\cap B _ i \\neq \\phi } \\lambda _ I ^ s . \\end{aligned} \\end{align*}"}
-{"id": "9488.png", "formula": "\\begin{align*} b _ { n , d } = b _ { n - 1 , d + r - 2 } + ( d - 1 ) \\cdot p _ { n - 1 , d - 2 } + d _ { n - 1 , d - 2 } + \\sum \\limits _ { i = 0 } ^ { n - 1 } \\sum \\limits _ { j = 0 } ^ { d - 2 } \\bigl ( p _ { n - i - 1 , d - j - 2 } \\cdot p _ { i , j } + 2 \\cdot s _ { n - i - 1 , d - j - 2 } \\cdot b _ { i , j } \\bigr ) . \\end{align*}"}
-{"id": "5499.png", "formula": "\\begin{align*} U _ m ( x ) = \\int _ 0 ^ x \\int _ t ^ \\infty y ^ m \\dd F ( y ) \\dd t , \\end{align*}"}
-{"id": "2128.png", "formula": "\\begin{align*} \\delta ^ { ( k + 1 ) } = \\mathfrak { D _ 2 } { x ^ { ( k + 1 ) } } \\end{align*}"}
-{"id": "5992.png", "formula": "\\begin{align*} E = \\begin{pmatrix} \\epsilon ^ 2 & \\xi ^ 2 \\\\ \\xi ^ 2 & \\epsilon ^ 2 \\end{pmatrix} , \\xi < \\epsilon < 1 . \\end{align*}"}
-{"id": "5650.png", "formula": "\\begin{align*} \\mathfrak { E } _ { W } ( \\gamma ) = \\int _ \\R \\bigg ( \\frac { 1 } { 2 } | \\dot { \\gamma } | ^ 2 ( t ) + W ( \\gamma ( t ) ) \\bigg ) \\d t , \\end{align*}"}
-{"id": "8864.png", "formula": "\\begin{align*} F ( D ) = F ( D ' ) . \\end{align*}"}
-{"id": "7469.png", "formula": "\\begin{align*} \\dfrac { \\partial } { \\partial z ^ { h } } & = \\dfrac { \\partial \\widetilde { z } ^ k } { \\partial z ^ h } \\dfrac { \\partial } { \\partial \\widetilde { z } k } + \\dfrac { \\partial M _ \\beta ^ \\alpha } { \\partial z ^ h } u ^ \\beta \\dfrac { \\partial } { \\partial \\widetilde { u } ^ \\alpha } , \\\\ \\dfrac { \\partial } { \\partial u ^ \\beta } & = M _ \\beta ^ \\alpha \\dfrac { \\partial } { \\partial \\widetilde { u } ^ \\alpha } , \\end{align*}"}
-{"id": "7938.png", "formula": "\\begin{align*} I _ { - } = i _ 1 \\cdots i _ { m - 1 } , \\end{align*}"}
-{"id": "7192.png", "formula": "\\begin{align*} c _ i ( x _ { n - 1 } , x _ n ) = \\frac { \\tilde { a } _ i } { 2 ^ s } \\left ( x _ { n + 1 } + \\sqrt { x _ { n - 1 } ^ 2 + x _ n ^ 2 } \\right ) ^ s . \\end{align*}"}
-{"id": "7805.png", "formula": "\\begin{align*} \\frac { d ^ { k } } { d x ^ { k } } R _ { v } ^ { k } f = \\frac { d } { d x } R _ { v } \\frac { d ^ { k - 1 } } { d x ^ { k - 1 } } R _ { v } ^ { k - 1 } f . \\end{align*}"}
-{"id": "4228.png", "formula": "\\begin{align*} E : = \\bigoplus _ { V _ i A } E ( V _ i ) \\end{align*}"}
-{"id": "326.png", "formula": "\\begin{align*} k _ 0 + k _ 1 + k _ 2 = l _ 1 + l _ 2 , l _ 1 \\le l _ 2 \\le \\tau \\ , l _ 1 , \\tau : = ( 1 + \\sqrt { 3 } ) / 2 = 1 . 3 6 6 0 2 5 4 0 \\cdots , \\end{align*}"}
-{"id": "286.png", "formula": "\\begin{align*} Z _ K ( 1 - s ) = Z _ K ( s ) , \\end{align*}"}
-{"id": "9323.png", "formula": "\\begin{align*} Q L _ { n } = L _ { n } + L _ { n + 1 } \\textbf { i } + L _ { n + 2 } \\textbf { j } + L _ { n + 3 } \\textbf { k } , \\end{align*}"}
-{"id": "6248.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } & = \\Omega ( \\mu ) + \\Delta ( \\sigma , \\mu ) + \\xi ( y , z , \\sigma , \\mu ) + f ( x , y , z , \\sigma , \\mu ) , \\\\ \\dot { y } & = \\sigma + \\eta ( y , z , \\sigma , \\mu ) + g ( x , y , z , \\sigma , \\mu ) , \\\\ \\dot { z } & = M ( \\mu ) z + \\zeta ( y , z , \\sigma , \\mu ) + h ( x , y , z , \\sigma , \\mu ) , \\end{aligned} \\end{align*}"}
-{"id": "5073.png", "formula": "\\begin{align*} & E _ 3 ( \\frac { B _ { 1 2 , 1 } } { b _ 1 - b _ 2 } ) - \\frac { B _ { 1 3 , 1 } } { b _ 1 - b _ 3 } [ \\frac { B _ { 1 2 , 1 } } { b _ 1 - b _ 2 } + \\frac { B _ { 2 3 , 3 } } { b _ 2 - b _ 3 } ] = 0 , \\\\ & E _ 2 ( \\frac { B _ { 1 2 , 1 } } { b _ 1 - b _ 2 } ) - [ ( \\frac { B _ { 1 2 , 1 } } { b _ 1 - b _ 2 } ) ^ 2 + \\frac { B _ { 1 3 , 1 } B _ { 2 3 , 2 } } { ( b _ 1 - b _ 3 ) ( b _ 2 - b _ 3 ) } ] = R _ { 1 2 1 2 } = b _ 1 b _ 2 + a _ 1 + a _ 2 . \\end{align*}"}
-{"id": "3503.png", "formula": "\\begin{align*} J ( u ( \\cdot ) ) = { \\displaystyle \\int \\limits _ { 0 } ^ { T } } f ( X ^ u { ( t ) } , u ( t ) ) d t + \\Psi ( X ^ u ( T ) ) , \\end{align*}"}
-{"id": "6623.png", "formula": "\\begin{align*} \\int _ \\Omega \\left | D g _ 2 - D g _ 3 \\right | ^ p \\ , d \\mu & = \\int _ { \\bigcup _ { j = 0 } ^ { k - 1 } D ( x _ j , r ) } \\left | D g _ 2 - D g _ 3 \\right | ^ p \\ , d \\mu \\\\ & \\leq K ^ p \\bigl [ g _ 2 - g _ 3 \\bigr ] ^ p _ { \\mathrm { L i p } } \\sum _ { j = 0 } ^ { k - 1 } \\mu \\left ( D ( x _ j , r ) \\right ) \\\\ & = K ^ p C ^ p \\pi k r ^ 2 \\ . \\end{align*}"}
-{"id": "8880.png", "formula": "\\begin{align*} T = S _ \\theta + ( \\cdot , \\overline \\chi \\theta ) u . \\end{align*}"}
-{"id": "3155.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\eta ^ { \\kappa } \\right ] & = \\frac { - 1 } { \\Gamma ( \\theta ) } \\int _ { 0 } ^ { \\infty } \\frac { \\partial } { \\partial u } \\mathbb { E } \\left [ e ^ { - u \\eta } \\right ] u ^ { \\theta - 1 } \\mathrm { d } u \\\\ & = \\frac { \\alpha \\beta } { \\Gamma ( \\theta ) \\left ( 1 - e ^ { - \\alpha } \\right ) } \\int _ { 0 } ^ { \\infty } \\exp \\left \\lbrace \\frac { - \\alpha u } { \\beta + u } \\right \\rbrace \\frac { u ^ { \\theta - 1 } } { ( \\beta + u ) ^ { 2 } } \\mathrm { d } u . \\end{align*}"}
-{"id": "3254.png", "formula": "\\begin{align*} [ Y ] = V ^ i \\cap X , \\end{align*}"}
-{"id": "7271.png", "formula": "\\begin{align*} \\sum _ { 1 \\le n \\le X } r ( n ) = \\frac { N ^ 2 } 2 + O ( N ) , \\end{align*}"}
-{"id": "3358.png", "formula": "\\begin{align*} \\mu ( M ) = \\bigl \\{ V \\subseteq X : ( \\exists C \\in \\Theta ^ c ) ( M \\subseteq C \\subseteq V ) \\bigr \\} . \\end{align*}"}
-{"id": "8827.png", "formula": "\\begin{align*} G _ { j , \\varpi _ p } ( z , \\overline { y } , m ) = 0 , \\end{align*}"}
-{"id": "6659.png", "formula": "\\begin{align*} f ' ( t ) t - f ( t ) \\leq \\left ( 1 + \\frac { 7 } { 2 } \\sqrt { \\eta } \\right ) f '' ( 0 ) t ^ 2 - \\frac { 1 - \\eta } { 2 } f '' ( 0 ) t ^ 2 \\leq ( 1 + 8 \\sqrt { \\eta } ) \\frac { f '' ( 0 ) } { 2 } t ^ 2 = \\Delta \\quad . \\end{align*}"}
-{"id": "8818.png", "formula": "\\begin{align*} _ { t ^ { m - 1 } } \\prod _ { i \\in I } \\frac { t ^ { N _ { i } } p ^ { - \\nu _ { i } } } { 1 - t ^ { N _ { i } } p ^ { - \\nu _ { i } } } & = \\sum _ { ( a _ { i } ) _ { i \\in I } \\in J _ { I , m } } p ^ { - \\sum _ { i \\in I } \\nu _ { i } ( a _ { i } + 1 ) } ; \\\\ _ { t ^ { m - 1 } } \\frac { 1 } { 1 - t } \\prod _ { i \\in I } \\frac { t ^ { N _ { i } } p ^ { - \\nu _ { i } } } { 1 - t ^ { N _ { i } } p ^ { - \\nu _ { i } } } & = \\sum _ { ( a _ { i } ) _ { i \\in I } \\in J ' _ { I , m } } p ^ { - \\sum _ { i \\in I } \\nu _ { i } ( a _ { i } + 1 ) } , \\end{align*}"}
-{"id": "7663.png", "formula": "\\begin{align*} ( x , y ) _ Q = Q ( x + y ) - Q ( x ) - Q ( y ) , x , y \\in L . \\end{align*}"}
-{"id": "6273.png", "formula": "\\begin{align*} \\nabla \\times ( F \\times G ) = [ ( G \\cdot \\nabla ) F - ( \\nabla \\cdot F ) G ] - [ ( F \\cdot \\nabla ) G - ( \\nabla \\cdot G ) F ] . \\end{align*}"}
-{"id": "7685.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } Q & 0 \\\\ 0 & Q \\end{array} \\right ] \\left [ \\begin{array} { c } \\dot { x } _ 1 ( t ) \\\\ \\dot { x } _ 2 ( t ) \\end{array} \\right ] = & \\left [ \\begin{array} { c c } 0 & Q \\\\ - Q L Q ^ { \\top } Q & - Q L Q ^ { \\top } Q \\end{array} \\right ] \\left [ \\begin{array} { c } x _ 1 ( t ) \\\\ x _ 2 ( t ) \\end{array} \\right ] \\\\ & + \\left [ \\begin{array} { c } 0 \\\\ Q \\end{array} \\right ] w ( t ) \\ , , \\end{align*}"}
-{"id": "7258.png", "formula": "\\begin{align*} g = ( d , e ) , d = g x , e = g y \\end{align*}"}
-{"id": "873.png", "formula": "\\begin{align*} \\underline { b } \\leq \\sum _ { i = 1 } ^ { N _ n } \\lambda _ { n , k } ( i ) ^ 2 \\leq \\overline { b } \\end{align*}"}
-{"id": "3178.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } _ { \\geqslant 0 } } f ( y ) \\rho _ { t } ( \\mathrm { d } y ) = \\lambda _ { t } ^ { - 1 } \\int _ { 0 } ^ { t } \\int _ { \\lbrace z > 1 \\rbrace } \\left ( \\int _ { \\mathbb { R } _ { \\geqslant 0 } } f ( y ) m _ { \\alpha ( z , s ) , \\beta ( z , s ) } ( \\mathrm { d } y ) \\right ) \\nu ( \\mathrm { d } z ) \\mathrm { d } s . \\end{align*}"}
-{"id": "2533.png", "formula": "\\begin{align*} \\begin{aligned} K _ \\varrho g & = e ^ { \\alpha \\vartheta ( 0 , x , \\xi ) / 2 } \\int _ { \\R ^ { 3 } } m _ 0 ^ { 1 / 2 } ( \\xi ) \\tilde { k } ( \\xi , \\xi _ * ) m _ 0 ^ { - 1 / 2 } ( \\xi _ * ) e ^ { - \\alpha \\vartheta ( 0 , x , \\xi _ * ) / 2 } g ( t , x , \\xi _ * ) d \\xi _ * + \\varpi \\chi _ R ( \\xi ) g ( \\xi ) \\ , . \\end{aligned} \\end{align*}"}
-{"id": "8368.png", "formula": "\\begin{align*} r ^ { [ i ] } ( m ) = \\# \\{ x \\in \\Lambda ^ { [ i ] } : Q ( x ) = m \\} \\end{align*}"}
-{"id": "8394.png", "formula": "\\begin{align*} \\frac { \\partial g _ { i j } } { \\partial t } & = - 2 H \\cdot h _ { i j } \\\\ [ 5 p t ] \\frac { \\partial H } { \\partial t } & = \\Delta H + H \\cdot h _ { p q } h _ { p q } \\\\ [ 5 p t ] \\frac { \\partial | h | ^ 2 } { \\partial t } & = \\Delta | h | ^ 2 - 2 | \\nabla h | ^ 2 + 2 R _ 1 \\\\ [ 5 p t ] \\frac { \\partial | H | ^ 2 } { \\partial t } & = \\Delta | H | ^ 2 - 2 | \\nabla H | ^ 2 + 2 R _ 2 \\end{align*}"}
-{"id": "3122.png", "formula": "\\begin{align*} & \\min _ { x _ 1 , \\ldots , x _ N \\in \\mathbb { R } ^ { n } } \\Big \\{ f _ 1 ( x _ 1 ) + \\cdots + f _ N ( x _ N ) \\Big \\} \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ x _ 1 + \\cdots + x _ N = 0 . \\end{align*}"}
-{"id": "3078.png", "formula": "\\begin{align*} B ( t , u ) : = \\nabla _ x F ( t , u ) - A ( t ) u \\ , . \\end{align*}"}
-{"id": "919.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ d \\left | \\frac { \\partial ^ 2 ( g \\circ \\Phi _ \\beta ) } { \\partial x _ i \\partial x _ j } ( x ) \\right | \\leq \\| g '' \\| _ \\infty + 2 \\beta \\| g ' \\| _ \\infty \\end{align*}"}
-{"id": "689.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\cfrac { c _ n ^ 2 a _ n ^ 0 } { 1 + c _ n a _ n ^ 0 } = 0 . \\end{align*}"}
-{"id": "7824.png", "formula": "\\begin{align*} \\int _ { S _ { r _ j } } | \\frac { \\partial v } { \\partial r } v | e ^ { - 2 \\rho } d x = o ( 1 ) . \\end{align*}"}
-{"id": "6683.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\Delta } \\left ( 1 - ( 1 - s ) ^ p \\right ) ^ { \\frac { n - 1 } { p } } \\mathrm { d } s = \\int _ 0 ^ { \\Delta } \\left ( p ( 1 - \\sigma ( s ) ) ^ { p - 1 } s \\right ) ^ { \\frac { n - 1 } { p } } \\mathrm { d } s \\leq \\int _ 0 ^ { \\Delta } ( p s ) ^ { \\frac { n - 1 } { p } } \\mathrm { d } s = \\frac { 1 } { n - 1 + p } ( p \\Delta ) ^ { \\frac { n - 1 + p } { p } } \\end{align*}"}
-{"id": "2027.png", "formula": "\\begin{align*} t _ 1 ^ 2 g \\Big ( \\frac { t _ 2 } { t _ 1 ^ 2 } \\Big ) \\frac { g ( t _ 2 ^ 2 ) } { g ( t _ 2 ) } + t _ 1 t _ 2 ^ 2 g \\Big ( \\frac { t _ 1 } { t _ 2 ^ 2 } \\Big ) \\frac { g \\big ( \\frac { 1 } { t _ 1 ^ 2 } \\big ) } { g \\big ( \\frac { 1 } { t _ 1 } \\big ) } + t _ 2 g ( t _ 1 t _ 2 ) \\frac { g \\big ( \\frac { t _ 1 ^ 2 } { t _ 2 ^ 2 } \\big ) } { g \\big ( \\frac { t _ 1 } { t _ 2 } \\big ) } = 0 \\end{align*}"}
-{"id": "4758.png", "formula": "\\begin{align*} { \\varepsilon } = F ( X , Y , Z ) \\ , { \\Delta \\sqrt { - \\det g ^ { i j } } } / { ( { L } _ 1 \\ , { L } _ 2 \\ , { L } _ 3 ) } , \\end{align*}"}
-{"id": "2630.png", "formula": "\\begin{align*} \\widehat { V _ a } ( \\omega ) = \\left \\{ \\begin{aligned} & \\frac 1 { 2 | \\omega | } & & a = 0 , \\\\ & \\frac { 2 \\sin ( \\tfrac { a \\pi } 2 ) \\Gamma ( 1 - a ) } { | 2 \\pi \\omega | ^ { 1 - a } } & & 0 < a < 1 \\end{aligned} \\right \\} | \\omega | > 0 , \\end{align*}"}
-{"id": "65.png", "formula": "\\begin{align*} \\frac { \\partial ( e e ^ * ) } { \\partial w ^ * } = \\frac { \\partial ( D D ^ * - D \\textbf { X } ^ { H } w - \\textbf { w } ^ { H } \\textbf { X X } ^ { * } + \\textbf { w } ^ { H } \\textbf { X X } ^ { H } \\textbf { w } ) } { \\partial w ^ { * } } \\end{align*}"}
-{"id": "9678.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d U } { d \\sigma _ j } = r _ j ( U ) , \\ , \\ , j = 2 , 3 , 4 , \\\\ U | _ { \\sigma _ j = 0 } = U _ a , \\end{cases} \\end{align*}"}
-{"id": "8608.png", "formula": "\\begin{align*} \\alpha ( f * \\Phi \\otimes \\Psi ) [ v , w ] & = \\sum _ { \\gamma } ( f * \\Phi ) ( v \\gamma ) \\otimes m ( v ) \\gamma \\Psi ( \\gamma ^ { - 1 } w ) \\\\ & = \\sum _ { \\gamma , \\varepsilon } f ( \\varepsilon ) \\varepsilon \\Phi ( \\varepsilon ^ { - 1 } v \\gamma ) \\otimes m ( v ) \\gamma \\Psi ( \\gamma ^ { - 1 } w ) \\\\ & = \\sum _ { \\varepsilon } f ( \\varepsilon ) \\varepsilon \\alpha ( \\Phi \\otimes \\Psi ) [ \\varepsilon ^ { - 1 } v , w ] = f * \\alpha ( \\Phi \\otimes \\Psi ) [ v , w ] \\end{align*}"}
-{"id": "887.png", "formula": "\\begin{align*} \\gamma _ { n , k } ( i , j ) = \\left \\{ \\begin{array} { l l } 1 / \\sqrt { N _ n } & | j - i | = k , \\\\ 0 & . \\end{array} \\right . \\end{align*}"}
-{"id": "1867.png", "formula": "\\begin{align*} f _ j ' ( x ) : = f _ j ( x ) - \\frac { m - \\varepsilon } { d } \\end{align*}"}
-{"id": "5239.png", "formula": "\\begin{align*} V ^ u ( 0 ) \\cap E ^ s ( 0 , z ) = \\{ 0 \\} z \\geq \\tau . \\end{align*}"}
-{"id": "4155.png", "formula": "\\begin{align*} Z _ { N , J } ^ n + \\sum _ { \\tau = 1 } ^ { n - 1 } n ^ { n - \\tau } Z _ { i , J } ^ \\tau \\leq 1 + \\sum _ { \\tau = 1 } ^ { n - 2 } n ^ { \\tau } . \\end{align*}"}
-{"id": "2161.png", "formula": "\\begin{align*} \\int _ { I ( s ) } | \\tilde { w } ( \\xi , s ) | ^ 2 \\rho _ d \\ , d \\xi = O ( e ^ { - 2 \\theta ' s } ) \\end{align*}"}
-{"id": "1593.png", "formula": "\\begin{align*} j _ i ' = \\begin{cases} 0 , & 1 \\leq i \\leq n - 3 \\\\ 1 , & i = n - 2 \\\\ 1 6 q + 5 & i = n - 1 \\\\ 1 & i = n , \\end{cases} \\end{align*}"}
-{"id": "5473.png", "formula": "\\begin{align*} { \\mathrm { B } } _ \\rho ^ { - 1 } q ( x ) = x ^ { 1 - \\rho } \\frac { \\dd } { \\dd x } [ x ^ \\rho q ( x ) ] , q \\in \\mathcal { P } _ { r , \\rho } ^ { 1 } . \\end{align*}"}
-{"id": "7638.png", "formula": "\\begin{align*} X X ^ * = \\Big ( \\sum _ { j = 1 } ^ k c _ j x _ j ^ 2 \\Big ) I _ n , \\end{align*}"}
-{"id": "2995.png", "formula": "\\begin{align*} U ( t ) = \\int _ 0 ^ t S ^ b _ { t - s } \\beta _ s d s + \\int _ 0 ^ t S ^ b _ { t - s } \\sigma _ s d W ( s ) \\end{align*}"}
-{"id": "1579.png", "formula": "\\begin{align*} j _ k ' = \\begin{cases} j _ k , & 1 \\leq k \\leq n - \\nu ( m ) - 1 ; \\\\ j _ { n - \\nu ( m ) } + p , & k = n - \\nu ( m ) ; \\\\ 0 , & n - \\nu ( m ) < k < n ; \\\\ \\sum _ { k = n - \\nu ( m ) + 1 } ^ n ( 2 ^ { n - k + 1 } - 1 ) j _ k - ( 2 ^ { \\nu ( m ) + 1 } - 1 ) p , & k = n , \\end{cases} \\end{align*}"}
-{"id": "8092.png", "formula": "\\begin{align*} \\lim _ { z \\to p } \\frac { \\kappa _ D ( z ; X ) } { \\kappa _ { U \\cap D } ( z ; X ) } = 1 , \\ ; X \\in \\mathbb C ^ n , X \\neq 0 , \\end{align*}"}
-{"id": "3606.png", "formula": "\\begin{align*} { \\rm T r } \\big ( K _ N M ^ \\ell K _ N M ^ \\ell K _ N \\big ) = \\sum _ { k = 0 } ^ { N - 1 } \\sum _ { \\gamma : ( 0 , k ) \\rightarrow ( 2 \\ell , k ) , \\ ; \\gamma ( \\ell ) < N } w ( \\gamma ) . \\end{align*}"}
-{"id": "9445.png", "formula": "\\begin{align*} \\sum _ { 2 ^ { 2 ^ N } \\leq n _ k \\leq 2 ^ { 2 ^ { N + 1 } } } \\frac { 1 } { \\log n _ k } = O ( 1 ) . \\end{align*}"}
-{"id": "4230.png", "formula": "\\begin{align*} 0 \\to \\bigoplus _ { i = 0 } ^ \\infty A \\xrightarrow { d } \\bigoplus _ { i = 0 } ^ \\infty A \\to 0 \\end{align*}"}
-{"id": "3751.png", "formula": "\\begin{align*} I _ m = I _ { k , 0 } + L _ k y , , \\end{align*}"}
-{"id": "9928.png", "formula": "\\begin{align*} \\| \\partial _ { x _ i } \\partial _ { x _ k } P _ { 2 t } \\phi \\| _ { \\infty } = \\| \\partial _ { x _ i } \\partial _ { x _ k } P _ { t } P _ { t } \\phi \\| _ { \\infty } = \\| \\partial _ { x _ i } P _ { t } ( \\partial _ { x _ k } P _ { t } \\phi ) \\| _ { \\infty } & \\leq c ( t ) \\| \\partial _ { x _ k } P _ { t } \\phi \\| _ { \\infty } \\\\ & \\leq c ( t ) ^ 2 \\| \\phi \\| _ { \\infty } . \\end{align*}"}
-{"id": "7185.png", "formula": "\\begin{align*} \\langle z _ j , \\zeta \\rangle \\geq 0 \\qquad \\zeta \\in T _ { \\psi _ j } \\mathfrak { H } _ { 1 + s } = T _ h \\mathfrak { H } _ { 1 + s } , \\end{align*}"}
-{"id": "551.png", "formula": "\\begin{align*} \\Gamma = \\{ ( z _ 1 , 0 , \\dots , 0 ) \\mid | z _ 1 | = 1 \\} \\end{align*}"}
-{"id": "9926.png", "formula": "\\begin{align*} X _ t - X _ t ^ { ( n ) } = \\int _ 0 ^ t \\left ( b ( s , X _ { s - } ) - b ( s , X _ { \\eta _ n ( s ) - } ^ { ( n ) } ) \\right ) d s , \\end{align*}"}
-{"id": "3937.png", "formula": "\\begin{align*} u _ j ^ { n + 1 } = \\sum \\limits _ { l = 0 } ^ { n } ( w _ { n - l } - w _ { n - l + 1 } ) u _ j ^ l + w _ n u _ j ^ 0 + \\nu \\alpha _ 0 ( u _ j ^ { n + 1 } ) _ { x x } - \\frac { \\alpha _ 0 } { 3 } \\left ( u _ j ^ n ( u _ j ^ { n + 1 } ) _ x + ( u _ j ^ n u _ j ^ { n + 1 } ) _ x \\right ) . \\end{align*}"}
-{"id": "4040.png", "formula": "\\begin{align*} d g ^ { - 1 } _ { g ( p ) } W _ 1 + c \\mu _ 1 W _ 1 = W _ 1 \\ \\ \\mathrm { a n d } \\\\ d g ^ { - 1 } _ { g ( p ) } W _ 2 + c \\mu _ 2 W _ 2 = W _ 2 . \\end{align*}"}
-{"id": "10.png", "formula": "\\begin{align*} G ^ { C } _ { \\sigma } ( C _ 1 - C _ 2 ) = \\frac { 1 } { 2 \\pi \\sigma ^ 2 } e x p \\left ( - \\frac { ( C _ { 1 } - C _ { 2 } ) ( C _ { 1 } - C _ { 2 } ) ^ { * } } { 2 \\sigma ^ 2 } \\right ) \\end{align*}"}
-{"id": "1446.png", "formula": "\\begin{align*} d _ 3 ( 1 ) & = 0 , & d _ 3 ( x _ 3 ) & = v _ 1 x _ 3 ^ 2 , \\\\ d _ 3 ( y _ 1 ) & = 0 , & d _ 3 ( x _ 3 y _ 1 ) & = v _ 1 x _ 3 ^ 2 y _ 1 , \\\\ d _ 3 ( z _ 1 ) & = v _ 1 x _ 3 z _ 1 , & d _ 3 ( x _ 3 z _ 1 ) & = 0 . \\end{align*}"}
-{"id": "10048.png", "formula": "\\begin{align*} \\mathcal { H } _ F = \\{ \\vec { \\tau } \\in F _ \\C : \\vec { v } \\mbox { i s t o t a l l y p o s i t i v e } \\} . \\end{align*}"}
-{"id": "8282.png", "formula": "\\begin{align*} \\mathrm { d i v } ( \\psi ( f ) ) = \\sum _ { \\substack { m > 0 \\\\ \\mu \\in V _ \\Z ^ \\vee / V _ \\Z } } c ( - m , \\mu ) \\cdot \\mathcal { Z } ( m , \\mu ) . \\end{align*}"}
-{"id": "784.png", "formula": "\\begin{align*} k = B n + r \\ \\textrm { w i t h } \\ B \\ge 1 , \\ 0 \\le \\vert r \\vert \\le \\frac { n - 1 } { 2 } . \\end{align*}"}
-{"id": "5788.png", "formula": "\\begin{align*} - \\langle \\Delta _ { p } u , \\varphi \\rangle = \\int _ { \\Omega } \\vert D u \\vert ^ { p - 2 } D u \\cdot \\nabla \\varphi \\ ; d x , \\varphi \\in C _ { 0 } ^ { \\infty } ( \\Omega ) . \\end{align*}"}
-{"id": "8828.png", "formula": "\\begin{align*} E ^ y _ { m , p } ( f _ j ) = \\sum _ { i = 1 } ^ { s _ j } a _ { i , p , y } m ^ { \\beta _ { i j } } p ^ { - \\lambda _ { i j } m } \\ 1 1 _ { A _ { i j } } ( m ) . \\end{align*}"}
-{"id": "4015.png", "formula": "\\begin{align*} \\omega ( X , Y ) : = \\mathrm { d e t } [ X , Y , \\eta ] , \\end{align*}"}
-{"id": "6745.png", "formula": "\\begin{align*} \\bar { E } \\Big [ \\Big ( \\mathbb { P } ( \\Theta > N ^ { \\gamma } t ) \\Big ) ^ { 2 } \\Big ] = \\Bigg ( 1 - \\frac { 1 } { N ^ { \\gamma } } \\Bigg ) \\Bigg ( 1 - \\frac { 1 } { N ^ { \\gamma } } \\Bigg ) ^ { 2 | G ^ { * } | } + o ( N ) . \\end{align*}"}
-{"id": "1976.png", "formula": "\\begin{align*} \\Vert ( v _ { s } , v _ { u } ) \\Vert ^ { 2 } & = \\Vert v _ { s } \\Vert ^ { 2 } + \\Vert v _ { u } \\Vert ^ { 2 } - 2 \\theta _ { p } \\Vert v _ { s } \\Vert \\Vert v _ { u } \\Vert \\geq \\Vert v _ { s } \\Vert ^ { 2 } + \\Vert v _ { u } \\Vert ^ { 2 } + 2 ( \\mu _ { i } - 1 ) \\Vert v _ { s } \\Vert \\Vert v _ { u } \\Vert \\\\ & \\geq \\mu _ { i } ( \\Vert v _ { s } \\Vert ^ { 2 } + \\Vert v _ { u } \\Vert ^ { 2 } ) . \\end{align*}"}
-{"id": "8872.png", "formula": "\\begin{align*} k _ { \\theta , \\lambda } ( z ) = \\frac { 1 - \\overline { \\theta ( \\lambda ) } \\theta ( z ) } { 1 - \\overline \\lambda z } \\ \\ \\ \\ k _ { \\ast \\theta , \\lambda } ( z ) = \\frac { \\theta ( z ) - \\theta ( \\lambda ) } { z - \\lambda } , \\ \\ \\ z \\in \\mathbb D . \\end{align*}"}
-{"id": "8985.png", "formula": "\\begin{gather*} D ^ { ( n ) } _ { q , t } ( d ) D ^ { ( n ) } _ q ( u _ 0 , u _ 1 , u _ 2 , u _ 3 ; t ) = D ^ { ( n ) } _ q ( u _ 0 + d q / 2 , u _ 1 + d q / 2 , u _ 2 + d q / 2 , u _ 3 + d q / 2 ; t ) D ^ { ( n ) } _ { q , t } ( d ) . \\end{gather*}"}
-{"id": "4649.png", "formula": "\\begin{align*} r _ { k , \\ell } ( x ) = \\min \\{ n \\in \\mathbb { N } : ( T | I _ k ) ^ n x \\in I _ \\ell \\} \\end{align*}"}
-{"id": "7081.png", "formula": "\\begin{align*} H P ( \\varphi : ( F , w _ F ) \\circlearrowleft , \\Gamma ) & \\simeq H M ( M _ \\varphi , \\mathfrak { s } _ \\Gamma ) \\\\ H P ( \\varphi : ( F , w _ F ) \\circlearrowleft , \\Gamma ) & \\simeq H M ( M _ \\varphi , \\mathfrak { s } _ \\Gamma , c _ - ) \\end{align*}"}
-{"id": "660.png", "formula": "\\begin{align*} ( D _ t - \\Delta ) ( P \\widetilde \\nabla ^ k \\widetilde A - \\nabla ^ k A ) & = ( ( D _ t - \\Delta ) P ) \\widetilde \\nabla ^ k \\widetilde A - 2 g ^ { i j } ( \\nabla _ i P ) ( \\nabla _ j \\widetilde \\nabla ^ k \\widetilde A ) \\\\ & + P ( ( D _ t - \\Delta ) \\widetilde \\nabla ^ k \\widetilde A ) - ( D _ t - \\Delta ) \\nabla ^ k A . \\end{align*}"}
-{"id": "517.png", "formula": "\\begin{align*} \\frac { H ( S ) } { n } & \\geq \\frac { n - m _ 1 - m _ 2 } { n } - \\delta = 1 - H _ b ( q * p _ A ) - 2 \\delta \\end{align*}"}
-{"id": "2401.png", "formula": "\\begin{align*} 0 = a _ 2 ^ 2 - a _ 3 ( 3 a _ 3 + a _ 4 ) = \\frac { R _ 2 } { a _ 2 ^ 2 } . \\end{align*}"}
-{"id": "6363.png", "formula": "\\begin{align*} \\Phi ( z ) : = \\int \\frac { d \\mu ( x ) } { z - x } , \\end{align*}"}
-{"id": "3671.png", "formula": "\\begin{align*} a _ i \\leqslant \\sum _ { j = 1 } ^ { n + 1 } b _ j ( i = 1 , \\dots , n ) . \\end{align*}"}
-{"id": "2221.png", "formula": "\\begin{align*} \\widetilde \\Gamma _ h = \\left \\{ w \\in \\mathbb C ^ n : \\left | h _ i \\left ( \\dfrac 1 { w _ 1 } , \\ldots , \\dfrac 1 { w _ n } \\right ) \\right | = \\varepsilon _ i , i = 1 , \\ldots , n \\right \\} \\end{align*}"}
-{"id": "8631.png", "formula": "\\begin{align*} \\omega ( \\tau _ k + s ) = \\omega ( \\tau _ k ) + x _ k ( s ) , ~ ~ \\omega ( \\tau _ k + 1 + s ) = \\omega ( \\tau _ k ) + x _ k ( 1 ) + x _ { k + 1 } ( s ) , \\end{align*}"}
-{"id": "5040.png", "formula": "\\begin{align*} x _ 0 ^ * + \\theta z _ 0 ^ * - p ^ * ( x ^ * + \\theta z ^ * ) h _ 0 ^ * = - T \\left ( \\sum _ { n \\in \\N } ( t _ n + \\theta \\tau _ n - p ^ * ( x ^ * + \\theta z ^ * ) s _ n ) r _ n e _ n \\right ) . \\end{align*}"}
-{"id": "8905.png", "formula": "\\begin{align*} \\begin{aligned} & \\lim _ k \\bigl ( \\| X J _ { \\theta , 1 } ^ { - 1 } U _ \\mu ^ { n _ k } p \\| ^ 2 - \\| p X J _ { \\theta , 1 } ^ { - 1 } \\chi ^ { - n _ k } \\| ^ 2 \\bigr ) = 0 \\\\ \\ \\ & \\lim _ k \\bigl ( \\| X J _ { \\theta , 1 } ^ { - 1 } U _ \\mu ^ { - n _ k } p \\| ^ 2 - \\| p X J _ { \\theta , 1 } ^ { - 1 } \\chi ^ { - n _ k } \\| ^ 2 \\bigr ) = 0 \\end{aligned} \\end{align*}"}
-{"id": "7066.png", "formula": "\\begin{align*} \\sigma ( L _ { \\lambda A } ) = \\left \\{ \\frac { 1 + \\lambda \\alpha _ j } { 1 + \\beta _ k } \\colon \\alpha _ j \\in \\sigma ( A ) , \\beta _ k \\in \\sigma ( - \\Delta ; B ^ N ) \\right \\} . \\end{align*}"}
-{"id": "2497.png", "formula": "\\begin{align*} \\begin{aligned} \\mathrm { P } _ { 0 } \\psi _ { j } \\left ( \\left | \\eta \\right | \\right ) & = \\sum _ { l = 0 } ^ { 2 } \\beta _ { j l } \\left ( \\left | \\eta \\right | \\right ) E _ { l } ^ { 1 } , \\ : j = 0 , 1 , 2 , \\\\ \\mathrm { P } _ { 0 } \\psi _ { 3 } \\left ( \\left | \\eta \\right | \\right ) & = \\beta _ { 3 3 } \\left ( \\left | \\eta \\right | \\right ) E _ { 3 } ^ { 1 } , \\\\ \\mathrm { P } _ { 0 } \\psi _ { 4 } \\left ( \\left | \\eta \\right | \\right ) & = \\beta _ { 4 4 } \\left ( \\left | \\eta \\right | \\right ) E _ { 4 } ^ { 1 } . \\end{aligned} \\end{align*}"}
-{"id": "161.png", "formula": "\\begin{align*} E _ 2 ^ { p , q } = H ^ p ( G ' , H ^ q ( K _ { A S L } , \\mathbf Z / p \\mathbf Z ) ) = 0 , \\ q \\geq 1 , p \\geq 2 \\end{align*}"}
-{"id": "456.png", "formula": "\\begin{align*} - i \\psi '' _ \\omega ( 0 ) = \\theta ' ( y _ \\omega ) u _ 1 \\otimes u _ 1 + \\frac { \\omega } { y _ \\omega } \\sum _ { j = 2 } ^ m u _ j \\otimes u _ j , \\end{align*}"}
-{"id": "3339.png", "formula": "\\begin{align*} \\alpha = \\frac { r _ s ^ { ( K ) } - r _ s ^ { ( K + 1 ) } } { r _ { s } ^ { ( K ) } - r _ { s - 1 } ^ { ( K ) } } . \\end{align*}"}
-{"id": "8695.png", "formula": "\\begin{align*} a _ { n , k } : = \\frac { c _ k ( 1 - q ^ n ) ^ { k } } { \\sqrt { v ( 1 - q ^ n ) } } \\in \\R . \\end{align*}"}
-{"id": "648.png", "formula": "\\begin{align*} \\bar \\nabla _ { J } ^ k \\dot \\gamma & = \\bar \\nabla _ { \\dot \\gamma } \\bar \\nabla ^ { k - 1 } _ { J } J + \\sum _ { i = 0 } ^ { k - 2 } \\bar \\nabla _ J ^ i ( \\bar R ( \\dot \\gamma , J ) \\bar \\nabla _ { J } ^ { k - 2 - i } J ) \\\\ & = \\bar \\nabla _ { \\dot \\gamma } \\bar \\nabla ^ { k - 1 } _ { J } J + \\sum ( \\bar \\nabla ^ { i _ r } \\bar R ) ^ { i _ r ' } * ( \\dot \\gamma ) ^ j * ( \\bar \\nabla _ { J } ^ { k _ p } J ) ^ { k _ p ' } * \\big ( \\bar \\nabla _ { \\dot \\gamma } \\bar \\nabla _ { J } ^ { l } { J } \\big ) ^ { l ' } , \\end{align*}"}
-{"id": "4048.png", "formula": "\\begin{align*} l _ e ( \\sigma ) = \\int _ a ^ b | | \\sigma ' ( t ) | | _ e d t . \\end{align*}"}
-{"id": "1188.png", "formula": "\\begin{align*} \\sum \\limits _ { i , j = 1 } ^ { n } a _ { i j } \\ , u _ { x _ j x _ i } = | \\nabla u | ^ { 2 - p } \\sum \\limits _ { i , j = 1 } ^ { n } \\frac { \\partial \\mathcal { A } _ i } { \\partial \\eta _ j } ( \\nabla u ) u _ { x _ j x _ i } \\ , = \\ , 0 \\mbox { a t } \\ , \\ , x _ 0 , y _ 0 , z _ 0 . \\end{align*}"}
-{"id": "10033.png", "formula": "\\begin{align*} \\det ( \\Lambda _ { h } \\otimes _ \\Z \\Q ) & = \\mathrm { r a t } ( \\nu ( h ) ) ^ { 1 - n } \\cdot \\det ( \\Lambda \\otimes _ \\Z \\Q ) , \\\\ \\det ( L _ { 1 , h } \\otimes _ \\Z \\Q ) & = \\mathrm { r a t } ( \\nu ( h ) ) ^ { 1 - n } \\cdot \\det ( L _ 1 \\otimes _ \\Z \\Q ) , \\end{align*}"}
-{"id": "5361.png", "formula": "\\begin{align*} N ^ c _ { t , t ' , t '' } : = \\begin{cases} 1 & \\ | t - t ' | + 1 \\leq t '' \\leq \\{ t + t ' - 1 , 2 c - t - t ' - 1 \\} \\ \\\\ & t + t ' + t '' \\ \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "1741.png", "formula": "\\begin{align*} \\langle f \\ , \\sqrt { d \\mu } , g \\ , \\sqrt { d \\nu } \\rangle : = \\int _ { \\Lambda ^ \\infty } \\overline { f } g \\ , \\Big ( \\sqrt { \\frac { d \\mu } { d ( \\mu + \\nu ) } } \\ \\sqrt { \\frac { d \\nu } { d ( \\mu + \\nu ) } } \\Big ) { { d ( \\mu + \\nu ) } } . \\end{align*}"}
-{"id": "9114.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { i = 0 } ^ { n } \\dfrac { 1 } { \\binom { n + 1 } { i } } \\sum _ { \\mathrm { c a r d } ( I ) = i } \\left ( \\prod _ { j \\in I } x _ { j } \\right ) \\cdot f _ { n + 1 - i } \\left ( \\prod _ { k \\in \\left \\{ 1 , \\ldots , n + 1 \\right \\} \\setminus I } x _ { k } \\right ) = 0 \\ ; \\left ( x _ 1 , \\ldots , x _ { n + 1 } \\in R \\right ) \\end{align*}"}
-{"id": "1343.png", "formula": "\\begin{align*} { \\rm D } \\vartheta _ c ( y ) ^ * = \\left ( \\begin{array} { c } - c ^ { - 1 } Y + c ^ { - 1 } { \\rm D } \\theta _ c ( F ( x _ c ( y ) ) + Y / c ) ) ^ * \\\\ h ( x _ c ( y ) ) \\\\ - c ^ { - 1 } \\lambda + c ^ { - 1 } \\Pi _ { K ^ * } ( \\lambda - c g ( x _ c ( y ) ) ) \\end{array} \\right ) \\ , , y = ( Y , \\mu , \\lambda ) \\in \\mathbb { B } _ { \\delta _ 0 } ( \\overline { y } ) \\ , . \\end{align*}"}
-{"id": "3126.png", "formula": "\\begin{align*} \\bar { K } : = \\frac { 2 ( N - 1 ) R ^ 2 ( x ^ 0 ) } { \\epsilon } \\left ( 1 + \\log \\frac { 1 } { \\rho } \\right ) + 2 . \\end{align*}"}
-{"id": "6915.png", "formula": "\\begin{align*} \\mu ( F _ i ( A ) ) & = \\left ( \\frac { p _ { m + 1 } } { 2 } \\right ) \\mu ( A ) \\\\ R ( F _ i ( A ) ) & = \\left ( \\frac { 1 - p _ { m + 1 } } { 2 } \\right ) R ( A ) \\end{align*}"}
-{"id": "5547.png", "formula": "\\begin{align*} a _ { n } = \\int _ { 0 } ^ { 1 } f \\left ( t \\right ) w _ { n } \\left ( t \\right ) d t \\end{align*}"}
-{"id": "979.png", "formula": "\\begin{align*} b _ n ( t ) & = \\sum _ { i = 1 } ^ n K _ h ( t _ { i - 1 } - t ) \\int _ { t _ { i - 1 } } ^ { t _ i } \\{ \\sigma ^ 2 ( s ) - \\sigma ^ 2 ( t ) \\} d s + \\sigma ^ 2 ( t ) \\left \\{ \\frac { 1 } { n } \\sum _ { i = 1 } ^ n K _ h ( t _ { i - 1 } - t ) - \\int _ { - \\infty } ^ \\infty K _ h ( s - t ) d s \\right \\} \\\\ & = : \\mathbb { I } _ n ( t ) + \\mathbb { I I } _ n ( t ) . \\end{align*}"}
-{"id": "10018.png", "formula": "\\begin{align*} c _ f ^ + ( m , \\mu ) = \\begin{cases} c _ 0 ^ + ( m ) & \\mbox { i f } \\mu = 0 \\\\ 0 & \\mbox { o t h e r w i s e } \\end{cases} \\end{align*}"}
-{"id": "2695.png", "formula": "\\begin{align*} f ( \\alpha e _ 1 + e _ 2 ) & = \\alpha ( x e _ 1 + y e _ 2 ) + z e _ 1 + w e _ 2 \\\\ & = ( x \\alpha + z ) e _ 1 + ( y \\alpha + w ) e _ 2 . \\end{align*}"}
-{"id": "9095.png", "formula": "\\begin{align*} R \\ni x \\longmapsto B ( x , y ^ { \\ast } ) = \\sum _ { i = 1 } ^ { n - 1 } \\binom { n } { i } d ^ { i } ( x ) d ^ { n - i } ( y ^ { \\ast } ) \\end{align*}"}
-{"id": "3351.png", "formula": "\\begin{align*} \\lim _ { K \\rightarrow \\infty } \\frac { 1 } { \\psi _ 2 ( N , K , s ) } & \\geq \\frac { N } { \\lambda + \\epsilon } \\sum _ { i = 0 } ^ { \\infty } \\left ( \\frac { ( 1 - ( \\lambda + \\epsilon ) ) ( N - 1 ) } { \\lambda + \\epsilon } \\right ) ^ i \\\\ & = \\frac { N } { N ( \\lambda + \\epsilon ) - ( N - 1 ) } . \\end{align*}"}
-{"id": "2862.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { P _ { n } } { p _ { n } } \\right ) ^ { k - 1 } \\mid V _ { n , r } \\mid ^ { k } < \\infty , \\textnormal { f o r } \\quad { r = 1 , 2 , 3 , 4 . } \\end{align*}"}
-{"id": "8514.png", "formula": "\\begin{align*} & \\sum _ { i = 1 } ^ l | \\alpha _ i | ^ p + \\sum _ { j = 1 } ^ s \\left | \\sum _ { i = 1 } ^ l a _ { j \\ , i } \\alpha _ i + v _ j \\right | ^ p + \\sum _ { j = 1 } ^ m \\left | \\sum _ { i = 1 } ^ l \\bar a _ { j \\ , i } \\alpha _ i + u _ j \\right | ^ p \\to \\min \\\\ & \\begin{cases} \\alpha \\in \\mathbb { Z } ^ l \\setminus \\{ 0 \\} \\\\ | | \\alpha | | _ 1 \\leq C . \\\\ \\end{cases} \\end{align*}"}
-{"id": "3602.png", "formula": "\\begin{align*} w \\Big ( ( n , m ) \\rightarrow ( n + 1 , m + 1 ) \\Big ) & = a _ { m + 1 } ^ N , \\\\ w \\Big ( ( n , m ) \\rightarrow ( n + 1 , m ) \\Big ) & = b _ { m } ^ N , \\\\ w \\Big ( ( n , m ) \\rightarrow ( n + 1 , m - 1 ) \\Big ) & = a _ { m } ^ N . \\end{align*}"}
-{"id": "7698.png", "formula": "\\begin{align*} Y _ { i i } = \\frac { ( u _ { i j } - u _ { i k } ) ^ 2 } { 2 \\lambda ^ 2 _ i } \\ , . \\end{align*}"}
-{"id": "9955.png", "formula": "\\begin{align*} \\overline { { \\cal N } } ( u ) - V _ 1 = \\overline { { \\cal N } } ( \\varphi ( u ) ) - V _ 2 \\end{align*}"}
-{"id": "296.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } F _ { s _ { \\tilde S ^ * _ { 0 , \\lfloor n t _ 1 \\rfloor } } , s _ { \\tilde S ^ * _ { \\lfloor n t _ 1 \\rfloor + 1 , \\lfloor n t _ 2 \\rfloor } } } ( u _ 1 , u _ 2 ) = \\lim _ { n \\to \\infty } F _ { s _ { \\tilde S ^ * _ { 0 , \\lfloor n t _ 1 \\rfloor } } } ( u _ 1 ) \\lim _ { n \\to \\infty } F _ { s _ { \\tilde S ^ * _ { { \\lfloor n t _ 1 \\rfloor } + 1 , \\lfloor n t _ 2 \\rfloor } } } ( u _ 2 ) \\end{align*}"}
-{"id": "9937.png", "formula": "\\begin{align*} I _ 6 : = \\left | \\int _ { T _ { i - 1 } } ^ t \\int _ { | y | \\geq 1 } H _ i ( s , y ) \\ , \\nu ( d y ) \\ , d s \\right | ^ p \\end{align*}"}
-{"id": "5165.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 2 } U _ { i } ( x , y ) = 0 , \\quad \\forall \\ , ( x , y ) \\in \\mathcal { S } . \\end{align*}"}
-{"id": "5657.png", "formula": "\\begin{align*} | \\dot { u } | _ { L ^ 2 ( \\Omega , \\R ^ n ) } ( t ) = \\| \\partial _ { x _ 1 } u ( t , \\cdot ) \\| _ { L ^ 2 ( \\Omega , \\R ^ n ) } \\end{align*}"}
-{"id": "6709.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\frac { t ^ { - } _ { N } } { N \\log N } = 1 \\end{align*}"}
-{"id": "8426.png", "formula": "\\begin{align*} \\sum _ { t = - \\infty } ^ { \\infty } q ^ { ( t + 1 ) ( \\frac { 1 } { 2 } - s ) } c _ { t , l } ( 1 ) = - q ^ { - l ( \\frac { 3 } { 2 } - s ) } \\zeta _ F ( 1 ) \\frac { 1 - q ^ { \\frac { 1 } { 2 } - s } } { 1 - q ^ { - \\frac { 1 } { 2 } - s } } . \\end{align*}"}
-{"id": "8435.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { t , l , v } ) = \\sum _ { \\mu \\in \\mathfrak { X } _ l } c _ { t , l } ( \\mu ) \\mu ( v ) . \\end{align*}"}
-{"id": "9524.png", "formula": "\\begin{align*} \\frac { \\partial g } { \\partial t } = - R \\ , g \\end{align*}"}
-{"id": "3835.png", "formula": "\\begin{align*} \\eta ^ B ( x ) : = \\left \\{ \\begin{array} { l l } N ( x , 0 ) & x \\notin B , \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "225.png", "formula": "\\begin{align*} \\chi _ a \\left ( h ( A _ G ) - h ( A _ { G ' } ) \\right ) \\chi _ b = \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma } \\mathrm { d } z \\ , h ( z ) \\chi _ a R _ z ( A _ { G } ) \\left ( \\chi _ G A \\chi _ { G ' \\setminus G } \\right ) R _ z ( A _ { G ' } ) \\chi _ b , \\end{align*}"}
-{"id": "1141.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { \\infty } \\frac { ( t ^ { q ^ i } - t ) z ^ { q ^ i } } { d _ i } = \\sum _ { i = 0 } ^ { \\infty } \\frac { z ^ { q ^ { i + 1 } } } { d _ i ^ q } \\end{align*}"}
-{"id": "8508.png", "formula": "\\begin{align*} E _ v : \\ ; y ^ 2 + \\bar { a } _ 1 x y + \\bar { a } _ 3 y = x ^ 3 + \\bar { a } _ 2 x ^ 2 + \\bar { a } _ 4 x + \\bar { a } _ 6 \\end{align*}"}
-{"id": "3003.png", "formula": "\\begin{align*} f ( t ) = e ^ { - \\alpha t } \\inf _ { x \\in F } u _ 0 ( x ) ^ 2 + \\int _ 0 ^ t \\kappa ( \\alpha ) e ^ { - \\alpha ( t - s ) } \\left ( 1 + ( t - s ) ^ { - \\frac { d _ s } { 2 } } \\right ) f ( s ) d s . \\end{align*}"}
-{"id": "1073.png", "formula": "\\begin{align*} \\mathcal O _ { S ' } = \\rho _ \\star \\mathcal O _ { T ' } \\mathop \\times _ { \\rho _ \\star \\mathcal O _ T } \\mathcal O _ S \\end{align*}"}
-{"id": "8082.png", "formula": "\\begin{align*} K _ t ^ \\ell = \\bigoplus _ { 1 \\le i _ 1 < i _ 2 < \\cdots < i _ t \\le n } A ( x _ { i _ 1 } ^ { \\ell } \\wedge x _ { i _ 2 } ^ { \\ell } \\wedge \\cdots \\wedge x _ { i _ t } ^ { \\ell } ) . \\end{align*}"}
-{"id": "9907.png", "formula": "\\begin{align*} e ( A ^ * _ s ) = \\sum _ { i \\in W _ s } e ( V _ i ) + \\sum _ { i j \\in W _ s ^ { ( 2 ) } } e ( V _ i , V _ j ) \\ , . \\end{align*}"}
-{"id": "3201.png", "formula": "\\begin{align*} G _ { t } ( g ) : = \\sigma \\int _ { 0 } ^ { t } \\frac { \\partial } { \\partial x } g ( s , X _ { s } ) \\sqrt { X _ { s } } \\mathrm { d } B _ { s } , t \\geqslant 0 , \\end{align*}"}
-{"id": "3993.png", "formula": "\\begin{align*} \\partial _ t ^ { \\nu _ n } p ^ { \\nu _ n } ( n , t ) = - \\lambda _ n p ^ { \\nu _ n } ( n , t ) + \\lambda _ { n - 1 } p ^ { \\nu _ { n - 1 } } ( n - 1 , t ) , \\ \\ 0 < \\nu _ n \\leq 1 , \\ \\lambda _ n > 0 , \\ n \\geq 1 , \\end{align*}"}
-{"id": "4573.png", "formula": "\\begin{align*} & x _ i ^ + ( z ) = \\theta ^ { - 1 } ( E _ { i } ( d ^ { - i } z ) ) \\ , , x _ i ^ - ( z ) = \\theta ^ { - 1 } ( F _ { i } ( d ^ { - i } z ) ) \\ , , \\\\ & \\phi ^ \\pm _ i ( z ) = \\theta ^ { - 1 } ( K ^ \\pm _ { i } ( d ^ { - i } z ) ) ( 1 \\le i \\le n - 1 ) . \\end{align*}"}
-{"id": "2698.png", "formula": "\\begin{align*} \\begin{cases} p ^ { n } { x } = - a _ 2 ( a ' _ 1 ) ^ { - 1 } y + p ^ { n } { w } \\\\ z = - a ' _ 3 ( a ' _ 1 ) ^ { - 1 } y . \\end{cases} \\end{align*}"}
-{"id": "6433.png", "formula": "\\begin{align*} \\mathcal { C } _ { \\mathcal { M } } \\left ( \\tau \\right ) \\overset { } { = } \\frac { 1 } { \\tau } \\int _ { 0 } ^ { \\tau } d s \\mathcal { V } _ { \\mathcal { M } } \\left ( s \\right ) \\end{align*}"}
-{"id": "9523.png", "formula": "\\begin{align*} I _ { X \\times _ Z Y } = I _ X \\times _ { I _ Z } I _ Y = f ^ * ( I _ X ) \\times _ { h ^ * ( I _ Z ) } g ^ * ( I _ Y ) . \\end{align*}"}
-{"id": "5210.png", "formula": "\\begin{align*} u _ t = u _ { x x } + Q S f ( u ) , \\end{align*}"}
-{"id": "1402.png", "formula": "\\begin{align*} \\rho ^ m _ 0 ( v ' ) = 1 \\otimes u . \\end{align*}"}
-{"id": "2069.png", "formula": "\\begin{align*} \\frac { \\partial f } { \\partial t } = \\tau ( f ) \\end{align*}"}
-{"id": "3613.png", "formula": "\\begin{align*} u _ \\lambda ( x ) = u ( x _ \\lambda ) \\ , . \\end{align*}"}
-{"id": "4680.png", "formula": "\\begin{align*} V _ { \\rm e f f } \\ = \\ \\frac { 3 ( { \\tilde V } _ 2 ^ 2 ) ^ 2 \\ , + \\ , 1 1 2 \\ , { \\tilde V } _ 3 ^ 2 } { 3 2 \\ , F _ 2 } \\ + \\ \\frac { ( d - 3 ) ( d - 5 ) \\ , { \\tilde V } _ 3 ^ 2 } { 7 2 F _ 1 } \\end{align*}"}
-{"id": "5293.png", "formula": "\\begin{align*} u ( x , 1 ) = 0 , b _ 1 u ( x , 0 ) + b _ 2 u _ x ( t , 0 ) & = d ( t ) , \\end{align*}"}
-{"id": "10127.png", "formula": "\\begin{align*} \\boldsymbol P _ k ( i ) = \\lambda ^ { - 1 } \\boldsymbol P _ k ( i - 1 ) - \\lambda ^ { - 1 } \\boldsymbol k _ k ( i ) \\boldsymbol x _ k ^ H ( i ) \\boldsymbol P _ k ( i - 1 ) \\end{align*}"}
-{"id": "9322.png", "formula": "\\begin{align*} Q F _ { n } = F _ { n } + F _ { n + 1 } \\textbf { i } + F _ { n + 2 } \\textbf { j } + F _ { n + 3 } \\textbf { k } \\end{align*}"}
-{"id": "2043.png", "formula": "\\begin{align*} f _ { n + 1 , k } ( z _ 1 , . . . , z _ n , \\varepsilon ) = - z _ 1 ^ { ( k + 1 ) i } f _ { n , k } ( \\varepsilon z _ 1 , z _ 2 , . . . , z _ n ) - . . . - z _ n ^ { ( k + 1 ) i } f _ { n , k } ( z _ 1 , . . . z _ { n - 1 } , \\varepsilon z _ n ) , \\end{align*}"}
-{"id": "2951.png", "formula": "\\begin{align*} \\sum _ { i \\in e } \\Theta ^ \\Delta _ \\epsilon ( H , e , i ) & \\left ( \\partial \\Theta ^ \\Delta _ { \\epsilon } ( H , i ) + \\epsilon \\log \\Theta ^ \\Delta _ { \\epsilon } ( H , e , i ) \\right ) \\\\ & = \\sum _ { i \\in e } \\Theta ^ \\Delta _ { \\epsilon ' } ( H , e , i ) \\left ( \\partial \\Theta ^ \\Delta _ { \\epsilon } ( H , i ) + \\epsilon \\log \\Theta ^ \\Delta _ { \\epsilon } ( H , e , i ) \\right ) , \\end{align*}"}
-{"id": "2913.png", "formula": "\\begin{align*} \\rho _ w = \\Pi _ { i = 1 } ^ n \\rho _ { e _ i } = \\rho _ { e _ n } \\circ \\cdots \\circ \\rho _ { e _ i } , \\end{align*}"}
-{"id": "6544.png", "formula": "\\begin{align*} C ( \\delta ) = \\bigcap _ { \\xi \\in \\mathrm { e x t } ( P ) } \\left \\{ y \\in \\mathbb { R } ^ { n } : \\langle y , \\xi \\rangle = 1 + \\frac { n | P ^ { \\circ } | _ n \\| \\xi \\| } { | F _ { \\xi } | _ { n - 1 } } \\delta \\right \\} \\cap \\bigcap _ { \\xi , \\xi ' \\in \\mathrm { e x t } ( P ) , \\xi \\neq \\xi ' } \\{ y \\in \\mathbb { R } ^ { n } : \\frac { 1 } { 2 } \\langle \\xi + \\xi ' , y \\rangle \\leq 1 \\} . \\end{align*}"}
-{"id": "9975.png", "formula": "\\begin{align*} \\alpha \\leq \\alpha _ { \\kappa } = \\bar { \\upsilon } _ { i } ^ { p } \\dfrac { \\frac { 1 } { \\gamma _ { i } ^ { \\mathsf { t h } } } - \\kappa \\sum _ { j \\neq i } ^ { L } \\mathbb { E } \\left [ \\mathsf { \\bar { L } } _ { j i } ^ { 2 } \\right ] } { 1 + \\sum _ { j \\neq i } ^ { L } \\mathbb { E } \\left [ \\mathsf { \\bar { L } } _ { j i } \\right ] } . \\end{align*}"}
-{"id": "7189.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } L _ a z _ \\infty = 0 & B _ 1 \\setminus { B ' } _ 1 ^ { ' , - } \\\\ z _ \\infty = 0 & { B ' } _ 1 ^ { , - } , \\end{array} \\right . \\end{align*}"}
-{"id": "9366.png", "formula": "\\begin{align*} D _ \\textrm { h r } ( k = 2 ) & \\approx 2 \\frac { C ( 2 , 2 , G _ { 2 } ) } { N } \\left ( \\int _ { \\mathbb { R } ^ 2 } \\sqrt { \\frac { \\mathrm { e } ^ { - \\frac { x ^ 2 + y ^ 2 } { \\sigma ^ 2 } } } { \\pi \\sigma ^ 2 } } \\ , \\mathrm { d } x \\ , \\mathrm { d } y \\right ) ^ 2 \\\\ & = \\frac { 2 \\pi \\sigma ^ 2 } { 3 N } . \\end{align*}"}
-{"id": "2795.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } \\vert \\omega _ { i , n } \\vert \\leq \\int _ { 0 } ^ { T } \\sum _ { i = 0 } ^ { n } \\bigg \\vert \\dfrac { \\frac { \\beta _ { i } } { t - t _ { i } } } { \\sum _ { i = 0 } ^ { n } \\frac { \\beta _ { i } } { t - t _ { i } } } \\bigg { \\vert } \\mathrm { d } t = \\int _ { 0 } ^ { T } \\Lambda _ { n } ( t ) \\mathrm { d } t , \\end{align*}"}
-{"id": "9446.png", "formula": "\\begin{align*} G _ N = \\sum _ { n = 2 ^ { 2 ^ N } + 1 } ^ { 2 ^ { 2 ^ { N + 1 } } } \\frac { \\psi _ 0 ( n ) \\varphi ( n ) } { n } > c \\end{align*}"}
-{"id": "8840.png", "formula": "\\begin{align*} \\mathcal { F } [ \\Psi _ v ] ( t , \\xi ) = \\frac { 1 } { 2 } \\lambda ^ { - 1 } \\chi _ 1 ( \\lambda ^ { - 1 } ( \\xi - \\xi _ v ) , \\lambda ^ { - 1 } \\xi _ v ) e ^ { \\frac { 1 } { 5 } i t \\xi ^ 5 } , \\end{align*}"}
-{"id": "1102.png", "formula": "\\begin{align*} \\tilde { n } _ { o , i } = \\sum _ { j \\in \\mathcal { M } _ i } \\beta _ j ^ { ( i ) } n _ { o , j } ^ { ( i ) } , \\end{align*}"}
-{"id": "2481.png", "formula": "\\begin{align*} \\xi \\cdot \\nabla _ { x } \\vartheta = \\frac { 1 0 } { 1 - \\gamma } \\delta \\frac { \\xi \\cdot x } { \\left \\langle x \\right \\rangle } \\left [ \\delta ( \\left \\langle x \\right \\rangle - M t ) \\right ] ^ { \\frac { 1 + \\gamma } { 1 - \\gamma } } . \\end{align*}"}
-{"id": "8293.png", "formula": "\\begin{align*} \\tau \\cdot ( 2 \\pi \\epsilon , a ) = ( 2 \\pi \\epsilon , a a _ \\tau ) \\end{align*}"}
-{"id": "8787.png", "formula": "\\begin{align*} \\textnormal { C o n t } ^ { \\geq m } ( f ) : = \\{ x \\in K [ [ t ] ] ^ { n } \\mid f ( x ) \\equiv 0 \\textnormal { m o d } t ^ { m } \\} \\end{align*}"}
-{"id": "8502.png", "formula": "\\begin{align*} \\abs { K _ { l _ 2 } } \\leq 2 \\zeta _ F ( 1 ) ^ 2 q ^ { - \\frac { a _ 1 } { 2 } - \\frac { a ( \\chi _ 1 \\chi _ 2 ^ { - 1 } ) } { 3 } } = 2 \\zeta _ F ( 1 ) ^ 2 q ^ { - \\frac { a _ 1 } { 2 } + \\frac { t } { 6 } } . \\end{align*}"}
-{"id": "2290.png", "formula": "\\begin{align*} \\frac { q - 1 + \\left | \\bigcup _ { i = 1 } ^ { q } T _ { i } \\right | - q \\cdot d _ { p } ( n ) } { p - 1 } \\end{align*}"}
-{"id": "2307.png", "formula": "\\begin{align*} & \\ ; ( \\| u _ * \\| _ { D ( A ) } ^ 2 + \\alpha ^ 2 \\| A ^ { 1 / 2 } u _ * \\| _ { L ^ 2 } ^ 2 ) ( t ) \\\\ & \\ ; + 2 \\nu \\int _ 0 ^ t ( \\| A ^ { s / 2 } u _ * \\| ^ 2 _ { D ( A ) } + \\alpha ^ 2 \\| A ^ { ( 1 + s ) / 2 } u _ * \\| ^ 2 _ { L ^ 2 } ) ( \\tau ) \\ , d \\tau \\\\ \\leq & \\ ; \\| u _ 0 \\| _ { D ( A ) } ^ 2 + \\alpha ^ 2 \\| A ^ { 1 / 2 } u _ 0 \\| _ { L ^ 2 } ^ 2 \\\\ & \\ ; + C \\int _ 0 ^ t ( \\| A u _ * \\| _ { L ^ 2 } \\| A u _ * \\| _ { D ( A ^ { s / 2 } ) } \\| A ^ { 1 / 2 } u _ * \\| _ { D ( A ^ { s / 2 } ) } ) ( \\tau ) \\ , d \\tau . \\end{align*}"}
-{"id": "4786.png", "formula": "\\begin{align*} { \\varepsilon } = F ( x ^ 0 , x ^ 1 , Y ) \\ , \\frac { \\Delta \\sqrt { - \\det g ^ { i j } } } { { L } _ 2 { L } _ 3 } , \\end{align*}"}
-{"id": "6703.png", "formula": "\\begin{align*} \\overline { \\frac { \\partial F } { \\partial \\theta ( u _ { 0 j } ) } } + \\overline { \\frac { \\partial F } { \\partial \\theta ( u _ { 0 0 } ) } } \\cdot ( - \\xi _ { j } ) = 0 , \\end{align*}"}
-{"id": "3427.png", "formula": "\\begin{align*} F _ 2 ( \\overline { L \\setminus K } ) \\cap F _ 2 ( \\overline { K \\setminus L } ) = \\varnothing . \\end{align*}"}
-{"id": "8172.png", "formula": "\\begin{align*} \\Vert v \\Vert _ { \\infty } = \\frac { \\Vert \\sigma ( v ) \\Vert _ { \\infty } } { \\Vert \\sigma ( v ) \\Vert _ 2 } . \\end{align*}"}
-{"id": "9433.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } n \\mathfrak { M } ( n ) ( \\log n ) ^ { 1 + \\epsilon } | { n } | _ { \\mathcal { D } } | | { n \\alpha } | | ^ { \\prime } = 0 , \\end{align*}"}
-{"id": "1465.png", "formula": "\\begin{align*} x _ 3 ^ 2 d _ u ( C \\langle 1 , y _ r , v _ s y _ s \\rangle ) & = d _ u ( x ^ 2 C \\langle 1 , y _ r , v _ s y _ s \\rangle ) \\\\ & \\subset d _ u ( D _ 1 / ( v _ t e _ t ) \\{ x _ 3 ^ 2 \\} \\oplus D _ r \\langle x _ 3 ^ 2 y _ r \\rangle ) = \\{ 0 \\} , \\end{align*}"}
-{"id": "3644.png", "formula": "\\begin{align*} C ( \\tilde P ) \\cap H _ t = t \\tilde P = t P + \\sigma ^ { \\vee } \\end{align*}"}
-{"id": "6341.png", "formula": "\\begin{align*} Q ( t ) = \\frac t { z _ 2 ^ * } - \\frac { z _ 1 ^ * - z _ 2 ^ * } { z _ 2 ^ * } \\log t . \\end{align*}"}
-{"id": "33.png", "formula": "\\begin{align*} \\frac { \\partial f } { \\partial z } = z ^ { * } \\frac { \\partial f } { \\partial z ^ { * } } = z \\end{align*}"}
-{"id": "9711.png", "formula": "\\begin{align*} \\begin{cases} & \\gamma _ 1 = K _ { 1 1 } \\alpha _ 5 + \\alpha _ 1 + \\beta _ 1 + O ( 1 ) \\Delta '' ( \\boldsymbol { \\alpha } ^ { * } , \\beta _ { 1 } ) , \\\\ & \\sigma ' _ { i } = K _ { 1 i } \\alpha _ 5 + \\alpha _ i + \\sigma _ i + O ( 1 ) \\Delta '' ( \\boldsymbol { \\alpha } ^ { * } , \\beta _ { 1 } ) , i = 2 , 3 , \\\\ & \\gamma _ 4 = \\alpha _ 4 + \\beta _ 4 , \\\\ & \\gamma _ 5 = K _ { 1 5 } \\alpha _ 5 + \\beta _ 5 + O ( 1 ) \\Delta '' ( \\boldsymbol { \\alpha } ^ { * } , \\beta _ { 1 } ) , \\end{cases} \\end{align*}"}
-{"id": "2076.png", "formula": "\\begin{align*} ( \\Delta _ H - \\partial _ t ) u \\geq 0 , \\ u ( p , t ) = \\phi ( p ) . \\end{align*}"}
-{"id": "8780.png", "formula": "\\begin{align*} F ( x , v ) = ( F _ { i n t } ( x ) + F _ { d i s s } ( x , v ) ) \\mathcal { H } ( x , v ) \\end{align*}"}
-{"id": "5190.png", "formula": "\\begin{align*} \\hat { K } _ { 1 } ( y , z ) = [ a _ { 1 } , y \\wedge a _ { 2 } ] , \\enskip \\hat { K } _ { 2 } ( x , z ) = [ x \\vee a _ { 1 } , a _ { 2 } ] , \\enskip \\hat { K } _ { 3 } ( x , y ) = [ b , 1 ] , \\end{align*}"}
-{"id": "8416.png", "formula": "\\begin{align*} L ( s , \\pi ) = \\begin{cases} L ( s , \\abs { \\cdot } ^ { \\frac { 1 } { 2 } } ) & \\chi = 1 , \\\\ 1 & \\chi \\neq 1 , \\end{cases} \\epsilon ( \\frac { 1 } { 2 } , \\pi ) = \\begin{cases} - 1 & \\chi = 1 , \\\\ \\epsilon ( \\frac { 1 } { 2 } , \\chi ) ^ 2 & \\chi \\neq 1 . \\end{cases} \\end{align*}"}
-{"id": "467.png", "formula": "\\begin{align*} a _ { k _ 1 , k _ 2 , \\pi / 2 } ^ { ( h ) } ( \\lambda ) = ( - 1 ) ^ { k _ 1 } i ^ { k _ 2 - n } \\left ( i \\frac { \\pi } { 2 } \\right ) ^ { n + k _ 1 + k _ 2 } \\frac { k _ 1 ! } { ( k _ 1 - h ) ! } \\lambda _ 1 ^ { k _ 1 - h } u _ 1 ^ { \\otimes h } + O \\left ( \\abs { \\lambda } ^ { k _ 1 - h + 1 } \\right ) \\end{align*}"}
-{"id": "6337.png", "formula": "\\begin{align*} \\varphi _ n ^ { [ a , b ] } : { \\vec p _ n } = ( p _ 1 , \\dots , p _ n ) \\mapsto { \\vec m _ n } = ( m _ 1 , \\dots , m _ n ) \\end{align*}"}
-{"id": "5115.png", "formula": "\\begin{align*} X _ { ( j , 1 ) } & \\cdots X _ { ( j , k ) } \\cdots X _ { ( j , \\kappa _ j ) } f ( p ) - X _ { ( j , 1 ) } \\cdots X ' _ { ( j , k ) } \\cdots X _ { ( j , \\kappa _ j ) } f ( p ) \\\\ & = X _ { ( j , 1 ) } \\cdots X _ { ( j , k - 1 ) } [ X _ { ( j , k - 1 ) } , X _ { ( j , k ) } - X ' _ { ( j , k ) } ] X _ { ( j , k + 1 ) } \\cdots X _ { ( j , \\kappa _ j ) } f ( p ) \\\\ & + \\cdots + [ X _ { ( j , 1 ) } , X _ { ( j , k ) } - X _ { ( j , k ) } ' ] X _ { ( j , 2 ) } \\cdots \\widehat { X _ { ( j , k ) } } \\cdots X _ { ( j , \\kappa _ j ) } f ( p ) \\end{align*}"}
-{"id": "751.png", "formula": "\\begin{align*} m ( a _ 1 , a _ 2 ) = m ( b _ 1 , b _ 2 ) + m ( a _ 1 - b _ 1 , a _ 2 - b _ 2 ) . \\end{align*}"}
-{"id": "5233.png", "formula": "\\begin{align*} \\Gamma ( \\gamma , V , t ^ * ) ( v ) = \\frac { d } { d t } \\omega ( v , B ( t ) v ) | _ { t = t ^ * } . \\end{align*}"}
-{"id": "4660.png", "formula": "\\begin{align*} \\mathcal { P } _ k = \\{ T ^ n I _ { k , j } : 0 \\le n < r _ { 0 , k } ( j ) , 1 \\le j \\le b \\} \\end{align*}"}
-{"id": "487.png", "formula": "\\begin{align*} p _ { 1 , k _ 1 , k _ 2 } ( x , t ) = \\frac { ( - 1 ) ^ { k _ 2 } \\pi ^ { k _ 1 + k _ 2 } } { 4 ^ n ( \\pi \\delta ) ^ { n + k _ 1 - \\frac { m + 1 } { 2 } } \\sqrt { 2 \\pi \\kappa ^ m } } e ^ { - \\frac { 1 } { 4 } d ( x , t ) ^ 2 } \\left [ 1 + O \\left ( \\delta + \\frac { 1 } { \\kappa } \\right ) \\right ] . \\end{align*}"}
-{"id": "724.png", "formula": "\\begin{align*} 1 5 \\Theta ^ 2 - 2 ( D - 8 d ) \\Theta + 4 d ^ 2 - D ^ 2 - 4 D = 0 . \\end{align*}"}
-{"id": "9570.png", "formula": "\\begin{align*} \\vert u ( x ) \\vert & \\le C \\int _ { D } \\frac { g ( y ) } { \\vert x - y \\vert ^ { n - s } } \\ , d y = C \\ , \\mathcal { I } _ { s } ( \\chi _ D g ) ( x ) \\end{align*}"}
-{"id": "3950.png", "formula": "\\begin{align*} D ^ { s } ( u v ) - u D ^ { s } v - v D ^ { s } u = c \\int \\frac { [ u ( x + y ) - u ( x ) ] [ v ( x + y ) - v ( x ) ] } { | y | ^ { n + s } } d y , 0 < s < 2 \\end{align*}"}
-{"id": "560.png", "formula": "\\begin{align*} L ( t , q , v ) : = \\max _ { p \\in T _ q ^ * Q } \\bigl ( p ( v ) - H ( t , q , p ) \\bigr ) , \\end{align*}"}
-{"id": "678.png", "formula": "\\begin{align*} K ( x , y ) + F ( x , y ) + \\int ^ x _ 0 K ( x , t ) F ( t , y ) d t = 0 , 0 \\leq y \\leq x \\leq \\pi , \\end{align*}"}
-{"id": "7178.png", "formula": "\\begin{align*} w _ k ^ r : = \\phi \\left ( w + ( \\theta - \\theta _ { j _ k } ) h \\right ) + ( 1 - \\phi ) z _ { j _ k } \\end{align*}"}
-{"id": "9718.png", "formula": "\\begin{align*} \\tilde { \\rho } - \\rho = - \\frac { \\rho } { u } ( \\tilde { u } - u ) + O ( h ^ 2 ) . \\end{align*}"}
-{"id": "2028.png", "formula": "\\begin{align*} g ( z ^ { - 1 } ) = - z ^ { - 1 } g ( z ) , ~ ~ ~ g ( 1 ) = 0 , ~ ~ ~ g ( q z ) = - z ^ { - 1 } g ( z ) . \\end{align*}"}
-{"id": "5370.png", "formula": "\\begin{align*} \\frac { S _ { ( a - 1 , 1 ) ( r ' , 1 ) } } { S _ { ( 1 , 1 ) ( r ' , 1 ) } } & = ( - 1 ) ^ { a } \\frac { \\sin \\left ( \\frac { \\pi b } { a } ( a - 1 ) r ' \\right ) \\sin \\left ( \\frac { \\pi a } { b } \\right ) } { \\sin \\left ( \\frac { \\pi b } { a } r ' \\right ) \\sin \\left ( \\frac { \\pi a } { b } \\right ) } = ( - 1 ) ^ { a + b r ' + 1 } \\end{align*}"}
-{"id": "5819.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } \\vert \\nabla v \\vert ^ { p - 2 } \\nabla v \\cdot \\nabla ( \\lambda _ t - \\lambda _ 0 ) \\ ; d x = \\int _ { \\R ^ n } v ^ { q } ( \\lambda _ t - \\lambda _ 0 ) \\ ; d \\sigma + \\int _ { \\R ^ n } ( \\lambda _ t - \\lambda _ 0 ) \\ ; d \\mu . \\end{align*}"}
-{"id": "1415.png", "formula": "\\begin{align*} \\iota _ 1 ^ * ( x _ 2 ) = s _ 1 ^ 2 + s _ 1 s _ 2 + s _ 2 ^ 2 , \\iota _ 1 ^ * ( x _ 3 ) = s _ 1 ^ 2 s _ 2 + s _ 1 s _ 2 ^ 2 , \\end{align*}"}
-{"id": "2026.png", "formula": "\\begin{align*} \\begin{array} { c c c } g ( z ) = ( 1 - z ) \\prod _ { i = 1 } ^ { \\infty } ( 1 - q ^ i ) ( 1 - q ^ i z ) ( 1 - q ^ i z ^ { - 1 } ) = \\sum _ { i \\in \\Z } ( - 1 ) ^ i z ^ i q ^ { \\frac { i ( i - 1 ) } { 2 } } , \\\\ \\\\ f _ 1 ( z _ 1 , z _ 2 ) = \\frac { z _ 2 g \\big ( \\frac { z _ 1 ^ 2 } { z _ 2 ^ 2 } \\big ) } { g \\big ( \\frac { z _ 1 } { z _ 2 } \\big ) } \\prod _ { i = 1 } ^ { \\infty } ( 1 - q ^ i ) = \\sum _ { i \\in \\Z } ( - 1 ) ^ i q ^ { \\frac { i ( 3 i - 1 ) } { 2 } } ( z _ 1 ^ { 3 i } z _ 2 ^ { 1 - 3 i } + z _ 1 ^ { 1 - 3 i } z _ 2 ^ { 3 i } ) \\end{array} \\end{align*}"}
-{"id": "5103.png", "formula": "\\begin{align*} \\mathop { \\textup { c a r d } } ( \\widehat { \\mathcal { L } } / \\mathcal { M } ^ \\perp ) = \\mathop { \\textup { c a r d } } \\mathcal { M } . \\end{align*}"}
-{"id": "5977.png", "formula": "\\begin{align*} \\beta = \\left ( 1 + \\frac { 4 \\varepsilon ^ 2 } { \\alpha ^ 2 } \\right ) ^ { 1 / 4 } , \\delta ^ 2 = \\frac { \\alpha ^ 2 } { 2 } ( \\beta ^ 2 - 1 ) . \\end{align*}"}
-{"id": "6362.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } \\frac { 1 } { n } \\log \\int _ 0 ^ 1 e ^ { - n V _ 1 ( x ) + ( n - i ) \\log ( x ( 1 - x ) ) } \\ , d x = - \\inf \\limits _ { y \\in ( 0 , 1 ) } W _ 1 ( y ) . \\end{align*}"}
-{"id": "6158.png", "formula": "\\begin{align*} g = V ^ 2 \\dd x _ 0 ^ 2 + f _ 1 \\dd x _ 1 ^ 2 + f _ 2 \\dd x _ 2 ^ 2 + f _ 3 \\dd x _ 3 ^ 2 \\end{align*}"}
-{"id": "7926.png", "formula": "\\begin{align*} \\dim _ H K _ { \\Delta } = \\dim _ B K _ { \\Delta } = \\frac { \\log 3 } { \\log 2 } . \\end{align*}"}
-{"id": "4713.png", "formula": "\\begin{align*} \\abs { \\det A _ s ^ i } ^ { - 1 } = \\max \\{ 1 , | s _ 1 | , \\ldots , | s _ { d - 1 } | \\} \\leq 1 + \\max \\{ | s _ 1 | , \\ldots , | s _ { d - 1 } | \\} \\lesssim 1 + \\| s \\| . \\end{align*}"}
-{"id": "3819.png", "formula": "\\begin{align*} \\inf _ { \\eta \\colon | \\eta | \\le k , \\eta ( 0 ) = 0 } \\P _ \\eta \\left ( G _ \\infty \\cap \\Lambda _ \\infty \\right ) \\ge p _ * ^ { k } \\ ; \\ ; \\ ; \\ ; \\forall \\ ; k \\ge 0 . \\end{align*}"}
-{"id": "6154.png", "formula": "\\begin{align*} \\omega _ 1 ^ 0 = \\dd x _ 0 \\wedge \\dd x _ 1 + \\dd x _ 2 \\wedge \\dd x _ 3 \\\\ \\omega _ 2 ^ 0 = \\dd x _ 0 \\wedge \\dd x _ 2 + \\dd x _ 3 \\wedge \\dd x _ 1 \\\\ \\omega _ 3 ^ 0 = \\dd x _ 0 \\wedge \\dd x _ 3 + \\dd x _ 1 \\wedge \\dd x _ 2 \\end{align*}"}
-{"id": "1470.png", "formula": "\\begin{align*} v _ 2 \\phi _ { r + 1 } ( x _ I ) x _ 3 ^ 2 = \\cdots = v _ r \\phi _ { r + 1 } ( x _ I ) x _ 3 ^ 2 = 0 , \\end{align*}"}
-{"id": "3177.png", "formula": "\\begin{align*} \\lambda _ { t } = \\int _ { 0 } ^ { t } \\int _ { \\lbrace z > 1 \\rbrace } \\left ( 1 - e ^ { - \\alpha ( z , s ) } \\right ) \\nu ( \\mathrm { d } z ) \\mathrm { d } s < \\infty . \\end{align*}"}
-{"id": "5434.png", "formula": "\\begin{align*} f _ i ( P _ k ( d ) ) = \\binom { d + 1 } { i + 1 } + \\binom { d } { i } - \\binom { d - k } { d - i } . \\end{align*}"}
-{"id": "1964.png", "formula": "\\begin{align*} \\mathcal { F } \\left ( ( - \\Delta ) ^ { \\beta / 2 } \\ , f \\right ) ( \\xi ) : = | \\xi | ^ \\beta \\ , \\widehat { f } ( \\xi ) , \\end{align*}"}
-{"id": "7933.png", "formula": "\\begin{align*} F ^ * ( a _ 1 , \\ldots , a _ m ) ( \\mu ) = \\sum _ { i = 1 } ^ m a _ i ( f _ i ) _ * ( \\mu ) . \\end{align*}"}
-{"id": "347.png", "formula": "\\begin{align*} & \\quad \\ 2 h ^ { i j } \\left ( \\nabla _ j R _ { \\nu l i } ^ { \\ \\ \\ l } + \\nabla _ l R _ { \\nu i j } ^ { \\ \\ \\ l } \\right ) \\\\ & = 2 \\left ( h _ { 1 1 } \\nabla _ 1 R _ { \\nu 2 1 2 } + h _ { 2 2 } \\nabla _ 2 R _ { \\nu 1 2 1 } + h _ { 1 1 } \\nabla _ 2 R _ { \\nu 1 1 2 } + h _ { 2 2 } \\nabla _ 1 R _ { \\nu 2 2 1 } \\right ) \\\\ & = 2 \\left ( h _ { 2 2 } - h _ { 1 1 } \\right ) \\left ( \\nabla _ 2 R _ { \\nu 1 2 1 } - \\nabla _ 1 R _ { \\nu 2 1 2 } \\right ) = 2 \\left ( h _ { 2 2 } - h _ { 1 1 } \\right ) ( \\nabla _ 2 R _ { \\nu 2 } - \\nabla _ 1 R _ { \\nu 1 } ) . \\end{align*}"}
-{"id": "4026.png", "formula": "\\begin{align*} X = \\alpha _ 1 V _ 1 + \\alpha _ 2 V _ 2 \\ \\mathrm { a n d } \\\\ Y = \\beta _ 1 V _ 1 + \\beta _ 2 V _ 2 . \\end{align*}"}
-{"id": "6540.png", "formula": "\\begin{align*} ( 1 - \\Delta ) e _ n = \\left ( 1 - \\left ( \\frac { n | P | _ n } { | ( B ^ { \\circ } - s ( B ^ { \\circ } ) ) ^ { \\circ } | _ { n - 1 } } \\right ) ^ { \\frac { 1 } { n } } \\delta ^ { 1 / n } \\right ) e _ n = \\left ( 1 - \\left ( \\frac { n | P | _ n } { | ( F - s ( F ) ) ^ { \\circ } | _ { n - 1 } \\| e _ n \\| } \\right ) ^ { \\frac { 1 } { n } } \\delta ^ { 1 / n } \\right ) e _ n . \\end{align*}"}
-{"id": "5209.png", "formula": "\\begin{align*} V _ { 2 } ^ { [ \\ell ^ { 1 } , \\ell ^ { 2 } ] } ( x ) \\coloneqq \\sup \\limits _ { \\tau _ { 2 } \\in \\mathcal { T } } M ^ { x } _ { 2 } ( D _ { [ \\ell ^ { 1 } , \\ell ^ { 2 } ] } , \\tau _ { 2 } ) = M ^ { x } _ { 2 } ( D _ { [ \\ell ^ { 1 } , \\ell ^ { 2 } ] } , D _ { [ r , 1 ] } ) , \\forall x \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "10008.png", "formula": "\\begin{align*} [ x _ 1 , x _ 2 ] = Q ( x _ 1 + x _ 2 ) - Q ( x _ 1 ) - Q ( x _ 2 ) \\end{align*}"}
-{"id": "5845.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ N ( - 1 ) ^ k \\binom { N } { k } g ^ k D ^ { N - 1 } ( g ^ { N - k } f ) = 0 , \\end{align*}"}
-{"id": "1877.png", "formula": "\\begin{align*} \\mathcal { Q } _ 1 : = \\Big \\{ \\mu \\in c a ^ + _ 1 ( \\mathbb { R } ^ d ) : \\overline { \\nu } _ j \\preceq _ 1 \\mu _ j \\preceq _ 1 \\underline { \\nu } _ j \\underline { \\pi } _ i \\leq \\mu ( A ^ i ) \\mbox { f o r a l l } i , j \\Big \\} . \\end{align*}"}
-{"id": "9341.png", "formula": "\\begin{align*} \\begin{aligned} J O _ { n } ^ { ( 3 ) } & = J _ { n } ^ { ( 3 ) } + \\sum _ { s = 1 } ^ { 7 } J _ { n + s } ^ { ( 3 ) } e _ { s } , \\ n \\geq 0 \\\\ & = J _ { n } ^ { ( 3 ) } + J _ { n + 1 } ^ { ( 3 ) } e _ { 1 } + J _ { n + 2 } ^ { ( 3 ) } e _ { 2 } + J _ { n + 3 } ^ { ( 3 ) } e _ { 3 } \\\\ & \\ \\ + J _ { n + 4 } ^ { ( 3 ) } e _ { 4 } + J _ { n + 5 } ^ { ( 3 ) } e _ { 5 } + J _ { n + 6 } ^ { ( 3 ) } e _ { 6 } + J _ { n + 7 } ^ { ( 3 ) } e _ { 7 } \\end{aligned} \\end{align*}"}
-{"id": "4496.png", "formula": "\\begin{align*} | \\psi ( s ) | = | \\phi ( u ) | = 1 . \\end{align*}"}
-{"id": "4795.png", "formula": "\\begin{align*} L = \\left ( \\begin{array} { c c c | c } - z _ 2 & 0 & \\phantom { - } z _ 4 & 0 \\\\ - z _ 1 & - z _ 3 & 0 & z _ { 5 } \\\\ - z _ 2 & \\phantom { - } z _ 3 & - z _ 4 & 0 \\\\ \\hline z _ 1 + 2 z _ 2 & 0 & 0 & - z _ { 5 } \\end{array} \\right ) \\end{align*}"}
-{"id": "7577.png", "formula": "\\begin{align*} \\pi _ 1 ( u , v ) & = \\sum _ { j = 0 } ^ \\infty \\ , ( u _ 0 + \\dots + u _ { j - 2 } ) \\cdot v _ j , \\\\ \\pi _ 2 ( u , v ) & = \\sum _ { j = 0 } ^ \\infty \\ , ( u _ { j - 1 } \\cdot v _ j + u _ j \\cdot v _ j + u _ j \\cdot v _ { j - 1 } ) , \\\\ \\pi _ 3 ( u , v ) & = \\sum _ { j = 0 } ^ \\infty \\ , u _ j \\cdot ( v _ 0 + \\dots + v _ { j - 2 } ) = \\pi _ 1 ( v , u ) . \\end{align*}"}
-{"id": "2353.png", "formula": "\\begin{align*} \\sum _ { n \\in \\N } \\alpha _ s ( g _ p p ^ n ) = \\alpha _ s ( g _ p ) \\sum _ { n \\in \\N } \\alpha _ s ( p ^ n ) , \\end{align*}"}
-{"id": "5020.png", "formula": "\\begin{align*} \\frac { | S | } { r _ { 1 } } = \\frac { k _ { 1 } | S | } { k _ { 1 } r _ { 1 } } \\le \\frac { | \\Gamma _ 2 ( S ) | - 1 } { r _ 2 - 1 } = \\frac { | \\Gamma _ 2 ( S ) | - 1 } { | L _ 2 \\setminus \\{ x \\} | } . \\end{align*}"}
-{"id": "8959.png", "formula": "\\begin{gather*} 1 = \\sum _ { w \\in W } { } ^ w h = \\sum _ { g \\in D _ x } { } ^ g \\bigg ( \\sum _ { w \\in D _ x \\setminus W } { } ^ w h \\bigg ) , \\end{gather*}"}
-{"id": "8998.png", "formula": "\\begin{gather*} { \\cal D } ^ { ( n ) } _ { q , t : * * } ( c ) \\ ! \\ ! \\prod _ { 1 \\le i \\le n } \\ ! \\ ! \\frac { ( ( v c d ) z _ i ; q ) _ \\infty } { ( ( v / c d ) z _ i ; q ) _ \\infty } \\ , { \\cal D } ^ { ( n ) } _ { q , t : * * } ( d ) = \\ ! \\ ! \\prod _ { 1 \\le i \\le n } \\ ! \\ ! \\frac { ( ( v d ) z _ i ; q ) _ \\infty } { ( ( v / d ) z _ i ; q ) _ \\infty } \\ , { \\cal D } ^ { ( n ) } _ { q , t : * * } ( c d ) \\ ! \\ ! \\prod _ { 1 \\le i \\le n } \\ ! \\ ! \\frac { ( ( v c ) z _ i ; q ) _ \\infty } { ( ( v / c ) z _ i ; q ) _ \\infty } \\end{gather*}"}
-{"id": "3965.png", "formula": "\\begin{align*} \\partial _ t ^ { \\nu _ n } p ^ { \\nu _ n } ( n , t ) = - \\lambda _ n p ^ { \\nu _ n } ( n , t ) + \\lambda _ { n - 1 } p ^ { \\nu _ { n - 1 } } ( n - 1 , t ) , \\ \\ 0 < \\nu _ n \\leq 1 , \\ n \\geq 1 , \\end{align*}"}
-{"id": "1424.png", "formula": "\\begin{align*} Q _ j ( x _ 3 x _ { 4 i } ) = x _ 3 ^ 2 \\partial _ j ( x _ { 4 i } ) , Q _ j ( x _ { 4 i _ { 1 } } x _ { 4 i _ { 2 } } ) = x _ 3 \\partial _ j ( x _ { 4 i _ { 1 } } x _ { 4 i _ { 2 } } ) . \\end{align*}"}
-{"id": "4244.png", "formula": "\\begin{align*} \\frac 1 { a ^ p } \\left ( \\frac { \\det ( B _ 0 ) } { \\det ( B _ { m + 1 } ) } \\right ) ^ 2 = \\frac { a ^ p } { \\lambda ^ { p - 1 } } \\left ( \\frac { \\det ( A _ { m + 1 } ) } { \\det ( A _ p ) } \\right ) ^ 2 \\end{align*}"}
-{"id": "2505.png", "formula": "\\begin{align*} \\begin{aligned} g ^ { - 1 } \\cdot \\chi _ { 0 } & = \\chi _ { 0 } , g ^ { - 1 } \\cdot \\chi _ { 4 } = \\chi _ { 4 } , \\\\ g ^ { - 1 } \\cdot \\chi _ { 1 } & = g ^ { - 1 } \\cdot \\mu ^ { 1 / 2 } \\xi _ { 1 } = \\mu ^ { 1 / 2 } \\left ( g \\xi \\right ) _ { 1 } = \\mu ^ { 1 / 2 } \\xi \\cdot \\frac { \\eta } { \\left | \\eta \\right | } = \\sum _ { j = 1 } ^ { 3 } \\frac { \\eta _ { j } } { \\left | \\eta \\right | } \\chi _ { j } , \\end{aligned} \\end{align*}"}
-{"id": "9763.png", "formula": "\\begin{align*} W ( U _ { h , \\theta } ( i h + , y ) ) = W ( U _ { h , \\theta } ( i h - , y _ { i , n } ) ) + G ( U _ { h , \\theta } ( i h - , y _ { i , n } ) ) h . \\end{align*}"}
-{"id": "1432.png", "formula": "\\begin{align*} M _ { 1 , 1 } \\cap \\mathrm { K e r } \\ , \\partial _ j = \\{ 0 \\} . \\end{align*}"}
-{"id": "2102.png", "formula": "\\begin{align*} u ^ a ( p , 0 ; s ) = \\phi ^ a ( p ; s ) . \\end{align*}"}
-{"id": "5232.png", "formula": "\\begin{align*} \\Sigma ( V ) = \\{ V ' \\in \\Lambda ( n ) : V \\cap V ' \\neq \\{ 0 \\} \\} . \\end{align*}"}
-{"id": "7844.png", "formula": "\\begin{align*} R _ { X _ { 1 } , X _ { 2 } } ( x _ { 1 } , x _ { 2 } ) = 1 - F _ { X _ { 1 } } ( x _ { 1 } ) - F _ { X _ { 2 } } ( x _ { 2 } ) + F _ { X _ { 1 } , X _ { 2 } } ( x _ { 1 } , x _ { 2 } ) . \\end{align*}"}
-{"id": "760.png", "formula": "\\begin{align*} & J ( u _ 1 ^ n , u _ 2 ^ n ) = J ( u _ 1 ^ n ( x ' , x _ 3 - y _ 1 ^ n ) , u _ 2 ^ n ( x ' , x _ 3 - y _ 1 ^ n ) ) \\\\ & = J ( u _ 1 , 0 ) + J ( u _ 1 ^ n ( x ' , x _ 3 - y _ 1 ^ n ) - u _ 1 , u _ 2 ^ n ( x ' , x _ 3 - y _ 1 ^ n ) ) + o _ n ( 1 ) \\\\ & = J ( u _ 1 , 0 ) + J ( u _ 1 ^ n ( x ' , x _ 3 - y _ 2 ^ n ) - u _ 1 ( x ' , x _ 3 + y _ 1 ^ n - y _ 2 ^ n ) , u _ 2 ^ n ( x ' , x _ 3 - y _ 2 ^ n ) ) + o _ n ( 1 ) \\\\ & = J ( u _ 1 , 0 ) + J ( 0 , \\tilde { u } _ 2 ) + J ( u _ 1 ^ n ( x ' , x _ 3 - y _ 2 ^ n ) - u _ 1 ( x ' , x _ 3 + y _ 1 ^ n - y _ 2 ^ n ) , u _ 2 ^ n ( x ' , x _ 3 - y _ 2 ^ n ) - \\tilde { u } _ 2 ) + o _ n ( 1 ) . \\end{align*}"}
-{"id": "6030.png", "formula": "\\begin{align*} & \\left | P _ { X ^ { n } Y ^ { n } } - \\pi _ { X ^ { n } Y ^ { n } } \\right | \\\\ & \\geq 1 - 4 \\exp \\left ( - n \\frac { \\frac { 1 } { n } \\Omega ^ { ( \\alpha , \\lambda ) } ( \\{ Q _ { i } \\} _ { i = 1 } ^ { n } ) - \\lambda \\alpha R } { 1 + ( 1 + \\bar { \\alpha } ) \\lambda } \\right ) . \\end{align*}"}
-{"id": "6442.png", "formula": "\\begin{align*} p _ { } ^ { \\prime } \\left ( x _ { 1 } , \\ldots , x _ { l } \\right ) = \\delta \\left ( x _ { l } - f _ { l } \\left ( x _ { 1 } , \\ldots , x _ { l - 1 } \\right ) \\right ) \\delta \\left ( x _ { l - 1 } - f _ { l - 1 } \\left ( x _ { 1 } , \\ldots , x _ { l - 2 } \\right ) \\right ) \\cdots \\delta \\left ( x _ { 2 } - f _ { 2 } \\left ( x _ { 1 } \\right ) \\right ) p _ { 1 } \\left ( x _ { 1 } \\right ) \\end{align*}"}
-{"id": "7524.png", "formula": "\\begin{gather*} u = \\frac { t w ^ 1 + w ^ 2 } { t ^ 2 + 1 } + \\frac { t ( x + \\mu ) } { t ^ 2 + 1 } - \\frac { y } { t ^ 2 + 1 } , \\\\ v = \\frac { - w ^ 1 + t w ^ 2 } { t ^ 2 + 1 } + \\frac { t y } { t ^ 2 + 1 } + \\frac { x - \\mu } { t ^ 2 + 1 } , \\end{gather*}"}
-{"id": "8935.png", "formula": "\\begin{gather*} s _ i ( x _ i , x _ j ) = ( - x _ i + \\mu _ { j i } ( x _ j ) , x _ j ) , \\\\ s _ j ( x _ i , x _ j ) = ( x _ i , - x _ j + \\mu _ { i j } ( x _ i ) ) . \\end{gather*}"}
-{"id": "6049.png", "formula": "\\begin{align*} \\lim _ { ( \\theta , Q _ { X Y U } ) \\to ( 0 , Q _ { X Y U } ' ) } \\frac { 1 } { \\theta } \\Omega ^ { ( \\alpha , \\theta ) } ( Q _ { X Y U } ) = R ^ { ( \\alpha ) } ( Q _ { X Y U } ' ) \\end{align*}"}
-{"id": "6741.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\frac { | F _ { 1 } ( N ) | } { N ^ { N ^ { \\gamma } t } } = 1 \\lim \\limits _ { N \\rightarrow \\infty } \\frac { | F _ { 2 } ( N ) | } { N ^ { N ^ { \\gamma } t } } = 0 . \\end{align*}"}
-{"id": "9772.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k - 1 } \\tilde { E } _ { i } ( h _ j , \\theta ) \\to 0 , \\end{align*}"}
-{"id": "6403.png", "formula": "\\begin{align*} \\mathcal { S } \\left ( \\theta ^ { \\prime } \\theta \\right ) = - \\int d x p \\left ( x | \\theta ^ { \\prime } \\right ) \\log \\left [ \\frac { p \\left ( x | \\theta ^ { \\prime } \\right ) } { p \\left ( x | \\theta \\right ) } \\right ] \\end{align*}"}
-{"id": "9346.png", "formula": "\\begin{align*} \\begin{aligned} J O _ { n + 1 } ^ { ( 3 ) } + J O _ { n } ^ { ( 3 ) } + 2 J O _ { n - 1 } ^ { ( 3 ) } & = \\sum _ { s = 0 } ^ { 7 } ( J _ { n + s + 1 } ^ { ( 3 ) } + J _ { n + s } ^ { ( 3 ) } + 2 J _ { n + s - 1 } ^ { ( 3 ) } ) e _ { s } \\\\ & = J _ { n + 2 } ^ { ( 3 ) } + \\sum _ { s = 1 } ^ { 7 } J _ { n + s + 2 } ^ { ( 3 ) } e _ { s } \\\\ & = J O _ { n + 2 } ^ { ( 3 ) } \\end{aligned} \\end{align*}"}
-{"id": "3258.png", "formula": "\\begin{align*} \\alpha _ 0 = \\alpha _ 0 ^ 0 + \\sum _ { l \\geq 1 } u ^ l \\alpha _ 0 ^ l . \\end{align*}"}
-{"id": "426.png", "formula": "\\begin{align*} H _ { k _ 1 , k _ 2 } ( R , t ) \\coloneqq & \\int _ { - \\pi } ^ { \\pi } e ^ { \\kappa \\cos ( s ) + R r ( - \\delta e ^ { i s } ) } \\phi _ { \\delta , k _ 1 , k _ 2 } ( s ) \\ , \\dd s = \\int _ { - \\pi } ^ \\pi e ^ { \\kappa q _ \\delta ( - i s ) } \\phi _ { \\delta , k _ 1 , k _ 2 } ( s ) \\ , \\dd s , \\end{align*}"}
-{"id": "3727.png", "formula": "\\begin{align*} \\left ( f _ 0 , \\frac { \\xi _ 1 } { \\xi _ 2 } , \\frac { \\xi _ 2 } { \\xi _ 3 } , \\frac { \\xi _ 3 } { \\xi _ 4 } , \\frac { \\xi _ 4 } { \\xi _ 5 } , \\frac { \\xi _ 5 } { \\xi _ 6 } , f _ 6 \\right ) = ( 0 , 1 , \\omega ^ { - 1 } , 1 , \\omega ^ { - 1 } , 1 , 0 ) . \\end{align*}"}
-{"id": "7343.png", "formula": "\\begin{align*} \\Vert \\Phi u \\Vert _ { X ^ { ( 4 , 3 ) } [ 0 , T ] } \\ ; & \\leqslant \\ ; C \\ , \\Big ( \\Vert f \\Vert _ { H ^ 1 ( \\R ^ 3 ) } + \\sum _ { i = 1 } ^ N \\| F _ i \\| _ { L ^ { s _ i } ( [ 0 , T ] , L ^ { p _ i } ( \\R ^ 3 ) } \\\\ & \\qquad \\qquad + \\sum _ { i = 1 } ^ N \\| G _ i \\| _ { L ^ { \\widetilde { s } _ i } ( [ 0 , T ] , L ^ { \\widetilde { p } _ i } ( \\R ^ 3 ) } + T ^ { \\theta } \\Vert u \\Vert _ { X ^ { ( 4 , 3 ) } [ 0 , T ] } \\Big ) \\ , . \\end{align*}"}
-{"id": "6940.png", "formula": "\\begin{align*} \\overline { \\omega ' } = a _ i t _ i ^ { n _ i } d t _ i , & \\\\ \\overline { \\omega '' } = b _ i t _ i ^ { n _ i } d t _ i , & \\textup { f o r } i = 2 , \\dots , d - 1 , \\end{align*}"}
-{"id": "3218.png", "formula": "\\begin{align*} \\varepsilon _ 2 \\leq \\varepsilon _ 1 , \\varepsilon _ 2 \\leq A _ 2 ^ { 1 / 2 } , A _ 2 = \\frac { 1 } { 6 4 C _ 2 ^ 2 C _ 3 ^ 2 } . \\end{align*}"}
-{"id": "10156.png", "formula": "\\begin{align*} { \\boldsymbol \\omega _ k } ^ { ( D ) } ( i ) = { \\boldsymbol S } _ { D _ k } ( i ) \\bar { \\boldsymbol \\omega } _ k ( i ) , \\end{align*}"}
-{"id": "9956.png", "formula": "\\begin{align*} \\overline { { \\cal N } } ( u ) - V _ 1 = V ( { \\cal G } ) - ( { \\cal N } ( u ) \\cup \\{ u \\} \\cup V _ 1 ) = V ( { \\cal G } ) - ( { \\cal N } ( u ) \\cup V _ 1 ) \\end{align*}"}
-{"id": "4413.png", "formula": "\\begin{align*} \\int \\nabla \\zeta \\cdot h \\ , \\ , d x = t \\int \\zeta ( \\cdot , x _ 3 = 0 ) \\ , \\nabla ' \\cdot m ' \\ , \\ , d x ' \\quad \\mbox { f o r a l l } \\ ; \\zeta . \\end{align*}"}
-{"id": "6561.png", "formula": "\\begin{align*} \\alpha _ { \\xi } = \\left ( \\frac { n \\frac { 2 ^ n } { n ! } } { \\frac { 2 ^ { n - 1 } } { ( n - 1 ) ! } \\cdot 1 } \\right ) ^ { 1 / n } = 2 ^ { 1 / n } . \\end{align*}"}
-{"id": "1350.png", "formula": "\\begin{align*} W ( H ) = P \\left ( \\Theta \\circ ( P ^ T H P ) \\right ) P ^ T \\forall \\ , H \\in { \\cal S } ^ p \\ , . \\end{align*} % \\end{align*}"}
-{"id": "200.png", "formula": "\\begin{align*} g ( \\sigma ) = g ( \\sigma \\cdot 1 ) = \\sigma \\cdot g ( 1 ) = \\sigma \\cdot f ( a ) = f ( \\sigma \\ast a ) = f \\circ h ( \\sigma ) . \\end{align*}"}
-{"id": "3314.png", "formula": "\\begin{align*} D ^ * ( r ) \\leq \\bar { D } ( r ) = \\alpha \\bar { D } ( r _ s ) + ( 1 - \\alpha ) \\bar { D } ( r _ { s + 1 } ) \\end{align*}"}
-{"id": "5462.png", "formula": "\\begin{align*} U _ x ( y ) = \\frac { U ( x y ) } { x ^ \\rho \\ell ( x ) } . \\end{align*}"}
-{"id": "70.png", "formula": "\\begin{align*} \\frac { \\partial ( e e ^ * ) } { \\partial w ^ * } = ( \\textbf { X } D ^ * - \\textbf { X X } ^ { H } \\textbf { w } ) \\end{align*}"}
-{"id": "2423.png", "formula": "\\begin{align*} D _ 3 = 0 , \\ D _ 1 < 0 , \\ D _ 3 ' \\neq 0 , \\ R _ { 1 3 3 } \\neq 0 , \\end{align*}"}
-{"id": "2199.png", "formula": "\\begin{align*} \\begin{cases} | l _ 1 | = r _ 1 , \\\\ \\ldots \\\\ | l _ n | = r _ n , \\end{cases} \\begin{cases} l _ k = 1 - a _ { k j _ { k } } z _ k , & \\ a _ { k j _ k } \\neq 0 , \\\\ l _ k = 1 / z _ k , & \\ a _ { k j _ k } = 0 . \\end{cases} \\end{align*}"}
-{"id": "9820.png", "formula": "\\begin{align*} f ( t ' ) = \\frac { ( 1 - p _ 0 ( t ' ) ) \\mu } { \\l } - \\frac { \\beta _ 2 \\mu } { ( \\alpha + \\beta _ 2 ) \\l } \\geq 0 , t ' \\leq t \\leq ( T ' _ { e _ 2 } \\wedge T _ 2 ) . \\end{align*}"}
-{"id": "6758.png", "formula": "\\begin{align*} L = \\bigl ( \\log | \\eta _ j | _ w \\bigr ) , \\end{align*}"}
-{"id": "2986.png", "formula": "\\begin{align*} - \\int _ { \\Omega } ( u - \\bar { u } ) \\Delta \\varphi & \\leq \\int _ { \\Omega } \\left ( \\bar { g } ( x , u ) - f h ( \\bar { u } ) \\right ) \\varphi \\\\ & = \\int _ { \\Omega } \\chi _ { \\{ u \\leq \\bar { u } \\} } \\left ( \\bar { g } ( x , u ) - f h ( \\bar { u } ) \\right ) \\varphi . \\end{align*}"}
-{"id": "7170.png", "formula": "\\begin{align*} \\| \\nabla u _ { x _ 0 , r _ k } \\| ^ 2 _ { L ^ 2 ( B _ 1 , | x _ n | ^ a ; \\R ^ n ) } = \\frac { D _ a ^ { x _ 0 } ( r _ k ) } { r _ k ^ { n + 1 } } \\stackrel { \\eqref { d : f r e q u e n c y } } { = } \\frac { H _ a ^ { x _ 0 } ( r _ k ) } { N _ a ^ { x _ 0 } ( r _ k ) \\ , r _ k ^ { n + 2 } } \\leq \\frac { H _ a ^ { x _ 0 } ( 1 - | x _ 0 | ) } { ( 1 + s ) \\ , ( 1 - | x _ 0 | ) ^ { n + 2 } } \\end{align*}"}
-{"id": "71.png", "formula": "\\begin{align*} E _ { D Y } [ G ^ { C } _ { \\sigma } ( e ) ( \\textbf { X } D ^ * - \\textbf { X X } ^ { H } \\textbf { w } ) ] = \\textbf { 0 } \\end{align*}"}
-{"id": "358.png", "formula": "\\begin{align*} U _ 0 = \\frac { ( R _ { 2 2 } - R _ { 1 1 } ) ^ 2 } { 4 | \\nabla f | ^ 2 H ^ 2 } = \\frac { ( R _ { 2 \\nu 2 \\nu } - R _ { 1 \\nu 1 \\nu } ) ^ 2 } { 4 | \\nabla f | ^ 2 H ^ 2 } \\in O ( r ^ { 2 b - 4 a } ) , \\end{align*}"}
-{"id": "7294.png", "formula": "\\begin{align*} I : = ( \\ell _ 1 \\cdots \\hat { \\ell _ i } \\cdots \\ell _ n \\ , | \\ , 1 \\leq i \\leq n ) . \\end{align*}"}
-{"id": "1569.png", "formula": "\\begin{align*} - \\nu ( C _ { j _ k } ( \\alpha ( k ) ) ) & = - \\nu \\left ( \\frac { \\alpha ( \\alpha - 1 ) \\dots ( \\alpha - j _ k + 1 ) } { j _ k ! } \\right ) \\\\ & = j _ k - s ( j _ k ) - \\nu ( ( \\alpha - j _ k + 1 ) \\dots ( \\alpha - 1 ) \\alpha ) \\\\ & = j _ k - s ( j _ k ) - ( \\nu ( ( - \\alpha - j - k + 1 ) ! ) - \\nu ( - \\alpha - 1 ) ) \\\\ & = - s ( j _ k ) - s ( - \\alpha - 1 ) + s ( - \\alpha - 1 + j _ k ) \\\\ & = - c ( j _ k , - \\alpha - 1 ) . \\end{align*}"}
-{"id": "9256.png", "formula": "\\begin{align*} s ( C ; c ) = s ( c ) = \\sum _ { u \\in N [ c ] } \\frac { 1 } { | I ( C ; u ) | } \\textrm { . } \\end{align*}"}
-{"id": "8947.png", "formula": "\\begin{gather*} \\begin{pmatrix} 0 & 4 b \\\\ 4 b ^ \\vee & 0 \\end{pmatrix} \\end{gather*}"}
-{"id": "4730.png", "formula": "\\begin{align*} { L } _ i = \\left ( L _ 0 ( x ^ 0 ) , \\alpha , \\beta , \\gamma \\right ) , \\alpha , \\beta , \\gamma - c o n s t , \\end{align*}"}
-{"id": "8460.png", "formula": "\\begin{align*} \\Delta = \\varpi ^ { 2 v ( b _ 2 ) } + 4 v ( b _ 1 + v x _ 2 ^ 2 \\zeta ) . \\end{align*}"}
-{"id": "2974.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\nabla u _ n \\nabla \\varphi = \\int _ { \\Omega } h _ n \\left ( u _ n + \\frac { 1 } { n } \\right ) f _ n \\varphi + \\int _ \\Omega \\mu _ n \\varphi , ~ \\forall \\varphi \\in C _ c ^ 1 ( \\bar \\Omega ) . \\end{align*}"}
-{"id": "3775.png", "formula": "\\begin{align*} \\P ^ \\rho \\left ( \\bar { X } _ 0 < - \\sqrt { L } \\right ) = \\P ^ { \\rho } \\left ( N ( z , 0 ) = 0 \\ ; \\forall \\ ; z \\in [ - \\sqrt { L } , 0 ] \\cap \\Z \\right ) \\le c e ^ { - \\rho \\sqrt { L } } . \\end{align*}"}
-{"id": "6406.png", "formula": "\\begin{align*} \\Delta l ^ { 2 } \\overset { } { = } g _ { \\mu \\nu } \\left ( \\theta \\right ) \\Delta \\theta ^ { \\mu } \\Delta \\theta ^ { \\nu } \\end{align*}"}
-{"id": "1404.png", "formula": "\\begin{align*} d _ r ( f ^ { * } ( 1 \\otimes u ) \\otimes 1 ) \\not = 0 . \\end{align*}"}
-{"id": "777.png", "formula": "\\begin{align*} B _ { k } ( x ) + B _ { k } ( x + \\frac { 1 } { 2 } ) = 2 ^ { 1 - k } B _ { k } ( 2 x ) \\end{align*}"}
-{"id": "1521.png", "formula": "\\begin{align*} \\displaystyle \\begin{cases} 2 , & \\ \\{ \\alpha _ { m , t } ( 1 ) \\} = 1 / 2 , \\\\ 1 , & \\ \\{ \\alpha _ { m , t } ( 1 ) \\} \\neq 1 / 2 . \\end{cases} \\end{align*}"}
-{"id": "1847.png", "formula": "\\begin{align*} F _ \\bullet = \\cdots \\to F _ 1 \\to F _ 0 \\to F _ { - 1 } \\to \\cdots , \\end{align*}"}
-{"id": "4892.png", "formula": "\\begin{align*} I = A A ^ * + B B ^ * = A ^ * A + C ^ * C . \\end{align*}"}
-{"id": "3787.png", "formula": "\\begin{align*} p _ \\circ \\ ; \\ ; N ( x , t ) = 0 , p _ \\bullet \\ ; \\ ; N ( x , t ) \\ge 1 , \\end{align*}"}
-{"id": "652.png", "formula": "\\begin{align*} ( \\nabla _ i \\nabla _ i P ) ( Z ) & = \\nabla _ i [ ( \\nabla _ i P ) ( Z ) ] - \\nabla P ( \\nabla _ { e _ i } e _ i , Z ) - ( \\nabla _ i P ) ( \\nabla _ i Z ) \\\\ & = \\nabla _ i ( \\nabla _ i ( P Z ) ) - \\nabla _ i ( P ( \\nabla _ i Z ) ) - \\nabla P ( \\nabla _ { e _ i } e _ i , Z ) - ( \\nabla _ i P ) ( \\nabla _ i Z ) \\\\ & = \\nabla _ i ( \\nabla _ i ( P Z ) ) \\\\ & = \\bar \\nabla _ { J _ i } \\bar \\nabla _ { J _ i } X ( 0 ) , \\end{align*}"}
-{"id": "9162.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } ( n + 1 - i ) a _ { n + 1 - i } \\left ( x ^ i A ( c x ^ { n + 1 - i } ) - x ^ { i + 1 } A ( c x ^ { n - i } ) \\right ) = 0 \\left ( x \\in K , c \\in K ^ * \\right ) . \\end{align*}"}
-{"id": "349.png", "formula": "\\begin{align*} & \\quad \\ 2 h ^ { i j } \\left ( \\nabla _ j R _ { \\nu l i } ^ { \\ \\ \\ l } + \\nabla _ l R _ { \\nu i j } ^ { \\ \\ \\ l } \\right ) = 2 S ( \\nabla _ 2 R _ { \\nu 2 } - \\nabla _ 1 R _ { \\nu 2 } ) \\\\ & = 2 S \\left ( \\nu ( S | \\nabla f | ) - 2 R i c ( \\nabla _ { \\nu } e _ 2 , e _ 2 ) + 2 R i c ( \\nabla _ { \\nu } e _ 1 , e _ 1 ) - S | \\nabla f | ^ 2 \\right ) . \\end{align*}"}
-{"id": "8974.png", "formula": "\\begin{gather*} \\lambda = w \\lambda _ 0 + j q ' \\end{gather*}"}
-{"id": "8960.png", "formula": "\\begin{gather*} \\dim \\big ( \\Gamma \\big ( X / W , ( M _ { k ( S ) } ) ^ W ( d ) \\big ) \\big ) \\le \\dim \\big ( \\Gamma \\big ( X / W , ( M _ s ) ^ W ( d ) \\big ) \\big ) \\\\ \\hphantom { \\dim \\big ( \\Gamma \\big ( X / W , ( M _ { k ( S ) } ) ^ W ( d ) \\big ) \\big ) } { } \\le \\sum _ i \\dim \\big ( \\Gamma \\big ( X / W , ( M ^ i _ s ) ^ W ( d ) \\big ) \\big ) \\\\ \\hphantom { \\dim \\big ( \\Gamma \\big ( X / W , ( M _ { k ( S ) } ) ^ W ( d ) \\big ) \\big ) } { } = \\sum _ i \\dim \\big ( \\Gamma \\big ( X / W , \\big ( M ^ i _ { k ( S ) } \\big ) ^ W ( d ) \\big ) \\big ) , \\end{gather*}"}
-{"id": "6590.png", "formula": "\\begin{align*} f _ n = \\psi _ { n } \\circ f \\circ \\varphi _ { n } ^ { - 1 } \\colon \\varphi _ { n } ( U _ n \\cap f ^ { - 1 } ( V _ n ) ) \\longrightarrow \\psi _ { n } ( f ( U _ n ) \\cap V _ n ) \\ . \\end{align*}"}
-{"id": "67.png", "formula": "\\begin{align*} \\frac { \\partial D \\textbf { X } ^ { H } w } { \\partial \\textbf { w } ^ * } = 0 \\end{align*}"}
-{"id": "3586.png", "formula": "\\begin{align*} \\aleph _ 0 \\leq ( K ) \\leq ( K ) = ( C ( K ) ) , \\end{align*}"}
-{"id": "4687.png", "formula": "\\begin{align*} 2 \\ , \\sum _ { i < j } ^ n \\bigg ( \\frac { m _ i + m _ j } { m _ i m _ j } \\bigg ) \\rho _ { i j } p ^ 2 _ { \\rho _ { i j } } + \\ , \\sum _ { i \\neq j , i \\neq k , j < k } ^ n \\ , \\frac { 2 } { m _ i } ( \\rho _ { i j } + \\rho _ { i k } - \\rho _ { j k } ) p _ { \\rho _ { i j } } p _ { \\rho _ { i k } } + V ( \\rho _ { i j } ) \\ = \\ E \\ , \\end{align*}"}
-{"id": "4686.png", "formula": "\\begin{align*} { \\cal H } _ r ^ { ( d ( n - 1 ) ) } + V ( \\rho _ { i j } ) \\ = \\ E \\ , \\end{align*}"}
-{"id": "7765.png", "formula": "\\begin{align*} | D _ { \\rho } u ( \\ell ) | ^ { k } \\leq \\bigg ( \\sum _ { i = 1 } ^ { N _ { \\rho } } | D _ { \\rho _ { i } } u ( \\ell _ { i } ) | \\bigg ) ^ { k } \\leq | N _ { \\rho } | ^ { k - 1 } \\sum _ { i = 1 } ^ { N _ { \\rho } } | D _ { \\rho _ { i } } u ( \\ell _ { i } ) | ^ { k } \\leq C | \\rho | ^ { k - 1 } \\sum _ { i = 1 } ^ { N _ { \\rho } } | D _ { \\rho _ { i } } u ( \\ell _ { i } ) | ^ { k } , \\end{align*}"}
-{"id": "5497.png", "formula": "\\begin{align*} f _ m ( s ) & = ( - 1 ) ^ { m + 1 } \\left [ \\widehat F ( s ) - \\sum _ { k = 0 } ^ m \\mu _ k \\frac { ( - s ) ^ k } { k ! } \\right ] , \\\\ g _ m ( s ) & = \\frac { \\dd ^ m } { \\dd s ^ m } f _ m ( s ) = \\mu _ m - ( - 1 ) ^ { m } \\widehat F ^ { ( m ) } ( s ) . \\end{align*}"}
-{"id": "1280.png", "formula": "\\begin{align*} \\mbox { C a p } _ { \\mathcal { A } } ( ( 1 - t ) E ( \\hat q ) + t E ( q ^ * ) ) = ( 1 - t ) ^ { n - p } \\mbox { C a p } _ { \\mathcal { A } } ( E ( \\hat q ) + s E ( q ^ * ) ) \\end{align*}"}
-{"id": "9918.png", "formula": "\\begin{align*} e ( C , B _ 2 ) = \\sum _ { x \\in B _ 2 } d _ C ( x ) \\le \\tfrac 1 2 ( \\ell - 1 ) | B _ 2 | + t \\ , , \\end{align*}"}
-{"id": "6167.png", "formula": "\\begin{align*} \\frac { 1 } { 3 ( A _ 1 A _ 2 A _ 3 ) ^ \\frac { 2 } { 3 } } \\left ( \\begin{array} { c c c } 2 & - \\frac { A _ 1 } { A _ 2 } & - \\frac { A _ 1 } { A _ 3 } \\\\ - \\frac { A _ 2 } { A _ 1 } & 2 & - \\frac { A _ 2 } { A _ 3 } \\\\ - \\frac { A _ 3 } { A _ 1 } & - \\frac { A _ 3 } { A _ 2 } & 2 \\end{array} \\right ) \\end{align*}"}
-{"id": "9287.png", "formula": "\\begin{align*} f ( x + \\epsilon ) - f ( x ) = f ( y + \\epsilon ) - f ( y ) 0 \\leq \\epsilon < \\delta \\end{align*}"}
-{"id": "1789.png", "formula": "\\begin{align*} d _ * : = \\left [ \\left ( \\frac { q - p } { r - p } \\right ) ^ { \\frac { q - p } { r - p } } - \\left ( \\frac { q - p } { r - p } \\right ) ^ { \\frac { r - p } { r - q } } \\right ] > 0 . \\end{align*}"}
-{"id": "9209.png", "formula": "\\begin{align*} [ z _ { i } \\otimes \\alpha _ { i } , z _ { j } \\otimes \\alpha _ { j } ] = z _ { i } \\diamond z _ { j } \\otimes \\frac { [ \\alpha _ { i } , \\alpha _ { j } ] } { 2 } + [ z _ { i } , z _ { j } ] \\otimes \\frac { \\alpha _ { i } \\circ \\alpha _ { j } } { 2 } . \\end{align*}"}
-{"id": "6321.png", "formula": "\\begin{align*} q _ { n } = \\ , \\frac { S - \\sum _ { i = 1 } ^ { n } a _ { i } b _ { i } } { a _ { n } } , \\ , \\ , n \\geq N . \\end{align*}"}
-{"id": "445.png", "formula": "\\begin{align*} \\Psi ( \\omega ) = \\begin{cases} \\frac { 1 } { 4 ^ n \\pi ^ { n + m } } \\sqrt { \\frac { ( 2 \\pi ) ^ m y _ \\omega ^ { m - 1 } \\sin ( y _ \\omega ) ^ 3 } { 2 \\omega ^ { m - 1 } ( \\sin ( y _ \\omega ) - y _ \\omega \\cos ( y _ \\omega ) ) } } , & \\\\ \\frac { ( 3 \\pi ) ^ { m / 2 } } { 4 ^ n \\pi ^ { n + m } } , & \\end{cases} \\end{align*}"}
-{"id": "3705.png", "formula": "\\begin{align*} \\begin{aligned} p _ 1 & = \\left ( x _ 1 , \\frac { u _ { 1 1 } } { v _ { 1 1 } } , t _ 2 , \\dots , t _ { n + 1 } \\right ) , \\\\ p _ k & = \\left ( t _ 1 , \\dots , t _ { k , k - 1 } , \\frac { v _ { k , k - 1 } } { u _ { k , k - 1 } } , \\frac { u _ { k , k } } { v _ { k , k } } , t _ { k + 1 } , \\dots , t _ { n + 1 } \\right ) . ( 1 < k \\leqslant n ) \\end{aligned} \\end{align*}"}
-{"id": "9392.png", "formula": "\\begin{align*} T _ { 2 ^ k } f & = \\phi _ k \\ast T f + \\sum _ { s \\ge 0 } ( \\delta _ 0 - \\phi _ k ) \\ast \\nu _ { k + s } \\ast f - \\phi _ k \\ast \\sum _ { s < 0 } \\nu _ { k + s } \\ast f \\\\ & : = T ^ 1 _ { k } f + T ^ 2 _ { k } f - T _ { k } ^ 3 f , \\end{align*}"}
-{"id": "8179.png", "formula": "\\begin{align*} \\underbrace { \\left ( \\begin{matrix} y & x \\\\ 0 & 1 \\end{matrix} \\right ) } _ { \\in B ( F _ { \\infty } ) } \\underbrace { \\left ( \\begin{matrix} \\theta _ i & 0 \\\\ 0 & 1 \\end{matrix} \\right ) } _ { = a ( \\theta _ i ) } , \\end{align*}"}
-{"id": "9021.png", "formula": "\\begin{align*} X = ( X ^ 1 , X ^ 2 ) \\in V ^ p _ 2 ( [ 0 , T ] ; \\mathbb { R } ^ { d } ) \\times V ^ { p / 2 } _ 2 ( [ 0 , T ] ; \\mathbb { R } ^ { d \\times d } ) \\end{align*}"}
-{"id": "1236.png", "formula": "\\begin{align*} D _ 1 = B ( w , s ) \\cap D \\cap \\{ v > t \\} . \\end{align*}"}
-{"id": "4329.png", "formula": "\\begin{align*} & \\Vert \\nabla ( n _ k - n _ \\ell ) \\Vert _ { L ^ 2 ( \\Omega ) } ^ 2 + \\big ( T ( \\varphi _ { p , k } ) ( n _ k - n _ \\ell ) , n _ k - n _ \\ell \\big ) = - \\big ( ( T ( \\varphi _ { p , k } ) - T ( \\varphi _ { p , \\ell } ) ) n _ \\ell , n _ k - n _ \\ell \\big ) \\ , . \\end{align*}"}
-{"id": "3734.png", "formula": "\\begin{align*} t _ j = \\frac { v _ { j , { j - 1 } } } { u _ { j , { j - 1 } } } \\frac { u _ { j j } } { v _ { j j } } = w _ { n + j } \\cdot c _ j , \\end{align*}"}
-{"id": "1405.png", "formula": "\\begin{align*} \\pi \\colon { S L _ 2 } ^ { n + 1 } \\to { S L _ 2 } ^ { n } , \\pi ( m _ 0 , m _ 1 , \\dots , m _ n ) = ( m _ 0 , m _ 1 , \\dots , m _ { n - 1 } ) . \\end{align*}"}
-{"id": "821.png", "formula": "\\begin{align*} - \\Delta \\hat { u } + \\frac { 2 } { \\tau } \\hat { u } & = \\frac { 2 } { \\tau } f \\quad \\partial \\Omega , \\\\ \\hat { u } & = 0 \\partial \\Omega . \\end{align*}"}
-{"id": "6472.png", "formula": "\\begin{align*} d s _ { 2 D u } ^ { 2 } = \\frac { 1 } { \\sigma ^ { 2 } } \\left ( d \\mu _ { x } ^ { 2 } + 4 d \\sigma ^ { 2 } \\right ) \\end{align*}"}
-{"id": "5917.png", "formula": "\\begin{align*} \\mu ^ N ( j _ 0 ) = \\frac 1 { E ^ N _ { j _ 0 } T ^ N _ { j _ 0 } } , \\end{align*}"}
-{"id": "7113.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\partial } { \\partial t } h _ { i j } = & ~ { \\Phi } ' \\dot { F } ^ { k l } \\nabla _ k \\nabla _ l h _ { i j } + { \\Phi } ' \\ddot { F } ^ { k l , m n } \\nabla _ i h _ { k l } \\nabla _ j h _ { m n } + { \\Phi } '' \\nabla _ i F \\nabla _ j F \\\\ & + { \\Phi } ' \\left ( \\dot { F } ^ { k l } h ^ p _ k h _ { p l } - K _ N \\dot { F } ^ { k l } g _ { k l } \\right ) h _ { i j } \\\\ & \\quad - ( \\Phi + \\Phi ' F ) h ^ k _ i h _ { k j } + K _ N ( \\Phi + \\Phi ' F ) g _ { i j } , \\end{aligned} \\end{align*}"}
-{"id": "4571.png", "formula": "\\begin{align*} ( 1 - q _ 1 ^ { - 1 } q _ 3 ) E ^ { ( 1 ) } _ { 0 | 1 , l } & = \\sum _ { j \\ge 0 } q _ 1 ^ { j - l } \\Bigl ( E _ { 0 , l - j } E _ { 1 , j } - ( q _ 1 + q _ 3 ) E _ { 0 , l - j - 1 } E _ { 1 , j + 1 } + q _ 1 q _ 3 E _ { 0 , l - j - 2 } E _ { 1 , j + 2 } \\Bigr ) \\\\ & + \\sum _ { j \\ge 1 } q _ 1 ^ { - j - l } \\Bigl ( q _ 1 q _ 3 E _ { 1 , - j } E _ { 0 , j + l } - ( q _ 1 + q _ 3 ) E _ { 1 , - j + 1 } E _ { 0 , l + j - 1 } + E _ { 1 , - j + 2 } E _ { 0 , j + l - 2 } \\Bigr ) \\ , , \\end{align*}"}
-{"id": "6700.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } s = \\sum _ { j = 1 } ^ { n - d } s _ { i _ j } , \\\\ - s ^ 2 / 2 = - \\sum _ { j = 1 } ^ { n - d } s _ { i _ j } ^ 2 / 2 + ( m - 2 ) ! \\cdot \\sum _ { j = 1 } ^ { n - d } \\big ( \\omega _ { H _ { i _ j } } , t ^ { m - 2 } \\big ) . \\end{array} \\right . \\end{align*}"}
-{"id": "5641.png", "formula": "\\begin{align*} \\mathfrak { L } _ { K } ( \\gamma ) : = \\int _ I K ( \\gamma ( t ) ) \\ , | \\dot { \\gamma } | ( t ) \\d t , \\gamma \\in A C _ { p l o c } ( I , X ) . \\end{align*}"}
-{"id": "7831.png", "formula": "\\begin{align*} P & = \\left ( \\begin{array} { c c c c c c } 1 & \\frac { q - 1 } { 2 } & \\frac { q - 1 } { 2 } \\\\ 1 & \\frac { - 1 + \\sqrt { - q } } { 2 } & \\frac { - 1 - \\sqrt { - q } } { 2 } \\\\ 1 & \\frac { - 1 - \\sqrt { - q } } { 2 } & \\frac { - 1 + \\sqrt { - q } } { 2 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "3685.png", "formula": "\\begin{align*} s = ( i _ 1 \\ ; \\dots \\ ; i _ { l _ 1 } ) ( i _ { l _ 1 + 1 } \\ ; \\dots \\ ; i _ { l _ 2 } ) \\dots ( i _ { l _ { r - 1 } + 1 } \\ ; \\dots \\ ; i _ { l _ r } ) , \\end{align*}"}
-{"id": "2023.png", "formula": "\\begin{align*} f _ { a , i } ( z _ 1 , z _ 2 ) = z _ 1 ^ i z _ 2 ^ i f _ a ( z _ 1 , z _ 2 ) \\end{align*}"}
-{"id": "6128.png", "formula": "\\begin{align*} \\mathbb { P } _ { c o u p l e } \\left ( W > i \\right ) \\leq \\sum _ { k = \\left [ \\log \\left ( i \\right ) \\right ] } ^ { \\left [ \\log \\left ( i c _ { t } \\right ) \\right ] - 1 } \\left | g \\left ( 2 ^ { k } \\right ) - g \\left ( 2 ^ { k + 1 } \\right ) \\right | + L \\left ( \\frac { i c _ { t } } { 4 } \\right ) \\left ( \\frac { i c _ { t } } { 4 } \\right ) ^ { - \\alpha } + \\left | g \\left ( \\frac { i c _ { t } } { 4 } \\right ) \\right | . \\end{align*}"}
-{"id": "9440.png", "formula": "\\begin{align*} N \\mathfrak { M } ( N ) \\asymp \\sum _ { n = 1 } ^ N \\mathfrak { M } ( n ) . \\end{align*}"}
-{"id": "5606.png", "formula": "\\begin{align*} G ( B ^ H _ T ) = G ( B ^ H _ 0 ) + \\int _ 0 ^ T D G ( B ^ H _ r ) d \\mathbf { B } ^ H _ r . \\end{align*}"}
-{"id": "6780.png", "formula": "\\begin{align*} \\overline { w } _ { \\lambda } = 2 \\ln { \\lambda } + 5 \\ln { 2 } - 4 \\pi \\sum \\limits _ { j < k } G ( \\xi _ j , \\xi _ k ) . \\end{align*}"}
-{"id": "4059.png", "formula": "\\begin{align*} \\bigcup _ { \\pi \\in \\Pi _ d } \\bigcap _ { k = 1 } ^ n M _ { k , \\pi ( k ) } = \\bigcap _ { k = 1 } ^ n \\bigcup _ { l = 1 } ^ r M _ { k , l } = [ 0 , 1 ] \\quad \\end{align*}"}
-{"id": "7527.png", "formula": "\\begin{gather*} w ^ 1 _ 1 + w ^ 1 w ^ 1 _ 2 - w ^ 1 _ { 2 2 } - \\frac { ( w ^ 2 ) ^ 2 } { z _ 2 } - \\frac 1 { z _ 2 ^ 3 } = 0 , \\\\ w ^ 2 _ 1 + w ^ 1 w ^ 2 _ 2 - w ^ 2 _ { 2 2 } + \\frac { w ^ 1 w ^ 2 } { z _ 2 } + 2 \\frac { w ^ 2 } { z _ 2 ^ 2 } = 0 . \\end{gather*}"}
-{"id": "4728.png", "formula": "\\begin{align*} { \\varepsilon } = F ( x ^ 0 , x ^ 1 , Z ) { \\Delta \\sqrt { - \\det g ^ { i j } } } \\end{align*}"}
-{"id": "1572.png", "formula": "\\begin{align*} v ( m , \\mathbf { j } ) = \\sum _ { k = 1 } ^ { n - \\nu ( m ) } [ ( n - k + 1 ) j _ k - s ( j _ k ) ] \\end{align*}"}
-{"id": "4088.png", "formula": "\\begin{align*} [ 0 , 2 \\pi ] \\ni \\theta \\mapsto V ( \\theta ) = V _ 1 \\cos \\theta + V _ 2 \\sin \\theta \\in T _ { \\eta ( p ) } \\partial B \\simeq T _ p M . \\end{align*}"}
-{"id": "2402.png", "formula": "\\begin{align*} p _ 1 ( x ) = \\frac { a _ 2 ( a _ 3 + a _ 2 x ) } { a _ 3 } , p _ 3 ( x ) = \\frac { ( a _ 3 + a _ 2 x ) ( a _ 1 a _ 2 + a _ 3 - a _ 2 x + a _ 3 x ^ 2 ) } { a _ 3 } . \\end{align*}"}
-{"id": "9806.png", "formula": "\\begin{align*} \\int _ { t = - T _ { e _ 1 } } ^ { T _ { e _ 2 } } f ( t ) d t = 1 . \\end{align*}"}
-{"id": "7563.png", "formula": "\\begin{gather*} \\frac { f _ { z z } } { f _ z - i g _ z } = - \\frac { f ^ * _ { z ^ * z ^ * } } { f ^ * _ { z ^ * } + i g ^ * _ { z ^ * } } = i \\lambda ^ 1 , \\frac { g _ { z z } } { f _ z - i g _ z } = - \\frac { g ^ * _ { z ^ * z ^ * } } { f ^ * _ { z ^ * } + i g ^ * _ { z ^ * } } = i \\lambda ^ 2 , \\end{gather*}"}
-{"id": "3544.png", "formula": "\\begin{align*} B _ { \\boldsymbol { t } } ^ { \\boldsymbol { \\varepsilon } } = B _ { \\boldsymbol { t } } \\cap B _ { \\boldsymbol { t + \\varepsilon } } , \\ \\ \\ \\boldsymbol { \\varepsilon } \\in \\mathcal { E } . \\end{align*}"}
-{"id": "9626.png", "formula": "\\begin{align*} \\frac { \\partial P _ { \\rm { t r } } \\big ( R _ { \\beta } , \\lambda _ { \\beta } , h _ \\beta \\big ) } { \\partial h _ \\beta } = 0 \\Leftrightarrow \\frac { \\partial P _ { \\rm { t r } , 1 } ( h _ \\beta / R _ { \\beta } ) } { \\partial h _ \\beta } = 0 . \\end{align*}"}
-{"id": "5038.png", "formula": "\\begin{align*} \\frac { x ^ * + \\theta z ^ * } { p ^ * ( x ^ * + \\theta z ^ * ) } = h _ 0 ^ * + \\sum _ { n \\in \\N } s _ n r _ n v _ n ^ * , \\end{align*}"}
-{"id": "967.png", "formula": "\\begin{align*} P \\left ( T ^ * _ n \\leq q _ n ^ Z ( 1 - \\alpha + \\varepsilon _ n ) | \\mathcal { F } ^ X \\right ) \\geq P \\left ( \\max _ { \\theta \\in \\mathcal { G } _ n } | Z _ n ( \\theta ) | \\leq q _ n ^ Z ( 1 - \\alpha + \\varepsilon _ n ) \\right ) - \\varepsilon _ n = 1 - \\alpha , \\end{align*}"}
-{"id": "5797.png", "formula": "\\begin{align*} | a + b | ^ { r } \\leq \\begin{cases} 2 ^ { r - 1 } \\left ( | a | ^ { r } + | b | ^ { r } \\right ) \\ ; \\ ; 1 \\leq r < \\infty , \\\\ | a | ^ { r } + | b | ^ { r } \\ ; \\ ; \\ ; \\ ; \\ ; 0 \\leq r < 1 , \\end{cases} \\end{align*}"}
-{"id": "9365.png", "formula": "\\begin{align*} C ( r , k , G _ { k } ) = \\frac { 1 } { k } \\frac { 1 } { V ^ { \\frac { r + k } { k } } } \\int _ { \\mathbb { R } _ n } | x ^ k - \\mathcal { Q } ( x ^ { k } ) | ^ 2 \\ , \\mathrm { d } x ^ k . \\end{align*}"}
-{"id": "8335.png", "formula": "\\begin{align*} \\langle \\alpha ( \\lambda ) , \\beta ( v ) \\rangle = - 2 \\pi i \\cdot [ \\lambda , v ] , \\end{align*}"}
-{"id": "7142.png", "formula": "\\begin{align*} e ^ * _ L \\leq \\alpha B + \\sum _ { r = 0 } ^ { R - 1 } \\sqrt { 2 [ U J ^ 2 + V _ r J g ^ * _ r + W J ] } . \\end{align*}"}
-{"id": "7992.png", "formula": "\\begin{align*} \\binom { n } { n / 4 c ^ { 2 } } \\exp ( - 3 n / ( 9 \\cdot 4 \\cdot 1 6 c ) ) \\leq \\Big ( 4 e c ^ 2 \\Big ) ^ { n / 4 c ^ { 2 } } \\exp ( - n / 2 0 0 c ) \\leq \\exp ( ( 3 + 2 \\log c - c / 5 0 ) n / 4 c ^ 2 ) = o ( n ^ { - 3 } ) , \\end{align*}"}
-{"id": "695.png", "formula": "\\begin{align*} \\kappa _ n = \\varphi ( \\pi , \\mu _ n ) = \\psi ^ { - 1 } ( 0 , \\mu _ n ) . \\end{align*}"}
-{"id": "6654.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\frac { | K | _ n - | K _ { \\delta } | _ n } { n | K | _ n \\delta ^ { \\frac { 2 } { n + 1 } } } = \\int _ { \\partial K } G ( x ) \\mathrm { d } n _ K ( x ) = \\| G \\| _ { 1 , n _ s } . \\end{align*}"}
-{"id": "623.png", "formula": "\\begin{align*} \\int _ { | z _ 1 | = r _ 1 } \\frac { f ( z _ 1 , z _ 2 ) \\ , d z _ 1 } { z _ 1 } = 2 \\pi i \\sum _ { \\alpha _ 2 \\in \\Z } a _ { ( 0 , \\alpha _ 2 ) } z _ 2 ^ { \\alpha _ 2 } , \\end{align*}"}
-{"id": "6558.png", "formula": "\\begin{align*} \\alpha _ { \\xi } = \\left ( \\frac { n 2 ^ n } { \\frac { n ^ n } { \\sqrt { n } ( n - 1 ) ! } \\sqrt { n } } \\right ) ^ { 1 / n } = 2 \\frac { \\sqrt [ n ] { n ! } } { n } \\ \\ \\ \\ \\beta _ \\xi = 2 ^ n . \\end{align*}"}
-{"id": "9143.png", "formula": "\\begin{align*} \\begin{array} { r c l } f _ { 3 } & = & D _ { 2 } \\\\ f _ { 2 } & = & - 3 D _ { 2 } - D _ { 1 } \\\\ f _ { 1 } & = & 3 D _ { 2 } + 2 D _ { 1 } + D _ { 0 } , \\end{array} \\end{align*}"}
-{"id": "3282.png", "formula": "\\begin{align*} \\begin{cases} a \\sin \\theta _ u - \\cos \\theta _ u = b , \\\\ \\sin ^ 2 \\theta _ u + \\cos ^ 2 \\theta _ u = 1 . \\end{cases} \\end{align*}"}
-{"id": "8159.png", "formula": "\\begin{align*} a _ 1 a _ 5 = 0 \\end{align*}"}
-{"id": "2261.png", "formula": "\\begin{align*} \\sum _ { k , m = 1 } ^ { \\infty } \\frac { 4 a _ 2 b _ 1 } { \\pi ^ 4 k ^ 2 m ^ 2 } = \\frac { a _ 2 b _ 1 } { 9 } \\end{align*}"}
-{"id": "8948.png", "formula": "\\begin{gather*} \\begin{pmatrix} 0 & m _ { 1 2 } \\big ( b \\mu _ { 2 1 } ^ { - 1 } - \\big ( b \\mu _ { 2 1 } ^ { - 1 } \\big ) ^ \\vee \\big ) \\mu _ { 2 1 } \\\\ - m _ { 1 2 } \\mu _ { 2 1 } ^ \\vee \\big ( b \\mu _ { 2 1 } ^ { - 1 } - \\big ( b \\mu _ { 2 1 } ^ { - 1 } \\big ) ^ \\vee \\big ) & 0 \\end{pmatrix} \\ ! , \\end{gather*}"}
-{"id": "3191.png", "formula": "\\begin{align*} \\left \\vert V ( y + z ) - V ( y ) \\right \\vert = \\log \\left ( 1 + \\frac { z } { 1 + y } \\right ) \\leqslant \\log ( 1 + z ) \\leqslant \\log ( 2 ) + \\log ( z ) . \\end{align*}"}
-{"id": "2840.png", "formula": "\\begin{align*} u _ { n } ^ { \\alpha } = \\frac { 1 } { A _ { n } ^ { \\alpha } } \\sum _ { v = 0 } ^ { n } A _ { n - v } ^ { \\alpha - 1 } s _ { v } ~ ~ ~ ~ \\end{align*}"}
-{"id": "7215.png", "formula": "\\begin{align*} u _ { k , r } : = \\frac { u _ k ( r \\cdot ) } { r } \\rightarrow v _ k \\end{align*}"}
-{"id": "1213.png", "formula": "\\begin{align*} r H _ 1 = s H _ 2 . \\end{align*}"}
-{"id": "3147.png", "formula": "\\begin{align*} \\widehat { m } _ { \\alpha , \\beta } ( u ) = \\exp \\left \\lbrace \\frac { \\alpha u } { \\beta - u } \\right \\rbrace , u \\in \\mathcal { U } . \\end{align*}"}
-{"id": "8760.png", "formula": "\\begin{align*} & B ( { \\bf f } ) ^ { \\left ( \\underset { { { \\{ m _ 1 , m \\} } } } { \\bf t } { { , s _ m } } , { \\bf s } \\right ) } \\\\ & = [ f _ 1 , f _ 2 ] ^ { ( { { s _ 2 } } , s _ 1 ) } \\\\ & = \\displaystyle A d _ { e ^ { { { s _ 2 } } f _ 2 } e ^ { s _ 1 f _ 1 } } \\left [ f _ 1 \\ , , \\ , f _ 2 \\right ] \\\\ & = e ^ { { { s _ 2 } } f _ 2 } e ^ { s _ 1 f _ 1 } \\left [ f _ 1 \\ , , \\ , f _ 2 \\right ] e ^ { - s _ 1 f _ 1 } e ^ { - { { s _ 2 } } f _ 2 } { { . } } \\end{align*}"}
-{"id": "2075.png", "formula": "\\begin{align*} \\int _ M H ( p , q , t ) d V _ q = 1 . \\end{align*}"}
-{"id": "3825.png", "formula": "\\begin{align*} p _ k \\ge p _ { * * } ^ k p _ { * * } : = p _ 0 \\tilde { p } \\end{align*}"}
-{"id": "7895.png", "formula": "\\begin{align*} u ( x , t ) = \\max _ { y \\in B _ t ( x ) } u _ 0 ( y ) = \\min \\{ t - x _ 1 , 0 \\} , \\end{align*}"}
-{"id": "9654.png", "formula": "\\begin{align*} \\omega = \\frac { 2 \\kappa } { \\Delta \\Phi _ { \\rm { s t } , \\max } ( t ) - \\xi _ { \\max } \\sum _ { \\beta = 1 } ^ { \\kappa } \\Lambda _ \\beta ( t ) } . \\end{align*}"}
-{"id": "6526.png", "formula": "\\begin{align*} n _ { P ^ \\circ } ( \\xi ) = \\frac { 1 } { n \\ | P ^ \\circ | _ n } \\ \\frac { 1 } { \\| \\xi \\| } \\ | F _ \\xi | _ { n - 1 } , \\end{align*}"}
-{"id": "215.png", "formula": "\\begin{align*} \\frac { Q _ 1 ( i ) } { m _ 1 } = \\frac { \\overline { P _ i ( 1 ) } } { k _ i } = \\cos \\frac { i \\cdot 2 t \\pi } { \\ell } . \\end{align*}"}
-{"id": "6418.png", "formula": "\\begin{align*} \\chi \\left ( \\xi \\right ) \\overset { } { = } \\frac { 1 } { 1 + \\xi } \\gamma \\left ( \\xi \\right ) \\overset { } { = } - \\frac { 1 } { 2 \\xi \\left ( 1 + \\xi \\right ) } \\end{align*}"}
-{"id": "8168.png", "formula": "\\begin{align*} \\omega ^ { \\star } ( t ) : = \\sup _ { s \\ge 0 } \\big ( \\omega ( s ) - s t \\big ) , t > 0 . \\end{align*}"}
-{"id": "8777.png", "formula": "\\begin{align*} \\mathcal { L } ( G _ 2 ) \\circ { C } = \\dfrac { 1 } { r _ 2 + 1 } ( I _ { n _ 2 } + r _ 2 \\mathcal { L } ( G _ 2 ) ) . \\end{align*}"}
-{"id": "8220.png", "formula": "\\begin{align*} t _ + = \\sum _ { \\ell \\in \\mathcal { L } ^ + } \\binom { k } { \\ell } \\Bigl ( U _ { i + \\ell } - \\frac { 1 } { 2 } ( U _ { i + \\ell } ) ^ 2 + \\frac { 1 } { 3 } ( U _ { i + \\ell } ) ^ 3 \\cdots \\Bigr ) \\\\ \\end{align*}"}
-{"id": "5070.png", "formula": "\\begin{align*} & B _ { k k , k } C _ { i , j } + b _ k [ C _ { i , j k } - C _ { k , j } \\frac { B _ { k i , k } } { b _ k - b _ i } - C _ { i , k } \\frac { B _ { k j , k } } { b _ k - b _ j } ] \\\\ & = - C _ i [ C _ { j , k } - C _ k \\frac { B _ { j k , k } } { b _ k - b _ j } ] - C _ j [ C _ { i , k } - C _ k \\frac { B _ { i k , k } } { b _ k - b _ i } ] . \\end{align*}"}
-{"id": "1548.png", "formula": "\\begin{gather*} \\alpha \\ = \\ \\begin{pmatrix} 2 & 0 & 0 & 0 & 0 & 0 & 0 & 3 \\end{pmatrix} ^ T . \\end{gather*}"}
-{"id": "3673.png", "formula": "\\begin{align*} P _ b [ n ] = P _ W [ n ] \\cap ( \\alpha _ W [ n ] \\otimes \\mathbb R ) ^ { - 1 } ( - b ) , \\end{align*}"}
-{"id": "7062.png", "formula": "\\begin{align*} \\Phi ( u , \\lambda ) = \\frac { 1 } { 2 } \\int \\limits _ { B ^ N } | \\nabla u ( x ) | ^ 2 d x - \\lambda \\int \\limits _ { B ^ N } F ( u ( x ) ) d x . \\end{align*}"}
-{"id": "5563.png", "formula": "\\begin{align*} \\bar { \\nu } ^ { T } = - \\tau ^ { 2 } \\bar { \\nu } ^ { T } \\left ( \\alpha I _ { 2 ^ { k } } + \\beta \\Lambda _ { \\bar { r } } \\right ) P ^ { 2 } + \\tau \\dot { x } _ { 0 } e _ { 1 } ^ { T } P + x _ { 0 } e _ { 1 } ^ { T } \\end{align*}"}
-{"id": "9056.png", "formula": "\\begin{align*} A _ { { \\mathbf x } } { \\mathcal H } ( w , w ' ) = ( a _ { 1 1 } ^ 1 { \\mathbf x } _ 1 \\alpha \\bar w _ 1 w _ 1 ' , a _ { 2 2 } ^ 2 { \\mathbf x } _ 2 ( \\gamma \\bar w _ 1 w _ 1 ' + \\delta \\bar w _ 2 w _ 2 ' ) ) . \\end{align*}"}
-{"id": "5453.png", "formula": "\\begin{align*} \\mathrm { A } _ \\rho p ( s ) = s ^ { \\rho } \\int _ 0 ^ \\infty e ^ { - s x } \\dd ( p ( x ) x ^ \\rho ) . \\end{align*}"}
-{"id": "8845.png", "formula": "\\begin{align*} \\chi _ 2 ( \\cdot , a ) : = \\chi e ^ { i R ( \\cdot , a ) } \\in \\mathcal { S } ( \\R ) \\end{align*}"}
-{"id": "4360.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { j = 1 } ^ \\infty M _ { 1 - b _ j } P _ \\alpha M _ { a _ j } \\Big \\| \\leq 2 ^ n + \\| P _ \\alpha \\| \\end{align*}"}
-{"id": "9369.png", "formula": "\\begin{align*} D = \\frac { 1 } { 2 \\pi } \\sum _ { \\forall n } \\iint _ { \\mathbb { C } _ n } | r \\mathrm { e } ^ { i \\phi } - r _ n \\mathrm { e } ^ { i \\phi _ n } | ^ 2 f ( r ) \\ , \\mathrm { d } r \\ , \\mathrm { d } \\phi , \\end{align*}"}
-{"id": "8441.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { - 2 , 1 , v } ) = 0 . \\end{align*}"}
-{"id": "3664.png", "formula": "\\begin{align*} G [ n ] : = \\{ ( t _ 1 , \\dots , t _ n , t _ { n + 1 } ) \\in ( \\mathbb C ^ * ) ^ { n + 1 } \\ ; | \\ ; t _ 1 \\dots t _ n t _ { n + 1 } = 1 \\} . \\end{align*}"}
-{"id": "7973.png", "formula": "\\begin{align*} d f ( T _ s ( t ) ) = T _ { s / 2 } ( t ) , \\ , \\ , \\ , d f ( J _ s ( t ) ) = \\frac { 1 } { 2 } J _ { s / 2 } ( t ) . \\end{align*}"}
-{"id": "7357.png", "formula": "\\begin{align*} { \\cal { P } } _ X = \\nabla _ X - \\frac { 1 } { n } \\widetilde { X } . \\displaystyle { \\not } D . \\end{align*}"}
-{"id": "8617.png", "formula": "\\begin{align*} h ( t , x ) = \\lambda ^ { - 1 } \\log u ( t , x ) - c _ 0 t , \\end{align*}"}
-{"id": "3041.png", "formula": "\\begin{align*} \\begin{cases} D ( B ) = \\{ q \\in Q \\mid \\exists w \\in H ( w , v ) _ H = b ( v , q ) , \\ \\forall v \\in V \\} , \\\\ ( B q , v ) _ H = b ( v , q ) , \\forall q \\in D ( B ) , \\forall v \\in V . \\end{cases} \\end{align*}"}
-{"id": "6860.png", "formula": "\\begin{align*} i ( A ) = \\# \\{ i _ j | i _ j = 0 , 3 \\} \\end{align*}"}
-{"id": "1439.png", "formula": "\\begin{align*} H ^ { * } ( B G _ n ; \\mathbb { Z } / 2 ) = \\mathbb { Z } / 2 [ x _ 2 ^ 2 , x _ 3 ^ 2 , x _ { 4 1 } , \\dots , x _ { 4 n } ] \\{ 1 , x _ 2 , x _ 3 , x _ 2 x _ 3 \\} . \\end{align*}"}
-{"id": "4513.png", "formula": "\\begin{align*} \\phi ( a ) = \\lim _ { n \\to \\infty } \\phi ( \\Delta _ n ^ * a \\Delta _ n ) \\end{align*}"}
-{"id": "7171.png", "formula": "\\begin{align*} \\sup _ k \\| u _ { x _ 0 , r _ k } \\| _ { L ^ 2 ( B _ 1 , \\mu _ a ) } \\leq C \\left ( \\sup _ k \\| u _ { x _ 0 , r _ k } \\| _ { L ^ 2 ( \\partial B _ 1 , | x _ n | ^ a \\ , \\mathcal { H } ^ { n - 1 } ) } + \\sup _ k \\| \\nabla u _ { x _ 0 , r _ k } \\| _ { L ^ 2 ( B _ 1 , \\mu _ a ; \\R ^ n ) } \\right ) < \\infty . \\end{align*}"}
-{"id": "4391.png", "formula": "\\begin{align*} M _ f ( C _ { z _ \\gamma } A _ m C _ { - z _ \\gamma } ) M _ g = C _ { z _ \\gamma } M _ { f \\circ \\tau _ { - z _ \\gamma } } A _ m M _ { g \\circ \\tau _ { - z _ \\gamma } } C _ { - z _ \\gamma } = 0 , \\end{align*}"}
-{"id": "8358.png", "formula": "\\begin{align*} \\mathrm { d i v } ( \\psi ( f ) ) = \\sum _ { \\substack { m > 0 \\\\ \\mu \\in V _ \\Z ^ \\vee / V _ \\Z } } c ( - m , \\mu ) \\cdot \\mathcal { Z } ( m , \\mu ) . \\end{align*}"}
-{"id": "4609.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { n = 0 } ^ { N - 1 } \\delta _ { S _ { f , B } ^ n ( x , t ) } \\end{align*}"}
-{"id": "1435.png", "formula": "\\begin{align*} f - \\partial _ J ( g _ k x _ { 4 n } ^ k + g _ { k + 1 } x _ { 4 n } ^ { k + 1 } + \\cdots ) = 0 \\end{align*}"}
-{"id": "9054.png", "formula": "\\begin{align*} \\Im b _ { 2 2 } ^ 2 = 0 . \\end{align*}"}
-{"id": "5261.png", "formula": "\\begin{align*} \\begin{aligned} u _ 1 ( \\lambda ^ * , z ) = u _ 4 ( \\lambda ^ * , z ) \\\\ u _ 2 ( \\lambda ^ * , z ) = u _ 3 ( \\lambda ^ * , z ) \\end{aligned} . \\end{align*}"}
-{"id": "9954.png", "formula": "\\begin{align*} { \\cal N } ( u ) \\cup V _ 1 = { \\cal N } ( \\varphi ( u ) ) \\cup V _ 2 \\end{align*}"}
-{"id": "933.png", "formula": "\\begin{align*} E [ g ( \\Phi _ \\beta ( Z ) ) ] \\leq E [ 1 _ { A ^ { 4 \\varepsilon } } ( \\Phi _ \\beta ( Z ) ) ] \\leq E [ 1 _ { A ^ { 5 \\varepsilon } } ( Z _ \\vee ) ] = P ( Z _ \\vee \\in A ^ { 5 \\varepsilon } ) . \\end{align*}"}
-{"id": "2825.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\ \\max _ { v , v ' \\in V _ { n + 1 } } \\left ( \\sum _ { w \\in V _ n } \\left | f _ { v w } ^ { ( n ) } - f _ { v ' w } ^ { ( n ) } \\right | \\right ) = 0 \\end{align*}"}
-{"id": "3597.png", "formula": "\\begin{align*} D ^ \\alpha _ { * a } y _ j ( x ) = D _ { * a } ^ \\alpha f ( x ) , j \\in \\{ 1 , 2 \\} . \\end{align*}"}
-{"id": "4148.png", "formula": "\\begin{align*} & \\sum _ { \\tau \\in [ t + 2 , n ] } Z _ { N , J } ^ { \\tau } + n Z _ { i _ { t + 1 } , J } ^ { t + 1 } + n ^ 2 Z _ { i _ t , J } ^ { t } \\leq 1 + n \\\\ & \\forall \\ i _ t , i _ { t + 1 } \\in N , \\ J \\subseteq V , \\ | J | = 2 , \\ t \\in [ n - 2 ] . \\end{align*}"}
-{"id": "8277.png", "formula": "\\begin{align*} g = z n ( x ) g _ { t ( g ) , l ( g ) , w } k . \\end{align*}"}
-{"id": "6260.png", "formula": "\\begin{align*} \\zeta _ 1 + \\zeta _ 2 + \\zeta _ 3 + \\zeta _ 4 + \\zeta _ 5 + \\zeta _ 6 = \\delta _ 1 \\alpha + \\delta _ 2 , \\ \\ \\zeta _ 1 , . . . , \\zeta _ 6 > 0 \\\\ \\frac 1 { \\theta _ 1 } + \\frac 1 { \\theta _ 2 } + \\frac 1 { \\theta _ 3 } + \\frac 1 { \\theta _ 4 } + \\frac 1 { \\theta _ 5 } + \\frac 1 { \\theta _ 6 } = 1 , \\\\ \\theta _ 1 = \\frac 2 { \\delta _ 1 } , \\ \\theta _ 3 = \\frac 2 { \\delta _ 2 } , \\ \\theta _ 5 = \\frac 2 { \\delta _ 3 } , \\ 1 < \\theta _ 2 , \\theta _ 4 , \\theta _ 6 < \\infty \\end{align*}"}
-{"id": "677.png", "formula": "\\begin{align*} F ( x , y ) = \\sum _ { n = 0 } ^ { \\infty } c _ n \\varphi _ 0 ( x , \\mu _ n ^ 0 ) \\varphi _ 0 ( y , \\mu _ n ^ 0 ) \\end{align*}"}
-{"id": "2467.png", "formula": "\\begin{align*} | g | _ { L _ { x } ^ { \\infty } } = \\sup _ { x \\in { \\mathbb { R } ^ { 3 } } } | g ( x ) | \\ , . \\end{align*}"}
-{"id": "5277.png", "formula": "\\begin{align*} & { ( a _ 1 + a _ 2 ) b _ 1 \\neq a _ 1 b _ 2 } , \\ d _ { 1 } \\in C ^ { 2 } ( \\mathbb { R } _ { \\geq 0 } ; \\mathbb { R } ) , \\\\ & u _ 0 \\in \\mathbb { H } ^ 2 _ { ( 0 ) } : = \\{ u \\in H ^ 2 ( 0 , 1 ) ; a _ 1 u ( 1 ) + a _ 2 u _ x ( 1 ) = 0 , \\\\ & \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; b _ 1 u ( 0 ) + b _ 2 u _ x ( 0 ) = 0 \\} . \\end{align*}"}
-{"id": "5269.png", "formula": "\\begin{align*} P = \\{ p \\in X \\mid p = g x \\hbox { f o r s o m e } g \\in G , x \\in P ' \\} \\end{align*}"}
-{"id": "1862.png", "formula": "\\begin{align*} \\mathcal { F } ^ { S , \\pi } _ { \\preceq _ 1 } ( F _ 1 ^ \\ast , \\dots , F _ d ^ \\ast ) : = \\bigg \\{ F : \\begin{array} { l } F \\mathbb { R } ^ d F _ i \\preceq _ 1 F _ i ^ \\ast i = 1 , \\dots , d \\\\ F ( s ) \\leq \\pi _ s s \\in S \\end{array} \\bigg \\} , \\end{align*}"}
-{"id": "6570.png", "formula": "\\begin{align*} | F _ { \\xi } | _ 1 = \\left \\| \\frac { 1 - \\varepsilon } { \\sqrt { 1 - \\varepsilon ^ 2 } } e _ 1 + e _ 2 - \\frac { 1 } { \\sqrt { 1 - \\varepsilon ^ 2 } } e _ 1 \\right \\| = \\frac { 1 } { \\sqrt { 1 - \\varepsilon ^ 2 } } \\end{align*}"}
-{"id": "9860.png", "formula": "\\begin{align*} \\lim _ { \\nu \\to \\infty } \\gamma ^ \\nu = 0 \\mbox { \\rm a n d } \\sum _ { \\nu = 0 } ^ \\infty \\gamma ^ \\nu \\ , = \\ , \\infty , \\end{align*}"}
-{"id": "4090.png", "formula": "\\begin{align*} k _ { M , p } ( \\theta ) : = k _ { M , p } ( V ( \\theta ) ) = \\lambda _ 1 ( \\cos \\theta ) ^ 2 + \\lambda _ 2 ( \\sin \\theta ) ^ 2 . \\end{align*}"}
-{"id": "1772.png", "formula": "\\begin{align*} U _ { \\gamma } : = \\ ; \\tilde { t } _ a \\tilde { t } _ { \\omega ( 0 , j ) } ^ * U _ { \\omega } t _ { \\omega ( 0 , j ) } t _ a ^ * . \\end{align*}"}
-{"id": "2276.png", "formula": "\\begin{align*} ( 1 - \\alpha ) ( x _ i - x _ { i - 1 } ) & = [ \\lambda _ 1 - \\alpha ( m + 2 ) ] ( x _ { i + 1 } - x _ i ) - ( 1 - \\alpha ) ( x _ { i + 2 } - x _ { i + 1 } ) = 0 , \\\\ [ 0 . 1 c m ] ( 1 - \\alpha ) ( x _ { i - 1 } - x _ { i - 2 } ) & = [ \\lambda _ 1 - \\alpha ( m + 2 ) ] ( x _ i - x _ { i - 1 } ) - ( 1 - \\alpha ) ( x _ { i + 1 } - x _ i ) = 0 , \\\\ [ 0 . 1 c m ] \\vdots \\\\ ( 1 - \\alpha ) ( x _ 2 - x _ 1 ) & = [ \\lambda _ 1 - \\alpha ( m + 2 ) ] ( x _ 3 - x _ 2 ) - ( 1 - \\alpha ) ( x _ 4 - x _ 3 ) = 0 , \\end{align*}"}
-{"id": "8235.png", "formula": "\\begin{align*} F = \\sum _ { s \\ge 0 } \\frac { a _ s ( r , j ) } { n ^ s } + \\sum _ i c _ i j ( j - 1 ) \\cdots ( j - z _ i + 1 ) \\frac { 1 } { n ^ { z _ i } r ^ { z _ i } } \\sum _ { s \\ge 0 } \\frac { a _ s ( r , j - z _ i ) } { n ^ s } \\end{align*}"}
-{"id": "7443.png", "formula": "\\begin{align*} a ^ * a = \\begin{pmatrix} t & 0 & 0 & 0 \\\\ z & 0 & 0 & 0 \\\\ y & 0 & 0 & 0 \\\\ x + t v & 0 & 0 & 0 \\end{pmatrix} \\cong \\begin{pmatrix} t & 0 & 0 & 0 \\\\ z & 0 & 0 & 0 \\\\ y & 0 & 0 & 0 \\\\ 0 & y & - z & t \\end{pmatrix} . \\end{align*}"}
-{"id": "6825.png", "formula": "\\begin{align*} L ( \\phi ) = h + \\sum \\limits _ { j = 1 } ^ 4 c _ j \\chi _ { R _ 1 , \\xi _ j } \\varphi _ { 0 , j } + c _ 0 \\textrm { i n } \\mathbb { S } ^ 2 _ { \\lambda } , \\end{align*}"}
-{"id": "9050.png", "formula": "\\begin{align*} a ( x , x ) = \\left ( a _ { 1 1 } ^ 1 x _ 1 ^ 2 + 2 a _ { 1 2 } ^ 1 x _ 1 x _ 2 + a _ { 2 2 } ^ 1 x _ 2 ^ 2 , a _ { 1 1 } ^ 2 x _ 1 ^ 2 + 2 a _ { 1 2 } ^ 2 x _ 1 x _ 2 + a _ { 2 2 } ^ 2 x _ 2 ^ 2 \\right ) , \\end{align*}"}
-{"id": "4190.png", "formula": "\\begin{gather*} g _ 2 ( \\Lambda ) = \\frac { 1 } { 1 6 r _ 1 ^ 2 } \\frac { 6 4 } { 3 } r _ 2 ^ 2 ( 1 + 3 r ^ 2 ) = \\frac { 1 } { 1 6 r _ 1 ^ 2 } 6 0 G _ 4 ( \\tau ) \\xrightarrow { \\ : q \\to 0 \\ : } \\frac { 1 } { 1 6 r _ 1 ^ 2 } \\frac { ( 2 \\pi ) ^ 4 } { 1 2 } . \\\\ \\textnormal { T h e r e f o r e , } r _ 2 \\xrightarrow { \\ : q \\to 0 \\ : } \\pi ^ 2 / 4 . \\end{gather*}"}
-{"id": "8731.png", "formula": "\\begin{align*} & \\Phi _ { \\varepsilon , \\delta } ( t , x , s , y ) = u ^ { \\varepsilon , \\delta } ( t , x ) - v _ { \\varepsilon , \\delta } ( s , y ) - \\varphi _ \\delta ( t , x , y ) \\quad \\\\ & \\varphi _ \\delta ( t , x , s , y ) = \\varphi ( t , x , y ) + | t - \\bar { t } | ^ 2 + | x - \\bar { x } | ^ 4 + | y - \\bar { y } | ^ 4 + \\frac { | t - s | ^ 2 } { \\delta } . \\end{align*}"}
-{"id": "4674.png", "formula": "\\begin{align*} r _ { j k } = | { \\bf r } _ j - { \\bf r } _ k | \\ , \\ \\end{align*}"}
-{"id": "173.png", "formula": "\\begin{align*} m \\star _ { \\gamma _ { \\vdash } } x = \\gamma _ { \\vdash } ( m \\star x ) . \\end{align*}"}
-{"id": "2743.png", "formula": "\\begin{align*} b ( \\cos \\theta , z ) = b _ 0 ( \\cos \\theta ) + b _ 1 ( \\cos \\theta ) z \\ , , \\end{align*}"}
-{"id": "9823.png", "formula": "\\begin{align*} p _ 0 ' ( t ) = p _ 1 ( t ) \\mu - p _ 0 ( t ) \\l f ( t ) , t ' \\leq t \\leq ( T ' _ { e _ 2 } \\wedge T _ 2 ) . \\end{align*}"}
-{"id": "6891.png", "formula": "\\begin{align*} \\mu ( A ) = \\mu _ 0 ^ { i ( A ) } \\mu _ 1 ^ { m - i ( A ) } = \\mu _ 0 ^ { i ( A ) } \\left ( \\frac { 1 - 3 \\mu _ 0 } { 6 } \\right ) ^ { m - i ( A ) } \\end{align*}"}
-{"id": "9228.png", "formula": "\\begin{align*} ( x _ { 1 } ^ { + } \\mid x _ { 2 } ^ { + } ) [ d , \\langle a _ { 1 } ^ { - } , a _ { 2 } ^ { - } \\rangle ] = ( x _ { 1 } ^ { + } \\mid x _ { 2 } ^ { + } ) ( \\langle d a _ { 1 } ^ { - } , a _ { 2 } ^ { - } \\rangle + \\langle a _ { 1 } ^ { - } , d a _ { 2 } ^ { - } \\rangle ) . \\end{align*}"}
-{"id": "8767.png", "formula": "\\begin{align*} \\Psi ( t _ 1 , \\sigma ) \\Psi ( t _ 1 , \\sigma ) ^ { - 1 } = I d _ M \\end{align*}"}
-{"id": "5959.png", "formula": "\\begin{align*} \\gamma ( t ) \\cdot \\vect { e } _ j ' ( t ) = 0 ( j = 2 , \\dots , n ) . \\end{align*}"}
-{"id": "9107.png", "formula": "\\begin{align*} 2 a _ { 2 } A ( x y ) + a _ { 1 } \\left ( x A ( y ) + y A ( x ) \\right ) = 0 \\left ( x , y \\in R \\right ) . \\end{align*}"}
-{"id": "3189.png", "formula": "\\begin{align*} X _ { t } = x + \\int _ { 0 } ^ { t } ( a - b X _ { s } ) \\mathrm { d } s + \\sigma \\int _ { 0 } ^ { t } \\sqrt { X _ { s } } \\mathrm { d } B _ { s } + \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { \\infty } z N ( \\mathrm { d } s , \\mathrm { d } z ) , t \\geqslant 0 , \\end{align*}"}
-{"id": "138.png", "formula": "\\begin{align*} h _ z ( x ) & = \\Phi ( \\langle a _ 0 , x \\rangle + a ' z + \\lambda b + \\mu c ) , \\\\ f _ z ( x ) & = \\Phi ( \\langle a _ 0 , x \\rangle + a ' z + b ) , \\\\ g _ z ( x ) & = \\Phi ( \\langle a _ 0 , x \\rangle + a ' z + c ) . \\end{align*}"}
-{"id": "8341.png", "formula": "\\begin{align*} W = ( \\Q k + \\Q \\ell ) ^ \\perp \\subset V \\end{align*}"}
-{"id": "6394.png", "formula": "\\begin{align*} P _ { } \\left ( x \\theta \\right ) = \\frac { \\exp \\left [ \\beta f \\left ( \\theta \\right ) \\right ] P _ { } \\left ( x \\theta \\right ) \\delta \\left ( x - x ^ { \\prime } \\right ) } { \\int d \\theta \\exp \\left [ \\beta f \\left ( \\theta \\right ) \\right ] P _ { } \\left ( x \\theta \\right ) } \\end{align*}"}
-{"id": "4561.png", "formula": "\\begin{align*} & \\theta ( E _ { 0 , 0 } ) = d ^ { - 1 } [ F _ { n - 1 , 0 } , \\ldots , [ F ^ - _ { 2 , 0 } , F ^ - _ { 1 , - 1 } ] _ q \\ldots ] _ q K _ 1 \\cdots K _ { n - 1 } \\ , , \\\\ & \\theta ( F _ { 0 , 0 } ) = d ( K _ 1 \\cdots K _ { n - 1 } ) ^ { - 1 } [ \\ldots [ E _ { 1 , 1 } , E _ { 2 , 0 } ] _ { q ^ { - 1 } } , \\ldots , E _ { n - 1 , 0 } ] _ { q ^ { - 1 } } \\ , . \\end{align*}"}
-{"id": "1392.png", "formula": "\\begin{align*} \\tilde U _ { a , b } ( x ) & = \\tilde U _ { b - 1 } \\circ \\cdots \\circ \\tilde U _ { a } ( x ) = \\tilde U _ { b - 1 } \\circ \\cdots \\circ \\tilde U _ { a + 1 } ( U _ a ( x - 1 ) + 1 ) \\\\ & = \\tilde U _ { b - 1 } \\circ \\cdots \\circ \\tilde U _ { a + 2 } ( U _ { a + 1 } \\circ U _ a ( x - 1 ) + 1 ) = U _ { a , b } ( x - 1 ) + 1 . \\end{align*}"}
-{"id": "1453.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\{ y _ 1 \\} = \\emptyset , & \\{ z _ 1 \\} = \\{ x _ { 4 1 } \\} , \\\\ \\{ y _ 2 \\} = \\emptyset , & \\{ z _ 2 \\} = \\emptyset , & \\{ e _ 2 \\} = \\{ x _ { 4 1 } ^ 2 \\} . \\end{array} \\end{align*}"}
-{"id": "8309.png", "formula": "\\begin{align*} \\mathrm { F J } _ \\alpha ^ { ( a a _ \\tau ) } ( \\psi ^ \\tau ) = \\tau \\big ( \\mathrm { F J } _ \\alpha ^ { ( a ) } ( \\psi ) \\big ) . \\end{align*}"}
-{"id": "6566.png", "formula": "\\begin{align*} | P ( \\varepsilon ) ^ { \\circ } | _ 2 = \\frac { 4 - 2 \\varepsilon } { \\sqrt { 1 - \\varepsilon ^ 2 } } . \\end{align*}"}
-{"id": "3202.png", "formula": "\\begin{align*} J _ { \\ast , t } ( V ) & : = \\int _ { 0 } ^ { t } \\int _ { \\lbrace z \\leqslant 1 \\rbrace } e ^ { c s } \\left ( V ( X _ { s - } + z ) - V ( X _ { s - } ) \\right ) \\widetilde { N } ( \\mathrm { d } s , \\mathrm { d } z ) , t \\geqslant 0 , \\\\ J _ { t } ^ { \\ast } ( V ) & : = \\int _ { 0 } ^ { t } \\int _ { \\lbrace z > 1 \\rbrace } e ^ { c s } \\left ( V ( X _ { s - } + z ) - V ( X _ { s - } ) \\right ) \\widetilde { N } ( \\mathrm { d } s , \\mathrm { d } z ) , t \\geqslant 0 . \\end{align*}"}
-{"id": "2973.png", "formula": "\\begin{align*} \\underset { s \\rightarrow 0 ^ + } { \\lim } ~ h ( s ) \\in ( 0 , \\infty ] ~ ~ \\underset { s \\rightarrow \\infty } { \\lim } h ( s ) = h ( \\infty ) < \\infty \\end{align*}"}
-{"id": "1206.png", "formula": "\\begin{align*} \\nabla _ X \\bar h ( X , t ) = \\bar x ( X , t ) \\end{align*}"}
-{"id": "2982.png", "formula": "\\begin{align*} \\int _ { \\left \\lbrace 1 < u _ n \\leq k + 1 \\right \\rbrace } | \\nabla T _ k ( G _ 1 ( u _ n ) ) | ^ 2 & = \\int _ { \\left \\lbrace 1 < u _ n \\leq k + 1 \\right \\rbrace } | \\nabla u _ n | ^ 2 \\\\ & \\geq \\int _ { \\left \\lbrace | \\nabla u _ n | > t , 1 < u _ n \\leq k + 1 \\right \\rbrace } | \\nabla u _ n | ^ 2 \\\\ & \\geq t ^ 2 m ( \\{ | \\nabla u _ n | > t , 1 < u _ n \\leq k + 1 \\} ) . \\end{align*}"}
-{"id": "2158.png", "formula": "\\begin{align*} U _ d ( r ( s ) ) \\sim c _ * , U _ d ' ( r ( s ) ) r ' ( s ) = O ( r ( s ) ^ { - 1 - \\delta } r ' ( s ) ) = O ( e ^ { - \\delta \\left ( \\frac { 1 } { 2 } - \\theta _ * \\right ) s } ) , \\end{align*}"}
-{"id": "9022.png", "formula": "\\begin{align*} \\delta y _ { s t } = f ( y _ s ) { X } ^ { 1 } _ { s t } + f _ { 2 } ( y _ s ) { X } ^ { 2 } _ { s t } + y ^ \\natural _ { s t } , y _ 0 = y _ { } , \\end{align*}"}
-{"id": "5088.png", "formula": "\\begin{align*} E _ G ( x + y , \\xi ) = E _ G ( x , \\xi ) E _ G ( y , \\xi ) , E _ G ( x , \\xi + \\zeta ) = E _ G ( x , \\xi ) E _ G ( x , \\zeta ) , \\end{align*}"}
-{"id": "9462.png", "formula": "\\begin{align*} \\omega ( z ; q ) & = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ n } { ( z q ; q ^ 2 ) _ { n + 1 } } . \\end{align*}"}
-{"id": "337.png", "formula": "\\begin{align*} H ( x ; n ) & = u ( x ; n ) \\cdot n ^ { \\gamma ( n ) } \\cdot \\Phi ( x ) ^ n \\cdot \\{ 1 + e ( x ; n ) \\} , \\\\ [ 1 m m ] | e ( x ; n ) | & \\le C _ 0 / n , { } ^ { \\forall } n \\ge N _ 0 , \\ , \\ , r _ 0 \\le { } ^ { \\forall } x \\le r _ 1 . \\end{align*}"}
-{"id": "1940.png", "formula": "\\begin{align*} q ( x ) - q _ B ( x ) = - \\sum _ { j = 2 } ^ \\infty \\widetilde { Q } _ j ( x ) . \\end{align*}"}
-{"id": "8962.png", "formula": "\\begin{gather*} \\dim \\big ( \\Gamma \\big ( X / W , ( M _ { k ( S ) } ) ^ W ( d ) \\big ) \\big ) = \\dim \\big ( \\Gamma \\big ( X / W , ( M _ s ) ^ W ( d ) \\big ) \\big ) \\end{gather*}"}
-{"id": "557.png", "formula": "\\begin{align*} d \\rho \\circ J _ t = \\lambda , \\end{align*}"}
-{"id": "3918.png", "formula": "\\begin{align*} \\begin{aligned} x ^ 3 \\circ ( x ^ { j _ 1 } + x ^ { j _ 2 } ) - x ^ 3 \\circ x ^ { j _ 2 } - x ^ 3 \\circ x ^ { j _ 1 } - 3 x ^ { 2 j _ 2 + j _ 1 } = 3 x ^ { 2 j _ 1 + j _ 2 } \\in \\langle F \\rangle \\end{aligned} \\end{align*}"}
-{"id": "4060.png", "formula": "\\begin{align*} \\int _ { \\Omega _ 1 } f ( t ) \\ , \\d t = - \\int _ { \\Omega _ 2 } f ( t ) \\ , \\d t = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "2115.png", "formula": "\\begin{align*} X _ t = X _ 0 + \\int _ 0 ^ t \\Theta ^ T X _ s d s + W _ t . \\end{align*}"}
-{"id": "5982.png", "formula": "\\begin{align*} \\beta = \\frac { \\gamma } { \\alpha } = \\frac { \\gamma } { \\sqrt { \\gamma ^ 2 - 2 \\varepsilon ^ 2 } } \\end{align*}"}
-{"id": "9222.png", "formula": "\\begin{align*} \\chi ( \\beta _ { 1 } , \\alpha \\beta _ { 2 } ) = \\eta ( \\chi ( \\alpha \\beta _ { 2 } , \\beta _ { 1 } ) ) = \\eta ( \\alpha \\chi ( \\beta _ { 2 } , \\beta _ { 1 } ) ) = \\eta ( \\chi ( \\beta _ { 2 } , \\beta _ { 1 } ) ) \\eta ( \\alpha ) = \\chi ( \\beta _ { 1 } , \\beta _ { 2 } ) \\eta ( \\alpha ) . \\end{align*}"}
-{"id": "2367.png", "formula": "\\begin{align*} \\Gamma = \\Big \\langle a _ 1 , b _ 1 , \\ldots , a _ g , b _ g , e _ 1 , \\ldots , e _ r , & \\ , p _ 1 , \\ldots , p _ s , h _ 1 , \\ldots , h _ t \\ \\Big \\vert \\ e _ 1 ^ { m _ 1 } = \\ldots = e _ r ^ { m _ r } \\\\ & = \\big ( \\prod _ { j = 1 } ^ g [ a _ j , b _ j ] \\big ) e _ 1 \\cdots e _ r p _ 1 \\cdots p _ s h _ 1 \\cdots h _ t = 1 \\Big \\rangle \\end{align*}"}
-{"id": "5909.png", "formula": "\\begin{align*} c : = P ( \\frac { S ^ { ( 1 ) } _ n } n \\ge r _ 1 , \\ n = 1 , 2 , \\cdots ) > 0 , \\end{align*}"}
-{"id": "3567.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\frac { \\chi ( n ) \\Lambda ( n ) } { n ^ s } = \\int _ 1 ^ { x } \\frac { 1 } { t ^ s } d \\psi _ { \\chi } ( t ) \\ll \\frac { 1 } { x ^ { s - 1 } } , \\end{align*}"}
-{"id": "7896.png", "formula": "\\begin{align*} \\max _ { B _ r ( z ) } u _ 0 + \\min _ { B _ r ( z ) } u _ 0 = - r < 0 = u _ 0 ( z ) \\end{align*}"}
-{"id": "2841.png", "formula": "\\begin{align*} A _ { n } ^ { \\alpha } = \\frac { ( \\alpha + 1 ) ( \\alpha + 2 ) . . . . ( \\alpha + n ) } { n ! } = O ( n ^ \\alpha ) , { A _ { - n } ^ \\alpha = 0 } { \\texttt { f o r } } n > 0 . \\ \\end{align*}"}
-{"id": "2525.png", "formula": "\\begin{align*} \\begin{aligned} I _ 3 & = \\int \\left ( \\nabla _ { x } h , [ \\nabla _ { \\xi } , \\mathcal { L } ] h \\right ) \\varrho \\ , m _ 0 \\\\ & = - \\int \\left | \\nabla _ { x } h \\right | ^ 2 \\varrho m _ 0 + \\underbrace { \\int ( \\partial _ { x _ { k } } h ) \\partial _ { i } [ ( \\partial _ { k } \\sigma ^ { i j } ) \\partial _ { j } h ] \\varrho \\ , m _ 0 } _ { = : T _ 1 } \\underbrace { - \\int h ( \\nabla _ { \\xi } \\psi , \\nabla _ { x } h ) \\varrho \\ , m _ 0 } _ { = : T _ 2 } . \\end{aligned} \\end{align*}"}
-{"id": "6925.png", "formula": "\\begin{align*} u ( x , 0 ) = & 0 \\\\ \\frac { \\partial u } { \\partial t } u ( x , 0 ) = & f ( x ) \\end{align*}"}
-{"id": "8586.png", "formula": "\\begin{align*} [ \\varphi _ + \\ast \\varphi _ - ( d ) , e ] = 0 \\mbox { \\ f o r \\ a n y \\ } d \\in C \\ast _ J C , \\end{align*}"}
-{"id": "1387.png", "formula": "\\begin{align*} q _ { t } & = \\Big ( 1 + \\int _ { t - 1 } ^ t e ^ { \\rho _ u } \\mu ( u ) d u \\Big ) ^ { - 1 } , p _ t = e ^ { \\rho _ t } q _ t , \\rho _ u : = \\rho ( t - 1 , u ) = \\int _ { t - 1 } ^ u ( \\mu ( v ) - \\lambda ( v ) ) d v . \\end{align*}"}
-{"id": "3724.png", "formula": "\\begin{align*} ( 1 \\ ; 2 \\ ; 3 ) ^ a ( 4 \\ ; 5 \\ ; 6 ) ^ b \\cdot \\frac { \\xi _ 3 } { \\xi _ 4 } = \\omega ^ { a - b } \\frac { \\xi _ 3 } { \\xi _ 4 } \\end{align*}"}
-{"id": "4636.png", "formula": "\\begin{align*} N : = \\sum _ { n = 0 } ^ { 2 ^ k - 1 } 1 _ { [ - B - C , B + C ] } \\left ( t + \\sum _ { i = 0 } ^ { n - 1 } f ( T ^ i x ) \\right ) \\end{align*}"}
-{"id": "4154.png", "formula": "\\begin{align*} \\max \\{ Z _ { I , J } ^ { t + h - 1 } : I \\subseteq N , J \\subseteq V , \\lvert I \\rvert = \\lvert J \\rvert = h = r \\} . \\end{align*}"}
-{"id": "3493.png", "formula": "\\begin{align*} \\int _ { \\beta \\alpha \\beta ^ { - 1 } } \\omega _ 1 \\omega _ 2 & = \\int _ \\alpha \\omega _ 1 \\omega _ 2 + \\left ( \\int _ \\beta \\omega _ 1 \\int _ \\alpha \\omega _ 2 - \\int _ \\alpha \\omega _ 1 \\int _ \\beta \\omega _ 2 \\right ) . \\end{align*}"}
-{"id": "7678.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } \\dot { x } _ 1 ( t ) \\\\ \\dot { x } _ 2 ( t ) \\end{array} \\right ] = \\left [ \\begin{array} { c c } 0 & I \\\\ - L & - L \\end{array} \\right ] \\left [ \\begin{array} { c } x _ 1 ( t ) \\\\ x _ 2 ( t ) \\end{array} \\right ] + \\left [ \\begin{array} { c } 0 \\\\ I \\end{array} \\right ] w ( t ) \\ , , \\end{align*}"}
-{"id": "462.png", "formula": "\\begin{align*} \\partial _ 1 ^ { 2 j } a _ { k _ 1 , k _ 2 } ( i y _ \\omega u _ 1 ) = ( - 1 ) ^ { k _ 1 + k _ 2 } i ^ { - 2 j } \\frac { k _ 2 ! } { ( k _ 2 - 2 j ) ! } y _ \\omega ^ { k _ 2 - 2 j } . \\end{align*}"}
-{"id": "2716.png", "formula": "\\begin{align*} & \\displaystyle | | h | | _ { H _ { x , v } ^ { s , r } } ^ 2 = \\sum _ { | m | \\leq r } \\ , | | \\partial ^ m h | | _ { H _ { x , v } ^ s } ^ 2 \\ , , \\qquad | | h | | _ { H _ { \\Lambda } ^ { s , r } } ^ 2 = \\sum _ { | m | \\leq r } \\ , | | \\partial ^ m h | | _ { H _ { \\Lambda } ^ s } ^ 2 \\ , , \\\\ [ 2 p t ] & \\displaystyle | | h | | _ { H _ { x } ^ { s , r } L _ v ^ 2 } ^ 2 = \\sum _ { | m | \\leq r } \\ , | | \\partial ^ m h | | _ { H _ { x } ^ s L _ v ^ 2 } ^ 2 \\ , . \\end{align*}"}
-{"id": "396.png", "formula": "\\begin{align*} x _ n = ( 1 + \\varepsilon ) \\sqrt { 2 \\ln \\Psi ( n ) } \\ll \\sqrt { ( p - 2 ) \\ln n } \\sim \\sqrt { 2 \\ln ( U _ { n p } ^ { - 1 } ) } . \\end{align*}"}
-{"id": "8180.png", "formula": "\\begin{align*} \\pi ^ { \\mathfrak { L } } ( g ) = \\omega _ { \\pi } ^ { - 1 } \\omega _ { \\pi } ^ { \\mathfrak { L } } ( \\det ( g ) ) \\pi ( g ) . \\end{align*}"}
-{"id": "4058.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } | h ( s ) | \\ , \\d s & = \\int _ { 0 } ^ { t _ { 1 } } | h ( s ) | \\ , \\d s + \\sum _ { k = 1 } ^ { n - 1 } \\int _ { t _ { k } } ^ { t _ { k + 1 } } | h ( s ) | \\ , \\d s + \\int _ { t _ { n } } ^ { 1 } | h ( s ) | \\ , \\d s \\\\ & \\ge | V h ( t _ { 1 } ) | + \\sum _ { k = 1 } ^ { n - 1 } | V h ( t _ { k + 1 } ) - V h ( t _ { k } ) | + | V h ( t _ { n } ) | \\\\ & = 1 + 2 ( n - 1 ) + 1 = 2 n . \\end{align*}"}
-{"id": "4920.png", "formula": "\\begin{align*} B B ^ * & = I - A A ^ * \\\\ C ^ * C & = I - A ^ * A . \\end{align*}"}
-{"id": "1876.png", "formula": "\\begin{align*} \\phi _ 1 ( f ) : = \\inf \\big \\{ \\pi ( f _ 1 , g _ 1 , \\dots , f _ d , g _ d , a ) : ( f _ 1 , g _ 1 , \\dots , f _ d , g _ d , a ) \\in \\Theta _ 1 ( f ) \\big \\} . \\end{align*}"}
-{"id": "716.png", "formula": "\\begin{align*} v _ + = \\frac { R T _ + } { M p _ + } , e _ + = \\frac { 3 R T _ + } { 2 M } , S _ + = \\frac { R } { M } \\left ( \\log p _ + - \\frac { 5 } { 2 } \\log T _ + \\right ) + \\end{align*}"}
-{"id": "3512.png", "formula": "\\begin{align*} 0 \\leq X ^ u { ( t _ i ) } , \\ i = 0 , 1 , \\cdots , n , \\end{align*}"}
-{"id": "221.png", "formula": "\\begin{align*} U _ { \\sigma } H U _ { \\sigma } = H ^ { R _ { \\sigma } } . \\end{align*}"}
-{"id": "8348.png", "formula": "\\begin{align*} \\mathcal { Z } ( m , \\mu ) ( S ) = \\{ x \\in V _ \\mu ( \\mathcal { A } _ S ) : Q ( x ) = m \\} \\end{align*}"}
-{"id": "1186.png", "formula": "\\begin{align*} 0 \\geq \\sum \\limits _ { i , j = 1 } ^ { n } \\left [ \\frac { ( 1 - K ) } { A ^ 2 } v _ { x _ i x _ j } ( y _ 0 ) + \\frac { K } { B ^ 2 } v _ { x _ i x _ j } ( z _ 0 ) - \\frac { 1 } { C ^ 2 } u _ { x _ i x _ j } ( x _ 0 ) \\right ] \\eta _ i \\eta _ j . \\end{align*}"}
-{"id": "520.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { R } } _ { } \\ ! = \\ ! & \\bigcup _ { P _ { U | \\widetilde { X } } } \\ ! \\Big \\{ \\left ( R _ s , R _ \\ell , R _ w \\right ) \\ ! \\colon \\ ! \\\\ & 0 \\leq R _ s \\leq I ( U ; Y ) , \\\\ & R _ \\ell \\geq I ( U ; X ) - I ( U ; Y ) , \\\\ & R _ w \\geq I ( U ; \\widetilde { X } ) - I ( U ; Y ) \\Big \\} , \\end{align*}"}
-{"id": "2830.png", "formula": "\\begin{align*} f _ { k i } ^ { ( n ) } = \\left \\{ \\begin{aligned} \\frac { k } { n + 1 } & , \\mbox { i f } i = k - 1 \\mbox { a n d } 0 < k \\leq n + 1 , \\\\ 1 - \\frac { k } { n + 1 } & , \\mbox { i f } i = k \\mbox { a n d } 0 \\leq k < n + 1 , \\\\ 0 & , \\mbox { o t h e r w i s e } . \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "8922.png", "formula": "\\begin{gather*} ( z , q ) Q ( z , q ) ^ t / 2 = k \\frac { z ^ 2 } { 2 } + \\frac { k ( k - 1 ) } { 2 } q z + \\frac { k ( 2 k - 1 ) ( k - 1 ) } { 6 } \\frac { q ^ 2 } { 2 } , \\end{gather*}"}
-{"id": "1123.png", "formula": "\\begin{align*} \\textrm { g a p } = \\frac { \\textrm { o p t i m u m - l o w e r b o u n d } } { \\textrm { o p t i m u m } } \\times 1 0 0 \\end{align*}"}
-{"id": "948.png", "formula": "\\begin{align*} E [ h ( U _ i + \\xi _ i V _ i ) ] & = \\frac { 1 } { 2 } \\sum _ { p , q = 1 } ^ d E \\left [ \\frac { \\partial ^ 2 h } { \\partial x _ p \\partial x _ q } ( U _ i ) V _ { p , i } V _ { q _ i } \\right ] \\\\ & \\quad + \\frac { 1 } { 2 } \\sum _ { p , q , r = 1 } ^ d \\int _ 0 ^ 1 ( 1 - t ) ^ 2 E \\left [ \\frac { \\partial ^ 3 h } { \\partial x _ p \\partial x _ q \\partial x _ r } ( U _ i + t \\xi _ i V _ i ) V _ { p , i } V _ { q , i } V _ { r , i } \\xi _ i ^ { 3 } \\right ] d t \\end{align*}"}
-{"id": "746.png", "formula": "\\begin{align*} I _ { \\mu , p } ( w _ n ) = I _ { \\mu , p } ( w _ n - u ) + I _ { \\mu , p } ( u ) + o _ n ( 1 ) , \\end{align*}"}
-{"id": "1050.png", "formula": "\\begin{align*} H ( P _ { k } ) < H ( P _ { k + 1 } ) = X _ { k + 1 } , \\vert P _ { k + 1 } ( \\zeta ) \\vert < \\vert P _ { k } ( \\zeta ) \\vert \\leq X _ { k + 1 } ^ { - \\widehat { w } _ { n } ( \\zeta ) + \\epsilon } . \\end{align*}"}
-{"id": "1212.png", "formula": "\\begin{align*} a _ { i j } = \\frac { 1 } { 2 } \\left [ \\frac { \\partial \\mathcal { A } _ i } { \\partial \\eta _ j } ( \\xi ) + \\frac { \\partial \\mathcal { A } _ j } { \\partial \\eta _ i } ( \\xi ) \\right ] \\mbox { f o r } \\ , \\ , 1 \\leq i , j \\leq n , \\end{align*}"}
-{"id": "5353.png", "formula": "\\begin{align*} I _ 4 = \\bar { \\eta } \\circ I _ 3 , \\end{align*}"}
-{"id": "3978.png", "formula": "\\begin{align*} p ^ { \\alpha _ 3 } ( 3 , t ) = - \\sum _ { k = 3 } ^ { \\infty } ( - \\lambda ) ^ k \\underset { \\Theta ^ { k } _ { 3 } } { \\sum } \\frac { t ^ { \\sum _ { j = 0 } ^ 3 k _ j \\alpha _ j } } { \\Gamma \\left ( \\sum _ { j = 0 } ^ 3 k _ j \\alpha _ j + 1 \\right ) } , \\end{align*}"}
-{"id": "9998.png", "formula": "\\begin{align*} \\langle \\widehat { \\phi } , g \\rangle _ { \\mathrm { P e t } } = \\int _ { \\Gamma _ 0 ( D ) \\backslash \\mathcal { H } } \\overline { g ( \\tau ) } \\cdot \\widehat { \\phi } ( \\tau ) \\ , \\frac { d u \\ , d v } { v ^ { 2 - n } } \\end{align*}"}
-{"id": "7928.png", "formula": "\\begin{align*} \\underline { \\dim } _ B A = \\mathop { \\underline { \\lim } } _ { \\epsilon \\to 0 } \\frac { \\log N _ { \\epsilon } ( A ) } { - \\log \\epsilon } , \\ , \\ , \\ , \\overline { \\dim } _ B A = \\mathop { \\overline \\lim } _ { \\epsilon \\to 0 } \\frac { \\log N _ { \\epsilon } ( A ) } { - \\log \\epsilon } . \\end{align*}"}
-{"id": "9603.png", "formula": "\\begin{align*} | \\langle f , g \\rangle | & = \\bigg | \\int _ { \\Bbb R ^ n } \\sum _ { k \\in \\Bbb Z } D _ k T ^ { - 1 } _ N ( f ) D ^ N _ k ( g ) d \\mu \\bigg | \\\\ & \\le \\sum _ { k \\in \\Bbb Z } \\| D _ k T ^ { - 1 } _ N ( f ) \\| _ { L ^ p _ \\mu } \\| D ^ N _ k ( g ) \\| _ { L ^ { p ' } _ \\mu } \\\\ & \\le \\bigg \\{ \\sum _ { k \\in \\Bbb Z } 2 ^ { k \\alpha q } \\| D _ k T ^ { - 1 } _ N ( f ) \\| _ { L ^ p _ \\mu } ^ q \\bigg \\} ^ { 1 / q } \\bigg \\{ \\sum _ { k \\in \\Bbb Z } 2 ^ { - k \\alpha q ' } \\| D ^ N _ k ( g ) \\| _ { L ^ { p ' } _ \\mu } ^ { q ' } \\bigg \\} ^ { 1 / q ' } . \\end{align*}"}
-{"id": "9315.png", "formula": "\\begin{align*} C : = \\{ ( d _ 1 , d _ 2 , d _ 3 , d _ 4 ) \\in D ^ 4 \\ : \\ d _ 1 \\neq d _ 2 , d _ 3 \\neq d _ 4 \\} . \\end{align*}"}
-{"id": "2364.png", "formula": "\\begin{align*} \\chi _ n ( p ) e _ j = e _ j + \\sum _ { k = 1 } ^ j { j \\choose k } e _ { j - k } \\end{align*}"}
-{"id": "1507.png", "formula": "\\begin{align*} Q _ { t , n } ( x ) = t _ { n } ( x ) + t _ { n + 1 } ( x ) \\textbf { i } + t _ { n + 2 } ( x ) \\textbf { j } + t _ { n + 3 } ( x ) \\textbf { k } \\end{align*}"}
-{"id": "9991.png", "formula": "\\begin{align*} Y _ n & ( T ) - Y _ n ( T - 1 ) \\\\ & = \\mathbb E _ 0 \\bigg [ \\exp \\big \\{ H _ { \\beta , T } ( W , B ) \\big \\} I _ n ( T , W _ T ) - \\exp \\big \\{ H _ { \\beta , T - 1 } ( W , B ) \\big \\} I _ n ( T , W _ T ) \\\\ & \\qquad + \\exp \\big \\{ H _ { \\beta , { T - 1 } } ( W , B ) \\big \\} \\Big ( I _ n ( T , W _ T ) - I _ n ( T - 1 , W _ { T - 1 } ) \\Big ) \\bigg ] \\\\ & = \\mathbb E _ 0 \\bigg [ \\exp \\big \\{ H _ { \\beta , T - 1 } ( W , B ) \\big \\} \\ , \\Big ( \\exp \\big \\{ H _ { \\beta , T - 1 , T } ( W , B ) \\big \\} - 1 \\Big ) \\ , I _ n ( T , W _ T ) \\bigg ] , \\end{align*}"}
-{"id": "2216.png", "formula": "\\begin{align*} F _ 1 \\left ( \\frac 1 { w _ 1 } , \\ldots , \\frac 1 { w _ n } , t \\right ) = \\left ( \\frac 1 { w _ 1 } \\right ) ^ { \\deg _ { w _ 1 } Q _ 1 } \\cdot \\ldots \\cdot \\left ( \\frac 1 { w _ n } \\right ) ^ { m _ { 1 n } } \\cdot \\left ( \\widetilde q _ 1 ( w ) + t \\cdot \\widetilde Q _ 1 ( w ) \\right ) , \\end{align*}"}
-{"id": "6137.png", "formula": "\\begin{align*} g _ { 3 } \\left ( x \\right ) = O \\left ( x ^ { - \\gamma } \\right ) , \\end{align*}"}
-{"id": "9275.png", "formula": "\\begin{align*} \\theta ^ { ( s , u ) } ( \\underline { \\tau } , \\underline { \\xi } ) : = ( t _ { N _ { u } + 1 } - u , \\tau _ { N _ u + 2 } , \\cdots , \\tau _ { N _ s } , s - t _ { N _ s } ; \\xi _ { N _ u + 1 } , \\cdots , \\xi _ { N _ s } ) \\end{align*}"}
-{"id": "912.png", "formula": "\\begin{align*} D ^ k F = \\sum _ { i _ 1 , \\dots , i _ k = 1 } ^ m \\frac { \\partial ^ k f } { \\partial x _ { i _ 1 } \\dots \\partial x _ { i _ k } } ( W ( h _ 1 ) , \\dots , W ( h _ m ) ) h _ i . \\end{align*}"}
-{"id": "3324.png", "formula": "\\begin{align*} \\tilde { D } ( r _ { K - 1 } ) \\geq \\frac { N } { 1 + N } = \\bar { D } ( r _ { K - 1 } ) \\end{align*}"}
-{"id": "1886.png", "formula": "\\begin{align*} f _ 2 ( x _ 2 ) + \\sum _ { i \\in I } a ^ i \\ 1 _ { A ^ i } ( x ) = f _ 2 ( x _ 2 ) + \\sum _ { i \\in I : B _ 1 \\leq A ^ i _ 1 } a ^ i \\ 1 _ { A ^ i } ( x ) \\geq 1 - s \\quad x \\in B \\cap S . \\end{align*}"}
-{"id": "7360.png", "formula": "\\begin{align*} \\displaystyle { \\not } D ^ 2 \\psi = - \\frac { n } { 4 ( n - 1 ) } { \\cal { R } } \\psi \\end{align*}"}
-{"id": "1140.png", "formula": "\\begin{align*} e ( z ) = \\sum _ { i = 0 } ^ { \\infty } \\frac { z ^ { q ^ i } } { d _ i } . \\end{align*}"}
-{"id": "9693.png", "formula": "\\begin{align*} \\gamma _ 1 = K _ b \\omega _ { k , k + 1 } + O ( 1 ) ( | \\omega _ { k , k + 1 } | ^ 2 + | U _ a - { U } _ 2 ^ { ( 0 ) } | ) , \\end{align*}"}
-{"id": "3570.png", "formula": "\\begin{align*} n = \\prod _ { p \\leq w } p \\ll e ^ { \\sum _ { p \\leq w } \\log p } . \\end{align*}"}
-{"id": "2876.png", "formula": "\\begin{align*} \\zeta ( 2 m + 1 ) + 2 \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ { 2 m + 1 } ( e ^ { 2 \\pi n } - 1 ) } = \\pi ^ { 2 m + 1 } 2 ^ { 2 m } \\sum _ { j = 0 } ^ { m + 1 } \\frac { ( - 1 ) ^ { j + 1 } B _ { 2 j } B _ { 2 m + 2 - 2 j } } { ( 2 j ) ! ( 2 m + 2 - 2 j ) ! } . \\end{align*}"}
-{"id": "5004.png", "formula": "\\begin{align*} \\left \\Vert \\lambda \\right \\Vert _ { \\dot { F } ^ { s , p } _ q ( \\Lambda ^ l \\R ^ d ) } = \\max _ { | I | = l } \\left \\Vert \\lambda _ I \\right \\Vert _ { \\dot { F } ^ { s , p } _ q ( \\R ^ d ) } \\end{align*}"}
-{"id": "1905.png", "formula": "\\begin{align*} \\psi _ { n + 1 } ( w ) = F _ { n } \\phi _ { n } ( r _ { n } ^ { - 1 } ( w ) ) + a _ { n } ( \\phi _ { n } ( r _ { n } ^ { - 1 } ( w ) ) , r _ { n } ^ { - 1 } ( w ) ) \\quad w \\in B _ { n + 1 } ^ { u } ( \\delta _ { n + 1 } ) . \\end{align*}"}
-{"id": "6279.png", "formula": "\\begin{align*} & \\| [ \\Delta _ q , u _ { \\leq { q - 1 } } \\cdot \\nabla ] b _ q \\| _ 2 \\\\ \\lesssim & \\| \\nabla u _ { \\leq q - 1 } \\| _ { \\infty } \\| b _ q \\| _ 2 \\left | \\lambda _ q ^ 3 \\int _ { \\R ^ 3 } | x - y | \\nabla h ( \\lambda _ q ( x - y ) ) \\ , d y \\right | \\\\ \\lesssim & \\| \\nabla u _ { \\leq q - 1 } \\| _ { \\infty } \\| b _ q \\| _ 2 . \\end{align*}"}
-{"id": "9994.png", "formula": "\\begin{align*} \\Big ( ( \\mathbb E _ 0 \\otimes \\mathbb E _ 0 ) \\Big [ ( I _ n ( T , W _ T ) I _ n ( T , W _ T ^ { \\prime } ) ) ^ 8 \\Big ] \\Big ) ^ { 1 / 8 } = O ( T ^ { | n | } ) . \\end{align*}"}
-{"id": "2277.png", "formula": "\\begin{align*} \\lambda _ 1 ( A _ { \\alpha } ( K _ m \\vee H ) ) & - \\lambda _ 1 ( A _ { \\alpha } ( K _ m \\vee P _ n ) ) \\\\ & \\geq 2 ( 1 - \\alpha ) ( x _ 2 x _ n - x _ 1 x _ 2 ) + \\alpha ( x _ n ^ 2 - x _ 1 ^ 2 ) \\\\ & = 0 . \\end{align*}"}
-{"id": "8558.png", "formula": "\\begin{align*} ( \\Delta _ H v ) ( 0 ) = N ( \\partial _ r ^ 2 v ^ \\sharp ) ( 0 ) = ( \\Delta v ^ * ) ( 0 ) . \\end{align*}"}
-{"id": "2482.png", "formula": "\\begin{align*} \\begin{aligned} \\left \\| w f ( t ) \\right \\| _ { H ^ 2 _ x L ^ 2 _ \\xi } ^ 2 & \\lesssim \\int _ { \\R ^ { 6 } } f ^ 2 \\Big ( w ^ 2 + \\left | \\nabla _ x w \\right | ^ 2 + \\left | D _ x ^ 2 w \\right | ^ 2 \\Big ) d \\xi d x + \\int _ { \\R ^ { 6 } } \\left | \\nabla f \\right | ^ 2 \\Big ( w ^ 2 + \\left | \\nabla _ x w \\right | ^ 2 \\Big ) d \\xi d x \\\\ & + \\int _ { \\R ^ { 6 } } \\left | D _ x ^ 2 f \\right | ^ 2 w ^ 2 d \\xi d x . \\end{aligned} \\end{align*}"}
-{"id": "830.png", "formula": "\\begin{align*} \\left \\langle \\mathcal { L } _ h v _ h , w _ h \\right \\rangle = \\left \\langle \\mathcal { L } v _ h , w _ h \\right \\rangle , \\mbox { f o r a l l } v _ h , \\ ; w _ h \\in \\widetilde { \\mathcal { V } } _ h . \\end{align*}"}
-{"id": "4431.png", "formula": "\\begin{align*} \\lefteqn { f _ T ( y ) - f _ T ( x ) - ( y - x ) _ 1 \\partial _ 1 f _ T ( x ) } \\\\ & = \\int _ 0 ^ 1 ( y _ 2 - x _ 2 ) \\partial _ 2 f _ T ( y _ 1 , s y _ 2 + ( 1 - s ) x _ 2 ) d s \\\\ & + \\int _ 0 ^ 1 ( y _ 1 - x _ 1 ) ^ 2 \\partial _ 1 ^ 2 f _ T ( s y _ 1 + ( 1 - s ) x _ 1 , x _ 2 ) \\ , ( 1 - s ) d s , \\end{align*}"}
-{"id": "572.png", "formula": "\\begin{align*} \\mathcal { A } _ { F } ( \\Omega ) = \\{ f : \\Omega \\rightarrow \\mathbb { C } \\ : f ^ { ( l ) } \\ \\ \\overline { \\Omega } , \\ \\ l \\in F \\} , \\end{align*}"}
-{"id": "7139.png", "formula": "\\begin{align*} d ^ * _ G \\leq \\frac { 1 } { R } \\sum _ { r = 0 } ^ { R - 1 } g ^ * _ r + \\frac { U J } { R } \\sum _ { r = 0 } ^ { R - 1 } \\frac { 1 } { V _ r } , \\end{align*}"}
-{"id": "1694.png", "formula": "\\begin{align*} \\frac { d ( \\mu _ x \\circ \\sigma _ { f _ j } ) } { d \\mu _ x } ( z ) = \\lim _ { n \\to \\infty } \\frac { \\mu _ { x } ( Z ( f _ j z _ n ) ) } { \\mu _ { x } ( Z ( z _ n ) ) } = \\lim _ { n \\to \\infty } \\frac { T _ { j + 1 , i _ 1 + 1 } T _ { i _ 1 + 1 , i _ 2 + 1 } \\cdots T _ { i _ { n - 1 } + 1 , i _ n + 1 } } { T _ { i _ 1 , i _ 2 } \\cdots T _ { i _ { n - 1 } , i _ n } } = T _ { j + 1 , i _ 1 + 1 } \\end{align*}"}
-{"id": "7785.png", "formula": "\\begin{align*} 0 = a ( \\dot { y } _ h , y _ h ) \\quad y _ h \\coloneq y _ h ( 0 ) . \\end{align*}"}
-{"id": "6221.png", "formula": "\\begin{align*} \\P ( \\widehat { \\tau _ 0 } < \\infty , \\widehat { X } ( \\widehat { \\tau _ 0 } - 1 ) = y , \\widehat { X } ( \\widehat { \\tau _ 0 } ) \\geq x , \\Delta C ^ i ( \\widehat { \\tau _ 0 } ) = x + 1 - y ) = \\P ( C ^ i ( 1 ) = x + 1 - y ) ~ , \\end{align*}"}
-{"id": "9613.png", "formula": "\\begin{align*} P _ { 0 } ( r , h _ \\beta ) = \\frac { 1 } { 1 + a { \\rm { e x p } } ( - b [ \\theta - a ] ) } , \\end{align*}"}
-{"id": "6836.png", "formula": "\\begin{align*} L ( \\frac { \\partial \\phi } { \\partial \\lambda } ) = - \\lambda ^ 2 \\left [ \\frac { \\partial ( \\frac { \\rho } { \\int _ { \\mathbb { S } ^ 2 } e ^ { w _ { \\lambda } } } e ^ { w _ { \\lambda } } ) } { \\partial \\lambda } \\phi + \\frac { \\partial S _ { \\rho } ( w _ { \\lambda } ) } { \\partial \\lambda } + \\frac { \\partial N ( \\phi ) } { \\partial \\lambda } - \\frac { \\partial c ' _ 0 } { \\partial \\lambda } \\right ] \\end{align*}"}
-{"id": "5375.png", "formula": "\\begin{align*} \\frac { S ^ { ( \\ell ) } _ { a \\omega , r \\omega } } { S ^ { ( \\ell ) } _ { 0 , r \\omega } } ( - 1 ) ^ { a r } = ( - 1 ) ^ { a + ( a + b ) ( r + 1 ) + 1 } , \\end{align*}"}
-{"id": "7580.png", "formula": "\\begin{align*} r _ \\Omega f ^ k g ^ k = r _ \\Omega ( \\eta _ 0 f ) ^ k ( \\eta _ 0 g ) ^ k + r _ \\Omega \\bigl ( ( \\eta _ 1 f ) ^ k ( \\eta _ 0 g ) ^ k + f ^ k ( \\eta _ 1 g ) ^ k \\bigr ) , \\end{align*}"}
-{"id": "4615.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { 2 ^ { k _ i } - 1 } 1 _ { [ - B , - B + b ] \\cup [ B - b , B ] } \\left ( t + \\sum _ { j = 0 } ^ { n - 1 } f ( T ^ j x ) \\right ) \\le 2 \\eta \\sum _ { n = 0 } ^ { 2 ^ { k _ i } - 1 } 1 _ { [ - B , B ] } \\left ( t + \\sum _ { j = 0 } ^ { n - 1 } f ( T ^ j x ) \\right ) \\end{align*}"}
-{"id": "3035.png", "formula": "\\begin{align*} \\begin{cases} ( u _ t ( t ) , v ) _ H + a ( u ( t ) , v ) + b ( v , p ( t ) ) = 0 , & \\forall v \\in V , \\\\ b ( u ( t ) , q ) = 0 , & \\forall q \\in Q , \\end{cases} \\end{align*}"}
-{"id": "7214.png", "formula": "\\begin{align*} \\frac { | B _ 1 | } { 2 } & \\leq W _ { A C } ( v _ 0 ; x , 0 + ) \\\\ & = \\lim _ { s \\searrow 0 + } \\frac { | B _ s ( x ) \\cap \\{ v _ 0 > 0 \\} | } { s ^ n } \\\\ & = \\lim _ { s \\searrow 0 + } \\frac { | B ' _ s ( x ' ) \\cap \\{ v _ * > 0 \\} | _ { n - 1 } } { s ^ { n - 1 } } \\\\ & = W _ { A C } ( v _ * ; x ' , 0 + ) . \\end{align*}"}
-{"id": "306.png", "formula": "\\begin{align*} \\P \\left ( Z _ d \\in B | Z _ 0 = z \\right ) \\ge ~ \\lambda _ d \\left ( B \\cap K \\right ) . \\end{align*}"}
-{"id": "3528.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c l } & \\displaystyle \\sum \\limits _ { i = 1 } ^ { n } - \\psi _ { x _ i } ( \\bar { X } ( t _ 1 ) , \\bar { X } ( t _ 2 ) , \\cdots , \\bar { X } ( t _ n ) ) y ( t _ { i } ) \\\\ = & \\displaystyle \\int \\limits _ { 0 } ^ { T } \\big { [ } f _ x ( \\bar { X } { ( t ) } , \\bar { u } ( t ) ) y ( t ) + p ( t ) ^ { } ( b ( \\bar { X } { ( t ) } , u ^ { \\varepsilon } ( t ) ) - b ( \\bar { X } { ( t ) } , \\bar { u } ( t ) ) ) \\big { ] } d t . \\end{array} \\end{align*}"}
-{"id": "4851.png", "formula": "\\begin{align*} h _ 0 = 0 , \\ ; h _ 1 = m , \\ ; h _ 2 = 2 m , \\ ; h _ 3 = 3 m , \\ ; h _ 4 = 4 m , \\ ; h _ 5 = c . \\end{align*}"}
-{"id": "5791.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta _ { p } u = \\omega \\ ; \\ ; \\mathbb { R } ^ n , \\\\ \\liminf \\limits _ { \\vert x \\vert \\rightarrow \\infty } u ( x ) = 0 \\end{cases} \\end{align*}"}
-{"id": "5075.png", "formula": "\\begin{align*} & F = b _ 1 Y + \\xi , \\ ; \\ ; X _ 2 = \\frac { B _ { 1 2 , 1 } } { b _ 1 - b _ 2 } Y + Y _ 2 , \\ ; \\ ; X _ 3 = \\frac { B _ { 1 3 , 1 } } { b _ 1 - b _ 3 } Y + Y _ 3 , \\\\ & T = a _ 1 Y + N - \\frac { B _ { 1 2 , 1 } } { b _ 1 - b _ 2 } Y _ 2 - \\frac { B _ { 1 3 , 1 } } { b _ 1 - b _ 3 } Y _ 3 - b _ 1 \\xi , \\\\ & Q = 2 a _ 1 + b _ 1 ^ 2 + ( \\frac { B _ { 1 2 , 1 } } { b _ 1 - b _ 2 } ) ^ 2 + ( \\frac { B _ { 1 3 , 1 } } { b _ 1 - b _ 3 } ) ^ 2 . \\end{align*}"}
-{"id": "7650.png", "formula": "\\begin{align*} \\Omega + \\Omega ^ { \\dagger } = 0 . \\end{align*}"}
-{"id": "4370.png", "formula": "\\begin{align*} C _ t : = \\sum _ { j = j _ 0 ( t ) } ^ \\infty M _ { \\varphi _ { j , t } } C _ { j , t } M _ { \\psi _ { j , t } } . \\end{align*}"}
-{"id": "7591.png", "formula": "\\begin{align*} \\prod _ { i = 0 } ^ { n - 1 } ( s - i ) - \\prod _ { i = 0 } ^ { n - 1 } \\left ( s - \\left ( n + i - \\frac { 1 } { n - i } \\right ) \\right ) . \\end{align*}"}
-{"id": "10013.png", "formula": "\\begin{align*} L ( \\tilde { g } , \\theta _ \\Lambda , s ) = \\Gamma \\left ( \\frac { s } { 2 } + n - 1 \\right ) \\sum _ { m \\ge 0 } \\frac { \\{ \\overline { \\tilde { c } ( m ) } , R _ \\Lambda ( m ) \\} } { ( 4 \\pi m ) ^ { \\frac { s } { 2 } + n - 1 } } . \\end{align*}"}
-{"id": "6982.png", "formula": "\\begin{align*} D f _ k ( B ) E _ { i j } = - \\frac { 1 } { k f _ k ( B ) ^ { k - 1 } } \\sum _ { \\mbox { \\tiny $ \\begin{array} { c } i \\leq j _ 1 < j _ 2 < . . . < j _ { k - 2 } \\leq n , \\\\ j _ 1 , . . . , j _ { k - 2 } \\neq i , j \\end{array} $ } } \\det { B _ { ( j , j _ 1 , . . . , j _ { k - 2 } ) ( i , j _ 1 , . . . , j _ { k - 2 } ) } } , \\end{align*}"}
-{"id": "6031.png", "formula": "\\begin{align*} \\underline { \\Omega } ^ { ( \\alpha , \\lambda ) } : = \\inf _ { n \\geq 1 } \\inf _ { \\{ Q _ { i } \\} _ { i = 1 } ^ { n } } \\frac { 1 } { n } \\Omega ^ { ( \\alpha , \\lambda ) } ( \\{ Q _ { i } \\} _ { i = 1 } ^ { n } ) . \\end{align*}"}
-{"id": "9259.png", "formula": "\\begin{align*} | C | > \\frac { 4 } { 1 1 } \\left ( n + \\frac { 3 } { 4 } \\right ) = 4 q _ 1 + 1 \\end{align*}"}
-{"id": "6811.png", "formula": "\\begin{align*} \\int _ { \\mathbb { S } ^ 2 _ { \\lambda _ n } } \\chi _ { R _ 1 , j } \\varphi _ { i , j } \\phi _ n = 0 \\textrm { f o r a l l } i = 0 , 1 , 2 , j = 1 , 2 , 3 , 4 . \\end{align*}"}
-{"id": "5025.png", "formula": "\\begin{align*} G ^ \\omega _ { k , p } ( E \\pm \\iota 0 ) \\phi = G ^ \\omega _ { p , k } ( E \\pm \\iota 0 ) ^ \\ast \\phi \\forall \\phi \\in k e r ( \\pm \\Im G _ { p , p } ^ \\omega ( E \\pm \\iota 0 ) ) , \\end{align*}"}
-{"id": "1932.png", "formula": "\\begin{align*} d ( g _ { i } ^ { n } \\circ h _ { i } ( x ) , g _ { i } ^ { n } \\circ h _ { i } ( y ) ) & \\leq d ( g _ { i } ^ { n } \\circ h _ { i } ( x ) , f _ { i } ^ { n } ( x ) ) + d ( f _ { i } ^ { n } ( x ) , f _ { i } ^ { n } ( y ) ) + d ( f _ { i } ^ { n } ( y ) , g _ { i } ^ { n } \\circ h _ { i } ( y ) ) \\\\ & = d ( h _ { i + n } \\circ f _ { i } ^ { n } ( x ) , f _ { i } ^ { n } ( x ) ) + d ( f _ { i } ^ { n } ( x ) , f _ { i } ^ { n } ( y ) ) + d ( f _ { i } ^ { n } ( y ) , h _ { i + n } \\circ f _ { i } ^ { n } ( y ) ) \\\\ & \\leq \\tilde { r } / 3 + \\tilde { r } / 3 + \\tilde { r } / 3 = \\tilde { r } . \\end{align*}"}
-{"id": "7573.png", "formula": "\\begin{align*} \\left ( \\frac { x _ 1 } { t ^ { m _ 1 } } \\right ) ^ 2 + \\dots + \\left ( \\frac { x _ n } { t ^ { m _ n } } \\right ) ^ 2 = 1 \\ , . \\end{align*}"}
-{"id": "9915.png", "formula": "\\begin{align*} 2 e ( G ) = \\sum _ { x \\in V ( G ) } d ( x ) \\le \\sum _ { i = 1 } ^ { k } | W _ i | \\cdot d ( z _ i ) \\ , . \\end{align*}"}
-{"id": "6105.png", "formula": "\\begin{align*} \\hat { f ^ { \\psi } } \\left ( s \\right ) = \\hat { \\Phi } _ { \\psi } \\hat { f } \\left ( s \\right ) \\sim \\sum _ { i = 0 } ^ { k } \\mu _ { i } \\frac { \\left ( - \\psi \\left ( s \\right ) \\right ) ^ { i } } { i ! } . \\end{align*}"}
-{"id": "9587.png", "formula": "\\begin{align*} \\| R _ N ( f ) \\| _ { \\dot { \\mathcal B } ^ { \\alpha , q } _ { p , \\mathcal F } } & = \\bigg \\{ \\sum _ { k \\in \\Bbb Z } \\Big ( 2 ^ { k \\alpha } \\big \\| D _ k R _ N ( f ) \\big \\| _ { L ^ p _ \\mu } \\Big ) ^ q \\bigg \\} ^ { 1 / q } \\\\ & = \\bigg \\{ \\sum _ { k \\in \\Bbb Z } \\bigg ( 2 ^ { k \\alpha } \\Big \\| D _ k R _ N \\Big ( \\sum _ { k ' \\in \\Bbb Z } T _ N ^ { - 1 } D _ { k ' } ^ N D _ { k ' } ( f ) \\Big ) \\Big \\| _ { L ^ p _ \\mu } \\bigg ) ^ q \\bigg \\} ^ { 1 / q } . \\end{align*}"}
-{"id": "524.png", "formula": "\\begin{align*} g _ x ( \\nabla f ( x ) , v ) \\ , = \\ , d f ( x ) [ v ] , x \\in M , \\ ; v \\in T _ x M . \\end{align*}"}
-{"id": "1183.png", "formula": "\\begin{align*} v \\left ( y _ 0 + \\frac { \\rho _ 1 } { A } \\eta \\right ) = v \\left ( z _ 0 + \\frac { \\rho _ 2 } { B } \\eta \\right ) . \\end{align*}"}
-{"id": "5274.png", "formula": "\\begin{align*} \\bigg ( \\int _ { c } ^ x u _ z ( z ) z \\bigg ) ^ 2 & \\leq \\bigg | ( x - c ) \\int _ { c } ^ x u ^ 2 _ z ( z ) z \\bigg | \\\\ & = ( x - c ) \\int _ { c } ^ x u ^ 2 _ z ( z ) z . \\end{align*}"}
-{"id": "6609.png", "formula": "\\begin{align*} \\liminf _ { J \\ni n \\to \\infty } \\int _ { \\Omega _ 0 } | D f _ { 0 } - D g _ { n , 0 } | ^ p \\ , d \\mu = 0 \\ . \\end{align*}"}
-{"id": "8850.png", "formula": "\\begin{align*} D _ N ^ 2 ( X _ N ) = \\left ( \\int _ 0 ^ \\pi V ( X _ N , \\phi ) \\sin ( \\phi ) \\ , \\dd \\phi \\right ) ^ { \\frac 1 2 } , \\end{align*}"}
-{"id": "5151.png", "formula": "\\begin{align*} G _ m \\vartheta _ m ( \\varphi , \\pi ) & = ( G _ m ^ - - G _ m ^ + ) \\vartheta _ m ( \\varphi , \\pi ) \\\\ & = ( 1 - \\chi ) A + \\chi A = A \\end{align*}"}
-{"id": "5774.png", "formula": "\\begin{align*} \\Vert u \\Vert _ { L ^ { p } ( \\Omega , d \\sigma ) } = \\left ( \\int _ { \\Omega } | u | ^ { p } \\ ; d \\sigma \\right ) ^ { \\frac { 1 } { p } } < \\infty . \\end{align*}"}
-{"id": "8680.png", "formula": "\\begin{align*} \\lbrace \\lbrace f , v \\rbrace , w \\rbrace + \\lbrace \\lbrace v , w \\rbrace , f \\rbrace + \\lbrace \\lbrace f , w \\rbrace , v \\rbrace & = \\lbrace \\phi ^ { - 1 } ( f ( \\phi ^ { - 1 } ( v ) ) ) , w \\rbrace + \\lbrace \\phi ^ { - 1 } ( f ( \\phi ^ { - 1 } ( w ) ) ) , v \\rbrace \\\\ & = [ f ( \\phi ^ { - 1 } ( v ) ) , w ] + [ f ( \\phi ^ { - 1 } ( w ) ) , v ] \\\\ & = f ( [ \\phi ^ { - 1 } ( v ) , \\phi ^ { - 1 } ( w ) ] ) + f ( [ \\phi ^ { - 1 } ( w ) , \\phi ^ { - 1 } ( v ) ] ) \\\\ & = 0 . \\end{align*}"}
-{"id": "7322.png", "formula": "\\begin{align*} Q ( x ) = ( \\rho _ j ( x ) ) _ { j = 1 } ^ q . \\end{align*}"}
-{"id": "9003.png", "formula": "\\begin{gather*} D ( c ) = { \\cal D } ^ { ( n ) } _ { q , t + q } ( - c - ( n - 1 ) q / 2 ) { \\cal D } ^ { ( n ) } _ { q , t } ( c ) , \\end{gather*}"}
-{"id": "5846.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ N & ( - 1 ) ^ k \\binom { N } { k } k g ^ { k - 1 } D ( g ) D ^ { N - 1 } ( g ^ { N - k } f ) \\\\ & = - \\sum _ { k = 0 } ^ N ( - 1 ) ^ k \\binom { N } { k } g ^ k D ^ N ( g ^ { N - k } f ) . \\end{align*}"}
-{"id": "5816.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } \\vert \\nabla v \\vert ^ { p } \\ ; d x = \\int _ { \\R ^ n } v ^ { 1 + q } \\ ; d \\sigma + \\int _ { \\R ^ n } v \\ ; d \\mu . \\end{align*}"}
-{"id": "7913.png", "formula": "\\begin{align*} e _ 1 = { x _ 1 + y _ 1 \\over 2 } - \\left [ x _ 1 + y _ 1 \\over 2 \\right ] \\leq { y _ 1 - x _ 1 \\over 2 } , \\end{align*}"}
-{"id": "8018.png", "formula": "\\begin{align*} { T _ { [ a ] } } f _ L ( x ) = \\sum _ { k = 1 } ^ { \\infty } { { \\Lambda } _ k \\ast f _ L ( x ) e ^ { 2 \\pi i \\langle \\mathbf { v } _ k , x \\rangle } } = \\sum _ { n = M } ^ { L } { d _ { { t _ n } } ( x ) } \\end{align*}"}
-{"id": "2678.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\int f d \\mu _ N = \\int f d \\mu \\end{align*}"}
-{"id": "8283.png", "formula": "\\begin{align*} \\psi ( f ) = \\frac { ( j ^ { [ 2 ] } ) ^ * \\psi ( f ^ { [ 2 ] } ) } { ( j ^ { [ 1 ] } ) ^ * \\psi ( f ^ { [ 1 ] } ) } \\end{align*}"}
-{"id": "7275.png", "formula": "\\begin{align*} u n = v m , ( u , m ) , ( v , n ) \\le T . \\end{align*}"}
-{"id": "6206.png", "formula": "\\begin{align*} \\begin{pmatrix} - 2 & 0 & 1 \\\\ 0 & - 2 & 1 \\\\ 1 & 1 & 2 k \\end{pmatrix} d = 4 + 8 k . \\end{align*}"}
-{"id": "5483.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd x } \\log U ( x ) = \\frac { U ' ( x ) } { U ( x ) } = \\frac { \\varepsilon ( x ) } { x } , \\end{align*}"}
-{"id": "4167.png", "formula": "\\begin{align*} & \\frac { 1 } { n } \\sum _ { k \\in N } e _ { k , g } ^ n + \\sum _ { \\tau = 0 } ^ { n - s - 2 } \\frac { 1 } { n } e ^ { n - \\tau - 1 } _ { i , g + \\tau + 1 } \\\\ & + \\left ( 1 - \\frac { 1 } { n } \\right ) \\left [ \\sum _ { \\tau = 0 } ^ { n - s - 2 } \\frac { 1 } { n ^ { n - \\tau - s - 1 } } e ^ { s } _ { i , g + \\tau + 1 } \\right ] + \\frac { 1 } { n } \\sum _ { \\tau = 1 } ^ { s - 1 } e _ { i , g + n - 1 - s + \\tau } ^ { s - \\tau } . \\end{align*}"}
-{"id": "6065.png", "formula": "\\begin{align*} \\delta _ u \\mathcal { S } ^ n = \\{ v \\in H ^ 1 _ 2 ( B ^ n , \\R ^ { n + 1 } ) \\ , \\lvert \\ , \\langle u ( x ) , v ( x ) \\rangle = 0 \\quad \\mbox { f o r a . e . } x \\in B ^ n \\} , \\end{align*}"}
-{"id": "8906.png", "formula": "\\begin{align*} H ^ 2 = \\mathcal K _ \\theta \\dotplus \\omega H ^ 2 . \\end{align*}"}
-{"id": "1180.png", "formula": "\\begin{align*} \\xi = \\frac { \\nabla v ( y _ 0 ) } { | \\nabla v ( y _ 0 ) | } = \\frac { \\nabla v ( z _ 0 ) } { | \\nabla v ( z _ 0 ) | } = \\frac { \\nabla u ( x _ 0 ) } { | \\nabla u ( x _ 0 ) | } . \\end{align*}"}
-{"id": "1537.png", "formula": "\\begin{gather*} | A - B | \\ = \\ \\mu ( A ^ * - B ^ * ) . \\end{gather*}"}
-{"id": "1047.png", "formula": "\\begin{align*} L _ { m , j } ^ { \\ast } ( \\tilde { q } ) \\leq L _ { P _ { 1 } } ^ { \\ast } ( \\tilde { q } ) = \\tilde { q } \\left ( \\frac { m + 1 } { m } \\cdot \\frac { 1 } { 1 + w } - \\frac { 1 } { m } \\right ) = \\tilde { q } \\cdot \\frac { m - w } { m ( 1 + w ) } . \\end{align*}"}
-{"id": "7164.png", "formula": "\\begin{align*} & ~ ~ Z ( a ^ * ) \\beta K - Z ( n ) \\beta K + 2 c _ { k , s _ n , u _ n } \\\\ & \\leq Z ( a ^ * ) \\beta K - Z ( n ) \\beta K + \\delta _ { n , b } = 0 , \\end{align*}"}
-{"id": "1360.png", "formula": "\\begin{align*} \\min \\ f ( x ) + \\theta ( F ( x ) ) { \\rm s . t . } \\ h ( x ) = 0 \\ , , \\ g ( x ) \\in { \\cal S } ^ p _ + \\ , , \\end{align*} % \\end{align*}"}
-{"id": "7292.png", "formula": "\\begin{align*} B ' : = \\left ( \\begin{array} { c c c c c c c } \\ell _ { 2 , 2 } & \\ldots & \\ell _ { 2 , n - 1 } \\\\ \\ell _ { 2 , 3 } & \\ldots & \\ell _ { 3 , n - 1 } \\end{array} \\right ) \\end{align*}"}
-{"id": "6093.png", "formula": "\\begin{align*} \\int _ { B ^ n } h ( x ) ^ 2 d x = \\int _ { B ^ n } \\cos ( g ( r ) ) ^ 2 { f _ { \\frak { e } _ k } ( x / r ) _ j } ^ 2 d x \\leq \\int _ { B ^ n } d x < \\infty . \\end{align*}"}
-{"id": "8723.png", "formula": "\\begin{align*} & u ^ { \\varepsilon , \\delta } ( \\hat { t } , \\hat { x } ) = u ^ \\varepsilon ( t ' , \\hat { x } ) - \\delta ^ { - 1 } | \\hat { t } - t ' | ^ 2 \\quad \\\\ & u ^ \\varepsilon ( t ' , \\hat { x } ) = u ( t ' , x ' ) - \\varepsilon ^ { - 1 } | \\hat { x } - x ' | ^ 2 . \\end{align*}"}
-{"id": "9146.png", "formula": "\\begin{align*} \\begin{array} { r c l } f _ { 2 } & = & D _ { 1 } \\\\ 2 D _ { 1 } + D _ { 0 } + f _ { 1 } & = & 0 , \\end{array} \\end{align*}"}
-{"id": "8734.png", "formula": "\\begin{align*} \\varphi _ { \\delta , \\sigma } ( t , x , s , y ) & = \\varphi ( t , x , y ) + | t - \\bar { t } | ^ 2 + | x - \\bar { x } | ^ 4 + | y - \\bar { y } | ^ 4 + \\frac { | t - s | ^ 2 } { \\delta } \\\\ & + | t - t _ \\delta | ^ 2 + | x - x _ \\delta | ^ 4 + | y - y _ \\delta | ^ 4 + a _ \\sigma t + \\zeta _ \\sigma \\cdot x - b _ \\sigma s - \\xi _ \\sigma \\cdot y \\end{align*}"}
-{"id": "568.png", "formula": "\\begin{align*} \\begin{array} { l } \\beta _ 1 = - b t _ 1 \\leq \\alpha _ 1 , \\\\ \\beta _ j = b ( t _ { j - 1 } - t _ j ) \\leq \\alpha _ j , \\mbox { f o r } j = 2 , \\ldots , m - 1 , \\\\ \\beta _ m = b t _ m \\leq \\alpha _ m . \\end{array} \\end{align*}"}
-{"id": "1504.png", "formula": "\\begin{align*} g ( y ) = \\sum _ { n = 0 } ^ { \\infty } t _ { n } ( x ) y ^ { n } = \\frac { 3 - 2 x ^ { 2 } y - x y ^ { 2 } } { 1 - x ^ { 2 } y - x y ^ { 2 } - y ^ { 3 } } , \\end{align*}"}
-{"id": "119.png", "formula": "\\begin{align*} \\boldsymbol { \\mathcal { P } } \\left ( - n + W ( x , n ) \\right ) & \\times \\boldsymbol { \\mathcal { Q } } \\left ( \\# ( x , n ) \\right ) \\to \\mathbb { R } ^ { \\Gamma \\cap \\left ( - n + \\mathrm { I n t } _ { r _ 0 + N \\sqrt { k } } W ( x , n ) \\right ) } \\\\ & ( p , q ) \\mapsto \\mathbb { G } ( p , q ) | _ { \\Gamma \\cap \\left ( - n + \\mathrm { I n t } _ { r _ 0 + N \\sqrt { k } } W ( x , n ) \\right ) } \\end{align*}"}
-{"id": "76.png", "formula": "\\begin{align*} m ( T ( \\sigma ~ 0 1 ~ 1 0 ~ w ~ 1 1 ~ \\tau ) ) & = m ( T ( \\sigma ~ 0 1 ~ \\underline { 1 0 } ~ \\underline { 0 1 } ~ w ' ~ 1 1 ~ \\tau ) ) \\\\ & = m ( T ( \\sigma ~ 0 1 ~ 0 1 ~ 1 0 ~ w ' ~ 1 1 ~ \\tau ) ) \\end{align*}"}
-{"id": "8867.png", "formula": "\\begin{align*} F _ k ( T ) = 1 + ( \\Delta ^ + - k ) + ( \\Delta ^ + - k ) ( \\Delta ^ + ) + ( \\Delta ^ + - k ) ( \\Delta ^ + ) ^ 2 + \\ldots + ( \\Delta ^ + - k ) ( \\Delta ^ + ) ^ { r - 1 } . \\end{align*}"}
-{"id": "8925.png", "formula": "\\begin{gather*} \\begin{pmatrix} a \\phi _ { d _ 0 , d _ 1 } & b \\phi _ { d _ 0 , d _ 2 } \\\\ \\phi _ { d _ 3 , d _ 1 } & \\phi _ { d _ 3 , d _ 2 } \\end{pmatrix} \\ ! \\colon \\ { \\cal E } _ { d _ 1 } \\times { \\cal E } _ { d _ 2 } \\to { \\cal E } _ { d _ 0 } \\times { \\cal E } _ { d _ 3 } \\end{gather*}"}
-{"id": "4904.png", "formula": "\\begin{align*} ( T - \\lambda I ) = \\begin{bmatrix} 0 & B \\\\ 0 & D \\end{bmatrix} \\end{align*}"}
-{"id": "2189.png", "formula": "\\begin{align*} f _ j ( z ) = z ^ { \\beta ^ j } + Q _ j ( z ) , j = 1 , \\ldots , n , \\end{align*}"}
-{"id": "3841.png", "formula": "\\begin{align*} \\tau = R _ { \\mathcal { I } } . \\end{align*}"}
-{"id": "943.png", "formula": "\\begin{align*} f _ { n , k } = \\sum _ { i , j = 1 } ^ N \\gamma _ { n , k } ( i , j ) e _ i \\otimes e _ j . \\end{align*}"}
-{"id": "4775.png", "formula": "\\begin{align*} { L } _ 0 = \\alpha , { L } _ 1 = \\frac 1 { 2 \\alpha } ( \\beta t _ 1 - \\alpha ^ 2 \\omega _ 1 + \\gamma ) , \\end{align*}"}
-{"id": "5631.png", "formula": "\\begin{align*} e ^ { \\rm t r n c } ( k ; K _ r , \\omega ) \\ , \\le \\ , c _ r \\ , \\left ( C ( 4 , \\omega ) \\ , C ( 4 \\ , r - 6 , \\omega ) \\right ) ^ { 1 / 4 } \\left ( \\sum _ { j = k + 1 } ^ \\infty \\xi _ j \\right ) \\left ( \\sum _ { i = 1 } ^ \\infty \\xi _ i \\right ) ^ { r - 3 / 2 } , \\end{align*}"}
-{"id": "4200.png", "formula": "\\begin{align*} \\nabla _ { H } ( \\frac { d x } { y } ) = - \\frac { x d x } { y } , \\ \\nabla _ { H } ( \\frac { x d x } { y } ) = 0 . \\end{align*}"}
-{"id": "1594.png", "formula": "\\begin{align*} j _ i '' = \\begin{cases} 0 , & 1 \\leq i \\leq n - 3 \\\\ 0 , & i = n - 2 \\\\ 1 6 q + 7 & i = n - 1 \\\\ 2 & i = n , \\end{cases} \\end{align*}"}
-{"id": "4544.png", "formula": "\\begin{align*} \\frac { 1 } { ( \\lambda - \\lambda _ 1 ^ { ( 0 ) } ) \\prod _ { n = 1 } ^ { N - 1 } ( \\lambda - \\lambda _ n ^ { ( 0 ) } ) } = \\\\ = \\frac { 1 } { ( \\lambda - \\lambda _ 1 ^ { ( 0 ) } ) } \\sum _ { n = 1 } ^ { N - 1 } \\frac { 1 } { \\lambda - \\lambda _ n ^ { ( 0 ) } } \\frac { 1 } { \\prod _ { n ' \\neq n } ( \\lambda _ { n } ^ { ( 0 ) } - \\lambda _ { n ' } ^ { ( 0 ) } ) } \\end{align*}"}
-{"id": "4637.png", "formula": "\\begin{align*} M : = \\sum _ { n = 0 } ^ { 2 ^ { k - 1 } - 1 } 1 _ { [ - B - C , B + C ] } \\left ( t + \\sum _ { i = 0 } ^ { n - 1 } f ( T ^ i x ) \\right ) \\end{align*}"}
-{"id": "5983.png", "formula": "\\begin{align*} P _ 1 = \\max \\left \\{ \\exp \\left ( \\left ( \\frac { \\gamma ^ 2 } { 2 } - \\varepsilon ^ 2 \\right ) L ^ 2 \\right ) , 1 \\right \\} , P _ 2 = \\max \\left \\{ \\exp \\left ( \\varepsilon ^ 2 L ^ 2 \\left ( \\frac { \\varepsilon ^ 2 } { \\gamma ^ 2 } - 1 \\right ) \\right ) , 1 \\right \\} . \\end{align*}"}
-{"id": "6475.png", "formula": "\\begin{align*} d s _ { 3 D c } ^ { 2 } = \\frac { 1 } { 1 - \\rho ^ { 2 } } \\frac { d \\mu _ { x } ^ { 2 } } { \\sigma _ { x } ^ { 2 } } + \\frac { 2 - \\rho ^ { 2 } } { 1 - \\rho ^ { 2 } } \\frac { d \\sigma _ { x } ^ { 2 } } { \\sigma _ { x } ^ { 2 } } + \\frac { 2 - \\rho ^ { 2 } } { 1 - \\rho ^ { 2 } } \\frac { d \\sigma _ { y } ^ { 2 } } { \\sigma _ { y } ^ { 2 } } - \\frac { 2 \\rho ^ { 2 } } { 1 - \\rho ^ { 2 } } \\frac { d \\sigma _ { x } d \\sigma _ { y } } { \\sigma _ { x } \\sigma _ { y } } \\end{align*}"}
-{"id": "546.png", "formula": "\\begin{align*} \\lambda _ 0 : = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ n \\bigl ( p _ j \\ , d q _ j - q _ j \\ , d p _ j \\bigr ) . \\end{align*}"}
-{"id": "5.png", "formula": "\\begin{align*} f ( t ) = \\frac { a _ { - 1 } } t + a _ 0 + a _ 1 t + a _ 2 t ^ 2 + \\cdots + a _ { n - 1 } t ^ { n - 1 } . \\end{align*}"}
-{"id": "8252.png", "formula": "\\begin{align*} \\abs { W _ { \\infty , s } ( a ( q ) n ( x ) a ( y ) ) } = \\abs { q y } _ { \\infty } ^ { \\frac { 1 } { 2 } } \\prod _ { \\nu } \\frac { \\abs { K _ { [ F _ { \\nu } \\colon \\R ] ( s + i t _ { \\nu } ) } ( [ F _ { \\nu } \\colon \\R ] 2 \\pi \\abs { q _ { \\nu } y _ { \\nu } } ) } } { \\abs { \\Gamma _ { \\nu } ( 2 ( s + i t _ { \\nu } ) + 1 ) } } . \\end{align*}"}
-{"id": "2067.png", "formula": "\\begin{align*} \\mathcal { L } _ t = \\Delta _ H - \\partial _ t \\end{align*}"}
-{"id": "3233.png", "formula": "\\begin{align*} ( Z _ 2 \\circ Z _ 1 ) _ \\ast ( \\alpha ) = \\langle Z _ 2 \\rangle _ \\ast \\circ \\langle Z _ 1 \\rangle _ \\ast ( \\alpha ) . \\end{align*}"}
-{"id": "9984.png", "formula": "\\begin{align*} \\| A _ N \\cdots A _ 1 \\| _ { 0 } ^ { r } \\leq d ^ { ( N + 1 ) ( d - 1 ) } C _ 1 \\left ( \\prod _ { i = 1 } ^ { N } { \\| A _ i \\| _ { 0 } } \\right ) ^ { r } \\cdot \\Lambda . \\end{align*}"}
-{"id": "1325.png", "formula": "\\begin{align*} \\rho ( P _ v ) = \\begin{bmatrix} \\pi ( P _ v ) & X _ v \\\\ Y _ v & Z _ v \\end{bmatrix} \\forall v \\in V , \\end{align*}"}
-{"id": "5139.png", "formula": "\\begin{align*} u = \\sum _ { \\l = 1 } ^ { k } z _ { 2 \\l - 1 } + v . \\end{align*}"}
-{"id": "1152.png", "formula": "\\begin{align*} \\ell _ 1 & = \\ell _ 0 \\\\ \\ell _ j & = \\frac { [ j ] _ x \\ell _ { j - 1 } \\ell _ { j - 2 } } { [ 1 ] _ x ^ { 2 ^ { j - 1 } } \\ell _ { j - 2 } + \\ell _ { j - 1 } } \\quad \\mbox { f o r $ j \\geq 2 $ } . \\end{align*}"}
-{"id": "2331.png", "formula": "\\begin{align*} H _ { i } ( \\textbf { f } , \\mathcal { A } ) = \\lim _ { n \\rightarrow + \\infty } \\frac { 1 } { n } H \\left ( \\bigvee _ { k = 0 } ^ { n - 1 } ( \\textbf { f } _ { i } ^ { \\ , k } ) ^ { - 1 } ( \\mathcal { A } ) \\right ) \\end{align*}"}
-{"id": "8103.png", "formula": "\\begin{align*} \\hat { g } _ { i , t } ( x ) = \\min \\{ 1 , \\delta ^ { - 1 } d ( x , X \\setminus t V _ i ) \\} . \\end{align*}"}
-{"id": "8100.png", "formula": "\\begin{align*} \\mu ( A ) = \\lim _ { j \\to \\infty } \\mu ( A _ { n _ j } ) \\geq \\lim _ { j \\to \\infty } \\mu ( B _ { n _ j } ) = \\mu ( B ) , \\end{align*}"}
-{"id": "9137.png", "formula": "\\begin{align*} f ( x ^ { 5 } ) - 5 x f ( x ^ { 4 } ) + 1 0 x ^ { 2 } f ( x ^ { 3 } ) - 1 0 x ^ { 3 } f ( x ^ { 2 } ) + 5 x ^ { 4 } f ( x ) = 0 \\left ( x \\in R \\right ) . \\end{align*}"}
-{"id": "5199.png", "formula": "\\begin{align*} \\frac { d } { { d } x } V _ { 1 } ^ { [ r , 1 ] } ( \\ell ) = f _ { 1 } ' ( \\ell ) = \\frac { g _ { 1 } ( r ) - f _ { 1 } ( \\ell ) } { r - \\ell } . \\end{align*}"}
-{"id": "7643.png", "formula": "\\begin{align*} B _ 0 & = I \\otimes I \\otimes I , \\ \\ \\ \\ \\ \\ \\ \\ B _ 4 = I \\otimes P \\otimes P , \\\\ B _ 1 & = I \\otimes I \\otimes P , \\ \\ \\ \\ \\ \\ \\ B _ 5 = P \\otimes I \\otimes P , \\\\ B _ 2 & = I \\otimes P \\otimes I , \\ \\ \\ \\ \\ \\ B _ 6 = P \\otimes P \\otimes I , \\\\ B _ 3 & = P \\otimes I \\otimes I , \\ \\ \\ \\ \\ \\ B _ 7 = P \\otimes P \\otimes P . \\end{align*}"}
-{"id": "6040.png", "formula": "\\begin{align*} R _ { \\mathrm { s h } } = R ^ { * } = C _ { \\mathsf { W y n e r } } ( X ; Y ) . \\end{align*}"}
-{"id": "4666.png", "formula": "\\begin{align*} { \\rm q d i s t } ( U , V ) : = \\sup \\limits _ { u \\in U } \\inf \\limits _ { v \\in V } \\| u - v \\| \\end{align*}"}
-{"id": "10038.png", "formula": "\\begin{align*} \\gamma _ { \\Q _ p } ( - D , \\psi _ p ) = \\begin{cases} ( p , - D ) _ p \\delta _ p & \\\\ 1 & \\end{cases} \\end{align*}"}
-{"id": "7886.png", "formula": "\\begin{align*} C _ k = - \\sup _ { 0 \\le p \\le - k / 2 } H ( p ) , \\end{align*}"}
-{"id": "9419.png", "formula": "\\begin{align*} C _ n ^ { ( 4 ) } = c ^ { \\rm ( i n ) } _ { n , 4 } + 2 \\cdot c ^ { \\rm ( o u t ) } _ { n , 2 } + \\sum _ { k = 1 } ^ n \\dfrac { n } { k } \\cdot \\mathfrak { d } ^ { \\rm ( i n ) } _ { n - k , k } + \\sum _ { k = 1 } ^ { n - 3 } \\sum _ { l = 1 } ^ { n - k - 2 } \\dfrac { 1 } { 2 } \\cdot \\dfrac { n } { k \\cdot l } \\cdot \\tilde { s } ^ { ( 2 ) } _ { n - k - l , k , l } , \\end{align*}"}
-{"id": "4067.png", "formula": "\\begin{align*} q ^ \\infty ( v ) = \\begin{cases} f ^ \\infty ( 1 ) , & \\ ; v = 1 , \\\\ - f ^ \\infty ( - 1 ) , & \\ ; v = - 1 , \\end{cases} \\end{align*}"}
-{"id": "9376.png", "formula": "\\begin{align*} P ( \\theta ) \\quad : \\ , = \\sum _ { \\{ j \\ , : \\ , \\alpha _ j \\le 2 m \\} } c _ j \\ , \\phi _ j ( \\theta ) \\qquad \\phi ( \\theta ) \\quad : \\ , = \\sum _ { \\{ j \\ , : \\ , \\alpha _ j > 2 m \\} } c _ j \\ , \\phi _ j ( \\theta ) \\ , . \\end{align*}"}
-{"id": "42.png", "formula": "\\begin{align*} \\hat { P } ( C _ { 1 } = C _ { 2 } ) = \\int \\limits _ { - \\infty } ^ { \\infty } \\int \\limits _ { - \\infty } ^ { \\infty } \\int \\limits _ { - \\infty } ^ { \\infty } \\int \\limits _ { - \\infty } ^ { \\infty } \\hat { f } _ { X Y Z S } ( x , y , z , s ) \\delta ( x - y ) \\delta ( z - s ) \\mathrm { d } x \\mathrm { d } y \\mathrm { d } z \\mathrm { d } s \\end{align*}"}
-{"id": "1849.png", "formula": "\\begin{align*} F _ { \\bullet , m } = \\cdots \\to F _ { 1 , m } \\to F _ { 0 , m } \\to F _ { - 1 , m } \\to \\cdots \\end{align*}"}
-{"id": "5644.png", "formula": "\\begin{align*} A _ K ( \\gamma ) : = \\sup \\ ; \\sum _ { i = 0 } ^ { N - 1 } \\ ; \\left ( \\inf \\limits _ { t _ i \\leq t \\leq t _ { i + 1 } } K ( \\gamma ( t ) ) \\right ) \\ d ( \\gamma ( t _ i ) , \\gamma ( t _ { i + 1 } ) ) \\in [ 0 , + \\infty ] , \\end{align*}"}
-{"id": "9522.png", "formula": "\\begin{align*} I _ { X / Y } = I _ X \\times _ { I _ Y } X . \\end{align*}"}
-{"id": "4956.png", "formula": "\\begin{align*} m _ { x y } & = m _ x m _ y , \\\\ h _ { \\{ x , y \\} } & = h _ x h _ y - h _ y h _ x , \\\\ h _ { x y } & = m _ y h _ x + m _ x h _ y , \\\\ m _ { \\{ x , y \\} } & = h _ x m _ y - m _ y h _ x = [ h _ x , m _ y ] , \\\\ m _ 1 & = 1 . \\end{align*}"}
-{"id": "7828.png", "formula": "\\begin{align*} n I _ n & = \\left ( \\frac { 1 } { \\sqrt { m + 1 } } A + i \\sqrt { \\frac { m } { m + 1 } } B \\right ) \\left ( \\frac { 1 } { \\sqrt { m + 1 } } A + i \\sqrt { \\frac { m } { m + 1 } } B \\right ) ^ * , \\end{align*}"}
-{"id": "6799.png", "formula": "\\begin{align*} J _ { \\rho } ( u ) = \\frac { 1 } { 2 } \\int _ { \\mathbb { S } ^ 2 } | \\nabla u | ^ 2 - \\rho \\ln { \\left ( \\int _ { \\mathbb { S } ^ 2 } e ^ { u } \\right ) } + \\frac { \\rho } { 4 \\pi } \\int _ { \\mathbb { S } ^ 2 } u \\end{align*}"}
-{"id": "5178.png", "formula": "\\begin{align*} g _ { 1 } ( r ) \\frac { x - \\ell _ { r } } { r - \\ell _ { r } } - g _ { 1 , [ r , 1 ] } ( x ) & = g _ { 1 } ( r ) \\left ( \\frac { x - \\ell _ { r } } { r - \\ell _ { r } } - \\frac { x } { r } \\right ) \\\\ & = g _ { 1 } ( r ) \\left ( \\frac { r ( x - \\ell _ { r } ) - x ( r - \\ell _ { r } ) } { r ( r - \\ell _ { r } ) } \\right ) \\\\ & = - g _ { 1 } ( r ) \\left ( \\frac { \\ell _ { r } } { r } \\right ) \\left ( \\frac { r - x } { r - \\ell _ { r } } \\right ) = - g _ { 1 , [ r , 1 ] } ( \\ell _ { r } ) \\frac { r - x } { r - \\ell _ { r } } . \\end{align*}"}
-{"id": "8461.png", "formula": "\\begin{align*} \\abs { W _ { \\pi } ( g _ { - n , 0 , v } ) } = \\abs { \\epsilon ( \\frac { 1 } { 2 } , \\tilde { \\pi } ) } = 1 . \\end{align*}"}
-{"id": "6036.png", "formula": "\\begin{align*} & \\Lambda _ { i } ^ { ( \\alpha , \\lambda ) } ( \\{ Q _ { j } \\} _ { j = 1 } ^ { i } ) \\\\ & = \\sum _ { x _ { i } , y _ { i } , u _ { i } , v _ { i } } P ^ { ( \\alpha , \\lambda ) } ( x _ { i } , y _ { i } , u _ { i } , v _ { i } ) g _ { Q _ { i } , P _ { i } } ^ { ( \\alpha , \\lambda ) } ( x _ { i } , y _ { i } | u _ { i } , v _ { i } ) . \\end{align*}"}
-{"id": "5795.png", "formula": "\\begin{align*} \\langle \\omega , u \\rangle = \\int _ { \\mathbb { R } ^ n } u \\ ; d \\omega = \\Vert u \\Vert _ { \\dot { W } _ { 0 } ^ { 1 , p } ( \\mathbb { R } ^ n ) } ^ { p } = \\Vert \\omega \\Vert _ { W ^ { - 1 , p ' } ( \\mathbb { R } ^ n ) } ^ { p ' } = \\Vert I _ { 1 } \\omega \\Vert _ { L ^ { p ' } ( \\mathbb { R } ^ n ) } ^ { p ' } = \\mathcal { E } _ { 1 , p } ( \\omega ) \\end{align*}"}
-{"id": "2659.png", "formula": "\\begin{align*} { \\rm S c a l } _ { \\tilde g } = { \\rm c o n s t } . \\end{align*}"}
-{"id": "6601.png", "formula": "\\begin{align*} \\lim _ { J _ 2 \\ni m \\to \\infty } \\int _ { \\varphi _ n ( \\mathsf { U } _ n ) } \\left | w _ { i j } ^ { ( m , n ) } \\right | ^ p \\ , d \\mu ( x ) = \\int _ { \\varphi _ n ( \\mathsf { U } _ n ) } \\left | u _ { i j } ^ { ( n ) } ( x ) \\right | ^ p \\ , d \\mu ( x ) \\ . \\end{align*}"}
-{"id": "5859.png", "formula": "\\begin{align*} \\Sigma ( E ) = \\left [ \\begin{array} { c c } \\sigma ^ 2 _ { 1 } & \\sigma _ { 1 2 } \\\\ \\sigma _ { 2 1 } & \\sigma ^ 2 _ 2 \\end{array} \\right ] . \\end{align*}"}
-{"id": "4691.png", "formula": "\\begin{align*} \\frac { d } { d r } \\bigg ( \\int _ { B _ r } \\varphi _ j \\ , d x \\bigg ) = \\int _ { \\partial B _ r } \\varphi _ j \\ , d \\mathcal { H } ^ { n - 1 } . \\end{align*}"}
-{"id": "8545.png", "formula": "\\begin{align*} P = a _ 1 ^ 3 a _ 2 ^ 2 + a _ 1 ^ 2 a _ 2 ^ 3 + ( a _ 1 ^ 2 a _ 2 + a _ 1 a _ 2 ^ 2 ) b c _ 1 + ( a _ 1 ^ 2 + a _ 2 ^ 2 ) c _ 1 ^ 2 + 2 a _ 1 ^ 2 a _ 2 ^ 2 c \\leqslant 0 , \\end{align*}"}
-{"id": "2341.png", "formula": "\\begin{align*} h = h _ 1 \\cdots h _ { b ( h ) } \\end{align*}"}
-{"id": "5941.png", "formula": "\\begin{align*} { \\rm A r f } ( V _ { \\phi } ) = \\frac { m ( m - 1 ) } { 2 } \\cdot \\mu ^ { - 1 } + ( m - 1 ) \\cdot \\lambda ^ { - 1 } + \\frac { \\mu } { \\lambda ^ 2 } ~ ~ ~ ~ ( { \\rm m o d } \\ ; N ) . \\end{align*}"}
-{"id": "4935.png", "formula": "\\begin{align*} D = \\left ( \\begin{array} { c c } 0 & D ^ 1 \\\\ D ^ 0 & 0 \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "7456.png", "formula": "\\begin{align*} \\mathbb { H } _ 3 \\rightarrow H \\cong \\mathbb { C } [ T ] : = \\mathbb { H } _ 3 [ T ] / ( t , t _ 1 ^ b , t _ 1 ^ c , T - t _ 0 ^ b , T - t _ 0 ^ c , T - t _ 0 ^ d ) . \\end{align*}"}
-{"id": "3001.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } ( t , x ) & = \\Delta _ b u ( t , x ) + f ( t , u ( t , x ) ) + g ( t , u ( t , x ) ) \\xi ( t , x ) , \\\\ u ( 0 , x ) & = u _ 0 ( x ) . \\end{align*}"}
-{"id": "6770.png", "formula": "\\begin{align*} G ( y , p ) = - \\frac { 1 } { 4 \\pi } \\ln { \\left ( \\frac { 4 | x | ^ 2 } { ( 1 + | x | ^ 2 ) } \\right ) } , \\end{align*}"}
-{"id": "8677.png", "formula": "\\begin{align*} \\left . \\frac { d ^ n } { d t ^ n } P ( t ) _ { i j } \\right | _ { t = 0 } = ( Q ^ n ) _ { i j } \\end{align*}"}
-{"id": "5806.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u = \\sigma u ^ { q } + \\mu \\ ; \\ ; \\Omega , \\\\ u = 0 \\ ; \\ ; \\ ; \\ ; \\partial \\Omega . \\end{cases} \\end{align*}"}
-{"id": "5798.png", "formula": "\\begin{align*} - \\Delta _ { p } w _ { 1 } = \\omega _ { 0 } \\ ; \\ ; \\R ^ n \\Vert w _ 1 \\Vert _ { \\dot { W } ^ { 1 , p } _ { 0 } ( \\R ^ n ) } ^ { p - 1 } = \\Vert \\omega _ { 0 } \\Vert _ { W ^ { - 1 , p } ( \\R ^ n ) } \\end{align*}"}
-{"id": "2673.png", "formula": "\\begin{align*} \\tilde { M } _ { f ^ { ( 1 ) } , 2 } = \\begin{pmatrix} a & b \\\\ \\bar { b } & c \\end{pmatrix} \\end{align*}"}
-{"id": "9032.png", "formula": "\\begin{align*} Y _ { a b c } = 2 \\nabla _ { [ a } \\Phi _ { b ] c } - 2 J _ { c [ a } S _ { b ] } + 2 J _ { a b } S _ c \\end{align*}"}
-{"id": "1571.png", "formula": "\\begin{align*} \\gamma ( m , k ) : = \\gamma ( k ) = \\begin{cases} - c ( j _ k , - \\alpha ( k ) - 1 ) , & \\alpha ( k ) < 0 ; \\\\ \\infty , & 0 \\leq \\alpha ( k ) \\leq j _ k ; \\\\ c ( j _ k , \\alpha ( k ) - j _ k ) , & \\alpha ( k ) > j _ k , \\end{cases} \\end{align*}"}
-{"id": "2165.png", "formula": "\\begin{align*} a ( s ) & = c _ d \\int _ { I ( s ) } w \\xi ^ { d - 1 } \\ , d \\xi + O ( e ^ { - d \\theta _ * s } ) \\\\ & = c _ d \\int _ { ( 1 + t ) ^ { \\frac { 1 } { 2 } - \\theta _ * } } ^ \\infty u _ * ( r , t ) U _ d ( r ) r ^ { d - 1 } \\ , d r + O ( e ^ { - d \\theta _ * s } ) \\\\ & = \\frac { c _ d } { c _ * } \\int _ { ( 1 + t ) ^ { \\frac { 1 } { 2 } - \\theta _ * } } ^ \\infty u _ * ( r , t ) \\nu _ d ( r ) r ^ { d - 1 } \\ , d r + o ( 1 ) \\end{align*}"}
-{"id": "6498.png", "formula": "\\begin{align*} \\varrho \\left ( k \\right ) \\overset { } { = } \\frac { 4 i \\left ( k _ { \\mathrm { o } } - i \\sigma _ { k \\mathrm { o } } ^ { 2 } R _ { \\mathrm { o } } \\right ) k ^ { 2 } f \\left ( k \\right ) } { \\sigma _ { k \\mathrm { o } } ^ { 2 } } , \\end{align*}"}
-{"id": "7901.png", "formula": "\\begin{align*} u ( y , t ) + u ( z , t ) \\geq 2 u _ 0 ( x _ t ) - 2 t L \\left ( { d ( x , x _ t ) \\over t } \\right ) = 2 u ( x , t ) . \\end{align*}"}
-{"id": "1699.png", "formula": "\\begin{align*} T _ { \\lambda \\nu } ( f ) ( x ) & = f _ { \\lambda \\nu } ( \\tau _ \\rho ( y ) ) \\cdot f ( y ) \\\\ & = \\begin{cases} 0 , & \\rho \\not = \\lambda \\nu \\\\ f _ { \\lambda \\nu } ( x ) \\cdot f ( y ) , & \\rho = \\lambda \\nu \\end{cases} \\\\ & = f _ \\lambda ( x ) \\cdot f _ \\nu ( \\tau _ \\nu ( y ) ) \\cdot f ( y ) \\end{align*}"}
-{"id": "7978.png", "formula": "\\begin{align*} \\cos \\alpha _ s ^ 0 & - \\cos \\alpha _ t ^ 0 \\\\ & \\le \\frac { a _ 2 ^ 2 + a _ 1 ^ 2 ( 1 + r ^ 2 ) - a _ 3 ^ 2 } { 2 a _ 2 a _ 1 ( 1 - r ^ 2 ) } - \\frac { a _ 2 ^ 2 + a _ 1 ^ 2 ( 1 - r ^ 2 ) - a _ 3 ^ 2 } { 2 a _ 2 a _ 1 ( 1 + r ^ 2 ) } \\\\ & = \\frac { r ^ 2 ( 2 a _ 1 ^ 2 + a _ 2 ^ 2 - a _ 3 ^ 2 ) } { a _ 1 a _ 2 ( 1 - r ^ 2 ) ( 1 + r ^ 2 ) } \\\\ & = \\frac { r ^ 2 } { 1 - r ^ 4 } \\left ( \\frac { 2 a _ 1 } { a _ 2 } + \\frac { a _ 2 } { a _ 1 } - \\frac { a _ 3 ^ 2 } { a _ 1 a _ 2 } \\right ) \\\\ & \\le C ( \\delta ) r ^ 2 . \\end{align*}"}
-{"id": "903.png", "formula": "\\begin{align*} \\widehat { \\sigma } _ n ^ 2 ( t ) : = \\sum _ { i = 1 } ^ n K _ h ( t _ { i - 1 } - t ) ( X _ { t _ i } - X _ { t _ { i - 1 } } ) ^ 2 , t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "5146.png", "formula": "\\begin{align*} A _ 1 = L e _ 1 + \\l Q , A _ 2 = \\l Q , A _ 3 = N e _ 2 + \\l Q . \\end{align*}"}
-{"id": "6537.png", "formula": "\\begin{align*} ( ( 1 - \\Delta ) e _ n + e _ n ^ { \\perp } ) \\cap \\bigcap _ { i = 1 } ^ k \\{ x \\in \\mathbb { R } ^ n : \\langle x , y _ i \\rangle \\leq 1 \\} \\end{align*}"}
-{"id": "6283.png", "formula": "\\begin{align*} \\Lambda _ r \\leq & \\Lambda _ r ( c _ r \\nu ) ^ { - m } \\Lambda _ r ^ { m ( \\frac 3 r - 1 ) } \\| u _ Q \\| _ r ^ m \\leq ( c _ r \\nu ) ^ { - m } \\Lambda _ r ^ { m / 2 + 1 } \\| u _ Q \\| _ 2 ^ m \\\\ \\leq & ( c _ r \\nu ) ^ { - m } \\| u _ Q \\| _ { \\dot H ^ { 1 / 2 + 1 / m } } ^ m . \\end{align*}"}
-{"id": "8108.png", "formula": "\\begin{align*} h _ { i , t } = H ^ { - 1 } \\hat { h } _ { i , t } \\end{align*}"}
-{"id": "4324.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { \\delta } { 2 } \\frac { d } { d t } \\| \\nabla N _ { q } \\| _ { L ^ { 2 } ( \\Omega ) } ^ { 2 } + \\frac { 1 } { 2 } \\| q \\| _ { L ^ { 2 } ( \\Omega ) } ^ { 2 } + \\delta \\| \\nabla q \\| _ { L ^ { 2 } ( \\Omega ) } ^ { 2 } \\\\ & \\leq C \\big ( 1 + \\sum _ { i = p , d } \\big ( \\| \\nabla \\mu _ { i } \\| _ { L ^ { 2 } ( \\Omega ) } ^ { 2 } + \\| \\varphi _ { i } \\| _ { L ^ { 2 } ( \\Omega ) } ^ { 2 } \\big ) \\big ) . \\end{aligned} \\end{align*}"}
-{"id": "6236.png", "formula": "\\begin{align*} C ( 1 ) = \\xi _ C ( i ) \\sim \\left ( \\begin{array} { c c c c c } 0 & 1 & 2 & 3 & \\ldots \\\\ p _ 0 & p _ 1 & p _ 2 & p _ 3 & \\ldots \\\\ \\end{array} \\right ) ~ ~ , \\end{align*}"}
-{"id": "6483.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } \\theta ^ { j } } { d \\tau ^ { 2 } } - \\omega _ { j } ^ { 2 } \\theta ^ { j } = 0 , \\forall j = 1 , \\ldots , l \\end{align*}"}
-{"id": "6428.png", "formula": "\\begin{align*} \\det \\left [ g _ { \\mu \\nu } \\left ( \\theta \\right ) \\right ] = g \\left ( \\theta ^ { 1 } \\theta ^ { n } \\right ) = { \\displaystyle \\prod \\limits _ { \\kappa = 1 } ^ { n } } g _ { \\kappa } \\left ( \\theta ^ { \\kappa } \\right ) \\end{align*}"}
-{"id": "3540.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l l } \\beta ^ { n , 0 , \\theta } = \\displaystyle \\frac { \\big { [ } J ( u ^ { n , \\theta } ( \\cdot ) ) + \\theta \\big { ] } ^ + } { J ^ { n , \\theta } ( u ^ { n , \\theta } ( \\cdot ) ) } , \\\\ \\beta ^ { n , i , \\theta } = \\displaystyle \\frac { - \\big { [ } - X ^ { n , \\theta } ( t _ i ) \\big { ] } ^ + } { J ^ { n , \\theta } ( u ^ { n , \\theta } ( \\cdot ) ) } , \\ i = 1 , 2 , \\cdots , n , \\\\ \\end{array} \\end{align*}"}
-{"id": "4364.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\inf _ { j \\in \\mathbb { N } } \\operatorname { d i s t } \\big ( \\operatorname { s u p p } ( \\varphi _ { j , t } ) , \\operatorname { s u p p } ( 1 - \\psi _ { j , t } ) \\big ) \\geq \\lim _ { t \\to 0 } \\frac { 2 } { t } = \\infty . \\end{align*}"}
-{"id": "554.png", "formula": "\\begin{align*} H = h ( \\rho ) \\forall \\rho \\geqslant 1 , \\end{align*}"}
-{"id": "2607.png", "formula": "\\begin{align*} H _ V ( t _ 1 , t _ 2 ) = \\overline { C _ c ^ \\infty ( t _ 1 , t _ 2 ) } ^ { \\| \\cdot \\| _ V } . \\end{align*}"}
-{"id": "5552.png", "formula": "\\begin{align*} w _ { n } \\left ( t \\right ) w _ { m } \\left ( t \\right ) = w _ { n \\oplus m } \\left ( t \\right ) \\end{align*}"}
-{"id": "1981.png", "formula": "\\begin{align*} \\sigma _ { i } : = \\max _ { p \\in M _ { i } } \\sigma _ { p } \\leq \\min \\left \\{ \\frac { ( \\lambda ^ { - 1 } - \\lambda ) \\alpha } { 2 ( 1 + \\alpha ) ^ { 2 } } , \\frac { \\lambda ^ { - 1 } ( 1 - \\alpha ) - ( 1 + \\alpha ) \\alpha } { 2 ( 1 + \\alpha ) } \\right \\} . \\end{align*}"}
-{"id": "9138.png", "formula": "\\begin{align*} a _ { 5 - i } = ( - 1 ) ^ { i } \\binom { 5 } { i } \\left ( i = 0 , 1 , 2 , 3 , 4 \\right ) . \\end{align*}"}
-{"id": "3782.png", "formula": "\\begin{align*} & \\P \\left ( h ^ T ( Y _ { R _ k } ) > T '' , B \\right ) = \\sum _ { y _ 1 , y _ 2 \\in \\Z ^ d \\times \\Z } \\P \\left ( h ^ T ( y _ 2 ) > T '' , Y _ { R _ k } = y _ 2 , Y _ { R _ { k - T ' } } = y _ 1 , B _ { y _ 1 } \\right ) . \\end{align*}"}
-{"id": "9244.png", "formula": "\\begin{align*} \\mathfrak { U } = \\left \\{ \\left [ \\begin{array} { c c } M & \\chi _ { \\underline { \\beta } } \\\\ \\underline { \\beta } ^ { t } & N \\end{array} \\right ] \\mid M \\in M _ { n } ( \\mathfrak { a } ) , \\ ( \\eta M ) ^ { t } + M = 0 , \\ \\underline { \\beta } \\in \\mathcal { B } ^ { n } , \\ N \\in \\mathfrak { U } ( \\chi ) \\right \\} , \\end{align*}"}
-{"id": "3995.png", "formula": "\\begin{align*} p ^ { \\nu _ 1 } _ { k - 1 } ( 1 , t ) = \\frac { ( - \\lambda _ 1 t ^ { \\nu _ 1 } ) ^ { k - 1 } } { \\Gamma ( ( k - 1 ) { \\nu _ 1 } + 1 ) } . \\end{align*}"}
-{"id": "9400.png", "formula": "\\begin{align*} M _ { \\Omega , t } f ( x ) & = \\frac 1 { t ^ n } \\int _ { | y | < t } \\Omega _ 0 ( y ' ) f ( x - y ) d y + C ( \\Omega , n ) \\frac 1 { t ^ n } \\int _ { | y | < t } f ( x - y ) d y \\\\ & : = M _ { \\Omega _ 0 , t } f + C ( \\Omega , n ) M _ t f , \\end{align*}"}
-{"id": "5618.png", "formula": "\\begin{align*} \\int _ 0 ^ T ( Y _ r , Y _ r ' ) d \\mathbf { B } _ r ^ { S t r a t } = \\int _ 0 ^ T ( Y _ r , Y _ r ' ) d \\mathbf { B } _ r ^ { I t o } + \\frac { 1 } { 2 } \\int _ 0 ^ T Y _ r ' d r . \\end{align*}"}
-{"id": "8395.png", "formula": "\\begin{align*} & R _ 1 = \\sum \\limits _ { \\alpha , \\beta } \\left ( \\sum \\limits _ { i , j } h _ { i j \\alpha } h _ { i j \\beta } \\right ) ^ 2 + \\sum \\limits _ { i , j , \\alpha , \\beta } \\left ( \\sum \\limits _ p h _ { i p \\alpha } h _ { j p \\beta } - h _ { j p \\alpha } h _ { i p \\beta } \\right ) ^ 2 \\\\ [ 5 p t ] & R _ 2 = \\sum \\limits _ { i , j } \\left ( \\sum \\limits _ { \\alpha } H _ { \\alpha } h _ { i j \\alpha } \\right ) ^ 2 . \\end{align*}"}
-{"id": "710.png", "formula": "\\begin{align*} R _ 1 \\cdot \\nabla \\lambda _ 1 = R _ 2 \\cdot \\nabla \\lambda _ 2 = \\frac { \\partial c } { \\partial \\rho } + \\frac { c } { \\rho } = \\frac { p _ { v v } } { 2 \\rho ^ 3 \\sqrt { - p _ v } } . \\end{align*}"}
-{"id": "1385.png", "formula": "\\begin{align*} \\hat q ( 1 ) = 1 , \\hat q ( x ) = ( 1 - P _ { 0 } ) ^ { - 1 } \\sum _ { k = 1 } ^ { x - 1 } P _ { k } \\hat q ( x - k ) , x \\ge 2 . \\end{align*}"}
-{"id": "182.png", "formula": "\\begin{align*} \\mathrm { T h } ( a ) + ^ { \\vdash } \\mathrm { T h } ( c ) = \\mathrm { T h } ( a + c ) \\subseteq \\mathrm { T h } ( b + c ) = \\mathrm { T h } ( b ) + ^ { \\vdash } \\mathrm { T h } ( c ) . \\end{align*}"}
-{"id": "7369.png", "formula": "\\begin{align*} { \\cal { L } } _ K \\psi = \\nabla _ K \\psi + \\frac { 1 } { 4 } d \\widetilde { K } . \\psi . \\end{align*}"}
-{"id": "3880.png", "formula": "\\begin{align*} g ( \\omega ; s ; k ) = \\pi ( - 1 ) ^ k \\int _ { 0 } ^ { \\infty } J _ { 2 k - 1 } ( x ) \\frac { f ( \\omega ; s ; x ) } { x } d x , \\end{align*}"}
-{"id": "3442.png", "formula": "\\begin{align*} U _ { \\varepsilon } ^ { - 1 } ( A ) & : = \\{ \\omega \\in \\Omega \\ , : \\ , x \\in A h ( \\omega , x ) \\in B _ { \\varepsilon } ( 0 ) \\} \\\\ & = \\bigcup _ { x \\in A } \\{ \\omega \\in \\Omega \\ , : \\ , h ( \\omega , x ) \\in B _ { \\varepsilon } ( 0 ) \\} \\end{align*}"}
-{"id": "5695.png", "formula": "\\begin{align*} \\overline { { v } _ n } ( s ) = \\begin{cases} P [ { v } _ n ] ( s ) & s \\geq 0 , \\\\ \\mathcal R _ n ( P [ { v } _ n ] ( - s ) ) & s < 0 , \\end{cases} \\end{align*}"}
-{"id": "9194.png", "formula": "\\begin{align*} \\mathfrak { g } : = V ( \\omega _ { 1 } + \\omega _ { n - 1 } ) , \\ V : = V ( \\omega _ { 1 } ) , \\ S : = V ( 2 \\omega _ { 1 } ) , \\ \\Lambda : = V ( \\omega _ { 2 } ) T : = V ( 0 ) . \\end{align*}"}
-{"id": "9312.png", "formula": "\\begin{align*} f ( x + \\epsilon ) - f ( x ) = f ( y + \\epsilon ) - f ( y ) 0 \\leq \\epsilon < \\delta . \\end{align*}"}
-{"id": "10020.png", "formula": "\\begin{align*} [ \\widehat { \\theta } ( g ) : \\mathcal { Y } _ \\mathrm { s m } ] = - \\deg _ \\C ( \\mathcal { Y } _ \\mathrm { s m } ) \\cdot L ' ( \\tilde { g } , \\theta _ \\Lambda , 0 ) . \\end{align*}"}
-{"id": "106.png", "formula": "\\begin{align*} K _ 0 = \\sup _ { t \\in \\mathbb { R } ^ k } \\sum _ { \\lambda \\in \\Gamma _ 1 } | \\chi _ 0 ( t - \\lambda ) | . \\end{align*}"}
-{"id": "3317.png", "formula": "\\begin{align*} \\tilde { r } _ i = \\frac { 1 } { 1 + N + N ^ 2 + \\cdots + N ^ { K - i } } , i = 1 , \\cdots , K - 1 . \\end{align*}"}
-{"id": "9203.png", "formula": "\\begin{align*} [ \\lambda ' \\otimes e ' , x ^ { + } \\otimes a ^ { - } ] & = \\lambda ' \\diamond x ^ { + } \\otimes \\frac { [ e ' , a ^ { - } ] _ { E ' } } { 2 } + [ \\lambda ' , x ^ { + } ] \\otimes \\frac { ( e ' \\circ a ^ { - } ) _ { C ' } } { 2 } , \\\\ { } [ \\lambda ' \\otimes e ' , x ^ { - } \\otimes a ^ { + } ] & = \\lambda ' \\diamond x ^ { - } \\otimes \\frac { [ e ' , a ^ { + } ] _ { C ' } } { 2 } + [ \\lambda ' , x ^ { - } ] \\otimes \\frac { ( e ' \\circ a ^ { + } ) _ { E ' } } { 2 } . \\end{align*}"}
-{"id": "5204.png", "formula": "\\begin{align*} \\Gamma _ { 1 } ( x , y ) & = f _ { 1 } ( x ) + f _ { 1 } ' ( x ) ( y - x ) - g _ { 1 } ( y ) \\\\ \\Gamma _ { 2 } ( x , y ) & = f _ { 2 } ( y ) + f _ { 2 } ' ( y ) ( x - y ) - g _ { 2 } ( x ) \\end{align*}"}
-{"id": "397.png", "formula": "\\begin{align*} \\sigma _ n ^ 2 = { \\rm V a r } ( S _ n ) = \\sum _ { i = 1 } ^ { \\infty } b _ { n i } ^ { 2 } \\sim c _ \\alpha n ^ { 3 - 2 \\alpha } L ^ { 2 } ( n ) , \\end{align*}"}
-{"id": "8207.png", "formula": "\\begin{align*} M ( L , j , \\delta ) = \\sum _ { \\alpha _ 1 , \\alpha _ 2 \\in \\mathcal { P } ( L ) } \\sharp \\left \\{ \\gamma \\in \\Gamma ( i , \\alpha _ 1 ^ j \\alpha _ 2 ^ j ) \\colon u _ v ( \\gamma _ v P _ v , P _ v ) \\leq \\delta _ v v \\right \\} . \\end{align*}"}
-{"id": "7971.png", "formula": "\\begin{align*} \\Pi _ { m = 0 } ^ { \\infty } \\left ( 1 - \\nu ^ { 2 m } r ^ 2 \\right ) \\frac { a _ { i } } { a _ { j } } < \\frac { a _ { I , i } } { a _ { I , j } } < \\Pi _ { m = 1 } ^ { \\infty } \\left ( 1 + \\nu ^ { 2 m } r ^ 2 \\right ) \\frac { a _ { i } } { a _ { j } } . \\end{align*}"}
-{"id": "5449.png", "formula": "\\begin{align*} f ^ * ( \\lambda ) = \\limsup _ { x \\to \\infty } \\frac { f ( \\lambda x ) } { f ( x ) } , f _ * ( \\lambda ) = \\liminf _ { x \\to \\infty } \\frac { f ( \\lambda x ) } { f ( x ) } . \\end{align*}"}
-{"id": "5079.png", "formula": "\\begin{align*} & d \\Phi = [ \\frac { 9 b _ 1 b _ 2 C _ 3 ^ 2 } { ( b _ 1 - b _ 3 ) ^ 2 ( b _ 2 - b _ 3 ) ^ 2 } - R _ { 1 2 1 2 } - R _ { 1 3 1 3 } - R _ { 2 3 2 3 } ] \\omega _ 1 \\wedge \\omega _ 2 \\wedge \\omega _ 3 \\\\ & + [ \\frac { 9 b _ 2 b _ 3 C _ 1 ^ 2 } { ( b _ 1 - b _ 2 ) ^ 2 ( b _ 1 - b _ 3 ) ^ 2 } + \\frac { 9 b _ 1 b _ 3 C _ 2 ^ 2 } { ( b _ 1 - b _ 2 ) ^ 2 ( b _ 2 - b _ 3 ) ^ 2 } ] \\omega _ 1 \\wedge \\omega _ 2 \\wedge \\omega _ 3 . \\end{align*}"}
-{"id": "8023.png", "formula": "\\begin{align*} & \\Big ( \\int _ { 0 } ^ 1 { \\int _ { \\Omega } { \\Big \\Vert \\Big ( \\sum _ { l = M } ^ { L - M } { \\big | \\sum _ { n = 1 } ^ { L - l } { 2 ^ { - { s } { t _ n } } { \\Gamma } _ { { t _ l } } \\ast h _ { { n + l } } ^ { \\omega , t } } \\big | ^ q } \\Big ) ^ { { 1 } / { q } } \\Big \\Vert _ { L ^ p } ^ p } d \\lambda } d t \\Big ) ^ { { 1 } / { p } } \\\\ & \\gtrsim L ^ { - ( { s } + d - { d } / { q } ) / ( 2 d ) } \\big ( \\log { L } \\big ) ^ { { 1 } / { 2 } } . \\end{align*}"}
-{"id": "2488.png", "formula": "\\begin{align*} \\left ( - i \\xi _ { 1 } \\left | \\eta \\right | + L \\right ) \\psi _ { j } \\left ( \\left | \\eta \\right | \\right ) = \\varrho _ { j } \\left ( \\left | \\eta \\right | \\right ) \\psi _ { j } \\left ( \\left | \\eta \\right | \\right ) , \\end{align*}"}
-{"id": "946.png", "formula": "\\begin{align*} U _ { p , i } = \\sum _ { \\begin{subarray} { c } j , k = 1 \\\\ j \\neq i , k \\neq i \\end{subarray} } ^ N \\gamma _ p ( j , k ) W ^ { ( i ) } _ j W ^ { ( i ) } _ k , V _ { p , i } = 2 \\sum _ { j = 1 } ^ N \\gamma _ p ( i , j ) W ^ { ( i ) } _ { j } \\end{align*}"}
-{"id": "9318.png", "formula": "\\begin{align*} f ( x ) = ( f _ 1 ( x ) , \\ldots , f _ m ( x ) ) x \\in \\R ^ n . \\end{align*}"}
-{"id": "9769.png", "formula": "\\begin{align*} \\tilde { E } _ { i } ( h , \\theta ) = & \\int \\limits _ { y _ { i } + 2 n _ { i - 1 } s } ^ { y _ { i } } \\big ( W ( U _ { h , \\theta } ( i h - , y ) ) - W ( U _ { h , \\theta _ i } ( i h - , y ) ) \\big ) d y \\\\ & + \\int \\limits _ { y _ { i } + ( 2 n _ { i - 1 } + d _ i ) s } ^ { y _ { i } + 2 n _ { i - 1 } s } W ( U _ { h , \\theta } ( i h - , y ) ) d y - \\int \\limits _ { y _ { i } + 2 ( n _ { i - 1 } - 1 ) s } ^ { y _ { i } + 2 n _ { i - 1 } s } W ( U _ { h , \\theta _ i } ( i h - , y ) ) d y . \\end{align*}"}
-{"id": "3040.png", "formula": "\\begin{align*} b ( v , p ( t ) ) = - ( u _ t ( t ) , v ) _ H + a ( u ( t ) , v ) , \\forall v \\in V , \\end{align*}"}
-{"id": "2732.png", "formula": "\\begin{align*} & \\displaystyle B ( | v - v _ { \\ast } | , \\cos \\theta , z ) = \\phi ( | v - v _ { \\ast } | ) \\ , b ( \\cos \\theta , z ) , \\phi ( \\xi ) = C _ { \\phi } \\ , \\xi ^ { \\gamma } , \\ , \\gamma \\in [ 0 , 1 ] , \\\\ [ 4 p t ] & \\displaystyle \\forall \\eta \\in [ - 1 , 1 ] , | b ( \\eta , z ) | \\leq C _ b , \\ , | \\partial _ { \\eta } b ( \\eta , z ) | \\leq C _ b , \\ , \\ , | \\partial _ z ^ k b ( \\eta , z ) | \\leq C _ b ^ { \\ast } , \\ , \\forall \\ , 0 \\leq k \\leq r \\ , . \\end{align*}"}
-{"id": "8877.png", "formula": "\\begin{align*} \\begin{aligned} & J _ { \\theta , c } \\in \\mathcal L ( \\mathcal K _ \\theta , L ^ 2 ( \\mathbb T , \\sigma _ c ) ) , \\\\ & ( J _ { \\theta , c } f ) ( \\zeta ) = f ( \\zeta ) \\ \\ \\ \\zeta \\in \\mathbb T \\sigma _ c , \\ \\ f \\in \\mathcal K _ \\theta , \\end{aligned} \\end{align*}"}
-{"id": "4267.png", "formula": "\\begin{align*} \\mathrm { d i v } ( ( x - a ) ^ p ( x - a _ p ) ) & = p ( Q _ 1 ) + p ( - Q _ 1 ) + ( Q _ p ) + ( - Q _ p ) - ( 2 p + 2 ) ( \\infty ) \\\\ & = \\mathrm { d i v } ( \\alpha \\overline { \\alpha } ) \\end{align*}"}
-{"id": "8456.png", "formula": "\\begin{align*} A _ 2 = \\left ( \\begin{matrix} \\Re ( \\frac { b } { x ^ 2 } ) & \\zeta \\Im ( \\frac { b } { x ^ 2 } ) \\\\ \\zeta \\Im ( \\frac { b } { x ^ 2 } ) & \\zeta \\Re ( \\frac { b } { x ^ 2 } ) \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "4559.png", "formula": "\\begin{align*} & \\theta ^ { - 1 } ( H _ { i , 1 } ) = - ( - d ) ^ { - i } [ [ \\cdots [ [ \\cdots [ F _ { 0 , 0 } , F _ { n - 1 , 0 } ] _ { q } , \\cdots , F _ { i + 1 , 0 } ] _ { q } , F _ { 1 , 0 } ] _ { q } , \\cdots , F _ { i - 1 , 0 } ] _ { q } , F _ { i , 0 } ] _ { q ^ { 2 } } \\ , , \\\\ & \\theta ^ { - 1 } ( H _ { i , - 1 } ) = - ( - d ) ^ { i } [ E _ { i , 0 } , [ E _ { i - 1 , 0 } , \\cdots , [ E _ { 1 , 0 } , [ E _ { i + 1 , 0 } , \\cdots , [ E _ { n - 1 , 0 } , E _ { 0 , 0 } ] _ { q ^ { - 1 } } , \\cdots ] _ { q ^ { - 1 } } ] _ { q ^ { - 1 } } , \\cdots ] _ { q ^ { - 1 } } ] _ { q ^ { - 2 } } \\ , \\end{align*}"}
-{"id": "51.png", "formula": "\\begin{align*} a = a ' = 2 b = \\frac { ( x _ { i } + y _ { i } ) } { 2 } b ' = \\frac { ( z _ { i } + s _ { i } ) } { 2 } c = \\frac { ( x _ { i } - y _ { i } ) ^ 2 + ( z _ { i } - s _ { i } ) ^ 2 } { 2 } \\end{align*}"}
-{"id": "5782.png", "formula": "\\begin{align*} - \\Delta _ { p } u = \\sigma u ^ { q } + \\mu \\ ; \\ ; \\Omega \\end{align*}"}
-{"id": "1082.png", "formula": "\\begin{align*} \\mathbf { H } = \\begin{bmatrix} \\alpha _ { 1 1 } & \\dots & \\alpha _ { N 1 } \\\\ \\vdots & \\ddots & \\vdots \\\\ \\alpha _ { 1 N } & \\dots & \\alpha _ { N N } \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "5951.png", "formula": "\\begin{align*} D _ { X } \\varphi ( Y ) - D _ { Y } \\varphi ( X ) - \\varphi ( [ X , Y ] ) = 0 \\end{align*}"}
-{"id": "8105.png", "formula": "\\begin{align*} B _ { i , k } = \\bigg ( \\bigcap _ { t \\in F ^ k } t S _ i \\bigg ) \\setminus \\bigcap _ { s \\in F ^ { k + 1 } } t S _ i . \\end{align*}"}
-{"id": "3335.png", "formula": "\\begin{align*} D ^ * ( r ) \\geq ( 1 - r ) \\sum _ { j = 0 } ^ { K + 1 - k } \\frac { 1 } { N ^ j } - r \\sum _ { j = 0 } ^ { K - k } \\frac { K + 1 - k - j } { N ^ j } \\end{align*}"}
-{"id": "5658.png", "formula": "\\begin{align*} \\mathfrak { E } _ { W } ( { v } , I ) = \\int _ I \\left ( \\frac 1 2 | \\dot { { v } } ( t ) | ^ 2 + W ( { v } ( t ) ) \\right ) \\d t . \\end{align*}"}
-{"id": "2178.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } w ( \\xi , s ) = c _ d m ( \\varphi ) e ^ { - \\frac { \\xi ^ 2 } { 4 } } . \\end{align*}"}
-{"id": "974.png", "formula": "\\begin{align*} & \\sup _ { t \\in [ a _ n , T - a _ n ] } \\left | \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\Psi _ h ( t _ { i - 1 } - t ) - \\int _ { - \\infty } ^ \\infty \\Psi _ h ( s ) d s \\right | \\\\ & \\leq \\sup _ { t \\in [ a _ n , T - a _ n ] } \\left \\{ \\sum _ { i = 1 } ^ n \\int _ { t _ { i - 1 } } ^ { t _ i } | \\Psi _ h ( t _ { i - 1 } - t ) - \\Psi _ h ( s - t ) | d s + \\int _ { - \\infty } ^ 0 | \\Psi _ h ( s - t ) | d s + \\int _ T ^ \\infty | \\Psi _ h ( s - t ) | d s \\right \\} \\\\ & = : \\mathbb { I } _ { n } + \\mathbb { I I } _ { n } + \\mathbb { I I I } _ { n } . \\end{align*}"}
-{"id": "6184.png", "formula": "\\begin{align*} \\mathcal { T } ' = 0 \\ ; \\ ; , \\ ; \\ ; \\mathcal { T } '' \\leq 0 \\end{align*}"}
-{"id": "6679.png", "formula": "\\begin{align*} G ( ( n ^ { - 1 / p } , \\dots , n ^ { - 1 / p } ) ) = c ( B _ p ^ n , n ) ( p - 1 ) ^ { \\frac { n - 1 } { n + 1 } } n ^ { \\frac { 2 n + p } { ( n + 1 ) p } } \\quad . \\end{align*}"}
-{"id": "3340.png", "formula": "\\begin{align*} \\left ( r _ s ^ { ( K ) } - r _ { s - 1 } ^ { ( K ) } \\right ) \\left ( \\bar { D } ( r _ { s } ^ { ( K + 1 ) } ) - \\bar { D } ( r _ s ^ { ( K ) } ) \\right ) = \\left ( r _ s ^ { ( K ) } - r _ s ^ { ( K + 1 ) } \\right ) \\left ( \\bar { D } ( r _ { s - 1 } ^ { ( K ) } ) - \\bar { D } ( r _ s ^ { ( K ) } ) \\right ) . \\end{align*}"}
-{"id": "8860.png", "formula": "\\begin{align*} V ( X _ N , \\phi ) \\ll \\left ( \\sin \\phi \\right ) ^ { d - 1 } N ^ 2 \\ , N ^ { - \\frac { d + 1 } { d } } = \\left ( \\sin \\phi \\right ) ^ { d - 1 } N ^ { 1 - \\frac { 1 } { d } } \\end{align*}"}
-{"id": "4747.png", "formula": "\\begin{align*} { L } _ i = \\left ( 0 , 0 , \\alpha , \\beta \\right ) . \\end{align*}"}
-{"id": "6579.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\d ( K _ n , K ) = 1 , \\end{align*}"}
-{"id": "9292.png", "formula": "\\begin{align*} D '' : = \\{ d \\in D ' \\ : \\ \\Delta _ { h } ^ { k - 1 } \\Delta _ { d } ^ 1 f ( x ) \\geq 0 \\hbox { f o r a l l } ( h , x ) \\in S _ { J , k - 1 } \\} . \\end{align*}"}
-{"id": "2838.png", "formula": "\\begin{align*} p _ 1 ( x + y ) & \\leq p _ 1 ( x ) + p _ 1 ( y ) - 2 \\delta _ 1 ( p _ 1 ( y ) p _ 1 ( u - v ) ) = p _ 1 ( x ) + p _ 1 ( y ) - 2 \\delta _ 1 ( p _ 1 ( \\tau x - y ) ) \\\\ & \\leq p _ 1 ( x ) + p _ 1 ( y ) - 2 \\delta _ 1 ( \\| \\tau x - y \\| / 2 ) \\leq p _ 1 ( x ) + p _ 1 ( y ) - 2 \\delta _ 1 ( \\| x - y \\| / 4 ) \\\\ & = p _ 1 ( x ) + p _ 1 ( y ) - 2 \\delta _ 1 ( t / 4 ) . \\end{align*}"}
-{"id": "2672.png", "formula": "\\begin{align*} c _ { \\ell , j } \\sim \\begin{cases} \\mathcal { N } _ \\R ( 0 , 1 ) & 2 ( j - \\ell ) \\equiv 0 \\pmod { N } f ( j , \\ell ) = f ( \\ell , j ) \\\\ \\mathcal { N } _ \\C ( 0 , 1 ) & \\end{cases} . \\end{align*}"}
-{"id": "9253.png", "formula": "\\begin{align*} \\widehat { \\mathcal { L } ( \\mathfrak { b } ) } : = ( \\mathfrak { g ^ { + } } \\otimes A ^ { - } ) \\oplus ( \\mathfrak { g } ^ { - } \\otimes A ^ { + } ) \\oplus \\cdots \\oplus ( \\Lambda ' \\otimes E ' ) \\oplus \\left \\{ \\mathfrak { b } , \\mathfrak { b } \\right \\} \\end{align*}"}
-{"id": "7984.png", "formula": "\\begin{align*} W : = \\Delta _ { I _ 0 } = g _ { I _ 0 } ( \\Delta ) = f _ { I _ 0 } \\circ \\cdots f _ { i _ 1 i _ 2 } \\circ f _ { i _ 1 } ( \\Delta ) . \\end{align*}"}
-{"id": "8331.png", "formula": "\\begin{align*} \\zeta _ \\mu = e ^ { 2 \\pi i [ \\mu , k ] } . \\end{align*}"}
-{"id": "1256.png", "formula": "\\begin{align*} \\hat E _ m : = M _ j ( E _ m ) \\mbox { a n d } \\hat E : = M _ j ( E ) . \\end{align*}"}
-{"id": "8626.png", "formula": "\\begin{align*} \\Omega _ T = \\{ \\omega : \\omega \\in \\C ( [ 0 , T ] ) , \\omega ( 0 ) = 0 \\} \\end{align*}"}
-{"id": "4965.png", "formula": "\\begin{align*} \\sum _ { j \\in \\Z } \\hat { \\Delta } ( 2 ^ j x ) = 1 , \\forall x \\in \\R ^ d \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "9634.png", "formula": "\\begin{align*} \\zeta _ \\tau | _ { \\tau _ { i - 1 } } ^ { \\tau _ { i } } = \\max \\left ( N ( \\tau _ { i - 1 } ) , N ( \\tau _ { i } ) \\right ) , \\end{align*}"}
-{"id": "1012.png", "formula": "\\begin{align*} B _ { i + 1 } = \\textup { m i n } \\left \\{ B _ i + ( 1 - \\beta _ i ) E _ h ^ i - \\beta _ i p _ s ^ { i } , B _ { m a x } \\right \\} , \\forall i , \\end{align*}"}
-{"id": "8401.png", "formula": "\\begin{align*} \\frac { \\partial \\varphi } { \\partial t } ( x , t ) = - F ( W ( x , t ) ) \\nu ( x , t ) = - f ( \\lambda ( W ( x , t ) ) ) \\nu ( x , t ) \\end{align*}"}
-{"id": "1873.png", "formula": "\\begin{align*} \\mu \\circ { \\max } ^ { - 1 } ( ( - \\infty , i ] ) = \\mu ( A ^ i ) = \\overline { \\pi } ^ i = \\nu _ { \\max } ( ( - \\infty , i ] ) \\end{align*}"}
-{"id": "8468.png", "formula": "\\begin{align*} \\abs { W _ { \\pi } ( g _ { t , l , v } ) } = \\begin{cases} q ^ { - ( t + k ) } & t \\geq - k , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "780.png", "formula": "\\begin{align*} T _ k ( x ) = \\frac { B _ { k + 1 } ( 2 x + 1 ) - B _ { k + 1 } ( x + 1 ) } { k + 1 } \\end{align*}"}
-{"id": "4938.png", "formula": "\\begin{align*} \\displaystyle { \\not } D ^ 0 _ k = ( \\displaystyle { \\not } D ^ 0 _ k ) - ( \\displaystyle { \\not } D ^ 1 _ k ) \\end{align*}"}
-{"id": "8802.png", "formula": "\\begin{align*} G _ { \\varpi _ { p } } ( z , m ) = 0 , \\end{align*}"}
-{"id": "7952.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ k \\lambda _ { I _ - j } ^ s = \\lambda ^ s _ { I _ - } . \\end{align*}"}
-{"id": "2930.png", "formula": "\\begin{align*} \\phi _ m = \\phi _ { n , m } \\circ \\phi _ n \\forall m > n . \\end{align*}"}
-{"id": "2003.png", "formula": "\\begin{align*} \\sum _ { \\alpha \\in \\Z } ( - 1 ) ^ { \\alpha } q ^ { \\frac { \\alpha ( 3 \\alpha - 1 ) } { 2 } } G ( t ^ { 3 \\alpha } z ) G ( t ^ { 1 - 3 \\alpha } z ) = 0 . \\end{align*}"}
-{"id": "4291.png", "formula": "\\begin{align*} \\underline { \\lambda } = \\left ( ( c , \\Delta _ { A , \\tilde { \\ell } } ) , ( c _ i , \\Delta _ { D , \\tilde { \\ell _ i } } ) _ { i = 1 } ^ { r } \\right ) , \\underline { \\lambda } ^ { \\vee } = \\left ( ( c , \\Delta _ { A , \\tilde { \\ell } } ^ { \\vee } ) , ( c _ i , \\Delta ^ { \\vee } _ { D , \\tilde { \\ell _ i } } ) _ { i = 1 } ^ { r } \\right ) . \\end{align*}"}
-{"id": "8304.png", "formula": "\\begin{align*} \\mathrm { w t } _ { - 2 } V = 0 , \\mathrm { w t } _ { - 1 } V = I , \\mathrm { w t } _ 0 V = I ^ \\perp , \\mathrm { w t } _ 1 V = V , \\end{align*}"}
-{"id": "62.png", "formula": "\\begin{align*} \\frac { \\partial J _ { M C C C } } { \\partial w ^ { * } } = \\frac { \\partial E _ { D Y } [ G ^ { C } _ { \\sigma \\ , \\sqrt { 2 } } ( e ) ] } { \\partial w ^ { * } } = E _ { D Y } \\left [ G ^ { C } _ { \\sigma \\ , \\sqrt { 2 } } ( e ) \\frac { \\partial ( e e ^ * ) } { \\partial w ^ { * } } \\right ] = \\textbf { 0 } \\end{align*}"}
-{"id": "7617.png", "formula": "\\begin{align*} E _ h ( Q _ R , \\lambda ) : = \\Big \\{ z \\in Q _ R : \\ , h ( z ) > \\lambda \\Big \\} . \\end{align*}"}
-{"id": "9308.png", "formula": "\\begin{align*} C _ 1 : = \\{ ( h , t , x ) : ( h , t , x ) \\in D , \\Delta ^ { k + 2 } _ h f _ x ( t ) \\geq 0 \\} \\end{align*}"}
-{"id": "6360.png", "formula": "\\begin{align*} I _ 1 ( p ) : = W _ 1 ( p ) - W _ 1 ( p _ 1 ^ * ) , \\ , p \\in ( 0 , 1 ) , I _ 1 ( p ) : = \\infty , \\ , p \\in \\{ 0 , 1 \\} , \\end{align*}"}
-{"id": "9749.png", "formula": "\\begin{align*} L ^ 1 ( J ) + L ^ 2 ( J ) - L ^ 1 ( I ) - L ^ 2 ( I ) & \\leq ( K ^ { * } _ { 1 1 } | K _ { 1 1 } | + K ^ { * } _ { 2 5 } | K _ { 1 5 } | - K ^ { * } _ { 1 5 } ) | \\alpha _ 5 | - \\sum \\limits _ { i = 2 , 3 , 4 } K ^ { * } _ { 1 i } | \\alpha _ i | + M \\Delta '' ( \\boldsymbol { \\alpha } ^ { * } , \\beta _ 1 ) . \\end{align*}"}
-{"id": "7560.png", "formula": "\\begin{gather*} { \\bf u } = A ( t ) { \\bf x } + { \\bf b } ( t ) , \\end{gather*}"}
-{"id": "8365.png", "formula": "\\begin{align*} \\tilde { \\Gamma } _ \\Phi ^ { ( a ) } = s ( a ) \\tilde { K } _ \\Phi s ( a ) ^ { - 1 } \\cap U _ \\Phi ( \\Q ) \\subset K _ \\Phi \\cap U _ \\Phi ( \\Q ) \\end{align*}"}
-{"id": "3721.png", "formula": "\\begin{align*} f _ 1 = f _ 2 = f _ 4 = f _ 5 = f _ 7 = f _ 8 = \\omega ^ { - 1 } \\mbox { a n d } f _ 6 = 0 . \\end{align*}"}
-{"id": "139.png", "formula": "\\begin{align*} & h _ { z } ( x ) = \\Phi ( \\langle a _ { z } , x \\rangle + \\lambda b _ { z } + \\mu c _ { z } ) , \\\\ & f _ { z } ( x ) = \\Phi ( \\langle a _ { z } , x \\rangle + b _ { z } ) , \\\\ & g _ { z } ( x ) = \\Phi ( \\langle a _ { z } , x \\rangle + c _ { z } ) . \\end{align*}"}
-{"id": "2112.png", "formula": "\\begin{align*} \\gamma _ p ( s + \\psi ( p ) ) = \\Phi ( p , s + \\psi ( p ) ) = \\Phi ( q , s + \\psi ( q ) ) = \\gamma _ { q } ( s + \\psi ( q ) ) \\end{align*}"}
-{"id": "2220.png", "formula": "\\begin{align*} \\widetilde { F _ i } = \\widetilde { F _ i } ( w , t ) = \\widetilde q _ i ( w ) + t \\cdot \\widetilde Q _ i ( w ) , i = 1 , \\ldots , n . \\end{align*}"}
-{"id": "512.png", "formula": "\\begin{align*} \\mathcal { R } _ { } \\ ! = \\ ! & \\bigcup _ { P _ { U | X } } \\bigcup _ { \\widehat { X } ( Y , U ) } \\ ! \\Big \\{ \\left ( R _ w , D \\right ) \\ ! \\colon \\\\ & R _ w \\geq I ( U ; X ) - I ( U ; Y ) , \\\\ & D \\geq E [ d ( X , \\widehat { X } ( Y , U ) ) ] \\Big \\} \\end{align*}"}
-{"id": "3961.png", "formula": "\\begin{align*} \\partial _ t ^ { { \\alpha _ n } } f ( t ) : = \\left \\{ \\begin{array} { l l } \\frac { 1 } { \\Gamma { ( 1 - { \\alpha _ n } ) } } \\int ^ t _ { 0 } ( t - s ) ^ { - { \\alpha _ n } } f ' ( s ) \\ , \\mathrm { d } s , \\ \\ 0 < { \\alpha _ n } < 1 , \\\\ \\\\ f ' ( t ) , \\ \\ { \\alpha _ n } = 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "3747.png", "formula": "\\begin{align*} B _ { k , 0 } = \\left \\{ [ - 2 \\mathfrak { R } L _ k , 3 \\mathfrak { R } L _ k ) ^ d \\times [ 0 , L _ k ) \\right \\} \\cap ( \\Z ^ d \\times \\Z ) . \\end{align*}"}
-{"id": "4111.png", "formula": "\\begin{align*} \\mathrm { h e s s } _ b D _ a ( V , V ) | _ p = 0 \\ \\Leftrightarrow \\ k _ { M , p } ( V ) = \\frac { 1 } { D _ a ( p ) } \\end{align*}"}
-{"id": "137.png", "formula": "\\begin{align*} & h _ { \\lambda z _ 1 + \\mu z _ 2 } ( x ) = \\Phi ( \\langle a _ { z _ 1 , z _ 2 } , x \\rangle + \\lambda b _ { z _ 1 , z _ 2 } + \\mu c _ { z _ 1 , z _ 2 } ) , \\\\ & f _ { z _ 1 } ( x ) = \\Phi ( \\langle a _ { z _ 1 , z _ 2 } , x \\rangle + b _ { z _ 1 , z _ 2 } ) , \\\\ & g _ { z _ 2 } ( x ) = \\Phi ( \\langle a _ { z _ 1 , z _ 2 } , x \\rangle + c _ { z _ 1 , z _ 2 } ) . \\end{align*}"}
-{"id": "3508.png", "formula": "\\begin{align*} X ^ { u ^ 0 } ( s ) = x _ 0 + \\displaystyle \\int _ 0 ^ { s } b ( X ^ { u ^ 0 } ( t ) , u ^ 0 ( t ) ) d t \\end{align*}"}
-{"id": "2269.png", "formula": "\\begin{align*} \\nabla _ i \\nabla _ j H ^ \\alpha = & \\Delta h ^ \\alpha _ { i j } - g ^ { p q } \\Big ( \\nabla _ i R _ { j p } + \\nabla _ j R _ { i p } - \\nabla _ p R _ { i j } \\Big ) X ^ \\alpha _ q \\\\ & + 2 g ^ { k p } g ^ { l q } R _ { i k j l } h ^ \\alpha _ { p q } - g ^ { p q } R _ { i p } h ^ \\alpha _ { j q } - g ^ { p q } R _ { j p } h ^ \\alpha _ { i q } . \\end{align*}"}
-{"id": "2203.png", "formula": "\\begin{align*} \\bigl | q _ i ( z ) \\bigr | > \\bigl | t \\cdot Q _ i ( z ) \\bigr | , \\ i = 1 , \\ldots , n \\end{align*}"}
-{"id": "5514.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty ( 1 - e ^ { - s y } ) \\nu ( \\dd y ) = \\int _ 0 ^ \\infty e ^ { - s y } s \\overline \\nu ( y ) \\dd y = s \\widehat U ( s ) , \\end{align*}"}
-{"id": "5545.png", "formula": "\\begin{align*} W _ { H } = \\left [ \\begin{array} { c c c c c c c c } 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\ 1 & 1 & 1 & 1 & - 1 & - 1 & - 1 & - 1 \\\\ 1 & 1 & - 1 & - 1 & 1 & 1 & - 1 & - 1 \\\\ 1 & 1 & - 1 & - 1 & - 1 & - 1 & 1 & 1 \\\\ 1 & - 1 & 1 & - 1 & 1 & - 1 & 1 & - 1 \\\\ 1 & - 1 & 1 & - 1 & - 1 & - 1 & - 1 & 1 \\\\ 1 & - 1 & - 1 & 1 & 1 & - 1 & - 1 & 1 \\\\ 1 & - 1 & - 1 & 1 & - 1 & 1 & 1 & - 1 \\end{array} \\right ] \\end{align*}"}
-{"id": "7250.png", "formula": "\\begin{align*} d \\nu _ { \\pi } ^ { \\psi } / d \\mu _ { \\pi } = m u l t ( \\pi , \\textit { c - I n d } _ { U } ^ { G } \\psi ) , \\end{align*}"}
-{"id": "7505.png", "formula": "\\begin{align*} \\| \\widetilde { \\omega } _ { N , t } ^ k \\| _ { \\mathrm { H S } } = \\sqrt { N } \\ , \\| W _ N ^ k \\| _ 2 \\end{align*}"}
-{"id": "2893.png", "formula": "\\begin{align*} n \\in W _ e ^ { X ' } \\exists t \\ , \\forall s \\geq t \\ , ( \\Phi _ { e ' } ^ X ( n , s ) = 1 ) . \\end{align*}"}
-{"id": "4578.png", "formula": "\\begin{align*} A = [ [ \\cdots [ F _ { 2 , 1 } , F _ { 3 , 0 } ] _ { q } , \\cdots , F _ { n - 2 , 0 } ] _ q , F _ { n - 1 , 0 } ] _ { q } \\ , . \\end{align*}"}
-{"id": "8173.png", "formula": "\\begin{align*} V _ { \\pi } = \\mathcal { B } ( \\chi _ + , \\chi _ - ) . \\end{align*}"}
-{"id": "3958.png", "formula": "\\begin{align*} \\frac { \\partial Z _ { j } } { \\partial x _ S } = \\frac { { \\tilde g } _ S ' ( x _ S ) } { { \\tilde g } _ S ( x _ S ) } \\langle e _ { j \\ell } , e _ S \\rangle Z _ j \\end{align*}"}
-{"id": "5167.png", "formula": "\\begin{align*} \\forall x \\in [ 0 , a ] \\colon & \\begin{cases} f _ { 1 } ( x ) = 0 , \\\\ g _ { 2 } ( x ) = x ( x - 1 ) . \\end{cases} \\\\ \\forall y \\in [ b , 1 ] \\colon & \\begin{cases} f _ { 2 } ( y ) = 0 , \\\\ g _ { 1 } ( y ) = y ( 1 - y ) . \\end{cases} \\end{align*}"}
-{"id": "7272.png", "formula": "\\begin{align*} \\frac { 1 6 s ^ 2 } { N ^ 4 } \\sum _ { 1 \\le n , m \\le X } r ( n ) r ( m ) + 4 s ^ 2 = \\frac { 1 6 s ^ 2 } { N ^ 2 } \\sum _ { 1 \\le n \\le X } r ( n ) + O ( N ^ { - 1 } ) . \\end{align*}"}
-{"id": "2994.png", "formula": "\\begin{align*} d U ( t ) & = \\Delta _ b U ( t ) d t + \\beta _ t d t + \\sigma _ t d W ( t ) t \\in [ 0 , T ] , \\\\ U ( 0 ) & = 0 , \\end{align*}"}
-{"id": "5297.png", "formula": "\\begin{align*} & a _ 1 = a , a _ 2 = 1 , b _ 1 = b , b _ 2 = 1 , \\\\ & d _ 1 ( t ) = d ( t ) , d _ 2 ( t ) = 0 , M _ 1 = - \\bigg ( \\frac { m ^ 2 } { 4 \\mu } + n \\bigg ) , M _ 2 = 0 . \\end{align*}"}
-{"id": "8390.png", "formula": "\\begin{align*} \\widehat { \\mathrm { d i v } } ( \\psi ( f ) ) = ( \\mathrm { d i v } ( \\psi ( f ) ) , - \\log \\| \\psi ( f ) \\| ^ 2 ) = \\sum _ { \\substack { m > 0 \\\\ \\mu \\in V _ \\Z ^ \\vee / V _ \\Z } } c ( - m , \\mu ) \\cdot \\widehat { \\mathcal { Z } } ( m , \\mu ) . \\end{align*}"}
-{"id": "4581.png", "formula": "\\begin{align*} A & = ( - d ) ^ { n - 3 } [ [ \\cdots [ F _ { n - 1 , 1 } , F _ { n - 2 , 0 } ] _ { q } , \\cdots , F _ { 3 , 0 } ] _ { q } , F _ { 2 , 0 } ] _ { q } \\ , . \\end{align*}"}
-{"id": "4452.png", "formula": "\\begin{align*} \\lim _ { \\ell \\downarrow 0 } \\big ( [ \\sigma v _ \\ell - \\sigma v ] _ { \\frac { 3 } { 4 } - \\epsilon } + [ \\sigma ^ 2 F ^ \\ell - \\sigma ^ 2 F ] _ { - \\frac { 3 } { 4 } - \\epsilon } \\big ) = 0 , \\end{align*}"}
-{"id": "6769.png", "formula": "\\begin{align*} G ( y , y ' ) = - \\frac { 1 } { 2 \\pi } \\log { | y - y ' | } , \\end{align*}"}
-{"id": "6938.png", "formula": "\\begin{align*} y ^ 2 + ( - x ^ 3 - 1 ) y = - x ^ 5 - 3 x ^ 4 + 2 x ^ 2 + 2 x - 2 . \\end{align*}"}
-{"id": "6726.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\mathbb { P } \\Big ( S _ { N } > t N \\Big ) = e ^ { - t } . \\end{align*}"}
-{"id": "547.png", "formula": "\\begin{align*} X _ 0 : = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ n \\left ( p _ j \\frac { \\partial } { \\partial p _ j } + q _ j \\frac { \\partial } { \\partial q _ j } \\right ) \\ , = \\ , \\frac 1 2 \\ , r \\frac { \\partial } { \\partial r } . \\end{align*}"}
-{"id": "3391.png", "formula": "\\begin{align*} G _ { t , s } = \\rho _ U F ^ t + \\rho _ V F ^ s + ( 1 - \\rho _ U ) ( 1 - \\rho _ V ) F \\end{align*}"}
-{"id": "6121.png", "formula": "\\begin{align*} \\boldsymbol { \\tau } = \\left \\{ \\tau _ { n } > 0 : n \\in \\mathbb { Z } _ { + } \\right \\} , \\end{align*}"}
-{"id": "1919.png", "formula": "\\begin{align*} \\Vert \\textbf { F } \\Vert = \\sup _ { \\Vert ( \\zeta _ { i } ) \\Vert _ { \\infty } = 1 } \\Vert \\textbf { F } _ { i } ( \\zeta _ { i } ) \\Vert _ { \\infty } \\leq \\sup _ { i \\in \\mathbb { Z } } \\Vert D f _ { i } \\Vert . \\end{align*}"}
-{"id": "1428.png", "formula": "\\begin{align*} e _ k ( t _ 1 , \\dots , t _ { k } ) = \\begin{vmatrix} t _ 1 & \\cdots & t _ k \\\\ t _ 1 ^ 2 & \\cdots & t _ k ^ 2 \\\\ \\vdots & & \\vdots \\\\ t _ 1 ^ { 2 ^ { k - 1 } } & \\cdots & t _ k ^ { 2 ^ { k - 1 } } \\end{vmatrix} . \\end{align*}"}
-{"id": "5559.png", "formula": "\\begin{align*} \\dot { x } = - \\alpha \\int _ { 0 } ^ { t } \\bar { \\nu } ^ { T } \\bar { w } _ { 2 ^ { k } } \\left ( t _ { 1 } \\right ) d t _ { 1 } - \\beta \\int _ { 0 } ^ { t } \\bar { r } ^ { T } M _ { 2 ^ { k } } \\bar { \\nu } d t _ { 1 } + \\dot { x } _ { 0 } , t \\in [ 0 , \\tau ] \\end{align*}"}
-{"id": "1315.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\| \\varphi _ n ( a ) \\| \\leq \\limsup _ { n \\to \\infty } \\| \\varphi _ n ( 1 ) \\| \\| a \\| = 0 . \\end{align*}"}
-{"id": "4764.png", "formula": "\\begin{align*} { \\varepsilon } = F ( x ^ 2 , x ^ 3 , Y ) \\ , { \\Delta \\sqrt { - \\det g ^ { i j } } } / { { L } _ 1 ( x ^ 1 ) } , \\end{align*}"}
-{"id": "8556.png", "formula": "\\begin{align*} \\lim _ { t \\to + 0 } \\int _ { { \\bf R } ^ N } u ( y , t ) \\phi ( y ) \\ , d y = \\int _ { { \\bf R } ^ N } \\phi ( y ) \\ , d \\mu ( y ) , \\qquad \\phi \\in C _ 0 ( { \\bf R } ^ N ) . \\end{align*}"}
-{"id": "10125.png", "formula": "\\begin{align*} { \\boldsymbol { \\omega } } _ k ( I ) = \\boldsymbol S _ { D _ k } ( I ) { \\boldsymbol { \\bar { \\omega } } } _ k ( I ) . \\end{align*}"}
-{"id": "3711.png", "formula": "\\begin{align*} I _ { \\mathbf t ( \\tilde \\gamma ) } = \\{ i \\ , | \\ , t _ i ( \\tilde \\gamma ) = 0 \\} = \\{ i _ 1 < \\dots < i _ r \\} . \\end{align*}"}
-{"id": "9282.png", "formula": "\\begin{align*} \\boldsymbol { E } [ \\sum _ { i = 1 } ^ { T - 1 } \\sum _ { j = i + 1 } ^ T f ( j - i ) X _ i X _ j ] & \\leq \\boldsymbol { E } [ \\sum _ { i = 1 } ^ { T - 1 } \\sum _ { j = i + 1 } ^ T f ( j - i ) X _ i ] \\\\ & \\leq \\left ( \\boldsymbol { E } [ \\sum _ { i = 1 } ^ { T - 1 } X _ i ^ 2 ] \\boldsymbol { E } [ \\sum _ { i = 1 } ^ { T - 1 } \\left ( \\sum _ { j = i } ^ T f ( j - i ) \\right ) ^ 2 ] \\right ) ^ \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "7518.png", "formula": "\\begin{gather*} w ^ i w ^ 1 _ i - w ^ 1 _ { i i } - 2 \\kappa w ^ 2 - \\kappa z _ 1 = 0 , \\\\ w ^ i w ^ 2 _ i - w ^ 2 _ { i i } + 2 \\kappa w ^ 1 - \\kappa z _ 2 = 0 . \\end{gather*}"}
-{"id": "826.png", "formula": "\\begin{align*} \\mathcal { L } u = 0 , u ( x , 0 ) = u _ 0 , \\quad \\mbox { f o r e x a m p l e : } \\mathcal { L } = \\partial _ t - \\Delta . \\end{align*}"}
-{"id": "2640.png", "formula": "\\begin{align*} q _ i | Q | = \\sum \\limits _ { j = 1 } ^ { n } q _ j f o r i = 1 , \\ldots , n . \\end{align*}"}
-{"id": "4993.png", "formula": "\\begin{align*} \\Delta = \\sum _ { | \\gamma | = a } \\partial ^ { \\gamma } \\Delta ^ { ( \\gamma ) } . \\end{align*}"}
-{"id": "1986.png", "formula": "\\begin{align*} \\sup _ { \\lambda / n \\leq s \\leq 1 - \\lambda / n } \\frac { n ^ { \\eta } \\left \\vert \\beta _ { n } \\left ( s \\right ) - B _ { n } \\left ( s \\right ) \\right \\vert } { \\left [ s \\left ( 1 - s \\right ) \\right ] ^ { 1 / 2 - \\eta } } = O _ { \\mathbb { P } } \\left ( 1 \\right ) , \\end{align*}"}
-{"id": "3342.png", "formula": "\\begin{align*} \\left [ \\frac { \\binom { K - 2 } { s - 1 } } { L _ s ^ { ( K ) } } - \\frac { \\binom { K - 2 } { s - 2 } } { L _ { s - 1 } ^ { ( K ) } } \\right ] \\left [ \\frac { D _ s ^ { ( K + 1 ) } } { L _ s ^ { ( K + 1 ) } } - \\frac { D _ s ^ { ( K ) } } { L _ s ^ { ( K ) } } \\right ] = \\left [ \\frac { \\binom { K - 2 } { s - 1 } } { L _ s ^ { ( K ) } } - \\frac { \\binom { K - 1 } { s - 1 } } { L _ { s } ^ { ( K + 1 ) } } \\right ] \\left [ \\frac { D _ { s - 1 } ^ { ( K ) } } { L _ { s - 1 } ^ { ( K ) } } - \\frac { D _ s ^ { ( K ) } } { L _ s ^ { ( K ) } } \\right ] , \\end{align*}"}
-{"id": "3843.png", "formula": "\\begin{align*} \\partial ^ + \\mathcal { P } _ t ( y ) = \\{ z \\in \\Z ^ 2 \\setminus \\mathcal { P } _ t ( y ) \\colon \\ , z - ( 1 , 0 ) \\in \\mathcal { P } _ t ( y ) \\} . \\end{align*}"}
-{"id": "6941.png", "formula": "\\begin{align*} G _ n : = \\pi _ 1 ( M _ n , m _ n ) \\end{align*}"}
-{"id": "2443.png", "formula": "\\begin{align*} ( \\alpha \\beta ) ^ { \\alpha } = 1 - 3 \\epsilon . \\end{align*}"}
-{"id": "9202.png", "formula": "\\begin{align*} [ x ^ { + } \\otimes a ^ { - } , s \\otimes c ] & = x ^ { + } \\diamond s \\otimes \\frac { [ a ^ { - } , c ] _ { C } } { 2 } + [ x ^ { + } , s ] \\otimes \\frac { ( a ^ { - } \\circ c ) _ { E } } { 2 } , \\\\ { } [ x ^ { - } \\otimes a ^ { + } , s \\otimes c ] & = x ^ { - } \\diamond s \\otimes \\frac { [ a ^ { + } , c ] _ { E } } { 2 } + [ x ^ { - } , s ] \\otimes \\frac { ( a ^ { + } \\circ c ) _ { C } } { 2 } . \\end{align*}"}
-{"id": "4002.png", "formula": "\\begin{align*} p ^ { \\nu _ { m + 1 } } ( m + 1 , t ) & = ( - 1 ) ^ { m } \\frac { \\lambda _ 1 } { \\lambda _ { m + 1 } } \\sum _ { k = m } ^ { \\infty } ( - 1 ) ^ k \\underset { \\Lambda ^ { k } _ { m + 1 } } { \\sum } \\frac { \\lambda _ 1 ^ { k _ 1 } \\lambda _ 2 ^ { k _ 2 } \\ldots \\lambda _ { m + 1 } ^ { k _ { m + 1 } } t ^ { \\sum _ { j = 1 } ^ { m + 1 } k _ j \\nu _ j } } { \\Gamma \\left ( \\sum _ { j = 1 } ^ { m + 1 } k _ j \\nu _ j + 1 \\right ) } , \\end{align*}"}
-{"id": "9719.png", "formula": "\\begin{align*} \\tilde { T } - T = \\frac { ( \\gamma - 1 ) ( u ^ 2 - R T ) } { R \\rho u ( u ^ 2 - \\gamma R T ) } q _ 0 \\rho \\phi ( T ) Z h + O ( h ^ 2 ) \\geq 0 , \\end{align*}"}
-{"id": "7706.png", "formula": "\\begin{align*} D ^ 2 _ B ( j ) & = \\sum _ { k = 0 } ^ { N - 1 } d ^ 2 _ B ( j , k ) = \\sum _ { k = 0 } ^ { N - 1 } \\sum _ { i = 1 } ^ { N - 1 } \\frac { 1 } { \\lambda _ i ^ 2 } ( u _ { i j } - u _ { i k } ) ^ 2 \\\\ & = \\sum _ { i = 1 } ^ { N - 1 } \\sum _ { k = 0 } ^ { N - 1 } \\frac { u ^ 2 _ { i j } - 2 u _ { i j } u _ { i k } + u _ { i k } ^ 2 } { \\lambda _ i ^ 2 } \\\\ & = N \\sum _ { i = 1 } ^ { N - 1 } \\frac { u ^ 2 _ { i j } } { \\lambda _ i ^ 2 } + \\sum _ { i = 1 } ^ { N - 1 } \\frac { 1 } { \\lambda _ i ^ 2 } \\ , . \\end{align*}"}
-{"id": "9495.png", "formula": "\\begin{align*} T ( z , t ) \\cdot A ( z , t ) = t \\ , z \\left [ z \\ , D ( z , t ) - T _ 2 ( t ) \\right ] , \\end{align*}"}
-{"id": "4622.png", "formula": "\\begin{align*} \\frac { 1 } { | W _ i | } \\sum _ { n \\in W _ i } g ( S _ { f , B } ^ n ( x , t ) ) = \\frac { 1 } { | W _ i | } \\sum _ { n \\in U _ i } g ( S _ { f , B } ^ n ( x , t ) ) \\end{align*}"}
-{"id": "4588.png", "formula": "\\begin{align*} d \\omega = \\theta \\wedge \\omega . \\end{align*}"}
-{"id": "9037.png", "formula": "\\begin{align*} A H ( w , w ' ) = H ( B w , w ' ) + H ( w , B w ' ) \\end{align*}"}
-{"id": "6490.png", "formula": "\\begin{align*} p _ { B } ^ { } \\left ( x _ { B } | \\mu _ { B } \\right ) = \\frac { 1 } { \\mu _ { B } } \\exp \\left ( - \\frac { x _ { B } } { \\mu _ { B } } \\right ) , \\end{align*}"}
-{"id": "8482.png", "formula": "\\begin{align*} \\abs { W _ { \\pi } ( g _ { t , l , v } ) } = \\zeta _ F ( 1 ) q ^ { - \\frac { l } { 2 } } \\abs { \\sum _ { \\mu \\in \\mathfrak { X } _ l } \\epsilon ( \\frac { 1 } { 2 } , \\mu ^ { - 1 } \\chi ^ { - 1 } ) \\mu ( - v ) } . \\end{align*}"}
-{"id": "443.png", "formula": "\\begin{align*} p _ { 1 , k _ 1 , k _ 2 } ( x , t ) = \\frac { 1 } { ( 4 \\pi ) ^ n ( 2 \\pi ) ^ { m } } h _ { k _ 1 , k _ 2 } \\left ( R , t \\right ) \\end{align*}"}
-{"id": "7642.png", "formula": "\\begin{align*} T = \\big < A _ 1 , \\ldots , A _ { 2 ^ n - 1 } \\big > . \\end{align*}"}
-{"id": "1226.png", "formula": "\\begin{align*} \\mathcal { A } _ { i } = \\frac { \\partial f } { \\partial \\eta _ i } ( \\eta ) , \\ , \\ , 1 \\leq i \\leq n , \\ , \\mbox { w h e r e } \\ , \\ , f ( t \\eta ) = t ^ { p } f ( \\eta ) \\ , \\ , \\mbox { w h e n } \\ , \\ , t > 0 , \\ , \\ , \\eta \\in \\mathbb { R } ^ { n } \\setminus \\{ 0 \\} \\end{align*}"}
-{"id": "2024.png", "formula": "\\begin{align*} \\sum _ { \\substack { j \\in \\Z \\\\ 1 \\leq a \\leq l } } v _ { a , j , s } ( z _ 1 ^ { s - 2 j - k _ a } z _ 2 ^ j z _ 3 ^ j f _ a ( z _ 2 , z _ 3 ) + z _ 2 ^ { s - 2 j - k _ a } z _ 3 ^ j z _ 1 ^ j f _ a ( z _ 3 , z _ 1 ) + z _ 3 ^ { s - 2 j - k _ a } z _ 1 ^ j z _ 2 ^ j f _ a ( z _ 1 , z _ 2 ) ) = 0 . \\end{align*}"}
-{"id": "5064.png", "formula": "\\begin{align*} g = \\frac { 1 } { t ^ 2 } \\left [ 4 H _ u ^ 2 - 3 ( K _ u - 1 ) \\right ] ( I _ { \\mathbb { R } ^ 1 } + t ^ 2 I _ u ) = \\left [ 4 H _ u ^ 2 - 3 ( K _ u - 1 ) \\right ] ( I _ { \\mathbb { H } ^ 1 } + I _ u ) , \\end{align*}"}
-{"id": "8333.png", "formula": "\\begin{align*} \\zeta _ \\mu ^ { \\mathrm { r e c } ( a ) } = e ^ { 2 \\pi i [ \\tilde { \\mu } , k ] } = \\zeta _ \\mu . \\end{align*}"}
-{"id": "9636.png", "formula": "\\begin{align*} I ( \\Delta h _ \\beta ( \\tau _ { i } , l ) ) = \\begin{cases} 1 , \\ , \\ , \\Delta h _ \\beta ( \\tau _ { i } , l ) \\geq 0 \\\\ 0 , \\ , \\ , \\Delta h _ \\beta ( \\tau _ { i } , l ) < 0 . \\end{cases} \\end{align*}"}
-{"id": "393.png", "formula": "\\begin{align*} S _ r & = \\sum _ { n = 3 } ^ \\infty \\frac { 1 } { n ( \\ln n ) ^ r } \\mathbb P \\left ( | S _ n | > ( 1 + \\varepsilon ) \\sigma _ n \\sqrt { 2 ( 1 - r ) \\ln \\ln n } \\right ) \\\\ & \\propto \\sum _ { n = 3 } ^ \\infty \\frac { 1 } { n \\sqrt { \\ln \\ln n } ( \\ln n ) ^ { ( 1 + \\varepsilon ) ^ 2 ( 1 - r ) + r } } \\\\ & = \\sum _ { n = 3 } ^ \\infty \\frac { 1 } { n \\sqrt { \\ln \\ln n } ( \\ln n ) ^ { 1 + ( 2 \\varepsilon + \\varepsilon ^ 2 ) ( 1 - r ) } } . \\end{align*}"}
-{"id": "6856.png", "formula": "\\begin{align*} I = \\bigcup _ { i = 0 } ^ 3 F _ i ( I ) \\end{align*}"}
-{"id": "1276.png", "formula": "\\begin{align*} k ( t ) = \\mbox { C a p } _ { \\mathcal { A } } ( E _ 1 + t E _ 2 ) ^ { - 1 / ( n - p ) } \\sum _ { i = 1 } ^ m c _ i ( \\hat q _ i + a t ) \\end{align*}"}
-{"id": "3582.png", "formula": "\\begin{align*} h + s + \\ell ( \\sigma ) + ( 2 - k ) t = 1 . \\end{align*}"}
-{"id": "8404.png", "formula": "\\begin{align*} \\frac { d } { d t } \\psi = \\psi ^ 2 ( 1 + \\alpha - C \\rho _ - ( t _ 0 ) ^ { 1 + \\frac 1 \\alpha } \\psi ^ { \\frac 1 \\alpha } ) , t \\leq t _ 0 , \\end{align*}"}
-{"id": "4470.png", "formula": "\\begin{align*} \\sum _ { k \\not = 0 } \\langle | \\ell \\frac { \\partial } { \\partial \\ell } F _ T ^ \\ell ( k ) | ^ 2 \\rangle \\lesssim \\min \\{ ( T ^ \\frac { 1 } { 3 } ) ^ { - \\frac { 3 } { 2 } } , \\ell ^ 2 ( T ^ \\frac { 1 } { 3 } ) ^ { - \\frac { 7 } { 2 } } \\} . \\end{align*}"}
-{"id": "7309.png", "formula": "\\begin{align*} s _ { ( n + 2 - i ) ( n + 2 - i ) } = \\frac { ( - 1 ) ^ { n + i + 1 } } { c _ n ( n + 2 ) ^ n } \\frac { t _ 1 ^ { n + 2 } - t _ { n + 2 } } { s _ { i i } } , \\ 1 \\leq i \\leq m \\ , . \\end{align*}"}
-{"id": "5794.png", "formula": "\\begin{align*} - \\Delta _ { p } u = \\omega \\ ; \\ ; \\mathbb { R } ^ n \\end{align*}"}
-{"id": "8929.png", "formula": "\\begin{gather*} \\sum _ { x \\in \\ker \\phi _ { N , 1 } } x = 0 \\end{gather*}"}
-{"id": "5680.png", "formula": "\\begin{align*} \\begin{cases} ( - \\log f ) ' \\geq ( - \\log E ) ' - \\log f > - \\log E & p _ 0 = 2 ; \\\\ ( f ^ { 1 - p _ 0 / 2 } ) ' > ( E ^ { 1 - p _ 0 / 2 } ) ' f ^ { 1 - p _ 0 / 2 } > E ^ { 1 - p _ 0 / 2 } & p _ 0 > 2 . \\end{cases} \\end{align*}"}
-{"id": "4323.png", "formula": "\\begin{align*} - \\Delta N _ { q } ( t ) = q ( t ) \\Omega , N _ { q } ( t ) = 0 \\Gamma . \\end{align*}"}
-{"id": "2489.png", "formula": "\\begin{align*} \\big < \\Lambda g , \\Lambda g \\big > _ { \\xi } & = \\big | \\nabla _ { \\xi } \\cdot \\big [ \\sigma \\nabla _ { \\xi } g \\big ] \\big | ^ { 2 } _ { L ^ { 2 } _ { \\xi } } + | \\psi ( \\xi ) g | ^ { 2 } _ { L ^ { 2 } _ { \\xi } } \\\\ & + 2 \\big < \\psi ( \\xi ) , \\left ( \\nabla _ { \\xi } g , \\sigma \\nabla _ { \\xi } g \\right ) \\big > _ { \\xi } + 2 \\big < g , \\left ( \\nabla _ { \\xi } \\psi ( \\xi ) , \\sigma \\nabla _ { \\xi } g \\right ) \\big > _ { \\xi } \\ , . \\end{align*}"}
-{"id": "4661.png", "formula": "\\begin{align*} \\left | \\sum _ { n = 0 } ^ { N - 1 } 1 _ { [ a _ { i - 1 } , a _ i ) } ( T ^ n x ) - \\sum _ { n = 0 } ^ { N - 1 } 1 _ { [ a _ { i - 1 } , a _ i ) } ( T ^ n y ) \\right | & \\le \\frac { 4 } { c _ 2 } \\sqrt { N } \\sqrt { x _ i } + \\frac { x _ i } { | I _ k | } E _ 3 \\max \\{ 1 , ( | I _ k | N ) ^ { \\zeta _ 3 } \\} \\\\ & \\le \\frac { 4 \\sqrt { N } } { c _ 2 } + E _ 3 \\rho _ 2 \\sqrt { N } \\max \\left \\{ 1 , N ^ { \\frac { \\zeta _ 3 } 2 } \\right \\} \\end{align*}"}
-{"id": "2791.png", "formula": "\\begin{align*} ( \\mathcal { P } _ { n } f ) ( t ) = \\frac { \\sum _ { i = 0 } ^ { n - d } \\lambda _ { i } ( t ) p _ { i } ( t ) } { \\sum _ { i = 0 } ^ { n - d } \\lambda _ { i } ( t ) } , \\end{align*}"}
-{"id": "9879.png", "formula": "\\begin{align*} \\omega ( ( a , b , 0 ) , ( c , d , 1 ) ) & = \\omega ( ( a , b ) , ( c , d ) ) = \\exp \\left ( - \\frac { \\sqrt { - 1 } \\theta } { 2 } ( a d - b c ) \\right ) , \\\\ \\omega ( ( c , d , 1 ) , ( a , b , 0 ) ) & = \\omega ( ( c , d ) , ( - a - b , a ) ) = \\exp \\left ( - \\frac { \\sqrt { - 1 } \\theta } { 2 } ( c a + a d + b d ) \\right ) . \\end{align*}"}
-{"id": "9075.png", "formula": "\\begin{align*} f ( x + y ) = f ( x ) + f ( y ) \\left ( x , y \\in P \\right ) \\end{align*}"}
-{"id": "9191.png", "formula": "\\begin{align*} L = ( \\mathfrak { g } \\otimes A ) \\oplus ( V \\otimes B ) \\oplus ( V ' \\otimes B ' ) \\oplus ( S \\otimes C ) \\oplus ( S ' \\otimes C ' ) \\oplus ( \\Lambda \\otimes E ) \\oplus ( \\Lambda ' \\otimes E ' ) \\oplus D \\end{align*}"}
-{"id": "5295.png", "formula": "\\begin{align*} u _ x ( t , 1 ) & = \\left ( \\frac { m } { 2 \\mu } - a \\right ) u ( t , 1 ) , \\\\ u _ x ( t , 0 ) & = \\left ( \\frac { m } { 2 \\mu } - b \\right ) u ( t , 0 ) + d ( t ) , \\end{align*}"}
-{"id": "1988.png", "formula": "\\begin{align*} \\sup _ { c n ^ { - 1 } \\log n \\leq s < t } \\frac { \\left \\vert \\alpha _ { n } \\left ( s ; t \\right ) - \\widetilde { B } \\left ( s ; t \\right ) \\right \\vert } { s ^ { \\nu } } = o _ { \\mathbb { P } } \\left ( 1 \\right ) , \\end{align*}"}
-{"id": "1306.png", "formula": "\\begin{align*} T _ w : = T _ { s _ 1 } T _ { s _ 2 } \\cdots T _ { s _ k } , \\end{align*}"}
-{"id": "5258.png", "formula": "\\begin{align*} \\tilde { D } ( \\lambda ) = - \\det \\left [ \\begin{array} { c c } \\Omega ( U ( \\lambda , z ) , V ( \\lambda , z ) ) & \\Omega ( U ( \\lambda , z ) , a _ u ( \\lambda , z ) ) \\\\ \\Omega ( a _ s ( \\lambda , z ) , V ( \\lambda , z ) ) & \\Omega ( a _ s ( \\lambda , z ) , a _ u ( \\lambda , z ) ) \\end{array} \\right ] . \\end{align*}"}
-{"id": "9673.png", "formula": "\\begin{align*} W ( U ) _ x + H ( U ) _ y = 0 . \\end{align*}"}
-{"id": "1166.png", "formula": "\\begin{align*} T _ { k , i } = S _ { k - 2 , k - 2 - i } ( D _ { 2 } , D _ { 3 } , \\ldots , D _ { k - 1 } ) , \\end{align*}"}
-{"id": "2942.png", "formula": "\\begin{align*} \\varrho ( H _ n ) : = \\max _ { v \\in V ( H ) } \\partial \\theta _ n ( v ) , \\end{align*}"}
-{"id": "7233.png", "formula": "\\begin{align*} ( \\mathcal { W } _ { \\psi } f ) ( g ) = ( \\psi * _ { U } f ) ( g ) . \\end{align*}"}
-{"id": "6053.png", "formula": "\\begin{align*} \\left . \\frac { \\partial \\Omega ^ { ( \\alpha , \\theta ) } ( Q _ { X Y U } ) } { \\partial \\theta } \\right | _ { \\theta = 0 } = R ^ { ( \\alpha ) } ( Q _ { X Y U } ) & > 0 , \\\\ \\frac { \\partial ^ { 2 } \\Omega ^ { ( \\alpha , \\theta ) } ( Q _ { X Y U } ) } { \\partial \\theta ^ { 2 } } & \\leq 0 , \\end{align*}"}
-{"id": "207.png", "formula": "\\begin{align*} { \\vdash } = { \\models _ { \\mathbf { M o d } ( { \\vdash } ) } } = { \\models ' _ { \\mathbf { M o d } ( { \\vdash } ) } } . \\end{align*}"}
-{"id": "2051.png", "formula": "\\begin{align*} T M = H M \\oplus \\mathbb { R } T \\end{align*}"}
-{"id": "5958.png", "formula": "\\begin{align*} \\gamma ' ( t ) \\cdot \\vect { e } _ j ( t ) = 0 ( j = 2 , \\dots , n ) . \\end{align*}"}
-{"id": "9699.png", "formula": "\\begin{align*} \\gamma _ 1 = \\beta _ 1 + K _ { b 0 } \\omega _ { k } + K _ { b 2 } \\alpha _ 2 + K _ { b 3 } \\alpha _ 3 + K _ { b 5 } \\alpha _ 5 , \\end{align*}"}
-{"id": "1984.png", "formula": "\\begin{align*} F ^ { s } _ { p } = \\bigcap _ { n = 0 } ^ { \\infty } D _ { \\textbf { \\textit { g } } ^ { n } ( p ) } \\textbf { \\textit { g } } ^ { - n } ( \\overline { K _ { \\alpha , \\textbf { \\textit { f } } , \\textbf { \\textit { g } } ^ { n } ( p ) } ^ { s } } ) \\quad F ^ { u } _ { p } = \\bigcap _ { n = 0 } ^ { \\infty } D _ { \\textbf { \\textit { g } } ^ { - n } ( p ) } \\textbf { \\textit { g } } ^ { n } ( \\overline { K _ { \\alpha , \\textbf { \\textit { f } } , \\textbf { \\textit { g } } ^ { - n } ( p ) } ^ { u } } ) . \\end{align*}"}
-{"id": "4166.png", "formula": "\\begin{align*} & & \\left ( - 1 + \\frac { 1 } { n } \\right ) e _ { i , g } ^ { n - 1 } + \\frac { 1 } { n } e ^ { n - 1 } _ { i , g + 1 } + \\frac { 1 } { n } e _ { i , g + 2 } ^ { n - 2 } - e _ { i , g + 1 } ^ { n - 2 } \\\\ & + \\left ( 1 - \\frac { 1 } { n } \\right ) \\left [ \\frac { 1 } { n ^ 2 } e _ { i , g + 1 } ^ { n - 3 } + \\frac { 1 } { n } e _ { i , g + 2 } ^ { n - 3 } \\right ] - e _ { i , g + 2 } ^ { n - 3 } + \\left ( - 1 + \\frac { 1 } { n } \\right ) \\sum _ { \\tau = 3 } ^ { n - 2 } e _ { i , g + \\tau } ^ { n - 1 - \\tau } . \\end{align*}"}
-{"id": "6688.png", "formula": "\\begin{align*} | H _ p ^ n ( \\Delta ) | _ n = \\frac { 1 } { n } \\left ( 1 + \\Delta - \\frac { 1 } { ( 1 + \\Delta ) ^ { \\frac { 1 } { p - 1 } } } \\right ) \\left ( 1 - \\frac { 1 } { ( 1 + \\Delta ) ^ { \\frac { p } { p - 1 } } } \\right ) ^ { \\frac { n - 1 } { p } } | B _ p ^ { n - 1 } | _ { n - 1 } - \\left | C _ p ^ n \\left ( 1 - \\frac { 1 } { ( 1 + \\Delta ) ^ { \\frac { 1 } { p - 1 } } } \\right ) \\right | _ n . \\end{align*}"}
-{"id": "3948.png", "formula": "\\begin{align*} \\Lambda ( \\mathcal { S } \\cap F ( m ) , r \\Gamma ( r ) ^ { \\sigma } ) = 0 . \\end{align*}"}
-{"id": "2816.png", "formula": "\\begin{align*} \\frac { \\partial I ^ { ( 1 ) } } { \\partial \\tau } [ \\tau , b ( \\tau ) ] = - \\lim _ { x \\rightarrow b ( \\tau ) } \\frac { \\partial I ^ { ( 2 ) } } { \\partial \\tau } [ \\tau , x ] . \\end{align*}"}
-{"id": "5042.png", "formula": "\\begin{align*} p ( x ) = \\| x \\| _ X + \\| R ( x ) \\| _ { \\ell _ 1 } ( x \\in X ) \\end{align*}"}
-{"id": "6192.png", "formula": "\\begin{align*} \\widehat { \\omega } _ i ( t ) & : = ( G _ t ^ { - 1 } ) ^ * \\omega _ i ( t ) \\\\ & = f _ i ( G _ t ^ { - 1 } ( y ) , t ) \\frac { v _ t } { 2 \\pi } \\dd y \\wedge \\dd x _ i + \\frac { 1 } { 2 } \\epsilon _ { i j k } \\dd x _ j \\wedge \\dd x _ k \\\\ & = \\frac { v _ t } { 2 \\pi } \\hat f _ i ( y , t ) \\dd y \\wedge \\dd x _ i + \\frac { 1 } { 2 } \\epsilon _ { i j k } \\dd x _ j \\wedge \\dd x _ k \\end{align*}"}
-{"id": "9641.png", "formula": "\\begin{align*} \\hat { \\Psi } ( \\tau _ { i } ) = \\begin{cases} \\left ( \\Psi ( \\tau _ { i } ) , \\Psi _ { \\rm { R } } \\right ) , ~ & N ( \\tau _ { i - 1 } ) \\geq N ( \\tau _ { i } ) \\\\ \\Psi ( \\tau _ { i } ) , ~ & N ( \\tau _ { i - 1 } ) < N ( \\tau _ { i } ) , \\end{cases} \\end{align*}"}
-{"id": "8382.png", "formula": "\\begin{align*} \\vartheta ^ { [ i ] } ( \\tau , z , g ) = \\vartheta ( \\tau , z , g ) \\cdot \\vartheta ^ { [ i ] } ( \\tau ) , \\end{align*}"}
-{"id": "1461.png", "formula": "\\begin{align*} d _ { 2 ^ { r + 2 } - 1 } ( x ) = v _ { r + 1 } Q _ { r + 1 } ( x ) \\end{align*}"}
-{"id": "7755.png", "formula": "\\begin{align*} f ( u , b , x ) = b ^ { - 1 } ( x ^ { - 1 } ) ^ w b u x u ^ { - 1 } . \\end{align*}"}
-{"id": "8183.png", "formula": "\\begin{align*} \\phi _ { \\circ } ( g \\eta _ { \\mathfrak { L } } ) = \\omega _ { \\pi } ( \\omega _ { \\pi } ^ { \\mathfrak { L } } ) ^ { - 1 } ( \\det ( g ) ) \\phi _ { \\circ } ^ { \\mathfrak { L } } ( g ) . \\end{align*}"}
-{"id": "3677.png", "formula": "\\begin{align*} w _ i = { } ^ t \\ ! ( \\underbrace { 1 , \\dots , 1 } _ { \\scriptsize ( i - 1 ) } , \\frac { n - i + 1 } { n + 1 } , 0 , \\dots , 0 ) \\end{align*}"}
-{"id": "1973.png", "formula": "\\begin{align*} \\Vert ( v _ { s } , v _ { u } ) \\Vert _ { 1 } = \\sqrt { { \\Vert v _ { s } \\Vert _ { 1 } } ^ { 2 } + { \\Vert v _ { u } \\Vert _ { 1 } } ^ { 2 } } , \\end{align*}"}
-{"id": "6369.png", "formula": "\\begin{align*} \\mu _ { p ^ * _ 1 , p ^ * _ 2 } ( \\{ x \\} ) = - \\lim _ { y \\to 0 + } y \\Im \\Phi _ { p ^ * _ 1 , p ^ * _ 2 } ( x + i y ) . \\end{align*}"}
-{"id": "8295.png", "formula": "\\begin{align*} F ^ { 1 } H _ \\C = 0 , F ^ 0 H _ \\C = z H _ \\C , F ^ { - 1 } H _ \\C = H _ \\C . \\end{align*}"}
-{"id": "5338.png", "formula": "\\begin{align*} \\Gamma _ R = \\mathfrak { g } \\otimes R \\oplus \\C { \\bf k } \\end{align*}"}
-{"id": "450.png", "formula": "\\begin{align*} \\Upsilon ( x , t ) = \\sum _ { j = 0 } ^ { k _ 1 / 2 } b _ { k _ 1 , k _ 2 , j } \\frac { \\left ( \\omega - \\frac { \\pi } { 2 } \\right ) ^ { k _ 1 - 2 j } } { | x | ^ { 2 j } } + O \\left ( \\sum _ { j = 0 } ^ { k _ 1 / 2 } \\frac { \\left ( \\omega - \\frac { \\pi } { 2 } \\right ) ^ { k _ 1 - 2 j + 1 } } { | x | ^ { 2 j } } + \\frac { 1 } { | x | ^ { k _ 1 + 2 } } \\right ) ; \\end{align*}"}
-{"id": "7123.png", "formula": "\\begin{align*} g _ { i j } ^ X = & \\frac 1 { 1 - | Y | ^ 2 } \\left ( g _ { i j } ^ Y + \\frac { \\langle Y , \\partial _ i Y \\rangle \\langle Y , \\partial _ j Y \\rangle } { ( 1 - | Y | ^ 2 ) } \\right ) . \\end{align*}"}
-{"id": "7295.png", "formula": "\\begin{align*} \\varphi = \\left ( \\begin{array} { r r r r } \\ell _ 1 & & & \\\\ - \\ell _ 2 & \\ell _ 2 & & \\\\ & - \\ell _ 3 & \\ddots & \\\\ & & \\ddots & \\ell _ { n - 1 } \\\\ & & & - \\ell _ n \\end{array} \\right ) \\end{align*}"}
-{"id": "991.png", "formula": "\\begin{align*} \\mathfrak { d } _ 0 ( t , t ' ) = a _ 3 \\sqrt { \\frac { 1 } { n ^ 2 } \\sum _ { i = 1 } ^ n \\{ K _ h ( t _ { i - 1 } - t ) - K _ h ( t _ { i - 1 } - t ' ) \\} ^ 2 } , t , t ' \\in [ 0 , T ] \\end{align*}"}
-{"id": "2138.png", "formula": "\\begin{align*} \\| w ( s ) \\| _ { L ^ 2 ( { \\bf R } _ + , \\rho _ d \\ , d \\xi ) } + \\| w ( s ) \\| _ { C ^ 2 ( K ) } = O ( e ^ { - s } ) \\quad \\mbox { a s } s \\to \\infty . \\end{align*}"}
-{"id": "7941.png", "formula": "\\begin{align*} \\bar V ^ { n } : = \\bigcup _ { I \\in \\mathcal I ^ n } \\bar V _ I . \\end{align*}"}
-{"id": "1679.png", "formula": "\\begin{align*} \\mu _ 2 ( Z ( u ) ) = \\mu _ 2 ( Z ( w ) ) = \\mu _ 2 ( Z ( v ) ) = 1 . \\end{align*}"}
-{"id": "9289.png", "formula": "\\begin{align*} \\Delta ^ 0 f ( x ) : = f ( x ) , \\end{align*}"}
-{"id": "5474.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { u ( r ^ n z ) } { ( r ^ n z ) ^ { \\rho - 1 } \\ell ( r ^ n z ) } = p _ 0 ( z ) z \\in C _ { p _ 0 } , p _ 0 \\in \\mathcal { P } _ { r } , \\end{align*}"}
-{"id": "6213.png", "formula": "\\begin{align*} X ( t ) = c t - \\sum _ { i = 1 } ^ { N ( t ) } Y _ i ~ , ~ ~ t \\geq 0 ~ , \\end{align*}"}
-{"id": "8776.png", "formula": "\\begin{align*} \\mathcal { L } ( G _ 1 ) \\circ { B } = I _ { n _ 1 } - \\dfrac { A ( G _ 1 ) } { r _ 1 + 1 } = \\dfrac { 1 } { r _ 1 + 1 } ( I _ { n _ 1 } + r _ 1 \\mathcal { L } ( G _ 1 ) ) . \\end{align*}"}
-{"id": "7551.png", "formula": "\\begin{gather*} \\psi _ t + \\psi _ x \\psi _ y - \\psi _ { x x } - \\psi _ { y y } = 0 . \\end{gather*}"}
-{"id": "5727.png", "formula": "\\begin{align*} \\mathfrak { E } _ { W } ( { v } ) = \\mathfrak { E } _ { W } ( { v } , ( - \\infty , s _ 0 ] ) + \\mathfrak { E } _ { W } ( { v } , [ s _ 0 , + \\infty ) ) \\geq \\mathfrak { E } _ { W } ( { v } , ( - \\infty , s _ 0 ] ) + d _ K ( B ( a ^ - , \\varepsilon _ 1 ) , a ^ + ) . \\end{align*}"}
-{"id": "3858.png", "formula": "\\begin{align*} \\ell ( x ^ * ) & = f ( x ^ * ) \\geq f ( x ^ k ) \\\\ & \\geq f ( x ^ k ) + \\sum \\limits _ { i = 1 } ^ m \\lambda _ i ^ * g _ i ( x ^ k ) + \\sum \\limits _ { i = 1 } ^ p \\mu _ i ^ * h _ i ( x ^ k ) + \\sum \\limits _ { i = 1 } ^ n \\gamma _ i ^ * x _ i ^ k = \\ell ( x ^ k ) . \\end{align*}"}
-{"id": "5301.png", "formula": "\\begin{align*} \\int _ { v } ^ { u } B ( r ) \\mathrm { d r } = B ( c ) ( u - v ) . \\end{align*}"}
-{"id": "3386.png", "formula": "\\begin{align*} \\R ^ n F _ b ( x , y ) = \\left ( f _ { b , n } ( x ) - a ( x ) b ^ { 2 ^ n } y E _ n ( x , y ) , x \\right ) \\end{align*}"}
-{"id": "3020.png", "formula": "\\begin{align*} \\bar { W } _ B = \\begin{bmatrix} - W _ 1 & W _ 2 \\end{bmatrix} . \\end{align*}"}
-{"id": "5666.png", "formula": "\\begin{align*} \\begin{cases} E '' = c E ^ { p _ 0 - 1 } , \\\\ E ( s _ 0 ) = \\varepsilon _ 0 , \\\\ \\lim \\limits _ { s \\to + \\infty } E ( s ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "3116.png", "formula": "\\begin{align*} M _ \\tau ^ { \\sigma - s s t } ( X , Q , I ) = Z ^ { \\theta - s s t } / / G . \\end{align*}"}
-{"id": "7917.png", "formula": "\\begin{align*} u ( x , k ) = 0 , u ( y , k ) \\leq u _ 0 ( ( 0 , 3 k ) ) = 3 k . \\end{align*}"}
-{"id": "6568.png", "formula": "\\begin{align*} \\alpha _ 1 : = \\alpha _ { \\xi } = \\left ( \\frac { 2 \\cdot 2 ( 1 + \\varepsilon ) \\cdot \\sqrt { 1 - \\varepsilon ^ 2 } } { 2 \\cdot \\frac { \\sqrt { 1 - \\varepsilon ^ 2 } } { 1 - \\varepsilon } \\cdot 1 } \\right ) ^ { 1 / 2 } = \\sqrt { 2 } \\cdot \\sqrt { 1 - \\varepsilon ^ 2 } \\end{align*}"}
-{"id": "792.png", "formula": "\\begin{align*} a _ 1 = \\frac { \\rho + 1 } { 2 } \\log ( 2 x ) + 2 \\log { y } \\textrm { i f } x \\in A \\end{align*}"}
-{"id": "7299.png", "formula": "\\begin{align*} \\varphi = \\left ( \\begin{array} { c c c c c c c c c c c } z & 0 & 0 & \\ldots & 0 & 0 & 0 \\\\ - c _ 1 & z & 0 & \\ldots & 0 & 0 & 0 \\\\ 0 & - c _ 2 & z & \\ldots & 0 & 0 & 0 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots \\\\ 0 & 0 & 0 & \\ldots & z & 0 & 0 \\\\ 0 & 0 & 0 & \\ldots & - c _ { n - 3 } & z & 0 \\\\ 0 & 0 & 0 & \\ldots & 0 & - c _ { n - 2 } & z \\\\ 0 & 0 & 0 & \\ldots & 0 & 0 & - c _ { n - 1 } \\end{array} \\right ) \\end{align*}"}
-{"id": "5756.png", "formula": "\\begin{align*} f ( x , y ) = \\sum _ { ( a , b ) \\in E } { f ( a , b ) \\ast \\lambda _ { a , b } ( x , y ) } . \\end{align*}"}
-{"id": "1805.png", "formula": "\\begin{align*} X ' _ k : = \\frac { F _ k ( X ) } { X _ k } . \\end{align*}"}
-{"id": "2455.png", "formula": "\\begin{align*} m ^ * ( \\delta ( [ \\rho , \\nu ^ k \\rho ] ) ) = \\sum _ { i = - 1 } ^ k \\delta ( [ \\nu ^ { i + 1 } \\rho , \\nu ^ k \\rho ] ) \\otimes \\delta ( [ \\rho , \\nu ^ { i } \\rho ] ) , \\end{align*}"}
-{"id": "8485.png", "formula": "\\begin{align*} \\abs { W _ { \\pi } ( g _ { t , l , v } ) } = 1 l = 0 t = - n l \\geq n t = - 2 l . \\end{align*}"}
-{"id": "7552.png", "formula": "\\begin{gather*} u = \\psi _ y , v = \\psi _ x . \\end{gather*}"}
-{"id": "6731.png", "formula": "\\begin{align*} \\sum \\limits _ { 0 < | u - s | \\leq N ^ { 1 + \\delta } } \\mathbb { P } ( A _ { u } \\cap A _ { s } ) & = \\sum \\limits _ { k = 1 } ^ { N ^ { 1 + \\delta } } ( t ( N ) - k + 1 ) \\mathbb { P } ( \\sigma ^ { \\eta } ( k ) \\in V , \\sigma ( 0 ) \\in V ) \\\\ & \\leq \\sum \\limits _ { k = 1 } ^ { N ^ { 1 + \\delta } } t ( N ) \\nu ( V ) = N ^ { 1 + \\delta } t ( N ) \\nu ( V ) . \\\\ \\end{align*}"}
-{"id": "2947.png", "formula": "\\begin{align*} \\int ( \\partial \\Theta - t ) ^ + d \\mu \\int ( \\partial \\Theta - t ) ^ + d \\nu \\geq \\int f \\partial \\Theta d \\nu - t \\int f d \\nu . \\end{align*}"}
-{"id": "3815.png", "formula": "\\begin{align*} \\begin{array} { l c l } H ^ { ( t ) } _ { + } & : = & \\inf \\{ s \\ge 0 \\colon \\ ; X ^ { ( \\hat { Z } , t ) } _ s - \\hat { Z } = \\ell _ L \\} , \\\\ H ^ { ( t ) } _ { - } & : = & \\inf \\{ s \\ge 0 \\colon \\ ; X ^ { ( \\hat { Z } , t ) } _ s - \\hat { Z } = - \\ell _ L + 1 \\} \\end{array} \\end{align*}"}
-{"id": "470.png", "formula": "\\begin{align*} \\partial ^ { k _ 1 + 1 } _ 1 a _ { k _ 1 , k _ 2 , \\pi / 2 } ( 0 ) = ( - 1 ) ^ { k _ 1 } i ^ { k _ 2 - n } \\left ( i \\frac { \\pi } { 2 } \\right ) ^ { n + k _ 1 + k _ 2 - 1 } ( k _ 1 + 1 ) ! ( n + k _ 1 + k _ 2 ) \\end{align*}"}
-{"id": "4143.png", "formula": "\\begin{align*} R ( t , d , 0 ) = \\frac { n - \\lvert I _ t \\rvert } { n } R ( t - 1 , d , 0 ) + \\frac { \\lvert I _ t \\rvert } { n } R ( t - 1 , d , 1 ) . \\end{align*}"}
-{"id": "7025.png", "formula": "\\begin{align*} C _ 3 = C _ 3 ( n , s ) = \\int _ { \\mathbb { R } } { \\frac { 1 } { ( 1 + z _ n ^ 2 ) ^ { \\frac { n + 2 s } { 2 } } } d z _ n } . \\end{align*}"}
-{"id": "1710.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\in f ( \\eta ) \\Lambda ^ n } t _ \\lambda P ( \\sigma _ \\lambda ^ { - 1 } ( Z ( \\eta ) ) ) t _ \\lambda ^ * = P ( Z ( \\eta ) ) ; \\end{align*}"}
-{"id": "6176.png", "formula": "\\begin{align*} \\Gamma _ { 0 0 } ^ { \\ ; \\ ; 0 } = \\frac { V ' } { V } \\ ; , \\ ; \\Gamma _ { i i } ^ { \\ ; \\ ; 0 } = - \\frac { f _ i ' } { 2 V ^ 2 } \\ ; , \\ ; \\ ; \\Gamma _ { i 0 } ^ { \\ ; \\ ; i } = \\frac { f _ i ' } { 2 f _ i } \\ ; \\ ; \\ ; ; \\ ; \\ ; i = 1 , 2 , 3 \\end{align*}"}
-{"id": "5078.png", "formula": "\\begin{align*} E _ 1 ( Q ) = 0 , E _ 2 ( Q ) = 2 \\frac { B _ { 1 2 , 1 } } { b _ 1 - b _ 2 } Q , E _ 3 ( Q ) = 2 \\frac { B _ { 1 3 , 1 } } { b _ 1 - b _ 3 } Q . \\end{align*}"}
-{"id": "1185.png", "formula": "\\begin{align*} v ( y _ 0 + \\frac { \\rho _ 1 } { A } \\eta ) - u ( x ) \\leq \\mathbf { v } ( x ) - u ( x ) \\leq \\mathbf { v } ( x _ 0 ) - u ( x _ 0 ) = v ( y _ 0 ) - u ( x _ 0 ) . \\end{align*}"}
-{"id": "5127.png", "formula": "\\begin{align*} \\frac { \\sum _ { | \\beta | \\leq N } ( p _ 0 ) _ { j , \\beta } \\prod _ { ( j ' , \\alpha ) \\in S } ( \\partial ^ \\alpha f _ { j ' } ( x ) ) ^ { \\beta _ { j ' , \\alpha } } } { \\sum _ { | \\beta | \\leq N } ( q _ 0 ) _ { j , \\beta } \\prod _ { ( j ' , \\alpha ) \\in S } ( \\partial ^ \\alpha f _ { j ' } ( x ) ) ^ { \\beta _ { j ' , \\alpha } } } = ( y _ 0 ) _ j , j = 1 , \\ldots , n , \\end{align*}"}
-{"id": "7162.png", "formula": "\\begin{align*} \\sum _ { m = r J + 1 } ^ { ( r + 1 ) J } y ( m ) \\leq \\sqrt { 2 [ U J ^ 2 + V _ r J g ^ * _ r + W J ] } . \\end{align*}"}
-{"id": "1766.png", "formula": "\\begin{align*} x = e h f g e h f g \\dots = h f g e h f g e \\dots = h g e e h g e e \\dots . \\end{align*}"}
-{"id": "858.png", "formula": "\\begin{align*} \\widehat { \\vartheta } _ n = \\mathrm { a r g } \\max _ { \\theta \\in \\mathcal { G } _ n } | U _ n ( \\theta ) | \\end{align*}"}
-{"id": "5037.png", "formula": "\\begin{align*} x ^ * = x _ 0 ^ * + \\sum _ { n \\in \\N } t _ n r _ n v _ n ^ * , z ^ * = z _ 0 ^ * + \\sum _ { n \\in \\N } \\tau _ n r _ n v _ n ^ * . \\end{align*}"}
-{"id": "3365.png", "formula": "\\begin{align*} p \\wedge \\bigvee _ { i \\in I } m _ i = \\bigvee _ { i \\in I } ( p \\wedge m _ i ) . \\end{align*}"}
-{"id": "1323.png", "formula": "\\begin{align*} E _ { i , j } & : = S _ { e _ { i + 1 } } S _ { e _ { i + 2 } } \\cdots S _ { e _ { j } } , i < j ; \\\\ E _ { i , j } & : = E _ { j , i } ^ * , i > j ; ~ \\\\ E _ { i , i } & : = E _ { i , 1 } E _ { 1 , i } = P _ i . \\end{align*}"}
-{"id": "3559.png", "formula": "\\begin{align*} L ( s , \\Psi \\Lambda ) = \\sum _ { n \\geq 1 } \\frac { \\Psi ( n ) \\Lambda ( n ) } { n ^ s } , \\end{align*}"}
-{"id": "3564.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { \\Lambda ( p n ) } { ( p n ) ^ s } = \\sum _ { k \\geq 0 } \\frac { \\log p } { p ^ { ( 1 + k ) s } } \\end{align*}"}
-{"id": "4014.png", "formula": "\\begin{align*} \\langle X , Y \\rangle _ p : = \\langle d u ^ { - 1 } _ { \\eta ( p ) } X , Y \\rangle , \\end{align*}"}
-{"id": "1198.png", "formula": "\\begin{align*} & v _ 1 ( y _ 0 + \\rho \\eta ) = v _ 1 ( y _ 0 ) + A _ 1 \\rho + A _ 2 \\rho ^ { 2 } + o ( \\rho ^ { 2 } ) , \\\\ & v _ 2 ( z _ 0 + \\rho \\eta ) = v _ 2 ( z _ 0 ) + B _ 1 \\rho + B _ 2 \\rho ^ { 2 } + o ( \\rho ^ { 2 } ) , \\\\ & u ( x _ 0 + \\rho \\eta ) = u ( x _ 0 ) + C _ 1 \\rho + C _ 2 \\rho ^ { 2 } + o ( \\rho ^ { 2 } ) \\end{align*}"}
-{"id": "9004.png", "formula": "\\begin{gather*} { \\cal S } ^ { ( n ) } _ { \\eta ' , x _ 1 , \\dots , x _ m ; q , t } ( 0 , v + ( d - a ) ( \\delta + ( n - 1 ) f ) ) \\\\ \\phantom { { \\cal S } ^ { ( n ) } _ { \\eta ' , x _ 1 , \\dots , x _ m ; q , t } } { } = { \\cal S } ^ { ( n ) } _ { \\eta ' + ( n - 1 ) t , x _ 1 , \\dots , x _ m ; q , 0 } ( 0 , v + ( d - a ) ( \\delta + ( n - 1 ) f ) ) \\end{gather*}"}
-{"id": "810.png", "formula": "\\begin{align*} T _ { k } ( x ) = Q _ { k } ( x ) - x ^ k + ( 2 x - 1 ) ^ k + ( 2 x ) ^ k \\end{align*}"}
-{"id": "9270.png", "formula": "\\begin{align*} 2 \\Re \\langle \\mathcal { L } _ { \\varepsilon } X , X \\rangle = 2 \\Re \\langle X , \\mathcal { L } _ { \\varepsilon } X \\rangle = \\langle \\mathcal { L } _ { \\varepsilon } X , X \\rangle + \\langle X , \\mathcal { L } _ { \\varepsilon } X \\rangle \\leq 0 , \\end{align*}"}
-{"id": "2885.png", "formula": "\\begin{align*} \\mathcal { P } ( y ) = - \\frac { 1 } { 2 } \\zeta ( - k ) . \\end{align*}"}
-{"id": "5229.png", "formula": "\\begin{align*} \\frac { d } { d z } \\Omega ( Y _ 1 , Y _ 2 ) = e ^ { c z } \\left ( c \\omega ( Y _ 1 , Y _ 2 ) + \\frac { d } { d z } \\omega ( Y _ 1 , Y _ 2 ) \\right ) = 0 . \\end{align*}"}
-{"id": "8111.png", "formula": "\\begin{align*} \\dim ( [ \\partial _ A W _ 1 ] _ { R _ { \\overline { W } , E } } ) \\leq \\dim ( F \\partial _ A W _ 1 ) = \\dim ( \\partial _ A W _ 1 ) < \\dim ( A ) . \\end{align*}"}
-{"id": "4162.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { k \\in N } e _ { k , g } ^ n + \\left ( 1 - \\frac { 1 } { n } \\right ) e _ { i , g } ^ { n - 1 } + \\sum _ { \\tau = 1 } ^ { n - 2 } e _ { i , g + \\tau } ^ { n - 1 - \\tau } . \\end{align*}"}
-{"id": "1529.png", "formula": "\\begin{align*} s = \\begin{cases} \\lfloor \\alpha _ { m , t } ( - 1 ) \\rfloor , & \\ \\ \\{ \\alpha _ { m , t } ( - 1 ) \\} < 1 / 2 ; \\\\ \\lfloor \\alpha _ { m , t } ( - 1 ) \\rfloor \\ \\ \\lfloor \\alpha _ { m , t } ( - 1 ) \\rfloor + 1 , & \\ \\ \\{ \\alpha _ { m , t } ( - 1 ) \\} = 1 / 2 ; \\\\ \\lfloor \\alpha _ { m , t } ( - 1 ) \\rfloor + 1 , & \\ \\ \\{ \\alpha _ { m , t } ( - 1 ) \\} > 1 / 2 . \\end{cases} \\end{align*}"}
-{"id": "5495.png", "formula": "\\begin{align*} \\widehat F ( s ) = \\int _ 0 ^ \\infty e ^ { - s x } \\dd F ( x ) \\end{align*}"}
-{"id": "6463.png", "formula": "\\begin{align*} p \\left ( x _ { 2 j - 1 } x _ { 2 j } | \\mu _ { 2 j - 1 } \\sigma _ { 2 j - 1 } \\right ) \\overset { } { = } \\frac { 1 } { 2 \\pi \\sigma _ { 2 j - 1 } ^ { 2 } } \\exp \\left [ - \\frac { \\left ( x _ { 2 j - 1 } - \\mu _ { 2 j - 1 } \\right ) ^ { 2 } + \\left [ x _ { 2 j } - \\mu _ { 2 j } \\left ( \\mu _ { 2 j - 1 } \\sigma _ { 2 j - 1 } \\right ) \\right ] ^ { 2 } } { 2 \\sigma _ { 2 j - 1 } ^ { 2 } } \\right ] \\end{align*}"}
-{"id": "5303.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ { 2 } \\sum _ { l = 1 } ^ n \\left ( L _ { i , j } ( \\mathbf { u } ) \\xi _ { j , l } \\right ) \\xi _ { l , i } \\geq ( L _ { 1 } ) _ \\# | \\xi _ 1 | ^ 2 + ( L _ { 2 } ) _ \\# | \\xi _ 2 | ^ 2 , \\end{align*}"}
-{"id": "8661.png", "formula": "\\begin{align*} \\begin{aligned} & \\exp \\Big \\{ \\int _ 0 ^ 1 \\Big ( \\int _ 0 ^ \\infty \\ 1 _ { \\{ | x - y + \\omega _ { X _ 0 } ( u _ 1 ) - \\omega _ { Y _ 0 } ( u _ 1 + u _ 2 ) | \\leq 1 \\} } d u _ 1 \\Big ) d u _ 2 \\Big \\} \\\\ & \\leq \\int _ 0 ^ 1 \\left ( \\exp \\Big \\{ \\int _ 0 ^ \\infty 1 _ { \\{ | x - y + \\omega _ { X _ 0 } ( u _ 1 ) - \\omega _ { Y _ 0 } ( u _ 1 + u _ 2 ) | \\leq 1 \\} } d u _ 1 \\Big \\} \\right ) d u _ 2 . \\end{aligned} \\end{align*}"}
-{"id": "5081.png", "formula": "\\begin{align*} 2 d \\Phi - d \\Psi = \\{ [ ( b _ 1 - b _ 2 ) ^ 2 - 2 ] R _ { 1 2 1 2 } + [ ( b _ 2 - b _ 3 ) ^ 2 - 2 ] R _ { 2 3 2 3 } + [ ( b _ 1 - b _ 3 ) ^ 2 - 2 ] R _ { 1 3 1 3 } \\} d v _ g . \\end{align*}"}
-{"id": "2904.png", "formula": "\\begin{align*} \\lim _ { s _ 1 } \\lim _ { s _ 2 } c ( k , s _ 1 , s _ 2 ) = i \\Leftrightarrow \\exists t _ 1 ( \\forall s _ 1 \\geq t _ 1 \\ , \\forall t _ 2 \\geq s _ 1 ) ( \\exists s _ 2 \\geq t _ 2 ) ~ c ( k , s _ 1 , s _ 2 ) = i . \\end{align*}"}
-{"id": "4418.png", "formula": "\\begin{align*} m _ 2 ( x _ 1 + 1 , x _ 2 ) = m _ 2 ( x _ 1 , x _ 2 + 1 ) = m _ 2 ( x _ 1 , x _ 2 ) , \\end{align*}"}
-{"id": "8316.png", "formula": "\\begin{align*} \\vartheta ( \\tau , \\gamma z , \\gamma g h ; \\varphi ) = \\vartheta ( \\tau , z , g ; \\varphi \\circ h ^ { - 1 } ) \\end{align*}"}
-{"id": "7600.png", "formula": "\\begin{align*} U ^ { - \\frac 1 2 } \\left ( U _ 2 U ^ { - 1 } U _ 1 \\right ) U ^ { - \\frac 1 2 } = ( U ^ { - \\frac 1 2 } U _ 2 U ^ { - \\frac 1 2 } ) ( U ^ { - \\frac 1 2 } U _ 1 U ^ { - \\frac 1 2 } ) \\end{align*}"}
-{"id": "2312.png", "formula": "\\begin{align*} & \\ ; \\| A ^ { ( r _ j - s ) / 2 } u _ j \\| _ { L ^ 2 } ^ 2 + \\| A ^ { ( r _ { j + 1 } - s ) / 2 } w _ j ( t _ { j + 1 } ) \\| _ { L ^ 2 } ^ 2 + \\int _ { t _ j } ^ \\infty \\| A ^ { r _ { j + 1 } / 2 } w _ j \\| _ { L ^ 2 } ^ 2 \\ , d \\tau \\\\ \\leq & \\ ; C ( \\delta ^ { - \\frac { j ( 2 s - 1 ) } { s } } + 1 ) ( \\| u _ 0 \\| ^ 2 _ { D ( A ) } + 1 ) ^ { j } \\| u _ 0 \\| ^ 2 _ { D ( A ) } , \\end{align*}"}
-{"id": "3484.png", "formula": "\\begin{align*} \\Big ( \\frac { 1 } { 2 \\tau } g \\Big ) ^ { \\prime \\prime } = - \\frac { \\tau ^ { \\prime \\prime } } { 2 \\tau ^ 2 } g . \\end{align*}"}
-{"id": "8245.png", "formula": "\\begin{align*} \\abs { E ( s , g ) } = \\abs { E ^ { \\mathfrak { L } } ( s , a ( \\theta _ i ) g ' ) } . \\end{align*}"}
-{"id": "3077.png", "formula": "\\begin{align*} t _ c : = \\inf \\left \\{ t \\in ( 0 , T ) : \\ , \\lambda _ 1 ( t ) = 0 \\right \\} . \\end{align*}"}
-{"id": "4738.png", "formula": "\\begin{align*} \\beta ^ 2 c _ 0 + 2 \\beta \\gamma f _ 0 + \\gamma ^ 2 d _ 0 = 0 . \\end{align*}"}
-{"id": "7347.png", "formula": "\\begin{align*} ( q ( \\alpha ) , r ( \\alpha ) ) \\ ; : = \\ ; \\begin{cases} \\quad \\ ; ( + \\infty , 2 ) & \\ ; \\ ; \\alpha \\in ( 0 , 2 ] \\\\ \\Big ( \\frac { 6 } { \\alpha - 2 } , \\frac { 1 8 } { 1 3 - 2 \\alpha } \\Big ) & \\ ; \\ ; \\alpha \\in ( 2 , 3 ) \\\\ \\end{cases} \\end{align*}"}
-{"id": "11.png", "formula": "\\begin{align*} \\hat { V } ^ { C } _ { \\sigma } ( C _ { 1 } , C _ { 2 } ) = \\frac { 1 } { 2 \\pi \\sigma ^ { 2 } } \\frac { 1 } { N } \\sum \\limits _ { n = 1 } ^ N e x p \\left ( - \\frac { ( x _ { n } - y _ { n } ) ^ { 2 } + ( z _ { n } - s _ { n } ) ^ { 2 } } { 2 \\sigma ^ 2 } \\right ) \\end{align*}"}
-{"id": "4368.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\Big \\| B _ t A - \\sum _ { j = j _ 0 ( t ) } ^ \\infty M _ { \\varphi _ { j , t } } \\Big \\| = 0 \\end{align*}"}
-{"id": "1648.png", "formula": "\\begin{align*} \\frac { d ( \\mu \\circ \\tau _ { \\lambda _ 1 } \\circ \\tau _ { \\lambda _ 2 } ) } { d ( \\mu \\circ \\tau _ { \\lambda _ 2 } ) } = \\Phi _ { \\tau _ { \\lambda _ 1 } } \\circ \\tau _ { \\lambda _ 2 } . \\end{align*}"}
-{"id": "53.png", "formula": "\\begin{align*} = \\int \\int \\frac { 1 } { N } \\sum \\limits _ { i = 1 } ^ N \\left ( \\frac { 1 } { 2 \\pi \\theta ^ 2 } \\right ) \\left ( \\frac { 1 } { 2 \\pi 4 \\theta ^ 2 } \\right ) e x p \\left ( - \\frac { ( u _ 1 - b ) ^ { 2 } } { 2 \\theta ^ 2 } \\right ) e x p \\left ( - \\frac { ( u _ 2 - b ' ) ^ { 2 } } { 2 \\theta ^ { 2 } } \\right ) e x p \\left ( - \\frac { c } { 4 \\theta ^ 2 } \\right ) \\mathrm { d } u _ 1 \\mathrm { d } u _ 2 \\end{align*}"}
-{"id": "9096.png", "formula": "\\begin{align*} A ^ { \\ast } ( x ) = A \\left ( x , \\ldots , x \\right ) \\left ( x \\in G \\right ) . \\end{align*}"}
-{"id": "3395.png", "formula": "\\begin{align*} \\beta \\circ G _ { s , 0 } \\circ \\alpha ^ { - 1 } \\circ \\gamma ( x ) & = \\beta \\circ F _ { b ^ * } \\circ \\alpha ^ { - 1 } ( x , s ) \\\\ & = ( \\beta _ x ( \\phi _ { b ^ * } ( \\alpha _ x ^ { - 1 } ( x ) , \\alpha _ y ^ { - 1 } ( x , s ) ) , \\alpha _ x ^ { - 1 } ( x ) ) , \\beta _ y ( \\alpha _ x ^ { - 1 } ( x ) ) ) \\end{align*}"}
-{"id": "1746.png", "formula": "\\begin{align*} \\langle S _ \\lambda ^ * ( f \\sqrt { d \\mu } ) , g \\sqrt { d \\nu } \\rangle & = \\int _ { Z ( s ( \\lambda ) ) } \\overline { f \\circ \\sigma _ \\lambda ( x ) } g ( x ) \\sqrt { \\frac { d ( \\mu \\circ \\sigma _ \\lambda ) } { d ( ( \\mu \\circ \\sigma _ \\lambda ) + \\nu ) } } \\sqrt { \\frac { d \\nu } { d ( ( \\mu \\circ \\sigma _ \\lambda ) + \\nu ) } } d ( ( \\mu \\circ \\sigma _ \\lambda ) + \\nu ) . \\end{align*}"}
-{"id": "6812.png", "formula": "\\begin{align*} \\tilde { V } \\left ( \\frac { y } { \\lambda } \\right ) = \\sum \\limits _ { j = 1 } ^ 4 f _ 0 \\left ( a \\frac { | \\Pi _ { \\xi _ j } ( y ) | } { \\lambda } \\right ) \\end{align*}"}
-{"id": "1754.png", "formula": "\\begin{align*} \\int _ { Z ( \\eta ) } \\nu _ { f \\sqrt { d \\mu } } \\ , d \\mu = \\nu _ { f \\sqrt { d \\mu } } ( Z ( \\eta ) ) = \\langle ( { \\chi _ { Z ( \\eta ) } } f ) \\ , \\sqrt { d \\mu } , f \\sqrt { d \\mu } \\rangle = \\int \\chi _ { Z ( \\eta ) } \\cdot | f | ^ 2 d \\mu \\ ; = \\int _ { Z ( \\eta ) } | f | ^ 2 \\ , d \\mu . \\end{align*}"}
-{"id": "4517.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| \\Delta _ n a \\Delta _ n ^ * \\| = | \\psi ( a ) | . \\end{align*}"}
-{"id": "2060.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ M | \\rho ( u ( x , t ) ) | ^ 2 = & 2 \\int _ M \\sum _ { a , b } ( \\rho ( u ) ) ^ a \\rho ^ a _ b ( u ) \\frac { \\partial u ^ b } { \\partial t } = 2 \\int _ M \\sum _ a ( \\rho ( u ) ) ^ a \\Delta _ H ( \\rho ( u ) ) ^ a \\\\ = & - 2 \\int _ M \\sum _ a | \\nabla _ H ( \\rho ( u ) ) ^ a | ^ 2 \\leq 0 \\end{align*}"}
-{"id": "2832.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { n } { a _ n + n } < \\infty . \\end{align*}"}
-{"id": "1034.png", "formula": "\\begin{align*} \\widehat { w } _ { n } ( \\zeta ) \\leq \\sigma _ { n } : = \\frac { 2 n - 1 + \\sqrt { 2 n ^ 2 - 2 n + 1 } } { 2 } . \\end{align*}"}
-{"id": "7180.png", "formula": "\\begin{align*} \\int _ { B _ 1 \\setminus \\overline { B _ r } } & | \\nabla z _ { j _ k } | ^ 2 \\ , d \\mu _ a \\leq 2 \\ , ( 1 - r ) \\ , ( n + 1 ) . \\end{align*}"}
-{"id": "2150.png", "formula": "\\begin{align*} F _ N ^ j ( r , t ) = F _ d ^ j ( r , t ) : = \\int _ 0 ^ { r } s ^ { 1 - d } [ \\nu _ d ( s ) ] ^ { - 1 } \\left ( \\int _ 0 ^ s \\tau ^ { d - 1 } \\nu _ d ( \\tau ) ( \\partial _ t ^ { j + 1 } u _ * ) ( \\tau , t ) \\ , d \\tau \\right ) \\ , d s , \\end{align*}"}
-{"id": "929.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ d \\left | \\frac { \\partial ^ 2 \\varphi } { \\partial x _ i x _ j } ( x ) \\right | \\leq \\| g '' \\| _ \\infty + 2 \\beta \\| g ' \\| _ \\infty \\end{align*}"}
-{"id": "4529.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| \\pi ( \\Delta _ n ) ^ * ( \\Pi ( a ) - \\pi ( a ) ) \\pi ( \\Delta _ n ) \\| = 0 \\end{align*}"}
-{"id": "4434.png", "formula": "\\begin{align*} ( T ^ { 1 / 3 } ) ^ { \\alpha + \\beta } [ u ] _ \\alpha [ f ] _ \\beta \\gtrsim \\begin{cases} \\| \\lceil u , ( \\cdot ) _ T \\rceil f \\| & \\textrm { i f } \\alpha \\in ( 0 , 1 ] , \\\\ \\| \\big ( \\lceil u , ( \\cdot ) _ T \\rceil - \\partial _ 1 u \\lceil x _ 1 , ( \\cdot ) _ T \\rceil \\big ) f \\| & \\textrm { i f } \\alpha \\in ( 1 , \\frac 3 2 ) . \\end{cases} \\end{align*}"}
-{"id": "2935.png", "formula": "\\begin{align*} \\sum _ { e ' \\ni i ' } \\theta ' _ \\epsilon ( e ' , i ' ) \\geq \\sum _ { e \\ni i } \\theta ' _ \\epsilon ( \\phi ( e ) , i ' ) . \\end{align*}"}
-{"id": "1027.png", "formula": "\\begin{align*} \\widehat { w } _ { 2 } ( \\zeta ) \\leq \\frac { 3 + \\sqrt { 5 } } { 2 } = 2 . 6 1 8 0 \\ldots . \\end{align*}"}
-{"id": "6719.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\bar { E } ( \\mathbb { P } ( \\Theta > N ^ { \\gamma } t ) ) = e ^ { - t } . \\end{align*}"}
-{"id": "882.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { N _ n } E \\left [ \\max _ { 1 \\leq k \\leq d _ n } \\left | \\sum _ { j = 1 } ^ { N _ n } \\gamma _ { n , k } ( i , j ) W ^ { ( i ) } _ j \\right | ^ 3 \\right ] \\leq 5 ^ { 3 / 2 } a ^ 3 \\log ^ { 3 / 2 } ( 2 d _ n - 1 + \\sqrt { e } ) \\sum _ { i = 1 } ^ { N _ n } \\max _ { 1 \\leq k \\leq d _ n } \\left ( \\sum _ { j = 1 } ^ { N _ n } \\gamma _ { n , k } ( i , j ) ^ 2 \\right ) ^ { 3 / 2 } \\end{align*}"}
-{"id": "9590.png", "formula": "\\begin{align*} \\| R _ N ( f ) \\| _ { \\dot { \\mathcal B } ^ { \\alpha , q } _ { p , \\mathcal F } } & \\le \\bigg \\{ \\sum _ { k } \\bigg ( \\sum _ { k ' } \\sum _ { m = 0 } ^ \\infty \\sum _ { | k _ 0 - \\ell _ 0 | > N } \\cdots \\sum _ { | k _ m - \\ell _ m | > N } \\\\ & \\qquad 2 ^ { k \\alpha } \\big \\| D _ k D _ { k _ 0 } D _ { \\ell _ 0 } D _ { k _ 1 } D _ { \\ell _ 1 } \\cdots D _ { k _ m } D _ { \\ell _ m } D ^ N _ { k ' } \\big \\| _ { L ^ p _ \\mu \\mapsto L ^ p _ \\mu } \\| D _ { k ' } ( f ) \\| _ { L ^ p _ \\mu } \\bigg ) ^ q \\bigg \\} ^ { 1 / q } . \\end{align*}"}
-{"id": "6773.png", "formula": "\\begin{align*} \\int _ { \\mathbb { S } ^ 2 } e ^ { U _ { \\lambda , p } } = 8 \\pi , \\end{align*}"}
-{"id": "4726.png", "formula": "\\begin{align*} \\alpha ^ 2 a _ 0 + 2 \\alpha \\beta b _ 0 + 2 \\alpha \\gamma c _ 0 + \\beta ^ 2 d _ 0 + 2 \\beta \\gamma e _ 0 + \\gamma ^ 2 f _ 0 = 0 . \\end{align*}"}
-{"id": "9367.png", "formula": "\\begin{align*} R _ \\textrm { h r } = \\log _ 2 \\left ( \\frac { 2 \\pi \\sigma ^ 2 } { 3 D } \\right ) . \\end{align*}"}
-{"id": "9594.png", "formula": "\\begin{align*} \\big \\langle f , D _ { k } ( g ) \\big \\rangle = \\big \\langle D _ { k } ( f ) , g \\big \\rangle \\qquad \\ g \\in \\dot { \\mathcal B } ^ { \\alpha , q } _ { p , \\mathcal F } , \\ f \\in \\big ( \\dot { \\mathcal B } ^ { \\alpha , q } _ { p , \\mathcal F } \\big ) ' . \\end{align*}"}
-{"id": "5681.png", "formula": "\\begin{align*} u \\circ \\mathcal R _ 2 = \\mathcal R _ n \\circ u , \\end{align*}"}
-{"id": "1923.png", "formula": "\\begin{align*} \\textbf { F } ( \\Gamma ^ { 1 } _ { j } ) = \\Gamma ^ { 1 } _ { j + 1 } \\textbf { F } ( \\Gamma ^ { 2 } _ { j } ) = \\Gamma ^ { 2 } _ { j + 1 } j \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "142.png", "formula": "\\begin{align*} 4 \\det ( N ) = ( N ) ^ 2 ; \\ ( M N ) = a \\cdot ( N ) + b z ; \\ 4 \\det ( M N ) = ( M N ) ^ 2 . \\end{align*}"}
-{"id": "1575.png", "formula": "\\begin{align*} V ( \\mathbf { j } ) = v ( \\mathbf { j } ) + \\gamma ( n ) . \\end{align*}"}
-{"id": "5427.png", "formula": "\\begin{align*} \\left \\langle x - \\alpha e , y - \\overline { \\beta } e \\right \\rangle = \\left \\langle x , y \\right \\rangle - \\alpha \\left \\langle e , y \\right \\rangle - \\beta \\left \\langle x , e \\right \\rangle + \\alpha \\beta , \\end{align*}"}
-{"id": "6952.png", "formula": "\\begin{align*} \\sigma _ k ( \\lambda ) = \\sum _ { 1 \\leq i _ 1 < i _ 2 < . . . < i _ k \\leq n } { \\lambda _ { i _ 1 } \\lambda _ { i _ 2 } . . . \\lambda _ { i _ k } } \\end{align*}"}
-{"id": "4634.png", "formula": "\\begin{align*} \\sum _ { n = 2 ^ { k - 1 } } ^ { 2 ^ k - 1 } 1 _ { [ - B + C , B - C ] } \\left ( t + \\sum _ { i = 0 } ^ { n - 1 } f ( T ^ i x ) \\right ) \\ge \\beta \\tau \\sum _ { n = 0 } ^ { 2 ^ k - 1 } 1 _ { [ - B - C , B + C ] } \\left ( t + \\sum _ { i = 0 } ^ { n - 1 } f ( T ^ i x ) \\right ) \\end{align*}"}
-{"id": "4253.png", "formula": "\\begin{align*} \\omega _ { \\theta , 1 } = \\left ( \\begin{array} { c c } 0 & \\frac { t - a } { \\lambda - 1 } \\\\ 0 & 0 \\\\ \\end{array} \\right ) \\cdot \\mathrm { d } \\log \\frac { t - \\lambda } { t - 1 } , \\end{align*}"}
-{"id": "8329.png", "formula": "\\begin{align*} H ( f _ 0 ) = \\sum _ { \\substack { m \\in \\Q _ { > 0 } \\\\ \\lambda \\in V _ { 0 \\Z } ^ \\vee / V _ { 0 \\Z } } } c _ 0 ( - m , \\lambda ) \\cdot H ( m , \\lambda ) . \\end{align*}"}
-{"id": "3217.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } \\bigl ( \\| \\phi _ k - \\phi _ { k - 1 } \\| _ { E _ 1 ( T ) } + \\| \\phi _ k - \\phi _ { k - 1 } \\| _ { Y _ 1 ( T ) \\cap Z _ 1 ( T ) } \\bigr ) < \\infty . \\end{align*}"}
-{"id": "8708.png", "formula": "\\begin{align*} ( \\partial _ t ^ \\alpha u ) ( t ) = \\frac { \\alpha } { \\Gamma ( 1 - \\alpha ) } \\int _ { - \\infty } ^ t \\frac { u ( t ) - \\tilde { u } ( s ) } { ( t - s ) ^ { \\alpha + 1 } } d s = : M _ t [ u ] , \\end{align*}"}
-{"id": "3576.png", "formula": "\\begin{align*} \\mathcal { X } \\otimes \\mathcal { Y } = \\{ X \\times Y \\colon X \\in \\mathcal { X } , Y \\in \\mathcal { Y } \\} . \\end{align*}"}
-{"id": "1472.png", "formula": "\\begin{align*} \\mathrm { I m } \\ , d _ { 2 ^ { r + 2 } - 1 } = ( v _ { r + 1 } e _ { r + 1 } ) \\{ x _ 3 ^ 2 \\} \\oplus D _ r \\langle v _ { r + 1 } x _ 3 ^ 2 y _ { r + 1 } , v _ { r + 1 } x _ 3 z _ { r + 1 } \\rangle , \\end{align*}"}
-{"id": "1430.png", "formula": "\\begin{align*} \\partial _ 2 \\cdots \\partial _ { \\ell + 1 } ( x _ I ) = e _ { \\ell } ( x _ { 4 i _ 1 } , \\dots , x _ { 4 i _ \\ell } ) ^ 2 . \\end{align*}"}
-{"id": "5027.png", "formula": "\\begin{align*} \\tilde { g } ^ \\omega _ { \\lambda , z } ( x ) & = \\det ( C _ n G ^ { \\omega , \\lambda } _ { p , n , n } ( z ) - x I ) \\\\ & = \\det ( C _ n G ^ \\omega _ { n , n } ( z ) - \\lambda C _ n G ^ \\omega _ { n , p } ( z ) ( I + \\lambda C _ p G ^ \\omega _ { p , p } ( z ) ) ^ { - 1 } C _ p G ^ \\omega _ { p , n } ( z ) - x I ) \\\\ & = \\frac { p _ l ^ \\omega ( z , \\lambda ) x ^ l + p _ { l - 1 } ^ \\omega ( z , \\lambda ) x ^ { l - 1 } + \\cdots + p _ 0 ^ \\omega ( z , \\lambda ) } { \\det ( C _ p ^ { - 1 } + \\lambda G ^ \\omega _ { n , n } ( z ) ) } , \\end{align*}"}
-{"id": "2722.png", "formula": "\\begin{align*} | | h | | _ { H _ { x , v } ^ { s } H _ z ^ r } ^ 2 = \\int _ { I _ z } \\ , | | h | | _ { H _ { x , v } ^ { s , r } } ^ 2 \\ , \\pi ( z ) d z \\leq | | h | | _ { H _ { x , v } ^ { s , r } L _ z ^ { \\infty } } ^ 2 \\ , \\int _ { I _ z } \\pi ( z ) d z \\leq C _ { I } ^ 2 \\ , e ^ { - 2 \\tau _ s t } \\ , , \\end{align*}"}
-{"id": "9490.png", "formula": "\\begin{align*} d _ { n , d } = d _ { n - 1 , d + r - 2 } - \\sum \\limits _ { i = 1 } ^ { r - 2 } q ^ { ( i ) } _ { n - 1 , d + r - 2 - i } + \\dfrac { d ( d - 1 ) } { 2 } s _ { n - 1 , d - 2 } + 2 \\sum \\limits _ { i = 0 } ^ { n - 1 } \\sum \\limits _ { j = 0 } ^ { d - 2 } s _ { n - i - 1 , d - j - 2 } \\cdot d _ { i , j } . \\end{align*}"}
-{"id": "2180.png", "formula": "\\begin{align*} e ^ { \\frac { d s } { 2 } + s } ( \\partial _ t u _ * ) ( e ^ { \\frac { s } { 2 } } \\xi , e ^ s - 1 ) = e ^ { \\frac { d s } { 2 } + s } ( \\partial _ t u _ * ) ( 0 , e ^ s - 1 ) + O ( \\xi ^ 2 ) \\end{align*}"}
-{"id": "9420.png", "formula": "\\begin{align*} M _ n ^ { ( 4 ) } = m ^ { \\rm ( i n ) } _ { n , 4 } + 2 \\cdot m ^ { \\rm ( o u t ) } _ { n , 2 } + \\sum _ { k = 1 } ^ n \\dfrac { n } { k } \\cdot \\tilde { p } _ { n - k , k } . \\end{align*}"}
-{"id": "2450.png", "formula": "\\begin{align*} \\vartheta _ { C _ 1 ^ * } . \\vartheta _ { C _ { 3 } ^ * } & = y , \\\\ \\vartheta _ { C _ i ^ * } . \\vartheta _ { C _ { i } ^ * } & = \\vartheta _ { 2 C _ i ^ * } , \\\\ \\vartheta _ { C _ i ^ * } . \\vartheta _ { C _ { i + 1 } ^ * } & = \\vartheta _ { C _ i ^ * + C _ { i + 1 } ^ * } , \\end{align*}"}
-{"id": "9099.png", "formula": "\\begin{align*} \\Delta ^ { n } _ { y } A ^ { \\ast } ( x ) = n ! A ^ { \\ast } ( y ) . \\end{align*}"}
-{"id": "10019.png", "formula": "\\begin{align*} f = \\sum _ { m > 0 } c ^ + _ 0 ( - m ) f _ m , \\end{align*}"}
-{"id": "5412.png", "formula": "\\begin{align*} 4 ( B ^ 2 - A C ) k ^ 3 - 1 2 B k ^ 2 + 9 k - 4 A = 0 , \\end{align*}"}
-{"id": "3238.png", "formula": "\\begin{align*} \\sum _ { r = - \\infty } ^ { + \\infty } M ^ { r } H ^ { 2 r + k } ( X ) . \\end{align*}"}
-{"id": "255.png", "formula": "\\begin{align*} \\widehat { \\chi } _ { L , m , n } : = \\chi _ { L , n , \\mathcal { M } _ { m , n } } \\chi _ { \\{ x _ { m + 1 } , . . . , x _ { d } \\in [ 0 , 1 ] \\} } \\end{align*}"}
-{"id": "8510.png", "formula": "\\begin{align*} E : \\ ; y ^ 2 = f ( x ) , \\end{align*}"}
-{"id": "964.png", "formula": "\\begin{align*} E [ | U ^ * _ n ( \\theta ) | ^ 2 | \\mathcal { F } ^ X ] = \\sum _ { I \\in \\Pi ^ 1 _ n , J \\in \\Pi ^ 2 _ n } X ^ 1 ( I ) ^ 2 X ^ 2 ( J ) ^ 2 K ( I , J _ { - \\theta } ) \\in \\overline { \\mathcal { P } } _ 4 ( \\mathbb { H } ) \\end{align*}"}
-{"id": "8602.png", "formula": "\\begin{align*} \\gamma ( i , j ) : = ( \\gamma ( i ) , t ^ { g } _ { i } ( \\gamma ) ( j ) ) . \\end{align*}"}
-{"id": "2917.png", "formula": "\\begin{align*} \\rho _ w = \\prod _ { i = 1 } ^ n \\rho _ { e _ i } = \\rho _ { e _ 1 } \\cdots \\rho _ { e _ n } = \\rho _ { e _ n } \\circ \\cdots \\circ \\rho _ { e _ 1 } \\end{align*}"}
-{"id": "3501.png", "formula": "\\begin{align*} J ( u ( \\cdot ) ) = E \\big { [ } \\displaystyle \\int _ 0 ^ T f ( X ( t ) , u ( t ) ) d t + \\Phi ( X ( \\gamma _ 1 ) , X ( \\gamma _ 2 ) , \\cdots , X ( \\gamma _ N ) ) \\big { ] } , \\end{align*}"}
-{"id": "7927.png", "formula": "\\begin{align*} B ( x , r ) \\subset \\displaystyle \\bigcup _ { i = 1 } ^ { C } B ( x _ i , r / 2 ) \\end{align*}"}
-{"id": "1273.png", "formula": "\\begin{align*} \\mu ( K ) = \\int _ { \\mathbf { g } ^ { - 1 } ( K ) } f ( \\nabla U ) \\ , d \\mathcal { H } ^ { n - 1 } \\mbox { w h e n e v e r } K \\subset \\mathbb { S } ^ { n - 1 } \\ , \\ , \\mbox { i s a B o r e l s e t } \\end{align*}"}
-{"id": "1595.png", "formula": "\\begin{align*} j _ i ''' = \\begin{cases} 0 , & 1 \\leq i \\leq n - 3 \\\\ 0 , & i = n - 2 \\\\ 1 6 q + 6 & i = n - 1 \\\\ 5 & i = n , \\end{cases} \\end{align*}"}
-{"id": "1313.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| \\phi _ n ( a ) - \\pi ( a ) \\| = 0 ~ a \\in A . \\end{align*}"}
-{"id": "8492.png", "formula": "\\begin{align*} \\abs { W _ { \\pi } ( g _ { t , l , v } ) } \\leq \\begin{cases} 1 & l < a _ 2 t = - n , \\\\ q ^ { - \\frac { t + n } { 2 } } & l = a _ 2 - n \\leq t < - a _ 1 , \\\\ q ^ { \\frac { l - a _ 2 } { 2 } } & l > a _ 2 t = - a _ 1 - l , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "8746.png", "formula": "\\begin{align*} & F _ \\varepsilon ( t , x , w , p , X ) = \\min \\{ F ( t , x ' , w , p , X ) \\mid x ' \\in B _ { x , \\varepsilon } \\} \\quad \\\\ & G ^ \\varepsilon ( t , x , w , p , X ) = \\max \\{ G ( t , x ' , w , p , X ) \\mid x ' \\in B _ { x , \\varepsilon } \\} . \\end{align*}"}
-{"id": "247.png", "formula": "\\begin{align*} \\chi _ { L , \\pi } : = \\chi _ { [ 0 , L ] ^ d } \\chi _ { \\{ x _ { \\pi ( 1 ) } \\leq . . . \\leq x _ { \\pi ( d ) } \\} } \\end{align*}"}
-{"id": "6197.png", "formula": "\\begin{align*} \\lambda _ 1 & : = 2 [ \\mathcal { O } _ X ] - [ \\mathcal { O } _ { \\Sigma } ] - 2 [ \\mathcal { O } _ L ] + [ \\mathcal { O } _ P ] \\\\ \\lambda _ 2 & : = 4 [ \\mathcal { O } _ X ] - 2 [ \\mathcal { O } _ H ] - [ \\mathcal { O } _ S ] - 5 [ \\mathcal { O } _ L ] + 5 [ \\mathcal { O } _ P ] . \\end{align*}"}
-{"id": "7376.png", "formula": "\\begin{align*} \\hat { H } _ { G e n } = - \\sum _ { n = 0 } ^ { N } \\sum _ { m = 1 } ^ { N / 2 } \\Delta _ { m } \\Big ( \\sigma _ { n } ^ { + } \\sigma _ { n + m } ^ { - } + \\sigma _ { n } ^ { + } \\sigma _ { n - m } ^ { - } \\Big ) , \\end{align*}"}
-{"id": "8995.png", "formula": "\\begin{gather*} \\Gamma { \\cal S } ^ { ( n ) } _ { 2 c ; q , t } ( 0 , d ( s + f ) ) _ { q = t = 0 } \\cong \\Gamma { \\cal S } ^ { ( n ) } _ { - 2 c ; q , t } ( 0 , d ( s + f ) ) _ { q = t = 0 } . \\end{gather*}"}
-{"id": "1002.png", "formula": "\\begin{align*} \\sup _ { t \\in [ a _ n , T - a _ n ] } \\left | \\widehat { z } _ n ( t ) - z _ n ( t ) \\right | = O _ p \\left ( \\frac { \\log ^ { 3 / 2 } n } { ( n h _ n ) ^ { 1 / 4 } } \\right ) . \\end{align*}"}
-{"id": "10032.png", "formula": "\\begin{align*} L _ h = L _ { 1 , h } \\oplus \\Lambda _ h \\end{align*}"}
-{"id": "3374.png", "formula": "\\begin{align*} \\left \\langle \\left \\langle \\cdot , \\cdot \\right \\rangle \\right \\rangle : \\mathbb { C } l _ { ( n + m ) } \\times \\mathbb { C } l _ { ( n + m ) } & \\rightarrow \\mathbb { C } l _ { ( n + m ) } \\\\ ( \\xi _ { 1 } , \\xi _ { 2 } ) & \\mapsto \\left \\langle \\left \\langle \\xi _ { 1 } , \\xi _ { 2 } \\right \\rangle \\right \\rangle = \\tau ( \\xi _ { 2 } ) \\xi _ { 1 } . \\end{align*}"}
-{"id": "787.png", "formula": "\\begin{align*} \\left \\vert ( 2 x ) ^ r \\cdot \\left ( \\frac { ( 2 x ) ^ B } { y } \\right ) ^ n - 1 \\right \\vert = \\frac { s } { y ^ n } . \\end{align*}"}
-{"id": "5400.png", "formula": "\\begin{align*} K : = \\langle ( 1 , 3 ) , ( 4 , 6 ) , ( 7 , 9 ) \\rangle \\leq G . \\end{align*}"}
-{"id": "6368.png", "formula": "\\begin{align*} \\Phi _ { p ^ * _ 1 , p ^ * _ 2 } ( z ) : = \\int \\frac { d \\mu _ { p ^ * _ 1 , p ^ * _ 2 } ( x ) } { z - x } \\end{align*}"}
-{"id": "1436.png", "formula": "\\begin{align*} \\partial _ j ( f _ { k , k } ) = 0 . \\end{align*}"}
-{"id": "6612.png", "formula": "\\begin{align*} \\int _ { \\Omega _ 0 } \\bigl | w _ { i j } ^ { ( n ) } ( x ) \\bigr | ^ p \\ , d \\mu ( x ) & = \\int _ { \\Omega _ 0 } \\bigl | \\sum _ { k = 1 } ^ d u _ { i k } ( \\phi _ n ( x ) ) v _ { k j } ^ { ( n ) } ( x ) \\bigr | ^ p \\ , d \\mu ( x ) \\\\ & = \\int _ { \\Omega _ 0 } \\bigl | \\sum _ { k = 1 } ^ d u _ { i k } ( y ) v _ { k j } ^ { ( n ) } ( \\phi _ n ^ { - 1 } ( y ) ) \\bigr | ^ p J _ { \\phi _ n ^ { - 1 } } ( y ) \\ , d \\mu ( y ) \\ , \\end{align*}"}
-{"id": "8038.png", "formula": "\\begin{align*} \\| \\ , \\mbox { \\boldmath $ u $ } ( \\cdot , t ) \\ , \\| _ { \\mbox { } _ { \\scriptstyle L ^ { q } ( \\mathbb { R } ^ { 3 } ) } } \\ ; \\ ! = \\ ; \\Bigl \\{ \\ , \\sum _ { i \\ , = \\ , 1 } ^ { 3 } \\int _ { \\mathbb { R } ^ { 3 } } \\ ! | \\ : u _ { i } ( x , t ) \\ , | ^ { q } \\ , d x \\ , \\Bigr \\} ^ { \\ ! \\ ! \\ : \\ ! 1 / q } \\end{align*}"}
-{"id": "756.png", "formula": "\\begin{align*} \\int _ { \\R ^ 3 } ( x _ 1 ^ 2 + x _ 2 ^ 2 ) | u ^ * | ^ 2 \\ , d x = \\int _ { \\R ^ 2 } ( x _ 1 ^ 2 + x _ 2 ^ 2 ) \\int _ { \\R } | u ^ * | ^ 2 \\ , d x _ 3 d x ' = \\int _ { \\R ^ 3 } ( x _ 1 ^ 2 + x _ 2 ^ 2 ) | u | ^ 2 \\ , d x , \\end{align*}"}
-{"id": "4420.png", "formula": "\\begin{align*} ( - \\partial _ 1 ^ 2 - | \\partial _ 1 | ^ { - 1 } \\partial _ 2 ^ 2 ) u + P ( u \\partial _ 2 R u ) + \\frac { 1 } { 2 } \\partial _ 2 R u ^ 2 - \\frac { 1 } { 2 } P ( u \\partial _ 1 R u ^ 2 ) = \\sigma P \\xi , \\end{align*}"}
-{"id": "6015.png", "formula": "\\begin{align*} F ^ { ( \\alpha , \\theta ) } ( R ) & : = \\frac { \\Omega ^ { ( \\alpha , \\theta ) } - \\theta \\alpha R } { 1 + ( 5 - 3 \\alpha ) \\theta } , \\\\ * F ( R ) & : = \\sup _ { ( \\alpha , \\theta ) \\in [ 0 , 1 ] \\times [ 0 , \\infty ) } F ^ { ( \\alpha , \\theta ) } ( R ) . \\end{align*}"}
-{"id": "1558.png", "formula": "\\begin{align*} C _ { j _ k } ( \\alpha ( k ) ) = \\frac { \\alpha ( \\alpha - 1 ) ( \\alpha - 2 ) \\cdots ( \\alpha - ( j _ k - 1 ) ) } { j _ k ! } . \\end{align*}"}
-{"id": "4806.png", "formula": "\\begin{align*} \\mathcal { A } ( M ) : = \\{ a \\in M \\mid a M \\} . \\end{align*}"}
-{"id": "6687.png", "formula": "\\begin{align*} T _ { t _ 0 } ( t ) = f ( t _ 0 ) + f ' ( t _ 0 ) ( t - t _ 0 ) = \\left ( 1 - t _ 0 ^ p \\right ) ^ { \\frac { 1 } { p } - 1 } ( 1 - t _ 0 ^ { p - 1 } t ) \\quad . \\end{align*}"}
-{"id": "9309.png", "formula": "\\begin{align*} C _ 2 : = \\{ ( h , t , x ) : ( h , t , x ) \\in D , \\Delta ^ { k + 2 } _ h f _ x ( t ) \\leq 0 \\} . \\end{align*}"}
-{"id": "3802.png", "formula": "\\begin{align*} \\mathbb { P } ^ { \\rho _ \\infty } \\Big ( X ^ 0 _ n \\geq \\hat { v } n \\Big ) \\leq c ^ { - 1 } \\exp \\big ( - c ( \\log n ) ^ { \\gamma } \\big ) \\end{align*}"}
-{"id": "7814.png", "formula": "\\begin{align*} \\liminf _ { r \\to \\infty } ~ r ^ { \\mu } [ M ( r ) ^ 2 + N ( r ) ^ 2 ] = \\infty . \\end{align*}"}
-{"id": "8875.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - \\theta ( z ) } = \\int _ { \\mathbb T } \\frac { 1 } { 1 - z \\overline \\zeta } \\mu ( \\zeta ) , \\ \\ z \\in \\mathbb D , \\end{align*}"}
-{"id": "4815.png", "formula": "\\begin{align*} m \\theta ( r ) = \\theta ( m r ) = \\theta ( n s ) = n \\theta ( s ) . \\end{align*}"}
-{"id": "3712.png", "formula": "\\begin{align*} \\frac { \\xi _ j } { \\xi _ { s ( j ) } } = \\alpha . \\end{align*}"}
-{"id": "1549.png", "formula": "\\begin{gather*} A y \\ = \\ \\begin{bmatrix} 3 D \\\\ D \\\\ \\vdots \\\\ 3 D \\\\ D \\end{bmatrix} \\pmod { 4 D } . \\end{gather*}"}
-{"id": "8858.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac 1 N \\sum _ { i , j = 1 } ^ N P _ { n _ 0 } ^ { ( d ) } ( \\langle \\mathbf { x } _ i , \\mathbf { x } _ j \\rangle ) = \\infty . \\end{align*}"}
-{"id": "3999.png", "formula": "\\begin{align*} p ^ { \\nu _ 3 } _ { k - 1 } ( 3 , t ) = ( - 1 ) ^ { k - 1 } \\underset { \\Lambda ^ { k - 1 } _ { 3 } } { \\sum } \\frac { \\lambda _ 1 ^ { k _ 1 + 1 } \\lambda _ 2 ^ { k _ 2 } \\lambda _ 3 ^ { k _ 3 - 1 } t ^ { k _ 1 \\nu _ 1 + k _ 2 \\nu _ 2 + k _ 3 \\nu _ 3 } } { \\Gamma \\left ( k _ 1 \\nu _ 1 + k _ 2 \\nu _ 2 + k _ 3 \\nu _ 3 + 1 \\right ) } . \\end{align*}"}
-{"id": "5094.png", "formula": "\\begin{align*} & \\int _ { \\widehat { G } / \\mathcal { L } ^ \\perp } p _ \\psi ( \\xi ) d \\xi = \\int _ { \\widehat { G } / \\mathcal { L } ^ \\perp } \\ ! \\Big ( \\sum _ { m \\in \\mathcal { L } ^ \\perp } | \\widehat { \\psi } ( \\xi + m ) | ^ 2 \\Big ) d \\xi \\\\ & = \\int _ { \\widehat { G } } | \\widehat { \\psi } ( \\zeta ) | ^ 2 d \\zeta = \\| \\widehat { \\psi } \\| _ { \\textup { L } ^ 2 ( \\widehat { G } ) } ^ 2 = \\| \\psi \\| _ { \\textup { L } ^ 2 ( G ) } ^ 2 < + \\infty . \\end{align*}"}
-{"id": "8447.png", "formula": "\\begin{align*} \\chi ( 1 + z \\varpi ^ { \\alpha } ) = \\psi ( z b _ { \\xi } \\varpi ^ { \\alpha - a ( \\chi ) } ) z \\in \\mathcal { O } . \\end{align*}"}
-{"id": "3123.png", "formula": "\\begin{align*} p _ { i j } : = \\frac { \\frac { 1 } { L _ i } + \\frac { 1 } { L _ j } } { ( N - 1 ) \\sum _ { t = 1 } ^ N \\frac { 1 } { L _ t } } , ~ ~ ~ ( i , j ) \\in E . \\end{align*}"}
-{"id": "5707.png", "formula": "\\begin{align*} - z '' + \\nabla W ( z ) = 0 , \\end{align*}"}
-{"id": "4810.png", "formula": "\\begin{align*} \\mathcal { L } ( M ) : = \\{ \\mathsf { L } ( x ) \\mid x \\in M \\} . \\end{align*}"}
-{"id": "7395.png", "formula": "\\begin{align*} \\sum \\omega _ i t _ i = 0 \\end{align*}"}
-{"id": "1884.png", "formula": "\\begin{align*} \\pi ( \\hat { f } _ 1 , f _ 2 , a ) = \\nu _ 1 ( F _ 1 ^ { i _ 1 - 1 } ) + \\pi ( f _ 1 , f _ 2 , a ) = \\nu _ 1 ( F _ 1 ^ { i _ 1 - 1 } ) + \\phi _ 0 ( \\ 1 _ { B \\setminus S } ) , \\end{align*}"}
-{"id": "770.png", "formula": "\\begin{align*} ( x + 1 ) ^ k + ( x + 2 ) ^ k + . . . + ( l x ) ^ k = y ^ n , x , y \\geq 1 , n \\in \\mathbb { Z } , n \\geq 2 \\end{align*}"}
-{"id": "6188.png", "formula": "\\begin{align*} ( a ^ 2 + b ^ 2 & + c ^ 2 ) ( A ^ 2 + B ^ 2 + C ^ 2 ) - ( a A + b B + c C ) ^ 2 \\\\ & - 2 ( a ^ 2 + b ^ 2 + c ^ 2 ) ( a ^ 2 A + b ^ 2 B + c ^ 2 C ) + 2 ( a ^ 3 + b ^ 3 + c ^ 3 ) ( a A + b B + c C ) \\\\ & = 3 ( b A - a B ) ^ 2 - 2 ( a - b ) ( 2 a + b ) ( a + 2 b ) ( b A - a B ) \\\\ & = 3 \\left \\{ ( b A - a B ) - \\frac { 1 } { 3 } ( a - b ) ( 2 a + b ) ( a + 2 b ) \\right \\} ^ 2 - \\frac { 1 } { 3 } ( a - b ) ^ 2 ( 2 a + b ) ^ 2 ( a + 2 b ) ^ 2 \\end{align*}"}
-{"id": "4519.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\psi ( \\Delta _ n ) = 1 \\end{align*}"}
-{"id": "6486.png", "formula": "\\begin{align*} \\theta _ { } \\left ( \\omega \\right ) = \\frac { 2 \\omega } { \\Omega _ { } ^ { 2 } } \\underset { 0 } { \\overset { \\Omega _ { } } { \\int } } \\theta _ { } \\left ( \\omega \\right ) d \\omega = 1 \\Omega _ { } = \\gamma \\Omega \\gamma \\in \\mathbb { R } \\end{align*}"}
-{"id": "2683.png", "formula": "\\begin{align*} E ( X _ { \\C } ) : = \\mathrm { E n d } _ { \\mathrm { H d g } } ( T ( X _ { \\C } ) ) \\otimes _ { \\Z } { \\Q } \\end{align*}"}
-{"id": "175.png", "formula": "\\begin{align*} Z \\setminus ( Z ' \\cup ( Y ' \\cap Z ) ) & = ( Z \\setminus Z ' ) \\cap ( Z \\setminus ( Y ' \\cap Z ) ) = ( Z \\setminus Z ' ) \\cap ( Z \\setminus Y ' ) \\subseteq Y \\cap ( Z \\setminus Y ' ) \\subseteq Y \\setminus Y ' \\subseteq X . \\end{align*}"}
-{"id": "7829.png", "formula": "\\begin{align*} S ( \\mathcal { A } _ { m } ) & = S ( I _ q \\otimes \\mathcal { J } _ { m - 1 } + Q \\otimes \\mathcal { A } _ { m - 1 } ) \\\\ & = S ( I _ q \\otimes ( J _ q \\otimes \\mathcal { A } _ { m - 2 } ) ) + S ( Q ) S ( \\mathcal { A } _ { m - 1 } ) \\\\ & = S ( J _ q ) S ( I _ q ) S ( \\mathcal { A } _ { m - 2 } ) \\\\ & = q ^ 3 \\ , S ( \\mathcal { A } _ { m - 2 } ) . \\end{align*}"}
-{"id": "9632.png", "formula": "\\begin{align*} P _ { \\rm { c u } } = \\gamma _ { \\rm { t r } } ^ * ( t ) P _ { \\rm { t r } } ( h _ { \\beta , 1 } ^ * ( t ) ) , \\end{align*}"}
-{"id": "4652.png", "formula": "\\begin{align*} | I _ k | = | I _ { k , 1 } | + | I _ { k , 2 } | + \\cdots + | I _ { k , b } | \\le | I _ { k , j } | + ( b - 1 ) D _ 2 | I _ { k , j } | = ( 1 + ( b - 1 ) D _ 2 ) | I _ { k , j } | \\end{align*}"}
-{"id": "3086.png", "formula": "\\begin{align*} \\begin{cases} \\dot { w } _ \\epsilon ( s ) = - \\nabla _ x F ( t _ \\epsilon + \\epsilon s , w _ \\epsilon ( s ) ) \\\\ w _ \\epsilon ( 0 ) = u _ \\epsilon ( t _ \\epsilon ) . \\end{cases} \\end{align*}"}
-{"id": "8387.png", "formula": "\\begin{align*} F _ { m , \\mu } ^ + ( \\tau ) = \\left ( \\frac { \\phi _ { \\mu } + \\phi _ { - \\mu } } { 2 } \\right ) \\cdot q ^ { - m } + O ( 1 ) , \\end{align*}"}
-{"id": "7623.png", "formula": "\\begin{align*} & \\int _ { Q _ R } f ^ l \\ , d \\mu \\leq ( c _ 0 \\lambda _ 0 M ) ^ l \\mu ( Q _ R ) \\\\ & + \\left [ ( M ^ l - 1 ) ( c _ 0 \\lambda _ 0 ) ^ l \\mu \\big ( E _ f ( Q _ { 2 R } , c _ 0 \\lambda _ 0 ) \\big ) + M ^ l \\int _ { Q _ { 2 R } } g ^ l \\ , d \\nu + M ^ l \\frac { M ^ l - 1 } { ( M ^ { \\frac { l } { \\hat p } } - 1 ) ^ { \\hat p } } \\Big ( \\int _ { Q _ { 2 R } } \\hat g ^ { \\frac { l } { \\hat p } } \\ , d \\hat \\nu \\Big ) ^ { \\hat p } \\right ] \\sum _ { j = 1 } ^ \\infty ( \\alpha M ^ l ) ^ j . \\end{align*}"}
-{"id": "9966.png", "formula": "\\begin{align*} \\bar { \\gamma } _ { k i } ^ { s } = \\frac { \\rho _ { k i i } ^ { 2 } \\bar { \\upsilon } _ { k i } ^ { s } } { \\rho _ { k i i } { \\displaystyle \\sum _ { j = 1 } ^ { L } \\alpha _ { j } \\mathbb { E } \\left [ \\mathsf { \\Gamma } _ { j i } \\right ] + \\bar { \\upsilon } _ { k i } ^ { s } \\sum _ { j \\neq i } ^ { L } \\rho _ { k j i } ^ { 2 } } } \\end{align*}"}
-{"id": "6244.png", "formula": "\\begin{align*} { \\boldsymbol \\Lambda } _ { \\rm b } \\left ( \\mu _ { \\rm b } \\right ) { \\boldsymbol \\Psi } _ { \\rm b } \\bar { \\bf H } _ { \\rm b } ^ H + \\bar { \\bf H } _ { \\rm b } ^ H \\left ( { \\bf Q } _ { \\rm s } - \\gamma { \\bf Q } _ { \\rm n } \\right ) \\bar { \\bf H } _ { \\rm b } { \\boldsymbol \\Psi } _ { \\rm b } \\bar { \\bf H } _ { \\rm b } ^ H = { \\bf 0 } . \\end{align*}"}
-{"id": "4488.png", "formula": "\\begin{align*} \\psi ( x _ 0 ) = 1 > | \\phi ( x _ 0 ) | \\end{align*}"}
-{"id": "5985.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\underbrace { \\frac { ( \\sqrt { 2 } \\varepsilon ^ 2 L ) ^ { n } } { \\gamma ^ n \\sqrt { n ! } } } _ { t _ { n } } . \\end{align*}"}
-{"id": "797.png", "formula": "\\begin{align*} \\varLambda _ { r } = b _ { 2 } \\log \\alpha _ { 2 } - b _ { 1 } \\log \\alpha _ { 1 } , \\end{align*}"}
-{"id": "1931.png", "formula": "\\begin{align*} \\Vert \\textbf { G } _ { i } ( 0 _ { i } ) ( p ) \\Vert & = \\Vert _ { p } ^ { - 1 } \\circ g _ { i } \\circ _ { f _ { i } ^ { - 1 } ( p ) } ( 0 _ { f _ { i } ^ { - 1 } ( p ) } ) \\Vert = \\Vert _ { p } ^ { - 1 } ( g _ { i } f _ { i } ^ { - 1 } ( p ) ) \\Vert \\\\ & = d ( g _ { i } f _ { i } ^ { - 1 } ( p ) , p ) = d ( g _ { i } f _ { i } ^ { - 1 } ( p ) , f _ { i } f _ { i } ^ { - 1 } ( p ) ) < \\xi ^ { \\prime } \\end{align*}"}
-{"id": "8622.png", "formula": "\\begin{align*} \\int _ { \\R ^ { d + 1 } } \\Phi _ { t , x , B } ( s ' , y ' ) d W ( s ' , y ' ) = \\left ( \\int _ { - \\infty } ^ r + \\int _ r ^ \\infty \\right ) \\int _ { \\R ^ d } \\Phi _ { t , x , B } ( s ' , y ' ) d W ( s ' , y ' ) , \\end{align*}"}
-{"id": "1925.png", "formula": "\\begin{align*} D f _ { j } ( \\Pi _ { j } ( \\Gamma _ { j } ^ { r } ) ) = \\Pi _ { j + 1 } ( \\Gamma _ { j + 1 } ^ { r } ) j \\in \\mathbb { Z } r = 1 , 2 . \\end{align*}"}
-{"id": "1764.png", "formula": "\\begin{align*} W f = f \\sqrt { d \\mu _ \\pi } . \\end{align*}"}
-{"id": "3048.png", "formula": "\\begin{align*} \\lambda ( w _ h , v _ h ) _ H + a ( w _ h , v _ h ) = ( g , v _ h ) _ H , \\forall v _ h \\in V _ { h , \\sigma } , \\end{align*}"}
-{"id": "144.png", "formula": "\\begin{align*} \\Lambda M ^ { \\sigma } = M \\Lambda \\end{align*}"}
-{"id": "9960.png", "formula": "\\begin{align*} \\big ( V ( { \\cal C } ) \\cup { \\cal N } _ s ( u ) \\big ) - V _ 1 = \\big ( V ( { \\cal C } ) \\cup { \\cal N } _ s ( \\varphi ( u ) ) \\big ) - V _ 2 . \\end{align*}"}
-{"id": "3777.png", "formula": "\\begin{align*} p _ \\star : = p _ { \\bullet } ( k _ \\star ) = \\inf _ { k \\ge k _ \\star } \\alpha ( k , x _ \\bullet ) > 0 , \\end{align*}"}
-{"id": "6055.png", "formula": "\\begin{align*} & \\lim _ { ( \\theta , Q _ { X Y U } ) \\to ( 0 , Q _ { X Y U } ' ) } \\frac { \\partial \\Omega ^ { ( \\alpha , \\theta ) } ( Q _ { X Y U } ) } { \\partial \\theta } \\\\ & = \\sum _ { x , y , u } Q _ { X Y U } ' ( x , y , u ) \\omega _ { Q _ { X Y U } ' } ^ { ( \\alpha ) } ( X , Y | U ) \\\\ & = R ^ { ( \\alpha ) } ( Q _ { X Y U } ' ) . \\end{align*}"}
-{"id": "1145.png", "formula": "\\begin{align*} y _ 2 ( x ^ 2 + x + 1 ) & = ( y ^ 2 + y ) ( x ^ 2 + x + y ^ 2 + y ) + ( y ^ 2 + y ) ( x + 1 ) \\\\ & = ( y ^ 2 + y ) ( y ^ 2 + y + x ^ 2 + 1 ) \\\\ & = ( y ^ 2 + y ) ( x ^ 3 + x ^ 2 + x ) . \\end{align*}"}
-{"id": "5937.png", "formula": "\\begin{align*} \\Phi ( \\alpha _ 1 , \\ldots , \\alpha _ { d - 1 } ) ~ = ~ \\left [ \\begin{array} { c c c c l } 1 & \\alpha _ 1 & & & \\\\ & 1 & \\alpha _ 2 & & \\\\ & & \\ddots & \\ddots & \\\\ & & & 1 & \\alpha _ { d - 1 } \\\\ & & & & ~ ~ 1 \\end{array} \\right ] , \\end{align*}"}
-{"id": "8126.png", "formula": "\\begin{gather*} T _ i ' = \\bigcup \\{ F _ k c : 1 \\leq k \\leq n , \\ , c \\in C _ k , F _ k c \\subseteq T _ i \\} , \\\\ T _ i '' = \\bigcap _ { s \\in F } s ^ { - 1 } T _ i . \\end{gather*}"}
-{"id": "6807.png", "formula": "\\begin{align*} L ( \\phi ) = h + \\sum \\limits _ { j = 1 } ^ 4 c _ j \\chi _ { R _ 1 , j } \\varphi _ { 0 , j } \\textrm { i n } \\mathbb { S } ^ 2 _ { \\lambda } . \\end{align*}"}
-{"id": "6950.png", "formula": "\\begin{align*} \\lambda = ( \\lambda _ 1 , \\lambda _ 2 , . . . , \\lambda _ n ) \\in \\overline { \\Gamma _ k } . \\end{align*}"}
-{"id": "4535.png", "formula": "\\begin{align*} f _ { n - 1 } \\circ \\dots \\circ f _ 1 \\circ f _ 0 \\ , = \\ , h _ n ^ { - 1 } \\circ f ^ n \\circ h \\end{align*}"}
-{"id": "7914.png", "formula": "\\begin{align*} 2 f ( z ) = x _ 1 + y _ 1 + x _ 2 + y _ 2 \\leq | x _ 1 | + | y _ 1 | + | x _ 2 | + | y _ 2 | = f ( x ) + f ( y ) . \\end{align*}"}
-{"id": "7809.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial r } \\int _ { S _ r } < X , \\nabla r > e ^ { - 2 \\rho } d x = \\int _ { S _ r } ( { \\rm d i v } X - 2 \\rho ^ { \\prime } < X , \\nabla r > ) e ^ { - 2 \\rho } d x . \\end{align*}"}
-{"id": "5557.png", "formula": "\\begin{align*} x = \\sum _ { n = 0 } ^ { 2 ^ { k } - 1 } \\nu _ { n } w _ { n } \\left ( t \\right ) = \\bar { \\nu } ^ { T } \\bar { w } _ { 2 ^ { k } } \\end{align*}"}
-{"id": "8693.png", "formula": "\\begin{align*} | 1 - a _ n z | = \\sqrt { 1 - 2 a _ n \\Re z + a _ n ^ 2 | z | ^ 2 } = 1 - a _ n \\Re z + o ( a _ n ) \\leq 1 - \\frac { a _ n \\Re z } 2 = 1 - \\frac { \\Re z } { 2 | z | } a _ n | z | , \\end{align*}"}
-{"id": "6446.png", "formula": "\\begin{align*} d s ^ { 2 } = \\frac { 1 } { \\sigma ^ { 2 } \\left ( 1 - \\rho ^ { 2 } \\right ) } d \\mu ^ { 2 } + \\frac { 4 } { \\sigma ^ { 2 } } d \\sigma ^ { 2 } \\end{align*}"}
-{"id": "8803.png", "formula": "\\begin{align*} \\overline { F } _ { \\varpi _ { p } } = \\textbf { 1 } _ { \\overline { X } _ { p } } \\end{align*}"}
-{"id": "8857.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac 1 N \\sum _ { i , j = 1 } ^ N P _ n ^ { ( d ) } ( \\langle \\mathbf { x } _ i , \\mathbf { x } _ j \\rangle ) = 0 . \\end{align*}"}
-{"id": "9503.png", "formula": "\\begin{align*} \\left . D ( z , t ) \\right | _ { z = z _ a ( t ) } = \\dfrac { \\left . \\biggl [ - \\dfrac { 1 } { 2 } \\ , z \\ , \\dfrac { \\partial ^ 2 A ( z , t ) } { \\partial z ^ 2 } - z ^ 2 \\ , \\dfrac { \\partial ^ 3 A ( z , t ) } { \\partial z ^ 3 } - t \\ , \\dfrac { \\partial \\tilde { Q } ^ { ( 1 ) } ( z , t ) } { \\partial z } - t \\ , Q ^ { ( 2 ) } ( z , t ) - t \\ , z \\ , \\dfrac { \\partial Q ^ { ( 2 ) } ( z , t ) } { \\partial z } \\biggr ] \\right | _ { z = z _ a ( t ) } } { \\left . \\dfrac { \\partial A ( z , t ) } { \\partial z } \\right | _ { z = z _ a ( t ) } } . \\end{align*}"}
-{"id": "269.png", "formula": "\\begin{align*} \\begin{cases} \\sum _ { j = 0 } ^ { k } t _ j = 1 , \\\\ \\sum _ { j = 0 } ^ { k } t _ j x _ j = s , \\end{cases} \\iff \\begin{cases} t _ 0 + t _ { k } = a \\\\ { - } t _ 0 { + } t _ { k } = b , \\end{cases} \\end{align*}"}
-{"id": "4556.png", "formula": "\\begin{align*} & [ E _ { i , k + 1 } , E _ { i + 1 , l } ] _ { q ^ { - 1 } } = q _ 1 [ E _ { i , k } , E _ { i + 1 , l + 1 } ] _ { q } \\ , , \\\\ & [ F _ { i , k + 1 } , F _ { i + 1 , l } ] _ { q } = q _ 3 ^ { - 1 } [ F _ { i , k } , F _ { i + 1 , l + 1 } ] _ { q ^ { - 1 } } \\ , , \\end{align*}"}
-{"id": "585.png", "formula": "\\begin{align*} M _ k : = \\underset { x \\in I } { \\sup } \\left | g ^ { ( k ) } ( x ) \\right | = \\underset { z \\in L } { \\sup } \\left | f ^ { ( k ) } ( z ) \\right | , \\end{align*}"}
-{"id": "3288.png", "formula": "\\begin{align*} \\Delta \\hat \\varphi _ \\xi ( y ) = - 2 K ( y _ \\xi ^ { - 1 } ( y ) ) e ^ { \\hat \\varphi _ \\xi ( y ) } \\hbox { f o r } y \\in B _ { 2 r _ 0 } ( 0 ) . \\end{align*}"}
-{"id": "4668.png", "formula": "\\begin{align*} \\min \\limits _ { x \\in F ^ { - 1 } ( y ) } \\| x _ n - x \\| = \\| A ^ \\dagger y _ n - A ^ \\dagger y \\| \\end{align*}"}
-{"id": "5508.png", "formula": "\\begin{align*} 1 - \\widehat F ( s ) & \\sim s ^ \\alpha \\sum _ { m = - \\infty } ^ \\infty \\left ( 1 - \\exp \\left [ 2 ^ { ( m - \\{ \\alpha \\log _ 2 s ^ { - 1 } \\} ) / \\alpha } \\right ] \\right ) 2 ^ { - m + \\{ \\alpha \\log _ 2 s ^ { - 1 } \\} } \\\\ & = : s ^ \\alpha q _ 0 ( s ) \\end{align*}"}
-{"id": "5706.png", "formula": "\\begin{align*} \\mathcal { C } ( z ) = \\{ z ( \\cdot - m ) \\ ; : \\ ; m \\in \\R \\} . \\end{align*}"}
-{"id": "8718.png", "formula": "\\begin{align*} & f ^ \\varepsilon ( z ) = \\sup _ { z ' \\in D } \\{ f ( z ' ) - \\varepsilon ^ { - 1 } | z - z ' | ^ 2 \\} \\quad \\\\ & f _ \\varepsilon ( z ) = \\inf _ { z ' \\in D } \\{ f ( z ' ) + \\varepsilon ^ { - 1 } | z - z ' | ^ 2 \\} . \\end{align*}"}
-{"id": "5730.png", "formula": "\\begin{align*} 0 = G ( t , m ( t ) ) : = \\int _ \\R z ' ( s ) \\cdot ( \\gamma ( t , s + m ( t ) ) - z ( s ) ) \\d s . \\end{align*}"}
-{"id": "2016.png", "formula": "\\begin{align*} \\sum _ { \\substack { j \\in \\Z \\\\ 1 \\leq a \\leq l } } p _ { a , j , s } x _ { i + s - 2 j - k _ a } y ^ 0 _ { a , i + j } = 0 , ~ ~ ~ i \\in \\Z \\end{align*}"}
-{"id": "10084.png", "formula": "\\begin{align*} \\theta ( X , Y ) = ( \\nabla _ { X } \\phi ) ( Y ) + ( \\nabla _ { X } \\psi ) ( Y ) - \\psi ( X ) \\psi ( Y ) - \\phi ( X ) \\phi ( Y ) - \\psi ( X ) \\phi ( Y ) - \\phi ( X ) \\psi ( Y ) , \\end{align*}"}
-{"id": "8213.png", "formula": "\\begin{align*} m _ j = \\exp \\biggl ( \\sum _ { s \\ge 4 } \\frac { \\varepsilon _ s } { 2 s } ( - \\hat { x } ) ^ s \\biggr ) M _ j \\end{align*}"}
-{"id": "1542.png", "formula": "\\begin{align*} C _ { A ^ * + B ^ * } & \\ = \\ \\bigcup _ { 1 \\leq i \\leq k , 1 \\leq j \\leq \\ell } C _ { I _ i ^ * + J _ j ^ * } \\\\ & \\ = \\ \\bigcup _ { 1 \\leq i \\leq k , 1 \\leq j \\leq \\ell } I _ i ^ * + J _ j ^ * \\\\ & \\ = \\ A ^ * + B ^ * . \\end{align*}"}
-{"id": "9947.png", "formula": "\\begin{align*} u _ t + ( u ^ m ) _ x + \\left [ u ^ { a _ 1 } ( u ^ { b _ 1 } ) _ { x x } \\right ] _ { x } + \\delta \\left [ u ^ { a _ 2 } ( u ^ { b _ 2 } ) _ { 4 x } \\right ] _ { x } = 0 , a _ 1 , a _ 2 \\geq 0 , \\ ; m , b _ 1 , b _ 2 \\geq 2 \\end{align*}"}
-{"id": "10145.png", "formula": "\\begin{align*} \\boldsymbol S _ { D _ k } ( 0 ) = \\boldsymbol R _ k ^ { - 1 / 2 } ( i ) \\boldsymbol \\Phi _ k \\boldsymbol { \\omega } _ k ( 0 ) , \\end{align*}"}
-{"id": "8772.png", "formula": "\\begin{align*} & ( x , y ) \\Psi _ { [ [ X _ 1 , X _ 2 ] , X _ 3 ] } ^ { ( f _ 1 , f _ 2 , f _ 3 ) } ( t _ 1 , t _ 2 , t _ 3 ) = ( x , y ) \\\\ & + \\Big ( t _ 1 ^ 3 t _ 2 ^ 3 \\left ( e ^ { - t _ 3 } - 1 \\right ) x + t _ 1 t _ 2 ^ 2 \\left ( e ^ { t _ 3 } - 1 \\right ) \\left ( t _ 1 t _ 2 - 1 \\right ) y , - t _ 1 ^ 2 t _ 2 e ^ { - t _ 3 } \\left ( e ^ { t _ 3 } - 1 \\right ) \\left ( t _ 1 ^ 2 t _ 2 ^ 2 + t _ 1 t _ 2 + 1 \\right ) x + t _ 1 ^ 3 t _ 2 ^ 3 \\left ( e ^ { t _ 3 } - 1 \\right ) y \\Big ) \\end{align*}"}
-{"id": "1326.png", "formula": "\\begin{align*} \\rho ( S _ e S _ e ^ * ) = \\begin{bmatrix} \\pi ( S _ e S _ e ^ * ) & X ' _ e \\\\ Y ' _ e & Z ' _ e \\end{bmatrix} \\forall e \\in E . \\end{align*}"}
-{"id": "855.png", "formula": "\\begin{align*} X ^ 1 _ t = x ^ 1 _ 0 + \\sigma _ 1 B ^ 1 _ t , X ^ 2 _ t = x ^ 2 _ 0 + \\sigma _ 2 B ^ 2 _ { t - \\vartheta } , t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "6720.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\bar { \\mathbb { P } } \\Big ( | \\mathbb { P } ( \\Theta > N ^ { \\gamma } t ) - e ^ { - t } | > \\epsilon \\Big ) = 0 . \\end{align*}"}
-{"id": "1254.png", "formula": "\\begin{align*} \\mathbf { g } _ { E } ( x ) = - \\frac { \\nabla u ( x ) } { | \\nabla u ( x ) | } \\end{align*}"}
-{"id": "2398.png", "formula": "\\begin{align*} & a _ 2 - a _ 1 a _ 6 = 2 a _ 3 + a _ 4 + a _ 1 ( 3 a _ 2 + a _ 5 ) + a _ 6 = R _ 2 = 0 , \\\\ & D _ 1 \\leq 0 , \\ a _ 6 \\left ( 3 a _ 2 - a _ 1 ( 3 a _ 3 + a _ 4 ) + a _ 5 \\right ) R _ { 1 1 3 } \\neq 0 . \\end{align*}"}
-{"id": "7108.png", "formula": "\\begin{align*} \\sum _ { k , l = 1 } ^ n \\ddot { f } ^ { k l } y _ k y _ l + 2 \\sum _ { k = 1 } ^ n \\frac { \\dot { f } ^ k } { \\kappa _ k } y _ k ^ 2 ~ \\geq ~ 2 f ^ { - 1 } ( \\sum _ { k = 1 } ^ n \\dot { f } ^ k y _ k ) ^ 2 \\end{align*}"}
-{"id": "5128.png", "formula": "\\begin{align*} \\begin{cases} i \\partial _ t u + \\Delta u = | u | ^ 2 u \\\\ u | _ { t = 0 } = u _ 0 \\in H ^ s ( \\R ^ 3 ) , \\end{cases} ( t , x ) \\in \\R \\times \\R ^ 3 . \\end{align*}"}
-{"id": "6016.png", "formula": "\\begin{align*} F ( R ) = 0 . \\end{align*}"}
-{"id": "2436.png", "formula": "\\begin{align*} \\frac { n - c _ 1 } { c _ 2 } - k \\frac { n } { d } = \\frac { n / d - b _ 1 } { b _ 2 } . \\end{align*}"}
-{"id": "3615.png", "formula": "\\begin{align*} { \\operatorname { C a p } _ p } ( \\Gamma ) = { \\operatorname { C a p } _ p } ( R _ { \\lambda } ( \\Gamma ) ) . \\end{align*}"}
-{"id": "4720.png", "formula": "\\begin{align*} g ^ { i j } S _ { , i } S _ { , j } = m ^ 2 , \\end{align*}"}
-{"id": "9680.png", "formula": "\\begin{align*} R _ i ( U _ a ) : \\ , \\ , d p = c ^ 2 d \\rho , \\ , \\ , d u = - \\lambda _ i d v , \\ , \\ , \\rho ( \\lambda _ i u - v ) d v = d p , \\ , \\ , d Z = 0 . \\end{align*}"}
-{"id": "6884.png", "formula": "\\begin{align*} \\frac { d } { d x } \\left ( P ( x ) \\frac { d u } { d x } \\right ) + Q ( x ) u & = 0 \\\\ \\frac { d } { d x } \\left ( P _ 1 ( x ) \\frac { d u _ 1 } { d x } \\right ) + Q _ 1 ( x ) u _ 1 & = 0 \\end{align*}"}
-{"id": "8885.png", "formula": "\\begin{align*} X U _ { ( \\theta ) c } = T X . \\end{align*}"}
-{"id": "9429.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { m } \\frac { \\varphi ( n _ k ) } { n _ k } \\geq c m ~ ~ m \\in \\N ~ ~ c > 0 , \\end{align*}"}
-{"id": "4886.png", "formula": "\\begin{align*} M ^ k = \\bigcup _ { \\underline { k } : \\sum i k _ i = k } M ^ k _ { \\underline { k } } , \\end{align*}"}
-{"id": "7414.png", "formula": "\\begin{align*} \\begin{aligned} \\tau _ 1 ^ { \\beta } & = ( t _ 2 + 2 t _ 3 ) / 3 , \\\\ \\tau _ 2 ^ { \\beta } & = ( t _ 2 - t _ 3 ) / 3 , \\\\ \\tau _ 3 ^ { \\beta } & = - ( 2 t _ 2 + t _ 3 ) / 3 , \\end{aligned} \\begin{aligned} \\tau _ 1 ^ { \\gamma } & = ( t _ 4 + 2 t _ 5 ) / 3 , \\\\ \\tau _ 2 ^ { \\gamma } & = ( t _ 4 - t _ 5 ) / 3 , \\\\ \\tau _ 3 ^ { \\gamma } & = - ( 2 t _ 4 + t _ 5 ) / 3 , \\end{aligned} \\begin{aligned} \\tau _ 1 ^ { \\delta } & = t _ 1 / 2 , \\\\ \\tau _ 2 ^ { \\delta } & = - t _ 1 / 2 . \\end{aligned} \\end{align*}"}
-{"id": "4456.png", "formula": "\\begin{align*} \\sum _ { k \\not = 0 } \\exp ( - T d ^ 3 ( k , 0 ) ) d ^ 2 ( k , 0 ) = T ^ { - \\frac 2 3 } \\sum _ { k \\not = 0 } \\exp ( - T d ^ 3 ( k , 0 ) ) \\big ( T ^ { \\frac 1 3 } d ( k , 0 ) \\big ) ^ 2 \\lesssim ( T ^ \\frac { 1 } { 3 } ) ^ { - \\frac { 9 } , { 2 } } \\end{align*}"}
-{"id": "564.png", "formula": "\\begin{align*} \\widehat { H } ( \\mathbf { Q } ) : = \\{ ( - v _ { Q _ 1 } ( h ) , \\ldots , - v _ { Q _ m } ( h ) ) \\in \\mathbb { Z } ^ m \\ : \\ h \\in R _ { \\mathbf { Q } } \\backslash \\{ 0 \\} \\} , \\end{align*}"}
-{"id": "817.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\left ( \\frac { 1 } { 2 } \\int _ { \\Omega } | \\vec { u } | ^ 2 \\ , d \\vec { x } \\right ) = - \\int _ { \\Omega } \\nu | \\nabla \\vec { u } | ^ 2 \\ , d \\vec { x } , \\end{align*}"}
-{"id": "4079.png", "formula": "\\begin{align*} { M } _ \\varphi ( \\Theta ) : = \\left \\{ ( x , u , v ) \\in \\mathbb { R } ^ { n _ { x } } \\times U \\times \\mathbb { R } ^ { n _ { x } } : \\varphi ( x , u , v ) + \\Theta \\in K _ { \\varphi } \\right \\} , \\end{align*}"}
-{"id": "398.png", "formula": "\\begin{align*} \\ln ( U _ { n p } ^ { - 1 } ) & = \\ln \\bigg [ ( \\sum _ { i } b _ { n i } ^ { 2 } ) ^ { p / 2 } / \\sum _ { i } b _ { n i } ^ { p } \\bigg ] \\\\ & \\gg \\ln [ n ^ { p ( 3 - 2 \\alpha ) / 2 } L ^ { p } ( n ) / ( n ^ { 1 + p ( 1 - \\alpha ) } L ^ p ( n ) ) ] \\\\ & \\sim \\frac { 1 } { 2 } ( p - 2 ) \\ln n . \\end{align*}"}
-{"id": "8672.png", "formula": "\\begin{align*} ( P ^ { n + 1 } ) _ { \\ell k } & = \\sum _ j ( P ^ n ) _ { \\ell j } ( P ) _ { j k } \\\\ & = \\sum _ { j \\in A } ( P ^ n ) _ { \\ell j } ( P ) _ { j k } + ( P ^ n ) _ { \\ell \\ell } ( P ) _ { \\ell k } , \\end{align*}"}
-{"id": "8824.png", "formula": "\\begin{align*} | c _ { I , \\chi } ^ { 0 } | & = \\Big | \\sum _ { a \\in \\overset { \\circ } { \\overline { E } } _ { I } \\cap \\overline { h } ^ { - 1 } ( 0 ) } \\Omega _ { \\chi } ( a ) \\Big | \\leq \\sum _ { a \\in \\overset { \\circ } { \\overline { E } } _ { I } \\cap \\overline { h } ^ { - 1 } ( 0 ) } 1 = \\# \\big ( \\overset { \\circ } { \\overline { E } } _ { I } \\cap \\overline { h } ^ { - 1 } ( 0 ) \\big ) \\\\ & \\leq \\# \\big ( \\overset { \\circ } { \\overline { E } } _ { I } \\big ) \\leq D _ { I } p ^ { n - \\# ( I ) } . \\end{align*}"}
-{"id": "3428.png", "formula": "\\begin{align*} f ( [ - 3 , - 2 ] \\cup [ 2 , 3 ] ) \\cap F _ 2 ( \\overline D _ 2 ) = \\{ f ( 2 ) , f ( - 2 ) \\} . \\end{align*}"}
-{"id": "8222.png", "formula": "\\begin{align*} \\biggl [ \\frac { 1 } { n ^ d } \\biggr ] ( t _ + - t _ - ) & = \\biggl [ \\frac { 1 } { n ^ d } \\biggr ] \\sum _ \\ell \\binom { k } { \\ell } ( - 1 ) ^ { k + \\ell } \\Bigl ( U _ { i + \\ell } - \\frac { 1 } { 2 } ( U _ { i + \\ell } ) ^ 2 + \\frac { 1 } { 3 } ( U _ { i + \\ell } ) ^ 3 \\cdots \\Bigr ) \\\\ & = \\begin{cases} \\hfil 0 & d < k - 1 \\\\ \\displaystyle \\frac { ( k - 2 ) ! } { r ^ { k - 1 } } & d = k - 1 \\end{cases} \\end{align*}"}
-{"id": "4207.png", "formula": "\\begin{align*} \\dot { \\Omega } _ i = \\Omega _ i ( \\Omega _ j + \\Omega _ k ) - \\Omega _ j \\Omega _ k , \\end{align*}"}
-{"id": "9735.png", "formula": "\\begin{align*} m : = \\sup _ { k \\geq 0 } \\Big \\{ \\frac { | y _ { k + 1 } - y _ { k } | } { h } \\Big \\} < \\delta _ 0 . \\end{align*}"}
-{"id": "10030.png", "formula": "\\begin{align*} L _ { 1 , h } = L _ { 1 \\Q } \\cap \\det ( h ) \\widehat L _ 1 \\end{align*}"}
-{"id": "7135.png", "formula": "\\begin{align*} r ( m , n ) = W \\log _ 2 \\left ( 1 + \\frac { P _ { \\mathrm { t x } } H _ { m , n } } { \\sigma ^ 2 + I _ { m , n } } \\right ) , \\end{align*}"}
-{"id": "3607.png", "formula": "\\begin{align*} \\int \\eta _ { k + 1 } ( \\d x _ { k + 1 } | x _ 1 , \\ldots , x _ { k } ) & = \\frac 1 { N - k } \\int \\| K _ { H _ { N - k } } \\psi _ { k + 1 } \\| ^ 2 \\mu ( \\d x _ { k + 1 } ) \\\\ & = \\frac 1 { N - k } \\int \\| K _ { H _ { N - k } } ( x _ { k + 1 } , \\cdot ) \\| ^ 2 \\mu ( \\d x _ { k + 1 } ) \\\\ & = \\frac 1 { N - k } \\int K _ { H _ { N - k } } ( x _ { k + 1 } , x _ { k + 1 } ) \\mu ( \\d x _ { k + 1 } ) \\\\ & = \\frac 1 { N - k } \\mathrm { T r } ( K _ { N - k } ) = 1 . \\end{align*}"}
-{"id": "5689.png", "formula": "\\begin{align*} \\mathcal { K } ( { v } ) = \\begin{cases} \\sqrt { \\mathfrak { E } _ { W } ( { v } ) - d _ K ( a ^ - , a ^ + ) } & { v } \\in H ^ 1 _ { l o c } ( \\R , \\R ^ n ) , \\\\ + \\infty & \\end{cases} \\end{align*}"}
-{"id": "9303.png", "formula": "\\begin{align*} A _ k : = \\{ x \\in B : \\dim \\pi _ k ( A _ x ) \\geq 1 \\} . \\end{align*}"}
-{"id": "4369.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\Big \\| P _ \\alpha B _ t A | _ { F _ \\alpha ^ p } - \\sum _ { j = j _ 0 ( t ) } ^ \\infty P _ \\alpha M _ { \\varphi _ { j , t } } | _ { F _ \\alpha ^ p } \\Big \\| = 0 . \\end{align*}"}
-{"id": "6497.png", "formula": "\\begin{align*} \\psi ^ { } \\left ( k _ { 1 } , k _ { 2 } , t \\right ) = \\left ( N \\right ) ^ { - 1 / 2 } \\left [ \\psi _ { 1 } \\left ( k _ { 1 } \\right ) \\psi _ { 2 } \\left ( k _ { 2 } \\right ) e ^ { - i \\hbar \\left ( k _ { 1 } ^ { 2 } + k _ { 2 } ^ { 2 } \\right ) t / \\left ( 2 m \\right ) } + \\varepsilon \\psi _ { \\mathrm { s c a t } } \\left ( k _ { 1 } , k _ { 2 } , t \\right ) \\right ] , \\end{align*}"}
-{"id": "9331.png", "formula": "\\begin{align*} \\left ( j _ { n - 3 } ^ { ( 3 ) } \\right ) ^ { 2 } + 3 J _ { n } ^ { ( 3 ) } j _ { n } ^ { ( 3 ) } = 4 ^ { n } , \\end{align*}"}
-{"id": "10016.png", "formula": "\\begin{align*} f _ m ^ + ( \\tau ) = \\phi _ 0 \\cdot q ^ { - m } + O ( 1 ) , \\end{align*}"}
-{"id": "4321.png", "formula": "\\begin{align*} X : = \\big \\{ ( q , n ) : \\| q \\| _ { L ^ { 2 } ( 0 , T _ { 0 } ; H ^ { 1 } ( \\Omega ) ) } + \\| n \\| _ { L ^ { 2 } ( 0 , T _ { 0 } ; L ^ { 2 } ( \\Omega ) ) } \\leq R , \\ ; 0 \\leq n \\leq 1 \\Omega \\times ( 0 , T _ { 0 } ) \\big \\} , \\end{align*}"}
-{"id": "4839.png", "formula": "\\begin{align*} D ^ 2 F _ p - \\left ( \\tfrac { 2 } { 3 } \\frac { 1 } { H ( p ) } { D H } _ p + \\Lambda ( p ) D F _ p \\right ) D F _ p = 0 . \\end{align*}"}
-{"id": "630.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n = 0 } ^ { \\infty } a _ n z ^ n . \\end{align*}"}
-{"id": "4052.png", "formula": "\\begin{align*} ( \\pi _ 0 ) _ \\# \\eta & = \\mu , & ( \\pi _ \\infty ) _ \\# \\eta & = \\nu , \\end{align*}"}
-{"id": "9396.png", "formula": "\\begin{align*} \\phi _ { k + j } \\ast \\mathbb D _ j f ( x ) - \\mathbb E _ { k + j } \\mathbb D _ { j } f ( x ) & = \\int _ { | y | \\le 2 ^ { k + j } } \\phi _ { k + j } ( y ) [ \\mathbb D _ j f ( x - y ) - \\mathbb D _ j f ( x ) ] d y \\\\ & \\quad + \\sum _ { d \\ge 1 } \\int _ { E _ { k , j , d } } \\phi _ { k + j } ( y ) [ \\mathbb D _ j f ( x - y ) - \\mathbb D _ j f ( x ) ] d y \\\\ & : = I _ 0 ( x ) + \\sum _ { d \\ge 1 } I _ d ( x ) , \\end{align*}"}
-{"id": "1391.png", "formula": "\\begin{align*} \\{ \\hat U _ { - b , - a } ( x ) \\le y \\} & = \\{ \\hat U _ { - a - 1 } \\circ \\cdots \\circ \\hat U _ { - b } ( x ) \\le y \\} = \\{ V _ { a } \\circ \\cdots \\circ V _ { b - 1 } ( x ) \\le y \\} \\\\ & = \\{ V _ { a + 1 } \\circ \\cdots \\circ V _ { b - 1 } ( x ) \\le U _ a ( y ) \\} = \\ldots = \\{ x \\le U _ { b - 1 } \\circ \\cdots \\circ U _ a ( y ) \\} = \\{ x \\le U _ { a , b } ( y ) \\} . \\end{align*}"}
-{"id": "6676.png", "formula": "\\begin{align*} 1 \\geq \\left \\langle \\lambda ( \\delta ) x , \\left ( 1 + ( 1 + \\tilde { \\Psi } _ y ( \\delta ) ) \\frac { \\tilde { c } ( S , n ) } { G ( x ) } \\delta ^ { \\frac { 2 } { n + 1 } } \\right ) y \\right \\rangle = \\lambda ( \\delta ) \\left ( 1 + ( 1 + \\tilde { \\Psi } _ y ( \\delta ) ) \\frac { \\tilde { c } ( S , n ) } { G ( x ) } \\delta ^ { \\frac { 2 } { n + 1 } } \\right ) \\quad . \\end{align*}"}
-{"id": "3019.png", "formula": "\\begin{align*} \\norm { ( W _ 1 - W _ 2 ) ^ * x } ^ 2 & = \\ < ( W _ 1 - W _ 2 ) ^ * x , ( W _ 1 - W _ 2 ) ^ * x \\ > \\\\ & = \\ < x , ( W _ 1 - W _ 2 ) ( W _ 1 - W _ 2 ) ^ * x \\ > = \\ < x , 0 \\ > = 0 \\end{align*}"}
-{"id": "1698.png", "formula": "\\begin{align*} T _ \\lambda T _ \\nu ( f ) & = T _ \\lambda ( f _ \\nu \\cdot ( f \\circ \\tau ^ { d ( \\nu ) } ) ) = f _ \\lambda \\cdot ( f _ \\nu \\circ \\tau ^ { d ( \\lambda ) } ) \\cdot ( f \\circ \\tau ^ { d ( \\lambda ) } \\circ \\tau ^ { d ( \\nu ) } ) ; \\\\ T _ { \\lambda \\nu } ( f ) & = f _ { \\lambda \\nu } \\cdot ( f \\circ \\tau ^ { d ( \\lambda \\nu ) } ) , \\end{align*}"}
-{"id": "9108.png", "formula": "\\begin{align*} a _ { 2 } A ( x y ) - a _ { 2 } \\left ( x A ( y ) + y A ( x ) \\right ) = 0 \\left ( x , y \\in R \\right ) \\end{align*}"}
-{"id": "4495.png", "formula": "\\begin{align*} s = p + ( I - p ) s ( I - p ) . \\end{align*}"}
-{"id": "2461.png", "formula": "\\begin{align*} L ( a , \\Delta _ k ) \\rtimes \\sigma = L ( a + ( \\Delta _ k ) ; \\sigma ) + L ( a ; \\delta ( \\Delta _ { k } ; \\sigma ) ) . \\end{align*}"}
-{"id": "4881.png", "formula": "\\begin{gather*} \\prod _ { m = 1 } ^ { \\infty } \\left ( \\prod _ { n = 1 } ^ { \\infty } \\frac { 1 } { 1 - ( t ^ m ) ^ n } \\right ) ^ { \\mu ( m ) } = \\prod _ { k = 1 } ^ { \\infty } \\left ( \\prod _ { m \\vert k } \\frac { 1 } { 1 - t ^ k } \\right ) ^ { \\mu ( m ) } \\\\ = \\prod _ { k = 1 } ^ { \\infty } \\left ( \\frac { 1 } { 1 - t ^ k } \\right ) ^ { \\sum _ { m \\vert k } \\mu ( m ) } = \\prod _ { k = 1 } ^ { \\infty } \\left ( \\frac { 1 } { 1 - t ^ k } \\right ) ^ { \\sum _ { m \\vert k } \\mu ( m ) } = \\frac { 1 } { 1 - t } , \\end{gather*}"}
-{"id": "7969.png", "formula": "\\begin{align*} r : = | \\Delta | . \\end{align*}"}
-{"id": "5271.png", "formula": "\\begin{align*} I ( p , q ) : = \\{ f \\in C ( [ 0 , 1 ] , M _ p ( \\C ) \\otimes M _ q ( \\C ) \\mid f ( 0 ) \\in M _ p ( \\C ) \\otimes \\C \\hbox { a n d } f ( 1 ) \\in \\C \\otimes M _ q ( \\C ) \\} . \\end{align*}"}
-{"id": "8434.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { t , l , v } ) = \\sum _ { \\mu \\in \\mathfrak { X } _ l } c _ { t , l } ( \\mu ) \\mu ( v ) . \\end{align*}"}
-{"id": "1915.png", "formula": "\\begin{align*} d ^ { 1 } ( \\phi , \\psi ) = d ^ { 0 } ( \\phi , \\psi ) + \\max _ { 1 \\leq i \\leq l _ { 1 } } \\sup _ { u \\in U _ { x _ { i } } } \\Vert D _ { u } ( \\tilde { \\phi } _ { x _ { i } } ) - D _ { u } ( \\tilde { \\psi } _ { x _ { i } } ) \\Vert + \\max _ { 1 \\leq i \\leq l _ { 2 } } \\sup _ { v \\in V _ { y _ { i } } } \\Vert D _ { v } ( \\hat { \\phi } _ { y _ { i } } ) - D _ { v } ( \\hat { \\psi } _ { y _ { i } } ) \\Vert , \\end{align*}"}
-{"id": "9252.png", "formula": "\\begin{align*} D _ { \\alpha _ { 1 } , \\alpha _ { 2 } } . \\left \\{ \\beta _ { 1 } , \\beta _ { 2 } \\right \\} : = \\left \\{ D _ { \\alpha _ { 1 } , \\alpha _ { 2 } } \\beta _ { 1 } , \\beta _ { 2 } \\right \\} + \\left \\{ \\beta _ { 1 } , D _ { \\alpha _ { 1 } , \\alpha _ { 2 } } \\beta _ { 2 } \\right \\} \\end{align*}"}
-{"id": "9317.png", "formula": "\\begin{align*} Z : = \\{ ( a , b ) \\in [ 0 , 1 ] ^ 2 \\ : \\ a < b \\} . \\end{align*}"}
-{"id": "3522.png", "formula": "\\begin{align*} u ^ { \\varepsilon } ( t ) = \\left \\{ \\begin{array} [ c ] { c l } \\bar { u } ( t ) , & t \\in \\lbrack 0 , T ] \\backslash E _ { \\varepsilon } , \\\\ u ( t ) , & t \\in E _ { \\varepsilon } , \\end{array} \\right . \\end{align*}"}
-{"id": "8357.png", "formula": "\\begin{align*} f ( \\tau ) = \\sum _ { \\substack { m \\in \\Q \\\\ m \\gg - \\infty } } c ( m ) \\cdot q ^ m \\in M ^ ! _ { 1 - \\frac { n } { 2 } } ( \\overline { \\rho } _ { V _ \\Z } ) \\end{align*}"}
-{"id": "463.png", "formula": "\\begin{align*} p _ { 1 , k _ 1 , k _ 2 } ( x , t ) = \\sum _ { h = 0 } ^ N \\frac { 1 } { ( 2 h + 1 ) ! } | t | ^ { 2 h + 1 } p _ { 1 , k _ 1 , k _ 2 + 2 h + 1 } ( x , 0 ) + O \\left ( \\abs { t } ^ { 2 N + 3 } p _ { 1 , k _ 1 , k _ 2 + 2 N + 3 } ( x , 0 ) \\right ) \\end{align*}"}
-{"id": "5772.png", "formula": "\\begin{align*} _ { p } ( E ) = \\inf \\big \\lbrace \\Vert \\nabla u \\Vert ^ { p } _ { L ^ { p } ( \\R ^ n ) } \\colon u \\geq 1 \\ ; \\ ; \\ ; \\ ; E , \\ , \\ , u \\in C _ { 0 } ^ { \\infty } ( \\R ^ n ) \\big \\rbrace . \\end{align*}"}
-{"id": "7713.png", "formula": "\\begin{align*} d _ B ( j , k ) = \\frac { \\sqrt { 2 } } { N } \\ , . \\end{align*}"}
-{"id": "930.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ d E \\left [ \\frac { \\partial \\varphi } { \\partial x _ j } ( \\sqrt { t } F + \\sqrt { 1 - t } Z ) \\frac { Z _ j } { \\sqrt { 1 - t } } \\right ] = \\sum _ { i , j = 1 } ^ d E \\left [ \\frac { \\partial ^ 2 \\varphi } { \\partial x _ i \\partial x _ j } ( \\sqrt { t } F + \\sqrt { 1 - t } Z ) \\mathfrak { C } ( i , j ) \\right ] . \\end{align*}"}
-{"id": "7222.png", "formula": "\\begin{align*} \\mathbb { F } _ K ( T ) = \\inf \\{ \\mathbf { M } _ K ( S ) + \\mathbf { M } _ K ( R ) : T = S + \\partial R , \\ ; \\ ; \\ ; S , R \\ ; { \\rm { a r e \\ ; i n t e g r a l \\ ; c u r r e n t s } } \\} . \\end{align*}"}
-{"id": "8848.png", "formula": "\\begin{align*} \\sigma \\left ( C ( \\mathbf { x } , \\phi ) \\right ) = \\gamma _ d \\int _ 0 ^ \\phi \\sin ( \\theta ) ^ { d - 1 } \\dd \\theta \\asymp \\phi ^ { d } \\quad \\phi \\to 0 , \\end{align*}"}
-{"id": "2513.png", "formula": "\\begin{align*} - \\int h ( \\xi \\cdot \\nabla _ x h ) \\varrho \\ , m _ 0 = \\frac { 1 } { 2 } \\int \\left ( \\xi \\cdot \\nabla _ x ( \\varrho \\ , m _ 0 ) \\right ) h ^ 2 , \\end{align*}"}
-{"id": "3514.png", "formula": "\\begin{align*} J ^ { n } ( \\bar { u } ( \\cdot ) ) = \\frac { 1 } { n } \\leq \\inf _ { u \\in \\mathcal { U } [ 0 , T ] } J ^ { n } ( u ( \\cdot ) ) + \\frac { 1 } { n } . \\end{align*}"}
-{"id": "8918.png", "formula": "\\begin{gather*} f ( z _ 1 , \\dots , z _ { i - 1 } , z _ i + 1 , z _ { i + 1 } , \\dots , z _ n ) = f ( z _ 1 , \\dots , z _ n ) , \\\\ f ( z _ 1 , \\dots , z _ { i - 1 } , z _ i + \\tau , z _ { i + 1 } , \\dots , z _ n ) = C _ i e \\bigg ( { - } \\sum _ j Q _ { i j } z _ j \\bigg ) f ( z _ 1 , \\dots , z _ n ) , \\end{gather*}"}
-{"id": "2326.png", "formula": "\\begin{align*} \\delta _ { \\Z } ( t ) = \\begin{cases} 1 & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "1018.png", "formula": "\\begin{align*} \\mathbf { T } = \\frac { 1 } { 2 } \\begin{bmatrix} 1 & 1 \\\\ 1 & 1 \\end{bmatrix} \\end{align*}"}
-{"id": "4254.png", "formula": "\\begin{align*} \\omega _ { \\nabla , 1 } = \\left ( \\begin{array} { c c } 0 & \\left ( \\frac { t - a } { \\lambda - 1 } \\right ) ^ p \\\\ 0 & 0 \\\\ \\end{array} \\right ) \\cdot \\mathrm { d } \\log \\frac { t - \\lambda } { t - 1 } \\end{align*}"}
-{"id": "5981.png", "formula": "\\begin{align*} \\beta = \\left ( 1 + \\frac { 4 \\varepsilon ^ 2 } { \\alpha ^ 2 } \\right ) ^ { 1 / 4 } = \\left ( 1 + \\frac { 4 \\varepsilon ^ 2 } { \\gamma ^ 2 - 2 \\varepsilon ^ 2 } \\right ) ^ { 1 / 4 } \\end{align*}"}
-{"id": "5419.png", "formula": "\\begin{align*} \\Sigma ^ { \\infty \\rho } _ { + } J ^ { \\sigma } X \\simeq \\left ( \\bigvee _ { i = 0 } ^ { \\infty } N _ { e } ^ { C _ { 2 } } ( i _ { e } ^ { \\ast } X ) ^ { \\wedge i } \\right ) \\wedge ( S ^ { 0 } \\vee X ) . \\end{align*}"}
-{"id": "359.png", "formula": "\\begin{align*} X _ { j , k } = \\sum _ { r , s \\in \\mathbb { Z } } a _ { r , s } \\xi _ { j - r , k - s } , \\end{align*}"}
-{"id": "4837.png", "formula": "\\begin{align*} a ' = H _ { x x } , \\ ; b ' = H _ { y y } , \\ ; c ' = H _ { z z } , \\ ; f ' = H _ { y z } , \\ ; g ' = H _ { x z } , \\ ; h ' = H _ { x y } . \\end{align*}"}
-{"id": "10091.png", "formula": "\\begin{align*} ' \\tilde { R } ( X , Y , Z , U ) = ' R ( X , Y , Z , U ) + \\lambda \\{ \\pi ( X ) \\pi ( Z ) g ( Y , U ) - \\pi ( Y ) \\pi ( Z ) g ( X , U ) \\} , \\end{align*}"}
-{"id": "3785.png", "formula": "\\begin{align*} & \\sum _ { j \\in J } p _ n ( 0 , x _ j ) = \\sum _ { i \\in I } \\sum _ { j \\colon x _ j \\in C _ i } p _ n ( 0 , x _ j ) \\le \\sum _ { i \\in I } \\rho L ^ d p _ n ( 0 , z _ i ) \\\\ & \\le \\rho \\sum _ { i \\in I } \\sum _ { x \\in C _ i } | p _ n ( 0 , x ) - p _ n ( 0 , z _ i ) | + \\rho . \\end{align*}"}
-{"id": "9486.png", "formula": "\\begin{align*} t _ { n , d } = t _ { n - 1 , d + r - 2 } + d _ { n - 1 , d - 2 } + 2 \\sum \\limits _ { i = 0 } ^ { n - 1 } \\sum \\limits _ { j = 0 } ^ { d - 2 } s _ { n - i - 1 , d - j - 2 } \\cdot t _ { i , j } , n \\geq 1 , d \\geq 1 , \\end{align*}"}
-{"id": "48.png", "formula": "\\begin{align*} \\hat { V } ^ { C } _ { \\sigma } ( C _ { 1 } , C _ { 2 } ) = \\int \\limits _ { - \\infty } ^ { \\infty } \\int \\limits _ { - \\infty } ^ { \\infty } \\frac { 1 } { N } \\sum \\limits _ { i = 1 } ^ N \\frac { 1 } { ( 4 \\pi ^ { 2 } \\sigma ^ { 4 } ) } e x p \\left ( - \\frac { 1 } { 2 \\sigma ^ { 2 } } ( a ( u _ 1 - b ) ^ { 2 } + a ' ( u _ 2 - b ' ) ^ { 2 } + c \\right ) \\mathrm { d } u _ 1 \\mathrm { d } u _ 2 \\end{align*}"}
-{"id": "609.png", "formula": "\\begin{align*} \\dim L _ q ^ n ( m ) = \\sum _ r \\dim C _ q ^ n ( r ) d _ { r , m } . \\end{align*}"}
-{"id": "9389.png", "formula": "\\begin{align*} \\int _ { \\mathbf S ^ { n - 1 } } \\Omega ( y ' ) d \\sigma ( y ' ) = 0 . \\end{align*}"}
-{"id": "6390.png", "formula": "\\begin{align*} \\int d x d \\theta f \\left ( \\theta \\right ) P \\left ( x \\theta \\right ) \\overset { } { = } \\left \\langle f \\left ( \\theta \\right ) \\right \\rangle = F \\end{align*}"}
-{"id": "9208.png", "formula": "\\begin{align*} \\alpha _ { 1 } \\alpha _ { 2 } : = \\frac { [ \\alpha _ { 1 } , \\alpha _ { 2 } ] } { 2 } + \\frac { \\alpha _ { 1 } \\circ \\alpha _ { 2 } } { 2 } \\end{align*}"}
-{"id": "2066.png", "formula": "\\begin{align*} S ^ p _ k ( \\Delta _ H , \\Omega ) = \\big \\{ \\psi \\in L ^ p ( \\Omega ) \\big | X _ { i _ 1 } \\dots X _ { i _ s } \\psi \\in L ^ p ( \\Omega ) , \\ s \\leq k \\mbox { a n d } X _ { i _ j } \\in \\{ X _ \\alpha , X _ { m + \\alpha } \\} _ { \\alpha = 1 } ^ m \\big \\} \\end{align*}"}
-{"id": "411.png", "formula": "\\begin{align*} \\int _ { V } e ^ { i R f ( \\lambda ) } g ( \\lambda ) \\ , \\dd \\lambda = e ^ { i R f ( 0 ) } \\sqrt { \\frac { ( 2 \\pi i ) ^ m } { R ^ m \\det f '' ( 0 ) } } \\sum _ { j = 0 } ^ k \\frac { L _ { j , f } g } { R ^ j } + O \\left ( \\frac { 1 } { R ^ { \\frac { m } { 2 } + k + 1 } } \\right ) \\end{align*}"}
-{"id": "9589.png", "formula": "\\begin{align*} D _ k ( R _ N ) ^ { m + 1 } D ^ N _ { k ' } & = D _ k \\sum _ { | k _ 0 - \\ell _ 0 | > N } D _ { k _ 0 } D _ { \\ell _ 0 } \\sum _ { | k _ 1 - \\ell _ 1 | > N } D _ { k _ 1 } D _ { \\ell _ 1 } \\cdots \\sum _ { | k _ m - \\ell _ m | > N } D _ { k _ m } D _ { \\ell _ m } D ^ N _ { k ' } \\\\ & = \\sum _ { | k _ 0 - \\ell _ 0 | > N } \\cdots \\sum _ { | k _ m - \\ell _ m | > N } D _ k D _ { k _ 0 } D _ { \\ell _ 0 } D _ { k _ 1 } D _ { \\ell _ 1 } \\cdots D _ { k _ m } D _ { \\ell _ m } D ^ N _ { k ' } . \\end{align*}"}
-{"id": "7954.png", "formula": "\\begin{align*} \\sum _ { I \\in \\mathcal S } \\lambda _ I ^ s = \\sum _ { I \\in \\mathcal S ' } \\lambda _ I ^ s = 1 \\end{align*}"}
-{"id": "5873.png", "formula": "\\begin{align*} 0 & \\leq \\langle M ' , X ' \\rangle = \\langle M , X \\rangle - y _ { \\omega } X _ { \\omega } \\\\ & = \\langle F _ { \\emptyset } , X \\rangle - \\sum _ { \\omega \\in \\Omega _ k ^ d } y _ { \\omega } b _ { \\omega } + \\sum _ { \\omega \\in \\Omega _ k ^ d } y _ { \\omega } \\epsilon _ { \\omega } - X _ { \\omega } y _ { \\omega } , \\end{align*}"}
-{"id": "8569.png", "formula": "\\begin{align*} \\overline { B _ { H _ 0 } ( x _ 0 , 1 ) } \\subset \\bigcup _ { i = 1 } ^ n \\bigcup _ k \\overline { B _ { H _ 0 } ( y _ { i , k } , \\sqrt { t } ) } , \\end{align*}"}
-{"id": "7596.png", "formula": "\\begin{align*} z = W y . \\end{align*}"}
-{"id": "5391.png", "formula": "\\begin{align*} a _ 1 \\cdot a _ 4 = \\frac { 1 } { 2 ^ 3 } ( a _ 1 + a _ 4 - a _ { 1 0 } ) . \\end{align*}"}
-{"id": "7944.png", "formula": "\\begin{align*} W : = V _ { I _ 0 } , \\end{align*}"}
-{"id": "574.png", "formula": "\\begin{align*} w _ n = \\alpha + 2 t \\frac { z _ n - \\alpha } { | z _ n - \\alpha | } = \\alpha + 2 t h _ n \\rightarrow a + 2 t h . \\end{align*}"}
-{"id": "1551.png", "formula": "\\begin{gather*} \\mathcal { D } \\ : = \\ ( \\mathcal { A } \\pm \\mathcal { A } ) \\ \\Delta \\ ( \\mathcal { C } \\pm \\mathcal { C } ) \\ = \\ ( \\mathcal { A } \\pm \\mathcal { A } ) \\setminus ( \\mathcal { C } \\pm \\mathcal { C } ) . \\end{gather*}"}
-{"id": "10136.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } { D } _ k \\bigg ( \\boldsymbol S _ { D _ k } ( i ) , \\bar { \\boldsymbol \\omega } _ k ( i ) \\bigg ) = { D } _ k \\bigg ( \\boldsymbol S _ { D _ k } ^ { \\rm o p t } , \\bar { \\boldsymbol \\omega } _ k ^ { \\rm o p t } \\bigg ) . \\end{align*}"}
-{"id": "7627.png", "formula": "\\begin{align*} \\liminf _ { k \\to \\infty } \\int _ { Q _ 3 } | D u ^ k - D w ^ k | ^ p \\ , d z = 0 \\end{align*}"}
-{"id": "9619.png", "formula": "\\begin{align*} P _ { \\rm { t r } , \\rm { u s e r } , \\xi } \\big ( r , h _ \\beta \\big ) = L _ \\xi ( r , h _ \\beta ) N _ 0 W \\left ( 2 ^ { C / W } - 1 \\right ) . \\end{align*}"}
-{"id": "8418.png", "formula": "\\begin{align*} \\epsilon ( \\frac { 1 } { 2 } , \\pi ) = \\gamma \\epsilon ( \\frac { 1 } { 2 } , \\xi ) , \\end{align*}"}
-{"id": "2823.png", "formula": "\\begin{align*} \\Delta _ { \\infty } ^ { ( n ) } = \\bigcap _ { m = 1 } ^ { \\infty } \\Delta _ m ^ { ( n ) } . \\end{align*}"}
-{"id": "4921.png", "formula": "\\begin{align*} 1 = \\pi ( I ) = \\pi ( A ) ^ * \\pi ( A ) . \\end{align*}"}
-{"id": "213.png", "formula": "\\begin{align*} A _ 1 = \\begin{bmatrix} C _ 1 & 0 & \\dots & 0 \\\\ 0 & C _ 2 & \\dots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\ldots & C _ N \\end{bmatrix} \\end{align*}"}
-{"id": "6410.png", "formula": "\\begin{align*} d l _ { \\xi \\rightarrow } ^ { 2 } = g _ { \\mu \\nu } \\left ( \\theta \\right ) \\left ( \\Delta \\theta ^ { \\mu } - d \\theta ^ { \\mu } \\right ) \\left ( \\Delta \\theta ^ { \\nu } - d \\theta ^ { \\nu } \\right ) \\end{align*}"}
-{"id": "3096.png", "formula": "\\begin{align*} \\alpha \\sigma v _ T = [ [ & { n + 1 } , n + 2 , \\ldots , { n + i } , { n + i + 1 } , \\ldots , { n + j - 1 } , { i + 1 } , { n + j } , \\ldots , { 2 n - 1 } ] , \\\\ & 1 , 2 , \\ldots , i , { i + 2 } , \\ldots , n ] . \\end{align*}"}
-{"id": "3974.png", "formula": "\\begin{align*} p ^ { \\alpha _ 1 } ( 1 , t ) = - \\sum _ { k = 1 } ^ { \\infty } ( - \\lambda ) ^ k \\underset { \\Theta ^ { k } _ { 1 } } { \\sum } \\frac { t ^ { k _ 0 \\alpha _ 0 + k _ 1 \\alpha _ 1 } } { \\Gamma \\left ( k _ 0 \\alpha _ 0 + k _ 1 \\alpha _ 1 + 1 \\right ) } , \\end{align*}"}
-{"id": "9898.png", "formula": "\\begin{align*} d ( x ) = \\sum _ { y \\in V } w ( x , y ) \\end{align*}"}
-{"id": "8019.png", "formula": "\\begin{align*} \\big \\Vert { T _ { [ a ] } f _ L } \\big \\Vert _ { F _ p ^ { { s } , q } } & = \\Big \\Vert \\Big ( \\sum _ { l = 0 } ^ { \\infty } { 2 ^ { { s } l q } \\Big | \\sum _ { n = M } ^ { L } { \\phi _ l \\ast d _ { { t _ n } } } \\Big | ^ q } \\Big ) ^ { { 1 } / { q } } \\Big \\Vert _ { L ^ p } \\\\ & \\gtrsim \\Big \\Vert \\Big ( \\sum _ { l = M } ^ { L - M } { 2 ^ { { s } { t _ l } q } \\Big | \\sum _ { n = M } ^ { L } { ( \\phi _ { { t _ l } } + \\phi _ { { t _ l } + 1 } ) \\ast d _ { { t _ n } } } \\Big | ^ q } \\Big ) ^ { { 1 } / { q } } \\Big \\Vert _ { L ^ p } . \\end{align*}"}
-{"id": "4744.png", "formula": "\\begin{align*} \\alpha ^ 2 a _ 0 + 2 \\alpha \\beta b _ 0 + \\beta ^ 2 c _ 0 - \\gamma = 0 . \\end{align*}"}
-{"id": "5296.png", "formula": "\\begin{align*} & w _ t = \\mu w _ { x x } - \\bigg ( \\frac { m ^ 2 } { 4 \\mu } + n \\bigg ) w , \\\\ & w _ x ( t , 1 ) = - a w ( t , 1 ) , \\\\ & w _ x ( t , 0 ) = - b w ( t , 0 ) + d ( t ) . \\end{align*}"}
-{"id": "5150.png", "formula": "\\begin{align*} q _ { m , j , \\alpha } ( [ f ] _ { m , j } ) = \\inf \\big \\{ p _ \\alpha ( g ) : g \\in [ f ] _ { m , j } \\big \\} \\end{align*}"}
-{"id": "7922.png", "formula": "\\begin{align*} \\begin{aligned} & \\varphi : ( 0 , \\infty ) \\to ( 0 , \\infty ) \\ , \\ , \\rm { i s \\ , \\ , i n c r e a s i n g \\ , \\ , w i t h } \\ , \\ , \\lim _ { x \\to + 0 } \\varphi ( x ) = 0 ; \\\\ & \\displaystyle { \\int _ 1 ^ { \\infty } \\varphi ( a \\nu ^ { x } ) \\ , d x < \\infty } \\ , \\ , \\rm { f o r \\ , \\ , s o m e \\ , \\ , c o n s t a n t s } \\ , \\ , a > 0 \\ , \\ , \\rm { a n d } \\ , \\ , 0 < \\nu < 1 . \\end{aligned} \\end{align*}"}
-{"id": "7047.png", "formula": "\\begin{align*} g ( t ) = \\frac { 1 - \\sum _ { 1 \\leq i < j \\leq n - 1 } { t _ i t _ j \\sigma _ i \\sigma _ j } } { \\sigma _ n ( t _ 1 \\sigma _ 1 + t _ 2 \\sigma _ 2 + . . . + t _ { n - 1 } \\sigma _ { n - 1 } ) } , \\end{align*}"}
-{"id": "8909.png", "formula": "\\begin{align*} \\lim _ N \\frac { 1 } { N + 1 } \\sum _ { n = 0 } ^ N U _ { ( \\theta ) 1 } ^ { - n } \\varphi ( U _ { ( \\theta ) c } ) U _ { ( \\theta ) 1 } ^ n x \\end{align*}"}
-{"id": "5734.png", "formula": "\\begin{align*} \\gamma ( t ) \\in \\mathcal { C } : = \\{ { v } : \\R \\to \\R ^ n \\pm ( s - s ^ \\pm ) \\geq 0 , \\ , | { v } ( s ) - a ^ \\pm | \\leq E ( s ) \\} \\subset X , \\end{align*}"}
-{"id": "3262.png", "formula": "\\begin{align*} ( \\alpha _ 0 ( 2 ) , \\omega ) _ X = ( \\alpha _ 0 ^ 0 , \\omega ) _ X . \\end{align*}"}
-{"id": "179.png", "formula": "\\begin{align*} \\mathrm { T h } _ { \\vdash } ( X ) = \\bigcup \\{ \\mathrm { T h } _ { \\vdash } ( x ) : x \\in X \\} . \\end{align*}"}
-{"id": "31.png", "formula": "\\begin{align*} \\frac { \\partial f } { \\partial z } ( c ) = \\frac { 1 } { 2 } \\left ( \\frac { \\partial f } { \\partial x } ( c ) - j \\frac { \\partial f } { \\partial y } ( c ) \\right ) \\end{align*}"}
-{"id": "2094.png", "formula": "\\begin{align*} E _ R ( \\psi ) = \\int _ M e _ R ( \\psi ) \\theta \\wedge ( d \\theta ) ^ m \\end{align*}"}
-{"id": "7300.png", "formula": "\\begin{align*} \\{ x ^ { a _ i } y ^ { b _ i } z ^ i , \\ , 1 \\leq i \\leq n - 2 , \\ , a _ i + b _ i = n - 1 - i \\} \\end{align*}"}
-{"id": "2447.png", "formula": "\\begin{align*} \\delta = \\frac { t \\lceil \\alpha k \\rceil } { k } . \\end{align*}"}
-{"id": "8859.png", "formula": "\\begin{align*} \\frac { V ( X _ N , \\phi _ N ) } { N \\sigma ( C ( \\cdot , \\phi _ N ) ) } = \\sum _ { n = 1 } ^ \\infty Z ( d , n ) \\frac { a _ n ( \\phi _ N ) ^ 2 } { \\sigma ( C ( \\cdot , \\phi _ N ) ) } \\frac 1 N \\sum _ { i , j = 1 } ^ N P _ n ^ { ( d ) } ( \\langle \\mathbf { x } _ i , \\mathbf { x } _ j \\rangle ) \\to 0 . \\end{align*}"}
-{"id": "9830.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - \\nu } { { n - i } \\choose { \\nu } } A _ i = - q ^ { \\frac { n } { 2 } - \\nu } \\sum _ { i = 0 } ^ \\nu { { n - i } \\choose { n - \\nu } } A _ i \\qquad ( \\nu = 0 , 1 , \\cdots , n ) . \\end{align*}"}
-{"id": "3267.png", "formula": "\\begin{align*} G ( \\theta ) = \\begin{cases} G _ { } , & \\ , \\ , \\ , \\ , \\ , \\theta < | \\theta _ m | , \\\\ G _ { } , & , \\end{cases} \\end{align*}"}
-{"id": "3786.png", "formula": "\\begin{align*} E ^ { \\mathbb { Q } } \\big [ \\exp \\{ L G _ J ( z ) \\} \\big ] = \\smash { \\prod _ { j \\in J } } E ^ { \\mathbb { Q } } \\big [ \\exp \\{ \\xi _ j L p _ n ( x _ j , z ) \\} \\big ] = \\prod _ { \\smash { j \\in J } } \\Big ( 1 - L p _ n ( x _ j , z ) \\Big ) ^ { - 1 } . \\end{align*}"}
-{"id": "4272.png", "formula": "\\begin{align*} \\gamma _ n = ( - 1 ) ^ { m + n } \\sum _ { \\begin{array} { c } i + j = n - m \\\\ 0 \\leq i , j \\leq m \\\\ \\end{array} } { m \\choose i } { m \\choose j } \\lambda ^ { m - j } . \\end{align*}"}
-{"id": "9170.png", "formula": "\\begin{align*} \\tilde { f } _ 3 ( x ^ 3 ) + x \\tilde { f } _ 2 ( x ^ 2 ) + x ^ 2 \\tilde { f } _ 1 ( x ) = 0 \\left ( x \\in K \\right ) , \\end{align*}"}
-{"id": "1422.png", "formula": "\\begin{align*} M _ n & = \\mathbb { Z } / 2 [ x _ 2 ^ 2 , x _ 3 , x _ { 4 1 } , \\dots , x _ { 4 n } ] , \\\\ L _ n & = \\mathbb { Z } / 2 [ x _ 2 ^ 2 , x _ 3 , x _ { 4 1 } ^ 2 , \\dots , x _ { 4 n } ^ 2 ] , \\end{align*}"}
-{"id": "8380.png", "formula": "\\begin{align*} - 2 \\log \\| \\tilde { \\psi } _ g ( f ^ { [ i ] } ) \\| = \\widetilde { \\Theta } ^ \\mathrm { r e g } _ g ( f ^ { [ i ] } ) . \\end{align*}"}
-{"id": "4612.png", "formula": "\\begin{align*} U _ i & = \\{ r ( n ) : \\ell _ i < n < 2 ^ { k _ i } \\textup { a n d } T _ f ^ n ( x , t ) \\in X _ { B - b } \\} \\\\ U _ i ' & = \\{ r _ i ( n ) : \\ell _ i < n < 2 ^ { k _ i } \\textup { a n d } T _ f ^ n ( x , t ) \\in X _ { B - b } \\} \\end{align*}"}
-{"id": "10112.png", "formula": "\\begin{align*} \\boldsymbol R _ k ( i ) = \\mathbb { E } [ \\boldsymbol x _ k ( i ) \\boldsymbol x _ k ^ H ( i ) ] , \\end{align*}"}
-{"id": "6092.png", "formula": "\\begin{align*} h ( x ) = \\cos ( g ( r ) ) f _ { \\frak { e } _ k } ( x / r ) _ j \\quad \\mbox { a n d } i ( x ) = \\sin ( g ( r ) ) f _ { \\frak { e } _ k } ( x / r ) _ j \\end{align*}"}
-{"id": "6848.png", "formula": "\\begin{align*} \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } f _ t ( x _ t ) - \\inf _ { x \\in X } \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } f _ t ( x ) \\leq r ( T ) , \\lim _ { T \\to \\infty } r ( T ) = 0 , \\end{align*}"}
-{"id": "2984.png", "formula": "\\begin{align*} - \\int _ { \\Omega } u \\Delta \\varphi = \\int _ { \\Omega } h ( u ) f \\varphi + \\int _ { \\Omega } \\varphi d \\mu , ~ \\forall \\varphi \\in C _ 0 ^ 2 ( { \\bar { \\Omega } } ) . \\end{align*}"}
-{"id": "7205.png", "formula": "\\begin{align*} W ( x , 0 + ) = \\lim _ { s \\rightarrow 0 } W _ p ( x , r ) , \\end{align*}"}
-{"id": "4083.png", "formula": "\\begin{align*} \\begin{array} { l } { \\lambda \\in N _ { K _ \\varphi } ( \\varphi ( \\bar x , \\bar u , \\bar v ) ) , } ( \\alpha , 0 , 0 ) \\in \\partial \\langle \\lambda , \\varphi \\rangle ( \\bar x , \\bar u , \\bar v ) + \\{ 0 \\} \\times N _ { U } ( \\bar u ) \\times \\{ 0 \\} \\end{array} \\Longrightarrow \\alpha = 0 \\end{align*}"}
-{"id": "7659.png", "formula": "\\begin{align*} & \\beta _ { \\mu } \\equiv i F _ { \\mu 0 } , \\beta _ { 0 } = i F _ { 0 0 } = 0 , \\\\ [ 1 e x ] & \\zeta _ { \\mu } \\equiv \\widetilde { F } _ { \\mu 0 } , \\quad \\ ; \\zeta _ { 0 } = \\widetilde { F } _ { 0 0 } \\neq 0 . \\end{align*}"}
-{"id": "836.png", "formula": "\\begin{align*} F = \\{ g _ f ^ + : f \\in E _ 0 \\cup E _ + \\} \\cup \\{ g _ f ^ - : f \\in E _ 0 \\cup E _ + \\} . \\end{align*}"}
-{"id": "9893.png", "formula": "\\begin{align*} e ( G ) = \\frac { e ( G _ t ) } { t ^ 2 } \\le \\frac { r - 1 } r \\cdot \\frac { n ^ 2 + n / t } 2 + \\frac { \\delta n ^ 2 } 2 \\ , . \\end{align*}"}
-{"id": "1599.png", "formula": "\\begin{align*} V ( \\mathbf { j } ''' ) & = 3 j ''' _ { n - 2 } - s ( j ''' _ { n - 2 } ) + 2 j ''' _ { n - 1 } - s ( j ''' _ { n - 1 } ) - c ( j ''' _ n , - \\alpha _ { \\mathbf { j } ''' } ( n ) - 1 ) \\\\ & = 3 ( 0 ) - s ( 0 ) + 2 ( 1 6 q + 6 ) - s ( 1 6 q + 6 ) - c ( 5 , 8 q ) \\\\ & = 3 2 q + 1 2 - s ( q ) - s ( 6 ) - c ( 5 , 8 q ) \\\\ & = 3 2 q + 1 0 - s ( q ) . \\end{align*}"}
-{"id": "4514.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| \\Delta _ n a \\Delta _ n ^ * \\| = | \\psi ( a ) | \\end{align*}"}
-{"id": "6784.png", "formula": "\\begin{align*} w _ { \\lambda , k } ( y ) = \\int _ { \\mathbb { S } ^ 2 } G ( y , y ' ) e ^ { U _ { \\lambda , \\xi _ k } } \\eta _ { R _ 0 , \\xi _ k } d H ^ 2 ( y ' ) \\end{align*}"}
-{"id": "1696.png", "formula": "\\begin{align*} \\frac { d \\mu } { d ( \\mu \\circ \\tau _ \\lambda ^ { - 1 } ) } | _ { R _ \\lambda } & = \\frac { d ( \\mu \\circ \\tau _ \\lambda \\circ \\tau ^ n ) } { d ( \\mu \\circ \\tau _ \\lambda ^ { - 1 } \\circ \\tau _ \\lambda \\circ \\tau ^ n ) } | _ { R _ \\lambda } = \\left . \\frac { d ( \\mu \\circ \\tau _ \\lambda ) } { d \\mu } \\circ \\tau ^ n \\right | _ { R _ \\lambda } \\\\ & = \\Phi _ \\lambda \\circ \\tau ^ { n } | _ { R _ \\lambda } , \\end{align*}"}
-{"id": "2879.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { n ^ { N - 2 h } } { e ^ { n ^ { N } x } - 1 } & = J ( x ) + R _ { 0 } + R _ { 1 } + R _ { \\frac { N - 2 h + 1 } { N } } + \\sum _ { j = 1 } ^ { \\left \\lfloor h / N \\right \\rfloor } R _ { - ( 2 j - 1 ) } , \\end{align*}"}
-{"id": "4265.png", "formula": "\\begin{align*} P _ { \\theta ' } ( t ) = \\frac { f _ o \\cdot \\mathrm { d } h _ o - h _ o \\cdot \\mathrm { d } f _ o } { \\mathrm { d } \\log \\frac { t - \\lambda } { t - 1 } } + f _ o ^ 2 \\cdot \\left ( \\frac { t - a } { \\lambda - 1 } \\right ) ^ p . \\end{align*}"}
-{"id": "1957.png", "formula": "\\begin{align*} c _ { k } = C C _ { 2 } ^ { k - 1 } \\prod _ { i = k } ^ { j - 1 } \\frac { 1 } { ( 1 + r _ { i } ) ^ { \\gamma } } , \\end{align*}"}
-{"id": "238.png", "formula": "\\begin{align*} f ( A ) = \\frac { 1 } { 2 \\pi } \\int _ { \\R ^ 2 } \\mathrm { d } x \\mathrm { d } y \\ , \\omega _ f ( x , y ) R _ { x + i y } ( A ) , \\end{align*}"}
-{"id": "6157.png", "formula": "\\begin{align*} A _ i = 1 + \\phi _ i '' , \\ ; \\ ; V = ( A _ 1 A _ 2 A _ 3 ) ^ \\frac { 1 } { 3 } , \\ ; \\ ; f _ i = \\frac { A _ i } { V } \\end{align*}"}
-{"id": "3399.png", "formula": "\\begin{align*} \\tilde f ( \\zeta ) = \\frac { 1 } { 2 } \\left ( f ( \\zeta ) + \\overline { f ( \\bar \\zeta ) } \\right ) \\end{align*}"}
-{"id": "6175.png", "formula": "\\begin{align*} \\frac { 1 } { C } \\leq f _ i \\leq C , \\ ; \\ ; i = 1 , 2 , 3 . \\end{align*}"}
-{"id": "10059.png", "formula": "\\begin{align*} w = \\left ( \\begin{smallmatrix} 0 & - 1 \\\\ 1 & 0 \\end{smallmatrix} \\right ) , n ( b ) = \\left ( \\begin{smallmatrix} 1 & b \\\\ 0 & 1 \\end{smallmatrix} \\right ) , \\end{align*}"}
-{"id": "5031.png", "formula": "\\begin{align*} t _ { \\Gamma } ( x _ 1 , x _ 2 , y _ 1 , \\dots , y _ n ) = [ \\dots \\ ! [ [ x _ 1 , x _ 2 ] , g _ { e _ 1 } ] , \\dots \\dots , g _ { e _ k } ] , \\end{align*}"}
-{"id": "900.png", "formula": "\\begin{align*} q _ n ^ * ( 1 - \\alpha ) = \\inf \\left \\{ z \\in \\mathbb { R } : P \\left ( T _ n ^ * \\leq z | \\mathcal { F } ^ X \\right ) \\geq 1 - \\alpha \\right \\} , \\end{align*}"}
-{"id": "7720.png", "formula": "\\begin{align*} d ^ 2 _ B ( j , k ) = \\sum ^ { N - 2 } _ { n = 1 } \\frac { ( u _ { n j } - u _ { n k } ) ^ 2 } { 1 ^ 2 } + \\frac { ( u _ { N - 1 , j } - u _ { N - 1 , k } ) ^ 2 } { N ^ 2 } . \\end{align*}"}
-{"id": "2658.png", "formula": "\\begin{align*} P ( s u p p ( T ) ) = s u p p ( P _ \\ast ( T ) ) . \\end{align*}"}
-{"id": "9774.png", "formula": "\\begin{align*} \\int | \\sum _ { i = 1 } ^ { k - 1 } \\tilde { E } _ { i } ( h , \\theta ) | ^ 2 d \\theta = \\sum _ { i = 1 } ^ { [ x / h ] } \\int | \\tilde { E } _ { i } ( h , \\theta ) | ^ 2 d \\theta \\leq C x ( \\frac { s } { h } ) ^ 2 h . \\end{align*}"}
-{"id": "942.png", "formula": "\\begin{align*} F _ { n , k } = \\sum _ { i , j = 1 } ^ { N _ n } \\gamma _ { n , k } ( i , j ) I ^ { W _ n } _ 2 ( e _ i \\otimes e _ j ) = I ^ { W _ n } _ 2 ( f _ { n , k } ) , \\end{align*}"}
-{"id": "1653.png", "formula": "\\begin{align*} \\tau _ { f _ 1 } ( x ) = \\frac { 1 - x } { 2 } , \\tau _ { f _ 2 } ( x ) = \\frac { 2 - x } { 2 } \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\tau _ { e } ( x ) = - x + 1 . \\end{align*}"}
-{"id": "5315.png", "formula": "\\begin{align*} \\\\ \\int _ \\Omega \\mathbf { j } ( \\theta , \\phi ) \\cdot \\nabla w \\mathrm { d x } = \\int _ { \\Gamma _ { \\rm N } } g w \\mathrm { d s } , \\end{align*}"}
-{"id": "4633.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { N _ j } 1 _ { [ - B - t , B - t ] } \\left ( \\ ; \\sum _ { i = 0 } ^ { n - 1 } f ( T ^ i x ) \\right ) > ( N _ j ) ^ \\gamma \\end{align*}"}
-{"id": "7823.png", "formula": "\\begin{align*} \\liminf _ { r } \\int _ { S _ r } | \\frac { \\partial v } { \\partial r } v | e ^ { - 2 \\rho } d x = 0 . \\end{align*}"}
-{"id": "4144.png", "formula": "\\begin{align*} R ( \\alpha , ( I _ t ) , J ) = \\frac { n - \\lvert I _ n \\rvert } { n } R ( n , | J | , 0 ) + \\frac { \\lvert I _ n \\rvert } { n } R ( n , | J | , 1 ) . \\end{align*}"}
-{"id": "7382.png", "formula": "\\begin{align*} \\frac { \\Z _ { N _ { f } } ^ { d } ( f _ 0 , | d | ) } { \\Z _ { N _ { f } } ( f _ 0 ) } = \\lim _ { ( N - N _ f ) \\rightarrow \\infty , N _ f \\rightarrow \\infty } 1 + \\mathcal { O } ( e ^ { - c ( N + N _ f ) } ) , \\end{align*}"}
-{"id": "6326.png", "formula": "\\begin{align*} { \\mathcal { L } } _ s ( u ) = \\int _ { \\R ^ n } | ( { - \\Delta } ) ^ { \\frac { s } { 2 } } u ( x ) | ^ p d x \\end{align*}"}
-{"id": "4664.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } \\| F ^ { - 1 } ( y _ n ) - F ^ { - 1 } ( y ) \\| = 0 \\ , . \\end{align*}"}
-{"id": "8537.png", "formula": "\\begin{align*} T _ 1 h ( x ) = \\int _ { \\mathbb { R } ^ 3 } e ^ { i x \\cdot \\xi } a ( x , \\xi ) h ( \\xi ) d \\xi = \\int _ { \\mathbb { R } ^ 3 } K ( x , \\xi ) h ( \\xi ) d \\xi . \\end{align*}"}
-{"id": "3847.png", "formula": "\\begin{align*} X \\coloneqq \\{ x \\in \\R ^ n \\mid g ( x ) \\leq 0 , \\ h ( x ) = 0 \\} \\subseteq \\R ^ n \\end{align*}"}
-{"id": "1691.png", "formula": "\\begin{align*} z _ n : = e f _ { i _ 1 } e f _ { i _ 2 } e f _ { i _ 3 } \\ldots f _ { i _ n } \\end{align*}"}
-{"id": "2800.png", "formula": "\\begin{align*} \\mathcal { B } _ { n } ( t ) = \\sum _ { i = 0 } ^ { n } \\mathcal { B } _ { i } \\mathcal { L } _ { i } ( t ) . \\end{align*}"}
-{"id": "9129.png", "formula": "\\begin{align*} \\sum _ { \\nu \\in \\left \\{ 1 , \\ldots , n \\right \\} \\setminus \\left \\{ i , j \\right \\} } x ^ { p _ { \\nu } } f _ { \\nu } ( x ^ { q _ { \\nu } } ) + x ^ { p _ { j } } \\widetilde { f } ( x ^ { q _ { j } } ) = 0 , \\end{align*}"}
-{"id": "6637.png", "formula": "\\begin{align*} \\int _ { 2 D } | D f | ^ 2 \\ ; & = \\ ; \\int _ { 2 D \\setminus \\bigcup _ { k = m - 1 } ^ { \\infty } Q _ k } | D f | ^ 2 + \\int _ { \\bigcup _ { k = m - 1 } ^ { \\infty } Q _ k } | D f | ^ 2 \\\\ & \\leq \\ ; { \\mathrm { A r e a } ( 2 D ) } + C \\sum _ { k = m - 1 } ^ { \\infty } k ^ { 2 p } { \\mathrm { A r e a } ( Q _ k ) } \\\\ & = \\ ; { \\mathrm { A r e a } ( 2 D ) } + C \\sum _ { k = m - 1 } ^ { \\infty } k ^ { 2 p } 2 ^ { - 2 k } . \\\\ \\end{align*}"}
-{"id": "8149.png", "formula": "\\begin{align*} \\bigg [ \\bigsqcup _ { i = 1 } ^ n A _ i \\times \\{ i \\} \\bigg ] + \\bigg [ \\bigsqcup _ { i = 1 } ^ m B _ i \\times \\{ i \\} \\bigg ] = \\bigg [ \\bigg ( \\bigsqcup _ { i = 1 } ^ n A _ i \\times \\{ i \\} \\bigg ) \\sqcup \\bigg ( \\bigsqcup _ { i = n + 1 } ^ { n + m } B _ i \\times \\{ i \\} \\bigg ) \\bigg ] . \\end{align*}"}
-{"id": "405.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( S _ { n } ^ { ( \\varepsilon x \\sigma _ { n } ) } \\geq x \\sigma _ { n } \\right ) = ( 1 - \\Phi ( x ) ) ( 1 + o ( 1 ) ) . \\end{align*}"}
-{"id": "3964.png", "formula": "\\begin{align*} \\tilde { p } ^ { \\beta _ n } ( n , s ) = \\int _ 0 ^ { \\infty } p ^ { \\beta _ n } ( n , t ) e ^ { - s t } \\ , \\mathrm { d } t = \\frac { \\lambda ^ n s ^ { \\beta _ n - 1 } } { \\prod _ { k = 0 } ^ n ( s ^ { \\beta _ k } + \\lambda ) } , \\ \\ s > 0 . \\end{align*}"}
-{"id": "4480.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| \\pi ( \\Delta _ n ) ^ * ( \\Pi ( a ) - \\pi ( a ) ) \\pi ( \\Delta _ n ) \\| = 0 \\end{align*}"}
-{"id": "6007.png", "formula": "\\begin{align*} \\mathcal { T } _ { \\epsilon } ^ { n } ( Q _ { Y X } | x ^ { n } ) : = \\left \\{ y ^ { n } \\in \\mathcal { Y } ^ { n } : ( x ^ { n } , y ^ { n } ) \\in \\mathcal { T } _ { \\epsilon } ^ { n } ( Q _ { X Y } ) \\right \\} . \\end{align*}"}
-{"id": "2059.png", "formula": "\\begin{align*} \\sum _ a ( \\rho ( u ) ) ^ a \\Delta _ H ( \\rho ( u ) ) ^ a = \\sum _ { a , b } ( \\rho ( u ) ) ^ a \\rho ^ a _ b ( u ) \\partial _ t u ^ b . \\end{align*}"}
-{"id": "690.png", "formula": "\\begin{align*} \\psi ( \\pi , \\mu ) = 1 , \\psi ' ( \\pi , \\mu ) = - \\cot \\beta . \\end{align*}"}
-{"id": "9077.png", "formula": "\\begin{align*} f ( p ) = p ' , \\end{align*}"}
-{"id": "3838.png", "formula": "\\begin{align*} \\bar { X } _ n = \\min \\{ S ^ { z , i } _ n \\colon \\ , z \\in \\Z , i \\leq N ( z , 0 ) \\xi ( z , i , n ) = 1 \\} , \\end{align*}"}
-{"id": "1531.png", "formula": "\\begin{align*} \\begin{cases} \\max \\left ( F ( \\lfloor \\alpha _ { m , t } ( 3 ) \\rfloor ) , G ( \\lfloor \\frac { t - 1 } { 2 } \\rfloor ) \\right ) , & \\ \\ \\{ \\alpha _ { m , t } ( 3 ) \\} \\leq 1 / 2 ; \\\\ \\max \\left ( F ( \\lfloor \\alpha _ { m , t } ( 3 ) \\rfloor + 1 ) , G ( \\lfloor \\frac { t - 1 } { 2 } \\rfloor ) \\right ) , & \\ \\ \\{ \\alpha _ { m , t } ( 3 ) \\} > 1 / 2 . \\end{cases} \\end{align*}"}
-{"id": "2009.png", "formula": "\\begin{align*} x _ i z _ { i + 1 } ( q ) - x _ { i + 2 } z _ i ( q ) = \\sum _ { \\beta \\in \\Z } g _ { \\beta } ( q ) x _ { j - 2 \\beta } z _ { j + 1 + \\beta } ( q ) , ~ ~ ~ i \\in \\Z . \\end{align*}"}
-{"id": "7388.png", "formula": "\\begin{align*} \\frac { \\Z _ { N _ f } ^ { d } ( f , | d | ) } { \\Z _ { N _ f } ( f ) } = \\frac { D _ { N _ { f } } ^ { d } ( f , | d | ) } { D _ { N _ { f } } ( f ) } = \\det ( I - \\mathcal { K } ) , \\end{align*}"}
-{"id": "284.png", "formula": "\\begin{align*} D ^ { 1 / 2 } _ { 0 + } x ( t ) = - x ( t ) + Q ( t ) x ( t ) + g ( t ) , \\end{align*}"}
-{"id": "618.png", "formula": "\\begin{align*} f ( w ) = \\sum _ { k \\ge - 1 } a _ k w ^ k \\end{align*}"}
-{"id": "7437.png", "formula": "\\begin{align*} x & : = x ' + t _ a ( z + y + t _ b + t _ c - t _ d + t _ a ^ 2 ) & u & : = - t _ b & t & : = 2 t _ a \\\\ v & : = - ( z + y + t _ b + t _ c - t _ d + t _ a ^ 2 ) / 2 & w & : = - t _ c & \\end{align*}"}
-{"id": "1959.png", "formula": "\\begin{align*} \\| \\widetilde { K } _ { j , \\textbf { r } } ( \\widehat { f _ 1 } , \\dots , \\widehat { f _ j } ) \\| _ { L ^ 2 _ \\alpha } \\leq \\sum _ { k = 1 } ^ j \\| \\widetilde { K } ^ k _ { j , \\textbf { r } } ( \\widehat { f _ 1 } , \\dots , \\widehat { f _ j } ) \\| _ { L ^ 2 _ \\alpha } , \\end{align*}"}
-{"id": "5348.png", "formula": "\\begin{align*} \\langle Y ' ( v , x ) w ' , w \\rangle = \\langle w ' , Y ^ { \\circ } _ W ( v , x ) w \\rangle \\end{align*}"}
-{"id": "2754.png", "formula": "\\begin{align*} e ^ K : = P _ { K } h - h ^ K = \\sum _ { k = 1 } ^ { K } \\ , ( \\hat h _ k ( t , x , v ) - h _ k ( t , x , v ) ) \\ , \\psi _ k ( z ) : = \\sum _ { k = 1 } ^ { K } \\ , e _ k ( t , x , v ) \\ , \\psi _ k ( z ) \\ , , \\end{align*}"}
-{"id": "1720.png", "formula": "\\begin{align*} S _ \\lambda = W ^ * t _ \\lambda W \\quad \\quad \\end{align*}"}
-{"id": "5445.png", "formula": "\\begin{align*} f ( x ) = x ^ \\rho \\ell ( x ) p ( x ) , x > 0 , \\end{align*}"}
-{"id": "3657.png", "formula": "\\begin{align*} \\bar \\delta ^ { ( n ) } = \\begin{pmatrix} 1 & 1 & \\cdots & 1 \\\\ 0 & 1 & \\cdots & 1 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "8484.png", "formula": "\\begin{align*} \\abs { W _ { \\pi } ( g _ { t , l , v } ) } = \\begin{cases} q ^ { \\frac { n - l } { 2 } } & v ( b _ { \\chi } - v ^ { - 1 } ) \\geq n - l , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "5350.png", "formula": "\\begin{align*} \\overline { W } = \\prod _ { n \\in \\C } W _ { [ n ] } , \\end{align*}"}
-{"id": "8711.png", "formula": "\\begin{align*} \\partial _ t ^ \\alpha u + F ( t , x , u , \\nabla u , \\nabla ^ 2 u ) = 0 \\quad \\end{align*}"}
-{"id": "7052.png", "formula": "\\begin{align*} \\tilde { \\lambda } _ n = h ( t ) ^ { - 1 / 2 } \\lambda _ n . \\end{align*}"}
-{"id": "2475.png", "formula": "\\begin{align*} \\mathbb { G } ^ { t } _ { S } f _ { 0 } = f ( t ) - \\mathbb { G } ^ { t } _ { L } f _ { 0 } . \\end{align*}"}
-{"id": "8346.png", "formula": "\\begin{align*} V _ { \\Z _ \\ell } = V _ \\Z \\otimes \\Z _ \\ell \\qquad \\mbox { a n d } V ^ \\vee _ { \\Z _ \\ell } = V ^ \\vee _ \\Z \\otimes \\Z _ \\ell , \\end{align*}"}
-{"id": "6764.png", "formula": "\\begin{align*} [ l : k ] h ( \\mu ) = [ l : k ] \\sum _ { v | \\infty } \\log ^ + | \\mu | _ { w _ v } = \\sum _ { v | \\infty } \\log ^ + | \\beta | _ v = h ( \\beta ) . \\end{align*}"}
-{"id": "5928.png", "formula": "\\begin{align*} \\begin{aligned} & s ^ I ( N , r _ 1 , \\cdots , r _ n ) = \\frac 1 N \\int _ { R ^ N } ( \\sum _ { i = 1 } ^ N z _ i ) \\pi ^ { N ; I } ( d z ) ; \\\\ & s ^ D ( N , m _ 1 , \\cdots , m _ n ) = \\frac 1 N \\int _ { R ^ N } ( \\sum _ { i = 1 } ^ N z _ i ) \\hat \\pi ^ { N ; D } ( d z ) ; \\\\ & s ^ { D _ 1 } ( N , m _ 1 , \\cdots , m _ n ) = \\frac 1 N \\int _ { R ^ N } ( \\sum _ { i = 1 } ^ N z _ i ) \\hat \\pi ^ { N ; D _ 1 } ( d z ) . \\end{aligned} \\end{align*}"}
-{"id": "2389.png", "formula": "\\begin{align*} a _ 6 = 3 a _ 3 + a _ 4 = 4 a _ 2 + a _ 5 = 3 a _ 1 a _ 5 + 2 a _ 4 = 0 . \\end{align*}"}
-{"id": "322.png", "formula": "\\begin{align*} f _ 0 ( n ) & = q _ 0 ( n ) \\cdot f _ 1 ( n ) + r _ 1 ( n ) \\cdot f _ 2 ( n ) , \\\\ f _ 1 ( n ) & = q _ 1 ( n ) \\cdot f _ 2 ( n ) + r _ 2 ( n ) \\cdot f _ 0 ( n + 1 ) , \\\\ f _ 2 ( n ) & = q _ 2 ( n ) \\cdot f _ 0 ( n + 1 ) + r _ 0 ( n + 1 ) \\cdot f _ 1 ( n + 1 ) , \\end{align*}"}
-{"id": "7289.png", "formula": "\\begin{align*} B _ 1 = \\left ( \\begin{array} { c c c c } t _ 1 & t _ 2 & \\ldots & t _ u \\\\ \\ell _ { 1 , 2 } & \\ell _ { 2 , 2 } & \\ldots & \\ell _ { u , 2 } \\\\ \\ell _ { 1 , 3 } & \\ell _ { 2 , 3 } & \\ldots & \\ell _ { u , 3 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "3378.png", "formula": "\\begin{align*} \\xi ( X ) : = \\left \\langle \\left \\langle X \\cdot \\varphi , \\varphi \\right \\rangle \\right \\rangle , ~ ~ \\forall X \\in T M . \\end{align*}"}
-{"id": "3872.png", "formula": "\\begin{align*} f ( x ^ * ) \\geq \\lim _ { k \\to \\infty } f ( x ^ k ) = f ( \\bar x ) > f ( x ^ * ) . \\end{align*}"}
-{"id": "1827.png", "formula": "\\begin{align*} E ^ 0 _ { D , f , h } \\left ( e ^ { \\Phi ^ h _ { D } ( \\tilde { f } ) } \\right ) = \\frac { E ^ 0 _ { D , f , 0 } \\left ( e ^ { h \\Phi ^ 0 _ { D } ( 1 _ D ) } e ^ { \\Phi ^ 0 _ { D } ( \\tilde { f } ) } \\right ) } { E ^ 0 _ { D , f , 0 } \\left ( e ^ { h \\Phi ^ 0 _ { D } ( 1 _ D ) } \\right ) } . \\end{align*}"}
-{"id": "3100.png", "formula": "\\begin{align*} H _ { 0 , \\Gamma } ^ 1 ( \\Omega ) : = \\left \\{ u \\in H ^ 1 ( \\Omega ) : u | _ \\Gamma = 0 \\right \\} \\ni u \\mapsto \\int _ \\Omega | \\nabla u | ^ 2 \\d x . \\end{align*}"}
-{"id": "4114.png", "formula": "\\begin{align*} \\Delta f : = \\mathrm { d i v } _ { \\nabla } ( \\mathrm { g r a d } _ h f ) , \\end{align*}"}
-{"id": "8257.png", "formula": "\\begin{align*} \\frac { L _ { \\nu } ( 2 s , \\chi _ 1 \\chi _ 2 ^ { - 1 } ) } { L _ { \\nu } ( 2 s + 1 , \\chi _ 1 \\chi _ 2 ^ { - 1 } ) } = \\sqrt { \\pi } \\frac { \\Gamma ( s + i t _ { \\nu } ) } { \\Gamma ( s + i t _ { \\nu } + \\frac { 1 } { 2 } ) } \\ll _ { \\sigma } ( 1 + \\abs { t + t _ { \\nu } } _ { \\nu } ) ^ { - \\frac { 1 } { 2 } } \\ll \\abs { T _ { s , \\nu } } _ { \\nu } ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "7804.png", "formula": "\\begin{align*} \\frac { d ^ { r } } { d x ^ { r } } R _ { v } ^ { r } f \\left ( x \\right ) = \\frac { d } { d x } R _ { v } \\frac { d ^ { r - 1 } } { d x ^ { r - 1 } } R _ { v } ^ { r - 1 } f \\left ( x \\right ) , x \\in \\mathbb { T } . \\end{align*}"}
-{"id": "1052.png", "formula": "\\begin{align*} H ( P _ { k + 1 } ) \\leq X _ { k } = H ( P _ { k } ) ^ { ( w _ { n } ( \\zeta ) + \\epsilon ) / ( \\widehat { w } _ { n } ( \\zeta ) - \\epsilon ) } . \\end{align*}"}
-{"id": "2039.png", "formula": "\\begin{align*} f _ { n , k } ( q z _ 1 , z _ 2 , . . . , z _ n ) = ( - 1 ) ^ { k - 1 } q ^ { - \\frac { ( k + 1 ) ( k - 2 ) } { 2 } } z _ 1 ^ { - k ^ 2 + 1 } z _ 2 ^ { k + 1 } . . . z _ n ^ { k + 1 } f _ { n , k } ( z _ 1 , . . . , z _ n ) . \\end{align*}"}
-{"id": "3149.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } _ { \\geqslant 0 } } x m _ { \\alpha , \\beta } ( \\mathrm { d } x ) = \\left . \\frac { \\partial } { \\partial u } \\widehat { m } _ { \\alpha , \\beta } ( u ) \\right \\vert _ { u = 0 } = \\frac { \\alpha } { \\beta } . \\end{align*}"}
-{"id": "8610.png", "formula": "\\begin{align*} \\partial _ t u = \\frac 1 2 \\Delta u + \\lambda V ( t , x ) u , \\ \\ x \\in \\R ^ d , d \\geq 3 . \\end{align*}"}
-{"id": "9977.png", "formula": "\\begin{align*} X _ { k i } = \\alpha { \\displaystyle \\sum _ { j = 1 } ^ { L } \\mathbb { E } \\left [ \\mathsf { \\bar { L } } _ { j i } \\right ] \\left ( 1 + \\frac { \\sigma ^ { 2 } } { P _ u } + \\sum _ { j \\neq i } ^ { L } \\bar { \\ell } _ { k j i } \\right ) + \\sum _ { j \\neq i } ^ { L } \\bar { \\ell } _ { k j i } ^ { 2 } } - \\frac { 1 } { \\gamma _ { i } ^ { \\mathsf { t h } } } . \\end{align*}"}
-{"id": "685.png", "formula": "\\begin{align*} \\cot \\beta = \\cot \\beta _ 0 + \\sum _ { n = 0 } ^ { \\infty } ( a _ n ^ 0 - a _ n ) \\cfrac { \\varphi ^ 2 _ 0 ( \\pi , \\mu _ n ^ 0 ) } { ( a _ n ^ 0 ) ^ 2 } . \\end{align*}"}
-{"id": "1646.png", "formula": "\\begin{align*} \\bigcup _ { \\lambda \\in \\Lambda ^ m } R _ \\lambda = \\bigcup _ { \\lambda = \\lambda _ 1 \\lambda _ 2 \\in \\Lambda ^ m } \\tau _ { \\lambda _ 1 } ( R _ { \\lambda _ 2 } ) = \\bigcup _ { d ( \\lambda _ 1 ) = \\ell } \\tau _ { \\lambda _ 1 } \\left ( \\cup _ { \\lambda _ 2 \\in s ( \\lambda _ 1 ) \\Lambda ^ { e _ j } } R _ { \\lambda _ 2 } \\right ) \\\\ \\end{align*}"}
-{"id": "3713.png", "formula": "\\begin{align*} \\left ( \\frac { \\xi _ 1 } { \\xi _ 2 } , \\frac { \\xi _ 2 } { \\xi _ 3 } , \\dots , \\frac { \\xi _ { n - 1 } } { \\xi _ n } \\right ) = ( f _ 1 , \\dots , f _ { n - 1 } ) \\end{align*}"}
-{"id": "3667.png", "formula": "\\begin{align*} b _ { G H H } = l _ { G H H } ( 0 ) = \\begin{pmatrix} \\frac { 1 } { n + 1 } \\\\ \\vdots \\\\ \\frac { n } { n + 1 } \\end{pmatrix} . \\end{align*}"}
-{"id": "1176.png", "formula": "\\begin{align*} \\hat u ( z ) = u ( F ( z ) ) \\ , \\ , \\mbox { w h e n e v e r } \\ , \\ , F ( z ) \\in O \\ , \\ , \\mbox { i s } \\ , \\ , \\mathcal { A } \\mbox { - h a r m o n i c i n } \\ , \\ , F ^ { - 1 } ( O ) . \\end{align*}"}
-{"id": "1739.png", "formula": "\\begin{align*} \\mathcal { H } ( \\Lambda ^ \\infty ) = \\{ ( f , \\mu ) \\mid \\mu \\ ; \\ ; , \\ ; f \\in L ^ 2 ( \\Lambda ^ \\infty , \\mu ) \\} \\big / \\sim . \\end{align*}"}
-{"id": "5033.png", "formula": "\\begin{align*} \\sigma _ n ^ t ( x _ 1 , \\dots , x _ n ) : = t ( \\sigma _ { \\lceil n / 2 \\rceil } ^ t ( x _ 1 , \\dots , x _ { \\lceil n / 2 \\rceil } ) , \\sigma _ { \\lfloor n / 2 \\rfloor } ^ t ( x _ { \\lceil n / 2 \\rceil + 1 } , \\dots , x _ n ) ) \\end{align*}"}
-{"id": "1196.png", "formula": "\\begin{align*} \\xi = \\frac { \\nabla v _ 1 ( y _ 0 ) } { | \\nabla v _ 1 ( y _ 0 ) | } = \\frac { \\nabla v _ 2 ( z _ 0 ) } { | \\nabla v _ 2 ( z _ 0 ) | } = \\frac { \\nabla u ( x _ 0 ) } { | \\nabla u ( x _ 0 ) | } \\end{align*}"}
-{"id": "3106.png", "formula": "\\begin{align*} \\Theta ^ { \\mu } = \\sum _ { i = 0 } ^ { m } h _ { i , 0 } = \\frac { 1 } { 1 - p } \\left [ 1 - \\frac { \\alpha \\beta p ^ { m + 1 } } { 1 - p + \\alpha \\beta p } \\right ] h _ { 0 , 0 } . \\end{align*}"}
-{"id": "2119.png", "formula": "\\begin{align*} \\mathcal { C } _ { \\Gamma } ( f ) ( z ) : = \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma } \\frac { f ( \\zeta ) } { \\zeta - z } \\ , d \\zeta . \\end{align*}"}
-{"id": "728.png", "formula": "\\begin{align*} \\frac { a ^ 2 \\theta _ - + ( U _ I - u _ - ) ^ 2 } { a ^ 2 \\theta _ + + U _ I ^ 2 } = \\frac { U _ I - u _ - } { U _ I } . \\end{align*}"}
-{"id": "10025.png", "formula": "\\begin{align*} \\lambda _ q = \\overline { c ( q ) } \\cdot \\begin{cases} - q ^ { 1 - \\frac { n } 2 } & \\\\ \\delta _ q q ^ { \\frac { 1 - n } 2 } & \\end{cases} \\end{align*}"}
-{"id": "7410.png", "formula": "\\begin{align*} t : = t _ 0 , T _ 0 ^ { \\beta } : = t _ 1 ^ 2 / 4 , T _ 0 ^ { \\gamma } : = t _ 2 ^ 2 / 4 , T _ 0 ^ { \\delta } : = t _ 3 ^ 2 / 4 \\end{align*}"}
-{"id": "526.png", "formula": "\\begin{align*} \\emptyset = U _ { - 1 } \\subset U _ 0 \\subset \\dots \\subset U _ { d - 1 } \\subset U _ d = M \\end{align*}"}
-{"id": "6185.png", "formula": "\\begin{align*} \\frac { V ' } { V } = \\frac { 1 } { \\overline { \\mathcal { T } } V ^ 2 } \\sum _ { i = 1 } ^ 3 ( \\log f _ i ) ' ( \\log f _ i ) '' \\end{align*}"}
-{"id": "3488.png", "formula": "\\begin{align*} f _ 1 f _ 2 f _ 3 & = 2 | z _ 1 | ^ 2 | z _ 2 | ^ 2 | z _ 3 | ^ 2 \\\\ & + | z _ 1 | ^ 2 ( z _ 2 ^ 2 \\bar z _ 3 ^ 2 + \\bar z _ 2 ^ 2 z _ 3 ^ 2 ) + | z _ 2 | ^ 2 ( z _ 1 ^ 2 \\bar z _ 3 ^ 2 + \\bar z _ 1 ^ 2 z _ 3 ^ 2 ) + | z _ 3 | ^ 2 ( z _ 1 ^ 2 \\bar z _ 2 ^ 2 + \\bar z _ 1 ^ 2 z _ 2 ^ 2 ) . \\end{align*}"}
-{"id": "5832.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\| w _ n \\| _ { L ^ 6 ( \\R ) } ^ 6 = & \\sum _ { j = 1 } ^ { J ^ * } \\| \\phi ^ j \\| _ { L ^ 6 ( \\R ) } ^ 6 . \\end{align*}"}
-{"id": "2032.png", "formula": "\\begin{align*} \\sum _ { \\beta \\in \\Z } b _ { \\beta } x _ { j + k \\beta } y _ { j + 1 - \\beta } = 0 , ~ ~ ~ j \\in \\Z . \\end{align*}"}
-{"id": "2198.png", "formula": "\\begin{align*} \\Gamma _ h = \\{ z \\in \\mathbb C ^ n \\colon | h _ i ( z ) | = r _ i , \\ r _ i > 0 , \\ i = 1 , \\ldots , n \\} . \\end{align*}"}
-{"id": "7668.png", "formula": "\\begin{align*} \\zeta _ s = \\zeta _ 1 ^ { m _ 1 + m _ 2 } \\zeta _ 2 ^ { m _ 2 + m _ 3 } \\cdots \\zeta _ { s - 1 } ^ { m _ { s - 1 } + m _ s } . \\end{align*}"}
-{"id": "6477.png", "formula": "\\begin{align*} \\sigma _ { x y } = \\rho \\Sigma ^ { 2 } \\sigma _ { x } \\sigma _ { y } = \\Sigma ^ { 2 } \\in \\mathbb { R } _ { 0 } ^ { + } \\end{align*}"}
-{"id": "6914.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\{ \\Phi _ { 1 } , \\Phi _ { 2 } , \\Phi _ { 3 } , \\Phi _ { 4 } , \\Phi _ { 5 } \\} & = \\{ 0 , 3 , 3 , \\frac { 3 } { 2 } , \\frac { 1 5 } { 2 } \\} \\\\ \\lim _ { r \\to \\infty } \\{ \\Phi _ { 1 } , \\Phi _ { 2 } , \\Phi _ { 3 } , \\Phi _ { 4 } , \\Phi _ { 5 } \\} & = \\{ 0 , \\frac { 3 } { 2 } , \\frac { 9 } { 2 } , 6 , 9 \\} \\end{align*}"}
-{"id": "4739.png", "formula": "\\begin{align*} { L } _ i = \\left ( 0 , \\alpha , \\beta , \\gamma \\right ) . \\end{align*}"}
-{"id": "5760.png", "formula": "\\begin{align*} D = D ( G , R , n , \\delta ) = \\left \\{ \\{ n | \\det ( G ) | \\} ^ { - 1 / p } ( a ^ T G R + \\delta ^ T ) : a \\in \\mathbb { Z } ^ p \\right \\} \\cap [ 0 , 1 ] ^ p . \\end{align*}"}
-{"id": "8972.png", "formula": "\\begin{gather*} c _ w + f _ w { } ^ r f _ w ^ { - 1 } h _ r \\zeta _ { r , w } ^ { - 1 } c _ { r w } = 0 \\in { \\cal Z } _ w ( [ X ^ r ] ) | _ { [ X ^ r ] } . \\end{gather*}"}
-{"id": "8317.png", "formula": "\\begin{align*} \\vartheta ( \\tau , z , g ) = \\sum _ { \\mu \\in V _ \\Z ^ \\vee / V _ \\Z } \\vartheta ( \\tau , z , g , \\phi _ \\mu ) \\cdot \\phi _ \\mu , \\end{align*}"}
-{"id": "5010.png", "formula": "\\begin{align*} \\psi ^ { ( \\gamma ) } ( \\xi ) : = \\eta ( \\xi ) \\left ( \\frac { 1 } { m ! } \\int _ { 0 } ^ { 1 } ( 1 - t ) ^ m \\partial _ { \\gamma } \\psi ( t \\xi ) \\ , d t \\right ) + c _ { \\gamma } ( 1 - \\eta ( \\xi ) ) \\frac { \\xi ^ { \\gamma } } { | \\xi | ^ { 2 ( m + 1 ) } } \\psi ( \\xi ) . \\end{align*}"}
-{"id": "2763.png", "formula": "\\begin{align*} \\sum _ { Q - Q _ 1 \\in \\Delta ^ - } \\Big ( \\big \\lfloor w ( p Q ) \\big \\rfloor - \\big \\lfloor w ( Q ) \\big \\rfloor - w ( p Q ) + w ( Q ) \\Big ) \\geq 0 \\textrm { \\ f o r e v e r y \\ } Q _ 1 = \\sum \\limits _ { i = 1 } ^ n m _ i \\mathbf V _ i \\in \\Lambda _ { \\Delta } . \\end{align*}"}
-{"id": "9149.png", "formula": "\\begin{align*} f _ { 5 - i } = ( - 1 ) ^ { i } \\sum _ { k = 0 } ^ { i } \\binom { 5 - i + k } { k } D _ { 4 - i + k } \\end{align*}"}
-{"id": "7674.png", "formula": "\\begin{align*} B = 2 ^ { k _ 1 } J _ 1 \\bot \\cdots \\bot 2 ^ { k _ r } J _ r , \\end{align*}"}
-{"id": "3237.png", "formula": "\\begin{align*} M ^ i H ^ { 2 i + k } ( X ) = \\left \\{ \\begin{array} { c c c } M ^ i H ^ { 2 i + k } ( X ) & \\mbox { f o r } & i \\in [ 0 , 2 d i m ( X ) - k ] \\\\ 0 & \\mbox { f o r } & 2 i + k \\not \\in [ 0 , 2 d i m ( X ) ] \\\\ H ^ { 2 i + k } ( X ; \\mathbb Q ) & \\mbox { f o r } & 2 i + k \\in [ 0 , k ] \\cup [ 2 d i m ( X ) - k , 2 d i m ( X ) ] \\end{array} \\right . \\end{align*}"}
-{"id": "6863.png", "formula": "\\begin{align*} R ( I _ { k } ^ { ( m ) } ) = \\frac { q ^ { i ( k ) } ( 1 - q ) ^ { m - i ( k ) } } { 2 ^ { m } } . \\end{align*}"}
-{"id": "4119.png", "formula": "\\begin{align*} \\mathrm { d i v } _ { \\nabla } ( \\mathrm { g r a d } _ h \\rho ) = b ( - \\nabla _ { V _ 1 } V , V _ 1 ) + b ( - \\nabla _ { V _ 2 } V , V _ 2 ) = b ( \\rho \\lambda _ 1 V _ 1 - V _ 1 , V _ 1 ) + b ( \\rho \\lambda _ 2 V _ 2 - V _ 2 , V _ 2 ) = \\\\ = \\rho \\lambda _ 1 + \\rho \\lambda _ 2 - 2 = 2 ( H \\rho - 1 ) , \\end{align*}"}
-{"id": "3346.png", "formula": "\\begin{align*} \\frac { \\binom { K - 2 } { s - 1 } } { \\binom { K - 2 } { s - 1 } + \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K - 1 } { s + i } ( N - 1 ) ^ i N } & \\geq \\frac { \\binom { K - 2 } { s - 2 } } { \\binom { K - 2 } { s - 2 } + \\sum _ { i = 0 } ^ { K - s } \\binom { K - 1 } { s + i - 1 } ( N - 1 ) ^ i N } . \\end{align*}"}
-{"id": "7284.png", "formula": "\\begin{align*} \\phi = \\left ( \\begin{array} { c c c c | c c c c c } x + a _ { 1 } & & & & & & \\\\ & x + a _ { 2 } & & & & & \\\\ & & \\ddots & & & & & \\\\ & & & x + a _ { u } & & & \\\\ \\hline & & & \\\\ & & & \\\\ & & & \\end{array} \\right ) , \\end{align*}"}
-{"id": "7738.png", "formula": "\\begin{align*} K _ N ( l ) & = G _ N ( l ) - G _ N ( l - 1 ) \\\\ & = Y _ N ( l ) + ( l - 1 ) G _ N ( 2 ) - ( 2 l - 3 ) G _ N ( 1 ) , \\end{align*}"}
-{"id": "1516.png", "formula": "\\begin{align*} S ^ { n } ( x ) = \\left [ \\begin{array} { c c c } x ^ { 2 } & x & 1 \\\\ 1 & 0 & 0 \\\\ 0 & 1 & 0 \\end{array} \\right ] ^ { n } = \\left [ \\begin{array} { c c c } T _ { n + 1 } ( x ) & x T _ { n } ( x ) + T _ { n - 1 } ( x ) & T _ { n } ( x ) \\\\ T _ { n } ( x ) & x T _ { n - 1 } ( x ) + T _ { n - 2 } ( x ) & T _ { n - 1 } ( x ) \\\\ T _ { n - 1 } ( x ) & x T _ { n - 2 } ( x ) + T _ { n - 3 } ( x ) & T _ { n - 2 } ( x ) \\end{array} \\right ] , \\end{align*}"}
-{"id": "4042.png", "formula": "\\begin{align*} ( 1 - c \\mu _ 1 ) ( 1 - c \\lambda _ 1 ) = 1 \\ \\ \\mathrm { o r } \\ \\ ( 1 - c \\mu _ 1 ) ( 1 - c \\lambda _ 2 ) = 1 \\ \\ \\mathrm { a n d } \\\\ ( 1 - c \\mu _ 2 ) ( 1 - c \\lambda _ 1 ) = 1 \\ \\ \\mathrm { o r } \\ \\ ( 1 - c \\mu _ 2 ) ( 1 - c \\lambda _ 2 ) = 1 . \\end{align*}"}
-{"id": "8121.png", "formula": "\\begin{align*} \\hat { h } _ i = \\sum _ { k = 0 } ^ n \\sum _ { t \\in B _ { i , k } } \\frac { k } { n } \\alpha _ t ( g _ i ) \\end{align*}"}
-{"id": "9500.png", "formula": "\\begin{align*} P _ 2 ( t ) = a \\ , z _ a ( t ) + 2 \\ , z _ a ^ 2 ( t ) \\left . \\dfrac { \\partial S ( z , t ) } { \\partial z } \\right | _ { z _ a ( t ) } , \\end{align*}"}
-{"id": "8750.png", "formula": "\\begin{align*} \\phi ^ 0 ( E ) & : = \\frac { \\pi } { 1 + \\frac { 1 } { M } } \\log \\left ( \\frac { 1 } { M + 1 } P _ f ^ 2 + H _ f - E \\right ) \\\\ \\phi ^ I ( E ) & : = \\int \\ ! d p \\ , d q \\ : a _ p ^ * \\ : \\frac { 1 } { \\frac { 1 } { M } ( P _ f + p + q ) ^ 2 + H _ f + p ^ 2 + q ^ 2 - E } \\ : a _ q . \\end{align*}"}
-{"id": "6988.png", "formula": "\\begin{align*} B \\mapsto M = M ( B ) = D f _ k ( B ) , \\end{align*}"}
-{"id": "8385.png", "formula": "\\begin{align*} 0 = \\sum _ { \\substack { m \\ge 0 \\\\ \\mu \\in V _ \\Z ^ \\vee / V _ \\Z } } c ( - m , \\mu ) \\cdot a ( m , \\mu ) \\end{align*}"}
-{"id": "5942.png", "formula": "\\begin{align*} g _ i ( x ) & = f _ i ( x ) & & \\forall i \\in V ( H ) \\\\ g _ u ( x ) & = x _ u + ( x _ a + 1 ) x _ u \\\\ & = x _ u \\land x _ a \\\\ g _ v ( x ) & = x _ v + x _ a ( x _ v + 1 ) \\\\ & = x _ v \\lor x _ a . \\end{align*}"}
-{"id": "5030.png", "formula": "\\begin{align*} & ( H _ { \\tau , l } \\psi ) _ x = E \\psi _ x = 0 \\\\ \\Rightarrow \\qquad & \\psi _ { P x } + t _ x \\psi _ x = 0 \\\\ \\Rightarrow \\qquad & \\psi _ { P x } = 0 , \\end{align*}"}
-{"id": "1383.png", "formula": "\\begin{align*} \\{ ( x , y ) \\in \\mathbb N _ 0 ^ 2 : V ( x ) \\le y \\} = \\{ ( x , y ) \\in \\mathbb N _ 0 ^ 2 : U ( y ) \\ge x \\} , \\end{align*}"}
-{"id": "9757.png", "formula": "\\begin{align*} \\mathcal { H } ( V _ A ^ { ( k ) } , A ( x ) ) = \\mathcal { H } ( V _ { A , 0 } , A ( x ) ) + \\mathcal { H } _ e ( V _ A ^ { ( k - 1 ) } , V _ { A , 0 } , A ( x ) , A ( 0 ) , Z _ A ^ { ( k - 1 ) } ) . \\end{align*}"}
-{"id": "171.png", "formula": "\\begin{align*} N \\star ' X = \\{ m \\star x : m \\in N , \\ x \\in X \\} . \\end{align*}"}
-{"id": "3307.png", "formula": "\\begin{align*} I ( W _ 1 , \\dots , W _ K ; Q _ 1 ^ { [ k ] } , \\dots , Q _ N ^ { [ k ] } ) = 0 . \\end{align*}"}
-{"id": "2079.png", "formula": "\\begin{align*} u ^ a ( p , 0 ) = \\phi ^ a ( p ) \\end{align*}"}
-{"id": "8891.png", "formula": "\\begin{align*} J _ { \\omega , 1 } ^ { - 1 } | \\varphi | ^ 2 = \\frac { 1 - \\omega } { 1 - \\overline c \\theta } . \\end{align*}"}
-{"id": "5957.png", "formula": "\\begin{align*} \\gamma ( t ) \\cdot \\vect { e } _ j ( t ) = 0 ( j = 2 , \\dots , n ) . \\end{align*}"}
-{"id": "8115.png", "formula": "\\begin{align*} D = S _ 1 \\partial _ A O _ 1 \\cup \\dots \\cup S _ q \\partial _ A O _ q , \\end{align*}"}
-{"id": "9393.png", "formula": "\\begin{align*} G _ s f ( x ) & = \\bigg ( \\sum _ { k \\in \\Bbb Z } \\big | [ ( \\delta _ 0 - \\phi _ k ) \\ast \\nu _ { s + k } ] \\ast \\sum _ { l \\in \\Bbb Z } \\Delta _ { l - k } ^ 2 f ( x ) \\big | ^ 2 \\bigg ) ^ { 1 / 2 } \\\\ & \\le \\sum _ { l \\in \\Bbb Z } \\bigg ( \\sum _ { k \\in \\Bbb Z } | \\Delta _ { l - k } [ ( \\delta _ 0 - \\phi _ k ) \\ast \\nu _ { s + k } ] \\ast \\Delta _ { l - k } f ( x ) | ^ 2 \\bigg ) ^ { 1 / 2 } : = \\sum _ { l \\in \\Bbb Z } G _ s ^ { l } f ( x ) . \\end{align*}"}
-{"id": "7128.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\tilde { X } = ~ - F _ * ( \\mathcal { \\tilde { W } } ) \\tilde { \\nu } \\end{align*}"}
-{"id": "8523.png", "formula": "\\begin{align*} & w _ { 1 : l } ^ \\top x \\to \\min \\\\ & \\begin{cases} G _ { * \\ , 1 : l } x \\equiv \\gamma \\ , ( \\ , S ) \\\\ h _ { 1 : l } x \\leq \\eta \\\\ x \\in \\mathbb { Z } _ + ^ l , \\end{cases} \\end{align*}"}
-{"id": "1846.png", "formula": "\\begin{align*} \\mathfrak { F } _ { \\mathcal { B } } = \\{ \\varepsilon _ \\alpha \\colon 0 \\to M _ \\alpha \\xrightarrow { f _ \\alpha } P _ \\alpha \\to M _ \\alpha \\to 0 \\mbox { : } \\alpha \\leq \\sigma \\} \\end{align*}"}
-{"id": "1583.png", "formula": "\\begin{align*} V ( \\mathbf { j } ' ) & = \\sum _ { k = 1 } ^ { n - 1 } [ ( n - k + 1 ) j _ k - s ( j _ k ) ] - c ( j _ n , - \\alpha ( n ) - 1 ) \\\\ & = \\frac { 2 ( m - 1 ) } { 3 } - s \\left ( \\frac { m - 1 } { 3 } \\right ) - c ( 2 , - \\alpha ( n ) - 1 ) \\\\ & = \\frac { 2 ( m - 1 ) } { 3 } - s \\left ( \\frac { 2 ( m - 1 ) } { 3 } \\right ) . \\end{align*}"}
-{"id": "7725.png", "formula": "\\begin{align*} D ^ 2 _ B ( j ) & = \\frac { 1 } { 3 6 0 } ( N ^ 4 + 1 0 N ^ 2 - 1 1 ) \\ , , \\\\ C _ B ( j ) & = \\frac { 3 6 0 N } { N ^ 4 + 1 0 N ^ 2 - 1 1 } \\ , . \\end{align*}"}
-{"id": "211.png", "formula": "\\begin{align*} f ( e [ \\gamma _ { 1 } , \\dots , \\gamma _ { n } ] ) = f ( e ( \\gamma _ { 1 } ) ) \\ast \\cdots \\ast f ( e ( \\gamma _ { n } ) ) , \\end{align*}"}
-{"id": "4601.png", "formula": "\\begin{align*} 0 & = a \\xi + x + \\frac 1 2 \\beta ( x , y ) \\xi + \\nabla _ x y \\\\ & = \\left ( a + \\frac 1 2 \\beta ( x , y ) \\right ) \\xi + \\left ( x + \\nabla _ x y \\right ) , \\end{align*}"}
-{"id": "2068.png", "formula": "\\begin{align*} S ^ p _ k ( \\mathcal { L } _ t , W ) = \\big \\{ \\psi \\in L ^ p ( \\Omega ) \\big | \\partial _ t ^ i X _ { i _ 1 } \\dots X _ { i _ s } \\psi \\in L ^ p ( W ) , \\ 2 i + s \\leq k \\big \\} . \\end{align*}"}
-{"id": "5482.png", "formula": "\\begin{align*} \\liminf _ { x \\to \\infty } \\frac { U ( x ) } { \\ell ( x ) } \\geq \\frac { k } { 2 } \\liminf _ { x \\to \\infty } \\frac { 1 } { \\ell ( x ) } \\int _ { \\varepsilon x } ^ x \\frac { \\ell ( y ) } { y } \\dd y = \\frac { k } { 2 } \\log \\varepsilon ^ { - 1 } . \\end{align*}"}
-{"id": "10080.png", "formula": "\\begin{align*} N ( k ) : p { \\longrightarrow } N _ { p } ( k ) = \\left \\lbrace Z \\in { T _ { p } M } : R ( X , Y ) Z = k \\left [ g ( Y , Z ) X - g ( X , Z ) Y \\right ] \\right \\rbrace \\end{align*}"}
-{"id": "1719.png", "formula": "\\begin{align*} W ^ * \\pi ( f ) W = M _ f W M _ f W ^ * = \\pi ( f ) . \\end{align*}"}
-{"id": "3458.png", "formula": "\\begin{align*} P _ h f = \\sum _ { i = 1 } ^ { d } f _ { h , i } \\psi _ i , \\end{align*}"}
-{"id": "2580.png", "formula": "\\begin{align*} \\left ( T _ u ^ { * } \\left ( H _ u T _ u - H _ v T _ v \\right ) \\right ) ( z ) & = \\left ( T _ u ^ { * } \\left ( H _ u - H _ v \\right ) T _ u \\right ) ( z ) - \\left ( T _ u ^ { * } H _ v \\left ( T _ v - T _ u \\right ) \\right ( z ) . \\end{align*}"}
-{"id": "7141.png", "formula": "\\begin{align*} d ^ * _ L \\leq \\frac { 1 } { R } \\sum _ { r = 0 } ^ { R - 1 } g ^ * _ r + \\frac { U J + W } { R } \\sum _ { r = 0 } ^ { R - 1 } \\frac { 1 } { V _ r } . \\end{align*}"}
-{"id": "6795.png", "formula": "\\begin{align*} e ^ { w _ { \\lambda } } \\leq \\sum \\limits _ { k = 1 } ^ 4 e ^ { U _ { \\lambda , \\xi _ k } } \\left [ 1 + \\theta _ { \\lambda } ( y ) \\right ] , \\end{align*}"}
-{"id": "7845.png", "formula": "\\begin{align*} R _ { X _ { 1 } , X _ { 2 } } ( x _ { 1 } , x _ { 2 } ) = \\left \\{ \\begin{array} { l } R _ { 1 } ( x _ { 1 } , x _ { 2 } ) \\ \\ \\ \\ 0 < x _ { 1 } < x _ { 2 } \\\\ R _ { 2 } ( x _ { 1 } , x _ { 2 } ) \\ \\ \\ \\ 0 < x _ { 2 } < x _ { 1 } \\\\ R _ { 3 } ( x , x ) \\ \\ \\ \\ \\ \\ \\ \\ x _ { 1 } = x _ { 2 } = x , \\end{array} \\right . \\end{align*}"}
-{"id": "6942.png", "formula": "\\begin{align*} M _ { \\infty } : = \\textnormal { t e l } ( M _ 0 \\xrightarrow { \\cdot m _ 1 } M _ 1 \\xrightarrow { \\cdot m _ 1 } M _ 2 \\xrightarrow { \\cdot m _ 1 } \\dots ) \\ , , \\end{align*}"}
-{"id": "984.png", "formula": "\\begin{align*} & \\frac { M _ n ( t ) } { \\sigma ^ 2 ( t ) \\mathfrak { s } _ n ( t ) } - F _ n ( t ) \\\\ & = \\frac { M _ n ( t ) - \\sigma ^ 2 ( ( t - \\ell h ) _ + ) M ^ 0 _ n ( t ) } { \\sigma ^ 2 ( t ) \\mathfrak { s } _ n ( t ) } + \\frac { \\{ \\sigma ^ 2 ( ( t - \\ell h ) _ + ) - \\sigma ^ 2 ( t ) \\} M ^ 0 _ n ( t ) } { \\sigma ^ 2 ( t ) \\mathfrak { s } _ n ( t ) } \\\\ & = : \\mathbf { I } _ n ( t ) + \\mathbf { I I } _ n ( t ) , \\end{align*}"}
-{"id": "4940.png", "formula": "\\begin{align*} ( M ) = \\prod _ { i = 1 } ^ n \\frac { x _ i } { 1 - e ^ { - x _ i } } \\end{align*}"}
-{"id": "9448.png", "formula": "\\begin{align*} = \\sum _ { n = 2 ^ { 2 ^ N } + 1 } ^ { 2 ^ { 2 ^ { N + 1 } } } \\left ( \\psi ( n ) - \\psi ( n + 1 ) \\right ) \\sum _ { j = 2 ^ { 2 ^ N } + 1 } ^ n \\frac { \\varphi ( j ) } { j | { j } | _ { \\mathcal { D } } } + \\psi ( 2 ^ { 2 ^ { N + 1 } } + 1 ) \\sum _ { j = 2 ^ { 2 ^ N } + 1 } ^ { 2 ^ { 2 ^ { N + 1 } } } \\frac { \\varphi ( j ) } { j | { j } | _ { \\mathcal { D } } } . \\end{align*}"}
-{"id": "9537.png", "formula": "\\begin{align*} \\widehat { \\phi } _ { x y } ^ * ( f ) = f ( x + h , x _ 2 , \\ldots , x _ n ) = \\left ( f + \\frac { \\partial { f } } { \\partial { x } } \\cdot h + \\ldots + \\frac { 1 } { i ! } \\frac { \\partial ^ i { f } } { \\partial { x ^ i } } \\cdot h ^ i + \\ldots \\right ) ( x , x _ 2 \\ldots , x _ n ) \\end{align*}"}
-{"id": "8657.png", "formula": "\\begin{align*} \\begin{aligned} \\pi [ \\tau _ 2 < \\infty | X _ { \\tau _ 1 } , Y _ { \\tau _ 1 } ] = \\pi \\Big [ \\int _ { \\tau _ 1 } ^ \\infty 1 _ { \\{ | x + \\omega _ { X _ 0 } ( s ) - y - \\omega _ { Y _ 0 } ( s ) | \\leq 1 \\} } d s > K | X _ { \\tau _ 1 } , Y _ { \\tau _ 1 } \\Big ] , \\end{aligned} \\end{align*}"}
-{"id": "7681.png", "formula": "\\begin{align*} D ^ 2 _ B ( j ) = \\sum _ { k \\in V } d _ B ^ 2 ( j , k ) \\ , . \\end{align*}"}
-{"id": "9014.png", "formula": "\\begin{align*} \\| f \\| _ { \\dot { B } _ { p , r } ^ { s } } = \\left \\{ \\aligned & \\Big ( \\sum _ { j \\in \\mathbb { Z } } 2 ^ { j r s } \\| \\dot { \\Delta } _ { j } f \\| _ { L ^ { p } } ^ { r } \\Big ) ^ { \\frac { 1 } { r } } , \\forall \\ r < \\infty , \\\\ & \\sup _ { j \\in \\mathbb { Z } } 2 ^ { j s } \\| \\dot { \\Delta } _ { j } f \\| _ { L ^ { p } } , \\forall \\ r = \\infty . \\\\ \\endaligned \\right . \\end{align*}"}
-{"id": "561.png", "formula": "\\begin{align*} \\Phi x = x + \\sum _ y \\nu ( x , y ) \\ , y , \\end{align*}"}
-{"id": "2558.png", "formula": "\\begin{align*} | a | \\leq | c | + | c - a | = | c | + | p _ 0 - c | = | p _ 0 | , \\end{align*}"}
-{"id": "167.png", "formula": "\\begin{align*} ( \\mathfrak { X } \\uplus \\mathfrak { Y } ) ( a ) = \\mathfrak { X } ( a ) + \\mathfrak { Y } ( a ) , a \\in A . \\end{align*}"}
-{"id": "741.png", "formula": "\\begin{align*} m _ { \\mu , p } ( a ) : = \\inf _ { u \\in S ( a ) } I _ { \\mu , p } ( u ) . \\end{align*}"}
-{"id": "5076.png", "formula": "\\begin{align*} E _ 1 ( F ) = 0 , ~ ~ E _ 2 ( F ) = ( b _ 1 - b _ 2 ) X _ 2 , ~ ~ E _ 3 ( F ) = ( b _ 1 - b _ 3 ) X _ 3 . \\end{align*}"}
-{"id": "8489.png", "formula": "\\begin{align*} G ( \\varpi ^ { - a _ 1 } , \\mu \\chi _ 1 ) = \\zeta _ F ( 1 ) q ^ { - \\frac { a _ 1 } { 2 } } \\mu ( - b _ 1 ) \\epsilon ( \\frac { 1 } { 2 } , \\chi _ 1 ^ { - 1 } ) . \\end{align*}"}
-{"id": "9548.png", "formula": "\\begin{align*} \\ell ( w ) = | \\{ \\alpha \\in \\Phi ^ + \\ , : \\ , w ( \\alpha ) \\in \\Phi ^ - \\} | , \\end{align*}"}
-{"id": "1115.png", "formula": "\\begin{align*} \\sigma _ { \\mathcal { Z } _ s ^ { ( i ) } } ^ 2 = \\frac { \\sum _ { j \\in \\mathcal { M } } \\left ( \\beta _ j ^ { ( i ) } \\right ) ^ 2 | \\alpha _ { i j } | ^ 2 \\sigma _ { Z _ { s , j } ^ { ( i ) } } ^ 2 } { \\sum _ { j \\in \\mathcal { M } } \\beta _ j ^ { ( i ) } | \\alpha _ { i j } | ^ 2 } , \\ , \\sigma _ { \\mathcal { Z } _ 0 ^ { ( i ) } } ^ 2 = \\sum _ { j \\in \\mathcal { M } } \\left ( \\beta _ j ^ { ( i ) } \\right ) ^ 2 \\sigma _ { Z _ { 0 , j } ^ { ( i ) } } ^ 2 , \\end{align*}"}
-{"id": "1858.png", "formula": "\\begin{align*} \\Big ( \\sum _ { i = 1 } ^ d F ^ \\ast _ i ( x _ i ) - ( d - 1 ) \\Big ) ^ + \\leq F ( x _ 1 , \\dots , x _ d ) \\leq \\min _ { i = 1 , \\dots , d } F ^ \\ast _ i ( x _ i ) , \\end{align*}"}
-{"id": "5512.png", "formula": "\\begin{align*} 1 - \\varphi ( t ) = t ^ \\alpha \\tilde \\ell ( 1 / t ) h ( t ) , \\end{align*}"}
-{"id": "4958.png", "formula": "\\begin{align*} h _ { x _ { i _ 0 } } ( h _ { x _ 1 } ^ { d _ 1 } \\cdots h _ { x _ { i _ 0 } } ^ { d _ { i _ 0 } } \\cdots h _ { x _ { n } } ^ { d _ n } ) = h _ { x _ 1 } ^ { d _ 1 } \\cdots h _ { x _ 0 } ^ { d _ { i _ 0 } + 1 } \\cdots h _ { x _ { n } } ^ { d _ n } + f \\end{align*}"}
-{"id": "3384.png", "formula": "\\begin{align*} Q _ { F , p _ 0 , p _ 1 , q } ( F ' ) = \\psi _ { F ' } ( c _ { F ' } ) \\end{align*}"}
-{"id": "1706.png", "formula": "\\begin{align*} \\int _ X \\sum _ { \\lambda : d ( \\lambda ) = n } \\phi \\cdot \\frac { 1 } { | f _ \\lambda \\circ \\tau _ \\lambda | ^ 2 } \\ , d \\mu = \\sum _ { \\lambda : \\ , d ( \\lambda ) = n } \\int _ { D _ { s ( \\lambda ) } } \\phi \\frac { 1 } { | f _ \\lambda \\circ \\tau _ \\lambda | ^ 2 } d \\mu = \\sum _ { \\lambda : \\ , d ( \\lambda ) = n } \\int _ { R _ \\lambda } ( \\phi \\circ \\tau ^ n ) \\frac { 1 } { | f _ \\lambda | ^ 2 } d ( \\mu \\circ ( \\tau _ \\lambda ) ^ { - 1 } ) . \\end{align*}"}
-{"id": "2140.png", "formula": "\\begin{align*} \\partial _ s w + { \\mathcal L } _ d w + \\tilde { V } ( \\xi , s ) w = 0 \\quad \\mbox { f o r } \\xi \\in [ 0 , \\infty ) , \\ , \\ , s > 0 , \\end{align*}"}
-{"id": "6180.png", "formula": "\\begin{align*} d ( p , q ) = \\frac { 1 } { 2 } \\int _ { S ^ 1 _ { \\bar x } } V ( x _ 0 ) \\dd x _ 0 \\end{align*}"}
-{"id": "7878.png", "formula": "\\begin{align*} u ( x , t ) = \\inf _ { a \\in B _ t ( x ) } u _ 0 ( a ) \\end{align*}"}
-{"id": "3737.png", "formula": "\\begin{align*} \\sum _ { x \\in \\Z ^ d } \\alpha ( k , x ) = 1 . \\end{align*}"}
-{"id": "7136.png", "formula": "\\begin{align*} d _ t ( m , n ) = \\frac { \\lambda _ 0 } { r ( m , n ) } . \\end{align*}"}
-{"id": "1159.png", "formula": "\\begin{align*} p _ k ( w ) = \\sum _ { i = 0 } ^ k A _ { k , i } w ^ { 2 ^ i } \\end{align*}"}
-{"id": "4817.png", "formula": "\\begin{align*} 3 ^ { f ( n + 1 ) - f ( n ) } - 2 ^ n = 3 ^ { 2 n + 1 } - 2 ^ n > 1 4 = 2 \\max \\{ 3 , 5 , F ( S _ n ) \\} . \\end{align*}"}
-{"id": "4901.png", "formula": "\\begin{align*} T = \\begin{bmatrix} R & B \\\\ C & D \\end{bmatrix} \\end{align*}"}
-{"id": "6194.png", "formula": "\\begin{align*} \\widehat { \\underline { \\omega } } ( t ) = ( G _ t ^ { - 1 } ) ^ * \\underline { \\omega } ( t ) \\xrightarrow { C ^ \\infty } _ { t \\to \\infty } \\underline { \\omega } ^ 0 \\end{align*}"}
-{"id": "6981.png", "formula": "\\begin{align*} D f _ k ( B ) E _ { i i } = \\frac { 1 } { k f _ k ( B ) ^ { k - 1 } } \\sum _ { \\mbox { \\tiny $ \\begin{array} { c } i \\leq j _ 1 < j _ 2 < . . . < j _ { k - 1 } \\leq n , \\\\ j _ 1 , . . . , j _ { k - 1 } \\neq i \\end{array} $ } } \\det { B _ { ( j _ 1 , . . . , j _ { k - 1 } ) } } , \\end{align*}"}
-{"id": "8975.png", "formula": "\\begin{gather*} \\sum _ { j \\ge 0 } \\sum _ { \\lambda \\in W \\lambda _ 0 } [ w \\lambda _ 0 = u + j q ' ] \\end{gather*}"}
-{"id": "317.png", "formula": "\\begin{align*} v & = ( v _ 1 , v _ 2 , \\dots , . . . , v _ k ) : = G _ { R _ k ( z ) } ^ { - 1 } ( R _ k ( u ) ) \\\\ & = \\left ( u _ { 0 } - z _ { 0 } \\frac { u _ k } { z _ k } , u _ 1 - z _ 1 \\frac { u _ k } { z _ k } , \\dots , u _ { k - 1 } - z _ { k - 1 } \\frac { u _ k } { z _ k } , u _ { k + 1 } - z _ { k + 1 } \\frac { u _ k } { z _ k } , \\dots , u _ { d } - z _ { d } \\frac { u _ k } { z _ k } \\right ) . \\end{align*}"}
-{"id": "9896.png", "formula": "\\begin{align*} Y ( i ) = \\bigl \\{ v \\in V _ a \\colon | N ( v ) \\cap V _ i | \\le \\bigl ( \\tfrac 1 2 + \\tfrac \\theta 2 \\bigr ) | V _ i | \\bigr \\} \\end{align*}"}
-{"id": "6549.png", "formula": "\\begin{align*} \\alpha _ { \\xi } = \\left ( \\frac { n | P | _ n } { | ( F _ { \\xi } - s ( F _ { \\xi } ) ) ^ { \\circ } | _ { n - 1 } \\| \\xi \\| } \\right ) ^ { 1 / n } , \\end{align*}"}
-{"id": "9439.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ N \\frac { \\varphi ( n ) \\psi ( n ) } { n | { n } | _ { \\mathcal { D } _ 1 } | { n } | _ { \\mathcal { D } _ 2 } \\cdots | n | _ { \\mathcal { D } _ m } } \\geq c \\sum _ { n = 1 } ^ N \\frac { \\psi ( n ) } { | n | _ { \\mathcal { D } _ 1 } | n | _ { \\mathcal { D } _ 2 } \\cdots | n | _ { \\mathcal { D } _ m } } . \\end{align*}"}
-{"id": "2056.png", "formula": "\\begin{align*} d \\iota ( \\tau ( f ) ) = \\tau ( u ) - t r a c e _ { G _ \\theta } ( \\nabla d P ) ( d _ H u , d _ H u ) . \\end{align*}"}
-{"id": "1255.png", "formula": "\\begin{align*} \\hat u _ m ( x ) : = u _ m ( M _ j ^ { - 1 } x ) \\mbox { a n d } \\hat u ( x ) : = u ( M _ j ^ { - 1 } x ) \\end{align*}"}
-{"id": "10024.png", "formula": "\\begin{align*} \\epsilon _ Q ( g ) = \\prod _ { \\substack { q \\mid Q \\\\ } } \\chi ^ n _ Q ( Q / q ) \\cdot \\lambda _ q , \\end{align*}"}
-{"id": "8373.png", "formula": "\\begin{align*} \\mathrm { d i v } ( \\psi ( f ^ { [ i ] } ) = \\sum _ { \\substack { m > 0 \\\\ \\mu \\in V _ \\Z ^ \\vee / V _ \\Z } } c ^ { [ i ] } ( - m , \\mu ) \\cdot \\mathcal { Z } ^ { [ i ] } ( m , \\mu ) . \\end{align*}"}
-{"id": "8160.png", "formula": "\\begin{align*} a _ 2 a _ 4 = 0 , \\end{align*}"}
-{"id": "4717.png", "formula": "\\begin{align*} a b c \\leq \\frac { a ^ p } { p } + \\frac { b ^ { p ' } c ^ { p ' } } { p ' } \\leq \\frac { a ^ p } { p } + \\frac { 1 } { p ' } \\left ( \\frac { ( b ^ { p ' } ) ^ { q / p ' } } { q / p ' } + \\frac { ( c ^ { p ' } ) ^ { r / p ' } } { r / p ' } \\right ) = \\frac { a ^ p } { p } + \\frac { b ^ q } { q } + \\frac { c ^ r } { r } , \\end{align*}"}
-{"id": "8700.png", "formula": "\\begin{align*} b ^ 2 _ { n , k - 1 } & \\leq a ^ 2 _ { n , \\lfloor \\zeta q ^ { - n } \\rfloor + k - \\lfloor n ^ \\delta q ^ { - n } \\rfloor } \\textrm { a n d } \\\\ F _ { n , k } & \\geq \\tilde { F } _ { n , k + \\lfloor n ^ { \\alpha } q ^ { - n } \\rfloor } \\textrm { f o r a l l } n \\geq n _ 0 . \\end{align*}"}
-{"id": "5428.png", "formula": "\\begin{align*} \\beta : = \\frac { \\alpha \\overline { \\left \\langle y , e \\right \\rangle } } { \\alpha - \\left \\langle x , e \\right \\rangle } . \\end{align*}"}
-{"id": "1042.png", "formula": "\\begin{align*} L _ { P } ^ { \\ast } ( q ) = \\max \\left \\{ \\log H ( P ) - \\frac { q } { m } , \\log \\vert P ( \\zeta ) \\vert + q \\right \\} \\end{align*}"}
-{"id": "7261.png", "formula": "\\begin{align*} B _ 1 = \\sum _ { \\substack { u \\le n , v \\le m \\\\ ( u , n ) , ( v , m ) \\le T \\\\ u / n = v / m } } \\frac { 2 s } { N n } = \\frac { 2 s } N \\frac { A ( n , m ) } n \\end{align*}"}
-{"id": "2518.png", "formula": "\\begin{align*} \\left [ \\nabla _ \\xi , \\mathcal { L } \\right ] h = - \\nabla _ { x } h + \\partial _ i \\left [ ( \\partial _ k \\sigma ^ { i j } ) \\partial _ j h \\right ] - ( \\nabla _ { \\xi } \\psi ) h . \\end{align*}"}
-{"id": "2035.png", "formula": "\\begin{align*} g \\Big ( \\frac { z _ 1 ^ k } { z _ 2 . . . z _ { k + 1 } } \\Big ) f ( z _ 2 , . . . , z _ { k + 1 } ) + . . . + g \\Big ( \\frac { z _ { k + 1 } ^ k } { z _ 1 . . . z _ k } \\Big ) f ( z _ 1 , . . . , z _ k ) = 0 . \\end{align*}"}
-{"id": "6271.png", "formula": "\\begin{align*} \\Delta _ q ( u \\cdot \\nabla v ) = & \\sum _ { | q - p | \\leq 2 } \\Delta _ q ( u _ { \\leq { p - 2 } } \\cdot \\nabla v _ p ) + \\sum _ { | q - p | \\leq 2 } \\Delta _ q ( u _ { p } \\cdot \\nabla v _ { \\leq { p - 2 } } ) \\\\ & + \\sum _ { p \\geq q - 2 } \\Delta _ q ( \\tilde u _ p \\cdot \\nabla v _ p ) , \\end{align*}"}
-{"id": "9796.png", "formula": "\\begin{align*} \\lambda _ x [ A ] = \\sum \\nolimits _ { u \\in U } { \\theta \\circ \\kappa ( - x ^ { - t } A , u ) } \\pi ( u ) . \\end{align*}"}
-{"id": "4840.png", "formula": "\\begin{align*} - 2 ^ 7 \\cdot 3 ^ { 1 1 } \\cdot 5 \\cdot 7 \\cdot y ^ { 1 8 } \\cdot \\left ( 4 x - y \\right ) \\cdot \\left ( 1 4 x ^ 2 - 7 x y + 2 y ^ 2 \\right ) = 0 . \\end{align*}"}
-{"id": "275.png", "formula": "\\begin{align*} t ^ { 1 - \\alpha } \\int _ { 0 } ^ { t } ( t - \\tau ) ^ { \\alpha - 1 } | | | E _ { \\alpha , \\alpha } ( ( t - \\tau ) ^ { \\alpha } A ) | | | \\tau ^ { \\alpha - 1 } d \\tau & \\leq C _ { 1 } t ^ { 1 - \\alpha } \\int _ { 0 } ^ { t } ( t - \\tau ) ^ { \\alpha - 1 } \\tau ^ { \\alpha - 1 } d \\tau \\\\ & = C _ { 1 } t ^ { 1 - \\alpha } t ^ { 2 \\alpha - 1 } B ( \\alpha , \\alpha ) \\\\ & = C _ { 1 } t ^ { \\alpha } B ( \\alpha , \\alpha ) \\\\ & \\leq C _ { 1 } T ^ { \\alpha } B ( \\alpha , \\alpha ) , \\end{align*}"}
-{"id": "9728.png", "formula": "\\begin{align*} & ( \\Phi _ 1 ^ { ( 1 ) } ( \\tilde { \\gamma } _ 1 ; V _ a + \\widetilde { \\mathcal { V } } ( V _ a , Z _ { a } h ) Z _ { a } h ) , \\Phi _ 1 ^ { ( 2 ) } ( \\tilde { \\gamma } _ 1 ; V _ a + \\widetilde { \\mathcal { V } } ( V _ a , Z _ { a } h ) Z _ { a } h ) ) \\cdot \\textbf { n } _ { k } \\\\ & = ( \\Phi _ 1 ^ { ( 1 ) } ( \\gamma _ 1 ; V _ a ) , \\Phi _ 1 ^ { ( 2 ) } ( \\gamma _ 1 ; V _ a ) ) \\cdot \\textbf { n } _ { k } . \\end{align*}"}
-{"id": "6893.png", "formula": "\\begin{align*} c ( F _ { j _ 1 k _ 1 } \\circ . . . \\circ F _ { j _ m k _ m } ( x ) , F _ { j _ 1 k _ 1 } \\circ . . . \\circ F _ { j _ m k _ m } ( y ) ) = \\frac { 1 } { r _ 0 ^ { i ( A ) } } \\left ( \\frac { 1 } { r _ 1 ^ { m - i ( A ) } } \\right ) c ( x , y ) \\end{align*}"}
-{"id": "3676.png", "formula": "\\begin{align*} \\square _ { n ^ 2 } \\cap ( \\alpha _ W [ n ] L [ n ] ) ^ { - 1 } ( - b ) = R _ 1 [ n ] \\times \\dots \\times R _ n [ n ] . \\end{align*}"}
-{"id": "1243.png", "formula": "\\begin{align*} ( \\breve b _ i ) _ { k j } ( x ) = f _ { \\eta _ k \\eta _ j } ( \\nabla \\breve v _ i ( x ) ) \\mbox { f o r } \\ , \\ , 1 \\leq k , j \\leq n . \\end{align*}"}
-{"id": "4367.png", "formula": "\\begin{align*} B _ t A & = \\sum _ { j = j _ 0 ( t ) } ^ \\infty M _ { \\psi _ { j , t } } B _ { j , t } A M _ { \\varphi _ { j , t } } M _ { \\psi _ { j , t } } + \\sum _ { j = j _ 0 ( t ) } ^ \\infty M _ { \\psi _ { j , t } } B _ { j , t } [ M _ { \\varphi _ { j , t } } , A ] M _ { \\psi _ { j , t } } \\\\ & + \\sum _ { j = j _ 0 ( t ) } ^ \\infty M _ { \\psi _ { j , t } } B _ { j , t } M _ { \\varphi _ { j , t } } A M _ { 1 - \\psi _ { j , t } } \\end{align*}"}
-{"id": "6328.png", "formula": "\\begin{align*} \\widehat { ( { - \\Delta } ) ^ { s } u } ( \\xi ) = | \\xi | ^ { 2 s } \\hat u ( \\xi ) \\ , . \\end{align*}"}
-{"id": "5098.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| g _ { F _ n } \\ ! \\ast q - q \\| _ { \\textup { L } ^ 1 ( \\widehat { \\mathcal { L } } ) } = 0 . \\end{align*}"}
-{"id": "4294.png", "formula": "\\begin{align*} 4 \\langle b _ i , b _ j \\rangle M _ 2 = D _ i \\langle \\Delta _ 1 \\rangle _ { 0 , 1 } \\cdot D _ j \\langle \\Delta _ 2 \\rangle _ { 0 , 1 } - \\langle \\Delta _ 1 \\rangle _ { 0 , 1 } \\cdot D _ i D _ j \\langle \\Delta _ 2 \\rangle _ { 0 , 1 } \\end{align*}"}
-{"id": "5131.png", "formula": "\\begin{align*} u = z _ 1 + v . \\end{align*}"}
-{"id": "4671.png", "formula": "\\begin{align*} F ( v + w ) = F ( v ) + B ( v , w ) , v \\in M , \\ , w \\in N , \\end{align*}"}
-{"id": "8837.png", "formula": "\\begin{align*} \\lambda : = t ^ { - \\frac { 1 } { 2 } } | v | ^ { - \\frac { 3 } { 8 } } , \\end{align*}"}
-{"id": "8594.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { n } } ( - 2 \\pi i x ) ^ { \\alpha } T a ( x ) \\ , d x = ( ( - 2 \\pi i x ) ^ { \\alpha } T a ) \\ , \\ , \\widehat { } \\ , \\ , ( 0 ) = 0 , \\ , \\ , \\ , \\textit { f o r a l l } \\ , \\ , \\ , | \\alpha | \\leq d . \\end{align*}"}
-{"id": "14.png", "formula": "\\begin{align*} V _ { \\sigma } ^ C ( C _ 1 , C _ 2 ) = \\frac { 1 } { 2 \\pi \\sigma } - \\frac { 1 } { 2 \\pi \\sigma } R [ C _ 1 , C _ 2 ] + h _ { \\sigma ^ 4 } \\end{align*}"}
-{"id": "4777.png", "formula": "\\begin{align*} V ^ i { L } _ i { } ^ 2 = 0 , i = 0 . . . 3 , \\end{align*}"}
-{"id": "8592.png", "formula": "\\begin{align*} x = Q ^ { - 1 } S ^ { - 1 } S _ { n \\ , n } t \\mod S _ { n \\ , n } , t \\in \\mathbb { Z } ^ n . \\end{align*}"}
-{"id": "454.png", "formula": "\\begin{align*} \\psi _ \\omega \\ : = \\phi _ \\omega ( \\ , \\cdot \\ , + i y _ \\omega u _ 1 ) - \\phi _ \\omega ( i y _ \\omega u _ 1 ) \\end{align*}"}
-{"id": "5877.png", "formula": "\\begin{align*} \\begin{aligned} & \\lim _ { n \\to \\infty } \\frac 1 n \\log P ( \\frac { S ^ { ( i ) } _ n } n \\ge r ) = \\lim _ { n \\to \\infty } \\frac 1 n \\log P ( \\frac { S ^ { ( i ) } _ n } n > r ) = - I _ i ( r ) , \\ \\mu _ i \\le r < x ^ + _ i ; \\\\ & \\lim _ { n \\to \\infty } \\frac 1 n \\log P ( \\frac { S ^ { ( i ) } _ n } n \\le r ) = \\lim _ { n \\to \\infty } \\frac 1 n \\log P ( \\frac { S ^ { ( i ) } _ n } n < r ) = - I _ i ( r ) , \\ x ^ - _ i < r \\le \\mu _ i . \\end{aligned} \\end{align*}"}
-{"id": "6454.png", "formula": "\\begin{align*} R _ { } ^ { } ( \\rho ) \\overset { } { = } \\frac { \\mathcal { C } _ { \\mathcal { M } } ( \\tau ) } { \\mathcal { C } _ { \\mathcal { M } } \\left . ( \\tau ) \\right \\vert _ { \\rho = 0 } } = \\sqrt { 1 + 2 \\rho } \\end{align*}"}
-{"id": "1840.png", "formula": "\\begin{align*} F _ { \\bullet , m } = \\cdots \\to F _ { 1 , m } \\to F _ { 0 , m } \\to F _ { - 1 , m } \\to \\cdots . \\end{align*}"}
-{"id": "5645.png", "formula": "\\begin{align*} K _ r ( x ) : = \\inf \\{ K ( y ) \\ ; : \\ ; d ( x , y ) \\leq r \\} . \\end{align*}"}
-{"id": "1664.png", "formula": "\\begin{align*} R _ { e } = ( 0 , \\frac { a } { 2 } ) , R _ f = ( a , 1 ) , R _ g = ( \\frac { a } { 2 } , a ) . \\end{align*}"}
-{"id": "7606.png", "formula": "\\begin{align*} g ^ * _ A ( p _ 1 ^ A ) : = \\sup _ x \\left \\{ p _ 1 ^ A \\cdot x - g _ A ( x ) \\right \\} = p _ 1 ^ A \\cdot x _ 1 - g ( x _ 1 ) . \\end{align*}"}
-{"id": "419.png", "formula": "\\begin{align*} h _ { k _ 1 , k _ 2 } ( R , { t } ) = e ^ { - \\frac { 1 } { 4 } d ( x , t ) ^ 2 } \\int _ \\R e ^ { i R \\psi _ \\omega ( \\lambda ) } a _ { k _ 1 , k _ 2 } ( \\lambda + i y _ \\omega ) \\ , \\dd \\lambda . \\end{align*}"}
-{"id": "6269.png", "formula": "\\begin{align*} & h = \\mathcal F ^ { - 1 } \\varphi , \\tilde h = \\mathcal F ^ { - 1 } \\chi , \\\\ & u _ q : = \\Delta _ q u = \\mathcal F ^ { - 1 } ( \\varphi ( \\lambda _ q ^ { - 1 } \\xi ) \\mathcal F u ) = \\lambda _ q ^ n \\int h ( \\lambda _ q y ) u ( x - y ) d y , \\mbox { f o r } q \\geq 0 , \\\\ & u _ { - 1 } = \\mathcal F ^ { - 1 } ( \\chi ( \\xi ) \\mathcal F u ) = \\int \\tilde h ( y ) u ( x - y ) d y , \\end{align*}"}
-{"id": "9234.png", "formula": "\\begin{align*} A _ { 1 } \\otimes A _ { 2 } & = ( s y m ( A _ { 1 } ) \\oplus s k e w ( A _ { 1 } ) ) \\otimes ( s y m ( A _ { 2 } ) \\oplus s k e w ( A _ { 2 } ) ) = s y m ( A _ { 1 } ) \\otimes s y m ( A _ { 2 } ) \\\\ & \\oplus s k e w ( A _ { 1 } ) \\otimes s k e w ( A _ { 2 } ) \\oplus s k e w ( A _ { 1 } ) \\otimes s y m ( A _ { 2 } ) \\oplus s y m ( A _ { 1 } ) \\otimes s k e w ( A _ { 2 } ) . \\end{align*}"}
-{"id": "1978.png", "formula": "\\begin{align*} \\tilde { g } _ { p } & = \\tau _ { f ( p ) } \\circ _ { f ( p ) } ^ { - 1 } \\circ g _ { i } \\circ _ { p } \\circ \\tau _ { p } ^ { - 1 } : B _ { p } ( 0 , \\beta _ { i } ) \\rightarrow B _ { f ( p ) } ( 0 , \\delta _ { i + 1 } ) \\\\ \\quad \\tilde { g } _ { p } ^ { - 1 } & = \\tau _ { p } \\circ _ { p } ^ { - 1 } \\circ g _ { i } ^ { - 1 } \\circ _ { f ( p ) } \\circ \\tau _ { f ( p ) } ^ { - 1 } : B _ { f ( p ) } ( 0 , \\beta _ { i + 1 } ) \\rightarrow B _ { p } ( 0 , \\delta _ { i } ) , \\end{align*}"}
-{"id": "7747.png", "formula": "\\begin{align*} \\sum ^ { 2 N - 1 } _ { n = 1 } \\frac { 1 } { \\sin ^ 2 \\frac { n \\pi } { 2 N } } = 2 \\sum ^ { N - 1 } _ { n = 1 } \\frac { 1 } { \\sin ^ 2 \\frac { n \\pi } { 2 N } } + 1 \\ , . \\end{align*}"}
-{"id": "7559.png", "formula": "\\begin{gather*} u = \\frac { t x - \\mu y } { t ^ 2 + \\mu ^ 2 } , v = \\frac { \\mu x + t y } { t ^ 2 + \\mu ^ 2 } , \\end{gather*}"}
-{"id": "9565.png", "formula": "\\begin{align*} | Q _ j | \\ , \\asymp \\ , 2 ^ j \\ , j ^ { d - 1 } \\ , , \\qquad | Q _ { [ J ] } | \\ , \\asymp \\ , \\sum _ { j = 0 } ^ J 2 ^ j \\ , j ^ { d - 1 } \\ , \\asymp \\ , 2 ^ J \\ , J ^ { d - 1 } \\ , . \\end{align*}"}
-{"id": "7438.png", "formula": "\\begin{align*} f : = - x ^ 2 - u y ^ 2 - 2 v y z - w z ^ 2 - ( u w - v ^ 2 ) t ^ 2 . \\end{align*}"}
-{"id": "8562.png", "formula": "\\begin{align*} \\partial _ t u + \\partial \\psi ( u ) \\ni 0 \\ \\mbox { i n } \\mathcal H , 0 < t < \\infty , u ( 0 ) = \\phi , \\end{align*}"}
-{"id": "7796.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { w \\in \\mathsf { s i b } _ { i _ j } ( v _ G | u _ { i _ j } ) } \\varPhi ( f _ 1 \\otimes \\cdots \\otimes f _ k ) ( w ) = 0 , v _ G \\in \\mathsf { s i b } _ { F , G } ( u ) , ~ G = F \\setminus \\{ i _ j \\} , ~ j = 1 , \\ldots , k . \\end{align*}"}
-{"id": "4017.png", "formula": "\\begin{align*} F ( p , t ) = F _ t ( p ) = p + t g ( p ) \\eta ( p ) , \\end{align*}"}
-{"id": "8854.png", "formula": "\\begin{align*} a _ n ( \\phi ) = \\gamma _ d \\int _ 0 ^ \\phi P _ n ^ { ( d ) } ( \\cos ( \\theta ) ) \\sin ( \\theta ) ^ { d - 1 } \\ , \\dd \\theta = \\frac { \\gamma _ d } d \\sin ( \\phi ) ^ d P _ { n - 1 } ^ { ( d + 2 ) } ( \\cos ( \\phi ) ) , n \\geq 1 . \\end{align*}"}
-{"id": "5203.png", "formula": "\\begin{align*} S _ { 1 } ' ( \\ell _ { * } ) = S _ { 2 } ' ( r _ { * } ) = \\left ( \\frac { g _ { 1 } ' ( r _ { * } ) - f _ { 1 } ' ( \\ell _ { * } ) } { f _ { 1 } '' ( \\ell _ { * } ) ( r _ { * } - \\ell _ { * } ) } \\right ) \\left ( \\frac { f _ { 2 } ' ( r _ { * } ) - g _ { 2 } ' ( \\ell _ { * } ) } { f _ { 2 } '' ( r _ { * } ) ( r _ { * } - \\ell _ { * } ) } \\right ) . \\end{align*}"}
-{"id": "631.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { \\alpha _ 1 , \\alpha _ 2 \\ge 0 } a _ { \\alpha } z _ 1 ^ { \\alpha _ 1 } z _ 2 ^ { \\alpha _ 2 } . \\end{align*}"}
-{"id": "10088.png", "formula": "\\begin{align*} \\tilde { R } ( X , Y ) Z = R ( X , Y ) Z + \\lambda \\{ \\pi ( X ) \\pi ( Z ) Y - \\pi ( Y ) \\pi ( Z ) X \\} . \\end{align*}"}
-{"id": "1066.png", "formula": "\\begin{align*} \\log H _ { k } - \\frac { q _ { k } } { m } = L _ { P _ { k } } ^ { \\ast } ( q _ { k } ) , \\end{align*}"}
-{"id": "3210.png", "formula": "\\begin{align*} \\sum _ { \\alpha , \\beta = 0 } ^ n | h ^ { \\alpha \\beta } ( t , x ) | \\leq \\frac { 1 } { 2 } , \\forall ( t , x ) \\in [ 0 , T ] \\times { \\mathbb R } ^ n . \\end{align*}"}
-{"id": "2038.png", "formula": "\\begin{align*} g ( q z ) = - z ^ { - 1 } g ( z ) ; ~ ~ ~ g ( z ) = 0 ~ ~ z = q ^ i , ~ i \\in \\Z , \\end{align*}"}
-{"id": "3960.png", "formula": "\\begin{align*} \\partial _ t ^ { \\alpha _ n } p ^ { \\alpha _ n } ( n , t ) = - \\lambda ( p ^ { \\alpha _ n } ( n , t ) - p ^ { \\alpha _ { n - 1 } } ( n - 1 , t ) ) , \\ \\ 0 < \\alpha _ n \\leq 1 , \\ n \\geq 0 , \\end{align*}"}
-{"id": "7188.png", "formula": "\\begin{align*} z _ \\infty = a _ 0 h + \\left ( \\sum _ { i = 1 } ^ { n - 2 } a _ i x _ i \\right ) \\left ( \\sqrt { x _ { n - 1 } ^ 2 + x _ n ^ 2 } + x _ { n - 1 } \\right ) ^ s , \\end{align*}"}
-{"id": "2046.png", "formula": "\\begin{align*} H ( q z _ 1 , . . . , z _ n ) = q ^ { k ( k + 1 ) } z _ 1 ^ { 2 k ^ 2 + 2 k } z _ 2 ^ { k ^ 2 + k } . . . z _ n ^ { k ^ 2 + k } H ( z _ 1 , . . . , z _ n ) . \\end{align*}"}
-{"id": "9881.png", "formula": "\\begin{align*} d \\psi ( g ^ { y _ 1 } g _ 1 ( g ^ { y _ 2 } ) ^ { - 1 } , g ^ { y _ 2 } g _ 2 ( g ^ { y _ 0 } ) ^ { - 1 } ) = \\omega _ 0 ( g ^ { y _ 1 } g _ 1 ( g ^ { y _ 2 } ) ^ { - 1 } , g ^ { y _ 2 } g _ 2 ( g ^ { y _ 0 } ) ^ { - 1 } ) . \\end{align*}"}
-{"id": "3336.png", "formula": "\\begin{align*} D ^ * ( r ) \\geq \\max _ { i \\in \\{ 2 , \\cdots , K + 1 \\} } ( 1 - r ) \\sum _ { j = 0 } ^ { K + 1 - i } \\frac { 1 } { N ^ j } - r \\sum _ { j = 0 } ^ { K - i } \\frac { K + 1 - k - j } { N ^ j } \\end{align*}"}
-{"id": "2856.png", "formula": "\\begin{align*} \\bar { \\Delta } A _ { n } ( s ) & = \\sum _ { v = 0 } ^ { n } \\hat { a } _ { n v } a _ { v } . \\end{align*}"}
-{"id": "6168.png", "formula": "\\begin{align*} \\partial _ t V = \\frac { 1 } { 3 } \\mathcal { T } V \\end{align*}"}
-{"id": "5351.png", "formula": "\\begin{align*} \\pi _ { n - m } ( I ( w _ { ( 1 ) } \\otimes w _ { ( 2 ) } ) ) = 0 \\ ; \\ ; \\ ; \\mbox { f o r } \\ ; m \\in \\N \\ ; \\mbox { s u f f i c i e n t l y l a r g e } , \\end{align*}"}
-{"id": "1759.png", "formula": "\\begin{align*} T ( f \\sqrt { d \\mu } ) = F _ \\mu \\ , f \\ , \\sqrt { d \\mu } \\end{align*}"}
-{"id": "5662.png", "formula": "\\begin{align*} \\mathcal { E } ( { u } ) = \\begin{cases} \\int _ \\R \\left [ \\int _ \\R \\left ( \\frac 1 2 | \\nabla { u } ( x _ 1 , x _ 2 ) | ^ 2 + W ( { u } ( x _ 1 , x _ 2 ) ) \\right ) \\d x _ 1 - d _ K ( a ^ - , a ^ + ) \\right ] \\d x _ 2 & { u } \\in H ^ 1 _ { l o c } ( \\R ^ 2 , \\R ^ n ) , \\\\ + \\infty & \\end{cases} \\end{align*}"}
-{"id": "6880.png", "formula": "\\begin{align*} \\lim _ { p \\to 1 , m \\to \\infty } \\left ( \\frac { 4 } { p q } \\right ) ^ { m } \\Phi _ { 1 } ^ { m } ( 2 ( 1 - \\sqrt { q } ) ) = \\infty \\end{align*}"}
-{"id": "2617.png", "formula": "\\begin{align*} \\pi \\tan ( \\tfrac { a \\pi } 2 ) \\tilde u ( t ) - S \\tilde u ( t ) = \\tilde g ( t ) = t ^ { 1 - a } D ^ { 1 - a } f ( t ) 0 < t < 1 . \\end{align*}"}
-{"id": "9139.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { 4 } a _ { 5 - i } x ^ { i } f ( x ^ { 5 - i } ) = 0 \\left ( x \\in R \\right ) . \\end{align*}"}
-{"id": "696.png", "formula": "\\begin{align*} a _ n = | \\varphi ( \\pi , \\mu _ n ) | | \\dot { \\Phi } ( \\mu _ n ) | , \\end{align*}"}
-{"id": "3158.png", "formula": "\\begin{align*} \\psi ( t , u ) = \\frac { u e ^ { - b t } } { 1 - \\tfrac { \\sigma ^ { 2 } u } { 2 b } \\left ( 1 - e ^ { - b t } \\right ) } . \\end{align*}"}
-{"id": "4334.png", "formula": "\\begin{align*} d \\mu _ { \\nu } ( z ) = \\Big ( \\frac { \\nu } { \\pi } \\Big ) ^ n e ^ { - \\nu | z | ^ 2 } d z \\end{align*}"}
-{"id": "7430.png", "formula": "\\begin{align*} \\beta : = a _ 4 a _ 4 ^ * + R _ 1 e _ 4 \\ , \\ , \\gamma : = a _ 4 B a _ 4 ^ * - S _ 2 e _ 4 . \\end{align*}"}
-{"id": "4539.png", "formula": "\\begin{align*} d ( F _ j ( h ( x _ i ) ) , G _ j ( x _ i ) ) < K \\zeta \\quad F _ j ( h ( x _ 1 ) ) = F _ j ( h ( x _ 2 ) ) \\end{align*}"}
-{"id": "8069.png", "formula": "\\begin{align*} B _ { 3 , 4 } & = \\mathcal { V } ( 1 - a _ 3 a _ 4 ^ { - 1 } ) \\\\ B _ { 1 , 2 , 3 } & = \\mathcal { V } ( 1 - a _ 1 a _ 2 ^ { - 1 } a _ 3 ^ 7 ) \\\\ B _ { 1 , 2 , 4 } & = \\mathcal { V } ( 1 - a _ 1 a _ 2 ^ { - 1 } a _ 4 ^ 7 ) \\\\ B _ { 1 , 2 , 3 , 4 } & = \\mathcal { V } ( 1 - a _ 1 a _ 2 ^ { - 1 } a _ 3 ^ 7 , 1 - a _ 3 a _ 4 ^ { - 1 } ) . \\end{align*}"}
-{"id": "6779.png", "formula": "\\begin{align*} \\tilde { w } _ { \\lambda } = \\sum \\limits _ { k = 1 } ^ 4 w _ { \\lambda , k } , \\end{align*}"}
-{"id": "7072.png", "formula": "\\begin{align*} \\pi _ 1 ( \\nabla _ u \\Phi ( u _ 1 , u _ 2 , \\lambda ) ) = 0 , \\end{align*}"}
-{"id": "9418.png", "formula": "\\begin{align*} \\begin{aligned} \\bar { \\tau } _ n ^ { ( r ) } & = \\dfrac { \\tilde { \\tau } _ n ^ { ( r ) } } { 2 } + \\dfrac { 1 } { 4 n } \\sum \\limits _ { ( 2 d + 1 ) \\mid n } \\phi ( 2 d + 1 ) \\cdot \\left [ C ^ { ( r ) } _ { 2 n / ( 2 d + 1 ) } + M ^ { ( r ) } _ { 2 n / ( 2 d + 1 ) } \\right ] + \\\\ & + \\dfrac { 1 } { 4 n } \\sum \\limits _ { ( 2 d + 1 ) \\mid n } \\phi ( 2 d + 1 ) \\cdot \\kappa ^ { ( r ) } _ { n / ( 2 d + 1 ) } + \\dfrac { 1 } { n } \\sum \\limits _ { 2 d \\mid n } \\phi ( 2 d ) \\cdot \\kappa ^ { ( r ) } _ { n / ( 2 d ) } . \\end{aligned} \\end{align*}"}
-{"id": "3791.png", "formula": "\\begin{align*} W = \\Big \\{ w : \\Z \\to \\Z \\colon \\ , | w ( i + 1 ) - w ( i ) | \\le 1 \\ ; \\ ; \\forall \\ ; i \\in \\Z \\Big \\} . \\end{align*}"}
-{"id": "5032.png", "formula": "\\begin{align*} s _ { \\Gamma } ( x _ 1 , x _ 2 , y _ 1 , \\dots , y _ n ) = s _ { k + 2 } ( x _ 1 , x _ 2 , g _ { e _ 1 } , \\dots \\dots , g _ { e _ k } ) . \\end{align*}"}
-{"id": "5179.png", "formula": "\\begin{align*} V _ { 1 } ^ { [ r , 1 ] } ( x ) - g _ { 1 , [ r , 1 ] } ( x ) & = M ^ { x } _ { 1 } ( D _ { [ 0 , \\ell _ { r } ] } , D _ { [ r , 1 ] } ) - g _ { 1 , [ r , 1 ] } ( x ) \\\\ & = \\begin{cases} f _ { 1 } ( x ) - g _ { 1 , [ r , 1 ] } ( x ) , & \\forall x \\le \\ell _ { r } \\\\ f _ { 1 } ( \\ell _ { r } ) \\frac { r - x } { r - \\ell _ { r } } + g _ { 1 } ( r ) \\frac { x - \\ell _ { r } } { r - \\ell _ { r } } - g _ { 1 , [ r , 1 ] } ( x ) , & \\forall \\ell _ { r } < x \\le r \\end{cases} \\end{align*}"}
-{"id": "3112.png", "formula": "\\begin{align*} Q _ 0 ' & = \\left \\{ v _ 1 , \\ldots , v _ N , w _ 1 , \\ldots , w _ N \\right \\} \\\\ Q _ 1 ' & = \\left \\{ \\varphi _ { k l } : v _ k \\to w _ l \\mid k , l = 1 , \\ldots , N \\right \\} . \\end{align*}"}
-{"id": "2013.png", "formula": "\\begin{align*} z _ i = \\sum _ { \\substack { \\alpha _ 1 , . . . , \\alpha _ { k + 1 } \\in \\Z , \\\\ \\alpha _ 1 \\leq . . . \\leq \\alpha _ { k + 1 } , \\\\ \\alpha _ 1 + . . . + \\alpha _ { k + 1 } = \\frac { k ( k + 1 ) } { 2 } } } a _ { \\alpha _ 1 , . . . , \\alpha _ { k + 1 } } x _ { i + \\alpha _ 1 } x _ { i + \\alpha _ 2 } . . . x _ { i + \\alpha _ { k + 1 } } , ~ ~ ~ i \\in \\Z \\end{align*}"}
-{"id": "2295.png", "formula": "\\begin{align*} \\mathcal { U } ^ \\alpha ( u _ 1 , u _ 2 ) = \\alpha ^ 2 ( 1 - \\alpha ^ 2 \\Delta ) ^ { - 1 } \\mathrm { d i v } \\ , [ \\nabla u _ 1 \\cdot \\nabla u _ 2 ^ T + \\nabla u _ 1 \\cdot \\nabla u _ 2 - \\nabla u _ 1 ^ T \\cdot \\nabla u _ 2 ] , \\end{align*}"}
-{"id": "7557.png", "formula": "\\begin{gather*} f _ t + \\frac i 2 ( f _ z ) ^ 2 = i \\lambda \\end{gather*}"}
-{"id": "158.png", "formula": "\\begin{align*} 1 \\to U L \\cap \\Delta = [ \\Delta , \\Delta ] \\to \\Delta \\to D \\cap \\Delta = \\Delta ^ { a b } \\to 1 \\end{align*}"}
-{"id": "3129.png", "formula": "\\begin{align*} c ( T ^ * ) = e ( T ) + f _ 4 ( T ) + \\varepsilon ^ * & = ( 2 t + 3 \\left \\lfloor { k } / { 2 } \\right \\rfloor - 1 6 + \\varepsilon ) + ( t - 3 \\left \\lfloor { k } / { 2 } \\right \\rfloor + 1 0 - \\varepsilon ) + \\varepsilon ^ * \\\\ & = ( 3 n + 9 \\left \\lfloor { k } / { 2 } \\right \\rfloor + 3 \\varepsilon - 4 2 - \\varepsilon ^ * ) / { 2 } \\\\ & \\ge ( 3 n + 9 \\left \\lfloor { k } / { 2 } \\right \\rfloor + 3 \\varepsilon - 4 3 ) / { 2 } , \\end{align*}"}
-{"id": "8120.png", "formula": "\\begin{align*} B _ { i , k } = \\bigg ( \\bigcap _ { t \\in F ^ k } t S _ i \\bigg ) \\setminus \\bigcap _ { t \\in F ^ { k + 1 } } t S _ i . \\end{align*}"}
-{"id": "6019.png", "formula": "\\begin{align*} & \\inf _ { P _ { X } , Q _ { X } : \\left | P _ { X } - Q _ { X } \\right | \\geq \\epsilon } D _ { 0 } ( P _ { X } \\| Q _ { X } ) \\\\ & = \\inf _ { P _ { X } , Q _ { X } : \\left | P _ { X } - Q _ { X } \\right | = \\epsilon } D _ { 0 } ( P _ { X } \\| Q _ { X } ) \\\\ & = 0 . \\end{align*}"}
-{"id": "1861.png", "formula": "\\begin{align*} \\mathcal { F } ^ { S , \\pi } _ { \\preceq _ 0 } ( F _ 1 ^ \\ast , \\dots , F _ d ^ \\ast ) : = \\bigg \\{ c F : \\begin{array} { l } c \\in [ 0 , 1 ] F \\mathbb { R } ^ d c F _ i \\preceq _ 0 F _ i ^ \\ast \\\\ i = 1 , \\dots , d c F ( s ) \\leq \\pi _ s s \\in S \\end{array} \\bigg \\} , \\end{align*}"}
-{"id": "9002.png", "formula": "\\begin{gather*} F ( \\vec { z } ; c ; q , t ) = G ( c - \\vec { z } ; q , t ) H ( c ; q , t ) , \\end{gather*}"}
-{"id": "1967.png", "formula": "\\begin{align*} \\widehat { Q _ { 2 } ( g _ \\beta ) } ( \\eta ) = P ( S _ r ( g _ \\beta ) ) ( \\eta ) + i \\pi S ( g _ \\beta ) ( \\eta ) , \\end{align*}"}
-{"id": "265.png", "formula": "\\begin{align*} c _ 1 k \\left ( \\frac { c _ 0 m _ { - 2 k } } { \\nu } \\right ) ^ { 1 / k } \\le N , k \\le \\frac { N } { 2 } . \\end{align*}"}
-{"id": "9482.png", "formula": "\\begin{align*} \\sum _ { s = 0 } ^ N \\frac { q ^ s ( q ; q ) _ { N + s } } { ( q ^ 2 ; q ^ 2 ) _ s } = ( q ^ 2 ; q ^ 2 ) _ N . \\end{align*}"}
-{"id": "1574.png", "formula": "\\begin{align*} v ( \\mathbf { j } ) : = v ( m , \\mathbf { j } ) = \\sum _ { k = 1 } ^ { n - 1 } [ ( n - k + 1 ) j _ k - s ( j _ k ) ] . \\end{align*}"}
-{"id": "4304.png", "formula": "\\begin{align*} T _ { \\Delta } = - \\sum _ { i , j = 1 } ^ { 4 } \\left ( G ^ { - 1 } \\right ) _ { i j } T _ { b _ i } T _ { b _ j } \\end{align*}"}
-{"id": "10106.png", "formula": "\\begin{align*} { d _ k ( i ) } = { \\boldsymbol { \\omega } } _ 0 ^ H { \\boldsymbol x _ k ( i ) } + { n _ k ( i ) } , ~ ~ ~ i = 1 , 2 , \\ldots , \\textrm { I } , \\end{align*}"}
-{"id": "856.png", "formula": "\\begin{align*} H _ 0 : \\rho = 0 \\qquad H _ 1 : \\rho \\neq 0 . \\end{align*}"}
-{"id": "5494.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { U ( r ^ { - n } z ) } { ( r ^ { - n } z ) ^ { \\rho } \\ell ( r ^ { - n } z ) } = p ( z ) z \\in C _ { p } , p \\in \\mathcal { P } _ { r } . \\end{align*}"}
-{"id": "4737.png", "formula": "\\begin{align*} \\alpha + \\beta a _ 0 + \\gamma b _ 0 = 0 , \\alpha , \\beta , \\gamma - c o n s t , \\end{align*}"}
-{"id": "4607.png", "formula": "\\begin{align*} [ 0 , 1 ) ^ { d + 1 } \\times \\mathbb { R } ^ { d + 1 } \\times [ 0 , 1 ) \\times \\mathbb { R } & \\to [ 0 , 1 ) \\times \\mathbb { R } \\\\ ( x _ 1 , \\dots , x _ { d + 1 } , y _ 1 , \\dots , y _ { d + 1 } , x , t ) & \\mapsto \\left ( T x , t + \\sum _ { i = 0 } ^ d y _ i \\cdot 1 _ { [ x _ 0 + \\cdots + x _ i , x _ 0 + \\cdots + x _ { i + 1 } ) } ( x ) \\right ) \\end{align*}"}
-{"id": "9832.png", "formula": "\\begin{align*} W ( x , y ) = \\sum _ { i = 0 } ^ n A _ i x ^ { 2 ( n - i ) } y ^ { 2 i } \\qquad ( A _ 0 = 1 ) . \\end{align*}"}
-{"id": "9781.png", "formula": "\\begin{align*} G _ m V : = \\begin{cases} 0 & ( m \\ll 0 ) \\\\ V & ( m \\gg 0 ) . \\end{cases} \\end{align*}"}
-{"id": "4628.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { N - 1 } 1 _ { ( - \\epsilon , \\epsilon ) } \\left ( \\sum _ { i = 0 } ^ n f ( T ^ i x ) \\right ) > N ^ { 1 - \\gamma - \\epsilon } \\end{align*}"}
-{"id": "4064.png", "formula": "\\begin{align*} \\mathcal Q _ j = \\bigl \\{ \\{ x \\in Q ; \\ , 2 ^ j x - z \\in Q \\} ; \\ , z \\in \\Z ^ d \\bigr \\} . \\end{align*}"}
-{"id": "7308.png", "formula": "\\begin{align*} & \\langle \\omega _ i , \\omega _ j \\rangle = 0 , { \\rm \\ i f } \\ i + j \\leq n + 1 \\ , , \\\\ & \\langle \\omega _ 1 , \\omega _ { n + 1 } \\rangle = ( - ( n + 2 ) ) ^ n \\frac { c _ n } { t _ 1 ^ { n + 2 } - t _ { n + 2 } } , { \\rm \\ w h e r e } \\ c _ n \\ { \\rm i s \\ a \\ c o n s t a n t } \\ , , \\\\ & \\langle \\omega _ j , \\omega _ { n + 2 - j } \\rangle = ( - 1 ) ^ { j - 1 } \\langle \\omega _ 1 , \\omega _ { n + 1 } \\rangle , { \\rm \\ f o r } \\ j = 1 , 2 , \\ldots , n + 1 \\ , , \\end{align*}"}
-{"id": "316.png", "formula": "\\begin{align*} K _ 0 = \\left \\{ ( v _ 1 , \\dots , v _ d ) \\in \\mathcal { S } _ d : s ( 1 - t ) ^ { d - 1 } - \\delta \\le \\frac { v _ j } { 1 - \\sum _ { l = j + 1 } ^ d v _ l } \\le t , j = 1 , 2 , . . . , d \\right \\} . \\end{align*}"}
-{"id": "6801.png", "formula": "\\begin{align*} \\frac { \\rho } { 4 \\pi } \\int _ { \\mathbb { S } ^ 2 } w _ { \\lambda } = ( 3 2 \\pi + \\epsilon ) \\overline { w } _ { \\lambda } \\end{align*}"}
-{"id": "425.png", "formula": "\\begin{align*} \\tilde \\phi _ { k _ 1 , k _ 2 } ( R , \\xi ) \\coloneqq \\begin{cases} e ^ { R \\ , r ( - \\xi ) } \\frac { ( \\pi \\xi ) ^ { n + k _ 1 } \\cos ( \\pi \\xi ) ^ { k _ 1 } ( 1 - \\xi ) ^ { n + k _ 1 + k _ 2 } } { \\sin ( \\pi \\xi ) ^ { n + k _ 1 } } , & \\\\ 1 , & \\end{cases} \\end{align*}"}
-{"id": "2629.png", "formula": "\\begin{gather*} { \\overline \\rho } ( t ) = \\gamma \\frac { \\cos ( \\tfrac { a \\pi } 2 ) } { \\pi a } ( t _ 2 - t ) ^ { \\tfrac { 1 + a } 2 } t ^ { - \\tfrac { 1 - a } 2 } , { \\overline C } = \\frac { 1 + a } { 2 a } \\gamma t _ 2 , t _ 2 = \\bigg [ \\frac \\gamma { 2 \\pi } \\frac { \\cos ( \\tfrac { a \\pi } 2 ) } { a ^ 2 \\Gamma ( a ) } \\Gamma \\Big ( \\frac { 1 + a } 2 \\Big ) ^ 2 \\bigg ] ^ { - \\tfrac 1 { 1 + a } } . \\end{gather*}"}
-{"id": "490.png", "formula": "\\begin{align*} p _ { 1 , k _ 1 , k _ 2 } ( x , t ) = \\frac { ( - 1 ) ^ { k _ 2 } \\pi ^ { k _ 1 + k _ 2 } } { { 2 ^ { 2 n + \\frac { m - 1 } { 2 } } } ( n + k _ 1 - 1 ) ! } \\abs { t } ^ { n + k _ 1 - 1 - \\frac { m - 1 } { 2 } } e ^ { - \\frac { 1 } { 4 } d ( x , t ) ^ 2 } \\left [ 1 + O \\left ( \\kappa + \\frac { 1 } { \\abs { t } } \\right ) \\right ] . \\end{align*}"}
-{"id": "2144.png", "formula": "\\begin{align*} \\partial _ t v _ j & = \\frac { 1 } { r ^ { N - 1 } \\nu ( r ) } \\partial _ r ( r ^ { N - 1 } \\nu ( r ) \\partial _ r v _ j ) \\\\ & = \\frac { 1 } { r ^ { N + k - 1 } r ^ { - k } \\nu ( r ) } \\partial _ r ( r ^ { N + k - 1 } r ^ { - k } \\nu ( r ) \\partial _ r v _ j ) , r > 0 , \\ , \\ , t > 0 , \\end{align*}"}
-{"id": "4890.png", "formula": "\\begin{align*} \\norm P T P ^ \\perp \\norm = \\norm P ^ \\perp T P \\norm . \\end{align*}"}
-{"id": "6999.png", "formula": "\\begin{align*} y _ j = z _ j , \\ \\ j = 1 , 2 , . . . , n - 2 , \\end{align*}"}
-{"id": "8308.png", "formula": "\\begin{align*} \\mathrm { r e c } ( a _ \\tau ) = \\tau | _ { \\Q ^ { a b } } . \\end{align*}"}
-{"id": "6943.png", "formula": "\\begin{align*} Y _ n : = G _ n ' \\backslash \\tilde { M } _ n \\end{align*}"}
-{"id": "3368.png", "formula": "\\begin{align*} \\nabla _ { X } ^ { \\Sigma ^ { a d \\mathbb { C } } } \\varphi = - \\frac { 1 } { 2 } \\sum _ { i = 1 } e _ { i } \\cdot B ( X , e _ { i } ) \\cdot \\varphi + \\frac { 1 } { 2 } \\textbf { i } ~ A ^ { l } ( X ) \\cdot \\varphi , ~ ~ \\forall X \\in T M . \\end{align*}"}
-{"id": "6916.png", "formula": "\\begin{align*} \\mu ( F _ i ( A ) ) & = \\left ( \\frac { 1 - p _ { m + 1 } } { 2 } \\right ) \\mu ( A ) \\\\ R ( F _ i ( A ) ) & = \\left ( \\frac { p _ { m + 1 } } { 2 } \\right ) R ( A ) \\end{align*}"}
-{"id": "6484.png", "formula": "\\begin{align*} \\theta ^ { j } \\left ( \\tau \\right ) \\overset { \\tau \\rightarrow \\infty } { \\approx } \\Xi _ { j } e ^ { \\omega _ { j } \\tau } \\Xi _ { j } \\in \\mathbb { R } , \\forall j = 1 , \\ldots , l \\end{align*}"}
-{"id": "7278.png", "formula": "\\begin{align*} S _ 3 = N ^ { - 3 } \\sum \\limits _ { m \\leqslant n \\leqslant X } r ( n ) r ( m ) \\frac { A ( m , n ) } { n } . \\end{align*}"}
-{"id": "7110.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } X ( x , t ) = ~ - \\Phi ( x , t ) \\nu ( x , t ) , \\end{align*}"}
-{"id": "10057.png", "formula": "\\begin{align*} a _ F ( 0 , \\mu ) = - 2 \\phi _ \\mu ( 0 ) \\Lambda ' ( 0 , \\chi _ E ) - \\Lambda ( 0 , \\chi _ E ) \\cdot \\frac { d } { d s } \\Big ( \\prod _ \\mathfrak { p } M _ \\mathfrak { p } ( s , \\phi _ \\mu ) \\Big ) \\Big | _ { s = 0 } , \\end{align*}"}
-{"id": "2211.png", "formula": "\\begin{align*} F _ i \\left ( \\frac 1 { w _ 1 } , \\ldots , \\frac 1 { w _ n } , t \\right ) = \\left ( \\frac 1 { w _ 1 } \\right ) ^ { m _ { i 1 } } \\cdot \\ldots \\cdot \\left ( \\frac 1 { w _ n } \\right ) ^ { m _ { i n } } \\cdot \\left ( \\widetilde q _ i ( w ) + t \\cdot \\widetilde Q _ i ( w ) \\right ) , \\end{align*}"}
-{"id": "819.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } - \\nabla \\cdot \\nu \\nabla u = 0 , \\mbox { o n } \\Omega , u = 0 , \\mbox { o n } \\partial \\Omega . \\end{align*}"}
-{"id": "285.png", "formula": "\\begin{align*} c = \\frac { 2 ^ { r _ 1 } ( 2 \\pi ) ^ { r _ 2 } h R } { w \\sqrt { D } } , \\end{align*}"}
-{"id": "9117.png", "formula": "\\begin{align*} 2 f _ 2 ( x ) + f _ 1 ( x ) = 0 \\end{align*}"}
-{"id": "9770.png", "formula": "\\begin{align*} \\tilde { E } _ { i } ( h , \\theta ) = & \\int \\limits _ { y _ { i } + 2 n _ { i - 1 } s } ^ { y _ { i } } \\big ( W ( U _ { h , \\theta } ( i h - , y ) ) - W ( U _ { h , \\theta _ i } ( i h - , y ) ) \\big ) d y \\\\ & - \\int \\limits _ { y _ { i } + 2 n _ { i - 1 } s } ^ { y _ { i } + ( 2 n _ { i - 1 } + d _ i ) s } W ( U _ { h , \\theta } ( i h - , y ) ) d y + \\int \\limits _ { y _ { i } + 2 n _ { i - 1 } s } ^ { y _ { i } + 2 ( n _ { i - 1 } + 1 ) s } W ( U _ { h , \\theta _ i } ( i h + , y ) ) d y . \\end{align*}"}
-{"id": "10015.png", "formula": "\\begin{align*} f ^ + ( \\tau ) = \\sum _ { \\substack { m \\in \\Q \\\\ m \\gg - \\infty } } c _ f ^ + ( m ) \\cdot q ^ m , \\end{align*}"}
-{"id": "4138.png", "formula": "\\begin{align*} D _ X \\eta = X ( K ^ { \\frac { 1 } { 4 } } ) \\xi + K ^ { \\frac { 1 } { 4 } } D _ X \\xi + D _ X Z = h ( Z , X ) \\xi + K ^ { \\frac { 1 } { 4 } } D _ X \\xi + \\nabla _ X Z + h ( X , Z ) \\eta , \\end{align*}"}
-{"id": "3004.png", "formula": "\\begin{align*} \\frac { \\partial x } { \\partial t } ( \\zeta , t ) = & \\left ( \\sum _ { k = 0 } ^ N P _ k \\frac { \\partial ^ k } { \\partial \\zeta ^ k } \\right ) ( { \\cal H } ( { \\zeta } ) x ( \\zeta , t ) ) , \\zeta \\in J , t \\ge 0 , \\\\ x ( \\zeta , 0 ) = & x _ 0 ( \\zeta ) , \\end{align*}"}
-{"id": "2516.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\int \\left | \\nabla _ \\xi h \\right | ^ 2 \\varrho \\ , m _ { 0 } = \\int ( \\nabla _ \\xi h , \\mathcal { L } \\nabla _ \\xi h ) \\varrho \\ , m _ { 0 } + \\int ( \\nabla _ \\xi h , \\left [ \\nabla _ \\xi , \\mathcal { L } \\right ] h ) \\varrho \\ , m _ { 0 } \\ , . \\end{align*}"}
-{"id": "6874.png", "formula": "\\begin{align*} \\lambda = \\lim _ { m \\to \\infty } \\left ( \\frac { 4 } { p q } \\right ) ^ { m } \\lambda _ { m } \\end{align*}"}
-{"id": "2949.png", "formula": "\\begin{align*} \\norm { \\gamma } _ 1 : = \\sum _ { k \\geq 2 } \\gamma ( k ) , \\end{align*}"}
-{"id": "2072.png", "formula": "\\begin{gather*} ( \\partial _ t - \\Delta _ H ) H ( p , q , t ) = 0 , \\\\ \\lim _ { t \\rightarrow 0 } H ( p , q , t ) = \\delta _ q ( p ) \\end{gather*}"}
-{"id": "2445.png", "formula": "\\begin{align*} \\beta = e ^ { 2 } ( \\gamma ( 3 ) ) ^ { | T | + 4 } \\left ( 4 \\left ( ^ { | T | + 7 } _ { \\phantom { q w } 3 } \\right ) + 4 \\right ) ( | T | + 4 ) ! , \\end{align*}"}
-{"id": "6265.png", "formula": "\\begin{align*} | I _ 1 + I _ 2 + I _ 3 + I _ 4 | \\leq & C ( \\nu ^ { - 1 } + \\mu ^ { - 1 } ) \\left ( \\| u ^ \\eta \\| _ \\infty + \\| b ^ \\eta \\| _ \\infty \\right ) \\left ( \\| U \\| _ 2 ^ 2 + \\| B \\| _ 2 ^ 2 \\right ) \\\\ & + \\frac 1 4 \\nu \\| \\nabla U \\| _ 2 ^ 2 + \\frac 1 4 \\mu \\| \\nabla B \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "3795.png", "formula": "\\begin{align*} \\mu = \\sum _ { z \\in \\Z ^ d } P _ z \\end{align*}"}
-{"id": "19.png", "formula": "\\begin{align*} E [ \\hat { V } ^ C _ { N , \\sigma } ( C _ 1 , C _ 2 ) ] = V ^ C _ \\sigma ( C _ 1 , C _ 2 ) \\end{align*}"}
-{"id": "955.png", "formula": "\\begin{align*} \\left \\{ 1 - \\left ( \\sum _ { m = 1 } ^ M | \\rho _ m | \\right ) ^ 2 \\right \\} \\underline { v } \\leq E [ F _ n ( \\theta ) ^ 2 ] \\leq \\left \\{ 1 + \\left ( \\sum _ { m = 1 } ^ M | \\rho _ m | \\right ) ^ 2 \\right \\} \\overline { v } \\end{align*}"}
-{"id": "5058.png", "formula": "\\begin{align*} ( B _ { i j } ) = d i a g \\{ b _ 1 , b _ 2 , b _ 3 \\} , ~ ~ ~ b _ 1 < b _ 2 < b _ 3 . \\end{align*}"}
-{"id": "3640.png", "formula": "\\begin{align*} \\sum _ { i , k } c _ i ' u c _ k u ^ { - 1 } \\otimes t _ k t _ i ' = 1 = \\sum _ { i , k } u c _ k u ^ { - 1 } c _ i ' \\otimes t _ i ' t _ k . \\end{align*}"}
-{"id": "2784.png", "formula": "\\begin{align*} u ( t ) = g ( t , u ( t ) ) + \\int _ { 0 } ^ { t } \\frac { 1 } { ( t - s ) ^ { \\alpha } } k _ { 1 } ( t , s , u ( t ) , u ( s ) ) \\mathrm { d } s + \\int _ { 0 } ^ { t } k _ { 2 } ( t , s , u ( t ) , u ( s ) ) \\mathrm { d } s , \\end{align*}"}
-{"id": "861.png", "formula": "\\begin{align*} \\langle D F _ i , - D L ^ { - 1 } F _ j \\rangle _ H = \\sum _ { k , l = 1 } ^ d \\gamma _ { i k } \\gamma _ { j l } \\langle e _ k , e _ l \\rangle _ H = \\sum _ { k = 1 } ^ d \\gamma _ { i k } \\gamma _ { j k } = \\Sigma ( i , j ) , \\end{align*}"}
-{"id": "7109.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n \\dot { f } ^ k \\kappa _ k ^ 2 \\geq ~ f ^ 2 / n . \\end{align*}"}
-{"id": "1332.png", "formula": "\\begin{align*} \\omega _ G ( p , m ) = \\psi _ G ^ { - \\frac { d } { 2 } } \\left ( \\frac { \\psi _ G } { \\Xi _ G ( p , m ) } \\right ) ^ { N - | G | \\frac { d } { 2 } } \\nu _ G = : f _ G ( p , m ) \\nu _ G \\end{align*}"}
-{"id": "3771.png", "formula": "\\begin{align*} D ^ \\omega _ \\ell ( \\sigma ) : = \\sum _ { k = 0 } ^ { \\ell - 1 } d ^ \\omega ( y + ( \\sigma ( k ) , k ) ) \\end{align*}"}
-{"id": "7549.png", "formula": "\\begin{gather*} \\mathrm D _ x R ^ 1 + \\mathrm D _ y R ^ 2 - ( \\mathrm D _ t + u \\mathrm D _ x + v \\mathrm D _ y - \\mathrm D _ x ^ 2 - \\mathrm D _ y ^ 2 ) R ^ 3 - ( R ^ 3 ) ^ 2 = - 2 ( u _ x v _ y - u _ y v _ x ) = 0 , \\end{gather*}"}
-{"id": "8559.png", "formula": "\\begin{align*} u ( x , t ) : = ( 4 \\pi t ) ^ { - \\frac { N } { 2 } } \\exp \\left ( - \\frac { H _ 0 ( x ) ^ 2 } { 4 t } \\right ) \\int _ 0 ^ \\infty I \\left ( \\frac { H _ 0 ( x ) r } { 2 t } \\right ) \\exp \\left ( - \\frac { r ^ 2 } { 4 t } \\right ) \\varphi ^ \\sharp ( r ) r ^ { N - 1 } \\ , d r \\end{align*}"}
-{"id": "4280.png", "formula": "\\begin{align*} \\begin{aligned} \\phi \\left ( \\frac { z } { c \\tau + d } , \\frac { a \\tau + b } { c \\tau + d } \\right ) & = ( c \\tau + d ) ^ k e \\left ( \\frac { c z ^ t L z } { c \\tau + d } \\right ) \\phi ( z , \\tau ) \\\\ \\phi \\left ( z + \\lambda \\tau + \\mu , \\tau \\right ) & = e \\left ( - \\lambda ^ t L \\lambda \\tau - 2 \\lambda ^ t L z \\right ) \\phi ( z , \\tau ) \\end{aligned} \\end{align*}"}
-{"id": "5051.png", "formula": "\\begin{align*} \\Phi ( \\rho ) _ { k l } = \\sum _ i \\rho _ { k l } A ^ { ( i ) } _ { k } \\overline { A ^ { ( i ) } _ { l } } = \\rho _ { k l } \\sum _ i A ^ { ( i ) } _ { k } \\overline { A ^ { ( i ) } _ { l } } . \\end{align*}"}
-{"id": "6757.png", "formula": "\\begin{align*} | \\ | _ w = \\| \\ \\| _ w ^ { d _ w / d } , \\end{align*}"}
-{"id": "3208.png", "formula": "\\begin{align*} & \\partial _ t ^ 2 \\phi - \\Delta \\phi + h ( \\phi ) \\Delta \\phi = F ( \\partial \\phi ) ( t , x ) \\in ( 0 , \\infty ) \\times { \\mathbb R } ^ 4 \\\\ & \\phi ( 0 , x ) = f ( x ) \\in H ^ { s } _ { { \\rm r a d } } ( { \\mathbb R } ^ 4 ) , \\partial _ { t } \\phi ( 0 , x ) = g ( x ) \\in H ^ { s - 1 } _ { { \\rm r a d } } ( { \\mathbb R } ^ 4 ) \\end{align*}"}
-{"id": "3028.png", "formula": "\\begin{align*} A f = \\frac { \\partial } { \\partial \\zeta } f \\end{align*}"}
-{"id": "7764.png", "formula": "\\begin{align*} | \\ell ' + \\rho ' - m | \\leq | \\ell ' + t _ { 0 } \\rho ' - m | + | ( 1 - t _ { 0 } ) \\rho ' | = t _ { 0 } | \\rho ' | + ( 1 - t _ { 0 } ) | \\rho ' | = | \\rho ' | . \\end{align*}"}
-{"id": "549.png", "formula": "\\begin{align*} \\lambda | _ { [ 1 + \\infty ) \\times \\partial M } = \\rho \\ , \\alpha . \\end{align*}"}
-{"id": "4827.png", "formula": "\\begin{align*} a _ { n _ { j + 1 } } > \\prod _ { i = 1 } ^ { j } a _ { n _ i } \\end{align*}"}
-{"id": "4918.png", "formula": "\\begin{align*} Z = \\begin{bmatrix} U _ 0 & 0 \\\\ 0 & \\tau \\end{bmatrix} . \\end{align*}"}
-{"id": "3443.png", "formula": "\\begin{align*} U _ { \\varepsilon } ^ { - 1 } ( A \\cap \\Q ^ d ) & = \\bigcup _ { x \\in A \\cap \\Q ^ d } \\{ \\omega \\in \\Omega \\ , : \\ , h ( \\omega , x ) \\in B _ { \\varepsilon } ( 0 ) \\} \\\\ & = \\bigcup _ { x \\in A \\cap \\Q ^ d } h ( \\cdot , x ) ^ { - 1 } ( B _ { \\varepsilon } ( 0 ) ) \\in \\tilde { \\mathcal { F } } . \\end{align*}"}
-{"id": "6643.png", "formula": "\\begin{align*} f ( t ) = \\left | \\Phi _ t ( x ) - \\Phi _ t ( y ) \\right | & = \\left | x - y + \\int _ 0 ^ t X ( \\Phi _ s ( x ) ) - X ( \\Phi _ s ( y ) ) \\ , d s \\right | \\\\ & \\leq | x - y | + \\int _ 0 ^ t M \\left | \\Phi _ s ( x ) - \\Phi _ s ( y ) \\right | \\ , d s \\\\ & = | x - y | + \\int _ 0 ^ t M f ( s ) \\ , d s \\end{align*}"}
-{"id": "1250.png", "formula": "\\begin{align*} D \\cap B ( w , 3 2 \\hat r ) = D _ 1 \\cap B ( w , 3 2 \\hat r ) . \\end{align*}"}
-{"id": "1345.png", "formula": "\\begin{align*} \\overline { M } = \\overline { P } \\Lambda \\overline { P } ^ T , \\end{align*}"}
-{"id": "9780.png", "formula": "\\begin{align*} & Z _ A = \\bar { Z } _ { 0 } \\exp ( - \\frac { 1 } { \\bar { \\rho } _ { 0 } \\bar { u } _ { 0 } A ( 0 ) } \\int _ 0 ^ x A ( \\tau ) \\rho _ A \\phi ( T _ A ) d \\tau ) . \\end{align*}"}
-{"id": "8795.png", "formula": "\\begin{align*} Z _ { \\Phi } ( \\mathfrak { p } , \\chi , s , f ) = q ^ { - n } \\sum _ { \\substack { I \\subset T , \\\\ \\forall i \\in I : d \\mid N _ { i } } } c _ { I , \\Phi , \\chi } \\prod _ { i \\in I } \\frac { ( q - 1 ) q ^ { - N _ { i } s - \\nu _ { i } } } { 1 - q ^ { - N _ { i } s - \\nu _ { i } } } , \\end{align*}"}
-{"id": "5216.png", "formula": "\\begin{align*} Y ' ( z ) = A ( \\lambda , z ) Y ( z ) , \\end{align*}"}
-{"id": "5669.png", "formula": "\\begin{align*} \\begin{cases} | { v } ( s ^ - ) | = E ( s ^ - ) , & \\\\ | { v } ( s ^ + ) | = E ( s ^ + ) & s ^ + < + \\infty , \\\\ | { v } ( s ) | > E ( s ) & s \\in I . \\end{cases} \\end{align*}"}
-{"id": "7078.png", "formula": "\\begin{align*} F _ { A } ^ { + } - ( \\Psi ^ { \\dag } \\tau \\Psi - i \\varpi ) = 0 \\mathcal { D } _ { A } ^ + \\Psi = 0 , \\end{align*}"}
-{"id": "573.png", "formula": "\\begin{align*} w _ n = \\left ( 1 - \\frac { 2 t } { | z _ n - \\alpha | } \\right ) \\alpha + \\frac { 2 t } { | z _ n - \\alpha | } z _ n \\in \\Omega \\end{align*}"}
-{"id": "4055.png", "formula": "\\begin{align*} \\mathbf { I } _ \\alpha ( \\eta ) = \\bar { \\mathbf { E } } _ \\alpha ( \\eta ) . \\end{align*}"}
-{"id": "2048.png", "formula": "\\begin{align*} E _ H ( f ) = \\frac { 1 } { 2 } \\int _ M \\langle G _ \\theta , f ^ * h \\rangle \\theta \\wedge ( d \\theta ) ^ m = \\frac { 1 } { 2 } \\int _ M | d f \\circ \\pi _ H | ^ 2 \\theta \\wedge ( d \\theta ) ^ m \\end{align*}"}
-{"id": "9155.png", "formula": "\\begin{align*} \\begin{array} { r c l } 1 5 f & = & 2 D _ { 1 } + 9 D _ { 2 } \\\\ 3 g & = & - D _ { 1 } - 3 D _ { 2 } \\\\ 3 h & = & 2 D _ { 1 } + 3 D _ { 2 } , \\end{array} \\end{align*}"}
-{"id": "2846.png", "formula": "\\begin{align*} A _ { n } ( s ) = \\sum _ { v = 0 } ^ { n } a _ { n v } s _ { v } , n = 0 , 1 , . . . \\end{align*}"}
-{"id": "1563.png", "formula": "\\begin{align*} \\nu ( B ( k ) ) & = \\nu ( \\beta ( \\beta - 2 ^ { n - k + 1 } ) ( \\beta - 2 \\cdot 2 ^ { n - k + 1 } ) \\cdots ( \\beta - ( j _ k - 1 ) 2 ^ { n - k + 1 } ) ) \\\\ & = \\nu ( \\beta ) + \\nu ( \\beta - 2 ^ { n - k + 1 } ) + \\nu ( \\beta - 2 \\cdot 2 ^ { n - k + 1 } ) + \\cdots + \\nu ( \\beta - ( j _ k - 1 ) 2 ^ { n - k + 1 } ) \\end{align*}"}
-{"id": "7132.png", "formula": "\\begin{align*} \\partial _ t s = ~ \\sqrt { ( 1 + s ^ 2 + | \\bar { \\nabla } s | ^ 2 ) ( 1 + s ^ 2 ) } F _ { * } ( \\mathcal { W } _ X ^ { - 1 } ) \\end{align*}"}
-{"id": "1511.png", "formula": "\\begin{align*} G _ { T } ( y ) & = \\frac { Q _ { T , 0 } ( x ) + ( Q _ { T , 1 } ( x ) - x ^ { 2 } Q _ { T , 0 } ( x ) ) y + ( Q _ { T , 2 } ( x ) - x ^ { 2 } Q _ { T , 1 } ( x ) - x Q _ { T , 0 } ( x ) ) y ^ { 2 } } { 1 - x ^ { 2 } y - x y ^ { 2 } - y ^ { 3 } } \\\\ & = \\frac { y + \\textbf { i } + ( x ^ { 2 } + x y + y ^ { 2 } ) \\textbf { j } + ( x ^ { 4 } + x + x ^ { 3 } y + y + x ^ { 2 } y ^ { 2 } ) \\textbf { k } } { 1 - x ^ { 2 } y - x y ^ { 2 } - y ^ { 3 } } . \\end{align*}"}
-{"id": "6383.png", "formula": "\\begin{align*} a _ n \\big ( ( m _ 1 ^ { ( n ) } , \\ldots , m _ k ^ { ( n ) } ) - ( m _ 1 ^ * , \\ldots , m _ k ^ * ) \\big ) = a _ n \\big ( \\varphi _ k ^ { [ a , b ] } ( p _ 1 ^ { ( n ) } , \\ldots , p _ k ^ { ( n ) } ) - \\varphi _ k ^ { [ a , b ] } ( \\vec { y } ^ * ) \\big ) \\end{align*}"}
-{"id": "1222.png", "formula": "\\begin{align*} u _ 2 ( x ) = u _ 1 ( a x + b ) \\mbox { w h e n e v e r } u _ 2 ( x ) < t \\ , \\ , \\mbox { a n d } \\ , \\ , t < t _ 0 . \\end{align*}"}
-{"id": "1605.png", "formula": "\\begin{align*} l y + { l \\choose 2 } y ^ 2 = 0 \\ , . \\end{align*}"}
-{"id": "4354.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { j = 1 } ^ \\infty M _ { a _ j } P _ \\alpha M _ { 1 - b _ j } \\Big \\| \\to 0 \\end{align*}"}
-{"id": "4767.png", "formula": "\\begin{align*} \\mbox { } + 2 V ^ 2 { L } _ 2 ' + 2 V ^ 3 { L } _ 3 ' = 0 . \\end{align*}"}
-{"id": "4395.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\nu ( \\hat { A } _ { x _ j } ) = \\inf \\{ \\nu ( \\hat { A } _ x ) ; \\ x \\in \\mathcal { M } \\setminus \\mathbb { C } ^ n \\} . \\end{align*}"}
-{"id": "4249.png", "formula": "\\begin{align*} \\mathcal G _ { i j } = \\Phi ( \\mathcal { F } _ { i j } ) \\exp ( \\mathcal G ^ \\Delta _ { i j } ) \\end{align*}"}
-{"id": "4816.png", "formula": "\\begin{align*} p \\bigg ( \\frac { q ^ { f ( n + 1 ) } } { p ^ { n + 1 } } a \\bigg ) = q ^ { f ( n ) } a + \\big ( q ^ { f ( n + 1 ) - f ( n ) } - p ^ n \\big ) \\frac { q ^ { f ( n ) } } { p ^ n } a \\in S + M . \\end{align*}"}
-{"id": "1806.png", "formula": "\\begin{align*} X ' _ k : = \\frac { F _ k ( X ) } { \\prod _ { j \\in J } X _ j } . \\end{align*}"}
-{"id": "3431.png", "formula": "\\begin{align*} \\| f \\| _ { C ( [ 0 , T ] ; H ) } = \\sup _ { t \\in [ 0 , T ] } \\| f ( t ) \\| _ H . \\end{align*}"}
-{"id": "1021.png", "formula": "\\begin{align*} \\iint _ Q \\vec u \\cdot ( \\partial _ t \\vec \\phi + \\Delta \\vec \\phi ) + \\vec u \\cdot ( \\vec u \\cdot \\vec \\nabla \\vec \\phi ) + \\vec f \\cdot \\vec \\phi \\ , d t \\ , d x = 0 . \\end{align*}"}
-{"id": "6923.png", "formula": "\\begin{align*} u ( x , t ) = \\sum _ { \\lambda _ { j } } e ^ { - \\lambda _ { j } t } u _ { j } ( x ) \\int u _ { j } ( y ) f ( y ) d \\mu ( y ) \\end{align*}"}
-{"id": "5828.png", "formula": "\\begin{align*} w _ n = D _ { \\| v _ n \\| _ { \\dot { H } ^ 1 ( \\R ) } } v _ n . \\end{align*}"}
-{"id": "9529.png", "formula": "\\begin{align*} M ^ p _ { l o c } ( \\R ^ n ) : = \\large \\{ f \\in L ^ 1 _ { l o c } ( \\R ^ n ) | f \\in M ^ p ( \\Omega ) \\Omega \\large \\} \\end{align*}"}
-{"id": "3401.png", "formula": "\\begin{align*} z ( t ) = z ( 0 ) - \\int _ 0 ^ t x ( s ) \\dot y ( s ) d s , t \\in [ - 1 , 1 ] . \\end{align*}"}
-{"id": "3754.png", "formula": "\\begin{align*} \\rho _ { k + 1 } = \\left \\{ \\begin{array} { l l } \\rho _ k ( 1 + L _ { k } ^ { - 1 / 1 6 } ) & \\\\ \\rho _ k ( 1 - L _ { k } ^ { - 1 / 1 6 } ) & \\end{array} \\right . \\end{align*}"}
-{"id": "9657.png", "formula": "\\begin{align*} N _ { \\rm { s } , 1 } ( t ) = \\frac { \\Lambda _ \\beta ^ 2 ( t ) \\left ( \\log d - \\log \\delta \\right ) } { 2 \\left ( \\Delta \\Phi _ { \\rm { s t } , \\max } ( t ) - \\xi _ { \\max } \\Lambda _ \\beta ( t ) \\right ) ^ 2 } , \\end{align*}"}
-{"id": "1914.png", "formula": "\\begin{align*} d ^ { 0 } ( \\phi , \\psi ) = \\max \\left \\{ \\max _ { x \\in X _ { 1 } } d _ { 2 } ( \\phi ( x ) , \\psi ( x ) ) , \\max _ { y \\in X _ { 2 } } d _ { 1 } ( \\phi ^ { - 1 } ( y ) , \\psi ^ { - 1 } ( y ) ) \\right \\} . \\end{align*}"}
-{"id": "10000.png", "formula": "\\begin{align*} E _ \\Phi = \\varphi ^ \\mathrm { s p } ( E ) \\subset \\C , \\end{align*}"}
-{"id": "9281.png", "formula": "\\begin{align*} \\boldsymbol { E } [ \\sum _ { i = 1 } ^ T X _ i ] = \\boldsymbol { E } [ X _ 1 ] \\boldsymbol { E } [ T ] \\leq \\boldsymbol { E } [ T ] . \\end{align*}"}
-{"id": "3156.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } _ { \\geqslant 0 } } x ^ { \\kappa } m _ { \\alpha , \\beta } ( \\mathrm { d } x ) = \\frac { \\alpha \\beta } { \\Gamma ( \\theta ) } \\int _ { 0 } ^ { \\infty } \\exp \\left \\lbrace \\frac { - \\alpha u } { \\beta + u } \\right \\rbrace \\frac { u ^ { \\theta - 1 } } { ( \\beta + u ) ^ { 2 } } \\mathrm { d } u , u \\in \\mathbb { R } _ { \\geqslant 0 } . \\end{align*}"}
-{"id": "148.png", "formula": "\\begin{align*} \\Lambda N ^ { \\sigma } = N \\Lambda \\end{align*}"}
-{"id": "155.png", "formula": "\\begin{align*} \\begin{pmatrix} a I & b I \\\\ c I & d I \\end{pmatrix} \\end{align*}"}
-{"id": "6196.png", "formula": "\\begin{align*} \\begin{pmatrix} - 2 & 0 \\\\ 0 & - 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "4490.png", "formula": "\\begin{align*} 1 = \\| \\Theta \\| = \\| \\Theta _ 1 \\| + \\| \\Theta _ 2 \\| . \\end{align*}"}
-{"id": "1565.png", "formula": "\\begin{align*} C _ { j _ k } ( \\alpha ( k ) ) & = \\frac { \\alpha ( \\alpha - 1 ) ( \\alpha - 2 ) \\cdots ( \\alpha - ( j _ k - 1 ) ) } { j _ k ! } \\\\ & = \\frac { \\beta ( \\beta - 2 ^ { n - k + 1 } ) ( \\beta - 2 \\cdot 2 ^ { n - k + 1 } ) \\cdots ( \\beta - ( j _ k - 1 ) 2 ^ { n - k + 1 } ) } { 2 ^ { j _ k ( n - k + 1 ) } j _ k ! } \\\\ & = \\frac { B ( k ) } { 2 ^ { j _ k ( n - k + 1 ) } j _ k ! } \\end{align*}"}
-{"id": "6592.png", "formula": "\\begin{align*} f _ 0 ^ k = \\varphi _ 0 \\circ f ^ k \\circ \\varphi _ 0 ^ { - 1 } \\colon \\varphi _ 0 ( U _ 0 \\cap f ^ { - k } ( U _ 0 ) ) \\longrightarrow \\varphi _ 0 ( f ^ k ( U _ 0 ) \\cap U _ 0 ) \\end{align*}"}
-{"id": "5866.png", "formula": "\\begin{align*} a _ 0 = 1 , \\ , \\ , a _ 8 = 1 3 0 , \\ , \\ , a _ { 1 2 } = 1 2 0 , \\ , \\ , a _ { 1 6 } = 5 , \\ , \\ , . \\end{align*}"}
-{"id": "4902.png", "formula": "\\begin{align*} V \\otimes I \\simeq \\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix} . \\end{align*}"}
-{"id": "37.png", "formula": "\\begin{align*} E _ { D Y } [ G ^ { C } _ { \\sigma \\ , \\sqrt { 2 } } ( e ) \\ , d ^ { * } \\textbf { X } ] = E _ { D Y } [ G ^ { C } _ { \\sigma \\ , \\sqrt { 2 } } ( e ) \\textbf { X X } ^ { H } ] \\ , \\textbf { w } \\end{align*}"}
-{"id": "4181.png", "formula": "\\begin{align*} p ( \\eta , \\zeta ) = \\eta ^ k + a _ 1 ( \\zeta ) \\eta ^ { k - 1 } + \\dots a _ k ( \\zeta ) = 0 , \\end{align*}"}
-{"id": "6411.png", "formula": "\\begin{align*} S \\left [ \\theta _ { } + d \\theta | \\theta _ { } \\right ] = - \\frac { 1 } { 2 } g _ { \\mu \\nu } \\left ( \\theta \\right ) d \\theta ^ { \\mu } d \\theta ^ { \\nu } = - \\frac { 1 } { 2 } d l _ { \\rightarrow \\xi } ^ { 2 } \\end{align*}"}
-{"id": "7671.png", "formula": "\\begin{align*} G = ( n _ 1 , \\ldots , n _ r ; m _ 1 , \\ldots , m _ r ; \\zeta _ 1 , \\ldots , \\zeta _ r ) \\end{align*}"}
-{"id": "6721.png", "formula": "\\begin{align*} \\mathbb { P } \\Big ( \\{ I ( 1 ) , \\dots , I ( 2 l ) \\} \\in J _ { l } \\Big ) = \\frac { | J _ { l } | } { N ^ { 2 l } } . \\end{align*}"}
-{"id": "9288.png", "formula": "\\begin{align*} Y : = \\{ ( d , x ) \\in D \\times \\R \\ : \\ \\exists e \\in D \\ e \\geq d \\wedge ( e , x ) \\in X \\} . \\end{align*}"}
-{"id": "1060.png", "formula": "\\begin{align*} \\log H ( P _ { k } ) - \\frac { q _ { k } } { 2 n - 2 } = \\log \\vert P _ { k - 1 } ( \\zeta ) \\vert + q _ { k } , \\end{align*}"}
-{"id": "9443.png", "formula": "\\begin{align*} \\sum _ { n \\in \\N : G _ n \\geq 3 } \\frac { \\log G _ n } { n \\cdot \\log \\log G _ n } = \\infty , \\end{align*}"}
-{"id": "1458.png", "formula": "\\begin{align*} D _ r / ( v _ t e _ t ) = D _ r = \\mathbb { Z } / 2 [ x _ 2 ^ 2 , x _ 3 ^ 2 , x _ { 4 1 } ^ 2 , \\dots , x _ { 4 n } ^ 2 , v _ { r + 1 } , \\dots , v _ m ] . \\end{align*}"}
-{"id": "4602.png", "formula": "\\begin{align*} T x = x - \\sum _ { j < i } \\lambda _ j + \\sum _ { \\pi j < \\pi i } \\lambda _ j \\end{align*}"}
-{"id": "7530.png", "formula": "\\begin{gather*} w ^ i w ^ 1 _ i - w ^ 1 _ { i i } - 2 \\kappa w ^ 2 + \\alpha z _ 1 = 0 , \\\\ w ^ i w ^ 2 _ i - w ^ 2 _ { i i } + 2 \\kappa w ^ 1 + \\alpha z _ 2 + \\beta = 0 , \\end{gather*}"}
-{"id": "5624.png", "formula": "\\begin{align*} \\forall \\ r \\geq 2 \\ \\ \\Sigma ^ r _ j = \\Omega _ { R _ r } ( \\Sigma ^ { r - 1 } _ j ) \\ , , \\end{align*}"}
-{"id": "2170.png", "formula": "\\begin{align*} \\| \\hat { w } ( s ) \\| _ { L ^ 2 ( { \\bf R } _ + , \\rho _ d \\ , d \\xi ) } = O ( e ^ { - \\theta ' s } ) , \\| w ( s ) \\| _ { L ^ 2 ( { \\bf R } _ + , \\rho _ d \\ , d \\xi ) } = O ( e ^ { - \\theta ' s } ) , \\end{align*}"}
-{"id": "4667.png", "formula": "\\begin{align*} { \\rm d i m } \\ , \\mathcal { R } ( A ) = \\infty \\ , , \\end{align*}"}
-{"id": "7888.png", "formula": "\\begin{align*} F ( x , y , z , t , r , s ) = 2 u ( z , t ) - u ( x , r ) - u ( y , s ) - k d ( z , m ( x , y ) ) ^ 2 , \\end{align*}"}
-{"id": "9177.png", "formula": "\\begin{align*} \\mathrm { M A } _ { \\mathbb { C } } ( \\psi _ \\phi ) ( \\exp ( x ) ) = \\frac { 1 } { 2 ^ { r + n } } \\mathrm { M A } _ { \\mathbb { R } } ( \\psi _ \\phi ) ( x ) \\frac { 1 } { \\mathbf J ( x ) } \\prod _ { \\alpha \\in \\Phi _ + } \\langle \\alpha , \\nabla \\psi _ \\phi ( x ) \\rangle ^ 2 . \\end{align*}"}
-{"id": "641.png", "formula": "\\begin{align*} \\bar \\nabla _ { \\dot \\gamma } \\bar \\nabla _ { \\dot \\gamma } J + \\bar R ( \\dot \\gamma , J ) \\dot \\gamma = 0 . \\end{align*}"}
-{"id": "844.png", "formula": "\\begin{align*} \\mathbb { E } ( | B _ { E _ t } - B _ { E _ s } | ) & = \\mathbb { E } ( | \\int _ { E _ s } ^ { E _ t } d B _ z | ) \\\\ & = \\int _ 0 ^ { + \\infty } \\int _ 0 ^ { + \\infty } \\mathbb { E } ( | B _ { r _ 1 } - B _ { r _ 2 } | ) h ( r _ 1 , t ) h ( r _ 2 , s ) d r _ 1 d r _ 2 \\\\ & = \\int _ 0 ^ { + \\infty } \\int _ 0 ^ { + \\infty } \\sqrt { r _ 1 - r _ 2 } h ( r _ 1 , t ) h ( r _ 2 , s ) d r _ 1 d r _ 2 \\\\ & = \\mathbb { E } ( \\sqrt { E _ t - E _ s } ) \\leq \\sqrt { \\mathbb { E } ( E _ t - E _ s ) } \\\\ & \\leq C ( \\beta , 1 ) ^ { \\frac { 1 } { 2 } } | t - s | ^ { \\beta / 2 } , \\end{align*}"}
-{"id": "1692.png", "formula": "\\begin{align*} \\frac { d ( \\mu _ x \\circ \\sigma _ g ) } { d \\mu _ x } ( z ) = \\lim _ { n \\to \\infty } \\frac { \\mu _ { x } ( Z ( g z _ n ) ) } { \\mu _ { x } ( Z ( z _ n ) ) } . \\end{align*}"}
-{"id": "7763.png", "formula": "\\begin{align*} r _ { 0 } = \\min _ { \\ell _ { 1 } \\in \\L , \\ell _ { 2 } \\in \\L \\setminus \\{ \\ell _ { 1 } \\} } | \\ell _ { 1 } - \\ell _ { 2 } | > 0 , \\end{align*}"}
-{"id": "570.png", "formula": "\\begin{align*} \\max \\left \\{ \\sum _ { j = 1 } ^ m \\left \\lfloor \\frac { \\alpha _ j } { b } \\right \\rfloor + 1 , 0 \\right \\} . \\end{align*}"}
-{"id": "5024.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\downarrow 0 } f ( \\epsilon ) ( I - \\lambda C _ p G ^ { \\omega , \\lambda } _ { p , p , p } ( E \\pm \\iota \\epsilon ) ) ( I + \\lambda C _ p G ^ \\omega _ { p , p } ( E \\pm \\iota \\epsilon ) ) = 0 \\\\ \\Rightarrow \\left ( \\lim _ { \\epsilon \\downarrow 0 } f ( \\epsilon ) C _ p G ^ { \\omega , \\lambda } _ { p , p , p } ( E \\pm \\iota \\epsilon ) C _ p \\right ) ( C _ p ^ { - 1 } + \\lambda G ^ \\omega _ { p , p } ( E \\pm \\iota 0 ) ) = 0 , \\end{align*}"}
-{"id": "627.png", "formula": "\\begin{align*} a _ { ( 0 , - \\alpha _ 2 ) } = \\frac { 1 } { 4 \\pi ^ 2 } \\int _ 0 ^ { 2 \\pi } \\int _ 0 ^ { 2 \\pi } f ( r _ 1 e ^ { i \\theta _ 1 } , r _ 2 e ^ { i \\theta _ 2 } ) r _ 2 ^ { \\alpha _ 2 } e ^ { i \\alpha _ 2 \\theta _ 2 } \\ , d \\theta _ 1 \\ , d \\theta _ 2 , \\end{align*}"}
-{"id": "5310.png", "formula": "\\begin{align*} \\widetilde { B } ( \\cdot , \\theta _ M ( t ) ) = B ( \\cdot , \\theta ^ { m - 1 } ) + \\frac { t - t _ { m - 1 , M } } { \\tau } \\left ( B ( \\cdot , \\theta ^ m ) - B ( \\cdot , \\theta ^ { m - 1 } ) \\right ) \\end{align*}"}
-{"id": "1872.png", "formula": "\\begin{align*} \\mathcal { Q } = \\Big \\{ \\mu \\in c a _ 1 ^ + ( \\mathbb { R } ^ d ) : \\mu _ 1 = \\nu _ 1 , \\ , \\dots , \\ , \\mu _ d = \\nu _ d \\mu \\circ { \\max } ^ { - 1 } = \\nu _ { \\max } \\Big \\} , \\end{align*}"}
-{"id": "2704.png", "formula": "\\begin{align*} 2 \\phi _ 1 ( x ) \\phi _ 2 ( x ) = ( \\phi _ { 1 } ( x ) + \\phi _ 2 ( x ) ) ^ { 2 } - ( \\phi _ { 1 } ( x ^ 2 ) + \\phi _ 2 ( x ^ 2 ) ) \\in E ^ { * } , \\end{align*}"}
-{"id": "8544.png", "formula": "\\begin{align*} + ( ( a _ 1 + a _ 2 ) c _ 1 ^ 2 - a _ 1 ^ 2 a _ 2 ^ 2 + a _ 1 a _ 2 b c _ 1 ) ^ 2 = 0 . \\end{align*}"}
-{"id": "2052.png", "formula": "\\begin{align*} g _ \\theta = G _ \\theta + \\theta \\otimes \\theta \\end{align*}"}
-{"id": "7509.png", "formula": "\\begin{align*} i \\varepsilon \\partial _ t \\mathcal { U } ( t ; s ) = h _ H ( t ) \\mathcal { U } ( t ; s ) , \\end{align*}"}
-{"id": "8156.png", "formula": "\\begin{align*} a _ 1 a _ 7 = 0 \\mbox { a n d } a _ 2 a _ 6 = 0 , \\end{align*}"}
-{"id": "7975.png", "formula": "\\begin{align*} \\sin b \\sin c \\cos \\alpha _ { + } & = \\cos a - \\cos b \\cos c \\\\ \\sinh b \\sinh c \\cos \\alpha _ { - } & = \\cosh b \\cosh c - \\cosh a \\\\ 2 b c \\cos \\alpha & = b ^ 2 + c ^ 2 - a ^ 2 , \\end{align*}"}
-{"id": "9905.png", "formula": "\\begin{align*} \\tfrac { 3 r - 5 } { 3 r - 2 } m n \\le \\sum _ { j = 0 } ^ t e ( V _ i , V _ j ) \\ , . \\end{align*}"}
-{"id": "6359.png", "formula": "\\begin{align*} \\det D \\varphi _ { 2 n } ^ \\mathbb { R } = \\prod \\limits _ { k = 1 } ^ n \\prod \\limits _ { j = 1 } ^ { k - 1 } \\beta _ j ^ 2 = \\prod \\limits _ { j = 1 } ^ { n - 1 } \\prod \\limits _ { k = j + 1 } ^ n \\beta _ j ^ 2 = \\prod \\limits _ { j = 1 } ^ { n - 1 } \\beta _ j ^ { 2 n - 2 j } . \\end{align*}"}
-{"id": "9509.png", "formula": "\\begin{align*} v ' - n ' + f ' = 2 . \\end{align*}"}
-{"id": "8318.png", "formula": "\\begin{align*} \\phi _ { \\mu } \\otimes \\phi _ \\nu \\mapsto \\begin{cases} 1 & \\mbox { i f } \\mu = \\nu \\\\ 0 & \\mbox { o t h e r w i s e } \\end{cases} \\end{align*}"}
-{"id": "8654.png", "formula": "\\begin{align*} \\sum _ { i = T _ k } ^ { T _ { k + 1 } - 1 } \\max _ { s \\in [ 0 , 1 ] } | X _ i ( s ) | \\end{align*}"}
-{"id": "8560.png", "formula": "\\begin{align*} \\exp \\left ( \\frac { z } { 2 } ( \\xi + \\xi ^ { - 1 } ) \\right ) \\ , \\xi ^ { - 1 } & = \\sum _ { n = 0 } ^ \\infty \\sum _ { m = 0 } ^ \\infty \\frac { 1 } { n ! } \\frac { 1 } { m ! } \\left ( \\dfrac { z } { 2 } \\right ) ^ { n + m } \\xi ^ { n - m - 1 } \\end{align*}"}
-{"id": "5853.png", "formula": "\\begin{align*} R ^ N \\left ( \\frac f \\varphi \\right ) = \\frac { ( - 1 ) ^ { N } } { \\varphi ^ { N + 1 } } \\sum _ { k = 0 } ^ N ( - 1 ) ^ k \\binom { N + 1 } { k } \\varphi ^ k R ^ N ( \\varphi ^ { N - k } f ) \\end{align*}"}
-{"id": "8091.png", "formula": "\\begin{align*} \\lim _ { z \\to p } \\inf \\{ g _ D ( z , w ) : w \\in D \\setminus U \\} = 0 . \\end{align*}"}
-{"id": "8945.png", "formula": "\\begin{gather*} s _ i ( x _ 1 , \\dots , x _ n ) = \\bigg ( x _ 1 , \\dots , x _ { i - 1 } , - x _ i + \\sum _ { j \\ne i } \\mu _ { j i } ( x _ j ) + u _ i , x _ { i + 1 } , \\dots , x _ n \\bigg ) ; \\end{gather*}"}
-{"id": "5451.png", "formula": "\\begin{align*} f ( x ) = x ^ \\rho \\ell ( x ) p ( x ) . \\end{align*}"}
-{"id": "4593.png", "formula": "\\begin{align*} N _ J ( A , y ) & = [ J A , J y ] - [ A , y ] - J ( [ J A , y ] + [ A , J y ] ) \\\\ & = - D y - J D J y \\\\ & = - D ' y - J ' D ' J ' y \\\\ & = 0 , \\end{align*}"}
-{"id": "6639.png", "formula": "\\begin{align*} \\mathrm { A r e a } ( 2 D ) = \\pi \\left ( \\sqrt { 2 } \\cdot 2 ^ { - m + 2 } \\right ) < 6 4 \\pi | z - w | ^ 2 \\ , \\end{align*}"}
-{"id": "6733.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } | \\mathbb { P } ( R _ { N } ^ { \\eta } > t ( N ) ) - \\mathbb { P } ( R _ { N } > t ( N ) ) | = 0 . \\end{align*}"}
-{"id": "4704.png", "formula": "\\begin{align*} v _ { r } ( \\alpha , s , t ) = v _ { r } ( \\alpha ) : = \\begin{cases} 1 , & \\alpha = \\infty , \\\\ | \\alpha | ^ { - r } , & \\alpha \\in [ - 1 , 1 ] ^ \\ast , \\end{cases} \\end{align*}"}
-{"id": "586.png", "formula": "\\begin{align*} \\left | f ^ { ( l ) } ( p ) \\right | = \\left | g ^ { ( l ) } ( 0 ) \\right | \\leqslant M _ l \\leqslant C ( \\alpha _ 1 , l , \\alpha _ 2 ) \\cdot \\max \\{ M _ { \\alpha _ 1 } , M _ { \\alpha _ 2 } \\} \\end{align*}"}
-{"id": "5898.png", "formula": "\\begin{align*} \\begin{aligned} & E \\tau ^ { N , i } \\approx e ^ { N \\min \\big ( I _ i ( r _ i ) , \\thinspace I _ i ( r _ { i + 1 } ) \\big ) } , \\ 1 \\le i \\le l - 1 ; \\\\ & E \\tau ^ { N , l , l } \\approx e ^ { N I _ l ( r _ l ) } ; \\ \\ E \\tau ^ { N , 0 } \\approx e ^ { N I _ 0 ( r _ 1 ) } . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "3781.png", "formula": "\\begin{align*} \\tau = R _ { \\mathcal { I } } . \\end{align*}"}
-{"id": "8078.png", "formula": "\\begin{align*} ( w y ) ^ q & = ( w \\wedge y ) ^ q \\left ( \\sum _ { i = 1 } ^ k z _ i \\right ) ^ q \\\\ & = ( w \\wedge y ) ^ q \\sum _ { i = 1 } ^ k z _ i ^ q + ( w \\wedge y ) ^ q \\sum _ \\ell f _ \\ell \\end{align*}"}
-{"id": "9776.png", "formula": "\\begin{align*} Q _ { h , \\theta } ( \\Lambda ) = \\begin{cases} Q ( \\Lambda ) & , \\\\ | \\omega _ k | + \\sum _ { i = 2 } ^ { 5 } | \\alpha _ i | & , \\\\ \\sum _ { i = 1 } ^ { 4 } | \\beta _ i | & , \\\\ \\sum _ { i = 2 } ^ { 5 } | \\alpha _ i | & . \\end{cases} \\end{align*}"}
-{"id": "4.png", "formula": "\\begin{align*} z = P _ { k } ( x ) + \\lambda u , \\lambda \\in K \\end{align*}"}
-{"id": "6768.png", "formula": "\\begin{align*} - \\Delta _ { g ( y ) } G ( y , y ' ) = \\delta _ { y ' } ( y ) - \\frac { 1 } { 4 \\pi } \\textrm { i n } \\mathbb { S } ^ 2 , \\end{align*}"}
-{"id": "7955.png", "formula": "\\begin{align*} \\mu = \\sum _ { I \\in \\mathcal S } \\lambda _ I ^ s ( g _ I ) _ * ( \\mu ) , \\end{align*}"}
-{"id": "2037.png", "formula": "\\begin{align*} \\sum _ { \\beta \\in \\Z } b _ { \\beta } ( z _ 1 ^ { k \\beta } z _ 2 ^ { - \\beta } . . . z _ { k + 1 } ^ { - \\beta } f ( z _ 2 , . . . , z _ { k + 1 } ) + . . . + z _ 1 ^ { - \\beta } . . . z _ k ^ { - \\beta } z _ { k + 1 } ^ { k \\beta } f ( z _ 1 , . . . , z _ k ) ) = 0 . \\end{align*}"}
-{"id": "7846.png", "formula": "\\begin{align*} h ( x _ { 1 } , x _ { 2 } ) = \\frac { f _ { X _ { 1 } , X _ { 2 } } ( x _ { 1 } , x _ { 2 } ) } { R _ { X _ { 1 } , X _ { 2 } } ( x _ { 1 } , x _ { 2 } ) } . \\end{align*}"}
-{"id": "761.png", "formula": "\\begin{align*} m ( a _ 1 , a _ 2 ) = J ( u _ 1 ^ * , 0 ) + J ( 0 , \\tilde { u } _ 2 ^ * ) . \\end{align*}"}
-{"id": "3881.png", "formula": "\\begin{align*} h ( \\omega ; s ; r ) = \\pi \\int _ { 0 } ^ { \\infty } k _ 0 ( x , i r ) \\frac { f ( \\omega ; s ; x ) } { x } d x . \\end{align*}"}
-{"id": "8271.png", "formula": "\\begin{align*} M ( L , j , \\delta ) = \\sharp \\mathfrak { M } ( L , j , \\delta ) . \\end{align*}"}
-{"id": "8068.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c | c | c } \\chi _ 1 & \\chi _ 2 & \\chi _ 3 \\end{array} \\right ) = \\left ( \\begin{array} { c c c } 1 & 1 & 2 \\\\ 0 & 7 & 7 \\end{array} \\right ) \\\\ \\left ( \\begin{array} { c | c | c } \\chi _ 1 ' & \\chi _ 2 ' & \\chi _ 3 ' \\end{array} \\right ) = \\left ( \\begin{array} { c c c } 1 & 2 & 3 \\\\ 0 & 7 & 7 \\end{array} \\right ) \\end{align*}"}
-{"id": "9105.png", "formula": "\\begin{align*} a _ { n + 1 - i } = ( - 1 ) ^ { i } \\binom { n + 1 } { i } \\left ( i = 0 , \\ldots , n \\right ) ; \\end{align*}"}
-{"id": "1852.png", "formula": "\\begin{align*} \\mathcal { Q } & : = \\mathcal { G F } _ { \\mathcal B } ( R ) , & \\mathcal { R } ' & : = \\mathcal { G C } _ { \\mathcal B } ( R ) , & \\mathcal { Q } ' & : = \\mathcal { F } ( R ) , & \\mathcal { R } & : = \\mathcal { C } ( R ) . \\end{align*}"}
-{"id": "5582.png", "formula": "\\begin{align*} \\Delta \\left ( \\alpha , \\beta \\right ) = A _ { 0 } - A _ { 1 } + A _ { 2 } + \\ldots + \\left ( - 1 \\right ) ^ { n } A _ { n } \\end{align*}"}
-{"id": "754.png", "formula": "\\begin{align*} J ( w _ 1 , w _ 2 ) = m ( c _ 1 , c _ 2 ) . \\end{align*}"}
-{"id": "2657.png", "formula": "\\begin{align*} P _ \\ast ( T ) = S _ a . \\end{align*}"}
-{"id": "8095.png", "formula": "\\begin{align*} \\mathcal H \\tilde \\rho ( z ) ( X ) = \\mathcal H \\rho ( z , \\varphi ( z ) ) ( X , 0 ) \\geq c | | X | | ^ 2 \\end{align*}"}
-{"id": "3332.png", "formula": "\\begin{align*} I \\left ( W _ { k : K } ; W _ { k - 1 } | W _ { 1 : k - 2 } , Z \\right ) & = H \\left ( W _ { k - 1 } | W _ { 1 : k - 2 } , Z \\right ) - H \\left ( W _ { k - 1 } | W _ { 1 : k - 2 } , W _ { k : K } , Z \\right ) \\\\ & = \\left ( L - L r \\right ) - \\left ( L - L r \\right ) \\\\ & = 0 \\end{align*}"}
-{"id": "8148.png", "formula": "\\begin{align*} \\bigsqcup _ { i = 1 } ^ n \\bigsqcup _ { j = 1 } ^ { J _ i } s _ { i , j } A _ { i , j } \\times \\{ k _ { i , j } \\} = \\bigsqcup _ { i = 1 } ^ m B _ i \\times \\{ i \\} . \\end{align*}"}
-{"id": "2049.png", "formula": "\\begin{align*} \\Delta _ H f ^ i + \\Gamma ^ i _ { j k } \\langle \\nabla _ H f ^ j , \\nabla _ H f ^ k \\rangle = 0 , \\end{align*}"}
-{"id": "3330.png", "formula": "\\begin{align*} \\bar { D } ( r ) = \\alpha \\bar { D } ( r _ s ) + ( 1 - \\alpha ) \\bar { D } ( r _ { s + 1 } ) , \\end{align*}"}
-{"id": "1645.png", "formula": "\\begin{align*} \\mu ( R _ \\lambda \\cap R _ \\nu ) = \\mu ( \\tau _ { \\lambda _ 1 } ( R _ { \\lambda _ 2 } \\cap R _ { \\nu _ 2 } ) ) & = \\int _ { R _ { \\lambda _ 2 } \\cap R _ { \\nu _ 2 } } 1 \\ , d ( \\mu \\circ \\tau _ { \\lambda _ 1 } ) = \\int _ { R _ { \\lambda _ 2 } \\cap R _ { \\nu _ 2 } } \\Phi _ { \\lambda _ 1 } \\ , d \\mu = 0 . \\end{align*}"}
-{"id": "2483.png", "formula": "\\begin{align*} \\left \\| w f ( t ) \\right \\| _ { H ^ 2 _ x L ^ 2 _ \\xi } ^ 2 \\lesssim \\int _ { \\R ^ { 6 } } \\left ( f ^ 2 + \\left | \\nabla _ x f \\right | ^ 2 + \\left | D _ x ^ 2 f \\right | ^ 2 \\right ) \\varrho \\ , d \\xi d x = \\left \\| f ( t ) \\right \\| ^ 2 _ { H ^ 2 _ x L ^ 2 _ \\xi ( \\varrho ) } . \\end{align*}"}
-{"id": "6734.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\mathbb { P } ( R _ { N } \\geq \\beta _ { N } ) = e ^ { - 1 } . \\end{align*}"}
-{"id": "6815.png", "formula": "\\begin{align*} e ^ { w _ { \\lambda } } \\tilde { V } \\leq C \\sum \\limits _ { j = 1 } ^ 4 \\frac { 1 } { | z _ { \\xi _ j } | ^ 4 } \\cdot \\frac { ( 1 + | x _ { \\xi _ j } | ^ 2 ) ^ 2 } { 4 } . \\end{align*}"}
-{"id": "6859.png", "formula": "\\begin{align*} \\mu ( I _ { 4 k + j } ^ { ( m + 1 ) } ) = \\begin{cases} \\frac { p } { 2 } \\mu ( I _ { k } ^ { ( m ) } ) & j = 0 , 3 , \\\\ \\frac { 1 - p } { 2 } \\mu ( I _ { k } ^ { ( m ) } ) & j = 1 , 2 . \\end{cases} \\end{align*}"}
-{"id": "3318.png", "formula": "\\begin{align*} D ^ * ( r ) = ( 1 - r ) \\left ( 1 + \\frac { 1 } { N } + \\cdots + \\frac { 1 } { N ^ { K - 1 } } \\right ) - r \\left ( ( K - 1 ) + \\frac { K - 2 } { N } + \\cdots + \\frac { 1 } { N ^ { K - 2 } } \\right ) \\end{align*}"}
-{"id": "8706.png", "formula": "\\begin{align*} u ( t ) = f ( t ) + ( k \\ast A u ) \\quad \\end{align*}"}
-{"id": "4442.png", "formula": "\\begin{align*} \\| ( u f _ t ) _ T \\| = & \\| \\big ( ( u f _ t ) _ { T - t } \\big ) _ { t } \\| \\lesssim \\| ( u f _ t ) _ { T - t } \\| \\leq \\| ( u f _ t ) _ { T - t } - u f _ { T } \\| + \\| u f _ { T } \\| \\\\ & \\lesssim [ u ] _ \\alpha [ f ] _ \\beta ( T ^ { 1 / 3 } ) ^ { \\alpha + \\beta } + \\| u \\| [ f ] _ \\beta T ^ { \\beta / 3 } \\lesssim ( [ u ] _ \\alpha + \\| u \\| ) [ f ] _ \\beta T ^ { \\beta / 3 } . \\end{align*}"}
-{"id": "5540.png", "formula": "\\begin{align*} \\mathbb { \\rho } _ { 1 , 2 } = \\frac { T r a c e \\left ( M \\right ) \\pm \\sqrt { \\left ( T r a c e \\left ( M \\right ) \\right ) ^ { 2 } - 4 } } { 2 } \\end{align*}"}
-{"id": "9083.png", "formula": "\\begin{align*} f ( x ^ { 2 } ) = 2 x f ( x ) \\left ( x \\in R \\right ) \\end{align*}"}
-{"id": "9692.png", "formula": "\\begin{align*} \\textbf { n } _ { k } = \\frac { ( - y _ { k + 1 } + y _ k , x _ { k + 1 } - x _ k ) } { \\sqrt { ( y _ { k + 1 } - y _ k ) ^ 2 + ( x _ { k + 1 } - x _ k ) ^ 2 } } = ( - \\sin ( \\omega _ { k , k + 1 } ) , \\cos ( \\omega _ { k , k + 1 } ) ) . \\end{align*}"}
-{"id": "8696.png", "formula": "\\begin{align*} b _ { n , 0 } ^ 2 = \\max _ { k = 0 , 1 , \\dots } a _ { n , k } ^ 2 \\leq q ^ { n / 2 \\cdot ( 1 \\wedge \\gamma ) } . \\end{align*}"}
-{"id": "2081.png", "formula": "\\begin{align*} u ^ a _ 0 ( p , t ) = \\int _ M H ( p , q , t ) \\phi ^ a ( q ) d V _ q . \\end{align*}"}
-{"id": "8994.png", "formula": "\\begin{gather*} { \\cal D } ^ { ( n ) } _ { q , t } ( c + \\tau ) = T _ \\omega ( \\tau ) { \\cal D } ^ { ( n ) } _ { q , t } ( c ) = { \\cal D } ^ { ( n ) } _ { q , t } ( c ) T _ \\omega ( \\tau ) . \\end{gather*}"}
-{"id": "7445.png", "formula": "\\begin{align*} x ' & = c _ { 0 0 } ( c _ { 1 0 } + d _ { 1 0 } + T _ 0 ^ \\beta ) + c _ { 1 0 } t , \\\\ y & = - c _ { 0 1 } ( c _ { 1 0 } + d _ { 1 0 } + T _ 0 ^ \\beta ) + c _ { 0 0 } t , \\\\ z & = - ( c _ { 1 0 } + d _ { 1 0 } + T _ 0 ^ \\beta ) . \\end{align*}"}
-{"id": "8258.png", "formula": "\\begin{align*} \\chi _ { \\infty } ( \\alpha ) = \\prod _ { \\nu } \\left ( \\frac { \\alpha _ { \\nu } } { \\abs { \\alpha _ { \\nu } } } \\right ) ^ { p _ { \\nu } } \\abs { \\alpha _ { \\nu } } _ { \\nu } ^ { i q _ { \\nu } } , \\end{align*}"}
-{"id": "7921.png", "formula": "\\begin{align*} V = V _ \\alpha : = ( \\beta K ) ^ { 1 \\over \\beta - 1 } = \\left ( { \\alpha \\over \\alpha - 1 } K \\right ) ^ { \\alpha - 1 } , \\end{align*}"}
-{"id": "417.png", "formula": "\\begin{align*} \\abs * { a _ { k _ 1 , k _ 2 } ( \\lambda + i y ) } = \\frac { \\abs * { \\lambda + i y } ^ { n + k _ 1 + k _ 2 } ( \\sinh ( \\lambda ) ^ 2 + \\cos ( y ) ^ 2 ) ^ { \\frac { k _ 1 } { 2 } } } { ( \\sinh ( \\lambda ) ^ 2 + \\sin ( y ) ^ 2 ) ^ { \\frac { n + k _ 1 } { 2 } } } . \\end{align*}"}
-{"id": "4374.png", "formula": "\\begin{align*} A _ z = \\widetilde { C } _ z A \\widetilde { C } _ { - z } . \\end{align*}"}
-{"id": "8314.png", "formula": "\\begin{align*} \\overline { \\rho } _ { V _ \\Z } ( \\gamma ) \\cdot \\varphi = \\overline { ( \\rho _ { V _ \\Z } ( \\gamma ) \\cdot \\overline { \\varphi } ) } , \\end{align*}"}
-{"id": "1032.png", "formula": "\\begin{align*} \\underline { h } _ { k } ^ { n , \\zeta } = ( h _ { k - 1 , 0 } , \\ldots , h _ { k - 1 , n } , h _ { k , 0 } , \\ldots , h _ { k , n } , h _ { k + 1 , 0 } , \\ldots , h _ { k + 1 , n } ) \\in \\mathbb { Z } ^ { 3 n + 3 } , k \\geq 2 , \\end{align*}"}
-{"id": "9018.png", "formula": "\\begin{align*} \\left \\{ \\aligned & \\partial _ { t } \\theta + ( u \\cdot \\nabla ) \\theta + \\Lambda _ { x _ { 1 } } ^ { 2 \\alpha } \\theta + \\Lambda _ { x _ { 2 } } ^ { 2 \\beta } \\theta = 0 , \\\\ & u = \\mathcal { R } ^ { \\perp } \\theta , \\\\ & \\theta ( x , 0 ) = \\theta _ { 0 } ( x ) . \\endaligned \\right . \\end{align*}"}
-{"id": "9855.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int \\hat { f } ( t ) e ^ { i x t } d t = f ( x ) . \\end{align*}"}
-{"id": "7416.png", "formula": "\\begin{align*} a = a _ 0 a _ 1 , a ^ * = a _ 1 ^ * a _ 0 ^ * , b = a _ 3 ^ * a _ 3 , c = a _ 5 ^ * a _ 5 , d = a _ 1 ^ * a _ 1 . \\end{align*}"}
-{"id": "5926.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { \\sum _ { s \\in A _ { } } L ( s ) Q ^ N ( s ) } { \\sum _ { s \\in A } L ( s ) Q ^ N ( s ) } = 1 . \\end{align*}"}
-{"id": "7217.png", "formula": "\\begin{align*} \\gamma _ { y } ( t ) = \\rho _ { x , y } ( t z ) ^ { - 1 } \\gamma _ { x } ( t ) \\rho _ { x , y } ( z ) \\end{align*}"}
-{"id": "4587.png", "formula": "\\begin{align*} g _ i = \\exp ( - f _ i ) \\ , g | _ { U _ i } \\end{align*}"}
-{"id": "2594.png", "formula": "\\begin{align*} \\frac 1 { \\phi _ a ( t ) } S ( \\phi _ a f ) ( t ) = \\frac { A + B t } { \\phi _ a ( t ) } + R ( t ) , \\end{align*}"}
-{"id": "1491.png", "formula": "\\begin{align*} 1 + \\cfrac { z g '' ( z ) } { g ' ( z ) } = 1 - 2 z \\cfrac { c u ' ( z ) + v ' ( z ) } { c u ( z ) + v ( z ) } . \\end{align*}"}
-{"id": "6355.png", "formula": "\\begin{align*} \\det D \\varphi _ { 2 n } ^ \\mathbb { R } = \\prod \\limits _ { k = 1 } ^ n \\left ( \\frac { \\partial m _ { 2 k - 1 } } { \\alpha _ k } \\cdot \\frac { \\partial m _ { 2 k } } { \\beta _ k } \\right ) . \\end{align*}"}
-{"id": "7704.png", "formula": "\\begin{align*} \\Theta _ { i i } = \\left [ \\begin{array} { c c } X _ { i i } & \\Psi _ { i i } \\\\ \\Psi _ { i i } & Y _ { i i } \\end{array} \\right ] \\ , . \\end{align*}"}
-{"id": "1117.png", "formula": "\\begin{align*} X _ { i j } = \\frac { - 1 } { k - 1 } + \\frac { k } { k - 1 } Y _ { i j } . \\end{align*}"}
-{"id": "4708.png", "formula": "\\begin{align*} I _ s ( t ) \\lesssim \\abs { \\det A _ s ^ i } \\sum _ { i = 1 } ^ 4 \\prod _ { j = 1 } ^ 3 ( 1 + \\abs { ( A _ s ^ i t ) _ j } ) ^ { - k } \\end{align*}"}
-{"id": "5664.png", "formula": "\\begin{align*} | \\dot { \\gamma } | ( t ) = \\| \\partial _ t \\gamma ( t , \\cdot ) \\| _ { L ^ 2 ( \\R ) } a . e . \\end{align*}"}
-{"id": "2531.png", "formula": "\\begin{align*} \\left \\| h ( t ) \\right \\| ^ 2 _ { L ^ 2 ( \\varrho \\ , m _ 1 ) } + \\left ( t \\left \\| \\nabla _ \\xi h ( t ) \\right \\| ^ 2 _ { L ^ 2 ( \\varrho \\ , m _ 0 ) } + t ^ 3 \\left \\| \\nabla _ { x } h ( t ) \\right \\| ^ 2 _ { L ^ 2 ( \\varrho \\ , m _ 0 ) } \\right ) \\lesssim \\mathcal { F } ( t , h ( t ) ) \\leq \\mathcal { F } ( 0 , h ( 0 ) ) = \\left \\| h ( 0 ) \\right \\| ^ 2 _ { L ^ 2 ( \\varrho \\ , m _ 1 ) } . \\end{align*}"}
-{"id": "8500.png", "formula": "\\begin{align*} K _ { l _ 2 } = K ( \\chi _ 1 \\otimes \\chi _ 2 , ( \\varpi ^ { t + l _ 2 } , \\varpi ^ { - l _ 2 } ) , v \\varpi ^ { - l } ) \\end{align*}"}
-{"id": "8420.png", "formula": "\\begin{align*} \\sum _ { t = - \\infty } ^ { \\infty } q ^ { ( t + a ( \\mu \\pi ) ) ( \\frac { 1 } { 2 } - s ) } c _ { t , l } ( \\mu ) = \\omega _ { \\pi } ( - 1 ) \\epsilon ( \\frac { 1 } { 2 } , \\mu \\pi ) ^ { - 1 } G ( \\varpi ^ { - l } , \\mu ^ { - 1 } ) . \\end{align*}"}
-{"id": "1763.png", "formula": "\\begin{align*} f F _ \\mu \\sqrt { d \\mu } = g F _ \\nu \\sqrt { d \\nu } , \\end{align*}"}
-{"id": "4122.png", "formula": "\\begin{align*} \\varphi '' ( s ) = f ( s ) \\varphi ' ( s ) + h ( s ) \\varphi ( s ) , \\end{align*}"}
-{"id": "7198.png", "formula": "\\begin{align*} \\sum _ j \\xi _ j = 1 , \\end{align*}"}
-{"id": "5093.png", "formula": "\\begin{align*} & \\Big \\| \\sum _ { k \\in \\Gamma _ n } T _ k \\varphi \\Big \\| _ { \\textup { L } ^ 2 ( G ) } ^ 2 = \\Big \\| \\sum _ { k \\in \\Gamma _ { n - 1 } } T _ k \\varphi + T _ m \\varphi \\Big \\| _ { \\textup { L } ^ 2 ( G ) } ^ 2 \\\\ & = \\Big \\| \\sum _ { k \\in \\Gamma _ { n - 1 } } T _ k \\varphi \\Big \\| _ { \\textup { L } ^ 2 ( G ) } ^ 2 + \\| T _ m \\varphi \\| _ { \\textup { L } ^ 2 ( G ) } ^ 2 = \\Big \\| \\sum _ { k \\in \\Gamma _ { n - 1 } } T _ k \\varphi \\Big \\| _ { \\textup { L } ^ 2 ( G ) } ^ 2 + \\| \\varphi \\| _ { \\textup { L } ^ 2 ( G ) } ^ 2 . \\end{align*}"}
-{"id": "1573.png", "formula": "\\begin{align*} V ( m , \\mathbf { j } ) : = V ( \\mathbf { j } ) = - \\nu \\left ( \\prod _ { k = 1 } ^ { n } C _ { j _ k } ( \\alpha ( k ) ) \\right ) . \\end{align*}"}
-{"id": "5213.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } u \\\\ v \\end{array} \\right ) _ z = \\left ( \\begin{array} { c } S v \\\\ - c v - Q f ( u ) \\end{array} \\right ) . \\end{align*}"}
-{"id": "4171.png", "formula": "\\begin{align*} r Z _ { N , J } ^ n + n Z _ { I , J } ^ { n - 1 } + \\sum _ { \\tau = 1 } ^ { n - 2 } n ^ { n - \\tau } Z _ { i , J } ^ { \\tau } \\leq r + \\sum _ { \\tau = 1 } ^ { n - 2 } n ^ { \\tau } . \\end{align*}"}
-{"id": "7950.png", "formula": "\\begin{align*} \\tilde K : = K _ { I _ 0 } . \\end{align*}"}
-{"id": "828.png", "formula": "\\begin{align*} u _ h = w _ h + \\varphi _ h , \\mbox { w h e r e } , \\left \\langle \\mathcal { L } \\varphi _ h , \\mathcal { L } v _ h \\right \\rangle = - \\left \\langle \\mathcal { L } w _ h , \\mathcal { L } v _ h \\right \\rangle . \\end{align*}"}
-{"id": "10034.png", "formula": "\\begin{align*} \\eta ( \\alpha h ) = \\chi _ \\eta ( \\alpha ) \\eta ( h ) . \\end{align*}"}
-{"id": "5813.png", "formula": "\\begin{align*} & \\int _ { \\R ^ n } \\vert \\nabla u \\vert ^ { p - 2 } \\nabla u \\cdot \\nabla \\phi \\ ; d x \\\\ = & \\int _ { \\R ^ n } u ^ { q } \\phi \\ ; d \\sigma + \\int _ { \\R ^ n } \\phi \\ ; d \\mu , \\phi \\in \\dot { W _ { 0 } } ^ { 1 , p } ( \\R ^ n ) , \\end{align*}"}
-{"id": "4828.png", "formula": "\\begin{align*} \\prod _ { g \\in G _ 0 } g ^ { m \\mathsf { v } _ g ( X ) } = \\theta ( r ) ^ m = \\theta ( s ) ^ n = \\prod _ { g \\in G _ 0 } g ^ { n \\mathsf { v } _ g ( X ) } . \\end{align*}"}
-{"id": "2129.png", "formula": "\\begin{align*} d ^ * ( \\mathfrak { D } f _ 1 , \\mathfrak { D } f _ 2 ) & = s u p \\{ | \\mathfrak { D } f _ 1 - \\mathfrak { D } f _ 2 | ~ u , v \\in U \\} \\\\ & = \\frac { 2 5 } { 3 6 } \\leq \\frac { 2 } { 3 } \\times \\frac { 1 1 } { 6 } \\\\ & = \\lambda d ^ * ( f _ 1 , f _ 2 ) . \\end{align*}"}
-{"id": "4299.png", "formula": "\\begin{align*} k + g - 1 + n = \\deg ( \\alpha ) + \\sum _ { i = 1 } ^ { n } \\deg ( \\gamma _ i ) \\end{align*}"}
-{"id": "3291.png", "formula": "\\begin{align*} V ( x ) = \\sum _ { j = 1 } ^ k m _ j P U _ j ( x ) , x \\in S , \\end{align*}"}
-{"id": "113.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { l _ 0 } \\omega _ l ( n ) \\leq u _ 0 ( n ) = \\frac { \\left | \\mathrm { I n t } _ { L _ 0 } W ( x , n ) \\right | } { A } , \\end{align*}"}
-{"id": "289.png", "formula": "\\begin{align*} d _ \\infty ( x ^ * , y ^ * ) = \\sup _ { \\alpha \\in [ 0 , 1 ] } d _ H \\bigl ( C _ \\alpha ( x ^ * ) , C _ \\alpha ( y ^ * ) \\bigr ) , \\ ; x ^ * , y ^ * \\in \\mathcal F _ { } ( \\R ^ d ) . \\end{align*}"}
-{"id": "5071.png", "formula": "\\begin{align*} b _ k C _ { i , j k } = - \\frac { 4 } { 3 } \\frac { C _ i C _ j C _ k } { b _ i b _ j b _ k } = - \\frac { 4 } { 3 } \\frac { C _ 1 C _ 2 C _ 3 } { b _ 1 b _ 2 b _ 3 } . \\end{align*}"}
-{"id": "8206.png", "formula": "\\begin{align*} k ( u ( \\gamma P , P ) ) = \\prod _ v k _ v ( u _ v ( \\gamma _ v P _ v , P _ v ) ) P _ v = n ( x _ v ) a ( y _ v ) . i _ v . \\end{align*}"}
-{"id": "7701.png", "formula": "\\begin{align*} M ^ { \\top } \\Sigma + \\Sigma M + Z = 0 \\ , , \\end{align*}"}
-{"id": "2388.png", "formula": "\\begin{align*} a _ 4 + 4 a _ 6 = 4 a _ 3 + a _ 4 = 4 a _ 2 + a _ 5 = a _ 1 = 0 , \\end{align*}"}
-{"id": "3700.png", "formula": "\\begin{align*} \\mathbf t ( p _ 1 ) = \\dots = \\mathbf t ( p _ n ) \\in \\mathbb A ^ { n + 1 } , \\end{align*}"}
-{"id": "7114.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } X ( x , t ) = ~ - F ( \\mathcal { W } ( x , t ) ) ^ { \\alpha } \\nu ( x , t ) \\end{align*}"}
-{"id": "10155.png", "formula": "\\begin{align*} { J _ { \\omega } ( { \\boldsymbol \\omega } ) } = \\sum _ { k = 1 } ^ { N } { \\mathbb { E } | { d _ k ( i ) } - { \\boldsymbol x _ k ^ H ( i ) } { \\boldsymbol \\omega } | ^ 2 } . \\end{align*}"}
-{"id": "9627.png", "formula": "\\begin{align*} \\frac { \\partial P _ { \\rm { t r } , 1 } ( h _ { \\beta , 1 } ) } { \\partial h _ { \\beta , 1 } } = 0 \\Leftrightarrow \\int _ 0 ^ 1 & \\Big \\{ 2 h _ { \\beta , 1 } \\big ( \\eta _ 1 + P _ { 0 } ( r , h _ { \\beta , 1 } ) ( \\eta _ 0 - \\eta _ 1 ) \\big ) + \\\\ & \\big ( r ^ 2 + h _ { \\beta , 1 } ^ 2 \\big ) \\big ( \\eta _ 1 + \\frac { \\partial P _ { 0 } ( r , h _ { \\beta , 1 } ) } { \\partial h _ { \\beta , 1 } } ( \\eta _ 0 - \\eta _ 1 ) \\big ) \\Big \\} r { \\rm { d } } r = 0 , \\end{align*}"}
-{"id": "5418.png", "formula": "\\begin{align*} J ^ { \\sigma } _ n ( X ) : = \\coprod _ { k = 0 } ^ { n } X ^ { \\times k } / \\sim , \\end{align*}"}
-{"id": "2736.png", "formula": "\\begin{align*} \\epsilon ^ 2 \\partial _ t \\partial ^ { l } g + \\epsilon v \\cdot \\nabla _ x ( \\partial ^ { l } g ) = \\partial ^ { l } \\mathcal L ( g ) \\ , . \\end{align*}"}
-{"id": "6670.png", "formula": "\\begin{align*} \\frac { \\| x ^ { \\delta } \\| _ 2 } { \\| x \\| _ 2 } = \\frac { \\langle x ^ { \\delta } , u ( x ' ) \\rangle } { \\langle x , u ( x ' ) \\rangle } \\geq \\frac { \\langle x ' + \\Delta ^ { x ' } ( \\delta ) u ( x ' ) , u ( x ' ) \\rangle } { \\langle x ' , u ( x ' ) \\rangle } = 1 + \\frac { \\Delta ^ { x ' } ( \\delta ) } { \\langle x ' , u ( x ' ) \\rangle } \\quad . \\end{align*}"}
-{"id": "6017.png", "formula": "\\begin{align*} & \\inf _ { P _ { X } , Q _ { X } : \\left | P _ { X } - Q _ { X } \\right | \\geq \\epsilon } D _ { 1 + s } ( P _ { X } \\| Q _ { X } ) \\\\ & = \\inf _ { P _ { X } , Q _ { X } : \\left | P _ { X } - Q _ { X } \\right | = \\epsilon } D _ { 1 + s } ( P _ { X } \\| Q _ { X } ) \\\\ & = \\inf _ { q \\in [ 0 , 1 - \\epsilon ] } d _ { 1 + s } ( q + \\epsilon \\| q ) , \\end{align*}"}
-{"id": "1926.png", "formula": "\\begin{align*} \\Pi _ { j } ( \\Gamma _ { j } ^ { 1 } ) & = \\{ \\zeta \\in \\Gamma ( M _ { i } ) : \\limsup _ { n \\rightarrow \\infty } \\Vert D \\textbf { \\textit { f } } ^ { n } ( \\zeta ) \\Vert _ { \\Gamma _ { j + n } } ^ { 1 / n } \\leq \\lambda _ { 1 } \\} \\\\ \\Pi _ { j } ( \\Gamma _ { j } ^ { 2 } ) & = \\{ \\zeta \\in \\Gamma ( M _ { i } ) : \\limsup _ { n \\rightarrow \\infty } \\Vert D \\textbf { \\textit { f } } ^ { - n } ( \\zeta ) \\Vert _ { \\Gamma _ { j - n } } ^ { 1 / n } \\leq \\lambda _ { 2 } \\} . \\end{align*}"}
-{"id": "2916.png", "formula": "\\begin{align*} u [ w ] : = \\{ [ ( a , w ) ] \\colon a \\in A _ { s } \\} \\end{align*}"}
-{"id": "2591.png", "formula": "\\begin{align*} C _ 0 ^ \\alpha ( [ 0 , 1 ] ) & : = \\big \\{ f \\in C ^ \\alpha ( [ 0 , 1 ] ) : f ( 0 ) = f ( 1 ) = 0 \\big \\} , \\\\ C ^ { k , \\alpha } ( [ 0 , 1 ] ) & : = \\big \\{ f \\in C ^ k ( [ 0 , 1 ] ) : f ^ { ( k ) } \\in C ^ \\alpha ( [ 0 , 1 ] ) \\big \\} , \\end{align*}"}
-{"id": "3199.png", "formula": "\\begin{align*} g ( t , X _ { t } ) - g ( 0 , X _ { 0 } ) = \\int _ { 0 } ^ { t } ( \\mathcal { L } g ) ( s , X _ { s } ) \\mathrm { d } s + \\int _ { 0 } ^ { t } g _ { s } ' ( s , X _ { s } ) \\mathrm { d } s + M _ { t } ( g ) , t \\geqslant 0 , \\end{align*}"}
-{"id": "6509.png", "formula": "\\begin{align*} \\mathcal { P } \\approx 1 - \\frac { 4 \\rho k _ { \\mathrm { o } } ^ { 2 } \\left ( 2 k _ { \\mathrm { o } } ^ { 2 } + \\sigma _ { k _ { \\mathrm { o } } } ^ { 2 } \\right ) R _ { \\mathrm { o } } L ^ { 3 } } { 3 } = 1 - \\frac { 8 \\mu V \\left ( 2 k _ { \\mathrm { o } } ^ { 2 } + \\sigma _ { k _ { \\mathrm { o } } } ^ { 2 } \\right ) R _ { \\mathrm { o } } L ^ { 3 } } { 3 \\hbar ^ { 2 } } , \\end{align*}"}
-{"id": "4423.png", "formula": "\\begin{align*} [ f ] _ \\beta : = \\inf \\{ | c | + [ g ] _ { \\beta + 1 } + [ h ] _ { \\beta + \\frac { 3 } { 2 } } \\ , : \\ , f = c + \\partial _ 1 g + \\partial _ 2 h \\} \\end{align*}"}
-{"id": "8138.png", "formula": "\\begin{align*} B _ { k , l , c , q } = F ^ { Q - q } B _ { k , l , c , Q } \\setminus F ^ { Q - q - 1 } B _ { k , l , c , Q } . \\end{align*}"}
-{"id": "2976.png", "formula": "\\begin{align*} - \\Delta v _ n & = f _ n h _ n \\left ( v _ n + \\frac { 1 } { n } \\right ) \\\\ & \\leq f _ { n + 1 } h _ n \\left ( v _ n + \\frac { 1 } { n + 1 } \\right ) ~ ~ \\Omega , \\\\ v _ n & = 0 ~ ~ \\partial \\Omega . \\end{align*}"}
-{"id": "6287.png", "formula": "\\begin{align*} & \\int _ { \\R ^ 3 } A \\times B \\cdot \\nabla C \\ , d x = - \\int _ { \\R ^ 3 } \\nabla \\times ( A \\times B ) \\cdot C \\ , d x \\\\ = & - \\int _ { \\R ^ 3 } \\left [ ( \\nabla \\cdot B + B \\cdot \\nabla ) A \\right ] \\cdot C \\ , d x + \\int _ { \\R ^ 3 } \\left [ ( \\nabla \\cdot A + A \\cdot \\nabla ) B \\right ] \\cdot C \\ , d x . \\end{align*}"}
-{"id": "6044.png", "formula": "\\begin{align*} R _ { \\mathrm { s h } } = R ^ { * } = \\lim _ { k \\to \\infty } \\frac { 1 } { \\alpha _ { m _ { k } } } R ^ { ( \\alpha _ { m _ { k } } ) } . \\end{align*}"}
-{"id": "3058.png", "formula": "\\begin{align*} K \\subset P _ T : = \\{ ( x , y , z ) \\in \\Omega \\mid ( x , y , 0 ) \\in T \\} , \\end{align*}"}
-{"id": "4473.png", "formula": "\\begin{align*} \\int _ { [ 0 , 1 ) ^ 2 } ( f _ n - A ) \\zeta \\ , d x & = - \\int _ { [ 0 , 1 ) ^ 2 } ( g _ n \\partial _ 1 \\zeta + h _ n \\partial _ 2 \\zeta ) \\ , d x \\\\ & \\stackrel { n \\to \\infty } { \\to } - \\int _ { [ 0 , 1 ) ^ 2 } ( g \\partial _ 1 \\zeta + h \\partial _ 2 \\zeta ) \\ , d x = \\int _ { [ 0 , 1 ) ^ 2 } ( f - A ) \\zeta \\ , d x . \\end{align*}"}
-{"id": "8239.png", "formula": "\\begin{align*} \\chi \\left ( \\left ( \\begin{matrix} a & b \\\\ 0 & d \\end{matrix} \\right ) \\right ) = \\chi _ 1 ( a ) \\chi _ 2 ( d ) . \\end{align*}"}
-{"id": "9948.png", "formula": "\\begin{align*} v _ t + \\left [ ( v + v _ { x x } ) ^ n \\right ] _ x = 0 , n \\geq 2 \\end{align*}"}
-{"id": "8871.png", "formula": "\\begin{align*} \\Bigg \\lfloor \\frac { \\Big \\lceil \\frac { \\lceil d \\rceil } { 2 } \\Big \\rceil } { 2 } \\Bigg \\rfloor = \\Bigg \\lfloor \\frac { \\lceil d \\rceil + 1 } { 4 } \\Bigg \\rfloor . \\end{align*}"}
-{"id": "5725.png", "formula": "\\begin{align*} F ( m _ 0 ) = \\inf F = d _ { L ^ 2 } ( { v } , \\mathcal { C } ( z ^ - ) \\cup \\mathcal { C } ( z ^ + ) ) \\leq \\delta . \\end{align*}"}
-{"id": "1214.png", "formula": "\\begin{align*} \\xi = e _ n = ( 0 , \\dots , 0 , 1 ) \\mbox { a n d } u ( x ) = u _ 1 ( y ) = u _ 2 ( z ) = t \\end{align*}"}
-{"id": "1844.png", "formula": "\\begin{align*} E ^ { + + } \\rightarrowtail ( R ^ { ( I ) } ) ^ + = ( R ^ + ) ^ I . \\end{align*}"}
-{"id": "1522.png", "formula": "\\begin{align*} \\displaystyle F ( x ) = - x ^ 2 \\cdot ( m ^ 2 + 2 m ) / { 2 } + x \\cdot ( m ^ 2 t + m t + 3 m ) / { 2 } - m t \\end{align*}"}
-{"id": "9242.png", "formula": "\\begin{align*} A ^ { + } & = s y m ( e _ { 1 } \\mathfrak { a } e _ { 1 } \\oplus e _ { 2 } \\mathfrak { a } e _ { 2 } ) , A ^ { - } = s k e w ( e _ { 1 } \\mathfrak { a } e _ { 1 } \\oplus e _ { 2 } \\mathfrak { a } e _ { 2 } ) , B = \\mathcal { B } e _ { 2 } , B ' = \\mathcal { B } e _ { 1 } , \\\\ E & = s y m ( e _ { 1 } \\mathfrak { a } e _ { 2 } ) , C = s k e w ( e _ { 1 } \\mathfrak { a } e _ { 2 } ) , E ' = s y m ( e _ { 2 } \\mathfrak { a } e _ { 1 } ) , C ' = s k e w ( e _ { 2 } \\mathfrak { a } e _ { 1 } ) , \\end{align*}"}
-{"id": "7769.png", "formula": "\\begin{align*} C ' _ { 4 } & : = \\max \\{ \\ , | y _ { 0 } ( n _ { 1 2 } ) | , | u ( n _ { 1 2 } ) | \\ , | \\ , n _ { 1 2 } \\in \\L \\cap D _ { R _ { 1 } + R _ { 2 } } \\} \\end{align*}"}
-{"id": "9038.png", "formula": "\\begin{align*} H ( w , c ( w ' , w ' ) ) = 2 i H ( \\Phi ( H ( w ' , w ) ) , w ' ) \\end{align*}"}
-{"id": "6179.png", "formula": "\\begin{align*} R = - \\frac { 1 } { 2 V ^ 2 } \\sum _ { i = 1 } ^ 3 \\frac { f _ i '^ 2 } { f _ i ^ 2 } = - \\frac { 1 } { 2 } \\mathcal { T } \\end{align*}"}
-{"id": "4065.png", "formula": "\\begin{align*} \\nu _ { ( x , t ) } = \\sigma _ { ( x , t ) } = \\delta _ { \\{ w ( x , t ) \\} } \\end{align*}"}
-{"id": "603.png", "formula": "\\begin{align*} ( V _ q ^ { \\otimes n } : \\Delta _ q ( m ) ) = \\binom { n } { r } - \\binom { n } { r - 1 } \\end{align*}"}
-{"id": "10040.png", "formula": "\\begin{align*} Q _ \\mu = \\prod _ { \\substack { p \\mid D \\\\ \\mu _ p \\neq 0 } } p , \\end{align*}"}
-{"id": "233.png", "formula": "\\begin{align*} F _ M = \\sup _ { x \\in \\R } | f ( x + i M ) | & \\leq \\sum _ { l = 0 } ^ { N } \\frac { \\mathbb { E } \\left [ \\Vert \\chi _ a A ^ l \\chi _ b \\Vert \\right ] } { 2 ^ { l + 1 } ( \\| A \\| _ { \\infty } + 1 ) ^ { l + 1 } } + \\sum _ { l = N + 1 } ^ { \\infty } \\frac { \\mathbb { E } \\left [ \\Vert \\chi _ a A ^ l \\chi _ b \\Vert \\right ] } { 2 ^ { l + 1 } ( \\| A \\| _ { \\infty } + 1 ) ^ { l + 1 } } \\\\ & = : I _ 1 + I _ 2 \\end{align*}"}
-{"id": "1471.png", "formula": "\\begin{align*} d _ { 2 ^ { r + 2 } - 1 } ( \\phi _ r ( x _ I ) ) & = v _ { r + 1 } \\phi _ { r + 1 } ( x _ I ) x _ 3 , \\\\ d _ { 2 ^ { r + 2 } - 1 } ( \\phi _ r ( x _ I ) x _ 3 ) & = v _ { r + 1 } \\phi _ { r + 1 } ( x _ I ) x _ 3 ^ 2 . \\end{align*}"}
-{"id": "3099.png", "formula": "\\begin{align*} \\beta _ { 3 , 4 } = S ^ { 4 3 2 } \\oplus S ^ { 4 ^ 2 1 } \\oplus S ^ { 4 2 1 ^ 3 } \\oplus S ^ { 4 3 1 ^ 2 } \\oplus S ^ { 4 2 ^ 2 1 } . \\end{align*}"}
-{"id": "7902.png", "formula": "\\begin{align*} \\delta _ { x , t } = \\begin{cases} \\delta _ { x _ t } \\wedge d ( x _ t , x ) & \\\\ \\delta _ { x _ t } & \\end{cases} \\end{align*}"}
-{"id": "8432.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { t , l , v } ) = \\begin{cases} \\epsilon ( \\frac { 1 } { 2 } , \\tilde { \\pi } ) & l = 0 t = - k , \\\\ \\gamma q ^ { - \\frac { t } { 2 } } K ( \\xi ^ { - 1 } , \\Omega ^ { \\frac { t } { f } } , v \\varpi ^ { - l } ) & l = \\frac { n } { 2 } - k \\leq t < 0 , l \\neq 0 , \\frac { n } { 2 } t = - k , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "9741.png", "formula": "\\begin{align*} U _ { h , \\theta } | _ { \\Omega _ { h , k } ^ { ( i ) } } \\in O _ { \\epsilon } ( { U } _ i ^ { ( 0 ) } ) , \\ , \\ , i = 1 , 2 , \\sigma _ { j } ^ { ( k ) } \\in O _ { \\hat { \\epsilon } } ( \\sigma _ { j 0 } ) , \\ , \\ , j = 2 , 3 , \\gamma _ { 4 } ^ { ( k ) } \\in O _ { \\hat { \\epsilon } } ( 0 ) . \\end{align*}"}
-{"id": "2847.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { P _ { n } } { p _ { n } } \\right ) ^ { k - 1 } \\left | \\bar { \\Delta } A _ { n } ( s ) \\right | ^ { k } < \\infty . \\end{align*}"}
-{"id": "5475.png", "formula": "\\begin{align*} U ( x ) = \\int _ 0 ^ x u ( y ) \\dd y \\sim x ^ \\rho \\ell ( x ) p ( x ) x \\to \\infty , \\end{align*}"}
-{"id": "8125.png", "formula": "\\begin{align*} G = \\bigsqcup _ { k = 1 } ^ n \\bigsqcup _ { c \\in C _ k } F _ k c . \\end{align*}"}
-{"id": "2680.png", "formula": "\\begin{align*} P ^ x _ \\omega ( X _ 0 = x ) = 1 \\end{align*}"}
-{"id": "2793.png", "formula": "\\begin{align*} { Q } _ { n } [ f ] = \\sum _ { i = 0 } ^ { n } \\omega _ { i , n } f ( t _ { i } ) , \\end{align*}"}
-{"id": "1014.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( Q _ j \\mid Q _ 1 , \\ldots , Q _ { j - 1 } \\right ) = \\mathbb { P } \\left ( Q _ j \\mid Q _ { j - 1 } \\right ) , \\ ; \\ ; j = 2 , \\ldots , N + 1 \\end{align*}"}
-{"id": "4549.png", "formula": "\\begin{align*} a _ { i , j } ( r ) = \\frac { [ r ] } { r } \\times \\left ( ( q ^ r + q ^ { - r } ) \\delta ^ { ( n ) } _ { i , j } - d ^ r \\delta ^ { ( n ) } _ { i , j - 1 } - d ^ { - r } \\delta ^ { ( n ) } _ { i , j + 1 } \\right ) , \\end{align*}"}
-{"id": "68.png", "formula": "\\begin{align*} \\frac { \\partial \\textbf { w } ^ { H } \\textbf { X } D ^ { * } } { \\partial \\textbf { w } ^ * } = \\frac { \\partial \\textbf { X } ^ T \\textbf { w } ^ * D ^ { * } } { \\partial \\textbf { w } ^ * } = \\textbf { X } D ^ * \\end{align*}"}
-{"id": "8997.png", "formula": "\\begin{gather*} { \\cal D } ^ { ( n ) } _ { q , t : * } ( q ^ { - 1 / 2 } ) = \\sum _ { \\sigma \\in \\{ \\pm 1 \\} ^ n } \\prod _ { 1 \\le i \\le n } \\frac { z _ i ^ { \\sigma _ i } } { 1 - z _ i ^ { 2 \\sigma _ i } } \\prod _ { 1 \\le i < j \\le n } \\frac { 1 - t z _ i ^ { \\sigma _ i } z _ j ^ { \\sigma _ j } } { 1 - z _ i ^ { \\sigma _ i } z _ j ^ { \\sigma _ j } } \\prod _ { 1 \\le i \\le n } T _ i ^ { \\sigma _ i / 2 } . \\end{gather*}"}
-{"id": "6857.png", "formula": "\\begin{align*} F _ i ( x ) = \\frac { 1 } { 4 } x + \\frac { i } { 4 } \\end{align*}"}
-{"id": "9528.png", "formula": "\\begin{align*} | | f | | _ { M ^ p ( \\Omega ) } = \\sup \\ , \\large \\{ | K | ^ { - ( 1 - p ) / p } \\int _ K | f | \\ , d x : K \\subset \\Omega , 0 < | K | < \\infty \\large \\} \\end{align*}"}
-{"id": "8232.png", "formula": "\\begin{align*} \\prod _ i ( 1 + x _ i ) ^ { e _ i } = e ^ { \\sum e _ i \\ln ( 1 + x _ i ) } = 1 + \\Bigl ( \\sum e _ i \\ln ( 1 + x _ i ) \\Bigr ) + \\frac { 1 } { 2 ! } \\Bigl ( \\sum e _ i \\ln ( 1 + x _ i ) \\Bigr ) ^ 2 + \\cdots \\end{align*}"}
-{"id": "4985.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\kappa } \\| \\partial _ i ( f - F ) \\| _ { \\dot { F } ^ { \\alpha - 1 , p } _ q } \\leq \\delta \\| f \\| _ { \\dot { F } ^ { \\alpha , p } _ q } + D _ { \\delta } \\min \\left ( \\eta _ { \\delta } , \\frac { \\delta } { D _ { \\delta } } \\right ) \\| f \\| _ { \\dot { F } ^ { \\alpha , p } _ q } \\leq 2 \\delta \\| f \\| _ { \\dot { F } ^ { \\alpha , p } _ q } , \\end{align*}"}
-{"id": "159.png", "formula": "\\begin{align*} E _ 2 ^ { p , q } = H ^ p ( \\Delta ^ { a b } , H ^ q ( [ \\Delta , \\Delta ] , \\mathbf Z / n \\mathbf Z ) ) = 0 , \\ p \\geq 2 , q \\geq 0 \\end{align*}"}
-{"id": "1293.png", "formula": "\\begin{align*} \\sqrt [ 3 ] { 1 / 9 } \\ - \\ \\sqrt [ 3 ] { 2 / 9 } \\ + \\ \\sqrt [ 3 ] { 4 / 9 } \\ & = \\ \\sqrt [ 3 ] { \\sqrt [ 3 ] { 2 } - 1 } \\\\ [ 1 . 2 e x ] \\sqrt [ 3 ] { \\cos \\frac { 2 \\pi } 9 } \\ + \\ \\sqrt [ 3 ] { \\cos \\frac { 4 \\pi } 9 } \\ - \\ \\sqrt [ 3 ] { \\cos \\frac { \\pi } 9 } \\ & = \\ \\sqrt [ 3 ] { \\frac { 3 } { 2 } \\left ( \\sqrt [ 3 ] { 9 } - 2 \\right ) } \\end{align*}"}
-{"id": "351.png", "formula": "\\begin{align*} R i c ( \\nabla _ { \\nu } e _ i , e _ i ) & = R i c \\left ( - h _ { i i } e _ i + \\frac { e _ i ( R ) } { 2 | \\nabla f | } \\partial _ 0 , e _ i \\right ) = - h _ { i i } R _ { i i } + \\frac { e _ i ( R ) } { 2 | \\nabla f | } R _ { 0 i } \\\\ & = - h _ { i i } R _ { i i } + \\frac { ( e _ i ( R ) ) ^ 2 } { 4 | \\nabla f | ^ 3 } , \\end{align*}"}
-{"id": "1378.png", "formula": "\\begin{align*} \\underline { { \\cal H } } _ c : = \\left [ \\begin{array} { c c } \\underline { { \\cal E } } _ c & 0 \\\\ 0 & { \\overline { \\sigma } } ^ { - 1 } \\overline { \\eta } ^ { - 1 } I _ { n _ 3 } \\end{array} \\right ] \\ , , \\overline { { \\cal H } } _ c : = \\left [ \\begin{array} { c c } \\overline { { \\cal E } } _ c & 0 \\\\ 0 & { \\underline { \\sigma } } ^ { - 1 } \\underline { \\eta } ^ { - 1 } I _ { n _ 3 } \\end{array} \\right ] \\ , . \\end{align*}"}
-{"id": "7514.png", "formula": "\\begin{gather*} [ P ^ t , D ] = 2 P ^ t , [ D , \\Pi ] = 2 \\Pi , [ P ^ t , \\Pi ] = D , \\\\ [ P ^ x , D ] = P ^ x , [ P ^ y , D ] = P ^ y , [ P ^ x , \\Pi ] = G ^ x , [ P ^ y , \\Pi ] = G ^ y , \\\\ [ P ^ t , G ^ x ] = P ^ x , [ P ^ t , G ^ y ] = P ^ y , [ D , G ^ x ] = G ^ x , [ D , G ^ y ] = G ^ y , \\\\ [ P ^ x , J ] = P ^ y , [ P ^ y , J ] = - P ^ x , [ G ^ x , J ] = G ^ y , [ G ^ y , J ] = - G ^ x . \\end{gather*}"}
-{"id": "2964.png", "formula": "\\begin{align*} R _ k & = \\mathtt { I } ( x _ k ; y _ { _ k } ) \\\\ & = \\log \\bigg ( 1 + \\frac { \\eta _ k p _ k | h _ { k k } | ^ { 2 } } { \\sigma ^ { 2 } _ k + \\eta _ k ( \\varrho ^ { 2 } _ k + \\sum _ { \\substack { j = 1 \\\\ j \\neq k } } ^ { K } p _ j | h _ { k j } | ^ { 2 } ) } \\bigg ) , \\ \\forall k \\in \\mathcal { K } , \\\\ E _ k & = ( 1 - \\eta _ k ) \\bigg ( \\sum _ { j = 1 } ^ { K } p _ j | h _ { k j } | ^ { 2 } + \\varrho ^ { 2 } _ k \\bigg ) , \\forall k \\in \\mathcal { K } , \\end{align*}"}
-{"id": "1527.png", "formula": "\\begin{align*} \\displaystyle r = \\begin{cases} \\lfloor \\alpha _ { m , t } ( 3 ) \\rfloor , & \\ \\ \\{ \\alpha _ { m , t } ( 3 ) \\} < 1 / 2 ; \\\\ \\lfloor \\alpha _ { m , t } ( 3 ) \\rfloor \\ \\ \\lfloor \\alpha _ { m , t } ( 3 ) \\rfloor + 1 , & \\ \\ \\{ \\alpha _ { m , t } ( 3 ) \\} = 1 / 2 ; \\\\ \\lfloor \\alpha _ { m , t } ( 3 ) \\rfloor + 1 , & \\ \\ \\{ \\alpha _ { m , t } ( 3 ) \\} > 1 / 2 . \\end{cases} \\end{align*}"}
-{"id": "4493.png", "formula": "\\begin{align*} \\xi = \\xi _ 1 e _ 1 + \\xi _ 2 e _ 2 \\end{align*}"}
-{"id": "439.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq \\abs { \\alpha } \\leq n + k _ 1 - 1 } O \\left ( I _ { n + k _ 1 - 1 - \\alpha _ 2 } ( \\kappa ) \\kappa ^ { \\alpha _ 1 } \\delta ^ { \\abs { \\alpha } } \\right ) & = \\sum _ { 1 \\leq \\abs { \\alpha } \\leq n + k _ 1 - 1 } O \\left ( \\kappa ^ { n + k _ 1 - 2 + 2 \\alpha _ 1 } \\delta \\right ) \\\\ & = O \\left ( \\kappa ^ { n + k _ 1 - 2 } \\delta \\right ) . \\end{align*}"}
-{"id": "410.png", "formula": "\\begin{align*} d ( x , t ) \\coloneqq \\begin{cases} \\abs { x } \\frac { y _ \\omega } { \\sin ( y _ \\omega ) } & \\\\ \\abs { x } & \\\\ \\sqrt { 4 \\pi \\abs { t } } & \\end{cases} \\end{align*}"}
-{"id": "2521.png", "formula": "\\begin{align*} T _ 2 = - T _ { 4 3 } , T _ 3 = - T _ { 4 4 } , \\end{align*}"}
-{"id": "3290.png", "formula": "\\begin{align*} \\log ( 8 \\mu _ j ^ 2 ) = - 2 \\log ( 2 m _ j ^ 2 ) + 8 \\pi H ( \\xi _ j , \\xi _ j ) + 8 \\pi \\sum _ { i = 1 , i \\ne j } ^ k m _ i m _ j ^ { - 1 } G ( \\xi _ i , \\xi _ j ) , \\quad j = 1 , \\dots , k \\end{align*}"}
-{"id": "5389.png", "formula": "\\begin{align*} a \\cdot x ^ 3 & = ( b \\cdot c ^ { a c } ) ^ 3 \\\\ b \\cdot x ^ 3 & = ( ( a c ) ^ 3 ) ^ { b c a c a c } \\\\ a b \\cdot x ^ 3 & = c ^ { a c a b c a c a c } . \\end{align*}"}
-{"id": "6629.png", "formula": "\\begin{align*} [ f _ n - g _ n ] _ { W ^ { 1 , p } , \\varphi _ n ( K _ n ) } & \\leq [ f _ n - g _ n ] _ { W ^ { 1 , p } , \\varphi _ n ( \\mathsf { U } _ n ) } \\\\ & = \\left ( \\sum _ { B \\in \\mathcal { B } _ n } \\int _ { \\varphi _ n ( B ) } | D f _ n - D g _ n | ^ p \\ , d \\mu \\right ) ^ { \\frac { 1 } { p } } \\\\ & \\leq \\sum _ { B \\in \\mathcal { B } _ n } \\left ( \\int _ { \\varphi _ n ( B ) } | D f _ n - D g _ n | ^ p \\ , d \\mu \\right ) ^ { \\frac { 1 } { p } } \\ . \\end{align*}"}
-{"id": "9950.png", "formula": "\\begin{align*} { \\cal N } ( u ) - V _ { 1 , j } = { \\cal N } ( u ) - V _ 1 = { \\cal N } ( \\varphi ( u ) ) - V _ 2 = { \\cal N } ( \\varphi _ j ( u ) ) - V _ { 2 , j } \\end{align*}"}
-{"id": "4384.png", "formula": "\\begin{align*} S _ \\gamma : = M _ { \\chi _ { B ( 0 , R ) } } \\Big ( C _ { z _ \\gamma } ( A P _ \\alpha + Q _ \\alpha ) C _ { - z _ \\gamma } - ( A _ { x } P _ \\alpha + Q _ \\alpha ) \\Big ) ( A _ { x } ^ { - 1 } P _ \\alpha + Q _ \\alpha ) \\end{align*}"}
-{"id": "6823.png", "formula": "\\begin{align*} \\langle L ( \\phi ) , \\eta _ { R _ 3 , \\xi _ j } \\varphi _ { 0 , j } \\rangle = \\langle \\phi , L ( \\eta _ { R _ 3 , \\xi _ j } \\varphi _ { 0 , j } ) \\rangle . \\end{align*}"}
-{"id": "811.png", "formula": "\\begin{align*} v _ { 2 } ( T _ { k } ( x ) ) = v _ { 2 } ( Q _ { k } ( x ) ) \\end{align*}"}
-{"id": "380.png", "formula": "\\begin{align*} \\sum _ { s \\in \\mathbb { Z } , r < - 2 n } | b _ { n , r , s } | ^ p = O \\big ( n ^ { p ( 2 - \\beta ) + 2 } L ^ p ( n ) \\big ) . \\end{align*}"}
-{"id": "7798.png", "formula": "\\begin{align*} \\left \\Vert f \\right \\Vert _ { p , \\gamma } = \\underset { G \\in L _ { p ^ { \\prime } , \\gamma ^ { \\prime } } } { \\sup } \\left \\{ \\left \\vert \\langle f , G \\rangle \\right \\vert : \\left \\Vert G \\right \\Vert _ { p ^ { \\prime } , \\gamma ^ { \\prime } } \\leq 1 \\right \\} . \\end{align*}"}
-{"id": "3572.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { \\left ( \\chi _ 0 ( n ) - \\chi ( n ) \\right ) \\Lambda ( n ) } { n ^ 2 } = \\sum _ { n \\geq 1 } \\frac { \\chi _ 0 ( n ) \\Lambda ( n ) } { n ^ 2 } - \\sum _ { n \\geq 1 } \\frac { \\chi ( n ) \\Lambda ( n ) } { n ^ 2 } . \\end{align*}"}
-{"id": "9911.png", "formula": "\\begin{align*} \\tfrac 1 2 \\alpha _ 1 ^ 2 + \\tfrac 1 2 \\alpha _ 2 ^ 2 + \\tfrac 1 2 ( \\alpha _ 1 + \\alpha _ 2 ) ^ 2 + \\sum _ { i = 3 } ^ { r } \\alpha _ i ^ 2 \\le \\tfrac { 3 r - 5 } { 3 r - 2 } - \\Bigl ( 2 \\sum _ { i < j } a _ i a _ j - a _ 1 a _ 2 \\Bigr ) \\ , , \\end{align*}"}
-{"id": "846.png", "formula": "\\begin{align*} E _ s : = T - \\bar E _ { T - s } \\end{align*}"}
-{"id": "646.png", "formula": "\\begin{align*} | \\bar \\nabla _ { \\dot \\gamma } \\bar \\nabla _ { \\dot \\gamma } \\bar \\nabla _ J ^ k J | & \\le C ^ { l o c } _ { k } d ^ 2 \\widehat P _ { k - 1 } ( | \\partial _ \\tau p | , \\cdots , | D _ \\tau ^ { k - 1 } \\partial _ \\tau p | ) \\\\ & + \\widehat Q _ { k - 1 } ( | P \\partial _ \\tau \\tilde p - \\partial _ \\tau p | , \\cdots , | P D _ \\tau ^ { k - 1 } \\partial _ \\tau \\tilde p - D _ \\tau ^ { k - 1 } \\partial _ \\tau p | ) + B _ 0 d ^ 2 | \\bar \\nabla _ J ^ k J | , \\end{align*}"}
-{"id": "8511.png", "formula": "\\begin{align*} E _ d : \\ ; d y ^ 2 = f ( x ) . \\end{align*}"}
-{"id": "4333.png", "formula": "\\begin{align*} T _ f = P _ \\alpha M _ f . \\end{align*}"}
-{"id": "5206.png", "formula": "\\begin{align*} S _ { 1 } ' ( \\ell _ { * } ) = \\psi _ { 1 } ' ( \\psi _ { 2 } ( \\ell _ { * } ) ) \\psi _ { 2 } ' ( \\ell _ { * } ) & = \\psi _ { 1 } ' ( r _ { * } ) \\psi _ { 2 } ' ( \\ell _ { * } ) \\\\ & = \\left ( \\frac { g _ { 1 } ' ( r _ { * } ) - f _ { 1 } ' ( \\ell _ { * } ) } { f _ { 1 } '' ( \\ell _ { * } ) ( r _ { * } - \\ell _ { * } ) } \\right ) \\left ( \\frac { f _ { 2 } ' ( r _ { * } ) - g _ { 2 } ' ( \\ell _ { * } ) } { f _ { 2 } '' ( r _ { * } ) ( r _ { * } - \\ell _ { * } ) } \\right ) \\\\ & = \\psi _ { 2 } ' ( \\psi _ { 1 } ( r _ { * } ) ) \\psi _ { 1 } ' ( r _ { * } ) \\\\ & = S _ { 2 } ' ( r _ { * } ) . \\end{align*}"}
-{"id": "6655.png", "formula": "\\begin{align*} | \\{ y \\in K : \\langle y , u ( x ) \\rangle \\geq \\langle x , u ( x ) \\rangle - \\Delta _ x ( \\delta ) \\} | _ n = \\delta | K | _ n \\quad . \\end{align*}"}
-{"id": "9786.png", "formula": "\\begin{align*} h _ 1 \\cdots h _ { m - \\ell } \\otimes y _ 1 \\wedge \\cdots \\wedge y _ \\ell \\mapsto \\frac { 1 } { ( 2 \\pi \\i ) ^ \\ell } \\left ( \\prod _ { i = 1 } ^ { m - \\ell } h _ i \\right ) \\cdot \\frac { d y _ 1 } { y _ 1 } \\wedge \\cdots \\wedge \\frac { d y _ \\ell } { y _ \\ell } \\end{align*}"}
-{"id": "6304.png", "formula": "\\begin{align*} \\partial _ t L & = [ A , L ] & \\mbox { w h e r e } & & L & = - \\partial _ x ^ 2 + u & \\mbox { a n d } & & A & = 4 \\partial _ x ^ 3 - 3 u \\partial _ x - 3 \\partial _ x u . \\end{align*}"}
-{"id": "7757.png", "formula": "\\begin{align*} \\Gamma : = \\{ x \\in \\R ^ 2 ~ | ~ x _ 2 = \\hat { x } _ 2 , ~ x _ 1 \\geq \\hat { x } _ 1 \\} \\end{align*}"}
-{"id": "6005.png", "formula": "\\begin{align*} C _ { \\mathsf { W y n e r } } ( X ; Y ) = \\min _ { P _ { X Y W } : \\ , P _ { X Y } = \\pi _ { X Y } , \\ , X - W - Y } I ( X Y ; W ) . \\end{align*}"}
-{"id": "2923.png", "formula": "\\begin{align*} R = \\{ ( a , a ' ) \\in A ^ 2 \\colon a \\not = a ' , a , a ' \\in u \\mbox { f o r s o m e } u \\in U \\} . \\end{align*}"}
-{"id": "9691.png", "formula": "\\begin{align*} & \\omega _ { k , k + 1 } = \\arctan \\big ( \\frac { y _ { k + 1 } - y _ k } { x _ { k + 1 } - x _ k } \\big ) , \\omega _ k = \\omega _ { k , k + 1 } - \\omega _ { k - 1 , k } , \\omega _ { - 1 , 0 } = 0 , \\\\ & g _ { k , h } ( x ) = y _ k + ( x - x _ k ) \\tan ( \\omega _ { k , k + 1 } ) , x \\in [ x _ k , x _ { k + 1 } ) , \\\\ & \\Omega _ { k , h } = \\{ ( x , y ) : x \\in [ x _ k , x _ { k + 1 } ) , y < g _ { k , h } ( x ) \\} , \\Omega _ { h } = \\bigcup _ { k \\ge 0 } \\Omega _ { k , h } , \\\\ & \\Gamma _ { k } = \\{ ( x , y ) : x \\in [ x _ k , x _ { k + 1 } ) , y = g _ { k , h } ( x ) \\} . \\end{align*}"}
-{"id": "2901.png", "formula": "\\begin{align*} \\lim _ { s _ 2 } \\cdots \\lim _ { s _ { n - 1 } } c ( k , s _ 1 , s _ 2 , \\ldots , s _ { n - 1 } ) = 0 , \\end{align*}"}
-{"id": "4755.png", "formula": "\\begin{align*} { \\varepsilon } = F ( x ^ 0 , x ^ 1 , Z ) \\ , \\Delta \\sqrt { - \\det g ^ { i j } } , \\end{align*}"}
-{"id": "2835.png", "formula": "\\begin{align*} \\omega [ - k _ n + 1 , l _ n - 1 ] = \\omega _ n [ - k _ n + 1 , l _ n - 1 ] \\end{align*}"}
-{"id": "8816.png", "formula": "\\begin{align*} Z ( p , \\chi _ { } , 0 ) & = p ^ { - n } \\sum _ { I \\subset T } c _ { I , \\chi _ { } } ^ { 0 } \\prod _ { i \\in I } \\frac { ( p - 1 ) p ^ { - \\nu _ { i } } } { 1 - p ^ { - \\nu _ { i } } } ; \\\\ Z ( p , \\chi _ { } , s ) & = p ^ { - n } \\sum _ { I \\subset T } c _ { I , \\chi _ { } } ^ { 0 } \\prod _ { i \\in I } \\frac { ( p - 1 ) t ^ { N _ { i } } p ^ { - \\nu _ { i } } } { 1 - t ^ { N _ { i } } p ^ { - \\nu _ { i } } } . \\end{align*}"}
-{"id": "534.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial s } + J ( u ) \\left ( \\frac { \\partial u } { \\partial t } - X _ { H _ t } ( u ) \\right ) = 0 , \\end{align*}"}
-{"id": "2702.png", "formula": "\\begin{align*} E ( C ) : = \\mathrm { E n d } ( C ) \\otimes _ { \\Z } { \\Q } \\end{align*}"}
-{"id": "6076.png", "formula": "\\begin{align*} \\tfrac { d } { d t } \\left ( \\begin{smallmatrix} \\Psi \\\\ \\Psi ' \\end{smallmatrix} \\right ) = \\left ( \\begin{smallmatrix} \\Psi ' \\\\ - ( n - 2 ) \\Psi ' + \\frak { e } _ k \\sin ( 2 \\Psi ) \\end{smallmatrix} \\right ) . \\end{align*}"}
-{"id": "4429.png", "formula": "\\begin{align*} ( f _ t ) _ T = f _ { t + T } \\quad \\mbox { f o r a l l } \\ ; t , T > 0 \\end{align*}"}
-{"id": "879.png", "formula": "\\begin{align*} W ^ { ( i ) } _ j = \\left \\{ \\begin{array} { l l } Y _ j & j \\leq i , \\\\ G _ j & j > i . \\end{array} \\right . \\end{align*}"}
-{"id": "9081.png", "formula": "\\begin{align*} f _ { 0 } ( p ) = p ^ { f } ( x ) = \\sum _ { k = 0 } ^ { n } f ( a _ { k } ) x ^ { k } . \\end{align*}"}
-{"id": "9732.png", "formula": "\\begin{align*} \\begin{cases} & \\tilde { \\gamma } _ i = \\gamma _ i + O ( 1 ) Z _ { a } h + O ( 1 ) | \\gamma _ 4 | h , \\ , \\ , i = 1 , 5 , \\\\ & \\tilde { \\sigma } _ j = \\sigma _ j + O ( 1 ) Z _ { a } h + O ( 1 ) | \\gamma _ 4 | h , \\ , \\ , j = 2 , 3 , \\\\ & \\tilde { \\gamma } _ 4 = ( 1 - { \\phi ( T _ b ) h } / { u _ b } ) \\gamma _ 4 + O ( 1 ) Z _ { a } h . \\end{cases} \\end{align*}"}
-{"id": "339.png", "formula": "\\begin{align*} \\rho _ 0 : = \\sum _ { i \\in I _ 0 } \\sigma _ i - \\sum _ { j \\in J _ 0 } \\tau _ j , \\rho _ 0 ^ + : = \\sum _ { i \\in I _ 0 ^ + } \\sigma _ i - \\sum _ { j \\in J _ 0 ^ + } \\tau _ j , \\rho _ 0 = - \\rho _ 0 ^ + , \\end{align*}"}
-{"id": "7619.png", "formula": "\\begin{align*} & \\mu \\Big ( E _ f ( Q _ { \\rho _ { m + 1 } } , c _ 0 \\lambda _ 0 M ^ { m + 1 } ) \\Big ) \\\\ & \\leq \\alpha \\left [ \\mu \\Big ( E _ f ( Q _ { \\rho _ m } , c _ 0 \\lambda _ 0 M ^ m ) \\Big ) + \\nu \\Big ( E _ g ( Q _ { \\rho _ m } , c _ 0 \\lambda _ 0 M ^ m ) \\Big ) + \\hat \\nu \\Big ( E _ { \\hat g } ( Q _ { \\rho _ m } , c _ 0 \\lambda _ 0 M ^ m ) \\Big ) ^ { \\hat p } \\right ] \\forall m = 0 , 1 , \\dots \\end{align*}"}
-{"id": "2090.png", "formula": "\\begin{align*} \\sup _ { M \\times [ 0 , T _ 0 ] } \\sum _ a \\big ( | u ^ a _ k - u ^ a _ l | + | \\nabla _ H u ^ a _ k - \\nabla _ H u ^ a _ l | \\big ) \\leq K ^ 2 C _ 3 C _ \\beta e ^ { 2 C _ 1 } | | d \\phi | | ^ 2 _ { C ^ 0 } \\sum _ { i = k + 1 } ^ l \\bigg ( \\frac { 1 } { 2 } \\bigg ) ^ { i - 2 } . \\end{align*}"}
-{"id": "5625.png", "formula": "\\begin{align*} ( \\phi , \\psi ) \\hat { \\otimes } ( y , y ^ { n - 1 } ) = \\left ( \\begin{bmatrix} \\phi & y I _ m \\\\ - y ^ { n - 1 } I _ m & \\psi \\end{bmatrix} , \\begin{bmatrix} \\psi & - y I _ m \\\\ y ^ { n - 1 } I _ m & \\phi \\end{bmatrix} \\right ) \\ , . \\end{align*}"}
-{"id": "6141.png", "formula": "\\begin{align*} \\mathbb { P } _ { c o u p l e } \\left ( n ^ { - \\frac { 1 } { \\alpha } } \\sum _ { i = 1 } ^ { M _ { 1 } \\left ( n \\right ) } Z > n ^ { - c ' } \\right ) & \\leq n ^ { \\left ( c ' - \\frac { 1 } { \\alpha } \\right ) \\xi } n c ' \\log n \\mathbb { E } \\left ( Z ^ { \\xi } \\right ) \\\\ & + o \\left ( n ^ { - c } \\right ) . \\end{align*}"}
-{"id": "4895.png", "formula": "\\begin{align*} X ^ * X = \\begin{bmatrix} x _ 1 & 0 \\\\ 0 & x _ 2 \\end{bmatrix} , \\ \\ \\ \\ \\ Y ^ * Y = \\begin{bmatrix} y _ 1 & 0 \\\\ 0 & y _ 2 \\end{bmatrix} , \\end{align*}"}
-{"id": "6876.png", "formula": "\\begin{align*} \\Phi _ i ( \\lambda _ { n , p } ^ { ( m - 1 ) } ) & = \\lambda _ { n + ( i - 1 ) ( s + 1 ) , p } ^ { ( m ) } \\\\ & = \\lambda _ { n + ( i - 1 ) ( s + 1 ) , q } ^ { ( m ) } \\end{align*}"}
-{"id": "178.png", "formula": "\\begin{align*} \\mathbf { R } ^ { [ 0 , 1 ] ^ { \\mathit { F m } _ { \\mathcal { L } } } } = \\langle [ 0 , 1 ] ^ { \\mathit { F m } _ { \\mathcal { L } } } , \\leq , \\vee , \\emptyset \\rangle , \\end{align*}"}
-{"id": "8926.png", "formula": "\\begin{gather*} \\begin{pmatrix} \\phi _ { d _ 1 , d _ 0 } & - b ( d _ 2 / d _ 0 ) \\phi _ { d _ 1 , d _ 3 } \\\\ - \\phi _ { d _ 2 , d _ 0 } & a ( d _ 1 / d _ 0 ) \\phi _ { d _ 2 , d _ 3 } \\end{pmatrix} \\ ! \\colon \\ { \\cal E } _ { d _ 0 } \\times { \\cal E } _ { d _ 3 } \\to { \\cal E } _ { d _ 1 } \\times { \\cal E } _ { d _ 2 } \\end{gather*}"}
-{"id": "6400.png", "formula": "\\begin{align*} P _ { } \\left ( \\theta \\right ) = P _ { } \\left ( \\theta \\left \\vert x ^ { \\prime } \\right . \\right ) \\end{align*}"}
-{"id": "7476.png", "formula": "\\begin{align*} L ^ { \\ : \\sigma } _ { \\alpha \\gamma } = L ^ { \\ : \\sigma } _ { \\gamma \\alpha } , \\end{align*}"}
-{"id": "7887.png", "formula": "\\begin{align*} 2 u ( m ( \\hat { x } , \\hat { y } ) , \\hat { t } ) - u ( \\hat { x } , \\hat { t } ) - u ( \\hat { y } , \\hat { t } ) = : 2 \\eta > 0 \\end{align*}"}
-{"id": "7838.png", "formula": "\\begin{align*} f ( x _ { i } ) = \\alpha ( \\theta _ { i } + \\theta _ { 3 } ) ( a + b x _ { i } ) \\eta ^ { \\alpha - 1 } ( x _ { i } ) e ^ { - \\eta ^ { \\alpha } ( x _ { i } ) } \\left ( 1 - e ^ { - \\eta ^ { \\alpha } ( x _ { i } ) } \\right ) ^ { \\theta _ { i } + \\theta _ { 3 } - 1 } , \\ \\ i = 1 , 2 . \\end{align*}"}
-{"id": "3502.png", "formula": "\\begin{align*} \\dot { X } ^ u ( s ) = b ( X ^ u { ( s ) } , u ( s ) ) , s \\in ( 0 , T ] , \\end{align*}"}
-{"id": "2062.png", "formula": "\\begin{align*} \\frac { \\partial f } { \\partial t } = \\tau ( f ) \\end{align*}"}
-{"id": "7144.png", "formula": "\\begin{align*} \\Delta _ 1 ( m ) = L ( q ( m + 1 ) ) - L ( q ( m ) ) , \\end{align*}"}
-{"id": "604.png", "formula": "\\begin{align*} ( M : T _ q ( m ) ) = ( M : \\Delta _ q ( m ) ) m . \\end{align*}"}
-{"id": "9070.png", "formula": "\\begin{align*} 0 \\rightarrow \\Omega _ X \\otimes \\omega _ X \\rightarrow \\Omega _ X ( l o g D _ 1 , l o g D _ 2 , l o g D _ 3 ) \\otimes \\omega _ X \\rightarrow \\bigoplus _ { i = 1 } ^ 3 \\mathcal { O } _ { D _ i } ( K _ X ) \\rightarrow 0 . \\end{align*}"}
-{"id": "3481.png", "formula": "\\begin{align*} u ( t , x ) = ( x ^ 2 - x ^ 3 ) w ( t ) + \\pi ^ { - 2 } \\sin ( \\pi x ) \\end{align*}"}
-{"id": "9076.png", "formula": "\\begin{align*} f ( x y ) = f ( x ) y + x f ( y ) \\left ( x , y \\in P \\right ) . \\end{align*}"}
-{"id": "3982.png", "formula": "\\begin{align*} p ^ { \\beta _ n } ( n , t ) = p ^ { \\beta _ n } ( n , 0 ) - \\lambda ( I _ t ^ { \\beta _ n } p ^ { \\beta _ n } ( n , t ) - I _ t ^ { \\beta _ { n - 1 } } p ^ { \\beta _ { n - 1 } } ( n - 1 , t ) ) , \\ \\ 0 < \\beta _ n \\leq 1 , \\ \\lambda > 0 , \\ n \\geq 0 , \\end{align*}"}
-{"id": "5423.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } ( - \\Delta + q ) h = v & \\mbox { i n } \\ ; \\Omega , \\\\ h ( x ) = v _ - ( x ) , & x \\in \\partial \\omega _ { - , \\theta } \\times \\R . \\end{array} \\right . \\end{align*}"}
-{"id": "7280.png", "formula": "\\begin{align*} T = \\log ^ 2 X . \\end{align*}"}
-{"id": "4996.png", "formula": "\\begin{align*} | \\partial ^ { \\gamma } \\partial _ { x ' } ( U _ { m } V _ { m } ) | \\lesssim | V _ m ( \\partial ^ { \\gamma } \\partial _ { x ' } U _ { m } ) | + \\sum _ { \\ell = 0 } ^ { a } | ( \\partial _ x ^ \\ell U _ m ) ( \\partial _ x ^ { a + 1 - \\ell } V _ { m } ) | . \\end{align*}"}
-{"id": "1322.png", "formula": "\\begin{align*} \\pi ( a ) \\xi = P _ { H _ \\gamma } \\pi ( a ) | _ { H _ \\gamma } \\xi = \\pi _ \\gamma ( a ) \\xi = P _ { H _ \\gamma } \\varphi ( a ) P _ { H _ \\gamma } \\xi = P _ { H _ \\gamma } \\varphi ( a ) \\xi . \\end{align*}"}
-{"id": "6033.png", "formula": "\\begin{align*} \\lambda = \\frac { \\theta } { 1 + 2 \\bar { \\alpha } \\theta } . \\end{align*}"}
-{"id": "1848.png", "formula": "\\begin{align*} S ^ 0 ( B ) \\otimes ^ . F _ \\bullet = \\cdots \\to S ^ 0 ( B ) \\otimes ^ . F _ 1 \\to S ^ 0 ( B ) \\otimes ^ . F _ 0 \\to S ^ 0 ( B ) \\otimes ^ . F _ { - 1 } \\to \\cdots , \\end{align*}"}
-{"id": "9165.png", "formula": "\\begin{align*} \\Phi ( x _ { 1 } , \\ldots , x _ { n + 1 } ) = \\sum _ { i = 0 } ^ { n } \\dfrac { 1 } { \\binom { n + 1 } { i } } \\sum _ { \\mathrm { c a r d } ( I ) = i } \\left ( \\prod _ { j \\in I } x _ { j } \\right ) \\cdot f _ { n + 1 - i } \\left ( \\prod _ { k \\in \\left \\{ 1 , \\ldots , n + 1 \\right \\} \\setminus I } x _ { k } \\right ) \\left ( x _ { 1 } , \\ldots , x _ { n + 1 } \\in K \\right ) . \\end{align*}"}
-{"id": "7331.png", "formula": "\\begin{align*} R ( e _ A , e _ B ) ( e _ 1 \\times e _ 2 ) & = \\big ( R ( e _ A , e _ B ) e _ 1 \\big ) \\times e _ 2 + e _ 1 \\times \\big ( R ( e _ A , e _ B ) e _ 2 \\big ) ~ , \\end{align*}"}
-{"id": "3852.png", "formula": "\\begin{align*} I ^ = _ { g , 0 } ( x ^ * , \\lambda ^ * , d ) \\coloneqq \\left \\{ i \\in I _ { g 0 } ( x ^ * , \\lambda ^ * ) \\mid \\nabla g _ i ( x ^ * ) ^ T d _ x = 0 \\right \\} , \\\\ I ^ < _ { g , = } ( x ^ * , \\lambda ^ * , d ) \\coloneqq \\left \\{ i \\in I _ { g 0 } ( x ^ * , \\lambda ^ * ) \\mid \\nabla g _ i ( x ^ * ) ^ T d _ x < 0 \\right \\} . \\end{align*}"}
-{"id": "565.png", "formula": "\\begin{align*} \\begin{array} { l } \\beta _ 1 = - a i + b ( a + 1 - m - t _ 1 ) \\leq \\alpha _ 1 , \\\\ \\beta _ j = i + b ( t _ { j - 1 } - t _ j ) \\leq \\alpha _ j , \\mbox { f o r } j = 2 , \\ldots , m - 1 , \\\\ \\beta _ m = i + b t _ m \\leq \\alpha _ m . \\end{array} \\end{align*}"}
-{"id": "10006.png", "formula": "\\begin{align*} [ ( \\mathrm { E x c } , - \\log ( D ) ) : \\mathcal { Y } _ \\mathrm { s m } ] = 0 , \\end{align*}"}
-{"id": "4536.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n - 1 } \\phi ( F _ j ( x ) ) & = \\sum _ { j = 0 } ^ { n - 1 } \\phi \\circ h _ j \\circ f ^ j ( h _ 0 ^ { - 1 } ( x ) ) \\\\ & = \\sum _ { i = 0 } ^ { N - 1 } \\sum _ { \\ell = 0 } ^ { \\big [ \\frac { n } { N } \\big ] } \\sum _ { 0 \\le \\ell N + i \\le n - 1 } ( \\phi \\circ h _ i ) ( f ^ { \\ell N + i } ( y ) ) \\\\ & = \\sum _ { i = 0 } ^ { N - 1 } \\sum _ { \\ell = 0 } ^ { \\big [ \\frac { n } { N } \\big ] } \\sum _ { 0 \\le \\ell N + i \\le n - 1 } \\psi _ i ( f ^ { \\ell N } ( y ) ) \\end{align*}"}
-{"id": "6345.png", "formula": "\\begin{align*} W _ 1 ( z ) : = V _ 1 ( z ) - \\log z W _ 2 ( z ) : = V _ 2 ( z ) - \\log z \\end{align*}"}
-{"id": "4864.png", "formula": "\\begin{align*} \\log ( 1 + x ) = x - \\frac { 1 } { 2 } x ^ 2 \\quad ( 1 + x \\in G ^ m _ r ) . \\end{align*}"}
-{"id": "8814.png", "formula": "\\begin{align*} | E _ { p , m } ^ { 0 } ( f ) | = \\Big | \\sum _ { i = 1 } ^ { s } a _ { i , p } m ^ { \\beta _ { i } } p ^ { - \\lambda _ { i } m } \\ 1 1 _ { A _ { i } } ( m ) \\Big | \\leq s C _ { 0 } m ^ { n - 1 } p ^ { - m \\sigma } , \\end{align*}"}
-{"id": "5357.png", "formula": "\\begin{align*} ( W _ { \\lambda } ) _ { [ n + k ] } = 0 \\ ; \\ ; \\ ; \\mbox { f o r $ k \\in \\Z $ s u f f i c i e n t l y n e g a t i v e } ; \\end{align*}"}
-{"id": "1241.png", "formula": "\\begin{align*} \\vartheta = \\delta ^ { - 1 } \\nabla \\breve v _ i ( x + \\delta e _ l ) \\mbox { a n d } \\upsilon = \\delta ^ { - 1 } \\nabla \\breve v _ i ( x ) . \\end{align*}"}
-{"id": "7675.png", "formula": "\\begin{align*} B ^ { [ j ] } = \\left ( \\begin{array} { @ { } c | c @ { } } B _ j & 0 \\\\ \\hline 0 & 2 ^ { k _ j } u _ 1 \\end{array} \\right ) , B = \\left ( \\begin{array} { @ { } c | c @ { } } B ^ { [ j ] } & 0 \\\\ \\hline 0 & \\begin{matrix} 2 ^ { k _ j + 1 } u _ 2 & 0 \\\\ 0 & \\ddots \\end{matrix} \\end{array} \\right ) . \\end{align*}"}
-{"id": "3935.png", "formula": "\\begin{align*} \\frac { \\partial ^ \\gamma } { \\partial t ^ \\gamma } u ( x , t ) | _ { ( x _ j , t _ { n + 1 } ) } = \\frac { \\partial ^ \\gamma } { \\partial t ^ \\gamma } u _ j ^ { n + 1 } = \\frac { 1 } { \\Delta t ^ \\gamma \\Gamma [ 2 - \\gamma ] } \\sum \\limits _ { l = 0 } ^ { n } w _ l ( u _ j ^ { n - l + 1 } - u _ j ^ { n - l } ) , \\end{align*}"}
-{"id": "7980.png", "formula": "\\begin{align*} d \\exp _ { \\gamma _ 2 ( s ) } ( ( V _ 1 ) _ { \\dot \\sigma _ s ( 0 ) } ) = \\dot \\gamma _ 3 ( s ) , \\ , \\ , d \\exp _ { \\gamma _ 3 ( s ) } ( ( V _ 2 ) _ { \\dot \\sigma _ s ^ { - } ( 0 ) } ) = \\dot \\gamma _ 2 ( s ) . \\end{align*}"}
-{"id": "8471.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { t , l , v } ) = \\delta _ { \\Delta \\in F ^ { 2 \\times } } q ^ { \\frac { \\Delta } { 4 } } \\sum _ { \\pm } \\gamma _ F ( - 1 \\pm \\sqrt { \\Delta } , \\rho ) \\gamma _ F ( \\Delta \\pm \\sqrt { \\Delta } , l - \\frac { v ( \\Delta ) } { 2 } ) \\chi ^ 2 ( - \\frac { 1 } { 2 v } \\pm \\frac { \\sqrt { \\Delta } } { 2 v } ) \\psi ( \\varpi ^ { - \\frac { n } { 2 } } ( \\frac { \\Delta - 3 } { 4 v } ) ) , \\end{align*}"}
-{"id": "6239.png", "formula": "\\begin{align*} | x + y | ^ 2 & \\le ( | x | + | y | ) ^ 2 = | x | ^ 2 + | y | ^ 2 + 2 | x | . | y | \\\\ | x + y | ^ 2 & \\ge ( | x | - | y | ) ^ 2 = | x | ^ 2 + | y | ^ 2 - 2 | x | . | y | \\end{align*}"}
-{"id": "852.png", "formula": "\\begin{align*} \\partial _ { [ s , T ) } ^ { \\beta } h ( r , s ) = \\partial _ r h ( r , s ) . \\end{align*}"}
-{"id": "9141.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } a _ i x ^ { p _ i } f ( x ^ { q _ i } ) = 0 \\end{align*}"}
-{"id": "4257.png", "formula": "\\begin{align*} \\mathcal G _ { 1 2 } = \\left ( \\begin{array} { c c } 1 & g \\\\ 0 & \\frac { t ^ p } { ( t - 1 ) ^ p } \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "342.png", "formula": "\\begin{align*} R i c + \\mathrm { H e s s } ( f ) = \\mu g \\end{align*}"}
-{"id": "6998.png", "formula": "\\begin{align*} y = z _ 1 \\tilde { e } _ 1 + z _ 2 \\tilde { e } _ 2 + . . . + z _ { n - 1 } \\tilde { e } _ { n - 1 } , \\end{align*}"}
-{"id": "5163.png", "formula": "\\begin{align*} \\begin{cases} M ^ { x } _ { 1 } ( \\tau ^ { * } _ { 1 } , \\tau ^ { * } _ { 2 } ) = \\sup \\limits _ { \\tau _ { 1 } \\in \\mathcal { T } } M ^ { x } _ { 1 } ( \\tau _ { 1 } , \\tau ^ { * } _ { 2 } ) \\\\ M ^ { x } _ { 2 } ( \\tau ^ { * } _ { 1 } , \\tau ^ { * } _ { 2 } ) = \\sup \\limits _ { \\tau _ { 2 } \\in \\mathcal { T } } M ^ { x } _ { 2 } ( \\tau ^ { * } _ { 1 } , \\tau _ { 2 } ) . \\end{cases} \\end{align*}"}
-{"id": "9764.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k - 1 } \\int \\limits _ { y _ { i , s } } ^ { y _ { i } } G ( U _ { h , \\theta } ( i h - , y _ { i , n } ) ) h d y \\to \\int \\limits _ { 0 } ^ { x } \\int \\limits _ { \\chi ( \\tau ) } ^ { g ( \\tau ) } G ( U ( \\tau , y ) ) d y d \\tau , \\end{align*}"}
-{"id": "8531.png", "formula": "\\begin{align*} \\mathcal { T } ^ * _ { ( j , - l ) } \\mathcal { T } _ { ( k , - m ) } g ( \\xi ) = \\int \\mathcal { K } _ { ( j , - l ) , ( k , - m ) } ( \\xi , \\eta ) g ( \\eta ) d \\eta \\end{align*}"}
-{"id": "9255.png", "formula": "\\begin{align*} N ( u ) = \\{ u \\pm d _ 1 , u \\pm d _ 2 , \\ldots , u \\pm d _ k \\} \\end{align*}"}
-{"id": "8205.png", "formula": "\\begin{align*} K _ { } ( h , g ) = 0 \\end{align*}"}
-{"id": "1269.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\mbox { C a p } _ { \\mathcal { A } } ( \\hat E _ l ) = \\mbox { C a p } _ { \\mathcal { A } } ( \\hat E ) . \\end{align*}"}
-{"id": "9782.png", "formula": "\\begin{align*} L _ 2 = \\sum _ { m \\in \\Z } G _ m L ' ( - m \\{ 0 \\} ) \\end{align*}"}
-{"id": "6869.png", "formula": "\\begin{align*} - \\Delta _ { m + 1 } f _ { m + 1 } ( x ) = \\lambda _ { m + 1 } f _ { m + 1 } ( x ) \\forall \\ x \\in V _ { m + 1 } \\setminus V _ 0 . \\end{align*}"}
-{"id": "6182.png", "formula": "\\begin{align*} \\Omega _ r = \\{ ( y , x _ 1 , x _ 2 , x _ 3 ) \\in \\mathbb { R } ^ 4 | \\ ; \\left | y - \\bar y \\right | < r , \\left | x _ i - \\bar x _ i \\right | < r , i = 1 , 2 , 3 \\} . \\end{align*}"}
-{"id": "8521.png", "formula": "\\begin{align*} | | \\binom { \\alpha } { y } | | _ \\infty \\leq n \\Delta \\end{align*}"}
-{"id": "5090.png", "formula": "\\begin{align*} E _ { \\mathbb { R } ^ d } \\colon \\mathbb { R } ^ d \\times \\mathbb { R } ^ d \\to \\textup { S } ^ 1 , E _ { \\mathbb { R } ^ d } ( x , \\xi ) = e ^ { 2 \\pi i x \\cdot \\xi } , \\end{align*}"}
-{"id": "154.png", "formula": "\\begin{align*} \\begin{pmatrix} x & y \\\\ r y ^ { \\sigma } & x ^ { \\sigma } \\end{pmatrix} \\end{align*}"}
-{"id": "3925.png", "formula": "\\begin{align*} X : = \\sum _ { i = 1 } ^ m x _ i , \\hat { X } : = \\sum _ { i = 1 } ^ m ( - 1 ) ^ { i + 1 } x _ i . \\end{align*}"}
-{"id": "2589.png", "formula": "\\begin{align*} ( S f ) ( t ) : = \\int _ 0 ^ 1 \\frac { f ( s ) } { t - s } \\ , d s 0 < t < 1 . \\end{align*}"}
-{"id": "1513.png", "formula": "\\begin{align*} \\sum _ { l = 0 } ^ { n } Q _ { T , l } ( x ) = \\frac { 1 } { \\delta ( x ) } \\left ( Q _ { T , n + 2 } ( x ) + ( 1 - x ^ { 2 } ) Q _ { T , n + 1 } ( x ) + Q _ { T , n } ( x ) - \\omega ( x ) \\right ) , \\end{align*}"}
-{"id": "7095.png", "formula": "\\begin{align*} k ^ A _ { i + 1 / 2 } = \\frac { ( v - w ) + ( v + w ) } { 2 } = v = k ^ + + \\tau ^ A , \\end{align*}"}
-{"id": "3464.png", "formula": "\\begin{align*} \\mathbf { u _ h ^ n } \\colon \\Omega \\to \\R ^ d , \\omega \\mapsto \\mathbf { u _ h ^ n } ( \\omega ) = \\big ( \\alpha _ { h , i } ^ { n } ( \\omega ) \\big ) _ { i \\in \\{ 1 , \\dots , d \\} } \\end{align*}"}
-{"id": "5767.png", "formula": "\\begin{align*} - \\Delta _ { p } u = \\sigma u ^ { q } + \\mu \\ ; \\ ; \\mathbb { R } ^ n \\end{align*}"}
-{"id": "7071.png", "formula": "\\begin{align*} \\nabla _ u \\Phi ( u , \\lambda ) = 0 \\end{align*}"}
-{"id": "5068.png", "formula": "\\begin{align*} & B _ { 2 3 , 3 1 } = \\frac { 3 b _ 2 B _ { 3 3 , 1 } - 3 b _ 3 B _ { 2 2 , 1 } } { ( b _ 3 - b _ 2 ) ^ 2 } C _ 2 + \\frac { 3 b _ 3 } { b _ 2 - b _ 3 } [ C _ { 2 , 1 } - \\frac { B _ { 1 2 , 1 } } { b _ 1 - b _ 2 } C _ 1 ] + B _ { 3 1 , 3 } \\frac { B _ { 1 2 , 1 } } { b _ 1 - b _ 2 } , \\\\ & B _ { 2 3 , 1 3 } = ( B _ { 2 2 , 1 } - B _ { 3 3 , 1 } ) \\frac { B _ { 2 3 , 3 } } { b _ 2 - b _ 3 } + ( B _ { 1 1 , 2 } - B _ { 3 3 , 2 } ) \\frac { B _ { 1 3 , 3 } } { b _ 1 - b _ 3 } , \\end{align*}"}
-{"id": "6370.png", "formula": "\\begin{align*} \\mu _ { p ^ * _ 1 , p ^ * _ 2 } ( \\{ a \\} ) = \\frac { p _ 2 ^ * - p _ 1 ^ * + \\lvert p _ 2 ^ * - p _ 1 ^ * \\rvert } { 2 p _ 2 ^ * } = \\begin{cases} 0 , & \\quad p _ 1 ^ * \\geq p _ 2 ^ * \\\\ 1 - \\frac { p _ 1 ^ * } { p _ 2 ^ * } , & p _ 1 ^ * < p _ 2 ^ * \\end{cases} . \\end{align*}"}
-{"id": "4534.png", "formula": "\\begin{align*} | \\phi _ 0 ( y ) | < 1 = \\phi _ 0 ( x _ 0 ) \\end{align*}"}
-{"id": "8325.png", "formula": "\\begin{align*} \\mathrm { L i g h t C o n e } ( V _ { 0 \\R } ) = \\{ w \\in V _ { 0 \\R } : Q ( w ) < 0 \\} \\end{align*}"}
-{"id": "8294.png", "formula": "\\begin{align*} F ^ 2 V _ \\C = 0 , F ^ 1 V _ \\C = \\C z , F ^ 0 V _ \\C = ( \\C z ) ^ \\perp , F ^ { - 1 } V _ \\C = V _ \\C , \\end{align*}"}
-{"id": "1833.png", "formula": "\\begin{align*} m \\| f \\| ^ 2 & \\le \\left \\langle S _ { U \\Gamma \\Lambda T } f , f \\right \\rangle = \\sum _ { i \\in I } \\left \\langle \\Lambda _ i T f , \\Gamma _ i U f \\right \\rangle \\\\ & \\le \\big ( \\sum _ { i \\in I } \\| \\Lambda _ i T f \\| ^ 2 \\big ) ^ { 1 / 2 } \\big ( \\sum _ { i \\in I } \\| \\Gamma _ i U f \\| ^ 2 \\big ) ^ { 1 / 2 } \\\\ & \\le \\sqrt { B _ U } \\| f \\| \\big ( \\sum _ { i \\in I } \\| \\Lambda _ i T f \\| ^ 2 \\big ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "2311.png", "formula": "\\begin{align*} & \\ ; \\| A ^ { ( r _ k - s ) / 2 } u _ k \\| _ { L ^ 2 } ^ 2 \\\\ \\leq & \\ ; C \\left ( \\| A ^ { ( r _ k - s ) / 2 } e ^ { - \\delta \\nu A ^ s } u _ { k - 1 } \\| _ { L ^ 2 } ^ 2 + \\| A ^ { ( r _ k - s ) / 2 } w _ { k - 1 } ( t _ k ) \\| _ { L ^ 2 } ^ 2 \\right ) \\\\ \\leq & \\ ; C \\left [ ( \\nu \\delta ) ^ { - \\frac { 2 s - 1 } { s } } \\| A ^ { ( r _ { k - 1 } - s ) / 2 } u _ { k - 1 } \\| _ { L ^ 2 } ^ 2 + \\| A ^ { ( r _ k - s ) / 2 } w _ { k - 1 } ( t _ k ) \\| _ { L ^ 2 } ^ 2 \\right ] . \\end{align*}"}
-{"id": "8892.png", "formula": "\\begin{align*} \\begin{aligned} | \\kappa _ n | ^ 2 = & | \\kappa _ { - n } | ^ 2 + 2 \\operatorname { R e } \\bigl ( \\theta \\overline \\kappa _ { - n } \\overline { \\widehat \\mu ( - n ) } ( 1 - \\overline \\theta ) \\bigr ) + | 1 - \\theta | ^ 2 | \\widehat \\mu ( - n ) | ^ 2 \\\\ & \\ \\ \\mathbb T \\ m ) , \\ \\ \\ n \\in \\mathbb N . \\end{aligned} \\end{align*}"}
-{"id": "209.png", "formula": "\\begin{align*} F / \\theta = \\{ [ a _ { 1 } / \\theta , \\dots , a _ { n } / \\theta ] : [ a _ { 1 } , \\dots , a _ { n } ] \\in F \\} . \\end{align*}"}
-{"id": "5757.png", "formula": "\\begin{align*} \\alpha ( x , y ) \\ast \\lambda _ { a , b } ( x , y ) = \\alpha ( a , b ) \\ast \\lambda _ { a , b } ( x , y ) . \\end{align*}"}
-{"id": "5720.png", "formula": "\\begin{align*} M ( { v } ) : = \\left \\{ m \\in \\R \\ ; : \\ ; \\| { v } - z ^ \\pm ( \\cdot - m ) \\| _ { L ^ 2 } = d _ { L ^ 2 } ( { v } , \\mathcal { C } ( z ^ + ) \\cup \\mathcal { C } ( z ^ - ) ) \\right \\} , \\end{align*}"}
-{"id": "2323.png", "formula": "\\begin{align*} \\begin{pmatrix} J _ 1 & \\\\ & J _ 2 & \\\\ & & \\ddots \\\\ & & & J _ { r } \\end{pmatrix} . \\end{align*}"}
-{"id": "6106.png", "formula": "\\begin{align*} \\frac { 1 - f ^ { \\psi } \\left ( s \\right ) } { \\psi \\left ( s \\right ) } = \\frac { 1 - f ^ { \\psi } \\left ( s \\right ) } { s } \\frac { s } { \\psi \\left ( s \\right ) } , \\end{align*}"}
-{"id": "1076.png", "formula": "\\begin{align*} \\int u _ k ( \\mathbf { r } , z ) u _ { k ' } ^ * ( \\mathbf { r } , z ) \\mathrm { d } \\mathbf { r } = \\begin{cases} 1 , & k = k ' \\\\ 0 , & k \\neq k ' \\end{cases} . \\end{align*}"}
-{"id": "9408.png", "formula": "\\begin{align*} \\operatorname { E } \\bigl [ f ( \\eta ) \\bigr ] = \\operatorname { E } \\biggl [ \\operatorname { E } \\bigl [ f ( \\eta ^ { [ \\lambda ] } ( Q _ L , \\eta ) ) \\bigr ] \\biggr ] . \\end{align*}"}
-{"id": "3411.png", "formula": "\\begin{align*} I _ n = [ 1 , 1 + 1 / n ] \\cup [ - 1 - 1 / n , - 1 ] . \\end{align*}"}
-{"id": "1277.png", "formula": "\\begin{align*} \\lim _ { \\tau \\to 0 } \\mbox { C a p } _ { \\mathcal { A } } ( E _ 1 + \\tau E _ 2 ) = \\mbox { C a p } _ { \\mathcal { A } } ( E _ 1 ) = 1 . \\end{align*}"}
-{"id": "3851.png", "formula": "\\begin{align*} \\nabla f ( x ^ * ) + \\sum _ { i = 1 } ^ m \\lambda ^ * _ i \\nabla g _ i ( x ^ * ) + \\sum _ { i = 1 } ^ p \\mu ^ * _ i \\nabla h _ i ( x ^ * ) + \\sum _ { i = 1 } ^ n \\gamma ^ * _ i e _ i & = 0 , \\\\ \\lambda ^ * _ i \\geq 0 , \\lambda ^ * _ i \\cdot g _ i ( x ^ * ) & = 0 \\forall i = 1 , \\dots , m , \\\\ \\gamma ^ * _ i & = 0 \\forall i \\in I _ { \\pm 0 } ( x ^ * , y ^ * ) . \\end{align*}"}
-{"id": "4706.png", "formula": "\\begin{align*} I _ 0 ( s ) & \\lesssim \\int \\limits _ { \\R ^ d } \\biggl ( \\int \\limits _ { \\R ^ d } \\prod _ { i = 1 } ^ d \\left [ ( 1 + | x _ i + t _ i | ) ^ { - k } ( 1 + | ( S _ { - s } x ) _ i | ) ^ { - k } \\right ] \\ , \\mathrm { d } x \\biggr ) \\ , \\mathrm { d } t = : \\int \\limits _ { \\R ^ 3 } I _ s ( t ) ^ { q _ 0 } \\ , \\mathrm { d } t \\end{align*}"}
-{"id": "7751.png", "formula": "\\begin{align*} \\left ( [ u _ 1 ( \\vec { x } ) ^ N , u _ 2 ( \\vec { x } ) ^ N ] \\neq 1 \\right ) \\wedge \\left ( \\bigwedge _ { j = 1 } ^ a r _ j ( \\vec { x } ) = 1 \\right ) \\wedge \\left ( \\bigwedge _ { i \\leq b } w _ i ( \\vec { x } ) \\neq 1 ) \\right ) . \\end{align*}"}
-{"id": "3655.png", "formula": "\\begin{align*} \\Phi ' _ W [ n ] : T _ W [ n ] : = ( \\mathbb C ^ * ) ^ { 2 n + 1 } \\to ( ( \\mathbb A ^ 2 ) ^ n \\times \\mathbb A ^ { n + 1 } ) \\times ( ( \\mathbb P ^ 1 ) ^ n ) ^ n ) \\end{align*}"}
-{"id": "10056.png", "formula": "\\begin{align*} a _ F ( 0 , \\mu ) = \\begin{cases} - 2 \\Lambda ' ( 0 , \\chi _ E ) & \\mbox { i f $ \\mu = 0 $ } \\\\ 0 & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "9388.png", "formula": "\\begin{align*} T _ \\varepsilon f ( x ) = \\int _ { | y | > \\varepsilon } \\frac { \\Omega ( y ' ) } { | y | ^ n } f ( x - y ) d y , \\end{align*}"}
-{"id": "1296.png", "formula": "\\begin{align*} \\textstyle s _ k \\ = \\ \\frac { 1 } { 3 } \\left ( \\left ( \\frac { 3 + B } { 2 } \\right ) \\ + \\ \\sqrt { 2 7 + B ^ 2 } \\ \\cos \\left ( \\frac { k \\pi } { 3 } + \\frac { 1 } { 3 } \\arctan \\frac { 3 \\sqrt { 3 } } { B } \\right ) \\right ) \\end{align*}"}
-{"id": "5901.png", "formula": "\\begin{align*} \\sum _ { L = 0 } ^ \\infty P ( \\sigma ^ { ( e ) } _ { i } > 2 L ) \\le \\sum _ { L = 1 } ^ \\infty P ( \\sigma ^ { ( e ) } _ { i } \\ge L ) = E \\sigma ^ { ( e ) } _ { i } \\approx e ^ { N \\min \\big ( I _ i ( r _ i ) , \\thinspace I _ i ( r _ { i + 1 } ) \\big ) } . \\end{align*}"}
-{"id": "7266.png", "formula": "\\begin{align*} B _ 2 & \\le 2 \\int _ 0 ^ { y / ( m , n ) } ( m , n ) \\min \\Bigl ( \\frac { 2 s } { N n } , \\frac s N \\Bigl ( \\frac 1 n + \\frac 1 m \\Bigr ) - \\frac { ( m , n ) t } { m n } \\Bigr ) \\d t \\\\ & = 2 \\int _ 0 ^ y \\min \\Bigl ( \\frac { 2 s } { N n } , \\frac s N \\Bigl ( \\frac 1 n + \\frac 1 m \\Bigr ) - \\frac { h } { m n } \\Bigr ) \\d h . \\end{align*}"}
-{"id": "9865.png", "formula": "\\begin{align*} \\lim _ { \\nu \\to \\infty } \\| d ( x ^ \\nu ) \\| = 0 \\end{align*}"}
-{"id": "2538.png", "formula": "\\begin{align*} \\left \\| \\mathbb { G } _ { L ; 0 } ( x , t ) \\right \\| _ { L ^ 2 _ \\xi } \\leq C _ N \\left [ \\sum _ { j = 1 } ^ { 3 } ( 1 + t ) ^ { - 1 / 2 } \\Big ( 1 + \\frac { | x - a _ { j } t | ^ { 2 } } { 1 + t } \\Big ) ^ { - N } + e ^ { - c t } \\right ] . \\end{align*}"}
-{"id": "6882.png", "formula": "\\begin{align*} \\lim _ { p \\to 0 } \\{ \\Phi _ { 1 } , \\Phi _ { 2 } , \\Phi _ { 3 } , \\Phi _ { 4 } \\} & = \\{ 0 , 2 , 2 , 4 \\} \\\\ \\lim _ { p \\to 1 } \\{ \\Phi _ { 1 } , \\Phi _ { 2 } , \\Phi _ { 3 } , \\Phi _ { 4 } \\} & = \\{ 0 , 2 , 2 , 4 \\} \\end{align*}"}
-{"id": "9802.png", "formula": "\\begin{align*} k = k _ 0 . \\end{align*}"}
-{"id": "8791.png", "formula": "\\begin{align*} Z _ { \\Phi } ( K _ { \\mathfrak { p } } , \\chi , s , f ) : = \\int _ { K _ { \\mathfrak { p } } ^ { n } } \\Phi ( x ) \\chi \\big ( ( f ( x ) ) \\big ) | f ( x ) | ^ { s } | d x | . \\end{align*}"}
-{"id": "2089.png", "formula": "\\begin{align*} X _ k ( t ) = \\sup _ { M \\times [ 0 , t ] } \\sum _ a | u ^ a _ k - u ^ a _ { k - 1 } | + \\sup _ { M \\times [ 0 , t ] } \\sum _ a | \\nabla _ H u ^ a _ k - \\nabla _ H u ^ a _ { k - 1 } | . \\end{align*}"}
-{"id": "2802.png", "formula": "\\begin{align*} \\vert ( \\mathcal { P } _ { n } \\mathcal { B } ) ( t ) - \\mathcal { B } _ { n } ( t ) \\vert & = \\Big \\vert \\sum _ { i = 0 } ^ { n } \\mathcal { L } _ { i } ( t ) ( \\mathcal { B } ( t _ { i } ) - \\mathcal { B } _ { i } ) \\Big \\vert \\\\ & \\leq \\sum _ { i = 0 } ^ { n } \\vert \\mathcal { L } _ { i } ( t ) \\vert \\vert \\mathcal { B } ( t _ { i } ) - \\mathcal { B } _ { i } \\vert , \\\\ \\end{align*}"}
-{"id": "94.png", "formula": "\\begin{align*} \\langle \\mathcal T ( \\chi ) , f \\otimes \\omega \\otimes g \\rangle = \\langle \\tilde { \\chi } , f \\otimes \\omega \\otimes g \\rangle , \\end{align*}"}
-{"id": "8551.png", "formula": "\\begin{align*} \\partial _ t u = \\Delta _ H \\ , u , x \\in { \\bf R } ^ N , t > 0 , \\end{align*}"}
-{"id": "7904.png", "formula": "\\begin{align*} \\begin{aligned} d \\left ( \\gamma \\left ( a + b \\over 2 \\right ) , \\gamma \\left ( a ' + b ' \\over 2 \\right ) \\right ) & \\leq \\left | { a + b \\over 2 } - { a ' + b ' \\over 2 } \\right | \\\\ & \\leq { 1 \\over 2 } | a - a ' | + { 1 \\over 2 } | b - b ' | = { 1 \\over 2 } d ( x , x ' ) + { 1 \\over 2 } d ( y , y ' ) . \\end{aligned} \\end{align*}"}
-{"id": "922.png", "formula": "\\begin{align*} \\frac { a _ { k + 1 } ( M ) } { a _ k ( M ) } = \\frac { C ^ { 2 / q } } { M ^ { 2 / q } } \\left ( \\frac { 2 k + 2 } { 2 k } \\right ) ^ k \\frac { 2 k + 2 } { q ( k + 1 ) } \\to \\frac { C ^ { 2 / q } } { M ^ { 2 / q } } \\cdot e \\cdot \\frac { 2 } { q } \\end{align*}"}
-{"id": "9579.png", "formula": "\\begin{align*} \\mathcal D = \\mathcal D ( n , p , s ) = 2 \\pi ^ { \\frac { n - 1 } 2 } \\frac { \\Gamma ( \\frac { 1 + s p } 2 ) } { \\Gamma ( \\frac { n + s p } 2 ) } \\int _ 0 ^ 1 \\left | 1 - r ^ { ( s p - 1 ) / p } \\right | ^ p \\frac { d r } { ( 1 - r ) ^ { 1 + s p } } \\ , , \\end{align*}"}
-{"id": "3639.png", "formula": "\\begin{gather*} B _ q T _ 1 T _ 2 = B _ q , \\ \\ T _ 1 T _ 2 B _ q ' = B _ q ' , \\\\ T _ 1 T _ 2 L _ q ' = L _ q ' T , \\end{gather*}"}
-{"id": "921.png", "formula": "\\begin{align*} E \\left [ \\exp \\left ( \\left ( \\frac { | Y | } { M \\Lambda } \\right ) ^ { 2 / q } \\right ) \\right ] & = \\sum _ { k = 0 } ^ \\infty \\frac { 1 } { k ! } E \\left [ \\left ( \\frac { | Y | } { M \\Lambda } \\right ) ^ { 2 k / q } \\right ] \\leq 1 + \\sum _ { k = 1 } ^ \\infty C ^ { 2 k / q } \\frac { ( 2 k / q ) ^ k } { k ! M ^ { 2 k / q } } . \\end{align*}"}
-{"id": "5891.png", "formula": "\\begin{align*} P \\big ( \\cap _ { n = 1 } ^ { 2 M } \\big \\{ Z _ n ^ { N , i } \\in \\{ 0 , - 1 \\} \\big \\} \\big | \\cup _ { n = 1 } ^ M \\{ Z _ { 2 n } ^ { N , i } = - 1 \\} \\big ) \\ge P \\big ( \\cap _ { n = 1 } ^ { 2 M } \\big \\{ Z _ n ^ { N , i } \\in \\{ 0 , - 1 \\} \\big \\} \\big ) . \\end{align*}"}
-{"id": "9016.png", "formula": "\\begin{align*} \\left \\{ \\aligned & \\partial _ { t } \\theta ^ { N } + \\mathcal { J } _ { N } ( \\mathcal { J } _ { N } u ^ { N } \\cdot \\nabla \\mathcal { J } _ { N } \\theta ^ { N } ) + \\Lambda _ { x _ { 1 } } ^ { 2 \\alpha } \\mathcal { J } _ { N } \\theta ^ { N } + \\Lambda _ { x _ { 2 } } ^ { 2 \\beta } \\mathcal { J } _ { N } \\theta ^ { N } = 0 , \\\\ & u ^ { N } = \\mathcal { R } ^ { \\perp } \\theta ^ { N } , \\\\ & \\theta ^ { N } ( x , 0 ) = \\mathcal { J } _ { N } \\theta _ { 0 } ( x ) . \\endaligned \\right . \\end{align*}"}
-{"id": "1479.png", "formula": "\\begin{align*} 2 \\sqrt { x } - ( 1 + \\alpha ) \\tan \\sqrt { x } = 0 \\end{align*}"}
-{"id": "6976.png", "formula": "\\begin{align*} M _ { i i } = \\frac { 1 } { k f _ k ( B ) ^ { k - 1 } } \\sum _ { \\mbox { \\tiny $ \\begin{array} { c } i \\leq j _ 1 < j _ 2 < . . . < j _ { k - 1 } \\leq n , \\\\ j _ 1 , . . . , j _ { k - 1 } \\neq i \\end{array} $ } } \\det { B _ { ( j _ 1 , . . . , j _ { k - 1 } ) } } , \\end{align*}"}
-{"id": "8400.png", "formula": "\\begin{align*} R _ 2 = | \\mathring { h } | ^ 2 | H | ^ 2 + \\frac { 1 } { n } | H | ^ 4 , \\end{align*}"}
-{"id": "5021.png", "formula": "\\begin{align*} H ^ \\omega = \\Delta _ { \\mathcal { B } } + \\sum _ { x \\in J } \\omega _ x \\chi _ { \\tilde { \\Lambda } ( x ) } \\end{align*}"}
-{"id": "3440.png", "formula": "\\begin{align*} \\big ( f ( t , x ) , x \\big ) & = \\big ( f ( t , x ) - f ( t , 0 ) , x - 0 \\big ) + \\big ( f ( t , 0 ) , x \\big ) \\\\ & \\le \\nu | x | ^ 2 + g ( t ) | x | \\end{align*}"}
-{"id": "3494.png", "formula": "\\begin{align*} \\alpha = \\sum _ { r > \\ell } \\sum _ { i _ 1 , \\ldots , i _ r } c _ { i _ 1 \\ldots i _ r } ( \\gamma _ { i _ 1 } - 1 ) \\cdots ( \\gamma _ { i _ r } - 1 ) \\end{align*}"}
-{"id": "5431.png", "formula": "\\begin{align*} f _ i ( P _ { k , \\ell } ^ m ) = \\binom { k + \\ell + m + 2 } { i + 1 } - \\binom { \\ell + m + 1 } { i - k } - \\binom { k + m + 1 } { i - \\ell } + \\binom { m + 1 } { i - k - \\ell } \\qquad \\textrm { f o r } - 1 \\le i \\le k + \\ell + m . \\end{align*}"}
-{"id": "1588.png", "formula": "\\begin{align*} v ( \\mathbf { g } ' ) - v ( \\mathbf { g } ) & = 2 + 2 ( g _ { n - 1 } + p ) - s ( g _ { n + 1 } + p ) - 3 g _ { n - 2 } + 1 - 2 g _ { n - 1 } + s ( g _ { n - 1 } ) \\\\ & \\geq 2 p - 3 g _ { n - 2 } - \\lceil \\log _ 2 ( p ) \\rceil + 3 \\\\ & \\geq \\frac { 1 1 p } { 7 } - \\lceil \\log _ 2 ( p ) \\rceil + \\frac { 1 2 } { 7 } \\\\ & > 0 . \\end{align*}"}
-{"id": "3304.png", "formula": "\\begin{align*} H ( W _ 1 ) = \\dots = H ( W _ K ) = L , H ( W _ 1 , \\dots , W _ K ) = H ( W _ 1 ) + \\dots + H ( W _ K ) . \\end{align*}"}
-{"id": "4710.png", "formula": "\\begin{align*} \\abs { \\det A _ s ^ i } = | \\lambda | = \\left \\{ \\begin{array} { l l } 1 , & | s _ 1 | , | s _ 2 | \\leq 1 , \\\\ | s _ 2 | ^ { - 1 } , & | s _ 1 | \\leq 1 , | s _ 2 | > 1 , \\\\ | s _ 1 | ^ { - 1 } , & | s _ 1 | > 1 , | s _ 2 | \\leq | s _ 1 | , \\\\ | s _ 2 | ^ { - 1 } , & | s _ 1 | > 1 , | s _ 2 | > | s _ 1 | \\end{array} \\right \\} = \\max \\{ 1 , | s _ 1 | , | s _ 2 | \\} ^ { - 1 } . \\end{align*}"}
-{"id": "2562.png", "formula": "\\begin{align*} | c | ^ { 2 } - | z | ^ { 2 } & = r ^ { 2 } - | p _ { 0 } - z | ^ { 2 } \\ , , \\end{align*}"}
-{"id": "8427.png", "formula": "\\begin{align*} \\sum _ { t = - \\infty } ^ { \\infty } q ^ { ( t + a ( \\mu \\pi ) ) ( \\frac { 1 } { 2 } - s ) } c _ { t , l } ( \\mu ) = G ( \\varpi ^ { - l } , \\mu ^ { - 1 } ) \\epsilon ( \\frac { 1 } { 2 } , \\mu ^ { - 1 } \\chi _ 2 ^ { - 1 } ) \\frac { L ( s , \\abs { \\cdot } ^ c ) } { L ( 1 - s , \\abs { \\cdot } ^ { - c } ) } . \\end{align*}"}
-{"id": "5407.png", "formula": "\\begin{align*} a \\cdot x ^ 3 & = ( ( a b c ) ^ 3 ) ^ { b c a b c a b c a } \\\\ b \\cdot x ^ 3 & = c ^ { a c b c a c a c } \\\\ a b \\cdot x ^ 3 & = ( a c ) ^ 3 . \\end{align*}"}
-{"id": "6574.png", "formula": "\\begin{align*} G _ c ( P ( \\varepsilon ) ) = \\max \\left [ c ( 8 - 4 \\varepsilon ) , \\sqrt { 2 } \\cdot \\sqrt { 1 - \\varepsilon ^ 2 } - c \\cdot \\frac { 4 - 2 \\varepsilon } { 1 - \\varepsilon } \\right ] \\end{align*}"}
-{"id": "4698.png", "formula": "\\begin{align*} \\int _ { B _ 1 } \\varphi ( x ) \\ , d x & = \\int _ 0 ^ 1 \\int _ { \\partial B _ t } \\varphi ( y ) \\ , d \\mathcal { H } ^ { n - 1 } ( y ) \\ , d t = \\int _ 0 ^ 1 t ^ { n + 1 } \\int _ { \\partial B _ 1 } \\varphi ( y ) \\ , d \\mathcal { H } ^ { n - 1 } ( y ) \\\\ & = \\frac { 1 } { n + 2 } \\int _ { \\partial B _ 1 } \\varphi ( y ) \\ , d \\mathcal { H } ^ { n - 1 } ( y ) . \\end{align*}"}
-{"id": "7554.png", "formula": "\\begin{gather*} \\psi ^ 1 _ t + \\psi ^ 1 \\psi ^ 1 _ x - \\psi ^ 1 _ { x x } = 0 , \\\\ \\psi ^ 0 _ t + \\psi ^ 1 \\psi ^ 0 _ x - \\psi ^ 0 _ { x x } = 0 . \\end{gather*}"}
-{"id": "8946.png", "formula": "\\begin{gather*} Q = \\begin{pmatrix} a & b \\\\ b ^ \\vee & c \\end{pmatrix} \\ ! , \\end{gather*}"}
-{"id": "7718.png", "formula": "\\begin{align*} d _ B ( 0 , j ) & = \\sqrt { \\frac { N - 1 } { N } } , \\ \\ \\ \\ \\ j = 1 , 2 , \\cdots , N - 1 , \\end{align*}"}
-{"id": "313.png", "formula": "\\begin{align*} \\tau _ 1 & = 0 , \\\\ \\eta _ i & = \\inf \\{ n > \\tau _ i : \\ Z _ n \\ge Z _ { n - 1 } \\} , \\\\ \\tau _ { i + 1 } & = \\inf \\{ n > \\eta _ { i } : \\ Z _ n < Z _ { n - 1 } \\} \\end{align*}"}
-{"id": "5961.png", "formula": "\\begin{align*} \\sigma ( t ) : = \\int _ 0 ^ { t } \\eta ( \\tau ) \\ , d \\tau \\left ( \\eta ( \\tau ) : = a _ 1 ( \\tau ) \\vect { e } ( \\tau ) + \\sum _ { j = 2 } ^ n a _ j ( \\tau ) \\vect { e } _ j ( \\tau ) \\right ) \\end{align*}"}
-{"id": "992.png", "formula": "\\begin{align*} \\mathfrak { d } _ n ( s , t ) = c _ 0 \\sqrt { E [ | F _ n ( s ) - F _ n ( t ) | ^ 2 ] } , s , t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "7242.png", "formula": "\\begin{align*} = \\int _ { \\hat { G } } ^ { } \\int _ { U } ^ { } \\psi _ { n } ( u ) ( \\int _ { G } ^ { } \\theta _ { \\pi } ( g ) f ( u ^ { - 1 } g ) d g ) d u d \\mu _ { \\pi } . \\end{align*}"}
-{"id": "3073.png", "formula": "\\begin{align*} \\nabla _ x F ( t , u ( t ) ) = 0 ; \\end{align*}"}
-{"id": "7571.png", "formula": "\\begin{align*} \\pi ( u , v ) = \\lim _ { k \\to \\infty } \\cal F ^ { - 1 } ( \\psi _ k \\cal F u ) \\cdot \\cal F ^ { - 1 } ( \\psi _ k \\cal F v ) , \\end{align*}"}
-{"id": "5379.png", "formula": "\\begin{align*} \\psi ( t ^ g ) = \\psi ( t ) ^ { \\varphi ( g ) } . \\end{align*}"}
-{"id": "6481.png", "formula": "\\begin{align*} d s ^ { 2 } = \\left [ 1 - \\Phi \\left ( \\theta \\right ) \\right ] \\delta _ { a b } d \\theta ^ { a } d \\theta ^ { b } \\Phi \\left ( \\theta \\right ) = \\overset { l } { \\underset { k = 1 } { \\sum } } u _ { k } \\left ( \\theta ^ { k } \\right ) \\end{align*}"}
-{"id": "8002.png", "formula": "\\begin{align*} \\mathfrak { S } _ { [ a ] } ^ { f a r } f : = \\sum _ { k , j : | j - k | \\geq 3 } { T _ { [ a _ { j , k } ] } f } . \\end{align*}"}
-{"id": "3555.png", "formula": "\\begin{align*} \\chi _ 0 ( n ) = \\left \\{ \\begin{array} { l l } 1 & \\gcd ( n , q ) = 1 , \\\\ 0 & \\gcd ( n , q ) \\neq 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "1065.png", "formula": "\\begin{align*} L _ { P _ { k - 1 } } ^ { \\ast } ( q _ { k } ) = L _ { P _ { k } } ^ { \\ast } ( q _ { k } ) = \\log H ( P _ { k } ) - \\frac { q _ { k } } { m } = \\log \\vert P _ { k - 1 } ( \\zeta ) \\vert + q _ { k } . \\end{align*}"}
-{"id": "8361.png", "formula": "\\begin{align*} H _ { \\widehat { \\Z } } = \\widehat { \\Z } \\ell \\oplus \\widehat { \\Z } \\ell _ * \\subset h V _ { \\widehat { \\Z } } , \\end{align*}"}
-{"id": "2682.png", "formula": "\\begin{align*} H _ r \\le \\tau _ r \\le \\sum _ { \\ell = 0 } ^ { \\hat \\tau _ r } T _ \\ell . \\end{align*}"}
-{"id": "6987.png", "formula": "\\begin{align*} D f _ k ( B ) B = f _ k ( B ) . \\end{align*}"}
-{"id": "6047.png", "formula": "\\begin{align*} \\widehat { R } ^ { ( \\alpha , \\theta ) } ( Q _ { X Y U } ) : = \\begin{cases} \\frac { 1 } { \\theta } \\Omega ^ { ( \\alpha , \\theta ) } ( Q _ { X Y U } ) , & \\theta > 0 \\\\ R ^ { ( \\alpha ) } ( Q _ { X Y U } ) , & \\theta = 0 \\end{cases} \\end{align*}"}
-{"id": "9126.png", "formula": "\\begin{align*} \\Phi ( x _ { 1 } , \\ldots , x _ { l } ) = \\sum _ { k = 1 } ^ { n } \\dfrac { 1 } { \\binom { l } { p _ { k } } } \\sum _ { \\mathrm { c a r d } ( I ) = p _ { k } } \\left ( \\prod _ { j \\in \\left \\{ 1 , \\ldots , l \\right \\} \\setminus I } x _ { j } \\right ) \\cdot f _ { k } \\left ( \\prod _ { i \\in I } x _ { i } \\right ) \\left ( x _ { 1 } , \\ldots , x _ l \\in R \\right ) , \\end{align*}"}
-{"id": "6900.png", "formula": "\\begin{align*} - \\Delta _ { m + 1 } u _ { m + 1 } ( x ) = \\lambda _ { m + 1 } u _ { m + 1 } ( x ) \\forall \\ x \\in V _ { m + 1 } \\setminus V _ 0 \\end{align*}"}
-{"id": "4844.png", "formula": "\\begin{align*} h _ 0 = 0 , \\ ; h _ 1 = m , \\ ; h _ 2 = l , \\ ; h _ 3 = 2 m , \\ ; h _ 4 = m + l , \\ ; h _ 5 = 2 l . \\end{align*}"}
-{"id": "2225.png", "formula": "\\begin{align*} | w _ j - a _ { j i _ { j } } | = \\varepsilon | w _ j | | w _ j - a _ { j i _ { j } } | ^ 2 = \\varepsilon ^ 2 | w _ j | ^ 2 . \\end{align*}"}
-{"id": "814.png", "formula": "\\begin{align*} S _ { k } ( 2 x ) \\equiv 2 S _ { k } ( x ) \\pmod { 3 ^ d } , . \\end{align*}"}
-{"id": "73.png", "formula": "\\begin{align*} \\textbf { w } = \\left [ \\sum _ { n = 1 } ^ { N } G ^ { C } _ { \\sigma } ( e _ { n } ) \\textbf { X } _ { n } \\textbf { X } _ { n } ^ { H } \\right ] ^ { - 1 } \\left [ \\sum _ { n = 1 } ^ { N } G ^ { C } _ { \\sigma } ( e _ { n } ) \\textit { D } _ { n } ^ { * } \\ , \\textbf { X } _ { n } \\right ] \\end{align*}"}
-{"id": "92.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ \\infty | a _ i ( s , t ) | ^ 2 < C \\mbox { a n d } \\sum _ { i = 1 } ^ \\infty | b _ i ( s , t ) | ^ 2 < C ( s , t ) \\end{align*}"}
-{"id": "1697.png", "formula": "\\begin{align*} T _ \\lambda ( f ) = f _ \\lambda \\cdot ( f \\circ \\tau ^ { d ( \\lambda ) } ) \\end{align*}"}
-{"id": "3784.png", "formula": "\\begin{align*} p _ n ( 0 , z _ i ) = \\max _ { x \\in C _ i } p _ n ( 0 , x ) . \\end{align*}"}
-{"id": "7742.png", "formula": "\\begin{align*} & \\mathrm { R e } \\big ( T _ N ( l ) \\big ) = \\frac { 1 } { 4 } \\bigg ( F _ N ( l ) - 2 F _ N ( l - 1 ) + F _ N ( l - 2 ) \\\\ & \\qquad - F _ N ( 2 ) + 2 F _ N ( 1 ) \\bigg ) \\\\ & = \\frac { 1 } { 4 } \\Big [ \\big ( E _ N ( l ) - E _ N ( l - 1 ) \\big ) - \\big ( E _ N ( 2 ) - E _ N ( 1 ) \\big ) \\Big ] \\end{align*}"}
-{"id": "6587.png", "formula": "\\begin{align*} g = \\left \\{ \\begin{array} { l l } f \\circ \\varphi _ 0 ^ { - 1 } \\circ \\phi \\circ \\varphi _ 0 & \\ \\mbox { i n } \\ E _ M \\\\ f & \\ \\mbox { e l s e w h e r e } \\end{array} \\right . \\ . \\end{align*}"}
-{"id": "5965.png", "formula": "\\begin{align*} S _ 1 & : = \\{ ( t , w _ 2 , \\dots , w _ n ) \\in U \\ , ; \\ , a ( t ) = \\kappa _ \\gamma ( t ) = 0 \\} , \\\\ S _ 2 & : = \\{ ( t , w _ 2 , \\dots , w _ n ) \\in U \\ , ; \\ , \\hat \\rho ( t , w _ 2 , \\dots , w _ n ) = 0 , \\kappa _ \\gamma ( t ) \\neq 0 \\} , \\end{align*}"}
-{"id": "9576.png", "formula": "\\begin{align*} w ( x ) = \\delta _ { \\partial D } ^ { ( q / p ) ( n - s p + \\beta ) - n } ( x ) \\ , . \\end{align*}"}
-{"id": "4613.png", "formula": "\\begin{align*} | U _ i | \\ge \\beta \\sum _ { n = 0 } ^ { 2 ^ { k _ i } - 1 } 1 _ { [ - B + b , B - b ] } \\left ( t + \\sum _ { j = 0 } ^ { n - 1 } f ( T ^ j x ) \\right ) \\end{align*}"}
-{"id": "4078.png", "formula": "\\begin{align*} \\iint \\langle { \\boldsymbol { \\nu } } _ { ( x , t ) } , A _ i ( \\lambda ) \\rangle \\partial _ t \\varphi _ { i } \\ , d x \\ , d t + \\iint \\langle { \\boldsymbol { \\nu } } _ { ( x , t ) } , F _ { i , \\alpha } ( \\lambda ) \\rangle \\partial _ \\alpha \\varphi _ { i } d x \\ , d t + \\int \\langle \\boldsymbol { \\nu } _ { ( x , 0 ) } , A _ i \\rangle \\varphi _ i ( x , 0 ) d x = 0 \\end{align*}"}
-{"id": "7864.png", "formula": "\\begin{align*} M ( x , y ) = B _ { d ( x , y ) / 2 } ( x ) \\cap B _ { d ( x , y ) / 2 } ( y ) . \\end{align*}"}
-{"id": "2424.png", "formula": "\\begin{align*} D _ 3 = D _ 3 ' = 0 , \\ D _ 1 < 0 , \\end{align*}"}
-{"id": "637.png", "formula": "\\begin{align*} \\nabla _ i ^ F Y ^ \\alpha = \\partial _ i Y ^ \\alpha + \\Gamma ^ \\alpha _ { \\beta \\gamma } F ^ \\beta _ i Y ^ \\gamma , \\ \\ \\ Y \\in \\Gamma ( M , N ) . \\end{align*}"}
-{"id": "2674.png", "formula": "\\begin{align*} \\tilde { M } _ { f ^ { ( 2 ) } , 2 } = \\begin{pmatrix} a & b \\\\ \\bar { b } & a \\end{pmatrix} \\end{align*}"}
-{"id": "5615.png", "formula": "\\begin{align*} \\mathbb { E } [ | J ( s , t , x , y ) | ^ p ] & = \\mathbb { E } \\biggl | \\int _ s ^ t \\int _ 0 ^ 1 f ( u \\phi ( s , r , x ) + ( 1 - u ) \\phi ( s , r , y ) ) d u d B _ r \\biggr | ^ p \\\\ & \\leq C ( p ) \\mathbb { E } \\biggl ( \\int _ s ^ t \\biggl \\{ \\int _ 0 ^ 1 f ( u \\phi ( s , r , x ) + ( 1 - u ) \\phi ( s , r , y ) ) d u \\biggr \\} ^ 2 d r \\biggr ) ^ \\frac { p } { 2 } \\\\ & \\leq C ( p , | | f | | _ { \\infty } ) | t - s | ^ { \\frac { p } { 2 } } \\\\ & \\leq C ( p , | | f | | _ { \\infty } ) | s - \\tilde { s } | ^ { \\frac { p } { 2 } } . \\end{align*}"}
-{"id": "4959.png", "formula": "\\begin{align*} H H ^ { ( s ) } \\subseteq H ^ { ( s + 1 ) } + \\sum _ { r = 0 } ^ { s - 1 } M ^ { ( s - r ) ( w ) } H ^ r . \\end{align*}"}
-{"id": "2968.png", "formula": "\\begin{align*} \\left ( R ^ s _ 1 , R ^ s _ 2 \\right ) \\in \\mathcal { R } ^ s = & \\Bigg \\{ \\left ( [ R _ 1 - \\tilde { R } _ { _ 1 } ] ^ { + } , [ R _ 2 - \\tilde { R } _ { _ 2 } ] ^ { + } \\right ) , \\\\ & \\left ( [ R _ 1 - \\hat { R } _ { _ 1 } ] ^ { + } , [ R _ 2 - \\hat { R } _ { _ 2 } ] ^ { + } \\right ) \\Bigg \\} , \\end{align*}"}
-{"id": "7253.png", "formula": "\\begin{align*} r ( n ) = \\# \\{ ( a , b ) \\in A ^ 2 : a - b = n , a \\ne b \\} \\end{align*}"}
-{"id": "2725.png", "formula": "\\begin{align*} & \\displaystyle \\rho _ { \\infty } = \\int _ { \\Omega \\times { \\mathbb R ^ d } } \\ , f ( v ) \\ , d x d v , u _ { \\infty } = \\frac { 1 } { \\rho _ { \\infty } } \\ , \\int _ { \\Omega \\times { \\mathbb R ^ d } } \\ , v f ( v ) \\ , d x d v , \\\\ [ 2 p t ] & \\displaystyle T _ { \\infty } = \\frac { 1 } { N \\rho _ { \\infty } } \\int _ { \\Omega \\times { \\mathbb R ^ d } } \\ , | u _ { \\infty } - v | ^ 2 \\ , f ( v ) \\ , d x d v , \\end{align*}"}
-{"id": "9858.png", "formula": "\\begin{align*} \\textstyle { \\lim \\limits _ { j \\to + \\infty } \\frac { \\varphi ( \\gamma ( t _ j ) ) } { t _ j } = d \\varphi ( v ) . } \\end{align*}"}
-{"id": "1438.png", "formula": "\\begin{align*} \\partial _ j ( f _ { k , k + 2 ^ { j _ 1 - 1 } - 1 } ) = 0 . \\end{align*}"}
-{"id": "6611.png", "formula": "\\begin{align*} \\lim _ { J _ 2 \\ni n \\to \\infty } \\int _ { \\Omega _ 0 } \\bigl | w _ { i j } ^ { ( n ) } ( x ) \\bigr | ^ p \\ , d \\mu ( x ) = \\int _ { \\Omega _ 0 } | u _ { i j } ( x ) | ^ p \\ , d \\mu ( x ) \\ . \\end{align*}"}
-{"id": "5257.png", "formula": "\\begin{align*} \\frac { d } { d \\lambda } \\left [ D ( \\lambda ) C ( \\lambda ) \\right ] | _ { \\lambda = \\lambda ^ * } = D ' ( \\lambda ^ * ) C ( \\lambda ^ * ) , \\end{align*}"}
-{"id": "2103.png", "formula": "\\begin{align*} v ^ a _ s ( p , t ) = - \\int _ 0 ^ t \\int _ M H ( p , q , t - \\lambda ) G ^ a _ s ( q , \\lambda ) d V _ q d \\lambda + \\Phi _ s ^ a ( p , t ) , \\end{align*}"}
-{"id": "4909.png", "formula": "\\begin{align*} T = - \\beta ^ { - 1 } F = - \\beta ^ { - 1 } ( 0 + F ) + 0 I \\end{align*}"}
-{"id": "2204.png", "formula": "\\begin{align*} J _ \\gamma ( t ) = \\frac 1 { ( 2 \\pi \\sqrt { - 1 } ) ^ n } \\int \\limits _ { \\Gamma _ h } \\frac 1 { z ^ { \\gamma + I } } \\cdot \\frac { d F } { F } = \\frac 1 { ( 2 \\pi \\sqrt { - 1 } ) ^ n } \\int \\limits _ { \\Gamma _ h } \\frac { 1 } { z _ 1 ^ { \\gamma _ 1 + 1 } \\cdot \\ldots \\cdot z _ n ^ { \\gamma _ n + 1 } } \\cdot \\frac { d F _ 1 } { F _ 1 } \\wedge \\ldots \\wedge \\frac { d F _ n } { F _ n } , \\end{align*}"}
-{"id": "1220.png", "formula": "\\begin{align*} A \\nabla _ X ( h _ 1 ) _ t = B \\nabla _ { X } ( h _ 2 ) _ t \\mbox { o r e q u i v a l e n t l y t h a t } \\nabla _ X \\log \\left ( \\frac { ( h _ 1 ) _ t } { ( h _ 2 ) _ t } \\right ) = 0 . \\end{align*}"}
-{"id": "9033.png", "formula": "\\begin{align*} ( \\nabla _ a \\nabla _ a - \\nabla _ b \\nabla _ a ) \\ ! \\left [ \\ ! \\begin{array} c \\sigma \\\\ \\mu _ d \\\\ \\rho \\end{array} \\ ! \\right ] = 2 J _ { a b } \\ ! \\left [ \\ ! \\begin{array} { c } \\rho \\\\ J _ d { } ^ e \\mu _ e \\\\ - \\sigma \\end{array} \\ ! \\right ] \\end{align*}"}
-{"id": "3506.png", "formula": "\\begin{align*} \\mathcal { U } ^ 0 [ 0 , s ] = \\big { \\{ } u ( \\cdot ) : X ^ u ( s ) = K _ 0 \\big { \\} } = \\emptyset , \\end{align*}"}
-{"id": "6281.png", "formula": "\\begin{align*} \\Lambda _ r \\leq & \\Lambda _ r ( c _ r \\nu ) ^ { - m } \\Lambda _ r ^ { m ( \\frac 3 r - 1 ) } \\| u _ Q \\| _ r ^ m \\leq ( c _ r \\nu ) ^ { - m } \\Lambda _ r ^ { m / 2 + 1 } \\| u _ Q \\| _ 2 ^ m \\\\ \\leq & ( c _ r \\nu ) ^ { - m } \\| u _ Q \\| _ { \\dot H ^ { 1 / 2 + 1 / m } } ^ m . \\end{align*}"}
-{"id": "3088.png", "formula": "\\begin{align*} \\underline { u } ^ * = \\lim _ { s \\to + \\infty } \\underline { w } ( s ) \\quad \\mbox { a n d } \\quad \\overline { u } ^ * = \\lim _ { s \\to + \\infty } \\overline { w } ( s ) \\ , . \\end{align*}"}
-{"id": "6356.png", "formula": "\\begin{align*} \\int x ^ k P _ { k - 1 } ( x ) \\ , d \\mu ( x ) = m _ { 2 k - 1 } + \\sum \\limits _ { i = 0 } ^ { 2 k - 2 } \\lambda _ i m _ i \\end{align*}"}
-{"id": "7049.png", "formula": "\\begin{align*} \\tilde { \\eta } _ j = \\frac { 1 } { t } \\eta _ j , j = 1 , 2 , . . . , n - 1 ; \\end{align*}"}
-{"id": "7923.png", "formula": "\\begin{align*} \\varphi ( y ) = y ^ { \\alpha } , \\ , \\ , \\ , \\varphi ( y ) = - ( \\log y ) ^ { - 1 - \\frac { 2 } { 2 n + 1 } } . \\end{align*}"}
-{"id": "4530.png", "formula": "\\begin{align*} \\Pi ( a ) - \\pi ( a ) = \\Pi ( a - s ) - \\pi ( a - s ) . \\end{align*}"}
-{"id": "3870.png", "formula": "\\begin{align*} s _ k ' ( \\tau _ k ) = \\frac { s _ k ( 1 ) - s _ k ( 0 ) } { 1 - 0 } = 0 . \\end{align*}"}
-{"id": "7337.png", "formula": "\\begin{align*} \\mathcal { R } \\ ; : = \\ ; \\big \\{ A \\in \\widetilde { \\mathcal { A } } _ 1 \\textrm { o r } A \\in \\widetilde { \\mathcal { A } } _ 2 \\ : | \\ : \\partial _ t A _ j \\in L ^ 1 _ { \\mathrm { l o c } } ( \\mathbb { R } , L ^ { b _ j } ( \\mathbb { R } ^ 3 , \\mathbb { R } ^ 3 ) ) , \\ , j = 1 , 2 \\big \\} \\ , . \\end{align*}"}
-{"id": "9046.png", "formula": "\\begin{align*} d ( D _ 4 ) \\le 1 7 < 2 3 = n ^ 2 - 2 \\end{align*}"}
-{"id": "5764.png", "formula": "\\begin{align*} L ( G R ) \\cap \\left ( [ - m , m ] ^ { p - 1 } \\times [ 0 , c _ 1 m ^ { - p + 1 } ] \\right ) = \\emptyset , \\end{align*}"}
-{"id": "212.png", "formula": "\\begin{align*} a \\sqsubseteq f ( \\mathfrak { C \\uplus } e ' ( \\Gamma ) ) = f ( \\mathfrak { C } ) \\otimes f ( e ' ( \\Gamma ) ) = f ( \\mathfrak { C } ) \\otimes f ( e ' ( \\varphi _ { 1 } ) ) \\otimes \\cdots \\otimes f ( e ' ( \\varphi _ { n } ) ) . \\end{align*}"}
-{"id": "4875.png", "formula": "\\begin{align*} \\Lambda _ x = A , \\ , \\ , \\Gamma _ x = B , \\ , \\ , \\mathrm { L } _ x = C . \\end{align*}"}
-{"id": "2265.png", "formula": "\\begin{align*} \\frac { 1 } { ( e ^ { 2 t k } - 1 ) ^ 2 } = - \\frac { 1 } { e ^ { 2 t k } - 1 } - \\frac { 1 } { 2 k } \\cdot \\frac { \\partial } { \\partial t } \\left [ \\frac { 1 } { e ^ { 2 t k } - 1 } \\right ] . \\end{align*}"}
-{"id": "8752.png", "formula": "\\begin{align*} \\beta ( u ) : = \\min \\left \\{ 1 , \\frac { ( M + 1 - u ) ( M + 2 ) } { M ^ 2 + 3 M + 1 - u } \\right \\} , \\end{align*}"}
-{"id": "7296.png", "formula": "\\begin{align*} 2 \\sum _ { i = 1 } ^ r m _ i ^ 2 + \\sum _ { i = 1 } ^ r m _ i = n ( 2 n - 1 ) - 3 ( n - 1 ) = 2 n ^ 2 - 4 n + 3 . \\end{align*}"}
-{"id": "7556.png", "formula": "\\begin{gather*} \\psi _ t + \\psi _ x \\psi _ y = 0 , \\psi _ { x x } + \\psi _ { y y } = 0 \\end{gather*}"}
-{"id": "2372.png", "formula": "\\begin{align*} m \\big ( h p ^ n \\big ) = m \\ll \\frac 1 { \\log n } \\end{align*}"}
-{"id": "38.png", "formula": "\\begin{align*} \\textbf { w } = \\left [ \\sum _ { n = 1 } ^ { N } G ^ { C } _ { \\sigma \\ , \\sqrt { 2 } } ( e _ { n } ) \\textbf { X } _ { n } \\textbf { X } _ { n } ^ { H } \\right ] ^ { - 1 } \\left [ \\sum _ { n = 1 } ^ { N } G ^ { C } _ { \\sigma \\ , \\sqrt { 2 } } ( e _ { n } ) \\textit { d } _ { n } ^ { * } \\ , \\textbf { X } _ { n } \\right ] \\end{align*}"}
-{"id": "5521.png", "formula": "\\begin{align*} b ( 3 n ) & = b ( n ) \\\\ b ( 3 n + 1 ) & = \\sqrt { 2 } \\cdot b ( n ) + b ( n + 1 ) \\\\ b ( 3 n + 2 ) & = b ( n ) + \\sqrt { 2 } \\cdot b ( n + 1 ) . \\end{align*}"}
-{"id": "3186.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ e ^ { c Z _ { t } ^ { 1 } } \\right ] = \\exp \\left \\lbrace \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { 1 } \\left ( e ^ { z \\psi ( s , c ) } - 1 \\right ) \\nu _ { 1 } ( \\mathrm { d } z ) \\mathrm { d } s \\right \\rbrace < \\infty , c \\in \\mathbb { R } , \\end{align*}"}
-{"id": "1802.png", "formula": "\\begin{align*} u _ { n + 1 } = - \\frac 1 { u _ n } - \\frac 1 { u _ { n - 1 } } + \\frac { \\alpha } { u _ { n } } \\end{align*}"}
-{"id": "9167.png", "formula": "\\begin{align*} g _ 4 ( c x ^ 4 ) + x g _ 3 ( c x ^ 3 ) + x ^ 2 g _ 2 ( c x ^ 2 ) + x ^ 3 g _ 1 ( c x ) = x ^ 4 ( g _ 1 + g _ 2 + g _ 3 + g _ 4 ) ( c ) \\left ( x \\in K \\right ) , \\end{align*}"}
-{"id": "6189.png", "formula": "\\begin{align*} \\Lambda ^ i = \\Lambda _ { { \\phi ( t _ i ) } } = \\sup _ { \\mathbb { T } ^ 4 \\times \\mathbb { T } ^ 3 } ( | | _ { g _ \\phi } ^ 2 + | \\nabla T | _ { g _ \\phi } ^ 2 ) ^ \\frac { 1 } { 2 } ( \\cdot , t _ i ) \\to \\infty , \\end{align*}"}
-{"id": "9513.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ l ( d - r _ i ) = k , 1 \\leq r _ 1 \\leq r _ 2 \\leq \\ldots \\leq r _ l \\end{align*}"}
-{"id": "1068.png", "formula": "\\begin{align*} \\tau _ { k } = q _ { k } \\cdot \\frac { m + 1 } { m ( 1 + \\widehat { w } _ { n } ( \\zeta ) ) } . \\end{align*}"}
-{"id": "4512.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | \\phi ( \\Delta _ n ) | = 1 \\end{align*}"}
-{"id": "9353.png", "formula": "\\begin{align*} j O _ { n + 3 } ^ { ( 3 ) } - 3 J O _ { n + 3 } ^ { ( 3 ) } = 2 j O _ { n } ^ { ( 3 ) } , \\end{align*}"}
-{"id": "3468.png", "formula": "\\begin{align*} \\| P _ h v - v \\| _ H = \\mathrm { d i s t } _ H ( v , V _ h ) , v \\in H . \\end{align*}"}
-{"id": "5945.png", "formula": "\\begin{align*} f ' _ v ( x ) & = x _ v & & v , \\\\ f ' _ u & \\equiv \\begin{cases} \\phi & f _ u \\equiv \\phi \\\\ \\psi & f _ u \\equiv \\psi \\end{cases} & & u . \\end{align*}"}
-{"id": "6638.png", "formula": "\\begin{align*} | f ( z ) - f ( w ) | \\ ; \\leq \\ ; C _ 3 \\left ( { \\mathrm { A r e a } ( 2 D ) } + C \\sum _ { k = m - 1 } ^ { \\infty } k ^ { 2 p } 2 ^ { - 2 k } \\right ) ^ { \\frac { 1 } { 2 } } \\end{align*}"}
-{"id": "4623.png", "formula": "\\begin{align*} Y _ i = \\{ t ( n ) : \\ell _ i < n < 2 ^ { k _ i } \\textup { a n d } T _ f ^ n ( x , t ) \\in X _ { B + b } \\} \\end{align*}"}
-{"id": "576.png", "formula": "\\begin{align*} u _ n - v _ n = \\frac { z _ n - \\alpha } { | z _ n - \\alpha | } - \\frac { z _ n - \\beta } { | z _ n - \\beta | } \\rightarrow 0 , \\end{align*}"}
-{"id": "3774.png", "formula": "\\begin{align*} \\bar { X } _ n = \\max \\{ S ^ { z , i } _ n \\colon \\ , \\eta ( z , i , n ) = 1 \\} , \\end{align*}"}
-{"id": "3717.png", "formula": "\\begin{align*} M ( \\tilde \\gamma ) _ k = \\{ m _ k , m _ k + 1 , \\dots , m _ { k + 1 } - 1 \\} \\end{align*}"}
-{"id": "9754.png", "formula": "\\begin{align*} U _ h ( k h + , - \\infty ) = \\lim \\limits _ { y \\to - \\infty } U _ 0 ( y ) . \\end{align*}"}
-{"id": "3061.png", "formula": "\\begin{align*} \\gamma _ z ( f ) : = f ( z ) \\end{align*}"}
-{"id": "6054.png", "formula": "\\begin{align*} \\lim _ { Q _ { X Y U } \\to Q _ { X Y U } ' } R ^ { ( \\alpha ) } ( Q _ { X Y U } ) = R ^ { ( \\alpha ) } ( Q _ { X Y U } ' ) . \\end{align*}"}
-{"id": "6469.png", "formula": "\\begin{align*} d s _ { 3 D u } ^ { 2 } = \\frac { 1 } { \\sigma _ { x } ^ { 2 } } \\left ( d \\mu _ { x } ^ { 2 } + 2 d \\sigma _ { x } ^ { 2 } \\right ) + \\frac { 2 } { \\sigma _ { y } ^ { 2 } } d \\sigma _ { y } ^ { 2 } . \\end{align*}"}
-{"id": "5775.png", "formula": "\\begin{align*} \\| u \\| _ { W ^ { 1 , p } ( \\Omega ) } = \\| u \\| _ { L ^ { p } ( \\Omega ) } + \\| \\nabla u \\| _ { L ^ { p } ( \\Omega ) } . \\end{align*}"}
-{"id": "1262.png", "formula": "\\begin{align*} ( 1 - \\hat u _ m ) ( x ) & \\leq v ^ * ( x ) \\\\ & \\leq c \\frac { v ^ * ( \\hat x + r _ j e _ n / 2 ) } { v ( r _ j e _ n / 2 ) } v ( x - \\hat x ) \\\\ & \\leq c ( | x - \\hat x | / r _ j ) ^ { 1 - \\eta } \\end{align*}"}
-{"id": "4281.png", "formula": "\\begin{align*} \\alpha \\left ( \\frac { z } { c \\tau + d } , \\frac { a \\tau + b } { c \\tau + d } \\right ) & = ( c \\tau + d ) \\alpha ( z , \\tau ) - c z \\\\ \\alpha \\left ( z + \\lambda \\tau + \\mu , \\tau \\right ) & = \\alpha ( z , \\tau ) + \\lambda \\end{align*}"}
-{"id": "4494.png", "formula": "\\begin{align*} | \\xi _ 1 | = | \\xi _ 2 | = 1 / \\sqrt { 2 } . \\end{align*}"}
-{"id": "1635.png", "formula": "\\begin{align*} \\tau ^ n ( \\rho ) = \\rho x _ j \\cdots x _ { i - 1 } ( n , d ( \\rho ) + ( i - j ) ( 1 , \\ldots , 1 ) ) \\in G _ i . \\end{align*}"}
-{"id": "9120.png", "formula": "\\begin{align*} ( n + 1 - i ) f _ { n + 1 - i } + ( i + 1 ) f _ { n + 1 - ( i + 1 ) } = ( - 1 ) ^ { i } \\sum _ { k = 0 } ^ { i } \\binom { n - i + k } { k } \\widetilde { D } _ { ( n - 1 ) - i + k } , \\end{align*}"}
-{"id": "2383.png", "formula": "\\begin{align*} f ( \\theta ) & = - a _ 3 \\cos ^ 3 \\theta + ( 3 a _ 2 + a _ 5 ) \\cos ^ 2 \\theta \\sin \\theta \\\\ & \\phantom { = } + ( 2 a _ 3 + a _ 4 + a _ 6 ) \\cos \\theta \\sin ^ 2 \\theta - a _ 2 \\sin ^ 3 \\theta , \\\\ g ( \\theta ) & = a _ 2 \\cos ^ 3 \\theta + ( 3 a _ 3 + a _ 4 ) \\cos ^ 2 \\theta \\sin \\theta \\\\ & \\phantom { = } - ( 3 a _ 2 + a _ 5 ) \\cos \\theta \\sin ^ 2 \\theta - a _ 6 \\sin ^ 3 \\theta , \\end{align*}"}
-{"id": "2602.png", "formula": "\\begin{align*} ( f , g ) _ V = \\int _ { \\R } ( V * f ) g \\end{align*}"}
-{"id": "8567.png", "formula": "\\begin{align*} \\| z \\| _ { L ^ { 2 \\kappa } ( Q _ 0 ) } & \\le \\epsilon ^ k \\| z \\| _ { L ^ { 2 \\kappa } ( Q _ k ) } + C \\epsilon ^ { - ( 1 - \\theta ) / \\theta } \\sum _ { n = 0 } ^ { k - 1 } \\epsilon ^ n D _ n ^ { 1 / 2 \\theta } \\| z \\| _ { L ^ 1 ( Q _ { n + 1 } ) } \\\\ & \\le \\epsilon ^ k \\| z \\| _ { L ^ { 2 \\kappa } ( Q _ k ) } + C \\epsilon ^ { - ( 1 - \\theta ) / \\theta } \\sum _ { n = 0 } ^ { k - 1 } \\epsilon ^ n \\left [ 2 ^ { 2 n } ( \\rho ^ { - 2 } + t ^ { - 1 } ) \\right ] ^ { 1 / 2 \\theta } \\| z \\| _ { L ^ 1 ( Q _ { n + 1 } ) } \\end{align*}"}
-{"id": "7026.png", "formula": "\\begin{align*} \\eta _ 0 = 2 ( 1 - s ) C _ 1 C _ 2 \\epsilon _ 1 ^ s , \\end{align*}"}
-{"id": "3239.png", "formula": "\\begin{align*} \\mathcal M _ k ( X ) = \\sum _ { r = 0 } ^ { n - k } M ^ { r } H ^ { 2 r + k } ( X ) . \\end{align*}"}
-{"id": "7394.png", "formula": "\\begin{align*} \\Phi ^ + _ 2 = \\begin{pmatrix} x + v t & - y & - z & - t \\\\ u y + 2 z v & x - v t & u t & - z \\\\ w z & - w t & x + v t & y \\\\ u w t & w z & - u y - 2 v z & x - v t \\end{pmatrix} , \\end{align*}"}
-{"id": "9628.png", "formula": "\\begin{align*} \\frac { \\partial P _ { 0 } ( r , h _ { \\beta , 1 } ) } { \\partial h _ { \\beta , 1 } } = \\frac { 1 8 0 b r P _ { 0 } ( r , h _ { \\beta , 1 } ) } { \\pi ( r ^ 2 + h _ { \\beta , 1 } ^ 2 ) } ( 1 - P _ { 0 } ( r , h _ { \\beta , 1 } ) ) . \\end{align*}"}
-{"id": "3860.png", "formula": "\\begin{align*} & ( x ^ k - x ^ * ) ^ T \\left ( \\nabla ^ 2 f ( \\xi ^ k ) + \\sum \\limits _ { i = 1 } ^ n \\lambda _ i ^ * \\nabla ^ 2 g _ i ( \\xi ^ k ) + \\sum \\limits _ { i = 1 } ^ p \\mu _ i ^ * \\nabla ^ 2 h _ i ( \\xi ^ k ) \\right ) ( x ^ k - x ^ * ) \\\\ = \\ & ( x ^ k - x ^ * ) ^ T \\nabla ^ 2 \\ell ( \\xi ^ k ) ( x ^ k - x ^ * ) \\ = \\ 2 ( \\ell ( x ^ k ) - \\ell ( x ^ * ) ) \\ \\leq \\ 0 \\end{align*}"}
-{"id": "210.png", "formula": "\\begin{align*} { \\vdash } = { \\models _ { \\mathbf { M o d } ^ \\ast ( { \\vdash } ) } } = { \\models ' _ { \\mathbf { M o d } ^ \\ast ( { \\vdash } ) } } . \\end{align*}"}
-{"id": "668.png", "formula": "\\begin{align*} & \\{ \\lambda ' _ i ( x ) \\} _ { i \\in S _ n } = E i g ^ { \\neq 0 } ( \\bar { A } _ n ( x ) ) = \\{ \\lambda \\in E i g ( \\bar { A } _ n ( x ) ) : \\lambda \\neq 0 \\} \\\\ & = E i g ^ { \\neq 0 } ( \\bar { B } _ n ( x ) ) = \\{ \\lambda \\in E i g ( \\bar { B } _ n ( x ) ) : \\lambda \\neq 0 \\} \\end{align*}"}
-{"id": "5168.png", "formula": "\\begin{align*} f _ { 1 } ( x ) & = \\begin{cases} x ( a - x ) , & x \\in [ 0 , a ] \\\\ ( 1 - x ) ( a - x ) , & x \\in ( a , 1 ] , \\end{cases} \\\\ g _ { 1 } ( x ) & = \\begin{cases} x ( b - x ) , & x \\in [ 0 , b ) \\\\ ( 1 - x ) ( b - x ) , & x \\in [ b , 1 ] , \\end{cases} \\\\ f _ { 2 } & = - g _ { 1 } , g _ { 2 } = - f _ { 1 } , h _ { 1 } = - h _ { 2 } . \\end{align*}"}
-{"id": "5354.png", "formula": "\\begin{align*} Y _ t ( v , x ) = \\sum _ { n \\in Z } ( v \\otimes t ^ n ) x ^ { - n - 1 } \\in ( V \\otimes \\C [ t , t ^ { - 1 } ] ) [ [ x , x ^ { - 1 } ] ] . \\end{align*}"}
-{"id": "7649.png", "formula": "\\begin{align*} S ' = \\Big < s , s _ 1 , \\ldots , & s _ { 2 ^ n - 4 } \\colon \\ , s ^ 2 = 1 , \\ s _ { \\alpha } ^ 2 = - 1 , \\\\ & s s _ { \\alpha } = - s _ { \\alpha } s , \\ s _ { \\alpha } s _ { \\beta } = - s _ { \\beta } s _ { \\alpha } , \\ \\ 1 \\le \\alpha \\neq \\beta \\le 2 ^ n - 4 \\Big > . \\end{align*}"}
-{"id": "5069.png", "formula": "\\begin{align*} b _ k C _ { i , j } = - C _ i C _ j , i \\neq j , i \\neq k , k \\neq j . \\end{align*}"}
-{"id": "575.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } w _ n + \\frac { 1 } { 2 } \\left ( 2 \\alpha + 2 t h - w _ n \\right ) = \\alpha + t h \\in \\Omega . \\end{align*}"}
-{"id": "9651.png", "formula": "\\begin{align*} & \\mathbb { E } \\left [ \\hat { \\Phi } _ { \\rm { s t } , \\beta } ( t ) - \\Phi _ { \\rm { s t } , \\beta } ( t ) \\right ] = \\Lambda _ \\beta ( t ) \\mathbb { E } \\left [ \\left ( \\hat { \\lambda } _ { \\beta } - \\lambda _ { \\beta } \\right ) ^ 2 \\right ] , \\end{align*}"}
-{"id": "301.png", "formula": "\\begin{align*} R _ j ( u ) & = ( u _ 0 , u _ 1 , \\dots , u _ { j - 1 } , u _ { j + 1 } , u _ { j + 2 } , \\dots , u _ d ) , j = 0 , 1 , \\dots , d ; \\\\ G _ z ( u ) & = \\left ( u _ 0 z _ 1 + u _ 1 , u _ 0 z _ 2 + u _ 2 , \\dots , u _ 0 z _ d + u _ d \\right ) . \\end{align*}"}
-{"id": "5934.png", "formula": "\\begin{align*} \\forall u , v \\in V & & ( u , v ) = u \\ , B \\ , v ^ { { \\rm t r } } . \\end{align*}"}
-{"id": "4547.png", "formula": "\\begin{align*} & \\deg e _ i = - \\deg f _ i = 0 ( 1 \\le i \\le n - 1 ) , & \\deg q ^ h = 0 ( h \\in P ) , \\\\ & \\deg \\ , e _ 0 ^ { ( l ) } = - \\deg \\ , f _ 0 ^ { ( l ) } = 1 , & \\deg Z _ r = r ( r \\neq 0 ) \\ , . \\end{align*}"}
-{"id": "3722.png", "formula": "\\begin{align*} \\left ( f _ 0 , \\frac { \\xi _ 1 } { \\xi _ 2 } , \\frac { \\xi _ 2 } { \\xi _ 3 } , \\frac { \\xi _ 3 } { \\xi _ 4 } , \\frac { \\xi _ 4 } { \\xi _ 5 } , \\frac { \\xi _ 5 } { \\xi _ 6 } , \\frac { \\xi _ 6 } { \\xi _ 7 } , \\frac { \\xi _ 7 } { \\xi _ 8 } , \\frac { \\xi _ 8 } { \\xi _ 9 } , f _ 9 \\right ) = \\left ( 0 , \\omega ^ { - 1 } , \\omega ^ { - 1 } , f _ 3 , \\omega ^ { - 1 } , \\omega ^ { - 1 } , 0 , \\omega ^ { - 1 } , \\omega ^ { - 1 } , 0 \\right ) . \\end{align*}"}
-{"id": "6340.png", "formula": "\\begin{align*} & a _ m \\to 0 , \\ b _ m \\to \\infty , p ^ * _ { 1 , m } , p _ { 2 , m } ^ * \\to 0 \\ \\\\ & p ^ * _ { i , m } b _ m \\to z _ i ^ * , i = 1 , 2 , \\end{align*}"}
-{"id": "7014.png", "formula": "\\begin{align*} g ( \\epsilon ) = ( \\frac { n - 2 } { n - 1 } \\epsilon + \\frac { h ( \\epsilon ) } { 2 } ) ^ { - 1 / 2 } . \\end{align*}"}
-{"id": "9012.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\| \\nabla \\theta ( t ) \\| _ { L ^ { 2 } } ^ { 2 } + \\| \\Lambda _ { x _ { 1 } } ^ { \\alpha } \\nabla \\theta \\| _ { L ^ { 2 } } ^ { 2 } + \\| \\Lambda _ { x _ { 2 } } ^ { \\beta } \\nabla \\theta \\| _ { L ^ { 2 } } ^ { 2 } = & \\int _ { \\mathbb { R } ^ { 2 } } { ( u \\cdot \\nabla ) \\theta \\Delta \\theta \\ , d x } \\\\ = & \\mathcal { H } _ { 1 } + \\mathcal { H } _ { 2 } + \\mathcal { H } _ { 3 } + \\mathcal { H } _ { 4 } , \\end{align*}"}
-{"id": "954.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { N _ n } \\left ( \\max _ { 1 \\leq k \\leq d _ n } \\sum _ { j = 1 } ^ { N _ n } \\gamma _ { n , k } ( i , j ) ^ 2 \\right ) ^ { 3 / 2 } \\leq \\left ( \\max _ { 1 \\leq i \\leq N _ n } \\sqrt { \\Lambda _ i } \\right ) \\sum _ { i = 1 } ^ { N _ n } \\Lambda _ i , \\end{align*}"}
-{"id": "5045.png", "formula": "\\begin{align*} f & = q - 4 0 9 6 \\ , q ^ 2 + 9 7 9 5 6 \\ , q ^ 3 + 1 6 7 7 7 2 1 6 \\ , q ^ 4 + 3 4 1 0 0 5 3 5 0 \\ , q ^ 5 - 4 0 1 2 2 7 7 7 6 \\ , q ^ 6 + \\cdots \\\\ g & = q + 4 0 9 6 \\ , q ^ 2 + ( 2 0 4 8 - a / 2 ) \\ , q ^ 3 + 1 6 7 7 7 2 1 6 \\ , q ^ 4 + ( 4 3 1 8 4 8 3 7 4 + 1 6 2 \\ , a ) \\ , q ^ 5 + \\cdots \\\\ g ' & = q + 4 0 9 6 \\ , q ^ 2 + ( 2 0 4 8 + a / 2 ) \\ , q ^ 3 + 1 6 7 7 7 2 1 6 \\ , q ^ 4 + ( 4 3 1 8 4 8 3 7 4 - 1 6 2 \\ , a ) \\ , q ^ 5 + \\cdots \\ , , \\end{align*}"}
-{"id": "5358.png", "formula": "\\begin{align*} \\langle w _ { ( 4 ) } ' , \\mathcal { Y } _ 1 ( w _ { ( 1 ) } , x _ 1 ) \\mathcal { Y } _ 2 ( w _ { ( 2 ) } , x _ 2 ) w _ { ( 3 ) } \\rangle _ { W _ 4 } | _ { x _ 1 = z _ 1 , \\ x _ 2 = z _ 2 } \\end{align*}"}
-{"id": "360.png", "formula": "\\begin{align*} S _ n = \\sum _ { r , s \\in \\mathbb { Z } } b _ { n , r , s } \\xi _ { - r , - s } , \\end{align*}"}
-{"id": "117.png", "formula": "\\begin{align*} \\frac { \\left | W ( x , n ) \\cup \\partial _ { N \\sqrt { k } } W ( x , n ) \\right | } { N ^ k } \\dim C P & + \\# ( x , n ) \\cdot \\dim C Q \\\\ & < \\frac { \\rho _ 1 \\dots \\rho _ k } { 2 } \\left | \\mathrm { I n t } _ { r _ 2 + N \\sqrt { k } } W ( x , n ) \\right | . \\end{align*}"}
-{"id": "8685.png", "formula": "\\begin{align*} ( x + \\lambda ) ^ { [ 2 p ] } & = \\Big ( \\frac { 1 } { 2 } P ( x + \\lambda , x + \\lambda ) \\Big ) ^ { [ p ] } \\\\ & = P ( x , \\lambda ) ^ { [ p ] } \\\\ & = ( \\lambda x ) ^ { [ p ] } \\end{align*}"}
-{"id": "9902.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\tfrac { k _ i + 2 } { 6 } \\le \\tfrac { r - 1 } { 2 } \\end{align*}"}
-{"id": "5793.png", "formula": "\\begin{align*} \\langle \\mu , u \\rangle = \\int _ { \\Omega } u \\ ; d \\mu , \\end{align*}"}
-{"id": "6621.png", "formula": "\\begin{align*} \\| D f ^ { - 1 } - D g ^ { - 1 } \\| _ { L ^ 1 ( \\Omega ^ * ) } \\ ; & = \\ ; \\| D ( f ^ { - 1 } | _ { \\mathcal { O } ^ * } ) - D \\phi ^ { - 1 } \\| _ { L ^ 1 ( \\mathcal { O } ^ * ) } \\\\ & \\leq \\ ; c _ 1 \\left [ \\mathcal { E } _ 1 ( f ^ { - 1 } | _ { \\mathcal { O } ^ * } ) + \\mathcal { E } _ 1 ( \\phi ^ { - 1 } ) \\right ] \\\\ & \\leq \\ ; c _ 1 \\left [ \\mathcal { E } _ 1 ( f ^ { - 1 } | _ { \\mathcal { O } ^ * } ) + 4 \\mathcal { E } _ p ( \\phi ) ^ { \\frac { 1 } { p } } \\right ] \\ . \\end{align*}"}
-{"id": "2308.png", "formula": "\\begin{align*} & \\ ; \\| A ^ { ( r _ k - 1 + s ) / 2 } w _ k ( t ) \\| _ { L ^ 2 } ^ 2 + \\int _ { t _ k } ^ { t } \\| A ^ { ( r _ k - 1 ) / 2 + s } w _ k \\| _ { L ^ 2 } ^ 2 \\\\ \\leq & \\ ; C \\int _ { t _ k } ^ { t } \\| A ^ { ( r _ k - 1 ) / 2 } f ( u _ * , u _ * ) \\| _ { L ^ 2 } ^ 2 \\ , d \\tau \\\\ \\leq & \\ ; C \\int _ { t _ k } ^ t \\| u _ * \\| _ { D ( A ) } ^ 2 ( \\| A ^ { s / 2 } u _ * \\| _ { L ^ 2 } ^ 2 + \\| A ^ { r _ k / 2 } u _ * \\| _ { L ^ 2 } ^ 2 ) \\ , d \\tau . \\end{align*}"}
-{"id": "8241.png", "formula": "\\begin{align*} \\hat { E } ( s , g ) = E _ { \\hat { v } ^ { \\circ } } ( s , g ) \\end{align*}"}
-{"id": "5773.png", "formula": "\\begin{align*} \\| u \\| _ { \\dot { H } ^ { \\alpha } ( \\R ^ n ) } = \\| f \\| _ { L ^ { 2 } ( \\R ^ n ) } . \\end{align*}"}
-{"id": "6742.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\bar { E } \\Big ( \\mathbb { P } ( \\Theta > N ^ { \\gamma } t ) \\Big ) = \\lim \\limits _ { N \\rightarrow \\infty } \\Bigg ( 1 - \\frac { 1 } { N ^ { \\gamma } } \\Bigg ) ^ { N ^ { \\gamma } t + 1 } = e ^ { - t } . \\end{align*}"}
-{"id": "972.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } P ( T _ n ^ * > T _ n ) \\leq P \\left ( - \\max _ { 1 \\leq m \\leq M } | \\rho _ m | \\Sigma ( \\theta _ m ) \\geq 0 \\right ) = 0 \\end{align*}"}
-{"id": "9777.png", "formula": "\\begin{align*} \\tilde { L } _ { 1 , h _ j } ( X _ { k } ^ * - ) = O ( 1 ) ( d L ^ { b } _ { h _ j , \\theta } ( \\Lambda _ { b , k - 1 } ^ { * } ) + d Q _ { h _ j , \\theta } ( \\Lambda _ { k - 1 } ^ { * } ) + ( e ^ { - l X _ { k - 1 } ^ * } - e ^ { - l X _ { k } ^ * } ) \\| Z _ { 0 } \\| _ { \\infty } ) , \\end{align*}"}
-{"id": "6944.png", "formula": "\\begin{align*} [ a ] [ z ] : = [ ( a \\oplus e _ { k - m } ) ( z \\oplus e _ { k - l } ) ] \\ , , \\end{align*}"}
-{"id": "4394.png", "formula": "\\begin{align*} \\nu ( \\hat { A } _ y ) = \\inf \\{ \\nu ( \\hat { A } _ x ) ; x \\in \\mathcal { M } \\setminus \\mathbb { C } ^ n \\} . \\end{align*}"}
-{"id": "90.png", "formula": "\\begin{align*} \\chi ( s , t ) = \\sum _ { i = 1 } ^ { \\infty } a _ i ( s ) b _ i ( t ) , \\ \\ \\ s , t \\in G , \\end{align*}"}
-{"id": "2374.png", "formula": "\\begin{align*} \\Gamma = \\bigcup _ { h \\in R } \\Lambda h . \\end{align*}"}
-{"id": "672.png", "formula": "\\begin{align*} ( \\overline { B ( \\lambda _ i ( x _ 0 ) , \\frac { \\beta } { 2 } ) } \\setminus B ( \\lambda _ i ( x _ 0 ) , \\frac { \\beta } { 4 } ) ) \\bigcap E i g ( \\bar { A } ( x ) ) = \\emptyset , \\forall x \\in ( x _ 0 - \\alpha _ { i , \\beta } ( x _ 0 ) , x _ 0 + \\alpha _ { i , \\beta } ( x _ 0 ) ) \\end{align*}"}
-{"id": "731.png", "formula": "\\begin{align*} \\int _ { \\R ^ 3 } | u _ 1 | ^ 2 \\ , d x = a _ 1 > 0 , \\int _ { \\R ^ 3 } | u _ 2 | ^ 2 \\ , d x = a _ 2 > 0 , \\end{align*}"}
-{"id": "9299.png", "formula": "\\begin{align*} f ( x ) = ( f _ 1 ( x ) , \\ldots , f _ m ( x ) ) x \\in \\R ^ k . \\end{align*}"}
-{"id": "1466.png", "formula": "\\begin{align*} d _ u ( C \\langle 1 , y _ r , v _ s y _ s \\rangle ) = \\{ 0 \\} \\end{align*}"}
-{"id": "6421.png", "formula": "\\begin{align*} d l _ { \\rightarrow \\xi } + d l _ { \\xi \\rightarrow } = \\frac { 1 } { 1 + \\xi } \\Delta \\theta + \\frac { \\xi } { 1 + \\xi } \\Delta \\theta = \\Delta \\theta \\end{align*}"}
-{"id": "208.png", "formula": "\\begin{align*} F / \\theta = \\{ \\langle a _ { 1 } / \\theta , \\dots , a _ { n } / \\theta \\rangle : \\langle a _ { 1 } , \\dots , a _ { n } \\rangle \\in F \\} . \\end{align*}"}
-{"id": "8649.png", "formula": "\\begin{align*} \\begin{aligned} \\pi [ E _ N ] \\leq & \\pi [ A ( N ) ] + \\pi [ E _ N \\cap A ( N ) ^ c ] \\leq & \\sum _ { k = N } ^ \\infty \\pi [ A _ k ] + \\pi [ E _ N \\cap A ( N ) ^ c ] . \\end{aligned} \\end{align*}"}
-{"id": "3071.png", "formula": "\\begin{align*} C ( t ) : = \\{ u \\in X : \\ , \\nabla _ x F ( t , u ) = 0 \\} \\end{align*}"}
-{"id": "1488.png", "formula": "\\begin{align*} { \\rm R e } \\Big ( \\cfrac { z w _ 2 ' ( z ) } { w _ 2 ( z ) } \\Big ) \\geq \\sqrt { c _ \\alpha } r \\cot ( \\sqrt { c _ \\alpha } r ) \\mbox { f o r $ | z | = r < 1 . $ } \\end{align*}"}
-{"id": "2752.png", "formula": "\\begin{align*} | | g | | _ { H _ { x , v } ^ { s } L _ z ^ 2 } ^ 2 : = \\int _ { I _ z } \\ , | | g | | _ { H _ { x , v } ^ s } ^ 2 \\pi ( z ) \\ , d z . \\end{align*}"}
-{"id": "2813.png", "formula": "\\begin{align*} \\frac { d \\widehat { P } } { d t } + \\Big ( \\frac { \\sigma ^ 2 } { 2 } ( \\omega ^ { 2 } + \\omega ) - r \\omega - r \\Big ) \\widehat { P } = \\frac { - r K } { \\omega } ( \\mathcal { B } ( t ) ) ^ { \\omega } . \\end{align*}"}
-{"id": "9790.png", "formula": "\\begin{align*} V _ Q = \\left \\{ \\big { ( } ( z _ 0 , z _ 1 , z _ 2 ) , [ w _ 0 : w _ 1 ] \\big { ) } \\in U _ Q \\times \\P ^ 1 \\mid z _ 0 w _ 1 - z _ 1 w _ 0 = 0 \\right \\} . \\end{align*}"}
-{"id": "9296.png", "formula": "\\begin{align*} Y _ z : = \\{ x \\in I \\ : \\ \\dim ( W _ z ) _ x = 1 \\} . \\end{align*}"}
-{"id": "997.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { x \\in \\mathbb { R } } P \\left ( \\left | \\sup _ { t \\in [ a _ n , T - a _ n ] } | Z _ n ( t ) | - x \\right | \\leq c _ 1 v _ n + 8 \\varepsilon \\right ) = O ( v _ n \\sqrt { \\log n } ) + O \\left ( \\frac { \\log n } { ( n h ) ^ { 1 / 6 } } \\right ) . \\end{align*}"}
-{"id": "9020.png", "formula": "\\begin{align*} \\Delta _ { k } \\theta ( t _ { 1 } ) - \\Delta _ { k } \\theta ( t _ { 2 } ) = & \\int _ { t _ { 1 } } ^ { t _ { 2 } } { \\frac { d } { d \\tau } \\Delta _ { k } \\theta ( \\tau ) \\ , d \\tau } = - \\int _ { t _ { 1 } } ^ { t _ { 2 } } { \\Delta _ { k } [ ( u \\cdot \\nabla ) \\theta + \\Lambda _ { x _ { 1 } } ^ { 2 \\alpha } \\theta + \\Lambda _ { x _ { 2 } } ^ { 2 \\beta } \\theta ] ( \\tau ) \\ , d \\tau } . \\end{align*}"}
-{"id": "1812.png", "formula": "\\begin{align*} F _ { \\beta _ i ( t ) } \\mid _ { \\alpha _ { i , 1 } ( t ) \\leftarrow \\frac { \\alpha _ { i - 1 , 2 } ( t ) \\beta _ { i + 1 } ( t ) \\beta _ { i + 1 } ( t + 1 ) } { \\alpha _ { i , 1 } ( t + 1 ) } } = \\frac { \\left ( \\begin{aligned} & \\beta _ { i - 1 } ( t ) \\alpha _ { i - 1 , 2 } ( t ) \\beta _ { i + 1 } ( t ) \\beta _ { i + 1 } ( t + 1 ) + \\\\ & + \\alpha _ { i , 1 } ( t + 1 ) \\alpha _ { i - 1 , 1 } ( t ) \\beta _ { i + 1 } ( t ) \\end{aligned} \\right ) } { \\alpha _ { i , 1 } ( t + 1 ) } . \\end{align*}"}
-{"id": "6867.png", "formula": "\\begin{align*} - \\Delta _ { m } ^ { ( p ) } f \\left ( \\frac { k } { 4 ^ { m } } \\right ) = \\lambda f \\left ( \\frac { k } { 4 ^ { m } } \\right ) . \\end{align*}"}
-{"id": "6546.png", "formula": "\\begin{align*} \\left | \\left \\{ x \\in \\mathbb { R } ^ n : \\left \\langle x , \\frac { \\zeta } { \\| \\zeta \\| } \\right \\rangle \\geq \\frac { 1 } { \\| \\zeta \\| } - \\Delta \\right \\} \\cap P \\right | _ n = \\Delta | F _ { \\zeta } | _ { n - 1 } ( 1 + T _ { \\zeta } ( \\Delta ) ) . \\end{align*}"}
-{"id": "767.png", "formula": "\\begin{align*} S _ { k } ( x ) = y ^ { 2 n } , \\ \\mbox { i n p o s i t i v e i n t e g e r s } \\ \\ x , y , n \\ \\mbox { w i t h } \\ n > 2 \\end{align*}"}
-{"id": "8765.png", "formula": "\\begin{align*} x [ [ f _ 1 , f _ 2 ] , f _ 3 ] ^ { ( t _ 1 , { { s _ 3 } } , s _ 1 , s _ 2 ) } & = x A d _ { e ^ { { { s _ 3 } } f _ 3 } \\Psi ( t _ 1 , s _ 2 ) } \\Big ( \\left [ [ f _ 1 , f _ 2 ] , f _ 3 \\right ] + \\displaystyle \\int _ 0 ^ { s _ 2 } \\Big [ A d _ { e ^ { \\tau f _ 2 } } [ f _ 2 , [ f _ 1 , f _ 2 ] ] , f _ 3 \\Big ] d \\tau \\\\ & + \\displaystyle \\int _ 0 ^ { s _ 1 } \\Big [ A d _ { e ^ { s _ 2 f _ 2 } e ^ { \\sigma f _ 1 } } [ f _ 1 , [ f _ 1 , f _ 2 ] ] , f _ 3 \\Big ] d \\sigma \\Big ) . \\end{align*}"}
-{"id": "9648.png", "formula": "\\begin{align*} & \\Lambda _ \\beta ( t ) = \\frac { S _ \\beta } { 4 \\pi E _ { \\rm { b } } } \\sqrt { P _ { \\rm { c u } } \\left ( 2 ^ { C / W } - 1 \\right ) P _ { \\rm { t r } , 1 } ( h _ { \\beta , 1 } ^ * ) / \\lambda ^ 3 _ { \\beta } } . \\end{align*}"}
-{"id": "6207.png", "formula": "\\begin{align*} \\begin{pmatrix} - 2 & 0 & a \\\\ 0 & - 2 & b \\\\ a & b & c \\end{pmatrix} . \\end{align*}"}
-{"id": "1382.png", "formula": "\\begin{align*} \\boldsymbol U = \\{ U _ { a , b } \\} _ { \\infty < a \\le b < \\infty } , \\end{align*}"}
-{"id": "705.png", "formula": "\\begin{align*} p = ( n _ { \\rm a } + n _ { \\rm i } + n _ { \\rm e } ) k T = ( 1 + \\alpha ) n _ { \\rm p } k T = ( 1 + \\alpha ) \\frac { \\rho } { m _ { \\rm p } } k T . \\end{align*}"}
-{"id": "8399.png", "formula": "\\begin{align*} \\frac { \\partial f _ 0 } { \\partial t } = \\ , & \\Delta f _ 0 + \\frac { 4 H } { | H | ^ 2 } \\left \\langle \\nabla | H | , \\nabla f _ 0 \\right \\rangle - \\frac { 2 } { | H | ^ 2 } \\left [ | \\nabla h | ^ 2 - \\left ( \\frac { 1 } { n } + f _ 0 \\right ) | \\nabla H | ^ 2 \\right ] \\\\ [ 5 p t ] & + \\frac { 2 } { | H | ^ 2 } \\left [ R _ 1 - \\left ( \\frac { 1 } { n } + f _ 0 \\right ) R _ 2 \\right ] - 4 n K f _ 0 . \\end{align*}"}
-{"id": "3882.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\omega ( n ) \\tau ( n ) \\tau ( n + l ) , l \\geq 1 , \\tau ( n ) = \\sum _ { d | n } 1 , \\end{align*}"}
-{"id": "5501.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { u _ m ( r ^ n z ) } { ( r ^ n z ) ^ { - \\beta } \\ell ( r ^ n z ) } = p _ { 0 , m } ( z ) z \\in C _ { p _ { 0 , m } } , \\end{align*}"}
-{"id": "2207.png", "formula": "\\begin{align*} \\frac { \\partial ^ { | | \\beta ( \\alpha ( J ) | | } } { \\partial z ^ { \\beta ( \\alpha , J ) } } = \\frac { \\partial ^ { m _ { 1 j _ { 1 } } ( \\alpha _ { j _ { 1 } } + 1 ) - 1 + \\ldots + m _ { n j _ { n } } ( \\alpha _ { j _ { n } } + 1 ) - 1 } } { \\partial z _ 1 ^ { m _ { 1 j _ { 1 } } ( \\alpha _ { j _ { 1 } } + 1 ) - 1 } \\ldots \\partial z _ n ^ { m _ { n j _ { n } } ( \\alpha _ { j _ { n } } + 1 ) - 1 } } . \\end{align*}"}
-{"id": "5576.png", "formula": "\\begin{align*} S _ { n } \\approx 1 - \\delta ^ { 2 } \\sum _ { i = 0 } ^ { n - 1 } \\left [ S _ { i } \\sum \\limits _ { j = i + 1 } ^ { n } \\xi _ { j } \\right ] \\end{align*}"}
-{"id": "2021.png", "formula": "\\begin{align*} f _ a ( z _ 1 , z _ 2 ) = f _ a ( z _ 2 , z _ 1 ) . \\end{align*}"}
-{"id": "4179.png", "formula": "\\begin{align*} | \\phi | = 1 - \\frac { k } { 2 r } + O ( r ^ { - 2 } ) \\ ; \\ ; r \\rightarrow \\infty , \\end{align*}"}
-{"id": "9894.png", "formula": "\\begin{align*} 2 e ( G ) = \\sum _ { i = 1 } ^ r e ( B _ i , V ) \\ , . \\end{align*}"}
-{"id": "4855.png", "formula": "\\begin{align*} d ( 1 2 d - 2 7 ) + 3 d ( d - 2 ) & = s + 3 0 \\sum \\delta _ p + \\sum _ I ( 4 m _ p + 4 l _ p - 1 5 ) + \\sum _ J ( 1 0 m _ p + c _ p - 1 5 ) , \\\\ d ( 1 2 d - 2 7 ) & = s + 2 4 \\sum \\delta _ p + \\sum _ I ( 3 m _ p + 3 l _ p - 1 2 ) + \\sum _ J ( 7 m _ p + c _ p - 1 2 ) . \\end{align*}"}
-{"id": "180.png", "formula": "\\begin{align*} \\mathrm { T h } _ \\vdash ( x ) + ^ { \\vdash } \\mathrm { T h } _ \\vdash ( y ) = \\mathrm { T h } _ \\vdash ( x + y ) . \\end{align*}"}
-{"id": "8936.png", "formula": "\\begin{gather*} \\begin{pmatrix} 2 r _ i & - \\psi _ { j i } \\\\ - \\psi _ { j i } ^ \\vee & 2 r _ j \\end{pmatrix} \\ ! , \\end{gather*}"}
-{"id": "850.png", "formula": "\\begin{align*} \\partial _ s h ( r , s ) = - D _ { [ s , T ) } ^ { 1 - \\beta } [ \\partial _ r h ( r , s ) ] - \\delta _ T ( r ) \\delta _ T ( s ) . \\end{align*}"}
-{"id": "2768.png", "formula": "\\begin{align*} h ( P _ 1 ) = h ( P _ 2 ) + ( k _ 1 - k _ 2 ) ( p - 1 ) , \\end{align*}"}
-{"id": "4732.png", "formula": "\\begin{align*} { \\varepsilon } = F ( X , Y , Z ) { \\Delta \\sqrt { - \\det g ^ { i j } } } / { ( \\alpha + \\beta a _ 0 + \\gamma b _ 0 ) } , \\end{align*}"}
-{"id": "6722.png", "formula": "\\begin{align*} \\dfrac { | J _ { l } | } { N ^ { 2 l } } \\leq \\frac { 3 ! \\ N ^ { 2 l - 3 } + N \\ | J _ { l - 1 } | } { N ^ { 2 l } } \\leq \\frac { 3 ! \\ N ^ { 2 l - 3 } + N \\ ( 2 N ^ { 2 l - 4 } ) } { N ^ { 2 l } } = \\frac { 8 } { N ^ { 3 } } . \\end{align*}"}
-{"id": "8797.png", "formula": "\\begin{align*} \\mathbb { A } : = \\mathbb { Z } \\Big [ \\mathbb { L } , \\mathbb { L } ^ { - 1 } , \\Big ( \\frac { 1 } { 1 - \\mathbb { L } ^ { - { i } } } \\Big ) _ { i > 0 } \\Big ] . \\end{align*}"}
-{"id": "6202.png", "formula": "\\begin{align*} \\begin{pmatrix} 2 & 0 & 0 & 0 \\\\ 0 & 2 & 0 & 0 \\\\ 0 & 0 & - 4 & - 1 - 2 n \\\\ 0 & 0 & - 1 - 2 n & - 2 ( 1 + n ^ 2 ) \\end{pmatrix} \\end{align*}"}
-{"id": "5194.png", "formula": "\\begin{align*} \\sup _ { y \\in [ a _ { 1 } , r ) } \\hat { U } _ { 2 } ( \\ell ^ { 1 } , y , r ) = \\sup _ { y \\in [ a _ { 1 } , a _ { 2 } ] } \\hat { U } _ { 2 } ( \\ell ^ { 1 } , y , r ) & = \\hat { U } _ { 2 } ( \\ell ^ { 1 } , \\ell ^ { 2 } , r ) , \\end{align*}"}
-{"id": "3678.png", "formula": "\\begin{align*} u = \\ ! \\ ! \\mbox { \\phantom { $ \\Big ( $ } } ^ t \\ ! \\left ( - \\frac { n ^ 2 + n - 1 } { n + 1 } , - \\frac { n ^ 2 - 3 } { n + 1 } , \\dots , - \\frac { n + 3 } { n + 1 } , - \\frac { 1 } { n + 1 } , \\right ) . \\end{align*}"}
-{"id": "7820.png", "formula": "\\begin{align*} F ( m , r _ j , s ) + \\alpha \\int _ { S _ { r _ j } } r ^ { s - 1 } v _ m ^ 2 e ^ { - 2 \\rho } d x = o ( 1 ) . \\end{align*}"}
-{"id": "2853.png", "formula": "\\begin{align*} \\bar { a } _ { n v } = \\sum _ { i = v } ^ { n } a _ { n i } , n , v = 0 , 1 , . . . \\bar { \\Delta } a _ { n v } = a _ { n v } - a _ { n - 1 , v } , a _ { - 1 , 0 } = 0 \\end{align*}"}
-{"id": "9611.png", "formula": "\\begin{align*} N _ \\beta ( t ) = \\frac { S _ \\beta } { \\pi R ^ 2 _ { \\beta } ( t ) } , \\end{align*}"}
-{"id": "1349.png", "formula": "\\begin{align*} { \\rm a f f } \\left ( { \\cal C } _ { { \\cal S } _ + ^ p } ( \\overline { M } ) \\right ) = \\left \\{ B \\in { \\cal S } ^ p \\ , | \\ , P _ { \\beta } ^ T B P _ { \\gamma } = 0 , \\ P _ { \\gamma } ^ T B P _ { \\gamma } = 0 \\right \\} . \\end{align*}"}
-{"id": "5325.png", "formula": "\\begin{align*} \\hat { \\mathfrak { g } } _ { \\pm } = \\mathfrak { g } \\otimes t ^ { \\pm 1 } \\C [ t ^ { \\pm 1 } ] \\end{align*}"}
-{"id": "498.png", "formula": "\\begin{align*} V _ 1 ( x , t ) & \\sim - \\frac { R } { 4 } \\frac { 1 } { \\delta ^ 2 } + \\frac { R } { 2 } \\frac { 1 } { \\delta ^ 2 } = \\frac { \\pi } { 4 } \\abs * { t } \\asymp d ( x , t ) ^ 2 . \\end{align*}"}
-{"id": "1557.png", "formula": "\\begin{align*} b _ m = - \\frac { 1 } { m } \\sum _ { J } \\prod _ { k = 1 } ^ { n } C _ { j _ k } ( \\alpha ( k ) ) \\end{align*}"}
-{"id": "1122.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & x _ { i j } & x _ { i h } \\\\ x _ { i j } & 1 & x _ { j h } \\\\ x _ { i h } & x _ { j h } & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "4553.png", "formula": "\\begin{align*} & [ E _ i ( z ) , F _ j ( w ) ] = \\frac { \\delta _ { i , j } } { q - q ^ { - 1 } } ( \\delta \\bigl ( C \\frac { w } { z } \\bigr ) K _ i ^ + ( w ) - \\delta \\bigl ( C \\frac { z } { w } \\bigr ) K _ i ^ - ( z ) ) \\ , . \\end{align*}"}
-{"id": "2519.png", "formula": "\\begin{align*} \\left | \\nabla _ { \\xi } \\varrho \\right | = \\left | \\alpha \\varrho \\nabla _ { \\xi } \\vartheta ( 0 , x , \\xi ) \\right | \\lesssim \\alpha \\varrho \\left < \\xi \\right > , \\end{align*}"}
-{"id": "2613.png", "formula": "\\begin{align*} \\mathcal C _ 0 ( t ^ k ) = \\frac k { \\pi ^ 2 \\phi _ 0 ( t ) } \\bigg [ S ( \\phi _ 0 ( t ) t ^ { k - 1 } ) + \\frac { 1 } { 2 \\log 2 } \\int _ 0 ^ 1 \\frac { s ^ k } { \\phi _ 0 ( s ) } \\ , d s \\bigg ] . \\end{align*}"}
-{"id": "9426.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } n | n | _ { p } | n | _ { q } \\| n \\alpha \\| = 0 . \\end{align*}"}
-{"id": "9427.png", "formula": "\\begin{align*} \\sum _ { n \\in \\N } ( \\log n ) ^ k \\psi ( n ) = \\infty . \\end{align*}"}
-{"id": "2155.png", "formula": "\\begin{align*} & | u _ * ( | x | , t ) | \\le C t ^ { - \\gamma - \\frac { d } { 2 } } , \\\\ & | F _ N ^ 0 ( | x | , t ) | \\le C t ^ { - \\gamma - \\frac { d } { 2 } - 1 } | x | ^ 2 \\le C \\epsilon ^ 2 t ^ { - \\gamma - \\frac { d } { 2 } } , \\\\ & | ( \\partial _ r F _ N ^ 0 ) ( | x | , t ) | \\le C t ^ { - \\gamma - \\frac { d } { 2 } - 1 } | x | \\le C \\epsilon t ^ { - \\gamma - \\frac { d } { 2 } - \\frac { 1 } { 2 } } , \\end{align*}"}
-{"id": "4694.png", "formula": "\\begin{align*} \\Delta v = 1 \\qquad \\mathrm { o n } \\ , \\ , \\R ^ n . \\end{align*}"}
-{"id": "1003.png", "formula": "\\begin{align*} \\widetilde { Z } _ n ^ * ( t ) = \\frac { 1 } { \\mathfrak { s } _ n ( t ) } \\sum _ { i = 1 } ^ n K _ h ( t _ { i - 1 } - t ) ( X _ { t _ { i } } - X _ { t _ { i - 1 } } ) ^ 2 \\xi _ i ^ * , t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "9652.png", "formula": "\\begin{align*} \\varepsilon ( d , N _ { \\rm { s } , \\beta } ( t ) , \\delta ) = \\sqrt { \\frac { 1 } { 2 N _ { \\rm { s } , \\beta } ( t ) } \\left ( \\log d + \\log \\frac { 1 } { \\delta } \\right ) } , \\end{align*}"}
-{"id": "1956.png", "formula": "\\begin{align*} T _ { j , j } ( f ) : = ( i \\pi d _ { j - 1 } + P _ { j - 1 } ) T _ { j , j - 1 } ( r _ { j - 1 } ) ( f ) & = \\prod _ { i = 1 } ^ { j - 1 } ( i \\pi d _ i + P _ i ) \\widetilde { S } _ { j , \\mathbf { r } } ( f ) \\\\ & = \\widehat { \\widetilde { Q } _ j ( f ) } . \\end{align*}"}
-{"id": "9226.png", "formula": "\\begin{align*} \\frac { 1 } { n } e _ { 2 } \\otimes ( [ \\alpha _ { 2 } , \\alpha _ { 1 } ] _ { A ^ { - } } \\alpha _ { 3 } - \\langle \\alpha _ { 2 } , \\alpha _ { 1 } \\rangle \\alpha _ { 3 } ) = \\frac { 1 } { 2 } e _ { 2 } \\otimes ( [ \\alpha _ { 3 } , \\alpha _ { 2 } ] _ { A ^ { - } } \\alpha _ { 1 } + ( \\alpha _ { 3 } \\circ \\alpha _ { 2 } ) _ { A ^ { + } } \\alpha _ { 1 } - [ \\alpha _ { 3 } , \\alpha _ { 1 } ] _ { C } \\alpha _ { 2 } - ( \\alpha _ { 3 } \\circ \\alpha _ { 1 } ) _ { E } \\alpha _ { 2 } ) , \\end{align*}"}
-{"id": "6450.png", "formula": "\\begin{align*} R _ { } ^ { } ( \\rho ) \\overset { } { = } \\frac { \\mathcal { C } _ { \\mathcal { M } } ( \\tau ) } { \\mathcal { C } _ { \\mathcal { M } } \\left . ( \\tau ) \\right \\vert _ { \\rho = 0 } } = \\sqrt { 3 } \\sqrt { \\frac { 1 + \\rho } { 3 + \\rho } } \\end{align*}"}
-{"id": "8429.png", "formula": "\\begin{align*} \\sum _ { t = - \\infty } ^ { \\infty } q ^ { t ( \\frac { 1 } { 2 } - s ) } c _ { t , l } ( \\mu ) = \\omega _ { \\pi } ( - 1 ) G ( \\varpi ^ { - l } , \\mu ^ { - 1 } ) \\frac { L ( s , \\abs { \\cdot } ^ c ) L ( s , \\abs { \\cdot } ^ { - c } ) } { L ( 1 - s , \\abs { \\cdot } ^ c ) L ( 1 - s , \\abs { \\cdot } ^ { - c } ) } . \\end{align*}"}
-{"id": "2212.png", "formula": "\\begin{align*} \\widetilde q _ i = ( w _ 1 - a _ { i 1 } ) ^ { m _ { i 1 } } \\cdot \\ldots \\cdot ( w _ n - a _ { i n } ) ^ { m _ { i n } } , \\end{align*}"}
-{"id": "5939.png", "formula": "\\begin{align*} { \\rm A r f } ( V _ { \\phi } ) & = \\sum _ { i = 1 } ^ m \\phi ( e _ i ) \\phi ( f _ i ) ~ ~ ~ { \\rm m o d } ~ N , \\end{align*}"}
-{"id": "9363.png", "formula": "\\begin{align*} \\lambda ( x , y ) & = \\frac { N } { 2 \\pi \\sigma ^ 2 } \\mathrm { e } ^ { - \\frac { x ^ 2 + y ^ 2 } { 2 \\sigma ^ 2 } } . \\end{align*}"}
-{"id": "1811.png", "formula": "\\begin{align*} \\sigma ( h _ 0 , \\dots , h _ { n - 2 } , h _ { n - 1 } ) : = ( 0 , h _ 0 , \\dots , h _ { n - 2 } ) . \\end{align*}"}
-{"id": "5250.png", "formula": "\\begin{align*} L _ c P = \\lambda P . \\end{align*}"}
-{"id": "242.png", "formula": "\\begin{align*} f ^ { T } _ { n } : = h \\big ( g ( H ^ { T } ) _ { \\mathbb { R } _ { \\geq 0 } ^ n \\times \\R ^ { d - n } } \\big ) \\end{align*}"}
-{"id": "7038.png", "formula": "\\begin{align*} \\sigma _ 1 + \\sigma _ 2 + . . . + \\sigma _ { n - 1 } = 2 \\epsilon = 2 \\eta _ n . \\end{align*}"}
-{"id": "8491.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { t , l , v } ) = \\zeta _ F ( 1 ) ^ { - 1 } q ^ { - \\frac { a _ 1 + t } { 2 } } q ^ { s ( l _ 1 - l _ 2 ) } \\epsilon ( \\frac { 1 } { 2 } , \\chi _ 1 ^ { - 1 } ) G ( \\varpi ^ { - t - a _ 1 } - \\varpi ^ { - l } v b _ 1 , \\chi _ 2 ) . \\end{align*}"}
-{"id": "9859.png", "formula": "\\begin{align*} x ^ { \\nu + 1 } \\ , = \\ , x ^ \\nu + \\gamma ^ \\nu d ( x ^ \\nu ) . \\end{align*}"}
-{"id": "300.png", "formula": "\\begin{align*} Z \\overset { d } { = } ( 1 - \\xi ) Z + \\xi \\Theta . \\end{align*}"}
-{"id": "8755.png", "formula": "\\begin{align*} \\norm { ( \\frac { 1 } { M } P _ f ^ 2 + H _ f - \\mu ) ^ { - 1 } a ^ * ( \\chi _ n - \\chi _ \\lambda ) } & \\leq \\norm { ( H _ f - \\mu ) ^ { - 1 } a ^ * ( \\chi _ n - \\chi _ \\lambda ) } \\\\ & = \\norm { a ( \\chi _ n - \\chi _ \\lambda ) ( H _ f - \\mu ) ^ { - 2 } a ^ * ( \\chi _ n - \\chi _ \\lambda ) } ^ { 1 / 2 } , \\end{align*}"}
-{"id": "7425.png", "formula": "\\begin{align*} a : = a _ 0 a _ 1 a _ 2 a _ 3 , a ^ * : = a _ 3 ^ * a _ 2 ^ * a _ 1 ^ * a _ 0 ^ * , d : = a _ 3 ^ * a _ 3 , b : = a _ 5 ^ * a _ 5 , c : = a _ 7 ^ * a _ 7 . \\end{align*}"}
-{"id": "510.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n E [ d ( X _ i , \\widehat { X } _ i ( Y ^ n , W ) ) ] \\end{align*}"}
-{"id": "223.png", "formula": "\\begin{align*} c _ { m , n } & : = \\frac { ( - 1 ) ^ { m - n } 2 ^ { m } d ! } { ( m - n ) ! ( d - m ) ! } , \\\\ \\widehat { \\chi } _ { m , n } & : = \\chi _ { \\R _ { \\geq 0 } ^ d } \\chi _ { \\{ x _ 1 \\leq . . . \\leq x _ { n } \\} } \\chi _ { \\{ x _ n \\geq x _ { n + 1 } , . . . , x _ { m } \\} } \\chi _ { \\{ x _ { m + 1 } , . . . , x _ { d } \\in [ 0 , 1 ] \\} } . \\end{align*}"}
-{"id": "5974.png", "formula": "\\begin{align*} \\mathrm { e } ^ { 2 s t - t ^ 2 } = \\sum _ { n \\geq 0 } \\frac { t ^ n } { n ! } h _ n ( s ) \\end{align*}"}
-{"id": "6343.png", "formula": "\\begin{align*} \\varphi _ n ^ { \\mathbb { R } _ + } : { \\vec z _ n } = ( z _ 1 , \\dots , z _ n ) \\mapsto { \\vec m _ n } = ( m _ 1 , \\dots , m _ n ) \\end{align*}"}
-{"id": "8741.png", "formula": "\\begin{align*} E ( t , x , u , \\partial _ t ^ \\alpha u , \\nabla u , \\nabla ^ 2 u ) = 0 \\quad \\end{align*}"}
-{"id": "7994.png", "formula": "\\begin{align*} \\widehat { a } ( \\eta , \\xi ) = 0 C \\big ( | \\eta + \\xi | + 1 \\big ) \\leq | \\xi | \\end{align*}"}
-{"id": "7656.png", "formula": "\\begin{align*} \\lambda = - 1 . \\end{align*}"}
-{"id": "8088.png", "formula": "\\begin{align*} \\hat { \\pi } ( x _ { i _ 1 } ^ { \\ell + 1 } \\wedge \\cdots \\wedge x _ { i _ t } ^ { \\ell + 1 } ) = ( x _ { i _ 1 } \\cdots x _ { i _ t } ) ( x _ { i _ 1 } ^ \\ell \\wedge \\cdots \\wedge x _ { i _ t } ^ \\ell ) . \\end{align*}"}
-{"id": "9668.png", "formula": "\\begin{align*} ( u , v ) \\cdot \\textbf { n } = 0 \\ , \\ , \\Gamma . \\end{align*}"}
-{"id": "5692.png", "formula": "\\begin{align*} | { v } ( t ) - a ^ - | = | { v } ( - t ) - a ^ + | \\geq \\varepsilon _ 1 . \\end{align*}"}
-{"id": "3118.png", "formula": "\\begin{align*} \\frac { \\sigma ( \\gamma _ 0 , \\gamma _ 1 \\circ \\gamma _ 2 ) } { \\sigma ( \\gamma _ 0 \\circ \\gamma _ 1 , \\gamma _ 2 ) } \\ ; = \\ ; \\frac { \\sigma ( \\gamma _ 0 , \\gamma _ 1 ) } { \\sigma ( \\gamma _ 1 , \\gamma _ 2 ) } \\ , . \\end{align*}"}
-{"id": "114.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { l _ 0 } \\omega _ l ( p - m _ l ) = v _ 0 ( p ) = \\alpha _ 3 \\left ( d \\left ( n , \\partial ( x ) \\right ) \\right ) . \\end{align*}"}
-{"id": "2464.png", "formula": "\\begin{align*} | g | _ { L _ { \\xi } ^ { 2 } ( m ) } = \\Big ( \\int _ { { \\mathbb { R } } ^ { 3 } } | g | ^ { 2 } m d \\xi \\Big ) ^ { 1 / 2 } \\ , . \\end{align*}"}
-{"id": "310.png", "formula": "\\begin{align*} \\P ( \\Theta = E _ j \\ | \\ Z = z ) = p _ j ( z ) , j = 0 , 1 , . . . , d , \\end{align*}"}
-{"id": "2529.png", "formula": "\\begin{align*} \\begin{aligned} T _ 2 & = - \\int h \\left ( \\nabla _ { \\xi } \\psi , \\nabla _ { x } h \\right ) \\varrho \\ , m _ 0 = - \\frac { 1 } { 2 } \\int \\left ( \\nabla _ { \\xi } \\psi , \\nabla _ { x } h ^ 2 \\right ) \\varrho \\ , m _ 0 \\\\ & = \\frac { 1 } { 2 } \\int h ^ 2 \\left [ \\frac { ( \\nabla _ { \\xi } \\psi , \\nabla _ { x } \\varrho ) } { \\varrho } \\right ] \\varrho \\ , m _ 0 , \\end{aligned} \\end{align*}"}
-{"id": "3864.png", "formula": "\\begin{align*} & \\nabla f ( x ^ * ) + \\sum \\limits _ { i = 1 } ^ m \\lambda _ i ^ * \\nabla g _ i ( x ^ * ) + \\sum \\limits _ { i = 1 } ^ p \\mu _ i ^ * \\nabla h _ i ( x ^ * ) + \\sum \\limits _ { i = 1 } ^ n \\gamma _ i ^ * e _ i = 0 , \\\\ & g _ i ( x ^ * ) \\leq 0 , \\ \\lambda _ i ^ * \\geq 0 , \\ \\lambda _ i ^ * g _ i ( x ^ * ) = 0 , \\quad \\forall i = 1 , \\dots , m , \\\\ & h _ i ( x ^ * ) = 0 , \\quad \\forall i = 1 , \\dots , p , \\\\ & \\gamma _ i ^ * = 0 , \\quad \\forall i \\in I _ { \\pm 0 } ( x ^ * , y ^ * ) . \\end{align*}"}
-{"id": "2821.png", "formula": "\\begin{align*} \\mu ( X _ w ^ { ( n ) } ) = h _ w ^ { ( n ) } p _ w ^ { ( n ) } = : q _ w ^ { ( n ) } . \\end{align*}"}
-{"id": "8349.png", "formula": "\\begin{align*} A _ z ( \\C ) = g H _ \\Z \\backslash H _ \\C / z H _ \\C . \\end{align*}"}
-{"id": "5920.png", "formula": "\\begin{align*} \\mathcal { R } : \\{ s ' \\in A _ { } : r _ { s ' } = r \\} \\stackrel { } { \\to } \\{ s ' \\in A _ { } : r _ { s ' } = r - 1 \\} , \\ \\ r \\ge 3 , \\end{align*}"}
-{"id": "541.png", "formula": "\\begin{align*} H ( \\phi ^ t _ H ( x ) ) = H ( x ) . \\end{align*}"}
-{"id": "3300.png", "formula": "\\begin{align*} \\int _ 1 ^ \\infty ( x - 1 ) ^ { \\mu - 1 } ( x ^ 2 + \\beta ) ^ \\nu d x = \\frac { \\Gamma ( \\mu ) \\Gamma ( - \\mu - 2 \\nu ) } { \\Gamma ( - 2 \\nu ) } { _ 2 F _ 1 } \\left ( - \\frac { \\mu } { 2 } - \\nu , \\frac { 1 - \\mu } { 2 } - \\nu ; \\frac { 1 } { 2 } - \\nu ; - \\beta \\right ) . \\end{align*}"}
-{"id": "9635.png", "formula": "\\begin{align*} \\Omega _ { \\rm { m } } ( \\tau _ { i } ) = \\sum _ { l = 1 } ^ { \\zeta _ \\tau | _ { \\tau _ { i - 1 } } ^ { \\tau _ { i } } } P _ { \\rm { h } } \\frac { d _ \\beta ( \\tau _ { i } , l ) } { v _ { \\rm { h } } } + I ( \\Delta h _ \\beta ( \\tau _ { i } , l ) ) P _ { \\rm { a } } \\frac { \\Delta h _ \\beta ( \\tau _ { i } , l ) } { v _ { \\rm { a } } } - \\big ( 1 - I ( \\Delta h _ \\beta ( \\tau _ { i } , l ) ) \\big ) P _ { \\rm { d } } \\frac { \\Delta h _ \\beta ( \\tau _ { i } , l ) } { v _ { \\rm { d } } } , \\end{align*}"}
-{"id": "516.png", "formula": "\\begin{align*} \\frac { \\log | \\mathcal { W } | } { n } & \\ ! = \\ ! \\frac { m _ 2 } { n } \\ ! = \\ ! H _ b ( q * p _ A ) \\ ! - \\ ! H _ b ( q ) + 2 \\delta \\end{align*}"}
-{"id": "5126.png", "formula": "\\begin{align*} F ( & x , y , p , q , s ) : = \\\\ & \\left ( \\left ( \\sum _ { ( 1 , \\beta ) \\in P } ( p _ { 1 , \\beta } - y _ 1 q _ { 1 , \\beta } ) s ^ \\beta , \\ldots , \\sum _ { ( n , \\beta ) \\in P } ( p _ { n , \\beta } - y _ n q _ { n , \\beta } ) s ^ \\beta \\right ) \\right . , y , p , q , \\\\ & \\left . \\vphantom { \\sum _ { ( 1 , \\beta ) \\in P } } \\{ s _ { j , \\alpha } - \\partial ^ \\alpha f _ j ( x ) \\} _ { ( j , \\alpha ) \\in S } \\right ) . \\end{align*}"}
-{"id": "3425.png", "formula": "\\begin{align*} K \\cup L = \\overline D _ 2 , K \\cap L = \\overline D _ 2 \\cap \\bigl ( ( \\overline V ^ + _ 1 \\setminus W ^ + _ 1 ) \\cup ( \\overline V ^ - _ 1 \\setminus W ^ - _ 1 ) \\bigr ) . \\end{align*}"}
-{"id": "1690.png", "formula": "\\begin{align*} \\mu _ x ( Z ( i _ 1 i _ 2 \\cdots i _ n ) ) = T _ { i _ 1 , i _ 2 } T _ { i _ 2 , i _ 3 } \\cdots T _ { i _ { n - 1 } , i _ n } , \\end{align*}"}
-{"id": "8430.png", "formula": "\\begin{align*} W _ { \\pi } \\left ( \\left ( \\begin{matrix} z & 0 \\\\ 0 & z \\end{matrix} \\right ) n ( x ) g _ { t , l , v } k \\right ) = \\omega _ { \\pi } ( z ) \\psi ( x ) W _ { \\pi } ( g _ { t , l , v } ) k \\in K _ { 1 } ( n ) . \\end{align*}"}
-{"id": "5888.png", "formula": "\\begin{align*} P ( \\cup _ { n = 1 } ^ M \\{ Z _ { 2 n } ^ { N , i } = - 1 \\} ) = 1 - ( 1 - a _ N ) ^ M . \\end{align*}"}
-{"id": "9961.png", "formula": "\\begin{align*} \\big ( V ( { \\cal C } ) \\cup { \\cal N } _ s ( u ) \\cup \\{ p ( u ) \\} \\big ) - V _ 1 = \\big ( V ( { \\cal C } ) \\cup { \\cal N } _ s ( \\varphi ( u ) ) \\cup \\{ p ( \\varphi ( u ) ) \\} \\big ) - V _ 2 . \\end{align*}"}
-{"id": "2409.png", "formula": "\\begin{align*} p _ 1 ( x ) = - a _ 6 \\left ( x \\pm \\frac { 1 } { \\sqrt { 2 } } \\right ) ^ 3 , p _ 3 ( x ) = \\frac { 9 \\sqrt { 2 } } { 8 } a _ 6 \\left ( x \\pm \\frac { 1 } { \\sqrt { 2 } } \\right ) . \\end{align*}"}
-{"id": "1703.png", "formula": "\\begin{align*} U ( f ) ( x ) \\ ; = \\ ; \\sqrt { g _ 2 ( x ) } \\cdot f ( x ) , \\ ; f \\in \\ ; L ^ 2 ( X , \\mu ) , \\end{align*}"}
-{"id": "8518.png", "formula": "\\begin{align*} \\bar \\sigma ( l , v , u , C ) = \\min \\{ f ( \\bar v , \\bar u , z ) : z \\in \\mathbb { Z } , \\ , | z | \\leq C , \\ , \\bar v _ i = v _ i + b _ { i \\ , l } z , \\ , \\bar u _ i = u _ i + \\bar b _ { i \\ , l } z \\} , \\end{align*}"}
-{"id": "2924.png", "formula": "\\begin{align*} u [ g ] : = [ A _ s \\times \\{ g \\} ] = \\{ [ ( a , g ) ] \\colon a \\in A _ s \\} , \\end{align*}"}
-{"id": "3521.png", "formula": "\\begin{align*} \\begin{array} [ c ] { r l } - d { p } ( t ) & = \\{ b _ x ( \\bar { X } { ( t ) } , \\bar { u } ( t ) ) ^ { } p ( t ) - f _ x ( \\bar { X } { ( t ) } , \\bar { u } ( t ) ) \\} d t , \\ t \\in ( t _ { i - 1 } , t _ { i } ) , \\\\ p ( t _ { i } ) & = - \\psi _ { x _ i } ( \\bar { X } ( t _ 1 ) , \\bar { X } ( t _ 2 ) , \\cdots , \\bar { X } ( t _ n ) ) + p ( t _ { i } ^ { + } ) , i = 1 , 2 , \\cdots , n , \\end{array} \\end{align*}"}
-{"id": "9935.png", "formula": "\\begin{align*} F ( s + h ) - F ( s ) = \\int _ X ( \\phi ( s + h , x ) - \\phi ( s , x ) ) \\ , \\mu ( d x ) & = \\int _ X \\int _ { s } ^ { s + h } \\partial _ r \\phi ( r , x ) \\ , d r \\ , \\mu ( d x ) \\\\ & = \\int _ s ^ { s + h } \\int _ X \\partial _ r \\phi ( r , x ) \\ , \\mu ( d x ) \\ , d r \\end{align*}"}
-{"id": "5659.png", "formula": "\\begin{align*} \\mathcal { Z } : = \\{ { z } \\in \\mathcal { S } ( a ^ - , a ^ + ) \\ ; : \\ ; \\forall { v } \\in \\mathcal { S } ( a ^ - , a ^ + ) , \\ , \\mathfrak { E } _ { W } ( { z } ) \\leq \\mathfrak { E } _ { W } ( { v } ) \\} , \\end{align*}"}
-{"id": "5709.png", "formula": "\\begin{align*} ( A ( z ) v \\ , ; \\ , v ) _ { L ^ 2 } = \\int _ \\R \\left ( | v ' ( s ) | ^ 2 + D ^ 2 W ( z ( s ) ) ( v ( s ) , v ( s ) ) \\right ) \\d s , \\end{align*}"}
-{"id": "2582.png", "formula": "\\begin{align*} ( V * \\rho ) ( t ) : = \\int _ { [ 0 , 1 ] } V ( t - s ) \\ , d \\rho ( s ) \\rho \\in \\mathcal P ( [ 0 , 1 ] ) \\end{align*}"}
-{"id": "2030.png", "formula": "\\begin{align*} x _ i y ^ 0 _ { i + 1 } - x _ { i + k } y ^ 0 _ i = 0 , ~ ~ ~ i \\in \\Z . \\end{align*}"}
-{"id": "3994.png", "formula": "\\begin{align*} p ^ { \\nu _ n } ( n , t ) = p ^ { \\nu _ n } ( n , 0 ) + I _ t ^ { \\nu _ n } ( - \\lambda _ n p ^ { \\nu _ n } ( n , t ) + \\lambda _ { n - 1 } p ^ { \\nu _ { n - 1 } } ( n - 1 , t ) ) , \\ \\ n \\geq 1 . \\end{align*}"}
-{"id": "511.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n E [ d ( X _ i , \\widehat { X } _ i ( Y ^ n , W ) ) ] \\leq D + \\epsilon . \\end{align*}"}
-{"id": "8506.png", "formula": "\\begin{align*} y ^ 2 z + a _ 1 x y z + a _ 3 y z ^ 2 = x ^ 3 + a _ 2 x ^ 2 z + a _ 4 x z ^ 2 + a _ 6 z ^ 3 , \\end{align*}"}
-{"id": "869.png", "formula": "\\begin{align*} E [ f ( F ) ] - E [ f ( Z ^ * ) ] = \\sum _ { i , j = 1 } ^ d E [ \\partial ^ 2 _ { i , j } U _ 0 f ( F ) ( C ^ { i j } - \\langle D F _ i , - D L ^ { - 1 } F _ j \\rangle _ H ) ] . \\end{align*}"}
-{"id": "9375.png", "formula": "\\begin{align*} v ( r , \\theta ) = C r ^ { 3 / 2 } h _ e ( \\theta ) + c _ 0 r ^ { 3 / 2 } u _ 0 ( \\theta ) + r ^ 2 \\phi ( \\theta ) , \\end{align*}"}
-{"id": "769.png", "formula": "\\begin{align*} ( x - 1 ) ^ k + x ^ k + ( x + 1 ) ^ k = y ^ n x , y , n \\in \\mathbb { Z } , n \\geq 2 , \\end{align*}"}
-{"id": "3316.png", "formula": "\\begin{align*} D ^ * ( r ) \\geq \\tilde { D } ( r ) = \\max _ { i \\in \\{ 2 , \\cdots , K + 1 \\} } ( 1 - r ) \\sum _ { j = 0 } ^ { K + 1 - i } \\frac { 1 } { N ^ j } - r \\sum _ { j = 0 } ^ { K - i } \\frac { K + 1 - i - j } { N ^ j } , \\end{align*}"}
-{"id": "5541.png", "formula": "\\begin{align*} \\Delta \\left ( \\alpha , \\beta \\right ) = T r a c e \\left ( M \\right ) \\end{align*}"}
-{"id": "6740.png", "formula": "\\begin{align*} F _ { 1 } ( N ) \\ ! = \\ ! \\Big \\{ \\ ! ( i _ { 1 } , \\ ! \\ ! \\dots , \\ ! \\ ! i _ { N ^ { \\gamma } t } ) \\ ! \\in \\ ! \\{ 1 , \\ ! \\dots , \\ ! N \\} ^ { N ^ { \\gamma } t } ; l = 1 , \\ ! \\dots , \\ ! N ^ { \\gamma } t : \\ ! \\eta ^ { i _ { 1 } , \\ ! \\dots , \\ ! i _ { l } } \\ ! \\notin \\ ! \\{ \\eta , \\ ! \\eta ^ { i _ { 1 } } , \\ ! \\dots , \\ ! \\eta ^ { i _ { 1 } , \\ ! \\dots , \\ ! i _ { l - 1 } } \\} \\ ! \\Big \\} \\end{align*}"}
-{"id": "5502.png", "formula": "\\begin{align*} u _ m ( x ) = x ^ m \\overline F ( x ) + m \\int _ x ^ \\infty y ^ { m - 1 } \\overline F ( y ) \\dd y . \\end{align*}"}
-{"id": "3801.png", "formula": "\\begin{align*} \\widetilde { Y } ^ y _ 0 = y \\widetilde { Y } ^ y _ { t + 1 } = \\widetilde { Y } ^ y _ t + ( g ( \\theta _ { \\widetilde { Y } ^ y _ t } ( \\omega , U ) ) , 1 ) , \\ ; \\ ; \\ ; t \\in \\Z _ + . \\end{align*}"}
-{"id": "6476.png", "formula": "\\begin{align*} \\sigma _ { x y } = \\rho \\sigma _ { x } \\sigma _ { y } \\end{align*}"}
-{"id": "9098.png", "formula": "\\begin{align*} \\Delta _ { y _ { 1 } , \\ldots , y _ { m } } A ^ { \\ast } ( x ) = \\left \\{ \\begin{array} { r c l } 0 & & m > n \\\\ n ! A ( y _ { 1 } , \\ldots , y _ { m } ) & & m = n . \\end{array} \\right . \\end{align*}"}
-{"id": "4320.png", "formula": "\\begin{align*} q ( 0 ) = 0 \\Omega , q = \\Delta q = 0 , \\ ; n = 1 \\Gamma . \\end{align*}"}
-{"id": "6371.png", "formula": "\\begin{align*} Q ' ( t ) = 2 H _ \\mu ( t ) , \\end{align*}"}
-{"id": "8952.png", "formula": "\\begin{gather*} D _ w = \\sum _ { 1 \\le i \\le n } { } ^ { s _ 1 \\cdots s _ { i - 1 } } [ X ^ { s _ i } ] = \\sum _ { 1 \\le i \\le n } [ X ^ { s _ 1 \\cdots s _ { i - 1 } s _ i s _ { i - 1 } \\cdots s _ 1 } ] . \\end{gather*}"}
-{"id": "1356.png", "formula": "\\begin{align*} \\psi ^ * ( Y ) = 2 \\displaystyle \\sum _ { i = 1 } ^ { r } \\langle Q ^ T _ { a _ i } Y Q _ { a _ i } , Q _ { a _ i } ^ T H ( X - \\varpi _ i I ) ^ { \\dag } H Q _ { a _ i } \\rangle , \\end{align*}"}
-{"id": "3897.png", "formula": "\\begin{align*} r = \\rho ^ { 2 / ( \\beta + 2 ) } , \\end{align*}"}
-{"id": "889.png", "formula": "\\begin{align*} E \\left [ \\left ( \\int _ { - \\infty } ^ \\infty f ( t ) d B ^ 1 _ t \\right ) \\left ( \\int _ { - \\infty } ^ \\infty g ( t ) d B ^ 2 _ t \\right ) \\right ] = \\sum _ { m = 1 } ^ M \\rho _ m \\int _ { - \\infty } ^ \\infty f ( t ) g ( t + \\theta _ m ) d t \\end{align*}"}
-{"id": "7336.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { A } } _ 1 \\ ; : = \\ ; \\left \\{ A = A ( t , x ) \\left | \\ ! \\begin{array} { c } \\mathrm { d i v } _ x A = 0 \\ ; \\textrm { f o r a . e . } \\ ; t \\in \\mathbb { R } , \\\\ A = A _ 1 + A _ 2 \\textrm { s u c h t h a t , f o r $ j \\in \\{ 1 , 2 \\} $ , } \\\\ A _ j \\in L ^ { a _ j } _ \\mathrm { l o c } ( \\mathbb { R } , L ^ { b _ j } ( \\mathbb { R } ^ 3 , \\mathbb { R } ^ 3 ) ) \\\\ a _ j \\in ( 4 , + \\infty ] , b _ j \\in ( 3 , 6 ) , \\frac { 2 } { \\ , a _ j } + \\frac { 3 } { \\ , b _ j } < 1 \\end{array} \\ ! \\ ! \\right . \\right \\} \\end{align*}"}
-{"id": "488.png", "formula": "\\begin{align*} \\phi _ \\delta ( s ) = \\pi [ \\sigma ( s ) - 1 ] + 2 \\pi \\delta \\left [ \\rho ( \\delta ) - \\sqrt { \\sigma ( s ) } \\rho \\left ( \\delta ( s ) \\right ) \\right ] . \\end{align*}"}
-{"id": "3337.png", "formula": "\\begin{align*} r _ s ^ { ( K + 1 ) } & = \\alpha r _ { s - 1 } ^ { ( K ) } + ( 1 - \\alpha ) r _ s ^ { ( K ) } , \\\\ \\bar { D } ( r _ s ^ { ( K + 1 ) } ) & = \\alpha \\bar { D } ( r _ { s - 1 } ^ { ( K ) } ) + ( 1 - \\alpha ) \\bar { D } ( r _ s ^ { ( K ) } ) , \\end{align*}"}
-{"id": "2361.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } p ^ { - n } . z = x _ p . \\end{align*}"}
-{"id": "1454.png", "formula": "\\begin{align*} E _ 2 & = C \\{ 1 , x _ { 4 1 } \\} \\oplus D _ 0 \\{ x _ { 3 } ^ 2 , x _ { 3 } ^ 2 x _ { 4 1 } , x _ 3 , x _ 3 x _ { 4 1 } \\} \\\\ E _ 4 & = C \\{ 1 , v _ 0 x _ { 4 1 } \\} \\oplus D _ 1 \\{ x _ 3 ^ 2 \\} \\oplus D _ 1 \\{ x _ 3 x _ { 4 1 } \\} , \\\\ E _ 8 & = C \\{ 1 , v _ 0 x _ { 4 1 } \\} \\oplus D _ 1 / ( v _ 2 x _ { 4 1 } ^ 2 ) \\{ x _ 3 ^ 2 \\} \\end{align*}"}
-{"id": "3475.png", "formula": "\\begin{align*} k \\sum _ { n = 1 } ^ N \\big \\| \\Xi ^ n \\big \\| _ V ^ 2 \\le \\big ( 1 + \\| P _ h \\| _ { \\L ( V ) } \\big ) ^ 2 k \\sum _ { n = 1 } ^ N \\mathrm { d i s t } _ V ( u ( t _ n ) , V _ h ) ^ 2 . \\end{align*}"}
-{"id": "10043.png", "formula": "\\begin{align*} A _ 0 ( \\C ) = \\mathfrak { a } _ { 0 \\R } / g \\mathfrak { a } _ 0 , A ( \\C ) = \\mathfrak { a } _ { \\R } / g \\mathfrak { a } . \\end{align*}"}
-{"id": "5930.png", "formula": "\\begin{align*} y ' ( a ) = 0 , y ' ( b ) = 0 . \\end{align*}"}
-{"id": "8894.png", "formula": "\\begin{align*} X J _ { \\theta , 1 } ^ { - 1 } U _ \\mu ^ n p = g \\psi _ n + g p \\kappa _ n , \\ \\ n \\in \\mathbb Z . \\end{align*}"}
-{"id": "8876.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - \\overline c \\theta ( z ) } = \\int _ { \\mathbb T } \\frac { 1 } { 1 - z \\overline \\zeta } \\sigma _ c ( \\zeta ) , \\ \\ z \\in \\mathbb D . \\end{align*}"}
-{"id": "8259.png", "formula": "\\begin{align*} \\lambda _ { \\infty } = \\chi _ 1 \\chi _ 2 ^ { - 1 } \\big \\vert _ { F _ { \\infty } ^ { \\times } } = \\prod _ { \\nu } \\abs { \\cdot } _ { \\nu } ^ { 2 t _ { \\nu } } \\end{align*}"}
-{"id": "10128.png", "formula": "\\begin{align*} \\boldsymbol t _ k ( i ) = \\frac { \\lambda ^ { - 1 } \\boldsymbol Q _ { \\bar { \\boldsymbol \\omega } _ k } ( i - 1 ) \\bar { \\boldsymbol \\omega } _ k ( i - 1 ) } { 1 + \\lambda ^ { - 1 } \\bar { \\boldsymbol \\omega } _ k ^ H ( i - 1 ) \\boldsymbol Q _ { \\bar { \\boldsymbol \\omega } _ k } ( i - 1 ) \\bar { \\boldsymbol \\omega } _ k ( i - 1 ) } . \\end{align*}"}
-{"id": "3229.png", "formula": "\\begin{align*} i _ \\ast = ( P _ Y ) _ \\ast ( \\langle c \\rangle \\cap \\langle G _ i \\rangle ) . \\end{align*}"}
-{"id": "9536.png", "formula": "\\begin{align*} \\widehat { \\phi } _ { x y } ^ * ( x ) = y , \\widehat { \\phi } _ { x y } ^ * ( x _ i ) = x _ i \\quad \\mbox { f o r } \\ i > 1 , \\widehat { \\phi } _ { x y } ^ * ( m ) = m \\ \\mbox { f o r } m \\in M . \\end{align*}"}
-{"id": "231.png", "formula": "\\begin{align*} \\{ z \\in \\C : \\ m < \\Im ( z ) < M \\} = : S _ { m , M } \\ni z \\mapsto f ( z ) : = \\mathbb { E } \\left [ \\chi _ a R _ z ( A ) \\chi _ b \\right ] , \\end{align*}"}
-{"id": "9082.png", "formula": "\\begin{align*} f ( p ( a ) ) = p ^ { f } ( a ) + f ( a ) p ' ( a ) . \\end{align*}"}
-{"id": "1601.png", "formula": "\\begin{align*} \\sum _ { n \\geqslant 0 } \\mu ^ { \\Sigma _ n } ( V ^ { \\times n } ) \\ , t ^ n = \\mu \\Big ( \\ , \\prod \\limits _ { i \\geqslant 1 } Z _ { m o t } ( V , t ^ i ) \\Big ) \\ , . \\end{align*}"}
-{"id": "5732.png", "formula": "\\begin{align*} 0 = \\int _ \\R z ' ( s ) \\cdot [ \\partial _ t \\gamma ( t , s + m ( t ) ) + m ' ( t ) \\ , \\partial _ s \\gamma ( t , s + m ( t ) ) ] \\d s = \\int _ \\R z ' ( s - m ( t ) ) \\cdot [ \\partial _ t \\gamma ( t , s ) + m ' ( t ) \\ , \\partial _ s \\gamma ( t , s ) ] \\d s . \\end{align*}"}
-{"id": "9483.png", "formula": "\\begin{align*} \\omega ( z ; q ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ n } { ( z q ; q ^ 2 ) _ n } & = \\sum _ { n = 1 } ^ { \\infty } \\frac { z ^ { n - 1 } q ^ n } { ( q ; q ^ 2 ) _ n } , \\\\ \\nu ' ( z ; q ) = \\sum _ { n = 0 } ^ { \\infty } ( - z q ; q ^ 2 ) _ n \\ , q ^ n & = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ { n ^ 2 + n } } { ( q ; q ^ 2 ) _ { n + 1 } } . \\end{align*}"}
-{"id": "7402.png", "formula": "\\begin{align*} \\mathbb { H } _ { \\Gamma } : = \\frac { \\mathbb { C } [ t _ 0 , t _ 1 ] } { ( t _ 0 + t _ 1 ) } \\cong \\mathbb { C } [ t ] . \\end{align*}"}
-{"id": "5614.png", "formula": "\\begin{align*} S _ t & = ( x - y ) \\exp \\biggl [ \\int _ s ^ t \\int _ 0 ^ 1 f ' ( Z _ r ^ u ) d u d r \\biggr ] \\\\ & = ( x - y ) \\exp \\biggl [ \\int _ 0 ^ 1 \\int _ s ^ t f ' ( Z _ r ^ u ) d r d u \\biggr ] \\end{align*}"}
-{"id": "2696.png", "formula": "\\begin{align*} a ' _ 1 { \\beta } ^ 2 + a _ 2 \\beta + p ^ { 2 n } a ' _ 3 = 0 . \\end{align*}"}
-{"id": "6473.png", "formula": "\\begin{align*} \\mathcal { S } _ { \\mathcal { M } } ^ { 2 D u } \\left ( \\tau \\right ) \\overset { \\tau \\rightarrow \\infty } { \\approx } \\left [ \\left ( \\frac { \\lambda _ { + } } { \\lambda _ { + } ^ { \\prime } } \\right ) \\cdot \\mathcal { S } _ { \\mathcal { M } } ^ { 3 D u } \\left ( \\tau \\right ) \\right ] \\frac { \\lambda _ { + } } { \\lambda _ { + } ^ { \\prime } } = \\frac { 1 } { \\sqrt { 2 } } < 1 \\end{align*}"}
-{"id": "7478.png", "formula": "\\begin{align*} \\Phi = i h _ { \\alpha \\bar { \\beta } } \\big ( \\mathcal { Z } ^ \\alpha \\wedge \\mathcal { Z } ^ { \\bar { \\beta } } + \\delta \\mathcal { V } ^ \\alpha \\wedge \\delta \\mathcal { V } ^ { \\bar { \\beta } } \\big ) = \\Phi ^ h + \\Phi ^ v . \\end{align*}"}
-{"id": "7709.png", "formula": "\\begin{align*} D _ R ( j ) = \\sum _ { k \\in V } d _ R ( j , k ) \\ , , \\end{align*}"}
-{"id": "6136.png", "formula": "\\begin{align*} \\psi ' \\left ( s \\right ) = - s - \\left ( \\mu _ { 1 } ^ { m } \\right ) ^ { - 1 } \\int _ { 0 } ^ { \\infty } \\left ( e ^ { - s y } - 1 \\right ) f \\left ( d y \\right ) , \\end{align*}"}
-{"id": "2940.png", "formula": "\\begin{align*} t - \\sum _ { k ' \\ge 2 } \\sum _ { i = 1 } ^ { \\Gamma ( k ' ) } \\left [ 1 - X _ { k ' , i , 1 } ^ + - \\dots - X _ { k ' , i , k ' - 1 } ^ + \\right ] _ 0 ^ 1 , \\end{align*}"}
-{"id": "5089.png", "formula": "\\begin{align*} E _ G ( - x , \\xi ) = \\overline { E _ G ( x , \\xi ) } = E _ G ( x , \\xi ) ^ { - 1 } = E _ G ( x , - \\xi ) \\end{align*}"}
-{"id": "7951.png", "formula": "\\begin{align*} \\sum _ { I \\in \\mathcal S } \\lambda _ I ^ s = 1 . \\end{align*}"}
-{"id": "7782.png", "formula": "\\begin{align*} 0 < \\beta ( X _ h , M _ h ) : = \\inf _ { x _ h \\in X _ h } \\sup _ { y _ h \\in M _ h } \\frac { b ( x _ h , y _ h ) } { \\Vert x _ h \\Vert _ X \\Vert y _ h \\Vert _ Y } \\end{align*}"}
-{"id": "5570.png", "formula": "\\begin{align*} \\bar { \\Gamma } _ { P } = \\left ( I _ { 2 ^ { k } } + \\frac { \\tau ^ { 2 } \\alpha } { 2 ^ { 2 k } } \\bar { P } ^ { 2 } + \\frac { \\tau ^ { 2 } \\beta } { 2 ^ { 2 k } } \\bar { P } \\bar { \\Lambda } _ { \\bar { r } } \\bar { P } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "6899.png", "formula": "\\begin{align*} \\mu _ { 0 } ( r ) r _ { 0 } ( r ) = L ( r ) = \\frac { 2 r ( r + 2 ) } { ( 2 r + 1 ) ( 9 r ^ { 2 } + 2 6 r + 1 5 ) } \\end{align*}"}
-{"id": "6595.png", "formula": "\\begin{align*} \\phi ^ k _ { n } = \\psi _ n \\circ \\varphi _ { k + 1 } ^ { - 1 } \\circ \\phi ^ k \\circ \\varphi _ { k + 1 } \\circ \\psi _ n ^ { - 1 } \\mbox { a n d } \\phi _ { n } = \\varphi _ n \\circ \\varphi _ 0 ^ { - 1 } \\circ \\phi \\circ \\varphi _ 0 \\circ \\varphi _ n ^ { - 1 } \\ . \\end{align*}"}
-{"id": "35.png", "formula": "\\begin{align*} \\frac { \\partial ( e e ^ * ) } { \\partial \\textbf { w } ^ * } = \\frac { \\partial ( d - \\textbf { w } ^ { H } \\textbf { X } ) ( d ^ * - \\textbf { w } ^ { T } \\textbf { X } ^ * ) } { \\partial \\textbf { w } ^ * } = ( - d ^ * \\textbf { X } + \\textbf { X X } ^ { H } \\textbf { w } ) \\end{align*}"}
-{"id": "2873.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { n ^ { N - 2 h } } { e ^ { n ^ { N } x } - 1 } = P ( x ) + S ( x ) , \\end{align*}"}
-{"id": "2195.png", "formula": "\\begin{align*} h _ i ( z ) = \\begin{cases} q _ i ( z ) , & \\mbox { i f } \\ a _ { i j } \\neq 0 \\ \\mbox { f o r e a c h } \\ j ; \\\\ q _ i ( z ) \\cdot \\frac { 1 } { z _ { j _ 1 } } \\cdot \\ldots \\cdot \\frac { 1 } { z _ { j _ k } } , & \\mbox { i f } \\ a _ { i { j _ 1 } } = \\ldots = a _ { i { j _ k } } = 0 . \\end{cases} \\end{align*}"}
-{"id": "3454.png", "formula": "\\begin{align*} \\dot { g } ( t ) = \\begin{cases} - 1 , & t \\in [ i p , ( i + 1 ) p ) , \\ i \\in \\{ 0 , \\dots , 2 ^ K - 1 \\} \\\\ 1 , & t \\in [ i p , ( i + 1 ) p ) , \\ i \\in \\{ 0 , \\dots , 2 ^ K - 1 \\} . \\end{cases} \\end{align*}"}
-{"id": "1732.png", "formula": "\\begin{align*} W f = h f ; \\end{align*}"}
-{"id": "8861.png", "formula": "\\begin{align*} V ( X _ { N } , \\phi _ { N } ) & \\ll \\left ( \\frac { \\sin \\phi _ { N } } { \\phi _ { N } } \\right ) ^ { d - 1 } \\left ( \\phi _ { N } \\right ) ^ { d - 1 } N ^ { 1 - \\frac { 1 } { d } } \\\\ & = \\left ( t \\ , N ^ { - \\frac { 1 } { d } } \\right ) ^ { d - 1 } N ^ { 1 - \\frac { 1 } { d } } \\\\ & = t ^ { d - 1 } \\end{align*}"}
-{"id": "4101.png", "formula": "\\begin{align*} p - q = g ( q ) \\eta ( p ) + V ( q ) , \\end{align*}"}
-{"id": "8008.png", "formula": "\\begin{align*} & \\sup _ { R \\in \\mathcal { D } _ { \\mu } } \\Big ( \\frac { 1 } { | R | } \\int _ R { \\sum _ { k = \\max { ( 0 , \\mu - 5 ) } } ^ { \\infty } { 2 ^ { ( m + s ) k q } \\big | \\Pi ^ * _ k f ( x ) \\big | ^ q } } d x \\Big ) ^ { 1 / q } \\\\ & \\lesssim \\sup _ { 0 \\leq k \\leq \\mu - 1 } { \\big \\Vert 2 ^ { k ( s + m ) } \\Pi _ k f \\big \\Vert _ { L ^ { \\infty } } } + \\sup _ { R \\in \\mathcal { D } _ { \\mu } } \\Big ( \\frac { 1 } { | R | } \\int _ R \\sum _ { k = \\mu } ^ { \\infty } { 2 ^ { ( s + m ) k q } \\big | \\Pi _ k f ( x ) \\big | ^ q } \\Big ) ^ { 1 / q } \\end{align*}"}
-{"id": "2470.png", "formula": "\\begin{align*} \\Vert g \\Vert _ { L _ { \\sigma } ^ { 2 } } ^ { 2 } = \\int _ { { \\mathbb { R } ^ { 3 } } } | g | _ { L _ { \\sigma } ^ { 2 } } ^ { 2 } d x \\ , , \\quad \\Vert g \\Vert _ { L _ { \\sigma } ^ { 2 } ( m ) } ^ { 2 } = \\int _ { { \\mathbb { R } ^ { 3 } } } | g | _ { L _ { \\sigma } ^ { 2 } ( m ) } ^ { 2 } d x \\ , . \\end{align*}"}
-{"id": "1492.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { n H ( c _ n ) ( 1 + \\theta ) / ( 1 - \\theta ) } } \\sum _ { k = 1 } ^ n X _ k \\to \\N ( 0 , 1 ) \\end{align*}"}
-{"id": "1663.png", "formula": "\\begin{align*} \\tau _ { d _ 0 c _ 0 } ( x ) = \\tau _ { d _ 0 } \\circ \\tau _ { c _ 0 } ( x ) = \\frac { - 3 \\tau _ { c _ 0 } ( x ) + 2 } { 3 } = - ( \\frac { x } { 2 } + \\frac { 1 } { 2 } ) + \\frac { 2 } { 3 } = - \\frac { x } { 2 } + \\frac { 1 } { 6 } \\end{align*}"}
-{"id": "8197.png", "formula": "\\begin{align*} x _ { \\alpha } = \\frac { \\overline { \\lambda ( ( \\alpha ) ) } } { \\abs { \\lambda ( ( \\alpha ) ) } } . \\end{align*}"}
-{"id": "6305.png", "formula": "\\begin{align*} L \\psi ( x ) & = \\lambda \\psi ( x ) , & T \\psi ( x ) & = \\psi ( x + \\gamma ) = \\mu \\psi ( x ) . \\end{align*}"}
-{"id": "10109.png", "formula": "\\begin{align*} \\sum \\limits _ { l } c _ { k l } = 1 , l \\in \\mathcal { N } _ k \\forall k . \\end{align*}"}
-{"id": "750.png", "formula": "\\begin{align*} J ( u _ 1 ^ n , u _ 2 ^ n ) & = J ( u _ 1 , u _ 2 ) + J ( v _ 1 ^ n , v _ 2 ^ n ) + o _ n ( 1 ) \\\\ & \\geq J ( u _ 1 , u _ 2 ) + m ( a _ 1 - b _ 1 , a _ 2 - b _ 2 ) + o _ n ( 1 ) . \\end{align*}"}
-{"id": "2105.png", "formula": "\\begin{align*} V _ s ( t ) = \\sup _ { M \\times [ 0 , t ] } \\sum _ a \\big ( | v ^ a _ s | + | \\nabla _ H v ^ a _ s | \\big ) . \\end{align*}"}
-{"id": "3912.png", "formula": "\\begin{align*} \\frac { \\langle H \\rangle ^ { d , \\ell } _ { m i n } } { E ^ { d , \\ell } _ { 0 } } = \\frac { ( \\ell + \\frac { d - 1 } { 2 } ) ^ 2 } { ( \\ell + \\frac { d } { 2 } ) } \\Big ( \\frac { \\Gamma ( \\ell + \\frac { d - 1 } { 2 } ) } { \\Gamma ( \\ell + \\frac { d } { 2 } ) } \\Big ) ^ 2 . \\end{align*}"}
-{"id": "4412.png", "formula": "\\begin{align*} \\int _ { \\mbox { s p a c e } } \\nabla \\zeta \\cdot h \\ , \\ , d x = { - } \\int _ { \\mbox { s a m p l e } } \\nabla \\zeta \\cdot m \\ , \\ , d x \\quad \\mbox { f o r a l l t e s t f u n c t i o n s } \\ ; \\zeta . \\end{align*}"}
-{"id": "5599.png", "formula": "\\begin{align*} \\Pi _ k ( n ) = \\frac { q ^ n } { n } \\frac { 1 } { ( k - 1 ) ! } \\sum _ { \\substack { n _ 1 + \\ldots + n _ { k - 1 } \\leq n - 1 \\\\ n _ i \\geq 1 } } \\frac { 1 } { n _ 1 \\ldots n _ { k - 1 } } ( 1 + O ( k n / q ) ) \\end{align*}"}
-{"id": "6618.png", "formula": "\\begin{align*} U _ x = \\ ; v _ y \\circ \\phi ^ { - 1 } \\cdot J _ { \\phi ^ { - 1 } } \\ \\ \\ & , \\ \\ \\ U _ y = - u _ y \\circ \\phi ^ { - 1 } \\cdot J _ { \\phi ^ { - 1 } } \\ \\ , \\\\ V _ x = - v _ x \\circ \\phi ^ { - 1 } \\cdot J _ { \\phi ^ { - 1 } } \\ \\ \\ & , \\ \\ \\ V _ y = \\ ; u _ x \\circ \\phi ^ { - 1 } \\cdot J _ { \\phi ^ { - 1 } } \\ \\ . \\end{align*}"}
-{"id": "3221.png", "formula": "\\begin{align*} \\mathcal H _ k ( X ) \\cap \\sum _ { r \\in [ 0 , k ] \\cup [ 2 n - k , 2 n ] } H ^ r ( X ; \\mathbb Q ) = \\sum _ { r \\in [ 0 , k ] \\cup [ 2 n - k , 2 n ] } H ^ r ( X ; \\mathbb Q ) . \\end{align*}"}
-{"id": "3381.png", "formula": "\\begin{align*} \\boldsymbol { \\kappa } _ F = \\frac { 1 } { 2 } \\frac { \\log b } { \\log \\sigma } \\end{align*}"}
-{"id": "403.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( S _ { n } \\geq x \\sigma _ { n } \\right ) = ( 1 - \\Phi ( x ) ) ( 1 + o ( 1 ) ) . \\end{align*}"}
-{"id": "3750.png", "formula": "\\begin{align*} B _ m = B _ { k , 0 } + L _ k y . \\end{align*}"}
-{"id": "3430.png", "formula": "\\begin{align*} \\begin{cases} \\dot { u } ( t ) = f ( t , u ( t ) ) , t \\in ( 0 , T ] , \\\\ u ( 0 ) = u _ 0 , \\end{cases} \\end{align*}"}
-{"id": "4376.png", "formula": "\\begin{align*} P _ \\alpha C _ z = \\widetilde { C } _ z P _ \\alpha . \\end{align*}"}
-{"id": "599.png", "formula": "\\begin{align*} \\frac { L _ { \\Gamma } ' ( s , \\rho ) } { L _ { \\Gamma } ( s , \\rho ) } = \\sum _ { \\overline { \\gamma } \\in \\overline { \\Gamma } _ { p } } \\sum _ { m = 1 } ^ { \\infty } \\ell ( \\gamma ) \\chi _ { \\rho } ( \\gamma ^ { m } ) \\frac { e ^ { - s m \\ell ( \\gamma ) } } { 1 - e ^ { - m \\ell ( \\gamma ) } } . \\end{align*}"}
-{"id": "3867.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ m \\lambda _ i ^ k g _ i ( x ^ * ) + \\sum \\limits _ { i = 1 } ^ p \\mu _ i ^ k h _ i ( x ^ * ) + \\sum \\limits _ { i = 1 } ^ n \\gamma _ i ^ k x _ i ^ * = 0 , \\\\ \\sum \\limits _ { i = 1 } ^ m \\lambda _ i ^ * g _ i ( x ^ k ) + \\sum \\limits _ { i = 1 } ^ p \\mu _ i ^ * h _ i ( x ^ k ) + \\sum \\limits _ { i = 1 } ^ n \\gamma _ i ^ * x _ i ^ k = 0 , \\end{align*}"}
-{"id": "3269.png", "formula": "\\begin{align*} d ^ c ( u , k ) = \\frac { M ^ c ( u , k ) } { Q } v _ u . \\end{align*}"}
-{"id": "5173.png", "formula": "\\begin{align*} \\left \\{ x \\in \\mathcal { D } \\colon \\left ( \\tfrac { f } { \\varphi } \\right ) ( x ) \\ge \\alpha \\right \\} = \\left \\{ x \\in \\mathcal { D } \\colon f ( x ) \\ge \\alpha \\varphi ( x ) \\right \\} = \\left \\{ x \\in \\mathcal { D } \\colon F _ { \\alpha } ( x ) \\ge 0 \\right \\} . \\end{align*}"}
-{"id": "8753.png", "formula": "\\begin{align*} \\left ( \\frac { M } { M + 1 } - \\alpha ( M ) \\right ) \\log \\left ( \\frac { \\mu } { E _ B } \\right ) - \\sqrt { \\frac { \\lambda } { - \\mu } } - \\sqrt { \\frac { \\lambda } { \\lambda - \\mu } } - \\alpha ( M ) \\log \\left ( E _ B \\left ( \\frac { 1 } { \\mu } - \\frac { 1 } { \\lambda } \\right ) \\right ) - \\alpha ( M ) = 0 \\end{align*}"}
-{"id": "5600.png", "formula": "\\begin{align*} X _ t = x _ 0 + \\int _ 0 ^ t f ( X _ s ) d s + \\int _ 0 ^ t \\sigma ( X _ s ) d { B } ^ H _ s \\end{align*}"}
-{"id": "6588.png", "formula": "\\begin{align*} g _ n = \\left \\{ \\begin{array} { l l } f _ n \\circ \\phi _ n & \\ \\mbox { i n } \\ \\varphi _ n ( E _ M ) \\\\ f _ n & \\ \\mbox { e l s e w h e r e } \\end{array} \\right . \\ . \\end{align*}"}
-{"id": "234.png", "formula": "\\begin{align*} I _ 2 \\leq \\sum _ { l = N + 1 } ^ { \\infty } \\frac { \\| A \\| _ { \\infty } ^ l } { 2 ^ { l + 1 } ( \\| A \\| _ { \\infty } + 1 ) ^ { l + 1 } } \\leq \\frac { 1 } { 2 ^ N } . \\end{align*}"}
-{"id": "9340.png", "formula": "\\begin{align*} J Q _ { n } ^ { ( 3 ) } = \\sum _ { s = 0 } ^ { 3 } J _ { n + s } ^ { ( 3 ) } e _ { s } = J _ { n } ^ { ( 3 ) } + \\sum _ { s = 1 } ^ { 3 } J _ { n + s } ^ { ( 3 ) } e _ { s } , \\ ( J _ { n } ^ { ( 3 ) } \\textbf { 1 } = J _ { n } ^ { ( 3 ) } ) \\end{align*}"}
-{"id": "6658.png", "formula": "\\begin{align*} f ' ( t ) t - f ( t ) \\geq \\left ( 1 - \\frac { 3 } { 2 } \\sqrt { \\eta } \\right ) f '' ( 0 ) t ^ 2 - \\frac { 1 + \\eta } { 2 } f '' ( 0 ) t ^ 2 \\geq \\left ( 1 - 4 \\sqrt { \\eta } \\right ) \\frac { f '' ( 0 ) } { 2 } t ^ 2 = \\Delta \\quad . \\end{align*}"}
-{"id": "7733.png", "formula": "\\begin{align*} d ^ 2 _ B ( j , k ) & = \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N - 1 } \\frac { | e ^ { i 2 j \\phi _ n } - e ^ { i 2 k \\phi _ n } | ^ 2 } { 4 ( 1 - \\cos 2 \\phi _ n ) ^ 2 } = \\frac { 1 } { 2 } G _ N ( j - k ) , \\end{align*}"}
-{"id": "1798.png", "formula": "\\begin{align*} \\frac { \\partial H } { \\partial a } ( a , b ) = ( 1 - b ) \\ln ( a ) + ( 1 - b ) + \\ln ( b ) . \\end{align*}"}
-{"id": "9293.png", "formula": "\\begin{align*} \\Delta _ { d , h } ^ k f _ z ( x ) = \\Delta _ { h } ^ { k - 1 } \\Delta _ { d } ^ 1 f _ z ( x ) \\geq 0 . \\end{align*}"}
-{"id": "3323.png", "formula": "\\begin{align*} \\bar { D } ( r _ { K - 1 } ) = \\frac { N } { 1 + N } , \\end{align*}"}
-{"id": "1596.png", "formula": "\\begin{align*} \\alpha _ { \\mathbf { j } } ( n ) & = - ( 8 q + 3 ) < 0 \\\\ \\alpha _ { \\mathbf { j } ' } ( n ) & = - ( 8 q + 3 ) < 0 \\\\ \\alpha _ { \\mathbf { j } '' } ( n ) & = - ( 8 q + 1 ) < 0 . \\end{align*}"}
-{"id": "2496.png", "formula": "\\begin{align*} \\left ( \\mathcal { R } \\cdot \\psi _ { 0 } \\right ) \\left ( - \\left | \\eta \\right | \\right ) & = \\psi _ { 1 } \\left ( \\left | \\eta \\right | \\right ) , \\\\ \\left ( \\mathcal { R } \\cdot \\psi _ { j } \\right ) \\left ( - \\left | \\eta \\right | \\right ) & = \\psi _ { j } \\left ( \\left | \\eta \\right | \\right ) , \\mbox { f o r } j = 2 , 3 , 4 . \\end{align*}"}
-{"id": "9416.png", "formula": "\\begin{align*} | \\mathrm { A n n } ' _ { n + 1 } ( k ) | = | \\mathrm { A n n } _ { n } ( k ) | . \\end{align*}"}
-{"id": "4675.png", "formula": "\\begin{align*} { \\cal P } _ { N } \\ = \\ \\langle u _ 1 ^ { p _ 1 } u _ 2 ^ { p _ 2 } u _ 3 ^ { p _ 3 } \\ldots u _ M ^ { p _ M } \\vert \\ 0 \\le p _ 1 + p _ 2 + p _ 3 + \\ldots + p _ M \\le N \\rangle \\ . \\end{align*}"}
-{"id": "7847.png", "formula": "\\begin{align*} h _ { X _ { 1 } , X _ { 2 } } ( x _ { 1 } , x _ { 2 } ) = \\left \\{ \\begin{array} { l } h _ { 1 } ( x _ { 1 } , x _ { 2 } ) \\ \\ \\ \\ \\ \\ 0 < x _ { 1 } < x _ { 2 } \\\\ h _ { 2 } ( x _ { 1 } , x _ { 2 } ) \\ \\ \\ \\ \\ \\ 0 < x _ { 2 } < x _ { 1 } \\\\ h _ { 3 } ( x , x ) \\ \\ \\ \\ \\ \\ \\ \\ x _ { 1 } = x _ { 2 } = x , \\end{array} \\right . \\end{align*}"}
-{"id": "3603.png", "formula": "\\begin{align*} B : = \\Big \\{ ( n , m ) \\in \\N ^ 2 : 0 \\leq n \\leq \\ell , N - \\ell \\leq m < N + \\ell \\Big \\} . \\end{align*}"}
-{"id": "7817.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial r } \\int _ { S _ r } \\langle X , \\nabla r \\rangle e ^ { - 2 \\rho } d x = \\int _ { S _ r } ( { \\rm d i v } X - 2 \\rho ^ { \\prime } \\langle X , \\nabla r \\rangle ) e ^ { - 2 \\rho } d x . \\end{align*}"}
-{"id": "10157.png", "formula": "\\begin{align*} \\bar { \\boldsymbol \\omega _ k } ( i ) & = \\bar { \\boldsymbol R } _ k ^ { - 1 } ( i ) \\bar { \\boldsymbol p } _ k ( i ) \\\\ & = \\Big ( { \\boldsymbol S } _ { D _ k } ^ H ( i ) { \\boldsymbol R } _ k ( i ) { \\boldsymbol S } _ { D _ k } ( i ) \\Big ) ^ { - 1 } { \\boldsymbol S } _ { D _ k } ^ H ( i ) { \\boldsymbol p } _ k ( i ) . \\end{align*}"}
-{"id": "259.png", "formula": "\\begin{align*} \\exists \\delta > 0 : \\ , m _ \\delta < \\infty , m _ 0 = \\rho ( T ^ * ) = 1 . \\end{align*}"}
-{"id": "2729.png", "formula": "\\begin{align*} N ( \\mathcal L ) = \\{ M , \\ , v _ 1 M , \\ , \\cdots , v _ d M , \\ , | v | ^ 2 M \\} : = \\{ \\varphi _ 1 , \\cdots , \\varphi _ n \\} , n = d + 2 \\ , , \\end{align*}"}
-{"id": "9181.png", "formula": "\\begin{align*} \\min _ { 2 P } \\hat u = \\hat u ( O ) = 0 . \\end{align*}"}
-{"id": "3094.png", "formula": "\\begin{align*} & | ( \\psi \\circ g ) ' ( x ) - ( \\psi \\circ g ) ' ( y ) | \\\\ = & | \\psi ' ( g ( x ) ) g ' ( x ) - \\psi ' ( g ( y ) ) g ' ( x ) + \\psi ' ( g ( y ) ) g ' ( x ) - \\psi ' ( g ( y ) ) g ' ( y ) | \\\\ \\leq & | \\psi ' ( g ( x ) ) - \\psi ' ( g ( y ) ) | \\cdot | g ' ( x ) | + | \\psi ' ( g ( y ) ) | \\cdot | g ' ( x ) - g ' ( y ) | , \\end{align*}"}
-{"id": "4716.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } g } { \\mathrm { d } x } ( x ) = - 2 e ^ { \\frac { 2 } { ( x - 1 ) ( x - 3 ) } } \\end{align*}"}
-{"id": "3033.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ z u = 0 , u _ 3 = 0 , & \\Gamma _ u , \\\\ U = 0 , & \\Gamma _ b , \\\\ & \\Gamma _ l , \\end{cases} \\end{align*}"}
-{"id": "6796.png", "formula": "\\begin{align*} e ^ { w _ { \\lambda } ( \\Pi ^ { - 1 } _ { \\xi _ k } ( \\lambda z ) ) } = e ^ { U _ { \\lambda , \\xi _ k } } \\big [ 1 + 4 \\pi \\lambda ^ 2 \\sum \\limits _ { j , j \\neq k } z \\nabla ^ 2 _ x G ( \\Pi ^ { - 1 } _ { \\xi _ k } ( x ) , \\xi _ j ) \\vert _ { x = 0 } z ^ T \\end{align*}"}
-{"id": "4232.png", "formula": "\\begin{align*} y ( n ) & = \\sum _ { l = 0 } ^ { L } x ( n - l ) h _ l + z ( n ) = x ( n ) h _ 0 + \\cdots + z ( n ) , \\end{align*}"}
-{"id": "8991.png", "formula": "\\begin{gather*} { \\cal D } ^ { ( n ) } _ { q , t } ( c ) D ^ { ( n ) } _ q ( u _ 0 , u _ 1 , u _ 2 , u _ 3 ; t ) = D ^ { ( n ) } _ q ( u _ 0 - c , u _ 1 - c , u _ 2 - c , u _ 3 - c ; t ) { \\cal D } ^ { ( n ) } _ { q , t } ( c ) . \\end{gather*}"}
-{"id": "7098.png", "formula": "\\begin{align*} \\begin{aligned} p _ { t t } & = ( k p _ x ) _ { x t } = ( k p _ x ) _ { t x } = ( k _ t p _ x + k p _ { t x } ) _ x = ( k _ t p _ x + k ( k p _ x ) _ { x x } ) _ x \\\\ & = k _ t p _ { x x } + k _ { t x } p _ x + k _ x ( k p _ x ) _ { x x } + k ( k p _ x ) _ { x x x } , \\end{aligned} \\end{align*}"}
-{"id": "8052.png", "formula": "\\begin{align*} \\delta ( \\lambda ( A ) , \\lambda ( B ) ) & = \\| \\log \\lambda ( A ) - \\log \\lambda ( B ) \\| _ 2 = \\| \\lambda ( \\log A ) - \\lambda ( \\log B ) \\| _ 2 \\\\ & \\leq \\| \\log A - \\log B \\| _ 2 \\leq \\delta ( A , B ) . \\end{align*}"}
-{"id": "9250.png", "formula": "\\begin{align*} D = \\langle \\mathfrak { b } , \\mathfrak { b } \\rangle = \\langle A ^ { + } , A ^ { + } \\rangle + \\langle A ^ { - } , A ^ { - } \\rangle + \\langle B , B ' \\rangle + \\langle C , C ' \\rangle + \\langle E , E ' \\rangle . \\end{align*}"}
-{"id": "2637.png", "formula": "\\begin{align*} ( - 1 ) ^ { i + j } \\det L ( D ) _ { i , j } = \\det L ( D ) _ { i , i } = c _ i . \\end{align*}"}
-{"id": "7455.png", "formula": "\\begin{align*} \\dim A _ { c o n } = 9 \\dim \\frac { A _ { c o n } } { [ A _ { c o n } , A _ { c o n } ] } = 5 \\end{align*}"}
-{"id": "6075.png", "formula": "\\begin{align*} E ( \\varphi ) = c \\int _ { - \\infty } ^ 0 ( \\Psi ' ( t ) ^ 2 + 2 \\mathfrak { e } _ k \\sin ^ 2 ( \\Psi ( t ) ) ) e ^ { ( n - 2 ) t } d t , \\end{align*}"}
-{"id": "59.png", "formula": "\\begin{align*} G ^ { C } _ { \\sigma } ( C _ 1 - C _ 2 ) = \\frac { 1 } { 2 \\pi \\sigma ^ 2 } e x p \\left ( - \\frac { ( C _ { 1 } - C _ { 2 } ) ( C _ { 1 } - C _ { 2 } ) ^ { * } } { 2 \\sigma ^ 2 } \\right ) = \\frac { 1 } { 2 \\pi \\sigma ^ 2 } e x p \\left ( - \\frac { ( x _ { n } - y _ { n } ) ^ { 2 } + ( z _ { n } - s _ { n } ) ^ { 2 } } { 2 \\sigma ^ 2 } \\right ) \\end{align*}"}
-{"id": "8145.png", "formula": "\\begin{align*} S _ k '' & = \\bigsqcup _ { l = 1 } ^ L \\bigsqcup _ { m = 1 } ^ M \\bigsqcup _ { i = 1 } ^ n \\bigsqcup _ { c \\in C _ { k , l , m } ^ { ( i ) } } B _ { k , l , c , Q } c . \\end{align*}"}
-{"id": "3929.png", "formula": "\\begin{align*} U ( x , t ) = \\sum \\limits _ { i = - 1 } ^ { N - 1 } c _ i ( t ) T B _ i ^ 4 ( x ) , \\end{align*}"}
-{"id": "1997.png", "formula": "\\begin{align*} \\lim _ { b \\rightarrow \\infty } \\mathbb { P } \\left \\{ T _ { n 2 } \\geq D _ { n , v } \\left ( b \\right ) \\right \\} = 0 . \\end{align*}"}
-{"id": "2955.png", "formula": "\\begin{align*} \\nabla ( \\tilde { f } \\Theta ) ( H , e , i ) & = \\frac { 1 } { | e | } \\sum _ { j \\in e } \\tilde { f } ( H , e , j ) \\Theta ( H , e , j ) \\\\ & = \\frac { 1 } { | e | } \\sum _ { j \\in e } f ( H , j ) \\Theta ( H , e , j ) \\\\ & \\stackrel { ( a ) } { \\geq } \\frac { 1 } { | e | } \\min _ { j \\in e } f ( H , j ) \\\\ & = \\tilde { f } _ ( H , e , i ) , \\end{align*}"}
-{"id": "298.png", "formula": "\\begin{align*} \\delta _ 1 ( u ) = y , ~ ~ \\delta _ 1 ( v ) = x ~ ~ ~ ~ \\delta _ 1 ( T ) = 1 \\end{align*}"}
-{"id": "8210.png", "formula": "\\begin{align*} \\Delta d ( i ) = d ( i + 1 ) - d ( i ) \\end{align*}"}
-{"id": "251.png", "formula": "\\begin{align*} U _ { \\pi _ 0 \\circ \\pi } \\chi _ { ( \\R _ { \\geq 0 } ^ n \\times \\R ^ { d - n } ) } U _ { \\pi _ 0 \\circ \\pi } ^ * = \\chi _ { ( \\R ^ n _ { \\geq 0 } \\times \\R ^ { d - n } ) } . \\end{align*}"}
-{"id": "2082.png", "formula": "\\begin{align*} u ^ a _ k ( p , t ) = - \\int _ 0 ^ t \\int _ M H ( p , q , t - s ) F ^ a _ { k - 1 } ( q , s ) d V _ q d s + u ^ a _ 0 ( p , t ) , \\end{align*}"}
-{"id": "2229.png", "formula": "\\begin{align*} J _ \\gamma ( t ) = \\frac { ( - 1 ) ^ n } { ( 2 \\pi \\sqrt { - 1 } ) ^ n } \\int \\limits _ { \\widetilde \\Gamma _ h } w _ 1 ^ { \\gamma _ 1 + 1 } \\cdot \\ldots \\cdot w _ n ^ { \\gamma _ n + 1 } \\cdot \\frac { d \\widetilde { F _ 1 } } { \\widetilde { F _ 1 } } \\wedge \\ldots \\wedge \\frac { d { \\widetilde F _ n } } { \\widetilde { F _ n } } . \\end{align*}"}
-{"id": "5003.png", "formula": "\\begin{align*} R : = \\frac { \\kappa + 1 } { \\min ( 1 , \\alpha a _ \\alpha ) } \\sigma \\end{align*}"}
-{"id": "8134.png", "formula": "\\begin{align*} \\bigg ( \\sum _ { l = 1 } ^ L | T _ l | \\bigg ) M n \\leq \\beta | S | . \\end{align*}"}
-{"id": "659.png", "formula": "\\begin{align*} \\partial _ t \\eta = \\frac { b } { ( a - b t ) ^ 2 } \\rho ^ 2 \\geq \\frac { b } { 8 n \\lambda } | \\nabla \\eta | ^ 2 . \\end{align*}"}
-{"id": "493.png", "formula": "\\begin{align*} \\gamma _ 1 = \\eta _ 1 + 2 \\eta _ 2 , \\gamma _ 2 = \\mu _ 1 + 2 \\mu _ 2 , \\sum _ { h = 1 } ^ { \\abs { \\mu } } h \\beta _ h = \\abs { \\mu } . \\end{align*}"}
-{"id": "2058.png", "formula": "\\begin{align*} \\rho ( p ) = p - P ( p ) . \\end{align*}"}
-{"id": "8298.png", "formula": "\\begin{align*} \\delta = v w \\in C ( V ) . \\end{align*}"}
-{"id": "3138.png", "formula": "\\begin{align*} \\mathcal { U } : = \\left \\lbrace u \\in \\mathbb { C } \\thinspace : \\thinspace \\mathrm { R e } \\thinspace u \\leqslant 0 \\right \\rbrace . \\end{align*}"}
-{"id": "5132.png", "formula": "\\begin{align*} v ( t ) = - i \\int _ 0 ^ t S ( t - t ' ) | v + z _ 1 | ^ 2 ( v + z _ 1 ) ( t ' ) d t ' , \\end{align*}"}
-{"id": "2881.png", "formula": "\\begin{align*} P _ 1 ( x ) : = P _ 1 ( x ; N , h ) : = P ( x ) + \\sum _ { j = 1 } ^ { \\left \\lfloor \\frac { h } { N } \\right \\rfloor } R _ { - ( 2 j - 1 ) } . \\end{align*}"}
-{"id": "920.png", "formula": "\\begin{align*} \\sum _ { i , j , k = 1 } ^ d \\left | \\frac { \\partial ^ 3 ( g \\circ \\Phi _ \\beta ) } { \\partial x _ i \\partial x _ j \\partial x _ k } ( x ) \\right | \\leq \\| g ''' \\| _ \\infty + 6 \\beta \\| g '' \\| _ \\infty + 6 \\beta ^ 2 \\| g ' \\| _ \\infty \\end{align*}"}
-{"id": "3716.png", "formula": "\\begin{align*} \\{ 1 , \\dots , n \\} = \\coprod _ k M ( \\tilde \\gamma ) _ k . \\end{align*}"}
-{"id": "4110.png", "formula": "\\begin{align*} k _ { M , p } ( X ) = - \\frac { h ( X , X ) } { b ( X , X ) } , \\end{align*}"}
-{"id": "197.png", "formula": "\\begin{align*} f ( \\sigma \\ast ^ { \\boldsymbol { R } } a ) = \\sigma \\ast ^ { \\boldsymbol { S } } f ( a ) \\leq ^ { \\boldsymbol { S } } \\sigma \\ast ^ { \\boldsymbol { S } } f ( b ) = f ( \\sigma \\ast ^ { \\boldsymbol { R } } b ) . \\end{align*}"}
-{"id": "4012.png", "formula": "\\begin{align*} h ( X , Y ) = \\frac { \\langle D _ X Y , \\xi \\rangle } { \\langle \\eta , \\xi \\rangle } = - \\frac { \\langle Y , d \\xi _ p X \\rangle } { \\langle \\eta , \\xi \\rangle } = - \\frac { \\langle d u ^ { - 1 } _ { \\eta ( p ) } T , d \\eta _ p X \\rangle } { \\langle \\eta , \\xi \\rangle } , \\end{align*}"}
-{"id": "8375.png", "formula": "\\begin{align*} \\psi ( f ) = \\frac { ( j ^ { [ 2 ] } ) ^ * \\psi ( f ^ { [ 2 ] } ) } { ( j ^ { [ 1 ] } ) ^ * \\psi ( f ^ { [ 1 ] } ) } . \\end{align*}"}
-{"id": "3886.png", "formula": "\\begin{align*} \\Psi _ k ( x ) : = x ^ k \\frac { \\Gamma ( k - 1 / 4 ) \\Gamma ( k + 1 / 4 ) } { \\Gamma ( 2 k ) } F \\left ( k - \\frac { 1 } { 4 } , k + \\frac { 1 } { 4 } , 2 k ; x \\right ) . \\end{align*}"}
-{"id": "8874.png", "formula": "\\begin{align*} U _ { ( \\theta ) c } = S _ \\theta + c ( \\cdot , \\overline \\chi \\theta ) . \\end{align*}"}
-{"id": "3875.png", "formula": "\\begin{align*} E X _ + ^ p = \\frac { \\Gamma ( p + 1 ) } { \\pi } \\int _ 0 ^ \\infty \\Re \\frac { \\phi ( u ) - 1 } { ( i u ) ^ { p + 1 } } d u . \\end{align*}"}
-{"id": "2616.png", "formula": "\\begin{align*} \\tilde g ( t ) : = t ^ { 1 - a } D ^ { 1 - a } f ( t ) \\in C ( [ 0 , 1 ] ) . \\end{align*}"}
-{"id": "10028.png", "formula": "\\begin{align*} \\Lambda _ h = \\Lambda _ \\Q \\cap h \\widehat \\Lambda . \\end{align*}"}
-{"id": "5964.png", "formula": "\\begin{align*} f ( t , w _ 2 , \\dots , w _ n ) = \\hat \\sigma ( t ) + \\sum _ { j = 2 } ^ n w _ j \\vect { e } _ j ( t ) \\left ( \\hat \\sigma ( t ) : = \\int _ 0 ^ t a ( \\tau ) \\gamma ' ( \\tau ) d \\tau \\right ) \\end{align*}"}
-{"id": "9219.png", "formula": "\\begin{align*} [ z \\otimes \\alpha , [ u _ { 1 } \\otimes b _ { 1 } , u _ { 2 } \\otimes b _ { 2 } ] ] = [ [ z \\otimes \\alpha , u _ { 1 } \\otimes b _ { 1 } ] , u _ { 2 } \\otimes b _ { 2 } ] + [ u _ { 1 } \\otimes b _ { 1 } , [ z \\otimes \\alpha , u _ { 2 } \\otimes b _ { 2 } ] ] . \\end{align*}"}
-{"id": "4185.png", "formula": "\\begin{gather*} \\tilde { \\eta } ^ 2 = 4 \\tilde { \\zeta } ^ 3 - g _ 2 ( \\Lambda ) \\tilde { \\zeta } - g _ 3 ( \\Lambda ) , \\ ; \\textrm { w h e r e } , \\\\ \\tilde { \\eta } = \\eta ( 4 / r _ 1 ) ^ { 1 / 2 } , \\quad \\tilde { \\zeta } = \\zeta - \\tfrac { r _ 2 } { 3 r _ 1 } , \\\\ g _ 2 ( \\Lambda ) = 6 0 G _ 4 ( \\Lambda ) = 1 2 ( r _ 2 / 3 r _ 1 ) ^ 2 + 4 \\quad \\textrm { a n d } , \\\\ g _ 3 ( \\Lambda ) = 1 4 0 G _ 6 ( \\Lambda ) = 8 ( r _ 2 / 3 r _ 1 ) ^ 3 + 4 ( r _ 2 / 3 r _ 1 ) . \\end{gather*}"}
-{"id": "9325.png", "formula": "\\begin{align*} j _ { 0 } = 2 , \\ j _ { 1 } = 1 , \\ j _ { n + 1 } = j _ { n } + 2 j _ { n - 1 } , \\ n \\geq 1 . \\end{align*}"}
-{"id": "7195.png", "formula": "\\begin{align*} a ( u , v ) = ( f , v ) \\qquad \\forall \\ , v \\in H ^ 1 _ 0 ( \\O ) , \\end{align*}"}
-{"id": "3988.png", "formula": "\\begin{align*} p ^ { \\beta _ 3 } _ { k - 1 } ( 3 , t ) = - ( - \\lambda ) ^ { k - 1 } \\underset { \\Omega ^ { k - 1 } _ { 3 } } { \\sum } \\frac { t ^ { \\sum _ { j = 0 } ^ 3 k _ j \\beta _ j } } { \\Gamma \\left ( \\sum _ { j = 0 } ^ 3 k _ j \\beta _ j + 1 \\right ) } . \\end{align*}"}
-{"id": "1830.png", "formula": "\\begin{align*} \\sum _ { i \\in I } \\left \\langle U ^ * \\Gamma ^ * _ i \\Gamma _ i T f , f \\right \\rangle & = \\sum _ { i \\in I } \\left \\langle U ^ * ( \\Gamma ^ * _ i \\Gamma _ i - \\Lambda ^ * _ i \\Lambda _ i ) T f , f \\right \\rangle + \\sum _ { i \\in I } \\left \\langle U ^ * \\Lambda ^ * _ i \\Lambda _ i T f , f \\right \\rangle \\\\ & \\le R \\| f \\| ^ 2 + B \\| f \\| ^ 2 = ( R + B ) \\| f \\| ^ 2 . \\end{align*}"}
-{"id": "7860.png", "formula": "\\begin{align*} m ( x ^ { \\ast } ) = \\left ( m _ { X } ( x ) , m _ { 1 2 } ( x _ { 1 } | x _ { 2 } ) , m _ { 2 1 } ( x _ { 2 } | x _ { 1 } ) \\right ) , \\end{align*}"}
-{"id": "8076.png", "formula": "\\begin{align*} B _ { h } = \\left ( I - ( I - I _ { 2 h , h } B _ { 2 h } I _ { h , 2 h } A _ h ) ( I - S _ h A _ h ) \\right ) ( A _ h ) ^ { - 1 } , \\end{align*}"}
-{"id": "7586.png", "formula": "\\begin{align*} f _ { n , I } ( X ) : = f _ { G _ { { n } , I } } ( X ) = \\binom { n } { I } _ X \\prod _ { j = \\min I + 1 } ^ n ( 1 - X ^ j ) \\in \\Z [ X ] , \\end{align*}"}
-{"id": "4989.png", "formula": "\\begin{align*} h - \\tilde { h } = \\sum _ { j } U _ j V _ j . \\end{align*}"}
-{"id": "1676.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ \\infty | ( - 1 ) ^ { \\ell _ i } 2 \\gamma _ i | , \\ ; \\ ; i . e . \\ ; \\sum _ { i = 1 } ^ { \\infty } | \\gamma _ i | ; \\end{align*}"}
-{"id": "2685.png", "formula": "\\begin{align*} G ( \\Q ) = \\{ \\ , x \\in { E ^ { \\times } } \\mid x c ( x ) = 1 \\ , \\} , \\end{align*}"}
-{"id": "785.png", "formula": "\\begin{align*} 2 ( 2 x - 1 ) ^ { k } & > ( x + 1 ) ^ k + ( x + 2 ) ^ k + \\cdots + ( 2 x - 1 ) ^ k = y ^ n - ( 2 x ) ^ k = y ^ n - ( 2 x ) ^ { B n } \\\\ & = ( y - ( 2 x ) ^ B ) ( y ^ { n - 1 } + \\cdots + ( 2 x ) ^ { B ( n - 1 ) } ) \\ge ( 2 x ) ^ { B ( n - 1 ) } . \\end{align*}"}
-{"id": "1654.png", "formula": "\\begin{align*} \\mu ( X \\setminus ( R _ { f _ 1 } \\cup R _ { f _ 2 } ) ) = 0 \\quad \\mu ( X \\setminus R _ { e } ) = 0 . \\end{align*}"}
-{"id": "3561.png", "formula": "\\begin{align*} \\Lambda ( n ) = \\left \\{ \\begin{array} { l l } \\log p & n = p ^ k , k \\geq 1 , \\\\ 0 & n \\neq p ^ k , k \\geq 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "3301.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty x ^ { \\rho - 1 } ( 1 + x ) ^ { \\sigma - 1 } { _ 2 F _ 1 } ( \\alpha , \\beta ; \\gamma ; - x ) d x = \\frac { \\Gamma ( \\rho ) \\Gamma ( \\alpha - \\sigma - \\rho + 1 ) } { \\Gamma ( \\alpha - \\sigma + 1 ) } { _ 3 F _ 2 } ( \\alpha , \\gamma - \\beta , \\rho ; \\gamma , \\alpha - \\sigma + 1 ; 1 ) . \\end{align*}"}
-{"id": "9328.png", "formula": "\\begin{align*} 3 J _ { n } ^ { ( 3 ) } + j _ { n } ^ { ( 3 ) } = 2 ^ { n + 1 } , \\end{align*}"}
-{"id": "5580.png", "formula": "\\begin{align*} \\ddot { x } + q \\left ( t \\right ) x = 0 q \\left ( \\tau + t \\right ) = q \\left ( t \\right ) \\end{align*}"}
-{"id": "2438.png", "formula": "\\begin{align*} \\ell c _ 1 c _ 2 ( q + 1 ) = q ( c _ 1 + c _ 2 ) . \\end{align*}"}
-{"id": "253.png", "formula": "\\begin{align*} U _ \\mathcal { M } \\{ f _ { n } - f _ { n - 1 } \\} U _ \\mathcal { M } ^ * = f ^ { T _ \\mathcal { M } } _ { n } - f ^ { T _ \\mathcal { M } } _ { n - 1 } . \\end{align*}"}
-{"id": "3652.png", "formula": "\\begin{align*} \\sigma [ n ] ^ { \\vee } = \\begin{pmatrix} - 1 & 1 & 0 & \\cdots & 0 \\\\ 1 & 0 & 1 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 1 & 0 & 0 & \\cdots & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "6870.png", "formula": "\\begin{align*} \\lambda _ m x _ 0 = \\left ( \\frac { 4 } { p q } \\right ) ^ m ( 2 x _ 0 - x _ 1 - x _ 1 ' ) \\end{align*}"}
-{"id": "8291.png", "formula": "\\begin{align*} g N _ \\Z = g N _ { \\widehat { \\Z } } \\cap N \\end{align*}"}
-{"id": "7519.png", "formula": "\\begin{gather*} w ^ i w ^ 1 _ i - w ^ 1 _ { i i } = 0 , \\\\ w ^ i w ^ 2 _ i - w ^ 2 _ { i i } + 1 = 0 . \\end{gather*}"}
-{"id": "9468.png", "formula": "\\begin{align*} S _ { n - 1 } ( 2 ) & = q ^ { - n } \\big ( - S _ { n } ( 1 ) + S _ { n - 1 } ( 1 ) + q ^ { n } ( q ; q ^ 2 ) _ { n } \\big ) . \\end{align*}"}
-{"id": "9470.png", "formula": "\\begin{align*} S _ n ( 1 ) = ( 1 + q - q ^ { 2 n } ) S _ { n - 1 } ( 1 ) - q ( 1 - q ^ { 2 n - 2 } ) S _ { n - 2 } ( 1 ) . \\end{align*}"}
-{"id": "8073.png", "formula": "\\begin{align*} A _ h ^ P = \\sum _ { j = 1 } ^ d \\left ( I _ N ^ { \\otimes ( j - 1 ) } \\otimes T _ h ^ P \\otimes I _ N ^ { \\otimes ( d - j ) } \\right ) + c I _ N ^ { \\otimes d } \\in \\mathbb { R } ^ { N ^ d \\times N ^ d } . \\end{align*}"}
-{"id": "3732.png", "formula": "\\begin{align*} d _ { l , i } = \\mu _ { k _ l + \\beta _ { l , i } } = \\dots = \\mu _ { k _ l + \\beta _ { l , i + 1 } - 1 } . \\end{align*}"}
-{"id": "7069.png", "formula": "\\begin{align*} C I _ { S O ( N ) } ( \\{ \\tilde { u } _ 0 \\} , - \\nabla \\Psi ^ n _ { \\pm } ) = C I _ { S O ( N ) } ( \\{ \\tilde { u } _ 0 \\} , - \\nabla \\Psi ^ { n _ 0 } _ { \\pm } ) . \\end{align*}"}
-{"id": "5162.png", "formula": "\\begin{align*} \\begin{cases} U _ 1 ( s ^ { * } ) = \\sup \\limits _ { ( s _ 1 , s _ 2 ^ * ) \\in \\mathcal C } U _ 1 ( s _ 1 , s _ 2 ^ * ) , \\\\ U _ 2 ( s ^ { * } ) = \\sup \\limits _ { ( s _ 1 ^ * , s _ 2 ) \\in \\mathcal C } U _ 2 ( s _ 1 ^ * , s _ 2 ) . \\end{cases} \\end{align*}"}
-{"id": "7048.png", "formula": "\\begin{align*} \\tilde { M } = D f _ 2 ( \\tilde { B } ) = d i a g \\{ \\tilde { \\eta } _ 1 , \\tilde { \\eta } _ 2 , . . . , \\tilde { \\eta } _ n \\} \\end{align*}"}
-{"id": "8694.png", "formula": "\\begin{align*} | v ( 1 - a _ n z ) | & \\sim | 2 a _ n z | ^ { - \\gamma } L ( 1 / a _ n ) , \\\\ v ( | 1 - a _ n z | ) & = v \\left ( 1 - a _ n \\frac { 1 - | 1 - a _ n z | } { a _ n } \\right ) \\sim 2 ^ { - \\gamma } ( 1 - | 1 - a _ n z | ) ^ { - \\gamma } L ( 1 / a _ n ) \\end{align*}"}
-{"id": "8299.png", "formula": "\\begin{align*} \\psi ( x , y ) = \\mathrm { T r d } ( x \\delta y ^ * ) \\end{align*}"}
-{"id": "4216.png", "formula": "\\begin{align*} \\mathbb { E } [ \\mathrm { t r } ( A ^ k ) ] = \\tau [ C _ k ( A ) ] = \\sum _ { \\pi \\in \\mathcal { P } ( k ) } \\tau ^ 0 [ C _ k ^ { \\pi } ( A ) ] , \\end{align*}"}
-{"id": "7877.png", "formula": "\\begin{align*} u ( x , t ) = \\inf _ { a \\in B _ { V t } } \\left \\{ u _ 0 ( a ) + t L \\left ( { d ( a , x ) \\over t } \\right ) \\right \\} \\end{align*}"}
-{"id": "7440.png", "formula": "\\begin{align*} \\Psi _ 2 ^ + : = \\begin{pmatrix} - x + t v & - y & z & - t \\\\ 2 v z + u y & - x - t v & - t u & - z \\\\ - w z & t w & - x + t v & - y \\\\ t u w & w z & 2 v z + u y & - x - t v \\end{pmatrix} . \\end{align*}"}
-{"id": "2805.png", "formula": "\\begin{align*} P : C \\big ( ( 0 , \\infty ) \\big ) \\rightarrow & C \\left ( ( 0 , T ] \\times ( 0 , \\infty ) \\right ) \\\\ \\mathcal { B } \\mapsto & P ( \\mathcal { B } ) = P ( t , S ) . \\end{align*}"}
-{"id": "2507.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t f + \\xi \\cdot \\nabla _ { x } f = \\Lambda f + K f , \\\\ f ( 0 , x , \\xi ) = f _ 0 ( x , \\xi ) , \\end{cases} \\end{align*}"}
-{"id": "2556.png", "formula": "\\begin{align*} | E _ { 0 } | = 2 | E _ 1 | \\geq \\sqrt { 2 } \\pi ( 1 - \\epsilon ) ^ 2 - 2 \\epsilon ( 1 - \\epsilon ) > \\pi , \\end{align*}"}
-{"id": "368.png", "formula": "\\begin{align*} D _ { n t } = \\sum _ { r , s \\in \\mathbb { Z } } b _ { n , r , s } ^ { 2 \\eta } b _ { n , r , s } ^ { t - 2 \\eta } \\leq \\bigg ( \\sum _ { r , s \\in \\mathbb { Z } } b _ { n , r , s } ^ { 2 } \\bigg ) ^ { \\eta } \\bigg ( \\sum _ { r , s \\in \\mathbb { Z } } b _ { n , r , s } ^ { ( t - 2 \\eta ) / ( 1 - \\eta ) } \\bigg ) ^ { 1 - \\eta } . \\end{align*}"}
-{"id": "1730.png", "formula": "\\begin{align*} T _ \\lambda \\ , W ( f ) = W \\ , S _ { \\lambda } ( f ) , \\ f ^ T _ \\lambda ( h \\circ \\sigma ^ n ) ( f \\circ \\sigma ^ n ) = h f ^ S _ \\lambda ( f \\circ \\sigma ^ n ) . \\end{align*}"}
-{"id": "7463.png", "formula": "\\begin{align*} & \\gamma \\beta \\gamma + \\gamma ^ 2 \\beta + \\gamma \\beta ^ 3 = T e _ 4 , \\\\ & \\gamma \\beta \\gamma - \\beta ^ 2 \\gamma \\beta - \\beta \\gamma \\beta ^ 2 - \\beta ^ 5 = T e _ 4 , \\\\ & \\gamma \\beta \\gamma + \\beta \\gamma ^ 2 + \\beta ^ 3 \\gamma = T e _ 4 . \\end{align*}"}
-{"id": "9578.png", "formula": "\\begin{align*} g ( y ) = \\biggl ( \\int _ { B ( y , \\tau \\delta _ { \\partial D } ( y ) ) } \\frac { \\vert u ( y ) - u ( z ) \\vert ^ p } { \\vert y - z \\vert ^ { n + s p } } \\ , d z \\biggr ) ^ { 1 / p } \\ , . \\end{align*}"}
-{"id": "3043.png", "formula": "\\begin{align*} V _ { h , \\sigma } : = \\{ v _ h \\in V _ h \\mid b ( v _ h , q _ h ) = 0 , \\ \\forall q _ h \\in Q _ h \\} . \\end{align*}"}
-{"id": "6006.png", "formula": "\\begin{align*} \\mathcal { T } _ { \\epsilon } ^ { n } ( Q _ { X } ) & : = \\big \\{ x ^ { n } \\in \\mathcal { X } ^ { n } : \\\\ & \\left | T _ { x ^ { n } } ( x ) - Q _ { X } ( x ) \\right | \\leq \\epsilon Q _ { X } ( x ) , \\forall x \\in \\mathcal { X } \\big \\} . \\end{align*}"}
-{"id": "6430.png", "formula": "\\begin{align*} \\mathcal { V } _ { \\mathcal { M } } \\left ( s \\right ) \\overset { } { = } \\int _ { \\mathcal { D } _ { \\theta } ^ { } } d \\mathcal { V } _ { \\mathcal { M } } = \\int _ { \\mathcal { D } _ { \\theta } ^ { } } \\sqrt { g \\left ( \\theta \\right ) } d ^ { n } \\theta \\mathbf { = } \\int d \\theta ^ { 1 } \\int d \\theta ^ { 2 } \\int \\sqrt { g \\left ( \\theta ^ { 1 } \\theta ^ { n } \\right ) } d \\theta ^ { n } \\end{align*}"}
-{"id": "2347.png", "formula": "\\begin{align*} \\big ( \\big ( p ^ { - n } g _ p ^ { - 1 } \\big ) ' ( z ) \\big ) ^ s = \\left ( \\left ( n c \\big ( c ( g _ { 2 2 } z - g _ { 1 2 } ) + d ( - g _ { 2 1 } z + g _ { 1 1 } ) \\big ) - g _ { 2 1 } z + g _ { 1 1 } \\right ) ^ { - 2 } \\right ) ^ s \\end{align*}"}
-{"id": "6969.png", "formula": "\\begin{align*} D f ( B ) E _ { i j } = \\frac { 1 } { n } ( \\det { B } ) ^ { \\frac { 1 } { n } - 1 } b _ { i j } ^ * , \\end{align*}"}
-{"id": "3696.png", "formula": "\\begin{align*} \\langle v _ { I , j } , m _ s \\rangle - d _ { I , j } = \\sum _ { i \\in I } ( s u ) _ i + \\frac { j ( j + 1 ) } 2 \\frac { n } { n + 1 } - \\# ( I ) \\cdot ( n - j ) . \\end{align*}"}
-{"id": "1366.png", "formula": "\\begin{align*} B _ { ( \\chi , \\chi ' ) } ( Q ) : = \\Big ( \\mbox { v e c } ( Q _ { \\chi } ^ T { \\cal J } _ { x _ 1 } F ( \\overline { x } ) Q _ { \\chi ' } ) \\ , \\ , \\cdots \\ , \\ , \\mbox { v e c } ( Q _ { \\chi } ^ T { \\cal J } _ { x _ n } F ( \\overline { x } ) Q _ { \\chi ' } ) \\Big ) \\ , \\end{align*} % \\end{align*}"}
-{"id": "4696.png", "formula": "\\begin{align*} \\Psi _ v ( r ) = \\Psi _ v ( 1 ) = \\int _ { B _ 1 } v \\ , d x . \\end{align*}"}
-{"id": "408.png", "formula": "\\begin{align*} X ^ \\gamma = \\frac { \\partial ^ { \\abs { \\gamma } } } { \\partial x ^ { \\gamma _ 1 } \\partial t ^ { \\gamma _ 2 } } \\end{align*}"}
-{"id": "6234.png", "formula": "\\begin{align*} X ( n ) = - C ( n ) + Z ( n ) ~ , ~ n \\geq 1 ~ , \\end{align*}"}
-{"id": "522.png", "formula": "\\begin{align*} f \\circ \\varphi ^ { - 1 } ( y _ 1 , \\dots , y _ d ) \\ , = \\ , f ( x ) - ( y _ 1 ^ 2 + \\dots + y _ k ^ 2 ) + ( y _ { k + 1 } ^ 2 + \\dots + y _ d ^ 2 ) . \\end{align*}"}
-{"id": "8609.png", "formula": "\\begin{align*} v _ { i } : C _ { c } ( \\Gamma , X ) & \\to C _ { c } ( g _ { i } \\Gamma , X ) \\subset C _ { c } ( \\Gamma g ^ { - 1 } \\Gamma , X ) , ( v _ { i } \\Phi ) ( g _ { i } \\xi ) : = g _ { i } \\Phi ( \\xi ) , \\end{align*}"}
-{"id": "4197.png", "formula": "\\begin{align*} ( t _ 1 ( \\tau ) , t _ 2 ( \\tau ) , t _ 3 ( \\tau ) ) : = ( \\frac { 2 \\pi i } { 1 2 } E _ 2 ( \\tau ) , \\ 1 2 ( \\frac { 2 \\pi i } { 1 2 } ) ^ 2 E _ 4 ( \\tau ) , 8 ( \\frac { 2 \\pi i } { 1 2 } ) ^ 3 E _ 6 ( \\tau ) ) , \\end{align*}"}
-{"id": "2020.png", "formula": "\\begin{align*} f _ a ( \\lambda z _ 1 , \\lambda z _ 2 ) = \\lambda ^ { k _ a } f _ a ( z _ 1 , z _ 2 ) . \\end{align*}"}
-{"id": "4290.png", "formula": "\\begin{align*} \\underline { \\eta } = \\big ( ( \\eta _ 1 , \\delta _ 1 ) , \\ldots , ( \\eta _ { l ( \\eta ) } , \\delta _ { l ( \\eta ) } ) \\big ) , \\delta _ i \\in H ^ { \\ast } ( D ) , \\end{align*}"}
-{"id": "4389.png", "formula": "\\begin{align*} \\nu _ t ( A | _ F ) : = \\inf _ { w \\in \\mathbb { C } ^ n } \\nu ( A | _ { F \\cap B ( w , r _ t ) } ) . \\end{align*}"}
-{"id": "8033.png", "formula": "\\begin{align*} \\big \\Vert g _ L \\big \\Vert _ { L ^ p } & \\lesssim \\Big \\Vert \\Big ( \\sum _ { n = M } ^ { L } { b _ n ^ 2 2 ^ { - 2 t _ n d ( 1 - 1 / p ) } | \\phi _ { t _ n } | ^ 2 } \\Big ) ^ { 1 / 2 } \\Big \\Vert _ { L ^ p } \\\\ & \\leq \\Big ( \\sum _ { n = M } ^ { L } { b _ n ^ p 2 ^ { - { t _ n } d ( p - 1 ) } \\big \\Vert \\phi _ { { t _ n } } \\big \\Vert _ { L ^ p } ^ p } \\Big ) ^ { { 1 } / { p } } \\approx \\Big ( \\sum _ { n = M } ^ { L } { b _ n ^ p } \\Big ) ^ { { 1 } / { p } } . \\end{align*}"}
-{"id": "6077.png", "formula": "\\begin{align*} V = \\Psi '^ 2 - 2 \\frak { e } _ k \\sin ^ 2 \\Psi . \\end{align*}"}
-{"id": "848.png", "formula": "\\begin{align*} v ( t , x ) = \\inf _ { u \\in \\mathcal { U } } J ( t , x , u ) . \\end{align*}"}
-{"id": "2329.png", "formula": "\\begin{align*} \\Phi ( 0 , a , y , s ) = \\frac { 2 \\pi ^ s | a | ^ { s - \\frac 1 2 } } { \\Gamma ( s ) } y ^ { \\frac 1 2 - s } K _ { s - \\frac 1 2 } ( 2 \\pi | a | y ) . \\end{align*}"}
-{"id": "555.png", "formula": "\\begin{align*} h ( 1 ) = h ' ( 1 ) = 0 \\mbox { a n d } \\lim _ { \\rho \\rightarrow + \\infty } h ' ( \\rho ) = + \\infty . \\end{align*}"}
-{"id": "6835.png", "formula": "\\begin{align*} \\langle L ( \\phi ) , \\eta _ { R _ 3 , \\xi _ j } \\varphi _ { 0 , j } \\rangle = \\langle L ( \\phi ) , \\eta _ { R _ 3 , \\xi _ k } \\varphi _ { 0 , k } \\rangle , \\end{align*}"}
-{"id": "6235.png", "formula": "\\begin{align*} Z ( 1 ) = \\xi _ Z ( i ) \\sim \\left ( \\begin{array} { c c c c c } \\ldots & - 2 & - 1 & 0 & 1 \\\\ \\ldots & q _ 2 & q _ 1 & q _ 0 & \\rho \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "515.png", "formula": "\\begin{align*} W = X _ { q } ^ { n } \\mathbf { H } _ 2 ^ T \\end{align*}"}
-{"id": "8.png", "formula": "\\begin{align*} V ^ { C } _ { \\sigma } ( C _ { 1 } , C _ { 2 } ) = E _ { C _ { 1 } C _ { 2 } } [ K ( C _ { 1 } , C _ { 2 } ) ] \\end{align*}"}
-{"id": "9045.png", "formula": "\\begin{align*} { \\mathcal H } ( w , w ' ) : = ( \\alpha \\bar w _ 1 w _ 1 ' + \\beta \\bar w _ 2 w _ 2 ' + \\beta \\bar w _ 3 w _ 3 ' , \\gamma \\bar w _ 1 w _ 1 ' + \\delta \\bar w _ 2 w _ 2 ' + \\delta \\bar w _ 3 w _ 3 ' ) . \\end{align*}"}
-{"id": "10027.png", "formula": "\\begin{align*} \\epsilon _ Q ( g ) = \\prod _ { q \\mid Q } \\big ( - q ^ { 1 - \\frac { n } { 2 } } c ( q ) \\big ) = \\pm 1 . \\end{align*}"}
-{"id": "8520.png", "formula": "\\begin{align*} & - c _ \\alpha ^ \\top b _ { 1 : k } + c _ \\alpha ^ \\top \\alpha - c _ \\beta ^ \\top \\beta \\to \\min \\\\ & \\begin{cases} B \\beta - A \\alpha + y = \\hat b \\\\ \\bar B \\beta - \\bar A \\alpha \\leq \\hat b _ d \\\\ \\alpha \\in \\mathbb { Z } _ + ^ k , \\ , y \\in \\mathbb { Z } _ + ^ s , \\ , \\beta \\in \\mathbb { Z } ^ s , \\\\ \\end{cases} \\end{align*}"}
-{"id": "7567.png", "formula": "\\begin{gather*} u = 6 \\frac x { y ^ 2 } + \\theta _ { y y } - \\frac 3 y \\theta _ y + \\frac 3 { y ^ 2 } \\theta , \\end{gather*}"}
-{"id": "9263.png", "formula": "\\begin{align*} V _ \\xi = \\sum _ { \\nu \\in \\sigma ( \\mathcal { N } ) } V _ \\xi ( \\nu ) P _ \\nu . \\end{align*}"}
-{"id": "2890.png", "formula": "\\begin{align*} [ \\sigma ] = \\{ p \\in k ^ { \\omega } : \\sigma \\prec p \\} \\end{align*}"}
-{"id": "86.png", "formula": "\\begin{align*} \\Gamma ( \\lambda ( s ) ) = \\lambda ( s ) \\otimes \\lambda ( s ) , \\ \\ \\ s \\in G . \\end{align*}"}
-{"id": "6904.png", "formula": "\\begin{align*} \\lambda = \\lim _ { m \\to \\infty } \\left ( \\frac { 1 } { \\mu _ { 0 } r _ { 0 } } \\right ) ^ { m } \\lambda _ { m } \\end{align*}"}
-{"id": "4463.png", "formula": "\\begin{align*} \\langle | F ^ \\ell ( k ) | ^ 2 \\rangle & \\lesssim \\sum _ { \\stackrel { k ' + k '' = k } { k ' , k '' \\not = 0 } } d ^ { - 4 } ( k ' , 0 ) d ^ { - 1 } ( k '' , 0 ) , \\\\ \\langle | \\ell \\frac { \\partial } { \\partial \\ell } F ^ \\ell ( k ) | ^ 2 \\rangle & \\lesssim \\sum _ { \\stackrel { k ' + k '' = k } { k ' , k '' \\not = 0 } } { \\min } ^ 2 \\{ 1 , \\ell ( d ( k ' , 0 ) + d ( k '' , 0 ) ) \\} d ^ { - 4 } ( k ' , 0 ) d ^ { - 1 } ( k ' , 0 ) . \\end{align*}"}
-{"id": "4645.png", "formula": "\\begin{align*} r ( k ) = \\frac { \\delta } { 3 \\cdot 2 ^ k } \\end{align*}"}
-{"id": "7412.png", "formula": "\\begin{align*} \\beta = b - t _ 1 / 2 , \\gamma = c - t _ 1 / 2 , \\delta = d - t _ 1 / 2 \\end{align*}"}
-{"id": "5630.png", "formula": "\\begin{align*} c _ r \\ , : = \\ , \\left [ \\frac 1 { 2 r - 1 } + \\frac { ( r - 1 ) ^ 2 } { 2 r - 3 } \\right ] ^ { 1 / 2 } \\ , \\frac 1 { ( r - 1 ) ! } . \\end{align*}"}
-{"id": "5847.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ N & ( - 1 ) ^ k \\binom { N } { k } k g ^ { k - 1 } D ( g ) D ^ { N - 1 } ( g ^ { N + 1 - k } f ) \\\\ & = - \\sum _ { k = 0 } ^ N ( - 1 ) ^ k \\binom { N } { k } g ^ k D ^ N ( g ^ { N + 1 - k } f ) . \\end{align*}"}
-{"id": "4865.png", "formula": "\\begin{align*} \\log \\left ( \\left ( \\exp ( x ) , \\exp ( y ) \\right ) \\right ) = \\left [ x , y \\right ] , \\end{align*}"}
-{"id": "2990.png", "formula": "\\begin{align*} \\{ e _ R \\ | \\ R = ( r _ 2 , r _ 3 , \\ldots ) l ( R ) = r _ 2 + r _ 3 + \\cdots = i \\} \\end{align*}"}
-{"id": "755.png", "formula": "\\begin{align*} m ( a _ 1 , a _ 2 ) = m ( b _ 1 , b _ 2 ) + m ( c _ 1 , c _ 2 ) + m ( a _ 1 - b _ 1 - c _ 1 , a _ 2 - b _ 2 - c _ 2 ) . \\end{align*}"}
-{"id": "5548.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t } w _ { 0 } \\left ( \\tau _ 1 \\right ) d \\tau _ 1 = t \\end{align*}"}
-{"id": "5046.png", "formula": "\\begin{align*} \\lambda _ 5 ( F _ 1 ) & = 7 1 1 9 8 1 9 0 0 + 1 5 5 5 2 0 0 \\sqrt { 1 0 6 7 0 5 } = a _ 5 ( f ) + a _ 5 ( g ) \\\\ \\lambda _ 5 ( F _ 2 ) & = 7 1 1 9 8 1 9 0 0 - 1 5 5 5 2 0 0 \\sqrt { 1 0 6 7 0 5 } = a _ 5 ( f ) + a _ 5 ( g ' ) . \\end{align*}"}
-{"id": "4957.png", "formula": "\\begin{align*} \\delta ( f ) ( = \\delta _ { \\eta } ( f ) ) = \\sum _ { j = 1 } ^ n \\frac { \\partial \\{ f , x _ j \\} } { \\partial x _ j } \\end{align*}"}
-{"id": "8310.png", "formula": "\\begin{align*} \\mathrm { F J } _ \\alpha ^ { ( a a _ \\tau ) } ( \\psi ) = \\tau \\big ( \\mathrm { F J } _ \\alpha ^ { ( a ) } ( \\psi ) \\big ) \\end{align*}"}
-{"id": "9049.png", "formula": "\\begin{align*} d ( D _ 6 ) \\le 1 3 < 1 4 = n ^ 2 - 2 \\end{align*}"}
-{"id": "7670.png", "formula": "\\begin{align*} & a _ 1 = \\cdots = a _ { n _ 1 } < a _ { n _ 1 + 1 } , \\\\ & a _ { n _ 1 } < a _ { n _ 1 + 1 } = \\cdots = a _ { n _ 1 + n _ 2 } < a _ { n _ 1 + n _ 2 + 1 } , \\\\ & \\cdots \\\\ & a _ { n _ 1 + \\cdots + n _ { r - 1 } } < a _ { n _ 1 + \\cdots + n _ { r - 1 } + 1 } = \\cdots = a _ { n _ 1 + \\cdots + n _ r } \\end{align*}"}
-{"id": "8376.png", "formula": "\\begin{align*} \\psi ( f ) = \\bigotimes _ { \\substack { m > 0 \\\\ \\mu \\in V _ \\Z ^ \\vee / V _ \\Z } } s ( m , \\mu ) ^ { \\otimes c ( - m , \\mu ) } \\end{align*}"}
-{"id": "9874.png", "formula": "\\begin{align*} e _ i ( x ) = \\sum _ { j _ 1 < j _ 2 < \\cdots < j _ i } x _ { j _ { 1 } } \\cdots x _ { j _ { i } } \\end{align*}"}
-{"id": "6921.png", "formula": "\\begin{align*} u ( x , 0 ) = f ( x ) \\end{align*}"}
-{"id": "5758.png", "formula": "\\begin{align*} \\sigma ( x , y ) = \\sum \\nolimits _ { i = 1 } ^ { | T | } q _ i \\lambda _ { a _ i , b _ i } ( x , y ) , \\end{align*}"}
-{"id": "6594.png", "formula": "\\begin{align*} g _ n = \\left \\{ \\begin{array} { l l } \\phi ^ 0 _ { n } \\circ f _ n \\circ \\phi _ { n } & \\mbox { i n } \\ \\varphi _ n ( E _ M ) \\\\ \\phi ^ k _ { n } \\circ f _ n & \\mbox { i n } \\ \\varphi _ n ( f ^ { - 1 } B ^ { k + 1 } _ M ) , \\ 0 < k < k _ 0 \\\\ f _ n & \\mbox { e l s e w h e r e } \\end{array} \\right . \\ , \\end{align*}"}
-{"id": "2619.png", "formula": "\\begin{align*} - S u = f ' ( 0 , 1 ) . \\end{align*}"}
-{"id": "5868.png", "formula": "\\begin{align*} a _ 1 = 1 , \\ , \\ , a _ { 8 } = 1 2 6 , \\ , \\ , a _ { 1 0 } = 1 6 , \\ , \\ , a _ { 1 2 } = 9 6 , \\ , \\ , a _ { 1 4 } = 1 6 , \\ , \\ , a _ { 1 6 } = 1 , \\ , \\ , , \\end{align*}"}
-{"id": "822.png", "formula": "\\begin{align*} - \\Delta u + \\frac { 2 } { \\tau } u = \\frac { 2 } { \\tau } u _ n , \\Omega , u = 0 \\partial \\Omega , \\end{align*}"}
-{"id": "10108.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } { \\boldsymbol \\psi } _ k ( i ) = { \\boldsymbol \\omega } _ k ( i - 1 ) + { \\mu } _ k { \\boldsymbol x _ k ( i ) } \\big [ { d _ k ( i ) } - { \\boldsymbol \\omega } _ k ^ H ( i - 1 ) { \\boldsymbol x _ k ( i ) } \\big ] ^ * , \\\\ \\ \\\\ { \\boldsymbol { \\omega } } _ k ( i ) = \\sum \\limits _ { l \\in \\mathcal { N } _ k } c _ { k l } \\boldsymbol \\psi _ l ( i ) , \\end{array} \\right . \\end{align*}"}
-{"id": "8218.png", "formula": "\\begin{align*} \\alpha _ 0 = \\biggl ( \\Bigl ( 1 + t _ + + \\frac { 1 } { 2 } t _ + ^ 2 + \\cdots \\Bigr ) - \\Bigl ( 1 + t _ - + \\frac { 1 } { 2 } t _ - ^ 2 + \\cdots \\Bigr ) \\biggr ) \\end{align*}"}
-{"id": "5470.png", "formula": "\\begin{align*} \\mathrm { B } _ \\rho p ( x ) = x ^ { - \\rho } \\int _ 0 ^ x y ^ { \\rho - 1 } p ( y ) \\dd y . \\end{align*}"}
-{"id": "2654.png", "formula": "\\begin{align*} P _ \\ast ( T _ { \\tilde \\Psi } ) = S _ a + b K \\end{align*}"}
-{"id": "3890.png", "formula": "\\begin{align*} f ( \\omega ; s ; x ) = \\frac { 1 } { 2 \\pi i } \\int _ { ( \\delta ) } \\frac { \\Gamma ( 1 / 2 - \\alpha / 2 ) } { \\Gamma ( \\alpha / 2 ) } \\widetilde { \\omega } ( \\alpha ) \\left ( \\frac { x } { 4 l } \\right ) ^ { \\alpha + s - 1 } d \\alpha , \\end{align*}"}
-{"id": "338.png", "formula": "\\begin{align*} \\sum _ { i \\in I _ 0 ^ + } \\bar { \\lambda } _ i = \\sum _ { j \\in J _ 0 ^ + } \\bar { \\mu } _ j . \\end{align*}"}
-{"id": "1367.png", "formula": "\\begin{align*} A ( Q , P ) = U [ \\Sigma ( Q , P ) \\ , \\ , \\ , \\ , \\ , 0 ] R ^ T \\ , , \\end{align*}"}
-{"id": "5626.png", "formula": "\\begin{align*} d ( M _ 2 ) = \\begin{pmatrix} t ^ 3 \\\\ t ^ 5 \\\\ t ^ 2 \\end{pmatrix} : M _ 1 \\oplus N _ 1 \\oplus M _ 2 \\to M _ 2 \\ , . \\end{align*}"}
-{"id": "4569.png", "formula": "\\begin{align*} E ^ { ( 1 ) } _ { 0 | 1 , l } = \\sum _ { j \\in \\Z } q _ 1 ^ { j - l } \\bigl ( E _ { 0 , - j + l } E _ { 1 , j } - q _ 1 ^ { - 1 } E _ { 0 , - j + l - 1 } E _ { 1 , j + 1 } \\bigr ) \\ , . \\end{align*}"}
-{"id": "2297.png", "formula": "\\begin{align*} \\partial _ t v + u \\cdot \\nabla v + \\nabla u ^ T v + \\nabla \\tilde { p } = - \\nu A ^ s v \\end{align*}"}
-{"id": "6011.png", "formula": "\\begin{align*} & T _ { 1 + s } ( \\pi _ { X Y } ) \\\\ & \\quad : = \\inf \\left \\{ R : \\ ; \\lim _ { n \\to \\infty } D _ { 1 + s } ( P _ { X ^ { n } Y ^ { n } } \\| \\pi _ { X ^ { n } Y ^ { n } } ) = 0 \\right \\} , \\\\ & \\widetilde { T } _ { 1 + s } ( \\pi _ { X Y } ) \\\\ & \\quad : = \\inf \\Big \\{ R : \\ ; \\lim _ { n \\to \\infty } \\frac { 1 } { n } D _ { 1 + s } ( P _ { X ^ { n } Y ^ { n } } \\| \\pi _ { X ^ { n } Y ^ { n } } ) = 0 \\Big \\} . \\end{align*}"}
-{"id": "7492.png", "formula": "\\begin{align*} \\partial ^ { * h } : \\mathcal { A } _ { p , q } ( H \\mathcal { T } E ) \\rightarrow \\mathcal { A } _ { p , q - 1 } ( H \\mathcal { T } E ) , ( \\partial ^ h \\Psi , \\Phi ) = ( \\Psi , \\partial ^ { * h } \\Psi ) , \\\\ \\bar { \\partial } ^ { * h } : \\mathcal { A } _ { p , q } ( H \\mathcal { T } E ) \\rightarrow \\mathcal { A } _ { p - 1 , q } ( H \\mathcal { T } E ) , ( \\bar { \\partial } ^ h \\Psi , \\Phi ) = ( \\Psi , \\bar { \\partial } ^ { * h } \\Psi ) , \\end{align*}"}
-{"id": "1562.png", "formula": "\\begin{align*} \\nu ( \\beta ) & = \\nu ( m - 2 ^ n j _ 1 - 2 ^ { n - 1 } j _ 2 - \\cdots - 2 ^ { n - k + 2 } j _ { k - 1 } ) \\\\ & = \\nu ( m - ( 2 ^ n j _ 1 - 2 ^ { n - 1 } j _ 2 - \\cdots - 2 ^ { n - k + 2 } j _ { k - 1 } ) ) \\\\ & = \\nu ( m ) , \\end{align*}"}
-{"id": "6322.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } p _ { n + 1 } ^ { x } - p _ { n } ^ { x } = 0 . \\end{align*}"}
-{"id": "6727.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\mathbb { P } \\Big ( \\Gamma _ { 1 } > t N \\Big ) = \\lim \\limits _ { N \\rightarrow \\infty } \\Big ( 1 - \\frac { 1 } { N } \\Big ) ^ { N t } = e ^ { - t } \\end{align*}"}
-{"id": "5279.png", "formula": "\\begin{align*} \\| F ( t _ 1 , u _ 1 ) - F ( t _ 2 , u _ 2 ) \\| \\leq & L ( | t _ 1 - t _ 2 | ^ { \\vartheta } + \\| u _ 1 - u _ 2 \\| _ { \\alpha } ) , \\\\ & \\forall ( t _ i , u _ i ) \\in V , i = 1 , 2 . \\end{align*}"}
-{"id": "7906.png", "formula": "\\begin{align*} d ( x , O ) + d ( x ' , O ) = d ( x , x ' ) d ( y , O ) + d ( y ' , O ) = d ( y , y ' ) . \\end{align*}"}
-{"id": "9231.png", "formula": "\\begin{align*} [ x ^ { + } \\otimes 1 ^ { - } , y ^ { + } \\otimes a ^ { - } ] = [ x ^ { + } , y ^ { + } ] \\otimes a ^ { + } [ x ^ { + } \\otimes 1 ^ { - } , y ^ { - } \\otimes a ^ { + } ] = [ x ^ { + } , y ^ { - } ] \\otimes a ^ { - } . \\end{align*}"}
-{"id": "10050.png", "formula": "\\begin{align*} \\Phi \\left ( \\left ( \\begin{matrix} a & b \\\\ & a ^ { - 1 } \\end{matrix} \\right ) g , s \\right ) = \\chi _ E ( a ) | a | ^ { s + 1 } \\Phi ( g , s ) . \\end{align*}"}
-{"id": "8976.png", "formula": "\\begin{gather*} \\omega ( z _ 1 , \\dots , z _ n ) = ( q / 2 - z _ n , \\dots , q / 2 - z _ 1 ) , \\end{gather*}"}
-{"id": "2448.png", "formula": "\\begin{align*} \\alpha _ i & = e _ i - e _ { i + 1 } 1 \\leq i \\leq 8 , \\\\ \\alpha _ 0 & = l - e _ 1 - e _ 2 - e _ 3 . & \\end{align*}"}
-{"id": "3794.png", "formula": "\\begin{align*} N ( y ) : = \\omega ( W _ { \\{ y \\} } ) . \\end{align*}"}
-{"id": "5472.png", "formula": "\\begin{align*} \\mathrm { B } _ \\rho p ( x ) = r ^ { - \\rho \\{ \\log _ r x \\} } \\left [ \\frac { 1 } { r ^ \\rho - 1 } \\int _ 1 ^ r s ^ { \\rho - 1 } p ( s ) \\dd s + \\int _ 1 ^ { r ^ { \\{ \\log _ r x \\} } } s ^ { \\rho - 1 } p ( s ) \\dd s \\right ] , \\end{align*}"}
-{"id": "9822.png", "formula": "\\begin{align*} g _ 1 ( t , p ) = \\begin{cases} - \\beta _ 1 t + \\alpha \\mathbb { E } \\left ( \\left [ \\sum \\limits _ { j = 1 } ^ { N _ p } X _ j \\ ! - \\ ! T _ 1 \\ ! - \\ ! t \\right ] ^ + \\right ) + \\beta _ 2 \\mathbb { E } \\left ( \\left [ \\sum \\limits _ { j = 1 } ^ { N _ p } X _ j - T _ 1 \\right ] ^ + \\right ) , & t < 0 , \\\\ \\beta _ 2 t + ( \\alpha + \\beta _ 2 ) \\mathbb { E } \\left ( \\left [ \\sum \\limits _ { j = 1 } ^ { N _ p } X _ j - T _ 1 - t \\right ] ^ + \\right ) , & t > 0 . \\end{cases} \\end{align*}"}
-{"id": "2396.png", "formula": "\\begin{align*} p _ 3 ( x ) = a _ 3 - a _ 5 x + \\left ( - 2 a _ 3 - a _ 4 - a _ 1 a _ 5 \\right ) x ^ 2 . \\end{align*}"}
-{"id": "2898.png", "formula": "\\begin{align*} c ( k , s _ 1 , \\ldots , s _ { n - 1 } ) = 1 \\end{align*}"}
-{"id": "2920.png", "formula": "\\begin{align*} \\textstyle \\alpha _ i : = \\bigcap _ { a \\in d _ i } \\alpha _ a , \\end{align*}"}
-{"id": "6523.png", "formula": "\\begin{align*} K ^ { \\delta } = \\{ x \\in \\mathbb { R } ^ n : | \\mathrm { c o n v } [ K , x ] | _ n \\leq ( 1 + \\delta ) | K | _ n \\} . \\end{align*}"}
-{"id": "7043.png", "formula": "\\begin{align*} \\tilde { \\sigma } _ j = t _ j \\sigma _ j , j = 1 , 2 , . . . , n - 1 \\end{align*}"}
-{"id": "4658.png", "formula": "\\begin{align*} \\left | \\sum _ { n = 0 } ^ { N - 1 } 1 _ { I _ { k , j } } ( T ^ n x ) - 1 _ { I _ { k , j } } ( T ^ n y ) \\right | < E _ 3 \\max \\{ 1 , ( | I _ { k , j } | N ) ^ { \\zeta _ 3 } \\} \\end{align*}"}
-{"id": "2814.png", "formula": "\\begin{align*} \\widehat { P } ( t , \\omega ) = A ( \\omega ) e ^ { - \\frac { 1 } { 2 } \\sigma ^ { 2 } q ( \\omega ) t } + \\frac { r K } { \\omega } \\int _ { t } ^ { T } ( \\mathcal { B } ( s ) ) ^ { \\omega } e ^ { \\frac { 1 } { 2 } \\sigma ^ { 2 } q ( \\omega ) ( s - t ) } \\mathrm { d } s , \\end{align*}"}
-{"id": "9740.png", "formula": "\\begin{align*} \\begin{cases} & U _ { k , n } ^ 0 : = U _ h ( k h - , y _ { k , n } ) , \\\\ & W ( \\widetilde { U } _ { k , n } ^ 0 ) : = W ( U _ { k , n } ^ 0 ) + G ( U _ { k , n } ^ 0 ) h . \\end{cases} \\end{align*}"}
-{"id": "2887.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 \\pi i } \\left [ \\int _ { \\lambda - i T } ^ { \\lambda + i T } + \\int _ { \\lambda + i T } ^ { \\mu + i T } + \\int _ { \\mu + i T } ^ { \\mu - i T } + \\int _ { \\mu - i T } ^ { \\lambda - i T } \\right ] \\Gamma ( s ) \\zeta ( s ) \\zeta ( s + 2 m + 1 ) y ^ { - s } \\ , { \\rm d } s \\\\ & = R _ { - 2 m } + R _ { 0 } + R _ { 1 } + \\sum _ { i = 0 } ^ m R _ { - ( 2 i + 1 ) } . \\end{align*}"}
-{"id": "3298.png", "formula": "\\begin{align*} { _ 2 F _ 1 } ( \\alpha , \\beta ; \\gamma ; x ) = ( 1 - x ) ^ { \\gamma - \\alpha - \\beta } { _ 2 F _ 1 } ( \\gamma - \\alpha , \\gamma - \\beta ; \\gamma ; x ) . \\end{align*}"}
-{"id": "7936.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { \\infty } \\log ( 1 + \\varphi ( \\nu ^ i | V | ) ) \\le \\sum _ { i = 0 } ^ { \\infty } \\varphi ( \\nu ^ i | V | ) < \\infty . \\end{align*}"}
-{"id": "9373.png", "formula": "\\begin{align*} \\lambda ( \\mu ) = \\mu ( \\mu + d - 2 ) , \\end{align*}"}
-{"id": "453.png", "formula": "\\begin{align*} \\abs { a _ { k _ 1 , k _ 2 } ( \\lambda ) } = \\frac { \\abs { \\lambda } ^ { n + k _ 1 } \\left ( \\sinh \\left ( \\Re \\sqrt { \\lambda ^ 2 } \\right ) ^ 2 + \\cos \\left ( \\Im \\sqrt { \\lambda ^ 2 } \\right ) ^ 2 \\right ) ^ { k _ 1 / 2 } } { \\left ( \\sinh \\left ( \\Re \\sqrt { \\lambda ^ 2 } \\right ) ^ 2 + \\sin \\left ( \\Im \\sqrt { \\lambda ^ 2 } \\right ) ^ 2 \\right ) ^ { ( n + k _ 1 ) / 2 } } \\abs * { ( \\lambda , u _ 1 ) } ^ { k _ 2 } , \\end{align*}"}
-{"id": "6681.png", "formula": "\\begin{align*} H _ p ^ n ( \\Delta ) = \\mathrm { c o n v } [ B _ p ^ n , ( 1 + \\Delta ) e _ 1 ] \\backslash B _ p ^ n . \\end{align*}"}
-{"id": "669.png", "formula": "\\begin{align*} \\sharp E i g ^ { \\neq 0 } ( A _ n ( t ) ) = \\sharp E i g ^ { \\neq 0 } ( B _ n ( t ) ) = s ( n ) , ~ \\forall t \\in S ^ 1 \\end{align*}"}
-{"id": "8026.png", "formula": "\\begin{align*} \\int _ { \\Omega } { \\Big ( \\sum _ { Q \\in \\mathcal { V } _ n ( l , P ) } { \\theta _ Q ( \\omega ) } \\Big ) ^ { { q } / { 2 } } } d \\lambda & \\geq \\sum _ { R \\in \\mathcal { V } _ { n } ( l , P ) } { \\int _ { \\Omega ( P , R , l , n ) } { \\Big ( \\sum _ { Q \\in \\mathcal { V } _ n ( l , P ) } { \\theta _ Q ( \\omega ) } \\Big ) ^ { { q } / { 2 } } } d \\lambda } \\\\ & = \\sum _ { R \\in \\mathcal { V } _ n ( l , P ) } { \\lambda ( \\Omega ( P , R , l , n ) ) } \\geq 2 ^ { { t _ n } d } \\frac { 1 } { L } \\Big ( 1 - \\frac { 1 } { L } \\Big ) ^ { 2 ^ { { t _ n } d } } \\end{align*}"}
-{"id": "9837.png", "formula": "\\begin{align*} \\varphi _ 3 ( x , y ) = x ^ 3 - 9 x y ^ 2 . \\end{align*}"}
-{"id": "9716.png", "formula": "\\begin{align*} \\tilde { v } = v . \\end{align*}"}
-{"id": "2421.png", "formula": "\\begin{align*} D _ 1 = 0 , \\ D _ 3 < 0 , \\ D _ 1 ' \\neq 0 , \\ R _ { 1 1 3 } \\neq 0 , \\end{align*}"}
-{"id": "3056.png", "formula": "\\begin{align*} \\begin{cases} ( u _ t , v ) _ H + a ( u , v ) + b ( v , p ) = \\langle f , v \\rangle _ { V , V ' } , & \\forall v \\in V , \\\\ b ( u , q ) = 0 , & \\forall q \\in Q , \\end{cases} \\end{align*}"}
-{"id": "5238.png", "formula": "\\begin{align*} \\Gamma ( E ^ u ( \\lambda , \\cdot ) , V , z ^ * ) ( \\zeta ) = \\omega ( \\zeta , A ( \\lambda , z ^ * ) \\zeta ) , \\end{align*}"}
-{"id": "7837.png", "formula": "\\begin{align*} F _ { X _ { i } } ( x _ { i } ) = \\left ( 1 - e ^ { - \\eta ^ { \\alpha } ( x _ { i } ) } \\right ) ^ { \\theta _ { i } + \\theta _ { 3 } } , \\ \\ i = 1 , 2 . \\end{align*}"}
-{"id": "8072.png", "formula": "\\begin{align*} ( X + Y ) \\otimes Z = ( X \\otimes Z ) + ( Y \\otimes Z ) , ( X _ 1 \\otimes X _ 2 ) ( Y _ 1 \\otimes Y _ 2 ) = ( X _ 1 Y _ 1 \\otimes X _ 2 Y _ 2 ) . \\end{align*}"}
-{"id": "4209.png", "formula": "\\begin{align*} \\mathtt { T _ \\Omega } : = \\{ ( \\Omega _ 1 , \\Omega _ 2 , \\Omega _ 3 ) \\in \\mathbb { R } ^ 3 | \\Omega _ 1 \\neq \\Omega _ 2 \\neq \\Omega _ 3 \\} , \\end{align*}"}
-{"id": "2109.png", "formula": "\\begin{align*} ( \\Delta _ H - \\partial _ t ) w ^ a _ s = Q ^ a _ s , \\mbox { w i t h } w ^ a _ s ( p , 0 ) = \\partial _ s \\phi ^ a _ s ( p ) , \\end{align*}"}
-{"id": "4511.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( \\phi ( \\Delta _ n ^ * a \\Delta _ n ) - \\phi ( \\Delta _ n ^ * ) \\phi ( a ) \\phi ( \\Delta _ n ) ) = 0 . \\end{align*}"}
-{"id": "6292.png", "formula": "\\begin{align*} | I _ { 2 1 2 } + I _ { 4 1 2 } | = & \\left | \\sum _ { q \\geq - 1 } \\left ( \\lambda _ q ^ { 2 r } - \\lambda _ q ^ { 2 s } \\right ) \\int _ { \\R ^ 3 } b _ { \\leq { q - 2 } } \\cdot \\nabla u _ q b _ q \\ , d x \\right | \\end{align*}"}
-{"id": "2379.png", "formula": "\\begin{align*} \\frac { 1 } { ( n + w ) ^ r } = \\frac { 1 } { \\Gamma ( r ) } \\int _ 0 ^ \\infty e ^ { - t ( n + w ) } t ^ { r - 1 } \\ , d t \\end{align*}"}
-{"id": "7085.png", "formula": "\\begin{align*} f ( a , b ) = ( - a , - b , a b ; a b ) _ { \\infty } . \\end{align*}"}
-{"id": "8601.png", "formula": "\\begin{align*} m _ { k } : = \\# \\{ ( i , j ) : g _ { i } h _ { j } \\Gamma = g _ { i ( k ) } h _ { j ( k ) } \\Gamma \\} \\quad \\textnormal { a n d } \\Gamma g ^ { - 1 } \\Gamma h ^ { - 1 } \\Gamma = \\bigsqcup _ { k = 1 } ^ { K } \\Gamma g _ { i ( k ) } h _ { j ( k ) } \\Gamma , \\end{align*}"}
-{"id": "5236.png", "formula": "\\begin{align*} \\Gamma ( \\gamma _ 1 , \\gamma _ 2 , t ^ * ) = \\Gamma ( \\gamma _ 1 , \\gamma _ 2 ( t ^ * ) , t ^ * ) - \\Gamma ( \\gamma _ 2 , \\gamma _ 1 ( t ^ * ) , t ^ * ) \\end{align*}"}
-{"id": "9662.png", "formula": "\\begin{align*} ( e , p , T ) = ( e ( \\rho , S ) , p ( \\rho , S ) , T ( \\rho , S ) ) . \\end{align*}"}
-{"id": "5082.png", "formula": "\\begin{align*} \\int _ { M ^ 3 } \\{ [ ( b _ 1 - b _ 2 ) ^ 2 - 2 ] R _ { 1 2 1 2 } + [ ( b _ 2 - b _ 3 ) ^ 2 - 2 ] R _ { 2 3 2 3 } + [ ( b _ 1 - b _ 3 ) ^ 2 - 2 ] R _ { 1 3 1 3 } \\} d v _ g = 0 . \\end{align*}"}
-{"id": "8225.png", "formula": "\\begin{align*} \\biggl [ \\frac { 1 } { n ^ s } \\biggr ] t _ + = \\biggl [ \\frac { 1 } { n ^ s } \\biggr ] t _ - 1 \\le s \\le d - 1 \\end{align*}"}
-{"id": "940.png", "formula": "\\begin{align*} E \\left [ \\exp \\left \\{ \\frac { c _ q } { 2 } \\left ( \\frac { | \\Delta _ { i , j } | } { \\sigma _ { i , j } } \\right ) ^ { 2 / q } \\right \\} - 1 \\right ] = \\int _ 0 ^ \\infty P \\left ( \\exp \\left \\{ \\frac { c _ q } { 2 } \\left ( \\frac { | \\Delta _ { i , j } | } { \\sigma _ { i , j } } \\right ) ^ { 2 / q } \\right \\} - 1 > u \\right ) d u , \\end{align*}"}
-{"id": "4981.png", "formula": "\\begin{align*} \\tilde { g } = \\sum _ { c = 0 } ^ { R - 1 } \\sum _ { j \\equiv c ( \\textrm { m o d } R ) } g _ j \\prod _ { \\substack { j ' > j \\\\ j ' \\equiv c ( \\textrm { m o d } R ) } } ( 1 - G _ { j ' } ) \\textrm { w i t h } G _ j : = \\sum _ { \\substack { t > 0 \\\\ t \\equiv 0 ( \\textrm { m o d } R ) } } 2 ^ { - \\alpha t } \\omega _ { j - t } . \\end{align*}"}
-{"id": "7986.png", "formula": "\\begin{align*} g _ J : = h _ J \\circ \\cdots \\circ h _ { j _ 1 j _ 2 } \\circ h _ { j _ 1 } , \\end{align*}"}
-{"id": "1834.png", "formula": "\\begin{align*} \\| S _ { T \\Gamma \\Lambda U } f \\| & = \\| S _ { T \\Gamma \\Lambda U } f - S _ { T \\Lambda T } f + S _ { T \\Lambda T } f \\| \\\\ & \\ge \\| S _ { T \\Lambda T } f \\| - \\| S _ { T \\Gamma \\Lambda U } f - S _ { T \\Lambda T } f \\| \\\\ & \\ge ( A - \\lambda ) \\| f \\| . \\end{align*}"}
-{"id": "7628.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\Big [ \\| u ^ k - w ^ k \\| _ { L ^ p ( Q _ \\frac 7 2 ) } + \\| u ^ k - w ^ k \\| _ { L ^ 2 ( Q _ \\frac 7 2 ) } \\Big ] = 0 . \\end{align*}"}
-{"id": "3097.png", "formula": "\\begin{align*} & [ [ x _ 1 , x _ 2 , \\ldots , x _ n ] , y _ { 1 } , \\ldots , y _ { n - 1 } ] \\\\ = & \\sum _ { i = 1 } ^ { n } [ [ x _ 1 , x _ { 2 } , \\ldots , x _ { i - 1 } , y _ { 1 } , x _ { i + 1 } , \\ldots , x _ { n } ] , x _ { i } , y _ { 2 } , \\ldots , y _ { n - 1 } ] \\end{align*}"}
-{"id": "1457.png", "formula": "\\begin{align*} \\mathbb { Z } _ { ( 2 ) } [ x _ 2 ^ 2 , x _ { 4 1 } , \\dots , x _ { 4 , m + 1 } ] = H ^ { * } ( B G _ { m + 1 } ; \\mathbb { Z } _ { ( 2 ) } ) / ( x _ 3 ) , \\end{align*}"}
-{"id": "2701.png", "formula": "\\begin{align*} \\frac { { a ^ 2 _ 2 } - 4 p ^ { 2 n } a ' _ 1 a ' _ 3 } { a '^ 2 _ 1 } t ^ 2 + ( \\mathrm { t e r m s \\ o f \\ d e g r e e \\ \\leq 1 \\ w i t h \\ c o e f f i c i e n t s \\ i n } \\ \\Z _ { ( p ) } ) = 0 . \\end{align*}"}
-{"id": "9010.png", "formula": "\\begin{align*} A ' _ { 1 } + B _ { 1 } = & A ' + A + B + e + \\sigma \\\\ \\leq & \\Big [ l ( t ) + m ( t ) \\ln ( A + e ) + n ( t ) \\big ( \\ln ( A + B + e ) \\big ) ^ { \\alpha } \\Big ] ( A + e ) + A + e + \\sigma + f ( t ) \\\\ = & \\Big [ l ( t ) + m ( t ) \\ln ( A _ { 1 } - \\sigma ) + n ( t ) \\big ( \\ln ( B _ { 1 } - \\sigma ) \\big ) ^ { \\alpha } \\Big ] ( A _ { 1 } - \\sigma ) + A _ { 1 } + f ( t ) \\\\ \\leq & \\Big [ 1 + l ( t ) + m ( t ) \\ln A _ { 1 } + n ( t ) \\big ( \\ln B _ { 1 } \\big ) ^ { \\alpha } \\Big ] A _ { 1 } + f ( t ) . \\end{align*}"}
-{"id": "72.png", "formula": "\\begin{align*} E _ { D Y } [ G ^ { C } _ { \\sigma } ( e ) \\textbf { X } \\ , D ^ { * } ] = E _ { D Y } [ G ^ { C } _ { \\sigma } ( e ) \\textbf { X X } ^ { H } ] \\ , \\textbf { w } \\end{align*}"}
-{"id": "7587.png", "formula": "\\begin{align*} \\ell ( w ) - \\ell ( \\widetilde w ) = \\binom { n + 1 } { 2 } - \\binom { i + 1 } { 2 } \\mbox { f o r a l l } w \\in B _ n ^ { \\{ i \\} ^ { \\textup { c } } } \\mbox { w i t h } w ( n ) < 0 . \\end{align*}"}
-{"id": "58.png", "formula": "\\begin{align*} \\hat { V } ^ { C } _ { \\sigma } ( C _ { 1 } , C _ { 2 } ) = E [ G _ { \\sigma } ^ { C } ( C _ 1 - C _ 2 ) ] \\end{align*}"}
-{"id": "2131.png", "formula": "\\begin{align*} U '' + \\frac { N - 1 } { r } U ' - V ( r ) U = 0 \\quad \\mbox { i n } \\quad ( 0 , \\infty ) \\end{align*}"}
-{"id": "1362.png", "formula": "\\begin{align*} \\psi ( t ; c , a , b , c _ 0 ) : = a - \\frac { 1 } { c } t + \\frac { t ^ 2 } { b + ( c - c _ 0 ) t } \\ , , t \\in [ 0 , 1 ] \\ , . \\end{align*}"}
-{"id": "7432.png", "formula": "\\begin{align*} \\mathbb { H } _ 6 : = \\mathbb { C } [ t , T _ { \\delta } ^ 0 , T _ { \\delta } ^ 1 , T _ { \\delta } ^ 2 , T _ { \\delta } ^ 3 , T _ { \\gamma } ^ 0 , T _ { \\gamma } ^ 1 , T _ { \\beta } ^ 0 ] . \\end{align*}"}
-{"id": "423.png", "formula": "\\begin{align*} p _ { 1 , k _ 1 , k _ 2 } ( x , t ) = \\frac { ( - 1 ) ^ { k _ 2 } \\pi ^ { k _ 1 + k _ 2 } } { 4 ^ { n } ( \\pi \\delta ) ^ { n + k _ 1 - 1 } \\sqrt { 2 \\pi \\kappa } } e ^ { - \\frac { 1 } { 4 } d ( x , t ) ^ 2 } \\left [ 1 + O \\left ( \\frac { 1 } { \\kappa } + \\delta \\right ) \\right ] . \\end{align*}"}
-{"id": "545.png", "formula": "\\begin{align*} X _ { \\lambda } = \\rho \\frac { \\partial } { \\partial \\rho } . \\end{align*}"}
-{"id": "6608.png", "formula": "\\begin{align*} \\| f _ { 0 } - g _ { n , 0 } \\| _ { \\mathbb { W } ^ { 1 , p } ( \\Omega _ 0 , \\mathbb { R } ^ d ) } & = \\| f _ { 0 } - g _ { n , 0 } \\| _ { C ^ 0 ( \\Omega _ 0 ) } + \\left ( \\int _ { \\Omega _ 0 } | D f _ { 0 } - D g _ { n , 0 } | ^ p \\ , d \\mu \\right ) ^ { \\frac { 1 } { p } } \\ . \\end{align*}"}
-{"id": "1755.png", "formula": "\\begin{align*} F _ \\mu = F _ { \\mu \\circ \\sigma _ \\lambda ^ { - 1 } } \\circ \\sigma _ \\lambda , \\ \\mu - a . e . \\end{align*}"}
-{"id": "1546.png", "formula": "\\begin{gather*} \\begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & - 1 \\\\ 1 & 1 & 2 & 2 & 1 & 1 & 1 & 1 & - 1 \\\\ 2 & 2 & 4 & 3 & 2 & 2 & 2 & 1 & - 1 \\\\ 3 & 2 & 6 & 4 & 3 & 3 & 3 & 1 & - 1 \\\\ 4 & 4 & 1 0 & 7 & 5 & 5 & 6 & 1 & - 1 \\\\ 5 & 4 & 1 0 & 7 & 5 & 5 & 6 & 1 & - 1 \\\\ 6 & 5 & 1 3 & 9 & 7 & 7 & 8 & 1 & - 1 \\\\ 7 & 7 & 1 6 & 1 1 & 8 & 9 & 1 0 & 1 & - 1 \\end{bmatrix} . \\end{gather*}"}
-{"id": "9925.png", "formula": "\\begin{align*} X _ t ^ { ( n ) } : = X _ 0 + \\int _ 0 ^ t b ( s , X _ { \\eta _ n ( s ) - } ^ { ( n ) } ) \\ , d s + \\int _ 0 ^ t g ( s , X _ { \\eta _ n ( s ) - } ^ { ( n ) } ) \\ , d L _ s , t \\in [ 0 , T ] \\end{align*}"}
-{"id": "2151.png", "formula": "\\begin{align*} | \\tilde { V } ( \\xi , s ) | \\le C \\xi ^ { - 2 } | e ^ { \\frac { s } { 2 } } \\xi | ^ { - \\theta } \\le C \\exp \\left [ - \\frac { \\theta } { 2 } s + ( 2 + \\theta ) \\theta _ * s \\right ] = C e ^ { - \\frac { \\theta } { 4 } s } \\end{align*}"}
-{"id": "8012.png", "formula": "\\begin{align*} \\frac { 1 } { | Q | } \\int _ Q { \\sum _ { k = 3 } ^ { \\infty } { 2 ^ { k m } \\mathfrak { M } _ { \\sigma , 2 ^ k } \\Pi ^ * _ k f ( x ) } } d x & \\lesssim \\frac { 1 } { | Q | } \\sum _ { P \\in \\mathcal { A } _ Q } { \\int _ { 3 P } { \\sum _ { k = 3 } ^ { \\infty } { 2 ^ { k m } \\mathfrak { M } _ { \\sigma , 2 ^ k } \\Pi _ k ^ * f ( x ) } } d x } \\\\ & \\lesssim \\sup _ { R \\in \\mathcal { D } _ 2 } { \\frac { 1 } { | R | } \\int _ R { \\sum _ { k = 3 } ^ { \\infty } { 2 ^ { k m } \\mathfrak { M } _ { \\sigma , 2 ^ k } \\Pi _ k ^ * f ( x ) } } d x } . \\end{align*}"}
-{"id": "5536.png", "formula": "\\begin{align*} \\Phi \\left ( t , t _ { 0 } \\right ) = P ^ { - 1 } \\left ( t \\right ) e ^ { B \\left ( t - t _ { 0 } \\right ) } P \\left ( t _ { 0 } \\right ) \\end{align*}"}
-{"id": "2501.png", "formula": "\\begin{align*} \\beta _ { 2 1 } \\left ( \\left | \\eta \\right | \\right ) & = \\beta _ { 2 0 } \\left ( - \\left | \\eta \\right | \\right ) , \\quad \\beta _ { 2 2 } \\left ( \\left | \\eta \\right | \\right ) = \\beta _ { 2 2 } \\left ( - \\left | \\eta \\right | \\right ) , \\\\ \\beta _ { 3 3 } \\left ( \\left | \\eta \\right | \\right ) & = \\beta _ { 3 3 } \\left ( - \\left | \\eta \\right | \\right ) , \\quad \\beta _ { 4 4 } \\left ( \\left | \\eta \\right | \\right ) = \\beta _ { 4 4 } \\left ( - \\left | \\eta \\right | \\right ) . \\end{align*}"}
-{"id": "3941.png", "formula": "\\begin{align*} u ( a , t ) = \\theta _ 1 ( t ) , ~ ~ u ( b , t ) = \\theta _ 2 ( t ) \\end{align*}"}
-{"id": "5800.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta _ { p } w _ { j } = \\sigma w ^ { q } _ { j - 1 } + \\mu \\ ; \\ ; \\mathbb { R } ^ n , \\\\ w _ { j } \\in L ^ { 1 + q } ( \\R ^ n , d \\sigma ) \\cap \\dot { W } ^ { 1 , p } _ { 0 } ( \\R ^ n ) , \\\\ \\sup _ { j \\in \\N } \\Vert w _ j \\Vert _ { \\dot { W } _ { 0 } ^ { 1 , p } ( \\R ^ n ) } < \\infty , \\\\ w _ { j - 1 } ^ { q } d \\sigma + d \\mu \\in W ^ { - 1 , p ' } ( \\R ^ n ) , \\\\ 0 < w _ { j - 1 } \\leq w _ { j } \\leq v \\ ; \\ ; \\end{cases} \\end{align*}"}
-{"id": "80.png", "formula": "\\begin{align*} \\mathcal T ( \\chi _ 1 \\otimes \\chi _ 2 ) = ( \\chi _ 1 \\otimes 1 ) ( 1 \\otimes \\chi _ 2 ) \\end{align*}"}
-{"id": "2031.png", "formula": "\\begin{align*} y _ i = \\sum _ { \\substack { \\alpha _ 1 , . . . , \\alpha _ k \\in \\Z , \\\\ \\alpha _ 1 \\leq . . . \\leq \\alpha _ k , \\\\ \\alpha _ 1 + . . . + \\alpha _ k = \\frac { k ( k - 1 ) } { 2 } } } a _ { \\alpha _ 1 , . . . , \\alpha _ k } x _ { i + \\alpha _ 1 } x _ { i + \\alpha _ 2 } . . . x _ { i + \\alpha _ k } , ~ ~ ~ i \\in \\Z \\end{align*}"}
-{"id": "9229.png", "formula": "\\begin{align*} d ( \\frac { [ a _ { 1 } ^ { - } , a _ { 2 } ^ { - } ] _ { A ^ { - } } } { 2 } + \\frac { ( a _ { 1 } ^ { - } \\circ a _ { 2 } ^ { - } ) _ { A ^ { + } } } { 2 } ) & = \\frac { [ d a _ { 1 } ^ { - } , a _ { 2 } ^ { - } ] _ { A ^ { - } } } { 2 } + \\frac { ( d a _ { 1 } ^ { - } \\circ a _ { 2 } ^ { - } ) _ { A ^ { + } } } { 2 } + \\frac { [ a _ { 1 } ^ { - } , d a _ { 2 } ^ { - } ] _ { A ^ { - } } } { 2 } + \\frac { ( a _ { 1 } ^ { - } \\circ d a _ { 2 } ^ { - } ) _ { A ^ { + } } } { 2 } , \\end{align*}"}
-{"id": "4182.png", "formula": "\\begin{align*} ( k + 1 ) ^ 2 - 1 - ( k - 1 ) ^ 2 = 4 k - 1 . \\end{align*}"}
-{"id": "5880.png", "formula": "\\begin{align*} \\begin{aligned} & P ( Z ^ { N , i } _ 1 = 1 ) \\le P \\big ( \\max \\big ( \\frac { S ^ { ( i ) } _ { 0 , N } } N , \\frac { S ^ { ( i ) } _ { 1 , N + 1 } } N , \\cdots , \\frac { S ^ { ( i ) } _ { N - 1 , 2 N - 1 } } N \\big ) \\ge r _ { i + 1 } \\big ) \\le \\\\ & \\sum _ { j = 0 } ^ { N - 1 } P ( \\frac { S ^ { ( i ) } _ { j , N + j } } N \\ge r _ { i + 1 } ) \\approx N e ^ { - N I _ i ( r _ { i + 1 } ) } \\approx e ^ { - N I _ i ( r _ { i + 1 } ) } . \\end{aligned} \\end{align*}"}
-{"id": "1757.png", "formula": "\\begin{align*} g \\sqrt { d \\zeta } = \\begin{cases} \\frac { g \\sqrt { h } } { | g | } \\sqrt { d \\mu } \\ ; \\in \\mathcal { L } ^ 2 ( \\mu ) , & g \\not = 0 \\\\ 0 , & g = 0 \\end{cases} \\end{align*}"}
-{"id": "5011.png", "formula": "\\begin{align*} \\mathfrak { M } f : = \\sup _ { j \\in \\mathbb { Z } } | f | * k _ j \\end{align*}"}
-{"id": "6528.png", "formula": "\\begin{align*} Z = C _ { \\xi } ^ * \\cap \\left \\{ x \\in \\mathbb { R } ^ n : \\left \\langle x , \\frac { \\xi } { \\| \\xi \\| } \\right \\rangle \\leq \\xi \\right \\} \\end{align*}"}
-{"id": "7855.png", "formula": "\\begin{align*} h _ { X } ( x ) = \\frac { \\alpha ( \\theta _ { 1 } + \\theta _ { 2 } + \\theta _ { 3 } ) ( a + b x ) \\eta ^ { \\alpha - 1 } ( x ) e ^ { - \\eta ^ { \\alpha } ( x ) } \\left ( 1 - e ^ { - \\eta ^ { \\alpha } ( x ) } \\right ) ^ { - 1 } } { \\left ( 1 - e ^ { - \\eta ^ { \\alpha } ( x ) } \\right ) ^ { \\theta _ { 1 } + \\theta _ { 2 } + \\theta _ { 3 } - 1 } - 1 } , \\end{align*}"}
-{"id": "8122.png", "formula": "\\begin{align*} \\varphi _ k ( a ) = \\theta _ k ( q _ k ^ { 1 / 2 } a q _ k ^ { 1 / 2 } ) . \\end{align*}"}
-{"id": "1841.png", "formula": "\\begin{align*} \\mathcal { G F } ( R ) & = { } ^ \\perp ( \\mathcal { C } ( R ) \\cap ( \\mathcal { P G F } ( R ) ) ^ \\perp ) \\end{align*}"}
-{"id": "3470.png", "formula": "\\begin{align*} E ^ n = P _ h E ^ n + ( I - P _ h ) E ^ n = : \\Theta ^ n + \\Xi ^ n . \\end{align*}"}
-{"id": "9.png", "formula": "\\begin{align*} V ^ { C } _ { \\sigma } ( C _ { 1 } , C _ { 2 } ) = E _ { C _ { 1 } C _ { 2 } } [ K ( C _ { 1 } , C _ { 2 } ) ] = E _ { C _ { 1 } C _ { 2 } } [ K ( C _ { 2 } , C _ { 1 } ) ] = V ^ { C } _ { \\sigma } ( C _ { 2 } , C _ { 1 } ) \\end{align*}"}
-{"id": "5331.png", "formula": "\\begin{align*} W ( N ) = \\{ w \\in W \\ ; | \\ ; Q _ N w = 0 \\} . \\end{align*}"}
-{"id": "30.png", "formula": "\\begin{align*} J _ { M C C C } = V ^ { C } _ { \\sigma } ( D , Y ) = E _ { D Y } [ G ^ { C } _ { \\sigma } ( D - \\textbf { w } ^ { H } \\textbf { X } ) ] = E _ { D Y } [ G ^ { C } _ { \\sigma } ( e ) ] \\end{align*}"}
-{"id": "1693.png", "formula": "\\begin{align*} \\lim _ n \\frac { \\mu _ { x } ( Z ( g z _ n ) ) } { \\mu _ { x } ( Z ( z _ n ) ) } = \\lim _ n \\frac { \\mu _ { x } ( Z ( e f _ { i _ 1 - 1 } e f _ { i _ 2 - 1 } \\ldots f _ { i _ n - 1 } e ) ) } { \\mu _ { x } ( Z ( e f _ { i _ 1 } e f _ { i _ 2 } \\ldots f _ { i _ n } ) ) } = \\lim _ n \\frac { T _ { i _ 1 - 1 , i _ 2 - 1 } \\cdots T _ { i _ { n - 1 } - 1 , i _ n - 1 } } { T _ { i _ 1 , i _ 2 } \\cdots T _ { i _ { n - 1 } , i _ n } } , \\end{align*}"}
-{"id": "6593.png", "formula": "\\begin{align*} g = \\left \\{ \\begin{array} { l l } \\varphi _ 1 ^ { - 1 } \\circ \\phi ^ 0 \\circ \\varphi _ { 1 } \\circ f \\circ \\varphi _ { 0 } ^ { - 1 } \\circ \\phi \\circ \\varphi _ 0 & \\mbox { i n } \\ E _ M \\\\ \\varphi _ { k + 1 } ^ { - 1 } \\circ \\phi ^ k \\circ \\varphi _ { k + 1 } \\circ f & \\mbox { i n } \\ f ^ { - 1 } ( B ^ { k + 1 } _ M ) , \\ 0 < k < k _ 0 \\\\ f & \\mbox { e l s e w h e r e } \\end{array} \\right . \\ . \\end{align*}"}
-{"id": "141.png", "formula": "\\begin{align*} M N = \\begin{pmatrix} a x + b z & a y + b w \\\\ a z & a w \\end{pmatrix} \\end{align*}"}
-{"id": "9190.png", "formula": "\\begin{align*} & \\frac { 1 4 4 } { \\pi ^ 2 } \\log ^ 3 ( T / 5 ) \\log ^ 3 ( T ) \\left ( \\frac { \\pi ^ 2 } { 1 4 4 \\log ^ 2 ( T / 5 ) } + \\frac { 0 . 1 0 5 2 } { \\log ^ 3 ( T / 5 ) } - E ( T ) \\right ) \\\\ & = \\log ( T / 5 ) \\log ^ 3 ( T ) - \\log ( T ) \\log ^ 3 ( T / 5 ) + \\frac { 1 4 4 } { \\pi ^ 2 } 0 . 1 0 5 2 \\log ^ 3 ( T ) - 6 . 5 9 1 2 5 \\log ^ 3 ( T / 5 ) \\\\ & = 2 \\log ( 5 ) \\log ( T / 5 ) \\log ( T ) \\log ( T / \\sqrt { 5 } ) + \\frac { 1 4 4 } { \\pi ^ 2 } 0 . 1 0 5 2 \\log ^ 3 ( T ) - 6 . 5 9 1 2 5 \\log ^ 3 ( T / 5 ) . \\end{align*}"}
-{"id": "6285.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ 3 } h ( \\nabla \\times G ) \\times F \\ , d x = & - \\int _ { \\mathbb R ^ 3 } ( \\nabla h \\times G ) \\times F \\ , d x + \\int _ { \\mathbb R ^ 3 } h G \\times ( \\nabla \\times F ) \\ , d x \\\\ & + \\int _ { \\mathbb R ^ 3 } h G \\cdot \\nabla F \\ , d x . \\end{align*}"}
-{"id": "10113.png", "formula": "\\begin{align*} \\boldsymbol P _ { D _ k } ( i ) = \\mathbb { E } [ d _ k ^ * ( i ) ( \\boldsymbol x _ k ( i ) \\bar { \\boldsymbol \\omega } _ k ^ H ( i ) + \\gamma { \\boldsymbol I } _ { M , D } ) ] \\end{align*}"}
-{"id": "508.png", "formula": "\\begin{align*} \\mathcal { R } _ { } \\ ! = \\ ! & \\bigcup _ { P _ { U | X } } \\ ! \\Big \\{ \\left ( R _ s , R _ \\ell , R _ w \\right ) \\ ! \\colon \\ ! \\\\ & 0 \\leq R _ s \\leq I ( U ; Y ) , \\\\ & R _ \\ell \\geq I ( U ; X ) - I ( U ; Y ) , \\\\ & R _ w \\geq I ( U ; X ) \\Big \\} \\end{align*}"}
-{"id": "5627.png", "formula": "\\begin{align*} & 0 \\to B _ 1 \\to M _ 1 ^ 2 \\oplus N _ 2 \\to A _ 1 \\to 0 & \\\\ & 0 \\to B _ 2 \\to M _ 1 \\oplus M _ 2 ^ 2 \\to A _ 2 \\to 0 & \\\\ & 0 \\to D _ i \\to M _ 1 \\oplus M _ 2 \\oplus N _ 1 \\oplus N _ 2 \\to C _ i \\to 0 \\ \\ ( i = 1 , 2 ) & \\\\ & 0 \\to Y _ 1 \\to M _ 1 \\oplus M _ 2 ^ 2 \\oplus N _ 1 \\oplus N _ 2 ^ 2 \\to X _ 1 \\to 0 & \\\\ & 0 \\to Y _ 2 \\to M _ 1 ^ 2 \\oplus M _ 2 \\oplus N _ 2 ^ 2 \\to X _ 2 \\to 0 & \\end{align*}"}
-{"id": "2970.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { 2 } p _ j | h _ { k j } | ^ { 2 } \\geq \\prod _ { j = 1 } ^ { 2 } \\left ( \\frac { p _ j | h _ { k j } | ^ { 2 } } { c _ { k j } } \\right ) ^ { c _ { k j } } , \\end{align*}"}
-{"id": "4639.png", "formula": "\\begin{align*} \\sum _ { n = 2 ^ { k - 1 } } ^ { 2 ^ k - 1 } 1 _ { [ - B , B ] } \\left ( t + \\sum _ { i = 0 } ^ { n - 1 } g ( T ^ i x ) \\right ) \\ge \\beta \\tau \\sum _ { n = 0 } ^ { 2 ^ k - 1 } 1 _ { [ - B , B ] } \\left ( t + \\sum _ { i = 0 } ^ { n - 1 } g ( T ^ i x ) \\right ) \\end{align*}"}
-{"id": "373.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( S _ { n } \\geq x \\right ) = ( 1 + o ( 1 ) ) \\sum _ { r , s \\in \\mathbb { Z } } \\mathbb { P } ( b _ { n , r , s } \\xi _ { 0 } \\geq x ) \\end{align*}"}
-{"id": "1261.png", "formula": "\\begin{align*} v ( t x ) = v ( t e _ n ) \\ , v ( x ) \\mbox { w h e n e v e r } \\ , \\ , x \\in \\mathbb { R } ^ n \\ , \\ , \\mbox { a n d } \\ , \\ , t \\in ( 0 , \\infty ) . \\end{align*}"}
-{"id": "20.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty , \\sigma \\rightarrow 0 } E [ \\hat { V } ^ C _ { N , \\sigma } ( C _ 1 , C _ 2 ) ] = P ( C _ 1 = C _ 2 ) = p _ E ( 0 ) \\end{align*}"}
-{"id": "598.png", "formula": "\\begin{align*} \\frac { L _ { \\Gamma } ' ( s , \\rho ) } { L _ { \\Gamma } ( s , \\rho ) } & = - \\frac { d } { d s } \\sum _ { \\overline { \\gamma } \\in \\overline { \\Gamma } _ { p } } \\sum _ { k = 0 } ^ { \\infty } \\sum _ { m = 1 } ^ { \\infty } \\frac { 1 } { m } \\chi _ { \\rho } ( \\gamma ^ { m } ) e ^ { - ( s + k ) m \\ell ( \\gamma ) } \\\\ & = \\sum _ { \\overline { \\gamma } \\in \\overline { \\Gamma } _ { p } } \\sum _ { k = 0 } ^ { \\infty } \\sum _ { m = 1 } ^ { \\infty } \\ell ( \\gamma ) \\chi _ { \\rho } ( \\gamma ^ { m } ) e ^ { - ( s + k ) m \\ell ( \\gamma ) } . \\end{align*}"}
-{"id": "6134.png", "formula": "\\begin{align*} \\mathbb { P } _ { c o u p l e } \\left ( \\sum _ { i = 1 } ^ { M _ { 1 } \\left ( n \\right ) } a _ { n } \\left | W _ { i } - U _ { i } \\right | > \\frac { \\epsilon } { 2 } \\right ) < \\frac { \\epsilon } { 4 } . \\end{align*}"}
-{"id": "3216.png", "formula": "\\begin{align*} \\partial _ t ^ 2 \\phi ^ { ( i ) } - \\Delta \\phi ^ { ( i ) } + h ( { \\tilde \\phi } ^ { ( i ) } ) \\Delta \\phi ^ { ( i ) } = F ( \\partial { \\tilde \\phi ^ { ( i ) } } ) \\mbox { i n } \\ , \\ , S _ T \\end{align*}"}
-{"id": "1331.png", "formula": "\\begin{align*} \\psi _ G = \\psi _ { \\gamma } \\psi _ { G / \\gamma } + R _ \\gamma , \\phi _ G = \\psi _ \\gamma \\phi _ { G / \\gamma } + R _ \\gamma ' , \\end{align*}"}
-{"id": "2278.png", "formula": "\\begin{align*} \\mathcal { N } _ S & : = \\sum _ { m \\ge 2 } ( ( 2 m - 3 ) N _ { 2 m } - N ' _ { 2 m } ) + \\sum _ { m \\ge 2 } ( ( 2 m - 2 ) N _ { 2 m + 1 } ) - 1 \\\\ & = \\sum _ { \\ell \\ge 3 } ( \\ell - 3 ) N _ { \\ell } - \\sum _ { m \\ge 2 } N ' _ { 2 m } - 1 . \\end{align*}"}
-{"id": "3277.png", "formula": "\\begin{align*} r _ { u , k } \\cos \\theta = r _ { u , k } ( \\boldsymbol { x } ) + r _ u \\cos \\hat { \\theta } , \\ , \\ , r _ { u , k } \\sin \\theta = r _ u \\sin \\hat { \\theta } , \\end{align*}"}
-{"id": "3429.png", "formula": "\\begin{align*} c = \\max \\left \\{ | \\tilde h ' ( z ) | : z \\in \\overline D _ 1 \\right \\} \\end{align*}"}
-{"id": "7759.png", "formula": "\\begin{align*} \\sum _ { \\ell \\in \\L _ { 0 } } \\chi _ { \\ell } ( y ; x ) & = 1 , \\\\ \\left | \\frac { \\partial ^ { k } \\chi _ { \\ell } ( y ; x ) } { \\partial y ( j _ { 1 } ) \\cdots \\partial y ( j _ { k } ) } \\right | & \\leq C e ^ { - \\gamma | x - y ( \\ell ) | } \\prod _ { 1 \\leq i \\leq k } e ^ { - \\gamma | x - y ( j _ { i } ) | } , \\end{align*}"}
-{"id": "4670.png", "formula": "\\begin{align*} [ F ( x ) ] ( s ) : = \\int \\limits _ 0 ^ s \\ ; \\ ; x ( s - t ) \\ , x ( t ) \\ , d t , 0 \\le s \\le 1 . \\end{align*}"}
-{"id": "6042.png", "formula": "\\begin{align*} \\widetilde { Q } _ { X Y } & = \\pi _ { X Y } , \\\\ \\widetilde { Q } _ { X Y | U } & = \\widetilde { Q } _ { X | U } \\widetilde { Q } _ { Y | U } . \\end{align*}"}
-{"id": "7041.png", "formula": "\\begin{align*} ( 1 - s ) \\prod { \\lambda _ j } \\int _ { \\bar { y } \\in \\mathbb { R } ^ { n - 1 } } { \\frac { u ( y _ 1 , y _ 2 , . . . , y _ { n - 1 } , 0 ) - u ( 0 ) } { ( \\lambda _ 1 ^ 2 y _ 1 ^ 2 + . . . + \\lambda _ { n - 1 } ^ 2 y _ { n - 1 } ^ 2 ) ^ { \\frac { n + 2 s - 1 } { 2 } } } d \\bar { y } } = - A < 0 . \\end{align*}"}
-{"id": "6890.png", "formula": "\\begin{align*} 3 \\mu _ 0 + 6 \\mu _ 1 = 1 \\end{align*}"}
-{"id": "4941.png", "formula": "\\begin{align*} ( M ) = 1 + \\frac { 1 } { 2 } c _ 1 + \\frac { 1 } { 1 2 } \\left ( c _ 1 ^ 2 + c _ 2 \\right ) + \\frac { 1 } { 2 4 } c _ 1 c _ 2 + . . . \\end{align*}"}
-{"id": "7515.png", "formula": "\\begin{gather*} \\mathcal T _ * P ^ t = \\varepsilon \\tilde P ^ t , \\mathcal T _ * P ^ x = \\varepsilon \\varepsilon ' a \\tilde P ^ x , \\mathcal T _ * P ^ y = \\varepsilon a \\tilde P ^ y , \\mathcal T _ * G ^ x = \\varepsilon ' a \\tilde G ^ x , \\mathcal T _ * G ^ y = a \\tilde G ^ y , \\\\ \\mathcal T _ * D = \\tilde D , \\mathcal T _ * \\Pi = \\varepsilon \\tilde \\Pi , \\mathcal T _ * J = \\varepsilon ' \\tilde J , \\end{gather*}"}
-{"id": "2197.png", "formula": "\\begin{align*} \\tilde a _ J = \\begin{cases} \\bigl ( 1 / a _ { 1 j _ { 1 } } , \\ldots , 1 / a _ { n j _ { n } } \\bigr ) , & \\ a _ { k j _ { k } } \\neq 0 \\ \\ k = 1 , \\ldots , n , \\\\ \\bigl ( 1 / a _ { 1 j _ { 1 } } , \\ldots , \\infty _ { [ i _ 1 ] } , \\ldots , \\infty _ { [ i _ k ] } , \\ldots , 1 / a _ { n j _ { n } } \\bigr ) , & \\ a _ { i _ 1 { j _ { i _ 1 } } } = \\ldots = a _ { i _ k { j _ { i _ k } } } = 0 , \\end{cases} \\end{align*}"}
-{"id": "9479.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } q ^ n ( - z q ^ { n + 1 } ; q ) _ { n } ( - z q ^ { 2 n + 2 } ; q ^ 2 ) _ { \\infty } = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ { n ^ 2 + n } } { ( q ; q ^ 2 ) _ { n + 1 } } . \\end{align*}"}
-{"id": "4289.png", "formula": "\\begin{align*} t _ { \\lambda \\ast } = \\exp A ( \\lambda ) \\end{align*}"}
-{"id": "1167.png", "formula": "\\begin{align*} T _ { k + 1 , 0 } & = D _ k T _ { k , 0 } \\\\ T _ { k + 1 , i } & = T ^ 2 _ { k , i - 1 } + D _ k T _ { k , i } \\\\ T _ { k + 1 , k - 1 } & = 1 \\end{align*}"}
-{"id": "1197.png", "formula": "\\begin{align*} A = | \\nabla v _ 1 ( y _ 0 ) | , \\ , \\ , B = | \\nabla v _ 2 ( z _ 0 ) | , \\ , \\ , C = | \\nabla u ( x _ 0 ) | , \\ , \\ , a = | x _ 0 - y _ 0 | , \\ , \\ , b = | x _ 0 - z _ 0 | . \\end{align*}"}
-{"id": "6534.png", "formula": "\\begin{align*} \\bigcap _ { i = 1 } ^ k \\{ x \\in \\mathbb { R } ^ n : \\langle x , y _ i \\rangle \\leq 1 \\} = \\{ ( \\lambda \\bar { x } , 1 - \\lambda ) : \\lambda \\geq 0 , \\ \\bar { x } \\in B \\} . \\end{align*}"}
-{"id": "518.png", "formula": "\\begin{align*} \\frac { I ( X ^ n ; W ) } { n } & \\stackrel { ( a ) } { = } \\frac { H ( W ) } { n } \\leq \\frac { \\log | \\mathcal { W } | } { n } = \\frac { m _ 2 } { n } \\\\ & = H _ b ( q * p _ A ) - H _ b ( q ) + 2 \\delta . \\end{align*}"}
-{"id": "6865.png", "formula": "\\begin{align*} \\mathcal { E } _ { m } ( f ) & = \\sum _ { k = 0 } ^ { 4 ^ { m } - 1 } c ( I _ { k } ^ { ( m ) } ) \\left ( f \\left ( \\frac { k } { 4 ^ { m } } \\right ) - f \\left ( \\frac { k + 1 } { 4 ^ m } \\right ) \\right ) ^ { 2 } , \\\\ \\mathcal { E } ( f ) & = \\lim _ { m \\to \\infty } \\mathcal { E } _ m . \\end{align*}"}
-{"id": "3887.png", "formula": "\\begin{align*} k _ 0 ( x , v ) : = \\frac { 1 } { 2 \\cos { \\pi ( 1 / 2 + v ) } } \\left ( J _ { 2 v } ( x ) - J _ { - 2 v } ( x ) \\right ) , \\end{align*}"}
-{"id": "5994.png", "formula": "\\begin{gather*} S ( x ) = x e _ 0 + y e _ 1 , S ( y ) = x e _ 1 + y e _ 0 , S ( z ) = z . \\end{gather*}"}
-{"id": "5811.png", "formula": "\\begin{align*} d \\omega : = u ^ { q } d \\sigma + d \\mu = v ^ { q } d \\sigma + d \\mu . \\end{align*}"}
-{"id": "4445.png", "formula": "\\begin{align*} \\lceil u , \\phi _ \\ell * \\rceil f & = u f _ \\ell - ( u f ) _ \\ell = \\int _ 0 ^ 1 u \\omega _ \\ell ^ t * f _ { t \\ell ^ 3 } - \\omega _ \\ell ^ t * ( u f ) _ { t \\ell ^ 3 } \\ , d t \\\\ & = \\int _ 0 ^ 1 \\lceil u , \\omega _ \\ell ^ t * \\psi _ { t \\ell ^ 3 } * \\rceil f \\ , d t \\stackrel { \\eqref { o p e r a } } { = } \\int _ 0 ^ 1 \\Big ( \\omega _ \\ell ^ t * \\lceil u , \\psi _ { t \\ell ^ 3 } * \\rceil f + \\lceil u , \\omega _ \\ell ^ t * \\rceil f _ { t \\ell ^ 3 } \\Big ) d t . \\end{align*}"}
-{"id": "1613.png", "formula": "\\begin{align*} Z ( \\lambda ) = \\{ x \\in \\Lambda ^ \\infty \\ , : \\ , x ( 0 , d ( \\lambda ) ) = \\lambda \\} , \\end{align*}"}
-{"id": "3778.png", "formula": "\\begin{align*} \\widehat { v } _ \\star : = v _ \\star \\wedge \\tfrac 1 2 \\bar { v } : = \\tfrac 1 3 \\widehat { v } _ \\star . \\end{align*}"}
-{"id": "8596.png", "formula": "\\begin{align*} \\Gamma g ^ { - 1 } \\Gamma = \\bigsqcup _ { i = 1 } ^ { d } g _ i \\Gamma , g _ { i } = \\delta _ i g ^ { - 1 } , \\Gamma = \\bigsqcup _ { i = 1 } ^ { d } \\delta _ { i } \\Gamma ^ { g } , \\end{align*}"}
-{"id": "7037.png", "formula": "\\begin{align*} \\eta _ j = \\frac { 1 } { 2 } ( \\sum _ i { \\sigma _ i } - \\sigma _ j ) . \\end{align*}"}
-{"id": "8370.png", "formula": "\\begin{align*} f ^ { [ i ] } ( \\tau ) = \\sum _ { \\substack { m \\in \\Q \\\\ m \\gg - \\infty } } c ^ { [ i ] } ( m ) \\cdot q ^ m \\in M ^ ! _ { - 1 1 - \\frac { n } { 2 } } ( \\overline { \\rho } _ { V ^ { [ i ] } _ \\Z } ) \\end{align*}"}
-{"id": "9892.png", "formula": "\\begin{align*} | N ( x ) \\cup N ( y ) | = d ( x ) + d ( y ) - | N ( x ) \\cap N ( y ) | \\le \\tfrac 1 r \\bigl ( d ( x ) + d ( y ) \\bigr ) + \\tfrac { r - 1 } { r + 1 } n \\ , , \\end{align*}"}
-{"id": "5676.png", "formula": "\\begin{align*} \\frac { f '' } { f } = | \\sigma ' | ^ 2 + \\frac { \\nabla W ( f \\sigma ) \\cdot f \\sigma } { f ^ 2 } \\geq c _ 0 f ^ { p _ 0 - 2 } \\{ f > E \\} . \\end{align*}"}
-{"id": "1887.png", "formula": "\\begin{align*} \\phi _ 0 ( \\ 1 _ B ) & = s \\nu _ 1 ( F _ 1 ^ { i _ 1 - 1 } ) + \\pi ( f _ 1 , f _ 2 , a ) \\\\ & = s \\nu _ 1 ( F _ 1 ^ { i _ 1 - 1 } ) + \\pi ( f _ 1 , f _ 2 , \\tilde { a } ) + \\pi ( 0 , 0 , \\bar { a } ) \\\\ & \\geq \\min _ { s \\in [ 0 , 1 ] } \\Big \\{ s \\nu _ 1 ( F _ 1 ^ { i _ 1 - 1 } ) + s \\phi _ 0 ( \\ 1 _ { B \\setminus S } ) + ( 1 - s ) \\eta ( 1 _ B ) \\Big \\} . \\end{align*}"}
-{"id": "3723.png", "formula": "\\begin{align*} ( 1 \\ ; 2 \\ ; 3 ) \\cdot \\frac { \\xi _ 3 } { \\xi _ 4 } = \\frac { \\xi _ 1 } { \\xi _ 4 } = \\frac { \\xi _ 1 } { \\xi _ 2 } \\frac { \\xi _ 2 } { \\xi _ 3 } \\frac { \\xi _ 3 } { \\xi _ 4 } = \\omega ^ { - 2 } \\frac { \\xi _ 3 } { \\xi _ 4 } = \\omega f _ 3 \\end{align*}"}
-{"id": "7059.png", "formula": "\\begin{align*} ( A + B ) ^ n = \\sum _ { i = 0 } ^ n \\binom { n } { i } A ^ i B ^ { n - i } \\ , . \\end{align*}"}
-{"id": "9914.png", "formula": "\\begin{align*} d _ 1 ( v ) \\ge \\tfrac { 3 r - 5 } { 3 r - 2 } n - | B _ 2 | - \\tfrac { 5 / 3 } { 3 r - 2 } n - \\sum _ { i = 4 } ^ r | B _ i | \\ , . \\end{align*}"}
-{"id": "288.png", "formula": "\\begin{align*} d _ p ( x ^ * , y ^ * ) = \\left ( \\int _ 0 ^ 1 d _ H \\bigl ( C _ \\alpha ( x ^ * ) , C _ \\alpha ( y ^ * ) \\bigr ) ^ p d \\alpha \\right ) ^ { 1 \\slash p } , \\end{align*}"}
-{"id": "5395.png", "formula": "\\begin{align*} x : = a , \\ , y : = b ^ { c a c a c } \\textrm { a n d } z : = ( ( a c ) ^ 3 ) ^ b \\end{align*}"}
-{"id": "8269.png", "formula": "\\begin{align*} R ( f _ { u r } ) v _ { } ^ { \\circ } ( i r ) = \\abs { A _ L ( r , T _ 0 ) } ^ 2 v _ { } ^ { \\circ } ( i r ) , \\end{align*}"}
-{"id": "550.png", "formula": "\\begin{align*} \\# ( \\phi _ H ( Q ) \\cap Q ) \\ , \\geqslant \\ , \\sum _ { k = 1 } ^ n b _ k ( Q ) . \\end{align*}"}
-{"id": "620.png", "formula": "\\begin{align*} \\tilde I ( x ) = \\int _ 0 ^ 1 r ^ { x } e ^ { - \\frac { 2 } { r } } \\ , d r \\end{align*}"}
-{"id": "194.png", "formula": "\\begin{align*} f ^ { \\ast } ( a ) = f ^ { - 1 } ( \\{ x : x \\leq f ( a ) \\} ) \\end{align*}"}
-{"id": "5382.png", "formula": "\\begin{align*} ( \\psi ( t _ 1 ) - \\psi ( t _ 0 t _ 1 ) ) \\cdot ( \\psi ( t _ 2 ) - \\psi ( t _ 0 t _ 2 ) ) = - \\frac { 1 } { 2 ^ 2 } ( \\psi ( t _ 1 t _ 2 ) - \\psi ( t _ 0 t _ 1 t _ 2 ) ) . \\end{align*}"}
-{"id": "8682.png", "formula": "\\begin{align*} a E ( e _ m ) + b F ( e _ 0 ) = 0 . \\end{align*}"}
-{"id": "1642.png", "formula": "\\begin{align*} \\tau _ \\lambda \\circ \\tau _ \\alpha = \\tau _ \\nu \\circ \\tau _ \\beta . \\end{align*}"}
-{"id": "4820.png", "formula": "\\begin{align*} n a _ j = a _ s + \\sum _ { i = 1 } ^ k \\alpha _ i a _ i \\end{align*}"}
-{"id": "10039.png", "formula": "\\begin{align*} \\gamma _ p = \\left ( \\frac { - 1 } { p } \\right ) ^ { n / 2 } \\mathrm { i n v } _ p ( V _ p ) . \\end{align*}"}
-{"id": "2803.png", "formula": "\\begin{align*} P _ { n } ( t , S ) = ~ & p ( t , S ) + \\int _ { 0 } ^ { t } r K e ^ { - r ( t - \\xi ) } \\aleph ( - d _ { 2 } ( S , t - \\xi , \\mathcal { B } _ { n } ( \\xi ) ) ) \\mathrm { d } \\xi \\\\ & - \\int _ { 0 } ^ { t } \\delta S e ^ { - \\delta ( t - \\xi ) } \\aleph ( - d _ { 1 } ( S , t - \\xi , \\mathcal { B } _ { n } ( \\xi ) ) ) \\mathrm { d } \\xi , \\end{align*}"}
-{"id": "10089.png", "formula": "\\begin{align*} \\tilde { S } ( Y , Z ) = S ( Y , Z ) - \\lambda ( n - 1 ) \\pi ( Y ) \\pi ( Z ) \\end{align*}"}
-{"id": "9517.png", "formula": "\\begin{align*} \\tilde { b } _ n ^ { ( d ) } = \\dfrac { 1 } { d \\cdot n } \\sum \\limits _ { m \\ , | \\ , d \\cdot n } \\phi ( m ) \\ , f ( d \\cdot n , m ) . \\end{align*}"}
-{"id": "2767.png", "formula": "\\begin{align*} h ( \\SS ^ \\star ) : = \\sum \\limits _ { Q \\in \\SS ^ \\star } \\big \\lfloor w ( p Q ) \\big \\rfloor - \\big \\lfloor w ( Q ) \\big \\rfloor . \\end{align*}"}
-{"id": "9148.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { 4 } x ^ { i } f _ { 5 - i } ( x ^ { 5 - i } ) = 0 \\left ( x \\in R \\right ) . \\end{align*}"}
-{"id": "10137.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } { D } _ k \\bigg ( \\boldsymbol S _ { D _ k } ( i ) , \\bar { \\boldsymbol \\omega } _ k ( i ) \\bigg ) = { D } _ k \\bigg ( \\boldsymbol S _ { D _ k } ^ { \\rm o p t } , \\bar { \\boldsymbol \\omega } _ k ^ { \\rm o p t } \\bigg ) . \\end{align*}"}
-{"id": "7137.png", "formula": "\\begin{align*} e ( m , n ) = \\frac { P _ { \\mathrm { t x } } \\lambda _ 0 } { r ( m , n ) } . \\end{align*}"}
-{"id": "3030.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 w } { \\partial t ^ 2 } ( \\zeta , t ) = \\frac { 1 } { \\rho ( \\zeta ) } \\frac { \\partial } { \\partial \\zeta } \\left ( T ( \\zeta ) \\frac { \\partial w } { \\partial \\zeta } ( \\zeta , t ) \\right ) , t \\ge 0 , \\zeta \\in [ 0 , \\infty ) , \\end{align*}"}
-{"id": "1163.png", "formula": "\\begin{align*} [ 1 ] _ w & = w ^ 2 + w \\\\ [ k ] _ w & = w ^ { 2 ^ k } + w . \\end{align*}"}
-{"id": "2780.png", "formula": "\\begin{align*} K - \\mathcal { B } ( t ) = p ^ { E } ( t , \\mathcal { B } ( t ) ) + & \\int _ { 0 } ^ { t } [ r K e ^ { - r ( t - s ) } \\aleph ( - d _ { 2 } ( \\mathcal { B } ( t ) , t - s , \\mathcal { B } ( s ) ) ) \\\\ - & \\delta \\mathcal { B } ( t ) e ^ { - \\delta ( t - s ) } \\aleph ( - d _ { 1 } ( \\mathcal { B } ( t ) , t - s , \\mathcal { B } ( s ) ) ) ] \\mathrm { d } s , \\end{align*}"}
-{"id": "8679.png", "formula": "\\begin{align*} P ( u , v ) ( w ) + P ( v , w ) ( u ) + P ( w , u ) ( v ) = 0 \\forall u , v , w \\in \\gg _ 1 . \\end{align*}"}
-{"id": "5998.png", "formula": "\\begin{gather*} x . a = x a x , y . a = y a y , z . a = a , \\ a \\in H _ 8 . \\end{gather*}"}
-{"id": "4564.png", "formula": "\\begin{align*} & A _ { - r } = \\eta _ r ^ { - 1 } C ^ { - r } ( H _ { 0 , - r } + \\sum _ { i = 1 } ^ { n - 1 } q _ 3 ^ { i r } H _ { i , - r } ) \\ , , A _ { r } = - \\eta _ r ^ { - 1 } ( H _ { 0 , r } + \\sum _ { i = 1 } ^ { n - 1 } q _ 3 ^ { ( n - i ) r } H _ { i , r } ) \\ , , \\\\ & B _ { - r } = - \\eta _ r ^ { - 1 } ( H _ { 0 , - r } + \\sum _ { i = 1 } ^ { n - 1 } q _ 3 ^ { - ( n - i ) r } H _ { i , - r } ) \\ , , B _ r = \\eta _ r ^ { - 1 } C ^ { r } ( H _ { 0 , r } + \\sum _ { i = 1 } ^ { n - 1 } q _ 3 ^ { - i r } H _ { i , r } ) \\ , . \\end{align*}"}
-{"id": "7517.png", "formula": "\\begin{gather*} u = w ^ 1 \\cos \\tau - w ^ 2 \\sin \\tau - \\kappa y , \\\\ v = w ^ 1 \\sin \\tau + w ^ 2 \\cos \\tau + \\kappa x , \\end{gather*}"}
-{"id": "8037.png", "formula": "\\begin{align*} \\nabla \\cdot \\mbox { \\boldmath $ u $ } ( \\cdot , t ) \\ , = \\ , 0 , \\end{align*}"}
-{"id": "1251.png", "formula": "\\begin{align*} D \\cap B ( w , 4 \\hat r ) = D _ 1 \\cap B ( w , 4 \\hat r ) = D _ 2 \\cap B ( w , 4 \\hat r ) . \\end{align*}"}
-{"id": "4044.png", "formula": "\\begin{align*} \\lambda : = \\max _ { p \\in M } \\left ( \\max _ { V \\in S _ p } \\Big ( k _ { M , p } ( V ) \\Big ) \\right ) \\end{align*}"}
-{"id": "8529.png", "formula": "\\begin{align*} \\mathcal { T } _ { ( j , - l ) } f ( x ) = \\int _ { \\mathbb { R } ^ 3 } e ^ { i x \\cdot \\xi } a _ { ( j , - l ) } ( x , \\xi ) f ( \\xi ) d \\xi \\end{align*}"}
-{"id": "4979.png", "formula": "\\begin{align*} h _ j ( x ) : = ( 1 - \\zeta _ j ( x ) ) \\Delta _ j f ( x ) , g _ j ( x ) : = \\zeta _ j ( x ) \\Delta _ j f ( x ) . \\end{align*}"}
-{"id": "1828.png", "formula": "\\begin{align*} E _ { U _ { \\mathcal { C } } } ( e ^ { S _ { \\mathcal { C } , h } \\mu ^ 0 _ { \\mathcal { C } } ( \\tilde { f } ) } ) = \\frac { \\cosh \\left ( h \\mu ^ 0 _ { \\mathcal { C } } ( D ) + \\mu ^ 0 _ { \\mathcal { C } } ( \\tilde { f } ) \\right ) } { \\cosh \\left ( h \\mu ^ 0 _ { \\mathcal { C } } ( D ) \\right ) } , \\end{align*}"}
-{"id": "7363.png", "formula": "\\begin{align*} { \\cal { R } } = - 4 \\lambda ^ 2 n ( n - 1 ) . \\end{align*}"}
-{"id": "4359.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { j = 1 } ^ \\infty P _ \\alpha M _ { a _ j } f \\Big \\| _ { \\infty , \\alpha } & \\leq \\| P _ \\alpha \\| \\Big \\| \\sum _ { j = 1 } ^ \\infty M _ { a _ j } f \\Big \\| _ { \\infty , \\alpha } \\\\ & \\leq \\| P _ \\alpha \\| \\| f \\| _ { \\infty , \\alpha } \\end{align*}"}
-{"id": "3060.png", "formula": "\\begin{align*} \\kappa ( m ) = - \\Phi _ K ( m ) \\cdot m , m \\in M . \\end{align*}"}
-{"id": "5575.png", "formula": "\\begin{align*} b _ { 0 } = c _ { 0 } \\approx 1 \\end{align*}"}
-{"id": "4849.png", "formula": "\\begin{align*} y z = a t ^ { 2 m } + a _ b t ^ b + \\cdots , x z = t ^ m , x y = a t ^ { 3 m } + a _ b t ^ { b + m } + \\cdots . \\end{align*}"}
-{"id": "8488.png", "formula": "\\begin{align*} \\epsilon ( \\frac { 1 } { 2 } , \\mu ^ { - 1 } \\chi _ 1 ^ { - 1 } ) = \\epsilon ( \\frac { 1 } { 2 } , \\chi _ 1 ^ { - 1 } ) \\mu ( - b _ 1 ) . \\end{align*}"}
-{"id": "1442.png", "formula": "\\begin{align*} \\mathrm { K e r } \\ , Q _ 0 / \\mathrm { I m } \\ , Q _ 0 & = \\mathbb { Z } / 2 [ x _ 2 ^ 2 , x _ { 4 1 } , \\dots , x _ { 4 n } ] . \\end{align*}"}
-{"id": "9507.png", "formula": "\\begin{align*} \\tilde { \\tau } _ n ^ { ( r ) } = \\dfrac { 1 } { 2 n } \\biggl [ \\tilde { s } _ { [ 2 ^ 4 ] } ( n ) + 2 \\cdot \\tilde { s } _ { [ 2 , 4 ^ 2 ] } ( n / 2 ) + 2 \\cdot \\tilde { s } _ { [ 3 ^ 3 ] } ( 2 n / 3 ) + 2 \\cdot \\tilde { s } _ { [ 2 , 3 , 6 ] } ( n / 3 ) + \\sum _ { L | n } J _ 2 ( L ) \\cdot \\tau ^ { ( r ) } ( n / L ) \\biggr ] . \\end{align*}"}
-{"id": "4719.png", "formula": "\\begin{align*} g ^ { i j } S _ { , i } S _ { , j } = 0 , \\end{align*}"}
-{"id": "4341.png", "formula": "\\begin{align*} e ^ { \\alpha \\langle z , w \\rangle - \\beta | z | ^ 2 - \\gamma | w | ^ 2 } = e ^ { | z | ^ 2 ( \\alpha \\sqrt { \\frac { \\beta } { \\gamma } } - 2 \\beta ) } . \\end{align*}"}
-{"id": "8709.png", "formula": "\\begin{align*} & M _ t [ u - v ] ( t , x ) \\\\ & = \\frac { ( u - v ) ( t , x ) - ( u - v ) ( 0 , x ) } { t ^ \\alpha \\Gamma ( 1 - \\alpha ) } + \\frac { \\alpha } { \\Gamma ( 1 - \\alpha ) } \\int _ 0 ^ t \\frac { ( u - v ) ( t , x ) - ( u - v ) ( s , x ) } { ( t - s ) ^ \\alpha } d s . \\end{align*}"}
-{"id": "8389.png", "formula": "\\begin{align*} f = \\sum _ { \\substack { m > 0 \\\\ \\mu \\in V _ \\Z ^ \\vee / V _ \\Z } } c ( - m , \\mu ) \\cdot F _ { m , \\mu } . \\end{align*}"}
-{"id": "1687.png", "formula": "\\begin{align*} \\mathcal { K } _ N = \\prod _ { i = 1 } ^ \\infty \\Z _ N = \\{ ( i _ 1 i _ 2 \\dots ) \\ , : \\ , i _ n \\in \\Z _ N , \\ ; \\ ; n = 1 , 2 , \\dots \\} . \\end{align*}"}
-{"id": "7945.png", "formula": "\\begin{align*} h _ i : = f _ { I _ 0 i } : \\bar W \\to \\bar W _ { i } , \\end{align*}"}
-{"id": "4189.png", "formula": "\\begin{align*} \\frac { 1 - I } { 2 7 } = \\frac { r ^ 4 ( \\frac { 1 } { 4 } + r ^ 2 ) } { ( 1 + 3 r ^ 2 ) ^ 3 } , \\quad \\textnormal { w i t h } r = \\frac { r _ 1 } { r _ 2 } . \\end{align*}"}
-{"id": "783.png", "formula": "\\begin{align*} T '' _ { k } ( x ) = k ( k + 1 ) ( { 4 B _ { k - 1 } ( 2 x + 1 ) - B _ { k - 1 } ( x + 1 ) } ) . \\end{align*}"}
-{"id": "7786.png", "formula": "\\begin{align*} b ' ( x _ h ; \\xi _ h , y _ h ) = 0 \\quad \\xi _ h \\in X _ h . \\end{align*}"}
-{"id": "1743.png", "formula": "\\begin{align*} S _ \\lambda ^ { u n i v } ( f \\ , \\sqrt { d \\mu } ) : = ( f \\circ \\sigma ^ n ) \\ , \\sqrt { d ( \\mu \\circ \\sigma _ \\lambda ^ { - 1 } ) } , \\end{align*}"}
-{"id": "4237.png", "formula": "\\begin{align*} & c _ 1 ( E ' _ { f _ 1 } ) = c _ 1 ( E _ 0 ) , \\\\ & c _ 2 ( E ' _ { f _ 1 } ) \\cdot [ H ] ^ { n - 2 } = c _ 2 ( E _ 0 ) \\cdot [ H ] ^ { n - 2 } . \\\\ \\end{align*}"}
-{"id": "7161.png", "formula": "\\begin{align*} & q ( ( r + 1 ) J + 1 ) = \\sqrt { 2 \\Delta _ J ( r J ) } \\\\ \\leq & \\sqrt { 2 [ U J ^ 2 + V _ r J g ^ * _ r + W J ] } . \\end{align*}"}
-{"id": "3936.png", "formula": "\\begin{align*} \\frac { \\partial ^ \\gamma } { \\partial t ^ \\gamma } u ( x _ j , t _ { n + 1 } ) = \\frac { 1 } { \\alpha _ 0 } \\left [ u _ j ^ { n + 1 } + \\sum \\limits _ { l = 0 } ^ { n } ( w _ { n - l + 1 } - w _ { n - l } ) u _ j ^ l - w _ n u _ j ^ 0 \\right ] , \\end{align*}"}
-{"id": "4000.png", "formula": "\\begin{align*} p ^ { \\nu _ 3 } ( 3 , t ) = \\frac { \\lambda _ 1 } { \\lambda _ 3 } \\sum _ { k = 2 } ^ { \\infty } ( - 1 ) ^ k \\underset { \\Lambda ^ { k } _ { 3 } } { \\sum } \\frac { \\lambda _ 1 ^ { k _ 1 } \\lambda _ 2 ^ { k _ 2 } \\lambda _ 3 ^ { k _ 3 } t ^ { k _ 1 \\nu _ 1 + k _ 2 \\nu _ 2 + k _ 3 \\nu _ 3 } } { \\Gamma \\left ( k _ 1 \\nu _ 1 + k _ 2 \\nu _ 1 + k _ 3 \\nu _ 3 + 1 \\right ) } , \\end{align*}"}
-{"id": "7816.png", "formula": "\\begin{align*} \\Delta v _ m - ( \\frac { 2 m } { r } + 2 \\rho ^ { \\prime } ) \\frac { \\partial v _ m } { \\partial r } + ( \\frac { m ( m + 1 ) } { r ^ 2 } + \\frac { m } { r } ( 2 \\rho ^ { \\prime } - \\Delta r ) - V _ 0 - V _ 1 - V _ 2 + \\lambda ) v _ m = 0 . \\end{align*}"}
-{"id": "9042.png", "formula": "\\begin{align*} \\varphi ^ 1 _ 2 = 0 , \\ , \\ , \\beta \\varphi ^ 2 _ 2 = 0 , \\ , \\ , \\gamma \\varphi ^ 1 _ 1 = 0 , \\ , \\ , \\delta \\varphi ^ 2 _ 1 = 0 \\end{align*}"}
-{"id": "6135.png", "formula": "\\begin{align*} \\mathcal { T } \\left ( J _ { i } ^ { n } \\left ( \\omega \\right ) , a _ { n } W _ { i } \\left ( \\omega \\right ) \\right ) & = x _ { t } \\\\ \\mathcal { T } \\left ( J _ { i } ^ { n } \\left ( \\omega \\right ) , a _ { n } U _ { i } \\left ( \\omega \\right ) \\right ) & = y _ { t } \\\\ \\left | B _ { n } ^ { X } [ 0 , T - \\frac { \\epsilon } { 2 } ] \\right | & = M . \\end{align*}"}
-{"id": "1247.png", "formula": "\\begin{align*} c ^ { - 1 } | \\xi | ^ 2 | x - w | ^ { \\frac { ( p - 2 ) ( 1 - n ) } { p - 1 } } \\leq \\sum _ { i , j = 1 } ^ n b _ { i j } ( x ) \\xi _ i \\xi _ j \\leq c | \\xi | ^ 2 | x - w | ^ { \\frac { ( p - 2 ) ( 1 - n ) } { p - 1 } } \\end{align*}"}
-{"id": "5478.png", "formula": "\\begin{align*} \\limsup _ { \\varepsilon \\downarrow 0 } \\limsup _ { n \\to \\infty } \\frac { U ( r ^ n z \\varepsilon ) } { ( r ^ n z ) ^ \\rho \\ell ( r ^ n z ) } = 0 . \\end{align*}"}
-{"id": "606.png", "formula": "\\begin{align*} \\dim L _ q ( m ) = \\sum _ r a _ r d _ { r , m } . \\end{align*}"}
-{"id": "1271.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m c _ i \\ , \\xi _ i = 0 . \\end{align*}"}
-{"id": "1607.png", "formula": "\\begin{align*} \\Lambda ^ { \\operatorname { m i n } } ( \\lambda , \\eta ) : = \\{ ( \\alpha , \\beta ) \\in \\Lambda \\times \\Lambda \\ , : \\ , \\lambda \\alpha = \\eta \\beta , \\ ; d ( \\lambda \\alpha ) = d ( \\lambda ) \\vee d ( \\eta ) \\} . \\end{align*}"}
-{"id": "5135.png", "formula": "\\begin{align*} v ( t ) = - i \\int _ 0 ^ t S ( t - t ' ) \\big \\{ \\N ( v + z _ 1 + z _ 3 ) - \\N ( z _ 1 ) \\big \\} ( t ' ) d t ' . \\end{align*}"}
-{"id": "6910.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 , m \\to \\infty } \\left ( \\frac { 1 } { \\mu _ { 0 } r _ { 0 } } \\right ) ^ { m } \\Phi ^ { m } _ { 1 } \\left ( \\frac { 3 ( 2 + 3 r - \\sqrt { 8 r + 9 r ^ { 2 } } ) } { 2 ( 1 + r ) } \\right ) = \\infty \\end{align*}"}
-{"id": "7729.png", "formula": "\\begin{align*} G _ N ( 1 ) & = \\frac { 1 } { N } \\sum ^ { N - 1 } _ { n = 1 } \\frac { 1 - \\cos 2 \\phi _ n } { ( 1 - \\cos 2 \\phi _ n ) ^ 2 } \\\\ & = \\frac { 1 } { 2 N } \\sum ^ { N - 1 } _ { n = 1 } \\frac { 1 } { \\sin ^ 2 \\phi _ n } = \\frac { N } { 6 } - \\frac { 1 } { 6 N } \\end{align*}"}
-{"id": "8278.png", "formula": "\\begin{align*} [ x , y ] = Q ( x + y ) - Q ( x ) - Q ( y ) \\end{align*}"}
-{"id": "6714.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\mathbb { P } ( S _ { N } = \\Gamma _ { 1 } ) = 1 . \\end{align*}"}
-{"id": "9550.png", "formula": "\\begin{align*} \\sum _ { \\tau \\in W } x ^ { \\ell _ { \\Delta ' } ( \\tau ) } y ^ { L _ { \\Delta ' } ( \\tau ) } = \\sum _ { \\tau \\in W } x ^ { \\ell _ { \\Delta } ( w ^ { - 1 } \\tau w ) } y ^ { L _ { \\Delta } ( w ^ { - 1 } \\tau w ) } = \\sum _ { \\sigma \\in W } x ^ { \\ell _ { \\Delta } ( \\sigma ) } y ^ { L _ { \\Delta } ( \\sigma ) } . \\end{align*}"}
-{"id": "665.png", "formula": "\\begin{align*} \\lambda _ i ' ( 0 ) = \\lambda _ { \\sigma ' ( i ) } ' ( 1 ) , \\forall i \\in S _ n \\end{align*}"}
-{"id": "9703.png", "formula": "\\begin{align*} \\begin{cases} & V _ b = \\tilde { \\Phi } _ 5 ( \\alpha _ 5 ; \\mathcal { F } ( \\sigma _ 3 , \\sigma _ 2 ; \\tilde { \\Phi } _ 1 ( \\alpha _ 1 ; V _ a ) ) ) , \\\\ & Z _ b = Z _ a + \\alpha _ 4 , \\end{cases} \\end{align*}"}
-{"id": "2566.png", "formula": "\\begin{align*} \\sin ^ { 2 } \\gamma = 1 - \\cos ^ { 2 } \\gamma > 1 - ( 1 - d _ { 0 } / 4 ) ^ { 2 } = d _ { 0 } / 2 - d _ { 0 } ^ { 2 } / 4 > d ^ 2 _ { 0 } / 4 \\end{align*}"}
-{"id": "4683.png", "formula": "\\begin{align*} { \\cal H } _ r ^ { ( d ( n - 1 ) ) } \\ = \\ p _ { { \\bf r } _ i ^ { ( F ) } } p _ { { \\bf r } _ i ^ { ( F ) } } . \\end{align*}"}
-{"id": "1178.png", "formula": "\\begin{align*} \\mbox { C a p } _ \\mathcal { A } ( B ( z , R ) ) = c _ 1 R ^ { n - p } \\end{align*}"}
-{"id": "2344.png", "formula": "\\begin{align*} c = c _ 1 + 4 ( d _ 0 - 1 ) . \\end{align*}"}
-{"id": "6303.png", "formula": "\\begin{align*} \\ell _ s = \\sup \\left \\{ \\gamma < s \\ , : \\ , ( \\forall x < \\gamma ) ( \\exists y < s ) \\ , \\ , \\psi ( \\alpha , x , y ) \\right \\} . \\end{align*}"}
-{"id": "4099.png", "formula": "\\begin{align*} h ( V _ 1 , V _ 1 ) = - \\frac { \\langle V _ 1 , d \\xi _ p V _ 1 \\rangle } { \\langle \\eta , \\xi \\rangle } = - \\frac { \\langle V _ 1 , d u ^ { - 1 } _ { \\eta ( p ) } \\circ d \\eta _ p V _ 1 \\rangle } { \\langle \\eta , \\xi \\rangle } = - \\frac { \\lambda _ 1 \\langle V _ 1 , d u ^ { - 1 } _ { \\eta ( p ) } V _ 1 \\rangle } { \\langle \\eta , \\xi \\rangle } , \\end{align*}"}
-{"id": "9963.png", "formula": "\\begin{align*} \\mathbf { y } { } _ { i } ^ { [ n ] } = \\sum _ { j = 1 } ^ { L } \\sum _ { k = 1 } ^ { K _ { j } } \\sqrt { \\rho _ { k j i } } h { } _ { k j i } ^ { [ n ] } \\mathbf { q } { } _ { k j } + \\mathbf { n } { } _ { i } ^ { [ n ] } \\end{align*}"}
-{"id": "10044.png", "formula": "\\begin{align*} \\chi _ E ( b ) = \\prod _ v \\epsilon _ v = 1 . \\end{align*}"}
-{"id": "2932.png", "formula": "\\begin{align*} F _ \\epsilon ( H , i ) : = \\lim _ { \\Delta \\rightarrow \\infty } F _ \\epsilon ^ \\Delta ( H , i ) . \\end{align*}"}
-{"id": "5805.png", "formula": "\\begin{align*} \\mu \\in \\dot { W } ^ { - 1 , p ' } ( \\Omega ) . \\end{align*}"}
-{"id": "6905.png", "formula": "\\begin{align*} \\Phi _ { 1 } ( x ) = \\frac { 2 r ( 2 + r ) } { ( 1 + 2 r ) ( 1 5 + 2 6 r + 9 r ^ { 2 } ) } x + O ( x ^ { 2 } ) \\end{align*}"}
-{"id": "7338.png", "formula": "\\begin{align*} \\frac { 2 } { s } + \\frac { 3 } { p } = \\frac { 7 } { 2 } , \\begin{cases} \\frac { 1 } { 2 } \\leqslant \\frac { 1 } { p } \\leqslant 1 & 2 \\leqslant r < 3 \\\\ \\frac { 1 } { 2 } \\leqslant \\frac { 1 } { p } < \\frac { 1 } { r } + \\frac { 2 } { 3 } & 3 \\leqslant r \\leqslant 6 \\ , . \\end{cases} \\end{align*}"}
-{"id": "6237.png", "formula": "\\begin{align*} X ( n ) = - C ( n ) + Z ( n ) ~ , ~ n \\geq 1 ~ , \\end{align*}"}
-{"id": "4695.png", "formula": "\\begin{align*} \\Psi _ v ( r ) : = \\frac { 1 } { r ^ { n + 2 } } \\int _ { B _ r } \\big ( | \\nabla v | ^ 2 + 2 v \\big ) \\ , d x - \\frac { 2 } { r ^ { n + 3 } } \\int _ { \\partial B _ r } v ^ 2 \\ , d \\mathcal { H } ^ { n - 1 } . \\end{align*}"}
-{"id": "8637.png", "formula": "\\begin{align*} \\rho = \\int _ { \\Omega \\times \\Omega } e ^ { I ( x , y ) } \\Psi ( y ) \\pi ( d x ) \\pi ( d y ) , \\end{align*}"}
-{"id": "1500.png", "formula": "\\begin{align*} t _ { n } ( x ) = x ^ { 2 } t _ { n - 1 } ( x ) + x t _ { n - 2 } ( x ) + t _ { n - 3 } ( x ) , \\end{align*}"}
-{"id": "2965.png", "formula": "\\begin{align*} \\mathtt { I } ( \\mathbf { x } _ \\mathcal { S } ; \\mathbf { y } _ ) = \\sum _ { k \\in \\mathcal { S } } \\mathtt { I } ( x _ k ; \\mathbf { y } _ | \\mathbf { x } _ { k + 1 } ^ { | \\mathcal { S } | } ) : = \\sum _ { k \\in \\mathcal { S } } R _ { _ k } . \\end{align*}"}
-{"id": "8647.png", "formula": "\\begin{align*} \\pi [ E _ N ] < \\frac 1 4 , ~ ~ \\hbox { w h e r e } ~ ~ E _ N = \\big \\{ \\min _ { s \\geq T _ N } | x + \\omega _ { X _ 0 } ( s ) - y - \\omega _ { Y _ 0 } ( s ) | \\leq 1 \\big \\} . \\end{align*}"}
-{"id": "9090.png", "formula": "\\begin{align*} d ^ { k } ( x y ) = \\sum _ { i = 0 } ^ { k } \\binom { k } { i } d ^ { i } ( x ) d ^ { k - i } ( y ) \\left ( x , y \\in R \\right ) , \\end{align*}"}
-{"id": "8916.png", "formula": "\\begin{align*} \\begin{aligned} X ^ \\ast T ^ \\ast T X & - X ^ \\ast X \\\\ & = | 1 - c | ^ 2 \\int _ { \\mathbb T } | \\varphi | ^ 2 { } \\nu ( \\cdot , \\overline \\chi \\theta ) \\overline \\chi \\theta + ( 1 - c ) ( \\cdot , \\overline \\chi \\theta ) f + ( 1 - \\overline c ) ( \\cdot , f ) \\overline \\chi \\theta . \\end{aligned} \\end{align*}"}
-{"id": "3322.png", "formula": "\\begin{align*} \\tilde { D } ( r _ { K - 2 } ) & \\geq ( 1 - r _ { K - 2 } ) \\sum _ { j = 0 } ^ { 1 } \\frac { 1 } { N ^ j } - r _ { K - 2 } \\sum _ { j = 0 } ^ { 0 } \\frac { 1 - j } { N ^ j } \\\\ & = \\left ( \\frac { N ^ 2 + ( K - 2 ) N } { N ^ 2 + ( K - 2 ) N + ( K - 2 ) } \\right ) \\left ( 1 + \\frac { 1 } { N } \\right ) - \\frac { K - 2 } { N ^ 2 + ( K - 2 ) N + ( K - 2 ) } \\\\ & = \\frac { N ^ 2 + ( K - 1 ) N } { N ^ 2 + ( K - 2 ) N + ( K - 2 ) } \\\\ & = \\bar { D } ( r _ { K - 2 } ) \\end{align*}"}
-{"id": "5323.png", "formula": "\\begin{align*} [ a \\otimes t ^ m , b \\otimes t ^ n ] = [ a , b ] \\otimes t ^ { m + n } + ( a , b ) m \\delta _ { m + n , 0 } { \\bf k } , \\end{align*}"}
-{"id": "5702.png", "formula": "\\begin{align*} \\int _ { \\R ^ 2 } - { u } \\cdot \\Delta \\psi + \\nabla W ( { u } ) \\cdot \\psi = 0 . \\end{align*}"}
-{"id": "1473.png", "formula": "\\begin{align*} \\mathrm { K e r } \\ , d _ { 2 ^ { r + 1 } - 1 } \\cap \\left ( D _ 1 / ( v _ t e _ t ) \\{ x _ 3 ^ 2 \\} \\oplus D _ r \\langle x _ 3 ^ 2 y _ r , x _ 3 z _ r \\rangle \\right ) \\\\ = D _ 1 / ( v _ t e _ t ) \\{ x _ 3 ^ 2 \\} \\oplus D _ r \\langle x _ { 3 } ^ 2 y _ { r + 1 } , x _ 3 z _ { r + 1 } , \\rangle . \\end{align*}"}
-{"id": "5109.png", "formula": "\\begin{align*} \\underline { u } = \\eta w . \\end{align*}"}
-{"id": "7441.png", "formula": "\\begin{align*} - f = x ^ 2 + u y ^ 2 + 2 v y z + w z ^ 2 + ( u w - v ^ 2 ) t ^ 2 . \\end{align*}"}
-{"id": "9088.png", "formula": "\\begin{align*} f ( x y ) = x f ( y ) + f ( x ) y \\end{align*}"}
-{"id": "4582.png", "formula": "\\begin{align*} [ A , F _ { 0 , - 1 } ] _ { q } & \\equiv ( - d ) ^ { n - 3 } [ [ \\cdots [ F _ { n - 1 , 1 } , F _ { n - 2 , 0 } ] _ { q } , \\cdots , F _ { 2 , 0 } ] _ { q } , F _ { 0 , - 1 } ] _ q \\\\ & = ( - d ) ^ { n - 3 } [ \\cdots [ [ F _ { n - 1 , 1 } , F _ { 0 , - 1 } ] _ q , F _ { n - 2 , 0 } ] _ { q } , \\cdots , F _ { 2 , 0 } ] _ { q } \\\\ & = ( - d ) ^ { n - 2 } [ \\cdots [ [ F _ { 0 , 0 } , F _ { n - 1 , 0 } ] _ q , F _ { n - 2 , 0 } ] _ { q } , \\cdots , F _ { 2 , 0 } ] _ { q } \\ , . \\end{align*}"}
-{"id": "2139.png", "formula": "\\begin{align*} G _ d ( r , t ) : = u _ * ( r , t ) - \\left [ u _ * ( 0 , t ) + ( \\partial _ t u _ * ) ( 0 , t ) F _ d ( r ) \\right ] \\quad \\mbox { f o r } r \\in [ 0 , \\infty ) , \\ , \\ , t > 0 , \\end{align*}"}
-{"id": "7323.png", "formula": "\\begin{align*} | V _ { x x _ j } | \\geq | V _ x | \\mathfrak { s } _ j = ( \\prod _ { i = 1 } ^ n \\mathfrak { s } _ i ^ { m _ i } ) \\mathfrak { s } _ j . \\end{align*}"}
-{"id": "7226.png", "formula": "\\begin{align*} \\tilde F _ { 0 , \\lambda } ( \\tilde R _ \\lambda ) & = F ( \\tilde R _ { \\lambda } ) + \\lambda \\mathbb { F } ( \\tilde R _ { \\lambda } - \\Sigma ) ^ { 2 } \\\\ & \\leq \\liminf _ { i \\to \\infty } \\{ F ( \\tilde R _ { \\delta _ i , \\lambda } ) + \\lambda ( \\mathbb { F } ( \\tilde R _ { \\delta _ i , \\lambda } - \\Sigma ) - \\tilde \\eta _ { \\delta _ i } ) ^ 2 \\} \\\\ & = \\liminf _ { i \\to \\infty } \\tilde F _ { \\delta _ i , \\lambda } ( \\tilde R _ { \\delta _ i , \\lambda } ) \\ , , \\end{align*}"}
-{"id": "8353.png", "formula": "\\begin{align*} \\widetilde { \\Lambda } = \\Lambda + \\langle x \\rangle \\subset V _ 0 ( \\mathcal { A } _ \\eta ) \\end{align*}"}
-{"id": "1240.png", "formula": "\\begin{align*} \\mathcal { M } ( x ) = \\left \\{ \\left [ t _ 1 | \\nabla \\breve v _ 1 ( x ) | + t _ 2 | \\nabla \\breve v _ 2 ( x ) | \\right ] ^ { 2 p - 6 } \\ , \\sum _ { i , j = 1 } ^ n \\ , \\left [ t _ 1 | ( \\breve v _ 1 ( x ) ) _ { x _ i x _ j } | + t _ 2 | ( \\breve v _ 2 ( x ) ) _ { x _ i x _ j } | \\right ] ^ 2 \\right \\} . \\end{align*}"}
-{"id": "9543.png", "formula": "\\begin{align*} \\frac { d \\mu _ T ' } { d \\lambda } ( x ) = \\Phi _ T ( x ) . \\end{align*}"}
-{"id": "2877.png", "formula": "\\begin{align*} P _ { 1 } ( x ) = P _ { 1 } ( x ; N , h ) = P ( x ) + ( - 1 ) ^ { h + 1 } 2 ^ { 2 h - 1 } \\pi ^ { 2 h } \\sum _ { j = 1 } ^ { \\left \\lfloor \\frac { h } { N } \\right \\rfloor } \\left ( \\frac { - 1 } { 4 \\pi ^ 2 } \\right ) ^ { j N } \\frac { B _ { 2 j } B _ { 2 h - 2 j N } x ^ { 2 j - 1 } } { ( 2 j ) ! ( 2 h - 2 j N ) ! } . \\end{align*}"}
-{"id": "9436.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\mathcal { M } ( N ) } t _ k \\leq c _ 3 N \\mathcal { M } ( N ) , \\end{align*}"}
-{"id": "2775.png", "formula": "\\begin{align*} M ( \\vec w , k - 1 ) = \\begin{pmatrix} c _ { w _ 1 } M _ { 1 1 } & \\cdots & c _ { w _ 1 + ( k - 1 ) p } M _ { 1 k } \\\\ c _ { w _ 1 - 1 } M _ { 2 1 } & \\cdots & c _ { w _ 1 + ( k - 1 ) p - 1 } M _ { 2 k } \\\\ \\vdots & \\ddots & \\vdots \\\\ c _ { w _ 1 - k + 1 } M _ { k 1 } & \\cdots & c _ { w _ 1 + ( k - 1 ) ( p - 1 ) } M _ { k k } \\end{pmatrix} . \\end{align*}"}
-{"id": "5539.png", "formula": "\\begin{align*} \\det \\left ( \\mathbb { \\rho } I _ { 2 } - M \\right ) = \\mathbb { \\rho } ^ { 2 } - T r a c e \\left ( M \\right ) \\mathbb { \\rho } + 1 \\end{align*}"}
-{"id": "2774.png", "formula": "\\begin{align*} c _ { w _ 1 } = \\frac { 1 } { w _ 1 ! } \\not \\equiv 0 \\pmod p . \\end{align*}"}
-{"id": "3531.png", "formula": "\\begin{align*} J ^ { \\theta } ( \\bar { u } ( \\cdot ) ) = \\theta \\leq \\inf _ { u \\in \\mathcal { U } [ 0 , T ] } J ^ { \\theta } ( u ( \\cdot ) ) + \\theta . \\end{align*}"}
-{"id": "3379.png", "formula": "\\begin{align*} \\mathcal M = [ p _ 1 , r ^ 0 ] ^ s \\cup F ^ { - 1 } \\left ( [ p _ 1 , r ^ 0 ] ^ s \\right ) \\cup F ^ { - 2 } \\left ( [ p _ 1 , r ^ 0 ] ^ s \\right ) \\end{align*}"}
-{"id": "3200.png", "formula": "\\begin{align*} M _ { t } ( g ) & : = \\sigma \\int _ { 0 } ^ { t } \\sqrt { X _ { s } } g _ { x } ' ( s , X _ { s } ) \\mathrm { d } B _ { s } + \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { \\infty } \\left ( g ( s , X _ { s - } + z ) - g ( s , X _ { s - } ) \\right ) \\widetilde { N } ( \\mathrm { d } s , \\mathrm { d } z ) \\\\ & \\thickspace = G _ { t } ( g ) + J _ { t } ( g ) , \\quad t \\geqslant 0 . \\end{align*}"}
-{"id": "6759.png", "formula": "\\begin{align*} z _ w = | W _ v ( l / k ) | ^ { - 1 } \\sum _ { x | v } \\log | \\mu | _ x - \\log | \\mu | _ w . \\end{align*}"}
-{"id": "8838.png", "formula": "\\begin{align*} \\Psi _ v ( t , x ) : = \\chi \\left ( \\lambda ( x - v t ) \\right ) e ^ { i \\phi ( t , x ) } , \\end{align*}"}
-{"id": "10122.png", "formula": "\\begin{align*} \\bar { \\boldsymbol \\omega } _ k ( i ) = \\bar { \\boldsymbol \\omega } _ k ( i - 1 ) + \\mu ( i ) e _ k ^ * ( i ) \\bar { \\boldsymbol x } _ k ( i ) , \\end{align*}"}
-{"id": "5748.png", "formula": "\\begin{align*} \\psi _ { x , y , z } ( a , b ) \\psi _ { x , y - 1 , z - 1 } ( a , b ) = \\psi _ { x , y - 1 , z } ( a , b ) \\psi _ { x , y , z - 1 } ( a , b ) + \\psi _ { x + 1 , y - 1 , z - 1 } ( a , b ) \\psi _ { x - 1 , y , z } ( a , b ) , \\end{align*}"}
-{"id": "1721.png", "formula": "\\begin{align*} S _ \\lambda S _ \\lambda ^ * ( f ) & = W ^ * t _ \\lambda t _ \\lambda ^ * W ( f ) = W ^ * \\pi ( \\chi _ { Z ( \\lambda ) } ) \\pi ( f ) \\xi \\\\ & = W ^ * \\pi ( \\chi _ { Z ( \\lambda ) } \\cdot f ) \\xi = W ^ * W ( \\chi _ { Z ( \\lambda ) } \\cdot f ) \\\\ & = \\chi _ { Z ( \\lambda ) } \\cdot f . \\end{align*}"}
-{"id": "8703.png", "formula": "\\begin{align*} u _ { n , k } & : = \\min _ { \\lfloor \\zeta q ^ { - n } \\rfloor \\leq i \\leq \\lfloor \\zeta q ^ { - n } \\rfloor + \\lfloor n ^ \\alpha q ^ { - n } \\rfloor + k } \\frac { L ( i ) i ^ { \\gamma - 1 } ( 1 - q ^ n ) ^ { 2 i } } { \\Gamma ( \\gamma ) v ( 1 - q ^ n ) } , \\\\ o _ { n , k } & : = \\max _ { j \\geq \\lfloor n ^ \\delta q ^ { - n } \\rfloor + k } \\frac { L ( j ) j ^ { \\gamma - 1 } ( 1 - q ^ n ) ^ { 2 j } } { \\Gamma ( \\gamma ) v ( 1 - q ^ n ) } . \\end{align*}"}
-{"id": "5856.png", "formula": "\\begin{align*} \\theta ^ { X Y } _ { i j } = 0 \\textrm { ( f o r $ \\forall $ $ i $ , $ j $ ) } . \\end{align*}"}
-{"id": "6715.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\mathbb { P } \\Big ( \\Big | \\{ \\xi ( 0 ) , \\dots , \\xi ( [ N ^ { \\gamma } ] ) \\} \\Big | = [ N ^ { \\gamma } ] + 1 \\Big ) = 1 . \\end{align*}"}
-{"id": "5060.png", "formula": "\\begin{align*} b _ k B _ { i k , j } - b _ j B _ { i j , k } = 2 ( B _ { i j } C _ k - B _ { i k } C _ j ) . \\end{align*}"}
-{"id": "7389.png", "formula": "\\begin{align*} \\mathcal { K } ( z , w ) = z ^ { - N _ f } \\frac { p _ { N _ f } ( z ) p ^ { * } _ { N _ f } ( w ) - p ^ { * } _ { N _ f } ( z ) p _ { N _ f } ( z ) } { 1 - z ^ { - 1 } ( w ) } \\sqrt { v _ N ( z ) v _ N ( w ) e ^ { V ( z ) } e ^ { V ( w ) } } , \\end{align*}"}
-{"id": "6458.png", "formula": "\\begin{align*} \\mathcal { F } \\left ( \\rho \\right ) \\overset { } { = } \\frac { 1 } { 2 ^ { \\frac { 5 } { 2 } } } \\sqrt { \\frac { 4 \\left ( 4 - \\rho ^ { 2 } \\right ) } { \\left ( 2 - 2 \\rho ^ { 2 } \\right ) ^ { 2 } } } \\left ( \\frac { 2 + \\rho } { 4 \\left ( 1 - \\rho ^ { 2 } \\right ) } \\right ) ^ { - \\frac { 3 } { 2 } } . \\end{align*}"}
-{"id": "6954.png", "formula": "\\begin{align*} ( \\det { D ^ 2 u } ) ^ { 1 / n } ( x ) = g ( x ) , \\end{align*}"}
-{"id": "8153.png", "formula": "\\begin{align*} a _ 1 a _ { k - 1 } ^ q a _ { n - k } ^ { q ^ k } + a _ k a _ { n - 1 } ^ q a _ { n - k + 1 } ^ { q ^ k } = b _ 1 b _ { k - 1 } ^ q b _ { n - k } ^ { q ^ k } + b _ k b _ { n - 1 } ^ q b _ { n - k + 1 } ^ { q ^ k } . \\end{align*}"}
-{"id": "8383.png", "formula": "\\begin{align*} f ( \\tau ) = \\sum _ { \\substack { m \\in \\Q \\\\ m \\gg - \\infty } } c ( m ) \\cdot q ^ m \\in M ^ ! _ { 1 - \\frac { n } { 2 } } ( \\overline { \\rho } _ { V _ \\Z } ) \\end{align*}"}
-{"id": "9432.png", "formula": "\\begin{align*} \\mathfrak { M } ( n ) = \\sum _ { n _ { k } \\leq n } \\frac { \\varphi ( n _ { k } ) } { n _ k } . \\end{align*}"}
-{"id": "4630.png", "formula": "\\begin{align*} E _ N = \\left \\{ x \\in [ 0 , 1 ) : \\sum _ { n = 0 } ^ { N - 1 } 1 _ { ( - \\epsilon , \\epsilon ) } \\left ( \\sum _ { i = 0 } ^ n f ( T ^ i x ) \\right ) > N ^ { 1 - \\gamma - \\epsilon } \\right \\} \\end{align*}"}
-{"id": "6361.png", "formula": "\\begin{align*} & \\limsup \\limits _ { n \\to \\infty } \\frac { 1 } { n } \\log \\int _ U e ^ { - n V _ 1 ( x ) + ( n - i ) \\log ( x ( 1 - x ) ) } \\ , d x \\le { } \\limsup \\limits _ { n \\to \\infty } \\frac { 1 } { n } \\log \\int _ 0 ^ 1 e ^ { - i V _ 1 ( x ) - ( n - i ) W ^ U } \\ , d x = - W ^ U . \\end{align*}"}
-{"id": "9814.png", "formula": "\\begin{align*} p _ 0 ( - T _ { e _ 1 } ) = 1 , p _ k ( - T _ { e _ 1 } ) = 0 , \\forall k \\in \\{ 1 , 2 , . . . , N _ { \\max } \\} . \\end{align*}"}
-{"id": "5142.png", "formula": "\\begin{align*} u = \\sum _ { \\l = 1 } ^ \\infty \\zeta _ { 2 \\l - 1 } + v . \\end{align*}"}
-{"id": "1530.png", "formula": "\\begin{align*} s = \\begin{cases} \\lfloor \\alpha _ { m , t } ( - 1 ) \\rfloor , & \\ \\ \\{ \\alpha _ { m , t } ( - 1 ) \\} < 1 / 2 ; \\\\ \\lfloor \\alpha _ { m , t } ( - 1 ) \\rfloor \\ \\ \\lfloor \\alpha _ { m , t } ( - 1 ) \\rfloor + 1 , & \\ \\ \\{ \\alpha _ { m , t } ( - 1 ) \\} = 1 / 2 ; \\\\ \\lfloor \\alpha _ { m , t } ( - 1 ) \\rfloor + 1 , & \\ \\ \\{ \\alpha _ { m , t } ( - 1 ) \\} > 1 / 2 . \\end{cases} \\end{align*}"}
-{"id": "7911.png", "formula": "\\begin{align*} e _ 2 = { x _ 2 + y _ 2 \\over 2 } - \\left [ { x _ 2 + y _ 2 \\over 2 } \\right ] . \\end{align*}"}
-{"id": "10055.png", "formula": "\\begin{align*} \\epsilon _ \\mathfrak { p } = \\begin{cases} 1 & \\mbox { i f $ \\mathfrak { p } $ i s i n e r t i n $ E $ } \\\\ 0 & \\mbox { i f $ \\mathfrak { p } $ i s r a m i f i e d i n $ E $ } . \\end{cases} \\end{align*}"}
-{"id": "5019.png", "formula": "\\begin{align*} \\frac { | S | } { r _ { 1 } } \\le \\frac { | \\Gamma _ 2 ( S ) | } { k _ 1 r _ 1 + 1 } \\le \\frac { | \\Gamma _ 2 ( S ) | } { k _ 1 r _ 1 } = \\frac { | \\Gamma _ 2 ( S ) | } { r _ { 2 } - 1 } = \\frac { | \\Gamma _ 2 ( S ) | } { | L _ 2 \\setminus \\{ x \\} | } . \\end{align*}"}
-{"id": "8747.png", "formula": "\\begin{align*} & E _ { \\varepsilon , \\delta _ j } ( t _ j , x _ j , u ^ { \\varepsilon , \\delta _ j } ( t _ j , x _ j ) , J [ u ^ { \\varepsilon , \\delta _ j } ] ( t _ j , x _ j ) + K _ { ( 0 , t _ j ) } [ u ^ { \\varepsilon , \\delta _ j } ] ( t _ j , x _ j ) , p _ j , X _ j ) \\\\ & = E _ * ( t ' _ j , x ' _ j , u ^ { \\varepsilon , \\delta _ j } ( t _ j , x _ j ) , J [ u ^ { \\varepsilon , \\delta _ j } ] ( t _ j , x _ j ) + K _ { ( 0 , t _ j ) } [ u ^ { \\varepsilon , \\delta _ j } ] ( t _ j , x _ j ) , p _ j , X _ j ) . \\end{align*}"}
-{"id": "5636.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - 1 } | t _ { i + 1 } - t _ i | < \\delta \\quad \\Longrightarrow \\quad \\sum _ { i = 0 } ^ { n - 1 } d ( \\gamma ( t _ { i + 1 } ) , \\gamma ( t _ i ) ) < \\varepsilon . \\end{align*}"}
-{"id": "5485.png", "formula": "\\begin{align*} \\frac { U ( b x ) - U ( a x ) } { x ^ \\rho \\ell ( x ) } = \\int _ a ^ b \\frac { u ( s x ) } { x ^ { \\rho - 1 } \\ell ( x ) } \\dd s \\end{align*}"}
-{"id": "6187.png", "formula": "\\begin{align*} ( \\log f _ 1 ) ' & = a , ( \\log f _ 2 ) ' = b , ( \\log f _ 3 ) ' = c = - ( a + b ) \\\\ ( \\log f _ 1 ) '' & = A , ( \\log f _ 2 ) '' = B , ( \\log f _ 3 ) '' = C = - ( A + B ) \\end{align*}"}
-{"id": "7802.png", "formula": "\\begin{align*} \\Omega _ { k } \\left ( f , v \\right ) _ { p , \\gamma } : = \\left \\Vert \\left ( \\mathbb { I } - \\mathfrak { T } _ { v } \\right ) ^ { k } f \\right \\Vert _ { p , \\gamma } , v > 0 . \\end{align*}"}
-{"id": "7153.png", "formula": "\\begin{align*} \\sum _ { m = r J + 1 } ^ { ( r + 1 ) J } e _ G ^ * ( m ) \\leq \\alpha B / R + \\sqrt { 2 ( U J ^ 2 + V _ r J g ^ * _ r ) } , \\end{align*}"}
-{"id": "7630.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\int _ { Q _ 3 } { | D u ^ k - D w ^ k | ^ p d z } = \\lim _ { k \\to \\infty } \\int _ { Q _ 3 } { | D m ^ k | ^ p d z } = 0 . \\end{align*}"}
-{"id": "4419.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 m _ 2 \\ , \\ , d x _ 1 = 0 \\quad \\mbox { f o r a l l } \\ ; x _ 2 . \\end{align*}"}
-{"id": "7213.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow 0 } \\frac { | B _ s ( y ) \\cap \\{ v _ 0 > 0 \\} | } { | B _ s | } = \\frac { 1 } { | B _ 1 | } \\lim _ { s \\rightarrow 0 } W _ { A C } ( v _ 0 ; y , s ) , \\end{align*}"}
-{"id": "9438.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { \\psi ( n ) } { | { n } | _ { \\mathcal { D } _ 1 } | { n } | _ { \\mathcal { D } _ 2 } \\cdots | n | _ { \\mathcal { D } _ m } } = \\infty . \\end{align*}"}
-{"id": "5284.png", "formula": "\\begin{align*} & v _ t - \\mu v _ { x x } = \\tilde { f } ( t , x , v , v _ x ) , \\\\ & a _ 1 v ( t , 1 ) + a _ 2 v _ x ( t , 1 ) = 0 , \\\\ & b _ 1 v ( t , 0 ) + b _ 2 v _ x ( t , 0 ) = 0 , \\\\ & v ( 0 , x ) = v _ 0 ( x ) , \\end{align*}"}
-{"id": "2499.png", "formula": "\\begin{align*} \\left ( \\mathcal { R } \\cdot \\mathrm { P } _ { 0 } \\psi _ { 0 } \\right ) \\left ( - \\left | \\eta \\right | \\right ) = \\mathrm { P } _ { 0 } \\mathcal { R } \\cdot \\psi _ { 0 } \\left ( - \\left | \\eta \\right | \\right ) = \\mathrm { P } _ { 0 } \\psi _ { 1 } \\left ( \\left | \\eta \\right | \\right ) . \\end{align*}"}
-{"id": "2270.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq i < j \\leq n } \\left | \\begin{array} { c c } \\alpha d _ i & ( 1 - \\alpha ) a _ { i , j } \\\\ ( 1 - \\alpha ) a _ { j , i } & \\alpha d _ j \\end{array} \\right | = \\sum _ { 1 \\leq i < j \\leq n } \\left | \\begin{array} { c c } \\alpha d _ i & ( 1 - \\alpha ) a _ { i , j } \\\\ ( 1 - \\alpha ) a _ { j , i } & \\alpha d _ j \\end{array} \\right | , \\end{align*}"}
-{"id": "5581.png", "formula": "\\begin{align*} \\Delta \\left ( \\alpha , \\beta \\right ) = 2 - A _ { 1 } + A _ { 2 } + \\ldots + \\left ( - 1 \\right ) ^ { n } A _ { n } \\end{align*}"}
-{"id": "6255.png", "formula": "\\begin{align*} u _ t + u \\cdot \\nabla u - b \\cdot \\nabla b + \\nabla p = \\nu \\Delta u , \\\\ b _ t + u \\cdot \\nabla b - b \\cdot \\nabla u + \\eta \\nabla \\times ( ( \\nabla \\times b ) \\times b ) = - \\mu ( - \\Delta ) ^ \\alpha b , \\\\ \\nabla \\cdot u = 0 , \\end{align*}"}
-{"id": "757.png", "formula": "\\begin{align*} J ( u _ 1 ^ * , u _ 2 ^ * ) = m ( b _ 1 , b _ 2 ) \\ \\ \\ \\ J ( w _ 1 ^ * , w _ 2 ^ * ) = m ( c _ 1 , c _ 2 ) . \\end{align*}"}
-{"id": "2134.png", "formula": "\\begin{align*} d : = N + 2 A , \\rho _ d ( \\xi ) : = \\xi ^ { d - 1 } e ^ { \\frac { \\xi ^ 2 } { 4 } } , \\psi _ d ( \\xi ) : = c _ d e ^ { - \\frac { \\xi ^ 2 } { 4 } } . \\end{align*}"}
-{"id": "2012.png", "formula": "\\begin{align*} x _ i y _ { i + 1 } - x _ { i + k + 1 } y _ i = 0 , ~ ~ ~ i \\in \\Z . \\end{align*}"}
-{"id": "7615.png", "formula": "\\begin{align*} \\tilde \\varphi ( x , t ) : = \\varphi ( \\bar x + \\theta \\lambda ^ { \\frac { p - 2 } { 2 } } x , \\ , \\bar t + \\theta ^ 2 t ) . \\end{align*}"}
-{"id": "6408.png", "formula": "\\begin{align*} \\mathcal { S } \\left [ \\theta | \\theta _ { } \\right ] \\overset { } { = } - \\int d x p ( x | \\theta ) \\log \\left [ \\frac { p ( x | \\theta ) } { p ( x | \\theta _ { } ) } \\right ] \\end{align*}"}
-{"id": "8933.png", "formula": "\\begin{gather*} s _ i \\circ \\iota _ j = \\iota _ j + ( s _ i - 1 ) \\circ \\iota _ j = \\iota _ j - \\iota _ i \\circ \\nu _ i \\circ \\iota ' _ i \\circ \\iota _ j . \\end{gather*}"}
-{"id": "4153.png", "formula": "\\begin{align*} r \\sum _ { \\tau = t + h - q } ^ n Z _ { N , J } ^ { \\tau } + n Z ^ { t + h - q - 1 } _ { I , J } & + \\sum _ { \\tau = t } ^ { t + h - q - 2 } n ^ { t + h - q - \\tau } Z ^ { \\tau } _ { i _ { \\tau } , J } \\leq r ( q + 1 ) + \\sum _ { \\tau = 1 } ^ { h - q - 1 } n ^ { h - q - \\tau } . \\end{align*}"}
-{"id": "5388.png", "formula": "\\begin{align*} K : = \\langle a , b , x ^ 3 \\rangle \\end{align*}"}
-{"id": "3571.png", "formula": "\\begin{align*} 0 = \\sum _ { n \\leq x } \\frac { \\Psi ( n ) \\Lambda ( n ) } { n ^ s } = \\sum _ { n \\leq x } \\frac { \\Lambda ( n ) } { n ^ s } \\left ( \\frac { \\varphi ( p - 1 ) } { p - 1 } \\sum _ { d \\ , | \\ , p - 1 } \\frac { \\mu ( d ) } { \\varphi ( d ) } \\sum _ { ( \\chi ) = d } \\chi ( n ) \\right ) , \\end{align*}"}
-{"id": "7660.png", "formula": "\\begin{align*} \\mathcal B _ i : = \\{ B \\in { [ m ] \\choose b } : B \\cap [ b - a + 3 + i ] = [ b - a + 2 + i ] \\} , \\\\ \\mathcal A _ i : = \\{ A \\in { [ m ] \\choose a } : A \\cap [ b - a + 3 + i ] = \\{ b - a + 3 + i \\} , \\end{align*}"}
-{"id": "8712.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ n \\lambda _ i \\partial _ t ^ { \\alpha _ i } u + F ( t , x , u , \\nabla u , \\nabla ^ 2 u ) = 0 , \\end{align*}"}
-{"id": "4804.png", "formula": "\\begin{align*} A _ { ( F \\setminus \\{ m + 1 \\} , B ) } = L _ { ( F , B \\cup \\{ m + 1 \\} ) } . \\end{align*}"}
-{"id": "5252.png", "formula": "\\begin{align*} u ( \\lambda ^ * , z ) = v ( \\lambda ^ * , z ) : = P ( z ) , \\end{align*}"}
-{"id": "3621.png", "formula": "\\begin{align*} w _ \\lambda ( x ) \\ , : = \\ , ( u - u _ \\lambda ) ( x ) , x \\in \\overline \\Omega _ { \\lambda } \\setminus R _ \\lambda ( \\Gamma ) . \\end{align*}"}
-{"id": "7055.png", "formula": "\\begin{align*} z _ j = y _ j , j = 1 , 2 , . . . , n - 1 , \\end{align*}"}
-{"id": "7939.png", "formula": "\\begin{gather*} d ( x _ 1 ^ i , x ) \\le d ( x _ 1 ^ i , z ) + d ( z , x ) \\le ( 2 c _ 2 + 1 ) \\rho . \\end{gather*}"}
-{"id": "3888.png", "formula": "\\begin{align*} S ( m , n _ 1 , n _ 2 ; q ) = \\sum _ { d | ( m , n _ 1 , q ) } d S \\left ( 1 , \\frac { m n _ 1 } { d ^ 2 } , n _ 2 ; \\frac { q } { d } \\right ) . \\end{align*}"}
-{"id": "2034.png", "formula": "\\begin{align*} f ( \\lambda z _ 1 , . . . , \\lambda z _ k ) = \\lambda ^ { \\frac { k ( k + 1 ) } { 2 } } f ( z _ 1 , . . . , z _ k ) . \\end{align*}"}
-{"id": "1934.png", "formula": "\\begin{align*} \\widehat { \\widetilde { Q } _ { j } ( q ) } ( \\xi ) : = \\chi ( \\xi ) \\widehat { { Q } _ { j } ( q ) } ( \\xi ) , \\end{align*}"}
-{"id": "4905.png", "formula": "\\begin{align*} \\alpha I + \\beta T + \\gamma T ^ * + \\delta T ^ * T + F = 0 . \\end{align*}"}
-{"id": "4240.png", "formula": "\\begin{align*} M = \\coprod _ { ( d _ 2 , d _ 1 ) } M ( d _ 2 , d _ 1 ) \\end{align*}"}
-{"id": "3034.png", "formula": "\\begin{align*} V = \\{ v \\in H ^ 1 ( \\Omega ) ^ 2 \\mid v | _ { \\Gamma _ b } = 0 , \\} . \\end{align*}"}
-{"id": "1051.png", "formula": "\\begin{align*} X _ { k } = H ( P _ { k } ) ^ { ( w _ { n } ( \\zeta ) + \\epsilon ) / ( \\widehat { w } _ { n } ( \\zeta ) - \\epsilon ) } . \\end{align*}"}
-{"id": "9147.png", "formula": "\\begin{align*} f ( x ^ { 5 } ) + x g ( x ^ { 4 } ) + x ^ { 4 } h ( x ) = 0 \\end{align*}"}
-{"id": "5564.png", "formula": "\\begin{align*} \\bar { \\nu } ^ { T } = \\left ( \\tau \\dot { x } _ { 0 } e _ { 1 } ^ { T } P + x _ { 0 } e _ { 1 } ^ { T } \\right ) \\Gamma \\end{align*}"}
-{"id": "5553.png", "formula": "\\begin{align*} \\ddot { z } + g \\left ( t \\right ) z = 0 z \\left ( 0 \\right ) = a \\dot { z } \\left ( 0 \\right ) = b \\end{align*}"}
-{"id": "3045.png", "formula": "\\begin{align*} ( A _ h u _ h , v _ h ) _ H = a ( u _ h , v _ h ) , \\forall u _ h , v _ h \\in V _ { h , \\sigma } . \\end{align*}"}
-{"id": "5402.png", "formula": "\\begin{align*} a c & = ( 1 , 3 , 9 , 8 , 5 , 4 ) ( 2 , 6 , 7 ) \\\\ b \\cdot c ^ { a c } & = ( 1 , 8 , 6 ) ( 2 , 4 , 9 ) ( 1 0 , 1 1 ) \\\\ a b \\cdot c ^ { a c } & = ( 1 , 9 , 6 , 4 , 8 , 2 ) ( 3 , 5 ) ( 1 0 , 1 1 ) \\\\ \\end{align*}"}
-{"id": "3997.png", "formula": "\\begin{align*} p ^ { \\nu _ 2 } _ { k - 1 } ( 2 , t ) = - ( - 1 ) ^ { k - 1 } \\underset { \\Lambda ^ { k - 1 } _ { 2 } } { \\sum } \\frac { \\lambda _ 1 ^ { k _ 1 + 1 } \\lambda _ 2 ^ { k _ 2 - 1 } t ^ { k _ 1 \\nu _ 1 + k _ 2 \\nu _ 2 } } { \\Gamma \\left ( k _ 1 \\nu _ 1 + k _ 2 \\nu _ 2 + 1 \\right ) } . \\end{align*}"}
-{"id": "5492.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { U ( r ^ n z ) } { ( r ^ n z ) ^ { \\rho } \\ell ( r ^ n z ) } = p ( z ) z \\in C _ { p } , p \\in \\mathcal { P } _ { r } . \\end{align*}"}
-{"id": "4669.png", "formula": "\\begin{align*} [ F ( x ) ] ( s ) : = \\begin{cases} \\int \\limits _ 0 ^ s \\ ; \\ ; x ( s - t ) \\ , x ( t ) \\ , d t , 0 \\le s \\le 1 , \\\\ \\int \\limits _ { s - 1 } ^ 1 \\ ; x ( s - t ) \\ , x ( t ) \\ , d t , 1 < s \\le 2 . \\end{cases} \\end{align*}"}
-{"id": "9846.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } \\frac { 2 } { 1 + t ^ 2 } e ^ { i x t } d t = e ^ { - | x | } . \\end{align*}"}
-{"id": "9394.png", "formula": "\\begin{align*} f ( x ) = - \\sum _ j \\mathbb D _ j f ( x ) \\ x \\in { \\Bbb R ^ n } . \\end{align*}"}
-{"id": "5186.png", "formula": "\\begin{align*} T ( w , \\bar { x } , \\bar { y } ) = \\left ( \\frac { f _ { 1 } ' ( \\bar { x } ) - g _ { 1 } ' ( \\bar { y } ) } { f _ { 1 } '' ( \\bar { x } ) ( \\bar { y } - \\bar { x } ) } \\right ) \\left ( \\frac { g _ { 2 } ' ( w ) - f _ { 2 } ' ( \\bar { y } ) } { f _ { 2 } '' ( \\bar { y } ) ( \\bar { y } - w ) } \\right ) . \\end{align*}"}
-{"id": "7335.png", "formula": "\\begin{align*} \\mathcal { A } _ 1 \\ ; & : = \\ ; \\widetilde { \\mathcal { A } } _ 1 \\cap \\mathcal { R } \\\\ \\mathcal { A } _ 2 \\ ; & : = \\ ; \\widetilde { \\mathcal { A } } _ 2 \\cap \\mathcal { R } \\ , , \\end{align*}"}
-{"id": "6902.png", "formula": "\\begin{align*} \\lambda _ { m + 1 } x _ { 0 } & = \\frac { 3 } { 2 } ( 2 x _ { 0 } - y _ { 0 , 1 } ( x _ { 0 } , x _ { 1 } , x _ { 2 } , \\lambda _ { m + 1 } , r ) - y _ { 0 , 2 } ( x _ { 0 } , x _ { 1 } , x _ { 2 } , \\lambda _ { m + 1 } , r ) ) \\\\ & + \\frac { 3 } { 2 } ( 2 x _ { 0 } - y _ { 2 , 0 } ( x _ { 1 } ' , x _ { 2 } ' , x _ { 0 } , \\lambda _ { m + 1 } , r ) - y _ { 2 , 1 } ( x _ { 1 } ' , x _ { 2 } ' , x _ { 0 } , \\lambda _ { m + 1 } , r ) ) \\end{align*}"}
-{"id": "2687.png", "formula": "\\begin{align*} \\widetilde { \\chi } : = \\chi ^ { - 1 } _ { \\iota } \\chi _ { \\iota \\circ c } \\chi ^ { - 1 } _ { \\mathrm { c y c } } . \\end{align*}"}
-{"id": "5100.png", "formula": "\\begin{align*} \\{ 0 \\} = \\mathcal { M } _ 0 \\subset \\mathcal { M } _ 1 \\subset \\mathcal { M } _ 2 \\subset \\cdots \\quad \\bigcup _ { n = 0 } ^ { \\infty } \\mathcal { M } _ n = \\mathcal { L } . \\end{align*}"}
-{"id": "6059.png", "formula": "\\begin{align*} F & = \\sup _ { ( \\alpha , \\theta ) \\in [ 0 , 1 ] \\times [ 0 , \\infty ) } F ^ { ( \\alpha , \\theta ) } ( R ) = 0 . \\end{align*}"}
-{"id": "1204.png", "formula": "\\begin{align*} \\{ u ^ * \\geq t \\} = \\{ u \\geq t \\} . \\end{align*}"}
-{"id": "4381.png", "formula": "\\begin{align*} \\Big \\| \\Big ( C _ { z _ \\gamma } ( A P _ \\alpha + Q _ \\alpha ) C _ { - z _ \\gamma } & - ( A _ { x } P _ \\alpha + Q _ \\alpha ) \\Big ) M _ { \\chi _ { B ( 0 , R ) } } \\Big \\| \\\\ & = \\Big \\| \\Big ( C _ { z _ \\gamma } A P _ \\alpha C _ { - z _ \\gamma } - A _ { x } P _ \\alpha ) M _ { \\chi _ { B ( 0 , R ) } } \\Big \\| \\\\ & = \\Big \\| ( \\widetilde { C } _ { z _ \\gamma } A \\widetilde { C } _ { - z _ \\gamma } - A _ { x } ) P _ \\alpha M _ { \\chi _ { B ( 0 , R ) } } \\Big \\| \\\\ & \\to 0 \\end{align*}"}
-{"id": "2645.png", "formula": "\\begin{align*} \\overrightarrow { E } ( D ) & = \\{ ( 1 , 2 ) , ( 2 , 3 ) , \\ldots , ( n - 1 , n ) , ( n , 1 ) \\} , \\\\ \\overleftarrow { E } ( D ) & \\subseteq \\{ ( n , n - 1 ) , ( n - 1 , n - 2 ) , \\ldots , ( 2 , 1 ) , ( 1 , n ) \\} . \\end{align*}"}
-{"id": "2760.png", "formula": "\\begin{align*} L ^ * _ f ( \\chi , s ) ^ { ( - 1 ) ^ { n - 1 } } = \\prod \\limits _ { j = 0 } ^ { n } C ^ * _ f ( \\chi , { p ^ { m ( f ) j } } s ) ^ { ( - 1 ) ^ j \\binom { n } { j } } . \\end{align*}"}
-{"id": "2738.png", "formula": "\\begin{align*} \\displaystyle \\epsilon ^ 2 \\partial _ t \\bigg ( | | \\partial ^ m g | | _ { L ^ 2 _ { x , v } } ^ 2 + \\sum _ { j = 0 } ^ { m - 1 } C _ { j m } \\ , | | \\partial ^ j g | | _ { L ^ 2 _ { x , v } } ^ 2 \\bigg ) \\leq \\begin{cases} - 2 \\lambda \\ , | | g ^ { \\perp } | | _ { \\Lambda } ^ 2 \\ , , m = 0 , \\\\ [ 4 p t ] - \\lambda \\ , | | \\partial ^ m g ^ { \\perp } | | _ { \\Lambda } ^ 2 \\ , , m \\geq 1 . \\end{cases} \\end{align*}"}
-{"id": "7999.png", "formula": "\\begin{align*} { \\Big ( \\frac { 1 } { | P | } \\int _ P { \\sum _ { n = \\mu } ^ { \\infty } { 2 ^ { s n q } \\Big ( \\sum _ { k = n - h } ^ { \\infty } { | \\Pi _ n f _ k ( x ) | } \\Big ) ^ q } } d x \\Big ) ^ { 1 / q } } . \\end{align*}"}
-{"id": "650.png", "formula": "\\begin{align*} | g - \\tilde g | & = | h ( F _ * , F _ * ) - h ( \\widetilde F _ * , \\widetilde F _ * ) | \\\\ & = | h ( F _ * , F _ * ) - h ( P \\widetilde F _ * , P \\widetilde F _ * ) | \\\\ & = | h ( P \\widetilde F _ * - F _ * , P \\widetilde F _ * ) + h ( F _ * , P \\widetilde F _ * - F _ * ) | \\\\ & \\le C ( | F _ * | + | P \\widetilde F _ * | ) | P \\widetilde F _ * - F _ * | . \\end{align*}"}
-{"id": "5776.png", "formula": "\\begin{align*} \\| u \\| _ { \\dot { W } ^ { 1 , p } _ { 0 } ( \\Omega ) } = \\| \\nabla u \\| _ { L ^ { p } ( \\Omega ) } . \\end{align*}"}
-{"id": "8359.png", "formula": "\\begin{align*} W = \\Q \\ell + \\Q \\ell _ * \\subset V . \\end{align*}"}
-{"id": "4087.png", "formula": "\\begin{align*} k _ { M , p } ( X ) = \\frac { \\langle d u ^ { - 1 } _ { \\eta ( p ) } X , d \\eta _ p X \\rangle } { \\langle d u ^ { - 1 } _ { \\eta ( p ) } X , X \\rangle } . \\end{align*}"}
-{"id": "4650.png", "formula": "\\begin{align*} B _ k ( i , j ) = \\sum _ { n = 0 } ^ { r _ { k , k + 1 } ( j ) - 1 } 1 _ { I _ { k , i } } \\Big ( ( T | I _ k ) ^ n I _ { k + 1 , j } \\Big ) \\end{align*}"}
-{"id": "7682.png", "formula": "\\begin{align*} C _ B ( j ) = \\left ( \\frac { 1 } { N } D ^ 2 _ B ( j ) \\right ) ^ { - 1 } \\ , . \\end{align*}"}
-{"id": "5141.png", "formula": "\\begin{align*} u = \\sum _ { \\l = 1 } ^ k \\zeta _ { 2 \\l - 1 } + v \\end{align*}"}
-{"id": "9097.png", "formula": "\\begin{align*} A ^ { \\ast } ( k x ) = k ^ { n } A ^ { \\ast } ( x ) \\left ( x \\in G \\right ) . \\end{align*}"}
-{"id": "8337.png", "formula": "\\begin{align*} \\ell _ * ^ { ( a ) } = k ^ { ( a ) } - Q ( k ^ { ( a ) } ) \\ell ^ { ( a ) } . \\end{align*}"}
-{"id": "8016.png", "formula": "\\begin{align*} a ( x , \\xi ) : = \\sum _ { k = 1 } ^ { \\infty } { \\widehat { { \\Lambda } } ( { \\xi } / { 2 ^ k } ) e ^ { 2 \\pi i \\langle \\mathbf { v } _ k , x \\rangle } } . \\end{align*}"}
-{"id": "8424.png", "formula": "\\begin{align*} \\sum _ { t = - \\infty } ^ { \\infty } q ^ { ( t + a ( \\mu \\pi ) ) ( \\frac { 1 } { 2 } - s ) } c _ { t , l } ( \\mu ) = \\zeta _ F ( 1 ) q ^ { - ( l - a ( \\mu ) ) ( \\frac { 3 } { 2 } - s ) - \\frac { a ( \\mu ) } { 2 } } \\epsilon ( \\frac { 1 } { 2 } , \\mu ) ^ { - 1 } . \\end{align*}"}
-{"id": "3981.png", "formula": "\\begin{align*} \\mathrm { P r } \\{ X _ 1 > t \\} = \\mathrm { P r } \\{ N _ 1 ( t , \\lambda ) = 0 \\} = E _ { \\alpha _ 0 } ( - \\lambda t ^ { \\alpha _ 0 } ) , \\end{align*}"}
-{"id": "5742.png", "formula": "\\begin{align*} \\phi _ { x , y , z } ( a , b ) \\phi _ { x , y - 1 , z - 1 } ( a , b ) = \\phi _ { x , y - 1 , z } ( a , b ) \\phi _ { x , y , z - 1 } ( a , b ) + \\phi _ { x + 1 , y - 1 , z - 1 } ( a , b ) \\phi _ { x - 1 , y , z } ( a , b ) . \\end{align*}"}
-{"id": "8812.png", "formula": "\\begin{align*} D _ { p } = \\sum _ { j = 1 } ^ { r } m _ { j } ^ { \\beta _ { i _ { 0 } } } p ^ { ( \\sigma - \\lambda _ { i _ { 0 } } ) m _ { j } } D _ { j , i _ { 0 } , p } = \\sum _ { j = 1 } ^ { r } m _ { j } ^ { \\beta _ { i _ { 0 } } } D _ { j , i _ { 0 } , p } . \\end{align*}"}
-{"id": "9553.png", "formula": "\\begin{align*} { ( - 1 ) ^ { \\ell ( \\check \\sigma ) } x _ 1 ^ { o ( \\check \\sigma ) } x _ 2 ^ { e ( \\check \\sigma ) } y ^ { o ( \\check \\sigma ) } z ^ { e ( \\check \\sigma ) } } = x _ 1 z ^ { \\lfloor \\frac { n - 1 } { 2 } \\rfloor } { ( - 1 ) ^ { \\ell ( \\sigma ) } x _ 1 ^ { e ( \\sigma ) } x _ 2 ^ { o ( \\sigma ) } y ^ { o ( \\sigma ) } z ^ { e ( \\sigma ) } } \\end{align*}"}
-{"id": "9307.png", "formula": "\\begin{align*} D : = \\{ ( h , t , x ) : h \\geq 0 , ( t , x ) \\in W , t + k h \\leq b \\} . \\end{align*}"}
-{"id": "6545.png", "formula": "\\begin{align*} \\langle y _ 0 , \\xi ' \\rangle = \\langle y _ 0 , \\xi + \\xi ' \\rangle - \\langle y _ 0 , \\xi \\rangle \\leq 2 - 1 = 1 . \\end{align*}"}
-{"id": "600.png", "formula": "\\begin{align*} \\bigcup _ { j = 1 } ^ { n } \\{ g _ { j } \\gamma _ { k } ^ { n ( \\gamma _ { k } ) } g _ { j } ^ { - 1 } : k \\in \\mathbb { N } \\} \\end{align*}"}
-{"id": "482.png", "formula": "\\begin{align*} p ^ { ( m ) } _ { 1 , k _ 1 , 0 } ( x , t ) = \\int _ \\R p ^ { ( m + 1 ) } _ { 1 , k _ 1 , 0 } ( x , ( t , t _ { m + 1 } ) ) \\ , \\dd t _ { m + 1 } , \\end{align*}"}
-{"id": "2476.png", "formula": "\\begin{align*} w ( x , t ) = e ^ { \\frac { \\left \\langle x \\right \\rangle - M t } { D } } , \\end{align*}"}
-{"id": "321.png", "formula": "\\begin{align*} b _ i - a _ j \\ , \\not \\in \\ , \\Z _ { \\le 0 } , i , j = 1 , 2 . \\end{align*}"}
-{"id": "1520.png", "formula": "\\begin{align*} \\displaystyle \\alpha _ { m , t } ( x ) : = \\frac { m t + t + x } { 2 ( m + 2 ) } \\end{align*}"}
-{"id": "7541.png", "formula": "\\begin{gather*} u = \\frac { \\varphi ^ 1 ( \\omega ) } y , v = \\frac { \\varphi ^ 2 ( \\omega ) } y \\quad \\mbox { w i t h } \\omega = \\frac x y . \\end{gather*}"}
-{"id": "9591.png", "formula": "\\begin{align*} & 2 ^ { k \\alpha } = 2 ^ { ( k - k _ 0 ) \\alpha } 2 ^ { ( k _ 0 - \\ell _ 0 ) \\alpha } 2 ^ { ( \\ell _ 0 - k _ 1 ) \\alpha } \\cdots 2 ^ { ( \\ell _ { m - 1 } - k _ m ) \\alpha } 2 ^ { ( k _ m - \\ell _ m ) \\alpha } 2 ^ { ( \\ell _ m - k ' - j ) \\alpha } 2 ^ { ( k ' + j ) \\alpha } \\end{align*}"}
-{"id": "7451.png", "formula": "\\begin{align*} y e _ 4 = a ^ * a \\gamma + \\gamma a ^ * a - t \\gamma z e _ 4 = a ^ * a \\beta + \\beta a ^ * a - t \\beta . \\end{align*}"}
-{"id": "5220.png", "formula": "\\begin{align*} \\mathrm { R e } \\sqrt { c ^ 2 + 4 ( \\lambda - \\nu _ i ) } \\geq \\sqrt { \\mathrm { R e } ( c ^ 2 + 4 ( \\lambda - \\nu _ i ) ) } > \\sqrt { c ^ 2 } = - c , \\end{align*}"}
-{"id": "9583.png", "formula": "\\begin{align*} \\Sigma _ s \\cong { \\bigwedge } ^ { s - 1 } \\Sigma _ s , \\end{align*}"}
-{"id": "1200.png", "formula": "\\begin{align*} | x | ^ { - n / 2 } \\ , \\left ( \\int _ { B ( x , | x | / 4 ) } | \\nabla \\bar e | ^ 2 \\ , d y \\right ) ^ { 1 / 2 } \\ , \\leq c \\ , | x | ^ { - 1 } \\max _ { B ( x , | x | / 2 ) } \\ , \\bar e = o \\left ( | x | ^ { \\frac { 1 - n } { p - 1 } } \\right ) \\ , \\ , \\mbox { a s } \\ , \\ , x \\to \\infty , \\end{align*}"}
-{"id": "5313.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\| \\widetilde { B } ( \\theta _ M ( t ) ) - B ( \\theta _ M ) \\| _ { 2 , \\Omega } ^ 2 \\mathrm { d t } \\leq C \\left ( \\tau ^ { 1 / \\ell + 1 / \\ell ' } + \\tau ^ { 1 / 2 + 1 / 2 } \\right ) = C \\tau , \\end{align*}"}
-{"id": "8720.png", "formula": "\\begin{align*} \\psi ( \\hat { s } , \\hat { y } ) - \\psi ( \\hat { s } - \\tau , \\hat { y } ) - \\varphi ( \\hat { s } , \\hat { y } ) + \\varphi ( \\hat { s } - \\tau , \\hat { y } ) & = \\lambda ' - \\psi ( \\hat { s } - \\tau , \\hat { y } ) + \\varphi ( \\hat { s } - \\tau , \\hat { y } ) \\\\ & \\le - \\psi ( \\hat { s } - \\tau , \\hat { y } ) + w ^ * ( \\hat { s } - \\tau , \\hat { y } ) \\\\ & \\le 0 \\end{align*}"}
-{"id": "7434.png", "formula": "\\begin{align*} x ' = a \\beta \\gamma a ^ * , \\ , \\ , y = a \\gamma a ^ * , \\ , \\ , z = a \\beta a ^ * \\end{align*}"}
-{"id": "45.png", "formula": "\\begin{align*} \\hat { f } _ { X ^ { 1 } , X ^ { 2 } , . . . X ^ { L } } ( x ^ { 1 } , x ^ { 2 } , . . . , x ^ { L } ) = \\frac { 1 } { N } \\sum \\limits _ { n = 1 } ^ N \\prod \\limits _ { l = 1 } ^ L G _ { \\sigma } ( x ^ { l } - x ^ { l } _ { n } ) \\end{align*}"}
-{"id": "2435.png", "formula": "\\begin{align*} \\min \\Delta ( H ) = \\gcd \\Delta ( H ) . \\end{align*}"}
-{"id": "6145.png", "formula": "\\begin{align*} H \\wr L : = \\bigoplus _ { i \\in L } H \\rtimes L , \\end{align*}"}
-{"id": "6743.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } | G ^ { * } | - N ^ { \\gamma } t = 0 . \\end{align*}"}
-{"id": "8351.png", "formula": "\\begin{align*} R _ \\Lambda ( m , \\mu ) = \\{ \\lambda \\in \\mu + \\Lambda : Q ( \\lambda ) = m \\} \\end{align*}"}
-{"id": "5768.png", "formula": "\\begin{align*} \\left ( - \\Delta \\right ) ^ { \\alpha } u = \\sigma u ^ { q } + \\mu \\ ; \\ ; \\mathbb { R } ^ n , \\end{align*}"}
-{"id": "7589.png", "formula": "\\begin{align*} f _ { K _ n , I } ( X ) = b _ { n , \\max I } ( X ) f _ { \\max I , I \\setminus \\{ \\max I \\} } ( X ) . \\end{align*}"}
-{"id": "7545.png", "formula": "\\begin{gather*} \\psi = C _ 1 + C _ 2 \\arctan \\omega \\quad \\mbox { i f } A = 0 , \\\\ [ 1 e x ] \\psi = C _ 1 e ^ { - \\alpha \\arctan \\omega } + C _ 2 e ^ { \\alpha \\arctan \\omega } \\quad \\mbox { w i t h } \\alpha = \\sqrt A \\quad \\mbox { i f } A > 0 , \\\\ [ 1 e x ] \\psi = C _ 1 \\cos ( \\beta \\arctan \\omega ) + C _ 2 \\sin ( \\beta \\arctan \\omega ) \\quad \\mbox { w i t h } \\beta = \\sqrt { - A } \\quad \\mbox { i f } A < 0 . \\end{gather*}"}
-{"id": "1283.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\nu _ j = \\nu \\mbox { w e a k l y w h e r e $ \\nu $ i s t h e c a p a c i t a r y m e a s u r e f o r $ E . $ } \\end{align*}"}
-{"id": "4966.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } P ( x ) \\Delta ( x ) \\ , d x = 0 . \\end{align*}"}
-{"id": "7506.png", "formula": "\\begin{align*} i \\varepsilon \\partial _ t \\ , \\mathcal { U } ^ * ( t ; 0 ) \\ , \\omega _ { N , t } ^ k \\ , \\mathcal { U } ( t ; 0 ) = - \\mathcal { U } ^ * ( t ; 0 ) \\left [ V * ( \\rho _ t - \\rho _ t ^ k ) , \\omega _ { N , t } ^ k \\right ] \\mathcal { U } ( t ; 0 ) , \\end{align*}"}
-{"id": "7330.png", "formula": "\\begin{align*} \\left \\langle - J \\left ( \\sum _ { i = 1 } ^ p R ^ \\perp ( e _ i , f _ i ) ( V ) \\right ) , V \\right \\rangle \\geq c _ 0 \\ , | V | ^ 2 \\end{align*}"}
-{"id": "6931.png", "formula": "\\begin{align*} H ^ 1 _ d = H ^ 2 _ d \\odot H ^ 2 _ d , \\end{align*}"}
-{"id": "6218.png", "formula": "\\begin{align*} \\P \\big ( \\sup _ { 0 \\leq s \\leq t } \\widehat { X } ( s ) > 0 ~ | ~ \\widehat { X } ( t ) \\big ) = 1 - \\big ( - \\frac { \\widehat { X } ( t ) } { c t } \\big ) ~ . \\end{align*}"}
-{"id": "6245.png", "formula": "\\begin{align*} \\left [ { \\bf I } _ { N _ { \\rm T } } \\ , \\ , { \\bf 0 } \\right ] \\bar { \\bf H } _ { \\rm b } ^ H & = { \\bf I } _ { N _ { \\rm T } } \\\\ \\left [ { \\bf I } _ { N _ { \\rm T } } \\ , \\ , { \\bf 0 } \\right ] { \\boldsymbol \\Lambda } _ { \\rm b } \\left ( \\mu _ { \\rm b } \\right ) & = \\mu _ { \\rm b } \\left [ { \\bf I } _ { N _ { \\rm T } } \\ , \\ , { \\bf 0 } \\right ] = \\mu _ { \\rm b } \\left ( \\bar { \\bf H } _ { \\rm b } - \\left [ { \\bf 0 } _ { N _ { \\rm T } } \\ , \\ , \\hat { \\bf h } _ { \\rm b } \\right ] \\right ) . \\end{align*}"}
-{"id": "584.png", "formula": "\\begin{align*} M _ k = \\underset { x \\in I } { \\sup } \\left | f ^ { ( k ) } ( x ) \\right | , \\end{align*}"}
-{"id": "6223.png", "formula": "\\begin{align*} \\P ( \\widehat { \\tau } _ 0 < \\infty , & \\widehat { X } ( \\widehat { \\tau } _ 0 - 1 ) = y , \\widehat { X } ( \\widehat { \\tau } _ 0 ) \\geq x , \\textrm { t h e n e w s u p r e m u m w a s c a u s e d b y t h e p r o c e s s $ C $ } ) \\\\ & = \\P ( C ( 1 ) \\geq x + 1 - y ) \\cdot \\P ( Z ( 1 ) = - 1 ) + \\P ( C ( 1 ) \\geq x - y ) \\cdot \\P ( Z ( 1 ) \\geq 0 ) ~ . \\end{align*}"}
-{"id": "9684.png", "formula": "\\begin{align*} { U } _ 2 ^ { ( 0 ) } = ( { u } _ 2 ^ { ( 0 ) } , 0 , { p } _ 2 ^ { ( 0 ) } , { \\rho } _ 2 ^ { ( 0 ) } , 0 ) ^ \\top = ( { u } _ 1 ^ { ( 0 ) } e ^ { \\sigma _ { 2 0 } } , 0 , { p } _ 1 ^ { ( 0 ) } , { \\rho } _ 1 ^ { ( 0 ) } e ^ { \\sigma _ { 3 0 } } , 0 ) ^ \\top . \\end{align*}"}
-{"id": "1785.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d \\omega } \\| R ( \\omega _ 0 , \\cdot ) \\| _ 2 ^ 2 = ( L ^ + ( S _ 0 ) , R _ 0 ) _ 2 . \\end{align*}"}
-{"id": "6160.png", "formula": "\\begin{align*} ( Q _ { i j } ) = \\frac { 1 } { ( A _ 1 A _ 2 A _ 3 ) ^ \\frac { 1 } { 3 } } \\left ( \\begin{array} { c c c } A _ 1 & 0 & 0 \\\\ 0 & A _ 2 & 0 \\\\ 0 & 0 & A _ 3 \\end{array} \\right ) = \\left ( \\begin{array} { c c c } f _ 1 & 0 & 0 \\\\ 0 & f _ 2 & 0 \\\\ 0 & 0 & f _ 3 \\end{array} \\right ) \\end{align*}"}
-{"id": "9277.png", "formula": "\\begin{align*} \\mathbb { E } \\big ( f _ m \\circ \\theta ^ { ( s _ m , s _ { m - 1 } ) } \\cdots f _ 1 \\circ \\theta ^ { ( s _ 2 , s _ { 1 } ) } \\big ) = \\mathbb { E } \\big ( f _ m \\circ \\theta ^ { ( s _ m , s _ { m - 1 } ) } \\big ) \\cdots \\mathbb { E } \\big ( f _ 1 \\circ \\theta ^ { ( s _ 2 , s _ { 1 } ) } \\big ) \\end{align*}"}
-{"id": "5870.png", "formula": "\\begin{align*} A _ 0 = 1 , \\ , A _ 4 = 5 , \\ , A _ 8 = 2 5 0 , \\ , A _ { 1 2 } = 2 5 0 , \\ , A _ { 1 6 } = 5 , \\ , A _ { 2 0 } = 1 . \\end{align*}"}
-{"id": "571.png", "formula": "\\begin{align*} H _ { F } ^ { \\infty } ( \\Omega ) = \\{ f : \\Omega \\rightarrow \\mathbb { C } \\ : f ^ { ( l ) } \\ \\ \\Omega , \\ \\ l \\in F \\} , \\end{align*}"}
-{"id": "7458.png", "formula": "\\begin{align*} \\begin{aligned} & a a ^ * a = - a \\beta ^ 3 , a ^ * a a ^ * = - \\beta ^ 3 a ^ * , \\beta ^ 3 = \\gamma ^ 3 , \\\\ & a ^ * a = ( \\beta + \\gamma ) ^ 2 - \\beta ^ 3 , ( \\beta + \\gamma ) a ^ * = 0 , a ( \\beta + \\gamma ) = 0 . \\end{aligned} \\end{align*}"}
-{"id": "1740.png", "formula": "\\begin{align*} [ ( f , \\mu ) ] : = f \\ , \\sqrt { d \\mu } . \\end{align*}"}
-{"id": "3405.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\frac { 1 } { 2 \\pi } u _ j d \\theta = \\delta _ { \\pi / 2 } + \\delta _ { - \\pi / 2 } \\end{align*}"}
-{"id": "3012.png", "formula": "\\begin{align*} u = \\begin{bmatrix} u _ 1 \\\\ \\vdots \\\\ u _ N \\\\ \\end{bmatrix} v = \\begin{bmatrix} v _ 1 \\\\ \\vdots \\\\ v _ N \\\\ \\end{bmatrix} , \\end{align*}"}
-{"id": "6520.png", "formula": "\\begin{align*} \\Delta \\mathcal { C } ^ { 2 } \\overset { } { = } \\left [ \\mathcal { C } ^ { } \\right ] ^ { 2 } - \\left [ \\mathcal { C } ^ { } \\right ] ^ { 2 } \\mathcal { C } _ { } ^ { 2 } \\overset { } { = } \\left [ \\mathcal { C } ^ { } \\right ] ^ { 2 } + \\left [ \\mathcal { C } ^ { } \\right ] ^ { 2 } \\end{align*}"}
-{"id": "2918.png", "formula": "\\begin{align*} G [ \\ast , s ] : = \\bigcup _ { s ' \\in S } G [ s ' , s ] \\ ; \\mbox { a n d } \\ ; G [ s , \\ast ] : = \\bigcup _ { s ' \\in S } G [ s , s ' ] , \\end{align*}"}
-{"id": "5273.png", "formula": "\\begin{align*} & a _ 1 u ( t , 1 ) + a _ 2 u _ x ( t , 1 ) = 0 , \\\\ & b _ 1 u ( t , 0 ) + b _ 2 u _ x ( t , 0 ) = d _ 1 ( t ) , \\\\ & u ( 0 , x ) = u _ 0 ( x ) , \\end{align*}"}
-{"id": "7880.png", "formula": "\\begin{align*} u _ 0 ( x ) = x _ 3 \\end{align*}"}
-{"id": "9160.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } a _ { n + 1 - i } x ^ { i } A \\left ( x ^ { n + 1 - i } \\right ) = 0 \\left ( x \\in K \\right ) . \\end{align*}"}
-{"id": "8287.png", "formula": "\\begin{align*} U _ \\Phi ( \\C ) / ( g K _ \\Phi g ^ { - 1 } \\cap U _ \\Phi ( \\Q ) ) = U _ \\Phi ( \\C ) / \\mathrm { r a t } ( \\bar { \\nu } _ \\Phi ( \\bar { g } ) ) \\cdot \\Gamma _ \\Phi \\end{align*}"}
-{"id": "2953.png", "formula": "\\begin{align*} \\norm { \\partial \\Theta ^ \\Delta _ \\epsilon - \\partial \\Theta ^ \\Delta _ { \\epsilon ' } } _ 2 ^ 2 + \\epsilon D ( \\Theta ^ \\Delta _ { \\epsilon ' } \\Vert \\Theta ^ \\Delta _ { \\epsilon } ) + \\epsilon ' D ( \\Theta ^ \\Delta _ \\epsilon \\Vert \\Theta ^ \\Delta _ { \\epsilon ' } ) = ( \\epsilon - \\epsilon ' ) ( H ( \\Theta ^ \\Delta _ \\epsilon ) - H ( \\Theta ^ \\Delta _ { \\epsilon ' } ) ) . \\end{align*}"}
-{"id": "2064.png", "formula": "\\begin{align*} \\mathcal { L } = \\sum _ { i = 1 } ^ k Y _ i ^ 2 + Y _ 0 . \\end{align*}"}
-{"id": "4282.png", "formula": "\\begin{align*} f ( z , \\tau ) = \\sum _ { d } D _ { \\zeta _ 1 } ^ { d _ 1 } \\cdots D _ { \\zeta _ n } ^ { d _ n } f _ d ( z , \\tau ) . \\end{align*}"}
-{"id": "3276.png", "formula": "\\begin{align*} \\ ! \\ ! R ^ c ( u , k ) = \\frac { 1 } { v _ u t _ u ^ c } \\int _ { r _ { u , k } ( \\boldsymbol { x } ) } ^ { r _ { u , k } ( \\boldsymbol { x } ' ) } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! w \\log \\left ( 1 + \\frac { \\beta P _ t \\psi r _ { u , k } ^ { - \\alpha } } { w N _ 0 } \\right ) d r _ { u , k } , \\end{align*}"}
-{"id": "958.png", "formula": "\\begin{align*} \\widetilde { \\Xi } _ n ( \\theta ) = \\left ( \\begin{array} { c c } O & \\Xi _ n ( \\theta ) \\\\ \\Xi _ n ( \\theta ) ^ \\top & O \\end{array} \\right ) . \\end{align*}"}
-{"id": "1364.png", "formula": "\\begin{align*} \\max _ { t \\in [ 0 , 1 ] } \\psi ( t ; c , a , b , c _ 0 ) = \\max \\Big \\{ \\psi ( 0 ; c , a , b , c _ 0 ) , \\ , \\psi ( 1 ; c , a , b , c _ 0 ) \\Big \\} \\ , . \\end{align*}"}
-{"id": "1318.png", "formula": "\\begin{align*} \\rho ( S _ e ) = \\begin{bmatrix} \\pi ( S _ e ) & X _ e \\\\ Y _ e & Z _ e \\end{bmatrix} e \\in E . \\end{align*}"}
-{"id": "3052.png", "formula": "\\begin{align*} \\begin{cases} \\lambda ( \\phi , \\zeta ) _ H + a ( \\phi , \\zeta ) + b ( \\phi , \\eta ) = ( \\phi , w - w _ h ) _ H , & \\forall \\phi \\in V , \\\\ b ( \\zeta , \\psi ) = 0 , & \\forall \\psi \\in Q . \\end{cases} \\end{align*}"}
-{"id": "692.png", "formula": "\\begin{align*} \\Psi ( \\mu ) : = \\psi ( 0 , \\mu ) \\cot \\alpha + \\psi ' ( 0 , \\mu ) = 0 . \\end{align*}"}
-{"id": "4039.png", "formula": "\\begin{align*} ( 1 - c \\lambda _ 1 ) V _ 1 = d g _ p V _ 1 \\ \\ \\mathrm { a n d } \\\\ ( 1 - c \\lambda _ 2 ) V _ 2 = d g _ p V _ 2 , \\end{align*}"}
-{"id": "9412.png", "formula": "\\begin{align*} a _ I = a _ { \\tau ' \\cdot I } I \\in N , \\end{align*}"}
-{"id": "5118.png", "formula": "\\begin{align*} \\left . \\frac { d \\mu } { d t } \\right | _ q \\lesssim C _ \\mu \\left [ \\sum _ { j _ 1 , \\ldots , j _ Q = 1 } ^ d \\left | \\sum _ { i _ 1 , \\ldots , i _ Q = 1 } ^ d { \\mathcal A } _ { q } ( M _ { j _ 1 i _ 1 } \\partial _ { t _ { i _ 1 } } , \\ldots , M _ { j _ Q i _ Q } \\partial _ { t _ { i _ Q } } ) \\right | ^ 2 \\right ] ^ { \\frac { d } { 2 Q } } \\end{align*}"}
-{"id": "628.png", "formula": "\\begin{align*} c _ { ( 0 , \\alpha _ 2 ) } ^ 2 = C \\left ( \\tilde I ( 2 \\alpha _ 2 + 1 ) \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "4405.png", "formula": "\\begin{align*} \\lim _ { | z | \\to \\infty } \\operatorname { O s c } _ z ^ r ( f ) = 0 r > 0 \\Leftrightarrow \\lim _ { | z | \\to \\infty } \\operatorname { O s c } _ z ^ r ( f ) = 0 r > 0 . \\end{align*}"}
-{"id": "2552.png", "formula": "\\begin{align*} h ( \\Omega ) : = \\inf \\left \\{ P ( A ) / | A | \\ , : \\ A \\subset \\Omega , \\ | A | > 0 \\right \\} \\ , . \\end{align*}"}
-{"id": "9921.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\phi ( z _ 1 ^ 2 ) & = & c _ 1 z _ 1 + c _ 0 z _ 2 \\\\ \\phi ( z _ 1 z _ 2 ) & = & - c _ 2 z _ 1 - c _ 1 z _ 2 \\\\ \\phi ( z _ 2 ^ 2 ) & = & c _ 3 z _ 1 + c _ 2 z _ 2 . \\end{array} \\end{align*}"}
-{"id": "333.png", "formula": "\\begin{align*} \\phi ( x ) : = - \\log \\Phi ( x ) \\end{align*}"}
-{"id": "5425.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to + \\infty } \\int _ \\Omega q u _ 2 u _ 1 d x = 0 . \\end{align*}"}
-{"id": "8164.png", "formula": "\\begin{align*} a _ 1 a _ 2 ^ q a _ 5 ^ { q ^ 3 } + a _ 3 a _ 7 ^ q a _ 6 ^ { q ^ 3 } = 0 . \\end{align*}"}
-{"id": "7287.png", "formula": "\\begin{align*} B ' : = \\left ( \\begin{array} { c c c c c c c } \\ell _ { u + 1 , 2 } & \\ldots & \\ell _ { n - 1 , 2 } \\\\ \\ell _ { u + 1 , 3 } & \\ldots & \\ell _ { n - 1 , 3 } \\end{array} \\right ) \\end{align*}"}
-{"id": "813.png", "formula": "\\begin{align*} Q _ { 4 } ( x ) = 2 ^ 3 r , 2 \\nmid r \\end{align*}"}
-{"id": "8833.png", "formula": "\\begin{align*} \\| u \\| _ { Z ^ { s , b } _ T } : = \\| u \\| _ { Y ^ { s , b } _ T } + \\| \\Lambda u \\| _ { Y ^ { 0 , b } _ T } . \\end{align*}"}
-{"id": "509.png", "formula": "\\begin{align*} \\mathcal { R } _ { } \\ ! = \\ ! & \\bigcup _ { q \\in [ 0 , 0 . 5 ] } \\ ! \\Big \\{ \\left ( R _ s , R _ \\ell , R _ w \\right ) \\ ! \\colon \\ ! \\\\ & 0 \\leq R _ s \\leq 1 - H _ b ( q * p _ A ) , \\\\ & R _ \\ell \\geq H _ b ( q * p _ A ) - H _ b ( q ) , \\\\ & R _ w \\geq H _ b ( q * p _ A ) - H _ b ( q ) \\Big \\} . \\end{align*}"}
-{"id": "9387.png", "formula": "\\begin{align*} V _ \\rho ( \\mathcal F ) ( x ) = \\| \\{ F _ t ( x ) \\} _ { t \\in \\mathbb { R } } \\| _ { V _ \\rho } , \\rho \\ge 1 . \\end{align*}"}
-{"id": "430.png", "formula": "\\begin{align*} - R - \\pi \\abs { t } + \\kappa q _ \\delta ( s ) = i R \\ , \\phi _ \\omega ( \\pi i ( 1 - \\delta e ^ { - s } ) ) \\end{align*}"}
-{"id": "1455.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\{ y _ 1 \\} = \\{ x _ { 4 1 } x _ { 4 2 } \\} , & \\{ z _ 1 \\} = \\{ x _ { 4 1 } , x _ { 4 2 } \\} , \\\\ \\{ y _ 2 \\} = \\emptyset , & \\{ z _ 2 \\} = \\{ x _ { 4 1 } ^ 2 x _ { 4 2 } + x _ { 4 1 } x _ { 4 2 } ^ 2 \\} , & \\{ e _ 2 \\} = \\{ x _ { 4 1 } ^ 2 , x _ { 4 2 } ^ 2 \\} , \\\\ \\{ y _ 3 \\} = \\emptyset , & \\{ z _ 3 \\} = \\emptyset , & \\{ e _ 3 \\} = \\{ x _ { 4 1 } ^ 2 x _ { 4 2 } ^ 4 + x _ { 4 1 } ^ 4 x _ { 4 2 } ^ 2 \\} . \\end{array} \\end{align*}"}
-{"id": "583.png", "formula": "\\begin{align*} \\mathcal { A } _ { F } ( \\Omega ) = \\{ f \\in H ( \\Omega ) : f ^ { ( l ) } \\in \\mathcal { A } ( \\Omega ) , \\ \\ l \\in F \\} . \\end{align*}"}
-{"id": "9773.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int _ { - 1 } ^ { 1 } \\tilde { E } _ { i } ( h , \\theta ) d \\theta _ i = 0 . \\end{align*}"}
-{"id": "2325.png", "formula": "\\begin{align*} \\Phi ( r , a , y , s ) = \\begin{cases} y ^ { 1 + r - 2 s } \\cfrac { \\left ( ( - 1 ) ^ r + 1 \\right ) \\Gamma \\left ( \\frac { r + 1 } { 2 } \\right ) \\Gamma \\left ( s - \\frac { r + 1 } { 2 } \\right ) } { 2 \\Gamma ( s ) } & \\\\ [ 4 m m ] \\cfrac { 2 \\pi ^ s } { i ^ r \\Gamma ( s ) } y ^ { \\frac 1 2 - s } \\cfrac { \\partial ^ r } { \\partial a ^ r } \\left ( | a | ^ { s - \\frac 1 2 } K _ { s - \\frac 1 2 } \\big ( 2 \\pi | a | y \\big ) \\right ) & \\end{cases} \\end{align*}"}
-{"id": "2712.png", "formula": "\\begin{align*} \\Pi _ { \\mathcal L } ( h ) = \\sum _ { i = 1 } ^ { n } \\ , \\left ( \\int _ { \\mathbb R ^ d } h \\varphi _ i \\ , d v \\right ) \\varphi _ i , \\end{align*}"}
-{"id": "6132.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\sum _ { i = 1 } ^ { M _ { 2 } \\left ( n \\right ) } a _ { n } W _ { i } \\leq \\frac { \\epsilon } { 2 } \\right ) & < \\frac { \\epsilon } { 8 } , \\\\ \\mathbb { P } \\left ( \\sum _ { i = 1 } ^ { M _ { 2 } \\left ( n \\right ) } a _ { n } U _ { i } \\leq \\frac { \\epsilon } { 2 } \\right ) & < \\frac { \\epsilon } { 8 } . \\end{align*}"}
-{"id": "1586.png", "formula": "\\begin{align*} V ( \\mathbf { j } ''' ) & = \\sum _ { k = 1 } ^ { n - 1 } [ ( n - k + 1 ) j _ k ''' - s ( j _ k ''' ) ] - c ( j _ n ''' , - \\alpha ( n ) - 1 ) \\\\ & = 3 \\cdot 1 - s ( 1 ) + \\frac { 2 ( m - 7 ) } { 3 } - s \\left ( \\frac { m - 7 } { 3 } \\right ) \\\\ & = \\frac { 2 ( m - 1 ) } { 3 } - 2 - s \\left ( \\frac { m - 1 } { 3 } - 2 \\right ) \\\\ & = \\frac { 2 ( m - 1 ) } { 3 } - s \\left ( \\frac { m - 1 } { 3 } \\right ) - 1 . \\end{align*}"}
-{"id": "4662.png", "formula": "\\begin{align*} F ( x ) = y \\ , \\end{align*}"}
-{"id": "1750.png", "formula": "\\begin{align*} \\nu _ { T ( x ) } ( Z ( \\lambda ) ) = \\langle ( S ^ { u n i v } _ \\lambda ( S ^ { u n i v } _ \\lambda ) ^ * ) T ( x ) , T ( x ) \\rangle = \\langle ( S ^ { u n i v } _ \\lambda ( S ^ { u n i v } _ \\lambda ) ^ * ) x , T ^ * T ( x ) \\rangle . \\end{align*}"}
-{"id": "4618.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { 2 ^ { k _ i } - 1 } 1 _ { [ - B - b , - B + b ] \\cup [ B - b , B + b ] } \\left ( t + \\sum _ { j = 0 } ^ { n - 1 } f ( T ^ j x ) \\right ) \\le 2 \\eta \\sum _ { n = 0 } ^ { 2 ^ { k _ i } - 1 } 1 _ { [ - B - b , B + b ] } \\left ( t + \\sum _ { j = 0 } ^ { n - 1 } f ( T ^ j x ) \\right ) \\end{align*}"}
-{"id": "5368.png", "formula": "\\begin{align*} h _ { r , 1 } = \\frac { ( r b - a ) ^ 2 - ( a - b ) ^ 2 } { 4 a b } \\end{align*}"}
-{"id": "5306.png", "formula": "\\begin{align*} \\int _ \\Omega \\sigma ( u ) \\nabla \\phi \\cdot \\nabla w \\mathrm { d x } = \\int _ \\Omega \\sigma ( u ) \\alpha _ \\mathrm { S } ( u ) \\nabla u \\cdot \\nabla w \\mathrm { d x } + \\int _ { \\Gamma _ \\mathrm { N } } g w \\mathrm { d s } , \\qquad \\forall w \\in V , \\end{align*}"}
-{"id": "7857.png", "formula": "\\begin{align*} h _ { 2 1 } ( x _ { 2 } | x _ { 1 } ) = \\frac { f _ { X _ { 2 } } ( x _ { 2 } ) \\left ( 1 - e ^ { - \\eta ^ { \\alpha } ( x _ { 2 } ) } \\right ) ^ { \\theta _ { 3 } } } { 1 - \\left ( 1 - e ^ { - \\eta ^ { \\alpha } ( x _ { 2 } ) } \\right ) ^ { \\theta _ { 2 } } } . \\end{align*}"}
-{"id": "9734.png", "formula": "\\begin{align*} \\tilde { \\gamma } _ { i } ( \\gamma _ { 5 } , \\sigma _ 3 , \\sigma _ 2 , \\gamma _ 1 , Z _ { a } h , Z _ { b } h , V _ a ) & = O ( 1 ) | Z _ a - Z _ b | h + \\tilde { \\gamma } _ { i } ( \\gamma _ { 5 } , \\sigma _ 3 , \\sigma _ 2 , \\gamma _ 1 , Z _ { a } h , Z _ { a } h , V _ a ) \\\\ & = O ( 1 ) | Z _ a - Z _ b | h + O ( 1 ) Z _ { a } h + \\tilde { \\gamma } _ { i } ( \\gamma _ { 5 } , \\sigma _ 3 , \\sigma _ 2 , \\gamma _ 1 , 0 , 0 , V _ a ) \\\\ & = \\gamma _ i + O ( 1 ) Z _ { a } h + O ( 1 ) | \\gamma _ 4 | h . \\end{align*}"}
-{"id": "9665.png", "formula": "\\begin{align*} u _ i > c _ i > 0 , \\ , \\ , 0 \\leq Z _ i \\leq 1 , \\ , \\ , \\lim \\limits _ { y \\to - \\infty } Z _ 1 ( y ) = 0 , \\end{align*}"}
-{"id": "4644.png", "formula": "\\begin{align*} \\sum _ { n = 2 ^ { k - 1 } } ^ { 2 ^ k - 1 } 1 _ { [ - B , B ] } \\left ( t + \\sum _ { i = 0 } ^ { n - 1 } h ( T ^ i x ) \\right ) \\ge \\beta \\tau \\sum _ { n = 0 } ^ { 2 ^ k - 1 } 1 _ { [ - B , B ] } \\left ( t + \\sum _ { i = 0 } ^ { n - 1 } h ( T ^ i x ) \\right ) \\end{align*}"}
-{"id": "6790.png", "formula": "\\begin{align*} ( w _ { \\lambda , k } - W _ { \\lambda , k } ) ( \\xi _ k ) = O ( \\lambda ^ 2 ) . \\end{align*}"}
-{"id": "4759.png", "formula": "\\begin{align*} \\alpha ^ 2 \\omega _ 1 = \\beta t _ 1 + \\gamma , \\mbox { $ \\alpha , \\beta , \\gamma $ - - c o n s t } . \\end{align*}"}
-{"id": "957.png", "formula": "\\begin{align*} \\mathbb { H } = \\left \\{ \\int _ { - \\infty } ^ \\infty f ( t ) d B ^ 1 _ t + \\int _ { - \\infty } ^ \\infty g ( t ) d B ^ 2 _ t : f , g \\in L ^ 2 ( \\mathbb { R } ) \\right \\} . \\end{align*}"}
-{"id": "3353.png", "formula": "\\begin{align*} \\lambda = \\frac { r + ( N - 1 ) } { N ( 1 - r ) } . \\end{align*}"}
-{"id": "6789.png", "formula": "\\begin{align*} w _ o ( y ) = 8 \\pi G ( y , \\xi _ k ) - 4 \\lambda ^ 2 \\ln { \\lambda } . \\end{align*}"}
-{"id": "8924.png", "formula": "\\begin{gather*} \\phi _ { d _ 0 , d _ 2 } \\circ \\phi _ { d _ 2 , d _ 3 } \\circ \\phi _ { d _ 3 , d _ 1 } \\circ \\phi _ { d _ 1 , d _ 0 } = \\phi _ { d _ 0 , d _ 3 } \\circ \\phi _ { d _ 3 , d _ 0 } = [ d _ 3 / d _ 0 ] , \\end{gather*}"}
-{"id": "4337.png", "formula": "\\begin{align*} ( P _ \\alpha f ) ( z ) = \\int _ { \\mathbb { C } ^ n } e ^ { \\alpha \\langle z , w \\rangle } f ( w ) d \\mu _ { \\alpha } ( w ) \\end{align*}"}
-{"id": "10058.png", "formula": "\\begin{align*} M _ \\mathfrak { p } ( s , \\phi _ \\mu ) = c _ \\mathfrak { p } \\cdot \\frac { L _ \\mathfrak { p } ( s + 1 , \\chi _ E ) } { L _ \\mathfrak { p } ( s , \\chi _ E ) } \\cdot W _ { 0 , \\mathfrak { p } } ( s , \\Phi _ \\mu ) \\end{align*}"}
-{"id": "1888.png", "formula": "\\begin{align*} \\phi _ 0 ( \\ 1 _ { \\hat { B } } ) = \\min _ { s \\in \\{ 0 , 1 \\} } \\Big \\{ s \\nu _ 1 ( F _ 1 ^ { i _ 2 - 1 } ) + s \\phi _ 0 ( \\ 1 _ { \\hat { B } \\setminus \\hat { S } } ) + \\eta ( \\ 1 _ { \\hat { B } } ) \\Big \\} . \\end{align*}"}
-{"id": "3469.png", "formula": "\\begin{align*} \\| Q _ h v - v \\| _ V = \\mathrm { d i s t } _ V ( v , V _ h ) , v \\in V . \\end{align*}"}
-{"id": "9036.png", "formula": "\\begin{align*} A H ( w , w ' ) = H ( B w , B w ' ) \\end{align*}"}
-{"id": "6232.png", "formula": "\\begin{align*} \\P ( \\widehat { \\tau _ 0 } < \\infty , \\widehat { X } ( \\widehat { \\tau _ 0 } - 1 ) = y , \\widehat { X } ( \\widehat { \\tau _ 0 } ) \\geq x , \\Delta C ^ i ( \\widehat { \\tau _ 0 } ) = x + 1 - y ) = \\P ( C ^ i ( 1 ) = x + 1 - y ) ~ . \\end{align*}"}
-{"id": "10029.png", "formula": "\\begin{align*} \\langle x , y \\rangle _ { h } = \\frac { 1 } { \\mathrm { r a t } ( \\nu ( h ) ) } \\cdot \\langle x , y \\rangle . \\end{align*}"}
-{"id": "5857.png", "formula": "\\begin{align*} \\theta ^ { X Y } _ { 1 2 } = \\theta ^ { X Y } _ { 2 1 } = 0 , \\end{align*}"}
-{"id": "7186.png", "formula": "\\begin{align*} \\langle z _ j , \\zeta \\rangle = 0 \\qquad \\zeta \\in T _ h \\mathfrak { H } _ { 1 + s } . \\end{align*}"}
-{"id": "8495.png", "formula": "\\begin{align*} - \\frac { b _ 1 } { \\varpi ^ { a _ 1 } } \\log _ F ( 1 + ( y + \\varpi ^ { \\kappa } x ) \\varpi ^ { a _ 1 - l } ) = \\sum _ { j \\geq 1 } \\frac { ( - 1 ) ^ { j } b _ 1 } { j } ( y + \\varpi ^ { \\kappa } x ) ^ j \\varpi ^ { ( j - 1 ) a _ 1 - j l } . \\end{align*}"}
-{"id": "612.png", "formula": "\\begin{align*} [ W ^ { \\otimes n } ] = \\sum _ { i = 0 } ^ n ( - 1 ) ^ i \\binom { n } { i } [ V ^ { \\otimes 2 ( n - i ) } ] . \\end{align*}"}
-{"id": "9101.png", "formula": "\\begin{align*} B ( x , y ) = A ( x y ) - A ( x ) y - x A ( y ) . \\end{align*}"}
-{"id": "7535.png", "formula": "\\begin{gather*} \\varphi ^ 1 \\varphi ^ 1 _ \\omega - \\varphi ^ 1 _ { \\omega \\omega } - \\frac { ( \\varphi ^ 2 ) ^ 2 } { \\omega } - \\frac 1 { \\omega ^ 3 } = 0 , \\\\ \\varphi ^ 1 \\varphi ^ 2 _ \\omega - \\varphi ^ 2 _ { \\omega \\omega } + \\frac { \\varphi ^ 1 \\varphi ^ 2 } { \\omega } + 2 \\frac { \\varphi ^ 2 } { \\omega ^ 2 } = 0 . \\end{gather*}"}
-{"id": "3098.png", "formula": "\\begin{align*} [ [ a _ 1 , & a _ 2 , \\dots , a _ n ] , b _ 1 , \\dots , b _ { n - 1 } ] - \\\\ & \\sum _ { i = 1 } ^ { n } [ [ b _ 1 , \\dots , b _ { i - 1 } , a _ { i } , b _ { i } , \\dots , b _ { n - 1 } ] , a _ 1 , \\dots , \\hat a _ { i } , \\dots , a _ n ] , \\end{align*}"}
-{"id": "9235.png", "formula": "\\begin{align*} ( i ) \\ ; \\zeta ( f , g ) = - \\zeta ( \\mathfrak { g } , f ) , \\quad ( i i ) \\ \\zeta ( [ f , g ] , h ) + \\zeta ( [ g , h ] , f ) + \\zeta ( [ h , f ] , g ) = 0 . \\end{align*}"}
-{"id": "6396.png", "formula": "\\begin{align*} \\Delta \\left ( x ^ { \\prime } \\beta \\right ) \\overset { } { = } \\int d \\theta \\exp \\left [ \\beta f \\left ( \\theta \\right ) \\right ] P _ { } \\left ( x ^ { \\prime } \\theta \\right ) \\end{align*}"}
-{"id": "47.png", "formula": "\\begin{align*} \\hat { V } ^ { C } _ { \\sigma } ( C _ { 1 } , C _ { 2 } ) = \\int \\limits _ { - \\infty } ^ { \\infty } \\int \\limits _ { - \\infty } ^ { \\infty } \\frac { 1 } { N } \\sum \\limits _ { n = 1 } ^ N G _ { \\sigma } ( u _ { 1 } - x _ { n } ) \\ , G _ { \\sigma } ( u _ { 1 } - y _ { n } ) \\ , G _ { \\sigma } ( u _ { 2 } - z _ { n } ) \\ , G _ { \\sigma } ( u _ { 2 } - s _ { n } ) \\mathrm { d } u _ { 1 } \\mathrm { d } u _ { 2 } \\end{align*}"}
-{"id": "4120.png", "formula": "\\begin{align*} X = X ( \\rho ) \\eta + \\rho D _ X \\eta + \\nabla _ X V + h ( X , V ) \\eta , \\end{align*}"}
-{"id": "5417.png", "formula": "\\begin{align*} R _ { \\ast } = H _ \\ast ( i _ e ^ { \\ast } X ; \\mathbb F _ 2 ) \\end{align*}"}
-{"id": "4838.png", "formula": "\\begin{align*} \\Psi & = - \\begin{vmatrix} 0 & H _ x & H _ y & H _ z \\\\ H _ x & a & h & g \\\\ H _ y & h & b & f \\\\ H _ z & g & f & c \\end{vmatrix} . \\end{align*}"}
-{"id": "9024.png", "formula": "\\begin{align*} ( \\delta g ) _ { s t } ( \\varphi ) = ( \\delta \\mu ) _ { s t } ( \\varphi ) + g _ s ( \\{ A ^ { 1 , * } _ { s t } + A ^ { 2 , * } _ { s t } \\} \\varphi ) + g ^ \\natural _ { s t } ( \\varphi ) , \\end{align*}"}
-{"id": "7313.png", "formula": "\\begin{align*} x _ j x _ i = \\left \\{ \\begin{aligned} & x _ i x _ j + t , & & j = 2 \\ell , i = 2 \\ell - 1 \\\\ & x _ i x _ j , & & . \\end{aligned} \\right . \\end{align*}"}
-{"id": "7116.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } F = & ~ \\Phi ' \\dot { F } ^ { k l } \\nabla _ k \\nabla _ l F + \\Phi '' \\dot { F } ^ { k l } \\nabla _ k F \\nabla _ l F \\\\ & \\quad + \\Phi \\left ( \\dot { F } ^ { i j } h _ i ^ k h _ { k j } + K _ N \\dot { F } ^ { i j } g _ { i j } \\right ) . \\end{align*}"}
-{"id": "323.png", "formula": "\\begin{align*} f ( n ) & : = f _ i ( ( n - i ) / 3 ) , \\\\ q ( n ) & : = q _ i ( ( n - i ) / 3 ) , \\mbox { f o r } n \\equiv i \\mod 3 , i = 0 , 1 , 2 . \\\\ r ( n ) & : = r _ i ( ( n - i ) / 3 ) . \\end{align*}"}
-{"id": "965.png", "formula": "\\begin{align*} \\xi _ j = \\sum _ { k = 1 } ^ N \\gamma ( j , k ) \\eta _ k = W _ 0 \\left ( \\sum _ { k = 1 } ^ N \\gamma ( j , k ) e _ k \\right ) , \\end{align*}"}
-{"id": "4893.png", "formula": "\\begin{align*} \\min ( \\norm B \\norm , \\norm C \\norm ) < 1 = \\max ( \\norm B \\norm , \\norm C \\norm ) . \\end{align*}"}
-{"id": "9498.png", "formula": "\\begin{align*} \\sigma _ { 2 n } ^ { ( 4 ) } = \\dfrac { 2 \\cdot 3 ^ n \\cdot ( 2 n ) ! } { n ! \\cdot ( n + 2 ) ! } \\end{align*}"}
-{"id": "6821.png", "formula": "\\begin{align*} \\norm { \\phi } _ { \\infty } \\leq C \\left [ \\norm { h } _ { \\ast } + \\sum \\limits _ { j = 1 } ^ 4 | c _ j | \\right ] , \\end{align*}"}
-{"id": "6710.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\mathbb { P } \\Big ( \\sigma ^ { \\eta } ( t ( N ) ) \\neq \\sigma ^ { \\varsigma } ( t ( N ) ) \\Big ) = 0 , \\end{align*}"}
-{"id": "6229.png", "formula": "\\begin{align*} C = C ^ 1 + C ^ 2 ~ . \\end{align*}"}
-{"id": "4036.png", "formula": "\\begin{align*} X - c D _ X \\eta = D _ X g + D _ X w , \\end{align*}"}
-{"id": "2096.png", "formula": "\\begin{align*} e ( \\psi ) = e _ H ( \\psi ) + e _ R ( \\psi ) . \\end{align*}"}
-{"id": "3171.png", "formula": "\\begin{align*} \\mathrm { d } Z _ { t } ^ { i } = - b Z _ { t } ^ { i } \\mathrm { d } t + \\sigma \\sqrt { Z _ { t } ^ { i } } \\mathrm { d } B _ { t } + \\mathrm { d } J _ { t } ^ { i } , t \\geqslant 0 , Z _ { 0 } ^ { i } = 0 , \\end{align*}"}
-{"id": "1983.png", "formula": "\\begin{align*} K _ { \\alpha } ^ { s } & = \\{ ( v , w ) \\in \\mathbb { R } ^ { k } \\oplus \\mathbb { R } ^ { d - k } : \\Vert w \\Vert < \\alpha \\Vert v \\Vert \\} ; \\\\ K _ { \\alpha } ^ { u } & = \\{ ( v , w ) \\in \\mathbb { R } ^ { k } \\oplus \\mathbb { R } ^ { d - k } : \\Vert v \\Vert < \\alpha \\Vert w \\Vert \\} . \\end{align*}"}
-{"id": "8931.png", "formula": "\\begin{gather*} ( a \\phi _ { d , d _ 1 } \\ , { \\times } \\ , b \\phi _ { d , d _ 2 } ) ^ * { \\cal P } _ { { \\cal E } _ d } \\ , { \\cong } \\ , ( 1 \\ , { \\times } \\ , a \\phi _ { d _ 1 , d } b \\phi _ { d , d _ 2 } ) ^ * { \\cal P } _ { { \\cal E } _ d } { = } ( 1 \\ , { \\times } \\ , a b c \\phi _ { d _ 1 , d _ 2 } ) ^ * { \\cal P } _ { { \\cal E } _ d } \\ , { \\cong } \\ , \\big ( ( 1 \\ , { \\times } \\ , \\phi _ { d _ 1 , d _ 2 } ) ^ * { \\cal P } _ { { \\cal E } _ d } \\big ) ^ { a b c } \\end{gather*}"}
-{"id": "5783.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\vert \\nabla u \\vert ^ { p - 2 } \\nabla u \\cdot \\nabla \\varphi \\ ; d x = \\int _ { \\Omega } \\varphi u ^ { q } \\ ; d \\sigma + \\int _ { \\Omega } \\varphi \\ ; d \\mu , \\varphi \\in C _ { 0 } ^ { \\infty } ( \\Omega ) . \\end{align*}"}
-{"id": "6468.png", "formula": "\\begin{align*} p _ { 3 D u } \\left ( x y | \\mu _ { x } \\sigma _ { x } \\sigma _ { y } \\right ) \\overset { } { = } \\frac { 1 } { 2 \\pi \\sigma _ { x } \\sigma _ { y } } \\exp \\left [ - \\frac { 1 } { 2 \\sigma _ { x } ^ { 2 } } \\left ( x - \\mu _ { x } \\right ) ^ { 2 } - \\frac { 1 } { 2 \\sigma _ { y } ^ { 2 } } y ^ { 2 } \\right ] , \\end{align*}"}
-{"id": "4098.png", "formula": "\\begin{align*} \\langle d u ^ { - 1 } _ { \\eta ( p ) } V _ 1 , V _ 2 \\rangle = \\langle V _ 1 , d u ^ { - 1 } _ { \\eta ( p ) } V _ 2 \\rangle = 0 , \\end{align*}"}
-{"id": "4214.png", "formula": "\\begin{align*} M _ 2 ( H _ 7 ) - M _ 2 ( H _ 8 ) & = \\sum \\limits _ { j = 1 } ^ { t } d _ { H _ 7 } ( v ) d _ { H _ 7 } ( v _ { j } ) + d _ { H _ 7 } ( v _ 0 ) d _ { H _ 7 } ( v _ 1 ) \\\\ & - \\sum \\limits _ { j = 3 } ^ { t } d _ { H _ 8 } ( v ) d _ { H _ 8 } ( v _ { j } ) - d _ { H _ 8 } ( v ) d _ { H _ 8 } ( v _ { 1 } ) - d _ { H _ 8 } ( v _ 0 ) d _ { H _ 8 } ( v _ 2 ) \\\\ & = \\sum \\limits _ { j = 3 } ^ { t } d _ { H _ 7 } ( v _ { j } ) + d _ { H _ 7 } ( v ) + ( d _ { H _ 7 } ( v ) - 2 ) d _ { H _ 7 } ( v _ 2 ) + 5 > 0 . \\end{align*}"}
-{"id": "8057.png", "formula": "\\begin{align*} J ^ T G _ w ( A _ 1 ^ r , \\dots , A _ m ^ r ) J & = G _ \\omega ( J ^ T A _ 1 ^ r J , \\dots , J ^ T A _ m ^ r J ) \\\\ & \\le G _ w ( \\alpha _ 1 ^ { 2 r } A _ 1 ^ { - r } , \\dots , \\alpha _ m ^ { 2 r } A _ m ^ { - r } ) \\\\ & = ( \\alpha _ 1 ^ { w _ 1 } \\cdots \\alpha _ m ^ { w _ m } ) ^ { 2 r } G _ w ( A _ 1 ^ r , \\dots , A _ m ^ r ) ^ { - 1 } , \\end{align*}"}
-{"id": "8958.png", "formula": "\\begin{gather*} D _ { w _ 0 w _ I w ^ { - 1 } } \\big ( \\vec { T } \\big ) + { } ^ { w _ 0 w _ I w ^ { - 1 } } D _ w \\big ( \\vec { T } \\big ) = D _ { w _ 0 W _ I } \\big ( \\vec { T } \\big ) \\end{gather*}"}
-{"id": "8725.png", "formula": "\\begin{align*} \\max _ { [ 0 , T ] \\times \\Omega } ( u ^ \\varepsilon - \\tilde { \\varphi } ) = ( u ^ \\varepsilon - \\tilde { \\varphi } ) ( t ' , \\hat { x } ) . \\end{align*}"}
-{"id": "5538.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } \\dot { z } _ { 1 } \\\\ \\dot { z } _ { 2 } \\end{array} \\right ] = \\left [ \\begin{array} { c c } 0 & 1 \\\\ - 1 & 0 \\end{array} \\right ] \\left [ \\begin{array} { c c } \\left ( \\alpha + \\beta p \\left ( t \\right ) \\right ) & 0 \\\\ 0 & 1 \\end{array} \\right ] \\left [ \\begin{array} { c } z _ { 1 } \\\\ z _ { 2 } \\end{array} \\right ] \\end{align*}"}
-{"id": "7480.png", "formula": "\\begin{align*} d \\mathcal { V } = \\dfrac { 1 } { ( 2 m ) ! } \\Phi ^ { 2 m } = i ^ { 2 m ^ 2 } h ^ 2 \\ : \\mathcal { Z } \\wedge \\bar { \\mathcal { Z } } \\wedge \\delta \\mathcal { V } \\wedge \\overline { \\delta \\mathcal { V } } , \\end{align*}"}
-{"id": "3921.png", "formula": "\\begin{align*} \\begin{aligned} 3 ^ { l _ 1 + 1 } x ^ { r + 6 } & + a ( r ) x ^ { 1 5 } + a ( r ) x ^ { 3 3 } + 3 ^ { 2 l _ 1 + 1 } x ^ { 2 r + 3 } \\\\ & + 2 \\cdot 3 ^ { l _ 1 + 1 } a ( r ) x ^ { 1 8 + r } + 2 \\cdot 3 ^ { l _ 1 + 1 } b ( r ) x ^ { 3 6 + r } \\in \\langle F \\rangle . \\end{aligned} \\end{align*}"}
-{"id": "3637.png", "formula": "\\begin{align*} A _ { \\lambda _ 0 + \\tau } & = \\{ x \\in \\Omega _ { \\lambda _ 0 + \\tau } \\setminus K \\ : \\ | \\nabla u _ { \\lambda _ 0 + \\tau } ( x ) | < \\dot C | \\nabla u ( x ) | \\} , \\\\ B _ { \\lambda _ 0 + \\tau } & = \\{ x \\in \\Omega _ { \\lambda _ 0 + \\tau } \\setminus K \\ : \\ | \\nabla u _ { \\lambda _ 0 + \\tau } ( x ) | \\geq \\dot C | \\nabla u ( x ) | \\} , \\end{align*}"}
-{"id": "5147.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ 3 \\| u _ j ( 0 ) \\| _ { H ^ s } \\sim \\l ^ { \\frac { 9 } { 2 } + s } L ^ s N ^ s . \\end{align*}"}
-{"id": "4222.png", "formula": "\\begin{align*} d _ z ( f ^ n ) & \\le d ^ { n - 1 } d _ z ( f ) + \\sum _ { k = 1 } ^ { n - 1 } d ^ { n - 1 - k } \\frac { d } { 2 } \\\\ & \\le d ^ { n - 1 } ( \\frac { d } { 2 } - 1 ) + \\frac { d ^ n } { 2 } \\frac { 1 } { d - 1 } ( 1 - \\frac { 1 } { d ^ { n - 1 } } ) \\\\ & < \\frac { d ^ n } { 2 } . \\end{align*}"}
-{"id": "7983.png", "formula": "\\begin{align*} | J _ s ^ T ( t ) | = | t a _ 3 \\cos \\beta _ s - ( 1 - t ) a _ 2 \\cos \\alpha _ s | . \\end{align*}"}
-{"id": "7634.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\int _ { Q _ 2 } { | D w ^ k - D v ^ k | ^ p d z } = \\lim _ { k \\to \\infty } \\int _ { Q _ 2 } { | D m ^ k | ^ p d z } = 0 . \\end{align*}"}
-{"id": "8726.png", "formula": "\\begin{align*} J [ u ^ \\varepsilon ] ( t ' , \\hat { x } ) + K _ { ( 0 , t ' ) } [ u ^ \\varepsilon ] ( t ' , \\hat { x } ) = \\frac { \\alpha } { \\Gamma ( 1 - \\alpha ) } \\int _ 0 ^ \\infty ( u ^ \\varepsilon ( t ' , \\hat { x } ) - \\bar { u } ^ \\varepsilon ( t ' - \\tau , \\hat { x } ) ) \\frac { d \\tau } { \\tau ^ { \\alpha + 1 } } , \\end{align*}"}
-{"id": "2453.png", "formula": "\\begin{align*} c _ r ( k ) = \\binom { 4 3 4 } { 2 8 9 } \\binom { 1 0 1 } { k - 2 9 0 } \\end{align*}"}
-{"id": "1850.png", "formula": "\\begin{align*} X _ m = { \\rm K e r } ( F ^ 0 \\to F ^ { - 1 } ) _ m = Z _ 0 ( F _ { \\bullet , m } ) \\end{align*}"}
-{"id": "4437.png", "formula": "\\begin{align*} \\tilde \\psi _ { 2 T } = { 2 ^ { 2 / 3 } } \\tilde \\psi _ T * \\psi _ T . \\end{align*}"}
-{"id": "4829.png", "formula": "\\begin{align*} _ { G _ 0 } ( \\theta ( a ) ) = _ { G _ 0 } ( \\theta ( a ' ) ) \\end{align*}"}
-{"id": "2868.png", "formula": "\\begin{align*} \\frac { y } { h } = \\frac { y - x ^ * _ 2 } { x ^ * _ 1 } \\Rightarrow h = \\frac { y x ^ * _ 1 } { y - x ^ * _ 2 } \\ . \\end{align*}"}
-{"id": "283.png", "formula": "\\begin{align*} D _ { 0 + } ^ { 1 / 2 } x ( t ) = - x ( t ) + Q ( t ) x ( t ) + g ( t ) , \\end{align*}"}
-{"id": "3275.png", "formula": "\\begin{align*} F _ { t ^ c } ( t _ 0 ) & = \\frac { 1 } { \\pi - \\theta _ k } \\bigg ( \\arccos \\left ( \\frac { r _ u ^ { } } { v _ u t _ 0 } \\right ) \\\\ & + \\min \\bigg \\lbrace \\arccos \\left ( \\frac { r _ u ^ { } } { r _ { u , k } ( \\boldsymbol { x } ) } \\right ) , \\arccos \\left ( \\frac { r _ u ^ { } } { v _ u t _ 0 } \\right ) \\bigg \\rbrace \\bigg ) . \\end{align*}"}
-{"id": "1809.png", "formula": "\\begin{align*} Y _ { t } Y _ { t + 1 0 } & = Y _ { t + 3 } Y _ { t + 7 } Z _ { t } Z _ { t + 1 } Z _ { t + 2 } + Y _ { t + 2 } ^ 2 Y _ { t + 5 } ^ 2 Y _ { t + 8 } ^ 2 , \\\\ Z _ { t } Z _ { t + 1 } Z _ { t + 2 } Z _ { t + 3 } & = Y _ { t + 2 } Y _ { t + 3 } Y _ { t + 5 } Y _ { t + 6 } Y _ { t + 8 } Y _ { t + 9 } + Y _ { t + 1 } Y _ { t + 4 } Y _ { t + 7 } Y _ { t + 1 0 } . \\end{align*}"}
-{"id": "7368.png", "formula": "\\begin{align*} \\nabla _ X \\omega = \\frac { 1 } { p + 1 } i _ X d \\omega - \\frac { 1 } { n - p + 1 } \\widetilde { X } \\wedge \\delta \\omega \\end{align*}"}
-{"id": "4880.png", "formula": "\\begin{align*} \\left ( \\sum _ { k = 0 } p ( k ) t ^ k \\right ) \\left ( \\sum _ { k = 0 } a _ k t ^ k \\right ) = 1 . \\end{align*}"}
-{"id": "1650.png", "formula": "\\begin{align*} S _ f \\xi ( x ) = \\chi _ { R _ f } ( x ) ( \\Phi _ { \\tau _ f } ( \\tau ^ { e _ i } ( x ) ) ^ { - 1 / 2 } \\xi ( \\tau ^ { e _ i } ( x ) ) . \\end{align*}"}
-{"id": "3764.png", "formula": "\\begin{align*} B _ { y , L } = y + \\{ [ - 2 \\mathfrak { R } L , 3 \\mathfrak { R } L ) ^ d \\times [ 0 , L ) \\} \\cap ( \\mathbb { Z } ^ { d } \\times \\mathbb { Z } ) \\end{align*}"}
-{"id": "8615.png", "formula": "\\begin{align*} \\partial _ t u = \\frac 1 2 \\Delta u + \\lambda \\dot { W } _ \\psi ( t , x ) u , \\ \\ x \\in \\R ^ d , d \\geq 3 . \\end{align*}"}
-{"id": "4318.png", "formula": "\\begin{align*} \\begin{aligned} 2 \\delta \\| \\nabla f _ { i } \\| _ { L ^ { 2 } ( \\Omega ) } ^ { 2 } + \\| f _ { i } \\| _ { L ^ { 2 } ( \\Omega ) } ^ 2 & \\leq \\| \\varphi _ { i , 0 } \\| _ { L ^ { 2 } ( \\Omega ) } ^ 2 , \\\\ 2 \\delta \\| \\Delta f _ { i } \\| _ { L ^ { 2 } ( \\Omega ) } ^ 2 + \\| \\nabla f _ { i } \\| _ { L ^ { 2 } ( \\Omega ) } ^ 2 & \\leq \\| \\nabla \\varphi _ { i , 0 } \\| _ { L ^ { 2 } ( \\Omega ) } ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "5304.png", "formula": "\\begin{align*} R _ 1 = \\left ( \\frac { 2 \\mathcal { R } } { \\min \\left \\lbrace ( L _ { 1 } ) _ \\# , ( L _ { 2 } ) _ \\# / 2 \\right \\rbrace } \\right ) ^ { 1 / 2 } + 1 . \\end{align*}"}
-{"id": "6833.png", "formula": "\\begin{align*} \\phi = T \\left ( - \\lambda ^ 2 \\left [ S _ { \\rho } ( w _ { \\lambda } ) + N ( \\phi ) \\right ] \\right ) \\equiv A ( \\phi ) . \\end{align*}"}
-{"id": "2145.png", "formula": "\\begin{align*} [ v ( r ) ^ { - 1 } U ( r ) ] ' = O ( r ^ { - 1 - \\delta } ) \\quad \\mbox { a s } r \\to \\infty . \\end{align*}"}
-{"id": "8996.png", "formula": "\\begin{gather*} \\hat { D } = { \\cal D } ^ { ( n ) } _ { q , t } ( c + l q / 2 ) D { \\cal D } ^ { ( n ) } _ { q , t } ( - c ) . \\end{gather*}"}
-{"id": "1490.png", "formula": "\\begin{align*} { \\rm R e } \\Big ( \\cfrac { z _ 0 w ' _ 2 ( z _ 0 ) } { w _ 2 ( z _ 0 ) } \\Big ) = { \\rm R e } \\Big ( \\cfrac { \\sqrt { c } z _ 0 \\cos ( \\sqrt { c } z _ 0 ) } { \\sin ( \\sqrt { c } z _ 0 ) } \\Big ) \\le \\cfrac { \\alpha + 1 } { 2 } . \\end{align*}"}
-{"id": "6692.png", "formula": "\\begin{align*} \\pm ( 1 - ( 1 + \\Phi _ 1 ( \\delta ) ) c _ 1 \\delta ^ { \\frac { p } { n - 1 + p } } ) e _ { i } \\in & \\partial ( B _ p ^ n ) _ { \\delta } , 1 \\leq i \\leq n , \\\\ \\left ( 1 - ( 1 + \\Phi _ 2 ( \\delta ) ) c _ 2 \\delta ^ { \\frac { 2 } { n + 1 } } \\right ) \\frac { 1 } { \\sqrt [ p ] { n } } \\sum _ { i = 1 } ^ n \\pm e _ i \\in & \\partial ( B _ p ^ n ) _ { \\delta } \\end{align*}"}
-{"id": "5217.png", "formula": "\\begin{align*} A _ \\infty ( \\lambda ) = \\lim \\limits _ { z \\rightarrow \\pm \\infty } A ( \\lambda , z ) , \\end{align*}"}
-{"id": "3102.png", "formula": "\\begin{align*} \\int _ \\Omega ( \\partial _ { 1 2 } u ) ^ 2 \\d x = \\int _ \\Omega ( \\partial _ { 1 1 } u ) ( \\partial _ { 2 2 } u ) \\d x . \\end{align*}"}
-{"id": "2883.png", "formula": "\\begin{align*} 2 \\textup { R e } \\left ( \\frac { e ^ { i u v } } { \\exp { \\left ( a e ^ { - i u } \\right ) } - 1 } \\right ) = 2 \\textup { R e } \\left ( \\frac { \\cos ( u v ) + i \\sin ( u v ) } { e ^ { a \\cos ( u ) } \\left ( \\cos ( a \\sin ( u ) ) - i \\sin ( a \\sin ( u ) ) \\right ) - 1 } \\right ) . \\end{align*}"}
-{"id": "2468.png", "formula": "\\begin{align*} \\mathbb { P } ( \\xi ) u = \\frac { u \\cdot \\xi } { | \\xi | ^ { 2 } } \\xi \\ , . \\end{align*}"}
-{"id": "9610.png", "formula": "\\begin{align*} \\| H ( f ) \\| _ { \\dot { B } ^ { \\alpha , q } _ { p , \\mathcal F } } & \\lesssim \\bigg \\{ \\sum _ { k \\in \\Bbb Z } \\Big ( 2 ^ { k \\alpha } \\sum _ { k ' \\in \\Bbb Z } 2 ^ { - | k - k ' | \\varepsilon ' } \\| D _ { k ' } T _ N ^ { - 1 } ( f ) \\| _ { L ^ p _ \\mu } \\Big ) ^ q \\bigg \\} ^ { 1 / q } \\\\ & \\lesssim \\bigg \\{ \\sum _ { k ' \\in \\Bbb Z } \\Big ( 2 ^ { k ' \\alpha } \\| D _ { k ' } T _ N ^ { - 1 } ( f ) \\| _ { L ^ p _ \\mu } \\Big ) ^ q \\bigg \\} ^ { 1 / q } . \\end{align*}"}
-{"id": "4679.png", "formula": "\\begin{align*} \\Gamma \\ = \\ ( F _ 1 F _ 2 ) ^ { - 1 / 4 } ( { \\tilde V } _ 4 ^ 2 ) ^ { 1 - d / 4 } \\ = \\ F _ 1 ^ { \\frac { 3 - d } { 4 } } \\ , F _ 2 ^ { - \\frac { 1 } { 4 } } \\ , \\end{align*}"}
-{"id": "9456.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } p _ { \\omega } ( n ) & = q \\ , \\omega ( q ) , \\sum _ { n = 0 } ^ { \\infty } p _ { \\nu } ( n ) = \\nu ( - q ) , \\end{align*}"}
-{"id": "7497.png", "formula": "\\begin{align*} ( \\bar { \\partial } ^ { * h } \\circ \\bar { \\partial } ^ h \\Phi ) _ { A _ p \\overline { B } _ q } & = - h ^ { \\bar { \\varepsilon } \\gamma } \\bigg ( ( \\delta _ \\gamma \\circ \\delta _ { \\bar { \\varepsilon } } ) ( \\phi _ { A _ p \\overline { B } _ q } ) - \\sum _ i ( - 1 ) ^ i ( \\delta _ \\gamma \\circ \\delta _ { \\bar { \\beta } _ i } ) \\big ( \\phi _ { A _ p \\bar { \\varepsilon } \\bar { \\beta } _ 1 \\dots \\hat { \\bar { \\beta } } _ i \\dots \\bar { \\beta } _ q } \\big ) \\bigg ) \\end{align*}"}
-{"id": "8883.png", "formula": "\\begin{align*} u = \\omega + \\frac { 1 } { \\overline { g _ 1 ( 0 ) } } ( c - \\theta ) g . \\end{align*}"}
-{"id": "3545.png", "formula": "\\begin{align*} \\partial \\Lambda = \\{ \\boldsymbol { s } \\notin \\Lambda : \\exists _ { \\boldsymbol { t } \\in \\Lambda } \\ \\boldsymbol { s } \\in B _ { \\boldsymbol { t } } \\} \\cup \\{ \\boldsymbol { t } \\in \\Lambda : \\exists _ { \\boldsymbol { s } \\in \\Lambda ^ c } \\ \\boldsymbol { t } \\in B _ { \\boldsymbol { s } } \\setminus B _ { \\boldsymbol { s + \\boldsymbol { 1 } } } \\} \\end{align*}"}
-{"id": "4377.png", "formula": "\\begin{align*} ( T _ f ) _ z = T _ { f \\circ \\tau _ z } . \\end{align*}"}
-{"id": "8451.png", "formula": "\\begin{align*} \\left ( \\begin{matrix} 0 & b \\\\ b & 0 \\end{matrix} \\right ) = \\left ( \\begin{matrix} - 1 & 1 \\\\ 1 & 1 \\end{matrix} \\right ) \\left ( \\begin{matrix} \\frac { b } { 2 } & 0 \\\\ 0 & \\frac { b } { 2 } \\end{matrix} \\right ) \\left ( \\begin{matrix} - 1 & 1 \\\\ 1 & 1 \\end{matrix} \\right ) \\end{align*}"}
-{"id": "6299.png", "formula": "\\begin{align*} \\{ - n ; ( \\tilde { \\varepsilon } , \\tilde { g } , 0 , 0 ) ; ( \\alpha _ 1 , \\beta _ 1 ) , \\{ ( \\alpha _ i , \\alpha _ i - \\beta _ i ) \\} _ { i = 1 } ^ n \\} . \\end{align*}"}
-{"id": "1971.png", "formula": "\\begin{align*} d ( x , y ) = \\begin{cases} \\min \\{ 1 , d _ { i } ( x , y ) \\} & \\mbox { i f } x , y \\in M _ { i } \\\\ 1 & \\mbox { i f } x \\in M _ { i } , y \\in M _ { j } i \\neq j . \\end{cases} \\end{align*}"}
-{"id": "6201.png", "formula": "\\begin{align*} Q ( x , y ) & = \\begin{vmatrix} - 2 & 0 & k x + m y \\\\ 0 & - 2 & l x + n y \\\\ k x + m y & l x + n y & 2 x y \\end{vmatrix} \\\\ & = 8 x y + 2 ( k x + m y ) ^ 2 + 2 ( l x + n y ) ^ 2 \\\\ & = ( 2 k ^ 2 + 2 l ^ 2 ) x ^ 2 + ( 8 + 4 k m + 4 l n ) x y + ( 2 m ^ 2 + 2 n ^ 2 ) y ^ 2 . \\end{align*}"}
-{"id": "4136.png", "formula": "\\begin{align*} \\lambda _ 1 = \\lambda _ 2 = - \\frac { 1 } { \\langle \\eta , \\xi \\rangle } . \\end{align*}"}
-{"id": "2208.png", "formula": "\\begin{align*} { \\sum _ J } ' \\sum _ { \\alpha } ( - t ) ^ { | | \\alpha | | + \\| \\beta \\| + n } \\frac { ( - 1 ) ^ { s ( J ) } } { \\beta ( \\alpha , J ) ! \\cdot a _ J ^ { \\beta + I } } \\cdot \\frac { \\partial ^ { | | \\beta ( \\alpha , J ) | | } } { \\partial z ^ { \\beta ( \\alpha , J ) } } \\left [ \\frac { \\Delta ( t ) } { z _ 1 ^ { \\gamma _ 1 + 1 } \\cdot \\ldots \\cdot z _ n ^ { \\gamma _ n + 1 } } \\cdot \\frac { Q ^ { \\alpha } } { q ^ { \\alpha + I } ( J ) } \\right ] _ { z = \\tilde a _ J } . \\end{align*}"}
-{"id": "6165.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t \\omega _ i = \\dd \\tau _ i , & i = 1 , 2 , 3 \\\\ \\underline \\omega | _ { t = 0 } = \\underline \\omega ( 0 ) . \\end{array} \\right . \\end{align*}"}
-{"id": "9851.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 } { 2 \\pi } \\int \\hat { f } ( y ) e ^ { i x y } d y , \\ ; a . e . \\end{align*}"}
-{"id": "9176.png", "formula": "\\begin{align*} M _ { \\alpha _ { ( i ) } } ( x ) = \\frac { 1 } { 2 } \\langle \\alpha _ { ( i ) } , \\nabla \\psi ( x ) \\rangle \\begin{pmatrix} \\coth \\alpha _ { ( i ) } ( x ) & \\sqrt { - 1 } \\\\ - \\sqrt { - 1 } & \\coth \\alpha _ { ( i ) } ( x ) \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "8904.png", "formula": "\\begin{align*} \\begin{aligned} & \\lim _ k \\bigl ( \\| X _ 1 J _ { \\theta , 1 } ^ { - 1 } U _ \\mu ^ { n _ k } p \\| ^ 2 - \\| p X _ 1 J _ { \\theta , 1 } ^ { - 1 } \\chi ^ { - n _ k } \\| ^ 2 \\bigr ) = 0 \\\\ \\ \\ & \\lim _ k \\bigl ( \\| X _ 1 J _ { \\theta , 1 } ^ { - 1 } U _ \\mu ^ { - n _ k } p \\| ^ 2 - \\| p X _ 1 J _ { \\theta , 1 } ^ { - 1 } \\chi ^ { - n _ k } \\| ^ 2 \\bigr ) = 0 \\end{aligned} \\end{align*}"}
-{"id": "3299.png", "formula": "\\begin{align*} { _ 2 F _ 1 } ( \\alpha , \\beta ; \\gamma ; 1 - x ) = \\frac { \\Gamma ( \\gamma ) } { \\Gamma ( \\gamma - \\beta ) \\Gamma ( \\beta ) } \\int _ 0 ^ \\infty t ^ { \\beta - 1 } ( 1 + t ) ^ { \\alpha - \\gamma } ( 1 + x t ) ^ { - \\alpha } d t . \\end{align*}"}
-{"id": "9380.png", "formula": "\\begin{align*} \\frac { 2 } { \\alpha + 2 m - 1 / 2 } - ( 1 - \\kappa _ { \\alpha , 2 m - 1 / 2 } ) \\frac { 2 } { 4 m - 1 } = 0 . \\end{align*}"}
-{"id": "6256.png", "formula": "\\begin{align*} u ( x , 0 ) = u _ 0 ( x ) , b ( x , 0 ) = b _ 0 ( x ) , \\nabla \\cdot u _ 0 = \\nabla \\cdot b _ 0 = 0 . \\end{align*}"}
-{"id": "5294.png", "formula": "\\begin{align*} u _ t = \\mu u _ { x x } - m u _ { x } - n u , \\end{align*}"}
-{"id": "4753.png", "formula": "\\begin{align*} { \\varepsilon } = F ( x ^ 0 , Y , Z ) \\ , { \\Delta \\sqrt { - \\det g ^ { i j } } } / { ( \\alpha f _ 1 + \\beta ) } , \\end{align*}"}
-{"id": "2544.png", "formula": "\\begin{align*} \\left \\{ \\rho + \\mathrm { P } _ 0 \\xi _ 1 + i \\eta \\mathrm { P } _ 0 \\xi _ 1 \\left [ L - i \\eta \\mathrm { P } _ 1 \\xi _ 1 - i \\eta \\rho \\right ] ^ { - 1 } \\mathrm { P } _ 1 \\xi _ 1 \\right \\} ( \\mathrm { P } _ 0 e ) = 0 . \\end{align*}"}
-{"id": "3349.png", "formula": "\\begin{align*} \\lim _ { K \\rightarrow \\infty } \\frac { 1 } { \\psi _ 2 ( N , K , s ) } & \\leq \\frac { N } { \\lambda } \\sum _ { i = 0 } ^ { \\infty } \\left ( \\frac { ( 1 - \\lambda ) ( N - 1 ) } { \\lambda } \\right ) ^ i \\\\ & = \\frac { N } { \\lambda } \\cdot \\frac { 1 } { 1 - \\frac { ( 1 - \\lambda ) ( N - 1 ) } { \\lambda } } \\\\ & = \\frac { N } { N \\lambda - ( N - 1 ) } . \\end{align*}"}
-{"id": "111.png", "formula": "\\begin{align*} & \\forall m \\in \\mathrm { I n t } _ { E + 2 r _ 1 \\sqrt { k } } W _ 0 ( x , n ) \\setminus \\left ( n + L \\mathbb { Z } ^ k \\right ) : \\ > \\nu ( x , m ) = 0 , \\\\ & \\forall \\ , m \\in \\left ( n + L \\mathbb { Z } ^ k \\right ) \\cap \\mathrm { I n t } _ { E + \\sqrt { k } } W _ 0 ( x , n ) : \\ > \\nu ( x , m ) = 1 . \\end{align*}"}
-{"id": "6096.png", "formula": "\\begin{align*} \\int _ { \\lvert x \\lvert = \\epsilon } \\lvert h ( x ) \\eta ( x ) \\nu _ i ( x ) \\lvert d S \\leq c \\lvert \\lvert \\eta \\lvert \\lvert _ { \\infty } \\omega _ { n } \\epsilon ^ { n - 1 } . \\end{align*}"}
-{"id": "1680.png", "formula": "\\begin{align*} \\mu _ 2 ( Z ( \\eta ) ) : = \\begin{cases} \\prod _ { 0 \\leq n \\leq N } ( 1 / 2 + \\alpha _ { 2 n } ^ \\eta ) , & \\ ; \\ ; r ( \\eta ) \\in \\{ u , w \\} \\\\ \\prod _ { 0 \\leq n \\leq N - 1 } ( 1 / 2 + \\alpha _ { 2 n + 1 } ^ \\eta ) , & \\ ; \\ ; r ( \\eta ) = v \\end{cases} \\end{align*}"}
-{"id": "85.png", "formula": "\\begin{align*} \\lambda ( f ) = L _ f ( W ) ; \\end{align*}"}
-{"id": "3991.png", "formula": "\\begin{align*} p ^ { \\beta _ { m + 1 } } ( m + 1 , t ) & = ( - 1 ) ^ { m + 1 } \\sum _ { k = m + 1 } ^ { \\infty } ( - \\lambda ) ^ k \\underset { \\Omega ^ { k } _ { m + 1 } } { \\sum } \\frac { t ^ { \\sum _ { j = 0 } ^ { m + 1 } k _ j \\beta _ j } } { \\Gamma \\left ( \\sum _ { j = 0 } ^ { m + 1 } k _ j \\beta _ j + 1 \\right ) } , \\end{align*}"}
-{"id": "5180.png", "formula": "\\begin{align*} g _ { 1 , [ r , 1 ] } ( x ) & = \\begin{cases} g _ { 1 } ( r ) \\cdot \\frac { x } { r } , & \\forall x \\in [ 0 , r ) \\\\ g _ { 1 } ( x ) , & \\forall x \\in [ r , 1 ] , \\end{cases} \\end{align*}"}
-{"id": "9055.png", "formula": "\\begin{align*} B _ { { \\mathbf x } } ( w ) = \\displaystyle \\frac { 1 } { 2 } ( 0 , b _ { 2 1 } ^ 2 { \\mathbf x } _ 2 w _ 1 + b _ { 2 2 } ^ 2 { \\mathbf x } _ 2 w _ 2 ) . \\end{align*}"}
-{"id": "8214.png", "formula": "\\begin{align*} \\hat { x } M _ j = M _ { j - 1 } \\end{align*}"}
-{"id": "8687.png", "formula": "\\begin{align*} ( a , b ) ^ { [ 2 p ] } & = \\Big ( \\frac { 1 } { 2 } P ( ( a , b ) , ( a , b ) ) \\Big ) ^ { [ p ] } \\\\ & = \\begin{pmatrix} - a b & a ^ 2 \\\\ - b ^ 2 & a b \\end{pmatrix} ^ { [ p ] } . \\end{align*}"}
-{"id": "10099.png", "formula": "\\begin{align*} ^ { \\prime } { \\tilde { P } } ( X , Y , Z , U ) = ^ { \\prime } \\tilde { R } ( X , Y , Z , U ) - \\frac { 1 } { n - 1 } \\{ \\tilde { S } ( Y , Z ) g ( X , U ) - \\tilde { S } ( X , Z ) g ( Y , U ) \\} \\end{align*}"}
-{"id": "1554.png", "formula": "\\begin{align*} z _ { n } = z _ { n - 1 } ^ 2 + c \\end{align*}"}
-{"id": "1485.png", "formula": "\\begin{align*} h ( x \\sqrt { c } ) = 2 x \\sqrt { c } - ( 1 + \\alpha ) \\tan ( x \\sqrt { c } ) \\geq 0 \\mbox { f o r $ 0 \\leq x \\sqrt { c } \\leq \\sqrt { c _ \\alpha } $ } \\end{align*}"}
-{"id": "1745.png", "formula": "\\begin{align*} P ( A ) ( f \\ , \\sqrt { d \\mu } ) = ( \\chi _ A \\cdot f ) \\ , \\sqrt { d \\mu } , \\end{align*}"}
-{"id": "7826.png", "formula": "\\begin{align*} d _ { } = \\lim _ { \\rho \\rightarrow \\infty } \\frac { C _ { } ( \\rho ) } { \\log ( \\rho ) } , \\end{align*}"}
-{"id": "2321.png", "formula": "\\begin{align*} \\Gamma _ \\infty = \\{ p _ k \\mid k \\in \\Z \\} . \\end{align*}"}
-{"id": "7625.png", "formula": "\\begin{align*} \\int _ { V } | g | ^ l \\ , d \\nu & = l \\int _ 0 ^ \\infty t ^ { l - 1 } \\nu \\big ( \\{ V : | g | > t \\} \\big ) \\ , d t \\geq l \\sum _ { i = 0 } ^ \\infty \\int _ { \\alpha _ 0 M ^ { i - 1 } } ^ { \\alpha _ 0 M ^ i } t ^ { l - 1 } \\nu \\big ( \\{ V : | g | > t \\} \\big ) \\ , d t \\\\ & \\geq \\sum _ { i = 0 } ^ \\infty \\big [ ( \\alpha _ 0 M ^ i ) ^ l - ( \\alpha _ 0 M ^ { i - 1 } ) ^ l \\big ] \\ , \\nu \\big ( \\{ V : | g | > \\alpha _ 0 M ^ i \\} \\big ) . \\end{align*}"}
-{"id": "6073.png", "formula": "\\begin{align*} \\Phi ^ { '' } ( r ) + ( n - 1 ) \\tfrac { \\Phi ' ( r ) } { r } - \\mathfrak { e } _ k \\tfrac { \\sin ( 2 \\Phi ( r ) ) } { r ^ 2 } = 0 . \\end{align*}"}
-{"id": "4397.png", "formula": "\\begin{align*} \\sup _ { x \\in \\mathcal { M } \\setminus \\mathbb { C } ^ n } \\| \\hat { A } _ x ^ { - 1 } \\| = \\sup _ { x \\in \\mathcal { M } \\setminus \\mathbb { C } ^ n } \\frac { 1 } { \\nu ( \\hat { A } _ x ) } = \\frac { 1 } { \\nu ( \\hat { A } _ y ) } < \\infty , \\end{align*}"}
-{"id": "5283.png", "formula": "\\begin{align*} \\| u _ x \\| ^ 2 \\geq u ^ 2 ( t , 0 ) - 2 \\| u \\| ^ 2 , \\\\ \\| u _ x \\| ^ 2 \\geq \\| u \\| ^ 2 - 2 u ^ 2 ( t , 1 ) . \\end{align*}"}
-{"id": "5409.png", "formula": "\\begin{align*} A x ^ 4 + 2 B x ^ 2 y ^ 2 + A y ^ 4 + C z ^ 4 - t ^ 2 = 0 , \\end{align*}"}
-{"id": "9210.png", "formula": "\\begin{align*} [ z _ { 1 } \\otimes \\alpha _ { 1 } , [ z _ { 2 } \\otimes \\alpha _ { 2 } , z _ { 3 } \\otimes \\alpha _ { 3 } ] ] = [ [ z _ { 1 } \\otimes \\alpha _ { 1 } , z _ { 2 } \\otimes \\alpha _ { 2 } ] , z _ { 3 } \\otimes \\alpha _ { 3 } ] + [ z _ { 2 } \\otimes \\alpha _ { 2 } , [ z _ { 1 } \\otimes \\alpha _ { 1 } , z _ { 3 } \\otimes \\alpha _ { 3 } ] ] . \\end{align*}"}
-{"id": "9751.png", "formula": "\\begin{align*} F _ { c } ( J _ k ) = F ( J _ k ) + K _ z \\sum _ { j = k + 1 } ^ { \\infty } e ^ { - l j h } h \\| Z _ { 0 } \\| _ { \\infty } , \\end{align*}"}
-{"id": "26.png", "formula": "\\begin{align*} \\hat { V } ^ { C } _ { \\sigma } ( C _ { 1 } , C _ { 2 } ) = \\frac { 1 } { 2 \\pi \\sigma ^ { 2 } } \\frac { 1 } { N } \\sum \\limits _ { n = 1 } ^ N e x p \\left ( - \\frac { | C _ 1 | ^ 2 + | C _ 2 | ^ 2 } { 2 \\sigma ^ 2 } - \\frac { | C _ 1 | | C _ 2 | c o s ( \\theta - \\phi ) } { \\sigma ^ 2 } \\right ) \\end{align*}"}
-{"id": "2018.png", "formula": "\\begin{align*} \\sum _ { \\substack { j \\in \\Z \\\\ 1 \\leq a \\leq l } } v _ { a , j , s } x _ { i + s - 2 j - k _ a } y _ { a , i + j } = 0 , ~ ~ ~ i \\in \\Z \\end{align*}"}
-{"id": "8169.png", "formula": "\\begin{align*} \\omega ( t ) = \\inf _ { s > 0 } \\big ( \\omega ^ { \\star } ( s ) + s t \\big ) , t > 0 . \\end{align*}"}
-{"id": "7070.png", "formula": "\\begin{align*} C I _ { S O ( N ) } ( \\{ \\tilde { u } _ 0 \\} , - \\nabla \\Psi ^ { \\nu } _ { \\pm } ) = C I _ { S O ( N ) } ( \\{ \\tilde { u } _ 0 \\} , - \\nabla \\xi ^ { { \\nu } } _ { \\lambda _ 0 \\pm \\varepsilon } ) . \\end{align*}"}
-{"id": "2352.png", "formula": "\\begin{align*} \\Phi _ m ( s , \\lambda , w ) \\coloneqq \\frac { 1 } { m ! } \\frac { \\partial ^ m \\Phi } { \\partial \\lambda ^ m } ( s , \\lambda , w ) = \\sum _ { n = m } ^ \\infty { n \\choose m } \\frac { \\lambda ^ { n - m } } { ( n + w ) ^ s } . \\end{align*}"}
-{"id": "5174.png", "formula": "\\begin{align*} \\begin{cases} U _ { 1 } ( x , r ) \\le U _ { 1 } ( \\ell , r ) , \\forall x \\in [ 0 , r \\wedge a ] , \\\\ U _ { 2 } ( \\ell , y ) \\le U _ { 2 } ( \\ell , r ) , \\forall y \\in [ \\ell \\vee b , 1 ] . \\end{cases} \\end{align*}"}
-{"id": "1361.png", "formula": "\\begin{align*} { \\cal T } ^ { { \\rm l i n } } ( \\overline X ) = \\{ H \\in { \\cal S } ^ q : \\theta ' ( \\overline X ; H ) = - \\theta ' ( \\overline X ; - H ) \\} = \\{ H \\in { \\cal S } ^ q : Q _ { b } ^ T H Q _ { b } = 0 \\} . \\end{align*}"}
-{"id": "9752.png", "formula": "\\begin{align*} U _ h | _ { \\Omega _ { h , k } ^ { ( i ) } } \\in O _ { \\epsilon } ( { U } _ i ^ { ( 0 ) } ) , \\ , \\ , i = 1 , 2 , | \\sigma ^ { ( k ) } _ { j } - \\sigma _ { j 0 } | < \\hat { \\epsilon } , \\ , \\ , j = 2 , 3 , | \\gamma _ { 4 } ^ { ( k ) } | < \\hat { \\epsilon } . \\end{align*}"}
-{"id": "2254.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { k - 1 } b _ j s _ { k - j } + k b _ k = 0 , b _ 0 = 1 , k \\geqslant 1 . \\end{align*}"}
-{"id": "7818.png", "formula": "\\begin{align*} \\liminf _ { r \\to \\infty } \\int _ { S _ { r } } r ^ { 1 - 2 m } | \\frac { \\partial v _ m } { \\partial r } v _ m | e ^ { - 2 \\rho } d x = 0 . \\end{align*}"}
-{"id": "5143.png", "formula": "\\begin{align*} u = \\zeta _ \\infty + v . \\end{align*}"}
-{"id": "3101.png", "formula": "\\begin{align*} \\lambda _ 1 ^ \\Gamma = \\min _ { u \\in H _ { 0 , \\Gamma } ^ 1 ( \\Omega ) } \\frac { \\int _ \\Omega | \\nabla u | ^ 2 \\d x } { \\int _ \\Omega | u | ^ 2 \\d x } . \\end{align*}"}
-{"id": "4616.png", "formula": "\\begin{align*} | U _ i ' | \\ge \\beta ( 1 - 2 \\eta ) \\sum _ { n = 0 } ^ { 2 ^ { k _ i } - 1 } 1 _ { [ - B , B ] } \\left ( t + \\sum _ { j = 0 } ^ { n - 1 } f ( T ^ j z _ i ) \\right ) \\end{align*}"}
-{"id": "4766.png", "formula": "\\begin{align*} \\alpha V ^ 1 ( 2 { L } _ 1 + \\alpha \\omega _ 1 ) + V ^ 2 ( { L } _ 2 { } ^ 2 + \\alpha ^ 2 \\omega _ 2 ) + V ^ 3 ( { L } _ 3 { } ^ 2 + \\alpha ^ 2 \\omega _ 3 ) = 0 , \\end{align*}"}
-{"id": "7405.png", "formula": "\\begin{align*} \\mathbb { H } _ { \\Gamma } : = \\frac { \\mathbb { C } [ t _ 0 , t _ 1 , t _ 2 , t _ 3 , t _ 4 , t _ 5 , t _ 6 , t _ 7 ] } { ( t _ 0 + 2 t _ 1 + 4 t _ 2 + 3 t _ 3 + 2 t _ 4 + 3 t _ 5 + 2 t _ 6 + t _ 7 ) } \\end{align*}"}
-{"id": "2950.png", "formula": "\\begin{align*} \\Theta _ \\epsilon ^ \\Delta ( H , e , i ) = \\begin{cases} \\theta ^ { H ^ \\Delta } _ \\epsilon ( e , i ) & \\mbox { i f $ e \\in E ( H ^ \\Delta ) $ } , \\\\ 0 & , \\end{cases} \\end{align*}"}
-{"id": "3184.png", "formula": "\\begin{align*} \\alpha ( z , s ) = \\frac { 2 b z } { \\sigma ^ { 2 } \\left ( e ^ { b s } - 1 \\right ) } \\geqslant \\frac { 2 b } { \\sigma ^ { 2 } \\left ( e ^ { b t } - 1 \\right ) } . \\end{align*}"}
-{"id": "3172.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ e ^ { u Z _ { t } ^ { i } } \\right ] = \\exp \\left \\lbrace \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { \\infty } \\left ( e ^ { z \\psi ( s , u ) } - 1 \\right ) \\nu _ { i } ( \\mathrm { d } z ) \\mathrm { d } s \\right \\rbrace , i = 1 , 2 , \\ ( t , u ) \\in \\mathbb { R } _ { \\geqslant 0 } \\times \\mathcal { U } . \\end{align*}"}
-{"id": "2318.png", "formula": "\\begin{align*} & \\ ; \\| I _ 4 + I _ 5 + I _ 6 \\| _ { L ^ 2 } \\\\ \\leq & \\ ; C \\| A ^ { - \\beta s } ( I d - e ^ { - h \\nu A ^ s } ) \\| _ { \\mathcal { L } ( L ^ 2 ) } \\| A ^ { \\beta s } e ^ { - t \\nu A ^ s } \\| _ { \\mathcal { L } ( L ^ 2 ) } \\| A ^ { 1 - s / 2 } f ( w _ 1 , w _ 2 ) ( t ) \\| _ { L ^ 2 } \\\\ & \\ ; + C \\| ( I d - e ^ { - 2 h \\nu A ^ s } ) \\| _ { \\mathcal { L } ( L ^ 2 ) } \\| A ^ { 1 - s / 2 } ( f ( w _ 1 , w _ 2 ) ( t + h ) - f ( w _ 1 , w _ 2 ) ( t ) ) \\| _ { L ^ 2 } \\\\ \\leq & \\ ; C h ^ \\beta t ^ { - ( \\beta + 1 / 2 ) } R _ 1 R _ 2 , \\end{align*}"}
-{"id": "4819.png", "formula": "\\begin{align*} n q & = \\mathsf { d } ( r ) \\alpha _ k r ^ k + \\sum _ { i \\neq k } \\mathsf { d } ( r ) \\alpha _ i r ^ i \\\\ & = \\mathsf { n } ( r ) r ^ { k - 1 } + ( \\alpha _ k - 1 ) \\mathsf { d } ( r ) r ^ k + \\sum _ { i \\neq k } \\mathsf { d } ( r ) \\alpha _ i r ^ i \\\\ & = \\mathsf { n } ( r ) + \\big ( s + ( \\alpha _ k - 1 ) \\mathsf { d } ( r ) r ^ k + \\sum _ { i \\neq k } \\mathsf { d } ( r ) \\alpha _ i r ^ i \\big ) , \\end{align*}"}
-{"id": "6739.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } E \\Bigg ( \\frac { R _ { N } } { \\beta _ { N } } \\Bigg ) = \\int _ { 0 } ^ { \\infty } \\lim \\limits _ { N \\rightarrow \\infty } \\mathbb { P } \\Bigg ( \\frac { R _ { N } } { \\beta _ { N } } > t \\Bigg ) d t = 1 . \\end{align*}"}
-{"id": "6415.png", "formula": "\\begin{align*} d \\theta _ { \\mu } = \\chi \\Delta \\theta _ { \\mu } \\end{align*}"}
-{"id": "4126.png", "formula": "\\begin{align*} \\frac { d ^ 2 g } { d \\theta ^ 2 } + g = g ^ { - 3 } . \\end{align*}"}
-{"id": "6133.png", "formula": "\\begin{align*} B _ { n } ^ { Y } [ a , b ] & = \\left \\{ j : \\sum _ { i = 1 } ^ { j } a _ { n } W _ { i } : \\in [ a , b ] \\right \\} \\\\ B _ { n } ^ { X } [ a , b ] & = \\left \\{ j : \\sum _ { i = 1 } ^ { j } a _ { n } U _ { i } : \\in [ a , b ] \\right \\} , \\end{align*}"}
-{"id": "7241.png", "formula": "\\begin{align*} = \\int _ { \\hat { G } } ^ { } \\int _ { G } ^ { } \\int _ { U } ^ { } \\theta _ { \\pi } ( g ) f ( u ^ { - 1 } g ) \\psi _ { n } ( u ) d u d g d \\mu _ { \\pi } . \\end{align*}"}
-{"id": "2130.png", "formula": "\\begin{align*} d ^ * ( \\mathfrak { D } f _ 1 , \\mathfrak { D } f _ 2 ) & = s u p \\{ | \\mathfrak { D } f _ 1 - \\mathfrak { D } f _ 2 | ~ u , v \\in U \\} \\\\ & = \\frac { 8 } { 9 } \\leq \\frac { 2 } { 3 } \\times \\frac { 4 } { 3 } \\\\ & = \\lambda d ^ * ( f _ 1 , f _ 2 ) \\end{align*}"}
-{"id": "6571.png", "formula": "\\begin{align*} | ( F _ { \\xi } - s ( F _ { \\xi } ) ) ^ { \\circ } | _ 1 = 2 \\cdot \\left ( \\frac { 1 } { 2 \\cdot \\sqrt { 1 - \\varepsilon ^ 2 } } \\right ) ^ { - 1 } = 4 \\cdot \\sqrt { 1 - \\varepsilon ^ 2 } . \\end{align*}"}
-{"id": "6045.png", "formula": "\\begin{align*} \\lim _ { \\theta \\downarrow 0 } \\frac { 1 } { \\theta } \\Omega ^ { ( \\alpha , \\theta ) } = R ^ { ( \\alpha ) } , \\end{align*}"}
-{"id": "7899.png", "formula": "\\begin{align*} \\max _ { B _ \\delta ( x _ t ) } u _ 0 = u _ 0 ( y ) . \\end{align*}"}
-{"id": "7959.png", "formula": "\\begin{align*} \\mu _ m = \\sum _ { I \\in \\mathcal S , \\alpha \\in A _ I } \\lambda _ { I J _ { \\alpha } } ^ s ( g _ { I J _ { \\alpha } } ) _ * ( \\mu _ m ) . \\end{align*}"}
-{"id": "4642.png", "formula": "\\begin{align*} A = 2 ( d + 1 ) \\frac { 1 0 \\delta + 6 c _ 1 } { 3 \\delta + 3 c _ 1 } \\max \\left \\{ 1 , \\frac { 8 D } { 3 \\xi } \\right \\} \\end{align*}"}
-{"id": "1211.png", "formula": "\\begin{align*} w ( \\hat x ) = - \\frac { ( 1 - K ) } { A ^ 2 } u _ 1 ( y + \\hat x ) - \\frac { K } { B ^ 2 } u _ 2 ( z + \\hat x ) \\ , + \\frac { 1 } { C ^ 2 } u ( x + \\hat x ) , \\end{align*}"}
-{"id": "863.png", "formula": "\\begin{align*} \\Delta = E \\left [ \\max _ { 1 \\leq i , j \\leq d } | V ^ { i j } - \\langle D F _ i , - D L ^ { - 1 } F _ j \\rangle _ H | \\right ] . \\end{align*}"}
-{"id": "5309.png", "formula": "\\begin{align*} \\Theta _ M ( \\cdot , t ) = \\theta ^ { m - 1 } + ( t - t _ { m - 1 , M } ) \\frac { \\theta ^ m - \\theta ^ { m - 1 } } { \\tau } . \\end{align*}"}
-{"id": "8513.png", "formula": "\\begin{align*} [ 2 ] P = P + P = \\left ( \\frac { ( 3 x ^ 2 + a ) ^ 2 - 8 x y ^ 2 } { 4 y ^ 2 } , \\frac { F _ { a , b } ( x ) } { ( 2 y ) ^ 3 } \\right ) , \\end{align*}"}
-{"id": "5154.png", "formula": "\\begin{align*} G _ m ^ \\pm F & = E _ m ^ \\pm F ' + m ^ { - 2 } E _ m ^ \\pm ( d \\delta + \\delta d + m ^ 2 ) F '' \\\\ & = E _ m ^ \\pm F ' + m ^ { - 2 } F '' . \\end{align*}"}
-{"id": "2504.png", "formula": "\\begin{align*} \\phi ^ { e } \\left ( s \\right ) = \\mathcal { A } \\left ( s ^ { 2 } \\right ) , \\quad \\frac { \\phi ^ { o } \\left ( s \\right ) } { s } = \\mathcal { B } \\left ( s ^ { 2 } \\right ) \\end{align*}"}
-{"id": "2101.png", "formula": "\\begin{align*} K ( p , 0 ; s ) = \\phi ( p ; s ) \\end{align*}"}
-{"id": "9688.png", "formula": "\\begin{gather*} V _ b = \\tilde { \\Phi } ( \\gamma _ 5 , \\gamma _ 3 , \\gamma _ 2 , \\gamma _ 1 ; V _ a ) , Z _ b = Z _ a + \\gamma _ 4 , \\\\ V _ b = \\tilde { \\Phi } ( \\beta _ 5 , \\beta _ 3 , \\beta _ 2 , \\beta _ 1 ; V _ m ) , Z _ b = Z _ m + \\beta _ 4 , \\\\ V _ m = \\tilde { \\Phi } ( \\alpha _ 5 , \\alpha _ 3 , \\alpha _ 2 , \\alpha _ 1 ; V _ a ) , Z _ m = Z _ a + \\alpha _ 4 . \\end{gather*}"}
-{"id": "6254.png", "formula": "\\begin{align*} \\dot { u } = U ( u , v ) , \\dot { v } = V ( v ) \\end{align*}"}
-{"id": "3536.png", "formula": "\\begin{align*} \\left | \\beta ^ { 0 , \\theta } \\right | ^ 2 + \\displaystyle \\sum _ { j = 1 } ^ n \\left | \\beta ^ { j , \\theta } \\right | ^ 2 = 1 . \\end{align*}"}
-{"id": "3986.png", "formula": "\\begin{align*} p ^ { \\beta _ 2 } _ { k - 1 } ( 2 , t ) = ( - \\lambda ) ^ { k - 1 } \\underset { \\Omega ^ { k - 1 } _ { 2 } } { \\sum } \\frac { t ^ { k _ 0 \\beta _ 0 + k _ 1 \\beta _ 1 + k _ 2 \\beta _ 2 } } { \\Gamma \\left ( k _ 0 \\beta _ 0 + k _ 1 \\beta _ 1 + k _ 2 \\beta _ 2 + 1 \\right ) } . \\end{align*}"}
-{"id": "2227.png", "formula": "\\begin{align*} \\left | w _ j - \\frac { a _ { j i _ { j } } } { 1 - \\varepsilon ^ 2 } \\right | ^ 2 = \\frac { \\varepsilon ^ 2 \\cdot | a _ { j i _ { j } } | ^ 2 } { ( 1 - \\varepsilon ^ 2 ) ^ 2 } , j = 1 , \\ldots , n . \\end{align*}"}
-{"id": "1503.png", "formula": "\\begin{align*} G ( y ) = \\sum _ { n = 0 } ^ { \\infty } T _ { n } ( x ) y ^ { n } = \\frac { y } { 1 - x ^ { 2 } y - x y ^ { 2 } - y ^ { 3 } } \\end{align*}"}
-{"id": "5153.png", "formula": "\\begin{align*} \\Lambda ^ { ( \\Sigma ' ) } _ m ( O ' _ m ) = \\widetilde { T } _ m \\Lambda ^ { ( \\Sigma ) } _ m ( O ' _ m ) \\end{align*}"}
-{"id": "1837.png", "formula": "\\begin{align*} ( N \\otimes _ R X ) _ m : = N \\otimes _ R X _ m \\end{align*}"}
-{"id": "1843.png", "formula": "\\begin{align*} P ^ i _ { \\alpha + 1 } & : = P ^ i _ { \\alpha } \\oplus P ^ i _ { \\alpha , \\alpha + 1 } & , \\\\ P ^ i _ \\beta & : = \\varinjlim _ { \\alpha < \\beta } P ^ i _ \\alpha & . \\end{align*}"}
-{"id": "5715.png", "formula": "\\begin{align*} \\mathcal { K } ( { v } ) = 0 \\Longleftrightarrow { v } \\in \\Sigma : = \\{ \\mathcal { C } ( z ^ - ) , \\mathcal { C } ( z ^ + ) \\} ; \\end{align*}"}
-{"id": "8315.png", "formula": "\\begin{align*} f ( \\tau ) = \\sum _ { \\substack { m \\in \\Q \\\\ m \\gg - \\infty } } c ( m ) \\cdot q ^ m \\in M ^ ! _ { 1 - \\frac { n } { 2 } } ( \\overline { \\rho } _ { V _ \\Z } ) \\end{align*}"}
-{"id": "1784.png", "formula": "\\begin{align*} R _ \\omega '' ( x ) - G ' ( R _ \\omega ( x ) ) - \\omega R _ \\omega ( x ) = 0 R _ \\omega ' ( 0 ) = 0 , R _ \\omega ( 0 ) = R _ * ( \\omega ) . \\end{align*}"}
-{"id": "4177.png", "formula": "\\begin{align*} \\omega \\left ( F ( \\rho _ 1 S _ 1 , \\ldots \\rho _ k S _ k ) \\right ) \\leq \\omega \\left ( F ( \\rho _ 1 S _ 1 , \\ldots \\rho _ { k - 1 } S _ { k - 1 } , 0 ) \\right ) \\left [ 1 + 2 \\sum _ { p _ k = 1 } ^ { m _ k } \\rho _ k ^ { p _ k } \\cos \\frac { \\pi } { \\left [ \\frac { m _ k } { p _ k } \\right ] + 2 } \\right ] . \\end{align*}"}
-{"id": "2000.png", "formula": "\\begin{align*} G ( z ) = \\sum _ { i = - \\infty } ^ { \\infty } z ^ i x _ i \\end{align*}"}
-{"id": "8303.png", "formula": "\\begin{align*} U _ \\Phi = W _ \\Phi = \\mathrm { k e r } ( \\nu _ \\Phi : Q _ \\Phi \\to \\mathbb { G } _ m ) , \\end{align*}"}
-{"id": "3632.png", "formula": "\\begin{align*} \\Lambda _ 0 = \\{ a < \\lambda < 0 : u \\leq u _ { t } \\ , \\ , \\ , \\ , \\ , \\ , \\Omega _ t \\setminus R _ t ( \\Gamma ) \\ , \\ , \\ , \\} \\end{align*}"}
-{"id": "7572.png", "formula": "\\begin{align*} \\pi _ \\Omega ( u , v ) = \\lim _ { k \\to \\infty } r _ \\Omega ( \\cal F ^ { - 1 } ( \\psi _ k \\cal F u ' ) \\cdot \\cal F ^ { - 1 } ( \\psi _ k \\cal F v ' ) ) , \\end{align*}"}
-{"id": "4361.png", "formula": "\\begin{align*} A _ j ^ k = \\{ z \\in \\operatorname { s u p p } ( a _ j ) ; j \\Lambda _ 1 ( z ) \\} \\end{align*}"}
-{"id": "10037.png", "formula": "\\begin{align*} \\gamma _ { \\Q _ p } ( D ^ n , \\psi _ p ) \\gamma _ { \\Q _ p } ( \\psi ) ^ { 2 n } & = \\gamma _ { \\Q _ p } ( D , \\psi _ p ) ^ n ( D , D ) _ p ^ { \\frac { n ( n - 1 ) } 2 } \\gamma _ { \\Q _ p } ( - 1 , \\psi _ p ) ^ { - n } \\\\ & = \\gamma _ { \\Q _ p } ( - D , \\psi _ p ) ^ n ( - D , - 1 ) _ p ^ n ( D , D ) _ p ^ { \\frac { n ( n - 1 ) } 2 } . \\end{align*}"}
-{"id": "1329.png", "formula": "\\begin{align*} \\begin{bmatrix} \\pi ( S _ e S _ e ^ * ) & 0 \\\\ 0 & * \\end{bmatrix} = \\rho ( S _ e S _ e ^ * ) = \\rho ( S _ e ) \\rho ( S _ e ) ^ * = \\begin{bmatrix} \\pi ( S _ e ) \\pi ( S _ e ) ^ * + X _ e X _ e ^ * & * \\\\ * & * \\end{bmatrix} , \\end{align*}"}
-{"id": "8106.png", "formula": "\\begin{align*} \\hat { h } _ { i , t } = \\frac { k } { n } \\alpha _ t ( g _ i ) \\end{align*}"}
-{"id": "4375.png", "formula": "\\begin{align*} C _ z M _ f C _ { - z } = M _ { f \\circ \\tau _ z } . \\end{align*}"}
-{"id": "1737.png", "formula": "\\begin{align*} S _ \\lambda S _ \\lambda ^ * f = \\chi _ { Z ( \\lambda ) } \\cdot f \\end{align*}"}
-{"id": "8090.png", "formula": "\\begin{align*} \\sup \\{ | f ( z ) | : z \\in U \\setminus D \\} < 1 = \\lim _ { z \\to p } f ( z ) . \\end{align*}"}
-{"id": "1055.png", "formula": "\\begin{align*} \\Omega _ { k } : = \\left \\{ P _ { k - 1 } , T P _ { k - 1 } , \\ldots , T ^ { d _ { k } - 1 } P _ { k - 1 } , P _ { k } , T P _ { k } , \\ldots , T ^ { d _ { k - 1 } - 1 } P _ { k } \\right \\} , \\end{align*}"}
-{"id": "8028.png", "formula": "\\begin{align*} & \\Big [ \\int _ 0 ^ 1 \\int _ { \\Omega } \\sup _ { R \\in \\mathcal { D } _ { \\mu } } \\Big ( \\frac { 1 } { | R | } \\int _ { R } \\sum _ { k = \\mu } ^ { \\infty } { 2 ^ { s k q } \\big | \\Pi _ k T _ { [ a ] } f _ L ^ { \\omega , t , \\mu } ( x ) \\big | ^ q } d x \\Big ) d \\lambda d t \\Big ] ^ { 1 / q } \\\\ & \\gtrsim L ^ { - ( s + d - d / q ) / 2 d } \\big ( \\log { L } \\big ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "1898.png", "formula": "\\begin{align*} G ( x ) & \\le c F ( s ) + \\sum _ { i = 1 } ^ d \\big ( c F _ i ( x _ i ) - c F _ i ( s _ i ) \\big ) ^ + \\\\ & \\le \\pi _ s + \\sum _ { i = 1 } ^ d \\big ( F ^ \\ast _ i ( x _ i ) - F ^ \\ast _ i ( s _ i ) \\big ) ^ + , \\end{align*}"}
-{"id": "6331.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big | _ { t = 0 } F ( t \\eta ) = L ( \\eta ) , \\end{align*}"}
-{"id": "7.png", "formula": "\\begin{align*} \\dim _ K L ( D _ S ) = \\dim _ { K ' } L ( D _ { K ' S } ) . \\end{align*}"}
-{"id": "2895.png", "formula": "\\begin{align*} \\tau _ s ^ { j + 1 } ( m ) = \\left \\{ \\begin{array} { l l } \\tau _ s ^ j ( m ) & m < | \\tau _ s ^ j | \\\\ j ^ * & m \\geq | \\tau _ s ^ j | \\delta _ s ^ j ( m ) = 1 \\\\ 0 & m \\geq | \\tau _ s ^ j | \\delta _ s ^ j ( m ) = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "7002.png", "formula": "\\begin{align*} \\lambda _ 1 \\leq \\lambda _ 2 \\leq . . . \\leq \\lambda _ n = \\epsilon ^ { - 1 / 2 } , \\end{align*}"}
-{"id": "1024.png", "formula": "\\begin{align*} \\partial _ t \\vert \\vec u \\vert ^ 2 & = \\Delta ( \\vert \\vec u \\vert ^ 2 ) - 2 \\vert \\vec \\nabla \\otimes \\vec u \\vert ^ 2 \\\\ & - 2 \\lim _ { \\epsilon \\rightarrow 0 } \\left ( \\vec u _ \\epsilon \\cdot \\varphi _ \\epsilon * ( \\vec u \\cdot \\vec \\nabla \\vec u ) + \\lim _ { \\eta \\rightarrow 0 } ( ( p * \\xi _ { \\eta , \\epsilon } ) \\vec u _ { \\epsilon , \\eta } ) \\right ) + 2 \\vec u \\cdot \\vec f . \\end{align*}"}
-{"id": "4326.png", "formula": "\\begin{align*} & \\lambda _ \\ell \\eta ^ k _ \\ell + \\sum _ { j = 1 } ^ { k } \\big ( T ( \\varphi _ { p , k } ) y _ j , y _ \\ell \\big ) \\eta _ j ^ k = - \\big ( T ( \\varphi _ { p , k } ) , y _ \\ell \\big ) \\ , , \\ell = 1 , \\cdots k \\ , , \\end{align*}"}
-{"id": "9842.png", "formula": "\\begin{align*} ( \\hat { f } ) ^ { \\prime } ( y ) = ( - i x f ( x ) ) \\hat { \\ , } ( y ) . \\end{align*}"}
-{"id": "1233.png", "formula": "\\begin{align*} \\lim \\limits _ { \\substack { y \\to x \\\\ \\ , y \\in \\Gamma ( x ) } } \\nabla u ( y ) = \\nabla u ( x ) \\mbox { e x i s t s f o r $ \\mathcal { H } ^ { n - 1 } $ a l m o s t e v e r y $ x \\in E . $ } \\end{align*}"}
-{"id": "1451.png", "formula": "\\begin{align*} & \\{ \\phi _ r ( x _ { I } ) \\ ; | \\ ; \\ell ( I ) - r + 1 = 0 \\} , \\\\ & \\{ \\phi _ r ( x _ { I } ) \\ ; | \\ ; \\ell ( I ) - r + 1 > 0 \\mbox { a n d $ \\ell ( I ) - r + 1 $ i s e v e n } \\} , \\\\ & \\{ \\phi _ r ( x _ { I } ) \\ ; | \\ ; \\ell ( I ) - r + 1 > 0 \\mbox { a n d $ \\ell ( I ) - r + 1 $ i s o d d } \\} , \\end{align*}"}
-{"id": "6671.png", "formula": "\\begin{align*} \\frac { \\kappa _ S ( x ) ^ { \\frac { 1 } { n + 1 } } } { \\langle x , u _ S ( x ) \\rangle } = \\left [ \\frac { \\kappa _ { S ^ { \\circ } } ( y ) ^ { \\frac { 1 } { n + 1 } } } { \\langle y , u _ { S ^ { \\circ } } ( y ) \\rangle } \\right ] ^ { - 1 } \\quad . \\end{align*}"}
-{"id": "8582.png", "formula": "\\begin{align*} \\sup _ { 0 < t < S _ \\lambda } \\int _ { { \\bf R } ^ N } e ^ { - \\lambda H _ 0 ( y ) ^ 2 / ( 1 - 4 \\lambda t ) } | u _ { m , n } ( y , t ) | \\ , d y \\le C I _ { m , n } \\end{align*}"}
-{"id": "3898.png", "formula": "\\begin{align*} D = \\frac { 2 ( \\beta + d ) } { \\beta + 2 } \\end{align*}"}
-{"id": "5504.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\int _ { r ^ n z } ^ \\infty y ^ { - m - 1 } u _ m ( y ) \\dd y } { ( r ^ n z ) ^ { - m - \\beta } \\ell ( r ^ n z ) } = { \\mathrm { B } } _ { - m - \\beta } p _ { 0 , m } ( z ) . \\end{align*}"}
-{"id": "6056.png", "formula": "\\begin{align*} \\Omega ^ { ( \\alpha , \\theta ) } ( Q _ { X Y U } ) & \\leq \\theta \\mathbb { E } _ { Q _ { X Y U } } \\Big [ \\omega _ { Q _ { X Y U } } ^ { ( \\alpha ) } ( X , Y | U ) \\Big ] \\\\ & = \\theta R ^ { ( \\alpha ) } ( Q _ { X Y U } ) . \\end{align*}"}
-{"id": "8190.png", "formula": "\\begin{align*} S _ 1 ^ 4 = \\abs { a } _ { \\infty } ^ 2 \\sum _ { q \\in a \\imath ^ { - 1 } \\cap B ( \\abs { a } R ) } \\abs { q } _ { \\infty } ^ { - 2 } \\abs { W _ { \\infty } ( a ( q a ^ { - 1 } y ) ) } ^ 4 . \\end{align*}"}
-{"id": "4270.png", "formula": "\\begin{align*} a _ p = \\frac { 1 } { a ^ p } \\left ( \\frac { \\beta _ 0 } { \\beta _ { m + 1 } } \\right ) ^ 2 . \\end{align*}"}
-{"id": "9992.png", "formula": "\\begin{align*} ( \\mathbb E _ 0 & \\otimes \\mathbb E _ 0 ) \\bigg [ \\mathbf E \\Big [ \\exp \\Big \\{ \\beta \\int _ 0 ^ { T - 1 } \\int _ { \\R ^ d } \\big ( \\phi ( y - W _ s ) + \\phi ( y - W _ s ^ { \\prime } ) \\big ) \\dot B ( s , \\d y ) \\d s - \\beta ^ 2 ( T - 1 ) V ( 0 ) \\Big \\} \\Big ] \\bigg ] \\\\ & = ( \\mathbb E _ 0 \\otimes \\mathbb E _ 0 ) \\bigg [ \\exp \\Big \\{ \\beta ^ 2 \\int _ 0 ^ { T - 1 } V ( W _ s - W _ s ^ { \\prime } ) \\d s \\Big \\} \\bigg ] \\end{align*}"}
-{"id": "7464.png", "formula": "\\begin{align*} [ s _ 1 , f s _ 2 ] _ E = f [ s _ 1 , s _ 2 ] _ E + \\rho _ E ( s _ 1 ) ( f ) s _ 2 \\end{align*}"}
-{"id": "418.png", "formula": "\\begin{align*} \\psi _ \\omega \\ : = \\phi _ \\omega ( \\ , \\cdot \\ , + i y _ \\omega ) - \\phi _ \\omega ( i y _ \\omega ) \\end{align*}"}
-{"id": "8745.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t ^ \\alpha v + G ( t , x , v , \\nabla v , \\nabla ^ 2 v ) = 0 \\quad & \\\\ \\nabla v \\cdot n ( x ) - \\rho = 0 \\quad & \\end{cases} \\end{align*}"}
-{"id": "4503.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| \\phi _ 0 ^ { n } f \\| = | f ( x _ 0 ) | , f \\in C ( X ) . \\end{align*}"}
-{"id": "9166.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { 3 } x ^ { i } f _ { 4 - i } \\left ( x ^ { 4 - i } \\right ) = 0 \\left ( x \\in K \\right ) . \\end{align*}"}
-{"id": "9230.png", "formula": "\\begin{align*} L = ( \\mathfrak { g } \\otimes A ) \\oplus ( V \\otimes B ) \\oplus ( V ' \\otimes B ' ) \\oplus ( S \\otimes C ) \\oplus ( S ' \\otimes C ' ) \\oplus ( \\Lambda \\otimes E ) \\oplus ( \\Lambda ' \\otimes E ' ) \\oplus D \\end{align*}"}
-{"id": "3854.png", "formula": "\\begin{align*} \\begin{aligned} & F ( \\nu ( t ) , t ) = 0 , \\qquad \\det D _ v ( \\nu ( t ) , t ) \\not = 0 , \\\\ & \\nu ' ( t ) = - \\left ( D _ v F ( \\nu ( t ) , t ) \\right ) ^ { - 1 } D _ t F ( \\nu ( t ) , t ) . \\end{aligned} \\end{align*}"}
-{"id": "394.png", "formula": "\\begin{align*} S = \\sum _ { n = 1 6 } ^ \\infty \\frac { 1 } { n \\ln n } \\mathbb P \\left ( | S _ n | > ( 1 + \\varepsilon ) \\sigma _ n \\sqrt { 2 \\ln \\ln \\ln n } \\right ) . \\end{align*}"}
-{"id": "4977.png", "formula": "\\begin{align*} \\zeta _ j : = \\left \\{ \\begin{array} { l l } \\displaystyle \\zeta \\left ( \\frac { 2 ^ { \\alpha j } \\omega _ j } { \\sum _ { k < j , k \\equiv j ( \\textrm { m o d } R ) } 2 ^ { \\alpha k } \\omega _ k } \\right ) & \\mbox { i f } \\sum _ { k < j , k \\equiv j ( \\textrm { m o d } R ) } 2 ^ { \\alpha k } \\omega _ k \\not \\equiv 0 , \\\\ 0 & \\mbox { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "1814.png", "formula": "\\begin{align*} \\Phi ^ { a , h } _ D : = a ^ { 1 5 / 8 } \\sum _ { x \\in D ^ a } \\sigma _ x \\delta _ x , \\end{align*}"}
-{"id": "7536.png", "formula": "\\begin{gather*} \\varphi ^ 1 \\varphi ^ 1 _ \\omega - \\varphi ^ 1 _ { \\omega \\omega } - \\frac { ( \\varphi ^ 2 ) ^ 2 } { \\omega } - \\frac 1 { \\omega ^ 3 } - \\frac \\omega 4 = 0 , \\\\ \\varphi ^ 1 \\varphi ^ 2 _ \\omega - \\varphi ^ 2 _ { \\omega \\omega } + \\frac { \\varphi ^ 1 \\varphi ^ 2 } { \\omega } + 2 \\frac { \\varphi ^ 2 } { \\omega ^ 2 } = 0 . \\end{gather*}"}
-{"id": "582.png", "formula": "\\begin{align*} H _ { F } ^ { \\infty } ( \\Omega ) = \\{ f \\in H ( \\Omega ) : f ^ { ( l ) } \\in H ^ { \\infty } ( \\Omega ) , \\ \\ l \\in F \\} \\end{align*}"}
-{"id": "6058.png", "formula": "\\begin{align*} \\lim _ { \\theta \\to 0 } F ^ { ( \\alpha , \\theta ) } = 0 . \\end{align*}"}
-{"id": "1335.png", "formula": "\\begin{align*} \\rho ^ * \\nu _ G & = \\rho ^ * \\big ( \\sum _ { i = 1 } ^ N ( - 1 ) ^ i x _ i d x _ 1 \\wedge \\ldots \\wedge \\widehat { d x _ i } \\wedge \\ldots \\wedge d x _ N \\big ) \\\\ & = \\prod _ { i = 1 } ^ m ( y _ i ^ * ) ^ { | E ( G _ i ) | - 1 } \\nu _ G . \\end{align*}"}
-{"id": "2910.png", "formula": "\\begin{align*} k \\in Z ^ c \\Leftrightarrow f ( \\langle n _ \\sigma , k \\rangle ) = 1 \\Leftrightarrow f _ { y , d } ( \\lambda ) = 1 . \\end{align*}"}
-{"id": "371.png", "formula": "\\begin{align*} L _ { n p } \\leq ( 2 C _ { p } ^ { \\prime } + 1 ) \\mathbb { E } | \\xi _ { 0 } | ^ { p } \\sum _ { r , s } | b _ { n , r , s } | ^ { p } = ( 2 C _ { p } ^ { \\prime } + 1 ) D _ { n p } \\mathbb { E } | \\xi _ { 0 } | ^ { p } . \\end{align*}"}
-{"id": "2110.png", "formula": "\\begin{align*} L ( F ) = \\sup _ { p \\in M } \\int _ 0 ^ 1 | \\partial _ s F ( p , s ) | d s . \\end{align*}"}
-{"id": "2071.png", "formula": "\\begin{align*} \\liminf _ { t \\rightarrow \\delta } | | d f ( \\cdot , t ) | | _ { C ^ 0 ( M ) } = + \\infty \\end{align*}"}
-{"id": "3271.png", "formula": "\\begin{align*} E ^ s = P ^ s \\frac { T } { T _ s } , \\end{align*}"}
-{"id": "7454.png", "formula": "\\begin{align*} A _ { c o n } : = A / A e _ 0 A \\cong \\frac { \\mathbb { C } \\langle \\beta , \\gamma \\rangle } { \\langle \\gamma ^ 3 - \\beta ^ 2 , \\beta \\gamma + \\gamma \\beta \\rangle } \\end{align*}"}
-{"id": "6810.png", "formula": "\\begin{align*} L ( \\phi _ n ) = h _ n \\textrm { i n } \\mathbb { S } ^ 2 _ { \\lambda } , \\end{align*}"}
-{"id": "6423.png", "formula": "\\begin{align*} \\Delta \\theta \\overset { } { = } d \\theta + \\left ( \\Delta \\theta - d \\theta \\right ) \\end{align*}"}
-{"id": "4525.png", "formula": "\\begin{align*} f ( \\lambda ) = \\langle f , k _ \\lambda \\rangle _ { H ^ 2 _ d } , f \\in H ^ 2 _ d . \\end{align*}"}
-{"id": "4250.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } \\Phi _ 1 } { p } \\left ( \\mathrm { d } \\log \\frac { t - \\lambda } { t - 1 } \\otimes _ { \\Phi } 1 \\right ) = \\mathrm { d } \\log \\frac { t - \\lambda } { t - 1 } , \\end{align*}"}
-{"id": "9452.png", "formula": "\\begin{align*} \\omega ( q ) & : = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { 2 n ^ 2 + 2 n } } { ( q ; q ^ 2 ) _ { n + 1 } ^ 2 } , \\nu ( q ) : = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ 2 + n } } { ( - q ; q ^ 2 ) _ { n + 1 } } , \\end{align*}"}
-{"id": "9724.png", "formula": "\\begin{align*} \\tilde { \\Phi } ( \\tilde { \\boldsymbol { \\gamma } } ^ { * } ; V _ a + \\widetilde { \\mathcal { V } } ( V _ a , Z _ { a } h ) Z _ { a } h ) = \\tilde { \\Phi } ( \\boldsymbol { \\gamma } ^ { * } ; V _ a ) + \\widetilde { \\mathcal { V } } ( V _ b , Z _ { b } h ) Z _ { b } h , \\end{align*}"}
-{"id": "4287.png", "formula": "\\begin{align*} \\zeta ^ { \\beta } = \\exp ( z \\cdot \\beta ) = \\prod _ { i = 1 } ^ { n } \\zeta _ i ^ { b _ i \\cdot \\beta } \\end{align*}"}
-{"id": "4109.png", "formula": "\\begin{align*} k _ { M , p } ( X ) = - \\frac { h ( X , X ) \\langle \\eta , \\xi \\rangle } { \\langle d u ^ { - 1 } _ { \\eta ( p ) } X , X \\rangle } , \\end{align*}"}
-{"id": "2621.png", "formula": "\\begin{align*} \\exists \\ , C \\geq 0 : h _ { \\overline \\rho } = C ( 0 , 1 ) . \\end{align*}"}
-{"id": "5061.png", "formula": "\\begin{align*} & B _ { 1 2 , 3 } = B _ { 1 3 , 2 } = 0 , \\\\ & B _ { i j , i } = \\frac { 3 b _ i } { b _ j - b _ i } C _ j , ~ ~ B _ { i i , j } = \\frac { b _ i - b _ k } { b _ j - b _ i } C _ j , ~ ~ i \\neq j , j \\neq k , i \\neq k . \\end{align*}"}
-{"id": "2719.png", "formula": "\\begin{align*} h ( 0 , x , v , z ) = h _ { i n } ( x , v , z ) , \\end{align*}"}
-{"id": "5616.png", "formula": "\\begin{align*} | L _ r - L _ u | & = | \\psi ( r , t , X _ r ) - \\psi ( u , t , X _ u ) | \\\\ & = | \\psi ( r , t , X _ r ) - \\psi ( r , t , \\psi ( u , r , X _ u ) ) | \\\\ & \\leq | | \\psi | | _ { 1 , R } | X _ r - \\psi ( u , r , X _ u ) | , \\end{align*}"}
-{"id": "3053.png", "formula": "\\begin{align*} ( w - w _ h , v ) _ H = ( w - w _ h , v - v _ h ) _ H + ( w - w _ h , v _ h ) _ H = : I _ 1 + I _ 2 . \\end{align*}"}
-{"id": "9868.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\displaystyle \\delta ^ 2 N & \\overset { ( a ) } { < } & \\frac { 1 } { \\gamma ^ { \\nu _ I } \\frac { \\eta c } { 4 } } \\left ( f ( x ^ 0 ) - f ^ m + \\frac { 1 } { T ^ { 0 } } g _ + ^ M - \\frac { 1 } { T ^ { 0 } } g _ + ^ M + \\frac { 1 } { T ^ { \\nu _ I } } g _ + ^ M \\right ) \\overset { ( b ) } { = } \\frac { 8 } { ( \\eta c ) ^ 2 T ^ { \\nu _ I } } \\max _ i \\{ L _ { \\nabla g _ i } \\} ( f ( x ^ 0 ) - f ^ m + \\frac { 1 } { T ^ { \\nu _ I } } g _ + ^ M ) \\end{array} \\end{align*}"}
-{"id": "2273.png", "formula": "\\begin{align*} \\Gamma _ { A _ { \\alpha } ( G _ 1 ) } ( x - \\alpha n _ 2 ) & = \\frac { n _ 1 } { x - \\alpha n _ 2 - r _ 1 } , \\\\ [ 0 . 3 c m ] \\Gamma _ { A _ { \\alpha } ( G _ 2 ) } ( x - \\alpha n _ 1 ) & = \\frac { n _ 2 } { x - \\alpha n _ 1 - r _ 2 } . \\end{align*}"}
-{"id": "3729.png", "formula": "\\begin{align*} \\gamma = \\sum _ k \\mu _ k ( p _ { m _ k } , p ' _ { m _ k } ) , \\end{align*}"}
-{"id": "10014.png", "formula": "\\begin{align*} \\mathbb { L } _ 0 = ( \\mathbb { L } _ { 0 , \\infty } , \\mathbb { L } _ { 0 , f } ) , \\mathbb { L } = ( \\mathbb { L } _ \\infty , \\mathbb { L } _ f ) \\end{align*}"}
-{"id": "2692.png", "formula": "\\begin{align*} W ( - 1 ) ^ { \\varphi = p } \\otimes _ { \\Z _ p } { W } \\rightarrow W \\end{align*}"}
-{"id": "4562.png", "formula": "\\begin{align*} & ( q - q ^ { - 1 } ) [ E _ { n - 1 } ( q _ 3 ^ { n / 2 - 1 } z _ { n - 1 } ) \\cdots E _ 1 ( q _ 3 ^ { - n / 2 + 1 } z _ 1 ) , F _ 1 ( w ) ] \\\\ & = E _ { n - 1 } ( q _ 3 ^ { n / 2 - 1 } z _ { n - 1 } ) \\cdots E _ { 2 } ( q _ 3 ^ { - n / 2 + 2 } z _ { 2 } ) \\Bigl ( \\delta \\bigl ( C q _ 3 ^ { n / 2 - 1 } \\frac { w } { z _ 1 } \\bigr ) K ^ + _ 1 ( w ) - \\delta \\bigl ( C q _ 3 ^ { - n / 2 + 1 } \\frac { z _ 1 } { w } \\bigr ) K ^ - _ 1 ( q _ 3 ^ { - n / 2 + 1 } z _ 1 ) \\Bigr ) \\ , . \\end{align*}"}
-{"id": "8366.png", "formula": "\\begin{align*} \\tilde { K } _ p = G ( \\Q _ p ) \\cap C ( V _ { \\Z _ p } ) ^ \\times , \\end{align*}"}
-{"id": "4884.png", "formula": "\\begin{align*} \\lambda ^ 2 _ m ( t ) : = \\prod _ { k = 1 } ^ { \\infty } ( 1 - t ) ^ { - m } = \\prod _ { k = 1 } ^ { \\infty } Z ( m , t ^ k ) = \\prod _ { k = 1 } ^ { \\infty } \\lambda _ 1 ( m , t ^ k ) \\end{align*}"}
-{"id": "6401.png", "formula": "\\begin{align*} \\mathcal { M } \\overset { } { = } \\left \\{ p \\left ( x | \\theta \\right ) : \\theta = \\left ( \\theta ^ { 1 } \\theta ^ { n } \\right ) \\in \\mathcal { D } _ { \\theta } \\right \\} \\end{align*}"}
-{"id": "2797.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } \\vert \\omega _ { i , n } \\vert \\leq T \\Lambda _ { n } , \\end{align*}"}
-{"id": "6804.png", "formula": "\\begin{align*} S ^ { ' } _ { \\rho } ( w _ { \\lambda } ) ( u ) = \\mathcal { L } \\left ( u - \\frac { \\int _ { \\mathbb { S } ^ 2 } e ^ { w _ { \\lambda } } u } { \\int _ { \\mathbb { S } ^ 2 } e ^ { w _ { \\lambda } } } \\right ) . \\end{align*}"}
-{"id": "2824.png", "formula": "\\begin{align*} F _ n ^ T ( \\Delta _ { \\infty } ^ { ( n + 1 ) } ) = \\Delta _ { \\infty } ^ { ( n ) } , n \\geq 1 . \\end{align*}"}
-{"id": "324.png", "formula": "\\begin{align*} f ( n ) = q ( n ) \\cdot f ( n + 1 ) + r ( n + 1 ) \\cdot f ( n + 2 ) , n \\in \\Z _ { \\ge 0 } . \\end{align*}"}
-{"id": "10012.png", "formula": "\\begin{align*} \\theta _ \\Lambda ( \\tau ) = \\sum _ { m \\in \\Q } R _ \\Lambda ( m ) q ^ m \\in M _ { n - 1 } ( \\omega _ \\Lambda ^ \\vee ) \\end{align*}"}
-{"id": "486.png", "formula": "\\begin{gather*} \\delta ' = \\delta \\left ( \\frac { t _ { m + 1 } } { | t | } \\right ) , \\kappa ' = \\kappa \\left ( \\frac { t _ { m + 1 } } { \\abs { t } } \\right ) . \\end{gather*}"}
-{"id": "8074.png", "formula": "\\begin{align*} \\Omega _ { 2 h } ^ D = \\{ x _ i = 2 i h \\ | \\ i = 0 , \\ldots , n / 2 \\} , \\mbox { a n d } \\Omega _ { 2 h } ^ P = \\{ x _ i = 2 i h \\ | \\ i = 0 , \\ldots , n \\} . \\end{align*}"}
-{"id": "485.png", "formula": "\\begin{align*} \\frac { \\partial ^ { j } } { \\partial \\abs { t } ^ j } \\sqrt { \\abs { t } ^ 2 + t _ { m + 1 } ^ 2 } = \\sum _ { \\ell _ 1 + 2 \\ell _ 2 = j } \\frac { j ! } { \\ell ! } { ( - 1 ) ^ { \\abs { \\ell } } } \\left ( { - \\frac { 1 } { 2 } } \\right ) _ { \\abs { \\ell } } ( \\abs { t } ^ 2 + t _ { m + 1 } ^ 2 ) ^ { \\frac { 1 } { 2 } - \\abs { \\ell } } ( 2 \\abs { t } ) ^ { \\ell _ 1 } . \\end{align*}"}
-{"id": "6493.png", "formula": "\\begin{align*} p _ { A ^ { \\prime } } ^ { } \\left ( x _ { A } ^ { \\prime } | \\mu _ { A } ^ { \\prime } \\right ) = \\frac { \\pi x _ { A } ^ { \\prime } } { 2 \\mu _ { A } ^ { \\prime 2 } } \\exp \\left ( - \\frac { \\pi x _ { A } ^ { \\prime 2 } } { 4 \\mu _ { A } ^ { \\prime 2 } } \\right ) , \\end{align*}"}
-{"id": "6447.png", "formula": "\\begin{align*} R _ { } ^ { } ( \\rho ) \\overset { } { = } \\frac { \\mathcal { C } _ { \\mathcal { M } } \\left ( \\tau \\right ) } { \\mathcal { C } _ { \\mathcal { M } } \\left . \\left ( \\tau \\right ) \\right \\vert _ { \\rho = 0 } } = \\sqrt { 1 + \\rho } \\end{align*}"}
-{"id": "8571.png", "formula": "\\begin{align*} & \\partial _ t h = \\ell s ^ { \\ell - 1 } H _ 0 ( y ) ^ 2 \\ge \\ell T _ 2 ^ { \\ell + \\sigma - 1 } s ^ { - \\sigma } H _ 0 ( y ) ^ 2 , \\\\ & C s ^ { - \\sigma } H ( \\nabla h ) ^ 2 = 4 C ( 1 + s ^ \\ell ) ^ 2 s ^ { - \\sigma } H _ 0 ( y ) ^ 2 \\le 4 C ( 1 + T _ 2 ^ \\ell ) ^ 2 s ^ { - \\sigma } H _ 0 ( y ) ^ 2 \\le \\partial _ t h , \\end{align*}"}
-{"id": "581.png", "formula": "\\begin{align*} \\mathcal { A } _ { F } ( \\Omega ) = \\{ f \\in H ( \\Omega ) : f ^ { ( l ) } \\in \\mathcal { A } ( \\Omega ) , \\ \\ l \\in F \\} . \\end{align*}"}
-{"id": "3684.png", "formula": "\\begin{align*} Z ^ { ( n ) \\circ } : = Z ^ { ( n ) } \\times _ C C ^ { \\circ } \\to C ^ { \\circ } \\end{align*}"}
-{"id": "6919.png", "formula": "\\begin{align*} \\int u _ { j } ( y ) ^ { 2 } d \\mu ( y ) = 1 \\end{align*}"}
-{"id": "6881.png", "formula": "\\begin{align*} \\lim _ { p \\to 0 } y _ { 1 } ( x _ { 1 } , x _ { 2 } , p , \\Phi _ { 1 } ( 2 ( 1 - \\sqrt { 1 - p } ) ) ) & = \\frac { x _ { 1 } + x _ { 2 } } { 2 } \\\\ \\lim _ { p \\to 0 } z ( x _ { 1 } , x _ { 2 } , p , \\Phi _ { 1 } ( 2 ( 1 - \\sqrt { 1 - p } ) ) ) & = \\frac { x _ { 1 } + x _ { 2 } } { 2 } \\\\ \\lim _ { p \\to 0 } y _ { 2 } ( x _ { 1 } , x _ { 2 } , p , \\Phi _ { 1 } ( 2 ( 1 - \\sqrt { 1 - p } ) ) ) & = \\frac { x _ { 1 } + x _ { 2 } } { 2 } \\end{align*}"}
-{"id": "9442.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ N \\frac { \\varphi ( n ) \\psi ( n ) } { n | { n } | _ { \\mathcal { D } } } \\ , = \\ , \\sum _ { n = 1 } ^ N \\left ( \\psi ( n ) - \\psi ( n + 1 ) \\right ) \\sum _ { m = 1 } ^ n \\frac { \\varphi ( m ) } { m | { m } | _ { \\mathcal { D } } } + \\psi ( N + 1 ) \\sum _ { m = 1 } ^ N \\frac { \\varphi ( m ) } { m | { m } | _ { \\mathcal { D } } } . \\end{align*}"}
-{"id": "5333.png", "formula": "\\begin{align*} { w _ { ( 1 ) } } _ { n + m , k } ^ { \\mathcal { Y } } w _ { ( 2 ) } = 0 \\ ; \\ ; \\mbox { f o r } \\ ; m \\in { \\mathbb N } \\ ; \\mbox { s u f f i c i e n t l y l a r g e } . \\end{align*}"}
-{"id": "5365.png", "formula": "\\begin{align*} S _ { X , Y } = ( X ) ( Y ) \\forall \\ Y . \\end{align*}"}
-{"id": "945.png", "formula": "\\begin{align*} R _ { 1 } & = \\sum _ { i = 1 } ^ { N } E \\left [ \\max _ { 1 \\leq k \\leq d } \\left | \\sum _ { j = 1 } ^ { N } \\gamma _ { k } ( i , j ) W ^ { ( i ) } _ j \\right | ^ 3 \\right ] ( E [ | Y _ i | ^ { 3 } ] + E [ | G _ i | ^ 3 ] ) , \\\\ R _ { 2 } & = \\max _ { 1 \\leq k , l \\leq d } \\left | \\mathfrak { C } ( k , l ) - E [ Q _ { k } ( G ) Q _ { l } ( G ) ] \\right | , \\\\ R _ { 3 } & = \\max _ { 1 \\leq k \\leq d } \\sqrt { E [ Q _ { k } ( G ) ^ 4 ] - 3 E [ Q _ { k } ( G ) ^ 2 ] ^ 2 } . \\end{align*}"}
-{"id": "7613.png", "formula": "\\begin{align*} Q _ { r } ^ \\lambda ( \\bar z ) : = \\left \\{ \\begin{array} { l c l l } B _ r ( \\bar x ) \\times ( \\bar t - \\lambda ^ { 2 - p } r ^ 2 , \\bar t + \\lambda ^ { 2 - p } r ^ 2 ) & p \\geq 2 , \\\\ B _ { \\lambda ^ { \\frac { p - 2 } { 2 } } r } ( \\bar x ) \\times ( \\bar t - r ^ 2 , \\bar t + r ^ 2 ) & 1 < p < 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "1290.png", "formula": "\\begin{align*} \\mathbf { m } ( t ) = \\mbox { C a p } _ { \\mathcal { A } } ( E _ t ) ^ { \\frac { 1 } { n - p } } . \\end{align*}"}
-{"id": "176.png", "formula": "\\begin{align*} K ( \\mathbf { L } ) = \\langle K ( L ) , \\leq , \\vee , 0 \\rangle , \\end{align*}"}
-{"id": "2604.png", "formula": "\\begin{align*} H _ V ( \\R ) : = \\overline { L ^ 2 ( \\R ) } ^ { \\| \\cdot \\| _ V } \\end{align*}"}
-{"id": "1897.png", "formula": "\\begin{align*} F ( x ) - F ( s ) \\le \\sum _ { i = 1 } ^ d \\big ( F _ i ( x _ i ) - F _ i ( s _ i ) \\big ) ^ + . \\end{align*}"}
-{"id": "3690.png", "formula": "\\begin{align*} D _ { \\tilde P _ W [ n ] } = \\sum ( - d _ { I , j } ) D _ { I , j } . \\end{align*}"}
-{"id": "6898.png", "formula": "\\begin{align*} - \\Delta ( u \\circ F _ { j j } ) & = \\frac { 1 } { r _ 0 \\mu _ 0 } ( - \\Delta u ) \\circ F _ { j j } \\\\ - \\Delta ( u \\circ F _ { j k } ) & = \\frac { 1 } { r _ 1 \\mu _ 1 } ( - \\Delta u ) \\circ F _ { j k } & & j \\neq k \\end{align*}"}
-{"id": "3212.png", "formula": "\\begin{align*} f _ k ( x ) = \\rho _ { k } * f ( x ) , g _ k ( x ) = \\rho _ { k } * g ( x ) , k \\ge 0 \\ . \\end{align*}"}
-{"id": "4939.png", "formula": "\\begin{align*} \\widehat { A } ( M ) = \\prod _ { i = 1 } ^ n \\frac { x _ i / 2 } { ( x _ i / 2 ) } . \\end{align*}"}
-{"id": "2980.png", "formula": "\\begin{align*} \\gamma \\int _ \\Omega \\nabla u _ n . \\nabla T _ k ( u _ n ) T _ k ^ { \\gamma - 1 } ( u _ n ) = \\int _ { \\Omega } h _ n \\left ( u _ n + \\frac { 1 } { n } \\right ) f _ n T _ k ^ \\gamma ( u _ n ) + \\int _ { \\Omega } T _ k ^ \\gamma ( u _ n ) \\mu _ n . \\end{align*}"}
-{"id": "8112.png", "formula": "\\begin{align*} \\varphi ( x ) = \\{ s \\in F : s x \\in [ x ] _ { R _ { W , E } } \\} , \\end{align*}"}
-{"id": "4161.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { \\tau = 0 } ^ { n - 2 } e _ { i , j + \\tau } ^ { n - \\tau - 1 } \\end{align*}"}
-{"id": "319.png", "formula": "\\begin{align*} & \\frac { v _ { 1 } } { \\displaystyle 1 - \\sum _ { l = 2 } ^ { d } v _ l } \\ge v _ 1 = \\displaystyle u _ 0 - z _ 0 \\frac { u _ k } { z _ k } \\ge ( 1 - t ) ^ d - \\delta . \\end{align*}"}
-{"id": "1390.png", "formula": "\\begin{align*} \\{ x : V ( x ) = y \\} = \\{ x : U ( y - 1 ) < x \\le U ( y ) \\} , y \\ge 1 , \\end{align*}"}
-{"id": "617.png", "formula": "\\begin{align*} B ( z , w ) = \\sum _ { k \\ge - 1 } \\frac { k + 2 } { \\pi } z ^ k \\bar w ^ k \\end{align*}"}
-{"id": "6673.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\frac { 1 } { \\delta ^ { \\frac { 2 } { n + 1 } } } \\cdot \\frac { \\| x ^ { \\delta } \\| _ 2 - \\| x \\| _ 2 } { \\| x \\| _ 2 } = \\frac { n ^ { \\frac { 2 } { n + 1 } } ( n + 1 ) ^ { \\frac { 2 } { n + 1 } } } { 2 } \\left ( \\frac { | K | _ n } { | B _ 2 ^ { n - 1 } | _ { n - 1 } } \\right ) ^ { \\frac { 2 } { n + 1 } } \\frac { \\kappa ( x ) ^ { \\frac { 1 } { n + 1 } } } { \\langle x , u ( x ) \\rangle } = n ^ { \\frac { 2 } { n + 1 } } G ( x ) \\quad , \\end{align*}"}
-{"id": "5525.png", "formula": "\\begin{align*} b ( m ) = b ( 3 k + 2 ) & \\leqslant h ( 3 k + 1 ) + ( \\sqrt { 2 } + 1 ) \\lfloor \\log _ 3 ( 3 k + 1 ) \\rfloor \\\\ & \\leqslant h ( 3 k + 2 ) + ( \\sqrt { 2 } + 1 ) \\lfloor \\log _ 3 ( 3 k + 2 ) \\rfloor . \\end{align*}"}
-{"id": "2433.png", "formula": "\\begin{align*} v ^ i ( x _ 1 ^ \\ast , y _ 1 ^ \\ast ) & = \\sum \\limits _ { x _ 2 , \\dots , x _ { k - 1 } \\in X } p ( x _ { k - 1 } | x _ { k - 2 } , \\mu _ { k - 2 } ^ \\ast ) \\dots p ( x _ 2 | x _ 1 ^ \\ast , \\mu _ 1 ^ \\ast ) \\\\ & \\sum \\limits _ { \\substack { x _ k \\in X \\\\ y _ k \\in C _ i } } p ( x _ k | x _ { k - 1 } , \\mu _ { k - 1 } ^ \\ast ) q ( y _ k | y _ { k - 1 } ^ \\ast , \\nu ( y _ { k - 1 } ^ \\ast ) ) v ^ i ( x _ k , y _ k ) . \\end{align*}"}
-{"id": "3132.png", "formula": "\\begin{align*} & ( q + 1 ) ( t - 2 ) ( m + 1 ) [ ( 2 ( k - 1 ) - 5 ] + ( 2 - q ) ( t - 2 ) m [ ( 2 ( k - 1 ) - 5 ] \\\\ & = ( t - 2 ) ( 2 k - 7 ) [ ( q + 1 + 2 - q ) m + q + 1 ] \\\\ & = ( t - 2 ) ( 2 k ^ 2 - 5 k - 7 ) \\\\ & \\ge k ^ 2 - k - 2 \\end{align*}"}
-{"id": "5531.png", "formula": "\\begin{align*} x \\left ( \\tau + t \\right ) = \\rho x \\left ( t \\right ) \\end{align*}"}
-{"id": "1804.png", "formula": "\\begin{align*} X ' _ k : = \\frac { \\prod _ { i \\to k } X _ i + \\prod _ { k \\to j } X _ j } { X _ k } , \\end{align*}"}
-{"id": "985.png", "formula": "\\begin{align*} & \\sqrt { n h } \\{ M _ n ( t ) - \\sigma ^ 2 ( ( t - \\ell h ) _ + ) M ^ 0 _ n ( t ) \\} \\\\ & = 2 \\sqrt { n h } \\sum _ { i = 1 } ^ n K _ h ( t _ { i - 1 } - t ) \\int _ { t _ { i - 1 } } ^ { t _ i } \\int _ { t _ { i - 1 } } ^ s \\{ \\sigma ( r ) - \\sigma ( ( t - \\ell h ) _ + ) \\} d B _ r \\sigma ( s ) d B _ s \\\\ & \\quad + 2 \\sqrt { n h } \\sum _ { i = 1 } ^ n K _ h ( t _ { i - 1 } - t ) \\int _ { t _ { i - 1 } } ^ { t _ i } \\int _ { t _ { i - 1 } } ^ s \\sigma ( ( t - \\ell h ) _ + ) d B _ r \\{ \\sigma ( s ) - \\sigma ( ( t - \\ell h ) _ + ) d B _ s \\\\ & = : \\mathbf { A } _ n ( t ) + \\mathbf { B } _ n ( t ) . \\end{align*}"}
-{"id": "1274.png", "formula": "\\begin{align*} \\mu ( \\{ \\xi _ i \\} ) = c _ i = \\int _ { F _ i } f ( \\nabla U ) \\ , d \\mathcal { H } ^ { n - 1 } \\mbox { f o r } 1 \\leq i \\leq m . \\end{align*}"}
-{"id": "2656.png", "formula": "\\begin{align*} P ( s u p p ( T ) ) = s u p p ( P _ \\ast ( T ) ) . \\end{align*}"}
-{"id": "915.png", "formula": "\\begin{align*} E [ F \\delta ( u ) ] = E [ \\langle D F , u \\rangle _ H ] . \\end{align*}"}
-{"id": "8932.png", "formula": "\\begin{gather*} C ' = \\bigcup _ k Q ^ { - 1 } B ^ * \\cap l ^ { - k } C . \\end{gather*}"}
-{"id": "720.png", "formula": "\\begin{align*} \\Gamma \\left ( \\frac { T _ + } { T _ - } \\right ) & = \\alpha _ - \\frac { T _ + } { T _ - } + 2 \\alpha _ - \\left ( \\frac { 2 T _ - } { T _ + } + \\frac { T _ { \\rm i } } { T _ + } \\right ) \\frac { T _ + } { T _ - } + 3 \\left ( 1 - \\frac { T _ + } { T _ - } \\right ) > 0 . \\end{align*}"}
-{"id": "162.png", "formula": "\\begin{align*} & z = x + i y , \\\\ & w _ \\tau = u _ \\tau + i v _ \\tau , 1 \\leq \\tau \\leq 2 k - 2 , \\\\ & \\zeta _ \\sigma = \\xi _ \\sigma + i \\eta _ \\sigma , 1 \\leq \\sigma \\leq k , \\end{align*}"}
-{"id": "8542.png", "formula": "\\begin{align*} d s ^ 2 = 2 e ^ { v ( x , y ) } ( d x ^ 2 + d y ^ 2 ) \\end{align*}"}
-{"id": "2279.png", "formula": "\\begin{align*} K ( f ) ( p ) : = - f _ { x y } ( p ) ^ 3 f _ { x x x } ( p ) + & 3 f _ { x x } ( p ) f _ { x y } ( p ) ^ 2 f _ { x x y } ( p ) \\\\ & - 3 f _ { x x } ( p ) ^ 2 f _ { x y } ( p ) f _ { x y y } ( p ) + f _ { x x } ( p ) ^ 3 f _ { y y y } ( p ) . \\end{align*}"}
-{"id": "2054.png", "formula": "\\begin{align*} E _ H ( f ) = \\int _ M e _ H ( f ) \\theta \\wedge ( d \\theta ) ^ m \\end{align*}"}
-{"id": "9767.png", "formula": "\\begin{align*} d _ { i } = \\frac { s ^ { ( i - 1 ) } h - ( y _ i - y _ { i - 1 } ) } { s } . \\end{align*}"}
-{"id": "3933.png", "formula": "\\begin{align*} u \\frac { \\partial u } { \\partial x } | _ { ( x _ j , t _ n ) } = \\frac { 1 } { 3 } \\left [ u _ j ^ n ( u _ x ) _ j ^ { n + 1 } + ( u _ j ^ n u _ j ^ { n + 1 } ) _ x \\right ] \\end{align*}"}
-{"id": "1288.png", "formula": "\\begin{align*} \\nabla k ( \\nabla h ( x ) ) = x / h ( x ) \\mbox { w h i l e } k ( \\nabla h ) = 1 . \\end{align*}"}
-{"id": "1649.png", "formula": "\\begin{align*} \\Phi _ \\lambda = \\Phi _ { \\tau _ { \\lambda _ 1 } \\circ \\tau _ { \\lambda _ 2 } } = \\frac { d ( \\mu \\circ \\tau _ { \\lambda _ 1 } \\circ \\tau _ { \\lambda _ 2 } ) } { d ( \\mu \\circ \\tau _ { \\lambda _ 2 } ) } \\ , \\frac { d ( \\mu \\circ \\tau _ { \\lambda _ 2 } ) } { d \\mu } = ( \\Phi _ { \\tau _ { \\lambda _ 1 } } \\circ \\tau _ { \\lambda _ 2 } ) ( \\Phi _ { \\tau _ { \\lambda _ 1 } } ) , \\end{align*}"}
-{"id": "6567.png", "formula": "\\begin{align*} | ( F _ { \\xi } - s ( F _ { \\xi } ) ) ^ { \\circ } | _ 1 = 2 \\cdot \\left ( \\frac { | F _ { \\xi } | _ 1 } { 2 } \\right ) ^ { - 1 } = 2 \\frac { \\sqrt { 1 - \\varepsilon ^ 2 } } { 1 - \\varepsilon } . \\end{align*}"}
-{"id": "4627.png", "formula": "\\begin{align*} \\left | \\sum _ { n = 0 } ^ { N - 1 } f ( T ^ n x ) \\right | < N ^ \\gamma \\end{align*}"}
-{"id": "3093.png", "formula": "\\begin{align*} t ^ * : = \\min \\left \\{ t \\in ( 0 , T ) : \\ , \\int _ { 0 } ^ t A ( s ) \\ , \\mathrm { d } s = 0 \\right \\} \\ , . \\end{align*}"}
-{"id": "693.png", "formula": "\\begin{align*} \\varphi ( x , \\mu _ n ) = \\kappa _ n \\psi ( x , \\mu _ n ) . \\end{align*}"}
-{"id": "9520.png", "formula": "\\begin{align*} a _ { v , k } ^ { ( m , d ) } = \\sum \\limits _ { l \\mid m } \\sum \\limits _ { t = k - \\tilde { d } + 1 } ^ { k + \\tilde { d } - 1 } c _ { k , t } \\cdot a _ { v - \\tilde { d } , t } ^ { ( l , \\tilde { d } ) } , \\end{align*}"}
-{"id": "9123.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } ( - 1 ) ^ { i } \\binom { n + 1 } { i } x ^ { i } A \\left ( x ^ { n + 1 - i } \\right ) = 0 \\left ( x \\in R \\right ) . \\end{align*}"}
-{"id": "2509.png", "formula": "\\begin{align*} \\begin{aligned} \\psi ( \\xi ) & \\equiv \\varpi \\chi _ { R } ( | \\xi | ) + \\frac { 1 } { 4 } ( \\xi , \\sigma \\xi ) - \\frac { 1 } { 2 } \\nabla _ \\xi \\cdot \\left [ \\sigma \\xi \\right ] \\\\ & = \\varpi \\chi _ { R } ( | \\xi | ) + \\left \\{ \\frac { 3 } { 4 } \\lambda _ 1 ( \\xi ) \\left | \\xi \\right | ^ 2 - \\lambda _ 2 ( \\xi ) - \\frac { 1 } { 2 } \\lambda _ 1 ( \\xi ) \\right \\} . \\end{aligned} \\end{align*}"}
-{"id": "729.png", "formula": "\\begin{align*} D _ - = \\frac { u _ - ^ 2 } { a ^ 2 \\theta _ - } . \\end{align*}"}
-{"id": "1553.png", "formula": "\\begin{align*} \\mu ( \\mathcal { D } ) \\ < \\ \\frac { 8 k ^ 2 } { N } \\ \\leq \\ \\varepsilon . \\end{align*}"}
-{"id": "4315.png", "formula": "\\begin{align*} F _ 0 ( s , r ) = s \\log s + r \\log r + ( 1 - s - r ) \\log ( 1 - s - r ) \\end{align*}"}
-{"id": "3518.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l l } & - d { p } ^ { 1 0 } ( t ) = \\{ p ^ { 1 0 } ( t ) - 2 \\beta ^ 0 \\bar { X } ^ { 1 0 } { ( t ) } \\} d t , \\ t \\in ( t _ { i - 1 } , t _ { i } ) , \\\\ & p ^ { 1 0 } ( t _ { i } ) = - 2 \\beta ^ 0 \\bar { X } ^ { 1 0 } ( t _ { 1 0 } ) 1 _ { i = 1 0 } ( i ) - \\beta ^ i + p ^ { 1 0 } ( t _ { i } ^ { + } ) , i = 1 , 2 , \\ldots , 1 0 , \\end{array} \\end{align*}"}
-{"id": "2175.png", "formula": "\\begin{align*} G _ d ( r , t ) = \\int _ 0 ^ r s ^ { 1 - d } [ \\nu _ d ( s ) ] ^ { - 1 } \\left ( \\int _ 0 ^ s \\tau ^ { d - 1 } \\nu _ d ( \\tau ) F _ d ^ 1 ( \\tau , t ) \\ , d \\tau \\right ) \\ , d s . \\end{align*}"}
-{"id": "2969.png", "formula": "\\begin{align*} \\frac { \\psi _ k - \\sigma ^ 2 _ k + \\eta _ k \\sum _ { j = 1 } ^ { 2 } p _ j | h _ { k j } | ^ { 2 } } { \\sum _ { j = 1 } ^ { 2 } p _ j | h _ { k j } | ^ { 2 } } \\leq 1 , \\quad \\forall k \\in \\{ 1 , 2 \\} . \\end{align*}"}
-{"id": "881.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { N _ n } E \\left [ \\max _ { 1 \\leq k \\leq d _ n } \\left | \\sum _ { j = 1 } ^ { N _ n } \\gamma _ { n , k } ( i , j ) W ^ { ( i ) } _ j \\right | ^ 3 \\right ] \\leq 2 d _ n ^ { 3 / p } ( p - 1 ) ^ { 3 / 2 } \\sup _ { i \\in \\mathbb { N } } \\| Y _ i \\| _ p ^ 3 \\sum _ { i = 1 } ^ { N _ n } \\max _ { 1 \\leq k \\leq d _ n } \\left ( \\sum _ { j = 1 } ^ { N _ n } \\gamma _ { n , k } ( i , j ) ^ 2 \\right ) ^ { 3 / 2 } \\end{align*}"}
-{"id": "4322.png", "formula": "\\begin{align*} - \\Delta ( n _ { a } - n _ { b } ) + ( T ( \\varphi _ { p , a } ) - T ( \\varphi _ { p , b } ) ) n _ { a } + T ( \\varphi _ { p , b } ) ( n _ { a } - n _ { b } ) = 0 , \\end{align*}"}
-{"id": "2181.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } s w ( \\xi , s ) = \\frac { 1 } { c _ * } \\left [ \\int _ 0 ^ \\infty w ( r , 0 ) U _ d ( r ) r \\ , d r \\right ] e ^ { - \\frac { \\xi ^ 2 } { 4 } } = 2 m ( \\varphi ) \\psi _ d ( \\xi ) \\end{align*}"}
-{"id": "2595.png", "formula": "\\begin{align*} S ( \\phi _ a ' f ) = \\frac { 1 - a } 2 \\bigg [ S \\Big ( \\frac { \\phi _ a ( t ) f ( t ) } t \\Big ) - S \\Big ( \\frac { \\phi _ a ( t ) f ( t ) } { 1 - t } \\Big ) \\bigg ] . \\end{align*}"}
-{"id": "2298.png", "formula": "\\begin{align*} \\partial _ t u ^ N + \\nu A ^ s u ^ N + \\mathcal { P } _ N \\mathcal { P } ^ \\alpha [ u ^ N \\cdot \\nabla u ^ N + \\mathcal { U } ^ \\alpha ( u ^ N , u ^ N ) ] = 0 , u ^ N | _ { t = 0 } = \\mathcal { P } _ N u _ 0 . \\end{align*}"}
-{"id": "7106.png", "formula": "\\begin{align*} \\frac { \\partial \\tilde { s } } { \\partial t } ~ = ~ F _ * ^ { \\alpha } ( \\bar { \\nabla } _ i \\bar { \\nabla } _ j \\tilde { s } + \\tilde { s } \\bar { g } _ { i j } ) - \\tilde { s } \\end{align*}"}
-{"id": "9370.png", "formula": "\\begin{align*} r _ n ^ { ( k + 1 ) } & = \\frac { \\iint _ { \\ , \\mathbb { C } _ n ^ { ( k ) } } r \\cos { ( \\phi - \\phi _ n ) } f ( r ) \\ , \\mathrm { d } r \\ , \\mathrm { d } \\phi } { \\iint _ { \\ , \\mathbb { C } _ n ^ { ( k ) } } f ( r ) \\ , \\mathrm { d } r \\ , \\mathrm { d } \\phi } . \\end{align*}"}
-{"id": "1162.png", "formula": "\\begin{align*} B _ { k , 0 } & = D _ { k - 1 } B _ { k - 1 , 0 } = D _ { k - 1 } D _ { k - 2 } \\cdots D _ 2 \\\\ B _ { k , i } & = D _ { k - 1 } B _ { k - 1 , i } + B ^ 2 _ { k - 1 , i - 1 } \\\\ B _ { k , k - 1 } & = B _ { k - 1 , k - 2 } = \\cdots = B _ { 2 , 1 } = 1 . \\end{align*}"}
-{"id": "5359.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ M z _ 2 ^ { r _ k } ( z _ 1 - z _ 2 ) ^ { s _ k } ( \\log z _ 2 ) ^ { i _ k } ( \\log ( z _ 1 - z _ 2 ) ) ^ { j _ k } f _ k ( \\frac { z _ 1 - z _ 2 } { z _ 2 } ) \\end{align*}"}
-{"id": "3264.png", "formula": "\\begin{align*} ( \\mathcal J ^ { 2 n - p } _ k ( X ) ) ^ \\vee = ( L _ 0 ( X ) ) ^ { \\vee } \\oplus \\cdots \\oplus ( L _ { [ { p - 1 \\over 2 } ] } ( X ) ) ^ { \\vee } . \\end{align*}"}
-{"id": "9597.png", "formula": "\\begin{align*} g _ { k , M } ( x ) & = \\sum _ { j = 1 } ^ { N _ J } \\int _ { Q _ j } D _ k ( x , y ) g ( y ) d \\mu ( y ) \\\\ & = \\sum _ { j = 1 } ^ { N _ J } \\int _ { Q _ j } [ D _ k ( x , y ) - D _ k ( x , y _ j ) ] g ( y ) d \\mu ( y ) \\\\ & \\quad + \\sum _ { j = 1 } ^ { N _ J } D _ k ( x , y _ j ) \\int _ { Q _ j } g ( y ) d \\mu ( y ) \\\\ & : = g ^ 1 _ { k , M , J } ( x ) + g ^ 2 _ { k , M , J } ( x ) , \\end{align*}"}
-{"id": "6420.png", "formula": "\\begin{align*} d l _ { \\xi \\rightarrow } ^ { 2 } \\overset { } { = } \\frac { \\xi ^ { 2 } } { \\left ( 1 + \\xi \\right ) ^ { 2 } } \\Delta \\theta ^ { 2 } \\end{align*}"}
-{"id": "918.png", "formula": "\\begin{align*} L ^ { - 1 } F = - \\sum _ { q = 1 } ^ \\infty \\frac { 1 } { q } J _ q F , F \\in L ^ 2 ( W ) . \\end{align*}"}
-{"id": "5443.png", "formula": "\\begin{align*} \\mathcal { P } _ { r } = \\Big \\{ p : ( 0 , \\infty ) \\to ( 0 , \\infty ) : & \\ , \\inf _ { x \\in [ 1 , r ] } p ( x ) > 0 , \\ p \\\\ & p ( x r ) = p ( x ) , \\ \\forall x > 0 \\Big \\} . \\end{align*}"}
-{"id": "494.png", "formula": "\\begin{align*} V _ s = - \\frac { 1 } { 4 } \\frac { \\abs * { \\nabla _ \\mathcal { H } p _ s } ^ 2 } { p _ s ^ 2 } - \\frac { 1 } { 2 } \\frac { \\mathcal { L } p _ s } { p _ s } = - \\frac { 1 } { 4 } \\frac { \\sum _ { j = 1 } ^ { 2 n } ( X _ j p _ s ) ^ 2 } { p _ s ^ 2 } { + \\frac { 1 } { 2 } \\frac { \\sum _ { j = 1 } ^ { 2 n } X _ j ^ 2 p _ s } { p _ s } . } \\end{align*}"}
-{"id": "3658.png", "formula": "\\begin{align*} D _ { p o s } = \\sum _ I D _ { e _ I } , D _ { n e g } = \\sum _ I D _ { - e _ I } . \\end{align*}"}
-{"id": "3144.png", "formula": "\\begin{align*} \\mathrm { d } X _ { t } ^ { x } = ( a - b X _ { t } ^ { x } ) \\mathrm { d } t + \\sigma \\sqrt { X _ { t } ^ { x } } \\mathrm { d } B _ { t } + \\mathrm { d } J _ { t } , t \\geqslant 0 , X _ { 0 } ^ { x } = x \\in \\mathbb { R } _ { \\geqslant 0 } . \\end{align*}"}
-{"id": "6508.png", "formula": "\\begin{align*} \\Sigma = 4 \\pi \\left \\vert f \\left ( k _ { \\mathrm { o } } \\right ) \\right \\vert ^ { 2 } \\approx \\frac { 4 \\pi \\rho ^ { 2 } k _ { \\mathrm { o } } ^ { 4 } L ^ { 6 } } { 9 } = \\frac { 1 6 \\pi \\mu ^ { 2 } V ^ { 2 } L ^ { 6 } } { 9 \\hbar ^ { 4 } } \\approx 4 \\pi a _ { \\mathrm { s } } ^ { 2 } . \\end{align*}"}
-{"id": "5897.png", "formula": "\\begin{align*} \\begin{aligned} & \\Pi _ N \\nu ^ { N , D } ( 0 ) = 1 ; \\\\ & \\Pi _ N \\nu ^ { N , D } ( k ) = \\prod _ { i = 1 } ^ k \\frac { p ^ { N ; D } _ { i - 1 , i } } { p ^ { N ; D } _ { i , i - 1 } } , \\ k = 1 , \\cdots , l , \\\\ & \\ \\Pi _ N = 1 + \\sum _ { k = 1 } ^ l \\prod _ { i = 1 } ^ k \\frac { p ^ { N ; D } _ { i - 1 , i } } { p ^ { N ; D } _ { i , i - 1 } } . \\end{aligned} \\end{align*}"}
-{"id": "4298.png", "formula": "\\begin{align*} \\underline { \\nu } = ( \\nu _ i , \\Delta _ { E , \\ell _ i } ) _ { i = 1 } ^ { m } , \\underline { \\nu } ^ { \\vee } = ( \\nu _ i , \\Delta ^ { \\vee } _ { E , \\ell _ i } ) _ { i = 1 } ^ { m } , \\end{align*}"}
-{"id": "7399.png", "formula": "\\begin{align*} a = a _ 0 a _ 1 a _ 2 , a ^ * = a _ 2 ^ * a _ 1 ^ * a _ 0 ^ * , b = a _ 3 ^ * a _ 3 , c = a _ 6 ^ * a _ 6 . \\end{align*}"}
-{"id": "5035.png", "formula": "\\begin{align*} n + 1 - \\lfloor \\frac n k \\rfloor < n - \\frac n k + 2 \\leq ( 1 - \\frac 1 k + \\epsilon ) n = c n . \\end{align*}"}
-{"id": "8409.png", "formula": "\\begin{align*} \\frac { n C - H | h | ^ 2 } { n H ^ { 1 - \\sigma } } - | h | ^ 2 f _ { \\sigma } \\geq \\frac { n f _ { \\sigma } H | h | ^ 2 + \\frac { 1 } { 2 } f _ { \\sigma } H ^ 3 } { n H } - | h | ^ 2 f _ { \\sigma } = f _ { \\sigma } \\frac { H ^ 2 } { 2 n } , \\end{align*}"}
-{"id": "1062.png", "formula": "\\begin{align*} \\log H _ { k } - \\frac { q _ { k } } { 2 n - 2 } = L _ { P _ { k } } ^ { \\ast } ( q _ { k } ) . \\end{align*}"}
-{"id": "8237.png", "formula": "\\begin{align*} \\omega = \\left ( \\begin{matrix} 0 & 1 \\\\ - 1 & 0 \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "217.png", "formula": "\\begin{align*} \\log \\det \\left ( ( a _ { j - k } ) _ { j , k = 1 } ^ { L } \\right ) = L ( \\log a ) _ 0 + \\sum _ { l = 1 } ^ \\infty l ( \\log a ) _ l ( \\log a ) _ { - l } + \\hbox { o } ( 1 ) ( L \\to \\infty ) . \\end{align*}"}
-{"id": "2156.png", "formula": "\\begin{align*} \\frac { d } { d s } \\int _ { I ( s ) } | w ( \\xi , s ) | ^ 2 \\rho _ d \\ , d \\xi & \\le - 2 \\int _ { I ( s ) } | \\partial _ \\xi w | ^ 2 \\rho _ d \\ , d \\xi + d \\int _ { I ( s ) } | w ( \\xi , s ) | ^ 2 \\rho _ d \\ , d \\xi \\\\ & + C e ^ { - \\frac { \\theta } { 4 } s } \\int _ { I ( s ) } | w ( \\xi , s ) | ^ 2 \\rho _ d \\ , d \\xi + O ( e ^ { - 2 \\gamma s } e ^ { - d \\theta _ * s } ) \\end{align*}"}
-{"id": "2610.png", "formula": "\\begin{align*} V _ k * \\rho _ \\varepsilon = \\mathcal F ^ { - 1 } ( \\widehat { V _ k } \\widehat { \\rho _ \\varepsilon } ) \\R \\int _ { \\R } ( V _ k * \\mu _ \\delta ) \\rho _ \\varepsilon = \\int _ { \\R } \\widehat { V _ k } \\widehat { \\mu _ \\delta } \\overline { \\widehat { \\rho _ \\varepsilon } } \\end{align*}"}
-{"id": "3773.png", "formula": "\\begin{align*} \\eta ( z , i , n + 1 ) = \\left \\{ \\begin{array} { l l } 1 & \\begin{array} { l } i \\le N ( z , 0 ) \\\\ z ' \\in \\Z , i ' \\in \\N \\eta ( z ' , i ' , n ) = 1 , S ^ { z ' , i ' } _ n = S ^ { z , i } _ n , \\end{array} \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "8398.png", "formula": "\\begin{align*} \\frac { \\partial | H | ^ 2 } { \\partial t } & = \\Delta | H | ^ 2 - 2 | \\nabla H | ^ 2 + 2 n K | H | ^ 2 + 2 R _ 2 \\\\ \\frac { \\partial | h | ^ 2 } { \\partial t } & = \\Delta | h | ^ 2 - 2 | \\nabla h | ^ 2 + 2 R _ 1 + 4 K | H | ^ 2 - 2 n K | h | ^ 2 , \\end{align*}"}
-{"id": "4846.png", "formula": "\\begin{align*} x ^ 2 = t ^ { 2 m } , y ^ 2 = a ^ 2 t ^ { 2 l } + \\cdots , z ^ 2 = 1 , \\end{align*}"}
-{"id": "5999.png", "formula": "\\begin{gather*} x . g _ 1 = g _ 2 , x . g _ 2 = g _ 1 , y . g _ 1 = g _ 2 , y . g _ 2 = g _ 1 . \\end{gather*}"}
-{"id": "3190.png", "formula": "\\begin{align*} \\mathbb { E } _ { x } \\left [ ( D _ { t } ) ^ { 2 } \\right ] = \\sigma ^ { 2 } \\int _ { 0 } ^ { t } \\mathbb { E } _ { x } \\left [ X _ { s } \\left ( 1 + X _ { s } \\right ) ^ { - 2 } \\right ] \\mathrm { d } s \\leqslant \\sigma ^ { 2 } \\int _ { 0 } ^ { t } \\mathbb { E } _ { x } \\left [ \\left ( 1 + X _ { s } \\right ) ^ { - 1 } \\right ] \\mathrm { d } s \\leqslant t \\sigma ^ { 2 } < \\infty , \\end{align*}"}
-{"id": "3504.png", "formula": "\\begin{align*} J ( \\bar { u } ( \\cdot ) ) = \\underset { u ( \\cdot ) \\in \\mathcal { U } [ 0 , T ] } { \\inf } J ( u ( \\cdot ) ) \\end{align*}"}
-{"id": "1948.png", "formula": "\\begin{align*} \\lVert P _ B \\rVert _ { L ^ p _ { \\alpha ' } } \\le 2 \\sum ^ { \\infty } _ { j = 0 } 2 ^ { j } \\lVert { P ^ { 2 ^ { - j } } } \\rVert _ { L ^ p _ { \\alpha ' } } & \\le 2 \\delta ^ { - \\varepsilon / s } M \\sum ^ { \\infty } _ { j = 0 } 2 ^ { - j \\varepsilon / s } \\\\ & \\le C ( \\delta , \\alpha , \\alpha ' , \\tau , p ) M . \\end{align*}"}
-{"id": "130.png", "formula": "\\begin{align*} u _ h '' ( 1 - t , \\lambda x + \\mu y ) = u _ f '' ( 1 - t , x ) = u _ g '' ( 1 - t , y ) = 0 , \\end{align*}"}
-{"id": "8296.png", "formula": "\\begin{align*} \\| z \\| ^ 2 = - \\frac { [ z , \\overline { z } ] } { 4 \\pi e ^ \\gamma } \\end{align*}"}
-{"id": "4659.png", "formula": "\\begin{align*} \\left | \\sum _ { n = 0 } ^ { N - 1 } 1 _ { T ^ s I _ { k , j } } ( T ^ n x ) - 1 _ { T ^ s I _ { k , j } } ( T ^ n y ) \\right | < E _ 3 \\max \\{ 1 , ( | I _ { k , j } | N ) ^ { \\zeta _ 3 } \\} \\end{align*}"}
-{"id": "2992.png", "formula": "\\begin{align*} d U ( t ) = \\Delta _ b U ( t ) d t + d W ( t ) , t \\in [ 0 , T ] \\end{align*}"}
-{"id": "2430.png", "formula": "\\begin{align*} v ^ i ( x _ 1 ^ \\ast , y _ 1 ^ \\ast ) & = \\sum \\limits _ { \\substack { x _ 2 \\in X \\\\ y _ 2 \\in C _ i } } p ( x _ 2 | x _ 1 ^ \\ast , \\mu _ 1 ^ \\ast ) q ( y _ 2 | y _ 1 ^ \\ast , \\nu ( y _ 1 ^ \\ast ) ) v ^ i ( x _ 2 , y _ 2 ) . \\end{align*}"}
-{"id": "5245.png", "formula": "\\begin{align*} Q = [ 0 , \\lambda _ \\mathrm { m a x } ] \\times [ - \\infty , \\tau ] . \\end{align*}"}
-{"id": "91.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ \\infty | a _ i ( s , t ) | ^ 2 < C \\mbox { a n d } \\sum _ { i = 1 } ^ \\infty | b _ i ( s , t ) | ^ 2 < C ( s , t ) \\not \\in M . \\end{align*}"}
-{"id": "9132.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } x ^ { i } f _ { n + 1 - i } ( x ^ { n + 1 - i } ) = 0 \\left ( x \\in \\mathbb { R } \\right ) . \\end{align*}"}
-{"id": "3968.png", "formula": "\\begin{align*} u _ 0 = f \\ \\ \\ \\ \\mathrm { a n d } \\ \\ \\ \\ u _ n = L ( u _ { n - 1 } ) + A _ { n - 1 } ( u _ 0 , u _ 1 , \\ldots , u _ { n - 1 } ) , \\ \\ n \\geq 1 . \\end{align*}"}
-{"id": "5487.png", "formula": "\\begin{align*} \\int _ a ^ b v _ z ( s ) \\dd s = b ^ \\rho p ( b z ) - a ^ \\rho p ( a z ) \\end{align*}"}
-{"id": "7286.png", "formula": "\\begin{align*} B = \\left ( \\begin{array} { c c c c c c c } t _ 1 & t _ 2 & \\ldots & t _ u & 0 & \\ldots & 0 \\\\ \\ell _ { 1 , 2 } & \\ell _ { 2 , 2 } & \\ldots & \\ell _ { u , 2 } & \\ell _ { u + 1 , 2 } & \\ldots & \\ell _ { n - 1 , 2 } \\\\ \\ell _ { 1 , 3 } & \\ell _ { 2 , 3 } & \\ldots & \\ell _ { u , 3 } & \\ell _ { u + 1 , 3 } & \\ldots & \\ell _ { n - 1 , 3 } \\end{array} \\right ) , \\end{align*}"}
-{"id": "5152.png", "formula": "\\begin{align*} q _ { m , j , \\alpha } \\big ( [ f ] _ { m , j } \\big ) = \\inf \\big \\{ p _ \\alpha ( g ) : g \\in [ f ] _ { m , j } \\big \\} \\end{align*}"}
-{"id": "81.png", "formula": "\\begin{align*} | E _ { w _ 0 } ( \\psi _ \\alpha ) - E _ { w _ 0 } ( \\psi ) | = | F _ { \\psi _ { \\alpha } } ( w _ 0 ) - F _ { \\psi } ( w _ 0 ) | < \\epsilon / 3 , \\ \\ \\ \\alpha \\geq \\alpha _ 0 . \\end{align*}"}
-{"id": "4021.png", "formula": "\\begin{align*} \\omega _ t ( X ^ t , Y ^ t ) = \\mathrm { d e t } [ X ^ t , Y ^ t , \\eta _ t ] = ( 1 + t g \\lambda _ 1 ) ( 1 + t g \\lambda _ 2 ) \\mathrm { d e t } [ X , Y , \\eta _ t ] + t X ( g ) ( 1 + t g \\lambda _ 2 ) \\mathrm { d e t } [ \\eta , Y , \\eta _ t ] + \\\\ + t Y ( g ) ( 1 + t g \\lambda _ 1 ) \\mathrm { d e t } [ \\eta , X , \\eta _ t ] , \\end{align*}"}
-{"id": "1006.png", "formula": "\\begin{align*} E [ | \\widetilde { Z } _ n ^ * ( s ) - \\widetilde { Z } _ n ^ * ( t ) | ^ 2 | \\mathcal { F } ^ X ] & = \\frac { 2 } { 3 } \\sum _ { i = 1 } ^ n \\left \\{ \\frac { K _ h ( t _ { i - 1 } - s ) } { \\mathfrak { s } _ n ( s ) } - \\frac { K _ h ( t _ { i - 1 } - t ) } { \\mathfrak { s } _ n ( t ) } \\right \\} ^ 2 ( X _ { t _ { i } } - X _ { t _ { i - 1 } } ) ^ 4 \\\\ & \\leq c _ 2 r _ n ^ 2 \\frac { n h } { h ^ 2 } \\cdot ( n h ) ^ 2 \\cdot \\frac { ( s - t ) ^ 2 } { n h ^ 3 } = c _ 2 r _ n ^ 2 \\frac { n ^ 2 } { h ^ 2 } ( s - t ) ^ 2 \\end{align*}"}
-{"id": "2302.png", "formula": "\\begin{align*} | \\langle A ^ 2 u ^ N , \\mathcal { U } ^ \\alpha ( u ^ N , u ^ N ) \\rangle | \\\\ \\leq C \\| u ^ N \\| _ { D ( A ) } ^ 2 \\| u ^ N \\| _ { D ( A ^ { 1 + s / 2 } ) } . \\end{align*}"}
-{"id": "9480.png", "formula": "\\begin{align*} \\sum _ { s = 0 } ^ { N } \\frac { q ^ { \\binom { N - s + 1 } { 2 } } } { ( q ^ 2 ; q ^ 2 ) _ s ( q ^ { N + s + 1 } ; q ) _ { N - s + 1 } } = \\frac { 1 } { ( q ; q ^ 2 ) _ { N + 1 } } . \\end{align*}"}
-{"id": "6552.png", "formula": "\\begin{align*} a \\geq ( 1 - \\Lambda \\delta ( 1 - t ( \\delta ) ) ) \\max _ { \\xi \\in \\mathrm { e x t } ( P ) } ( 1 + \\beta _ { \\xi } \\delta ' ) = ( 1 - \\Lambda \\delta ( 1 - t ( \\delta ) ) ) ( 1 + \\beta c _ o \\delta ^ { 1 / n } ) . \\end{align*}"}
-{"id": "4260.png", "formula": "\\begin{align*} \\iota ( e _ 2 ) = ( e _ { 2 1 } \\otimes _ \\Phi 1 , e _ { 2 2 } \\otimes _ \\Phi 1 ) \\left ( \\begin{array} { c } ( h + f g ) \\cdot \\left ( \\frac { t - 1 } { t } \\right ) ^ { m + 1 } \\\\ \\\\ f \\cdot \\left ( \\frac { t } { t - 1 } \\right ) ^ { m } \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "461.png", "formula": "\\begin{align*} \\partial ^ { 2 \\alpha } a _ { k _ 1 , k _ 2 } ( i y _ \\omega u _ 1 ) = O \\left ( y _ \\omega ^ { k _ 2 - 2 j + 2 } \\right ) = O \\left ( y _ \\omega ^ { k _ 2 - 2 j + 1 } \\right ) , \\end{align*}"}
-{"id": "6603.png", "formula": "\\begin{align*} J _ { \\phi _ { m , n } ^ { - 1 } } ( y ) = \\det D \\phi _ { m , n } ^ { - 1 } ( y ) \\leq d ! \\ , \\| D \\phi _ { m , n } ^ { - 1 } ( y ) \\| _ { C ^ 0 } ^ d \\leq d ! \\ , K ^ d \\ . \\end{align*}"}
-{"id": "1963.png", "formula": "\\begin{align*} R _ \\theta ( q ) ( x ) : = e ^ { - i k \\theta \\cdot x } R _ k \\left ( e ^ { i k \\theta \\cdot ( \\cdot ) } q ( \\cdot ) \\right ) ( x ) . \\end{align*}"}
-{"id": "1634.png", "formula": "\\begin{align*} \\lambda \\nu = \\rho _ 1 x _ { j _ 1 } x _ { j _ 1 + 1 } \\dots x _ { i - 1 } = \\rho _ 2 x _ { j _ 2 } x _ { j _ 2 + 1 } \\dots x _ { i - 1 } \\end{align*}"}
-{"id": "6968.png", "formula": "\\begin{align*} D f ( B ) B = \\lim _ { \\epsilon \\to 0 } { \\frac { f ( B + \\epsilon B ) - f ( B ) } { \\epsilon } } = f ( B ) . \\end{align*}"}
-{"id": "6846.png", "formula": "\\begin{align*} \\min _ x \\left \\{ f ( x ) : \\sup _ { u ^ i \\in U ^ i } f ^ i ( x , u ^ i ) \\leq 0 , \\ i = 1 , \\ldots , m , x \\in X \\right \\} . \\end{align*}"}
-{"id": "7940.png", "formula": "\\begin{gather*} d ( y , x ) \\le d ( y , x _ 1 ^ i ) + d ( x _ 1 ^ i , x ) < ( c _ 1 + 2 c _ 2 + 1 ) \\rho . \\end{gather*}"}
-{"id": "878.png", "formula": "\\begin{align*} Q _ { n , k } ( \\xi ) : = \\sum _ { i , j = 1 } ^ { N _ n } \\gamma _ { n , k } ( i , j ) \\xi _ i \\xi _ j , k = 1 , \\dots , d _ n \\end{align*}"}
-{"id": "7883.png", "formula": "\\begin{align*} \\begin{aligned} & \\varphi ( d ( z , m ( x ' , y ) ) ) - \\varphi ( d ( z , m ( x , y ) ) ) \\\\ & \\ = \\varphi ' ( d ( z , m ( x , y ) ) ) ( d ( z , m ( x ' , y ) ) - d ( z , m ( x , y ) ) ) + o ( | d ( z , m ( x ' , y ) ) - d ( z , m ( x , y ) ) | ) \\\\ & \\ \\le { 1 \\over 2 } \\varphi ' ( d ( z , m ( x , y ) ) ) d ( x ' , x ) + o ( d ( x ' , x ) ) \\end{aligned} \\end{align*}"}
-{"id": "3714.png", "formula": "\\begin{align*} f _ i \\cdot f _ { i + 1 } \\cdots f _ { j - 1 } = \\frac { u _ { i , i } } { v _ { i , i } } \\frac { v _ { i + 1 , i } } { u _ { i + 1 , i } } \\cdot \\frac { u _ { i + 1 , i + 1 } } { v _ { i + 1 , i + 1 } } \\frac { v _ { i + 2 , i + 1 } } { u _ { i + 2 , i + 1 } } \\cdots \\frac { u _ { j - 1 , j - 1 } } { v _ { j - 1 , j - 1 } } \\frac { v _ { j , j - 1 } } { u _ { j , j - 1 } } = 1 . \\end{align*}"}
-{"id": "6457.png", "formula": "\\begin{align*} d s _ { } ^ { 2 } = \\frac { 1 } { \\sigma ^ { 2 } } \\left [ \\frac { 1 } { 1 - \\rho ^ { 2 } } \\left ( d \\mu _ { x } ^ { 2 } + d \\mu _ { x } ^ { 2 } + 2 \\rho d \\mu _ { x } d \\mu _ { y } \\right ) + 4 d ^ { 2 } \\sigma \\right ] \\end{align*}"}
-{"id": "9602.png", "formula": "\\begin{align*} \\| D _ k ( x , \\cdot ) \\| _ { \\dot { \\mathcal B } ^ { \\alpha , q } _ { p , \\mathcal F } } & = \\bigg \\{ \\sum _ { j \\in \\Bbb Z } \\Big ( 2 ^ { j \\alpha } \\| D _ j ( D _ k ( x , \\cdot ) ) \\| _ { L ^ p _ \\mu } \\Big ) ^ q \\bigg \\} ^ { 1 / q } \\\\ & \\le C \\frac 1 { V _ k ( x ) ^ { 1 - 1 / p } } \\bigg \\{ \\sum _ { j \\in \\Bbb Z } 2 ^ { j \\alpha q - | j - k | \\varepsilon q } \\bigg \\} ^ { 1 / q } \\\\ & \\le C 2 ^ { k \\alpha } \\frac 1 { V _ k ( x ) ^ { 1 - 1 / p } } , \\end{align*}"}
-{"id": "444.png", "formula": "\\begin{align*} p _ { 1 , k _ 1 , k _ 2 } ( x , t ) = \\frac { 1 } { | x | ^ m } e ^ { - \\frac { 1 } { 4 } d ( x , t ) ^ 2 } \\Psi ( \\omega ) \\Upsilon ( x , t ) \\end{align*}"}
-{"id": "3848.png", "formula": "\\begin{align*} \\mathcal { X } : = \\{ x \\in X \\mid \\| x \\| _ 0 \\leq \\kappa \\} \\neq \\emptyset . \\end{align*}"}
-{"id": "191.png", "formula": "\\begin{align*} \\mathfrak { X } \\ast \\Gamma = \\sigma _ { 1 } ( \\Gamma ) \\uplus \\dots \\uplus \\sigma _ { n } ( \\Gamma ) . \\end{align*}"}
-{"id": "7120.png", "formula": "\\begin{align*} \\frac { Q _ 1 } { \\alpha F ^ { - \\alpha - 2 } } = & ~ \\ddot { F } ^ { k l , m n } \\nabla _ 1 h _ { k l } \\nabla _ 1 h _ { m n } - ( \\alpha + 1 ) F ^ { - 1 } ( \\nabla _ 1 F ) ^ 2 \\\\ & + ( \\alpha + 1 ) F ^ { - 2 } \\dot { f } ^ { k } ( \\nabla _ k F ) ^ 2 \\kappa _ 1 + 2 \\sum _ { k = 1 } ^ n \\sum _ { l = 2 } ^ n \\frac { \\dot { f } ^ k } { \\kappa _ l - \\kappa _ 1 } ( \\nabla _ k h _ { 1 l } ) ^ 2 ~ \\geq ~ 0 . \\end{align*}"}
-{"id": "4868.png", "formula": "\\begin{align*} P _ d ( t ) = \\begin{cases} w _ d ( t ) - E _ d ( t ) & d > 1 \\\\ t - 1 & d = 1 . \\end{cases} \\end{align*}"}
-{"id": "1911.png", "formula": "\\begin{align*} \\P ^ n \\supset \\P ^ { n - 1 } \\ldots \\supset \\P ^ 1 \\supset \\P ^ 0 = \\mathfrak q \\end{align*}"}
-{"id": "1041.png", "formula": "\\begin{align*} L _ { m , j } ^ { \\ast } ( q ) : = q \\psi _ { m , j } ^ { \\ast } ( q ) , q = \\log Q , \\end{align*}"}
-{"id": "8577.png", "formula": "\\begin{align*} & \\sup _ { x \\in \\Omega } \\int _ 0 ^ t \\int _ { \\Omega \\ , \\cap \\ , B _ { H _ 0 } ( x , 1 ) } e ^ { - h ( y , s ) } H ( \\nabla u ( y , s ) ) \\ , d y \\ , d s \\\\ & \\qquad \\le C _ 1 t ^ { \\sigma ' } \\sup _ { x \\in \\Omega } \\int _ { \\Omega \\ , \\cap \\ , B _ { H _ 0 } ( x , 1 ) } e ^ { - H _ 0 ( y ) ^ 2 } | \\phi ( y ) | \\ , d y \\end{align*}"}
-{"id": "956.png", "formula": "\\begin{align*} A _ n ( \\theta ) = \\left ( \\begin{array} { c c } 0 & K _ n ( \\theta ) \\\\ K _ n ( \\theta ) ^ \\top & 0 \\end{array} \\right ) , K _ n ( \\theta ) = ( \\sqrt { n } K ( I , J _ { - \\theta } ) / 2 ) _ { I \\in \\Pi ^ 1 _ n , J \\in \\Pi ^ 2 _ n } . \\end{align*}"}
-{"id": "6264.png", "formula": "\\begin{align*} U _ t + u \\cdot \\nabla U - b \\cdot \\nabla B + U \\cdot \\nabla u ^ \\eta - B \\cdot \\nabla b ^ \\eta + \\nabla \\pi & = \\nu \\Delta U , \\\\ B _ t + u \\cdot \\nabla B - b \\cdot \\nabla U + U \\cdot \\nabla b ^ \\eta - B \\cdot \\nabla u ^ \\eta - \\eta \\nabla \\times ( ( \\nabla \\times b ^ \\eta ) \\times b ^ \\eta ) & = \\mu \\Delta B , \\\\ \\nabla \\cdot U = 0 , \\ \\ \\ \\nabla \\cdot B & = 0 . \\end{align*}"}
-{"id": "7819.png", "formula": "\\begin{align*} \\int _ { S _ { r _ j } } r ^ { 1 - 2 m } | \\frac { \\partial v _ m } { \\partial r } v _ m | e ^ { - 2 \\rho } d x = o ( 1 ) . \\end{align*}"}
-{"id": "2169.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty u _ * ( r , t ) \\nu _ d ( r ) r ^ { d - 1 } \\ , d r = \\int _ 0 ^ \\infty u _ * ( r , t ) \\nu ( r ) r ^ { N - 1 } \\ , d r = \\int _ 0 ^ \\infty \\varphi _ * ( r ) \\nu ( r ) r ^ { N - 1 } \\ , d r . \\end{align*}"}
-{"id": "1540.png", "formula": "\\begin{gather*} C _ { A ^ * + B ^ * } \\ : = \\ \\bigcup _ { n \\in ( A ^ * + B ^ * ) \\cap \\mathbb { Z } } \\overline { B } _ { \\frac { 1 } { 2 } } ( n ) \\end{gather*}"}
-{"id": "9119.png", "formula": "\\begin{align*} 0 = \\sum _ { i = 0 } ^ { n - 1 } \\dfrac { 1 } { \\binom { n - 1 + 1 } { i } } \\sum _ { \\mathrm { c a r d } ( I ) = i } \\left ( \\prod _ { j \\in I } x _ { j } \\right ) \\cdot \\tilde { f } _ { ( n - 1 ) + 1 - i } \\left ( \\prod _ { k \\in \\left \\{ 1 , \\ldots , n \\right \\} \\setminus I } x _ { k } \\right ) , \\end{align*}"}
-{"id": "5579.png", "formula": "\\begin{align*} S _ { n } \\approx 1 - \\delta \\sum _ { i = 0 } ^ { n - 1 } \\left [ S _ { i } \\int \\limits _ { \\tau - \\left ( n + 1 \\right ) \\delta } ^ { \\tau - \\left ( i + 1 \\right ) \\delta } \\left ( \\alpha + \\beta p \\left ( t \\right ) \\right ) d t \\right ] \\end{align*}"}
-{"id": "6334.png", "formula": "\\begin{align*} \\int _ \\Omega \\phi ( t ) = \\int _ \\Omega \\phi ( 0 ) \\forall t \\in [ 0 , T ] \\ , . \\end{align*}"}
-{"id": "5091.png", "formula": "\\begin{align*} \\widehat { f } \\colon \\widehat { G } \\to \\mathbb { C } , \\widehat { f } ( \\xi ) : = \\int _ G f ( x ) \\overline { E _ G ( x , \\xi ) } d x ; \\xi \\in \\widehat { G } \\end{align*}"}
-{"id": "2097.png", "formula": "\\begin{align*} 0 \\geq \\frac { d } { d t } E _ H ( f _ t ) \\geq \\frac { d } { d t } E _ H ( f _ { t _ 0 } ) = - | | \\tau ( f _ { t _ 0 } ) | | _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "2622.png", "formula": "\\begin{align*} 0 \\leq m : = \\operatorname * { e s s \\ , i n f } _ { ( 0 , 1 ) } h _ { \\overline \\rho } < \\sup _ { ( 0 , 1 ) } h _ { \\overline \\rho } \\leq \\infty . \\end{align*}"}
-{"id": "8227.png", "formula": "\\begin{align*} \\biggl [ \\frac { 1 } { n ^ 2 } \\biggr ] \\ln ( F _ s ) = - \\frac { 1 } { 1 2 } \\frac { s \\bigl ( 3 r ^ 2 s - 3 r ^ 2 - 1 2 r s - 2 s ^ 2 + 1 2 r + 9 s - 7 \\bigr ) } { r ^ 2 } \\end{align*}"}
-{"id": "7288.png", "formula": "\\begin{align*} \\tilde { A } B \\tilde { C } = \\left ( \\begin{array} { c c c | c c c c c c } t _ 1 & \\ldots & t _ u & \\boldsymbol 0 \\\\ \\ast & \\ldots & \\ast & A B ' C \\end{array} \\right ) , \\end{align*}"}
-{"id": "9257.png", "formula": "\\begin{align*} C _ q = \\{ 1 1 i + j \\mid 0 \\leq i \\leq q - 1 j \\in \\{ 0 , 1 , 4 , 5 \\} \\} \\end{align*}"}
-{"id": "5966.png", "formula": "\\begin{align*} \\iota ( e ( \\alpha ) ) = \\alpha . \\end{align*}"}
-{"id": "2550.png", "formula": "\\begin{align*} \\theta ( E ) ( x ) : = \\lim _ { r \\to 0 ^ + } \\frac { | E \\cap B _ r ( x ) | } { \\omega _ n r ^ n } \\end{align*}"}
-{"id": "9106.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ { n + 1 } i a _ { i } \\right ) \\cdot A ( x ) = 0 \\left ( x \\in R \\right ) , \\end{align*}"}
-{"id": "7422.png", "formula": "\\begin{align*} t : = ( 4 t _ 0 + 3 t _ 1 + 2 t _ 2 + t _ 3 ) / 5 R _ 1 : = ( 3 t _ 5 + 6 t _ 8 + 4 t _ 7 + 2 t _ 6 ) / 5 = - ( t _ 0 + 2 t _ 1 + 3 t _ 2 + 4 t _ 3 + 5 t _ 4 ) / 5 \\end{align*}"}
-{"id": "6276.png", "formula": "\\begin{align*} K ( q ) = \\begin{cases} \\lambda _ q , q \\leq 0 ; \\\\ \\lambda _ q ^ { - 1 } , q > 0 . \\end{cases} \\end{align*}"}
-{"id": "1665.png", "formula": "\\begin{align*} \\mu ( D _ v \\setminus ( R _ { e } \\cup R _ { g } ) ) = 0 D _ w = R _ { f } . \\end{align*}"}
-{"id": "9508.png", "formula": "\\begin{align*} \\sum \\limits _ { x \\in V ( \\mathfrak { M } ) } \\deg ( x ) = 2 n ' \\end{align*}"}
-{"id": "4563.png", "formula": "\\begin{align*} C = q _ 3 ^ { n / 2 } \\ , . \\end{align*}"}
-{"id": "6409.png", "formula": "\\begin{align*} d l _ { \\rightarrow \\xi } ^ { 2 } = g _ { \\mu \\nu } \\left ( \\theta \\right ) d \\theta ^ { \\mu } d \\theta ^ { \\nu } \\end{align*}"}
-{"id": "1089.png", "formula": "\\begin{align*} f _ { n _ { i } } ( n _ { i } ) = \\frac { m _ { c , i } ^ { n _ { i } } } { ( 1 + m _ { c , i } ) ^ { n _ { i } + 1 } } \\ , \\mathrm { e x p } \\left ( - \\frac { m _ { s , i } } { 1 + m _ { c , i } } \\right ) L _ { n _ { d } } \\left ( - \\frac { m _ { s , i } } { m _ { c , i } ( 1 + m _ { c , i } ) } \\right ) , \\end{align*}"}
-{"id": "2978.png", "formula": "\\begin{align*} \\int _ \\Omega | \\nabla T _ k ( u _ n ) | ^ 2 \\leq C k . \\end{align*}"}
-{"id": "715.png", "formula": "\\begin{align*} \\eta _ { \\alpha \\alpha } \\eta _ T ^ 2 - 2 \\eta _ { \\alpha T } \\eta _ { \\alpha } \\eta _ T + \\eta _ { T T } \\eta _ { \\alpha } ^ 2 & = \\frac { 2 ( 1 + \\alpha ) } { \\alpha ^ 2 ( 1 - \\alpha ) ^ 2 } \\frac { T _ { \\rm i } } { T } \\left \\{ - \\left [ 1 - 3 \\alpha - \\frac { 5 } { 2 } \\alpha ^ 2 ( 1 - \\alpha ) ^ 2 \\right ] \\frac { T _ { \\rm i } } { T } + 4 \\left [ 1 + \\frac { 5 } { 4 } \\alpha ( 1 - \\alpha ) \\right ] ^ { \\ ! 2 } \\right \\} , \\end{align*}"}
-{"id": "4263.png", "formula": "\\begin{align*} \\nabla ( v _ { 1 1 } , v _ { 1 2 } ) = ( v _ { 1 1 } , v _ { 1 2 } ) \\cdot \\upsilon _ { \\nabla , 1 } \\end{align*}"}
-{"id": "4255.png", "formula": "\\begin{align*} A _ { \\theta , 1 2 } = \\left ( \\begin{array} { c c } 0 & \\frac { t - a } { ( t - 1 ) ( t - \\lambda ) } \\\\ 0 & 0 \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "7099.png", "formula": "\\begin{align*} \\begin{aligned} & k _ t = k _ p p _ t = k _ p ( k p _ x ) _ x = k _ p ^ 2 p _ x ^ 2 + k k _ p p _ { x x } , \\\\ & k _ { t x } = 2 k _ p k _ { p p } p _ x ^ 3 + ( 3 k _ p ^ 2 + k k _ { p p } ) p _ x p _ { x x } + k k _ p p _ { x x x } . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "1924.png", "formula": "\\begin{align*} \\Gamma ( M _ { j } ) = \\Pi _ { j } ( \\Gamma ^ { 1 } _ { j } ) = \\Pi _ { j } ( \\Gamma ^ { 1 } _ { j } ) \\oplus \\Pi _ { j } ( \\Gamma ^ { 2 } _ { j } ) . \\end{align*}"}
-{"id": "3917.png", "formula": "\\begin{align*} \\begin{cases} n \\equiv f _ 0 , \\dots , d - 2 \\pmod d , \\\\ n - g \\equiv 0 , \\dots , f _ 0 \\pmod d . \\end{cases} \\end{align*}"}
-{"id": "2914.png", "formula": "\\begin{align*} a \\approx a ' \\ ; \\mbox { i f f } \\ ; a ' = \\rho _ w ( a ) \\mbox { f o r s o m e w a l k $ w $ f r o m $ s $ t o $ s ' $ . } \\end{align*}"}
-{"id": "6533.png", "formula": "\\begin{align*} B = F ^ { \\circ } = \\bigcap _ { i = 1 } ^ k \\{ \\bar { x } \\in \\mathbb { R } ^ { n - 1 } : \\langle \\bar { x } , \\bar { y } _ i \\rangle \\leq 1 \\} \\end{align*}"}
-{"id": "6707.png", "formula": "\\begin{align*} \\begin{cases} \\ U ( t + 1 ) < \\frac { 1 } { 2 } , & \\sigma ^ { \\eta } _ { i } ( t + 1 ) = + 1 , \\ \\sigma ^ { \\varsigma } _ { i } ( t + 1 ) = + 1 ; \\\\ \\ U ( t + 1 ) \\geq \\frac { 1 } { 2 } , & \\sigma ^ { \\eta } _ { i } ( t + 1 ) = - 1 , \\ \\sigma ^ { \\varsigma } _ { i } ( t + 1 ) = - 1 . \\end{cases} \\end{align*}"}
-{"id": "4349.png", "formula": "\\begin{align*} A _ m : = \\sum _ { j = 1 } ^ \\infty M _ { \\varphi _ { j , \\frac { 1 } { m } } } A M _ { \\psi _ { j , \\frac { 1 } { m } } } \\end{align*}"}
-{"id": "1265.png", "formula": "\\begin{align*} t _ 2 / 2 \\leq t _ 1 < t _ 2 \\mbox { a n d } \\rho = 2 t _ 2 \\left ( \\mbox { d i a m } ( E _ 1 ) + \\mbox { d i a m } ( E _ 2 ) \\right ) \\end{align*}"}
-{"id": "3486.png", "formula": "\\begin{align*} f _ 1 ( z ) = z _ 1 \\bar z _ 2 + \\bar z _ 1 z _ 2 , f _ 2 ( z ) = z _ 2 \\bar z _ 3 + \\bar z _ 2 z _ 3 , f _ 3 ( z ) = z _ 3 \\bar z _ 1 + \\bar z _ 3 z _ 1 . \\end{align*}"}
-{"id": "7611.png", "formula": "\\begin{align*} d : = \\left \\{ \\begin{array} { l c l l } \\frac { p } { 2 } & p \\geq 2 , \\\\ \\frac { 2 p } { ( n + 2 ) p - 2 n } & \\frac { 2 n } { n + 2 } < p < 2 \\end{array} \\right . \\mbox { a n d } \\hat p : = \\frac { p ( n + 2 ) - n } { p ( n + 1 ) - n } . \\end{align*}"}
-{"id": "4135.png", "formula": "\\begin{align*} K ^ { \\frac { 1 } { 4 } } \\xi + Z = \\langle \\eta , \\xi \\rangle \\xi - \\lambda _ 1 \\langle \\eta , \\xi \\rangle \\langle \\eta , E _ 1 \\rangle E _ 1 - \\lambda _ 2 \\langle \\eta , \\xi \\rangle \\langle \\eta , E _ 2 \\rangle E _ 2 = \\\\ = \\langle \\eta , \\xi \\rangle \\xi + \\langle \\eta , E _ 1 \\rangle E _ 1 + \\langle \\eta , E _ 2 \\rangle E _ 2 . \\end{align*}"}
-{"id": "987.png", "formula": "\\begin{align*} \\mathfrak { d } ^ j ( t , t ' ) = a _ 0 | t - t ' | ^ \\gamma , t , t ' \\in [ u _ j , u _ { j + 1 } ] . \\end{align*}"}
-{"id": "237.png", "formula": "\\begin{align*} \\chi _ G \\left ( R _ z ( A _ G ) - R _ z ( A _ { G ' } ) \\right ) \\chi _ { G ' } = R _ z ( A _ { G } ) \\left ( \\chi _ G A \\chi _ { G ' \\setminus G } \\right ) R _ z ( A _ { G ' } ) . \\end{align*}"}
-{"id": "1716.png", "formula": "\\begin{align*} \\pi ( \\chi _ { Z ( \\lambda ) } ) = \\int _ { \\Lambda ^ \\infty } \\chi _ { Z ( \\lambda ) } \\ , ( x ) \\ , d P ( x ) = P ( Z ( \\lambda ) ) = t _ \\lambda t ^ * _ \\lambda . \\end{align*}"}
-{"id": "6127.png", "formula": "\\begin{align*} \\epsilon _ { i } ^ { 1 } & = \\sup _ { j \\geq i } \\sup _ { 1 \\leq \\lambda \\leq 2 } \\left | \\frac { L \\left ( j \\lambda \\right ) } { L \\left ( j \\right ) } - 1 \\right | \\\\ \\epsilon _ { i } ^ { 2 } & = \\sup _ { t \\geq i } \\left | \\frac { g \\left ( t \\right ) } { L \\left ( t \\right ) t ^ { - \\alpha } } \\right | . \\end{align*}"}
-{"id": "6657.png", "formula": "\\begin{align*} f ' ( t ) \\leq & \\frac { f ( d t ) - f ( t ) } { d t - t } \\leq \\frac { \\frac { 1 + \\eta } { 2 } f '' ( 0 ) d ^ 2 t ^ 2 - \\frac { 1 - \\eta } { 2 } f '' ( 0 ) t ^ 2 } { d t - t } = \\left ( \\frac { 1 + d } { 2 } + \\eta \\frac { 1 + d ^ 2 } { 2 ( d - 1 ) } \\right ) f '' ( 0 ) t \\\\ = & \\left ( 1 + \\sqrt { \\eta } + \\frac { \\sqrt { \\eta } } { 2 } + \\eta + \\eta ^ { 3 / 2 } \\right ) f '' ( 0 ) t \\leq \\left ( 1 + \\frac { 7 } { 2 } \\sqrt { \\eta } \\right ) f '' ( 0 ) t \\quad . \\end{align*}"}
-{"id": "4677.png", "formula": "\\begin{align*} { V _ { \\rm e f f } } \\ = \\ \\frac { 3 } { 8 } \\ \\frac { ( m _ 1 + m _ 2 + m _ 3 ) } { F _ 2 } \\ + \\frac { ( d - 2 ) ( d - 4 ) } { 2 } \\ \\frac { P _ m } { m _ 1 m _ 2 m _ 3 \\ , F _ { 1 } } \\ , \\end{align*}"}
-{"id": "7636.png", "formula": "\\begin{align*} & E ( \\lambda ) : = \\Big \\{ z \\in Q _ 3 : \\ , z \\mbox { i s a L e b e s g u e p o i n t o f $ | D u | $ a n d } | D u ( z ) | > \\lambda \\Big \\} , \\\\ & E _ h ( \\lambda ) : = \\big \\{ z \\in Q _ 3 : h > \\lambda \\big \\} , \\mbox { a n d } E _ { \\hat f } ( \\lambda ) : = \\big \\{ z \\in Q _ 3 : \\hat f > \\lambda \\big \\} . \\end{align*}"}
-{"id": "1417.png", "formula": "\\begin{align*} Q _ 0 ( x _ 3 ) & = 0 , & Q _ 1 ( x _ 3 ) & = x _ 3 ^ 2 , & Q _ 2 ( x _ 3 ) & = x _ 2 ^ 2 x _ 3 ^ 2 , \\\\ Q _ 0 ( x _ 4 ) & = 0 , & Q _ 1 ( x _ 4 ) & = x _ 3 x _ 4 , & Q _ 2 ( x _ 4 ) & = x _ 3 x _ 4 ^ 2 + x _ 2 ^ 2 x _ 3 x _ 4 . \\end{align*}"}
-{"id": "3274.png", "formula": "\\begin{align*} r ^ c _ u = v _ u t _ u ^ c = \\frac { y _ u - x _ u \\tan \\theta _ 0 } { \\tan \\theta _ 0 \\cos \\theta _ u - \\sin \\theta _ u } . \\end{align*}"}
-{"id": "9990.png", "formula": "\\begin{align*} \\mathbf E [ Y _ n ( T ) | \\mathcal { F } _ S ] & = \\mathbb E _ 0 \\Big [ \\mathbf E \\big [ \\exp \\big \\{ H _ { \\beta , T } ( W , B ) \\big \\} \\big | \\mathcal { F } _ S \\big ] I _ n ( T , W _ T ) \\Big ] \\\\ & = \\mathbb E _ 0 \\Big [ \\exp \\big \\{ H _ { \\beta , S } ( W , B ) \\big \\} \\mathbb E _ 0 \\big [ I _ n ( T , W _ T ) \\big | \\mathcal { G } _ S \\big ] \\Big ] \\\\ & = Y _ n ( S ) . \\end{align*}"}
-{"id": "2286.png", "formula": "\\begin{align*} d ( S ) & = \\sharp \\Delta _ { \\Z } - 4 - \\bigl \\{ \\sharp V ( S ) - 1 - \\sum _ { k = 1 } ^ N ( \\sharp V ( \\Delta _ k ) - 3 ) \\bigr \\} \\\\ & = \\sharp \\Delta _ { \\Z } - \\sharp V ( S ) - 3 + \\sum _ { k = 1 } ^ N ( \\sharp V ( \\Delta _ k ) - 3 ) \\\\ & \\ge - 3 + \\sum _ { k = 1 } ^ N ( \\sharp V ( \\Delta _ k ) - 3 ) \\\\ & = \\sum _ { m \\ge 3 } ( m - 3 ) N _ m - 3 . \\end{align*}"}
-{"id": "295.png", "formula": "\\begin{align*} s _ { M _ { t , n } ^ * } \\ , ( \\alpha , u ) = & \\frac { 1 } { \\tilde \\sigma \\sqrt n } s _ { L _ { n t } ^ * } ( \\alpha , u ) \\\\ = & \\frac { 1 } { \\tilde \\sigma \\sqrt n } \\left ( s _ { S _ { \\lfloor n t \\rfloor } } ( \\alpha , u ) + ( n t - \\lfloor n t \\rfloor ) s _ { X _ { \\lfloor n t \\rfloor + 1 } ^ * } ( \\alpha , u ) \\right ) . \\end{align*}"}
-{"id": "6691.png", "formula": "\\begin{align*} u ( x ) = \\frac { 1 } { ( \\sum _ { i = 1 } ^ n | x _ i | ^ { 2 p - 2 } ) ^ { \\frac { 1 } { 2 } } } \\left ( \\mathrm { s i g n } ( x _ i ) | x _ i | ^ { p - 1 } \\right ) _ { i = 1 } ^ n \\quad , \\end{align*}"}
-{"id": "4432.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ 2 } \\partial _ 1 ^ j \\partial _ 2 ^ l \\psi _ T = 0 & \\quad \\mbox { p r o v i d e d } \\ ; ( j , l ) \\not = ( 0 , 0 ) , \\\\ \\int _ { \\mathbb { R } ^ 2 } x _ 1 \\partial _ 1 ^ j \\partial _ 2 ^ l \\psi _ T = 0 & \\quad \\mbox { p r o v i d e d } \\ ; ( j , l ) \\not = ( 0 , 0 ) , ( 1 , 0 ) . \\end{align*}"}
-{"id": "1514.png", "formula": "\\begin{align*} \\sum _ { l = 0 } ^ { n } Q _ { T , l } ( x ) & = \\sum _ { l = 0 } ^ { n } T _ { l } ( x ) + \\textbf { i } \\sum _ { l = 0 } ^ { n } T _ { l + 1 } ( x ) + \\textbf { j } \\sum _ { l = 0 } ^ { n } T _ { l + 2 } ( x ) + \\textbf { k } \\sum _ { l = 0 } ^ { n } T _ { l + 3 } ( x ) \\\\ & = ( T _ { 0 } ( x ) + T _ { 1 } ( x ) + \\cdots + T _ { n } ( x ) ) + \\textbf { i } ( T _ { 1 } ( x ) + T _ { 2 } ( x ) + \\cdots + T _ { n + 1 } ( x ) ) \\\\ & \\ \\ + \\textbf { j } ( T _ { 2 } ( x ) + T _ { 3 } ( x ) + \\cdots + T _ { n + 2 } ( x ) ) + \\textbf { k } ( T _ { 3 } ( x ) + T _ { 4 } ( x ) + \\cdots + T _ { n + 3 } ( x ) ) . \\end{align*}"}
-{"id": "2385.png", "formula": "\\begin{align*} A ( \\theta ) = g ( \\theta ) ( a _ 1 g ( \\theta ) - f ( \\theta ) ) , B ( \\theta ) = f ( \\theta ) - 2 a _ 1 g ( \\theta ) - g ' ( \\theta ) . \\end{align*}"}
-{"id": "287.png", "formula": "\\begin{align*} s _ { x ^ * \\oplus y ^ * } ( \\alpha , u ) = s _ { x ^ * } ( \\alpha , u ) + s _ { y ^ * } ( \\alpha , u ) \\end{align*}"}
-{"id": "2186.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\ , t ^ { \\frac { N + 2 A } { 2 } } \\frac { \\left [ e ^ { - t L _ k } \\phi ^ { k , i } \\right ] ( | x | ) } { U ( | x | ) } = 0 \\quad \\mbox { i n } L ^ \\infty ( B ( 0 , R ) ) \\end{align*}"}
-{"id": "1135.png", "formula": "\\begin{align*} M _ p \\sim \\begin{bmatrix} 1 & b _ p \\\\ 0 & 1 \\end{bmatrix} , M _ m \\sim \\begin{bmatrix} 1 & - b _ m \\\\ 0 & 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "6088.png", "formula": "\\begin{align*} u ( x ) = \\left ( \\begin{smallmatrix} R _ { g ( r ) } f _ { \\frak { e } _ k } ( x / r ) \\sin ( \\Phi ( r ) ) \\\\ \\cos ( \\Phi ( r ) ) \\end{smallmatrix} \\right ) , \\end{align*}"}
-{"id": "123.png", "formula": "\\begin{align*} & \\Phi ^ { - 1 } \\bigg ( \\int h \\ , d \\gamma _ n \\bigg ) - \\lambda \\Phi ^ { - 1 } \\bigg ( \\int f \\ , d \\gamma _ n \\bigg ) - \\mu \\Phi ^ { - 1 } \\bigg ( \\int g \\ , d \\gamma _ n \\bigg ) \\\\ & = C ( 1 , 0 , 0 ) = \\mathbf { E } \\bigg [ C ( 1 - t , X _ t , Y _ t ) \\ , e ^ { - \\frac { 1 } { 2 } \\int _ 0 ^ t \\| \\nabla u _ h ( 1 - s , \\lambda X _ s + \\mu Y _ s ) \\| ^ 2 d s } \\bigg ] . \\end{align*}"}
-{"id": "5077.png", "formula": "\\begin{align*} & E _ 1 ( Y _ 1 ) = - T , ~ ~ E _ 2 ( Y _ 1 ) = 0 , ~ ~ E _ 3 ( Y _ 1 ) = 0 , \\\\ & E _ 1 ( T ) = Q Y _ 1 , ~ ~ E _ 2 ( T ) = \\frac { B _ { 1 2 , 1 } } { b _ 1 - b _ 2 } T , ~ ~ E _ 3 ( T ) = \\frac { B _ { 1 3 , 1 } } { b _ 1 - b _ 3 } T . \\end{align*}"}
-{"id": "5182.png", "formula": "\\begin{align*} \\frac { V _ { 1 } ^ { [ r , 1 ] } ( x ) - g _ { 1 , [ r , 1 ] } ( x ) } { r - x } = \\begin{cases} U _ { 1 } ( x , r ) , & \\forall x \\le \\ell _ { r } \\\\ U _ { 1 } ( \\ell _ { r } , r ) , & \\forall \\ell _ { r } < x < r . \\end{cases} \\end{align*}"}
-{"id": "7561.png", "formula": "\\begin{gather*} A _ t + A ^ 2 = 0 , { \\bf b } _ t + A { \\bf b } = 0 . \\end{gather*}"}
-{"id": "5761.png", "formula": "\\begin{align*} \\psi ( D ) = \\left [ \\{ n ( n - 1 ) \\} ^ { - 1 } \\sum _ { 1 \\leq i < j \\leq n } \\frac { 1 } { \\prod _ { k = 1 } ^ p ( x _ { i , k } - x _ { j , k } ) ^ 2 } \\right ] ^ { 1 / p } . \\end{align*}"}
-{"id": "6868.png", "formula": "\\begin{align*} - \\Delta _ m f ( z ) & = \\left ( \\frac { 4 } { p q } \\right ) ^ m ( 2 f ( z ) - f ( y _ 1 ) - f ( y _ 2 ) ) & & i ( [ z , y _ 2 ] ) = i ( [ y _ 1 , z ] ) , \\\\ - \\Delta _ m f ( z ) & = \\left ( \\frac { 4 } { p q } \\right ) ^ m ( 2 f ( z ) - 2 q f ( y _ 1 ) - 2 p f ( y _ 2 ) ) & & i ( [ z , y _ 2 ] ) = i ( [ y _ 1 , z ] ) + 1 , \\\\ - \\Delta _ m f ( z ) & = \\left ( \\frac { 4 } { p q } \\right ) ^ m ( 2 f ( z ) - 2 p f ( y _ 1 ) - 2 q f ( y _ 2 ) ) & & i ( [ z , y _ 2 ] ) = i ( [ y _ 1 , z ] ) - 1 . \\end{align*}"}
-{"id": "1966.png", "formula": "\\begin{align*} g _ \\beta ( x ) : = ( \\phi * \\phi ) ( x ) G _ \\beta ( x ) , \\end{align*}"}
-{"id": "896.png", "formula": "\\begin{align*} V _ n ( \\theta ) = n \\sum _ { I \\in \\Pi ^ 1 _ n , J \\in \\Pi ^ 2 _ n } \\left ( \\int _ { I } \\sigma _ 1 ( t ) ^ 2 d t \\right ) \\left ( \\int _ { J } \\sigma _ 2 ( t ) ^ 2 d t \\right ) K ( I , J _ { - \\theta } ) . \\end{align*}"}
-{"id": "3167.png", "formula": "\\begin{align*} \\Delta _ t ( u ) = \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { \\infty } \\left ( e ^ { z \\psi ( s , u ) } - 1 \\right ) \\nu ( \\mathrm { d } z ) \\mathrm { d } s , u \\in \\mathcal { U } , \\end{align*}"}
-{"id": "2988.png", "formula": "\\begin{align*} \\| u _ n - u \\| _ { L ^ 1 ( \\Omega ) } & \\leq \\| u _ n - u \\| _ { W _ 0 ^ { 1 , q } ( \\Omega ) } \\\\ & \\leq \\| ( \\bar { g } ( x , v _ n ) + \\mu ) - ( \\bar { g } ( x , v ) + \\mu ) \\| _ { \\mathcal { M } ( \\Omega ) } \\\\ & = \\| \\bar { g } ( x , v _ n ) - \\bar { g } ( x , v ) \\| _ { L ^ 1 ( \\Omega ) } \\rightarrow 0 , ~ ~ n \\rightarrow \\infty . \\end{align*}"}
-{"id": "109.png", "formula": "\\begin{align*} \\mathcal { F } ( \\chi _ 1 ) ( 0 ) = \\int _ { \\mathbb { R } ^ k } \\chi _ 1 ( t ) \\ , d t _ 1 \\dots d t _ k = 1 . \\end{align*}"}
-{"id": "5687.png", "formula": "\\begin{align*} X : = \\{ { v } = z ^ + + { w } \\ ; : \\ ; { w } \\in L ^ 2 ( \\R , \\R ^ n ) { w } _ 1 ( t ) = - { w } _ 1 ( - t ) t \\in \\R \\} , \\end{align*}"}
-{"id": "3744.png", "formula": "\\begin{align*} \\theta _ y w ( i ) : = w ( i - n ) + x , \\ ; \\ ; \\ ; i \\in \\Z . \\end{align*}"}
-{"id": "7827.png", "formula": "\\begin{align*} y ' _ { 1 } = g x _ s . \\end{align*}"}
-{"id": "193.png", "formula": "\\begin{align*} \\sigma \\ast ^ { \\delta } \\delta ( a ) = \\delta ( \\sigma \\ast a ) \\leq ^ { \\delta } \\delta ( \\sigma \\ast b ) = \\sigma \\ast ^ { \\delta } \\delta ( b ) . \\end{align*}"}
-{"id": "9013.png", "formula": "\\begin{align*} \\left \\{ \\aligned & \\partial _ { t } \\widetilde { \\theta } + ( u ^ { ( 1 ) } \\cdot \\nabla ) \\widetilde { \\theta } + \\Lambda _ { x _ { 1 } } ^ { 2 \\alpha } \\widetilde { \\theta } + \\Lambda _ { x _ { 2 } } ^ { 2 \\beta } \\widetilde { \\theta } = - ( \\widetilde { u } \\cdot \\nabla ) \\theta ^ { ( 2 ) } , \\\\ & \\widetilde { \\theta } ( x , 0 ) = 0 . \\endaligned \\right . \\end{align*}"}
-{"id": "9848.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } \\hat { f } ( y ) e ^ { i x y } d y \\end{align*}"}
-{"id": "2182.png", "formula": "\\begin{align*} & \\lim _ { t \\to \\infty } t ^ { \\frac { N + A } { 2 } } v \\left ( t ^ { \\frac { 1 } { 2 } } y , t \\right ) = c _ d m ( \\varphi ^ { 0 , 1 } ) | y | ^ A e ^ { - \\frac { | y | ^ 2 } { 4 } } \\ , \\ , \\ , \\mbox { i n } \\ , \\ , \\ , L ^ 2 ( { \\bf R } ^ N , e ^ { | y | ^ 2 / 4 } \\ , d y ) \\ , \\cap \\ , L ^ \\infty ( K ) , \\\\ & \\lim _ { t \\to \\infty } \\ , t ^ { \\frac { N + 2 A } { 2 } } \\frac { v ( x , t ) } { U ( | x | ) } = \\frac { c _ d } { c _ * } m ( \\varphi ^ { 0 , 1 } ) \\ , \\ , \\ , \\mbox { i n } \\ , \\ , \\ , L ^ \\infty ( B ( 0 , R ) ) . \\end{align*}"}
-{"id": "4788.png", "formula": "\\begin{align*} & \\displaystyle \\lim _ { a \\to 0 } \\sup _ { | Q | = a } \\frac { 1 } { | Q | } \\int _ Q | b ( x ) - b _ Q | \\ , d x = 0 , \\\\ & \\displaystyle \\lim _ { a \\to \\infty } \\sup _ { | Q | = a } \\frac { 1 } { | Q | } \\int _ Q | b ( x ) - b _ Q | \\ , d x = 0 , \\quad \\quad { } \\\\ & \\displaystyle \\lim _ { | y | \\to \\infty } \\frac { 1 } { | Q | } \\int _ Q | b ( x + y ) - b _ Q | \\ , d x = 0 , \\mbox { f o r e a c h } Q . \\end{align*}"}
-{"id": "6772.png", "formula": "\\begin{align*} \\Delta V _ { \\lambda } + e ^ { V _ { \\lambda } } = 0 . \\end{align*}"}
-{"id": "5345.png", "formula": "\\begin{align*} W _ 1 \\circ _ C W _ 2 ( 1 ) = Z ^ 1 = \\hom _ { \\C } ( W _ 1 \\otimes W _ 2 / G _ 1 \\cdot ( W _ 1 \\otimes W _ 2 ) , \\C ) \\end{align*}"}
-{"id": "2790.png", "formula": "\\begin{align*} \\beta _ { i } = ( - 1 ) ^ { i } , i = 0 , 1 , \\ldots , n , \\end{align*}"}
-{"id": "7461.png", "formula": "\\begin{align*} \\Phi : = a \\beta a ^ * + a \\gamma a ^ * - \\beta ^ { i + 1 } - \\gamma ^ { j + 1 } - ( - \\beta - \\gamma ) ^ { k + 1 } . \\end{align*}"}
-{"id": "2543.png", "formula": "\\begin{align*} \\mathrm { R e } \\left \\langle \\left [ L - i \\eta \\mathrm { P } _ 1 \\xi _ 1 \\right ] g , \\bar { g } \\right \\rangle = \\left \\langle L g , \\bar { g } \\right \\rangle \\leq - \\nu \\Vert g \\Vert ^ 2 \\end{align*}"}
-{"id": "10147.png", "formula": "\\begin{align*} \\boldsymbol { \\omega } _ k ( i ) = \\sum _ { l \\in \\mathcal { N } _ k } c _ { k l } \\boldsymbol Q _ l ( i ) \\boldsymbol Q _ l ( i - 1 ) \\ldots \\boldsymbol Q _ l ( 1 ) \\boldsymbol \\omega _ l ( 0 ) , \\end{align*}"}
-{"id": "3321.png", "formula": "\\begin{align*} \\bar { D } ( r _ { K - 2 } ) & = \\frac { \\sum _ { i = 0 } ^ { 1 } \\binom { K } { K - 1 + i } ( N - 1 ) ^ i N } { \\binom { K - 2 } { K - 3 } + \\sum _ { i = 0 } ^ { 1 } \\binom { K - 1 } { K - 2 + i } ( N - 1 ) ^ i N } \\\\ & = \\frac { N ^ 2 + ( K - 1 ) N } { N ^ 2 + ( K - 2 ) N + ( K - 2 ) } , \\end{align*}"}
-{"id": "2485.png", "formula": "\\begin{align*} \\left ( - i \\xi \\cdot \\eta + L \\right ) e _ { j } \\left ( \\eta \\right ) = \\varrho _ { j } \\left ( \\eta \\right ) e _ { j } \\left ( \\eta \\right ) . \\end{align*}"}
-{"id": "8820.png", "formula": "\\begin{align*} - \\sum _ { i \\in I } \\nu _ { i } ( a _ { i } + 1 ) & = - \\sum _ { i \\in I } N _ { i } \\sigma _ { i } ( a _ { i } + 1 ) \\\\ & \\leq - \\sum _ { i \\in I } N _ { i } ( a _ { i } + 1 ) ( \\sigma _ { i } - c _ 0 ( f ) ) - m c _ 0 ( f ) \\\\ & \\leq - m c _ 0 ( f ) . \\end{align*}"}
-{"id": "870.png", "formula": "\\begin{align*} | E [ f ( F ) ] - E [ f ( Z ^ * ) ] | & \\leq E \\left [ \\max _ { 1 \\leq i , j \\leq d } | C ^ { i j } - \\langle D F _ i , - D L ^ { - 1 } F _ j \\rangle _ H | \\sum _ { i , j = 1 } ^ p | \\partial ^ 2 _ { i , j } U _ 0 f ( F ) | \\right ] \\\\ & \\lesssim \\beta \\delta ^ { - 1 } E \\left [ \\max _ { 1 \\leq i , j \\leq d } | C ^ { i j } - \\langle D F _ i , - D L ^ { - 1 } F _ j \\rangle _ H | \\right ] . \\end{align*}"}
-{"id": "6313.png", "formula": "\\begin{align*} A _ m ( \\lambda ) = z _ m ^ { d _ m } ( \\lambda ) + a _ m . \\end{align*}"}
-{"id": "305.png", "formula": "\\begin{align*} \\P \\left ( Z _ d \\in B \\ | \\ Z _ 0 = z \\right ) \\ge ~ \\lambda _ d \\left ( B \\cap K \\right ) . \\end{align*}"}
-{"id": "7317.png", "formula": "\\begin{align*} Y : = \\bigcup _ { i \\geq 0 } v _ i ^ U , \\end{align*}"}
-{"id": "2303.png", "formula": "\\begin{align*} \\partial _ t u ^ \\varepsilon + \\nu \\eta _ \\varepsilon * \\eta _ \\varepsilon * A ^ s u ^ \\varepsilon + \\mathcal { P } ^ \\alpha \\eta _ \\varepsilon * [ ( \\eta _ \\varepsilon * u ^ \\varepsilon ) \\cdot \\nabla ( \\eta _ \\varepsilon * u ^ \\varepsilon ) + \\mathcal { U } ^ \\alpha ( \\eta _ \\varepsilon * u ^ \\varepsilon , \\eta _ \\varepsilon * u ^ \\varepsilon ) ] = 0 \\end{align*}"}
-{"id": "8785.png", "formula": "\\begin{align*} \\iint _ { \\R ^ { 2 } \\times \\R ^ { 2 } } f ^ N ( t , x , v ) \\ , d x d v = \\iint _ { \\R ^ { 2 } \\times \\R ^ { 2 } } f ^ N _ { 0 } ( x , v ) \\ , d x d v = : \\mathcal { M } _ 0 , \\end{align*}"}
-{"id": "3761.png", "formula": "\\begin{align*} v _ \\infty : = v _ { \\hat { k } } \\prod _ { k \\ge \\hat { k } } ( 1 - L _ k ^ { - 1 / 1 6 } ) \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "263.png", "formula": "\\begin{align*} d = \\Big \\lfloor \\frac { A | I _ N | } { 4 } \\Big \\rfloor , \\end{align*}"}
-{"id": "87.png", "formula": "\\begin{align*} \\sigma ( h ) ( s , t ) = h ( t , s ) , \\mbox { f o r a l m o s t a l l } ( s , t ) \\in G \\times G , \\ \\ h \\in L ^ { \\infty } ( G \\times G ) , \\end{align*}"}
-{"id": "3350.png", "formula": "\\begin{align*} \\frac { 1 } { \\psi _ 2 ( N , K , s ) } & \\geq N \\sum _ { i = 0 } ^ { \\epsilon K } \\frac { ( K - 1 ) ( K - 1 - s ) ( K - 2 - s ) \\cdots ( K - i - s ) } { s ( s + 1 ) ( s + 2 ) \\cdots ( s + i ) } ( N - 1 ) ^ i \\\\ & \\geq N \\sum _ { i = 0 } ^ { \\epsilon K } \\frac { ( K - 1 ) ( K - \\epsilon K - s ) ^ { i } } { ( s + \\epsilon K ) ^ { i + 1 } } ( N - 1 ) ^ i \\\\ & = N \\sum _ { i = 0 } ^ { \\epsilon K } \\frac { ( 1 - \\frac { 1 } { K } ) ( ( 1 - ( \\lambda + \\epsilon ) ) ^ { i } } { ( \\lambda + \\epsilon ) ^ { i + 1 } } ( N - 1 ) ^ i . \\end{align*}"}
-{"id": "8388.png", "formula": "\\begin{align*} f ( \\tau ) = \\sum _ { \\substack { m \\in \\Q \\\\ \\mu \\in V _ \\Z ^ \\vee / V _ \\Z } } c ( m , \\mu ) \\cdot q ^ m \\in M ^ ! _ { 1 - \\frac { n } { 2 } } ( \\overline { \\rho } _ { V _ \\Z } ) . \\end{align*}"}
-{"id": "4776.png", "formula": "\\begin{align*} { \\varepsilon } = F ( x ^ 1 , Y , Z ) \\ , { \\Delta \\sqrt { - \\det g ^ { i j } } } / { { L } _ 2 } , \\end{align*}"}
-{"id": "3803.png", "formula": "\\begin{align*} \\begin{array} { l c l } \\bar { X } ^ { ( x , t ) } _ 0 & : = & x , \\\\ \\bar { X } ^ { ( x , t ) } _ { s + 1 } & : = & \\bar { X } ^ { ( x , t ) } _ { s } + \\left \\{ \\begin{array} { l l } 1 & U _ { ( \\bar { X } ^ { ( x , t ) } _ s , s + t ) } \\le p _ \\circ , \\\\ - 1 & \\end{array} \\right . s \\in \\Z _ + . \\end{array} \\end{align*}"}
-{"id": "9725.png", "formula": "\\begin{align*} V _ { k } = \\Phi _ 1 ( \\gamma _ 1 ; V _ a ) , ( u _ { k } , v _ { k } ) \\cdot \\textbf { n } _ { k } = 0 . \\end{align*}"}
-{"id": "8223.png", "formula": "\\begin{align*} \\biggl [ \\frac { 1 } { n ^ d } \\biggr ] ( t _ + ^ 2 - t _ - ^ 2 ) = 0 d \\le k - 1 \\end{align*}"}
-{"id": "3728.png", "formula": "\\begin{align*} ( p _ { m _ k } , p ' _ { m _ k } ) = ( p _ { m _ k + 1 } , p ' _ { m _ k + 1 } ) = \\dots = ( p _ { m _ { k + 1 } - 1 } , p ' _ { m _ { k + 1 } - 1 } ) \\neq ( p _ { m _ { k + 1 } } , p ' _ { m _ k + 1 } ) \\end{align*}"}
-{"id": "7201.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ N \\xi _ { k , p } ( u _ { k , p } ) _ \\nu ^ 2 ( x ) = \\xi _ { 0 , p } ( x ) \\end{align*}"}
-{"id": "4460.png", "formula": "\\begin{align*} G ( k ) : = \\frac { | k _ 1 | } { | k _ 1 | ^ 3 + k _ 2 ^ 2 } \\quad \\mbox { a n d } G _ \\ell ( k ) : = \\phi ( \\ell k _ 1 , \\ell ^ \\frac { 3 } { 2 } k _ 2 ) G ( k ) ; \\end{align*}"}
-{"id": "7614.png", "formula": "\\begin{align*} [ g ] _ h ( x , t ) : = \\left \\{ \\begin{array} { l c l l } \\frac { 1 } { h } \\int _ t ^ { t + h } g ( x , s ) \\ , d s & \\qquad t \\in ( \\bar t - \\theta , \\bar t + \\theta - h ] , \\\\ 0 & t > \\bar t + \\theta - h . \\end{array} \\right . \\end{align*}"}
-{"id": "9546.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ { M - 1 } \\prod _ { l = 1 } ^ { \\infty } ( 1 - e ^ { 2 \\pi i k / M } Y q ^ l q ^ { - \\{ k s / M \\} } ) & = \\prod _ { k ' } \\prod _ { l = 1 } ^ { \\infty } ( 1 - e ^ { 2 \\pi i k ' t ' / m } Y ^ r q ^ { r l } q ^ { - k ' r / m } ) \\\\ & = \\prod _ { l = 0 } ^ { \\infty } ( 1 - Y ^ r q ^ { l r / m } \\eta ^ { - l } ) / \\prod _ { l = 0 } ^ { \\infty } ( 1 - Y q ^ l ) . \\end{align*}"}
-{"id": "3085.png", "formula": "\\begin{align*} t _ \\epsilon : = \\min \\{ t \\in [ 0 , T ] : \\ , \\| u _ \\epsilon ( t ) \\| = \\mu \\} \\ , . \\end{align*}"}
-{"id": "2647.png", "formula": "\\begin{align*} \\dfrac { 1 } { K + 1 } + \\sum _ { i = 3 } ^ { n } \\left \\{ \\dfrac { c _ i } { K + 1 } \\right \\} \\geq \\dfrac { 1 } { K + 1 } + \\dfrac { K - 1 } { K + 1 } + \\dfrac { K } { K + 1 } > 1 , \\end{align*}"}
-{"id": "4172.png", "formula": "\\begin{align*} \\frac { 1 } { n } e _ { k , j } ^ n + \\frac { 1 } { n } \\left ( 1 - \\frac { 1 } { n } \\right ) \\sum _ { k ' \\in I } e _ { k ' , j } ^ { n - 1 } + \\frac { 1 } { n } \\sum _ { \\tau = 1 } ^ { n - 2 } e _ { i , j + \\tau } ^ { n - 1 - \\tau } . \\end{align*}"}
-{"id": "9575.png", "formula": "\\begin{align*} \\bigg ( \\int _ D \\lvert u ( x ) \\rvert ^ q \\delta _ { \\partial D } ^ { ( q / p ) ( n - s p + \\beta ) - n } ( x ) \\ , d x \\bigg ) ^ { p / q } \\le C \\int _ D \\int _ { B ( x , \\tau \\delta _ { \\partial D } ( x ) ) } \\frac { \\lvert u ( x ) - u ( y ) \\rvert ^ p } { \\lvert x - y \\rvert ^ { n + s p } } \\ , d y \\ , \\delta _ { \\partial D } ^ \\beta ( x ) \\ , d x \\ , , \\end{align*}"}
-{"id": "2100.png", "formula": "\\begin{align*} \\frac { \\partial K } { \\partial t } = \\tau ( K ) \\end{align*}"}
-{"id": "616.png", "formula": "\\begin{align*} \\left ( \\int _ { \\Omega } | f | ^ 2 \\delta ^ { - 2 s } \\right ) ^ { 1 / 2 } \\le \\left ( \\int _ { \\Omega } | f | ^ { r } \\right ) ^ { 1 / r } \\left ( \\int _ { \\Omega } \\delta ^ { - 2 s \\cdot \\frac { r } { r - 2 } } \\right ) ^ { \\frac { r - 2 } { 2 r } } = C \\| f \\| _ { L ^ { r } } , \\end{align*}"}
-{"id": "5779.png", "formula": "\\begin{align*} \\| \\mu \\| _ { \\dot { H } ^ { - \\alpha } ( \\R ^ n ) } = \\sup \\frac { | \\langle \\mu , u \\rangle | } { \\| u \\| _ { \\dot { H } ^ { \\alpha } ( \\R ^ n ) } } < + \\infty , \\end{align*}"}
-{"id": "2660.png", "formula": "\\begin{align*} { \\rm S c a l } _ { ( \\xi , a , p ) } ( g ) : = f ^ { 2 } _ { ( \\xi , \\omega , a ) } { \\rm S c a l } _ g - 2 ( p - 1 ) f _ { ( \\xi , \\omega , a ) } \\Delta _ g f _ { ( \\xi , \\omega , a ) } - p ( p - 1 ) | \\xi | ^ { 2 } _ g , \\end{align*}"}
-{"id": "7370.png", "formula": "\\begin{align*} L _ { \\omega } \\psi = ( i _ { X ^ a } \\omega ) . \\nabla _ { X _ a } \\psi + \\frac { p } { 2 ( p + 1 ) } d \\omega . \\psi . \\end{align*}"}
-{"id": "6512.png", "formula": "\\begin{align*} d s _ { \\mathcal { M } ^ { } } ^ { 2 } = \\frac { 1 } { \\sigma ^ { 2 } } \\left ( \\frac { 1 } { 1 - \\rho ^ { 2 } } d \\mu _ { x } ^ { 2 } + \\frac { 1 } { 1 - \\rho ^ { 2 } } d \\mu _ { y } ^ { 2 } - \\frac { 2 \\rho } { 1 - \\rho ^ { 2 } } d \\mu _ { x } d \\mu _ { y } + 4 d \\sigma ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "8509.png", "formula": "\\begin{align*} L ( E , s ) = \\prod _ { v \\nmid N } ^ { } \\left ( 1 - \\frac { a _ v } { q _ v ^ { s } } + \\frac { q _ v } { { q _ v } ^ { 2 s } } \\right ) ^ { - 1 } \\times \\prod _ { v \\mid N } ^ { } \\left ( 1 - \\frac { 1 } { q _ v ^ { s } } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "7392.png", "formula": "\\begin{align*} f = x ^ 2 + u y ^ 2 + 2 v y z + w z ^ 2 + ( u w - v ^ 2 ) t ^ 2 . \\end{align*}"}
-{"id": "1179.png", "formula": "\\begin{align*} v ( y _ 0 ) = v ( z _ 0 ) . \\end{align*}"}
-{"id": "4091.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } k _ { M , p } ( \\theta ) \\ d \\theta = \\frac { \\lambda _ 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } ( \\cos \\theta ) ^ 2 d \\theta + \\frac { \\lambda _ 2 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } ( \\sin \\theta ) ^ 2 d \\theta = \\frac { \\lambda _ 1 + \\lambda _ 2 } { 2 } = H , \\end{align*}"}
-{"id": "673.png", "formula": "\\begin{gather*} - y '' + q ( x ) y = \\mu y , x \\in ( 0 , \\pi ) , \\ \\mu \\in \\mathbb { C } , \\\\ y ( 0 ) \\cot \\alpha + y ' ( 0 ) = 0 , \\alpha \\in ( 0 , \\pi ) , \\\\ y ( \\pi ) \\cot \\beta + y ' ( \\pi ) = 0 , \\beta \\in ( 0 , \\pi ) , \\end{gather*}"}
-{"id": "5660.png", "formula": "\\begin{align*} \\mathcal { S } ( b ^ - , b ^ + ) = \\left \\{ { v } \\in H ^ 1 _ { l o c } ( \\R , \\R ^ n ) \\ ; : \\ ; \\lim \\limits _ { t \\to \\pm \\infty } { v } ( t ) = b ^ \\pm \\right \\} . \\end{align*}"}
-{"id": "6632.png", "formula": "\\begin{align*} A _ 1 & = \\left ( \\int _ { \\varphi _ n ( E _ M ) } | D f _ n - D h _ n | ^ p \\ , d \\mu \\right ) ^ { \\frac { 1 } { p } } \\\\ & \\leq \\| \\mathrm { i d } - D \\phi _ n ^ 0 \\circ f _ n \\| _ { L ^ \\infty } \\left ( \\int _ { \\varphi _ n ( E _ M ) } | D f _ n | ^ p \\ , d \\mu \\right ) ^ { \\frac { 1 } { p } } \\ , \\end{align*}"}
-{"id": "5943.png", "formula": "\\begin{align*} g _ i ( x ) & = f _ i ( x ) & & \\forall i \\in V ( H ) \\\\ g _ u ( x ) & = x _ u + ( x _ a + 1 ) x _ u \\\\ & = x _ u \\land x _ a \\\\ g _ v ( x ) & = x _ v + ( x _ u + 1 ) x _ v \\\\ & = x _ v \\land x _ u . \\end{align*}"}
-{"id": "3756.png", "formula": "\\begin{align*} \\log \\rho _ k \\geq \\log \\rho _ { \\hat { k } } - 2 \\sum _ { i = \\hat { k } } ^ \\infty L _ i ^ { - 1 / 1 6 } = \\log ( \\iota _ { \\hat { k } } ^ { - 1 } \\rho _ { \\hat { k } } ) > - \\infty . \\end{align*}"}
-{"id": "3180.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ f \\left ( Z _ { t } ^ { 2 } \\right ) \\right ] = \\int _ { \\mathbb { R } _ { \\geqslant 0 } } f ( y ) \\mu _ { Z _ { t } ^ { 2 } } ( \\mathrm { d } y ) = e ^ { - \\lambda _ { t } } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\lambda _ { t } ^ { n } } { n ! } \\int _ { \\mathbb { R } _ { \\geqslant 0 } } f ( y ) \\rho _ { t } ^ { \\ast n } ( \\mathrm { d } y ) < \\infty . \\end{align*}"}
-{"id": "3042.png", "formula": "\\begin{align*} ( A u , v ) _ { L ^ 2 } = ( \\nabla u , \\nabla v ) _ { L ^ 2 } , u , v \\in H ^ 1 _ 0 ( \\Omega ) , \\end{align*}"}
-{"id": "4567.png", "formula": "\\begin{align*} \\theta ^ { - 1 } ( E _ i ( z ) ) v = 0 , \\theta ^ { - 1 } ( K ^ { \\pm } _ i ( z ) ) v = P _ i ^ \\pm ( z ) v ( 0 \\le i \\le n - 1 ) \\ , . \\end{align*}"}
-{"id": "736.png", "formula": "\\begin{align*} u ^ * ( x ) = u ^ * ( x ' , x _ 3 ) : = \\int _ { 0 } ^ { \\infty } \\chi _ { \\{ | u ( x ' , y ) | > t \\} ^ * } ( x _ 3 ) \\ , d t , \\end{align*}"}
-{"id": "7645.png", "formula": "\\begin{align*} E & = \\Big ( y , x _ 1 C ^ { ( 1 ) } , \\ldots , x _ { 2 ^ { n - 3 } - 1 } C ^ { ( 2 ^ { n - 3 } - 1 ) } , z A , x _ { 2 ^ { n - 3 } - 1 } C ^ { ( 2 ^ { n - 3 } - 1 ) } _ { \\overline { R } } , \\ldots , x _ 1 C ^ { ( 1 ) } _ { \\overline { R } } \\Big ) , \\\\ F & = \\Big ( y , x _ 1 D ^ { ( 1 ) } , \\ldots , x _ { 2 ^ { n - 3 } - 1 } D ^ { ( 2 ^ { n - 3 } - 1 ) } , z B , x _ { 2 ^ { n - 3 } - 1 } D ^ { ( 2 ^ { n - 3 } - 1 ) } _ { \\overline { R } } , \\ldots , x _ 1 D ^ { ( 1 ) } _ { \\overline { R } } \\Big ) , \\end{align*}"}
-{"id": "820.png", "formula": "\\begin{align*} \\frac { u _ { n + 1 } - u _ n } { \\tau } = \\frac { \\Delta u _ { n + 1 } + \\Delta u _ n } { 2 } \\end{align*}"}
-{"id": "6435.png", "formula": "\\begin{align*} \\mathcal { S } _ { } ^ { } = - \\int d X \\rho \\left ( X \\right ) \\log _ { 2 } \\left [ \\rho \\left ( X \\right ) \\Delta v \\right ] = - \\sum _ { j } p _ { j } \\log _ { 2 } p _ { j } \\approx h _ { } t \\end{align*}"}
-{"id": "6414.png", "formula": "\\begin{align*} \\left \\{ \\left [ 1 + 2 \\gamma \\left ( \\xi ^ { 2 } - 1 \\right ) \\right ] d \\theta _ { \\mu } + 2 \\gamma \\Delta \\theta _ { \\mu } \\right \\} \\delta \\left ( d \\theta ^ { \\mu } \\right ) = 0 \\end{align*}"}
-{"id": "5158.png", "formula": "\\begin{align*} D _ { A } \\coloneqq \\inf \\{ t \\ge 0 \\colon X _ { t } \\in A \\} = \\inf \\{ t > 0 \\colon X _ { t } \\in A \\} \\end{align*}"}
-{"id": "7948.png", "formula": "\\begin{align*} h _ { j _ 1 \\cdot \\cdot j _ { \\ell } } : = f _ { I j _ 1 \\cdot \\cdot j _ { \\ell } } : \\bar W _ { j _ 1 \\cdot \\cdot j _ { \\ell - 1 } } \\to \\bar W _ { j _ 1 \\cdot \\cdot j _ { \\ell } } , \\end{align*}"}
-{"id": "1126.png", "formula": "\\begin{align*} Z ( s , A _ { \\Phi } ^ 0 ) : = \\sum _ { \\chi } a _ { \\chi } Z ( s , \\chi ) , Z ( s , \\chi ) : = \\frac { L ' ( s , \\chi ) } { L ( s , \\chi ) } + \\frac { 1 } { 2 } \\log f _ { \\chi } . \\end{align*}"}
-{"id": "5016.png", "formula": "\\begin{align*} \\int _ { \\| y \\| \\geq 4 \\| x \\| } \\sum _ { 2 ^ j \\| x \\| \\geq \\| r \\| } | k _ j ( y - x ) - k _ j ( y ) | d y & \\leq \\sum _ { 2 ^ j \\| x \\| \\geq \\| r \\| } 2 \\int _ { \\| y \\| \\geq 2 \\| x \\| } k _ j ( y ) d y \\\\ & = \\sum _ { 2 ^ j \\| x \\| \\geq \\| r \\| } 2 \\int _ { \\| y \\| \\geq 2 \\cdot 2 ^ j \\| x \\| } \\varphi ( y + r ) d y \\\\ & \\lesssim \\sum _ { 2 ^ j \\| x \\| \\geq \\| r \\| } \\frac { 1 } { 2 ^ j \\| x \\| } \\\\ & \\lesssim \\frac { 1 } { \\| r \\| } \\lesssim 1 . \\end{align*}"}
-{"id": "6664.png", "formula": "\\begin{align*} S _ 2 = \\{ \\xi \\in S : \\langle \\xi - x , u ( x ) \\rangle \\geq - ( 1 + \\varepsilon ) \\Delta \\} \\quad . \\end{align*}"}
-{"id": "2395.png", "formula": "\\begin{align*} a _ 1 a _ 6 - a _ 2 = a _ 1 a _ 5 - a _ 3 + a _ 4 + a _ 6 = 0 , \\\\ a _ 1 ( 3 a _ 3 + a _ 4 ) - 3 a _ 2 - a _ 5 = a _ 1 a _ 2 + a _ 3 = 0 . \\end{align*}"}
-{"id": "8015.png", "formula": "\\begin{align*} \\big | T _ { [ d _ k ] } f ( x ) - T _ { [ d _ k ] } f ( y ) \\big | & = \\Big | \\int _ { \\mathbb { R } ^ d } { \\big ( U _ k ( x , z ) - U _ k ( y , z ) \\big ) \\Pi ^ * _ k f ( z ) } d z \\Big | \\\\ & \\leq \\big \\Vert \\Pi ^ * _ k f \\big \\Vert _ { L ^ { \\infty } } \\int _ { \\mathbb { R } ^ d } { \\big | U _ k ( x , z ) - U _ k ( y , z ) \\big | } d z . \\end{align*}"}
-{"id": "1278.png", "formula": "\\begin{align*} \\left . \\frac { d } { d t } k ( t ) \\right | _ { t = \\tau } < 0 \\mbox { f o r } \\ , \\ , \\tau \\in ( 0 , t _ 0 ] . \\end{align*}"}
-{"id": "9435.png", "formula": "\\begin{align*} N \\mathcal { M } ( N ) \\asymp \\sum _ { n = 1 } ^ N \\mathcal { M } ( n ) . \\end{align*}"}
-{"id": "8552.png", "formula": "\\begin{align*} & \\mbox { I f $ u $ i s a s o l u t i o n o f \\eqref { e q : 1 . 4 } , t h e n $ k u $ a n d $ u ( k x , k ^ 2 t ) $ a r e a l s o s o l u t i o n s o f \\eqref { e q : 1 . 4 } } \\\\ & \\mbox { f o r a n y $ k \\in { \\bf R } $ } \\end{align*}"}
-{"id": "6958.png", "formula": "\\begin{align*} \\delta ( u , x , y ) = u ( x + y ) - 2 u ( x ) + u ( x - y ) . \\end{align*}"}
-{"id": "1992.png", "formula": "\\begin{align*} \\sup _ { \\lambda / n \\leq s < t } \\frac { n ^ { \\eta } \\left \\vert \\beta _ { n } \\left ( s ; t \\right ) - B _ { n } \\left ( s \\right ) \\right \\vert } { s ^ { 1 / 2 - \\eta } } = O _ { \\mathbb { P } } \\left ( 1 \\right ) \\end{align*}"}
-{"id": "185.png", "formula": "\\begin{align*} \\delta _ { 1 } ( a ) = R a \\in R \\end{align*}"}
-{"id": "3248.png", "formula": "\\begin{align*} ( \\alpha _ a , \\omega ) _ X = ( \\alpha , \\omega ) _ X . \\end{align*}"}
-{"id": "9316.png", "formula": "\\begin{align*} d _ 1 - d _ 2 = g _ r ( d _ 3 ) - g _ r ( d _ 4 ) = f ( r + d _ 3 ) - f ( r + d _ 4 ) . \\end{align*}"}
-{"id": "9359.png", "formula": "\\begin{align*} \\left ( j O _ { n } ^ { ( 3 ) } \\right ) ^ { 2 } + 3 J O _ { n + 3 } ^ { ( 3 ) } \\cdot j O _ { n + 3 } ^ { ( 3 ) } = 4 ^ { n + 3 } \\underline { \\alpha } ^ { 2 } + \\frac { 3 \\cdot 2 ^ { n + 1 } } { 4 9 } \\left ( 2 5 \\underline { \\alpha } \\cdot \\widehat { \\epsilon _ { n } } - 3 1 \\widehat { \\epsilon _ { n } } \\cdot \\underline { \\alpha } \\right ) , \\end{align*}"}
-{"id": "149.png", "formula": "\\begin{align*} x ^ { \\sigma } = x ; w ^ { \\sigma } = w ; \\lambda _ { 1 1 } y ^ { \\sigma } = \\lambda _ { 2 2 } y ; \\lambda _ { 2 2 } z ^ { \\sigma } = \\lambda _ { 1 1 } z \\end{align*}"}
-{"id": "4047.png", "formula": "\\begin{align*} d ( p , q ) : = \\inf _ { \\sigma } l ( \\sigma ) , \\end{align*}"}
-{"id": "978.png", "formula": "\\begin{align*} \\widehat { \\sigma } ^ 2 _ n ( t ) - \\sigma ^ 2 ( t ) = M _ n ( t ) + b _ n ( t ) . \\end{align*}"}
-{"id": "10052.png", "formula": "\\begin{align*} \\mathcal { L } ( s , \\tilde { g } ) = \\Lambda ( s + 1 , \\chi _ E ) \\cdot \\langle E ( s ) , \\tilde { g } \\rangle _ \\mathrm { P e t } \\end{align*}"}
-{"id": "7282.png", "formula": "\\begin{align*} \\varphi = \\left ( \\begin{array} { c c c c c c c c c c c } z & 0 & 0 & \\ldots & 0 & 0 & 0 \\\\ - c _ 1 & z & 0 & \\ldots & 0 & 0 & 0 \\\\ 0 & - c _ 2 & z & \\ldots & 0 & 0 & 0 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots \\\\ 0 & 0 & 0 & \\ldots & z & 0 & 0 \\\\ 0 & 0 & 0 & \\ldots & - c _ { n - 3 } & z & 0 \\\\ 0 & 0 & 0 & \\ldots & 0 & - c _ { n - 2 } & z \\\\ 0 & 0 & 0 & \\ldots & 0 & 0 & - c _ { n - 1 } \\end{array} \\right ) \\end{align*}"}
-{"id": "8846.png", "formula": "\\begin{align*} \\gamma ( t , v ) : = \\int _ { \\R } u ( t , x ) \\overline { \\Psi } _ v ( t , x ) d x , \\end{align*}"}
-{"id": "8107.png", "formula": "\\begin{align*} \\hat { h } _ { i , t } ( x ) = \\alpha _ t ( g _ i ) ( x ) \\geq \\hat { g } _ { i , t } ( x ) = 1 . \\end{align*}"}
-{"id": "7115.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } S _ { i j } = a ^ { k l } \\nabla _ k \\nabla _ l S _ { i j } + u ^ k \\nabla _ k S _ { i j } + N _ { i j } , \\end{align*}"}
-{"id": "7578.png", "formula": "\\begin{align*} \\pi _ \\psi ( u , v ) = \\pi _ 1 ( u , v ) + \\pi _ 2 ( u , v ) + \\pi _ 3 ( u , v ) , \\end{align*}"}
-{"id": "8699.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } I _ { n , k } = \\sum _ { k = 1 } ^ { k _ 0 ( n ) } I _ { n , k } + \\sum _ { k = k _ 0 ( n ) + 1 } ^ { \\lfloor n ^ 2 q ^ { - n } \\rfloor } I _ { n , k } + \\sum _ { k = \\lceil n ^ 2 q ^ { - n } \\rceil } ^ { \\infty } I _ { n , k } . \\end{align*}"}
-{"id": "3954.png", "formula": "\\begin{align*} g ^ { * } _ { \\lambda } ( u ) ( x ) = \\Bigl [ \\int _ { 0 } ^ { \\infty } \\int _ { \\mathbb { R } ^ { n } } \\bigl ( \\frac { t } { t + | y | } \\bigr ) ^ { \\lambda n } t ^ { 1 - n } | \\partial _ { t , x } U ( x - y , t ) | ^ { 2 } d y d t \\Bigr ] ^ { \\frac 1 2 } \\end{align*}"}
-{"id": "4866.png", "formula": "\\begin{align*} \\log \\left ( \\exp ( x ) \\cdot \\exp ( y ) \\right ) = x + y + \\frac { 1 } { 2 } \\left [ x , y \\right ] . \\end{align*}"}
-{"id": "5001.png", "formula": "\\begin{align*} \\sum _ { r \\in \\Z } \\dots = \\sum _ { r \\leq 0 } \\dots + \\sum _ { 0 < r < a _ { \\alpha } t } \\dots + \\sum _ { r \\geq a _ { \\alpha } t } \\dots , \\end{align*}"}
-{"id": "6087.png", "formula": "\\begin{align*} D _ t = \\left ( \\begin{smallmatrix} \\cos t & - \\sin t \\\\ \\sin t & \\cos t \\end{smallmatrix} \\right ) . \\end{align*}"}
-{"id": "9696.png", "formula": "\\begin{align*} \\frac { \\partial \\gamma _ 1 } { \\partial \\omega _ { k , k + 1 } } { \\big | _ { \\{ \\omega _ { k , k + 1 } = 0 , U _ a = { U } _ 2 ^ { ( 0 ) } \\} } } = \\frac { { u } _ 2 ^ { ( 0 ) } } { \\kappa _ 1 ( { U } _ 2 ^ { ( 0 ) } ) } > 0 . \\end{align*}"}
-{"id": "6067.png", "formula": "\\begin{align*} \\Phi ^ { '' } ( r ) + ( n - 1 ) \\tfrac { \\Phi ' ( r ) } { r } - \\mathfrak { e } _ 1 \\tfrac { \\sin ( 2 \\Phi ( r ) ) } { r ^ 2 } = 0 , \\end{align*}"}
-{"id": "1084.png", "formula": "\\begin{align*} \\varLambda _ i = \\mu \\ , \\bigg | \\rho _ i \\alpha _ { i i } + \\sum \\limits _ { k \\in \\mathcal { N } , k \\neq i } \\rho _ k \\alpha _ { k i } \\bigg | ^ 2 , \\end{align*}"}
-{"id": "2380.png", "formula": "\\begin{align*} \\varphi _ { \\lambda , w } ( t ) \\sim \\sum _ { n = - ( m + 1 ) } ^ \\infty a _ n ( \\lambda , w ) t ^ n \\end{align*}"}
-{"id": "5486.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { u ( r ^ { n _ k } z s ) } { ( r ^ { n _ k } z ) ^ { \\rho - 1 } \\ell ( r ^ { n _ k } z ) } = v _ z ( s ) s \\in C _ { v _ z } . \\end{align*}"}
-{"id": "9409.png", "formula": "\\begin{align*} \\sum _ { d \\colon d \\mid k } \\left ( \\alpha _ { n + 1 } ( d , k ) + \\beta _ { n + 1 } ( d , k ) \\right ) = { n \\choose k } { n + k \\choose k } \\end{align*}"}
-{"id": "5655.png", "formula": "\\begin{align*} \\int _ a ^ b F _ n ( \\varphi ( t ) ) \\d t = \\sum _ k \\lambda ^ k _ n \\mathcal { L } ^ 1 \\big ( \\{ t \\ , : \\ , a \\leq t \\leq b , \\ , t ^ k _ n \\leq \\varphi ( t ) \\leq t ^ { k + 1 } _ n \\} \\big ) = \\sum _ k \\lambda ^ k _ n \\big | ( G ( t ^ { k + 1 } _ n ) \\wedge b ) - ( G ( t ^ k _ n ) \\vee a ) \\big | . \\end{align*}"}
-{"id": "7281.png", "formula": "\\begin{align*} N = \\lfloor M \\log _ 3 ( M ) \\rfloor , K = \\log M ( \\log \\log M ) ^ C \\log _ 4 M . \\end{align*}"}
-{"id": "8338.png", "formula": "\\begin{align*} K _ p = G ( \\Q _ p ) \\cap C ( V _ { \\Z _ p } ) ^ \\times . \\end{align*}"}
-{"id": "7898.png", "formula": "\\begin{align*} u ( x , t ) = u _ 0 ( x _ t ) - t L \\left ( { d ( x , x _ t ) \\over t } \\right ) . \\end{align*}"}
-{"id": "2601.png", "formula": "\\begin{align*} ( D ^ a R _ \\ell ) ( t ) = \\frac 1 { \\Gamma ( 1 - a ) } \\frac d { d t } \\int _ 0 ^ t \\frac { R _ \\ell ( t - r ) } { r ^ a } \\ , d r = \\bigg [ \\frac { R _ \\ell ( 0 ) } { \\Gamma ( 1 - a ) t ^ a } + \\int _ 0 ^ t \\frac { R _ \\ell ' ( t - r ) } { r ^ a } \\ , d r \\bigg ] . \\end{align*}"}
-{"id": "9571.png", "formula": "\\begin{align*} g ( y ) : = \\biggl ( \\int _ { B ( y , \\tau \\delta _ { \\partial D } ( y ) ) } \\frac { \\vert u ( y ) - u ( z ) \\vert ^ p } { \\vert y - z \\vert ^ { n + s p } } \\ , d z \\biggr ) ^ { 1 / p } \\end{align*}"}
-{"id": "3463.png", "formula": "\\begin{align*} \\mathbf { g } ( \\omega , \\mathbf { x } ) = 0 \\mathbf { g } ( \\omega , \\mathbf { y } ) = 0 \\end{align*}"}
-{"id": "9909.png", "formula": "\\begin{align*} \\alpha _ i = \\begin{cases} a _ i - \\frac { 2 } { 3 r - 2 } & \\\\ a _ i - \\frac { 3 } { 3 r - 2 } & \\end{cases} \\end{align*}"}
-{"id": "4858.png", "formula": "\\begin{align*} s = 6 ( 2 d - 5 ) - \\sum _ I ( 4 m _ p + 4 l _ p - 1 5 ) - \\sum _ J ( 1 0 m _ p + c _ p - 1 5 ) . \\end{align*}"}
-{"id": "483.png", "formula": "\\begin{align*} p ^ { ( m ) } _ { 1 , k _ 1 , k _ 2 } ( x , t ) = \\int _ \\R \\frac { \\partial ^ { k _ 2 } } { \\partial \\abs { t } ^ { k _ 2 } } p ^ { ( m + 1 ) } _ { 1 , k _ 1 , 0 } ( x , ( t , t _ { m + 1 } ) ) \\ , \\dd t _ { m + 1 } . \\end{align*}"}
-{"id": "1734.png", "formula": "\\begin{align*} { \\tilde { \\mathcal { S } } } : = \\Big \\{ \\sum _ i ^ { n } a _ { i } \\chi _ { { B _ i } } \\ | \\ \\ n \\in \\N , B _ i \\in \\sigma ( \\mathcal { \\tilde { R } } ) , a _ i \\in \\C \\Big \\} \\end{align*}"}
-{"id": "3119.png", "formula": "\\begin{align*} U ( a ) U ( b ) U ( a + b ) ^ { - 1 } = e ^ { \\imath ( e / \\hbar ) \\ , \\Phi ( 0 , a , b ) } \\ , , \\end{align*}"}
-{"id": "2850.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { m } \\frac { | s _ { n } | ^ { k } } { n } = O ( X _ { m } ) m \\rightarrow \\infty , \\end{align*}"}
-{"id": "9551.png", "formula": "\\begin{align*} ( - 1 ) ^ { \\ell ( \\tilde \\sigma ) } x _ 1 ^ { o ( \\tilde \\sigma ) } x _ 2 ^ { e ( \\tilde \\sigma ) } y ^ { o ( \\tilde \\sigma ) } z ^ { e ( \\tilde \\sigma ) } = x _ 1 ^ { 1 - \\delta ( n ) } x _ 2 ^ { \\delta ( n ) } y ^ { \\lceil \\frac { n - 1 } { 2 } \\rceil } z ^ { \\lfloor \\frac { n - 1 } { 2 } \\rfloor } ( - 1 ) ^ { n - 1 } ( - 1 ) ^ { \\ell ( \\sigma ) } x _ 1 ^ { o ( \\sigma ) } x _ 2 ^ { e ( \\sigma ) } y ^ { o ( \\sigma ) } z ^ { e ( \\sigma ) } , \\end{align*}"}
-{"id": "6086.png", "formula": "\\begin{align*} \\Phi ( r ) = \\Psi _ n ( \\tau + \\ln r ) , \\quad \\mbox { f o r a l l } r \\in [ 0 , 1 ] , \\quad \\Psi _ n ( \\tau ) = \\rho , \\end{align*}"}
-{"id": "1418.png", "formula": "\\begin{align*} Q _ j ( x _ 3 ) & = x _ 2 ^ { 2 ^ { j - 1 } } Q _ { j - 1 } x _ 3 + x _ 3 ^ { 2 ^ { j - 1 } } Q _ { j - 2 } x _ 3 , \\\\ Q _ j ( x _ 4 ) & = x _ 3 x _ { 4 } ^ { 2 ^ { j - 1 } } + x _ 2 ^ { 2 ^ { j - 1 } } Q _ { j - 1 } x _ 4 + x _ 3 ^ { 2 ^ { j - 1 } } Q _ { j - 2 } x _ 4 . \\end{align*}"}
-{"id": "6708.png", "formula": "\\begin{align*} D _ { N } ^ { \\eta , \\varsigma } ( t ) = \\frac { 1 } { 2 N } \\sum _ { i = 1 } ^ { N } | \\sigma _ { i } ^ { \\eta } ( t ) - \\sigma _ { i } ^ { \\varsigma } ( t ) | . \\end{align*}"}
-{"id": "6306.png", "formula": "\\begin{align*} \\partial _ x \\big ( \\begin{smallmatrix} \\psi _ 1 \\\\ \\psi _ 2 \\end{smallmatrix} \\big ) & = U \\big ( \\begin{smallmatrix} \\psi _ 1 \\\\ \\psi _ 2 \\end{smallmatrix} \\big ) & & & U & = \\big ( \\begin{smallmatrix} 0 & 1 \\\\ u - \\lambda & 0 \\end{smallmatrix} \\big ) . \\end{align*}"}
-{"id": "5712.png", "formula": "\\begin{align*} v _ 1 \\sim v _ 2 \\Longleftrightarrow ( v _ 1 = v _ 2 ) ( v _ 1 , v _ 2 \\in \\mathcal { C } ( z ^ + ) ) ( v _ 1 , v _ 2 \\in \\mathcal { C } ( z ^ - ) ) . \\end{align*}"}
-{"id": "8866.png", "formula": "\\begin{align*} n = \\sum _ { i = 1 } ^ r ( \\Delta ^ + ) ^ i = \\frac { ( \\Delta ^ + ) ^ { r + 1 } - 1 } { \\Delta ^ + - 1 } . \\end{align*}"}
-{"id": "8470.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { t , l , v } ) = q ^ { \\frac { n } { 1 2 } } \\gamma _ F ( - 2 v , l ) \\chi ^ { - 1 } ( 4 v ^ 2 ) \\psi ( \\frac { 3 } { 4 v } \\varpi ^ { - \\frac { n } { 2 } } ) _ { \\psi } ( - 1 6 b v ^ 3 \\varpi ^ { \\lceil \\frac { l } { 2 } \\rceil + 2 \\{ \\frac { l } { 2 } \\} - 3 \\lfloor \\frac { 1 } { 2 } \\lceil \\frac { l } { 2 } \\rceil \\rfloor } ; \\Delta \\varpi ^ { - \\lfloor \\frac { l } { 2 } \\rfloor - \\lfloor \\frac { 1 } { 2 } \\lceil \\frac { l } { 2 } \\rceil \\rfloor } ) , \\end{align*}"}
-{"id": "5749.png", "formula": "\\begin{align*} \\frac { \\psi _ { x , y - 1 , z } ( a , b ) } { \\psi _ { x , y - 1 , z - 1 } ( a , b ) } \\frac { \\psi _ { x , y , z - 1 } ( a , b ) } { \\psi _ { x , y , z } ( a , b ) } + \\frac { \\psi _ { x + 1 , y - 1 , z - 1 } ( a , b ) } { \\psi _ { x , y - 1 , z - 1 } ( a , b ) } \\frac { \\psi _ { x - 1 , y , z } ( a , b ) } { \\psi _ { x , y , z } ( a , b ) } = 1 . \\end{align*}"}
-{"id": "8911.png", "formula": "\\begin{align*} \\begin{aligned} \\| f _ N \\| _ { L ^ 1 ( \\nu ) } & = \\frac { 1 } { N + 1 } \\sum _ { n = 0 } ^ N \\int _ { \\mathbb T } | W ^ n \\gamma | ^ 2 \\nu \\\\ & = \\frac { 1 } { N + 1 } \\sum _ { n = 0 } ^ N \\| W ^ n \\gamma \\| _ { L ^ 2 ( \\nu ) } ^ 2 = \\frac { 1 } { N + 1 } \\sum _ { n = 0 } ^ N \\| \\gamma \\| _ { L ^ 2 ( \\nu ) } ^ 2 = \\| \\gamma \\| _ { L ^ 2 ( \\nu ) } ^ 2 \\end{aligned} \\end{align*}"}
-{"id": "7269.png", "formula": "\\begin{align*} \\sum \\limits _ { m \\le n } A ( m , n ) \\leqslant \\sum \\limits _ { \\substack { b \\vert n \\\\ \\frac { n } { b } \\leqslant T } } b \\frac { n } { b } = \\sum \\limits _ { \\substack { c \\vert n \\\\ c \\leqslant T } } n . \\end{align*}"}
-{"id": "3622.png", "formula": "\\begin{align*} A _ \\lambda = \\{ x \\in \\Omega _ \\lambda \\ : \\ | \\nabla u _ \\lambda ( x ) | < \\dot C | \\nabla u ( x ) | \\} B _ \\lambda = \\{ x \\in \\Omega _ \\lambda \\ : \\ | \\nabla u _ \\lambda ( x ) | \\geq \\dot C | \\nabla u ( x ) | \\} . \\end{align*}"}
-{"id": "8912.png", "formula": "\\begin{align*} \\lim _ N \\frac { 1 } { N + 1 } \\sum _ { n = 0 } ^ N ( U _ { ( \\theta ) 1 } ^ { - n } \\varphi ( U _ { ( \\theta ) c } ) U _ { ( \\theta ) 1 } ^ n x , x ) = \\lim _ N \\int _ { \\mathbb T } \\varphi f _ N \\nu = \\int _ { \\mathbb T } \\varphi f \\nu \\end{align*}"}
-{"id": "6142.png", "formula": "\\begin{align*} \\mathbb { P } _ { c o u p l e } \\left ( \\overline { S } _ { \\left \\{ 1 , . . . , M _ { 2 } \\left ( n \\right ) \\right \\} } > n ^ { - c } \\right ) & \\leq M _ { 2 } \\left ( n \\right ) n ^ { 2 c } n ^ { - 1 } \\mathbb { E } \\left ( J _ { 1 } ^ { 2 } \\right ) \\\\ & \\leq n ^ { - \\alpha c ' + 2 c } c \\log n \\mathbb { E } \\left ( J _ { 1 } ^ { 2 } \\right ) \\\\ & = n ^ { - c } c \\log n \\mathbb { E } \\left ( J _ { 1 } ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "7053.png", "formula": "\\begin{align*} z _ j = \\frac { y _ j } { | y _ n | } \\sqrt { t h ( t ) } , j = 1 , 2 , . . . , n - 1 \\end{align*}"}
-{"id": "1657.png", "formula": "\\begin{align*} & a _ 0 b _ 0 = d _ 0 c _ 0 , a _ 1 b _ 1 = d _ 1 c _ 1 , a _ 1 b _ 0 = d _ 1 c _ 0 , \\\\ & a _ 0 b _ 1 = d _ 0 c _ 1 , c _ 0 d _ 0 = b _ 1 a _ 1 , c _ 1 d _ 1 = b _ 0 a _ 0 \\end{align*}"}
-{"id": "944.png", "formula": "\\begin{align*} Q _ k ( \\xi ) = \\sum _ { i , j = 1 } ^ N \\gamma _ k ( i , j ) \\xi _ i \\xi _ j , k = 1 , \\dots , d \\end{align*}"}
-{"id": "1620.png", "formula": "\\begin{align*} \\gamma _ z ( t _ \\lambda ) = z ^ { d ( \\lambda ) } t _ \\lambda , \\end{align*}"}
-{"id": "5242.png", "formula": "\\begin{align*} \\lim \\limits _ { z \\rightarrow \\infty } E ^ u ( \\lambda , z ) = U ( \\lambda ) . \\end{align*}"}
-{"id": "6909.png", "formula": "\\begin{align*} W ( \\lambda ) = \\frac { N ( \\lambda ) } { \\lambda ^ \\alpha } \\end{align*}"}
-{"id": "7462.png", "formula": "\\begin{align*} \\mathbb { H } _ 5 \\rightarrow H \\cong \\mathbb { C } [ T ] : = \\mathbb { H } _ 5 [ T ] / ( t , T ^ { \\delta } _ 2 , T ^ { \\delta } _ 1 , T _ 3 , T _ 2 , T - T _ 0 ^ { \\delta } , T - T _ 1 , T - T _ 0 ) . \\end{align*}"}
-{"id": "1647.png", "formula": "\\begin{align*} \\int _ { X } \\chi _ E ( x ) \\ , d ( \\mu \\circ \\tau _ { \\lambda _ 1 } \\circ \\tau _ { \\lambda _ 2 } ) = \\int _ X \\chi _ E ( x ) ( \\Phi _ { \\tau _ { \\lambda _ 1 } } \\circ \\tau _ { \\lambda _ 2 } ) ( x ) \\ , d ( \\mu \\circ \\tau _ { \\lambda _ 2 } ) ( x ) . \\end{align*}"}
-{"id": "5844.png", "formula": "\\begin{align*} A = \\sqrt { 2 ( B ^ 2 + 1 ) C _ 0 + 1 } , \\end{align*}"}
-{"id": "3855.png", "formula": "\\begin{align*} \\nu ' ( 0 ) & = - \\left ( D _ v F ( \\nu ( 0 ) , 0 ) \\right ) ^ { - 1 } \\cdot D q ( x ^ * ) \\cdot d _ x \\\\ & = - \\left ( D _ v F ( \\nu ( 0 ) , 0 ) \\right ) ^ { - 1 } \\cdot \\underbrace { { \\begin{pmatrix*} [ l ] \\nabla g _ { \\mathcal { A } ( x ^ * ) } ( x ^ * ) ^ T \\\\ \\nabla h ( x ^ * ) ^ T \\\\ e _ { \\mathcal { B } ( x ^ * , y ^ * ) } ^ T \\end{pmatrix*} } \\cdot d _ x } _ { = 0 } \\ = 0 , \\end{align*}"}
-{"id": "5009.png", "formula": "\\begin{align*} \\psi ( \\xi ) = \\eta ( \\xi ) \\sum _ { | \\gamma | = m + 1 } \\xi ^ \\gamma \\left ( \\frac { 1 } { m ! } \\int _ { 0 } ^ { 1 } ( 1 - t ) ^ m \\partial _ { \\gamma } \\psi ( t \\xi ) \\ , d t \\right ) + \\frac { 1 - \\eta ( \\xi ) } { | \\xi | ^ { 2 ( m + 1 ) } } \\psi ( \\xi ) \\sum _ { | \\gamma | = m + 1 } c _ { \\gamma } \\xi ^ { 2 \\gamma } . \\end{align*}"}
-{"id": "3944.png", "formula": "\\begin{align*} \\frac { \\partial ^ \\gamma u } { \\partial t ^ \\gamma } + u \\frac { \\partial u } { \\partial x } - \\nu \\frac { \\partial ^ 2 u } { \\partial x ^ 2 } = 0 , \\ , \\ , \\ , \\ , \\ , \\ , a \\leq b \\leq 3 , \\ , \\ , \\ , \\ , \\ , 0 < \\gamma < 1 , \\ , \\ , \\ , \\ , t > 0 . \\end{align*}"}
-{"id": "7731.png", "formula": "\\begin{align*} F _ N ( 1 ) & = \\frac { 1 } { N } \\sum ^ { N - 1 } _ { n = 1 } \\frac { 1 - \\cos \\phi _ n } { ( 1 - \\cos \\phi _ n ) ^ 2 } \\\\ & = \\frac { 1 } { 2 N } \\sum ^ { N - 1 } _ { n = 1 } \\frac { 1 } { \\sin ^ 2 \\phi _ n / 2 } = \\frac { N } { 3 } - \\frac { 1 } { 3 N } \\end{align*}"}
-{"id": "1800.png", "formula": "\\begin{align*} x _ 1 x _ 1 ' & = ( x _ 2 + x _ 3 ) \\left ( \\frac 1 { x _ 3 ' } \\right ) ^ { - 1 } = x _ 1 x _ 2 ; \\\\ x _ 2 x _ 2 ' & = ( x _ 3 ) \\left ( \\frac 1 { x _ 1 ' } \\right ) ^ { - 1 } = x _ 2 x _ 3 ; \\\\ x _ 3 x _ 3 ' & = ( x _ 1 ) \\left ( \\frac 1 { x _ 1 ' } + \\frac 1 { x _ 2 ' } \\right ) ^ { - 1 } = \\frac { x _ 1 x _ 2 x _ 3 } { x _ 2 + x _ 3 } . \\\\ \\end{align*}"}
-{"id": "1996.png", "formula": "\\begin{align*} \\widetilde { \\alpha } _ { n } \\left ( s ; t \\right ) - \\widetilde { \\beta } _ { n } \\left ( \\frac { t _ { n } } { n } , \\frac { k } { n } \\right ) = O _ { \\mathbb { P } } \\left ( n ^ { - 1 / 2 } \\right ) . \\end{align*}"}
-{"id": "790.png", "formula": "\\begin{align*} \\rho \\vert \\log \\alpha _ { 2 } \\vert - \\log \\vert \\alpha _ { 2 } \\vert + 2 D { \\rm h } ( \\alpha _ { 2 } ) = ( \\rho + 1 ) \\log { 2 x } . \\end{align*}"}
-{"id": "3694.png", "formula": "\\begin{align*} \\Lambda _ { b , k } = \\left \\{ \\sum ( \\langle v _ { I , j } , m \\rangle - k d _ { I , j } ) D _ { I , j } \\ ; | \\ ; m \\in k P _ b [ n ] \\right \\} . \\end{align*}"}
-{"id": "1977.png", "formula": "\\begin{align*} \\Vert ( v _ { s } , v _ { u } ) \\Vert _ { 1 } ^ { 2 } & = \\Vert v _ { s } \\Vert _ { 1 } ^ { 2 } + \\Vert v _ { u } \\Vert _ { 1 } ^ { 2 } \\leq ( \\frac { \\lambda + \\varepsilon } { \\varepsilon } c ) ^ { 2 } ( \\Vert v _ { s } \\Vert ^ { 2 } + \\Vert v _ { u } \\Vert ^ { 2 } ) \\leq \\frac { 1 } { \\mu _ { i } } ( \\frac { \\lambda + \\varepsilon } { \\varepsilon } c ) ^ { 2 } \\Vert ( v _ { s } , v _ { u } ) \\Vert ^ { 2 } . \\end{align*}"}
-{"id": "9278.png", "formula": "\\begin{align*} { \\sigma } _ \\varepsilon ^ { ( s , u ) } : = \\Theta \\circ \\theta ^ { ( s , u ) } . \\end{align*}"}
-{"id": "6071.png", "formula": "\\begin{align*} \\ddot r ( t ) + ( p _ 1 \\cot t - p _ 2 \\tan t ) \\dot r ( t ) - \\tfrac { 1 } { 2 } \\left ( \\tfrac { \\lambda _ 1 } { \\sin ^ 2 t } - \\tfrac { \\lambda _ 2 } { \\cos ^ 2 t } \\right ) \\sin 2 r ( t ) = 0 , \\end{align*}"}
-{"id": "1602.png", "formula": "\\begin{align*} \\{ X \\} ^ G = \\{ X \\smallsetminus Z \\} ^ G + \\{ Z \\} ^ G \\ , , \\end{align*}"}
-{"id": "9377.png", "formula": "\\begin{align*} h ( r , \\theta ) = r ^ { 2 m } P ( \\theta ) + M \\ , r ^ { 2 m } h _ { 2 m } ( \\theta ) - M \\ , r ^ { \\alpha } h _ { 2 m } ( \\theta ) + r ^ { \\alpha } \\phi ( \\theta ) . \\end{align*}"}
-{"id": "9679.png", "formula": "\\begin{align*} & C _ 2 ( U _ a ) : \\ , \\ , U = ( u _ a e ^ { \\sigma _ 2 } , v _ a e ^ { \\sigma _ 2 } , p _ a , \\rho _ a , Z _ a ) ^ \\top , \\\\ & C _ 3 ( U _ a ) : \\ , \\ , U = ( u _ a , v _ a , p _ a , \\rho _ a e ^ { \\sigma _ 3 } , Z _ a ) ^ \\top , \\\\ & C _ 4 ( U _ a ) : \\ , \\ , U = ( u _ a , v _ a , p _ a , \\rho _ a , Z _ a + \\sigma _ 4 ) ^ \\top . \\end{align*}"}
-{"id": "2123.png", "formula": "\\begin{align*} \\int _ { \\R } f _ j ( x ) \\ , d x = 0 . \\end{align*}"}
-{"id": "1434.png", "formula": "\\begin{align*} f - \\partial _ J ( g _ k x _ { 4 n } ^ k ) = f _ { k + 1 , k + 1 } x _ { 4 n } ^ { k + 1 } + f _ { k + 1 , k + 2 } x _ { 4 n } ^ { k + 2 } + \\cdots \\end{align*}"}
-{"id": "21.png", "formula": "\\begin{align*} v a r [ \\hat { V } ^ C _ { N , \\sigma } ( C _ 1 , C _ 2 ) ] = N ^ { - 1 } v a r [ G _ { \\sigma } ( C _ 1 - C _ 2 ) ] \\end{align*}"}
-{"id": "4234.png", "formula": "\\begin{align*} a \\star g : = \\sum _ { i = 0 } ^ { f - 1 } \\sigma ^ { i } ( a ) g _ i \\end{align*}"}
-{"id": "4537.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac 1 n \\sum _ { j = 0 } ^ { n - 1 } \\phi ( F _ j ( x ) ) = \\int \\phi \\ ; d \\Big ( \\frac 1 N \\sum _ { i = 0 } ^ { N - 1 } ( h _ i ) _ * \\mu \\Big ) \\end{align*}"}
-{"id": "5149.png", "formula": "\\begin{align*} f = \\sum _ { i = 1 } ^ { k } h _ i \\cdot \\big ( - \\i \\mathcal { G } _ m ( \\psi _ i , \\psi ' _ i ) , 0 , \\psi _ i \\otimes \\psi ' _ i - \\psi ' _ i \\otimes \\psi _ i , 0 , 0 , \\dots \\big ) \\cdot \\tilde { h } _ i \\end{align*}"}
-{"id": "7230.png", "formula": "\\begin{align*} h ( 1 ) = \\int _ { \\hat { G } ^ { } } \\Theta _ { \\pi } ( h ) d \\mu _ { \\pi } \\\\ , \\forall h \\in \\mathcal { C } _ { c } ( G ) , \\end{align*}"}
-{"id": "6783.png", "formula": "\\begin{align*} w _ { \\lambda , k } ( y ) - w _ { \\lambda , k } ( \\xi _ k ) = \\int _ { \\mathbb { S } ^ 2 } \\left [ G ( y , y ' ) - G ( \\xi _ k , y ' ) \\right ] e ^ { U _ { \\lambda , \\xi _ k } } \\eta _ { R _ 0 , \\xi _ k } d H ^ 2 ( y ' ) \\end{align*}"}
-{"id": "9833.png", "formula": "\\begin{align*} W ( x , y ) = \\sum _ { i = 0 } ^ n A _ i x ^ { 2 ( n - i ) + 1 } y ^ { 2 i } \\qquad ( A _ 0 = 1 ) . \\end{align*}"}
-{"id": "4004.png", "formula": "\\begin{align*} \\exp _ { A } ^ { \\prime } ( y ) = A ( \\exp _ { A } ( y ) ) , \\exp _ { A } ( 0 ) = 1 \\ . \\end{align*}"}
-{"id": "5685.png", "formula": "\\begin{align*} v _ { s y m } ( t ) = \\begin{cases} { v } ( t _ 0 + t ) & t > 0 , \\\\ \\mathcal R _ n ( { v } ( t _ 0 - t ) ) & t \\leq 0 . \\end{cases} \\end{align*}"}
-{"id": "3293.png", "formula": "\\begin{align*} L ( \\phi ) = - [ R + N ( \\phi ) ] \\qquad , \\end{align*}"}
-{"id": "8009.png", "formula": "\\begin{align*} & \\sup _ { R \\in \\mathcal { D } _ { \\mu } } { \\Big ( \\frac { 1 } { | R | } \\int _ R { \\sum _ { k = 0 } ^ { \\mu - 4 } 2 ^ { ( k - \\mu ) ( N - \\epsilon - s ) q } 2 ^ { k ( m + s ) q } \\big ( \\mathfrak { M } _ { \\sigma , 2 ^ k } \\Pi ^ * _ k f ( x ) \\big ) ^ q } d x \\Big ) ^ { 1 / q } } \\\\ & \\lesssim \\sup _ { 0 \\leq k \\leq \\mu - 3 } \\big \\Vert 2 ^ { k ( m + s ) } \\Pi _ k f \\big \\Vert _ { L ^ { \\infty } } \\leq \\sup _ { 0 \\leq k \\leq \\mu - 1 } \\big \\Vert 2 ^ { k ( m + s ) } \\Pi _ k f \\big \\Vert _ { L ^ { \\infty } } \\end{align*}"}
-{"id": "7622.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ \\infty M ^ { l i } I _ i & \\leq \\sum _ { i = 0 } ^ \\infty M ^ { l i } \\nu \\big ( E _ g ( Q _ { 2 R } , c _ 0 \\lambda _ 0 M ^ i ) \\big ) + \\Big [ \\sum _ { i = 0 } ^ \\infty M ^ { \\frac { l } { \\hat p } i } \\hat \\nu \\big ( E _ { \\hat g } ( Q _ { 2 R } , c _ 0 \\lambda _ 0 M ^ i ) \\big ) \\Big ] ^ { \\hat p } . \\end{align*}"}
-{"id": "6935.png", "formula": "\\begin{align*} \\| \\Gamma _ { \\mu _ j f , \\ , \\Xi } \\| \\leq \\| \\hat { \\nu } _ j \\| _ { L ^ 1 } \\| \\Gamma _ { f , \\ , \\Xi } \\| , j = 1 , 2 . \\end{align*}"}
-{"id": "1377.png", "formula": "\\begin{align*} \\underline { { \\cal E } } _ c : = \\left ( \\overline { \\sigma } \\overline { \\eta } I _ { n _ 2 } + ( c - c _ 0 ) D _ c \\right ) ^ { - 1 } \\ , , \\overline { { \\cal E } } _ c : = \\left ( \\underline { \\sigma } \\underline { \\eta } I _ { n _ 2 } + ( c - c _ 0 ) D _ c \\right ) ^ { - 1 } \\end{align*} % \\end{align*}"}
-{"id": "6378.png", "formula": "\\begin{align*} g ( x ) : = \\exp \\big \\{ { - } W _ 1 '' ( z _ 1 ^ * ) x ^ 2 / 4 \\big \\} ( z _ 1 ^ * - \\varepsilon ) ^ { - ( 2 i - 1 ) } . \\end{align*}"}
-{"id": "7238.png", "formula": "\\begin{align*} \\psi _ { n } ( u ) = \\psi ( u ) \\mathbf { 1 } _ { K _ { n } } ( u ) \\quad ( u \\in U ) \\end{align*}"}
-{"id": "7550.png", "formula": "\\begin{gather*} u _ x = v _ y . \\end{gather*}"}
-{"id": "8888.png", "formula": "\\begin{align*} \\operatorname { c a r d } \\{ w \\in \\mathbb D \\ : \\ \\theta ( w ) = \\theta ( \\lambda ) \\} > 2 , \\end{align*}"}
-{"id": "629.png", "formula": "\\begin{align*} B ( z , w ) = \\sum _ { n \\ge - 1 } \\frac { n + 2 } { \\pi } z ^ n \\bar w ^ n = \\frac { 1 } { \\pi } \\cdot \\frac { 1 } { z \\bar w } \\cdot \\frac { 1 } { ( 1 - z \\bar w ) ^ 2 } . \\end{align*}"}
-{"id": "7079.png", "formula": "\\begin{align*} \\varpi = 2 r \\omega - i F _ { A _ K } ^ + , \\end{align*}"}
-{"id": "8155.png", "formula": "\\begin{align*} a _ 1 a _ 7 ^ q = \\delta \\end{align*}"}
-{"id": "8794.png", "formula": "\\begin{align*} \\overline { f } \\circ \\overline { h } = \\overline { u } \\prod _ { i \\in T _ { a } } \\overline { g } _ { i } ^ { N _ { i } } , \\end{align*}"}
-{"id": "8515.png", "formula": "\\begin{align*} \\sigma ( 1 , v , u , C ) = \\min \\{ | z | ^ p + \\sum _ { i = 1 } ^ s \\left | a _ { i \\ , 1 } z + v _ i \\right | ^ p + \\sum _ { i = 1 } ^ m \\left | \\bar a _ { i \\ , 1 } z + u _ i \\right | ^ p : z \\in \\mathbb { Z } , \\ , | z | \\leq C \\} . \\end{align*}"}
-{"id": "6913.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\{ b _ { 1 } , b _ { 2 } , b _ { 3 } , b _ { 4 } , b _ { 5 } , b _ { 6 } , b _ { 7 } \\} & = \\{ 3 , 3 , \\frac { 1 5 } { 2 } , 3 , \\frac { 9 } { 2 } , 9 , 9 \\} \\\\ \\lim _ { r \\to \\infty } \\{ b _ { 1 } , b _ { 2 } , b _ { 3 } , b _ { 4 } , b _ { 5 } , b _ { 6 } , b _ { 7 } \\} & = \\{ 0 , 9 , 9 , \\frac { 3 } { 2 } , \\frac { 9 } { 2 } , 6 , 9 \\} \\end{align*}"}
-{"id": "5530.png", "formula": "\\begin{align*} \\rho ^ { 2 } - \\left ( x _ { 1 } \\left ( \\tau \\right ) + \\dot { x } _ { 2 } \\left ( \\tau \\right ) \\right ) \\rho + 1 = 0 \\end{align*}"}
-{"id": "8733.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } F ( t ' _ j , x ' _ j , u ^ { \\varepsilon , \\delta _ j } ( t _ j , x _ j ) , p _ j , X _ j ) & = F ( \\bar { t } , \\bar { x } ' , u ^ \\varepsilon ( \\bar { t } , \\bar { x } ) , \\nabla _ x \\varphi ( \\bar { t } , \\bar { x } , \\bar { y } ) , X ) \\\\ & \\ge F _ \\varepsilon ( \\bar { t } , \\bar { x } , u ^ \\varepsilon ( \\bar { t } , \\bar { x } ) , \\nabla _ x \\varphi ( \\bar { t } , \\bar { x } , \\bar { y } ) , X ) . \\end{align*}"}
-{"id": "6738.png", "formula": "\\begin{align*} E \\Bigg ( \\frac { R _ { N } } { \\beta _ { N } } \\Bigg ) = \\int _ { 0 } ^ { \\infty } \\mathbb { P } \\Bigg ( \\frac { R _ { N } } { \\beta _ { N } } > t \\Bigg ) d t . \\end{align*}"}
-{"id": "4565.png", "formula": "\\begin{align*} & \\mathcal { K } = q ^ { - { \\Lambda } _ 1 + { \\Lambda } _ { n - 1 } } . \\end{align*}"}
-{"id": "4882.png", "formula": "\\begin{align*} \\sum _ { m \\vert k } \\mu ( m ) = \\begin{cases} 1 & k = 1 , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "3604.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\left | \\iint x ^ m y ^ n Q _ N ( \\d x , \\d y ) - \\iint x ^ m y ^ n \\breve Q _ N ( \\d x , \\d y ) \\right | = 0 . \\end{align*}"}
-{"id": "5824.png", "formula": "\\begin{align*} v ( r ) = \\sum _ { n = 1 } ^ \\infty \\frac { g _ n ( \\o ) } { z _ n } e _ n ( r ) , r ^ 2 = x ^ 2 + y ^ 2 , \\end{align*}"}
-{"id": "147.png", "formula": "\\begin{align*} N = \\begin{pmatrix} x & y \\\\ z & w \\end{pmatrix} \\end{align*}"}
-{"id": "8704.png", "formula": "\\begin{align*} \\partial _ t ^ \\alpha u + F ( t , x , u , \\nabla u , \\nabla ^ 2 u ) = 0 \\quad \\end{align*}"}
-{"id": "7094.png", "formula": "\\begin{align*} \\begin{aligned} & k _ p ^ 2 = k ^ 2 p ^ { - 4 } , \\\\ & k k _ { p p } = k ^ 2 ( p ^ { - 4 } - 2 p ^ { - 3 } ) = k _ p ^ 2 ( 1 - 2 p ) , \\end{aligned} \\end{align*}"}
-{"id": "2289.png", "formula": "\\begin{align*} \\sigma _ { a , b } = \\prod _ { i = 1 } ^ { p ^ { a - 1 } } ( i + b p ^ { a } , i + b p ^ { a } + p ^ { a - 1 } , \\dots , i + b p ^ { a } + ( p - 1 ) p ^ { a - 1 } ) . \\end{align*}"}
-{"id": "5665.png", "formula": "\\begin{align*} P [ { v } ] ( s ) : = \\begin{dcases} a ^ + + E ( s ) \\ , \\frac { { v } ( s ) - a ^ + } { | { v } ( s ) - a ^ + | } & s > s _ 0 | { v } ( s ) - a ^ + | > E ( s ) , \\\\ { v } ( s ) & \\end{dcases} \\end{align*}"}
-{"id": "6327.png", "formula": "\\begin{align*} \\div ( | \\nabla u | ^ { n - 2 } \\nabla u ) = \\Omega | \\nabla u | ^ { n - 2 } \\nabla u \\end{align*}"}
-{"id": "562.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial s } + J ( u ) \\left ( \\frac { \\partial u } { \\partial t } - X _ { H _ t } ( u ) \\right ) = 0 , \\end{align*}"}
-{"id": "7633.png", "formula": "\\begin{align*} \\int _ { Q _ 3 } m _ + ( x , t ) \\ , d x d t = 0 , \\end{align*}"}
-{"id": "6089.png", "formula": "\\begin{align*} E ( u ) = \\tfrac { \\omega _ { n } } { 2 } \\int _ 0 ^ 1 \\left ( \\Phi ' ( r ) ^ 2 + ( \\tfrac { 2 \\frak { e } _ k } { r ^ 2 } + g ' ( r ) ^ 2 ) \\sin ^ 2 \\Phi ( r ) \\right ) r ^ { n - 1 } d r . \\end{align*}"}
-{"id": "7090.png", "formula": "\\begin{align*} r ' = ( r ' _ 1 , r ' _ 2 , r ' _ 7 , r ' _ { 1 4 } ) = ( 3 , 0 , 0 , 0 ) . \\end{align*}"}
-{"id": "9225.png", "formula": "\\begin{align*} \\alpha _ { 1 } ( \\alpha _ { 2 } \\alpha _ { 3 } ) = [ \\alpha _ { 1 } , \\alpha _ { 2 } ] _ { A ^ { - } } \\alpha _ { 3 } - \\varepsilon \\alpha _ { 2 } ( \\alpha _ { 3 } \\alpha _ { 1 } ) - \\frac { 1 } { n } [ [ \\alpha _ { 1 } , \\alpha _ { 2 } ] _ { A ^ { - } } , \\alpha _ { 3 } ] + \\frac { 2 } { n } \\langle \\alpha _ { 1 } , \\alpha _ { 2 } \\rangle \\alpha _ { 3 } . \\end{align*}"}
-{"id": "8200.png", "formula": "\\begin{align*} K _ f ( g _ 1 , g _ 2 ) = \\sum _ { \\gamma \\in Z ( F ) \\setminus G ( F ) } f ( g _ 1 ^ { - 1 } \\gamma g _ 2 ) . \\end{align*}"}
-{"id": "8761.png", "formula": "\\begin{gather*} f _ 1 ( x , y ) = A _ 1 \\left ( \\begin{array} { c } x \\\\ y \\end{array} \\right ) , f _ 2 ( x , y ) = A _ 2 \\left ( \\begin{array} { c } x \\\\ y \\end{array} \\right ) \\end{gather*}"}
-{"id": "5282.png", "formula": "\\begin{align*} \\frac { 1 } { \\mu } \\langle \\mathcal { A } u , u \\rangle = & - \\frac { a _ 1 } { a _ 2 } u ^ 2 ( t , 1 ) - \\| u _ x \\| ^ 2 \\\\ \\leq & \\frac { 1 } { 2 } \\times 2 \\| u _ x \\| ^ 2 - \\| u _ x \\| ^ 2 = 0 . \\end{align*}"}
-{"id": "9124.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } a _ { n + 1 - i } x ^ { i } A \\left ( x ^ { n + 1 - i } \\right ) = 0 \\end{align*}"}
-{"id": "6543.png", "formula": "\\begin{align*} \\left \\{ y \\in \\mathbb { R } ^ { n } : \\langle y , \\xi \\rangle = 1 + \\frac { n | P ^ { \\circ } | _ n \\| \\xi \\| } { | F _ { \\xi } | _ { n - 1 } } \\delta \\right \\} \\end{align*}"}
-{"id": "8739.png", "formula": "\\begin{align*} \\mathcal { F } _ \\sigma : & = F _ \\varepsilon ( t _ \\sigma , x _ \\sigma , u ^ \\varepsilon ( t _ \\sigma , x _ \\sigma ) , \\nabla _ x \\varphi ( t _ \\sigma , x _ \\sigma , y _ \\sigma ) , X _ \\sigma ) \\\\ & - F ^ \\varepsilon ( t _ \\sigma , y _ \\sigma , v _ \\varepsilon ( t _ \\sigma , y _ \\sigma ) , - \\nabla _ y \\varphi ( t _ \\sigma , x _ \\sigma , y _ \\sigma ) , - Y _ \\sigma ) . \\end{align*}"}
-{"id": "4842.png", "formula": "\\begin{align*} \\sum _ { p \\in C } w _ p ( Q ) = ( r + 1 ) ( \\deg Q + r g - r ) . \\end{align*}"}
-{"id": "1970.png", "formula": "\\begin{align*} \\langle \\cdot , \\cdot \\rangle | _ { M _ { i } } = \\langle \\cdot , \\cdot \\rangle _ { i } \\quad i \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "8678.png", "formula": "\\begin{align*} [ x , P ( u , v ) ] = P ( x ( u ) , v ) + P ( u , x ( v ) ) \\forall x \\in \\gg _ 0 , ~ \\forall u , v \\in \\gg _ 1 , \\end{align*}"}
-{"id": "3916.png", "formula": "\\begin{align*} \\begin{cases} n \\equiv - 1 , 0 , \\dots , f _ 0 - 1 \\pmod d , \\\\ n - g \\equiv f _ 0 + 1 , \\dots , d - 1 \\pmod d . \\end{cases} \\end{align*}"}
-{"id": "6997.png", "formula": "\\begin{align*} \\tilde { e } _ { n - 1 } = ( 0 , 0 , . . . , 0 , \\cos \\theta , - \\sin \\theta ) . \\end{align*}"}
-{"id": "4723.png", "formula": "\\begin{align*} \\alpha ^ 2 a _ 0 + 2 \\alpha \\beta b _ 0 + \\beta ^ 2 d _ 0 + 2 \\alpha \\gamma c _ 0 + 2 \\beta \\gamma e _ 0 + \\gamma ^ 2 f _ 0 = - { L } _ 0 { } ^ 2 . \\end{align*}"}
-{"id": "4242.png", "formula": "\\begin{align*} \\varphi ( \\varphi ( x ) ) = \\varphi ( x ^ { q ^ m } ) = \\varphi ( x ) ^ { q ^ m } = x ^ { q ^ { 2 m } } \\end{align*}"}
-{"id": "4504.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| \\phi _ 0 ^ { n } f \\| = | f ( x _ 0 ) | . \\end{align*}"}
-{"id": "6220.png", "formula": "\\begin{align*} X ( n ) = n - C ( n ) ~ , ~ ~ n \\geq 0 ~ , \\end{align*}"}
-{"id": "7239.png", "formula": "\\begin{align*} ( \\psi _ { n } * _ { U } f ) ( 1 ) = \\int _ { \\hat { G } } ^ { } \\Theta _ { \\pi } ( \\psi _ { n } * _ { U } f ) d \\mu _ { \\pi } . \\end{align*}"}
-{"id": "9672.png", "formula": "\\begin{align*} & W ( U ) = ( \\rho u , \\rho u ^ 2 + p , \\rho u v , \\rho u ( \\frac { u ^ 2 + v ^ 2 } { 2 } + \\frac { \\gamma p } { ( \\gamma - 1 ) \\rho } ) , \\rho u Z ) ^ \\top , \\\\ & H ( U ) = ( \\rho v , \\rho u v , \\rho v ^ 2 + p , \\rho v ( \\frac { u ^ 2 + v ^ 2 } { 2 } + \\frac { \\gamma p } { ( \\gamma - 1 ) \\rho } ) , \\rho v Z ) ^ \\top , \\\\ & G ( U ) = ( 0 , 0 , 0 , q _ 0 \\rho Z \\phi ( T ) , - \\rho \\phi ( T ) Z ) ^ \\top . \\end{align*}"}
-{"id": "9721.png", "formula": "\\begin{align*} V _ b = \\tilde { \\Phi } ( \\gamma _ 5 , \\gamma _ 3 , \\gamma _ 2 , \\gamma _ 1 ; V _ a ) , Z _ b = Z _ a + \\gamma _ 4 . \\end{align*}"}
-{"id": "3546.png", "formula": "\\begin{align*} \\delta _ { \\boldsymbol { t } } ( U ) = \\sum _ { \\boldsymbol { \\varepsilon } \\in \\mathcal { E } } ( - 1 ) ^ { \\abs { \\boldsymbol { \\varepsilon } } } I ( S _ { B _ { \\boldsymbol { t } } ^ { \\boldsymbol { \\varepsilon } } } \\in U ) . \\end{align*}"}
-{"id": "3415.png", "formula": "\\begin{align*} D ' = \\vartheta _ n ( \\Omega _ n ) \\end{align*}"}
-{"id": "670.png", "formula": "\\begin{align*} & | | U _ { 2 m ( n ) } '' ( t ) P _ { m ( n ) } A ( t ) P _ { m ( n ) } U _ { 2 m ( n ) } ''^ * ( t ) - P _ { m ( n ) } B ( t ) P _ { m ( n ) } | | \\\\ & \\leq \\frac { 1 0 } { n } + | | P _ { m ( n ) } U _ { 2 m ( n ) } '' ( t ) P _ { m ( n ) } A ( t ) P _ { m ( n ) } U _ { 2 m ( n ) } ''^ * ( t ) P _ { m ( n ) } - P _ { m ( n ) } B ( t ) P _ { m ( n ) } | | \\\\ & = \\frac { 1 0 } { n } + | | U _ { m ( n ) } ' ( t ) A ( t ) U _ { m ( n ) } '^ * ( t ) - P _ { m ( n ) } B ( t ) P _ { m ( n ) } | | \\leq \\frac { 1 9 } { n } \\end{align*}"}
-{"id": "224.png", "formula": "\\begin{align*} \\widehat { \\Sigma } _ g : = [ \\inf \\Sigma _ { g ( H ) } , \\sup \\Sigma _ { g ( H ) } ] , \\end{align*}"}
-{"id": "4184.png", "formula": "\\begin{align*} \\eta = i \\frac { \\pi } { 2 } \\zeta \\textnormal { a n d } \\eta = - i \\frac { \\pi } { 2 } \\zeta . \\end{align*}"}
-{"id": "1954.png", "formula": "\\begin{align*} r _ i \\in ( 1 - \\delta , 1 + \\delta ) a _ i = 1 , r _ i \\in ( 0 , \\infty ) a _ i = 0 . \\end{align*}"}
-{"id": "10065.png", "formula": "\\begin{align*} \\frac { n \\cdot [ \\widehat { \\mathcal { Z } } ( f _ m ) : \\mathcal { Y } _ \\mathrm { b i g } ] } { \\deg _ \\C ( \\mathcal { Y } _ \\mathrm { b i g } ) } = - 2 c _ { f _ m } ^ + ( 0 , 0 ) \\cdot \\frac { \\Lambda ' ( 0 , \\chi _ E ) } { \\Lambda ( 0 , \\chi _ E ) } - \\frac { d } { d s } \\langle E ( s ) , \\xi ( f _ m ) \\rangle _ \\mathrm { P e t } \\big | _ { s = 0 } , \\end{align*}"}
-{"id": "5086.png", "formula": "\\begin{align*} & [ ( b _ 1 - b _ 2 ) ^ 2 - 2 ] R _ { 1 2 1 2 } + [ ( b _ 2 - b _ 3 ) ^ 2 - 2 ] R _ { 2 3 2 3 } + [ ( b _ 1 - b _ 3 ) ^ 2 - 2 ] R _ { 1 3 1 3 } \\\\ & = \\frac { 3 } { 2 } | \\tilde { A } | ^ 2 - \\frac { 3 } { 2 } [ | R i c | ^ 2 - \\frac { 1 } { 3 } R ^ 2 ] - \\frac { 1 } { 9 } . \\end{align*}"}
-{"id": "6530.png", "formula": "\\begin{align*} \\frac { \\| \\xi _ { \\delta } \\| } { \\| \\xi \\| } = 1 - \\left ( \\frac { n | P | _ { n } } { | ( F _ { \\xi } - s ( F _ { \\xi } ) ) ^ { \\circ } | _ { n - 1 } \\| \\xi \\| } \\right ) ^ { 1 / n } \\delta ^ { 1 / n } . \\end{align*}"}
-{"id": "1817.png", "formula": "\\begin{align*} \\langle \\Phi ^ { a , 1 } ( 1 _ Q ) \\rangle _ { a , 1 } = a ^ { 1 5 / 8 } \\mathcal { N } ( a ) \\langle \\sigma _ 0 \\rangle _ { 1 , H = a ^ { 1 5 / 8 } } = H ^ { - 1 / 1 5 } a ^ 2 \\mathcal { N } ( a ) \\langle \\sigma _ 0 \\rangle _ { 1 , H } . \\end{align*}"}
-{"id": "4124.png", "formula": "\\begin{align*} \\frac { k _ e } { g } = | h | = [ \\varphi ' , \\varphi ] ^ 2 = [ \\varphi ' , \\langle \\varphi , \\xi \\rangle \\xi + \\langle \\varphi ' , \\varphi \\rangle \\varphi ' ] ^ 2 = \\langle \\varphi , \\xi \\rangle ^ 2 = g ^ 2 . \\end{align*}"}
-{"id": "2861.png", "formula": "\\begin{align*} \\bar { \\Delta } V _ { n } & = \\frac { a _ { n n } P _ { n } \\lambda _ { n } } { n p _ { n } } s _ { n } + \\sum _ { v = 1 } ^ { n - 1 } \\frac { P _ { v } \\lambda _ { v } } { v p _ { v } } \\Delta _ { v } ( \\hat { a } _ { n v } ) s _ { v } + \\sum _ { v = 1 } ^ { n - 1 } \\hat { a } _ { n , v + 1 } \\lambda _ { v } \\Delta \\left ( \\frac { P _ { v } } { v p _ { v } } \\right ) s _ { v } + \\sum _ { v = 1 } ^ { n - 1 } \\hat { a } _ { n , v + 1 } \\frac { P _ { v + 1 } } { ( v + 1 ) p _ { v + 1 } } \\Delta \\lambda _ { v } s _ { v } \\\\ \\bar { \\Delta } V _ { n } & = V _ { n , 1 } + V _ { n , 2 } + V _ { n , 3 } + V _ { n , 4 } . \\end{align*}"}
-{"id": "3134.png", "formula": "\\begin{align*} m = | A _ 1 | + \\sum _ { \\ell \\geq 2 } { | A _ \\ell | } \\leq | A _ 1 | + \\sum _ { \\ell \\geq 2 } | A _ \\ell | / 2 = | A _ 1 | / { 2 } + \\sum _ { \\ell \\geq 1 } { \\ell } { | A _ \\ell | } / 2 \\leq ( 2 k - 5 ) q / ( k - 2 ) , \\end{align*}"}
-{"id": "2941.png", "formula": "\\begin{align*} Q _ k = F _ { P , t } ^ { ( k ) } ( \\{ Q _ { l } \\} _ { l \\geq 2 } ) . \\end{align*}"}
-{"id": "3009.png", "formula": "\\begin{align*} P _ 1 = S ^ { - 1 } \\Delta S = S ^ { - 1 } \\begin{bmatrix} \\Lambda & 0 \\\\ 0 & \\Theta \\end{bmatrix} S , \\end{align*}"}
-{"id": "199.png", "formula": "\\begin{align*} f \\circ h ( a ) & = ( \\hat { f } \\circ f ^ { \\ast } ) \\circ h ( a ) = \\hat { f } \\circ ( f ^ { \\ast } \\circ h ) ( a ) = \\hat { f } \\circ ( \\hat { f } ^ { - 1 } \\circ i \\circ \\hat { g } \\circ g ^ { \\ast } ) ( a ) \\\\ & = ( \\hat { f } \\circ \\hat { f } ^ { - 1 } ) \\circ i \\circ \\hat { g } \\circ g ^ { \\ast } ( a ) = i \\circ \\hat { g } \\circ g ^ { \\ast } ( a ) = i \\circ g ( a ) = g ( a ) . \\end{align*}"}
-{"id": "9283.png", "formula": "\\begin{align*} \\boldsymbol { E } [ \\sum _ { i = 1 } ^ { T - 1 } \\sum _ { j = i + 1 } ^ T X _ i X _ j ] \\leq ( \\boldsymbol { E } [ T ] ) ^ { \\frac { 1 } { 2 } } ( \\boldsymbol { E } [ T ^ 3 ] ) ^ \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "55.png", "formula": "\\begin{align*} \\hat { V } ^ { C } _ { \\sigma } ( C _ { 1 } , C _ { 2 } ) = \\left ( \\frac { 1 } { 2 \\pi 4 \\theta ^ 2 } \\right ) \\frac { 1 } { N } \\sum \\limits _ { i = 1 } ^ N e x p \\left ( - \\frac { ( x _ { i } - y _ { i } ) ^ 2 + ( z _ { i } - s _ { i } ) ^ 2 } { 8 \\theta ^ 2 } \\right ) \\end{align*}"}
-{"id": "9057.png", "formula": "\\begin{align*} a _ { 1 1 } ^ 1 = 0 , \\ , \\ , a _ { 2 2 } ^ 2 = 0 , \\ , \\ , b _ { 2 1 } ^ 2 = 0 , \\ , \\ , \\Re b _ { 2 2 } ^ 2 = 0 . \\end{align*}"}
-{"id": "9801.png", "formula": "\\begin{align*} C _ { d a } = B _ { d a } - \\beta B _ { d s } + \\frac { 1 } { 2 } \\beta ^ 2 B _ { d \\bar a } \\quad C _ { d s } = B _ { d s } + \\beta B _ { d \\bar a } \\quad s = \\bar s = n + 1 . \\end{align*}"}
-{"id": "9069.png", "formula": "\\begin{align*} z ^ 2 _ 1 = x _ 1 + b _ 1 z _ 2 , z _ 2 ^ 2 = x _ 2 + b _ 2 z _ 1 . \\end{align*}"}
-{"id": "587.png", "formula": "\\begin{align*} M _ { \\alpha _ j } = \\underset { z \\in L } { \\sup } \\left | f ^ { ( \\alpha _ j ) } ( z ) \\right | \\leqslant \\underset { z \\in \\Omega } { \\sup } \\left | f ^ { ( \\alpha _ j ) } ( z ) \\right | < + \\infty . \\end{align*}"}
-{"id": "9008.png", "formula": "\\begin{align*} \\left \\{ \\aligned & \\partial _ { t } \\theta + ( u \\cdot \\nabla ) \\theta + \\mu \\Lambda ^ { 2 \\alpha } \\theta = 0 , \\\\ & \\theta ( x , 0 ) = \\theta _ { 0 } ( x ) , \\endaligned \\right . \\end{align*}"}
-{"id": "1629.png", "formula": "\\begin{align*} Z ( \\lambda _ i ) = \\bigsqcup _ { j \\in \\N } Z ( \\lambda _ i ) \\cap Z ( \\eta _ j ) , \\end{align*}"}
-{"id": "6247.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } & = \\Omega ( \\mu ) + \\Delta ( \\sigma , \\mu ) + \\xi ( y , z , \\sigma , \\mu ) , \\\\ \\dot { y } & = \\sigma + \\eta ( y , z , \\sigma , \\mu ) , \\\\ \\dot { z } & = M ( \\mu ) z + \\zeta ( y , z , \\sigma , \\mu ) , \\end{aligned} \\end{align*}"}
-{"id": "827.png", "formula": "\\begin{align*} u = \\varphi + w _ h , \\quad \\mbox { w h e r e } \\left \\langle \\mathcal { L } \\varphi , v \\right \\rangle = - \\left \\langle \\mathcal { L } w _ h , v \\right \\rangle , \\end{align*}"}
-{"id": "4105.png", "formula": "\\begin{align*} k _ { M , p } ( X ) = \\mathrm { h e s s } _ b g ( X , X ) | _ p . \\end{align*}"}
-{"id": "8790.png", "formula": "\\begin{align*} c _ { 0 } ( f ) = \\inf _ { m \\geq 1 } \\frac { \\textnormal { c o d i m } \\textnormal { C o n t } _ { 0 } ^ { \\geq m } ( f ) } { m } . \\end{align*}"}
-{"id": "9686.png", "formula": "\\begin{align*} \\tilde { \\Phi } ( \\alpha _ 5 , \\alpha _ 3 , \\alpha _ 2 , \\alpha _ 1 ; V _ a ) = \\tilde { \\Phi } _ 5 ( \\alpha _ 5 ; \\tilde { \\Phi } _ 3 ( \\alpha _ 3 ; \\tilde { \\Phi } _ 2 ( \\alpha _ 2 ; \\tilde { \\Phi } _ 1 ( \\alpha _ 1 ; V _ a ) ) ) ) ) , \\end{align*}"}
-{"id": "9849.png", "formula": "\\begin{align*} ( f g ) \\hat { \\ , } ( x ) = \\frac { 1 } { 2 \\pi } ( \\hat { f } \\ast \\hat { g } ) ( x ) . \\end{align*}"}
-{"id": "5651.png", "formula": "\\begin{align*} \\mathfrak { E } _ { W } ( \\gamma ) = \\inf \\{ \\mathfrak { E } _ { W } ( \\sigma ) \\ ; : \\ ; \\sigma \\in A C _ { p l o c } ( \\R , X ) , \\ , \\sigma : x ^ - \\mapsto x ^ + \\} = d _ K ( x ^ - , x ^ + ) . \\end{align*}"}
-{"id": "8988.png", "formula": "\\begin{gather*} \\bigg | \\sum _ { \\vec { k } } c _ k \\prod _ i T _ i ^ { k _ i } \\bigg | : = \\max _ { \\vec { k } : c _ k \\ne 0 } \\exp \\bigg ( { - } \\sum _ i k _ i \\bigg ) . \\end{gather*}"}
-{"id": "7542.png", "formula": "\\begin{gather*} ( 1 + \\omega ^ 2 ) \\varphi ^ i _ { \\omega \\omega } + 4 \\omega \\varphi ^ i _ \\omega + 2 \\varphi ^ i + \\varphi ^ 2 ( \\omega \\varphi ^ i _ \\omega + \\varphi ^ i ) - \\varphi ^ 1 \\varphi ^ i _ \\omega = 0 , i = 1 , 2 . \\end{gather*}"}
-{"id": "5743.png", "formula": "\\begin{align*} \\frac { \\phi _ { x , y - 1 , z } ( a , b ) } { \\phi _ { x , y - 1 , z - 1 } ( a , b ) } \\frac { \\phi _ { x , y , z - 1 } ( a , b ) } { \\phi _ { x , y , z } ( a , b ) } + \\frac { \\phi _ { x + 1 , y - 1 , z - 1 } ( a , b ) } { \\phi _ { x , y - 1 , z - 1 } ( a , b ) } \\frac { \\phi _ { x - 1 , y , z } ( a , b ) } { \\phi _ { x , y , z } ( a , b ) } = 1 . \\end{align*}"}
-{"id": "544.png", "formula": "\\begin{align*} \\lambda = \\rho \\alpha , \\end{align*}"}
-{"id": "4063.png", "formula": "\\begin{align*} \\int _ E f & = \\int _ E f ( t ) \\ , \\d t - \\frac { 1 } { 2 } \\int _ { E \\cup E ^ c } f ( t ) \\ , \\d t \\\\ & = \\frac { 1 } { 2 } \\int _ E f ( t ) \\ , \\d t - \\frac { 1 } { 2 } \\int _ { E ^ c } f ( t ) \\ , \\d t \\\\ & \\le \\frac { 1 } { 2 } \\int _ { 0 } ^ { 1 } | f ( t ) | \\ , \\d t \\le \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "4873.png", "formula": "\\begin{align*} \\Lambda _ x ( \\mathcal { I } _ \\varphi ) = \\{ \\ell \\in X : \\mathfrak { u } ( \\ell ) > 0 \\} . \\end{align*}"}
-{"id": "7017.png", "formula": "\\begin{align*} d z = d y \\frac { 1 } { | y _ n | ^ { n - 1 } } ( \\sqrt { \\epsilon } g ( \\epsilon ) ) ^ { n - 1 } , \\end{align*}"}
-{"id": "9593.png", "formula": "\\begin{align*} \\langle f , T ^ { - 1 } _ N D _ k ^ N D _ k ( g ) \\rangle = \\langle ( D _ k ) ^ * ( D _ k ^ N ) ^ * ( T ^ { - 1 } _ N ) ^ * ( f ) , g \\rangle = \\langle D _ k D _ k ^ N T ^ { - 1 } _ N ( f ) , g \\rangle . \\end{align*}"}
-{"id": "5371.png", "formula": "\\begin{align*} S _ { ( a - 1 , 1 ) ( r ' , 1 ) } = \\dim ( M _ { a - 1 , 1 } ) \\dim ( M _ { r ' , 1 } ) \\end{align*}"}
-{"id": "841.png", "formula": "\\begin{align*} \\langle D _ { ( 0 , t ] } ^ { 1 - \\beta } u , f \\rangle = \\int _ 0 ^ T \\int _ { \\R ^ d } u ( t , x ) D _ { [ t , T ) } ^ { 1 - \\beta } f ( t , x ) d x d t , \\\\ \\langle D _ { [ t , T ) } ^ { 1 - \\beta } u , f \\rangle = \\int _ 0 ^ T \\int _ { \\R ^ d } u ( t , x ) D _ { ( 0 , t ] } ^ { 1 - \\beta } f ( t , x ) d x d t , \\end{align*}"}
-{"id": "8524.png", "formula": "\\begin{align*} \\tilde { b } ( \\cos \\theta ) = 4 ( \\cos \\theta ) b ( \\frac { v - v _ * } { | v - v _ * | } \\cdot \\frac { v ' - v ' _ * } { | v ' - v ' _ * | } ) . \\end{align*}"}
-{"id": "6831.png", "formula": "\\begin{align*} L ( \\phi ) = - \\lambda ^ 2 \\left [ S _ { \\rho } ( w _ { \\lambda } ) + N ( \\phi ) \\right ] + \\sum \\limits _ { j = 1 } ^ 4 c _ j \\chi _ { R _ 1 , j } \\varphi _ { 0 , j } + c _ 0 \\textrm { i n } \\mathbb { S } ^ 2 _ { \\lambda } , \\end{align*}"}
-{"id": "3642.png", "formula": "\\begin{align*} \\Big \\{ s _ { m } ^ { ( n ) } : = \\frac { m } { n - 1 } \\ , L \\Big \\} _ { m = 0 } ^ { n - 1 } \\subset [ 0 , L ] \\ , , \\end{align*}"}
-{"id": "1901.png", "formula": "\\begin{align*} \\Vert ( v _ { s } , v _ { u } ) \\Vert _ { i } = \\begin{cases} \\sqrt { a ^ { 2 i } \\Vert v _ { s } \\Vert ^ { 2 } + b ^ { 2 i } \\Vert v _ { u } \\Vert ^ { 2 } } & \\mbox { i f } i \\geq 0 \\\\ \\Vert ( v _ { s } , v _ { u } ) \\Vert & \\mbox { i f } i < 0 , \\\\ \\end{cases} \\end{align*}"}
-{"id": "1561.png", "formula": "\\begin{align*} - \\nu ( b _ m ) = 1 + V ( \\mathbf { j } _ { \\max } ) . \\end{align*}"}
-{"id": "6451.png", "formula": "\\begin{align*} d s ^ { 2 } = \\frac { 3 - 4 \\rho } { ( 1 - 2 \\rho ^ { 2 } ) \\sigma ^ { 2 } } d \\mu ^ { 2 } + \\frac { 6 } { \\sigma ^ { 2 } } d \\sigma ^ { 2 } \\end{align*}"}
-{"id": "5766.png", "formula": "\\begin{align*} - \\Delta _ { p } u = \\sigma u ^ { q } + \\mu \\ ; \\ ; \\mathbb { R } ^ n \\end{align*}"}
-{"id": "656.png", "formula": "\\begin{align*} D _ t ( P \\widetilde F _ i - F _ i ) & = ( D _ t P ) \\widetilde F _ i + P D _ t \\widetilde F _ i - D _ t F _ i \\\\ & = ( D _ t P ) \\widetilde F _ i + P \\widetilde \\nabla _ i \\widetilde H - \\nabla _ i H \\\\ & = ( D _ t P ) \\widetilde F _ i + ( \\tilde g ^ { k l } - g ^ { k l } ) ( P \\widetilde \\nabla _ i \\widetilde A _ { k l } ) + g ^ { k l } ( P \\widetilde \\nabla _ i \\widetilde A _ { k l } - \\nabla _ i A _ { k l } ) . \\end{align*}"}
-{"id": "8487.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { t , l , v } ) = \\zeta _ F ( 1 ) ^ { - 2 } q ^ { - \\frac { t } { 2 } } q ^ { s ( l _ 1 - l _ 2 ) } \\sum _ { \\mu \\in \\mathfrak { X } _ l } G ( \\varpi ^ { - a _ 1 } , \\mu \\chi _ 1 ) G ( \\varpi ^ { - l _ 2 } , \\mu \\chi _ 2 ) G ( v \\varpi ^ { - l } , \\mu ^ { - 1 } ) . \\end{align*}"}
-{"id": "4383.png", "formula": "\\begin{align*} \\Big \\| M _ { \\chi _ { B ( 0 , R ) } } \\Big ( C _ { z _ \\gamma } ( A P _ \\alpha + Q _ \\alpha ) C _ { - z _ \\gamma } & - ( A _ { x } P _ \\alpha + Q _ \\alpha ) \\Big ) \\Big \\| \\\\ & = \\Big \\| M _ { \\chi _ { B ( 0 , R ) } } P _ \\alpha ( C _ { z _ \\gamma } A C _ { - z _ \\gamma } - A _ { x } ) P _ \\alpha \\Big \\| \\\\ & \\to 0 \\end{align*}"}
-{"id": "3375.png", "formula": "\\begin{align*} \\left \\langle \\left \\langle ( g \\otimes s ) \\xi _ { 1 } , ( g \\otimes s ) \\xi _ { 2 } \\right \\rangle \\right \\rangle & = s \\overline { s } \\tau ( \\xi _ { 2 } ) \\tau ( g ) g \\xi _ { 1 } = \\tau ( \\xi _ { 2 } ) \\xi _ { 1 } = \\left \\langle \\left \\langle \\xi _ { 1 } , \\xi _ { 2 } \\right \\rangle \\right \\rangle , \\\\ g \\otimes s & \\in S p i n _ { ( n + m ) } ^ { \\mathbb { C } } \\subset \\mathbb { C } l _ { ( n + m ) } , \\end{align*}"}
-{"id": "9569.png", "formula": "\\begin{align*} \\mathcal { I } _ \\alpha ( f ) ( x ) = \\int _ { \\R ^ n } \\frac { f ( y ) } { \\lvert x - y \\rvert ^ { n - \\alpha } } \\ , d y \\ , , x \\in \\mathbb { R } ^ n \\ , . \\end{align*}"}
-{"id": "9053.png", "formula": "\\begin{align*} \\Im ( b _ { 2 1 } ^ 1 + b _ { 2 2 } ^ 2 ) = 0 . \\end{align*}"}
-{"id": "5918.png", "formula": "\\begin{align*} \\mu ^ N ( j ) = \\frac 1 { \\sum _ { s \\in A } L ( s ) Q ^ N ( s ) } . \\end{align*}"}
-{"id": "5674.png", "formula": "\\begin{align*} \\int _ I \\frac 1 2 ( f ' ) ^ 2 + W ( f \\sigma ) = \\mathfrak { E } _ { W } ( f \\sigma , I ) - \\frac 1 2 \\int _ I ( \\sigma ' f ) ^ 2 \\leq \\mathfrak { E } _ { W } ( E \\sigma , I ) - \\frac 1 2 \\int _ I | \\sigma ' | ^ 2 E ^ 2 = \\int _ { s _ 0 } ^ { + \\infty } \\frac 1 2 ( E ' ) ^ 2 + W ( E \\sigma ) . \\end{align*}"}
-{"id": "1028.png", "formula": "\\begin{align*} \\widehat { w } _ { 3 } ( \\zeta ) \\leq 3 + \\sqrt { 2 } = 4 . 4 1 4 2 \\ldots . \\end{align*}"}
-{"id": "2726.png", "formula": "\\begin{align*} & \\displaystyle B ( | v - v _ { \\ast } | , \\cos \\theta ) = \\phi ( | v - v _ { \\ast } | ) \\ , b ( \\cos \\theta ) , \\phi ( \\xi ) = C _ { \\phi } \\ , \\xi ^ { \\gamma } , \\ , \\gamma \\in [ 0 , 1 ] , \\\\ [ 2 p t ] & \\displaystyle \\forall \\eta \\in [ - 1 , 1 ] , | b ( \\eta ) | \\leq C _ b , | b ^ { \\prime } ( \\eta ) | \\leq C _ b \\ , , \\end{align*}"}
-{"id": "9092.png", "formula": "\\begin{align*} d ^ { n + 1 } ( x y ) - x d ^ { n + 1 } ( y ) - y d ^ { n + 1 } ( x ) = \\sum _ { i = 1 } ^ { n } \\binom { n + 1 } { i } d ^ { i } ( x ) d ^ { n + 1 - i } ( y ) \\left ( x , y \\in R \\right ) . \\end{align*}"}
-{"id": "8819.png", "formula": "\\begin{align*} - \\sum _ { i \\in I } \\nu _ { i } ( a _ { i } + 1 ) & = - \\sum _ { i \\in I } N _ { i } \\sigma _ { i } ( a _ { i } + 1 ) \\\\ & = - \\sum _ { i \\in I } N _ { i } ( a _ { i } + 1 ) ( \\sigma _ { i } - c _ 0 ( f ) ) - ( m - 1 ) c _ 0 ( f ) \\\\ & \\leq - ( m - 1 ) c _ 0 ( f ) , \\end{align*}"}
-{"id": "8805.png", "formula": "\\begin{align*} E _ { m , p } ^ { 0 } ( f ) = \\sum _ { i = 1 } ^ { s } a _ { i , p } m ^ { \\beta _ { i } } p ^ { - \\lambda _ { i } m } \\ 1 1 _ { A _ { i } } ( m ) \\end{align*}"}
-{"id": "5746.png", "formula": "\\begin{align*} \\frac { f _ { x , y - 1 , z } ( \\textbf { a } ) f _ { x , y , z - 1 } ( \\textbf { a } ) } { f _ { x , y - 1 , z - 1 } ( \\textbf { a } ) f _ { x , y , z } ( \\textbf { a } ) } = 1 . \\end{align*}"}
-{"id": "8116.png", "formula": "\\begin{align*} [ x ] _ { R _ { C , E } } \\subseteq [ x ] _ { R _ { U _ 1 , E } } = S _ i x . \\end{align*}"}
-{"id": "7087.png", "formula": "\\begin{align*} \\psi ( q ) & = a + q b + q ^ 3 c , \\\\ \\psi ^ { 2 } ( q ^ { 5 } ) & = a b + q ^ { 5 } c ^ { 2 } . \\end{align*}"}
-{"id": "5872.png", "formula": "\\begin{align*} \\langle X , F _ { \\omega } \\rangle = b _ { \\omega } + \\epsilon _ { \\omega } , \\end{align*}"}
-{"id": "7345.png", "formula": "\\begin{align*} \\Vert \\Phi u - \\Phi v \\Vert _ { X ^ { ( 4 , 3 ) } [ 0 , T ] } \\ ; & = \\ ; \\Vert \\Phi ( u - v ) \\Vert _ { X ^ { ( 4 , 3 ) } [ 0 , T ] } \\ ; \\leqslant \\ ; C T ^ { \\theta } \\Vert u - v \\Vert _ { X ^ { ( 4 , 3 ) } [ 0 , T ] } \\\\ & < \\ ; \\frac 1 2 \\Vert u - v \\Vert _ { X ^ { ( 4 , 3 ) } [ 0 , T ] } \\ , , \\end{align*}"}
-{"id": "691.png", "formula": "\\begin{align*} \\Phi ( \\mu ) : = \\varphi ( \\pi , \\mu ) \\cot \\beta + \\varphi ' ( \\pi , \\mu ) = 0 , \\end{align*}"}
-{"id": "9538.png", "formula": "\\begin{align*} \\int \\limits _ A \\deg _ B ( x ) \\ , d \\lambda ( x ) = \\int \\limits _ B \\deg _ A ( x ) \\ , d \\lambda ( x ) . \\end{align*}"}
-{"id": "3241.png", "formula": "\\begin{align*} \\sum _ { i = o d d } H _ a ^ i ( \\cdot ; \\mathbb Q ) = \\mathcal N _ 1 . \\end{align*}"}
-{"id": "6170.png", "formula": "\\begin{align*} A _ i ' = f _ i ' V + f _ i V ' \\ ; \\ ; \\ ; , \\ ; \\ ; \\ ; A _ i '' = f _ i '' + 2 f _ i ' V ' + f _ i V '' \\end{align*}"}
-{"id": "1108.png", "formula": "\\begin{align*} \\beta _ j ^ { ( i ) } = \\upsilon \\ , \\frac { | \\alpha _ { i j } | ^ 2 } { | \\alpha _ { i j } | ^ 2 \\sigma _ { Z _ { s , j } ^ { ( i ) } } ^ 2 + \\frac { N } { \\mu P _ t } \\sigma _ { Z _ { 0 , j } ^ { ( i ) } } ^ 2 } , \\end{align*}"}
-{"id": "9731.png", "formula": "\\begin{align*} & \\tilde { V } _ b = \\tilde { \\Phi } _ 5 ( \\tilde { \\gamma } _ 5 , \\mathcal { F } ( \\tilde { \\sigma } _ 3 , \\tilde { \\sigma } _ 2 ; \\tilde { \\Phi } _ 1 ( \\tilde { \\gamma } _ 1 ; \\tilde { V } _ a ) ) ) , \\tilde { Z } _ b = \\tilde { Z } _ a + \\tilde { \\gamma } _ 4 . \\end{align*}"}
-{"id": "7467.png", "formula": "\\begin{align*} \\widetilde { e } _ \\alpha = W ^ \\beta _ \\alpha e _ \\beta , \\end{align*}"}
-{"id": "7259.png", "formula": "\\begin{align*} E = N ^ 2 + 2 \\sum _ { n \\le X } r ( n ) ^ 2 . \\end{align*}"}
-{"id": "2615.png", "formula": "\\begin{align*} S \\Big ( \\frac { u ( t ) } { t ^ a } \\Big ) = S \\Big ( \\frac { ( 1 - t ) ^ { a } ( \\phi _ a u ) ( t ) } { [ t ( 1 - t ) ] ^ { ( a + 1 ) / 2 } } \\Big ) \\in w L ^ { \\tfrac 2 { 1 + a } } ( 0 , 1 ) \\cap L _ { } ^ \\infty ( 0 , 1 ) . \\end{align*}"}
-{"id": "7010.png", "formula": "\\begin{align*} C _ 4 = C _ 4 ( n , s , \\eta _ 0 , L , S C ) \\end{align*}"}
-{"id": "8949.png", "formula": "\\begin{gather*} \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} \\ ! , \\begin{pmatrix} 2 & - \\mu _ { 2 1 } \\\\ - \\mu _ { 2 1 } ^ \\vee & \\mu _ { 2 1 } ^ \\vee \\mu _ { 2 1 } \\end{pmatrix} \\ ! , \\begin{pmatrix} 0 & 0 \\\\ 0 & 1 \\end{pmatrix} \\ ! . \\end{gather*}"}
-{"id": "5320.png", "formula": "\\begin{align*} w \\left ( f ( A ) X + X \\bar { f } ( A ) \\right ) & \\leq \\left ( { \\frac { 2 } { d _ { A } ^ { 2 } } } \\int _ { 0 } ^ { 2 \\pi } d \\mu ( \\alpha ) \\right ) w ( X - A X A ^ { \\ast } ) \\\\ & = \\left ( { \\frac { 2 } { d _ { A } ^ { 2 } } } f ( 0 ) \\right ) w ( X - A X A ^ { \\ast } ) \\\\ & = { \\frac { 2 } { d _ { A } ^ { 2 } } } w ( X - A X A ^ { \\ast } ) . \\end{align*}"}
-{"id": "2750.png", "formula": "\\begin{align*} | \\widetilde S _ { k i } | \\leq \\xi \\ , | | z | | _ { L ^ { \\infty } } \\langle | \\psi _ k | , | \\psi _ i | \\rangle _ { L _ z ^ 2 } \\leq \\xi \\ , C _ z \\ , | | \\psi _ k | | _ { L ^ 2 _ z } \\ , | | \\psi _ i | | _ { L ^ 2 _ z } = C _ 2 ( \\xi ) , \\qquad k \\neq i , \\end{align*}"}
-{"id": "4894.png", "formula": "\\begin{align*} U = \\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix} . \\end{align*}"}
-{"id": "6948.png", "formula": "\\begin{align*} ( a \\oplus e _ n ) ( z \\oplus e _ n ) = ( e _ n \\oplus a ) ( z \\oplus e _ n ) = ( e _ n z ) \\oplus ( a e _ n ) = z \\oplus a \\ , . \\end{align*}"}
-{"id": "8402.png", "formula": "\\begin{align*} \\frac { d } { d s } F ( A + s B ) | _ { s = 0 } = \\dot { F } ^ { i j } | _ A B _ { i j } \\\\ [ 5 p t ] \\frac { d ^ 2 } { d s ^ 2 } F ( A + s B ) | _ { s = 0 } = \\ddot { F } ^ { i j , k l } | _ A B _ { i j } B _ { k l } \\end{align*}"}
-{"id": "36.png", "formula": "\\begin{align*} E _ { D Y } [ G ^ { C } _ { \\sigma \\ , \\sqrt { 2 } } ( e ) ( - d ^ * \\textbf { X } + \\textbf { X X } ^ { H } \\textbf { w } ) ] = \\textbf { 0 } \\end{align*}"}
-{"id": "3530.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l l } & - d { p } ( t ) = \\{ b _ x ( \\bar { X } { ( t ) } , \\bar { u } ( t ) ) ^ { } p ( t ) - \\beta ^ 0 f _ x ( \\bar { X } { ( t ) } , \\bar { u } ( t ) ) \\} d t , \\ t \\in ( t _ { i - 1 } , t _ { i } ) , \\\\ & p ( t _ { i } ) = - \\beta ^ 0 \\Psi _ { x } ( \\bar { X } ( t _ n ) 1 _ { i = n } ( i ) - \\beta ^ i + p ( t _ { i } ^ { + } ) , \\ i = 1 , 2 , \\ldots , n , \\end{array} \\end{align*}"}
-{"id": "3371.png", "formula": "\\begin{align*} \\nabla ^ { \\Sigma ^ { a d \\mathbb { C } } } = \\nabla ^ { \\Sigma ^ { \\mathbb { C } } M \\otimes \\Sigma ^ { \\mathbb { C } } \\nu } : = \\nabla ^ { \\Sigma ^ { \\mathbb { C } } M } \\otimes I d + I d \\otimes \\nabla ^ { \\Sigma ^ { \\mathbb { C } } \\nu } . \\end{align*}"}
-{"id": "4479.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| \\Delta _ n a \\Delta _ n ^ * \\| = | \\psi ( a ) | \\end{align*}"}
-{"id": "2869.png", "formula": "\\begin{align*} \\tilde { P } _ C = \\{ x \\in \\R ^ n _ + \\ , | \\ , x _ i + \\displaystyle \\sum _ { j \\in [ n ] \\setminus \\{ i \\} } 2 x _ j \\geq 2 , \\ \\forall i \\in [ n ] \\} . \\end{align*}"}
-{"id": "3433.png", "formula": "\\begin{align*} L ^ p ( 0 , T ; H ) = \\left \\{ u : [ 0 , T ] \\to H \\ , : \\ , u \\| u \\| _ { L ^ p ( 0 , T ; H ) } < \\infty \\right \\} \\end{align*}"}
-{"id": "5516.png", "formula": "\\begin{align*} \\varphi ( \\mu t ) = f ( \\varphi ( t ) ) . \\end{align*}"}
-{"id": "7508.png", "formula": "\\begin{align*} X _ t ( x , v ) & = x + \\int _ 0 ^ t \\frac { V _ s ( x , v ) } { \\sqrt { 1 + V _ s ^ 2 ( x , v ) } } d s , V _ t ( x , v ) = v - \\int _ 0 ^ t \\nabla ( V * \\widetilde { \\rho } _ s ) ( X _ s ( x , v ) ) d s \\end{align*}"}
-{"id": "6945.png", "formula": "\\begin{align*} j _ n ( \\gamma , t ) = \\alpha _ t \\gamma : I \\rightarrow M _ { \\infty } \\ , . \\end{align*}"}
-{"id": "3220.png", "formula": "\\begin{align*} \\mathcal H _ k ( X ) \\subset \\sum _ { r = 0 } ^ { r = n - k } H ^ { 2 r + k } ( X ; \\mathbb Q ) . \\end{align*}"}
-{"id": "4760.png", "formula": "\\begin{align*} { L } _ i = \\Big ( \\alpha , 0 , { L } _ 2 ( x ^ 2 ) , { L } _ 3 ( x ^ 3 ) \\Big ) , \\end{align*}"}
-{"id": "5529.png", "formula": "\\begin{align*} \\ddot { x } + \\lambda q \\left ( t \\right ) x = 0 \\end{align*}"}
-{"id": "9836.png", "formula": "\\begin{align*} \\varphi _ 4 ( x , y ) = x ^ 4 - 6 x ^ 2 y ^ 2 + y ^ 4 . \\end{align*}"}
-{"id": "1157.png", "formula": "\\begin{align*} p _ k ( a ) = \\rho _ { a , k } = \\begin{cases} 0 , & \\mbox { i f $ \\deg ( a ) < k $ } \\cr 1 , & \\mbox { i f $ \\deg ( a ) = k $ . } \\end{cases} \\end{align*}"}
-{"id": "9059.png", "formula": "\\begin{align*} A _ { { \\mathbf x } } { \\mathcal H } ( w , w ' ) = ( a _ { 1 1 } ^ 1 { \\mathbf x } _ 1 \\alpha \\bar w _ 1 w _ 1 ' , a _ { 2 2 } ^ 2 { \\mathbf x } _ 2 ( \\gamma \\bar w _ 1 w _ 1 ' + \\delta \\bar w _ 2 w _ 2 ' + \\delta \\bar w _ 3 w _ 3 ' ) ) . \\end{align*}"}
-{"id": "8716.png", "formula": "\\begin{align*} & u ^ \\varepsilon ( t , x ) = \\sup _ { x ' \\in \\Omega } \\{ u ( t , x ' ) - \\varepsilon ^ { - 1 } | x - x ' | ^ 2 \\} \\quad \\\\ & v _ \\varepsilon ( t , x ) = \\inf _ { x ' \\in \\Omega } \\{ v ( t , x ' ) + \\varepsilon ^ { - 1 } | x - x ' | ^ 2 \\} \\end{align*}"}
-{"id": "6616.png", "formula": "\\begin{align*} \\rho ( f , g ) = \\| f - g \\| _ { \\mathbb { W } ^ { 1 , p } ( \\Omega , \\mathbb { R } ^ 2 ) } + \\| f ^ { - 1 } - g ^ { - 1 } \\| _ { \\mathbb { W } ^ { 1 , 1 } ( \\Omega ^ * , \\mathbb { R } ^ 2 ) } \\ . \\end{align*}"}
-{"id": "8261.png", "formula": "\\begin{align*} A _ L ( r _ 1 , r _ 2 ) = \\frac { 2 \\tilde { w } ( 1 ) } { \\sharp C l _ F ^ { \\mathfrak { q } } } L ( 1 + o _ { B , \\delta } ( 1 ) ) \\end{align*}"}
-{"id": "6172.png", "formula": "\\begin{align*} \\frac { \\partial \\log { f _ i } } { \\partial t } = \\frac { 1 } { V ^ { 2 } } \\Big \\{ ( \\log f _ i ) ^ { \\prime \\prime } + \\Big ( ( \\log f _ i ) ^ { \\prime } \\Big ) ^ { 2 } - \\frac { 1 } { 2 } ( \\log f _ i ) ^ { \\prime } ( \\log V ^ { 2 } ) ^ { \\prime } \\Big \\} - \\frac { 1 } { 3 } \\mathcal { T } \\end{align*}"}
-{"id": "7327.png", "formula": "\\begin{align*} V _ { ( 1 , 0 ) } & = \\left ( ( p ^ { - 1 } \\mathbb { Z } _ p / \\mathbb { Z } _ p ) \\times ( p ^ { - 1 } \\mathbb { Z } _ p / \\mathbb { Z } _ p ) \\times ( \\mathbb { Z } _ p / \\mathbb { Z } _ p ) \\right ) ( e _ 1 ) \\\\ V _ { ( 1 , 1 ) } & = \\left ( ( p ^ { - 1 } \\mathbb { Z } _ p / \\mathbb { Z } _ p ) \\times ( p ^ { - 2 } \\mathbb { Z } _ p / \\mathbb { Z } _ p ) \\times ( p ^ { - 1 } \\mathbb { Z } _ p / \\mathbb { Z } _ p ) \\right ) ( e _ 1 e _ 2 ) \\end{align*}"}
-{"id": "9415.png", "formula": "\\begin{align*} a = | \\{ I \\in N _ { \\mathcal { M } } \\mid \\sigma _ 1 \\tau \\cdot I = I \\} | 2 b = | \\{ I \\in N _ { \\mathcal { M } } \\mid \\sigma _ 1 \\tau \\cdot I \\neq I \\} | . \\end{align*}"}
-{"id": "2415.png", "formula": "\\begin{align*} R _ 2 = ( a _ 2 ^ 2 - 3 a _ 3 ^ 2 - a _ 3 a _ 4 ) \\left ( 4 a _ 2 ^ 2 + ( 2 a _ 3 + a _ 4 ) ^ 2 \\right ) . \\end{align*}"}
-{"id": "9200.png", "formula": "\\begin{align*} [ x \\otimes 1 , y \\otimes b ] = x . y \\otimes b \\end{align*}"}
-{"id": "5810.png", "formula": "\\begin{align*} \\int _ { \\Omega } u ^ { 1 + q } \\ ; d \\sigma = \\int _ { \\Omega } u \\ ; d \\nu = \\langle \\nu , u \\rangle < + \\infty . \\end{align*}"}
-{"id": "3748.png", "formula": "\\begin{align*} M _ k = \\{ k \\} \\times ( \\Z ^ d \\times \\Z ) , \\ ; \\ ; \\ ; k \\geq 0 , \\end{align*}"}
-{"id": "9227.png", "formula": "\\begin{align*} [ d , [ x _ { 1 } ^ { + } \\otimes a _ { 1 } ^ { - } , x _ { 2 } ^ { + } \\otimes a _ { 2 } ^ { - } ] ] = [ [ d , x _ { 1 } ^ { + } \\otimes a _ { 1 } ^ { - } ] , x \\otimes a _ { 2 } ^ { - } ] + [ x _ { 1 } ^ { + } \\otimes a _ { 1 } ^ { - } , [ d , x _ { 2 } ^ { + } \\otimes a _ { 2 } ^ { - } ] ] \\end{align*}"}
-{"id": "1552.png", "formula": "\\begin{align*} \\mu ( \\mathcal { D } ) & \\ \\leq \\ \\sum _ { i , j } \\mu \\left ( ( J _ i \\pm J _ j ) \\setminus ( C _ i \\pm C _ j ) \\right ) \\\\ & \\ \\leq \\ k ^ 2 \\max _ { i , j } \\mu \\left ( ( J _ i \\pm J _ j ) \\setminus ( C _ i \\pm C _ j ) \\right ) . \\end{align*}"}
-{"id": "9173.png", "formula": "\\begin{align*} R i c ( \\omega _ \\phi ) - \\omega _ \\phi = \\sqrt { - 1 } \\partial \\bar \\partial \\log ( 1 - \\theta _ X ( \\omega _ \\phi ) ) . \\end{align*}"}
-{"id": "3246.png", "formula": "\\begin{align*} \\int _ V i _ ! i ^ \\ast ( \\theta ) \\wedge \\phi = \\int _ V \\theta \\omega _ Z \\wedge \\phi . \\end{align*}"}
-{"id": "6554.png", "formula": "\\begin{align*} \\beta _ { \\xi _ 1 } = \\max _ { \\zeta \\in \\mathrm { e x t } ( P ) } \\beta _ { \\zeta } \\alpha _ { \\xi _ 2 } - c _ 0 \\beta _ { \\xi _ 2 } = \\max _ { \\zeta \\in \\mathrm { e x t } ( P ) } [ \\alpha _ { \\zeta } - c _ 0 \\beta _ { \\zeta } ] . \\end{align*}"}
-{"id": "2561.png", "formula": "\\begin{align*} | c | \\ge | p ^ { * } | - r = 1 - r - d ^ { * } \\ge 1 - r - d _ { 0 } / 2 \\ , . \\end{align*}"}
-{"id": "9981.png", "formula": "\\begin{align*} ( f _ { j } ) ^ { r } = \\sum _ { \\alpha , \\beta , \\ell } { p ^ { \\alpha , \\beta } _ { j , \\ell } T ^ { \\ell } _ { \\alpha , \\beta } } . \\end{align*}"}
-{"id": "8729.png", "formula": "\\begin{align*} & \\Phi _ \\varepsilon ( t , x , y ) : = u ^ \\varepsilon ( t , x ) - v _ \\varepsilon ( t , y ) - \\varphi ( t , x , y ) \\quad \\\\ & \\Phi _ { \\varepsilon , \\delta } ( t , x , s , y ) : = u ^ { \\varepsilon , \\delta } ( t , x ) - v _ { \\varepsilon , \\delta } ( s , y ) - \\varphi ( t , x , y ) - \\frac { | t - s | ^ 2 } { \\delta } . \\end{align*}"}
-{"id": "101.png", "formula": "\\begin{align*} \\beta = \\inf _ t \\frac { \\beta _ t } { t } \\end{align*}"}
-{"id": "477.png", "formula": "\\begin{align*} p _ { 1 , k _ 1 , k _ 2 } ^ { ( m ) } ( x , t ) = \\sum _ { k = 1 } ^ { \\frac { m - 1 } { 2 } } \\frac { c _ { m , k } ( - 1 ) ^ k } { ( 2 \\pi ) ^ { \\frac { m - 1 } { 2 } } } \\sum _ { r = 0 } ^ { k _ 2 } \\binom { k _ 2 } { r } \\frac { ( - 1 ) ^ r ( m - 1 - k ) _ r } { | t | ^ { m - 1 - k + r } } p _ { 1 , k _ 1 , k _ 2 + k - r } ^ { ( 1 ) } ( x , | t | ) , \\end{align*}"}
-{"id": "8236.png", "formula": "\\begin{align*} [ j ^ k n ^ { - h } ] \\ , \\ln ( F ) & = 0 , k \\ge h + 2 \\\\ [ j ^ { h + 1 } n ^ { - h } ] \\ , \\ln ( F ) & = \\frac { 1 } { ( h + 1 ) h } \\biggl ( \\frac { 1 } { r ^ h } - 2 \\biggr ) \\end{align*}"}
-{"id": "4998.png", "formula": "\\begin{align*} H _ j : = \\sum _ { \\substack { j ' < j \\\\ j ' \\equiv j ( \\textrm { m o d } R ) } } g _ { j ' } \\prod _ { \\substack { j ' < j '' < j \\\\ j '' \\equiv j ( \\textrm { m o d } R ) } } ( 1 - G _ { j '' } ) . \\end{align*}"}
-{"id": "1644.png", "formula": "\\begin{align*} \\mu \\left ( \\bigcup _ { v \\in \\Lambda ^ 0 } D _ v \\right ) & = \\sum _ { v \\in \\Lambda ^ 0 } \\mu ( D _ v ) = \\sum _ { v \\in \\Lambda ^ 0 } \\sum _ { \\lambda \\in v \\Lambda ^ { e _ i } } \\mu ( R _ \\lambda ) = \\mu ( X ) \\end{align*}"}
-{"id": "6329.png", "formula": "\\begin{align*} \\frac { 1 } { p } = \\frac { 1 } { p _ 1 } + \\frac { 1 } { p _ 2 } , \\frac { 1 } { q } = \\frac { 1 } { q _ 1 } + \\frac { 1 } { q _ 2 } , \\end{align*}"}
-{"id": "5278.png", "formula": "\\begin{align*} \\frac { a _ 1 } { a _ 2 } & \\geq - \\frac { 1 } { 2 } , & \\ \\ a _ 2 \\neq 0 \\ \\ \\ \\ b _ 2 = 0 , \\\\ \\frac { b _ 1 } { b _ 2 } & \\leq \\frac { 1 } { 2 } , & \\ \\ a _ 2 = 0 \\ \\ \\ \\ b _ 2 \\neq 0 . \\end{align*}"}
-{"id": "5717.png", "formula": "\\begin{align*} m ' ( t ) = \\frac { ( \\partial _ t \\gamma ( t , \\cdot ) , \\partial _ s \\gamma ( t , \\cdot ) ) _ { L ^ 2 ( \\R ) } } { \\| \\partial _ s \\gamma ( t , \\cdot ) \\| _ { L ^ 2 ( \\R ) } } \\quad \\end{align*}"}
-{"id": "3089.png", "formula": "\\begin{align*} \\| u _ \\epsilon ( t ^ \\eta _ \\epsilon ) - \\varphi ( t ^ * ) \\| = \\| u _ \\epsilon ( t ^ \\eta _ \\epsilon ) - u ^ * \\| < \\eta . \\end{align*}"}
-{"id": "1493.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { [ n t ] } Y _ { n , k } = \\sum _ { k = 1 } ^ { [ n t ] } X _ { n , k } + \\sum _ { k = 1 } ^ { [ n t ] } \\xi _ { n , k } . \\end{align*}"}
-{"id": "6527.png", "formula": "\\begin{align*} n _ { P } ( \\xi ) = \\frac { 1 } { n \\ | P | _ n } \\ \\| \\xi \\| \\ | ( F _ { \\xi } - s ( F _ { \\xi } ) ) ^ { \\circ } | _ { n - 1 } . \\end{align*}"}
-{"id": "4883.png", "formula": "\\begin{align*} \\lambda ^ 1 _ m ( t ) : = ( 1 - t ) ^ { - m } = \\prod _ { n = 1 } ^ { \\infty } Z ( m , t ^ n ) ^ { \\mu ( n ) } = \\sum _ { k \\geq 0 } \\sum _ { \\underline { k } : \\sum _ i i k _ i = k } \\prod _ i T _ { k _ i } ( m ) t ^ k \\end{align*}"}
-{"id": "8272.png", "formula": "\\begin{align*} \\log ( \\abs { T } _ { \\infty } ) = \\sum _ { \\nu } [ F _ { \\nu } : \\R ] \\log ( \\abs { t _ { \\nu } + T _ 0 } ) \\ll \\log ( t _ 0 ) , \\end{align*}"}
-{"id": "9273.png", "formula": "\\begin{align*} \\Big | \\mathbb { P } _ \\rho ^ \\varepsilon ( \\{ \\mathcal { M } _ { s _ j \\varepsilon ^ { - 2 } } = \\nu _ j : j = 1 , \\dots , n \\} ) - \\mathbb { P } _ { \\pi _ { \\rho } } ( \\{ Y _ { s _ j } = \\nu _ j , j = 1 , \\dots , n \\} ) \\Big | & \\leq C n \\varepsilon , \\end{align*}"}
-{"id": "1738.png", "formula": "\\begin{align*} \\mu < < m , \\ \\nu < < m , \\ \\hbox { a n d } \\ ; \\ ; f \\sqrt { \\frac { d \\mu } { d m } } = g \\sqrt { \\frac { d \\nu } { d m } } \\ ; \\ ; \\ , L ^ 2 ( \\Lambda ^ \\infty , m ) . \\end{align*}"}
-{"id": "2575.png", "formula": "\\begin{align*} B = \\left [ \\begin{array} { c c c c } ( 0 , 1 , 8 ) & ( \\infty ) & ( 0 ) & ( \\infty ) \\\\ ( \\infty ) & ( 8 , 1 2 ) & ( 0 , 4 ) & ( \\infty ) \\\\ ( \\infty ) & ( 5 ) & ( \\infty ) & ( 4 , 9 , 1 0 ) \\end{array} \\right ] . \\end{align*}"}
-{"id": "205.png", "formula": "\\begin{align*} \\Gamma \\vdash \\Delta , \\ f ( \\Gamma ) \\in F _ { s } , f ( \\Delta ) = \\mathfrak { X } . \\end{align*}"}
-{"id": "6781.png", "formula": "\\begin{align*} w _ { \\lambda } = \\tilde { w } _ { \\lambda } + \\overline { w } _ { \\lambda } . \\end{align*}"}
-{"id": "2152.png", "formula": "\\begin{align*} 0 < \\theta < \\min \\{ 1 , d , d ^ { - 1 } \\} , \\sigma : = \\left ( \\frac { 1 } { 2 } - \\theta _ * \\right ) ( 1 + \\delta ) - \\frac { 1 } { 2 } > \\theta _ * > 0 . \\end{align*}"}
-{"id": "3826.png", "formula": "\\begin{align*} \\tau : = \\inf \\{ n \\in \\N \\colon N ( \\bar { X } _ n + 1 , n ) + N ( \\bar { X } _ n + 2 , n ) \\ge 1 \\} . \\end{align*}"}
-{"id": "2929.png", "formula": "\\begin{align*} \\phi _ { n , m } = \\phi _ { m - 1 , m } \\circ \\dots \\circ \\phi _ { n , n + 1 } . \\end{align*}"}
-{"id": "9185.png", "formula": "\\begin{align*} R _ { } & = \\frac { N m } { N ( p _ 1 - p _ 2 ) } \\\\ & = \\frac { N ^ { K - 1 } ( N - T ) } { ( N ^ K - T ^ K ) - T ^ { K - M } ( N ^ M - T ^ M ) } \\\\ & = \\frac { 1 - \\frac { T } { N } } { 1 - ( \\frac { T } { N } ) ^ { K - M } } \\\\ & = \\left ( 1 + \\frac { T } { N } + \\cdots + \\left ( \\frac { T } { N } \\right ) ^ { K - M - 1 } \\right ) ^ { - 1 } . \\\\ \\end{align*}"}
-{"id": "9061.png", "formula": "\\begin{align*} \\Im b _ { 1 1 } ^ 1 = 0 . \\end{align*}"}
-{"id": "3796.png", "formula": "\\begin{align*} g ( \\omega , U ) = \\begin{cases} 1 , & \\\\ - 1 , & , \\end{cases} \\end{align*}"}
-{"id": "2065.png", "formula": "\\begin{align*} \\Delta _ H = \\sum _ { \\alpha = 1 } ^ m X _ { \\alpha } ^ 2 + \\sum _ { \\alpha = 1 } ^ m X _ { m + \\alpha } ^ 2 + \\sum _ { \\alpha = 1 } ^ m ( \\nabla _ { X _ { \\alpha } } X _ \\alpha + \\nabla _ { X _ { m + \\alpha } } X _ { m + \\alpha } ) , \\end{align*}"}
-{"id": "7879.png", "formula": "\\begin{align*} \\partial _ t u + | \\nabla u | = 0 \\end{align*}"}
-{"id": "2029.png", "formula": "\\begin{align*} \\sum _ { \\alpha \\in \\Z } ( - 1 ) ^ { \\alpha } q ^ { \\frac { \\alpha ( 3 \\alpha - 1 ) } { 2 } } x _ { i + 3 \\alpha } x _ { i + 1 - 3 \\alpha } = 0 , ~ i \\in \\Z . \\end{align*}"}
-{"id": "7479.png", "formula": "\\begin{align*} ( \\Phi ^ h ) ^ m & = i ^ m ( - 1 ) ^ { \\frac { m ( m - 1 ) } { 2 } } m ! \\ : h \\ : \\mathcal { Z } ^ 1 \\wedge \\dots \\wedge \\mathcal { Z } ^ m \\wedge \\mathcal { Z } ^ { \\bar { 1 } } \\wedge \\dots \\wedge \\mathcal { Z } ^ { \\bar { m } } , \\\\ ( \\Phi ^ v ) ^ m & = i ^ m ( - 1 ) ^ { \\frac { m ( m - 1 ) } { 2 } } m ! \\ : h \\ : \\delta \\mathcal { V } ^ 1 \\wedge \\dots \\wedge \\delta \\mathcal { V } ^ m \\wedge \\delta \\mathcal { V } ^ { \\bar { 1 } } \\wedge \\dots \\wedge \\delta \\mathcal { V } ^ { \\bar { m } } , \\end{align*}"}
-{"id": "10138.png", "formula": "\\begin{align*} \\boldsymbol R _ k ( 0 ) \\boldsymbol S _ { D _ k } ( 0 ) \\bar { \\boldsymbol R } _ { \\bar { \\boldsymbol \\omega } _ k } ( 0 ) = \\boldsymbol P _ { D _ k } ( 0 ) = \\boldsymbol p _ k ( 0 ) \\boldsymbol { \\bar { \\omega } } _ k ^ H ( 0 ) + \\delta { \\boldsymbol \\Upsilon } , \\end{align*}"}
-{"id": "8035.png", "formula": "\\begin{align*} \\mbox { \\boldmath $ u $ } _ t \\ ; \\ ! + \\ , \\mbox { \\boldmath $ u $ } \\cdot \\nabla \\mbox { \\boldmath $ u $ } \\ , + \\ ; \\ ! \\nabla \\ : \\ ! { P } \\ ; = \\ ; ( \\ : \\ ! \\mu + \\chi \\ : \\ ! ) \\ , \\Delta \\mbox { \\boldmath $ u $ } \\ , + \\ , \\chi \\ , \\nabla \\times { \\bf w } , \\end{align*}"}
-{"id": "4714.png", "formula": "\\begin{align*} q _ 0 + d q _ 1 & = \\frac { q - 1 } { d } + ( d - 1 ) q + 1 = 1 + q \\left ( d - 1 + \\frac 1 d \\right ) - \\frac 1 d \\\\ & = d - \\left ( d - 1 + \\frac 1 d \\right ) + q \\left ( d - 1 + \\frac 1 d \\right ) = d + ( q - 1 ) \\left ( d - 1 + \\frac 1 d \\right ) > d \\end{align*}"}
-{"id": "7521.png", "formula": "\\begin{gather*} w ^ i w ^ 1 _ i - w ^ 1 _ { i i } - 2 \\hat \\kappa w ^ 2 - \\left ( \\kappa ^ 2 + \\frac 1 4 \\right ) z _ 1 = 0 , \\\\ w ^ i w ^ 2 _ i - w ^ 2 _ { i i } + 2 \\hat \\kappa w ^ 1 - \\left ( \\kappa ^ 2 + \\frac 1 4 \\right ) z _ 2 = 0 . \\end{gather*}"}
-{"id": "4259.png", "formula": "\\begin{align*} \\alpha _ { i } = ( - 1 ) ^ { i } \\cdot \\det A _ i \\end{align*}"}
-{"id": "7737.png", "formula": "\\begin{align*} \\mathrm { R e } ( H _ N ( l ) ) = \\frac { l ^ 2 } { 2 N } - \\frac { l } { N } - \\frac { l } { 2 } + 1 \\ , . \\end{align*}"}
-{"id": "5436.png", "formula": "\\begin{align*} \\widetilde { N } _ 3 = \\left ( \\begin{array} { l l l l } 1 & 4 & 6 & 4 \\\\ 1 & 5 & 8 & 5 \\\\ 1 & 5 & 9 & 6 \\\\ \\end{array} \\right ) \\quad \\rightsquigarrow { N } _ 3 = \\left ( \\begin{array} { l l l l } 1 & 4 & 6 & 4 \\\\ 0 & 1 & 2 & 1 \\\\ 0 & 0 & 1 & 1 \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "3487.png", "formula": "\\begin{align*} f ^ 3 = \\sum _ { j = 1 } ^ 3 f _ j ^ 3 + 3 \\sum _ { j = 1 } ^ 3 \\sum _ { k \\neq j } f _ j f _ k ^ 2 + 6 f _ 1 f _ 2 f _ 3 . \\end{align*}"}
-{"id": "5900.png", "formula": "\\begin{align*} \\sum _ { L = 0 } ^ \\infty P ( \\tau ^ { N , i } \\ge 2 L N ) \\ge \\sum _ { m = 0 } ^ \\infty ( \\frac m { 2 N } + 1 ) P ( \\tau ^ { N , i } = m ) = 1 + \\frac 1 { 2 N } E \\tau ^ { N , i } . \\end{align*}"}
-{"id": "8878.png", "formula": "\\begin{align*} ( J _ { \\theta , c } ^ { - 1 } \\gamma ) ( z ) = ( 1 - \\overline c \\theta ( z ) ) \\int _ { \\mathbb T } \\frac { \\gamma ( \\zeta ) } { 1 - z \\overline \\zeta } \\sigma _ c ( \\zeta ) , \\ \\ z \\in \\mathbb D , \\ \\ \\gamma \\in L ^ 2 ( \\sigma _ c ) \\end{align*}"}
-{"id": "8683.png", "formula": "\\begin{align*} x ^ { [ p ] } = \\Big ( \\frac { K ( x , x ) } { 2 } \\Big ) ^ { \\frac { p - 1 } { 2 } } x \\forall x \\in \\gg _ 0 . \\end{align*}"}
-{"id": "8137.png", "formula": "\\begin{align*} B _ { k , l , c , Q } = \\bigcap _ { s \\in F ^ Q } s T _ { k , l , c } ' \\end{align*}"}
-{"id": "6225.png", "formula": "\\begin{align*} \\P ( R ( i ) > 0 ~ \\textrm { f o r e a c h $ 1 \\leq i \\leq n $ } ~ | ~ R ( n ) = k ) = \\frac { k } { n } ~ . \\end{align*}"}
-{"id": "1994.png", "formula": "\\begin{align*} \\sup _ { \\lambda / n \\leq s < t } \\frac { n ^ { \\eta } \\left \\vert \\widetilde { \\beta } _ { n } \\left ( s ; t \\right ) - \\widetilde { B } _ { n } \\left ( s \\right ) \\right \\vert } { s ^ { 1 / 2 - \\eta } } = O _ { \\mathbb { P } } \\left ( 1 \\right ) , \\end{align*}"}
-{"id": "6027.png", "formula": "\\begin{align*} & \\liminf _ { n \\to \\infty } s R - \\frac { 1 } { n } s D _ { 1 + s } ( P _ { W ^ { n } X ^ { n } Y ^ { n } } \\| P _ { W ^ { n } } \\pi _ { X ^ { n } Y ^ { n } } ) \\\\ & \\qquad + \\frac { 1 } { n } s D _ { 1 + s } ( P _ { X ^ { n } Y ^ { n } } \\| \\pi _ { X ^ { n } Y ^ { n } } ) \\\\ & = s R - s \\left ( \\frac { ( 1 - \\epsilon ) ^ { 2 } } { 1 + \\epsilon ' } I _ { Q } ( X Y ; W ) + \\frac { 4 \\epsilon } { 1 - \\epsilon ' } H _ { Q } ( X Y ) \\right ) . \\end{align*}"}
-{"id": "6034.png", "formula": "\\begin{align*} g _ { Q _ { i } , P _ { i } } ^ { ( \\alpha , \\lambda ) } ( x _ { i } , y _ { i } | u _ { i } , v _ { i } ) & : = \\exp \\Big ( - \\lambda \\omega _ { Q _ { i } , P _ { i } } ^ { ( \\alpha ) } ( x _ { i } , y _ { i } | u _ { i } , v _ { i } ) \\Big ) , \\end{align*}"}
-{"id": "7715.png", "formula": "\\begin{align*} L ^ { s t a r } _ N = \\left ( \\begin{array} { c c c c c } { N - 1 } & - 1 & { \\cdots } & - 1 & - 1 \\\\ - 1 & 1 & { \\cdots } & 0 & 0 \\\\ { \\vdots } & { \\vdots } & { \\ddots } & { \\vdots } & { \\vdots } \\\\ - 1 & 0 & { \\cdots } & 1 & 0 \\\\ - 1 & 0 & { \\cdots } & 0 & 1 \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "5193.png", "formula": "\\begin{align*} i ) & a _ { 1 } \\le \\ell ^ { 1 } \\le \\ell ^ { 2 } \\le a _ { 2 } , \\\\ i i ) & \\hat { U } _ { 1 } ( x , \\ell ^ { 2 } , r ) \\le \\hat { U } _ { 1 } ( \\ell ^ { 1 } , \\ell ^ { 2 } , r ) , \\forall x \\in ( 0 , r ) , \\\\ i i i ) & \\hat { U } _ { 2 } ( \\ell ^ { 1 } , y , r ) \\le \\hat { U } _ { 2 } ( \\ell ^ { 1 } , \\ell ^ { 2 } , r ) , \\forall y \\in [ 0 , r ) . \\end{align*}"}
-{"id": "8605.png", "formula": "\\begin{align*} \\gamma ^ { k } _ { n } ( i ( k , \\ell ) , j ( k , \\ell ) ) = \\gamma ^ { k ' } _ { n ' } ( i ( k ' , \\ell ' ) , j ( k ' , \\ell ' ) ) \\end{align*}"}
-{"id": "7949.png", "formula": "\\begin{align*} g _ J : = h _ J \\circ \\cdots \\circ h _ { j _ 1 j _ 2 } \\circ h _ { j _ 1 } . \\end{align*}"}
-{"id": "932.png", "formula": "\\begin{align*} P ( F _ \\vee \\in A ) \\leq P ( \\Phi _ \\beta ( F ) \\in A ^ { \\varepsilon } ) = E [ 1 _ { A ^ { \\varepsilon } } ( \\Phi _ \\beta ( F ) ) ] . \\end{align*}"}
-{"id": "1912.png", "formula": "\\begin{align*} \\Pi = S _ 1 \\sqcup S _ 2 \\sqcup S _ 3 , S _ 1 = T _ 1 , S _ 2 = \\sqcup _ { j \\in J _ 2 } T _ j , S _ 3 = \\sqcup _ { j \\in J _ 3 } T _ j . \\end{align*}"}
-{"id": "8147.png", "formula": "\\begin{align*} X \\setminus \\bigsqcup _ { k = 1 } ^ K S _ k '' V _ k \\prec \\bigsqcup _ { k = 1 } ^ K S _ k ^ \\sharp V _ k . \\end{align*}"}
-{"id": "7651.png", "formula": "\\begin{align*} \\Lambda = \\frac { 1 } { 2 } \\ , c . \\end{align*}"}
-{"id": "10154.png", "formula": "\\begin{align*} { J _ { \\boldsymbol \\omega _ k ( i ) } ( { \\boldsymbol \\omega _ k ( i ) } ) } = { \\mathbb { E } | { d _ k ( i ) } - { \\boldsymbol x _ k ^ H ( i ) } { \\boldsymbol \\omega _ k ( i ) } } | ^ 2 . \\end{align*}"}
-{"id": "2636.png", "formula": "\\begin{align*} N = \\left \\{ ( w _ 1 , w _ 2 , \\dots , w _ { n } ) \\in ( \\textbf { B } ( 0 , 1 ) ) ^ n : | w _ 1 \\wedge \\dots \\wedge w _ n | = 0 \\right \\} \\end{align*}"}
-{"id": "8528.png", "formula": "\\begin{align*} a ( x , \\xi ) = \\int _ { \\omega \\in S ^ 2 } e ^ { - i ( x \\cdot \\omega ) ( \\xi \\cdot \\omega ) } b _ \\delta ( \\cos \\theta ) d \\Omega ( \\omega ) . \\end{align*}"}
-{"id": "1094.png", "formula": "\\begin{align*} n _ { o , i } = m _ { s , i } + \\sqrt { m _ { s , i } } Z _ { s , i } + Z _ { 0 , i } , \\end{align*}"}
-{"id": "3194.png", "formula": "\\begin{align*} ( \\mathcal { J } _ { \\ast } V ) ( y ) & = \\int _ { \\lbrace z \\leqslant 1 \\rbrace } \\left ( V ( y + z ) - V ( y ) \\right ) \\nu ( \\mathrm { d } z ) , \\\\ ( \\mathcal { J } ^ { \\ast } V ) ( y ) & = \\int _ { \\lbrace z > 1 \\rbrace } \\left ( V ( y + z ) - V ( y ) \\right ) \\nu ( \\mathrm { d } z ) . \\end{align*}"}
-{"id": "9995.png", "formula": "\\begin{align*} \\mathbf E \\big [ M _ { \\tau + N , n } ^ 2 \\big ] & = \\sum _ { S = 1 } ^ { N } S ^ { - | n | } \\mathbf E \\big [ ( Y _ n ( \\tau + S ) - Y _ n ( \\tau + S - 1 ) ) ^ 2 \\big ] \\\\ & + \\sum _ { S = 1 } ^ { N } \\sum _ { S \\neq R = 1 } ^ { N } S ^ { - | n | / 2 } R ^ { - | n | / 2 } \\mathbf E \\big [ ( Y _ n ( \\tau + S ) - Y _ n ( \\tau + S - 1 ) ) ( Y _ n ( \\tau + R ) - Y _ n ( \\tau + R - 1 ) ) \\big ] \\\\ & = \\sum _ { S = 1 } ^ { N } S ^ { - | n | } \\mathbf E \\big [ ( Y _ n ( \\tau + S ) - Y _ n ( \\tau + S - 1 ) ) ^ 2 \\big ] \\end{align*}"}
-{"id": "2779.png", "formula": "\\begin{align*} C ( t , S ) & = S - K , S = \\mathcal { B } ( t ) , 0 \\leq t < T , \\\\ \\frac { \\partial C } { \\partial S } ( t , S ) & = 1 , S = \\mathcal { B } ( t ) , 0 \\leq t < T , \\\\ C ( T , S ) & = \\max \\{ 0 , S - K \\} , \\lim _ { S \\rightarrow 0 } C ( t , S ) = 0 . \\end{align*}"}
-{"id": "4523.png", "formula": "\\begin{align*} | \\chi ( t _ \\omega ) | < 1 = \\omega ( t _ \\omega ) \\end{align*}"}
-{"id": "3871.png", "formula": "\\begin{align*} x _ i ^ * \\neq 0 \\Longrightarrow x _ i \\neq 0 \\Longrightarrow y _ i = 0 \\end{align*}"}
-{"id": "7118.png", "formula": "\\begin{align*} Q _ 1 : = & \\alpha F ^ { - \\alpha - 2 } \\ddot { F } ^ { k l , m n } \\nabla _ 1 h _ { k l } \\nabla _ 1 h _ { m n } - \\alpha ( \\alpha + 1 ) F ^ { - \\alpha - 3 } ( \\nabla _ 1 F ) ^ 2 \\\\ & + \\alpha ( \\alpha + 1 ) F ^ { - \\alpha - 4 } \\dot { F } ^ { k l } \\nabla _ k F \\nabla _ l F \\kappa _ 1 \\\\ & \\quad + 2 \\alpha F ^ { - \\alpha - 1 } \\sup _ { \\Lambda } \\dot { F } ^ { k l } \\left ( 2 \\Lambda _ k ^ p \\nabla _ l G _ { 1 p } - \\Lambda _ k ^ p \\Lambda _ l ^ q G _ { p q } \\right ) ~ \\geq ~ 0 \\end{align*}"}
-{"id": "9371.png", "formula": "\\begin{align*} r _ n ^ { ( k + 1 ) } & = \\frac { \\iint _ { \\ , \\mathbb { C } _ n ^ { ( k ) } } g ( \\phi _ n , x , y ) f \\left ( \\sqrt { x ^ 2 + y ^ 2 } \\right ) \\ , \\mathrm { d } x \\ , \\mathrm { d } y } { \\iint _ { \\ , \\mathbb { C } _ n ^ { ( k ) } } \\frac { 1 } { \\sqrt { x ^ 2 + y ^ 2 } } f \\left ( \\sqrt { x ^ 2 + y ^ 2 } \\right ) \\ , \\mathrm { d } x \\ , \\mathrm { d } y } , \\end{align*}"}
-{"id": "2829.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { h _ { v } ^ { ( n + 1 ) } } { \\rho _ { \\alpha } ^ { ( n ) } } = \\sum _ { \\mathcal { E } _ { \\beta } \\preceq \\mathcal { E } _ { \\alpha } } \\left ( \\sum _ { w \\in \\mathcal { E } _ { \\beta } } a _ { v w } h _ w ^ { ( 1 ) } \\right ) \\end{align*}"}
-{"id": "9853.png", "formula": "\\begin{align*} { \\displaystyle \\int \\hat { f } g = \\int f \\hat { g } } . \\end{align*}"}
-{"id": "9821.png", "formula": "\\begin{align*} f ( T ' _ { e _ 2 } ) = 0 . \\end{align*}"}
-{"id": "653.png", "formula": "\\begin{align*} D _ t ^ k P \\big | _ { t = 0 } = 0 \\end{align*}"}
-{"id": "413.png", "formula": "\\begin{align*} \\int _ { V } e ^ { - R f ( \\lambda ) } g ( \\lambda ) \\ , \\dd \\lambda = \\sqrt { \\frac { ( 2 \\pi ) ^ m } { R ^ m \\det f '' ( 0 ) } } \\sum _ { j = 0 } ^ k \\frac { L _ { j , f } g } { R ^ j } + O \\left ( \\frac { 1 } { R ^ { \\frac { m } { 2 } + k + 1 } } \\right ) , \\end{align*}"}
-{"id": "589.png", "formula": "\\begin{align*} \\underset { z \\in \\Omega } { \\sup } \\left | f ^ { ( l ) } ( z ) \\right | \\leqslant \\underset { z \\in \\Omega } { \\sup } \\left | f ^ { ( \\alpha ) } ( z ) \\right | \\cdot ( ( \\Omega ) ) ^ { a - l } + \\sum _ { k = l } ^ { a - 1 } \\left | f ^ { ( k ) } ( z _ o ) \\right | \\cdot ( ( \\Omega ) ) ^ { k - l } . \\end{align*}"}
-{"id": "8882.png", "formula": "\\begin{align*} ( ( S _ \\theta - \\lambda ) ^ { - 1 } f , \\overline \\chi \\theta ) = - \\frac { f ( \\lambda ) } { \\theta ( \\lambda ) } \\end{align*}"}
-{"id": "6102.png", "formula": "\\begin{align*} \\lambda _ { 1 , 2 } = - ( n - 1 ) \\pm \\sqrt { ( n - 2 ) ^ 2 - 2 - c ^ 2 } . \\end{align*}"}
-{"id": "2393.png", "formula": "\\begin{align*} q ( x ) = & - \\Big ( 2 a _ 1 a _ 2 + 4 a _ 3 + a _ 4 \\Big ) + \\Big ( 1 2 a _ 2 + 3 a _ 5 - a _ 1 ( 6 a _ 3 + 2 a _ 4 ) \\Big ) x \\\\ & + \\Big ( 8 a _ 3 + 3 a _ 4 + 4 a _ 6 + a _ 1 ( 6 a _ 2 + 2 a _ 5 ) \\Big ) x ^ 2 - \\Big ( 4 a _ 2 + a _ 5 - 2 a _ 1 a _ 6 \\Big ) x ^ 3 . \\end{align*}"}
-{"id": "9220.png", "formula": "\\begin{align*} [ z \\otimes \\alpha , ( E _ { 1 , 3 } + E _ { 3 , 1 } ) \\otimes \\frac { 1 } { 2 } [ b _ { 1 } , b _ { 2 } ] _ { C } + [ z \\otimes \\alpha , ( E _ { 1 , 3 } - E _ { 3 , 1 } ) \\otimes \\frac { 1 } { 2 } ( b _ { 1 } \\circ b _ { 2 } ) _ { E } ] = [ \\varepsilon e _ { 2 } \\otimes \\alpha b _ { 1 } , u _ { 2 } \\otimes b _ { 2 } ] . \\end{align*}"}
-{"id": "9385.png", "formula": "\\begin{align*} \\| \\mathfrak { a } \\| _ { V _ \\rho } = \\sup \\big ( | a _ { t _ 0 } | + \\sum _ { k \\geq 1 } | a _ { t _ k } - a _ { t _ { k - 1 } } | ^ \\rho \\big ) ^ { \\frac { 1 } { \\rho } } , \\end{align*}"}
-{"id": "2292.png", "formula": "\\begin{align*} & \\ ; \\partial _ t ( 1 - \\alpha ^ 2 \\Delta ) u + u \\cdot \\nabla ( 1 - \\alpha ^ 2 \\Delta ) u - \\alpha ^ 2 \\nabla u ^ T \\cdot \\Delta u + \\nabla p = - \\nu ( 1 - \\alpha ^ 2 \\Delta ) A ^ s u , \\\\ & \\ ; \\mathrm { d i v } \\ , u = 0 , u | _ { t = 0 } = u _ 0 . \\end{align*}"}
-{"id": "2242.png", "formula": "\\begin{align*} s _ i = m _ { 1 i } ( k _ 1 + 1 ) + \\ldots + m _ { n i } ( k _ n + 1 ) . \\end{align*}"}
-{"id": "4266.png", "formula": "\\begin{align*} \\left \\{ \\sum \\limits _ { \\ell = m + 1 } ^ { \\infty } a _ { \\ell } \\cdot \\left ( \\frac { t } { t - 1 } \\right ) ^ \\ell + \\sum \\limits _ { \\ell = m + 1 } ^ { \\infty } b _ { \\ell } \\cdot \\left ( \\frac { t } { t - \\lambda } \\right ) ^ \\ell | a _ { \\ell } , b _ { \\ell } \\in k \\right \\} \\end{align*}"}
-{"id": "3252.png", "formula": "\\begin{align*} \\mathcal J _ k ^ { 2 n - p } ( X ) = L _ 0 ( X ) \\oplus L _ 1 ( X ) \\cdots \\oplus L _ { [ { p - 1 \\over 2 } ] } ( X ) . \\end{align*}"}
-{"id": "3203.png", "formula": "\\begin{align*} ( \\mathcal { A } V ) ( y ) = ( \\mathcal { D } V ) ( y ) + ( \\mathcal { J } V ) ( y ) \\leqslant - c V ( y ) + M , y \\in \\mathbb { R } _ { \\geqslant 0 } . \\end{align*}"}
-{"id": "2004.png", "formula": "\\begin{align*} \\psi _ l ( x _ { i _ 1 } . . . x _ { i _ l } ) = \\sum _ { \\sigma \\in S _ l } z _ { \\sigma ( 1 ) } ^ { i _ 1 } . . . z _ { \\sigma ( l ) } ^ { i _ l } \\end{align*}"}
-{"id": "6319.png", "formula": "\\begin{align*} \\dot { \\theta } _ 1 & = ( \\ell _ 1 + 1 ) \\Hat { C } _ 1 \\theta _ 1 \\ , , & \\dot { s } & = \\tfrac { \\ell _ 1 + 1 } { N } s \\ ; \\Re \\ , \\bigl ( C _ 1 e ^ { - ( \\ell _ 1 + 1 ) \\Bar { \\theta } _ 1 } \\bigr ) = - \\tfrac { ( \\ell _ 1 + 1 ) \\Hat { C } _ 1 } { N } s . \\end{align*}"}
-{"id": "7126.png", "formula": "\\begin{align*} \\partial _ t s = ~ \\sqrt { ( 1 - s ^ 2 - | \\bar { \\nabla } s | ^ 2 ) ( 1 - s ^ 2 ) } F _ { * } ^ { \\alpha } ( \\mathcal { W } _ X ^ { - 1 } ) \\end{align*}"}
-{"id": "6123.png", "formula": "\\begin{align*} \\mathcal { A } & = \\cup _ { \\psi \\in \\mathfrak { B } _ { s ^ { \\alpha } } } \\Phi _ { \\psi } \\left ( \\mathfrak { M } \\right ) , \\\\ \\mathcal { B } & = \\Phi _ { s ^ { \\alpha } } \\left ( \\mathfrak { M } \\right ) . \\end{align*}"}
-{"id": "4907.png", "formula": "\\begin{align*} T = \\lambda ( R + L ) + \\mu I . \\end{align*}"}
-{"id": "5858.png", "formula": "\\begin{align*} A = \\left [ \\begin{array} { c c } A _ { 1 1 } & A _ { 1 2 } \\\\ A _ { 2 1 } & A _ { 2 2 } \\end{array} \\right ] , \\end{align*}"}
-{"id": "2888.png", "formula": "\\begin{align*} R _ { - 2 m } & = ( - 1 ) ^ m \\frac { \\zeta ( 2 m + 1 ) y ^ { 2 m } } { 2 ^ { 2 m + 1 } \\pi ^ { 2 m } } , R _ 0 = - \\frac { 1 } { 2 } \\zeta ( 2 m + 1 ) , R _ 1 = ( - 1 ) ^ { m } \\frac { ( 2 \\pi ) ^ { 2 m + 2 } B _ { 2 m + 2 } } { 2 ( 2 m + 2 ) ! y } , \\\\ \\sum _ { i = 0 } ^ m R _ { - ( 2 i + 1 ) } & = ( - 1 ) ^ { m + 1 } \\sum _ { i = 0 } ^ { m } \\frac { ( - 1 ) ^ i B _ { 2 i + 2 } B _ { 2 m - 2 i } ( 2 \\pi ) ^ { 2 m - 2 i } y ^ { 2 i + 1 } } { 2 ( 2 i + 2 ) ! ( 2 m - 2 i ) ! } . \\end{align*}"}
-{"id": "1476.png", "formula": "\\begin{align*} ( D _ r \\langle x _ 3 ^ 2 y _ r , x _ 3 z _ r \\rangle ) ^ { * , - u + 1 } = \\{ 0 \\} . \\end{align*}"}
-{"id": "4307.png", "formula": "\\begin{align*} M _ + ( z ) + M _ + ^ t ( - z ) = 0 , \\end{align*}"}
-{"id": "6347.png", "formula": "\\begin{align*} \\det D \\varphi _ { 2 n - 1 } ^ \\mathbb { R } = \\prod \\limits _ { j = 1 } ^ { n - 1 } \\beta _ j ^ { 2 n - 2 j - 1 } . \\end{align*}"}
-{"id": "8507.png", "formula": "\\begin{align*} y ^ 2 + a _ 1 x y + a _ 3 y = x ^ 3 + a _ 2 x ^ 2 + a _ 4 x + a _ 6 . \\end{align*}"}
-{"id": "5925.png", "formula": "\\begin{align*} \\sum _ { s \\in A - A _ { } } L ( s ) Q ^ N ( s ) \\le C e ^ { - \\min ( c _ 0 , c _ 1 ) N } \\big ( \\sum _ { s \\in A _ { } : r _ s = 2 } L ( s ) Q ^ N ( s ) + 1 \\big ) . \\end{align*}"}
-{"id": "9616.png", "formula": "\\begin{align*} { X } _ { \\beta } ^ { \\rm { r } } [ k ] = \\begin{cases} { X } _ { \\beta } [ k ] , & k = 0 , 4 , 2 8 , 5 6 , N - 4 , N - 2 8 , N - 5 6 \\\\ 0 , & . \\end{cases} \\end{align*}"}
-{"id": "2370.png", "formula": "\\begin{align*} 1 = \\det ( \\chi ' _ 4 ( p _ 1 ) ) = \\det ( \\varrho ( p _ 1 ) ) = \\tau ^ { 5 } \\det ( \\chi ' _ 4 ( p _ 1 ) ) = \\tau ^ 5 , \\end{align*}"}
-{"id": "7966.png", "formula": "\\begin{align*} | \\Delta | = \\max _ { 1 \\le i \\le 3 } a _ i , \\end{align*}"}
-{"id": "9326.png", "formula": "\\begin{align*} J _ { n + 3 } ^ { ( 3 ) } = J _ { n + 2 } ^ { ( 3 ) } + J _ { n + 1 } ^ { ( 3 ) } + 2 J _ { n } ^ { ( 3 ) } , \\ J _ { 0 } ^ { ( 3 ) } = 0 , \\ J _ { 1 } ^ { ( 3 ) } = J _ { 2 } ^ { ( 3 ) } = 1 , \\ n \\geq 0 , \\end{align*}"}
-{"id": "3169.png", "formula": "\\begin{align*} \\lim _ { u \\to - \\infty } \\frac { \\partial } { \\partial u } \\left ( e ^ { z \\psi ( s , u ) } - 1 \\right ) = \\exp \\left \\lbrace \\frac { - 2 b z } { \\sigma ^ { 2 } \\left ( e ^ { b s } - 1 \\right ) } \\right \\rbrace \\lim _ { u \\to - \\infty } \\frac { z e ^ { - b s } } { \\left ( 1 - \\frac { \\sigma ^ { 2 } u } { 2 b } \\left ( 1 - e ^ { - b s } \\right ) \\right ) ^ { 2 } } = 0 . \\end{align*}"}
-{"id": "5410.png", "formula": "\\begin{align*} ( x ^ 3 + y ^ 3 ) z + A x ^ 2 y ^ 2 + 2 B x y z ^ 2 + C z ^ 4 - t ^ 2 = 0 . \\end{align*}"}
-{"id": "9882.png", "formula": "\\begin{align*} \\omega _ 0 ( g , h ) - \\psi ( g ) - \\psi ( h ) + \\psi ( g h ) = \\tilde { \\omega } _ 0 ( g , h ) ( g , h \\in C _ G ( x ) ) . \\end{align*}"}
-{"id": "1903.png", "formula": "\\begin{align*} \\tilde { f } _ { n } ( v , w ) = ( a _ { n } ( v , w ) + F _ { n } ( v ) , b _ { n } ( v , w ) + F _ { n } ( w ) ) , \\end{align*}"}
-{"id": "8050.png", "formula": "\\begin{align*} \\int _ D \\ , K \\ , d M & = \\lim _ { s \\to \\infty } ( \\alpha _ + ( s ) + \\alpha _ { - } ( s ) + \\angle _ p ( D ) - \\pi ) \\\\ & = \\angle _ p ( D ) = 2 ( \\pi - t _ { * } ) \\le \\int _ M K \\ , d M , \\end{align*}"}
-{"id": "9501.png", "formula": "\\begin{align*} \\tilde { Q } ^ { ( 1 ) } ( z , t ) \\cdot A ( z , t ) = \\dfrac { 1 } { 2 } \\left ( z - t - A ( z , t ) \\right ) - t \\ , z \\ , Q ^ { ( 1 ) } _ 1 ( t ) , \\end{align*}"}
-{"id": "3462.png", "formula": "\\begin{align*} R = R ( \\omega ) = \\frac { q } { 2 } + \\sqrt { \\frac { q ^ 2 } { 4 } + k \\frac { M ^ 2 } { 4 \\mu } } \\in ( 0 , \\infty ) . \\end{align*}"}
-{"id": "6844.png", "formula": "\\begin{align*} \\omega ( z ) = \\beta _ x \\omega _ x ( x ) + \\beta _ y \\omega _ y ( y ) \\mbox { a n d } z _ \\omega = [ x _ { \\omega _ x } ; y _ { \\omega _ y } ] , \\end{align*}"}
-{"id": "8419.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { t , l , v } ) = \\sum _ { \\mu \\in \\mathfrak { X } _ l } c _ { t , l } ( \\mu ) \\mu ( v ) . \\end{align*}"}
-{"id": "7641.png", "formula": "\\begin{align*} \\omega _ { 1 , 3 } & = ( j k x _ 1 ) ( \\bar { k } x _ 3 ) + ( j x _ 2 ) ( x _ 3 ) + ( k x _ 3 ) ( \\overline { j k } x _ 1 ) + ( x _ 3 ) ( \\bar { j } x _ 2 ) \\\\ & = j k \\bar { k } x _ 1 x _ 3 + j x _ 2 x _ 3 + k \\overline { j k } x _ 3 x _ 1 + \\bar { j } x _ 3 x _ 2 \\\\ & = j x _ 1 x _ 3 + j x _ 2 x _ 3 - j x _ 1 x _ 3 - j x _ 2 x _ 3 \\ \\ \\ \\ \\ \\ { \\rm c o m m u t i n g \\ v a r i a b l e s } \\\\ & = j ( x _ 1 x _ 3 - x _ 1 x _ 3 ) + j ( x _ 2 x _ 3 - x _ 2 x _ 3 ) \\\\ & = 0 . \\end{align*}"}
-{"id": "5851.png", "formula": "\\begin{align*} \\frac 1 N \\sum _ { k = 0 } ^ N & ( - 1 ) ^ k \\binom { N } { k } g ^ k D ^ N ( g ^ { N + 1 - k } f ) \\\\ & - \\frac 1 N \\sum _ { k = 0 } ^ N ( - 1 ) ^ k \\binom { N } { k } g ^ { k + 1 } D ^ N ( g ^ { N - k } f ) = 0 . \\end{align*}"}
-{"id": "1797.png", "formula": "\\begin{align*} H ( a , b ) = \\ln ( M ^ { 1 / y } ) = a ( 1 - b ) \\ln ( a ) + ( a - 1 ) \\ln ( b ) . \\end{align*}"}
-{"id": "5739.png", "formula": "\\begin{align*} ( x ) _ n = \\begin{cases} x ( x + 1 ) \\dotsc ( x + n - 1 ) & \\\\ 1 & \\\\ \\frac { 1 } { ( x - 1 ) ( x - 2 ) \\dotsc ( x + n ) } & \\end{cases} \\end{align*}"}
-{"id": "1979.png", "formula": "\\begin{align*} \\tilde { f } _ { p } ( v , w ) & = ( ( \\tilde { f _ { p } } ) _ { 1 } ( v , w ) , ( \\tilde { f _ { p } } ) _ { 2 } ( v , w ) ) = ( \\tilde { a } _ { p } ( v , w ) + \\tilde { F } _ { p } ( v ) , \\tilde { b } _ { p } ( v , w ) + \\tilde { F } _ { p } ( w ) ) , \\end{align*}"}
-{"id": "6462.png", "formula": "\\begin{align*} p _ { } \\left ( x | \\theta \\right ) = { \\displaystyle \\prod \\limits _ { j = 1 } ^ { l } } p \\left ( x _ { 2 j - 1 } x _ { 2 j } | \\mu _ { 2 j - 1 } \\sigma _ { 2 j - 1 } \\right ) \\end{align*}"}
-{"id": "5987.png", "formula": "\\begin{align*} f _ 1 ( x ) & = e ^ x \\sin ( 2 \\pi x ) + \\frac { 1 } { x ^ 2 + 1 } , x \\in [ - 1 , 1 ] , \\\\ f _ 2 ( x ) & = \\sin \\left ( \\frac { x } { 2 } \\right ) - 2 \\cos ( x ) + 4 \\sin ( \\pi x ) , x \\in [ - 4 , 4 ] . \\end{align*}"}
-{"id": "566.png", "formula": "\\begin{align*} \\begin{array} { l } \\beta _ 1 = - a i + b ( a + 1 - m - \\sum _ { j = 2 } ^ m d _ j ) \\leq \\alpha _ 1 \\\\ \\beta _ j = i + b d _ j \\leq \\alpha _ j \\mbox { f o r } j = 2 , \\ldots , m . \\end{array} \\end{align*}"}
-{"id": "8690.png", "formula": "\\begin{align*} c _ n ^ 2 = \\frac { n ^ { \\gamma - 1 } } { \\Gamma ( \\gamma ) } L ( n ) \\end{align*}"}
-{"id": "1409.png", "formula": "\\begin{align*} H ^ { * } ( B G _ 1 ; \\mathbb { Z } / 2 ) = \\mathbb { Z } / 2 [ x _ 2 , x _ 3 , x _ 4 ] \\end{align*}"}
-{"id": "9697.png", "formula": "\\begin{align*} & V _ m = \\tilde { \\Phi } ( \\alpha _ 5 , \\alpha _ 3 , \\alpha _ 2 ; V _ a ) , Z _ m = Z _ a + \\alpha _ 4 , \\\\ & U _ { k - 1 } = \\Phi _ 1 ( \\beta _ 1 ; U _ m ) , ( u _ { k - 1 } , v _ { k - 1 } ) \\cdot \\textbf { n } _ { k - 1 } = 0 . \\end{align*}"}
-{"id": "9804.png", "formula": "\\begin{align*} C ( t ) = \\begin{cases} - t \\beta _ 1 + \\alpha w ( t ) + \\beta _ 2 \\mathbb { E } \\left ( \\left [ \\sum \\limits _ { i = 1 } ^ { N ( t ) } X _ i + t \\right ] ^ + \\right ) , & t < 0 , \\\\ ( \\alpha + \\beta _ 2 ) w ( t ) + \\beta _ 2 t , & t \\geq 0 . \\end{cases} \\end{align*}"}
-{"id": "8178.png", "formula": "\\begin{align*} a ( v ) \\omega = \\omega \\left ( \\begin{matrix} 1 & 0 \\\\ 0 & v \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "9091.png", "formula": "\\begin{align*} d ^ { n + 1 } ( x y ) = \\sum _ { i = 0 } ^ { n + 1 } \\binom { n + 1 } { i } d ^ { i } ( x ) d ^ { n + 1 - i } ( y ) \\left ( x , y \\in R \\right ) , \\end{align*}"}
-{"id": "6367.png", "formula": "\\begin{align*} \\lim _ { y \\to 0 + } \\sqrt { ( x + i y - \\alpha ) ^ 2 - 4 \\beta } = \\begin{cases} - \\sqrt { ( x - \\alpha ) ^ 2 - 4 \\beta } & , x < \\alpha - 2 \\sqrt { \\beta } \\\\ i \\sqrt { 4 \\beta - ( x - \\alpha ) ^ 2 } & , x \\in [ \\alpha - 2 \\sqrt { \\beta } , \\alpha + 2 \\sqrt { \\beta } ] \\\\ \\sqrt { ( x - \\alpha ) ^ 2 - 4 \\beta } & , x > \\alpha + 2 \\sqrt { \\beta } \\end{cases} . \\end{align*}"}
-{"id": "5372.png", "formula": "\\begin{align*} L ( \\ell , r \\omega ) \\otimes L ( 1 , t \\omega ) \\cong \\bigoplus _ { \\substack { s = 0 \\\\ s + t + r \\ } } ^ { a + b - 2 } L ( \\ell + 1 , s \\omega ) \\otimes M _ { r + 1 , s + 1 } \\end{align*}"}
-{"id": "3525.png", "formula": "\\begin{align*} \\begin{array} [ c ] { r l } & \\hat { J } ( u ^ { \\varepsilon } ( \\cdot ) ) - \\hat { J } ( \\bar { u } ( \\cdot ) ) \\\\ = & \\displaystyle \\sum \\limits _ { i = 1 } ^ { n } \\psi _ { x _ i } ( \\bar { X } ( t _ 1 ) , \\bar { X } ( t _ 2 ) , \\cdots , \\bar { X } ( t _ n ) ) y ( t _ { i } ) \\\\ & + E { \\displaystyle \\int \\limits _ { 0 } ^ { T } } \\{ f _ x ( \\bar { X } { ( t ) } , \\bar { u } ( t ) ) y ( t ) + f ( \\bar { X } { ( t ) } , u ^ { \\varepsilon } ( t ) ) - f ( \\bar { X } { ( t ) } , \\bar { u } ( t ) ) \\} d t + o ( \\varepsilon ) . \\end{array} \\end{align*}"}
-{"id": "2943.png", "formula": "\\begin{align*} \\varrho ( H _ n ) = \\max _ { S \\subseteq V ( H ) } \\frac { | E _ H ( S ) | } { | S | } , \\end{align*}"}
-{"id": "1232.png", "formula": "\\begin{align*} \\mathcal { H } ^ { n - 1 } ( \\mathbf { g } ^ { - 1 } ( K ) ) = \\mu ( K ) \\mbox { w h e n e v e r $ K \\subset \\mathbb { S } ^ { n - 1 } $ i s a B o r e l s e t . } \\end{align*}"}
-{"id": "8079.png", "formula": "\\begin{align*} x _ j ^ \\ell m = \\left ( \\prod _ { i = 1 } ^ k ( x _ j y _ i ) ^ { p _ i } \\right ) x _ j ^ { \\ell - q _ k } . \\end{align*}"}
-{"id": "4783.png", "formula": "\\begin{align*} { \\varepsilon } = F ( x ^ 0 , X , Y ) \\ , \\Delta \\sqrt { - \\det g ^ { i j } } / { \\prod _ { i \\neq 0 } { L } _ i } , \\end{align*}"}
-{"id": "4269.png", "formula": "\\begin{align*} \\beta = \\left ( \\begin{array} { c } \\beta _ 0 \\\\ \\beta _ 1 \\\\ \\cdots \\\\ \\beta _ { m + 1 } \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "9828.png", "formula": "\\begin{align*} W ^ { \\sigma _ q } ( x , y ) = - W ( x , y ) \\end{align*}"}
-{"id": "1930.png", "formula": "\\begin{align*} D _ { v } [ D _ { q } ( f _ { i - 1 } ) - _ { p } ^ { - 1 } \\circ g _ { i - 1 } \\circ _ { q } ] & = D _ { v } [ D _ { 0 _ { q } } ( _ { p } ^ { - 1 } \\circ f _ { i - 1 } \\circ _ { q } ) - _ { p } ^ { - 1 } \\circ g _ { i - 1 } \\circ _ { q } ] \\\\ & = D _ { 0 _ { q } } ( _ { p } ^ { - 1 } \\circ f _ { i - 1 } \\circ _ { q } ) - D _ { v } ( _ { p } ^ { - 1 } \\circ g _ { i - 1 } \\circ _ { q } ) . \\end{align*}"}
-{"id": "6748.png", "formula": "\\begin{align*} Y _ { n } = \\begin{cases} X _ { n } , & \\delta _ { n } = 0 \\\\ X ' _ { n } , & \\delta _ { n } = 1 \\\\ \\end{cases} , & & n = 1 , \\dots , N . \\end{align*}"}
-{"id": "5477.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { U ( r ^ n z ) - U ( r ^ n z \\varepsilon ) } { ( r ^ n z ) ^ \\rho \\ell ( r ^ n z ) } = \\int _ \\varepsilon ^ 1 t ^ { \\rho - 1 } p _ 0 ( t z ) \\ , \\dd t . \\end{align*}"}
-{"id": "10069.png", "formula": "\\begin{align*} N = \\epsilon N \\oplus \\overline { \\epsilon } N , \\end{align*}"}
-{"id": "4509.png", "formula": "\\begin{align*} \\begin{bmatrix} \\phi ( \\Delta _ n ^ * \\Delta _ n ) - \\phi ( \\Delta _ n ) ^ * \\phi ( \\Delta _ n ) & \\phi ( \\Delta _ n ^ * a ^ * ) - \\phi ( \\Delta _ n ) ^ * \\phi ( a ) ^ * \\\\ \\phi ( a \\Delta _ n ) - \\phi ( a ) \\phi ( \\Delta _ n ) & \\phi ( a a ^ * ) - \\phi ( a ) \\phi ( a ) ^ * \\end{bmatrix} \\geq 0 . \\end{align*}"}
-{"id": "6122.png", "formula": "\\begin{align*} L f \\left ( x , z \\right ) = \\sum _ { y \\sim x } \\tau _ { z } ^ { - 1 } p _ { \\tau _ { z } } \\left ( y \\right ) \\left ( f \\left ( y , z + 1 \\right ) - f \\left ( x , z \\right ) \\right ) . \\end{align*}"}
-{"id": "1592.png", "formula": "\\begin{align*} V ( \\mathbf { j } ' ) = V ( \\mathbf { j } '' ) = V ( \\mathbf { j } ''' ) = \\frac { 2 ( m - 1 ) } { 3 } - s \\left ( \\frac { 2 ( m - 1 ) } { 3 } \\right ) - 1 . \\end{align*}"}
-{"id": "2036.png", "formula": "\\begin{align*} f _ i ( z _ 1 , . . . , z _ k ) = z _ 1 ^ { i - 1 } . . . z _ k ^ { i - 1 } f ( z _ 1 , . . . , z _ k ) \\end{align*}"}
-{"id": "5491.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { u ( r ^ n z ) } { ( r ^ n z ) ^ { \\rho - 1 } \\ell ( r ^ { n } z ) } = p _ 0 ( z ) z \\in C _ { p _ 0 } , p _ 0 \\in \\mathcal { P } _ { r } , \\end{align*}"}
-{"id": "8216.png", "formula": "\\begin{align*} \\alpha _ 0 = \\Biggl ( \\prod _ { \\ell \\in \\mathcal { L } ^ + } \\bigl ( ( 1 + \\hat { H _ { i + \\ell } } ) ( 1 + K _ { i + \\ell } ) \\bigr ) ^ { \\binom { k } { \\ell } } - \\prod _ { \\ell \\in \\mathcal { L } ^ - } \\bigl ( ( 1 + \\hat { H _ { i + \\ell } } ) ( 1 + K _ { i + \\ell } ) \\bigr ) ^ { \\binom { k } { \\ell } } \\Biggr ) \\end{align*}"}
-{"id": "6320.png", "formula": "\\begin{align*} \\dot { \\theta } _ m & = ( \\ell _ m + 1 ) \\Hat { C } _ m \\Bar { r } _ m ^ { - ( \\ell _ m + 1 ) } \\theta _ m \\ , , & \\dot { r } _ m & = \\tfrac { ( \\ell _ m + 1 ) } { r ^ { \\ell _ m + 1 } } \\Hat { C } _ m ( r _ m - \\Bar { r } _ m ) . \\end{align*}"}
-{"id": "7251.png", "formula": "\\begin{align*} \\begin{cases} 1 , & x \\le 2 0 , \\\\ ( \\log x ) ^ { - 1 } ( \\log \\log x ) ^ { - C } , & x > 2 0 . \\end{cases} \\end{align*}"}
-{"id": "614.png", "formula": "\\begin{align*} \\dim L _ W ^ n ( s ) = \\sum _ { i = 0 } ^ n ( - 1 ) ^ i \\binom { n } { i } c _ { 2 ( n - i ) , s } . \\end{align*}"}
-{"id": "5161.png", "formula": "\\begin{align*} \\mathcal { C } = \\{ ( x , y ) \\in [ 0 , a ] \\times [ b , 1 ] \\colon x \\le y \\} . \\end{align*}"}
-{"id": "7964.png", "formula": "\\begin{align*} \\sum _ { J \\in \\mathcal S } \\ , \\lambda _ { J } ^ s = 1 . \\end{align*}"}
-{"id": "2317.png", "formula": "\\begin{align*} \\| I _ 3 \\| _ { L ^ 2 } \\leq C \\int _ { t - h } ^ { t } ( t - \\tau ) ^ { - 1 } \\frac { ( t - \\tau ) ^ \\beta } { \\tau ^ { \\beta + 1 / 2 } } R _ 1 R _ 2 \\leq C R _ 1 R _ 2 h ^ { \\beta } t ^ { - ( \\beta + 1 / 2 ) } , \\end{align*}"}
-{"id": "904.png", "formula": "\\begin{align*} \\mathfrak { s } _ n ( t ) = \\sqrt { \\frac { 2 } { n ^ 2 } \\sum _ { i = 1 } ^ n K _ h ( t _ { i - 1 } - t ) ^ 2 } \\end{align*}"}
-{"id": "4597.png", "formula": "\\begin{align*} [ \\Lambda _ { k , \\pi } , \\Lambda _ { k , \\pi } ] & = \\{ ( 0 , r , s _ 1 , t _ 1 , \\ldots , s _ n , t _ n ) \\in \\Lambda _ { k , \\pi } \\ , : r \\in \\Z ; \\ ; \\\\ & s _ j = t _ j = 0 \\ , ( j = 1 , \\ldots , p ) ; \\ , s _ j , t _ j \\in 2 \\Z \\ , ( j = p + 1 , \\ldots , n ) \\} , \\end{align*}"}
-{"id": "8209.png", "formula": "\\begin{align*} \\overline { m } _ i = \\frac { v ! } { ( v - 2 i ) ! \\ , i ! \\ , 2 ^ i } \\end{align*}"}
-{"id": "8590.png", "formula": "\\begin{align*} I \\cap J = 0 . \\end{align*}"}
-{"id": "336.png", "formula": "\\begin{align*} \\chi ( x ) : = \\prod _ { j \\in J } m _ j ( x ) ^ { \\tau _ j } - \\prod _ { i \\in I } l _ i ( x ) ^ { \\sigma _ i } = 0 , x \\in ( r _ 0 , \\ , r _ 1 ) , \\end{align*}"}
-{"id": "8568.png", "formula": "\\begin{align*} & \\| z \\| _ { L ^ { 2 \\kappa } ( [ \\Omega \\ , \\cap \\ , B _ { H _ 0 } ( x , \\rho ) ] \\times ( \\frac { t } { 4 } , t ) ) } = \\| z \\| _ { L ^ { 2 \\kappa } ( Q _ 0 ) } \\\\ & \\le C [ \\rho ^ { - 2 } + t ^ { - 1 } ] ^ { 1 / 2 \\theta } \\| z \\| _ { L ^ 1 ( Q _ \\infty ) } = C [ \\rho ^ { - 2 } + t ^ { - 1 } ] ^ { 1 / 2 \\theta } \\| z \\| _ { L ^ 1 ( B _ { H _ 0 } ( x , 2 \\rho ) \\times ( \\frac { t } { 8 } , t ) ) } \\end{align*}"}
-{"id": "8764.png", "formula": "\\begin{align*} x [ f _ 1 , f _ 2 ] = x [ f _ 1 , f _ 2 ] ^ { ( t _ 2 , s _ 1 ) } = - x [ f _ 2 , f _ 1 ] ^ { ( t _ 2 , s _ 1 ) } = - x [ f _ 2 , f _ 1 ] \\end{align*}"}
-{"id": "6716.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\mathbb { P } ( R _ { N } > t ( N ) ) = 1 . \\end{align*}"}
-{"id": "4453.png", "formula": "\\begin{align*} ( F , v , w ) \\mapsto \\Phi ( F , v , w ) : = & - { ( - \\partial _ 1 ^ 2 - | \\partial _ 1 | ^ { - 1 } \\partial _ 2 ^ 2 ) } ^ { - 1 } { P } \\Big ( F + v \\partial _ 2 R w + w \\partial _ 2 R v + w \\partial _ 2 R w \\\\ & + \\partial _ 2 \\frac { 1 } { 2 } R ( w + v ) ^ 2 - ( w + v ) \\partial _ 1 \\frac { 1 } { 2 } R ( w + v ) ^ 2 ) \\Big ) ; \\end{align*}"}
-{"id": "6217.png", "formula": "\\begin{align*} \\P ( \\widehat { \\tau } _ 0 < \\infty , \\widehat { X } ( \\widehat { \\tau } _ 0 - ) \\in d y , \\widehat { X } ( \\widehat { \\tau } _ 0 ) \\in d x , \\Delta C ( \\widehat { \\tau } _ 0 ) = \\Delta C _ i ( \\widehat { \\tau } _ 0 ) ) = \\frac { 1 } { c } \\Lambda _ i ( - y + d x ) d y ~ , \\end{align*}"}
-{"id": "8992.png", "formula": "\\begin{gather*} { \\cal D } ^ { ( n ) } _ { q , t } ( d ) = D ^ { ( n ) } _ q ( - d \\pm ( t _ 0 - c ) ; t ) { \\cal D } ^ { ( n ) } _ { q , t } ( d + q / 2 ) \\prod _ { 1 \\le i \\le n } \\vartheta ( z _ i \\pm ( t _ 0 - c ) ) ^ { - 1 } , \\end{gather*}"}
-{"id": "3627.png", "formula": "\\begin{align*} c ( x ) : = d ^ p \\frac { f ( u ( d x + x _ 0 ) ) } { u ^ { p - 1 } ( d x + x _ 0 ) } . \\end{align*}"}
-{"id": "2707.png", "formula": "\\begin{align*} d _ { u } ( e _ { w u ^ { k + 1 } } ) & = e _ { w u ^ { k + 2 } } + e _ { w u ^ { k } } - 2 e _ { w u ^ { k + 1 } } \\\\ & = e _ { w u ^ { k + 2 } } + ( \\epsilon _ { k - 1 } + a _ { k - 1 } e _ { w } ) - 2 ( \\epsilon _ { k } + a _ { k } e _ { w } ) \\\\ & = e _ { w u ^ { k + 2 } } - ( 2 a _ { k } - a _ { k - 1 } ) e _ { w } + \\epsilon _ { k - 1 } - 2 \\epsilon _ { k } \\\\ & = \\epsilon _ { k + 1 } + \\epsilon _ { k - 1 } - 2 \\epsilon _ { k } \\end{align*}"}
-{"id": "2749.png", "formula": "\\begin{align*} \\frac { 1 } { \\epsilon ^ 2 } \\ , \\sum _ { k = 1 } ^ { K } \\ , k ^ { 2 q } \\ , \\langle \\mathcal L ( \\sum _ { i = 1 } ^ { K } \\widetilde S _ { k i } h _ i ) , \\ , h _ k \\rangle _ { L ^ 2 _ { x , v } } \\ , . \\end{align*}"}
-{"id": "7407.png", "formula": "\\begin{align*} \\mathbb { H } _ { \\Gamma } : = \\frac { \\mathbb { C } [ t _ 0 , t _ 1 ] } { ( t _ 0 + t _ 1 ) } \\cong \\mathbb { C } [ t ] . \\end{align*}"}
-{"id": "442.png", "formula": "\\begin{align*} { \\tilde I _ \\nu ( s ) = \\frac { e ^ s } { s ^ \\nu \\sqrt { 2 \\pi s } } \\left [ 1 + O \\left ( \\frac { 1 } { s } \\right ) \\right ] \\mbox { f o r $ s \\to + \\infty $ . } } \\end{align*}"}
-{"id": "7569.png", "formula": "\\begin{gather*} z ( z - 1 ) Y _ { z z } + \\big ( \\alpha z ( z - 1 ) + ( \\beta + 1 ) ( z - 1 ) + ( \\gamma + 1 ) z \\big ) Y _ z \\\\ \\qquad { } + \\frac 1 2 \\big ( \\alpha ( \\beta + 1 ) ( z - 1 ) + \\alpha ( \\gamma + 1 ) z + 2 \\delta z + ( \\beta + 1 ) ( \\gamma + 1 ) + 2 \\eta - 1 \\big ) Y = 0 , \\\\ Y ( 0 ) = 1 , Y _ z ( 0 ) = \\frac 1 2 \\left ( \\frac { 2 \\eta - 1 } { \\beta + 1 } + \\gamma + 1 - \\alpha \\right ) . \\end{gather*}"}
-{"id": "7211.png", "formula": "\\begin{align*} v _ r ( y ) : = \\frac { v ( x + r y ) } { r } \\rightarrow v _ 0 ( y ) . \\end{align*}"}
-{"id": "591.png", "formula": "\\begin{align*} Z _ { \\Gamma } ( s ) : = \\prod _ { \\overline { \\gamma } \\in \\overline { \\Gamma } _ { p } } \\prod _ { k = 0 } ^ { \\infty } \\left ( 1 - e ^ { - ( s + k ) \\ell ( \\gamma ) } \\right ) , \\end{align*}"}
-{"id": "6672.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\frac { 1 } { \\delta ^ { \\frac { 2 } { n + 1 } } } \\cdot \\frac { \\| x \\| _ 2 - \\| x _ { \\delta } \\| _ 2 } { \\| x \\| _ 2 } = \\frac { ( n + 1 ) ^ { \\frac { 2 } { n + 1 } } } { 2 } \\left ( \\frac { | K | _ n } { | B _ 2 ^ { n - 1 } | _ { n - 1 } } \\right ) ^ { \\frac { 2 } { n + 1 } } \\frac { \\kappa ( x ) ^ { \\frac { 1 } { n + 1 } } } { \\langle x , u ( x ) \\rangle } = G ( x ) , \\end{align*}"}
-{"id": "6602.png", "formula": "\\begin{align*} & \\int _ { \\varphi _ n ( \\mathsf { U } _ n ) } \\left | w _ { i j } ^ { ( m , n ) } ( x ) \\right | ^ p \\ , d \\mu ( x ) = \\int _ { \\varphi _ n ( \\mathsf { U } _ n ) } \\Bigl | \\sum _ { 1 \\leq k \\leq d } u _ { i k } ^ { ( n ) } ( \\phi _ { m , n } ( x ) ) v _ { k j } ^ { ( m , n ) } ( x ) \\ , \\Bigr | ^ p \\ , d \\mu ( x ) \\\\ & = \\int _ { \\varphi _ n ( \\mathsf { U } _ n ) } \\Bigl | \\sum _ { 1 \\leq k \\leq d } u _ { i k } ^ { ( n ) } ( y ) v _ { k j } ^ { ( m , n ) } ( \\phi _ { m , n } ^ { - 1 } ( y ) ) \\ , \\Bigr | ^ p J _ { \\phi _ { m , n } ^ { - 1 } } ( y ) \\ , d \\mu ( y ) \\ , \\end{align*}"}
-{"id": "2945.png", "formula": "\\begin{align*} \\varrho ( H ) = \\max _ { S \\subseteq V ( H ) , S \\neq \\emptyset } \\frac { | E _ H ( S ) | } { | S | } , \\end{align*}"}
-{"id": "1045.png", "formula": "\\begin{align*} L _ { m , m + 1 } ^ { \\ast } ( q ) \\geq - \\sum _ { j = 1 } ^ { m } L _ { m , j } ^ { \\ast } ( q ) - C ( k ) \\geq - m L _ { m , m } ^ { \\ast } ( q ) - C ( m ) . \\end{align*}"}
-{"id": "7647.png", "formula": "\\begin{align*} \\Omega = \\Big \\{ E ' _ j - E '' _ j , E ' _ j + E '' _ j , F ' _ j - F '' _ j , F ' _ j + F '' _ j ; \\ \\ 1 \\le j \\le 2 ^ { n - 2 } \\Big \\} \\end{align*}"}
-{"id": "352.png", "formula": "\\begin{align*} - R i c ( \\nabla _ { \\nu } e _ 2 , e _ 2 ) + R i c ( \\nabla _ { \\nu } e _ 1 , e _ 1 ) & = h _ { 2 2 } R _ { 2 2 } - \\frac { ( e _ 2 ( R ) ) ^ 2 } { 4 | \\nabla f | ^ 3 } - h _ { 1 1 } R _ { 1 1 } + \\frac { ( e _ 1 ( R ) ) ^ 2 } { 4 | \\nabla f | ^ 3 } . \\end{align*}"}
-{"id": "663.png", "formula": "\\begin{align*} U _ 1 ( 0 ) e _ i = U _ 1 ( 1 ) e _ { \\sigma ( i ) } , \\forall i \\in \\mathbb { N } U _ 2 ( 0 ) e _ i = U _ 2 ( 1 ) e _ { \\sigma ( i ) } , \\forall i \\in \\mathbb { N } \\end{align*}"}
-{"id": "2757.png", "formula": "\\begin{align*} \\alpha _ 1 ( x ' ) , \\alpha _ 2 ( y ' ) , \\alpha _ 3 ( z ) & \\Leftrightarrow f _ 2 ( x ^ { - 1 } ) , f _ 2 ( y ^ { - 1 } \\xi ^ { - 1 } ) , f _ 1 ( z ^ { - 1 } \\xi ^ { 3 k - 1 } ) \\\\ & \\Leftrightarrow x ^ { - 1 } y ^ { - 1 } \\xi ^ { - 1 } = z ^ { - 1 } \\xi ^ { 3 k - 1 } \\\\ & \\Leftrightarrow x y \\xi ^ { 3 k } = z \\\\ & \\Leftrightarrow x ' * y ' = z . \\end{align*}"}
-{"id": "282.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } | | | Q ( t ) | | | = 0 , \\end{align*}"}
-{"id": "8893.png", "formula": "\\begin{align*} \\psi _ n ( z ) = ( 1 - \\theta ( z ) ) ( \\chi ^ n , f _ z ) _ { L ^ 2 ( \\mu ) } . \\end{align*}"}
-{"id": "4997.png", "formula": "\\begin{align*} C _ { \\alpha } ( T ) = \\begin{cases} C 2 ^ { ( 1 - \\alpha ) T } & \\textrm { i f } \\alpha < 1 , \\\\ C T & \\textrm { i f } \\alpha = 1 , \\\\ C & \\textrm { i f } \\alpha > 1 . \\end{cases} \\end{align*}"}
-{"id": "2983.png", "formula": "\\begin{align*} \\int _ \\Omega \\nabla u _ n . \\nabla T _ 1 ( u _ n ) T _ 1 ^ { \\gamma - 1 } ( u _ n ) & = \\int _ \\Omega h _ n \\left ( u _ n + \\frac { 1 } { n } \\right ) f _ n T _ 1 ^ { \\gamma } ( u _ n ) + \\int _ \\Omega T _ 1 ^ { \\gamma } ( u _ n ) \\mu _ n \\\\ & \\leq C . \\end{align*}"}
-{"id": "1831.png", "formula": "\\begin{align*} \\sum _ { i \\in I } \\sum _ { j \\in J _ i } \\left \\langle \\Gamma _ { i j } \\Lambda _ i T f , \\Gamma _ { i j } \\Lambda _ i U f \\right \\rangle & = \\sum _ { i \\in I } \\sum _ { j \\in J _ i } \\left \\langle \\Gamma ^ * _ { i j } \\Gamma _ { i j } \\Lambda _ i T f , \\Lambda _ i U f \\right \\rangle \\\\ & \\le \\sum _ { i \\in I } D _ i \\left \\langle \\Lambda _ i T f , \\Lambda _ i U f \\right \\rangle \\le D B \\| f \\| ^ 2 . \\end{align*}"}
-{"id": "4964.png", "formula": "\\begin{align*} X = Y + \\nabla u . \\end{align*}"}
-{"id": "747.png", "formula": "\\begin{align*} m _ { \\mu , p } ( \\theta a ) \\leq I _ { \\mu , p } ( v _ n ) & = \\frac { \\theta } { 2 } \\int _ { \\R ^ 2 } | \\nabla w _ n | ^ 2 + ( x _ 1 ^ 2 + x _ 2 ^ 2 ) | w _ n | ^ 2 \\ , d x - \\theta ^ { \\frac p 2 } \\frac { \\mu } { p } \\int _ { \\R ^ 3 } | w _ n | ^ p \\ , d x \\\\ & = \\theta I _ { \\mu , p } ( u _ n ) + \\frac { \\mu } { p } \\left ( \\theta - \\theta ^ { \\frac p 2 } \\right ) \\int _ { \\R ^ 3 } | u _ n | ^ p \\ , d x . \\end{align*}"}
-{"id": "3465.png", "formula": "\\begin{align*} U ^ n _ h = \\sum _ { i = 1 } ^ { d } \\alpha _ { h , i } ^ { n } \\psi _ i , n \\in \\{ 0 , \\dots , N \\} , \\end{align*}"}
-{"id": "5377.png", "formula": "\\begin{align*} ( u , v \\cdot w ) = ( u \\cdot v , w ) \\end{align*}"}
-{"id": "7957.png", "formula": "\\begin{align*} \\sum _ { | J | = m } \\lambda ^ s _ J = 1 , \\end{align*}"}
-{"id": "3587.png", "formula": "\\begin{align*} ( \\tau ) > ( K ) = \\max ( | D | , | \\mathcal { C } | ) \\geq \\aleph _ 0 . \\end{align*}"}
-{"id": "4008.png", "formula": "\\begin{align*} q = \\exp _ { A } ( u - K _ p ( u ) + \\log _ { A } p ) , u \\in L ^ { 1 } ( \\mu ) \\ , \\ q \\in \\mathcal P \\ , \\end{align*}"}
-{"id": "4888.png", "formula": "\\begin{align*} Z _ { c a t } ( \\mu ( M ) , t ) & = \\prod _ { n = 1 } ^ { \\infty } \\left ( \\frac { 1 } { 1 - t ^ n } \\right ) ^ { \\mu ( M ) } = \\prod _ { n = 1 } ^ { \\infty } \\mu \\left ( \\frac { 1 } { 1 - t ^ n } \\right ) ^ { \\mu ( M ) } \\\\ & = \\prod _ { n = 1 } ^ { \\infty } \\mu \\left ( \\left ( \\frac { 1 } { 1 - t ^ n } \\right ) ^ { [ M ] } \\right ) = \\prod _ { n = 1 } ^ { \\infty } \\mu ( Z _ { m o t } ( M , t ^ n ) ) . \\end{align*}"}
-{"id": "6652.png", "formula": "\\begin{align*} \\d ( C _ 1 , C _ 2 ) = \\sup _ { u \\in \\mathbb { S } ^ { n - 1 } } \\max \\left [ \\frac { r _ { C _ 1 } ( u ) } { r _ { C _ 2 } ( u ) } , \\frac { r _ { C _ 2 } ( u ) } { r _ { C _ 1 } ( u ) } \\right ] = \\inf \\left \\{ a \\geq 1 : \\frac { 1 } { a } C _ 1 \\subseteq C _ 2 \\subseteq a C _ 1 \\right \\} \\quad . \\end{align*}"}
-{"id": "5468.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\widehat U _ { r ^ n z } ( s ) = s ^ { - \\rho } q ( s / z ) . \\end{align*}"}
-{"id": "3141.png", "formula": "\\begin{align*} X _ { t } = e ^ { - b t } \\left ( X _ { 0 } + a \\int _ { 0 } ^ { t } e ^ { b s } \\mathrm { d } s + \\sigma \\int _ { 0 } ^ { t } e ^ { b s } \\sqrt { X _ { s } } \\mathrm { d } B _ { s } + \\int _ { 0 } ^ { t } e ^ { b s } \\mathrm { d } J _ { s } \\right ) , t \\geqslant 0 . \\end{align*}"}
-{"id": "4227.png", "formula": "\\begin{align*} s \\circ r = e , r \\circ s = 1 _ I . \\end{align*}"}
-{"id": "1702.png", "formula": "\\begin{align*} T _ \\lambda ^ * T _ \\lambda ( f ) & = T _ \\lambda ^ * ( f _ \\lambda \\cdot ( f \\circ \\tau ^ { d ( \\lambda ) } ) ) = \\frac { \\chi _ { D _ \\lambda } \\cdot ( ( f _ \\lambda \\cdot ( f \\circ \\tau ^ { d ( \\lambda ) } ) ) \\circ \\tau _ \\lambda ) } { f _ \\lambda \\circ \\tau _ \\lambda } \\\\ & = \\chi _ { D _ \\lambda } f = T _ v ( f ) , \\end{align*}"}
-{"id": "1146.png", "formula": "\\begin{align*} e ( x z ) & = \\rho _ x \\left ( e ( z ) \\right ) \\\\ & = x e ( z ) + [ 1 ] _ x e ^ 2 ( z ) + e ^ 4 ( z ) \\end{align*}"}
-{"id": "2044.png", "formula": "\\begin{align*} \\begin{array} { c c } f _ { n + 1 , k } ( z _ 1 , . . . , z _ n , \\eta ) = ( - 1 ) ^ { i - 1 } z _ 1 ^ i . . . z _ n ^ i \\eta ^ i q ^ { - \\frac { i ( i - 1 ) } { 2 } } f _ { n , k } ( z _ 1 , . . . , z _ n ) - \\\\ \\\\ - z _ 1 ^ { k i } f _ { n , k } ( \\eta z _ 1 , z _ 2 , . . . , z _ n ) - . . . - z _ n ^ { k i } f _ { n , k } ( z _ 1 , . . . z _ { n - 1 } , \\eta z _ n ) \\end{array} \\end{align*}"}
-{"id": "1414.png", "formula": "\\begin{align*} \\pi ' ( \\Delta ( \\eta ) ) , \\ ; \\pi ' ( \\Delta ( \\zeta ) ) , \\ ; \\pi ' ( \\Gamma _ i ( \\xi ) ) , \\mbox { ( $ i = 1 , \\dots , n $ ) } , \\end{align*}"}
-{"id": "1597.png", "formula": "\\begin{align*} V ( \\mathbf { j } ' ) & = 3 j _ { n - 2 } ' - s ( j _ { n - 2 } ' ) + 2 j _ { n - 1 } ' - s ( j _ { n - 1 } ' ) - c ( j _ n , - \\alpha _ { \\mathbf { j } ' } ( n ) - 1 ) \\\\ & = 3 ( 1 ) - s ( 1 ) + 2 ( 1 6 q + 5 ) - s ( 1 6 q + 5 ) - c ( 1 , 8 q + 2 ) \\\\ & = 3 2 q + 1 2 - s ( q ) - s ( 5 ) \\\\ & = 3 2 q + 1 0 - s ( q ) . \\end{align*}"}
-{"id": "5803.png", "formula": "\\begin{align*} \\Vert w _ { j } \\Vert _ { \\dot { W } ^ { 1 , p } _ { 0 } ( \\R ^ n ) } ^ { p } = \\Vert \\omega _ { j - 1 } \\Vert _ { W ^ { - 1 , p ' } ( \\R ^ n ) } ^ { p ' } \\leq \\int _ { \\R ^ n } w _ { j } w _ { j - 1 } ^ { q } \\ ; d \\sigma + \\Vert \\mu \\Vert _ { W ^ { - 1 , p ' } ( \\R ^ n ) } ^ { p ' } . \\end{align*}"}
-{"id": "708.png", "formula": "\\begin{align*} e = \\frac 3 2 a ^ 2 ( 1 + \\alpha ) T + a ^ 2 T _ { \\rm i } \\alpha , H = \\frac 5 2 a ^ 2 ( 1 + \\alpha ) T + a ^ 2 T _ { \\rm i } \\alpha , \\end{align*}"}
-{"id": "3344.png", "formula": "\\begin{align*} \\frac { \\binom { K - 2 } { s - 1 } } { \\binom { K - 2 } { s - 1 } + \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K - 1 } { s + i } ( N - 1 ) ^ i N } & \\geq \\frac { \\binom { K - 1 } { s - 1 } } { \\binom { K - 1 } { s - 1 } + \\sum _ { i = 0 } ^ { K - s } \\binom { K } { s + i } ( N - 1 ) ^ i N } . \\end{align*}"}
-{"id": "9638.png", "formula": "\\begin{align*} \\Psi ( \\tau _ { i - 1 } ) = \\left ( \\psi ( 1 , \\tau _ { i - 1 } ) , \\cdots , \\psi ( N ( \\tau _ { i - 1 } ) , \\tau _ { i - 1 } ) \\right ) , ~ i = 1 , \\cdots , N _ \\tau . \\end{align*}"}
-{"id": "4975.png", "formula": "\\begin{align*} T ( x ) : = \\min \\left ( 1 , \\| x \\| ^ { - ( d + 1 ) } \\right ) \\end{align*}"}
-{"id": "3240.png", "formula": "\\begin{align*} H _ a ^ { 2 p - 1 } ( X ; \\mathbb C ) = T J _ a ^ p \\oplus \\overline { T J _ a ^ p } . \\end{align*}"}
-{"id": "6037.png", "formula": "\\begin{align*} Q _ { X _ { i } Y _ { i } U _ { i } V _ { i } } ( x _ { i } , y _ { i } , u _ { i } , v _ { i } ) & = P ^ { ( \\alpha , \\lambda ) } ( x _ { i } , y _ { i } , u _ { i } , v _ { i } ) . \\end{align*}"}
-{"id": "8327.png", "formula": "\\begin{align*} c _ 0 ( m , \\lambda ) = \\sum _ { \\substack { \\mu \\in h V _ { \\Z } ^ \\vee / h V _ { \\Z } \\\\ \\mu \\sim \\lambda } } c ( m , h ^ { - 1 } \\mu ) . \\end{align*}"}
-{"id": "1413.png", "formula": "\\begin{align*} \\pi ' \\colon { S L _ 2 } ^ { n + 1 } \\to { S L _ 2 } ^ { n + 1 } / \\mu _ 2 = G _ n \\end{align*}"}
-{"id": "4202.png", "formula": "\\begin{gather*} ( T _ 1 , T _ 2 , T _ 3 ) \\rightarrow ( T , - 4 \\sum _ { 1 \\leq i < j \\leq 3 } ( T - T _ i ) ( T - T _ j ) , 4 ( T - T _ 1 ) ( T - T _ 2 ) ( T - T _ 3 ) ) , \\\\ \\textnormal { w h e r e } T = \\frac { T _ 1 + T _ 2 + T _ 3 } { 3 } \\end{gather*}"}
-{"id": "2605.png", "formula": "\\begin{align*} ( \\zeta , \\eta ) _ V = \\int _ \\R ( T \\zeta ) ( T \\eta ) , \\end{align*}"}
-{"id": "1987.png", "formula": "\\begin{align*} \\sup _ { \\lambda / n \\leq s \\leq 1 - \\lambda / n } \\frac { n ^ { \\nu } \\left \\vert \\alpha _ { n } \\left ( s \\right ) - B _ { n } \\left ( s \\right ) \\right \\vert } { \\left [ s \\left ( 1 - s \\right ) \\right ] ^ { 1 / 2 - \\nu } } = O _ { \\mathbb { P } } \\left ( 1 \\right ) , \\end{align*}"}
-{"id": "5551.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t } \\bar { w } _ { m } \\left ( t \\right ) = P ^ { \\left ( m \\right ) } \\bar { w } _ { m } \\left ( t \\right ) \\end{align*}"}
-{"id": "5593.png", "formula": "\\begin{align*} A _ { n } = \\int _ { 0 } ^ { \\tau } d t _ { 1 } \\int _ { 0 } ^ { t _ { 1 } } d t _ { 2 } \\ldots \\int _ { 0 } ^ { t _ { \\bar { n } - 1 } } \\left ( \\tau - t _ { 1 } + t _ { n } \\right ) \\left ( t _ { 1 } - t _ { 2 } \\right ) \\ldots \\left ( t _ { n - 1 } - t _ { n } \\right ) q \\left ( t _ { 1 } \\right ) q \\left ( t _ { 2 } \\right ) \\ldots q \\left ( t _ { n } \\right ) d t _ { n } \\end{align*}"}
-{"id": "4356.png", "formula": "\\begin{align*} \\| f \\| _ { \\infty , \\alpha } : = \\underset { z \\in \\mathbb { C } ^ n } { \\operatorname { e s s s u p } } | f ( z ) | e ^ { - \\frac { \\alpha } { 2 } | z | ^ 2 } . \\end{align*}"}
-{"id": "7321.png", "formula": "\\begin{align*} s ( x ) = \\prod \\left \\{ s _ j ^ { \\rho _ j ( x ) \\vee 0 } \\mid j \\in \\{ 1 , \\dots , q \\} \\right \\} \\end{align*}"}
-{"id": "5769.png", "formula": "\\begin{align*} - \\Delta u = \\sigma u ^ { q } + \\mu \\ ; \\ ; \\Omega , \\end{align*}"}
-{"id": "7028.png", "formula": "\\begin{align*} g ( \\epsilon ) = ( \\frac { 1 } { 4 \\epsilon } + \\frac { n - 2 } { 2 ( n - 1 ) } \\epsilon ) ^ { - 1 / 2 } , \\end{align*}"}
-{"id": "3107.png", "formula": "\\begin{align*} \\Theta ^ { \\mu } = \\frac { 2 ( 1 - 2 p ) } { ( 1 - 2 p ) ( W + 1 ) + p W ( 1 - ( 2 p ) ^ m ) } , \\end{align*}"}
-{"id": "5765.png", "formula": "\\begin{align*} ( 2 \\epsilon _ 1 ) ( 2 b _ 1 ) ( 2 b _ 3 ) ^ { p - 2 } = ( 2 \\epsilon _ 2 ) ( 2 b _ 2 ) ( 2 b _ 3 ) ^ { p - 2 } = 2 ^ p . \\end{align*}"}
-{"id": "6358.png", "formula": "\\begin{align*} \\frac { \\partial m _ { 2 k } } { \\partial \\beta _ k } = \\beta _ 1 \\cdots \\beta _ { k - 1 } , \\end{align*}"}
-{"id": "8566.png", "formula": "\\begin{align*} \\partial _ t h \\ge \\ell T _ 1 ^ { \\ell - 1 } H _ 0 ( x ) ^ 2 , H ( \\nabla h ) ^ 2 = 4 H _ 0 ( x ) ^ 2 ( 1 + t ^ \\ell ) ^ 2 \\le 1 6 H _ 0 ( x ) ^ 2 , \\end{align*}"}
-{"id": "2886.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\sigma _ { - ( 2 m + 1 ) } ( n ) e ^ { - n y } = \\frac { 1 } { 2 \\pi i } \\int _ { \\lambda - i \\infty } ^ { \\lambda + i \\infty } \\Gamma ( s ) \\zeta ( s ) \\zeta ( s + 2 m + 1 ) y ^ { - s } \\ , { \\rm d } s . \\end{align*}"}
-{"id": "3574.png", "formula": "\\begin{align*} n ( p ) \\leq x = O ( ( \\log p ) ( \\log \\log p ) ) , \\end{align*}"}
-{"id": "9745.png", "formula": "\\begin{align*} U _ h | _ { I \\cap ( \\Omega _ { h , k - 1 } ^ { ( i ) } \\cup \\Omega _ { h , k } ^ { ( i ) } ) } \\in O _ { \\epsilon } ( { U } _ i ^ { ( 0 ) } ) , i = 1 , 2 ; | \\sigma ^ { I } _ { j } - \\sigma _ { j 0 } | < \\hat { \\epsilon } , j = 2 , 3 ; | \\gamma _ { 4 } ^ { I } | < \\hat { \\epsilon } , \\end{align*}"}
-{"id": "5140.png", "formula": "\\begin{align*} \\zeta _ 3 = z _ 3 \\in X ^ { 2 s - } _ , \\zeta _ 5 = z _ 5 \\in X ^ { \\frac 5 2 s - } _ , \\zeta _ 7 \\in X ^ { \\frac { 1 1 } { 4 } s - } _ . \\end{align*}"}
-{"id": "7574.png", "formula": "\\begin{align*} \\Phi _ j ( \\xi ) = \\Psi _ j ( \\xi ) - \\Psi _ { j - 1 } ( \\xi ) , \\quad j \\in \\Bbb Z \\ , . \\end{align*}"}
-{"id": "8983.png", "formula": "\\begin{gather*} D ^ { ( n ) } _ { q , t } ( d ) = \\sum _ { \\sigma \\in \\{ \\pm 1 \\} ^ n } \\prod _ { 1 \\le i \\le n } \\frac { 1 } { \\vartheta ( 2 \\sigma _ i z _ i ; q ) _ d } \\prod _ { 1 \\le i < j \\le n } \\frac { \\vartheta ( t + \\sigma _ i z _ i + \\sigma _ j z _ j ; q ) _ d } { \\vartheta ( \\sigma _ i z _ i + \\sigma _ i z _ j ; q ) _ d } \\prod _ { 1 \\le i \\le n } T _ i ^ { - d / 2 } . \\end{gather*}"}
-{"id": "5834.png", "formula": "\\begin{align*} A _ 1 ( x ) : & \\ ! = - \\frac 1 2 Q ( x ) - x Q ' ( x ) , A _ 2 ( x ) : = - \\frac 3 2 Q ' ( x ) - x Q '' ( x ) , \\\\ & A _ 3 ( x ) : = - \\frac 5 2 Q '' ( x ) - x Q ''' ( x ) . \\end{align*}"}
-{"id": "6461.png", "formula": "\\begin{align*} \\sigma _ { 2 j } = \\sigma _ { 2 j - 1 } \\mu _ { 2 j } = \\mu _ { 2 j } \\left ( \\mu _ { 2 j - 1 } \\sigma _ { 2 j - 1 } \\right ) j = 1 , . . . , l , \\end{align*}"}
-{"id": "2987.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( u - \\bar { u } ) ^ + & \\leq \\int _ { \\Omega } \\chi _ { \\{ u \\leq \\bar { u } \\} } \\left ( \\bar { g } ( x , u ) - f h ( \\bar { u } ) \\right ) ( s i g n _ + ( u - \\bar { u } ) ) \\varphi \\\\ & = 0 . \\end{align*}"}
-{"id": "10075.png", "formula": "\\begin{align*} [ x , \\overline { x } ] \\cdot \\psi _ 0 ( a , \\overline { a } ) = \\psi ( x a , \\overline { x a } ) \\end{align*}"}
-{"id": "6971.png", "formula": "\\begin{align*} M = M ( B ) = D f ( B ) = \\frac { 1 } { n } ( \\det { B } ) ^ { \\frac { 1 } { n } - 1 } b _ { i j } ^ * . \\end{align*}"}
-{"id": "6382.png", "formula": "\\begin{align*} \\frac { d m _ i } { d z _ r } ( z _ 1 ^ * , z _ 1 ^ * , \\ldots ) = ( z _ 1 ^ * ) ^ { i - 1 } \\frac { d m _ i } { d z _ r } ( 1 , 1 , \\ldots ) = ( z _ 1 ^ * ) ^ { i - 1 } \\left ( \\binom { 2 i } { i - r } - \\binom { 2 i } { i - r - 1 } \\right ) . \\end{align*}"}
-{"id": "5823.png", "formula": "\\begin{align*} u ( x ) = \\sum _ { n \\neq 0 } \\frac { g _ n ( \\o ) } { 2 \\pi n } e ^ { 2 \\pi i n x } , \\end{align*}"}
-{"id": "1437.png", "formula": "\\begin{align*} f _ { k , k } + \\partial _ { j _ 1 } f _ { k , k + 2 ^ { j _ 1 - 1 } - 1 } = 0 . \\end{align*}"}
-{"id": "9401.png", "formula": "\\begin{align*} S _ { 2 } ( \\mathcal { M } _ { \\Omega _ 0 } f ) ( x ) & = \\Big ( \\sum _ { j \\in \\mathbb { Z } } | V _ { 2 , j } ( \\mathcal { M } _ { \\Omega _ 0 } f ) ( x ) | ^ 2 \\Big ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "5212.png", "formula": "\\begin{align*} u _ t = u _ { z z } + c u _ z + Q S f ( u ) , \\end{align*}"}
-{"id": "3027.png", "formula": "\\begin{align*} W _ B \\Sigma W _ B ^ * & = \\frac { 1 } { 4 } \\begin{bmatrix} I + L & I - L \\end{bmatrix} \\Sigma \\begin{bmatrix} I + L & I - L \\end{bmatrix} ^ * = \\frac { 1 } { 4 } \\begin{bmatrix} I - L & I + L \\end{bmatrix} \\begin{bmatrix} I + L ^ * & I - L ^ * \\end{bmatrix} \\\\ & = \\frac { 1 } { 4 } ( ( I - L ) ( I + L ^ * ) + ( I + L ) ( I - L ^ * ) ) = \\frac { 1 } { 4 } ( 2 I - 2 L L ^ * ) = 0 . \\end{align*}"}
-{"id": "2184.png", "formula": "\\begin{align*} & \\lim _ { t \\to \\infty } \\ , t ^ { \\frac { N + A } { 2 } } u _ m \\left ( t ^ { \\frac { 1 } { 2 } } y , t \\right ) = 0 \\quad \\mbox { i n } L ^ 2 ( { \\bf R } ^ N , e ^ { | y | ^ 2 / 4 } \\ , d y ) \\ , \\cap \\ , L ^ \\infty ( K ) , \\\\ & \\lim _ { t \\to \\infty } \\ , t ^ { \\frac { N + 2 A } { 2 } } \\frac { u _ m ( x , t ) } { U ( | x | ) } = 0 \\qquad \\ , \\mbox { i n } L ^ \\infty ( B ( 0 , R ) ) , \\end{align*}"}
-{"id": "8486.png", "formula": "\\begin{align*} \\abs { W _ { \\pi } ( g _ { t , a _ i , v } ) } = q ^ { - \\frac { t + n } { 2 } } t \\geq - a _ 1 i = 1 , 2 . \\end{align*}"}
-{"id": "7784.png", "formula": "\\begin{align*} \\Phi ( \\xi _ h ) = \\frac 1 2 \\Vert \\varrho _ h ( \\xi _ h ) \\Vert _ Y ^ 2 . \\end{align*}"}
-{"id": "6849.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { T } \\theta _ t f _ t ( x _ t ) - \\inf _ { x \\in X } \\sum _ { t = 1 } ^ { T } \\theta _ t f _ t ( x ) \\leq r ( T ) , \\lim _ { T \\to \\infty } r ( T ) = 0 . \\end{align*}"}
-{"id": "8070.png", "formula": "\\begin{align*} - u '' ( x ) + c u ( x ) = f ( x ) , \\ \\ x \\in \\Omega ^ D , u ( 0 ) = u ( 1 ) = 0 . \\end{align*}"}
-{"id": "6252.png", "formula": "\\begin{align*} \\begin{gathered} \\sigma = v ( \\omega _ 0 , \\mu _ 0 , \\chi _ 0 ) , \\omega = \\omega _ 0 + u ( \\omega _ 0 , \\mu _ 0 , \\chi _ 0 ) , \\\\ \\mu = \\mu _ 0 + w ( \\omega _ 0 , \\mu _ 0 , \\chi _ 0 ) , \\chi = \\chi _ 0 + W ( \\omega _ 0 , \\mu _ 0 , \\chi _ 0 ) \\end{gathered} \\end{align*}"}
-{"id": "8452.png", "formula": "\\begin{align*} S _ { \\chi } ( 1 , u , m ) = \\begin{cases} 2 \\zeta _ F ( 1 ) q ^ { - \\frac { m } { 2 } } \\Re ( \\alpha ( u ) ) & u \\in \\mathcal { O } ^ { \\times 2 } , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "9233.png", "formula": "\\begin{align*} \\varphi ( a b ) & = \\varphi ( \\frac { a \\circ b } { 2 } + \\frac { [ a , b ] } { 2 } ) = \\varphi ( \\frac { a \\circ b } { 2 } ) + \\varphi ( \\frac { [ a , b ] } { 2 } ) = ( \\frac { a \\circ b } { 4 } ) ^ { + } + ( \\frac { a \\circ b } { 4 } ) ^ { - } + ( \\frac { [ a , b ] } { 4 } ) ^ { + } + ( \\frac { [ a , b ] } { 4 } ) ^ { - } \\\\ & = \\frac { 1 } { 4 } ( a ^ { + } a ^ { + } + a ^ { + } a ^ { - } + a ^ { - } a ^ { + } + a ^ { - } a ^ { - } ) = ( \\frac { a ^ { + } + a ^ { - } } { 2 } ) ( \\frac { a ^ { + } + a ^ { - } } { 2 } ) = \\varphi ( a ) \\varphi ( b ) , \\end{align*}"}
-{"id": "1972.png", "formula": "\\begin{align*} \\Vert D _ { p } ( \\textbf { \\textit { f } } _ { i } ^ { n } ) ( v _ { s } ) \\Vert \\leq c \\lambda ^ { n } \\Vert v _ { s } \\Vert \\quad s = 1 , \\dots , k + 1 . \\end{align*}"}
-{"id": "3519.png", "formula": "\\begin{align*} 0 \\leq X ^ u ( t _ i ) , \\ \\ i = 0 , 1 , \\cdots , n , \\end{align*}"}
-{"id": "6417.png", "formula": "\\begin{align*} \\xi ^ { 2 } \\chi ^ { 2 } - \\left ( 1 - \\chi \\right ) ^ { 2 } = 0 \\end{align*}"}
-{"id": "1035.png", "formula": "\\begin{align*} \\sigma _ { 2 } = 2 . 6 1 8 0 \\ldots , \\sigma _ { 4 } = 6 , \\sigma _ { 6 } = 9 . 4 0 5 1 \\ldots , \\sigma _ { 8 } = 1 2 . 8 1 5 0 \\ldots . \\end{align*}"}
-{"id": "553.png", "formula": "\\begin{align*} \\frac { d } { d t } K ( \\phi ^ t ( z ) ) \\Big | _ { t = 0 } & = d K ( z ) [ X _ { H _ 0 } ( z ) ] = - \\omega _ 0 ( X _ K ( z ) , X _ { H _ 0 } ( z ) ) = \\omega _ 0 ( X _ { H _ 0 } ( z ) , X _ K ( z ) ) \\\\ & = - d H _ 0 ( z ) [ X _ K ( z ) ] . \\end{align*}"}
-{"id": "8494.png", "formula": "\\begin{align*} \\Delta = ( 1 + v b _ 1 - v b _ 2 \\varpi ^ { a _ 1 - a _ 2 } ) ^ 2 - 4 v b _ 1 = ( 1 - v b ) ^ 2 - 4 v b _ 2 \\varpi ^ { a _ 1 - a _ 2 } . \\end{align*}"}
-{"id": "2410.png", "formula": "\\begin{align*} a _ 2 = a _ 6 = 3 a _ 3 + a _ 4 = 0 , \\ 4 a _ 3 ^ 2 - 4 a _ 1 a _ 3 a _ 5 \\geq a _ 5 ^ 2 > 0 , \\ a _ 3 - a _ 1 a _ 5 \\neq 0 . \\end{align*}"}
-{"id": "1244.png", "formula": "\\begin{align*} \\mbox { o s c } ( \\rho ) : = M ( \\rho ) - m ( \\rho ) \\mbox { f o r } 0 < \\rho < t . \\end{align*}"}
-{"id": "3309.png", "formula": "\\begin{align*} H \\left ( W _ k | Z , \\mathbb { H } , Q _ 1 ^ { [ k ] } , \\dots , Q _ N ^ { [ k ] } , A _ 1 ^ { [ k ] } , \\dots , A _ N ^ { [ k ] } \\right ) = o ( L ) , \\end{align*}"}
-{"id": "2479.png", "formula": "\\begin{align*} \\begin{aligned} T _ 2 & \\lesssim \\alpha \\int _ { \\R ^ { 3 } } \\left ( \\int _ { \\R ^ { 3 } } e ^ { - c | \\xi | ^ 2 } | u ( t , x , \\xi ) | ^ 2 d \\xi \\right ) ^ { 2 } \\ , \\textbf { 1 } _ { H _ { 0 } \\ , \\cup H _ { - } } \\ , d x \\\\ & \\lesssim \\alpha \\int _ { H _ { 0 } \\ , \\cup H _ { - } } e ^ { - c | \\xi | ^ 2 } | u | ^ 2 d \\xi d x . \\end{aligned} \\end{align*}"}
-{"id": "3980.png", "formula": "\\begin{align*} p ^ { \\alpha _ { m + 1 } } ( m + 1 , t ) & = ( - 1 ) ^ { m + 1 } \\sum _ { k = m + 1 } ^ { \\infty } ( - \\lambda ) ^ k \\underset { \\Theta ^ { k } _ { m + 1 } } { \\sum } \\frac { t ^ { \\sum _ { j = 0 } ^ { m + 1 } k _ j \\alpha _ j } } { \\Gamma \\left ( \\sum _ { j = 0 } ^ { m + 1 } k _ j \\alpha _ j + 1 \\right ) } , \\end{align*}"}
-{"id": "9993.png", "formula": "\\begin{align*} \\Big ( ( \\mathbb E _ 0 \\otimes \\mathbb E _ 0 ) \\Big [ ( I _ n ( T , W _ T ) I _ n ( T , W _ T ^ { \\prime } ) ) ^ 8 \\Big ] \\Big ) ^ { 1 / 8 } = \\Big ( \\mathbb E _ 0 \\Big [ I _ n ( T , W _ T ) ^ 8 \\Big ] \\Big ) ^ { 1 / 4 } . \\end{align*}"}
-{"id": "3402.png", "formula": "\\begin{align*} z _ { 1 } ( \\zeta ) = z _ 0 ( 0 ) - \\int _ 0 ^ \\zeta x _ { 1 } ( t ) \\dot y _ { 1 } ( t ) \\ , d t . \\end{align*}"}
-{"id": "2001.png", "formula": "\\begin{align*} G ( z ) G ( t z ) . . . G ( t ^ k z ) = 0 \\end{align*}"}
-{"id": "5691.png", "formula": "\\begin{align*} \\mathfrak { L } _ { \\mathcal { K } } ( \\gamma ) = \\int _ 0 ^ 1 \\mathcal { K } ( \\gamma ( t ) ) | \\dot { \\gamma } | ( t ) \\d t = \\int _ 0 ^ 1 \\mathcal { K } ( \\gamma ( t , \\cdot ) ) \\| \\partial _ t \\gamma ( t , \\cdot ) \\| _ { L ^ 2 ( \\R ) } \\d t . \\end{align*}"}
-{"id": "3438.png", "formula": "\\begin{align*} u _ n \\leq a \\exp \\Big ( \\sum _ { j = 1 } ^ { n - 1 } b _ j \\Big ) , n \\in \\{ 1 , \\dots , N \\} , \\end{align*}"}
-{"id": "5986.png", "formula": "\\begin{align*} E _ { \\mathrm { t e n s o r } } = \\begin{pmatrix} \\epsilon ^ 2 & 0 \\\\ 0 & \\epsilon ^ 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "2746.png", "formula": "\\begin{align*} | S _ { m n k } | \\leq ( C _ b + \\xi C _ z ) \\ , | | \\psi _ n | | _ { L ^ { \\infty } } \\ , \\langle | \\psi _ m | , \\ , | \\psi _ k | \\rangle _ { L ^ 2 _ z } \\leq ( C _ b + \\xi C _ z ) \\ , | | \\psi _ n | | _ { L ^ { \\infty } } \\ , | | \\psi _ m | | _ { L ^ 2 _ z } \\ , | | \\psi _ k | | _ { L ^ 2 _ z } = \\widetilde C \\ , n ^ p \\ , , \\end{align*}"}
-{"id": "10009.png", "formula": "\\begin{align*} \\overline { \\omega } _ L ( \\gamma ) \\phi = \\overline { \\omega _ L ( \\gamma ) \\overline { \\phi } } . \\end{align*}"}
-{"id": "1314.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| \\phi _ n ( a ) - \\pi ( a ) \\| = 0 ~ a \\in A . \\end{align*}"}
-{"id": "6565.png", "formula": "\\begin{align*} \\sqrt { 1 - \\varepsilon ^ 2 } y _ 1 + \\varepsilon y _ 2 = 1 \\ \\ \\ \\ \\sqrt { 1 - \\varepsilon ^ 2 } y _ 1 + \\varepsilon y _ 2 = 1 , \\end{align*}"}
-{"id": "7547.png", "formula": "\\begin{gather*} - \\mathrm D _ 0 \\gamma ^ i - \\mathrm D _ j ( w ^ j \\gamma ^ i ) - \\mathrm D _ j \\mathrm D _ j \\gamma ^ i + w ^ j _ { \\delta _ i } \\gamma ^ j = 0 \\end{gather*}"}
-{"id": "3250.png", "formula": "\\begin{align*} ( \\beta _ Y , \\omega _ Y ) _ Y = ( \\alpha _ Y , \\omega _ Y ) _ Y . \\end{align*}"}
-{"id": "4388.png", "formula": "\\begin{align*} \\nu ( A | _ F ) : = \\inf \\{ \\| A f \\| ; f \\in L _ \\alpha ^ p , \\| f \\| = 1 , \\operatorname { s u p p } f \\subseteq F \\} \\end{align*}"}
-{"id": "4533.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| \\pi ( \\Delta _ n ) ^ * ( \\Pi ( a ) - \\pi ( a ) ) \\pi ( \\Delta _ n ) \\| = 0 . \\end{align*}"}
-{"id": "4217.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ { \\mathbb { T } } z ^ { k } \\overline { z } ^ l d z = \\delta _ { k l } . \\end{align*}"}
-{"id": "2532.png", "formula": "\\begin{align*} K _ \\varrho = \\left ( \\varrho \\ , m _ 0 \\right ) ^ { 1 / 2 } K \\left ( \\varrho \\ , m _ 0 \\right ) ^ { - 1 / 2 } . \\end{align*}"}
-{"id": "4518.png", "formula": "\\begin{align*} \\psi = \\frac { 1 } { 2 } ( \\phi _ 1 + \\phi _ 2 ) \\end{align*}"}
-{"id": "816.png", "formula": "\\begin{align*} \\frac { \\partial E _ { t o t a l } } { \\partial t } = - \\mathcal { D } , \\end{align*}"}
-{"id": "6531.png", "formula": "\\begin{align*} e _ n ^ { \\perp } \\cap \\bigcap _ { i = 1 } ^ k \\{ x \\in \\mathbb { R } ^ n : \\langle x , y _ i \\rangle \\leq 1 \\} \\end{align*}"}
-{"id": "8903.png", "formula": "\\begin{align*} \\begin{aligned} \\lim _ j \\| T ^ { - n _ { k _ j } } X J _ { \\theta , 1 } ^ { - 1 } p \\| ^ 2 & = \\lim _ j ( \\| T ^ { - n _ { k _ j } } X J _ { \\theta , 1 } ^ { - 1 } p \\| ^ 2 - \\| p X J _ { \\theta , 1 } ^ { - 1 } \\chi ^ { - n _ { k _ j } } \\| ^ 2 ) + L ( p ) \\\\ & = \\| R ^ { - 1 } X _ 1 P _ { L ^ 2 ( \\mu _ { } ) } p \\| ^ 2 + F ( p ) + L ( p ) . \\end{aligned} \\end{align*}"}
-{"id": "1641.png", "formula": "\\begin{align*} \\tau _ \\eta : = \\tau _ { \\eta _ 1 } \\circ \\tau _ { \\eta _ 2 } \\circ \\cdots \\circ \\tau _ { \\eta _ \\ell } . \\end{align*}"}
-{"id": "8607.png", "formula": "\\begin{align*} & f * \\Psi ( v ) : = \\sum _ { \\gamma } f ( \\gamma ) \\gamma \\Psi ( \\gamma ^ { - 1 } v ) , \\Psi * f ( v ) : = \\sum _ { \\gamma } \\Psi ( v \\gamma ) m ( v \\gamma ) f ( \\gamma ^ { - 1 } ) . \\end{align*}"}
-{"id": "1590.png", "formula": "\\begin{align*} \\alpha _ { \\mathbf { j } '' } ( n ) & = m / 2 - 2 ^ { n - 1 } j _ 1 - 2 ^ { n - 2 } j _ 2 - \\cdots - 2 j _ { n - 1 } \\\\ & = m / 2 - 2 ( m - 1 ) / 3 = ( 4 - m ) / 6 , \\end{align*}"}
-{"id": "1305.png", "formula": "\\begin{align*} [ T _ s \\Delta ( \\mu ) ] = [ \\Delta ( s \\cdot \\mu ) ] \\end{align*}"}
-{"id": "1246.png", "formula": "\\begin{align*} b _ { i j } ( x ) = \\int _ 0 ^ 1 f _ { \\eta _ i \\eta _ j } ( t \\nabla F ( x ) + ( 1 - t ) \\nabla \\mathcal { G } ( x ) ) d t \\mbox { f o r } \\ , \\ , 1 \\leq i , j \\leq n . \\end{align*}"}
-{"id": "6932.png", "formula": "\\begin{align*} \\Psi _ { f * \\psi _ n } ( \\varphi ) ( x ) & = \\int _ { \\R ^ d } \\int _ { \\R ^ d } f ( ( x + y ) - z ) \\psi _ n ( z ) \\ , d z ~ \\varphi ( y ) \\ , d y \\\\ & = \\int _ { \\R ^ d } f ( x + z ) \\int _ { \\R ^ d } \\psi _ n ( y - z ) \\varphi ( y ) \\ , d y \\ , d z = \\Psi _ f ( \\widetilde { \\psi } _ n * \\varphi ) ( x ) , \\end{align*}"}
-{"id": "9996.png", "formula": "\\begin{align*} \\d u _ t = \\frac 1 2 \\Delta u _ t \\d t + \\beta \\ , u _ t \\ , \\d B _ t , \\end{align*}"}
-{"id": "9197.png", "formula": "\\begin{align*} \\mathcal { W } ( L \\downarrow \\mathfrak { g } ) = \\mathcal { W } ( L \\downarrow \\mathfrak { k } ) \\downarrow \\mathfrak { g } \\subseteq B C _ { r } \\cup \\{ 0 \\} \\downarrow \\mathfrak { g } = \\Gamma _ { \\mathfrak { k } } \\downarrow \\mathfrak { g } = \\Gamma _ { \\mathfrak { g } } = \\Theta _ { r } . \\end{align*}"}
-{"id": "2349.png", "formula": "\\begin{align*} M = ( m _ { i j } ) _ { i , j = 1 , \\ldots , d } \\end{align*}"}
-{"id": "3813.png", "formula": "\\begin{align*} \\widehat { N } _ L ( x ) : = \\sum _ { z \\notin [ x - 2 \\ell _ L , x + 2 \\ell _ L ] } \\ ; \\sum _ { i \\le N ( z , 0 ) } \\mathbf { 1 } _ { \\{ \\exists \\ ; s \\in [ 0 , L ^ \\alpha ] \\colon \\ , S ^ { z , i } _ s \\in [ x - \\ell _ L , x + \\ell _ L ] \\} } \\end{align*}"}
-{"id": "624.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } f ( r _ 1 e ^ { i \\theta _ 1 } , z _ 2 ) \\ , d \\theta _ 1 = \\sum _ { \\alpha _ 2 \\in \\Z } a _ { ( 0 , \\alpha _ 2 ) } z _ 2 ^ { \\alpha _ 2 } . \\end{align*}"}
-{"id": "1770.png", "formula": "\\begin{align*} t _ { \\eta } ^ * P ( \\{ \\omega \\} ) t _ { \\eta } = 0 . \\end{align*}"}
-{"id": "9562.png", "formula": "\\begin{align*} \\mu = \\frac { n - 1 } { \\beta } \\in ( 0 , 1 ) \\ , , \\end{align*}"}
-{"id": "9265.png", "formula": "\\begin{align*} { \\sigma } _ \\varepsilon ^ { ( 0 , s ) } ( E _ 1 \\cap E _ 2 ) = { \\sigma } _ \\varepsilon ^ { ( u , s ) } ( E _ 1 ) { \\sigma } _ \\varepsilon ^ { ( 0 , u ) } ( E _ 2 ) . \\end{align*}"}
-{"id": "7621.png", "formula": "\\begin{align*} \\mu \\Big ( E _ f ( Q _ R , c _ 0 \\lambda _ 0 M ^ k ) \\Big ) \\leq \\alpha ^ k \\mu \\big ( E _ f ( Q _ { 2 R } , c _ 0 \\lambda _ 0 ) \\big ) + \\sum _ { i = 0 } ^ { k - 1 } \\alpha ^ { k - i } I _ i \\forall k \\geq 1 , \\end{align*}"}
-{"id": "6980.png", "formula": "\\begin{align*} D f _ k ( B ) A = \\lim _ { \\epsilon \\to 0 } { \\frac { f _ k ( B + \\epsilon A ) - f _ k ( B ) } { \\epsilon } } . \\end{align*}"}
-{"id": "7870.png", "formula": "\\begin{align*} d ( x , w ) \\le d ( x , y ) + d ( y , w ) = d ( w , z ) + d ( y , w ) = d ( y , z ) . \\end{align*}"}
-{"id": "8674.png", "formula": "\\begin{align*} ( P ) _ { 1 1 } & = P _ { 1 1 } \\\\ ( P ^ 2 ) _ { 1 1 } & = P _ { 1 1 } ^ 2 + P _ { 1 2 } P _ { 2 1 } \\\\ ( P ^ 3 ) _ { 1 1 } & = P _ { 1 1 } ^ 3 + 2 P _ { 1 1 } P _ { 1 2 } P _ { 2 1 } + P _ { 1 2 } P _ { 2 1 } P _ { 2 2 } \\\\ & \\cdots \\\\ ( P ^ n ) _ { 1 1 } & = \\cdots . \\end{align*}"}
-{"id": "8085.png", "formula": "\\begin{align*} \\deg ( m ) = \\deg ( \\hat { x } _ F ^ \\ell ) + \\alpha . \\end{align*}"}
-{"id": "2590.png", "formula": "\\begin{align*} \\big ( S _ { t _ 1 } ^ { t _ 2 } f \\big ) ( t ) : = \\int _ { t _ 1 } ^ { t _ 2 } \\frac { f ( s ) } { t - s } \\ , d s t \\in ( t _ 1 , t _ 2 ) . \\end{align*}"}
-{"id": "3619.png", "formula": "\\begin{align*} \\psi _ { \\varepsilon } ( x ) : = g ( T ( \\varphi _ { \\varepsilon } ( x ) ) ) . \\end{align*}"}
-{"id": "2993.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } ( t , x ) = \\Delta _ b u ( t , x ) + u ( t , x ) \\xi ( t , x ) , ( t , x ) \\in [ 0 , T ] \\times F , \\end{align*}"}
-{"id": "3185.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } _ { \\geqslant 0 } } y ^ { \\kappa } m _ { \\alpha ( z , s ) , \\beta ( z , s ) } ( \\mathrm { d } y ) \\geqslant c _ { 3 } \\left ( \\frac { \\alpha ( z , s ) } { \\beta ( z , s ) } \\right ) ^ { \\kappa } = c _ { 3 } z ^ { \\kappa } e ^ { - \\kappa b s } , s \\in [ 0 , t ] , \\ z > 1 . \\end{align*}"}
-{"id": "327.png", "formula": "\\begin{align*} l _ i > k _ i > 0 = k _ 0 , i = 1 , \\dots , p , \\end{align*}"}
-{"id": "8604.png", "formula": "\\begin{align*} \\bigsqcup _ { k = 1 } ^ { K } \\bigsqcup _ { \\ell = 1 } ^ { m _ { k } } \\Gamma z _ { ( k , \\ell ) } \\Gamma = \\bigsqcup _ { k = 1 } ^ { K } \\bigsqcup _ { \\ell = 1 } ^ { m _ { k } } \\bigsqcup _ { n = 1 } ^ { d _ { k } } \\gamma ^ { k } _ { n } g _ { i ( k , \\ell ) } h _ { j ( k , \\ell ) } \\Gamma . \\end{align*}"}
-{"id": "1460.png", "formula": "\\begin{align*} d _ { 2 ^ { r + 2 } - 1 } ( x _ 1 ) = v _ { r + 1 } x _ 1 ^ { 2 ^ { r + 2 } } = v _ { r + 1 } Q _ { r + 1 } ( x _ 1 ) \\end{align*}"}
-{"id": "3478.png", "formula": "\\begin{align*} w ( i P ) = \\begin{cases} i P ^ 2 , & i \\in \\{ 0 , \\dots , 2 ^ K \\} , \\\\ 0 , & i \\in \\{ 0 , \\dots , 2 ^ K \\} , \\end{cases} \\end{align*}"}
-{"id": "3804.png", "formula": "\\begin{align*} G ^ { ( x , t ) } _ { T } : = \\left \\{ X _ s ^ { ( x , t ) } = \\bar { X } _ s ^ { ( x , t ) } \\ \\forall s \\in [ 0 , T ] \\right \\} \\end{align*}"}
-{"id": "8244.png", "formula": "\\begin{align*} \\eta _ { \\chi _ 1 , \\chi _ 2 , i t } ( \\mathfrak { m } ) = \\sum _ { \\mathfrak { a b } = \\mathfrak { m } } \\chi _ 1 ( i t ) ( \\mathfrak { a } ) \\chi _ 2 ( - i t ) ( \\mathfrak { b } ) \\end{align*}"}
-{"id": "3359.png", "formula": "\\begin{align*} \\mu ( p ) = \\bigcap _ { x \\leq p } \\mu ( x ) . \\end{align*}"}
-{"id": "4954.png", "formula": "\\begin{align*} L _ { f } \\left ( t \\right ) : = \\frac { M - t } { M - m } f \\left ( m \\right ) + \\frac { t - m } { M - m } f \\left ( M \\right ) . \\end{align*}"}
-{"id": "3905.png", "formula": "\\begin{align*} E _ 0 ^ \\ell = - \\frac { m e ^ 4 } { 2 \\hbar ^ 2 } \\frac { 1 } { ( \\ell + 1 ) ^ 2 } . \\end{align*}"}
-{"id": "6827.png", "formula": "\\begin{align*} \\int _ { \\mathbb { S } ^ 2 _ { \\lambda } } h + \\sum \\limits _ { j = 1 } ^ 4 \\int _ { \\mathbb { S } ^ 2 _ { \\lambda } } c _ j \\chi _ { R _ 1 , j } \\varphi _ { 0 , j } + \\frac { 4 \\pi c _ 0 } { \\lambda ^ 2 } = 0 . \\end{align*}"}
-{"id": "4474.png", "formula": "\\begin{align*} | \\phi ( x ) | < 1 = \\phi ( \\xi ) \\end{align*}"}
-{"id": "3581.png", "formula": "\\begin{align*} \\sum ( \\sigma _ i - 1 ) + \\sum ( \\sigma ' _ { i ' } - 1 ) + \\sum \\tau _ j + \\sum \\tau _ { j ' } & = ( s F _ k + t E ) \\cdot C _ k + ( s F _ k + t E ) \\cdot E - \\ell ( \\sigma ) - \\ell ( \\sigma ' ) \\\\ & = 2 s - k t - \\ell ( \\sigma ) - \\ell ( \\sigma ' ) \\end{align*}"}
-{"id": "1181.png", "formula": "\\begin{align*} A = | \\nabla v ( y _ 0 ) | , \\ , \\ , B = | \\nabla v ( z _ 0 ) | , \\ , \\ , C = | \\nabla u ( x _ 0 ) | , \\ , \\ , a = | x _ 0 - y _ 0 | , \\ , \\ , b = | x _ 0 - z _ 0 | . \\end{align*}"}
-{"id": "6095.png", "formula": "\\begin{align*} \\int _ { \\epsilon < \\lvert x \\lvert < 1 } \\tfrac { \\partial } { \\partial x _ i } ( h \\eta ) d x = \\int _ { \\lvert x \\lvert = \\epsilon } h ( x ) \\eta ( x ) \\nu _ i ( x ) d S , \\end{align*}"}
-{"id": "7687.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } \\dot { z } _ 1 ( t ) \\\\ \\dot { z } _ 2 ( t ) \\end{array} \\right ] = & V ^ { \\top } K V \\left [ \\begin{array} { c } z _ 1 ( t ) \\\\ z _ 2 ( t ) \\end{array} \\right ] + \\left [ \\begin{array} { c } 0 \\\\ Q \\end{array} \\right ] w ( t ) \\ , , \\end{align*}"}
-{"id": "3362.png", "formula": "\\begin{align*} \\bigl ( \\bigvee T \\bigr ) ( m ) = \\bigvee _ { \\tau \\in T } \\tau ( m ) , \\\\ \\intertext { a n d } \\bigl ( \\bigwedge T \\bigr ) ( m ) = \\bigcap _ { \\tau \\in T } \\tau ( m ) . \\end{align*}"}
-{"id": "1539.png", "formula": "\\begin{gather*} A ^ * - B ^ * \\ = \\ \\left [ a _ 1 - b _ 2 - \\frac { 1 } { 2 } , a _ 2 - b _ 1 + \\frac { 1 } { 2 } \\right ] . \\end{gather*}"}
-{"id": "7428.png", "formula": "\\begin{align*} \\gamma \\beta \\gamma + \\gamma ^ 2 \\beta + \\gamma \\beta ^ 3 = ( R _ 1 ^ 2 - R _ 2 - S _ 2 ) \\gamma \\beta + ( R _ 1 R _ 2 - R _ 1 S _ 2 - R _ 3 ) \\gamma - R _ 3 S _ 2 e _ 4 \\end{align*}"}
-{"id": "7377.png", "formula": "\\begin{align*} f _ { G e n } ( z ) = e ^ { t V _ { G e n } ( z ) } \\quad , V _ { G e n } ( z ) = \\sum _ { m = 1 } ^ { \\frac { N } { 2 } } \\Delta _ m \\Big ( z ^ { m } + z ^ { - m } \\Big ) , \\end{align*}"}
-{"id": "181.png", "formula": "\\begin{align*} \\mathbf { T h } _ { \\vdash } = \\langle \\mathrm { T h } ^ { p } ( \\vdash ) , \\subseteq , + ^ { \\vdash } , \\mathrm { T h } ( 0 ) \\rangle \\end{align*}"}
-{"id": "1088.png", "formula": "\\begin{align*} m _ { c , i } = { 2 \\mu \\sigma _ { c , i } ^ 2 } \\end{align*}"}
-{"id": "884.png", "formula": "\\begin{align*} \\max _ { 1 \\leq i \\leq N _ n } \\sum _ { j = 1 } ^ { N _ n } \\bar { \\gamma } _ { n } ( i , j ) ^ 2 \\leq \\| \\bar { \\Gamma } _ n \\| ^ 2 _ \\mathrm { s p } . \\end{align*}"}
-{"id": "8438.png", "formula": "\\begin{align*} \\sum _ { \\mu \\in \\mathfrak { X } _ l } \\mu ( y _ 1 y _ 2 y _ 3 ^ { - 1 } ) = \\begin{cases} \\sharp \\mathfrak { X } _ l & y _ 1 y _ 2 y _ 3 ^ { - 1 } \\in 1 + \\varpi ^ l \\mathcal { O } , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "1120.png", "formula": "\\begin{align*} & \\sum \\limits _ { \\{ h , f \\} \\in C _ q } x _ { h f } + x _ { i _ 1 j _ 1 } & = \\sum \\limits _ { \\{ h , f \\} \\in C _ q : f < i _ 1 } x _ { h f } + \\sum \\limits _ { \\{ h , i _ 1 \\} \\in C _ q } x _ { h i _ 1 } + \\sum \\limits _ { \\{ h , j _ 1 \\} \\in C _ q } x _ { h j _ 1 } \\end{align*}"}
-{"id": "2239.png", "formula": "\\begin{align*} \\Re = \\{ K = ( k _ 1 , \\ldots , k _ n ) \\colon \\ \\gamma _ i \\ \\ \\| K \\| < \\gamma _ i + 2 , \\ i = 1 , \\ldots , n \\} . \\end{align*}"}
-{"id": "5255.png", "formula": "\\begin{align*} \\begin{aligned} D ( \\lambda ) & = e ^ { 2 c z } \\det \\left [ u _ 1 ( \\lambda , z ) , u _ 2 ( \\lambda , z ) , u _ 3 ( \\lambda , z ) , u _ 4 ( \\lambda , z ) \\right ] \\\\ & = - \\det \\left [ \\begin{array} { c c } \\Omega ( u _ 1 ( \\lambda , z ) , u _ 3 ( \\lambda , z ) ) & \\Omega ( u _ 1 ( \\lambda , z ) , u _ 4 ( \\lambda , z ) ) \\\\ \\Omega ( u _ 2 ( \\lambda , z ) , u _ 3 ( \\lambda , z ) ) & \\Omega ( u _ 2 ( \\lambda , z ) , u _ 4 ( \\lambda , z ) ) \\end{array} \\right ] \\end{aligned} . \\end{align*}"}
-{"id": "7210.png", "formula": "\\begin{align*} \\lim _ { s \\searrow 0 } \\frac { | B _ s ( x ) \\cap \\{ v > 0 \\} | } { s ^ n } = W _ { A C } ( v ; x , 0 + ) \\end{align*}"}
-{"id": "1071.png", "formula": "\\begin{align*} \\lambda ( v ) = \\sum _ { i = 1 } ^ n \\ell ( e _ i ) . \\end{align*}"}
-{"id": "654.png", "formula": "\\begin{align*} \\bar \\nabla _ { \\dot \\gamma } \\bar \\nabla _ { J _ t } ^ k X & = \\sum _ { i = 0 } ^ { k - 1 } \\bar \\nabla _ { J _ t } ^ i ( \\bar R ( J _ t , \\dot \\gamma ) \\bar \\nabla _ { J _ t } ^ { k - 1 - i } X ) \\\\ & = \\sum _ { j + i + l + m = k - 1 } ( \\bar \\nabla _ { J _ t } ^ j \\bar R ) * ( \\bar \\nabla _ { J _ t } ^ i J _ t ) * ( \\bar \\nabla _ { J _ t } ^ l \\dot \\gamma ) * ( \\bar \\nabla _ { J _ t } ^ { m } X ) . \\end{align*}"}
-{"id": "5690.png", "formula": "\\begin{align*} \\mathcal { K } ( { v } ) = 0 \\Longleftrightarrow { v } \\in \\Sigma : = \\{ z ^ - , z ^ + \\} ; \\end{align*}"}
-{"id": "4834.png", "formula": "\\begin{align*} s = 6 ( 2 d + 5 g - 5 ) - \\sum _ I ( 4 m _ p + 4 l _ p - 1 5 ) - \\sum _ J ( 1 0 m _ p + c _ p - 1 5 ) . \\end{align*}"}
-{"id": "2108.png", "formula": "\\begin{align*} \\sum _ a \\big ( | v _ s ^ a | ^ 2 + | \\nabla v ^ a _ s | ^ 2 \\big ) ( p , t ) \\leq & e ^ { C _ 9 '' ( t - \\frac { T _ 3 } { 2 } ) } \\sup _ { q \\in M } \\sum _ a \\bigg ( | v _ s ^ a ( q , \\frac { T _ 3 } { 2 } ) | ^ 2 + | \\nabla v ^ a _ s ( q , \\frac { T _ 3 } { 2 } ) | ^ 2 \\bigg ) \\\\ = & O ( s ) , \\end{align*}"}
-{"id": "1469.png", "formula": "\\begin{align*} v _ 2 \\phi _ r ( x _ I ) x _ 3 = \\cdots = v _ r \\phi _ r ( x _ I ) x _ 3 = 0 \\end{align*}"}
-{"id": "4972.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\kappa } \\| \\partial _ i ( f _ J - F _ J ) \\| _ { \\dot { F } ^ { \\alpha - 1 , p } _ q } \\leq \\delta \\| f _ J \\| _ { \\dot { F } ^ { \\alpha , p } _ q } , \\end{align*}"}
-{"id": "3074.png", "formula": "\\begin{align*} \\nabla _ x F ( t , 0 ) = 0 , \\forall t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "7332.png", "formula": "\\begin{align*} | | f | | _ { V ^ 2 ( D _ T ) } & = \\sup _ { t \\in [ T - 1 , T ] } | | f | | _ { L ^ 2 ( B _ t ) } + | | f | | _ { L ^ 2 ( D _ T ) } ~ . \\end{align*}"}
-{"id": "7449.png", "formula": "\\begin{align*} c _ { 0 0 } & = - B _ { 0 0 } / B _ { 0 1 } , & c _ { 0 1 } & = 1 / B _ { 0 1 } , \\\\ c _ { 1 0 } & = - B _ { 0 0 } ^ 2 / B _ { 0 1 } + B _ { 0 1 } T _ 0 ^ \\gamma , & d _ { 1 0 } & = - B _ { 0 0 } t - B _ { 0 0 } ^ 2 + B _ { 0 0 } ^ 2 / B _ { 0 1 } + B _ { 0 1 } D _ { 1 0 } . \\end{align*}"}
-{"id": "4165.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { k \\in N } e _ { k , g } ^ n & + \\frac { 1 } { n } e ^ { n - 1 } _ { i , g + 1 } + \\frac { 1 } { n } e _ { i , g + 2 } ^ { n - 2 } \\\\ & + \\left ( 1 - \\frac { 1 } { n } \\right ) \\left [ \\frac { 1 } { n ^ 2 } e _ { i , g + 1 } ^ { n - 3 } + \\frac { 1 } { n } e _ { i , g + 2 } ^ { n - 3 } \\right ] + \\frac { 1 } { n } \\sum _ { \\tau = 1 } ^ { n - 4 } e _ { i , g + \\tau + 2 } ^ { n - \\tau - 3 } . \\end{align*}"}
-{"id": "1431.png", "formula": "\\begin{align*} \\partial _ j ( \\alpha x _ { 4 1 } ) = \\alpha x _ { 4 1 } ^ { 2 ^ { j - 1 } } , \\end{align*}"}
-{"id": "1054.png", "formula": "\\begin{align*} \\underline { v } _ { k } = ( a _ { k , 0 } , \\ldots , a _ { k , n / 2 - 1 } , b _ { k , 0 } , \\ldots , b _ { k , n / 2 - 1 } , c _ { k , 0 } , \\ldots , c _ { k , n / 2 - 1 } ) \\in \\mathbb { Z } ^ { 3 n / 2 } . \\end{align*}"}
-{"id": "6257.png", "formula": "\\begin{align*} & \\frac 1 2 \\frac { d } { d t } \\sum _ { q \\geq - 1 } \\left ( \\lambda _ q ^ { 2 s } \\| u _ q \\| _ 2 ^ 2 + \\lambda _ q ^ { 2 s } \\| b _ q \\| _ 2 ^ 2 \\right ) \\\\ \\leq & - \\nu \\sum _ { q \\geq - 1 } \\lambda _ q ^ { 2 s + 2 } \\| u _ q \\| _ 2 ^ 2 - \\mu \\sum _ { q \\geq - 1 } \\lambda _ q ^ { 2 s + 2 \\alpha } \\| b _ q \\| _ 2 ^ 2 + I _ 1 + I _ 2 + I _ 3 + I _ 4 + I _ 5 , \\end{align*}"}
-{"id": "7699.png", "formula": "\\begin{align*} \\phi ( t ) = e _ j ^ { \\top } Q ^ { \\top } [ I _ { N - 1 } | 0 _ { N - 1 } ] z ( t ) = e _ j ^ { \\top } Q ^ { \\top } z _ 1 ( t ) \\ , . \\end{align*}"}
-{"id": "3242.png", "formula": "\\begin{align*} \\mathcal A _ { \\tau , k } ( X ) = \\sum _ { p = - \\infty } ^ { + \\infty } A ^ { p } _ \\tau H ^ { 2 p + k } ( X ) . \\end{align*}"}
-{"id": "6494.png", "formula": "\\begin{align*} p _ { B ^ { \\prime } } ^ { } \\left ( x _ { B } ^ { \\prime } | \\mu _ { B } ^ { \\prime } , \\sigma _ { B } ^ { \\prime } \\right ) = \\frac { 1 } { \\sqrt { 2 \\pi \\sigma _ { B } ^ { \\prime 2 } } } \\exp \\left ( - \\frac { \\left ( x _ { B } ^ { \\prime } - \\mu _ { B } ^ { \\prime } \\right ) ^ { 2 } } { 2 \\sigma _ { B } ^ { \\prime 2 } } \\right ) , \\end{align*}"}
-{"id": "481.png", "formula": "\\begin{align*} p _ { 1 , k _ 1 , k _ 2 } ^ { ( m ) } ( x , t ) = \\frac { ( - 1 ) ^ { \\frac { m - 1 } { 2 } } } { ( 2 \\pi ) ^ { \\frac { m - 1 } { 2 } } } \\abs { t } ^ { - \\frac { m - 1 } { 2 } } p _ { 1 , k _ 1 , k _ 2 + \\frac { m - 1 } { 2 } } ^ { ( 1 ) } ( x , t ) \\left [ 1 + O \\left ( \\frac { 1 } { \\abs { t } } \\right ) \\right ] . \\end{align*}"}
-{"id": "3394.png", "formula": "\\begin{align*} \\alpha ^ { - 1 } ( x , y ) = ( \\alpha _ x ^ { - 1 } ( x ) , \\alpha _ y ^ { - 1 } ( x , y ) ) , \\beta ^ { - 1 } ( x , y ) = ( \\beta _ x ^ { - 1 } ( x , y ) , \\beta _ y ^ { - 1 } ( y ) ) \\end{align*}"}
-{"id": "5843.png", "formula": "\\begin{align*} f ( 0 ) = 0 , f ' ( 0 ) = A , f ( \\tfrac 1 2 ) = 0 . \\end{align*}"}
-{"id": "3975.png", "formula": "\\begin{align*} p ^ { \\alpha _ 2 } _ { k - 1 } ( 2 , t ) = ( - \\lambda ) ^ { k - 1 } \\underset { \\Theta ^ { k - 1 } _ { 2 } } { \\sum } \\frac { t ^ { k _ 0 \\alpha _ 0 + k _ 1 \\alpha _ 1 + k _ 2 \\alpha _ 2 } } { \\Gamma \\left ( k _ 0 \\alpha _ 0 + k _ 1 \\alpha _ 1 + k _ 2 \\alpha _ 2 + 1 \\right ) } . \\end{align*}"}
-{"id": "1526.png", "formula": "\\begin{align*} r = \\begin{cases} \\lfloor \\alpha _ { m , t } ( 1 ) \\rfloor , & \\ \\ \\{ \\alpha _ { m , t } ( 1 ) \\} < 1 / 2 ; \\\\ \\lfloor \\alpha _ { m , t } ( 1 ) \\rfloor \\ \\ \\lfloor \\alpha _ { m , t } ( 1 ) \\rfloor + 1 , & \\ \\ \\{ \\alpha _ { m , t } ( 1 ) \\} = 1 / 2 ; \\\\ \\lfloor \\alpha _ { m , t } ( 1 ) \\rfloor + 1 , & \\ \\ \\{ \\alpha _ { m , t } ( 1 ) \\} > 1 / 2 . \\end{cases} \\end{align*}"}
-{"id": "7494.png", "formula": "\\begin{align*} \\Box ^ h = \\bar { \\partial } ^ h \\circ \\bar { \\partial } ^ { * h } + \\bar { \\partial } ^ { * h } \\circ \\bar { \\partial } ^ h . \\end{align*}"}
-{"id": "7672.png", "formula": "\\begin{align*} \\widetilde F ( B , X ) = \\sum _ { i = 0 } ^ { a _ 1 } X ^ { i - ( a _ 1 / 2 ) } . \\end{align*}"}
-{"id": "2896.png", "formula": "\\begin{gather*} U = \\{ \\langle \\langle k , m \\rangle , \\sigma \\rangle : \\langle m , \\sigma \\rangle \\in V _ k \\} \\\\ V = \\{ \\langle 0 , \\sigma \\rangle : \\forall k \\ , \\forall \\tau \\succeq \\sigma \\ , \\exists m \\ , \\exists \\rho \\succeq \\tau \\ , \\langle m , \\rho \\rangle \\in U _ k \\} . \\end{gather*}"}
-{"id": "267.png", "formula": "\\begin{align*} \\Sigma _ { k } = \\{ ( t _ 0 , \\dots , t _ { k } ) : \\ ; \\forall j \\ t _ j \\ge 0 \\sum _ { j = 0 } ^ { k } t _ j = 1 \\} , \\end{align*}"}
-{"id": "3388.png", "formula": "\\begin{align*} F ' ( b _ 1 , \\ldots , b _ { d + 1 } ) | T _ { 0 , N _ { d } - 1 } ( b _ 1 , \\ldots , b _ { d + 1 } ) = F ( b _ 1 , \\ldots , b _ { d } ) | T _ { 0 , N _ { d } - 1 } ( b _ 1 , \\ldots , b _ { d } ) \\end{align*}"}
-{"id": "10150.png", "formula": "\\begin{align*} \\boldsymbol Q _ l ( i ) = \\textrm { d i a g } ( \\underbrace { 1 \\ldots 1 } _ D \\underbrace { 0 \\ldots 0 } _ { M - D } ) + O _ l \\big ( \\epsilon _ l ( i ) \\big ) , \\end{align*}"}
-{"id": "8666.png", "formula": "\\begin{align*} \\zeta _ { T _ n } = c _ 1 T _ n + c _ 2 + o ( 1 ) \\end{align*}"}
-{"id": "3866.png", "formula": "\\begin{align*} g _ i ( x ^ * ) \\not = 0 \\ \\Rightarrow \\ g _ i ( x ^ k ) \\not = 0 \\ \\Rightarrow \\ \\lambda _ i ^ k = 0 , \\\\ x _ i ^ * \\not = 0 \\ \\Rightarrow \\ x _ i ^ k \\not = 0 \\ \\Rightarrow \\ \\gamma _ i ^ k = 0 , \\\\ \\lambda _ i ^ * \\not = 0 \\ \\Rightarrow \\ \\lambda _ i ^ k \\not = 0 \\ \\Rightarrow \\ g _ i ( x ^ k ) = 0 , \\\\ \\gamma _ i ^ * \\not = 0 , \\ \\Rightarrow \\ \\gamma _ i ^ k \\not = 0 \\ \\Rightarrow \\ x _ i ^ k = 0 , \\end{align*}"}
-{"id": "1340.png", "formula": "\\begin{align*} \\vartheta _ c ( y ) = L _ c ( x _ c ( y ) , y ) \\ , , y \\in \\mathbb { B } _ { \\delta _ 0 } ( \\overline { y } ) \\ , . \\end{align*} % \\end{align*}"}
-{"id": "4070.png", "formula": "\\begin{align*} \\begin{aligned} \\eta ( v , F , Z , w ) & = \\frac { 1 } { 2 } | v | ^ 2 + G ( F , Z , w ) , \\\\ q _ \\alpha & = v _ i \\ , \\frac { \\partial G } { \\partial \\Xi ^ A } ( \\Xi ) \\frac { \\partial \\Phi ^ A } { \\partial F _ { i \\alpha } } ( F ) . \\end{aligned} \\end{align*}"}
-{"id": "5837.png", "formula": "\\begin{align*} | J _ \\psi ( x , \\alpha ) | \\le C _ d \\Big ( 1 + \\sup _ { k = 1 , \\dots , d } \\| \\nabla t _ k ( x ) \\| \\Big ) ^ d \\end{align*}"}
-{"id": "4912.png", "formula": "\\begin{align*} T ^ * = - \\alpha \\gamma ^ { - 1 } ( I - \\alpha ^ { - 1 } F ) + 0 I \\end{align*}"}
-{"id": "5693.png", "formula": "\\begin{align*} \\forall t > 0 , \\ , | { v } _ n ( t ) - a ^ - | = | { v } _ n ( - t ) - a ^ + | \\geq \\eta \\forall a \\in \\Sigma \\setminus \\{ a ^ - , a ^ + \\} , \\ , \\forall t \\in \\R , \\ , | { v } _ n ( t ) - a | \\geq \\eta . \\end{align*}"}
-{"id": "7629.png", "formula": "\\begin{align*} \\int _ { Q _ \\frac 7 2 } m _ + ( x , t ) \\ , d x d t = 0 , \\end{align*}"}
-{"id": "2315.png", "formula": "\\begin{align*} \\| I _ 1 \\| _ { L ^ 2 } \\leq & \\ ; C \\int _ 0 ^ { t - h } h ^ { \\beta ' } ( t - \\tau ) ^ { - ( 1 + \\beta ' ) } \\frac { ( t - \\tau ) ^ \\beta } { \\tau ^ { \\beta + 1 / 2 } } R _ 1 R _ 2 \\ , d \\tau \\\\ = & \\ ; C h ^ { \\beta ' } R _ 1 R _ 2 \\int _ 0 ^ { t / 2 } + \\int _ { t / 2 } ^ { t - h } ( t - \\tau ) ^ { - ( 1 + \\beta ' - \\beta ) } \\tau ^ { - ( \\beta + 1 / 2 ) } \\ , d \\tau \\\\ \\leq & \\ ; C R _ 1 R _ 2 h ^ { \\beta } t ^ { - ( \\beta + 1 / 2 ) } , \\end{align*}"}
-{"id": "7105.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } s ( z , t ) ~ = ~ F _ * ^ { \\alpha } ( \\bar { \\nabla } _ i \\bar { \\nabla } _ j s ( z , t ) + s ( z , t ) \\bar { g } _ { i j } ) , ( z , t ) \\in \\mathbb { S } ^ n \\times [ 0 , T ) , \\end{align*}"}
-{"id": "3143.png", "formula": "\\begin{align*} \\left ( \\mathcal { D } f \\right ) ( x ) & = ( a - b x ) \\frac { \\partial f ( x ) } { \\partial x } + \\frac { 1 } { 2 } \\sigma ^ { 2 } x \\frac { \\partial ^ { 2 } f ( x ) } { \\partial x ^ { 2 } } , \\\\ \\left ( \\mathcal { J } f \\right ) ( x ) & = \\int _ { 0 } ^ { \\infty } \\left ( f ( x + z ) - f ( x ) \\right ) \\nu ( \\mathrm { d } z ) , \\end{align*}"}
-{"id": "3824.png", "formula": "\\begin{align*} \\tilde { p } : = \\inf _ { z \\in B } P ( S ^ { z , 1 } A ' ) > 0 . \\end{align*}"}
-{"id": "2136.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } w ( \\xi , s ) = m ( \\varphi ) \\psi _ d ( \\xi ) \\quad \\mbox { i n } L ^ 2 ( { \\bf R } _ + , \\rho _ d \\ , d \\xi ) \\ , \\cap \\ , C ^ 2 ( K ) \\end{align*}"}
-{"id": "7080.png", "formula": "\\begin{align*} \\frac { d } { d s } ( B , \\Phi ) = - \\big ( * _ 3 ( F _ B - \\Phi ^ \\dag \\tau \\Phi + i \\mu ) , D _ B \\Phi \\big ) . \\end{align*}"}
-{"id": "3638.png", "formula": "\\begin{gather*} R _ q T _ 1 T _ 2 = T _ 2 T _ 1 R _ q , \\\\ A _ q ^ { - 1 } T ^ t A _ q T = 1 = T A _ q ^ { - 1 } T ^ t A _ q , \\\\ L _ q T _ 1 T _ 2 = T L _ q , \\end{gather*}"}
-{"id": "6725.png", "formula": "\\begin{align*} \\mathbb { P } \\Bigg ( \\min \\limits _ { 2 \\leq l \\leq \\frac { N ^ { 1 + \\delta } } { 2 } } \\Gamma _ { l } > N ^ { 1 + \\delta } \\Bigg ) & = 1 - \\mathbb { P } \\Bigg ( \\min \\limits _ { 2 \\leq l \\leq \\frac { N ^ { 1 + \\delta } } { 2 } } \\Gamma _ { l } \\leq N ^ { 1 + \\delta } \\Bigg ) \\\\ & \\geq 1 - N ^ { 1 + \\delta } \\Bigg ( \\frac { 2 } { N ^ { 2 } } \\Bigg ) - \\sum \\limits _ { l = 3 } ^ { \\frac { N ^ { 1 + \\delta } } { 2 } } \\frac { 8 N ^ { 1 + \\delta } } { N ^ { 3 } } \\end{align*}"}
-{"id": "1995.png", "formula": "\\begin{align*} A _ { n , \\nu } \\left ( t \\right ) : = \\sup _ { \\widetilde { U } _ { 1 : n } \\leq s < \\widetilde { U } _ { t _ { n } : n } } \\frac { n ^ { \\nu } \\left \\vert \\widetilde { \\alpha } _ { n } \\left ( s ; t \\right ) - \\widetilde { B } _ { n } \\left ( s \\right ) \\right \\vert } { s ^ { 1 / 2 - \\nu } } = O _ { \\mathbb { P } } \\left ( 1 \\right ) , \\end{align*}"}
-{"id": "840.png", "formula": "\\begin{align*} X _ t = Y _ { E _ t } . \\end{align*}"}
-{"id": "1576.png", "formula": "\\begin{align*} j _ k ' = \\begin{cases} j _ k , & k \\neq i , i + 1 , n \\\\ j _ i - r , & k = i \\\\ j _ { i + 1 } + p , & k = i + 1 \\\\ j _ n + q , & k = n \\end{cases} \\end{align*}"}
-{"id": "9297.png", "formula": "\\begin{align*} V _ z : = \\{ x \\in U _ z \\ : \\ \\forall \\delta , \\epsilon > 0 \\ ( x - \\delta , x + \\delta ) \\times ( 0 , \\epsilon ) \\cap W _ z \\neq \\emptyset \\} . \\end{align*}"}
-{"id": "4516.png", "formula": "\\begin{align*} | \\psi ( a ) | = \\lim _ { n \\to \\infty } | \\psi ( \\Delta _ n ^ * a \\Delta _ n ) | \\leq \\liminf _ { n \\to \\infty } \\| \\Delta _ n ^ * a \\Delta _ n \\| \\end{align*}"}
-{"id": "9877.png", "formula": "\\begin{align*} \\omega _ 0 ( g , h ) + \\xi ( g h ) = \\omega _ 0 ( h , g ) + \\xi ( h g ) \\end{align*}"}
-{"id": "145.png", "formula": "\\begin{align*} a ^ { \\sigma } = a ; \\ \\lambda _ { 2 1 } = 0 ; \\ b ^ { \\sigma } = \\lambda _ { 2 2 } \\lambda _ { 1 1 } ^ { - 1 } b \\end{align*}"}
-{"id": "8950.png", "formula": "\\begin{gather*} f _ 0 + f _ 1 ( s - 1 ) = ( f _ 0 - f _ 1 - { } ^ s f _ 1 ) + ( 1 + s ) { } ^ s f _ 1 , \\end{gather*}"}
-{"id": "6425.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } \\theta ^ { \\kappa } } { d \\tau ^ { 2 } } + \\Gamma _ { \\mu \\nu } ^ { \\kappa } \\frac { d \\theta ^ { \\mu } } { d \\tau } \\frac { d \\theta ^ { \\nu } } { d \\tau } = 0 \\end{align*}"}
-{"id": "9748.png", "formula": "\\begin{align*} L ^ 1 ( J ) + L ^ 2 ( J ) - L ^ 1 ( I ) - L ^ 2 ( I ) & \\leq ( K ^ { * } _ { 1 1 } | K _ { 2 1 } | + K ^ { * } _ { 2 5 } | K _ { 2 5 } | - 1 ) | \\beta _ 1 | - \\sum \\limits _ { i = 2 , 3 , 4 } K ^ { * } _ { 2 i } | \\beta _ i | + M \\Delta ' ( \\alpha _ 5 , \\boldsymbol { \\beta } ^ { * } ) . \\end{align*}"}
-{"id": "5642.png", "formula": "\\begin{align*} d _ K ( x ^ - , x ^ + ) : = \\inf \\left \\{ \\mathfrak { L } _ { K } ( \\gamma ) \\ ; : \\ ; \\gamma \\in A C _ { p l o c } ( I , X ) \\gamma : x ^ - \\mapsto x ^ + \\right \\} \\in [ 0 , + \\infty ] . \\end{align*}"}
-{"id": "8195.png", "formula": "\\begin{align*} R ( f _ { \\nu } ) w _ { \\nu } ^ { \\circ } = c _ { \\nu } ( \\pi _ { \\nu } ) w _ { \\nu } ^ { \\circ } . \\end{align*}"}
-{"id": "9397.png", "formula": "\\begin{align*} | \\Omega ( x ' ) | & = \\bigg [ | \\Omega ( x ' ) | - \\frac 1 { \\omega _ { n - 1 } } \\int _ { \\mathbf S ^ { n - 1 } } | \\Omega ( y ' ) | d \\sigma ( y ' ) \\bigg ] + \\frac 1 { \\omega _ { n - 1 } } \\int _ { \\mathbf S ^ { n - 1 } } | \\Omega ( y ' ) | d \\sigma ( y ' ) \\\\ & : = \\Omega _ 0 ( x ' ) + C ( \\Omega , n ) , \\end{align*}"}
-{"id": "1154.png", "formula": "\\begin{align*} \\rho _ a \\rho _ x & = \\left ( \\sum _ { k = 0 } ^ d \\rho _ { a , k } \\tau ^ k \\right ) ( x + [ 1 ] _ x \\tau + \\tau ^ 2 ) \\\\ & = \\sum _ { k = 0 } ^ d \\left ( x ^ { 2 ^ k } \\rho _ { a , k } \\tau ^ k + [ 1 ] _ x ^ { 2 ^ k } \\rho _ { a , k } \\tau ^ { k + 1 } + \\rho _ { a , k } \\tau ^ { k + 2 } \\right ) . \\end{align*}"}
-{"id": "8890.png", "formula": "\\begin{align*} u = ( \\overline c - 1 ) J _ { \\omega , 1 } ^ { - 1 } | \\varphi | ^ 2 + 1 . \\end{align*}"}
-{"id": "7277.png", "formula": "\\begin{align*} \\psi ( x ) = \\begin{cases} 1 , & x \\le 2 0 , \\\\ ( \\log x ) ^ { - 1 } ( \\log \\log x ) ^ { - C } , & x > 2 0 . \\end{cases} \\end{align*}"}
-{"id": "9311.png", "formula": "\\begin{align*} F _ i : = \\{ ( t , x ) \\in W : \\exists \\delta > 0 ( \\delta , t , x ) \\in E _ i \\} . \\end{align*}"}
-{"id": "8347.png", "formula": "\\begin{align*} V _ \\mu ( \\mathcal { A } _ S ) = \\bigcap _ \\ell V _ { \\mu _ \\ell } ( \\mathcal { A } _ S ) . \\end{align*}"}
-{"id": "6918.png", "formula": "\\begin{align*} \\langle f , g \\rangle = \\int a ( y ) b ( y ) d \\mu ( y ) \\end{align*}"}
-{"id": "5493.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { u ( r ^ { - n } z ) } { ( r ^ { - n } z ) ^ { \\rho - 1 } \\ell ( r ^ { - n } z ) } = p _ 0 ( z ) z \\in C _ { p _ 0 } , p _ 0 \\in \\mathcal { P } _ { r } , \\end{align*}"}
-{"id": "4425.png", "formula": "\\begin{align*} \\ell ^ { \\alpha + \\beta } [ u ] _ \\alpha [ f ] _ \\beta \\gtrsim \\begin{cases} \\| \\lceil u , ( \\cdot ) _ \\ell \\rceil f \\| & \\textrm { i f } \\alpha \\in ( 0 , 1 ] , \\\\ \\| \\big ( \\lceil u , ( \\cdot ) _ \\ell \\rceil - \\partial _ 1 u \\lceil x _ 1 , ( \\cdot ) _ \\ell \\rceil \\big ) f \\| & \\textrm { i f } \\alpha \\in ( 1 , \\frac 3 2 ) , \\end{cases} \\end{align*}"}
-{"id": "4647.png", "formula": "\\begin{align*} \\left | \\sum _ { n = 0 } ^ { N - 1 } f ( T ^ n x ) \\right | \\le N ^ \\gamma \\end{align*}"}
-{"id": "10062.png", "formula": "\\begin{align*} W _ { 0 , \\mathfrak { p } } ( s , \\Phi _ \\mu ) = \\frac { L _ \\mathfrak { p } ( s , \\chi _ E ) } { L _ \\mathfrak { p } ( s + 1 , \\chi _ E ) } \\end{align*}"}
-{"id": "3390.png", "formula": "\\begin{align*} F ^ s = F \\circ \\alpha ^ { - 1 } \\circ J _ s \\circ \\alpha \\end{align*}"}
-{"id": "7362.png", "formula": "\\begin{align*} \\nabla _ X \\psi = \\lambda \\widetilde { X } . \\psi \\end{align*}"}
-{"id": "4605.png", "formula": "\\begin{align*} \\mathcal { C } _ d = \\left \\{ ( x _ 1 , \\dots , x _ { d + 1 } , y _ 1 , \\dots , y _ { d + 1 } ) \\in ( 0 , 1 ) ^ { d + 1 } \\times \\mathbb { R } ^ { d + 1 } : \\begin{aligned} x _ 1 y _ 1 + \\cdots + x _ { d + 1 } y _ { d + 1 } = 0 \\\\ x _ 1 + \\cdots + x _ { d + 1 } = 1 \\\\ y _ i \\ne y _ { i + 1 } \\textrm { f o r a l l } 1 \\le i \\le d \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "867.png", "formula": "\\begin{align*} ( \\partial _ i U _ 0 f ) ( x ) & = \\int _ 0 ^ 1 \\frac { 1 } { 2 \\sqrt { t } } E \\left [ \\partial _ i f ( \\sqrt { t } x + \\sqrt { 1 - t } Z ^ * ) \\right ] d t , \\\\ ( \\partial _ { i , j } ^ 2 U _ 0 f ) ( x ) & = \\int _ 0 ^ 1 \\frac { 1 } { 2 } E \\left [ \\partial _ { i , j } f ( \\sqrt { t } x + \\sqrt { 1 - t } Z ^ * ) \\right ] d t \\end{align*}"}
-{"id": "1191.png", "formula": "\\begin{align*} E _ l = \\{ x : d ( x , E ) \\leq 1 / l \\} , \\ , \\ , l = 1 , 2 , \\dots , \\end{align*}"}
-{"id": "4343.png", "formula": "\\begin{align*} \\Big ( \\sum _ { j = 1 } ^ k x _ j \\Big ) ^ p \\leq k ^ p \\sum _ { j = 1 } ^ k x _ j ^ p . \\end{align*}"}
-{"id": "4960.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { d - 1 } \\| \\partial _ i ( f - F ) \\| _ { L ^ d } \\leq \\delta \\| f \\| _ { \\dot { W } ^ { 1 , d } } , \\end{align*}"}
-{"id": "3780.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { F } _ k : = & \\Big \\{ B \\in \\sigma ( \\omega , U ) \\colon \\ , \\\\ & \\forall \\ , y \\in \\Z ^ d \\times \\Z , \\ , \\exists \\ , B _ { y } \\in \\mathcal { F } _ { y } B \\cap \\{ Y _ { R _ k } = y \\} = B _ { y } \\cap \\{ Y _ { R _ k } = y \\} \\Big \\} , \\end{aligned} \\end{align*}"}
-{"id": "9291.png", "formula": "\\begin{align*} X _ { z , d } : = \\Big ( ( a , b - d ) \\setminus U _ { z , d } \\Big ) \\cup \\{ b - d \\} . \\end{align*}"}
-{"id": "2735.png", "formula": "\\begin{align*} \\epsilon ^ 2 \\partial _ t \\bigg ( \\sum _ { m = 0 } ^ { r } \\widetilde C _ { m , r + 1 } \\ , | | \\partial ^ m g | | _ { L ^ 2 _ { x , v } } ^ 2 \\bigg ) \\leq - \\lambda \\ , | | g ^ { \\perp } | | _ { \\Lambda ^ { r } } ^ 2 \\ , , \\end{align*}"}
-{"id": "5735.png", "formula": "\\begin{align*} \\begin{cases} { u } ( x _ 1 , x _ 2 ) \\to z ^ - ( x _ 1 - c ^ - ) & x _ 2 \\to - \\infty , x _ 1 ; \\\\ { u } ( x _ 1 , x _ 2 ) \\to z ^ + ( x _ 1 - c ^ + ) & x _ 2 \\to + \\infty , x _ 1 . \\end{cases} \\end{align*}"}
-{"id": "6222.png", "formula": "\\begin{align*} X ( n ) = - C ( n ) + Z ( n ) ~ , ~ ~ n \\geq 0 ~ . \\end{align*}"}
-{"id": "7842.png", "formula": "\\begin{align*} f _ { X _ { i } \\mid X _ { j } } ( x _ { i } \\mid x _ { j } ) = \\left \\{ \\begin{array} { l } f _ { X _ { i } \\mid X _ { j } } ^ { ( 1 ) } ( x _ { i } \\mid x _ { j } ) \\ \\ \\ \\ \\ x _ { i } > x _ { j } > 0 \\\\ f _ { X _ { i } \\mid X _ { j } } ^ { ( 2 ) } ( x _ { i } \\mid x _ { j } ) \\ \\ \\ \\ x _ { j } > x _ { i } > 0 \\\\ f _ { X _ { i } \\mid X _ { j } } ^ { ( 3 ) } ( x _ { i } \\mid x _ { j } ) \\ \\ \\ \\ x _ { i } = x _ { j } > 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "6429.png", "formula": "\\begin{align*} \\eta _ { \\left ( \\mathcal { M } g \\right ) } \\left ( \\theta ^ { 1 } \\theta ^ { n } \\right ) \\overset { } { = } \\sqrt { g \\left ( \\theta \\right ) } \\end{align*}"}
-{"id": "6514.png", "formula": "\\begin{align*} \\mathcal { S } _ { \\mathcal { M } ^ { } } \\left ( \\tau ; \\rho \\right ) = \\lambda _ { \\mathcal { M } } \\tau - \\log \\left ( \\lambda _ { \\mathcal { M } } \\tau \\right ) + \\frac { 1 } { 2 } \\log \\left ( \\frac { 1 - \\rho } { 1 + \\rho } \\right ) . \\end{align*}"}
-{"id": "10002.png", "formula": "\\begin{align*} \\Lambda ( s , \\chi _ E ) = \\left | \\frac { D _ E } { D _ F } \\right | ^ { \\frac { s } { 2 } } \\Gamma _ \\R ( s + 1 ) ^ { [ F : \\Q ] } L ( s , \\chi _ E ) \\end{align*}"}
-{"id": "5677.png", "formula": "\\begin{align*} c _ 0 f ^ { p _ 0 - 2 } ( \\bar s ) \\leq \\frac { f '' ( \\bar s ) } { f ( \\bar s ) } \\leq \\frac { E '' ( \\bar s ) } { E ( \\bar s ) } = c E ^ { p _ 0 - 2 } ( \\bar s ) \\leq c f ^ { p _ 0 - 2 } ( \\bar s ) , \\end{align*}"}
-{"id": "7537.png", "formula": "\\begin{gather*} \\varphi ^ 1 \\varphi ^ 1 _ \\omega - \\varphi ^ 1 _ { \\omega \\omega } - \\frac { ( \\varphi ^ 2 ) ^ 2 } { \\omega } - \\frac 1 { \\omega ^ 3 } + \\omega = 0 , \\\\ \\varphi ^ 1 \\varphi ^ 2 _ \\omega - \\varphi ^ 2 _ { \\omega \\omega } + \\frac { \\varphi ^ 1 \\varphi ^ 2 } { \\omega } + 2 \\frac { \\varphi ^ 2 } { \\omega ^ 2 } = 0 . \\end{gather*}"}
-{"id": "1813.png", "formula": "\\begin{align*} \\lim _ { H \\downarrow 0 } \\frac { \\mathcal { M } ( H ) } { H ^ { 1 / 1 5 } } = B \\in ( 0 , \\infty ) . \\end{align*}"}
-{"id": "8966.png", "formula": "\\begin{gather*} { \\cal Z } _ w : = { \\cal Z } _ { s _ 1 } \\otimes { } ^ { s _ 1 } { \\cal Z } _ { s _ 2 } \\otimes { } ^ { s _ 1 s _ 2 } { \\cal Z } _ { s _ 3 } \\otimes \\cdots \\otimes { } ^ { s _ 1 \\cdots s _ { m - 1 } } { \\cal Z } _ { s _ m } . \\end{gather*}"}
-{"id": "4464.png", "formula": "\\begin{align*} \\langle | F ^ \\ell ( k ) | ^ 2 \\rangle = \\sum _ { k ' + k '' = k } & \\big ( ( G _ \\ell ( k ' ) \\tilde G _ \\ell ( k '' ) ) ^ 2 + ( G _ \\ell ( k ' ) \\tilde G _ \\ell ( k '' ) ) ( G _ \\ell ( k '' ) \\tilde G _ \\ell ( k ' ) ) \\big ) \\\\ & \\times \\langle | \\xi ( k ' ) | ^ 2 | \\xi ( k '' ) | ^ 2 \\rangle . \\end{align*}"}
-{"id": "2338.png", "formula": "\\begin{align*} A _ 1 \\cdots A _ n = A _ 1 \\cdots A _ { i } A ^ k A _ { i + k + 1 } \\cdots A _ n \\end{align*}"}
-{"id": "10142.png", "formula": "\\begin{align*} \\boldsymbol { \\bar { \\omega } } _ k ( 1 ) = \\bigg ( \\boldsymbol S _ { D _ k } ^ H ( 0 ) \\boldsymbol R _ k ( 0 ) \\boldsymbol S _ { D _ k } ( 0 ) \\bigg ) ^ { - 1 } \\boldsymbol S _ { D _ k } ^ H ( 0 ) \\boldsymbol p _ k ( 0 ) . \\end{align*}"}
-{"id": "7727.png", "formula": "\\begin{align*} d _ B ( j , k ) & = \\Bigg ( \\frac { j } { 6 } + \\frac { j ^ 2 } { 2 } - \\frac { j ^ 2 } { 4 N } + \\frac { j ^ 3 } { 3 } - \\frac { j ^ 3 } { 2 N } - \\frac { j ^ 4 } { 4 N } \\\\ & - \\frac { k } { 6 } - j k + \\frac { j k } { 2 N } + \\frac { j ^ 2 k } { 2 N } + \\frac { k ^ 2 } { 2 } - \\frac { k ^ 2 } { 4 N } \\\\ & - j k ^ 2 + \\frac { j k ^ 2 } { 2 N } + \\frac { j ^ 2 k ^ 2 } { 2 N } + \\frac { 2 k ^ 3 } { 3 } - \\frac { k ^ 3 } { 2 N } - \\frac { k ^ 4 } { 4 N } \\Bigg ) ^ { \\frac { 1 } { 2 } } \\ , . \\end{align*}"}
-{"id": "2585.png", "formula": "\\begin{align*} h _ \\rho = C ( 0 , 1 ) . \\end{align*}"}
-{"id": "2053.png", "formula": "\\begin{gather*} T _ { \\nabla ^ M } ( X , Y ) = 2 d \\theta ( X , Y ) T \\mbox { a n d } T _ { \\nabla ^ M } ( T , J _ b X ) + J _ b T _ { \\nabla ^ M } ( T , X ) = 0 . \\end{gather*}"}
-{"id": "3656.png", "formula": "\\begin{align*} z _ { 1 1 } z _ { 1 2 } = z _ { 2 1 } z _ { 2 2 } = \\dots = z _ { n 1 } z _ { n 2 } \\end{align*}"}
-{"id": "4862.png", "formula": "\\begin{align*} \\xi ( s , t ) = - 2 ^ { 2 4 } \\cdot 3 ^ { 1 2 } \\cdot 5 ^ 2 \\cdot 7 ^ 4 \\cdot s ^ { 1 7 } t ^ { 1 0 } ( 1 9 2 s ^ 3 + 1 6 8 0 s ^ 2 t + 5 2 7 5 s t ^ 2 + 5 2 5 0 t ^ 3 ) . \\end{align*}"}
-{"id": "5185.png", "formula": "\\begin{align*} - g _ { 2 } ' ( \\ell _ { * } ) = f _ { 1 } ' ( \\ell _ { * } ) & = \\frac { g _ { 1 } ( r _ { * } ) - f _ { 1 } ( \\ell _ { * } ) } { r _ { * } - \\ell _ { * } } \\\\ & = \\frac { [ - f _ { 2 } ( r _ { * } ) + g _ { 2 } ( \\ell _ { * } ) ] } { r _ { * } - \\ell _ { * } } = - f _ { 2 } ' ( r _ { * } ) = g _ { 1 } ' ( r _ { * } ) . \\end{align*}"}
-{"id": "3213.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } \\bigl ( \\| \\nabla f _ k - \\nabla f _ { k - 1 } \\| _ { L ^ 2 ( { \\mathbb R } ^ 4 ) } + \\| g _ k - g _ { k - 1 } \\| _ { L ^ 2 ( { \\mathbb R } ^ 4 ) } \\bigr ) < \\infty . \\end{align*}"}
-{"id": "7961.png", "formula": "\\begin{align*} \\sum _ { \\alpha \\in A _ I } \\lambda _ { J _ { \\alpha } } ^ s = 1 . \\end{align*}"}
-{"id": "10148.png", "formula": "\\begin{align*} \\boldsymbol Q _ l ( i ) = \\boldsymbol \\Lambda _ l ^ { 2 i } \\boldsymbol { \\omega } _ l ( 0 ) \\big ( \\boldsymbol { \\omega } _ l ^ H ( 0 ) \\boldsymbol \\Lambda _ l ^ { 4 i - 2 } \\boldsymbol { \\omega } _ l ( 0 ) \\big ) ^ { - 1 } \\boldsymbol \\omega _ l ^ H ( 0 ) \\boldsymbol \\Lambda _ l ^ { 2 i - 2 } . \\end{align*}"}
-{"id": "9683.png", "formula": "\\begin{align*} & \\Phi _ 2 ( \\sigma _ 2 ; U _ a ) = ( u _ { a } e ^ { \\sigma _ 2 } , v _ { a } e ^ { \\sigma _ 2 } , p _ a , \\rho _ a , Z _ a ) , \\\\ & \\Phi _ 3 ( \\sigma _ 3 ; U _ a ) = ( u _ a , v _ a , p _ a , \\rho _ a e ^ { \\sigma _ 3 } , Z _ a ) , \\\\ & \\Phi _ 4 ( \\alpha _ 4 ; U _ a ) = ( u _ a , v _ a , p _ a , \\rho _ a , Z _ a + \\alpha _ 4 ) . \\end{align*}"}
-{"id": "3529.png", "formula": "\\begin{align*} J ( u ( \\cdot ) ) = \\big { [ } { \\displaystyle \\int \\limits _ { 0 } ^ { T } } f ( X { ( t ) } , u ( t ) ) d t + \\Psi ( X ( T ) ) \\big { ] } , \\end{align*}"}
-{"id": "7548.png", "formula": "\\begin{gather*} R ^ 1 : = u _ t + u u _ x + v u _ y - u _ { x x } - u _ { y y } = 0 , \\\\ R ^ 2 : = v _ t + u v _ x + v v _ y - v _ { x x } - v _ { y y } = 0 , \\\\ R ^ 3 : = u _ x + v _ y = 0 \\end{gather*}"}
-{"id": "120.png", "formula": "\\begin{align*} q _ n = \\sum _ { k = 1 } ^ { n } \\binom { n } { k } \\binom { n } { k - 1 } \\left ( \\frac { k - 1 } { n } \\right ) ^ { k \\alpha } . \\end{align*}"}
-{"id": "2870.png", "formula": "\\begin{align*} a _ { j , N } = \\cos \\left ( \\frac { \\pi j } { 2 N } \\right ) , b _ { j , N } = \\sin \\left ( \\frac { \\pi j } { 2 N } \\right ) , A _ { N } ( y ) = \\pi \\left ( 2 \\pi y \\right ) ^ { 1 / N } . \\end{align*}"}
-{"id": "5511.png", "formula": "\\begin{align*} \\varphi ( t ) = \\psi ( h ( t ) t ^ \\alpha ) , t \\geq 0 , \\end{align*}"}
-{"id": "8268.png", "formula": "\\begin{align*} f _ { \\nu } ( g _ { \\nu } ) = k _ { \\nu } ( u _ { \\nu } ( g _ { \\nu } . i _ { \\nu } , i _ { \\nu } ) ) , \\end{align*}"}
-{"id": "4629.png", "formula": "\\begin{align*} \\left | \\left \\{ 1 \\le L \\le N : \\sum _ { m = 0 } ^ N 1 _ { ( - \\epsilon , \\epsilon ) } \\left ( \\ , \\sum _ { n = 0 } ^ m f ( T ^ i T ^ L x ) \\right ) > N ^ { 1 - \\gamma - \\epsilon } \\right \\} \\right | > ( 1 - \\eta ) N \\end{align*}"}
-{"id": "1819.png", "formula": "\\begin{align*} P _ { \\Lambda _ L , \\eta , h } ^ { a } ( \\sigma ) = \\frac { 1 } { Z ^ a _ { \\Lambda _ L , \\eta , h } } e ^ { \\beta _ c \\sum _ { \\{ u , v \\} } \\sigma _ u \\sigma _ v + \\beta _ c \\sum _ { \\{ u , v \\} : u \\in \\Lambda _ L ^ a , v \\in \\partial _ { e x } \\Lambda _ L ^ a } \\sigma _ u \\eta _ v + a ^ { 1 5 / 8 } h \\sum _ { u \\in \\Lambda ^ a _ { L } } \\sigma _ u } , \\end{align*}"}
-{"id": "10074.png", "formula": "\\begin{align*} \\| a \\| ^ 2 = ( 2 \\pi ) ^ { - 1 } | \\psi _ 0 ( a , \\overline { a } ) | . \\end{align*}"}
-{"id": "9516.png", "formula": "\\begin{align*} \\tilde { b } _ n ^ { ( d ) } = \\dfrac { 1 } { | G | } \\sum \\limits _ { g \\in G } | { \\rm F i x } ( g ) | . \\end{align*}"}
-{"id": "6749.png", "formula": "\\begin{align*} E ( \\bar { Y } _ { N } ) = E ( E ( \\bar { Y } _ { N } | \\delta _ { 1 } ^ { N } ) ) = \\sum \\limits _ { n = 1 } ^ { N } \\frac { 1 } { N } \\Bigg [ \\Big ( \\frac { 1 } { 2 } + \\theta \\Big ) u _ { n } + \\frac { 1 } { 2 } ( 1 - u _ { n } ) \\Bigg ] = \\theta \\frac { U _ { N } } { N } + \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "4799.png", "formula": "\\begin{align*} - z _ 2 x _ 1 + z _ 3 x _ 2 & = 0 , \\\\ x _ 1 + x _ 2 - z _ 1 & = 0 , \\\\ z _ 3 x _ 2 - z _ 4 x _ 3 + z _ 5 & = 0 . \\end{align*}"}
-{"id": "4309.png", "formula": "\\begin{align*} D _ i \\Omega = r _ { M _ i } \\Omega \\end{align*}"}
-{"id": "5411.png", "formula": "\\begin{align*} C z ^ 4 + 2 B x ^ 2 z ^ 2 + 2 x ^ 3 z + A x ^ 4 = 0 \\end{align*}"}
-{"id": "8869.png", "formula": "\\begin{align*} | S | = n - \\min \\{ k , \\Delta \\} = \\max \\{ n - k , n - \\Delta \\} . \\end{align*}"}
-{"id": "3863.png", "formula": "\\begin{align*} \\nabla f ( x ^ * ) + \\sum \\limits _ { i = 1 } ^ m \\lambda _ i ^ * g _ i ( x ^ * ) + \\sum \\limits _ { i = 1 } ^ p \\mu _ i ^ * \\nabla h _ i ( x ^ * ) + \\sum \\limits _ { i = 1 } ^ n \\gamma _ i ^ * e _ i = 0 . \\end{align*}"}
-{"id": "10070.png", "formula": "\\begin{align*} \\| b \\| _ s ^ 2 = \\left | \\frac { \\psi ( b , \\overline { b } ) } { 2 \\pi i } \\right | , \\end{align*}"}
-{"id": "4825.png", "formula": "\\begin{align*} _ { G _ 0 } ( X ) : = \\{ g \\in G _ 0 \\mid \\mathsf { v } _ g ( X ) > 0 \\} . \\end{align*}"}
-{"id": "25.png", "formula": "\\begin{align*} \\hat { V } ^ { C } _ { \\sigma } ( R _ 1 , R _ 2 ) \\sqrt { 2 \\pi } \\sigma = \\frac { 1 } { \\sqrt { 2 \\pi } \\sigma } \\frac { 1 } { N } \\sum \\limits _ { n = 1 } ^ N e x p \\left ( - \\frac { ( x _ { n } - y _ { n } ) ^ { 2 } } { 2 \\sigma ^ 2 } \\right ) \\end{align*}"}
-{"id": "7260.png", "formula": "\\begin{align*} A ( n , m ) = \\sum _ { \\substack { u \\le n , v \\le m \\\\ \\frac u n = \\frac v m \\\\ ( u , n ) , ( v , m ) \\le T } } 1 . \\end{align*}"}
-{"id": "2796.png", "formula": "\\begin{align*} \\Lambda _ { n } = \\sup _ { t \\in [ a , b ] } \\Lambda _ { n } ( t ) , \\end{align*}"}
-{"id": "1662.png", "formula": "\\begin{align*} \\tau _ { a _ 0 b _ 0 } ( x ) = \\tau _ { a _ 0 } \\circ \\tau _ { b _ 0 } ( x ) = \\frac { 3 \\tau _ { b _ 0 } ( x ) - 1 } { 3 } = ( - \\frac { x } { 2 } + \\frac { 1 } { 2 } ) - \\frac { 1 } { 3 } = - \\frac { x } { 2 } + \\frac { 1 } { 6 } \\end{align*}"}
-{"id": "10094.png", "formula": "\\begin{align*} S ( Y , Z ) = \\lambda ( n - 1 ) \\pi ( Y ) \\pi ( Z ) , \\end{align*}"}
-{"id": "6118.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( d _ { J _ { 1 } } \\left ( \\left ( A _ { \\cdot } ^ { n } , D _ { \\cdot } ^ { n } \\right ) , f _ { \\cdot } ^ { n } \\right ) > \\epsilon | A _ { \\delta ' , \\tilde { T } } ^ { n } \\cap B _ { \\delta , \\tilde { T } } ^ { n } , \\left ( A _ { \\cdot } ^ { n } , D _ { \\cdot } ^ { n } \\right ) , \\left \\{ E _ { T } ^ { n } > \\tilde { T } \\right \\} \\right ) = 0 . \\end{align*}"}
-{"id": "8045.png", "formula": "\\begin{align*} | s , p | + | p , r | = | s , q | + | q , r | . \\end{align*}"}
-{"id": "7068.png", "formula": "\\begin{align*} \\dim \\ker \\nabla ^ 2 F ( u _ 0 ) = \\dim \\Gamma ( u _ 0 ) . \\end{align*}"}
-{"id": "5809.png", "formula": "\\begin{align*} \\langle \\nu , u \\rangle = \\lim _ { j \\rightarrow \\infty } \\langle \\nu , \\varphi _ { j } \\rangle \\leq \\lim _ { j \\rightarrow \\infty } \\int _ { \\Omega } \\nabla u \\cdot \\nabla \\varphi _ { j } \\ ; d x = \\int _ { \\Omega } | \\nabla u | ^ { 2 } \\ ; d x < + \\infty . \\end{align*}"}
-{"id": "1444.png", "formula": "\\begin{align*} D \\{ 1 , y _ 0 \\} \\oplus D _ 0 \\{ x _ 3 , x _ 3 y _ 0 \\} = C \\{ 1 , y _ 0 \\} \\oplus D _ 0 \\{ x _ 3 , x _ 3 y _ 0 , x _ 3 ^ 2 , x _ 3 ^ 2 y _ 0 \\} \\end{align*}"}
-{"id": "4013.png", "formula": "\\begin{align*} k _ { M , p } ( V ) : = \\frac { \\langle d u ^ { - 1 } _ { \\eta ( p ) } V , d \\eta _ p V \\rangle } { \\langle d u ^ { - 1 } _ { \\eta ( p ) } V , V \\rangle } . \\end{align*}"}
-{"id": "8873.png", "formula": "\\begin{align*} S _ \\theta f = \\chi f - ( f , \\overline \\chi \\theta ) \\theta , \\ \\ \\ f \\in \\mathcal K _ \\theta . \\end{align*}"}
-{"id": "5442.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { f ( x r ^ n ) } { ( x r ^ n ) ^ \\rho \\ell ( x r ^ n ) } = p ( x ) , x \\in C _ p , \\end{align*}"}
-{"id": "7491.png", "formula": "\\begin{align*} ( \\partial ^ h \\Psi ) _ { A _ { p + 1 } \\overline { B } _ q } & = \\sum _ { i = 1 } ^ { p + 1 } ( - 1 ) ^ { i - 1 } \\nabla _ { \\mathcal { X } _ { \\alpha _ i } } \\psi _ { \\alpha _ 1 \\dots \\hat { \\alpha } _ i \\dots \\alpha _ { p + 1 } \\overline { B } _ q } , \\\\ ( \\bar { \\partial } ^ h \\Psi ) _ { A _ p \\overline { B } _ { q + 1 } } & = ( - 1 ) ^ p \\sum _ { i = 1 } ^ { q + 1 } ( - 1 ) ^ { i - 1 } \\nabla _ { \\mathcal { X } _ { \\bar { \\beta } _ i } } \\psi _ { A _ p \\bar { \\beta } _ 1 \\dots \\hat { \\bar { \\beta } } _ i \\dots \\bar { \\beta } _ { q + 1 } } , \\end{align*}"}
-{"id": "7088.png", "formula": "\\begin{align*} a = f ( q ^ { 1 0 } , q ^ { 1 5 } ) , b = f ( q ^ { 5 } , q ^ { 2 0 } ) , c = \\psi ( q ^ { 2 5 } ) . \\end{align*}"}
-{"id": "1751.png", "formula": "\\begin{align*} \\nu _ { T ( x ) } ( Z ( \\lambda ) ) ^ 2 \\leq \\Vert ( S ^ { u n i v } _ \\lambda ( S ^ { u n i v } _ \\lambda ) ^ * ) x \\Vert ^ 2 \\ \\Vert T ^ * T x \\Vert ^ 2 = \\nu _ { x } ( Z ( \\lambda ) ) ^ 2 \\ \\Vert T ^ * T x \\Vert ^ 2 , \\end{align*}"}
-{"id": "4317.png", "formula": "\\begin{align*} ( s - S ) \\log s = \\begin{cases} > 0 & s < S , \\\\ < 0 & s > S , \\\\ = 0 & s = S , \\end{cases} ( s - S ) \\log s \\to \\begin{cases} 0 & s \\to 1 , \\\\ \\infty & s \\to 0 . \\end{cases} \\end{align*}"}
-{"id": "3382.png", "formula": "\\begin{align*} P _ { F : p _ 0 , p _ 1 } = \\frac { \\log | \\lambda _ 0 ^ s | } { \\log | \\lambda _ 1 ^ u | } \\end{align*}"}
-{"id": "3080.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } u _ \\epsilon ( 0 ) = 0 \\quad \\mbox { a n d } \\liminf _ { \\epsilon \\to 0 } \\frac { \\| u _ \\epsilon ( 0 ) \\| } { \\epsilon ^ \\alpha } > 0 \\ , . \\end{align*}"}
-{"id": "719.png", "formula": "\\begin{align*} \\Gamma ( \\chi ) = ( 1 + \\alpha _ - ) \\frac { T _ - } { T _ + } \\chi ^ 2 + 2 \\left [ 2 ( 1 + \\alpha _ - ) \\frac { T _ - } { T _ + } + \\frac { T _ { \\rm i } } { T _ + } \\alpha _ - - 2 \\right ] \\chi - 1 = 0 . \\end{align*}"}
-{"id": "2731.png", "formula": "\\begin{align*} & \\displaystyle \\mathcal F ^ { + } ( h , h ) = \\int _ { \\mathbb R ^ { d } \\times S ^ { d - 1 } } \\ , \\phi ( | v - v _ { \\ast } | ) \\ , b ( \\cos \\theta ) M _ { \\ast } \\ , h _ { \\ast } ^ { \\prime } \\ , h ^ { \\prime } \\ , d v _ { \\ast } d \\sigma \\ , , \\\\ [ 2 p t ] & \\displaystyle \\mathcal F ^ { - } ( h , h ) = - \\int _ { \\mathbb R ^ { d } \\times S ^ { d - 1 } } \\ , \\phi ( | v - v _ { \\ast } | ) \\ , b ( \\cos \\theta ) M _ { \\ast } \\ , h _ { \\ast } \\ , h \\ , d v _ { \\ast } d \\sigma \\ , . \\end{align*}"}
-{"id": "5610.png", "formula": "\\begin{align*} \\int _ 0 ^ t D G ( X _ r ) d \\mathbf { X } _ r & = \\int _ 0 ^ t D G ( X _ r ) f ( X _ r ) d r + \\int _ 0 ^ t D G ( X _ r ) \\sigma ( X _ r ) d \\mathbf { B } ^ H _ r \\\\ & = \\int _ 0 ^ t D G ( X _ r ) f ( X _ r ) d r + \\mathbf { B } ^ H _ t \\end{align*}"}
-{"id": "8139.png", "formula": "\\begin{align*} h _ { k , l , c , i , j } = \\sum _ { q = 1 } ^ Q \\sum _ { t \\in B _ { k , l , c , q } } \\frac { q } { Q } u _ { t \\Lambda _ { k , i , j } ( c ) c ^ { - 1 } t ^ { - 1 } } \\alpha _ { t c } ( h _ k ) . \\end{align*}"}
-{"id": "8444.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { t , l , v } ) = \\omega _ { \\pi } ( - v ^ { - 1 } ) \\chi _ 2 ( \\varpi ^ { - t - 2 l } ) \\zeta _ F ( 1 ) ^ { - 1 } q ^ { - \\frac { t } { 2 } } \\int _ { 1 + \\varpi ^ l \\mathcal { O } } \\omega _ { \\pi } ( x ) \\psi ( - x v ^ { - 1 } \\varpi ^ { t + l } ) d ^ { \\times } x . \\end{align*}"}
-{"id": "7316.png", "formula": "\\begin{align*} \\widehat { q } ^ k ( \\psi ^ { - 1 } \\sigma \\psi ) ( x _ i ) = \\sum _ { j \\in I } a _ j \\widehat { q } ^ { \\ell _ j } \\xi _ j , \\end{align*}"}
-{"id": "5804.png", "formula": "\\begin{align*} - \\Delta _ { p } u = u ^ { q } \\sigma + \\mu \\ ; \\ ; \\Omega , \\end{align*}"}
-{"id": "9989.png", "formula": "\\begin{align*} H _ { \\beta , T } ( W , B ) = \\beta \\int _ 0 ^ T \\int _ { \\mathbb { R } ^ d } \\phi ( y - W _ s ) \\dot B ( s , \\d y ) \\d s - \\frac { \\beta ^ 2 T } { 2 } V ( 0 ) \\end{align*}"}
-{"id": "6816.png", "formula": "\\begin{align*} \\norm { \\phi } _ i = \\sup \\limits _ { \\bigcup \\limits _ { j = 1 } ^ 4 \\lambda ^ { - 1 } \\left ( \\Pi _ { \\xi _ j } ( B ( 0 , R _ 2 ) ) \\right ) } \\abs { \\phi } . \\end{align*}"}
-{"id": "9639.png", "formula": "\\begin{align*} \\Psi _ { \\rm { R } } = \\begin{cases} \\left ( \\psi _ { \\rm { R } } ( 1 ) , \\cdots , \\psi _ { \\rm { R } } ( | N ( \\tau _ i ) - N ( \\tau _ { i + 1 } ) | ) \\right ) , ~ & N ( \\tau _ i ) \\neq N ( \\tau _ { i + 1 } ) \\\\ \\quad \\quad \\quad \\quad \\quad \\ , \\ , \\ , \\ , \\varnothing , ~ & N ( \\tau _ i ) = N ( \\tau _ { i + 1 } ) . \\end{cases} \\end{align*}"}
-{"id": "2587.png", "formula": "\\begin{align*} \\overline \\rho = \\mathcal C _ a f _ { \\overline \\rho } . \\end{align*}"}
-{"id": "8360.png", "formula": "\\begin{align*} s ( a ) = a ^ { - 1 } \\ell _ * \\ell + \\ell \\ell _ * \\in C ^ + ( W ) ^ \\times . \\end{align*}"}
-{"id": "3653.png", "formula": "\\begin{align*} \\begin{pmatrix} - 1 & - 1 & \\cdots & - 1 & - 1 \\\\ 0 & 0 & \\cdots & 0 & 0 \\\\ 1 & 0 & \\cdots & 0 & 0 \\\\ 1 & 1 & \\ddots & 0 & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 1 & 1 & \\cdots & 1 & 0 \\\\ 1 & 1 & \\cdots & 1 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "6212.png", "formula": "\\begin{align*} \\begin{pmatrix} - 2 & 0 & 1 \\\\ 0 & - 2 & 1 \\\\ 1 & 1 & 2 k \\end{pmatrix} d = 4 + 8 k . \\end{align*}"}
-{"id": "9368.png", "formula": "\\begin{align*} R _ \\textrm { e c h r } & = \\log _ 2 \\left ( \\frac { \\pi \\sqrt { \\mathrm { e } } \\sigma ^ 2 } { 3 D } \\right ) . \\end{align*}"}
-{"id": "1652.png", "formula": "\\begin{align*} f _ 1 e = e f _ 2 \\quad e f _ 1 = f _ 2 e . \\end{align*}"}
-{"id": "3423.png", "formula": "\\begin{align*} f ( [ 1 , 2 ] \\cup [ - 2 , - 1 ] ) \\cap \\Gamma _ 1 = f ( [ 1 , 2 ] \\cup [ - 2 , - 1 ] ) \\cap F _ 1 ( \\overline D _ 1 ) = \\{ f ( 1 ) , f ( - 1 ) \\} . \\end{align*}"}
-{"id": "3477.png", "formula": "\\begin{align*} b \\colon \\R \\to \\R , b ( x ) = \\begin{cases} | x | ^ { \\tilde { p } - 2 } x , & | x | \\leq R , \\\\ R ^ { \\tilde { p } - 2 } x , & | x | > R , \\end{cases} \\end{align*}"}
-{"id": "5640.png", "formula": "\\begin{align*} \\sigma : \\varphi ( I ) \\to X \\ , \\sigma ( \\varphi ( t ) ) = \\gamma ( t ) \\end{align*}"}
-{"id": "8024.png", "formula": "\\begin{align*} \\Big [ \\sum _ { l = M } ^ { L - M } { \\sum _ { P \\in \\mathcal { Q } ( l ) } { \\int _ { P } { \\int _ { \\Omega } { \\Big ( \\sum _ { n = 1 } ^ { L - l } { 2 ^ { - 2 { s } { t _ n } } \\sum _ { Q \\in \\mathcal { V } _ n ( l , P ) } { \\theta _ Q ( \\omega ) \\big | { \\Gamma } _ { { t _ l } } \\ast \\chi _ Q ( x ) \\big | ^ 2 } } \\Big ) ^ { { q } / { 2 } } } d \\lambda } d x } } \\Big ] ^ { { 1 } / { q } } . \\end{align*}"}
-{"id": "9701.png", "formula": "\\begin{align*} & \\varphi _ { k , k - 1 } ( \\gamma _ 1 , \\beta _ 1 , \\omega _ k , \\alpha _ 2 , \\alpha _ 3 , \\alpha _ 5 ) \\\\ : = & \\Big ( \\Phi _ { 1 } ^ { ( 1 ) } ( \\gamma _ 1 ; V _ a ) , \\Phi _ { 1 } ^ { ( 2 ) } ( \\gamma _ 1 ; V _ a ) \\Big ) \\cdot \\textbf { n } _ { k } - \\Big ( \\Phi _ { 1 } ^ { ( 1 ) } ( \\beta _ 1 ; \\tilde { \\Phi } ( \\alpha _ 5 , \\alpha _ 3 , \\alpha _ 2 ; V _ a ) ) , \\Phi _ { 1 } ^ { ( 2 ) } ( \\beta _ 1 ; \\tilde { \\Phi } ( \\alpha _ 5 , \\alpha _ 3 , \\alpha _ 2 ; V _ a ) ) \\Big ) \\cdot \\textbf { n } _ { k - 1 } . \\end{align*}"}
-{"id": "2626.png", "formula": "\\begin{align*} V * ( \\rho _ 1 - \\rho _ 2 ) = C _ 1 - C _ 2 ( 0 , 1 ) . \\end{align*}"}
-{"id": "2390.png", "formula": "\\begin{align*} p ( x ) = p _ 1 ( x ) p _ 2 ( x ) \\end{align*}"}
-{"id": "1063.png", "formula": "\\begin{align*} \\log H _ { k + 1 } - \\frac { q _ { k } } { 2 n - 2 } = L _ { R _ { k + 1 } } ^ { \\ast } ( q _ { k } ) . \\end{align*}"}
-{"id": "6200.png", "formula": "\\begin{align*} \\begin{pmatrix} - 2 & 0 & k & m \\\\ 0 & - 2 & l & n \\\\ k & l & 0 & 1 \\\\ m & n & 1 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "4807.png", "formula": "\\begin{align*} \\phi \\colon \\mathsf { Z } ( M ) \\to M \\ \\ \\ \\ \\phi ( a ) = a \\ \\ \\ \\ a \\in \\mathcal { A } ( M ) \\end{align*}"}
-{"id": "3914.png", "formula": "\\begin{align*} \\left ( - \\frac { \\hbar ^ 2 } { 2 m } \\left ( \\frac { d ^ 2 } { d \\rho ^ 2 } + \\frac { 3 } { \\rho } \\frac { d } { d \\rho } - \\frac { \\cal { L } ( \\cal { L } + 2 ) } { \\rho ^ 2 } \\right ) + K \\rho ^ 2 - { \\cal E } \\right ) R ( \\rho ) = 0 , \\end{align*}"}
-{"id": "2652.png", "formula": "\\begin{align*} h ^ * _ i = \\begin{cases} 2 , & \\mbox { i f } \\begin{cases} 2 ( k + 1 ) - \\alpha \\leq i \\leq 2 ( k + 1 ) - 1 , \\ , \\\\ 3 ( k + 1 ) - \\alpha - \\beta \\leq i \\leq 3 ( k + 1 ) - \\alpha - 1 , \\ , \\\\ 4 ( k + 1 ) - 2 \\alpha - \\beta \\leq i \\leq 4 ( k + 1 ) - \\alpha - \\beta - 1 ; \\ , \\\\ \\end{cases} \\\\ 1 , & \\mbox { o t h e r w i s e . } \\\\ \\end{cases} \\end{align*}"}
-{"id": "1689.png", "formula": "\\begin{align*} \\lambda \\ , T _ x = \\lambda , T _ x \\ , e = e . \\end{align*}"}
-{"id": "4898.png", "formula": "\\begin{align*} \\det { \\left ( U ^ { ^ { * } } T U \\right ) _ { _ { N E } } } & = \\left ( \\frac { - 2 \\gamma ^ 2 \\sigma ^ 2 } { \\beta + 6 } \\right ) \\left ( \\frac { 1 } { \\frac { e ^ { i \\theta } \\sigma ^ 2 } { \\gamma ^ 2 } } + \\frac { e ^ { i \\theta } \\sigma ^ 2 } { \\gamma ^ 2 } - \\beta \\right ) , \\end{align*}"}
-{"id": "5335.png", "formula": "\\begin{align*} ( v + O ( V ) ) \\cdot w = o ( v ) w \\end{align*}"}
-{"id": "3926.png", "formula": "\\begin{align*} \\frac { \\partial ^ \\gamma u } { \\partial t ^ \\gamma } + u \\frac { \\partial u } { \\partial x } - \\nu \\frac { \\partial ^ 2 u } { \\partial x ^ 2 } = 0 , \\ , \\ , \\ , \\ , \\ , \\ , a \\leq x \\leq b , \\ , \\ , \\ , \\ , \\ , 0 < \\gamma < 1 , \\ , \\ , \\ , \\ , t > 0 . \\end{align*}"}
-{"id": "8270.png", "formula": "\\begin{align*} \\mathfrak { M } ( L , j , \\delta ) = \\bigg \\{ \\gamma \\in \\Gamma ( i , \\alpha _ 1 ^ j \\alpha _ 2 ^ j ) \\colon u _ { \\nu } ( \\gamma _ { \\nu } . P _ { \\nu } , P _ { \\nu } ) \\leq \\delta _ { \\nu } \\nu \\alpha _ 1 , \\alpha _ 2 \\in \\mathcal { P } ( L ) \\bigg \\} \\end{align*}"}
-{"id": "6605.png", "formula": "\\begin{align*} \\liminf _ { J ' \\ni m \\to \\infty } \\int _ { \\Omega _ n ^ * } \\left | D f _ n ^ { - 1 } - D g _ { m , n } ^ { - 1 } \\right | ^ { p ^ * } \\ , d \\mu \\ ; = \\ ; 0 \\ . \\end{align*}"}
-{"id": "6644.png", "formula": "\\begin{align*} f ( t ) & = \\left | ( \\Phi _ t ( x ) - x ) - ( \\Phi _ t ( y ) - y ) \\right | \\\\ & \\leq \\int _ 0 ^ t \\left | X ( \\Phi _ s ( x ) ) - X ( \\Phi _ s ( y ) ) \\right | \\ , d s \\\\ & \\leq \\int _ 0 ^ t M \\left | \\Phi _ s ( x ) - \\Phi _ s ( y ) \\right | \\ , d s \\\\ & \\leq M \\int _ 0 ^ t | x - y | + \\left | ( \\Phi _ s ( x ) - x ) - ( \\Phi _ s ( y ) - y ) \\right | \\ , d s \\\\ & = M t | x - y | + \\int _ 0 ^ t M f ( s ) \\ , d s \\ . \\end{align*}"}
-{"id": "6296.png", "formula": "\\begin{align*} \\left \\{ b ; \\varepsilon , g , ( f + s , k _ 1 + k _ 3 ) , ( t , k _ 2 ) ; \\{ ( \\alpha _ i , \\beta _ i ) \\} _ { i = 1 } ^ n \\right \\} \\end{align*}"}
-{"id": "5383.png", "formula": "\\begin{align*} G = \\langle a , b , c \\mid a ^ 2 , b ^ 2 , c ^ 2 , ( a b ) ^ 2 , ( a c ) ^ 6 , ( b c ) ^ 6 , ( a b c ) ^ 6 , ( a \\cdot b ^ c ) ^ { r _ 1 } , ( a b \\cdot b ^ c ) ^ { r _ 2 } , ( a b \\cdot a ^ c ) ^ { r _ 3 } \\rangle \\end{align*}"}
-{"id": "5124.png", "formula": "\\begin{align*} { \\mathcal A } _ p & ( ( X _ \\lambda ) _ { \\lambda \\in \\Lambda _ { d , n } } ) = \\det _ V \\left ( X _ { ( 1 , 1 ) } f _ V ( p ) \\wedge \\cdots \\wedge X _ { ( j _ * , 1 ) } \\cdots X _ { ( j _ * , \\kappa _ { n } - 1 ) } f _ V ( p ) \\right ) \\\\ & \\times \\det _ W \\left ( X _ { ( j _ * + 1 , 1 ) } \\cdots X _ { ( j _ * + 1 , \\kappa _ { n } ) } f _ W ( p ) \\wedge \\cdots \\wedge X _ { ( n , 1 ) } \\cdots X _ { ( n - 1 , \\kappa _ { n } ) } f _ W ( p ) \\right ) . \\end{align*}"}
-{"id": "4382.png", "formula": "\\begin{align*} R _ \\gamma : & = ( A _ { x } ^ { - 1 } P _ \\alpha + Q _ \\alpha ) \\Big ( C _ { z _ \\gamma } ( A P _ \\alpha + Q _ \\alpha ) C _ { - z _ \\gamma } - ( A _ { x } P _ \\alpha + Q _ \\alpha ) \\Big ) M _ { \\chi _ { B ( 0 , R ) } } \\\\ & = ( A _ { x } ^ { - 1 } P _ \\alpha + Q _ \\alpha ) C _ { z _ \\gamma } ( A P _ \\alpha + Q _ \\alpha ) C _ { - z _ \\gamma } M _ { \\chi _ { B ( 0 , R ) } } - M _ { \\chi _ { B ( 0 , R ) } } \\end{align*}"}
-{"id": "2172.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } \\left \\| w ( s ) - m ( \\varphi ) \\psi _ d \\right \\| _ { C ^ 2 ( \\{ R ^ { - 1 } \\le | y | \\le R \\} ) } = 0 , \\lim _ { s \\to \\infty } \\left \\| ( \\partial _ s w ) ( s ) \\right \\| _ { C ^ 2 ( \\{ R ^ { - 1 } \\le | y | \\le R \\} ) } = 0 . \\end{align*}"}
-{"id": "9115.png", "formula": "\\begin{align*} f _ { n + 1 - i } = ( - 1 ) ^ { i } \\sum _ { k = 0 } ^ { i } \\binom { n + 1 - i + k } { k } D _ { n - i + k } \\left ( i = 0 , \\ldots , n \\right ) , \\end{align*}"}
-{"id": "5567.png", "formula": "\\begin{align*} \\Delta \\left ( \\alpha , \\beta \\right ) = e ^ { T } \\left [ \\Gamma + P \\Gamma P ^ { - 1 } \\right ] \\bar { w } _ { 2 ^ { k } } \\left ( \\tau \\right ) \\end{align*}"}
-{"id": "497.png", "formula": "\\begin{align*} V _ 1 ( x , t ) & \\sim - \\frac { R } { 4 } \\frac { y _ \\omega ^ { 2 } } { \\sin ( y _ \\omega ) ^ { 2 } } + \\frac { R } { 2 } \\frac { y _ \\omega ^ { 2 } } { \\sin ( y _ \\omega ) ^ { 2 } } = \\frac { R } { 4 } \\frac { y _ \\omega ^ { 2 } } { \\sin ( y _ \\omega ) ^ { 2 } } \\asymp d ( x , t ) ^ 2 \\end{align*}"}
-{"id": "4778.png", "formula": "\\begin{align*} V ^ i ( { L } _ i P _ { , i } + 2 { L } _ i ' ) = 0 . \\end{align*}"}
-{"id": "676.png", "formula": "\\begin{gather*} \\mu _ n ( q , \\alpha _ 0 , \\beta ) = \\mu _ n ( q _ 0 , \\alpha _ 0 , \\beta _ 0 ) , \\\\ a _ n ( q , \\alpha _ 0 , \\beta ) \\geq a _ n ( q _ 0 , \\alpha _ 0 , \\beta _ 0 ) , \\end{gather*}"}
-{"id": "7293.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r \\mu _ i = 3 ( s - 1 ) { \\rm a n d } \\sum _ { i = 1 } ^ r \\mu _ i ^ 2 = s ( s - 1 ) . \\end{align*}"}
-{"id": "9079.png", "formula": "\\begin{align*} f ( p q ) = p f ( q ) + q f ( p ) \\end{align*}"}
-{"id": "6585.png", "formula": "\\begin{align*} g _ n = \\psi _ n \\circ g \\circ \\varphi _ n ^ { - 1 } \\colon \\varphi _ n ( U _ n \\cap g ^ { - 1 } ( V _ n ) ) \\longrightarrow \\psi _ n ( g ( U _ n ) \\cap V _ n ) \\ . \\end{align*}"}
-{"id": "3710.png", "formula": "\\begin{align*} w _ j = w _ j ( \\tilde \\gamma _ 1 ) ( j = 1 , \\dots , n ) \\end{align*}"}
-{"id": "6491.png", "formula": "\\begin{align*} P ^ { } \\left ( x _ { A } , x _ { B } | \\mu _ { A } , \\mu _ { B } \\right ) = \\frac { 1 } { \\mu _ { A } \\mu _ { B } } \\exp \\left [ - \\left ( \\frac { x _ { A } } { \\mu _ { A } } + \\frac { x _ { B } } { \\mu _ { B } } \\right ) \\right ] , \\end{align*}"}
-{"id": "7771.png", "formula": "\\begin{align*} u _ { 1 } ( \\ell ) : = \\left \\{ \\begin{array} { l l } u ( \\ell ) & | \\ell | > R _ { 1 } , \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "9362.png", "formula": "\\begin{align*} \\lambda ( x ^ k ) = N \\frac { f ( x ^ k ) ^ { \\frac { k } { k + r } } } { \\int _ { \\mathbb { R } ^ k } f ( x ^ k ) ^ { \\frac { k } { k + r } } \\mathrm { d } x ^ k } , \\end{align*}"}
-{"id": "8356.png", "formula": "\\begin{align*} K _ p = G ( \\Q _ p ) \\cap C ( V _ { \\Z _ p } ) ^ \\times \\end{align*}"}
-{"id": "1083.png", "formula": "\\begin{align*} \\mathbf { H } ' = \\begin{bmatrix} | \\alpha _ { 1 1 } | ^ 2 & \\dots & | \\alpha _ { N 1 } | ^ 2 \\\\ \\vdots & \\ddots & \\vdots \\\\ | \\alpha _ { 1 N } | ^ 2 & \\dots & | \\alpha _ { N N } | ^ 2 \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "6301.png", "formula": "\\begin{align*} \\tau : \\mathbb { T } ^ 2 \\times \\mathbb { S } ^ 1 \\to \\mathbb { T } ^ 2 \\times \\mathbb { S } ^ 1 , \\tau ( p , z ) = ( \\sigma ( p ) , \\bar { z } ) . \\end{align*}"}
-{"id": "981.png", "formula": "\\begin{align*} F _ n ( t ) = M _ n ^ 0 ( t ) / \\mathfrak { s } _ n ( t ) \\end{align*}"}
-{"id": "3127.png", "formula": "\\begin{align*} \\hat { K } : = \\frac { N - 1 } { \\mu _ f } \\log \\left ( \\frac { f ( x ^ 0 ) - f ^ \\ast } { \\epsilon \\rho } \\right ) . \\end{align*}"}
-{"id": "4584.png", "formula": "\\begin{align*} q q _ 1 ^ 2 \\tilde \\theta ^ { - 1 } \\bigl ( \\tilde H _ { 0 , 1 } \\bigr ) \\equiv F _ { 1 , 0 } F _ { 0 , 0 } - ( 1 + q _ 1 q _ 3 ^ { - 1 } - q _ 1 ^ { 2 } ) F _ { 0 , 0 } F _ { 1 , 0 } + q _ 1 F _ { 0 , 1 } F _ { 1 , - 1 } \\ , . \\end{align*}"}
-{"id": "527.png", "formula": "\\begin{align*} \\partial _ k \\circ \\partial _ { k + 1 } = 0 \\end{align*}"}
-{"id": "1733.png", "formula": "\\begin{align*} { \\mathcal { S } } : = \\Big \\{ \\sum _ { i = 1 } ^ { n } a _ { i } t _ { \\lambda _ i } t _ { \\lambda _ i } ^ * \\chi _ X \\ | \\ n \\in \\N , \\lambda _ i \\in \\Lambda , a _ i \\in \\C \\Big \\} = \\Big \\{ \\sum _ { i = 1 } ^ { n } a _ { i } \\chi _ { R _ { \\lambda _ i } } \\ | \\ \\ n \\in \\N , \\lambda _ i \\in \\Lambda , a _ i \\in \\C \\Big \\} \\end{align*}"}
-{"id": "2098.png", "formula": "\\begin{align*} \\frac { \\partial f } { \\partial t } = \\tau ( f ) \\end{align*}"}
-{"id": "7482.png", "formula": "\\begin{align*} \\Delta ^ h f = \\dfrac { 1 } { h } \\delta _ \\alpha \\big [ h h ^ { \\bar { \\gamma } \\alpha } ( \\delta _ { \\bar { \\gamma } } f ) \\big ] - \\big [ h ^ { \\bar { \\gamma } \\alpha } ( \\delta _ { \\bar { \\gamma } } f ) \\big ] \\mathcal { C } _ \\alpha \\end{align*}"}
-{"id": "635.png", "formula": "\\begin{align*} \\lVert h _ { T _ 0 ^ c } \\rVert _ { 2 , p } ^ p & \\leq \\omega \\lVert h _ { T _ 0 } \\rVert _ { 2 , p } ^ p + ( 1 - \\omega ) \\lVert h _ { \\tilde { T } \\cap T _ 0 ^ c } \\rVert _ { 2 , p } ^ p + ( 1 - \\omega ) \\lVert h _ { \\tilde { T } ^ c \\cap T _ 0 ^ c } \\rVert _ { 2 , p } ^ p \\\\ & = \\omega \\lVert h _ { T _ 0 } \\rVert _ { 2 , p } ^ p + ( 1 - \\omega ) \\lVert h _ S \\rVert _ { 2 , p } ^ p , \\end{align*}"}
-{"id": "10021.png", "formula": "\\begin{align*} R _ Q = \\left ( \\begin{matrix} \\alpha & \\beta \\\\ \\frac { D } { Q } \\gamma & Q \\delta \\end{matrix} \\right ) \\in \\Gamma _ 0 ( D / Q ) \\end{align*}"}
-{"id": "5036.png", "formula": "\\begin{align*} 2 = | f ( \\widetilde { x } + \\widetilde { y } ) | & = | f ( x + y ) + [ T ^ * f ] ( u + v ) | \\\\ & \\leq \\| f \\| _ { X ^ * } \\| x + y \\| + \\| T ^ * f \\| _ Y \\| u + v \\| \\\\ & \\leq 2 ( \\| f \\| _ { X ^ * } + \\| T ^ * f \\| ) = 2 , \\end{align*}"}
-{"id": "8855.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ N P _ n ^ { ( d ) } ( \\langle \\mathbf { x } _ i , \\mathbf { x } _ j \\rangle ) \\geq 0 \\end{align*}"}
-{"id": "3649.png", "formula": "\\begin{align*} x = \\frac { t _ 1 \\dots t _ { n + 1 } } { s } \\mbox { a n d } y = s . \\end{align*}"}
-{"id": "2350.png", "formula": "\\begin{align*} \\left ( m _ { i j } \\tau _ s ^ \\C ( h ) \\right ) _ { i , j = 1 , \\ldots , d } . \\end{align*}"}
-{"id": "9343.png", "formula": "\\begin{align*} J O _ { n } ^ { ( 3 ) } \\pm j O _ { n } ^ { ( 3 ) } = \\sum _ { s = 0 } ^ { 7 } ( J _ { n + s } ^ { ( 3 ) } \\pm j _ { n + s } ^ { ( 3 ) } ) e _ { s } . \\end{align*}"}
-{"id": "4274.png", "formula": "\\begin{align*} \\mathsf { Z } _ { \\alpha } ( u , \\mathbf { z } _ 2 , q _ 2 ) = \\Big [ \\mathsf { Z } ( t , u , \\mathbf { z } _ 1 , \\mathbf { z } _ 2 , q _ 1 , q _ 2 ) \\Big ] _ { t ^ k q ^ { W _ 1 \\cdot \\alpha } e ( \\mathbf { z } _ 1 \\cdot \\alpha ) } . \\end{align*}"}
-{"id": "645.png", "formula": "\\begin{align*} \\bar \\nabla _ { \\dot \\gamma } \\bar \\nabla _ { \\dot \\gamma } \\nabla _ J ^ k J = - \\bar R ( \\dot \\gamma , \\bar \\nabla _ J ^ k J ) \\dot \\gamma + \\sum ( \\bar \\nabla ^ { i _ r } \\bar R ) ^ { i _ r ' } * ( \\dot \\gamma ) ^ j * ( \\bar \\nabla _ J ^ { k _ p } J ) ^ { k _ p ' } * ( \\bar \\nabla _ { \\dot \\gamma } \\bar \\nabla _ J ^ { l _ q } J ) ^ { l _ q ' } , \\end{align*}"}
-{"id": "5221.png", "formula": "\\begin{align*} \\mu _ i ( \\lambda ) + \\mu _ { i + n } ( \\lambda ) = - c . \\end{align*}"}
-{"id": "2078.png", "formula": "\\begin{align*} ( \\Delta _ H - \\partial _ t ) u _ \\sigma ( p _ 1 , t _ 1 ) = \\sigma > 0 . \\end{align*}"}
-{"id": "1124.png", "formula": "\\begin{align*} \\Phi ^ c = \\sum _ { \\sigma \\in \\Phi ^ c } \\sigma . \\end{align*}"}
-{"id": "2954.png", "formula": "\\begin{align*} \\int ( \\partial \\Theta - t ) ^ + d \\mu \\geq \\int ( \\partial \\Theta - t ) f d \\mu = \\int f \\partial \\Theta d \\mu - t \\int f d \\mu . \\end{align*}"}
-{"id": "9043.png", "formula": "\\begin{align*} 2 i { \\mathcal H } ( \\Phi ( { \\mathcal H } ( w ' , w ) ) , w ' ) = \\left ( 0 , 2 i \\gamma \\delta \\bar \\varphi ^ 2 _ 2 \\bar w _ 1 w _ 1 ' w _ 2 ' + 2 i \\delta ^ 2 \\bar \\varphi ^ 2 _ 2 \\bar w _ 2 ( w _ 2 ' ) ^ 2 \\right ) . \\end{align*}"}
-{"id": "2939.png", "formula": "\\begin{align*} \\Phi _ \\mu ( t ) : = \\int ( \\partial \\Theta - t ) ^ + d \\mu . \\end{align*}"}
-{"id": "2563.png", "formula": "\\begin{align*} | c | ^ { 2 } - r ^ { 2 } & = | z | ^ { 2 } - | p _ { 0 } - z | ^ { 2 } \\\\ & = | p _ { 0 } | \\cdot ( | z | - | p _ { 0 } - z | ) \\\\ & = | p _ { 0 } | \\cdot ( | p _ { 0 } | - 2 | p _ { 0 } - z | ) \\\\ & = ( 1 - d _ { 0 } ) ( 1 - d _ { 0 } - 2 | p _ { 0 } - z | ) \\ , . \\end{align*}"}
-{"id": "5270.png", "formula": "\\begin{align*} \\| r _ { \\overline { g } } ^ { ( l ) } a - a r _ { \\overline { g } } ^ { ( l ) } \\| = \\epsilon / 2 + \\| f _ { \\overline { g } } ^ { ( l ) } a - a f _ { \\overline { g } } ^ { ( l ) } \\| < \\epsilon , \\end{align*}"}
-{"id": "9670.png", "formula": "\\begin{align*} \\begin{cases} ( \\rho u A ( x ) ) _ x = 0 , \\\\ ( ( \\rho u ^ 2 + p ) A ( x ) ) _ x = A ' ( x ) p , \\\\ \\big ( ( e + \\frac { 1 } { 2 } u ^ 2 + \\frac { p } { \\rho } ) \\rho u A ( x ) \\big ) _ x = q _ 0 A ( x ) \\rho \\phi ( T ) Z , \\\\ ( \\rho u Z A ( x ) ) _ x = - A ( x ) \\rho \\phi ( T ) Z . \\end{cases} \\end{align*}"}
-{"id": "1223.png", "formula": "\\begin{align*} | \\tau \\nabla u _ 2 ( x ) + ( 1 - \\tau ) \\nabla u _ 1 ( a x + b ) | \\not = 0 \\mbox { w h e n } \\ , \\ , x \\in O \\ , \\ , \\mbox { a n d } \\ , \\ , \\tau \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "6649.png", "formula": "\\begin{align*} K _ { \\delta } = \\bigcap \\limits _ { | K \\cap H ^ { - } | _ n \\leq \\delta | K | _ n } H ^ + , \\end{align*}"}
-{"id": "3797.png", "formula": "\\begin{align*} H \\big ( ( \\omega , U ) , z \\big ) = 1 _ { \\{ g ( \\omega , U ) = z \\} } . \\end{align*}"}
-{"id": "4715.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } ^ n } { \\mathrm { d } x ^ n } \\bigl ( ( f \\circ g ) ( x ) \\bigr ) = \\sum \\limits _ { k = 1 } ^ n \\frac { \\mathrm { d } ^ k f } { \\mathrm { d } x ^ k } ( g ( x ) ) B _ { n , k } \\Bigl ( \\frac { \\mathrm { d } g } { \\mathrm { d } x } ( x ) , \\frac { \\mathrm { d } ^ 2 g } { \\mathrm { d } x ^ 2 } ( x ) , \\ldots , \\frac { \\mathrm { d } ^ { ( n - k + 1 ) } g } { \\mathrm { d } x ^ { ( n - k + 1 ) } } ( x ) \\Bigr ) \\end{align*}"}
-{"id": "7246.png", "formula": "\\begin{align*} = \\int _ { \\hat { G } } ^ { } \\int _ { G } ^ { } f ( x ) ( \\bar { \\psi _ { n } } * _ { U } \\theta _ { \\pi } ) ( x ) d x d \\mu _ { \\pi } . \\end{align*}"}
-{"id": "9161.png", "formula": "\\begin{align*} \\Phi ( x , \\ldots , x , c x ) - x \\Phi ( x , \\ldots , x , c ) = \\dfrac { 1 } { n + 1 } \\sum _ { i = 0 } ^ { n } ( n + 1 - i ) a _ { n + 1 - i } \\left ( x ^ i A ( c x ^ { n + 1 - i } ) - x ^ { i + 1 } A ( c x ^ { n - i } ) \\right ) , \\end{align*}"}
-{"id": "8066.png", "formula": "\\begin{align*} p & = ( 1 , 1 ) = H _ 1 \\cap H _ 2 \\cap H _ 3 \\\\ q & = ( 1 , \\zeta _ 3 ^ 2 ) = H _ 1 \\cap H _ 3 \\cap H _ 4 \\\\ r & = ( 1 , \\zeta _ 3 ) \\subset H _ 1 \\cap H _ 3 \\\\ s & = ( \\zeta _ 3 , 1 ) = H _ 2 \\cap H _ 4 \\\\ t & = ( 1 , - \\zeta _ 3 ) \\subset H _ 1 \\cap H _ 4 \\end{align*}"}
-{"id": "3704.png", "formula": "\\begin{align*} \\tilde \\gamma _ 1 = ( p _ 1 , p _ 2 , \\dots , p _ n ) \\in W _ 1 \\times _ { \\mathbb A ^ { n + 1 } } W _ 2 \\times _ { \\mathbb A ^ { n + 1 } } \\cdots \\times _ { \\mathbb A ^ { n + 1 } } W _ n . \\end{align*}"}
-{"id": "3328.png", "formula": "\\begin{align*} D ( r _ s ) = \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K } { s + 1 + i } ( N - 1 ) ^ i N . \\end{align*}"}
-{"id": "1608.png", "formula": "\\begin{align*} \\operatorname { M C E } ( \\lambda , \\eta ) : = \\{ \\lambda \\alpha \\ , : \\ , ( \\alpha , \\beta ) \\in \\Lambda ^ { \\operatorname { m i n } } ( \\lambda , \\eta ) \\} = \\{ \\eta \\beta \\ , : \\ , ( \\alpha , \\beta ) \\in \\Lambda ^ { \\operatorname { m i n } } ( \\lambda , \\eta ) \\} . \\end{align*}"}
-{"id": "2442.png", "formula": "\\begin{align*} \\beta = e ^ { 2 } ( \\gamma _ { l } ) ^ { h + l } 2 ( 2 h + l + 1 ) ( h + l ) ! , \\end{align*}"}
-{"id": "10121.png", "formula": "\\begin{align*} \\boldsymbol S _ { D _ k } ( i ) & = \\boldsymbol S _ { D _ k } ( i - 1 ) + \\eta ( i ) e _ k ^ * ( i ) \\boldsymbol x _ k ( i ) \\bar { \\boldsymbol \\omega } _ k ^ H ( i - 1 ) \\\\ & + \\eta ( i ) \\big ( \\gamma d _ k ^ * ( i ) { \\boldsymbol I } _ { M , D } - \\delta { \\boldsymbol x } _ k ( i ) { \\boldsymbol x } _ { k } ^ H ( i ) { \\boldsymbol S } _ { D _ k } ( i - 1 ) \\big ) , \\end{align*}"}
-{"id": "9604.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } D _ k ( f _ m ) ( x ) = D _ k ( f ) ( x ) . \\end{align*}"}
-{"id": "9596.png", "formula": "\\begin{align*} \\langle f , D _ { k } ( g ) \\rangle = \\lim _ { M \\to \\infty } \\langle f , g _ { k , M } \\rangle . \\end{align*}"}
-{"id": "320.png", "formula": "\\begin{align*} q ( n ) : = q _ i ( ( n - i ) / 3 ) , r ( n ) : = r _ i ( ( n - i ) / 3 ) , \\mbox { f o r } n \\equiv i \\mod 3 , i = 0 , 1 , 2 , \\end{align*}"}
-{"id": "5833.png", "formula": "\\begin{align*} \\sup _ { j = 1 , \\dots , J ^ * } \\| \\phi ^ j \\| _ { \\dot H ^ 1 ( \\R ) } ^ 2 \\leq \\sum _ { j = 1 } ^ { J ^ * } \\| \\phi ^ j \\| _ { \\dot H ^ 1 ( \\R ) } ^ 2 \\leq 1 , \\end{align*}"}
-{"id": "6936.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\rightarrow 0 ^ + } \\frac { e ^ { \\varepsilon z \\cdot \\nu } } { \\| E _ \\varepsilon \\| ^ 2 } \\int e ^ { 2 \\varepsilon y \\cdot \\nu } \\chi _ { \\Xi } ( z + y ) \\chi _ { \\Xi } ( y ) \\ , d y = 1 \\end{align*}"}
-{"id": "4734.png", "formula": "\\begin{align*} \\beta ^ 2 c _ 0 + 2 \\beta \\gamma f _ 0 + \\gamma ^ 2 d _ 0 = 0 . \\end{align*}"}
-{"id": "3360.png", "formula": "\\begin{align*} \\mu ( m ) \\subseteq \\bigcup _ { p \\in \\mu ( m ) } \\bigcap _ { x \\leq p } \\mu ( x ) = \\bigcup _ { p \\in \\mu ( m ) } \\mu ( p ) \\subseteq \\mu ( m ) \\Longrightarrow \\mu ( m ) = \\bigcup _ { p \\in \\mu ( m ) } \\mu ( p ) , \\end{align*}"}
-{"id": "5181.png", "formula": "\\begin{align*} g _ { 1 } ( r ) \\frac { x - \\ell _ { r } } { r - \\ell _ { r } } - g _ { 1 , [ r , 1 ] } ( x ) & = g _ { 1 } ( r ) \\left ( \\frac { x - \\ell _ { r } } { r - \\ell _ { r } } - \\frac { x } { r } \\right ) \\\\ & = g _ { 1 } ( r ) \\left ( \\frac { r ( x - \\ell _ { r } ) - x ( r - \\ell _ { r } ) } { r ( r - \\ell _ { r } ) } \\right ) \\\\ & = - g _ { 1 } ( r ) \\left ( \\frac { \\ell _ { r } } { r } \\right ) \\left ( \\frac { r - x } { r - \\ell _ { r } } \\right ) = - g _ { 1 , [ r , 1 ] } ( \\ell _ { r } ) \\frac { r - x } { r - \\ell _ { r } } . \\end{align*}"}
-{"id": "367.png", "formula": "\\begin{align*} \\mathbb { P } ( S _ { n } \\geq x ) = \\big ( 1 + o ( 1 ) \\big ) \\sum _ { r , s \\in \\mathbb { Z } } \\mathbb { P } ( b _ { n , r , s } \\xi _ { - r , - s } \\geq x ) + \\mathbb { P } ( S _ { n } ^ { ( \\varepsilon x ) } \\geq x ) \\end{align*}"}
-{"id": "6630.png", "formula": "\\begin{align*} \\left ( \\int _ { \\varphi _ n ( E _ M ) } | D h _ n | ^ p \\ , d \\mu \\right ) ^ { \\frac { 1 } { p } } & = \\left ( \\int _ { \\varphi _ n ( E _ M ) } | D \\phi _ n ^ 0 \\circ f _ n | ^ p | D f _ n | ^ p \\ , d \\mu \\right ) ^ { \\frac { 1 } { p } } \\\\ & \\leq \\| D \\phi _ n ^ { 0 } \\| _ { L ^ \\infty } \\left ( \\int _ { \\varphi _ n ( E _ M ) } | D f _ n | ^ p \\ , d \\mu \\right ) ^ { \\frac { 1 } { p } } \\ . \\end{align*}"}
-{"id": "8060.png", "formula": "\\begin{align*} G ( \\Lambda ( \\lambda _ * \\mu _ 1 , \\dots , \\lambda _ * \\mu _ m ) ) = G \\bigl ( G ( \\lambda ( A _ { 1 i _ 1 } ) , \\dots , \\lambda ( A _ { m i _ m } ) ) : i _ 1 , \\dots , i _ m \\bigr ) . \\end{align*}"}
-{"id": "5737.png", "formula": "\\begin{align*} c ^ \\pm = \\lim \\limits _ { x _ 2 \\to \\pm \\infty } m ( x _ 2 ) . \\end{align*}"}
-{"id": "7262.png", "formula": "\\begin{align*} B _ 2 = \\sum _ { \\substack { u \\le n , v \\le m \\\\ ( u , n ) , ( v , m ) \\le T \\\\ u / n \\ne v / m } } \\mu \\Bigl ( \\Bigl ( \\frac { u - s / N } n , \\frac { u + s / N } n \\Bigr ) \\cap \\Bigl ( \\frac { v - s / N } m , \\frac { v + s / N } m \\Bigr ) \\Bigr ) . \\end{align*}"}
-{"id": "9827.png", "formula": "\\begin{align*} W ( x , y ) = x ^ n + \\sum _ { i = d } ^ n A _ i x ^ { n - i } y ^ i \\in { \\bf C } [ x , y ] ( A _ d \\ne 0 ) \\end{align*}"}
-{"id": "8030.png", "formula": "\\begin{align*} \\big \\Vert f _ L \\big \\Vert _ { F _ p ^ { { s } , q } } & \\lesssim \\Big [ \\int _ { \\mathbb { R } ^ d } { \\Big ( \\sum _ { n = M } ^ { L } { 2 ^ { { d } { t _ n } q / p } \\frac { 1 } { ( 1 + 2 ^ { { t _ n } } | x | ) ^ { { \\sigma } q } } } \\Big ) ^ { { p } / { q } } } d x \\Big ] ^ { { 1 } / { p } } \\\\ & \\leq \\Big ( \\int _ { \\mathbb { R } ^ d } { \\sum _ { n = M } ^ { L } { 2 ^ { d { t _ n } } \\frac { 1 } { ( 1 + 2 ^ { { t _ n } } | x | ) ^ { { \\sigma } p } } } } d x \\Big ) ^ { { 1 } / { p } } \\lesssim L ^ { { 1 } / { p } } . \\end{align*}"}
-{"id": "8968.png", "formula": "\\begin{gather*} c _ { w ' w W _ I } = { } ^ { w ' } c _ { w W _ I } \\end{gather*}"}
-{"id": "901.png", "formula": "\\begin{align*} P ( w ^ 1 _ I = 1 ) = P ( w ^ 1 _ I = - 1 ) = P ( w ^ 2 _ J = 1 ) = P ( w ^ 2 _ J = - 1 ) = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "899.png", "formula": "\\begin{align*} U _ n ^ * ( \\theta ) = \\sum _ { I \\in \\Pi ^ 1 _ n , J \\in \\Pi ^ 2 _ n } \\left ( w ^ 1 _ I X ^ 1 ( I ) \\right ) \\left ( w ^ 2 _ J X ^ 2 ( J ) \\right ) K ( I , J _ { - \\theta } ) . \\end{align*}"}
-{"id": "5387.png", "formula": "\\begin{align*} a ^ G = ( 1 , 2 ) ( 3 , 4 ) ^ G , \\ , b ^ G = ( 5 , 6 ) ^ G , \\ , ( a b ) ^ G = ( 1 , 2 ) ( 3 , 4 ) ( 5 , 6 ) ^ G . \\end{align*}"}
-{"id": "9606.png", "formula": "\\begin{align*} \\bigg | \\int _ { \\Bbb R ^ n } \\ k _ j ( u , v ) D _ { k ' } ^ \\# ( v , y ) d \\mu ( v ) \\bigg | & = \\bigg | \\int _ { \\Bbb R ^ n } \\ k _ j ( u , v ) [ D _ { k ' } ^ \\# ( v , y ) - D _ { k ' } ^ \\# ( u , y ) ] d \\mu ( v ) \\bigg | \\\\ & \\lesssim \\frac { 2 ^ { - | j - k ' | \\varepsilon } } { V _ { - k ' } ( y ) } \\int _ { \\Bbb R ^ n } \\ | k _ j ( u , v ) | d \\mu ( v ) \\\\ & \\le \\frac { 2 ^ { - | j - k ' | \\varepsilon } } { V _ { - k ' } ( y ) } . \\end{align*}"}
-{"id": "3760.png", "formula": "\\begin{align*} v _ { k + 1 } = v _ k ( 1 - L _ k ^ { - 1 / 1 6 } ) , k \\ge \\hat { k } . \\end{align*}"}
-{"id": "8744.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t ^ \\alpha u + F ( t , x , u , \\nabla u , \\nabla ^ 2 u ) = 0 \\quad & \\\\ \\nabla u \\cdot n ( x ) + \\rho = 0 \\quad & \\end{cases} \\end{align*}"}
-{"id": "4106.png", "formula": "\\begin{align*} \\langle p - q , \\xi ( p ) \\rangle = g ( q ) \\langle \\eta ( p ) , \\xi ( p ) \\rangle . \\end{align*}"}
-{"id": "748.png", "formula": "\\begin{align*} J ( u _ 1 ^ n , u _ 2 ^ n ) \\geq \\frac { \\Lambda _ 0 a _ 1 } { 2 } + m _ { \\mu _ 2 , p _ 2 } ( a _ 2 ) + o _ n ( 1 ) , \\end{align*}"}
-{"id": "6026.png", "formula": "\\begin{align*} & D _ { 1 + s } ( P _ { X ^ { n } Y ^ { n } | U _ { n } } \\| \\pi _ { X ^ { n } Y ^ { n } } | P _ { U _ { n } } ) \\\\ & = \\frac { 1 } { s } \\log \\mathbb { E } _ { U _ { n } } \\bigg [ \\sum _ { x ^ { n } , y ^ { n } } P _ { X ^ { n } Y ^ { n } | U _ { n } } ( x ^ { n } , y ^ { n } | U _ { n } ) \\\\ & \\qquad \\times \\left ( \\frac { P _ { X ^ { n } Y ^ { n } | U _ { n } } ( x ^ { n } , y ^ { n } | U _ { n } ) } { \\pi _ { X ^ { n } Y ^ { n } } ( x ^ { n } , y ^ { n } ) } \\right ) ^ { s } \\bigg ] , \\end{align*}"}
-{"id": "7753.png", "formula": "\\begin{align*} ( x u x ^ { - 1 } ) w ( b u x u ^ { - 1 } x ^ { - 1 } ) = [ x , u ] ( u w b ) [ x , u ] ^ { - 1 } \\in [ \\gamma ^ \\Gamma \\cup ( \\gamma ^ { - 1 } ) ^ \\Gamma ] ^ { C _ 1 } . \\end{align*}"}
-{"id": "7946.png", "formula": "\\begin{align*} \\bar W _ { i } : = h _ i ( \\bar W ) \\subset \\bar W . \\end{align*}"}
-{"id": "9051.png", "formula": "\\begin{align*} a _ { 1 2 } ^ 1 = 0 , a _ { 2 2 } ^ 1 = 0 , a _ { 1 1 } ^ 2 = 0 , a _ { 1 2 } ^ 2 = 0 . \\end{align*}"}
-{"id": "2860.png", "formula": "\\begin{align*} \\bar { \\Delta } V _ { n } & = \\sum _ { v = 1 } ^ { n - 1 } \\Delta _ { v } \\left ( \\frac { \\hat { a } _ { n v } P _ { v } \\lambda _ { v } } { v p _ { v } } \\right ) \\sum _ { r = 1 } ^ { v } a _ { r } + \\frac { \\hat { a } _ { n n } P _ { n } \\lambda _ { n } } { n p _ { n } } \\sum _ { r = 1 } ^ { n } a _ { r } \\\\ \\bar { \\Delta } V _ { n } & = \\sum _ { v = 1 } ^ { n - 1 } \\Delta _ { v } \\left ( \\frac { \\hat { a } _ { n v } P _ { v } \\lambda _ { v } } { v p _ { v } } \\right ) s _ { v } + \\frac { \\hat { a } _ { n n } P _ { n } \\lambda _ { n } } { n p _ { n } } s _ { n } , \\end{align*}"}
-{"id": "4692.png", "formula": "\\begin{align*} \\| G _ j - G \\| _ { L ^ 1 ( ( 0 , 1 ) ) } & = \\int _ 0 ^ 1 \\Big | \\int _ { B _ r } ( \\varphi _ j - \\varphi ) \\ , d x \\Big | \\ , d r \\leq \\| \\varphi _ j - \\varphi \\| _ { L ^ 1 ( B _ 1 ) } \\xrightarrow { j \\to \\infty } 0 , \\\\ \\| G _ j ' - G ' \\| _ { L ^ 1 ( ( 0 , 1 ) ) } & = \\int _ 0 ^ 1 \\Big | \\int _ { \\partial B _ r } ( \\varphi _ j - \\varphi ) \\ , d \\mathcal { H } ^ { n - 1 } \\Big | \\ , d r \\leq \\| \\varphi _ j - \\varphi \\| _ { L ^ 1 ( B _ 1 ) } \\xrightarrow { j \\to \\infty } 0 , \\end{align*}"}
-{"id": "2996.png", "formula": "\\begin{align*} d Y ( t ) & = \\Delta _ b Y ( t ) d t + f ( t , Y ( t ) ) d t + g ( t , Y ( t ) ) d W ( t ) , t \\in [ 0 , T ] , \\\\ Y ( 0 ) & = Y _ 0 \\end{align*}"}
-{"id": "1394.png", "formula": "\\begin{align*} j ( v ' ) = \\sum _ { I , J } \\dfrac { \\beta _ { I J } } { \\alpha _ { I J } } v _ { I } y _ { J } \\end{align*}"}
-{"id": "5360.png", "formula": "\\begin{align*} S _ { X , Y } = _ { X \\boxtimes Y } \\left ( c _ { Y , X } \\circ c _ { X , Y } \\right ) \\end{align*}"}
-{"id": "5319.png", "formula": "\\begin{align*} w \\left ( \\left [ \\begin{array} { c c } 0 & A X B ^ { \\ast } \\\\ B Y A ^ { \\ast } & 0 \\end{array} \\right ] \\right ) & = w ( C S C ^ { \\ast } ) \\\\ & \\leq \\Vert C \\Vert ^ { 2 } w ( S ) \\\\ & = \\max \\{ | | A | | ^ { 2 } , | | B | | ^ { 2 } \\} w \\left ( \\left [ \\begin{array} { c c } 0 & X \\\\ Y & 0 \\end{array} \\right ] \\right ) \\end{align*}"}
-{"id": "4129.png", "formula": "\\begin{align*} \\omega ( E _ 1 , E _ 2 ) = \\mathrm { d e t } [ E _ 1 , E _ 2 , \\eta ] = \\langle \\eta , \\xi \\rangle , \\end{align*}"}
-{"id": "7015.png", "formula": "\\begin{align*} J _ 1 = g ( \\epsilon ) ^ { n - 1 } \\epsilon ^ { - 1 / 2 } \\int _ { \\mathbb { R } ^ n } { \\frac { u ( \\bar { y } , y _ n ) - u ( \\bar { y } , 0 ) } { ( g ( \\epsilon ) ^ 2 | \\bar { y } | ^ 2 + \\frac { 1 } { \\epsilon } y _ n ^ 2 ) ^ { \\frac { n + 2 s } { 2 } } } d y } \\leq \\epsilon ^ s C _ 1 C _ 2 , \\end{align*}"}
-{"id": "4140.png", "formula": "\\begin{align*} \\max \\Biggl \\{ \\sum _ { i \\in N } \\sum _ { j \\in V } w _ { i j } z _ { i j } : \\exists ~ ( x , y ) \\geq 0 \\eqref { e q : o b m _ l p _ p o l 1 } \\eqref { e q : o b m _ l p _ p o l 2 } z _ { i j } = \\sum _ { t \\in [ n ] } \\sum _ { S \\subseteq V \\backslash j } x _ { i , j } ^ { t , S } \\Biggr \\} , \\end{align*}"}
-{"id": "5340.png", "formula": "\\begin{align*} ( W _ 1 \\otimes \\cdots \\otimes W _ m ) ^ * = \\hom ( W _ 1 \\otimes \\cdots \\otimes W _ m , \\C ) \\end{align*}"}
-{"id": "8439.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { - 2 , l , v } ) = q \\zeta _ F ( 1 ) ^ { - 2 } \\int _ { \\mathcal { O } ^ { \\times } } \\chi ( y _ 1 ) \\psi ( y _ 1 \\varpi ^ { - 1 } ) G ( \\varpi ^ { - 1 } + y _ 1 v \\varpi ^ { - l } , \\chi ) d ^ { \\times } y _ 1 \\end{align*}"}
-{"id": "7263.png", "formula": "\\begin{align*} 0 < | h | < ( n + m ) \\frac s N , \\frac u n - \\frac v m = \\frac h { m n } . \\end{align*}"}
-{"id": "4444.png", "formula": "\\begin{align*} \\| f _ \\ell \\| & = \\| f * \\int _ 0 ^ 1 \\omega ^ { t } _ \\ell * \\psi _ { t \\ell ^ 3 } d t \\| \\leq \\int _ 0 ^ 1 \\| \\omega ^ { t } _ \\ell * f _ { t \\ell ^ 3 } \\| d t \\leq \\int _ 0 ^ 1 \\| f _ { t \\ell ^ 3 } \\| \\| \\omega ^ { t } _ \\ell \\| _ { L ^ 1 } \\ , d t \\lesssim [ f ] _ \\beta \\int _ 0 ^ 1 ( t ^ { \\frac 1 3 } \\ell ) ^ \\beta \\ , d t \\lesssim [ f ] _ \\beta \\ell ^ \\beta , \\end{align*}"}
-{"id": "9682.png", "formula": "\\begin{align*} s _ i = \\frac { u _ a v _ a + ( - 1 ) ^ { \\frac { i + 3 } { 4 } } \\hat { c } _ a \\sqrt { u _ a ^ 2 + v _ a ^ 2 - \\hat { c } _ a ^ 2 } } { u _ a ^ 2 - \\hat { c } _ a ^ 2 } , \\end{align*}"}
-{"id": "4885.png", "formula": "\\begin{align*} ( A ( t ) ) ^ m = \\prod _ { n = 1 } ^ { \\infty } ( 1 - t ^ n ) ^ { - r _ n m } . \\end{align*}"}
-{"id": "8773.png", "formula": "\\begin{align*} \\begin{array} { l } \\Phi _ { \\mathcal { L } ( G ^ { ( R ) } \\otimes { \\{ G _ 1 , G _ 2 \\} } ) } ( x ) = \\det { B _ 1 } \\\\ \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; = \\det ( x I _ { n _ 2 } - \\mathcal { L } ( G _ 2 ) \\circ { C } ) ^ { m } \\cdot \\det ( x I _ { n _ 1 } - \\mathcal { L } ( G _ 1 ) \\circ { B } ) ^ { n } \\cdot \\det { S } , \\end{array} \\end{align*}"}
-{"id": "8378.png", "formula": "\\begin{align*} \\tilde { \\psi } _ g ( f ^ { [ i ] } ) = \\psi _ g ( f ^ { [ i ] } ) \\otimes \\bigotimes _ { m > 0 } \\bigotimes _ { \\substack { \\lambda \\in \\Lambda ^ { [ i ] } \\\\ Q ( \\lambda ) = m } } ( \\mathrm { o b s t } ^ { a n } _ \\lambda ) ^ { \\otimes - c ^ { [ i ] } ( - m , 0 ) } . \\end{align*}"}
-{"id": "4118.png", "formula": "\\begin{align*} X = \\rho D _ X \\eta + \\nabla _ X V . \\end{align*}"}
-{"id": "7468.png", "formula": "\\begin{align*} \\widetilde { \\rho } ^ k _ \\alpha = W ^ \\beta _ \\alpha \\rho ^ h _ \\beta \\dfrac { \\partial \\widetilde { z } ^ k } { \\partial z ^ h } , \\end{align*}"}
-{"id": "8665.png", "formula": "\\begin{align*} \\zeta _ { r k } = c _ 1 r k + c _ 2 + o ( 1 ) , \\end{align*}"}
-{"id": "1969.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { \\beta / 2 } ( f g ) = ( - \\Delta ) ^ { \\tilde { \\beta } / 2 } ( - \\Delta ) ^ k ( f g ) . \\end{align*}"}
-{"id": "7348.png", "formula": "\\begin{align*} b _ 1 * 2 * M _ 1 \\ ; = \\ ; b _ 2 * 2 * M _ 2 \\ ; = \\ ; 1 \\ , , \\end{align*}"}
-{"id": "75.png", "formula": "\\begin{align*} s ( \\alpha , \\beta ) = \\sum _ { i \\in A } 2 i + \\sum _ { i \\in B } ( 2 i + 1 ) = \\beta + 2 \\sum _ { i = 0 } ^ { \\alpha + \\beta - 1 } i = ( \\alpha + \\beta ) ^ 2 - ( \\alpha + \\beta ) + \\beta = ( \\alpha + \\beta ) ^ 2 - \\alpha . \\end{align*}"}
-{"id": "1874.png", "formula": "\\begin{align*} g : = \\sum _ { i \\in I } a ^ i \\ 1 _ { ( - \\infty , i ] } \\end{align*}"}
-{"id": "5647.png", "formula": "\\begin{align*} A _ K ( \\gamma ) \\geq \\sum _ { i = 0 } ^ { n - 2 } \\left ( \\inf \\limits _ { s \\in [ t ^ \\delta _ i , t ^ \\delta _ { i + 1 } ] } K ( \\gamma ( s ) ) \\right ) \\ d \\left ( \\gamma ( t ^ \\delta _ i ) , \\gamma ( t ^ \\delta _ { i + 1 } ) \\right ) . \\end{align*}"}
-{"id": "3675.png", "formula": "\\begin{align*} H _ i : c _ { i 1 } + c _ { i 2 } + \\dots + c _ { i n } = \\frac { i n } { n + 1 } ( i = 1 , \\dots , n ) . \\end{align*}"}
-{"id": "5790.png", "formula": "\\begin{align*} \\lim _ { j \\rightarrow \\infty } \\int _ { \\Omega } \\varphi \\ ; d \\mu [ u _ j ] = \\int _ { \\Omega } \\varphi \\ ; d \\mu [ u ] \\end{align*}"}
-{"id": "3913.png", "formula": "\\begin{align*} \\frac { \\langle H \\rangle ^ { k , { \\ell } } _ { m i n } } { { \\cal E } ^ { k , \\ell } _ { 0 } } = \\frac { \\sqrt { ( \\ell + k + \\frac { 1 } { 2 } ) } } { ( \\ell + k ) } \\Big ( \\frac { \\Gamma ( \\ell + k + \\frac { 1 } { 2 } ) } { \\Gamma ( \\ell + k ) } \\Big ) . \\end{align*}"}
-{"id": "3139.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ e ^ { u J _ { t } } \\right ] = \\exp \\left \\lbrace t \\int _ { 0 } ^ { \\infty } \\left ( e ^ { u z } - 1 \\right ) \\nu ( \\mathrm { d } z ) \\right \\rbrace , \\quad ( t , u ) \\in \\mathbb { R } _ { \\geqslant 0 } \\times \\mathcal { U } , \\end{align*}"}
-{"id": "4643.png", "formula": "\\begin{align*} E _ B ( q , ( x , t ) ) = \\bigcap _ { k \\ge q } \\{ e \\in \\mathcal { C } _ { d , D } : \\textup { o n e o f } \\ref { f r : 3 } , \\ref { f r : 4 } , \\ref { f r : 5 } \\textup { a t } \\beta \\tau \\textup { , f a i l s f o r } ( x , t ) \\textup { a n d } e \\} \\end{align*}"}
-{"id": "40.png", "formula": "\\begin{align*} G S N R = 1 0 l o g _ { 1 0 } \\left ( \\frac { 1 } { \\gamma M } \\sum _ { t = 1 } ^ { M } | s ( t ) ^ 2 | \\right ) \\end{align*}"}
-{"id": "1310.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\phi _ n ( g ) = g ~ ~ g \\in \\mathcal { G } ~ ~ \\implies ~ ~ \\lim _ { n \\to \\infty } \\phi _ n ( a ) = a ~ ~ a \\in A \\end{align*}"}
-{"id": "3699.png", "formula": "\\begin{align*} \\tilde \\gamma _ 1 = ( p _ 1 , \\dots , p _ n ) \\in W [ n ] = X [ n ] \\times _ { \\mathbb A ^ { n + 1 } } \\dots \\times _ { \\mathbb A ^ { n + 1 } } X [ n ] , \\end{align*}"}
-{"id": "6280.png", "formula": "\\begin{align*} \\frac { d } { d t } \\sum _ { q \\geq - 1 } \\lambda _ q ^ { 2 s } \\left ( \\| u _ q \\| _ 2 ^ 2 + \\| b _ q \\| _ 2 ^ 2 \\right ) \\lesssim & - \\sum _ { q \\geq - 1 } \\lambda _ q ^ { 2 s } \\left ( \\nu \\| \\nabla u _ q \\| _ 2 ^ 2 + \\mu \\| \\nabla b _ q \\| _ 2 ^ 2 \\right ) \\\\ & + f ( t ) \\sum _ { q \\geq - 1 } \\lambda _ q ^ { 2 s } \\| u _ q \\| _ 2 ^ 2 + Q f ( t ) \\sum _ { q \\geq - 1 } \\lambda _ q ^ { 2 s } \\| b _ q \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "1545.png", "formula": "\\begin{gather*} J \\ = \\ \\left \\{ \\left [ - \\frac { 1 } { 4 } , 2 + \\frac { 1 } { 4 } \\right ] , \\left [ 4 - \\frac { 1 } { 4 } , 5 + \\frac { 1 } { 4 } \\right ] , \\left [ 9 - \\frac { 1 } { 4 } , 9 + \\frac { 1 } { 4 } \\right ] , \\left [ 1 2 - \\frac { 1 } { 4 } , 1 4 + \\frac { 1 } { 4 } \\right ] \\right \\} . \\end{gather*}"}
-{"id": "3752.png", "formula": "\\begin{align*} p _ k ( \\rho ) = \\sup _ { m \\in M _ k } \\mathbb { P } ^ { \\rho } ( A _ m ) , k \\ge \\hat { k } . \\end{align*}"}
-{"id": "1462.png", "formula": "\\begin{align*} E _ u = E _ { 2 ^ { r + 1 } } = C \\langle 1 , y _ r , v _ { s } y _ { s } \\rangle \\oplus D _ 1 / ( v _ t e _ t ) \\{ x _ 3 ^ 2 \\} \\oplus D _ r \\langle x _ 3 ^ 2 y _ r , x _ { 3 } z _ { r } \\rangle . \\end{align*}"}
-{"id": "4465.png", "formula": "\\begin{align*} \\langle | F ^ \\ell ( k ) | ^ 2 \\rangle \\le 2 \\sum _ { k ' + k '' = k } G _ \\ell ^ 2 ( k ' ) \\tilde G _ \\ell ^ 2 ( k '' ) \\langle | \\xi ( k ' ) | ^ 2 | \\xi ( k '' ) | ^ 2 \\rangle , \\end{align*}"}
-{"id": "5812.png", "formula": "\\begin{align*} - \\Delta _ { p } u = - \\Delta _ { p } v = \\omega \\ ; \\ ; \\R ^ n . \\end{align*}"}
-{"id": "3389.png", "formula": "\\begin{align*} R _ i = ( P _ { G _ * , p _ { m _ i } ^ * , p _ { n _ i } ^ * } , Q _ { G _ * , p _ { m _ i } ^ * , p _ { n _ i } ^ * , q _ i ^ * } ) \\end{align*}"}
-{"id": "9875.png", "formula": "\\begin{align*} b ( a _ 0 \\otimes \\cdots \\otimes a _ n ) = \\sum _ { i = 0 } ^ { n - 1 } ( - 1 ) ^ i a _ 0 \\otimes \\cdots \\otimes a _ i a _ { i + 1 } \\otimes \\cdots \\otimes a _ n + ( - 1 ) ^ n a _ n a _ 0 \\otimes a _ 1 \\otimes \\cdots \\otimes a _ { n - 1 } , \\end{align*}"}
-{"id": "5490.png", "formula": "\\begin{align*} { \\mathrm { B } } ^ { - 1 } _ \\rho q ( x ) = - x ^ { 1 - \\rho } \\frac { \\dd } { \\dd x } [ x ^ \\rho q ( x ) ] , q \\in \\mathcal { P } _ { r , \\rho } ^ { 1 } . \\end{align*}"}
-{"id": "3124.png", "formula": "\\begin{align*} K : = \\frac { 2 ( N - 1 ) R ^ 2 ( x ^ 0 ) } { \\epsilon } \\left [ 1 + \\log \\left ( \\frac { \\tilde { R } ^ 2 ( x ^ 0 ) } { 2 R ^ 2 ( x ^ 0 ) \\rho } \\right ) \\right ] - N + 3 . \\end{align*}"}
-{"id": "5633.png", "formula": "\\begin{align*} \\left ( \\gamma : I \\to \\R ^ d \\right ) \\mapsto \\mathfrak { E } _ { W } ( \\gamma ) : = \\int _ I \\left ( \\frac { 1 } { 2 } | \\dot { \\gamma } | ^ 2 ( t ) + W ( \\gamma ( t ) ) \\right ) \\d t \\end{align*}"}
-{"id": "3670.png", "formula": "\\begin{align*} \\tilde P _ b [ n ] : = \\tilde P _ W [ n ] \\cap ( \\alpha _ W [ n ] \\otimes \\mathbb R ) ^ { - 1 } ( - b ) \\end{align*}"}
-{"id": "9904.png", "formula": "\\begin{align*} | W _ 0 | \\le \\sum _ { x \\in V ( G ) } ( q ( x ) - 6 ) = 2 ( 3 r - 5 ) n - 2 \\bigl ( d ( v _ 1 ) + d ( v _ 2 ) \\bigr ) + 3 \\bigl ( d ( v _ 3 ) + \\ldots + d ( v _ r ) \\bigr ) \\end{align*}"}
-{"id": "2206.png", "formula": "\\begin{align*} \\beta ( \\alpha , J ) ! = \\prod _ p \\bigl ( m _ { p j _ { p } } ( \\alpha _ { j _ p } + 1 ) - 1 \\bigr ) ! \\end{align*}"}
-{"id": "6718.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\mathbb { P } \\Bigg ( \\frac { R _ { N } } { \\beta _ { N } } > t \\Bigg ) = e ^ { - t } . \\end{align*}"}
-{"id": "7131.png", "formula": "\\begin{align*} \\partial _ t Y = ~ \\sqrt { ( 1 + | Y | ^ 2 ) ( 1 + \\langle N , Y \\rangle ^ 2 ) } F ^ { - 1 } ( \\mathcal { W } ^ X ) N , \\end{align*}"}
-{"id": "3799.png", "formula": "\\begin{align*} \\mathbb { P } ^ { \\rho } \\Big ( X ^ 0 _ n \\leq v _ \\star n \\Big ) & \\leq \\P ^ { \\rho } \\Big ( \\exists \\Big ) \\\\ & \\leq c ^ { - 1 } \\exp \\big ( - c ( \\log n ) ^ { 3 / 2 } \\big ) \\end{align*}"}
-{"id": "727.png", "formula": "\\begin{align*} \\frac { a ^ 2 \\theta _ + + V _ + ^ 2 } { a ^ 2 \\theta _ - + V _ - ^ 2 } = \\frac { \\rho _ - } { \\rho _ + } = \\frac { V _ + } { V _ - } . \\end{align*}"}
-{"id": "3446.png", "formula": "\\begin{align*} U _ 0 ( \\omega ) : = \\bigcap _ { j \\in \\N } U _ { \\frac { 1 } { j } } ( \\omega ) \\end{align*}"}
-{"id": "5224.png", "formula": "\\begin{align*} \\frac { d } { d z } \\omega ( Y _ 1 , Y _ 2 ) = - c \\ , \\omega ( Y _ 1 , Y _ 2 ) . \\end{align*}"}
-{"id": "4011.png", "formula": "\\begin{align*} D _ X Y = \\nabla _ X Y + h ( X , Y ) \\eta , \\end{align*}"}
-{"id": "7474.png", "formula": "\\begin{align*} C ^ { \\ : \\gamma } _ { \\alpha \\beta } = C ^ { \\ : \\gamma } _ { \\beta \\alpha } \\end{align*}"}
-{"id": "1416.png", "formula": "\\begin{align*} { Q } _ j ( x _ 3 ) = \\alpha _ { j 0 } x _ 3 ^ 2 , { Q } _ j ( x _ 4 ) = x _ 3 \\left ( \\sum _ { k = 0 } ^ { j - 1 } \\alpha _ { j k } x _ 4 ^ { 2 ^ k } \\right ) \\end{align*}"}
-{"id": "9743.png", "formula": "\\begin{align*} F ( J ) = L ( J ) + K Q ( J ) , \\end{align*}"}
-{"id": "5899.png", "formula": "\\begin{align*} \\begin{aligned} & \\{ \\tau ^ { N , i } \\ge 2 L N \\} = \\{ Z ^ { N , i } _ n = 0 , \\ \\ n = 1 , \\cdots , 2 L \\} = \\\\ & \\{ \\sigma ^ { ( e ) } _ { i } > 2 L , \\sigma ^ { ( o ) } _ { i } > 2 L - 1 \\} . \\end{aligned} \\end{align*}"}
-{"id": "732.png", "formula": "\\begin{align*} \\int _ { \\R ^ 3 } | u _ 1 | ^ 2 \\ , d x = a _ 1 , \\int _ { \\R ^ 3 } | u _ 2 | ^ 2 \\ , d x = a _ 2 . \\end{align*}"}
-{"id": "1308.png", "formula": "\\begin{align*} \\dim \\mathrm { H o m } ( \\theta _ { \\mathtt { 1 3 5 } } ^ { \\mathrm { o n } } \\theta _ { \\mathtt { 2 4 } } L ( \\mathtt { 1 2 1 4 5 4 } ) , \\theta _ { \\mathtt { 1 3 5 } } ^ { \\mathrm { o n } } L ( \\mathtt { 1 2 1 4 5 4 3 } ) ) = 1 . \\end{align*}"}
-{"id": "3624.png", "formula": "\\begin{align*} w _ \\lambda ( x ) \\ , : = \\ , ( u - u _ \\lambda ) ( x ) , x \\in \\overline \\Omega \\setminus ( \\Gamma \\cup R _ \\lambda ( \\Gamma ) ) . \\end{align*}"}
-{"id": "8075.png", "formula": "\\begin{align*} B _ { T G } = \\left ( I - ( I - I _ { 2 h , h } ( A _ { 2 h } ) ^ { - 1 } I _ { h , 2 h } A _ h ) ( I - S _ h A _ h ) \\right ) ( A _ h ) ^ { - 1 } , \\end{align*}"}
-{"id": "8449.png", "formula": "\\begin{align*} G ( A \\varpi ^ { - s } , B ) = \\int _ { \\mathcal { O } ^ n } \\psi ( { } ^ { t } x A x \\varpi ^ { - s } + B \\cdot x ) d x A \\in G L _ n ( \\mathcal { O } ) B \\in F ^ n , \\end{align*}"}
-{"id": "6295.png", "formula": "\\begin{align*} \\left \\{ b ; \\varepsilon , g , ( f , k _ 1 ) , ( t , k _ 2 ) , ( s , k _ 3 ) ; \\{ ( \\alpha _ i , \\beta _ i ) \\} _ { i = 1 } ^ n ; ( r _ 1 , r _ 2 , \\ldots , r _ { s - k _ { 3 } } ) ; ( q _ 1 , q _ 2 , \\ldots , q _ { k _ 3 } ) \\right \\} . \\end{align*}"}
-{"id": "966.png", "formula": "\\begin{align*} \\mathcal { E } _ n = \\left \\{ \\sup _ { x \\in \\mathbb { R } } \\left | P \\left ( \\max _ { \\theta \\in \\mathcal { G } _ n } | Z _ n ( \\theta ) | \\leq x \\right ) - P \\left ( T ^ * _ n \\leq x | \\mathcal { F } ^ X \\right ) \\right | \\leq \\varepsilon _ n \\right \\} . \\end{align*}"}
-{"id": "2313.png", "formula": "\\begin{align*} & \\ ; \\| A ^ { 1 + s / 2 } u _ * ( t _ { K + 2 } ) \\| _ { L ^ 2 } ^ 2 \\\\ \\leq & \\ ; C ( \\delta ^ { - 1 } + 1 ) ( \\| u _ 0 \\| ^ 2 _ { D ( A ) } + 1 ) ^ { K + 1 } \\| u _ 0 \\| ^ 2 _ { D ( A ) } \\\\ \\leq & \\ ; C ( \\varepsilon ^ { - 1 } + 1 ) ( \\| u _ 0 \\| ^ 2 _ { D ( A ) } + 1 ) ^ { K + 1 } \\| u _ 0 \\| ^ 2 _ { D ( A ) } , \\end{align*}"}
-{"id": "2510.png", "formula": "\\begin{align*} K f = \\widetilde { K } f + \\varpi \\chi _ R ( | \\xi | ) f . \\end{align*}"}
-{"id": "1026.png", "formula": "\\begin{align*} w _ { n } ( \\zeta ) = \\widehat { w } _ { n } ( \\zeta ) = \\min \\{ d - 1 , n \\} . \\end{align*}"}
-{"id": "9934.png", "formula": "\\begin{align*} f ( \\lambda ) = \\int _ { ( 0 , \\infty ) } ( 1 - e ^ { - \\lambda r } ) \\ , \\mu ( d r ) & \\geq \\int _ { ( 0 , \\lambda ^ { - 1 } ) } ( 1 - e ^ { - \\lambda r } ) \\ , \\mu ( d r ) \\\\ & \\geq c \\int _ { ( 0 , \\lambda ^ { - 1 } ) } ( 1 - e ^ { - \\lambda r } ) \\ , \\frac { d r } { r ^ { 1 + \\rho } } . \\end{align*}"}
-{"id": "2536.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { G } _ { L ; 0 } ( x , t ) = & \\sum _ { j = 1 } ^ { 3 } \\int _ { | \\eta | < \\delta } e ^ { i \\eta x + \\sigma ( \\eta ) t } \\vert e _ j ( \\eta ) \\rangle \\langle e _ j ( \\eta ) \\vert d \\eta \\\\ = & \\sum _ { j = 1 } ^ { 3 } \\int _ { | \\eta | < \\delta } e ^ { i \\eta ( x - a _ j t ) - A _ j t | \\eta | ^ 2 + O ( 1 ) | \\eta | ^ 3 t } \\left ( \\vert e _ j ( 0 ) \\rangle \\langle e _ j ( 0 ) \\vert + \\varepsilon _ j ( \\eta ) \\right ) d \\eta , \\end{aligned} \\end{align*}"}
-{"id": "642.png", "formula": "\\begin{align*} \\partial _ \\tau p = \\sum J _ i ( 0 ) e _ i ( 0 ) P ^ { - 1 } \\partial _ \\tau p = \\sum J _ i ( 0 ) e _ i ( 1 ) . \\end{align*}"}
-{"id": "5215.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } p \\\\ q \\end{array} \\right ) ' = \\left ( \\begin{array} { c c } 0 & S \\\\ \\lambda S ^ { - 1 } - Q f ' ( \\hat { u } ) & - c I \\end{array} \\right ) \\left ( \\begin{array} { c } p \\\\ q \\end{array} \\right ) . \\end{align*}"}
-{"id": "4580.png", "formula": "\\begin{align*} A & = ( - d ) [ [ \\cdots [ F _ { 3 , 1 } , F _ { 2 , 0 } ] _ { q } , \\cdots , F _ { n - 2 , 0 } ] _ q , F _ { n - 1 , 0 } ] _ { q } \\\\ & = ( - d ) [ [ [ \\cdots [ F _ { 3 , 1 } , F _ { 4 , 0 } ] _ { q } , \\cdots , F _ { n - 2 , 0 } ] _ q , F _ { n - 1 , 0 } ] _ { q } , F _ { 2 , 0 } ] _ { q } \\ , , \\end{align*}"}
-{"id": "1864.png", "formula": "\\begin{align*} \\mathcal { Q } : = \\Big \\{ \\mu \\in c a _ 1 ^ + ( \\mathbb { R } ^ d ) : \\mu _ 1 = \\nu _ 1 , \\ , \\dots , \\ , \\mu _ d = \\nu _ d \\underline { \\pi } ^ i \\leq \\mu ( A ^ i ) \\leq \\overline { \\pi } ^ i , i \\in I \\Big \\} \\end{align*}"}
-{"id": "7319.png", "formula": "\\begin{align*} U _ + : = \\bigcap _ { i = 0 } ^ \\infty x ^ i U x ^ { - i } , U _ - : = \\bigcap _ { i = 0 } ^ \\infty x ^ { - i } U x ^ { i } , \\end{align*}"}
-{"id": "1053.png", "formula": "\\begin{align*} \\Omega : = \\left \\{ P , T P , \\ldots , T ^ { b - 1 } P , Q , T Q , \\ldots , T ^ { a - 1 } Q \\right \\} , \\end{align*}"}
-{"id": "9410.png", "formula": "\\begin{align*} f ( n + 1 , k ) = \\sum _ { d \\mid \\mathrm { g c d } ( n + 1 , k ) } g ( \\frac { n + 1 } { d } , \\frac { k } { d } ) \\boldsymbol { \\mu } ( ( \\frac { n + 1 } { d } , \\frac { k } { d } ) , ( n + 1 , k ) ) , \\end{align*}"}
-{"id": "8084.png", "formula": "\\begin{align*} \\deg ( m ) = \\deg ( \\hat { x } _ F ^ \\ell ) + \\alpha . \\end{align*}"}
-{"id": "6596.png", "formula": "\\begin{align*} \\left [ g _ n \\right ] _ { \\alpha , \\varphi _ { n } ( E _ M ) } & \\leq \\left [ \\phi ^ { 0 } _ { n } \\right ] _ { \\mathrm { L i p } , \\psi _ n \\circ f ( E _ M ) } \\left [ f _ n \\right ] _ { \\alpha , \\varphi _ { n } ( E _ M ) } \\left [ \\phi _ { n } \\right ] _ { \\mathrm { L i p } , \\varphi _ { n } ( E _ M ) } ^ \\alpha \\ . \\end{align*}"}
-{"id": "1275.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\mbox { C a p } _ { \\mathcal { A } } ( E _ l ) = \\mbox { C a p } _ { \\mathcal { A } } ( E _ 1 ) . \\end{align*}"}
-{"id": "4963.png", "formula": "\\begin{align*} \\left \\| \\sup _ { j \\in \\Z } ( 2 ^ { \\alpha j } \\omega _ j ) \\right \\| _ { L ^ p ( \\R ^ d ) } ^ p & = \\int _ { \\R ^ d } \\sup _ { j \\in \\Z } ( 2 ^ { \\alpha j } \\omega _ j ( x ) ) ^ p d x \\\\ & \\leq \\sum _ { j \\in \\Z } \\int _ { \\R ^ d } ( 2 ^ { \\alpha j } \\omega _ j ( x ) ) ^ p d x \\\\ & = \\sum _ { j \\in \\Z } \\sum _ { r \\in \\Z ^ d } ( e ^ { - | r '' | - 2 ^ { - \\sigma } | r ' | } ) ^ p ( 2 ^ { \\alpha j } ) ^ p \\int _ { \\R ^ d } | \\Delta _ j f ( x - 2 ^ { - j } r ) | ^ p d x . \\end{align*}"}
-{"id": "700.png", "formula": "\\begin{align*} \\cot \\beta = \\cot \\beta _ 0 + \\sum _ { n = 0 } ^ { \\infty } \\cfrac { | \\kappa _ n ^ 0 | - | \\kappa _ n | } { | \\dot \\Phi ( \\mu _ n ^ 0 ) | } . \\end{align*}"}
-{"id": "9489.png", "formula": "\\begin{align*} s _ { n , d } = s _ { n - 1 , d + r - 2 } + \\sum \\limits _ { i = 0 } ^ { n - 1 } \\sum \\limits _ { j = 0 } ^ { d - 2 } s _ { n - i - 1 , d - j - 2 } \\cdot s _ { i , j } , \\end{align*}"}
-{"id": "4673.png", "formula": "\\begin{align*} \\Vert a _ n S w _ n + b _ n w _ n \\Vert \\ge c \\Vert w _ n \\Vert , n = 1 , 2 , \\ldots \\end{align*}"}
-{"id": "6211.png", "formula": "\\begin{align*} \\begin{pmatrix} - 2 & 0 & 1 \\\\ 0 & - 2 & 0 \\\\ 1 & 0 & 2 k \\end{pmatrix} \\begin{pmatrix} - 2 & 0 & 0 \\\\ 0 & - 2 & 1 \\\\ 0 & 1 & 2 k \\end{pmatrix} . \\end{align*}"}
-{"id": "4610.png", "formula": "\\begin{align*} \\left | \\alpha - \\frac { 1 } { N - M } \\sum _ { n = M } ^ { N - 1 } a _ n \\right | < \\epsilon \\end{align*}"}
-{"id": "7892.png", "formula": "\\begin{align*} u ( x , t ) = \\min \\{ t - x , 0 \\} \\end{align*}"}
-{"id": "2991.png", "formula": "\\begin{align*} U ( t ) = \\int _ 0 ^ t S ^ b _ { t - s } \\beta _ s d s + \\int _ 0 ^ t S ^ b _ { t - s } \\sigma _ s d W ( s ) \\end{align*}"}
-{"id": "9674.png", "formula": "\\begin{align*} \\lambda _ i = \\frac { u v + ( - 1 ) ^ { \\frac { i + 3 } { 4 } } c \\sqrt { u ^ 2 + v ^ 2 - c ^ 2 } } { u ^ 2 - c ^ 2 } , \\ , \\ , i = 1 , 5 , \\lambda _ j = \\frac { v } { u } , \\ , \\ , j = 2 , 3 , 4 . \\end{align*}"}
-{"id": "7834.png", "formula": "\\begin{align*} F _ { X } ( x ) = \\left ( 1 - e ^ { - \\eta ^ { \\alpha } ( x ) } \\right ) ^ { \\theta } ; \\ \\ x \\geq 0 , \\end{align*}"}
-{"id": "8844.png", "formula": "\\begin{align*} ( 1 + i ) \\sqrt { \\frac { \\pi } { 2 } } = \\int _ { \\R } e ^ { i ( \\eta - \\sqrt { 2 } ( \\zeta + \\frac { z } { 4 } ) ) ^ 2 } d \\eta \\end{align*}"}
-{"id": "7058.png", "formula": "\\begin{align*} P _ 1 + P _ 2 = \\prod { \\tilde { \\lambda } _ j } \\int _ { \\mathbb { R } ^ n } { \\frac { u ( y ) - u ( 0 ) } { ( \\tilde { \\lambda } _ 1 ^ 2 y _ 1 ^ 2 + . . . + \\tilde { \\lambda } _ n ^ 2 y _ n ^ 2 ) ^ { ( n + 2 s ) / 2 } } d y } \\geq \\frac { \\eta _ 0 } { 1 - s } > 0 , \\end{align*}"}
-{"id": "7708.png", "formula": "\\begin{align*} d _ R ( j , k ) = L ^ { \\dagger } _ { j j } + L ^ { \\dagger } _ { k k } - 2 L ^ { \\dagger } _ { j k } = \\sum _ { i = 1 } ^ { N - 1 } \\frac { 1 } { \\lambda _ i } ( u _ { i j } - u _ { i k } ) ^ 2 \\ , . \\end{align*}"}
-{"id": "4371.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\Big \\| A C _ t - \\sum _ { j = j _ 0 ( t ) } ^ \\infty M _ { \\varphi _ { j , t } } \\Big \\| = \\lim _ { t \\to 0 } \\Big \\| C _ t ^ \\ast A ^ \\ast - \\sum _ { j = j _ 0 ( t ) } ^ \\infty M _ { \\varphi _ { j , t } } \\Big \\| = 0 \\end{align*}"}
-{"id": "8323.png", "formula": "\\begin{align*} \\ell _ * = k - Q ( k ) \\ell . \\end{align*}"}
-{"id": "8534.png", "formula": "\\begin{align*} \\mathcal { K } _ { ( j , - l ) , ( k , - m ) } ( \\xi , \\eta ) = ( 2 ^ { j } ) ^ 3 \\mathcal { K } _ { ( 0 , j - l ) , ( 1 , j - m ) } ( \\tilde { \\xi } , \\tilde { \\eta } ) . \\end{align*}"}
-{"id": "9128.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } \\dfrac { 1 } { \\binom { l } { p _ { k } } } \\sum _ { \\mathrm { c a r d } ( I ) = p _ { k } } \\left ( \\prod _ { j \\in \\left \\{ 1 , \\ldots , l \\right \\} \\setminus I } x _ { j } \\right ) \\cdot f _ { k } \\left ( \\prod _ { i \\in I } x _ { i } \\right ) = 0 . \\end{align*}"}
-{"id": "6626.png", "formula": "\\begin{align*} g _ n = \\left \\{ \\begin{array} { l l } \\phi _ n ^ 0 \\circ f _ n \\circ \\phi _ n & \\mbox { i n } \\ \\varphi _ n ( E _ M ) \\\\ \\phi _ n ^ k \\circ f _ n & \\mbox { i n } \\ \\varphi _ n ( f ^ { - 1 } ( B _ M ^ { k + 1 } ) ) , \\ \\mbox { f o r } \\ 0 < k < k _ 0 \\\\ f _ n & \\mbox { e l s e w h e r e } \\end{array} \\right . \\ , \\end{align*}"}
-{"id": "8355.png", "formula": "\\begin{align*} K _ 0 = \\Z _ p ^ \\times \\cdot K _ 0 ^ p . \\end{align*}"}
-{"id": "9889.png", "formula": "\\begin{align*} 2 e ( G ) & = \\bigl [ \\bigl ( \\tfrac { 4 } { ( 3 r - 2 ) ^ 2 } + \\tfrac { 4 } { 3 r - 2 } \\delta - \\delta ^ 2 \\bigr ) + \\tfrac { 3 ( r - 2 ) } { 3 r - 2 } \\delta + \\tfrac { 9 ( r - 2 ) ( r - 3 ) + 2 4 ( r - 2 ) } { ( 3 r - 2 ) ^ 2 } - o ( 1 ) \\bigr ] n ^ 2 \\\\ & = \\bigl ( \\tfrac { 3 r - 5 } { 3 r - 2 } + \\delta - \\delta ^ 2 - o ( 1 ) \\bigr ) n ^ 2 \\ , . \\end{align*}"}
-{"id": "3959.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d t } } p ( n , t ) = - \\lambda ( p ( n , t ) - p ( n - 1 , t ) ) , \\ \\ n \\geq 0 , \\end{align*}"}
-{"id": "907.png", "formula": "\\begin{align*} Z _ n ^ * ( t ) = \\frac { 1 } { \\widehat { \\mathfrak { s } } _ n ( t ) } \\sum _ { i = 1 } ^ n K _ h ( t _ { i - 1 } - t ) ( X _ { t _ { i } } - X _ { t _ { i - 1 } } ) ^ 2 \\xi _ i ^ * , t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "9650.png", "formula": "\\begin{align*} \\hat { R } ^ { * } _ { \\beta } ( t ) = { R } ^ { * } _ { \\beta } + \\frac { \\partial { R } _ { \\beta } } { \\partial \\lambda _ { \\beta } } \\Big | _ { \\lambda _ { \\beta } } \\left ( \\hat { \\lambda } _ { \\beta } - \\lambda _ { \\beta } \\right ) + O \\left ( \\left ( \\hat { \\lambda } _ { \\beta } - \\lambda _ { \\beta } \\right ) ^ 2 \\right ) . \\end{align*}"}
-{"id": "4128.png", "formula": "\\begin{align*} \\omega _ h ( E _ 1 , E _ 2 ) = \\left ( \\frac { \\lambda _ 1 \\lambda _ 2 } { \\langle \\eta , \\xi \\rangle ^ 2 } \\right ) ^ { \\frac { 1 } { 2 } } = \\frac { ( \\lambda _ 1 \\lambda _ 2 ) ^ { \\frac { 1 } { 2 } } } { \\langle \\eta , \\xi \\rangle } . \\end{align*}"}
-{"id": "1408.png", "formula": "\\begin{align*} H ^ { * } ( B G _ n ; \\mathbb { Z } / 2 ) = H ^ { * } ( B G _ { n - 1 } ; \\mathbb { Z } / 2 ) \\otimes H ^ { * } ( B { S L _ 2 } ; \\mathbb { Z } / 2 ) \\end{align*}"}
-{"id": "3770.png", "formula": "\\begin{align*} J _ { \\check { k } } : = \\bigcup _ { k \\geq \\check { k } } \\bigcup _ { l = 1 } ^ { L _ { k + 2 } / L _ k } \\lbrace l L _ k \\rbrace \\subset \\N . \\end{align*}"}
-{"id": "877.png", "formula": "\\begin{align*} \\sup _ { x _ 1 , \\dots , x _ { d _ n } \\in \\mathbb { R } } \\left | P \\left ( \\bigcap _ { k = 1 } ^ { d _ n } \\left \\{ F _ { n , k } \\leq x _ k \\right \\} \\right ) - P \\left ( \\bigcap _ { k = 1 } ^ { d _ n } \\left \\{ Z _ { n , k } \\leq x _ k \\right \\} \\right ) \\right | \\to 0 . \\end{align*}"}
-{"id": "5438.png", "formula": "\\begin{align*} - \\binom { k } { i - d + k } + \\binom { d - k + 1 } { i - k + 1 } , \\end{align*}"}
-{"id": "540.png", "formula": "\\begin{align*} \\eta ^ { 0 , 1 } = \\eta - J \\circ \\eta \\circ j . \\end{align*}"}
-{"id": "3356.png", "formula": "\\begin{align*} \\bar { D } ( \\tilde { r } _ i ) & \\leq \\frac { N \\left ( 1 - \\frac { 1 } { 1 + N + \\cdots + N ^ i } \\right ) ^ 2 } { ( N - 1 ) + \\frac { 1 } { 1 + N + \\cdots + N ^ i } } \\\\ & = \\frac { N ( 1 + N + N ^ 2 + \\cdots + N ^ i - 1 ) ^ 2 } { ( N - 1 ) ( 1 + N + \\cdots + N ^ i ) ^ 2 + ( 1 + N + \\cdots + N ^ i ) } \\\\ & = \\frac { N ^ 2 ( 1 + N + \\cdots + N ^ { i - 1 } ) ^ 2 } { N ^ { i } ( 1 + N + \\cdots + N ^ i ) } . \\end{align*}"}
-{"id": "6956.png", "formula": "\\begin{align*} L _ M u ( x ) = t r a c e ( M D ^ 2 u ( x ) ) = \\Delta ( u \\circ \\sqrt { M } ) ( x ) , \\end{align*}"}
-{"id": "3977.png", "formula": "\\begin{align*} p ^ { \\alpha _ 3 } _ { k - 1 } ( 3 , t ) = - ( - \\lambda ) ^ { k - 1 } \\underset { \\Theta ^ { k - 1 } _ { 3 } } { \\sum } \\frac { t ^ { k _ 0 \\alpha _ 0 + k _ 1 \\alpha _ 1 + k _ 2 \\alpha _ 2 + k _ 3 \\alpha _ 3 } } { \\Gamma \\left ( k _ 0 \\alpha _ 0 + k _ 1 \\alpha _ 1 + k _ 2 \\alpha _ 2 + k _ 3 \\alpha _ 3 + 1 \\right ) } . \\end{align*}"}
-{"id": "6930.png", "formula": "\\begin{align*} \\| \\Theta _ f \\| = \\| \\hat f \\| _ { L ^ \\infty } . \\end{align*}"}
-{"id": "3376.png", "formula": "\\begin{gather*} \\sum \\nolimits ^ { \\mathbb { C } } Q \\times \\sum \\nolimits ^ { \\mathbb { C } } Q \\rightarrow \\mathbb { C } l _ { ( n + m ) } \\\\ ( \\varphi _ { 1 } , \\varphi _ { 2 } ) = ( [ p , [ \\varphi _ { 1 } ] ] , [ p , [ \\varphi _ { 2 } ] ] ) \\mapsto \\left \\langle \\left \\langle \\lbrack \\varphi _ { 1 } ] , [ \\varphi _ { 2 } ] \\right \\rangle \\right \\rangle = \\tau ( [ \\varphi _ { 2 } ] ) [ \\varphi _ { 1 } ] , \\end{gather*}"}
-{"id": "8114.png", "formula": "\\begin{align*} \\dim ( \\partial _ A S _ i \\varphi ^ { - 1 } ( S _ i ) ) & = \\dim ( S _ i \\partial _ A \\varphi ^ { - 1 } ( S _ i ) ) \\\\ & = \\dim ( \\partial _ A \\varphi ^ { - 1 } ( S _ i ) ) \\leq \\dim ( [ \\partial _ A W _ 1 ] _ { R _ { \\overline { W } , E } } ) < \\dim ( A ) \\end{align*}"}
-{"id": "9852.png", "formula": "\\begin{align*} F ( x + i u ) = P _ u * f ( x ) = \\frac { 1 } { \\pi } \\int _ { \\mathbb { R } } \\frac { u f ( s ) } { u ^ 2 + ( x - s ) ^ 2 } d s . \\end{align*}"}
-{"id": "6143.png", "formula": "\\begin{align*} g ^ 2 = 4 g , \\ ; g h = g k = g \\ell = 2 g , \\ ; h ^ 2 = 2 h , \\ ; k ^ 2 = 2 k , \\ ; \\ell ^ 2 = 2 \\ell , \\ ; h k = h \\ell = k \\ell = g . \\end{align*}"}
-{"id": "3292.png", "formula": "\\begin{align*} \\| h \\| _ * = \\sup _ { x \\in S } \\rho ( x ) ^ { - 1 } | h ( x ) | , \\end{align*}"}
-{"id": "4988.png", "formula": "\\begin{align*} V _ j : = \\sum _ { j ' < j } h _ { j ' } \\prod _ { j ' < j '' < j } ( 1 - U _ { j '' } ) . \\end{align*}"}
-{"id": "5452.png", "formula": "\\begin{align*} f ^ * ( \\lambda ) = \\lambda ^ \\rho \\sup _ { x \\in [ 1 , r ] } \\frac { p ( \\lambda x ) } { p ( x ) } , \\end{align*}"}
-{"id": "539.png", "formula": "\\begin{align*} ( d u - X _ H \\otimes \\beta ) ^ { 0 , 1 } = 0 , \\end{align*}"}
-{"id": "1092.png", "formula": "\\begin{align*} n _ { o , i } = n _ { i } + n _ { t h , i } , \\end{align*}"}
-{"id": "6277.png", "formula": "\\begin{align*} d _ { u , q } = \\lambda _ q ^ { 1 / 2 } \\| u _ q \\| _ 2 , \\\\ d _ u ^ 2 = \\left \\{ d _ { u , q } ^ 2 \\right \\} _ { q \\geq - 1 } . \\end{align*}"}
-{"id": "6155.png", "formula": "\\begin{align*} \\Omega = \\sum _ { i = 1 } ^ 3 A _ i \\dd x _ 0 \\wedge \\dd x _ i + B _ 1 \\dd x _ 2 \\wedge \\dd x _ 3 + B _ 2 \\dd x _ 3 \\wedge \\dd x _ 1 + B _ 3 \\dd x _ 1 \\wedge \\dd x _ 2 \\end{align*}"}
-{"id": "8930.png", "formula": "\\begin{gather*} ( a \\phi _ { d , d _ 1 } ) ^ * \\hat { \\cal L } _ { 1 , { \\cal E } _ d } = [ a ] ^ * \\phi _ { \\gcd ( d , d _ 1 ) , d _ 1 } ^ * \\phi _ { d , \\gcd ( d , d _ 1 ) } ^ * \\hat { \\cal L } _ { 1 , { \\cal E } _ d } = \\hat { \\cal L } _ { 1 , { \\cal E } _ { d _ 1 } } ^ { a ^ 2 \\gcd ( d , d _ 1 ) ^ 2 / d _ 1 d } , \\end{gather*}"}
-{"id": "1795.png", "formula": "\\begin{align*} M ( x , y , z ) = \\frac { z ^ { x - y } x ^ { y - z } } { y ^ { x - z } } = \\left ( \\frac { z } { y } \\right ) ^ { x - y } \\left ( \\frac { x } { y } \\right ) ^ { y - z } = \\left [ \\left ( \\frac { z } { y } \\right ) ^ { \\frac { x } { y } - 1 } \\left ( \\frac { x } { y } \\right ) ^ { 1 - \\frac { z } { y } } \\right ] ^ y . \\end{align*}"}
-{"id": "9557.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in B _ n } { ( - 1 ) ^ { \\ell ( \\sigma ) } x ^ { L _ { o o e } ( \\sigma ) } } = ( 1 - x ^ { \\left \\lceil \\frac { n } { 2 } \\right \\rceil } ) \\prod _ { i = i } ^ { n - 1 } ( 1 - x ^ i ) , \\end{align*}"}
-{"id": "8048.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } | \\gamma _ { \\pm t _ * } ( s ) , \\sigma _ { \\pi } ( s ) | / s = 0 . \\end{align*}"}
-{"id": "2458.png", "formula": "\\begin{align*} \\Delta _ 1 \\cup \\ldots \\cup \\ \\Delta _ k = \\{ \\nu ^ { \\alpha + c ' + 1 } \\rho , \\ldots , \\nu ^ { \\alpha + n - 1 } \\rho , \\nu ^ { \\alpha + n } \\rho \\} . \\end{align*}"}
-{"id": "818.png", "formula": "\\begin{align*} \\frac { \\partial \\vec { u } } { \\partial t } + \\nabla p - \\nabla \\cdot \\nu \\nabla \\vec { u } & = 0 , \\\\ \\nabla \\cdot \\vec { u } & = 0 . \\end{align*}"}
-{"id": "908.png", "formula": "\\begin{align*} q _ n ^ * ( 1 - \\alpha ) = \\inf \\left \\{ z \\in \\mathbb { R } : P \\left ( \\sup _ { t \\in [ a _ n , T - a _ n ] } | Z _ n ^ * ( t ) | \\leq z \\Big | \\mathcal { F } ^ X \\right ) \\geq 1 - \\alpha \\right \\} . \\end{align*}"}
-{"id": "2125.png", "formula": "\\begin{align*} A _ 2 I _ { \\ell } \\cap A _ 2 I _ m = \\emptyset . \\end{align*}"}
-{"id": "343.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } H = \\lambda \\left ( \\Delta H + H ( | A | ^ 2 + R _ { \\nu \\nu } ) \\right ) + H \\Delta \\lambda + 2 \\langle \\nabla \\lambda , \\nabla H \\rangle . \\end{align*}"}
-{"id": "3069.png", "formula": "\\begin{align*} \\dot w ( s ) = - \\nabla _ x F ( t ^ * , w ( s ) ) \\end{align*}"}
-{"id": "9187.png", "formula": "\\begin{align*} \\frac { 1 } { \\pi } \\sum _ { n = 1 } ^ \\infty \\frac { \\Lambda ( n ) } { n } \\hat { h } \\ ! \\left ( \\frac { \\log n } { \\pi } \\right ) . \\end{align*}"}
-{"id": "7603.png", "formula": "\\begin{align*} U _ 2 ( x _ 1 + x _ 2 ) = U x _ 1 , U _ 1 ( x _ 1 + x _ 2 ) = U x _ 2 , U _ 1 , U _ 2 \\ge 0 , U _ 1 + U _ 2 = U . \\end{align*}"}
-{"id": "9449.png", "formula": "\\begin{align*} \\psi ( n ) - \\psi ( n + 1 ) = \\frac { O ( 1 ) } { n ^ 2 \\mathfrak { M } ( n ) \\log n } , \\end{align*}"}
-{"id": "3822.png", "formula": "\\begin{align*} p _ k : = \\inf _ { | \\eta | = k , \\eta ( z ) = 0 \\ , \\forall \\ , z \\le 0 } \\P _ \\eta ( G _ \\infty \\cap \\Lambda _ \\infty ) . \\end{align*}"}
-{"id": "227.png", "formula": "\\begin{align*} \\mu _ n ( \\chi _ a h ( B _ G ) ) & = \\sqrt { \\lambda _ n ( \\chi _ a | h | ^ 2 ( B _ G ) \\chi _ a ) } . \\end{align*}"}
-{"id": "1525.png", "formula": "\\begin{align*} | \\lambda | = \\sum _ { x \\in \\beta ( \\lambda ) } { x } - \\binom { | \\beta ( \\lambda ) | } { 2 } . \\end{align*}"}
-{"id": "5034.png", "formula": "\\begin{align*} q _ { n + 1 } ( x _ 0 , \\dots , x _ { 2 ^ { n + 1 } - 1 } ) = q _ 2 ( q _ n ( x _ 0 , \\dots , x _ { 2 ^ n - 2 } ) , x _ { 2 ^ n - 1 } , q _ n ( x _ { 2 ^ n } , \\dots , x _ { 2 ^ { n + 1 } - 2 } ) ) . \\end{align*}"}
-{"id": "9951.png", "formula": "\\begin{align*} { \\cal N } ( u ) - V _ 1 = { \\cal N } ( \\zeta ( u ) ) - V _ 3 \\end{align*}"}
-{"id": "5701.png", "formula": "\\begin{align*} \\tilde { \\gamma } ( t , s ) = \\begin{dcases} a ^ + + E ( s ) \\ , \\frac { \\gamma ( t , s ) - a ^ + } { | \\gamma ( t , s ) - a ^ + | } & s > s ^ + | \\gamma ( t , s ) - a ^ + | > E ( s ) , \\\\ a ^ - + E ( s ) \\ , \\frac { \\gamma ( t , s ) - a ^ - } { | \\gamma ( t , s ) - a ^ - | } & s < s ^ - | \\gamma ( t , s ) - a ^ - | > E ( s ) , \\\\ \\gamma ( t , s ) & \\end{dcases} \\end{align*}"}
-{"id": "7367.png", "formula": "\\begin{align*} [ \\omega _ 1 , \\omega _ 2 ] _ { S N } = i _ { X ^ a } \\omega _ 1 \\wedge \\nabla _ { X _ a } \\omega _ 2 + ( - 1 ) ^ { p q } i _ { X ^ a } \\omega _ 2 \\wedge \\nabla _ { X _ a } \\omega _ 1 . \\end{align*}"}
-{"id": "3120.png", "formula": "\\begin{align*} \\Phi _ x ( 0 , a , b ) = \\int _ { ( 0 , a , b ) } \\sum _ { i , j = 1 } ^ d B _ { i j } ( x - s ) d s _ i \\wedge d s _ j \\ , = \\ , \\oint _ { \\partial ( 0 , a , b ) } A ( x - s ) d s \\ , . \\end{align*}"}
-{"id": "8215.png", "formula": "\\begin{align*} m _ j & = \\frac { n ^ j r ^ j } { j ! } ( 1 + \\hat { H _ j ) } \\\\ \\hat { H _ j } & = \\sum _ { h = 1 } ^ { j - 1 } \\frac { \\hat { a _ h } ( r , j , \\{ \\epsilon _ i \\} ) } { n ^ h } \\end{align*}"}
-{"id": "1253.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ n b ^ * _ { i j } ( x , \\tau ) \\xi _ i \\xi _ j \\approx | \\xi | ^ 2 \\ , ( | \\nabla u ( x , t ) | + | \\nabla u ( x , \\tau ) | ) ^ { p - 2 } \\end{align*}"}
-{"id": "2681.png", "formula": "\\begin{align*} X ^ \\xi _ \\ell : = X _ { T _ 0 + \\cdots + T _ \\ell } . \\end{align*}"}
-{"id": "4459.png", "formula": "\\begin{align*} v _ \\ell ( k ) = G _ \\ell ( k ) \\xi ( k ) , k \\in ( 2 \\pi \\Z ) ^ 2 , \\end{align*}"}
-{"id": "4863.png", "formula": "\\begin{align*} \\xi ( s , t ) = - 2 ^ { 2 5 } \\cdot 3 ^ { 1 3 } \\cdot 5 ^ { 5 } \\cdot 7 ^ { 5 } \\cdot s ^ { 1 7 } t ^ { 1 3 } . \\end{align*}"}
-{"id": "7143.png", "formula": "\\begin{align*} \\sum _ { t = r J + 1 } ^ { ( r + 1 ) J } y ( m ) \\leq q ( ( r + 1 ) J + 1 ) - q ( r J + 1 ) , \\end{align*}"}
-{"id": "1119.png", "formula": "\\begin{align*} & \\sum \\limits _ { \\{ h , f \\} \\in C _ q } x _ { h f } + x _ { i _ 1 j _ 1 } \\ge 1 , \\end{align*}"}
-{"id": "64.png", "formula": "\\begin{align*} \\frac { \\partial ( e e ^ * ) } { \\partial w ^ * } = \\frac { \\partial ( D - \\textbf { w } ^ { H } \\textbf { X } ) ( D ^ * - \\textbf { w } ^ { T } \\textbf { X } ^ * ) } { \\partial w ^ * } \\end{align*}"}
-{"id": "4010.png", "formula": "\\begin{align*} K _ { p } \\left ( u \\right ) \\geq \\int w u \\ d \\mu - K _ p ^ * ( w ) \\forall u \\in L ^ 1 ( \\mu ) , \\forall w \\in L ^ { \\infty } ( \\mu ) \\\\ K _ { p } \\left ( u \\right ) = \\int w u \\ d \\mu - K _ p ^ * ( w ) \\iff w \\in \\partial K _ { p } \\left ( u \\right ) . \\end{align*}"}
-{"id": "7654.png", "formula": "\\begin{align*} \\{ A ^ { ( 1 ) } _ { k } \\ ! , H ^ { ( 1 ) } _ { B } \\} = 0 \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\{ B ^ { ( 2 ) } _ { m } \\ ! , H ^ { ( 2 ) } _ { A } \\} = 0 . \\end{align*}"}
-{"id": "8766.png", "formula": "\\begin{align*} x \\Psi _ B ^ { { \\bf f } } ( { \\bf t } ) - x = \\int _ 0 ^ { t _ { m } } x \\Psi _ B ^ { { \\bf f } } \\left ( \\underset { m } { \\bf t } , { s _ m } \\right ) V _ B ^ { { \\bf f } } \\left ( \\underset { m } { \\bf t } , { s _ m } \\right ) \\ , d s _ m \\ , , \\end{align*}"}
-{"id": "9454.png", "formula": "\\begin{align*} \\omega ( z ; q ) : = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ { 2 n ^ 2 + 2 n } } { ( q ; q ^ 2 ) _ { n + 1 } ( z q ; q ^ 2 ) _ { n + 1 } } , \\nu ( z ; q ) : = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ 2 + n } } { ( - z q ; q ^ 2 ) _ { n + 1 } } . \\end{align*}"}
-{"id": "6488.png", "formula": "\\begin{align*} \\vec { B } _ { } = B _ { \\perp } \\hat { B } _ { _ { \\perp } } \\end{align*}"}
-{"id": "5523.png", "formula": "\\begin{align*} b ( m ) = b ( 3 k + 1 ) & = \\sqrt { 2 } \\cdot b ( k ) + b ( k + 1 ) \\\\ & \\leqslant ( \\sqrt { 2 } + 1 ) \\cdot \\max \\{ b ( k ) , b ( k + 1 ) \\} \\\\ & \\leqslant ( \\sqrt { 2 } + 1 ) \\left ( h ( k + 1 ) + \\lfloor \\log _ 3 ( k + 1 ) \\rfloor \\right ) , \\end{align*}"}
-{"id": "5344.png", "formula": "\\begin{align*} ( a \\otimes g ) \\cdot \\lambda = ( a \\otimes f ) \\cdot \\lambda . \\end{align*}"}
-{"id": "6268.png", "formula": "\\begin{align*} \\chi ( \\xi ) = \\begin{cases} 1 , \\ \\ \\mbox { f o r } | \\xi | \\leq \\frac { 3 } { 4 } \\\\ 0 , \\ \\ \\mbox { f o r } | \\xi | \\geq 1 . \\end{cases} \\end{align*}"}
-{"id": "1474.png", "formula": "\\begin{align*} E _ u = E ^ { e v e n , * } \\oplus E ^ { o d d , * } . \\end{align*}"}
-{"id": "9068.png", "formula": "\\begin{align*} z ^ 2 _ 1 = x _ 1 , z _ 2 ^ 2 = x _ 2 . \\end{align*}"}
-{"id": "4038.png", "formula": "\\begin{align*} p - c \\eta ( p ) = g ( p ) , \\ \\ p \\in M . \\end{align*}"}
-{"id": "3537.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l l } \\beta ^ { 0 } = \\displaystyle \\lim _ { k \\to \\infty } \\beta ^ { 0 , \\theta _ k } , \\\\ \\beta ^ { j } = \\displaystyle \\lim _ { k \\to \\infty } \\beta ^ { j , \\theta _ k } , \\\\ \\end{array} \\end{align*}"}
-{"id": "1095.png", "formula": "\\begin{align*} \\sigma _ { Z _ { s , i } } ^ 2 = 1 + 2 m _ { c , i } , \\sigma _ { Z _ { 0 , i } } ^ 2 = m _ { c , i } + m _ { c , i } ^ 2 + \\sigma _ { t h } ^ 2 , \\end{align*}"}
-{"id": "5754.png", "formula": "\\begin{align*} \\frac { { \\rm V o l } ( C \\cap B _ \\varepsilon ( p ) ) } { | B _ 1 ^ n | \\varepsilon ^ n } = \\frac { 1 } { n | B _ 1 ^ n | } \\int _ \\Gamma r ^ { 1 - n } \\langle \\overline { \\nabla } r , \\nu _ C \\rangle \\ , d V _ \\Gamma . \\end{align*}"}
-{"id": "7872.png", "formula": "\\begin{align*} d ( ( \\theta _ x , x _ 3 ) , ( \\theta _ y , y _ 3 ) ) = \\min _ { n \\in \\Z } \\sqrt { ( \\theta _ y - \\theta _ x + 2 \\pi n ) ^ 2 + ( y _ 3 - x _ 3 ) ^ 2 } . \\end{align*}"}
-{"id": "2671.png", "formula": "\\begin{align*} B _ 0 = \\begin{pmatrix} c _ 0 & c _ 1 \\\\ c _ 1 & d _ 0 \\end{pmatrix} B _ 1 = \\begin{pmatrix} c _ 2 & c _ 3 \\\\ c _ 1 & d _ 2 \\end{pmatrix} , \\end{align*}"}
-{"id": "8246.png", "formula": "\\begin{align*} \\mathbf { H } ( s ) \\ni h \\mapsto \\tilde { h } = [ \\omega _ { \\pi } ^ { - 1 } \\omega _ { \\pi } ^ { \\mathfrak { L } } ] ( \\det ( \\cdot ) ) h \\in \\mathbf { H } _ { \\mathfrak { L } } ( s ) . \\end{align*}"}
-{"id": "9306.png", "formula": "\\begin{align*} A _ i : = \\{ x \\in B : \\dim \\pi _ i ( A _ x ) \\geq 1 \\} . \\end{align*}"}
-{"id": "9615.png", "formula": "\\begin{align*} x _ { \\beta } ^ { \\rm { r } } [ n ] = \\frac { \\gamma _ { \\rm { r } , \\beta } } { N } \\sum _ { k = 0 } ^ { N - 1 } { X } _ { \\beta } ^ { \\rm { r } } [ k ] \\exp ( 2 \\pi j k n / N ) , \\end{align*}"}
-{"id": "9125.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } x ^ { p _ { k } } f _ { k } ( x ^ { q _ { k } } ) = 0 \\left ( x \\in R \\right ) , \\end{align*}"}
-{"id": "3133.png", "formula": "\\begin{align*} c ( T ^ * ) = e ( T ^ { \\prime } ) + f _ 3 ( T ) + r & = 3 ( k ^ 2 - 2 k - 3 ) ( t - 2 ) + 2 ( t - 2 ) + r \\\\ & = ( 3 k ^ 2 - 6 k - 7 ) ( t - 2 ) + r \\\\ & = ( 3 k ^ 2 - 6 k - 7 ) \\frac { n - r - 2 } { k ^ 2 - k - 2 } + r \\\\ & = \\left ( \\frac { k - 3 } { k - 2 } + \\frac { 2 } { 3 ( k + 1 ) ( k - 2 ) } \\right ) ( 3 n - 6 ) - \\frac { 2 k ^ 2 - 5 k - 5 } { k ^ 2 - k - 2 } r , \\end{align*}"}
-{"id": "5919.png", "formula": "\\begin{align*} \\begin{aligned} & Q ^ N ( \\mathcal { M } ^ { - 1 } ( \\{ s \\} ) ) < \\Big ( \\sum _ { \\gamma = 1 } ^ { r _ s } \\binom { r _ s } { \\gamma } ( 2 ^ { l - 1 } ) ^ \\gamma e ^ { - c _ 0 \\gamma N } \\Big ) Q ^ N ( s ) = \\\\ & \\Big ( ( 1 + 2 ^ { l - 1 } e ^ { - c _ 0 N } ) ^ { r _ s } - 1 \\Big ) Q ^ N ( s ) , \\ s \\in A _ { } . \\end{aligned} \\end{align*}"}
-{"id": "9584.png", "formula": "\\begin{align*} \\langle f , g \\rangle = \\sum _ { k \\in \\Bbb Z } \\langle T _ N ^ { - 1 } D _ k D _ k ^ N ( f ) , g \\rangle = \\sum _ { k \\in \\Bbb Z } \\langle D _ k D _ k ^ N T _ N ^ { - 1 } ( f ) , g \\rangle , \\qquad \\forall \\ g \\in \\dot { \\mathcal B } ^ { \\alpha , q } _ { p , \\mathcal F } . \\end{align*}"}
-{"id": "6050.png", "formula": "\\begin{align*} & Q _ { X Y U } ^ { ( \\alpha , \\theta ) } ( x , y , u ) \\\\ & : = \\frac { Q _ { X Y U } ( x , y , u ) \\exp \\Big ( - \\theta \\omega _ { Q _ { X Y U } } ^ { ( \\alpha ) } ( x , y | u ) \\Big ) } { \\sum _ { x , y , u } Q _ { X Y U } ( x , y , u ) \\exp \\Big ( - \\theta \\omega _ { Q _ { X Y U } } ^ { ( \\alpha ) } ( x , y | u ) \\Big ) } . \\end{align*}"}
-{"id": "294.png", "formula": "\\begin{align*} M _ { t , n } ^ * = L _ { n t } ^ * \\slash ( \\tilde \\sigma \\sqrt n ) . \\end{align*}"}
-{"id": "10131.png", "formula": "\\begin{align*} \\bar { \\boldsymbol k } _ k ( i ) = \\frac { \\lambda ^ { - 1 } \\bar { \\boldsymbol \\Phi } _ k ( i - 1 ) \\bar { \\boldsymbol x } _ k ( i ) } { 1 + \\lambda ^ { - 1 } \\bar { \\boldsymbol x } _ k ^ H ( i ) \\bar { \\boldsymbol \\Phi } _ k ( i - 1 ) \\bar { \\boldsymbol x } _ k ( i ) } . \\end{align*}"}
-{"id": "10129.png", "formula": "\\begin{align*} \\boldsymbol Q _ { \\bar { \\boldsymbol \\omega } _ k } ( i ) = \\lambda ^ { - 1 } \\boldsymbol Q _ { \\bar { \\boldsymbol \\omega } _ k } ( i - 1 ) - \\lambda ^ { - 1 } \\boldsymbol t _ k ( i ) \\bar { \\boldsymbol \\omega } _ k ^ H ( i - 1 ) \\boldsymbol Q _ { \\bar { \\boldsymbol \\omega } _ k } ( i - 1 ) . \\end{align*}"}
-{"id": "4387.png", "formula": "\\begin{align*} r _ t : = \\operatorname { d i a m } ( \\operatorname { s u p p } \\varphi _ { j , t } ) = \\frac { 8 \\sqrt { 2 n } } { t } , \\end{align*}"}
-{"id": "1883.png", "formula": "\\begin{align*} \\hat { f } _ 1 ^ i : = \\begin{cases} f _ 1 ^ i , & i \\neq i _ 1 - 1 \\\\ 1 , & , \\end{cases} \\end{align*}"}
-{"id": "3032.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ z u = 0 , & \\Gamma _ u : = G \\times \\{ 0 \\} , \\\\ u = 0 , & \\Gamma _ b : = G \\times \\{ - D \\} , \\\\ & \\Gamma _ l : = \\partial G \\times ( - D , 0 ) . \\end{cases} \\end{align*}"}
-{"id": "5686.png", "formula": "\\begin{align*} \\begin{cases} \\int _ \\R ( u ( x _ 1 , x _ 2 ) - z ^ + ( x _ 1 ) ) ^ 2 \\d x _ 1 < + \\infty & x _ 2 \\in \\R ; \\\\ \\int _ \\R ( u ( x _ 1 , x _ 2 ) - z ^ \\pm ( x _ 1 ) ) ^ 2 \\d x _ 1 \\to 0 & x _ 2 \\to \\pm \\infty . \\end{cases} \\end{align*}"}
-{"id": "5604.png", "formula": "\\begin{align*} X _ t = x _ 0 + \\int _ 0 ^ t f ( X _ r ) d r + W _ { \\int _ 0 ^ t \\sigma ^ 2 ( X _ r ) d r } \\end{align*}"}
-{"id": "4279.png", "formula": "\\begin{align*} \\widehat { C } _ 2 ( \\tau ) = \\widehat { C } _ 2 ( \\tau , \\nu ) = C _ 2 ( \\tau ) + \\nu \\end{align*}"}
-{"id": "9224.png", "formula": "\\begin{align*} \\alpha _ { 1 } ( \\alpha _ { 2 } \\alpha _ { 3 } ) - \\frac { 1 } { 2 } ( 2 - \\frac { 4 } { n } ) [ \\alpha _ { 1 } , \\alpha _ { 2 } ] _ { A ^ { - } } \\alpha _ { 3 } - \\frac { 2 } { n } \\langle \\alpha _ { 1 } , \\alpha _ { 2 } \\rangle \\alpha _ { 3 } - \\alpha _ { 2 } ( \\alpha _ { 1 } \\alpha _ { 3 } ) = 0 . \\end{align*}"}
-{"id": "4756.png", "formula": "\\begin{align*} \\Omega \\alpha ^ 2 + V ^ \\mu { { L } _ \\mu } ^ 2 = 0 , \\end{align*}"}
-{"id": "280.png", "formula": "\\begin{align*} I _ { 2 , 2 } ( h ) & = \\sup _ { t \\in \\lbrack 0 , t _ { 0 } ] } \\Vert f ( t , 0 ) \\Vert \\int _ { 0 } ^ { t _ { 0 } } \\left ( ( t _ { 0 } - \\tau ) ^ { \\alpha - 1 } - ( t _ { 0 } + h - \\tau ) ^ { \\alpha - 1 } \\right ) d \\tau \\\\ & \\leq \\sup _ { t \\in \\lbrack 0 , t _ { 0 } ] } \\Vert f ( t , 0 ) \\Vert \\times \\frac { h ^ { \\alpha } } { \\alpha } . \\end{align*}"}
-{"id": "2118.png", "formula": "\\begin{align*} f _ I : = \\frac { 1 } { | I | } \\int _ I f ( y ) \\ , d y \\end{align*}"}
-{"id": "9908.png", "formula": "\\begin{align*} e ( X _ 1 , X _ 2 ) = \\sum _ { i \\in W _ 1 } \\sum _ { j \\in W _ 2 } e ( V _ i \\cap X _ 1 , V _ j \\cap X _ 2 ) \\ , . \\end{align*}"}
-{"id": "8762.png", "formula": "\\begin{gather*} A _ 1 = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & 0 \\\\ \\end{array} \\right ) , A _ 2 = \\left ( \\begin{array} { c c } 0 & 1 \\\\ 1 & 0 \\\\ \\end{array} \\right ) . \\end{gather*}"}
-{"id": "764.png", "formula": "\\begin{align*} \\widetilde { P } _ { g } = \\{ ( y , \\omega ) \\in Y \\times \\mbox { \\rm M a p } ( I , Z ) \\mid \\omega ( 0 ) = g ( y ) \\} . \\end{align*}"}
-{"id": "9581.png", "formula": "\\begin{align*} J [ v ] : = \\iint _ { \\R ^ n _ + \\times \\R ^ n _ + } \\frac { \\lvert v ( x ) - v ( y ) \\rvert ^ p } { \\lvert x - y \\rvert ^ { n + s p } } ( x _ n y _ n ) ^ { ( s p - 1 ) / 2 } \\ , d y \\ , d x \\ , , \\end{align*}"}
-{"id": "1423.png", "formula": "\\begin{align*} \\partial _ j ( x _ 3 ) & = 0 , \\\\ \\partial _ j ( x _ { 4 i } ) & = \\sum _ { k = 1 } ^ { j - 1 } \\alpha _ { j k } x _ { 4 i } ^ { 2 ^ k } , \\end{align*}"}
-{"id": "9506.png", "formula": "\\begin{align*} T ( x , t ) \\cdot A ( x , t ) = t \\ , x \\ , \\left [ x ^ 2 \\ , D ( x , t ) - T _ 1 ( t ) \\right ] , \\end{align*}"}
-{"id": "9423.png", "formula": "\\begin{align*} c ^ { \\rm ( i n ) } _ { n , d } = c ^ { \\rm ( i n ) } _ { n - 2 , d + 2 } + c ^ { \\rm ( o u t ) } _ { n - 1 , d - 1 } + 2 \\cdot c ^ { \\rm ( o u t ) } _ { n - 2 , d } + \\sum \\limits _ { i = 0 } ^ { n - 2 } \\sum \\limits _ { j = 0 } ^ { d - 2 } \\bigl ( 2 \\cdot \\tilde { s } _ { i , j } \\cdot c ^ { \\rm ( i n ) } _ { n - 2 - i , d - 2 - j } + d _ { i , j } \\cdot d _ { n - 2 - i , d - 2 - j } \\bigr ) . \\end{align*}"}
-{"id": "1659.png", "formula": "\\begin{align*} \\tau ^ { e _ 1 } ( x ) = \\begin{cases} \\tau _ { a _ 0 } ^ { - 1 } ( x ) \\quad \\ ; \\ ; x \\in R _ { a _ 0 } \\\\ \\tau _ { a _ 1 } ^ { - 1 } ( x ) \\quad \\ ; \\ ; x \\in R _ { a _ 1 } \\\\ \\tau _ { c _ 0 } ^ { - 1 } ( x ) \\quad \\ ; \\ ; x \\in R _ { c _ 0 } \\\\ \\tau _ { c _ 1 } ^ { - 1 } ( x ) \\quad \\ ; \\ ; x \\in R _ { c _ 1 } \\end{cases} \\end{align*}"}
-{"id": "3630.png", "formula": "\\begin{align*} h _ \\varepsilon ( t ) = \\begin{cases} \\frac { G _ \\varepsilon ( t ) } { t } & \\ , \\ , t > 0 \\\\ 0 & \\ , \\ , t = 0 , \\end{cases} \\end{align*}"}
-{"id": "2713.png", "formula": "\\begin{align*} \\Pi _ { \\mathcal G } ( h ) = \\sum _ { i = 1 } ^ { n } \\ , \\left ( \\int _ { \\mathbb T ^ d \\times { \\mathbb R ^ d } } h \\varphi _ i \\ , d x d v \\right ) \\varphi _ i , \\end{align*}"}
-{"id": "1585.png", "formula": "\\begin{align*} \\mathbf { j } ' & = ( 0 , 0 , \\dots , \\frac { m - 1 } { 3 } , 2 ) \\\\ \\mathbf { j } '' & = ( 0 , 0 , \\dots , \\frac { m - 1 } { 3 } - 1 , 5 ) \\\\ \\mathbf { j } ''' & = ( 0 , 0 , \\dots , 1 , \\frac { m - 1 } { 3 } - 2 , 1 ) . \\end{align*}"}
-{"id": "6724.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\mathbb { P } \\Big ( \\Gamma _ { 1 } < N ^ { 1 + \\delta } \\Big ) & = \\lim \\limits _ { N \\rightarrow \\infty } 1 - \\mathbb { P } \\Big ( \\Gamma _ { 1 } \\geq N ^ { 1 + \\delta } \\Big ) \\\\ & = \\lim \\limits _ { N \\rightarrow \\infty } 1 - \\Big ( 1 - \\frac { 1 } { N } \\Big ) ^ { N ^ { 1 + \\delta } } = 1 . \\end{align*}"}
-{"id": "6399.png", "formula": "\\begin{align*} P _ { } \\left ( \\theta \\left \\vert x \\right . \\right ) = \\frac { P _ { } \\left ( \\theta \\right ) P _ { } \\left ( x \\left \\vert \\theta \\right . \\right ) } { P _ { } \\left ( x \\right ) } \\end{align*}"}
-{"id": "89.png", "formula": "\\begin{align*} \\Gamma _ { \\rm o p } ( a ) ( s , t ) = a ( t s ) , \\mbox { f o r a l m o s t a l l } ( s , t ) \\in G \\times G , \\ \\ a \\in L ^ { \\infty } ( G ) . \\end{align*}"}
-{"id": "4624.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { | W _ i | } { | Y _ i | } = \\mu _ { f , B + b , ( x , t ) } ( X _ B ) \\ge 1 - \\frac { \\eta } { 2 } \\end{align*}"}
-{"id": "3103.png", "formula": "\\begin{align*} \\int _ \\Omega | \\nabla v | ^ 2 \\d x = \\lambda _ 1 ^ \\Gamma \\int _ \\Omega v ^ 2 \\d x , \\end{align*}"}
-{"id": "1728.png", "formula": "\\begin{align*} f ^ S _ { \\lambda } = \\frac { h \\circ \\sigma ^ n } { h } f ^ T _ { \\lambda } \\quad \\end{align*}"}
-{"id": "9765.png", "formula": "\\begin{align*} \\tilde { E } _ { i } ( h , \\theta ) = \\int \\limits _ { y _ { i - 1 , s } + s ^ { ( i - 1 ) } h } ^ { y _ { i } } W ( U _ { h , \\theta } ( i h - , y ) ) d y - \\int \\limits _ { y _ { i , s } } ^ { y _ { i } } W ( U _ { h , \\theta _ i } ( i h - , y ) ) d y , \\end{align*}"}
-{"id": "2833.png", "formula": "\\begin{align*} g _ { v w } ^ { ( n + m , n ) } \\geq \\prod _ { s = n } ^ { \\infty } \\frac { a _ s } { a _ s + s } = C _ n \\to 1 \\mbox { a s } n \\rightarrow \\infty , \\end{align*}"}
-{"id": "3701.png", "formula": "\\begin{align*} \\tilde \\gamma = ( \\tilde \\gamma _ 1 , \\tilde \\gamma _ 2 ) \\in W [ n ] \\times \\mathbb A ^ n . \\end{align*}"}
-{"id": "2741.png", "formula": "\\begin{align*} | | g | | _ { L _ { x , v } ^ { 2 , r \\ast } } : = \\sum _ { m = 0 } ^ { r } \\widetilde C _ { m , r + 1 } \\ , | | \\partial ^ m g | | _ { L ^ 2 _ { x , v } } , \\end{align*}"}
-{"id": "7148.png", "formula": "\\begin{align*} & \\sum _ { m = r J + 1 } ^ { ( r + 1 ) J } ( q ( m ) - q ( r J + 1 ) ) y ( m ) \\\\ \\leq & \\sum _ { m = r J + 1 } ^ { ( r + 1 ) J } ( m - ( r J + 1 ) ) y ^ 2 _ { m a x } \\\\ = & \\frac { J ( J - 1 ) } { 2 } y ^ 2 _ { m a x } \\leq J ( J - 1 ) U . \\end{align*}"}
-{"id": "2214.png", "formula": "\\begin{align*} \\deg _ { w _ j } \\widetilde Q _ i < m _ { i j } , \\ i , j = 1 , \\ldots , n . \\end{align*}"}
-{"id": "1839.png", "formula": "\\begin{align*} D ^ 0 ( I ) \\otimes ^ . F _ \\bullet = \\cdots \\to D ^ 0 ( I ) \\otimes ^ . F _ 1 \\to D ^ 0 ( I ) \\otimes ^ . F _ 0 \\to D ^ 0 ( I ) \\otimes ^ . F _ { - 1 } \\to \\cdots \\end{align*}"}
-{"id": "8603.png", "formula": "\\begin{align*} \\omega : \\bigsqcup _ { k = 1 } ^ { K } \\bigsqcup _ { \\ell = 1 } ^ { m _ { k } } \\Gamma z _ { ( k , \\ell ) } \\Gamma & \\to \\Gamma g ^ { - 1 } \\Gamma \\times _ { \\Gamma } \\Gamma h ^ { - 1 } \\Gamma \\\\ \\gamma z _ { ( k , \\ell ) } \\delta & \\mapsto [ \\gamma g _ { i ( k , \\ell ) } , h _ { j ( k , \\ell ) } \\delta ] , \\end{align*}"}
-{"id": "7173.png", "formula": "\\begin{align*} \\mathfrak { H } _ { 1 + s } : = \\{ \\l \\ , h _ e \\ , \\ , : \\ , \\ , e \\in \\mathbb { S } ^ { n - 2 } , \\l \\in [ 0 , + \\infty ) \\} \\subset H ^ 1 _ { l o c } ( \\R ^ n , \\mu _ a ) . \\end{align*}"}
-{"id": "6800.png", "formula": "\\begin{align*} - \\rho \\ln { \\left ( \\int _ { \\mathbb { S } ^ 2 } e ^ { w _ { \\lambda } } \\right ) } = - ( 3 2 \\pi + \\epsilon ) \\ln { ( 3 2 \\pi - 8 9 6 \\pi \\lambda ^ 2 \\ln { \\lambda } + O ( \\lambda ^ 2 ) ) } \\end{align*}"}
-{"id": "2724.png", "formula": "\\begin{align*} \\mathcal M ( \\rho _ { \\infty } , u _ { \\infty } , T _ { \\infty } ) = \\frac { \\rho _ { \\infty } } { ( 2 \\pi T _ { \\infty } ) ^ { N / 2 } } \\ , \\exp \\left ( - \\frac { | u _ { \\infty } - v | ^ 2 } { 2 T _ { \\infty } } \\right ) , \\end{align*}"}
-{"id": "2246.png", "formula": "\\begin{align*} y _ j ^ { m _ j } + \\sum _ { \\lambda \\in \\Lambda ^ { ( j ) } \\cup \\{ 0 \\} } x _ { \\lambda } ^ { ( j ) } y ^ \\lambda = 0 , \\lambda _ 1 + \\ldots + \\lambda _ n < m _ j , j = 1 , \\ldots , n , \\end{align*}"}
-{"id": "5533.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } \\dot { z } _ { 1 } \\\\ \\dot { z } _ { 2 } \\end{array} \\right ] = \\left [ \\begin{array} { c c } 0 & 1 \\\\ - \\left ( \\alpha + \\beta p \\left ( t \\right ) \\right ) & 0 \\end{array} \\right ] \\left [ \\begin{array} { c } z _ { 1 } \\\\ z _ { 2 } \\end{array} \\right ] \\end{align*}"}
-{"id": "2238.png", "formula": "\\begin{align*} \\beta ( K , J ) = \\bigl ( m _ { 1 j _ { 1 } } ( k _ { j _ { 1 } } + 1 ) - 1 , \\ldots , m _ { n j _ { n } } ( k _ { j _ { n } } + 1 ) - 1 \\bigr ) , \\end{align*}"}
-{"id": "10087.png", "formula": "\\begin{align*} \\tilde { \\nabla } _ { X } \\xi = \\frac { 1 } { n + 1 } \\{ n X - \\pi ( X ) \\xi \\} . \\end{align*}"}
-{"id": "3333.png", "formula": "\\begin{align*} I ( W _ { k : K } ; Z | W _ { 1 : k - 1 } ) & = H \\left ( Z | W _ { 1 : k - 1 } \\right ) - H ( Z | W _ { 1 : K } ) \\\\ & = \\left ( K - k + 1 \\right ) L r \\end{align*}"}
-{"id": "4123.png", "formula": "\\begin{align*} h = \\frac { \\langle \\varphi '' , \\xi \\rangle } { \\langle \\varphi , \\xi \\rangle } , \\end{align*}"}
-{"id": "5653.png", "formula": "\\begin{align*} \\mathfrak { E } _ { W } ( \\gamma ) = \\mathfrak { L } _ { K } ( \\gamma ) = \\mathfrak { L } _ { K } ( \\gamma _ 0 ) = d _ K ( x ^ - , x ^ + ) \\leq \\inf \\{ \\mathfrak { E } _ { W } ( \\sigma ) \\ ; : \\ ; \\sigma \\in A C _ { p l o c } ( \\R , X ) , \\ , \\gamma : x ^ - \\mapsto x ^ + \\} . \\end{align*}"}
-{"id": "49.png", "formula": "\\begin{align*} \\hat { V } ^ { C } _ { \\sigma } ( C _ { 1 } , C _ { 2 } ) = \\int \\limits _ { - \\infty } ^ { \\infty } \\int \\limits _ { - \\infty } ^ { \\infty } \\frac { 1 } { N } \\sum \\limits _ { i = 1 } ^ N \\frac { 1 } { ( 4 \\pi ^ { 2 } \\sigma ^ { 4 } ) } e x p \\left ( - \\frac { 1 } { 2 \\sigma ^ { 2 } } a ( u _ 1 - b ) ^ { 2 } \\right ) e x p \\left ( - \\frac { 1 } { 2 \\sigma ^ { 2 } } a ' ( u _ 2 - b ' ) ^ { 2 } \\right ) e x p \\left ( - \\frac { c } { 2 \\sigma ^ { 2 } } \\right ) \\mathrm { d } u _ 1 \\mathrm { d } u _ 2 \\end{align*}"}
-{"id": "1825.png", "formula": "\\begin{align*} & \\mathbb { E } ^ a _ { Q , w , 0 } \\left ( \\cosh \\left ( ( t + h ) A _ 0 \\right ) 1 _ { \\{ 0 \\longleftrightarrow \\partial _ { i n } Q ^ a \\} } \\prod _ { i \\geq 1 } \\cosh ( h A _ i ) \\right ) \\\\ & = \\sum _ { \\Gamma } \\cosh \\left ( \\left ( t + h \\right ) | \\Gamma | \\right ) \\mathbb { P } ^ a _ { Q , w , 0 } ( \\mathcal { C } _ 0 = \\Gamma ) \\mathbb { E } ^ a _ { Q , w , 0 } \\left ( \\prod _ { i \\geq 1 } \\cosh ( h A _ i ) | \\mathcal { C } _ 0 = \\Gamma \\right ) , \\end{align*}"}
-{"id": "372.png", "formula": "\\begin{align*} \\mathbb { P } ( S _ n ^ { ( \\varepsilon x ) } \\geq x ) = ( 1 - \\Phi ( x ) ) ( 1 + o ( 1 ) ) . \\end{align*}"}
-{"id": "4946.png", "formula": "\\begin{align*} p ^ x + p ^ y = z ^ { 2 n } \\end{align*}"}
-{"id": "5753.png", "formula": "\\begin{align*} \\frac { { \\rm V o l } ( \\Sigma \\cap B _ \\varepsilon ( p ) ) } { | B ^ n _ 1 | \\varepsilon ^ n } \\leq \\frac { 1 } { n | B ^ n _ 1 | } \\int _ \\Gamma r ^ { 1 - n } \\langle \\overline { \\nabla } r , \\nu _ \\Sigma \\rangle \\ , d V _ \\Gamma , ~ | B _ 1 ^ n | : = { \\rm V o l } ( B ^ n _ 1 ( 0 ) ) . \\end{align*}"}
-{"id": "8473.png", "formula": "\\begin{align*} G \\left ( \\frac { \\varpi ^ { - \\rho } } { 2 } \\left ( \\begin{matrix} - b & - b \\\\ - b & - b \\end{matrix} \\right ) , \\varpi ^ { r + \\frac { t } { 2 } } \\left ( \\begin{matrix} x \\\\ - \\frac { b } { v x } \\end{matrix} \\right ) \\right ) = \\gamma _ F ( - \\frac { b } { 2 } , \\rho ) q ^ { - \\frac { \\rho } { 2 } } \\end{align*}"}
-{"id": "164.png", "formula": "\\begin{align*} ( \\mathfrak { X } \\uplus \\mathfrak { Y } ) ( a ) = \\mathfrak { X } ( a ) + \\mathfrak { Y } ( a ) . \\end{align*}"}
-{"id": "1210.png", "formula": "\\begin{align*} 0 \\geq \\sum \\limits _ { i , j = 1 } ^ { n } \\left [ \\frac { ( 1 - K ) } { A ^ 2 } ( u _ 1 ) _ { x _ i x _ j } ( y ) + \\frac { K } { B ^ 2 } ( u _ 2 ) _ { x _ i x _ j } ( z ) - \\frac { 1 } { C ^ 2 } u _ { x _ i x _ j } ( x ) \\ , \\right ] \\eta _ i \\eta _ j \\end{align*}"}
-{"id": "1079.png", "formula": "\\begin{align*} \\varphi ( \\mathbf { r } , z ) = \\sum \\limits _ { i = - \\infty } ^ { + \\infty } \\sum \\limits _ { k \\in \\mathcal { N } } ^ { } \\rho _ k \\alpha _ { k i } u _ i ( \\mathbf { r } , z ) , \\end{align*}"}
-{"id": "465.png", "formula": "\\begin{align*} 4 ^ j L _ { j , \\psi _ \\omega } a _ { k _ 1 , k _ 2 , \\omega } = b _ { k _ 1 , k _ 2 , j } \\left ( \\omega - \\frac { \\pi } { 2 } \\right ) ^ { k _ 1 - 2 j } + O \\left ( \\left ( \\omega - \\frac { \\pi } { 2 } \\right ) ^ { k _ 1 - 2 j + 1 } \\right ) \\end{align*}"}
-{"id": "842.png", "formula": "\\begin{align*} X _ t - X _ s & = \\int _ s ^ t d X _ z = \\int _ s ^ t b ( X _ z , z ) d E _ z + \\int _ s ^ t \\sigma ( X _ z , z ) d B _ { E _ z } . \\end{align*}"}
-{"id": "2272.png", "formula": "\\begin{align*} \\Gamma _ { M } ( x ) = \\frac { n } { x - t } . \\end{align*}"}
-{"id": "6642.png", "formula": "\\begin{align*} \\int _ { r _ 1 } ^ { r _ 2 } b _ 2 ( \\xi ) \\ , d \\xi = \\int _ { r _ 1 } ^ { r _ 2 } ( r _ 2 - \\xi ) ^ 2 ( \\xi - r _ 1 ) ^ 2 \\ , d \\xi = \\int _ 0 ^ { r } ( r - \\zeta ) ^ 2 \\zeta ^ 2 \\ , d \\zeta = \\frac { r ^ 5 } { 3 0 } \\end{align*}"}
-{"id": "3527.png", "formula": "\\begin{align*} \\begin{array} [ c ] { r l } & \\hat { J } ( u ^ { \\varepsilon } ( t ) ) - \\hat { J } ( \\bar { u } ( t ) ) \\\\ = & \\displaystyle \\sum _ { i = 1 } ^ { n } \\psi _ { x _ i } ( \\bar { X } ( t _ 1 ) , \\bar { X } ( t _ 2 ) , \\cdots , \\bar { X } ( t _ n ) ) y ( t _ i ) \\\\ & + { \\displaystyle \\int \\limits _ { 0 } ^ { T } } \\{ f _ x ( \\bar { X } { ( t ) } , \\bar { u } ( t ) ) y ( t ) + f ( \\bar { X } { ( t ) } , u ^ { \\varepsilon } ( t ) ) - f ( \\bar { X } { ( t ) } , \\bar { u } ( t ) ) \\} d t + o ( \\varepsilon ) . \\\\ \\end{array} \\end{align*}"}
-{"id": "9900.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\tfrac { k _ i + 1 } { 2 k _ i } \\alpha _ i ^ 2 \\cdot \\sum _ { i = 1 } ^ m \\tfrac { 2 k _ i } { k _ i + 1 } \\ge \\bigl ( \\sum _ { i = 1 } ^ m \\alpha _ i \\bigr ) ^ 2 = 1 \\end{align*}"}
-{"id": "9048.png", "formula": "\\begin{align*} { \\mathcal H } ( w , w ' ) : = ( \\bar w w ' , \\bar w w ' , 0 ) . \\end{align*}"}
-{"id": "9121.png", "formula": "\\begin{align*} f _ { n + 1 - ( i + 1 ) } = ( - 1 ) ^ { i } \\left \\{ \\left ( \\sum _ { k = 0 } ^ { i } \\binom { n - k } { i - k } \\dfrac { \\widetilde { D } _ { n - ( k + 1 ) } } { k + 1 } \\right ) - \\binom { n + 1 } { i + 1 } f _ { n + 1 } \\right \\} , \\end{align*}"}
-{"id": "7018.png", "formula": "\\begin{align*} C _ 1 = C _ 1 ( s , L , S C ) = \\int _ { \\mathbb { R } } { \\frac { \\max \\{ 2 L | z _ n | , S C | z _ n | ^ 2 \\} } { | z _ n | ^ { 1 + 2 s } } d z _ n } , \\end{align*}"}
-{"id": "6061.png", "formula": "\\begin{align*} E ( \\psi ) = c \\int _ 0 ^ { s } ( \\Phi ' ( r ) ^ 2 + 2 \\mathfrak { e } \\tfrac { \\varphi ^ 2 ( \\Phi ( r ) ) } { f ( r ) ^ 2 } ) f ( r ) ^ { n - 1 } d r , \\end{align*}"}
-{"id": "5954.png", "formula": "\\begin{align*} ( \\nu ^ { \\delta } ) _ { u _ j } = 0 , ( f ^ { \\delta } ) _ { u _ 1 } \\cdot ( f ^ { \\delta } ) _ { u _ j } = 0 , - ( \\nu ^ { \\delta } ) _ { u _ 1 } = \\alpha ( f ^ { \\delta } ) _ { u _ 1 } \\end{align*}"}
-{"id": "5369.png", "formula": "\\begin{align*} h _ { r - 1 , 1 } - h _ { r + 1 , 1 } = \\frac { ( b ( r - 1 ) - a ) ^ 2 - ( b ( r + 1 ) - a ) ^ 2 } { 4 a b } = - \\frac { r b } { a } \\mod \\Z . \\end{align*}"}
-{"id": "1532.png", "formula": "\\begin{gather*} A + A \\ : = \\ \\{ a _ 1 + a _ 2 : a _ 1 , a _ 2 \\in A \\} , \\\\ A - A \\ : = \\ \\{ a _ 1 - a _ 2 : a _ 1 , a _ 2 \\in A \\} . \\end{gather*}"}
-{"id": "356.png", "formula": "\\begin{align*} D = \\frac { H ^ { \\frac { 2 + \\sigma } { 2 } } } { 2 } \\left ( H ( \\lambda _ { , 2 2 } - \\lambda _ { , 1 1 } ) + 2 ( H _ { , 2 } \\lambda _ { , 2 } - H _ { , 1 } \\lambda _ { , 1 } ) \\right ) \\sqrt { U _ \\sigma } , \\end{align*}"}
-{"id": "2925.png", "formula": "\\begin{align*} u [ g ] : = [ A _ { \\iota _ 2 ( g ) } \\times \\{ g \\} ] = \\{ [ ( a , g ) ] \\colon a \\in A _ { \\iota _ 2 ( g ) } \\} . \\end{align*}"}
-{"id": "5421.png", "formula": "\\begin{align*} P _ { - \\rho } E _ { \\rho } F = F , F \\in L ^ 2 ( \\Omega ) , \\end{align*}"}
-{"id": "3558.png", "formula": "\\begin{align*} \\varphi ( n ) = n \\prod _ { d \\ , | \\ , n } ( 1 - 1 / p ) = n \\sum _ { d \\ , | \\ , n } \\frac { \\mu ( d ) } { d } . \\end{align*}"}
-{"id": "3153.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ e ^ { - u \\eta } \\right ] & = \\left ( 1 - e ^ { - \\alpha } \\right ) ^ { - 1 } \\left ( \\widehat { m } _ { \\alpha , \\beta } ( - u ) - e ^ { - \\alpha } \\right ) \\\\ & = \\left ( 1 - e ^ { - \\alpha } \\right ) ^ { - 1 } \\left ( \\exp \\left \\lbrace \\frac { - \\alpha u } { \\beta + u } \\right \\rbrace - \\exp \\left \\lbrace - \\alpha \\right \\rbrace \\right ) . \\end{align*}"}
-{"id": "4130.png", "formula": "\\begin{align*} \\langle \\eta , \\xi \\rangle = ( \\lambda _ 1 \\lambda _ 2 ) ^ { \\frac { 1 } { 4 } } . \\end{align*}"}
-{"id": "4256.png", "formula": "\\begin{align*} \\mathcal G ^ \\Delta _ { 1 2 } = \\Phi ( A _ { \\theta , 1 2 } ) \\cdot z _ { 1 2 } \\end{align*}"}
-{"id": "4393.png", "formula": "\\begin{align*} \\widetilde { C } _ { - w } \\widetilde { C } _ { z _ \\gamma } A \\widetilde { C } _ { - z _ \\gamma } \\widetilde { C } _ { w } & = \\widetilde { C } _ { z _ \\gamma - w } A \\widetilde { C } _ { - ( z _ \\gamma - w ) } \\to A _ y \\end{align*}"}
-{"id": "1421.png", "formula": "\\begin{align*} \\alpha _ { j , j - 1 } & = 1 , \\\\ \\alpha _ { j , j - 2 } & = \\alpha _ { j - 1 , j - 2 } x _ 2 ^ { 2 ^ { j - 1 } } , \\\\ \\alpha _ { j , k } & = \\alpha _ { j - 1 , k } x _ 2 ^ { 2 ^ { j - 1 } } + \\alpha _ { j - 2 , k } x _ 3 ^ { 2 ^ { j - 1 } } \\end{align*}"}
-{"id": "5996.png", "formula": "\\begin{gather*} G ( H _ 8 ) = \\{ 1 , g _ 1 , g _ 2 , z \\} , \\end{gather*}"}
-{"id": "5332.png", "formula": "\\begin{align*} W ( \\infty ) = \\bigcup _ { N \\geq 1 } W ( N ) . \\end{align*}"}
-{"id": "10159.png", "formula": "\\begin{align*} { \\boldsymbol S } _ { D _ k } ( i ) \\bar { \\boldsymbol \\omega _ k } ( i ) & = { \\boldsymbol S } _ { D _ k } ( i ) \\Lambda _ D ^ { - 1 } { \\boldsymbol \\Phi } _ D ^ H { \\boldsymbol p } _ k ( i ) \\\\ & = { \\boldsymbol \\Phi } _ D \\Lambda _ D ^ { - 1 } { \\boldsymbol \\Phi } _ D ^ H { \\boldsymbol p } _ k ( i ) \\\\ & = { \\boldsymbol \\omega _ k } ^ { ( D ) } ( i ) , \\end{align*}"}
-{"id": "9541.png", "formula": "\\begin{align*} \\frac { w _ a } { y _ a } \\ , \\frac { y _ a } { w _ a } & + \\sum _ { i = a + 1 } ^ m \\Bigl ( \\frac { w _ i } { y _ i } - \\frac { w _ { i - 1 } } { y _ { i - 1 } } \\Bigr ) \\frac { y _ i } { w _ i } \\le 1 + \\sum _ { i = a + 1 } ^ m \\int \\limits _ { w _ { i - 1 } / y _ { i - 1 } } ^ { w _ i / y _ i } \\frac 1 t \\ , d t \\\\ & = 1 + \\int \\limits _ { w _ a / y _ a } ^ { w _ m / y _ m } \\frac 1 t \\ , d t \\le 1 + \\int \\limits _ { \\sigma ^ * } ^ 1 \\frac 1 t \\ , d t = 1 + \\ln \\frac 1 { \\sigma ^ * } . \\end{align*}"}
-{"id": "6760.png", "formula": "\\begin{align*} \\bigl | \\log | \\gamma | _ w - z _ w \\bigr | & = \\Bigl | \\log | \\gamma \\mu | _ w - | W _ v ( l / k ) | ^ { - 1 } \\sum _ { x | v } \\log | \\mu | _ x \\Bigr | \\\\ & = \\Bigl | \\log | \\gamma \\mu | _ w - | W _ v ( l / k ) | ^ { - 1 } \\sum _ { x | v } \\log | \\gamma \\mu | _ x \\Bigr | . \\end{align*}"}
-{"id": "3049.png", "formula": "\\begin{align*} \\begin{cases} \\lambda ( w , v ) _ H + a ( w , v ) + b ( v , \\pi ) = ( g , v ) _ H , & \\forall v \\in V , \\\\ b ( w , q ) = 0 , & \\forall q \\in Q , \\end{cases} \\end{align*}"}
-{"id": "7655.png", "formula": "\\begin{align*} \\sigma ( P x ) = \\sigma ( x ) , \\ ; \\mbox { f o r a n y } \\ ; x \\in V \\end{align*}"}
-{"id": "2111.png", "formula": "\\begin{align*} \\frac { \\partial f } { \\partial t } = \\tau ( f ) , \\mbox { a n d } f ( p , 0 ) = \\phi ( p ) \\end{align*}"}
-{"id": "4283.png", "formula": "\\begin{align*} Q _ { E _ 8 } = \\begin{pmatrix} 2 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ - 1 & 2 & - 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & - 1 & 2 & - 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & - 1 & 2 & - 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & - 1 & 2 & - 1 & 0 & - 1 \\\\ 0 & 0 & 0 & 0 & - 1 & 2 & - 1 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & - 1 & 2 & 0 \\\\ 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "9843.png", "formula": "\\begin{align*} ( f ^ { \\prime } ) \\hat ( y ) = i y \\hat { f } ( y ) . \\end{align*}"}
-{"id": "279.png", "formula": "\\begin{align*} I _ { 1 } ( h ) & \\leq \\sup _ { t \\in \\lbrack t _ { 0 } , t _ { 0 } + h ] } \\Vert f ( t , \\xi ( t ) ) \\Vert \\int _ { t _ { 0 } } ^ { t _ { 0 } + h } ( t _ { 0 } + h - \\tau ) ^ { \\alpha - 1 } \\ ; d \\tau \\\\ & = \\sup _ { t \\in \\lbrack t _ { 0 } , t _ { 0 } + h ] } \\Vert f ( t , \\xi ( t ) ) \\Vert \\times \\frac { h ^ { \\alpha } } { \\alpha } . \\end{align*}"}
-{"id": "3901.png", "formula": "\\begin{align*} { \\cal V } = - E \\left ( \\frac { 2 } { \\beta + 2 } \\right ) ^ 2 \\rho ^ { - \\frac { 2 \\beta } { \\beta + 2 } } . \\end{align*}"}
-{"id": "6224.png", "formula": "\\begin{align*} \\P ( \\xi _ i \\leq r _ i ~ \\textrm { f o r e a c h $ 1 \\leq i \\leq n $ } ~ ) = \\P ( \\xi _ i \\leq r _ { \\sigma ( i ) } ~ \\textrm { f o r e a c h $ 1 \\leq i \\leq n $ } ~ ) ~ . \\end{align*}"}
-{"id": "9334.png", "formula": "\\begin{align*} j _ { n } ^ { ( 3 ) } = \\frac { 8 } { 7 } 2 ^ { n } + \\frac { 3 + 2 i \\sqrt { 3 } } { 7 } \\omega _ { 1 } ^ { n } + \\frac { 3 - 2 i \\sqrt { 3 } } { 7 } \\omega _ { 2 } ^ { n } , \\end{align*}"}
-{"id": "9399.png", "formula": "\\begin{align*} \\Omega ( x ' ) & = \\bigg [ \\Omega ( x ' ) - \\frac 1 { \\omega _ { n - 1 } } \\int _ { \\mathbf S ^ { n - 1 } } \\Omega ( y ' ) d \\sigma ( y ' ) \\bigg ] + \\frac 1 { \\omega _ { n - 1 } } \\int _ { \\mathbf S ^ { n - 1 } } \\Omega ( y ' ) d \\sigma ( y ' ) : = \\Omega _ 0 ( x ' ) + C ( \\Omega , n ) , \\end{align*}"}
-{"id": "3817.png", "formula": "\\begin{align*} \\Lambda ^ { ( x , t ) } _ T : = \\left \\{ \\bar { X } ^ { ( x , t ) } _ s - x \\ge \\frac 1 2 v _ \\circ s \\ , \\ ; \\forall \\ , s \\in [ 0 , T ] \\right \\} . \\end{align*}"}
-{"id": "2284.png", "formula": "\\begin{align*} \\sharp \\Delta _ { \\Z } - 4 & = \\sharp V ( S ) - 1 - \\sum _ { k = 1 } ^ N ( \\sharp V ( \\Delta _ k ) - 3 ) \\\\ & = \\sharp V ( S ) - 1 - N _ 4 ' . \\end{align*}"}
-{"id": "7431.png", "formula": "\\begin{align*} \\mathbb { H } _ { \\Gamma } : = \\frac { \\mathbb { C } [ t _ 0 , t _ 1 , t _ 2 , t _ 3 , t _ 4 , t _ 5 , t _ 6 , t _ 7 , t _ 8 ] } { ( t _ 0 + 2 t _ 1 + 3 t _ 2 + 4 t _ 3 + 5 t _ 4 + 3 t _ 5 + 2 t _ 6 + 4 t _ 7 + 6 t _ 8 ) } . \\end{align*}"}
-{"id": "2215.png", "formula": "\\begin{align*} F _ 1 \\left ( \\frac 1 { w _ 1 } , \\ldots , \\frac 1 { w _ n } , t \\right ) = q _ 1 \\left ( \\frac 1 { w _ 1 } , \\ldots , \\frac 1 { w _ n } \\right ) + t \\cdot Q _ 1 \\left ( \\frac 1 { w _ 1 } , \\ldots , \\frac 1 { w _ n } \\right ) , \\end{align*}"}
-{"id": "3159.png", "formula": "\\begin{align*} \\mathrm { d } Y _ { t } ^ { x } = ( a - b Y _ { t } ^ { x } ) \\mathrm { d } t + \\sqrt { Y _ { t } ^ { x } } \\mathrm { d } B _ { t } , t \\geqslant 0 , Y _ { 0 } ^ { x } = x \\in \\mathbb { R } _ { \\geqslant 0 } . \\end{align*}"}
-{"id": "3015.png", "formula": "\\begin{align*} \\Re \\ < A x , x \\ > = \\Re \\ < ( A - P _ 0 ) x , x \\ > _ { L ^ 2 } + \\Re \\ < P _ 0 x , x \\ > _ { L ^ 2 } \\leq 0 \\ ; . \\end{align*}"}
-{"id": "8910.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { N + 1 } \\sum _ { n = 0 } ^ N ( U _ { ( \\theta ) 1 } ^ { - n } \\varphi ( U _ { ( \\theta ) c } ) U _ { ( \\theta ) 1 } ^ n x , x ) & = \\frac { 1 } { N + 1 } \\sum _ { n = 0 } ^ N ( W ^ { - n } \\varphi ( U _ \\nu ) W ^ n \\gamma , \\gamma ) \\\\ & = \\int _ { \\mathbb T } \\varphi \\frac { 1 } { N + 1 } \\sum _ { n = 0 } ^ N | W ^ n \\gamma | ^ 2 \\nu . \\end{aligned} \\end{align*}"}
-{"id": "1794.png", "formula": "\\begin{align*} M ( x , y , z ) = y ^ { z - x } z ^ { x - y } x ^ { y - z } , D : = \\{ 0 < z \\leq y \\leq x \\} . \\end{align*}"}
-{"id": "9607.png", "formula": "\\begin{align*} \\bigg | \\int _ { \\Bbb R ^ n } \\ k _ j ( u , v ) D _ { k ' } ^ \\# ( v , y ) d \\mu ( v ) \\bigg | & = \\bigg | \\int _ { S ( y , A _ 0 2 ^ { 3 + k ' } ) } \\ [ k _ j ( u , v ) - k _ j ( u , y ) ] D _ { k ' } ^ \\# ( v , y ) d \\mu ( v ) \\bigg | \\\\ & \\lesssim \\int _ { S ( y , A _ 0 2 ^ { 3 + k ' } ) } \\ \\frac { 1 } { V _ { - j } ( u ) } | T _ j ( v ) - T _ j ( y ) | ^ { \\gamma } | D _ { k ' } ^ \\# ( v , y ) | d \\mu ( v ) . \\end{align*}"}
-{"id": "3161.png", "formula": "\\begin{align*} \\mathrm { d } Z _ { t } = - b Z _ { t } \\mathrm { d } t + \\sigma \\sqrt { Z _ { t } } \\mathrm { d } B _ { t } + \\mathrm { d } J _ { t } , t \\geqslant 0 , Z _ { 0 } = 0 , \\end{align*}"}
-{"id": "8587.png", "formula": "\\begin{align*} \\varphi _ + \\ast \\varphi _ - ( d ) \\in ( 1 - e ) M ( B ) ( 1 - e ) \\mbox { \\ f o r \\ a n y \\ } d \\in C \\ast _ J C \\mbox { \\ s u c h \\ t h a t \\ } d J = 0 . \\end{align*}"}
-{"id": "8017.png", "formula": "\\begin{align*} f _ L ( x ) : = \\sum _ { k = M } ^ { L } { 2 ^ { - { s } { t _ k } } { \\Lambda } _ { { t _ k } } \\ast \\big ( g _ { { t _ k } } e ^ { - 2 \\pi i \\langle \\mathbf { v } _ { { t _ k } } , \\cdot \\rangle } \\big ) ( x ) } . \\end{align*}"}
-{"id": "1224.png", "formula": "\\begin{align*} \\psi ( z ) = \\frac { e ^ { - N | z - \\hat { w } | ^ { 2 } } - e ^ { - N } } { e ^ { - N / 4 } - e ^ { - N } } \\end{align*}"}
-{"id": "6008.png", "formula": "\\begin{align*} D ( P _ { X } \\| Q _ { X } ) & : = \\sum _ { x \\in \\mathrm { s u p p } ( P _ { X } ) } P _ { X } ( x ) \\log \\frac { P _ { X } ( x ) } { Q _ { X } ( x ) } \\\\ * D _ { 1 + s } ( P _ { X } \\| Q _ { X } ) & : = \\frac { 1 } { s } \\log \\sum _ { x \\in \\mathrm { s u p p } ( P _ { X } ) } P _ { X } ( x ) ^ { 1 + s } Q _ { X } ( x ) ^ { - s } , \\end{align*}"}
-{"id": "6333.png", "formula": "\\begin{align*} \\omega = L ( \\eta ) . \\end{align*}"}
-{"id": "3422.png", "formula": "\\begin{align*} f \\bigl ( [ 1 - \\delta , 1 + \\delta ] \\cup [ - 1 - \\delta , - 1 + \\delta ] \\bigr ) \\cap \\Gamma _ 1 = \\{ f ( 1 ) , f ( - 1 ) \\} . \\end{align*}"}
-{"id": "4852.png", "formula": "\\begin{align*} s = 6 ( 2 d + 5 g - 5 ) - \\sum _ { I } ( 4 m _ p + 4 l _ p - 1 5 ) - \\sum _ { J } ( 1 0 m _ p + c _ p - 1 5 ) . \\end{align*}"}
-{"id": "7931.png", "formula": "\\begin{align*} \\mu _ 0 ( A ) = a _ 1 \\mu _ 0 ( f _ 1 ^ { - 1 } ( A ) ) + \\cdots + a _ m \\mu _ 0 ( f _ m ^ { - 1 } ( A ) ) \\end{align*}"}
-{"id": "8717.png", "formula": "\\begin{align*} & F _ \\varepsilon ( t , x , w , p , X ) = \\min \\{ F ( t , x ' , w , p , X ) \\mid x ' \\in \\overline { B ( x , M \\varepsilon ^ { 1 / 2 } ) } \\} , \\quad \\\\ & F ^ \\varepsilon ( t , x , w , p , X ) = \\max \\{ F ( t , x ' , w , p , X ) \\mid x ' \\in \\overline { B ( x , M \\varepsilon ^ { 1 / 2 } ) } \\} . \\end{align*}"}
-{"id": "835.png", "formula": "\\begin{align*} \\sum _ { a ' \\leq i < b ' } \\pi _ { N } ( d ' ) _ i = \\sum _ { a \\leq i < b } \\pi _ { N } ( d ' ) _ i . \\end{align*}"}
-{"id": "8363.png", "formula": "\\begin{align*} \\psi ( f ) = ( 2 \\pi i ) ^ { c ( 0 , 0 ) } \\Psi ( 2 f ) \\end{align*}"}
-{"id": "7770.png", "formula": "\\begin{align*} \\| D u \\| _ { \\ell ^ { 2 } _ { \\mathcal { N } } ( \\L \\setminus B _ { R } ) } = \\bigg ( \\sum _ { | \\ell | > R } \\sum _ { \\rho \\in \\mathcal { N } ( \\ell ) - \\ell } \\big | D _ \\rho u ( \\ell ) \\big | ^ 2 \\bigg ) ^ { 1 / 2 } < \\varepsilon . \\end{align*}"}
-{"id": "2648.png", "formula": "\\begin{align*} c _ i = \\begin{cases} 3 , & \\mbox { i f } i \\in S _ 1 , \\\\ 2 , & \\mbox { i f } i \\in S _ 2 \\cup S _ 3 , \\\\ 1 , & \\mbox { i f } i \\in [ n ] \\setminus S . \\end{cases} \\end{align*}"}
-{"id": "8655.png", "formula": "\\begin{align*} \\begin{aligned} \\tau _ k = \\min \\Big \\{ n > \\tau _ { k - 1 } : \\int _ { \\tau _ { k - 1 } } ^ { n + 1 } 1 _ { \\{ | x + \\omega _ { X _ 0 } ( s ) - y - \\omega _ { Y _ 0 } ( s ) | \\leq 1 \\} } d s > K \\Big \\} , \\ \\ k \\geq 1 , \\end{aligned} \\end{align*}"}
-{"id": "9403.png", "formula": "\\begin{align*} \\mathcal V _ { \\xi ' } ( x ) = \\mathcal V _ { \\{ x , x _ 1 , x _ 2 , \\ldots , x _ 6 \\} } ( x ) . \\end{align*}"}
-{"id": "7424.png", "formula": "\\begin{align*} T _ 3 & = R _ 2 + S _ 2 + R _ 1 S _ 1 = R _ 2 + S _ 2 - R _ 1 ^ 2 \\\\ T _ 2 & = R _ 3 + R _ 2 S _ 1 + R _ 1 S _ 2 = R _ 3 - R _ 1 R _ 2 + R _ 1 S _ 2 \\\\ T _ 1 & = - R _ 3 R _ 1 + R _ 2 S _ 2 \\\\ T _ 0 & = R _ 3 S _ 2 . \\end{align*}"}
-{"id": "434.png", "formula": "\\begin{align*} { p _ { 1 , k _ 1 , k _ 2 } ( x , t ) = \\frac { ( - 1 ) ^ { k _ 2 } \\pi ^ { k _ 1 + k _ 2 } } { 4 ^ n ( \\pi \\delta ) ^ { n + k _ 1 - 1 } } e ^ { - \\frac { 1 } { 4 } d ( x , t ) ^ 2 } e ^ { - \\kappa } I _ { { n + k _ 1 - 1 } } ( \\kappa ) \\left [ 1 + O ( \\delta ) \\right ] . } \\end{align*}"}
-{"id": "6120.png", "formula": "\\begin{align*} L f \\left ( x \\right ) = \\sum _ { y \\sim x } w _ { x , y } \\left ( f \\left ( y \\right ) - f \\left ( x \\right ) \\right ) . \\end{align*}"}
-{"id": "3979.png", "formula": "\\begin{align*} p ^ { \\alpha _ { m + 1 } } _ { 1 } ( m + 1 , t ) & = - \\lambda I _ t ^ { \\alpha _ { m + 1 } } \\left ( p ^ { \\alpha _ { m + 1 } } _ { 0 } ( m + 1 , t ) - p ^ { \\alpha _ { m } } _ { 0 } ( m , t ) \\right ) = 0 , \\\\ p ^ { \\alpha _ { m + 1 } } _ { 2 } ( m + 1 , t ) & = - \\lambda I _ t ^ { \\alpha _ { m + 1 } } \\left ( p ^ { \\alpha _ { m + 1 } } _ { 1 } ( m + 1 , t ) - p ^ { \\alpha _ { m } } _ { 1 } ( m , t ) \\right ) = 0 . \\end{align*}"}
-{"id": "7406.png", "formula": "\\begin{align*} \\mathbb { H } _ { \\Gamma } : = \\frac { \\mathbb { C } [ t _ 0 , t _ 1 , t _ 2 , t _ 3 , t _ 4 , t _ 5 , t _ 6 , t _ 7 , t _ 8 ] } { ( t _ 0 + 2 t _ 1 + 3 t _ 2 + 4 t _ 3 + 5 t _ 4 + 3 t _ 5 + 2 t _ 6 + 4 t _ 7 + 6 t _ 8 ) } \\end{align*}"}
-{"id": "3827.png", "formula": "\\begin{align*} p _ k : = \\inf _ { | \\eta | = k } \\ , \\P _ \\eta ( G _ \\infty \\cap \\Lambda _ \\infty ) = \\inf _ { y \\in \\Z ^ 2 } \\ , \\inf _ { | \\eta | = k } \\ , \\P _ \\eta ( G ^ y _ \\infty \\cap \\Lambda ^ y _ \\infty ) , \\end{align*}"}
-{"id": "8842.png", "formula": "\\begin{align*} R ( z , a ) : = - \\frac { 3 } { 3 2 } \\frac { z ^ 3 } { a } \\int _ 0 ^ 1 ( 1 - \\theta ) ^ 2 \\left ( - \\theta \\frac { z } { a } + 1 \\right ) ^ { - \\frac { 7 } { 4 } } d \\theta . \\end{align*}"}
-{"id": "1501.png", "formula": "\\begin{align*} T _ { n } ( x ) & = \\frac { \\alpha ^ { n + 1 } ( x ) } { ( \\alpha ( x ) - \\omega _ { 1 } ( x ) ) ( \\alpha ( x ) - \\omega _ { 2 } ( x ) ) } - \\frac { \\omega ^ { n + 1 } ( x ) } { ( \\alpha ( x ) - \\omega _ { 1 } ( x ) ) ( \\omega _ { 1 } ( x ) - \\omega _ { 2 } ( x ) ) } \\\\ & \\ \\ + \\frac { \\omega _ { 2 } ^ { n + 1 } ( x ) } { ( \\alpha ( x ) - \\omega _ { 2 } ( x ) ) ( \\omega _ { 1 } ( x ) - \\omega _ { 2 } ( x ) ) } , \\ n \\geq 0 \\end{align*}"}
-{"id": "4558.png", "formula": "\\begin{align*} \\deg E _ { i , k } = \\deg F _ { i , k } = k , \\deg H _ { i , r } = r , \\deg C = \\deg q ^ h = 0 . \\end{align*}"}
-{"id": "2444.png", "formula": "\\begin{align*} \\delta = \\frac { \\lceil \\alpha k \\rceil ( h + l ) } { k } . \\end{align*}"}
-{"id": "3809.png", "formula": "\\begin{align*} B _ L : = \\{ \\sigma _ { k + 1 } - \\sigma _ k \\le L ^ { \\alpha } \\ ; \\forall \\ ; k = 0 , \\ldots , K \\} , \\end{align*}"}
-{"id": "679.png", "formula": "\\begin{align*} q ( x ) = q _ 0 ( x ) + 2 \\cfrac { d } { d x } K ( x , x ) , \\end{align*}"}
-{"id": "6489.png", "formula": "\\begin{align*} p _ { A } ^ { } \\left ( x _ { A } | \\mu _ { A } \\right ) = \\frac { 1 } { \\mu _ { A } } \\exp \\left ( - \\frac { x _ { A } } { \\mu _ { A } } \\right ) , \\end{align*}"}
-{"id": "9313.png", "formula": "\\begin{align*} ( - a + d _ 1 ) - ( - a + d _ 2 ) = - a ' + f ( d _ 3 ) - \\big ( - a ' + f ( d _ 4 ) \\big ) \\end{align*}"}
-{"id": "4193.png", "formula": "\\begin{align*} \\nabla \\begin{pmatrix} \\alpha \\\\ \\omega \\end{pmatrix} = A \\begin{pmatrix} \\alpha \\\\ \\omega \\end{pmatrix} , \\end{align*}"}
-{"id": "8527.png", "formula": "\\begin{align*} \\mathcal { T } f ( x ) = \\int _ { \\mathbb { R } ^ 3 } e ^ { i x \\cdot \\xi } a ( x , \\xi ) f ( \\xi ) d \\xi , \\end{align*}"}
-{"id": "6035.png", "formula": "\\begin{align*} \\exp \\Big ( - \\Omega ^ { ( \\alpha , \\lambda ) } ( \\{ Q _ { i } \\} _ { i = 1 } ^ { n } ) \\Big ) = \\prod _ { i = 1 } ^ { n } \\Lambda _ { i } ^ { ( \\alpha , \\lambda ) } ( \\{ Q _ { j } \\} _ { j = 1 } ^ { i } ) . \\end{align*}"}
-{"id": "7507.png", "formula": "\\begin{align*} \\begin{cases} \\dot { X } _ t ( x , v ) = \\frac { V _ t ( x , v ) } { \\sqrt { 1 + V _ t ^ 2 ( x , v ) } } \\\\ \\dot { V } _ t ( x , v ) = - \\nabla ( V * \\widetilde { \\rho } _ t ) ( X _ t ( x , v ) ) \\end{cases} \\begin{cases} \\dot { X } ^ k _ t ( x , v ) = \\frac { V ^ k _ t ( x , v ) } { \\sqrt { 1 + V ^ k _ t ( x , v ) ^ 2 } } \\\\ \\dot { V } ^ k _ t ( x , v ) = - \\nabla ( V * \\widetilde { \\rho } ^ k _ t ) ( X ^ k _ t ( x , v ) ) \\end{cases} , \\end{align*}"}
-{"id": "6539.png", "formula": "\\begin{align*} \\Delta = \\left ( \\frac { n | P | _ n } { | B | _ { n - 1 } } \\right ) ^ { \\frac { 1 } { n } } \\delta ^ { 1 / n } . \\end{align*}"}
-{"id": "2569.png", "formula": "\\begin{align*} \\frac { \\eta } 2 \\le \\sin ( \\eta ) = \\frac { | p - h | } { | p - p _ 0 | } = \\frac { | p - p _ 0 | } { 2 r } < \\frac { d _ 0 } { 2 4 r } < \\frac { d _ 0 } { 8 } \\ , . \\end{align*}"}
-{"id": "8806.png", "formula": "\\begin{align*} E _ { m , p } ^ { 0 } ( f ) = \\sum _ { i = 1 } ^ { r } a _ { i , p } m ^ { \\beta _ { i } } p ^ { - \\lambda _ { i } m } . \\end{align*}"}
-{"id": "8055.png", "formula": "\\begin{align*} \\d ( A ) & = \\lambda ^ { 1 / 2 } ( A ^ { 1 / 2 } J ^ T A J A ^ { 1 / 2 } ) \\le \\lambda ^ { 1 / 2 } ( A ^ { 1 / 2 } J ^ T B J A ^ { 1 / 2 } ) \\\\ & = \\lambda ^ { 1 / 2 } ( B ^ { 1 / 2 } J A J ^ T B ^ { 1 / 2 } ) \\le \\lambda ^ { 1 / 2 } ( B ^ { 1 / 2 } J B J ^ T B ^ { 1 / 2 } ) = \\d ( B ) . \\end{align*}"}
-{"id": "6711.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\bigg | \\mathbb { P } ( \\sigma ^ { + } ( t ( N ) ) = \\eta ) - \\frac { 1 } { 2 ^ { N } } \\bigg | = 0 , \\end{align*}"}
-{"id": "4681.png", "formula": "\\begin{align*} \\Gamma \\ = \\ F _ 1 ^ { \\frac { n - 1 - d } { 4 } } \\ , F _ 2 ^ { - \\frac { 1 } { 4 } } \\ , \\end{align*}"}
-{"id": "9787.png", "formula": "\\begin{align*} \\psi ^ { \\rm H d g } _ g : = \\big { ( } ( s ( C ^ { \\bullet , \\bullet } ) , W ) , ( s ( A ^ { \\bullet , \\bullet } ) , F , W ) , \\alpha _ 0 \\big { ) } \\end{align*}"}
-{"id": "475.png", "formula": "\\begin{align*} \\partial _ 1 ^ { k _ 1 + 1 } \\partial _ h ^ 2 \\left [ ( \\psi _ { \\pi / 2 } - P _ { 2 , 0 } \\psi _ { \\pi / 2 } ) a _ { k _ 1 , k _ 2 , \\pi / 2 } \\right ] ( 0 ) = \\frac { 2 } { \\pi } ( - 1 ) ^ { k _ 1 } { i ^ { k _ 2 - n } \\left ( i \\frac { \\pi } { 2 } \\right ) ^ { n + k _ 1 + k _ 2 } ( k _ 1 + 1 ) ! } . \\end{align*}"}
-{"id": "3553.png", "formula": "\\begin{align*} \\Gamma ( E ) = \\{ ( a , E ( a ) ) \\mid a \\in A \\} \\end{align*}"}
-{"id": "1684.png", "formula": "\\begin{align*} \\Lambda ^ \\infty _ { 2 N } = \\{ x \\in \\Lambda ^ \\infty _ { 2 N } \\ , : \\ , r ( x ) = v \\} \\sqcup \\{ x \\in \\Lambda ^ \\infty _ { 2 N } \\ , : \\ , r ( x ) \\ne v \\} . \\end{align*}"}
-{"id": "5997.png", "formula": "\\begin{gather*} x v _ 1 = v _ 1 , y v _ 1 = v _ 2 , z v _ 1 = - v _ 1 , x v _ 2 = - v _ 2 , y v _ 2 = v _ 1 , z v _ 2 = - v _ 2 . \\end{gather*}"}
-{"id": "4348.png", "formula": "\\begin{align*} M _ { \\varphi _ { j , t } } B M _ { 1 - \\psi _ { j , t } } = 0 . \\end{align*}"}
-{"id": "1111.png", "formula": "\\begin{align*} E F F \\ , \\left ( \\mathrm { d B } \\right ) = 1 0 \\ , \\mathrm { l o g } _ { 1 0 } \\left \\{ \\frac { \\mathrm { V a r } [ \\zeta _ i ] } { \\left ( E [ \\zeta _ i ] \\right ) ^ 2 } \\right \\} , \\end{align*}"}
-{"id": "7163.png", "formula": "\\begin{align*} \\sum _ { m = r J + 1 } ^ { ( r + 1 ) J } d _ L ^ * ( m ) \\leq J g ^ * _ r + \\frac { U J ^ 2 + W J } { V _ r } . \\end{align*}"}
-{"id": "4022.png", "formula": "\\begin{align*} \\left . \\frac { \\partial } { \\partial t } \\eta _ t ( p ) \\right | _ { t = 0 } \\in \\mathrm { s p a n } \\{ X ( p ) , Y ( p ) \\} = T _ p M \\ , , \\end{align*}"}
-{"id": "8541.png", "formula": "\\begin{align*} g ( v ) - g ( v ' ) = ( v - v ' ) \\cdot \\nabla g ( v ' ) + | v - v ' | ^ 2 \\int _ 0 ^ 1 ( 1 - \\kappa ) D ^ 2 g ( v ' + \\kappa ( v - v ' ) ) \\cdot ( \\frac { v - v ' } { | v - v ' | } , \\frac { v - v ' } { | v - v ' | } ) d \\kappa , \\end{align*}"}
-{"id": "1267.png", "formula": "\\begin{align*} ( t _ 1 - t _ 2 ) ^ { - 1 } [ \\mbox { C a p } _ { \\mathcal { A } } ( E _ 1 + t _ 1 E _ 2 ) - \\mbox { C a p } _ { \\mathcal { A } } ( E _ 1 + t _ 2 E _ 2 ) ] = T _ 1 + T _ 2 \\end{align*}"}
-{"id": "3348.png", "formula": "\\begin{align*} \\psi _ 1 ( N , K , s ) & = \\frac { \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K } { s + 1 + i } ( N - 1 ) ^ i } { \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K - 1 } { s + i } ( N - 1 ) ^ i } \\\\ & = \\frac { \\sum _ { i = 0 } ^ { K - 1 - s } \\frac { K } { s + 1 + i } \\binom { K - 1 } { s + i } ( N - 1 ) ^ i } { \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K - 1 } { s + i } ( N - 1 ) ^ i } \\\\ & \\leq \\frac { \\sum _ { i = 0 } ^ { K - 1 - s } \\frac { K } { s } \\binom { K - 1 } { s + i } ( N - 1 ) ^ i } { \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K - 1 } { s + i } ( N - 1 ) ^ i } = \\frac { 1 } { \\lambda } . \\end{align*}"}
-{"id": "7224.png", "formula": "\\begin{align*} \\mbox { $ \\partial S = \\Sigma - T $ a n d $ \\mathbf { M } ( S ) = \\mathbb { F } ( \\Sigma - T ) $ . } \\end{align*}"}
-{"id": "9819.png", "formula": "\\begin{align*} p _ 0 ( t ' ) = 1 - \\sum _ { k = 1 } ^ { \\infty } p _ k ( t ' ) . \\end{align*}"}
-{"id": "3966.png", "formula": "\\begin{align*} \\tilde { p } ^ { \\nu _ n } ( n , s ) = \\int _ 0 ^ { \\infty } p ^ { \\nu _ n } ( n , t ) e ^ { - s t } \\ , \\mathrm { d } t = \\frac { s ^ { \\nu _ 1 - 1 } \\prod _ { k = 1 } ^ { n - 1 } \\lambda _ k } { \\prod _ { k = 1 } ^ n ( s ^ { \\nu _ k } + \\lambda _ k ) } , \\ \\ s > 0 . \\end{align*}"}
-{"id": "2459.png", "formula": "\\begin{align*} \\Delta _ { k + 1 } \\cup \\cdots \\cup \\ \\Delta _ l = \\{ \\nu ^ { \\alpha } \\rho , \\nu ^ { \\alpha + 1 } \\rho , \\ldots , \\nu ^ { \\alpha + c ' } \\rho \\} , \\end{align*}"}
-{"id": "6613.png", "formula": "\\begin{align*} \\bigl \\| f _ 0 ^ { - 1 } - g _ { n , 0 } ^ { - 1 } \\bigr \\| _ { \\mathbb { W } ^ { 1 , p ^ * } ( \\Omega _ 1 , \\mathbb { R } ^ d ) } & = \\bigl \\| f _ 0 ^ { - 1 } - g _ { n , 0 } ^ { - 1 } \\bigr \\| _ { C ^ 0 ( \\Omega _ 1 ) } + \\left ( \\int _ { \\Omega _ 1 } \\left | D f _ 0 ^ { - 1 } - D g _ { n , 0 } ^ { - 1 } \\right | ^ { p ^ * } \\ , d \\mu \\right ) ^ { \\frac { 1 } { p ^ * } } \\ . \\end{align*}"}
-{"id": "3456.png", "formula": "\\begin{align*} u \\in \\mathcal { W } ( 0 , T ) = \\big \\{ v \\in L ^ 2 ( 0 , T ; V ) \\ , : \\ , \\dot { v } \\in L ^ 2 ( 0 , T ; V ^ * ) \\big \\} \\end{align*}"}
-{"id": "4146.png", "formula": "\\begin{align*} \\left ( 1 - \\frac { | I | } { n } \\right ) ^ { t - 1 } + \\sum _ { \\tau \\leq t - 1 } \\frac { \\lvert I \\rvert } { n } \\left ( 1 - \\frac { | I | } { n } \\right ) ^ { \\tau - 1 } = 1 . \\end{align*}"}
-{"id": "1266.png", "formula": "\\begin{align*} x _ k ( Z ) = \\nabla h _ k ( Z ) \\mbox { w h e n e v e r } \\ , \\ , Z \\in \\mathbb { S } ^ { n - 1 } . \\end{align*}"}
-{"id": "6412.png", "formula": "\\begin{align*} \\xi d l _ { \\rightarrow \\xi } = d l _ { \\xi \\rightarrow } \\end{align*}"}
-{"id": "2179.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } t ^ { \\frac { d } { 2 } } u _ * ( 0 , t ) = \\frac { c _ d } { c _ * } m ( \\varphi ) . \\end{align*}"}
-{"id": "950.png", "formula": "\\begin{align*} P ( Q _ \\vee ( Y ) \\in A ) \\leq P ( \\Phi _ \\beta ( Q ( Y ) ) \\in A ^ { \\varepsilon } ) = E [ 1 _ { A ^ { \\varepsilon } } ( \\Phi _ \\beta ( Q ( Y ) ) ) ] . \\end{align*}"}
-{"id": "971.png", "formula": "\\begin{align*} P \\left ( T _ n < q _ n ^ * ( 1 - \\alpha ) \\right ) & \\leq P \\left ( P ( T _ n ^ * \\leq T _ n | \\mathcal { F } ^ X ) < 1 - \\alpha \\right ) \\leq \\alpha ^ { - 1 } P ( T _ n ^ * > T _ n ) . \\end{align*}"}
-{"id": "2971.png", "formula": "\\begin{align*} B = \\sum _ { j = 1 } ^ { 2 } p _ j | h _ { k j } | ^ { 2 } - \\prod _ { j = 1 } ^ { 2 } \\left ( \\frac { p _ j | h _ { k j } | ^ { 2 } } { c _ { k j } } \\right ) ^ { c _ { k j } } . \\end{align*}"}
-{"id": "5590.png", "formula": "\\begin{align*} U ^ { - 1 } = \\left [ \\begin{array} { c c c c c c } \\ast & \\ast & \\cdots & \\ast & \\ast & a _ { n - 1 } \\\\ 0 & \\ast & \\cdots & \\ast & \\ast & a _ { n - 2 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots \\\\ 0 & 0 & \\cdots & \\ast & \\ast & a _ { 2 } \\\\ 0 & 0 & \\cdots & 0 & \\ast & a _ { 1 } \\\\ 0 & 0 & \\cdots & 0 & 0 & a _ { 0 } \\end{array} \\right ] \\end{align*}"}
-{"id": "6119.png", "formula": "\\begin{align*} T _ { i } ^ { n , 1 } & = \\inf \\left \\{ T ^ { n } \\cap [ t _ { i } , t _ { i + 1 } ) \\right \\} \\\\ T _ { i } ^ { n , 2 } & = \\sup \\left \\{ T ^ { n } \\cap [ t _ { i } , t _ { i + 1 } ) \\right \\} , \\end{align*}"}
-{"id": "6633.png", "formula": "\\begin{align*} A _ 3 & = \\left ( \\int _ { \\varphi _ n ( E _ M ) } | D h _ n \\circ \\phi _ n - D h _ n \\circ \\phi _ n D \\phi _ n | ^ p \\ , d \\mu \\right ) ^ { \\frac { 1 } { p } } \\\\ & \\leq \\| \\mathrm { i d } - D \\phi _ n \\| _ { L ^ \\infty } \\| D \\phi _ n ^ 0 \\| _ { L ^ \\infty } \\left ( \\int _ { \\varphi _ n ( E _ M ) } | D f _ n | ^ p \\ , d \\mu \\right ) ^ { \\frac { 1 } { p } } \\ . \\end{align*}"}
-{"id": "5227.png", "formula": "\\begin{align*} J A + A ^ T J = - c J . \\end{align*}"}
-{"id": "1184.png", "formula": "\\begin{align*} \\rho _ 2 = \\rho _ 1 + \\frac { B } { B _ 1 } \\left ( \\frac { A _ 2 } { A ^ { 2 } } - \\frac { B _ 2 } { B ^ { 2 } } \\right ) \\rho _ 1 ^ { 2 } + o ( \\rho ^ { 2 } _ 1 ) . \\end{align*}"}
-{"id": "4410.png", "formula": "\\begin{align*} ( T _ f ) _ { z _ \\gamma } = T _ { f \\circ \\tau _ { z _ \\gamma } } \\overset { s } { \\to } T _ { x ( f ) } = ( T _ f ) _ x \\end{align*}"}
-{"id": "6927.png", "formula": "\\begin{align*} u ( x , t ) = \\sum _ { \\lambda _ { j } } \\frac { \\sin { t \\sqrt { \\lambda _ { j } } } } { \\sqrt { \\lambda _ { j } } } u _ { j } ( x ) \\int u _ { j } ( y ) f ( y ) d \\mu ( y ) \\end{align*}"}
-{"id": "2506.png", "formula": "\\begin{align*} \\mathrm { P } _ { 0 } e _ { 2 } \\left ( \\eta \\right ) & = g ^ { - 1 } \\cdot \\mathrm { P } _ { 0 } \\psi _ { 2 } \\left ( \\left | \\eta \\right | \\right ) \\\\ & = \\mathtt { a } _ { 2 , 1 } ^ { 0 } ( \\left | \\eta \\right | ^ { 2 } ) \\chi _ { 0 } + i \\mathtt { a } _ { 2 , 2 } ^ { 1 } ( \\left | \\eta \\right | ^ { 2 } ) \\sum _ { j = 1 } ^ { 3 } \\eta _ { j } \\chi _ { j } + \\mathtt { a } _ { 2 , 1 } ^ { 4 } ( \\left | \\eta \\right | ^ { 2 } ) \\chi _ { 4 } , \\end{align*}"}
-{"id": "2406.png", "formula": "\\begin{align*} R _ { 1 1 3 } = 9 a _ 6 ^ 2 \\left ( a _ 5 + a _ 1 \\left ( - a _ 3 + a _ 1 a _ 5 + 4 a _ 6 + 3 a _ 1 ^ 2 a _ 6 \\right ) \\right ) ^ 2 p _ 1 ' ( x _ 0 ) . \\end{align*}"}
-{"id": "3038.png", "formula": "\\begin{align*} V _ \\sigma : = \\{ v \\in V \\mid b ( v , q ) = 0 , \\ \\forall q \\in Q \\} . \\end{align*}"}
-{"id": "268.png", "formula": "\\begin{align*} \\int _ { \\Sigma _ { k } } F \\Big ( t _ 0 x _ 0 + t _ 1 x _ 1 + \\dots t _ { k } x _ { k } \\Big ) d \\sigma = \\int _ { - 1 } ^ { 1 } F ( s ) g _ { k } ( s ) d s . \\end{align*}"}
-{"id": "7794.png", "formula": "\\begin{align*} & x y z \\approx z y x , \\ , x ^ 2 y \\approx 0 ; \\\\ & x y z \\approx y z x , \\ , x ^ 2 y \\approx 0 ; \\\\ & x y z \\approx y x z , \\ , x y z t \\approx x z t y , \\ , x y ^ 2 \\approx 0 ; \\\\ & x y z \\approx x z y , \\ , x y z t \\approx y z x t , \\ , x ^ 2 y \\approx 0 . \\end{align*}"}
-{"id": "3689.png", "formula": "\\begin{align*} ( 0 \\ ; 1 ) : N _ X = \\mathbb Z ^ 2 \\to \\mathbb Z , \\mbox { a n d } ( 1 \\ ; 1 \\ ; \\cdots \\ ; 1 ) : N _ { \\mathbb A ^ { n + 1 } } = \\mathbb Z ^ { n + 1 } \\to \\mathbb Z . \\end{align*}"}
-{"id": "6963.png", "formula": "\\begin{align*} f ( D ^ 2 u ) = ( \\det { D ^ 2 u } ) ^ { 1 / n } = \\inf _ { M \\in \\mathcal { M } } { L _ M u } , \\end{align*}"}
-{"id": "3516.png", "formula": "\\begin{align*} J ( u ( \\cdot ) ) = \\int _ 0 ^ 1 \\left ( X ^ u ( t ) \\right ) ^ 2 d t + \\left ( X ^ u ( 1 ) \\right ) ^ 2 \\end{align*}"}
-{"id": "8768.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Big ( x \\Psi _ B ^ { { \\bf f } } ( t _ 1 , \\ldots , t _ { m - 1 } , t ) ^ { - 1 } \\Big ) = - x V _ B ^ { { \\bf f } } ( t _ 1 , \\ldots , t _ { m - 1 } , t ) \\Psi _ B ^ { { \\bf f } } ( t _ 1 , \\ldots , t _ { m - 1 } , t ) ^ { - 1 } \\ , . \\end{align*}"}
-{"id": "7714.png", "formula": "\\begin{align*} D ^ 2 _ B ( j ) & = ( N - 1 ) \\cdot \\frac { 2 } { N ^ 2 } = \\frac { 2 ( N - 1 ) } { N ^ 2 } \\ , , \\\\ C _ B ( j ) & = \\frac { N ^ 3 } { 2 ( N - 1 ) } \\ , . \\end{align*}"}
-{"id": "6975.png", "formula": "\\begin{align*} f _ k ( D ^ 2 u ) = \\inf _ { M \\in \\mathcal { M } _ k } \\{ t r a c e ( M D ^ 2 u ) \\} \\\\ . \\end{align*}"}
-{"id": "32.png", "formula": "\\begin{align*} \\frac { \\partial f } { \\partial z ^ { * } } ( c ) = \\frac { 1 } { 2 } \\left ( \\frac { \\partial f } { \\partial x } ( c ) + j \\frac { \\partial f } { \\partial y } ( c ) \\right ) \\end{align*}"}
-{"id": "4426.png", "formula": "\\begin{align*} \\lefteqn { ( - \\partial _ 1 ^ 2 - | \\partial _ 1 | ^ { - 1 } \\partial _ 2 ^ 2 ) w } \\\\ & + P \\big ( \\sigma ^ 2 F + \\sigma v \\partial _ 2 R w + \\sigma w \\partial _ 2 R v + w \\partial _ 2 R w \\big ) \\\\ & + \\frac { 1 } { 2 } \\partial _ 2 R ( w + \\sigma v ) ^ 2 - \\frac { 1 } { 2 } P ( ( w + \\sigma v ) \\partial _ 1 R ( w + \\sigma v ) ^ 2 ) = 0 \\end{align*}"}
-{"id": "6606.png", "formula": "\\begin{align*} \\liminf _ { J ' \\ni m \\to \\infty } \\int _ { \\psi _ n ( \\mathsf { V } _ n ) } | D f _ n ^ { - 1 } - D g _ { m , n } ^ { - 1 } | ^ { p ^ * } \\ , d \\mu \\ ; = \\ ; 0 \\ . \\end{align*}"}
-{"id": "7365.png", "formula": "\\begin{align*} ( \\psi \\overline { \\psi } ) _ p = ( \\psi , e _ { a _ p . . . a _ 2 a _ 1 } . \\psi ) e ^ { a _ 1 a _ 2 . . . a _ p } . \\end{align*}"}
-{"id": "5219.png", "formula": "\\begin{align*} \\mathrm { R e } \\sqrt { a + b i } = \\frac { 1 } { \\sqrt { 2 } } \\sqrt { \\sqrt { a ^ 2 + b ^ 2 } + a } \\end{align*}"}
-{"id": "9730.png", "formula": "\\begin{align*} & V _ b = \\tilde { \\Phi } _ 5 ( \\gamma _ 5 , \\mathcal { F } ( \\sigma _ 3 , \\sigma _ 2 ; \\tilde { \\Phi } _ 1 ( \\gamma _ 1 ; V _ a ) ) ) , Z _ b = Z _ a + \\gamma _ 4 . \\end{align*}"}
-{"id": "2287.png", "formula": "\\begin{align*} d ( S ) & = \\sharp \\Delta _ { \\Z } - 4 - \\{ \\sharp V ( S ) - 1 - 5 \\} \\\\ & = \\sharp \\Delta _ { \\Z } - \\sharp V ( S ) + 2 \\\\ & \\ge 4 . \\end{align*}"}
-{"id": "4378.png", "formula": "\\begin{align*} ( C _ { z } M _ f C _ { - { z } } g ) ( w ) & = e ^ { \\alpha \\langle w , { z } \\rangle - \\frac { \\alpha } { 2 } | { z } | ^ 2 } ( M _ f C _ { - { z } } g ) ( w - z ) \\\\ & = e ^ { \\alpha \\langle w , z \\rangle - \\frac { \\alpha } { 2 } | z | ^ 2 } f ( w - z ) ( C _ { - z } g ) ( w - z ) \\\\ & = f ( w - z ) g ( w ) \\end{align*}"}
-{"id": "9623.png", "formula": "\\begin{align*} \\sum _ { \\beta = 1 } ^ { \\kappa } N _ \\beta ( t ) \\bigg ( P _ { \\rm { t r } } \\big ( R _ { \\beta } , \\lambda _ { \\beta } , h _ \\beta \\big ) + P _ { \\rm { c u } } \\bigg ) = E _ { \\rm { b } } \\sum _ { \\beta = 1 } ^ { \\kappa } \\Phi _ { \\rm { s t } , \\beta } ( t ) . \\end{align*}"}
-{"id": "8362.png", "formula": "\\begin{align*} \\widehat { \\Z } ^ \\times = \\nu _ \\Phi ( s ( \\widehat { \\Z } ^ \\times ) ) \\subset \\nu _ \\Phi ( K _ \\Phi ) \\subset \\nu ( K ) , \\end{align*}"}
-{"id": "8742.png", "formula": "\\begin{align*} & u ^ \\varepsilon ( t , x ) = \\sup _ { x ' \\in \\overline { \\Omega } } \\{ u ( t , x ' ) - \\varepsilon ^ { - 1 } | x - x ' | ^ 2 \\} \\quad \\\\ & v _ \\varepsilon ( t , x ) = \\inf _ { x ' \\in \\overline { \\Omega } } \\{ v ( t , x ' ) + \\varepsilon ^ { - 1 } | x - x ' | ^ 2 \\} , \\end{align*}"}
-{"id": "4485.png", "formula": "\\begin{align*} \\psi = \\frac { 1 } { 2 } ( \\phi _ 1 + \\phi _ 2 ) \\end{align*}"}
-{"id": "5336.png", "formula": "\\begin{align*} a ( v \\otimes x ) = a \\cdot v \\otimes x + v \\otimes a x \\end{align*}"}
-{"id": "7700.png", "formula": "\\begin{align*} Z = \\left [ \\begin{array} { c c } Q e _ j ( Q e _ j ) ^ { \\top } & 0 \\\\ 0 & 0 \\end{array} \\right ] \\ , . \\end{align*}"}
-{"id": "6557.png", "formula": "\\begin{align*} | ( F _ { \\xi } - s ( F _ { \\xi } ) ) ^ { \\circ } | _ { n - 1 } = \\frac { 1 } { | F _ \\xi | _ { n - 1 } } \\cdot \\frac { n ^ n } { ( ( n - 1 ) ! ) ^ 2 } = \\frac { n ^ n } { \\sqrt { n } ( n - 1 ) ! } . \\end{align*}"}
-{"id": "4034.png", "formula": "\\begin{align*} \\Delta _ b g ( p ) = \\mathrm { t r } _ b ( \\mathrm { h e s s } _ b g | _ p ) = \\mathrm { t r } _ b ( - h ) = \\mathrm { t r } ( d \\eta _ p ) = 2 H ( p ) . \\end{align*}"}
-{"id": "104.png", "formula": "\\begin{align*} \\Gamma = \\left \\{ \\left ( \\frac { t _ 1 } { \\rho _ 1 } , \\dots , \\frac { t _ k } { \\rho _ k } \\right ) \\middle | \\ , t _ 1 , \\dots , t _ k \\in \\mathbb { Z } \\right \\} . \\end{align*}"}
-{"id": "1103.png", "formula": "\\begin{align*} n _ { o , j } ^ { ( i ) } = \\mu \\rho _ i ^ 2 | \\alpha _ { i j } | ^ 2 + \\sqrt { \\mu \\rho _ i ^ 2 | \\alpha _ { i j } | ^ 2 } Z _ { s , j } ^ { ( i ) } + Z _ { 0 , j } ^ { ( i ) } , \\end{align*}"}
-{"id": "9338.png", "formula": "\\begin{align*} N r ^ { 2 } ( p \\cdot q ) = N r ^ { 2 } ( p ) N r ^ { 2 } ( q ) \\ \\textrm { a n d } \\ ( p \\cdot q ) ^ { - 1 } = q ^ { - 1 } \\cdot p ^ { - 1 } . \\end{align*}"}
-{"id": "3610.png", "formula": "\\begin{align*} \\Omega _ \\lambda = \\{ x \\in \\Omega : x _ 1 < \\lambda \\} \\end{align*}"}
-{"id": "6838.png", "formula": "\\begin{align*} \\frac { \\partial \\phi } { \\partial \\lambda } = \\tilde { \\psi } + \\sum \\limits _ { j = 1 } ^ 4 b _ j \\eta _ { R _ 3 , \\xi _ j } \\varphi _ { 0 , j } . \\end{align*}"}
-{"id": "6788.png", "formula": "\\begin{align*} w _ i ( \\Pi ^ { - 1 } _ { \\xi _ k } ( \\lambda z ) ) = - 4 \\ln { \\lambda } - 4 \\ln { 2 } - 4 \\lambda ^ 2 \\ln { \\lambda } + \\ln { \\left ( \\frac { 1 } { ( 1 + | z | ^ 2 ) ^ 2 } \\right ) } + 2 \\ln { ( 1 + \\lambda ^ 2 | z | ^ 2 ) } , \\end{align*}"}
-{"id": "3932.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle { u _ j ^ n = a _ 1 c _ { j - 1 } ^ n + a _ 2 c _ { j } ^ n + a _ 1 c _ { j + 1 } ^ n } \\\\ \\displaystyle { ( u _ j ^ n ) _ x = - a _ 3 c _ { j - 1 } ^ n + a _ 3 c _ { j + 1 } ^ n } \\\\ \\displaystyle { ( u _ j ^ n ) _ { x x } = a _ 4 c _ { j - 1 } ^ n + a _ 5 c _ { j } ^ n + a _ 4 c _ { j + 1 } ^ n , } \\end{cases} \\end{align*}"}
-{"id": "6466.png", "formula": "\\begin{align*} \\Sigma \\left ( \\rho _ { k } , \\lambda _ { k } , \\alpha _ { \\pm } \\right ) \\overset { } { = } - \\frac { \\Xi _ { k } } { 4 \\lambda _ { k } } \\frac { 1 + \\sqrt { \\Delta \\left ( \\rho _ { k } \\right ) } } { 1 - \\sqrt { \\Delta \\left ( \\rho _ { k } \\right ) } } \\sqrt { \\frac { 2 \\alpha _ { - } \\left ( \\rho _ { k } \\right ) } { \\alpha _ { + } \\left ( \\rho _ { k } \\right ) } } > 0 \\forall \\rho \\in \\lbrack 0 , 1 ) \\end{align*}"}
-{"id": "9175.png", "formula": "\\begin{align*} \\mathrm { H e s s } _ { \\mathbb { C } } ( \\psi ) ( \\exp ( x ) ) = \\begin{pmatrix} \\frac { 1 } { 4 } \\mathrm { H e s s } _ { \\mathbb { R } } ( \\psi ) ( x ) & 0 & & & 0 \\\\ 0 & M _ { \\alpha _ { ( 1 ) } } ( x ) & & & 0 \\\\ 0 & 0 & \\ddots & & \\vdots \\\\ \\vdots & \\vdots & & \\ddots & 0 \\\\ 0 & 0 & & & M _ { \\alpha _ { ( \\frac { n - r } { 2 } ) } } ( x ) \\\\ \\end{pmatrix} , \\end{align*}"}
-{"id": "3022.png", "formula": "\\begin{align*} W _ B \\Sigma W _ B ^ * & = S \\begin{bmatrix} I + V & I - V \\end{bmatrix} \\begin{bmatrix} 0 & I \\\\ I & 0 \\end{bmatrix} ( S \\begin{bmatrix} I + V & I - V \\end{bmatrix} ) ^ * \\\\ & = S ( 2 I - 2 V V ^ * ) S ^ * = 0 . \\end{align*}"}
-{"id": "1832.png", "formula": "\\begin{align*} \\sum _ { i \\in I } \\sum _ { j \\in J _ i } \\left \\langle \\Gamma _ { i j } \\Lambda _ i T f , \\Gamma _ { i j } \\Lambda _ i U f \\right \\rangle & = \\sum _ { i \\in I } \\sum _ { j \\in J _ i } \\left \\langle \\Gamma ^ * _ { i j } \\Gamma _ { i j } \\Lambda _ i T f , \\Lambda _ i U f \\right \\rangle \\\\ & \\ge \\sum _ { i \\in I } C _ i \\left \\langle \\Lambda _ i T f , \\Lambda _ i U f \\right \\rangle \\ge C A \\| f \\| ^ 2 . \\end{align*}"}
-{"id": "9357.png", "formula": "\\begin{align*} 4 9 ( j _ { n + r } ^ { ( 3 ) } J _ { n + s } ^ { ( 3 ) } ) & = 2 ^ { 2 n + 4 + r + s } - 2 ^ { n + r + 3 } ( a \\omega _ { 1 } ^ { n + s } + b \\omega _ { 2 } ^ { n + s } ) + 3 \\cdot 2 ^ { n + s + 1 } ( a \\omega _ { 1 } ^ { n + r } + b \\omega _ { 2 } ^ { n + r } ) \\\\ & \\ \\ - 3 ( a ^ { 2 } \\omega _ { 1 } ^ { 2 n + r + s } + b ^ { 2 } \\omega _ { 2 } ^ { 2 n + r + s } ) - 7 ( \\omega _ { 1 } ^ { r } \\omega _ { 2 } ^ { s } + \\omega _ { 1 } ^ { s } \\omega _ { 2 } ^ { r } ) , \\end{align*}"}
-{"id": "5879.png", "formula": "\\begin{align*} \\begin{aligned} & P ( Z ^ { N , i } _ 1 = 1 ) \\approx e ^ { - N I _ i ( r _ { i + 1 } ) } ; \\\\ & P ( Z ^ { N , i } _ 1 = - 1 ) \\approx e ^ { - N I _ i ( r _ i ) } ; \\\\ & P ( Z ^ { N , i } _ 1 = - 1 1 ) \\lessapprox e ^ { - N \\big ( I _ i ( r _ i ) + I _ i ( r _ { i + 1 } ) \\big ) } . \\end{aligned} \\end{align*}"}
-{"id": "8740.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t ^ \\alpha u + F ( t , x , u , \\nabla u , \\nabla ^ 2 u ) = 0 \\quad & \\\\ \\nabla u \\cdot n ( x ) = 0 \\quad & \\end{cases} \\end{align*}"}
-{"id": "2765.png", "formula": "\\begin{align*} \\sum _ { Q - Q _ 1 \\in \\Delta ^ - } w ( Q ) = \\sum _ { Q - Q _ 1 \\in \\Delta ^ - } w ( Q _ 1 + \\eta ( Q - Q _ 1 ) ) . \\end{align*}"}
-{"id": "928.png", "formula": "\\begin{align*} \\Psi ' ( t ) = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ d E \\left [ \\frac { \\partial \\varphi } { \\partial x _ j } ( \\sqrt { t } F + \\sqrt { 1 - t } Z ) \\left ( \\frac { F _ j } { \\sqrt { t } } - \\frac { Z _ j } { \\sqrt { 1 - t } } \\right ) \\right ] \\end{align*}"}
-{"id": "1229.png", "formula": "\\begin{align*} k [ h ( X ) \\ , \\nabla h ( X ) ] \\ , \\ , \\nabla k [ h ( X ) \\nabla h ( X ) ] = h ( X ) \\ , k [ \\nabla h ( X ) ] ( X / h ( X ) ) = X . \\end{align*}"}
-{"id": "5993.png", "formula": "\\begin{gather*} x ^ 2 = y ^ 2 = z ^ 2 = 1 , x z = z x , z y = y z , x y z = y x . \\end{gather*}"}
-{"id": "5420.png", "formula": "\\begin{align*} Y = C _ { 2 + } \\wedge C X \\cup X ^ { \\ast C _ 2 } . \\end{align*}"}
-{"id": "459.png", "formula": "\\begin{align*} a _ { k _ 1 , k _ 2 } ^ { ( h ) } ( \\lambda ) = ( - 1 ) ^ { k _ 1 } i ^ { k _ 2 } \\frac { k _ 2 ! } { ( k _ 2 - h ) ! } \\lambda _ 1 ^ { k _ 2 - h } u _ 1 ^ { \\otimes h } + O \\left ( \\abs { \\lambda } ^ { k _ 2 - h + 2 } \\right ) \\end{align*}"}
-{"id": "7016.png", "formula": "\\begin{align*} z _ n = y _ n , z _ j = \\frac { y _ j } { | y _ n | } \\sqrt { \\epsilon } g ( \\epsilon ) , j = 1 , 2 , . . . , n - 1 . \\end{align*}"}
-{"id": "5063.png", "formula": "\\begin{align*} g = \\rho ^ 2 I = \\frac { n } { n - 1 } ( | I I | ^ 2 - n H ^ 2 ) I = \\left ( 4 H _ u ^ 2 - 3 K _ u \\right ) ( I _ { \\mathbb { R } ^ 1 } + I _ u ) , \\end{align*}"}
-{"id": "8689.png", "formula": "\\begin{align*} \\lbrace x , \\lbrace x , x \\rbrace \\rbrace = 0 \\forall x \\in \\gg _ 1 \\end{align*}"}
-{"id": "7891.png", "formula": "\\begin{align*} \\begin{aligned} F ( x , y , z , t , r , s ) & \\le C d ( z , m ( x , y ) ) - k d ( z , m ( x , y ) ) ^ 2 + C ( t + r + s ) \\\\ & \\le { C ^ 2 \\over 4 k } + 3 C T < \\infty \\end{aligned} \\end{align*}"}
-{"id": "7255.png", "formula": "\\begin{align*} 0 \\le n - \\Phi ( n ) = \\sum _ { \\substack { u \\le n : \\\\ ( u , n ) > T } } 1 = \\sum _ { \\substack { d \\mid n : \\\\ d > T } } \\phi \\left ( \\frac { n } { d } \\right ) \\le \\sum _ { \\substack { d \\mid n : \\\\ d > T } } \\frac n d , \\end{align*}"}
-{"id": "9614.png", "formula": "\\begin{align*} \\bar { L } ( r , h _ \\beta ) & = P _ { 0 } ( r , h _ \\beta ) L _ 0 ( r , h _ \\beta ) + \\left ( 1 - P _ { 0 } ( r , h _ \\beta ) \\right ) L _ 1 ( r , h _ \\beta ) \\\\ & = \\underbrace { ( 4 \\pi f / c ) ^ 2 \\left ( r ^ 2 + h _ \\beta ^ 2 \\right ) } _ { \\rm { F S P L } } \\ , \\underbrace { \\big ( \\eta _ 1 + P _ { 0 } ( r , h _ \\beta ) ( \\eta _ 0 - \\eta _ 1 ) \\big ) } _ { \\rm { a v e r a g e \\ , e x c e s s i v e \\ , p a t h \\ , l o s s } } . \\end{align*}"}
-{"id": "9413.png", "formula": "\\begin{align*} a _ { i , j } = a _ { i ' , j } b _ { i , k } = b _ { i ' , k } i \\equiv i ' \\pmod { \\frac { d } { \\ell } } \\end{align*}"}
-{"id": "8707.png", "formula": "\\begin{align*} \\partial _ t ^ \\alpha u + A u = 0 \\end{align*}"}
-{"id": "177.png", "formula": "\\begin{align*} C _ { x } = \\{ c \\in K ( L ) : c \\leq x \\} . \\end{align*}"}
-{"id": "5878.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\limsup _ { n \\to \\infty } \\frac { X ^ { N ; I } _ n } n = \\lim _ { N \\to \\infty } \\liminf _ { n \\to \\infty } \\frac { X ^ { N ; I } _ n } n = \\begin{cases} \\mu _ 0 , \\ \\ I _ 0 ( r _ 1 ) > I _ 1 ( r _ 1 ) ; \\\\ \\mu _ 1 , \\ \\ I _ 1 ( r _ 1 ) > I _ 0 ( r _ 1 ) . \\end{cases} \\end{align*}"}
-{"id": "2188.png", "formula": "\\begin{align*} \\frac { 4 \\cdot ( N / 2 ) ^ 3 } { r \\cdot ( N / r ) ^ 3 } = r ^ 2 / 2 \\end{align*}"}
-{"id": "1080.png", "formula": "\\begin{align*} y _ i & = \\int \\bigg | \\sum \\limits _ { k \\in \\mathcal { N } } ^ { } \\rho _ k \\alpha _ { k i } u _ i ( \\mathbf { r } , z ) \\bigg | ^ 2 \\ , d \\mathbf { r } \\\\ & = \\bigg | \\sum \\limits _ { k \\in \\mathcal { N } } ^ { } \\rho _ k \\alpha _ { k i } \\bigg | ^ 2 , \\end{align*}"}
-{"id": "4331.png", "formula": "\\begin{align*} t _ { A } ( { \\bf x } , { \\bf y } ; z ) & = z \\cdot \\prod _ { a \\in A - \\max ( A ) } ( z + { \\bf y } _ { \\leq a } ^ { A } + { \\bf x } _ { > a } ^ { A } ) , \\\\ s _ { A } ( { \\bf x } , { \\bf y } ; z ) & = \\prod _ { a \\in A } ( z + { \\bf y } _ { \\leq a } ^ { A } + { \\bf x } _ { > a } ^ { A } ) . \\end{align*}"}
-{"id": "4688.png", "formula": "\\begin{align*} D _ 6 \\ = \\ c _ { 6 } ( m = 1 ) \\ F _ 1 \\ F _ 2 \\ , \\ c _ 6 ( m = 1 ) = \\ 6 \\ , c _ 1 \\ , c _ 2 \\ , . . . c _ { 5 } \\ , \\end{align*}"}
-{"id": "3257.png", "formula": "\\begin{align*} \\biggl ( i _ ! ( \\alpha _ { i , Y } ^ a ) , \\omega \\biggr ) _ { X } = ( \\alpha , u ^ i , \\omega ) _ { X } = \\psi ^ i ( u ^ i \\omega ) , \\end{align*}"}
-{"id": "5327.png", "formula": "\\begin{align*} \\widehat { M } _ { \\ell } = U ( \\hat { \\mathfrak { g } } ) \\otimes _ { U ( \\mathfrak { g } \\oplus \\C { \\bf k } \\oplus \\hat { \\mathfrak { g } } _ { + } ) } M , \\end{align*}"}
-{"id": "7843.png", "formula": "\\begin{align*} f _ { X _ { i } \\mid X _ { j } } ( x _ { i } \\mid x _ { j } ) \\ = \\frac { f _ { X _ { i } , X _ { j } } ( x _ { i } , x _ { j } ) } { f _ { X _ { j } } ( x _ { j } ) \\ } , ( i \\neq j = 1 , 2 ) . \\end{align*}"}
-{"id": "5230.png", "formula": "\\begin{align*} \\Omega ( Y _ 1 , Y _ 2 ) = \\lim \\limits _ { z \\rightarrow \\infty } e ^ { c z } \\omega ( Y _ 1 , Y _ 2 ) = 0 . \\end{align*}"}
-{"id": "7225.png", "formula": "\\begin{align*} & F ( \\Sigma ) - F ( T ) = \\int _ M 1 d \\| \\Sigma \\| - \\int _ { \\R ^ d } 1 d \\| T \\| \\\\ & \\leq \\int _ { \\R ^ d } \\langle \\omega ( x ) ; \\tau _ \\Sigma ( x ) \\rangle d \\| \\Sigma \\| ( x ) - \\int _ { \\R ^ d } \\langle \\omega ( x ) ; \\vec { T } ( x ) \\rangle d \\| T \\| ( x ) \\\\ & = \\langle S ; d \\omega \\rangle \\leq \\| d \\omega \\| _ \\infty \\mathbf { M } ( S ) = \\| d \\omega \\| _ \\infty \\mathbb { F } ( \\Sigma - T ) . \\end{align*}"}
-{"id": "8724.png", "formula": "\\begin{align*} & u ( t ' , x ' ) - \\varepsilon ^ { - 1 } | \\hat { x } - x ' | ^ 2 - \\delta ^ { - 1 } | \\hat { t } - t ' | ^ 2 - \\varphi ( \\hat { t } , \\hat { x } ) \\\\ & = ( u ^ { \\varepsilon , \\delta } - \\varphi ) ( \\hat { t } , \\hat { x } ) \\\\ & \\ge u ^ { \\varepsilon , \\delta } ( t - t ' + \\hat { t } , x - x ' + \\hat { x } ) - \\varphi ( t - t ' + \\hat { t } , x - x ' + \\hat { x } ) \\\\ & \\ge u ( t , x ) - \\varepsilon ^ { - 1 } | \\hat { x } - x ' | ^ 2 - \\delta ^ { - 1 } | \\hat { t } - t ' | ^ 2 - \\varphi ( t - t ' + \\hat { t } , x - x ' + \\hat { x } ) \\end{align*}"}
-{"id": "2418.png", "formula": "\\begin{align*} p _ 1 ( x ) = x ( a _ 3 - a _ 5 x - a _ 6 x ^ 2 ) , p _ 3 ( x ) = ( 1 + a _ 1 x ) ( a _ 3 - a _ 5 x - a _ 6 x ^ 2 ) . \\end{align*}"}
-{"id": "8151.png", "formula": "\\begin{align*} a _ 0 = b _ 0 , \\end{align*}"}
-{"id": "8096.png", "formula": "\\begin{align*} \\begin{array} { c c c } f _ 0 = X _ 0 ^ { d - 3 } X _ 1 ( X _ 0 ^ 2 - X _ 1 ^ 2 ) , & & f _ 2 = X _ 0 ^ { d - 3 } X _ 2 ( X _ 1 ^ 2 - X _ 2 ^ 2 ) , \\\\ f _ 1 = X _ 0 ^ { d - 3 } X _ 2 ( X _ 0 ^ 2 - X _ 1 ^ 2 ) , & & f _ 3 = X _ 1 ^ { d - 3 } X _ 2 ( X _ 1 ^ 2 - X _ 2 ^ 2 ) . \\end{array} \\end{align*}"}
-{"id": "4718.png", "formula": "\\begin{align*} \\nabla ^ i T _ { i j } = 0 , i , j , k = 0 . . . 3 , \\end{align*}"}
-{"id": "2755.png", "formula": "\\begin{align*} \\mathcal F ( h , h ) - \\mathcal F ( h ^ K , h ^ K ) = \\mathcal F ( h - h ^ K , h ) + \\mathcal F ( h ^ K , h - h ^ K ) , \\end{align*}"}
-{"id": "6083.png", "formula": "\\begin{align*} \\pi + \\arctan \\lambda _ { n , k } ^ + < \\Theta _ n ( t ) < \\pi \\quad \\mbox { a n d } \\quad \\lim _ { t \\rightarrow \\infty } \\Theta _ n ( t ) = \\pi + \\arctan \\lambda _ { n , k } ^ + . \\end{align*}"}
-{"id": "4621.png", "formula": "\\begin{align*} W _ i & = \\{ s ( n ) : \\ell < n < 2 ^ { k _ i } \\textup { a n d } T _ f ^ n ( x , t ) \\in X _ B \\} \\\\ V _ i & = \\{ s _ i ( n ) : \\ell _ i < n < 2 ^ { k _ i } \\textup { a n d } T _ f ^ n ( z _ i , t ) \\in X _ B \\} \\end{align*}"}
-{"id": "9829.png", "formula": "\\begin{align*} \\sigma _ q = \\frac { 1 } { \\sqrt { q } } \\left ( \\begin{array} { r r } 1 & q - 1 \\\\ 1 & - 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "2359.png", "formula": "\\begin{align*} \\left ( \\left ( p ^ { - n } \\right ) ' ( z ) \\right ) ^ s = \\left ( c ( c z + d ) \\right ) ^ { - 2 s } \\left ( n + \\frac { 1 } { c ( c z + d ) } \\right ) ^ { - 2 s } . \\end{align*}"}
-{"id": "9294.png", "formula": "\\begin{align*} S _ z : = \\{ ( x , h ) \\in I _ z \\times \\R _ { > 0 } \\ : \\ ( h , \\dots , h , x ) \\in S _ { I _ z , k + 2 } \\} . \\end{align*}"}
-{"id": "740.png", "formula": "\\begin{align*} & \\mathcal { L } ^ 3 ( \\{ x \\in \\R ^ 3 : ( u \\ast v ) ( x ) > t \\} ) \\\\ & = \\mathcal { L } ^ 3 ( \\{ x \\in \\R ^ 3 : | u ( x ) | > t \\} ) + \\mathcal { L } ^ 3 ( \\{ x \\in \\R ^ 3 : | v ( x ) | > t \\} ) ; \\end{align*}"}
-{"id": "9118.png", "formula": "\\begin{align*} f _ 2 ( x _ 1 x _ 2 ) - x _ 1 f _ 2 ( x _ 2 ) - x _ 2 f _ { 2 } ( x _ 1 ) = 0 . \\end{align*}"}
-{"id": "9630.png", "formula": "\\begin{align*} R ^ * _ { \\beta } ( t ) = \\sqrt [ 4 ] { \\frac { P _ { \\rm { c u } } } { \\lambda _ { \\beta } ( t ) \\left ( 2 ^ { C / W } - 1 \\right ) P _ { \\rm { t r } , 1 } ( h _ { \\beta , 1 } ^ * ( t ) ) } } \\end{align*}"}
-{"id": "4386.png", "formula": "\\begin{align*} B _ \\gamma \\hat { A } M _ { \\chi _ { B ( w _ \\gamma , R ) } } = B _ \\gamma \\hat { A } M _ { \\chi _ { B ( 0 , R ) } \\circ \\tau _ { w _ \\gamma } } = M _ { \\chi _ { B ( 0 , R ) } \\circ \\tau _ { w _ \\gamma } } = M _ { \\chi _ { B ( w _ \\gamma , R ) } } , \\end{align*}"}
-{"id": "2606.png", "formula": "\\begin{align*} T \\zeta : = \\mathcal F ^ { - 1 } ( v \\widehat \\zeta ) , v : = \\sqrt { \\widehat V } . \\end{align*}"}
-{"id": "9264.png", "formula": "\\begin{align*} { \\sigma } _ \\varepsilon ^ { ( s , u ) } { \\sigma } _ \\varepsilon ^ { ( u , v ) } = { \\sigma } _ \\varepsilon ^ { ( s , v ) } , \\end{align*}"}
-{"id": "6079.png", "formula": "\\begin{align*} \\left ( \\begin{smallmatrix} q ' \\\\ p ' \\end{smallmatrix} \\right ) = \\left ( \\begin{smallmatrix} p \\\\ - ( n - 2 ) p - 2 \\frak { e } _ k \\sin ( q ) \\end{smallmatrix} \\right ) = : V ( q , p ) . \\end{align*}"}
-{"id": "2679.png", "formula": "\\begin{align*} \\sum _ { m \\geq 1 } ( \\mu ^ { ( 2 m ) } ) ^ { - \\frac { 1 } { 2 m } } = \\infty , \\end{align*}"}
-{"id": "3010.png", "formula": "\\begin{align*} \\begin{bmatrix} I + V & I - V \\end{bmatrix} \\begin{psmallmatrix} x \\\\ y \\end{psmallmatrix} & = \\begin{bmatrix} I + V & I - V \\end{bmatrix} \\begin{bmatrix} I - V \\\\ - I - V \\end{bmatrix} l = 0 \\end{align*}"}
-{"id": "9976.png", "formula": "\\begin{align*} \\alpha _ { \\kappa } = \\dfrac { \\frac { 1 } { \\gamma _ { i } ^ { \\mathsf { t h } } } - \\sum _ { j \\neq i } ^ { L } \\mathbb { E } \\left [ \\mathsf { \\bar { L } } _ { j i } ^ { 2 } \\right ] } { \\left ( 1 + \\sum _ { j \\neq i } ^ { L } \\mathbb { E } \\left [ \\mathsf { \\bar { L } } _ { j i } \\right ] \\right ) \\left ( 1 + { \\sum _ { j \\neq i } ^ { L } } \\mathbb { E } \\left [ \\mathsf { \\bar { L } } _ { j i } \\right ] + \\frac { \\sigma ^ { 2 } } { P _ u } \\right ) } . \\end{align*}"}
-{"id": "1445.png", "formula": "\\begin{align*} Q _ 1 ( x _ 3 ) = x _ 3 ^ 2 , Q _ 1 ( x _ { 4 i } ) = x _ 3 x _ { 4 i } , \\end{align*}"}
-{"id": "838.png", "formula": "\\begin{align*} \\mathbb { E } ( E _ t ^ \\gamma ) = C ( \\beta , \\gamma ) t ^ { \\beta \\gamma } \\end{align*}"}
-{"id": "3643.png", "formula": "\\begin{align*} \\tilde { S } _ 0 ^ { ( n ) } & : = \\{ 0 \\} \\times [ - 1 , 1 ] , \\\\ \\tilde { S } _ m ^ { ( n ) } ( \\delta ) & : = \\overline { S _ m ^ { ( n ) } ( \\delta ) } \\quad \\ ; m \\in \\{ 1 , 2 , \\ldots , n - 2 \\} , \\\\ \\tilde { S } _ { n - 1 } ^ { ( n ) } & : = \\{ L \\} \\times [ - 1 , 1 ] , \\end{align*}"}
-{"id": "4789.png", "formula": "\\begin{align*} \\mathcal R ^ k _ 1 ( f , g ) ( x ) : = \\ , & \\iint _ { \\R ^ { 2 n } } \\frac { x ^ k - y ^ k } { ( | x - y | ^ 2 + | x - z | ^ 2 ) ^ { n + 1 / 2 } } \\ , f ( y ) g ( z ) \\ , d y d z , \\\\ \\mathcal R ^ k _ 2 ( f , g ) ( x ) : = \\ , & \\iint _ { \\R ^ { 2 n } } \\frac { x ^ k - z ^ k } { ( | x - y | ^ 2 + | x - z | ^ 2 ) ^ { n + 1 / 2 } } \\ , f ( y ) g ( z ) \\ , d y d z . \\end{align*}"}
-{"id": "2739.png", "formula": "\\begin{align*} \\epsilon ^ 2 \\partial _ t \\bigg ( \\sum _ { j = 0 } ^ p C _ { j , p + 1 } ^ { \\prime } \\ , | | \\partial ^ j g | | _ { L ^ 2 _ { x , v } } ^ 2 \\bigg ) \\leq - \\lambda \\ , | | g ^ { \\perp } | | _ { \\Lambda ^ { p } } ^ 2 \\ , , \\end{align*}"}
-{"id": "6660.png", "formula": "\\begin{align*} S _ 1 = S \\cap \\{ \\xi \\in \\mathbb { R } ^ n : \\langle \\xi - x , u ( x ) \\rangle \\leq - ( 1 + \\varepsilon ) \\Delta \\} \\end{align*}"}
-{"id": "1718.png", "formula": "\\begin{align*} W ( f ) = \\pi ( f ) \\xi \\end{align*}"}
-{"id": "7297.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r m _ { i } ^ 2 + \\sum _ { i = 1 } ^ r m _ i = n ^ 2 - n \\end{align*}"}
-{"id": "4591.png", "formula": "\\begin{align*} N _ \\phi ( x , y ) = [ \\phi x , \\phi y ] + \\phi ^ 2 [ x , y ] - \\phi ( [ \\phi x , y ] + [ x , \\phi y ] ) . \\end{align*}"}
-{"id": "3877.png", "formula": "\\begin{align*} M T ( s ) = \\sigma _ { - s } ( l ^ 2 ) \\frac { \\zeta ( 2 s ) } { \\zeta ( 1 + s ) } \\int _ 0 ^ { \\infty } \\omega ( x ) d x , \\end{align*}"}
-{"id": "8917.png", "formula": "\\begin{align*} J _ { \\theta , c } f = J _ { \\theta , c } ( \\overline \\chi | \\varphi | ^ 2 ) ( U _ { ( \\theta ) c } ) { } = ( \\overline \\chi | \\varphi | ^ 2 ) ( U _ \\nu ) J _ { \\theta , c } { } = \\overline \\chi | \\varphi | ^ 2 . \\end{align*}"}
-{"id": "7077.png", "formula": "\\begin{align*} \\cdots \\to H F ^ - ( M , \\mathfrak { s } ) \\to H F ^ \\infty ( M , \\mathfrak { s } ) \\to H F ^ + ( M , \\mathfrak { s } ) \\to \\cdots \\\\ \\cdots \\to H F ^ - ( M , \\mathfrak { s } ) \\to H F ^ - ( M , \\mathfrak { s } ) \\to \\widehat { H F } ( M , \\mathfrak { s } ) \\to \\cdots \\end{align*}"}
-{"id": "1464.png", "formula": "\\begin{align*} d _ u ( D _ 1 / ( v _ t e _ t ) \\{ x _ 3 ^ 2 \\} \\oplus D _ r \\langle x _ 3 ^ 2 y _ r \\rangle ) = \\{ 0 \\} . \\end{align*}"}
-{"id": "252.png", "formula": "\\begin{align*} \\chi _ { L , n , \\mathcal { M } } : = \\chi _ { [ 0 , L ] ^ d } \\chi _ { \\{ x _ 1 \\leq . . . \\leq x _ { n } \\} } \\chi _ { \\{ \\forall t \\in \\mathcal { M } : \\ , x _ n \\geq x _ t \\} } . \\end{align*}"}
-{"id": "6960.png", "formula": "\\begin{align*} \\epsilon _ 0 = \\epsilon _ 0 ( \\eta _ 0 , n , s , L , S C ) > 0 \\end{align*}"}
-{"id": "7427.png", "formula": "\\begin{align*} \\begin{aligned} & t a = a \\delta , \\delta a ^ * = a ^ * t , \\\\ & a a ^ * = ( t ^ 4 - T ^ { \\delta } _ 2 t ^ 2 - T _ 1 ^ { \\delta } t - T _ 0 ^ { \\delta } ) e _ 0 , a ^ * a = \\delta ^ 4 - T ^ { \\delta } _ 2 \\delta ^ 2 - T _ 1 ^ { \\delta } \\delta - T _ 0 ^ { \\delta } e _ 4 , \\\\ & a _ 4 a _ 4 ^ * - \\delta = \\frac { t _ 1 + 2 t _ 2 + 3 t _ 3 + 4 t _ 4 } { 4 } e _ 4 = - R _ 1 - \\frac { t } { 5 } e _ 4 , \\\\ & a _ 4 ^ * a _ 4 + B + C + R _ 1 = 0 , B ^ 2 + R _ 1 B + S _ 2 = 0 , \\\\ & C ^ 3 - R _ 1 C ^ 2 + R _ 2 C - R _ 3 = 0 . \\end{aligned} \\end{align*}"}
-{"id": "3769.png", "formula": "\\begin{align*} \\mathcal { B } _ { \\check { k } } : = \\bigcap _ { k \\geq \\check { k } } \\bigcap _ { m \\in M ' _ k } A _ m ^ c \\end{align*}"}
-{"id": "4970.png", "formula": "\\begin{align*} \\left \\Vert f \\right \\Vert _ { \\dot { F } ^ { \\alpha , p } _ q } \\simeq \\sum _ { i = 1 } ^ d \\left \\Vert \\partial _ i f \\right \\Vert _ { \\dot { F } ^ { \\alpha - 1 , p } _ q } . \\end{align*}"}
-{"id": "4897.png", "formula": "\\begin{align*} | \\det ( Y ) | ^ 2 & = \\det ( Y ^ * Y ) \\\\ & = y _ 1 y _ 2 \\\\ & = \\frac { 1 } { 2 } \\left ( ( y _ 1 + y _ 2 ) ^ 2 - ( y _ 1 ^ 2 + y _ 2 ^ 2 ) \\right ) \\\\ & = \\frac { 1 } { 2 } \\left ( ( x _ 1 + x _ 2 ) ^ 2 - ( x _ 1 ^ 2 + x _ 2 ^ 2 ) \\right ) \\\\ & = x _ 1 x _ 2 \\\\ & = \\det ( X ^ * X ) \\\\ & = | \\det ( X ) | ^ 2 , \\end{align*}"}
-{"id": "8738.png", "formula": "\\begin{align*} K _ { ( 0 , \\lambda ) } [ \\psi _ \\lambda ] ( T ) & = \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } \\left ( \\int _ { T - \\lambda } ^ T \\frac { \\partial _ t \\psi _ \\lambda ( s ) } { ( t - s ) ^ \\alpha } d s - \\frac { \\psi _ \\lambda ( T ) - \\psi _ \\lambda ( T - \\lambda ) } { \\lambda ^ \\alpha } \\right ) \\\\ & = \\frac { a } { 2 \\lambda ^ \\alpha } \\left ( \\Gamma ( 1 + \\alpha ) - \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } \\right ) . \\end{align*}"}
-{"id": "6955.png", "formula": "\\begin{align*} \\inf _ { M \\in \\mathcal { M } } { L _ M u } ( x ) = g ( x ) , \\end{align*}"}
-{"id": "2751.png", "formula": "\\begin{align*} & \\displaystyle \\chi _ { k i } = \\begin{cases} 0 , \\widetilde S _ { k i } = 0 , \\\\ [ 2 p t ] 1 , \\qquad \\widetilde S _ { k i } \\neq 0 , \\end{cases} \\end{align*}"}
-{"id": "4180.png", "formula": "\\begin{align*} ( u ^ j D _ j - i \\phi ) s = 0 , \\end{align*}"}
-{"id": "7179.png", "formula": "\\begin{align*} \\limsup _ k \\int _ { B _ 1 } | \\nabla w _ k ^ r | ^ 2 \\ , d \\mu _ a \\leq \\int _ { B _ r } | \\nabla w | ^ 2 \\ , d \\mu _ a + 3 \\int _ { B _ R \\setminus \\overline { B _ r } } | \\nabla w | ^ 2 \\ , d \\mu _ a + 4 \\limsup _ k \\ , \\int _ { B _ 1 \\setminus \\overline { B _ r } } | \\nabla z _ { j _ k } | ^ 2 \\ , d \\mu _ a \\end{align*}"}
-{"id": "7496.png", "formula": "\\begin{align*} ( \\bar { \\partial } ^ h \\circ \\bar { \\partial } ^ { * h } \\Phi ) _ { A _ p \\overline { B } _ q } & = - \\sum _ i ( - 1 ) ^ { i - 1 } h ^ { \\bar { \\varepsilon } \\gamma } ( \\delta _ { \\bar { \\beta } _ i } \\circ \\delta _ \\gamma ) \\big ( \\phi _ { A _ p \\bar { \\varepsilon } \\bar { \\beta } _ 1 \\dots \\hat { \\bar { \\beta } } _ i \\dots \\bar { \\beta } _ q } \\big ) . \\end{align*}"}
-{"id": "3831.png", "formula": "\\begin{align*} \\inf _ { \\substack { \\eta \\colon \\ , | \\eta | \\le \\delta \\log L , \\\\ \\eta ( z ) = 0 \\ , \\forall \\ , z \\in [ - \\ell _ L , \\ell _ L ] } } \\ , \\P _ \\eta \\left ( \\bigcup _ { i \\in [ 0 , M _ L - 1 ] } G ^ { ( 0 , T _ i ) } _ { T _ 1 } \\cap \\Lambda ^ { ( 0 , T _ i ) } _ { T _ 1 } \\right ) \\ge 1 - c e ^ { - c ^ { - 1 } L ^ { \\varepsilon } } . \\end{align*}"}
-{"id": "4693.png", "formula": "\\begin{align*} \\Theta = \\Theta ( s , p , n , t _ 0 ) = \\left \\{ \\begin{array} { l l } p & \\textrm { i f } \\ , \\ , p > n \\\\ \\frac { n p } { n - ( s - t _ 0 ) p } & \\textrm { i f } \\ , \\ , p \\leq n . \\end{array} \\right . \\end{align*}"}
-{"id": "1711.png", "formula": "\\begin{align*} L H S = t _ \\lambda P ( \\eta ) t _ \\lambda ^ * = t _ \\lambda t _ \\eta t _ \\eta ^ * t _ \\lambda ^ * . \\end{align*}"}
-{"id": "1787.png", "formula": "\\begin{align*} L ( s ) : = 1 2 G ( s ) - 7 s G ' ( s ) + s ^ 2 G '' ( s ) \\geq 0 \\Omega . \\end{align*}"}
-{"id": "7821.png", "formula": "\\begin{align*} \\int _ { r > t } r ^ { 1 - 2 m } [ | \\nabla v _ m | ^ 2 - q _ 0 v _ m ^ 2 ] e ^ { - 2 \\rho } d x = - \\frac { 1 } { 2 } \\frac { d } { d t } ( t ^ { 1 - 2 m } \\int _ { S _ t } v _ m ^ 2 e ^ { - 2 \\rho } d x ) - \\frac { 1 } { 2 } \\int _ { S _ t } r ^ { - 2 m } ( 2 m - 1 ) v _ m ^ 2 e ^ { - 2 \\rho } d x \\end{align*}"}
-{"id": "2217.png", "formula": "\\begin{align*} \\widetilde q _ 1 = ( w _ 1 ) ^ { \\deg _ { w _ 1 } Q _ 1 } \\cdot ( w _ 2 - a _ { 1 2 } ) ^ { m _ { 1 2 } } \\cdot \\ldots \\cdot ( w _ n - a _ { 1 n } ) ^ { m _ { 1 n } } , \\end{align*}"}
-{"id": "4950.png", "formula": "\\begin{align*} f \\left ( M + m - \\sum \\limits _ { i = 1 } ^ { n } { { { w } _ { i } } { { x } _ { i } } } \\right ) \\le f \\left ( M \\right ) + f \\left ( m \\right ) - \\sum \\limits _ { i = 1 } ^ { n } { { { w } _ { i } } f \\left ( { { x } _ { i } } \\right ) } , \\end{align*}"}
-{"id": "2788.png", "formula": "\\begin{align*} \\mathcal { L } _ { i } ( t ) = \\frac { \\frac { \\beta _ { i } } { t - t _ { i } } } { \\sum _ { i = 0 } ^ { n } \\frac { \\beta _ { i } } { t - t _ { i } } } , i = 0 , 1 , \\cdots , n . \\end{align*}"}
-{"id": "8836.png", "formula": "\\begin{align*} u = u ^ + + u ^ - , u ^ { \\pm } : = P ^ { \\pm } u . \\end{align*}"}
-{"id": "8835.png", "formula": "\\begin{align*} U ( t , y ) : = t ^ { \\frac { 1 } { 5 } } u ( t , t ^ { \\frac { 1 } { 5 } } y ) . \\end{align*}"}
-{"id": "9350.png", "formula": "\\begin{align*} N r ^ { 2 } ( J O _ { n } ^ { ( 3 ) } ) & = ( J _ { n } ^ { ( 3 ) } ) ^ { 2 } + ( J _ { n + 1 } ^ { ( 3 ) } ) ^ { 2 } + \\cdots + ( J _ { n + 7 } ^ { ( 3 ) } ) ^ { 2 } \\\\ & = \\frac { 1 } { 4 9 } \\left ( ( 2 ^ { n + 1 } - ( a \\omega _ { 1 } ^ { n } + b \\omega _ { 2 } ^ { n } ) ) ^ { 2 } + \\cdots + ( ( 2 ^ { n + 8 } - ( a \\omega _ { 1 } ^ { n + 7 } + b \\omega _ { 2 } ^ { n + 7 } ) ) ^ { 2 } \\right ) , \\end{align*}"}
-{"id": "7307.png", "formula": "\\begin{align*} \\omega _ i \\in F ^ { n + 1 - i } \\setminus F ^ { n + 2 - i } , i = 1 , 2 , \\ldots , n + 1 , \\end{align*}"}
-{"id": "6826.png", "formula": "\\begin{align*} \\langle L ( \\phi ) , \\eta _ { R _ 3 , \\xi _ j } \\varphi _ { 0 , j } \\rangle = \\langle h , \\eta _ { R _ 3 , \\xi _ j } \\varphi _ { 0 , j } \\rangle + c _ j \\int _ { \\mathbb { S } ^ 2 _ { \\lambda } } \\chi _ { R _ 1 , j } | \\varphi _ { 0 , j } | ^ 2 + c _ 0 \\int _ { \\mathbb { S } ^ 2 _ { \\lambda } } \\chi _ { R _ 1 , j } \\varphi _ { 0 , j } . \\end{align*}"}
-{"id": "1566.png", "formula": "\\begin{align*} - \\nu ( C _ { j _ k } ( \\alpha ( k ) ) & = - \\nu \\left ( \\frac { B ( k ) } { 2 ^ { j _ k ( n - k + 1 ) } j _ k ! } \\right ) \\\\ & = - ( \\nu ( B ( k ) - \\nu ( 2 ^ { j _ k ( n - k + 1 ) } j _ k ! ) ) \\\\ & = ( n - k + 1 ) j _ k + j _ k - s ( j _ k ) - \\nu ( B ( k ) ) \\\\ & = ( n - k + 1 ) j _ k - s ( j _ k ) \\end{align*}"}
-{"id": "2084.png", "formula": "\\begin{align*} z _ k ( t ) = \\sup _ { M \\times [ 0 , t ] } \\sum _ a | \\nabla _ H u ^ a | \\end{align*}"}
-{"id": "7396.png", "formula": "\\begin{align*} s _ i ( t _ j ) : = \\left \\{ \\begin{array} { c c } - t _ i & \\\\ t _ j + k t _ i & \\\\ t _ j & \\end{array} \\right . \\end{align*}"}
-{"id": "5890.png", "formula": "\\begin{align*} \\{ Z _ { 2 n } ^ { N , i } = - 1 \\} = \\big ( \\cap _ { m = ( 2 n - 1 ) N } ^ { 2 n N - 1 } \\{ \\frac { S _ { m , N + m } ^ { ( i ) } } N < r _ { i + 1 } \\} \\big ) \\cap \\big ( \\cup _ { m = ( 2 n - 1 ) N } ^ { 2 n N - 1 } \\{ \\frac { S _ { m , N + m } ^ { ( i ) } } N < r _ i \\} \\big ) . \\end{align*}"}
-{"id": "7452.png", "formula": "\\begin{align*} & \\gamma ^ 2 \\beta = \\beta \\gamma ^ 2 , \\gamma ^ 2 \\delta = \\delta \\gamma ^ 2 , a a ^ * a = a \\gamma ^ 2 , a ^ * a a ^ * = \\gamma ^ 2 a ^ * , \\\\ & \\beta ^ 2 = - a ^ * a \\gamma - \\gamma a ^ * a + \\gamma ^ 3 , a ^ * a + \\beta + \\gamma + \\delta - \\gamma ^ 2 / 2 = 0 , \\\\ & \\delta ^ 2 = \\gamma ^ 4 / 4 + \\gamma ^ 2 + a ^ * a \\beta + \\beta a ^ * a - \\gamma ^ 2 \\beta \\end{align*}"}
-{"id": "5465.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\widehat U ( s / ( r ^ n z ) ) } { ( r ^ n z ) ^ \\rho \\ell ( r ^ n z ) } = \\widehat V _ z ( s ) \\end{align*}"}
-{"id": "4502.png", "formula": "\\begin{align*} | \\phi _ 0 ( y ) | < \\phi _ 0 ( x _ 0 ) = 1 \\end{align*}"}
-{"id": "4701.png", "formula": "\\begin{align*} \\rho ( t ) : = \\sum _ { j = h } ^ \\infty \\frac { 1 } { j ^ { \\frac { a } { 2 } } } \\textrm { i f } t \\in [ 2 ^ { - h } , 2 ^ { - h + 1 } ) . \\end{align*}"}
-{"id": "6907.png", "formula": "\\begin{align*} N ( x ) = \\# \\{ \\lambda | \\lambda \\leq x , \\lambda \\} \\end{align*}"}
-{"id": "8442.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { t , l , v } ) = q ^ { - \\frac { t } { 2 } } \\zeta _ F ( 1 ) ^ { - 2 } \\int _ { \\mathcal { O } ^ { \\times } } \\chi ( y _ 1 ) \\psi ( y _ 1 \\varpi ^ { \\frac { t } { 2 } } ) G ( \\varpi ^ { \\frac { t } { 2 } } + y _ 1 v \\varpi ^ { - l } , \\chi ) d ^ { \\times } y _ 1 . \\end{align*}"}
-{"id": "8865.png", "formula": "\\begin{align*} F _ k ( D ) \\leq | S | = | \\{ v \\} | + \\max \\{ 0 , d ^ + ( v ) - k \\} = \\max \\{ 1 , d ^ + ( v ) - k + 1 \\} . \\end{align*}"}
-{"id": "2334.png", "formula": "\\begin{align*} d s ^ 2 = \\frac { d x ^ 2 + d y ^ 2 } { y ^ 2 } . \\end{align*}"}
-{"id": "2378.png", "formula": "\\begin{align*} \\Phi _ m ( s , \\lambda , w ) = \\sum _ { n = 0 } ^ m \\binom { k } { n } \\lambda ^ { k - n } \\Phi _ { m - n } ( s , \\lambda , w + k ) + \\sum _ { n = 0 } ^ { k - 1 } \\binom { n } { m } \\frac { \\lambda ^ { n - m } } { ( n + w ) ^ s } . \\end{align*}"}
-{"id": "4302.png", "formula": "\\begin{align*} \\mathsf { N } ^ X _ { g , \\ell _ 1 F _ 1 + \\ell _ 2 F _ 2 } = \\begin{cases} 1 2 \\delta _ { \\ell _ 1 0 } \\frac { \\sigma ( \\ell _ 2 ) } { \\ell _ 2 } + 1 2 \\delta _ { \\ell _ 2 0 } \\frac { \\sigma ( \\ell _ 1 ) } { \\ell _ 1 } & g = 1 \\\\ 0 & g \\neq 1 . \\end{cases} \\end{align*}"}
-{"id": "5007.png", "formula": "\\begin{align*} \\psi = \\sum _ { | \\gamma | = m + 1 } \\xi ^ { \\gamma } \\psi ^ { ( \\gamma ) } . \\end{align*}"}
-{"id": "6003.png", "formula": "\\begin{gather*} E _ 4 = e _ 1 \\end{gather*}"}
-{"id": "8939.png", "formula": "\\begin{gather*} s _ 0 ( x _ 0 , x _ 1 , x _ 2 ) = ( x _ 1 + x _ 2 - x _ 0 , x _ 1 , x _ 2 ) , \\\\ s _ 1 ( x _ 0 , x _ 1 , x _ 2 ) = ( x _ 0 , x _ 0 + x _ 2 - x _ 1 , x _ 2 ) , \\\\ s _ 2 ( x _ 0 , x _ 1 , x _ 2 ) = ( x _ 0 , x _ 1 , x _ 0 + x _ 1 - x _ 2 ) . \\end{gather*}"}
-{"id": "9704.png", "formula": "\\begin{align*} \\varphi _ c ( \\alpha _ 5 , \\sigma _ 3 , \\sigma _ 2 , \\alpha _ 1 , { V } _ a , V _ b ) = \\tilde { \\Phi } _ 5 ( \\alpha _ 5 ; \\mathcal { F } ( \\sigma _ 3 , \\sigma _ 2 ; \\tilde { \\Phi } _ 1 ( \\alpha _ 1 ; { V } _ a ) ) ) - V _ b . \\end{align*}"}
-{"id": "9631.png", "formula": "\\begin{align*} h _ \\beta ^ * ( t ) = R ^ * _ { \\beta } ( t ) h _ { \\beta , 1 } ^ * ( t ) , \\end{align*}"}
-{"id": "6937.png", "formula": "\\begin{align*} q ( i ) = \\begin{cases} i - 4 , & 4 \\leq i \\leq 7 \\\\ i , & . \\end{cases} \\end{align*}"}
-{"id": "1865.png", "formula": "\\begin{align*} \\mathcal { Q } _ 0 : = \\Big \\{ \\mu \\in c a _ { \\leq 1 } ^ + ( \\mathbb { R } ^ d ) : \\mu _ 1 \\preceq _ 0 \\nu _ 1 , \\ , \\dots , \\ , \\mu _ d \\preceq _ 0 \\nu _ d \\mu ( A ^ i ) \\leq \\overline { \\pi } ^ i , i \\in I \\Big \\} , \\end{align*}"}
-{"id": "96.png", "formula": "\\begin{align*} \\mathcal T \\circ ( \\Gamma \\otimes \\Gamma _ { \\rm o p } ( \\chi ) ) ( s , r , t ) = ( \\Gamma \\otimes \\Gamma _ { \\rm o p } ) ( \\chi ) ( s , r , r , t ) = \\chi _ r ( s , t ) , \\end{align*}"}
-{"id": "7929.png", "formula": "\\begin{align*} \\dim _ B A = \\lim _ { \\epsilon \\to 0 } \\frac { \\log N _ { \\epsilon } ( A ) } { - \\log \\epsilon } \\end{align*}"}
-{"id": "8781.png", "formula": "\\begin{align*} G ( x , v ) = g ( x ) - v , \\end{align*}"}
-{"id": "9215.png", "formula": "\\begin{align*} z _ { 1 } ( z _ { 2 } u ) \\otimes \\alpha _ { 1 } ( \\alpha _ { 2 } b ) - ( z _ { 1 } \\circ z _ { 2 } ) u \\otimes \\frac { [ \\alpha _ { 1 } , \\alpha _ { 2 } ] } { 2 } b - [ z _ { 1 } , z _ { 2 } ] u \\otimes \\frac { \\alpha _ { 1 } \\circ \\alpha _ { 2 } } { 2 } b = 0 . \\end{align*}"}
-{"id": "3352.png", "formula": "\\begin{align*} r _ s \\rightarrow r & = \\lim _ { K \\rightarrow \\infty } \\frac { \\binom { K - 2 } { s - 1 } } { \\binom { K - 2 } { s - 1 } + \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K - 1 } { s + i } ( N - 1 ) ^ i N } \\\\ & = \\lim _ { K \\rightarrow \\infty } \\frac { \\psi _ 2 ( N , K , s ) } { \\psi _ 2 ( N , K , s ) + 1 } \\\\ & = \\frac { N \\lambda - ( N - 1 ) } { N \\lambda + 1 } . \\end{align*}"}
-{"id": "1913.png", "formula": "\\begin{align*} \\langle \\cdot , \\cdot \\rangle | _ { M _ { i } } = \\langle \\cdot , \\cdot \\rangle _ { i } \\quad i \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "5048.png", "formula": "\\begin{align*} & \\tau _ { T _ i , R _ i } ^ C = \\textstyle \\min \\left \\lbrace \\left ( 1 - t _ e - t _ { d , T _ i } \\right ) \\ , \\log _ 2 \\left ( 1 + \\frac { \\theta \\eta _ { _ { T _ i } } P _ 0 \\rho _ { _ { T _ i } } H _ { T _ i } ^ 2 \\ , t _ e } { \\sigma ^ 2 \\left ( 1 - t _ e - t _ { d , T _ i } \\right ) } \\right ) \\right . , \\\\ & \\qquad \\qquad \\qquad \\ ; \\ ; \\textstyle \\left . t _ { d , T _ i } \\ , \\log _ 2 \\left ( 1 + { P _ 0 \\ , \\phi _ { _ { R _ i } } H _ { R _ i } } { \\sigma ^ { - 2 } } \\right ) \\right \\rbrace . \\end{align*}"}
-{"id": "3654.png", "formula": "\\begin{align*} W [ n ] = ( X [ n ] / \\mathbb A ^ { n + 1 } ) ^ n = X [ n ] \\times _ { \\mathbb A ^ { n + 1 } } \\dots \\times _ { \\mathbb A ^ { n + 1 } } X [ n ] . \\end{align*}"}
-{"id": "9713.png", "formula": "\\begin{align*} \\begin{cases} & \\tilde { \\rho } \\tilde { u } = \\rho u , \\\\ & \\tilde { \\rho } \\tilde { u } ^ 2 + \\tilde { p } = \\rho u ^ 2 + p , \\\\ & \\tilde { \\rho } \\tilde { u } \\tilde { v } = \\rho u v , \\\\ & ( \\tilde { \\rho } \\tilde { E } + \\tilde { p } ) \\tilde { u } = ( \\rho E + p ) u + q _ 0 \\rho \\phi ( T ) Z h , \\\\ & \\tilde { \\rho } \\tilde { u } \\tilde { Z } = \\rho u Z - \\rho \\phi ( T ) Z h . \\end{cases} \\end{align*}"}
-{"id": "8887.png", "formula": "\\begin{align*} S _ \\omega X - X S _ \\theta = ( \\cdot , \\overline \\chi \\theta ) \\cdot ( \\theta g - \\overline { g _ 1 ( 0 ) } \\omega ) . \\end{align*}"}
-{"id": "3894.png", "formula": "\\begin{align*} h ( \\omega ; 1 ; r ) = \\sqrt { \\pi } \\int _ { 0 } ^ { \\infty } \\omega ( 2 \\cosh { \\xi } ) \\cos { ( 2 r \\xi ) } d \\xi , \\end{align*}"}
-{"id": "1044.png", "formula": "\\begin{align*} \\left \\vert \\sum _ { j = 1 } ^ { m + 1 } L _ { m , j } ^ { \\ast } ( q ) \\right \\vert \\leq C ( m ) \\end{align*}"}
-{"id": "7064.png", "formula": "\\begin{align*} \\Phi _ A ( u , \\lambda ) = \\frac { 1 } { 2 } \\int \\limits _ { B ^ N } | \\nabla u ( x ) | ^ 2 d x - \\frac { \\lambda } { 2 } \\int \\limits _ { B ^ N } ( A u ( x ) , u ( x ) ) d x . \\end{align*}"}
-{"id": "5817.png", "formula": "\\begin{align*} \\lambda _ { t } ( x ) : = \\left [ ( 1 - t ) v ^ { p } ( x ) + t u ^ { p } ( x ) \\right ] ^ { \\frac { 1 } { p } } , t \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "712.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\rho _ + V _ + = \\rho _ - V _ - , \\\\ p _ + + \\rho _ + V _ + ^ 2 = p _ - + \\rho _ - V _ - ^ 2 , \\\\ H _ + + \\frac { 1 } { 2 } V _ + ^ 2 = H _ - + \\frac { 1 } { 2 } V _ - ^ 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "2087.png", "formula": "\\begin{align*} T _ 1 = \\min \\bigg \\{ \\big ( \\frac { 1 } { 8 } K ^ { - 2 } e ^ { - C _ 1 } C _ 3 ^ { - 1 } C _ \\beta ^ { - 1 } | | d \\phi | | _ { C _ 0 } ^ { - 1 } \\big ) ^ { \\frac { 1 } { \\beta } } , 1 \\bigg \\} \\leq 1 , \\end{align*}"}
-{"id": "5861.png", "formula": "\\begin{align*} \\Phi _ G = \\frac { 1 } { 2 } \\log \\frac { \\left | \\Sigma ( E ) \\right | } { \\left | \\Sigma ( E ' ) \\right | } , \\end{align*}"}
-{"id": "2137.png", "formula": "\\begin{align*} m ( \\varphi ) : = \\frac { c _ d } { c _ * } \\int _ 0 ^ \\infty \\varphi ( r ) U ( r ) r ^ { N - 1 } \\ , d r . \\end{align*}"}
-{"id": "7076.png", "formula": "\\begin{align*} J ' _ { l + \\frac { N - 2 } { 2 } } ( x ) - \\frac { N - 2 } { 2 x } J _ { l + \\frac { N - 2 } { 2 } } ( x ) = 0 . \\end{align*}"}
-{"id": "5683.png", "formula": "\\begin{align*} \\mathcal { Z } _ { s y m } ( a ^ - , a ^ + ) = \\{ { z } \\in \\mathcal { S } _ { s y m } ( a ^ - , a ^ + ) \\ ; : \\ ; \\forall { v } \\in \\mathcal { S } _ { s y m } ( a ^ - , a ^ + ) , \\ , \\mathfrak { E } _ { W } ( { z } ) \\leq \\mathfrak { E } _ { W } ( { v } ) \\} . \\end{align*}"}
-{"id": "519.png", "formula": "\\begin{align*} & P _ { S W X ^ n Y ^ n } ( s , w , x ^ n , y ^ n ) \\\\ & \\qquad = \\widetilde { P } _ { S W | X ^ n } ( s , w | x ^ n ) \\prod _ { i = i } ^ n P _ X ( x _ i ) P _ { Y | X } ( y _ i | x _ i ) . \\end{align*}"}
-{"id": "8898.png", "formula": "\\begin{align*} \\lim _ k \\| g \\psi _ { n _ k } \\| ^ 2 = \\int _ { \\mathbb T } | g | ^ 2 | J _ { \\theta , 1 } ^ { - 1 } ( p \\gamma ) - p J _ { \\theta , 1 } ^ { - 1 } \\gamma | ^ 2 m \\end{align*}"}
-{"id": "8913.png", "formula": "\\begin{align*} \\begin{aligned} \\lim _ N & \\frac { 1 } { N + 1 } \\sum _ { n = 0 } ^ N ( U _ { ( \\theta ) 1 } ^ { - n } \\varphi ( U _ { ( \\theta ) c } ) U _ { ( \\theta ) 1 } ^ n x , x ) = ( \\varphi ( U _ { ( \\theta ) 1 } ) x , x ) \\\\ & = ( J _ { \\theta , 1 } \\varphi ( U _ { ( \\theta ) 1 } ) x , J _ { \\theta , 1 } x ) = ( \\varphi ( U _ \\mu ) J _ { \\theta , 1 } x , J _ { \\theta , 1 } x ) = \\int \\varphi | J _ { \\theta , 1 } x | ^ 2 \\mu \\end{aligned} \\end{align*}"}
-{"id": "7707.png", "formula": "\\begin{align*} \\dot { x } ( t ) = - L x ( t ) + w ( t ) \\ , , \\end{align*}"}
-{"id": "1896.png", "formula": "\\begin{align*} G ( x _ 1 , \\dots , x _ d ) = c F ( x _ 1 , \\dots , x _ d ) \\le c \\min _ { i = 1 , \\dots , d } F _ i ( x _ i ) = \\min _ { i = 1 , \\dots , d } c F _ i ( x _ i ) \\le \\min _ { i = 1 , \\dots , d } F ^ \\ast _ i ( x _ i ) , \\end{align*}"}
-{"id": "4210.png", "formula": "\\begin{align*} g _ 2 ( \\tau ) = ( - 2 \\pi i ) ^ 2 t _ 2 ( \\tau ) , g _ 3 ( \\tau ) = ( - 2 \\pi i ) ^ 3 t _ 3 ( \\tau ) , \\end{align*}"}
-{"id": "39.png", "formula": "\\begin{align*} B _ { K } = \\begin{bmatrix} A _ K & A _ { K - 1 } \\end{bmatrix} \\begin{bmatrix} w _ 1 \\\\ w _ 2 \\end{bmatrix} \\end{align*}"}
-{"id": "7096.png", "formula": "\\begin{align*} k ^ H _ { i + 1 / 2 } = \\frac { v ^ 2 - w ^ 2 } { v } = v - \\frac { w ^ 2 } { v } = k ^ + + \\tau ^ H , \\end{align*}"}
-{"id": "8595.png", "formula": "\\begin{align*} I _ { \\alpha } f ( x ) = \\int _ { \\mathbb { R } ^ { n } } \\frac { 1 } { \\left \\vert x - y \\right \\vert ^ { n - \\alpha } } f ( y ) d y , \\end{align*}"}
-{"id": "7051.png", "formula": "\\begin{align*} \\tilde { \\lambda } _ j = \\sqrt { t } \\lambda _ j , j = 1 , 2 , . . . , n - 1 ; \\end{align*}"}
-{"id": "3426.png", "formula": "\\begin{align*} F _ 2 = e ^ { - h _ 2 } f \\colon \\overline D _ 2 \\to \\C ^ 2 _ * \\end{align*}"}
-{"id": "8053.png", "formula": "\\begin{align*} Q A ^ { 1 / 2 } J A ^ { 1 / 2 } Q ^ T & = \\begin{bmatrix} D ^ { 1 / 2 } & 0 \\\\ 0 & D ^ { 1 / 2 } \\end{bmatrix} M J M ^ T \\begin{bmatrix} D ^ { 1 / 2 } & 0 \\\\ 0 & D ^ { 1 / 2 } \\end{bmatrix} \\\\ & = \\begin{bmatrix} D ^ { 1 / 2 } & 0 \\\\ 0 & D ^ { 1 / 2 } \\end{bmatrix} J \\begin{bmatrix} D ^ { 1 / 2 } & 0 \\\\ 0 & D ^ { 1 / 2 } \\end{bmatrix} = \\begin{bmatrix} 0 & D \\\\ - D & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "3611.png", "formula": "\\begin{align*} x _ \\lambda = R _ \\lambda ( x ) = ( 2 \\lambda - x _ 1 , x _ 2 , \\ldots , x _ n ) \\end{align*}"}
-{"id": "6896.png", "formula": "\\begin{align*} - \\Delta _ m ^ { ( r ) } u ( x ) & = \\frac { 1 } { \\int \\Psi _ m ^ { ( x ) } d \\mu } \\sum _ { x \\sim y } c ^ { ( m ) } ( x , y ) ( u ( x ) - u ( y ) ) & & x \\in V _ m \\setminus V _ 0 \\\\ - \\Delta ^ { ( r ) } u ( x ) & = \\lim _ { m \\to \\infty } - \\Delta _ m ^ { ( r ) } u ( x ) & & x \\in V _ m \\setminus V _ 0 \\end{align*}"}
-{"id": "8170.png", "formula": "\\begin{align*} k : = ( 2 C E F + D ) n m : = 4 L k . \\end{align*}"}
-{"id": "6576.png", "formula": "\\begin{align*} G ( P ( \\varepsilon ) ) = G _ { c _ 0 } ( P ( \\varepsilon ) ) = 2 \\ \\sqrt { 2 } \\cdot \\frac { ( 1 + \\varepsilon ) ^ { 1 / 2 } ( 1 - \\varepsilon ) ^ { 3 / 2 } } { 3 - 2 \\varepsilon } . \\end{align*}"}
-{"id": "9237.png", "formula": "\\begin{align*} z _ { 1 } = E _ { 1 , 2 } + \\varepsilon _ { 1 } E _ { 2 , 1 } , \\ z _ { 2 } = E _ { 2 , 3 } + \\varepsilon _ { 2 } E _ { 3 , 2 } z _ { 3 } = E _ { 3 , 1 } + \\varepsilon _ { 3 } E _ { 1 , 3 } \\varepsilon _ { i } = \\pm 1 . \\end{align*}"}
-{"id": "9563.png", "formula": "\\begin{align*} e ( A _ n , f ) \\ , : = \\ , \\| S f - A _ n ( f ) \\| _ G \\ , . \\end{align*}"}
-{"id": "1616.png", "formula": "\\begin{align*} \\mu ( Z ( \\lambda ) ) = \\rho ( \\Lambda ) ^ { - d ( \\lambda ) } \\mu ( Z ( s ( \\lambda ) ) ) \\quad \\ ; \\ ; \\lambda \\in \\Lambda , \\end{align*}"}
-{"id": "3789.png", "formula": "\\begin{align*} q _ 0 : = P ( S ^ { 0 , 1 } _ 1 = 0 ) \\in [ 0 , 1 ) . \\end{align*}"}
-{"id": "3294.png", "formula": "\\begin{align*} L ( \\phi ) = \\Delta _ g \\phi + K \\phi - { 1 \\over | S | } \\int _ S K \\phi \\ , d v _ g , \\end{align*}"}
-{"id": "8089.png", "formula": "\\begin{align*} ( \\theta _ 1 , \\ldots , \\theta _ { i - 1 } ) M : \\theta _ i = 0 \\end{align*}"}
-{"id": "5675.png", "formula": "\\begin{align*} \\mathfrak { L } _ { K } ( f ) : = \\int _ I \\sqrt { 2 W ( f ( s ) \\sigma ( s ) ) } | f ' ( s ) | \\d s \\leq \\int _ I \\frac 1 2 ( f ' ( s ) ) ^ 2 + W ( f \\sigma ( s ) ) \\d s , \\end{align*}"}
-{"id": "1778.png", "formula": "\\begin{align*} i \\partial _ t \\phi ( t , x ) + \\partial _ { x x } ^ 2 \\phi ( t , x ) - F ' ( \\phi ( t , x ) ) = 0 , \\end{align*}"}
-{"id": "8334.png", "formula": "\\begin{align*} \\Gamma _ \\Phi ^ { ( a ) } = \\mathrm { r a t } ( \\nu _ \\Phi ( s ( a ) ) ) \\cdot \\big ( K _ \\Phi \\cap U _ \\Phi ( \\Q ) \\big ) = \\mathrm { r a t } ( a ) \\cdot \\Gamma _ \\Phi \\end{align*}"}
-{"id": "4417.png", "formula": "\\begin{align*} \\ell _ 1 = h _ { e x t } ^ { - \\frac { 1 } { 2 } } d , \\ell _ 2 = h _ { e x t } ^ { - \\frac { 3 } { 4 } } ( t \\ , d ) ^ \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "7404.png", "formula": "\\begin{align*} \\mathbb { H } _ { \\Gamma } : = \\frac { \\mathbb { C } [ t _ 0 , t _ 1 , t _ 2 , t _ 3 , t _ 4 , t _ 5 , t _ 6 ] } { ( t _ 0 + 2 t _ 1 + t _ 2 + 2 t _ 3 + t _ 4 + 2 t _ 5 + 3 t _ 6 ) } \\end{align*}"}
-{"id": "2843.png", "formula": "\\begin{align*} P _ { n } = \\sum _ { v = 0 } ^ { \\infty } p _ { v } \\rightarrow \\infty { a s } { n } \\rightarrow \\infty , ( P _ { - i } = p _ { - i } = 0 , ~ ~ i \\geq 1 ) . \\end{align*}"}
-{"id": "5578.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\delta \\sum _ { i = \\ell } ^ { n } \\xi _ { i } \\approx \\int \\limits _ { \\tau - \\left ( n + 1 \\right ) \\delta } ^ { \\tau - \\left ( \\ell + 1 \\right ) \\delta } \\left ( \\alpha + \\beta p \\left ( t \\right ) \\right ) d t \\end{align*}"}
-{"id": "2559.png", "formula": "\\begin{align*} \\frac { P ( E ) } { | E | } & \\le \\frac { P ( F ) } { | F | } = \\frac { P ( E , \\R ^ { 2 } \\setminus \\overline { B _ { r } ( z ) } ) + m ' ( r ) } { | E | + m ( r ) } \\\\ & = \\frac { P ( E ) - P ( B _ { r } ( z ) \\setminus E ) + 2 m ' ( r ) } { | E | + m ( r ) } \\end{align*}"}
-{"id": "7833.png", "formula": "\\begin{align*} I _ q \\otimes \\mathcal { J } _ { m - 1 } ^ { ( q ) } & = I _ q \\otimes J _ q \\otimes \\mathcal { A } _ { m - 2 } ^ { ( q ) } , \\\\ Q \\otimes \\mathcal { A } _ { m - 1 } ^ { ( q ) } & = ( A _ 1 - A _ 2 ) \\otimes \\mathcal { A } _ { m - 1 } ^ { ( q ) } , \\end{align*}"}
-{"id": "9064.png", "formula": "\\begin{align*} a _ { 1 1 } ^ 1 = 0 , \\ , \\ , a _ { 2 2 } ^ 2 = 0 , \\ , \\ , \\Re b _ { 1 1 } ^ 1 = 0 , \\ , \\ , b _ { 1 2 } ^ 1 = 0 , \\ , \\ , b _ { 1 3 } ^ 1 = 0 . \\end{align*}"}
-{"id": "934.png", "formula": "\\begin{align*} \\varepsilon = \\Delta ^ { 1 / 3 } ( 1 \\vee a _ d \\vee \\log ^ { 1 / 2 } ( 1 / \\Delta ) ) ^ { - 1 / 3 } ( 2 \\log d ) ^ { 1 / 3 } . \\end{align*}"}
-{"id": "3460.png", "formula": "\\begin{align*} \\mathbf { u _ { h } ^ { n } } ( \\omega ) : = \\big ( \\alpha _ { h , i } ^ { n } ( \\omega ) \\big ) _ { i \\in \\{ 1 , \\dots , d \\} } \\mathbf { f _ h } ( t ) : = \\big ( f _ { h , i } ( t ) \\big ) _ { i \\in \\{ 1 , \\dots , d \\} } \\end{align*}"}
-{"id": "8973.png", "formula": "\\begin{gather*} \\sum _ { j \\ge 0 } \\sum _ { 1 \\le i \\le n + 1 } [ z _ i = u + j q ] . \\end{gather*}"}
-{"id": "5053.png", "formula": "\\begin{align*} C ^ { \\otimes N } = \\bigoplus _ { i _ 1 , \\ldots , i _ N = 1 } ^ m T _ { k _ { i _ 1 } \\ldots k _ { i _ N } } , \\end{align*}"}
-{"id": "5187.png", "formula": "\\begin{align*} \\left | \\frac { F _ 1 ' ( x ) } { F _ 1 '' ( x ) } \\right | = \\left | x - \\tfrac { a } { 4 } \\right | \\leq \\tfrac { 1 } { 4 } . \\end{align*}"}
-{"id": "3319.png", "formula": "\\begin{align*} D ^ * ( r ) = \\left \\{ \\begin{array} { l l } ( 1 - r ) \\left ( 1 + \\frac { 1 } { N } \\right ) - r , & \\frac { K - 2 } { ( N + 1 ) K + N ^ 2 - 2 N - 2 } \\leq r \\leq \\frac { 1 } { 1 + N } \\\\ 1 - r , & \\qquad \\frac { 1 } { 1 + N } \\leq r \\leq 1 \\end{array} \\right . \\end{align*}"}
-{"id": "1636.png", "formula": "\\begin{align*} S _ \\lambda ( \\delta _ \\eta ) ( \\nu ) & = \\begin{cases} 0 , & \\nu \\not \\in R _ \\lambda \\\\ \\delta _ \\eta ( \\tau ^ { d ( \\lambda ) } ( \\nu ) ) , & \\end{cases} \\end{align*}"}
-{"id": "5102.png", "formula": "\\begin{align*} v _ { \\{ 0 , 1 , 2 \\} } & = \\textstyle \\frac { 1 } { 6 } v _ { \\{ 0 , 1 \\} } + \\frac { 1 } { 6 } v _ { \\{ 0 , 2 \\} } + \\frac { 1 } { 6 } v _ { \\{ 0 , 3 \\} } + \\frac { 1 } { 2 } v _ { \\{ 0 , 1 , 2 , 3 \\} } \\\\ & = \\textstyle \\frac { 1 } { 3 } v _ { \\{ 0 \\} } + \\frac { 2 } { 3 } v _ { \\{ 0 , 1 , 2 , 3 \\} } . \\end{align*}"}
-{"id": "8614.png", "formula": "\\begin{align*} V ( t , x ) = \\dot { W } _ \\psi ( t , x ) = \\int \\psi ( x - y ) d W ( t , y ) . \\end{align*}"}
-{"id": "4994.png", "formula": "\\begin{align*} \\Delta _ m ( x ) = 2 ^ { m d } \\Delta ( 2 ^ m x ) = \\sum _ { | \\gamma | = a } 2 ^ { m d } [ \\partial ^ { \\gamma } \\Delta ^ { ( \\gamma ) } ] ( 2 ^ m x ) = 2 ^ { - m a } \\sum _ { | \\gamma | = a } \\partial ^ { \\gamma } [ ( \\Delta ^ { ( \\gamma ) } ) _ m ] ( x ) \\end{align*}"}
-{"id": "1291.png", "formula": "\\begin{align*} \\mathbf { m } ' ( 0 ) & = \\mbox { C a p } _ { \\mathcal { A } } ( E _ 0 ) ^ { \\frac { 1 } { n - p } - 1 } [ \\mbox { C a p } _ { \\mathcal { A } } ( E _ 1 ) - \\mbox { C a p } _ { \\mathcal { A } } ( E _ 0 ) ] \\\\ & = \\mathbf { m } ( 0 ) ^ { 1 - n + p } [ \\mathbf { m } ( 1 ) ^ { n - p } - \\mathbf { m } ( 0 ) ^ { n - p } ] . \\end{align*}"}
-{"id": "7520.png", "formula": "\\begin{gather*} u = \\frac 1 { \\sqrt { | t | } } ( w ^ 1 \\cos \\tau - w ^ 2 \\sin \\tau ) + \\frac x { 2 t } - \\kappa \\frac y t , \\\\ v = \\frac 1 { \\sqrt { | t | } } ( w ^ 1 \\sin \\tau + w ^ 2 \\cos \\tau ) + \\frac y { 2 t } + \\kappa \\frac x t , \\end{gather*}"}
-{"id": "5489.png", "formula": "\\begin{align*} \\mathrm { B } _ \\rho p ( x ) = x ^ { - \\rho } \\int _ x ^ \\infty y ^ { \\rho - 1 } p ( y ) \\dd y . \\end{align*}"}
-{"id": "9811.png", "formula": "\\begin{align*} p _ k ' ( t ) = - p _ k ( t ) ( \\mu + \\l f ( t ) ) + p _ { k - 1 } ( t ) \\l f ( t ) + p _ { k + 1 } ( t ) \\mu , - T _ { e _ 1 } \\leq t \\leq T _ { e _ 2 } . \\end{align*}"}
-{"id": "1564.png", "formula": "\\begin{align*} - \\nu ( C _ { j _ k } ( \\alpha ( k ) ) = ( n - k + 1 ) j _ k - s ( j _ k ) \\end{align*}"}
-{"id": "6674.png", "formula": "\\begin{align*} \\frac { \\kappa ( x ) ^ { \\frac { 1 } { n + 1 } } } { \\langle x , u ( x ) \\rangle } = \\left [ \\frac { \\kappa ( y ) ^ { \\frac { 1 } { n + 1 } } } { \\langle y , u ( y ) \\rangle } \\right ] ^ { - 1 } \\quad . \\end{align*}"}
-{"id": "7457.png", "formula": "\\begin{align*} x = w - 1 / 2 ( z ^ 2 + T y ) \\end{align*}"}
-{"id": "54.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { \\sqrt { 2 \\pi } \\theta } \\right ) \\int \\limits _ { - \\infty } ^ { \\infty } e x p \\left ( - \\frac { ( u _ 1 - b ) ^ 2 } { 2 \\theta ^ 2 } \\right ) \\mathrm { d } u _ 1 \\left ( \\frac { 1 } { \\sqrt { 2 \\pi } \\theta } \\right ) \\int \\limits _ { - \\infty } ^ { \\infty } e x p \\left ( - \\frac { ( u _ 2 - b ' ) ^ 2 } { 2 \\theta ^ 2 } \\right ) \\mathrm { d } u _ 2 = 1 \\end{align*}"}
-{"id": "9110.png", "formula": "\\begin{align*} \\tilde { a } _ { ( n - 1 ) + 1 - i } = \\binom { n - 1 + 1 } { i } \\left ( \\dfrac { a _ { n + 1 - ( i + 1 ) } } { \\binom { n + 1 } { i + 1 } } + \\dfrac { a _ { n + 1 - i } } { \\binom { n + 1 } { i } } \\right ) = 0 \\left ( 0 \\leq i \\leq n - 1 \\right ) \\end{align*}"}
-{"id": "1640.png", "formula": "\\begin{align*} \\phi ( [ \\xi ^ i _ \\mu ] _ x ) : = [ \\xi ^ j _ { \\mu \\lambda _ { i , j } } ] _ y . \\end{align*}"}
-{"id": "4551.png", "formula": "\\begin{align*} & E _ i ( z ) = \\sum _ { k \\in \\Z } E _ { i , k } z ^ { - k } , F _ i ( z ) = \\sum _ { k \\in \\Z } F _ { i , k } z ^ { - k } , \\\\ & K ^ { \\pm } _ i ( z ) = K _ i ^ { \\pm 1 } \\bar { K } ^ { \\pm } _ i ( z ) \\ , , \\bar { K } ^ { \\pm } _ i ( z ) = \\exp \\Bigl ( \\pm ( q - q ^ { - 1 } ) \\sum _ { r > 0 } H _ { i , \\pm r } z ^ { \\mp r } \\Bigr ) \\ , . \\end{align*}"}
-{"id": "8963.png", "formula": "\\begin{gather*} \\sum _ { \\alpha \\in \\Phi ^ - ( W ) \\setminus \\Phi ^ - ( W _ J ) } \\ ! \\ ! \\ ! \\ ! T _ \\alpha - \\ ! \\ ! \\ ! \\sum _ { \\alpha \\in \\Phi ^ - ( W ) \\setminus \\Phi ^ - ( W _ I ) } \\ ! \\ ! \\ ! \\ ! T _ { w \\alpha } = \\sum _ { \\alpha \\in \\Phi ^ - ( W ) } ( T _ \\alpha - T _ { w \\alpha } ) - \\sum _ { \\alpha \\in \\Phi ^ - ( W _ J ) } \\ ! \\ ! \\ ! \\ ! T _ \\alpha + \\sum _ { \\alpha \\in \\Phi ^ - ( W _ I ) } \\ ! \\ ! \\ ! \\ ! T _ { w \\alpha } . \\end{gather*}"}
-{"id": "4603.png", "formula": "\\begin{align*} c _ 3 = \\inf \\{ n \\eta ( n ) : n \\in \\mathbb { N } \\} > 0 \\end{align*}"}
-{"id": "2540.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } e ^ { i ( x - a _ j t ) \\eta - A _ j t | \\eta | ^ 2 } d \\eta = \\frac { e ^ { - \\frac { ( x - a _ j t ) ^ 2 } { 4 A _ j t } } } { \\sqrt { 4 \\pi A _ j t } } , \\end{align*}"}
-{"id": "4203.png", "formula": "\\begin{align*} d s ^ 2 = ( a b c ) ^ 2 d \\rho ^ 2 + a ^ 2 ( \\sigma _ 1 ) ^ 2 + b ^ 2 ( \\sigma _ 2 ) ^ 2 + c ^ 2 ( \\sigma _ 3 ) ^ 2 , \\end{align*}"}
-{"id": "3900.png", "formula": "\\begin{align*} { \\cal L } = \\frac { 2 } { \\beta + 2 } \\ell \\end{align*}"}
-{"id": "7119.png", "formula": "\\begin{align*} 2 \\dot { F } ^ { k l } & \\left ( 2 \\Lambda _ k ^ p \\nabla _ l G _ { 1 p } - \\Lambda _ k ^ p \\Lambda _ l ^ q G _ { p q } \\right ) \\\\ & = 2 \\sum _ { k = 1 } ^ n \\sum _ { p = 2 } ^ n \\dot { f } ^ k \\left ( 2 \\Lambda _ k ^ p \\nabla _ k G _ { 1 p } - ( \\Lambda _ k ^ p ) ^ 2 G _ { p p } \\right ) \\\\ & = 2 \\sum _ { k = 1 } ^ n \\sum _ { p = 2 } ^ n \\dot { f } ^ k \\left ( \\frac { ( \\nabla _ k G _ { 1 p } ) ^ 2 } { G _ { p p } } - \\left ( \\Lambda _ k ^ p - \\frac { \\nabla _ k G _ { 1 p } } { G _ { p p } } \\right ) ^ 2 G _ { p p } \\right ) . \\end{align*}"}
-{"id": "2872.png", "formula": "\\begin{align*} f _ { j } ( x ; n , N , h ) = \\frac { \\cos \\left ( 2 A _ { N } \\left ( \\frac { n } { x } \\right ) b _ { j , N } + \\frac { \\pi ( 2 h - 1 ) j } { 2 N } \\right ) - e ^ { - 2 A _ { N } \\left ( \\frac { n } { x } \\right ) a _ { j , N } } \\cos \\left ( \\frac { \\pi ( 2 h - 1 ) j } { 2 N } \\right ) } { \\cosh \\left ( 2 A _ { N } \\left ( \\frac { n } { x } \\right ) a _ { j , N } \\right ) - \\cos \\left ( 2 A _ { N } \\left ( \\frac { n } { x } \\right ) b _ { j , N } \\right ) } . \\end{align*}"}
-{"id": "9912.png", "formula": "\\begin{align*} \\tfrac 1 2 \\alpha _ 1 ^ 2 + \\tfrac 1 2 \\alpha _ 2 ^ 2 + \\tfrac 1 2 ( \\alpha _ 1 + \\alpha _ 2 ) ^ 2 + \\sum _ { i = 3 } ^ { r } \\alpha _ i ^ 2 \\le 2 \\rho \\ , , \\end{align*}"}
-{"id": "1078.png", "formula": "\\begin{align*} \\alpha _ { k i } = \\int \\varphi _ k ( \\mathbf { r } , z ) u _ i ^ * ( \\mathbf { r } , z ) \\mathrm { d } \\mathbf { r } . \\end{align*}"}
-{"id": "3315.png", "formula": "\\begin{align*} \\hat { D } ( r ) = ( 1 - r ) \\left ( 1 + \\frac { 1 } { N } + \\dots + \\frac { 1 } { N ^ { K - 1 } } \\right ) \\end{align*}"}
-{"id": "10046.png", "formula": "\\begin{align*} [ ( \\mathrm { E x c } , - \\log ( D ) ) : \\mathcal { Y } _ \\mathrm { b i g } ] = 0 , \\end{align*}"}
-{"id": "9927.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } u ( t , x ) + A _ x u ( t , x ) + b ( t , x ) \\cdot \\nabla _ x u ( t , x ) = - b ( t , x ) , u ( T , x ) = 0 \\end{align*}"}
-{"id": "1938.png", "formula": "\\begin{align*} \\chi ( \\xi ) \\widehat { q } _ B ( \\xi ) = \\chi ( \\xi ) \\widehat { q } ( \\xi ) + \\sum _ { j = 2 } ^ { \\infty } \\widehat { \\widetilde Q _ { j } ( q ) } ( \\xi ) , \\end{align*}"}
-{"id": "3223.png", "formula": "\\begin{align*} { C \\mathcal D ' ( X ) \\over E \\mathcal D ' ( X ) } = \\sum _ i H ^ { i } ( X ; \\mathbb C ) . \\end{align*}"}
-{"id": "6735.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\mathbb { P } ( R _ { N } \\geq \\beta _ { N } ) = e ^ { - 1 } . \\end{align*}"}
-{"id": "10132.png", "formula": "\\begin{align*} \\bar { \\boldsymbol \\Phi } _ k ( i ) = \\lambda ^ { - 1 } \\bar { \\boldsymbol \\Phi } _ k ( i - 1 ) - \\lambda ^ { - 1 } \\bar { \\boldsymbol k } _ k ( i ) \\bar { \\boldsymbol x } _ k ^ H ( i ) \\bar { \\boldsymbol \\Phi } _ k ( i - 1 ) . \\end{align*}"}
-{"id": "4075.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t A ( u ) + \\partial _ \\alpha F _ \\alpha ( u ) = 0 , \\end{aligned} \\end{align*}"}
-{"id": "3833.png", "formula": "\\begin{align*} \\P _ \\eta \\left ( \\bigcap _ { i = 0 } ^ { \\lfloor M _ L \\rfloor - 1 } \\left ( G ^ { ( 0 , T _ i ) } _ { T _ 1 } \\cap \\Lambda ^ { ( 0 , T _ i ) } _ { T _ 1 } \\right ) ^ c \\cap \\{ \\eta _ { T _ i } ( 0 ) = 0 \\} \\right ) & \\le \\left ( 1 - L ^ { \\delta \\log p _ { * } } \\right ) ^ { \\lfloor M _ L \\rfloor } \\\\ & \\le c e ^ { - \\frac { 1 } { c } L ^ { \\varepsilon _ * } } \\end{align*}"}
-{"id": "52.png", "formula": "\\begin{align*} = \\int \\limits _ { - \\infty } ^ { \\infty } \\int \\limits _ { - \\infty } ^ { \\infty } \\frac { 1 } { N } \\sum \\limits _ { i = 1 } ^ N \\frac { 1 } { ( 4 \\pi ^ { 2 } \\sigma ^ { 4 } ) } e x p \\left ( - \\frac { ( u _ 1 - b ) ^ { 2 } } { \\sigma ^ { 2 } } \\right ) e x p \\left ( - \\frac { ( u _ 2 - b ' ) ^ { 2 } } { \\sigma ^ { 2 } } \\right ) e x p \\left ( - \\frac { c } { 2 \\sigma ^ { 2 } } \\right ) \\mathrm { d } u _ 1 \\mathrm { d } u _ 2 \\end{align*}"}
-{"id": "9705.png", "formula": "\\begin{align*} & V _ m = \\tilde { \\Phi } _ 5 ( \\alpha _ 5 ; \\mathcal { F } ( \\sigma _ 3 , \\sigma _ 2 ; \\tilde { \\Phi } _ 1 ( \\alpha _ 1 ; V _ a ) ) ) , Z _ m = Z _ a + \\alpha _ 4 , \\\\ & V _ b = \\tilde { \\Phi } ( \\beta _ 5 , \\beta _ 3 , \\beta _ 2 , \\beta _ 1 ; V _ m ) , Z _ b = Z _ m + \\beta _ 4 , \\\\ & V _ b = \\tilde { \\Phi } _ 5 ( \\gamma _ 5 ; \\mathcal { F } ( \\sigma ' _ 3 , \\sigma ' _ 2 ; \\tilde { \\Phi } _ 1 ( \\gamma _ 1 ; V _ a ) ) ) , Z _ b = Z _ a + \\gamma _ 4 , \\end{align*}"}
-{"id": "7863.png", "formula": "\\begin{align*} | \\nabla _ z d ( z , m ( x , y ) ) | = 1 \\end{align*}"}
-{"id": "363.png", "formula": "\\begin{align*} \\sum _ { r , s \\in \\mathbb { Z } } b _ { n , r , s } ^ { 2 } \\le | \\Gamma _ n | \\sum _ { r , s \\in \\mathbb { Z } } \\sum _ { ( j , k ) \\in \\Gamma _ n } a ^ 2 _ { j + r , k + s } = | \\Gamma _ n | ^ 2 \\sum _ { r , s \\in \\mathbb { Z } } a _ { r , s } ^ { 2 } . \\end{align*}"}
-{"id": "8758.png", "formula": "\\begin{align*} ( H _ N - E ) \\psi = ( H _ 0 - E ) ( \\psi - R _ E ^ * w _ \\psi ) . \\end{align*}"}
-{"id": "452.png", "formula": "\\begin{align*} h _ { k _ 1 , k _ 2 } ( R , t ) = \\int _ { \\R ^ m } e ^ { i R \\phi _ \\omega ( \\lambda + i y u _ 1 ) } a _ { k _ 1 , k _ 2 } ( \\lambda + i y u _ 1 ) \\ , \\dd \\lambda . \\end{align*}"}
-{"id": "2251.png", "formula": "\\begin{align*} s _ i = \\sigma _ { i , \\ldots , i } = \\sum _ { j = 1 } ^ \\infty \\frac { 1 } { z _ { j 1 } ^ { i } \\cdot \\ldots \\cdot z _ { j n } ^ { i } } , i \\geqslant 1 . \\end{align*}"}
-{"id": "6278.png", "formula": "\\begin{align*} [ \\Delta _ q , u _ { \\leq { q - 1 } } \\cdot \\nabla ] b _ q = & \\lambda _ q ^ 3 \\int _ { \\R ^ 3 } h ( \\lambda _ q ( x - y ) ) \\left ( u _ { \\leq q - 1 } ( y ) - u _ { \\leq q - 1 } ( x ) \\right ) \\nabla b _ q ( y ) \\ , d y \\\\ = & - \\lambda _ q ^ 3 \\int _ { \\R ^ 3 } \\nabla h ( \\lambda _ q ( x - y ) ) \\left ( u _ { \\leq q - 1 } ( y ) - u _ { \\leq q - 1 } ( x ) \\right ) b _ q ( y ) \\ , d y . \\end{align*}"}
-{"id": "638.png", "formula": "\\begin{align*} \\frac { \\partial F } { \\partial t } = H , \\ \\ \\ F ( 0 , \\cdot ) = F _ 0 ( \\cdot ) . \\end{align*}"}
-{"id": "3397.png", "formula": "\\begin{align*} Q _ { F _ * , p _ m ^ * , p _ n ^ * , q ^ * } ( G _ { 0 , s } ) = \\psi _ s ( c ( s ) ) \\end{align*}"}
-{"id": "3018.png", "formula": "\\begin{align*} 2 \\Re \\ < A x , x \\ > _ { L ^ 2 } & = 2 \\Re \\ < f _ { \\partial , x } , e _ { \\partial , x } \\ > _ { H ^ N } + 2 \\langle P _ 0 x , x \\rangle \\\\ & \\le 2 \\ < l , ( - I + V ^ * V ) l \\ > _ { H ^ N } \\leq 0 . \\end{align*}"}
-{"id": "18.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\hat { f _ { \\sigma } } _ { X Y Z S } ( x , y , z , s ) = f _ { \\sigma X Y Z S } ( x , y , z , s ) \\end{align*}"}
-{"id": "4344.png", "formula": "\\begin{align*} \\zeta = \\{ [ - 3 , 3 ) ^ { 2 n } + \\sigma \\subset \\mathbb { R } ^ { 2 n } ; \\sigma \\in 6 \\mathbb { Z } ^ { 2 n } \\} \\end{align*}"}
-{"id": "3800.png", "formula": "\\begin{align*} g ( \\omega , U ) = \\begin{cases} - 1 , & \\\\ 1 , & . \\end{cases} \\end{align*}"}
-{"id": "4275.png", "formula": "\\begin{align*} \\mathsf { Z } ( t , u , \\mathbf { z } _ 1 , \\mathbf { z } _ 2 , q _ 1 , q _ 2 ) = \\sum _ { k = 0 } ^ { \\infty } \\mathsf { Z } _ k ( u , \\mathbf { z } _ 1 , \\mathbf { z } _ 2 , q _ 1 , q _ 2 ) t ^ k . \\end{align*}"}
-{"id": "1774.png", "formula": "\\begin{align*} U _ \\gamma ^ * U _ \\gamma & = t _ a t _ { \\omega ( 0 , j ) } ^ * t _ { \\omega ( 0 , j ) } t _ a ^ * P ( \\{ \\gamma \\} ) = t _ a t _ a ^ * P ( \\{ \\gamma \\} ) \\\\ & = P ( Z ( a ) ) P ( \\{ \\gamma \\} ) = P ( \\{ \\gamma \\} ) . \\end{align*}"}
-{"id": "4980.png", "formula": "\\begin{align*} \\tilde { h } : = \\sum _ { j = - \\infty } ^ { \\infty } h _ j \\prod _ { j ' > j } ( 1 - U _ { j ' } ) \\textrm { w i t h } U _ j = ( 1 - \\zeta _ j ) \\omega _ j \\end{align*}"}
-{"id": "8504.png", "formula": "\\begin{align*} \\Delta = ( 1 - v b _ { \\chi _ 1 \\chi _ 2 ^ { - 1 } } \\varpi ^ { a _ 1 - a ( \\chi _ 1 \\chi _ 2 ^ { - 1 } ) } ) ^ 2 - 4 v b _ 2 = 1 - 2 v ( b _ 1 + b _ 2 ) + v ^ 2 b _ { \\chi _ 1 \\chi _ 2 ^ { - 1 } } ^ 2 \\varpi ^ { 2 a _ 1 - 2 a ( \\chi _ 1 \\chi _ 2 ^ { - 1 } ) } . \\end{align*}"}
-{"id": "4325.png", "formula": "\\begin{align*} & \\varphi _ { i , k } ( t ) = \\sum _ { j = 1 } ^ { k } \\alpha _ { i , j } ^ { k } ( t ) w _ j , \\mu _ { i , k } ( t ) = \\sum _ { j = 1 } ^ k \\beta _ { i , j } ^ { k } ( t ) w _ j , \\\\ & q _ k ( t ) = \\sum _ { j = 1 } ^ { k } \\gamma _ j ^ k ( t ) y _ j , n _ k ( t ) = 1 + \\sum _ { j = 1 } ^ { k } \\eta _ j ^ k ( t ) y _ j , \\end{align*}"}
-{"id": "6792.png", "formula": "\\begin{align*} - \\Delta _ g \\tilde { W } _ { \\lambda , k } = \\tilde { r } _ { k , \\lambda } ( x ) , \\end{align*}"}
-{"id": "8628.png", "formula": "\\begin{align*} \\{ \\omega _ s \\} = ( x _ 0 , \\ldots , x _ { N + 1 } ) . \\end{align*}"}
-{"id": "9972.png", "formula": "\\begin{align*} \\bar { \\gamma } _ { k i } ^ { s , p } = \\frac { \\bar { \\upsilon } _ { k i } ^ { s , p } } { { \\displaystyle \\sum _ { j = 1 } ^ { L } \\alpha _ { j } \\mathbb { E } \\left [ \\mathsf { \\bar { L } } _ { j i } \\right ] + \\bar { \\upsilon } _ { k i } ^ { s , p } \\sum _ { j \\neq i } ^ { L } \\frac { \\ell _ { k j i } ^ { 2 } } { \\ell _ { k j j } ^ { 2 } } } } \\end{align*}"}
-{"id": "1882.png", "formula": "\\begin{align*} \\eta ( \\ 1 _ B ) : = & \\ , \\inf \\Big \\{ \\pi ( 0 , f _ 2 , a ) : f _ 2 ( x _ 2 ) + \\sum _ { i \\in I } a ^ i \\ 1 _ { A ^ i } ( x ) = 1 \\ x \\in B f _ 2 , a ^ i \\geq 0 \\Big \\} \\\\ = & \\ , \\min \\Big \\{ \\nu _ 2 ( ( - \\infty , B _ 2 ] ) , \\min _ { i \\in I : B _ 1 \\leq A _ 1 ^ i } \\Big \\{ \\bar { \\pi } ^ i + \\nu _ 2 ( ( A ^ i _ 2 , B _ 2 ] ) \\Big \\} \\Big \\} . \\end{align*}"}
-{"id": "3255.png", "formula": "\\begin{align*} ( \\alpha _ i , u ^ i , \\omega ) _ X = ( \\alpha _ { i , Y } , \\omega _ { Y } ) _ { Y } \\end{align*}"}
-{"id": "4286.png", "formula": "\\begin{align*} \\langle T _ { + } ( \\alpha ) , \\alpha ' \\rangle = \\langle \\alpha , T _ { - } ( \\alpha ' ) \\rangle , \\alpha , \\alpha ' \\in H ^ { \\ast } ( X ) \\end{align*}"}
-{"id": "5952.png", "formula": "\\begin{align*} R ^ D ( X , Y ) \\xi : = D _ X D _ Y \\xi - D _ Y D _ X \\xi - D _ { [ X , Y ] } \\xi . \\end{align*}"}
-{"id": "8201.png", "formula": "\\begin{align*} K _ f = K _ { } + K _ { } + K _ { } . \\end{align*}"}
-{"id": "5080.png", "formula": "\\begin{align*} & d \\Psi = - [ ( b _ 1 - b _ 2 ) ^ 2 R _ { 1 2 1 2 } + ( b _ 1 - b _ 3 ) ^ 2 R _ { 1 3 1 3 } + ( b _ 2 - b _ 3 ) ^ 2 R _ { 2 3 2 3 } ] d v _ g \\\\ & + [ \\frac { 1 8 b _ 1 b _ 2 C _ 3 ^ 2 } { ( b _ 1 - b _ 3 ) ^ 2 ( b _ 2 - b _ 3 ) ^ 2 } + \\frac { 1 8 b _ 2 b _ 3 C _ 1 ^ 2 } { ( b _ 1 - b _ 2 ) ^ 2 ( b _ 1 - b _ 3 ) ^ 2 } + \\frac { 1 8 b _ 1 b _ 3 C _ 2 ^ 2 } { ( b _ 1 - b _ 2 ) ^ 2 ( b _ 2 - b _ 3 ) ^ 2 } ] d v _ g , \\end{align*}"}
-{"id": "9759.png", "formula": "\\begin{align*} w ^ { ( n ) } = - \\frac { 1 } { \\rho _ { A , 0 } u _ { A , 0 } A ( 0 ) } \\int _ 0 ^ x A ( \\tau ) \\rho _ A ^ { ( n ) } \\phi ( T _ A ^ { ( n ) } ) d \\tau . \\end{align*}"}
-{"id": "7485.png", "formula": "\\begin{align*} \\int _ E ( \\nabla _ { \\mathcal { X } _ \\alpha } Z ^ \\alpha - Z ^ \\alpha L _ \\alpha ) d \\mathcal { V } = 0 , \\int _ E ( \\nabla _ { \\mathcal { X } _ { \\bar { \\alpha } } } \\overline { Z ^ \\alpha } - \\overline { Z ^ \\alpha L _ \\alpha } ) d \\mathcal { V } = 0 . \\end{align*}"}
-{"id": "8530.png", "formula": "\\begin{align*} a _ { ( j , - l ) } ( x , \\xi ) = \\gamma _ { j } ( x ) \\gamma _ { - l } ( \\xi ) a ( x , \\xi ) . \\end{align*}"}
-{"id": "3927.png", "formula": "\\begin{align*} u ( x , 0 ) = \\phi ( x ) \\end{align*}"}
-{"id": "9360.png", "formula": "\\begin{align*} \\left ( j O _ { n } ^ { ( 3 ) } \\right ) ^ { 2 } - 9 \\left ( J O _ { n } ^ { ( 3 ) } \\right ) ^ { 2 } = \\frac { 2 ^ { n + 1 } } { 7 } ( 2 \\underline { \\alpha } ^ { 2 } + 3 ( \\underline { \\alpha } \\cdot \\widehat { \\epsilon _ { n } } + \\widehat { \\epsilon _ { n } } \\cdot \\underline { \\alpha } ) ) , \\end{align*}"}
-{"id": "4069.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\partial v _ i } { \\partial t } & = \\frac { \\partial } { \\partial x ^ \\alpha } \\left ( \\frac { \\partial G } { \\partial \\Xi ^ A } ( \\Xi ) \\frac { \\partial \\Phi ^ A } { \\partial F _ { i \\alpha } } ( F ) \\right ) , \\\\ \\frac { \\partial \\Xi ^ A } { \\partial t } & = \\frac { \\partial } { \\partial x ^ \\alpha } \\left ( \\frac { \\partial \\Phi ^ A } { \\partial F _ { i \\alpha } } ( F ) v _ i \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "448.png", "formula": "\\begin{align*} \\Upsilon ( x , t ) = \\sum _ { j = 0 } ^ { ( k _ 2 - 1 ) / 2 } c _ { k _ 1 , k _ 2 , j } \\frac { \\omega ^ { k _ 2 - 2 j } } { | x | ^ { 2 j } } + O \\left ( \\sum _ { j = 0 } ^ { ( k _ 2 + 1 ) / 2 } \\frac { \\omega ^ { k _ 2 - 2 j + 1 } } { | x | ^ { 2 j } } \\right ) ; \\end{align*}"}
-{"id": "9260.png", "formula": "\\begin{align*} | C | > \\frac { 4 } { 1 1 } \\left ( n + \\frac { 3 } { 4 } \\right ) = 4 q _ 2 + 2 + \\frac { 1 } { 1 1 } \\end{align*}"}
-{"id": "2728.png", "formula": "\\begin{align*} & \\displaystyle \\mathcal F ( h , h ) = M ^ { - 1 } \\left [ \\mathcal Q ( M h , M h ) + \\mathcal Q ( M h , M h ) \\right ] \\\\ [ 2 p t ] & \\displaystyle \\qquad = \\int _ { \\mathbb R ^ d \\times \\mathbb S ^ { d - 1 } } \\phi ( | v - v _ { \\ast } | ) b ( \\cos \\theta ) M _ { \\ast } ( h _ { \\ast } ^ { \\prime } h ^ { \\prime } - h _ { \\ast } h ) \\ , d v _ { \\ast } d \\sigma \\ , . \\end{align*}"}
-{"id": "5566.png", "formula": "\\begin{align*} \\begin{tabular} { l l l } $ x _ { 1 } = e _ { 1 } ^ { T } \\Gamma \\bar { w } _ { 2 ^ { k } } \\left ( t \\right ) $ , & $ x _ { 0 } = 1 $ , & $ \\dot { x } _ { 0 } = 0 $ \\\\ $ x _ { 2 } = \\tau e _ { 1 } ^ { T } P \\Gamma \\bar { w } _ { 2 ^ { k } } \\left ( t \\right ) $ & $ x _ { 0 } = 0 $ & $ \\dot { x } _ { 0 } = 1 $ \\end{tabular} \\end{align*}"}
-{"id": "4461.png", "formula": "\\begin{align*} F ^ \\ell ( k ) = \\sum _ { k ' + k '' = k } G _ \\ell ( k ' ) \\tilde G _ \\ell ( k '' ) \\xi ( k ' ) \\xi ( k '' ) . \\end{align*}"}
-{"id": "6656.png", "formula": "\\begin{align*} f ' ( t ) \\geq & \\frac { f ( t ) - f ( t c ) } { t - t c } \\geq \\frac { \\frac { 1 - \\eta } { 2 } f '' ( 0 ) t ^ 2 - \\frac { 1 + \\eta } { 2 } f '' ( 0 ) c ^ 2 t ^ 2 } { ( 1 - c ) t } = \\left ( \\frac { 1 + c } { 2 } - \\eta \\frac { 1 + c ^ 2 } { 2 ( 1 - c ) } \\right ) f '' ( 0 ) t \\\\ \\geq & \\left ( \\frac { 1 + c } { 2 } - \\frac { \\eta } { 1 - c } \\right ) f '' ( 0 ) t = \\left ( 1 - \\sqrt { \\eta } - \\frac { 1 } { 2 } \\sqrt { \\eta } \\right ) f '' ( 0 ) t = \\left ( 1 - \\frac { 3 } { 2 } \\sqrt { \\eta } \\right ) f '' ( 0 ) t \\quad . \\end{align*}"}
-{"id": "9133.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } x ^ { i } f _ { n + 1 - i } \\left ( x ^ { n + 1 - i } \\right ) = 0 \\left ( x \\in R \\right ) \\end{align*}"}
-{"id": "1643.png", "formula": "\\begin{align*} \\mu ( D _ v \\setminus \\cup _ { g \\in v \\Lambda ^ { e _ i } } R _ { g } ) = 0 . \\end{align*}"}
-{"id": "6581.png", "formula": "\\begin{align*} d ( x , x ' ) \\leq \\max _ { 1 \\leq m \\leq n } \\mathrm { d i a m } ( \\Omega _ m ) = \\kappa \\min _ { 1 \\leq j < k \\leq n } \\mathrm { d i s t } _ H ( \\Omega _ j , \\Omega _ k ) \\leq \\kappa d ( x , y ) \\end{align*}"}
-{"id": "1151.png", "formula": "\\begin{align*} b _ 1 & = b _ 0 \\\\ b _ j & = \\frac { [ 1 ] _ x ^ { 2 ^ { j - 1 } } b _ { j - 1 } + b _ { j - 2 } } { [ j ] _ x } \\quad \\mbox { f o r $ j \\geq 2 $ . } \\end{align*}"}
-{"id": "8020.png", "formula": "\\begin{align*} ( \\phi _ { { t _ l } } + \\phi _ { { t _ l } + 1 } ) \\ast d _ { { t _ n } } ( x ) = 2 ^ { - { s } { t _ n } } ( \\phi _ { { t _ l } } + \\phi _ { { t _ l } + 1 } ) \\ast g _ { { t _ n } } ( x ) \\end{align*}"}
-{"id": "2494.png", "formula": "\\begin{align*} \\left ( - i \\xi _ { 1 } \\left ( - \\left | \\eta \\right | \\right ) + L \\right ) \\left ( \\mathcal { R } \\cdot \\psi _ { j } \\right ) = \\varrho _ { j } \\left ( \\left | \\eta \\right | \\right ) \\left ( \\mathcal { R } \\cdot \\psi _ { j } \\right ) , \\end{align*}"}
-{"id": "3703.png", "formula": "\\begin{align*} x \\cdot \\frac { u _ 1 } { v _ 1 } = t _ 1 , \\frac { v _ { k - 1 } } { u _ { k - 1 } } \\frac { u _ k } { v _ k } = t _ k , \\mbox { a n d } \\frac { v _ n } { u _ n } \\cdot y = t _ { n + 1 } . \\end{align*}"}
-{"id": "8671.png", "formula": "\\begin{align*} ( P ^ { n + 1 } ) _ { i k } & = \\sum _ j ( P ^ n ) _ { i j } ( P ) _ { j k } \\\\ & = \\sum _ { j \\in A } ( P ^ n ) _ { i j } ( P ) _ { j k } + ( P ^ n ) _ { i \\ell } ( P ) _ { \\ell k } \\end{align*}"}
-{"id": "8583.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } u _ { m , n _ m } ( x , t ) = u ( x , t ) , \\lim _ { m \\to \\infty } \\nabla u _ { m , n _ m } ( x , t ) = \\nabla u ( x , t ) , \\end{align*}"}
-{"id": "8957.png", "formula": "\\begin{gather*} D _ w \\big ( \\vec { T } \\big ) = \\sum _ { \\alpha \\in \\Phi ^ + ( W ) \\cap w \\Phi ^ - ( W ) } \\big ( [ X ^ { r _ \\alpha } ] - T _ \\alpha \\big ) . \\end{gather*}"}
-{"id": "5311.png", "formula": "\\begin{align*} \\int _ \\Omega \\sigma ( \\theta _ M ) \\nabla \\phi _ M \\cdot \\nabla w \\mathrm { d x } + \\int _ \\Omega \\sigma ( \\theta _ M ) \\alpha _ \\mathrm { S } ( \\theta _ M ) \\nabla \\theta _ M \\cdot \\nabla w \\mathrm { d x } = \\int _ { \\Gamma _ \\mathrm { N } } g w \\mathrm { d s } , \\end{align*}"}
-{"id": "2008.png", "formula": "\\begin{align*} z _ i ( q ) = x _ i x _ { i + 1 } + q ^ 4 a _ 1 x _ { i - 1 } x _ { i + 2 } + q ^ { 1 2 } a _ 2 x _ { i - 2 } x _ { i + 3 } + . . . \\end{align*}"}
-{"id": "5884.png", "formula": "\\begin{align*} \\sigma _ i ^ { ( e ) } = \\inf \\big \\{ 2 n \\ge 2 : Z _ { 2 n } ^ { N , i } \\neq 0 \\} \\big \\} , \\ \\ \\sigma _ i ^ { ( o ) } = \\inf \\big \\{ 2 n - 1 \\ge 1 , Z _ { 2 n - 1 } ^ { N , i } \\neq 0 \\} \\big \\} . \\end{align*}"}
-{"id": "2837.png", "formula": "\\begin{align*} \\frac { p _ 1 ( x + y ) } { p _ 1 ( x ) } & = p _ 1 \\left ( \\frac { x } { p _ 1 ( x ) } + \\frac { y } { p _ 1 ( x ) } \\right ) \\\\ & = p _ 1 \\left ( u + \\tau v \\right ) = p _ 1 \\left ( ( 1 - \\tau ) u + \\tau ( u + v ) \\right ) \\leq ( 1 - \\tau ) + \\tau p _ 1 ( u + v ) \\\\ & \\leq ( 1 - \\tau ) + 2 \\tau ( 1 - \\delta _ 1 ( p _ 1 ( u - v ) ) ) = 1 + \\tau - 2 \\tau \\delta _ 1 ( p _ 1 ( u - v ) ) , \\end{align*}"}
-{"id": "7229.png", "formula": "\\begin{align*} \\hat { f } ( \\pi ) = \\int _ { G } ^ { } f ( x ) \\pi ( x ) d x , \\end{align*}"}
-{"id": "7447.png", "formula": "\\begin{align*} ( a ) = \\begin{pmatrix} 1 , 0 \\end{pmatrix} , \\ ; ( a ^ * ) = \\begin{pmatrix} A _ 0 ^ * \\\\ A _ 1 ^ * \\end{pmatrix} , \\ ; ( \\beta ) = \\begin{pmatrix} B _ { 0 0 } & B _ { 0 1 } \\\\ B _ { 1 0 } & B _ { 1 1 } \\\\ \\end{pmatrix} , \\ ; ( \\gamma ) = \\begin{pmatrix} 0 & 1 \\\\ C _ { 1 0 } & C _ { 1 1 } \\\\ \\end{pmatrix} , \\ ; ( \\delta ) = \\begin{pmatrix} D _ { 0 0 } & D _ { 0 1 } \\\\ D _ { 1 0 } & D _ { 1 1 } \\\\ \\end{pmatrix} \\end{align*}"}
-{"id": "4336.png", "formula": "\\begin{align*} \\| f \\| _ { p , \\alpha } ^ p : = \\int _ { \\mathbb { C } ^ n } | f ( z ) | ^ p d \\mu _ { p \\alpha / 2 } ( z ) . \\end{align*}"}
-{"id": "1190.png", "formula": "\\begin{align*} \\mbox { t r a c e } \\left ( ( a _ { i j } ) \\cdot ( w _ { x _ i x _ j } ( 0 ) ) \\right ) = \\sum _ { i , j = 1 } ^ n a _ { i j } \\ , w _ { x _ i x _ j } ( 0 ) < 0 . \\end{align*}"}
-{"id": "8713.png", "formula": "\\begin{align*} K _ { ( 0 , \\hat { t } ) } [ \\varphi _ \\sigma ] ( \\hat { t } , \\hat { x } ) & = K _ { ( 0 , \\rho ) } [ \\varphi _ \\sigma ] ( \\hat { t } , \\hat { x } ) + K _ { ( \\rho , \\hat { t } ) } [ \\varphi _ \\sigma ] ( \\hat { t } , \\hat { x } ) \\\\ & \\ge K _ { ( 0 , \\rho ) } [ \\varphi ] ( \\hat { t } , \\hat { x } ) + K _ { ( \\rho , \\hat { t } ) } [ \\varphi _ \\sigma ] ( \\hat { t } , \\hat { x } ) \\end{align*}"}
-{"id": "5450.png", "formula": "\\begin{align*} f ( x ) = \\begin{cases} n , & x \\in [ ( 1 + n ^ { - 1 } ) 2 ^ n , ( 1 + 2 n ^ { - 1 } ) 2 ^ n ] , \\\\ 1 , & \\end{cases} \\end{align*}"}
-{"id": "1496.png", "formula": "\\begin{align*} \\varphi _ t ( x ) = - t \\log | J ^ u T ( x ) | , \\ - \\infty < t < \\infty , \\end{align*}"}
-{"id": "7925.png", "formula": "\\begin{align*} \\dim _ H K = \\dim _ B K = s , \\end{align*}"}
-{"id": "5447.png", "formula": "\\begin{align*} & \\mathcal { P } _ { r , \\rho } = \\Big \\{ p : ( 0 , \\infty ) \\to ( 0 , \\infty ) \\ , : \\ , p \\in \\mathcal { P } _ { r } , x ^ { \\rho } p ( x ) \\Big \\} , \\ \\rho \\geq 0 , \\\\ & \\mathcal { P } _ { r , \\rho } = \\Big \\{ p : ( 0 , \\infty ) \\to ( 0 , \\infty ) \\ , : \\ , p \\in \\mathcal { P } _ { r } , x ^ { \\rho } p ( x ) \\Big \\} , \\ \\rho < 0 . \\end{align*}"}
-{"id": "7841.png", "formula": "\\begin{align*} E ( X _ { i } ^ { r } \\ ) = \\sum _ { j = 0 } ^ { \\infty } \\sum _ { k = 0 } ^ { \\alpha - 1 } \\sum _ { l = 0 } ^ { \\infty } \\zeta _ { j , k } ^ { ( l ) } ( a + b \\Theta ) \\Gamma ( \\Theta ) , \\ \\end{align*}"}
-{"id": "9159.png", "formula": "\\begin{align*} \\left | A _ l ( c x ^ { n + 1 - i } ) - f ( c x ^ { n + 1 - i } ) \\right | < \\varepsilon ( i = 0 , \\ldots , n + 1 ) \\end{align*}"}
-{"id": "8412.png", "formula": "\\begin{align*} G ( x , \\mu ) = \\int _ { \\mathcal { O } ^ { \\times } } \\psi ( x y ) \\mu ( y ) d ^ { \\times } y = \\begin{cases} 1 & \\\\ - \\zeta _ F ( 1 ) q ^ { - 1 } & \\\\ \\zeta _ F ( 1 ) q ^ { - \\frac { a ( \\mu ) } { 2 } } \\epsilon ( \\frac { 1 } { 2 } , \\mu ^ { - 1 } ) \\mu ^ { - 1 } ( x ) & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "3334.png", "formula": "\\begin{align*} D ( r ) \\geq L ( 1 - r ) \\sum _ { j = 0 } ^ { K + 1 - k } \\frac { 1 } { N ^ j } - L r \\sum _ { j = 0 } ^ { K - k } \\frac { K + 1 - k - j } { N ^ j } - o ( L ) , \\end{align*}"}
-{"id": "1304.png", "formula": "\\begin{align*} [ \\Delta ( \\mu ) : L ( \\nu ) ] \\not = 0 \\quad \\Leftrightarrow \\nu \\preceq \\mu \\quad \\Leftrightarrow \\Delta ( \\nu ) \\subset \\Delta ( \\mu ) . \\end{align*}"}
-{"id": "3971.png", "formula": "\\begin{align*} p ^ { \\alpha _ 0 } _ { k - 1 } ( 0 , t ) = \\frac { ( - \\lambda t ^ { \\alpha _ 0 } ) ^ { k - 1 } } { \\Gamma ( ( k - 1 ) { \\alpha _ 0 } + 1 ) } . \\end{align*}"}
-{"id": "7894.png", "formula": "\\begin{align*} u ( x , t ) = \\sup _ { a \\in X } \\left \\{ u _ 0 ( a ) - t L \\left ( { d ( a , x ) \\over t } \\right ) \\right \\} \\end{align*}"}
-{"id": "9145.png", "formula": "\\begin{align*} 3 f ( x ^ { 2 } ) + 2 x g ( x ) = 0 \\left ( x \\in R \\right ) . \\end{align*}"}
-{"id": "4850.png", "formula": "\\begin{align*} w _ p ( 2 ) & = \\sum _ { i = 0 } ^ 5 ( h _ i - i ) \\\\ & = 4 m + 4 l - 1 5 . \\end{align*}"}
-{"id": "4784.png", "formula": "\\begin{align*} \\alpha a _ 0 + \\beta b _ 0 + \\gamma = 0 , \\alpha a _ 1 + \\beta b _ 1 + \\gamma = 0 . \\end{align*}"}
-{"id": "2620.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 - \\log | t - s | \\ , u ( s ) \\ , d s = f ( t ) + C t \\in ( 0 , 1 ) \\end{align*}"}
-{"id": "297.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } F _ { s _ { \\tilde S ^ * _ { 0 , \\lfloor n t _ 1 \\rfloor } } , s _ { \\tilde S ^ * _ { \\lfloor n t _ 1 \\rfloor + 1 , \\lfloor n t _ 2 \\rfloor } } } ( u _ 1 , u _ 2 ) = e ^ { - u _ 1 ^ 2 t _ 1 \\slash 2 } e ^ { - u _ 2 ^ 2 ( t _ 2 - t _ 1 ) \\slash 2 } . \\end{align*}"}
-{"id": "7329.png", "formula": "\\begin{align*} U = U _ 0 U _ 1 \\dots U _ q \\end{align*}"}
-{"id": "626.png", "formula": "\\begin{align*} \\frac { 1 } { 4 \\pi ^ 2 } \\int _ 0 ^ { 2 \\pi } \\int _ 0 ^ { 2 \\pi } \\frac { f ( r _ 1 e ^ { i \\theta _ 1 } , r _ 2 e ^ { i \\theta _ 2 } ) \\ , d \\theta _ 1 \\ , d \\theta _ 2 } { ( r _ 2 e ^ { i \\theta _ 2 } ) ^ { \\alpha _ 2 } } = a _ { ( 0 , \\alpha _ 2 ) } . \\end{align*}"}
-{"id": "3755.png", "formula": "\\begin{align*} \\log \\rho _ k = \\log \\rho _ { \\hat { k } } + \\sum _ { i = \\hat { k } } ^ k \\log ( 1 + L _ i ^ { - 1 / 1 6 } ) \\leq \\log \\rho _ { \\hat { k } } + \\sum _ { i = \\hat { k } } ^ \\infty L _ i ^ { - 1 / 1 6 } < \\log ( \\iota _ { \\hat { k } } \\rho _ { \\hat { k } } ) < \\infty \\end{align*}"}
-{"id": "9262.png", "formula": "\\begin{align*} \\| \\mathcal { O } \\| _ { 2 , } : = \\underset { \\| X \\| _ { 2 } = 1 } { \\sup } \\| \\mathcal { O } ( X ) \\| _ { 2 } . \\end{align*}"}
-{"id": "6467.png", "formula": "\\begin{align*} \\Delta \\left ( \\rho _ { k } \\right ) \\overset { } { = } 1 + 4 \\rho _ { k } ^ { 2 } \\end{align*}"}
-{"id": "2899.png", "formula": "\\begin{align*} \\exists v \\forall s _ { m + 1 } \\geq v \\big ( \\lim _ { s _ { m + 2 } } \\cdots \\lim _ { s _ { n - 1 } } c ( k , s _ 1 , \\ldots , s _ { n - 1 } ) = 1 \\big ) \\end{align*}"}
-{"id": "2579.png", "formula": "\\begin{align*} ( T _ u ^ { * } H _ u T _ u - T _ v ^ { * } H _ v T _ v ) ( z ) & = \\left ( T _ u ^ { * } \\left ( H _ u T _ u - H _ v T _ v \\right ) \\right ) ( z ) + \\left ( \\left ( T _ u ^ { * } - T _ v ^ { * } \\right ) H _ v T _ v \\right ) ( z ) \\\\ & = \\left ( T _ u ^ { * } \\left ( H _ u T _ u - H _ v T _ v \\right ) \\right ) ( z ) + \\mathcal { O } \\left ( | z | ^ { - 3 } \\right ) | z | \\\\ & = \\left ( T _ u ^ { * } \\left ( H _ u T _ u - H _ v T _ v \\right ) \\right ) ( z ) + \\mathcal { O } \\left ( | z | ^ { - 2 } \\right ) . \\end{align*}"}
-{"id": "9714.png", "formula": "\\begin{align*} \\tilde { T } \\ge T \\ge T _ 0 > 0 , \\tilde { V } - V = O ( 1 ) Z h , 0 \\leq \\tilde { Z } \\le e ^ { - l h } Z \\leq 1 , \\end{align*}"}
-{"id": "6064.png", "formula": "\\begin{align*} \\langle d u , d v \\rangle = \\sum _ { i = 1 } ^ n \\langle \\partial _ i u , \\partial _ i v \\rangle , \\end{align*}"}
-{"id": "2709.png", "formula": "\\begin{align*} x _ { w } & = d _ { u } ( w ' u ^ { k - 1 } ) - p ( x _ { w ' u ^ { k - 2 } } ) + 2 p ( x _ { w ' u ^ { k - 1 } } ) - \\dim ( w ' u ^ { k - 2 } ) + 2 \\dim ( w ' u ^ { k - 1 } ) - \\dim ( w ' u ^ { k } ) \\\\ & = d _ { u } ( w ' u ^ { k - 1 } ) - p ( x _ { w ' u ^ { k - 2 } } ) + 2 p ( x _ { w ' u ^ { k - 1 } } ) \\end{align*}"}
-{"id": "8202.png", "formula": "\\begin{align*} c _ { \\Psi } = \\delta _ { \\Psi } c _ { \\Psi , } \\prod _ { \\nu } c _ { \\nu } ( \\pi _ { \\Psi , \\nu } ) . \\end{align*}"}
-{"id": "6210.png", "formula": "\\begin{align*} \\begin{pmatrix} - 2 & 0 & 1 \\\\ 0 & - 2 & n \\\\ 1 & n & 0 \\end{pmatrix} n \\in \\Z . \\end{align*}"}
-{"id": "6994.png", "formula": "\\begin{align*} \\tilde { e } ( \\theta ) = ( 0 , 0 , . . . , 0 , \\sin \\theta , \\cos \\theta ) , \\end{align*}"}
-{"id": "10060.png", "formula": "\\begin{align*} W _ { 0 , \\mathfrak { p } } ( s , \\Phi _ \\mu ) = \\int _ { F _ \\mathfrak { p } } \\Phi _ { \\mu , \\mathfrak { p } } \\left ( w n ( b ) , s \\right ) \\ , d b \\end{align*}"}
-{"id": "2365.png", "formula": "\\begin{align*} 2 g - 2 + s + t + \\sum _ { j = 1 } ^ r \\left ( 1 - \\frac 1 { m _ j } \\right ) > 0 . \\end{align*}"}
-{"id": "9973.png", "formula": "\\begin{align*} \\kappa _ { \\tau } = \\frac { - B _ { i } + \\sqrt { B _ { i } ^ { 2 } + \\frac { 4 A _ { i } } { \\gamma _ { i } ^ { \\mathsf { t h } } } } } { 2 A _ { i } } . \\end{align*}"}
-{"id": "4080.png", "formula": "\\begin{align*} M ( \\Theta ) : = \\{ ( x , u ) \\in \\mathbb { R } ^ { n _ { x } } \\times U : \\Phi ( x , u ) + \\Theta \\in \\Omega \\} , \\end{align*}"}
-{"id": "773.png", "formula": "\\begin{align*} B _ { q } ( x ) = \\sum _ { i = 0 } ^ { q } { \\binom { q } { i } } B _ { i } x ^ { q - i } , \\end{align*}"}
-{"id": "1909.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\log d ( \\textbf { \\textit { f } } _ { 0 } ^ { - n } ( q ) , \\textbf { \\textit { f } } _ { 0 } ^ { - n } ( p ) ) & \\leq \\frac { 1 } { n } \\log \\frac { 2 } { \\Delta _ { 0 } } + \\frac { \\log d ( q , p ) } { n } + \\frac { 1 } { n } \\sum _ { k = - 1 } ^ { - n } \\log \\tau _ { k } . \\end{align*}"}
-{"id": "4439.png", "formula": "\\begin{align*} \\left | ( \\lceil x _ 1 , ( \\cdot ) _ { 2 T } \\rceil f ) ( x ) \\right | = \\left | { 2 ^ { 2 / 3 } } ( 2 T ) ^ { 1 / 3 } \\int _ { \\mathbb { R } ^ 2 } \\tilde \\psi _ T ( x - y ) f _ T ( y ) d y \\right | \\leq { 2 T ^ { 1 / 3 } \\| f _ T \\| \\int _ { \\R ^ 2 } | \\tilde \\psi | \\ , d x } . \\end{align*}"}
-{"id": "4626.png", "formula": "\\begin{align*} \\sum _ { n = 2 ^ { k - 1 } } ^ { 2 ^ k - 1 } 1 _ { [ - B , B ] } \\left ( t + \\sum _ { i = 0 } ^ { n - 1 } f ( T ^ i x ) \\right ) \\ge \\beta \\sum _ { n = 0 } ^ { 2 ^ k - 1 } 1 _ { [ - B , B ] } \\left ( t + \\sum _ { i = 0 } ^ { n - 1 } f ( T ^ i x ) \\right ) \\end{align*}"}
-{"id": "8238.png", "formula": "\\begin{align*} \\lambda _ { \\nu } ( s ) = \\begin{cases} \\frac { 1 } { 4 } + ( t _ { \\nu } + s ) ^ { 2 } & \\\\ 1 + 4 ( t _ { \\nu } + s ) ^ 2 & \\end{cases} \\end{align*}"}
-{"id": "5549.png", "formula": "\\begin{align*} \\begin{array} { c c } a _ { 0 } = \\int _ { 0 } ^ { 1 } t w _ { 0 } \\left ( t \\right ) d t = \\frac { 1 } { 2 } & a _ { 1 } = \\int _ { 0 } ^ { 1 } t w _ { 1 } \\left ( t \\right ) d t = - \\frac { 1 } { 4 } \\\\ a _ { 2 } = \\int _ { 0 } ^ { 1 } t w _ { 2 } \\left ( t \\right ) d t = - \\frac { 1 } { 8 } & a _ { 3 } = \\int _ { 0 } ^ { 1 } t w _ { 3 } \\left ( t \\right ) d t = 0 \\end{array} \\end{align*}"}
-{"id": "4409.png", "formula": "\\begin{align*} | ( f \\circ \\tau _ { z _ \\gamma } ) ( 0 ) - ( f \\circ \\tau _ { z _ \\gamma } ) ( w ) | = | f ( z _ \\gamma ) - f ( z _ \\gamma - w ) | \\leq \\operatorname { O s c } _ { z _ \\gamma } ^ r ( f ) \\to 0 \\end{align*}"}
-{"id": "9198.png", "formula": "\\begin{align*} \\Theta _ { n } ^ { + } & = \\{ \\omega _ { 1 } + \\omega _ { n - 1 } = \\varepsilon _ { 1 } - \\varepsilon _ { n } , \\ \\omega _ { 1 } = \\varepsilon _ { 1 } , \\ \\omega _ { n - 1 } = - \\varepsilon _ { n } , \\\\ & \\ \\ \\ \\ \\ \\ 2 \\omega _ { 1 } = 2 \\varepsilon _ { 1 } , \\ 2 \\omega _ { n - 1 } = - 2 \\varepsilon _ { n } , \\ \\omega _ { 2 } = \\varepsilon _ { 1 } + \\varepsilon _ { 2 } , \\ \\omega _ { n - 2 } = - \\varepsilon _ { n - 1 } - \\varepsilon _ { n } , \\ 0 \\} . \\end{align*}"}
-{"id": "219.png", "formula": "\\begin{align*} U _ j H _ { \\omega } U _ j ^ * = H _ { T _ j \\omega } = : H ^ { T _ j } _ { \\omega } . \\end{align*}"}
-{"id": "3279.png", "formula": "\\begin{align*} \\widehat { A o B } & = \\frac { 1 } { 2 } A B = \\frac { 1 } { 2 } \\left [ 2 \\pi - A o B \\right ] \\\\ & = \\frac { 1 } { 2 } \\left [ 2 \\pi - \\left ( \\frac { 2 \\pi } { N _ k } - \\theta _ k \\right ) \\right ] = \\left ( 1 - \\frac { 1 } { N _ k } \\right ) \\pi + \\frac { \\theta _ k } { 2 } . \\end{align*}"}
-{"id": "6174.png", "formula": "\\begin{align*} L ( h ) = U _ { 0 } \\frac { \\partial h } { \\partial t } - h ^ { \\prime \\prime } + A _ { 0 } ( h ^ { \\prime } , h ) + \\sum _ { k = 1 } ^ { m } B _ { k } \\cdot \\int _ { 0 } ^ { t } A _ { k } ( h ^ { \\prime \\prime } , h ^ { \\prime } , h ) \\dd s , \\end{align*}"}
-{"id": "4774.png", "formula": "\\begin{align*} ( p \\alpha - q \\beta ) \\ , t _ 3 = 0 , p , q - c o n s t . \\end{align*}"}
-{"id": "4872.png", "formula": "\\begin{align*} \\mathfrak { u } ( \\ell ) : = \\lim _ { k \\to \\infty } \\| \\{ n : x _ n \\in U _ k \\} \\| _ \\varphi \\end{align*}"}
-{"id": "9142.png", "formula": "\\begin{align*} f ( x ^ { 3 } ) + x ^ { 2 } g ( x ) = 0 \\left ( x \\in R \\right ) . \\end{align*}"}
-{"id": "2664.png", "formula": "\\begin{align*} \\kappa = \\frac { 1 + b ^ 2 } { 2 b } . \\end{align*}"}
-{"id": "8576.png", "formula": "\\begin{align*} \\sup _ { x \\in \\Omega } \\int _ { \\Omega \\ , \\cap \\ , B _ { H _ 0 } ( x , 1 ) } F ( | z ( y , t ) | ) \\ , d y \\le M + C L ^ { 1 + \\delta } \\int _ 0 ^ t s ^ { \\sigma - 1 - \\frac { \\delta N } { 2 } } \\ , d s + C L \\int _ 0 ^ t s ^ { - \\sigma } \\ , d s \\le 2 M \\end{align*}"}
-{"id": "4540.png", "formula": "\\begin{align*} \\Big | \\frac 1 n \\sum _ { j = 0 } ^ { n - 1 } \\phi ( F _ j ( x ) ) - \\frac 1 n \\sum _ { j = 0 } ^ { n - 1 } \\phi ( f ^ j ( h ^ { - 1 } ( x ) ) ) \\Big | & = \\Big | \\frac 1 n \\sum _ { j = 0 } ^ { n - 1 } \\phi _ j ( f ^ j ( h ^ { - 1 } ( x ) ) ) - \\frac 1 n \\sum _ { j = 0 } ^ { n - 1 } \\phi ( f ^ j ( h ^ { - 1 } ( x ) ) ) \\Big | \\\\ & \\le \\frac 1 n \\sum _ { j = 0 } ^ { n - 1 } \\| \\phi _ j - \\phi \\| _ { C ^ 0 } \\end{align*}"}
-{"id": "8900.png", "formula": "\\begin{align*} \\begin{aligned} \\lim _ k ( g \\psi _ { n _ k } , g p \\kappa _ { n _ k } ) & = \\bigl ( g ( J _ { \\theta , 1 } ^ { - 1 } ( p \\gamma ) - p J _ { \\theta , 1 } ^ { - 1 } \\gamma ) , g p J _ { \\theta , 1 } ^ { - 1 } \\gamma \\bigr ) \\\\ & = ( g J _ { \\theta , 1 } ^ { - 1 } ( p \\gamma ) , g p J _ { \\theta , 1 } ^ { - 1 } \\gamma ) - \\| g p J _ { \\theta , 1 } ^ { - 1 } \\gamma \\| ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "10026.png", "formula": "\\begin{align*} c _ Q ( m ) & = \\epsilon _ Q ( g ) \\chi _ Q ^ n ( m ) c ( m ) & & \\\\ c _ Q ( m ) & = \\epsilon _ Q ( g ) \\chi _ { D / Q } ^ n ( m ) \\overline { c ( m ) } & & \\\\ c _ Q ( m _ 1 m _ 2 ) & = \\epsilon _ Q ( g ) ^ { - 1 } c _ Q ( m _ 1 ) c _ Q ( m _ 2 ) & & \\end{align*}"}
-{"id": "10105.png", "formula": "\\begin{align*} \\tilde { \\nabla } _ { X } Y = { \\nabla } _ { X } Y + \\frac { n - 1 } { 2 ( n + 1 ) } \\{ \\eta ( Y ) X + \\eta ( X ) Y \\} + \\frac { 1 } { 2 } \\{ \\eta ( Y ) X - \\eta ( X ) Y \\} , \\end{align*}"}
-{"id": "5963.png", "formula": "\\begin{align*} f ( t , w _ 2 + b _ 2 ( t ) , \\dots , w _ n + b _ n ( t ) ) & = \\sigma ( t ) + \\sum _ { j = 2 } ^ n ( w _ j + b _ j ( t ) ) \\vect { e } _ j ( t ) \\\\ & = \\hat \\sigma ( t ) + \\sum _ { j = 2 } ^ n w _ j \\vect { e } _ j ( t ) , \\end{align*}"}
-{"id": "344.png", "formula": "\\begin{align*} B & = 4 h ^ { i j } h _ { j l } R _ { \\ m i \\ } ^ { l \\ \\ \\ m } - 4 h ^ { i j } h ^ { l m } R _ { i l j m } - 2 h ^ { i j } ( \\nabla _ j R _ { \\nu l i } ^ { \\ \\ \\ l } + \\nabla _ l R _ { \\nu i j } ^ { \\ \\ \\ l } ) , \\quad \\mbox { a n d } \\\\ D & = h ^ { i j } \\left ( H \\lambda _ { , i j } + 2 H _ { , i } \\lambda _ { , j } \\right ) - \\frac { 1 } { 2 } H ( H \\Delta \\lambda + 2 \\langle \\nabla H , \\nabla \\lambda \\rangle ) . \\end{align*}"}
-{"id": "8212.png", "formula": "\\begin{align*} [ j ^ k n ^ { - h } ] \\ , \\ln \\biggl ( 1 + H _ j \\biggr ) = [ j ^ k n ^ { - h } ] \\ , \\ln \\biggl ( 1 + \\sum _ { s = 1 } ^ { j - 1 } \\frac { a _ s ( r , j ) } { n ^ s } \\biggr ) & = 0 , k \\ge h + 2 \\\\ [ j ^ { h + 1 } n ^ { - h } ] \\ , \\ln \\biggl ( 1 + H _ j \\biggr ) = [ j ^ { h + 1 } n ^ { - h } ] \\ , \\ln \\biggl ( 1 + \\sum _ { s = 1 } ^ { j - 1 } \\frac { a _ s ( r , j ) } { n ^ s } \\biggr ) & = \\frac { 1 } { ( h + 1 ) h } \\biggl ( \\frac { 1 } { r ^ h } - 2 \\biggr ) \\end{align*}"}
-{"id": "4663.png", "formula": "\\begin{align*} A x = y \\ , . \\end{align*}"}
-{"id": "3709.png", "formula": "\\begin{align*} \\begin{aligned} f _ 0 & = w _ { n + 1 } = x _ 1 , \\\\ f _ k & = w _ { k } w _ { n + k + 1 } = \\frac { u _ { k , k } } { v _ { k , k } } \\frac { v _ { k + 1 , k } } { u _ { k + 1 , k } } , ( 0 < k < n ) \\\\ f _ n & = w _ n w _ { n + 1 } = \\frac { u _ { n n } } { v _ { n n } } \\cdot t _ { n + 1 } . \\end{aligned} \\end{align*}"}
-{"id": "1714.png", "formula": "\\begin{align*} \\overline { } _ { \\lambda \\in \\Lambda } \\{ t _ \\lambda t _ \\lambda ^ * \\xi \\} = \\mathcal { H } . \\end{align*}"}
-{"id": "3214.png", "formula": "\\begin{align*} \\partial _ t ^ 2 \\phi - \\Delta \\phi + h ( t , x ) \\Delta \\phi = F \\mbox { i n } \\ , \\ , S _ T , \\end{align*}"}
-{"id": "340.png", "formula": "\\begin{align*} \\nu _ p ( L _ 1 ) = \\vert S _ 1 : N _ { S _ 1 } ( P _ 1 ) \\vert = \\frac { \\vert S _ 1 \\vert _ { p ' } } { \\vert N _ { S _ 1 } ( P _ 1 ) \\vert _ { p ' } } . \\end{align*}"}
-{"id": "532.png", "formula": "\\begin{align*} \\xi ( t ) : = d \\phi _ H ^ t ( x ( 0 ) ) \\ , \\xi _ 0 , \\end{align*}"}
-{"id": "5598.png", "formula": "\\begin{align*} & ( q - 1 ) \\frac { q } { q - 1 } \\frac { q ^ { h } } { n } \\frac { ( \\log n ) ^ { k - 1 } } { ( k - 1 ) ! } \\left ( H \\left ( \\frac { k - 1 } { \\log n } \\right ) + O _ A \\left ( \\frac { k } { ( \\log n ) ^ 2 } \\right ) \\right ) \\\\ & = \\frac { q ^ { h + 1 } } { n } \\frac { ( \\log n ) ^ { k - 1 } } { ( k - 1 ) ! } \\left ( H \\left ( \\frac { k - 1 } { \\log n } \\right ) + O _ A \\left ( \\frac { k } { ( \\log n ) ^ 2 } \\right ) \\right ) . \\end{align*}"}
-{"id": "495.png", "formula": "\\begin{align*} \\mathcal { L } p _ 1 ( x , t ) & = - R \\ , p _ { 1 , 2 , 0 } ( x , t ) - n \\ , p _ { 1 , 1 , 0 } ( x , t ) - R \\ , p _ { 1 , 0 , 2 } ( x , t ) { + \\frac { R } { \\abs { t } } ( m - 1 ) p _ { 1 , 0 , 1 } ( x , t ) } . \\end{align*}"}
-{"id": "3418.png", "formula": "\\begin{align*} P _ j = \\{ z = x + \\imath y \\in \\C : j - 1 \\le x \\le j , \\ | y | < b _ j \\} \\Subset S \\end{align*}"}
-{"id": "3266.png", "formula": "\\begin{align*} L ( u , k ) = 2 0 \\log _ { 1 0 } \\left ( \\frac { 4 \\pi r _ 0 } { \\lambda } \\right ) \\ ! + \\ ! 1 0 \\alpha \\log _ { 1 0 } \\left ( \\frac { r _ { u , k } } { r _ 0 } \\right ) \\ ! + \\ ! \\chi , \\end{align*}"}
-{"id": "299.png", "formula": "\\begin{align*} \\delta _ 2 ( u ) = y T , ~ ~ \\delta _ 2 ( v ) = x T ~ ~ ~ ~ \\delta _ 1 ( T ) = 1 . \\end{align*}"}
-{"id": "4545.png", "formula": "\\begin{align*} C ^ 2 = q _ 3 ^ n . \\end{align*}"}
-{"id": "7004.png", "formula": "\\begin{align*} \\mu _ 1 = \\mu _ 1 ( n , s , L , S C ) \\end{align*}"}
-{"id": "100.png", "formula": "\\begin{align*} C _ { W , S } = ( K - M ) ^ { - 1 } . \\end{align*}"}
-{"id": "758.png", "formula": "\\begin{align*} \\int _ { \\R ^ 3 } ( x _ 1 ^ 2 + x _ 2 ^ 2 ) | u _ i ^ * \\ast w _ i ^ * | ^ 2 \\ , d x & = \\int _ { \\R ^ 2 } ( x _ 1 ^ 2 + x _ 2 ^ 2 ) \\int _ { \\R } | u _ i ^ * \\ast w _ i ^ * | ^ 2 \\ , d x _ 3 d x ' \\\\ & = \\int _ { \\R ^ 3 } ( x _ 1 ^ 2 + x _ 2 ^ 2 ) ( | u _ i ^ * | ^ 2 + | w _ i ^ * | ^ 2 ) \\ , d x \\end{align*}"}
-{"id": "6190.png", "formula": "\\begin{align*} | ( { g ^ \\infty } ) | _ { g ^ \\infty } ( p _ \\infty ) = 1 , \\end{align*}"}
-{"id": "7795.png", "formula": "\\begin{align*} Z & = \\{ \\mathbf w \\in F \\mid \\ \\ \\mathbf w \\approx 0 \\ \\ \\mathbf N \\} , \\\\ L & = \\{ \\mathbf w \\in F \\mid \\ \\ \\mathbf w \\ \\ \\mathbf w \\notin Z \\} , \\\\ S & = \\{ \\mathbf w \\in F \\mid \\mathbf w \\notin Z \\cup L \\} . \\end{align*}"}
-{"id": "4319.png", "formula": "\\begin{align*} \\varphi _ { i } ( 0 ) = \\varphi _ { 0 , i , \\delta } \\Omega . \\end{align*}"}
-{"id": "290.png", "formula": "\\begin{align*} \\rho _ p ( f , g ) = \\| f - g \\| _ p \\ , . \\end{align*}"}
-{"id": "6541.png", "formula": "\\begin{align*} \\left \\{ y \\in \\mathbb { R } ^ { n } : \\langle y , \\xi \\rangle = 1 + \\frac { n | P ^ { \\circ } | _ n \\| \\xi \\| } { | F _ { \\xi } | _ { n - 1 } } \\delta \\right \\} \\end{align*}"}
-{"id": "1614.png", "formula": "\\begin{align*} A _ i \\kappa ^ { \\Lambda } = \\rho _ i \\ , \\kappa ^ { \\Lambda } , \\end{align*}"}
-{"id": "126.png", "formula": "\\begin{align*} x ^ h ( s ) & = \\int _ 0 ^ s b _ 1 ( 1 - u , x ^ h ( u ) , y ^ h ( u ) ) \\ , d u + h ( s ) , \\\\ y ^ h ( s ) & = \\int _ 0 ^ s b _ 2 ( 1 - u , x ^ h ( u ) , y ^ h ( u ) ) \\ , d u + h ( s ) . \\end{align*}"}
-{"id": "4928.png", "formula": "\\begin{align*} \\mathrm { r a n k } \\ , \\begin{bmatrix} t _ { N - 1 , N - 2 } & Q _ { N - 1 } T P ^ \\perp \\\\ t _ { N , N - 2 } & Q _ N T P ^ \\perp \\\\ \\vdots & \\vdots \\\\ t _ { N - 3 , N - 2 } & Q _ { N - 3 } T P ^ \\perp \\end{bmatrix} = \\mathrm { r a n k } \\ , R T R ^ \\perp = 1 , \\end{align*}"}
-{"id": "3385.png", "formula": "\\begin{align*} \\lambda _ n ^ \\pm ( b ) ^ 2 - \\mathsf { T } _ n ( b ) \\lambda _ n ^ \\pm ( b ) + \\mathsf { D } _ n ( b ) = 0 \\end{align*}"}
-{"id": "8061.png", "formula": "\\begin{align*} E ^ { p , q } _ 2 ( M ) = \\check H ^ p ( T ; R ^ q j _ * \\Z _ M ) \\end{align*}"}
-{"id": "3725.png", "formula": "\\begin{align*} ( p _ 1 , p ' _ 1 ) = ( p _ 2 , p ' _ 2 ) , \\ ; ( p _ 3 , p ' _ 3 ) = ( p _ 4 , p ' _ 4 ) , \\ ; ( p _ 5 , p ' _ 5 ) = ( p _ 6 , p ' _ 6 ) \\end{align*}"}
-{"id": "3517.png", "formula": "\\begin{align*} 1 \\leq X ^ u ( t _ i ) , \\ i = 0 , 1 , \\cdots , 1 0 . \\end{align*}"}
-{"id": "1105.png", "formula": "\\begin{align*} m _ { c , j } ^ { ( i ) } = \\frac { \\mu P _ t } { N } \\sum _ { k \\in \\mathcal { N } , k \\neq i } E [ | \\alpha _ { k j } | ^ 2 ] . \\end{align*}"}
-{"id": "7558.png", "formula": "\\begin{gather*} f = a z - \\frac i 2 a ^ 2 t + i \\Lambda ( t ) + F ( a ) , z - i a t + F _ a ( a ) = 0 , \\end{gather*}"}
-{"id": "9441.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\mathcal { M } ( N ) } \\varphi ( n _ k ) \\leq 2 N . \\end{align*}"}
-{"id": "7024.png", "formula": "\\begin{align*} d z = d y ( \\sqrt { \\epsilon } g ( \\epsilon ) | \\bar { y } | ) ^ { - 1 } , \\end{align*}"}
-{"id": "377.png", "formula": "\\begin{align*} a _ { r , s } = ( | r | + | s | ) ^ { - \\beta } L ( | r | + | s | ) b \\Big ( \\frac { r } { \\sqrt { r ^ 2 + s ^ 2 } } , \\ , \\frac { s } { \\sqrt { r ^ 2 + s ^ 2 } } \\Big ) \\end{align*}"}
-{"id": "4915.png", "formula": "\\begin{align*} T = \\lambda ( V + L ) + \\mu I . \\end{align*}"}
-{"id": "8345.png", "formula": "\\begin{align*} \\psi ( x v , y v ) = - Q ( v ) \\cdot \\psi ( x , y ) \\end{align*}"}
-{"id": "6993.png", "formula": "\\begin{align*} \\inf _ { M \\in \\mathcal { M } _ k } { \\{ - C _ { n , s } ^ { - 1 } ( - \\Delta ) ^ { s } ( u \\circ \\sqrt { M } ) ( x ) , \\lambda _ { m i n } ( M ) = \\epsilon \\} } \\to \\infty . \\end{align*}"}
-{"id": "7499.png", "formula": "\\begin{align*} \\partial _ t W _ t + \\frac { v } { \\sqrt { 1 + v ^ 2 } } \\cdot \\nabla _ x W _ t + \\nabla ( V * \\rho _ t ) \\cdot \\nabla _ v W _ t = 0 \\end{align*}"}
-{"id": "6702.png", "formula": "\\begin{align*} \\mathcal { F } _ l = \\mathcal { F } _ 0 \\big ( \\gamma _ { l } \\big ) \\end{align*}"}
-{"id": "8403.png", "formula": "\\begin{align*} \\rho _ - & = \\sup \\left \\{ r \\ , | \\ , B _ r ( y ) \\subset \\Omega \\ , \\ , y \\in \\mathbb { R } ^ { n + 1 } \\right \\} \\\\ \\rho _ + & = \\inf \\left \\{ r \\ , | \\ , \\Omega \\subset B _ r ( y ) \\ , \\ , y \\in \\mathbb { R } ^ { n + 1 } \\right \\} . \\end{align*}"}
-{"id": "4911.png", "formula": "\\begin{align*} T ^ * = - \\gamma ^ { - 1 } ( 0 + F ) + 0 I \\end{align*}"}
-{"id": "2095.png", "formula": "\\begin{align*} E ( \\psi ) = \\int _ M e ( \\psi ) = E _ H ( \\psi ) + E _ R ( \\psi ) , \\end{align*}"}
-{"id": "273.png", "formula": "\\begin{align*} D _ { 0 _ { + } } ^ { \\alpha } x ( t ) = f ( t , x ( t ) ) , \\end{align*}"}
-{"id": "412.png", "formula": "\\begin{align*} L _ { j , f } g = i ^ { - j } \\sum _ { \\mu = 0 } ^ { 2 j } \\frac { ( f '' ( 0 ) ^ { - 1 } \\partial , \\partial ) ^ { \\mu + j } [ ( f - P _ { 2 , 0 } f ) ^ \\mu g ] ( 0 ) } { 2 ^ { \\mu + j } \\mu ! ( \\mu + j ) ! } . \\end{align*}"}
-{"id": "2608.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\rho : = \\langle \\rho , \\varphi \\rangle , \\end{align*}"}
-{"id": "8664.png", "formula": "\\begin{align*} \\zeta _ { m _ 2 k } = c _ 1 m _ 2 k + c _ 2 + o ( 1 ) \\end{align*}"}
-{"id": "4072.png", "formula": "\\begin{align*} F = \\langle \\nu , \\lambda _ F \\rangle \\ , , v = \\langle \\nu , \\lambda _ v \\rangle \\ , , \\Xi = \\langle \\nu , \\lambda _ X i \\rangle \\ , . \\end{align*}"}
-{"id": "1013.png", "formula": "\\begin{align*} Q _ i \\triangleq \\left \\{ \\begin{array} { l l } B _ 1 , & i = 1 \\\\ ( E _ h ^ { i - 1 } , B _ i ) , & i = 2 , \\ldots , N \\\\ B _ { N + 1 } , & i = N + 1 \\end{array} \\right . \\end{align*}"}
-{"id": "905.png", "formula": "\\begin{align*} Z _ n ( t ) = \\frac { 1 } { \\mathfrak { s } _ n ( t ) } \\sum _ { i = 1 } ^ n K _ h ( t _ { i - 1 } - t ) z _ i ^ n , t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "6513.png", "formula": "\\begin{align*} \\mathcal { C } _ { \\mathcal { M } ^ { } } \\left ( \\tau ; \\rho \\right ) = \\frac { 8 } { \\lambda _ { \\mathcal { M } } } \\sqrt { \\frac { 1 - \\rho } { 1 + \\rho } } \\left [ - \\frac { 3 } { 4 } \\lambda _ { \\mathcal { M } } + \\frac { 1 } { 4 } \\frac { \\sinh \\left ( \\lambda _ { \\mathcal { M } } \\tau \\right ) } { \\tau } + \\frac { \\tanh \\left ( \\frac { 1 } { 2 } \\lambda _ { \\mathcal { M } } \\tau \\right ) } { \\tau } \\right ] . \\end{align*}"}
-{"id": "1776.png", "formula": "\\begin{align*} Y = \\tilde { P } ( \\{ \\omega \\} ) X P ( \\{ \\omega \\} ) : \\operatorname { R a n g e } P ( \\{ \\omega \\} ) \\to \\operatorname { R a n g e } \\tilde { P } ( \\{ \\omega \\} ) . \\end{align*}"}
-{"id": "3505.png", "formula": "\\begin{align*} X ( t _ i ) \\in \\mathbb { S } , \\ i = 0 , 1 , \\cdots , n . \\end{align*}"}
-{"id": "2961.png", "formula": "\\begin{align*} y _ k & = h _ { k k } x _ k + \\sum _ { \\substack { j = 1 \\\\ j \\neq k } } ^ { K } h _ { k j } x _ j + w _ k , \\forall k \\in \\mathcal { K } , \\\\ \\mathbf { y } _ { } & = \\sum _ { j = 1 } ^ { K } \\mathbf { h } _ { j } x _ j + \\mathbf { w } _ { } , \\end{align*}"}
-{"id": "387.png", "formula": "\\begin{align*} \\sigma _ n ^ 2 = \\sum _ { r , s \\in \\mathbb { Z } } b _ { n , r , s } ^ 2 = c _ \\beta n ^ { 6 - 2 \\beta } L ^ 2 ( n ) \\end{align*}"}
-{"id": "996.png", "formula": "\\begin{align*} \\varepsilon ^ { - 2 } \\frac { c _ 2 } { \\sqrt { n h } } \\log ^ 2 n = O \\left ( \\frac { \\log n } { ( n h ) ^ { 1 / 6 } } \\right ) . \\end{align*}"}
-{"id": "3663.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 & \\cdots & \\cdots & 0 & 0 & 0 \\\\ - 1 & 1 & \\cdots & \\cdots & 0 & 0 & 0 \\\\ 0 & - 1 & \\ddots & & 0 & 0 & 0 \\\\ \\vdots & \\vdots & \\ddots & \\ddots & \\vdots & \\vdots & \\vdots \\\\ 0 & 0 & & \\ddots & 1 & 0 & 0 \\\\ 0 & 0 & \\cdots & \\cdots & - 1 & 1 & 0 \\\\ 0 & 0 & \\cdots & \\cdots & 0 & 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "8658.png", "formula": "\\begin{align*} x + \\omega _ { X _ 0 } ( s ) - y - \\omega _ { Y _ 0 } ( s ) = x + \\omega _ { X _ 0 } ( \\tau _ 1 ) + [ \\omega _ { X _ 0 } ( s ) - \\omega _ { X _ 0 } ( \\tau _ 1 ) ] - y - \\omega _ { Y _ 0 } ( \\tau _ 1 ) - [ \\omega _ { Y _ 0 } ( s ) - \\omega _ { Y _ 0 } ( \\tau _ 1 ) ] . \\end{align*}"}
-{"id": "4454.png", "formula": "\\begin{align*} \\Psi ( v , w ) : = & v \\partial _ 2 R w + w \\partial _ 2 R v + w \\partial _ 2 R w \\\\ & + \\partial _ 2 \\frac { 1 } { 2 } R ( w + v ) ^ 2 - ( w + v ) \\partial _ 1 \\frac { 1 } { 2 } R ( w + v ) ^ 2 \\end{align*}"}
-{"id": "8670.png", "formula": "\\begin{align*} ( P ^ { n + 1 } ) _ { k i } & = \\sum _ j ( P ) _ { k j } ( P ^ { n } ) _ { j i } \\\\ & = \\sum _ { j \\in A } ( P ) _ { k j } ( P ^ { n } ) _ { j i } + ( P ) _ { k \\ell } ( P ^ { n } ) _ { \\ell i } , \\end{align*}"}
-{"id": "2305.png", "formula": "\\begin{align*} & \\ ; ( \\| u _ * \\| _ { L ^ 2 } ^ 2 + \\alpha ^ 2 \\| A ^ { 1 / 2 } u _ * \\| _ { L ^ 2 } ^ 2 ) ( t ) \\\\ & \\ ; + 2 \\nu \\int _ 0 ^ t ( \\| A ^ { s / 2 } u _ * \\| ^ 2 _ { L ^ 2 } + \\alpha ^ 2 \\| A ^ { ( 1 + s ) / 2 } u _ * \\| ^ 2 _ { L ^ 2 } ) ( \\tau ) \\ , d \\tau \\\\ \\leq & \\ ; \\| u _ 0 \\| _ { L ^ 2 } ^ 2 + \\alpha ^ 2 \\| A ^ { 1 / 2 } u _ 0 \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "2.png", "formula": "\\begin{align*} L _ 1 ( x , y ) Q _ 2 ( x , y ) = L _ 2 ( x , y ) Q _ 1 ( x , y ) . \\end{align*}"}
-{"id": "2471.png", "formula": "\\begin{align*} \\Vert g \\Vert _ { L _ { x } ^ { \\infty } L _ { \\xi } ^ { 2 } } = \\sup _ { x \\in { \\mathbb { R } ^ { 3 } } } | g | _ { L _ { \\xi } ^ { 2 } } \\ , , \\quad \\Vert g \\Vert _ { L _ { x } ^ { 1 } L _ { \\xi } ^ { 2 } } = \\int _ { { \\mathbb { R } ^ { 3 } } } | g | _ { L _ { \\xi } ^ { 2 } } d x \\ , . \\end{align*}"}
-{"id": "4568.png", "formula": "\\begin{align*} & - \\frac { q q _ 1 ^ { - 1 } q _ 3 } { 1 - q _ 1 ^ { - 1 } q _ 3 } \\tilde { E } _ 0 ( z ) = E ^ { ( 1 ) } _ { 0 | 1 } ( z ) = \\Bigl ( 1 - \\frac { z } { z ' } \\Bigr ) E _ { 0 } ( q _ 1 z ' ) E _ 1 ( z ) \\Bigl | _ { z ' = z } \\ , , \\\\ & ( 1 - q _ 1 ^ 2 ) \\tilde { F } _ 0 ( z ) = F ^ { ( 1 ) } _ { 0 | 1 } ( z ) = \\Bigl ( 1 - \\frac { z ' } { z } \\Bigr ) F _ 1 ( z ) F _ { 0 } ( q _ 1 z ' ) \\Bigl | _ { z ' = z } \\ , , \\\\ & \\tilde K ^ \\pm _ 0 ( z ) = K ^ \\pm _ 0 ( q _ 1 z ) K _ 1 ^ \\pm ( z ) \\ , . \\end{align*}"}
-{"id": "6599.png", "formula": "\\begin{align*} \\liminf _ { J \\ni m \\to \\infty } \\int _ { \\varphi _ n ( \\mathsf { U } _ n ) } | D f _ n - D g _ { m , n } | ^ p \\ , d \\mu = 0 \\ . \\end{align*}"}
-{"id": "458.png", "formula": "\\begin{align*} 4 ^ j L _ { j , \\psi _ \\omega } a _ { k _ 1 , k _ 2 , \\omega } = c _ { k _ 1 , k _ 2 , j } \\omega ^ { k _ 2 - 2 j } + O \\left ( \\omega ^ { k _ 2 - 2 j + 1 } \\right ) \\end{align*}"}
-{"id": "793.png", "formula": "\\begin{align*} \\sigma = 0 . 9 0 7 5 5 \\ \\textrm { a n d } \\ \\lambda = 0 . 9 0 7 5 5 \\log { \\rho } , \\end{align*}"}
-{"id": "3026.png", "formula": "\\begin{align*} A f = \\frac { \\partial } { \\partial \\zeta } f \\end{align*}"}
-{"id": "8676.png", "formula": "\\begin{align*} P _ { 3 3 } = \\left [ ( P ^ { 3 } ) _ { 4 2 } - ( P ^ { 2 } ) _ { 4 2 } P _ { 2 2 } - ( P ^ { 2 } ) _ { 4 2 } P _ { 4 4 } - P _ { 4 8 } P _ { 7 2 } P _ { 7 8 } \\right ] / ( P ^ { 2 } ) _ { 4 2 } . \\end{align*}"}
-{"id": "2572.png", "formula": "\\begin{align*} H = \\left [ \\begin{array} { c c c c } H _ { 0 0 } & H _ { 0 1 } & \\cdots & H _ { 0 ( n - 1 ) } \\\\ H _ { 1 0 } & H _ { 1 1 } & \\cdots & H _ { 1 ( n - 1 ) } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ H _ { ( m - 1 ) 0 } & H _ { ( m - 1 ) 1 } & \\cdots & H _ { ( m - 1 ) ( n - 1 ) } \\\\ \\end{array} \\right ] , \\end{align*}"}
-{"id": "3845.png", "formula": "\\begin{align*} f ( i _ r ) = \\prod _ { s = 0 } ^ { r - 1 } \\frac { 2 ( \\ell - 2 s ) - 1 } { 2 ( \\ell - 2 s ) + 1 } . \\end{align*}"}
-{"id": "2149.png", "formula": "\\begin{align*} U _ d ( r ) \\sim c _ * \\log r , U _ d ' ( r ) = O ( r ^ { - 1 } ) \\quad \\mbox { a s $ r \\to \\infty $ i n c a s e o f ( $ \\mbox { S } _ * $ ) } . \\end{align*}"}
-{"id": "4487.png", "formula": "\\begin{align*} \\psi ( x _ 0 ) = 1 > | \\Lambda ( x _ 0 ) | \\end{align*}"}
-{"id": "250.png", "formula": "\\begin{align*} \\mathcal { S } _ n ^ d ( \\vec { k } , l ) : = \\{ \\pi \\in \\mathcal { S } ^ d : \\ , ( \\pi ( 1 ) , . . . , \\pi ( n ) ) = ( k _ 1 , . . . , k _ { n - 1 } , l ) \\} \\subseteq \\mathcal { S } ^ d . \\end{align*}"}
-{"id": "5253.png", "formula": "\\begin{align*} \\Gamma ( E ^ u ( \\cdot , \\tau ) , E ^ s ( \\cdot , \\tau ) , \\lambda ^ * ) ( \\xi ) = \\int _ { - \\infty } ^ { \\infty } e ^ { c z } \\ , \\omega ( P , A _ \\lambda P ) \\ , d z , \\end{align*}"}
-{"id": "369.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( S _ { n } \\geq x _ { n } \\sigma _ { n } \\right ) = ( 1 + o ( 1 ) ) \\bigg [ x _ n ^ { - t } \\sum _ { r , s \\in \\mathbb { Z } } b _ { n , r , s } ^ t h \\Big ( \\frac { x _ n } { b _ { n , r , s } } \\Big ) + 1 - \\Phi ( x _ { n } ) \\bigg ] . \\end{align*}"}
-{"id": "2413.png", "formula": "\\begin{align*} a _ 2 ^ 2 + a _ 3 a _ 6 = a _ 2 ( 3 a _ 3 + a _ 4 - 3 a _ 6 ) - a _ 5 a _ 6 = a _ 2 ( 3 a _ 2 + a _ 5 ) + a _ 6 ( 2 a _ 3 + a _ 4 + a 6 ) = 0 . \\end{align*}"}
-{"id": "8025.png", "formula": "\\begin{align*} \\Big [ \\sum _ { l = M } ^ { L - M } { 2 ^ { - { t _ l } d } \\sum _ { P \\in \\mathcal { Q } ( l ) } { \\int _ { \\Omega } { \\Big ( \\sum _ { n = 1 } ^ { L - l } { 2 ^ { - 2 ( { s } + d ) { t _ n } } \\sum _ { Q \\in \\mathcal { V } _ n ( l , P ) } { \\theta _ Q ( \\omega ) } } \\Big ) ^ { { q } / { 2 } } } d \\lambda } } \\Big ] ^ { { 1 } / { q } } \\end{align*}"}
-{"id": "6487.png", "formula": "\\begin{align*} V \\left ( x \\right ) = - \\frac { \\Omega ^ { 2 } x ^ { 2 } } { 2 } \\end{align*}"}
-{"id": "3575.png", "formula": "\\begin{align*} y _ { i } = x _ { 1 i } \\beta _ { 1 } + x _ { 2 i } \\beta _ { 2 } + . . . + x _ { p i } \\beta _ { p } + \\varepsilon _ { i } , \\ i = 1 , \\ldots , n , \\end{align*}"}
-{"id": "7358.png", "formula": "\\begin{align*} \\nabla _ X \\psi = \\frac { 1 } { n } \\widetilde { X } . \\displaystyle { \\not } D \\psi . \\end{align*}"}
-{"id": "8123.png", "formula": "\\begin{align*} C ( X ) \\rtimes G \\stackrel { \\varphi } { \\longrightarrow } \\bigoplus _ { k = 0 } ^ d D _ { k , 0 } \\oplus \\cdots \\oplus D _ { k , m _ k } \\stackrel { \\psi } { \\longrightarrow } C ( X ) \\rtimes G \\end{align*}"}
-{"id": "1327.png", "formula": "\\begin{align*} P \\rho ( P _ v ) ^ * ( 1 - P ) \\rho ( P _ v ) P & = P \\rho ( P _ v ) P - P \\rho ( P _ v ) P \\rho ( P _ v ) P \\\\ & = \\pi ( P _ v ) - \\pi ( P _ v ) \\pi ( P _ v ) = 0 , \\end{align*}"}
-{"id": "7125.png", "formula": "\\begin{align*} ( \\mathcal { W } _ X ^ { - 1 } ) _ { i } ^ j = & ( h ^ { - 1 } _ X ) ^ { j k } g _ { k i } ^ X \\\\ = & ( h ^ { - 1 } _ Y ) ^ { k j } \\left ( g _ { k i } ^ Y + \\frac { \\langle Y , \\partial _ i Y \\rangle \\langle Y , \\partial _ k Y \\rangle } { ( 1 - | Y | ^ 2 ) } \\right ) \\sqrt { \\frac { 1 - \\langle N , Y \\rangle ^ 2 } { 1 - | Y | ^ 2 } } . \\end{align*}"}
-{"id": "6793.png", "formula": "\\begin{align*} w _ { \\lambda } ( \\Pi ^ { - 1 } _ { \\xi _ k } ( \\lambda z ) ) = \\ln { \\left ( \\frac { 8 } { \\lambda ^ 2 ( 1 + | z | ^ 2 ) ^ 2 } \\right ) } - 1 6 \\lambda ^ 2 \\ln { \\lambda } + 2 \\ln { ( 1 + \\lambda ^ 2 | z | ^ 2 ) } - \\ln { 4 } \\end{align*}"}
-{"id": "9364.png", "formula": "\\begin{align*} & \\iint _ { \\sqrt { x ^ 2 + y ^ 2 } \\leqslant r _ n } \\lambda ( x , y ) \\ , \\mathrm { d } x \\ , \\mathrm { d } y = n \\\\ & \\Rightarrow \\int _ 0 ^ { r _ n } \\frac { N } { 2 \\pi \\sigma ^ 2 } \\mathrm { e } ^ { - \\frac { r ^ 2 } { 2 \\sigma ^ 2 } } 2 \\pi r \\ , \\mathrm { d } r = n \\\\ & \\Rightarrow N ( 1 - \\mathrm { e } ^ { - \\frac { r _ n ^ 2 } { 2 \\sigma ^ 2 } } ) = n \\\\ & \\Rightarrow r _ n = \\sigma \\sqrt { 2 \\ln \\left ( \\frac { N } { N - n } \\right ) } . \\end{align*}"}
-{"id": "2106.png", "formula": "\\begin{align*} \\sup _ { M \\times [ 0 , T _ 3 ] } \\sum _ a \\big ( | v ^ a _ s | + | \\nabla _ H v ^ a _ s | \\big ) = V _ s ( T _ 3 ) \\leq 2 e ^ { C _ 1 } | | \\phi _ { s + s _ 0 } - \\phi _ { s _ 0 } | | _ { C ^ 1 } = O ( s ) , \\end{align*}"}
-{"id": "1020.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N \\beta _ i \\log _ 2 \\left ( 1 + \\frac { \\left | g ^ i _ { s s } \\right | p _ s ^ i } { \\sigma _ n ^ 2 + \\left ( \\left | \\hat { g } ^ i _ { p s } \\right | ^ 2 + 2 \\varepsilon \\left | \\hat { g } ^ i _ { p s } \\right | + \\varepsilon ^ 2 \\right ) p _ p } \\right ) \\end{align*}"}
-{"id": "6641.png", "formula": "\\begin{align*} | f ( x ) - f ( y ) | & \\leq \\sum _ { k = 0 } ^ \\infty | f ( z _ k ) - f ( z _ { k + 1 } ) | \\\\ & \\leq \\sum _ { k = 0 } ^ \\infty [ f ] _ { \\mathrm { L i p } , \\mathrm { c l } ( \\Omega _ { r _ k } ) } | z _ { k } - z _ { k + 1 } | \\\\ & \\leq \\left ( \\max _ { r = 0 , 1 \\ldots } [ f ] _ { \\mathrm { L i p } , \\mathrm { c l } ( \\Omega _ r ) } \\right ) \\sum _ { k = 0 } ^ \\infty | z _ { k } - z _ { k + 1 } | \\\\ & = \\left ( \\max _ { r = 0 , 1 , \\ldots } [ f ] _ { \\mathrm { L i p } , \\mathrm { c l } ( \\Omega _ r ) } \\right ) | x - y | \\end{align*}"}
-{"id": "5148.png", "formula": "\\begin{align*} L _ m \\eta ( m ' ) - y = L _ m ( \\eta ( m ' ) - x ) + ( L _ m x - y ) \\in \\frac { 1 } { 2 } V + \\frac { 1 } { 2 } V \\subset V \\end{align*}"}
-{"id": "3338.png", "formula": "\\begin{align*} \\bar { D } ( r _ s ^ { ( K + 1 ) } ) - \\bar { D } ( r _ s ^ { ( K ) } ) = \\alpha \\left ( \\bar { D } ( r _ { s - 1 } ^ { ( K ) } ) - \\bar { D } ( r _ s ^ { ( K ) } ) \\right ) . \\end{align*}"}
-{"id": "8619.png", "formula": "\\begin{align*} \\Phi _ { t , x , B } ( s , y ) : = \\int _ 0 ^ t \\phi ( t - r - s ) \\psi ( x + B _ r - y ) d r , \\end{align*}"}
-{"id": "1218.png", "formula": "\\begin{align*} 0 = \\sum _ { j = 1 } ^ { n - 1 } \\bar h _ { X _ i X _ j } \\ , \\frac { \\partial X _ j } { \\partial x _ n } + \\bar h _ { X _ i t } \\ , \\frac { \\partial t } { \\partial x _ n } = \\sum \\limits _ { j = 1 } ^ { n - 1 } \\bar h _ { X _ i X _ j } \\ , \\frac { \\partial X _ j } { \\partial x _ n } - \\frac { \\bar h _ { X _ i t } } { \\bar h _ t } . \\end{align*}"}
-{"id": "4532.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| \\pi ( \\Delta _ n ) ^ * \\Pi ( a ) \\pi ( \\Delta _ n ) \\| = 0 = \\lim _ { n \\to \\infty } \\| \\pi ( \\Delta _ n ) ^ * \\pi ( a ) \\pi ( \\Delta _ n ) \\| \\end{align*}"}
-{"id": "7510.png", "formula": "\\begin{gather*} u = - 2 \\frac { \\phi _ x } \\phi , v = - 2 \\frac { \\phi _ y } \\phi . \\end{gather*}"}
-{"id": "3142.png", "formula": "\\begin{align*} \\left ( \\mathcal { A } f \\right ) ( x ) = ( a - b x ) \\frac { \\partial f ( x ) } { \\partial x } + \\frac { 1 } { 2 } \\sigma ^ { 2 } x \\frac { \\partial ^ { 2 } f ( x ) } { \\partial x ^ { 2 } } + \\int _ { 0 } ^ { \\infty } \\left ( f ( x + z ) - f ( x ) \\right ) \\nu ( \\mathrm { d } z ) , \\end{align*}"}
-{"id": "376.png", "formula": "\\begin{align*} A : = \\sum _ { r , s \\in \\mathbb { Z } } | a _ { r , s } | < \\infty , \\ ; \\ ; a : = \\sum _ { r , s \\in \\mathbb { Z } } a _ { r , s } \\ne 0 , \\end{align*}"}
-{"id": "6068.png", "formula": "\\begin{align*} \\ddot \\rho ( r ) + ( n - 1 ) \\cot r \\dot \\rho ( r ) = \\frak { e } _ k \\tfrac { \\sin ( 2 \\rho ( r ) ) } { \\sin ^ 2 r } . \\end{align*}"}
-{"id": "2348.png", "formula": "\\begin{align*} c ( g _ { 2 2 } z - g _ { 1 2 } ) + d ( - g _ { 2 1 } z + g _ { 1 1 } ) \\not = 0 \\end{align*}"}
-{"id": "5850.png", "formula": "\\begin{align*} \\frac 1 N \\sum _ { k = 0 } ^ N & ( - 1 ) ^ k \\binom { N } { k } g ^ k D ^ N ( g ^ { N + 1 - k } f ) \\\\ & - \\frac { N + 1 } N g \\sum _ { k = 0 } ^ N ( - 1 ) ^ k \\binom { N } { k } g ^ k D ^ N ( g ^ { N - k } f ) \\\\ & + \\sum _ { k = 0 } ^ N ( - 1 ) ^ k \\binom { N } { k } g ^ { k + 1 } D ^ N ( g ^ { N - k } f ) = 0 . \\end{align*}"}
-{"id": "10114.png", "formula": "\\begin{align*} \\bar { \\boldsymbol R } _ { \\bar { \\boldsymbol \\omega } _ k } ( i ) = \\boldsymbol { \\bar { \\omega } } _ k ( i ) \\boldsymbol { \\bar { \\omega } } _ k ^ H ( i ) + \\delta { \\boldsymbol I } _ { D } . \\end{align*}"}
-{"id": "9572.png", "formula": "\\begin{align*} B = B \\biggl ( x _ 0 , \\frac { \\delta _ { \\partial D } ( x _ 0 ) } { 1 6 M c ^ 2 _ 1 } \\biggr ) \\ , . \\end{align*}"}
-{"id": "8770.png", "formula": "\\begin{align*} x \\Psi _ { B } ^ { \\bf f } ( { \\bf t } ) = x + \\int _ 0 ^ { t _ 1 } \\cdots \\int _ 0 ^ { t _ m } x \\Psi _ { B } ^ { \\bf f } \\left ( \\underset { m } { \\bf t } , s _ m \\right ) { B } ( { \\bf f } ) ^ { \\left ( \\underset { { { \\{ m _ 1 , m \\} } } } { \\bf t } { { , s _ m } } , { \\bf { s } } \\right ) } d s _ 1 \\dots d s _ m { { . } } \\ , \\ , \\ , \\end{align*}"}
-{"id": "6536.png", "formula": "\\begin{align*} e _ n ^ { \\perp } \\cap \\bigcap _ { i = 1 } ^ k \\{ x \\in \\mathbb { R } ^ n : \\langle x , y _ i \\rangle \\leq 1 \\} \\end{align*}"}
-{"id": "1341.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } Y _ c ( y ) \\\\ \\mu _ c ( y ) \\\\ \\lambda _ c ( y ) \\end{array} \\right ) : = \\left ( \\begin{array} { c } { \\rm D } \\theta _ c ( F ( x _ c ( y ) ) + Y / c ) ^ * \\\\ \\mu + c h ( x _ c ( y ) ) \\\\ \\Pi _ { K ^ * } ( \\lambda - c g ( x _ c ( y ) ) ) \\end{array} \\right ) \\ , . \\end{align*}"}
-{"id": "6834.png", "formula": "\\begin{align*} \\mathcal { A } \\sum \\limits _ { j = 1 } ^ 4 c _ j = - \\frac { 4 \\pi c _ 0 } { \\lambda ^ 2 } . \\end{align*}"}
-{"id": "1598.png", "formula": "\\begin{align*} V ( \\mathbf { j } '' ) & = 3 j '' _ { n - 2 } - s ( j '' _ { n - 2 } ) + 2 j '' _ { n - 1 } - s ( j '' _ { n - 1 } ) - c ( j '' _ n , - \\alpha _ { \\mathbf { j } '' } ( n ) - 1 ) \\\\ & = 3 ( 0 ) - s ( 0 ) + 2 ( 1 6 q + 7 ) - s ( 1 6 q + 7 ) - c ( 2 , 8 q + 2 ) \\\\ & = 3 2 q + 1 4 - s ( q ) - s ( 7 ) - c ( 2 , 8 q + 2 ) \\\\ & = 3 2 q + 1 0 - s ( q ) \\end{align*}"}
-{"id": "5558.png", "formula": "\\begin{align*} \\dot { x } = - \\alpha \\int _ { 0 } ^ { t } x \\left ( t _ { 1 } \\right ) d t _ { 1 } - \\beta \\int _ { 0 } ^ { t } p \\left ( t _ { 1 } \\right ) x \\left ( t _ { 1 } \\right ) d t _ { 1 } + \\dot { x } _ { 0 } \\end{align*}"}
-{"id": "4685.png", "formula": "\\begin{align*} { \\cal H } _ r ^ { ( d ( n - 1 ) ) } = \\ 2 \\ , \\sum _ { i < j } ^ n \\bigg ( \\frac { m _ i + m _ j } { m _ i m _ j } \\bigg ) \\rho _ { i j } p ^ 2 _ { \\rho _ { i j } } \\ + \\ 2 \\ , \\sum _ { i \\neq j , i \\neq k , j < k } ^ n \\ , \\frac { 1 } { m _ i } \\ , ( \\rho _ { i j } + \\rho _ { i k } - \\rho _ { j k } ) p _ { \\rho _ { i j } } p _ { \\rho _ { i k } } \\ + \\ \\Omega \\ , \\end{align*}"}
-{"id": "4665.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } { \\rm q d i s t } ( F ^ { - 1 } ( y _ n ) , F ^ { - 1 } ( y ) ) = 0 \\ , , \\end{align*}"}
-{"id": "3686.png", "formula": "\\begin{align*} \\frac { \\xi _ { i _ { l _ { k - 1 } + 1 } } } { \\xi _ { i _ { l _ { k - 1 } + 2 } } } = \\frac { \\xi _ { i _ { l _ { k - 1 } + 2 } } } { \\xi _ { i _ { l _ { k - 1 } + 3 } } } = \\cdots = \\frac { \\xi _ { i _ { l _ k - 1 } } } { \\xi _ { i _ { l _ k } } } = \\alpha _ k \\end{align*}"}
-{"id": "3.png", "formula": "\\begin{align*} y ^ 2 + B ( x ) y + C ( x ) = 0 \\end{align*}"}
-{"id": "9286.png", "formula": "\\begin{align*} \\boldsymbol { E } [ \\sum _ { j = 1 } ^ { T - 2 } \\sum _ { k = j + 1 } ^ { T - 1 } \\sum _ { l = k + 1 } ^ T X _ j X _ k X _ l ] \\leq ( \\boldsymbol { E } [ T ] ) ^ { \\frac { 1 } { 4 } } ( \\boldsymbol { E } [ T ^ 3 ] ) ^ \\frac { 1 } { 4 } ( \\boldsymbol { E } [ T ^ 4 ] ) ^ \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "8455.png", "formula": "\\begin{align*} _ { \\psi } ( a ; b ) = q ^ { \\frac { v ( a ) } { 3 } } \\int _ { \\mathcal { O } } \\psi ( a x ^ 3 + b x ) d x . \\end{align*}"}
-{"id": "8938.png", "formula": "\\begin{gather*} s _ i ( x _ 1 , \\dots , x _ n ) = \\bigg ( x _ 1 , \\dots , x _ { i - 1 } , - x _ i + \\sum _ { j \\ne i } \\mu _ { j i } ( x _ j ) , x _ { i + 1 } , \\dots , x _ n \\bigg ) . \\end{gather*}"}
-{"id": "6776.png", "formula": "\\begin{align*} - \\Delta _ g w _ { \\lambda , k } = e ^ { U _ { \\lambda , \\xi _ k } } \\eta _ { R _ 0 , \\xi _ k } - m _ 0 , \\end{align*}"}
-{"id": "8964.png", "formula": "\\begin{gather*} \\sum _ { \\alpha \\in \\Phi ^ - ( W ) } ( T _ \\alpha - T _ { w \\alpha } ) = \\ ! \\ ! \\sum _ { \\alpha \\in \\Phi ^ + ( W ) \\cap w \\Phi ^ - ( W ) } \\ ! \\ ! \\ ! \\ ! ( T _ { - \\alpha } - T _ \\alpha ) , \\end{gather*}"}
-{"id": "1497.png", "formula": "\\begin{align*} \\kappa ( t ) : = \\limsup _ { n \\to \\infty } \\frac { 1 } { n } \\log U _ n ( t ) . \\end{align*}"}
-{"id": "4999.png", "formula": "\\begin{align*} g - \\tilde { g } = \\sum _ { j } G _ j H _ j . \\end{align*}"}
-{"id": "6547.png", "formula": "\\begin{align*} P _ { \\delta } \\subseteq \\bigcap _ { \\zeta \\in \\mathrm { e x t } ( P ^ { \\circ } ) } \\left \\{ x \\in \\mathbb { R } ^ n : \\langle \\zeta , x \\rangle \\leq 1 - \\Lambda \\delta ( 1 - t ( \\delta ) ) \\right \\} = ( 1 - \\Lambda \\delta ( 1 - t ( \\delta ) ) ) P . \\end{align*}"}
-{"id": "7274.png", "formula": "\\begin{align*} S _ 2 = N ^ { - 3 } \\sum _ { m , n \\le X } r ( m ) r ( n ) \\frac { A ( m , n ) } { \\sqrt { m n } } . \\end{align*}"}
-{"id": "2439.png", "formula": "\\begin{align*} \\ell \\left ( \\frac { n - 2 q } { q } \\right ) = \\frac { m - c _ 1 - c _ 2 } { c _ 1 c _ 2 } \\end{align*}"}
-{"id": "7310.png", "formula": "\\begin{align*} \\{ \\overline { a } , \\overline { b } \\} = \\overline { \\hbar ^ { - 1 } ( a b - b a ) } \\end{align*}"}
-{"id": "4068.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 y } { \\partial t ^ 2 } = \\nabla \\cdot S ( \\nabla y ) , \\end{align*}"}
-{"id": "9304.png", "formula": "\\begin{align*} h _ 1 ( u ' ) + h _ 2 ( v ' ) = h _ 1 ( u ) + h _ 2 ( v ) . \\end{align*}"}
-{"id": "4116.png", "formula": "\\begin{align*} \\Delta \\rho = 2 ( H \\rho - 1 ) . \\end{align*}"}
-{"id": "994.png", "formula": "\\begin{align*} z _ n ( t ) & = \\frac { \\widehat { \\sigma } ^ 2 _ n ( t ) - \\sigma ^ 2 ( t ) } { \\sigma ^ 2 ( t ) \\mathfrak { s } _ n ( t ) } , t \\in [ 0 , T ] , \\\\ e _ n & = \\sup _ { t \\in [ a _ n , T - a _ n ] } \\left | \\frac { M _ n ( t ) } { \\sigma ^ 2 ( t ) \\mathfrak { s } _ n ( t ) } - F _ n ( t ) \\right | , \\\\ v _ n & = \\sqrt { n h ^ { 1 + 2 \\gamma } } + ( n h ) ^ { - 1 / 2 } . \\end{align*}"}
-{"id": "6243.png", "formula": "\\begin{align*} { \\boldsymbol \\Lambda } _ { \\rm b } \\left ( \\mu _ { \\rm b } \\right ) { \\boldsymbol \\Psi } _ { \\rm b } + \\bar { \\bf H } _ { \\rm b } ^ H \\left ( { \\bf Q } _ { \\rm s } - \\gamma { \\bf Q } _ { \\rm n } \\right ) \\bar { \\bf H } _ { \\rm b } { \\boldsymbol \\Psi } _ { \\rm b } = { \\bf 0 } . \\end{align*}"}
-{"id": "5820.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } \\frac { u ^ p v ^ q } { v ^ { p - 1 } } - u ^ { 1 + q } \\ ; d \\sigma & = \\int _ { \\R ^ n } \\frac { u ^ p v ^ q - u ^ { 1 + q } v ^ { p - 1 } } { v ^ { p - 1 } } \\ ; d \\sigma \\\\ & = \\int _ { \\R ^ n } \\frac { u ^ { 1 + q } v ^ { q } ( u ^ { p - 1 - q } - v ^ { p - 1 - q } ) } { v ^ { p - 1 } } \\ ; d \\sigma \\\\ & \\geq 0 \\end{align*}"}
-{"id": "7677.png", "formula": "\\begin{align*} Q = \\left ( \\begin{array} { c c c } m _ 1 & \\frac { t _ 3 } { 2 } & \\frac { t _ 2 } { 2 } \\\\ \\frac { t _ 3 } { 2 } & m _ 2 & \\frac { t _ 1 } { 2 } \\\\ \\frac { t _ 2 } { 2 } & \\frac { t _ 1 } { 2 } & m _ 3 \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "4869.png", "formula": "\\begin{align*} m _ x ( t ) = t ^ { d _ 1 } \\cdot \\prod _ { i = 1 } ^ { d _ 2 } \\varphi _ i ( t ) ^ { l _ i } \\cdot \\prod _ { i = 1 } ^ { d _ 3 } \\theta _ i ( t ) ^ { r _ i } , \\end{align*}"}
-{"id": "8266.png", "formula": "\\begin{align*} \\tilde { w } ( 1 - i r _ 1 + i r _ 2 ) L ^ { 1 - i r _ 1 + i r _ 2 } = \\tilde { w } ( 1 ) L ( 1 + o ( 1 ) ) . \\end{align*}"}
-{"id": "9526.png", "formula": "\\begin{align*} \\bar u ( x , t ) : = e ^ { \\gamma t } u ( e ^ { \\beta t } x , t ) , \\beta > 0 , \\ , \\ , \\gamma = \\frac { 2 \\beta } { 1 - m } . \\end{align*}"}
-{"id": "5054.png", "formula": "\\begin{align*} \\mathrm { T r } ( X \\Phi ( Y ) ) = \\mathrm { T r } ( \\Phi ^ { \\dagger } ( X ) Y ) . \\end{align*}"}
-{"id": "1875.png", "formula": "\\begin{align*} \\Theta _ 1 ( f ) : = \\Big \\{ ( f _ 1 , g _ 1 , \\dots , f _ d , g _ d , a ) : \\sum _ { j = 1 } ^ d \\big ( f _ j ( x _ j ) - g _ j ( x _ j ) ) + \\sum _ { i \\in I } a ^ i 1 _ { A ^ i } ( x ) \\geq f ( x ) , \\forall x \\in \\mathbb { R } ^ d \\Big \\} , \\end{align*}"}
-{"id": "6990.png", "formula": "\\begin{align*} M = D f _ k ( B ) = d i a g \\{ \\lambda _ 1 , \\lambda _ 2 , . . . , \\lambda _ n \\} \\end{align*}"}
-{"id": "4491.png", "formula": "\\begin{align*} \\Theta = \\| \\Theta _ 1 \\| \\frac { \\Theta _ 1 } { \\| \\Theta _ 1 \\| } + \\| \\Theta _ 2 \\| \\frac { ( - \\Theta _ 2 ) } { \\| \\Theta _ 2 \\| } \\end{align*}"}
-{"id": "4724.png", "formula": "\\begin{align*} { L } _ i = ( 0 , \\alpha , \\beta , \\gamma ) . \\end{align*}"}
-{"id": "3278.png", "formula": "\\begin{align*} f _ D ( D ) = \\frac { 2 } { \\pi \\sqrt { 4 a ^ 2 - D ^ 2 } } , \\end{align*}"}
-{"id": "7027.png", "formula": "\\begin{align*} \\epsilon _ 1 = ( \\frac { \\eta _ 0 } { 2 ( 1 - s ) C _ 1 C _ 2 } ) ^ { 1 / s } , \\end{align*}"}
-{"id": "3421.png", "formula": "\\begin{align*} F = ( F _ 1 , F _ 2 ) = e ^ { - h } ( f _ 1 , f _ 2 ) : \\overline { D } \\to \\C ^ 2 _ * . \\end{align*}"}
-{"id": "129.png", "formula": "\\begin{align*} u _ f ( 1 - t , x ) = u _ g ( 1 - t , y ) \\end{align*}"}
-{"id": "7813.png", "formula": "\\begin{align*} \\Delta u + \\lambda u = 0 . \\end{align*}"}
-{"id": "9005.png", "formula": "\\begin{align*} & { \\rm r e g } _ { S _ 1 } ( S _ 1 / J _ 1 ) = { \\rm r e g } _ { S _ 1 } ( S _ 1 / J ( G \\setminus N _ G [ x _ 1 ] ) ^ { ( k ) } S _ 1 ) + k { \\rm d e g } ( u _ 1 ) \\\\ & \\leq ( k - 1 ) { \\rm d e g } ( J ( G \\setminus N _ G [ x _ 1 ] ) ) + | V ( G \\setminus N _ G [ x _ 1 ] ) | - 2 + k { \\rm d e g } _ G ( x _ 1 ) \\\\ & \\leq ( k - 1 ) ( { \\rm d e g } ( J ( G ) ) - { \\rm d e g } _ G ( x _ 1 ) ) + n - { \\rm d e g } _ G ( x _ 1 ) - 1 - 2 + k { \\rm d e g } _ G ( x _ 1 ) \\\\ & < ( k - 1 ) { \\rm d e g } ( J ( G ) ) + n - 2 . \\end{align*}"}
-{"id": "4916.png", "formula": "\\begin{align*} \\alpha I + \\beta T + \\gamma T ^ * + \\frac { 1 } { 2 } T ^ * T + F = 0 . \\end{align*}"}
-{"id": "521.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { R } } _ { } \\ ! = \\ ! & \\bigcup _ { P _ { U | \\widetilde { X } } } \\ ! \\Big \\{ \\left ( R _ s , R _ \\ell , R _ w \\right ) \\ ! \\colon \\ ! \\\\ & 0 \\leq R _ s \\leq I ( U ; Y ) , \\\\ & R _ \\ell \\geq I ( U ; X ) - I ( U ; Y ) , \\\\ & R _ w \\geq I ( U ; \\widetilde { X } ) \\Big \\} \\end{align*}"}
-{"id": "3645.png", "formula": "\\begin{align*} S ( \\tilde P ) = \\mathbb C [ C ( \\tilde P ) \\cap ( M \\times \\mathbb Z ) ] , \\end{align*}"}
-{"id": "1205.png", "formula": "\\begin{align*} - \\frac { \\nabla \\bar u } { | \\nabla \\bar u | } \\ , \\ , \\mbox { i s a 1 - 1 m a p p i n g f r o m } \\ , \\ , \\{ \\bar u = t \\} \\ , \\ , \\mbox { o n t o } \\ , \\ , \\mathbb { S } ^ { n - 1 } \\end{align*}"}
-{"id": "5829.png", "formula": "\\begin{align*} w _ n ( x ) = \\sum _ { j = 1 } ^ J \\phi ^ j ( x - x _ n ^ j ) + r ^ J _ n ( x ) \\end{align*}"}
-{"id": "865.png", "formula": "\\begin{align*} P ( Z \\in A ) \\leq P ( \\Phi _ \\beta ( F ) \\in A ^ { e _ \\beta } ) = E [ 1 _ { A ^ { e _ \\beta } } ( \\Phi _ \\beta ( F ) ) ] . \\end{align*}"}
-{"id": "6258.png", "formula": "\\begin{align*} \\frac 1 { \\delta _ 1 } + \\frac 1 { \\delta _ 2 } + \\frac 1 { \\delta _ 3 } + \\frac 1 { \\delta _ 4 } = \\delta \\alpha , \\ \\ 0 < \\delta _ 1 , \\delta _ 2 , \\delta _ 3 , \\delta _ 4 < 1 \\\\ \\frac 1 { \\theta _ 1 } + \\frac 1 { \\theta _ 2 } + \\frac 1 { \\theta _ 3 } + \\frac 1 { \\theta _ 4 } = 1 , \\ \\ \\theta _ 1 = \\theta _ 3 = \\frac 2 { \\delta } , \\ \\ 1 < \\theta _ 2 , \\theta _ 4 < \\infty . \\end{align*}"}
-{"id": "6961.png", "formula": "\\begin{align*} \\epsilon _ 0 = \\sqrt { \\frac { n } { n - 1 } } C _ 4 ^ { 1 / s } ( \\frac { \\mu _ 0 } { \\mu _ 1 } ) ^ { \\frac { 1 } { s } } , \\end{align*}"}
-{"id": "4990.png", "formula": "\\begin{align*} h - \\tilde { h } & = \\sum _ { j = - \\infty } ^ { \\infty } h _ j \\left ( 1 - \\prod _ { j ' > j } ( 1 - U _ { j ' } ) \\right ) \\\\ & = \\sum _ j h _ j \\sum _ { j ^ { \\prime } > j } U _ { j ^ { \\prime } } \\prod _ { j < j ^ { \\prime \\prime } < j ^ { \\prime } } \\left ( 1 - U _ { j ^ { \\prime \\prime } } \\right ) \\\\ & = \\sum _ { j ^ { \\prime } } U _ { j ^ { \\prime } } \\sum _ { j < j ^ { \\prime } } h _ j \\prod _ { j < j ^ { \\prime \\prime } < j ^ { \\prime } } \\left ( 1 - U _ { j ^ { \\prime \\prime } } \\right ) = \\sum _ j U _ j V _ j . \\end{align*}"}
-{"id": "7373.png", "formula": "\\begin{align*} L _ { \\omega } \\psi = ( i _ { X ^ a } \\omega ) . \\nabla _ { X _ a } \\psi + \\frac { p } { 2 ( p + 1 ) } d \\omega . \\psi + \\frac { p } { 2 ( n - p + 1 ) } \\delta \\omega . \\psi \\end{align*}"}
-{"id": "1621.png", "formula": "\\begin{align*} \\mu ( X \\setminus \\cup _ i R _ i ) = 0 , \\quad \\quad \\mu ( R _ i \\cap R _ j ) = 0 \\ ; \\ ; . \\end{align*}"}
-{"id": "10095.png", "formula": "\\begin{align*} \\tilde { R } ( X , Y ) Z = P ( X , Y ) Z , \\end{align*}"}
-{"id": "7568.png", "formula": "\\begin{gather*} u = 1 2 ( 4 \\pm \\sqrt 6 ) \\frac { y ^ 2 + ( 1 8 \\pm 8 \\sqrt 6 ) t } { ( y ^ 2 + 1 0 \\lambda _ \\pm t ) ^ 2 } x + | t | ^ { \\nu + 3 / 2 } \\exp \\left ( - \\frac { y ^ 2 } { 4 t } \\right ) \\frac { C _ 1 y + C _ 2 | t | ^ { 1 / 2 } } { ( y ^ 2 + 1 0 \\lambda _ \\pm t ) ^ 2 } \\\\ \\phantom { u = } \\times \\mathop { \\rm H e u n C } \\left ( \\frac 5 2 \\lambda _ \\pm , - \\frac 1 2 , - 5 , \\frac 5 8 \\lambda _ \\pm ( 4 \\nu + 1 ) , - \\frac 5 2 \\lambda _ \\pm \\nu - \\frac { 5 9 } 8 \\mp \\frac { 2 9 } 8 \\sqrt 6 , \\frac { - y ^ 2 } { 1 0 \\lambda _ \\pm t } \\right ) , \\end{gather*}"}
-{"id": "7667.png", "formula": "\\begin{align*} & ( \\phi _ i ' , \\phi _ j ' ) + 4 e _ i e _ j ( \\widetilde \\phi _ n , \\widetilde \\phi _ n ) \\\\ & = ( \\psi _ i , \\psi _ j ) + d _ i ( \\psi _ { i _ 0 } , \\psi _ j ) + d _ j ( \\psi _ { i _ 0 } , \\psi _ i ) + d _ i d _ j ( \\psi _ { i _ 0 } , \\psi _ { i _ 0 } ) \\end{align*}"}
-{"id": "4524.png", "formula": "\\begin{align*} | \\theta ( q ( t _ \\omega ) ) | < 1 = \\widehat \\omega ( q ( t _ \\omega ) ) \\end{align*}"}
-{"id": "8047.png", "formula": "\\begin{align*} \\xi _ t = \\cos t \\cdot \\xi _ 0 + \\sin t \\cdot \\eta _ 0 \\end{align*}"}
-{"id": "1953.png", "formula": "\\begin{align*} f _ m ( x ) : = R _ k ( ( q R _ k ) ^ { m - 1 } ( q ( \\cdot ) e ^ { i k \\theta \\cdot ( \\cdot ) } ) ) ( x ) . \\end{align*}"}
-{"id": "4094.png", "formula": "\\begin{align*} \\lambda _ 1 ( \\cos ^ 2 \\theta _ 0 + \\cos ^ 2 \\theta _ 1 ) + \\lambda _ 2 ( \\sin ^ 2 \\theta _ 0 + \\sin ^ 2 \\theta _ 1 ) = \\lambda _ 1 + \\lambda _ 2 , \\end{align*}"}
-{"id": "4522.png", "formula": "\\begin{align*} | \\chi ( t _ \\omega ) | < 1 = \\omega ( t _ \\omega ) \\end{align*}"}
-{"id": "3766.png", "formula": "\\begin{align*} \\chi ^ g _ \\sigma ( \\omega , U ) : = \\frac { 1 } { L } \\sum _ { i = n } ^ { n + L - 1 } g _ { ( \\sigma ( i ) , i ) } ( \\omega , U ) . \\end{align*}"}
-{"id": "4001.png", "formula": "\\begin{align*} p ^ { \\nu _ { m + 1 } } _ { 1 } ( m + 1 , t ) & = I _ t ^ { \\nu _ { m + 1 } } \\left ( - \\lambda _ { m + 1 } p ^ { \\nu _ { m + 1 } } _ { 0 } ( m + 1 , t ) + \\lambda _ { m } p ^ { \\nu _ { m } } _ { 0 } ( m , t ) \\right ) = 0 , \\\\ p ^ { \\nu _ { m + 1 } } _ { 2 } ( m + 1 , t ) & = I _ t ^ { \\nu _ { m + 1 } } \\left ( - \\lambda _ { m + 1 } p ^ { \\nu _ { m + 1 } } _ { 1 } ( m + 1 , t ) + \\lambda _ { m } p ^ { \\nu _ { m } } _ { 1 } ( m , t ) \\right ) = 0 . \\end{align*}"}
-{"id": "7998.png", "formula": "\\begin{align*} \\Vert T _ { [ a ] } f \\Vert _ { F _ { p } ^ { s , q } } \\gtrsim \\sum _ { k = 1 } ^ { \\infty } { 2 ^ { - s k } | v _ k | } \\end{align*}"}
-{"id": "1855.png", "formula": "\\begin{align*} \\mathcal { G F } _ { \\mathcal B } ( R ) \\cap \\mathcal { G C } _ { \\mathcal B } ( R ) = \\mathcal { F } ( R ) \\cap \\mathcal { C } ( R ) \\end{align*}"}
-{"id": "6584.png", "formula": "\\begin{align*} f _ n = \\psi _ n \\circ f \\circ \\varphi _ n ^ { - 1 } \\colon \\varphi _ n ( U _ n \\cap f ^ { - 1 } ( V _ n ) ) \\longrightarrow \\psi _ n ( f ( U _ n ) \\cap V _ n ) \\ . \\end{align*}"}
-{"id": "7779.png", "formula": "\\begin{align*} D _ { R } : = \\{ ( x _ { 1 } , x _ { 2 } ) \\in \\mathbb { R } ^ { 2 } \\ , | \\ , ( x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } ) ^ { 1 / 2 } < R \\ , \\} . \\end{align*}"}
-{"id": "1302.png", "formula": "\\begin{align*} 2 \\sqrt { 6 } \\cos \\frac { 1 1 \\pi } { 3 6 } \\ + \\ 6 \\cos \\frac { 1 0 \\pi } { 3 6 } \\ - \\ \\left ( 3 \\sqrt { 2 } + \\sqrt { 6 } \\right ) \\cos \\frac { \\pi } { 3 6 } & = 0 \\\\ [ 1 . 2 e x ] ( \\sqrt { 3 } - \\sqrt { 7 } ) \\cos \\frac { \\pi } { 4 2 } \\ - \\ 2 \\sqrt { 7 } \\cos \\frac { 2 5 \\pi } { 4 2 } \\ - \\ 8 \\cos \\frac { \\pi } { 4 2 } \\cos \\frac { 2 5 \\pi } { 4 2 } \\ & = \\ 3 . \\end{align*}"}
-{"id": "9502.png", "formula": "\\begin{align*} \\tilde { Q } ^ { ( 2 ) } ( z , t ) \\cdot A ( z , t ) = t \\ , z \\ , \\left [ 2 \\ , S ( z , t ) - Q ^ { ( 2 ) } _ 2 ( t ) \\right ] . \\end{align*}"}
-{"id": "4248.png", "formula": "\\begin{align*} h _ { i j } ( \\theta \\otimes _ \\Phi 1 ) ( \\Phi ^ * e _ { i , \\cdot } ) = ( \\Phi ^ * e _ { i , \\cdot } ) \\cdot \\mathcal G ^ \\Delta _ { i j } \\end{align*}"}
-{"id": "9756.png", "formula": "\\begin{align*} Z _ A ^ { ( k - 1 ) } = Z _ { A , 0 } \\exp ( - \\frac { 1 } { \\rho _ { A , 0 } u _ { A , 0 } A ( 0 ) } \\int _ 0 ^ x A ( \\tau ) \\rho _ A ^ { ( k - 1 ) } \\phi ( T _ A ^ { ( k - 1 ) } ) d \\tau ) . \\end{align*}"}
-{"id": "4802.png", "formula": "\\begin{align*} x _ { 1 } = & \\frac { z _ 7 z _ 2 } { z _ 2 + z _ 1 } , x _ { 2 } = \\frac { z _ 1 z _ 7 } { z _ 2 + z _ 1 } , x _ { 3 } = \\frac { z _ 1 z _ 2 z _ 7 + ( z _ 1 + z _ 2 ) z _ 8 } { z _ 4 \\left ( z _ 2 + z _ 1 \\right ) } , \\\\ x _ { 4 } = & \\frac { z _ 3 \\left ( z _ 1 z _ 2 z _ 7 + ( z _ 1 + z _ 2 ) z _ 8 \\right ) } { z _ 4 \\left ( z _ 2 + z _ 1 \\right ) z _ 5 } , x _ { 5 } = \\frac { 2 \\ , z _ 1 z _ 2 z _ 3 z _ 7 + ( z _ 1 + z _ 2 ) ( 2 \\ , z _ 3 z _ 8 + z _ 4 z _ 9 ) } { z _ 6 z _ 4 \\left ( z _ 2 + z _ 1 \\right ) } . \\end{align*}"}
-{"id": "1610.png", "formula": "\\begin{align*} \\operatorname { O b j } ( \\Omega _ k ) = \\N ^ k , \\operatorname { M o r } ( \\Omega _ k ) = \\{ ( p , q ) \\in \\N ^ k \\times \\N ^ k \\ , : \\ , p \\le q \\} . \\end{align*}"}
-{"id": "4056.png", "formula": "\\begin{align*} \\mathbf { I } _ \\alpha ( \\eta ) = \\mathbf { E } _ \\alpha ( \\eta ) . \\end{align*}"}
-{"id": "1623.png", "formula": "\\begin{align*} D _ v = \\cup _ { \\lambda \\in v \\Lambda ^ m } R _ \\lambda \\end{align*}"}
-{"id": "9812.png", "formula": "\\begin{align*} f ( T _ { e _ 2 } ) = 0 \\textrm { o r , e q u i v a l e n t l y , } p _ { 0 } ( T _ { e _ 2 } ) = \\frac { \\alpha } { \\alpha + \\beta _ 2 } . \\end{align*}"}
-{"id": "7597.png", "formula": "\\begin{align*} W ' = \\begin{bmatrix} \\begin{matrix} z _ 1 & z _ 2 \\\\ z _ 2 & - z _ 1 \\end{matrix} & 0 \\\\ 0 & 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "2555.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i = 1 } ^ { \\infty } \\sum _ { j = 1 } ^ { 2 ^ { i - 1 } } f _ { \\delta _ { i } } ( x - \\mu ^ { i } _ { j } ) \\ , , \\end{align*}"}
-{"id": "8606.png", "formula": "\\begin{align*} \\langle \\Phi , \\Psi \\rangle ( \\delta ) : = \\sum _ { v \\in V } m ( v ) ^ { - 1 } \\langle \\Phi ( v ) , \\Psi ( v \\delta ) \\rangle . \\end{align*}"}
-{"id": "6195.png", "formula": "\\begin{align*} \\phi ( t ) = \\dd t ^ 1 \\wedge \\dd t ^ 2 \\wedge \\dd t ^ 3 - \\dd t ^ 1 \\wedge \\omega _ 1 ( t ) - \\dd t ^ 2 \\wedge \\omega _ 2 ( t ) - \\dd t ^ 3 \\wedge \\omega _ 3 ( t ) \\end{align*}"}
-{"id": "7075.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l c l } - \\triangle u & = & \\beta u & & B ^ N \\\\ \\frac { \\partial u } { \\partial \\nu } & = & 0 & & S ^ { N - 1 } . \\end{array} \\right . \\end{align*}"}
-{"id": "7003.png", "formula": "\\begin{align*} \\mu _ 0 = \\mu _ 0 ( \\eta _ 0 , n , s , L , S C ) \\end{align*}"}
-{"id": "3990.png", "formula": "\\begin{align*} p ^ { \\beta _ { m + 1 } } _ { 1 } ( m + 1 , t ) & = - \\lambda \\left ( I _ t ^ { \\beta _ { m + 1 } } p ^ { \\beta _ { m + 1 } } _ { 0 } ( m + 1 , t ) - I _ t ^ { \\beta _ { m } } p ^ { \\beta _ { m } } _ { 0 } ( m , t ) \\right ) = 0 , \\\\ p ^ { \\beta _ { m + 1 } } _ { 2 } ( m + 1 , t ) & = - \\lambda \\left ( I _ t ^ { \\beta _ { m + 1 } } p ^ { \\beta _ { m + 1 } } _ { 1 } ( m + 1 , t ) - I _ t ^ { \\beta _ { m } } p ^ { \\beta _ { m } } _ { 1 } ( m , t ) \\right ) = 0 . \\end{align*}"}
-{"id": "6615.png", "formula": "\\begin{align*} \\mu \\left \\{ x : \\left | \\varphi _ n ( x ) \\right | > 0 \\right \\} = \\mu \\left \\{ x : \\left | \\mathrm { i d } _ { \\mathbb { R } ^ d } - D \\phi _ n ^ { - 1 } ( f _ 0 ^ { - 1 } ( x ) ) \\right | > 0 \\right \\} \\leq \\mu ( f _ 0 ( E _ n ^ 2 ) ) \\end{align*}"}
-{"id": "95.png", "formula": "\\begin{align*} ( \\Gamma \\otimes \\Gamma _ { \\rm o p } ) ( \\chi ) = \\hat { \\chi } , \\ \\ \\mbox { m . a . e . o n } ( G \\times G ) \\times ( G \\times G ) . \\end{align*}"}
-{"id": "4163.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { k \\in N } e _ { k , g } ^ n + \\frac { 1 } { n } e ^ { n - 1 } _ { i , g + 1 } + \\frac { 1 } { n } \\left ( 1 - \\frac { 1 } { n } \\right ) e _ { i , g + 1 } ^ { n - 2 } + \\frac { 1 } { n } \\sum _ { \\tau = 1 } ^ { n - 3 } e _ { i , g + 1 + \\tau } ^ { n - \\tau - 2 } . \\end{align*}"}
-{"id": "7104.png", "formula": "\\begin{align*} f _ * ( x _ 1 , \\cdots , x _ n ) = f ( x _ 1 ^ { - 1 } , \\cdots , x _ n ^ { - 1 } ) ^ { - 1 } \\end{align*}"}
-{"id": "9943.png", "formula": "\\begin{align*} u ^ c ( x , t ) \\sim ( x - x _ c ) ^ { \\frac 2 { n - 1 } } H ( x - x _ c ) \\textrm { a s } x \\rightarrow x _ c . \\end{align*}"}
-{"id": "8264.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { ( 5 ) } L ^ s \\tilde { w } ( s ) \\frac { L ' ( s - i r _ 1 - i r _ 2 , \\xi \\chi _ 1 \\chi _ 2 ^ { - 1 } ) } { L ( s - i r _ 1 - i r _ 2 , \\xi \\chi _ 1 \\chi _ 2 ^ { - 1 } ) } d s = \\frac { 1 } { 2 \\pi i } \\int _ { \\mathcal { C } _ 1 } L ^ s \\tilde { w } ( s ) \\frac { L ' ( s - i r _ 1 - i r _ 2 , \\xi \\chi _ 1 \\chi _ 2 ^ { - 1 } ) } { L ( s - i r _ 1 - i r _ 2 , \\xi \\chi _ 1 \\chi _ 2 ^ { - 1 } ) } d s . \\end{align*}"}
-{"id": "6562.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\frac { | P | _ n - | P _ { \\delta } | _ n } { \\delta \\mathrm { l n } ( \\delta ) ^ { n - 1 } } = \\frac { \\operatorname { f l } _ { n } ( P ) } { n ! n ^ { n - 1 } } , \\end{align*}"}
-{"id": "8423.png", "formula": "\\begin{align*} \\frac { L ( s , \\abs { \\cdot } ^ { \\frac { 1 } { 2 } } ) } { L ( 1 - s , \\abs { \\cdot } ^ { \\frac { 1 } { 2 } } ) } = - q ^ { - \\frac { 3 } { 2 } + s } + \\zeta _ F ( 2 ) ^ { - 1 } \\sum _ { a = 0 } ^ { \\infty } q ^ { - \\frac { a } { 2 } - a s } . \\end{align*}"}
-{"id": "983.png", "formula": "\\begin{align*} \\left \\| \\sup _ { t \\in [ 0 , T ] } \\left | \\frac { M _ n ( t ) } { \\sigma ^ 2 ( t ) \\mathfrak { s } _ n ( t ) } - F _ n ( t ) \\right | \\right \\| _ { \\psi _ 1 } = O \\left ( h ^ \\gamma ( \\log n ) ^ 2 \\right ) \\end{align*}"}
-{"id": "5138.png", "formula": "\\begin{align*} u = z _ 1 + z _ 3 + z _ 5 + z _ 7 + v \\end{align*}"}
-{"id": "2143.png", "formula": "\\begin{align*} \\partial _ t ^ j u _ * \\in C ^ { 2 , 1 } ( [ { \\bf R } ^ N \\setminus \\{ 0 \\} ] \\times ( 0 , \\infty ) ) , j = 0 , 1 , 2 , \\dots . \\end{align*}"}
-{"id": "3924.png", "formula": "\\begin{align*} T ^ * \\cap \\theta ( T ^ * ) = \\{ \\} \\end{align*}"}
-{"id": "4103.png", "formula": "\\begin{align*} \\mathrm { h e s s } _ h f ( X , Y ) : = X ( Y f ) - ( \\bar { \\nabla } _ X Y ) f , \\end{align*}"}
-{"id": "5883.png", "formula": "\\begin{align*} \\begin{aligned} & P ( \\frac { S ^ { ( i ) } _ { \\tau ^ { N , i } , N + \\tau ^ { N , i } } } N < r _ i ) \\approx e ^ { - N \\big ( I _ i ( r _ i ) - I _ i ( r _ { i + 1 } ) \\big ) ^ + } ; \\\\ & P ( \\frac { S ^ { ( i ) } _ { \\tau ^ { N , i } , N + \\tau ^ { N , i } } } N \\ge r _ { i + 1 } ) \\approx e ^ { - N \\big ( I _ i ( r _ { i + 1 } ) - I _ i ( r _ i ) \\big ) ^ + } . \\end{aligned} \\end{align*}"}
-{"id": "7084.png", "formula": "\\begin{align*} f ( a , b ) : = \\sum _ { n = - \\infty } ^ { \\infty } a ^ { n ( n + 1 ) / 2 } b ^ { n ( n - 1 ) / 2 } , \\end{align*}"}
-{"id": "2539.png", "formula": "\\begin{align*} \\nabla _ { x } ^ k \\mathbb { G } _ { L ; 0 } ( x , t ) = \\sum _ { j = 1 } ^ { 3 } \\int _ { | \\eta | < \\delta } \\eta ^ k e ^ { i \\eta x + \\sigma ( \\eta ) t } \\vert e _ j ( \\eta ) \\rangle \\langle e _ j ( \\eta ) \\vert d \\eta . \\end{align*}"}
-{"id": "4526.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| \\pi ( \\Delta _ n ) ^ * \\Pi ( a ) \\pi ( \\Delta _ n ) \\| = 0 \\end{align*}"}
-{"id": "3920.png", "formula": "\\begin{align*} \\begin{aligned} 3 ^ { l _ 1 + 1 } x ^ { r + 6 } & + 3 a ( r ) x ^ { 2 1 } + 3 b ( r ) x ^ { 3 9 } + 3 a ( r ) ^ 2 x ^ { 3 3 } + 2 \\cdot 3 a ( r ) b ( r ) x ^ { 5 1 } + 3 b ( r ) ^ 2 x ^ { 6 9 } \\\\ & + 3 \\cdot 3 ^ { 2 l _ 1 } x ^ { 2 r + 3 } + 2 \\cdot 3 ^ { l _ 1 + 1 } a ( r ) x ^ { 1 8 + r } + 2 \\cdot 3 ^ { l _ 1 + 1 } b ( r ) x ^ { 3 6 + r } \\in \\langle F \\rangle . \\end{aligned} \\end{align*}"}
-{"id": "1109.png", "formula": "\\begin{align*} \\zeta _ i = \\frac { \\mu P _ t } { N } \\sum _ { j \\in \\mathcal { M } _ i } \\frac { | \\alpha _ { i j } | ^ 4 } { | \\alpha _ { i j } | ^ 2 \\sigma _ { Z _ { s , j } ^ { ( i ) } } ^ 2 + \\frac { N } { \\mu P _ t } \\sigma _ { Z _ { 0 , j } ^ { ( i ) } } ^ 2 } , \\end{align*}"}
-{"id": "1381.png", "formula": "\\begin{align*} U _ { a , b } = U _ { b - 1 } \\circ U _ { b - 2 } \\circ \\cdots \\circ U _ { a } , a < b , U _ { a , a } ( x ) \\equiv x . \\end{align*}"}
-{"id": "8386.png", "formula": "\\begin{align*} f ( \\tau ) = \\sum _ { \\substack { m \\in \\Q \\\\ \\mu \\in V _ \\Z ^ \\vee / V _ \\Z } } c ( m , \\mu ) \\cdot q ^ m \\in M ^ ! _ { 1 - \\frac { n } { 2 } } ( \\overline { \\rho } _ { V _ \\Z } ) . \\end{align*}"}
-{"id": "102.png", "formula": "\\begin{align*} \\bigcap _ { m = 1 } ^ \\infty \\{ f : X \\to \\mathbb { R } ^ n | \\ , \\} . \\end{align*}"}
-{"id": "1625.png", "formula": "\\begin{align*} \\sigma _ \\lambda ( x ) = \\lambda x , \\end{align*}"}
-{"id": "8953.png", "formula": "\\begin{gather*} { \\cal H } _ { \\langle s _ i \\rangle } ( X ) { \\cal H } _ { \\langle s _ j \\rangle } ( X ) \\cdots = { \\cal H } _ { \\langle s _ j \\rangle } ( X ) { \\cal H } _ { \\langle s _ i \\rangle } ( X ) \\cdots \\end{gather*}"}
-{"id": "7190.png", "formula": "\\begin{align*} z _ \\infty = a _ 0 h + \\left ( \\sum _ { i = 1 } ^ { n - 2 } a _ i x _ i \\right ) \\left ( \\sqrt { x _ { n - 1 } ^ 2 + x _ n ^ 2 } + x _ { n - 1 } \\right ) ^ s , \\end{align*}"}
-{"id": "6164.png", "formula": "\\begin{align*} \\omega _ i ( 0 ) = \\omega ^ 0 _ i - \\dd I _ i \\dd \\phi _ i ( 0 ) \\end{align*}"}
-{"id": "2237.png", "formula": "\\begin{align*} J _ \\gamma ( t ) = \\sum _ { K \\in \\Re } ( - t ) ^ { | | K | | + n } \\sum _ J \\frac { ( - 1 ) ^ { s ( J ) } } { \\beta ( K , J ) ! } \\cdot \\frac { \\partial ^ { | | \\beta ( K , J ) | | } } { \\partial w ^ \\beta ( K , J ) } \\left [ \\widetilde \\Delta \\cdot w _ 1 ^ { \\gamma _ 1 + 1 } \\cdot \\ldots \\cdot w _ n ^ { \\gamma _ n + 1 } \\cdot \\frac { \\widetilde Q ^ K } { \\widetilde q ^ { K + I } ( J ) } \\right ] _ { w = a _ J } , \\ \\end{align*}"}
-{"id": "4402.png", "formula": "\\begin{align*} \\| A P _ \\alpha | _ { \\mathbb { C } ^ n \\setminus B ( 0 , s ) } \\| = \\| A P _ \\alpha - A P _ \\alpha M _ { \\chi _ { B ( 0 , s ) } } \\| > \\sup _ { x \\in \\mathcal { M } \\setminus \\mathbb { C } ^ n } \\| A _ x P _ \\alpha \\| + \\varepsilon \\end{align*}"}
-{"id": "281.png", "formula": "\\begin{align*} D _ { 0 _ { + } } ^ { \\alpha } x ( t ) = A x ( t ) + Q \\left ( t \\right ) x ( t ) + g ( t ) , \\end{align*}"}
-{"id": "8757.png", "formula": "\\begin{align*} V ( \\psi - R _ E ^ * w _ \\psi ) = \\phi ( E ) w _ \\psi . \\end{align*}"}
-{"id": "6441.png", "formula": "\\begin{align*} p _ { } ^ { \\prime } \\left ( x _ { 1 } , \\ldots , x _ { l } \\right ) = p _ { l } \\left ( x _ { l } | x _ { 1 } , \\ldots , x _ { l - 1 } \\right ) p _ { l - 1 } \\left ( x _ { l - 1 } | x _ { 1 } , \\ldots , x _ { l - 2 } \\right ) \\cdots p _ { 2 } \\left ( x _ { 2 } | x _ { 1 } \\right ) p _ { 1 } \\left ( x _ { 1 } \\right ) \\end{align*}"}
-{"id": "5876.png", "formula": "\\begin{align*} \\begin{aligned} & I _ i ( \\mu _ i ) = 0 ; \\\\ & I _ i : [ \\mu _ i , x _ i ^ + ) \\to [ 0 , \\infty ) \\ ; \\\\ & I _ i : ( x _ i ^ - , \\mu _ i ] \\to [ 0 , \\infty ) \\ . \\end{aligned} \\end{align*}"}
-{"id": "7350.png", "formula": "\\begin{align*} ( + \\infty , b _ i ) * \\Big ( q , \\frac { b _ i r } { b _ i - r } \\Big ) \\ ; = \\ ; ( q , r ) , i \\in \\{ 1 , 2 \\} \\ , . \\end{align*}"}
-{"id": "3549.png", "formula": "\\begin{align*} \\sum _ { \\boldsymbol { t } \\in \\Lambda } \\delta _ { \\boldsymbol { t } } ( U ) = \\sum _ { \\boldsymbol { t } \\in B _ { \\boldsymbol { t } _ 0 } \\cap \\Lambda } \\delta _ { \\boldsymbol { t } } ( U ) . \\end{align*}"}
-{"id": "5097.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\mathop { \\textup { c a r d } } ( F _ n \\triangle ( k + F _ n ) ) } { \\mathop { \\textup { c a r d } } F _ n } = 0 . \\end{align*}"}
-{"id": "506.png", "formula": "\\begin{align*} N _ 0 ( J , t ) = \\left \\{ ( 0 , 0 , \\ldots , 0 ) , ( 1 , 0 , \\ldots , 0 ) , ( 2 , 0 , \\ldots , 0 ) , \\ldots , ( 0 , 0 , \\ldots , 1 ) , ( 0 , 0 , \\ldots , 2 ) \\right \\} . \\end{align*}"}
-{"id": "3193.png", "formula": "\\begin{align*} \\sup _ { y \\mathbb { \\in R } _ { \\geqslant 0 } } \\left \\vert ( \\mathcal { D } V ) ( y ) \\right \\vert = \\sup _ { y \\mathbb { \\in R } _ { \\geqslant 0 } } \\left \\vert ( a - b y ) ( 1 + y ) ^ { - 1 } - \\frac { \\sigma ^ { 2 } } { 2 } y ( 1 + y ) ^ { - 2 } \\right \\vert < \\infty . \\end{align*}"}
-{"id": "5723.png", "formula": "\\begin{align*} F ' ( m _ 0 ) = 2 ( z ' ( \\cdot - m _ 0 ) \\ , , \\ , ( { v } - z ( \\cdot - m _ 0 ) ) ) _ { L ^ 2 } = 0 . \\end{align*}"}
-{"id": "4557.png", "formula": "\\begin{align*} & [ E _ { i , k + 1 } , E _ { i + 1 , l - 1 } ] _ { q ^ { - 2 } } - ( q _ 1 + q _ 3 ) [ E _ { i , k } , E _ { i + 1 , l } ] + q _ 1 q _ 3 [ E _ { i , k - 1 } , E _ { i + 1 , l + 1 } ] _ { q ^ { 2 } } = 0 \\ , , \\\\ & [ F _ { i + 1 , l - 1 } , F _ { i , k + 1 } ] _ { q ^ { - 2 } } - ( q _ 1 + q _ 3 ) [ F _ { i + 1 , l } , F _ { i , k } ] + q _ 1 q _ 3 [ F _ { i + 1 , l + 1 } , F _ { i , k - 1 } ] _ { q ^ { 2 } } = 0 \\ , . \\end{align*}"}
-{"id": "1700.png", "formula": "\\begin{align*} \\overline { f _ \\lambda ( \\tau _ \\lambda ( x ) ) } \\Phi _ \\lambda ( x ) = \\big ( f _ \\lambda ( \\tau _ \\lambda ( x ) ) \\big ) ^ { - 1 } . \\end{align*}"}
-{"id": "7154.png", "formula": "\\begin{align*} \\sum _ { m = r J + 1 } ^ { ( r + 1 ) J } d _ G ^ * ( m ) \\leq J g ^ * _ r + \\frac { U J ^ 2 } { V _ r } . \\end{align*}"}
-{"id": "7477.png", "formula": "\\begin{align*} \\mathcal { G } = h _ { \\alpha \\bar { \\beta } } \\mathcal { Z } ^ \\alpha \\otimes \\bar { \\mathcal { Z } } ^ \\beta + h _ { \\alpha \\bar { \\beta } } \\delta \\mathcal { V } ^ \\alpha \\otimes \\delta \\bar { \\mathcal { V } } ^ \\beta . \\end{align*}"}
-{"id": "4448.png", "formula": "\\begin{align*} & \\partial _ 1 u _ T ( x _ 1 ) = \\int _ { \\R } \\partial _ 1 G _ T ( y _ 1 ) ( u ( x _ 1 - y _ 1 ) - u ( x _ 1 ) ) \\ , d y _ 1 \\\\ \\Rightarrow \\ , & \\| \\partial _ 1 u _ T \\| \\leq [ u ] _ \\alpha \\int _ \\R | y _ 1 | ^ \\alpha | \\partial _ 1 G _ T | \\ , \\ , d y _ 1 \\stackrel { \\eqref { e q : e s t G } } { \\lesssim } T ^ { \\alpha - 1 } [ u ] _ \\alpha \\end{align*}"}
-{"id": "84.png", "formula": "\\begin{align*} \\langle x , y \\rangle _ \\varphi = \\varphi ( y ^ * x ) \\end{align*}"}
-{"id": "5044.png", "formula": "\\begin{align*} F _ 1 = & \\ , 4 3 9 \\chi _ { 2 4 , 2 } ^ { ( 1 ) } + ( 1 1 4 8 4 7 + 6 5 0 \\sqrt { 1 0 6 7 0 5 } ) \\ , \\chi _ { 2 4 , 2 } ^ { ( 2 ) } \\\\ F _ 2 = & \\ , 4 3 9 \\chi _ { 2 4 , 2 } ^ { ( 1 ) } + ( 1 1 4 8 4 7 - 6 5 0 \\sqrt { 1 0 6 7 0 5 } ) \\ , \\chi _ { 2 4 , 2 } ^ { ( 2 ) } \\end{align*}"}
-{"id": "6677.png", "formula": "\\begin{align*} | \\mathrm { c o n v } [ \\Delta e _ n + ( 1 - \\eta ) \\sqrt { 2 \\Delta } \\mathcal { E } , \\zeta - \\Delta e _ n ] | _ n = ( 1 - \\eta ) ^ { n - 1 } \\frac { ( 2 \\Delta ) ^ { \\frac { n + 1 } { 2 } } } { n } | \\mathcal { E } | _ { n - 1 } \\end{align*}"}
-{"id": "4967.png", "formula": "\\begin{align*} \\mathcal { Z } ( \\R ^ d ) = \\{ f \\in \\mathcal { S } ( \\R ^ d ) : \\partial ^ \\gamma \\hat { f } ( 0 ) = 0 , \\ , \\forall \\gamma \\in \\N ^ d \\} . \\end{align*}"}
-{"id": "513.png", "formula": "\\begin{align*} \\mathbf { H } = \\begin{bmatrix} \\mathbf { H } _ 1 \\\\ [ 0 . 3 e m ] \\mathbf { H } _ 2 \\end{bmatrix} \\end{align*}"}
-{"id": "4891.png", "formula": "\\begin{align*} \\mathrm { r a n k } \\ , P T P ^ \\perp = \\mathrm { r a n k } \\ , P ^ \\perp T P . \\end{align*}"}
-{"id": "2962.png", "formula": "\\begin{align*} \\mathbf { y } _ { } & = \\sum _ { j = 1 } ^ { K } \\mathbf { h } _ { j } x _ j + \\mathbf { w } _ { } + \\mathbf { n } _ { } , \\end{align*}"}
-{"id": "3821.png", "formula": "\\begin{align*} N ^ { B , \\xi } ( x , t ) : = \\sum _ { z \\notin B } \\ ; \\sum _ { 1 \\le i \\le \\eta ( z ) } \\mathbf { 1 } _ { \\{ S ^ { z , i } _ t = x \\} } + \\sum _ { z \\in B } \\ ; \\sum _ { \\xi ( z ) < i \\le \\eta ( z ) } \\mathbf { 1 } _ { \\{ S ^ { z , i } _ t = x \\} } . \\end{align*}"}
-{"id": "7040.png", "formula": "\\begin{align*} C _ 4 = C _ 4 ( n , s , L , S C , \\eta _ 0 ) = \\frac { C } { 2 } = \\frac { 1 } { 2 } \\sqrt { ( 1 - \\frac { \\mu _ 0 } { 2 \\mu _ 1 } ) ^ { \\frac { - 2 } { n + 2 s } } - 1 } . \\end{align*}"}
-{"id": "1851.png", "formula": "\\begin{align*} F ^ 0 : = \\bigoplus _ { m \\in \\mathbb { Z } } D ^ { m + 1 } ( F _ m ) \\end{align*}"}
-{"id": "6198.png", "formula": "\\begin{align*} \\begin{cases} \\chi ( \\mathcal { O } _ X , \\kappa ) = \\chi ( \\mathcal { O } _ X ( 1 ) , \\kappa ) = \\chi ( \\mathcal { U } ^ * _ X , \\kappa ) = \\chi ( \\mathcal { U } ^ * _ X ( 1 ) , \\kappa ) = 0 \\\\ \\chi ( \\lambda _ 1 , \\kappa ) = \\chi ( \\lambda _ 2 , \\kappa ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "1399.png", "formula": "\\begin{align*} \\alpha ' ( \\alpha '' f ^ { * } ( v ' ) - f ^ { * } ( v '' ) ) = 0 . \\end{align*}"}
-{"id": "1355.png", "formula": "\\begin{align*} \\| Q _ { b } ^ T [ H ] Q _ { b } \\| _ * = \\langle Q _ { b } ^ T [ H ] Q _ { b } , { \\rm D i a g } \\ , ( w _ { b } ) \\rangle . \\end{align*}"}
-{"id": "8463.png", "formula": "\\begin{align*} \\sharp S ' \\leq \\begin{cases} 2 q ^ { \\frac { 2 l + t } { 2 } } & \\frac { 2 l + t } { 2 } < \\lfloor \\frac { r - 1 } { 2 } \\rfloor , \\\\ 2 q ^ { \\lfloor \\frac { r - 1 } { 2 } \\rfloor } & \\frac { 2 l + t } { 2 } \\geq \\lfloor \\frac { r - 1 } { 2 } \\rfloor , \\\\ 0 & , \\end{cases} \\end{align*}"}
-{"id": "8189.png", "formula": "\\begin{align*} g _ { \\nu } ( k _ { \\nu } ) = \\min \\left ( T _ { \\nu } ^ { \\frac { 1 } { 6 } } , \\abs { \\frac { a T _ { \\nu } } { y _ { \\nu } k _ { \\nu } } } ^ { \\frac { 1 } { 4 } } \\right ) \\end{align*}"}
-{"id": "6144.png", "formula": "\\begin{align*} g : = [ G / 1 ] \\quad \\textrm { a n d } \\quad s : = \\sum _ { H \\le G , | H | = p } [ G / H ] \\ , , \\end{align*}"}
-{"id": "5962.png", "formula": "\\begin{align*} \\vect { e } _ j ' ( t ) = - \\mu _ j ( t ) \\vect { e } ( t ) \\qquad ( j = 2 , \\dots , n ) . \\end{align*}"}
-{"id": "5023.png", "formula": "\\begin{align*} & ( I - \\lambda C _ p G ^ { \\omega , \\lambda } _ { p , p , p } ( z ) ) ( I + \\lambda C _ p G ^ \\omega _ { p , p } ( z ) ) = I , \\\\ & G ^ { \\omega , \\lambda } _ { p , n , m } ( z ) = G ^ \\omega _ { n , m } ( z ) - \\lambda G ^ \\omega _ { n , p } ( z ) C _ p G ^ \\omega _ { p , m } ( z ) \\\\ & \\qquad \\qquad \\qquad \\qquad + \\lambda ^ 2 G ^ \\omega _ { n , p } ( z ) C _ p G ^ { \\omega , \\lambda } _ { p , p , p } ( z ) C _ p G ^ \\omega _ { p , m } ( z ) . \\end{align*}"}
-{"id": "1319.png", "formula": "\\begin{align*} 1 = \\sum _ { v \\in W } P _ v + \\sum _ { v \\in V \\setminus W } P _ v = \\sum _ { v \\in W } S _ { e _ v } ^ * S _ { e _ v } + \\sum _ { v \\in V \\setminus W } \\sum _ { r ( e ) = v } S _ e S _ e ^ * . \\end{align*}"}
-{"id": "5914.png", "formula": "\\begin{align*} \\begin{aligned} & i _ 0 = 0 , i _ 1 = 1 , i _ 2 = 2 , \\cdots , i _ j = j , i _ { j + 1 } = j - 1 , i _ { j + 2 } = j , i _ { j + 3 } = j + 1 , \\cdots , i _ { j + k + 2 } = j + k , \\\\ & i _ { j + k + 3 } = j + k - 1 , \\ 2 \\le j \\le l \\ \\ j + k \\le l , \\end{aligned} \\end{align*}"}
-{"id": "4246.png", "formula": "\\begin{align*} \\omega _ { \\nabla , i } = \\frac { \\mathrm { d } \\Phi _ i } { p } ( \\omega _ { \\theta , i } \\otimes _ \\Phi 1 ) . \\end{align*}"}
-{"id": "7209.png", "formula": "\\begin{align*} v _ k ( x ) = \\lim _ { r \\rightarrow 0 } \\frac { u ( r x ) } { r } \\end{align*}"}
-{"id": "5673.png", "formula": "\\begin{align*} - f '' + f | \\sigma ' | ^ 2 + \\nabla W ( f \\sigma ) \\cdot \\sigma = 0 J : = \\{ f > E \\} . \\end{align*}"}
-{"id": "5979.png", "formula": "\\begin{align*} \\alpha \\beta = \\gamma . \\end{align*}"}
-{"id": "5074.png", "formula": "\\begin{align*} & E _ 2 ( \\frac { B _ { 1 3 , 1 } } { b _ 1 - b _ 3 } ) - \\frac { B _ { 1 2 , 1 } } { b _ 1 - b _ 2 } [ \\frac { B _ { 1 3 , 1 } } { b _ 1 - b _ 3 } - \\frac { B _ { 2 3 , 2 } } { b _ 2 - b _ 3 } ] = 0 , \\\\ & E _ 3 ( \\frac { B _ { 1 3 , 1 } } { b _ 1 - b _ 3 } ) - [ ( \\frac { B _ { 1 3 , 1 } } { b _ 1 - b _ 3 } ) ^ 2 - \\frac { B _ { 1 2 , 1 } B _ { 2 3 , 3 } } { ( b _ 1 - b _ 2 ) ( b _ 2 - b _ 3 ) } ] = R _ { 1 3 1 3 } = b _ 1 b _ 3 + a _ 1 + a _ 3 . \\end{align*}"}
-{"id": "4986.png", "formula": "\\begin{align*} \\hat { \\Delta } ( \\xi ) \\hat { K } ( \\xi ) = \\hat { \\Delta } ( \\xi ) . \\end{align*}"}
-{"id": "5882.png", "formula": "\\begin{align*} \\tau ^ { N , i } = \\inf \\big \\{ n \\ge 0 : \\frac { S ^ { ( i ) } _ { n , N + n } } N \\not \\in [ r _ i , r _ { i + 1 } ) \\} . \\end{align*}"}
-{"id": "1133.png", "formula": "\\begin{align*} b _ p = \\frac { 2 - a - a ^ { - 1 } } { i - j } , b _ m = \\frac { - 2 - a - a ^ { - 1 } } { i - j } . \\end{align*}"}
-{"id": "7175.png", "formula": "\\begin{align*} \\delta W ^ { \\underline { 0 } } _ { 1 + s } ( 1 , \\psi ) [ \\psi ] = 0 , \\end{align*}"}
-{"id": "567.png", "formula": "\\begin{align*} \\max \\left \\{ \\left \\lfloor \\frac { \\alpha _ 1 + i a - b ( a + 1 - m ) } { b } \\right \\rfloor + \\sum _ { j = 2 } ^ m \\left \\lfloor \\frac { \\alpha _ j - i } { b } \\right \\rfloor + 1 , 0 \\right \\} . \\end{align*}"}
-{"id": "6640.png", "formula": "\\begin{align*} \\sum _ { k = m - 1 } ^ { \\infty } k ^ { 2 p } 2 ^ { - 2 k } \\ ; < \\ ; C _ 4 m ^ { 2 p } 2 ^ { - 2 m } \\ ; < \\ ; C _ 5 | z - w | ^ 2 \\left ( \\log { \\frac { 1 } { | z - w | } } \\right ) ^ { 2 p } \\ . \\end{align*}"}
-{"id": "12.png", "formula": "\\begin{align*} V _ { \\sigma } ^ C ( C _ 1 , C _ 2 ) = \\frac { 1 } { 2 \\pi \\sigma } - \\frac { 1 } { 2 \\pi \\sigma } E [ \\frac { ( X - Y ) ^ 2 + ( Z - S ) ^ 2 } { 2 \\sigma ^ 2 } ] + h _ { \\sigma ^ 4 } \\end{align*}"}
-{"id": "4247.png", "formula": "\\begin{align*} h _ { i j } ( f \\cdot \\mathrm { d } t _ { i j } \\otimes _ \\Phi 1 ) = \\Phi ( f ) \\cdot \\frac { \\Phi _ i ( t _ { i j } ) - \\Phi _ j ( t _ { i j } ) } { p } , \\end{align*}"}
-{"id": "6330.png", "formula": "\\begin{align*} P _ 0 \\ , \\exp ( U ) - I = ( P _ 0 - I ) \\exp ( U ) + I - \\exp ( U ) \\in L ^ { \\frac { n q _ 1 } { n - q _ 1 s } } ( \\R ^ n , \\R ^ { N \\times N } ) . \\end{align*}"}
-{"id": "4733.png", "formula": "\\begin{align*} \\alpha + \\beta a _ 0 + \\gamma b _ 0 = 0 , \\alpha , \\beta , \\gamma - c o n s t , \\end{align*}"}
-{"id": "3691.png", "formula": "\\begin{align*} d _ { I , j } = \\min ( \\{ \\langle v _ { I , j } , u \\rangle \\ ; | \\ ; u \\in P _ W [ n ] \\} \\cup \\{ 0 \\} ) . \\end{align*}"}
-{"id": "5129.png", "formula": "\\begin{align*} \\phi ^ { \\omega } : = \\sum _ { n \\in \\mathbb { Z } ^ 3 } g _ n ( \\o ) \\psi ( D - n ) \\phi , \\end{align*}"}
-{"id": "3261.png", "formula": "\\begin{align*} \\omega = u ^ { n - p } \\omega ^ { p } \\end{align*}"}
-{"id": "3566.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\frac { \\Lambda ( n ) } { n ^ s } \\sum _ { 1 < d \\mid p - 1 } \\frac { \\mu ( d ) } { \\varphi ( d ) } \\sum _ { ( \\chi ) = d } \\chi ( n ) = O \\left ( \\frac { 2 ^ { \\omega ( p - 1 ) } } { x ^ { \\sigma - 1 } } \\right ) . \\end{align*}"}
-{"id": "5745.png", "formula": "\\begin{align*} \\frac { \\Phi _ { x , y - 1 , z } ( \\textbf { a } ) } { \\Phi _ { x , y - 1 , z - 1 } ( \\textbf { a } ) } \\frac { \\Phi _ { x , y , z - 1 } ( \\textbf { a } ) } { \\Phi _ { x , y , z } ( \\textbf { a } ) } + \\frac { \\Phi _ { x + 1 , y - 1 , z - 1 } ( \\textbf { a } ) } { \\Phi _ { x , y - 1 , z - 1 } ( \\textbf { a } ) } \\frac { \\Phi _ { x - 1 , y , z } ( \\textbf { a } ) } { \\phi _ { x , y , z } ( \\textbf { a } ) } = 1 . \\end{align*}"}
-{"id": "3899.png", "formula": "\\begin{align*} { \\cal E } = - ( \\frac { 2 } { \\beta + 2 } ) ^ 2 K \\end{align*}"}
-{"id": "9195.png", "formula": "\\begin{align*} \\mathcal { W } ( L \\downarrow \\mathfrak { k } ) = \\mathcal { W } ( L \\downarrow \\mathfrak { g } ) \\downarrow \\mathfrak { k } \\subseteq \\Theta _ { n } \\downarrow \\mathfrak { k } = \\Gamma _ { \\mathfrak { g } } \\downarrow \\mathfrak { k } = \\Gamma _ { \\mathfrak { k } } = B C _ { r } \\cup \\{ 0 \\} . \\end{align*}"}
-{"id": "6889.png", "formula": "\\begin{align*} S G = \\bigcup _ { 0 \\leq j , k \\leq 2 } F _ { j k } ( S G ) \\end{align*}"}
-{"id": "9793.png", "formula": "\\begin{align*} p _ { r \\bar r } = - \\frac { 1 } { 2 } \\sum _ { r < l < \\bar r } t _ { r l } t _ { r \\ , \\bar l } \\end{align*}"}
-{"id": "7029.png", "formula": "\\begin{align*} g ( \\epsilon _ 1 ) ^ { 2 s } = ( \\frac { 1 } { 4 \\epsilon _ 1 } + \\frac { n - 2 } { 2 ( n - 1 ) } \\epsilon _ 1 ) ^ { - s } , \\end{align*}"}
-{"id": "7159.png", "formula": "\\begin{align*} & = V C \\left ( \\sum _ { n \\neq a ^ { * } } \\mathbb { E } [ S _ n ] + \\mathbb { E } [ S _ { a ^ { * } } ] \\right ) \\\\ & \\leq V C \\left ( 2 \\sum _ { n \\neq a ^ { * } } \\mathbb { E } [ S _ n ] + 1 \\right ) \\leq V C \\left ( 2 \\sum _ { n \\neq a ^ { * } } \\mathbb { E } [ \\theta _ { n , K } ] + 1 \\right ) \\\\ & \\leq V C \\left ( 2 \\sum _ { n \\neq a ^ { * } } \\left [ \\frac { 8 \\ln K } { \\delta ^ 2 _ n } + 1 + \\frac { \\pi ^ 2 } { 3 } \\right ] + 1 \\right ) . \\end{align*}"}
-{"id": "5266.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c } 2 & 1 \\\\ 1 & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "7020.png", "formula": "\\begin{align*} J _ 2 = g ( \\epsilon ) ^ { - 2 s } C _ 3 \\int _ { \\mathbb { R } ^ { n - 1 } } { \\frac { u ( \\bar { z } , 0 ) - u ( 0 ) } { | \\bar { z } | ^ { n + 2 s - 1 } } d \\bar { z } } , \\end{align*}"}
-{"id": "5160.png", "formula": "\\begin{align*} K _ { 1 } ( y ) & = [ 0 , y \\wedge a ] , \\\\ K _ { 2 } ( x ) & = [ x \\vee b , 1 ] , \\end{align*}"}
-{"id": "7653.png", "formula": "\\begin{align*} \\alpha _ { k l } + \\alpha ^ { \\ast } _ { l k } = 0 . \\end{align*}"}
-{"id": "1534.png", "formula": "\\begin{gather*} A \\ = \\ [ 0 , 1 ] _ \\mathbb { Z } \\cup [ 3 , 5 ] _ \\mathbb { Z } \\cup [ 7 , 7 ] _ \\mathbb { Z } \\cup [ 9 , 1 0 ] _ \\mathbb { Z } . \\end{gather*}"}
-{"id": "2070.png", "formula": "\\begin{align*} f ( p , 0 ) = \\phi ( p ) \\end{align*}"}
-{"id": "1917.png", "formula": "\\begin{align*} \\Vert ( v _ { s } , v _ { u } ) \\Vert _ { \\star } = \\max \\{ \\Vert v _ { s } \\Vert _ { \\ast } , \\Vert v _ { u } \\Vert _ { \\ast } \\} , ( v _ { s } , v _ { u } ) \\in E _ { p } ^ { s } \\oplus E _ { p } ^ { u } , p \\in \\mathcal { M } . \\end{align*}"}
-{"id": "2218.png", "formula": "\\begin{align*} \\widetilde Q _ 1 = w _ 1 ^ { \\deg _ { w _ 1 } Q _ 1 } \\cdot w _ 2 ^ { m _ { 1 2 } } \\cdot \\ldots \\cdot w _ n ^ { m _ { 1 n } } \\cdot Q _ 1 \\left ( \\dfrac 1 { w _ 1 } , \\ldots \\dfrac 1 { w _ n } \\right ) . \\end{align*}"}
-{"id": "8021.png", "formula": "\\begin{align*} \\big \\Vert f _ L \\big \\Vert _ { F _ p ^ { { s } , q } } \\lesssim \\Big [ \\int _ { \\mathbb { R } ^ d } { \\Big ( \\sum _ { n = M } ^ { L } { \\big ( \\mathfrak { M } _ { { \\sigma } , 2 ^ { t _ n } } { g _ { { t _ n } } } ( x ) \\big ) ^ q } \\Big ) ^ { { p } / { q } } } d x \\Big ] ^ { { 1 } / { p } } , \\end{align*}"}
-{"id": "4653.png", "formula": "\\begin{align*} | I _ k | = | I _ { k , 1 } | + | I _ { k , 2 } | + \\cdots + | I _ { k , b } | \\ge | I _ { k , j } | + \\frac { b - 1 } { D _ 2 } | I _ { k , j } | = \\left ( 1 + \\frac { b - 1 } { D _ 2 } \\right ) | I _ { k , j } | \\end{align*}"}
-{"id": "2632.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\frac { \\log | t - s | } { \\phi _ 0 ( t ) } \\ , u ( s ) \\ , d s \\ , d t = \\int _ 0 ^ 1 \\bigg ( \\int _ 0 ^ 1 \\frac { \\log | t - s | } { \\phi _ 0 ( t ) } \\ , d t \\bigg ) \\ , u ( s ) \\ , d s . \\end{align*}"}
-{"id": "5116.png", "formula": "\\begin{align*} f ( t ) - f ( t _ 0 ) = \\sum _ { 0 < | \\alpha | \\leq \\ell } \\frac { ( t - t _ 0 ) ^ \\alpha } { \\alpha ! } \\partial _ t ^ \\alpha f ( t _ 0 ) + \\sum _ { | \\beta | = \\ell + 1 } \\frac { ( t - t _ 0 ) ^ \\beta } { \\beta ! } R ^ \\beta _ { t _ 0 } ( t ) , \\end{align*}"}
-{"id": "8306.png", "formula": "\\begin{align*} I H & = \\mathrm { S p a n } _ \\Q \\{ \\ell x : \\ell \\in I , \\ , x \\in H \\} \\\\ I ^ 2 H & = \\mathrm { S p a n } _ \\Q \\{ \\ell \\ell ' x : \\ell , \\ell ' \\in I , \\ , x \\in H \\} . \\end{align*}"}
-{"id": "2854.png", "formula": "\\begin{align*} \\hat { a } _ { 0 0 } = \\bar { a } _ { 0 0 } = a _ { 0 0 } , \\hat { a } _ { n v } = \\bar { \\Delta } \\bar { a } _ { n v } , n = 1 , 2 , . . . \\end{align*}"}
-{"id": "3743.png", "formula": "\\begin{align*} p _ \\bullet ( k ) : = \\inf _ { \\ell \\ge k } \\alpha ( \\ell , x _ \\bullet ) , k \\in \\Z _ + . \\end{align*}"}
-{"id": "5588.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } w _ { 3 } \\left ( t \\right ) \\\\ w _ { 2 } \\left ( t \\right ) \\\\ w _ { 1 } \\left ( t \\right ) \\\\ w _ { 0 } \\left ( t \\right ) \\end{array} \\right ] = \\left [ \\begin{array} { c c c c } 0 & 0 & 0 & 1 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 1 & 0 & 0 & 0 \\end{array} \\right ] \\left [ \\begin{array} { c } w _ { 0 } \\left ( t \\right ) \\\\ w _ { 1 } \\left ( t \\right ) \\\\ w _ { 2 } \\left ( t \\right ) \\\\ w _ { 3 } \\left ( t \\right ) \\end{array} \\right ] \\end{align*}"}
-{"id": "5105.png", "formula": "\\begin{align*} g _ \\mathcal { M } ( \\tau ) & = \\sum _ { k \\in \\mathcal { M } } e ( k _ 0 + k - k _ 0 , \\tau ) = e ( k _ 0 , \\tau ) \\sum _ { k \\in \\mathcal { M } } e ( k - k _ 0 , \\tau ) \\\\ & = e ( k _ 0 , \\tau ) \\sum _ { m \\in \\mathcal { M } - k _ 0 = \\mathcal { M } } e ( m , \\tau ) = \\underbrace { e ( k _ 0 , \\tau ) } _ { \\neq 1 } g _ \\mathcal { M } ( \\tau ) , \\end{align*}"}
-{"id": "8391.png", "formula": "\\begin{align*} \\frac { \\partial \\varphi } { \\partial t } ( x , t ) = \\textbf { H } ( x , t ) , \\end{align*}"}
-{"id": "9666.png", "formula": "\\begin{align*} \\phi ( T ) = T ^ { \\mu } e ^ { - \\mathcal { E } / R T } , \\end{align*}"}
-{"id": "1357.png", "formula": "\\begin{align*} \\psi ^ * ( Y ) = 2 \\displaystyle \\sum _ { i \\ne s } \\displaystyle \\frac { 1 } { \\varpi _ i } \\langle Q ^ T _ { b _ s } Y Q _ { b _ s } - I _ { | b _ s | } , Q _ { b _ s } ^ T H Q _ { a _ i } Q _ { a _ i } ^ T H Q _ { b _ s } \\rangle . \\end{align*}"}
-{"id": "6384.png", "formula": "\\begin{align*} I ( x ) : = \\inf \\{ \\| H y \\| _ 2 ^ 2 / 2 \\mid ( D \\varphi _ k ^ { [ a , b ] } ( \\vec { y } ^ * ) ) y = x \\} = \\| H D \\varphi _ k ^ { [ a , b ] } ( \\vec { y } ^ * ) ^ { - 1 } x \\| _ 2 ^ 2 / 2 . \\end{align*}"}
-{"id": "357.png", "formula": "\\begin{align*} \\frac { 2 | A | ^ 2 - H ^ 2 } { H ^ 2 } \\in O ( r ^ { - a \\sigma } ) = O \\left ( r ^ { 6 a - \\frac { 8 a ^ 2 } { b } } \\right ) . \\end{align*}"}
-{"id": "8142.png", "formula": "\\begin{align*} \\varphi ( b ) = h \\psi ( b ) \\end{align*}"}
-{"id": "4457.png", "formula": "\\begin{align*} \\| f _ T \\| & \\le \\Big ( \\int _ { \\R ^ 2 } | \\psi _ \\frac { T } { 2 } | ^ \\frac { p } { p - 1 } \\Big ) ^ \\frac { p - 1 } { p } \\Big ( \\int _ { [ 0 , 1 ) ^ 2 } | f _ \\frac { T } { 2 } | ^ p \\Big ) ^ \\frac { 1 } { p } \\\\ \\stackrel { ( \\ref { f 2 5 } ) } { = } & \\big ( ( T ^ \\frac { 1 } { 3 } ) ^ \\frac { 5 } { 2 } \\big ) ^ { - \\frac { 1 } { p } } \\Big ( \\int _ { \\R ^ 2 } | \\psi | ^ \\frac { p } { p - 1 } \\Big ) ^ \\frac { p - 1 } { p } \\Big ( \\int _ { [ 0 , 1 ) ^ 2 } | f _ \\frac { T } { 2 } | ^ p \\Big ) ^ \\frac { 1 } { p } , \\end{align*}"}
-{"id": "7631.png", "formula": "\\begin{align*} \\liminf _ { k \\to \\infty } \\int _ { Q _ 2 } | D w ^ k - D v ^ k | ^ p \\ , d z = 0 \\end{align*}"}
-{"id": "1407.png", "formula": "\\begin{align*} \\sigma ( m _ 0 , \\dots , m _ { n - 1 } ) = ( m _ 0 , \\dots , m _ { n - 1 } , m _ { n - 1 } ) . \\end{align*}"}
-{"id": "2527.png", "formula": "\\begin{align*} T _ { 1 1 2 3 } = - T _ { 1 1 3 } , T _ { 1 1 2 4 } = - T _ { 1 1 4 } , \\end{align*}"}
-{"id": "1397.png", "formula": "\\begin{align*} j ( \\alpha '' v ' - v '' ) = \\sum _ { I , J } \\dfrac { \\alpha '' \\beta _ { I J } } { \\alpha _ { I J } } v _ I y _ J \\end{align*}"}
-{"id": "2017.png", "formula": "\\begin{align*} y _ { a , i } = x _ i x _ { i + k _ a } + \\sum _ { j \\in \\Z \\setminus \\{ 0 , - k _ a \\} } u _ { a , j } x _ { i - j } x _ { i + k _ a + j } \\end{align*}"}
-{"id": "8063.png", "formula": "\\begin{align*} | a _ { 1 , 1 } | & = \\frac { m ( 2 , 3 ) } { m ( 1 , 2 ) } = \\frac { 5 } { 2 } \\\\ | a _ { 2 , 1 } | & = \\frac { m ( 1 , 3 ) } { m ( 1 , 2 ) } = \\frac { 3 } { 2 } \\end{align*}"}
-{"id": "5244.png", "formula": "\\begin{align*} \\tau = \\max \\{ \\tau _ { \\lambda _ 1 } , \\dots , \\tau _ { \\lambda _ k } \\} . \\end{align*}"}
-{"id": "794.png", "formula": "\\begin{align*} \\rho = \\begin{cases} 9 . 6 & \\mbox { i f } x = 2 , 3 , 6 , 7 \\\\ 9 . 3 & \\mbox { i f } x = 1 0 , 1 1 . \\\\ \\end{cases} \\end{align*}"}
-{"id": "2545.png", "formula": "\\begin{align*} \\rho \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ \\end{pmatrix} + \\begin{pmatrix} a _ 1 & 0 & 0 \\\\ 0 & a _ 2 & 0 \\\\ 0 & 0 & a _ 3 \\\\ \\end{pmatrix} + i \\eta \\mathrm { A } _ { i j } ( \\eta , \\rho ) , \\end{align*}"}
-{"id": "5176.png", "formula": "\\begin{align*} V _ { 2 } ^ { [ 0 , \\ell ] } ( x ) \\coloneqq \\sup \\limits _ { \\tau _ { 2 } \\in \\mathcal { T } } M ^ { x } _ { 2 } ( D _ { [ 0 , \\ell ] } , \\tau _ { 2 } ) = M ^ { x } _ { 2 } ( D _ { [ 0 , \\ell ] } , D _ { [ r _ { \\ell } , 1 ] } ) , \\forall x \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "1627.png", "formula": "\\begin{align*} \\Phi _ { \\sigma _ \\lambda } ( x ) = \\rho ( \\Lambda ) ^ { - d ( \\lambda ) } . \\end{align*}"}
-{"id": "5008.png", "formula": "\\begin{align*} \\forall \\xi \\in \\R ^ d , \\psi ( \\xi ) = \\sum _ { | \\gamma | = m + 1 } \\xi ^ \\gamma \\left ( \\frac { 1 } { m ! } \\int _ { 0 } ^ { 1 } ( 1 - t ) ^ m \\partial _ { \\gamma } \\psi ( t \\xi ) \\ , d t \\right ) . \\end{align*}"}
-{"id": "7265.png", "formula": "\\begin{align*} B _ 2 & \\le \\sum _ { \\substack { 0 < | h | < y \\\\ ( m , n ) \\mid h } } ( m , n ) \\min \\Bigl ( \\frac { 2 s } { N n } , \\frac s N \\Bigl ( \\frac 1 n + \\frac 1 m \\Bigr ) - \\frac { | h | } { m n } \\Bigr ) \\\\ & = \\sum _ { 0 < | t | < y / ( m , n ) } ( m , n ) \\min \\Bigl ( \\frac { 2 s } { N n } , \\frac s N \\Bigl ( \\frac 1 n + \\frac 1 m \\Bigr ) - \\frac { ( m , n ) | t | } { m n } \\Bigr ) . \\end{align*}"}
-{"id": "7539.png", "formula": "\\begin{gather*} \\varphi ^ 1 _ \\omega = \\varphi ^ 2 _ \\omega = 0 . \\end{gather*}"}
-{"id": "9027.png", "formula": "\\begin{align*} R _ { a b } { } ^ c { } _ d = \\delta _ a { } ^ c \\Phi _ { b d } - \\delta _ b { } ^ c \\Phi _ { a d } + J _ { a d } \\Phi _ { b e } J ^ { c e } - J _ { b d } \\Phi _ { a e } J ^ { c e } + 2 J _ { a b } \\Phi _ { d e } J ^ { c e } , \\end{align*}"}
-{"id": "4608.png", "formula": "\\begin{align*} G _ 3 ( B ) = \\{ ( f , ( x , t ) ) \\in \\mathcal { C } _ d \\times X _ B : ( x , t ) \\textrm { i s g e n e r i c f o r a n } S _ { f , B } \\textrm { i n v a r i a n t p r o b a b i l i t y m e a s u r e o n } X _ B \\} \\end{align*}"}
-{"id": "2495.png", "formula": "\\begin{align*} \\varrho _ { 0 } \\left ( - \\left | \\eta \\right | \\right ) & = \\varrho _ { 1 } \\left ( \\left | \\eta \\right | \\right ) , \\\\ \\varrho _ { j } \\left ( - \\left | \\eta \\right | \\right ) & = \\varrho _ { j } \\left ( \\left | \\eta \\right | \\right ) , \\mbox { f o r } j = 2 , 3 , 4 , \\end{align*}"}
-{"id": "8977.png", "formula": "\\begin{gather*} \\Gamma { \\cal H } ^ { ( 1 ) } _ { \\eta ' , x _ 1 , \\dots , x _ m ; q } ( v , w ) \\cong \\begin{pmatrix} \\Gamma { \\cal S } ^ { ( 1 ) } _ { \\eta ' , x _ 1 , \\dots , x _ m ; q } ( v , w ) & \\Gamma { \\cal S } ^ { ( 1 ) } _ { \\eta ' , x _ 1 , \\dots , x _ m ; q } ( v - 2 f , w ) \\\\ \\Gamma { \\cal S } ^ { ( 1 ) } _ { \\eta ' , x _ 1 , \\dots , x _ m ; q } ( v , w - 2 f ) & \\Gamma { \\cal S } ^ { ( 1 ) } _ { \\eta ' , x _ 1 , \\dots , x _ m ; q } ( v - 2 f , w - 2 f ) \\end{pmatrix} \\ ! , \\end{gather*}"}
-{"id": "3016.png", "formula": "\\begin{align*} \\Re \\langle A x , x \\rangle & = \\Phi _ 1 ( x ) ^ * Q \\Phi _ 1 ( x ) - \\Phi _ 0 ( x ) ^ * Q \\Phi _ 0 ( x ) + \\Re \\langle P _ 0 x , x \\rangle . \\end{align*}"}
-{"id": "3792.png", "formula": "\\begin{align*} \\Omega = \\Big \\{ \\omega = \\sum _ i \\delta _ { w _ i } ; \\ ; w _ i \\in W \\omega ( W _ { \\{ y \\} } ) < \\infty y \\in \\Z ^ d \\times \\Z \\Big \\} , \\end{align*}"}
-{"id": "1494.png", "formula": "\\begin{align*} \\R ( x ) = \\min \\{ m \\ge 1 : T ^ m x \\in M \\} \\end{align*}"}
-{"id": "168.png", "formula": "\\begin{align*} ( \\mathfrak { X } \\setminus \\mathfrak { Y } ) ( a ) = \\max \\bigl \\{ \\mathfrak { X } ( a ) - \\mathfrak { Y } ( a ) , 0 \\bigr \\} , a \\in A , \\end{align*}"}
-{"id": "3539.png", "formula": "\\begin{align*} u ^ { n , \\theta , \\rho } ( t ) = \\left \\{ \\begin{array} [ c ] { c l } { u } ^ { n , \\theta } ( t ) , & t \\in \\lbrack 0 , T ] \\backslash \\ E _ { \\rho } , \\\\ u , & t \\in E _ { \\rho } , \\end{array} \\right . \\end{align*}"}
-{"id": "8662.png", "formula": "\\begin{align*} \\ell ( u _ 2 , x , y , X _ 0 , Y _ 0 ) = \\int _ 0 ^ \\infty 1 _ { \\{ | x - y + \\omega _ { X _ 0 } ( u _ 1 ) - \\omega _ { Y _ 0 } ( u _ 1 + u _ 2 ) | \\leq 1 \\} } d u _ 1 \\end{align*}"}
-{"id": "7977.png", "formula": "\\begin{align*} \\cos \\alpha _ s ^ 0 & = \\frac { a _ 2 ^ 2 + ( a _ 1 ( s ) / s ) ^ 2 - a _ 3 ^ 2 } { 2 a _ 2 ( a _ 1 ( s ) / s ) } \\\\ \\cos \\alpha _ t ^ 0 & = \\frac { a _ 2 ^ 2 + ( a _ 1 ( t ) / t ) ^ 2 - a _ 3 ^ 2 } { 2 a _ 2 ( a _ 1 ( t ) / t ) } , \\end{align*}"}
-{"id": "5343.png", "formula": "\\begin{align*} Z ^ { \\infty } = \\bigcup _ { N \\in \\Z _ { + } } Z ^ N . \\end{align*}"}
-{"id": "7212.png", "formula": "\\begin{align*} W _ { A C } ( v _ 0 ; y , s ) = \\lim _ { s \\searrow 0 } W _ { A C } ( v _ r ; y , s ) = \\lim _ { s \\searrow 0 } W _ { A C } ( v ; x + r y , s r ) , \\end{align*}"}
-{"id": "9481.png", "formula": "\\begin{align*} \\sum _ { s = 0 } ^ { N } \\frac { q ^ { \\binom { N - s + 1 } { 2 } } ( q ; q ) _ { N + s } } { ( q ^ 2 ; q ^ 2 ) _ s } = ( q ^ 2 ; q ^ 2 ) _ N . \\end{align*}"}
-{"id": "1116.png", "formula": "\\begin{align*} R ~ = ~ R _ 0 ~ \\subset ~ R _ 1 ~ \\subset \\cdots \\subset ~ R _ { h } ~ \\subset ~ R _ { h + 1 } ~ = ~ V \\end{align*}"}
-{"id": "9530.png", "formula": "\\begin{align*} u ^ m _ r f ^ m _ r = m ^ 2 u ^ { m - 1 } u _ r f ^ { m - 1 } f _ r \\quad \\end{align*}"}
-{"id": "88.png", "formula": "\\begin{align*} \\Gamma ( a ) ( s , t ) = a ( s t ) , \\mbox { f o r a l m o s t a l l } ( s , t ) \\in G \\times G , \\ \\ a \\in L ^ { \\infty } ( G ) . \\end{align*}"}
-{"id": "694.png", "formula": "\\begin{gather*} \\mu _ n ( q , \\alpha , \\beta ) = \\mu _ n ( q _ 0 , \\alpha _ 0 , \\beta _ 0 ) , \\\\ \\kappa _ n ( q , \\alpha , \\beta ) = \\kappa _ n ( q _ 0 , \\alpha _ 0 , \\beta _ 0 ) , \\end{gather*}"}
-{"id": "6153.png", "formula": "\\begin{align*} \\partial _ t \\phi = \\Delta _ \\phi \\phi \\end{align*}"}
-{"id": "2858.png", "formula": "\\begin{align*} n X _ { n } \\beta _ { n } & = O ( 1 ) , \\\\ \\sum _ { n = 1 } ^ { \\infty } \\beta _ { n } X _ n & < \\infty . \\end{align*}"}
-{"id": "9089.png", "formula": "\\begin{align*} d ^ { 0 } = \\mathrm { i d } , d ^ { n } = d \\circ d ^ { n - 1 } \\left ( n \\in \\mathbb { N } \\right ) . \\end{align*}"}
-{"id": "1544.png", "formula": "\\begin{gather*} y _ 1 - 0 \\ = \\ 1 - x _ 2 \\\\ y _ 1 - 0 \\ = \\ x _ 2 - y _ 1 \\\\ 1 - x _ 2 \\ = \\ x _ 2 - y _ 1 . \\end{gather*}"}
-{"id": "243.png", "formula": "\\begin{align*} \\Lambda _ { L / 2 } ( \\sigma ) : = \\{ x \\in \\R ^ d : \\ , x _ { \\sigma } \\in [ - L , 0 ] ^ d \\} , \\sigma \\in \\mathcal { R } ^ d , \\end{align*}"}
-{"id": "7001.png", "formula": "\\begin{align*} y _ n = - z _ { n - 1 } \\sin \\theta . \\end{align*}"}
-{"id": "9857.png", "formula": "\\begin{align*} { \\rm d e g } ( X ) = \\sum \\limits _ { j = 0 } ^ r k _ X ^ { \\infty } ( X _ j ) \\cdot { \\rm d e g } ( X _ j ) . \\end{align*}"}
-{"id": "6928.png", "formula": "\\begin{align*} \\| \\Gamma _ f \\| = \\| b \\| _ { L ^ \\infty } . \\end{align*}"}
-{"id": "4373.png", "formula": "\\begin{align*} ( C _ z f ) ( w ) = f ( w - z ) e ^ { \\alpha \\langle w , z \\rangle - \\frac { \\alpha } { 2 } | z | ^ 2 } . \\end{align*}"}
-{"id": "8285.png", "formula": "\\begin{align*} \\mathrm { g r } _ k ( V ) = \\mathrm { w t } _ k V / \\mathrm { w t } _ { k - 1 } V \\end{align*}"}
-{"id": "4305.png", "formula": "\\begin{align*} \\Omega _ { 0 , 3 } ^ 0 ( u , v , w ) = \\eta ( u * v , w ) . \\end{align*}"}
-{"id": "4093.png", "formula": "\\begin{align*} k _ { M , p } ( X ) + k _ { M , p } ( Y ) = \\lambda _ 1 ( \\cos ^ 2 \\theta _ 0 + \\sin ^ 2 \\theta _ 0 ) + \\lambda _ 2 ( \\cos ^ 2 \\theta _ 0 + \\sin ^ 2 \\theta _ 0 ) = \\lambda _ 1 + \\lambda _ 2 = 2 H . \\end{align*}"}
-{"id": "5101.png", "formula": "\\begin{align*} \\mathcal { L } & : = \\bigoplus _ { i \\in \\mathbb { N } } \\mathbb { Z } _ { n _ i } = \\mathbb { Z } _ { n _ 1 } \\oplus \\mathbb { Z } _ { n _ 2 } \\oplus \\mathbb { Z } _ { n _ 3 } \\oplus \\cdots \\\\ & \\ , = \\big \\{ ( a _ j ) _ { j = 1 } ^ { \\infty } \\in { \\textstyle \\prod _ { j = 1 } ^ { \\infty } \\mathbb { Z } _ { n _ j } } : a _ j \\big \\} , \\end{align*}"}
-{"id": "8043.png", "formula": "\\begin{align*} 2 V _ 1 / 3 & \\le \\mathcal H ^ n ( M - B ( p , R _ M / 2 + \\delta ) ) \\\\ & \\le \\mathcal H ^ n ( A _ T ) \\\\ & \\le C _ 1 ( n , D , v ) ( \\mathcal H ^ { n - 1 } ( B ( \\Uparrow _ p ^ q , \\pi / 2 + \\epsilon ) ) - \\mathcal H ^ { n - 1 } ( B ( \\Uparrow _ p ^ q , \\pi / 2 - \\epsilon ) ) ) \\\\ & \\le C _ 1 ( n , D , v ) \\mathcal H ^ { n - 1 } ( \\mathbb S ^ { n - 1 } ) \\theta _ n ( \\epsilon ) \\\\ & \\le V _ 1 / 2 . \\end{align*}"}
-{"id": "7503.png", "formula": "\\begin{align*} \\| \\omega _ N - \\omega _ N ^ k \\| _ { \\mathrm { H S } } = \\sqrt { N } \\| W _ N - W _ N ^ k \\| _ { L ^ 2 } \\leq C \\sqrt { \\frac { N } { k } } \\| W _ N \\| _ { H ^ 1 } . \\end{align*}"}
-{"id": "1725.png", "formula": "\\begin{align*} S _ \\lambda ( f ) & = W ^ * t _ \\lambda W ( f ) = W ^ * t _ \\lambda \\pi ( f ) \\xi \\\\ & = W ^ * \\pi ( f \\circ \\sigma ^ n ) t _ \\lambda \\xi = W ^ * \\pi ( f \\circ \\sigma ^ n ) W W ^ * t _ \\lambda \\xi \\\\ & = ( f \\circ \\sigma ^ n ) \\cdot f _ \\lambda . \\end{align*}"}
-{"id": "9924.png", "formula": "\\begin{align*} \\partial _ t u ( t , x ) + A _ x u ( t , x ) + b ( t , x ) \\cdot \\nabla _ x u ( t , x ) = - b ( t , x ) ; \\end{align*}"}
-{"id": "4648.png", "formula": "\\begin{align*} r _ { k , \\ell } ( j ) = \\min \\{ n \\in \\mathbb { N } : ( T | I _ k ) ^ n I _ { \\ell , j } \\subset I _ \\ell \\} \\end{align*}"}
-{"id": "1736.png", "formula": "\\begin{align*} \\tilde { s } = \\sum _ i ^ { n } a _ { i } \\chi _ { { B _ i } } , n \\in \\N , B _ i \\in \\sigma ( \\mathcal { \\tilde { R } } ) , a _ i \\not = 0 . \\end{align*}"}
-{"id": "1727.png", "formula": "\\begin{align*} \\frac { d \\mu _ S } { d \\mu _ T } = | h | ^ 2 \\end{align*}"}
-{"id": "128.png", "formula": "\\begin{align*} & \\frac { \\partial C ( 1 - t , x , y ) } { \\partial x } = \\frac { \\partial C ( 1 - t , x , y ) } { \\partial y } = 0 , \\\\ & \\frac { \\partial ^ 2 } { \\partial x ^ 2 } C ( 1 - t , x , y ) = \\frac { \\partial ^ 2 } { \\partial y ^ 2 } C ( 1 - t , x , y ) = - \\frac { \\partial ^ 2 } { \\partial x \\partial y } C ( 1 - t , x , y ) \\end{align*}"}
-{"id": "707.png", "formula": "\\begin{align*} \\frac { n _ { \\rm i } n _ { \\rm e } } { n _ { \\rm a } } = \\frac { 2 Z _ { \\rm i } } { Z _ { \\rm a } } \\frac { ( 2 \\pi m _ { \\rm e } k T ) ^ { \\frac { 3 } { 2 } } } { h ^ 3 } \\ , e ^ { - \\frac { T _ { \\rm i } } { T } } . \\end{align*}"}
-{"id": "8804.png", "formula": "\\begin{align*} \\alpha _ { i , \\varpi _ p } ( \\xi , m ) = \\eta & \\Leftrightarrow \\vee _ { j \\in J } ( \\phi _ { i j } ( \\xi ) \\wedge \\eta _ { i j } ( \\eta , m ) ) ; \\\\ \\beta _ { i , \\varpi _ p } ( \\xi , m ) = \\nu & \\Leftrightarrow \\vee _ { j \\in J } ( \\theta _ { i j } ( \\xi ) \\wedge \\nu _ { i j } ( \\nu , m ) ) ; \\\\ ( \\xi , m , \\varsigma ) \\in V _ { i , \\varpi _ { p } } & \\Leftrightarrow \\vee _ { j \\in J } ( \\varsigma _ { i j } ( \\varsigma , \\xi ) \\wedge \\tau _ { i j } ( m ) ) . \\end{align*}"}
-{"id": "4057.png", "formula": "\\begin{align*} \\int _ \\Omega \\frac { | \\nabla u ( z ) | } { | x - z | ^ { d - 1 } } \\ , \\d z & = \\int _ { 0 } ^ { \\infty } \\int _ { \\{ | \\nabla u | > s \\} } | x - z | ^ { 1 - d } \\ , \\d z \\ , \\d s \\cr & \\le \\int _ { 0 } ^ { \\infty } \\bigl [ \\mu _ { | \\nabla u | } ( s ) \\bigr ] ^ { \\frac { 1 } { d } } \\ , \\d s = \\frac { 1 } { d } \\| \\nabla u \\| _ { d , 1 ( \\Omega ) } \\end{align*}"}
-{"id": "9768.png", "formula": "\\begin{align*} \\tilde { E } _ { i } ( h , \\theta ) = & \\int \\limits _ { y _ { i } + 2 n _ { i - 1 } s } ^ { y _ { i } } \\big ( W ( U _ { h , \\theta } ( i h - , y ) ) - W ( U _ { h , \\theta _ i } ( i h - , y ) ) \\big ) d y + \\int \\limits _ { y _ { i } + ( 2 n _ { i - 1 } + d _ i ) s } ^ { y _ { i } + 2 n _ { i - 1 } s } W ( U _ { h , \\theta } ( i h - , y ) ) d y . \\end{align*}"}
-{"id": "1822.png", "formula": "\\begin{align*} E ^ a _ { Q , + , h } ( \\sigma _ 0 ) & = \\mathbb { P } ^ a _ { Q , \\bar { w } , h } ( 0 \\longleftrightarrow g ) \\geq \\mathbb { P } ^ a _ { Q , \\bar { w } , h } ( 0 \\overset { a \\mathbb { Z } ^ 2 } \\longleftrightarrow \\partial _ { i n } Q ^ a ) \\\\ & \\geq \\mathbb { P } ^ a _ { Q , w , h } ( 0 \\overset { a \\mathbb { Z } ^ 2 } \\longleftrightarrow \\partial _ { i n } Q ^ a ) = \\tilde { \\mathbb { P } } ^ a _ { Q , w , h } ( 0 \\longleftrightarrow \\partial _ { i n } Q ^ a ) . \\end{align*}"}
-{"id": "613.png", "formula": "\\begin{align*} \\dim C _ W ^ n ( s ) = \\sum _ { i = 0 } ^ n ( - 1 ) ^ i \\binom { n } { i } ( \\binom { 2 ( n - i ) } { n - i - s / 2 } - \\binom { 2 ( n - i ) } { n - i - 1 - s / 2 } ) . \\end{align*}"}
-{"id": "4398.png", "formula": "\\begin{align*} \\sigma _ { e s s } ( A ) : = \\{ \\lambda \\in \\mathbb { C } ; \\ A - \\lambda \\} \\end{align*}"}
-{"id": "6906.png", "formula": "\\begin{align*} \\frac { 1 } { \\mu _ { 0 } r _ { 0 } } = \\frac { ( 1 + 2 r ) ( 1 5 + 2 6 r + 9 r ^ { 2 } ) } { 2 r ( 2 + r ) } \\end{align*}"}
-{"id": "3859.png", "formula": "\\begin{align*} \\ell ( x ^ k ) & = \\ell ( x ^ * ) + \\nabla \\ell ( x ^ * ) ^ T ( x ^ k - x ^ * ) + \\frac { 1 } { 2 } ( x ^ k - x ^ * ) ^ T \\nabla ^ 2 \\ell ( \\xi ^ k ) ( x ^ k - x ^ * ) \\end{align*}"}
-{"id": "9169.png", "formula": "\\begin{align*} f _ 4 ( x ^ 4 ) - 4 x f _ 4 ( x ^ 3 ) + 6 x ^ 2 f _ 4 ( x ^ 2 ) - 4 x ^ 3 f _ 4 ( x ) = 0 \\left ( x \\in K \\right ) \\end{align*}"}
-{"id": "9094.png", "formula": "\\begin{align*} B ( x , y ) = \\sum _ { i = 1 } ^ { n - 1 } \\binom { n } { i } d ^ { i } ( x ) d ^ { n - i } ( y ) \\left ( x , y \\in R \\right ) \\end{align*}"}
-{"id": "2183.png", "formula": "\\begin{align*} c _ d m ( \\varphi ^ { 0 , 1 } ) & = \\frac { c _ d ^ 2 } { c _ * } \\int _ 0 ^ \\infty \\varphi ^ { 0 , 1 } ( r ) U ( r ) r ^ { N - 1 } \\ , d r = \\frac { c _ d ^ 2 } { c _ * } | { \\bf S } ^ { N - 1 } | ^ { - 1 } \\int _ { { \\bf R } ^ N } \\varphi ^ { 0 , 1 } ( | x | ) U ( | x | ) \\ , d x \\\\ & = \\frac { c _ d ^ 2 } { c _ * } | { \\bf S } ^ { N - 1 } | ^ { - 1 } \\int _ { { \\bf R } ^ N } \\varphi ( x ) U ( | x | ) \\ , d x = M ( \\varphi ) . \\end{align*}"}
-{"id": "4833.png", "formula": "\\begin{align*} \\mathcal { C } ^ * ( M , D ) : = \\big \\{ [ x ] ^ D _ M \\mid x \\in ( D \\setminus D ^ \\times ) \\cup \\{ 1 \\} \\big \\} \\end{align*}"}
-{"id": "6851.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { T } t \\left ( \\frac { 1 } { \\gamma _ t } V _ { x _ t } ( x ) - \\frac { 1 } { \\gamma _ t } V _ { x _ { t + 1 } } ( x ) - \\alpha V _ { x _ t } ( x ) + \\frac { \\gamma _ t G ^ 2 } { 2 } \\right ) \\leq \\frac { G ^ 2 T } { \\alpha } - \\frac { \\alpha \\ , T ( T + 1 ) } { 2 } V _ { x _ { T + 1 } } ( x ) \\leq \\frac { G ^ 2 T } { \\alpha } . \\end{align*}"}
-{"id": "5643.png", "formula": "\\begin{align*} d _ K ( x ^ - , x ^ + ) = \\inf \\left \\{ \\mathfrak { L } _ { K } ( \\gamma ) \\ ; : \\ ; \\gamma \\in A C _ { p l o c } ( [ 0 , 1 ] , X ) \\gamma : x ^ - \\mapsto x ^ + \\mathrm { I m } ( \\gamma ) \\subset F \\right \\} . \\end{align*}"}
-{"id": "8228.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ k \\binom { k } { \\ell } ( - 1 ) ^ { \\ell + k } \\ell ^ d = \\begin{cases} \\hfil 0 & d < k \\\\ k ! & d = k \\end{cases} \\end{align*}"}
-{"id": "5621.png", "formula": "\\begin{align*} { \\lambda ' _ a } & = { \\lambda _ a } \\left ( { 1 - \\Pr \\left [ \\mathrm { n o \\ s e r v i n g \\ M T D } \\right ] } \\right ) \\\\ & = { \\lambda _ a } \\left ( { 1 - { { \\left ( { 1 + { { 3 . 5 } ^ { - 1 } } { \\lambda _ u } / { \\lambda _ a } } \\right ) } ^ { - 3 . 5 } } } \\right ) . \\end{align*}"}
-{"id": "9354.png", "formula": "\\begin{align*} j O _ { n } ^ { ( 3 ) } + j O _ { n + 1 } ^ { ( 3 ) } = 3 J O _ { n + 2 } ^ { ( 3 ) } , \\end{align*}"}
-{"id": "7783.png", "formula": "\\begin{align*} \\lambda a ( y _ h , \\eta _ h ) + b ( x _ h ; \\eta _ h ) = F ( \\eta _ h ) , b ' ( x _ h ; \\xi _ h , y _ h ) = 0 , a ( y _ h , y _ h ) = 1 . \\end{align*}"}
-{"id": "206.png", "formula": "\\begin{align*} \\mathfrak { X } \\uplus \\mathfrak { C } = f ( \\Delta ) \\uplus \\mathfrak { C } = f ' ( \\Delta ) \\uplus f ' ( \\Pi ) \\in G _ { s + 1 } . \\end{align*}"}
-{"id": "9527.png", "formula": "\\begin{align*} M ^ p ( \\Omega ) : = \\large \\{ f \\in L ^ 1 _ { l o c } ( \\Omega ) | \\ , \\ , \\exists C \\int _ K | f | \\ , d x \\le C \\ , | K | ^ { ( p - 1 ) / p } | K | < \\infty \\large \\} \\end{align*}"}
-{"id": "1149.png", "formula": "\\begin{align*} d _ 1 & = d _ 0 ^ 2 = 1 \\\\ d _ j & = \\frac { [ j ] _ x d _ { j - 1 } ^ 2 d _ { j - 2 } ^ 4 } { [ 1 ] _ x d _ { j - 2 } ^ 4 + d _ { j - 1 } ^ 2 } \\quad \\mbox { f o r $ j \\geq 2 $ . } \\end{align*}"}
-{"id": "4396.png", "formula": "\\begin{align*} \\nu ( \\hat { A } _ y ) & \\leq \\nu ( \\hat { A } _ y | _ { B ( 0 , 4 r _ { t _ k } ) } ) = \\lim _ \\gamma \\nu ( \\hat { A } _ { y _ { j _ \\gamma } } | _ { B ( 0 , 4 r _ { t _ k } ) } ) \\\\ & \\leq \\lim _ \\gamma \\nu ( \\hat { A } _ { x _ { j _ \\gamma } } ) + \\frac { 1 } { 2 ^ { k - 1 } } = \\lim _ { j \\to \\infty } \\nu ( \\hat { A } _ { x _ j } ) + \\frac { 1 } { 2 ^ { k - 1 } } . \\end{align*}"}
-{"id": "5017.png", "formula": "\\begin{align*} f ( i ) = \\left \\lceil \\frac { r _ { 0 } ( 1 + i ) } { r _ { 2 } - r _ { 0 } } \\right \\rceil - \\left \\lfloor \\frac { r _ { 0 } i } { r _ { 1 } - r _ { 0 } } \\right \\rfloor \\mbox { ; o r } \\end{align*}"}
-{"id": "2500.png", "formula": "\\begin{align*} \\beta _ { 1 0 } \\left ( \\left | \\eta \\right | \\right ) = \\beta _ { 0 1 } \\left ( - \\left | \\eta \\right | \\right ) , \\quad \\beta _ { 1 1 } \\left ( \\left | \\eta \\right | \\right ) = \\beta _ { 0 0 } \\left ( - \\left | \\eta \\right | \\right ) , \\quad \\beta _ { 1 2 } \\left ( \\left | \\eta \\right | \\right ) = \\beta _ { 0 2 } \\left ( - \\left | \\eta \\right | \\right ) . \\end{align*}"}
-{"id": "2386.png", "formula": "\\begin{align*} R _ 1 = ( 4 a _ 2 + a _ 5 ) ^ 2 + ( 3 a _ 3 + a _ 4 + a _ 6 ) ^ 2 , \\end{align*}"}
-{"id": "1510.png", "formula": "\\begin{align*} ( 1 - & x ^ { 2 } y - x y ^ { 2 } - y ^ { 3 } ) G _ { T } ( y ) \\\\ & = Q _ { T , 0 } ( x ) + ( Q _ { T , 1 } ( x ) - x ^ { 2 } Q _ { T , 0 } ( x ) ) y + ( Q _ { T , 2 } ( x ) - x ^ { 2 } Q _ { T , 1 } ( x ) - x Q _ { T , 0 } ( x ) ) y ^ { 2 } \\\\ & \\ \\ + \\sum _ { n = 3 } ^ { \\infty } ( Q _ { T , n } ( x ) - x ^ { 2 } Q _ { T , n - 1 } ( x ) - x Q _ { T , n - 2 } ( x ) - Q _ { T , n - 3 } ( x ) ) y ^ { n } . \\end{align*}"}
-{"id": "5825.png", "formula": "\\begin{align*} ( p - 2 ) d \\Delta Q + 4 Q ^ { p - 1 } - ( 4 + ( p - 2 ) ( 2 - d ) ) Q = 0 \\end{align*}"}
-{"id": "5292.png", "formula": "\\begin{align*} u _ t = { \\mu u _ { x x } + \\alpha u } - \\beta | u | ^ 2 u - \\gamma | u | ^ 4 u + \\lambda u _ x , \\end{align*}"}
-{"id": "7371.png", "formula": "\\begin{align*} L _ { \\omega } \\psi = - ( - 1 ) ^ p \\lambda p \\omega . \\psi + \\frac { p } { 2 ( p + 1 ) } d \\omega . \\psi . \\end{align*}"}
-{"id": "4033.png", "formula": "\\begin{align*} \\Delta _ b f ( p ) : = \\mathrm { t r } _ b \\left ( \\mathrm { h e s s } _ b f | _ p \\right ) , \\ \\ p \\in M . \\end{align*}"}
-{"id": "6866.png", "formula": "\\begin{align*} \\mathcal { E } ( u , v ) = - \\int f v d \\mu \\end{align*}"}
-{"id": "3225.png", "formula": "\\begin{align*} j ^ \\ast ( \\omega _ { Z \\subset X } ) = \\omega _ { W \\subset Z } . \\end{align*}"}
-{"id": "9932.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\frac { f ( r ) } { \\log ( 1 + r ) } = \\infty , \\end{align*}"}
-{"id": "1100.png", "formula": "\\begin{align*} u _ i ( r , \\phi , 0 ) = \\sqrt { \\frac { 2 } { \\pi | i | ! } } \\frac { 1 } { w _ 0 } \\left ( \\frac { \\sqrt { 2 } r } { w _ 0 } \\right ) ^ { | i | } L ^ i _ 0 \\left ( \\frac { 2 r ^ 2 } { w _ 0 ^ 2 } \\right ) \\mathrm { e x p } \\left ( \\frac { - r ^ 2 } { w _ 0 ^ 2 } \\right ) \\mathrm { e x p } \\left ( - j i \\phi \\right ) , \\end{align*}"}
-{"id": "9249.png", "formula": "\\begin{gather*} [ v \\otimes b , v ' \\otimes b ' ] = [ \\left [ \\begin{array} { c c } 0 & v \\otimes b \\\\ v ^ { t } \\otimes b & 0 \\end{array} \\right ] , \\left [ \\begin{array} { c c } 0 & v ' \\otimes b ' \\\\ v '^ { t } \\otimes b ' & 0 \\end{array} \\right ] ] = \\left [ \\begin{array} { c c } v ( v ' ) ^ { t } \\otimes b b ' - v ' v ^ { t } \\otimes b ' b & 0 \\\\ 0 & ( v ) ^ { t } v ' \\otimes [ b , b ' ] _ { A ^ { - } } \\end{array} \\right ] . \\end{gather*}"}
-{"id": "7728.png", "formula": "\\begin{align*} D ^ 2 _ B ( j ) = & \\frac { 1 } { 3 0 } ( N ^ 4 - 1 0 j ( j + 1 ) N ^ 2 \\\\ & + 1 0 j ( 2 j + 1 ) ( j + 1 ) N - 1 0 j ^ 2 ( j + 1 ) ^ 2 - 1 ) \\ , , \\\\ C _ B ( j ) = & \\textstyle \\frac { 3 0 N } { N ^ 4 - 1 0 j ( j + 1 ) N ^ 2 + 1 0 j ( 2 j + 1 ) ( j + 1 ) N - 1 0 j ^ 2 ( j + 1 ) ^ 2 - 1 } \\ , . \\end{align*}"}
-{"id": "7435.png", "formula": "\\begin{align*} x ' , \\ , y , \\ , z , \\ , t _ a : = 2 t e _ 0 , \\ , t _ b : = T ^ { \\beta } _ 0 e _ 0 , \\ , t _ c : = T ^ { \\gamma } _ 0 e _ 0 , \\ , t _ d : = T ^ { \\delta } _ 0 e _ 0 \\end{align*}"}
-{"id": "2638.png", "formula": "\\begin{align*} c ( D ) = \\sum _ { i = 1 } ^ n \\det L ( D ) _ { i , i } . \\end{align*}"}
-{"id": "9771.png", "formula": "\\begin{align*} \\tilde { E } _ { i } ( h , \\theta ) = & \\int \\limits _ { y _ { i } + 2 n _ { i - 1 } s } ^ { y _ { i } } \\big ( W ( U _ { h , \\theta } ( i h - , y ) ) - W ( U _ { h , \\theta _ i } ( i h - , y ) ) \\big ) d y - \\int \\limits _ { y _ { i } + 2 n _ { i - 1 } s } ^ { y _ { i } + ( 2 n _ { i - 1 } + d _ i ) s } W ( U _ { h , \\theta } ( i h - , y ) ) d y . \\end{align*}"}
-{"id": "8289.png", "formula": "\\begin{align*} \\Sigma _ { \\Phi _ 1 } = \\{ \\sigma \\in \\Sigma _ \\Phi : \\sigma \\subset C _ { \\Phi _ 1 } ^ * \\} . \\end{align*}"}
-{"id": "7166.png", "formula": "\\begin{align*} \\mathfrak { A } _ g : = \\{ v \\in H ^ 1 ( B _ 1 ^ + , \\mu _ a ) \\ , : \\ , v \\geq 0 \\ , \\it { o n } \\ , B ' _ 1 , v = g \\ , \\it { o n } \\ , ( \\partial { B _ 1 } ) ^ + \\} , \\end{align*}"}
-{"id": "9505.png", "formula": "\\begin{align*} \\begin{aligned} & t x ^ 3 a ^ 2 - ( x - t ) a + ( x - t ) - t x S _ 1 ( t ) = 0 , \\\\ & 2 t x ^ 3 a + t - x = 0 , \\\\ & 3 t x ^ 2 a ^ 2 - a + 1 - t S _ 1 ( t ) = 0 \\end{aligned} \\end{align*}"}
-{"id": "455.png", "formula": "\\begin{align*} h _ { k _ 1 , k _ 2 } ( R , t ) = e ^ { - \\frac { 1 } { 4 } d ( x , t ) ^ 2 } \\int _ \\R e ^ { i R \\psi _ \\omega ( \\lambda ) } a _ { k _ 1 , k _ 2 } ( \\lambda + i y _ \\omega u _ 1 ) \\ , \\dd \\lambda . \\end{align*}"}
-{"id": "4750.png", "formula": "\\begin{align*} a _ 1 = f _ 1 = 0 . \\end{align*}"}
-{"id": "3569.png", "formula": "\\begin{align*} \\frac { 1 } { x } \\sum _ { n \\leq x } \\omega ( n ) = \\log \\log n + B _ 1 + ( \\gamma - 1 ) / \\log n + O \\left ( e ^ { - c \\sqrt { \\log n } } \\right ) , \\end{align*}"}
-{"id": "7900.png", "formula": "\\begin{align*} d ( z _ t , x ) = d ( x _ t , x ) - \\delta \\end{align*}"}
-{"id": "5953.png", "formula": "\\begin{align*} - \\rho \\nu _ { u _ 1 } = f _ { u _ 1 } , \\nu _ { u _ j } = 0 , \\nu _ { u _ 1 } \\cdot f _ { u _ j } = 0 \\end{align*}"}
-{"id": "602.png", "formula": "\\begin{align*} \\# \\{ \\zeta \\in \\mathcal { R } _ { X _ { N } } : \\vert \\delta - \\zeta \\vert < \\varepsilon \\} \\geq \\left ( c _ { 0 } N \\right ) ^ { r } = c _ { 0 } ^ { r } [ \\Gamma : \\Gamma _ { N } ] , \\end{align*}"}
-{"id": "330.png", "formula": "\\begin{align*} \\alpha ^ { ( \\nu ) } _ i ( n ) : = \\alpha _ i ( n ) + \\sigma _ i ( \\lceil r _ { \\nu } n \\rceil - r _ { \\nu } n ) \\not \\in \\Z _ { \\le 0 } + | \\sigma _ i | \\ , \\Z _ { \\le - \\nu } , i \\in I _ { \\nu } , \\ , \\ , \\nu = 0 , 1 , \\ , \\ , n \\in \\N , \\end{align*}"}
-{"id": "10076.png", "formula": "\\begin{align*} 4 \\pi e ^ \\gamma \\cdot \\| x \\| ^ 2 \\cdot \\| a \\| ^ 2 = ( 2 \\pi ) ^ { - 1 } \\cdot | \\psi ( x a , \\overline { x a } ) | = \\| x a \\| ^ 2 , \\end{align*}"}
-{"id": "1749.png", "formula": "\\begin{align*} \\nu _ { y } ( Z ( \\lambda ) ) : = \\langle y , ( S ^ { u n i v } _ \\lambda ( S ^ { u n i v } _ \\lambda ) ^ * ) y \\rangle , \\end{align*}"}
-{"id": "7689.png", "formula": "\\begin{align*} P _ i = \\left [ \\begin{array} { c c } 0 & 1 \\\\ - \\lambda _ i & - \\lambda _ i \\end{array} \\right ] \\ , . \\end{align*}"}
-{"id": "8543.png", "formula": "\\begin{align*} 2 e ^ { v ( x ) } = a _ 1 \\left ( 1 - \\frac { a _ 1 - a _ 2 } { a _ 1 } \\mathrm { s n } ^ { 2 } \\left ( x \\sqrt { a _ 1 + a _ 3 } , \\frac { a _ 1 - a _ 2 } { a _ 1 + a _ 3 } \\right ) \\right ) \\end{align*}"}
-{"id": "5587.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } w _ { 2 } \\left ( t \\right ) \\\\ w _ { 3 } \\left ( t \\right ) \\\\ w _ { 0 } \\left ( t \\right ) \\\\ w _ { 1 } \\left ( t \\right ) \\end{array} \\right ] = \\left [ \\begin{array} { c c c c } 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\end{array} \\right ] \\left [ \\begin{array} { c } w _ { 0 } \\left ( t \\right ) \\\\ w _ { 1 } \\left ( t \\right ) \\\\ w _ { 2 } \\left ( t \\right ) \\\\ w _ { 3 } \\left ( t \\right ) \\end{array} \\right ] \\end{align*}"}
-{"id": "1147.png", "formula": "\\begin{align*} [ 1 ] _ x e ^ 2 ( z ) + e ^ 4 ( z ) = [ 1 ] _ x \\sum _ { i = 0 } ^ \\infty a _ i ^ 2 z ^ { 2 ^ { i + 1 } } + \\sum _ { i = 0 } ^ \\infty a _ i ^ 4 z ^ { 2 ^ { i + 2 } } . \\end{align*}"}
-{"id": "3207.png", "formula": "\\begin{align*} ( \\mathcal { D } V ) ( x ) & = ( a - b x ) V ' ( x ) + \\frac { \\sigma ^ { 2 } x } { 2 } V '' ( x ) \\\\ & = - b \\kappa x ^ { \\kappa } + \\kappa x ^ { \\kappa - 1 } \\left ( a + \\frac { \\sigma ^ { 2 } ( \\kappa - 1 ) } { 2 } \\right ) \\leqslant - b \\kappa x ^ { \\kappa } + c _ { 3 } \\end{align*}"}
-{"id": "4141.png", "formula": "\\begin{align*} \\max _ { z \\geq 0 } ~ & \\sum _ { i \\in N } \\sum _ { j \\in V } w _ { i j } z _ { i j } \\\\ & \\sum _ { j \\in V } z _ { i j } \\leq T / n = 1 , i \\in N \\\\ & \\sum _ { i \\in N } z _ { i j } \\leq 1 , j \\in V . \\end{align*}"}
-{"id": "5136.png", "formula": "\\begin{align*} u = z _ 1 + z _ 3 + z _ 5 + v . \\end{align*}"}
-{"id": "6806.png", "formula": "\\begin{align*} \\tilde { L } ( u ) = \\Delta _ { z } u + \\frac { 8 } { ( 1 + | z | ^ 2 ) ^ 2 } u , \\end{align*}"}
-{"id": "7193.png", "formula": "\\begin{align*} D _ K \\cap { I _ { K , f } K ^ \\times / K ^ \\times } = \\overline { \\mathcal { O } _ K ^ \\times } K ^ \\times / K ^ \\times = \\overline { \\mathcal { O } _ F ^ \\times } F ^ \\times / F ^ \\times . \\end{align*}"}
-{"id": "2958.png", "formula": "\\begin{align*} \\theta _ i = \\frac { e ^ { - g _ i ( \\theta _ i ) } } { \\sum _ { j = 1 } ^ n e ^ { - g _ j ( \\theta _ j ) } + C } 1 \\leq i \\leq n , \\end{align*}"}
-{"id": "3636.png", "formula": "\\begin{align*} \\frac { 1 } { \\varrho } : = \\left ( 1 + | \\nabla u | ^ 2 + | \\nabla u _ { \\lambda _ 0 + \\tau } | ^ 2 \\right ) ^ { \\frac { 2 - p } { 2 } } \\in L ^ t ( \\Omega _ { \\lambda _ 0 + \\tau } ) , \\end{align*}"}
-{"id": "2564.png", "formula": "\\begin{align*} | c | ^ { 2 } - r ^ { 2 } \\ge ( 1 - r - d _ { 0 } / 2 ) ^ { 2 } - r ^ { 2 } = 1 + d _ { 0 } ^ { 2 } / 4 - 2 r - d _ { 0 } + d _ { 0 } r \\ , , \\end{align*}"}
-{"id": "8102.png", "formula": "\\begin{align*} B _ - & = \\{ x \\in X : d ( x , X \\setminus B ) > \\eta \\} , \\\\ A _ + & = \\{ x \\in X : d ( x , A ) \\leq \\eta \\} \\end{align*}"}
-{"id": "749.png", "formula": "\\begin{align*} \\| ( u _ 1 ^ n , u _ 2 ^ n ) \\| _ { \\dot H \\times \\dot H } ^ 2 & = \\| ( v _ 1 ^ n , v _ 2 ^ n ) \\| _ { \\dot H \\times \\dot H } ^ 2 + \\| ( u _ 1 , u _ 2 ) \\| _ { \\dot H \\times \\dot H } ^ 2 + o _ n ( 1 ) , \\\\ \\| ( u _ 1 ^ n , u _ 2 ^ n ) \\| _ { q } ^ q & = \\| ( v _ 1 ^ n , v _ 2 ^ n ) \\| _ { q } ^ q + \\| ( u _ 1 , u _ 2 ) \\| _ { q } ^ q + o _ n ( 1 ) \\ \\ \\ \\ 1 \\leq q < \\infty . \\end{align*}"}
-{"id": "7199.png", "formula": "\\begin{align*} \\sum _ j \\xi _ j ( u _ j ) _ \\nu ^ 2 ( x ) = 1 . \\end{align*}"}
-{"id": "8625.png", "formula": "\\begin{align*} \\Delta B ^ i _ { u , v } = B ^ i _ v - B ^ i _ u . \\end{align*}"}
-{"id": "8981.png", "formula": "\\begin{gather*} D ^ { ( n ) } _ q ( ( d + 1 ) q / 2 \\pm u ; t ) D ^ { ( n ) } _ { q , t } ( d ) \\ ! \\ ! \\ ! \\prod _ { 1 \\le i \\le n } \\ ! \\ ! \\ ! \\vartheta ( z _ i \\pm v ) = D ^ { ( n ) } _ q ( ( d + 1 ) q / 2 \\pm v ; t ) D ^ { ( n ) } _ { q , t } ( d ) \\ ! \\ ! \\ ! \\prod _ { 1 \\le i \\le n } \\ ! \\ ! \\ ! \\vartheta ( z _ i \\pm u ) . \\end{gather*}"}
-{"id": "7526.png", "formula": "\\begin{gather*} w ^ 1 _ 1 + w ^ 1 w ^ 1 _ 2 - \\frac { ( w ^ 2 ) ^ 2 } { z _ 2 } - w ^ 1 _ { 2 2 } - \\frac { w ^ 1 _ 2 } { z _ 2 } + \\frac { w ^ 1 } { z _ 2 ^ 2 } = 0 , \\\\ w ^ 2 _ 1 + w ^ 1 w ^ 2 _ 2 + \\frac { w ^ 1 w ^ 2 } { z _ 2 } - w ^ 2 _ { 2 2 } - \\frac { w ^ 2 _ 2 } { z _ 2 } + \\frac { w ^ 2 } { z _ 2 ^ 2 } = 0 . \\end{gather*}"}
-{"id": "8659.png", "formula": "\\begin{align*} \\begin{aligned} ( \\omega _ { X _ 0 } ( \\tau _ 1 + \\cdot ) - \\omega _ { X _ 0 } ( \\tau _ 1 ) , \\omega _ { Y _ 0 } ( \\tau _ 1 + \\cdot ) - \\omega _ { Y _ 0 } ( \\tau _ 1 ) ) \\stackrel { } { = } ( \\tilde { \\omega } _ { X _ { \\tau _ 1 } } ( \\cdot ) , \\tilde { \\omega } _ { Y _ { \\tau _ 1 } } ( \\cdot ) ) , \\end{aligned} \\end{align*}"}
-{"id": "4295.png", "formula": "\\begin{align*} 3 M _ 3 = M _ 1 \\cdot D ^ 2 M _ 2 - 4 D ^ 2 M _ 1 \\cdot M _ 2 + \\sum _ { i = 1 } ^ { 8 } \\left ( D _ i D M _ 1 \\cdot 2 D _ i M _ 2 - D _ i M _ 1 \\cdot D _ i D M _ 2 \\right ) \\end{align*}"}
-{"id": "3206.png", "formula": "\\begin{align*} V ' ( x ) = \\kappa x ^ { \\kappa - 1 } \\quad V '' ( x ) = \\kappa ( \\kappa - 1 ) x ^ { \\kappa - 2 } \\quad x \\geqslant 1 , \\end{align*}"}
-{"id": "7979.png", "formula": "\\begin{align*} Y _ 1 ( t ) & = d \\exp _ { \\gamma _ 2 ( s ) } ( t ( V _ 1 ) _ { t \\dot \\sigma _ s ( 0 ) } ) , \\ , \\ , Y _ 2 ( t ) = d \\exp _ { \\gamma _ 3 ( s ) } ( t ( V _ 2 ) _ { t \\dot \\sigma _ s ^ { - } ( 0 ) } ) , \\\\ Y _ 1 ^ { N } ( t ) & = d \\exp _ { \\gamma _ 2 ( s ) } ( t ( V _ 1 ^ N ) _ { t \\dot \\sigma _ s ( 0 ) } ) , \\ , \\ , Y _ 2 ^ { N } ( t ) = d \\exp _ { \\gamma _ 3 ( s ) } ( t ( V _ 2 ^ N ) _ { t \\dot \\sigma _ s ^ { - } ( 0 ) } ) , \\end{align*}"}
-{"id": "3447.png", "formula": "\\begin{align*} [ u | _ N ] ^ n : = u ( t _ n ) \\end{align*}"}
-{"id": "625.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ { | z _ 2 | = r _ 2 } \\int _ 0 ^ { 2 \\pi } \\frac { f ( r _ 1 e ^ { i \\theta _ 1 } , z _ 2 ) \\ , d \\theta _ 1 } { z _ 2 ^ { \\alpha _ 2 + 1 } } \\ , d z _ 2 = 2 \\pi i a _ { ( 0 , \\alpha _ 2 ) } , \\end{align*}"}
-{"id": "446.png", "formula": "\\begin{align*} \\Upsilon ( x , t ) = ( - 1 ) ^ { k _ 1 + k _ 2 } \\frac { y _ \\omega ^ { n + k _ 1 + k _ 2 } \\cos ( y _ \\omega ) ^ { k _ 1 } } { \\sin ( y _ \\omega ) ^ { n + k _ 1 } } + O \\left ( \\frac { 1 } { | x | ^ 2 } \\right ) ; \\end{align*}"}
-{"id": "4297.png", "formula": "\\begin{align*} \\underline { \\eta } = \\big ( ( \\eta _ 1 , \\delta _ 1 ) , \\ldots , ( \\eta _ { l ( \\eta ) } , \\delta _ { l ( \\eta ) } ) \\big ) , \\delta _ i \\in H ^ { \\ast } ( E ) . \\end{align*}"}
-{"id": "578.png", "formula": "\\begin{align*} f ( w ) = f ( z ) + { \\displaystyle \\int _ { [ z , w ] } f ' ( \\zeta ) d \\zeta } , \\end{align*}"}
-{"id": "7676.png", "formula": "\\begin{align*} B = \\left ( \\begin{array} { @ { } c | c @ { } } B _ j & 0 \\\\ \\hline 0 & \\begin{matrix} 2 ^ { k _ j } u _ 1 & 0 & 0 \\\\ 0 & 2 ^ { k _ j + 1 } u _ 2 & 0 \\\\ 0 & 0 & \\ddots \\end{matrix} \\end{array} \\right ) \\sim \\left ( \\begin{array} { @ { } c | c @ { } } B _ j & 0 \\\\ \\hline 0 & \\begin{matrix} 2 ^ { k _ j } u _ 1 ' & 0 & 0 \\\\ 0 & 2 ^ { k _ j + 1 } u _ 2 ' & 0 \\\\ 0 & 0 & \\ddots \\end{matrix} \\end{array} \\right ) = B ' . \\end{align*}"}
-{"id": "1320.png", "formula": "\\begin{align*} 1 & = \\sum _ { v \\in W } \\rho ( S _ { e _ v } ) ^ * \\rho ( S _ { e _ v } ) + \\sum _ { v \\in V \\setminus W } \\sum _ { r ( e ) = v } \\rho ( S _ e ) \\rho ( S _ e ) ^ * \\\\ & = \\sum _ { v \\in W } \\begin{bmatrix} \\pi ( S _ { e _ v } ) ^ * \\pi ( S _ { e _ v } ) + Y _ { e _ v } ^ * Y _ { e _ v } & * \\\\ * & * \\end{bmatrix} \\\\ & \\quad \\quad + \\sum _ { v \\in V \\setminus W } \\sum _ { r ( e ) = v } \\begin{bmatrix} \\pi ( S _ e ) \\pi ( S _ e ) ^ * + X _ e X _ e ^ * & * \\\\ * & * \\end{bmatrix} , \\end{align*}"}
-{"id": "1038.png", "formula": "\\begin{align*} \\widehat { w } _ { n } ( \\zeta ) \\leq \\mathcal { E } _ { n } ( \\Psi ) : = \\frac { \\Psi + 1 - \\sqrt { \\Psi ^ { 2 } - 4 \\Psi n + 8 n ^ { 2 } + 2 \\Psi - 1 2 n + 5 } } { 2 } . \\end{align*}"}
-{"id": "3739.png", "formula": "\\begin{align*} W = \\Big \\{ w : \\Z \\to \\Z ^ d \\colon \\ , | w ( i + 1 ) - w ( i ) | \\le 1 \\ ; \\ ; \\forall \\ ; i \\in \\Z \\Big \\} . \\end{align*}"}
-{"id": "1441.png", "formula": "\\begin{align*} \\mathrm { K e r } \\ , Q _ 0 & = \\mathbb { Z } / 2 [ x _ 2 ^ 2 , x _ { 4 1 } , \\dots , x _ { 4 n } ] \\oplus \\mathbb { Z } / 2 [ x _ 2 ^ 2 , x _ 3 ^ 2 , x _ { 4 1 } , \\dots , x _ { 4 n } ] \\{ x _ 3 , x _ 3 ^ 2 \\} , \\\\ \\mathrm { I m } \\ , Q _ 0 & = \\mathbb { Z } / 2 [ x _ 2 ^ 2 , x _ 3 ^ 2 , x _ { 4 1 } , \\dots , x _ { 4 n } ] \\{ x _ 3 , x _ 3 ^ 2 \\} \\end{align*}"}
-{"id": "9344.png", "formula": "\\begin{align*} \\overline { J O _ { n } ^ { ( 3 ) } } = J _ { n } ^ { ( 3 ) } - \\sum _ { s = 1 } ^ { 7 } J _ { n + s } ^ { ( 3 ) } e _ { s } , \\ \\overline { j O _ { n } ^ { ( 3 ) } } = j _ { n } ^ { ( 3 ) } - \\sum _ { s = 1 } ^ { 7 } j _ { n + s } ^ { ( 3 ) } e _ { s } , \\end{align*}"}
-{"id": "9945.png", "formula": "\\begin{align*} u _ t + A _ 1 u ^ { m - 1 } u _ x + A _ 2 u ^ { n - 1 } u _ { 3 x } + A _ 3 u ^ { n - 2 } u _ x u _ { 2 x } + A _ 4 u ^ { n - 3 } u _ x ^ 3 = 0 \\end{align*}"}
-{"id": "4656.png", "formula": "\\begin{align*} \\upsilon \\leq \\sum _ { i = 0 } ^ { N - 1 } 1 _ { I _ k } ( T ^ i x ) \\leq | C _ { \\max } ( B _ k \\cdots B _ { k + r } ) | + \\upsilon \\end{align*}"}
-{"id": "8591.png", "formula": "\\begin{align*} \\begin{cases} A x \\equiv 0 \\ , ( m o d \\ , S _ { n \\ , n } ) \\\\ 0 \\leq x \\leq S _ { n \\ , n } - 1 \\\\ x \\in \\mathbb { Z } ^ n \\\\ \\end{cases} . \\end{align*}"}
-{"id": "4121.png", "formula": "\\begin{align*} d \\eta _ p ( X ) = \\frac { 1 } { \\rho } X , \\end{align*}"}
-{"id": "2047.png", "formula": "\\begin{align*} \\Delta u ( x ) + n u ( x ) = u ^ { p _ 0 } ( x ) f ( x ) , x \\in \\mathbb S ^ n . \\end{align*}"}
-{"id": "2375.png", "formula": "\\begin{align*} \\Phi ( s , \\lambda , w ) = \\sum _ { n = 0 } ^ \\infty \\frac { \\lambda ^ n } { ( n + w ) ^ s } \\end{align*}"}
-{"id": "9102.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } ( - 1 ) ^ i \\sum _ { \\mathrm { c a r d } ( I ) = i } \\left ( \\prod _ { j \\in I } x _ { j } \\right ) \\cdot A \\left ( \\prod _ { k \\in \\left \\{ 1 , \\ldots , n + 1 \\right \\} \\setminus I } x _ { k } \\right ) = 0 ( x _ { 1 } , \\ldots , x _ { n + 1 } \\in R ) . \\end{align*}"}
-{"id": "8185.png", "formula": "\\begin{align*} B ( R ) = \\prod _ { \\nu } \\{ \\xi _ { \\nu } \\in F _ { \\nu } \\colon \\abs { \\xi _ { \\nu } } \\leq R _ { \\nu } \\} . \\end{align*}"}
-{"id": "10061.png", "formula": "\\begin{align*} \\int _ { F _ \\mathfrak { p } } \\Phi _ { \\mu , \\mathfrak { p } } \\left ( w n ( b ) , s \\right ) \\ , d b = \\int _ { F _ \\mathfrak { p } } \\Phi ^ \\sharp _ { \\mu , \\mathfrak { p } } \\left ( w n ( b ) , s \\right ) \\ , d b . \\end{align*}"}
-{"id": "9761.png", "formula": "\\begin{align*} \\mathcal { H } ( V _ A ^ { ( n ) } , A ( x ) ) - \\mathcal { H } ( V _ A ^ { ( n - 1 ) } , A ( x ) ) = & \\mathcal { H } _ e ( V _ A ^ { ( n - 1 ) } , V _ { A , 0 } , A ( x ) , A ( 0 ) , Z _ A ^ { ( n - 1 ) } ) \\\\ & - \\mathcal { H } _ e ( V _ A ^ { ( n - 2 ) } , V _ { A , 0 } , A ( x ) , A ( 0 ) , Z _ A ^ { ( n - 2 ) } ) . \\end{align*}"}
-{"id": "6631.png", "formula": "\\begin{align*} A _ 1 & = \\left ( \\int _ { \\varphi _ n ( E _ M ) } | D f _ n - D h _ n | ^ p \\ , d \\mu \\right ) ^ { \\frac { 1 } { p } } \\ , \\\\ A _ 2 & = \\left ( \\int _ { \\varphi _ n ( E _ M ) } | D h _ n - D h _ n \\circ \\phi _ n | ^ p \\ , d \\mu \\right ) ^ { \\frac { 1 } { p } } \\ , \\\\ A _ 3 & = \\left ( \\int _ { \\varphi _ n ( E _ M ) } | D h _ n \\circ \\phi _ n - D g _ n | ^ p \\ , d \\mu \\right ) ^ { \\frac { 1 } { p } } \\ . \\end{align*}"}
-{"id": "7050.png", "formula": "\\begin{align*} \\tilde { \\eta } _ n = \\frac { t _ 1 \\sigma _ 1 + t _ 2 \\sigma _ 2 + . . . + t _ { n - 1 } \\sigma _ { n - 1 } } { \\sigma _ 1 + \\sigma _ 2 + . . . + \\sigma _ { n - 1 } } = h ( t ) \\eta _ n \\end{align*}"}
-{"id": "9314.png", "formula": "\\begin{align*} d _ 1 - d _ 2 = ( - a + d _ 1 ) - ( - a + d _ 2 ) = - a ' + f ( d _ 3 ) - \\big ( - a ' + f ( d _ 4 ) \\big ) = f ( d _ 3 ) - f ( d _ 4 ) . \\end{align*}"}
-{"id": "10143.png", "formula": "\\begin{align*} \\boldsymbol { \\omega } _ k ( 1 ) = \\boldsymbol S _ { D _ k } ( 1 ) \\sum _ { l \\in \\mathcal { N } _ k } c _ { k l } \\boldsymbol { \\bar { \\omega } } _ l ( 1 ) . \\end{align*}"}
-{"id": "6173.png", "formula": "\\begin{align*} V ^ { 2 } ( x _ 0 , t ) = \\frac { 2 } { 3 } \\int ^ { t } _ { 0 } K \\dd s + V ^ { 2 } ( x _ { 0 } , 0 ) \\end{align*}"}
-{"id": "2894.png", "formula": "\\begin{align*} \\tau _ s ^ 0 ( m ) = \\left \\{ \\begin{array} { l l } \\tau _ s ( m ) & m < | \\tau _ s | \\\\ s + 1 & \\gamma _ s ( m ) = 1 m \\geq | \\tau _ s | \\\\ 0 & m \\geq | \\tau _ s | \\gamma _ s ( m ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "348.png", "formula": "\\begin{align*} \\nu ( S | \\nabla f | ) = \\nu ( R _ { 2 2 } - R _ { 1 1 } ) = \\nabla _ { \\nu } R _ { 2 2 } + 2 R i c ( \\nabla _ { \\nu } e _ 2 , e _ 2 ) - \\nabla _ { \\nu } R _ { 1 1 } - 2 R i c ( \\nabla _ { \\nu } e _ 1 , e _ 1 ) . \\end{align*}"}
-{"id": "6199.png", "formula": "\\begin{align*} \\phi ( \\lambda _ 1 ) = i ( f _ 1 ) , \\phi ( \\lambda _ 2 ) = i ( f _ 2 ) , \\phi ( \\langle \\lambda _ 1 , \\lambda _ 2 \\rangle ^ { \\perp } ) \\cong I _ { 0 , 2 } ( 2 ) ^ { \\perp } \\cong E _ 8 ^ 2 \\oplus U ^ 2 \\oplus I _ { 2 , 0 } ( 2 ) , \\end{align*}"}
-{"id": "4380.png", "formula": "\\begin{align*} B _ \\gamma \\hat { A } M _ { f \\circ \\tau _ { - z _ \\gamma } } = M _ { f \\circ \\tau _ { - z _ \\gamma } } = M _ { f \\circ \\tau _ { - z _ \\gamma } } \\hat { A } D _ \\gamma . \\end{align*}"}
-{"id": "1941.png", "formula": "\\begin{align*} { \\lambda q } - { q } _ { B } ( \\lambda ) = - \\sum _ { j = 2 } ^ { l - 1 } \\lambda ^ { j } { \\widetilde { Q } _ { j } ( q ) } - \\sum _ { j = l } ^ { \\infty } \\lambda ^ { j } { \\widetilde { Q } _ { j } ( q ) } , \\end{align*}"}
-{"id": "5862.png", "formula": "\\begin{align*} \\mathcal { C } _ k ( D ) : = \\{ C \\in \\mathcal { C } _ k \\ , \\ , | \\ , \\ , C \\supseteq D , \\ , | D | + 2 | C \\setminus D | \\leq k \\} , \\ , \\ , . \\end{align*}"}
-{"id": "1853.png", "formula": "\\begin{align*} \\mathcal { G F } _ { \\mathcal { B } } ( R ) \\cap \\mathcal { W } & = \\mathcal { F } ( R ) & & & \\mathcal { C } ( R ) \\cap \\mathcal { W } & = \\mathcal { G C } _ { \\mathcal { B } } ( R ) . \\end{align*}"}
-{"id": "3969.png", "formula": "\\begin{align*} \\partial _ t ^ { \\alpha _ n } p ^ { \\alpha _ n } ( n , t ) = - \\lambda ( p ^ { \\alpha _ n } ( n , t ) - p ^ { \\alpha _ { n - 1 } } ( n - 1 , t ) ) , \\ \\ 0 < \\alpha _ n \\leq 1 , \\ \\lambda > 0 , \\ n \\geq 0 , \\end{align*}"}
-{"id": "8547.png", "formula": "\\begin{align*} T = \\frac { 2 u \\left ( \\frac { \\pi } { 2 } \\right ) } { \\sqrt { a _ 1 + a _ 3 } } . \\end{align*}"}
-{"id": "1231.png", "formula": "\\begin{align*} G ( x ) = b ^ { \\frac { - 1 } { p - 1 } } \\hat { G } ( x ) = b ^ { \\frac { - 1 } { p - 1 } } h ( x ) ^ { \\frac { p - n } { p - 1 } } \\end{align*}"}
-{"id": "9496.png", "formula": "\\begin{align*} \\begin{aligned} & t z ^ 2 a ^ 2 - ( z - t ) a + ( z - t ) - t z S _ 2 ( t ) = 0 , \\\\ & 2 t z ^ 2 a + t - z = 0 , \\\\ & 2 t z a ^ 2 - a + 1 - t S _ 2 ( t ) = 0 \\end{aligned} \\end{align*}"}
-{"id": "2274.png", "formula": "\\begin{align*} \\lambda _ 1 ( A _ { \\alpha } ( K _ m \\vee H ) ) & - \\lambda _ 1 ( A _ { \\alpha } ( K _ m \\vee P _ n ) ) \\\\ & \\geq 2 ( 1 - \\alpha ) ( x _ { i - 1 } x _ { i + k + 1 } + x _ { i } x _ { i + k } - x _ { i - 1 } x _ i - x _ { i + k } x _ { i + k + 1 } ) \\\\ & = 2 ( 1 - \\alpha ) ( x _ { i + k } - x _ { i - 1 } ) ( x _ { i } - x _ { i + k + 1 } ) . \\end{align*}"}
-{"id": "5464.png", "formula": "\\begin{align*} \\widehat U _ x ( s ) = \\frac { \\widehat U ( s / x ) } { x ^ { \\rho } \\ell ( x ) } . \\end{align*}"}
-{"id": "9158.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n + 1 } ( - 1 ) ^ { i } \\binom { n + 1 } { i } x ^ { i } A \\left ( c x ^ { n + 1 - i } \\right ) = 0 \\left ( x \\in K , c \\in K ^ * \\right ) . \\end{align*}"}
-{"id": "1624.png", "formula": "\\begin{align*} S _ \\lambda \\xi ( x ) = \\chi _ { R _ \\lambda } ( x ) ( \\Phi _ { \\tau _ \\lambda } ( \\tau ^ { d ( \\lambda ) } ( x ) ) ) ^ { - 1 / 2 } \\xi ( \\tau ^ { d ( \\lambda ) } ( x ) ) . \\end{align*}"}
-{"id": "1899.png", "formula": "\\begin{align*} H ( x ) \\le \\pi _ s + \\sum _ { i = 1 } ^ d \\big ( F _ i ( x _ i ) - F _ i ( s _ i ) \\big ) ^ + . \\end{align*}"}
-{"id": "807.png", "formula": "\\begin{align*} v _ { 2 } ( 3 x + 1 ) = d \\end{align*}"}
-{"id": "3136.png", "formula": "\\begin{align*} \\mathrm { d } X _ { t } = ( a - b X _ { t } ) \\mathrm { d } t + \\sigma \\sqrt { X _ { t } } \\mathrm { d } B _ { t } + \\mathrm { d } J _ { t } , t \\geqslant 0 , X _ { 0 } \\geqslant 0 , \\end{align*}"}
-{"id": "1918.png", "formula": "\\begin{align*} F _ { i } = \\left ( \\begin{array} { c c c } 1 & 0 \\\\ n _ { i , k _ { i } } & 1 \\end{array} \\right ) \\left ( \\begin{array} { c c c } 1 & n _ { i , k _ { i } - 1 } \\\\ 0 & 1 \\end{array} \\right ) \\cdots \\left ( \\begin{array} { c c c } 1 & 0 \\\\ n _ { i , 2 } & 1 \\end{array} \\right ) \\left ( \\begin{array} { c c c } 1 & n _ { i , 1 } \\\\ 0 & 1 \\end{array} \\right ) , \\end{align*}"}
-{"id": "7690.png", "formula": "\\begin{align*} \\phi ( t ) & = ( e _ j - e _ k ) ^ { \\top } Q ^ { \\top } [ I _ { N - 1 } | 0 _ { N - 1 } ] z ( t ) \\\\ & = ( e _ j - e _ k ) ^ { \\top } Q ^ { \\top } z _ 1 ( t ) \\ , . \\end{align*}"}
-{"id": "1908.png", "formula": "\\begin{align*} V _ { n } ( \\delta _ { n } ) = \\{ ( \\phi _ { n } ^ { \\star } w , w ) : w \\in B _ { n } ^ { u } ( \\delta _ { n } ) \\} , \\end{align*}"}
-{"id": "2522.png", "formula": "\\begin{align*} \\int ( \\partial _ k \\sigma ^ { i j } ) ( \\partial _ j h ) ( \\partial _ { i k } h ) \\varrho \\ , m _ 0 = T _ 1 + T _ 2 + T _ 3 + T _ 4 = T _ 1 + T _ { 4 1 } + T _ { 4 2 } . \\end{align*}"}
-{"id": "531.png", "formula": "\\begin{align*} \\phi _ H ^ 1 ( x ) = x . \\end{align*}"}
-{"id": "230.png", "formula": "\\begin{align*} \\| \\chi _ a h ( B _ G ) \\| _ { p ' } ^ { p ' } & = \\sum _ { n \\in \\N } \\mu _ n ( \\chi _ a h ( B _ G ) ) ^ { p ' } \\leq C ^ { p ' } _ h \\sum _ { n \\in \\N } \\lambda _ n ( \\chi _ a \\chi _ G B ^ 2 \\chi _ G \\chi _ a ) ^ { p ' \\gamma _ h / 2 } \\\\ & \\leq C ^ { 2 p / \\gamma _ h } _ h \\| \\chi _ a B ^ 2 \\chi _ a \\| _ { p } ^ { p } . \\end{align*}"}
-{"id": "8984.png", "formula": "\\begin{gather*} D ^ { ( n ) } _ q ( ( d + 1 ) q / 2 \\pm u ; t ) D ^ { ( n ) } _ { q , t } ( d ) = D ^ { ( n ) } _ { q , t } ( d + 1 ) \\prod _ { 1 \\le i \\le n } \\vartheta ( z _ i \\pm u ) , \\\\ D ^ { ( n ) } _ { q , t } ( d ) D ^ { ( n ) } _ q ( - d q / 2 \\pm u ; t ) = \\prod _ { 1 \\le i \\le n } \\vartheta ( z _ i \\pm u ) D ^ { ( n ) } _ { q , t } ( d + 1 ) , \\end{gather*}"}
-{"id": "6879.png", "formula": "\\begin{align*} \\lim _ { p \\to 0 , m \\to \\infty } \\left ( \\frac { 4 } { p q } \\right ) ^ { m } \\Phi _ { 1 } ^ { m } ( 2 ( 1 - \\sqrt { q } ) ) = 4 \\end{align*}"}
-{"id": "5902.png", "formula": "\\begin{align*} \\begin{aligned} & P ( \\sigma ^ { ( e ) } _ { i } > 2 L , \\sigma ^ { ( o ) } _ { i } > 2 L - 1 ) = P ( Z ^ { N , i } _ n = 0 , \\ \\ n = 1 , \\cdots , 2 L ) \\ge \\\\ & 1 - 2 L P ( Z ^ { N , i } _ 1 \\neq 0 ) \\ge 1 - 2 L e ^ { \\epsilon N - N \\min \\big ( I _ i ( r _ i ) , \\thinspace I _ i ( r _ { i + 1 } ) \\big ) } . \\end{aligned} \\end{align*}"}
-{"id": "2979.png", "formula": "\\begin{align*} \\{ | \\nabla u _ n | \\geq t \\} & = \\{ | \\nabla u _ n | \\geq t , u _ n < k \\} \\cup \\{ | \\nabla u _ n | \\geq t , u _ n \\geq k \\} \\\\ & \\subset \\{ | \\nabla u _ n | \\geq t , u _ n < k \\} \\cup \\{ u _ n \\geq k \\} \\subset \\Omega . \\end{align*}"}
-{"id": "8243.png", "formula": "\\begin{align*} \\abs { E ( i T _ 0 , g ) } = \\abs { E ^ { \\mathfrak { L } } ( i T _ 0 , a ( \\theta _ i ) g ' ) } , \\end{align*}"}
-{"id": "6839.png", "formula": "\\begin{align*} J _ { \\rho } ( w _ { \\lambda } + \\phi ) = J _ { \\rho } ( w _ { \\lambda } ) + \\langle S _ { \\rho } ( w _ { \\lambda } + \\theta \\phi ) , \\phi \\rangle _ { \\mathbb { S } ^ 2 } , \\end{align*}"}
-{"id": "8899.png", "formula": "\\begin{align*} \\lim _ k \\| g \\psi _ { - n _ k } \\| ^ 2 = \\int _ { \\mathbb T } | g | ^ 2 | J _ { \\theta , 1 } ^ { - 1 } ( p \\overline \\gamma ) - p J _ { \\theta , 1 } ^ { - 1 } \\overline \\gamma | ^ 2 m . \\end{align*}"}
-{"id": "7383.png", "formula": "\\begin{align*} x = j ( \\tau , n ^ { - 1 } ) N _ { f } ^ { \\frac { 2 } { 3 } } , y = j ( \\tau ) N _ { f } ^ { \\frac { 2 } { 3 } } , \\end{align*}"}
-{"id": "8881.png", "formula": "\\begin{align*} \\overline \\chi \\theta = S _ \\theta ^ \\ast u + ( \\overline \\chi \\theta , S _ \\theta ^ \\ast u ) \\overline \\chi \\theta + \\| u \\| ^ 2 \\overline \\chi \\theta . \\end{align*}"}
-{"id": "853.png", "formula": "\\begin{align*} w ( t , x ) = \\int _ { \\mathbb { R } ^ { d } } S _ { \\beta , 1 } ( t , x - y ) w _ T ( y ) d y + \\int _ { t } ^ { T } \\int _ { \\mathbb { R } ^ { d } } G _ { \\beta } ( s - t , x - y ) \\ell ( s , y ) d y d s . \\end{align*}"}
-{"id": "7074.png", "formula": "\\begin{align*} \\left . \\begin{array} { r c l } \\chi _ G ( X ) + \\chi _ G ( Y ) & = & \\chi _ G ( X \\vee Y ) , \\\\ \\chi _ G ( X ) \\star \\chi _ G ( Y ) & = & \\chi _ G ( X \\wedge Y ) , \\end{array} \\right . \\end{align*}"}
-{"id": "2351.png", "formula": "\\begin{align*} \\Phi ( s , \\lambda , w ) \\coloneqq \\sum _ { n = 0 } ^ \\infty \\frac { \\lambda ^ n } { ( n + w ) ^ s } . \\end{align*}"}
-{"id": "1127.png", "formula": "\\begin{align*} \\begin{bmatrix} x ^ { - 1 } & 0 \\\\ 0 & x \\end{bmatrix} \\{ w B , B , n _ { - } ( i ) B \\} = \\{ w B , B , n _ { - } ( i x ^ 2 ) B \\} . \\end{align*}"}
-{"id": "4005.png", "formula": "\\begin{align*} d S _ A ( u ) [ h ] = A ( \\exp _ A ( u ) ) h \\ . \\end{align*}"}
-{"id": "5885.png", "formula": "\\begin{align*} \\begin{aligned} & P ( Z _ { \\sigma _ { i } ^ { ( * ) } } ^ { N , i } = - 1 ) \\approx e ^ { - N \\big ( I _ i ( r _ i ) - I _ i ( r _ { i + 1 } ) \\big ) } , \\\\ & P ( Z _ { \\sigma _ { i } ^ { ( * ) } } ^ { N , i } = - 1 1 ) \\lessapprox e ^ { - N I _ i ( r _ i ) } , \\\\ & \\ \\sigma _ { i } ^ { ( * ) } = \\sigma _ { i } ^ { ( e ) } \\ \\ \\sigma _ { i } ^ { ( * ) } = \\sigma _ { i } ^ { ( o ) } . \\end{aligned} \\end{align*}"}
-{"id": "6964.png", "formula": "\\begin{align*} L _ M u = t r a c e ( M D ^ 2 u ) = \\Delta ( u \\circ \\sqrt { M } ) . \\end{align*}"}
-{"id": "3992.png", "formula": "\\begin{align*} \\mathrm { P r } \\{ X _ 2 > t \\} = \\mathrm { P r } \\{ N _ 2 ( t , \\lambda ) = 0 \\} = E _ { \\beta _ 0 } ( - \\lambda t ^ { \\beta _ 0 } ) . \\end{align*}"}
-{"id": "9540.png", "formula": "\\begin{align*} \\frac { w _ i } { y _ i } = \\frac { w ( Y _ i ) } { | Y _ i \\setminus ( Y _ 1 \\cup \\dots \\cup Y _ { i - 1 } ) | } \\le \\frac { w ( A ) } { | A \\setminus ( Y _ 1 \\cup \\dots \\cup Y _ { i - 1 } ) | } . \\end{align*}"}
-{"id": "9321.png", "formula": "\\begin{align*} \\{ e _ { 0 } = \\textbf { 1 } , e _ { 1 } = \\textbf { i } , e _ { 2 } = \\textbf { j } , e _ { 3 } = \\textbf { k } , e _ { 4 } = \\textbf { e } , e _ { 5 } = \\textbf { i e } , e _ { 6 } = \\textbf { j e } , e _ { 7 } = \\textbf { k e } \\} , \\end{align*}"}
-{"id": "9815.png", "formula": "\\begin{align*} g _ 1 ( t ' , p ' ) = g _ 2 ( p ' ) . \\end{align*}"}
-{"id": "3562.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\frac { \\Lambda ( n ) \\chi _ 0 ( n ) } { n ^ s } = - \\frac { \\zeta ^ { ' } ( s ) } { \\zeta ( s ) } - \\sum _ { k \\geq 0 } \\frac { \\log p } { p ^ { ( 1 + k ) s } } + O \\left ( \\frac { \\log x } { x ^ { s - 1 } } \\right ) . \\end{align*}"}
-{"id": "6756.png", "formula": "\\begin{align*} r ( l / k ) = r ( l ) - r ( k ) . \\end{align*}"}
-{"id": "717.png", "formula": "\\begin{align*} A _ + \\sim \\widehat { A } _ + = \\sqrt { \\frac { T _ { + } ^ { 5 / 2 } } { \\kappa p _ { + } } } . \\end{align*}"}
-{"id": "6311.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\xi _ O & = \\dot { f } _ O ( \\lambda ) \\nu + f _ O ( \\lambda ) \\dot { \\nu } = \\frac { 2 \\dot { f } _ O ( \\lambda ) a ( \\lambda ) + f _ O ( \\lambda ) \\dot { a } ( \\lambda ) } { 2 \\nu } . \\end{align*}"}
-{"id": "3820.png", "formula": "\\begin{align*} \\eta ^ { B , \\xi } ( x ) : = \\left \\{ \\begin{array} { l l } \\eta ( x ) - \\xi ( x ) & x \\in B , \\\\ \\eta ( x ) & \\end{array} \\right . \\end{align*}"}
-{"id": "5911.png", "formula": "\\begin{align*} \\frac { \\sum _ { k = 1 } ^ m T ^ { N , 1 } _ k } { \\sum _ { k = 1 } ^ m ( T ^ { N , 1 } _ k + T ^ { N , 0 } _ k ) } \\ge \\frac { \\sum _ { k = 1 } ^ m ( N + \\tau ^ { N , 1 } _ k ) ( c ) _ k } { \\sum _ { k = 1 } ^ m ( N + \\tau ^ { N , 1 } _ k ) ( c ) _ k + \\tau ^ { N , 0 } _ k } \\ \\ \\end{align*}"}
-{"id": "9109.png", "formula": "\\begin{align*} A ( x y ) = x A ( y ) + y A ( x ) \\left ( x , y \\in R \\right ) . \\end{align*}"}
-{"id": "4288.png", "formula": "\\begin{align*} ( \\exp A ( \\lambda ) ) \\gamma = \\gamma + \\lambda \\cup \\pi ^ { \\ast } \\pi _ { \\ast } ( \\gamma ) - \\pi ^ { \\ast } \\pi _ { \\ast } ( \\lambda \\cup \\gamma ) - \\frac { 1 } { 2 } \\pi ^ { \\ast } \\left ( \\pi _ { \\ast } ( \\lambda ^ 2 ) \\cdot \\pi _ { \\ast } ( \\gamma ) \\right ) . \\end{align*}"}
-{"id": "3455.png", "formula": "\\begin{align*} \\begin{cases} \\dot { v } ( t ) = \\lambda v ( t ) + 1 , t \\in ( 0 , T ] , \\\\ v ( 0 ) = g ( 0 ) . \\end{cases} \\end{align*}"}
-{"id": "4870.png", "formula": "\\begin{align*} \\mathcal { I } = \\mathcal { I } _ \\varphi : = \\{ A \\subseteq \\mathbf { N } : \\| A \\| _ \\varphi = 0 \\} \\end{align*}"}
-{"id": "3183.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } _ { \\geqslant 0 } } y ^ { \\kappa } \\rho _ { t } ( \\mathrm { d } y ) = \\lambda _ { t } ^ { - 1 } \\int _ { 0 } ^ { t } \\int _ { \\lbrace z > 1 \\rbrace } \\left ( \\int _ { \\mathbb { R } _ { \\geqslant 0 } } y ^ { \\kappa } m _ { \\alpha ( z , s ) , \\beta ( z , s ) } ( \\mathrm { d } y ) \\right ) \\nu ( \\mathrm { d } z ) \\mathrm { d } s < \\infty . \\end{align*}"}
-{"id": "1606.png", "formula": "\\begin{align*} \\Lambda ^ n : = \\{ \\lambda \\in \\Lambda \\ , : \\ , d ( \\lambda ) = n \\} \\ \\end{align*}"}
-{"id": "4923.png", "formula": "\\begin{align*} \\xi = \\mathrm { r a n k } \\ , ( I - W ^ * W ) - \\mathrm { r a n k } \\ , ( R - W W ^ * ) \\end{align*}"}
-{"id": "4515.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\psi ( \\Delta _ n ) = 1 \\end{align*}"}
-{"id": "2074.png", "formula": "\\begin{align*} u ( p , t ) = \\int _ M H ( p , q , t ) \\phi ( q ) d V _ q + \\int _ 0 ^ t \\int _ M H ( p , q , t - s ) F ( q , s ) d V _ q d s \\end{align*}"}
-{"id": "7279.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\mu ( E _ { X _ j , s \\frac { N _ j } { N _ { j + 1 } } } ) \\ll \\sum _ { j = 1 } ^ \\infty j ^ { ( \\eta - 1 ) ( 1 + \\xi / 5 ) } < \\infty , \\end{align*}"}
-{"id": "9484.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ n } { ( - q ^ n ; q ) _ { n + 1 } ( - q ^ { 2 n + 2 } ; q ^ 2 ) _ { \\infty } } & = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n + 1 } } { ( - q ; q ^ 2 ) _ { n + 1 } } . \\end{align*}"}
-{"id": "2759.png", "formula": "\\begin{align*} C ^ * _ f ( \\chi , s ) : = \\Big ( \\prod \\limits _ { j = 0 } ^ { \\infty } L ^ * _ f ( \\chi , { p ^ { m ( f ) j } } s ) ^ { \\binom { n + j - 1 } { n - 1 } } \\Big ) ^ { { ( - 1 ) } ^ { n - 1 } } , \\end{align*}"}
-{"id": "9243.png", "formula": "\\begin{align*} [ \\lambda _ { 1 } \\beta _ { 1 } , \\cdots , \\lambda _ { n } \\beta _ { 1 } ] = [ \\eta ( \\beta _ { 1 } \\gamma _ { 1 } ) , \\cdots , \\eta ( \\beta _ { 1 } \\gamma _ { n } ) ] = [ \\gamma _ { 1 } \\beta _ { 1 } , \\cdots , \\gamma _ { n } \\beta _ { 1 } ] . \\end{align*}"}
-{"id": "5503.png", "formula": "\\begin{align*} \\frac { x ^ m \\overline F ( x ) } { u _ m ( x ) } = 1 - \\frac { m x ^ m } { u _ m ( x ) } \\int _ x ^ \\infty y ^ { - m - 1 } u _ m ( y ) \\dd y . \\end{align*}"}
-{"id": "10071.png", "formula": "\\begin{align*} p _ { a \\otimes b } ( e ) = \\psi ( \\overline { \\epsilon } a , e ) \\cdot \\overline { \\epsilon } b = - D \\psi ( a , e ) \\cdot b . \\end{align*}"}
-{"id": "8253.png", "formula": "\\begin{align*} f _ { \\nu } ( n _ { \\nu } ) = \\begin{cases} \\frac { \\abs { a } } { \\abs { y _ { \\nu } } } + 1 & \\\\ ( n _ { \\nu } + 1 ) \\left ( \\frac { \\abs { a } } { \\abs { y _ { \\nu } } } + 1 \\right ) ^ 2 & \\end{cases} \\end{align*}"}
-{"id": "5789.png", "formula": "\\begin{align*} - \\Delta _ { p } u = \\omega \\ ; \\ ; \\Omega \\end{align*}"}
-{"id": "2427.png", "formula": "\\begin{align*} D _ 1 = D _ 3 = 0 , \\ D _ 1 ' \\neq 0 , \\ D _ 3 ' \\neq 0 , \\ \\bar R _ { 1 1 3 } \\neq 0 , \\ \\bar R _ { 1 3 3 } \\neq 0 , \\end{align*}"}
-{"id": "2420.png", "formula": "\\begin{align*} p _ 1 ( x ) & = - a _ 6 ( x - a ) \\big ( ( x - b ) ^ 2 + d ^ 2 \\big ) , \\\\ p _ 3 ( x ) & = ( - a _ 1 a _ 6 + a _ 2 ) ( x - a ' ) \\big ( ( x - b ) ^ 2 + d ^ 2 \\big ) . \\\\ \\end{align*}"}
-{"id": "7030.png", "formula": "\\begin{align*} \\mu _ 0 = \\mu _ 0 ( n , s , \\eta _ 0 , L , S C ) = \\frac { \\eta _ 0 } { 2 ( 1 - s ) C _ 3 } g ( \\epsilon _ 1 ) ^ { 2 s } , \\end{align*}"}
-{"id": "7635.png", "formula": "\\begin{align*} u ^ \\eta _ { \\bar x , \\rho } = u ^ \\eta _ { \\bar x , \\rho } ( t ) = \\frac { \\int _ { B _ \\rho ( \\bar x ) } \\eta _ { \\bar x , \\rho } ( x ) ^ p u ( x , t ) \\ , d x } { \\int _ { B _ \\rho ( \\bar x ) } \\eta _ { \\bar x , \\rho } ^ p \\ , d x } . \\end{align*}"}
-{"id": "6970.png", "formula": "\\begin{align*} D f ( B ) A = D f ( B ) ( a _ { i j } E _ { i j } ) = a _ { i j } ( \\frac { 1 } { n } ( \\det { B } ) ^ { \\frac { 1 } { n } - 1 } b _ { i j } ^ * ) = t r a c e ( A M ^ T ) , \\end{align*}"}
-{"id": "5577.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } b _ { n } \\approx 1 - \\delta ^ { 2 } \\left [ S _ { 0 } \\left ( \\xi _ { 1 } + \\ldots + \\xi _ { n } \\right ) + S _ { 1 } \\left ( \\xi _ { 2 } + \\ldots + \\xi _ { n } \\right ) + \\ldots + S _ { n - 1 } \\left ( \\xi _ { n } \\right ) \\right ] \\end{align*}"}
-{"id": "1049.png", "formula": "\\begin{align*} ( w _ { m } ( \\zeta ) + 1 ) \\left ( \\frac { 1 } { m } + \\underline { \\psi } _ { m , 1 } ^ { \\ast } \\right ) = ( \\widehat { w } _ { m } ( \\zeta ) + 1 ) \\left ( \\frac { 1 } { m } + \\overline { \\psi } _ { m , 1 } ^ { \\ast } \\right ) = \\frac { m + 1 } { m } , \\end{align*}"}
-{"id": "4678.png", "formula": "\\begin{align*} \\Gamma ^ { - 1 } \\ , \\Delta _ { \\rm r a d } \\ , \\Gamma \\ = \\ { \\Delta _ { L B } } - { V _ { \\rm e f f } } \\ , \\end{align*}"}
-{"id": "6551.png", "formula": "\\begin{align*} G _ c ( P ) = \\max _ { \\xi \\in \\mathrm { e x t } ( P ) } [ a _ { \\xi } - c \\beta _ { \\xi } , c \\beta ] \\ \\ \\ \\ G ( P ) = \\min _ { c \\geq 0 } G _ c ( P ) . \\end{align*}"}
-{"id": "3329.png", "formula": "\\begin{align*} \\bar { D } ( r _ s ) = \\frac { D ( r _ s ) } { L ( s ) } = \\frac { \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K } { s + 1 + i } ( N - 1 ) ^ i N } { \\binom { K - 2 } { s - 1 } + \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K - 1 } { s } ( N - 1 ) ^ i N } . \\end{align*}"}
-{"id": "7290.png", "formula": "\\begin{align*} B '' _ { i , j } : = \\left ( \\begin{array} { c | c } D _ { i , j } & B ' \\end{array} \\right ) , \\end{align*}"}
-{"id": "8642.png", "formula": "\\begin{align*} \\begin{aligned} \\hat { \\pi } ( x , A ) & \\geq \\gamma \\pi ( A ) \\\\ \\end{aligned} \\end{align*}"}
-{"id": "7056.png", "formula": "\\begin{align*} z _ n = \\frac { y _ n } { ( \\lambda _ 1 ^ 2 y _ 1 ^ 2 + . . . + \\lambda _ { n - 1 } ^ 2 y _ { n - 1 } ^ 2 ) ^ { 1 / 2 } } \\frac { \\lambda _ n } { \\sqrt { t h ( t ) } } , \\end{align*}"}
-{"id": "6216.png", "formula": "\\begin{align*} X ( t ) = c t - C ( t ) ~ , ~ ~ t \\geq 0 ~ , \\end{align*}"}
-{"id": "7723.png", "formula": "\\begin{align*} \\lambda _ n & = 2 ( 1 - \\cos 2 \\phi _ n ) , n = 0 , 1 , 2 , \\cdots , N - 1 \\\\ u _ { n m } & = \\frac { 1 } { \\sqrt { N } } e ^ { i 2 m \\phi _ n } , n , m = 0 , 1 , \\cdots , N - 1 . \\end{align*}"}
-{"id": "8230.png", "formula": "\\begin{align*} \\biggl [ \\frac { 1 } { n ^ { k - 1 } } \\biggr ] \\ln ( F _ i ) = \\biggl [ \\frac { 1 } { n ^ { k - 1 } } \\biggr ] \\ln ( 1 + \\hat { H _ i } ) + \\biggl [ \\frac { 1 } { n ^ { k - 1 } } \\biggr ] \\ln ( 1 + K _ i ) \\end{align*}"}
-{"id": "5722.png", "formula": "\\begin{align*} \\begin{cases} F ' ( m ) = 2 ( z ' ( \\cdot - m ) \\ , , \\ , ( { v } - z ( \\cdot - m ) ) ) _ { L ^ 2 } \\ , \\\\ F '' ( m ) = 2 \\| z ' \\| _ { L ^ 2 } ^ 2 - 2 ( z '' ( \\cdot - m ) \\ , , \\ , { v } - z ( \\cdot - m ) ) _ { L ^ 2 } \\ . \\end{cases} \\end{align*}"}
-{"id": "10126.png", "formula": "\\begin{align*} \\boldsymbol k _ k ( i ) = \\frac { \\lambda ^ { - 1 } \\boldsymbol P _ k ( i - 1 ) \\boldsymbol x _ k ( i ) } { 1 + \\lambda ^ { - 1 } \\boldsymbol x _ k ^ H ( i ) \\boldsymbol P _ k ( i - 1 ) \\boldsymbol x _ k ( i ) } . \\end{align*}"}
-{"id": "8554.png", "formula": "\\begin{align*} \\partial _ t u = \\Delta u \\quad \\mbox { i n } \\quad { \\bf R } ^ N \\times ( 0 , 1 / 4 \\Lambda ) , u ( \\cdot , 0 ) = \\mu \\quad \\mbox { i n } \\quad { \\bf R } ^ N . \\end{align*}"}
-{"id": "4187.png", "formula": "\\begin{align*} E _ { 2 i } ( q ) : = 1 + b _ i \\sum _ { n = 1 } ^ \\infty \\left ( \\sum _ { d \\mid n } d ^ { 2 i - 1 } \\right ) q ^ { n } , \\ \\ i = 1 , 2 , 3 , \\end{align*}"}
-{"id": "8193.png", "formula": "\\begin{align*} g _ { \\nu } ( k _ { \\nu } ) = e ^ { - \\pi \\abs { y _ { \\nu } } R _ { \\nu } k _ { \\nu } } . \\end{align*}"}
-{"id": "1487.png", "formula": "\\begin{align*} | w ( r e ^ { i \\theta } ) | ^ 2 { \\rm R e } \\Big ( \\frac { r e ^ { i \\theta } w ' ( r e ^ { i \\theta } ) } { w ( r e ^ { i \\theta } ) } \\Big ) = r \\int _ 0 ^ r | w ' ( \\rho e ^ { \\theta } ) | ^ 2 d \\rho - r \\int _ 0 ^ r { \\rm R e } ( \\rho ^ 2 e ^ { 2 i \\theta } p ( \\rho e ^ { i \\theta } ) ) \\cfrac { | w ( \\rho e ^ { i \\theta } ) | ^ 2 } { \\rho ^ 2 } d \\rho . \\end{align*}"}
-{"id": "7840.png", "formula": "\\begin{align*} M _ { X _ { i } } = \\frac { 1 } { b } \\left ( - a + \\sqrt { a ^ { 2 } + 2 b \\left [ - \\ln \\left ( 1 - \\left ( \\frac { 1 } { 2 } \\right ) ^ { ( \\frac { 1 } { \\theta _ { i } + \\theta _ { 3 } } ) } \\right ) \\right ] ^ { \\frac { 1 } { \\alpha } } } \\right ) , \\ \\ \\ \\ i = 1 , 2 . \\end{align*}"}
-{"id": "1426.png", "formula": "\\begin{align*} Q _ j ( x _ 3 x _ { 4 i } ) = x _ 3 ^ 2 \\partial _ j ( x _ { 4 i } ) , Q _ j ( x _ { 4 i _ { 1 } } x _ { 4 i _ { 2 } } ) = x _ 3 \\partial _ j ( x _ { 4 i _ { 1 } } x _ { 4 i _ { 2 } } ) . \\end{align*}"}
-{"id": "1658.png", "formula": "\\begin{align*} R _ { a _ 0 } = \\big ( 0 , \\frac { 1 } { 3 } \\big ) , R _ { a _ 1 } = \\big ( \\frac { 2 } { 3 } , 1 \\big ) , R _ { c _ 0 } = \\big ( \\frac { 1 } { 2 } , \\frac { 2 } { 3 } \\big ) , \\quad \\ R _ { c _ 1 } = \\big ( \\frac { 1 } { 3 } , \\frac { 1 } { 2 } \\big ) . \\end{align*}"}
-{"id": "8751.png", "formula": "\\begin{align*} \\alpha ( M ) : = \\frac { 1 } { 2 ( M + 1 ) } + \\frac { 1 } { 2 } \\int \\limits _ 0 ^ 1 \\frac { 1 } { \\beta ( u ) ( M + 1 - u ) } \\ , d u , \\end{align*}"}
-{"id": "3066.png", "formula": "\\begin{align*} \\epsilon \\dot { u } _ \\epsilon ( t ) = - \\nabla _ x F ( t , u _ \\epsilon ( t ) ) , \\end{align*}"}
-{"id": "2737.png", "formula": "\\begin{align*} \\frac { \\epsilon ^ 2 } { 2 } \\partial _ t | | \\partial ^ l g | | _ { L ^ 2 _ { x , v } } ^ 2 = \\langle \\partial ^ { l } \\mathcal L ( g ) , \\ , \\partial ^ { l } g \\rangle _ { L ^ 2 _ { x , v } } , \\end{align*}"}
-{"id": "9944.png", "formula": "\\begin{align*} I _ 1 = \\int u d x , \\ : I _ \\omega = \\frac 1 { \\omega } \\int u ^ { \\omega } d x . \\end{align*}"}
-{"id": "2142.png", "formula": "\\begin{align*} \\left [ e ^ { - t L } \\varphi ^ { k , i } \\right ] ( x ) & = \\left [ e ^ { - t L _ k } \\phi ^ { k , i } \\right ] ( x ) \\ , Q _ { k , i } \\left ( \\frac { x } { | x | } \\right ) , \\\\ \\left [ e ^ { - t L } \\varphi \\right ] ( x ) & = \\sum _ { k = 0 } ^ \\infty \\sum _ { i = 1 } ^ { \\ell _ k } \\left [ e ^ { - t L _ k } \\phi ^ { k , i } \\right ] ( x ) \\ , Q _ { k , i } \\left ( \\frac { x } { | x | } \\right ) \\\\ & \\qquad \\mbox { i n $ L ^ 2 ( { \\bf R } ^ N ) \\cap L ^ \\infty ( { \\bf R } ^ N ) $ f o r a n y $ t > 0 $ } . \\end{align*}"}
-{"id": "3436.png", "formula": "\\begin{align*} \\P \\Big ( \\bigcap _ { n \\in I } A _ n \\Big ) = \\prod _ { n \\in I } \\P ( A _ n ) \\end{align*}"}
-{"id": "3251.png", "formula": "\\begin{align*} \\biggl ( i _ ! ( \\alpha _ { Y } ) , \\omega \\biggr ) _ { X } = ( \\alpha _ Y , \\omega _ Y ) _ Y = ( \\alpha , \\omega ) _ X . \\end{align*}"}
-{"id": "10085.png", "formula": "\\begin{align*} \\beta ( X , Y ) = ( \\nabla _ { X } \\psi ) ( Y ) - ( \\nabla _ { Y } \\psi ) ( X ) + ( \\nabla _ { X } \\phi ) ( Y ) - ( \\nabla _ { Y } \\phi ) ( X ) , \\end{align*}"}
-{"id": "9929.png", "formula": "\\begin{gather*} \\frac { \\partial } { \\partial t } u _ n ( t , x ) = A _ x u _ n ( t , x ) + g _ n ( t , x ) , u _ n ( 0 , x ) = 0 \\intertext { i . e . } u _ n ( t , x ) = \\int _ { ( 0 , t ) } ( A _ x u _ n ( s , x ) + g _ n ( s , x ) ) \\ , d s . \\end{gather*}"}
-{"id": "2699.png", "formula": "\\begin{align*} \\begin{cases} s ^ 2 - p ^ { 2 n } a ' _ 3 ( a ' _ 1 ) ^ { - 1 } t ^ 2 + b _ 1 s + b _ 2 = 0 \\\\ 2 s t - a _ 2 ( a ' _ 1 ) ^ { - 1 } t ^ 2 + b _ 1 t = 0 . \\end{cases} \\end{align*}"}
-{"id": "4160.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { k \\in N } & e _ { k , j + n - 2 } ^ n + \\frac { 1 } { n } \\sum _ { \\tau = 0 } ^ { n - s - 2 } e ^ { n - \\tau - 1 } _ { i , j + \\tau } \\\\ & + \\left ( 1 - \\frac { 1 } { n } \\right ) \\left [ \\sum _ { \\tau = 0 } ^ { n - s - 2 } \\frac { 1 } { n ^ { n - \\tau - s - 1 } } e ^ { s } _ { i , j + \\tau } \\right ] + \\frac { 1 } { n } \\sum _ { \\tau = 1 } ^ { s - 1 } e _ { i , j + n - s - 2 + \\tau } ^ { s - \\tau } \\end{align*}"}
-{"id": "5460.png", "formula": "\\begin{align*} \\limsup _ { x \\to \\infty } \\frac { U ( x ) } { x ^ \\rho \\ell ( x ) } = K < \\infty \\end{align*}"}
-{"id": "2946.png", "formula": "\\begin{align*} X _ v = t - \\sum _ { k = 2 } ^ { h ( \\Gamma _ v ) } \\sum _ { l = 1 } ^ { \\Gamma _ v ( k ) } \\left [ 1 - X _ { ( v , k , l , 1 ) } ^ + - \\dots - X _ { ( v , k , l , k - 1 ) } ^ + \\right ] _ 0 ^ 1 ; \\end{align*}"}
-{"id": "7122.png", "formula": "\\begin{align*} h _ { i j } ^ X = & \\frac { h _ { i j } ^ Y } { \\sqrt { ( 1 - | Y | ^ 2 ) ( 1 - \\langle N , Y \\rangle ^ 2 ) } } , \\end{align*}"}
-{"id": "2831.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\sum _ { u \\notin V _ { n , r } } f _ { s u } ^ { ( n ) } < \\infty . \\end{align*}"}
-{"id": "6887.png", "formula": "\\begin{align*} \\int _ A f _ i ( - \\Delta f _ j ) d \\mu - \\int _ A f _ j ( - \\Delta f _ i ) d \\mu & = \\sum _ { x \\in \\partial A } \\big ( ( f _ j \\partial _ n f _ i ) ( x ) - ( f _ i \\partial _ n f _ j ) ( x ) \\big ) \\end{align*}"}
-{"id": "5326.png", "formula": "\\begin{align*} \\hat { \\mathfrak { g } } = \\hat { \\mathfrak { g } } _ { - } \\oplus \\mathfrak { g } \\oplus \\C { \\bf k } \\oplus \\hat { \\mathfrak { g } } _ { + } . \\end{align*}"}
-{"id": "103.png", "formula": "\\begin{align*} \\forall n \\in \\mathbb { Z } ^ k : W ( x , n ) = W ( y , n ) \\end{align*}"}
-{"id": "3435.png", "formula": "\\begin{align*} X ^ { - 1 } ( B ) = \\{ \\omega \\in \\Omega \\ , : \\ , X ( \\omega ) \\in B \\} \\end{align*}"}
-{"id": "5272.png", "formula": "\\begin{align*} u _ t - \\mu u _ { x x } = f ( t , x , u , u _ x ) \\ \\ \\ \\ \\ \\ \\mathbb { R } _ { \\geq 0 } \\times ( 0 , 1 ) \\end{align*}"}
-{"id": "3557.png", "formula": "\\begin{align*} \\mu ( n ) = \\left \\{ \\begin{array} { l l } ( - 1 ) ^ t & n = p _ 1 p _ 2 \\cdots p _ t , v _ k = 1 k \\geq 1 , \\\\ 0 & n \\neq p _ 1 p _ 2 \\cdots p _ t , v _ k \\geq 2 k \\geq 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "845.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } d X _ s = f ( s , X _ s , u _ s ) d E _ s + \\sqrt { 2 \\nu } \\ , d B _ { E _ s } , s \\in ( t , T ] \\\\ X _ t = x . \\end{array} \\right . \\end{align*}"}
-{"id": "5200.png", "formula": "\\begin{align*} \\frac { d } { { d } x } V _ { 2 } ^ { [ 0 , \\ell ] } ( r ) = f _ { 2 } ' ( r ) = \\frac { f _ { 2 } ( r ) - g _ { 2 } ( \\ell ) } { r - \\ell } . \\end{align*}"}
-{"id": "8782.png", "formula": "\\begin{align*} \\iint _ { \\R ^ { 2 } \\times \\R ^ { 2 } } f ( t , x , v ) \\ , d x d v = \\iint _ { \\R ^ { 2 } \\times \\R ^ { 2 } } f _ { 0 } ( x , v ) \\ , d x d v = : \\mathcal { M } _ 0 \\end{align*}"}
-{"id": "8233.png", "formula": "\\begin{align*} \\prod _ { \\ell \\in \\mathcal { L } ^ + } ( 1 + U _ { i + \\ell } ) ^ { \\binom { k } { \\ell } } = 1 + t _ + + \\frac { 1 } { 2 } t _ + ^ 2 + \\frac { 1 } { 3 ! } t _ + ^ 3 \\cdots \\end{align*}"}
-{"id": "9600.png", "formula": "\\begin{align*} & \\sum _ { j = 1 } ^ { N _ J } D _ k ( f ) ( y _ j ) \\int _ { Q _ j } g ( y ) d \\mu ( y ) \\\\ & \\quad = \\sum _ { j = 1 } ^ { N _ J } \\int _ { Q _ j } D _ k ( f ) ( y ) g ( y ) d \\mu ( y ) \\\\ & \\qquad + \\int _ { \\Bbb R ^ n } \\bigg \\{ \\sum _ { j = 1 } ^ { N _ J } [ D _ k ( f ) ( y _ j ) - D _ k ( f ) ( y ) ] \\chi _ { { } _ { Q _ j } } \\bigg \\} g ( y ) d \\mu ( y ) . \\end{align*}"}
-{"id": "8081.png", "formula": "\\begin{align*} m _ z = z ^ { p _ 1 } \\left ( \\prod _ { i = 2 } ^ r z _ i ^ { p _ i } \\right ) z _ { r + 1 } ^ { \\ell - q _ r } y _ { r + 1 } ^ { p _ { r + 1 } - ( \\ell - q _ r ) } \\left ( \\ , \\ , \\prod _ { i = r + 2 } ^ k y _ i ^ { p _ i } \\right ) \\end{align*}"}
-{"id": "5905.png", "formula": "\\begin{align*} \\begin{aligned} & E S ^ { ( i ) } _ { \\tau ^ { N , i } + N } \\approx \\mu _ i e ^ { N \\min \\big ( I _ i ( r _ i ) , I _ i ( r _ { i + 1 } ) \\big ) } , \\ \\ 1 \\le i \\le l - 1 ; \\\\ & E S ^ { ( 0 ) } _ { \\tau ^ { N , 0 } + N } \\approx \\mu _ 0 e ^ { N I _ 0 ( r _ 1 ) } , \\ E S ^ { ( l ) } _ { \\tau ^ { N , l } + N } \\approx \\mu _ l e ^ { N I _ l ( r _ l ) } . \\end{aligned} \\end{align*}"}
-{"id": "5496.png", "formula": "\\begin{align*} \\widehat F ( s ) = \\sum _ { k = 0 } ^ m \\mu _ k \\frac { ( - s ) ^ k } { k ! } + o ( s ^ m ) s \\downarrow 0 , \\end{align*}"}
-{"id": "5056.png", "formula": "\\begin{align*} \\mathrm { T r } ( x \\Phi ( a ) ) & = \\mathrm { T r } ( \\Phi ^ { \\dagger } ( x ) a ) \\\\ & = \\frac { 1 } { n } \\mathrm { T r } ( \\Phi ^ { \\dagger } ( x ) ) \\mathrm { T r } ( a ) \\\\ & = \\mathrm { T r } ( x \\Phi ( I ) \\frac { \\mathrm { T r } ( a ) } { n } ) \\end{align*}"}
-{"id": "4175.png", "formula": "\\begin{align*} w ( T _ 1 , \\ldots , T _ n ) = \\omega ( T _ 1 ^ * \\otimes S _ 1 + \\cdots + T _ n ^ * \\otimes S _ n ) , \\end{align*}"}
-{"id": "1547.png", "formula": "\\begin{gather*} d \\ = \\ \\begin{pmatrix} - 1 & 2 & - 2 & 0 & 1 & 2 & - 2 & 0 \\end{pmatrix} ^ T . \\end{gather*}"}
-{"id": "7007.png", "formula": "\\begin{align*} M _ 0 = D f _ 2 ( I ) = \\sqrt { \\frac { n - 1 } { 2 n } } I \\in \\mathcal { M } _ 2 . \\end{align*}"}
-{"id": "5166.png", "formula": "\\begin{align*} g _ { 1 , [ y , 1 ] } ( x ) & = \\begin{cases} g _ { 1 } ( y ) \\cdot \\frac { x } { y } , & \\forall x \\in [ 0 , y ) \\\\ g _ { 1 } ( x ) , & \\forall x \\in [ y , 1 ] , \\end{cases} \\\\ g _ { 2 , [ 0 , x ] } ( y ) & = \\begin{cases} g _ { 2 } ( y ) , & \\forall y \\in [ 0 , x ] \\\\ g _ { 2 } ( x ) \\cdot \\frac { 1 - y } { 1 - x } , & \\forall y \\in ( x , 1 ] , \\end{cases} \\end{align*}"}
-{"id": "5476.png", "formula": "\\begin{align*} \\frac { U ( r ^ n z ) - U ( r ^ n z \\varepsilon ) } { ( r ^ n z ) ^ \\rho \\ell ( r ^ n z ) } = \\int _ { \\varepsilon } ^ 1 \\frac { u ( r ^ n z t ) } { ( r ^ n z t ) ^ { \\rho - 1 } \\ell ( r ^ n z t ) } t ^ { \\rho - 1 } \\frac { \\ell ( r ^ n z t ) } { \\ell ( r ^ n z ) } \\dd t . \\end{align*}"}
-{"id": "3915.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\Gamma ( n + 1 ) ^ 2 } { \\Gamma ( n + \\frac 3 2 ) \\Gamma ( n + \\frac 1 2 ) } = 1 . \\end{align*}"}
-{"id": "2789.png", "formula": "\\begin{align*} \\beta _ { i } = \\frac { 1 } { \\Pi _ { i \\neq j } ( t _ { i } - t _ { j } ) } , i = 0 , 1 , \\cdots , n , \\end{align*}"}
-{"id": "7356.png", "formula": "\\begin{align*} \\displaystyle { \\not } D \\psi = m \\psi . \\end{align*}"}
-{"id": "1427.png", "formula": "\\begin{align*} Q _ { r + 1 } ( \\phi _ r ( x _ I ) x _ 3 ^ s ) = \\phi _ { r + 1 } ( x _ I ) x _ 3 ^ { s + 1 } . \\end{align*}"}
-{"id": "9512.png", "formula": "\\begin{align*} a _ { n , 0 } ^ { ( d ) } = \\sum \\limits _ { t = 0 } ^ { d - 1 } { c _ { n , 0 , t } ^ { ( d ) } \\cdot a _ { n - 1 , t } ^ { ( d ) } } . \\end{align*}"}
-{"id": "9789.png", "formula": "\\begin{align*} Q ( x , y ) = \\varepsilon ( i + j - n ) \\int _ { D ( i ) } \\big { ( } ( 2 \\pi \\i ) ^ i x \\big { ) } \\wedge L ^ j C ( 2 \\pi \\i ) ^ i y \\end{align*}"}
-{"id": "124.png", "formula": "\\begin{align*} & \\Phi ^ { - 1 } \\bigg ( \\int h \\ , d \\gamma _ n \\bigg ) - \\lambda \\Phi ^ { - 1 } \\bigg ( \\int f \\ , d \\gamma _ n \\bigg ) - \\mu \\Phi ^ { - 1 } \\bigg ( \\int g \\ , d \\gamma _ n \\bigg ) \\\\ & = \\mathbf { E } \\bigg [ C ( 0 , X _ 1 , Y _ 1 ) \\ , e ^ { - \\frac { 1 } { 2 } \\int _ 0 ^ 1 \\| \\nabla u _ h ( 1 - s , \\lambda X _ s + \\mu Y _ s ) \\| ^ 2 d s } \\bigg ] \\ge 0 \\end{align*}"}
-{"id": "3063.png", "formula": "\\begin{align*} K _ z ( w ) = e ^ { 2 w \\overline { z } } . \\end{align*}"}
-{"id": "2502.png", "formula": "\\begin{align*} \\overline { \\beta _ { j k } \\left ( \\left | \\eta \\right | \\right ) } = \\beta _ { j k } \\left ( - \\left | \\eta \\right | \\right ) . \\end{align*}"}
-{"id": "8163.png", "formula": "\\begin{align*} a _ 1 ^ { q + 1 } a _ 6 ^ { q ^ 2 } + a _ 2 a _ 7 ^ { q + q ^ 2 } = 0 , \\end{align*}"}
-{"id": "7021.png", "formula": "\\begin{align*} J _ 2 = g ( \\epsilon ) ^ { n - 1 } \\epsilon ^ { - 1 / 2 } \\int _ { \\mathbb { R } ^ n } { \\frac { u ( \\bar { y } , 0 ) - u ( 0 ) } { ( g ( \\epsilon ) ^ 2 | \\bar { y } | ^ 2 + \\frac { 1 } { \\epsilon } y _ n ^ 2 ) ^ { \\frac { n + 2 s } { 2 } } } d y } . \\end{align*}"}
-{"id": "5424.png", "formula": "\\begin{align*} \\int _ \\Omega q u _ 2 u _ 1 d x = - \\int _ { \\Gamma \\setminus G } \\partial _ \\nu u u _ 1 d \\sigma ( x ) . \\end{align*}"}
-{"id": "8743.png", "formula": "\\begin{align*} & E _ { \\varepsilon , \\delta } ( t , x , w , l , p , X ) = \\min \\{ E _ * ( t ' , x ' , w , l , p , X ) \\mid | t - t ' | \\le M \\delta ^ { 1 / 2 } , x ' \\in B _ { x , \\varepsilon } \\} , \\\\ & E ^ { \\varepsilon , \\delta } ( t , x , w , l , p , X ) = \\max \\{ E ^ * ( t ' , x ' , w , l , p , X ) \\mid | t - t ' | \\le M \\delta ^ { 1 / 2 } , x ' \\in B _ { x , \\varepsilon } \\} , \\end{align*}"}
-{"id": "3650.png", "formula": "\\begin{align*} \\phi _ { \\tilde P [ n ] } : X [ n ] = X ( \\tilde P [ n ] ) \\to X \\times _ { \\mathbb A ^ 1 } \\mathbb A ^ { n + 1 } = X ( \\sigma [ n ] ) \\end{align*}"}
-{"id": "7429.png", "formula": "\\begin{align*} ( \\gamma + \\beta ^ 2 ) ^ 2 + \\beta \\gamma \\beta = ( R _ 1 ^ 2 - R _ 2 - S _ 2 ) ( \\gamma + \\beta ^ 2 ) + ( R _ 1 R _ 2 - R _ 1 S _ 2 - R _ 3 ) \\beta + ( R _ 1 R _ 3 - R _ 2 S _ 2 ) e _ 4 . \\end{align*}"}
-{"id": "3079.png", "formula": "\\begin{align*} t ^ { \\sigma , \\epsilon } : = \\inf \\left \\{ s \\in [ 0 , t ] : \\ , \\| u _ \\epsilon ( s ) \\| \\geq \\sigma \\right \\} , \\end{align*}"}
-{"id": "6290.png", "formula": "\\begin{align*} A _ 7 \\lesssim & \\sum _ { - 1 \\leq p \\leq Q _ 1 + 2 } \\| \\nabla u _ { \\leq p - 2 } \\| _ \\infty \\| b _ p \\| _ 2 \\sum _ { | p - q | \\leq 2 } \\lambda _ q ^ { 2 s } \\| b _ q \\| _ 2 \\\\ \\lesssim & Q _ 1 f ( t ) \\sum _ { - 1 \\leq p \\leq Q _ 1 + 2 } \\| b _ p \\| _ 2 \\sum _ { | p - q | \\leq 2 } \\lambda _ q ^ { 2 s } \\| b _ q \\| _ 2 \\\\ \\lesssim & Q _ 1 f ( t ) \\sum _ { q \\geq - 1 } \\lambda _ q ^ { 2 s } \\| b _ q \\| _ 2 ; \\end{align*}"}
-{"id": "2457.png", "formula": "\\begin{align*} M _ { G L } ^ * ( \\pi ) = \\sum x \\times \\tilde { y } . \\end{align*}"}
-{"id": "3793.png", "formula": "\\begin{align*} \\omega : = \\sum _ { z \\in \\Z } \\sum _ { i \\le \\eta ( z ) } \\delta _ { S ^ { z , i } } \\in \\Omega \\end{align*}"}
-{"id": "3893.png", "formula": "\\begin{align*} h ( \\omega ; s ; r ) = h _ 1 ( \\omega ; s ; r ) + h _ 1 ( \\omega ; s ; - r ) + h _ 2 ( \\omega ; s ; r ) , \\end{align*}"}
-{"id": "7598.png", "formula": "\\begin{align*} W _ 1 y = ( y + z ) / 2 = y _ 1 , W _ 2 y = ( y - z ) / 2 = y _ 2 . \\end{align*}"}
-{"id": "6496.png", "formula": "\\begin{align*} d s _ { } ^ { 2 } = d s _ { } ^ { 2 } + d s _ { } ^ { 2 } = \\frac { 4 } { \\mu _ { A } ^ { \\prime 2 } } d \\mu _ { A } ^ { \\prime 2 } + \\frac { 1 } { \\sigma _ { B } ^ { \\prime 2 } } d \\mu _ { B } ^ { \\prime 2 } + \\frac { 2 } { \\sigma _ { B } ^ { \\prime 2 } } d \\sigma _ { B } ^ { \\prime 2 } . \\end{align*}"}
-{"id": "5137.png", "formula": "\\begin{align*} z _ 5 \\ , \\ ! = \\ ! \\ , \\ < 3 1 > \\ , + \\ , \\ < 3 2 > \\ , + \\ , \\ < 3 3 > \\ , \\end{align*}"}
-{"id": "8597.png", "formula": "\\begin{align*} ( t _ { g } ( f ) \\Psi ) _ { i } ( \\delta ) = \\sum _ { \\gamma } g _ { i } ^ { - 1 } f ( \\gamma ) t _ { i } ( \\gamma ^ { - 1 } ) ^ { - 1 } \\Psi _ { \\gamma ^ { - 1 } ( i ) } ( t _ { i } ( \\gamma ^ { - 1 } ) \\delta ) . \\end{align*}"}
-{"id": "4192.png", "formula": "\\begin{align*} \\mathtt { T _ R } : = \\{ ( t _ 1 , t _ 2 , t _ 3 ) \\in \\C ^ 3 | \\Delta = t _ 2 ^ 3 - 2 7 t _ 3 ^ 2 \\neq 0 \\} . \\end{align*}"}
-{"id": "6761.png", "formula": "\\begin{align*} 2 [ l : k ] h ( \\gamma \\mu ) = [ k : l ] \\sum _ { w | \\infty } \\bigl | \\log | \\gamma \\mu | _ w \\bigr | + [ k : l ] \\sum _ { w \\nmid \\infty } \\bigl | \\log | \\gamma \\mu | _ w \\bigr | . \\end{align*}"}
-{"id": "619.png", "formula": "\\begin{align*} a _ { - 1 } = \\frac { 1 } { 2 \\pi i } \\int _ { | z | = r } f ( z ) \\ , d z , \\end{align*}"}
-{"id": "6288.png", "formula": "\\begin{align*} \\| u \\| _ \\infty \\leq C ( q ) \\left ( 1 + \\| u \\| _ { B M O } ( \\ln ( \\| u \\| _ { W ^ { 1 , q } } + \\| u \\| _ \\infty ) ) ^ { \\frac 1 2 } \\right ) \\\\ \\| \\nabla b \\| _ \\infty \\leq C ( q ) \\left ( 1 + \\| \\nabla b \\| _ { B M O } ( \\ln ( \\| \\nabla b \\| _ { W ^ { 1 , q } } + \\| u \\| _ \\infty ) ) ^ { \\frac 1 2 } \\right ) \\end{align*}"}
-{"id": "3192.png", "formula": "\\begin{align*} \\mathbb { E } _ { x } & \\left [ \\int _ { 0 } ^ { t } \\int _ { \\lbrace z > 1 \\rbrace } \\left \\vert V ( X _ { s - } + z ) - V ( X _ { s - } ) \\right \\vert \\nu ( \\mathrm { d } z ) \\mathrm { d } s \\right ] \\\\ & \\quad \\leqslant t \\int _ { \\lbrace z > 1 \\rbrace } \\left ( \\log ( 2 ) + \\log ( z ) \\right ) \\nu ( \\mathrm { d } z ) \\\\ & \\quad = t \\log ( 2 ) \\nu ( \\lbrace z > 1 \\rbrace ) + t \\int _ { \\lbrace z > 1 \\rbrace } \\log ( z ) \\nu ( \\mathrm { d } z ) < \\infty , t \\geqslant 0 , \\end{align*}"}
-{"id": "8343.png", "formula": "\\begin{align*} \\delta ^ 2 = - Q ( v ) Q ( w ) \\in \\Z _ { ( p ) } ^ \\times . \\end{align*}"}
-{"id": "7566.png", "formula": "\\begin{gather*} \\varphi _ { z z } = 6 \\big ( C _ 3 + \\wp ( z ; 0 , C _ 1 ) \\big ) \\varphi . \\end{gather*}"}
-{"id": "1666.png", "formula": "\\begin{align*} \\tau ^ 1 ( x ) = \\begin{cases} \\tau ^ { - 1 } _ { e } \\ ; \\ ; x \\in R _ { e } \\\\ \\tau ^ { - 1 } _ { f } \\ ; \\ ; x \\in R _ { f } \\\\ \\tau ^ { - 1 } _ g \\ ; \\ ; x \\in R _ { g } \\end{cases} \\end{align*}"}
-{"id": "2730.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ d } \\ , \\mathcal L ( h ) \\varphi _ i \\ , d v = 0 \\ , , i = 1 , \\cdots , n \\ , . \\end{align*}"}
-{"id": "3117.png", "formula": "\\begin{align*} ( H _ x \\psi ) ( g ) \\ ; : = \\ ; \\sum \\limits _ { h \\in K } t _ h ( g ^ { - 1 } x ) \\cdot \\psi ( g h ) + v ( g ^ { - 1 } x ) \\cdot \\psi ( g ) \\ , , \\end{align*}"}
-{"id": "4392.png", "formula": "\\begin{align*} \\widetilde { C } _ { w _ 1 } \\widetilde { C } _ { w _ 2 } = \\widetilde { C } _ { w _ 1 + w _ 2 } e ^ { \\frac { \\alpha } { 2 } ( \\langle w _ 2 , w _ 1 \\rangle - \\langle w _ 1 , w _ 2 \\rangle ) } \\end{align*}"}
-{"id": "686.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\left ( \\cfrac { 1 } { a _ n } - \\cfrac { 1 } { a _ n ^ 0 } \\right ) = 0 . \\end{align*}"}
-{"id": "431.png", "formula": "\\begin{align*} - R - \\pi \\abs { t } + \\kappa q _ \\delta ( \\sigma _ \\delta ) = i R \\phi _ \\omega ( i y _ \\omega ) = - \\frac { 1 } { 4 } d ( x , t ) ^ 2 . \\end{align*}"}
-{"id": "739.png", "formula": "\\begin{align*} \\mathcal { L } ^ 1 ( \\{ | ( u \\ast v ) ( x ' , y ) | > t \\} ) = \\mathcal { L } ^ 1 ( \\{ | u ( x ' , y ) | > t \\} ) + \\mathcal { L } ^ 1 ( \\{ | v ( x ' , y ) | > t \\} ) . \\end{align*}"}
-{"id": "3128.png", "formula": "\\begin{align*} \\sum _ { ( i , j ) \\in E } \\left [ \\frac { p _ { i j } } { L _ i + L _ j } ( e _ i - e _ j ) ( L _ i e _ i ^ T - L _ j e _ j ^ T ) \\otimes I _ n \\right ] x = \\frac { 1 } { N - 1 } x . \\end{align*}"}
-{"id": "5012.png", "formula": "\\begin{align*} \\varphi _ j ( x ) : = 2 ^ { j d } \\varphi ( 2 ^ j x ) , \\end{align*}"}
-{"id": "4427.png", "formula": "\\begin{align*} \\psi _ T ( k ) = \\exp ( - T ( | k _ 1 | ^ 3 + k _ 2 ^ 2 ) ) , \\forall k \\in \\R ^ 2 , \\end{align*}"}
-{"id": "2758.png", "formula": "\\begin{align*} \\Delta ^ { ( 0 ) } & = ( \\alpha _ 1 ( y H ) , \\alpha _ 2 ( H ) , \\alpha _ 3 ( y H ) ) \\\\ \\Delta ^ { ( 1 ) } & = ( \\alpha _ 1 ( y H ) , \\alpha _ 2 ( H x ) , \\alpha _ 3 ( y H x ) ) \\\\ \\Delta ^ { ( 2 ) } & = ( \\alpha _ 1 ( y H ) , \\alpha _ 2 ( H x ^ 2 ) , \\alpha _ 3 ( y H x ^ 2 ) ) \\end{align*}"}
-{"id": "8825.png", "formula": "\\begin{align*} z \\in V _ { f } \\Leftrightarrow \\exists y \\Big [ z = f ( y ) \\wedge \\dfrac { \\partial f } { \\partial x _ { 1 } } ( y ) = 0 \\wedge \\ldots \\wedge \\dfrac { \\partial f } { \\partial x _ { n } } ( y ) = 0 \\Big ] . \\end{align*}"}
-{"id": "2766.png", "formula": "\\begin{align*} w \\Big ( p Q - \\eta ( Q - Q _ 1 ) \\Big ) = ( p r _ j + p m _ j ) - ( p r _ j - \\lfloor p r _ j \\rfloor ) = p m _ j + \\lfloor p r _ j \\rfloor = \\big \\lfloor w ( p Q ) \\big \\rfloor . \\end{align*}"}
-{"id": "9009.png", "formula": "\\begin{align*} \\left \\{ \\aligned & \\partial _ { t } \\theta + ( { u } \\cdot \\nabla ) \\theta + \\mu \\Lambda _ { x _ { 1 } } ^ { 2 \\alpha } \\theta + \\nu \\Lambda _ { x _ { 2 } } ^ { 2 \\beta } \\theta = 0 , x = ( x _ { 1 } , x _ { 2 } ) \\in \\mathbb { R } ^ 2 , \\ , \\ , t > 0 , \\\\ & { u } = - \\nabla p - \\theta e _ { 2 } , \\\\ & \\nabla \\cdot { u } = 0 , \\\\ & \\theta ( x , 0 ) = \\theta _ { 0 } ( x ) . \\endaligned \\right . \\end{align*}"}
-{"id": "8010.png", "formula": "\\begin{align*} & \\sup _ { R \\in \\mathcal { D } _ { \\mu } } { \\Big ( \\frac { 1 } { | R | } \\int _ R { \\sum _ { k = \\max { ( 0 , \\mu - 4 ) } } ^ { \\infty } { 2 ^ { ( s + m ) k q } \\big | \\Pi _ k f ( x ) \\big | ^ q } } d x \\Big ) ^ { 1 / q } } \\\\ & \\lesssim \\sup _ { 0 \\leq k \\leq \\mu - 1 } \\big \\Vert 2 ^ { k ( m + s ) } \\Pi _ k f \\big \\Vert _ { L ^ { \\infty } } + \\sup _ { R \\in \\mathcal { D } _ { \\mu } } \\Big ( \\frac { 1 } { | R | } \\int _ R { \\sum _ { k = \\mu } ^ { \\infty } { 2 ^ { ( m + s ) k q } \\big | \\Pi _ k f ( x ) \\big | ^ q } } d x \\Big ) ^ { 1 / q } \\end{align*}"}
-{"id": "963.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\max _ { \\theta \\in \\mathcal { G } _ n } | E [ U _ n ( \\theta ) ] | \\geq \\liminf _ { n \\to \\infty } | E [ U _ n ( \\vartheta _ n ) ] | = \\rho _ { m ^ * } \\Sigma ( \\theta _ { m ^ * } ) = \\max _ { 1 \\leq m \\leq M } | \\rho _ m | \\Sigma ( \\theta _ { m } ) , \\end{align*}"}
-{"id": "823.png", "formula": "\\begin{align*} \\left | \\frac { \\| u ^ h _ { n + 1 } \\| ^ 2 - \\| u ^ h _ n \\| ^ 2 } { 2 \\tau } + \\| \\vec { V } ^ h _ { n + \\frac { 1 } { 2 } } \\| ^ 2 \\right | \\leq \\frac { C ( \\tau ) \\delta } { \\tau } h ^ p \\| u _ 0 \\| _ { H ^ { p + 1 } } ^ 2 , \\delta = \\max _ n \\delta _ n . \\end{align*}"}
-{"id": "7197.png", "formula": "\\begin{align*} \\int _ D \\phi \\ , d x = 1 , | \\phi | \\lesssim 1 | \\nabla \\phi ( x ) | \\lesssim 1 . \\end{align*}"}
-{"id": "1268.png", "formula": "\\begin{align*} E _ 0 = E _ 1 ^ { ( k ) } + t E ^ { ( k ) } _ 2 \\end{align*}"}
-{"id": "1067.png", "formula": "\\begin{align*} \\log H _ { k } \\leq \\frac { q _ { k } } { m } + q _ { k } \\cdot \\frac { m - \\widehat { w } _ { n } ( \\zeta ) } { m ( 1 + \\widehat { w } _ { n } ( \\zeta ) ) } + \\epsilon _ { 2 } q _ { k } = q _ { k } \\left ( \\frac { m + 1 } { m ( 1 + \\widehat { w } _ { n } ( \\zeta ) ) } + \\epsilon _ { 2 } \\right ) . \\end{align*}"}
-{"id": "3459.png", "formula": "\\begin{align*} U ^ n _ h = \\sum _ { i = 1 } ^ { d } \\alpha _ { h , i } ^ { n } \\psi _ i \\end{align*}"}
-{"id": "2574.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { k ^ { \\prime } - 1 } \\left ( B ^ { r ^ { } _ i } _ { m _ { i } n _ { i } } - B ^ { r ^ { \\prime } _ i } _ { m _ { i } n _ { i + 1 } } \\right ) = 0 \\mod N , \\end{align*}"}
-{"id": "8254.png", "formula": "\\begin{align*} S _ { 1 , \\nu } = \\sum _ { n _ { \\nu } \\geq 0 } f _ { \\nu } ( n _ { \\nu } ) \\abs { y _ { \\nu } q _ { \\nu } a ^ { - 1 } } _ { \\nu } ^ { 4 \\epsilon + 4 \\abs { \\sigma } - 1 } ( 1 + \\abs { y _ { \\nu } q _ { \\nu } a ^ { - 1 } } _ { \\nu } ) ^ { 2 + 4 \\epsilon } \\abs { W _ { \\nu , s } ( y _ { \\nu } q _ { \\nu } a ^ { - 1 } ) } ^ 4 . \\end{align*}"}
-{"id": "9592.png", "formula": "\\begin{align*} \\langle f , g \\rangle = \\bigg \\langle f , \\sum _ { k \\in \\Bbb Z } T ^ { - 1 } _ N D _ k ^ N D _ k ( g ) \\bigg \\rangle = \\sum _ { k \\in \\Bbb Z } \\langle f , T ^ { - 1 } _ N D _ k ^ N D _ k ( g ) \\rangle . \\end{align*}"}
-{"id": "5935.png", "formula": "\\begin{align*} \\forall v \\in V & & \\phi ( v ) = v \\ , \\Phi \\ , v ^ { { \\rm t r } } , \\end{align*}"}
-{"id": "7797.png", "formula": "\\begin{align*} F _ { f , G } \\left ( u \\right ) : = \\int \\nolimits _ { \\mathbb { T } } f \\left ( x + u \\right ) \\left \\vert G \\left ( x \\right ) \\right \\vert d x , u \\in \\mathbb { T } \\end{align*}"}
-{"id": "2296.png", "formula": "\\begin{align*} \\partial _ t ( 1 + \\alpha ^ 2 A ) u + \\mathcal { P } [ u \\cdot \\nabla ( 1 - \\alpha ^ 2 \\Delta ) u - \\alpha ^ 2 \\nabla u ^ T \\cdot \\Delta u ] = - \\nu ( 1 + \\alpha ^ 2 A ) A ^ s u . \\end{align*}"}
-{"id": "7745.png", "formula": "\\begin{align*} F _ N ( 1 ) & = \\frac { 1 } { N } \\sum ^ { N - 1 } _ { n = 1 } \\frac { 1 - \\cos \\phi _ n } { ( 1 - \\cos \\phi _ n ) ^ 2 } = \\frac { 1 } { 2 N } \\sum ^ { N - 1 } _ { n = 1 } \\frac { 1 } { \\sin ^ 2 \\phi _ n / 2 } \\ , , \\\\ F _ N ( 2 ) & = \\frac { 1 } { N } \\sum ^ { N - 1 } _ { n = 1 } \\frac { 1 - \\cos 2 \\phi _ n } { ( 1 - \\cos \\phi _ n ) ^ 2 } = \\frac { 2 } { N } \\sum ^ { N - 1 } _ { n = 1 } \\frac { \\cos ^ 2 \\phi _ n / 2 } { \\sin ^ 2 \\phi _ n / 2 } \\ , . \\end{align*}"}
-{"id": "17.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\sigma = 0 ; \\end{align*}"}
-{"id": "9211.png", "formula": "\\begin{align*} \\gamma ( a ^ { - } ) = - a ^ { - } , \\gamma ( a ^ { + } ) = a ^ { + } , \\gamma ( c ) = - c , \\gamma ( e ) = e , \\gamma ( c ' ) = - c ' , \\gamma ( e ' ) = e ' , \\end{align*}"}
-{"id": "5506.png", "formula": "\\begin{align*} & 1 - \\widehat F ( s ) \\sim s ^ \\alpha \\ell ( 1 / s ) q _ 0 ( s ) s \\downarrow 0 ; \\\\ & \\begin{cases} \\lim _ { n \\to \\infty } \\frac { ( r ^ n z ) ^ \\alpha } { \\ell ( r ^ n z ) } \\overline F ( r ^ n z ) = p ( z ) z \\in C _ p , & \\alpha \\in [ 0 , 1 ) , \\\\ \\lim _ { n \\to \\infty } \\ell ( r ^ n ) ^ { - 1 } \\int _ 0 ^ { r ^ n z } y \\dd F ( y ) = p ( z ) z \\in C _ p , & \\alpha = 1 . \\end{cases} \\end{align*}"}
-{"id": "1346.png", "formula": "\\begin{align*} { \\cal T } _ { { \\cal S } _ + ^ p } ( \\overline { M } _ + ) = \\{ B \\in { \\cal S } ^ p \\ , | \\ , B = \\Pi _ { { \\cal S } _ + ^ p } ^ { \\prime } ( \\overline { M } _ + ; B ) \\} = \\{ B \\in { \\cal S } ^ p \\ , | \\ , [ P _ \\beta \\ P _ \\gamma ] ^ T B [ P _ \\beta \\ P _ \\gamma ] \\succeq 0 \\} \\ , . \\end{align*} % \\end{align*}"}
-{"id": "5381.png", "formula": "\\begin{align*} R _ 1 : = a \\cdot b ^ c , \\ , R _ 2 : = a b \\cdot b ^ c , \\ , R _ 3 : = a b \\cdot a ^ c , \\ , R _ 4 : = c \\cdot b ^ { c a } , \\ , R _ 5 : = c ^ a \\cdot c ^ { b c } . \\end{align*}"}
-{"id": "196.png", "formula": "\\begin{align*} f ( a + ^ { \\boldsymbol { R } } c ) = f ( a ) + ^ { \\boldsymbol { S } } f ( c ) \\leq ^ { \\boldsymbol { S } } f ( b ) + ^ { \\boldsymbol { S } } f ( c ) = f ( b + ^ { \\boldsymbol { R } } c ) . \\end{align*}"}
-{"id": "9103.png", "formula": "\\begin{align*} \\mathbf { 1 } , \\ \\mathbf { 2 } = \\mathbf { 1 } + \\mathbf { 1 } , \\ldots , \\mathbf { n } = \\underbrace { \\mathbf { 1 } + \\ldots + \\mathbf { 1 } } _ { } \\end{align*}"}
-{"id": "1298.png", "formula": "\\begin{align*} \\textstyle \\sqrt [ 3 ] { a - c \\cdot t _ 1 } \\ + \\ \\sqrt [ 3 ] { a - c \\cdot t _ 2 } \\ + \\ \\sqrt [ 3 ] { a - c \\cdot t _ 3 } \\ = \\ \\sqrt [ 3 ] { \\left ( \\frac { 3 + B } { 2 } \\right ) \\ - \\ 6 \\ + \\ 3 \\sqrt [ 3 ] { \\frac { 2 7 + B ^ 2 } { 4 } } } , \\end{align*}"}
-{"id": "8705.png", "formula": "\\begin{align*} ( \\partial _ t ^ \\alpha u ) ( t , x ) = \\begin{cases} \\displaystyle \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } \\int _ 0 ^ t \\frac { \\partial _ t u ( s , x ) } { ( t - s ) ^ \\alpha } d s \\quad & \\\\ ( \\partial _ t u ) ( t , x ) \\quad & \\end{cases} \\end{align*}"}
-{"id": "6346.png", "formula": "\\begin{align*} m _ j ^ * = \\sum \\limits _ { i = 0 } ^ { \\lfloor \\frac { j - 1 } { 2 } \\rfloor } \\binom { j - 1 } { i } ( z _ 1 ^ * ) ^ { i + 1 } ( z _ 2 ^ * ) ^ i ( z _ 1 ^ * + z _ 2 ^ * ) ^ { j - 1 - i } \\frac { 1 } { i + 1 } \\binom { 2 i } { i } . \\end{align*}"}
-{"id": "2927.png", "formula": "\\begin{align*} C _ { t + 1 } & = C _ t \\cup \\{ e _ t , e _ { t , 1 } , \\dots , e _ { t , k _ t - 1 } \\} , \\\\ A _ { t + 1 } & = A _ t \\setminus \\{ e _ t , e _ { t , 1 } , \\dots , e _ { t , k _ t - 1 } \\} \\bigcup _ { j = 1 } ^ { k _ t - 1 } \\left ( \\Delta ^ { ( n ) } _ { u _ { t , j } } \\cap U _ t \\right ) , \\\\ U _ { t + 1 } & = U _ t \\setminus \\bigcup _ { j = 1 } ^ { k _ t - 1 } \\Delta ^ { ( n ) } _ { u _ { t , j } } , \\\\ N _ { t + 1 } & = N _ t \\cup \\{ u _ { t , 1 } , \\dots , u _ { t , k _ t - 1 } \\} . \\end{align*}"}
-{"id": "9218.png", "formula": "\\begin{align*} b ( \\alpha _ { 1 } \\alpha _ { 2 } ) = \\eta ( ( \\eta ( \\alpha _ { 2 } ) \\eta ( \\alpha _ { 1 } ) ) \\eta ( b ) ) = \\eta ( \\eta ( \\alpha _ { 2 } ) ( \\eta ( \\alpha _ { 1 } ) \\eta ( b ) ) ) = \\eta ( \\eta ( \\alpha _ { 2 } ) \\eta ( ( b \\alpha _ { 1 } ) ) ) = ( b \\alpha _ { 1 } ) \\alpha _ { 2 } . \\end{align*}"}
-{"id": "329.png", "formula": "\\begin{align*} \\sum _ { i \\in I } l _ i ( x ) = \\sum _ { j \\in J } m _ j ( x ) , { } ^ { \\forall } x \\in \\mathbb { R } . \\end{align*}"}
-{"id": "8895.png", "formula": "\\begin{align*} \\| X J _ { \\theta , 1 } ^ { - 1 } U _ \\mu ^ { - n } p \\| ^ 2 = \\| g \\psi _ { - n } \\| ^ 2 + 2 \\operatorname { R e } ( g \\psi _ { - n } , g p \\kappa _ { - n } ) + \\| g p \\kappa _ { - n } \\| ^ 2 , \\ \\ \\ n \\in \\mathbb N . \\end{align*}"}
-{"id": "4874.png", "formula": "\\begin{align*} \\mathcal { I } = \\{ A \\subseteq \\mathbf { N } : \\varphi ( A ) < \\infty \\} , \\end{align*}"}
-{"id": "345.png", "formula": "\\begin{align*} - \\frac { 1 } { H ^ { 2 + \\sigma } } B = | \\nabla f | ^ 2 \\frac { \\partial } { \\partial t } U _ { \\sigma } - \\left ( 2 C _ 0 - \\frac { 2 + \\sigma } { H } \\langle \\nabla H , \\nabla f \\rangle - 2 H | \\nabla f | \\right ) U _ { \\sigma } - \\frac { 8 L _ { 2 2 } } { H ^ { \\frac { 2 + \\sigma } { 2 } } | \\nabla f | ^ { 3 } } \\sqrt { U _ \\sigma } , \\end{align*}"}
-{"id": "2399.png", "formula": "\\begin{align*} 4 a _ 4 - 9 a _ 6 = 4 a _ 3 + 5 a _ 6 = 9 a _ 2 + a _ 5 = 9 a _ 1 a _ 6 + a _ 5 = 8 a _ 1 ^ 2 - 1 = 0 , \\ a _ 6 \\neq 0 . \\end{align*}"}
-{"id": "10086.png", "formula": "\\begin{align*} ( \\nabla _ { X } \\pi ) ( Y ) = \\nabla _ { X } \\pi ( Y ) - \\pi ( \\nabla _ { X } Y ) = 0 \\end{align*}"}
-{"id": "1803.png", "formula": "\\begin{align*} ( x _ 1 ' , x _ 2 ' , x _ 3 ' ) : = ( x _ { 2 } , x _ { 3 } , x _ 1 + x _ { 2 } - \\max ( x _ { 2 } , x _ { 3 } ) ) . \\end{align*}"}
-{"id": "1891.png", "formula": "\\begin{align*} \\min \\{ F _ i ^ \\ast ( x _ i ) : i = 1 , \\dots , d \\} \\wedge \\min \\Big \\{ \\pi _ s + \\sum _ { i = 1 } ^ d ( F _ i ^ \\ast ( x _ i ) - F _ i ^ \\ast ( s _ i ) ) ^ + : s \\in S \\Big \\} \\end{align*}"}
-{"id": "1384.png", "formula": "\\begin{align*} ( v ( 1 ) , v ( 2 ) , \\ldots ) & = ( \\xi _ { 1 } + 1 , \\underbrace { 0 , \\ldots , 0 } _ { \\eta _ 1 } , \\xi _ { 2 } + 1 , \\underbrace { 0 , \\ldots , 0 } _ { \\eta _ 2 } , \\ldots ) , \\end{align*}"}
-{"id": "6522.png", "formula": "\\begin{align*} K _ { \\delta } = \\bigcap \\limits _ { | K \\cap H ^ { - } | _ n \\leq \\delta | K | _ n } H ^ + , \\end{align*}"}
-{"id": "6263.png", "formula": "\\begin{align*} \\zeta _ 1 + \\zeta _ 2 + \\zeta _ 3 + \\zeta _ 4 = \\delta _ 1 \\alpha - 1 , \\ \\ \\zeta _ 1 , . . . , \\zeta _ 4 > 0 \\\\ \\frac 1 { \\theta _ 1 } + \\frac 1 { \\theta _ 2 } + \\frac 1 { \\theta _ 3 } + \\frac 1 { \\theta _ 4 } = 1 , \\ \\ \\theta _ 1 = \\frac 2 { \\delta _ 1 } , \\ \\ \\theta _ 3 = \\frac 2 { \\delta _ 2 } , \\ \\ 1 < \\theta _ 2 , \\theta _ 4 < \\infty , \\end{align*}"}
-{"id": "4492.png", "formula": "\\begin{align*} | \\Lambda ( x _ 0 ) | \\leq \\| \\Theta _ 1 \\| \\frac { | \\Theta _ 1 ( x _ 0 ) | } { \\| \\Theta _ 1 \\| } + \\| \\Theta _ 2 \\| \\frac { | \\Theta _ 2 ( x _ 0 ) | } { \\| \\Theta _ 2 \\| } < \\| \\Theta _ 1 \\| + \\| \\Theta _ 2 \\| = 1 . \\end{align*}"}
-{"id": "135.png", "formula": "\\begin{align*} \\hat h ( z ) = \\Phi ( a z + \\lambda b + \\mu c ) , \\hat f ( z ) = \\Phi ( a z + b ) , \\hat g ( z ) = \\Phi ( a z + c ) \\end{align*}"}
-{"id": "1907.png", "formula": "\\begin{align*} ( \\psi _ { n + 1 } ( x ) , x ) = ( F _ { n } \\phi _ { n } ( w ) + a _ { n } ( \\phi _ { n } w , w ) , F _ { n } ( w ) + b _ { n } ( \\phi _ { n } w , w ) ) = \\tilde { f } _ { n } ( \\phi _ { n } w , w ) . \\end{align*}"}
-{"id": "4824.png", "formula": "\\begin{align*} \\mathcal { B } ( G _ 0 ) : = \\{ X \\in \\mathcal { B } ( G ) \\mid \\mathsf { v } _ g ( X ) = 0 \\ \\emph { i f } \\ g \\notin G _ 0 \\} \\end{align*}"}
-{"id": "2514.png", "formula": "\\begin{align*} \\nabla _ x ( \\varrho \\ , m _ 0 ) = \\frac { 1 } { D } \\frac { x } { \\left < x \\right > } \\varrho \\ , m _ 0 , ( \\partial _ i \\sigma ^ { i j } ) \\partial _ j ( \\varrho \\ , m _ 0 ) = - \\lambda _ 1 ( \\xi ) \\xi _ j \\varrho \\partial _ j m _ 0 , \\partial _ { i j } ( \\varrho \\ , m _ 0 ) = \\varrho \\partial _ { i j } m _ 0 , \\end{align*}"}
-{"id": "9552.png", "formula": "\\begin{align*} { ( - 1 ) ^ { \\ell ( \\hat \\sigma ) } x _ 1 ^ { o ( \\hat \\sigma ) } x _ 2 ^ { e ( \\hat \\sigma ) } y ^ { o ( \\hat \\sigma ) } z ^ { e ( \\hat \\sigma ) } } = ( - 1 ) ^ { n - 1 } y ^ { \\lceil \\frac { n - 1 } { 2 } \\rceil } { ( - 1 ) ^ { \\ell ( \\sigma ) } x _ 1 ^ { e ( \\sigma ) } x _ 2 ^ { o ( \\sigma ) } y ^ { o ( \\sigma ) } z ^ { e ( \\sigma ) } } \\\\ \\end{align*}"}
-{"id": "8862.png", "formula": "\\begin{align*} N ^ { - 1 - \\frac { 1 } { d } + \\frac { 1 - \\alpha } { d } \\ , ( 2 s - d + 1 ) } \\ , \\sum _ { i , j = 1 } ^ { N } P _ { n } ^ { ( d ) } ( \\langle \\mathbf { x } _ { i } , \\mathbf { x } _ { j } \\rangle ) = \\mathcal { O } ( \\left ( \\Psi ( N ) \\right ) ^ { 2 s - d + 1 } ) \\end{align*}"}
-{"id": "3432.png", "formula": "\\begin{align*} \\| f \\| _ { C ^ { \\gamma } ( [ 0 , T ] ; H ) } = \\sup _ { t \\in [ 0 , T ] } \\| f ( t ) \\| _ H + \\sup _ { \\substack { s , t \\in [ 0 , T ] \\\\ s \\neq t } } \\frac { \\| f ( s ) - f ( t ) \\| _ H } { | s - t | ^ { \\gamma } } . \\end{align*}"}
-{"id": "1258.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } J _ m = ( n - p ) \\int _ { D ' } \\hat f ( \\nabla \\hat u ) d x \\end{align*}"}
-{"id": "5249.png", "formula": "\\begin{align*} Y ' ( z ) = A ( \\lambda , z ) Y ( z ) . \\end{align*}"}
-{"id": "6517.png", "formula": "\\begin{align*} \\frac { \\mathcal { C } _ { \\mathcal { M } ^ { } } \\left [ \\mathcal { D } _ { \\theta } ^ { } \\left ( \\tau ; \\rho \\right ) \\right ] } { \\mathcal { C } _ { \\mathcal { M } ^ { } } \\left [ \\mathcal { D } _ { \\theta } ^ { } \\left ( \\tau ; 0 \\right ) \\right ] } = \\sqrt { \\frac { 1 - \\rho } { 1 + \\rho } } . \\end{align*}"}
-{"id": "8352.png", "formula": "\\begin{align*} \\mathcal { Z } ( \\Lambda ) ( S ) = \\{ \\mbox { i s o m e t r i c e m b e d d i n g s } \\iota : \\Lambda \\to V _ 0 ( \\mathcal { A } _ S ) \\} \\end{align*}"}
-{"id": "9112.png", "formula": "\\begin{align*} a _ { n + 1 - ( i + 1 ) } = ( - 1 ) ^ { i + 1 } \\binom { n + 1 } { i + 1 } a _ { n + 1 } \\left ( 0 \\leq i \\leq n - 1 \\right ) . \\end{align*}"}
-{"id": "388.png", "formula": "\\begin{align*} & \\mathbb P \\left ( | S _ n | > ( 1 + \\varepsilon ) \\sigma _ n \\sqrt { 2 \\ln \\Psi ( n ) } \\right ) \\\\ & = 2 \\left ( 1 - \\Phi ( x _ n ) \\right ) \\left ( 1 + o ( 1 ) \\right ) \\\\ & = 2 ( 2 \\pi ) ^ { - 1 / 2 } \\frac { 1 } { ( 1 + \\varepsilon ) \\sqrt { 2 \\ln \\Psi ( n ) } } \\exp \\left ( - ( 1 + \\varepsilon ) ^ 2 \\ln \\Psi ( n ) \\right ) ( 1 + o ( 1 ) ) \\\\ & \\propto \\frac { 1 } { \\sqrt { \\ln \\Psi ( n ) } } \\big ( \\Psi ( n ) \\big ) ^ { - ( 1 + \\varepsilon ) ^ 2 } . \\end{align*}"}
-{"id": "2050.png", "formula": "\\begin{gather*} T _ { 1 , 0 } M \\cap T _ { 0 , 1 } M = 0 \\\\ [ \\Gamma ( T _ { 1 , 0 } M ) , \\Gamma ( T _ { 1 , 0 } M ) ] \\subset \\Gamma ( T _ { 1 , 0 } M ) \\end{gather*}"}
-{"id": "4032.png", "formula": "\\begin{align*} \\mathrm { h e s s } _ b f : = X ( Y f ) - ( \\hat { \\nabla } _ X Y ) f , \\end{align*}"}
-{"id": "174.png", "formula": "\\begin{align*} \\mathbf { R } ^ { \\wp ( A ) } = \\langle \\wp ( A ) , \\subseteq , \\cup , \\emptyset \\rangle . \\end{align*}"}
-{"id": "1370.png", "formula": "\\begin{align*} W _ 2 ( H _ 2 ) = P \\left ( \\Theta _ c \\circ ( P ^ T H _ 2 P ) \\right ) P ^ T , \\forall \\ , H _ 2 \\in { \\cal S } ^ p \\ , \\end{align*}"}
-{"id": "8446.png", "formula": "\\begin{align*} \\chi ( 1 + z \\varpi ^ { \\alpha } ) = \\psi \\left ( \\frac { b _ { \\chi } } { \\varpi ^ { a ( \\chi ) } } \\left ( z \\varpi ^ { \\alpha } - \\frac { z ^ 2 } { 2 } \\varpi ^ { 2 \\alpha } \\right ) \\right ) z \\in \\mathcal { O } . \\end{align*}"}
-{"id": "9031.png", "formula": "\\begin{align*} \\textstyle \\Phi _ { b d } = \\frac 1 { 2 ( n + 1 ) } R _ { a b } { } ^ a { } _ d = \\frac 1 { 4 ( n + 1 ) } J ^ { a e } R _ { a e } { } ^ c { } _ b J _ { c d } . \\end{align*}"}
-{"id": "3480.png", "formula": "\\begin{align*} f ( t , x ) & = ( x ^ 2 - x ^ 3 ) \\dot { w } ( t ) - ( 2 - 6 x ) w ( t ) + \\sin ( \\pi x ) + b ( ( x ^ 2 - x ^ 3 ) w ( t ) + \\pi ^ { - 2 } \\sin ( \\pi x ) ) , \\\\ u _ 0 ( x ) & = \\pi ^ { - 2 } \\sin ( \\pi x ) \\end{align*}"}
-{"id": "3367.png", "formula": "\\begin{align*} f ( g ( y ) \\wedge x ) = y \\wedge f ( x ) , \\quad x \\in X , y \\in Y . \\end{align*}"}
-{"id": "3680.png", "formula": "\\begin{align*} \\iota P ^ { ( n ) } = \\frac { n + 1 } { n + 2 } \\cdot Q ' ( P _ b [ n ] - u ) . \\end{align*}"}
-{"id": "2727.png", "formula": "\\begin{align*} & \\displaystyle \\Lambda ( h ) = \\nu ( v ) h , K ( h ) = \\mathcal L ^ { + } ( h ) - \\mathcal L ^ { \\ast } ( h ) , \\mathcal L ^ { \\ast } ( h ) = M [ ( h M ) \\ast \\phi ] , \\\\ [ 2 p t ] & \\displaystyle \\mathcal L ^ { + } ( h ) = \\int _ { \\mathbb R ^ d \\times \\mathbb S ^ { d - 1 } } \\phi ( | v - v _ { \\ast } | ) b ( \\cos \\theta ) \\ , [ h ^ { \\prime } M _ { \\ast } ^ { \\prime } + h _ { \\ast } ^ { \\prime } M ^ { \\prime } ] M _ { \\ast } \\ , d v _ { \\ast } d \\sigma \\ , . \\end{align*}"}
-{"id": "5617.png", "formula": "\\begin{align*} S y m ( \\mathbb { W } _ { s , t } ) = \\frac { 1 } { 2 } W _ { s , t } \\otimes W _ { s , t } , \\end{align*}"}
-{"id": "4860.png", "formula": "\\begin{align*} \\xi ( 1 , t ) = \\xi ( t ) = \\begin{vmatrix} v _ 2 ( C ) ( t ) \\\\ v _ 2 ( C ) ' ( t ) \\\\ v _ 2 ( C ) '' ( t ) \\\\ v _ 2 ( C ) ^ { 3 } ( t ) \\\\ v _ 2 ( C ) ^ { 4 } ( t ) \\\\ v _ 2 ( C ) ^ { 5 } ( t ) \\\\ \\end{vmatrix} . \\end{align*}"}
-{"id": "6684.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\Delta } \\left ( p ( 1 - \\sigma ( s ) ) ^ { p - 1 } s \\right ) ^ { \\frac { n - 1 } { p } } \\mathrm { d } s \\geq ( 1 - \\Delta ) ^ { \\frac { p - 1 } { p } ( n - 1 ) } \\int _ 0 ^ { \\Delta } ( p s ) ^ { \\frac { n - 1 } { p } } \\mathrm { d } s = ( 1 - \\Delta ) ^ { \\frac { p - 1 } { p } ( n - 1 ) } \\frac { 1 } { n - 1 + p } ( p \\Delta ) ^ { \\frac { n - 1 + p } { p } } \\quad . \\end{align*}"}
-{"id": "10133.png", "formula": "\\begin{align*} r _ H ( X , Y ) = \\max \\bigg \\{ \\sup _ { x \\in X } \\inf _ { y \\in Y } r ( x , y ) , \\sup _ { y \\in Y } \\inf _ { x \\in X } r ( x , y ) \\bigg \\} . \\end{align*}"}
-{"id": "1086.png", "formula": "\\begin{align*} f _ { \\varLambda _ i } ( \\varLambda _ i ) = \\frac { 1 } { m _ { c , i } } \\mathrm { e x p } \\left ( - \\frac { \\varLambda _ i + m _ { s , i } } { m _ { c , i } } \\right ) I _ 0 \\left ( \\frac { 2 \\sqrt { \\varLambda _ i m _ { s , i } } } { m _ { c , i } } \\right ) , \\end{align*}"}
-{"id": "9157.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } ( - 1 ) ^ { i } \\binom { n + 1 } { i } x ^ { i } A \\left ( x ^ { n + 1 - i } \\right ) = 0 \\left ( x \\in K \\right ) . \\end{align*}"}
-{"id": "3876.png", "formula": "\\begin{align*} E X _ + ^ p = \\frac { \\Gamma ( p + 1 ) } { \\pi } \\int _ 0 ^ \\infty \\Re \\frac { \\phi ( u ) - 1 - i u E X } { ( i u ) ^ { p + 1 } } d u , \\end{align*}"}
-{"id": "3174.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ e ^ { u Z _ { t } ^ { 2 } } \\right ] = e ^ { \\lambda _ { t } ( \\widehat { \\rho } _ { t } ( u ) - 1 ) } , \\quad ( t , u ) \\in \\mathbb { R } _ { > 0 } \\times \\mathcal { U } , \\end{align*}"}
-{"id": "5543.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } h w _ { n } \\left ( t \\right ) w _ { m } \\left ( t \\right ) d t = \\left \\{ \\begin{array} { c } h n = m \\\\ 0 n \\not = m \\end{array} \\right . \\end{align*}"}
-{"id": "9729.png", "formula": "\\begin{align*} \\tilde { \\gamma } _ { 1 } ( \\gamma _ 1 , Z _ { a } h , V _ a ) & = \\tilde { \\gamma } _ { 1 } ( \\gamma _ 1 , 0 , V _ a ) + \\tilde { \\gamma } _ { 1 } ( \\gamma _ 1 , Z _ { a } h , V _ a ) - \\tilde { \\gamma } _ { 1 } ( \\gamma _ 1 , 0 , V _ a ) = \\gamma _ 1 + O ( 1 ) Z _ { a } h . \\end{align*}"}
-{"id": "6872.png", "formula": "\\begin{align*} \\lambda _ { m } ( \\lambda _ { m + 1 } , p ) = \\frac { ( 4 - \\lambda _ { m + 1 } ) ( \\lambda _ { m + 1 } - 2 ) ^ 2 \\lambda _ { m + 1 } } { 4 p q } \\end{align*}"}
-{"id": "6669.png", "formula": "\\begin{align*} \\frac { \\| x _ { \\delta } \\| _ 2 } { \\| x \\| _ 2 } \\leq \\frac { \\langle x _ { \\delta } , u ( x ) \\rangle } { \\langle x , u ( x ) \\rangle } = \\frac { \\langle x - \\Delta ( x , \\delta ) u ( x ) , u ( x ) \\rangle } { \\langle x , ( u ( x ) ) \\rangle } = 1 - \\frac { \\Delta ( x , \\delta ) } { \\langle x , u ( x ) \\rangle } \\leq 1 - \\Gamma _ { \\min } ( \\delta ) \\quad , \\end{align*}"}
-{"id": "1881.png", "formula": "\\begin{align*} \\phi _ 0 ( \\ 1 _ B ) = \\min _ { s \\in \\{ 0 , 1 \\} } \\Big \\{ s \\nu _ 1 ( F _ 1 ^ { i _ 1 - 1 } ) + s \\phi _ 0 ( \\ 1 _ { B \\setminus S } ) + ( 1 - s ) \\eta ( \\ 1 _ B ) \\Big \\} , \\end{align*}"}
-{"id": "9577.png", "formula": "\\begin{align*} \\bigg ( \\int _ { \\R ^ n } \\mathcal { I } _ { s } ( \\chi _ D g ) ( x ) ^ q w ( x ) \\ , d x \\bigg ) ^ { p / q } & \\le C \\int _ { \\R ^ n } \\chi _ D ( y ) g ( y ) ^ p \\ , \\delta _ { \\partial D } ^ { \\beta } ( y ) \\ , d y \\\\ & = C \\int _ D \\int _ { B ( y , \\tau \\delta _ { \\partial D } ( y ) ) } \\frac { \\vert u ( y ) - u ( z ) \\vert ^ p } { \\vert y - z \\vert ^ { n + s p } } \\ , d z \\ , \\delta _ { \\partial D } ^ { \\beta } ( y ) \\ , d y \\ , . \\end{align*}"}
-{"id": "3928.png", "formula": "\\begin{align*} \\frac { \\partial ^ \\gamma } { \\partial t ^ \\gamma } u ( x , t ) = \\begin{cases} \\frac { 1 } { \\Gamma ( 1 - \\gamma ) } \\int \\limits _ 0 ^ t \\frac { \\partial u ( x , s ) } { \\partial s } ( t - s ) ^ { - \\gamma } d s , 0 < \\gamma < 1 \\\\ \\frac { \\partial u ( x , s ) } { \\partial s } , \\qquad \\qquad \\qquad \\qquad \\qquad \\gamma = 1 . \\end{cases} \\end{align*}"}
-{"id": "7852.png", "formula": "\\begin{align*} M _ { w } ( t _ { 1 } , t _ { 2 } ) = \\frac { 1 } { F ( t _ { 1 } , t _ { 2 } ) } \\int _ { 0 } ^ { t _ { 1 } } \\int _ { 0 } ^ { t _ { 2 } } F ( x _ { 1 } , x _ { 2 } ) \\ d x _ { 2 } d x _ { 1 } . \\end{align*}"}
-{"id": "556.png", "formula": "\\begin{align*} X _ H ( \\rho , x ) = h ' ( \\rho ) \\ , R _ { \\alpha } ( x ) , \\end{align*}"}
-{"id": "2972.png", "formula": "\\begin{align*} c ^ { \\star } _ { k j } = \\frac { p _ j | h _ { k j } | ^ { 2 } } { \\sum _ { j = 1 } ^ { K } p _ j | h _ { k j } | ^ { 2 } } . \\end{align*}"}
-{"id": "4781.png", "formula": "\\begin{align*} \\alpha a _ 0 + \\beta b _ 0 + \\gamma = 0 , \\mbox { $ \\alpha $ , $ \\beta $ , $ \\gamma $ - - c o n s t . } \\end{align*}"}
-{"id": "9567.png", "formula": "\\begin{align*} \\int _ D \\int _ { B ( x , \\tau \\delta _ { \\partial D } ( x ) ) } \\frac { \\lvert u ( x ) - u ( y ) \\rvert ^ p } { \\lvert x - y \\rvert ^ { n + s p } } \\ , d y \\ , \\delta _ { \\partial D } ^ \\beta ( x ) \\ , d x \\ge C \\bigg ( \\int _ D \\lvert u ( x ) \\rvert ^ q \\delta _ { \\partial D } ^ { ( q / p ) ( n - s p + \\beta ) - n } ( x ) \\ , d x \\bigg ) ^ { p / q } \\end{align*}"}
-{"id": "4853.png", "formula": "\\begin{align*} s = & \\ ; 6 \\cdot ( 2 \\cdot 5 + 5 \\cdot 0 - 5 ) - 1 7 - 1 0 - 1 \\\\ = & \\ ; 2 . \\end{align*}"}
-{"id": "1961.png", "formula": "\\begin{align*} \\| \\widetilde { K } ^ j _ { j , \\textbf { r } } ( \\widehat { f _ 1 } , \\dots , \\widehat { f _ j } ) \\| _ { L ^ 2 _ \\alpha } ^ 2 \\le C \\left ( \\prod _ { i = 1 } ^ { j - 1 } r _ i ^ { 2 \\lambda } \\right ) \\norm { f _ { j } } _ { W _ 2 ^ { \\beta , 2 } } ^ 2 \\prod _ { l = 1 } ^ { j - 1 } \\norm { f _ l } _ { W _ 2 ^ { ( n - 1 ) / 2 - \\lambda , 2 } } ^ 2 . \\end{align*}"}
-{"id": "2936.png", "formula": "\\begin{align*} f _ \\epsilon ( x _ 1 , \\dots , x _ k ) - f _ \\epsilon ( x ' _ 1 , \\dots , x ' _ k ) = \\sum _ { i = 1 } ^ k \\frac { e ^ { - \\frac { x ' _ i } { \\epsilon } } - e ^ { - \\frac { x _ i } { \\epsilon } } } { \\left ( 1 + \\sum _ { r = 1 } ^ k e ^ { - \\frac { x _ r } { \\epsilon } } \\right ) \\left ( 1 + \\sum _ { s = 1 } ^ k e ^ { - \\frac { x ' _ s } { \\epsilon } } \\right ) } . \\end{align*}"}
-{"id": "163.png", "formula": "\\begin{align*} ( \\mathfrak { X } \\vee \\mathfrak { Y } ) ( a ) = \\sup \\{ \\mathfrak { X } ( a ) , \\mathfrak { Y } ( a ) \\} \\quad ( \\mathfrak { X } \\wedge \\mathfrak { Y } ) ( a ) = \\inf \\{ \\mathfrak { X } ( a ) , \\mathfrak { Y } ( a ) \\} \\end{align*}"}
-{"id": "1138.png", "formula": "\\begin{align*} M _ p \\sim \\begin{bmatrix} 1 & b _ p \\\\ 0 & 1 \\end{bmatrix} , M _ m \\sim \\begin{bmatrix} 1 & - b _ m \\\\ 0 & 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "4822.png", "formula": "\\begin{align*} | X | = l = \\sum _ { g \\in G } \\mathsf { v } _ g ( X ) . \\end{align*}"}
-{"id": "10140.png", "formula": "\\begin{align*} \\bar { \\boldsymbol R } _ { \\bar { \\boldsymbol \\omega } _ k } ( 0 ) & = \\bigg ( \\boldsymbol S _ { D _ k } ^ H ( 0 ) \\boldsymbol R _ k ^ 2 ( 0 ) \\boldsymbol S _ { D _ k } ( 0 ) \\bigg ) ^ { - 1 } \\\\ & \\times \\boldsymbol S _ { D _ k } ( 0 ) \\boldsymbol R _ k ( 0 ) \\boldsymbol P _ { D _ k } ( 0 ) . \\end{align*}"}
-{"id": "2676.png", "formula": "\\begin{align*} \\nu _ A = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\delta \\left ( x - \\frac { \\lambda _ { i } } { \\sqrt { N } } \\right ) , \\end{align*}"}
-{"id": "864.png", "formula": "\\begin{align*} \\Phi _ \\beta ( x ) = \\beta ^ { - 1 } \\log \\left ( \\sum _ { j = 1 } ^ d e ^ { \\beta x _ j } \\right ) \\qquad ( x = ( x _ 1 , \\dots , x _ d ) ^ \\top \\in \\mathbb { R } ^ d ) . \\end{align*}"}
-{"id": "6991.png", "formula": "\\begin{align*} \\lambda _ i = \\frac { 1 } { k } ( \\sum _ { 1 \\leq i _ 1 < . . . < i _ { k - 1 } \\leq n , i _ j \\neq i } { \\sigma _ { i _ 1 } \\sigma _ { i _ 2 } . . . \\sigma _ { i _ { k - 1 } } } ) . \\end{align*}"}
-{"id": "3204.png", "formula": "\\begin{align*} ( \\mathcal { J } _ { \\ast } V ) ( y ) = \\int _ { \\lbrace z \\leqslant 1 \\rbrace } | V ( y + z ) - V ( y ) | \\nu ( \\mathrm { d } z ) \\leqslant ( 1 + c _ { 1 } ) \\int _ { \\lbrace z \\leqslant 1 \\rbrace } z \\nu ( \\mathrm { d } z ) < \\infty . \\end{align*}"}
-{"id": "9035.png", "formula": "\\begin{gather*} \\begin{pmatrix} 0 & 2 a _ 1 ^ 2 y _ 1 ( \\neq 0 ) & 2 a _ 2 ^ 2 y _ 2 ( \\neq 0 ) & 0 \\\\ 0 & 3 b _ 1 y _ 1 ^ 2 ( \\neq 0 ) & 3 b _ 2 y _ 2 ^ 2 ( \\neq 0 ) & b _ 3 y _ 1 y _ 2 ( \\neq 0 ) \\end{pmatrix} \\end{gather*}"}
-{"id": "1769.png", "formula": "\\begin{align*} t _ { \\omega ( 0 , n ) } ^ * P ( \\{ \\omega \\} ) t _ { \\omega ( 0 , n ) } \\ ; = \\ ; P ( \\{ \\sigma ^ n ( \\omega ) \\} ) . \\end{align*}"}
-{"id": "829.png", "formula": "\\begin{align*} \\mathcal { L } _ h \\Pi _ h = Q _ h \\mathcal { L } , \\end{align*}"}
-{"id": "3596.png", "formula": "\\begin{align*} \\frac { \\mathrm d ^ k } { \\mathrm d x ^ k } y _ j ( 0 ) = 0 , k = 0 , 1 , \\ldots , \\lceil \\alpha \\rceil - 1 , j \\in \\{ 1 , 2 \\} . \\end{align*}"}
-{"id": "3249.png", "formula": "\\begin{align*} \\alpha = u \\cup \\beta . \\end{align*}"}
-{"id": "4292.png", "formula": "\\begin{align*} \\big ( 1 \\boxtimes c _ 1 ( N _ { A / B _ 2 } ) \\big ) \\cdot ( \\mathrm { i d } \\boxtimes \\pi ^ { \\ast } \\pi _ { \\ast } ) \\Delta _ D & = \\big ( 1 \\boxtimes c _ 1 ( N _ { A / B _ 2 } ) \\big ) \\cdot \\Delta _ A \\\\ & = \\big ( c _ 1 ( N _ { A / B _ 2 } ) \\boxtimes 1 \\big ) \\cdot \\Delta _ A \\\\ & = - \\big ( c _ 1 ( N _ { A / B _ 1 } ) \\boxtimes 1 \\big ) \\cdot \\Delta _ A . \\end{align*}"}
-{"id": "6578.png", "formula": "\\begin{align*} \\Delta = \\left ( \\frac { n | S | _ n } { \\| \\xi \\| | B _ 2 ^ { n - 1 } | _ { n - 1 } } \\right ) ^ { 1 / n } \\delta ^ { 1 / n } . \\end{align*}"}
-{"id": "9236.png", "formula": "\\begin{align*} ( i ) \\ ; \\epsilon ( \\beta _ { 1 } , \\beta _ { 2 } ) = - \\epsilon ( \\beta _ { 2 } , \\beta _ { 1 } ) , \\quad ( i i ) \\ \\epsilon ( \\beta _ { 1 } \\beta _ { 2 } , \\beta _ { 3 } ) + \\epsilon ( \\beta _ { 2 } \\beta _ { 3 } , \\beta _ { 1 } ) + \\epsilon ( \\beta _ { 3 } \\beta _ { 1 } , \\beta _ { 2 } ) = 0 . \\end{align*}"}
-{"id": "9474.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ n } { ( z q ^ n ; q ) _ { n + 1 } ( z q ^ { 2 n + 2 } ; q ^ 2 ) _ { \\infty } } = \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ { n } } { ( z q ; q ^ 2 ) _ n } . \\end{align*}"}
-{"id": "4948.png", "formula": "\\begin{align*} p ^ x + p ^ y = w ^ 2 \\end{align*}"}
-{"id": "5678.png", "formula": "\\begin{align*} \\left ( \\frac 1 2 | f ' | ^ 2 - \\frac { c _ 0 } { p _ 0 } f ^ { p _ 0 } \\right ) ' = f ' ( f '' - c _ 0 f ^ { p _ 0 - 1 } ) \\leq 0 . \\end{align*}"}
-{"id": "4887.png", "formula": "\\begin{align*} [ M ^ k / S _ k ] = \\bigcup _ { \\underline { k } : \\sum i k _ i = k } [ M ^ k _ { \\underline { k } } / S _ k ] . \\end{align*}"}
-{"id": "597.png", "formula": "\\begin{align*} L _ { \\Gamma } ( s , \\rho ) : = \\prod _ { \\overline { \\gamma } \\in \\overline { \\Gamma } _ { p } } \\prod _ { k = 0 } ^ { \\infty } \\det \\left ( 1 - \\rho ( \\gamma ) e ^ { - ( s + k ) \\ell ( \\gamma ) } \\right ) . \\end{align*}"}
-{"id": "1023.png", "formula": "\\begin{align*} \\partial _ t \\vert \\vec u \\vert ^ 2 = \\Delta ( \\vert \\vec u \\vert ^ 2 ) - 2 \\vert \\vec \\nabla \\otimes \\vec u \\vert ^ 2 - ( ( \\vert \\vec u \\vert ^ 2 + 2 p ) \\vec u ) + 2 \\vec u \\cdot \\vec f . \\end{align*}"}
-{"id": "9649.png", "formula": "\\begin{align*} \\hat { \\Phi } _ { \\rm { s t } , \\beta } ( t ) = \\Phi ^ * _ { \\rm { s t } , \\beta } ( t ) + \\frac { 1 } { 2 } \\frac { \\partial ^ 2 \\Phi _ { \\rm { s t } , \\beta } ( t ) } { \\partial ^ 2 R _ { \\beta } } \\Big | _ { { R } ^ { * } _ { \\beta } } \\left [ \\hat { R } ^ { * } _ { \\beta } - { R } ^ { * } _ { \\beta } \\right ] ^ 2 + O \\left [ \\left ( \\hat { R } ^ { * } _ { \\beta } - { R } ^ { * } _ { \\beta } \\right ) ^ 3 \\right ] . \\end{align*}"}
-{"id": "1885.png", "formula": "\\begin{align*} \\phi _ 0 ( \\ 1 _ B ) = \\min _ { s \\in [ 0 , 1 ] } \\Big \\{ s \\nu _ 1 ( F _ 1 ^ { i _ 1 - 1 } ) + \\phi _ 0 ^ { \\setminus i _ 1 - 1 } ( \\ 1 _ B - s \\ 1 _ S ) \\Big \\} , \\end{align*}"}
-{"id": "5324.png", "formula": "\\begin{align*} [ a \\otimes t ^ m , { \\bf k } ] = 0 , \\end{align*}"}
-{"id": "5248.png", "formula": "\\begin{align*} 0 = \\mu ( \\alpha ) = \\mu ( \\alpha _ 1 ) + \\mu ( \\alpha _ 2 ) + \\mu ( \\alpha _ 3 ) + \\mu ( \\alpha _ 4 ) . \\end{align*}"}
-{"id": "5298.png", "formula": "\\begin{align*} B ( v ) = \\int _ 0 ^ v b ( \\cdot , z ) \\mathrm { d z } , \\end{align*}"}
-{"id": "2452.png", "formula": "\\begin{align*} c _ s ( k ) & = \\binom { 9 9 } { 6 6 } \\cdot \\binom { 4 3 5 } { k - 6 7 } + \\binom { 9 9 } { 6 6 } \\cdot \\binom { 4 3 5 } { k - 6 8 } \\\\ c _ r ( k ) & = \\binom { 1 0 0 } { k - 2 9 0 } \\cdot \\binom { 4 3 4 } { 2 8 9 } + \\binom { 1 0 0 } { k - 2 9 1 } \\cdot \\binom { 4 3 4 } { 2 8 9 } . \\end{align*}"}
-{"id": "5085.png", "formula": "\\begin{align*} & [ ( b _ 1 - b _ 2 ) ^ 2 - 2 ] R _ { 1 2 1 2 } + [ ( b _ 2 - b _ 3 ) ^ 2 - 2 ] R _ { 2 3 2 3 } + [ ( b _ 1 - b _ 3 ) ^ 2 - 2 ] R _ { 1 3 1 3 } \\\\ & = \\frac { 5 } { 9 } - \\frac { 1 6 } { 3 } t r ( A ) + \\frac { 1 5 } { 2 } t r ( A ) ^ 2 + \\frac { 3 } { 2 } | A | ^ 2 - \\frac { 3 } { 2 } | R i c | ^ 2 . \\end{align*}"}
-{"id": "1782.png", "formula": "\\begin{align*} R _ 0 '' - G ' ( R _ 0 ) - \\omega _ 0 R _ 0 = 0 . \\end{align*}"}
-{"id": "9586.png", "formula": "\\begin{align*} | \\psi ( 2 ^ k { \\bar \\rho } ( x , y ) ) - \\psi ( 2 ^ k { \\bar \\rho } ( x ' , y ) ) | & \\le C 2 ^ k ( { \\bar \\rho } ( x , x ' ) ) ^ \\varepsilon [ { \\bar \\rho } ( x , y ) + { \\bar \\rho } ( x ' , y ) ] ^ { 1 - \\varepsilon } \\\\ & \\le C 2 ^ k 2 ^ { - k ( 1 - \\varepsilon ) } ( { \\bar \\rho } ( x , x ' ) ) ^ \\varepsilon \\\\ & = C ( 2 ^ k { \\bar \\rho } ( x , x ' ) ) ^ \\varepsilon \\qquad \\ { \\bar \\rho } ( x , x ' ) \\le ( A _ 0 ) ^ 3 2 ^ { 5 - k } . \\end{align*}"}
-{"id": "3910.png", "formula": "\\begin{align*} \\lim _ { { \\cal L } \\to \\infty } \\frac { \\langle H \\rangle ^ { { \\cal L } } _ { m i n } } { { \\cal E } ^ { { \\cal L } } _ { 0 } } = \\lim _ { { \\cal L } \\to \\infty } \\frac { 2 \\sqrt { \\frac { { \\cal L } + 3 } { 2 } } } { ( { \\cal L } + 2 ) } \\Big ( \\frac { \\Gamma ( \\frac { { \\cal L } + 3 } { 2 } ) } { \\Gamma ( \\frac { \\cal L } { 2 } + 1 ) } \\Big ) = 1 . \\end{align*}"}
-{"id": "2528.png", "formula": "\\begin{align*} \\begin{aligned} T _ 1 & = T _ { 1 1 } + T _ { 1 2 } = T _ { 1 1 1 } + T _ { 1 1 2 } + T _ { 1 1 3 } + T _ { 1 1 4 } + T _ { 1 2 } \\\\ & = T _ { 1 1 1 } + T _ { 1 1 2 1 } + T _ { 1 1 2 2 } + T _ { 1 2 } , \\end{aligned} \\end{align*}"}
-{"id": "4168.png", "formula": "\\begin{align*} \\sum _ { \\tau = 0 } ^ { n - s - 2 } \\frac { 1 } { n } e ^ { n - \\tau - 1 } _ { i , g + \\tau + 1 } & + \\left ( 1 - \\frac { 1 } { n } \\right ) \\left [ \\sum _ { \\tau = 0 } ^ { n - s - 2 } \\frac { 1 } { n ^ { n - \\tau - s - 1 } } e ^ { s } _ { i , g + \\tau + 1 } \\right ] \\\\ & + \\frac { 1 } { n } \\sum _ { \\tau = 1 } ^ { s - 1 } e _ { i , g + n - 1 - s + \\tau } ^ { s - \\tau } - \\left ( 1 - \\frac { 1 } { n } \\right ) e _ { i , g } ^ { n - 1 } - \\sum _ { \\tau = 1 } ^ { n - 2 } e _ { i , g + \\tau } ^ { n - 1 - \\tau } , \\end{align*}"}
-{"id": "10103.png", "formula": "\\begin{align*} \\pi ( U ) S ( \\xi , V ) = \\frac { n + 1 } { n - 1 } \\{ \\pi ( U ) \\pi ( V ) S ( \\xi , \\xi ) - \\pi ( V ) S ( U , \\xi ) \\} . \\end{align*}"}
-{"id": "422.png", "formula": "\\begin{align*} \\Im \\ , \\psi ( \\lambda , y ) \\geq \\frac { \\lambda \\coth ( \\lambda ) - 1 } { 1 + \\frac { 1 } { \\sinh ( \\lambda ) ^ 2 } } > 0 \\end{align*}"}
-{"id": "7318.png", "formula": "\\begin{align*} V ( \\Gamma _ + ) : = \\bigcup _ { i \\geq 0 } \\nu _ i ^ U , E ( \\Gamma _ + ) : = \\bigcup _ { i \\geq 0 } ( \\nu _ i , \\nu _ { i + 1 } ) ^ U . \\end{align*}"}
-{"id": "5830.png", "formula": "\\begin{align*} \\lim _ { J \\to \\infty } \\limsup _ { n \\to \\infty } \\| r ^ J _ n \\| _ { L ^ 6 ( \\R ) } = 0 . \\end{align*}"}
-{"id": "4614.png", "formula": "\\begin{align*} | U _ i | \\ge \\beta ( 1 - 2 \\eta ) \\sum _ { n = 0 } ^ { 2 ^ { k _ i } - 1 } 1 _ { [ - B , B ] } \\left ( t + \\sum _ { j = 0 } ^ { n - 1 } f ( T ^ j x ) \\right ) \\end{align*}"}
-{"id": "888.png", "formula": "\\begin{align*} \\mathfrak { s } ( \\lambda ) = \\sum _ { m = 1 } ^ M \\rho _ m e ^ { - \\sqrt { - 1 } \\theta _ m \\lambda } , \\qquad \\lambda \\in \\mathbb { R } . \\end{align*}"}
-{"id": "4579.png", "formula": "\\begin{align*} [ A , F _ { 1 , 0 } ] _ { q } & = [ [ \\cdots [ [ F _ { 2 , 1 } , F _ { 1 , 0 } ] _ q , F _ { 3 , 0 } ] _ { q } , \\cdots , F _ { n - 2 , 0 } ] _ q , F _ { n - 1 , 0 } ] _ { q } \\ , \\\\ & = ( - d ) ^ { - 1 } [ [ \\cdots [ [ F _ { 1 , 1 } , F _ { 2 , 0 } ] _ { q } , F _ { 3 , 0 } ] _ { q } , \\cdots , F _ { n - 2 , 0 } ] _ q , F _ { n - 1 , 0 } ] _ { q } \\ , . \\end{align*}"}
-{"id": "1287.png", "formula": "\\begin{align*} \\mathcal { G } _ 0 = \\mathcal { G } _ 0 ( \\cdot , z _ t ) , \\mathcal { G } _ 1 = \\mathcal { G } _ 1 ( \\cdot , z _ t ) , \\mbox { a n d } \\mathcal { G } _ 2 = \\mathcal { G } _ 2 ( \\cdot , z _ t ) \\end{align*}"}
-{"id": "4301.png", "formula": "\\begin{align*} \\underline { \\eta } = ( \\eta _ i , \\Delta _ { E , \\ell _ i } ) _ { i = 1 } ^ { m } , \\underline { \\eta } ^ { \\vee } = ( \\eta _ i , \\Delta ^ { \\vee } _ { E , \\ell _ i } ) _ { i = 1 } ^ { m } , \\end{align*}"}
-{"id": "4768.png", "formula": "\\begin{align*} { L } _ 2 = \\sqrt { \\beta t _ 2 - \\alpha ^ 2 \\omega _ 2 + \\gamma } , { L } _ 3 = \\sqrt { \\beta t _ 3 - \\alpha ^ 2 \\omega _ 3 + \\gamma } . \\end{align*}"}
-{"id": "7417.png", "formula": "\\begin{align*} \\beta : = b - ( 2 t _ 3 + t _ 2 ) / 3 , \\gamma : = c - ( 2 t _ 5 + t _ 4 ) / 3 , \\delta : = d - t _ 1 / 2 \\end{align*}"}
-{"id": "8701.png", "formula": "\\begin{align*} \\binom { j } { k } \\leq \\frac { j ^ k } { k ! } . \\end{align*}"}
-{"id": "9621.png", "formula": "\\begin{align*} P _ { \\rm { t r } } \\big ( R _ { \\beta } , \\lambda _ { \\beta } , h _ \\beta \\big ) = \\lambda _ { \\beta } \\int _ 0 ^ { R _ { \\beta } } 2 \\pi r \\bar { P } _ { \\rm { t r } , \\rm { u s e r } } \\big ( r , h _ \\beta \\big ) \\ , d r . \\end{align*}"}
-{"id": "8421.png", "formula": "\\begin{align*} c _ { t , l } ( \\mu ) = \\begin{cases} \\omega _ { \\pi } ( - 1 ) \\frac { G ( \\varpi ^ { - l } , \\mu ^ { - 1 } ) } { \\epsilon ( \\frac { 1 } { 2 } , \\mu \\pi ) } & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "7446.png", "formula": "\\begin{align*} x ' & = c _ { 0 0 } z + c _ { 1 0 } t , y = - c _ { 0 1 } z + c _ { 0 0 } t , T ^ \\gamma _ 0 = c _ { 0 0 } ^ 2 + c _ { 0 1 } c _ { 1 0 } , \\\\ T ^ \\delta _ 0 & = ( 1 + c _ { 0 1 } ) ( z + c _ { 1 0 } + T ^ { \\beta } _ 0 ) + \\frac { 1 } { 4 } t ^ 2 + c _ { 0 0 } t + c _ { 0 0 } ^ 2 . \\end{align*}"}
-{"id": "1916.png", "formula": "\\begin{align*} d ^ { 2 } ( \\phi , \\psi ) = d ^ { 1 } ( \\phi , \\psi ) + \\max _ { 1 \\leq i \\leq l _ { 1 } } \\sup _ { u \\in U _ { x _ { i } } } \\Vert D _ { u } ^ { 2 } ( \\tilde { \\phi } _ { x _ { i } } ) - D _ { u } ^ { 2 } ( \\tilde { \\psi } _ { x _ { i } } ) \\Vert + \\max _ { 1 \\leq i \\leq l _ { 2 } } \\sup _ { v \\in V _ { y _ { i } } } \\Vert D _ { v } ^ { 2 } ( \\hat { \\phi } _ { y _ { i } } ) - D _ { v } ^ { 2 } ( \\hat { \\psi } _ { y _ { i } } ) \\Vert , \\end{align*}"}
-{"id": "4933.png", "formula": "\\begin{align*} Q ( x ) = x _ 1 ^ 2 + . . . + x _ p ^ 2 - x _ { p + 1 } ^ 2 - . . . - x _ { p + q } ^ 2 \\end{align*}"}
-{"id": "4156.png", "formula": "\\begin{align*} j + ( 0 , \\ldots , n - 2 ) : = ( j , \\ldots , j + n - 2 ) \\mod ( n - 1 ) . \\end{align*}"}
-{"id": "8636.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\Psi ( y ) \\pi ( d y ) = 1 . \\end{align*}"}
-{"id": "2921.png", "formula": "\\begin{align*} \\hat { G } _ s = \\bigcup \\bigl \\{ \\hat { G } [ \\ast , g ] \\colon \\iota _ 2 ( g ) = s \\bigr \\} \\end{align*}"}
-{"id": "6162.png", "formula": "\\begin{align*} * \\dd x _ 0 = V ^ { - 1 } \\dd x _ { 1 2 3 } \\ ; , \\ ; * \\dd x _ 1 = - f _ 1 ^ { - 1 } V \\dd x _ { 0 2 3 } \\ ; , \\ ; * \\dd x _ 2 = - f _ 2 ^ { - 1 } V \\dd x _ { 0 3 1 } \\ ; , \\ ; * \\dd x _ 3 = - f _ 3 ^ { - 1 } V \\dd x _ { 0 1 2 } . \\end{align*}"}
-{"id": "2306.png", "formula": "\\begin{align*} & \\ ; \\| A u _ * \\| _ { L ^ 2 } ^ 2 ( t ) + 2 \\nu \\int _ 0 ^ t \\| A ^ { 1 + s / 2 } u _ * \\| _ { L ^ 2 } ^ 2 \\ , d \\tau \\\\ = & \\ ; \\| A u _ 0 \\| _ { L ^ 2 } ^ 2 - 2 \\int _ 0 ^ t \\langle A ^ 2 u _ * , u _ * \\cdot \\nabla u _ * + \\mathcal { U } ^ \\alpha ( u _ * , u _ * ) \\rangle ( \\tau ) \\ , d \\tau . \\end{align*}"}
-{"id": "1221.png", "formula": "\\begin{align*} \\bar x _ 1 ( X , t ) = a \\bar x _ 2 ( X , t ) + b \\mbox { w h e n e v e r } X \\in \\mathbb { S } ^ { n - 1 } . \\end{align*}"}
-{"id": "8414.png", "formula": "\\begin{align*} N = \\left \\{ n ( x ) = \\left ( \\begin{matrix} 1 & x \\\\ 0 & 1 \\end{matrix} \\right ) \\colon x \\in F \\right \\} A = \\left \\{ a ( y ) = \\left ( \\begin{matrix} y & 0 \\\\ 0 & 1 \\end{matrix} \\right ) \\colon y \\in F ^ { \\times } \\right \\} . \\end{align*}"}
-{"id": "4330.png", "formula": "\\begin{align*} f ( \\vec { y } ; z ) = z \\cdot ( z + y _ { 1 } + n - 1 ) \\cdot ( z + y _ { 1 } + y _ { 2 } + n - 2 ) \\cdots ( z + y _ { 1 } + \\cdots + y _ { n - 1 } + 1 ) . \\end{align*}"}
-{"id": "1144.png", "formula": "\\begin{align*} y ^ 4 + y & = ( y ^ 2 + y ) ^ 2 + y ^ 2 + y \\\\ & = ( y ^ 2 + y ) ( y ^ 2 + y + 1 ) \\\\ & = ( y ^ 2 + y ) ( x ^ 3 + x ) \\\\ & = ( y ^ 2 + y ) ( x ^ 2 + x ) ( x + 1 ) \\end{align*}"}
-{"id": "23.png", "formula": "\\begin{align*} \\hat { V } ^ { C } _ { \\sigma } ( R _ 1 , R _ 2 ) \\sqrt { 2 \\pi } \\sigma = \\hat { V } _ { \\sigma } ( R _ 1 , R _ 2 ) \\end{align*}"}
-{"id": "379.png", "formula": "\\begin{align*} | b _ { n , r , s } | & \\le C \\sum _ { j , k = 1 } ^ n ( - j - r + | k + s | ) ^ { - \\beta } L ( - j - r + | k + s | ) \\\\ & = C \\sum _ { j , k = 1 } ^ n ( - j + n + R + | k + s | ) ^ { - \\beta } L ( - j + n + R + | k + s | ) \\\\ & \\propto n ^ 2 ( | r | + | s | ) ^ { - \\beta } L ( | r | + | s | ) . \\end{align*}"}
-{"id": "5729.png", "formula": "\\begin{align*} \\mathcal { K } ^ 2 ( { v } ) = \\frac 1 2 ( A ( z ) ( { v } - z ) , ( { v } - z ) ) _ { L ^ 2 } + \\frac 1 2 \\int _ \\R ( D ^ 2 W ( \\xi ) - D ^ 2 W ( z ) ) ( { v } - z , v - z ) \\d s , \\end{align*}"}
-{"id": "9348.png", "formula": "\\begin{align*} J O _ { n + 2 } ^ { ( 3 ) } + J O _ { n + 1 } ^ { ( 3 ) } + J O _ { n } ^ { ( 3 ) } = \\sum _ { s = 0 } ^ { 7 } ( J _ { n + s + 2 } ^ { ( 3 ) } + J _ { n + s + 1 } ^ { ( 3 ) } + J _ { n + s } ^ { ( 3 ) } ) e _ { s } . \\end{align*}"}
-{"id": "9424.png", "formula": "\\begin{align*} \\mathfrak { d } ^ { \\rm ( o u t ) } _ { n , d } = \\mathfrak { d } ^ { \\rm ( o u t ) } _ { n - 2 , d + 1 } + \\mathfrak { d } ^ { \\rm ( o u t ) } _ { n - 1 , d - 1 } + \\mathfrak { d } ^ { \\rm ( o u t ) } _ { n - 3 , d } + \\sum \\limits _ { i = 0 } ^ { n - 3 } \\mathfrak { d } ^ { \\rm ( o u t ) } _ { i , 1 } \\cdot \\mathfrak { d } ^ { \\rm ( o u t ) } _ { n - i - 3 , d - 1 } + \\sum \\limits _ { i = 0 } ^ { n - 2 } \\sum \\limits _ { j = 0 } ^ { d - 2 } \\tilde { s } _ { i , j } \\cdot \\mathfrak { d } ^ { \\rm ( o u t ) } _ { n - 2 - i , d - 2 - j } . \\end{align*}"}
-{"id": "6470.png", "formula": "\\begin{align*} \\sigma _ { x } \\sigma _ { y } = \\Sigma ^ { 2 } \\Sigma ^ { 2 } \\in \\mathbb { R } _ { 0 } ^ { + } \\end{align*}"}
-{"id": "2554.png", "formula": "\\begin{align*} F _ \\delta = \\{ ( x , y ) \\in \\R ^ 2 \\ , : \\ , | x | \\le \\delta , \\ | y | \\leq f _ \\delta ( x ) \\} \\ , , \\end{align*}"}
-{"id": "125.png", "formula": "\\begin{align*} d X _ t & = b _ 1 ( 1 - t , X _ t , Y _ t ) \\ , d t + d W _ t , \\\\ d Y _ t & = b _ 2 ( 1 - t , X _ t , Y _ t ) \\ , d t + d W _ t , \\end{align*}"}
-{"id": "5933.png", "formula": "\\begin{align*} \\forall u , v \\in V & & ( u , v ) = \\phi ( u + v ) + \\phi ( u ) + \\phi ( v ) . \\end{align*}"}
-{"id": "7702.png", "formula": "\\begin{align*} & K ^ { \\top } \\Theta + \\Theta K = - V Z V ^ { \\top } \\\\ & \\qquad = - \\left [ \\begin{array} { c | c | c } Z _ { 1 1 } & \\cdots & Z _ { 1 ( N - 1 ) } \\\\ \\hline \\vdots & \\ddots & \\vdots \\\\ \\hline Z _ { ( N - 1 ) 1 } & \\cdots & Z _ { ( N - 1 ) ( N - 1 ) } \\end{array} \\right ] \\ , , \\end{align*}"}
-{"id": "5613.png", "formula": "\\begin{align*} S _ t & = x - y + \\int _ s ^ t \\{ f ( \\phi ( s , r , x ) ) - f ( \\phi ( s , r , y ) ) \\} d r \\\\ & = x - y + \\int _ s ^ t \\int _ 0 ^ 1 f ' ( u \\phi ( s , r , x ) + ( 1 - u ) \\phi ( s , r , y ) ) ( \\phi ( s , r , x ) - \\phi ( s , r , y ) ) d u d r \\\\ & = x - y + \\int _ s ^ t \\biggl \\{ \\int _ 0 ^ 1 f ' ( Z _ r ^ u ) d u \\biggr \\} S _ r d r \\end{align*}"}
-{"id": "1112.png", "formula": "\\begin{align*} P _ \\mathrm { o u t } = \\mathrm { P r } \\left \\{ \\zeta _ i < \\zeta _ \\mathrm { t h } \\right \\} . \\end{align*}"}
-{"id": "6794.png", "formula": "\\begin{align*} w _ { \\lambda } ( y ) = 2 \\ln { \\lambda } + 5 \\ln { 2 } - 4 \\pi \\sum \\limits _ { j < k } G ( \\xi _ j , \\xi _ k ) \\end{align*}"}
-{"id": "1517.png", "formula": "\\begin{align*} Q _ { S } ( x ) = \\left [ \\begin{array} { c c c } Q _ { T , 4 } ( x ) & x Q _ { T , 3 } ( x ) + Q _ { T , 2 } ( x ) & Q _ { T , 3 } ( x ) \\\\ Q _ { T , 3 } ( x ) & x Q _ { T , 2 } ( x ) + Q _ { T , 1 } ( x ) & Q _ { T , 2 } ( x ) \\\\ Q _ { T , 2 } ( x ) & x Q _ { T , 1 } ( x ) + Q _ { T , 0 } ( x ) & Q _ { T , 1 } ( x ) \\end{array} \\right ] . \\end{align*}"}
-{"id": "3341.png", "formula": "\\begin{align*} L _ s ^ { ( K ) } & = \\binom { K - 2 } { s - 1 } + \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K - 1 } { s + i } ( N - 1 ) ^ i N , \\\\ D _ s ^ { ( K ) } & = \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K } { s + 1 + i } ( N - 1 ) ^ i N . \\end{align*}"}
-{"id": "2798.png", "formula": "\\begin{align*} \\Big \\vert \\int _ { a } ^ { b } f ( t ) \\mathrm { d } t - \\sum _ { i = 0 } ^ { n } \\omega _ { i , n } f _ { i } \\Big \\vert \\leq C h ^ { d + 1 } , \\end{align*}"}
-{"id": "402.png", "formula": "\\begin{align*} \\Lambda _ { n } ( u , v , \\varepsilon ) = \\frac { u } { \\sigma _ { n } ^ { 2 } } \\sum _ { j = 1 } ^ { k _ { n } } \\mathbb { E } \\big [ X _ { n j } ^ { 2 } I ( X _ { n j } \\leq - \\varepsilon \\sigma _ { n } / v ) \\big ] . \\end{align*}"}
-{"id": "5406.png", "formula": "\\begin{align*} K : = \\langle a , b , x ^ 3 \\rangle \\end{align*}"}
-{"id": "8384.png", "formula": "\\begin{align*} N ( V _ \\Z ) = N ( V _ \\Z ^ { [ 1 ] } ) \\cdot N ( V _ \\Z ^ { [ 2 ] } ) \\end{align*}"}
-{"id": "6310.png", "formula": "\\begin{align*} ( a , b ) & \\mapsto ( \\Hat { a } , \\Hat { b } ) = ( a p ^ 2 , b p ) & & & ( \\Hat { a } , \\Hat { b } ) & \\mapsto ( a , b ) = ( a / p ^ 2 , b / p ) . \\end{align*}"}
-{"id": "6249.png", "formula": "\\begin{align*} \\Omega _ + ( \\mu ) = \\Omega _ + ( \\mu ^ \\star ) , \\alpha _ + ( \\mu ) = \\alpha _ + ( \\mu ^ \\star ) , \\beta _ + ( \\mu ) = \\beta _ + ( \\mu ^ \\star ) \\end{align*}"}
-{"id": "7426.png", "formula": "\\begin{align*} \\delta = d - \\frac { t _ 1 + 2 t _ 2 + 3 t _ 3 } { 4 } , B = b + \\tau _ 3 , C = c + \\tau _ 5 \\end{align*}"}
-{"id": "3289.png", "formula": "\\begin{align*} \\Delta _ { \\tilde g } = e ^ { - \\varphi } \\Delta _ g . \\end{align*}"}
-{"id": "4590.png", "formula": "\\begin{align*} d \\omega = \\theta \\wedge \\omega . \\end{align*}"}
-{"id": "8041.png", "formula": "\\begin{align*} \\begin{aligned} & \\angle a c b \\le \\epsilon / 1 0 ; \\\\ & | \\angle b a c + \\angle a b c - \\pi | \\le \\epsilon / 1 0 . \\end{aligned} \\end{align*}"}
-{"id": "2041.png", "formula": "\\begin{align*} f _ { 1 , k } ( 1 ) = k , ~ ~ ~ f _ { 1 , k } ( \\varepsilon ) = 0 , ~ ~ ~ f _ { 1 , k } ( \\eta ) = ( - 1 ) ^ { i - 1 } \\eta ^ i q ^ { - \\frac { i ( i - 1 ) } { 2 } } \\end{align*}"}
-{"id": "8817.png", "formula": "\\begin{align*} _ { t ^ { m - 1 } } \\frac { ( t - p ) Z ( p , \\chi _ { } , s ) } { ( p - 1 ) ( 1 - t ) } = - _ { t ^ { m - 1 } } \\frac { Z ( p , \\chi _ { } , s ) } { p - 1 } - _ { t ^ { m - 1 } } \\frac { Z ( p , \\chi _ { } , s ) } { 1 - t } , \\end{align*}"}
-{"id": "4992.png", "formula": "\\begin{align*} \\| \\partial _ { x ' } ( h - \\tilde { h } ) \\| _ { \\dot { F } ^ { \\alpha - 1 , p } _ q } & = \\| \\| 2 ^ { ( \\alpha - 1 ) m } \\Delta _ m ( \\partial _ { x ' } ( h - \\tilde { h } ) ) \\| _ { \\ell ^ q ( m ) } \\| _ { L ^ p } \\\\ & = \\| \\| 2 ^ { ( \\alpha - 1 ) m } \\Delta _ m ( \\partial _ { x ' } \\sum _ { j \\in \\Z } U _ j V _ j ) \\| _ { \\ell ^ q ( m ) } \\| _ { L ^ p } \\\\ & = \\| \\| 2 ^ { ( \\alpha - 1 ) m } \\Delta _ m ( \\partial _ { x ' } \\sum _ { r \\in \\Z } U _ { r + m } V _ { r + m } ) \\| _ { \\ell ^ q ( m ) } \\| _ { L ^ p } . \\end{align*}"}
-{"id": "9810.png", "formula": "\\begin{align*} p _ 0 ' ( t ) = p _ 1 ( t ) \\mu - p _ 0 ( t ) \\l f ( t ) , - T _ { e _ 1 } \\leq t \\leq T _ { e _ 2 } . \\end{align*}"}
-{"id": "3424.png", "formula": "\\begin{align*} K = \\overline D _ 1 \\setminus ( W ^ + _ 1 \\cup W ^ - _ 1 ) , L = \\overline { D _ 2 \\setminus D _ 1 } \\cup ( \\overline D _ 1 \\cap ( \\overline V ^ + _ 1 \\cup \\overline V ^ - _ 1 ) ) . \\end{align*}"}
-{"id": "3911.png", "formula": "\\begin{align*} \\lim _ { \\ell \\to \\infty } \\frac { \\sqrt { \\ell + \\frac { 3 } { 2 } } } { ( \\ell + 1 ) } \\Big ( \\frac { \\Gamma ( \\ell + \\frac { 3 } { 2 } ) } { \\Gamma ( \\ell + 1 ) } \\Big ) = 1 . \\end{align*}"}
-{"id": "6495.png", "formula": "\\begin{align*} d s _ { } ^ { 2 } = d s _ { } ^ { 2 } + d s _ { } ^ { 2 } = \\frac { 1 } { \\mu _ { A } ^ { 2 } } d \\mu _ { A } ^ { 2 } + \\frac { 1 } { \\mu _ { B } ^ { 2 } } d \\mu _ { B } ^ { 2 } . \\end{align*}"}
-{"id": "7340.png", "formula": "\\begin{align*} \\frac { 2 } { s } + \\frac { 3 } { p } < \\frac { 7 } { 2 } , \\begin{cases} \\frac { 1 } { 2 } \\leqslant \\frac { 1 } { p } \\leqslant 1 & 2 \\leqslant r < 3 \\\\ \\frac { 1 } { 2 } \\leqslant \\frac { 1 } { p } < \\frac { 1 } { r } + \\frac { 2 } { 3 } & 3 \\leqslant r \\leqslant 6 \\ , . \\end{cases} \\end{align*}"}
-{"id": "2263.png", "formula": "\\begin{align*} \\frac 1 { { \\sin h } ^ 2 ( t k ) } = \\left ( \\frac { 2 } { e ^ { t k } - e ^ { - t k } } \\right ) ^ 2 = \\frac { 4 e ^ { 2 t k } } { ( e ^ { 2 t k } - 1 ) ^ 2 } . \\end{align*}"}
-{"id": "639.png", "formula": "\\begin{align*} \\partial _ t g _ { i j } & = - 2 g ^ { p q } h ( A _ { p q } , A _ { i j } ) , \\\\ \\partial _ t g ^ { i j } & = 2 g ^ { p q } g ^ { i k } g ^ { j l } h ( A _ { p q } , A _ { k l } ) , \\\\ \\partial _ t d \\mu & = - | H | ^ 2 d \\mu , \\\\ \\partial _ t \\Gamma _ { i j } ^ k & = - g ^ { k l } \\left ( \\nabla _ i h ( H , A _ { j l } ) + \\nabla _ j h ( H , A _ { i l } ) - \\nabla _ l h ( H , A _ { i j } ) \\right ) . \\end{align*}"}
-{"id": "1099.png", "formula": "\\begin{align*} \\overline { \\mathcal { C } } _ \\mathrm { \\infty } = E \\left [ \\sum _ { i \\in \\mathcal { N } } ^ { } \\mathcal { C } ^ \\mathrm { \\infty } _ { i | \\alpha _ { i i } } \\right ] . \\end{align*}"}
-{"id": "5596.png", "formula": "\\begin{align*} A _ z ( n ; \\chi ) & = \\sum _ { 0 \\leq a \\leq n } B _ z ( a , \\chi ) D _ z ( n - a , \\chi ) \\\\ & \\leq q ^ { n / 2 } \\sum _ { 0 \\leq a \\leq n } \\frac { | B _ z ( a , \\chi ) | } { q ^ { a / 2 } } \\binom { n - a + A m } { n - a } \\\\ & \\leq q ^ { n / 2 } \\binom { n + A m } { n } \\sum _ { 0 \\leq a \\leq n } \\frac { | B _ z ( a , \\chi ) | } { q ^ { a / 2 } } \\leq q ^ { n / 2 } \\binom { n + A m } { n } n ^ { c _ A } . \\end{align*}"}
-{"id": "3413.png", "formula": "\\begin{align*} \\theta _ n ( 1 + 1 / n ) = b , \\theta _ n ( - 1 - 1 / n ) = - b . \\end{align*}"}
-{"id": "2249.png", "formula": "\\begin{align*} \\frac { d \\ , f _ j ( z ) } { f _ j ( z ) } = \\frac { d \\prod \\limits _ { s = 1 } ^ \\infty f _ { j s } ( z ) } { \\prod \\limits _ { s = 1 } ^ \\infty f _ { j s } ( z ) } = \\sum _ { s = 1 } ^ \\infty \\frac { d \\ , f _ { j s } ( z ) } { f _ { j s } ( z ) } . \\end{align*}"}
-{"id": "3224.png", "formula": "\\begin{align*} \\mathcal D ^ p H ^ { 2 p + k } ( X ) = N ^ p H ^ { 2 p + k } ( X ) . \\end{align*}"}
-{"id": "1235.png", "formula": "\\begin{align*} \\int _ { \\mathbf { g } ^ { - 1 } ( K ) } | \\nabla u ( x ) | ^ p \\ , d \\mathcal { H } ^ { n - 1 } = \\mu ( K ) \\end{align*}"}
-{"id": "4236.png", "formula": "\\begin{align*} ( \\mathrm { E n d } ( E ) ) ^ { k , - k } : = \\bigoplus _ { p + q = n } ( E ^ { p , q } ) ^ { \\vee } \\otimes E ^ { p + k , q - k } \\end{align*}"}
-{"id": "0.png", "formula": "\\begin{align*} \\dim S ^ 2 = 2 \\dim S - 1 + \\gamma \\end{align*}"}
-{"id": "1550.png", "formula": "\\begin{gather*} m \\ = \\ A \\bigg ( \\frac { p } { D } + \\frac { k } { D } + 4 \\ell \\bigg ) \\end{gather*}"}
-{"id": "4790.png", "formula": "\\begin{align*} \\mathcal R ^ k ( F ) ( u ) : = \\ , & \\int _ { \\R ^ { 2 n } } \\frac { u ^ k - v ^ k } { | u - v | ^ { 2 n + 1 } } \\ , F ( v ) \\ , d v , \\end{align*}"}
-{"id": "2337.png", "formula": "\\begin{align*} D , \\ A _ 1 . D , \\ A _ 1 A _ 2 . D , \\ \\ldots , \\ A _ 1 A _ 2 \\cdots A _ n . D = g . D \\end{align*}"}
-{"id": "1286.png", "formula": "\\begin{align*} \\int _ { E } f ( \\nabla U _ + ) d \\mathcal { H } ^ { n - 1 } = \\infty . \\end{align*}"}
-{"id": "6924.png", "formula": "\\begin{align*} \\frac { \\partial ^ { 2 } u ( x , t ) } { \\partial t ^ { 2 } } = \\Delta _ { x } u ( x , t ) \\end{align*}"}
-{"id": "2740.png", "formula": "\\begin{align*} \\mathcal L _ { i } ( g ) = M \\int _ { \\mathbb R ^ d \\times S ^ { d - 1 } } \\phi ( | v - v _ { \\ast } | ) \\partial ^ { p + 1 - i } b \\ , \\mathcal M ( v _ { \\ast } ) \\left [ \\frac { \\partial ^ i g _ { \\ast } ^ { \\prime } } { M _ { \\ast } ^ { \\prime } } + \\frac { \\partial ^ i g ^ { \\prime } } { M ^ { \\prime } } - \\frac { \\partial ^ i g _ { \\ast } } { M _ { \\ast } } - \\frac { \\partial ^ i g } { M } \\right ] d v _ { \\ast } d \\sigma \\ , , \\end{align*}"}
-{"id": "5305.png", "formula": "\\begin{align*} R _ 2 ^ 2 = \\left ( \\frac { 2 } { \\min \\left \\lbrace ( L _ { 1 } ) _ \\# , ( L _ { 2 } ) _ \\# / 2 \\right \\rbrace } + \\frac { \\ell ' } { \\gamma _ \\# } \\right ) \\mathcal { R } . \\end{align*}"}
-{"id": "7304.png", "formula": "\\begin{align*} x _ 0 ^ { n + 2 } + x _ 1 ^ { n + 2 } + \\ldots + x _ { n + 1 } ^ { n + 2 } - ( n + 2 ) \\psi x _ 0 x _ 1 \\ldots x _ { n + 1 } = 0 . \\end{align*}"}
-{"id": "4727.png", "formula": "\\begin{align*} { L } _ i = ( 0 , \\alpha , \\beta , \\gamma ) . \\end{align*}"}
-{"id": "4919.png", "formula": "\\begin{align*} U = \\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix} . \\end{align*}"}
-{"id": "5708.png", "formula": "\\begin{align*} D ( A ( z ) ) = H ^ 2 ( \\R , \\R ^ n ) , A ( z ) { v } = - { v } '' + ( \\nabla ^ 2 W ( z ) { v } ^ T ) ^ T . \\end{align*}"}
-{"id": "9675.png", "formula": "\\begin{align*} & r _ i = \\kappa _ i ( - \\lambda _ i , 1 , \\rho ( \\lambda _ i u - v ) , \\frac { \\rho ( \\lambda _ i u - v ) } { c ^ 2 } , 0 ) ^ \\top , i = 1 , 5 ; \\\\ & r _ 2 = ( u , v , 0 , 0 , 0 ) ^ \\top , r _ 3 = ( 0 , 0 , 0 , \\rho , 0 ) ^ \\top , r _ 4 = ( 0 , 0 , 0 , 0 , 1 ) ^ \\top , \\end{align*}"}
-{"id": "2576.png", "formula": "\\begin{align*} D = \\left [ \\begin{array} { c c c c } ( 1 , 1 2 , 5 , 8 , 6 , 7 ) & ( \\infty ) & ( 0 ) & ( \\infty ) \\\\ ( \\infty ) & ( 9 , 4 ) & ( 9 , 4 ) & ( \\infty ) \\\\ ( \\infty ) & ( 5 ) & ( \\infty ) & ( 8 , 5 , 7 , 6 , 1 2 , 1 ) \\end{array} \\right ] , \\end{align*}"}
-{"id": "9816.png", "formula": "\\begin{align*} \\int _ { t = t ' } ^ { ( T ' _ { e _ 2 } \\wedge T _ 2 ) } f ( t ) d t = 1 - p ' . \\end{align*}"}
-{"id": "9834.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - [ \\frac { \\nu + 1 } { 2 } ] } { { 2 n - 2 i } \\choose { \\nu } } A _ i = - q ^ { n - \\nu } \\sum _ { i = 0 } ^ { [ \\frac { \\nu } { 2 } ] } { { 2 n - 2 i } \\choose { 2 n - \\nu } } A _ i \\qquad ( \\nu = 0 , 1 , \\cdots , 2 n ) , \\end{align*}"}
-{"id": "9620.png", "formula": "\\begin{align*} \\bar { P } _ { \\rm { t r } , \\rm { u s e r } } \\big ( r , h _ \\beta ) = \\bar { L } ( r , h _ \\beta \\big ) N _ 0 W \\left ( 2 ^ { C / W } - 1 \\right ) . \\end{align*}"}
-{"id": "6104.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow \\infty } \\frac { L \\left ( \\lambda x \\right ) } { L \\left ( x \\right ) } = 1 , \\end{align*}"}
-{"id": "1217.png", "formula": "\\begin{align*} A H _ 1 = B H _ 2 = C H \\mbox { a t $ ( X , t ) $ w h e n \\eqref { e q n 5 . 3 3 } h o l d s . } \\end{align*}"}
-{"id": "6051.png", "formula": "\\begin{align*} \\frac { \\partial \\Omega ^ { ( \\alpha , \\theta ) } ( Q _ { X Y U } ) } { \\partial \\theta } & = \\mathbb { E } _ { Q _ { X Y U } ^ { ( \\alpha , \\theta ) } } \\Big [ \\omega _ { Q _ { X Y U } } ^ { ( \\alpha ) } ( X , Y | U ) \\Big ] , \\end{align*}"}
-{"id": "8467.png", "formula": "\\begin{align*} S = \\{ x + \\alpha \\Omega ^ { r - 1 } \\colon x \\in S ' , \\alpha \\in \\mathfrak { O } / \\mathfrak { P } \\} \\subset ( \\mathfrak { O } / \\mathfrak { P } ^ r ) ^ { \\times } . \\end{align*}"}
-{"id": "2866.png", "formula": "\\begin{align*} \\tilde { P } _ P = \\{ x \\in \\R ^ 2 _ + \\ , | \\ , x _ 1 + M x _ 2 \\leq M , \\ M x _ 1 + x _ 2 \\leq M \\} . \\end{align*}"}
-{"id": "1433.png", "formula": "\\begin{align*} f = f _ { k , k } x _ { 4 n } ^ k + f _ { k , k + 1 } x _ { 4 n } ^ { k + 1 } + \\cdots \\in M _ { n , i } \\cap \\left ( \\bigcap _ { j \\in J } \\mathrm { K e r } \\ , \\partial _ j \\right ) , \\end{align*}"}
-{"id": "5263.png", "formula": "\\begin{align*} { \\rm A u t } ( X , \\varphi ) : = \\{ \\beta : X \\rightarrow X \\ : | \\ : \\beta \\hbox { i s a h o m e o m o r p h i s m a n d } \\beta \\circ \\varphi = \\varphi \\circ \\beta \\} . \\end{align*}"}
-{"id": "9982.png", "formula": "\\begin{align*} | f ( A _ 1 , \\dots , A _ N ) | \\leq e ^ H \\prod _ { i = 1 } ^ { N } { \\binom { d ^ 2 - 1 + \\lambda _ i } { \\lambda _ i } } \\| A _ 1 \\| _ { 0 } ^ { \\lambda _ 1 } \\cdots \\| A _ { N } \\| _ { 0 } ^ { \\lambda _ N } \\end{align*}"}
-{"id": "4258.png", "formula": "\\begin{align*} \\delta _ n = \\frac { \\lambda ^ p ( 1 - a ^ p ) - ( \\lambda ^ p - a ^ p ) \\lambda ^ n } { n } , \\end{align*}"}
-{"id": "8000.png", "formula": "\\begin{align*} a _ { j , k } ( x , \\xi ) : = \\phi _ j \\ast a ( \\cdot , \\xi ) ( x ) \\widehat { \\phi _ k } ( \\xi ) \\end{align*}"}
-{"id": "5052.png", "formula": "\\begin{align*} G \\boxtimes G = \\biguplus _ { i , j } ( G _ i \\boxtimes G _ j ) . \\end{align*}"}
-{"id": "7151.png", "formula": "\\begin{align*} q ( ( r + 1 ) J + 1 ) = \\sqrt { 2 \\Delta _ J ( r J ) } \\leq \\sqrt { 2 ( U J ^ 2 + V _ r J g ^ * _ r ) } . \\end{align*}"}
-{"id": "6837.png", "formula": "\\begin{align*} \\norm { \\sum \\limits _ { j = 1 } ^ 4 c ' _ j ( - | x _ { \\xi _ j } | ) \\chi ' _ { R _ 1 } ( | z _ { \\xi _ j } | ) \\varphi _ { 0 , j } + \\sum \\limits _ { j = 1 } ^ 4 c _ j \\chi _ { R _ 1 , j } \\frac { \\partial \\varphi _ { 0 , j } } { \\partial \\lambda } } _ { \\ast } \\leq C . \\end{align*}"}
-{"id": "4906.png", "formula": "\\begin{align*} \\alpha I + \\beta T + \\gamma T ^ * + F = 0 , \\end{align*}"}
-{"id": "218.png", "formula": "\\begin{align*} \\Omega \\ni \\omega \\mapsto H : = H _ { \\omega } \\in \\mathcal { L } _ { } ( L ^ 2 ( \\R ^ d ) ) \\end{align*}"}
-{"id": "808.png", "formula": "\\begin{align*} T _ { k } ( \\frac { 2 ^ d r - 1 } { 3 } ) = ( \\frac { 1 } { 3 } ) ^ k [ ( 2 ^ d r + 2 ) ^ k + ( 2 ^ d r + 5 ) ^ k + \\cdots + ( 3 2 ^ { d - 1 } r ) ^ k + \\cdots + ( 2 ^ { d + 1 } r - 2 ) ^ k ] \\end{align*}"}
-{"id": "7352.png", "formula": "\\begin{align*} \\big \\Vert \\ , | \\cdot | ^ { - \\alpha } * | u | ^ 2 \\big \\Vert _ { L ^ { \\infty } ( [ 0 , T ] , L ^ { q ( m ( \\alpha ) ) } ( \\mathbb { R } ^ 3 ) ) } \\ ; & \\lesssim \\ ; \\Vert u ^ 2 \\Vert _ { L ^ { \\infty } ( [ 0 , T ] , L ^ { m ( \\alpha ) } ( \\mathbb { R } ^ 3 ) ) } \\\\ & \\lesssim \\ ; \\Vert u \\Vert _ { L ^ { \\infty } ( [ 0 , T ] , H ^ 1 ( \\mathbb { R } ^ 3 ) ) } ^ 2 \\ , . \\end{align*}"}
-{"id": "1942.png", "formula": "\\begin{align*} \\widehat { Q _ 2 ( q ) } ( \\eta ) = ( i \\pi d + P ) S _ { r } ( q ) ( \\eta ) , \\end{align*}"}
-{"id": "6274.png", "formula": "\\begin{align*} [ \\Delta _ q , F \\times \\nabla \\times ] G = \\Delta _ q ( F \\times ( \\nabla \\times G ) ) - F \\times ( \\nabla \\times G _ q ) , \\end{align*}"}
-{"id": "9709.png", "formula": "\\begin{align*} \\sigma ' _ j & = \\sigma ' _ j ( \\delta _ 5 , \\delta _ 3 , \\delta _ 2 , \\delta _ 1 , \\sigma _ { 3 } , \\sigma _ { 2 } , \\alpha _ 1 ) - \\sigma ' _ j ( \\delta _ 5 , \\delta _ 3 , \\delta _ 2 , 0 , \\sigma _ { 3 } , \\sigma _ { 2 } , \\alpha _ 1 ) + \\sigma ' _ j ( \\delta _ 5 , \\delta _ 3 , \\delta _ 2 , 0 , \\sigma _ { 3 } , \\sigma _ { 2 } , \\alpha _ 1 ) \\\\ & = K _ { 2 j } \\delta _ 1 + \\delta _ j + \\sigma _ j , j = 2 , 3 , \\end{align*}"}
-{"id": "4379.png", "formula": "\\begin{align*} C _ { z } M _ f C _ { - z } = M _ { f \\circ \\tau _ { z } } \\end{align*}"}
-{"id": "6917.png", "formula": "\\begin{align*} f ( - \\Delta ) u = \\sum _ { j = 1 } ^ { \\infty } f ( \\lambda _ { j } ) \\langle u , u _ { j } \\rangle u _ { j } \\end{align*}"}
-{"id": "2462.png", "formula": "\\begin{align*} L f = 2 \\mu ^ { - 1 / 2 } Q ( \\mu , \\mu ^ { 1 / 2 } f ) \\ , . \\end{align*}"}
-{"id": "7183.png", "formula": "\\begin{align*} \\limsup _ { k \\to \\infty } I _ k ^ { ( 4 ) } \\leq \\limsup _ { k \\to \\infty } \\frac { c } { \\varepsilon ^ 2 } \\int _ { B _ { 1 - \\varepsilon } \\setminus B _ { 1 - 2 \\varepsilon } } | z _ \\infty - z _ { j _ k } | ^ 2 \\ , d \\mu _ a + O ( \\varepsilon ) \\leq O ( \\varepsilon ) . \\end{align*}"}
-{"id": "8902.png", "formula": "\\begin{align*} \\begin{aligned} \\lim _ j \\| T ^ { n _ { k _ j } } X J _ { \\theta , 1 } ^ { - 1 } p \\| ^ 2 & = \\lim _ j ( \\| T ^ { n _ { k _ j } } X J _ { \\theta , 1 } ^ { - 1 } p \\| ^ 2 - \\| p X J _ { \\theta , 1 } ^ { - 1 } \\chi ^ { - n _ { k _ j } } \\| ^ 2 ) + L ( p ) \\\\ & = \\| R X _ 1 P _ { L ^ 2 ( \\mu _ { } ) } p \\| ^ 2 + F ( p ) + L ( p ) \\end{aligned} \\end{align*}"}
-{"id": "8879.png", "formula": "\\begin{align*} J _ { \\theta , c } U _ { ( \\theta ) c } = U _ { \\sigma _ c } J _ { \\theta , c } . \\end{align*}"}
-{"id": "2196.png", "formula": "\\begin{align*} h _ i ( z ) = 0 , i = 1 , \\ldots , n \\end{align*}"}
-{"id": "9939.png", "formula": "\\begin{align*} u ^ c ( x , t ) = \\pm \\frac { 4 \\lambda } 3 \\cos ^ 2 \\left ( \\frac { x \\mp \\lambda t } { 4 } \\right ) H ( 2 \\pi - | x \\mp \\lambda t | ) \\end{align*}"}
-{"id": "7364.png", "formula": "\\begin{align*} \\psi \\otimes \\overline { \\psi } = ( \\psi , \\psi ) + ( \\psi , e _ a . \\psi ) e ^ a + ( \\psi , e _ { b a } . \\psi ) e ^ { a b } + . . . + ( \\psi , e _ { a _ p . . . a _ 2 a _ 1 } . \\psi ) e ^ { a _ 1 a _ 2 . . . a _ p } + . . . + ( - 1 ) ^ { \\lfloor { n / 2 } \\rfloor } ( \\psi , z . \\psi ) z \\end{align*}"}
-{"id": "9351.png", "formula": "\\begin{align*} N r ^ { 2 } ( J O _ { n } ^ { ( 3 ) } ) ) & = \\frac { 1 } { 4 9 } \\left ( \\begin{array} { c } \\left ( 2 ^ { n + 1 } - 2 \\right ) ^ { 2 } + \\left ( 2 ^ { n + 2 } + 3 \\right ) ^ { 2 } + \\left ( 2 ^ { n + 3 } - 1 \\right ) ^ { 2 } + \\left ( 2 ^ { n + 4 } - 2 \\right ) ^ { 2 } \\\\ + \\left ( 2 ^ { n + 5 } + 3 \\right ) ^ { 2 } + \\left ( 2 ^ { n + 6 } - 1 \\right ) ^ { 2 } + \\left ( 2 ^ { n + 7 } - 2 \\right ) ^ { 2 } + \\left ( 2 ^ { n + 8 } + 3 \\right ) ^ { 2 } \\end{array} \\right ) \\\\ & = \\frac { 1 } { 4 9 } \\left ( 2 1 8 4 5 \\cdot 2 ^ { 2 n + 2 } + 2 ^ { n + 1 0 } + 4 1 \\right ) . \\end{align*}"}
-{"id": "8618.png", "formula": "\\begin{align*} \\partial _ t h = \\frac 1 2 \\Delta h + \\frac 1 2 \\lambda | \\nabla h | ^ 2 + V ( t , x ) - c _ 0 , \\ \\ x \\in \\R ^ d , d \\geq 3 , \\end{align*}"}
-{"id": "24.png", "formula": "\\begin{align*} \\hat { V } ^ { C } _ { \\sigma } ( R _ 1 , R _ 2 ) \\sqrt { 2 \\pi } \\sigma = \\frac { 1 } { 2 \\pi \\sigma ^ { 2 } } \\frac { 1 } { N } \\sum \\limits _ { n = 1 } ^ N e x p \\left ( - \\frac { ( x _ { n } - y _ { n } ) ^ { 2 } + ( z _ { n } - s _ { n } ) ^ { 2 } } { 2 \\sigma ^ 2 } \\right ) \\sqrt { 2 \\pi } \\sigma \\end{align*}"}
-{"id": "6845.png", "formula": "\\begin{align*} \\min _ x \\left \\{ f ( x ) : ~ f ^ i ( x , u ^ i ) \\leq 0 , \\ \\forall i = 1 , \\ldots , m , ~ ~ x \\in X \\right \\} , \\end{align*}"}
-{"id": "6416.png", "formula": "\\begin{align*} \\chi \\left ( \\xi \\gamma \\right ) \\overset { } { = } \\frac { 1 } { \\left ( 1 - \\xi ^ { 2 } \\right ) - \\frac { 1 } { 2 \\gamma } } \\end{align*}"}
-{"id": "6474.png", "formula": "\\begin{align*} p _ { 3 D c } \\left ( x y | \\mu _ { x } \\sigma _ { x } \\sigma _ { y } ; \\rho \\right ) = \\dfrac { 1 } { 2 \\pi \\sigma _ { x } \\sigma _ { y } \\sqrt { 1 - \\rho ^ { 2 } } } \\exp \\left [ \\frac { - 1 } { 2 \\left ( 1 - \\rho ^ { 2 } \\right ) } \\left ( \\frac { \\left ( x - \\mu _ { x } \\right ) ^ { 2 } } { \\sigma _ { x } ^ { 2 } } + \\frac { y ^ { 2 } } { \\sigma _ { y } ^ { 2 } } - \\frac { 2 \\rho \\left ( x - \\mu _ { x } \\right ) y } { \\sigma _ { x } \\sigma _ { y } } \\right ) \\right ] \\end{align*}"}
-{"id": "1753.png", "formula": "\\begin{align*} \\nu _ { f \\sqrt { d \\mu } } ( Z ( \\eta ) ) : = \\langle ( S ^ { u n i v } _ \\eta ( S ^ { u n i v } _ \\eta ) ^ * ) f \\sqrt { d \\mu } , f \\sqrt { d \\mu } \\rangle . \\end{align*}"}
-{"id": "8132.png", "formula": "\\begin{align*} B & = \\bigg \\{ x \\in X : d \\bigg ( x , X \\setminus \\bigsqcup _ { i \\in I } S _ i '' V _ i \\bigg ) > \\eta \\bigg \\} , \\\\ A & = \\bigg \\{ x \\in X : d \\bigg ( x , X \\setminus \\bigsqcup _ { i \\in I } S _ i V _ i \\bigg ) \\leq \\eta \\bigg \\} \\end{align*}"}
-{"id": "6829.png", "formula": "\\begin{align*} S _ { \\rho } ( w _ { \\lambda } + \\phi ) = S _ { \\rho } ( w _ { \\lambda } ) + \\mathcal { L } \\left ( \\phi - \\frac { \\int _ { \\mathbb { S } ^ 2 } e ^ { w _ { \\lambda } } \\phi } { \\int _ { \\mathbb { S } ^ 2 } e ^ { w _ { \\lambda } } } \\right ) + N ( \\phi ) , \\end{align*}"}
-{"id": "9100.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } ( - 1 ) ^ i \\sum _ { \\mathrm { c a r d } ( I ) = i } \\left ( \\prod _ { j \\in I } x _ { j } \\right ) \\cdot A \\left ( \\prod _ { k \\in \\left \\{ 1 , \\ldots , n + 1 \\right \\} \\setminus I } x _ { k } \\right ) = 0 ( x _ { 1 } , \\ldots , x _ { n + 1 } \\in R ) \\end{align*}"}
-{"id": "8653.png", "formula": "\\begin{align*} B _ { k , X } = \\left \\{ \\sum _ { i = T _ k } ^ { T _ { k + 1 } - 1 } \\max _ { s \\in [ 0 , 1 ] } | X _ i ( s ) | > \\frac { k ^ \\alpha } { 3 } \\right \\} , \\ \\ B _ { k , Y } = \\left \\{ \\sum _ { i = T _ k } ^ { T _ { k + 1 } - 1 } \\max _ { s \\in [ 0 , 1 ] } | Y _ i ( s ) | > \\frac { k ^ \\alpha } { 3 } \\right \\} . \\end{align*}"}
-{"id": "9395.png", "formula": "\\begin{align*} b ( x ) = \\sum _ { ( j , \\beta ) \\in \\Lambda } [ f ( x ) - \\mathbb E _ j f ( x ) ] \\chi _ { Q _ { \\beta } ^ j } ( x ) : = \\sum _ { ( j , \\beta ) \\in \\Lambda } b _ { j , \\beta } ( x ) . \\end{align*}"}
-{"id": "1707.png", "formula": "\\begin{align*} ( \\tau ^ n ) ^ { - 1 } ( A ) = A , \\quad ( \\tau ^ n ) ^ { - 1 } ( B ) = B . \\end{align*}"}
-{"id": "389.png", "formula": "\\begin{align*} S _ { \\Psi } & = \\sum _ { n = m } ^ \\infty \\frac { 1 } { n h ( n ) } \\mathbb P \\left ( | S _ n | > ( 1 + \\varepsilon ) \\sigma _ n \\sqrt { 2 \\ln \\Psi ( n ) } \\right ) \\\\ & \\propto \\sum _ { n = m } ^ \\infty \\frac { 1 } { n h ( n ) \\sqrt { \\ln \\Psi ( n ) } \\Psi ( n ) ^ { ( 1 + \\varepsilon ) ^ 2 } } \\\\ & = \\sum _ { n = m } ^ \\infty \\frac { \\Psi ' ( n ) } { \\sqrt { \\ln \\Psi ( n ) } \\Psi ( n ) ^ { ( 1 + \\varepsilon ) ^ 2 } } . \\end{align*}"}
-{"id": "276.png", "formula": "\\begin{align*} D _ { 0 _ { + } } ^ { \\alpha } x ( t ) = f ( t , x ) \\end{align*}"}
-{"id": "6912.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } w _ { 0 } ( x _ { 0 } , x _ { 1 } , x _ { 2 } , r , \\Phi _ { 1 } ( b _ { 1 } ) ) & = \\frac { x _ { 0 } + x _ { 1 } + x _ { 2 } } { 3 } \\\\ \\lim _ { r \\to \\infty } z _ { 0 } ( x _ { 0 } , x _ { 1 } , x _ { 2 } , r , \\Phi _ { 1 } ( b _ { 1 } ) ) & = \\frac { x _ { 0 } + x _ { 1 } + x _ { 2 } } { 3 } \\\\ \\lim _ { r \\to \\infty } y _ { 0 , 1 } ( x _ { 0 } , x _ { 1 } , x _ { 2 } , r , \\Phi _ { 1 } ( b _ { 1 } ) ) & = \\frac { x _ { 0 } + x _ { 1 } + x _ { 2 } } { 3 } \\end{align*}"}
-{"id": "8572.png", "formula": "\\begin{align*} & \\int _ { \\Omega \\ , \\cap \\ , B _ { H _ 0 } ( x , 1 ) } F ( | z ( t ) | ) \\ , d y \\le \\int _ { \\Omega \\ , \\cap \\ , B _ { H _ 0 } ( x , 2 ) } F ( | z ( 0 ) | ) \\ , d y \\\\ & + C \\int _ 0 ^ t \\int _ { \\Omega \\ , \\cap \\ , B _ { H _ 0 } ( x , 2 ) } s ^ { \\sigma - 1 } | z | ^ { 1 + \\delta } \\ , d y \\ , d s + C \\int _ 0 ^ t \\int _ { \\Omega \\ , \\cap \\ , B _ { H _ 0 } ( x , 2 ) } s ^ { - \\sigma } F ( z ) \\ , d y \\ , d s \\end{align*}"}
-{"id": "2548.png", "formula": "\\begin{align*} \\left | \\int _ A H \\ , d x \\right | < P ( A ) \\ , \\forall A \\subsetneq \\Omega \\ , , \\left | \\int _ \\Omega H \\ , d x \\right | = P ( \\Omega ) \\end{align*}"}
-{"id": "6386.png", "formula": "\\begin{align*} P _ { } \\left ( \\theta \\right ) \\overset { } { = } \\int d x P _ { } \\left ( x \\theta \\right ) \\end{align*}"}
-{"id": "4651.png", "formula": "\\begin{align*} B _ { k , r } ( i , j ) = \\sum _ { n = 0 } ^ { r _ { k , k + r + 1 } ( j ) - 1 } 1 _ { I _ { k , i } } \\Big ( ( T | I _ k ) ^ n I _ { k + r + 1 , j } \\Big ) \\end{align*}"}
-{"id": "7854.png", "formula": "\\begin{align*} h ( x ^ { \\ast } ) = \\left ( h _ { X } ( x ) , h _ { 1 2 } ( x _ { 1 } | x _ { 2 } ) , h _ { 2 1 } ( x _ { 2 } | x _ { 1 } ) \\right ) , \\end{align*}"}
-{"id": "5605.png", "formula": "\\begin{align*} Z _ t = G ( x ) + \\int _ 0 ^ t \\tilde { f } ( Z _ r ) d r + B ^ H _ t , \\end{align*}"}
-{"id": "4577.png", "formula": "\\begin{align*} \\tilde \\theta ^ { - 1 } \\bigl ( \\tilde H _ { 0 , 1 } \\bigr ) & \\equiv - ( - \\tilde d ) ^ { - n + 2 } [ \\cdots [ \\tilde F _ { 1 , 1 } , \\tilde F _ { 2 , 0 } ] _ { q } , \\cdots , \\tilde F _ { n - 2 , 0 } ] _ { q } \\cdot \\tilde F _ { 0 , - 1 } \\ , \\\\ & = - ( - d ) ^ { - n + 1 } q _ 1 A \\cdot \\bigl ( F _ { 0 , - 1 } F _ { 1 , 0 } - q F _ { 1 , 0 } F _ { 0 , - 1 } \\bigr ) \\ , , \\end{align*}"}
-{"id": "4867.png", "formula": "\\begin{align*} E _ d ( t ) = \\begin{cases} \\frac { 1 } { d } \\displaystyle \\sum _ { \\substack { m \\mid d \\ : , \\ : m } } \\mu \\left ( m \\right ) ( t ^ { d / 2 m } - 1 ) & \\\\ 0 & ; \\end{cases} \\end{align*}"}
-{"id": "1036.png", "formula": "\\begin{align*} \\Psi ( n ) = \\max _ { \\zeta \\in \\mathbb { R } \\setminus { \\overline { \\mathbb { Q } } } } \\Psi ( n , \\zeta ) , \\qquad \\widetilde { \\Psi } ( n ) = \\max _ { \\zeta \\in \\mathbb { R } \\setminus { \\overline { \\mathbb { Q } } } } \\widetilde { \\Psi } ( n , \\zeta ) . \\end{align*}"}
-{"id": "2248.png", "formula": "\\begin{align*} f _ { j , s } ( z ) = \\bigl ( q _ { j , s } + Q _ { j , s } ( z ) \\bigr ) , \\end{align*}"}
-{"id": "4934.png", "formula": "\\begin{align*} D = e ^ a . \\nabla _ { X _ a } \\end{align*}"}
-{"id": "8956.png", "formula": "\\begin{gather*} \\sum _ { \\alpha \\in \\Phi ^ + ( W ) } \\ ! \\ ! T _ \\alpha - w \\bigg ( \\sum _ { \\alpha \\in \\Phi ^ + ( W ) } \\ ! \\ ! T _ \\alpha \\bigg ) = \\sum _ { \\alpha \\in \\Phi ^ + ( W ) } \\ ! \\ ! T _ \\alpha - \\sum _ { \\alpha \\in \\Phi ^ + ( W ) } \\ ! \\ ! T _ { w \\alpha } = \\sum _ { \\alpha \\in \\Phi ^ + ( W ) \\cap w \\Phi ^ - ( W ) } \\ ! \\ ! ( T _ \\alpha - T _ { - \\alpha } ) , \\end{gather*}"}
-{"id": "1626.png", "formula": "\\begin{align*} \\sigma ^ m ( x ) ( p , q ) = x ( m + p , m + q ) . \\end{align*}"}
-{"id": "989.png", "formula": "\\begin{align*} \\left \\| \\max _ { j = 0 , 1 , \\dots , \\lfloor T / h \\rfloor } \\sup _ { t \\in [ u _ j , u _ { j + 1 } ] } \\left | \\mathbf { A } ^ j _ n ( t ) \\right | \\right \\| _ { \\psi _ 1 } \\lesssim h ^ \\gamma ( \\log n ) ^ 2 \\end{align*}"}
-{"id": "916.png", "formula": "\\begin{align*} \\mathrm { D o m } ( L ) = \\left \\{ F \\in L ^ 2 ( W ) : \\sum _ { q = 1 } ^ \\infty q ^ 2 E [ \\| J _ q F \\| ^ 2 ] < \\infty \\right \\} . \\end{align*}"}
-{"id": "3832.png", "formula": "\\begin{align*} \\left ( 1 - L ^ { \\delta \\log p _ { * } } \\right ) \\ , \\P _ \\eta \\left ( \\bigcap _ { i = 0 } ^ { k } \\left ( G ^ { ( 0 , T _ i ) } _ { T _ 1 } \\cap \\Lambda ^ { ( 0 , T _ i ) } _ { T _ 1 } \\right ) ^ c \\cap \\{ \\eta _ { T _ i } ( 0 ) = 0 \\} \\right ) . \\end{align*}"}
-{"id": "4303.png", "formula": "\\begin{align*} \\sum _ { ( d _ 1 , d _ 2 ) > 0 } \\mathsf { N } _ { 1 , d _ 1 F _ 1 + d _ 2 F _ 2 } q _ 1 ^ { d _ 1 } q _ 2 ^ { d _ 2 } = 1 2 \\sum _ { m , n \\geq 1 } \\frac { q _ 1 ^ { m n } } { n } + 1 2 \\sum _ { m , n \\geq 1 } \\frac { q _ 2 ^ { m n } } { n } . \\end{align*}"}
-{"id": "1591.png", "formula": "\\begin{align*} \\alpha _ { \\mathbf { j } '' } ( n ) & = m / 2 - 2 ^ { n - 1 } j _ 1 - 2 ^ { n - 2 } j _ 2 - \\cdots - 2 j _ { n - 1 } \\\\ & = m / 2 - 2 ( m - 4 ) / 3 = ( 1 6 - m ) / 6 , \\end{align*}"}
-{"id": "8954.png", "formula": "\\begin{gather*} ( f _ 0 + f _ 1 ( s - 1 ) ) ( g _ 0 + ( s + 1 ) { } ^ s g _ 1 ) = f _ 0 g _ 0 + f _ 1 ( { } ^ s g _ 0 - g _ 0 ) + f _ 0 ( g _ 1 + { } ^ s g _ 1 ) \\\\ \\hphantom { ( f _ 0 + f _ 1 ( s - 1 ) ) ( g _ 0 + ( s + 1 ) { } ^ s g _ 1 ) = } { } + ( f _ 1 { } ^ s g _ 0 + f _ 0 g _ 1 ) ( s - 1 ) , \\end{gather*}"}
-{"id": "8620.png", "formula": "\\begin{align*} M _ { t , x , B } ( r ) : = \\int _ { - \\infty } ^ r \\int _ { \\R ^ d } \\Phi _ { t , x , B } ( s , y ) d W ( s , y ) , \\end{align*}"}
-{"id": "2394.png", "formula": "\\begin{align*} a _ 6 = a _ 5 = 3 a _ 3 + a _ 4 = a _ 2 = 0 , \\end{align*}"}
-{"id": "5949.png", "formula": "\\begin{align*} f _ { \\xi , \\alpha } ( t , v ) : = \\hat { \\sigma } ( t ) + v \\ , \\xi ( t ) \\left ( \\hat { \\sigma } ( t ) : = \\int _ 0 ^ t a ( \\tau ) \\ , \\xi ( \\tau ) d \\tau \\right ) \\end{align*}"}
-{"id": "622.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { \\alpha _ 1 \\ge 0 , \\alpha _ 2 \\in \\Z } a _ { \\alpha } z _ 1 ^ { \\alpha _ 1 } z _ 2 ^ { \\alpha _ 2 } . \\end{align*}"}
-{"id": "3583.png", "formula": "\\begin{align*} h + s - t + \\ell ( \\sigma ) = 1 . \\end{align*}"}
-{"id": "9285.png", "formula": "\\begin{align*} \\boldsymbol { E } [ \\sum _ { j = 1 } ^ { T - 2 } \\sum _ { k = j + 1 } ^ { T - 1 } \\sum _ { l = k + 1 } ^ T X _ j X _ k X _ l ] & \\leq \\boldsymbol { E } [ \\sum _ { j = 1 } ^ { T - 2 } \\sum _ { k = j + 1 } ^ { T - 1 } \\sum _ { l = k + 1 } ^ T X _ j X _ k ] \\\\ & \\leq \\left ( [ \\boldsymbol { E } [ \\sum _ { j = 1 } ^ { T - 2 } \\sum _ { k = j + 1 } ^ { T - 1 } X _ j ^ 2 X _ k ^ 2 ] \\right ) ^ { \\frac { 1 } { 2 } } \\left ( [ \\boldsymbol { E } [ \\sum _ { j = 1 } ^ { T - 2 } \\sum _ { k = j + 1 } ^ { T - 1 } ( T - k ) ^ 2 ] \\right ) ^ \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "4131.png", "formula": "\\begin{align*} \\eta = K ^ { \\frac { 1 } { 4 } } \\xi + Z , \\end{align*}"}
-{"id": "7905.png", "formula": "\\begin{align*} d ( z , z ' ) = d ( z , O ) + d ( z ' , O ) \\leq { 1 \\over 2 } d ( x , O ) + { 1 \\over 2 } d ( x ' , O ) + { 1 \\over 2 } d ( y , O ) + { 1 \\over 2 } d ( y ' , O ) . \\end{align*}"}
-{"id": "9207.png", "formula": "\\begin{align*} [ s ' \\otimes c ' , \\lambda \\otimes e ] = s ' \\diamond \\lambda \\otimes \\frac { [ c ' , e ] _ { A ^ { + } } } { 2 } + [ s ' , \\lambda ] \\otimes \\frac { ( c ' \\circ e ) _ { A ^ { - } } } { 2 } . \\end{align*}"}
-{"id": "256.png", "formula": "\\begin{align*} \\partial _ { \\R _ { \\geq 0 } ^ { n - 1 } \\times \\R ^ { d - ( n - 1 ) } } \\big ( \\R _ { \\geq 0 } ^ n \\times \\R ^ { d - n } \\big ) = \\R _ { > 0 } ^ { n - 1 } \\times \\{ 0 \\} \\times \\R ^ { d - n } . \\end{align*}"}
-{"id": "9750.png", "formula": "\\begin{align*} & L ( \\tilde { J _ { k } } ) \\leq L ( J _ { k } ) + M e ^ { - l k h } h \\| Z _ { 0 } \\| _ { \\infty } \\big ( L ( J _ k ) + 1 \\big ) , \\\\ & Q ( \\tilde { J _ { k } } ) \\leq Q ( J _ { k } ) + M e ^ { - l k h } h \\| Z _ { 0 } \\| _ { \\infty } \\big ( L ( J _ k ) + 1 \\big ) ^ 2 . \\end{align*}"}
-{"id": "5733.png", "formula": "\\begin{align*} 0 = \\int _ \\R ( z ' ( \\cdot - m ( t ) ) - \\partial _ s \\gamma ) \\cdot ( \\partial _ t \\gamma + m ' ( t ) \\ , \\partial _ s \\gamma ) \\d s + \\int _ \\R \\partial _ s \\gamma \\cdot ( \\partial _ t \\gamma + m ' ( t ) \\ , \\partial _ s \\gamma ) \\d s . \\end{align*}"}
-{"id": "2411.png", "formula": "\\begin{align*} p _ 3 \\left ( \\frac { 3 a _ 3 + a _ 4 } { a _ 5 } \\right ) = - \\frac { a _ 4 ( a _ 4 ^ 2 + a _ 5 ^ 2 ) } { a _ 5 ^ 2 } = 0 . \\end{align*}"}
-{"id": "8638.png", "formula": "\\begin{align*} \\Psi ( x ) = \\frac { 1 } { \\rho } \\int _ { \\Omega } e ^ { I ( x , y ) } \\Psi ( y ) \\pi ( d y ) , \\end{align*}"}
-{"id": "4421.png", "formula": "\\begin{align*} d ( x , y ) : = | x _ 1 - y _ 1 | + | x _ 2 - y _ 2 | ^ \\frac { 2 } { 3 } , x , y \\in { [ 0 , 1 ) ^ 2 } , \\end{align*}"}
-{"id": "2963.png", "formula": "\\begin{align*} y _ { _ k } & = \\sqrt { \\eta _ k } \\bigg ( h _ { k k } x _ k + \\sum _ { \\substack { j = 1 \\\\ j \\neq k } } ^ { K } h _ { k j } x _ j + w _ k \\bigg ) + n _ k , \\forall k \\in \\mathcal { K } , \\\\ y _ { _ k } & = \\sqrt { 1 - \\eta _ k } \\bigg ( \\sum _ { j = 1 } ^ { K } h _ { k j } x _ j + w _ k \\bigg ) , \\forall k \\in \\mathcal { K } , \\end{align*}"}
-{"id": "6854.png", "formula": "\\begin{align*} F _ j ( x ) = \\frac { 1 } { 2 } ( x - q _ j ) + \\frac { 1 } { 2 } q _ j \\end{align*}"}
-{"id": "7867.png", "formula": "\\begin{align*} d ( z _ 1 , z _ 2 ) \\le d ( z _ 1 , z _ 0 ) + d ( z _ 0 , z _ 2 ) = { 1 \\over 2 } d ( x , y ) \\end{align*}"}
-{"id": "4976.png", "formula": "\\begin{align*} \\omega _ j ( x ) : = \\left ( \\sum _ { r \\in \\Z ^ d } \\left [ T _ j | \\Delta _ j f | ( 2 ^ { - j } r ) E ( 2 ^ j x - r ) \\right ] ^ { p } \\right ) ^ { \\frac { 1 } { p } } , \\ x \\in \\R ^ d . \\end{align*}"}
-{"id": "3313.png", "formula": "\\begin{align*} D ^ * ( r _ s ) \\leq \\bar { D } ( r _ s ) = \\frac { \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K } { s + 1 + i } ( N - 1 ) ^ i N } { \\binom { K - 2 } { s - 1 } + \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K - 1 } { s + i } ( N - 1 ) ^ i N } \\end{align*}"}
-{"id": "8796.png", "formula": "\\begin{align*} c _ { I , \\Phi , \\chi } = \\underset { a \\in \\overset { \\circ } { \\overline { E } } _ { I } ( k _ { \\mathfrak { p } } ) } { \\sum } \\overline { \\Phi } ( \\overline { h } ( a ) ) \\Omega _ { \\chi } ( a ) . \\end{align*}"}
-{"id": "2387.png", "formula": "\\begin{align*} R _ 2 = a _ 3 a _ 6 l _ 0 ^ 2 + a _ 2 l _ 0 l _ 1 l _ 2 + a _ 2 ^ 2 ( l _ 1 + l _ 2 ) ( l _ 1 + l _ 3 ) , \\end{align*}"}
-{"id": "9271.png", "formula": "\\begin{align*} \\frac { d } { d t } W _ t = \\varepsilon ^ 2 Q _ { \\varepsilon } W _ t + \\frac { d } { d t } I _ t . \\end{align*}"}
-{"id": "4107.png", "formula": "\\begin{align*} - \\langle D _ X Y , \\xi \\rangle = X ( Y ( g ) ) \\langle \\eta , \\xi \\rangle , \\end{align*}"}
-{"id": "5744.png", "formula": "\\begin{align*} \\frac { \\phi _ { x , y - 1 , z } ( a , b ) } { \\phi _ { x , y - 1 , z - 1 } ( a , b ) } \\frac { \\phi _ { x , y , z - 1 } ( a , b ) } { \\phi _ { x , y , z } ( a , b ) } & + \\frac { \\phi _ { x + 1 , y - 1 , z - 1 } ( a , b ) } { \\phi _ { x , y - 1 , z - 1 } ( a , b ) } \\frac { \\phi _ { x - 1 , y , z } ( a , b ) } { \\phi _ { x , y , z } ( a , b ) } \\\\ & = \\frac { ( 2 x + y + z + 2 a + 2 b ) ( 2 y + 2 z + 2 a + 2 b - 1 ) } { ( 2 x + 2 y + 2 z + 2 a + 2 b - 1 ) ( x + y + z + 2 a + 2 b ) } \\\\ & + \\frac { x ( 2 x + 2 a + 2 b + 1 ) } { ( 2 x + 2 y + 2 z + 2 a + 2 b - 1 ) ( x + y + z + 2 a + 2 b ) } = 1 . \\end{align*}"}
-{"id": "1338.png", "formula": "\\begin{align*} \\nabla _ x L ( x , Y , \\mu , \\lambda ) = 0 , \\ , Y \\in \\partial \\theta ( F ( x ) ) , h ( x ) = 0 , g ( x ) \\in K , \\ , \\lambda \\in K ^ * \\ , { \\rm a n d } \\ , \\langle g ( x ) , \\lambda \\rangle = 0 , \\end{align*}"}
-{"id": "4284.png", "formula": "\\begin{align*} T _ + ( \\alpha ) = ( \\pi ^ { \\ast } \\pi _ { \\ast } \\alpha ) \\cup W , T _ - ( \\alpha ) = \\pi ^ { \\ast } \\pi _ { \\ast } ( \\alpha \\cup W ) , \\end{align*}"}
-{"id": "2266.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } \\frac 1 { { \\sinh } ^ 2 ( t k ) \\cdot { k ^ 2 } } = - \\frac { \\partial } { \\partial t } \\left [ \\sum _ { k = 1 } ^ \\infty \\frac 2 { k ^ 3 ( e ^ { 2 t k } - 1 ) } \\right ] \\end{align*}"}
-{"id": "5656.png", "formula": "\\begin{align*} E ( u ) = \\int _ \\R \\left ( \\frac 1 2 \\| \\partial _ { x _ 1 } u \\| ^ 2 _ { L ^ 2 ( \\Omega ) } + \\int _ { \\Omega } \\left ( \\frac 1 2 | \\nabla _ { x ' } u | ^ 2 + F ( x ' , u ) \\right ) \\d x ' \\right ) \\d x _ 1 . \\end{align*}"}
-{"id": "1856.png", "formula": "\\begin{align*} \\mathsf { D } ( \\mathcal { G F } _ { \\mathcal { B } } ( R ) ) : = \\frac { \\mathsf { K } ( \\mathcal { G F } _ { \\mathcal { B } } ( R ) ) } { \\widetilde { \\mathcal { G F } } _ { \\mathcal { B } } ( R ) } . \\end{align*}"}
-{"id": "7799.png", "formula": "\\begin{align*} \\lim _ { \\lambda ^ { - 1 } + \\tau \\rightarrow 0 } \\left \\Vert S _ { \\lambda , \\tau } f - f \\right \\Vert _ { p , \\gamma } = 0 . \\end{align*}"}
-{"id": "3343.png", "formula": "\\begin{align*} & \\binom { K - 2 } { s - 1 } D _ { s } ^ { ( K + 1 ) } L _ { s - 1 } ^ { ( K ) } + \\binom { K - 1 } { s - 1 } D _ { s - 1 } ^ { ( K ) } L _ { s } ^ { ( K ) } + \\binom { K - 2 } { s - 2 } D _ s ^ { ( K ) } L _ s ^ { ( K + 1 ) } \\\\ & = \\binom { K - 2 } { s - 1 } D _ { s - 1 } ^ { ( K ) } L _ { s } ^ { ( K + 1 ) } + \\binom { K - 2 } { s - 2 } D _ s ^ { ( K + 1 ) } L _ { s } ^ { ( K ) } + \\binom { K - 1 } { s - 1 } D _ s ^ { ( K ) } L _ { s - 1 } ^ { ( K ) } . \\end{align*}"}
-{"id": "6648.png", "formula": "\\begin{align*} J _ f ( x ) \\ ; = \\ ; \\mathrm { d e t } D f ( x ) \\ ; = \\ ; \\sum _ { \\sigma \\in S _ d } ( - 1 ) ^ { \\mathrm { s i g n } ( \\sigma ) } f _ { 1 \\sigma ( 1 ) } ( x ) f _ { 2 \\sigma ( 1 ) } ( x ) \\cdots f _ { d \\sigma ( d ) } ( x ) \\ . \\end{align*}"}
-{"id": "3219.png", "formula": "\\begin{align*} \\mathcal H _ k ( X ) = 0 . \\end{align*}"}
-{"id": "5394.png", "formula": "\\begin{align*} ( a _ 1 , a _ 4 \\cdot a _ 5 ) = \\frac { 1 } { 2 ^ 6 } \\neq ( a _ 1 \\cdot a _ 4 , a _ 5 ) \\end{align*}"}
-{"id": "601.png", "formula": "\\begin{align*} Z _ { \\Gamma _ { N } } ( s ) = \\prod _ { a \\in \\{ 0 , \\dots , N - 1 \\} ^ { r } } L _ { \\Gamma } \\left ( s , \\frac { 1 } { N } a \\right ) . \\end{align*}"}
-{"id": "1393.png", "formula": "\\begin{align*} G _ { m + 1 } = { S L _ p } ^ { m + 2 } / \\mu _ p , \\end{align*}"}
-{"id": "5524.png", "formula": "\\begin{align*} b ( m ) = b ( 3 k + 1 ) & \\leqslant h ( 3 k + 1 ) + ( \\sqrt { 2 } + 1 ) \\lfloor \\log _ 3 ( k + 1 ) \\rfloor + 1 \\\\ & \\leqslant h ( 3 k + 1 ) + ( \\sqrt { 2 } + 1 ) \\lfloor \\log _ 3 ( 3 k + 1 ) \\rfloor , \\end{align*}"}
-{"id": "611.png", "formula": "\\begin{align*} \\dim C _ W ^ n ( 0 ) = \\dim C _ w ^ { n - 1 } ( 2 ) \\dim C _ W ^ n ( s ) = \\dim C _ W ^ { n - 1 } ( s - 2 ) + \\dim C _ W ^ { n - 1 } ( s ) + \\dim C _ W ^ { n - 1 } ( s + 2 ) . \\end{align*}"}
-{"id": "4743.png", "formula": "\\begin{align*} { L } _ 1 = \\sqrt { { \\epsilon } ( - \\gamma - \\alpha ^ 2 a _ 1 - 2 \\alpha \\beta b _ 1 - \\beta ^ 2 c _ 1 ) } . \\end{align*}"}
-{"id": "9156.png", "formula": "\\begin{align*} V _ a = \\{ A | _ { K ^ * } \\ | \\ A ( x + y ) = A ( x ) + A ( y ) , \\ \\ x \\ \\textrm { a n d } \\ y \\in K \\} . \\end{align*}"}
-{"id": "436.png", "formula": "\\begin{align*} H _ { k _ 1 , k _ 2 } ( R , t ) = 2 \\pi \\sum _ { \\abs { \\alpha } \\leq N } I _ { { n + k _ 1 - 1 - \\alpha _ 2 } } ( \\kappa ) \\frac { \\partial ^ \\alpha \\tilde \\phi _ { k _ 1 , k _ 2 } ( 0 , 0 ) \\kappa ^ { \\alpha _ 1 } } { 2 ^ { \\alpha _ 1 } \\alpha ! } \\delta ^ { \\abs * { \\alpha } } + O \\left ( \\delta ^ { N + 1 } \\right ) \\end{align*}"}
-{"id": "7778.png", "formula": "\\begin{align*} \\widetilde { D } u ( \\ell ) : = ( \\widetilde { D } _ \\rho u ( \\ell ) ) _ { \\rho \\in \\L - \\ell } \\widetilde { D } _ \\rho u ( \\ell ) : = \\left \\{ \\begin{array} { l l } S ^ * D _ \\rho S u ( \\ell ) , & \\ell \\in \\Omega _ { \\Gamma } , \\\\ D _ \\rho u ( \\ell ) , & \\end{array} \\right . \\end{align*}"}
-{"id": "1317.png", "formula": "\\begin{align*} \\varphi ( a ) = \\varphi ( \\lim e _ n ^ { \\frac { 1 } { 2 } } a e _ n ^ { \\frac { 1 } { 2 } } ) \\leq \\| a \\| \\lim \\varphi ( e _ n ) = 0 . \\end{align*}"}
-{"id": "2219.png", "formula": "\\begin{align*} \\deg _ { w _ j } \\widetilde Q _ 1 < m _ { 1 j } , \\ j = 1 , \\ldots , n . \\end{align*}"}
-{"id": "438.png", "formula": "\\begin{align*} e ^ { - \\frac { 3 \\pi \\abs * { t } } { 2 } } = o \\left ( \\delta ^ { N + 2 - n - k _ 1 } e ^ { - R - \\pi \\abs * { t } } \\right ) \\end{align*}"}
-{"id": "3137.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\left \\Vert \\mathbf { P } ^ t ( x , \\cdot ) - \\pi \\right \\Vert _ { T V } = 0 , \\forall x \\in E , \\end{align*}"}
-{"id": "2863.png", "formula": "\\begin{align*} \\breve { u } _ j = \\left \\{ \\begin{array} { r l } u _ j , & \\ j \\in [ n ] \\setminus T \\\\ 0 , & \\ j \\in T \\end{array} \\right . . \\end{align*}"}
-{"id": "3110.png", "formula": "\\begin{align*} & T _ s = \\Gamma + + + + 2 \\delta , \\\\ & T _ c = \\Gamma + + \\delta , \\\\ & T _ { } = + + 2 + 4 \\delta , \\end{align*}"}
-{"id": "6020.png", "formula": "\\begin{align*} & \\inf _ { P _ { X } , Q _ { X } : \\left | P _ { X } - Q _ { X } \\right | = \\epsilon } D _ { 1 + s } ( P _ { X } \\| Q _ { X } ) \\geq \\frac { ( 1 + s ) \\epsilon ^ { 2 } } { 2 } . \\end{align*}"}
-{"id": "5595.png", "formula": "\\begin{align*} C _ z ( n ) & = \\sum _ { 0 \\leq a \\leq n } M _ z ( a ) D _ z ( n - a ) \\\\ & = q ^ n \\left [ \\sum _ { 0 \\leq a < n } \\frac { M _ z ( a ) } { q ^ a } \\frac { ( n - a ) ^ { z - 1 } } { \\Gamma ( z ) } + O _ A \\left ( \\sum _ { 0 \\leq a < n } \\frac { | M _ z ( a ) | } { q ^ a } ( n - a ) ^ { \\Re { z } - 2 } \\right ) + \\frac { M _ z ( n ) } { q ^ n } \\right ] . \\end{align*}"}
-{"id": "4139.png", "formula": "\\begin{align*} \\max \\Biggl \\{ \\sum _ { i \\in N } \\sum _ { j \\in V } \\sum _ { t \\in [ n ] } w ^ t _ { i j } z ^ t _ { i j } : \\exists ~ ( x , y ) \\geq 0 \\eqref { e q : o b m _ l p _ p o l 1 } \\eqref { e q : o b m _ l p _ p o l 2 } z ^ t _ { i j } = \\sum _ { S \\subseteq V \\backslash j } x _ { i , j } ^ { t , S } \\Biggr \\} , \\end{align*}"}
-{"id": "1177.png", "formula": "\\begin{align*} \\mathcal { A } ( \\eta ) = | \\eta | ^ { p - 2 } ( \\eta _ 1 , \\dots , \\eta _ n ) \\end{align*}"}
-{"id": "160.png", "formula": "\\begin{align*} g = \\begin{pmatrix} a & 0 \\\\ c & d \\\\ \\end{pmatrix} \\end{align*}"}
-{"id": "7008.png", "formula": "\\begin{align*} F _ { 2 , s } [ u ] ( 0 ) = F _ { 2 , s } ^ { \\epsilon _ 0 } [ u ] ( 0 ) , \\end{align*}"}
-{"id": "2586.png", "formula": "\\begin{align*} \\left . \\begin{array} { l } \\overline \\rho ( t ) = C _ 1 t ^ { - \\tfrac { 1 - a } 2 } + R _ 1 ( t ) \\\\ \\overline \\rho ( t ) = C _ 2 ( 1 - t ) ^ { - \\tfrac { 1 - a } 2 } + R _ 2 ( 1 - t ) \\end{array} \\right \\} \\end{align*}"}
-{"id": "6628.png", "formula": "\\begin{align*} \\| f _ n - g _ n \\| _ { C ^ 0 ( \\varphi _ n ( K _ n ) ) } = O ( \\epsilon ) = \\| f _ n ^ { - 1 } - g _ n ^ { - 1 } \\| _ { C ^ 0 ( \\psi _ n ( L _ n ) ) } \\ . \\end{align*}"}
-{"id": "5067.png", "formula": "\\begin{align*} ( A _ { i j } ) = d i a g \\{ a _ 1 , a _ 2 , a _ 3 \\} . \\end{align*}"}
-{"id": "4707.png", "formula": "\\begin{align*} \\int \\limits _ { \\R ^ 3 } I _ s ( t ) ^ { q _ 0 } \\ , \\mathrm { d } t \\lesssim ( 1 + \\| s \\| ) ^ { 1 - q _ 0 } \\int \\limits _ { \\R ^ d } \\prod _ { i = 1 } ^ d ( 1 + | t _ i | ) ^ { - k q _ 0 } \\ , \\mathrm { d } t \\end{align*}"}
-{"id": "6985.png", "formula": "\\begin{align*} M _ { i j } = D f _ k ( B ) E _ { i j } . \\end{align*}"}
-{"id": "875.png", "formula": "\\begin{align*} \\lambda _ { n , k } ( i ) = \\left \\{ \\begin{array} { l l } 1 / N _ n ^ { 1 / 4 } & i = k , \\\\ 1 / \\sqrt { N _ n } & . \\end{array} \\right . \\end{align*}"}
-{"id": "8187.png", "formula": "\\begin{align*} \\imath ^ { - 1 } \\cap B ( R ) = \\{ 0 \\} . \\end{align*}"}
-{"id": "4231.png", "formula": "\\begin{align*} d ( \\delta _ i ) = \\begin{cases} \\delta _ 0 & \\\\ \\delta _ { i - 1 } - a \\cdot \\delta _ i & \\end{cases} \\end{align*}"}
-{"id": "4570.png", "formula": "\\begin{align*} ( 1 - q _ 1 ^ { - 1 } q _ 3 ) E ^ { ( 1 ) } _ { 0 | 1 , l } = \\sum _ { j \\in \\Z } q _ 1 ^ { j - l } \\bigl ( E _ { 0 , - j + l } E _ { 1 , j } - ( q _ 1 + q _ 3 ) E _ { 0 , - j + l - 1 } E _ { 1 , j + 1 } + q _ 1 q _ 3 E _ { 0 , - j + l + 2 } E _ { 1 , j + 2 } \\bigr ) \\end{align*}"}
-{"id": "6130.png", "formula": "\\begin{align*} \\mathbb { P } _ { c o u p l e } \\left ( W > i \\right ) = o \\left ( L \\left ( i \\right ) i ^ { - \\alpha } \\right ) , \\end{align*}"}
-{"id": "588.png", "formula": "\\begin{align*} F ( z ) = { \\displaystyle \\int _ { [ \\alpha , z ] } f ( \\zeta ) d \\zeta } \\end{align*}"}
-{"id": "6398.png", "formula": "\\begin{align*} P _ { } \\left ( \\theta \\right ) = \\frac { P _ { } \\left ( \\theta \\right ) P _ { } \\left ( x ^ { \\prime } \\left \\vert \\theta \\right . \\right ) } { P _ { } \\left ( x ^ { \\prime } \\right ) } \\end{align*}"}
-{"id": "1974.png", "formula": "\\begin{align*} \\Vert v _ { s } \\Vert _ { 1 } = \\sum _ { n = 0 } ^ { \\infty } ( \\lambda + \\varepsilon ) ^ { - n } \\Vert D _ { p } ( \\textbf { \\textit { f } } ^ { n } ) v _ { s } \\Vert \\leq \\sum _ { n = 0 } ^ { \\infty } ( \\lambda + \\varepsilon ) ^ { - n } c \\lambda ^ { n } \\Vert v _ { s } \\Vert = \\frac { \\lambda + \\varepsilon } { \\varepsilon } c \\Vert v _ { s } \\Vert . \\end{align*}"}
-{"id": "4018.png", "formula": "\\begin{align*} A ( t ) : = \\int _ { \\bar { D } } \\omega _ t . \\end{align*}"}
-{"id": "10118.png", "formula": "\\begin{align*} \\bar { { \\boldsymbol { \\omega } } } _ k ( i ) = \\sum \\limits _ { l \\in \\mathcal { N } _ k } c _ { k l } \\boldsymbol { \\bar { \\psi } } _ l ( i ) , \\end{align*}"}
-{"id": "9839.png", "formula": "\\begin{align*} \\frac { P ( T ) } { ( 1 - T ) ( 1 - q T ) } ( y ( 1 - T ) + x T ) ^ n = \\cdots + \\frac { W ( x , y ) - x ^ n } { q - 1 } T ^ { n - d } + \\cdots . \\end{align*}"}
-{"id": "782.png", "formula": "\\begin{align*} T ' _ { k } ( x ) = ( k + 1 ) ( { 2 B _ { k } ( 2 x + 1 ) - B _ { k } ( x + 1 ) } ) \\end{align*}"}
-{"id": "8986.png", "formula": "\\begin{gather*} D ^ { ( n ) } _ { q , t } ( d ) D ^ { ( n ) } _ q ( - d q / 2 \\pm u ; t ) = \\prod _ { 1 \\le i \\le n } \\vartheta ( z _ i \\pm u ) D ^ { ( n ) } _ { q , t } ( d + 1 ) \\end{gather*}"}
-{"id": "6010.png", "formula": "\\begin{align*} & P _ { X ^ { n } Y ^ { n } } ( x ^ { n } , y ^ { n } ) \\\\ & \\qquad : = \\frac { 1 } { | { \\cal M } _ { n } | } \\sum _ { m \\in { \\cal M } _ { n } } P _ { X ^ { n } | M _ { n } } ( x ^ { n } | m ) P _ { Y ^ { n } | M _ { n } } ( y ^ { n } | m ) \\end{align*}"}
-{"id": "8297.png", "formula": "\\begin{align*} H _ { \\Z } = C ( V _ \\Z ) , \\end{align*}"}
-{"id": "4859.png", "formula": "\\begin{align*} 2 x ^ 2 + y ^ 2 + z ^ 2 + 3 x z = 0 . \\end{align*}"}
-{"id": "2332.png", "formula": "\\begin{align*} \\textbf { H } _ { i } ( \\textbf { \\textit { f } } ) = h ( \\phi ) , \\quad i \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "8312.png", "formula": "\\begin{align*} \\mathrm { S p i n } ( V ) = \\mathrm { k e r } \\big ( \\nu : G \\to \\mathbb { G } _ m ) , \\end{align*}"}
-{"id": "2812.png", "formula": "\\begin{align*} \\int _ { S ^ { * } } ^ { K } S ^ { - \\frac { 1 } { 2 \\sigma ^ { 2 } } \\left [ 2 \\delta - 2 r + 3 \\sigma ^ { 2 } + \\lambda ( p ) \\right ] } F ( S ) \\mathrm { d } S = 0 . \\end{align*}"}
-{"id": "2192.png", "formula": "\\begin{align*} f _ j ( z ) = \\bigl ( z ^ { \\beta ^ j } + Q _ j ( z ) \\bigr ) e ^ { P _ j } , j = 1 , \\ldots , n . \\end{align*}"}
-{"id": "9622.png", "formula": "\\begin{align*} \\Phi _ { \\rm { s t } , \\beta } ( t ) = \\frac { N _ \\beta ( t ) \\left ( P _ { \\rm { t r } } ( R _ { \\beta } , \\lambda _ { \\beta } , h _ \\beta ) + P _ { \\rm { c u } } \\right ) } { E _ { \\rm { b } } } . \\end{align*}"}
-{"id": "935.png", "formula": "\\begin{align*} \\varepsilon = \\Delta ^ { 1 / 3 } ( 1 \\vee a _ d \\vee \\log ^ { 1 / 2 } ( 1 / \\Delta ) ) ^ { - 1 / 3 } ( 2 \\log d ) ^ { 1 / 3 } \\end{align*}"}
-{"id": "9417.png", "formula": "\\begin{align*} \\bar { \\tau } _ n = \\dfrac { \\tilde { \\tau } _ n + \\ddot { \\tau } _ n } { 2 } , \\end{align*}"}
-{"id": "3046.png", "formula": "\\begin{align*} u ( t ) - u _ h ( t ) = \\frac { 1 } { 2 \\pi i } \\int _ \\Gamma e ^ { t \\lambda } \\left [ ( \\lambda + A ) ^ { - 1 } - ( \\lambda + A _ h ) P _ { h , \\sigma } \\right ] u _ 0 d \\lambda . \\end{align*}"}
-{"id": "7760.png", "formula": "\\begin{align*} r _ { 0 } = r _ { 0 } ( \\L ) : = \\min _ { \\ell _ { 1 } \\in \\L , \\ell _ { 2 } \\in \\L \\setminus \\ell _ { 1 } } | \\ell _ { 1 } - \\ell _ { 2 } | > 0 . \\end{align*}"}
-{"id": "6336.png", "formula": "\\begin{align*} p _ j : = \\frac { m _ j - m _ j ^ - } { m _ j ^ + - m _ j ^ - } . \\end{align*}"}
-{"id": "5092.png", "formula": "\\begin{align*} E _ H ( x , \\zeta ) = E _ G ( x , \\xi ) \\end{align*}"}
-{"id": "7391.png", "formula": "\\begin{align*} \\mathcal { A } _ l : = \\frac { \\mathbb { H } _ l Q _ l } { I _ l } . \\end{align*}"}
-{"id": "7302.png", "formula": "\\begin{gather*} s _ { n , j } = \\frac { a _ { n , j } - \\bar { a } _ n } { \\sqrt [ ] { \\sum _ { j = 1 } ^ n \\left ( a _ { n , j } - \\bar { a } _ n \\right ) ^ 2 } } , \\\\ S _ n = \\sum _ { j = 1 } ^ n s _ { n , j } T _ { n , j } = \\left [ Z _ { n , j } = s _ { n , j } T _ { n , j } \\right ] = \\sum _ { j = 1 } ^ { n } Z _ { n , j } . \\end{gather*}"}
-{"id": "4595.png", "formula": "\\begin{align*} D = \\left ( \\begin{array} { c c c c c c } 0 & & & & & \\\\ & 0 & - a _ 1 & & & \\\\ & a _ 1 & 0 & & & \\\\ & & & \\ddots & & \\\\ & & & & 0 & - a _ n \\\\ & & & & a _ n & 0 \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "9473.png", "formula": "\\begin{align*} S _ n ( 1 ) = ( q ^ 2 ; q ^ 2 ) _ n . \\end{align*}"}
-{"id": "1762.png", "formula": "\\begin{align*} F _ \\mu \\sqrt { d \\mu / d \\lambda } = F _ \\lambda \\sqrt { d \\mu / d \\lambda } , \\ \\ ( \\lambda - a . e . ) , F _ \\nu \\sqrt { d \\nu / d \\lambda } = F _ \\lambda \\sqrt { d \\nu / d \\lambda } , \\ \\ ( \\lambda - a . e . ) . \\end{align*}"}
-{"id": "1677.png", "formula": "\\begin{align*} \\Phi _ { f _ 1 } ( \\xi ) = \\frac { d ( \\mu \\circ \\sigma _ { f _ 1 } ) } { d \\mu } ( \\xi ) = \\lim _ { N \\to \\infty } \\frac { \\mu ( \\sigma _ { f _ 1 } ( Z ( \\lambda _ N ) ) } { \\mu ( Z ( \\lambda _ N ) ) } \\end{align*}"}
-{"id": "4077.png", "formula": "\\begin{align*} \\frac { | A ( u ) | } { \\eta ( u ) } = o ( 1 ) | u | \\to \\infty \\ , . \\end{align*}"}
-{"id": "2262.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } \\frac { a _ 3 } { 2 \\pi ^ 2 b _ 2 k ^ 4 } = \\frac { a _ 3 \\pi ^ 2 } { 1 8 0 b _ 2 } . \\end{align*}"}
-{"id": "5110.png", "formula": "\\begin{align*} \\overline { u } = M w + M ^ \\prime \\phi _ { 1 , s } \\end{align*}"}
-{"id": "9646.png", "formula": "\\begin{align*} \\hat { R } ^ { * } _ { \\beta } = \\sqrt [ 4 ] { \\frac { P _ { \\rm { c u } } } { \\hat { \\lambda } _ { \\beta } \\left ( 2 ^ { C / W } - 1 \\right ) P _ { \\rm { t r } , 1 } ( h _ { \\beta , 1 } ^ * ) } } . \\end{align*}"}
-{"id": "354.png", "formula": "\\begin{align*} \\partial _ 0 U _ { \\sigma } = \\partial _ 0 \\left ( \\frac { S ^ 2 } { H ^ { 2 + \\sigma } } \\right ) = \\frac { \\partial _ 0 S ^ 2 } { H ^ { 2 + \\sigma } } - \\frac { 2 + \\sigma } { H } ( \\partial _ 0 H ) U _ { \\sigma } , \\end{align*}"}
-{"id": "7565.png", "formula": "\\begin{gather*} \\tilde v _ t - \\tilde v _ { y y } = 0 , \\\\ w ^ 1 _ t - w ^ 1 _ { y y } + \\frac { ( w ^ 1 ) ^ 2 } { \\tilde v } = 0 , \\\\ w ^ 0 _ t - w ^ 0 _ { y y } + \\frac { w ^ 1 } { \\tilde v } w ^ 0 = 0 \\end{gather*}"}
-{"id": "2257.png", "formula": "\\begin{align*} \\sigma _ { ( 1 , 1 ) } = 4 a _ 2 b _ 1 - \\frac { a _ 3 b _ 2 } { ( b _ 2 - a _ 2 ) ^ 2 } \\end{align*}"}
-{"id": "2770.png", "formula": "\\begin{align*} \\prod _ { Q \\in \\Delta _ k ^ \\pm ( P _ 0 ) } \\widetilde { e } _ { p Q - \\tau ( Q ) } = 0 . \\end{align*}"}
-{"id": "6502.png", "formula": "\\begin{align*} k _ { r } & = \\frac { \\sqrt { 2 \\mu \\left ( \\mathcal { E } - V \\right ) } } { \\hbar } , \\ ; 0 < x < L \\\\ k _ { \\mathrm { o } } & = \\frac { \\sqrt { 2 \\mu \\mathcal { E } } } { \\hbar } , \\ ; x > L . \\end{align*}"}
-{"id": "7501.png", "formula": "\\begin{align*} \\partial _ t \\widetilde { W } _ { N , t } + \\frac { v } { \\sqrt { 1 + v ^ 2 } } \\cdot \\nabla _ x \\widetilde { W } _ { N , t } = \\nabla \\left ( V * \\widetilde { \\rho } _ t \\right ) \\cdot \\nabla _ v \\widetilde { W } _ { N , t } \\end{align*}"}
-{"id": "7234.png", "formula": "\\begin{align*} \\Theta _ { \\pi } ( f ) = \\int _ { G } ^ { } f ( g ) \\theta _ { \\pi } ( g ) d g , \\space ( f \\in \\mathcal { C } _ { c } ( G ) ) . \\end{align*}"}
-{"id": "1818.png", "formula": "\\begin{align*} \\lim _ { H \\downarrow 0 } \\frac { \\langle \\sigma _ 0 \\rangle _ { 1 , H } } { H ^ { 1 / 1 5 } } = \\lim _ { a \\downarrow 0 } \\langle \\Phi ^ { a , 1 } ( 1 _ Q ) \\rangle _ { a , 1 } = E ^ 0 _ { h = 1 } \\left ( \\Phi ^ { h = 1 } ( 1 _ Q ) \\right ) , \\end{align*}"}
-{"id": "476.png", "formula": "\\begin{align*} ( \\psi _ { \\pi / 2 } '' ( 0 ) ^ { - 1 } & \\partial , \\partial ) ^ { ( k _ 1 + 3 ) / 2 } \\left [ ( \\psi _ { \\pi / 2 } - P _ { 2 , 0 } \\psi _ { \\pi / 2 } ) a _ { k _ 1 , k _ 2 , \\pi / 2 } \\right ] ( 0 ) \\\\ & = ( - 1 ) ^ { k _ 1 } i ^ { k _ 2 - \\frac { k _ 1 + 1 } { 2 } } \\frac { ( k _ 1 + 1 ) ! } { 2 ^ { ( k _ 1 + 3 ) / 2 } } i ^ { - n } \\left ( i \\frac { \\pi } { 2 } \\right ) ^ { n + k _ 1 + k _ 2 - 1 } ( k _ 1 + 3 ) \\left [ \\frac { \\pi ^ 2 } { 1 2 } ( k _ 1 + 2 ) + m - 1 \\right ] \\end{align*}"}
-{"id": "131.png", "formula": "\\begin{align*} & u _ f ' ( 1 - t , x ) ( u _ h ' ( 1 - t , \\lambda x + \\mu y ) + u _ f ' ( 1 - t , x ) ) \\\\ & \\qquad \\quad + u _ f ( 1 - t , x ) ( u _ h '' ( 1 - t , \\lambda x + \\mu y ) + u _ f '' ( 1 - t , x ) ) = \\mbox { } \\\\ & u _ g ' ( 1 - t , y ) ( u _ h ' ( 1 - t , \\lambda x + \\mu y ) + u _ g ' ( 1 - t , y ) ) \\\\ & \\qquad \\quad + u _ g ( 1 - t , y ) ( u _ h '' ( 1 - t , \\lambda x + \\mu y ) + u _ g '' ( 1 - t , y ) ) \\end{align*}"}
-{"id": "3037.png", "formula": "\\begin{align*} \\begin{cases} ( u _ t ( t ) , v ) _ H + a ( u ( t ) , v ) + b ( v , p ( t ) ) = 0 , & \\forall v \\in V , \\\\ b ( u ( t ) , q ) = 0 , & \\forall q \\in Q , \\\\ u ( 0 ) = u _ 0 , \\end{cases} \\end{align*}"}
-{"id": "6894.png", "formula": "\\begin{align*} \\mathcal { E } _ m ( u ) & = \\sum _ { x \\sim y } c ^ { ( m ) } ( x , y ) | u ( x ) - u ( y ) | ^ 2 \\\\ \\mathcal { E } ( u ) & = \\lim _ { m \\to \\infty } \\mathcal { E } _ m ( u ) \\end{align*}"}
-{"id": "4441.png", "formula": "\\begin{align*} & - \\partial _ 1 u \\lceil x _ 1 , ( \\cdot ) _ { T } \\rceil f + \\big ( \\partial _ 1 u \\lceil x _ 1 , ( \\cdot ) _ { t } \\rceil f \\big ) _ { T - t } \\\\ & \\quad = - \\sum _ { \\tau = T / 2 ^ k , \\ , k = 1 , \\dots , n } \\big ( \\partial _ 1 u \\lceil x _ 1 , ( \\cdot ) _ { \\tau } \\rceil f _ { \\tau } + \\lceil \\partial _ 1 u , ( \\cdot ) _ { \\tau } \\rceil \\lceil x _ 1 , ( \\cdot ) _ { \\tau } \\rceil f \\big ) _ { T - 2 \\tau } . \\end{align*}"}
-{"id": "2934.png", "formula": "\\begin{align*} \\partial \\Theta _ \\epsilon ( H , i ) = F _ \\epsilon ( H , i ) . \\end{align*}"}
-{"id": "1168.png", "formula": "\\begin{align*} T _ { k + 1 , i } & = T ^ 2 _ { k , i - 1 } + D _ k T _ { k , i } \\cr & = S ^ 2 _ { k - 2 , k - 2 - ( i - 1 ) } ( D _ 2 , D _ 3 , \\ldots , D _ { k - 1 } ) + D _ k S _ { k - 2 , k - 2 - i } ( D _ 2 , D _ 3 , \\ldots , D _ { k - 1 } ) \\cr & = S ^ 2 _ { k - 2 , k - 1 - i } ( D _ 2 , D _ 3 , \\ldots , D _ { k - 1 } ) + D _ k S _ { k - 2 , k - 1 - i - 1 } ( D _ 2 , D _ 3 , \\ldots , D _ { k - 1 } ) \\cr & = S _ { k - 1 , k - 1 - i } ( D _ 2 , D _ 3 , \\ldots , D _ { k } ) . \\end{align*}"}
-{"id": "3181.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\left ( X _ { t } ^ { x } \\right ) ^ { \\kappa } \\right ] = \\int _ { \\mathbb { R } _ { \\geqslant 0 } } \\int _ { \\mathbb { R } _ { \\geqslant 0 } } ( y + z ) ^ { \\kappa } \\mu _ { Y _ { t } ^ { x } } ( \\mathrm { d } y ) \\mu _ { Z _ { t } } ( \\mathrm { d } z ) < \\infty . \\end{align*}"}
-{"id": "1347.png", "formula": "\\begin{align*} { \\rm l i n } \\left ( { \\cal T } _ { { \\cal S } _ + ^ p } ( \\overline { M } _ + ) \\right ) = \\{ B \\in { \\cal S } ^ p \\ , | \\ , [ P _ \\beta \\ P _ \\gamma ] ^ T B [ P _ \\beta \\ P _ \\gamma ] = 0 \\} \\ , . \\end{align*} % \\end{align*}"}
-{"id": "2684.png", "formula": "\\begin{align*} E ( X _ { \\C } ) : = \\mathrm { E n d } _ { \\mathrm { H d g } } ( T ( X _ { \\C } ) ) \\otimes _ { \\Z } { \\Q } \\end{align*}"}
-{"id": "5995.png", "formula": "\\begin{gather*} e _ 0 = \\frac { 1 } { 2 } ( 1 + z ) , e _ 1 = \\frac { 1 } { 2 } ( 1 - z ) . \\end{gather*}"}
-{"id": "4224.png", "formula": "\\begin{align*} d ( a \\cdot b ) = d ( a ) \\cdot b + ( - 1 ) ^ { i } \\cdot a \\cdot d ( b ) \\end{align*}"}
-{"id": "960.png", "formula": "\\begin{align*} | E [ U _ n ( \\theta ) ] | & = \\left | \\sum _ { I , J } K ( I , J _ { - \\theta } ) \\sum _ { m = 1 } ^ M \\rho _ m \\int _ { I \\cap J _ { - \\theta _ m } } \\sigma _ 1 ( t ) \\sigma _ 2 ( t + \\theta _ m ) d t \\right | \\\\ & \\leq \\max _ { 1 \\leq m \\leq M } | \\rho _ m | \\sum _ { I , J } \\int _ { I \\cap J _ { - \\theta _ m } } \\sigma _ 1 ( t ) \\sigma _ 2 ( t + \\theta _ m ) d t \\end{align*}"}
-{"id": "6023.png", "formula": "\\begin{align*} & \\inf \\left \\{ R : D _ { 1 + s } ( P _ { X ^ { n } Y ^ { n } | U _ { n } } \\| \\pi _ { X ^ { n } Y ^ { n } } | P _ { U _ { n } } ) \\rightarrow 0 \\right \\} \\\\ \\leq & \\inf _ { \\substack { \\{ P _ { W ^ { n } } , P _ { X ^ { n } | W ^ { n } } , P _ { Y ^ { n } | W ^ { n } } \\} _ { n = 1 } ^ { \\infty } : \\\\ D _ { 1 + s } ( P _ { X ^ { n } Y ^ { n } } \\| \\pi _ { X ^ { n } Y ^ { n } } ) \\rightarrow 0 } } \\\\ & \\qquad \\limsup _ { n \\to \\infty } \\frac { 1 } { n } D _ { 1 + s } ( P _ { X ^ { n } Y ^ { n } | W ^ { n } } \\| \\pi _ { X ^ { n } Y ^ { n } } | P _ { W ^ { n } } ) . \\end{align*}"}
-{"id": "7525.png", "formula": "\\begin{gather*} w ^ i w ^ 1 _ i - w ^ 1 _ { i i } - 2 w ^ 2 = 0 , \\\\ w ^ i w ^ 2 _ i - w ^ 2 _ { i i } + 2 w ^ 1 + 2 \\mu = 0 . \\end{gather*}"}
-{"id": "8006.png", "formula": "\\begin{align*} K _ k ( x , y ) = \\int _ { \\mathbb { R } ^ d } { b _ k ( x , \\xi ) e ^ { 2 \\pi i \\langle x - y , \\xi \\rangle } } d \\xi . \\end{align*}"}
-{"id": "5903.png", "formula": "\\begin{align*} E S ^ { ( i ) } _ { \\tau ^ { N , i } + N } = \\mu _ i ( E \\tau ^ { N , i } + N ) , \\ 0 \\le i \\le l . \\end{align*}"}
-{"id": "5871.png", "formula": "\\begin{align*} A _ 0 = 1 , \\ , A _ 4 = 5 , \\ , A _ 6 = 8 0 , \\ , A _ 8 = 2 5 0 , \\ , A _ { 1 0 } = 3 5 2 , \\ , A _ { 1 2 } = 2 5 0 , \\ , A _ { 1 4 } = 8 0 , \\ , A _ { 1 6 } = 5 , \\ , A _ { 2 0 } = 1 , \\end{align*}"}
-{"id": "7976.png", "formula": "\\begin{align*} 2 b c \\cos \\alpha _ { + } & = 2 b c \\cos \\alpha + O ( b ^ 3 c ) + O ( b c ^ 3 ) + O ( b ^ 2 c ^ 2 ) + O ( a ^ 4 ) \\\\ 2 b c \\cos \\alpha _ { - } & = 2 b c \\cos \\alpha + O ( b ^ 3 c ) + O ( b c ^ 3 ) + O ( b ^ 2 c ^ 2 ) + O ( a ^ 4 ) . \\end{align*}"}
-{"id": "9258.png", "formula": "\\begin{align*} | C | \\geq \\frac { 2 4 } { 6 5 } n = \\frac { 2 4 } { 6 5 } \\left ( 1 1 q + r \\right ) = 4 q + \\frac { 4 q + 2 4 r } { 6 5 } \\end{align*}"}
-{"id": "5609.png", "formula": "\\begin{align*} F ( X _ u ) X _ { u , v } & + D F ( X _ u ) \\mathbb { X } _ { u , v } \\\\ & - F ( X _ u ) f ( X _ u ) ( v - u ) - F ( X _ u ) \\sigma ( X _ u ) B _ { u , v } ^ H - \\{ F ( X _ u ) \\sigma ( X _ u ) \\} ' \\mathbb { B } ^ H _ { u , v } = o ( | v - u | ) . \\end{align*}"}
-{"id": "4583.png", "formula": "\\begin{align*} \\tilde \\theta ^ { - 1 } \\bigl ( \\tilde H _ { 0 , 1 } \\bigr ) & = \\theta ^ { - 1 } ( H _ { 0 , 1 } ) + q _ 1 \\theta ^ { - 1 } ( H _ { 1 , 1 } ) \\ , . \\end{align*}"}
-{"id": "5302.png", "formula": "\\begin{align*} \\mathcal { F } ( v , w ) = \\frac { 1 } { \\tau } \\int _ \\Omega f v \\mathrm { d x } + \\int _ \\Gamma H v \\mathrm { d s } + \\int _ { \\Gamma _ \\mathrm { N } } g w \\mathrm { d s } \\end{align*}"}
-{"id": "2651.png", "formula": "\\begin{align*} w ( b ) = \\left \\{ \\begin{alignedat} { 5 } & b , & & \\mbox { i f } & 0 & \\leq b \\leq 2 ( k + 1 ) - 1 , \\\\ & b - \\alpha , & & \\mbox { i f } & 2 ( k + 1 ) & \\leq b \\leq 3 ( k + 1 ) - 1 , \\\\ & b - \\alpha - \\beta , & & \\mbox { i f } & 3 ( k + 1 ) & \\leq b \\leq 4 ( k + 1 ) - 1 , \\\\ & b - 2 \\alpha - \\beta , & & \\mbox { i f } & 4 ( k + 1 ) & \\leq b \\leq 6 ( k + 1 ) - 1 . \\end{alignedat} \\right . \\end{align*}"}
-{"id": "4848.png", "formula": "\\begin{align*} x ^ 2 = t ^ { 2 m } , y ^ 2 = a ^ 2 t ^ { 4 m } + 2 a \\cdot a _ b t ^ { 2 m + b } + a _ b ^ 2 t ^ { 2 b } + \\cdots , z ^ 2 = 1 , \\end{align*}"}
-{"id": "1132.png", "formula": "\\begin{align*} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\begin{bmatrix} \\delta & x \\\\ 0 & \\delta \\end{bmatrix} \\begin{bmatrix} d & - b \\\\ - c & a \\end{bmatrix} = \\begin{bmatrix} \\delta - a c x & a ^ 2 x \\\\ - c ^ 2 x & \\delta + a c x \\end{bmatrix} . \\end{align*}"}
-{"id": "390.png", "formula": "\\begin{align*} S = \\sum _ { n = 3 } ^ \\infty \\frac { 1 } { n ( \\ln \\ln n ) ^ b } \\mathbb P \\left ( | S _ n | > ( 1 + \\varepsilon ) \\sigma _ n \\sqrt { 2 \\ln \\ln n } \\right ) . \\end{align*}"}
-{"id": "10036.png", "formula": "\\begin{align*} \\gamma _ Q = \\prod _ { q \\mid Q } \\gamma _ q . \\end{align*}"}
-{"id": "325.png", "formula": "\\begin{align*} x ( n ) = q ( n ) \\cdot x ( n + 1 ) + r ( n + 1 ) \\cdot x ( n + 2 ) , n \\in \\Z _ { \\ge 0 } . \\end{align*}"}
-{"id": "5638.png", "formula": "\\begin{align*} | \\dot { \\gamma } | ( t ) = \\lim \\limits _ { s \\to t } \\frac { d ( \\gamma ( t ) , \\gamma ( s ) ) } { | t - s | } , \\end{align*}"}
-{"id": "703.png", "formula": "\\begin{align*} \\psi _ j ( \\lambda ^ \\beta _ \\gamma ( b v _ \\delta ) ) & = \\psi _ j ( v _ \\gamma b v _ \\delta v _ { \\gamma ^ { - 1 } } ) \\\\ & = \\psi _ j ( \\beta _ \\gamma ( b ) v _ { \\gamma \\delta \\gamma ^ { - 1 } } ) \\\\ & = \\beta _ { \\gamma } ( b ) \\rho _ j ^ 2 ( u _ { \\gamma \\delta \\gamma ^ { - 1 } } ) \\\\ & = \\beta _ \\gamma ( b ) \\beta _ \\gamma ( \\rho _ j ^ 2 ( u _ \\gamma ) ) \\\\ & = \\beta _ \\gamma ( \\psi _ j ( b v _ \\delta ) ) , \\end{align*}"}
-{"id": "5172.png", "formula": "\\begin{align*} ( i ) & f _ { 1 } [ 0 , a ] [ a , 1 ] \\\\ ( i i ) & f _ { 2 } [ 0 , b ] [ b , 1 ] \\\\ ( i i i ) & b \\le a , f _ { i } < g _ { i } [ b , a ] i = 1 , 2 . \\end{align*}"}
-{"id": "1449.png", "formula": "\\begin{align*} \\phi _ r ( x _ I ) = \\partial _ 2 \\cdots \\partial _ r ( x _ I ) , \\end{align*}"}
-{"id": "9074.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } x ^ { p _ { k } } f _ { k } ( x ^ { q _ { k } } ) = 0 , \\end{align*}"}
-{"id": "5316.png", "formula": "\\begin{align*} \\int _ { Q _ T } T _ \\mathcal { M } ( \\phi _ M ) \\nabla \\theta _ M \\cdot \\mathbf { v } \\mathrm { d x } \\mathrm { d t } = - \\int _ { Q _ T [ | \\phi _ M | < \\mathcal { M } ] } \\theta _ M \\nabla \\phi _ M \\cdot \\mathbf { v } \\mathrm { d x } \\mathrm { d t } , \\end{align*}"}
-{"id": "1768.png", "formula": "\\begin{align*} t _ { \\lambda } P ( \\{ \\omega \\} ) t _ { \\lambda } ^ * \\ ; = \\ ; P ( \\{ \\lambda \\omega \\} ) , \\end{align*}"}
-{"id": "4646.png", "formula": "\\begin{align*} R ( k ) = r ( k ) + \\frac { 2 c _ 1 + 3 \\delta } { 2 ^ k } ( d + 1 ) \\max \\left \\{ 1 , \\frac { 8 D } { 3 \\xi } \\right \\} \\end{align*}"}
-{"id": "6699.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ n | x _ i | ^ { \\frac { p - 2 } { n + 1 } } = \\left ( \\sqrt [ n ] { \\prod _ { i = 1 } ^ n | x _ i | ^ p } \\right ) ^ { \\frac { n } { n + 1 } \\cdot \\frac { p - 2 } { p } } \\leq \\left ( \\frac { 1 } { n } \\left ( \\sum _ { i = 1 } ^ n | x | _ i ^ p \\right ) \\right ) ^ { \\frac { n } { n + 1 } \\cdot \\frac { p - 2 } { p } } = n ^ { - \\frac { n } { n + 1 } \\cdot \\frac { p - 2 } { p } } \\end{align*}"}
-{"id": "6375.png", "formula": "\\begin{align*} f _ n ( x ) : = \\frac { g _ n ( x ) } { c _ n } , \\end{align*}"}
-{"id": "2234.png", "formula": "\\begin{align*} f _ i ( z ) = 0 , i = 1 , \\ldots , n . \\end{align*}"}
-{"id": "3892.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } \\cos ( x ) x ^ { s - 1 } d x = \\Gamma ( s ) \\cos ( \\pi s / 2 ) = \\sqrt { \\pi } 2 ^ { s - 1 } \\frac { \\Gamma ( s / 2 ) } { \\Gamma ( 1 / 2 - s / 2 ) } . \\end{align*}"}
-{"id": "4978.png", "formula": "\\begin{align*} h : = \\sum _ { j = - \\infty } ^ { \\infty } h _ j , g : = \\sum _ { j = - \\infty } ^ { \\infty } g _ j , \\end{align*}"}
-{"id": "2358.png", "formula": "\\begin{align*} \\left ( \\left ( p ^ { - n } \\right ) ' ( z ) \\right ) ^ s & = \\left ( \\big ( n c ( c z + d ) + 1 \\big ) ^ { - 2 } \\right ) ^ s \\\\ & = e ^ { - 2 s i \\cdot \\varphi ( n , z ) } \\left ( \\left ( c ( c z + d ) \\right ) ^ { - 2 } \\right ) ^ s \\left ( \\left ( n + \\frac { 1 } { c ( c z + d ) } \\right ) ^ { - 2 } \\right ) ^ s \\end{align*}"}
-{"id": "2127.png", "formula": "\\begin{align*} x ^ { ( k + 1 ) } = a r g m i n _ { x } | \\mathfrak { D _ 1 } x + \\delta ^ { ( k ) } - T | \\end{align*}"}
-{"id": "4061.png", "formula": "\\begin{align*} V f ( x ) = \\int _ { 0 } ^ { x } f ( t ) \\ , \\d t = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "3095.png", "formula": "\\begin{align*} h ( g ( x ) ) & = g ( x ) + \\psi _ 0 \\circ g ( x ) = \\psi _ * ( x ) + a x + \\psi _ 0 \\circ g ( x ) \\\\ & = \\psi _ * ( x ) + a x + a S _ \\delta ( \\psi _ 0 ) ( x ) = a \\psi _ 0 ( x ) + a x = a h ( x ) . \\end{align*}"}
-{"id": "1543.png", "formula": "\\begin{gather*} y _ 1 + y _ 1 \\ = \\ 0 + x _ 2 \\\\ y _ 1 + 1 \\ = \\ x _ 2 + x _ 2 . \\end{gather*}"}
-{"id": "4773.png", "formula": "\\begin{align*} \\alpha ^ 2 \\omega _ 3 = \\beta t _ 3 + \\gamma , \\alpha , \\beta , \\gamma - c o n s t , \\end{align*}"}
-{"id": "8196.png", "formula": "\\begin{align*} R ( \\kappa _ { \\mathfrak { c } } ) w _ { } ^ { \\circ } = \\lambda ( \\mathfrak { c } ) w _ { } ^ { \\circ } . \\end{align*}"}
-{"id": "8644.png", "formula": "\\begin{align*} Z _ i = \\max _ { s \\in [ T _ i , T _ { i + 1 } ] } | B _ s - B _ { T _ i } | , ~ ~ i \\ge 0 , \\end{align*}"}
-{"id": "3109.png", "formula": "\\begin{align*} P _ s ^ { \\mu } = \\frac { \\sum _ { j \\in \\mathcal { J } } \\Theta _ { j } ^ { \\mu } \\prod _ { j ' \\in \\mathcal { J } \\backslash j } ( 1 - \\Theta _ { j ' } ^ { \\mu } ) } { P _ t ^ { \\mu } } . \\end{align*}"}
-{"id": "10063.png", "formula": "\\begin{align*} \\frac { n \\cdot [ \\widehat { \\mathcal { Z } } ( f ) : \\mathcal { Y } _ \\mathrm { b i g } ] } { \\deg _ \\C ( \\mathcal { Y } _ \\mathrm { b i g } ) } + 2 c _ f ^ + ( 0 , 0 ) \\frac { \\Lambda ' ( 0 , \\chi _ E ) } { \\Lambda ( 0 , \\chi _ E ) } = - \\frac { d } { d s } \\langle E ( s ) , \\xi ( f ) \\rangle _ \\mathrm { P e t } \\big | _ { s = 0 } . \\end{align*}"}
-{"id": "8381.png", "formula": "\\begin{align*} - 2 \\log \\| \\psi _ g ( f ) \\| = ( j ^ { [ 2 ] } ) ^ * \\widetilde { \\Theta } ^ \\mathrm { r e g } _ g ( f ^ { [ 2 ] } ) - ( j ^ { [ 1 ] } ) ^ * \\widetilde { \\Theta } ^ \\mathrm { r e g } _ g ( f ^ { [ 1 ] } ) . \\end{align*}"}
-{"id": "4183.png", "formula": "\\begin{align*} \\eta ^ 2 = r _ 1 \\zeta ^ 3 - r _ 2 \\zeta ^ 2 - r _ 1 \\zeta , \\ ; r _ i \\in \\mathbb { R } , r _ 1 \\ge 0 \\end{align*}"}
-{"id": "9221.png", "formula": "\\begin{align*} [ \\alpha , [ b _ { 1 } , b _ { 2 } ] _ { C } ] + \\alpha \\circ [ b _ { 1 } , b _ { 2 } ] _ { C } + [ \\alpha , ( b _ { 1 } \\circ b _ { 2 } ) _ { E } ] + \\alpha \\circ ( b _ { 1 } \\circ b _ { 2 } ) _ { E } = [ \\alpha b _ { 1 } , b _ { 2 } ] + \\alpha b _ { 1 } \\circ b _ { 2 } , \\end{align*}"}
-{"id": "5426.png", "formula": "\\begin{align*} \\cos \\Psi _ { x y } = \\frac { \\left \\vert \\left \\langle x , y \\right \\rangle \\right \\vert } { \\left \\Vert x \\right \\Vert \\left \\Vert y \\right \\Vert } , x , y \\neq 0 , \\end{align*}"}
-{"id": "3908.png", "formula": "\\begin{align*} \\tilde R ( \\rho ) = e ^ { - a \\rho ^ 4 } \\rho ^ { \\cal L } . \\end{align*}"}
-{"id": "2700.png", "formula": "\\begin{align*} ( a _ 2 ( a ' _ 1 ) ^ { - 1 } t - b _ 1 ) ^ 2 - 4 p ^ { 2 n } a ' _ 3 ( a ' _ 1 ) ^ { - 1 } t ^ 2 + 2 b _ 1 ( a _ 2 ( a ' _ 1 ) ^ { - 1 } t - b _ 1 ) + 4 b _ 2 = 0 . \\end{align*}"}
-{"id": "3305.png", "formula": "\\begin{align*} H ( Z ) = K L r . \\end{align*}"}
-{"id": "2928.png", "formula": "\\begin{align*} \\Theta ' _ \\epsilon ( H , e , i ) = \\frac { \\exp \\left ( - \\frac { \\partial \\Theta ' _ \\epsilon ( H , i ) } { \\epsilon } \\right ) } { \\sum _ { j \\in e } \\exp \\left ( - \\frac { \\partial \\Theta ' _ \\epsilon ( H , j ) } { \\epsilon } \\right ) } . \\end{align*}"}
-{"id": "4159.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { k \\in N } e _ { k , j + n - 2 } ^ n & + \\frac { 1 } { n } e ^ { n - 1 } _ { i , j } + \\frac { 1 } { n } e _ { i , j + 1 } ^ { n - 2 } \\\\ & + \\left ( 1 - \\frac { 1 } { n } \\right ) \\left [ \\frac { 1 } { n ^ 2 } e _ { i , j } ^ { n - 3 } + \\frac { 1 } { n } e _ { i , j + 1 } ^ { n - 3 } \\right ] + \\frac { 1 } { n } \\sum _ { \\tau = 1 } ^ { n - 4 } e _ { i , j + \\tau + 1 } ^ { n - \\tau - 3 } \\end{align*}"}
-{"id": "535.png", "formula": "\\begin{align*} \\dot { w } ( t ) = J _ 0 A ( - \\infty , t ) \\ , w ( t ) \\mbox { a n d } \\dot { w } ( t ) = J _ 0 A ( + \\infty , t ) \\ , w ( t ) \\end{align*}"}
-{"id": "1098.png", "formula": "\\begin{align*} \\overline { \\mathcal { C } } = E \\left [ \\sum _ { i \\in \\mathcal { N } } ^ { } \\mathcal { C } _ { i | \\alpha _ { i i } } \\right ] . \\end{align*}"}
-{"id": "8397.png", "formula": "\\begin{align*} | M _ t | \\leq V o l ( B _ { d _ t } ) = \\theta d _ t ^ n , \\end{align*}"}
-{"id": "5517.png", "formula": "\\begin{align*} Q ( s ) = \\lim _ { n \\to \\infty } \\frac { f _ n ( s ) - q } { \\gamma ^ n } \\end{align*}"}
-{"id": "1153.png", "formula": "\\begin{align*} \\rho _ x \\rho _ a & = ( x + [ 1 ] _ x \\tau + \\tau ^ 2 ) \\left ( \\sum _ { k = 0 } ^ d \\rho _ { a , k } \\tau ^ k \\right ) \\\\ & = \\sum _ { k = 0 } ^ d \\left ( x \\rho _ { a , k } \\tau ^ k + [ 1 ] _ x \\rho _ { a , k } ^ 2 \\tau ^ { k + 1 } + \\rho _ { a , k } ^ 4 \\tau ^ { k + 2 } \\right ) \\end{align*}"}
-{"id": "4117.png", "formula": "\\begin{align*} X = ( X \\rho ) \\eta + \\rho D _ X \\eta + D _ X V = ( X \\rho ) \\eta + \\rho D _ X \\eta + \\nabla _ X V + h ( X , V ) \\eta . \\end{align*}"}
-{"id": "7766.png", "formula": "\\begin{align*} | D _ { \\rho } v ( \\ell ) | & \\leq \\sum _ { i = 1 } ^ { N _ { \\rho } } | D _ { \\rho _ { i } } v ( \\ell _ { i } ) | \\leq N _ { \\rho } ^ { 1 / 2 } \\bigg ( \\sum _ { i = 1 } ^ { N _ { \\rho } } | D _ { \\rho _ { i } } v ( \\ell _ { i } ) | \\bigg ) ^ { 1 / 2 } \\leq C | \\rho | ^ { 1 / 2 } = : C _ { 0 } | \\ell - m | ^ { 1 / 2 } , \\end{align*}"}
-{"id": "4051.png", "formula": "\\begin{align*} c = \\inf _ { v \\in \\partial B } \\frac { | | v | | _ e } { | | v | | } \\ \\ ( > 0 ) , \\end{align*}"}
-{"id": "2966.png", "formula": "\\begin{align*} R ^ s _ 1 + R ^ s _ 2 & \\leq \\left [ \\underbrace { \\log \\bigg ( 1 + \\frac { \\eta _ 1 p _ 1 | h _ { 1 1 } | ^ { 2 } } { \\sigma ^ { 2 } _ 1 + \\eta _ 1 ( \\varrho ^ { 2 } _ 1 + p _ 2 | h _ { 1 2 } | ^ { 2 } ) } \\bigg ) } _ { R _ 1 } + \\underbrace { \\log \\bigg ( 1 + \\frac { \\eta _ 2 p _ 2 | h _ { 2 2 } | ^ { 2 } } { \\sigma ^ { 2 } _ 2 + \\eta _ 2 ( \\varrho ^ { 2 } _ 2 + p _ 1 | h _ { 2 1 } | ^ { 2 } ) } \\bigg ) } _ { R _ 2 } - \\sum _ { k = 1 } ^ { 2 } R _ { _ k } \\right ] ^ { + } , \\end{align*}"}
-{"id": "3919.png", "formula": "\\begin{align*} \\begin{aligned} x ^ 2 \\circ ( x ^ { 2 j _ 1 } + x ^ { j _ 2 } ) - x ^ 2 \\circ x ^ { 2 j _ 1 } - x ^ 2 \\circ x ^ { j _ 2 } = 2 x ^ { 2 j _ 1 + j _ 2 } \\in \\langle F \\rangle . \\end{aligned} \\end{align*}"}
-{"id": "5205.png", "formula": "\\begin{align*} i ) & \\psi _ { 1 } ( y ) \\in ( 0 , a ) \\implies \\Gamma _ { 1 } ( \\psi _ { 1 } ( y ) , y ) = 0 , y \\in \\mathcal { S } _ { 2 } \\\\ i i ) & \\psi _ { 2 } ( x ) \\in ( b , 1 ) \\implies \\Gamma _ { 2 } ( x , \\psi _ { 2 } ( x ) ) = 0 , x \\in \\mathcal { S } _ { 1 } \\end{align*}"}
-{"id": "9617.png", "formula": "\\begin{align*} \\lambda _ { \\beta } ( t ) = \\frac { 1 } { C S _ { \\rm { b s } } } \\ , x _ { \\beta } ^ { \\rm { r } } \\left [ \\left \\lfloor { t } / { \\mu } \\right \\rfloor \\right ] . \\end{align*}"}
-{"id": "6250.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } & = \\omega + \\xi ( y , z , \\sigma , \\omega , \\mu ) + f ( x , y , z , \\sigma , \\omega , \\mu ) , \\\\ \\dot { y } & = \\sigma + \\eta ( y , z , \\sigma , \\omega , \\mu ) + g ( x , y , z , \\sigma , \\omega , \\mu ) , \\\\ \\dot { z } & = M ( \\omega , \\mu ) z + \\zeta ( y , z , \\sigma , \\omega , \\mu ) + h ( x , y , z , \\sigma , \\omega , \\mu ) , \\end{aligned} \\end{align*}"}
-{"id": "9612.png", "formula": "\\begin{align*} L _ \\xi ( r , h _ \\beta ) = \\begin{cases} \\left ( 4 \\pi f / c \\right ) ^ 2 \\left ( r ^ 2 + h _ \\beta ^ 2 \\right ) \\ , \\eta _ { 0 } , & { \\xi = 0 } \\\\ \\left ( 4 \\pi f / c \\right ) ^ 2 \\left ( r ^ 2 + h _ \\beta ^ 2 \\right ) \\ , \\eta _ { 1 } , & { \\xi = 1 } , \\end{cases} \\end{align*}"}
-{"id": "1944.png", "formula": "\\begin{align*} P ( F _ r ) ( \\eta ) = & \\\\ = & \\int _ { | 1 - r | \\le \\delta _ \\eta } \\frac { F _ r - F _ 1 } { 1 - r } ( \\eta ) \\ , d r + \\int _ { \\delta _ \\eta < | 1 - r | < \\delta } \\frac { F _ r ( \\eta ) } { 1 - r } d r + \\int _ { \\delta \\le | 1 - r | } \\frac { F _ r ( \\eta ) } { 1 - r } \\ , d r \\\\ : = & \\ , P _ { A } ( \\eta ) + P _ { B } ( \\eta ) + P _ { C } ( \\eta ) , \\end{align*}"}
-{"id": "3633.png", "formula": "\\begin{align*} \\varphi _ \\varepsilon : = w ^ + _ { \\lambda _ 0 + \\tau } \\psi _ \\varepsilon ^ p \\chi _ { \\Omega _ { \\lambda _ 0 + \\tau } } . \\end{align*}"}
-{"id": "655.png", "formula": "\\begin{align*} ( P ^ * S | _ { \\widetilde F } - S | _ { F } ) ( a _ 1 , \\cdots , a _ p , b ^ 1 , \\cdots , b ^ q ) & = \\int _ 0 ^ 1 \\partial _ s \\big ( S ( a _ 1 ( s ) , \\cdots , a _ p ( s ) , b ^ 1 ( s ) , \\cdots , b ^ q ( s ) \\big ) d s \\\\ & = - \\int _ 0 ^ 1 ( \\bar \\nabla _ { \\dot \\gamma } S ) ( a _ 1 ( s ) , \\cdots , a _ p ( s ) , b ^ 1 ( s ) , \\cdots , b ^ q ( s ) ) d s \\end{align*}"}
-{"id": "6798.png", "formula": "\\begin{align*} \\int _ { \\mathbb { S } ^ 2 } e ^ { w _ { \\lambda } } = 4 \\int _ { B ( \\xi _ k , R _ 0 ) } e ^ { w _ { \\lambda } } + \\int _ { \\mathbb { S } ^ 2 \\setminus ( \\bigcup \\limits _ { k = 1 } ^ 4 B ( \\xi _ k , R _ 0 ) ) } e ^ { w _ { \\lambda } } \\end{align*}"}
-{"id": "5403.png", "formula": "\\begin{align*} K : = \\langle a , b , y \\rangle . \\end{align*}"}
-{"id": "9726.png", "formula": "\\begin{align*} \\tilde { V } _ { k } = \\Phi _ 1 ( \\tilde { \\gamma } _ 1 ; \\tilde { V } _ a ) , ( \\tilde { u } _ { k } , \\tilde { v } _ { k } ) \\cdot \\textbf { n } _ { k } = 0 . \\end{align*}"}
-{"id": "4925.png", "formula": "\\begin{align*} B B ^ * & = I - A A ^ * \\mbox { a n d } \\\\ C ^ * C & = I - A ^ * A \\end{align*}"}
-{"id": "8290.png", "formula": "\\begin{align*} \\Sigma _ \\Phi = \\{ \\sigma ^ \\prime \\cap C _ { \\Phi } ^ * : \\sigma ^ \\prime \\in \\Sigma ^ \\prime _ { \\Phi ^ \\prime } \\} \\end{align*}"}
-{"id": "9447.png", "formula": "\\begin{align*} \\lambda ( \\limsup { \\mathcal E } _ n ( C \\varepsilon \\psi _ 0 ) ) = 1 . \\end{align*}"}
-{"id": "8915.png", "formula": "\\begin{align*} \\begin{aligned} X ^ \\ast T ^ \\ast T X - X ^ \\ast X & = U _ { ( \\theta ) 1 } ^ \\ast X ^ \\ast X U _ { ( \\theta ) 1 } - X ^ \\ast X \\\\ & = ( K ^ \\ast + U _ { ( \\theta ) c } ^ \\ast ) | \\varphi | ^ 2 ( U _ { ( \\theta ) c } ) ( K + U _ { ( \\theta ) c } \\bigr ) - | \\varphi | ^ 2 ( U _ { ( \\theta ) c } ) \\\\ & = K ^ \\ast | \\varphi | ^ 2 ( U _ { ( \\theta ) c } ) K + ( \\overline \\chi | \\varphi | ^ 2 ) ( U _ { ( \\theta ) c } ) K + K ^ \\ast ( \\chi | \\varphi | ^ 2 ) ( U _ { ( \\theta ) c } ) . \\end{aligned} \\end{align*}"}
-{"id": "6920.png", "formula": "\\begin{align*} \\frac { \\partial u ( x , t ) } { \\partial t } = \\Delta _ { x } u ( x , t ) \\end{align*}"}
-{"id": "3830.png", "formula": "\\begin{align*} \\P _ \\eta \\left ( G _ \\infty \\cap \\Lambda _ \\infty \\right ) & = \\P _ \\eta ( \\tau = \\infty , \\Lambda _ \\infty ) + \\P _ \\eta ( \\tau < \\infty , G _ \\infty \\cap \\Lambda _ \\infty ) \\\\ & \\ge p _ \\bullet \\hat { p } p _ { * * } ^ { | \\eta | - 1 } \\left \\{ \\P _ \\eta ( \\tau = \\infty , \\Lambda _ \\infty ) + \\P _ \\eta ( \\tau < \\infty , \\Lambda _ \\infty ) \\right \\} = p _ { * * } ^ { | \\eta | } , \\end{align*}"}
-{"id": "7465.png", "formula": "\\begin{align*} \\rho _ E ( e _ \\alpha ) = \\rho _ \\alpha ^ k \\dfrac { \\partial } { \\partial z ^ k } , \\end{align*}"}
-{"id": "1324.png", "formula": "\\begin{align*} \\rho ( S _ e ) = \\begin{bmatrix} \\pi ( S _ e ) & X _ e \\\\ Y _ e & Z _ e \\end{bmatrix} \\forall e \\in E , \\end{align*}"}
-{"id": "5827.png", "formula": "\\begin{align*} \\tau _ { x _ 0 } f ( x ) = f ( x - x _ 0 ) , \\end{align*}"}
-{"id": "1259.png", "formula": "\\begin{align*} \\mu _ m : = \\sum _ { j = 0 } ^ N \\mu _ { j , m } \\mbox { a n d } \\mu : = \\sum _ { j = 0 } ^ N \\mu _ { j } . \\end{align*}"}
-{"id": "7920.png", "formula": "\\begin{align*} \\Phi ( x , y , z , t ) = 2 u ( z , t ) - u ( x , t ) - u ( y , t ) + \\alpha d ( z , M ( x , y ) ) - { \\sigma \\over T - t } \\end{align*}"}
-{"id": "4510.png", "formula": "\\begin{align*} | \\phi ( \\Delta _ n ^ * a ) - \\phi ( a ) \\phi ( \\Delta ^ * _ n ) | ^ 2 & = | \\phi ( a ^ * \\Delta _ n ) - \\phi ( a ^ * ) \\phi ( \\Delta _ n ) | ^ 2 \\\\ & \\leq \\| a \\| ^ 2 ( \\phi ( \\Delta _ n ^ * \\Delta _ n ) - \\phi ( \\Delta _ n ) ^ * \\phi ( \\Delta _ n ) ) . \\end{align*}"}
-{"id": "4447.png", "formula": "\\begin{align*} \\int _ { [ 0 , 1 ) ^ 2 } u f _ \\ell \\zeta \\ , d x & = \\int _ { [ 0 , 1 ) ^ 2 } \\big ( u f _ \\ell - ( u f ) _ \\ell \\big ) \\zeta \\ , d x + \\int _ { [ 0 , 1 ) ^ 2 } ( u f ) _ \\ell \\zeta \\ , d x \\\\ & \\stackrel { \\eqref { w 2 3 } } { \\lesssim } [ u ] _ \\alpha [ f ] _ \\beta \\ell ^ { \\alpha + \\beta } \\| \\zeta \\| _ { L ^ 1 } + \\langle u f , \\zeta * \\phi _ \\ell \\rangle _ { { \\cal D } ' , { \\cal D } } \\longrightarrow \\langle u f , \\zeta \\rangle _ { { \\cal D } ' , { \\cal D } } \\textrm { a s } \\ell \\to 0 , \\end{align*}"}
-{"id": "1017.png", "formula": "\\begin{align*} \\max _ { \\vec { p } _ s , \\pmb { \\vec { \\beta } } } & \\sum _ { i = 1 } ^ N R _ 1 ( \\beta _ i , p _ s ^ i ) , \\\\ & \\eqref { e q : c 1 _ o r i g } , \\ , \\eqref { e q : c 2 _ o r i g } , \\ , \\eqref { e q : c 4 _ o r i g } , \\ , \\eqref { e q : c 2 _ n e w } , \\end{align*}"}
-{"id": "1297.png", "formula": "\\begin{align*} b & = \\frac { 2 Q ^ 2 - 6 P R } { 2 \\sqrt { \\Delta } } \\\\ [ 1 . 2 e x ] d & = 1 - a \\end{align*}"}
-{"id": "3075.png", "formula": "\\begin{align*} A ( t ) : = \\nabla ^ 2 _ x F ( t , 0 ) , t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "5672.png", "formula": "\\begin{align*} \\int _ I | \\sigma ' | ^ 2 E ^ 2 = \\int _ I \\frac { E ^ 2 } { | { v } | ^ 2 } \\left ( { v } ' - ( { v } ' \\cdot \\frac { { v } } { | { v } | } ) \\frac { { v } } { | { v } | } \\right ) ^ 2 \\leq 4 \\int _ I | { v } ' | ^ 2 \\leq 8 \\mathfrak { E } _ { W } ( { v } ) < + \\infty . \\end{align*}"}
-{"id": "2851.png", "formula": "\\begin{align*} & P _ { n } = O ( n p _ { n } ) , \\\\ & P _ { n } \\Delta p _ { n } = O ( p _ { n } p _ { n + 1 } ) , \\end{align*}"}
-{"id": "248.png", "formula": "\\begin{align*} f _ { n , \\pi } ^ T : = h \\big ( U _ \\pi \\chi _ { ( \\R _ { \\geq 0 } ^ n \\times \\R ^ { d - n } ) } U _ { \\pi } ^ * g ( H ^ T ) U _ \\pi \\chi _ { ( \\R _ { \\geq 0 } ^ n \\times \\R ^ { d - n } ) } U _ { \\pi } ^ * \\big ) , \\end{align*}"}
-{"id": "9715.png", "formula": "\\begin{align*} \\tilde { u } - u = - \\frac { 1 } { \\rho u } ( \\tilde { p } - p ) . \\end{align*}"}
-{"id": "291.png", "formula": "\\begin{align*} j ( \\lambda \\odot x ^ * + \\mu \\odot y ^ * ) = \\lambda j ( x ^ * ) + \\mu j ( y ^ * ) , \\lambda , \\mu \\geq 0 . \\end{align*}"}
-{"id": "9086.png", "formula": "\\begin{align*} 2 f ( x y ) = 2 x f ( y ) + 2 y f ( x ) \\left ( x \\in R \\right ) , \\end{align*}"}
-{"id": "5929.png", "formula": "\\begin{align*} \\epsilon y '' + k y = f ( t ) , k > 0 , 0 < \\epsilon < < 1 , f \\in C ^ 3 \\left ( \\langle a , b \\rangle \\right ) \\end{align*}"}
-{"id": "5867.png", "formula": "\\begin{align*} y _ { \\omega _ t } & = \\frac { 1 } { | G | } \\sum _ { g \\in G } x \\circ g ( \\{ x , y \\} ) = \\frac { t ! ( 2 0 - t ) ! } { | G | } | \\{ ( u , v ) \\in C ^ 2 \\ , \\ , : \\ , \\ , d _ H ( u , v ) = t \\} | \\\\ & = \\frac { | \\{ ( u , v ) \\in C ^ 2 \\ , \\ , : \\ , \\ , d _ H ( u , v ) = t \\} | } { 2 ^ { 2 0 } \\binom { 2 0 } { t } } = \\frac { | C | a _ t } { 2 ^ { 2 0 } \\binom { 2 0 } { t } } , \\end{align*}"}
-{"id": "5223.png", "formula": "\\begin{align*} \\omega ( a , b ) : = \\langle a , J b \\rangle \\end{align*}"}
-{"id": "5944.png", "formula": "\\begin{align*} f _ { a _ 1 } & \\equiv \\phi & & ( f _ { a _ 1 } ( x ) = x _ { a _ 1 } \\lor x _ { a _ k } ) \\\\ c _ j & \\equiv a _ { b _ j } & & 1 \\le j \\le l \\\\ a _ { b _ j + 1 } & \\not \\equiv c _ j & & 1 \\le j \\le l \\\\ a _ { i + 1 } & \\equiv a _ i & & i \\le k - 1 , a _ i \\notin B . \\end{align*}"}
-{"id": "9983.png", "formula": "\\begin{align*} | f ( A _ 1 , \\dots , A _ N ) | = | c | \\prod _ { i = 1 } ^ { N } { \\prod _ { j = 1 } ^ { \\lambda _ { i } } { | A _ { i , \\ell _ { i , j } } } | } \\leq e ^ H \\prod _ { i = 1 } ^ { N } { \\| A _ i \\| _ { 0 } ^ { \\lambda _ { i } } } . \\end{align*}"}
-{"id": "9888.png", "formula": "\\begin{align*} e ( G ) = \\binom { r } { 2 } ( a n _ * ) ^ 2 + r e ( H ) = \\left ( \\frac { r - 1 } { r } + \\frac { d _ * } { r n _ * } \\right ) \\frac { ( a r n _ * ) ^ 2 } 2 \\ , . \\end{align*}"}
-{"id": "8756.png", "formula": "\\begin{align*} & \\int \\limits _ { \\lambda < p ^ 2 , q ^ 2 \\leq n } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! d p \\ , d q \\ , \\ : a _ q ^ * \\ : \\frac { 1 } { ( H _ f + p ^ 2 + q ^ 2 - \\mu ) ^ 2 } \\ : a _ p \\\\ & \\qquad = \\int \\limits _ 0 ^ \\infty \\ ! d s \\int \\limits _ 0 ^ \\infty \\ ! d t \\ ! \\int \\limits _ { \\lambda < p ^ 2 , q ^ 2 \\leq n } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! d p \\ , d q \\ , \\ : a _ q ^ * \\ : e ^ { - ( s + t ) q ^ 2 } e ^ { - ( s + t ) ( H _ f - \\mu ) } e ^ { - ( s + t ) p ^ 2 } \\ : a _ p . \\end{align*}"}
-{"id": "1171.png", "formula": "\\begin{align*} \\ell _ k = \\frac { D _ k d _ k } { ( d _ k + D _ k ^ 2 ) ( D _ { k - 1 } D _ { k - 2 } \\cdots D _ 2 ) } . \\end{align*}"}
-{"id": "4826.png", "formula": "\\begin{align*} \\mathcal { L } ( K ) = \\mathcal { L } ( \\mathcal { B } ( G _ 0 ) ) . \\end{align*}"}
-{"id": "2275.png", "formula": "\\begin{align*} \\lambda _ 1 ( A _ { \\alpha } ( K _ m \\vee H ) ) & - \\lambda _ 1 ( A _ { \\alpha } ( K _ m \\vee P _ n ) ) \\\\ & \\geq 2 ( 1 - \\alpha ) ( x _ { i + k } - x _ { i - 1 } ) ( x _ { i } - x _ { i + k + 1 } ) \\\\ & = 0 \\end{align*}"}
-{"id": "4638.png", "formula": "\\begin{align*} C = 6 d D A \\frac { c _ 1 } { c _ 2 } + 2 d D + 4 d A c _ 1 \\end{align*}"}
-{"id": "2900.png", "formula": "\\begin{align*} \\lim _ { s _ 2 } \\cdots \\lim _ { s _ { n - 1 } } c ( k , s _ 1 , s _ 2 , \\ldots , s _ { n - 1 } ) = 1 , \\end{align*}"}
-{"id": "4937.png", "formula": "\\begin{align*} \\displaystyle { \\not } D _ k = \\left ( \\begin{array} { c c } 0 & \\displaystyle { \\not } D ^ 1 _ k \\\\ \\displaystyle { \\not } D ^ 0 _ k & 0 \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "3355.png", "formula": "\\begin{align*} \\tilde { r } _ i = \\frac { 1 } { 1 + N + \\cdots + N ^ i } , i = 1 , \\cdots , K - 1 . \\end{align*}"}
-{"id": "5967.png", "formula": "\\begin{align*} s ( x ) = \\sum _ { k = 1 } ^ N \\alpha _ k \\phi _ k ( x ) = \\Phi ( x , X ^ { \\mathrm { c e n } } ) \\alpha , \\end{align*}"}
-{"id": "9272.png", "formula": "\\begin{align*} W _ { t } = e ^ { \\varepsilon ^ 2 t Q _ \\varepsilon } \\mathcal { P } + e ^ { \\varepsilon ^ 2 t Q _ \\varepsilon } \\mathcal { P } \\int _ 0 ^ t e ^ { - \\varepsilon ^ 2 u Q _ \\varepsilon } \\mathcal { P } \\frac { d } { d u } I _ u d u . \\end{align*}"}
-{"id": "7236.png", "formula": "\\begin{align*} \\Phi _ { \\pi } ^ { \\psi } ( f ) = \\int _ { G } ^ { } f ( g ) ( \\bar { \\psi } * _ { U } \\theta _ { \\pi } ) ( g ) d g , \\end{align*}"}
-{"id": "1354.png", "formula": "\\begin{align*} { \\cal C } _ \\theta ( Z ) = \\{ H \\in { \\cal S } ^ q : \\theta ' ( X ; H ) = \\langle Y , H \\rangle \\} . \\end{align*}"}
-{"id": "5546.png", "formula": "\\begin{align*} f \\left ( t \\right ) = \\sum _ { n = 0 } ^ { 2 ^ { k } - 1 } a _ { n } w _ { n } \\left ( t \\right ) = \\bar { a } ^ { T } \\bar { w } _ { 2 ^ { k } } \\left ( t \\right ) \\end{align*}"}
-{"id": "3972.png", "formula": "\\begin{align*} p ^ { \\alpha _ 0 } ( 0 , t ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { ( - \\lambda t ^ { \\alpha _ 0 } ) ^ { k } } { \\Gamma ( k { \\alpha _ 0 } + 1 ) } , \\end{align*}"}
-{"id": "9248.png", "formula": "\\begin{align*} [ x _ { 1 } ^ { + } \\otimes a _ { 1 } ^ { - } , x _ { 2 } ^ { + } \\otimes a _ { 2 } ^ { - } ] = x _ { 1 } ^ { + } \\circ x _ { 2 } ^ { + } \\otimes \\frac { [ a _ { 1 } ^ { - } , a _ { 2 } ^ { - } ] _ { A ^ { - } } } { 2 } + [ x _ { 1 } ^ { + } , x _ { 2 } ^ { + } ] \\otimes \\frac { ( a _ { 1 } ^ { - } \\circ a _ { 2 } ^ { - } ) _ { A ^ { + } } } { 2 } + ( x _ { 1 } ^ { + } \\mid x _ { 2 } ^ { + } ) I \\otimes [ a _ { 1 } ^ { - } , a _ { 2 } ^ { - } ] _ { A ^ { - } } . \\end{align*}"}
-{"id": "1216.png", "formula": "\\begin{align*} ( \\lambda B + ( 1 - \\lambda ) A ) ^ { 2 } H ^ { - 1 } = \\lambda B ^ 2 H _ { 1 } ^ { - 1 } + ( 1 - \\lambda ) A ^ 2 H _ { 2 } ^ { - 1 } . \\end{align*}"}
-{"id": "9669.png", "formula": "\\begin{align*} U = \\begin{cases} { U } _ 2 ^ { ( 0 ) } = ( { u } _ 2 ^ { ( 0 ) } , 0 , { p } _ 2 ^ { ( 0 ) } , { \\rho } _ 2 ^ { ( 0 ) } , 0 ) , { y } ^ { ( 0 ) } < y < 0 , \\\\ { U } _ 1 ^ { ( 0 ) } = ( { u } _ 1 ^ { ( 0 ) } , 0 , { p } _ 1 ^ { ( 0 ) } , { \\rho } _ 1 ^ { ( 0 ) } , 0 ) , y < { y } ^ { ( 0 ) } , \\end{cases} \\end{align*}"}
-{"id": "8036.png", "formula": "\\begin{align*} { \\bf w } _ t \\ ; \\ ! + \\ , \\mbox { \\boldmath $ u $ } \\cdot \\nabla \\mbox { \\bf w } \\ ; = \\ ; \\gamma \\ ; \\ ! \\ , \\Delta \\mbox { \\bf w } \\ , + \\ , \\nabla \\ : \\ ! ( \\ : \\ ! \\nabla \\cdot \\ ; { \\bf w } \\ : \\ ! ) \\ , + \\ , \\chi \\ , \\nabla \\times \\ , \\mbox { \\boldmath $ u $ } \\ , - \\ , 2 \\ , \\chi \\ , { \\bf w } , \\end{align*}"}
-{"id": "5904.png", "formula": "\\begin{align*} s ^ D ( N , r _ 1 , \\cdots , r _ l ) = \\frac { \\sum _ { i = 0 } ^ l \\nu ^ { N , D } ( i ) E S ^ { ( i ) } _ { \\tau ^ { N , i } + N } } { \\sum _ { i = 0 } ^ l \\nu ^ { N , D } ( i ) \\big ( E \\tau ^ { N , i } + N \\big ) } . \\end{align*}"}
-{"id": "1396.png", "formula": "\\begin{align*} v '' = \\sum _ { I , J } \\dfrac { \\alpha '' \\beta _ { I J } } { \\alpha _ { I J } } v _ I y _ { J } \\end{align*}"}
-{"id": "4955.png", "formula": "\\begin{align*} \\Delta ( x ) = x _ 1 \\otimes 1 + 1 \\otimes x _ 2 + y , \\end{align*}"}
-{"id": "2316.png", "formula": "\\begin{align*} \\| I _ 2 \\| _ { L ^ 2 } \\leq C \\int _ { t - h } ^ { t + h } ( t + h - \\tau ) ^ { - 1 } \\cdot \\frac { ( t + h - \\tau ) ^ { \\beta } } { \\tau ^ { - ( \\beta + 1 / 2 ) } } R _ 1 R _ 2 \\leq C R _ 1 R _ 2 h ^ { \\beta } t ^ { \\beta + 1 / 2 } , \\end{align*}"}
-{"id": "10049.png", "formula": "\\begin{align*} E ( \\vec { \\tau } , s , \\phi ) = \\sum _ { \\alpha \\in F } E _ \\alpha ( \\vec { v } , s , \\phi ) \\cdot q ^ \\alpha \\end{align*}"}
-{"id": "3719.png", "formula": "\\begin{align*} \\tau _ { i _ l } = \\lambda _ { i _ 1 } \\cdots \\lambda _ { i _ l } \\end{align*}"}
-{"id": "8763.png", "formula": "\\begin{gather*} [ A _ 1 , A _ 2 ] = \\left ( \\begin{array} { c c } 0 & - 1 \\\\ 1 & 0 \\\\ \\end{array} \\right ) , \\\\ [ A _ 1 , A _ 2 ] ^ { ( t _ 2 , s _ 1 ) } = \\left ( \\begin{array} { c c } - \\cosh ( s _ 1 ) \\sinh ( 2 t _ 2 ) & \\sinh ( s _ 1 ) - \\cosh ( s _ 1 ) \\cosh ( 2 t _ 2 ) \\\\ \\cosh ( s _ 1 ) \\cosh ( 2 t _ 2 ) + \\sinh ( s _ 1 ) & \\cosh ( s _ 1 ) \\sinh ( 2 t _ 2 ) \\\\ \\end{array} \\right ) . \\end{gather*}"}
-{"id": "4543.png", "formula": "\\begin{align*} \\frac { 1 } { \\prod _ { n = 1 } ^ N ( \\lambda - \\lambda _ n ^ { ( 0 ) } ) } & = \\sum _ { n = 1 } ^ N \\frac { 1 } { \\lambda - \\lambda _ n ^ { ( 0 ) } } \\frac { 1 } { \\frac { d } { d \\lambda } \\prod _ { n ' = 1 } ^ N ( \\lambda - \\lambda _ { n ' } ^ { ( 0 ) } ) \\vert _ { \\lambda = \\lambda _ n ^ { ( 0 ) } } } \\\\ & = \\sum _ { n = 1 } ^ { N } \\frac { 1 } { \\lambda - \\lambda _ n ^ { ( 0 ) } } \\frac { 1 } { \\prod _ { n ' \\neq n } ( \\lambda _ { n } ^ { ( 0 ) } - \\lambda _ { n ' } ^ { ( 0 ) } ) } \\end{align*}"}
-{"id": "5214.png", "formula": "\\begin{align*} L p = \\lambda p . \\end{align*}"}
-{"id": "6662.png", "formula": "\\begin{align*} f _ x ( s w ) \\leq \\frac { 1 } { 2 } \\left ( \\langle H f _ x ( 0 ) s w , s w \\rangle + \\frac { \\varepsilon ' } { 2 } \\| s w \\| ^ 2 \\right ) = \\frac { s ^ 2 } { 2 } \\left ( \\langle H f _ x ( 0 ) w , w \\rangle + \\frac { \\varepsilon ' } { 2 } \\right ) \\quad . \\end{align*}"}
-{"id": "9295.png", "formula": "\\begin{align*} V _ { 1 , z } & : = \\{ ( x , h ) \\in S _ z \\ : \\ \\Delta ^ { k + 2 } _ { ( h , \\dots , h ) } f _ z ( x ) \\geq 0 \\} \\\\ V _ { 2 , z } & : = \\{ ( x , h ) \\in S _ z \\ : \\ \\Delta ^ { k + 2 } _ { ( h , \\dots , h ) } f _ z ( x ) \\leq 0 \\} . \\end{align*}"}
-{"id": "6518.png", "formula": "\\begin{align*} \\mathcal { S } _ { \\mathcal { M } ^ { } } \\left ( \\tau ; \\rho \\right ) - \\mathcal { S } _ { \\mathcal { M } ^ { } } \\left ( \\tau ; 0 \\right ) = \\frac { 1 } { 2 } \\log \\left ( \\frac { 1 - \\rho } { 1 + \\rho } \\right ) . \\end{align*}"}
-{"id": "4157.png", "formula": "\\begin{align*} \\frac { 1 } { n } e _ { k , j } ^ n + \\frac { 1 } { n } \\left ( 1 - \\frac { 1 } { n } \\right ) e _ { i , j } ^ { n - 1 } + \\frac { 1 } { n } \\sum _ { \\tau = 1 } ^ { n - 2 } e _ { i , j + \\tau } ^ { n - 1 - \\tau } . \\end{align*}"}
-{"id": "6767.png", "formula": "\\begin{align*} u ( y ) = u ( T y ) , \\textrm { f o r a l l } T \\in T _ d \\textrm { a n d a n y } y \\in \\mathbb { S } ^ 2 \\end{align*}"}
-{"id": "2848.png", "formula": "\\begin{align*} \\bar { \\Delta } A _ { n } ( s ) = A _ { n } ( s ) - A _ { n - 1 } ( s ) . \\end{align*}"}
-{"id": "7495.png", "formula": "\\begin{align*} ( \\Box ^ h \\Phi ) _ { A _ p \\overline { B } _ q } = - h ^ { \\bar { \\varepsilon } \\gamma } \\bigg ( \\delta _ \\gamma \\circ \\delta _ { \\bar { \\varepsilon } } ( \\phi _ { A _ p \\overline { B } _ q } ) - \\sum _ i ( - 1 ) ^ { i - 1 } [ \\delta _ \\gamma , \\delta _ { \\bar { \\beta } _ i } ] \\phi _ { A _ p \\bar { \\varepsilon } \\bar { \\beta } _ 1 \\dots \\hat { \\bar { \\beta } } _ i \\dots \\bar { \\beta } _ q } \\bigg ) . \\end{align*}"}
-{"id": "3949.png", "formula": "\\begin{align*} \\frac 1 r = \\frac { 1 } { p _ { 1 } } + \\frac { 1 } { p _ { 2 } } = \\frac { 1 } { q _ { 1 } } + \\frac { 1 } { q _ { 2 } } , p _ { j } , q _ { j } \\in ( 1 , \\infty ] , s > \\max ( 0 , \\frac n r - n ) \\ \\ s \\in 2 \\mathbb { Z } ^ { + } . \\end{align*}"}
-{"id": "7150.png", "formula": "\\begin{align*} \\Delta _ J ( r J ) + V _ r \\sum _ { m = r J + 1 } ^ { ( r + 1 ) J } d _ G ^ * ( m ) \\leq U J ^ 2 + V _ r J g ^ * _ r , \\end{align*}"}
-{"id": "938.png", "formula": "\\begin{align*} E \\left [ \\exp \\left \\{ \\frac { c _ q } { 2 } \\left ( \\frac { | \\Delta _ { i , j } | } { \\sigma _ { i , j } } \\right ) ^ { 2 / q } \\right \\} - 1 \\right ] = \\int _ 0 ^ \\infty P \\left ( \\exp \\left \\{ \\frac { c _ q } { 2 } \\left ( \\frac { | \\Delta _ { i , j } | } { \\sigma _ { i , j } } \\right ) ^ { 2 / q } \\right \\} - 1 > u \\right ) d u , \\end{align*}"}
-{"id": "9727.png", "formula": "\\begin{align*} \\tilde { \\gamma } _ 1 = \\gamma _ 1 + O ( 1 ) Z _ { a } h . \\end{align*}"}
-{"id": "2085.png", "formula": "\\begin{align*} C _ 3 = \\sup _ { B ( N ) , a , b , c , d } \\big ( | P ^ a _ { b c } | + | P ^ a _ { b c d } | \\big ) . \\end{align*}"}
-{"id": "5318.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } = \\frac { 1 } { p } + \\frac { 1 } { q } > \\frac { 1 } { p } + \\frac { n } { 2 ( n + 2 ) } . \\end{align*}"}
-{"id": "6434.png", "formula": "\\begin{align*} \\mathcal { S } _ { \\mathcal { M } } \\left ( \\tau \\right ) \\overset { } { = } \\log \\left [ \\mathcal { C } _ { \\mathcal { M } } \\left ( \\tau \\right ) \\right ] \\end{align*}"}
-{"id": "2258.png", "formula": "\\begin{align*} \\frac { \\sin \\sqrt z } { \\sqrt z } = \\prod _ { k = 1 } ^ \\infty \\left ( 1 - \\frac z { k ^ 2 \\pi ^ 2 } \\right ) = \\sum _ { k = 0 } ^ \\infty \\frac { ( - 1 ) ^ k z ^ k } { ( 2 k + 1 ) ! } . \\end{align*}"}
-{"id": "4403.png", "formula": "\\begin{align*} \\| A + \\mathcal { K } ( F _ \\alpha ^ 2 ) \\| = \\max _ { x \\in \\mathcal { M } \\setminus \\mathbb { C } ^ n } \\| A _ x \\| . \\end{align*}"}
-{"id": "3025.png", "formula": "\\begin{align*} 2 { \\rm R e } \\ , \\langle A x , x \\rangle = 2 \\Re \\ < P _ 1 \\frac { d } { d \\zeta } x + P _ 0 x , x \\ > = - x ( 0 ) ^ * P _ 1 x ( 0 ) + 2 { \\rm R e } \\ , \\int _ 0 ^ \\infty x ( \\zeta ) ^ * P _ 0 x ( \\zeta ) \\ , d \\zeta . \\end{align*}"}
-{"id": "5122.png", "formula": "\\begin{align*} \\det & \\left ( X _ { \\kappa _ n , i _ { ( 1 , 1 ) } ' } f ( p ) \\wedge \\cdots \\wedge X _ { \\kappa _ n , i _ { ( n , 1 ) } } \\cdots X _ { \\kappa _ n , i _ { ( n , \\kappa _ n ) } ' } f ( p ) \\right ) \\\\ & = \\sum _ { i } c _ { i i ' } \\det \\left ( X _ { 1 , i _ { ( 1 , 1 ) } } f ( p ) \\wedge \\cdots \\wedge X _ { \\kappa _ n , i _ { ( n , 1 ) } } \\cdots X _ { 1 , i _ { ( n , \\kappa _ n ) } } f ( p ) \\right ) \\end{align*}"}
-{"id": "5467.png", "formula": "\\begin{align*} \\widehat U _ x ( s ) = \\frac { \\widehat U ( s / x ) } { x ^ \\rho \\ell ( x ) } \\sim s ^ { - \\rho } q ( s / x ) x \\to \\infty . \\end{align*}"}
-{"id": "3952.png", "formula": "\\begin{align*} D ^ { s } ( u v ) - u D ^ { s } v - v D ^ { s } u = c ( n , s ) \\cdot T _ { s } ( u , v ) , 0 < s < 2 \\end{align*}"}
-{"id": "7680.png", "formula": "\\begin{align*} v _ j = \\sum _ { i = 1 } ^ { N - 1 } \\frac { u _ { i j } } { \\lambda _ i } u _ i \\in \\mathbb { R } ^ N j = 0 , 1 , \\dots , N - 1 \\ , . \\end{align*}"}
-{"id": "2956.png", "formula": "\\begin{align*} ( \\partial \\Theta - t ) f = ( \\partial \\Theta - t ) ^ + \\mu . \\end{align*}"}
-{"id": "970.png", "formula": "\\begin{align*} P \\left ( T _ n < q _ n ^ * ( 1 - \\alpha ) \\right ) & \\geq P \\left ( T _ n < q _ n ^ Z ( 1 - \\alpha - \\varepsilon _ n ) \\right ) - P ( \\mathcal { E } _ n ^ c ) \\\\ & \\geq P \\left ( \\max _ { \\theta \\in \\mathcal { G } _ n } | Z _ n ( \\theta ) | < q _ n ^ Z ( 1 - \\alpha - \\varepsilon _ n ) \\right ) - 2 \\varepsilon _ n = 1 - \\alpha - 3 \\varepsilon _ n . \\end{align*}"}
-{"id": "9609.png", "formula": "\\begin{align*} | D _ k ^ \\# H _ j D _ { k ' } ^ \\# ( x , y ) | & \\lesssim \\frac { 2 ^ { - | j - k ' | \\varepsilon } } { V _ { - k ' } ( y ) } \\int _ { S ( x , 9 ( A _ 0 ) ^ 2 2 ^ j ) } \\frac { 2 ^ { - | j - k | \\varepsilon ' } } { V _ { - j } ( x ) } d \\mu ( v ) \\\\ & \\lesssim \\frac { 2 ^ { - | j - k ' | \\varepsilon } } { V _ { - k ' } ( y ) } 2 ^ { - | j - k | \\varepsilon ' } \\\\ & = \\frac { 2 ^ { - | k - k ' | \\varepsilon ' } } { V _ { - k ' } ( y ) } 2 ^ { - | j - k ' | ( \\varepsilon - \\varepsilon ' ) } . \\end{align*}"}
-{"id": "1249.png", "formula": "\\begin{align*} \\mathcal { G } ( x ) / 2 = \\mathcal { G } ( x ) - \\mathcal { G } ( y ) \\ , \\leq \\ , | \\nabla \\mathcal { G } ( \\hat w ) | \\ , | y - x | . \\end{align*}"}
-{"id": "3646.png", "formula": "\\begin{align*} \\tilde M _ { \\mathbb R , b } = \\{ ( m , t ) \\in M _ { \\mathbb R } \\times \\mathbb R \\ ; | \\ ; ( l _ b \\otimes \\mathbb R ) ( m ) = 0 \\} , \\end{align*}"}
-{"id": "1425.png", "formula": "\\begin{align*} \\phi _ r ( x ) = \\partial _ 2 \\cdots \\partial _ r ( x ) \\in M _ { n , i } , \\end{align*}"}
-{"id": "1136.png", "formula": "\\begin{align*} b _ p = \\frac { 2 - a - a ^ { - 1 } } { i - j } , b _ m = \\frac { - 2 - a - a ^ { - 1 } } { i - j } . \\end{align*}"}
-{"id": "4469.png", "formula": "\\begin{align*} \\sum _ { k ' \\not = 0 } \\frac { 1 } { d ^ 4 ( k ' , 0 ) } \\lesssim \\sum _ { k ' \\not = 0 } \\frac { 1 } { d ^ 3 ( k ' , 0 ) } \\lesssim 1 , \\end{align*}"}
-{"id": "3759.png", "formula": "\\begin{align*} \\chi ^ g _ \\sigma ( \\omega , U ) : = \\frac { 1 } { L _ k } \\sum _ { i = n } ^ { n + L _ k - 1 } g _ { ( \\sigma ( i ) , i ) } ( \\omega , U ) , \\end{align*}"}
-{"id": "9629.png", "formula": "\\begin{align*} \\Phi _ { \\rm { s t } , \\beta } ( t ) = \\frac { S _ \\beta } { \\pi E _ { \\rm { b } } } \\bigg ( \\frac { P _ { \\rm { c u } } } { R ^ 2 _ { \\beta } } + \\lambda _ { \\beta } \\left ( 2 ^ { C / W } - 1 \\right ) P _ { \\rm { t r } , 1 } ( h _ { \\beta , 1 } ^ * ) R ^ 2 _ { \\beta } \\bigg ) . \\end{align*}"}
-{"id": "5373.png", "formula": "\\begin{align*} L ( \\ell , ( r - 1 ) \\omega ) \\boxtimes _ { P ( 1 ) } L ( \\ell , ( r ' - 1 ) \\omega ) \\cong \\bigoplus _ { r '' = 1 } ^ { a - 1 } N ^ a _ { r , r ' , r '' } L ( \\ell , ( r '' - 1 ) \\omega ) . \\end{align*}"}
-{"id": "5195.png", "formula": "\\begin{align*} \\sup _ { z \\in ( \\ell ^ { 2 } , 1 ] } \\hat { U } _ { 3 } ( \\ell ^ { 1 } , \\ell ^ { 2 } , z ) = \\sup _ { z \\in [ b , 1 ] } \\hat { U } _ { 3 } ( \\ell ^ { 1 } , \\ell ^ { 2 } , z ) = \\hat { U } _ { 3 } ( \\ell ^ { 1 } , \\ell ^ { 2 } , r ) . \\end{align*}"}
-{"id": "3326.png", "formula": "\\begin{align*} r _ s = \\frac { \\binom { K - 2 } { s - 1 } } { \\binom { K - 2 } { s - 1 } + \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K - 1 } { s + i } ( N - 1 ) ^ i N } , \\end{align*}"}
-{"id": "7861.png", "formula": "\\begin{align*} | \\nabla u | ( x , t ) : = \\limsup _ { y \\to x } \\frac { | u ( y , t ) - u ( x , t ) | } { d ( x , y ) } \\end{align*}"}
-{"id": "1182.png", "formula": "\\begin{align*} v ( y _ 0 + \\rho \\eta ) & = v ( y _ 0 ) + A _ 1 \\rho + A _ 2 \\rho ^ { 2 } + o ( \\rho ^ { 2 } ) , \\\\ v ( z _ 0 + \\rho \\eta ) & = v ( z _ 0 ) + B _ 1 \\rho + B _ 2 \\rho ^ { 2 } + o ( \\rho ^ { 2 } ) , \\\\ u ( x _ 0 + \\rho \\eta ) & = u ( x _ 0 ) + C _ 1 \\rho + C _ 2 \\rho ^ { 2 } + o ( \\rho ^ { 2 } ) \\end{align*}"}
-{"id": "2319.png", "formula": "\\begin{align*} & \\ ; \\| \\tilde { w } _ j \\| _ { L ^ \\infty _ T ( D ( A ) ) \\cap L ^ 2 _ T ( D ( A ^ { 1 + s / 2 } ) ) } \\\\ \\leq & \\ ; C ( \\nu ) ( \\| f ( \\tilde { w } _ { j - 1 } , u ^ { ( j - 1 ) } ) \\| _ { L ^ 2 _ T D ( A ^ { 1 - s / 2 } ) } + \\| f ( u ^ { ( j - 2 ) } , \\tilde { w } _ { j - 1 } ) \\| _ { L ^ 2 _ T D ( A ^ { 1 - s / 2 } ) } ) \\\\ \\leq & \\ ; C _ 2 M \\| \\tilde { w } _ { j - 1 } \\| _ { L ^ \\infty _ T ( D ( A ) ) \\cap L ^ 2 _ T ( D ( A ^ { 1 + s / 2 } ) ) } , \\end{align*}"}
-{"id": "3387.png", "formula": "\\begin{align*} \\R ^ n F _ { b } ( p _ { 1 , n } ( b ) ) - p _ { 1 , n } ( b ) = 0 \\end{align*}"}
-{"id": "1775.png", "formula": "\\begin{align*} U _ { \\lambda \\omega } t _ { \\lambda } \\ ; = \\ ; \\tilde { t } _ { \\lambda } U _ { \\omega } t _ { \\lambda } ^ * t _ { \\lambda } . \\end{align*}"}
-{"id": "8135.png", "formula": "\\begin{align*} X \\setminus \\bigsqcup _ { k = 1 } ^ K S _ k V _ k \\prec \\bigsqcup _ { k = 1 } ^ K S _ k ' V _ k . \\end{align*}"}
-{"id": "5936.png", "formula": "\\begin{align*} \\Sigma _ { W _ 1 } \\cap \\Sigma _ { W _ 2 } = \\Sigma _ { U } . \\end{align*}"}
-{"id": "1835.png", "formula": "\\begin{align*} ( X \\otimes ^ . Y ) _ m : = \\bigoplus _ { k \\in \\mathbb { Z } } X _ k \\displaystyle \\operatorname * { \\otimes } _ R Y _ { m - k } \\end{align*}"}
-{"id": "6348.png", "formula": "\\begin{align*} \\det D \\varphi _ { 2 n } ^ \\mathbb { R } = \\prod \\limits _ { j = 1 } ^ { n - 1 } \\beta _ j ^ { 2 n - 2 j } . \\end{align*}"}
-{"id": "4711.png", "formula": "\\begin{align*} A _ s ^ i = \\begin{pmatrix} \\ast & \\ast & \\cdots & \\ast \\\\ & 1 & & \\\\ & & \\ddots & \\\\ & & & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "2523.png", "formula": "\\begin{align*} \\mathcal { L } ( f g ) = f \\mathcal { L } g + g \\mathcal { L } f + \\psi \\ , f \\ , g + 2 \\left ( \\nabla _ \\xi f , \\sigma \\nabla _ \\xi g \\right ) . \\end{align*}"}
-{"id": "791.png", "formula": "\\begin{align*} a _ 2 = ( \\rho + 1 ) \\log ( 2 x ) , \\end{align*}"}
-{"id": "9920.png", "formula": "\\begin{align*} 2 e ( G ) = \\sum _ { i = 1 } ^ r e ( B _ i , V ) \\le 2 \\sum _ { 1 \\le i < j \\le r } | B _ i | | B _ j | - | B _ 1 | | B _ 2 | + \\delta n \\sum _ { i = 1 } ^ r | B _ i | - \\delta ^ 2 n ^ 2 \\end{align*}"}
-{"id": "5201.png", "formula": "\\begin{align*} 0 = \\frac { { d } } { { d } x } \\left ( \\frac { f _ { 1 } ( x ) - g _ { 1 , [ r , 1 ] } ( x ) } { r - x } \\right ) \\bigg \\vert _ { x = \\ell } \\implies f _ { 1 } ' ( \\ell ) = \\frac { g _ { 1 } ( r ) - f _ { 1 } ( \\ell ) } { r - \\ell } . \\end{align*}"}
-{"id": "2902.png", "formula": "\\begin{align*} \\lim _ { s _ 1 } \\cdots \\lim _ { s _ { n - 1 } } c ( k , s _ 1 , \\ldots , s _ { n - 1 } ) = i . \\end{align*}"}
-{"id": "5705.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta { u } _ 1 + D _ 1 W ( { u } ) \\\\ - \\Delta { u } _ 2 + D _ 2 W ( { u } ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "6149.png", "formula": "\\begin{align*} Q = \\big ( Q _ { i j } \\big ) = \\Big ( \\frac { \\omega _ i \\wedge \\omega _ j } { 2 \\mu } \\Big ) = \\Big ( \\frac { 1 } { 2 } \\langle \\omega _ i , \\omega _ j \\rangle \\Big ) \\end{align*}"}
-{"id": "1228.png", "formula": "\\begin{align*} \\nabla k ( \\nabla h ( X ) ) \\ , = \\ , \\frac { | \\nabla k ( \\eta ) | X } { | X | } = \\frac { X } { h ( X ) } \\end{align*}"}
-{"id": "4987.png", "formula": "\\begin{align*} \\left \\vert g _ j ( x ) \\right \\vert \\lesssim \\zeta _ j ( x ) \\omega _ j ( x ) \\lesssim \\sum _ { \\substack { k < j \\\\ k \\equiv j ( \\textrm { m o d } R ) } } 2 ^ { \\alpha ( k - j ) } \\omega _ k = G _ j . \\end{align*}"}
-{"id": "7196.png", "formula": "\\begin{align*} \\int _ D \\zeta \\ , d x = 0 . \\end{align*}"}
-{"id": "6231.png", "formula": "\\begin{align*} X ( n ) = n - ( C ^ 1 + \\cdots + C ^ m ) ( n ) ~ , ~ n \\geq 0 ~ . \\end{align*}"}
-{"id": "5322.png", "formula": "\\begin{align*} \\mathfrak { g } = \\mathfrak { n } _ { - } \\oplus \\mathfrak { h } \\oplus \\mathfrak { n } _ { + } , \\end{align*}"}
-{"id": "216.png", "formula": "\\begin{align*} \\log \\det \\left ( ( a _ { j - k } ) _ { j , k = 1 } ^ { L } \\right ) = L ( \\log a ) _ 0 + \\hbox { o } ( L ) ( L \\to \\infty ) . \\end{align*}"}
-{"id": "4363.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty M _ { a _ j } P _ \\alpha M _ { 1 - b _ j } = \\sum _ { k = 1 } ^ N \\sum _ { l = 1 } ^ N \\sum _ { j = 1 } ^ \\infty M _ { a _ j \\chi _ { A _ j ^ k } } P _ \\alpha M _ { 1 - b _ j \\chi _ { B _ j ^ l } } \\end{align*}"}
-{"id": "9491.png", "formula": "\\begin{align*} q ^ { ( i ) } _ { n , d } = q ^ { ( i ) } _ { n - 1 , d + r - 2 } + i \\cdot s _ { n - 1 , d + i - 2 } + 2 \\sum \\limits _ { j = 0 } ^ { n - 1 } \\sum \\limits _ { k = 0 } ^ { d - 2 } q ^ { ( i ) } _ { j , k } \\cdot s _ { n - j - 1 , d - k - 2 } . \\end{align*}"}
-{"id": "1920.png", "formula": "\\begin{align*} \\Gamma ^ { t } ( \\mathcal { M } ) = \\{ ( \\sigma _ { i } ) _ { i \\in \\mathbb { Z } } \\in \\Gamma ( \\mathcal { M } ) : \\sigma _ { i } \\in \\Gamma ^ { t } ( M _ { i } ) i \\in \\mathbb { Z } \\} . \\end{align*}"}
-{"id": "962.png", "formula": "\\begin{align*} E [ U _ n ( \\vartheta _ n ) ] & = \\sum _ { I , J } K ( I , J _ { - \\vartheta _ n } ) \\rho _ { m ^ * } \\int _ { I \\cap J _ { - \\vartheta _ n } } \\sigma _ 1 ( t ) \\sigma _ 2 ( t + \\theta _ { m ^ * } ) d t + o ( 1 ) = \\rho _ { m ^ * } \\Sigma ( \\theta _ { m ^ * } ) + o ( 1 ) . \\end{align*}"}
-{"id": "3524.png", "formula": "\\begin{align*} \\begin{array} [ l ] { l } \\max _ { t \\in \\lbrack 0 , T ] } \\left \\vert y ^ { } ( t ) \\right \\vert = O ( \\varepsilon ^ { } ) , \\\\ \\max _ { t \\in \\lbrack 0 , T ] } \\left \\vert X ^ { \\varepsilon } ( t ) - \\bar { X } ( t ) - y ^ { } ( t ) \\right \\vert = o ( \\varepsilon ) , \\\\ \\end{array} \\end{align*}"}
-{"id": "6786.png", "formula": "\\begin{align*} w _ { \\lambda , k } ( \\Pi ^ { - 1 } _ { \\xi _ k } ( \\lambda z ) ) = - 4 \\ln { \\lambda } - 4 \\ln { 2 } - 4 \\lambda ^ 2 \\ln { \\lambda } + \\ln { \\left ( \\frac { 1 } { ( 1 + | z | ^ 2 ) ^ 2 } \\right ) } + 2 \\ln { ( 1 + \\lambda ^ 2 | z | ^ 2 ) } \\end{align*}"}
-{"id": "4178.png", "formula": "\\begin{align*} F = \\star D \\phi , \\textnormal { w i t h } \\\\ F : = \\mathrm { d } { A } + { A } \\wedge { A } \\textrm { a n d } D : = \\mathrm { d } + { A } , \\end{align*}"}
-{"id": "7473.png", "formula": "\\begin{align*} \\mathcal { T } _ { \\mathbb { C } } E = H \\mathcal { T } _ { \\mathbb { C } } E \\oplus V \\mathcal { T } _ { \\mathbb { C } } E \\oplus \\overline { H \\mathcal { T } _ { \\mathbb { C } } E } \\oplus \\overline { V \\mathcal { T } _ { \\mathbb { C } } E } \\end{align*}"}
-{"id": "3970.png", "formula": "\\begin{align*} p ^ { \\alpha _ n } ( n , t ) = p ^ { \\alpha _ n } ( n , 0 ) - \\lambda I _ t ^ { \\alpha _ n } ( p ^ { \\alpha _ n } ( n , t ) - p ^ { \\alpha _ { n - 1 } } ( n - 1 , t ) ) , \\ \\ n \\geq 0 . \\end{align*}"}
-{"id": "5629.png", "formula": "\\begin{align*} K _ \\beta ( x , y ) \\ , = \\ , \\frac { | x | ^ { 2 \\beta } + | y | ^ { 2 \\beta } - | x - y | ^ { 2 \\beta } } 2 , \\ \\ \\mbox { w h e r e $ \\beta \\in ( 0 , 1 ) $ . } \\end{align*}"}
-{"id": "5652.png", "formula": "\\begin{align*} \\varphi ' ( t ) = F ( \\varphi ( t ) ) , \\end{align*}"}
-{"id": "8940.png", "formula": "\\begin{gather*} s _ 0 ( x _ 0 , x _ 1 , x _ 2 ) = ( - x _ 0 , x _ 0 + x _ 1 , x _ 0 + x _ 2 ) , \\\\ s _ 1 ( x _ 0 , x _ 1 , x _ 2 ) = ( x _ 0 + x _ 1 , - x _ 1 , x _ 1 + x _ 2 ) , \\\\ s _ 2 ( x _ 0 , x _ 1 , x _ 2 ) = ( x _ 0 + x _ 2 , x _ 1 + x _ 2 , - x _ 2 ) , \\end{gather*}"}
-{"id": "43.png", "formula": "\\begin{align*} \\hat { P } ( C _ { 1 } = C _ { 2 } ) = \\int \\limits _ { - \\infty } ^ { \\infty } \\int \\limits _ { - \\infty } ^ { \\infty } \\ ! \\hat { f } _ { X Y Z S } ( x , y , z , s ) \\ , \\mathrm { d } u _ { 1 } \\mathrm { d } u _ { 2 } \\Big | _ { x = y = u _ { 1 } , z = s = u _ { 2 } } = \\int \\limits _ { - \\infty } ^ { \\infty } \\int \\limits _ { - \\infty } ^ { \\infty } \\ ! \\hat { f } _ { X Y Z S } ( u _ { 1 } , u _ { 1 } , u _ { 2 } , u _ { 2 } ) \\ , \\mathrm { d } u _ { 1 } \\mathrm { d } u _ { 2 } \\end{align*}"}
-{"id": "3620.png", "formula": "\\begin{align*} \\int _ { R _ \\lambda ( \\Omega ) } | \\nabla u _ \\lambda | ^ { p - 2 } ( \\nabla u _ \\lambda , \\nabla \\varphi ) \\ , d x \\ , = \\ , \\int _ { R _ \\lambda ( \\Omega ) } f ( u _ \\lambda ) \\varphi \\ , d x \\qquad \\forall \\varphi \\in C ^ { 1 } _ c ( R _ \\lambda ( \\Omega ) \\setminus R _ \\lambda ( \\Gamma ) ) \\ , . \\end{align*}"}
-{"id": "9384.png", "formula": "\\begin{align*} \\| u _ { x _ 1 , r } - p _ { x _ 1 } \\| _ { L ^ 1 ( \\partial B _ 1 ) } + \\| u _ { x _ 2 , r } - p _ { x _ 2 } \\| _ { L ^ 1 ( \\partial B _ 1 ) } & \\leq C \\ , ( - \\log ( r ) ) ^ { - \\frac { 1 - \\gamma } { 2 \\gamma } } = C \\big ( - \\log | x _ 1 - x _ 2 | \\big ) ^ { - \\frac { 1 - \\gamma } { 2 \\gamma } } \\end{align*}"}
-{"id": "6183.png", "formula": "\\begin{align*} \\frac { _ { g ( t ) } B _ { g ( t ) } ( p , r ) } { r ^ 4 } & \\geq \\frac { _ { g ( t ) } \\Omega _ { \\frac { r } { 2 \\sqrt { C } } } } { r ^ 4 } \\\\ & = \\frac { 1 } { r ^ 4 } \\int _ { \\Omega _ { \\frac { r } { 2 \\sqrt { C } } } } \\tilde f _ 1 \\tilde f _ 2 \\tilde f _ 3 \\dd y \\dd x _ 1 \\dd x _ 2 \\dd x _ 3 \\\\ & = \\frac { 1 } { C ^ 2 } \\end{align*}"}
-{"id": "5654.png", "formula": "\\begin{align*} \\varphi ' ( t ) = F ( \\varphi ( t ) ) \\quad J . \\end{align*}"}
-{"id": "1860.png", "formula": "\\begin{align*} & \\Big ( \\sum _ { i = 1 } ^ d F ^ \\ast _ i ( x _ i ) - ( d - 1 ) \\Big ) ^ + \\vee \\max \\Big \\{ \\pi _ s - \\sum _ { i = 1 } ^ d \\big ( F _ i ^ \\ast ( s _ i ) - F _ i ^ \\ast ( x _ i ) \\big ) ^ + : s \\in S \\Big \\} \\\\ & \\le F ( x _ 1 , \\dots , x _ d ) \\le \\min _ { i = 1 , \\dots , d } F ^ \\ast _ i ( x _ i ) \\wedge \\min \\Big \\{ \\pi _ s + \\sum _ { i = 1 } ^ d \\big ( F _ i ^ \\ast ( x _ i ) - F _ i ^ \\ast ( s _ i ) \\big ) ^ + : s \\in S \\Big \\} . \\end{align*}"}
-{"id": "8032.png", "formula": "\\begin{align*} ( \\ref { m a i n l o w e r } ) & \\gtrsim _ M \\Big [ \\sum _ { j = M } ^ { L - M } { \\Big ( \\sum _ { n = 1 } ^ { L - j } { 2 ^ { - { t _ n } ( { s } + d - { d } / { p } ) } } \\Big ) ^ p } \\Big ] ^ { { 1 } / { p } } \\\\ & \\geq \\Big [ \\sum _ { j = \\lfloor L / 3 \\rfloor } ^ { \\lfloor L / 2 \\rfloor } { \\Big ( \\sum _ { n = 1 } ^ { \\lfloor L / 3 \\rfloor } { 2 ^ { - { t _ n } ( { s } + d - { d } / { p } ) } } \\Big ) ^ p } \\Big ] ^ { { 1 } / { p } } \\gtrsim L ^ { { 1 } / { p } } L ^ { - ( { s } + d - { d } / { p } ) } \\log { L } \\end{align*}"}
-{"id": "1952.png", "formula": "\\begin{align*} \\left | \\partial _ r S _ { r } ( q ) ( \\eta ) \\right | \\le C K _ { r } ( \\widehat { q } , \\widehat { q } ) ( \\eta ) + C | \\eta | \\sum _ { i = 1 } ^ n K _ { r } ( \\widehat { x _ i q } , \\widehat { q } ) . \\end{align*}"}
-{"id": "1985.png", "formula": "\\begin{align*} \\Vert D _ { p } g ^ { n } ( v ) \\Vert & \\leq 2 \\Vert D _ { p } g ^ { n } ( v ) \\Vert ^ { \\star } \\leq 2 \\eta ^ { n } ( \\Vert v _ { s } \\Vert ^ { \\star } + \\Vert v _ { u } \\Vert ^ { \\star } ) \\leq 2 \\eta ^ { n } ( 1 + \\alpha ) \\Vert v _ { s } \\Vert ^ { \\star } \\\\ & \\leq 2 \\eta ^ { n } ( 1 + \\alpha ) \\frac { \\lambda + \\varepsilon } { \\varepsilon } c ( 1 - \\alpha \\frac { \\lambda + \\varepsilon } { \\varepsilon } c ) ^ { - 1 } \\Vert v \\Vert = c ^ { \\prime } \\eta ^ { n } \\Vert v \\Vert , \\end{align*}"}
-{"id": "7057.png", "formula": "\\begin{align*} C _ 3 = C _ 3 ( n , s ) = \\int _ { \\mathbb { R } } { \\frac { 1 } { ( 1 + z _ n ^ 2 ) ^ { ( n + 2 s ) / 2 } } d z _ n } \\end{align*}"}
-{"id": "860.png", "formula": "\\begin{align*} \\Delta = E \\left [ \\max _ { 1 \\leq i , j \\leq d } | \\mathfrak { C } ( i , j ) - \\langle D F _ i , - D L ^ { - 1 } F _ j \\rangle _ H | \\right ] . \\end{align*}"}
-{"id": "6572.png", "formula": "\\begin{align*} \\alpha _ 2 : = \\alpha _ { \\xi } = \\left ( \\frac { 2 \\cdot 2 ( 1 + \\varepsilon ) \\cdot \\sqrt { 1 - \\varepsilon ^ 2 } } { 4 \\cdot \\sqrt { 1 - \\varepsilon ^ 2 } } \\right ) ^ { 1 / 2 } = ( 1 + \\varepsilon ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "1857.png", "formula": "\\begin{align*} A ^ \\infty _ \\infty = \\cdots \\to \\cdot \\to \\cdot \\to \\cdot \\to \\cdots \\end{align*}"}
-{"id": "6465.png", "formula": "\\begin{align*} \\rho _ { 2 j - 1 } \\overset { } { = } \\frac { \\frac { \\partial \\mu _ { 2 j } } { \\partial \\mu _ { 2 j - 1 } } \\frac { \\partial \\mu _ { 2 j } } { \\partial \\sigma _ { 2 j - 1 } } } { \\left [ 1 + \\left ( \\frac { \\partial \\mu _ { 2 j } } { \\partial \\mu _ { 2 j - 1 } } \\right ) ^ { 2 } \\right ] ^ { \\frac { 1 } { 2 } } \\left [ 2 + \\frac { 1 } { 2 } \\left ( \\frac { \\partial \\mu _ { 2 j } } { \\partial \\sigma _ { 2 j - 1 } } \\right ) ^ { 2 } \\right ] ^ { \\frac { 1 } { 2 } } } , \\end{align*}"}
-{"id": "5247.png", "formula": "\\begin{align*} \\mu ( \\alpha ) = \\mu ( U ( 0 ) , E ^ s ( 0 , \\tau ) ) = 0 . \\end{align*}"}
-{"id": "5697.png", "formula": "\\begin{align*} \\overline { { v } } = z ^ + \\quad \\overline { { v } } = z ^ - . \\end{align*}"}
-{"id": "8519.png", "formula": "\\begin{align*} & c _ \\alpha ^ \\top \\alpha + c _ \\beta ^ \\top \\beta \\to \\max \\\\ & \\begin{cases} \\alpha \\leq b _ { 1 : k } \\\\ A \\alpha + B \\beta \\leq b _ { k + 1 : k + s } \\\\ \\bar A \\alpha + \\bar B \\beta \\leq b _ { d } \\\\ \\alpha \\in \\mathbb { Z } ^ k , \\ , \\beta \\in \\mathbb { Z } ^ s . \\\\ \\end{cases} \\\\ \\end{align*}"}
-{"id": "3730.png", "formula": "\\begin{align*} \\gamma _ l = \\sum _ { j = 0 } ^ { \\beta _ l - 1 } \\mu _ { k _ l + j } ( p _ { m _ { k _ l + j } } , p ' _ { m _ { k _ l + j } } ) . \\end{align*}"}
-{"id": "8778.png", "formula": "\\begin{align*} \\chi _ { G _ 1 } ( B ) = \\dfrac { n _ 1 } { x - \\dfrac { 1 } { r _ 1 + 1 } } ; \\ ; \\ ; \\chi _ { G _ 2 } ( C ) = \\dfrac { n _ 2 } { x - \\dfrac { 1 } { r _ 2 + 1 } } . \\end{align*}"}
-{"id": "3168.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial u } \\left ( \\Delta _ t ( u ) \\right ) = \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { \\infty } \\frac { \\partial } { \\partial u } \\left ( e ^ { z \\psi ( s , u ) } - 1 \\right ) \\nu ( \\mathrm { d } z ) \\mathrm { d } s , u \\in ( - \\infty , - 1 ] . \\end{align*}"}
-{"id": "4942.png", "formula": "\\begin{align*} p ^ x + p ^ y = z ^ 2 \\end{align*}"}
-{"id": "189.png", "formula": "\\begin{align*} \\pi \\cdot ( \\sigma + \\varepsilon ) = ( \\pi \\cdot \\sigma ) + ( \\pi \\cdot \\varepsilon ) \\quad ( \\sigma + \\varepsilon ) \\cdot \\pi = ( \\sigma \\cdot \\pi ) + ( \\varepsilon \\cdot \\pi ) . \\end{align*}"}
-{"id": "8393.png", "formula": "\\begin{align*} C _ 0 < \\left \\{ \\begin{aligned} & \\frac { 1 } { n - 1 } \\quad \\quad \\ , & & n \\geq 4 \\\\ [ 5 p t ] & \\frac { 4 } { 3 n } \\quad \\quad \\ , & & n = 2 , 3 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "4982.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\kappa } \\| \\partial _ i ( f - F ) \\| _ { \\dot { F } ^ { \\alpha - 1 , p } _ q } \\leq \\delta \\| f \\| _ { \\dot { F } ^ { \\alpha , p } _ q } + D _ { \\delta } \\| f \\| _ { \\dot { F } ^ { \\alpha , p } _ q } ^ 2 , \\end{align*}"}
-{"id": "1578.png", "formula": "\\begin{align*} v ( m , \\mathbf { j } ' ) - v ( m , \\mathbf { j } ) & = ( n - i ) p - ( n - i + 1 ) r + s ( j _ i ) - s ( j _ i - r ) + s ( j _ { i + 1 } ) - s ( j _ { i + 1 } + p ) \\\\ & \\geq ( n - i ) p - ( n - i + 1 ) r + 1 - s ( p ) \\\\ & > \\frac { n - i - 1 } { 2 } p - \\frac { n - i + 1 } { 2 } - \\lceil \\log _ 2 ( p ) \\rceil \\\\ & \\geq 0 \\end{align*}"}
-{"id": "5591.png", "formula": "\\begin{align*} \\mu _ { h } = \\alpha + \\frac { \\beta } { 2 } \\left ( p _ { 2 ^ { k } - h } + p _ { 2 ^ { k } - h + 1 } \\right ) \\end{align*}"}
-{"id": "5196.png", "formula": "\\begin{align*} { d } Z _ { t } = \\mu ( Z _ { t } ) { d } t + \\sigma ( Z _ { t } ) { d } W _ { t } , \\end{align*}"}
-{"id": "8667.png", "formula": "\\begin{align*} \\nu _ 1 ( A ) = \\int _ { \\Omega } \\nu _ 0 ( d x ) \\hat { \\pi } ( x , A ) . \\end{align*}"}
-{"id": "4355.png", "formula": "\\begin{align*} L _ \\alpha ^ \\infty : = \\{ f : \\mathbb { C } ^ n \\to \\mathbb { C } ; f \\| f \\| _ { \\infty , \\alpha } < \\infty \\} \\end{align*}"}
-{"id": "4761.png", "formula": "\\begin{align*} { \\varepsilon } = F ( x ^ 1 , Y , Z ) \\ , \\Delta \\sqrt { - \\det g ^ { i j } } / \\prod _ { \\mu \\neq 1 } { L } _ \\mu , \\end{align*}"}
-{"id": "3853.png", "formula": "\\begin{align*} \\mathcal { A } ( x ^ * ) \\coloneqq \\ & I _ { g + } ( x ^ * , \\lambda ^ * ) \\cup I _ { g 0 } ^ = ( x ^ * , \\lambda ^ * , d ) , \\\\ \\mathcal { B } ( x ^ * , y ^ * ) \\coloneqq \\ & I _ { 0 + } ( x ^ * , y ^ * ) \\cup I _ { 0 1 } ( x ^ * , y ^ * ) , \\\\ M \\coloneqq \\ & | \\mathcal { A } ( x ^ * ) | + p + | \\mathcal { B } ( x ^ * , y ^ * ) | \\end{align*}"}
-{"id": "1683.png", "formula": "\\begin{align*} \\xi \\equiv ( v , Q _ 1 , v , Q _ 3 , \\ldots ) Q _ { 2 i + 1 } = u _ j w _ j 1 \\leq j \\leq N . \\end{align*}"}
-{"id": "2490.png", "formula": "\\begin{align*} \\begin{aligned} 2 \\big < g , \\left ( \\nabla _ { \\xi } \\psi ( \\xi ) , \\sigma \\nabla _ { \\xi } g \\right ) \\big > _ { \\xi } & = \\int \\sigma ^ { i j } \\partial _ { i } \\psi \\partial _ { j } ( g ^ 2 ) d \\xi \\\\ & = - \\int \\nabla _ { \\xi } \\cdot [ \\sigma \\nabla _ { \\xi } \\psi ] g ^ 2 d \\xi \\\\ & = - \\int \\nabla _ { \\xi } \\cdot [ \\lambda _ 1 ( \\xi ) \\nabla _ { \\xi } \\psi ] g ^ 2 d \\xi \\ , , \\end{aligned} \\end{align*}"}
-{"id": "9791.png", "formula": "\\begin{align*} u _ { r s } = - \\varepsilon _ { r s } u _ { \\bar s \\ , \\bar r } - \\sum _ { r < l < s } \\varepsilon _ { l s } u _ { r l } u _ { \\bar s \\ , \\bar l } . \\end{align*}"}
-{"id": "4475.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| \\Phi _ n ( f ) - f \\| = 0 \\end{align*}"}
-{"id": "2493.png", "formula": "\\begin{align*} \\overline { \\varrho _ { j } \\left ( \\left | \\eta \\right | \\right ) } & = \\varrho _ { j } \\left ( - \\left | \\eta \\right | \\right ) , j = 0 , \\dots , 4 , \\\\ \\overline { \\psi _ { j } \\left ( \\left | \\eta \\right | \\right ) } & = \\psi _ { j } \\left ( - \\left | \\eta \\right | \\right ) , j = 0 , \\dots , 4 . \\end{align*}"}
-{"id": "7086.png", "formula": "\\begin{align*} \\psi ( q ) & : = f ( q , q ^ 3 ) = \\sum _ { n = 0 } ^ { \\infty } q ^ { n ( n + 1 ) / 2 } . \\end{align*}"}
-{"id": "6449.png", "formula": "\\begin{align*} d s ^ { 2 } = \\frac { 3 + \\rho } { ( 1 + \\rho ) \\sigma ^ { 2 } } d \\mu ^ { 2 } + \\frac { 6 } { \\sigma ^ { 2 } } d \\sigma ^ { 2 } \\end{align*}"}
-{"id": "4574.png", "formula": "\\begin{align*} & E _ { 0 , 0 } = [ x ^ - _ { n - 1 , 0 } , \\ldots , [ x ^ - _ { 2 , 0 } , x ^ - _ { 1 , - 1 } ] _ q \\ldots ] _ q K _ 1 \\cdots K _ { n - 1 } \\ , , \\\\ & F _ { 0 , 0 } = ( K _ 1 \\cdots K _ { n - 1 } ) ^ { - 1 } [ \\ldots [ x ^ + _ { 1 , 1 } , x ^ + _ { 2 , 0 } ] _ { q ^ { - 1 } } , \\ldots , x ^ + _ { n - 1 , 0 } ] _ { q ^ { - 1 } } \\ , . \\end{align*}"}
-{"id": "226.png", "formula": "\\begin{align*} C _ { B , p } : = \\sup _ { a \\in \\R ^ d } \\| \\chi _ a B ^ 2 \\chi _ a \\| _ p < \\infty \\end{align*}"}
-{"id": "5724.png", "formula": "\\begin{align*} | F ' ( m ) | \\leq 2 \\| z ' \\| _ { L ^ 2 } \\sqrt { F ( m ) } \\quad F '' ( m ) \\geq 2 \\| z ' \\| _ { L ^ 2 } ^ 2 - \\lambda ^ { - 2 } \\| z '' \\| ^ 2 _ { L ^ 2 } - \\lambda ^ { 2 } F ( m ) = \\| z ' \\| _ { L ^ 2 } ^ 2 - \\lambda ^ 2 F ( m ) . \\end{align*}"}
-{"id": "1747.png", "formula": "\\begin{align*} S _ \\lambda ^ * S _ \\lambda ( f \\sqrt { d \\mu } ) & = S _ \\lambda ^ * ( f \\circ \\sigma ^ \\ell \\sqrt { d ( \\mu \\circ \\sigma _ \\lambda ^ { - 1 } ) } ) = f \\circ \\sigma ^ \\ell \\circ \\sigma _ \\lambda \\sqrt { d ( \\mu \\circ \\sigma _ \\lambda ^ { - 1 } \\circ \\sigma _ \\lambda ) } \\\\ & = ( \\chi _ { Z ( s ( \\lambda ) ) } \\cdot f ) \\sqrt { d \\mu } \\\\ & = S _ { s ( \\lambda ) } ( f \\sqrt { d \\mu } ) , \\end{align*}"}
-{"id": "6150.png", "formula": "\\begin{align*} \\tau _ i = - \\sum _ { j = 1 } ^ 3 E _ j ( \\dd Q Q ^ { - 1 } ) _ { i j } , \\ ; \\ ; i = 1 , 2 , 3 \\end{align*}"}
-{"id": "1225.png", "formula": "\\begin{align*} c ( 1 - u ( z ) ) \\geq \\delta \\ , ( 1 - | \\hat { w } - z | ) \\ , = \\ , \\delta \\ , d ( z , \\partial B ( \\hat { w } , 1 ) ) \\end{align*}"}
-{"id": "10100.png", "formula": "\\begin{align*} ^ { \\prime } { \\tilde { P } } ( X , Y , Z , U ) = ^ { \\prime } { P } ( X , Y , Z , U ) , \\end{align*}"}
-{"id": "4006.png", "formula": "\\begin{align*} q ( u ) = \\exp _ { A } ( u - K _ { p } ( u ) + \\log _ { A } p ) \\end{align*}"}
-{"id": "8613.png", "formula": "\\begin{align*} \\partial _ t \\bar { u } = \\frac 1 2 \\Delta \\bar { u } , \\ \\ \\bar { u } ( 0 , x ) = u _ 0 ( x ) , \\end{align*}"}
-{"id": "895.png", "formula": "\\begin{align*} F _ n ( \\theta ) = \\sqrt { n } ( U _ n ( \\theta ) - E [ U _ n ( \\theta ) ] ) . \\end{align*}"}
-{"id": "2762.png", "formula": "\\begin{align*} w ( Q ) = \\max \\limits _ { 1 \\leq i \\leq n } \\{ z _ i \\} . \\end{align*}"}
-{"id": "1777.png", "formula": "\\begin{align*} E ( i ) ( ( 0 , n ) ) = \\lambda , \\ ; \\ ; \\lambda \\ ; \\ ; \\Lambda ^ n \\ ; \\ ; i \\ ; \\in \\ : K _ { \\lambda } . \\end{align*}"}
-{"id": "4024.png", "formula": "\\begin{align*} A ' ( 0 ) = \\int _ D g ( \\lambda _ 1 + \\lambda _ 2 ) \\omega = \\int _ D 2 g H \\omega , \\end{align*}"}
-{"id": "9840.png", "formula": "\\begin{align*} P ( T ) = - P \\left ( \\frac { 1 } { q T } \\right ) q ^ g T ^ { 2 g } . \\end{align*}"}
-{"id": "9971.png", "formula": "\\begin{align*} \\bar { \\upsilon } _ { i } ^ { p } = \\dfrac { 1 } { 1 + { \\displaystyle \\sum _ { j \\neq i } ^ { L } } \\kappa _ { j } \\mathbb { E } \\left [ \\mathsf { \\bar { L } } _ { j i } \\right ] + \\sigma ^ { 2 } / P _ u } \\end{align*}"}
-{"id": "9677.png", "formula": "\\begin{align*} C _ j ( U _ a ) : \\ , \\ , d p = 0 , v d u - u d v = 0 . \\end{align*}"}
-{"id": "2653.png", "formula": "\\begin{align*} \\tilde \\Psi _ \\ast ( H ^ 1 ( C ; \\mathbb Q ) ) = L . \\end{align*}"}
-{"id": "10139.png", "formula": "\\begin{align*} \\bar { \\boldsymbol R } _ k ( 0 ) \\boldsymbol { \\bar { \\omega } } _ k ( 1 ) & = \\boldsymbol S _ { D _ k } ^ H ( 0 ) \\boldsymbol R _ k ( 0 ) \\boldsymbol S _ { D _ k } ( 0 ) \\boldsymbol { \\bar { \\omega } } _ k ( 1 ) \\\\ & = \\boldsymbol S _ { D _ k } ( 0 ) { \\boldsymbol p } _ k ( 0 ) = \\bar { \\boldsymbol p } _ k ( 0 ) , \\end{align*}"}
-{"id": "951.png", "formula": "\\begin{align*} E [ g ( \\Phi _ \\beta ( Z ) ) ] \\leq E [ 1 _ { A ^ { 4 \\varepsilon } } ( \\Phi _ \\beta ( Z ) ) ] \\leq E [ 1 _ { A ^ { 5 \\varepsilon } } ( Z _ \\vee ) ] = P ( Z _ \\vee \\in A ^ { 5 \\varepsilon } ) , \\end{align*}"}
-{"id": "1816.png", "formula": "\\begin{align*} \\lim _ { H \\downarrow 0 } \\frac { \\langle \\sigma _ 0 \\rangle _ { 1 , H } } { H ^ { 1 / 1 5 } } = B . \\end{align*}"}
-{"id": "9152.png", "formula": "\\begin{align*} 5 f ( x ^ { 3 } ) + 2 g ( x ^ { 3 } ) + 3 x g ( x ^ { 2 } ) + 3 x ^ { 2 } h ( x ) = 0 . \\end{align*}"}
-{"id": "5287.png", "formula": "\\begin{align*} \\int _ { 0 } ^ 1 u _ t u x & - \\mu \\int _ { 0 } ^ 1 u _ { x x } u x = \\int _ { 0 } ^ 1 f ( t , x , u , u _ x ) u x \\\\ \\leq & \\int _ { 0 } ^ 1 \\big ( ( | d _ 2 ( t ) | + M _ 2 | u _ x | ) | u | + M _ 1 u ^ 2 \\big ) x : = I _ 1 , \\end{align*}"}
-{"id": "3228.png", "formula": "\\begin{align*} \\langle \\Delta _ V \\rangle = \\langle \\Delta _ V \\rangle \\cup u \\cup u ^ \\ast . \\end{align*}"}
-{"id": "7130.png", "formula": "\\begin{align*} h _ { i j } ^ X = & \\frac { h _ { i j } ^ Y } { \\sqrt { ( 1 + | Y | ^ 2 ) ( 1 + \\langle N , Y \\rangle ^ 2 ) } } , \\end{align*}"}
-{"id": "8476.png", "formula": "\\begin{align*} \\Delta = \\Delta ( v ) = 1 - 4 b v . \\end{align*}"}
-{"id": "7301.png", "formula": "\\begin{align*} L _ n = \\sum _ { j = 1 } ^ n ( a _ { n , j } - \\bar { a } _ n ) T _ { n , j } , \\bar { a } _ n = \\frac { 1 } { n } \\sum _ { j = 1 } ^ n a _ { n , j } ; \\end{align*}"}
-{"id": "5781.png", "formula": "\\begin{align*} \\langle \\Delta _ { p } u , \\varphi \\rangle = \\langle \\nabla \\cdot \\left ( \\vert \\nabla u \\vert ^ { p - 2 } \\nabla u \\right ) , \\varphi \\rangle = - \\int _ { \\Omega } \\vert \\nabla u \\vert ^ { p - 2 } \\nabla u \\cdot \\nabla \\varphi \\ ; d x . \\end{align*}"}
-{"id": "1756.png", "formula": "\\begin{align*} | g | \\sqrt { d \\zeta } = \\sqrt { h } \\sqrt { d \\mu } , \\ ; \\ ; \\ ; \\ ; | g | \\ , g \\sqrt { d \\zeta } = g \\ , \\sqrt { h } \\ , \\sqrt { d \\mu } . \\end{align*}"}
-{"id": "8011.png", "formula": "\\begin{align*} & \\frac { 1 } { | Q | } \\int _ Q { \\big | \\mathfrak { S } _ { [ a ] } ^ { n e a r } f ( x ) \\big | } d x \\lesssim \\frac { 1 } { | Q | } \\int _ Q { \\sum _ { k = 0 } ^ { \\infty } { 2 ^ { k m } \\mathfrak { M } _ { \\sigma , 2 ^ k } \\Pi ^ * _ k f ( x ) } } d x \\\\ & \\lesssim \\sup _ { 0 \\leq k \\leq 2 } { \\big \\Vert 2 ^ { k m } \\mathfrak { M } _ { \\sigma , 2 ^ k } \\Pi ^ * _ k f \\big \\Vert _ { L ^ { \\infty } } } + \\frac { 1 } { | Q | } \\int _ Q { \\sum _ { k = 3 } ^ { \\infty } { 2 ^ { k m } \\mathfrak { M } _ { \\sigma , 2 ^ k } \\Pi ^ * _ k f ( x ) } } d x \\end{align*}"}
-{"id": "8727.png", "formula": "\\begin{align*} u ^ \\varepsilon ( t ' , \\hat { x } ) - \\bar { u } ^ \\varepsilon ( t ' - \\tau , \\hat { x } ) & = u ^ { \\varepsilon , \\delta } ( \\hat { t } , \\hat { x } ) + \\delta ^ { - 1 } | \\hat { t } - t ' | ^ 2 - u ^ \\varepsilon ( 0 , \\hat { x } ) \\\\ & \\ge u ^ { \\varepsilon , \\delta } ( \\hat { t } , \\hat { x } ) - u ^ { \\varepsilon , \\delta } ( 0 , \\hat { x } ) . \\end{align*}"}
-{"id": "204.png", "formula": "\\begin{align*} \\mathcal { C } ^ { p } = \\Bigl \\{ \\bigcap \\{ C \\in \\mathcal C : x \\in C \\} : x \\in X \\Bigr \\} . \\end{align*}"}
-{"id": "2859.png", "formula": "\\begin{align*} \\bar { \\Delta } V _ { n } & = \\sum _ { v = 1 } ^ { n } \\hat { a } _ { n v } a _ { v } \\frac { P _ { v } \\lambda _ { v } } { v p _ { v } } . \\end{align*}"}
-{"id": "9006.png", "formula": "\\begin{align*} | C \\cup N _ G ( w ) | & = | C | + | N _ G ( w ) | \\geq \\frac { | V ( H ) | - | U | } { 2 } + | N _ G ( w ) | \\\\ & \\geq \\frac { | V ( H ) | + | N _ G ( w ) | + 1 } { 2 } = \\frac { | V ( G ) | } { 2 } , \\end{align*}"}
-{"id": "9850.png", "formula": "\\begin{align*} \\int _ { y - t } ^ { y + t } \\hat { D } _ { t } ( u ) d u = \\left \\{ \\begin{array} { l l } 2 t - | y | & \\mbox { i f $ | y | \\leq 2 t $ } \\\\ $ 0 $ & \\mbox { i f $ | y | > 2 t $ } \\end{array} \\right . = 2 t \\hat { K } _ { 2 t } ( y ) . \\end{align*}"}
-{"id": "3865.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ m \\lambda _ i ^ k g _ i ( x ^ k ) + \\sum \\limits _ { i = 1 } ^ p \\mu _ i ^ k h _ i ( x ^ k ) + \\sum \\limits _ { i = 1 } ^ n \\gamma _ i ^ k x _ i ^ k = 0 \\end{align*}"}
-{"id": "8027.png", "formula": "\\begin{align*} \\Big [ \\int _ 0 ^ 1 \\int _ { \\Omega } \\sup _ { R \\in \\mathcal { D } _ { \\mu } } \\Big ( \\frac { 1 } { | R | } \\int _ { R } \\sum _ { k = \\mu } ^ { \\infty } { 2 ^ { s k q } \\big | \\Pi _ k f _ L ^ { \\omega , t , \\mu } ( x ) \\big | ^ q } d x \\Big ) d \\lambda d t \\Big ] ^ { 1 / q } \\lesssim 1 \\end{align*}"}
-{"id": "6427.png", "formula": "\\begin{align*} d \\varphi ^ { a } \\overset { } { = } X _ { \\mu } ^ { a } d \\theta ^ { \\mu } X _ { \\mu } ^ { a } \\overset { } { = } \\frac { \\partial \\varphi ^ { a } } { \\partial \\theta ^ { \\mu } } \\end{align*}"}
-{"id": "6101.png", "formula": "\\begin{align*} D V _ { p _ i } = \\left ( \\begin{smallmatrix} 0 & 1 \\\\ \\pm ( ( 2 n - 1 ) + c ^ 2 ) & - ( 2 n - 2 ) \\end{smallmatrix} \\right ) , \\end{align*}"}
-{"id": "6246.png", "formula": "\\begin{align*} \\left ( \\mu _ { \\rm b } { \\bf I } _ { N _ { \\rm T } } + { \\bf Q } _ { \\rm s } - \\gamma _ { \\rm b } { \\bf Q } _ { \\rm n } \\right ) \\bar { \\bf H } _ { \\rm b } { \\boldsymbol \\Psi } _ { \\rm b } \\bar { \\bf H } _ { \\rm b } ^ H = \\mu _ { \\rm b } \\left [ { \\bf 0 } _ { N _ { \\rm T } } \\ , \\ , \\hat { \\bf h } _ { \\rm b } \\right ] { \\boldsymbol \\Psi } _ { \\rm b } \\bar { \\bf H } _ { \\rm b } ^ H . \\end{align*}"}
-{"id": "9431.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { \\psi ( n ) } { | { n } | _ { \\mathcal { D } _ 1 } | { n } | _ { \\mathcal { D } _ 2 } \\cdots | { n } | _ { \\mathcal { D } _ m } } = \\infty . \\end{align*}"}
-{"id": "8800.png", "formula": "\\begin{align*} G _ { \\varpi _ { p } } ( z , m ) = 0 , \\end{align*}"}
-{"id": "2827.png", "formula": "\\begin{align*} \\sum _ { u \\in V _ { n + m + 1 , j } } r _ { v u } ^ { ( n + m + 1 ) } = 1 . \\end{align*}"}
-{"id": "7991.png", "formula": "\\begin{align*} p | A | | V ( G ) \\setminus A | \\geq 3 p | A | n / 4 = 3 n / 1 6 c . \\end{align*}"}
-{"id": "3500.png", "formula": "\\begin{align*} X ( t _ i ) \\in \\mathbb { S } , \\ \\ i = 0 , 1 , \\cdots , n , \\end{align*}"}
-{"id": "3377.png", "formula": "\\begin{align*} \\nabla _ { X } ^ { \\Sigma ^ { a d \\mathbb { C } } } \\varphi = - \\frac { 1 } { 2 } \\sum _ { i = 1 } e _ { i } \\cdot B ( X , e _ { i } ) \\cdot \\varphi + \\frac { 1 } { 2 } i ~ A ^ { l } ( X ) \\cdot \\varphi , ~ ~ \\forall X \\in T M . \\end{align*}"}
-{"id": "3281.png", "formula": "\\begin{align*} \\ ! \\ ! \\ ! \\ ! F _ { t ^ c } ( t _ 0 ) \\ ! = \\ ! \\frac { 1 } { \\pi - \\theta _ k } \\ ! \\bigg ( \\ ! \\ ! \\arccos \\left ( \\frac { r _ u ^ { } } { v _ u t _ 0 } \\right ) \\ ! + \\ ! \\arccos \\left ( \\frac { r _ u ^ { } } { r _ { u , k } ( \\boldsymbol { x } ) } \\right ) \\ ! \\ ! \\bigg ) . \\end{align*}"}
-{"id": "10022.png", "formula": "\\begin{align*} W _ Q = \\begin{pmatrix} Q \\alpha & \\beta \\\\ D \\gamma & Q \\delta \\end{pmatrix} = R _ Q \\begin{pmatrix} Q \\\\ & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "8580.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { { \\bf R } ^ N } \\mu _ n ^ \\pm ( x ) \\psi ( x ) \\ , d x = \\int _ { { \\bf R } ^ N } \\psi ( x ) \\ , d \\mu ^ \\pm ( y ) \\quad \\mbox { f o r } \\quad \\psi \\in C _ 0 ( { \\bf R } ^ N ) . \\end{align*}"}
-{"id": "10042.png", "formula": "\\begin{align*} [ \\widehat { \\theta } ( g ) : \\mathcal { Y } _ { \\mathrm { s m } , h } ] = - \\deg _ \\C ( \\mathcal { Y } _ { \\mathrm { s m } , h } ) \\cdot \\frac { d } { d s } L ( \\tilde { g } , \\theta _ { \\Lambda _ h } , s ) \\big | _ { s = 0 } . \\end{align*}"}
-{"id": "2343.png", "formula": "\\begin{align*} \\sum _ { j \\in J _ p } t _ j \\leq \\ell ( c _ h ) = \\ell ( h ) , \\end{align*}"}
-{"id": "8692.png", "formula": "\\begin{align*} X _ a ( z ) : = \\frac { f ( 1 - a z ) } { \\sqrt { v ( 1 - a z ) } } , z \\in Q _ R . \\end{align*}"}
-{"id": "2778.png", "formula": "\\begin{align*} P ( t , S ) & = K - S , S = \\mathcal { B } ( t ) , 0 \\leq t < T , \\\\ \\frac { \\partial P } { \\partial S } ( t , S ) & = - 1 , S = \\mathcal { B } ( t ) , 0 \\leq t < T , \\\\ P ( T , S ) & = \\max \\{ 0 , K - S \\} , \\lim _ { S \\rightarrow \\infty } P ( t , S ) = 0 , \\end{align*}"}
-{"id": "7787.png", "formula": "\\begin{align*} \\Vert F - B x _ h ( t ) \\Vert _ { Y _ h ^ * } ^ 2 = a ( y _ h ( t ) , y _ h ( t ) ) = F ( y _ h ( t ) ) - b ( x _ h ( t ) ; y _ h ( t ) ) \\end{align*}"}
-{"id": "8056.png", "formula": "\\begin{align*} X = G ( \\mu ) \\Longleftrightarrow \\int _ { { \\Bbb P } } \\log X ^ { - 1 / 2 } A X ^ { - 1 / 2 } \\ , d \\mu ( A ) = 0 . \\end{align*}"}
-{"id": "10149.png", "formula": "\\begin{align*} \\boldsymbol { \\omega } _ k ^ H ( i ) \\boldsymbol { \\omega } _ k ( i - 1 ) = \\boldsymbol { \\omega } _ k ^ H ( i - 1 ) \\boldsymbol { \\omega } _ k ( i - 1 ) . \\end{align*}"}
-{"id": "9180.png", "formula": "\\begin{align*} \\theta _ { v } ( \\omega _ { \\phi } ) = v ^ i { \\frac { \\partial \\psi _ \\phi } { \\partial x ^ i } } + c _ v , \\end{align*}"}
-{"id": "9831.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ n A _ i x ^ { n - i } y ^ i = - \\frac { 1 } { q ^ { n / 2 } } \\sum _ { i = 0 } ^ n A _ i ( x + ( q - 1 ) y ) ^ { n - i } ( x - y ) ^ i \\end{align*}"}
-{"id": "307.png", "formula": "\\begin{align*} \\phi ( B ) = \\frac { \\lambda _ d \\left ( B \\cap K \\right ) } { \\lambda _ d \\left ( K \\right ) } . \\end{align*}"}
-{"id": "4598.png", "formula": "\\begin{align*} [ \\Lambda _ { k , \\frac { \\pi } { 2 } } , \\Lambda _ { k , \\frac { \\pi } { 2 } } ] & = \\{ ( 0 , r , s _ 1 , t _ 1 , \\ldots , s _ n , t _ n ) \\in \\Lambda _ { k , \\frac { \\pi } { 2 } } \\ , : r \\in \\Z , \\ ; s _ j = t _ j = 0 \\ , ( j = 1 , \\ldots , c ) ; \\\\ & s _ j , t _ j \\in 2 \\Z \\ , ( j = c + 1 , \\ldots , c + d ) ; \\ ; s _ j + t _ j \\in 2 \\Z \\ , ( j > c + d ) \\} , \\end{align*}"}
-{"id": "9154.png", "formula": "\\begin{align*} f _ { 3 } = D _ { 2 } , f _ { 2 } = - D _ { 1 } - 3 D _ { 2 } , f _ { 1 } = 2 D _ { 1 } + 3 D _ { 2 } , \\end{align*}"}
-{"id": "2783.png", "formula": "\\begin{align*} P ( t , S ) = p ^ { E } ( t , S ) + & \\int _ { 0 } ^ { t } r K e ^ { - r ( t - \\xi ) } \\aleph ( - d _ { 2 } ( S , t - \\xi , \\mathcal { B } ( \\xi ) ) ) \\mathrm { d } \\xi \\\\ - & \\int _ { 0 } ^ { t } \\delta S e ^ { - \\delta ( t - \\xi ) } \\aleph ( - d _ { 1 } ( S , t - \\xi , \\mathcal { B } ( \\xi ) ) ) \\mathrm { d } \\xi , \\end{align*}"}
-{"id": "5488.png", "formula": "\\begin{align*} v _ z ( s ) = \\frac { \\dd } { \\dd s } ( s ^ \\rho p ( s z ) ) = s ^ { \\rho - 1 } p _ 0 ( s z ) . \\end{align*}"}
-{"id": "6112.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( e ^ { - s a _ { n } W _ { 1 } } \\right ) & = \\mathbb { E } \\left [ e ^ { - U _ { 1 } \\left ( s a _ { n } b + \\int _ { 0 } ^ { \\infty } \\left ( 1 - e ^ { - s a _ { n } y } \\right ) \\mu \\left ( d y \\right ) \\right ) } \\right ] \\\\ & = \\mathbb { E } \\left [ e ^ { - n ^ { - 1 } U _ { 1 } \\left ( s a _ { n } n b + \\int _ { 0 } ^ { \\infty } \\left ( 1 - e ^ { - s y } \\right ) n \\mu \\left ( a _ { n } ^ { - 1 } d y \\right ) \\right ) } \\right ] , \\end{align*}"}
-{"id": "2897.png", "formula": "\\begin{align*} \\lim _ { s _ 1 } \\cdots \\lim _ { s _ { n - 1 } } ~ c ( k , s _ 1 , \\ldots , s _ { n - 1 } ) = i . \\end{align*}"}
-{"id": "7121.png", "formula": "\\begin{align*} X ~ = ~ \\frac { ( 1 , Y ) } { \\sqrt { 1 - | Y | ^ 2 } } . \\end{align*}"}
-{"id": "8022.png", "formula": "\\begin{align*} \\Big ( \\int _ { 0 } ^ 1 { \\int _ { \\Omega } { \\Big \\Vert \\Big ( \\sum _ { k = 1 } ^ { L } { ( \\mathfrak { M } _ { { \\sigma } , 2 ^ { t _ k } } h _ { k } ^ { \\omega , t } ) ^ q } \\Big ) ^ { { 1 } / { q } } \\Big \\Vert _ { L ^ p } ^ p } d \\lambda } d t \\Big ) ^ { { 1 } / { p } } \\lesssim _ { p , q } 1 \\end{align*}"}
-{"id": "558.png", "formula": "\\begin{align*} \\Delta \\rho = \\left \\| \\frac { \\partial u } { \\partial s } \\right \\| _ { J _ t } ^ 2 + \\rho \\ , h '' ( \\rho ) \\ , \\frac { \\partial \\rho } { \\partial s } \\end{align*}"}
-{"id": "3370.png", "formula": "\\begin{align*} \\Sigma ^ { \\mathbb { C } } Q \\mid _ M \\simeq \\Sigma ^ { \\mathbb { C } } M \\hat { \\otimes } \\Sigma ^ { \\mathbb { C } } \\nu ( M ) = : \\Sigma ^ { a d \\mathbb { C } } . \\end{align*}"}
-{"id": "1709.png", "formula": "\\begin{align*} P ( Z ( \\lambda ) ) & = \\sum _ { \\zeta \\in r ( \\lambda ) \\Lambda ^ m } \\sum _ { ( \\rho , \\xi ) \\in \\Lambda ^ { \\operatorname { m i n } } ( \\lambda , \\zeta ) } P ( Z ( \\lambda \\rho ) ) \\\\ & = \\sum _ { \\eta \\in s ( \\lambda ) \\Lambda ^ { m - d ( \\lambda ) } } P ( Z ( \\lambda \\eta ) ) \\\\ & = \\sum _ { i = 1 } ^ p \\sum _ { \\alpha _ { i j } \\in s ( \\eta _ i ) \\Lambda ^ { m - d ( \\eta _ i ) } } P ( Z ( \\eta _ i \\alpha _ { i j } ) ) \\\\ & = \\sum _ { i = 1 } ^ p P ( Z ( \\eta _ i ) ) , \\end{align*}"}
-{"id": "7488.png", "formula": "\\begin{align*} < \\Psi , \\Phi > = \\dfrac { 1 } { p ! q ! } \\psi _ { A _ p \\bar { B } _ q } \\overline { \\phi ^ { \\bar { A } _ p B _ q } } = \\sum \\psi _ { A _ p \\bar { B } _ q } \\overline { \\phi ^ { \\bar { A } _ p B _ q } } , \\end{align*}"}
-{"id": "4575.png", "formula": "\\begin{align*} \\bar u = ( - 1 ) ^ n q ^ { - \\sum _ { r = 0 } ^ { n - 1 } \\lambda ^ 0 _ { r , n - 1 } + n - 2 } u \\ , . \\end{align*}"}
-{"id": "1945.png", "formula": "\\begin{align*} \\lVert P _ A \\rVert _ { L ^ p _ \\alpha } & \\le \\delta ^ { 1 / 2 } \\int _ { | 1 - r | < \\delta } \\int ^ { 1 } _ { 0 } | 1 - r | ^ { - 1 / 2 } \\lVert \\partial _ u F _ { u } \\big | _ { u = u ( t ) } \\rVert _ { L ^ p _ { \\tau } } \\ , d t \\ , d r \\\\ & \\le 4 \\delta M . \\end{align*}"}
-{"id": "504.png", "formula": "\\begin{align*} N ( J , t ) = \\left \\{ X ^ { \\boldsymbol { b } } \\ ; | \\ ; \\epsilon _ j \\leq b _ j \\leq N _ j - 1 , \\ ; 1 \\leq j \\leq m , \\ ; \\mathrm { a n d } \\ ; \\prod _ { j = 1 } ^ m \\left ( b _ j + 1 - \\epsilon _ j \\right ) < t \\right \\} , \\end{align*}"}
-{"id": "5083.png", "formula": "\\begin{align*} & [ ( b _ 1 - b _ 2 ) ^ 2 - 2 ] R _ { 1 2 1 2 } + [ ( b _ 2 - b _ 3 ) ^ 2 - 2 ] R _ { 2 3 2 3 } + [ ( b _ 1 - b _ 3 ) ^ 2 - 2 ] R _ { 1 3 1 3 } \\\\ & = \\frac { 2 } { 9 } - \\frac { 1 0 } { 3 } t r ( A ) + 3 [ b _ 1 ^ 2 a _ 1 + b _ 2 ^ 2 a _ 2 + b _ 3 ^ 2 a _ 3 ] . \\end{align*}"}
-{"id": "10135.png", "formula": "\\begin{align*} \\bar { \\boldsymbol \\omega } _ k ( i ) \\in \\arg \\min _ { \\bar { \\boldsymbol \\omega } _ k ^ { \\rm o p t } \\in \\bar { \\underline { \\boldsymbol \\omega } } _ k ( i ) } { D } _ k \\bigg ( \\boldsymbol S _ { D _ k } ( i ) , \\bar { \\boldsymbol \\omega } _ k ^ { \\rm o p t } \\bigg ) \\ \\ \\ \\ \\textrm { f o r } \\ k = 1 , 2 , \\ldots , N \\end{align*}"}
-{"id": "4415.png", "formula": "\\begin{align*} \\langle h _ { r a n } ( x ' ) \\rangle = 0 , \\end{align*}"}
-{"id": "5755.png", "formula": "\\begin{align*} \\Pi _ { \\psi _ P ( q ) } ( \\psi _ P ( \\Gamma _ i ) ) \\leq c _ { n - 1 } = \\int _ { \\psi _ P ( \\Gamma _ i ) } | G K _ { \\psi _ P ( \\Gamma _ i ) } | d V _ { \\psi _ P ( \\Gamma _ i ) } , \\end{align*}"}
-{"id": "1780.png", "formula": "\\begin{align*} \\mathcal { G } _ \\lambda ( u ) : = \\{ z u ( \\cdot + y ) \\mid ( z , y ) \\in S ^ 1 \\times \\mathbb { R } \\} . \\end{align*}"}
-{"id": "6215.png", "formula": "\\begin{align*} \\vartheta ( u ) = \\P ( u + X ( t ) < 0 ~ , ~ ~ \\textrm { f o r s o m e $ t > 0 $ } ) ~ . \\end{align*}"}
-{"id": "4316.png", "formula": "\\begin{align*} \\nabla F _ { 0 } ( s , r ) \\cdot ( s - S , r - R ) ^ { \\top } & = ( s - S ) \\log s + ( r - R ) \\log r \\\\ & + ( ( R + S ) - ( r + s ) ) \\log ( 1 - ( r + s ) ) . \\end{align*}"}
-{"id": "6532.png", "formula": "\\begin{align*} ( \\alpha e _ n + e _ n ^ { \\perp } ) \\cap \\bigcap _ { i = 1 } ^ k \\{ x \\in \\mathbb { R } ^ n : \\langle x , y _ i \\rangle \\leq 1 \\} \\end{align*}"}
-{"id": "9011.png", "formula": "\\begin{align*} X ( t ) \\leq & e ^ { \\int _ { 0 } ^ { t } { \\big ( m ( s ) + C _ { 3 } n ( s ) \\big ) \\ , d s } } \\Big ( X ( 0 ) + \\int _ { 0 } ^ { t } { \\{ ( C _ { 1 } , \\alpha , \\beta , \\gamma ) + l ( s ) + f ( s ) \\} \\ , d s } \\Big ) \\\\ : = & C ( C _ { 1 } , l , m , n , f , \\alpha , \\beta , \\gamma , K , t ) . \\end{align*}"}
-{"id": "9301.png", "formula": "\\begin{align*} \\big ( g ( a _ 1 ' ) \\otimes g ( b ) \\big ) \\oplus g ( a _ 1 ' ) = \\big ( g ( a _ 2 ) \\otimes g ( b ) \\big ) \\oplus g ( a _ 2 ' ) \\end{align*}"}
-{"id": "4100.png", "formula": "\\begin{align*} \\frac { \\mathrm { d e t } ( h _ { i j } ) } { \\mathrm { d e t } ( b _ { i j } ) } = \\frac { h ( V _ 1 , V _ 1 ) h ( V _ 2 , V _ 2 ) } { b ( V _ 1 , V _ 1 ) b ( V _ 2 , V _ 2 ) } = \\frac { \\lambda _ 1 \\lambda _ 2 \\langle d u ^ { - 1 } _ { \\eta ( p ) } V _ 1 , V _ 1 \\rangle \\langle d u ^ { - 1 } _ { \\eta ( p ) } V _ 2 , V _ 2 \\rangle \\langle \\eta , \\xi \\rangle ^ { - 2 } } { \\langle d u ^ { - 1 } _ { \\eta ( p ) } V _ 1 , V _ 1 \\rangle \\langle d u ^ { - 1 } _ { \\eta ( p ) } V _ 2 , V _ 2 \\rangle \\langle \\eta , \\xi \\rangle ^ { - 2 } } = \\lambda _ 1 \\lambda _ 2 , \\end{align*}"}
-{"id": "10066.png", "formula": "\\begin{align*} \\sum _ { m \\ge 0 } c _ 0 ( - m ) \\cdot d ( m ) = 0 . \\end{align*}"}
-{"id": "9694.png", "formula": "\\begin{align*} \\varphi _ k ( \\gamma _ 1 , \\omega _ { k , k + 1 } ) = ( u , v ) \\cdot \\textbf { n } _ k = \\Big ( \\Phi _ { 1 } ^ { ( 1 ) } ( \\gamma _ 1 ; U _ a ) , \\Phi _ { 1 } ^ { ( 2 ) } ( \\gamma _ 1 ; U _ a ) \\Big ) \\cdot \\Big ( - \\sin ( \\omega _ { k , k + 1 } ) , \\cos ( \\omega _ { k , k + 1 } ) \\Big ) , \\end{align*}"}
-{"id": "3885.png", "formula": "\\begin{align*} t _ { j } ( n ) t _ { j } ( m ) = \\sum _ { d | ( m , n ) } t _ { j } \\left ( \\frac { n m } { d ^ 2 } \\right ) . \\end{align*}"}
-{"id": "4917.png", "formula": "\\begin{align*} W = X _ { n , 1 } \\oplus X _ { n , 2 } \\oplus \\cdots \\oplus X _ { n , n } . \\end{align*}"}
-{"id": "3182.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ Z _ { t } ^ { \\kappa } \\right ] = \\int _ { \\mathbb { R } _ { \\geqslant 0 } } z ^ { \\kappa } \\mu _ { Z _ { t } } ( \\mathrm { d } z ) \\leqslant \\int _ { \\mathbb { R } _ { \\geqslant 0 } } ( y + z ) ^ { \\kappa } \\mu _ { Z _ { t } } ( \\mathrm { d } z ) < \\infty . \\end{align*}"}
-{"id": "1316.png", "formula": "\\begin{align*} \\rho ( f ) = \\int f ( z ) d \\mu ( z ) , \\end{align*}"}
-{"id": "3547.png", "formula": "\\begin{align*} Z _ { \\boldsymbol { t } } = 0 \\ \\ \\ \\ \\boldsymbol { t } \\notin \\Lambda . \\end{align*}"}
-{"id": "188.png", "formula": "\\begin{align*} \\bigcap \\mathcal { X } = \\bigcup \\Bigl \\{ \\delta _ { \\mathcal { C } } ( x ) : x \\in \\bigcap \\mathcal { X } \\Bigr \\} \\in \\mathcal { C } ^ { \\mathit { B J } } . \\end{align*}"}
-{"id": "5938.png", "formula": "\\begin{align*} z = ( 1 , 0 , \\beta _ 1 , 0 , \\ldots , 0 , \\beta _ m ) , & & w h e r e ~ \\beta _ s = \\prod _ { i = 1 } ^ s \\left ( \\frac { \\alpha _ { 2 i - 1 } } { \\alpha _ { 2 i } } \\right ) ~ \\mbox { f o r } ~ s = 1 , \\ldots , m . \\end{align*}"}
-{"id": "5202.png", "formula": "\\begin{align*} \\psi _ { 1 } ' ( r _ { * } ) = \\frac { g _ { 1 } ' ( r _ { * } ) - f _ { 1 } ' ( \\ell _ { * } ) } { f _ { 1 } '' ( \\ell _ { * } ) ( r _ { * } - \\ell _ { * } ) } , \\psi _ { 2 } ' ( \\ell _ { * } ) = \\frac { f _ { 2 } ' ( r _ { * } ) - g _ { 2 } ' ( \\ell _ { * } ) } { f _ { 2 } '' ( r _ { * } ) ( r _ { * } - \\ell _ { * } ) } , \\end{align*}"}
-{"id": "7876.png", "formula": "\\begin{align*} | \\nabla ^ \\pm u | ( x , t ) : = \\limsup _ { y \\to x } \\frac { [ u ( y , t ) - u ( x , t ) ] _ \\pm } { d ( x , y ) } \\end{align*}"}
-{"id": "3828.png", "formula": "\\begin{align*} \\hat { p } : = P ( S ^ { 0 , 1 } A '' ) , A '' : = \\{ ( x , t ) \\in \\Z ^ 2 \\colon \\ , t \\ge 1 , x \\ge \\tfrac 1 2 v _ \\circ t \\} . \\end{align*}"}
-{"id": "766.png", "formula": "\\begin{align*} S _ { k } ( x ) = y ^ { n } \\ : , \\end{align*}"}
-{"id": "4780.png", "formula": "\\begin{align*} { \\varepsilon } = F ( X , Y , Z ) \\ , { \\Delta \\sqrt { - \\det g ^ { i j } } } / { ( { L } _ 0 \\ , { L } _ 1 \\ , { L } _ 2 \\ , { L } _ 3 ) } , \\end{align*}"}
-{"id": "9962.png", "formula": "\\begin{align*} y _ n = \\begin{cases} z _ { n - 2 } & n \\\\ z _ { n - 2 } + s _ { \\frac { n } { 2 } - 1 } & n \\\\ \\end{cases} \\end{align*}"}
-{"id": "8834.png", "formula": "\\begin{align*} & N _ 0 ^ { - s _ 0 } N _ 1 ^ { - s _ 1 } N _ 2 ^ { - s _ 2 } N _ 3 ^ { - s _ 3 } \\\\ & \\lesssim \\begin{cases} N _ 1 ^ { - s _ 2 ^ { \\ast } - s _ 3 ^ { \\ast } } N _ 2 ^ { - s _ 1 ^ { \\ast } } N _ 3 ^ { - s _ 0 ^ { \\ast } } , & N _ 0 \\sim N _ 1 \\gg N _ 2 \\ge N _ 3 , \\\\ N _ 1 ^ { - s _ 2 ^ { \\ast } - s _ 3 ^ { \\ast } } \\max ( N _ 0 , N _ 3 ) ^ { - s _ 1 ^ { \\ast } } \\min ( N _ 0 , N _ 3 ) ^ { - s _ 0 ^ { \\ast } } , & N _ 1 \\sim N _ 2 \\gg N _ 0 , N _ 3 , \\end{cases} \\end{align*}"}
-{"id": "9887.png", "formula": "\\begin{align*} \\alpha ( G ) = \\alpha ( H ) < a d _ * + 1 \\ , . \\end{align*}"}
-{"id": "2524.png", "formula": "\\begin{align*} I _ 2 = 0 . \\end{align*}"}
-{"id": "9192.png", "formula": "\\begin{align*} & [ \\Lambda \\otimes E , \\Lambda \\otimes E ] = [ \\Lambda ' \\otimes E ' , \\Lambda ' \\otimes E ' ] = 0 n = 6 ; \\\\ & [ \\Lambda \\otimes E , ( \\Lambda \\otimes E ) \\oplus ( V \\otimes B ) ] = [ \\Lambda ' \\otimes E ' , ( \\Lambda ' \\otimes E ' ) \\oplus ( V ' \\otimes B ' ) ] = 0 n = 5 . \\end{align*}"}
-{"id": "8944.png", "formula": "\\begin{gather*} \\sum _ { w \\in B _ n / B _ { n - 1 } } \\ ! \\ ! \\ ! w \\cdot \\frac { \\prod _ { 1 \\le i \\le 2 n - 2 } \\vartheta ( z _ n - y _ i ) \\prod _ { 1 \\le i \\le n } \\vartheta ( Y + z _ i ) \\prod _ { 1 \\le i < n } \\vartheta ( Y - z _ i ) } { \\vartheta ( z _ n ) \\prod _ { 1 \\le i < n } \\vartheta ( z _ n + z _ i ) \\vartheta ( z _ n - z _ i ) } \\\\ \\hphantom { \\sum _ { w \\in B _ n / B _ { n - 1 } } \\ ! \\ ! \\ ! } { } = \\frac { \\vartheta ( 2 Y ) } { \\vartheta ( Y ) } \\prod _ { 1 \\le i \\le 2 n - 2 } \\vartheta ( Y - y _ i ) . \\end{gather*}"}
-{"id": "580.png", "formula": "\\begin{align*} { \\displaystyle \\int _ { \\gamma } g ( \\zeta ) d \\zeta } = \\lim { \\displaystyle \\int _ { \\gamma } f _ n ' ( \\zeta ) d \\zeta } = 0 , \\end{align*}"}
-{"id": "6129.png", "formula": "\\begin{align*} \\sum _ { k = \\left [ \\log \\left ( i \\right ) \\right ] } ^ { \\left [ \\log \\left ( i c _ { t } \\right ) \\right ] - 1 } \\left | g \\left ( 2 ^ { k } \\right ) - g \\left ( 2 ^ { k + 1 } \\right ) \\right | = o \\left ( L \\left ( i \\right ) i ^ { - \\alpha } \\right ) . \\end{align*}"}
-{"id": "7937.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { \\infty } \\log ( 1 - \\varphi ( \\nu ^ i | V | ) ) \\ge - 2 \\sum _ { i = 0 } ^ { \\infty } \\varphi ( \\nu ^ i | V | ) > - \\infty . \\end{align*}"}
-{"id": "1481.png", "formula": "\\begin{align*} w _ 1 ( 0 ) = 1 , & ~ w _ 2 ( 0 ) = 0 ; \\\\ w _ 1 ' ( 0 ) = 0 , & ~ w _ 2 ' ( 0 ) = 1 . \\end{align*}"}
-{"id": "4440.png", "formula": "\\begin{align*} [ u ] _ \\alpha [ f ] _ \\beta ( T ^ { 1 / 3 } ) ^ { \\alpha } ( \\tau ^ { 1 / 3 } ) ^ { \\beta } \\gtrsim \\begin{cases} \\| \\lceil u , ( \\cdot ) _ T \\rceil f _ \\tau \\| & \\textrm { i f } \\alpha \\in ( 0 , 1 ] , \\\\ \\| \\big ( \\lceil u , ( \\cdot ) _ T \\rceil - \\partial _ 1 u \\lceil x _ 1 , ( \\cdot ) _ T \\rceil \\big ) f _ \\tau \\| & \\textrm { i f } \\alpha \\in ( 1 , \\frac 3 2 ) . \\end{cases} \\end{align*}"}
-{"id": "7748.png", "formula": "\\begin{align*} \\sum ^ { 2 N - 1 } _ { n = 1 } \\frac { \\cos ^ 2 \\frac { n \\pi } { 2 N } } { \\sin ^ 2 \\frac { n \\pi } { 2 N } } & = \\sum ^ { N - 1 } _ { n = 1 } \\frac { \\cos ^ 2 \\frac { n \\pi } { 2 N } } { \\sin ^ 2 \\frac { n \\pi } { 2 N } } + \\frac { \\cos ^ 2 \\frac { N \\pi } { 2 N } } { \\sin ^ 2 \\frac { N \\pi } { 2 N } } + \\sum ^ { 2 N - 1 } _ { n = N + 1 } \\frac { \\cos ^ 2 \\frac { n \\pi } { 2 N } } { \\sin ^ 2 \\frac { n \\pi } { 2 N } } \\\\ & = 2 \\sum ^ { N - 1 } _ { n = 1 } \\frac { \\cos ^ 2 \\frac { n \\pi } { 2 N } } { \\sin ^ 2 \\frac { n \\pi } { 2 N } } \\ , . \\end{align*}"}
-{"id": "1609.png", "formula": "\\begin{align*} A _ i ( v , w ) = | v \\Lambda ^ { e _ i } w | . \\end{align*}"}
-{"id": "6275.png", "formula": "\\begin{align*} [ \\Delta _ q , ( \\nabla \\times F ) \\times ] G = \\Delta _ q ( ( \\nabla \\times F ) \\times G ) - ( \\nabla \\times F ) \\times G _ q . \\end{align*}"}
-{"id": "8267.png", "formula": "\\begin{align*} \\frac { L ' ( s + i r _ 1 - i r _ 2 , \\xi ) } { L ( s + i r _ 1 - i r _ 2 , \\xi ) } \\ll \\begin{cases} \\log ( V ( T _ 0 ) ) & \\abs { t } \\leq 1 0 0 V ( T _ 0 ) , \\\\ \\log ( \\abs { t } ) & \\end{cases} \\end{align*}"}
-{"id": "10158.png", "formula": "\\begin{align*} { \\boldsymbol \\omega _ k } ( i ) & = { \\boldsymbol R } _ k ^ { - 1 } ( i ) { \\boldsymbol p } _ k ( i ) . \\end{align*}"}
-{"id": "6478.png", "formula": "\\begin{align*} p _ { 2 D c } \\left ( x y | \\mu _ { x } \\sigma ; \\rho \\right ) \\overset { } { = } \\dfrac { 1 } { 2 \\pi \\Sigma ^ { 2 } \\sqrt { 1 - \\rho ^ { 2 } } } \\exp \\left \\{ \\frac { - 1 } { 2 \\left ( 1 - \\rho ^ { 2 } \\right ) } \\left [ \\frac { \\left ( x - \\mu _ { x } \\right ) ^ { 2 } } { \\sigma ^ { 2 } } + \\frac { y ^ { 2 } \\sigma ^ { 2 } } { \\Sigma ^ { 4 } } - \\frac { 2 \\rho \\left ( x - \\mu _ { x } \\right ) y } { \\Sigma ^ { 2 } } \\right ] \\right \\} \\end{align*}"}
-{"id": "8234.png", "formula": "\\begin{align*} t _ + \\equiv \\sum _ { \\ell \\in \\mathcal { L } ^ + } \\binom { k } { \\ell } \\biggl ( U _ { i + \\ell } - \\frac { 1 } { 2 } ( U _ { i + \\ell } ) ^ 2 + \\frac { 1 } { 3 } ( U _ { i + \\ell } ) ^ 3 \\cdots \\biggr ) \\end{align*}"}
-{"id": "3157.png", "formula": "\\begin{align*} \\liminf _ { \\alpha \\to \\infty } I ( \\alpha ) & \\geqslant \\int _ { 0 } ^ { \\infty } \\liminf _ { \\alpha \\to \\infty } \\exp \\left \\lbrace \\frac { - \\alpha w } { \\alpha + w } \\right \\rbrace \\frac { w ^ { - \\kappa } } { \\left ( 1 + w / \\alpha \\right ) ^ { 2 } } \\mathrm { d } w \\\\ & = \\int _ { 0 } ^ { \\infty } \\exp \\left \\lbrace - w \\right \\rbrace w ^ { - \\kappa } \\mathrm { d } w = \\Gamma ( 1 - \\kappa ) > 0 . \\end{align*}"}
-{"id": "7460.png", "formula": "\\begin{align*} \\dim A _ { } = 2 7 \\dim A _ { } / [ A _ { } , A _ { } ] = 6 \\end{align*}"}
-{"id": "5057.png", "formula": "\\begin{align*} d C = 0 \\Leftrightarrow \\sum _ k ( B _ { i k } A _ { k j } - B _ { j k } A _ { k i } ) = 0 . \\end{align*}"}
-{"id": "2425.png", "formula": "\\begin{align*} D _ 1 = D _ 1 ' = D _ 3 = 0 , \\ R _ { 1 3 3 } \\neq 0 , \\end{align*}"}
-{"id": "480.png", "formula": "\\begin{align*} p _ { 1 , k _ 1 , k _ 2 } ^ { ( m ) } ( x , t ) = \\frac { ( - 1 ) ^ { k _ 2 } \\pi ^ { k _ 1 + k _ 2 } } { 2 ^ { n - k _ 1 + 1 + \\frac { m - 1 } { 2 } } } | t | ^ { n + k _ 1 - 1 - \\frac { m - 1 } { 2 } } e ^ { - \\frac { 1 } { 4 } d ( x , t ) ^ 2 } e ^ { - \\kappa \\rho ( \\delta ) } \\tilde I _ { n + k _ 1 - 1 } ( \\kappa \\rho ( \\delta ) ) \\left [ 1 + g ( \\abs { x } , | t | ) \\right ] , \\end{align*}"}
-{"id": "4528.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\pi ( \\Delta _ n ) ^ * \\Pi ( a ) \\pi ( \\Delta _ n ) = 0 . \\end{align*}"}
-{"id": "5039.png", "formula": "\\begin{align*} 0 & = x ^ * + \\theta z ^ * - p ^ * ( x ^ * + \\theta z ^ * ) \\frac { x ^ * + \\theta z ^ * } { p ^ * ( x ^ * + \\theta z ^ * ) } \\\\ & = ( x _ 0 ^ * + \\theta z _ 0 ^ * - p ^ * ( x ^ * + \\theta z ^ * ) h _ 0 ^ * ) + \\sum _ { n \\in \\N } ( t _ n + \\theta \\tau _ n - p ^ * ( x ^ * + \\theta z ^ * ) s _ n ) r _ n v _ n ^ * . \\end{align*}"}
-{"id": "6634.png", "formula": "\\begin{align*} [ f _ n - g _ n ] _ { W ^ { 1 , p } , \\varphi _ n ( K _ n ) } = O \\left ( \\sum _ { B \\in \\mathcal { B } _ n } \\left ( \\int _ { B } | D f _ n | ^ p \\ , d \\mu \\right ) ^ { \\frac { 1 } { p } } \\right ) = O ( \\epsilon ^ { \\frac { 1 } { p } } ) \\ . \\end{align*}"}
-{"id": "5130.png", "formula": "\\begin{align*} z _ 1 ( t ) : = S ( t ) \\phi ^ { \\omega } \\end{align*}"}
-{"id": "6376.png", "formula": "\\begin{align*} g _ n ( x ) : = \\exp \\big \\{ { - } n ( W _ 1 ( z _ 1 ^ * + \\tfrac { x } { \\sqrt { n } } ) - W _ 1 ( z _ 1 ^ * ) ) \\big \\} ( z _ 1 ^ * + \\tfrac { x } { \\sqrt { n } } ) ^ { - ( 2 i - 1 ) } 1 _ { \\left \\{ z _ 1 ^ * + \\tfrac { x } { \\sqrt { n } } > 0 \\right \\} } \\end{align*}"}
-{"id": "3087.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { s \\to - \\infty } \\frac { \\langle \\overline { w } ( s ) , e _ 1 \\rangle } { \\| \\overline { w } ( s ) \\| } = 1 , \\quad \\mbox { a n d } \\quad \\lim _ { s \\to - \\infty } \\frac { \\langle \\underline { w } ( s ) , e _ 1 \\rangle } { \\| \\underline { w } ( s ) \\| } = - 1 \\ , , \\end{align*}"}
-{"id": "3380.png", "formula": "\\begin{align*} \\mathcal M _ n = [ p _ { 1 , n } , r ^ 0 _ n ] ^ s \\cup F _ n ^ { - 1 } \\left ( [ p _ { 1 , n } , r ^ 0 _ n ] ^ s \\right ) \\cup F _ n ^ { - 2 } \\left ( [ p _ { 1 , n } , r ^ 0 _ n ] ^ s \\right ) \\end{align*}"}
-{"id": "5133.png", "formula": "\\begin{align*} z _ 3 ( t ) : = - i \\int _ 0 ^ t S ( t - t ' ) | z _ 1 | ^ 2 z _ 1 ( t ' ) d t ' . \\end{align*}"}
-{"id": "4134.png", "formula": "\\begin{align*} \\langle Z , E _ 1 \\rangle = - \\lambda _ 1 \\langle \\eta , \\xi \\rangle \\langle \\eta , E _ 1 \\rangle \\ \\ \\mathrm { a n d } \\\\ \\langle Z , E _ 2 \\rangle = - \\lambda _ 2 \\langle \\eta , \\xi \\rangle \\langle \\eta , E _ 2 \\rangle . \\end{align*}"}
-{"id": "9196.png", "formula": "\\begin{align*} L = ( \\mathfrak { k } \\otimes ( A \\oplus E \\oplus E ' ) ) \\oplus ( \\mathfrak { s } \\otimes ( A \\oplus C \\oplus C ' ) ) \\oplus ( V _ { \\mathfrak { k } } \\otimes ( B \\oplus B ' ) ) \\oplus D ' \\end{align*}"}
-{"id": "4466.png", "formula": "\\begin{align*} \\sum _ { \\stackrel { k ' + k '' = k } { k ' , k '' \\not = 0 } } d ^ { - 4 } ( k ' , 0 ) d ^ { - 1 } ( k '' , 0 ) & \\lesssim d ^ { - 1 } ( k , 0 ) \\end{align*}"}
-{"id": "5947.png", "formula": "\\begin{align*} V _ q & < \\sum _ { \\substack { t \\\\ t \\le L } } \\frac { 1 } { t ! } - ( 1 - \\beta ) \\sum _ { \\substack { t \\\\ t \\le L } } \\frac { 1 } { t ! } + \\gamma \\\\ & = \\sum _ { t = 0 } ^ L \\frac { ( - 1 ) ^ t } { t ! } + \\beta \\sum _ { \\substack { t \\\\ t \\le L } } \\frac { 1 } { t ! } + \\gamma \\\\ & < e ^ { - 1 } + \\alpha + e \\beta + \\gamma \\\\ & < e ^ { - 1 } + \\epsilon . \\end{align*}"}
-{"id": "3823.png", "formula": "\\begin{align*} p _ 0 = \\P _ 0 ( \\Lambda _ \\infty ) = P \\left ( \\bar { X } _ n \\ge \\tfrac 1 2 v _ \\circ n \\ ; \\forall \\ , n \\in \\Z _ + \\right ) > 0 . \\end{align*}"}
-{"id": "7409.png", "formula": "\\begin{align*} s _ i ( t _ j ) = \\left \\{ \\begin{array} { c c } - t _ i & , \\\\ t _ j & . \\end{array} \\right . \\end{align*}"}
-{"id": "1533.png", "formula": "\\begin{gather*} A _ t \\ : = \\ \\left \\{ \\sum _ { i = 1 } ^ k a _ j m ^ { i - 1 } : j \\in [ n ] , k \\in [ t ] \\right \\} . \\end{gather*}"}
-{"id": "8324.png", "formula": "\\begin{align*} V _ { 0 \\Z } = ( I ^ \\perp \\cap h V _ { \\Z } ) / ( I \\cap h V _ { \\Z } ) \\subset V _ 0 . \\end{align*}"}
-{"id": "6529.png", "formula": "\\begin{align*} F _ i \\subseteq H _ i : = \\{ x \\in \\mathbb { R } ^ n : \\langle x , y _ i \\rangle = 1 \\} . \\end{align*}"}
-{"id": "7736.png", "formula": "\\begin{align*} E _ 1 & \\equiv - \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N - 1 } \\sum _ { l ' = 2 } ^ { l - 1 } \\sum _ { l '' = 1 } ^ { l ' - 1 } e ^ { 2 i l '' \\phi _ n } \\\\ & = - \\frac { 1 } { N } \\sum _ { l ' = 2 } ^ { l - 1 } \\sum _ { l '' = 1 } ^ { l ' - 1 } \\left ( \\frac { 1 - e ^ { i 2 \\pi l '' } } { 1 - e ^ { i \\pi y '' / N } } - 1 \\right ) \\\\ & = \\frac { 1 } { N } \\sum _ { l ' = 2 } ^ { l - 1 } \\sum _ { l '' = 1 } ^ { l ' - 1 } 1 , \\end{align*}"}
-{"id": "6113.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\mathbb { \\int } _ { 0 } ^ { \\infty } h \\left ( y \\right ) n \\mu _ { 1 } \\left ( a _ { n } ^ { - 1 } d y \\right ) & = \\lim _ { n \\rightarrow \\infty } \\mathbb { \\int } _ { a } ^ { b } \\frac { \\partial h \\left ( y \\right ) } { \\partial y } n \\bar { \\mu } _ { 1 } \\left ( a _ { n } ^ { - 1 } y \\right ) d y \\\\ & = \\mathbb { \\int } _ { a } ^ { b } \\frac { \\partial h \\left ( y \\right ) } { \\partial y } \\frac { y ^ { - \\alpha } } { \\Gamma \\left ( 1 - \\alpha \\right ) } d y , \\end{align*}"}
-{"id": "2625.png", "formula": "\\begin{align*} \\overline \\rho = \\mathcal C _ a f _ { \\overline \\rho } = \\frac C { \\phi _ a } + R _ 3 , \\end{align*}"}
-{"id": "7298.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r m _ { i } ^ 2 = n ^ 2 - 3 n + 3 \\quad \\mbox { e } \\sum _ { i = 1 } ^ r m _ i = 2 n - 3 . \\end{align*}"}
-{"id": "3715.png", "formula": "\\begin{align*} \\begin{aligned} 1 & = \\frac { u _ { i , i } } { v _ { i , i } } \\cdot t _ { i + 1 } \\cdots t _ { j - 1 } \\cdot \\frac { v _ { j , j - 1 } } { u _ { j , j - 1 } } \\\\ & = \\frac { u _ { i , i } } { v _ { i , i } } \\left ( \\frac { v _ { i , i } } { u _ { i , i } } \\frac { u _ { i , i + 1 } } { v _ { i , i + 1 } } \\right ) \\cdots \\left ( \\frac { v _ { i , j - 2 } } { u _ { i , j - 2 } } \\frac { u _ { i , j - 1 } } { v _ { i , j - 1 } } \\right ) \\frac { v _ { j , j - 1 } } { u _ { j , j - 1 } } \\\\ & = \\frac { u _ { i , j - 1 } } { v _ { i , j - 1 } } \\frac { v _ { j , j - 1 } } { u _ { j , j - 1 } } , \\end{aligned} \\end{align*}"}
-{"id": "4071.png", "formula": "\\begin{align*} \\partial _ t \\eta + \\partial _ \\alpha q _ \\alpha = 0 . \\end{align*}"}
-{"id": "5405.png", "formula": "\\begin{align*} ( a \\cdot b ^ c ) ^ 6 = ( a b \\cdot b ^ c ) ^ 6 = ( a b \\cdot a ^ c ) ^ 6 = ( c \\cdot b ^ { c a } ) ^ { r _ 4 } = 1 \\end{align*}"}
-{"id": "8118.png", "formula": "\\begin{align*} [ x ] _ { R _ { C , E } } \\subseteq [ x ] _ { R _ { U _ 2 , E } } = S _ i x . \\end{align*}"}
-{"id": "3651.png", "formula": "\\begin{align*} \\tilde P [ n ] = P [ n ] + \\sigma [ n ] ^ { \\vee } \\end{align*}"}
-{"id": "7397.png", "formula": "\\begin{align*} \\dots \\rightarrow \\bigoplus _ { j \\in \\widetilde { \\Gamma } } \\bigoplus _ { l = 1 } ^ { v _ { i j } } P _ j ^ { ( d _ l ^ { i j } ) } \\rightarrow \\bigoplus _ { j \\in \\widetilde { \\Gamma } } \\bigoplus _ { l = 1 } ^ { u _ { i j } } P _ j ^ { ( d _ l ^ { i j } ) } \\rightarrow P _ i ^ { ( 0 ) } \\rightarrow s _ i ^ { ( 0 ) } \\rightarrow 0 \\end{align*}"}
-{"id": "2614.png", "formula": "\\begin{align*} \\frac { u ( t ) } { t ^ a } = \\frac { ( \\phi _ a u ) ( t ) } { \\phi _ a ( t ) t ^ a } \\in w L ^ { \\tfrac 2 { 1 + a } } ( 0 , 1 ) , \\end{align*}"}
-{"id": "3312.png", "formula": "\\begin{align*} r _ s = \\frac { \\binom { K - 2 } { s - 1 } } { \\binom { K - 2 } { s - 1 } + \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K - 1 } { s + i } ( N - 1 ) ^ i N } , \\end{align*}"}
-{"id": "7849.png", "formula": "\\begin{align*} r ( x _ { 1 } , x _ { 2 } ) = \\frac { f _ { X _ { 1 } , X _ { 2 } } ( x _ { 1 } , x _ { 2 } ) } { F _ { X _ { 1 } , X _ { 2 } } ( x _ { 1 } , x _ { 2 } ) } . \\end{align*}"}
-{"id": "995.png", "formula": "\\begin{align*} \\kappa _ 4 ( F _ n ( t ) ) & = \\frac { 1 } { \\mathfrak { s } _ n ^ 4 ( t ) } \\sum _ { i = 1 } ^ n K _ h ( t _ { i - 1 } - t ) ^ 4 \\kappa _ 4 ( ( B _ { t _ i } - B _ { t _ { i - 1 } } ) ^ 2 ) \\lesssim ( n h ) ^ 2 \\cdot \\frac { n h } { h ^ 4 } \\cdot \\frac { 1 } { n ^ 4 } = \\frac { 1 } { n h } , \\end{align*}"}
-{"id": "7679.png", "formula": "\\begin{align*} d ^ 2 _ B ( j , k ) = L ^ { 2 \\dagger } _ { j j } + L ^ { 2 \\dagger } _ { k k } - 2 L ^ { 2 \\dagger } _ { j k } = \\sum _ { i = 1 } ^ { N - 1 } \\frac { 1 } { \\lambda _ i ^ 2 } ( u _ { i j } - u _ { i k } ) ^ 2 \\ , . \\end{align*}"}
-{"id": "726.png", "formula": "\\begin{align*} D _ { + } = \\frac { u _ - ^ 2 } { a ^ 2 \\theta _ + } . \\end{align*}"}
-{"id": "5634.png", "formula": "\\begin{align*} \\partial _ t u ( t , x ) - \\varepsilon ^ 2 \\Delta u ( t , x ) + \\nabla W ( u ( t , x ) ) = 0 , x \\in \\Omega \\subset \\R ^ 2 , \\ , t > 0 , \\end{align*}"}
-{"id": "3889.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { L ^ { ( - ) } _ n ( s ) } { n ^ { \\alpha } } , L ^ { ( - ) } _ n ( s ) : = \\sum _ { c = 1 } ^ { \\infty } \\frac { a _ n ( c ) } { c ^ s } , \\Re { \\alpha } , \\Re { s } > 1 , \\end{align*}"}
-{"id": "7400.png", "formula": "\\begin{align*} a = a _ 0 a _ 1 a _ 2 a _ 3 , a ^ * = a _ 3 ^ * a _ 2 ^ * a _ 1 ^ * a _ 0 ^ * , b = a _ 4 a _ 4 ^ * , c = a _ 4 a _ 5 ^ * a _ 5 a _ 4 ^ * . \\end{align*}"}
-{"id": "6084.png", "formula": "\\begin{align*} \\Psi _ n = \\tfrac { 1 } { 2 } ( Q _ n + \\pi ) . \\end{align*}"}
-{"id": "169.png", "formula": "\\begin{align*} f ( \\mathfrak { X } ) = [ f ( a _ { 1 } ) , \\dots , f ( a _ { n } ) ] \\end{align*}"}
-{"id": "5498.png", "formula": "\\begin{align*} & q _ m = \\mathrm { B } _ { \\alpha - ( m - 1 ) } ^ { - 1 } \\mathrm { B } _ { \\alpha - ( m - 2 ) } ^ { - 1 } \\ldots \\mathrm { B } _ \\alpha ^ { - 1 } \\widetilde q _ m , \\ \\beta \\in [ 0 , 1 ] , \\ q _ 0 = \\widetilde q _ 0 , \\\\ & p _ { 0 , m } = \\mathrm { B } ^ { - 1 } _ { 1 - \\beta } \\mathrm { A } _ { 1 - \\beta } ^ { - 1 } q _ m , \\ \\beta \\in ( 0 , 1 ) , \\\\ & p = p _ { 0 , m } - m \\mathrm { B } _ { - m - \\beta } p _ { 0 , m } , \\ p _ { 0 , m } = p + m \\mathrm { B } _ { - \\beta } p , \\ \\beta \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "7803.png", "formula": "\\begin{align*} \\lim \\limits _ { v \\rightarrow 0 ^ { + } } \\Omega _ { k } \\left ( f , v \\right ) _ { p , \\gamma } = 0 , \\end{align*}"}
-{"id": "7788.png", "formula": "\\begin{align*} 2 \\ , a ( \\dot { y } _ h , y _ h ) = F ( \\dot { y } _ h ) - b ' ( x _ h ; \\xi _ h , y _ h ) - b ( x _ h ; \\dot { y } _ h ) . \\end{align*}"}
-{"id": "7801.png", "formula": "\\begin{align*} f \\left ( x \\right ) \\backsim \\frac { a _ { 0 } \\left ( f \\right ) } { 2 } + \\sum \\limits _ { k = 1 } ^ { \\infty } \\left ( a _ { k } \\left ( f \\right ) \\cos k x + b _ { k } \\left ( f \\right ) \\sin k x \\right ) = : \\sum \\limits _ { k = 0 } ^ { \\infty } A _ { k } \\left ( x , f \\right ) \\end{align*}"}
-{"id": "5713.png", "formula": "\\begin{align*} d _ X ( v _ 1 , v _ 2 ) = \\min \\bigg \\{ d _ { L ^ 2 } ( v _ 1 , v _ 2 ) \\ , ; \\ , d _ { L ^ 2 } ( v _ 1 , \\mathcal { C } ( z ^ - ) ) + d _ { L ^ 2 } ( v _ 2 , \\mathcal { C } ( z ^ - ) ) \\ , ; \\ , d _ { L ^ 2 } ( v _ 1 , \\mathcal { C } ( z ^ + ) ) + d _ { L ^ 2 } ( v _ 2 , \\mathcal { C } ( z ^ + ) ) \\bigg \\} , \\end{align*}"}
-{"id": "384.png", "formula": "\\begin{align*} \\sum _ { - 2 n \\le s , r \\le n } | b _ { n , r , s } | ^ p \\ll n ^ 2 n ^ { p ( 2 - \\beta ) } L ^ p ( n ) = n ^ { p ( 2 - \\beta ) + 2 } L ^ p ( n ) . \\end{align*}"}
-{"id": "382.png", "formula": "\\begin{align*} \\sum _ { r \\in \\mathbb { Z } , s < - 2 n } | b _ { n , r , s } | ^ p = O \\big ( n ^ { p ( 2 - \\beta ) + 2 } L ^ p ( n ) \\big ) . \\end{align*}"}
-{"id": "1334.png", "formula": "\\begin{align*} \\gamma _ m : = G _ m , \\ \\gamma _ { m - 1 } : = G _ { m - 1 } / G _ m \\ \\ldots \\ \\gamma _ 0 : = G / G _ 1 . \\end{align*}"}
-{"id": "9913.png", "formula": "\\begin{align*} \\Omega _ s = 6 e ( A ^ s _ 1 ) + 6 e ( A ^ s _ 2 ) + \\sum _ { i = 3 } ^ r e ( A ^ s _ i ) \\ , . \\end{align*}"}
-{"id": "1227.png", "formula": "\\begin{align*} \\nabla h ( X ) = \\eta ( X ) \\ , \\ , \\mbox { w h e r e } \\ , \\ , \\eta \\ , \\ , \\mbox { i s t h e p o i n t i n } \\ , \\ , \\{ k = 1 \\} \\ , \\ , \\mbox { w i t h } \\ , \\ , \\frac { X } { | X | } = \\frac { \\nabla k ( \\eta ) } { | \\nabla k ( \\eta ) | } . \\end{align*}"}
-{"id": "868.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ d | \\partial _ { i , j } ^ 2 ( U _ 0 f ) ( x ) | \\leq \\| g '' \\| _ \\infty + 2 \\| g ' \\| _ \\infty \\beta \\end{align*}"}
-{"id": "1767.png", "formula": "\\begin{align*} y = f g e h f g e h \\dots = g e h f g e h f \\dots = g h f f g h f f \\dots . \\end{align*}"}
-{"id": "2346.png", "formula": "\\begin{align*} p ^ { - n } g _ p ^ { - 1 } . z \\not = \\infty \\end{align*}"}
-{"id": "2541.png", "formula": "\\begin{align*} & - i \\eta \\mathrm { P } _ 0 \\xi _ 1 \\left ( \\left ( \\mathrm { P } _ 0 e \\right ) + \\left ( \\mathrm { P } _ 1 e \\right ) \\right ) = \\sigma ( \\mathrm { P } _ 0 e ) , \\\\ & - i \\eta \\mathrm { P } _ 1 \\xi _ 1 ( \\mathrm { P } _ 0 e ) - i \\eta \\mathrm { P } _ 1 \\xi _ 1 ( \\mathrm { P } _ 1 e ) + L ( \\mathrm { P } _ 1 e ) = \\sigma ( \\mathrm { P } _ 1 e ) . \\end{align*}"}
-{"id": "332.png", "formula": "\\begin{align*} \\Phi ( x ) & : = \\prod _ { i \\in I } l _ i ( x ) ^ { l _ i ( x ) } \\prod _ { j \\in J } m _ j ( x ) ^ { - m _ j ( x ) } , \\\\ u ( x ; n ) & : = ( 2 \\pi ) ^ { \\frac { | I | - | J | } { 2 } } \\prod _ { i \\in I } l _ i ( x ) ^ { \\alpha _ i ( n ) - \\frac { 1 } { 2 } } \\prod _ { j \\in J } m _ j ( x ) ^ { \\frac { 1 } { 2 } - \\beta _ j ( n ) } , \\end{align*}"}
-{"id": "2628.png", "formula": "\\begin{align*} \\tilde h _ { \\rho } ( \\tilde t ) = C [ t _ 1 , t _ 2 ] . \\end{align*}"}
-{"id": "3320.png", "formula": "\\begin{align*} r _ 1 = \\tilde { r } _ 1 = & \\frac { 1 } { 1 + N + N ^ 2 + \\cdots + N ^ { K - 1 } } , \\\\ r _ { K - 2 } = & \\frac { K - 2 } { ( N + 1 ) K + N ^ 2 - 2 N - 2 } , \\\\ r _ { K - 1 } = \\tilde { r } _ { K - 1 } = & \\frac { 1 } { 1 + N } . \\end{align*}"}
-{"id": "2822.png", "formula": "\\begin{align*} \\Delta _ m ^ { ( n ) } = F _ n ^ T \\cdot \\ldots \\cdot F _ { n + m - 1 } ^ T ( \\Delta ^ { ( n + m ) } ) \\end{align*}"}
-{"id": "8059.png", "formula": "\\begin{align*} G ( \\lambda _ * \\Lambda ( \\mu _ 1 , \\dots , \\mu _ m ) ) = G \\bigl ( \\lambda ( G ( A _ { 1 i _ 1 } , \\dots , A _ { m i _ m } ) ) : i _ 1 , \\dots , i _ m \\bigr ) , \\end{align*}"}
-{"id": "9778.png", "formula": "\\begin{align*} \\tilde { L } _ { 5 , h _ j } ( X _ { k } ^ * - ) = O ( 1 ) ( d L ^ { c } _ { h _ j , \\theta } ( \\Lambda _ { c , k - 1 } ^ { * } ) + d Q _ { h _ j , \\theta } ( \\Lambda _ { k - 1 } ^ { * } ) + ( e ^ { - l X _ { k - 1 } ^ * } - e ^ { - l X _ { k } ^ * } ) \\| Z _ { 0 } \\| _ { \\infty } ) , \\end{align*}"}
-{"id": "5480.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { r ^ n z \\ , u ( r ^ n z ) } { \\ell ( r ^ n z ) } = p _ 0 ( z ) z \\in C _ { p _ 0 } , \\end{align*}"}
-{"id": "8276.png", "formula": "\\begin{align*} \\mathfrak { X } _ { r _ 1 , r _ 2 } = \\{ \\mu \\in \\mathfrak { X } _ a ' \\colon a ( \\mu \\chi _ i ) = a - r _ i \\} . \\end{align*}"}
-{"id": "3302.png", "formula": "\\begin{align*} { _ 2 F _ 1 } ( \\alpha , \\beta ; \\gamma ; 1 ) = \\frac { \\Gamma ( \\gamma ) \\Gamma ( \\gamma - \\alpha - \\beta ) } { \\Gamma ( \\gamma - \\alpha ) \\Gamma ( \\gamma - \\beta ) } . \\end{align*}"}
-{"id": "1285.png", "formula": "\\begin{align*} 1 - U _ j ( x ) \\geq C _ 1 ^ { - 1 } \\ , | x '' | ^ { \\psi } \\mbox { f o r } \\ , \\ , x = ( x ' , x '' ) \\in B ( 0 , 4 \\rho ) \\ , \\ , \\mbox { a n d } \\ , \\ , C _ 1 \\ , t _ j \\leq | x '' | \\end{align*}"}
-{"id": "3808.png", "formula": "\\begin{align*} X _ L = \\sum _ { i = 0 } ^ { K - 1 } X _ { \\sigma _ { i + 1 } } - X _ { \\sigma _ i } + X _ L - X _ { \\sigma _ { K } } \\ge K L ^ \\beta - ( \\sigma _ { K + 1 } - \\sigma _ { K } ) . \\end{align*}"}
-{"id": "8007.png", "formula": "\\begin{align*} & \\Big ( \\frac { 1 } { | P | } \\int _ P { \\sum _ { k = \\mu } ^ { \\infty } { 2 ^ { s k q } \\Big | \\Pi _ k \\Big ( \\sum _ { j = 3 } ^ { \\infty } { T _ { [ c _ j ] } f } \\Big ) ( x ) \\Big | ^ q } } d x \\Big ) ^ { 1 / q } \\\\ & \\lesssim \\sup _ { R \\in \\mathcal { D } _ { \\mu } } { \\Big ( \\dfrac { 1 } { | R | } \\int _ R { \\sum _ { j = \\max { ( 3 , \\mu - 2 ) } } ^ { \\infty } { 2 ^ { s j q } \\big | T _ { [ c _ j ] } f ( x ) \\big | ^ q } } d x \\Big ) ^ { 1 / q } } . \\end{align*}"}
-{"id": "5157.png", "formula": "\\begin{align*} \\mathcal S _ 1 : = [ x _ \\ell , a ] , \\mathcal S _ 2 : = [ b , x _ r ] , \\end{align*}"}
-{"id": "4406.png", "formula": "\\begin{align*} \\sigma _ { e s s } ( T _ f ) = f ( \\partial \\mathbb { C } ^ n ) , \\end{align*}"}
-{"id": "6717.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\mathbb { P } ( R _ { N } \\leq t ( N ) ) = 1 . \\end{align*}"}
-{"id": "2721.png", "formula": "\\begin{align*} \\mathcal G _ { \\epsilon } ( g ) = \\frac { 1 } { \\epsilon ^ 2 } \\mathcal L ( g ) - \\frac { 1 } { \\epsilon } v \\cdot \\nabla _ x g \\end{align*}"}
-{"id": "1203.png", "formula": "\\begin{align*} | x | ^ { - n } \\ , \\left ( \\int _ { B ( \\hat x , | x | / ( 8 \\breve c ) ) } | \\nabla \\bar e _ { h , m } | ^ 2 \\ , \\ , d \\hat y \\right ) ^ { 1 / 2 } = o \\left ( | x | ^ { \\frac { 2 - n - p } { p - 1 } } \\right ) \\mbox { a s } | x | \\to \\infty . \\end{align*}"}
-{"id": "7632.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\Big [ \\| w ^ k - v ^ k \\| _ { L ^ p ( Q _ 3 ) } + \\| w ^ k - v ^ k \\| _ { L ^ 2 ( Q _ 3 ) } \\Big ] = 0 . \\end{align*}"}
-{"id": "3039.png", "formula": "\\begin{align*} \\begin{cases} D ( A ) = \\{ u \\in V _ \\sigma \\mid \\exists w \\in H _ \\sigma ( w , v ) _ H = a ( u , v ) , \\ \\forall v \\in V _ \\sigma \\} , \\\\ ( A u , v ) _ H = a ( u , v ) , \\forall u \\in D ( A ) , \\forall v \\in V _ \\sigma , \\end{cases} \\end{align*}"}
-{"id": "4952.png", "formula": "\\begin{align*} f \\left ( \\left ( M + m \\right ) { { 1 } _ { K } } - \\sum \\limits _ { i = 1 } ^ { n } { { { \\Phi } _ { i } } \\left ( { { A } _ { i } } \\right ) } \\right ) \\le \\left ( f \\left ( M \\right ) + f \\left ( m \\right ) \\right ) { { 1 } _ { K } } - \\sum \\limits _ { i = 1 } ^ { n } { { { \\Phi } _ { i } } \\left ( f \\left ( { { A } _ { i } } \\right ) \\right ) } . \\end{align*}"}
-{"id": "1207.png", "formula": "\\begin{align*} \\frac { X } { | X | } = - \\frac { \\nabla \\bar u ( \\bar x ) } { | \\nabla \\bar u ( \\bar x ) | } . \\end{align*}"}
-{"id": "7612.png", "formula": "\\begin{align*} & | \\xi - \\eta | ^ p = \\big ( 1 + | \\xi | + | \\xi - \\eta | \\big ) ^ { \\frac { p ( 2 - p ) } { 2 } } \\big ( 1 + | \\xi | + | \\xi - \\eta | \\big ) ^ { \\frac { p ( p - 2 ) } { 2 } } | \\xi - \\eta | ^ p \\\\ & \\leq \\tau \\ , 3 ^ { - p } \\big ( 1 + | \\xi | + | \\xi - \\eta | \\big ) ^ p + C _ p \\tau ^ { \\frac { p - 2 } { p } } \\big ( 1 + | \\xi | + | \\xi - \\eta | \\big ) ^ { p - 2 } | \\xi - \\eta | ^ 2 . \\end{align*}"}
-{"id": "4983.png", "formula": "\\begin{align*} \\tilde { f } : = \\frac { \\min \\left ( \\eta _ { \\delta } , \\frac { \\delta } { D _ { \\delta } } \\right ) } { \\left \\Vert f \\right \\Vert _ { \\dot { F } ^ { \\alpha , p } _ { q } } } f . \\end{align*}"}
-{"id": "4546.png", "formula": "\\begin{align*} Z _ r = \\sum _ { i = 0 } ^ { n - 1 } \\frac { q ^ { i r } + q ^ { ( n - i ) r } } { q ^ r - q ^ { - r } } h _ { i , r } . \\end{align*}"}
-{"id": "6455.png", "formula": "\\begin{align*} \\frac { R _ { } ^ { } ( \\rho ) } { R _ { } ^ { } ( \\rho ) } = \\sqrt { \\frac { 1 + 2 \\rho } { 1 + \\rho } } \\end{align*}"}
-{"id": "2782.png", "formula": "\\begin{align*} d _ { 1 } ( x , t , y ) = \\frac { \\log ( \\frac { x } { y } ) + ( r - \\delta + \\frac { \\sigma ^ { 2 } } { 2 } ) t } { \\sigma \\sqrt { t } } , d _ { 2 } ( x , t , y ) = d _ { 1 } ( x , t , y ) - \\sigma \\sqrt { t } . \\end{align*}"}
-{"id": "4443.png", "formula": "\\begin{align*} \\phi = \\int _ 0 ^ 1 \\omega ^ t * \\psi _ t d t \\quad \\mbox { a n d } \\int _ { \\R ^ 2 } ( 1 + d ^ 2 ( x , 0 ) ) | \\omega ^ t ( x ) | \\ , \\ , d x \\lesssim 1 , t \\in ( 0 , 1 ] , \\end{align*}"}
-{"id": "7100.png", "formula": "\\begin{align*} p _ { t t } = 4 k k _ p p _ { x x } ^ 2 + ( 3 k _ p k _ { p p } + k k _ { p p p } ) p _ x ^ 4 + 7 ( k _ p ^ 2 + k k _ { p p } ) p _ x ^ 2 p _ { x x } + 6 k k _ p p _ { x x x } p _ x + k ^ 2 p _ { x x x x } . \\end{align*}"}
-{"id": "7884.png", "formula": "\\begin{align*} \\begin{aligned} | \\nabla _ x \\varphi ( d ( z , m ( x , y ) ) ) | & = \\limsup _ { x ' \\to x } { | \\varphi ( d ( z , m ( x ' , y ) ) ) - \\varphi ( d ( z , m ( x , y ) ) ) | \\over d ( x ' , x ) } \\\\ & \\le { 1 \\over 2 } | \\varphi ' ( d ( z , m ( x , y ) ) ) | . \\end{aligned} \\end{align*}"}
-{"id": "2484.png", "formula": "\\begin{align*} \\begin{aligned} \\sup _ { ( x , t ) \\in \\mathbb { R } ^ 3 \\times \\mathbb { R } _ + } e ^ { \\alpha \\left ( \\delta ( \\left < \\left < x \\right > \\right > - M t ) \\right ) ^ { \\frac { 2 } { 1 - \\gamma } } } \\left | f ( t , x ) \\right | _ { L ^ 2 _ \\xi } \\lesssim ( 1 + t ) \\left \\| f _ 0 \\right \\| _ { L ^ { 2 } ( e ^ { 3 \\alpha \\left | \\xi \\right | ^ 2 } \\mathcal { M } ) } . \\end{aligned} \\end{align*}"}
-{"id": "4296.png", "formula": "\\begin{align*} ( k , g ) \\leq ( k ' , g ' ) \\Longleftrightarrow k < k ' \\big ( k = k ' g \\leq g ' \\big ) . \\end{align*}"}
-{"id": "2255.png", "formula": "\\begin{align*} \\begin{cases} f _ 1 ( z _ 1 , z _ 2 ) = ( 1 - a _ 2 z _ 2 ) ^ 2 + a _ 3 z _ 1 z ^ 2 _ 2 = 0 , \\\\ f _ 2 ( z _ 1 , z _ 2 ) = ( 1 - b _ 1 z _ 1 ) ^ 2 ( 1 - b _ 2 z _ 2 ) + b _ 3 z ^ 2 _ 1 z _ 2 = 0 \\end{cases} \\end{align*}"}
-{"id": "7553.png", "formula": "\\begin{gather*} \\mathcal T _ * P ^ t = \\varepsilon \\tilde P ^ t , \\mathcal T _ * P ^ x = a \\tilde P ^ x + c \\tilde P ^ y , \\mathcal T _ * P ^ y = b \\tilde P ^ x + d \\tilde P ^ y , \\\\ \\mathcal T _ * G ^ x = \\varepsilon a \\tilde G ^ x + \\varepsilon c \\tilde G ^ y , \\mathcal T _ * G ^ y = \\varepsilon b \\tilde G ^ x + \\varepsilon d \\tilde G ^ y , \\mathcal T _ * D = \\tilde D , \\mathcal T _ * \\Pi = \\varepsilon \\tilde \\Pi , \\end{gather*}"}
-{"id": "8371.png", "formula": "\\begin{align*} f = \\vartheta ^ { [ 2 ] } f ^ { [ 2 ] } - \\vartheta ^ { [ 1 ] } f ^ { [ 1 ] } \\end{align*}"}
-{"id": "3091.png", "formula": "\\begin{align*} \\begin{cases} \\epsilon \\dot { u } _ \\epsilon = \\displaystyle - \\frac { \\mathrm { d } F } { \\mathrm { d } x } ( t , u _ \\epsilon ( t ) ) , t \\in [ 0 , T ] , \\\\ \\displaystyle \\lim _ { \\epsilon \\to 0 } u _ \\epsilon ( 0 ) = 0 \\ , , \\end{cases} \\end{align*}"}
-{"id": "420.png", "formula": "\\begin{align*} \\psi _ \\omega ( 0 ) = \\psi ' _ \\omega ( 0 ) = 0 , i \\psi _ \\omega '' ( 0 ) < 0 . \\end{align*}"}
-{"id": "5264.png", "formula": "\\begin{align*} ( z _ n ) _ { n \\in \\N } \\mapsto ( g ( z _ 0 ) , g ( z _ 1 ) , g ( z _ 2 ) , \\ldots ) = ( g ( z _ 0 ) , z _ 0 , z _ 1 , \\ldots ) . \\end{align*}"}
-{"id": "7169.png", "formula": "\\begin{align*} u _ { x _ 0 , r } ( x ) : = \\frac { u ( x _ 0 + r x ) } { r ^ { 1 + s } } , \\end{align*}"}
-{"id": "5849.png", "formula": "\\begin{align*} - \\frac 1 N \\sum _ { k = 0 } ^ N & ( - 1 ) ^ k \\binom { N } { k } k g ^ { k - 1 } D ( g ) D ^ { N - 1 } ( g ^ { N + 1 - k } f ) \\\\ & + \\frac { N + 1 } N g \\sum _ { k = 0 } ^ N ( - 1 ) ^ k \\binom { N } { k } k g ^ { k - 1 } D ( g ) D ^ { N - 1 } ( g ^ { N - k } f ) . \\end{align*}"}
-{"id": "7996.png", "formula": "\\begin{align*} & \\sup _ { P \\in \\mathcal { D } _ { \\mu } } \\Big ( \\frac { 1 } { | P | } \\int _ P \\sum _ { k = \\mu } ^ { \\infty } { 2 ^ { s k q } \\big | \\Pi _ k T _ { [ a ] } f ( x ) \\big | ^ q } d x \\Big ) ^ { 1 / q } \\\\ & \\lesssim \\sup _ { 0 \\leq k \\leq \\mu - 1 } { \\Vert 2 ^ { k ( s + m ) } \\Pi _ k f \\Vert _ { L ^ { \\infty } } } + \\sup _ { R \\in \\mathcal { D } _ { \\mu } } \\Big ( \\frac { 1 } { | R | } \\int _ R \\sum _ { k = \\mu } ^ { \\infty } { 2 ^ { ( s + m ) k q } \\big | \\Pi _ k f ( x ) \\big | ^ q } d x \\Big ) ^ { 1 / q } . \\end{align*}"}
-{"id": "7219.png", "formula": "\\begin{align*} \\left < X , K _ { \\alpha } \\right > & = \\frac { B ( h _ { \\alpha _ { 0 } } , h _ { \\alpha _ { 0 } } ) } { 2 ( 2 \\pi i ) ^ { 2 } } B ( X , K _ { \\alpha } ) = \\frac { B ( h _ { \\alpha _ { 0 } } , h _ { \\alpha _ { 0 } } ) } { 2 ( 2 \\pi i ) } B ( X , H _ { \\alpha } ) \\\\ & = \\frac { B ( h _ { \\alpha _ { 0 } } , h _ { \\alpha _ { 0 } } ) } { B ( h _ { \\alpha } , h _ { \\alpha } ) } \\left ( \\frac { { \\alpha } ( X ) } { 2 \\pi i } \\right ) = d _ { \\alpha } \\left ( \\frac { \\alpha ( X ) } { 2 \\pi i } \\right ) . \\end{align*}"}
-{"id": "9515.png", "formula": "\\begin{align*} a _ { n - 1 , k + d - 1 } ^ { ( d ) } = \\frac { a _ { n , k } ^ { ( d ) } - \\sum \\limits _ { t = k - d + 1 } ^ { k + d - 2 } { c _ { n , k , t } ^ { ( d ) } \\cdot a _ { n - 1 , t } ^ { ( d ) } } } { c _ { n , k , k + d - 1 } ^ { ( d ) } } . \\end{align*}"}
-{"id": "3591.png", "formula": "\\begin{align*} D _ { * a } ^ \\alpha J ^ \\alpha _ a f ( x ) = f ( x ) \\end{align*}"}
-{"id": "6325.png", "formula": "\\begin{align*} { \\mathcal { L } } _ s ( u ) = \\int _ { \\R ^ n } | ( { - \\Delta } ) ^ { \\frac { s } { 2 } } u ( x ) | ^ p d x \\ , , \\end{align*}"}
-{"id": "6597.png", "formula": "\\begin{align*} D ( \\phi \\circ f ) ( x ) \\ ; = \\ ; D \\phi ( f ( x ) ) D \\phi ( x ) \\ \\ \\ \\textrm { f o r L e b e s g u e a . e . } \\ \\ \\ x \\in U \\ . \\end{align*}"}
-{"id": "3618.png", "formula": "\\begin{align*} T ( s ) : = \\max \\{ 0 ; \\min \\{ s ; 1 \\} \\} , \\ , s \\in \\mathbb { R } g ( s ) : = \\max \\{ 0 ; - 2 s + 1 \\} , \\ , s \\in \\mathbb { R } ^ + _ 0 . \\end{align*}"}
-{"id": "5594.png", "formula": "\\begin{align*} D _ z ( n ) = q ^ n \\binom { n + z - 1 } { n } = q ^ n \\frac { \\Gamma ( n + z ) } { \\Gamma ( n + 1 ) \\Gamma ( z ) } = q ^ n \\frac { n ^ { z - 1 } } { \\Gamma ( z ) } \\left ( 1 + O _ A \\left ( \\frac { 1 } { n } \\right ) \\right ) . \\end{align*}"}
-{"id": "3998.png", "formula": "\\begin{align*} p ^ { \\nu _ 2 } ( 2 , t ) = - \\frac { \\lambda _ 1 } { \\lambda _ 2 } \\sum _ { k = 1 } ^ { \\infty } ( - 1 ) ^ k \\underset { \\Lambda ^ { k } _ { 2 } } { \\sum } \\frac { \\lambda _ 1 ^ { k _ 1 } \\lambda _ 2 ^ { k _ 2 } t ^ { k _ 1 \\nu _ 1 + k _ 2 \\nu _ 2 } } { \\Gamma \\left ( k _ 1 \\nu _ 1 + k _ 2 \\nu _ 2 + 1 \\right ) } , \\end{align*}"}
-{"id": "1030.png", "formula": "\\begin{align*} \\widehat { w } _ { n } ( \\zeta ) \\leq \\theta _ { n } : = \\frac { 3 ( n - 1 ) + \\sqrt { n ^ { 2 } - 2 n + 5 } } { 2 } . \\end{align*}"}
-{"id": "2809.png", "formula": "\\begin{align*} p _ { t } & = p _ { x x } + \\Big [ \\rho - 1 + b ' ( t ) \\Big ] p _ { y } , y > 0 , t > 0 , \\\\ p ( 0 , y ) & = e ^ { y } - 1 , y \\geq 0 , \\\\ p ( t , 0 ) & = e ^ { \\rho t } - 1 , p _ { y } ( t , 0 ) = 0 , t > 0 . \\end{align*}"}
-{"id": "9633.png", "formula": "\\begin{align*} \\Phi ^ * _ { \\rm { s t } , \\beta } ( t ) = \\frac { 2 S _ \\beta } { \\pi E _ { \\rm { b } } } \\sqrt { \\lambda _ { \\beta } \\left ( 2 ^ { C / W } - 1 \\right ) P _ { \\rm { c u } } P _ { \\rm { t r } , 1 } ( h _ { \\beta , 1 } ^ * ) } , \\end{align*}"}
-{"id": "6389.png", "formula": "\\begin{align*} \\int d x d \\theta P \\left ( x \\theta \\right ) = 1 \\end{align*}"}
-{"id": "3007.png", "formula": "\\begin{align*} \\Phi : { \\mathcal W } ^ { N , 2 } ( J ; H ) \\rightarrow H ^ { N } , \\Phi ( x ) : = \\Phi _ 0 ( x ) . \\end{align*}"}
-{"id": "6381.png", "formula": "\\begin{align*} \\frac { d m _ i } { d z _ r } ( 1 , 1 , \\ldots ) = \\binom { 2 i } { i - r } - \\binom { 2 i } { i - r - 1 } \\end{align*}"}
-{"id": "3154.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\frac { \\partial } { \\partial u } \\mathbb { E } \\left [ e ^ { - u \\eta } \\right ] u ^ { \\theta - 1 } \\mathrm { d } u & = - \\int _ { 0 } ^ { \\infty } \\mathbb { E } \\left [ Y e ^ { - u \\eta } \\right ] u ^ { \\theta - 1 } \\mathrm { d } u \\\\ & = - \\mathbb { E } \\left [ \\int _ { 0 } ^ { \\infty } \\eta e ^ { - u \\eta } u ^ { \\theta - 1 } \\mathrm { d } u \\right ] = - \\mathbb { E } \\left [ \\Gamma ( \\theta ) \\eta ^ { 1 - \\theta } \\right ] , \\end{align*}"}
-{"id": "7584.png", "formula": "\\begin{align*} f _ { n , i } ( X ) : = f _ { G _ { n } , \\{ i \\} } ( X ) = \\binom { n } { i } _ { X } \\prod _ { j = i + 1 } ^ n ( 1 - X ^ j ) \\in \\Z [ X ] \\end{align*}"}
-{"id": "592.png", "formula": "\\begin{align*} \\Gamma _ { N } : = \\{ \\gamma \\in \\Gamma : [ \\gamma ] \\equiv 0 \\mod N \\} , \\end{align*}"}
-{"id": "7254.png", "formula": "\\begin{align*} \\Phi ( n ) = \\# \\{ u \\in [ n ] : ( u , n ) \\le T \\} \\end{align*}"}
-{"id": "4531.png", "formula": "\\begin{align*} \\Pi ( a ) - \\pi ( a ) = \\Pi ( a - \\psi ( a ) I ) - \\pi ( a - \\psi ( a ) I ) \\end{align*}"}
-{"id": "5222.png", "formula": "\\begin{align*} J = \\left ( \\begin{array} { c c } 0 & Q \\\\ - Q & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "8942.png", "formula": "\\begin{gather*} \\sum _ { w \\in C _ n / C _ { n - 1 } } \\ ! \\ ! \\ ! w \\cdot \\frac { \\prod _ { 1 \\le i \\le 2 n + 1 } \\vartheta ( z _ n - y _ i ) \\prod _ { 1 \\le i \\le n } \\vartheta ( Y + z _ i ) \\prod _ { 1 \\le i < n } \\vartheta ( Y - z _ i ) } { \\vartheta ( 2 z _ n ) \\prod _ { 1 \\le i < n } \\vartheta ( z _ n + z _ i ) \\vartheta ( z _ n - z _ i ) } \\\\ \\hphantom { \\sum _ { w \\in C _ n / C _ { n - 1 } } \\ ! \\ ! \\ ! } { } = \\prod _ { 1 \\le i \\le 2 n + 1 } \\vartheta ( Y - y _ i ) , \\end{gather*}"}
-{"id": "7789.png", "formula": "\\begin{align*} b ' ( D _ h ; X _ h , ( 1 - \\Pi _ h ) Y ) = 0 , \\end{align*}"}
-{"id": "2981.png", "formula": "\\begin{align*} \\{ | \\nabla u _ n | > t , u _ n > 1 \\} & = \\{ | \\nabla u _ n | > t , 1 < u _ n \\leq k + 1 \\} \\cup \\{ | \\nabla u _ n | > t , u _ n > k + 1 \\} \\\\ & \\subset \\{ | \\nabla u _ n | > t , 1 < u _ n \\leq k + 1 \\} \\cup \\{ u _ n > k + 1 \\} \\subset \\Omega . \\end{align*}"}
-{"id": "5518.png", "formula": "\\begin{align*} Q ( f ( s ) ) = \\gamma Q ( s ) s \\in [ 0 , 1 ) , \\end{align*}"}
-{"id": "10144.png", "formula": "\\begin{align*} \\boldsymbol R _ k ^ { - 1 / 2 } ( i ) \\boldsymbol p _ k ( i ) = \\boldsymbol \\Phi _ k \\boldsymbol \\Lambda _ k \\boldsymbol \\Phi _ k ^ H \\boldsymbol p _ k ( i ) , \\end{align*}"}
-{"id": "7111.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } g _ { i j } = ~ - 2 \\Phi h _ { i j } , \\end{align*}"}
-{"id": "6977.png", "formula": "\\begin{align*} M _ { i j } = - \\frac { 1 } { k f _ k ( B ) ^ { k - 1 } } \\sum _ { \\mbox { \\tiny $ \\begin{array} { c } i \\leq j _ 1 < j _ 2 < . . . < j _ { k - 2 } \\leq n , \\\\ j _ 1 , . . . , j _ { k - 2 } \\neq i , j \\end{array} $ } } \\det { B _ { ( j , j _ 1 , . . . , j _ { k - 2 } ) ( i , j _ 1 , . . . , j _ { k - 2 } ) } } , \\end{align*}"}
-{"id": "2688.png", "formula": "\\begin{align*} \\chi ' ( z _ 1 ) \\chi ' ( z _ 2 ) ^ { - 1 } = \\chi ' ( z _ 1 z ^ { - 1 } _ 2 ) = z _ 1 z ^ { - 1 } _ 2 . \\end{align*}"}
-{"id": "9931.png", "formula": "\\begin{gather*} X _ t ^ { ( n _ k ) } - x = \\int _ 0 ^ t b ( \\eta _ { n _ k } ( s ) , X _ { \\eta _ { n _ k } ( s ) } ^ { ( n _ k ) } ) \\ , d s + L _ t \\intertext { w e f i n d } X _ t - x = \\int _ 0 ^ t b ( s , X _ { s - } ) \\ , d s + L _ t . \\end{gather*}"}
-{"id": "5718.png", "formula": "\\begin{align*} \\frac { ( \\partial _ t \\gamma ( t , \\cdot ) , \\partial _ s \\gamma ( t , \\cdot ) ) _ { L ^ 2 } } { \\| \\partial _ s \\gamma ( t , \\cdot ) \\| _ { L ^ 2 } } \\leq \\| \\partial _ t \\gamma ( t , \\cdot ) \\| _ { L ^ 2 } = | \\dot { \\gamma } | ( t ) \\in L ^ 1 _ { l o c } ( I ) , \\end{align*}"}
-{"id": "735.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } | u _ i | ^ { p _ i } \\ , d x \\le C ( N , p _ i , a _ i ) \\| \\nabla u _ i \\| _ 2 ^ { \\frac { N ( p _ i - 2 ) } { 2 } } \\ \\ \\ \\ i = 1 , 2 , \\end{align*}"}
-{"id": "8146.png", "formula": "\\begin{align*} X \\setminus \\bigsqcup _ { k = 1 } ^ K S _ k V _ k \\prec \\bigsqcup _ { k = 1 } ^ K Q _ k V _ k . \\end{align*}"}
-{"id": "10005.png", "formula": "\\begin{align*} \\gamma * ( A _ 0 , A _ 1 , B , \\eta ) = ( A _ 0 , A _ 1 , B , \\gamma \\circ \\eta ) , \\end{align*}"}
-{"id": "1072.png", "formula": "\\begin{align*} M _ { S ' } ^ { \\rm g p } = \\rho _ \\star M _ { T ' } ^ { \\rm g p } \\mathop \\times _ { \\rho _ \\star M _ T ^ { \\rm g p } } M _ S ^ { \\rm g p } \\end{align*}"}
-{"id": "2889.png", "formula": "\\begin{align*} V ( m ) = ( - 1 ) ^ { m + 1 } \\left ( \\frac { y } { 2 \\pi } \\right ) ^ { 2 m } \\sum _ { n = 1 } ^ { \\infty } \\sigma _ { - ( 2 m + 1 ) } ( n ) e ^ { - \\frac { 4 \\pi ^ 2 n } { y } } . \\end{align*}"}
-{"id": "1371.png", "formula": "\\begin{align*} D _ c : = \\left [ \\begin{array} { c c c c c } I _ { m } & 0 & 0 & 0 & 0 \\\\ 0 & \\Sigma _ c & 0 & 0 & 0 \\\\ 0 & 0 & ( { \\widehat \\Theta _ c } ) _ { ( \\alpha , \\alpha ) } & 0 & 0 \\\\ 0 & 0 & 0 & ( { \\widehat \\Theta _ c } ) _ { ( \\beta , \\beta ) } & 0 \\\\ 0 & 0 & 0 & 0 & 2 I _ { | \\alpha | | \\beta | } \\end{array} \\right ] , \\end{align*} % \\end{align*}"}
-{"id": "2875.png", "formula": "\\begin{align*} S ( x ) = S ( x ; N , h ) = \\frac { ( - 1 ) ^ { h + 1 } } { N } \\left ( \\frac { 2 \\pi } { x } \\right ) ^ { \\frac { N - 2 h + 1 } { N } } \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ { \\frac { 2 h - 1 } { N } } } \\sum _ { j = 1 } ^ { \\frac { N } { 2 } } f _ { 2 j - 1 } ( x ; n , N , h ) \\end{align*}"}
-{"id": "4438.png", "formula": "\\begin{align*} 2 \\tilde \\psi _ T * \\psi _ T ( x ) & = \\int _ { \\R ^ 2 } \\frac { x _ 1 - y _ 1 } { T ^ { 1 / 3 } } \\psi _ T ( x - y ) \\psi _ T ( y ) \\ , d y + \\int _ { \\R ^ 2 } \\frac { y _ 1 } { T ^ { 1 / 3 } } \\psi _ T ( y ) \\psi _ T ( x - y ) \\ , d y \\\\ & = \\frac { x _ 1 } { T ^ { 1 / 3 } } \\psi _ { 2 T } ( x ) = 2 ^ { 1 / 3 } \\tilde \\psi _ { 2 T } ( x ) . \\end{align*}"}
-{"id": "6022.png", "formula": "\\begin{align*} & \\Gamma _ { 1 + s } ( P _ { W } , P _ { X | W } , \\pi _ { X } , R ) \\\\ & : = \\max \\left \\{ D _ { 1 + s } ( P _ { X | W } \\| \\pi _ { X } | P _ { W } ) - R , D _ { 1 + s } ( P _ { X } \\| \\pi _ { X } ) \\right \\} . \\end{align*}"}
-{"id": "2787.png", "formula": "\\begin{align*} \\mathcal ( { \\mathcal { P } } _ { n } f ) ( t ) = \\frac { \\sum _ { i = 0 } ^ { n } \\frac { \\beta _ { i } } { t - t _ { i } } f ( t _ { i } ) } { \\sum _ { i = 0 } ^ { n } \\frac { \\beta _ { i } } { t - t _ { i } } } = \\sum _ { i = 0 } ^ { n } f ( t _ { i } ) \\mathcal { L } _ { i } ( t ) , \\end{align*}"}
-{"id": "6388.png", "formula": "\\begin{align*} P _ { } \\left ( x \\theta \\right ) = P _ { } \\left ( x \\left \\vert \\theta \\right . \\right ) P _ { } \\left ( \\theta \\right ) \\end{align*}"}
-{"id": "1312.png", "formula": "\\begin{align*} \\iota ( x ) = ( x , x , \\dots ) + c _ 0 ( B ( H ) ) , \\end{align*}"}
-{"id": "466.png", "formula": "\\begin{align*} & a _ { k _ 1 , k _ 2 , \\pi / 2 } \\left ( \\lambda \\right ) = ( - 1 ) ^ { k _ 1 } i ^ { k _ 2 - n } \\left ( i \\frac { \\pi } { 2 } \\right ) ^ { n + k _ 1 + k _ 2 } \\lambda _ 1 ^ { k _ 1 } \\\\ & \\quad + ( - 1 ) ^ { k _ 1 } i ^ { k _ 2 - n } \\left ( i \\frac { \\pi } { 2 } \\right ) ^ { n + k _ 1 + k _ 2 - 1 } \\left ( ( n + k _ 1 + k _ 2 ) \\lambda _ 1 ^ { k _ 1 + 1 } + \\frac { k _ 1 } { 2 } \\lambda _ 1 ^ { k _ 1 - 1 } ( \\lambda ^ 2 - \\lambda _ 1 ^ 2 ) \\right ) + O \\left ( \\abs { \\lambda } ^ { k _ 1 + 2 } \\right ) . \\end{align*}"}
-{"id": "468.png", "formula": "\\begin{align*} y _ \\omega - \\frac { \\pi } { 2 } = \\frac { 1 } { 2 } \\left ( \\omega - \\frac { \\pi } { 2 } \\right ) + O \\left [ \\left ( \\omega - \\frac { \\pi } { 2 } \\right ) ^ 2 \\right ] \\end{align*}"}
-{"id": "4455.png", "formula": "\\begin{align*} \\ell \\frac { \\partial } { \\partial \\ell } [ \\phi ( \\ell k _ 1 , \\ell ^ \\frac { 3 } { 2 } k _ 2 ) ] = \\ell k _ 1 \\frac { \\partial \\phi } { \\partial k _ 1 } ( \\ell k _ 1 , \\ell ^ \\frac { 3 } { 2 } k _ 2 ) + \\frac { 3 } { 2 } \\ell ^ \\frac { 3 } { 2 } k _ 2 \\frac { \\partial \\phi } { \\partial k _ 2 } ( \\ell k _ 1 , \\ell ^ \\frac { 3 } { 2 } k _ 2 ) = : \\delta \\phi _ \\ell ( k ) \\end{align*}"}
-{"id": "4721.png", "formula": "\\begin{align*} g ^ { i j } ( S _ { , i } + A _ i ) ( S _ { , j } + A _ j ) = m ^ 2 , \\end{align*}"}
-{"id": "9459.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ { n } } { ( z q ^ n ; q ) _ { n + 1 } ( z q ^ { 2 n + 2 } ; q ^ 2 ) _ { \\infty } } = \\sum _ { n = 1 } ^ { \\infty } \\frac { z ^ { n - 1 } q ^ { n } } { ( q ; q ^ 2 ) _ n } . \\end{align*}"}
-{"id": "464.png", "formula": "\\begin{align*} p _ { 1 , k _ 1 , k _ 2 } ( x , t ) = \\sum _ { h = 0 } ^ N \\frac { 1 } { ( 2 h ) ! } | t | ^ { 2 h } p _ { 1 , k _ 1 , k _ 2 + 2 h } ( x , 0 ) + O \\left ( \\abs { t } ^ { 2 N + 2 } p _ { 1 , k _ 1 , k _ 2 + 2 N + 2 } ( x , 0 ) \\right ) . \\end{align*}"}
-{"id": "7471.png", "formula": "\\begin{align*} & \\rho _ { \\mathcal { T } } ( \\mathcal { Z } _ \\alpha ) = \\rho ^ k _ \\alpha \\partial _ k = : \\partial _ \\alpha , \\rho _ { \\mathcal { T } } ( \\mathcal { V } _ \\alpha ) = \\dot { \\partial } _ \\alpha , \\\\ & \\rho _ { \\mathcal { T } } ( \\mathcal { Z } _ { \\bar { \\alpha } } ) = \\rho ^ { \\bar { z } } _ { \\bar { \\alpha } } \\partial _ { \\bar { k } } = : \\partial _ { \\bar { \\alpha } } , \\rho _ { \\mathcal { T } } ( \\mathcal { V } _ { \\bar { \\alpha } } ) = \\dot { \\partial } _ { \\bar { \\alpha } } . \\end{align*}"}
-{"id": "7620.png", "formula": "\\begin{align*} \\mu \\Big ( E _ f ( Q _ { \\rho _ k } , c _ 0 \\lambda _ 0 M ^ k ) \\Big ) \\leq \\alpha ^ k \\mu \\big ( E _ f ( Q _ { 2 R } , c _ 0 \\lambda _ 0 ) \\big ) + \\sum _ { i = 0 } ^ { k - 1 } \\alpha ^ { k - i } \\Big [ \\nu \\Big ( E _ g ( Q _ { \\rho _ i } , c _ 0 \\lambda _ 0 M ^ i ) \\Big ) + \\hat \\nu \\Big ( E _ { \\hat g } ( Q _ { \\rho _ i } , c _ 0 \\lambda _ 0 M ^ i ) \\Big ) ^ { \\hat p } \\Big ] . \\end{align*}"}
-{"id": "1701.png", "formula": "\\begin{align*} T _ \\lambda ^ * f = \\frac { \\chi _ { D _ \\lambda } \\cdot ( f \\circ \\tau _ \\lambda ) } { f _ \\lambda \\circ \\tau _ \\lambda } . \\end{align*}"}
-{"id": "6464.png", "formula": "\\begin{align*} d s _ { } ^ { 2 } = { \\displaystyle \\sum \\limits _ { j = 1 } ^ { l } } \\frac { 1 } { \\sigma _ { 2 j - 1 } ^ { 2 } } \\left ( d \\mu _ { 2 j - 1 } ^ { 2 } + 2 \\rho _ { 2 j - 1 } d \\mu _ { 2 j - 1 } d \\sigma _ { 2 j - 1 } + 2 d \\sigma _ { 2 j - 1 } ^ { 2 } \\right ) \\end{align*}"}
-{"id": "5055.png", "formula": "\\begin{align*} \\Phi ^ { \\dagger } ( X ) = \\sum _ { i = 1 } ^ p A _ i ^ * X A _ i . \\end{align*}"}
-{"id": "7988.png", "formula": "\\begin{align*} \\P ( d _ { G } ( v ) \\leq M / n ) \\leq 2 \\exp ( - 5 0 \\log n / 1 2 ) = o ( n ^ { - 4 } ) . \\end{align*}"}
-{"id": "9254.png", "formula": "\\begin{align*} \\mathcal { L } ( \\mathfrak { b } , X ) : = ( \\mathfrak { g } \\otimes A ) \\oplus \\cdots \\oplus ( \\Lambda ' \\otimes E ' ) \\oplus \\prec \\mathfrak { b } , \\mathfrak { b } \\succ \\end{align*}"}
-{"id": "8208.png", "formula": "\\begin{align*} \\mathfrak { q } = \\prod _ { \\substack { u \\in \\mathcal { O } _ F ^ { \\times } / ( \\mathcal { O } _ F ^ { \\times } ) ^ 2 , \\\\ [ u ] \\neq [ 1 ] } } \\mathfrak { q } _ u \\end{align*}"}
-{"id": "2210.png", "formula": "\\begin{align*} F _ i ( z , t ) = \\bigl ( q _ i ( z ) + t \\cdot Q _ i ( z ) \\bigr ) , i = 1 , \\ldots , n . \\end{align*}"}
-{"id": "7355.png", "formula": "\\begin{align*} \\displaystyle { \\not } D \\psi = 0 \\end{align*}"}
-{"id": "6357.png", "formula": "\\begin{align*} \\frac { \\partial m _ { 2 k - 1 } } { \\partial \\alpha _ k } = \\frac { \\partial \\int x ^ k P _ { k - 1 } ( x ) \\ , d \\mu ( x ) } { \\partial \\alpha _ k } = \\beta _ 1 \\cdots \\beta _ { k - 1 } . \\end{align*}"}
-{"id": "9026.png", "formula": "\\begin{align*} ( \\nabla _ a \\nabla _ b - \\nabla _ b \\nabla _ a ) \\Sigma = 2 J _ { a b } \\Theta \\Sigma , \\end{align*}"}
-{"id": "936.png", "formula": "\\begin{align*} \\langle D F _ i , D F _ j \\rangle _ H = q ^ 2 \\sum _ { r = 1 } ^ q ( r - 1 ) ! \\binom { q - 1 } { r - 1 } ^ 2 I _ { 2 q - 2 r } ( f _ i \\widetilde { \\otimes } _ r f _ j ) . \\end{align*}"}
-{"id": "9744.png", "formula": "\\begin{align*} & K ^ { * } _ { 2 0 } > | K _ { b 0 } | , K ^ { * } _ { 2 i } > | K _ { b i } | , i = 2 , 3 , 5 , K ^ { * } _ { 2 4 } > C ^ { * } _ { 2 } , K ^ { * } _ { 1 4 } > C _ { 2 } ^ { * } , \\\\ & K ^ { * } _ { 1 1 } < \\frac { 1 - K ^ { * } _ { 2 5 } | K _ { 2 5 } | } { | K _ { 2 1 } | } , K ^ { * } _ { 1 5 } > K ^ { * } _ { 2 5 } | K _ { 1 5 } | + K ^ { * } _ { 1 1 } | K _ { 1 1 } | , \\end{align*}"}
-{"id": "8065.png", "formula": "\\begin{align*} P _ \\Delta ( t ) = \\sum _ { q = 0 } ^ r | \\mathcal { N } _ q | ( 1 + t ) ^ { r - q } t ^ q \\end{align*}"}
-{"id": "4697.png", "formula": "\\begin{align*} \\Phi ( 0 ^ + ) = \\Psi _ v ( 1 ) \\textrm { f o r s o m e } \\ , v \\ , 2 \\textrm { - h o m o g e n e o u s s o l u t i o n o f \\eqref { l a p l a c i a n o v = 1 } } . \\end{align*}"}
-{"id": "7705.png", "formula": "\\begin{align*} Y _ { i i } = \\frac { u ^ 2 _ { i j } } { 2 \\lambda ^ 2 _ i } \\ , . \\end{align*}"}
-{"id": "5667.png", "formula": "\\begin{align*} \\mathcal { C } ^ + : = \\{ { v } \\in L ^ 2 _ { l o c } ( \\R , \\R ^ n ) \\ ; : \\ ; s \\geq s _ 0 , \\ , | { v } ( s ) - a ^ + | \\leq E ( s ) \\} . \\end{align*}"}
-{"id": "3297.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty x ^ { \\lambda - 1 } ( 1 + \\alpha x ^ 2 ) ^ { - \\mu } ( 1 + \\beta x ^ 2 ) ^ { - \\nu } \\ , d x = \\frac { 1 } { 2 } \\alpha ^ { - \\frac { \\lambda } { 2 } } B \\left ( \\frac { \\lambda } { 2 } , \\mu + \\nu - \\frac { \\lambda } { 2 } \\right ) { _ 2 F _ 1 } \\left ( \\nu , \\frac { \\lambda } { 2 } ; \\mu + \\nu ; 1 - \\frac { \\beta } { \\alpha } \\right ) . \\end{align*}"}
-{"id": "3196.png", "formula": "\\begin{align*} V ( x + z ) - V ( x ) = \\log \\left ( 1 + \\frac { z } { 1 + x } \\right ) \\longrightarrow 0 \\quad x \\to \\infty . \\end{align*}"}
-{"id": "4497.png", "formula": "\\begin{align*} 1 = \\psi ( s ) > | \\phi ( s ) | \\end{align*}"}
-{"id": "8646.png", "formula": "\\begin{align*} \\ell ( x , y , X _ 0 , Y _ 0 ) = \\int _ 0 ^ \\infty 1 _ { \\{ | x + \\omega _ { X _ 0 } ( s ) - y - \\omega _ { Y _ 0 } ( s ) | \\leq 1 \\} } d s \\end{align*}"}
-{"id": "56.png", "formula": "\\begin{align*} \\hat { V } ^ { C } _ { \\sigma } ( C _ { 1 } , C _ { 2 } ) = \\left ( \\frac { 1 } { 2 \\pi ( \\sigma ' ) ^ 2 } \\right ) \\frac { 1 } { N } \\sum \\limits _ { i = 1 } ^ N e x p \\left ( - \\frac { ( x _ { i } - y _ { i } ) ^ 2 + ( z _ { i } - s _ { i } ) ^ 2 } { 2 ( \\sigma ' ) ^ 2 } \\right ) \\end{align*}"}
-{"id": "152.png", "formula": "\\begin{align*} \\sigma ^ 2 = 1 ; x = w ^ { \\sigma } ; w = x ^ { \\sigma } ; \\lambda _ { 2 1 } y ^ { \\sigma } = \\lambda _ { 1 2 } z ; \\lambda _ { 1 2 } z ^ { \\sigma } = \\lambda _ { 2 1 } y \\end{align*}"}
-{"id": "6847.png", "formula": "\\begin{align*} \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } f _ t ( x _ t ) - \\inf _ { x \\in X } \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } f _ t ( x ) . \\end{align*}"}
-{"id": "2141.png", "formula": "\\begin{align*} { \\mathcal L } _ d \\phi = \\mu \\phi \\quad \\mbox { i n } \\quad { \\bf R } _ + , \\phi \\in H ^ 1 ( { \\bf R } _ + , \\rho _ d ( \\xi ) \\ , d \\xi ) \\end{align*}"}
-{"id": "803.png", "formula": "\\begin{align*} T _ { k } ( x ) = S _ { k } ( 2 x ) - S _ { k } ( x ) , \\end{align*}"}
-{"id": "8226.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ k \\binom { k } { \\ell } ( - 1 ) ^ { \\ell + k } \\biggl [ \\frac { 1 } { n ^ { k - 1 } } \\biggr ] \\sum _ { m = 1 } ^ { k - 1 } ( - 1 ) ^ { m + 1 } \\frac { 1 } { m } \\bigl ( F _ { ( i + \\ell ) } - 1 \\bigr ) ^ m = \\frac { ( k - 2 ) ! } { r ^ { k - 1 } } \\end{align*}"}
-{"id": "7061.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l c l } - \\triangle u & = & \\lambda \\nabla F ( u ) & & B ^ N \\\\ \\frac { \\partial u } { \\partial \\nu } & = & 0 & & S ^ { N - 1 } . \\end{array} \\right . \\end{align*}"}
-{"id": "457.png", "formula": "\\begin{align*} L _ { j , \\psi _ \\omega } a _ { k _ 1 , k _ 2 , \\omega } = i ^ { - j } \\sum _ { \\mu = 0 } ^ { 2 j } \\frac { ( \\psi _ \\omega '' ( 0 ) ^ { - 1 } \\partial , \\partial ) ^ { \\mu + j } [ ( \\psi _ \\omega - P _ { 2 , 0 } \\psi _ \\omega ) ^ \\mu a _ { k _ 1 , k _ 2 , \\omega } ] ( 0 ) } { 2 ^ { \\mu + j } \\mu ! ( \\mu + j ) ! } . \\end{align*}"}
-{"id": "9402.png", "formula": "\\begin{gather*} R _ { i j k l } = \\bar { R } _ { i j k l } + ( h _ { i k } h _ { j l } - h _ { i l } h _ { j k } ) \\\\ h _ { i j , k } = h _ { i k , j } + \\bar { R } _ { \\vec { v } i j k } \\end{gather*}"}
-{"id": "473.png", "formula": "\\begin{align*} \\psi _ { \\pi / 2 } ''' ( 0 ) = \\pi u _ 1 \\otimes u _ 1 \\otimes u _ 1 + \\frac { 2 } { \\pi } \\sum _ { h = 2 } ^ m ( u _ 1 \\otimes u _ h \\otimes u _ h + u _ h \\otimes u _ 1 \\otimes u _ h + u _ h \\otimes u _ h \\otimes u _ 1 ) , \\end{align*}"}
-{"id": "7149.png", "formula": "\\begin{align*} & \\Delta _ J ( r J ) + V \\sum _ { m = r J + 1 } ^ { ( r + 1 ) J } D ( m , a _ m ^ * ) \\\\ \\leq & U J ^ 2 + V \\sum _ { m = r J + 1 } ^ { ( r + 1 ) J } D ( m , a _ m ^ * ) . \\end{align*}"}
-{"id": "82.png", "formula": "\\begin{align*} | E _ { w } ( \\psi ) | = | E _ { w } ( \\psi _ \\alpha ) - E _ { w } ( \\psi ) | = | F _ { \\psi _ { \\alpha } } ( w ) - F _ { \\psi } ( w ) | < \\epsilon , \\ \\ \\ \\alpha \\geq \\alpha _ 0 . \\end{align*}"}
-{"id": "8086.png", "formula": "\\begin{align*} m = y ^ { p _ 0 } \\prod _ { i = 1 } ^ \\ell y _ { S _ i } ^ { p _ i - p _ { i - 1 } } . \\end{align*}"}
-{"id": "3361.png", "formula": "\\begin{align*} \\mu ( m ) = \\bigvee _ { p \\in \\mu ( m ) } \\mu ( p ) = \\bigcup _ { p \\in \\mu ( m ) } \\mu ( p ) . \\end{align*}"}
-{"id": "9390.png", "formula": "\\begin{align*} T f ( x ) = \\lim _ { \\varepsilon \\rightarrow 0 ^ + } T _ \\varepsilon f ( x ) , \\ a . e . \\ x \\in \\mathbb R ^ n . \\end{align*}"}
-{"id": "698.png", "formula": "\\begin{align*} a _ n = | \\kappa _ n | | \\dot { \\Phi } ( \\mu _ n ) | \\end{align*}"}
-{"id": "274.png", "formula": "\\begin{align*} D _ { 0 _ { + } } ^ { \\alpha } x ( t ) = A x ( t ) + Q \\left ( t \\right ) x ( t ) + g ( t ) , \\end{align*}"}
-{"id": "374.png", "formula": "\\begin{align*} 1 - \\Phi ( x ) = o \\bigg ( \\sum _ { r , s \\in \\mathbb { Z } } \\mathbb { P } ( b _ { n , r , s } \\xi _ { 0 } \\geq x ) \\bigg ) . \\end{align*}"}
-{"id": "4489.png", "formula": "\\begin{align*} \\Theta = \\Theta _ 1 - \\Theta _ 2 \\end{align*}"}
-{"id": "220.png", "formula": "\\begin{align*} U _ { \\pi } H U _ { \\pi } ^ * = H ^ { P _ { \\pi } } . \\end{align*}"}
-{"id": "6352.png", "formula": "\\begin{align*} I ( x ) : = \\frac { 1 } { 2 } \\| \\operatorname { d i a g } ( W _ 1 '' ( y _ 1 ^ * ) , W _ 2 '' ( y _ 2 ^ * ) , W _ 1 '' ( y _ 1 ^ * ) , \\ldots ) ^ { 1 / 2 } D \\varphi _ k ^ { E } ( \\vec { y } ^ * ) ^ { - 1 } x \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "8555.png", "formula": "\\begin{align*} \\partial _ t u = \\Delta ( | u | ^ { m - 1 } u ) , x \\in { \\bf R } ^ N , \\ , \\ , t > 0 , u ( \\cdot , 0 ) = \\mu \\quad \\mbox { i n } \\quad { \\bf R } ^ N , \\end{align*}"}
-{"id": "5500.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { U _ m ( r ^ n z ) } { ( r ^ n z ) ^ { 1 - \\beta } \\ell ( r ^ n z ) } = p _ m ( z ) z \\in C _ { p _ m } , \\end{align*}"}
-{"id": "3479.png", "formula": "\\begin{align*} \\dot { w } ( t ) = \\begin{cases} i P , & t \\in [ ( i - 1 ) P , i P ) , \\ i \\in \\{ 1 , \\dots , 2 ^ K \\} \\\\ - ( i - 1 ) P , & t \\in [ ( i - 1 ) P , i P ) , \\ i \\in \\{ 1 , \\dots , 2 ^ K \\} . \\end{cases} \\end{align*}"}
-{"id": "9199.png", "formula": "\\begin{align*} & x . s = x s + s x ^ { t } x \\in s l _ { n } , \\ , s \\in S , \\\\ & x . \\lambda = x \\lambda + \\lambda x ^ { t } x \\in s l _ { n } , \\ , \\lambda \\in \\Lambda , \\\\ & x . s ' = - s ' x - x ^ { t } s ' x \\in s l _ { n } , \\ , s ' \\in S ' , \\\\ & x . \\lambda ' = - \\lambda ' x - x ^ { t } \\lambda ' x \\in s l _ { n } , \\ , \\lambda ' \\in \\Lambda ' . \\end{align*}"}
-{"id": "1824.png", "formula": "\\begin{align*} E ^ a _ { Q _ { 1 / 2 } , + , 0 } \\left ( m ^ a _ { Q _ { 1 / 2 } } e ^ { h m ^ a _ { Q _ { 1 / 2 } } } \\right ) & \\leq E ^ a _ { Q _ { 1 / 2 } , + , 0 } \\left ( e ^ { m ^ a _ { Q _ { 1 / 2 } } } e ^ { h m ^ a _ { Q _ { 1 / 2 } } } \\right ) \\\\ & = E ^ a _ { Q _ { 1 / 2 } , + , 0 } \\left ( e ^ { ( h + 1 ) m ^ a _ { Q _ { 1 / 2 } } } \\right ) . \\end{align*}"}
-{"id": "3483.png", "formula": "\\begin{align*} v \\big | _ { s = 0 } = - 2 f ' + H . \\end{align*}"}
-{"id": "4800.png", "formula": "\\begin{align*} L = \\left ( \\begin{array} { c c c | c } - z _ 2 & z _ 3 & 0 & 0 \\\\ - 1 & - 1 & 0 & z _ 1 \\\\ 0 & \\phantom { - } z _ 3 & - z _ 4 & z _ 5 \\\\ \\hline 1 + z _ 2 & 1 - 2 z _ 3 & z _ 4 & - z _ 1 - z _ { 5 } \\end{array} \\right ) \\end{align*}"}
-{"id": "3270.png", "formula": "\\begin{align*} \\gamma _ { } ( u , k ) = \\begin{cases} 1 , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , t _ { u , k } < t _ { } , \\\\ 0 , \\ , \\ , \\ , \\ , \\ , \\ , , \\end{cases} \\end{align*}"}
-{"id": "3697.png", "formula": "\\begin{align*} I _ { \\mathbf t } = \\{ i \\ ; | \\ ; t _ i = 0 \\} . \\end{align*}"}
-{"id": "2512.png", "formula": "\\begin{align*} \\frac { d } { d t } \\frac { 1 } { 2 } \\int h ^ 2 \\varrho \\ , m _ 0 = \\int h \\mathcal { L } h \\varrho \\ , m _ 0 = - \\int h ( \\xi \\cdot \\nabla _ x h ) \\varrho \\ , m _ 0 + \\int h \\nabla _ { \\xi } \\cdot [ \\sigma \\nabla _ { \\xi } h ] \\varrho \\ , m _ 0 - \\int \\psi ( \\xi ) h ^ 2 \\varrho \\ , m _ 0 . \\end{align*}"}
-{"id": "7264.png", "formula": "\\begin{align*} \\begin{cases} ( m , n ) & ( m , n ) \\mid h \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "4208.png", "formula": "\\begin{align*} \\tau = i \\rho \\textnormal { a n d } \\Omega _ j = i T _ j \\end{align*}"}
-{"id": "1359.png", "formula": "\\begin{align*} ( \\Delta _ { \\tau } ) _ { i j } = 1 - [ p _ { \\tau } ^ { [ 1 ] } ( \\Lambda ( Z _ { \\tau } ) ) ] _ { i j } , ( i , j ) \\in a \\times [ b _ S \\cup b _ L \\cup c ] \\mbox { o r } ( i , j ) \\in c \\times [ b _ U \\cup b _ S ] . \\end{align*}"}
-{"id": "9163.png", "formula": "\\begin{align*} r \\in \\mathbb { Q } \\longmapsto \\frac { m ( x + r ) } { x + r } = \\frac { m ( x ) + r } { x + r } \\end{align*}"}
-{"id": "9223.png", "formula": "\\begin{align*} D = \\langle \\mathfrak { b } , \\mathfrak { b } \\rangle = \\langle A ^ { + } , A ^ { + } \\rangle + \\langle A ^ { - } , A ^ { - } \\rangle + \\langle B , B ' \\rangle + \\langle C , C ' \\rangle + \\langle E , E ' \\rangle . \\end{align*}"}
-{"id": "9658.png", "formula": "\\begin{align*} M = \\begin{pmatrix} A & B \\\\ B ^ T & C \\end{pmatrix} \\in \\R ^ { N \\times N } , \\end{align*}"}
-{"id": "2511.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ t h = \\mathcal { L } h , \\quad \\mbox { w h e r e } \\mathcal { L } h = - \\xi \\cdot \\nabla _ x h + \\nabla _ { \\xi } \\cdot [ \\sigma \\nabla _ { \\xi } h ] - \\psi ( \\xi ) h , \\\\ & h ( 0 , x , \\xi ) = h _ 0 ( x , \\xi ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "3373.png", "formula": "\\begin{align*} \\nabla _ { X } ^ { \\Sigma ^ { a d \\mathbb { C } } } \\varphi = - \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { n } e _ i \\cdot B ( e _ i , X ) \\cdot \\varphi . \\end{align*}"}
-{"id": "2584.png", "formula": "\\begin{align*} ( f , g ) _ V : = \\int _ 0 ^ 1 ( V * f ) ( t ) g ( t ) \\ , d t \\end{align*}"}
-{"id": "7734.png", "formula": "\\begin{align*} H _ N ( l ) = \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N - 1 } \\frac { 1 - e ^ { 2 i l \\phi _ n } } { ( 1 - e ^ { 2 i \\phi _ n } ) ^ 2 } \\ , . \\end{align*}"}
-{"id": "1447.png", "formula": "\\begin{align*} \\mathrm { K e r } \\ , d _ 3 & = C \\{ 1 , y _ 1 , v _ 0 z _ 1 \\} \\oplus D _ 0 \\{ x _ 3 ^ 2 , x _ 3 ^ 2 y _ 1 , x _ 3 z _ 1 \\} , \\\\ \\mathrm { I m } \\ , d _ 3 & = D _ 0 \\{ v _ 1 x _ 3 ^ 2 , v _ 1 x _ 3 ^ 2 y _ 1 , v _ 1 x _ 3 z _ 1 \\} \\intertext { a n d } \\mathrm { K e r } \\ , d _ 3 / \\mathrm { I m } \\ , d _ 3 & = C \\{ 1 , y _ 1 , v _ 0 z _ 1 \\} \\oplus D _ 1 \\{ x _ 3 ^ 2 , x _ 3 ^ 2 y _ 1 , x _ 3 z _ 1 \\} . \\end{align*}"}
-{"id": "2019.png", "formula": "\\begin{align*} f _ { a } ( z _ 1 , z _ 2 ) = z _ 1 ^ { k _ a } + z _ 2 ^ { k _ a } + \\sum _ { j \\in \\Z \\setminus \\{ 0 , - k _ a \\} } u _ { a , j } ( z _ 1 ^ { - j } z _ 2 ^ { k _ a + j } + z _ 1 ^ { k _ a + j } z _ 2 ^ { - j } ) , ~ ~ ~ g _ { a , s } ( z ) = \\sum _ { j \\in \\Z } v _ { a , j , s } z ^ j . \\end{align*}"}
-{"id": "1518.png", "formula": "\\begin{align*} Q _ { S } ( x ) \\cdot \\left [ \\begin{array} { c c c } x ^ { 2 } & x & 1 \\\\ 1 & 0 & 0 \\\\ 0 & 1 & 0 \\end{array} \\right ] ^ { n } = \\left [ \\begin{array} { c c c } Q _ { T , n + 4 } ( x ) & P _ { T , n + 3 } ( x ) & Q _ { T , n + 3 } ( x ) \\\\ Q _ { T , n + 3 } ( x ) & P _ { T , n + 2 } ( x ) & Q _ { T , n + 2 } ( x ) \\\\ Q _ { T , n + 2 } ( x ) & P _ { T , n + 1 } ( x ) & Q _ { T , n + 1 } ( x ) \\end{array} \\right ] , \\end{align*}"}
-{"id": "5049.png", "formula": "\\begin{align*} \\tau _ { T _ i , R _ i } ^ D & = \\textstyle \\left ( 1 - t _ e \\right ) \\ , \\log _ 2 \\left ( 1 + \\frac { \\theta \\eta _ { _ { T _ i } } P _ 0 H _ { T _ i } G _ { T _ i , R _ i } \\ , t _ e } { \\sigma ^ 2 \\left ( 1 - t _ e \\right ) } \\right ) . \\end{align*}"}
-{"id": "7359.png", "formula": "\\begin{align*} \\nabla _ { X _ a } \\displaystyle { \\not } D \\psi = \\frac { n } { 2 } K _ a . \\psi \\end{align*}"}
-{"id": "8923.png", "formula": "\\begin{gather*} z _ k = \\vartheta ( z ; q ) _ k : = \\begin{cases} \\prod \\limits _ { 0 \\le i < k } \\vartheta ( i q + z ) , & k \\ge 0 , \\\\ \\prod \\limits _ { 1 \\le i \\le - k } \\vartheta ( - i q + z ) ^ { - 1 } , & k \\le 0 . \\end{cases} \\end{gather*}"}
-{"id": "8941.png", "formula": "\\begin{gather*} \\sum _ { w \\in S _ { n + 1 } / S _ n } \\ ! \\ ! \\ ! w \\cdot \\frac { \\prod _ { 1 \\le i \\le n + 2 } \\vartheta ( z _ { n + 1 } - y _ i ) \\prod _ { 1 \\le i \\le n } \\vartheta ( Y - z _ i ) } { \\prod _ { 1 \\le i \\le n } \\vartheta ( z _ { n + 1 } - z _ i ) } = \\prod _ { 1 \\le i \\le n + 2 } \\vartheta ( Y - y _ i ) , \\end{gather*}"}
-{"id": "8632.png", "formula": "\\begin{align*} \\begin{aligned} Q _ { k , k + 1 } & = \\int _ { \\tau _ k } ^ { \\tau _ { k } + 1 } \\int _ { \\tau _ k + 1 } ^ { \\tau _ { k } + 2 } R ( s - u , \\omega ( s ) - \\omega ( u ) ) d s d u \\\\ & = \\int _ 0 ^ 1 \\int _ 0 ^ 1 R ( s + 1 - u , \\omega ( \\tau _ k + 1 + s ) - \\omega ( \\tau _ k + u ) ) d s d u \\\\ & = \\int _ 0 ^ 1 \\int _ 0 ^ 1 R ( s + 1 - u , x _ k ( 1 ) + x _ { k + 1 } ( s ) - x _ k ( u ) ) d s d u . \\end{aligned} \\end{align*}"}
-{"id": "8710.png", "formula": "\\begin{align*} & J [ f ] ( t ) : = \\frac { f ( t ) - f ( 0 ) } { t ^ \\alpha \\Gamma ( 1 - \\alpha ) } \\quad \\\\ & K _ { ( 0 , t ) } [ f ] ( t ) : = \\frac { \\alpha } { \\Gamma ( 1 - \\alpha ) } \\int _ 0 ^ t ( f ( t ) - f ( t - \\tau ) ) \\frac { d \\tau } { \\tau ^ { \\alpha + 1 } } . \\end{align*}"}
-{"id": "7419.png", "formula": "\\begin{align*} a = a _ 0 a _ 1 a _ 2 , a ^ * = a _ 2 ^ * a _ 1 ^ * a _ 0 ^ * , b = a _ 3 ^ * a _ 3 , c = a _ 6 ^ * a _ 6 , d = a _ 2 ^ * a _ 2 . \\end{align*}"}
-{"id": "2336.png", "formula": "\\begin{align*} \\ell ( h ) = \\log N ( h ) . \\end{align*}"}
-{"id": "7873.png", "formula": "\\begin{align*} M ( ( 0 , 0 ) , ( \\pi , 0 ) ) = \\{ ( \\pi / 2 , 0 ) , ( 3 \\pi / 2 , 0 ) \\} . \\end{align*}"}
-{"id": "3749.png", "formula": "\\begin{align*} M _ { \\geq \\hat { k } } = \\bigcup _ { k \\geq \\hat { k } } M _ k \\end{align*}"}
-{"id": "3628.png", "formula": "\\begin{align*} c ( x ) = C d ^ p ( 1 + u ^ { q + 1 - p } ) , \\end{align*}"}
-{"id": "4007.png", "formula": "\\begin{align*} D _ { A } ( q \\Vert p ) & = \\frac { 1 } { \\int A \\left ( q \\right ) d \\mu } \\int \\left ( \\log _ { A } q - \\log _ { A } p \\right ) A \\left ( q \\right ) \\ d \\mu \\\\ & \\geq \\frac { 1 } { \\int A \\left ( q \\right ) d \\mu } \\int \\left ( q - p \\right ) \\ d \\mu = 0 . \\end{align*}"}
-{"id": "361.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( S _ { n } \\geq x _ { n } \\sigma _ { n } \\right ) = ( 1 - \\Phi ( x _ { n } ) ) ( 1 + o ( 1 ) ) n \\rightarrow \\infty ; \\end{align*}"}
-{"id": "9379.png", "formula": "\\begin{align*} C _ 1 = ( 4 m + d ) \\| h _ { 2 m } \\| _ { L ^ 2 ( \\partial B _ 1 ) } ^ 2 \\qquad C _ 2 = \\frac { \\lambda ( 2 m + 1 ) - \\lambda ( 2 m + \\frac 1 2 ) } { \\lambda ( 2 m + 1 ) } . \\end{align*}"}
-{"id": "8396.png", "formula": "\\begin{align*} f _ \\sigma = \\frac { | h | ^ 2 - \\frac { 1 } { n } | H | ^ 2 } { | H | ^ { 2 ( 1 - \\sigma ) } } , \\end{align*}"}
-{"id": "8969.png", "formula": "\\begin{gather*} \\sum _ { w ' \\in W _ J } \\sum _ { w \\in { } ^ J W } ( - 1 ) ^ { \\ell ( w w ' ) } w ^ { - 1 } c _ { W _ J w W _ I } w ^ { \\prime { - } 1 } = \\sum _ { w \\in W ^ J } \\sum _ { w ' \\in W _ J } ( - 1 ) ^ { \\ell ( w w ' ) } w c _ { W _ J w ^ { - 1 } W _ I } w ' \\end{gather*}"}
-{"id": "93.png", "formula": "\\begin{align*} | \\tilde { \\chi } _ n ( s , r , t ) | \\leq \\left ( \\sum _ { i = 1 } ^ n | a _ i ( s , r ) | ^ 2 \\right ) ^ { 1 / 2 } \\left ( \\sum _ { i = 1 } ^ n | b _ i ( r , t ) | ^ 2 \\right ) ^ { 1 / 2 } < C , \\end{align*}"}
-{"id": "4742.png", "formula": "\\begin{align*} { L } _ i = \\left ( 0 , { L } _ 1 ( x ^ 1 ) , \\alpha , \\beta \\right ) , \\alpha , \\beta , \\gamma - c o n s t , \\end{align*}"}
-{"id": "990.png", "formula": "\\begin{align*} \\left \\| \\max _ { j = 0 , 1 , \\dots , \\lfloor T / h \\rfloor } \\left | \\mathbf { A } _ n ( u _ j ) \\right | \\right \\| _ { \\psi _ 1 } \\lesssim h ^ \\gamma \\log n \\end{align*}"}
-{"id": "266.png", "formula": "\\begin{align*} \\sup _ I | f | & \\ge \\max _ { 0 \\le j \\le k } | f ( x _ j ) | = \\max _ { 0 \\le j \\le k } | P _ k ( x _ j ) | \\ge | a | 2 ^ { 1 - k } \\ge ( 2 M ) ^ { - k } . \\end{align*}"}
-{"id": "8286.png", "formula": "\\begin{align*} \\mathrm { w t } _ k V _ \\C = \\bigoplus _ { p + q \\le k } V ^ { ( p , q ) } , F ^ i V _ \\C = \\bigoplus _ { p \\ge i } V ^ { ( p , q ) } . \\end{align*}"}
-{"id": "4225.png", "formula": "\\begin{align*} f \\circ g = r ' \\circ s \\circ r \\circ s ' = r ' \\circ e \\circ s ' = r ' \\circ s ' \\circ r ' \\circ s ' = 1 _ { N ' } \\circ 1 _ { N ' } = 1 _ { N ' } , \\end{align*}"}
-{"id": "6548.png", "formula": "\\begin{align*} | S \\cap \\{ x \\in \\mathbb { R } ^ n : x _ 2 \\geq 1 - \\Delta \\} | _ n = \\frac { 1 } { n } ( 2 \\varepsilon \\Delta ) ^ { n - 1 } . \\end{align*}"}
-{"id": "9545.png", "formula": "\\begin{align*} \\Phi ^ 1 _ { \\phi ^ { - 1 } ( T ) } ( x ) = \\Phi ^ 2 _ T ( \\phi ( x ) ) \\end{align*}"}
-{"id": "5259.png", "formula": "\\begin{align*} E ^ s ( \\lambda ^ * , z ) = \\mathrm { s p } \\{ U , a _ s \\} = \\mathrm { s p } \\{ V , a _ u \\} = E ^ u ( \\lambda ^ * , z ) . \\end{align*}"}
-{"id": "6559.png", "formula": "\\begin{align*} G ( B _ { \\infty } ^ n ) = \\frac { \\alpha _ { \\xi } } { 2 } = \\frac { \\sqrt [ n ] { n ! } } { n } , \\end{align*}"}
-{"id": "6405.png", "formula": "\\begin{align*} g _ { \\mu \\nu } ^ { \\prime } \\left ( \\theta ^ { \\prime } \\right ) = \\int d x p ^ { \\prime } \\left ( x | \\theta ^ { \\prime } \\right ) \\partial _ { \\mu } ^ { \\prime } \\log p ^ { \\prime } \\left ( x | \\theta ^ { \\prime } \\right ) \\partial _ { \\nu } \\log p ^ { \\prime } \\left ( x | \\theta ^ { \\prime } \\right ) \\end{align*}"}
-{"id": "2460.png", "formula": "\\begin{align*} \\gamma = L ( \\Delta _ 1 , \\ldots , \\Delta _ k ; \\delta ( \\Delta _ { k + 1 } ; \\sigma ) ) \\end{align*}"}
-{"id": "6262.png", "formula": "\\begin{align*} | I _ { 2 } | \\leq & \\frac { \\mu } { 8 } \\sum _ { g \\geq - 1 } \\lambda _ q ^ { 2 s + 2 \\alpha } \\| b _ q \\| _ 2 ^ 2 + \\frac { \\nu } { 1 6 } \\sum _ { g \\geq - 1 } \\lambda _ q ^ { 2 s + 2 } \\| u _ q \\| _ 2 ^ 2 \\\\ & + C _ { \\nu , \\mu } \\left ( \\sum _ { q \\geq - 1 } \\lambda _ q ^ { 2 s } \\| b _ q \\| _ 2 ^ 2 \\right ) ^ { \\gamma _ 1 } + C _ { \\nu , \\mu } \\left ( \\sum _ { q \\geq - 1 } \\lambda _ q ^ { 2 s } \\| u _ q \\| _ 2 ^ 2 \\right ) ^ { \\gamma _ 2 } , \\end{align*}"}
-{"id": "6074.png", "formula": "\\begin{align*} \\Psi ^ { '' } ( t ) + ( n - 2 ) \\Psi ' ( t ) - \\mathfrak { e } _ k \\sin ( 2 \\Psi ( t ) ) = 0 . \\end{align*}"}
-{"id": "3310.png", "formula": "\\begin{align*} D ( r ) = \\sum _ { n = 1 } ^ N H \\left ( A _ n ^ { [ k ] } \\right ) . \\end{align*}"}
-{"id": "6452.png", "formula": "\\begin{align*} R _ { } ^ { } ( \\rho ) \\overset { } { = } \\frac { \\mathcal { C } _ { \\mathcal { M } } ( \\tau ) } { \\mathcal { C } _ { \\mathcal { M } } \\left . ( \\tau ) \\right \\vert _ { \\rho = 0 } } = \\sqrt { 3 } \\sqrt { \\frac { 1 - 2 \\rho ^ { 2 } } { 3 - 4 \\rho } } , \\end{align*}"}
-{"id": "5682.png", "formula": "\\begin{align*} \\mathcal { S } _ { s y m } ( a ^ - , a ^ + ) : = \\{ { v } \\in \\mathcal { S } ( a ^ - , a ^ + ) \\ ; : \\ ; t \\in \\R , \\ , { v } _ 1 ( - t ) = - { v } _ 1 ( t ) \\} , \\end{align*}"}
-{"id": "2426.png", "formula": "\\begin{align*} D _ 1 = D _ 3 = D _ 3 ' = 0 , \\ R _ { 1 1 3 } \\neq 0 , \\end{align*}"}
-{"id": "2919.png", "formula": "\\begin{align*} \\alpha _ a : = \\bigl \\{ e \\in E \\colon f _ a ( s ) \\in \\mathrm { d o m } ( \\rho _ e ) \\mbox { f o r } s = \\iota _ 1 ( e ) \\bigr \\} , \\end{align*}"}
-{"id": "2294.png", "formula": "\\begin{align*} \\partial _ t u + \\nu A ^ s u + \\mathcal { P } ^ \\alpha [ u \\cdot \\nabla u + \\mathcal { U } ^ \\alpha ( u , u ) ] = 0 , \\end{align*}"}
-{"id": "5556.png", "formula": "\\begin{align*} x = - \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { t _ { 1 } } \\left ( \\alpha + \\beta p \\left ( t _ { 2 } \\right ) \\right ) z \\left ( t _ { 2 } \\right ) d t _ { 2 } d t _ { 1 } + \\dot { x } _ { 0 } t + x _ { 0 } , 0 \\leqslant t _ 1 \\leqslant t \\leqslant \\tau \\end{align*}"}
-{"id": "5349.png", "formula": "\\begin{align*} Y ^ { \\circ } _ W ( v , x ) = Y _ W ( e ^ { x L ( 1 ) } ( - x ^ { - 2 } ) ^ { L _ 0 } v , x ^ { - 1 } ) , \\end{align*}"}
-{"id": "2015.png", "formula": "\\begin{align*} x _ i z _ { i + 1 } ( q ) - x _ { i + k + 1 } z _ i ( q ) = \\sum _ { \\beta \\in \\Z } g _ { \\beta } ( q ) x _ { j - ( k + 1 ) \\beta } z _ { j + \\beta } ( q ) , ~ ~ ~ i \\in \\Z . \\end{align*}"}
-{"id": "5759.png", "formula": "\\begin{align*} ~ F \\circ \\tilde { H } = G + \\gamma + q U . \\end{align*}"}
-{"id": "1260.png", "formula": "\\begin{align*} \\bar v _ l = \\frac { 1 - \\bar u _ l } { 1 - \\bar u _ l ( e _ n ) } \\mbox { f o r } \\ , \\ , l = 1 , 2 , \\dots . \\end{align*}"}
-{"id": "5363.png", "formula": "\\begin{align*} \\frac { S _ { ( r , s ) ( r ' , s ' ) } } { S _ { ( 1 , 1 ) ( r ' , s ' ) } } : = \\frac { _ { M _ { r , s } \\boxtimes _ { P ( 1 ) } M _ { r ' , s ' } } \\left ( c _ { M _ { r ' , s ' } , M _ { r , s } } \\circ c _ { M _ { r , s } , M _ { r ' , s ' } } \\right ) } { \\left ( M _ { r ' , s ' } \\right ) } , \\end{align*}"}
-{"id": "2957.png", "formula": "\\begin{align*} \\int f \\circ \\partial \\Theta ' d \\mu \\leq \\frac { 1 } { 2 } \\left ( \\int f \\circ \\partial \\Theta d \\mu + \\int f \\circ \\partial \\Theta _ 0 d \\mu \\right ) = \\int f \\circ \\partial \\Theta _ 0 d \\mu . \\end{align*}"}
-{"id": "3973.png", "formula": "\\begin{align*} p ^ { \\alpha _ 1 } _ { k - 1 } ( 1 , t ) = - ( - \\lambda ) ^ { k - 1 } \\underset { \\Theta ^ { k - 1 } _ { 1 } } { \\sum } \\frac { t ^ { k _ 0 \\alpha _ 0 + k _ 1 \\alpha _ 1 } } { \\Gamma \\left ( k _ 0 \\alpha _ 0 + k _ 1 \\alpha _ 1 + 1 \\right ) } . \\end{align*}"}
-{"id": "8247.png", "formula": "\\begin{align*} \\abs { [ \\pi ( s ) ( \\eta _ { \\mathfrak { L } } ) v ^ { \\circ } ( s ) ] ( g ) } = \\abs { [ \\pi ^ { \\mathfrak { L } } ( s ) ( \\eta _ { \\mathfrak { L } } ) \\tilde { v } ^ { \\circ } ( s ) ] ( g ) } = \\abs { [ v ^ { \\circ } _ { \\mathfrak { L } } ( s ) ] ( g ) } . \\end{align*}"}
-{"id": "3002.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } ( t , x ) & = \\Delta _ N u ( t , x ) + g ( u ( t , x ) ) \\xi ( t , x ) , \\\\ u ( 0 , x ) & = u _ 0 ( x ) . \\end{align*}"}
-{"id": "195.png", "formula": "\\begin{align*} \\hat { f } ( f ^ { \\ast } ( a ) ) = f ( a ) \\end{align*}"}
-{"id": "4604.png", "formula": "\\begin{align*} f = y _ 1 1 _ { [ 0 , x _ 1 ) } + \\cdots + y _ { d + 1 } 1 _ { [ x _ 1 + \\cdots + x _ d , 1 ) } \\end{align*}"}
-{"id": "5290.png", "formula": "\\begin{align*} I _ 2 \\leq & \\frac { | d _ 2 ( t ) | ^ 2 } { 2 \\varepsilon _ 2 } + \\frac { \\varepsilon _ 2 } { 2 } \\| u \\| ^ 2 + M _ 1 \\| u \\| ^ 2 + \\frac { M _ 2 } { 2 } ( u ^ 2 ( t , 1 ) - u ^ 2 ( t , 0 ) ) \\\\ = & \\frac { | d _ 2 ( t ) | ^ 2 } { 2 \\varepsilon _ 2 } + \\bigg ( \\frac { \\varepsilon _ 2 } { 2 } + M _ 1 \\bigg ) \\| u \\| ^ 2 + \\frac { M _ 2 } { 2 } ( u ^ 2 ( t , 1 ) - u ^ 2 ( t , 0 ) ) . \\end{align*}"}
-{"id": "9471.png", "formula": "\\begin{align*} & ( 1 + q - q ^ { 2 n } ) ( q ^ 2 ; q ^ 2 ) _ { n - 1 } - q ( 1 - q ^ { 2 n - 2 } ) ( q ^ 2 ; q ^ 2 ) _ { n - 2 } \\\\ & = ( q ^ 2 ; q ^ 2 ) _ n + q ( q ^ 2 ; q ^ 2 ) _ { n - 1 } - q ( q ^ 2 ; q ^ 2 ) _ { n - 1 } \\\\ & = ( q ^ 2 ; q ^ 2 ) _ n . \\end{align*}"}
-{"id": "7609.png", "formula": "\\begin{align*} A ^ t ( x , y ) = \\inf \\left \\{ \\int _ 0 ^ t L ( \\gamma , \\dot { \\gamma } ) d t : \\gamma ( 0 ) = x , \\ , \\gamma ( t ) = y \\right \\} . \\end{align*}"}
-{"id": "8171.png", "formula": "\\begin{align*} \\omega = \\left ( \\begin{matrix} 0 & 1 \\\\ - 1 & 0 \\end{matrix} \\right ) \\end{align*}"}
-{"id": "8581.png", "formula": "\\begin{align*} \\sup _ { \\tau < t < S _ { \\lambda ' } + \\tau } \\int _ { { \\bf R } ^ N } e ^ { - \\frac { 2 \\lambda ' H _ 0 ( y ) ^ 2 } { 1 - 4 \\lambda ' ( t - \\tau ) } } u _ { m , n } ( y , t ) ^ 2 \\ , d y & \\le C \\int _ { { \\bf R } ^ N } e ^ { - 2 \\lambda ' H _ 0 ( y ) ^ 2 } u _ { m , n } ( y , \\tau ) ^ 2 \\ , d y \\\\ & \\le C \\tau ^ { - \\frac { N } { 2 } } I _ { m , n } ^ 2 \\end{align*}"}
-{"id": "2804.png", "formula": "\\begin{align*} \\tilde { P } _ { n } ( t , S ) = ~ & p ( t , S ) + \\sum _ { i = 0 } ^ { n } r K e ^ { - r ( t - t _ { i } ) } \\aleph \\Big ( - d _ { 2 } ( S , t - t _ { i } , \\mathcal { B } _ { n } ( t _ { i } ) ) \\Big ) \\\\ & - \\sum _ { i = 0 } ^ { n } \\delta S e ^ { - \\delta ( t - t _ { i } ) } \\aleph \\Big ( - d _ { 1 } ( S , t - t _ { i } , \\mathcal { B } _ { n } ( t _ { i } ) ) \\Big ) . \\end{align*}"}
-{"id": "9276.png", "formula": "\\begin{align*} \\mathcal { A } : = \\Big \\{ ( \\underline { \\tau } , \\underline { \\xi } ) \\in \\Xi \\Big | N _ s ( \\underline { \\tau } ) < \\infty , \\ , s \\geq 0 \\Big \\} \\end{align*}"}
-{"id": "7481.png", "formula": "\\begin{align*} \\nabla _ { \\mathcal { X } _ \\alpha } Z ^ \\alpha & = \\delta _ \\alpha ( Z ^ \\alpha ) + Z ^ \\alpha L ^ { \\ : \\beta } _ { \\alpha \\beta } , \\\\ \\nabla _ { \\mathcal { V } _ \\alpha } V ^ \\alpha & = \\dot { \\partial } _ \\alpha ( V ^ \\alpha ) + V ^ \\alpha C ^ { \\ : \\beta } _ { \\alpha \\beta } . \\end{align*}"}
-{"id": "2756.png", "formula": "\\begin{align*} F _ 1 ( X , Y , Z ) & = ( X ^ m - Y ^ m ) ( Z ^ m - \\omega ^ 2 X ^ m ) ( Y ^ m - \\omega Z ^ m ) \\\\ F _ 2 ( X , Y , Z ) & = ( X ^ m - \\omega Y ^ m ) ( Z ^ m - X ^ m ) ( Y ^ m - \\omega ^ 2 Z ^ m ) \\\\ F _ 3 ( X , Y , Z ) & = ( X ^ m - \\omega ^ 2 Y ^ m ) ( Z ^ m - \\omega X ^ n ) ( Y ^ m - Z ^ m ) , \\end{align*}"}
-{"id": "7006.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { \\mathbb { R } ^ n } { \\frac { u ( y ) - u ( 0 ) } { | \\sqrt { M } ^ { - 1 } y | ^ { n + 2 s } } \\det { \\sqrt { M } ^ { - 1 } } d y } \\\\ & = \\prod _ { j = 1 } ^ { n } { \\lambda _ j } \\int _ { \\mathbb { R } ^ n } { \\frac { u ( y ) - u ( 0 ) } { | \\lambda _ 1 ^ 2 y _ 1 ^ 2 + . . . + \\lambda _ n ^ 2 y _ n ^ 2 | ^ { ( n + 2 s ) / 2 } } d y } \\\\ & = I _ 1 + I _ 2 \\\\ & \\geq \\frac { C _ 4 \\mu _ 0 } { 1 - s } \\epsilon ^ { - s } + 0 = \\frac { C _ 4 \\mu _ 0 } { 1 - s } \\epsilon ^ { - s } . \\end{aligned} \\end{align*}"}
-{"id": "8813.png", "formula": "\\begin{align*} | a _ { i _ { 0 } , p } | \\leq C _ { I } \\dfrac { \\sum _ { j = 1 } ^ { r } | D _ { j , i _ 0 , p } | } { | D _ { p } | } \\leq C _ { 0 } , \\end{align*}"}
-{"id": "5912.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\frac { \\sum _ { k = 1 } ^ m ( N + \\tau ^ { N , 1 } _ k ) ( c ) _ k } { \\sum _ { k = 1 } ^ m ( N + \\tau ^ { N , 1 } _ k ) ( c ) _ k + \\tau ^ { N , 0 } _ k } = \\frac { c ( N + E \\tau ^ { N , 1 } ) } { c ( N + E \\tau ^ { N , 1 } ) + E \\tau ^ { N , 0 } } \\ \\ \\end{align*}"}
-{"id": "7590.png", "formula": "\\begin{align*} I _ n \\left ( X ^ { - 1 } ; \\left ( X ^ i Z ) ^ { n - i } \\right ) _ { i = n - 1 } ^ { 0 } \\right ) = \\frac { 1 } { \\prod _ { j = 0 } ^ { n - 1 } ( 1 - X ^ j Z ) } . \\end{align*}"}
-{"id": "8199.png", "formula": "\\begin{align*} \\abs { y _ \\alpha } \\ll \\begin{cases} L & \\alpha = 1 , \\\\ 1 & \\alpha = \\alpha _ 1 ^ j \\alpha _ 2 ^ j j = 1 , 2 \\alpha _ 1 , \\alpha _ 2 \\in \\mathcal { P } ( L ) , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "699.png", "formula": "\\begin{align*} \\cot \\alpha = \\cot \\alpha _ 0 + \\sum _ { n = 0 } ^ { \\infty } \\cfrac { 1 } { | \\dot \\Phi ( \\mu _ n ^ 0 ) | } \\left ( \\cfrac { 1 } { | \\kappa _ n | } - \\cfrac { 1 } { | \\kappa _ n ^ 0 | } \\right ) , \\end{align*}"}
-{"id": "3474.png", "formula": "\\begin{align*} \\big \\| \\Xi ^ n \\big \\| _ V & = \\big \\| ( P _ h - I ) u ( t _ n ) \\big \\| _ V \\le \\big \\| P _ h ( I - Q _ h ) u ( t _ n ) \\big \\| _ V + \\big \\| ( Q _ h - I ) u ( t _ n ) \\big \\| _ V \\\\ & \\le \\big ( 1 + \\| P _ h \\| _ { \\L ( V ) } \\big ) \\| ( Q _ h - I ) u ( t _ n ) \\| = \\big ( 1 + \\| P _ h \\| _ { \\L ( V ) } \\big ) \\mathrm { d i s t } _ V ( u ( t _ n ) , V _ h ) , \\end{align*}"}
-{"id": "8799.png", "formula": "\\begin{align*} F _ { \\varpi _ { p } } = \\textbf { 1 } _ { X _ { p } } \\end{align*}"}
-{"id": "478.png", "formula": "\\begin{align*} p _ { 1 , k _ 1 , 0 } ^ { ( m ) } ( x , t ) = \\frac { ( - 1 ) ^ { \\frac { m - 1 } { 2 } } } { ( 2 \\pi ) ^ m ( 4 \\pi ) ^ n } \\int _ { 0 } ^ \\infty \\int _ { S ^ { m - 1 } } e ^ { i \\rho \\abs { t } ( \\sigma , u _ 1 ) } \\ , \\dd \\sigma \\ , e ^ { - R \\rho \\coth ( \\rho ) } a _ { k _ 1 , m - 1 } ( \\rho ) \\ , \\dd \\rho \\end{align*}"}
-{"id": "5352.png", "formula": "\\begin{align*} \\mbox { d i m } \\ ; \\mathcal { M } [ P ( z ) ] _ { W _ 1 W _ 2 } ^ { W _ 3 } = \\mathcal { N } _ { W _ 1 W _ 2 } ^ { W _ 3 } . \\end{align*}"}
-{"id": "7907.png", "formula": "\\begin{align*} \\min \\{ | x _ 1 - y _ 1 | , | x _ 2 - y _ 2 | \\} = 0 \\end{align*}"}
-{"id": "4770.png", "formula": "\\begin{align*} t _ 1 = 0 . \\end{align*}"}
-{"id": "3707.png", "formula": "\\begin{align*} \\pi \\sigma _ { 1 \\dots n } = \\delta ^ { ( n ) } \\end{align*}"}
-{"id": "3490.png", "formula": "\\begin{align*} \\int _ { \\alpha \\beta } \\omega _ 1 \\cdots \\omega _ k = \\sum _ { i = 0 } ^ k \\int _ { \\alpha } \\omega _ 1 \\cdots \\omega _ i \\int _ { \\beta } \\omega _ { i + 1 } \\cdots \\omega _ k . \\end{align*}"}
-{"id": "2291.png", "formula": "\\begin{align*} \\Omega \\subset \\mathbb { R } ^ n \\mbox { i s a s m o o t h b o u n d e d d o m a i n , o r } \\Omega = \\mathbb { R } ^ n , \\mbox { w i t h } n = 2 , 3 , \\end{align*}"}
-{"id": "3608.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta _ p u = f ( u ) & \\ , \\ , \\Omega \\setminus \\Gamma \\\\ u > 0 & \\ , \\ , \\Omega \\setminus \\Gamma \\\\ u = 0 & \\ , \\ , \\partial \\Omega , \\end{cases} \\end{align*}"}
-{"id": "1842.png", "formula": "\\begin{align*} \\mathcal { G F } _ { \\mathcal { B } } ( R ) = { } ^ \\perp ( \\mathcal { C } ( R ) \\cap ( \\mathcal { P G F } _ { \\mathcal { B } } ( R ) ) ^ \\perp ) . \\end{align*}"}
-{"id": "1037.png", "formula": "\\begin{align*} \\widehat { w } _ { n } ( \\zeta ) \\leq \\mathcal { D } _ { n } ( \\widetilde { \\Psi } ) : = \\frac { 2 \\widetilde { \\Psi } - n + 1 + \\sqrt { 4 \\widetilde { \\Psi } ^ { 2 } + 1 7 n ^ { 2 } - 1 6 \\widetilde { \\Psi } n + 8 \\widetilde { \\Psi } - 1 8 n + 5 } } { 2 } . \\end{align*}"}
-{"id": "9916.png", "formula": "\\begin{align*} e ( G ) \\le \\alpha ( G ) \\sum _ { i = 1 } ^ k d ( z _ i ) \\ , . \\end{align*}"}
-{"id": "5971.png", "formula": "\\begin{align*} s ( x _ i ^ { \\mathrm { c o l } } ) = f _ i i = 1 , \\ldots , N . \\end{align*}"}
-{"id": "1238.png", "formula": "\\begin{align*} k ( y ) ^ { p - 1 } = p \\frac { f ( \\nabla v ( y ) ) } { | \\nabla v ( y ) | } \\end{align*}"}
-{"id": "5623.png", "formula": "\\begin{align*} R _ r = S [ \\ ! [ y _ 1 , \\ldots , y _ r ] \\ ! ] / ( f + y _ 1 ^ { a _ 1 } + \\cdots y _ r ^ { a _ r } ) \\end{align*}"}
-{"id": "1619.png", "formula": "\\begin{align*} t _ \\lambda ^ * t _ \\eta = \\sum _ { ( \\alpha , \\beta ) \\in \\Lambda ^ { \\operatorname { m i n } } ( \\lambda , \\eta ) } t _ \\alpha t ^ * _ \\beta . \\end{align*}"}
-{"id": "753.png", "formula": "\\begin{align*} m ( a _ 1 - b _ 1 , a _ 2 - b _ 2 ) = m ( c _ 1 , c _ 2 ) + m ( a _ 1 - b _ 1 - c _ 1 , a _ 2 - b _ 2 - c _ 2 ) , \\end{align*}"}
-{"id": "4362.png", "formula": "\\begin{align*} B _ j ^ l = \\{ z \\in \\operatorname { s u p p } ( b _ j ) ; j \\Lambda _ 2 ( z ) \\} \\end{align*}"}
-{"id": "8588.png", "formula": "\\begin{align*} [ \\varphi _ \\pm ( c ) , e ] = 0 \\end{align*}"}
-{"id": "4399.png", "formula": "\\begin{align*} \\sigma _ { e s s } ( A ) = \\bigcup _ { x \\in \\mathcal { M } \\setminus \\mathbb { C } ^ n } \\sigma ( A _ x ) . \\end{align*}"}
-{"id": "9760.png", "formula": "\\begin{align*} | Z _ A ^ { ( n ) } - Z _ A ^ { ( n - 1 ) } | & = | Z _ { A , 0 } ( w ^ { ( n ) } - w ^ { ( n - 1 ) } ) \\int _ { 0 } ^ { 1 } \\mathrm { e x p } ( s w ^ { ( n ) } + ( 1 - s ) w ^ { ( n - 1 ) } ) d s | \\\\ & \\leq \\frac { Z _ { A , 0 } e ^ { - C _ * x } } { \\rho _ { A , 0 } u _ { A , 0 } A ( 0 ) } \\int _ 0 ^ x A ( \\tau ) | \\rho _ A ^ { ( n ) } \\phi ( T _ A ^ { ( n ) } ) - \\rho _ A ^ { ( n - 1 ) } \\phi ( T _ A ^ { ( n - 1 ) } ) | d \\tau \\\\ & \\leq O ( 1 ) \\delta _ 0 \\max \\limits _ { 0 \\leq \\tau \\leq x } ( | \\rho _ A ^ { ( n ) } - \\rho _ A ^ { ( n - 1 ) } | + | T _ A ^ { ( n ) } - T _ A ^ { ( n - 1 ) } | ) . \\end{align*}"}
-{"id": "4300.png", "formula": "\\begin{align*} ( k _ 1 , g _ 1 , n _ 1 ) < ( k _ 2 , g _ 2 , n _ 2 ) \\Longleftrightarrow & k _ 1 < k _ 2 \\\\ & \\big ( k _ 1 = k _ 2 g _ 1 < g _ 2 \\big ) \\\\ & \\big ( k _ 1 = k _ 2 g _ 1 = g _ 2 n _ 1 < n _ 2 \\big ) \\end{align*}"}
-{"id": "3448.png", "formula": "\\begin{align*} \\big \\| \\rho _ N ^ n ( u | _ N ) \\big \\| _ { L ^ 2 ( \\Omega ; \\R ^ d ) } & \\le \\big ( 1 + \\| u \\| _ { C ( [ 0 , T ] ; \\R ^ d ) } \\big ) \\Big ( 1 + T ^ { \\frac { 1 } { 2 } } \\big \\| L _ { K _ u } + g \\big \\| _ { L ^ 2 ( 0 , T ; \\R ) } \\Big ) \\\\ & \\times \\Big ( \\big \\| g \\big \\| _ { L ^ 2 ( t _ { n - 1 } , t _ n ; \\R ) } + \\big \\| L _ { K _ u } \\big \\| _ { L ^ 2 ( t _ { n - 1 } , t _ n ; \\R ) } \\Big ) k ^ { \\frac { 1 } { 2 } } \\end{align*}"}
-{"id": "2690.png", "formula": "\\begin{align*} L _ { \\infty } : = \\bigcup _ { n \\geq 0 } L ( \\pi ^ { { 1 / p } ^ n } _ L ) . \\end{align*}"}
-{"id": "5228.png", "formula": "\\begin{align*} \\begin{aligned} J A + A ^ T J & = \\left ( \\begin{array} { c c } \\lambda Q S ^ { - 1 } - f ' ( \\hat { u } ) & - c Q \\\\ 0 & - Q S \\end{array} \\right ) + \\left ( \\begin{array} { c c } - \\lambda S ^ { - 1 } Q + f ' ( \\hat { u } ) & 0 \\\\ c Q & S Q \\end{array} \\right ) \\\\ & = - c \\left ( \\begin{array} { c c } 0 & Q \\\\ - Q & 0 \\end{array} \\right ) = - c J . \\end{aligned} \\end{align*}"}
-{"id": "9469.png", "formula": "\\begin{align*} & q ^ { - n } \\big ( - S _ n ( 1 ) + S _ { n - 1 } ( 1 ) + q ^ n ( q ; q ^ 2 ) _ n \\big ) \\\\ & = q ^ { - n + 1 } \\big ( - S _ { n - 1 } ( 1 ) + S _ { n - 2 } ( 1 ) + q ^ { n - 1 } ( q ; q ^ 2 ) _ { n - 1 } \\big ) \\\\ & - q ^ { n - 1 } \\big ( S _ { n - 2 } ( 1 ) - q S _ { n - 1 } ( 1 ) + q ^ { n - 1 } ( 1 + q ) ( q ; q ^ 2 ) _ { n - 1 } \\big ) + q ^ { 2 n - 2 } ( q ; q ^ 2 ) _ { n - 1 } . \\end{align*}"}
-{"id": "1410.png", "formula": "\\begin{align*} H ^ { * } ( B G _ n ; \\mathbb { Z } / 2 ) = \\mathbb { Z } / 2 [ x _ 2 , x _ 3 , x _ { 4 1 } , \\dots , x _ { 4 n } ] \\end{align*}"}
-{"id": "8557.png", "formula": "\\begin{align*} & ( \\nabla v ) ( x ) = ( \\partial _ r v ^ \\sharp ) ( H _ 0 ( x ) ) \\nabla H _ 0 ( x ) , \\\\ & H ( ( \\nabla v ) ( x ) ) \\nabla _ \\xi H ( ( \\nabla v ) ( x ) ) = ( \\partial _ r v ^ \\sharp ) ( H _ 0 ( x ) ) \\nabla _ \\xi H ( \\nabla H _ 0 ( x ) ) = \\frac { ( \\partial _ r v ^ \\sharp ) ( H _ 0 ( x ) ) x } { H _ 0 ( x ) } , \\end{align*}"}
-{"id": "6203.png", "formula": "\\begin{align*} \\begin{pmatrix} 2 & 0 & 0 & 0 \\\\ 0 & 2 & 0 & 0 \\\\ 0 & 0 & - 2 ( k ^ 2 + l ^ 2 ) & 1 - 2 k m - 2 l n \\\\ 0 & 0 & 1 - 2 k m - 2 l n & - 2 ( m ^ 2 + n ^ 2 ) \\end{pmatrix} . \\end{align*}"}
-{"id": "6294.png", "formula": "\\begin{align*} \\alpha ( x , e ^ { i \\theta } ) : = ( - x , e ^ { - i \\theta } ) . \\end{align*}"}
-{"id": "5955.png", "formula": "\\begin{align*} \\vect { n } _ { u _ j } = \\psi _ { u _ j } \\nu _ { u _ 1 } + \\psi \\nu _ { u _ 1 u _ j } = \\psi ^ { - 1 } \\psi _ { u _ j } \\vect { n } \\end{align*}"}
-{"id": "6366.png", "formula": "\\begin{align*} f _ N ( z ) = \\frac { A _ N ( z ) } { B _ N ( z ) } = \\frac { 1 } { P _ N ( z ) } \\int \\frac { P _ N ( z ) } { z - t } \\ , d \\mu ( t ) = \\int \\frac { 1 } { z - t } \\ , d \\mu ( t ) , \\end{align*}"}
-{"id": "1064.png", "formula": "\\begin{align*} \\tau _ { k } = \\frac { q _ { k } } { 2 n - 2 } + q _ { k } \\cdot \\frac { 2 n - 2 - \\widehat { w } _ { n } ( \\zeta ) } { ( 2 n - 2 ) ( 1 + \\widehat { w } _ { n } ( \\zeta ) ) } = q _ { k } \\cdot \\frac { 2 n - 1 } { ( 2 n - 2 ) ( 1 + \\widehat { w } _ { n } ( \\zeta ) ) } . \\end{align*}"}
-{"id": "314.png", "formula": "\\begin{align*} & \\left \\{ \\eta _ i = k , \\ \\tau _ i = k - \\ell \\right \\} = \\left \\{ Z _ k \\ge Z _ { k - 1 } , \\ Z _ { k - 1 } < Z _ { k - 2 } < \\dots < Z _ { k - \\ell } , \\tau _ i = k - \\ell \\right \\} \\\\ & = \\left \\{ U _ k \\ge \\frac { L _ { k - 1 } } { L _ { k - 1 } + R _ { k - 1 } } , \\ U _ { k - 1 } < \\frac { L _ { k - 2 } } { L _ { k - 2 } + R _ { k - 2 } } , \\dots , U _ { k - \\ell + 1 } < \\frac { L _ { k - \\ell } } { L _ { k - \\ell } + R _ { k - \\ell } } , \\ \\tau _ i = k - \\ell \\right \\} . \\end{align*}"}
-{"id": "6205.png", "formula": "\\begin{align*} \\begin{pmatrix} - 2 & 0 & 1 \\\\ 0 & - 2 & 0 \\\\ 1 & 0 & 2 k \\end{pmatrix} \\begin{pmatrix} - 2 & 0 & 0 \\\\ 0 & - 2 & 1 \\\\ 0 & 1 & 2 k \\end{pmatrix} d = 2 + 8 k , \\end{align*}"}
-{"id": "3742.png", "formula": "\\begin{align*} \\mu = \\sum _ { z \\in \\Z ^ d } P _ z \\end{align*}"}
-{"id": "4328.png", "formula": "\\begin{align*} & \\Vert \\nabla n _ k \\Vert _ { L ^ 2 ( \\Omega ) } ^ 2 + \\big ( T ( \\varphi _ { p , n } ) n _ k , n _ k \\big ) = \\big ( T ( \\varphi _ { p , n } ) , n _ k \\big ) \\ , . \\end{align*}"}
-{"id": "8961.png", "formula": "\\begin{gather*} \\dim \\big ( \\Gamma \\big ( X / W , \\big ( M _ { k ( S ) } \\big ) ^ W ( d ) \\big ) \\big ) = \\sum _ i \\dim \\big ( \\Gamma \\big ( X / W , \\big ( M ^ i _ { k ( S ) } \\big ) ^ W ( d ) \\big ) \\big ) \\end{gather*}"}
-{"id": "9332.png", "formula": "\\begin{align*} \\left ( j _ { n } ^ { ( 3 ) } \\right ) ^ { 2 } - 9 \\left ( J _ { n } ^ { ( 3 ) } \\right ) ^ { 2 } = 2 ^ { n + 2 } j _ { n - 3 } ^ { ( 3 ) } . \\end{align*}"}
-{"id": "4763.png", "formula": "\\begin{align*} { L } _ i = \\left ( \\alpha , { L } _ 1 ( x ^ 1 ) , 0 , 0 \\right ) , { L } _ 1 = \\sqrt { \\beta t _ 1 - \\alpha ^ 2 \\omega _ 1 + \\gamma } . \\end{align*}"}
-{"id": "9964.png", "formula": "\\begin{align*} \\bar { \\gamma } _ { k i } = \\frac { \\rho _ { k i i } ^ { 2 } \\bar { \\upsilon } _ { k i } } { \\rho _ { k i i } { \\displaystyle \\sum _ { j = 1 } ^ { L } \\alpha _ { j } \\mathbb { E } \\left [ \\mathsf { \\Gamma } _ { j i } \\right ] + \\bar { \\upsilon } _ { k i } \\sum _ { j \\neq i } ^ { L } \\kappa _ { j } \\mathbb { E } \\left [ \\mathsf { \\Gamma } _ { j i } ^ { 2 } \\right ] } } \\end{align*}"}
-{"id": "8392.png", "formula": "\\begin{align*} \\frac { \\partial \\varphi } { \\partial t } ( x , t ) = - F ( W ( x , t ) ) \\nu ( x , t ) = - f ( \\lambda ( W ( x , t ) ) ) \\nu ( x , t ) , \\end{align*}"}
-{"id": "3104.png", "formula": "\\begin{align*} h _ { 0 , 0 } = ( 1 - p ) \\sum _ { j = 0 } ^ { m } h _ { j , 0 } + \\alpha h _ { \\hat { m } } . \\end{align*}"}
-{"id": "2405.png", "formula": "\\begin{align*} p _ 1 ( x _ 0 ) = \\frac { - R _ 2 \\left ( ( a _ 5 + 4 a _ 1 a _ 6 ) ^ 2 + ( a _ 3 - a _ 1 a _ 5 - 3 a _ 1 ^ 2 a _ 6 ) ^ 2 \\right ) } { a _ 6 ^ 2 \\left ( a _ 5 + a _ 1 \\left ( - a _ 3 + a _ 1 a _ 5 + 4 a _ 6 + 3 a _ 1 ^ 2 a _ 6 \\right ) \\right ) ^ 3 } . \\end{align*}"}
-{"id": "6736.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\Big | \\mathbb { P } ( R _ { N } > \\beta _ { N } t ) - \\mathbb { P } ( R _ { N } ^ { \\eta } > \\beta _ { N } t ) \\Big | = 0 . \\end{align*}"}
-{"id": "5416.png", "formula": "\\begin{align*} W _ { i , j , g H } = \\big \\{ ( v _ 1 , \\dots , v _ n ) \\mid v _ i = g v _ j \\big \\} \\subset ( V ^ H ) ^ n . \\end{align*}"}
-{"id": "4771.png", "formula": "\\begin{align*} { L } _ 0 = 0 , { L } _ 1 = L _ 1 ( x ^ 1 ) , { L } _ 2 = \\sqrt { \\beta t _ 2 } , { L } _ 3 = \\sqrt { \\beta t _ 3 } . \\end{align*}"}
-{"id": "9279.png", "formula": "\\begin{align*} { \\sigma } ^ { ( s , 0 ) } _ { \\varepsilon } \\Big ( \\big ( \\theta ^ { ( s , u ) } \\big ) ^ { - 1 } ( A ) \\cap \\big ( \\theta ^ { ( u , 0 ) } \\big ) ^ { - 1 } ( B ) \\Big ) = { \\sigma } _ \\varepsilon ^ { ( s , u ) } \\Big ( \\big ( \\theta ^ { ( s , u ) } \\big ) ^ { - 1 } ( A ) \\Big ) { \\sigma } _ \\varepsilon ^ { ( u , 0 ) } \\Big ( \\big ( \\theta ^ { ( u , 0 ) } \\big ) ^ { - 1 } ( B ) \\Big ) . \\end{align*}"}
-{"id": "4312.png", "formula": "\\begin{align*} Y _ { i , 0 } : = \\frac { 1 } { \\abs { \\Omega } } \\int _ { \\Omega } \\varphi _ { i , 0 } ( x ) \\ , d x \\end{align*}"}
-{"id": "10041.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\mu \\in \\Lambda ' / \\Lambda \\\\ Q _ \\mu \\mid Q } } R _ { \\Lambda } ( m / Q , \\mu ) = R _ { \\Lambda _ { { \\mathfrak q } ^ { - 1 } } } ( m , 0 ) = R _ { \\Lambda _ { { \\mathfrak q } } } ( m , 0 ) . \\end{align*}"}
-{"id": "5285.png", "formula": "\\begin{align*} \\| u ( t , \\cdot ) \\| \\leq & \\beta ( \\| { u _ 0 } \\| , t ) + \\gamma _ 1 ( \\| d _ 1 \\| _ { L ^ { \\infty } ( 0 , t ) } ) \\\\ & + \\gamma _ 2 ( \\| d _ 2 \\| _ { L ^ { \\infty } ( 0 , t ) } ) , \\ \\forall t \\geq 0 , \\end{align*}"}
-{"id": "5889.png", "formula": "\\begin{align*} \\{ Z ^ { N , i } _ n \\in \\{ 0 , - 1 \\} \\} = \\cap _ { m = ( n - 1 ) N } ^ { n N - 1 } \\{ \\frac { S ^ { ( i ) } _ { m , N + m } } N < r _ { i + 1 } \\} \\end{align*}"}
-{"id": "8600.png", "formula": "\\begin{align*} [ \\Gamma g ^ { - 1 } \\Gamma ] \\cdot [ \\Gamma h ^ { - 1 } \\Gamma ] : = \\sum _ { k = 1 } ^ { K } m _ { k } [ \\Gamma g _ { i ( k ) } h _ { j ( k ) } \\Gamma ] , \\end{align*}"}
-{"id": "6751.png", "formula": "\\begin{align*} E ( \\bar { V } _ { N } ) = E ( E ( \\bar { V } _ { N } | \\delta _ { 1 } ^ { N } ) ) = \\sum \\limits _ { n = 1 } ^ { N } \\frac { 1 } { N } \\Bigg [ ( M + \\Delta ) u _ { n } + M ( 1 - u _ { n } ) \\Bigg ] = \\Delta \\frac { U _ { N } } { N } + M . \\end{align*}"}
-{"id": "4306.png", "formula": "\\begin{align*} M = \\ , \\ldots + M _ { - 1 } z ^ { - 1 } + M _ 0 + M _ 1 z + \\ldots , \\end{align*}"}
-{"id": "3473.png", "formula": "\\begin{align*} \\Big ( k \\sum _ { n = 1 } ^ N \\big \\| E ^ n \\big \\| _ { L ^ 2 ( \\Omega ; V ) } ^ 2 \\Big ) ^ { \\frac { 1 } { 2 } } \\le \\Big ( k \\sum _ { n = 1 } ^ N \\big \\| \\Theta ^ n \\big \\| _ { L ^ 2 ( \\Omega ; V ) } ^ 2 \\Big ) ^ { \\frac { 1 } { 2 } } + \\Big ( k \\sum _ { n = 1 } ^ N \\big \\| \\Xi ^ n \\big \\| _ V ^ 2 \\Big ) ^ { \\frac { 1 } { 2 } } \\end{align*}"}
-{"id": "2441.png", "formula": "\\begin{align*} \\ell d c _ 1 c _ 2 n - 2 \\ell d c _ 1 c _ 2 q = q ^ 2 n - q ^ 2 d ( c _ 1 + c _ 2 ) \\end{align*}"}
-{"id": "10102.png", "formula": "\\begin{align*} S ( U , Y ) = \\pi ( U ) S ( \\xi , Y ) + \\pi ( Y ) S ( U , \\xi ) - \\pi ( U ) \\pi ( Y ) S ( \\xi , \\xi ) . \\end{align*}"}
-{"id": "7972.png", "formula": "\\begin{align*} L ( r ) : = e ^ { \\frac { 2 r ^ 2 } { 1 - \\nu ^ 2 } } . \\end{align*}"}
-{"id": "7172.png", "formula": "\\begin{align*} W ^ { x _ 0 } _ { 1 + s } ( r , u ) = \\frac { 1 } { r ^ n + 1 } \\int _ { B _ r ( x _ 0 ) } | \\nabla u | ^ 2 \\ , | x _ n | ^ a \\ , d x - \\frac { 1 + s } { r ^ { n + 2 } } \\int _ { \\partial B _ r ( x _ 0 ) } u ^ 2 \\ , | x _ n | ^ a \\ , d \\mathcal { H } ^ { n - 1 } . \\end{align*}"}
-{"id": "6965.png", "formula": "\\begin{align*} f ( A ) = \\inf _ { B \\in \\Gamma _ n } { \\{ D f ( B ) ( A - B ) + f ( B ) \\} } , \\end{align*}"}
-{"id": "3434.png", "formula": "\\begin{align*} \\| A \\| _ { \\L ( H , U ) } = \\sup _ { v \\in H , \\| v \\| _ H = 1 } \\| A v \\| _ { U } . \\end{align*}"}
-{"id": "6286.png", "formula": "\\begin{align*} \\| [ \\Delta _ q , F \\times \\nabla \\times ] G \\| _ r \\lesssim & \\| \\nabla F \\| _ \\infty \\| G \\| _ r \\lambda _ q ^ 3 \\int _ { \\R ^ 3 } | x - y | | \\nabla h ( \\lambda _ q ( x - y ) ) | \\ , d x \\\\ & + \\| \\nabla F \\| _ \\infty \\| G \\| _ r \\int _ { \\R ^ 3 } | h ( \\lambda _ q ( x - y ) ) | \\ , d x \\\\ \\lesssim & \\| \\nabla F \\| _ \\infty \\| G \\| _ r . \\end{align*}"}
-{"id": "5197.png", "formula": "\\begin{align*} H = \\{ M ^ { x } _ { \\rho } ( \\tau _ { 1 } ) \\ge M ^ { x } _ { \\rho } ( \\tau _ { 2 } ) \\} . \\end{align*}"}
-{"id": "8211.png", "formula": "\\begin{align*} M _ j & = \\frac { n ^ j r ^ j } { j ! } ( 1 + H _ j ) \\\\ H _ j & = \\sum _ { h = 1 } ^ { j - 1 } \\frac { a _ h ( r , j ) } { n ^ h } \\end{align*}"}
-{"id": "6108.png", "formula": "\\begin{align*} \\bar { f ^ { s ^ { \\alpha } } } \\left ( y \\right ) = \\frac { l \\left ( y \\right ) } { \\Gamma \\left ( 1 - \\alpha \\right ) } y ^ { - \\alpha } \\end{align*}"}
-{"id": "6728.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\mathbb { P } \\Big ( \\cup _ { l \\geq 2 } \\Big [ \\Gamma _ { l } < N ^ { 1 + \\delta } \\Big ] \\Big ) = 0 . \\end{align*}"}
-{"id": "6448.png", "formula": "\\begin{align*} p ( x | \\theta ) = \\frac { 1 } { \\sqrt { ( 2 \\pi ) ^ { n } \\det C } } \\exp \\left [ - \\frac { 1 } { 2 } ( x - \\mu ) ^ { t } \\cdot { C } ^ { - 1 } \\cdot ( x - \\mu ) \\right ] , \\end{align*}"}
-{"id": "3513.png", "formula": "\\begin{align*} 0 \\leq X ^ u { ( s _ i ) } , \\ i = 0 , 1 , \\cdots , n + 1 . \\end{align*}"}
-{"id": "9854.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } \\frac { 1 } { 2 \\pi } \\int \\hat { f } ( t ) e ^ { - \\epsilon | t | } e ^ { i x t } d t = f ( x ) . \\end{align*}"}
-{"id": "8174.png", "formula": "\\begin{align*} t _ { \\nu } = ( t _ { \\nu , 1 } - t _ { \\nu , 2 } ) / 2 s _ { \\nu } = t _ { \\nu , 1 } + t _ { \\nu , 2 } . \\end{align*}"}
-{"id": "973.png", "formula": "\\begin{align*} \\sup _ { t \\in [ a _ n , T - a _ n ] } \\left | \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\Psi _ h ( t _ { i - 1 } - t ) - \\int _ { - \\infty } ^ \\infty \\Psi ( s ) d s \\right | = O ( ( n h ) ^ { - 1 } ) \\end{align*}"}
-{"id": "6753.png", "formula": "\\begin{align*} a _ { i j } = \\left \\{ \\begin{array} { r l } 1 , & ( i , j ) \\in \\C { E } \\\\ 0 , & ( i , j ) \\notin \\C { E } \\end{array} \\right . \\quad \\forall \\ ; i , j \\in \\C { V } \\ ; . \\end{align*}"}
-{"id": "7903.png", "formula": "\\begin{align*} d ( x , y ) = \\| x - y \\| . \\end{align*}"}
-{"id": "1991.png", "formula": "\\begin{align*} \\sup _ { \\lambda / n \\leq s < t } \\frac { \\left \\vert \\alpha _ { n } \\left ( s ; t \\right ) - \\widehat { B } \\left ( s ; t \\right ) \\right \\vert } { s ^ { \\nu } } = o _ { \\mathbb { P } } \\left ( 1 \\right ) . \\end{align*}"}
-{"id": "7919.png", "formula": "\\begin{align*} | x _ i - y _ i | \\geq \\mu . \\end{align*}"}
-{"id": "3403.png", "formula": "\\begin{align*} z _ { j + 1 } ( \\zeta ) = z _ 0 ( 0 ) - \\int _ 0 ^ \\zeta x _ { j + 1 } ( t ) \\dot y _ { j + 1 } ( t ) \\ , d t , \\zeta \\in \\C , \\end{align*}"}
-{"id": "6564.png", "formula": "\\begin{align*} y _ 2 = 1 \\ \\ \\ \\ \\sqrt { 1 - \\varepsilon ^ 2 } y _ 1 + \\varepsilon y _ 2 = 1 , \\end{align*}"}
-{"id": "4611.png", "formula": "\\begin{align*} T _ f ^ n ( z _ i , t ) & = \\big ( T ^ n ( z _ i ) , t + f ( z _ i ) + \\cdots + f ( T ^ { n - 1 } z _ i ) \\big ) \\\\ & = \\big ( T ^ n ( z _ i ) , t + f ( x ) + \\cdots + f ( T ^ { n - 1 } x ) + f ( T ^ { \\ell _ i } z _ i ) - f ( T ^ { \\ell _ i } x ) \\big ) \\\\ & = V ^ v \\big ( T ^ n ( z _ i ) , t + f ( x ) + \\cdots + f ( T ^ { n - 1 } x ) \\big ) \\end{align*}"}
-{"id": "4221.png", "formula": "\\begin{align*} d _ z ( f ^ n ) & = d ^ { n - 1 } d _ z ( f ) + \\sum _ { k = 1 } ^ { n - 1 } d ^ { n - 1 - k } d _ { \\hat f ^ n ( z ) } ( f ) \\\\ & \\le d ^ { n - 1 } ( \\frac { d } { 2 } - 1 ) ( 1 + \\sum _ { k = 1 } ^ { n - 1 } d ^ { - k } ) \\\\ & < \\frac { d ^ n } { 2 } , \\end{align*}"}
-{"id": "381.png", "formula": "\\begin{align*} \\sum _ { r \\in \\mathbb { Z } , s > n } | b _ { n , r , s } | ^ p = O \\big ( n ^ { p ( 2 - \\beta ) + 2 } L ^ p ( n ) \\big ) , \\end{align*}"}
-{"id": "8982.png", "formula": "\\begin{gather*} D ^ { ( n ) } _ { q , t } ( d + 1 ) = D ^ { ( n ) } _ q ( ( d + 1 ) q / 2 \\pm u ; t ) D ^ { ( n ) } _ { q , t } ( d ) \\prod _ { 1 \\le i \\le n } \\vartheta ( z _ i \\pm u ) ^ { - 1 } , \\end{gather*}"}
-{"id": "1788.png", "formula": "\\begin{align*} A = \\emptyset ( A \\neq \\emptyset , V V ( \\inf ( A ) ) = \\sup ( V ) < + \\infty ) . \\end{align*}"}
-{"id": "1046.png", "formula": "\\begin{align*} \\max _ { 1 \\leq i \\leq j } H ( P _ { i } ) = H , \\max _ { 1 \\leq i \\leq j } \\vert P _ { i } ( \\zeta ) \\vert = H ^ { - w } . \\end{align*}"}
-{"id": "1420.png", "formula": "\\begin{align*} Q _ j ( x _ 4 ) & = x _ 3 x _ 4 ^ { 2 ^ { j - 1 } } + x _ { 2 } ^ { 2 ^ { j - 1 } } Q _ { j - 1 } ( x _ 4 ) + x _ { 3 } ^ { 2 ^ { j - 1 } } Q _ { j - 2 } ( x _ 4 ) \\\\ & = x _ 3 \\left ( x _ 4 ^ { 2 ^ { j - 1 } } + \\alpha _ { j - 1 , j - 2 } x _ 2 ^ { 2 ^ { j - 1 } } x _ 4 ^ { 2 ^ { j - 2 } } + \\sum _ { k = 0 } ^ { j - 3 } ( \\alpha _ { j - 1 , k } x _ 2 ^ { 2 ^ { j - 1 } } + \\alpha _ { j - 2 , k } x _ 3 ^ { 2 ^ { j - 1 } } ) x _ 4 ^ { 2 ^ k } \\right ) . \\end{align*}"}
-{"id": "5988.png", "formula": "\\begin{align*} f ^ { \\mathrm { c } } _ 1 = \\cos ( x ^ 2 ) , f ^ { \\mathrm { c } } _ 2 = \\cos ( 2 x ^ 2 ) , f ^ { \\mathrm { c } } _ 4 = \\cos ( 4 x ^ 2 ) . \\end{align*}"}
-{"id": "5256.png", "formula": "\\begin{align*} \\begin{aligned} E ^ s ( \\lambda , z ) = \\mathrm { s p } \\{ U ( \\lambda , z ) , a _ s ( \\lambda , z ) \\} \\\\ E ^ u ( \\lambda , z ) = \\mathrm { s p } \\{ V ( \\lambda , z ) , a _ u ( \\lambda , z ) \\} \\end{aligned} , \\end{align*}"}
-{"id": "2520.png", "formula": "\\begin{align*} \\begin{aligned} \\left | T _ 1 \\right | & \\lesssim \\int \\left < \\xi \\right > ^ { \\gamma } \\left | \\nabla _ { \\xi } h \\right | \\left | D ^ 2 _ \\xi h \\right | \\varrho \\ , m _ 0 \\\\ & \\lesssim \\varepsilon \\left \\| \\left < \\xi \\right > ^ { \\frac { \\gamma } { 2 } } \\left | D ^ 2 _ \\xi h \\right | \\right \\| _ { L ^ 2 ( \\varrho \\ , m _ 0 ) } + C ( \\varepsilon ) \\left \\| \\left < \\xi \\right > ^ { \\frac { \\gamma } { 2 } } \\left | \\nabla _ \\xi h \\right | \\right \\| _ { L ^ 2 ( \\varrho \\ , m _ 0 ) } . \\end{aligned} \\end{align*}"}
-{"id": "7848.png", "formula": "\\begin{align*} h _ { i } ( x _ { i } ) = \\frac { \\alpha ( \\theta _ { i } + \\theta _ { 3 } ) ( a + b x _ { i } ) \\eta ^ { \\alpha - 1 } ( x _ { i } ) e ^ { - \\eta ^ { \\alpha } ( x _ { i } ) } \\left ( \\Psi ( x _ { i } ) \\right ) ^ { \\theta _ { i } + \\theta _ { 3 } - 1 } } { 1 - \\left ( \\Psi ( x _ { i } ) \\right ) ^ { \\theta _ { i } + \\theta _ { 3 } } } , \\ i = 1 , 2 . \\end{align*}"}
-{"id": "2597.png", "formula": "\\begin{align*} \\Gamma ( a ) \\Gamma ( 1 - a ) & = \\frac \\pi { \\sin ( a \\pi ) } , & & \\\\ \\int _ 0 ^ { t } s ^ { \\alpha - 1 } ( t - s ) ^ { \\beta - 1 } \\ , d s & = \\frac { \\Gamma ( \\alpha ) \\Gamma ( \\beta ) } { \\Gamma ( \\alpha + \\beta ) } t ^ { \\alpha + \\beta - 1 } , & & t > 0 , \\end{align*}"}
-{"id": "4812.png", "formula": "\\begin{align*} m \\textbf { x } = \\pi ( \\theta ( m r ) ) = \\pi ( \\theta ( n s ) ) = n \\textbf { y } , \\end{align*}"}
-{"id": "8454.png", "formula": "\\begin{align*} S _ { \\chi } ( 1 , a \\varpi ^ l , m ) = 0 . \\end{align*}"}
-{"id": "5527.png", "formula": "\\begin{align*} \\ddot { x } + \\left ( \\alpha + \\beta p \\left ( t \\right ) \\right ) x = 0 p \\left ( t + \\tau \\right ) = p \\left ( t \\right ) \\end{align*}"}
-{"id": "7208.png", "formula": "\\begin{align*} v ^ r _ k ( x ) = \\frac { u _ k ( r x ) } { r } . \\end{align*}"}
-{"id": "9560.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in D _ n } { ( - 1 ) ^ { \\ell _ D ( \\sigma ) } x ^ { L _ { o e } } ( \\sigma ) } = 0 \\end{align*}"}
-{"id": "832.png", "formula": "\\begin{align*} k _ 1 ( n , m ) & = B ( n - 1 ) + \\underset { j = 1 } { \\overset { m - 1 } { \\sum } } ( - 1 ) ^ j \\binom { m - 1 } { j } B ( n - ( j + 1 ) ) \\\\ & = B ( n - 1 ) + \\underset { j = 2 } { \\overset { m } { \\sum } } ( - 1 ) ^ { j + 1 } \\binom { m - 1 } { j - 1 } B ( n - j ) \\\\ & = \\underset { j = 1 } { \\overset { m } { \\sum } } ( - 1 ) ^ { j + 1 } \\binom { m - 1 } { j - 1 } B ( n - j ) \\end{align*}"}
-{"id": "5308.png", "formula": "\\begin{align*} Z ^ { m } : = \\frac { B ( \\theta ^ { m } ) - B ( \\theta ^ { m - 1 } ) } { \\tau } . \\end{align*}"}
-{"id": "2960.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } g _ n ^ { - 1 } ( t ) = g ^ { - 1 } ( t ) . \\end{align*}"}
-{"id": "8336.png", "formula": "\\begin{align*} q _ { \\alpha ( \\lambda ) } ( v ) = e ^ { - 2 \\pi i [ \\lambda , v ] } . \\end{align*}"}
-{"id": "6765.png", "formula": "\\begin{align*} \\Delta _ g u + \\rho \\left ( \\frac { e ^ u } { \\int _ { \\mathbb { S } ^ 2 } e ^ u } - \\frac { 1 } { 4 \\pi } \\right ) = 0 , \\end{align*}"}
-{"id": "9655.png", "formula": "\\begin{align*} \\mathcal { L } \\left ( \\sum _ { \\beta = 1 } ^ { \\kappa } N _ { \\beta , \\rm { s } } ( t ) , \\omega \\right ) = \\sum _ { \\beta = 1 } ^ { \\kappa } & N _ { \\beta , \\rm { s } } ( t ) - \\omega \\left ( \\sum _ { \\beta = 1 } ^ { \\kappa } \\Delta \\Phi _ { \\rm { s t } , \\beta } ( t ) - \\Delta \\Phi _ { \\rm { s t } , \\max } ( t ) \\right ) . \\end{align*}"}
-{"id": "8512.png", "formula": "\\begin{align*} E : y ^ 2 = x ^ 3 + a x + b , \\ ; a , b \\in K . \\end{align*}"}
-{"id": "3408.png", "formula": "\\begin{align*} \\Gamma = \\langle \\tau _ x , \\tau _ y \\rangle \\cong \\Z _ 2 ^ 2 . \\end{align*}"}
-{"id": "5910.png", "formula": "\\begin{align*} \\frac { \\sum _ { k = 1 } ^ m T ^ { N , 1 } _ k } { \\sum _ { k = 1 } ^ m ( T ^ { N , 1 } _ k + T ^ { N , 0 } _ k ) } . \\end{align*}"}
-{"id": "2241.png", "formula": "\\begin{align*} p _ i = m _ { 1 i } + \\ldots + m _ { n i } - 1 + \\gamma _ i + 1 + ( m _ { 1 i } - 1 ) k _ 1 + \\ldots + ( m _ { n i } - 1 ) k _ n . \\end{align*}"}
-{"id": "5608.png", "formula": "\\begin{align*} & f ( X _ u ) ( v - u ) + \\sigma ( X _ u ) B ^ H _ { u , v } + D \\sigma ( X _ u ) \\sigma ( X _ u ) \\mathbb { B } ^ H _ { u , v } \\\\ & = \\int _ u ^ v f ( X _ r ) d r + \\int _ u ^ v \\sigma ( X _ r ) d \\mathbf { B } ^ H _ r + o ( | v - u | ) \\\\ & = X _ v - X _ u + o ( | v - u | ) . \\end{align*}"}
-{"id": "3940.png", "formula": "\\begin{align*} \\frac { \\partial ^ \\gamma u } { \\partial t ^ \\gamma } + u \\frac { \\partial u } { \\partial x } - \\nu \\frac { \\partial ^ 2 u } { \\partial x ^ 2 } = 0 , \\ , \\ , \\ , \\ , \\ , \\ , - 3 \\leq x \\leq 3 , \\ , \\ , \\ , \\ , \\ , 0 < \\gamma < 1 , \\ , \\ , \\ , \\ , t > 0 . \\end{align*}"}
-{"id": "1101.png", "formula": "\\begin{align*} \\Phi ( \\kappa ) = \\beta _ 1 C _ n ^ 2 \\left [ 1 + \\beta _ 2 ( \\kappa / \\kappa _ l ) - \\beta _ 3 ( \\kappa / \\kappa _ l ) ^ { 7 / 6 } \\right ] \\frac { \\mathrm { e x p } \\left ( - \\kappa ^ 2 / \\kappa _ l ^ 2 \\right ) } { ( \\kappa _ 0 ^ 2 + \\kappa ^ 2 ) ^ { 1 1 / 6 } } , \\end{align*}"}
-{"id": "6949.png", "formula": "\\begin{align*} f _ k ( D ^ 2 u ) ( x ) = ( \\sum _ { 1 \\leq i _ 1 < i _ 2 < . . . < i _ k \\leq n } { \\lambda _ { i _ 1 } \\lambda _ { i _ 2 } . . . \\lambda _ { i _ k } } ) ^ { 1 / k } , \\end{align*}"}
-{"id": "2187.png", "formula": "\\begin{align*} \\begin{cases} a \\odot b = \\cfrac { D + a b } { a + b } , a + b \\not = 0 \\cr a \\odot b = \\alpha , a + b = 0 \\end{cases} . \\end{align*}"}
-{"id": "5404.png", "formula": "\\begin{align*} a \\cdot y & = ( a b \\cdot c ^ { a c } ) ^ 3 \\in T \\\\ b \\cdot y & = ( ( a c ) ^ 3 ) ^ { b ( a c ) ^ 3 } \\in T \\\\ a b \\cdot y & = c ^ { a c a b c ( a c ) ^ 2 } \\in T . \\end{align*}"}
-{"id": "1333.png", "formula": "\\begin{align*} s _ \\gamma = d | \\gamma | - 2 N _ \\gamma . \\end{align*}"}
-{"id": "9073.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } x ^ { p _ { k } } f _ { k } ( x ^ { q _ { k } } ) = 0 , \\end{align*}"}
-{"id": "9564.png", "formula": "\\begin{align*} e ( A _ n , S ) \\ , : = \\ , \\sup _ { \\| f \\| _ F \\leq 1 } e ( A _ n , f ) \\ , . \\end{align*}"}
-{"id": "4779.png", "formula": "\\begin{align*} { L } _ i = \\sqrt { \\alpha a _ i + \\beta b _ i + \\gamma } , i , j \\mbox { = 0 . . . 3 . } \\end{align*}"}
-{"id": "3393.png", "formula": "\\begin{align*} \\alpha ( x , y ) = ( \\alpha _ x ( x ) , \\alpha _ y ( x , y ) ) , \\beta = ( \\beta _ x ( x , y ) , \\beta _ y ( y ) ) \\end{align*}"}
-{"id": "9305.png", "formula": "\\begin{align*} A ' : = \\{ ( p , q ) \\in V \\times \\R ^ 2 \\ : \\ ( p , g ( p ) , q ) \\in A \\} . \\end{align*}"}
-{"id": "6293.png", "formula": "\\begin{align*} \\delta _ 1 + \\delta _ 2 + \\delta _ 3 + \\delta _ 4 = \\delta , \\ \\ \\delta _ 1 , \\delta _ 2 , \\delta _ 3 , \\delta _ 4 > 0 \\\\ \\frac 1 { \\theta _ 1 } + \\frac 1 { \\theta _ 2 } + \\frac 1 { \\theta _ 3 } + \\frac 1 { \\theta _ 4 } = 1 , \\ \\ \\theta _ 1 = \\theta _ 3 = \\frac 2 { \\delta } , \\ \\ 1 < \\theta _ 2 , \\theta _ 4 < \\infty . \\end{align*}"}
-{"id": "5784.png", "formula": "\\begin{align*} - \\Delta _ { p } u = 0 \\ ; \\ ; \\Omega \\end{align*}"}
-{"id": "6324.png", "formula": "\\begin{align*} \\nu _ { l } ( f _ { k _ l } ) = \\mathbf { k l } ( f _ { k _ l } ) ( x _ l ) - E _ { s _ l } \\left [ \\mathbf { k l } ( f _ { k _ l } ) ( X _ l ) \\right ] \\end{align*}"}
-{"id": "4449.png", "formula": "\\begin{align*} \\lefteqn { ( - \\partial _ 1 ^ 2 - | \\partial _ 1 | ^ { - 1 } \\partial _ 2 ^ 2 ) w } \\\\ & + P \\Big ( F + v \\partial _ 2 R w + w \\partial _ 2 R v + w \\partial _ 2 R w + \\partial _ 2 \\frac { 1 } { 2 } R ( w + v ) ^ 2 - ( w + v ) \\partial _ 1 \\frac { 1 } { 2 } R ( w + v ) ^ 2 \\Big ) = 0 . \\end{align*}"}
-{"id": "9001.png", "formula": "\\begin{gather*} \\hat { D } = { \\cal D } ^ { ( n ) } _ { q , t } ( c + ( b + ( n - 1 ) a ) q / 2 ) D { \\cal D } ^ { ( n ) } _ { q , t } ( - c ) \\end{gather*}"}
-{"id": "6000.png", "formula": "\\begin{gather*} E _ 1 = \\frac { e _ 0 } { 4 } ( 1 - x + y - x y ) \\end{gather*}"}
-{"id": "7346.png", "formula": "\\begin{align*} ( q ( \\gamma ) , r ( \\gamma ) ) \\ ; : = \\ ; \\textstyle \\Big ( \\frac { 4 ( \\gamma + 1 ) } { \\gamma - 1 } , \\frac { 3 ( \\gamma + 1 ) } { \\gamma + 2 } \\Big ) \\end{align*}"}
-{"id": "4268.png", "formula": "\\begin{align*} ( x - a ) ^ p ( x - a _ p ) = \\alpha \\overline { \\alpha } = f ^ 2 - x ( x - 1 ) ( x - \\lambda ) g ^ 2 . \\end{align*}"}
-{"id": "7918.png", "formula": "\\begin{align*} \\max _ { B _ r ( z ) } f = f ( x ) > f ( z ) . \\end{align*}"}
-{"id": "4617.png", "formula": "\\begin{align*} | U _ i | \\ge \\sum _ { n = 2 ^ { k _ i - 1 } } ^ { 2 ^ { k _ i } - 1 } 1 _ { [ - B + b , B - b ] } \\left ( t + \\sum _ { j = 0 } ^ { n - 1 } f ( T ^ j x ) \\right ) \\ge \\beta \\sum _ { n = 0 } ^ { 2 ^ { k _ i } - 1 } 1 _ { [ - B + b , B - b ] } \\left ( t + \\sum _ { j = 0 } ^ { n - 1 } f ( T ^ j x ) \\right ) \\end{align*}"}
-{"id": "471.png", "formula": "\\begin{align*} \\partial ^ { k _ 1 - 1 } _ 1 \\partial _ h ^ 2 a _ { k _ 1 , k _ 2 , \\pi / 2 } ( 0 ) = ( - 1 ) ^ { k _ 1 } { i ^ { k _ 2 - n } \\left ( i \\frac { \\pi } { 2 } \\right ) ^ { n + k _ 1 + k _ 2 - 1 } k _ 1 ! } \\end{align*}"}
-{"id": "9903.png", "formula": "\\begin{align*} \\sum _ { x \\in V ( G ) } \\bigl ( 2 r - 2 - d _ K ( x ) \\bigr ) = \\sum _ { v \\in K } \\bigl ( 2 n - d ( v ) \\bigr ) \\le \\tfrac { ( 6 + \\alpha ) ( r - 1 ) } { 3 r - 2 } n < 2 n \\end{align*}"}
-{"id": "5198.png", "formula": "\\begin{align*} \\{ \\tau _ { 3 } \\le t \\} = ( \\{ \\tau _ { 1 } \\le t \\} \\cap H ) \\cup ( \\{ \\tau _ { 2 } \\le t \\} \\cap H ^ { c } ) \\in \\mathcal { F } _ { t } \\end{align*}"}
-{"id": "7384.png", "formula": "\\begin{align*} J _ { j } = \\sum _ { i = 0 } ^ { \\big [ \\frac { N - j } { 2 } \\big ] } B _ { 2 i + j } \\binom { 2 i + j } { i } , \\Delta _ m = \\sum _ { i = 0 } ^ { \\big [ \\frac { N - m } { 2 } \\big ] } B _ { 2 i + m } \\binom { 2 i + m } { i } . \\end{align*}"}
-{"id": "3533.png", "formula": "\\begin{align*} u ^ { \\theta , \\rho } ( t ) = \\left \\{ \\begin{array} [ c ] { c l } { u } ^ { \\theta } ( t ) , & t \\in \\lbrack 0 , T ] \\backslash E _ { \\rho } , \\\\ u , & t \\in E _ { \\rho } , \\end{array} \\right . \\end{align*}"}
-{"id": "1793.png", "formula": "\\begin{align*} \\frac { c ( r - 2 ) } { a ( p - 2 ) } t _ 1 ^ { \\frac { r - p } { q - p } } = \\frac { b ( q - 2 ) } { a ( p - 2 ) } t _ 1 - 1 \\end{align*}"}
-{"id": "8539.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\| W _ l f _ \\phi \\| ^ 2 _ { H ^ { \\frac { s } { 4 ( 4 + s ) } } _ x L ^ 2 _ v } d t \\lesssim \\eta ^ { - 8 } \\| f | _ { t = 0 } \\| _ { L ^ 2 } ^ 2 + \\eta ^ { 2 s } \\| f _ \\phi \\| ^ 2 _ { L ^ 2 ( [ 0 , T ] ; L ^ 2 _ x H ^ s _ v ) } + \\eta ^ { - 8 } ( \\| g \\| _ { L ^ 2 ( [ 0 , T ] ; L ^ 2 _ x H ^ { - 1 } _ v ) } ^ 2 + \\| f \\| _ { L ^ 2 ( [ 0 , T ] ; L ^ 2 ) } ^ 2 ) . \\end{align*}"}
-{"id": "1799.png", "formula": "\\begin{align*} x _ 1 ' = \\frac { x _ 1 x _ 2 x _ 3 } { x _ 1 x _ 3 } = x _ 2 ; x _ 2 ' = \\frac { x _ 1 x _ 2 x _ 3 } { x _ 1 x _ 2 } = x _ 3 ; x _ 3 ' = \\frac { x _ 1 x _ 2 x _ 3 } { x _ 2 x _ 3 + x _ 3 ^ 2 } = \\frac { x _ 1 x _ 2 } { x _ 2 + x _ 3 } . \\end{align*}"}
-{"id": "9952.png", "formula": "\\begin{align*} { \\cal N } ( v ) - V ( H _ 1 ) = { \\cal N } ( \\varphi ( v ) ) - V ( H _ 2 ) . \\end{align*}"}
-{"id": "2166.png", "formula": "\\begin{align*} \\sup _ { 0 \\le r \\le ( 1 + t ) ^ { \\frac { 1 } { 2 } - \\theta _ * } } | u _ * ( r , t ) | = O ( t ^ { - \\frac { d } { 2 } } ) \\end{align*}"}
-{"id": "1363.png", "formula": "\\begin{align*} \\min _ { t \\in [ 0 , 1 ] } \\psi ( t ; c , a , b , c _ 0 ) = a - \\frac { 1 } { c } \\frac { b } { ( \\sqrt { c } + \\sqrt { c _ 0 } ) ^ 2 } \\ , , \\end{align*}"}
-{"id": "3325.png", "formula": "\\begin{align*} D ^ * ( r ) = \\left \\{ \\begin{array} { l l } ( 1 - r ) \\left ( 1 + \\frac { 1 } { N } + \\frac { 1 } { N ^ 2 } \\right ) - r \\left ( 2 + \\frac { 1 } { N } \\right ) , & 0 \\leq r \\leq \\frac { 1 } { 1 + N + N ^ 2 } \\\\ ( 1 - r ) \\left ( 1 + \\frac { 1 } { N } \\right ) - r , & \\qquad \\frac { 1 } { 1 + N + N ^ 2 } \\leq r \\leq \\frac { 1 } { 1 + N } \\\\ 1 - r , & \\qquad \\frac { 1 } { 1 + N } \\leq r \\leq 1 \\end{array} \\right . \\end{align*}"}
-{"id": "6787.png", "formula": "\\begin{align*} W _ { \\lambda , k } = w _ i \\eta _ { \\lambda ^ { \\alpha } , \\xi _ k } + w _ o ( 1 - \\eta _ { \\lambda ^ { \\alpha } , \\xi _ k } ) , \\end{align*}"}
-{"id": "6152.png", "formula": "\\begin{align*} \\phi = \\dd t ^ 1 \\wedge \\dd t ^ 2 \\wedge \\dd t ^ 3 - \\dd t ^ 1 \\wedge \\omega _ 1 - \\dd t ^ 2 \\wedge \\omega _ 2 - \\dd t ^ 3 \\wedge \\omega _ 3 \\end{align*}"}
-{"id": "662.png", "formula": "\\begin{align*} & \\tilde { U } p ' = s - \\lim _ { F \\subset _ { f i n i t e } \\mathcal { P } } \\sum _ { p \\in F } \\tilde { z _ p } \\sigma ( p ) U p p ' = \\tilde { z } _ { p ' } \\sigma ( p ' ) U p ' \\\\ & = \\sigma ( p ' ) s - \\lim _ { F \\subset _ { f i n i t e } \\mathcal { P } } \\sum _ { p \\in F } \\tilde { z } _ p \\sigma ( p ) U p = \\sigma ( p ' ) \\tilde { U } , ~ ~ \\forall p ' \\in \\mathcal { P } , ~ \\end{align*}"}
-{"id": "2320.png", "formula": "\\begin{align*} & \\ ; \\| u _ * - e ^ { - t \\nu A ^ s } u _ 0 \\| _ { L ^ \\infty _ T ( D ( A ) ) \\cap L ^ 2 _ T ( D ( A ^ { 1 + s / 2 } ) ) } \\leq \\sum _ { j = 1 } ^ \\infty ( C _ 2 M ) ^ { j - 1 } C _ 1 M ^ 2 = : C _ 6 M ^ 2 , \\\\ & \\ ; u _ * - e ^ { - t \\nu A ^ s } u _ 0 \\in B ^ \\beta _ { \\sum _ { j = 1 } ^ \\infty ( C _ 3 M ) ^ { j + 1 } , T } = : B ^ \\beta _ { C _ 7 M ^ 2 , T } . \\end{align*}"}
-{"id": "1169.png", "formula": "\\begin{align*} e _ k ( w ) & = \\sum _ { i = 0 } ^ { k - 2 } T _ { k , i } [ 1 ] _ w ^ { 2 ^ i } \\cr & = \\sum _ { i = 0 } ^ { k - 2 } T _ { k , i } \\left ( w + w ^ 2 \\right ) ^ { 2 ^ i } \\cr & = T _ { k , k - 2 } w ^ { 2 ^ { k - 1 } } + \\sum _ { i = 1 } ^ { k - 2 } \\left ( T _ { k , i } + T _ { k , i - 1 } \\right ) w ^ { 2 ^ i } + T _ { k , 0 } w \\mbox { . } \\end{align*}"}
-{"id": "3585.png", "formula": "\\begin{align*} B _ + ( z , Q _ 1 ) : = \\sum _ { i > 0 } \\alpha _ i [ 1 ] Q _ 1 ^ { - i } z ^ i , B _ - ( z , Q _ 1 ) : = \\sum _ { i < 0 } \\alpha _ i [ 1 ] Q _ 1 ^ { - i } z ^ i \\end{align*}"}
-{"id": "1584.png", "formula": "\\begin{align*} V ( \\mathbf { j } ) & = \\sum _ { k = 1 } ^ { n - 1 } [ ( n - k + 1 ) j _ k - s ( j _ k ) ] - c ( j _ n , - \\alpha ( n ) - 1 ) \\\\ & \\leq \\sum _ { k = 1 } ^ { n - 1 } [ ( n - k + 1 ) j _ k - s ( j _ k ) ] ] = v ( \\mathbf { j } ) \\\\ & < v ( \\mathbf { j } ' ) = V ( \\mathbf { j } ' ) . \\end{align*}"}
-{"id": "5787.png", "formula": "\\begin{align*} D u = \\lim _ { k \\rightarrow \\infty } \\nabla \\left ( T _ k ( u ) \\right ) . \\end{align*}"}
-{"id": "3114.png", "formula": "\\begin{align*} Z = \\overline { \\bigcup _ { \\sigma \\in \\Sigma } B ^ { [ \\sigma - s s t ] } } . \\end{align*}"}
-{"id": "1004.png", "formula": "\\begin{align*} \\sup _ { t \\in [ 0 , T ] } \\left | \\widetilde { Z } _ n ^ * ( t ) \\right | = O _ { p } ( \\log ^ { 3 / 2 } n ) , w ( \\widetilde { Z } _ n ^ * ; n ^ { - 1 } ) = O _ { p } \\left ( \\frac { \\log ^ { 3 / 2 } n } { n h } \\right ) \\end{align*}"}
-{"id": "8927.png", "formula": "\\begin{gather*} \\sum _ { x \\in \\phi _ { N , 1 } ^ { - 1 } ( \\sigma _ N ) } x = N \\sigma _ 1 , \\end{gather*}"}
-{"id": "9015.png", "formula": "\\begin{align*} \\| f \\| _ { B _ { p , r } ^ { s } } = \\left \\{ \\aligned & \\Big ( \\sum _ { j \\geq - 1 } 2 ^ { j r s } \\| \\Delta _ { j } f \\| _ { L ^ { p } } ^ { r } \\Big ) ^ { \\frac { 1 } { r } } , \\forall \\ r < \\infty , \\\\ & \\sup _ { j \\geq - 1 } 2 ^ { j s } \\| \\Delta _ { j } f \\| _ { L ^ { p } } , \\forall \\ r = \\infty . \\\\ \\endaligned \\right . \\end{align*}"}
-{"id": "8732.png", "formula": "\\begin{align*} & \\Phi _ { \\varepsilon , \\delta , \\sigma } ( t , x , s , y ) : = u ^ { \\varepsilon , \\delta } ( t , x ) - v _ { \\varepsilon , \\delta } ( s , y ) - \\varphi _ { \\delta , \\sigma } ( t , x , s , y ) \\quad \\\\ & \\varphi _ { \\delta , \\sigma } ( t , x , s , y ) : = \\varphi _ \\delta ( t , x , s , y ) + a _ \\sigma t + \\zeta _ \\sigma \\cdot x - b _ \\sigma s - \\xi _ \\sigma \\cdot y \\end{align*}"}
-{"id": "4589.png", "formula": "\\begin{align*} \\theta = - \\frac { 1 } { n - 1 } ( \\delta \\omega ) \\circ J , \\end{align*}"}
-{"id": "3963.png", "formula": "\\begin{align*} p ^ { \\beta _ n } ( n , t ) = p ^ { \\beta _ n } ( n , 0 ) - \\lambda ( I _ t ^ { \\beta _ n } p ^ { \\beta _ n } ( n , t ) - I _ t ^ { \\beta _ { n - 1 } } p ^ { \\beta _ { n - 1 } } ( n - 1 , t ) ) , \\ \\ n \\geq 0 , \\end{align*}"}
-{"id": "2407.png", "formula": "\\begin{align*} & 8 a _ 5 ^ 2 - 8 1 a _ 6 ^ 2 = 4 a _ 4 - 9 a _ 6 = 9 a _ 1 a _ 6 + a _ 5 = 8 a _ 1 a _ 5 + 9 a _ 6 = 8 a _ 1 ^ 2 - 1 = 0 , \\\\ & 3 a _ 1 ^ 2 a _ 6 + a _ 1 a _ 5 + 2 a _ 3 + a _ 4 + a _ 6 = - a _ 1 a _ 6 + a _ 2 = 0 . \\end{align*}"}
-{"id": "2926.png", "formula": "\\begin{align*} \\pi _ { 0 } \\circ \\pi = \\pi _ { u } , \\end{align*}"}
-{"id": "1523.png", "formula": "\\begin{align*} \\displaystyle G ( x ) = - x ^ 2 \\cdot ( m ^ 2 + 2 m ) / { 2 } + x \\cdot ( m ^ 2 t + m t - m ) / { 2 } . \\end{align*}"}
-{"id": "7910.png", "formula": "\\begin{align*} z = \\begin{cases} \\big ( \\left [ { ( x _ 1 + y _ 1 ) / 2 } \\right ] , { ( x _ 2 + y _ 2 ) / 2 } - e _ 1 \\big ) , & \\\\ \\big ( { ( x _ 1 + y _ 1 ) / 2 } - e _ 2 , \\left [ { ( x _ 2 + y _ 2 ) / 2 } \\right ] \\big ) , & \\end{cases} \\end{align*}"}
-{"id": "1011.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( E _ h ^ i \\mid E _ h ^ 1 , E _ h ^ 2 , \\cdots , E _ h ^ { i - 1 } \\right ) = \\mathbb { P } \\left ( E _ h ^ i \\mid E _ h ^ { i - 1 } \\right ) , \\ ; \\ ; i = 2 , \\ldots , N + 1 \\end{align*}"}
-{"id": "2720.png", "formula": "\\begin{align*} \\partial _ t g + \\frac { 1 } { \\epsilon } v \\cdot \\nabla _ x g = \\frac { 1 } { \\epsilon ^ 2 } \\mathcal L ( g ) , \\end{align*}"}
-{"id": "2472.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle \\sigma ( \\xi ) = \\int _ { \\R ^ { 3 } } \\Phi ( \\xi - \\xi _ { * } ) \\mu ( \\xi _ { * } ) d \\xi _ { * } \\ , , \\end{array} \\end{align*}"}
-{"id": "5072.png", "formula": "\\begin{align*} \\omega _ { 1 2 } = \\frac { B _ { 1 2 , 1 } } { b _ 1 - b _ 2 } \\omega _ 1 , ~ \\omega _ { 1 3 } = \\frac { B _ { 1 3 , 1 } } { b _ 1 - b _ 3 } \\omega _ 1 , ~ \\omega _ { 2 3 } = \\frac { B _ { 2 3 , 2 } } { b _ 2 - b _ 3 } \\omega _ 2 + \\frac { B _ { 2 3 , 3 } } { b _ 2 - b _ 3 } \\omega _ 3 . \\end{align*}"}
-{"id": "2233.png", "formula": "\\begin{align*} J _ \\gamma ( t ) = \\frac { ( - 1 ) ^ n } { ( 2 \\pi \\sqrt { - 1 } ) ^ n } \\int \\limits _ { \\widetilde \\Gamma _ h } w _ 1 ^ { \\gamma _ 1 + 1 } \\ldots w _ n ^ { \\gamma _ n + 1 } \\cdot \\frac { d \\widetilde F _ 1 } { \\widetilde F _ 1 } \\wedge \\ldots \\wedge \\frac { d \\widetilde F _ n } { \\widetilde F _ n } \\end{align*}"}
-{"id": "5572.png", "formula": "\\begin{align*} \\alpha + \\beta p _ { n } = - \\frac { 2 ^ { 2 k + 2 } } { \\tau ^ { 2 } } \\end{align*}"}
-{"id": "407.png", "formula": "\\begin{align*} p _ s ( x , t ) = \\frac { 1 } { ( 4 \\pi ) ^ n ( 2 \\pi ) ^ m s ^ { n + m } } \\int _ { \\R ^ m } e ^ { \\frac { i } { s } ( \\lambda , u _ 1 ) \\abs { t } - \\frac { \\abs { x } ^ 2 } { 4 s } | \\lambda | \\coth ( | \\lambda | ) } \\left ( \\frac { | \\lambda | } { \\sinh | \\lambda | } \\right ) ^ n \\ , \\dd \\lambda . \\end{align*}"}
-{"id": "3064.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ \\varphi } = \\inf \\left \\{ \\lambda > 0 : \\rho _ { \\varphi } \\left ( \\frac { f } { \\lambda } \\right ) \\leqslant 1 \\right \\} \\end{align*}"}
-{"id": "9206.png", "formula": "\\begin{align*} [ s \\otimes c , \\lambda ' \\otimes e ' ] = s \\diamond \\lambda ' \\otimes \\frac { [ c , e ' ] _ { A ^ { + } } } { 2 } + [ s , \\lambda ' ] \\otimes \\frac { ( c \\circ e ' ) _ { A ^ { - } } } { 2 } . \\end{align*}"}
-{"id": "4239.png", "formula": "\\begin{align*} \\varphi ^ { m } ( E _ 0 , \\theta _ 0 ) = ( E , \\theta ) _ { m f _ 1 } \\otimes ( \\det ( E _ 0 ) ^ \\frac { 1 - p ^ { m f _ 1 } } { r } , 0 ) . \\end{align*}"}
-{"id": "6746.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\bar { E } \\Big [ \\Big ( \\mathbb { P } ( \\Theta > N ^ { \\gamma } t ) \\Big ) ^ { 2 } \\Big ] = e ^ { - 2 t } . \\end{align*}"}
-{"id": "2878.png", "formula": "\\begin{align*} \\zeta ( 2 m ) = ( - 1 ) ^ { m + 1 } \\frac { ( 2 \\pi ) ^ { 2 m } B _ { 2 m } } { 2 ( 2 m ) ! } \\end{align*}"}
-{"id": "2892.png", "formula": "\\begin{align*} x _ { F , n } ( k ) = \\begin{cases} 0 & k \\notin F k \\neq n \\\\ 1 & k \\in F \\\\ 2 & k = n \\end{cases} \\end{align*}"}
-{"id": "5956.png", "formula": "\\begin{align*} f ( t , w _ 2 , \\dots , w _ n ) = \\sigma ( t ) + w _ 2 \\vect { e } _ 2 ( t ) + \\cdots + w _ n \\vect { e } _ n ( t ) . \\end{align*}"}
-{"id": "7581.png", "formula": "\\begin{align*} \\rho ^ { ( t ) } _ { N , l } = \\sum _ { k = l + 1 } ^ { l + N } 2 ^ { - k t } \\rho _ k , \\omega ^ { ( t ) } _ { N } = \\sum _ { k = N + 1 } ^ { 2 N } 2 ^ { - k t } \\omega _ k , \\Omega ^ { ( t ) } _ { N } = \\sum _ { k = 2 } ^ { N + 1 } 2 ^ { - 2 ^ k t } \\omega _ { 2 ^ k } , \\end{align*}"}
-{"id": "1289.png", "formula": "\\begin{align*} \\left . \\frac { d } { d t } \\mbox { C a p } _ { \\mathcal { A } } ( E _ t ) \\right | _ { t = 0 } \\ , & = ( p - 1 ) \\int _ { \\mathbb { S } ^ { n - 1 } } ( h _ 1 ( \\xi ) - h _ 0 ( \\xi ) ) d \\mu ( \\xi ) \\\\ & = ( n - p ) [ \\mbox { C a p } _ { \\mathcal { A } } ( E _ 1 ) - \\mbox { C a p } _ { \\mathcal { A } } ( E _ 0 ) ] . \\end{align*}"}
-{"id": "3467.png", "formula": "\\begin{align*} \\mathrm { d i s t } _ V ( v , V _ h ) & : = \\inf _ { v _ h \\in V _ h } \\| v - v _ h \\| _ V . \\end{align*}"}
-{"id": "7248.png", "formula": "\\begin{align*} \\Phi _ { \\pi } ^ { \\psi } = \\psi * _ { U } \\Theta _ { \\pi } . \\end{align*}"}
-{"id": "9663.png", "formula": "\\begin{align*} p = R \\rho T , e = c _ v T , \\gamma = 1 + \\frac { R } { c _ v } > 1 , \\end{align*}"}
-{"id": "7916.png", "formula": "\\begin{align*} u _ 0 ( x ) = \\begin{cases} ( x _ 1 + 1 ) x _ 2 & , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "9178.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal C _ { W } = \\{ u | ~ u u - u _ G \\in C ^ { \\infty } ( \\overline { 2 P } ) u W \\} . \\end{aligned} \\end{align*}"}
-{"id": "8852.png", "formula": "\\begin{align*} V ( X _ N , \\phi _ N ) = o \\left ( N \\sigma ( C ( \\cdot , \\phi _ N ) ) \\right ) N \\to \\infty \\end{align*}"}
-{"id": "4506.png", "formula": "\\begin{align*} | \\phi ( \\Delta _ n ) | ^ 2 = \\phi ( \\Delta _ n ) ^ * \\phi ( \\Delta _ n ) \\leq \\phi ( \\Delta _ n ^ * \\Delta _ n ) \\leq 1 \\end{align*}"}
-{"id": "1292.png", "formula": "\\begin{align*} \\left . b _ 0 \\ , \\frac { d } { d t } \\mbox { C a p } _ { \\mathcal { A } } ( E _ t ) \\right | _ { t = 0 } \\ , = ( p - 1 ) \\int _ { \\mathbb { S } ^ { n - 1 } } ( h _ 1 ( \\xi ) - h _ 0 ( \\xi ) ) d \\mu ( \\xi ) = \\left . b _ 1 \\ , \\frac { d } { d t } \\mbox { C a p } _ { \\mathcal { A } } ( E _ t ) \\right | _ { t = 1 } . \\end{align*}"}
-{"id": "4971.png", "formula": "\\begin{align*} f _ J : = \\sum _ { j = - J } ^ J \\Delta _ j f . \\end{align*}"}
-{"id": "788.png", "formula": "\\begin{align*} \\varLambda _ r = \\begin{cases} \\ , \\ \\ , r \\log ( 2 x ) - n \\log { \\frac { y } { ( 2 x ) ^ B } } & , \\\\ \\ , | r | \\log ( 2 x ) - n \\log { \\frac { ( 2 x ) ^ B } { y } } & . \\end{cases} \\end{align*}"}
-{"id": "8851.png", "formula": "\\begin{align*} V ( X _ N , \\phi ) = o \\left ( N \\right ) N \\to \\infty \\end{align*}"}
-{"id": "8563.png", "formula": "\\begin{align*} & \\sup _ { t _ 1 < s < t } \\int _ { \\Omega \\ , \\cap \\ , B _ { H _ 0 } ( x , R _ 2 ) } z ( y , s ) ^ 2 \\zeta _ x ^ 2 \\ , d y + \\iint _ { Q _ 2 } H ( \\nabla ( z \\zeta _ x ) ) ^ 2 \\ , d y \\ , d s \\\\ & \\qquad \\le C [ ( R _ 2 - R _ 1 ) ^ { - 2 } + ( t _ 1 - t _ 2 ) ^ { - 1 } ] \\iint _ { Q _ 2 } z ( y , s ) ^ 2 \\ , d y \\ , d s \\end{align*}"}
-{"id": "5444.png", "formula": "\\begin{align*} X \\stackrel { \\mathcal { D } } { = } A X + B , \\end{align*}"}
-{"id": "1400.png", "formula": "\\begin{align*} \\alpha f ^ { * } ( v ' ) = \\alpha ' f ^ { * } ( v '' ) \\end{align*}"}
-{"id": "9182.png", "formula": "\\begin{align*} \\displaystyle { \\frac { 1 } { 2 } \\left ( z + \\frac { 1 } { z } \\right ) } = x , \\end{align*}"}
-{"id": "108.png", "formula": "\\begin{align*} \\Psi ( p , u ) = \\int _ { \\Lambda \\in \\mathrm { A d s } } \\psi _ \\Lambda ( u | _ \\Lambda ) \\ , d \\mu _ p ( \\Lambda ) . \\end{align*}"}
-{"id": "5831.png", "formula": "\\begin{align*} \\| w _ n \\| _ { L ^ 2 ( \\R ) } ^ 2 & = \\sum _ { j = 1 } ^ J \\| \\phi ^ j \\| _ { L ^ 2 ( \\R ) } ^ 2 + \\| r _ n ^ J \\| _ { L ^ 2 ( \\R ) } ^ 2 + o ( 1 ) , \\\\ \\| w _ n \\| _ { \\dot H ^ 1 ( \\R ) } ^ 2 & = \\sum _ { j = 1 } ^ J \\| \\phi ^ j \\| _ { \\dot H ^ 1 ( \\R ) } ^ 2 + \\| r _ n ^ J \\| _ { \\dot H ^ 1 ( \\R ) } ^ 2 + o ( 1 ) , \\end{align*}"}
-{"id": "4404.png", "formula": "\\begin{align*} \\operatorname { O s c } _ z ^ r ( f ) : = \\sup \\{ | f ( z ) - f ( w ) | ; w \\in \\mathbb { C } ^ n , | z - w | \\leq r \\} \\end{align*}"}
-{"id": "5635.png", "formula": "\\begin{align*} B _ d ( x , r ) = \\{ y \\in X \\ ; : \\ ; d ( y , x ) < r \\} \\overline { B } _ d ( x , r ) = \\{ y \\in X \\ ; : \\ ; d ( y , x ) \\leq r \\} . \\end{align*}"}
-{"id": "1724.png", "formula": "\\begin{align*} t _ \\lambda ^ * \\pi ( f ) = \\pi ( f \\circ \\sigma _ \\lambda ) t _ \\lambda ^ * . \\end{align*}"}
-{"id": "3578.png", "formula": "\\begin{align*} \\bar \\xi _ x = \\eta ( x ) \\cdot \\xi _ { u ( x ) } \\end{align*}"}
-{"id": "3987.png", "formula": "\\begin{align*} p ^ { \\beta _ 2 } ( 2 , t ) = \\sum _ { k = 2 } ^ { \\infty } ( - \\lambda ) ^ k \\underset { \\Omega ^ { k } _ { 2 } } { \\sum } \\frac { t ^ { k _ 0 \\beta _ 0 + k _ 1 \\beta _ 1 + k _ 2 \\beta _ 2 } } { \\Gamma \\left ( k _ 0 \\beta _ 0 + k _ 1 \\beta _ 1 + k _ 2 \\beta _ 2 + 1 \\right ) } , \\end{align*}"}
-{"id": "3687.png", "formula": "\\begin{align*} \\alpha _ 1 = \\dots = \\alpha _ r = 1 . \\end{align*}"}
-{"id": "2819.png", "formula": "\\begin{align*} [ e ( v _ 0 , w ) ] = \\bigcup _ { e ( w , v ) , v \\in V _ { n + 1 } } [ e ( v _ 0 , w ) , e ( w , v ) ] \\end{align*}"}
-{"id": "661.png", "formula": "\\begin{align*} | P ( D _ t - \\widetilde \\Delta ) & \\widetilde \\nabla ^ k \\widetilde A - ( D _ t - \\Delta ) \\nabla ^ k A | \\\\ & \\le C _ { k + 2 } \\left ( | P \\widetilde A - A | + \\sum _ { i = 0 } ^ { k } | \\nabla ( P \\widetilde \\nabla ^ i \\widetilde A - \\nabla ^ i A ) | + | \\Gamma - \\widetilde \\Gamma | + | P \\widetilde F _ * - F _ * | + | v | \\right ) . \\end{align*}"}
-{"id": "8786.png", "formula": "\\begin{align*} c _ { x } ( f ) = \\min _ { E : x \\in \\pi ( E ) } \\Big \\{ \\frac { \\nu } { N } \\Big \\} . \\end{align*}"}
-{"id": "4046.png", "formula": "\\begin{align*} l \\left ( \\sigma | _ { [ a , b ] } \\right ) : = l ( \\sigma ) : = \\int _ a ^ b | | \\sigma ' ( t ) | | d t . \\end{align*}"}
-{"id": "6397.png", "formula": "\\begin{align*} P _ { } \\left ( \\theta \\right ) = P _ { } \\left ( \\theta \\right ) P _ { } \\left ( x ^ { \\prime } \\left \\vert \\theta \\right . \\right ) \\frac { \\exp \\left [ \\beta f \\left ( \\theta \\right ) \\right ] } { \\Delta \\left ( x ^ { \\prime } \\beta \\right ) } \\end{align*}"}
-{"id": "9372.png", "formula": "\\begin{align*} r _ n ^ { ( k + 1 ) } & = \\frac { \\iint _ { \\ , \\mathbb { C } _ n ^ { ( k ) } } ( x \\cos { \\phi _ n } + y \\sin { \\phi _ n } ) \\ , \\mathrm { e } ^ { - \\frac { x ^ 2 + y ^ 2 } { \\sigma ^ 2 } } \\ , \\mathrm { d } x \\ , \\mathrm { d } y } { \\iint _ { \\ , \\mathbb { C } _ n ^ { ( k ) } } \\mathrm { e } ^ { - \\frac { x ^ 2 + y ^ 2 } { \\sigma ^ 2 } } \\ , \\mathrm { d } x \\ , \\mathrm { d } y } , \\\\ \\phi _ n & = 2 \\pi \\varphi n . \\end{align*}"}
-{"id": "1245.png", "formula": "\\begin{align*} \\breve v = T h _ 1 - h _ 2 \\mbox { i n } \\ , \\ , D \\cap B ( w , \\bar r ) . \\end{align*}"}
-{"id": "7472.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { X } } _ \\alpha & = W ^ \\beta _ \\alpha \\mathcal { X } _ \\beta , \\\\ \\widetilde { \\mathcal { V } } _ \\alpha & = W ^ \\beta _ \\alpha \\mathcal { V } _ \\beta . \\end{align*}"}
-{"id": "5347.png", "formula": "\\begin{align*} W ' = \\coprod _ { n \\in \\C } ( W _ { [ n ] } ) ^ * , \\end{align*}"}
-{"id": "1962.png", "formula": "\\begin{align*} \\begin{cases} 0 < \\lambda < 1 \\\\ 0 < \\lambda \\le \\frac { n - 1 } { 2 } \\end{cases} \\Longleftrightarrow \\begin{cases} \\beta + ( j - 2 ) < \\alpha < \\beta + ( j - 1 ) \\\\ \\beta + ( j - 1 ) - \\frac { n - 1 } { 2 } \\le \\alpha < \\beta + ( j - 1 ) . \\end{cases} \\end{align*}"}
-{"id": "5699.png", "formula": "\\begin{align*} \\int _ \\R W ( { v } ( t ) ) \\d t & \\geq \\int _ \\R W ( \\bar { v } ( s ) ) \\d s - | \\lambda | \\delta \\int _ \\R \\int _ \\R \\rho ( t - s ) \\left ( \\int _ { t - 2 \\delta } ^ { t + 2 \\delta } | { v } ' ( s ' ) | ^ 2 \\d s ' \\right ) \\d t \\d s \\\\ & = \\int _ \\R W ( \\bar { v } ( s ) ) \\d s - | \\lambda | \\delta \\int _ \\R \\int _ { - 2 \\delta } ^ { 2 \\delta } | { v } ' ( t + s ' ) | ^ 2 \\d s ' \\d t \\\\ & = \\int _ \\R W ( \\bar { v } ( s ) ) \\d s - 4 \\delta ^ 2 | \\lambda | \\int _ \\R | { v } ' ( s ' ) | ^ 2 \\d s ' . \\end{align*}"}
-{"id": "993.png", "formula": "\\begin{align*} E \\left [ \\sup _ { t \\in [ 0 , T ] } | Z _ n ( t ) | \\right ] = O ( \\sqrt { \\log n } ) , \\left \\| w ( Z _ n ; n ^ { - 1 } ) \\right \\| _ { \\psi _ 1 } = O \\left ( \\frac { \\sqrt { \\log n } } { n h } \\right ) \\end{align*}"}
-{"id": "2077.png", "formula": "\\begin{align*} u _ \\sigma \\big | _ { M \\times [ 0 , t _ 1 ) } < c , \\ u _ \\sigma ( p _ 1 , t _ 1 ) = \\max _ M u _ \\sigma ( p , t _ 1 ) = c \\end{align*}"}
-{"id": "3672.png", "formula": "\\begin{align*} \\pi ( v ) = \\begin{pmatrix} - a _ n + 1 \\\\ a _ 1 - a _ n \\\\ \\vdots \\\\ a _ { n - 1 } - a _ n \\\\ a _ n \\end{pmatrix} \\end{align*}"}
-{"id": "6117.png", "formula": "\\begin{align*} \\omega _ { f } ^ { \\tilde { T } } \\left ( \\delta \\right ) & = \\inf \\left \\{ \\max _ { 1 \\leq i \\leq m } \\theta _ { f } [ t _ { i - 1 } , t _ { i } ) : \\exists m \\geq 1 , \\right . \\\\ & \\left . 0 = t _ { 0 } < t _ { 1 } . . . < t _ { m } = \\tilde { T } t _ { i } - t _ { i - 1 } > \\delta i \\leq m \\right \\} , \\end{align*}"}
-{"id": "4831.png", "formula": "\\begin{align*} \\mathsf { v } _ { g _ 1 } ( A _ { m + 1 } ) = \\sum _ { i = 1 } ^ m \\alpha _ i \\mathsf { v } _ { g _ 1 } ( A _ i ) . \\end{align*}"}
-{"id": "4145.png", "formula": "\\begin{align*} \\sum _ { \\tau = t + 1 } ^ n Z _ { N , j } ^ { \\tau } + n z _ { i , j } ^ { t } \\leq 1 \\forall \\ i \\in N , \\ j \\in V , \\ t \\in [ n ] . \\end{align*}"}
-{"id": "3068.png", "formula": "\\begin{align*} \\dot { v } ( s ) = - \\nabla _ x F ( t _ i , v ( s ) ) , s \\in \\mathbb { R } . \\end{align*}"}
-{"id": "7697.png", "formula": "\\begin{align*} \\Theta _ { i i } = \\left [ \\begin{array} { c c } X _ { i i } & \\Psi _ { i i } \\\\ \\Psi _ { i i } & Y _ { i i } \\end{array} \\right ] \\ , . \\end{align*}"}
-{"id": "4436.png", "formula": "\\begin{align*} ( \\lceil x _ 1 , ( \\cdot ) _ T \\rceil f ) ( x ) = T ^ { 1 / 3 } \\int _ { \\mathbb { R } ^ 2 } \\tilde \\psi _ T ( x - y ) f ( y ) d y = T ^ { 1 / 3 } \\tilde \\psi _ T * f ( x ) , \\end{align*}"}
-{"id": "8589.png", "formula": "\\begin{align*} ( \\varphi _ + ( c ) - \\varphi _ - ( c ) ) \\varphi _ \\pm ( j ) = 0 \\end{align*}"}
-{"id": "496.png", "formula": "\\begin{align*} V _ 1 = - \\frac { R } { 4 } \\frac { p _ { 1 , 1 , 0 } ^ 2 + p _ { 1 , 0 , 1 } ^ 2 } { p _ { 1 , 0 , 0 } ^ 2 } + \\frac { R } { 2 } \\frac { p _ { 1 , 2 , 0 } + p _ { 1 , 0 , 2 } + \\frac { n } { R } { p _ { 1 , 1 , 0 } } - \\frac { m - 1 } { \\abs { t } } p _ { 1 , 0 , 1 } } { p _ { 1 , 0 , 0 } } . \\end{align*}"}
-{"id": "3006.png", "formula": "\\begin{align*} \\Phi : { \\mathcal W } ^ { N , 2 } ( J ; H ) \\rightarrow H ^ { 2 N } , \\Phi ( x ) : = [ \\Phi _ 1 ( x ) \\ ; \\Phi _ 0 ( x ) ] ^ T , \\end{align*}"}
-{"id": "8040.png", "formula": "\\begin{align*} V _ 1 : = V _ 1 ( n , D , v ) & : = \\min _ { M \\in \\mathcal A ( n , D , v ) } V _ 1 ( M ) \\\\ & = \\min _ { M \\in \\mathcal A } \\min _ { p \\in M } \\mathcal H ^ n ( M - B ( p , R _ M / 2 ) ) . \\end{align*}"}
-{"id": "5515.png", "formula": "\\begin{align*} U ( x ) = \\int _ 0 ^ x \\overline \\nu ( y ) \\dd y = x ^ { 1 - \\alpha } \\mathrm { B } _ { 1 - \\alpha } p ( x ) . \\end{align*}"}
-{"id": "10067.png", "formula": "\\begin{align*} [ \\widehat { \\theta } ( g ) : \\mathcal { Y } _ \\mathrm { b i g } ] = \\frac { - 1 } { n } \\cdot \\deg _ \\C ( \\mathcal { Y } _ \\mathrm { b i g } ) \\cdot \\frac { d } { d s } \\langle E ( s ) , \\tilde { g } \\rangle _ \\mathrm { P e t } \\big | _ { s = 0 } . \\end{align*}"}
-{"id": "6698.png", "formula": "\\begin{align*} \\frac { \\kappa ( x ) ^ { \\frac { 1 } { n + 1 } } } { \\langle x , u ( x ) \\rangle } = ( p - 1 ) ^ { \\frac { n - 1 } { n + 1 } } \\prod _ { i = 1 } ^ n | x _ i | ^ { \\frac { p - 2 } { n + 1 } } \\quad . \\end{align*}"}
-{"id": "2135.png", "formula": "\\begin{align*} w ( \\xi , s ) : = ( 1 + t ) ^ { \\frac { d } { 2 } } r ^ { - A } u ( r , t ) \\quad \\mbox { w i t h } \\xi = ( 1 + t ) ^ { - \\frac { 1 } { 2 } } r \\ge 0 , \\ , \\ , \\ , s = \\log ( 1 + t ) \\ge 0 . \\end{align*}"}
-{"id": "7097.png", "formula": "\\begin{align*} k ^ H _ { i + 1 / 2 } \\approx \\frac { 2 k _ i \\epsilon } { k _ i } = 2 \\epsilon < < \\frac { k _ i } { 2 } \\approx k ^ A _ { i + 1 / 2 } . \\end{align*}"}
-{"id": "796.png", "formula": "\\begin{align*} y ^ { r } \\left ( \\frac { y ^ { B } } { 2 x } \\right ) ^ { k } - 1 = \\frac { s } { x ^ k } , \\end{align*}"}
-{"id": "2600.png", "formula": "\\begin{align*} D ^ a \\bigg ( \\sum _ { k = 0 } ^ { \\ell } \\frac { b _ k } { k ! } t ^ k \\bigg ) = \\sum _ { k = 0 } ^ { \\ell } \\frac { b _ k } { \\Gamma ( k + 1 - a ) } t ^ { k - a } . \\end{align*}"}
-{"id": "3740.png", "formula": "\\begin{align*} \\Omega = \\Big \\{ \\omega = \\sum _ i \\delta _ { w _ i } ; \\ ; w _ i \\in W \\omega ( W _ { \\{ y \\} } ) < \\infty y \\in \\Z ^ d \\times \\Z \\Big \\} , \\end{align*}"}
-{"id": "7032.png", "formula": "\\begin{align*} A = \\int _ { 0 } ^ { \\infty } { \\int _ { \\{ x \\in \\partial B _ 1 ^ { n - 1 } ( 0 ) , u ( r x ) - u ( 0 ) > 0 \\} } { \\frac { u ( r x ) - u ( 0 ) } { r ^ { 1 + 2 s } } d x } d r } \\geq 0 , \\end{align*}"}
-{"id": "2342.png", "formula": "\\begin{align*} D _ j \\coloneqq \\bigcup _ { k = 0 } ^ { m _ j } p _ j ^ k . D \\end{align*}"}
-{"id": "927.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ d \\left | \\frac { \\partial ^ 2 \\varphi } { \\partial x _ i x _ j } ( x ) \\right | \\leq \\| g '' \\| _ \\infty + 2 \\beta \\| g ' \\| _ \\infty \\end{align*}"}
-{"id": "7408.png", "formula": "\\begin{align*} \\mathbb { H } _ { \\Gamma } : = \\frac { \\mathbb { C } [ t _ 0 , t _ 1 , t _ 2 , t _ 3 , t _ 4 ] } { ( t _ 0 + t _ 1 + t _ 2 + t _ 3 + 2 t _ 4 ) } . \\end{align*}"}
-{"id": "9477.png", "formula": "\\begin{align*} \\sum _ { s = 0 } ^ N \\frac { q ^ { 2 s } } { ( q ^ 2 ; q ^ 2 ) _ s ( q ^ { 1 + N + s } ; q ) _ { N - s + 1 } } = \\frac { 1 } { ( q ^ 2 ; q ^ 2 ) _ N } + \\frac { q ^ { N + 1 } } { ( q ; q ^ 2 ) _ { N + 1 } } . \\end{align*}"}
-{"id": "4037.png", "formula": "\\begin{align*} 0 = X \\langle w , \\xi \\rangle = \\langle D _ X w , \\xi \\rangle + \\langle w , D _ X \\xi \\rangle = \\langle w , D _ X \\xi \\rangle \\end{align*}"}
-{"id": "7036.png", "formula": "\\begin{align*} C _ 5 = 1 - \\frac { \\mu _ 0 } { 2 \\mu _ 1 } . \\end{align*}"}
-{"id": "6471.png", "formula": "\\begin{align*} p _ { 2 D u } \\left ( x y | \\mu _ { x } \\sigma \\right ) \\overset { } { = } \\frac { 1 } { 2 \\pi \\Sigma ^ { 2 } } \\exp \\left [ - \\frac { 1 } { 2 \\sigma ^ { 2 } } \\left ( x - \\mu _ { x } \\right ) ^ { 2 } - \\frac { \\sigma ^ { 2 } } { 2 \\Sigma ^ { 4 } } y ^ { 2 } \\right ] \\end{align*}"}
-{"id": "8801.png", "formula": "\\begin{align*} F _ { \\varpi _ { p } } = \\textbf { 1 } _ { X _ { p } } \\end{align*}"}
-{"id": "2091.png", "formula": "\\begin{align*} ( \\Delta _ H - \\partial _ t ) w = & 2 \\sum _ a ( u ^ a - v ^ a ) ( \\Delta _ H - \\partial _ t ) ( u ^ a - v ^ a ) + 2 \\sum _ a | \\nabla _ H u ^ a - \\nabla _ H v ^ a | ^ 2 \\\\ \\geq & - C _ 5 '' w \\end{align*}"}
-{"id": "1091.png", "formula": "\\begin{align*} u _ { i } & = m _ { s , i } + m _ { c , i } , \\\\ \\sigma _ { i } ^ 2 & = u _ { i } + m _ { c , i } ^ 2 + 2 m _ { s , i } m _ { c , i } . \\end{align*}"}
-{"id": "5014.png", "formula": "\\begin{align*} \\int _ { \\| y \\| \\geq 4 \\| x \\| } \\sum _ { 2 ^ j \\| x \\| \\leq 1 } | k _ j ( y - x ) - k _ j ( y ) | d y & \\leq \\int _ { \\R ^ d } \\sum _ { 2 ^ j \\| x \\| \\leq 1 } 2 ^ { j d } | \\varphi ( 2 ^ j ( y - x ) + r ) - \\varphi ( 2 ^ j y + r ) | d y \\\\ & = \\int _ { \\R ^ d } \\sum _ { 2 ^ j \\| x \\| \\leq 1 } | \\varphi ( y - 2 ^ j x + r ) - \\varphi ( y + r ) | d y \\\\ & \\lesssim \\sum _ { 2 ^ j \\| x \\| \\leq 1 } 2 ^ j \\| x \\| \\\\ & \\lesssim 1 . \\end{align*}"}
-{"id": "134.png", "formula": "\\begin{align*} V ( \\alpha x + ( 1 - \\alpha ) y ) & = \\liminf _ { t \\to 1 } \\Phi ^ { - 1 } ( Q _ { 1 - t } f ( \\alpha x + ( 1 - \\alpha ) y ) ) \\\\ & \\ge \\liminf _ { t \\to 1 } \\{ \\alpha \\Phi ^ { - 1 } ( Q _ { 1 - t } f ( x ) ) + ( 1 - \\alpha ) \\Phi ^ { - 1 } ( Q _ { 1 - t } f ( y ) ) \\} \\\\ & \\ge \\alpha V ( x ) + ( 1 - \\alpha ) V ( y ) \\end{align*}"}
-{"id": "5481.png", "formula": "\\begin{align*} \\liminf _ { x \\to \\infty } \\frac { x u ( x ) } { \\ell ( x ) } = : k > 0 . \\end{align*}"}
-{"id": "6424.png", "formula": "\\begin{align*} \\left \\Vert \\Delta \\theta \\right \\Vert = \\left \\Vert d \\theta \\right \\Vert + \\left \\Vert \\Delta \\theta - d \\theta \\right \\Vert \\end{align*}"}
-{"id": "2611.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 V _ a ( t - s ) u ( s ) \\ , d s = f ( t ) , 0 < t < 1 \\end{align*}"}
-{"id": "6979.png", "formula": "\\begin{align*} f _ k ( A ) = \\inf _ { B \\in \\Gamma _ k } { \\{ D f _ k ( B ) ( A - B ) + f _ k ( B ) \\} } . \\end{align*}"}
-{"id": "6666.png", "formula": "\\begin{align*} \\frac { 1 } { n } ( 2 + \\varepsilon ) \\left [ \\sqrt { \\frac { 2 \\Delta } { ( 1 - \\varepsilon ) } } ( 2 + \\varepsilon ) \\right ] ^ { n - 1 } | \\mathcal { E } _ x | _ { n - 1 } = \\frac { ( 2 \\Delta ) ^ { \\frac { n + 1 } { 2 } } | B _ 2 ^ { n - 1 } | } { \\kappa ( x ) ^ { 1 / 2 } } \\frac { ( 1 + \\frac { \\varepsilon } { 2 } ) ^ { n } } { n ( 1 - \\varepsilon ) ^ { \\frac { n - 1 } { 2 } } } \\quad . \\end{align*}"}
-{"id": "7177.png", "formula": "\\begin{align*} \\| z _ j \\| _ { H ^ 1 ( B _ 1 , \\mu _ a ) } = 1 . \\end{align*}"}
-{"id": "6939.png", "formula": "\\begin{align*} y ^ 2 = f ( x ) = x ^ { 2 g + 1 } + a _ 2 x ^ { 2 g - 1 } + a _ 3 x ^ { 2 g - 2 } + \\dots + a _ { 2 g + 1 } , \\end{align*}"}
-{"id": "6480.png", "formula": "\\begin{align*} \\mathcal { S } _ { \\mathcal { M } } ^ { 2 D c } \\left ( \\tau \\right ) = \\log \\left [ \\mathcal { C } _ { \\mathcal { M } } ^ { 2 D u } \\left ( \\tau \\right ) \\right ] \\overset { \\tau \\rightarrow \\infty } { \\approx } \\sigma _ { 0 } \\lambda _ { + } \\tau \\end{align*}"}
-{"id": "4096.png", "formula": "\\begin{align*} b ( X , Y ) : = \\frac { \\langle d u ^ { - 1 } _ { \\eta ( p ) } X , Y \\rangle } { \\langle \\eta ( p ) , \\xi ( p ) \\rangle } , \\end{align*}"}
-{"id": "6992.png", "formula": "\\begin{align*} F _ { k , s } [ u ] ( x ) = \\inf _ { M \\in \\mathcal { M } _ k } { \\{ - C _ { n , s } ^ { - 1 } ( - \\Delta ) ^ { s } ( u \\circ \\sqrt { M } ) ( x ) \\} } . \\end{align*}"}
-{"id": "4346.png", "formula": "\\begin{align*} \\widetilde { \\varphi } _ { 0 } ( x _ 1 , \\dots , x _ { 2 n } ) = \\phi ( x _ 1 ) \\cdot \\phi ( x _ 2 ) \\cdot \\hdots \\cdot \\phi ( x _ { 2 n } ) . \\end{align*}"}
-{"id": "1389.png", "formula": "\\begin{align*} ( \\tilde u ( 1 ) , \\tilde u ( 2 ) , \\ldots ) & = ( 1 , \\underbrace { 0 , \\ldots , 0 } _ { \\xi _ 1 } , \\eta _ { 1 } + 1 , \\underbrace { 0 , \\ldots , 0 } _ { \\xi _ 2 } , \\eta _ { 2 } + 1 , \\ldots ) . \\end{align*}"}
-{"id": "8204.png", "formula": "\\begin{align*} K _ { } ( g , g ) = \\frac { 1 } { 4 \\pi } \\sum _ { \\Psi \\in \\mathcal { B } _ { \\tilde { \\mathbf { H } } } } \\int _ { - \\infty } ^ { \\infty } c _ { \\Psi } ( i y ) \\abs { E _ { \\Psi } ( i y , g ) } ^ 2 d y . \\end{align*}"}
-{"id": "2624.png", "formula": "\\begin{align*} \\overline \\rho = \\mathcal C _ a f _ { \\overline \\rho } = \\mathcal C _ a ( A + B t ) + \\mathcal C _ a R . \\end{align*}"}
-{"id": "9300.png", "formula": "\\begin{align*} h ( t ) = & \\ , [ ( t + r ' - r ) ( t ) + \\beta ( g ( t + r ' - r ) ) ] \\\\ & - [ ( t + r ' - r ) ( r ) + \\beta ( g ( t + r ' - r ) ) ] \\\\ = & \\ , t ^ 2 + t ( r ' - 2 r ) - r ' r + r ^ 2 . \\end{align*}"}
-{"id": "1758.png", "formula": "\\begin{align*} T [ \\pi ( \\psi ) f ] \\sqrt { d \\mu _ \\pi } & = T \\pi _ { u n i v } ( \\psi ) ( f \\sqrt { d \\mu _ \\pi } ) = \\pi _ { u n i v } ( \\psi ) T ( f \\sqrt { d \\mu _ \\pi } ) \\\\ & = \\pi _ { u n i v } ( \\psi ) ( g \\sqrt { d \\mu _ \\pi } ) = [ \\pi ( \\psi ) ( g ) ] \\sqrt { d \\mu _ \\pi } = \\pi ( \\psi ) T \\left ( f \\sqrt { d \\mu _ \\pi } \\right ) , \\end{align*}"}
-{"id": "3850.png", "formula": "\\begin{align*} I _ { \\pm 0 } ( x ^ * , y ^ * ) & : = \\ \\{ i = 1 , \\dots , n \\mid x _ i ^ * \\not = 0 , \\ , y ^ * _ i = 0 \\} , \\\\ I _ { 0 0 } ( x ^ * , y ^ * ) & : = \\ \\{ i = 1 , \\dots , n \\mid x _ i ^ * = 0 , \\ , y ^ * _ i = 0 \\} , \\\\ I _ { 0 + } ( x ^ * , y ^ * ) & : = \\ \\{ i = 1 , \\dots , n \\mid x _ i ^ * = 0 , \\ , y ^ * _ i \\in ( 0 , 1 ) \\} , \\\\ I _ { 0 1 } ( x ^ * , y ^ * ) & : = \\ \\{ i = 1 , \\dots , n \\mid x _ i ^ * = 0 , \\ , y ^ * _ i = 1 \\} . \\end{align*}"}
-{"id": "4632.png", "formula": "\\begin{align*} \\sum _ { n = 2 ^ { k - 1 } } ^ { 2 ^ k - 1 } 1 _ { [ - B - t , B - t ] } \\left ( \\ ; \\sum _ { i = 0 } ^ { n - 1 } f ( T ^ i x ) \\right ) \\ge \\beta \\sum _ { n = 0 } ^ { 2 ^ k - 1 } 1 _ { [ - B - t , B - t ] } \\left ( \\ ; \\sum _ { i = 0 } ^ { n - 1 } f ( T ^ i x ) \\right ) . \\end{align*}"}
-{"id": "6151.png", "formula": "\\begin{align*} \\partial _ t \\omega _ i = \\dd \\tau _ i , \\ ; \\ ; i = 1 , 2 , 3 \\end{align*}"}
-{"id": "2535.png", "formula": "\\begin{align*} \\nabla _ { x } h ^ { ( 1 ) } ( t ) = e ^ { ( t - t _ { 1 } / 2 ) \\mathcal { L } } \\nabla _ { x } h ^ { ( 1 ) } ( t _ { 1 } / 2 ) + \\int _ { t _ { 1 } / 2 } ^ { t } e ^ { ( t - s ) \\mathcal { L } } K \\nabla _ { x } h ^ { ( 0 ) } ( s ) d s \\ , , \\end{align*}"}
-{"id": "4158.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { k \\in N } e _ { k , j + n - 2 } ^ n + \\frac { 1 } { n } e ^ { n - 1 } _ { i , j } + \\frac { 1 } { n } \\left ( 1 - \\frac { 1 } { n } \\right ) e _ { i , j } ^ { n - 2 } + \\frac { 1 } { n } \\sum _ { \\tau = 1 } ^ { n - 3 } e _ { i , j + \\tau } ^ { n - \\tau - 2 } \\end{align*}"}
-{"id": "2776.png", "formula": "\\begin{align*} M _ { 0 0 } = M ( \\vec w ' , k ) \\end{align*}"}
-{"id": "6099.png", "formula": "\\begin{align*} \\Psi ^ { '' } ( t ) + ( 2 n - 2 ) \\Psi ' ( t ) - ( ( 2 n - 1 ) + c ^ 2 ) \\tfrac { \\sin ( 2 \\Psi ( t ) ) } { 2 } = 0 . \\end{align*}"}
-{"id": "172.png", "formula": "\\begin{align*} x \\vdash _ { \\gamma } y y \\leq ^ { \\mathbf { L } } \\gamma ( x ) \\quad \\gamma _ { \\vdash } ( x ) = \\bigvee \\{ y \\in L : x \\vdash y \\} . \\end{align*}"}
-{"id": "4845.png", "formula": "\\begin{align*} h _ 0 = 0 , \\ ; h _ 1 = m , \\ ; h _ 2 = 2 m , \\ ; h _ 3 = 3 m , \\ ; h _ 4 = 4 m , \\ ; h _ 5 = c , \\end{align*}"}
-{"id": "1193.png", "formula": "\\begin{align*} E ( t ) = \\{ x : u ( x ) > t \\} \\ , \\ , \\mbox { i s c o n v e x } . \\end{align*}"}
-{"id": "3416.png", "formula": "\\begin{align*} S _ g = \\{ x + \\imath y : x \\in \\R , \\ \\ | y | < g ( x ) \\} . \\end{align*}"}
-{"id": "1675.png", "formula": "\\begin{align*} \\frac { \\mu ( \\sigma _ { f _ 1 } Z ( \\lambda _ { N } ) ) } { \\mu ( Z ( \\lambda _ { N } ) ) } \\ ; = \\ ; \\left ( ( \\frac { 1 } { 2 } - \\gamma _ 1 ) \\prod _ { i = 2 } ^ { N + 1 } [ 1 + ( - 1 ) ^ { m _ i } 2 \\gamma _ i ] \\right ) / \\left ( \\prod _ { i = 1 } ^ N [ 1 + ( - 1 ) ^ { \\ell _ i } 2 \\gamma _ i ] \\right ) . \\end{align*}"}
-{"id": "7221.png", "formula": "\\begin{align*} \\theta _ { \\alpha _ { 0 } } ^ { n } \\equiv \\prod _ { i = 1 } ^ { r } ( \\theta _ { \\alpha _ { i } } ^ { n d _ { i } } ) ^ { n _ { i } ^ { \\vee } } \\equiv 1 \\ ( J _ { n } ) . \\end{align*}"}
-{"id": "3698.png", "formula": "\\begin{align*} \\mathbf t ( \\gamma ) = ( t _ 1 ( \\gamma ) , \\dots , t _ { n + 1 } ( \\gamma ) ) \\in \\mathbb A ^ { n + 1 } . \\end{align*}"}
-{"id": "6156.png", "formula": "\\begin{align*} \\omega _ 1 = ( 1 + \\phi _ 1 '' ) \\dd x _ 0 \\wedge \\dd x _ 1 + \\dd x _ 2 \\wedge \\dd x _ 3 \\\\ \\omega _ 2 = ( 1 + \\phi _ 2 '' ) \\dd x _ 0 \\wedge \\dd x _ 2 + \\dd x _ 3 \\wedge \\dd x _ 1 \\\\ \\omega _ 3 = ( 1 + \\phi _ 3 '' ) \\dd x _ 0 \\wedge \\dd x _ 3 + \\dd x _ 1 \\wedge \\dd x _ 2 \\end{align*}"}
-{"id": "7398.png", "formula": "\\begin{align*} a = a _ 0 a _ 1 , a ^ * = a _ 1 ^ * a _ 0 ^ * , b = a _ 3 ^ * a _ 3 , c = a _ 5 ^ * a _ 5 . \\end{align*}"}
-{"id": "6241.png", "formula": "\\begin{align*} \\boldsymbol \\Sigma { \\bf Q } _ { \\rm s } = \\bar { \\bf H } _ { \\rm b } { \\boldsymbol \\Psi } _ { \\rm b } \\bar { \\bf H } _ { \\rm b } ^ H { \\bf Q } _ { \\rm s } \\end{align*}"}
-{"id": "9985.png", "formula": "\\begin{align*} \\frac { \\epsilon ' ( ( m _ { \\beta } - m _ { \\alpha - 1 } ) + 2 r ( m _ N ) ) + r \\log { C } } { m _ { \\beta } - m _ { \\alpha - 1 } } & = \\epsilon ' + \\frac { \\epsilon ' 2 r ( m _ N ) } { ( m _ { \\beta } - m _ { \\alpha - 1 } ) } + \\frac { r \\log { C } } { ( m _ { \\beta } - m _ { \\alpha - 1 } ) } \\\\ & \\leq \\epsilon ' + \\epsilon ' 6 N r + \\frac { r \\log { C } } { ( m _ { \\alpha } - m _ { \\alpha - 1 } ) } \\\\ & \\leq ( 2 + 6 N r ) \\epsilon ' = \\epsilon . \\end{align*}"}
-{"id": "5921.png", "formula": "\\begin{align*} \\begin{aligned} & Q ^ N ( s ) \\le e ^ { - c _ 1 N } Q ^ N ( \\mathcal { R } ( s ) ) , \\ s \\in \\{ s ' \\in A _ { } : r _ { s ' } \\ge 3 \\} , \\\\ & \\ c _ 1 = \\min _ { i = 1 , \\cdots , l - 1 } | I _ i ( r _ { i + 1 } ) - I _ i ( r _ i ) | , \\end{aligned} \\end{align*}"}
-{"id": "6482.png", "formula": "\\begin{align*} u _ { k } \\left ( \\theta ^ { k } \\right ) = - \\frac { 1 } { 2 } \\omega _ { k } ^ { 2 } \\left ( \\theta ^ { k } \\right ) ^ { 2 } \\theta ^ { k } = \\theta ^ { k } \\left ( s \\right ) \\end{align*}"}
-{"id": "8250.png", "formula": "\\begin{align*} [ M ( s ) v ^ { \\circ } ( s ) ] ( g ) = c ( s ) \\hat { v } ^ { \\circ } ( - s ) ( g ) . \\end{align*}"}
-{"id": "5413.png", "formula": "\\begin{align*} 4 C \\lambda ^ 3 - 4 ( B ^ 2 - A C ) \\lambda ^ 2 - 4 B \\lambda - 1 = 0 . \\end{align*}"}
-{"id": "1669.png", "formula": "\\begin{align*} \\sigma _ \\lambda ( x ) : = \\lambda x , \\sigma ^ n ( x ) ( p , q ) : = x ( p + n , q + n ) , \\end{align*}"}
-{"id": "79.png", "formula": "\\begin{align*} m ( T ( \\sigma ~ 0 1 ~ 1 0 ~ w ~ 0 0 ~ \\tau ) ) & = m ( T ( \\sigma ~ \\underline { 0 1 } ~ \\underline { 1 0 } ~ 1 0 ~ w ' ~ 0 0 ~ \\tau ) ) \\\\ & = m ( T ( \\sigma ~ 1 0 ~ 0 1 ~ 1 0 ~ w ' ~ 0 0 ~ \\tau ) ) \\end{align*}"}
-{"id": "1187.png", "formula": "\\begin{align*} a _ { i j } = \\frac { 1 } { 2 } \\left [ \\frac { \\partial \\mathcal { A } _ i } { \\partial \\eta _ j } ( \\xi ) ) + \\frac { \\partial \\mathcal { A } _ j } { \\partial \\eta _ i } ( \\xi ) ) \\right ] \\mbox { f o r } \\ , \\ , 1 \\leq i , j \\leq n , \\end{align*}"}
-{"id": "6754.png", "formula": "\\begin{align*} \\big \\{ \\omega _ i \\psi _ j : \\big \\} \\end{align*}"}
-{"id": "3681.png", "formula": "\\begin{align*} S [ n ] : = X [ n ] \\times \\mathbb A ^ { m - 1 } \\to X [ n ] \\to \\mathbb A ^ { n + 1 } . \\end{align*}"}
-{"id": "187.png", "formula": "\\begin{align*} \\delta ^ { \\mathit { B J } } ( X ) = \\bigcup \\{ \\delta ( x ) : x \\in X \\} \\end{align*}"}
-{"id": "9788.png", "formula": "\\begin{align*} ( \\mathrm { C } ( \\vartheta _ { \\Q } ) [ - 1 ] ) ^ k & = K ^ k _ \\infty \\oplus s ( C ^ { \\bullet , \\bullet } ) ^ { k - 1 } \\\\ & = \\widetilde { C } ^ { k , 0 } \\oplus \\bigoplus _ { p + q = k - 1 } C ^ { p , q } = \\bigoplus _ { p + q = k } \\widetilde { C } ^ { p , q } . \\end{align*}"}
-{"id": "4211.png", "formula": "\\begin{gather*} E _ i = \\sum _ j \\partial ( \\mu , j ) f ( \\mu , j ) ^ { + } \\varphi _ { ( i , \\min ( \\mu _ j ) ) } , \\\\ F _ i = \\sum _ j \\partial ( \\mu , j ) f ( \\mu , j ) ^ { - } \\varphi _ { ( i , \\min ( \\mu _ j ) ) } ^ { - 1 } . \\end{gather*}"}
-{"id": "3617.png", "formula": "\\begin{align*} \\operatorname { C a p } ^ D _ p ( \\mathcal A ) : = \\inf \\left \\{ \\int _ { D } | \\nabla \\varphi | ^ p d x < + \\infty \\ ; : \\ ; \\varphi \\in C ^ { \\infty } _ c ( D ) \\ \\ \\ \\ \\varphi \\geq \\chi _ { \\mathcal A } \\right \\} . \\end{align*}"}
-{"id": "5005.png", "formula": "\\begin{align*} | \\gamma | = \\gamma _ 1 + \\cdots + \\gamma _ d , \\partial ^ { \\gamma } = \\partial ^ { \\gamma _ 1 } _ 1 \\cdots \\partial ^ { \\gamma _ d } _ d . \\end{align*}"}
-{"id": "5015.png", "formula": "\\begin{align*} \\int _ { \\| y \\| \\geq 4 \\| x \\| } \\sum _ { 1 < 2 ^ j \\| x \\| < \\| r \\| } | k _ j ( y - x ) - k _ j ( y ) | d y & \\leq \\sum _ { 1 < 2 ^ j \\| x \\| < \\| r \\| } 2 \\int _ { \\R ^ d } k _ j ( y ) d y \\\\ & = 2 \\sum _ { 1 < 2 ^ j \\| x \\| < \\| r \\| } \\int _ { \\R ^ d } \\varphi ( y ) d y \\\\ & \\lesssim \\log \\| r \\| . \\end{align*}"}
-{"id": "5906.png", "formula": "\\begin{align*} \\nu ^ { N , D } ( i ) E S ^ { ( i ) } _ { \\tau ^ { N , i } + N } \\approx \\frac 1 { \\Pi _ N } \\mu _ i e ^ { N \\Lambda _ i } , \\ 0 \\le i \\le l . \\end{align*}"}
-{"id": "3626.png", "formula": "\\begin{align*} g ( x ) : = u ( d x + x _ 0 ) \\ B _ { \\frac { 1 } { 2 } } ( 0 ) , \\end{align*}"}
-{"id": "9084.png", "formula": "\\begin{align*} f ( z ^ { 2 } ) = 2 ( x + y ) f ( x + y ) = 2 \\left ( x f ( x ) + x f ( y ) + y f ( x ) + y f ( y ) \\right ) . \\end{align*}"}
-{"id": "8538.png", "formula": "\\begin{align*} \\partial _ t f ( t , x , v ) + v \\cdot \\nabla _ x f ( t , x , v ) = g ( t , x , v ) . \\end{align*}"}
-{"id": "6456.png", "formula": "\\begin{align*} p _ { } ( x , y | \\mu _ { x } , \\mu _ { y } , \\sigma ) = \\frac { \\exp \\left \\{ - \\frac { 1 } { 2 \\sigma ^ { 2 } ( 1 - \\rho ^ { 2 } ) } \\left [ ( x - \\mu _ { x } ) ^ { 2 } - 2 \\rho ( x - \\mu _ { x } ) ( y - \\mu _ { y } ) + ( y - \\mu _ { y } ) ^ { 2 } \\right ] \\right \\} } { 2 \\pi \\sigma ^ { 2 } \\sqrt { 1 - \\rho ^ { 2 } } } \\end{align*}"}
-{"id": "4338.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\int _ { \\mathbb { C } ^ n } | f _ j ( z ) g ( z ) | ^ p d \\mu _ { p \\alpha / 2 } ( z ) \\leq N \\| g \\| _ { p , \\alpha } ^ p . \\end{align*}"}
-{"id": "7035.png", "formula": "\\begin{align*} C = \\sqrt { ( 1 - \\frac { \\mu _ 0 } { 2 \\mu _ 1 } ) ^ { \\frac { - 2 } { n + 2 s } } - 1 } \\end{align*}"}
-{"id": "1582.png", "formula": "\\begin{align*} - \\alpha ( n ) - 1 & = - \\frac { j _ n - ( 1 + j _ 1 + \\dots + j _ { n - 1 } ) } { 2 } - 1 \\\\ & = - \\frac { 1 - \\frac { m - 1 } { 3 } } { 2 } - 1 \\\\ & = \\frac { \\frac { m - 1 } { 3 } - 1 } { 2 } - 1 \\\\ & = \\frac { \\frac { m - 1 } { 3 } - 3 } { 2 } \\\\ & = \\frac { m - 1 0 } { 6 } \\end{align*}"}
-{"id": "9899.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\tfrac { k _ i + 1 } { 2 k _ i } \\alpha _ i ^ 2 \\ge \\tfrac { 1 } { r - 1 } \\ , . \\end{align*}"}
-{"id": "5059.png", "formula": "\\begin{align*} S _ { i j , k } = - \\sum _ l ( B _ { i l , k } B _ { l j } + B _ { i l } B _ { l j , k } ) + A _ { i j , k } . \\end{align*}"}
-{"id": "3884.png", "formula": "\\begin{align*} t _ { j , k } ( n ) t _ { j , k } ( m ) = \\sum _ { d | ( m , n ) } t _ { j , k } \\left ( \\frac { n m } { d ^ 2 } \\right ) . \\end{align*}"}
-{"id": "5026.png", "formula": "\\begin{align*} ( H _ \\lambda - z ) ^ { - 1 } & = ( H - z ) ^ { - 1 } - \\lambda ( H - z ) ^ { - 1 } C _ 1 ( H _ \\lambda - z ) ^ { - 1 } \\\\ & = ( H - z ) ^ { - 1 } - \\lambda ( H - z ) ^ { - 1 } C _ 1 ( H - z ) ^ { - 1 } \\\\ & \\qquad + \\lambda ^ 2 ( H - z ) ^ { - 1 } C _ 1 ( H _ \\lambda - z ) ^ { - 1 } C _ 1 ( H _ \\lambda - z ) ^ { - 1 } , \\end{align*}"}
-{"id": "6453.png", "formula": "\\begin{align*} d s ^ { 2 } = \\frac { 3 } { ( 1 + 2 \\rho ) \\sigma ^ { 2 } } d \\mu ^ { 2 } + \\frac { 6 } { \\sigma ^ { 2 } } d \\sigma ^ { 2 } \\end{align*}"}
-{"id": "8516.png", "formula": "\\begin{align*} \\sigma ( l , v , u , C ) = \\min \\{ f ( \\bar v , \\bar u , z ) : z \\in \\mathbb { Z } , \\ , | z | \\leq C , \\ , \\bar v _ i = v _ i + a _ { i \\ , l } z , \\ , \\bar u _ i = u _ i + \\bar a _ { i \\ , l } z \\} , \\end{align*}"}
-{"id": "9532.png", "formula": "\\begin{align*} X _ n \\to X _ { n - 1 } \\to \\dots \\to X _ 1 \\to X _ 0 = X , \\end{align*}"}
-{"id": "7767.png", "formula": "\\begin{align*} | y ( \\ell ' ) - x | = | I y ( \\ell ' ) - I y ( x ' ) | \\leq \\frac { 3 | \\ell ' - x ' | } { 2 } \\leq \\frac { 3 C _ { 2 } } { 2 } . \\end{align*}"}
-{"id": "4772.png", "formula": "\\begin{align*} { \\varepsilon } = F ( x ^ 1 , Y , Z ) \\ , { \\Delta \\sqrt { - \\det g ^ { i j } } } / { ( { L } _ 2 \\ , { L } _ 3 ) } , \\end{align*}"}
-{"id": "3577.png", "formula": "\\begin{align*} \\eta ( x ) = \\left \\{ \\begin{array} { l l } 1 & x \\in U \\\\ \\frac 1 R d ( x , X \\setminus N ( U , R ) ) & x \\not \\in U . \\end{array} \\right . \\end{align*}"}
-{"id": "2480.png", "formula": "\\begin{align*} \\nabla _ { \\xi } \\vartheta = 0 \\quad \\hbox { o n } H _ { + } \\ , , \\end{align*}"}
-{"id": "7176.png", "formula": "\\begin{align*} W ^ { \\underline { 0 } } _ { 1 + s } ( 1 , \\psi ) = \\frac { 1 } { 2 } \\delta W ^ { \\underline { 0 } } _ { 1 + s } ( 1 , \\psi ) [ \\psi ] = 0 \\qquad \\forall \\psi \\in \\mathfrak { H } _ { 1 + s } . \\end{align*}"}
-{"id": "7593.png", "formula": "\\begin{align*} \\displaystyle \\int \\dfrac { F ( z ) ( z - x _ 0 ) } { z - x } k ( z ) d A _ z - \\int \\dfrac { F ( x ) ( z - x _ 0 ) } { z - x } k ( z ) d A _ z = 0 . \\end{align*}"}
-{"id": "424.png", "formula": "\\begin{align*} r ( \\lambda ) = \\begin{cases} 1 + \\frac { 1 } { \\lambda } - \\pi ( 1 + \\lambda ) \\cot ( \\pi \\lambda ) , & \\\\ 0 , & \\end{cases} \\\\ \\end{align*}"}
-{"id": "893.png", "formula": "\\begin{align*} U _ n ( \\theta ) = \\sum _ { I \\in \\Pi ^ 1 _ n , J \\in \\Pi ^ 2 _ n } X ^ 1 ( I ) X ^ 2 ( J ) K ( I , J _ { - \\theta } ) , \\end{align*}"}
-{"id": "6413.png", "formula": "\\begin{align*} \\delta \\left [ - \\frac { 1 } { 2 } g _ { \\mu \\nu } \\left ( \\theta \\right ) d \\theta ^ { \\mu } d \\theta ^ { \\nu } - \\gamma \\left ( \\xi ^ { 2 } d l _ { \\rightarrow \\xi } ^ { 2 } - d l _ { \\xi \\rightarrow } ^ { 2 } \\right ) \\right ] = 0 \\end{align*}"}
-{"id": "2363.png", "formula": "\\begin{align*} e _ j \\left ( \\begin{smallmatrix} x \\\\ y \\end{smallmatrix} \\right ) \\coloneqq x ^ j y ^ { n - j } , j = 0 , \\ldots , n . \\end{align*}"}
-{"id": "2449.png", "formula": "\\begin{align*} \\beta . C _ k & = C _ 2 ^ * ( C _ k ) + C _ { 4 } ^ * ( C _ k ) - \\sum _ { j = 1 } ^ 4 r _ j C _ j ^ * ( C _ k ) \\\\ & = \\left \\{ \\begin{array} { c l } - r _ k & k = 1 , 3 \\\\ 1 - r _ k & k = 2 , 4 . \\end{array} \\right . \\end{align*}"}
-{"id": "6653.png", "formula": "\\begin{align*} \\frac { 1 } { c _ n } \\ \\lim _ { \\delta \\rightarrow 0 } \\frac { | K | _ n - | K _ { \\delta / | K | _ n } | _ n } { \\delta ^ { \\frac { 2 } { n + 1 } } } = \\int _ { \\partial K } \\kappa ^ { \\frac { 1 } { n + 1 } } ( x ) \\mathrm { d } \\mu ( x ) \\quad , \\end{align*}"}
-{"id": "232.png", "formula": "\\begin{align*} F _ t : = \\sup _ { x \\in \\R } \\left \\Vert f ( x + i t ) \\right \\Vert \\leq \\frac { 1 } { t } . \\end{align*}"}
-{"id": "4793.png", "formula": "\\begin{align*} G & : = \\{ x : \\gamma _ 1 d ^ { ( i ) } _ k < | x - y _ k ^ { ( i ) } | < \\gamma _ 2 d _ k ^ { ( i ) } \\} , \\\\ G _ 1 & : = G \\setminus \\{ x : | x - y _ { k + m } ^ { ( i ) } | \\leq \\gamma _ 2 d _ { k + m } ^ { ( i ) } \\} , \\\\ G _ 2 & : = \\{ x : | x - y _ { k + m } ^ { ( i ) } | > \\gamma _ 2 d _ { k + m } ^ { ( i ) } \\} . \\end{align*}"}
-{"id": "5738.png", "formula": "\\begin{align*} \\lim \\limits _ { x _ 2 \\to \\pm \\infty } \\| { u } ( \\cdot , x _ 2 ) - z ^ \\pm ( \\cdot - c ^ \\pm ) \\| _ { L ^ 2 ( \\R ) } = 0 . \\end{align*}"}
-{"id": "7168.png", "formula": "\\begin{align*} \\Gamma _ { 1 + s } ( u ) : = \\{ x _ 0 \\in \\Gamma _ u \\ , \\ , : \\ , \\ , N _ a ^ { x _ 0 } ( 0 ^ + , u ) = 1 + s \\} . \\end{align*}"}
-{"id": "5532.png", "formula": "\\begin{align*} \\rho _ { 1 , 2 } = \\frac { \\Delta \\left ( \\alpha , \\beta \\right ) \\pm \\sqrt { \\Delta \\left ( \\alpha , \\beta \\right ) ^ { 2 } - 4 } } { 2 } \\end{align*}"}
-{"id": "1708.png", "formula": "\\begin{align*} T _ \\lambda W ( f ) = f _ \\lambda ( h \\circ \\tau ^ { d ( \\lambda ) } ) ( f \\circ \\tau ^ { d ( \\lambda ) } ) = f _ \\lambda ' \\ , h \\ , ( f \\circ \\tau ^ { d ( \\lambda ) } ) = W T _ \\lambda ' ( f ) . \\end{align*}"}
-{"id": "5119.png", "formula": "\\begin{align*} v = \\sum _ { j = 1 } ^ k c _ j v _ j \\end{align*}"}
-{"id": "8162.png", "formula": "\\begin{align*} a _ 1 a _ 7 = a _ 2 a _ 6 = a _ 3 a _ 5 = 0 . \\end{align*}"}
-{"id": "4311.png", "formula": "\\begin{align*} f ( [ A , B ] ) = [ f ( A ) , f ( B ) ] + A ( f ( B ) ) - B ( f ( A ) ) \\end{align*}"}
-{"id": "5663.png", "formula": "\\begin{align*} \\mathcal { E } ( { u } ) = \\int _ \\R \\left ( \\frac 1 2 \\| \\partial _ { x _ 2 } { u } ( \\cdot , x _ 2 ) \\| _ { L ^ 2 ( \\R ) } ^ 2 + \\mathcal { K } ( { u } ( \\cdot , x _ 2 ) ) ^ 2 \\right ) \\d x _ 2 , \\mathcal { K } ( { v } ) : = \\sqrt { \\mathfrak { E } _ { W } ( { v } ) - d _ K ( a ^ - , a ^ + ) } . \\end{align*}"}
-{"id": "3861.png", "formula": "\\begin{align*} & \\nabla f ( x ^ k ) + \\sum \\limits _ { i \\in I _ g ( x ^ k ) } \\lambda _ i ^ k \\nabla g _ i ( x ^ k ) + \\sum \\limits _ { i = 1 } ^ p \\mu _ i ^ k \\nabla h _ i ( x ^ k ) + \\sum \\limits _ { i \\in I _ 0 ( x ^ k ) } \\gamma _ i ^ k e _ i = 0 , \\\\ & \\lambda _ i ^ k \\geq 0 , \\lambda _ i ^ k g _ i ( x ^ k ) = 0 , \\quad \\forall i = 1 , \\dots , m , \\\\ & \\gamma _ i ^ k = 0 , \\quad \\forall i \\in I _ { \\pm 0 } ( x ^ k , y ^ k ) . \\end{align*}"}
-{"id": "8044.png", "formula": "\\begin{align*} { \\rm o b } _ \\infty ( x , y ) & = \\inf _ { R > R _ 0 } \\sup _ { | z p | > R / 2 } \\max \\left \\{ \\angle ( \\Uparrow _ x ^ y , \\uparrow _ x ^ z ) , \\angle ( \\Uparrow _ y ^ x , \\uparrow _ y ^ z ) \\right \\} - \\pi / 2 \\\\ & \\ge \\inf _ { R > R _ 0 } \\sup _ { R / 2 < | z p | \\le R } \\max \\left \\{ \\angle ( \\Uparrow _ x ^ y , \\uparrow _ x ^ z ) , \\angle ( \\Uparrow _ y ^ x , \\uparrow _ y ^ z ) \\right \\} - \\pi / 2 \\\\ & \\ge \\epsilon _ n ( v ) . \\end{align*}"}
-{"id": "3605.png", "formula": "\\begin{align*} ( K _ N M ^ \\ell K _ N M ^ \\ell K _ N ) P _ { k } ^ N = \\sum _ { m = 0 } ^ { N - 1 } \\left ( \\sum _ { \\gamma : ( 0 , k ) \\rightarrow ( \\ell , m ) , \\ ; \\gamma ( \\ell ) < N } w ( \\gamma ) \\right ) P _ { m } ^ N , \\end{align*}"}
-{"id": "15.png", "formula": "\\begin{align*} V ^ { C } _ { \\sigma } ( C _ { 1 } , C _ { 2 } ) = \\int \\limits _ { - \\infty } ^ { \\infty } \\int \\limits _ { - \\infty } ^ { \\infty } \\int \\limits _ { - \\infty } ^ { \\infty } \\int \\limits _ { - \\infty } ^ { \\infty } f _ { X Y Z S } ( x , y , z , s ) G _ { \\sigma } ( x - y ) G _ { \\sigma } ( z - s ) \\mathrm { d } x \\mathrm { d } y \\mathrm { d } z \\mathrm { d } s \\end{align*}"}
-{"id": "6039.png", "formula": "\\begin{align*} C _ { \\mathsf { W y n e r } } ( X ; Y ) & = R ^ { * } . \\end{align*}"}
-{"id": "802.png", "formula": "\\begin{align*} \\sigma = 0 . 9 0 7 5 5 \\ \\textrm { a n d } \\ \\lambda = 0 . 9 0 7 5 5 \\log { \\rho } , \\end{align*}"}
-{"id": "2710.png", "formula": "\\begin{align*} \\partial _ t g + \\frac { 1 } { \\epsilon ^ \\alpha } v \\cdot \\nabla _ x g = \\frac { 1 } { \\epsilon ^ { 1 + \\alpha } } \\mathcal L ( g ) \\ , , \\end{align*}"}
-{"id": "9941.png", "formula": "\\begin{align*} u ( x , t ) = 2 \\lambda \\cos ^ 2 \\left ( \\frac { x - \\lambda t } 2 \\right ) H \\left ( \\pi - | x - \\lambda t | \\right ) , \\end{align*}"}
-{"id": "8300.png", "formula": "\\begin{align*} \\mathrm { w t } _ { - 3 } V = 0 , \\mathrm { w t } _ { - 2 } V = \\mathrm { w t } _ { - 1 } V = I , \\mathrm { w t } _ 0 V = \\mathrm { w t } _ 1 V = I ^ \\perp , \\mathrm { w t } _ 2 V = V , \\end{align*}"}
-{"id": "7851.png", "formula": "\\begin{align*} r _ { i } ( x _ { i } ) = \\frac { \\alpha ( \\theta _ { i } + \\theta _ { 3 } ) ( a + b x _ { i } ) \\eta ^ { \\alpha - 1 } ( x _ { i } ) e ^ { - \\eta ^ { \\alpha } ( x _ { i } ) } } { \\left ( 1 - e ^ { - \\eta ^ { \\alpha } ( x _ { i } ) } \\right ) } , \\ i = 1 , 2 . \\end{align*}"}
-{"id": "5762.png", "formula": "\\begin{align*} G _ { , p } = \\left \\{ ( p + 1 ) - ( p + 1 ) ^ { 1 / 2 } \\right \\} I _ p - J _ p , \\end{align*}"}
-{"id": "1419.png", "formula": "\\begin{align*} Q _ j ( x _ 3 ) & = x _ { 2 } ^ { 2 ^ { j - 1 } } Q _ { j - 1 } ( x _ 3 ) + x _ { 3 } ^ { 2 ^ { j - 1 } } Q _ { j - 2 } ( x _ 3 ) \\\\ & = x _ 3 \\left ( \\alpha _ { j - 1 , 0 } x _ 2 ^ { 2 ^ { j - 1 } } + \\alpha _ { j - 2 , 0 } x _ 3 ^ { 2 ^ { j - 1 } } \\right ) , \\end{align*}"}
-{"id": "649.png", "formula": "\\begin{align*} D _ t ^ m H = Q ( g , h , F , m ) \\end{align*}"}
-{"id": "4729.png", "formula": "\\begin{align*} \\beta ^ 2 c _ 0 + 2 \\beta \\gamma f _ 0 + \\gamma ^ 2 d _ 0 + 2 ( \\alpha + \\beta a _ 0 + \\gamma b _ 0 ) { L } _ 0 = 0 , \\end{align*}"}
-{"id": "6984.png", "formula": "\\begin{align*} M _ { i i } = D f _ k ( B ) E _ { i i } , \\end{align*}"}
-{"id": "7859.png", "formula": "\\begin{align*} V ( x ) = \\frac { 1 } { 1 - \\left ( \\Psi ( x ) \\right ) ^ { \\theta _ { 1 } + \\theta _ { 2 } + \\theta _ { 3 } } } \\sum _ { i = 0 } ^ { \\infty } \\sum _ { j = 0 } ^ { \\alpha - 1 } \\sum _ { k = 0 } ^ { \\infty } \\vartheta _ { i , j } ^ { ( k ) } ( a \\Gamma ( \\alpha ^ { \\ast } , x ) + b ( \\Gamma ( \\alpha ^ { \\ast } + 1 , x ) ) . \\end{align*}"}
-{"id": "9645.png", "formula": "\\begin{align*} \\hat { \\Phi } _ { \\rm { s t } , \\beta } ( t ) = \\frac { S _ \\beta } { \\pi E _ { \\rm { b } } } \\bigg ( \\frac { P _ { \\rm { c u } } } { \\hat { R } ^ { * 2 } _ { \\beta } } + \\lambda _ { \\beta } \\left ( 2 ^ { C / W } - 1 \\right ) P _ { \\rm { t r } , 1 } ( h _ { \\beta , 1 } ^ * ) \\hat { R } ^ { * 2 } _ { \\beta } \\bigg ) , \\end{align*}"}
-{"id": "4327.png", "formula": "\\begin{align*} \\sum _ { j , l = 1 } ^ { k } a _ { j l } \\zeta _ j \\zeta _ l = \\Vert \\nabla \\phi \\Vert _ { L ^ 2 ( \\Omega ) } ^ 2 + \\big ( T ( \\varphi _ { p , k } ) \\phi , \\phi \\big ) \\geq \\Vert \\nabla \\phi \\Vert _ { L ^ 2 ( \\Omega ) } ^ 2 \\geq \\lambda _ 1 \\Vert \\phi \\Vert _ { L ^ 2 ( \\Omega ) } ^ 2 \\ , . \\end{align*}"}
-{"id": "6645.png", "formula": "\\begin{align*} \\varrho \\leq \\delta _ 1 \\leq \\kappa ( 1 + K _ 0 ^ 2 ) ^ { 1 / 2 } \\ell _ 0 \\leq \\kappa ( 1 + K _ 0 ^ 2 ) ^ { 1 / 2 } \\kappa \\ell = K \\ell \\ . \\end{align*}"}
-{"id": "6014.png", "formula": "\\begin{align*} & \\omega _ { Q _ { X Y U } } ^ { ( \\alpha ) } ( x , y | u ) : = \\bar { \\alpha } \\Bigg ( \\log \\frac { Q _ { X Y } ( x , y ) } { \\pi _ { X Y } ( x , y ) } \\\\ & + \\log \\frac { Q _ { X Y | U } ( x , y | u ) } { Q _ { X | U } ( x | u ) Q _ { Y | U } ( y | u ) } \\Bigg ) + \\alpha \\log \\frac { Q _ { X Y | U } ( x , y | u ) } { \\pi _ { X Y } ( x , y ) } . \\end{align*}"}
-{"id": "7866.png", "formula": "\\begin{align*} d ( z _ 1 , z _ 2 ) = { 1 \\over 2 } d ( x , y ) . \\end{align*}"}
-{"id": "8150.png", "formula": "\\begin{align*} \\frac { 1 } { | S _ i | } \\sum _ { j = 1 } ^ n | E _ { i , j } | \\leq \\frac { 1 } { | S _ i | } \\sum _ { k = 1 } ^ m | F _ { i , k } | , \\end{align*}"}
-{"id": "1790.png", "formula": "\\begin{align*} g ( t ) & : = 1 + \\frac { b ( q - 2 ) ( 6 - q ) } { a ( p - 2 ) ( 6 - p ) } t - \\frac { c ( r - 2 ) ( 6 - r ) } { a ( p - 2 ) ( 6 - p ) } t ^ { \\frac { r - p } { q - p } } \\\\ h ( t ) & : = 1 + \\frac { b ( q - 2 ) } { a ( p - 2 ) } t - \\frac { c ( r - 2 ) } { a ( p - 2 ) } t ^ { \\frac { r - p } { q - p } } . \\end{align*}"}
-{"id": "7243.png", "formula": "\\begin{align*} = \\int _ { \\hat { G } } ^ { } \\int _ { U } ^ { } ( \\int _ { G } ^ { } \\theta _ { \\pi } ( u x ) f ( x ) d x ) \\psi _ { n } ( u ) d u d \\mu _ { \\pi } . \\end{align*}"}
-{"id": "6209.png", "formula": "\\begin{align*} \\begin{pmatrix} - 2 & 0 & e \\\\ 0 & - 2 & 0 \\\\ e & 0 & 2 k \\end{pmatrix} . \\end{align*}"}
-{"id": "7761.png", "formula": "\\begin{align*} Q _ { 1 , i } & = \\bigg \\{ \\ , \\sum _ { j = 1 } ^ { d } n _ { j } A e _ { j } \\ , \\bigg | \\ , ( n _ { j } ) \\in \\Z ^ { d } , n _ { i } < - k n _ { j } \\in [ - k , k ] j \\neq i \\ , \\bigg \\} , \\\\ Q _ { 2 , i } & = \\bigg \\{ \\ , \\sum _ { j = 1 } ^ { d } n _ { j } A e _ { j } \\ , \\bigg | \\ , ( n _ { j } ) \\in \\Z ^ { d } , n _ { i } > k n _ { j } \\in [ - k , k ] j \\neq i \\ , \\bigg \\} . \\end{align*}"}
-{"id": "1600.png", "formula": "\\begin{align*} c ( j _ n , - \\alpha ( n ) - 1 ) = s _ ( j _ n ) + s ( - \\alpha ( n ) - 1 ) - s ( j _ n - \\alpha ( n ) - 1 ) , \\end{align*}"}
-{"id": "5268.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c c } 1 & - 1 & 0 \\\\ - 1 & 2 & - 1 \\\\ 0 & - 1 & 2 \\end{array} \\right ) \\hbox { a n d } B = \\left ( \\begin{array} { c c c } 2 & 0 & - 1 \\\\ 0 & 1 & 1 \\\\ - 1 & 1 & 2 \\end{array} \\right ) \\end{align*}"}
-{"id": "7513.png", "formula": "\\begin{gather*} u _ t + u u _ x - u _ { x x } - u _ { y y } = 0 , \\end{gather*}"}
-{"id": "3175.png", "formula": "\\begin{align*} \\rho _ { t } = \\lambda _ { t } ^ { - 1 } \\int _ { 0 } ^ { t } \\int _ { \\lbrace z > 1 \\rbrace } m _ { \\alpha ( z , s ) , \\beta ( z , s ) } \\nu ( \\mathrm { d } z ) \\mathrm { d } s , \\end{align*}"}
-{"id": "1242.png", "formula": "\\begin{align*} \\mathcal { L } _ i \\zeta = \\sum _ { k , j = 1 } ^ n \\left ( ( \\breve b _ i ) _ { k j } \\zeta _ { x _ j } \\right ) _ { x _ k } = 0 \\end{align*}"}
-{"id": "3115.png", "formula": "\\begin{align*} Z ^ { \\theta - s s t } = R ^ { \\theta - s s t } \\cap Z = B ^ { [ \\sigma - s s t ] } , \\end{align*}"}
-{"id": "3283.png", "formula": "\\begin{align*} \\theta _ { } = \\arcsin \\left ( \\frac { a b - \\frac { 1 } { 2 } \\sqrt { \\Delta } } { a ^ 2 + 1 } \\right ) , \\theta _ { } = \\arcsin \\left ( \\frac { a b + \\frac { 1 } { 2 } \\sqrt { \\Delta } } { a ^ 2 + 1 } \\right ) , \\end{align*}"}
-{"id": "6060.png", "formula": "\\begin{align*} \\tau ( \\varphi ) = 0 , \\end{align*}"}
-{"id": "3682.png", "formula": "\\begin{align*} \\tilde Z ^ { ( n ) } = \\tilde Z ^ { ( n ) \\prime } \\times \\mathbb A ^ { n } \\to \\tilde X ^ { ( n ) \\prime } \\times \\mathbb A ^ { n } = ( S / C ) ^ n . \\end{align*}"}
-{"id": "7815.png", "formula": "\\begin{align*} - \\Delta v + 2 \\rho ^ { \\prime } \\frac { \\partial v } { \\partial r } + ( V _ 1 + V _ 2 + V _ 0 ) v = \\lambda v , \\end{align*}"}
-{"id": "4725.png", "formula": "\\begin{align*} \\alpha a _ 0 + \\beta b _ 0 + \\gamma c _ 0 = 0 , \\alpha , \\beta , \\gamma - c o n s t , \\end{align*}"}
-{"id": "8990.png", "formula": "\\begin{gather*} D ^ { ( n ) } _ q ( - c \\pm u ; t ) { \\cal D } ^ { ( n ) } _ { q , t } ( c + q / 2 ) = { \\cal D } ^ { ( n ) } _ { q , t } ( c ) \\prod _ { 1 \\le i \\le n } \\vartheta ( z _ i \\pm u ) , \\\\ { \\cal D } ^ { ( n ) } _ { q , t } ( c ) D ^ { ( n ) } _ q ( c \\pm u ; t ) = \\prod _ { 1 \\le i \\le n } \\vartheta ( z _ i \\pm u ) { \\cal D } ^ { ( n ) } _ { q , t } ( c - q / 2 ) , \\end{gather*}"}
-{"id": "7692.png", "formula": "\\begin{align*} B = & \\left [ \\begin{array} { c } 0 \\\\ Q \\end{array} \\right ] \\\\ M = & \\left [ \\begin{array} { c c } 0 & I \\\\ - \\bar { \\Lambda } & - \\bar { \\Lambda } \\end{array} \\right ] \\\\ Z = & \\left [ \\begin{array} { c c } Q ( e _ j - e _ k ) ( Q ( e _ j - e _ k ) ) ^ { \\top } & 0 \\\\ 0 & 0 \\end{array} \\right ] \\ , . \\end{align*}"}
-{"id": "3236.png", "formula": "\\begin{align*} ( P _ { W } ^ { Y W } \\circ P _ { Y W } ^ { X Y W } ) _ \\ast ( \\beta ) = ( P _ { W } ^ { X Y W } ) _ \\ast ( \\beta ) . \\end{align*}"}
-{"id": "10130.png", "formula": "\\begin{align*} \\bar { \\boldsymbol p } _ k ( i ) = \\lambda \\bar { \\boldsymbol p } _ k ( i - 1 ) + d _ k ^ * ( i ) \\boldsymbol x _ k ( i ) . \\end{align*}"}
-{"id": "2903.png", "formula": "\\begin{align*} ( \\exists t _ 1 \\forall s _ 1 \\geq t _ 1 ) ( \\forall t _ 2 \\geq s _ 1 \\exists s _ 2 \\geq t _ 2 ) \\cdots c ( k , s _ 1 , \\ldots , s _ { n - 1 } ) = i . \\end{align*}"}
-{"id": "4229.png", "formula": "\\begin{align*} \\bar { S } ^ { - 1 } A : = A \\otimes _ { A ^ 0 } S ^ { - 1 } A ^ 0 . \\end{align*}"}
-{"id": "4541.png", "formula": "\\begin{align*} \\Big | \\sum _ { j = 0 } ^ { n - 1 } \\phi \\circ F _ j - \\sum _ { j = 0 } ^ { n - 1 } \\phi \\circ f ^ j \\circ h ^ { - 1 } \\Big | & \\le \\sum _ { j = 0 } ^ { n - 1 } \\| \\phi _ j - \\phi \\| _ { C ^ 0 } \\le | \\phi | _ \\alpha L ^ \\alpha \\sum _ { j = 0 } ^ { n - 1 } a _ j ^ \\alpha \\end{align*}"}
-{"id": "851.png", "formula": "\\begin{align*} \\partial _ { ( 0 , s ] } ^ { \\beta } \\bar h ( r , s ) = - \\partial _ r \\bar h ( r , s ) , \\end{align*}"}
-{"id": "1264.png", "formula": "\\begin{align*} \\bar { d } _ { i j } ( x ) = \\int _ 0 ^ 1 f _ { \\eta _ i \\eta _ j } ( s \\nabla u ( x , t _ 1 ) + ( 1 - s ) \\nabla u ( x , t _ 2 ) ) d s . \\end{align*}"}
-{"id": "7448.png", "formula": "\\begin{align*} x ' & = - B _ { 0 0 } ( B _ { 1 0 } + D _ { 1 0 } + T _ 0 ^ \\gamma ) + B _ { 0 1 } t T _ 0 ^ \\gamma \\\\ y & = - ( B _ { 1 0 } + D _ { 1 0 } + T _ 0 ^ \\gamma ) , \\\\ z & = - B _ { 0 1 } ( B _ { 1 0 } + D _ { 1 0 } + T _ 0 ^ \\gamma ) + B _ { 0 0 } t . \\end{align*}"}
-{"id": "5458.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { U ( r ^ n z ) } { ( r ^ n z ) ^ \\rho \\ell ( r ^ n z ) } = p ( z ) z \\in C _ p , p \\in \\mathcal { P } _ { r } , \\end{align*}"}
-{"id": "9862.png", "formula": "\\begin{align*} [ - N _ { \\alpha \\mathbb B ^ n _ \\infty } ( d ) ] \\cap N _ { K - \\hat x } ( d ) = \\{ 0 \\} \\end{align*}"}
-{"id": "9917.png", "formula": "\\begin{align*} d _ 2 ( u ) + d _ 2 ( v ) + d _ 2 ( w ) > 2 \\bigl ( \\tfrac 1 2 | B _ 2 | - \\tfrac { 1 / 1 5 } { 3 r - 2 } n \\bigr ) + \\tfrac { 1 / 3 } { 3 r - 2 } n = | B _ 2 | + \\tfrac { 1 / 5 } { 3 r - 2 } n \\ , . \\end{align*}"}
-{"id": "9193.png", "formula": "\\begin{align*} L = \\mathfrak { g } \\oplus ( V \\otimes B ) \\oplus ( V ' \\otimes B ' ) \\oplus D \\end{align*}"}
-{"id": "10090.png", "formula": "\\begin{align*} \\tilde { r } = r - \\lambda ( n - 1 ) . \\end{align*}"}
-{"id": "9085.png", "formula": "\\begin{align*} f ( z ^ { 2 } ) = f ( x ^ { 2 } + 2 x y + y ^ { 2 } ) = 2 x f ( x ) + 2 f ( x y ) + 2 y f ( y ) . \\end{align*}"}
-{"id": "8829.png", "formula": "\\begin{align*} \\phi ( t , x ) = - \\frac { 4 } { 5 } t ^ { - \\frac { 1 } { 4 } } | x | ^ { \\frac { 5 } { 4 } } + \\frac { \\pi } { 4 } . \\end{align*}"}
-{"id": "9753.png", "formula": "\\begin{align*} F _ { c } ( \\tilde { J } _ k ) - F _ { c } ( \\tilde { J } _ { k - 1 } ) & = F ( \\tilde { J } _ k ) - F ( \\tilde { J } _ { k - 1 } ) - K _ z e ^ { - l k h } h \\| Z _ { 0 } \\| _ { \\infty } \\\\ & \\leq ( M \\big ( F ( J _ k ) + 2 \\big ) ^ 2 - K _ z ) e ^ { - l k h } h \\| Z _ { 0 } \\| _ { \\infty } . \\end{align*}"}
-{"id": "1164.png", "formula": "\\begin{align*} e _ k ( w ) = \\sum _ { i = 0 } ^ { k - 2 } T _ { k , i } [ 1 ] _ w ^ { 2 ^ i } . \\end{align*}"}
-{"id": "440.png", "formula": "\\begin{align*} p _ { 1 , k _ 1 , k _ 2 } ( x , t ) = \\frac { \\pi ^ { k _ 1 + k _ 2 } ( - 1 ) ^ { k _ 2 } } { 4 ^ n ( n + k _ 1 - 1 ) ! } e ^ { - \\pi \\abs { t } - R } \\abs { t } ^ { n + k _ 1 - 1 } \\left [ 1 + O \\left ( \\frac { 1 } { \\abs { t } } + \\kappa + \\delta \\right ) \\right ] + O \\left ( e ^ { - \\frac { 3 \\pi \\abs { t } } { 2 } } \\right ) . \\end{align*}"}
-{"id": "4340.png", "formula": "\\begin{align*} | e ^ { \\alpha \\langle z , w \\rangle - \\beta | z | ^ 2 - \\gamma | w | ^ 2 } | = e ^ { \\alpha \\operatorname { R e } \\langle z , w \\rangle - \\beta | z | ^ 2 - \\gamma | w | ^ 2 } \\end{align*}"}
-{"id": "6342.png", "formula": "\\begin{align*} z _ k : = \\frac { m _ k - m _ k ^ - } { m _ { k - 1 } - m _ { k - 1 } ^ - } , k = 1 , \\dots , n , \\end{align*}"}
-{"id": "9886.png", "formula": "\\begin{align*} S = \\Bigl \\{ \\tfrac d n \\colon ( d , n ) \\in \\Omega \\Bigr \\} \\ , . \\end{align*}"}
-{"id": "6855.png", "formula": "\\begin{align*} \\Delta ( u \\circ F _ j ) = r _ j ^ { - 1 } ( \\Delta u ) \\circ F _ j \\end{align*}"}
-{"id": "1282.png", "formula": "\\begin{align*} E _ j \\subset \\bar B ( 0 , \\rho ) \\mbox { f o r $ j = 1 , 2 , \\dots , $ a n d s o m e } \\rho < \\infty . \\end{align*}"}
-{"id": "9707.png", "formula": "\\begin{align*} \\tilde { \\Phi } ( \\delta _ 5 , \\delta _ 3 , \\delta _ 2 , \\delta _ 1 ; V _ { \\widetilde { m } } ) = \\tilde { \\Phi } ( \\beta _ 5 , \\beta _ 3 , \\beta _ 2 , \\beta _ 1 ; \\tilde { \\Phi } _ 5 ( \\alpha _ 5 ; V _ { \\widetilde { m } } ) ) . \\end{align*}"}
-{"id": "1075.png", "formula": "\\begin{align*} \\sum _ j c _ j b _ j = 0 . \\end{align*}"}
-{"id": "633.png", "formula": "\\begin{align*} C _ 1 & = \\frac { 2 ^ { 2 / p - 1 } a ^ { 1 / 2 - 1 / p } \\left [ ( 1 + \\delta _ { a k } ) ^ { 1 / p } + ( 1 - \\delta _ { ( a + 1 ) k } ) ^ { 1 / p } \\right ] } { [ ( 1 - \\delta _ { ( a + 1 ) k } ) - a ^ { p / 2 - 1 } ( 1 + \\delta _ { a k } ) \\gamma ] ^ { 1 / p } } , \\\\ C _ 2 & = \\frac { 2 ^ { 1 / p } m ^ { 1 / p - 1 / 2 } ( 1 + a ^ { 1 / 2 - 1 / p } \\gamma ^ { 1 / p } ) } { [ ( 1 - \\delta _ { ( a + 1 ) k } ) - a ^ { p / 2 - 1 } ( 1 + \\delta _ { a k } ) \\gamma ] ^ { 1 / p } } . \\end{align*}"}
-{"id": "8640.png", "formula": "\\begin{align*} \\hat { \\pi } _ { 0 , 1 } ( X _ 0 , d x _ 1 ) \\left ( \\prod _ { k = 1 } ^ { N - 1 } \\hat { \\pi } ( x _ k , d x _ { k + 1 } ) \\right ) \\hat { \\pi } _ { N , N + 1 } ( x _ N , d x _ { N + 1 } ) . \\end{align*}"}
-{"id": "9093.png", "formula": "\\begin{align*} B ( x , y ) = \\sum _ { i = 1 } ^ { n } \\binom { n + 1 } { i } d ^ { i } ( x ) d ^ { n + 1 - i } ( y ) \\left ( x , y \\in R \\right ) \\end{align*}"}
-{"id": "6986.png", "formula": "\\begin{align*} D f _ k ( B ) A = a _ { i j } D f _ k ( B ) E _ { i j } = a _ { i j } M _ { i j } = t r a c e ( A M ^ T ) . \\end{align*}"}
-{"id": "6307.png", "formula": "\\begin{align*} \\partial _ t \\big ( \\begin{smallmatrix} \\psi _ 1 \\\\ \\psi _ 2 \\end{smallmatrix} \\big ) & = V \\big ( \\begin{smallmatrix} \\psi _ 1 \\\\ \\psi _ 2 \\end{smallmatrix} \\big ) & & & V & = \\big ( \\begin{smallmatrix} u _ x & - 4 \\lambda - 2 u \\\\ 4 \\lambda ^ 2 - 2 u \\lambda - 2 u ^ 2 + u _ { x x } & - u _ x \\end{smallmatrix} \\big ) . \\end{align*}"}
-{"id": "5237.png", "formula": "\\begin{align*} \\mu ( \\gamma _ 1 , \\gamma _ 2 ) = - n _ - ( \\Gamma ( \\gamma _ 1 , \\gamma _ 2 , a ) ) + \\sum \\limits _ { t ^ * \\in ( a , b ) } \\mathrm { s i g n } \\ , \\Gamma ( \\gamma _ 1 , \\gamma _ 2 , t ^ * ) + n _ + ( \\Gamma ( \\gamma _ 1 , \\gamma _ 2 , b ) ) , \\end{align*}"}
-{"id": "7868.png", "formula": "\\begin{align*} d ( x , y ) \\le d ( x , z _ 1 ) + d ( z _ 1 , z _ 2 ) + d ( z _ 2 , y ) = { 1 \\over 2 } d ( x , y ) + d ( z _ 1 , z _ 2 ) . \\end{align*}"}
-{"id": "9561.png", "formula": "\\begin{align*} Q _ 0 ( y ) = \\begin{cases} c ^ + | y | ^ { - \\sigma } \\ , , & y > M _ 0 \\ , , \\\\ [ 2 p t ] c ^ - | y | ^ { - \\sigma } \\ , , & y < - M _ 0 \\ , , \\end{cases} \\sigma > 1 , Q _ 0 ( y ) > 0 , \\int _ \\R Q _ 0 \\ , d y = 1 . \\end{align*}"}
-{"id": "7044.png", "formula": "\\begin{align*} \\tilde { \\sigma } _ n = g ( t ) \\sigma _ n . \\end{align*}"}
-{"id": "3226.png", "formula": "\\begin{align*} \\psi ^ \\ast ( \\eta _ Z ) = \\eta _ W . \\end{align*}"}
-{"id": "150.png", "formula": "\\begin{align*} N ' = \\begin{pmatrix} x ' & y ' \\\\ z ' & w ' \\end{pmatrix} \\end{align*}"}
-{"id": "8039.png", "formula": "\\begin{align*} \\| \\ , D \\mbox { \\boldmath $ u $ } ( \\cdot , t ) \\ , \\| _ { \\mbox { } _ { \\scriptstyle L ^ { q } ( \\mathbb { R } ^ { 3 } ) } } \\ ; \\ ! = \\ ; \\Bigl \\{ \\ , \\sum _ { i , \\ , j \\ , = \\ , 1 } ^ { 3 } \\int _ { \\mathbb { R } ^ { 3 } } \\ ! | \\ , D _ { \\ ! \\ ; \\ ! j } \\ ; \\ ! u _ { i } ( x , t ) \\ , | ^ { q } \\ , d x \\ , \\Bigr \\} ^ { \\ ! \\ ! \\ : \\ ! 1 / q } \\end{align*}"}
-{"id": "7411.png", "formula": "\\begin{align*} \\mathbb { H } _ 2 : = \\mathbb { C } [ t , T _ 0 ^ { \\beta } , T _ 0 ^ { \\gamma } , T _ 0 ^ { \\delta } ] . \\end{align*}"}
-{"id": "4599.png", "formula": "\\begin{align*} [ x , y ] = x \\cdot y - y \\cdot x \\end{align*}"}
-{"id": "2491.png", "formula": "\\begin{align*} | \\xi g | ^ { 2 } _ { L ^ { 2 } _ { \\xi } } \\leq C \\big < \\Lambda g , \\Lambda g \\big > _ { \\xi } & = \\big < L g - K g , L g - K g \\big > _ { \\xi } \\\\ & \\leq C _ { 1 } | L g | ^ { 2 } _ { L ^ { 2 } _ { \\xi } } + C _ { 2 } | g | ^ { 2 } _ { L ^ { 2 } _ { \\xi } } \\ , . \\end{align*}"}
-{"id": "2618.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 - \\log | t - s | \\ , u ( s ) \\ , d s = f ( t ) 0 < t < 1 \\end{align*}"}
-{"id": "2508.png", "formula": "\\begin{align*} \\Lambda f = \\nabla _ \\xi \\cdot \\left [ \\sigma \\nabla _ \\xi f \\right ] - \\psi ( \\xi ) f , \\end{align*}"}
-{"id": "3560.png", "formula": "\\begin{align*} \\Psi ( n ) = \\left \\{ \\begin{array} { l l } 1 & n , \\\\ 0 & n , \\end{array} \\right . \\end{align*}"}
-{"id": "8498.png", "formula": "\\begin{align*} 1 - v b _ 1 = \\beta \\varpi ^ { \\frac { a _ 1 - a _ 2 } { 2 } } . \\end{align*}"}
-{"id": "859.png", "formula": "\\begin{align*} \\Phi _ \\beta ( x ) = \\beta ^ { - 1 } \\log \\left ( \\sum _ { j = 1 } ^ d e ^ { \\beta x _ j } \\right ) \\qquad ( x = ( x _ 1 , \\dots , x _ d ) ^ \\top \\in \\mathbb { R } ^ d ) . \\end{align*}"}
-{"id": "1150.png", "formula": "\\begin{align*} x \\log ( z ) & = \\log \\left ( \\rho _ x ( z ) \\right ) \\\\ & = \\log \\left ( x z + [ 1 ] _ x z ^ 2 + z ^ 4 \\right ) \\\\ & = \\log ( x z ) + \\log ( [ 1 ] _ x z ^ 2 ) + \\log ( z ^ 4 ) , \\end{align*}"}
-{"id": "10047.png", "formula": "\\begin{align*} [ ( 0 , \\log ( D ) ) : \\mathcal { Y } _ \\mathrm { b i g } ] = \\deg _ \\C ( \\mathcal { Y } _ \\mathrm { b i g } ) \\cdot \\log ( D ) = [ ( \\mathrm { E x c } , 0 ) : \\mathcal { Y } _ \\mathrm { b i g } ] . \\end{align*}"}
-{"id": "9544.png", "formula": "\\begin{align*} \\int \\limits _ I \\overline { g } \\ , d \\mu _ T = \\int \\limits _ I \\Phi _ T ( x ) \\ , d \\lambda _ g . \\end{align*}"}
-{"id": "3846.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\lambda _ i a _ i + \\sum _ { i = 1 } ^ p \\mu _ i b _ i = 0 . \\end{align*}"}
-{"id": "4794.png", "formula": "\\begin{align*} L = \\left ( \\begin{array} { c | c } A & b \\\\ \\hline \\cdots & \\cdot \\end{array} \\right ) , \\quad L _ { i j } = \\begin{cases} a _ { i j } & i , j \\leq m \\\\ b _ i & i \\leq m j = m + 1 \\\\ - \\sum \\limits _ { k = 1 } ^ m a _ { k j } & i = m + 1 j \\leq m \\\\ - \\sum \\limits _ { k = 1 } ^ m b _ { k } & i = j = m + 1 , \\end{cases} \\end{align*}"}
-{"id": "7045.png", "formula": "\\begin{align*} \\frac { \\tilde { \\eta } _ j } { \\eta _ j } = \\frac { ( \\sum _ { 1 \\leq i \\leq n - 1 } { t _ i \\sigma _ i } ) - t _ j \\sigma _ j + g ( t ) \\sigma _ n } { ( \\sum { \\sigma _ i } ) - \\sigma _ j } = \\frac { 1 } { t } , j = 1 , 2 , . . . , n - 1 ; \\end{align*}"}
-{"id": "730.png", "formula": "\\begin{align*} \\frac { a ^ 2 \\theta _ \\sharp + U _ R ^ 2 } { a ^ 2 \\theta _ - + ( U _ R + u _ - ) ^ 2 } = \\frac { U _ R } { U _ R + u _ - } \\end{align*}"}
-{"id": "7483.png", "formula": "\\begin{align*} \\Delta ^ v f = \\dfrac { 1 } { h } \\dot { \\partial } _ \\alpha \\big [ h h ^ { \\bar { \\gamma } \\alpha } ( \\dot { \\partial } _ { \\bar { \\gamma } } f ) \\big ] + \\big [ h ^ { \\bar { \\gamma } \\alpha } ( \\dot { \\partial } _ { \\bar { \\gamma } } f ) \\big ] C _ \\alpha . \\end{align*}"}
-{"id": "999.png", "formula": "\\begin{align*} \\widehat { \\sigma } _ { k , n } ^ 2 ( t ) = \\sum _ { i = 1 } ^ n K _ h ( t _ { i - 1 } - t ) \\left ( \\int _ { t _ { i - 1 } } ^ { t _ i } \\sigma _ k ( s ) d B _ s \\right ) ^ 2 . \\end{align*}"}
-{"id": "4625.png", "formula": "\\begin{align*} \\left | \\frac { y _ { i + 1 } x _ { i + 1 } - \\zeta y _ i } { x _ { i + 1 } - \\zeta } - y _ { i + 1 } \\right | = \\left | \\frac { \\zeta y _ { i + 1 } - \\zeta y _ i } { x _ { i + 1 } - \\zeta } \\right | \\le \\frac { | \\zeta | 2 D } { | x _ { i + 1 } - \\zeta | } \\le \\frac { | \\zeta | 8 D } { 3 x _ { i + 1 } } \\le \\frac { | \\zeta | 8 D } { 3 \\xi } \\end{align*}"}
-{"id": "6542.png", "formula": "\\begin{align*} F _ { \\xi } ^ { \\eta } = \\{ z \\in F _ { \\xi } : \\mathrm { d i s t } ( z , \\partial F _ { \\xi } ) \\geq \\eta \\} \\end{align*}"}
-{"id": "9890.png", "formula": "\\begin{align*} k ( r - 1 ) - \\tbinom { r } { 2 } = \\tbinom { k } { 2 } - \\tbinom { r - k } { 2 } \\le \\tbinom { k } { 2 } \\ , . \\end{align*}"}
-{"id": "4073.png", "formula": "\\begin{align*} \\partial _ t v _ i - \\partial _ \\alpha \\big \\langle \\nu , \\frac { \\partial G } { \\partial \\Xi ^ A } ( \\lambda _ \\Xi ) \\ , \\frac { \\partial \\Phi ^ A } { \\partial F _ { i \\alpha } } ( \\lambda _ F ) \\big \\rangle = 0 \\\\ \\intertext { a n d f o r $ A = 1 , \\dots , 1 9 $ } \\partial _ t \\Phi ^ A ( F ) - \\partial _ \\alpha \\big ( \\frac { \\partial \\Phi ^ A } { \\partial F _ { i \\alpha } } ( F ) v _ i \\big ) = 0 \\end{align*}"}
-{"id": "5095.png", "formula": "\\begin{align*} e ( k , \\tau ) = E _ G ( k , \\xi ) \\end{align*}"}
-{"id": "8364.png", "formula": "\\begin{align*} \\mathrm { F J } ^ { ( a ) } ( \\tilde { \\psi } ( f ) ) = \\big ( \\kappa ^ { ( a ) } \\cdot q _ { \\alpha ( \\varrho ) } \\cdot \\mathrm { B P } ( f ) \\big ) ^ 2 , \\end{align*}"}
-{"id": "7882.png", "formula": "\\begin{align*} \\begin{aligned} | d ( z ' , m ( x ' , y ' ) ) - d ( z , m ( x , y ) ) | & \\le d ( z ' , z ) + d ( m ( x ' , y ' ) , m ( x , y ) ) \\\\ & \\le d ( z ' , z ) + { 1 \\over 2 } d ( x ' , x ) + { 1 \\over 2 } d ( y ' , y ) . \\end{aligned} \\end{align*}"}
-{"id": "8144.png", "formula": "\\begin{align*} f h _ { k , c , i , j } ^ * h _ { k , c , i , j } & = \\sum _ { q = 1 } ^ Q \\sum _ { t \\in B _ { k , l , c , q } } \\frac { q ^ 2 } { Q ^ 2 } f \\alpha _ { t c } ( h _ k ^ 2 ) \\end{align*}"}
-{"id": "1589.png", "formula": "\\begin{align*} v ( \\mathbf { j } ''' ) - v ( \\mathbf { g } ) & = 2 ( g _ { n - 1 } + p ) - s ( g _ { n - 1 } + p ) - 1 - 2 g _ { n - 1 } + s ( g _ { n - 1 } ) \\\\ & \\geq 2 p - 1 - s ( p ) \\\\ & > 0 . \\end{align*}"}
-{"id": "9866.png", "formula": "\\begin{align*} \\underset { _ { \\nu \\rightarrow \\infty } } \\liminf \\ ; \\ ; \\theta \\delta ^ \\alpha \\left ( \\sum _ { t = \\nu } ^ { i _ { \\nu } - 1 } \\gamma ^ { t } \\right ) ^ \\alpha > 0 . \\end{align*}"}
-{"id": "7716.png", "formula": "\\begin{align*} & \\lambda _ { 0 } = 0 \\ , \\\\ & \\lambda _ n = 1 \\ , , n = 1 , 2 , \\cdots , N - 2 \\ , , \\\\ & \\lambda _ { N - 1 } = N \\ , , \\end{align*}"}
-{"id": "7013.png", "formula": "\\begin{align*} h ( \\epsilon ) = \\frac { 1 - \\frac { 2 ( n - 2 ) } { n - 1 } \\epsilon ^ 2 } { 2 \\epsilon } , \\end{align*}"}
-{"id": "6951.png", "formula": "\\begin{align*} \\Gamma _ k = \\{ \\lambda \\in \\mathbb { R } ^ n , \\sigma _ l ( \\lambda ) > 0 , l = 1 , 2 , . . . , k \\} . \\end{align*}"}
-{"id": "2400.png", "formula": "\\begin{align*} p _ 3 ( x _ 1 ) = - \\frac { ( - a _ 2 ^ 2 + a _ 3 ( 3 a _ 3 + a _ 4 ) ) ( a _ 2 ^ 2 + ( 3 a _ 3 + a _ 4 ) ^ 2 ) } { ( 3 a _ 3 + a _ 4 ) ^ 3 } = 0 . \\end{align*}"}
-{"id": "8342.png", "formula": "\\begin{align*} V _ { \\Z _ p } = \\Z _ p k \\oplus \\Z _ p \\ell \\oplus W _ { \\Z _ p } . \\end{align*}"}
-{"id": "1673.png", "formula": "\\begin{align*} \\mu ( Z ( e g _ 1 \\cdots e g _ n e g _ { n + 1 } ) ) & + \\mu ( Z ( e g _ 1 \\cdots e g _ n e h _ { n + 1 } ) ) \\\\ & = \\mu ( Z ( e g _ 1 \\cdots e g _ n ) ) ( 1 / 2 + \\gamma _ { n + 1 } ) + \\mu ( Z ( e g _ 1 \\cdots e g _ n ) ) ( 1 / 2 - \\gamma _ { n + 1 } ) \\\\ & = \\mu ( Z ( e g _ 1 \\cdots e g _ n ) ) . \\end{align*}"}
-{"id": "5586.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } w _ { 1 } \\left ( t \\right ) \\\\ w _ { 0 } \\left ( t \\right ) \\\\ w _ { 3 } \\left ( t \\right ) \\\\ w _ { 2 } \\left ( t \\right ) \\end{array} \\right ] = \\left [ \\begin{array} { c c c c } 0 & 1 & 0 & 0 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 1 & 0 \\end{array} \\right ] \\left [ \\begin{array} { c } w _ { 0 } \\left ( t \\right ) \\\\ w _ { 1 } \\left ( t \\right ) \\\\ w _ { 2 } \\left ( t \\right ) \\\\ w _ { 3 } \\left ( t \\right ) \\end{array} \\right ] \\end{align*}"}
-{"id": "1199.png", "formula": "\\begin{align*} \\bar e ( x ) = o ( G ( x ) ) \\ , = o ( | x | ^ { \\frac { p - n } { p - 1 } } ) \\mbox { a s } \\ , \\ , | x | \\to \\infty . \\end{align*}"}
-{"id": "9549.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in W } x ^ { \\ell _ { \\Delta } ( \\sigma ) } y ^ { L _ { \\Delta } ( \\sigma ) } = \\sum _ { \\sigma \\in W } x ^ { \\ell _ { \\Delta ' } ( \\sigma ) } y ^ { L _ { \\Delta ' } ( \\sigma ) } . \\end{align*}"}
-{"id": "684.png", "formula": "\\begin{align*} \\cot \\alpha = \\cot \\alpha _ 0 + \\sum _ { n = 0 } ^ { \\infty } \\left ( \\cfrac { 1 } { a _ n } - \\cfrac { 1 } { a _ n ^ 0 } \\right ) , \\end{align*}"}
-{"id": "4016.png", "formula": "\\begin{align*} A ( M ) : = \\int _ M \\omega . \\end{align*}"}
-{"id": "5751.png", "formula": "\\begin{align*} \\int _ M G K _ M \\ , d V _ M = \\frac { 1 } { 2 } \\ , c _ n \\ , \\chi ( M ) . \\end{align*}"}
-{"id": "1301.png", "formula": "\\begin{align*} \\cos \\frac { 5 \\pi } { 1 8 } \\ = \\ 2 \\cos \\frac { \\pi } { 1 8 } - \\sqrt { 3 } \\cos \\frac { 4 \\pi } { 1 8 } . \\end{align*}"}
-{"id": "3047.png", "formula": "\\begin{align*} \\lambda ( w , v ) _ H + a ( w , v ) = ( g , v ) _ H , \\forall v \\in V _ \\sigma , \\end{align*}"}
-{"id": "429.png", "formula": "\\begin{align*} H _ { k _ 1 , k _ 2 } ( R , t ) = e ^ { \\kappa \\ , q _ \\delta ( \\sigma _ \\delta ) } \\int _ { - \\pi } ^ { \\pi } e ^ { i \\kappa F _ \\delta \\left ( s \\right ) } \\psi _ { \\delta , k _ 1 , k _ 2 } ( s ) \\ , \\dd s . \\end{align*}"}
-{"id": "8702.png", "formula": "\\begin{align*} j _ { n , k } : = \\arg \\max _ { z \\geq 0 } z ^ { k + \\gamma _ 2 } ( 1 - q ^ { n + 2 } ) ^ z = - \\frac { k + \\gamma _ 2 } { \\log ( 1 - q ^ { n + 2 } ) } \\leq ( k + \\gamma _ 2 ) q ^ { - ( n + 2 ) } \\end{align*}"}
-{"id": "4450.png", "formula": "\\begin{align*} \\lefteqn { ( - \\partial _ 1 ^ 2 - | \\partial _ 1 | ^ { - 1 } \\partial _ 2 ^ 2 ) w ^ \\ell } \\\\ & + P \\Big ( \\sigma ^ 2 F ^ \\ell + \\sigma v _ \\ell \\partial _ 2 R w ^ \\ell + \\sigma w ^ \\ell \\partial _ 2 R v _ \\ell + w ^ \\ell \\partial _ 2 R w ^ \\ell \\\\ & + \\partial _ 2 \\frac { 1 } { 2 } R ( w ^ \\ell + \\sigma v _ \\ell ) ^ 2 - ( w ^ \\ell + \\sigma v _ \\ell ) \\partial _ 1 \\frac { 1 } { 2 } R ( w ^ \\ell + \\sigma v _ \\ell ) ^ 2 \\Big ) = 0 . \\end{align*}"}
-{"id": "3308.png", "formula": "\\begin{align*} H ( A _ n ^ { [ k ] } | Q _ n ^ { [ k ] } , W _ 1 , \\dots , W _ K ) = 0 . \\end{align*}"}
-{"id": "8505.png", "formula": "\\begin{align*} \\delta = \\begin{cases} \\lfloor \\frac { r } { 2 } \\rfloor & v ( \\Delta ) \\geq r , \\\\ \\delta _ 0 & v ( \\Delta ) = 2 \\delta _ 0 < r , \\\\ \\end{cases} Y = \\begin{cases} 0 & v ( \\Delta ) \\geq r , \\\\ Y _ 0 & ( \\Delta ) _ 0 = Y _ 0 ^ 2 v ( \\Delta ) < r . \\end{cases} \\end{align*}"}
-{"id": "9058.png", "formula": "\\begin{align*} \\Im ( b _ { 1 1 } ^ 1 + b _ { 1 2 } ^ 2 + b _ { 1 3 } ^ 3 ) = 0 . \\end{align*}"}
-{"id": "4482.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\| ( \\pi ( a ) - \\Pi ( a ) ) E _ \\pi ( x , \\delta ) \\| = 0 , a \\in C ( X ) . \\end{align*}"}
-{"id": "1368.png", "formula": "\\begin{align*} \\underline { \\sigma } : = \\min \\left \\{ 1 , \\min _ { Q \\in { \\cal O } ( \\overline X ) , P \\in { \\cal O } ( \\overline M ) } \\min _ { 1 \\leq i \\leq n _ 2 } \\sigma _ i ^ { - 2 } ( A ( Q , P ) ) \\right \\} \\end{align*} a n d \\begin{align*} \\overline { \\sigma } : = \\max \\left \\{ 1 , \\max _ { Q \\in { \\cal O } ( \\overline X ) , P \\in { \\cal O } ( \\overline M ) } \\max _ { 1 \\leq i \\leq n _ 2 } \\sigma _ i ^ { - 2 } ( A ( Q , P ) ) \\right \\} \\ , . \\end{align*} % \\end{align*}"}
-{"id": "1604.png", "formula": "\\begin{align*} X = \\bigcup _ { \\lambda \\in \\Lambda } X _ { \\lambda } \\ , . \\end{align*}"}
-{"id": "57.png", "formula": "\\begin{align*} V ^ { C } _ { \\sigma } ( C _ { 1 } , C _ { 2 } ) = \\frac { 1 } { N } \\sum \\limits _ { n = 1 } ^ N G _ { \\sigma \\sqrt { 2 } } ( x _ { n } - y _ { n } ) \\ , G _ { \\sigma \\sqrt { 2 } } ( z _ { n } - s _ { n } ) \\end{align*}"}
-{"id": "7997.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { F _ { p } ^ { s + m , q } } \\lesssim \\Big ( \\sum _ { k = 1 } ^ { \\infty } { | v _ k | ^ { p } } \\Big ) ^ { 1 / p } \\quad \\Vert f \\Vert _ { F _ { p } ^ { s + m , q } } \\lesssim \\Big ( \\sum _ { k = 1 } ^ { \\infty } { | v _ k | ^ { q } } \\Big ) ^ { 1 / q } \\end{align*}"}
-{"id": "7897.png", "formula": "\\begin{align*} u ( \\xi , t ) = \\max _ { \\eta \\in X } \\left \\{ u _ 0 ( \\eta ) - t L \\left ( { d ( \\xi , \\eta ) \\over t } \\right ) \\right \\} \\end{align*}"}
-{"id": "10003.png", "formula": "\\begin{align*} \\frac { \\Lambda ' ( 0 , \\chi _ E ) } { \\Lambda ( 0 , \\chi _ E ) } = \\frac { L ' ( 0 , \\chi _ E ) } { L ( 0 , \\chi _ E ) } + \\frac { 1 } { 2 } \\log \\left | \\frac { D _ E } { D _ F } \\right | - \\frac { [ F : \\Q ] } { 2 } \\log ( 4 \\pi e ^ \\gamma ) , \\end{align*}"}
-{"id": "7273.png", "formula": "\\begin{align*} S _ 1 = \\frac { 1 } { N ^ 2 } \\sum _ { n \\le X } r ( n ) \\left ( 1 - \\frac { \\Phi ( n ) } { n } \\right ) \\end{align*}"}
-{"id": "5796.png", "formula": "\\begin{align*} u : = \\lim _ { j \\rightarrow \\infty } u _ j \\end{align*}"}
-{"id": "9186.png", "formula": "\\begin{align*} D ( \\hat { h } ) = \\frac 1 \\pi & \\sum _ { t = 3 } ^ \\infty \\frac { L ( 1 , \\chi _ d ) } { l } \\prod _ { p \\mid l } \\left ( 1 + \\left ( p - \\chi _ d ( p ) \\right ) \\frac { \\left ( p ^ \\infty , l \\right ) - 1 } { p - 1 } \\right ) \\hat { h } \\ ! \\left ( \\frac { 1 } { \\pi } \\log \\ ! \\left ( \\frac { t + \\sqrt { t ^ 2 - 4 } } { 2 } \\right ) \\right ) \\\\ & + \\sum _ { n = 1 } ^ \\infty \\frac { \\Lambda ( n ) } { n } \\hat { h } \\ ! \\left ( \\frac { \\log n } { \\pi } \\right ) \\end{align*}"}
-{"id": "8242.png", "formula": "\\begin{align*} E ^ { \\mathfrak { L } } ( s , g ) = E _ { v ^ { \\circ } _ { \\mathfrak { L } } } ( s , g ) . \\end{align*}"}
-{"id": "533.png", "formula": "\\begin{align*} g _ J ( \\xi , \\eta ) \\ , = \\ , \\omega ( J \\xi , \\eta ) \\end{align*}"}
-{"id": "4735.png", "formula": "\\begin{align*} { L } _ i = \\left ( L _ 0 ( x ^ 0 ) , \\alpha , \\beta , \\gamma \\right ) . \\end{align*}"}
-{"id": "4762.png", "formula": "\\begin{align*} \\alpha ^ 2 \\omega _ 2 = \\beta t _ 2 + \\gamma , \\alpha ^ 2 \\omega _ 3 = \\beta t _ 3 + \\gamma , \\end{align*}"}
-{"id": "4081.png", "formula": "\\begin{align*} D : = c l \\left \\{ \\cup _ { t \\in [ t _ 0 , t _ 1 ] } ( x _ * ( t ) , u _ * ( t ) ) \\right \\} \\end{align*}"}
-{"id": "8422.png", "formula": "\\begin{align*} \\sum _ { t = - \\infty } ^ { \\infty } q ^ { ( t + 1 ) ( \\frac { 1 } { 2 } - s ) } c _ { t , l } ( \\chi ^ { - 1 } ) = - \\omega _ { \\pi } ( - 1 ) \\frac { L ( s , \\abs { \\cdot } ^ { \\frac { 1 } { 2 } } ) } { L ( 1 - s , \\abs { \\cdot } ^ { \\frac { 1 } { 2 } } ) } G ( \\varpi ^ { - l } , \\chi ) \\end{align*}"}
-{"id": "6169.png", "formula": "\\begin{align*} \\partial _ t f _ i = \\frac { 1 } { V } ( \\frac { f _ i '' } { V } - \\frac { f _ i ' V ' } { V ^ 2 } ) - \\frac { 1 } { 3 } f _ i \\mathcal { T } \\ ; \\ ; , \\ ; \\ ; i = 1 , 2 , 3 \\end{align*}"}
-{"id": "5241.png", "formula": "\\begin{align*} \\begin{aligned} \\lim \\limits _ { z \\rightarrow - \\infty } E ^ u ( \\lambda , z ) = U ( \\lambda ) \\\\ \\lim \\limits _ { z \\rightarrow \\infty } E ^ s ( \\lambda , z ) = S ( \\lambda ) . \\end{aligned} \\end{align*}"}
-{"id": "350.png", "formula": "\\begin{align*} \\nabla _ { \\nu } e _ i = - h _ { i i } e _ i + [ | \\nabla f | \\partial _ 0 , e _ i ] = - h _ { i i } e _ i - e _ i ( | \\nabla f | ) \\partial _ 0 = - h _ { i i } e _ i + \\frac { e _ i ( R ) } { 2 | \\nabla f | } \\partial _ 0 . \\end{align*}"}
-{"id": "5896.png", "formula": "\\begin{align*} \\begin{aligned} & p ^ { N ; D } _ { i , i + 1 } \\approx e ^ { - N \\big ( I _ i ( r _ { i + 1 } ) - I _ i ( r _ i ) \\big ) ^ + } ; \\ i \\in \\{ 1 , \\cdots , l - 1 \\} ; \\\\ & p ^ { N ; D } _ { i , i - 1 } \\approx e ^ { - N \\big ( I _ i ( r _ i ) - I _ i ( r _ { i + 1 } ) \\big ) ^ + } ; \\ i \\in \\{ 1 , \\cdots , l - 1 \\} ; \\\\ & p ^ { N ; D } _ { 0 , 1 } = p ^ { N ; D } _ { l , l - 1 } = 1 . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "6069.png", "formula": "\\begin{align*} \\ddot r ( t ) + ( p _ 1 \\cot t - p _ 2 \\tan t ) \\dot r ( t ) - \\tfrac { 1 } { 2 } \\left ( \\tfrac { \\lambda _ 1 } { \\sin ^ 2 t } + \\tfrac { \\lambda _ 2 } { \\cos ^ 2 t } \\right ) \\sin 2 r ( t ) = 0 , \\end{align*}"}
-{"id": "1560.png", "formula": "\\begin{align*} \\epsilon ( m ) = \\begin{cases} 0 , & \\mbox { i f } m \\equiv 2 2 \\mod 2 4 ; \\\\ 1 , & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "5041.png", "formula": "\\begin{align*} [ R ( x ) ] ( n ) = r _ n v _ n ^ * ( x ) \\bigl ( n \\in \\N , \\ x \\in X \\bigr ) , \\end{align*}"}
-{"id": "4751.png", "formula": "\\begin{align*} { L } _ 0 = \\alpha \\sqrt { - a _ 0 } , L _ 1 = \\sigma - \\alpha b _ 1 , \\alpha , \\sigma - c o n s t . \\end{align*}"}
-{"id": "6828.png", "formula": "\\begin{align*} S _ { \\rho } ( w _ { \\lambda } + \\phi ) = 0 \\end{align*}"}
-{"id": "6438.png", "formula": "\\begin{align*} p \\left ( x | \\theta \\right ) = { \\displaystyle \\prod \\limits _ { k = 1 } ^ { l } } p \\left ( x _ { k } | \\mu _ { k } \\sigma _ { k } \\right ) p \\left ( x _ { k } | \\mu _ { k } \\sigma _ { k } \\right ) \\overset { } { = } \\frac { 1 } { \\sqrt { 2 \\pi \\sigma _ { k } ^ { 2 } } } \\exp \\left [ - \\frac { \\left ( x _ { k } - \\mu _ { k } \\right ) ^ { 2 } } { 2 \\sigma _ { k } ^ { 2 } } \\right ] \\end{align*}"}
-{"id": "41.png", "formula": "\\begin{align*} \\hat { V } ^ { C } _ { \\sigma ' } ( C _ 1 , C _ 2 ) = \\int \\limits _ { - \\infty } ^ { \\infty } \\int \\limits _ { - \\infty } ^ { \\infty } \\ ! \\hat { f _ { \\sigma } } _ { X Y Z S } ( x , y , z , s ) \\ , \\mathrm { d } u _ { 1 } \\mathrm { d } u _ { 2 } \\Big | _ { x = y = u _ { 1 } , z = s = u _ { 2 } } \\end{align*}"}
-{"id": "8141.png", "formula": "\\begin{align*} h = \\sum _ { k = 1 } ^ K \\sum _ { l = 1 } ^ L \\sum _ { m = 1 } ^ M \\sum _ { i = 1 } ^ n \\sum _ { c \\in C _ { k , l , m } ^ { ( i ) } } h _ { k , l , c , i , i } . \\end{align*}"}
-{"id": "9988.png", "formula": "\\begin{align*} \\partial _ t \\hat u = \\frac 1 2 \\mathrm { d i v } \\big ( \\mathrm { a _ \\beta } \\nabla \\hat u \\big ) \\end{align*}"}
-{"id": "8643.png", "formula": "\\begin{align*} T _ 0 = 0 , \\ \\ T _ i = \\inf \\{ j > T _ { i - 1 } : \\eta _ j = 1 \\} , \\ \\ i \\geq 1 . \\end{align*}"}
-{"id": "6266.png", "formula": "\\begin{align*} | I _ 5 | = & \\left | \\eta \\int _ { \\R ^ 3 } ( ( \\nabla \\times b ^ \\eta ) \\times b ^ \\eta ) \\cdot \\nabla \\times B \\ , d x \\right | \\\\ \\leq & C \\eta \\| \\nabla b ^ \\eta \\| _ \\infty \\| b ^ \\eta \\| _ 2 \\| \\nabla B \\| _ 2 \\\\ \\leq & C \\eta ^ 2 \\mu ^ { - 1 } \\| \\nabla b ^ \\eta \\| _ \\infty ^ 2 \\| b ^ \\eta \\| _ 2 ^ 2 + \\frac 1 { 4 } \\mu \\| \\nabla B \\| _ 2 ^ 2 \\end{align*}"}
-{"id": "4020.png", "formula": "\\begin{align*} A ( t ) = \\int _ D \\omega _ t ( X ^ t , Y ^ t ) \\ d x d y = \\int _ D \\mathrm { d e t } [ X ^ t , Y ^ t , \\eta _ t ] \\ d x d y . \\end{align*}"}
-{"id": "4030.png", "formula": "\\begin{align*} - b ( Y , d \\eta _ p X ) = h ( X , Y ) = - K ( p ) \\cdot h ( d \\eta _ p X , d \\eta _ p Y ) = K ( p ) \\cdot b ( d \\eta _ p Y , d \\eta _ p \\circ d \\eta _ p X ) \\end{align*}"}
-{"id": "8629.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { [ 0 , T ] ^ 2 } R ( s - u , \\omega ( s ) - \\omega ( u ) ) d s d u = \\sum _ { k , m = 0 } ^ { N + 1 } Q _ { k m } , ~ ~ Q _ { k m } = \\int _ { \\tau _ k } ^ { \\tau _ { k + 1 } } \\int _ { \\tau _ m } ^ { \\tau _ { m + 1 } } R ( s - u , \\omega ( s ) - \\omega ( u ) ) d s d u . \\end{aligned} \\end{align*}"}
-{"id": "5801.png", "formula": "\\begin{align*} d \\omega _ { j } : = w ^ { q } _ { j } d \\sigma + d \\mu , j \\in \\N . \\end{align*}"}
-{"id": "801.png", "formula": "\\begin{align*} a _ 2 = 2 \\cdot ( \\rho + 1 ) \\log { 2 x } . \\end{align*}"}
-{"id": "6706.png", "formula": "\\begin{align*} \\mathbb { P } _ { 0 } \\Big ( \\xi ( k + 1 ) = \\eta ^ { i } \\Big | \\xi ( k ) = \\eta \\Big ) = \\frac { 1 } { N } , \\forall k \\geq 0 ; \\eta \\in H _ { N } , \\end{align*}"}
-{"id": "1958.png", "formula": "\\begin{align*} & \\lVert \\partial _ { \\mathbf { r } } ^ { \\mathbf { a } ' } T _ { j , k + 1 } ( r _ { k } , \\dots , r _ { j - 1 } ) ( f ) \\rVert _ { L ^ { 2 } _ { \\alpha ' } } = \\\\ & \\lVert ( i \\pi d _ { k } + P _ { k } ) \\partial _ { \\mathbf { r } } ^ { \\mathbf { a } ' } T _ { k } ( r _ { k } , \\dots , r _ { j - 1 } ) ( f ) \\rVert _ { L ^ { 2 } _ { \\alpha ' } } = \\lVert ( i \\pi d _ { k } + P _ { k } ) F _ { r _ { k } } \\rVert _ { L ^ { 2 } _ { \\alpha ' } } \\le c _ { k } \\lVert f \\rVert ^ { j } _ { W ^ { \\beta , 2 } _ { 4 } } , \\end{align*}"}
-{"id": "7575.png", "formula": "\\begin{align*} u = \\sum _ { j = 0 } ^ \\infty \\ , \\cal F ^ { - 1 } \\Phi _ j \\cal F u \\ , , \\quad u \\in \\cal S ' \\ , . \\end{align*}"}
-{"id": "6555.png", "formula": "\\begin{align*} t ( \\lambda \\xi _ 1 + ( 1 - \\lambda ) z ) = \\mu ( 1 + \\beta _ { \\xi _ 1 } \\delta ' ) ^ { - 1 } \\xi _ 1 + ( 1 - \\mu ) z . \\end{align*}"}
-{"id": "4974.png", "formula": "\\begin{align*} E ( x ) : = e ^ { - ( 1 + \\| x _ { \\sigma } \\| ^ 2 ) ^ { \\frac { 1 } { 2 } } } . \\end{align*}"}
-{"id": "6349.png", "formula": "\\begin{align*} m _ j ^ * = \\sum \\limits _ { i = 0 } ^ { \\lfloor j / 2 \\rfloor } \\binom { j } { 2 i } ( \\beta ^ * ) ^ { i } ( \\alpha ^ * ) ^ { j - 2 i } \\frac { 1 } { i + 1 } \\binom { 2 i } { i } . \\end{align*}"}
-{"id": "5924.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { s \\in A - A _ { } } L ( s ) Q ^ N ( s ) \\le \\\\ & \\Big ( \\sum _ { s \\in A _ { } : r _ s = 2 } L ( s ) Q ^ N ( s ) \\Big ) \\sum _ { r = 2 } ^ \\infty \\big ( ( 1 + 2 ^ { l - 1 } e ^ { - c _ 0 N } ) ^ r - 1 \\big ) e ^ { - c _ 1 ( r - 2 ) N } K _ l ^ { r - 2 } + \\\\ & \\sum _ { r = 2 } ^ \\infty \\big ( ( 1 + 2 ^ { l - 1 } e ^ { - c _ 0 N } ) ^ r - 1 \\big ) e ^ { - c _ 1 ( r - 2 ) N } K _ l ^ { r - 2 } ( 2 ( l - 1 ) ( r - 2 ) ) . \\end{aligned} \\end{align*}"}
-{"id": "1248.png", "formula": "\\begin{align*} z = 0 \\mbox { a n d } \\mbox { d i a m } ( D ) = 1 . \\end{align*}"}
-{"id": "7982.png", "formula": "\\begin{align*} & \\langle Y _ 1 ( t ) , T _ s ( t ) \\rangle = t a _ 3 | T _ s | \\cos \\beta _ s , \\ , \\ , \\\\ & \\langle Y _ 2 ( t ) , T _ s ( t ) \\rangle = - ( 1 - t ) a _ 2 | T _ s | \\cos \\alpha _ s . \\end{align*}"}
-{"id": "262.png", "formula": "\\begin{align*} L = \\begin{cases} ( C k ) ^ k \\sqrt { m _ { - 2 k } } \\ , \\ell & k > 0 \\\\ C ( s ) \\sqrt { m _ { - s } } \\ , \\ell , & k = 0 . \\end{cases} \\end{align*}"}
-{"id": "7312.png", "formula": "\\begin{align*} \\varphi ( \\sigma ) : \\overline { A } \\longrightarrow \\overline { A } , \\ \\ \\varphi ( \\sigma ) ( z ) = \\widehat { \\Gamma } \\sigma ( z ' ) , \\end{align*}"}
-{"id": "5684.png", "formula": "\\begin{align*} \\inf \\{ \\mathfrak { E } _ { W } ( { v } ) \\ ; : \\ ; { v } \\in \\mathcal { S } _ { s y m } ( a ^ - , a ^ + ) \\} = d _ K ( u ^ - , u ^ + ) . \\end{align*}"}
-{"id": "5785.png", "formula": "\\begin{align*} _ { p } ( E ) : = \\inf \\big \\lbrace \\Vert \\nabla u \\Vert ^ { p } _ { L ^ { p } ( \\Omega ) } \\colon \\ , \\ , u \\geq 1 \\ ; \\ ; \\ ; \\ ; E , u \\in C _ { 0 } ^ { \\infty } ( \\Omega ) \\big \\rbrace . \\end{align*}"}
-{"id": "5648.png", "formula": "\\begin{align*} L _ { d _ K } ( \\gamma ) \\geq \\sum _ { i = 0 } ^ { n - 2 } K _ { \\omega ( 1 / n ) } ( \\gamma ( t _ i ^ \\delta ) ) \\ , d ( \\gamma ( t ^ \\delta _ i ) , \\gamma ( t ^ \\delta _ { i + 1 } ) ) , \\end{align*}"}
-{"id": "6094.png", "formula": "\\begin{align*} \\tfrac { \\partial } { \\partial x _ i } ( \\cos ( g ( r ) ) { f _ { \\frak { e } _ k } ( x / r ) _ j } ) = & - \\tfrac { \\sin ( g ( r ) ) } { r ^ 2 } x _ i { f _ { \\frak { e } _ k } ( x / r ) _ j } \\\\ & + \\sum _ { \\ell = 1 } ^ n \\tfrac { \\partial { f _ { \\frak { e } _ k } ( y ) _ j } } { \\partial y _ { \\ell } } ( \\tfrac { 1 } { r } \\delta _ { \\ell , i } - \\tfrac { x _ i x _ { \\ell } } { r ^ 3 } ) \\cos ( g ( r ) ) , \\end{align*}"}
-{"id": "2985.png", "formula": "\\begin{align*} - \\int _ { \\Omega } u \\Delta \\varphi = \\int _ { \\Omega } \\bar { g } ( x , u ) \\varphi + \\int _ \\Omega \\varphi d \\mu , ~ \\forall \\varphi \\in C _ 0 ^ 2 ( \\bar \\Omega ) . \\end{align*}"}
-{"id": "3296.png", "formula": "\\begin{align*} \\Delta \\psi + \\frac { 8 \\mu ^ { 2 } } { ( \\mu _ j ^ { 2 } + | z | ^ { 2 } ) ^ { 2 } } \\psi = 0 , \\mu _ j = \\mu _ j ^ * \\end{align*}"}
-{"id": "492.png", "formula": "\\begin{align*} p _ { 1 , k _ 1 , k _ 2 } ( x , t ) = \\frac { ( - 1 ) ^ { k _ 2 } \\pi ^ { k _ 1 + k _ 2 } } { 2 ^ { n - k _ 1 + 1 + \\frac { m - 1 } { 2 } } } \\abs { t } ^ { n + k _ 1 - \\frac { m + 1 } { 2 } } e ^ { - \\frac { 1 } { 4 } d ( x , t ) ^ 2 } e ^ { - \\kappa \\rho ( \\delta ) } \\tilde I _ { n + k _ 1 - 1 } ( \\kappa \\rho ( \\delta ) ) \\left [ 1 + g ( \\abs { x } , \\abs { t } ) \\right ] , \\end{align*}"}
-{"id": "8161.png", "formula": "\\begin{align*} a _ 1 ^ { q + 1 } a _ 4 ^ { q ^ 2 } + a _ 2 a _ 5 ^ { q + q ^ 2 } = \\delta ^ { q + 1 } . \\end{align*}"}
-{"id": "4600.png", "formula": "\\begin{align*} x \\cdot ( y \\cdot z ) - ( x \\cdot y ) \\cdot z = y \\cdot ( x \\cdot z ) - ( y \\cdot x ) \\cdot z , \\end{align*}"}
-{"id": "8896.png", "formula": "\\begin{align*} \\lim _ k \\widehat \\mu ( - n _ k ) = a . \\end{align*}"}
-{"id": "2205.png", "formula": "\\begin{align*} \\beta ( \\alpha , J ) = \\bigl ( m _ { 1 j _ { 1 } } ( \\alpha _ { j _ { 1 } } + 1 ) - 1 , \\ldots , m _ { n j _ { n } } ( \\alpha _ { j _ { n } } + 1 ) - 1 \\bigr ) , \\end{align*}"}
-{"id": "8307.png", "formula": "\\begin{align*} \\mathrm { F J } ^ { ( a ) } ( \\psi ) = \\sum _ { \\substack { \\alpha \\in \\Gamma _ \\Phi ^ \\vee ( 1 ) \\\\ \\langle \\alpha , \\sigma \\rangle \\ge 0 } } \\mathrm { F J } _ \\alpha ^ { ( a ) } ( \\psi ) \\cdot q _ \\alpha \\in \\C [ [ q _ \\alpha ] ] _ { \\substack { \\alpha \\in \\Gamma _ \\Phi ^ \\vee ( 1 ) \\\\ \\langle \\alpha , \\sigma \\rangle \\ge 0 } } . \\end{align*}"}
-{"id": "5646.png", "formula": "\\begin{align*} \\mathfrak { L } _ { K } ( \\gamma ) = \\int _ I K ( \\gamma ( t ) ) \\ , | \\dot { \\gamma } | ( t ) \\d t \\geq r \\inf _ B K = r K _ r ( x ) . \\end{align*}"}
-{"id": "3406.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } T [ u _ j ] ( z ) = T [ \\delta _ { \\pi / 2 } + \\delta _ { - \\pi / 2 } ] ( z ) = \\frac { i + z } { i - z } + \\frac { - i + z } { - i - z } = 2 \\frac { 1 - | z | ^ 4 - 2 \\imath \\Im ( z ^ 2 ) } { | 1 + z ^ 2 | ^ 2 } . \\end{align*}"}
-{"id": "6070.png", "formula": "\\begin{align*} E ( \\rho ) = c \\int _ 0 ^ { \\pi / 2 } ( r ' ( t ) ^ 2 + \\lambda _ { 1 } \\tfrac { \\sin ^ 2 r } { \\sin ^ 2 t } + \\lambda _ 2 \\tfrac { \\sin ^ 2 r } { \\cos ^ 2 t } ) \\sin ^ { p _ 1 } t \\cos ^ { p _ 2 } t d t , \\end{align*}"}
-{"id": "1097.png", "formula": "\\begin{align*} \\mathcal { C } ^ \\mathrm { \\infty } _ { i | \\alpha _ { i i } } = \\frac { 1 } { 2 } \\mathrm { l o g } \\left ( \\ ! \\frac { 1 } { 2 } \\gamma _ i + 2 \\ ! \\right ) - \\ ! \\frac { \\gamma _ i } { 2 } \\ ! - \\ ! 1 + \\frac { \\sqrt { \\gamma _ i ( \\gamma _ i + 4 ) } } { 2 } - \\sqrt { \\frac { \\pi } { 4 \\gamma _ i } } , \\end{align*}"}
-{"id": "10146.png", "formula": "\\begin{align*} \\boldsymbol { \\omega } _ k ( i ) & = \\sum _ { l \\in \\mathcal { N } _ k } c _ { k l } \\boldsymbol \\Lambda _ l ^ 2 \\boldsymbol { \\omega } _ l ( i - 1 ) \\big ( \\boldsymbol { \\omega } _ l ^ H ( i - 1 ) \\boldsymbol \\Lambda _ l ^ 2 \\boldsymbol { \\omega } _ l ( i - 1 ) \\big ) ^ { - 1 } \\\\ & \\times \\boldsymbol \\omega _ l ^ H ( i - 1 ) \\boldsymbol \\omega _ l ( i - 1 ) . \\end{align*}"}
-{"id": "2055.png", "formula": "\\begin{align*} \\tau ( u ) = & d \\iota ( \\tau ( f ) ) + t r a c e _ { G _ \\theta } ( \\nabla d \\iota ) ( d _ H f , d _ H f ) , \\\\ \\nabla d \\iota = & d P ( \\nabla d \\iota ) + \\nabla d P ( d \\iota , d \\iota ) , \\end{align*}"}
-{"id": "2061.png", "formula": "\\begin{align*} u ^ a ( p , 0 ) = \\phi ^ a ( p ) , \\mbox { f o r a l l } p \\in M . \\end{align*}"}
-{"id": "3461.png", "formula": "\\begin{align*} u _ { \\mathbf { x } } = \\sum _ { i = 1 } ^ { d } \\mathbf { x } _ i \\psi _ i v _ { \\mathbf { y } } = \\sum _ { i = 1 } ^ { d } \\mathbf { y } _ i \\psi _ i \\end{align*}"}
-{"id": "2190.png", "formula": "\\begin{align*} Q _ j ( z ) = \\sum _ { \\| \\alpha \\| > k _ j } a _ \\alpha ^ j z ^ \\alpha , \\end{align*}"}
-{"id": "7781.png", "formula": "\\begin{align*} | e _ { \\sigma } ( \\ell - \\rho ) | & = | D _ { \\sigma } u _ { 0 } ( \\ell - \\rho ) | \\leq | u _ { 0 } ( \\ell - \\rho + \\sigma ) | + | u _ { 0 } ( \\ell - \\rho ) | \\\\ & \\leq C \\b ( \\log ( 2 + | \\ell - \\rho | ) + | \\sigma | \\ , \\b ) \\leq C \\b ( \\log ( 1 + | \\rho | ) + | \\sigma | \\ , \\b ) , \\end{align*}"}
-{"id": "9978.png", "formula": "\\begin{align*} \\bar { \\psi } _ { k i } = \\frac { \\bar { \\upsilon } _ { k i } ^ { 2 } } { \\rho _ { k i i } } \\left ( \\rho _ { k i i } + \\sigma ^ { 2 } + \\sum _ { l \\neq \\{ i , j \\} } ^ { L } \\kappa _ { l } \\mathbb { E } \\left [ \\mathsf { \\Gamma } _ { l i } \\right ] \\right ) . \\end{align*}"}
-{"id": "4277.png", "formula": "\\begin{align*} \\nu \\left ( \\frac { a \\tau + b } { c \\tau + d } \\right ) = ( c \\tau + d ) ^ 2 \\nu ( \\tau ) + \\frac { c ( c \\tau + d ) } { 4 \\pi i } . \\end{align*}"}
-{"id": "3457.png", "formula": "\\begin{align*} ( P _ h v , w _ h ) _ H = ( v , w _ h ) _ H , w _ h \\in V _ h . \\end{align*}"}
-{"id": "687.png", "formula": "\\begin{align*} \\varphi ( \\pi , \\mu _ n ) = ( - 1 ) ^ n , n \\geq 0 . \\end{align*}"}
-{"id": "667.png", "formula": "\\begin{align*} & \\{ \\lambda ' _ i ( x ) \\} _ { i \\in S _ n } = E i g ^ { \\neq 0 } ( \\bar { \\Lambda } ' ( x ) ) = E i g ^ { \\neq 0 } ( V _ 1 ( x ) \\bar { \\Lambda } ' ( x ) V _ 1 ( x ) ^ * ) \\\\ & = E i g ^ { \\neq 0 } ( U _ 1 ( x ) V _ 1 ( x ) \\bar { \\Lambda } ' ( x ) V _ 1 ( x ) ^ * U _ 1 ( x ) ^ * ) , ~ ~ \\forall i \\in S _ n \\end{align*}"}
-{"id": "9723.png", "formula": "\\begin{align*} \\begin{cases} & \\tilde { \\gamma } _ i = \\gamma _ i + O ( 1 ) | \\boldsymbol { \\gamma } ^ { * } | Z _ { a } h + O ( 1 ) | \\gamma _ 4 | h , i = 1 , 2 , 3 , 5 , \\\\ & \\tilde { \\gamma } _ 4 = ( 1 - { \\phi ( T _ b ) h } / { u _ b } ) \\gamma _ 4 + O ( 1 ) | \\boldsymbol { \\gamma } ^ { * } | Z _ { a } h . \\end{cases} \\end{align*}"}
-{"id": "74.png", "formula": "\\begin{align*} b b b \\dots b b b = 1 a a a \\dots a a a { * } b b b \\dots b b b * = 1 a a a \\dots a a a \\end{align*}"}
-{"id": "2073.png", "formula": "\\begin{align*} ( \\partial _ t - \\Delta _ H ) u ( p , t ) = F ( p , t ) , u ( p , 0 ) = \\phi ( p ) \\end{align*}"}
-{"id": "4150.png", "formula": "\\begin{align*} \\lvert I _ 1 \\rvert \\sum _ { \\tau = t + 2 } ^ n Z _ { N , J } ^ { \\tau } + n Z _ { I _ 1 , J } ^ { t + 1 } + n ^ 2 Z _ { i _ 2 , J } ^ { t } \\leq \\lvert I _ 1 \\rvert + n . \\end{align*}"}
-{"id": "10117.png", "formula": "\\begin{align*} \\bar { \\boldsymbol p } _ k ( i ) = \\mathbb { E } [ d _ k ^ * ( i ) \\boldsymbol S _ { D _ k } ^ H ( i ) \\boldsymbol x _ k ( i ) ] = \\mathbb { E } [ d _ k ^ * ( i ) \\bar { \\boldsymbol x } _ k ( i ) ] . \\end{align*}"}
-{"id": "8980.png", "formula": "\\begin{gather*} D ^ { ( n ) } _ q ( u _ 0 , u _ 1 , \\dots , u _ { 2 d ' + 1 } ; t ) \\\\ { } = \\sum _ { \\sigma \\in \\{ \\pm 1 \\} ^ n } \\prod _ { 1 \\le i \\le n } \\frac { \\prod _ { 0 \\le r < 2 d ' + 2 } \\vartheta ( u _ r + \\sigma _ i z _ i ) } { \\vartheta ( 2 \\sigma _ i z _ i ) } \\prod _ { 1 \\le i < j \\le n } \\frac { \\vartheta ( t + \\sigma _ i z _ i + \\sigma _ j z _ j ) } { \\vartheta ( \\sigma _ i z _ i + \\sigma _ j z _ j ) } \\prod _ { 1 \\le i \\le n } T _ i ^ { \\sigma _ i / 2 } , \\end{gather*}"}
-{"id": "9510.png", "formula": "\\begin{align*} a _ { n , k } ^ { ( d ) } = \\sum \\limits _ { t = k - d + 1 } ^ { k + d - 1 } { c _ { n , k , t } ^ { ( d ) } \\cdot a _ { n - 1 , t } ^ { ( d ) } } , n > 0 , 0 \\le k \\le n ( d - 1 ) , \\end{align*}"}
-{"id": "6426.png", "formula": "\\begin{align*} d \\mathcal { V } _ { \\mathcal { M } } \\overset { } { = } \\left \\vert \\frac { \\partial \\varphi } { \\partial \\theta } \\right \\vert d \\theta ^ { 1 } d \\theta ^ { 2 } d \\theta ^ { n } \\end{align*}"}
-{"id": "8369.png", "formula": "\\begin{align*} V _ { \\Z } ^ { [ i ] } = V _ { \\Z } \\oplus \\Lambda ^ { [ i ] } \\qquad \\mbox { a n d } V ^ { [ i ] } = V \\oplus \\Lambda _ \\Q ^ { [ i ] } . \\end{align*}"}
-{"id": "7990.png", "formula": "\\begin{align*} n \\binom { n } { d / 1 0 } p ^ { d / 1 0 } \\leq n \\left ( \\frac { 1 0 e n p } { d } \\right ) ^ { d / 1 0 } \\leq n \\left ( \\frac { 3 0 e } { d n ^ { 1 / 8 } } \\right ) ^ { d / 1 0 } = o ( n ^ { 1 - d / 1 0 0 } ) = o ( n ^ { - 3 } ) . \\end{align*}"}
-{"id": "961.png", "formula": "\\begin{align*} E [ U _ n ( \\vartheta _ n ) ] & = \\sum _ { I , J } K ( I , J _ { - \\vartheta _ n } ) \\rho _ { m ^ * } \\int _ { I \\cap J _ { - \\theta _ { m ^ * } } } \\sigma _ 1 ( t ) \\sigma _ 2 ( t + \\theta _ { m ^ * } ) d t \\end{align*}"}
-{"id": "7450.png", "formula": "\\begin{align*} g : = x ^ 2 + y ^ 3 - t z ^ 2 - y t ^ 3 . \\end{align*}"}
-{"id": "1765.png", "formula": "\\begin{align*} W ( t _ \\lambda f ) & = W ( f _ \\lambda ( f \\circ \\sigma ^ { d ( \\lambda ) } ) ) = f _ \\lambda ( f \\circ \\sigma ^ { d ( \\lambda ) } ) \\sqrt { d \\mu _ \\pi } = ( f \\circ \\sigma ^ { d ( \\lambda ) } ) \\sqrt { | f _ \\lambda | ^ 2 d \\mu _ \\pi } \\\\ & = ( f \\circ \\sigma ^ { d ( \\lambda ) } ) \\sqrt { d [ \\mu _ \\pi \\circ ( \\sigma _ \\lambda ) ^ { - 1 } ] } = S _ \\lambda ^ { u n i v } ( f \\sqrt { d \\mu _ \\pi } ) = S _ \\lambda ^ { u n i v } W ( f ) \\end{align*}"}
-{"id": "6885.png", "formula": "\\begin{align*} - \\Delta u _ 1 & = \\lambda _ 1 u _ 1 \\\\ - \\Delta u _ 2 & = \\lambda _ 2 u _ 2 \\end{align*}"}
-{"id": "1821.png", "formula": "\\begin{align*} \\lim _ { a \\downarrow 0 } E ^ a _ { D , \\rho , 0 } \\left ( e ^ { h m ^ a _ D } \\right ) = E _ { D , \\rho , 0 } \\left ( e ^ { h m _ D } \\right ) , \\end{align*}"}
-{"id": "3776.png", "formula": "\\begin{align*} \\P ^ \\rho & \\bigg ( \\begin{array} { c } \\sigma : [ 0 , L ) \\cap \\Z \\to \\Z \\\\ ( 0 , L ) ( ( - L , 0 ) , L ) \\chi _ \\sigma ^ { g _ { L _ { \\hat { k } } } } < h _ \\infty / 2 \\end{array} \\bigg ) \\\\ & \\le c ^ { - 1 } e ^ { - c ( \\log L ) ^ { 3 / 2 } } , \\end{align*}"}
-{"id": "4372.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\Big \\| A P _ \\alpha C _ t | _ { F _ \\alpha ^ p } - \\sum _ { j = j _ 0 ( t ) } ^ \\infty P _ \\alpha M _ { \\varphi _ { j , t } } | _ { F _ \\alpha ^ p } \\Big \\| = 0 . \\end{align*}"}
-{"id": "9339.png", "formula": "\\begin{align*} \\begin{aligned} p \\cdot ( p \\cdot q ) & = p ^ { 2 } \\cdot q , \\\\ ( p \\cdot q ) \\cdot q & = p \\cdot q ^ { 2 } , \\\\ ( p \\cdot q ) \\cdot p & = p \\cdot ( q \\cdot p ) = p \\cdot q \\cdot p , \\end{aligned} \\end{align*}"}
-{"id": "6774.png", "formula": "\\begin{align*} U _ { \\lambda , p } ( y ) = V _ { \\lambda } ( x ) + 2 \\ln { ( 1 + | x | ^ 2 ) } - \\ln { 4 } . \\end{align*}"}
-{"id": "9736.png", "formula": "\\begin{align*} \\frac { s } { h } > \\max _ { j = 1 , 5 } \\Big ( \\sup _ { U \\in O _ { \\epsilon } ( U _ { 1 } ^ { ( 0 ) } ) \\cup O _ { \\epsilon } ( U _ { 2 } ^ { ( 0 ) } ) } | \\lambda _ j ( U ) | \\Big ) + m . \\end{align*}"}
-{"id": "3814.png", "formula": "\\begin{align*} \\lambda _ L ( x ) : = L ^ { - \\frac { 1 } { 1 6 } } \\sum _ { z \\notin [ x - 2 \\ell _ L , x + 2 \\ell _ L ] } P ( \\exists \\ ; s \\in [ 0 , L ^ \\alpha ] \\colon \\ , S ^ { z , 1 } \\in [ x - \\ell _ L , x + \\ell _ L ] ) , \\end{align*}"}
-{"id": "1139.png", "formula": "\\begin{align*} z \\mapsto f ( z ) = a _ 0 z + a _ 1 z ^ q + \\dotsb + a _ d z ^ { q ^ d } . \\end{align*}"}
-{"id": "6888.png", "formula": "\\begin{align*} \\{ F _ { j k } \\mid F _ { j k } = F _ j \\circ F _ k \\} \\end{align*}"}
-{"id": "762.png", "formula": "\\begin{align*} J ( u _ 1 ^ n , u _ 2 ^ n ) = J ( \\varphi _ 1 ^ n , \\varphi _ 2 ^ n ) & = J ( \\varphi _ 1 , \\varphi _ 2 ) + J ( \\varphi _ 1 ^ n - \\varphi _ 1 , \\varphi _ 2 ^ n - \\varphi _ 2 ) + o _ n ( 1 ) \\\\ & \\geq m ( a _ 1 , a _ 2 ) + \\frac 1 2 \\| ( \\varphi _ 1 ^ n - \\varphi _ 1 , \\varphi _ 2 ^ n - \\varphi _ 2 ) \\| ^ 2 _ { \\dot H \\times \\dot H } + o _ n ( 1 ) . \\end{align*}"}
-{"id": "414.png", "formula": "\\begin{align*} p _ { 1 , k _ 1 , k _ 2 } ( x , t ) = \\frac { 2 } { ( 4 \\pi ) ^ { n + 1 } } h _ { k _ 1 , k _ 2 } \\left ( R , t \\right ) \\end{align*}"}
-{"id": "3541.png", "formula": "\\begin{align*} \\begin{array} [ c ] { r l } & \\displaystyle \\sum _ { i = 1 } ^ { n } \\frac { \\beta ^ { n , i , \\theta } } { n } \\left [ X ^ { n , \\theta , \\rho } ( t _ i ) - X ^ { n , \\theta } ( t _ i ) \\right ] \\\\ = & \\displaystyle \\sum _ { i = 1 } ^ { n } \\int _ { t _ { i - 1 } } ^ { t _ i } \\frac { \\sum _ { j = 1 } ^ { n - i + 1 } \\beta ^ { n , j , \\theta } } { n } \\left [ b ( X ^ { n , \\theta , \\rho } ( t ) , u ^ { n , \\theta , \\rho } ( t ) ) - b ( X ^ { n , \\theta } ( t ) , u ^ { n , \\theta } ( t ) ) \\right ] d t . \\end{array} \\end{align*}"}
-{"id": "2691.png", "formula": "\\begin{align*} \\mathfrak { S } _ i : = W _ i [ [ u ] ] \\end{align*}"}
-{"id": "4924.png", "formula": "\\begin{align*} V = \\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix} \\end{align*}"}
-{"id": "2422.png", "formula": "\\begin{align*} D _ 1 = D _ 1 ' = 0 , \\ D _ 3 < 0 , \\end{align*}"}
-{"id": "2603.png", "formula": "\\begin{align*} \\| f \\| _ V ^ 2 = \\int _ \\R ( V * f ) f = \\int _ \\R \\widehat V \\big | \\widehat f \\big | ^ 2 \\geq 0 \\end{align*}"}
-{"id": "4521.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | \\phi ( \\Delta _ n ) | = \\lim _ { n \\to \\infty } | \\psi ( \\Delta _ n ) | = 1 \\end{align*}"}
-{"id": "7023.png", "formula": "\\begin{align*} z _ n = ( \\sqrt { \\epsilon } g ( \\epsilon ) ) ^ { - 1 } \\frac { y _ n } { | \\bar { y } | } , \\end{align*}"}
-{"id": "9493.png", "formula": "\\begin{align*} S ( z , t ) - 1 = \\dfrac { t } { z } \\left [ S ( z , t ) - 1 - \\sum \\limits _ { i = 1 } ^ { r / 2 - 1 } z ^ i \\ , S _ { 2 i } ( t ) \\right ] + t \\ , z \\ , S ^ 2 ( z , t ) . \\end{align*}"}
-{"id": "22.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty , \\sigma \\rightarrow 0 } N \\sigma \\ , v a r [ \\hat { V } ^ C _ { N , \\sigma } ( C _ 1 , C _ 2 ) ] = p _ { E } ( 0 ) \\int G ^ 2 _ { 1 } ( u ) d u \\end{align*}"}
-{"id": "4176.png", "formula": "\\begin{align*} \\rho _ k ^ { p _ k } w \\left ( C _ { ( \\alpha _ k ) } : \\ | \\alpha _ k | = p _ k \\right ) \\leq \\| C _ 0 \\| \\cos \\frac { \\pi } { \\left [ \\frac { m _ i } { p _ k } \\right ] + 2 } . \\end{align*}"}
-{"id": "9245.png", "formula": "\\begin{align*} \\mathfrak { u } = ( \\mathfrak { g } \\otimes A ) \\oplus ( V \\otimes B ) \\oplus ( V ' \\otimes B ' ) \\oplus \\ldots \\oplus ( \\Lambda ' \\otimes E ' ) \\oplus D \\end{align*}"}
-{"id": "293.png", "formula": "\\begin{align*} L _ t ^ * = S _ { \\lfloor t \\rfloor } \\oplus ( t - \\lfloor t \\rfloor ) \\odot X _ { \\lfloor t \\rfloor + 1 } ^ * t \\geq 0 , \\end{align*}"}
-{"id": "8633.png", "formula": "\\begin{align*} I ( x , y ) = \\lambda ^ 2 \\int _ 0 ^ 1 \\int _ 0 ^ 1 R ( s + 1 - u , y ( s ) + x ( 1 ) - x ( u ) ) d s d u . \\end{align*}"}
-{"id": "1406.png", "formula": "\\begin{align*} H ^ { * } ( B G _ 0 ; \\mathbb { Z } / 2 ) = \\mathbb { Z } / 2 [ x _ 2 , x _ 3 ] , \\end{align*}"}
-{"id": "7490.png", "formula": "\\begin{align*} ( \\partial ^ h \\Psi ) _ { A _ { p + 1 } \\overline { B } _ q } & = \\sum _ { i = 1 } ^ { p + 1 } ( - 1 ) ^ { i - 1 } \\delta _ { \\alpha _ i } ( \\psi _ { \\alpha _ 1 \\dots \\hat { \\alpha } _ i \\dots \\alpha _ { p + 1 } \\overline { B } _ q } ) , \\\\ ( \\bar { \\partial } ^ h \\Psi ) _ { A _ p \\overline { B } _ { q + 1 } } & = ( - 1 ) ^ p \\sum _ { i = 1 } ^ { q + 1 } ( - 1 ) ^ { i - 1 } \\delta _ { \\bar { \\beta } _ i } ( \\psi _ { A _ p \\bar { \\beta } _ 1 \\dots \\hat { \\bar { \\beta } } _ i \\dots \\bar { \\beta } _ { q + 1 } } ) . \\end{align*}"}
-{"id": "278.png", "formula": "\\begin{align*} x ( t ) = \\frac { x _ { 0 } } { t ^ { 1 - \\alpha } } + \\frac { 1 } { \\Gamma \\left ( \\alpha \\right ) } \\int _ { 0 } ^ { t } \\left ( t - \\tau \\right ) ^ { \\alpha - 1 } f \\left ( \\tau , x ( \\tau ) \\right ) d \\tau \\end{align*}"}
-{"id": "2948.png", "formula": "\\begin{align*} \\tilde { f } _ ( T , e , i ) = \\begin{cases} \\frac { 1 } { | e | } & F ( T , e , j ) > G ( T , e , j ) \\ , \\ , \\ , \\forall j \\in e , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "1879.png", "formula": "\\begin{align*} \\phi _ 0 ( 1 _ B ) = \\min \\Big \\{ \\nu _ 1 ( ( - \\infty , B _ 1 ] ) , \\nu _ 2 ( ( - \\infty , B _ 2 ] ) , \\min _ { i \\in I } \\Big \\{ \\bar { \\pi } ^ i + \\nu _ 1 ( ( A ^ i _ 1 , B _ 1 ] ) ) + \\nu _ 2 ( ( A ^ i _ 2 , B _ 2 ] ) \\Big \\} \\Big \\} . \\end{align*}"}
-{"id": "8722.png", "formula": "\\begin{align*} & F _ { \\varepsilon , \\delta } ( t , x , w , p , X ) = \\min \\{ F ( t ' , x ' , w , p , X ) \\mid | t - t ' | \\le M \\delta ^ { 1 / 2 } , x ' \\in \\overline { B ( x , M \\varepsilon ^ { 1 / 2 } ) } \\} , \\\\ & F ^ { \\varepsilon , \\delta } ( t , x , w , p , X ) = \\max \\{ F ( t ' , x ' , w , p , X ) \\mid | t - t ' | \\le M \\delta ^ { 1 / 2 } , x ' \\in \\overline { B ( x , M \\varepsilon ^ { 1 / 2 } ) } \\} , \\end{align*}"}
-{"id": "5321.png", "formula": "\\begin{align*} f ( A ) X \\bar { f } ( B ) & - f ( B ) X \\bar { f } ( A ) \\\\ & = \\int _ { 0 } ^ { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } \\Big [ \\left ( e ^ { i \\alpha } - A \\right ) ^ { - 1 } ( e ^ { i \\alpha } + A ) X ( e ^ { - i \\beta } + B ^ { \\ast } ) \\left ( e ^ { - i \\beta } - B ^ { \\ast } \\right ) ^ { - 1 } \\\\ & \\ , \\ , \\ , \\ , - \\left ( e ^ { i \\beta } - B \\right ) ^ { - 1 } ( e ^ { i \\beta } + B ) X ( e ^ { - i \\alpha } + A ^ { \\ast } ) \\left ( e ^ { - i \\alpha } - A ^ { \\ast } \\right ) ^ { - 1 } \\Big ] d \\mu ( \\alpha ) d \\mu ( \\beta ) . \\end{align*}"}
-{"id": "5246.png", "formula": "\\begin{align*} G ( \\lambda , z ) = ( E ^ u ( \\lambda , z ) , E ^ s ( \\lambda , \\tau ) ) , \\end{align*}"}
-{"id": "4635.png", "formula": "\\begin{align*} \\sum _ { n = 2 ^ { k - 1 } } ^ { 2 ^ k - 1 } 1 _ { [ - B - C , B + C ] } \\left ( t + \\sum _ { i = 0 } ^ { n - 1 } f ( T ^ i x ) \\right ) \\ge \\beta \\sum _ { n = 0 } ^ { 2 ^ k - 1 } 1 _ { [ - B - C , B + C ] } \\left ( t + \\sum _ { i = 0 } ^ { n - 1 } f ( T ^ i x ) \\right ) \\end{align*}"}
-{"id": "4505.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | \\phi ( \\Delta _ n ) | = 1 . \\end{align*}"}
-{"id": "7743.png", "formula": "\\begin{align*} \\mathrm { R e } ( T _ N ( l ) ) = \\frac { - 4 N ( l - 2 ) + 2 l ^ 2 - 4 l + ( - 1 ) ^ l - 1 } { 8 N } . \\end{align*}"}
-{"id": "5583.png", "formula": "\\begin{align*} A _ { n } = \\int _ { 0 } ^ { \\tau } d t _ { 1 } \\int _ { 0 } ^ { t _ { 1 } } d t _ { 2 } \\ldots \\int _ { 0 } ^ { t _ { \\bar { n } - 1 } } \\left ( \\tau - t _ { 1 } + t _ { n } \\right ) \\left ( t _ { 1 } - t _ { 2 } \\right ) \\ldots \\left ( t _ { n - 1 } - t _ { n } \\right ) q \\left ( t _ { 1 } \\right ) q \\left ( t _ { 2 } \\right ) \\ldots q \\left ( t _ { n } \\right ) d t _ { n } \\end{align*}"}
-{"id": "6171.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t f _ 1 = \\frac { 1 } { V } \\Big ( \\frac { f _ 1 '' } { V } - \\frac { f _ 1 ' V ' } { V ^ 2 } \\Big ) - \\frac { 1 } { 3 } f _ 1 \\mathcal { T } \\\\ \\\\ \\partial _ t f _ 2 = \\frac { 1 } { V } \\Big ( \\frac { f _ 2 '' } { V } - \\frac { f _ 2 ' V ' } { V ^ 2 } \\Big ) - \\frac { 1 } { 3 } f _ 2 \\mathcal { T } \\\\ \\\\ \\partial _ t V = \\frac { 1 } { 3 } \\mathcal { T } V \\end{array} \\right . \\end{align*}"}
-{"id": "6314.png", "formula": "\\begin{align*} \\Tilde { A } _ m ( z _ m ) = z _ m ^ { d _ m } + \\Tilde { a } _ { m , 1 } z _ m ^ { d _ m - 1 } + \\Tilde { a } _ { m , 2 } z _ m ^ { d _ m - 2 } + \\ldots + \\Tilde { a } _ { m , m } \\end{align*}"}
-{"id": "2952.png", "formula": "\\begin{align*} \\nabla \\left ( \\Theta ^ \\Delta _ \\epsilon \\partial \\Theta ^ \\Delta _ \\epsilon + \\epsilon \\Theta ^ \\Delta _ \\epsilon \\log \\Theta ^ \\Delta _ \\epsilon - \\Theta ^ \\Delta _ { \\epsilon ' } \\partial \\Theta ^ \\Delta _ { \\epsilon } \\right ) = \\epsilon \\nabla ( \\Theta ^ \\Delta _ { \\epsilon ' } \\log \\Theta ^ \\Delta _ \\epsilon ) . \\end{align*}"}
-{"id": "8406.png", "formula": "\\begin{align*} \\frac { \\partial \\widetilde { \\varphi } } { \\partial \\tau } ( x , \\tau ) = - \\frac { \\tilde K ^ { \\beta } ( x , t ) } { \\omega _ n ^ { - 1 } \\int _ { \\mathbb { S } ^ n } \\tilde K ^ { \\beta - 1 } d \\theta } \\nu ( x , \\tau ) + \\widetilde { \\varphi } ( x , \\tau ) . \\end{align*}"}
-{"id": "9437.png", "formula": "\\begin{align*} c n \\mathcal { M } ( n ) \\leq \\sum _ { j = 1 } ^ n \\frac { 1 } { | j | _ { \\mathcal { D } _ 1 } | j | _ { \\mathcal { D } _ 2 } \\cdots | j | _ { \\mathcal { D } _ m } } \\leq n \\mathcal { M } ( n ) . \\end{align*}"}
-{"id": "1779.png", "formula": "\\begin{align*} ( u , w ) _ { H ^ 1 ( \\mathbb { R } ; \\mathbb { C } ) } : = \\int _ { \\mathbb { R } } u ( x ) \\overline { w } ( x ) d x + \\int _ { \\mathbb { R } } u ' ( x ) \\cdot \\overline { w ' ( x ) } d x \\end{align*}"}
-{"id": "1541.png", "formula": "\\begin{gather*} C ^ * _ { A ^ * - B ^ * } \\ : = \\ \\bigcup _ { n \\in ( A ^ * - B ^ * ) \\cap \\mathbb { Z } } \\overline { B } _ { \\frac { 1 } { 2 } } ( n ) . \\end{gather*}"}
-{"id": "4478.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\psi ( \\Delta _ n ) = 1 \\end{align*}"}
-{"id": "2371.png", "formula": "\\begin{align*} \\mu _ \\pm = \\frac { a + d + c n \\pm \\sqrt { ( a + d + c n ) ^ 2 - 4 } } { 2 } . \\end{align*}"}
-{"id": "2547.png", "formula": "\\begin{align*} h ( \\Omega ) : = \\inf _ { E \\subset \\Omega } \\frac { P ( E ) } { | E | } \\ , , \\end{align*}"}
-{"id": "5984.png", "formula": "\\begin{align*} \\vert s ( x ) - s _ M ^ { \\gamma } ( x ) \\vert & \\leq \\underbrace { \\vphantom { \\sum _ { n = 1 } ^ { \\infty } } c \\mathcal { A } N P _ 1 P _ 2 } _ { C } \\underbrace { \\sum _ { n = M } ^ { \\infty } \\frac { ( \\sqrt { 2 } \\varepsilon ^ 2 L ) ^ { n } } { \\gamma ^ n \\sqrt { n ! } } } _ { T _ { M } } \\end{align*}"}
-{"id": "9861.png", "formula": "\\begin{align*} \\tilde g _ i ( d ; x ) \\ , \\geq \\ , g _ i ( x + d ) , \\ ; \\ ; \\forall i = 1 , \\ldots , m , \\ ; \\ ; \\forall d \\ , \\in \\ , K - x . \\end{align*}"}
-{"id": "7184.png", "formula": "\\begin{align*} \\| z _ \\infty \\| _ { H ^ 1 ( B _ 1 , \\mu _ a ) } = 1 . \\end{align*}"}
-{"id": "98.png", "formula": "\\begin{align*} p _ { \\mathbf { W } , \\mathbf { S } } ( W , S ) = \\left \\{ \\begin{array} { l l } \\frac { 1 } { ( K - M ) \\binom { K } { M } } , & W \\not \\in S , | S | = M , \\\\ 0 , & \\textrm { o t h e r w i s e } . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "5396.png", "formula": "\\begin{align*} x y & = ( a b ) ^ { c a c a c } \\in T , \\\\ x z & = c ^ { a c b } \\in T , \\\\ ( y z ) ^ 2 & = ( b \\cdot b ^ { c a c a c } ) ^ 3 \\in T . \\end{align*}"}
-{"id": "353.png", "formula": "\\begin{align*} - R i c ( \\nabla _ { \\nu } e _ 2 , e _ 2 ) + R i c ( \\nabla _ { \\nu } e _ 1 , e _ 1 ) = h _ { 2 2 } R _ { 2 2 } - h _ { 1 1 } R _ { 1 1 } - \\frac { 4 L _ { 2 2 } } { | \\nabla f | ^ { 3 } } = H S | \\nabla f | - \\frac { 4 L _ { 2 2 } } { | \\nabla f | ^ { 3 } } . \\end{align*}"}
-{"id": "9671.png", "formula": "\\begin{align*} W ( U ) _ x + H ( U ) _ y = G ( U ) , \\end{align*}"}
-{"id": "7544.png", "formula": "\\begin{align*} ( 1 + \\omega ^ 2 ) ^ 2 \\psi _ { \\omega \\omega } + ( \\varphi ^ 2 + 4 ) \\omega ( 1 + \\omega ^ 2 ) \\psi _ \\omega - A \\psi = 0 . \\end{align*}"}
-{"id": "1339.png", "formula": "\\begin{align*} \\vartheta _ c ( Y , \\mu , \\lambda ) : = \\min _ { x \\in \\mathbb { B } _ { \\varepsilon } ( \\overline { x } ) } L _ c ( x , Y , \\mu , \\lambda ) , ( Y , \\mu , \\lambda ) \\in { \\cal Z } \\times \\Re ^ m \\times { \\cal Y } \\ , . \\end{align*}"}
-{"id": "7930.png", "formula": "\\begin{align*} d _ { \\mathcal M } ( \\mu _ 1 , \\mu _ 2 ) = & \\sup \\left \\{ \\left | \\int _ { Y } \\phi \\ , d \\mu _ 1 - \\int _ { Y } \\phi \\ , d \\mu _ 2 \\right | : \\phi \\in \\mathrm { L i p } _ 1 ( Y ) , \\ , \\sup _ { x \\in Y } | \\phi ( x ) | \\le 1 \\right \\} , \\\\ d _ { \\mathcal M } ^ * ( \\mu _ 1 , \\mu _ 2 ) = & \\sup \\left \\{ \\left | \\int _ { Y } \\phi \\ , d \\mu _ 1 - \\int _ { Y } \\phi \\ , d \\mu _ 2 \\right | : \\phi \\in \\mathrm { L i p } _ 1 ( Y ) \\right \\} , \\\\ \\end{align*}"}
-{"id": "5670.png", "formula": "\\begin{align*} \\min \\left \\{ \\mathfrak { E } _ { W } ( f \\sigma , I ) \\ ; : \\ ; f \\in H ^ 1 ( I ) , \\ , f \\geq E , \\ , f ( s ^ - ) = E ( s ^ - ) , f ( s ^ + ) = E ( s ^ + ) s ^ + < + \\infty \\right \\} . \\end{align*}"}
-{"id": "1722.png", "formula": "\\begin{align*} f _ \\lambda = S _ \\lambda 1 = W ^ * t _ \\lambda \\xi . \\end{align*}"}
-{"id": "5107.png", "formula": "\\begin{align*} \\int _ { \\mathcal { M } _ { 2 i - 1 } ^ \\perp \\setminus \\mathcal { M } _ { 2 i } ^ \\perp } \\ ! g _ { \\Gamma _ n } ( \\tau ) d \\tau = 2 ^ { - i } \\end{align*}"}
-{"id": "2002.png", "formula": "\\begin{align*} G ( z ) G ( t z ) = 0 . \\end{align*}"}
-{"id": "1538.png", "formula": "\\begin{gather*} A + B \\ = \\ [ a _ 1 + b _ 1 , a _ 2 + b _ 2 ] _ \\mathbb { Z } , \\\\ A ^ * + B ^ * \\ = \\ \\left [ a _ 1 + b _ 1 - \\frac { 1 } { 2 } , a _ 2 + b _ 2 + \\frac { 1 } { 2 } \\right ] , \\\\ A - B \\ = \\ [ a _ 1 - b _ 2 , a _ 2 - b _ 1 ] _ \\mathbb { Z } , \\end{gather*}"}
-{"id": "7436.png", "formula": "\\begin{align*} ( x ' ) ^ 2 + 2 t _ a ( y + z - t _ d + t _ c + t _ b + t _ a ^ 2 ) x ' = ( y + z - t _ d + t _ c + t _ b + t _ a ^ 2 ) y z - 4 t _ a ^ 2 t _ b t _ c + t _ b y ^ 2 + t _ c z ^ 2 \\end{align*}"}
-{"id": "9422.png", "formula": "\\begin{align*} \\mathfrak { d } ^ { \\rm ( i n ) } _ { n , d } = \\mathfrak { d } ^ { \\rm ( i n ) } _ { n - 2 , d + 2 } + \\mathfrak { d } ^ { \\rm ( o u t ) } _ { n - 1 , d - 1 } + 2 \\cdot \\mathfrak { d } ^ { \\rm ( o u t ) } _ { n - 2 , d } + 2 \\sum \\limits _ { i = 0 } ^ { n - 2 } \\sum \\limits _ { j = 0 } ^ { d - 2 } \\tilde { s } _ { i , j } \\cdot \\mathfrak { d } ^ { \\rm ( i n ) } _ { n - 2 - i , d - 2 - j } . \\end{align*}"}
-{"id": "6524.png", "formula": "\\begin{align*} K ^ \\circ = \\{ y \\in \\mathbb { R } ^ n : \\langle x , y \\rangle \\leq 1 , x \\in K \\} . \\end{align*}"}
-{"id": "8305.png", "formula": "\\begin{align*} \\mathrm { w t } _ { - 3 } H = 0 , \\mathrm { w t } _ { - 2 } H = I ^ 2 H , \\mathrm { w t } _ { - 1 } H = I H , \\mathrm { w t } _ 0 H = H , \\end{align*}"}
-{"id": "9667.png", "formula": "\\begin{align*} U ( 0 , y ) = U _ 0 ( y ) = \\begin{cases} U _ 2 ( y ) , { y } ^ { ( 0 ) } < y < 0 , \\\\ U _ 1 ( y ) , y < { y } ^ { ( 0 ) } , \\end{cases} \\end{align*}"}
-{"id": "9785.png", "formula": "\\begin{align*} \\nabla ' ( v \\cdot g \\omega _ 2 ) = \\nabla ' ( v ) \\cdot g \\cdot \\omega _ 2 + v d ( g \\cdot \\omega _ 2 ) = v d ( g \\cdot \\omega _ 2 ) . \\end{align*}"}
-{"id": "6667.png", "formula": "\\begin{align*} \\left \\langle \\left ( 1 - \\frac { \\alpha } { \\langle x , u \\rangle } \\right ) \\xi - x , u \\right \\rangle = \\left ( 1 - \\frac { \\alpha } { \\langle x , u \\rangle } \\right ) \\langle \\xi - x , u \\rangle - \\alpha \\leq - \\alpha \\quad . \\end{align*}"}
-{"id": "7865.png", "formula": "\\begin{align*} d ( z , z ' ) = d ( z , x ) + d ( x , z ' ) \\le d ( x , y ) . \\end{align*}"}
-{"id": "1048.png", "formula": "\\begin{align*} \\beta ( q ) : = \\max \\{ \\log H - \\frac { q } { m } , w + q \\} , q > 0 . \\end{align*}"}
-{"id": "202.png", "formula": "\\begin{align*} f \\circ g ( \\sigma \\ast v ) = \\sigma \\ast ( f \\circ g ( v ) ) = \\sigma \\ast f ( \\mu ) = \\sigma \\ast ( \\mu \\ast v ) = \\sigma \\ast v . \\end{align*}"}
-{"id": "3417.png", "formula": "\\begin{align*} 1 \\le | f | ^ 2 = | f _ 1 | ^ 2 + | f _ 2 | ^ 2 < 2 \\ \\ \\ \\ \\overline S , \\end{align*}"}
-{"id": "9818.png", "formula": "\\begin{align*} p _ k ( t ' ) = \\sum _ { n = k } ^ { \\infty } \\frac { e ^ { - \\l p } ( \\l p ) ^ n } { n ! } \\frac { e ^ { - \\mu ( t ' + T _ 1 ) } ( \\mu ( t ' + T _ 1 ) ^ { n - k } ) } { ( n - k ! ) } , k \\geq 1 . \\end{align*}"}
-{"id": "1057.png", "formula": "\\begin{align*} H ( P _ { k - 1 } ) < H ( P _ { k } ) = H ( V _ { k , 0 } ) = H ( V _ { k , 1 } ) = \\cdots = H ( V _ { k , n - 2 } ) , k \\geq 1 . \\end{align*}"}
-{"id": "939.png", "formula": "\\begin{align*} E \\left [ \\exp \\left \\{ \\frac { c _ q } { 2 } \\left ( \\frac { | \\Delta _ { i , j } | } { \\sigma _ { i , j } } \\right ) ^ { 2 / q } \\right \\} - 1 \\right ] & = \\int _ 0 ^ \\infty P \\left ( \\frac { | \\Delta _ { i , j } | } { \\sigma _ { i , j } } > \\{ \\log ( ( 1 + u ) ^ { 2 / c _ q } ) \\} ^ { q / 2 } \\right ) d u \\\\ & \\leq \\int _ 0 ^ \\infty ( 1 + u ) ^ { - 2 } d u = 1 . \\end{align*}"}
-{"id": "6013.png", "formula": "\\begin{align*} & C _ { 1 + s } \\left ( X ; Y \\right ) : = \\min _ { P _ { X Y W } : \\ , P _ { X Y } = \\pi _ { X Y } , \\ , X - W - Y } \\\\ & \\qquad \\sum _ { w } P _ { W } \\left ( w \\right ) D _ { 1 + s } \\left ( P _ { X | W } ( \\cdot | w ) P _ { Y | W } ( \\cdot | w ) \\| P _ { X Y } \\right ) \\end{align*}"}
-{"id": "5437.png", "formula": "\\begin{align*} f _ i ( S _ k ( d ) ) = \\binom { d + 2 } { i + 1 } - \\binom { k + 1 } { i - d + k } - \\binom { d - k + 1 } { i - k } \\textrm { f o r } - 1 \\le i \\le d - 1 . \\end{align*}"}
-{"id": "4483.png", "formula": "\\begin{align*} \\psi ( s ) = 1 > | \\phi ( s ) | \\end{align*}"}
-{"id": "8034.png", "formula": "\\begin{align*} \\big \\Vert f _ L \\big \\Vert _ { B _ p ^ { s , q } } \\lesssim \\Big ( \\sum _ { n = M } ^ { L } { b _ n ^ { \\widetilde { p } } } \\Big ) ^ { { 1 } / { \\widetilde { p } } } \\end{align*}"}
-{"id": "9164.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n } x ^ { i } f _ { n + 1 - i } \\left ( x ^ { n + 1 - i } \\right ) = 0 \\left ( x \\in K \\right ) \\end{align*}"}
-{"id": "7065.png", "formula": "\\begin{align*} L _ { \\lambda A } u = \\sum \\limits _ { k = 1 } ^ { \\infty } \\sum \\limits _ { j = 1 } ^ m \\frac { 1 + \\lambda \\alpha _ j } { 1 + \\beta _ k } \\pi _ j ( u _ k ) \\cdot f _ j . \\end{align*}"}
-{"id": "866.png", "formula": "\\begin{align*} U _ 0 f ( x ) = \\int _ 0 ^ 1 \\frac { 1 } { 2 t } E \\left [ f ( \\sqrt { t } x + \\sqrt { 1 - t } Z ^ * ) - f ( Z ^ * ) \\right ] d t \\qquad ( x \\in \\mathbb { R } ^ d ) . \\end{align*}"}
-{"id": "3259.png", "formula": "\\begin{align*} ( \\alpha _ 0 ( 1 ) , \\omega ) _ X = ( \\sum _ { l \\geq 1 } ( u ^ l \\alpha _ l ) , \\omega ) _ X , \\end{align*}"}
-{"id": "2820.png", "formula": "\\begin{align*} X _ w ^ { ( n ) } = \\bigcup _ { e \\in E ( v _ 0 , w ) } [ e ] , \\ \\ w \\in V _ n . \\end{align*}"}
-{"id": "9525.png", "formula": "\\begin{align*} u _ { \\beta , \\lambda } ( 0 ) = \\lambda . \\end{align*}"}
-{"id": "5429.png", "formula": "\\begin{align*} \\alpha = \\left \\langle x , e \\right \\rangle + t t \\in \\mathbb { R } , t \\neq 0 . \\end{align*}"}
-{"id": "2799.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow 0 } \\mathcal { B } ( t ) & = \\mathcal { B } ( 0 ) = K , \\delta \\leq r , \\\\ \\lim _ { s \\rightarrow 0 } \\mathcal { B } ( t ) & = \\mathcal { B } ( 0 ) = ( \\frac { r } { \\delta } ) K , \\delta > r . \\end{align*}"}
-{"id": "6744.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\mathbb { P } \\Big ( \\{ \\eta , \\eta ^ { i _ { 1 } } , \\dots , \\eta ^ { i _ { 1 } , \\dots , i _ { N ^ { \\gamma } t } } \\} \\cap \\{ \\eta ^ { i ^ { * } _ { 1 } } , \\dots , \\eta ^ { i ^ { * } _ { 1 } , \\dots , i ^ { * } _ { N ^ { \\gamma } t } } \\} \\neq \\emptyset \\Big ) = 0 . \\end{align*}"}
-{"id": "847.png", "formula": "\\begin{align*} d Y _ s = f ( D _ s , Y _ s , \\bar u _ s ) d s + \\nu d B _ s , \\end{align*}"}
-{"id": "3692.png", "formula": "\\begin{align*} d _ { I , j } = \\min ( \\{ \\langle { \\ , } ^ t L [ n ] v _ { I , j } , \\tilde u \\rangle \\ ; | \\ ; \\tilde u \\in \\square _ { n ^ 2 } \\} \\cup \\{ 0 \\} ) . \\end{align*}"}
-{"id": "8428.png", "formula": "\\begin{align*} \\frac { L ( s , \\abs { \\cdot } ^ c ) } { L ( 1 - s , \\abs { \\cdot } ^ { - c } ) } = - q ^ { - 1 } \\chi _ 1 ( \\varpi ) ^ { - 1 } q ^ s + \\zeta _ F ( 1 ) ^ { - 1 } \\sum _ { a = 0 } ^ { \\infty } \\chi _ 1 ( \\varpi ^ a ) q ^ { - a s } . \\end{align*}"}
-{"id": "3396.png", "formula": "\\begin{align*} \\psi _ s ( Y ) = \\beta _ x ( \\phi _ { b ^ * } ( \\beta _ y ^ { - 1 } ( Y ) , \\alpha _ y ^ { - 1 } ( \\alpha _ x \\beta _ y ^ { - 1 } ( Y ) , s ) ) , \\beta _ y ^ { - 1 } ( Y ) ) \\end{align*}"}
-{"id": "9374.png", "formula": "\\begin{align*} \\kappa _ { \\alpha , \\mu } : = \\frac { \\alpha - \\mu } { \\alpha + \\mu + d - 2 } \\ , . \\end{align*}"}
-{"id": "8493.png", "formula": "\\begin{align*} \\abs { W _ { \\pi } ( g _ { t , a _ 1 , v } ) } = q ^ { - \\frac { t + 2 l } { 2 } } \\leq 1 . \\end{align*}"}
-{"id": "8863.png", "formula": "\\begin{align*} W = B ^ { m _ 1 } A ^ { n _ 1 } B ^ { m _ 2 } A ^ { n _ 2 } \\cdots B ^ { m _ k } A ^ { n _ k } , \\end{align*}"}
-{"id": "6538.png", "formula": "\\begin{align*} \\frac { 1 } { n } | B | _ { n - 1 } \\Delta ^ { n } = \\delta | P | _ n , \\end{align*}"}
-{"id": "3768.png", "formula": "\\begin{align*} M ' _ k : = \\lbrace m \\in M _ k : B _ m \\subset B _ { ( k + 2 , 0 ) } \\rbrace , \\end{align*}"}
-{"id": "7034.png", "formula": "\\begin{align*} C _ 5 = ( 1 + C ^ 2 ) ^ { - ( n + 2 s ) / 2 } \\end{align*}"}
-{"id": "4968.png", "formula": "\\begin{align*} \\forall k \\in \\Z , \\forall f \\in \\mathcal { Z } ^ { \\prime } , \\Delta _ k f = \\sum _ { j \\in \\Z } \\Delta _ k \\Delta _ j f = \\sum _ { | j - k | \\leq 1 } \\Delta _ k \\Delta _ j f . \\end{align*}"}
-{"id": "834.png", "formula": "\\begin{align*} \\sum _ { a \\leq j < b } v _ j = \\sum _ { a ' \\leq j < b ' } v _ j , \\end{align*}"}
-{"id": "5752.png", "formula": "\\begin{align*} \\int _ M K \\ , d V _ M = \\chi ( M ) . \\end{align*}"}
-{"id": "1950.png", "formula": "\\begin{align*} \\beta = \\alpha - 1 + \\lambda , \\end{align*}"}
-{"id": "2080.png", "formula": "\\begin{align*} ( \\Delta _ H - \\partial _ t ) u ^ a _ 0 = 0 , \\mbox { a n d } u ^ a _ 0 ( p , 0 ) = \\phi ^ a ( p ) \\end{align*}"}
-{"id": "8536.png", "formula": "\\begin{align*} a ( x , \\xi ) = \\int _ { S ^ 2 } e ^ { - i ( x \\cdot \\omega ) ( \\xi \\cdot \\omega ) } b _ \\delta ( \\cos \\theta ) d \\Omega ( \\omega ) . \\end{align*}"}
-{"id": "9733.png", "formula": "\\begin{align*} & \\tilde { \\Phi } _ 5 ( \\tilde { \\gamma } _ 5 , \\mathcal { F } ( \\tilde { \\sigma } _ 3 , \\tilde { \\sigma } _ 2 ; \\tilde { \\Phi } _ 1 ( \\tilde { \\gamma } _ 1 ; V _ a + \\widetilde { \\mathcal { V } } ( V _ a , Z _ { a } h ) Z _ { a } h ) ) ) \\\\ & = \\tilde { \\Phi } _ 5 ( \\gamma _ 5 , \\mathcal { F } ( \\sigma _ 3 , \\sigma _ 2 ; \\tilde { \\Phi } _ 1 ( \\gamma _ 1 ; V _ a ) ) ) + \\widetilde { \\mathcal { V } } ( V _ b , Z _ { b } h ) Z _ { b } h . \\end{align*}"}
-{"id": "8979.png", "formula": "\\begin{gather*} P _ d ( \\eta ' ; q , t ) : = - ( ( n - 1 ) t - ( d - 1 ) q + \\eta ' ) \\sum _ i z _ i ^ 2 / q . \\end{gather*}"}
-{"id": "5288.png", "formula": "\\begin{align*} & C _ 0 : = A ' _ 2 - 2 A ' _ 1 - \\bigg ( \\frac { \\varepsilon _ 2 } { 2 } + M _ 1 + \\frac { \\varepsilon _ 3 M _ 2 } { 2 } \\bigg ) > 0 , \\\\ & \\bigg ( \\frac { b _ 1 } { b _ 2 } + \\frac { \\varepsilon _ 1 } { 2 } \\bigg ) \\mu \\leq A ' _ 1 , \\ \\frac { M _ 2 } { 2 \\varepsilon _ 3 } \\leq A ' _ 3 . \\end{align*}"}
-{"id": "9497.png", "formula": "\\begin{align*} t = \\dfrac { \\sqrt { a - 1 } } { 3 a - 2 } , S _ 2 ( t ) = \\sqrt { a - 1 } ( 2 - a ) , z _ a ( t ) = \\dfrac { \\sqrt { a - 1 } } { a } . \\end{align*}"}
-{"id": "3951.png", "formula": "\\begin{align*} D ^ { s } u ( x ) = c ( n , s ) \\cdot \\lim _ { \\epsilon \\downarrow 0 } \\int _ { | x | > \\epsilon } \\frac { u ( x + y ) - u ( x ) } { | y | ^ { n + s } } d y . \\end{align*}"}
-{"id": "7924.png", "formula": "\\begin{align*} V _ I : = f _ { i _ n } \\circ \\cdots \\circ f _ { i _ 1 } ( V ) , \\ , \\ , \\ , f _ I : = f _ { i _ n } : \\bar V _ { I ' } \\to \\bar V _ { I } . \\end{align*}"}
-{"id": "5299.png", "formula": "\\begin{align*} \\Psi ( s ) : = B ( s ) s - \\int _ 0 ^ s B ( r ) \\mathrm { d r } = \\int _ 0 ^ s ( B ( s ) - B ( r ) ) \\mathrm { d r } . \\end{align*}"}
-{"id": "1637.png", "formula": "\\begin{align*} S _ \\lambda ( \\delta _ \\eta ) ( \\nu ) & = \\begin{cases} 0 , & \\nu \\not \\in R _ \\lambda \\tau ^ { d ( \\lambda ) } ( \\nu ) \\not \\in G _ i \\\\ \\delta _ \\eta ( \\tau ^ { d ( \\lambda ) } ( \\nu ) ) ) , & \\end{cases} \\\\ & = \\begin{cases} 1 , & \\lambda \\eta = \\nu x _ j \\cdots x _ { i - 1 } \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "7862.png", "formula": "\\begin{align*} \\varphi ( x , y , z ) = d ( z , m ( x , y ) ) ^ 2 \\end{align*}"}
-{"id": "10023.png", "formula": "\\begin{align*} g _ Q ( \\tau ) & = \\chi ^ n _ Q ( \\beta ) \\chi _ { D / Q } ^ n ( \\alpha ) \\cdot g | _ n W _ Q \\\\ & = \\sum _ { m > 0 } c _ Q ( m ) q ^ m , \\end{align*}"}
-{"id": "9987.png", "formula": "\\begin{align*} \\d u _ t = \\frac 1 2 \\Delta u _ t \\d t + \\beta \\ , u _ t \\ , \\d B _ t , \\end{align*}"}
-{"id": "8182.png", "formula": "\\begin{align*} [ \\sigma ^ { \\mathfrak { L } } ( v ) ] ( g ) = \\omega ^ { - 1 } _ { \\pi } \\omega _ { \\pi } ^ { \\mathfrak { L } } ( \\det ( g ) ) [ \\sigma ( v ) ] ( g ) . \\end{align*}"}
-{"id": "5710.png", "formula": "\\begin{align*} \\exists c _ 0 > 0 , \\ , \\forall v \\in L ^ 2 ( \\R , \\R ^ n ) , \\ , \\forall z \\in \\{ z ^ - , z ^ + \\} , ( v \\ , ; \\ , z ' ) _ { L ^ 2 } = 0 \\Longrightarrow ( A ( z ) v \\ , ; \\ , v ) _ { L ^ 2 } \\ge c _ 0 \\| v \\| _ { L ^ 2 } . \\end{align*}"}
-{"id": "1311.png", "formula": "\\begin{align*} \\rho ^ 1 ( a + \\lambda \\cdot 1 ) = \\rho ( a ) + \\lambda \\cdot I _ H \\quad ~ a \\in A , ~ \\lambda \\in \\mathbb C ; \\end{align*}"}
-{"id": "170.png", "formula": "\\begin{align*} ( m _ { 1 } \\cdot m _ { 2 } ) \\star a = m _ { 1 } \\star ( m _ { 2 } \\star a ) \\quad 1 \\star a = a . \\end{align*}"}
-{"id": "9999.png", "formula": "\\begin{align*} [ \\widehat { \\theta } ( g ) : \\mathcal { Y } _ \\mathrm { s m } ] = - \\deg _ \\C ( \\mathcal { Y } _ \\mathrm { s m } ) \\cdot \\frac { d } { d s } L ( \\tilde { g } , \\theta _ \\Lambda , s ) \\big | _ { s = 0 } . \\end{align*}"}
-{"id": "118.png", "formula": "\\begin{align*} \\bigcup _ { i = 1 } ^ k \\Delta _ i \\cup ( - \\Delta _ i ) , ( - \\Delta _ i = \\{ - \\xi | \\ , \\xi \\in \\Delta _ i \\} ) . \\end{align*}"}
-{"id": "3753.png", "formula": "\\begin{align*} \\iota _ { \\hat { k } } : = \\exp \\bigg \\{ 2 \\sum _ { k \\ge \\hat { k } } L _ k ^ { - 1 / 1 6 } \\bigg \\} \\in ( 0 , \\infty ) . \\end{align*}"}
-{"id": "4366.png", "formula": "\\begin{align*} B _ t : = \\sum _ { j = j _ 0 ( t ) } ^ \\infty M _ { \\psi _ { j , t } } B _ { j , t } M _ { \\varphi _ { j , t } } , \\end{align*}"}
-{"id": "4700.png", "formula": "\\begin{align*} \\int _ { \\partial B _ 1 } | u _ t - u _ \\varrho | \\ , d \\mathcal { H } ^ { n - 1 } & \\leq C \\left ( \\log \\frac { t } { \\varrho } \\right ) ^ \\frac { 1 } { 2 } \\ , | \\log t | ^ { - \\frac { a } { 2 } } \\left ( 1 + \\int _ 0 ^ { r _ 0 } \\frac { \\omega ( r ) \\ , | \\log r | ^ a } { r } \\ , d r \\right ) ^ \\frac { 1 } { 2 } \\leq C \\left ( \\log \\frac { t } { \\varrho } \\right ) ^ \\frac { 1 } { 2 } \\ , | \\log t | ^ { - \\frac { a } { 2 } } . \\end{align*}"}
-{"id": "9873.png", "formula": "\\begin{align*} X _ G = \\sum _ { \\kappa } x _ { \\kappa ( v _ 1 ) } x _ { \\kappa ( v _ 2 ) } \\cdots x _ { \\kappa ( v _ { N } ) } \\end{align*}"}
-{"id": "5975.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\frac { \\varepsilon ^ { 2 n } } { \\gamma ^ n n ! } y ^ n h _ n ( \\gamma x ) = \\exp \\left ( 2 \\varepsilon ^ 2 y x - \\frac { \\varepsilon ^ 4 y ^ 2 } { \\gamma ^ 2 } \\right ) . \\end{align*}"}
-{"id": "872.png", "formula": "\\begin{align*} F _ { n , k } = \\sum _ { i = 1 } ^ { N _ n } \\lambda _ { n , k } ( i ) ( \\eta _ i ^ 2 - 1 ) , \\end{align*}"}
-{"id": "9450.png", "formula": "\\begin{align*} \\psi ( n _ k ) - \\psi ( n _ k + 1 ) = \\frac { O ( 1 ) } { n _ k \\mathfrak { M } ^ 2 ( n _ k ) \\log n _ k } . \\end{align*}"}
-{"id": "5589.png", "formula": "\\begin{align*} M _ { 4 } \\left ( t \\right ) = \\left [ \\bar { w } _ { 4 } \\left ( t \\right ) , \\Lambda _ { 1 } \\bar { w } _ { 4 } \\left ( t \\right ) , \\Lambda _ { 2 } \\bar { w } _ { 4 } \\left ( t \\right ) , \\Lambda _ { 3 } \\bar { w } _ { 4 } \\left ( t \\right ) \\right ] \\end{align*}"}
-{"id": "6028.png", "formula": "\\begin{align*} D _ { 1 + s } ( P _ { X ^ { n } Y ^ { n } | U _ { n } = u _ { n } } \\| \\pi _ { X ^ { n } Y ^ { n } } ) \\rightarrow 0 \\end{align*}"}
-{"id": "8730.png", "formula": "\\begin{align*} & u ^ { \\varepsilon , \\delta } ( t _ \\delta , x _ \\delta ) = u ^ \\varepsilon ( t _ \\delta ' , x _ \\delta ) - \\delta ^ { - 1 } | t _ \\delta - t _ \\delta ' | ^ 2 \\quad \\\\ & v _ { \\varepsilon , \\delta } ( s _ \\delta , y _ \\delta ) = v _ \\varepsilon ( s _ \\delta ' , y _ \\delta ) + \\delta ^ { - 1 } | s _ \\delta - s _ \\delta ' | ^ 2 . \\end{align*}"}
-{"id": "1194.png", "formula": "\\begin{align*} \\breve \\mu ( \\bar B ( 0 , 1 ) ) = \\mbox { C a p } _ { \\mathcal { A } } ( \\bar B ( 0 , 1 ) ) = c _ 1 . \\end{align*}"}
-{"id": "2147.png", "formula": "\\begin{align*} U ( r ) = a v ^ + ( r ) + b v ^ - ( r ) + G ( r ) \\quad \\mbox { i n } \\quad ( 0 , \\infty ) . \\end{align*}"}
-{"id": "5807.png", "formula": "\\begin{align*} \\big | \\langle - \\Delta u , \\varphi \\rangle \\big | = \\bigg | \\int _ { \\Omega } \\nabla u \\cdot \\nabla \\varphi \\ ; d x \\bigg | \\leq \\| \\nabla u \\| _ { L ^ { 2 } ( \\Omega ) } \\| \\nabla \\varphi \\| _ { L ^ { 2 } ( \\Omega ) } . \\end{align*}"}
-{"id": "1904.png", "formula": "\\begin{align*} ( a _ { n } , b _ { n } ) = \\tilde { f } _ { n } - F _ { n } \\quad D ( \\tilde { f } _ { n } ) _ { x } - D ( \\tilde { f } _ { n } ) _ { 0 } = D ( a _ { n } , b _ { n } ) _ { x } \\quad x \\in B _ { n } ( \\delta _ { n } ) . \\end{align*}"}
-{"id": "5592.png", "formula": "\\begin{align*} A _ { n } = \\int _ { 0 } ^ { \\tau } d t _ { 1 } \\int _ { 0 } ^ { t _ { 1 } } d t _ { 2 } \\ldots \\int _ { 0 } ^ { t _ { \\bar { n } - 1 } } \\left ( \\tau - t _ { 1 } + t _ { n } \\right ) \\left ( t _ { 1 } - t _ { 2 } \\right ) \\ldots \\left ( t _ { n - 1 } - t _ { n } \\right ) q \\left ( t _ { 1 } \\right ) q \\left ( t _ { 2 } \\right ) \\ldots q \\left ( t _ { n } \\right ) d t _ { n } \\end{align*}"}
-{"id": "4251.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } \\Phi _ 2 } { p } \\left ( \\mathrm { d } \\log t \\otimes _ { \\Phi } 1 \\right ) = \\mathrm { d } \\log t . \\end{align*}"}
-{"id": "2271.png", "formula": "\\begin{align*} \\det ( S ) & = P _ { A _ { \\alpha } ( G _ 2 ) } ( x - \\alpha n _ 1 ) \\cdot \\left ( 1 - ( 1 - \\alpha ) ^ 2 \\Gamma _ { A _ { \\alpha } ( G _ 1 ) } ( x - \\alpha n _ 2 ) \\cdot \\mathbf { 1 } _ { n _ 2 } ^ T \\left ( ( x - \\alpha n _ 1 ) I _ { n _ 2 } - A _ { \\alpha } ( G _ 2 ) \\right ) ^ { - 1 } \\mathbf { 1 } _ { n _ 2 } \\right ) \\\\ & = P _ { A _ { \\alpha } ( G _ 2 ) } ( x - \\alpha n _ 1 ) \\cdot \\left ( 1 - ( 1 - \\alpha ) ^ 2 \\Gamma _ { A _ { \\alpha } ( G _ 1 ) } ( x - \\alpha n _ 2 ) \\Gamma _ { A _ { \\alpha } ( G _ 2 ) } ( x - \\alpha n _ 1 ) \\right ) . \\end{align*}"}
-{"id": "3198.png", "formula": "\\begin{align*} g _ { t } ' ( t , x ) & : = \\frac { \\partial } { \\partial t } g ( t , x ) = c e ^ { c t } V ( x ) , \\\\ g _ { x } ' ( t , x ) & : = \\frac { \\partial } { \\partial x } g ( t , x ) = \\begin{cases} \\kappa e ^ { c t } x ^ { \\kappa - 1 } , & x > 1 , \\\\ e ^ { c t } V ' ( x ) , & x \\in [ 0 , 1 ] , \\end{cases} \\\\ g _ { x } '' ( t , x ) & : = \\frac { \\partial ^ { 2 } } { \\partial x ^ { 2 } } g ( t , x ) = \\begin{cases} \\kappa ( \\kappa - 1 ) e ^ { c t } x ^ { \\kappa - 2 } , & x > 1 , \\\\ e ^ { c t } V '' ( x ) , & x \\in [ 0 , 1 ] . \\end{cases} \\end{align*}"}
-{"id": "5456.png", "formula": "\\begin{align*} g ( x ) = r ^ { \\rho } g ( x / r ) , x \\in C _ g , \\end{align*}"}
-{"id": "9573.png", "formula": "\\begin{align*} B _ i : = B \\biggl ( x _ i , \\frac { \\delta _ { \\partial D _ i } ( x _ i ) } { 1 6 M c ^ 2 _ 1 } \\biggr ) \\subset D _ i \\subset D \\ , . \\end{align*}"}
-{"id": "6818.png", "formula": "\\begin{align*} - \\Delta _ g \\tilde { g } _ j = \\frac { 2 \\lambda } { | x _ { \\xi _ j } | ^ 3 } + 2 \\textrm { i n } \\mathbb { S } ^ 2 \\setminus \\Pi _ { \\xi _ j } ( B ( 0 , R _ 2 ) ) , \\end{align*}"}
-{"id": "8934.png", "formula": "\\begin{gather*} ( s _ 1 s _ 2 - 1 ) = ( s _ 1 - 1 ) ( s _ 2 - 1 ) + ( s _ 1 - 1 ) + ( s _ 2 - 1 ) , \\end{gather*}"}
-{"id": "1440.png", "formula": "\\begin{align*} Q _ 0 ( x _ 2 ) = x _ 3 , \\ ; Q _ 0 ( x _ 2 x _ 3 ) = x _ 3 ^ 2 \\end{align*}"}
-{"id": "3947.png", "formula": "\\begin{align*} \\begin{aligned} \\iint _ { L ^ { k } ( r ) } | \\nabla u | ^ 2 d x d s & \\leq C r ^ { 2 q - 4 } m ( 6 r ) m ( r ) ^ { \\tau } \\sum _ i \\iint _ { Q ( x _ { i } , t _ { i } ; \\ , r ) \\cap \\tilde { K } ( r ) } | \\nabla u | ^ q d x d s \\\\ & = C r ^ { 2 q - 4 } m ( r ) ^ { \\tau + 1 } \\iint _ { \\widetilde { L } ( r ) } | \\nabla u | ^ q d x d s . \\end{aligned} \\end{align*}"}
-{"id": "3383.png", "formula": "\\begin{align*} \\beta ( l _ { F ' } ) \\subset \\{ x = 0 \\} , \\beta ( k _ { F ' } ) \\subset \\{ x = \\psi ( y ) \\} \\end{align*}"}
-{"id": "5432.png", "formula": "\\begin{align*} f _ i ( P _ { k , \\ell } ^ m ) = \\binom { k + \\ell + m + 2 } { i + 1 } - \\binom { \\ell + m + 1 } { i - k } - \\binom { k + m + 1 } { i - \\ell } + \\binom { m } { i - k - \\ell - 1 } + \\binom { m } { i - k - \\ell } . \\end{align*}"}
-{"id": "4936.png", "formula": "\\begin{align*} ( D ^ 0 ) = ( D ^ 0 ) - ( D ^ 1 ) \\end{align*}"}
-{"id": "5679.png", "formula": "\\begin{align*} \\frac { - f ' } { f ^ { p _ 0 / 2 } } \\geq \\sqrt { \\frac { 2 c _ 0 } { p _ 0 } } > \\sqrt { \\frac { 2 c } { p _ 0 } } = \\frac { - E ' } { E ^ { p _ 0 / 2 } } . \\end{align*}"}
-{"id": "3767.png", "formula": "\\begin{align*} p _ k ( \\rho ) = \\max _ { m \\in M _ k } \\P ^ { \\rho } \\left ( A _ m \\right ) \\le \\exp \\left \\{ - ( \\log L _ k ) ^ { \\gamma } \\right \\} \\ ; \\ ; \\ ; \\forall \\ ; k \\ge \\hat { k } . \\end{align*}"}
-{"id": "6081.png", "formula": "\\begin{align*} Q _ n ( t ) + i P _ n ( t ) = R _ n ( t ) e ^ { i \\Theta _ n ( t ) } . \\end{align*}"}
-{"id": "1279.png", "formula": "\\begin{align*} E _ 1 \\subset \\{ x = ( x ' , x '' ) : x ' = ( x _ 1 , \\dots , x _ k ) \\mbox { a n d } x '' = ( x _ { k + 1 } , \\dots x _ n ) = ( 0 , \\dots , 0 ) \\} = \\mathbb { R } ^ k . \\end{align*}"}
-{"id": "1019.png", "formula": "\\begin{align*} \\min _ { | \\Delta g _ { p s } ^ i | \\leq \\varepsilon } & \\sum _ { i = 1 } ^ { N } \\beta _ { i } \\log _ { 2 } \\left ( 1 + \\frac { \\left | g _ { s s } ^ { i } \\right | ^ { 2 } p _ s ^ { i } } { \\sigma _ n ^ 2 + \\left ( \\left | \\hat { g } _ { p s } ^ { i } + \\Delta g ^ i _ { p s } \\right | ^ { 2 } \\right ) p _ { p } } \\right ) \\end{align*}"}
-{"id": "6364.png", "formula": "\\begin{align*} A _ j ( z ) & = ( z - \\alpha _ j ) A _ { j - 1 } ( z ) - \\beta _ { j - 1 } A _ { j - 2 } ( z ) , \\\\ B _ j ( z ) & = ( z - \\alpha _ j ) B _ { j - 1 } ( z ) - \\beta _ { j - 1 } B _ { j - 2 } ( z ) \\end{align*}"}
-{"id": "3904.png", "formula": "\\begin{align*} \\langle H \\rangle ^ { \\ell } _ { m i n } = - \\frac { m e ^ 2 } { 2 \\hbar ^ 2 } \\frac { 1 } { ( \\ell + \\frac { 3 } { 2 } ) } \\Big ( \\frac { \\Gamma ( \\ell + 1 ) } { \\Gamma ( \\ell + \\frac { 3 } { 2 } ) } \\Big ) ^ 2 , \\end{align*}"}
-{"id": "7800.png", "formula": "\\begin{align*} D _ { n } f ( x ) : = \\frac { 1 } { \\pi } \\int \\nolimits _ { \\mathbb { T } } f ( x - t ) J _ { 2 , \\lfloor \\frac { n } { 2 } \\rfloor + 1 } ( t ) d t \\in \\mathcal { T } _ { n } \\end{align*}"}
-{"id": "5740.png", "formula": "\\begin{align*} [ x ] _ n = \\begin{cases} x ( x + 2 ) \\dotsc ( x + 2 ( n - 1 ) ) & \\\\ 1 & \\\\ \\frac { 1 } { ( x - 2 ) ( x - 4 ) \\dotsc ( x + 2 n ) } & \\end{cases} \\end{align*}"}
-{"id": "3179.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } _ { \\geqslant 0 } } f ( y ) \\rho _ { t } ^ { \\ast n } ( \\mathrm { d } y ) & = \\int _ { \\mathbb { R } _ { \\geqslant 0 } } \\cdots \\int _ { \\mathbb { R } _ { \\geqslant 0 } } f ( y _ { 1 } + \\cdots + y _ { n } ) \\rho _ { t } ( \\mathrm { d } y _ { 1 } ) \\cdots \\rho _ { t } ( \\mathrm { d } y _ { n } ) \\\\ & \\leqslant c _ { 1 } ^ { n } \\left ( \\int _ { \\mathbb { R } _ { \\geqslant 0 } } f ( y ) \\rho _ { t } ( \\mathrm { d } y ) \\right ) ^ { n } < \\infty , \\end{align*}"}
-{"id": "5842.png", "formula": "\\begin{align*} \\frac A B = \\sqrt { 2 C _ 0 } \\ , \\big ( 1 + O ( A ^ { - 2 } ) \\big ) \\frac B A = \\frac { 1 } { \\sqrt { 2 C _ 0 } } \\big ( 1 + O ( A ^ { - 2 } ) \\big ) \\end{align*}"}
-{"id": "5050.png", "formula": "\\begin{align*} m _ i = \\begin{cases} 0 , & \\tau _ { _ { T _ i , R _ i } } ^ { C ^ * } > \\tau _ { _ { T _ i , R _ i } } ^ D \\\\ 1 , & \\tau _ { _ { T _ i , R _ i } } ^ { C ^ * } \\le \\tau _ { _ { T _ i , R _ i } } ^ D . \\end{cases} \\end{align*}"}
-{"id": "4213.png", "formula": "\\begin{align*} M _ 1 ( H _ 7 ) - M _ 1 ( H _ 8 ) & = ( d _ { H _ 7 } ( v ) ) ^ 2 + ( d _ { H _ 7 } ( v _ 1 ) ) ^ 2 - ( d _ { H _ 8 } ( v ) ) ^ 2 - ( d _ { H _ 8 } ( v _ 1 ) ) ^ 2 \\\\ & = ( d _ { H _ 7 } ( v ) + d _ { H _ 8 } ( v ) ) + ( d _ { H _ 7 } ( v _ 1 ) + d _ { H _ 8 } ( v _ 1 ) ) \\\\ & \\geq 7 + 3 = 1 0 > 0 , \\end{align*}"}
-{"id": "3234.png", "formula": "\\begin{align*} \\beta = \\langle Z _ 1 \\otimes Y \\rangle \\cup \\langle X \\times Z _ 2 \\rangle \\cup ( \\alpha \\otimes \\langle \\times X \\times Y \\rangle ) . \\end{align*}"}
-{"id": "4276.png", "formula": "\\begin{align*} f \\left ( \\frac { a \\tau + b } { c \\tau + d } \\right ) = ( c \\tau + d ) ^ k f ( \\tau ) \\end{align*}"}
-{"id": "647.png", "formula": "\\begin{align*} P ^ 1 _ k ( x _ 0 , \\cdots , x _ k ) & = \\widehat P _ { k - 1 } ( x _ 0 , \\cdots , x _ { k - 1 } ) + x _ k , \\\\ Q ^ 1 _ k ( x _ 0 , \\cdots , x _ k ) & = \\widehat Q _ { k - 1 } ( x _ 0 , \\cdots , x _ { k - 1 } ) + 2 x _ k . \\end{align*}"}
-{"id": "1161.png", "formula": "\\begin{align*} e ^ 2 _ { k - 1 } ( w ) + D _ { k - 1 } \\cdot e _ { k - 1 } ( w ) & = \\left ( \\sum _ { i = 0 } ^ { k - 2 } B _ { k - 1 , i } w ^ { 2 ^ i } \\right ) ^ 2 + D _ { k - 1 } \\left ( \\sum _ { i = 0 } ^ { k - 2 } B _ { k - 1 , i } w ^ { 2 ^ i } \\right ) \\cr & = \\sum _ { i = 1 } ^ { k - 1 } B ^ 2 _ { k - 1 , i - 1 } w ^ { 2 ^ i } + \\sum _ { i = 0 } ^ { k - 2 } D _ { k - 1 } B _ { k - 1 , i } w ^ { 2 ^ i } . \\end{align*}"}
-{"id": "3084.png", "formula": "\\begin{align*} \\dot w ( s ) = - \\nabla _ x F ( t ^ * , w ( s ) ) \\ , , \\end{align*}"}
-{"id": "5337.png", "formula": "\\begin{align*} ( v \\otimes x ) a = v \\otimes x a \\end{align*}"}
-{"id": "3856.png", "formula": "\\begin{align*} \\nabla _ x \\ell ( x ^ * , \\lambda ^ * , \\mu ^ * , \\gamma ^ * ) = 0 . \\end{align*}"}
-{"id": "7605.png", "formula": "\\begin{align*} \\Phi _ C ( x , y ) = ( x , y + C x ) , \\Phi _ A ( x , y ) = \\left ( A ^ { - 1 } x , A ^ T y \\right ) . \\end{align*}"}
-{"id": "501.png", "formula": "\\begin{align*} V _ 1 ( x , t ) & \\sim - \\frac { R } { 4 } \\frac { \\frac { 1 } { \\delta ^ 2 } I _ { n } ( \\kappa ) ^ 2 } { I _ { n - 1 } ( \\kappa ) ^ 2 } + \\frac { R } { 2 } \\frac { \\frac { 1 } { \\delta ^ 2 } I _ { n + 1 } ( \\kappa ) + \\frac { n } { R \\delta } I _ { n } ( \\kappa ) } { I _ { n - 1 } ( \\kappa ) } \\\\ & = \\frac { \\pi \\abs * { t } } { 4 } \\frac { \\frac { 2 n } { \\kappa } I _ n ( \\kappa ) I _ { n - 1 } ( \\kappa ) + 2 I _ { n - 1 } ( \\kappa ) I _ { n + 1 } ( \\kappa ) - I _ n ( \\kappa ) ^ 2 } { I _ { n - 1 } ( \\kappa ) ^ 2 } \\asymp d ( x , t ) ^ 2 . \\end{align*}"}
-{"id": "7366.png", "formula": "\\begin{align*} \\nabla _ X \\omega = \\frac { 1 } { p + 1 } i _ X d \\omega \\end{align*}"}
-{"id": "6159.png", "formula": "\\begin{align*} \\mu = V \\dd x _ { 0 1 2 3 } \\end{align*}"}
-{"id": "9028.png", "formula": "\\begin{align*} \\textstyle J ^ { b c } J _ { [ b c } \\phi _ { d e f \\cdots g ] } = \\frac { 4 ( n - k ) } { ( k + 1 ) ( k + 2 ) } \\phi _ { d e f \\cdots g } + \\frac { k ( k - 1 ) } { ( k + 1 ) ( k + 2 ) } J _ { [ d e } \\phi _ { f \\cdots g ] b c } J ^ { b c } \\end{align*}"}
-{"id": "355.png", "formula": "\\begin{align*} \\frac { | \\nabla f | ^ 2 } { H ^ { 2 + \\sigma } } \\partial _ 0 S ^ 2 & = | \\nabla f | ^ 2 \\left ( \\partial _ 0 U _ { \\sigma } + \\frac { 2 + \\sigma } { H } ( \\partial _ 0 H ) U _ { \\sigma } \\right ) \\\\ & = | \\nabla f | ^ 2 ( \\partial _ 0 U _ { \\sigma } ) + \\frac { 2 + \\sigma } { H } \\langle \\nabla H , \\nabla f \\rangle U _ { \\sigma } . \\end{align*}"}
-{"id": "8157.png", "formula": "\\begin{align*} a _ 1 ^ { q + 1 } a _ 6 ^ { q ^ 2 } + a _ 2 a _ 7 ^ { q + q ^ 2 } = 0 . \\end{align*}"}
-{"id": "2694.png", "formula": "\\begin{align*} 2 p ^ n { a ' _ 1 } \\alpha ^ 2 + 2 a _ 2 \\alpha + 2 p ^ n { a ' _ 3 } = 0 . \\end{align*}"}
-{"id": "7965.png", "formula": "\\begin{align*} | \\sin ^ 2 \\beta ' / 2 - \\sin ^ 2 \\beta / 2 | & = | a - a ' | b ( b + c ) / a a ' c \\le b ( b + c ) \\epsilon / c a ' \\\\ & \\le \\frac { \\epsilon } { 1 - \\epsilon } \\frac { b ( b + c ) } { a } \\le \\frac { \\epsilon } { 1 - \\epsilon } 2 C ( \\delta ) ^ 2 , \\end{align*}"}
-{"id": "192.png", "formula": "\\begin{align*} \\boldsymbol { R } _ { \\delta } = \\langle \\delta ( R ) , \\subseteq , + ^ { \\delta } , \\delta ( 0 ) , \\ast ^ { \\delta } \\rangle , \\end{align*}"}
-{"id": "311.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { j = 0 } ^ d T _ j ( z ) \\lambda _ d \\mathcal { S } _ d , \\end{align*}"}
-{"id": "2661.png", "formula": "\\begin{align*} \\Pi _ { ( \\Psi , \\xi , k \\phi ) } ( x ) = 1 + \\frac { 1 } { 4 k } \\left [ { \\rm S c a l } ( \\phi ) + 2 \\Delta _ \\phi ( \\log \\Psi ( f _ { ( \\xi , \\phi ) } ) ) \\right ] + \\mathcal { O } \\left ( \\frac { 1 } { k ^ { 2 } } \\right ) . \\end{align*}"}
-{"id": "507.png", "formula": "\\begin{align*} \\mathcal { R } _ { } \\ ! = \\ ! & \\bigcup _ { P _ { U | X } } \\ ! \\Big \\{ \\left ( R _ s , R _ \\ell , R _ w \\right ) \\ ! \\colon \\ ! \\\\ & 0 \\leq R _ s \\leq I ( U ; Y ) , \\\\ & R _ \\ell \\geq I ( U ; X ) - I ( U ; Y ) , \\\\ & R _ w \\geq I ( U ; X ) - I ( U ; Y ) \\Big \\} \\end{align*}"}
-{"id": "4787.png", "formula": "\\begin{align*} [ b , T ] _ 1 ( f , g ) : = \\ , & T ( b f , g ) - b T ( f , g ) , \\\\ [ b , T ] _ 2 ( f , g ) : = \\ , & T ( f , b g ) - b T ( f , g ) , \\end{align*}"}
-{"id": "4592.png", "formula": "\\begin{align*} [ x , y ] = \\omega ( x , y ) J A + [ x , y ] _ W , \\end{align*}"}
-{"id": "5968.png", "formula": "\\begin{align*} \\tilde d _ { i , j } = \\gamma ^ { j _ 1 - j _ 2 } \\varepsilon ^ { 2 ( j _ 2 - j _ 1 ) } L ^ { j _ 2 - j _ 1 } \\sqrt { \\frac { j _ 1 ! } { j _ 2 ! } } \\sqrt { 2 ^ { j _ 2 - j _ 1 } } . \\end{align*}"}
-{"id": "2429.png", "formula": "\\begin{align*} v ^ i ( x _ 1 ^ \\ast , y _ 1 ^ \\ast ) & = \\max _ { \\mu _ 1 \\in \\Delta ( A ) } \\min _ { \\nu _ 1 \\in \\Delta ( B _ { y _ 1 ^ \\ast } ) } \\sum \\limits _ { \\substack { x _ 2 \\in X \\\\ y _ 2 \\in C _ i } } p ( x _ 2 | x _ 1 ^ \\ast , \\mu _ 1 ) q ( y _ 2 | y _ 1 ^ \\ast , \\nu _ 1 ) v ^ i ( x _ 2 , y _ 2 ) . \\end{align*}"}
-{"id": "7576.png", "formula": "\\begin{align*} \\pi _ \\psi ( u , v ) = \\lim _ { k \\to \\infty } \\cal F ^ { - 1 } ( \\psi _ k \\cal F u ) \\cdot \\cal F ^ { - 1 } ( \\psi _ k \\cal F v ) , \\end{align*}"}
-{"id": "7657.png", "formula": "\\begin{align*} \\beta _ { \\mu } \\beta _ { \\nu } \\beta _ { \\lambda } + \\beta _ { \\lambda } \\beta _ { \\nu } \\beta _ { \\mu } = \\delta _ { \\mu \\nu } \\beta _ { \\lambda } + \\delta _ { \\lambda \\nu } \\beta _ { \\mu } . \\end{align*}"}
-{"id": "2253.png", "formula": "\\begin{align*} c _ i = \\frac { 1 } { z _ { j 1 } ^ { i } \\cdot \\ldots \\cdot z _ { j n } ^ { i } } , \\ i \\geqslant 1 \\end{align*}"}
-{"id": "8749.png", "formula": "\\begin{align*} \\phi ( E ) = \\alpha + \\phi ^ 0 ( E ) + \\phi ^ I ( E ) , \\end{align*}"}
-{"id": "5881.png", "formula": "\\begin{align*} \\begin{aligned} & P \\big ( \\max \\big ( \\frac { S ^ { ( i ) } _ { 0 , N } } N , \\cdots , \\frac { S ^ { ( i ) } _ { N - 1 , 2 N - 1 } } N \\big ) \\ge r _ { i + 1 } \\big ) \\le N P ( \\frac { S ^ { ( i ) } _ { 0 , N } } N \\ge r _ { i + 1 } ) \\approx e ^ { - N I _ i ( r _ { i + 1 } ) } . \\end{aligned} \\end{align*}"}
-{"id": "2864.png", "formula": "\\begin{align*} P ^ 1 _ \\epsilon : = \\{ x \\in \\R ^ n _ + ~ | ~ ( v _ i ) ^ \\top x \\leq \\sigma _ { \\tilde { Q } } ( v _ i ) , \\forall i = 1 , \\hdots , \\ell , \\ \\ x _ i \\leq \\bar { M } , \\forall i = 1 , \\hdots , n \\} \\end{align*}"}
-{"id": "6365.png", "formula": "\\begin{align*} Q _ j ( z ) : = \\int \\frac { P _ { j } ( z ) - P _ { j } ( t ) } { z - t } \\ , d \\mu ( t ) \\end{align*}"}
-{"id": "5398.png", "formula": "\\begin{align*} a & : = ( 1 , 2 ) ( 4 , 5 ) \\\\ b & : = ( 4 , 5 ) ( 7 , 8 ) \\\\ c & : = ( 1 , 3 ) ( 4 , 6 ) ( 7 , 9 ) \\end{align*}"}
-{"id": "5916.png", "formula": "\\begin{align*} s ' = \\begin{cases} i _ 0 , i _ 1 , \\cdots , i _ { j - k } , i _ { j + k + 1 } , \\cdots , i _ m , \\ \\ s \\ \\ \\eqref { b e g i n n i n g 1 } , \\\\ i _ 0 , i _ 1 , i _ 2 , \\cdots , i _ j , i _ { j + 3 } , \\cdots , i _ m , \\ \\ s \\ \\ \\eqref { b e g i n n i n g 2 } , \\\\ i _ 0 , i _ 1 , \\cdots , i _ { l - k } , i _ { l + 2 j + k + 1 } , \\cdots , i _ m , \\ \\ s \\ \\ \\eqref { b e g i n n i n g 3 } . \\end{cases} \\end{align*}"}
-{"id": "2865.png", "formula": "\\begin{align*} \\max \\{ c ^ \\top x \\mid x \\in Q \\} = \\max \\{ c _ 1 x _ 1 + c _ 2 x _ 2 | ( x _ 1 , x _ 2 ) \\in Q | _ { \\R ^ 2 } \\} . \\end{align*}"}
-{"id": "1638.png", "formula": "\\begin{align*} S _ \\lambda ( \\delta _ \\eta ) = \\delta _ { \\tau _ \\lambda ( \\eta ) } . \\end{align*}"}
-{"id": "8546.png", "formula": "\\begin{align*} u ( \\theta ) = \\int _ { 0 } ^ { \\theta } \\frac { d \\phi } { \\sqrt { 1 - k ^ { 2 } \\mathrm { s i n } ^ { 2 } \\phi } } . \\end{align*}"}
-{"id": "8445.png", "formula": "\\begin{align*} \\chi ( 1 + z \\varpi ^ { \\kappa _ F } ) = \\psi \\left ( \\frac { b _ { \\chi } } { \\varpi ^ { a ( \\chi ) } } \\log _ F ( 1 + z \\varpi ^ { \\kappa _ F } ) \\right ) z \\in \\mathcal { O } . \\end{align*}"}
-{"id": "3197.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\lim _ { T \\to \\infty } \\frac { 1 } { T } \\int _ { 0 } ^ { T } f ( X _ { s } ) \\mathrm { d } s = \\int _ { \\mathbb { R } _ { \\geqslant 0 } } f ( x ) \\pi ( \\mathrm { d } x ) \\right ) = 1 . \\end{align*}"}
-{"id": "9434.png", "formula": "\\begin{align*} = \\sum _ { n = 1 } ^ N \\left ( \\psi ( n ) - \\psi ( n + 1 ) \\right ) \\sum _ { j = 1 } ^ n \\frac { \\varphi ( j ) } { j | { j } | _ { \\mathcal { D } _ 1 } | { j } | _ { \\mathcal { D } _ 2 } \\cdots | j | _ { \\mathcal { D } _ m } } + \\psi ( N + 1 ) \\sum _ { j = 1 } ^ N \\frac { \\varphi ( j ) } { j | { j } | _ { \\mathcal { D } _ 1 } | { j } | _ { \\mathcal { D } _ 2 } \\cdots | j | _ { \\mathcal { D } _ m } } . \\end{align*}"}
-{"id": "7664.png", "formula": "\\begin{align*} b _ { i j } = \\frac 1 2 ( \\psi _ i , \\psi _ j ) . \\end{align*}"}
-{"id": "6177.png", "formula": "\\begin{align*} R _ { 0 i 0 } ^ { \\ ; \\ ; \\ ; \\ ; \\ ; i } = \\frac { f _ i '' } { 2 f _ i } - \\frac { f _ i '^ 2 } { 4 f _ i ^ 2 } - \\frac { V ' f _ i ' } { 2 V f _ i } \\ ; \\ ; \\ ; ; \\ ; \\ ; i = 1 , 2 , 3 \\end{align*}"}
-{"id": "3706.png", "formula": "\\begin{align*} ( w _ 1 , \\dots , w _ n ; w _ { n + 1 } , \\dots , w _ { 2 n } ; w _ { 2 n + 1 } ) = \\left ( \\frac { u _ { 1 1 } } { v _ { 1 1 } } , \\dots , \\frac { u _ { n n } } { v _ { n n } } ; x _ 1 , \\frac { v _ { 2 1 } } { u _ { 2 1 } } , \\dots , \\frac { v _ { n , n - 1 } } { u _ { n , n - 1 } } ; t _ { n + 1 } \\right ) . \\end{align*}"}
-{"id": "5314.png", "formula": "\\begin{align*} \\int ^ t _ 0 Z ( \\varsigma ) \\mathrm { d \\varsigma } = B ( \\theta ) - B ( \\theta ^ { 0 } ) , \\end{align*}"}
-{"id": "292.png", "formula": "\\begin{align*} d _ p ( x ^ * , y ^ * ) = \\rho _ p ( j ( x ^ * ) , j ( y ^ * ) ) . \\end{align*}"}
-{"id": "9151.png", "formula": "\\begin{align*} \\begin{array} { r c l } f - D _ { 4 } & = & 0 \\\\ 5 \\ , D _ { 4 } + g + D _ { 3 } & = & 0 \\\\ - 1 0 \\ , D _ { 4 } - 4 \\ , D _ { 3 } - D _ { 2 } & = & 0 \\\\ 1 0 \\ , D _ { 4 } + 6 \\ , D _ { 3 } + 3 \\ , D _ { 2 } + D _ { 1 } & = & 0 \\\\ - 5 \\ , D _ { 4 } - 4 \\ , D _ { 3 } - 3 \\ , D _ { 2 } - 2 \\ , D _ { 1 } + h & = & 0 . \\end{array} \\end{align*}"}
-{"id": "5171.png", "formula": "\\begin{align*} V ^ A ( x ) = f ( x ) \\iff \\tilde V ^ A ( x ) = f ( x ) - g _ { A } ( x ) . \\end{align*}"}
-{"id": "7245.png", "formula": "\\begin{align*} = \\int _ { \\hat { G } } ^ { } \\int _ { G } ^ { } f ( x ) ( \\int _ { U } ^ { } \\bar { \\psi _ { n } } ( u ) \\theta _ { \\pi } ( u ^ { - 1 } x ) d u ) d x d \\mu _ { \\pi } , \\end{align*}"}
-{"id": "6139.png", "formula": "\\begin{align*} \\mathbb { P } _ { c o u p l e } \\left ( n ^ { - \\frac { 1 } { \\alpha } } \\sum _ { i = 1 } ^ { M _ { 1 } \\left ( n \\right ) } Z _ { i } > n ^ { - c ' } \\right ) & \\leq M _ { 1 } \\left ( n \\right ) \\mathbb { P } _ { c o u p l e } \\left ( Z > n ^ { \\frac { 1 } { \\alpha } - c ' } \\right ) \\\\ & + \\exp \\left ( 1 - \\frac { A \\left ( n \\right ) } { n ^ { \\left ( \\frac { 1 } { \\alpha } - c ' \\right ) } } - \\log \\left ( \\frac { n ^ { \\left ( \\frac { 1 } { \\alpha } - c ' \\right ) } } { A \\left ( n \\right ) } \\right ) \\right ) , \\end{align*}"}
-{"id": "1960.png", "formula": "\\begin{align*} \\beta : = \\alpha - ( j - 1 ) ( 1 - \\lambda ) . \\end{align*}"}
-{"id": "7381.png", "formula": "\\begin{align*} \\Z _ { N _ f } \\left ( f _ 0 \\right ) = \\prod _ { j = 0 } ^ { N _ f - 1 } \\frac { \\Gamma ( 2 + j ) } { \\Gamma ( 1 + j ) \\Gamma ( 2 ) } = M _ { N _ f } ( 0 , 0 , 1 ) , \\end{align*}"}
-{"id": "2381.png", "formula": "\\begin{align*} x ' & = a _ 1 x - y - a _ 3 x ^ 2 + ( 2 a _ 2 + a _ 5 ) x y + a _ 6 y ^ 2 , \\\\ y ' & = x + a _ 1 y + a _ 2 x ^ 2 + ( 2 a _ 3 + a _ 4 ) x y - a _ 2 y ^ 2 . \\end{align*}"}
-{"id": "815.png", "formula": "\\begin{align*} v _ { 3 } ( S _ { k } ( 2 x ) ) = v _ { 3 } ( S _ { k } ( x ) ) = \\gamma + 2 d - 1 \\end{align*}"}
-{"id": "3902.png", "formula": "\\begin{align*} \\left ( - \\frac { \\hbar ^ 2 } { 2 m } \\left ( \\frac { d ^ 2 } { d \\rho ^ 2 } + \\frac { ( D - 1 ) } { \\rho } \\frac { d } { d \\rho } - \\frac { \\cal { L } ( \\cal { L } + D - 2 ) } { \\rho ^ 2 } \\right ) + { \\cal V } ( \\rho ) - { \\cal E } \\right ) \\tilde R ( \\rho ) = 0 , \\end{align*}"}
-{"id": "9135.png", "formula": "\\begin{align*} f ( x ^ { 3 } ) + x f ( x ^ { 2 } ) - 2 x ^ { 2 } f ( x ) = 0 \\end{align*}"}
-{"id": "8221.png", "formula": "\\begin{align*} t _ - = \\sum _ { \\ell \\in \\mathcal { L } ^ - } \\binom { k } { \\ell } \\Bigl ( U _ { i + \\ell } - \\frac { 1 } { 2 } ( U _ { i + \\ell } ) ^ 2 + \\frac { 1 } { 3 } ( U _ { i + \\ell } ) ^ 3 \\cdots \\Bigr ) \\end{align*}"}
-{"id": "4435.png", "formula": "\\begin{align*} - ( \\lceil u , ( \\cdot ) _ T \\rceil f ) ( x ) & = \\int _ { \\mathbb { R } ^ 2 } \\psi _ T ( x - y ) ( u ( y ) - u ( x ) ) f ( y ) d y , \\\\ - ( \\lceil u , ( \\cdot ) _ T \\rceil - \\partial _ 1 u \\lceil x _ 1 , ( \\cdot ) _ T \\rceil ) f ( x ) & = \\int _ { \\mathbb { R } ^ 2 } \\psi _ T ( x - y ) ( u ( y ) - u ( x ) - ( y - x ) _ 1 \\partial _ 1 u ( x ) ) f ( y ) d y \\end{align*}"}
-{"id": "3839.png", "formula": "\\begin{align*} R _ k = \\inf \\{ n \\in { \\mathbb { Z } _ + } \\colon \\ , X _ n \\ge ( 1 - \\bar { v } ) k + \\bar { v } n \\} , k \\in \\N , \\end{align*}"}
-{"id": "6525.png", "formula": "\\begin{align*} \\d ( S _ 1 , S _ 2 ) = \\sup _ { u \\in \\mathbb { S } ^ { n - 1 } } \\max \\left [ \\frac { r _ { S _ 1 } ( u ) } { r _ { S _ 2 } ( u ) } , \\frac { r _ { S _ 2 } ( u ) } { r _ { S _ 1 } ( u ) } \\right ] = \\inf \\left \\{ a \\geq 1 : \\frac { 1 } { a } S _ 1 \\subseteq S _ 2 \\subseteq a S _ 1 \\right \\} . \\end{align*}"}
-{"id": "1365.png", "formula": "\\begin{align*} \\nabla _ x L ( \\overline { x } , \\overline Y , \\overline { \\mu } , \\overline { \\Gamma } ) = 0 , \\ \\overline Y \\in \\partial \\theta ( F ( \\overline x ) , \\ h ( \\overline { x } ) = 0 , \\ \\overline { \\Gamma } \\succeq 0 , \\ g ( \\overline { x } ) \\succeq 0 \\ , \\ , { \\rm a n d } \\ , \\ , \\langle \\overline { \\Gamma } , g ( \\overline { x } ) \\rangle = 0 . \\end{align*}"}
-{"id": "6354.png", "formula": "\\begin{align*} \\int x ^ k P _ k ( x ) \\ , d \\mu ( x ) & = \\beta _ 1 \\cdots \\beta _ k \\\\ \\int x ^ { k + 1 } P _ k ( x ) \\ , d \\mu ( x ) & = \\beta _ 1 \\cdots \\beta _ k ( \\alpha _ 1 + \\cdots + \\alpha _ { k + 1 } ) . \\end{align*}"}
-{"id": "6228.png", "formula": "\\begin{align*} C ^ i ( n ) = U ^ i ( 1 ) + \\cdots + U ^ i ( n ) ~ , ~ ~ n \\geq 1 ~ , \\end{align*}"}
-{"id": "7869.png", "formula": "\\begin{align*} d ( w , z ) \\le d ( w , y ) + d ( y , z ) = d ( w , y ) + d ( x , w ) = d ( x , y ) . \\end{align*}"}
-{"id": "1587.png", "formula": "\\begin{align*} g _ i ' & = 0 \\ \\ 1 \\leq i \\leq n - 3 \\\\ g _ { n - 2 } ' & = 1 \\\\ g _ { n - 1 } ' & = g _ { n - 1 } + p \\\\ g _ n ' & = g _ n + q . \\end{align*}"}
-{"id": "7152.png", "formula": "\\begin{align*} \\sum _ { m = r J + 1 } ^ { ( r + 1 ) J } y ( m ) \\leq \\sqrt { 2 ( U J ^ 2 + V _ r J g ^ * _ r ) } . \\end{align*}"}
-{"id": "6730.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\rightarrow \\infty } \\Big | \\mathbb { P } ( R _ { N } > N ^ { 1 + \\delta } ) - \\mathbb { P } ( R _ { N } ^ { \\eta } > N ^ { 1 + \\delta } ) \\Big | = 0 . \\end{align*}"}
-{"id": "1672.png", "formula": "\\begin{align*} \\mu ( Z ( e g _ 1 e g _ 2 e g _ 3 \\cdots e g _ n ) ) = \\prod _ { i = 1 } ^ n \\alpha _ i . \\end{align*}"}
-{"id": "2463.png", "formula": "\\begin{align*} | g | _ { L _ { \\xi } ^ { 2 } } = \\Big ( \\int _ { { \\mathbb { R } } ^ { 3 } } | g | ^ { 2 } d \\xi \\Big ) ^ { 1 / 2 } \\ , , \\end{align*}"}
-{"id": "5170.png", "formula": "\\begin{align*} V ^ { A } ( x ) & = \\bar { V } ^ { A } ( x ) = g _ { A } ( x ) + \\tilde V ^ A ( x ) . \\end{align*}"}
-{"id": "4125.png", "formula": "\\begin{align*} k _ e ^ { - 1 } = \\frac { d ^ 2 g } { d \\theta ^ 2 } + g , \\end{align*}"}
-{"id": "2534.png", "formula": "\\begin{align*} \\nabla _ { x } h ^ { ( 0 ) } ( t ) = e ^ { ( t - t _ { 0 } / 2 ) \\mathcal { L } } \\nabla _ { x } h ^ { ( 0 ) } ( t _ { 0 } / 2 ) \\ , , \\end{align*}"}
-{"id": "7830.png", "formula": "\\begin{align*} A _ j = \\sum _ { i = 0 } ^ d P _ { i j } E _ i . \\end{align*}"}
-{"id": "3369.png", "formula": "\\begin{align*} \\xi ( X ) : = \\left \\langle \\left \\langle X \\cdot \\varphi , \\varphi \\right \\rangle \\right \\rangle , ~ ~ \\forall X \\in T M . \\end{align*}"}
-{"id": "3253.png", "formula": "\\begin{align*} \\begin{array} { c c c } \\oplus _ { i = 1 } ^ { [ { p - 1 \\over 2 } ] } H ^ { p - 2 i } ( X ; \\mathbb Q ) & \\stackrel { \\sum _ i u ^ i } \\rightarrow & \\oplus _ { i = 1 } ^ { [ { p - 1 \\over 2 } ] } ( L _ i ( X ) ) ^ { \\vee } \\end{array} \\end{align*}"}
-{"id": "2122.png", "formula": "\\begin{align*} | y _ 0 - y _ l | \\sim M r y _ 0 \\in B _ 0 y _ l \\in B _ l , l = 1 , 2 , \\ldots , m , \\end{align*}"}
-{"id": "479.png", "formula": "\\begin{align*} p ^ { ( m ) } _ { 1 , k _ 1 , 0 } ( x , t ) & = \\sum _ { k = 1 } ^ { \\frac { m - 1 } { 2 } } \\frac { c _ { m , k } ( - 1 ) ^ k } { ( 2 \\pi ) ^ { \\frac { m - 1 } { 2 } } } \\frac { 1 } { | t | ^ { m - 1 - k } } p ^ { ( 1 ) } _ { 1 , k _ 1 , k } ( x , | t | ) \\end{align*}"}
-{"id": "5923.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { s \\in A _ { } : r _ s = r } L ( s ) Q ^ N ( s ) \\le e ^ { - c _ 1 ( r - 2 ) N } K _ l ^ { r - 2 } \\sum _ { s \\in A _ { } : r _ { s } = 2 } L ( s ) Q ^ N ( s ) + \\\\ & 2 ( l - 1 ) ( r - 2 ) e ^ { - c _ 1 ( r - 2 ) N } K _ l ^ { r - 2 } \\sum _ { s \\in A _ { } : r _ { s } = 2 } Q ^ N ( s ) . \\end{aligned} \\end{align*}"}
-{"id": "1682.png", "formula": "\\begin{align*} \\mu _ 2 \\circ \\sigma _ e ( Z ( \\lambda _ n ) ) & = \\mu _ 2 ( Z ( e \\lambda _ n b _ 0 ) ) + \\mu _ 2 ( Z ( e \\lambda _ n b _ 1 ) \\\\ & = 2 ^ { n + 1 } ( 1 + ( - 1 ) ^ { m _ Q } 2 \\delta _ 0 ) \\left ( \\prod _ { i = 1 } ^ n ( 1 + ( - 1 ) ^ { m _ i } 2 \\delta _ { 2 i } ) \\right ) \\left ( 1 / 2 + \\delta _ { 2 n + 2 } + 1 / 2 - \\delta _ { 2 n + 2 } \\right ) \\\\ & = 2 ^ { n + 1 } ( 1 + ( - 1 ) ^ { m _ Q } 2 \\delta _ 1 ) \\prod _ { i = 1 } ^ n ( 1 + ( - 1 ) ^ { m _ i } 2 \\delta _ { 2 i } ) . \\end{align*}"}
-{"id": "7684.png", "formula": "\\begin{align*} z _ 1 ( t ) = \\left [ ~ Q ~ | ~ 0 ~ \\right ] x ( t ) = Q x _ 1 ( t ) \\ , , \\end{align*}"}
-{"id": "6516.png", "formula": "\\begin{align*} \\mathcal { S } _ { } \\left ( t \\right ) = h ^ { } t . \\end{align*}"}
-{"id": "7439.png", "formula": "\\begin{align*} \\Psi _ 2 : = \\begin{pmatrix} - x - t v & y & - z & t \\\\ - u y - 2 v z & - x + t v & t u & z \\\\ w z & - t w & - x - t v & y \\\\ - t u w & - w z & - u y - 2 v z & - x + t v \\end{pmatrix} , \\end{align*}"}
-{"id": "3364.png", "formula": "\\begin{align*} \\mu ( p _ 0 ) = \\bigcap _ { u \\in T _ p } \\mu ( u ) \\Rightarrow p \\in \\mu ( p _ 0 ) \\Rightarrow p _ 0 \\in T _ p . \\end{align*}"}
-{"id": "5509.png", "formula": "\\begin{align*} X \\stackrel { \\mathcal { D } } { = } \\sum _ { i \\geq 1 } X _ i T _ i , \\end{align*}"}
-{"id": "78.png", "formula": "\\begin{align*} m ( T ( \\sigma ~ 0 1 ~ 1 0 ~ w ~ 0 0 ~ \\tau ) ) & = m ( T ( \\sigma ~ 0 1 ~ \\underline { 1 0 } ~ \\underline { 0 1 } ~ w ' ~ 0 0 ~ \\tau ) ) \\\\ & = m ( T ( \\sigma ~ 0 1 ~ 0 1 ~ 1 0 ~ w ' ~ 0 0 ~ \\tau ) ) \\end{align*}"}
-{"id": "9910.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r \\alpha _ i ^ 2 + \\alpha _ 1 \\alpha _ 2 = \\sum _ { i = 1 } ^ r a _ i ^ 2 + a _ 1 a _ 2 - \\sum _ { i = 1 } ^ r \\frac { 6 a _ i } { 3 r - 2 } + \\frac { 4 \\cdot 3 + 9 ( r - 2 ) } { ( 3 r - 2 ) ^ 2 } = \\sum _ { i = 1 } ^ r a _ i ^ 2 + a _ 1 a _ 2 - \\frac { 3 } { 3 r - 2 } \\ , . \\end{align*}"}
-{"id": "5978.png", "formula": "\\begin{align*} \\delta = \\varepsilon \\end{align*}"}
-{"id": "8599.png", "formula": "\\begin{align*} \\langle \\Phi , \\Psi \\rangle ( \\delta ) : = \\sum _ { \\xi \\in \\Gamma g ^ { - 1 } \\Gamma } \\xi ^ { - 1 } ( \\Phi ( \\xi ) ^ { * } \\Psi ( \\xi \\delta ) ) , \\end{align*}"}
-{"id": "7328.png", "formula": "\\begin{align*} U = U _ { x + } U _ { y + } U _ { x - } U _ { y - } , \\end{align*}"}
-{"id": "3231.png", "formula": "\\begin{align*} \\sum _ { r = - \\infty } ^ { + \\infty } N ^ r H ^ { 2 r + k } . \\end{align*}"}
-{"id": "5392.png", "formula": "\\begin{align*} ( a _ 1 , a _ 5 ) = ( a _ 4 , a _ 5 ) = \\frac { 1 } { 2 ^ 6 } . \\end{align*}"}
-{"id": "9428.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } n \\ , ( \\log n ) ^ { k + 1 } | n | _ { p _ 1 } \\cdots | n | _ { p _ k } \\| n \\alpha \\| = 0 \\ \\end{align*}"}
-{"id": "6.png", "formula": "\\begin{align*} \\lambda _ { - 1 } a _ { - 1 } + \\lambda _ 0 a _ 0 + \\cdots + \\lambda _ { n - 1 } a _ { n - 1 } = 0 . \\end{align*}"}
-{"id": "8322.png", "formula": "\\begin{align*} \\psi ( f ) = ( 2 \\pi i ) ^ { c ( 0 , 0 ) } \\Psi ( 2 f ) \\end{align*}"}
-{"id": "5415.png", "formula": "\\begin{align*} \\mathring { V } ^ H : = V ^ H - V ^ { \\partial H } . \\end{align*}"}
-{"id": "1955.png", "formula": "\\begin{align*} T _ { j , 1 } ( r _ { 1 } , \\dots , r _ { j - 1 } ) ( f ) : & = \\widetilde { S } _ { j , \\mathbf { r } } ( f ) , \\\\ T _ { j , k } ( r _ { k } , \\dots , r _ { j - 1 } ) ( f ) : & = ( i \\pi d _ { k - 1 } + P _ { k - 1 } ) T _ { j , k - 1 } ( r _ { k - 1 } , \\dots , r _ { j - 1 } ) ( f ) \\\\ & = \\prod _ { i = 1 } ^ { k - 1 } ( i \\pi d _ { i } + P _ { i } ) \\widetilde { S } _ { j , \\mathbf { r } } ( f ) , \\end{align*}"}
-{"id": "400.png", "formula": "\\begin{align*} S _ n & = \\sum _ { ( j , k ) \\in \\Gamma _ n } \\sum _ { r , s \\in \\mathbb Z } a _ { r , s } \\xi _ { j - r , k - s } \\\\ & = \\sum _ { ( j , k ) \\in \\Gamma _ n } \\sum _ { r , s \\in \\mathbb Z } a _ { j + r , k + s } \\xi _ { - r , - s } \\\\ & = \\sum _ { ( j , k ) \\in \\Gamma _ n } \\lim _ { m \\to \\infty } \\sum _ { r , s \\in [ - m , m ] } a _ { j + r , k + s } \\xi _ { - r , - s } \\\\ & = \\lim _ { m \\to \\infty } \\sum _ { ( j , k ) \\in \\Gamma _ n } \\sum _ { r , s \\in [ - m , m ] } a _ { j + r , k + s } \\xi _ { - r , - s } , \\end{align*}"}
-{"id": "4188.png", "formula": "\\begin{gather*} \\frac { 1 - I ( \\tau ) } { 2 7 } = \\frac { 6 4 } { j ( \\tau ) } = 2 ^ { 1 2 } 3 ^ 3 ( q - 7 4 4 q ^ 2 + \\dots ) \\quad \\textnormal { w h e r e , } \\\\ j ( \\tau ) = \\frac { 1 7 2 8 g _ 2 ^ 3 } { \\Delta } \\ ; \\textnormal { i s t h e K l e i n m o d u l a r f u n c t i o n } . \\end{gather*}"}
-{"id": "7256.png", "formula": "\\begin{align*} 0 \\le \\sum _ { n \\le X } \\Bigl ( 1 - \\frac { \\Phi ( n ) } n \\Bigr ) & \\le \\sum _ { n \\le X } \\sum _ { \\substack { d \\mid n : \\\\ d > T } } \\frac 1 d = \\sum _ { T < d \\le X } \\frac 1 d \\sum _ { \\substack { n \\le X : \\\\ d \\mid n } } 1 \\\\ & \\le X \\sum _ { T < d \\le X } d ^ { - 2 } \\ll \\frac { X } { T } . \\end{align*}"}
-{"id": "3563.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { \\Lambda ( n ) } { n ^ s } = - \\frac { \\zeta ^ { ' } ( s ) } { \\zeta ( s ) } , \\end{align*}"}
-{"id": "5555.png", "formula": "\\begin{align*} \\ddot { x } + \\left ( \\alpha + \\beta p \\left ( t \\right ) \\right ) x = 0 x \\left ( 0 \\right ) = x _ { 0 } , \\dot { x } \\left ( 0 \\right ) = \\dot { x } _ { 0 } \\end{align*}"}
-{"id": "8166.png", "formula": "\\begin{align*} f _ { \\mathrm { N M L } } ( { \\bf x } ^ n ; \\mathcal { M } ) \\buildrel \\rm d e f \\over = \\frac { f ( { \\bf x } ^ n ; \\hat { \\mu } ( { \\bf x } ^ n ) , \\hat { \\Sigma } ( { \\bf x } ^ n ) ) } { \\int _ { Y } f ( { \\bf y } ^ n ; \\hat { \\mu } ( { \\bf y } ^ n ) , \\hat { \\Sigma } ( { \\bf y } ^ n ) ) { \\rm d } { \\bf y } ^ n } . \\end{align*}"}
-{"id": "9876.png", "formula": "\\begin{align*} h _ n = \\sum _ { i = 0 } ^ n ( - 1 ) ^ { n + i + 1 } \\theta _ i \\colon C ^ x _ n ( \\C [ G ] ) \\to C ^ x _ { n + 1 } ( \\C [ G ] ) \\end{align*}"}
-{"id": "8275.png", "formula": "\\begin{align*} \\bigcup _ { i \\in I } U _ i = ( - 4 \\log ( \\abs { T } _ { \\infty } ) , 4 \\log ( \\abs { T } _ { \\infty } ) ) \\end{align*}"}
-{"id": "7673.png", "formula": "\\begin{align*} V _ i ' = H \\bot \\dots \\bot H \\bot \\underbrace { Y \\bot \\dots \\bot Y } _ { m _ i } \\end{align*}"}
-{"id": "2327.png", "formula": "\\begin{align*} u ( t ) = t + x + \\frac { d } { c } . \\end{align*}"}
-{"id": "1868.png", "formula": "\\begin{align*} \\phi ^ \\ast _ { C _ b } ( \\mu ) \\geq \\sup _ { m \\in \\mathbb { R } } \\big \\{ m \\mu ( \\mathbb { R } ^ d ) - m \\big \\} = + \\infty \\end{align*}"}
-{"id": "1695.png", "formula": "\\begin{align*} \\left . \\frac { d ( \\mu \\circ ( \\tau _ \\lambda ) ^ { - 1 } ) } { d \\mu } \\right | _ { R _ \\lambda } = \\frac 1 { ( \\Phi _ \\lambda \\circ \\tau ^ { n } ) | _ { R _ \\lambda } } . \\end{align*}"}
-{"id": "5886.png", "formula": "\\begin{align*} \\big \\{ \\frac { S ^ { ( i ) } _ { \\tau ^ { N , i } , N + \\tau ^ { N , i } } } N < r _ i \\Big \\} \\supset \\big ( \\cap _ { n = 1 } ^ { 2 M } \\big \\{ Z _ n ^ { N , i } \\in \\{ 0 , - 1 \\} \\big \\} \\big ) \\cap \\big ( \\cup _ { n = 1 } ^ M \\{ Z _ { 2 n } ^ { N , i } = - 1 \\} \\big ) . \\end{align*}"}
-{"id": "6043.png", "formula": "\\begin{align*} \\eqref { e q : - 2 } & \\geq D ( \\widetilde { Q } _ { X Y | U } \\| \\pi _ { X Y } | \\widetilde { Q } _ { U } ) \\\\ & = I ( \\widetilde { Q } _ { X Y | U } , \\widetilde { Q } _ { X Y } ) \\\\ & \\geq R ^ { * } . \\end{align*}"}
-{"id": "5821.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } \\left ( \\frac { u ^ p } { v ^ { p - 1 } } - u \\right ) \\ ; d \\mu = \\int _ { \\R ^ n } \\frac { u ^ { p } - u v ^ { p - 1 } } { v ^ { p - 1 } } \\ ; d \\mu \\geq 0 . \\end{align*}"}
-{"id": "3011.png", "formula": "\\begin{align*} W = \\begin{bmatrix} W _ 1 & W _ 2 \\end{bmatrix} & = \\frac { 1 } { 2 } ( W _ 1 + W _ 2 ) \\begin{bmatrix} I + V & I - V \\end{bmatrix} . \\end{align*}"}
-{"id": "8181.png", "formula": "\\begin{align*} v ^ { \\circ } _ { \\mathfrak { L } } = \\pi ( \\eta _ { \\mathfrak { L } } ) v ^ { \\circ } \\end{align*}"}
-{"id": "9959.png", "formula": "\\begin{align*} \\big ( V ( { \\cal C } ) \\cup { \\cal N } _ s ( u ) \\big ) - V _ { 1 s } = \\big ( V ( { \\cal C } ) \\cup { \\cal N } _ s ( \\varphi ( u ) ) \\big ) - V _ { 2 s } \\end{align*}"}
-{"id": "8344.png", "formula": "\\begin{align*} H _ { \\Z _ p } = H _ { \\Z _ p } ^ + \\oplus H _ { \\Z _ p } ^ - \\end{align*}"}
-{"id": "4836.png", "formula": "\\begin{align*} a = F _ { x x } , \\ ; b = F _ { y y } , \\ ; c = F _ { z z } , \\ ; f = F _ { y z } , \\ ; g = F _ { x z } , \\ ; h = F _ { x y } , \\end{align*}"}
-{"id": "1715.png", "formula": "\\begin{align*} \\pi ( f ) = \\int _ { \\Lambda ^ \\infty } f ( x ) d P ( x ) , \\end{align*}"}
-{"id": "4684.png", "formula": "\\begin{align*} { \\cal H } _ r ^ { ( d ( n - 1 ) ) } = p _ { { \\bf r } _ i ^ { ( F ) } } p _ { { \\bf r } _ i ^ { ( F ) } } = { \\cal H } _ { \\rm r a d } + \\Omega , \\end{align*}"}
-{"id": "6193.png", "formula": "\\begin{align*} \\widehat { \\omega } _ i ( t _ { \\alpha _ m } ) \\xrightarrow { C ^ k } _ { m \\to \\infty } \\frac { v _ \\infty } { 2 \\pi } c _ i \\dd y \\wedge \\dd x _ i + \\frac { 1 } { 2 } \\epsilon _ { i j k } \\dd x _ j \\wedge \\dd x _ k \\ ; , \\ ; \\ ; i = 1 , 2 , 3 . \\end{align*}"}
-{"id": "2339.png", "formula": "\\begin{align*} \\| \\chi ( p ^ m ) \\| \\ll _ p \\max _ { j = 1 } ^ r \\max _ { \\ell = 0 } ^ { k _ j - 1 } \\left \\{ | \\lambda _ j | ^ { m - \\ell } { m \\choose \\ell } \\right \\} . \\end{align*}"}
-{"id": "805.png", "formula": "\\begin{align*} T _ { k } ( m ) = S _ { k } ( 2 m ) - S _ { k } ( m ) \\end{align*}"}
-{"id": "1174.png", "formula": "\\begin{align*} \\mathcal { H } ^ { k } ( E ) = \\lim _ { \\delta \\to 0 } \\inf \\left \\{ \\sum _ { j } r _ j ^ { k } ; \\ , \\ , E \\subset \\cup B ( x _ j , r _ j ) , \\ , \\ , r _ j \\leq \\delta \\right \\} \\end{align*}"}
-{"id": "4458.png", "formula": "\\begin{align*} \\langle \\big ( \\int _ 0 ^ { \\ell _ 0 } \\| \\frac { \\partial } { \\partial \\ell } \\xi _ { \\ell , T } \\| d \\ell \\big ) ^ p \\rangle ^ \\frac { 1 } { p } & \\le \\int _ 0 ^ { \\ell _ 0 } \\langle \\| \\frac { \\partial } { \\partial \\ell } \\xi _ { \\ell , T } \\| ^ p \\rangle ^ \\frac { 1 } { p } d \\ell = \\int _ 0 ^ { \\ell _ 0 } \\langle \\| \\ell \\frac { \\partial } { \\partial \\ell } \\xi _ { \\ell , T } \\| ^ p \\rangle ^ \\frac { 1 } { p } \\frac { d \\ell } { \\ell } \\end{align*}"}
-{"id": "7415.png", "formula": "\\begin{align*} \\mathbb { H } _ 3 : = \\mathbb { H } ^ { W _ { C } } _ { \\Gamma } \\cong \\mathbb { C } [ t , T _ 0 ^ \\beta , T _ 1 ^ \\beta , T _ 0 ^ \\gamma , T _ 1 ^ \\gamma , T _ 0 ^ { \\delta } ] . \\end{align*}"}
-{"id": "4278.png", "formula": "\\begin{align*} C _ { k } ( \\tau ) = - \\frac { B _ k } { k \\cdot k ! } + \\frac { 2 } { k ! } \\sum _ { n \\geq 1 } \\sum _ { d | n } d ^ { k - 1 } q ^ n \\end{align*}"}
-{"id": "6600.png", "formula": "\\begin{align*} \\lim _ { J \\ni m \\to \\infty } \\left \\| w _ { i j } ^ { ( m , n ) } - u _ { i j } ^ { ( n ) } \\right \\| _ { L ^ p } = 0 \\ . \\end{align*}"}
-{"id": "7475.png", "formula": "\\begin{align*} L ^ { \\ : \\gamma } _ { \\alpha \\beta } = \\dot { \\partial } _ \\alpha N ^ \\gamma _ \\beta . \\end{align*}"}
-{"id": "2088.png", "formula": "\\begin{align*} \\sup _ { M \\times [ 0 , T _ 1 ] } \\sum _ a | \\nabla _ H u ^ a _ k | = z _ k ( T _ 1 ) \\leq 2 K e ^ { C _ 1 } | | d \\phi | | _ { C ^ 0 } . \\end{align*}"}
-{"id": "4102.png", "formula": "\\begin{align*} - X = ( X g ) \\eta + D _ X V = ( X g ) \\eta + \\nabla _ X V + h ( X , V ) \\eta . \\end{align*}"}
-{"id": "8648.png", "formula": "\\begin{align*} A _ k = \\{ | x + \\omega _ { X _ 0 } ( T _ k ) - y - \\omega _ { Y _ 0 } ( T _ k ) | \\leq k ^ \\alpha \\} , ~ ~ ~ A ( N ) = \\bigcup _ { k \\geq N } A _ k , \\end{align*}"}
-{"id": "2711.png", "formula": "\\begin{align*} f = \\mathcal M + \\epsilon M h \\ , , \\end{align*}"}
-{"id": "6344.png", "formula": "\\begin{align*} \\frac { V _ i ( z ) } { \\log z } \\geq 2 + \\varepsilon , \\ i = 1 , 2 \\end{align*}"}
-{"id": "8663.png", "formula": "\\begin{align*} \\zeta _ { r m _ 1 k } = c _ { 1 , r } m _ 1 k + c _ { 2 , r } + o ( 1 ) \\end{align*}"}
-{"id": "1892.png", "formula": "\\begin{align*} S = \\{ ( 0 , 0 ) , ( 0 , 2 ) , ( 2 , 0 ) , ( 1 , 1 ) \\} \\quad \\pi _ { ( 0 , 0 ) } = 0 \\pi _ { ( 0 , 2 ) } = \\pi _ { ( 2 , 0 ) } = \\pi _ { ( 1 , 1 ) } = 0 . 1 . \\end{align*}"}
-{"id": "8265.png", "formula": "\\begin{align*} \\frac { - 1 } { 2 \\pi i } \\int _ { ( 5 ) } L ^ s \\tilde { w } ( s ) \\frac { L ' ( s - i r _ 1 - i r _ 2 , \\xi \\chi _ 1 \\chi _ 2 ^ { - 1 } ) } { L ( s - i r _ 1 - i r _ 2 , \\xi \\chi _ 1 \\chi _ 2 ^ { - 1 } ) } d s = \\frac { - 1 } { 2 \\pi i } \\int _ { \\mathcal { C } _ 2 } L ^ s \\tilde { w } ( s ) \\frac { L ' ( s - i r _ 1 - i r _ 2 , \\xi \\chi _ 1 \\chi _ 2 ^ { - 1 } ) } { L ( s - i r _ 1 - i r _ 2 , \\xi \\chi _ 1 \\chi _ 2 ^ { - 1 } ) } d s . \\end{align*}"}
-{"id": "9204.png", "formula": "\\begin{align*} [ s \\otimes c , s ' \\otimes c ' ] = s \\circ s ' \\otimes \\frac { [ c , c ' ] _ { A ^ { - } } } { 2 } + [ s , s ' ] \\otimes \\frac { ( c \\circ c ' ) _ { A ^ { + } } } { 2 } + ( s \\mid s ' ) \\langle c , c ' \\rangle . \\end{align*}"}
-{"id": "6374.png", "formula": "\\begin{align*} \\mu _ { M P , z ^ * _ 1 , z ^ * _ 2 } ( \\{ 0 \\} ) = \\frac { z _ 2 ^ * - z _ 1 ^ * - \\lvert z _ 1 ^ * - z _ 2 ^ * \\rvert } { 2 z _ 2 ^ * } = \\begin{cases} 0 , & \\quad z _ 2 ^ * \\geq z _ 1 ^ * , \\\\ 1 - \\frac { z _ 1 ^ * } { z _ 2 ^ * } , & z _ 2 ^ * < z _ 1 ^ * . \\end{cases} \\end{align*}"}
-{"id": "8612.png", "formula": "\\begin{align*} \\nu _ 0 ^ 2 = \\int _ { \\R ^ { d + 1 } } R ( s , y ) d s d y . \\end{align*}"}
-{"id": "926.png", "formula": "\\begin{align*} \\Psi ' ( t ) = \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ d E \\left [ \\frac { \\partial ^ 2 \\varphi } { \\partial x _ i \\partial x _ j } ( \\sqrt { 1 - t } F + \\sqrt { t } Z ) ( \\mathfrak { C } ( i , j ) - \\langle D F _ i , - D L ^ { - 1 } F _ j \\rangle _ H ) \\right ] \\end{align*}"}
-{"id": "7719.png", "formula": "\\begin{align*} d _ B ( j , k ) = \\sqrt { 2 } , \\ \\ \\ \\ \\ \\ j , k = 1 , 2 , \\cdots , N - 1 ; j \\neq k \\ , . \\end{align*}"}
-{"id": "8154.png", "formula": "\\begin{align*} a _ 1 a _ 5 ^ q = \\delta \\end{align*}"}
-{"id": "3113.png", "formula": "\\begin{align*} W _ { i j } '' & = \\sum _ { k = 1 } ^ N \\phi _ { i k j } ( V ' _ { i k } \\otimes H _ { k j } ) \\\\ V _ { i j } '' & = \\left \\{ v \\in V _ { i j } \\mid \\phi _ { i j k } ( v \\otimes h ) \\in W _ { i k } '' \\mathrm { ~ f o r ~ a l l ~ } k \\mathrm { ~ a n d ~ f o r ~ a l l ~ } h \\in H _ { j k } \\right \\} . \\end{align*}"}
-{"id": "1143.png", "formula": "\\begin{align*} \\rho _ x & = x + x _ 1 \\tau + \\tau ^ 2 , \\\\ \\rho _ y & = y + y _ 1 \\tau + y _ 2 \\tau ^ 2 + \\tau ^ 3 \\end{align*}"}
-{"id": "6191.png", "formula": "\\begin{align*} G _ t : \\mathbb { T } ^ 4 & \\longrightarrow \\mathbb { T } ^ 4 \\\\ ( \\theta , x _ 1 , x _ 2 , x _ 3 ) & \\mapsto ( G _ t ( \\theta ) , x _ 1 , x _ 2 , x _ 3 ) \\end{align*}"}
-{"id": "4271.png", "formula": "\\begin{align*} \\Big ( x ( x - 1 ) ( x - \\lambda ) \\Big ) ^ { m } = \\gamma _ m x ^ m + \\gamma _ m x ^ { m + 1 } + \\cdots \\gamma _ { 3 m } x ^ { 3 m } , \\end{align*}"}
-{"id": "6107.png", "formula": "\\begin{align*} M _ { s } \\left ( f \\right ) = M _ { \\psi } \\left ( \\Phi _ { \\psi } f \\right ) , \\end{align*}"}
-{"id": "3661.png", "formula": "\\begin{align*} \\tilde P ^ { ( n ) } = \\sigma ^ { ( n ) \\vee } + \\iota P ^ { ( n ) } . \\end{align*}"}
-{"id": "5893.png", "formula": "\\begin{align*} P ( \\cup _ { n = 1 } ^ { [ \\frac 1 { b _ N } ] } \\{ Z _ { 2 n } ^ { N , i } = - 1 \\} ) \\ge 1 - e ^ { - a _ N [ \\frac 1 { b _ N } ] } \\ge \\frac { a _ N } { 2 b _ N } , \\ \\ N . \\end{align*}"}
-{"id": "6422.png", "formula": "\\begin{align*} d l _ { \\rightarrow } = \\Delta \\theta \\end{align*}"}
-{"id": "4089.png", "formula": "\\begin{align*} H _ e = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } k _ n ( \\theta ) \\ d \\theta , \\end{align*}"}
-{"id": "4142.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ n \\alpha _ t Z _ { I _ t , J } ^ t \\leq R ( \\alpha , ( I _ t ) , J ) , \\end{align*}"}
-{"id": "1838.png", "formula": "\\begin{align*} F _ \\bullet = \\cdots \\to F _ 1 \\to F _ 0 \\to F _ { - 1 } \\to \\cdots \\end{align*}"}
-{"id": "529.png", "formula": "\\begin{align*} X _ { H _ t } = - J _ 0 \\ 1 \\nabla H _ t , \\end{align*}"}
-{"id": "902.png", "formula": "\\begin{align*} X _ t = X _ 0 + \\int _ 0 ^ t \\sigma ( s ) d B _ s , t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "6063.png", "formula": "\\begin{align*} \\Phi ^ { '' } ( t ) + ( n - 2 ) \\Phi ' ( t ) - 2 \\mathfrak { e } \\varphi ( \\Phi ( t ) ) \\varphi ' ( \\Phi ( t ) ) = 0 \\end{align*}"}
-{"id": "3280.png", "formula": "\\begin{align*} F _ { t ^ c } ( t _ 0 ) = \\frac { 2 } { \\pi - \\theta _ k } \\arccos \\left ( \\frac { r _ u ^ { } } { v _ u t _ 0 } \\right ) . \\end{align*}"}
-{"id": "3131.png", "formula": "\\begin{align*} e ( T ' ) & = ( q + 1 ) ( t - 2 ) ( m + 1 ) [ ( 3 ( k - 1 ) - 6 ] + ( 2 - q ) ( t - 2 ) m [ 3 ( k - 1 ) - 6 ] \\\\ & = ( t - 2 ) ( 3 k - 9 ) [ ( q + 1 + 2 - q ) m + q + 1 ] \\\\ & = ( t - 2 ) ( 3 k - 9 ) ( 3 m + q + 1 ) \\\\ & = ( t - 2 ) ( 3 k - 9 ) ( k + 1 ) = 3 ( k ^ 2 - 2 k - 3 ) ( t - 2 ) . \\end{align*}"}
-{"id": "5113.png", "formula": "\\begin{align*} C _ 1 \\max _ { j = 1 , \\ldots , N } | p _ j ( \\mathcal A ) | ^ { \\frac { 1 } { d _ j } } \\leq \\inf _ { M \\in G } | | \\rho _ M \\mathcal A | | \\leq C _ 2 \\max _ { j = 1 , \\ldots , N } | p _ j ( \\mathcal A ) | ^ { \\frac { 1 } { d _ j } } \\mbox { f o r a l l } \\mathcal A \\in V . \\end{align*}"}
-{"id": "5940.png", "formula": "\\begin{align*} e _ i & = v _ { 2 i - 1 } + \\sum _ { j = 1 } ^ { i - 1 } \\left ( \\prod _ { \\ell = 1 } ^ j \\frac { \\alpha _ { 2 i - 2 \\ell } } { \\alpha _ { 2 i - 2 \\ell - 1 } } \\right ) v _ { 2 i - 2 j - 1 } & \\mbox { f o r } \\ ; 2 \\leq i \\leq m , \\end{align*}"}
-{"id": "5728.png", "formula": "\\begin{align*} \\mathcal { K } ( { v } ) ^ 2 = \\mathfrak { E } _ { W } ( { v } ) - \\mathfrak { E } _ { W } ( z ) = \\frac 1 2 \\int _ \\R | { v } ' ( s ) - z ' ( s ) | ^ 2 \\d s + \\frac 1 2 \\int _ \\R D ^ 2 W ( \\xi ( s ) ) ( { v } ( s ) - z ( s ) , { v } ( s ) - z ( s ) ) \\d s . \\end{align*}"}
-{"id": "5726.png", "formula": "\\begin{align*} \\| z ( \\cdot - m _ 1 ) - z ( \\cdot - m _ 2 ) \\| _ { L ^ 2 } \\leq \\| z ( \\cdot - m _ 1 ) - { v } \\| _ { L ^ 2 } + \\| { v } - z ( \\cdot - m _ 2 ) \\| _ { L ^ 2 } = \\sqrt { F ( m _ 1 ) } + \\sqrt { F ( m _ 2 ) } \\leq 2 \\sqrt { \\delta } . \\end{align*}"}
-{"id": "5384.png", "formula": "\\begin{align*} K : = \\langle a , b , ( a b c ) ^ 3 \\rangle \\leq G \\end{align*}"}
-{"id": "7579.png", "formula": "\\begin{align*} f ^ k g ^ k - f g = ( f ^ k - f ) ( g ^ k - g ) + ( f ^ k - f ) g + f ( g ^ k - g ) \\end{align*}"}
-{"id": "99.png", "formula": "\\begin{align*} C _ { W } = \\left \\lceil \\frac { K } { M + 1 } \\right \\rceil ^ { - 1 } . \\end{align*}"}
-{"id": "8311.png", "formula": "\\begin{align*} \\mathrm { c u s p } ^ { ( a a _ \\tau ) } = \\tau ( \\mathrm { c u s p } ^ { ( a ) } ) \\end{align*}"}
-{"id": "5399.png", "formula": "\\begin{align*} ( a c ) ^ 3 & = ( 7 , 9 ) \\\\ ( b c ) ^ 3 & = ( 1 , 3 ) \\\\ ( a b c ) ^ 3 & = ( 4 , 6 ) . \\end{align*}"}
-{"id": "433.png", "formula": "\\begin{align*} I _ \\nu ( s ) = \\frac { 1 } { 2 \\pi } \\int _ { - \\pi } ^ \\pi e ^ { s \\cos ( \\xi ) - i \\nu \\xi } \\ , \\dd \\xi , \\end{align*}"}
-{"id": "241.png", "formula": "\\begin{align*} \\begin{aligned} & \\left \\Vert \\mathbb { E } \\left [ \\chi _ a \\{ h ( A _ G ) - h ( A _ { G ' } ) \\} \\chi _ b \\right ] \\right \\Vert \\leq \\frac { 1 } { 2 \\pi } \\int _ { \\R ^ 2 } \\mathrm { d } x \\mathrm { d } y \\ , \\vert \\omega _ { h , n } ( x , y ) \\vert \\big \\Vert \\mathbb { E } \\big [ T _ { x + i y } ^ { a , b } ( G , G ' ) \\big ] \\big \\Vert . \\end{aligned} \\end{align*}"}
-{"id": "6646.png", "formula": "\\begin{align*} [ c _ 1 ^ - , c _ 1 ^ + ] = \\left [ f ( c _ 0 ^ { - } ) + \\tfrac { 1 } { 3 } ( f ( c _ 0 ^ { + } ) - f ( c _ 0 ^ { - } ) ) , f ( c _ 0 ^ { + } ) - \\tfrac { 1 } { 3 } ( f ( c _ 0 ^ { + } ) - f ( c _ 0 ^ { - } ) ) \\right ] \\end{align*}"}
-{"id": "7127.png", "formula": "\\begin{align*} ( \\mathcal { W } _ X ^ { - 1 } ) _ { i } ^ j = & ~ \\left ( \\bar { g } ^ { j q } + \\frac { \\langle \\bar { g } ^ { j a } \\bar { \\nabla } _ a s , \\bar { g } ^ { q b } \\bar { \\nabla } _ b s \\rangle } { 1 - s ^ 2 - | \\bar { \\nabla } s | ^ 2 } \\right ) \\tau _ { q i } \\sqrt { \\frac { 1 - s ^ 2 } { 1 - s ^ 2 - | \\bar { \\nabla } s | ^ 2 } } \\end{align*}"}
-{"id": "5852.png", "formula": "\\begin{align*} & D ^ N ( g ^ { N + 1 } f ) + ( - 1 ) ^ { N + 1 } g ^ { N + 1 } D ^ N ( f ) \\\\ & + \\sum _ { k = 1 } ^ N ( - 1 ) ^ k \\left [ \\binom { N } { k } + \\binom { N } { k - 1 } \\right ] g ^ k D ^ N ( g ^ { N + 1 - k } f ) = 0 , \\end{align*}"}
-{"id": "5114.png", "formula": "\\begin{align*} { \\mathcal A } _ p ( ( X _ \\lambda ) _ { \\lambda \\in \\Lambda _ { d , n } } ) : = & \\det ( X _ { ( 1 , 1 ) } f ( p ) \\wedge \\cdots \\wedge \\\\ & X _ { ( j , 1 ) } \\cdots X _ { ( j , \\kappa _ j ) } f ( p ) \\wedge \\cdots \\wedge X _ { ( n , 1 ) } \\cdots X _ { ( n , \\kappa _ n ) } f ( p ) ) . \\end{align*}"}
-{"id": "706.png", "formula": "\\begin{align*} p = a ^ 2 ( 1 + \\alpha ) \\rho T . \\end{align*}"}
-{"id": "2693.png", "formula": "\\begin{align*} \\begin{cases} a _ 1 = 2 p ^ n { a ' _ 1 } \\\\ a _ 3 = 2 p ^ n { a ' _ 3 } \\end{cases} \\end{align*}"}
-{"id": "8433.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { t , l , v } ) = \\begin{cases} - q ^ { - t - 1 } & \\\\ q ^ { - t - 2 l } & \\\\ q ^ { - t - 2 l } \\psi ( - \\varpi ^ { l + t } v ^ { - 1 } ) & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "9986.png", "formula": "\\begin{align*} \\partial _ t u ( t , x ) = \\frac { 1 } { 2 } \\Delta u ( t , x ) + u ( t , x ) \\eta ( t , x ) \\end{align*}"}
-{"id": "2734.png", "formula": "\\begin{align*} \\langle \\mathcal L ( h ) , \\ , g \\rangle _ { L ^ 2 _ { x , v } } = \\langle \\mathcal L ( h ) , \\ , g - \\Pi _ { \\mathcal L } ( g ) \\rangle _ { L ^ 2 _ { x , v } } = \\langle \\mathcal L ( h ^ { \\perp } ) , \\ , g ^ { \\perp } \\rangle _ { L ^ 2 _ { x , v } } \\leq C ^ { \\mathcal L } | | h ^ { \\perp } | | _ { \\Lambda } \\ , | | g ^ { \\perp } | | _ { \\Lambda } \\ , , \\end{align*}"}
-{"id": "7616.png", "formula": "\\begin{align*} \\bigcup _ { z \\in E } Q _ { r _ z } ^ \\lambda ( z ) \\subset \\bigcup _ { i = 1 } ^ \\infty Q _ { 5 r _ i } ^ \\lambda ( z _ i ) . \\end{align*}"}
-{"id": "4223.png", "formula": "\\begin{align*} A = \\bigoplus _ { n = - \\infty } ^ { \\infty } A ^ n \\end{align*}"}
-{"id": "5950.png", "formula": "\\begin{align*} f ^ { \\delta } : = f + \\delta \\nu , \\nu ^ { \\delta } : = \\nu . \\end{align*}"}
-{"id": "8517.png", "formula": "\\begin{align*} \\bar \\sigma ( 1 , v , u , C ) = \\min \\{ f ( \\bar v , \\bar u , z ) : z \\in \\mathbb { Z } , \\ , | z | \\leq C , \\ , \\bar v _ i = v _ i + b _ { i \\ , 1 } z , \\ , \\bar u _ i = u _ i + \\bar b _ { i \\ , 1 } z \\} , \\end{align*}"}
-{"id": "4709.png", "formula": "\\begin{align*} A _ s ^ i = \\begin{pmatrix} \\lambda & \\mu & \\nu \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} \\mbox { f o r s o m e } \\lambda , \\mu , \\nu \\in \\R \\mbox { d e p e n d i n g o n } s _ 1 , s _ 2 . \\end{align*}"}
-{"id": "839.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } d Y _ t = b ( D _ t , Y _ t ) d t + \\sigma ( D _ t , Y _ t ) d B _ t , \\\\ Y _ 0 = x _ 0 , \\ , D _ 0 = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "4446.png", "formula": "\\begin{align*} \\| \\lceil u , \\phi _ \\ell * \\rceil f \\| = \\| u f _ \\ell - ( u f ) _ \\ell \\| & \\lesssim [ u ] _ \\alpha [ f ] _ \\beta \\int _ 0 ^ 1 \\big ( ( t ^ \\frac { 1 } { 3 } \\ell ) ^ { \\alpha + \\beta } + \\ell ^ \\alpha ( t ^ \\frac { 1 } { 3 } \\ell ) ^ \\beta \\big ) d t , \\forall \\ell \\in ( 0 , 1 ] . \\end{align*}"}
-{"id": "3869.png", "formula": "\\begin{align*} \\nabla f ( x ^ * ) d _ x & = - \\left ( \\sum \\limits _ { i = 1 } ^ m \\lambda _ i ^ * \\nabla g _ i ( x ^ * ) + \\sum \\limits _ { i = 1 } ^ p \\mu _ i ^ * \\nabla h _ i ( x ^ * ) + \\sum \\limits _ { i = 1 } ^ n \\gamma _ i ^ * e _ i ^ T \\right ) d _ x \\\\ & = - \\nabla \\ell ( x ^ * , \\lambda ^ * , \\mu ^ * , \\gamma ^ * ) ) d _ x = 0 . \\end{align*}"}
-{"id": "7564.png", "formula": "\\begin{gather*} v = \\left ( - 2 \\frac { \\theta ^ 1 _ y } { \\theta ^ 1 } x + \\frac { \\theta ^ 0 } { \\theta ^ 1 } \\right ) _ y , v = - 2 \\frac { \\theta ^ 1 _ y } { \\theta ^ 1 } \\quad \\mbox { w i t h } \\theta ^ i = \\theta ^ i ( t , y ) \\colon \\ \\theta ^ i _ t = \\theta ^ i _ { y y } , i = 0 , 1 . \\end{gather*}"}
-{"id": "7810.png", "formula": "\\begin{align*} \\Delta v _ m - ( \\frac { 2 m } { r } + 2 \\rho ^ { \\prime } ) \\frac { \\partial v _ m } { \\partial r } + ( \\frac { m ( m + 1 ) } { r ^ 2 } + \\frac { m } { r } ( 2 \\rho ^ { \\prime } - \\Delta r ) - V _ 0 - V _ 1 - V _ 2 + \\lambda ) v _ m = 0 . \\end{align*}"}
-{"id": "2240.png", "formula": "\\begin{align*} J _ \\gamma ( t ) = \\sum _ { \\| K \\| \\geqslant 0 } ( - t ) ^ { \\| K \\| + n } \\sum _ J \\frac { ( - 1 ) ^ { s ( J ) } } { \\beta ( K , J ) ! } \\cdot \\frac { \\partial ^ { | | \\beta ( K , J ) | | } } { \\partial w ^ \\beta ( K , J ) } \\left [ \\widetilde \\Delta \\cdot w _ 1 ^ { \\gamma _ 1 + 1 } \\cdots w _ n ^ { \\gamma _ n + 1 } \\cdot \\frac { \\widetilde Q ^ K } { \\widetilde q ^ { K + I } ( J ) } \\right ] _ { w = a _ J } . \\end{align*}"}
-{"id": "3146.png", "formula": "\\begin{align*} I _ { 1 } ( r ) = \\frac { r } { 2 } \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( \\frac { 1 } { 4 } r ^ { 2 } \\right ) ^ { k } } { k ! ( k + 1 ) ! } , r \\in \\mathbb { R } . \\end{align*}"}
-{"id": "6253.png", "formula": "\\begin{align*} \\begin{aligned} \\omega + u ( \\omega , \\mu , \\chi ) & = \\Omega \\bigl ( \\mu + w ( \\omega , \\mu , \\chi ) \\bigr ) + \\Delta \\bigl ( v ( \\omega , \\mu , \\chi ) , \\ , \\mu + w ( \\omega , \\mu , \\chi ) \\bigr ) , \\\\ \\chi + W ( \\omega , \\mu , \\chi ) & = 0 \\end{aligned} \\end{align*}"}
-{"id": "6024.png", "formula": "\\begin{align*} & { \\cal B } : = \\Bigl \\{ ( T _ { W } , V _ { X | W } , V _ { Y | W } ) : \\forall ( w , x , y ) , \\\\ & \\frac { ( 1 - \\epsilon ) ^ { 2 } } { 1 + \\epsilon ' } \\leq \\frac { [ T _ { W } V _ { X | W } V _ { Y | W } ] ( w , x , y ) } { Q _ { W X Y } ( w , x , y ) } \\leq \\frac { ( 1 + \\epsilon ) ^ { 2 } } { 1 - \\epsilon ' } \\Bigr \\} . \\end{align*}"}
-{"id": "2168.png", "formula": "\\begin{align*} a _ \\infty = \\lim _ { s \\to \\infty } a ( s ) = \\lim _ { t \\to \\infty } \\frac { c _ d } { c _ * } \\int _ 0 ^ \\infty u _ * ( r , t ) \\nu _ d ( r ) r ^ { d - 1 } \\ , d r . \\end{align*}"}
-{"id": "9329.png", "formula": "\\begin{align*} j _ { n } ^ { ( 3 ) } - 3 J _ { n } ^ { ( 3 ) } = 2 j _ { n - 3 } ^ { ( 3 ) } , \\end{align*}"}
-{"id": "5526.png", "formula": "\\begin{align*} \\frac { { \\rm d } ^ 2 } { { \\rm d } x ^ 2 } H ( x ) & = \\frac { { \\rm d } ^ 2 } { { \\rm d } x ^ 2 } \\left \\{ - ( 2 x ) ^ { \\log _ 3 ( \\sqrt { 2 } + 1 ) } \\right \\} \\\\ & = - 2 ^ { \\log _ 3 ( \\sqrt { 2 } + 1 ) } \\log _ 3 ( \\sqrt { 2 } + 1 ) ( \\log _ 3 ( \\sqrt { 2 } + 1 ) - 1 ) x ^ { \\log _ 3 ( \\sqrt { 2 } + 1 ) - 2 } , \\end{align*}"}
-{"id": "2905.png", "formula": "\\begin{align*} f ^ i = f \\upharpoonright \\{ \\langle n , a \\rangle : n \\in \\omega \\wedge a < _ X i \\} . \\end{align*}"}
-{"id": "3497.png", "formula": "\\begin{align*} \\int _ { \\alpha \\gamma _ i \\alpha ^ { - 1 } } \\omega _ j \\omega _ i & = \\int _ { \\gamma _ i } \\omega _ j \\omega _ i + \\left ( \\int _ { \\alpha } \\omega _ j \\int _ { \\gamma _ i } \\omega _ i - \\int _ { \\gamma _ i } \\omega _ j \\int _ { \\alpha } \\omega _ i \\right ) \\\\ & = \\int _ { \\gamma _ i } \\omega _ j \\omega _ i + \\int _ { \\alpha } \\omega _ j \\end{align*}"}
-{"id": "5121.png", "formula": "\\begin{align*} X _ { j , i } = \\sum _ { i ' = 1 } ^ d c _ { j , i , i ' } X _ { j - 1 , i ' } \\end{align*}"}
-{"id": "988.png", "formula": "\\begin{align*} \\int _ 0 ^ { a _ 0 h ^ \\gamma } \\log N ( [ u _ j , u _ { j + 1 } ] , \\mathfrak { d } _ 0 , r ) d r \\lesssim \\int _ 0 ^ { a _ 0 h ^ \\gamma } \\log \\frac { 1 } { r ^ { 1 / \\gamma } } d r \\lesssim h ^ \\gamma \\log n , \\end{align*}"}
-{"id": "7599.png", "formula": "\\begin{align*} U _ 2 U ^ { - 1 } U _ 1 = ( U - U _ 1 ) U ^ { - 1 } ( U - U _ 2 ) = U - U _ 1 - U _ 2 + U _ 1 U ^ { - 1 } U _ 2 = U _ 1 U ^ { - 1 } U _ 2 , \\end{align*}"}
-{"id": "3489.png", "formula": "\\begin{align*} \\sum _ { g \\in G } a _ g g = \\sum _ { g \\in G \\setminus \\{ 1 \\} } a _ g ( g - 1 ) + \\sum _ { g \\in G } a _ g = \\sum _ { g \\in G \\setminus \\{ 1 \\} } a _ g ( g - 1 ) , \\end{align*}"}
-{"id": "615.png", "formula": "\\begin{align*} \\mathbf { F } & : A ^ 2 ( \\Omega ) \\to A ^ 2 ( \\Omega ) \\\\ \\mathbf { F } & ( g ) = \\mathbf { B } ( \\overline { g } ) . \\end{align*}"}
-{"id": "2063.png", "formula": "\\begin{align*} f ( p , 0 ) = \\phi ( p ) . \\end{align*}"}
-{"id": "704.png", "formula": "\\begin{align*} \\alpha ( v _ n ) & = \\left ( e ^ { 2 \\pi i r _ n ^ { ( 1 ) } / m _ 1 } u _ 1 ^ { r _ n ^ { ( 1 ) } } \\right ) \\cdots \\left ( e ^ { 2 \\pi i r _ n ^ { ( d ) } / m _ d } u _ d ^ { r _ n ^ { ( d ) } } \\right ) \\\\ & = \\left ( e ^ { 2 \\pi i / m _ 1 } u _ 1 ^ { r _ n ^ { ( 1 ) } } \\right ) \\cdots \\left ( e ^ { 2 \\pi i / m _ d } u _ d ^ { r _ n ^ { ( d ) } } \\right ) \\\\ & = e ^ { 2 \\pi i \\left ( \\frac { 1 } { m _ 1 } + \\cdots + \\frac { 1 } { m _ d } \\right ) } v _ n . \\end{align*}"}
-{"id": "3708.png", "formula": "\\begin{align*} ( \\pi \\sigma _ { 1 \\dots n } ) ^ { \\vee } = \\delta ^ { ( n ) \\ , \\vee } = \\begin{pmatrix} 1 & 0 & 0 & & & 0 & 0 \\\\ - 1 & 1 & 0 & & & 0 & 0 \\\\ 0 & - 1 & 1 & & & 0 & 0 \\\\ 0 & 0 & - 1 & \\ddots & & 0 & 0 \\\\ & & & \\ddots & \\ddots & & \\\\ 0 & 0 & 0 & & \\ddots & 1 & 0 \\\\ 0 & 0 & 0 & & & - 1 & 0 \\\\ 0 & 0 & 0 & & & 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "3306.png", "formula": "\\begin{align*} H \\left ( \\theta , \\mathbb { H } , W _ 1 , \\dots , W _ K \\right ) = H \\left ( \\theta \\right ) + H \\left ( \\mathbb { H } \\right ) + H ( W _ 1 ) + \\dots + H ( W _ K ) . \\end{align*}"}
-{"id": "6974.png", "formula": "\\begin{align*} f _ k ( D ^ 2 u ) = ( \\sum _ { 1 \\leq i _ 1 < i _ 2 < . . . < i _ k \\leq n } { \\lambda _ { i _ 1 } \\lambda _ { i _ 2 } . . . \\lambda _ { i _ k } } ) ^ { 1 / k } \\end{align*}"}
-{"id": "2834.png", "formula": "\\begin{align*} A _ v ^ { ( n + 1 ) } = A _ { w _ 1 } ^ { ( n ) } \\ldots A _ { w _ { | r ^ { - 1 } ( v ) | } } ^ { ( n ) } . \\end{align*}"}
-{"id": "1968.png", "formula": "\\begin{align*} \\lVert Q ( g _ l , & g _ k - g _ l , g _ k , \\dots , g _ k ) \\rVert _ X \\le \\\\ & = \\frac { 1 } { j ! } \\sum _ { 0 < a + b + c \\le j } N ( a , b , c ) \\lVert T ( a h ^ { - ( j - 1 ) } ( g _ l - g _ k ) + h ( b g _ l + c g _ k ) ) \\rVert _ X \\\\ & \\le C ( j ) \\left ( h ^ { - j ( j - 1 ) } \\lVert g _ l - g _ k \\rVert _ { Y } ^ j + h ^ { j } M ^ j \\right ) < { \\varepsilon } / { j } , \\end{align*}"}
-{"id": "7067.png", "formula": "\\begin{align*} \\sigma ( I d - L _ { \\lambda A } ) = \\left \\{ \\frac { \\beta _ k - \\lambda \\alpha _ j } { 1 + \\beta _ k } \\colon \\alpha _ j \\in \\sigma ( A ) , \\beta _ k \\in \\sigma ( - \\Delta ; B ^ N ) \\right \\} . \\end{align*}"}
-{"id": "3260.png", "formula": "\\begin{align*} \\omega = u ^ { n - p } \\omega _ { p } + u ^ { n - p + 1 } \\omega _ { p - 2 } + \\cdots + u ^ { n - p + [ { p \\over 2 } ] } \\omega _ { p - 2 [ { p \\over 2 } ] } . \\end{align*}"}
-{"id": "4027.png", "formula": "\\begin{align*} h ( d \\eta _ p X , d \\eta _ p Y ) = h ( \\alpha _ 1 \\lambda _ 1 V _ 1 + \\alpha _ 2 \\lambda _ 2 V _ 2 , \\beta _ 1 \\lambda _ 1 V _ 1 + \\beta _ 2 \\lambda _ 2 V _ 2 ) = \\alpha _ 1 \\beta _ 1 \\lambda _ 1 ^ 2 - \\alpha _ 2 \\beta _ 2 \\lambda _ 2 ^ 2 . \\end{align*}"}
-{"id": "7648.png", "formula": "\\begin{align*} A & = ( 1 , 1 , 1 , - , 1 , 1 , - , 1 ) , \\ \\ \\ \\ \\ \\ \\ \\ \\ B = ( 1 , 1 , 1 , - , - , - , 1 , - ) , \\\\ C & = ( 1 , i , - , 1 , - , i , \\overline { i } , - , i , i , 1 ) , \\ \\ D = ( 1 , 1 , \\overline { i } , \\overline { i } , \\overline { i } , 1 , 1 , i , - , 1 , - ) . \\end{align*}"}
-{"id": "270.png", "formula": "\\begin{align*} f [ x _ 0 , x _ 1 , \\dots , x _ k ] & = \\int _ { \\Sigma _ k } f ^ { ( k ) } \\Big ( t _ 0 x _ 0 + t _ 1 x _ 1 + \\dots t _ k x _ k \\Big ) d t \\\\ & \\ge \\frac 1 { k ! L ^ k } \\int _ { - 0 . 9 } ^ { 0 . 9 } f ^ { ( k ) } ( s ) d s \\\\ & \\ge \\frac 1 { L ^ k } . \\end{align*}"}
-{"id": "6712.png", "formula": "\\begin{align*} J _ { l } = \\big \\{ ( i _ { 1 } , \\dots , i _ { 2 l } ) \\in \\{ 1 , \\dots , N \\} ^ { 2 l } : g _ { 2 l } ( ( i _ { 1 } , \\dots , i _ { 2 l } ) ) = 1 \\big \\} , \\end{align*}"}
-{"id": "236.png", "formula": "\\begin{align*} F _ M & \\leq e ^ { - N \\log ( 2 ) } + e ^ { - \\frac { \\mu } { 2 N } | a - b | } \\sum _ { l = 1 } ^ { N } \\frac { ( B l ^ d ) ^ { l - 1 } } { 2 ^ { l + 1 } } \\\\ & \\leq e ^ { - N \\log ( 2 ) } + e ^ { - \\frac { \\mu } { 2 N } | a - b | } N B ^ N N ^ { d N } . \\end{align*}"}
-{"id": "7022.png", "formula": "\\begin{align*} z _ j = y _ j , \\ \\ j = 1 , 2 , . . . , n - 1 , \\end{align*}"}
-{"id": "7063.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l c l } - \\triangle u & = & \\lambda A u & & B ^ N \\\\ \\frac { \\partial u } { \\partial \\nu } & = & 0 & & S ^ { N - 1 } , \\end{array} \\right . \\end{align*}"}
-{"id": "8152.png", "formula": "\\begin{align*} a _ k a _ { n - k } ^ { q ^ k } = b _ k b _ { n - k } ^ { q ^ k } , \\end{align*}"}
-{"id": "3205.png", "formula": "\\begin{align*} ( \\mathcal { J } ^ { \\ast } V ) ( x ) & = \\int _ { \\lbrace z > 1 \\rbrace } | V ( y + z ) - V ( y ) | \\nu ( \\mathrm { d } z ) \\\\ & \\leqslant \\int _ { \\lbrace z > 1 \\rbrace } z ^ { \\kappa } \\nu ( \\mathrm { d } z ) + c _ { 2 } \\nu ( \\lbrace z > 1 \\rbrace ) < \\infty . \\end{align*}"}
-{"id": "681.png", "formula": "\\begin{align*} \\varphi ( \\pi , \\mu _ n ^ 0 ) \\cot \\beta + \\varphi ' ( \\pi , \\mu _ n ^ 0 ) = 0 , \\end{align*}"}
-{"id": "3614.png", "formula": "\\begin{align*} \\operatorname { C a p } _ p ( \\mathcal A ) : = \\inf \\left \\{ \\int _ { \\mathbb { R } ^ n } | \\nabla \\varphi | ^ p d x < + \\infty \\ ; : \\ ; \\varphi \\in C ^ { \\infty } _ c ( \\mathbb R ^ n ) \\ \\ \\ \\ \\varphi \\geq \\chi _ { \\mathcal A } \\right \\} , \\end{align*}"}
-{"id": "8847.png", "formula": "\\begin{align*} \\langle N _ { t \\Omega } ^ 2 \\rangle - \\langle N _ { t \\Omega } \\rangle ^ 2 = \\mathcal { O } ( t ^ { d - 1 } ) \\asymp \\mathrm { s u r f a c e } ( t \\Omega ) . \\end{align*}"}
-{"id": "5099.png", "formula": "\\begin{align*} ( g _ \\Gamma \\ast p _ \\psi ) ( \\sigma ) = \\int _ { \\widehat { \\mathcal { L } } } g _ \\Gamma ( \\sigma - \\tau ) p _ \\psi ( \\tau ) d \\tau \\leq C . \\end{align*}"}
-{"id": "6704.png", "formula": "\\begin{align*} ( \\eta ^ { j } ) _ { i } = \\begin{cases} \\eta _ { i } , & i \\neq j ; \\\\ - ( \\eta _ { i } ) , & i = j ; \\end{cases} , \\ i \\in \\{ 1 , \\dots , N \\} . \\end{align*}"}
-{"id": "1448.png", "formula": "\\begin{align*} E _ 2 & = D \\{ 1 , y _ 0 \\} \\oplus D _ 0 \\{ x _ 3 , x _ 3 y _ 0 \\} , \\\\ E _ 4 & = C \\langle 1 , y _ 1 , v _ 0 y _ 0 \\rangle \\oplus D _ 1 \\langle x _ 3 ^ 2 , x _ 3 ^ 2 y _ 1 , x _ 3 z _ 1 \\rangle . \\end{align*}"}
-{"id": "3005.png", "formula": "\\begin{align*} \\hat { W } _ B ( \\Phi ( { \\cal H } x ) ) ( \\cdot , t ) = 0 . \\end{align*}"}
-{"id": "1935.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta + q - k ^ 2 ) u = 0 \\\\ u ( x ) = e ^ { i k \\theta \\cdot x } + u _ s ( k , \\theta , x ) \\\\ \\lim _ { | x | \\to \\infty } ( \\frac { \\partial u _ s } { \\partial r } - i k u _ s ) ( x ) = o ( | x | ^ { - ( n - 1 ) / 2 } ) , \\end{cases} \\end{align*}"}
-{"id": "484.png", "formula": "\\begin{align*} \\frac { \\partial ^ { k _ 2 } } { \\partial \\abs { t } ^ { k _ 2 } } p ^ { ( m + 1 ) } _ { 1 , k _ 1 , 0 } ( x , ( t , t _ { m + 1 } ) ) = \\sum _ { h \\in \\mathfrak { I } ^ { k _ 2 } } \\frac { k _ 2 ! } { h ! } p _ { 1 , k _ 1 , \\abs { h } } ^ { ( m + 1 ) } ( x , ( t , t _ { m + 1 } ) ) \\prod _ { j = 1 } ^ { k _ 2 } \\left ( \\frac { 1 } { j ! } \\frac { \\partial ^ { j } } { \\partial \\abs { t } ^ j } \\sqrt { \\abs { t } ^ 2 + t _ { m + 1 } ^ 2 } \\right ) ^ { h _ j } , \\end{align*}"}
-{"id": "8697.png", "formula": "\\begin{align*} \\tilde { F } _ { n , k } : = \\sum _ { j = k } ^ \\infty a _ { n , j } ^ 2 F _ { n , k } : = \\sum _ { j = k } ^ \\infty b _ { n , j } ^ 2 , k = 0 , 1 , 2 , \\dots . \\end{align*}"}
-{"id": "5208.png", "formula": "\\begin{align*} V _ { 1 } ^ { [ r , 1 ] } ( x ) \\coloneqq \\sup \\limits _ { \\tau _ { 1 } \\in \\mathcal { T } } M ^ { x } _ { 1 } ( \\tau _ { 1 } , D _ { [ r , 1 ] } ) = M ^ { x } _ { 1 } ( D _ { [ \\ell ^ { 1 } , \\ell ^ { 2 } ] } , D _ { [ r , 1 ] } ) , \\forall x \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "5969.png", "formula": "\\begin{align*} f _ 2 ( x ) = \\sin \\left ( \\frac { x } { 2 } \\right ) - 2 \\cos ( x ) + 4 \\sin ( \\pi x ) , x \\in [ - 4 , 4 ] . \\end{align*}"}
-{"id": "2209.png", "formula": "\\begin{align*} Q _ i ( z ) = z _ 1 \\cdot \\ldots \\cdot z _ n \\sum _ { | \\alpha \\| \\geq 0 } C ^ i _ { \\alpha } z ^ { \\alpha } i = 1 , \\ldots , n , \\end{align*}"}
-{"id": "2609.png", "formula": "\\begin{gather*} V * \\rho = T ^ 2 \\rho ( 0 , 1 ) \\\\ ( \\mu , \\rho ) _ V = \\int _ 0 ^ 1 ( V * \\mu ) \\ , d \\rho . \\end{gather*}"}
-{"id": "7314.png", "formula": "\\begin{align*} \\psi : A _ { \\widehat { q } } \\to W _ n , \\ \\ \\ \\psi ( x _ i ) = \\left \\{ \\begin{aligned} & x _ i , & & i = 2 \\ell - 1 , \\\\ & \\widehat { q } x _ i , & & i = 2 \\ell \\end{aligned} \\right . \\end{align*}"}
-{"id": "6967.png", "formula": "\\begin{align*} f ( t B ) = t f ( B ) , \\end{align*}"}
-{"id": "3552.png", "formula": "\\begin{align*} \\delta _ { \\boldsymbol { t } } = \\sum _ { \\varepsilon \\in \\mathcal { E } } ( - 1 ) ^ { | \\varepsilon | } I ( A _ { B _ { \\boldsymbol { t } } ^ { \\varepsilon } } ) , \\end{align*}"}
-{"id": "6373.png", "formula": "\\begin{align*} H _ { \\mu } ( t ) & = \\lim _ { y \\to 0 + } \\Re \\Phi _ { \\mu } ( t + i y ) . \\end{align*}"}
-{"id": "718.png", "formula": "\\begin{align*} 0 < \\alpha _ - < \\widehat { A } _ + \\left ( \\frac { T _ { - } } { T _ { + } } \\right ) ^ { \\frac { 3 } { 4 } } e ^ { - \\frac { T _ { \\rm i } } { 2 T _ { - } } } = \\sqrt { \\frac { T _ { + } T _ { \\rm i } ^ { \\frac { 3 } { 2 } } } { \\kappa p _ { + } } } \\left ( \\frac { T _ { - } } { T _ { \\rm i } } \\right ) ^ { \\frac { 3 } { 4 } } e ^ { - \\frac { T _ { \\rm i } } { 2 T _ { - } } } . \\end{align*}"}
-{"id": "8131.png", "formula": "\\begin{align*} S _ { i , 1 } & = \\{ s \\in S _ i : s V _ i \\cap A \\neq \\emptyset \\} , \\\\ S _ { i , 2 } & = \\{ s \\in S _ i : s V _ i \\cap B _ - \\neq \\emptyset \\} , \\\\ S _ { i , 3 } & = \\{ s \\in S _ i : s V _ i \\cap C _ - \\neq \\emptyset \\} \\end{align*}"}
-{"id": "5120.png", "formula": "\\begin{align*} \\max _ { j = 1 , \\ldots , k } | c _ j | \\leq | | v | | \\leq \\sum _ { j = 1 } ^ k | c _ j | . \\end{align*}"}
-{"id": "9795.png", "formula": "\\begin{align*} \\hat V = \\{ \\chi _ A = \\theta \\circ \\tau _ A \\ , | \\ , A \\in V \\} . \\end{align*}"}
-{"id": "9919.png", "formula": "\\begin{align*} e ( C , B _ 2 ) = \\sum _ { k = 1 } ^ \\ell d _ 2 ( v _ k ) \\ge \\tfrac 1 2 \\ell | B _ 2 | - \\tfrac { 7 / 1 5 } { 3 r - 2 } n \\ , . \\end{align*}"}
-{"id": "6098.png", "formula": "\\begin{align*} \\int _ { B ^ n } \\tfrac { \\sin ^ 2 ( g ( r ) ) } { r ^ 4 } x _ i ^ 2 { f _ { \\frak { e } _ k } ( x / r ) _ j } ^ 2 d x \\leq \\int _ { B ^ n } \\tfrac { 1 } { r ^ 2 } d x = \\omega _ { n } \\int _ { 0 } ^ 1 r ^ { n - 3 } d r < \\infty \\end{align*}"}
-{"id": "2570.png", "formula": "\\begin{align*} | p - p _ 0 | = | q - q _ 0 | = \\frac { d _ { q _ 0 } } { 1 6 } . \\end{align*}"}
-{"id": "5484.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { u ( r ^ n z ) } { ( r ^ n z ) ^ { \\rho - 1 } \\ell ( r ^ n z ) } = p _ 0 ( z ) z \\in C _ { p _ 0 } . \\end{align*}"}
-{"id": "9891.png", "formula": "\\begin{align*} & \\sum _ { x y \\in Q ^ { ( 2 ) } } \\left ( \\tfrac { r - 1 } { r } \\bigl ( d ( x ) + d ( y ) \\bigr ) - \\tfrac { r - 1 } { r + 1 } n \\right ) = ( r - 1 ) \\sum _ { x \\in Q } d ( x ) - \\tbinom { r } { 2 } n \\\\ & = \\sum _ { k = 0 } ^ { r + 1 } \\left ( k ( r - 1 ) - \\tbinom { r } { 2 } \\right ) | Q _ k | \\le \\sum _ { k = 0 } ^ { r + 1 } \\tbinom { k } { 2 } | Q _ k | = \\sum _ { x y \\in Q ^ { ( 2 ) } } | N ( x ) \\cap N ( y ) | \\ , , \\end{align*}"}
-{"id": "8737.png", "formula": "\\begin{align*} \\lim _ { \\sigma \\to 0 } \\mathcal { F } _ \\sigma & = F ( T , \\tilde { x } , u ( T , \\tilde { x } ) , \\nabla _ x \\varphi ( \\tilde { x } , \\tilde { y } ) , \\nabla _ x ^ 2 \\varphi ( \\tilde { x } , \\tilde { y } ) ) \\\\ & - F ( T , \\tilde { y } , v ( T , \\tilde { y } ) , - \\nabla _ y \\varphi ( \\tilde { x } , \\tilde { y } ) , - \\nabla _ y ^ 2 \\varphi ( \\tilde { x } , \\tilde { y } ) ) ) . \\end{align*}"}
-{"id": "1351.png", "formula": "\\begin{align*} X = Q \\Lambda ( X ) Q ^ T , \\end{align*}"}
-{"id": "6032.png", "formula": "\\begin{align*} \\theta : = \\frac { \\lambda } { 1 - 2 \\bar { \\alpha } \\lambda } . \\end{align*}"}
-{"id": "759.png", "formula": "\\begin{align*} m ( b _ 1 + c _ 1 , b _ 2 + c _ 2 ) & \\leq J ( u _ 1 ^ * \\ast w _ 1 ^ * , u _ 2 ^ * \\ast w _ 2 ^ * ) < J ( u _ 1 ^ * , u _ 2 ^ * ) + J ( w _ 1 ^ * , w _ 2 ^ * ) \\\\ & = m ( b _ 1 , b _ 2 ) + m ( c _ 1 , c _ 2 ) . \\end{align*}"}
-{"id": "5694.png", "formula": "\\begin{align*} \\inf \\big \\{ W ( z ) \\ ; : \\ ; | z | \\leq C _ 0 , \\ , \\forall a \\in \\Sigma , \\ , | z - a | \\geq \\eta \\big \\} > \\frac { C _ 0 } { S } , \\quad C _ 0 : = \\sup _ n \\ ; ( \\| { v } _ n \\| _ { L ^ \\infty } + \\mathfrak { E } _ { W } ( { v } _ n ) ) . \\end{align*}"}
-{"id": "9588.png", "formula": "\\begin{align*} D _ k R _ N \\Big ( \\sum _ { k ' } T _ N ^ { - 1 } D ^ N _ { k ' } D _ { k ' } ( f ) \\Big ) ( x ) & = D _ k R _ N \\sum _ { k ' } \\sum _ { m = 0 } ^ \\infty ( R _ N ) ^ m D ^ N _ { k ' } D _ { k ' } ( f ) ( x ) \\\\ & = \\sum _ { k ' } \\sum _ { m = 0 } ^ \\infty D _ k ( R _ N ) ^ { m + 1 } D ^ N _ { k ' } D _ { k ' } ( f ) ( x ) \\end{align*}"}
-{"id": "9867.png", "formula": "\\begin{align*} W ( { x } ^ { i _ { \\nu } } ; \\tilde \\varepsilon ) \\le W ( { x } ^ { \\nu } ; \\tilde \\varepsilon ) - \\omega \\sum _ { t = \\nu } ^ { i _ { \\nu } - 1 } \\gamma ^ { t } \\| d ( { x } ^ { t } ) \\| ^ { 2 } \\le W ( { x } ^ { \\nu } ; \\tilde \\varepsilon ) - \\omega \\frac { \\delta ^ { 2 } } { 1 6 } \\sum _ { t = \\nu } ^ { i _ { \\nu } - 1 } \\gamma ^ { t } . \\end{align*}"}
-{"id": "2764.png", "formula": "\\begin{align*} \\big \\lfloor w ( Q ) \\big \\rfloor + w ( p Q ) = m _ j + p r _ j + p m _ j = w \\Big ( p Q + Q _ 1 \\Big ) . \\end{align*}"}
-{"id": "6419.png", "formula": "\\begin{align*} d l _ { \\rightarrow \\xi } ^ { 2 } \\overset { } { = } \\frac { 1 } { \\left ( 1 + \\xi \\right ) ^ { 2 } } \\Delta \\theta ^ { 2 } \\end{align*}"}
-{"id": "8110.png", "formula": "\\begin{align*} X _ 0 = A \\setminus [ \\partial _ A W _ 1 ] _ { R _ { \\overline { W } , E } } \\end{align*}"}
-{"id": "6690.png", "formula": "\\begin{align*} \\kappa ( x ) = ( p - 1 ) ^ { n - 1 } \\frac { \\prod _ { i = 1 } ^ n | x _ i | ^ { p - 2 } } { \\left ( \\sum _ { i = 1 } ^ n | x _ i | ^ { 2 p - 2 } \\right ) ^ { \\frac { n + 1 } { 2 } } } \\end{align*}"}
-{"id": "1603.png", "formula": "\\begin{align*} \\{ X \\} ^ G - \\{ Z \\} ^ G = \\{ \\widetilde { X } \\} ^ G - \\{ E \\} ^ G \\ , , \\{ \\varnothing \\} ^ G = 0 \\ , , \\end{align*}"}
-{"id": "1975.png", "formula": "\\begin{align*} \\Vert ( v _ { s } , v _ { u } ) \\Vert & \\leq \\Vert v _ { s } \\Vert + \\Vert v _ { u } \\Vert \\leq 2 ( \\Vert v _ { s } \\Vert ^ { 2 } + \\Vert v _ { u } \\Vert ^ { 2 } ) ^ { 1 / 2 } \\leq 2 ( \\Vert v _ { s } \\Vert ^ { 2 } _ { 1 } + \\Vert v _ { u } \\Vert ^ { 2 } _ { 1 } ) ^ { 1 / 2 } = 2 \\Vert ( v _ { s } , v _ { u } ) \\Vert _ { 1 } . \\end{align*}"}
-{"id": "9183.png", "formula": "\\begin{align*} P ( z ) = p _ { 2 n } \\prod _ { j = 1 } ^ { n } ( z - z _ j ) ( z - \\bar { z _ j } ) = p _ { 2 n } \\prod _ { j = 1 } ^ { n } ( z ^ 2 - ( z _ j + \\bar { z _ j } ) z + 1 ) . \\end{align*}"}
-{"id": "6124.png", "formula": "\\begin{align*} \\bar { X } _ { t } = B _ { E _ { t } } , \\end{align*}"}
-{"id": "8014.png", "formula": "\\begin{align*} \\frac { 1 } { | Q | } \\int _ Q { \\sum _ { k = \\mu + 1 } ^ { \\infty } { \\big | T _ { [ d _ k ] } f ( x ) \\big | } } d x & \\leq \\frac { 1 } { | Q | } \\sum _ { P \\in \\mathcal { B } _ Q } { \\int _ P { \\sum _ { k = \\mu + 1 } ^ { \\infty } { \\big | T _ { [ d _ k ] } f ( x ) \\big | } } d x } \\\\ & \\lesssim \\sup _ { P \\in \\mathcal { D } _ { \\mu } } \\frac { 1 } { | P | } \\int _ P { \\sum _ { k = \\mu + 1 } ^ { \\infty } { \\big | T _ { [ d _ k ] } f ( x ) \\big | } } d x \\end{align*}"}
-{"id": "1713.png", "formula": "\\begin{align*} t _ \\lambda P ( \\sigma _ \\lambda ^ { - 1 } ( Z ( \\eta ) ) ) = \\sum _ { ( \\rho , \\xi ) \\in \\Lambda ^ { m i n } ( \\lambda , \\eta ) } P ( \\sigma _ \\lambda ( Z ( \\rho ) ) ) t _ \\lambda = \\sum _ { ( \\rho , \\xi ) \\in \\Lambda ^ { \\operatorname { m i n } } ( \\lambda , \\eta ) } t _ \\lambda t _ \\rho t _ \\rho ^ * . \\end{align*}"}
-{"id": "8564.png", "formula": "\\begin{align*} & \\zeta _ x ( y , s ) : = \\frac { 1 } { R _ 2 - R _ 1 } \\min \\{ \\max \\{ R _ 2 - H _ 0 ( y - x ) , 0 \\} , R _ 2 - R _ 1 \\} \\\\ & \\qquad \\qquad \\qquad \\times \\frac { 1 } { t _ 1 - t _ 2 } \\min \\{ \\max \\{ s - t _ 2 , 0 \\} , t _ 1 - t _ 2 \\} \\end{align*}"}
-{"id": "1303.png", "formula": "\\begin{align*} \\mathfrak { g } \\ ; = \\ ; \\mathfrak { n } ^ - \\oplus \\mathfrak { h } \\oplus \\mathfrak { n } ^ + . \\end{align*}"}
-{"id": "1401.png", "formula": "\\begin{align*} H ^ { 2 i } ( B G ; \\mathbb { Z } _ { ( p ) } ) = \\bigoplus _ { k } H ^ { k } ( B \\mathbb { G } _ m ; \\mathbb { Z } _ { ( p ) } ) \\otimes H ^ { 2 i - k } ( B G _ { m + 1 } ; \\mathbb { Z } _ { ( p ) } ) \\end{align*}"}
-{"id": "1450.png", "formula": "\\begin{align*} M _ n = \\mathbb { Z } / 2 [ x _ 2 ^ 2 , x _ 3 , x _ { 4 1 } , \\dots , x _ { 4 n } ] . \\end{align*}"}
-{"id": "8231.png", "formula": "\\begin{align*} \\biggl [ \\frac { 1 } { n ^ { k - 1 } } \\biggr ] \\ln ( F _ i ) = \\frac { ( k - 2 ) ! } { k ! } \\frac { 1 } { r ^ { k - 1 } } i ^ k \\end{align*}"}
-{"id": "2304.png", "formula": "\\begin{align*} \\partial _ t u _ * + \\nu A ^ s u _ * = - \\mathcal { P } ^ \\alpha [ u _ * \\cdot \\nabla u _ * + \\mathcal { U } ^ \\alpha ( u _ * , u _ * ) ] \\mbox { i n } \\Omega \\times [ 0 , T ] , u _ * | _ { t = 0 } = u _ 0 . \\end{align*}"}
-{"id": "4657.png", "formula": "\\begin{align*} \\left | \\frac 1 N \\sum _ { i = 0 } ^ { N - 1 } 1 _ { I _ k } ( T ^ i x ) - \\frac { C _ j ( B _ k \\cdots B _ { k + r } ) } { | C _ j ( B _ k \\cdots B _ { k + r } ) | } \\right | \\end{align*}"}
-{"id": "925.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ d E \\left [ \\frac { \\partial \\varphi } { \\partial x _ i } ( \\sqrt { 1 - t } F + \\sqrt { t } Z ) \\frac { Z _ i } { \\sqrt { t } } \\right ] = \\sum _ { i , j = 1 } ^ d E \\left [ \\frac { \\partial ^ 2 \\varphi } { \\partial x _ i \\partial x _ j } ( \\sqrt { 1 - t } F + \\sqrt { t } Z ) \\mathfrak { C } ( i , j ) \\right ] . \\end{align*}"}
-{"id": "4655.png", "formula": "\\begin{align*} \\upsilon = \\sum _ { i = 0 } ^ M 1 _ { I _ k } ( T ^ i x ) = | C _ { i _ 1 } ( B _ k \\cdots B _ { k + r } ) | + \\cdots + | C _ { i _ { a - 1 } } ( B _ k \\cdots B _ { k + r } ) | \\end{align*}"}
-{"id": "2642.png", "formula": "\\begin{align*} \\frac { 1 } { q _ j } + \\sum _ { i \\neq j } \\left \\{ \\frac { q _ i } { q _ j } \\right \\} = 1 . \\end{align*}"}
-{"id": "5466.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\widehat U ( 1 / ( r ^ n z ) ) } { ( r ^ n z ) ^ \\rho \\ell ( r ^ n z ) } = q ( 1 / z ) , \\end{align*}"}
-{"id": "8167.png", "formula": "\\begin{align*} f _ { \\mathrm { N M L } } ( { \\bf x } ^ n ) \\buildrel \\rm d e f \\over = \\frac { f ( { \\bf x } ^ n ; \\hat { \\mu } ( { \\bf x } ^ n ) , \\hat { \\Sigma } ( { \\bf x } ^ n ) ) } { \\int _ { Y ( R , \\epsilon _ 1 , \\epsilon _ 2 ) } f ( { \\bf y } ^ n ; \\hat { \\mu } ( { \\bf y } ^ n ) , \\hat { \\Sigma } ( { \\bf y } ^ n ) ) { \\rm d } { \\bf y } ^ n } \\end{align*}"}
-{"id": "4085.png", "formula": "\\begin{align*} D _ X Y = \\nabla _ X Y + h ( X , Y ) \\eta , \\end{align*}"}
-{"id": "8522.png", "formula": "\\begin{align*} & w ^ \\top x \\to \\min \\\\ & \\begin{cases} G x \\equiv g \\ , ( \\ , S ) \\\\ h x \\leq h _ 0 \\\\ x \\in \\mathbb { Z } _ + ^ n , \\ , | | x | | _ { \\infty } \\leq n \\Delta , \\end{cases} \\end{align*}"}
-{"id": "3535.png", "formula": "\\begin{align*} \\begin{array} [ c ] { r l } - d { p } ^ { \\theta } ( t ) = & \\{ b _ x ( X ^ { \\theta } ( t ) , u ^ { \\theta } ( t ) ) ^ { } p ^ { \\theta } ( t ) - \\beta ^ { 0 , \\theta } f _ x ( X ^ { \\theta } ( t ) , u ^ { \\theta } ( t ) ) \\} d t , \\ t \\in ( t _ { i - 1 } , t _ { i } ) , \\\\ p ^ { \\theta } ( t _ { i } ) = & - \\beta ^ { 0 , \\theta } \\Psi _ x ( X ^ { \\theta } ( t _ n ) ) 1 _ { i = n } ( i ) - \\displaystyle { \\beta ^ { i , \\theta } } { } + p ( t _ { i } ^ { + } ) , i = 1 , \\ldots , n . \\end{array} \\end{align*}"}
-{"id": "5446.png", "formula": "\\begin{align*} \\widehat U ( s ) = \\int _ 0 ^ \\infty e ^ { - s x } \\dd U ( x ) \\end{align*}"}
-{"id": "3702.png", "formula": "\\begin{align*} \\begin{aligned} \\big ( & x , \\frac { u _ 1 } { v _ 1 } , t _ 2 , \\dots , t _ { n + 1 } \\big ) , & ( & k = 1 ) \\\\ \\big ( & t _ 1 , \\dots , t _ { k - 1 } , \\frac { v _ { k - 1 } } { u _ { k - 1 } } , \\frac { u _ k } { v _ k } , t _ { k + 1 } , \\dots , t _ { n + 1 } \\big ) , & ( & 1 < k < n + 1 ) \\\\ \\big ( & t _ 1 , \\dots , t _ n , \\frac { v _ n } { u _ n } , y ) . & ( & k = n + 1 ) \\end{aligned} \\end{align*}"}
-{"id": "9478.png", "formula": "\\begin{align*} \\sum _ { s = 0 } ^ N \\frac { q ^ { 2 s } ( q ; q ) _ { N + s } } { ( q ^ 2 ; q ^ 2 ) _ s } = ( q ; q ^ 2 ) _ { N + 1 } + q ^ { N + 1 } ( q ^ 2 ; q ^ 2 ) _ N . \\end{align*}"}
-{"id": "28.png", "formula": "\\begin{align*} ( z - s ) ^ 2 = ( | C _ 1 | s i n ( \\theta ) - | C _ 2 | s i n ( \\phi ) ) ^ 2 = | C _ 1 | ^ 2 s i n ( \\theta ) ^ 2 - 2 | C _ 1 | | C _ 2 | s i n ( \\theta ) s i n ( \\phi ) + | C _ 2 | ^ 2 s i n ( \\phi ) ^ 2 \\end{align*}"}
-{"id": "3135.png", "formula": "\\begin{align*} m & = | A _ 1 | + | A _ 2 | + \\sum _ { \\ell \\geq 3 } { | A _ \\ell | } \\leq | A _ 1 | + | A _ 2 | + \\sum _ { \\ell \\geq 3 } { \\ell } { | A _ \\ell | } / 3 \\\\ & = ( 2 | A _ 1 | + | A _ 2 | ) / { 3 } + \\sum _ { \\ell \\geq 1 } \\ell | A _ \\ell | / 3 = ( 2 | A _ 1 | + | A _ 2 | ) / { 3 } + { 2 q } / { 3 } \\leq { 5 q } / { 3 } , \\end{align*}"}
-{"id": "3065.png", "formula": "\\begin{align*} & ( a _ 1 ^ { k _ 1 } a _ 2 ^ { k _ 2 } b _ 1 ^ { \\ell _ 1 } b _ 2 ^ { \\ell _ 2 } [ a _ 1 , b _ 1 ] ^ { r _ 1 } [ a _ 1 , b _ 2 ] ^ { r _ 2 } [ a _ 2 , b _ 1 ] ^ { r _ 3 } [ a _ 2 , b _ 2 ] ^ { r _ 4 } ) ^ p = \\\\ & [ a _ 1 , b _ 1 ] ^ { k _ 1 + \\lambda _ 1 \\cdot k _ 2 } [ a _ 1 , b _ 2 ] ^ { k _ 2 } [ a _ 2 , b _ 1 ] ^ { \\ell _ 1 } [ a _ 2 , b _ 2 ] ^ { \\ell _ 1 + l _ 2 } . \\end{align*}"}
-{"id": "3580.png", "formula": "\\begin{align*} E = 2 \\sum _ { i } \\left ( ( 2 R _ i + 1 ) \\varepsilon _ i + \\frac 1 { R _ i } \\right ) . \\end{align*}"}
-{"id": "5317.png", "formula": "\\begin{align*} \\frac { \\beta } { 2 } = \\frac { 1 - \\beta } { 2 } + \\frac { \\beta } { 2 ^ * } = \\frac { 1 } { q } , ( 0 < \\beta < 1 ) , \\end{align*}"}
-{"id": "8897.png", "formula": "\\begin{align*} \\begin{aligned} \\psi _ { n _ k } \\to _ k J _ { \\theta , 1 } ^ { - 1 } ( p \\gamma ) - p J _ { \\theta , 1 } ^ { - 1 } \\gamma \\ \\ & \\ \\ \\psi _ { - n _ k } \\to _ k J _ { \\theta , 1 } ^ { - 1 } ( p \\overline \\gamma ) - p J _ { \\theta , 1 } ^ { - 1 } \\overline \\gamma \\\\ & m . \\end{aligned} \\end{align*}"}
-{"id": "9319.png", "formula": "\\begin{align*} f ( x ) = ( f _ 1 ( x ) , \\ldots , f _ m ( x ) ) x \\in \\R ^ n . \\end{align*}"}
-{"id": "1686.png", "formula": "\\begin{align*} \\alpha _ i = \\begin{cases} \\delta ^ j _ i , & Q _ i = u _ j \\\\ - \\delta ^ j _ i , & Q _ i = w _ j \\end{cases} \\mu _ { 2 N } ( Z ( \\eta ) ) = \\begin{cases} \\prod _ { i = 1 } ^ n \\frac { 1 + \\alpha _ { 2 i - 1 } } { 2 N } , & r ( \\eta ) = v \\\\ \\prod _ { i = 0 } ^ n \\frac { 1 + \\alpha _ { 2 i } } { 2 N } , & r ( \\eta ) \\not = v . \\end{cases} \\end{align*}"}
-{"id": "2404.png", "formula": "\\begin{align*} x _ 0 = \\frac { a _ 3 + a _ 1 ^ 2 a _ 6 } { a _ 5 + a _ 1 \\left ( - a _ 3 + a _ 1 a _ 5 + 4 a _ 6 + 3 a _ 1 ^ 2 a _ 6 \\right ) } . \\end{align*}"}
-{"id": "4416.png", "formula": "\\begin{align*} \\langle h _ { r a n } ( x ' ) h _ { r a n } ( y ' ) \\rangle = 0 \\quad \\mbox { f o r } \\ ; | x ' - y ' | \\gg \\ell . \\end{align*}"}
-{"id": "4746.png", "formula": "\\begin{align*} \\alpha ^ 2 A + 2 \\alpha \\beta B + \\beta ^ 2 C = 0 , \\alpha , \\beta - c o n s t . \\end{align*}"}
-{"id": "3811.png", "formula": "\\begin{align*} K : = \\inf \\{ k \\ge 0 \\colon \\ , | E _ { k + 1 } - E _ { k } | > 4 \\ell _ L \\} . \\end{align*}"}
-{"id": "1239.png", "formula": "\\begin{align*} \\nu _ m ( J ) = t _ m ^ { 1 - n } \\tau ( t _ m J ) \\mbox { w h e n e v e r $ J $ i s a B o r e l s u b s e t o f $ B ( 0 , 2 R ) . $ } \\end{align*}"}
-{"id": "4152.png", "formula": "\\begin{align*} r \\sum _ { \\tau = t + h - 1 } ^ { n } & Z _ { N , J } ^ { \\tau } + n Z _ { I , J } ^ { t + h - 2 } + \\sum _ { \\tau = t } ^ { t + h - 2 } n ^ { t + h - \\tau - 1 } Z _ { i _ { \\tau } , J } ^ { \\tau } \\leq 2 r + \\sum _ { \\tau = 1 } ^ { h - 2 } n ^ { \\tau } , \\\\ & \\forall \\ t \\in [ n - h ] , \\ J \\subseteq V , \\ l \\in [ n - 1 ] , \\ I \\subseteq N , \\ r \\in [ n - 1 ] , \\ i _ t , \\dotsc , i _ { t + h - 2 } \\in N . \\end{align*}"}
-{"id": "5505.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { ( r ^ n z ) ^ { m + \\beta } } { \\ell ( r ^ n z ) } \\overline F ( r ^ n z ) = p _ { 0 , m } ( z ) - m \\mathrm { B } _ { - m - \\beta } p _ { 0 , m } ( z ) z \\in C _ { p _ { 0 , m } } , \\end{align*}"}
-{"id": "3044.png", "formula": "\\begin{align*} \\begin{cases} ( u _ { h , t } ( t ) , v _ h ) _ H + a ( u _ h ( t ) , v _ h ) + b ( v _ h , p _ h ( t ) ) = 0 , & \\forall v _ h \\in V _ h , \\\\ b ( u _ h ( t ) , q _ h ) = 0 , & \\forall q _ h \\in Q _ h , \\\\ u _ h ( 0 ) = P _ { h , \\sigma } u _ 0 , \\end{cases} \\end{align*}"}
-{"id": "1612.png", "formula": "\\begin{align*} x = x ( 0 , m ) \\sigma ^ m ( x ) . \\end{align*}"}
-{"id": "9349.png", "formula": "\\begin{align*} J O _ { n + 2 } ^ { ( 3 ) } - 4 J O _ { n } ^ { ( 3 ) } & = \\sum _ { s = 0 } ^ { 7 } ( J _ { n + s + 2 } ^ { ( 3 ) } - 4 J _ { n + s } ^ { ( 3 ) } ) e _ { s } \\\\ & = J _ { n + 2 } ^ { ( 3 ) } - 4 J _ { n } ^ { ( 3 ) } + ( J _ { n + 3 } ^ { ( 3 ) } - 4 J _ { n + 1 } ^ { ( 3 ) } ) e _ { 1 } + \\cdots + ( J _ { n + 9 } ^ { ( 3 ) } - 4 J _ { n + 7 } ^ { ( 3 ) } ) e _ { 7 } \\\\ & = e _ { 1 } + e _ { 2 } + e _ { 4 } + e _ { 5 } + e _ { 7 } - 2 ( 1 + e _ { 3 } + e _ { 6 } ) \\end{align*}"}
-{"id": "2997.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } ( t , x ) & = \\Delta _ b u ( t , x ) + f ( t , u ( t , x ) ) + g ( t , u ( t , x ) ) \\xi ( t , x ) , \\\\ u ( 0 , x ) & = u _ 0 ( x ) \\end{align*}"}
-{"id": "1219.png", "formula": "\\begin{align*} - \\bar u _ { x _ n x _ n } & = - \\frac { 1 } { \\bar h _ t ^ { 2 } } \\sum \\limits _ { i = 1 } ^ { n - 1 } \\bar h _ { t X _ i } \\frac { \\partial X _ i } { \\partial x _ n } + \\frac { 1 } { \\bar h _ { t } ^ { 3 } } \\bar h _ { t t } \\\\ & = - \\frac { 1 } { \\bar h _ t ^ { 3 } } \\left [ \\langle \\ , \\nabla _ { X } \\bar h _ t \\ , ( \\bar h _ { X _ i X _ j } ) ^ { - 1 } , \\nabla _ { X } \\bar h _ t \\rangle - \\bar h _ { t t } \\right ] \\end{align*}"}
-{"id": "1403.png", "formula": "\\begin{align*} d _ r ( ( 1 \\otimes u ) \\otimes 1 ) \\not = 0 \\in E _ r ^ { 2 i + r , 1 - r } . \\end{align*}"}
-{"id": "5134.png", "formula": "\\begin{align*} u = z _ 1 + z _ 3 + v \\end{align*}"}
-{"id": "2828.png", "formula": "\\begin{align*} g _ { v w } ^ { ( n + 1 , 1 , j ) } = \\sum _ { u \\in V _ { n + 1 , j } } f _ { v u } ^ { ( n + 1 ) } g _ { u w } ^ { ( n , 1 , j ) } \\end{align*}"}
-{"id": "7750.png", "formula": "\\begin{align*} \\mathbf { q } : \\mathbf { G } = \\prod _ { w \\in S } G ( k _ w ) \\rightarrow \\prod _ { w \\in S } G ^ { a d } ( k _ w ) = \\prod _ { w \\in S } G ( k _ w ) / Z ( G ( k _ w ) ) = \\mathbf { G } / Z ( \\mathbf { G } ) \\end{align*}"}
-{"id": "9601.png", "formula": "\\begin{align*} \\langle f , D _ { k } ( g ) \\rangle & = \\lim _ { M \\to \\infty } \\lim _ { J \\to \\infty } \\sum _ { j = 1 } ^ { N _ J } \\int _ { Q _ j } D _ k ( f ) ( y ) g ( y ) d \\mu ( y ) \\\\ & = \\int _ { \\Bbb R ^ n } D _ k ( f ) ( y ) g ( y ) d \\mu ( y ) \\\\ & = \\langle D _ k ( f ) , g \\rangle . \\end{align*}"}
-{"id": "8004.png", "formula": "\\begin{align*} & \\Big ( \\frac { 1 } { | P | } \\int _ P { \\sum _ { k = \\mu } ^ { \\infty } { 2 ^ { s k q } \\Big | \\Pi _ k \\Big ( \\sum _ { n = 3 } ^ { \\infty } { T _ { [ b _ n ] } f } \\Big ) ( x ) \\Big | ^ q } } d x \\Big ) ^ { 1 / q } \\\\ & \\lesssim \\sup _ { R \\in \\mathcal { D } _ { \\mu } } { \\Big ( \\dfrac { 1 } { | R | } \\int _ R { \\sum _ { k = \\max { ( 3 , \\mu - 2 ) } } ^ { \\infty } { 2 ^ { s k q } \\big | T _ { [ b _ k ] } f ( x ) \\big | ^ q } } d x \\Big ) ^ { 1 / q } } . \\end{align*}"}
-{"id": "3659.png", "formula": "\\begin{align*} \\Big \\{ v _ s = \\begin{pmatrix} s ( 1 ) \\\\ s ( 2 ) \\\\ \\vdots \\\\ s ( n - 1 ) \\end{pmatrix} - \\begin{pmatrix} 1 \\\\ 2 \\\\ \\vdots \\\\ n - 1 \\end{pmatrix} \\ ; \\Big | \\ ; s \\in \\mathfrak S _ n \\Big \\} \\subset \\mathbb R ^ { n - 1 } . \\end{align*}"}
-{"id": "9535.png", "formula": "\\begin{align*} w _ 1 & : = \\phi _ x ^ * ( x ) = ( \\phi _ 1 \\phi _ x ) ^ * ( z _ 1 ) = ( \\phi _ 2 \\phi _ y ) ^ * ( z _ 1 ) = \\phi _ y ^ * ( y ) , \\\\ w _ i & : = \\phi _ x ^ * ( x _ i ) = \\phi _ y ^ * ( x _ i ) \\textrm { f o r $ i \\geq 2 $ } . \\end{align*}"}
-{"id": "7522.png", "formula": "\\begin{gather*} u = \\frac 1 { \\sqrt { t ^ 2 + 1 } } ( w ^ 1 \\cos \\tau - w ^ 2 \\sin \\tau ) + \\frac { t x } { t ^ 2 + 1 } - \\frac { \\kappa y } { t ^ 2 + 1 } , \\\\ v = \\frac 1 { \\sqrt { t ^ 2 + 1 } } ( w ^ 1 \\sin \\tau + w ^ 2 \\cos \\tau ) + \\frac { t y } { t ^ 2 + 1 } + \\frac { \\kappa x } { t ^ 2 + 1 } , \\end{gather*}"}
-{"id": "6575.png", "formula": "\\begin{align*} c _ 0 = \\frac { ( 1 + \\varepsilon ) ^ { 1 / 2 } \\ ( 1 - \\varepsilon ) ^ { 3 / 2 } } { \\sqrt { 2 } \\ ( 2 - \\varepsilon ) \\ ( 3 - 2 \\varepsilon ) } . \\end{align*}"}
-{"id": "7594.png", "formula": "\\begin{align*} \\dfrac { 1 } { z - x } = \\dfrac { 1 } { z - x _ 0 } + \\dfrac { x - x _ 0 } { ( z - x ) ( z - x _ 0 ) } \\end{align*}"}
-{"id": "8651.png", "formula": "\\begin{align*} \\sum _ { k = N } ^ \\infty \\pi [ A _ k ] < \\frac 1 8 . \\end{align*}"}
-{"id": "7624.png", "formula": "\\begin{align*} ( M ^ l - 1 ) \\Big ( \\frac { \\alpha _ 0 } { M } \\Big ) ^ l \\sum _ { i = 0 } ^ \\infty M ^ { l i } \\nu \\big ( \\{ V : | g | > \\alpha _ 0 M ^ i \\} \\big ) \\leq \\int _ { V } | g | ^ l \\ , d \\nu . \\end{align*}"}
-{"id": "10035.png", "formula": "\\begin{align*} \\gamma _ p = \\delta _ p ^ { - n } \\cdot ( D , p ) _ p ^ n \\cdot \\operatorname { i n v } _ p ( V _ p ) \\in \\{ \\pm 1 , \\pm i \\} , \\end{align*}"}
-{"id": "7129.png", "formula": "\\begin{align*} X ~ = ~ \\frac { ( 1 , Y ) } { \\sqrt { 1 + | Y | ^ 2 } } . \\end{align*}"}
-{"id": "4672.png", "formula": "\\begin{align*} & B ( v , \\cdot ) _ { \\vert N } = a S _ { \\vert N } + b T _ { \\vert N } = a S _ N + b I _ N : N \\to N , \\end{align*}"}
-{"id": "6485.png", "formula": "\\begin{align*} \\Omega = \\overset { l } { \\underset { i = 1 } { \\sum } } \\omega _ { i } , \\Xi _ { j } \\in \\mathbb { R } , \\forall j = 1 , \\ldots , l \\end{align*}"}
-{"id": "3600.png", "formula": "\\begin{align*} E : = \\Big \\{ ( n , k ) \\rightarrow ( n + 1 , m ) , n , k \\in \\N , 0 \\leq m \\leq k + 1 \\Big \\} , \\end{align*}"}
-{"id": "3165.png", "formula": "\\begin{align*} f _ { X _ { t } ^ { x } } ( y ) = \\int _ { \\mathbb { R } _ { \\geqslant 0 } } f _ { Y _ { t } ^ { x } } ( y - z ) \\mu _ { Z _ { t } } ( \\mathrm { d } z ) , y \\geqslant 0 , \\end{align*}"}
-{"id": "643.png", "formula": "\\begin{align*} D _ t v & = \\bar \\nabla _ { F _ t } \\dot \\gamma \\big | _ { s = 0 } = \\bar \\nabla _ { \\dot \\gamma } J _ t \\big | _ { s = 0 } , \\\\ \\bar \\nabla _ i v & = \\bar \\nabla _ { F _ i } \\dot \\gamma \\big | _ { s = 0 } = \\bar \\nabla _ { \\dot \\gamma } J _ i \\big | _ { s = 0 } \\end{align*}"}
-{"id": "10152.png", "formula": "\\begin{align*} { d _ k ( i ) } = { X _ k \\big ( { \\boldsymbol \\omega } _ 0 ( i ) \\big ) + n _ k ( i ) } , ~ ~ ~ k = 1 , 2 , \\ldots , 1 4 , \\end{align*}"}
-{"id": "4914.png", "formula": "\\begin{align*} 0 = | \\theta | ^ 2 i ( A - A ^ * ) + F _ 0 . \\end{align*}"}
-{"id": "1025.png", "formula": "\\begin{align*} M _ \\epsilon ( \\vec u ) = \\sum _ { k = 1 } ^ 3 \\int \\frac 1 { \\epsilon ^ 3 } \\partial _ k \\varphi ( \\frac y \\epsilon ) ( 2 A _ { k , \\epsilon } ( \\vec u ) - B _ { k , \\epsilon } ( \\vec u ) ) \\ , d y - C _ \\epsilon ( \\vec u ) \\end{align*}"}
-{"id": "771.png", "formula": "\\begin{align*} T _ { k } ( x ) = y ^ { n } \\end{align*}"}
-{"id": "3996.png", "formula": "\\begin{align*} p ^ { \\nu _ 1 } ( 1 , t ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { ( - \\lambda _ 1 t ^ { \\nu _ 1 } ) ^ { k } } { \\Gamma ( k { \\nu _ 1 } + 1 ) } , \\end{align*}"}
-{"id": "9330.png", "formula": "\\begin{align*} j _ { n + 1 } ^ { ( 3 ) } + j _ { n } ^ { ( 3 ) } = 3 J _ { n + 2 } ^ { ( 3 ) } , \\end{align*}"}
-{"id": "658.png", "formula": "\\begin{align*} P D _ t \\widetilde A - D _ t A = P \\widetilde \\Delta \\widetilde A - \\Delta A + ( I ) + ( I I ) + ( I I I ) + ( I V ) \\end{align*}"}
-{"id": "392.png", "formula": "\\begin{align*} S _ r & = \\sum _ { n = 3 } ^ \\infty \\frac { 1 } { n ( \\ln n ) ^ r } \\mathbb P \\left ( | S _ n | > ( 1 + \\varepsilon ) \\sigma _ n \\sqrt { 2 ( 1 - r ) \\ln \\ln n } \\right ) . \\end{align*}"}
-{"id": "4871.png", "formula": "\\begin{align*} \\| A \\| _ \\varphi = \\| A \\setminus \\{ 1 , \\ldots , n \\} \\| _ \\varphi . \\end{align*}"}
-{"id": "1508.png", "formula": "\\begin{align*} T _ { n } ( x ) = \\left ( \\begin{array} { c } \\frac { \\underline { \\alpha } \\alpha ^ { n + 1 } ( x ) } { ( \\alpha ( x ) - \\omega _ { 1 } ( x ) ) ( \\alpha ( x ) - \\omega _ { 2 } ( x ) ) } - \\frac { \\underline { \\omega _ { 1 } } \\omega _ { 1 } ^ { n + 1 } ( x ) } { ( \\alpha ( x ) - \\omega _ { 1 } ( x ) ) ( \\omega _ { 1 } ( x ) - \\omega _ { 2 } ( x ) ) } \\\\ + \\frac { \\underline { \\omega _ { 2 } } \\omega _ { 2 } ^ { n + 1 } ( x ) } { ( \\alpha ( x ) - \\omega _ { 2 } ( x ) ) ( \\omega _ { 1 } ( x ) - \\omega _ { 2 } ( x ) ) } \\end{array} \\right ) \\end{align*}"}
-{"id": "7942.png", "formula": "\\begin{align*} K = \\bigcap _ { n = 1 } ^ { \\infty } \\bar V ^ { n } . \\end{align*}"}
-{"id": "6001.png", "formula": "\\begin{gather*} E _ 2 = \\frac { e _ 0 } { 4 } ( 1 - x - y + x y ) \\end{gather*}"}
-{"id": "1628.png", "formula": "\\begin{align*} S _ \\lambda ^ * \\chi _ { Z ( \\eta ) } ( x ) = \\rho ( \\Lambda ) ^ { - d ( \\lambda ) / 2 } \\sum _ { ( \\alpha , \\beta ) \\in \\Lambda ^ { \\operatorname { m i n } } ( \\lambda , \\eta ) } \\chi _ { Z ( \\alpha ) } ( x ) . \\end{align*}"}
-{"id": "9647.png", "formula": "\\begin{align*} \\Delta \\Phi _ { \\rm { s t } , \\beta } ( t ) \\triangleq \\mathbb { E } \\left [ \\hat { \\Phi } _ { \\rm { s t } , \\beta } ( t ) - \\Phi _ { \\rm { s t } , \\beta } ( t ) \\right ] = \\Lambda _ \\beta ( t ) \\xi _ \\beta ( t ) . \\end{align*}"}
-{"id": "2267.png", "formula": "\\begin{align*} d ^ { \\circ } _ { 2 g } = \\frac { d ^ * _ { 2 g } } { 2 } + \\frac { d ^ | _ { 2 g } + d ^ { | | } _ { 2 g } } { 4 } , \\end{align*}"}
-{"id": "8771.png", "formula": "\\begin{align*} x \\Psi _ B ^ { { \\bf f } } ( t _ 1 , \\dots , t _ m ) = x + t _ 1 \\cdots t _ m B ( { \\bf f } ) ( x ) + o ( t _ 1 \\cdots t _ m ) \\end{align*}"}
-{"id": "1061.png", "formula": "\\begin{align*} H _ { k + 1 } \\leq H _ { k } ^ { \\nu + \\epsilon } , \\nu = \\frac { n - 1 } { \\widehat { w } _ { n } ( \\zeta ) - n } . \\end{align*}"}
-{"id": "7342.png", "formula": "\\begin{align*} \\begin{aligned} & \\| A _ i \\cdot A _ j \\ , u \\| _ { L ^ { s _ { i j } } ( [ 0 , T ] , L ^ { p _ { i j } } ( \\R ^ 3 ) ) } \\\\ & \\quad \\lesssim \\| A _ i \\| _ { L ^ { a _ i } ( [ 0 , T ] , L ^ { b _ i } ( \\R ^ 3 ) ) } \\ , \\| A _ j \\| _ { L ^ { a _ j } ( [ 0 , T ] , L ^ { b _ j } ( \\R ^ 3 ) ) } \\ , \\| u \\| _ { X ^ { ( 4 , 3 ) } [ 0 , T ] } \\ , . \\end{aligned} \\end{align*}"}
-{"id": "8967.png", "formula": "\\begin{gather*} { \\cal H } _ { W ; \\vec { T } + \\vec { T } ' ; \\gamma } ( X ) = { \\cal H } _ { W ; \\vec { T } ; \\gamma } ( X ) \\cap { \\cal H } _ { W ; \\vec { T } ' ; \\gamma } ( X ) . \\end{gather*}"}
-{"id": "7585.png", "formula": "\\begin{align*} b _ { n , i } ( X ) & = f _ { n , i } ( X ) + ( - 1 ) ^ { n - i } \\binom { n } { i } _ X ( 1 - X ) X ^ { \\binom { n + 1 } { 2 } - \\binom { i + 1 } { 2 } - 1 } \\\\ & = \\binom { n } { i } _ X \\left ( \\left ( \\prod _ { j = i + 1 } ^ n ( 1 - X ^ j ) \\right ) + ( - 1 ) ^ { n - i } ( 1 - X ) X ^ { \\binom { n + 1 } { 2 } - \\binom { i + 1 } { 2 } - 1 } \\right ) \\in \\Z [ X ] . \\end{align*}"}
-{"id": "4241.png", "formula": "\\begin{align*} m : = \\mathrm { m i n } \\{ \\lambda \\in \\Z \\ , | \\ , H ^ 0 ( \\P ^ 1 , V _ { \\theta } ( \\lambda ) ) \\neq 0 \\} \\end{align*}"}
-{"id": "9087.png", "formula": "\\begin{align*} f ( x y ) - x f ( y ) - f ( x ) y = B ( x , y ) \\left ( x , y \\in R \\right ) . \\end{align*}"}
-{"id": "1815.png", "formula": "\\begin{align*} S _ { \\mathcal { C } , h } = \\begin{cases} + 1 , & U _ { \\mathcal { C } } \\leq \\left ( 1 + \\tanh ( h \\mu ^ 0 _ { \\mathcal { C } } ( D ) ) \\right ) / 2 , \\\\ - 1 , & . \\end{cases} \\end{align*}"}
-{"id": "7693.png", "formula": "\\begin{align*} M ^ { \\top } \\Sigma + \\Sigma M + Z = 0 \\ , . \\end{align*}"}
-{"id": "2839.png", "formula": "\\begin{align*} \\| f \\| _ n ^ * = \\| f \\| _ { Y ^ * } + \\frac { 1 } { n } \\| f \\| _ { ( Y , p ) ^ * } \\end{align*}"}
-{"id": "2578.png", "formula": "\\begin{align*} \\left ( T _ u ^ { * } H _ u T _ u \\right ) ( z _ 0 ) = \\left ( T _ v ^ { * } H _ v T _ v \\right ) ( z _ 0 ) + \\mathcal { O } \\left ( | z _ 0 | ^ { - 2 } \\right ) = - 8 v ( z _ 0 ) + \\mathcal { O } \\left ( | z _ 0 | ^ { - 2 } \\right ) < 0 \\end{align*}"}
-{"id": "3162.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ e ^ { u Z _ { t } } \\right ] = \\exp \\left \\lbrace \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { \\infty } \\left ( e ^ { z \\psi ( s , u ) } - 1 \\right ) \\nu ( \\mathrm { d } z ) \\mathrm { d } s \\right \\rbrace , \\quad ( t , u ) \\in \\mathbb { R } _ { \\geqslant 0 } \\times \\mathcal { U } . \\end{align*}"}
-{"id": "9792.png", "formula": "\\begin{align*} u _ { r \\bar r } = - \\frac { 1 } { 2 } \\sum _ { r < l < \\bar r } u _ { r l } u _ { r \\ , \\bar l } \\end{align*}"}
-{"id": "1660.png", "formula": "\\begin{align*} R _ { d _ 0 } = \\big ( 0 , \\frac { 1 } { 3 } \\big ) , R _ { d _ 1 } = \\big ( \\frac { 2 } { 3 } , 1 \\big ) , R _ { b _ 0 } = \\big ( \\frac { 1 } { 3 } , \\frac { 1 } { 2 } \\big ) , R _ { b _ 1 } = \\big ( \\frac { 1 } { 2 } , \\frac { 2 } { 3 } \\big ) . \\end{align*}"}
-{"id": "8184.png", "formula": "\\begin{align*} \\psi _ { \\nu } ( x ) = \\begin{cases} e ( x ) & , \\\\ e ( x + \\overline { x } ) & \\end{cases} \\end{align*}"}
-{"id": "2546.png", "formula": "\\begin{align*} \\det \\left [ \\begin{pmatrix} \\rho + a _ 1 & 0 & 0 \\\\ 0 & \\rho + a _ 2 & 0 \\\\ 0 & 0 & \\rho + a _ 3 \\\\ \\end{pmatrix} + i \\eta \\mathrm { A } _ { i j } ( \\eta , \\rho ) \\right ] = 0 . \\end{align*}"}
-{"id": "5364.png", "formula": "\\begin{align*} \\frac { S _ { ( r , s ) ( r ' , s ' ) } } { S _ { ( 1 , 1 ) ( r ' , s ' ) } } = ( - 1 ) ^ { ( r + 1 ) s ' + ( s + 1 ) r ' } \\frac { \\sin \\left ( \\frac { \\pi b } { a } r r ' \\right ) \\sin \\left ( \\frac { \\pi a } { b } s s ' \\right ) } { \\sin \\left ( \\frac { \\pi b } { a } r ' \\right ) \\sin \\left ( \\frac { \\pi a } { b } s ' \\right ) } . \\end{align*}"}
-{"id": "7305.png", "formula": "\\begin{align*} \\eta : = \\frac { d x _ 1 \\wedge d x _ 2 \\wedge \\ldots \\wedge d x _ { n + 1 } } { d f _ z } \\ , \\ \\ \\ \\omega _ 1 : = \\frac { d x _ 1 \\wedge d x _ 2 \\wedge \\ldots \\wedge d x _ { n + 1 } } { d f _ { t _ 1 , t _ { n + 2 } } } \\ . \\end{align*}"}
-{"id": "969.png", "formula": "\\begin{align*} P \\left ( \\max _ { \\theta \\in \\mathcal { G } _ n } | Z _ n ( \\theta ) | \\leq q _ n ^ Z ( 1 - \\alpha - \\varepsilon _ n ) \\right ) = 1 - \\alpha - \\varepsilon _ n \\leq P ( T _ n ^ * \\leq z | \\mathcal { F } ^ X ) ( \\omega ) - \\varepsilon _ n \\leq P \\left ( \\max _ { \\theta \\in \\mathcal { G } _ n } | Z _ n ( \\theta ) | \\leq z \\right ) , \\end{align*}"}
-{"id": "1456.png", "formula": "\\begin{align*} E _ \\infty = C \\langle 1 , y _ m , v _ s y _ s \\rangle \\oplus D _ 1 / ( v _ t e _ t ) \\{ x _ 3 ^ 2 \\} \\oplus D _ m \\langle x _ 3 ^ 2 y _ m , x _ 3 z _ m \\rangle . \\end{align*}"}
-{"id": "4194.png", "formula": "\\begin{gather*} A = \\frac { 1 } { \\Delta } \\begin{pmatrix} - \\tfrac { 3 } { 2 } t _ 1 \\beta - \\tfrac { 1 } { 1 2 } d \\Delta & \\tfrac { 3 } { 2 } \\beta \\\\ \\Delta d t _ 1 - \\tfrac { 1 } { 6 } t _ 1 d - ( \\tfrac { 3 } { 2 } t _ 1 ^ 2 + \\tfrac { 1 } { 8 } t _ 2 ) \\beta & \\tfrac { 3 } { 2 } t _ 1 \\beta \\Delta + \\tfrac { 1 } { 1 2 } d \\Delta \\end{pmatrix} , \\\\ \\beta = 3 t _ 3 d t _ 2 - 2 t _ 2 d t _ 3 . \\end{gather*}"}
-{"id": "1528.png", "formula": "\\begin{align*} | \\mu ^ s | & = \\sum _ { j = 1 } ^ { s } \\left ( m ( t - 2 j ) + t \\binom { m } { 2 } \\right ) - \\binom { m s } { 2 } \\\\ & = - s ^ 2 \\cdot ( m ^ 2 + 2 m ) / { 2 } + s \\cdot ( m ^ 2 t + m t - m ) / { 2 } \\\\ & = - \\frac { m ^ 2 + 2 m } { 2 } \\cdot ( s - \\alpha _ { m , t } ( - 1 ) ) ^ 2 + \\frac { m ^ 2 + 2 m } { 2 } \\cdot \\alpha ^ 2 _ { m , t } ( - 1 ) . \\end{align*}"}
-{"id": "10007.png", "formula": "\\begin{align*} [ ( 0 , \\log ( D ) ) : \\mathcal { Y } _ \\mathrm { s m } ] = \\deg _ \\C ( \\mathcal { Y } _ \\mathrm { s m } ) \\cdot \\log ( D ) = [ ( \\mathrm { E x c } , 0 ) : \\mathcal { Y } _ \\mathrm { s m } ] . \\end{align*}"}
-{"id": "8621.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ t V ( t - s , x + B _ s ) d s = & \\int _ 0 ^ t \\left ( \\int _ { \\R ^ { d + 1 } } \\phi ( t - s - s ' ) \\psi ( x + B _ s - y ' ) d W ( s ' , y ' ) \\right ) d s \\\\ = & \\int _ { \\R ^ { d + 1 } } \\Phi _ { t , x , B } ( s ' , y ' ) d W ( s ' , y ' ) , \\end{aligned} \\end{align*}"}
-{"id": "5355.png", "formula": "\\begin{align*} Y _ { P ( z ) } ' ( v , x ) = \\tau _ { P ( z ) } ( Y _ t ( v , x ) ) . \\end{align*}"}
-{"id": "8194.png", "formula": "\\begin{align*} f _ { \\nu } ( g _ { \\nu } ) = k _ { \\nu } \\big ( u _ { \\nu } ( g _ { \\nu } . i _ { \\nu } , i _ { \\nu } ) \\big ) , \\end{align*}"}
-{"id": "4620.png", "formula": "\\begin{align*} | U _ i ' | \\ge \\beta ( 1 - 2 \\eta ) \\sum _ { n = 0 } ^ { 2 ^ { k _ i } - 1 } 1 _ { [ - B , B ] } \\left ( t + \\sum _ { j = 0 } ^ { n - 1 } f ( T ^ j z _ i ) \\right ) \\end{align*}"}
-{"id": "7232.png", "formula": "\\begin{align*} \\textit { c - I n d } _ { U } ^ { G } \\psi : = \\{ W : G \\rightarrow \\mathbb { C } : W ( u g ) = \\psi ( u ) W ( g ) \\} , \\end{align*}"}
-{"id": "143.png", "formula": "\\begin{align*} ( a \\cdot ( N ) ) ^ 2 = 4 a ^ 2 \\det ( N ) = 4 \\det ( M N ) = ( a \\cdot ( N ) + b z ) ^ 2 \\end{align*}"}
-{"id": "9758.png", "formula": "\\begin{align*} Z _ A ^ { ( k ) } = Z _ { A , 0 } \\mathrm { e x p } ( - \\frac { 1 } { \\rho _ { A , 0 } u _ { A , 0 } A ( 0 ) } \\int _ 0 ^ x A ( \\tau ) \\rho _ A ^ { ( k ) } \\phi ( T _ A ^ { ( k ) } ) d \\tau ) , \\end{align*}"}
-{"id": "5848.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ N & ( - 1 ) ^ k \\binom { N } { k } ( k + 1 ) g ^ k D ( g ) D ^ { N - 1 } ( g ^ { N - k } f ) \\\\ & + \\sum _ { k = 0 } ^ N ( - 1 ) ^ k \\binom { N } { k } g ^ { k + 1 } D ^ N ( g ^ { N - k } f ) = 0 . \\end{align*}"}
-{"id": "7987.png", "formula": "\\begin{align*} h _ { j _ 1 \\cdot \\cdot j _ { \\ell } } : = f _ { I j _ 1 \\cdot \\cdot j _ { \\ell } } : W _ { j _ 1 \\cdot \\cdot j _ { \\ell - 1 } } \\to W _ { j _ 1 \\cdot \\cdot j _ { \\ell } } , \\end{align*}"}
-{"id": "5218.png", "formula": "\\begin{align*} \\mu _ j ( \\lambda ) = \\frac { 1 } { 2 } \\left ( - c \\pm \\sqrt { c ^ 2 + 4 ( \\lambda - \\nu _ i ) } \\right ) , \\end{align*}"}
-{"id": "2005.png", "formula": "\\begin{align*} f g ( z _ 1 , . . . , z _ { \\alpha + \\beta } ) = \\frac { 1 } { \\alpha ! \\beta ! } \\sum _ { \\sigma \\in S _ { \\alpha + \\beta } } f ( z _ { \\sigma ( 1 ) } , . . . , z _ { \\sigma ( \\alpha ) } ) g ( z _ { \\sigma ( \\alpha + 1 ) } , . . . , z _ { \\sigma ( \\alpha + \\beta ) } ) \\end{align*}"}
-{"id": "3909.png", "formula": "\\begin{align*} { \\langle H \\rangle ^ { { \\cal L } } _ { m i n } } = 2 \\hbar \\omega \\sqrt { \\frac { { \\cal L } + 3 } { 2 } } \\Big ( \\frac { \\Gamma ( \\frac { { \\cal L } + 3 } { 2 } ) } { \\Gamma ( \\frac { \\cal L } { 2 } + 1 ) } \\Big ) , \\end{align*}"}
-{"id": "271.png", "formula": "\\begin{align*} ( \\Delta f ) ( n ) = f ( n + 1 ) - f ( n ) . \\end{align*}"}
-{"id": "9444.png", "formula": "\\begin{align*} G _ n = \\sum _ { k = 2 ^ { 2 ^ n } + 1 } ^ { 2 ^ { 2 ^ { n + 1 } } } \\frac { \\psi ( k ) \\varphi ( k ) } { k } . \\end{align*}"}
-{"id": "2093.png", "formula": "\\begin{align*} \\liminf _ { t \\rightarrow \\delta } | | d u ( t , \\cdot ) | | _ { C ^ 0 ( M ) } = \\infty . \\end{align*}"}
-{"id": "4749.png", "formula": "\\begin{align*} { \\varepsilon } = F ( X , Y , Z ) \\ , { \\Delta \\sqrt { - \\det g ^ { i j } } } / { \\big ( { L } _ 0 ( \\alpha f _ 1 + \\beta ) \\big ) } , \\end{align*}"}
-{"id": "9411.png", "formula": "\\begin{align*} N _ i = \\{ A _ { i , j } \\mid 1 \\leq j \\leq a \\} \\cup \\{ B _ { i , k } \\mid 1 \\leq k \\leq 2 b \\} \\end{align*}"}
-{"id": "3409.png", "formula": "\\begin{align*} \\phi = \\gamma \\circ \\phi \\circ \\gamma \\ \\ \\gamma \\in \\Gamma . \\end{align*}"}
-{"id": "2465.png", "formula": "\\begin{align*} \\left < f , g \\right > _ \\xi = \\int f ( \\xi ) \\overline { g ( \\xi ) } d \\xi . \\end{align*}"}
-{"id": "2817.png", "formula": "\\begin{align*} V ( t , x ) = \\sup _ { 0 \\leq \\tau \\leq T - t } E _ { t , x } \\left ( e ^ { - r \\tau } \\left ( K - X _ { t + \\tau } \\right ) ^ { + } \\right ) , \\end{align*}"}
-{"id": "8759.png", "formula": "\\begin{align*} \\frac { 1 } { M } \\left ( \\sum _ { i = 1 } ^ N i \\nabla _ { r _ i } \\right ) ^ 2 - \\sum _ { i = 1 } ^ N \\Delta _ { r _ i } - g \\sum _ { i = 1 } ^ N \\delta ( r _ i ) . \\end{align*}"}
-{"id": "3551.png", "formula": "\\begin{align*} \\delta _ { \\boldsymbol { s } } ( U ) = ( \\sum _ { \\boldsymbol { \\varepsilon } \\in \\mathcal { E } } ( - 1 ) ^ { \\abs { \\boldsymbol { \\varepsilon } } } ) I ( S _ { B _ { \\boldsymbol { s } } ^ { \\boldsymbol { 1 } } } \\in U ) = 0 . \\end{align*}"}
-{"id": "7881.png", "formula": "\\begin{align*} u ( x , t ) = x _ 3 - t \\end{align*}"}
-{"id": "5286.png", "formula": "\\begin{align*} \\| u ( t , \\cdot ) \\| \\leq & \\beta ( \\| { u _ 0 } \\| , t ) + \\theta _ 1 \\bigg ( \\int _ { 0 } ^ { t } \\gamma _ 1 ( | d _ 1 ( s ) | ) \\bigg ) \\\\ & + \\theta _ 2 \\bigg ( \\int _ { 0 } ^ { t } \\gamma _ 2 ( | d _ 2 ( s ) | ) \\bigg ) , \\ \\forall t \\geq 0 . \\end{align*}"}
-{"id": "6777.png", "formula": "\\begin{align*} 4 \\pi m _ 0 = \\int _ { \\mathbb { S } ^ 2 } e ^ { U _ { \\lambda , \\xi _ k } } \\eta _ { R _ 0 , \\xi _ k } , \\end{align*}"}
-{"id": "7268.png", "formula": "\\begin{align*} B _ 2 \\le \\frac { 4 s ^ 2 } { N ^ 2 } \\Bigl ( 1 - \\frac m n + \\frac m n + 1 - 1 \\Bigr ) = \\frac { 4 s ^ 2 } { N ^ 2 } . \\end{align*}"}
-{"id": "891.png", "formula": "\\begin{align*} r _ n : = \\max _ { \\nu = 1 , 2 } \\max _ { i = 0 , 1 , \\dots , n _ \\nu + 1 } ( t ^ \\nu _ i - t ^ \\nu _ { i - 1 } ) \\to 0 \\end{align*}"}
-{"id": "184.png", "formula": "\\begin{align*} \\mathcal { C } _ { \\vdash } = \\{ \\mathrm { T h } _ { \\vdash } ( a ) : a \\in R \\} \\quad { \\vdash _ { \\mathcal { C } } } = \\{ \\langle a , b \\rangle : b \\in \\delta _ { \\mathcal { C } } ( a ) \\} . \\end{align*}"}
-{"id": "7165.png", "formula": "\\begin{align*} ' _ b & = V C \\sum _ { n \\in \\mathcal { A } _ { m , b } } \\mathbb { E } [ S _ { m , n , b } ] \\\\ & \\leq C \\left ( 2 \\sum _ { n \\neq a _ b ^ { * } } \\mathbb { E } [ \\theta _ { n , b , K _ b } ] + 1 \\right ) \\\\ & \\leq C \\left ( 2 \\sum _ { n \\neq a _ b ^ { * } } \\left [ \\frac { 8 \\ln ( K _ b - u _ n ) } { \\delta ^ 2 _ { n , b } } + 1 + \\frac { \\pi ^ 2 } { 3 } \\right ] + 1 \\right ) . \\end{align*}"}
-{"id": "1900.png", "formula": "\\begin{align*} & E _ { p } ^ { s } = \\{ v \\in T _ { p } M _ { i } : ( \\Vert D ( \\textbf { \\textit { f } } ^ { n } _ { i } ) _ { p } ( v ) \\Vert ) _ { n \\geq 1 } \\} \\\\ \\quad \\quad & E _ { p } ^ { u } = \\{ v \\in T _ { p } M _ { i } : ( \\Vert D ( \\textbf { \\textit { f } } ^ { - n } _ { i } ) _ { p } ( v ) \\Vert ) _ { n \\geq 1 } \\} . \\end{align*}"}
-{"id": "4477.png", "formula": "\\begin{align*} \\psi ( s ) = 1 > | \\phi ( s ) | \\end{align*}"}
-{"id": "2998.png", "formula": "\\begin{align*} u ( t , x ) = & \\int _ F p ^ b _ t ( x , y ) u _ 0 ( y ) \\mu ( d y ) + \\int _ 0 ^ t \\int _ F p ^ b _ { t - s } ( x , y ) f ( s , u ( s , y ) ) \\mu ( d y ) d s \\\\ & + \\int _ 0 ^ t \\int _ F p ^ b _ { t - s } ( x , y ) g ( s , u ( s , y ) ) \\xi ( s , y ) \\mu ( d y ) d s \\end{align*}"}
-{"id": "1681.png", "formula": "\\begin{align*} \\sum _ { \\eta \\in s ( \\lambda ) \\Lambda ^ { ( 1 , 1 ) } } \\mu _ 2 ( Z ( \\lambda \\eta ) ) & = \\mu _ 2 ( Z ( \\lambda ) ) \\cdot \\begin{cases} ( 1 / 2 - \\delta _ { 2 N + 1 } ) + ( 1 / 2 + \\delta _ { 2 N + 1 } ) , & \\ ; \\ ; s ( \\lambda ) = v \\\\ ( 1 / 2 - \\delta _ { 2 N + 2 } ) + ( 1 / 2 + \\delta _ { 2 N + 2 } ) , & \\ ; \\ ; s ( \\lambda ) \\in \\{ u , w \\} . \\end{cases} \\\\ & = \\mu _ 2 ( Z ( \\lambda ) ) . \\end{align*}"}
-{"id": "9897.png", "formula": "\\begin{align*} w ( x , y ) = \\begin{cases} 0 & \\cr 1 & \\cr 2 & \\end{cases} \\end{align*}"}
-{"id": "5973.png", "formula": "\\begin{align*} h _ { n + 1 } ( x ) = 2 x h _ n ( x ) - 2 n h _ { n - 1 } ( x ) . \\end{align*}"}
-{"id": "4238.png", "formula": "\\begin{align*} \\chi ( X _ { 1 , \\bar { k } } , E ( m ) ) = \\Sigma ^ d _ { i = 0 } a _ i ( E ) { m + d - i \\choose d - i } . \\end{align*}"}
-{"id": "1927.png", "formula": "\\begin{align*} \\tilde { g } _ { p } = & \\ , _ { \\textbf { \\textit { f } } ( p ) } ^ { - 1 } \\circ \\mathcal { G } \\circ _ { p } : B ( 0 _ { p } , r ) \\rightarrow B ( 0 _ { \\mathcal { F } ( p ) } , \\varrho / 2 ) \\\\ \\quad \\tilde { g } _ { p } ^ { - 1 } & = _ { p } ^ { - 1 } \\circ \\mathcal { G } ^ { - 1 } \\circ _ { \\mathcal { F } ( p ) } : B ( 0 _ { \\mathcal { F } ( p ) } , r ) \\rightarrow B ( 0 _ { p } , \\varrho / 2 ) , \\end{align*}"}
-{"id": "9775.png", "formula": "\\begin{align*} & \\int \\limits _ { 0 } ^ { x } \\int \\limits _ { \\chi ( \\tau ) } ^ { g ( \\tau ) } \\rho \\phi ( T ) Z d y d \\tau - \\int \\limits _ 0 ^ x ( g ( \\tau ) - \\chi ( \\tau ) ) \\bar { \\rho } \\phi ( \\bar { T } ) \\bar { Z } d \\tau \\\\ = & \\int \\limits _ { 0 } ^ { x } \\int \\limits _ { \\chi ( \\tau ) } ^ { g ( \\tau ) } \\rho ( \\phi ( T ) - \\phi ( \\bar { T } ) ) Z d y d \\tau + \\int \\limits _ { 0 } ^ { x } \\int \\limits _ { \\chi ( \\tau ) } ^ { g ( \\tau ) } ( \\rho - \\bar { \\rho } ) \\phi ( \\bar { T } ) ( Z - \\bar { Z } ) d y d \\tau \\\\ = & O ( 1 ) \\delta _ * ^ 2 . \\end{align*}"}
-{"id": "9391.png", "formula": "\\begin{align*} M _ { \\Omega , t } f ( x ) = \\frac 1 { t ^ n } \\int _ { | y | < t } \\Omega ( y ' ) f ( x - y ) d y , \\end{align*}"}
-{"id": "6479.png", "formula": "\\begin{align*} d s _ { 2 D c } ^ { 2 } = \\frac { 1 } { \\sigma ^ { 2 } \\left ( 1 - \\rho ^ { 2 } \\right ) } d \\mu _ { x } ^ { 2 } + \\frac { 4 } { \\sigma ^ { 2 } \\left ( 1 - \\rho ^ { 2 } \\right ) } d \\sigma ^ { 2 } \\end{align*}"}
-{"id": "6335.png", "formula": "\\begin{align*} \\mathcal { M } _ n ( E ) : = \\big \\{ ( m _ 1 ( \\mu ) , \\dots , m _ n ( \\mu ) ) : \\mu \\in \\mathcal { P } ( E ) \\big \\} \\end{align*}"}
-{"id": "8593.png", "formula": "\\begin{align*} T f ( x ) = \\lim _ { \\epsilon \\rightarrow 0 ^ { + } } \\int _ { | y | > \\epsilon } \\frac { \\Omega ( y / | y | ) } { | y | ^ { n } } f ( x - y ) \\ , d y , \\ , \\ , \\ , \\ , x \\in \\mathbb { R } ^ { n } . \\end{align*}"}
-{"id": "8574.png", "formula": "\\begin{align*} B _ { H _ 0 } ( x , 2 ) \\subset \\bigcup _ { j = 1 } ^ M B _ { H _ 0 } ( x _ j , 1 ) \\end{align*}"}
-{"id": "6782.png", "formula": "\\begin{align*} w _ { \\lambda , k } ( \\xi _ k ) = \\int _ { \\mathbb { S } ^ 2 } G ( \\xi _ k , y ) \\left [ e ^ { U _ { \\lambda , \\xi _ k } } \\eta _ { R _ 0 , \\xi _ k } ( y ) - m _ 0 \\right ] d H ^ 2 ( y ) \\end{align*}"}
-{"id": "4947.png", "formula": "\\begin{align*} 2 ^ x + 2 ^ y = w ^ 2 \\end{align*}"}
-{"id": "3625.png", "formula": "\\begin{align*} \\hat \\varrho : = | \\nabla u | ^ { p - 2 } \\frac { 1 } { \\hat \\varrho } : = | \\nabla u | ^ { 2 - p } \\ , . \\end{align*}"}
-{"id": "3922.png", "formula": "\\begin{align*} \\begin{aligned} 3 ^ { l _ 1 + 1 } x ^ { r + 1 8 } & + 3 a ( r ) x ^ { 3 3 } + 3 b ( r ) x ^ { 5 1 } + 3 a ( r ) ^ 2 x ^ { 3 9 } + 2 \\cdot 3 a ( r ) b ( r ) x ^ { 5 7 } + 3 b ( r ) ^ 2 x ^ { 7 5 } \\\\ & + 3 \\cdot 3 ^ { 2 l _ 1 } x ^ { 2 r + 9 } + 2 \\cdot 3 ^ { l _ 1 + 1 } a ( r ) x ^ { 2 4 + r } + 2 \\cdot 3 ^ { l _ 1 + 1 } b ( r ) x ^ { 4 2 + r } \\in \\langle F \\rangle . \\end{aligned} \\end{align*}"}
-{"id": "9722.png", "formula": "\\begin{align*} \\tilde { V } _ b = \\tilde { \\Phi } ( \\tilde { \\gamma } _ 5 , \\tilde { \\gamma } _ 3 , \\tilde { \\gamma } _ 2 , \\tilde { \\gamma } _ 1 ; \\tilde { V } _ a ) , \\tilde { Z } _ b = \\tilde { Z } _ a + \\tilde { \\gamma } _ 4 . \\end{align*}"}
-{"id": "9310.png", "formula": "\\begin{align*} E _ i : = \\{ ( \\delta , t , x ) : \\delta \\geq 0 , ( t , x ) \\in W [ 0 , \\delta ] \\times [ t - \\delta , t + \\delta ] \\times \\{ x \\} \\subseteq C _ i \\} . \\end{align*}"}
-{"id": "3930.png", "formula": "\\begin{align*} T B ^ 4 _ i ( x ) = \\frac { 1 } { w } \\begin{cases} p ^ 3 ( x _ i ) & x \\in [ x _ i , x _ { i + 1 } ] \\\\ p ( x _ i ) ( p ( x _ i ) q ( x _ { i + 2 } ) + q ( x _ { i + 3 } ) p ( x _ { i + 1 } ) ) + q ( x _ { i + 4 } ) p ^ 2 ( x _ { i + 1 } ) , & x \\in [ x _ { i + 1 } , x _ { i + 2 } ] \\\\ q ( x _ { i + 4 } ) ( p ( x _ { i + 1 } ) q ( x _ { i + 3 } ) + q ( x _ { i + 4 } ) p ( x _ { i + 2 } ) ) + p ( x _ i ) q ^ 2 ( x _ { i + 3 } ) , & x \\in [ x _ { i + 2 } , x _ { i + 3 } ] \\\\ q ^ 3 ( x _ { i + 4 } ) , & x \\in [ x _ { i + 3 } , x _ { i + 4 } ] \\end{cases} \\end{align*}"}
-{"id": "1337.png", "formula": "\\begin{align*} [ e _ { \\tau } \\theta ] ( Z ) = \\inf _ { Z ' \\in \\mathbb { \\cal Z } } \\left \\{ \\theta ( Z ' ) + \\displaystyle \\frac { 1 } { \\tau } \\| Z ' - Z \\| ^ 2 \\right \\} . \\end{align*}"}
-{"id": "6443.png", "formula": "\\begin{align*} p _ { j } \\left ( x _ { j } \\right ) = \\int \\cdots \\int d x _ { 1 } \\cdots d x _ { j - 1 } d x _ { j + 1 } \\cdots d x _ { l } p _ { } ^ { \\prime } \\left ( x _ { 1 } , \\ldots , x _ { l } \\right ) \\end{align*}"}
-{"id": "5189.png", "formula": "\\begin{align*} \\hat { U } _ { 1 } ( x , y , z ) = \\frac { f _ { 1 } ( x ) - g _ { 1 , [ z , 1 ] } ( x ) } { x } , \\\\ \\hat { U } _ { 2 } ( x , y , z ) = \\frac { f _ { 1 } ( y ) - g _ { 1 , [ z , 1 ] } ( y ) } { z - y } , \\\\ \\hat { U } _ { 3 } ( x , y , z ) = \\frac { f _ { 2 } ( z ) - g _ { 2 , [ 0 , y ] } ( z ) } { z - y } , \\end{align*}"}
-{"id": "2355.png", "formula": "\\begin{align*} P _ M ( f ) ( z ) \\coloneqq ( 1 - Q _ M ) f ( z ) \\coloneqq f ( z ) - \\sum _ { k = 0 } ^ M \\frac { f ^ { ( k ) } ( x _ p ) } { k ! } ( z - x _ p ) ^ k . \\end{align*}"}
-{"id": "7592.png", "formula": "\\begin{align*} \\zeta _ { F _ { n , \\delta } , \\textup { t o p } } ( s ) & = \\prod _ { i = 0 } ^ { n - 1 } \\frac { s - 2 i } { s - 2 ( n + i + \\delta ) + 1 } , \\\\ \\zeta _ { G _ n , \\textup { t o p } } ( s ) & = \\prod _ { i = 0 } ^ { n - 1 } \\frac { s - i } { s - n - i } , \\\\ \\zeta _ { H _ n , \\textup { t o p } } ( s ) & = \\prod _ { i = 0 } ^ { n - 1 } \\frac { s - i } { s - \\frac { n + i + 1 } { 2 } } . \\end{align*}"}
-{"id": "9974.png", "formula": "\\begin{align*} 0 \\leq \\alpha \\leq \\alpha _ { \\tau } , \\quad \\alpha _ { \\tau } = \\kappa _ { \\tau } \\theta . \\end{align*}"}
-{"id": "1836.png", "formula": "\\begin{align*} ( X \\otimes Y ) _ m : = \\frac { ( X \\otimes ^ . Y ) _ m } { B _ m ( X \\otimes ^ . Y ) } , \\end{align*}"}
-{"id": "3152.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } _ { \\geqslant 0 } } x ^ { \\kappa } m _ { \\alpha , \\beta } ( \\mathrm { d } x ) \\geqslant \\left ( \\int _ { \\mathbb { R } _ { \\geqslant 0 } } x m _ { \\alpha , \\beta } ( \\mathrm { d } x ) \\right ) ^ { \\kappa } = \\left ( \\frac { \\alpha } { \\beta } \\right ) ^ { \\kappa } . \\end{align*}"}
-{"id": "6636.png", "formula": "\\begin{align*} | f ( z ) - f ( w ) | \\ ; & \\leq \\ ; C _ 1 \\left ( \\int _ { 2 D } | D f | ^ 2 \\ , \\right ) ^ { \\frac { 1 } { 2 } } \\\\ & \\leq \\ ; C _ 2 \\left ( m ^ { 2 p } \\mathrm { A r e a } ( 2 D ) \\right ) ^ { \\frac { 1 } { 2 } } \\ ; = \\ ; 2 C _ 2 \\sqrt { \\pi } m ^ p | z - w | \\ . \\end{align*}"}
-{"id": "906.png", "formula": "\\begin{align*} \\widehat { \\mathfrak { s } } _ n ^ 2 ( t ) : = \\frac { 2 } { 3 } \\sum _ { i = 1 } ^ n K _ h ( t _ { i - 1 } - t ) ^ 2 ( X _ { t _ i } - X _ { t _ { i - 1 } } ) ^ 4 . \\end{align*}"}
-{"id": "8130.png", "formula": "\\begin{align*} B _ - & = \\{ x \\in X : d ( x , X \\setminus \\tilde { B } ) > \\eta ' \\} , \\\\ A _ + & = \\{ x \\in X : d ( x , A ) \\leq \\eta ' \\} \\end{align*}"}
-{"id": "5117.png", "formula": "\\begin{align*} \\limsup _ { r \\rightarrow 0 ^ + } r ^ { - \\alpha Q } \\mu ( B _ r ( t _ 0 ) ) \\lesssim C _ \\mu \\left | \\left [ \\sum _ { j _ 1 , \\ldots , j _ Q = 1 } ^ d \\left | { \\mathcal A } _ { t _ 0 } ( \\partial _ { t _ { j _ 1 } } , \\ldots , \\partial _ { t _ { j _ Q } } ) \\right | ^ 2 \\right ] ^ { \\frac { 1 } { 2 } } \\right | ^ { \\alpha } \\end{align*}"}
-{"id": "3874.png", "formula": "\\begin{align*} \\phi ( u ) = \\phi ( u | \\alpha , \\beta , \\gamma , \\delta ) = E \\exp ( i u Z ) = \\exp ( - \\gamma ^ \\alpha [ | u | ^ \\alpha + i \\beta \\eta ( u , \\alpha ) ] + i u \\delta ) , \\end{align*}"}
-{"id": "2581.png", "formula": "\\begin{align*} V _ a ( r ) : = \\left \\{ \\begin{aligned} & | r | ^ { - a } & & 0 < a < 1 , \\\\ & - \\log | r | \\ : & & a = 0 \\end{aligned} \\right . \\end{align*}"}
-{"id": "1622.png", "formula": "\\begin{align*} \\Phi _ { \\sigma _ i } = \\frac { d ( \\mu \\circ \\sigma _ i ) } { d \\mu } \\end{align*}"}
-{"id": "1031.png", "formula": "\\begin{align*} \\theta _ { 4 } = 6 . 3 0 2 8 \\ldots , \\theta _ { 5 } = 8 . 2 3 6 1 \\ldots , \\theta _ { 1 0 } = 1 8 . 1 0 9 8 \\ldots , \\end{align*}"}
-{"id": "2264.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\left [ \\frac { 1 } { e ^ { t k } - 1 } \\right ] = - \\frac { 2 k e ^ { 2 t k } } { ( e ^ { 2 t k } - 1 ) ^ 2 } , \\end{align*}"}
-{"id": "2649.png", "formula": "\\begin{align*} h ^ * ( P _ D , z ) = \\sum _ { b = 0 } ^ { c ( D ) - 1 } x ^ { w ( b ) } , \\ ; w ( b ) = b - \\sum _ { i = 1 } ^ { n } \\left \\lfloor \\frac { c _ i b } { 6 ( k + 1 ) } \\right \\rfloor . \\end{align*}"}
-{"id": "7732.png", "formula": "\\begin{align*} F _ N ( 2 ) & = \\frac { 1 } { N } \\sum ^ { N - 1 } _ { n = 1 } \\frac { 1 - \\cos 2 \\phi _ n } { ( 1 - \\cos \\phi _ n ) ^ 2 } \\\\ & = \\frac { 1 } { 2 N } \\sum ^ { N - 1 } _ { n = 1 } \\frac { \\sin ^ 2 \\phi _ n } { \\sin ^ 4 \\phi _ n / 2 } = \\frac { 2 } { N } \\sum ^ { N - 1 } _ { n = 1 } \\frac { \\cos ^ 2 \\phi _ n / 2 } { \\sin ^ 2 \\phi _ n / 2 } \\\\ & = \\frac { 4 N } { 3 } - 2 + \\frac { 2 } { 3 N } \\end{align*}"}
-{"id": "7240.png", "formula": "\\begin{align*} = \\int _ { \\hat { G } } ^ { } \\int _ { G } ^ { } \\theta _ { \\pi } ( g ) ( \\psi _ { n } * _ { U } f ) ( g ) d g d \\mu _ { \\pi } . \\end{align*}"}
-{"id": "1.png", "formula": "\\begin{align*} - \\min v _ Q ( S _ { j + 1 } ) + \\min v _ Q ( S _ j ) = - \\min v _ Q ( S _ { i + 1 } ) + \\min v _ Q ( S _ i ) . \\end{align*}"}
-{"id": "9746.png", "formula": "\\begin{align*} & L _ 0 ( J ) - L _ 0 ( I ) = - | \\omega _ k | , \\\\ & L _ { 1 } ^ { 2 } ( J ) - L _ { 1 } ^ { 2 } ( I ) \\leq | K _ { b 0 } | | \\omega _ { k } | + \\sum \\limits _ { i = 2 , 3 , 5 } | K _ { b i } | | \\alpha _ i | , \\\\ & L _ { i } ^ { 2 } ( J ) - L _ { i } ^ { 2 } ( I ) = - | \\alpha _ i | , i = 2 , 3 , 4 , 5 . \\\\ & Q ( J ) - Q ( I ) \\leq ( | K _ { b 0 } | | \\omega _ { k } | + \\sum \\limits _ { i = 2 , 3 , 5 } | K _ { b i } | | \\alpha _ i | ) L ( I _ 0 ) . \\end{align*}"}
-{"id": "8377.png", "formula": "\\begin{align*} \\mathrm { d i v } ( \\psi ( f ) ) = \\sum _ { \\substack { m > 0 \\\\ \\mu \\in V _ \\Z ^ \\vee / V _ \\Z } } c ( - m , \\mu ) \\cdot \\mathcal { Z } ( m , \\mu ) . \\end{align*}"}
-{"id": "3150.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { ( k + 1 ) \\cdots ( k + n ) } { ( k + 2 - n ) \\cdots ( k + 1 ) } = 1 , \\end{align*}"}
-{"id": "5235.png", "formula": "\\begin{align*} \\mu ( \\gamma , V ) = - n _ - ( \\Gamma ( \\gamma , V , a ) ) + \\sum \\limits _ { t ^ * \\in ( a , b ) } \\mathrm { s i g n } \\ , \\Gamma ( \\gamma , V , t ^ * ) + n _ + ( \\Gamma ( \\gamma , V , b ) ) , \\end{align*}"}
-{"id": "9127.png", "formula": "\\begin{align*} \\Phi ^ { \\ast } ( x ) = \\Phi ( x , \\ldots , x ) = \\sum _ { k = 1 } ^ { n } x ^ { p _ { k } } f _ { k } ( x ^ { q _ { k } } ) = 0 \\left ( x \\in R \\right ) \\end{align*}"}
-{"id": "9023.png", "formula": "\\begin{align*} ( \\delta u ) _ { s t } ( \\varphi ) = u _ s ( \\{ A ^ { 1 , * } _ { s t } + A ^ { 2 , * } _ { s t } \\} \\varphi ) + u ^ \\natural _ { s t } ( \\varphi ) , u _ 0 = u _ { } . \\end{align*}"}
-{"id": "1336.png", "formula": "\\begin{align*} \\bar { R } _ t = \\exp \\left \\{ - \\int _ 0 ^ t \\langle \\bar { \\gamma } ( s ) , \\d W ( s ) \\rangle - \\frac { 1 } { 2 } \\int _ 0 ^ t | \\bar { \\gamma } ( s ) | ^ 2 \\d s \\right \\} , \\ \\ t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "5584.png", "formula": "\\begin{align*} \\ddot { x } + ( \\alpha + \\beta \\left ( \\cos \\left ( t \\right ) + \\cos \\left ( 2 t \\right ) \\right ) ) x = 0 \\end{align*}"}
-{"id": "1577.png", "formula": "\\begin{align*} ( 2 ^ { n - i } - 1 ) p + q = ( 2 ^ { n - i + 1 } - 1 ) r \\end{align*}"}
-{"id": "7237.png", "formula": "\\begin{align*} K _ { i } \\subset K _ { i + 1 } \\forall i \\in \\mathbb { N } \\cup K _ { i } = U . \\end{align*}"}
-{"id": "10064.png", "formula": "\\begin{align*} a ( m ) = \\sum _ { \\substack { \\alpha \\in F _ + \\\\ \\mathrm { T r } _ { F / \\Q } ( \\alpha ) = m } } a _ F ( \\alpha ) \\end{align*}"}
-{"id": "190.png", "formula": "\\begin{align*} \\mathfrak { X } \\cdot \\mathfrak { Y } = [ \\sigma _ { 1 } \\circ \\pi _ { 1 } , \\dots , \\sigma _ { 1 } \\circ \\pi _ { m } , \\dots , \\sigma _ { n } \\circ \\pi _ { 1 } , \\dots , \\sigma _ { n } \\circ \\pi _ { m } ] , \\end{align*}"}
-{"id": "644.png", "formula": "\\begin{align*} \\partial _ t d ^ 2 & = 2 h ( P \\widetilde F _ t - F _ t , v ) , \\\\ | \\nabla d | & \\le | P \\widetilde F _ * - F _ * | . \\end{align*}"}
-{"id": "5006.png", "formula": "\\begin{align*} \\phi = \\sum _ { | \\gamma | = m + 1 } \\partial ^ { \\gamma } \\phi ^ { ( \\gamma ) } . \\end{align*}"}
-{"id": "6620.png", "formula": "\\begin{align*} \\| D f - D g \\| _ { L ^ p ( \\Omega ) } \\ ; & = \\ ; \\| D ( f | _ { \\mathcal { O } } ) - D \\phi \\| _ { L ^ p ( \\mathcal { O } ) } \\\\ & \\leq \\| D ( f | _ { \\mathcal { O } } ) \\| _ { L ^ p ( \\mathcal { O } ) } + \\| D \\phi \\| _ { L ^ p ( \\mathcal { O } ) } \\\\ & \\leq \\ ; c _ p ^ { \\frac { 1 } { p } } \\left [ \\mathcal { E } _ p ( f | _ { \\mathcal { O } } ) ^ { \\frac { 1 } { p } } + \\mathcal { E } _ p ( \\phi ) ^ { \\frac { 1 } { p } } \\right ] \\ . \\end{align*}"}
-{"id": "9824.png", "formula": "\\begin{align*} p _ k ' ( t ) \\ ! = \\ ! - p _ k ( t ) ( \\mu \\ ! + \\ ! \\l f ( t ) ) \\ ! + \\ ! p _ { k - 1 } ( t ) \\l f ( t ) \\ ! + \\ ! p _ { k + 1 } ( t ) \\mu , t ' \\leq t \\leq ( T ' _ { e _ 2 } \\wedge T _ 2 ) . \\end{align*}"}
-{"id": "8937.png", "formula": "\\begin{gather*} \\deg ( \\psi _ { j i } ) = \\psi _ { j i } \\psi _ { j i } ^ \\vee = r _ i \\mu _ { j i } r _ j \\mu _ { i j } , \\end{gather*}"}
-{"id": "914.png", "formula": "\\begin{align*} D \\varphi ( F _ 1 , \\dots , F _ m ) = \\sum _ { i = 1 } ^ m \\frac { \\partial \\varphi } { \\partial x _ i } ( F _ 1 , \\dots , F _ m ) D F _ i . \\end{align*}"}
-{"id": "9066.png", "formula": "\\begin{align*} f _ 1 = a _ 1 y _ 1 ^ 2 + a _ 2 y _ 2 ^ 2 + a _ 3 y ^ 2 _ 3 + a _ 4 y _ 1 y _ 2 + a _ 5 y _ 1 y _ 3 - a _ 6 y _ 2 y _ 3 , \\end{align*}"}
-{"id": "1353.png", "formula": "\\begin{align*} Y = Q { \\rm D i a g } \\ , ( w ) Q ^ T = Q _ { a \\cup _ { b _ U } } Q _ { a \\cup _ { b _ U } } ^ T + Q _ { b _ S } { \\rm D i a g } \\ , ( w _ { b _ S } ) Q ^ T _ { b _ S } - Q _ { c \\cup _ { b _ L } } Q _ { c \\cup _ { b _ L } } ^ T \\end{align*}"}
-{"id": "7989.png", "formula": "\\begin{align*} \\frac { M n } { N - 2 n } = \\frac { 2 M } { n - 5 } \\leq 6 0 \\log n \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; ( 2 n - 3 d ) M / N \\geq 1 . 8 \\log n . \\end{align*}"}
-{"id": "9637.png", "formula": "\\begin{align*} \\bar { \\Phi } _ { \\rm { d n } } | ^ { T } _ 0 = \\frac { 1 } { T } \\int _ 0 ^ T \\sum _ { \\beta = 1 } ^ { \\kappa } \\Phi _ { \\rm { d n } , \\beta } ( t ) d t = \\frac { 1 } { T } \\left \\{ \\int _ 0 ^ T \\sum _ { \\beta = 1 } ^ { \\kappa } \\Phi _ { \\rm { s t } , \\beta } ( t ) d t + \\frac { 1 } { E _ { \\rm { b } } } \\sum _ { i = 0 } ^ { N _ \\tau } \\Omega _ { \\rm { m } } ( \\tau _ i ) \\right \\} , \\end{align*}"}
-{"id": "2557.png", "formula": "\\begin{align*} | p _ { 0 } | = \\min _ { p \\in \\Gamma } | p | < \\min \\{ | a | , | b | \\} \\ , . \\end{align*}"}
-{"id": "7012.png", "formula": "\\begin{align*} \\sigma _ 2 ( B ) = 2 \\epsilon h ( \\epsilon ) + \\frac { 2 ( n - 2 ) } { n - 1 } \\epsilon ^ 2 = 1 , \\end{align*}"}
-{"id": "6109.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow 0 ^ { + } } \\frac { s } { \\psi \\left ( s \\right ) } = 0 . \\end{align*}"}
-{"id": "229.png", "formula": "\\begin{align*} | B _ G | ^ { 2 \\gamma _ h } = ( \\chi _ G B \\chi _ G B \\chi _ G ) ^ { \\gamma _ h } \\leq ( \\chi _ G B ^ 2 \\chi _ G ) ^ { \\gamma _ h } \\end{align*}"}
-{"id": "6521.png", "formula": "\\begin{align*} \\mathcal { P } \\approx 1 - \\eta _ { \\mathcal { C } } \\cdot \\frac { \\left ( \\Delta \\mathcal { C } \\right ) ^ { 2 } } { \\mathcal { C } _ { } ^ { 2 } } \\eta _ { \\mathcal { C } } \\overset { } { = } \\frac { 8 } { 3 } k _ { \\mathrm { o } } ^ { 2 } \\left ( 2 k _ { \\mathrm { o } } ^ { 2 } + \\sigma _ { k _ { \\mathrm { o } } } ^ { 2 } \\right ) R _ { \\mathrm { o } } L ^ { 3 } . \\end{align*}"}
-{"id": "7092.png", "formula": "\\begin{align*} k ( p ) = \\exp ( - 1 / p ) . \\end{align*}"}
-{"id": "2116.png", "formula": "\\begin{align*} \\| v \\| _ q & = \\left ( \\sum \\limits _ { j = 1 } ^ p | v _ j | ^ q \\right ) ^ { \\frac { 1 } { q } } , q < \\infty ; \\\\ \\| v \\| _ { \\infty } & = \\sup \\limits _ { 1 \\leq j \\leq p } | v _ j | . \\end{align*}"}
-{"id": "2224.png", "formula": "\\begin{align*} \\left | 1 - a _ { j i _ { j } } \\dfrac 1 { w _ j } \\right | = \\varepsilon \\end{align*}"}
-{"id": "6038.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\Omega ^ { ( \\alpha , \\lambda ) } ( \\{ Q _ { i } \\} _ { i = 1 } ^ { n } ) & = - \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\log \\Lambda _ { i } ^ { ( \\alpha , \\lambda ) } ( \\{ Q _ { j } \\} _ { j = 1 } ^ { i } ) \\\\ * & \\geq \\frac { \\Omega ^ { ( \\alpha , \\theta ) } } { 1 + 2 \\bar { \\alpha } \\theta } . \\end{align*}"}
-{"id": "4554.png", "formula": "\\begin{align*} & [ E _ i ( z ) , E _ j ( w ) ] = 0 \\ , , [ F _ i ( z ) , F _ j ( w ) ] = 0 \\ , ( i \\not \\equiv j , j \\pm 1 ) \\ , , \\\\ & d _ { i , j } g _ { i , j } ( z , w ) E _ i ( z ) E _ j ( w ) + g _ { j , i } ( w , z ) E _ j ( w ) E _ i ( z ) = 0 , \\\\ & d _ { j , i } g _ { j , i } ( w , z ) F _ i ( z ) F _ j ( w ) + g _ { i , j } ( z , w ) F _ j ( w ) F _ i ( z ) = 0 . \\end{align*}"}
-{"id": "6238.png", "formula": "\\begin{align*} \\P ( \\widehat { \\tau } _ 0 < \\infty , & \\widehat { X } ( \\widehat { \\tau } _ 0 - 1 ) = y , \\widehat { X } ( \\widehat { \\tau } _ 0 ) \\geq x , \\textrm { $ C $ c a u s e d t h e j u m p o f t h e p r o c e s s $ \\widehat { X } $ o v e r t h e l e v e l $ 0 $ } ) \\\\ & = \\P ( C ( 1 ) \\geq x + 1 - y ) \\cdot \\P ( Z ( 1 ) = - 1 ) + \\P ( C ( 1 ) \\geq x - y ) \\cdot \\P ( Z ( 1 ) \\geq 0 ) ~ . \\end{align*}"}
-{"id": "3050.png", "formula": "\\begin{align*} \\begin{cases} \\lambda ( w _ h , v _ h ) _ H + a ( w _ h , v _ h ) + b ( v _ h , \\pi _ h ) = ( g , v _ h ) _ H , & \\forall v \\in V _ h , \\\\ b ( w _ h , q _ h ) = 0 , & \\forall q _ h \\in Q _ h , \\end{cases} \\end{align*}"}
-{"id": "401.png", "formula": "\\begin{align*} S _ n & = \\lim _ { m \\to \\infty } \\sum _ { r , s \\in [ - m , m ] } \\bigg ( \\sum _ { ( j , k ) \\in \\Gamma _ n } a _ { j + r , k + s } \\bigg ) \\xi _ { - r , - s } \\\\ & = \\sum _ { r , s \\in \\mathbb { Z } } \\bigg ( \\sum _ { ( j , k ) \\in \\Gamma _ n } a _ { j + r , k + s } \\bigg ) \\xi _ { - r , - s } = \\sum _ { r , s \\in \\mathbb { Z } } b _ { n , r , s } \\xi _ { - r , - s } . \\end{align*}"}
-{"id": "9361.png", "formula": "\\begin{align*} x _ n & = r _ n \\mathrm { e } ^ { i 2 \\pi \\varphi n } , \\ , n \\in \\{ 1 , 2 , \\ldots , N \\} , \\end{align*}"}
-{"id": "775.png", "formula": "\\begin{align*} B _ { 2 k + 1 } ( 0 ) = B _ { 2 k + 1 } ( 1 ) = B _ { 2 k + 1 } = 0 , k = 1 , 2 , . . . \\end{align*}"}
-{"id": "5714.png", "formula": "\\begin{align*} \\mathcal { K } ( [ { v } ] ) = \\begin{cases} \\sqrt { \\mathfrak { E } _ { W } ( { v } ) - d _ K ( a ^ - , a ^ + ) } & { v } \\in H ^ 1 _ { l o c } ( \\R , \\R ^ n ) , \\\\ + \\infty & \\end{cases} \\end{align*}"}
-{"id": "1411.png", "formula": "\\begin{align*} I = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} , \\ ; \\xi = \\begin{pmatrix} - 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} , \\ ; \\eta = \\begin{pmatrix} i & 0 \\\\ 0 & - i \\end{pmatrix} , \\ ; \\zeta = \\begin{pmatrix} 0 & i \\\\ i & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "2742.png", "formula": "\\begin{align*} \\partial _ t f + \\frac { 1 } { \\epsilon ^ { \\alpha } } v \\cdot \\nabla _ x f = \\frac { \\sigma ( z ) } { \\epsilon ^ { 1 + \\alpha } } \\ , \\nabla _ v \\cdot ( \\nabla _ v f + f v ) \\ , , \\end{align*}"}
-{"id": "8281.png", "formula": "\\begin{align*} f ( \\tau ) = \\sum _ { \\substack { m \\in \\Q \\\\ m \\gg - \\infty } } c ( m ) \\cdot q ^ m \\in M ^ ! _ { 1 - \\frac { n } { 2 } } ( \\overline { \\rho } _ { V _ \\Z } ) \\end{align*}"}
-{"id": "9936.png", "formula": "\\begin{align*} \\begin{gathered} | ( F - F _ k ) ( s , x + z ) - ( F - F _ k ) ( s , x ) | \\leq c _ k \\min \\{ 1 , | z | \\} \\\\ | ( F - F _ k ) ( s , x + z ) - ( F - F _ k ) ( s , x ) - \\nabla _ x ( F - F _ k ) ( s , x ) \\cdot z | \\leq C \\min \\{ 1 , | z | ^ { \\gamma } \\} \\end{gathered} \\end{align*}"}
-{"id": "69.png", "formula": "\\begin{align*} \\frac { \\partial \\textbf { w } ^ { H } \\textbf { X X } ^ { H } \\textbf { w } } { \\partial \\textbf { w } ^ * } = \\frac { \\partial \\textbf { w } ^ T \\textbf { X } ^ * \\textbf { X } ^ T \\textbf { w } ^ * } { \\partial \\textbf { w } ^ * } = \\textbf { X } \\textbf { X } ^ H \\textbf { w } \\end{align*}"}
-{"id": "3285.png", "formula": "\\begin{align*} T = \\Big \\{ z = s { a \\over 2 } + t { b \\over 2 } \\ : \\ s , t \\in \\Big ( - { 1 \\over 2 } , { 1 \\over 2 } \\Big ) \\Big \\} , \\end{align*}"}
-{"id": "6871.png", "formula": "\\begin{align*} \\lambda _ { m + 1 } x _ 0 = \\left ( \\frac { 4 } { p q } \\right ) ^ { m + 1 } ( 2 x _ 0 - y _ 2 ( x _ 1 , x _ 0 , \\lambda _ { m + 1 } , p ) - y _ 1 ( x _ 0 , x _ 1 ' , \\lambda _ { m + 1 } , p ) ) \\end{align*}"}
-{"id": "7093.png", "formula": "\\begin{align*} k _ t = ( 1 - C ( p ) ) | \\nabla k | ^ 2 + k \\Delta k , \\end{align*}"}
-{"id": "166.png", "formula": "\\begin{align*} ( \\mathfrak { X } \\vee \\mathfrak { Y } ) ( a ) = \\sup \\{ \\mathfrak { X } ( a ) , \\mathfrak { Y } ( a ) \\} \\quad ( \\mathfrak { X } \\wedge \\mathfrak { Y } ) ( a ) = \\inf \\{ \\mathfrak { X } ( a ) , \\mathfrak { Y } ( a ) \\} , \\end{align*}"}
-{"id": "8302.png", "formula": "\\begin{align*} I H = \\mathrm { S p a n } _ \\Q \\{ \\ell x : \\ell \\in I , \\ , x \\in H \\} . \\end{align*}"}
-{"id": "1890.png", "formula": "\\begin{align*} \\max _ { F \\in \\mathcal { F } ^ { S , \\pi } _ { \\preceq _ 1 } ( F _ 1 ^ \\ast , \\dots , F _ d ^ \\ast ) } F ( x ) = \\min _ { i = 1 , \\dots , d } F ^ \\ast _ i ( x _ i ) \\wedge \\min \\{ \\pi _ s \\colon s \\in S x \\leq s \\} , \\end{align*}"}
-{"id": "4961.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\kappa } \\| \\partial _ i ( f - F ) \\| _ { \\dot { F } ^ { \\alpha - 1 , p } _ q } \\leq \\delta \\| f \\| _ { \\dot { F } ^ { \\alpha , p } _ q } , \\end{align*}"}
-{"id": "8013.png", "formula": "\\begin{align*} \\frac { 1 } { | Q | } \\int _ Q { \\big | \\mathfrak { S } _ { [ a ] } ^ { n e a r } f ( x ) - \\big ( \\mathfrak { S } _ { [ a ] } ^ { n e a r } f \\big ) _ Q \\big | } d x & \\leq { \\frac { 1 } { | Q | } \\int _ Q { \\sum _ { k = 0 } ^ { \\infty } { \\Big | T _ { [ d _ k ] } f ( x ) - \\frac { 1 } { | Q | } \\int _ Q { T _ { [ d _ k ] } f ( y ) } d y \\Big | } } d x } \\\\ & \\lesssim I + I I \\end{align*}"}
-{"id": "7960.png", "formula": "\\begin{align*} \\mu _ m = & \\sum _ { J _ 1 , \\ldots , J _ { \\ell } \\in \\mathcal I ^ m } \\lambda _ { J _ 1 } ^ s \\cdots \\lambda _ { J _ { \\ell } } ^ s ( g _ { J _ { 1 } } \\circ \\cdots \\circ g _ { \\ell } ) _ * ( \\mu _ m ) \\\\ = & \\sum _ { I _ i \\in \\mathcal S , \\alpha _ i \\in A _ { I _ i } } \\lambda _ { I _ 1 J _ { \\alpha _ 1 } } ^ s \\cdots \\lambda _ { I _ { \\ell } J _ { \\alpha _ { \\ell } } } ^ s ( g _ { J _ { 1 } } \\circ \\cdots \\circ g _ { J _ { \\ell } } ) _ { * } ( \\mu _ m ) . \\end{align*}"}
-{"id": "8808.png", "formula": "\\begin{align*} \\Big | \\sum _ { i = 1 } ^ { r } a _ { i , p } m ^ { \\beta _ { i } } p ^ { ( \\sigma - \\lambda _ { i } ) m } \\Big | \\leq C _ { m } . \\end{align*}"}
-{"id": "437.png", "formula": "\\begin{align*} p _ { 1 , k _ 1 , k _ 2 } ( x , t ) \\ ! = \\ ! \\frac { ( - 1 ) ^ { k _ 2 } 2 \\pi ^ { k _ 2 - n } } { 4 ^ { n + 1 } \\delta ^ { n + k _ 1 - 1 } } e ^ { - R - \\pi \\abs * { t } } H _ { k _ 1 , k _ 2 } ( R , t ) + O \\left ( e ^ { - \\frac { 3 \\pi \\abs * { t } } { 2 } } \\right ) . \\end{align*}"}
-{"id": "8320.png", "formula": "\\begin{align*} \\Psi _ g ( f _ 1 + f _ 2 ) = \\Psi _ g ( f _ 1 ) \\otimes \\Psi _ g ( f _ 2 ) \\end{align*}"}
-{"id": "2686.png", "formula": "\\begin{align*} H ^ 2 _ { \\rm { \\acute { e } t } } ( X _ { \\overline { K } } , \\mathbb { A } _ { f } { ( 1 ) } ) : = ( \\prod _ { p } H ^ 2 _ { \\rm { \\acute { e } t } } ( X _ { \\overline { K } } , \\Z _ p ( 1 ) ) ) \\otimes _ { \\Z } { \\Q } , \\end{align*}"}
-{"id": "9383.png", "formula": "\\begin{align*} \\sum _ { \\{ j \\ , , \\ , \\alpha _ j < 2 m \\} } ( \\lambda ( \\alpha ) - \\lambda _ j ) c _ j ^ 2 & \\ge C _ 2 \\sum _ { \\{ j \\ , , \\ , \\alpha _ j < 2 m \\} } c _ j ^ 2 = { C _ 2 } \\| Q \\| _ { L ^ 2 ( \\partial B _ 1 ) } ^ { 2 } \\geq \\frac { C _ 2 } { C _ 1 } M ^ 2 , \\end{align*}"}
-{"id": "3021.png", "formula": "\\begin{align*} 0 = 2 \\Re \\ < A x , x \\ > = 2 \\ < l , ( - I + V ^ * V ) l ) \\ > _ { H ^ N } \\ ; l \\in H ^ N \\end{align*}"}
-{"id": "7853.png", "formula": "\\begin{align*} M _ { w } ( t _ { 1 } , t _ { 2 } ) = \\left \\{ \\begin{array} { l } M _ { w _ { 1 } } ( t _ { 1 } , t _ { 2 } ) \\ \\ \\ \\ \\ \\ \\ \\ \\ t _ { 1 } > t _ { 2 } > 0 \\\\ M _ { w _ { 2 } } ( t _ { 1 } , t _ { 2 } ) \\ \\ \\ \\ \\ \\ \\ \\ \\ 0 < t _ { 1 } < t _ { 2 } \\\\ M _ { w _ { 3 } } ( t , t ) \\ \\ \\ \\ \\ \\ \\ t _ { 1 } = t _ { 2 } = t , \\end{array} \\right . \\end{align*}"}
-{"id": "9378.png", "formula": "\\begin{align*} \\sum ( \\lambda _ j - \\lambda ( \\alpha ) ) c _ j ^ 2 & = \\sum \\lambda _ j c _ j ^ 2 - \\lambda ( \\alpha ) \\sum c _ j ^ 2 \\ge \\sum \\lambda _ j c _ j ^ 2 - \\frac { \\lambda _ \\alpha } { \\lambda ( 2 m + 1 ) } \\sum \\lambda _ j c _ j ^ 2 \\\\ & \\ge \\sum \\lambda _ j c _ j ^ 2 - \\frac { \\lambda ( 2 m + \\frac 1 2 ) } { \\lambda ( 2 m + 1 ) } \\sum \\lambda _ j c _ j ^ 2 \\ge C _ 2 \\sum \\lambda _ j c _ j ^ 2 = C _ 2 \\| \\nabla _ \\theta \\phi \\| _ { L ^ 2 ( \\partial B _ 1 ) } ^ { 2 } , \\end{align*}"}
-{"id": "9150.png", "formula": "\\begin{align*} \\begin{array} { r c l } f _ { 5 } - D _ { 4 } & = & 0 \\\\ 5 \\ , D _ { 4 } + f _ { 4 } + D _ { 3 } & = & 0 \\\\ - 1 0 \\ , D _ { 4 } - 4 \\ , D _ { 3 } + f _ { 3 } - D _ { 2 } & = & 0 \\\\ 1 0 \\ , D _ { 4 } + 6 \\ , D _ { 3 } + 3 \\ , D _ { 2 } + f _ { 2 } + D _ { 1 } & = & 0 \\\\ - 5 \\ , D _ { 4 } - 4 \\ , D _ { 3 } - 3 \\ , D _ { 2 } - 2 \\ , D _ { 1 } + f _ { 1 } - D _ { 0 } & = & 0 , \\end{array} \\end{align*}"}
-{"id": "5280.png", "formula": "\\begin{align*} \\frac { 1 } { \\mu } \\langle \\mathcal { A } u , u \\rangle = & \\int _ { 0 } ^ 1 u _ { x x } u x \\\\ = & u _ { x } ( t , 1 ) u ( t , 1 ) - u _ { x } ( t , 0 ) u ( t , 0 ) - \\| u _ x \\| ^ 2 , \\end{align*}"}
-{"id": "1077.png", "formula": "\\begin{align*} \\varphi _ k ( \\mathbf { r } , z ) = \\sum \\limits _ { i = - \\infty } ^ { + \\infty } \\alpha _ { k i } u _ i ( \\mathbf { r } , z ) , \\end{align*}"}
-{"id": "6694.png", "formula": "\\begin{align*} c _ 3 = \\left [ \\frac { n ( n - 1 + p ' ) } { p ' - 1 } \\cdot \\frac { | B _ { p ' } ^ n | _ n } { | B _ { p ' } ^ { n - 1 } | _ { n - 1 } } \\right ] ^ { \\frac { p ' } { n - 1 + p ' } } \\cdot \\frac { p ' - 1 } { p ' } , & c _ 4 = n ^ { \\frac { 2 } { n + 1 } } G _ { B _ { p ' } ^ n } ( ( n ^ { - 1 / p ' } , \\dots , n ^ { - 1 / p ' } ) ) \\quad . \\end{align*}"}
-{"id": "6282.png", "formula": "\\begin{align*} Q = C \\log \\Lambda _ r \\leq C ( r , \\nu , s ) \\left ( 1 + \\log \\| u _ Q \\| _ { H ^ s } \\right ) . \\end{align*}"}
-{"id": "3779.png", "formula": "\\begin{align*} R _ { k + 1 } & : = \\inf \\left \\{ n \\ge R _ k + 1 \\colon \\ , \\left ( X _ n - X _ { R _ k } \\right ) \\cdot e _ 1 > \\bar { v } ( n - R _ k ) \\right \\} . \\end{align*}"}
-{"id": "3896.png", "formula": "\\begin{align*} \\left ( - \\frac { \\hbar ^ 2 } { 2 m } \\left ( \\frac { d ^ 2 } { d r ^ 2 } + \\frac { ( d - 1 ) } { r } \\frac { d } { d r } - \\frac { \\ell ( \\ell + d - 2 ) } { r ^ 2 } \\right ) + V ( r ) - E \\right ) R ( r ) = 0 . \\end{align*}"}
-{"id": "9179.png", "formula": "\\begin{align*} \\left . { \\frac { \\partial } { \\partial s } } \\right | _ { s = 0 } ( I _ { X } - J _ { X } ) ( \\phi _ { t \\ , s } ) = - { \\frac { 1 } { 1 - t } } \\int _ M Y ( h _ t ) \\omega _ { \\phi _ t } ^ n - { \\frac { 1 } { 1 - t } } \\int _ M \\theta _ X ( \\omega _ { \\phi _ t } ) \\theta _ Y ( \\omega _ { \\phi _ t } ) \\omega _ { \\phi _ t } ^ n = 0 . \\end{align*}"}
-{"id": "7267.png", "formula": "\\begin{align*} B _ 2 & \\le 2 \\int _ 0 ^ { ( n - m ) s / N } \\frac { 2 s } { N n } \\d h + 2 \\int _ { ( n - m ) s / N } ^ { ( n + m ) s / N } \\Bigl ( \\frac s N \\Bigl ( \\frac 1 n + \\frac 1 m \\Bigr ) - \\frac h { m n } \\Bigr ) \\d h \\\\ & = \\frac { 4 s ^ 2 } { N ^ 2 } \\frac { n - m } n + \\frac { 4 s ^ 2 } { N ^ 2 } m \\Bigl ( \\frac 1 n + \\frac 1 m \\Bigr ) - \\frac 1 { m n } \\frac { s ^ 2 } { N ^ 2 } ( ( n + m ) ^ 2 - ( n - m ) ^ 2 ) , \\end{align*}"}
-{"id": "8920.png", "formula": "\\begin{gather*} \\vartheta ( z ; \\tau ) : = \\frac { ( e ( z / 2 ) - e ( - z / 2 ) ) \\prod _ { 1 \\le j } ( 1 - e ( j \\tau + z ) ) ( 1 - e ( j \\tau - z ) ) } { \\prod _ { 1 \\le j } ( 1 - e ( j \\tau ) ) ^ 2 } \\\\ \\phantom { \\vartheta ( z ; \\tau ) { : } } { } = \\frac { \\sum _ { k \\in 1 / 2 + \\Z } ( - 1 ) ^ { k - 1 / 2 } e ( k z + k ^ 2 \\tau / 2 ) } { \\sum _ { k \\in 1 / 2 + \\Z } ( - 1 ) ^ { k - 1 / 2 } k e ( k ^ 2 \\tau / 2 ) } \\end{gather*}"}
-{"id": "5000.png", "formula": "\\begin{align*} \\partial _ { } ^ \\gamma H _ m = \\sum _ { \\substack { m ' < m \\\\ m ' \\equiv m ( \\textrm { m o d } R ) } } \\left ( \\partial _ { } ^ { \\gamma } g _ { m ' } - \\sum _ { 0 < \\gamma ' \\leq \\gamma } c _ { \\gamma ' , \\gamma } \\partial _ { } ^ { \\gamma ' } G _ { m ' } \\partial _ { } ^ { \\gamma - \\gamma ' } H _ { m ' } \\right ) \\prod _ { \\substack { m ' < m '' < m \\\\ m '' \\equiv m ( \\textrm { m o d } R ) } } ( 1 - G _ { m '' } ) . \\end{align*}"}
-{"id": "121.png", "formula": "\\begin{align*} A = \\{ x \\in \\mathbb { R } ^ n : \\langle a , x \\rangle + b \\ge 0 \\} , B = \\{ x \\in \\mathbb { R } ^ n : \\langle a , x \\rangle + c \\ge 0 \\} \\end{align*}"}
-{"id": "4023.png", "formula": "\\begin{align*} \\left . \\frac { \\partial } { \\partial t } \\omega _ t ( X ^ t , Y ^ t ) \\right | _ { t = 0 } = g ( \\lambda _ 1 + \\lambda _ 2 ) \\mathrm { d e t } [ X , Y , \\eta ] . \\end{align*}"}
-{"id": "6762.png", "formula": "\\begin{align*} [ l : k ] \\log | \\mu | _ { w _ v } = \\log | \\beta | _ v . \\end{align*}"}
-{"id": "472.png", "formula": "\\begin{align*} ( \\psi _ { \\pi / 2 } '' ( 0 ) ^ { - 1 } \\partial , \\partial ) ^ { ( k _ 1 + 1 ) / 2 } & a _ { k _ 1 , k _ 2 , \\pi / 2 } ( 0 ) \\\\ & = ( - 1 ) ^ { k _ 1 } \\frac { i ^ { k _ 2 - n - \\frac { k _ 1 + 1 } { 2 } } } { 2 ^ { \\frac { k _ 1 + 1 } { 2 } } } \\left ( i \\frac { \\pi } { 2 } \\right ) ^ { n + k _ 1 + k _ 2 - 1 } ( k _ 1 + 1 ) ! ( n + k _ 1 + k _ 2 + m - 1 ) . \\end{align*}"}
-{"id": "789.png", "formula": "\\begin{align*} ( \\rho , \\mu ) = ( 7 . 7 , 0 . 5 7 ) . \\end{align*}"}
-{"id": "7790.png", "formula": "\\begin{align*} \\Vert x - \\tilde x _ h \\Vert _ X = \\inf _ { \\xi _ h \\in D _ h } \\Vert x - \\xi _ h \\Vert _ X \\leq \\Vert x - x _ h \\Vert _ X < \\varepsilon . \\end{align*}"}
-{"id": "8415.png", "formula": "\\begin{align*} g _ { t , l , v } = \\left ( \\begin{matrix} 0 & \\varpi ^ t \\\\ - 1 & - v \\varpi ^ { - l } \\end{matrix} \\right ) l _ n = \\min ( l , n - l ) \\end{align*}"}
-{"id": "7543.png", "formula": "\\begin{align*} \\varphi ^ 1 = - 2 ( 1 + \\omega ^ 2 ) \\frac { \\psi _ \\omega } \\psi \\quad \\mbox { w i t h } \\psi = \\psi ( \\omega ) \\end{align*}"}
-{"id": "9065.png", "formula": "\\begin{align*} 2 L _ i \\equiv D _ j + D _ k , D _ k + L _ k \\equiv L _ i + L _ j . \\{ i , j , k \\} = \\{ 1 , 2 , 3 \\} \\end{align*}"}
-{"id": "9969.png", "formula": "\\begin{align*} \\mathbf { \\Gamma } _ { j i } & = \\begin{cases} P _ u \\mathbf { I } _ { K _ { j } } , & j = i \\\\ P _ u \\bar { \\mathbf { L } } _ { j i } , & j \\neq i \\end{cases} , \\end{align*}"}
-{"id": "3983.png", "formula": "\\begin{align*} p ^ { \\beta _ 0 } _ { k } ( 0 , t ) = \\frac { ( - \\lambda t ^ { \\beta _ 0 } ) ^ { k } } { \\Gamma ( k \\beta _ 0 + 1 ) } , \\ \\ k \\geq 0 . \\end{align*}"}
-{"id": "8856.png", "formula": "\\begin{align*} g _ \\phi ( x ) = \\sum _ { n = 1 } ^ \\infty a _ n ( \\phi ) ^ 2 Z ( d , n ) P _ n ^ { ( d ) } ( x ) , - 1 \\leq x \\leq 1 , \\end{align*}"}
-{"id": "8821.png", "formula": "\\begin{align*} c _ { I , \\chi _ { } } ^ 0 = \\sum _ { a \\in \\overset { \\circ } { \\overline { E } } _ { I } \\cap \\overline { h } ^ { - 1 } ( 0 ) } \\Omega _ { \\chi _ { } } ( a ) = \\# \\big ( \\overset { \\circ } { \\overline { E } } _ { I } \\cap \\overline { h } ^ { - 1 } ( 0 ) \\big ) \\leq \\# \\big ( \\overset { \\circ } { \\overline { E } } _ { I } \\big ) \\leq D _ { I } p ^ { n - \\# I } . \\end{align*}"}
-{"id": "768.png", "formula": "\\begin{align*} ( x + 1 ) ^ k + ( x + 2 ) ^ k + . . . + ( x + d ) ^ k = y ^ { n } \\end{align*}"}
-{"id": "8279.png", "formula": "\\begin{align*} \\Z _ \\Omega = \\Z [ 1 / p : p \\in \\Omega ] . \\end{align*}"}
-{"id": "6908.png", "formula": "\\begin{align*} \\alpha = \\frac { \\log ( 9 ) } { \\log ( L ( r ) ^ { - 1 } ) } \\end{align*}"}
-{"id": "610.png", "formula": "\\begin{align*} \\dim L _ q ^ n ( m , s ) = \\sum _ r \\dim C _ q ^ n ( m , r ) d _ { r , s } \\end{align*}"}
-{"id": "9901.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\tfrac { k _ i } { k _ i + 1 } \\le \\tfrac { r - 1 } { 2 } \\ , . \\end{align*}"}
-{"id": "6791.png", "formula": "\\begin{align*} - \\Delta _ g W _ { \\lambda , k } ( y ) = - \\frac { ( 1 + | x | ^ 2 ) ^ 2 } { 4 } \\Delta _ { \\mathbb { R } ^ 2 } W _ { \\lambda , k } ( \\Pi ^ { - 1 } _ { \\xi _ k } ( x ) ) \\end{align*}"}
-{"id": "1580.png", "formula": "\\begin{align*} v ( m , \\mathbf { j } ' ) - v ( m , \\mathbf { j } ) & = ( n - \\nu ( m ) + 1 ) ( j _ { n - \\nu ( m ) } + p ) - s ( j _ { n - \\nu ( m ) } + p ) \\\\ & - ( n - \\nu ( m ) + 1 ) j _ { n - \\nu ( m ) } + s ( j _ { n - \\nu ( m ) } ) \\\\ & = ( n - \\nu ( m ) + 1 ) p + s ( j _ { n - \\nu ( m ) } ) - s ( j _ { n - \\nu ( m ) } + p ) \\\\ & = ( n - \\nu ( m ) + 1 ) p + c ( j _ { n - \\nu ( m ) } , p ) - s ( p ) \\\\ & \\geq ( n - \\nu ( m ) + 1 ) p - s ( p ) \\\\ & > 0 . \\end{align*}"}
-{"id": "1344.png", "formula": "\\begin{align*} M = P \\Lambda ( M ) P ^ T , \\end{align*}"}
-{"id": "8639.png", "formula": "\\begin{align*} \\hat { \\pi } ( x , d y ) = \\frac { e ^ { I ( x , y ) } \\Psi ( y ) \\pi ( d y ) } { \\rho \\Psi ( x ) } . \\end{align*}"}
-{"id": "1016.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { N + 1 } ( 1 - \\beta _ j ) E _ h ^ j = \\sum _ { j = 1 } ^ { N + 1 } \\beta _ j p _ s ^ j . \\end{align*}"}
-{"id": "5698.png", "formula": "\\begin{align*} \\lim \\limits _ { S \\to \\infty } \\limsup \\limits _ { n \\to \\infty } \\mathfrak { E } _ { W } ( { v } _ n , \\R \\setminus [ - S , S ] ) = 0 . \\end{align*}"}
-{"id": "4486.png", "formula": "\\begin{align*} \\psi ( x _ 0 ) = 1 > \\Lambda ( x _ 0 ) \\end{align*}"}
-{"id": "7688.png", "formula": "\\begin{align*} K = \\left [ \\begin{array} { c | c | c } P _ 1 & & \\\\ \\hline & \\ddots & \\\\ \\hline & & P _ { N - 1 } \\end{array} \\right ] , \\end{align*}"}
-{"id": "6046.png", "formula": "\\begin{align*} \\frac { 1 } { \\theta } \\Omega ^ { ( \\alpha , \\theta ) } = R ^ { ( \\alpha ) } + \\epsilon ^ { ( \\alpha , \\theta ) } , \\end{align*}"}
-{"id": "3392.png", "formula": "\\begin{align*} \\alpha ( V \\cap l _ s ' ) = \\{ y = 0 \\} , \\alpha ( V \\cap k _ s ' ) = \\{ y = | x | ^ a - s \\} \\end{align*}"}
-{"id": "1009.png", "formula": "\\begin{align*} \\Delta _ { j k } = \\frac { 2 } { 3 \\mathfrak { s } _ n ( s ^ n _ j ) \\mathfrak { s } _ n ( s ^ n _ k ) } \\sum _ { i = 1 } ^ n K _ h ( t _ { i - 1 } - s ^ n _ j ) K _ h ( t _ { i - 1 } - s ^ n _ k ) \\left \\{ ( X _ { t _ i } - X _ { t _ { i - 1 } } ) ^ 4 - E [ ( X _ { t _ i } - X _ { t _ { i - 1 } } ) ^ 4 ] \\right \\} , \\end{align*}"}
-{"id": "499.png", "formula": "\\begin{align*} I ' _ { \\nu } ( \\zeta ) = \\frac { I _ { \\nu - 1 } ( \\zeta ) + I _ { \\nu + 1 } ( \\zeta ) } { 2 } , \\end{align*}"}
-{"id": "9144.png", "formula": "\\begin{align*} 3 f ( x _ { 1 } x _ { 2 } x _ { 3 } ) + x _ { 1 } x _ { 2 } g ( x _ { 3 } ) + x _ { 1 } x _ { 3 } g ( x _ { 2 } ) + x _ { 2 } x _ { 3 } g ( x _ { 1 } ) = 0 \\left ( x _ { 1 } , x _ { 2 } , x _ { 3 } \\in R \\right ) . \\end{align*}"}
-{"id": "3013.png", "formula": "\\begin{align*} P _ u ( \\zeta ) : = \\sum _ { i = 0 } ^ N \\frac { u _ { i + 1 } } { i ! } ( \\zeta - 1 ) ^ { i } P _ v ( \\zeta ) : = \\sum _ { i = 0 } ^ N \\frac { v _ { i + 1 } } { i ! } \\zeta ^ { i } . \\end{align*}"}
-{"id": "10123.png", "formula": "\\begin{align*} { \\boldsymbol { \\bar { \\psi } } } _ k ( i ) = \\bar { \\boldsymbol \\omega } _ k ( i - 1 ) + \\mu ( i ) e _ k ^ * ( i ) \\bar { \\boldsymbol x } _ k ( i ) , \\end{align*}"}
-{"id": "6066.png", "formula": "\\begin{align*} \\delta _ u \\mathcal { S } ^ n _ 0 = \\delta _ u \\mathcal { S } ^ n \\cap \\mathring { H } ^ 1 _ 2 . \\end{align*}"}
-{"id": "4945.png", "formula": "\\begin{align*} k ^ 2 - 3 ^ { y - 2 e } = 1 \\end{align*}"}
-{"id": "1922.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\Vert ( \\textbf { F } | _ { \\Gamma ^ { 1 } } ) ^ { n } \\Vert ^ { 1 / n } = \\lambda _ { 1 } \\lim _ { n \\rightarrow \\infty } \\Vert ( \\textbf { F } | _ { \\Gamma ^ { 2 } } ) ^ { - n } \\Vert ^ { 1 / n } = \\lambda _ { 2 } ; \\end{align*}"}
-{"id": "1040.png", "formula": "\\begin{align*} \\underline { \\psi } _ { m , j } ^ { \\ast } = \\liminf _ { Q \\to \\infty } \\psi _ { m , j } ^ { \\ast } ( Q ) , \\overline { \\psi } _ { m , j } ^ { \\ast } = \\limsup _ { Q \\to \\infty } \\psi _ { m , j } ^ { \\ast } ( Q ) . \\end{align*}"}
-{"id": "1039.png", "formula": "\\begin{align*} \\mathcal { D } _ { n } ( 2 n - d ) = 2 n - d + o ( 1 ) , \\mathcal { E } _ { n } ( 2 n - d ) = 2 n - \\frac { d + 1 } { 2 } + o ( 1 ) , \\end{align*}"}
-{"id": "5112.png", "formula": "\\begin{align*} { \\mathcal A } ( M _ { j _ 1 i _ 1 } ^ { ( t ) } v _ { i _ 1 } , \\ldots , M _ { j _ k i _ k } ^ { ( t ) } v _ { i _ k } ) = t ^ { - k } { \\mathcal A } ( M _ { j _ 1 i _ 1 } v _ { i _ 1 } , \\ldots , M _ { j _ k i _ k } v _ { i _ k } ) \\end{align*}"}
-{"id": "8332.png", "formula": "\\begin{align*} ( 2 \\pi i ) ^ { c ( 0 , 0 ) / 2 } \\cdot \\mathrm { F J } ^ { ( a ) } ( \\Psi ( f ) ) = \\kappa ^ { ( a ) } A ^ { \\mathrm { r e c } ( a ) } q _ { \\alpha ( \\varrho ) } \\cdot \\mathrm { B P } ( f ) ^ { \\mathrm { r e c } ( a ) } . \\end{align*}"}
-{"id": "1480.png", "formula": "\\begin{align*} w '' + p ( z ) w = 0 . \\end{align*}"}
-{"id": "7740.png", "formula": "\\begin{align*} d ^ 2 _ B ( j , k ) = & \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N - 1 } \\frac { [ \\cos ( j + \\frac { 1 } { 2 } ) \\phi _ n - \\cos ( k + \\frac { 1 } { 2 } ) \\phi _ n ] ^ 2 } { 2 ( 1 - \\cos \\phi _ n ) ^ 2 } \\\\ = & \\frac { 1 } { 2 } \\Big ( F _ N ( j + k + 1 ) + F _ N ( j - k ) \\\\ & - \\frac { 1 } { 2 } F _ N ( 2 j + 1 ) - \\frac { 1 } { 2 } F _ N ( 2 k + 1 ) \\Big ) \\end{align*}"}
-{"id": "2157.png", "formula": "\\begin{align*} \\frac { d } { d s } w ( e ^ { - \\theta _ * s } , s ) & = \\frac { d } { 2 } w ( e ^ { - \\theta _ * s } , s ) + \\frac { U _ d ' ( r ( s ) ) } { U _ d ( r ( s ) ) } r ' ( s ) w ( e ^ { - \\theta _ * s } , s ) \\\\ & + e ^ { \\frac { d } { 2 } s } U _ d ( r ( s ) ) ( \\partial _ t u _ * ) ( 0 , t ( s ) ) t ' ( s ) \\\\ & + e ^ { \\frac { d } { 2 } s } U _ d ( r ( s ) ) [ ( \\partial _ r F _ N ^ 0 ) ( r ( s ) , t ( s ) ) r ' ( s ) + ( \\partial _ t F _ N ^ 0 ) ( r ( s ) , t ( s ) ) t ' ( s ) ] \\end{align*}"}
-{"id": "982.png", "formula": "\\begin{align*} \\| Y \\| _ { \\psi _ 1 } = \\inf \\{ C > 0 : \\psi _ 1 ( | Y | / C ) \\leq 1 \\} . \\end{align*}"}
-{"id": "346.png", "formula": "\\begin{align*} & \\quad \\ 4 h ^ { i j } h _ { j l } R _ { \\ m i \\ } ^ { l \\ \\ \\ m } - 4 h ^ { i j } h ^ { l m } R _ { i l j m } \\\\ & = 4 h _ { 2 2 } h _ { 2 2 } R _ { 2 1 2 1 } + 4 h _ { 1 1 } h _ { 1 1 } R _ { 1 2 1 2 } - 4 h _ { 1 1 } h _ { 2 2 } R _ { 1 2 1 2 } - 4 h _ { 2 2 } h _ { 1 1 } R _ { 2 1 2 1 } \\\\ & = 4 R _ { 1 2 1 2 } ( h _ { 2 2 } ^ 2 + h _ { 1 1 } ^ 2 - 2 h _ { 1 1 } h _ { 2 2 } ) = 4 R _ { 1 2 1 2 } S ^ 2 . \\end{align*}"}
-{"id": "9664.png", "formula": "\\begin{align*} \\Omega = \\{ ( x , y ) : y < g ( x ) , x > 0 \\} , \\Gamma = \\{ ( x , y ) : y = g ( x ) , x \\geq 0 \\} , \\end{align*}"}
-{"id": "9738.png", "formula": "\\begin{align*} P _ { k , n } = ( k h , y _ { k , n } ) , \\end{align*}"}
-{"id": "3631.png", "formula": "\\begin{align*} 0 < \\int _ C f ( u ) h _ \\varepsilon ( | \\nabla u | ) \\psi ^ 2 _ \\varepsilon \\ , d x & = \\ , \\int _ C | \\nabla u | ^ { p - 2 } ( \\nabla u , \\nabla | \\nabla u | ) h _ { \\varepsilon } ' ( | \\nabla u | ) \\psi _ \\varepsilon ^ 2 \\ , d x \\ , \\\\ & + 2 \\int _ C | \\nabla u | ^ { p - 2 } ( \\nabla u , \\nabla \\psi _ \\varepsilon ) h _ { \\varepsilon } ( | \\nabla u | ) \\psi _ \\varepsilon \\ , d x . \\end{align*}"}
-{"id": "3693.png", "formula": "\\begin{align*} d _ { I , j } = - ( n - j ) \\cdot \\# ( I ) . \\end{align*}"}
-{"id": "8101.png", "formula": "\\begin{align*} A _ { i , j } = \\bigg ( \\bigg ( A \\setminus \\bigsqcup _ { k = 0 } ^ { i - 1 } \\bigsqcup _ { l = 1 } ^ { j _ i } A _ { k , l } \\bigg ) \\cap C _ { i , j } \\bigg ) \\setminus ( C _ { i , 1 } \\cup \\cdots \\cup C _ { i , j - 1 } ) . \\end{align*}"}
-{"id": "1467.png", "formula": "\\begin{align*} d _ { 2 ^ { r + 2 } - 1 } ( x ) = v _ { r + 1 } Q _ { r + 1 } ( x ) \\in E _ { 2 ^ { r + 2 } - 1 } ^ { * , - 2 ^ { r + 2 } + 2 } \\end{align*}"}
-{"id": "9320.png", "formula": "\\begin{align*} z = - \\frac { a _ p } { r - 1 } r ^ p + \\sum _ { i = - \\infty } ^ { p - 1 } a _ { i } r ^ i \\end{align*}"}
-{"id": "8914.png", "formula": "\\begin{align*} K = ( 1 - c ) ( \\cdot , \\overline \\chi \\theta ) { } , \\end{align*}"}
-{"id": "5840.png", "formula": "\\begin{align*} f ( 0 ) = 1 , f ' ( 0 ) = 0 , f ' ( \\tfrac 1 2 ) = 0 . \\end{align*}"}
-{"id": "312.png", "formula": "\\begin{align*} \\mathcal { F } _ n & = \\sigma ( U _ 1 , . . . , U _ n , \\xi _ 1 , \\dots , \\xi _ n ) , \\\\ \\mathcal { G } _ n & = \\sigma ( U _ 1 , . . . , U _ n , \\xi _ 1 , \\dots \\xi _ { n - 1 } ) , \\end{align*}"}
-{"id": "1854.png", "formula": "\\begin{align*} \\mathcal { G F } _ { \\mathcal { B } } ( R ) \\cap ( \\mathcal { P G F } _ { \\mathcal { B } } ( R ) ) ^ \\perp = \\mathcal { F } ( R ) , \\end{align*}"}
-{"id": "2718.png", "formula": "\\begin{align*} \\left | \\langle \\partial ^ { m } \\partial _ l ^ j \\mathcal F ( h , h ) , \\ , f \\rangle _ { L ^ 2 _ { x , v } } \\right | \\leq \\begin{cases} & \\displaystyle \\mathcal G _ { x , v , z } ^ { s , m } ( h , h ) \\ , | | f | | _ { \\Lambda } \\ , , j \\neq 0 , \\\\ [ 2 p t ] & \\displaystyle \\mathcal G _ { x , z } ^ { s , m } ( h , h ) \\ , | | f | | _ { \\Lambda } \\ , , \\qquad j = 0 . \\end{cases} \\end{align*}"}
-{"id": "9063.png", "formula": "\\begin{align*} A _ { { \\mathbf x } } { \\mathcal H } ( w , w ' ) = ( a _ { 1 1 } ^ 1 { \\mathbf x } _ 1 ( \\alpha \\bar w _ 1 w _ 1 ' + \\beta \\bar w _ 2 w _ 2 ' + \\beta \\bar w _ 3 w _ 3 ' ) , a _ { 2 2 } ^ 2 { \\mathbf x } _ 2 ( \\delta \\bar w _ 2 w _ 2 ' + \\delta \\bar w _ 3 w _ 3 ' ) ) . \\end{align*}"}
-{"id": "61.png", "formula": "\\begin{align*} J _ { M C C C } = V ^ { C } _ { \\sigma } ( D , Y ) = E _ { D Y } [ G ^ { C } _ { \\sigma \\ , \\sqrt { 2 } } ( D - \\textbf { w } ^ { H } \\textbf { X } ) ] = E _ { D Y } [ G ^ { C } _ { \\sigma \\ , \\sqrt { 2 } } ( e ) ] \\end{align*}"}
-{"id": "4555.png", "formula": "\\begin{align*} & [ H _ { i , r } , E _ j ( z ) ] = a _ { i , j } ( r ) C ^ { - ( r + | r | ) / 2 } \\ , z ^ r E _ j ( z ) \\ , , \\\\ & [ H _ { i , r } , F _ j ( z ) ] = - a _ { i , j } ( r ) C ^ { - ( r - | r | ) / 2 } \\ , z ^ r F _ j ( z ) \\ , , \\\\ & [ H _ { i , r } , H _ { j , s } ] = \\delta _ { r + s , 0 } \\cdot a _ { i , j } ( r ) \\eta _ r \\ , , \\eta _ r = \\frac { C ^ r - C ^ { - r } } { q - q ^ { - 1 } } \\ , . \\end{align*}"}
-{"id": "5225.png", "formula": "\\begin{align*} \\Omega : = e ^ { c z } \\omega \\end{align*}"}
-{"id": "772.png", "formula": "\\begin{align*} T _ { k } ( x ) = ( x + 1 ) ^ k + ( x + 2 ) ^ k + . . . + ( 2 x ) ^ k \\end{align*}"}
-{"id": "3499.png", "formula": "\\begin{align*} J ( u ( \\cdot ) ) = \\displaystyle \\int _ 0 ^ T f ( X ( t ) , u ( t ) ) d t + \\Psi ( X ( T ) ) . \\end{align*}"}
-{"id": "4640.png", "formula": "\\begin{align*} t + \\sum _ { i = 0 } ^ { n - 1 } f ( T ^ i x ) \\in [ - B + C , B - C ] \\Rightarrow \\ , & t + \\sum _ { i = 0 } ^ { n - 1 } g ( T ^ i x ) \\in [ - B , B ] \\\\ t + \\sum _ { i = 0 } ^ { n - 1 } g ( T ^ i x ) \\in [ - B , B ] \\Rightarrow \\ , & t + \\sum _ { i = 0 } ^ { n - 1 } f ( T ^ i x ) \\in [ - B - C , B + C ] \\end{align*}"}
-{"id": "4082.png", "formula": "\\begin{align*} M _ { \\varphi } ( \\Theta ) : = \\left \\{ ( x , y , u , v ) \\in \\mathbb { R } ^ { n _ { x } } \\times \\mathbb { R } ^ { n _ { y } } \\times U \\times \\mathbb { R } ^ { n _ { x } + n _ { y } } : \\varphi ( x , y , u , v ) + \\Theta = 0 \\right \\} , \\end{align*}"}
-{"id": "4821.png", "formula": "\\begin{align*} M _ 1 = \\bigg \\langle \\frac { 1 } { p } \\ \\bigg { | } \\ p \\ \\bigg \\rangle , \\ M _ 2 = \\bigg \\langle \\frac { p - 1 } { p } \\ \\bigg { | } \\ p \\ \\bigg \\rangle , \\ \\ M _ 3 = \\bigg \\langle \\frac { p ^ 2 + 1 } { p } \\ \\bigg { | } \\ p \\ \\bigg \\rangle . \\end{align*}"}
-{"id": "3550.png", "formula": "\\begin{align*} I ( S _ { B _ { \\boldsymbol { t } _ 0 } } \\in U ) = \\sum _ { \\boldsymbol { t } \\in B _ { \\boldsymbol { t } _ 0 } } \\delta _ { \\boldsymbol { t } } ( U ) . \\end{align*}"}
-{"id": "5927.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } s ^ I ( N , r _ 1 ^ { ( N ) } ) = \\lim _ { N \\to \\infty } s ^ D ( N , r _ 1 ^ { ( N ) } ) = \\begin{cases} \\mu _ 0 , \\ \\ I _ 0 ( r _ 1 ) > I _ 1 ( r _ 1 ) , \\\\ \\mu _ 1 , \\ \\ I _ 1 ( r _ 1 ) > I _ 0 ( r _ 1 ) . \\end{cases} \\end{align*}"}
-{"id": "7378.png", "formula": "\\begin{align*} V _ { S e l b } ( z ) = \\frac { 1 } { t } \\log \\big ( z ^ { - t } ( 1 + z ) ^ { 2 t } \\big ) , \\Delta ^ { ( S e l b ) } _ m = \\sum _ { i = 0 } ^ { \\big [ \\frac { N - m } { 2 } \\big ] } \\frac { ( - 1 ) ^ { ( 2 i + m ) - 1 } } { 2 ^ { ( 2 i + m ) } ( 2 i + m ) } \\binom { 2 i + m } { i } . \\end{align*}"}
-{"id": "7908.png", "formula": "\\begin{align*} e _ 1 = { x _ 1 + y _ 1 \\over 2 } - \\left [ { x _ 1 + y _ 1 \\over 2 } \\right ] . \\end{align*}"}
-{"id": "6393.png", "formula": "\\begin{align*} P _ { } \\left ( x \\theta \\right ) = P _ { } \\left ( x \\theta \\right ) \\exp \\left [ - 1 + \\alpha + \\beta f \\left ( \\theta \\right ) + \\gamma \\left ( x \\right ) \\right ] \\end{align*}"}
-{"id": "976.png", "formula": "\\begin{align*} b _ n ( t ) = \\sum _ { i = 1 } ^ n K _ h ( t _ { i - 1 } - t ) \\int _ { t _ { i - 1 } } ^ { t _ i } \\sigma ^ 2 ( s ) d s - \\sigma ^ 2 ( t ) \\end{align*}"}
-{"id": "9885.png", "formula": "\\begin{align*} \\rho ( K _ { 2 r } ) = \\tfrac { 3 r - 5 } { 3 r - 2 } r \\ge 2 \\ , . \\end{align*}"}
-{"id": "5087.png", "formula": "\\begin{align*} \\int _ { M ^ 3 } \\{ | \\tilde { A } | ^ 2 + \\frac { 1 } { 3 } R ^ 2 - | R i c | ^ 2 - \\frac { 2 } { 2 7 } \\} d v _ g = 0 . \\end{align*}"}
-{"id": "5688.png", "formula": "\\begin{align*} d _ X ( { v } _ 1 , { v } _ 2 ) = \\| { v } _ 1 - { v } _ 2 \\| _ { L ^ 2 ( \\R , \\R ^ n ) } . \\end{align*}"}
-{"id": "1008.png", "formula": "\\begin{align*} \\lim _ { M \\to \\infty } \\limsup _ { n \\to \\infty } P \\left ( \\sup _ { t \\in [ a _ n , T - a _ n ] } | Z ^ * _ n ( t ) - \\widetilde { Z } ^ * _ n ( t ) | > M v _ n \\right ) = 0 . \\end{align*}"}
-{"id": "4995.png", "formula": "\\begin{align*} \\| \\| 2 ^ { ( \\alpha - 1 ) m } \\Delta _ m ( \\partial _ { x ' } ( U _ { m + r } V _ { m + r } ) ) \\| _ { \\ell ^ q ( m ) } \\| _ { L ^ p } \\lesssim 2 ^ { - ( \\alpha - 1 - a ) r } \\sum _ { | \\gamma | = a } \\| \\| 2 ^ { ( \\alpha - 1 - a ) m } ( \\partial ^ { \\gamma } \\partial _ { x ' } ( U _ { m } V _ { m } ) ) \\| _ { \\ell ^ q ( m ) } \\| _ { L ^ p } . \\end{align*}"}
-{"id": "8481.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { - 2 l , l , v } ) = \\chi ( x _ 0 ) ^ 2 \\psi ( ( x _ 0 - b \\varpi ^ { \\frac { k } { 2 } - a ( \\chi ) } ) \\varpi ^ { - \\frac { k } { 2 } } ) \\end{align*}"}
-{"id": "3437.png", "formula": "\\begin{align*} u _ n \\leq a + \\sum _ { j = 1 } ^ { n - 1 } b _ j u _ j , n \\in \\{ 1 , \\dots , N \\} . \\end{align*}"}
-{"id": "10092.png", "formula": "\\begin{align*} \\tilde { R } ( X , Y ) Z + \\tilde { R } ( Y , Z ) X + \\tilde { R } ( Z , X ) Y = 0 , \\end{align*}"}
-{"id": "7602.png", "formula": "\\begin{align*} U _ 1 = U ^ { \\frac 1 2 } W _ 1 U ^ { \\frac 1 2 } , U _ 2 = U ^ { \\frac 1 2 } W _ 2 U ^ { \\frac 1 2 } , \\end{align*}"}
-{"id": "3054.png", "formula": "\\begin{align*} \\lambda I _ 2 = - a ( w - w _ h , v _ h ) - b ( v _ h , \\pi - \\pi _ h ) . \\end{align*}"}
-{"id": "1234.png", "formula": "\\begin{align*} \\int _ { \\mathbf { g } ^ { - 1 } ( K ) } | \\nabla u ( x ) | ^ 2 \\ , d \\mathcal { H } ^ { n - 1 } = \\mu ( K ) \\end{align*}"}
-{"id": "5639.png", "formula": "\\begin{align*} L _ d ( \\gamma ) = \\int _ I | \\dot { \\gamma } | ( t ) \\d t . \\end{align*}"}
-{"id": "812.png", "formula": "\\begin{align*} Q _ { k } ( x ) = 2 ^ 4 t , 2 \\nmid t \\end{align*}"}
-{"id": "8262.png", "formula": "\\begin{align*} D ( s , \\xi ) = \\frac { L ( s - i r _ 1 + i r _ 2 , \\xi ) L ( s + i r _ 1 - i r _ 2 , \\xi ) L ( s + i r _ 1 + i r _ 2 , \\xi \\chi _ 1 ^ { - 1 } \\chi _ 2 ) L ( s - i r _ 1 - i r _ 2 , \\xi \\chi _ 1 \\chi _ 2 ^ { - 1 } ) } { L ( 2 s , \\xi ^ 2 ) E ^ { - 1 } ( s , \\xi ) } \\end{align*}"}
-{"id": "4066.png", "formula": "\\begin{align*} \\eta ^ \\infty = \\lim \\limits _ { \\lambda \\to \\pm \\infty } \\frac { | \\lambda v - k | } { | \\lambda | } = 1 \\end{align*}"}
-{"id": "10151.png", "formula": "\\begin{align*} \\boldsymbol { \\omega } _ k ( i ) = \\sum _ { l \\in \\mathcal { N } _ k } c _ { k l } \\bigg ( \\boldsymbol R _ l ^ { - 1 / 2 } ( i ) \\boldsymbol \\Phi _ { l , 1 } \\boldsymbol \\Lambda _ { l , 1 } \\boldsymbol \\Phi _ { l , 1 } ^ H \\boldsymbol p _ l ( i ) + O _ l \\big ( \\epsilon _ l ( i ) \\big ) \\bigg ) \\end{align*}"}
-{"id": "7466.png", "formula": "\\begin{align*} [ e _ \\alpha , e _ \\beta ] _ E = \\mathcal { C } ^ { \\ : \\gamma } _ { \\alpha \\beta } e _ \\gamma . \\end{align*}"}
-{"id": "8328.png", "formula": "\\begin{align*} H ( m , \\lambda ) = \\sum _ { \\substack { x \\in \\lambda + V _ { 0 \\Z } \\\\ Q ( x ) = m } } x ^ \\perp , \\end{align*}"}
-{"id": "6431.png", "formula": "\\begin{align*} \\mathcal { D } _ { \\theta } ^ { } \\overset { } { = } \\left \\{ \\theta ^ { \\kappa } \\left ( \\alpha \\right ) : \\theta ^ { \\kappa } \\left ( s _ { 0 } \\right ) \\leq \\theta ^ { \\kappa } \\leq \\theta ^ { \\kappa } \\left ( s _ { 0 } + s \\right ) \\right \\} \\end{align*}"}
-{"id": "6934.png", "formula": "\\begin{align*} \\Xi = \\R _ + \\times ( 0 , 1 ) . \\end{align*}"}
-{"id": "6771.png", "formula": "\\begin{align*} G ( y , y ' ) = - \\frac { 1 } { 4 \\pi } \\ln { \\left ( \\frac { 4 | x - x ' | ^ 2 } { ( 1 + | x | ^ 2 ) ( 1 + | x ' | ^ 2 ) } \\right ) } , \\end{align*}"}
-{"id": "7174.png", "formula": "\\begin{align*} \\delta W ^ { \\underline { 0 } } _ { 1 + s } ( 1 , \\psi ) [ \\phi ] = 2 \\int _ { B _ 1 } \\nabla \\psi \\cdot \\nabla \\phi \\ , d \\mu _ a - 2 ( 1 + s ) \\int _ { \\partial B _ 1 } \\psi \\ , \\phi \\ , | x _ n | ^ a \\ , d \\mathcal { H } ^ { n - 1 } . \\end{align*}"}
-{"id": "8565.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } \\int _ { \\Omega \\ , \\cap \\ , B _ { H _ 0 } ( x , R _ 2 ) } z ( y , t ) ^ 2 \\zeta _ x ( y , t ) ^ 2 \\ , d y + \\iint _ { Q _ 2 } z ^ 2 [ ( \\partial _ t h ) \\zeta _ x ^ 2 - \\zeta _ x \\partial _ t \\zeta _ x ] \\ , d y \\ , d s \\\\ & + \\frac { 1 } { 6 } \\iint _ { Q _ 2 } H ( \\nabla ( z \\zeta _ x ) ) ^ 2 \\ , d y \\ , d s \\le C \\iint _ { Q _ 2 } [ H ( \\nabla h ) ^ 2 \\zeta _ x ^ 2 + H ( \\nabla \\zeta _ x ) ^ 2 ] z ^ 2 \\ , d y \\ , d s . \\end{align*}"}
-{"id": "4722.png", "formula": "\\begin{align*} P = \\ln \\left | \\frac { { \\varepsilon } ^ 2 } { \\Delta ^ 2 \\det g ^ { i j } } \\right | , \\end{align*}"}
-{"id": "8575.png", "formula": "\\begin{align*} & \\sup _ { x \\in \\Omega } \\int _ { \\Omega \\ , \\cap \\ , B _ { H _ 0 } ( x , 1 ) } F ( | z ( y , t ) | ) \\ , d y \\\\ & \\le M I ( 0 ) + C \\int _ 0 ^ t s ^ { \\sigma - 1 } \\sup _ { x \\in \\Omega } \\| z ( s ) \\| _ { L ^ { 1 + \\delta } ( \\Omega \\ , \\cap \\ , B _ { H _ 0 } ( x , 1 ) ) } ^ { 1 + \\delta } \\ , d s + C \\int _ 0 ^ t s ^ { - \\sigma } I ( s ) \\ , d s \\\\ & \\le M + C \\int _ 0 ^ t s ^ { \\sigma - 1 - \\frac { \\delta N } { 2 } } I ( s ) ^ { 1 + \\delta } \\ , d s + C \\int _ 0 ^ t s ^ { - \\sigma } I ( s ) \\ , d s \\end{align*}"}
-{"id": "7204.png", "formula": "\\begin{align*} \\lim _ { p \\rightarrow \\ 8 } \\xi _ { k , p } = \\xi _ k \\end{align*}"}
-{"id": "6624.png", "formula": "\\begin{align*} \\int _ \\Omega | D g _ 3 - D g | ^ p \\ , d \\mu & = \\int _ { \\bigcup _ { j = 0 } ^ { k - 1 } D ( x _ j , r '' ) } | D g _ 3 - D g | ^ p \\ , d \\mu \\\\ & \\leq K [ g _ 3 - g ] _ { \\mathrm { L i p } } ^ p \\sum _ { j = 0 } ^ { k - 1 } \\mu ( D ( x _ j , r '' ) ) \\\\ & \\leq K ( 1 + c n ) ^ p \\pi k ( r '' ) ^ 2 \\ . \\end{align*}"}
-{"id": "5571.png", "formula": "\\begin{align*} \\Delta \\left ( \\alpha , \\beta \\right ) = e ^ { T } \\left ( \\bar { \\Gamma } + \\bar { \\Gamma } _ { P } \\right ) e _ { 2 ^ { k } } \\end{align*}"}
-{"id": "7762.png", "formula": "\\begin{align*} ( \\ell _ { 1 } , \\ell _ { 2 } ) \\in \\bigcup _ { i = 1 } ^ { d } \\big ( ( Q _ { 1 , i } \\times Q _ { 2 , i } ) \\cup ( Q _ { 2 , i } \\times Q _ { 1 , i } ) \\big ) . \\end{align*}"}
-{"id": "5826.png", "formula": "\\begin{align*} z _ n = \\pi \\bigg ( n - \\frac 1 4 \\bigg ) + O \\bigg ( \\frac { 1 } { n } \\bigg ) \\end{align*}"}
-{"id": "4913.png", "formula": "\\begin{align*} T = \\beta ^ { - 1 } ( R + i B ) - \\frac { \\alpha \\beta ^ { - 1 } } { 2 } I \\end{align*}"}
-{"id": "6625.png", "formula": "\\begin{align*} g = \\left \\{ \\begin{array} { l l } \\varphi _ { 1 } ^ { - 1 } \\circ \\phi ^ { 0 } \\circ \\varphi _ { 1 } \\circ f \\circ \\varphi _ { 0 } ^ { - 1 } \\circ \\phi \\circ \\varphi _ { 0 } & \\mbox { i n } \\ E _ M \\\\ \\varphi _ { k + 1 } ^ { - 1 } \\circ \\phi ^ { k } \\circ \\varphi _ { k + 1 } \\circ f & \\mbox { i n } \\ f ^ { - 1 } ( B _ M ^ { k + 1 } ) , \\ \\mbox { f o r } \\ 0 < k < k _ 0 \\\\ f & \\mbox { e l s e w h e r e } \\end{array} \\right . \\end{align*}"}
-{"id": "2124.png", "formula": "\\begin{align*} E _ { j _ \\ell } \\subset \\mathcal { J } \\subset \\bigcup _ { k = \\lfloor \\log _ 2 A _ 1 \\rfloor } ^ { \\infty } I _ { j _ \\ell } ^ { k } . \\end{align*}"}
-{"id": "6018.png", "formula": "\\begin{align*} d _ { 1 + s } ( p \\| q ) : = \\begin{cases} \\frac { 1 } { s } \\log ( p ^ { 1 + s } q ^ { - s } + \\bar { p } ^ { 1 + s } \\bar { q } ^ { - s } ) , & s \\geq - 1 , s \\neq 0 \\\\ p \\log \\frac { p } { q } + \\bar { p } \\log \\frac { \\bar { p } } { \\bar { q } } , & s = 0 \\end{cases} \\end{align*}"}
-{"id": "9817.png", "formula": "\\begin{align*} f ( t ) = \\frac { ( 1 - p _ 0 ( t ) ) \\mu } { \\l } - \\frac { \\beta _ 2 \\mu } { ( \\alpha + \\beta _ 2 ) \\l } , t ' \\leq t \\leq ( T ' _ { e _ 2 } \\wedge T _ 2 ) . \\end{align*}"}
-{"id": "4092.png", "formula": "\\begin{align*} k _ { M , p } ( X ) + k _ { M , p } ( Y ) = 2 H \\end{align*}"}
-{"id": "2655.png", "formula": "\\begin{align*} T : = T _ { \\tilde \\Psi } - [ t ] \\otimes b K , \\end{align*}"}
-{"id": "6004.png", "formula": "\\begin{gather*} m ( 0 ) = m ( 2 ) = 0 , m ( 1 ) = m ( 3 ) = 4 , m ( 4 ) = 2 . \\end{gather*}"}
-{"id": "4962.png", "formula": "\\begin{align*} \\omega _ j ( x ) = \\left [ \\sum _ { r \\in 2 ^ { - j } \\Z ^ d } \\left ( | \\Delta _ j f | ( r ) e ^ { - 2 ^ j | x '' - r '' | - 2 ^ { j - \\sigma } | x ' - r ' | } \\right ) ^ p \\right ] ^ { 1 / p } \\end{align*}"}
-{"id": "5696.png", "formula": "\\begin{align*} \\mathcal { C } : = \\{ { v } \\in \\mathcal { S } _ { s y m } ( a ^ - , a ^ + ) \\ ; : \\ ; \\mathfrak { E } _ { W } ( v ) \\leq C _ 0 \\mbox { a n d } \\forall t \\geq 2 S , \\ , | { v } ( t ) - a ^ + | = | { v } ( - t ) - a ^ - | \\leq E ( t ) \\} . \\end{align*}"}
-{"id": "9451.png", "formula": "\\begin{align*} \\sum _ { a } ^ { b } \\frac { 1 } { n \\log n } \\asymp \\int _ a ^ b \\frac { d x } { x \\log x } = \\log \\log b - \\log \\log a b > a > 1 . \\end{align*}"}
-{"id": "6186.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 3 & ( \\log f _ i ) ' ( \\log f _ i ) ''' - ( \\frac { V ' } { V } ) ' \\overline { \\mathcal { T } } V ^ 2 \\\\ & \\leq - \\sum _ { i = 1 } ^ 3 \\Big ( ( \\log f _ i ) '' \\Big ) ^ 2 + \\frac { 2 V ' } { V } \\sum _ { i = 1 } ^ 3 ( \\log f _ i ) ' ( \\log f _ i ) '' \\\\ & = - \\sum _ { i = 1 } ^ 3 \\Big ( ( \\log f _ i ) '' \\Big ) ^ 2 + \\frac { 2 } { \\overline { \\mathcal { T } } V ^ 2 } \\Big \\{ \\sum _ { i = 1 } ^ 3 ( \\log f _ i ) ' ( \\log f _ i ) '' \\Big \\} ^ 2 \\end{align*}"}
-{"id": "1398.png", "formula": "\\begin{align*} j ( f ^ { * } ( \\alpha '' v ' - v '' ) ) = 0 \\end{align*}"}
-{"id": "4414.png", "formula": "\\begin{align*} \\int \\partial _ 1 \\zeta h _ 1 + \\partial _ 3 \\zeta h _ 3 \\ , \\ , d x = t \\int \\zeta ( \\cdot , x _ 3 = 0 ) \\ , \\nabla ' \\cdot m ' \\ , \\ , d x ' \\quad \\mbox { f o r a l l } \\ ; \\zeta . \\end{align*}"}
-{"id": "3666.png", "formula": "\\begin{align*} [ u _ i : v _ i ] = \\left [ 1 : \\frac { t _ { i + 1 } \\dots t _ { n + 1 } } { s } \\right ] , \\end{align*}"}
-{"id": "593.png", "formula": "\\begin{align*} \\mathrm { a r e a } ( \\mathcal { N } ) = - 2 \\pi \\Xi ( X ) , \\end{align*}"}
-{"id": "9807.png", "formula": "\\begin{align*} - \\beta _ 1 + \\alpha w ' ( t ) + \\beta _ 2 \\left ( \\mathbb { E } \\left ( \\sum _ { i = 1 } ^ { N ( t ) } X _ i + t \\right ) ^ { + } \\right ) ^ { ' } = 0 , - T _ { e _ 1 } \\leq t < 0 , \\end{align*}"}
-{"id": "9466.png", "formula": "\\begin{align*} S _ n ( 1 ) & = S _ { n - 1 } ( 1 ) - q ^ n S _ { n - 1 } ( 2 ) + q ^ n ( q ; q ^ 2 ) _ n , \\\\ S _ { n - 1 } ( 2 ) & = S _ { n - 2 } ( 2 ) - q ^ { n - 1 } S _ { n - 2 } ( 3 ) + q ^ { 2 n - 2 } ( q ; q ^ 2 ) _ { n - 1 } , \\end{align*}"}
-{"id": "5393.png", "formula": "\\begin{align*} a _ 4 \\cdot a _ 5 = \\frac { 1 } { 2 ^ 6 } ( a _ 4 + a _ 5 - a _ 3 - a _ 6 + a _ 9 ) \\end{align*}"}
-{"id": "6775.png", "formula": "\\begin{align*} \\eta _ { t , \\xi } ( y ) = \\eta \\left ( \\frac { | \\Pi _ { \\xi } ( y ) | } { t } \\right ) , \\end{align*}"}
-{"id": "595.png", "formula": "\\begin{align*} I n d _ { K } ^ { H } \\pi : = \\{ f : H \\to V : f ( h k ) = \\pi ( k ^ { - 1 } ) f ( h ) k \\in K f \\in L ^ { 2 } _ { \\mu } ( H / K ) \\} . \\end{align*}"}
-{"id": "9267.png", "formula": "\\begin{align*} l _ { s } ^ T ( \\nu | \\underline { \\xi } ) : = \\frac { 1 } { N _ { s + T } - N _ s } \\sum _ { j = N _ s + 1 } ^ { N _ { s + T } } \\log f ( \\xi _ { j } | \\nu ) . \\end{align*}"}
-{"id": "563.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow - \\infty } u ( s , \\cdot ) = x \\mbox { a n d } \\lim _ { s \\rightarrow + \\infty } u ( s , \\cdot ) = y \\end{align*}"}
-{"id": "9327.png", "formula": "\\begin{align*} j _ { n + 3 } ^ { ( 3 ) } = j _ { n + 2 } ^ { ( 3 ) } + j _ { n + 1 } ^ { ( 3 ) } + 2 j _ { n } ^ { ( 3 ) } , \\ j _ { 0 } ^ { ( 3 ) } = 2 , \\ j _ { 1 } ^ { ( 3 ) } = 1 , \\ j _ { 2 } ^ { ( 3 ) } = 5 , \\ n \\geq 0 , \\end{align*}"}
-{"id": "5808.png", "formula": "\\begin{align*} d \\nu : = u ^ { q } d \\sigma \\in \\dot { W } ^ { - 1 , 2 } ( \\Omega ) \\cap \\mathcal { M } ^ { + } ( \\Omega ) . \\end{align*}"}
-{"id": "6563.png", "formula": "\\begin{align*} | T _ 1 | _ 2 = \\frac { | S _ 1 | _ 1 + | S _ 2 | _ 1 } { 2 } \\cdot \\sqrt { 1 - \\varepsilon ^ 2 } = \\frac { 2 + 2 \\varepsilon } { 2 } \\cdot \\sqrt { 1 - \\varepsilon ^ 2 } = ( 1 + \\varepsilon ) \\cdot \\sqrt { 1 - \\varepsilon ^ 2 } , \\end{align*}"}
-{"id": "5275.png", "formula": "\\begin{align*} \\| u \\| ^ 2 _ { L ^ 2 ( a , c ) } & = ( c - a ) \\| v \\| ^ 2 _ { L ^ 2 ( 0 , 1 ) } \\\\ & \\leq ( c - a ) v ^ 2 ( 0 ) + \\frac { c - a } { 2 } \\| v _ x \\| ^ 2 _ { L ^ 2 ( 0 , 1 ) } \\\\ & = ( c - a ) u ^ 2 ( c ) + \\frac { ( c - a ) ^ 2 } { 2 } \\| u _ x \\| ^ 2 _ { L ^ 2 ( a , c ) } . \\end{align*}"}
-{"id": "3849.png", "formula": "\\begin{align*} I _ g ( x ^ * ) & : = \\ \\{ i = 1 , \\dots , m \\mid g _ i ( x ^ * ) = 0 \\} , \\\\ I _ 0 ( x ^ * ) & : = \\ \\{ i = 1 , \\dots , n \\mid x ^ * _ i = 0 \\} . \\end{align*}"}
-{"id": "5435.png", "formula": "\\begin{align*} - \\binom { d - k } { d - i } + \\binom { d - k + 1 } { d - i } = \\binom { d - k } { d - i - 1 } \\end{align*}"}
-{"id": "6903.png", "formula": "\\begin{align*} \\frac { 5 ( 3 ^ { 2 m + 1 } - 3 ) } { 2 } - \\frac { 2 ( 3 ^ { 2 m } - 3 ) } { 2 } + 9 ^ { m } + 3 \\left ( \\frac { 3 ^ { 2 m } + 3 } { 2 } \\right ) + \\frac { 3 ^ { 2 m + 2 } - 3 } { 2 } & \\\\ = \\frac { 3 ^ { 2 m + 3 } - 3 } { 2 } & \\end{align*}"}
-{"id": "3511.png", "formula": "\\begin{align*} J ( u ( \\cdot ) ) = { \\displaystyle \\int \\limits _ { 0 } ^ { T } } f ( X ^ u { ( t ) } ) d t + \\Psi ( X ^ u ( T ) ) \\end{align*}"}
-{"id": "9656.png", "formula": "\\begin{align*} 2 = \\omega \\Lambda _ \\beta ( t ) \\sqrt { \\left ( \\log d - \\log \\delta \\right ) / \\left ( 2 N _ { \\beta , \\rm { s } } ( t ) \\right ) } , \\end{align*}"}
-{"id": "4538.png", "formula": "\\begin{align*} h ( x ) = \\bigcap _ { j = - \\infty } ^ { \\infty } F _ { j } ^ { - 1 } ( B _ { K \\zeta } ( G _ j ( x ) ) ) , \\end{align*}"}
-{"id": "10141.png", "formula": "\\begin{align*} \\boldsymbol S _ { D _ k } ( 0 ) & = \\boldsymbol R _ k ^ { - 1 } ( 0 ) \\boldsymbol P _ { D _ k } ( 0 ) \\bigg ( \\boldsymbol S _ { D _ k } ^ H ( 0 ) \\boldsymbol R _ k ( 0 ) \\boldsymbol P _ { D _ k } ( 0 ) \\bigg ) ^ { - 1 } \\\\ & \\times \\bigg ( \\boldsymbol S _ { D _ k } ^ H ( 0 ) \\boldsymbol R _ k ^ 2 ( 0 ) \\boldsymbol S _ { D _ k } ( 0 ) \\bigg ) ^ { - 1 } . \\end{align*}"}
-{"id": "10115.png", "formula": "\\begin{align*} \\boldsymbol { \\bar { \\omega } } _ k ( i ) = \\bar { \\boldsymbol R } _ k ^ { - 1 } ( i ) \\bar { \\boldsymbol p } _ k ( i ) , \\end{align*}"}
-{"id": "9421.png", "formula": "\\begin{align*} \\mathfrak { d } ^ { \\rm ( o u t ) } _ { n , d } = \\mathfrak { d } ^ { \\rm ( o u t ) } _ { n - 2 , d + 2 } + \\mathfrak { d } ^ { \\rm ( o u t ) } _ { n - 1 , d - 1 } + \\mathfrak { d } ^ { \\rm ( o u t ) } _ { n - 2 , d } + \\sum \\limits _ { i = 0 } ^ { n - 2 } \\mathfrak { d } ^ { \\rm ( o u t ) } _ { i , 1 } \\cdot \\mathfrak { d } ^ { \\rm ( o u t ) } _ { n - i - 2 , d - 1 } + \\sum \\limits _ { i = 0 } ^ { n - 2 } \\sum \\limits _ { j = 0 } ^ { d - 2 } \\tilde { s } _ { i , j } \\cdot \\mathfrak { d } ^ { \\rm ( o u t ) } _ { n - 2 - i , d - 2 - j } , \\end{align*}"}
-{"id": "4498.png", "formula": "\\begin{align*} | \\chi ( \\rho ( s ) ) | = \\| \\rho ( s ) \\| . \\end{align*}"}
-{"id": "3962.png", "formula": "\\begin{align*} \\tilde { p } ^ { \\alpha _ n } ( n , s ) = \\int _ 0 ^ { \\infty } p ^ { \\alpha _ n } ( n , t ) e ^ { - s t } \\ , \\mathrm { d } t = \\frac { \\lambda ^ n s ^ { \\alpha _ 0 - 1 } } { \\prod _ { k = 0 } ^ n ( s ^ { \\alpha _ k } + \\lambda ) } , \\ \\ s > 0 . \\end{align*}"}
-{"id": "7181.png", "formula": "\\begin{align*} I _ \\sigma = \\{ x \\in B _ 1 \\ , \\ , : \\ , \\ , \\mathrm { d i s t } ( x , B _ 1 ^ { ' , - } ) < \\sigma \\} \\end{align*}"}
-{"id": "1742.png", "formula": "\\begin{align*} W _ \\mu ( f ) = f \\sqrt { d \\mu } . \\end{align*}"}
-{"id": "8573.png", "formula": "\\begin{align*} I ( 0 ) = \\sup _ { x \\in \\Omega } \\int _ { \\Omega \\ , \\cap \\ , B _ { H _ 0 } ( x , 1 ) } | z ( y , 0 ) | \\ , d y = \\sup _ { x \\in \\Omega } \\int _ { \\Omega \\ , \\cap \\ , B _ { H _ 0 } ( x , 1 ) } e ^ { - H _ 0 ( y ) ^ 2 } | \\phi ( y ) | \\ , d y = 1 . \\end{align*}"}
-{"id": "375.png", "formula": "\\begin{align*} \\mathbb { P } \\big ( | S _ { n } | \\geq x _ { n } \\sigma _ { n } \\big ) = 2 ( 1 - \\Phi ( x _ { n } ) ) ( 1 + o ( 1 ) ) n \\rightarrow \\infty , \\end{align*}"}
-{"id": "1281.png", "formula": "\\begin{align*} C ^ { - 1 } \\leq \\mu _ j ( \\mathbb { S } ^ { n - 1 } ) \\leq C \\mbox { f o r } \\ , \\ , j = 1 , 2 , \\dots \\end{align*}"}
-{"id": "9661.png", "formula": "\\begin{align*} E = e + \\frac { 1 } { 2 } ( u ^ 2 + v ^ 2 ) + q _ 0 Z , \\end{align*}"}
-{"id": "7735.png", "formula": "\\begin{align*} & \\mathrm { R e } ( H _ N ( l ) ) \\\\ & = \\frac { 1 } { 4 } \\Big ( \\left [ \\left ( G _ N ( l ) - G _ N ( l - 1 ) \\big ) - \\big ( G _ N ( l - 1 ) - G _ N ( l - 2 ) \\right ) \\right ] \\\\ & \\ \\ \\ \\ \\ - \\left [ \\left ( G _ N ( 2 ) - G _ N ( 1 ) \\right ) - \\left ( G _ N ( 1 ) - G _ N ( 0 ) \\right ) \\right ] \\Big ) \\\\ & = \\frac { 1 } { 4 } \\left [ \\left ( K _ N ( l ) - K _ N ( l - 1 ) \\right ) - \\left ( K _ N ( 2 ) - K _ N ( 1 ) \\right ) \\right ] \\end{align*}"}
-{"id": "8987.png", "formula": "\\begin{gather*} D ^ { ( n ) } _ { q , t } ( d ) D ^ { ( n ) } _ q ( - d q / 2 \\pm u ; t ) | _ { u = z _ i } = 0 \\end{gather*}"}
-{"id": "4422.png", "formula": "\\begin{align*} \\phi _ \\ell ( x _ 1 , x _ 2 ) : = \\frac { 1 } { \\ell ^ { \\frac { 5 } { 2 } } } \\phi ( \\frac { x _ 1 } { \\ell } , \\frac { x _ 2 } { \\ell ^ \\frac { 3 } { 2 } } ) , x \\in \\R ^ 2 , \\end{align*}"}
-{"id": "9690.png", "formula": "\\begin{align*} \\Delta _ j ( \\alpha _ { j } , \\beta _ { j } ) = \\left \\{ \\begin{array} { l l } 0 , & \\mbox { $ \\alpha _ j \\ge 0 $ , $ \\beta _ j \\ge 0 $ } , \\\\ | \\alpha _ j | | \\beta _ j | , & \\mbox { o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "897.png", "formula": "\\begin{align*} \\Sigma ( \\theta ) = \\left \\{ \\begin{array} { l l } \\int _ 0 ^ { T - \\theta } \\sigma _ 1 ( t ) \\sigma _ 2 ( t + \\theta ) d t & \\theta \\geq 0 , \\\\ \\int _ { 0 } ^ { T + \\theta } \\sigma _ 1 ( t - \\theta ) \\sigma _ 2 ( t ) d t & \\theta < 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "3857.png", "formula": "\\begin{align*} d _ x ^ T \\left ( \\nabla ^ 2 f ( x ^ * ) + \\sum \\limits _ { i = 1 } ^ m \\lambda ^ * _ i \\nabla ^ 2 g _ i ( x ^ * ) + \\sum \\limits _ { i = 1 } ^ p \\mu ^ * _ i \\nabla ^ 2 h _ i ( x ^ * ) \\right ) d _ x > 0 , \\end{align*}"}
-{"id": "4468.png", "formula": "\\begin{align*} \\sum _ { \\stackrel { k ' + k '' = k } { k ' , k '' \\not = 0 } } \\frac { 1 } { d ^ 4 ( k ' , 0 ) d ( k '' , 0 ) } & \\lesssim \\frac { 1 } { d ( k , 0 ) } , \\\\ \\sum _ { \\stackrel { k ' + k '' = k } { k ' , k '' \\not = 0 } } \\frac { 1 } { d ^ 2 ( k ' , 0 ) d ( k '' , 0 ) } & \\lesssim d ( k , 0 ) , \\\\ \\sum _ { \\stackrel { k ' + k '' = k } { k ' , k '' \\not = 0 } } \\frac { d ( k '' , 0 ) } { d ^ 4 ( k ' , 0 ) } & \\lesssim d ( k , 0 ) . \\end{align*}"}
-{"id": "9261.png", "formula": "\\begin{align*} | C | > \\frac { 4 } { 1 1 } \\left ( n + \\frac { 3 } { 4 } \\right ) = 4 q _ 3 + 3 + \\frac { 2 } { 1 1 } \\end{align*}"}
-{"id": "2666.png", "formula": "\\begin{align*} \\begin{pmatrix} c _ { 0 , 0 } & c _ { 0 , 1 } & \\cdots & c _ { 0 , N - 1 } \\\\ c _ { 1 , 0 } & c _ { 1 , 1 } & \\cdots & c _ { 1 , N - 1 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ c _ { N - 1 , 0 } & c _ { N - 1 , 1 } & \\cdots & c _ { N - 1 , N - 1 } \\\\ \\end{pmatrix} , \\end{align*}"}
-{"id": "7968.png", "formula": "\\begin{align*} f _ 1 ( \\varphi ( t , s ) ) = \\varphi ( t , s / 2 ) . \\end{align*}"}
-{"id": "5155.png", "formula": "\\begin{align*} \\lim _ { m \\to 0 } G _ m ^ { \\pm } \\delta d F & = \\lim _ { m \\to 0 } E _ m ^ { \\pm } \\delta d F = E _ 0 ^ { \\pm } \\delta d F \\\\ & = F - E _ 0 ^ { \\pm } d \\delta F \\end{align*}"}
-{"id": "3565.png", "formula": "\\begin{align*} - \\frac { \\zeta ^ { ' } ( 2 ) } { \\zeta ( 2 ) } - \\sum _ { k \\geq 0 } \\frac { \\log p } { p ^ { ( 1 + k ) 2 } } + O \\left ( \\frac { \\log x } { x } \\right ) = \\kappa _ 2 ( p ) + O \\left ( \\frac { \\log x } { x } \\right ) , \\end{align*}"}
-{"id": "6312.png", "formula": "\\begin{align*} \\xi _ m ^ 2 = A _ m \\end{align*}"}
-{"id": "5454.png", "formula": "\\begin{align*} s ^ { - \\rho } q ( s ) = \\int _ 0 ^ \\infty e ^ { - s x } \\dd g ( x ) . \\end{align*}"}
-{"id": "9554.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in W ( \\Phi ( C _ n ) ) } { ( - 1 ) ^ { \\ell ( \\sigma ) } x ^ { L _ { \\phi ( C _ n ) } ( \\sigma ) } } = ( 1 - x ^ { \\left \\lceil \\frac { n } { 2 } \\right \\rceil } ) \\prod _ { i = 1 } ^ { \\left \\lceil \\frac { n } { 2 } \\right \\rceil } ( 1 - x ^ { 2 i } ) ^ 2 . \\end{align*}"}
-{"id": "1459.png", "formula": "\\begin{align*} \\mathrm { K e r } \\ , \\pi ^ { i } _ { i + 1 } = ( v _ { i + 1 } x _ 3 ^ 2 ) = D _ { i } / ( e _ { i + 1 } , v _ { i + 2 } e _ { i + 2 } , \\dots , v _ r e _ r ) \\{ v _ { i + 1 } x _ 3 ^ 2 \\} \\end{align*}"}
-{"id": "5188.png", "formula": "\\begin{align*} \\tilde f _ 2 ( y ) & = f _ 2 ( y ) , y \\in \\{ y _ 0 \\} \\cup N _ 0 ^ c , \\\\ \\tilde f _ 2 ' ( y _ 0 ) & = f _ 2 ' ( y _ 0 ) , \\\\ \\tilde f _ 2 '' ( y _ 0 ) & = \\epsilon , \\end{align*}"}
-{"id": "7890.png", "formula": "\\begin{align*} h ( t , r , s ) = { 1 \\over T - t } + { 1 \\over T - r } + { 1 \\over T - s } \\ge 0 . \\end{align*}"}
-{"id": "8087.png", "formula": "\\begin{align*} \\deg ( m ) _ i = p _ 0 \\deg ( y ) _ i + \\sum _ { m = 1 } ^ \\ell ( p _ m - p _ { m - 1 } ) \\deg ( y _ { S _ m } ) _ i = p _ 0 + \\sum _ { m = 1 } ^ j ( p _ m - p _ { m - 1 } ) = p _ j . \\end{align*}"}
-{"id": "831.png", "formula": "\\begin{align*} | \\left \\langle \\mathcal { L } u , u \\right \\rangle - \\left \\langle \\mathcal { L } u _ h , u _ h \\right \\rangle | = \\inf _ { w _ h } \\left \\{ | \\left \\langle \\mathcal { L } u , u \\right \\rangle - \\left \\langle \\mathcal { L } u _ h , w _ h \\right \\rangle | , \\ ; w _ h ( \\cdot , 0 ) = u _ 0 \\right \\} . \\end{align*}"}
-{"id": "4053.png", "formula": "\\begin{align*} [ A ] & = \\{ \\gamma \\in \\Gamma ^ 1 : \\gamma \\cap A \\neq \\emptyset \\} , & \\abs * { A } _ \\eta & = \\eta ( [ A ] ) , \\end{align*}"}
-{"id": "2599.png", "formula": "\\begin{align*} ( D _ { t _ 1 } ^ a f ) ( t ) : = \\frac 1 { \\Gamma ( 1 - a ) } \\frac d { d t } \\int _ { t _ 1 } ^ t \\frac { f ( s ) } { ( t - s ) ^ a } \\ , d s , t > t _ 1 . \\end{align*}"}
-{"id": "4594.png", "formula": "\\begin{align*} ( t _ 2 , g _ 2 ) = & ( a k , \\gamma ) ( t _ 1 , g _ 1 ) \\\\ = & ( a k + t _ 1 , \\gamma \\phi ( a ) ^ k g _ 1 ) . \\end{align*}"}
-{"id": "1506.png", "formula": "\\begin{align*} Q _ { T , n } ( x ) = T _ { n } ( x ) + T _ { n + 1 } ( x ) \\textbf { i } + T _ { n + 2 } ( x ) \\textbf { j } + T _ { n + 3 } ( x ) \\textbf { k } \\end{align*}"}
-{"id": "3211.png", "formula": "\\begin{align*} \\phi ( 0 , \\cdot ) = f \\in H ^ 1 ( { \\mathbb R } ^ n ) , \\partial _ t \\phi ( 0 , \\cdot ) = g \\in L ^ 2 ( { \\mathbb R } ^ n ) . \\end{align*}"}
-{"id": "1252.png", "formula": "\\begin{align*} b ^ * _ { i j } ( x , \\tau ) = \\int _ 0 ^ 1 f _ { \\eta _ i \\eta _ j } ( s \\nabla u ( x , t ) + ( 1 - s ) \\nabla u ( x , \\tau ) ) \\ , d s \\mbox { f o r } \\ , \\ , 1 \\leq i , j \\leq n . \\end{align*}"}
-{"id": "953.png", "formula": "\\begin{align*} E \\left [ \\max _ { 1 \\leq k \\leq d _ n } \\left | \\sum _ { j = 1 } ^ N \\gamma _ k ( i , j ) W ^ { ( i ) } _ j \\right | ^ 3 \\right ] \\leq \\left ( 5 a ^ 2 \\log ( 2 d _ n - 1 + \\sqrt { e } ) \\max _ { 1 \\leq k \\leq d _ n } \\sum _ { j = 1 } ^ { N _ n } \\gamma _ { n , k } ( i , j ) ^ 2 \\right ) ^ { 3 / 2 } \\end{align*}"}
-{"id": "949.png", "formula": "\\begin{align*} E [ h ( Q ( Y ) ) ] - E [ h ( Q ( G ) ) ] & = \\sum _ { i = 1 } ^ N \\left ( E [ h ( Q ( W ^ { ( i ) } ) ) ] - E [ h ( Q ( W ^ { ( i - 1 ) } ) ) ] \\right ) , \\end{align*}"}
-{"id": "5479.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { U ( r ^ n z ) } { ( r ^ n z ) ^ \\rho \\ell ( r ^ n z ) } = \\int _ 0 ^ 1 t ^ { \\rho - 1 } p _ 0 ( t z ) \\dd t = z ^ { - \\rho } \\int _ 0 ^ z s ^ { \\rho - 1 } p _ 0 ( s ) \\dd s = \\mathrm { B } _ \\rho p _ 0 ( z ) . \\end{align*}"}
-{"id": "3495.png", "formula": "\\begin{align*} \\int _ { \\gamma _ i } \\omega _ j \\omega _ k & = \\int _ { \\phi ( \\gamma _ i ) } \\phi ( \\omega _ j ) \\phi ( \\omega _ k ) \\\\ & = ( - 1 ) ^ { \\sigma _ j + \\sigma _ k } \\left ( ( - 1 ) ^ { \\sigma _ i } \\int _ { \\gamma _ i } \\omega _ j \\omega _ k + a _ { i j k } \\right ) . \\end{align*}"}
-{"id": "5211.png", "formula": "\\begin{align*} Q = \\mathrm { d i a g } \\{ d _ 1 , \\dots , d _ n \\} , \\end{align*}"}
-{"id": "5562.png", "formula": "\\begin{align*} \\bar { \\nu } ^ { T } \\bar { w } _ { 2 ^ { k } } \\left ( t \\right ) = \\left ( - \\tau ^ { 2 } \\bar { \\nu } ^ { T } \\left ( \\alpha I _ { 2 ^ { k } } + \\beta \\Lambda _ { \\bar { r } } \\right ) P ^ { 2 } + \\tau \\dot { x } _ { 0 } e _ { 1 } ^ { T } P + x _ { 0 } e _ { 1 } ^ { T } \\right ) \\bar { w } _ { 2 ^ { k } } \\left ( t \\right ) \\end{align*}"}
-{"id": "6911.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty , m \\to \\infty } \\left ( \\frac { 1 } { \\mu _ { 0 } r _ { 0 } } \\right ) ^ { m } \\Phi ^ { m } _ { 1 } \\left ( \\frac { 3 ( 2 + 3 r - \\sqrt { 8 r + 9 r ^ { 2 } } ) } { 2 ( 1 + r ) } \\right ) = 9 \\end{align*}"}
-{"id": "2568.png", "formula": "\\begin{align*} \\frac { | p - h | } { | p - p _ 0 | } = \\frac { | p - p _ { 0 } | } { 2 r } \\ , , \\end{align*}"}
-{"id": "8099.png", "formula": "\\begin{align*} B _ n & = \\{ x \\in X : d ( x , X \\setminus B ) > 1 / n \\} , \\\\ A _ n & = \\{ x \\in X : d ( x , A ) \\leq 1 / n \\} \\end{align*}"}
-{"id": "5569.png", "formula": "\\begin{align*} \\bar { \\Gamma } = \\left ( I _ { 2 ^ { k } } + \\frac { \\tau ^ { 2 } } { 2 ^ { 2 k } } \\left ( \\alpha I _ { 2 ^ { k } } + \\beta \\bar { \\Lambda } _ { \\bar { r } } \\right ) \\bar { P } ^ { 2 } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "1786.png", "formula": "\\begin{align*} \\frac { d \\lambda } { d \\omega } ( \\omega _ 0 ) = - \\frac { 2 R _ * ' ( \\omega _ 0 ) } { R _ * ( \\omega _ 0 ) ^ 5 } \\int _ 0 \\sp 1 \\frac { \\theta ^ 2 ( K ( R _ * ( \\omega _ 0 ) ) - K ( \\theta R _ * ( \\omega _ 0 ) ) ) } { ( \\Psi ( \\theta , R _ * ( \\omega _ 0 ) , \\omega _ 0 ) ) ^ { 3 / 2 } } d \\theta \\geq 0 , \\end{align*}"}
-{"id": "6750.png", "formula": "\\begin{align*} V _ { n } = \\begin{cases} W _ { n } , & \\delta _ { n } = 0 \\\\ W ' _ { n } , & \\delta _ { n } = 1 \\\\ \\end{cases} , & & n = 1 , \\dots , N \\end{align*}"}
-{"id": "6308.png", "formula": "\\begin{align*} \\big [ \\partial _ t - V , \\partial _ x - U \\big ] & = 0 . \\end{align*}"}
-{"id": "6843.png", "formula": "\\begin{align*} L _ { X , Y } : = L _ X \\Omega _ X + 2 G _ X \\sqrt { \\Omega _ X \\log ( m ) } , \\end{align*}"}
-{"id": "7943.png", "formula": "\\begin{align*} \\bar V _ { I _ 0 } : = g _ { I _ 0 } ( \\bar V ) = f _ { I _ 0 } \\circ \\cdots f _ { i _ 1 i _ 2 } \\circ f _ { i _ 1 } ( \\bar V ) , \\ , \\ , \\ , K _ { I _ 0 } : = K \\cap V _ { I _ 0 } . \\end{align*}"}
-{"id": "909.png", "formula": "\\begin{align*} \\mathcal { P } _ q ( \\mathbb { H } ) = \\{ \\varphi ( \\xi _ 1 , \\dots , \\xi _ m ) : ; \\xi _ 1 , \\dots , \\xi _ m \\in \\mathbb { H } ; m \\in \\mathbb { N } \\} \\end{align*}"}
-{"id": "7249.png", "formula": "\\begin{align*} \\Theta _ { \\pi } d \\nu _ { \\pi } ^ { \\psi } = \\Phi _ { \\pi } ^ { \\psi } d \\mu _ { \\pi } . \\end{align*}"}
-{"id": "6233.png", "formula": "\\begin{align*} \\P ( \\widehat { \\tau _ 0 } < \\infty , \\widehat { X } ( \\widehat { \\tau _ 0 } ) \\geq x , \\Delta C ^ i ( \\widehat { \\tau _ 0 } ) \\geq x + 1 ) = \\P ( C ^ i ( 1 ) \\geq x + 1 ) ~ . \\end{align*}"}
-{"id": "1792.png", "formula": "\\begin{align*} g ( t ) & : = 1 - \\frac { b ( q - 2 ) ( 6 - q ) } { a ( p - 2 ) ( 6 - p ) } t + \\frac { c ( r - 2 ) ( 6 - r ) } { a ( p - 2 ) ( 6 - p ) } t ^ { \\frac { r - p } { q - p } } \\\\ h ( t ) & : = 1 - \\frac { b ( q - 2 ) } { a ( p - 2 ) } t + \\frac { c ( r - 2 ) } { a ( p - 2 ) } t ^ { \\frac { r - p } { q - p } } . \\end{align*}"}
-{"id": "9284.png", "formula": "\\begin{align*} \\boldsymbol { E } [ \\sum _ { i = 1 } ^ { T - 1 } \\sum _ { j = i + 1 } ^ T f ( j - i ) X _ i X _ j ] \\leq \\boldsymbol { E } [ T ] \\sum _ { k = 1 } ^ \\infty f ( k ) . \\end{align*}"}
-{"id": "744.png", "formula": "\\begin{align*} m _ { \\mu , p } ( a ) = I _ { \\mu , p } ( u _ n ) + o _ n ( 1 ) & = \\frac 1 2 \\int _ { \\R ^ 3 } | \\nabla u _ n | ^ 2 + ( x _ 1 ^ 2 + x _ 2 ^ 2 ) | u _ n | ^ 2 \\ , d x + o _ n ( 1 ) \\\\ & \\geq \\frac { a \\Lambda _ 0 } { 2 } + o _ n ( 1 ) , \\end{align*}"}
-{"id": "8313.png", "formula": "\\begin{align*} g V _ \\Z = g V _ { \\widehat { \\Z } } \\cap V . \\end{align*}"}
-{"id": "3934.png", "formula": "\\begin{align*} u \\frac { \\partial u } { \\partial x } | _ { ( x _ j , t _ n ) } = \\frac { 1 } { 3 } u _ j ^ { n + 1 } ( u _ x ) _ j ^ n + \\frac { 2 } { 3 } u _ j ^ n ( u _ x ) _ j ^ { n + 1 } \\end{align*}"}
-{"id": "6705.png", "formula": "\\begin{align*} \\sigma _ { i } ( t , \\omega ) = \\begin{cases} \\sigma _ { i } ( t - 1 , \\omega ) , & I ( t , \\omega ) \\neq i ; \\\\ + 1 , & I ( t , \\omega ) = i ; \\ U ( t , \\omega ) < \\frac { 1 } { 2 } ; \\\\ - 1 , & I ( t , \\omega ) = i ; \\ U ( t , \\omega ) \\geq \\frac { 1 } { 2 } . \\end{cases} \\end{align*}"}
-{"id": "7339.png", "formula": "\\begin{align*} \\frac { 2 } { s } + \\frac { 3 } { p } = \\frac { 5 } { 2 } , \\qquad \\frac { 1 } { 2 } \\leqslant \\frac { 1 } { p } < \\frac { 1 } { r } + \\frac { 1 } { 3 } \\ , . \\end{align*}"}
-{"id": "1559.png", "formula": "\\begin{align*} - \\nu ( b _ { m } ) = \\Bigl \\lfloor \\frac { 2 } { 3 } ( m + 1 ) \\Bigr \\rfloor - s \\left ( \\Bigl \\lfloor \\frac { 2 } { 3 } ( m + 1 ) \\Bigr \\rfloor \\right ) + \\epsilon ( m ) , \\end{align*}"}
-{"id": "3195.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } ( \\mathcal { D } V ) ( x ) = \\lim _ { x \\to \\infty } \\left [ ( a - b x ) ( 1 + x ) ^ { - 1 } - \\frac { \\sigma ^ { 2 } } { 2 } x ( 1 + x ) ^ { - 2 } \\right ] = - b . \\end{align*}"}
-{"id": "9706.png", "formula": "\\begin{align*} \\begin{cases} & \\gamma _ 1 = K _ { 2 1 } \\beta _ 1 + \\alpha _ 1 + O ( 1 ) \\Delta ' ( \\alpha _ { 5 } , \\boldsymbol { \\beta } ^ { * } ) , \\\\ & \\sigma ' _ { i } = K _ { 2 i } \\beta _ 1 + \\beta _ i + \\sigma _ i + O ( 1 ) \\Delta ' ( \\alpha _ { 5 } , \\boldsymbol { \\beta } ^ { * } ) , i = 2 , 3 , \\\\ & \\gamma _ 4 = \\alpha _ 4 + \\beta _ 4 , \\\\ & \\gamma _ 5 = K _ { 2 5 } \\beta _ 1 + \\alpha _ 5 + \\beta _ 5 + O ( 1 ) \\Delta ' ( \\alpha _ { 5 } , \\boldsymbol { \\beta } ^ { * } ) , \\end{cases} \\end{align*}"}
-{"id": "10097.png", "formula": "\\begin{align*} \\tilde { R } ( X , Y ) \\xi = \\lambda \\{ \\pi ( X ) Y - \\pi ( Y ) X \\} . \\end{align*}"}
-{"id": "6323.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { L } n _ l \\mathbf { K L } ( s _ l , \\bar { f } _ { k _ l } ) \\leq \\inf _ { f \\in \\mathcal { F } _ m } \\sum _ { l = 1 } ^ { L } n _ l \\mathbf { K L } ( s _ l , f _ { k _ l } ) + \\delta . \\end{align*}"}
-{"id": "3953.png", "formula": "\\begin{align*} | T _ { s } ( u , v ) | \\le \\mathcal { D } _ { s _ { 1 } } [ u ] \\mathcal { D } _ { s _ { 2 } } [ v ] , s = s _ { 1 } + s _ { 2 } , s \\in ( 0 , 2 ) , \\ s _ { j } \\in ( 0 , 1 ) \\end{align*}"}
-{"id": "3111.png", "formula": "\\begin{align*} P _ { \\mathcal { E } } ^ { ( \\sigma , \\rho ) } ( m ) = \\sum _ { i \\in Q _ 0 } \\sum _ { j = 1 } ^ N \\sigma _ { i j } \\chi ( \\mathcal { E } _ i \\otimes L _ j ^ m ) , m \\in \\N . \\end{align*}"}
-{"id": "7661.png", "formula": "\\begin{align*} \\mathcal D _ r : = \\big \\{ D \\in { [ n ] \\choose k } : | D \\cap [ 2 r + 1 ] | \\ge r + 1 \\big \\} , \\ \\ \\ \\ \\ \\ \\ \\ r = 1 , \\ldots , k - 1 . \\end{align*}"}
-{"id": "6763.png", "formula": "\\begin{align*} [ l : k ] \\log ^ + | \\mu | _ { w _ v } = \\log ^ + | \\beta | _ v . \\end{align*}"}
-{"id": "5970.png", "formula": "\\begin{align*} s ( x ) = \\sum _ { k = 1 } ^ N \\alpha _ k \\phi _ k ( x ) , \\end{align*}"}
-{"id": "3076.png", "formula": "\\begin{align*} t ^ * : = \\min \\left \\{ t \\in ( 0 , T ) : \\ , \\int _ { 0 } ^ t \\lambda _ 1 ( s ) \\ , \\mathrm { d } s = 0 \\right \\} . \\end{align*}"}
-{"id": "5869.png", "formula": "\\begin{align*} D = \\begin{matrix} 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 \\\\ 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 \\\\ 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 \\\\ 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 \\\\ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 \\end{matrix} , \\end{align*}"}
-{"id": "1158.png", "formula": "\\begin{align*} A _ { < k } & : = \\{ a \\in A \\colon \\deg ( a ) < k \\} \\\\ A _ { k } & : = \\{ a \\in A \\colon \\deg ( a ) = k \\} \\intertext { a n d t h e p o l y n o m i a l } e _ k ( w ) & = \\prod _ { a \\in A _ { < k } } ( w + a ) . \\end{align*}"}
-{"id": "537.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial s } + J _ t ( u ) \\left ( \\frac { \\partial u } { \\partial t } - X _ { H _ { s , t } } ( u ) \\right ) = 0 \\end{align*}"}
-{"id": "9640.png", "formula": "\\begin{align*} \\hat { \\Psi } ( \\tau _ { i - 1 } ) = \\begin{cases} \\Psi ( \\tau _ { i - 1 } ) , ~ & N ( \\tau _ { i - 1 } ) \\geq N ( \\tau _ { i } ) \\\\ \\left ( \\Psi ( \\tau _ { i - 1 } ) , \\Psi _ { \\rm { R } } \\right ) , ~ & N ( \\tau _ { i - 1 } ) < N ( \\tau _ { i } ) \\end{cases} \\end{align*}"}
-{"id": "8405.png", "formula": "\\begin{align*} \\frac { \\partial \\varphi } { \\partial t } ( x , t ) = - K ^ { \\beta } ( x , t ) \\nu ( x , t ) \\end{align*}"}
-{"id": "6553.png", "formula": "\\begin{align*} & \\frac { 1 } { ( 1 - \\alpha _ { \\xi } \\delta ^ { 1 / n } ) ( 1 + \\beta _ { \\xi } \\delta ' ) } = \\frac { 1 } { ( 1 - \\alpha _ { \\xi } \\delta ^ { 1 / n } ) ( 1 + \\beta _ { \\xi } c _ 0 \\delta ^ { 1 / n } ) } = 1 + ( \\alpha _ { \\xi } - c _ 0 \\beta ) \\delta ^ { 1 / n } + o ( \\delta ^ { 1 / n } ) \\\\ \\leq & 1 + G _ { c _ 0 } ( P ) \\delta ^ { 1 / n } + o ( \\delta ^ { 1 / n } ) \\leq 1 + G ( P ) \\delta ^ { 1 / n } + o ( \\delta ^ { 1 / n } ) . \\end{align*}"}
-{"id": "6204.png", "formula": "\\begin{align*} \\begin{pmatrix} 2 & 0 & 2 k & 2 m \\\\ 0 & 2 & 2 l & 2 n \\\\ 2 k & 2 l & 0 & 1 \\\\ 2 m & 2 n & 1 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "3942.png", "formula": "\\begin{align*} u ( x , 0 ) = \\frac { \\mu + \\sigma + ( \\sigma - \\mu ) \\exp [ \\frac { \\mu } { \\nu } ( x - \\lambda ) ] } { 1 + \\exp [ \\frac { \\mu } { \\nu } ( x - \\lambda ) ] } \\end{align*}"}
-{"id": "2252.png", "formula": "\\begin{align*} f ( w ) = 1 + b _ 1 w + \\ldots + b _ k w ^ k + \\ldots \\end{align*}"}
-{"id": "10110.png", "formula": "\\begin{align*} \\bar { \\boldsymbol x } _ k ( i ) = \\boldsymbol S _ { D _ k } ^ H ( i ) \\boldsymbol x _ k ( i ) , \\end{align*}"}
-{"id": "9942.png", "formula": "\\begin{align*} u ( x , t ) = u _ 0 \\cos ( x ) H \\left ( \\frac \\pi 2 - | x | \\right ) , \\end{align*}"}
-{"id": "6607.png", "formula": "\\begin{align*} \\mu \\left ( \\left \\{ x : \\left | \\sigma _ { m , n } ( x ) \\right | > 0 \\right \\} \\right ) & = \\mu \\left ( \\left \\{ x : \\left | \\mathrm { i d } _ { \\mathrm { R } ^ d } - D \\phi _ { m , n } ^ { - 1 } ( f _ { n } ^ { - 1 } ( x ) ) \\right | > 0 \\right \\} \\right ) \\\\ & \\leq \\mu \\left ( f _ { n } ( \\varphi _ { n } ( E _ { m , M } ) ) \\right ) \\ . \\end{align*}"}
-{"id": "1705.png", "formula": "\\begin{align*} \\sum _ { d ( \\lambda ) = n } \\frac { 1 } { | f _ \\lambda \\circ \\tau _ \\lambda | ^ 2 } = \\frac { d \\mu \\circ ( \\tau ^ { { n } } ) ^ { - 1 } } { d \\mu } . \\end{align*}"}
-{"id": "4712.png", "formula": "\\begin{align*} I _ s ( t ) \\lesssim \\min ( s ) \\sum _ { i = 1 } ^ { 2 ^ { d - 1 } } \\prod _ { j = 1 } ^ d ( 1 + | ( A ^ i _ s t ) _ j | ) ^ { - k } \\end{align*}"}
-{"id": "5980.png", "formula": "\\begin{align*} \\varepsilon ^ 2 = \\frac { \\gamma ^ 2 - \\alpha ^ 2 } { 2 } \\implies \\alpha = \\sqrt { \\gamma ^ 2 - 2 \\varepsilon ^ 2 } . \\end{align*}"}
-{"id": "5104.png", "formula": "\\begin{align*} \\mathop { \\textup { c a r d } } \\mathcal { M } = | \\widehat { \\mathcal { L } } | / | \\mathcal { M } ^ \\perp | = 1 / | \\mathcal { M } ^ \\perp | , \\end{align*}"}
-{"id": "7722.png", "formula": "\\begin{align*} D ^ 2 _ B ( j ) & = \\frac { N - 1 } { N } + ( N - 2 ) \\cdot 2 = \\frac { 2 N ^ 2 - 3 N - 1 } { N } \\ , , \\\\ C _ B ( j ) & = \\frac { N ^ 2 } { 2 N ^ 2 - 3 N - 1 } \\ , . \\end{align*}"}
-{"id": "2083.png", "formula": "\\begin{align*} ( \\Delta _ H - \\partial _ t ) u ^ a _ { k } = F ^ a _ { k - 1 } \\mbox { a n d } u ^ a _ k ( p , 0 ) = \\phi ^ a ( p ) . \\end{align*}"}
-{"id": "5716.png", "formula": "\\begin{align*} | \\dot { \\gamma } | ( t ) = \\| \\partial _ t \\gamma ( t , \\cdot ) \\| _ { L ^ 2 ( \\R ) } \\quad \\gamma ( t ) \\notin \\Sigma . \\end{align*}"}
-{"id": "9342.png", "formula": "\\begin{align*} \\begin{aligned} j O _ { n } ^ { ( 3 ) } & = j _ { n } ^ { ( 3 ) } + \\sum _ { s = 1 } ^ { 7 } j _ { n + s } ^ { ( 3 ) } e _ { s } , \\ n \\geq 0 \\\\ & = j _ { n } ^ { ( 3 ) } + j _ { n + 1 } ^ { ( 3 ) } e _ { 1 } + j _ { n + 2 } ^ { ( 3 ) } e _ { 2 } + j _ { n + 3 } ^ { ( 3 ) } e _ { 3 } \\\\ & \\ \\ + j _ { n + 4 } ^ { ( 3 ) } e _ { 4 } + j _ { n + 5 } ^ { ( 3 ) } e _ { 5 } + j _ { n + 6 } ^ { ( 3 ) } e _ { 6 } + j _ { n + 7 } ^ { ( 3 ) } e _ { 7 } , \\end{aligned} \\end{align*}"}
-{"id": "6926.png", "formula": "\\begin{align*} u ( x , t ) \\big | _ { \\partial \\Omega } = 0 \\end{align*}"}
-{"id": "6338.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ n ( p _ j ( 1 - p _ j ) ) ^ { n - j } = \\exp \\Big [ \\sum _ { j = 1 } ^ n ( n - j ) \\log ( p _ j ( 1 - p _ j ) ) \\Big ] . \\end{align*}"}
-{"id": "2753.png", "formula": "\\begin{align*} h ^ { e } = h - h ^ { K } : = \\underbrace { h - P _ { K } h } _ { R ^ K } \\ , + \\underbrace { P _ { K } h - h ^ K } _ { e ^ K } \\ , , \\end{align*}"}
-{"id": "9494.png", "formula": "\\begin{align*} A ^ 2 ( z , t ) = B ( z , t ) . \\end{align*}"}
-{"id": "4814.png", "formula": "\\begin{align*} m ( t _ 1 , \\mathbf { v } ) = m \\theta ( r ) = n \\theta ( s ) = n ( t _ 2 , \\mathbf { v } ) \\end{align*}"}
-{"id": "1773.png", "formula": "\\begin{align*} \\| P ( \\{ \\omega \\} ) t _ { \\omega ( 0 , j ) } t _ a ^ * ( \\xi ) - t _ { \\omega ( 0 , N { \\bf 1 } ) } t _ A ^ * ( \\xi ) \\| < \\varepsilon \\end{align*}"}
-{"id": "5123.png", "formula": "\\begin{align*} \\frac { 1 } { n ! } \\sum _ { i _ 1 , \\ldots , i _ n = 1 } ^ m | \\det [ A ] _ { i _ 1 \\cdots i _ n } | ^ 2 \\leq \\left [ \\frac { 1 } { n } \\sum _ { j = 1 } ^ n \\sum _ { i = 1 } ^ m | A _ { j i } | ^ 2 \\right ] ^ n \\end{align*}"}
-{"id": "122.png", "formula": "\\begin{align*} Q _ t f ( x ) & = \\int f ( \\sqrt { t } y ) e ^ { \\langle x , y \\rangle / \\sqrt { t } - \\| x \\| ^ 2 / 2 t } d \\gamma _ n ( y ) \\\\ & = e ^ { - \\| x \\| ^ 2 / 2 t } \\sum _ { n = 0 } ^ \\infty \\frac { t ^ { - n / 2 } } { n ! } \\int f ( \\sqrt { t } y ) \\langle x , y \\rangle ^ n d \\gamma _ n ( y ) \\end{align*}"}
-{"id": "3623.png", "formula": "\\begin{align*} \\int _ { R _ \\lambda ( \\Omega ) } | \\nabla u _ \\lambda | ^ { p - 2 } ( \\nabla u _ \\lambda , \\nabla \\varphi ) \\ , d x \\ , = \\ , \\int _ { R _ \\lambda ( \\Omega ) } f ( u _ \\lambda ) \\varphi \\ , d x \\qquad \\forall \\varphi \\in C ^ { 1 } _ c ( R _ \\lambda ( \\Omega ) \\setminus R _ \\lambda ( \\Gamma ) ) \\ , . \\end{align*}"}
-{"id": "8807.png", "formula": "\\begin{align*} | E _ { m , p } ^ { 0 } | = \\Big | \\sum _ { i = 1 } ^ { r } a _ { i , p } m ^ { \\beta _ { i } } p ^ { - \\lambda _ { i } m } \\Big | \\leq C _ { m } p ^ { - \\sigma m } . \\end{align*}"}
-{"id": "2466.png", "formula": "\\begin{align*} | g | _ { L _ { x } ^ { 2 } } = \\Big ( \\int _ { { \\mathbb { R } ^ { 3 } } } | g | ^ { 2 } d x \\Big ) ^ { 1 / 2 } \\ , . \\end{align*}"}
-{"id": "3736.png", "formula": "\\begin{align*} \\theta ( t , \\kappa ) : = \\sup _ s \\{ - { \\rm s g n } ( s - t ) ( { \\dot P } ( s ; b ) - { \\dot P } ( t ; b ) ) - \\kappa | s - t | \\} \\end{align*}"}
-{"id": "6318.png", "formula": "\\begin{align*} ( 2 \\ell \\ ! + \\ ! 1 ) ( w \\ ! - \\ ! 2 ) \\Bar { q } ( w ) \\ ! - \\ ! w \\Bar { q } ( w ) \\ ! - \\ ! 2 w ( w \\ ! - \\ ! 2 ) \\Bar { q } ' ( w ) & = \\binom { - \\frac { 1 } { 2 } } { \\ell } ( - 2 ) ^ { \\ell + 1 } ( 2 \\ell + 1 ) \\\\ & = \\frac { 1 \\cdot 3 \\cdots ( 2 \\ell \\ ! - \\ ! 1 ) ( 2 \\ell \\ ! + \\ ! 1 ) } { \\ell ! } ( - 2 ) . \\end{align*}"}
-{"id": "503.png", "formula": "\\begin{align*} \\delta _ { \\boldsymbol { a } } : = \\prod _ { j = 1 } ^ m \\left ( N _ j - \\epsilon _ j - a _ j \\right ) , \\end{align*}"}
-{"id": "883.png", "formula": "\\begin{align*} \\gamma _ { n , k } ( i , j ) = \\left \\{ \\begin{array} { l l } 1 / \\sqrt { N _ n } & | j - i | = k , \\\\ 0 & . \\end{array} \\right . \\end{align*}"}
-{"id": "7320.png", "formula": "\\begin{align*} U = U _ 0 U _ 1 \\cdots U _ q , \\end{align*}"}
-{"id": "2376.png", "formula": "\\begin{align*} \\Phi _ m ( s , \\lambda , w ) = \\frac { 1 } { m ! } \\frac { \\partial ^ m \\Phi } { \\partial \\lambda ^ m } ( s , \\lambda , w ) = \\sum _ { n = m } ^ \\infty { n \\choose m } \\frac { \\lambda ^ { n - m } } { ( n + w ) ^ s } ( m \\in \\N _ 0 ) \\end{align*}"}
-{"id": "6878.png", "formula": "\\begin{align*} N ( x ) = \\# \\{ \\lambda \\ | \\ \\lambda \\leq x , \\lambda \\} \\end{align*}"}
-{"id": "1688.png", "formula": "\\begin{align*} \\mu _ { \\mathcal { C } } ( Z ( I ) ) : = \\lambda _ { i _ 1 } T _ { i _ 1 , i _ 2 } \\cdots T _ { i _ { n - 1 } , i _ n } , I = i _ 1 \\ldots i _ n . \\end{align*}"}
-{"id": "5542.png", "formula": "\\begin{align*} \\ddot { x } + ( \\alpha + \\beta \\left ( \\cos \\left ( t \\right ) + \\cos \\left ( 2 t \\right ) \\right ) ) x = 0 \\end{align*}"}
-{"id": "7453.png", "formula": "\\begin{align*} & \\gamma ^ 2 \\beta = \\beta \\gamma ^ 2 , a a ^ * a = a \\gamma ^ 2 , a ^ * a a ^ * = \\gamma ^ 2 a ^ * , \\\\ & \\beta ^ 2 = - a ^ * a \\gamma - \\gamma a ^ * a + \\gamma ^ 3 , \\gamma ^ 3 - \\beta ^ 2 - \\beta \\gamma - \\gamma \\beta - \\gamma a ^ * a - a ^ * a \\gamma = 0 . \\end{align*}"}
-{"id": "666.png", "formula": "\\begin{align*} & U _ 1 ( 1 ) V _ 1 ( 1 ) \\bar { \\Lambda } ' ( 1 ) V _ 1 ( 1 ) ^ * U _ 1 ( 1 ) ^ * U _ 1 ( 0 ) e _ i \\\\ & = U _ 1 ( 1 ) V _ 1 ( 1 ) \\bar { \\Lambda } ' ( 1 ) V _ 1 ( 1 ) ^ * e _ { \\sigma ( i ) } \\\\ & = U _ 1 ( 1 ) V _ 1 ( 1 ) \\bar { \\Lambda } ' ( 1 ) e _ { \\sigma ' ( i ) } \\\\ & = U _ 1 ( 1 ) V _ 1 ( 1 ) \\lambda ' _ { \\sigma ' ( i ) } ( 1 ) e _ { \\sigma ' ( i ) } \\\\ & = U _ 1 ( 1 ) \\lambda ' _ { \\sigma ' ( i ) } ( 1 ) e _ { \\sigma ( i ) } \\\\ & = \\lambda ' _ { i } ( 0 ) U _ 1 ( 0 ) e _ i , ~ ~ ~ \\forall i \\in S _ n \\end{align*}"}
-{"id": "2571.png", "formula": "\\begin{align*} B = \\left [ \\begin{array} { c c c c } \\vec { B } _ { 0 0 } & \\vec { B } _ { 0 1 } & \\cdots & \\vec { B } _ { 0 ( n - 1 ) } \\\\ \\vec { B } _ { 1 0 } & \\vec { B } _ { 1 1 } & \\cdots & \\vec { B } _ { 1 ( n - 1 ) } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\vec { B } _ { ( m - 1 ) 0 } & \\vec { B } _ { ( m - 1 ) 1 } & \\cdots & \\vec { B } _ { ( m - 1 ) ( n - 1 ) } \\\\ \\end{array} \\right ] , \\end{align*}"}
-{"id": "843.png", "formula": "\\begin{align*} \\mathbb { E } ( | E _ t - E _ s | ) = \\mathbb { E } ( E _ t - E _ s ) = C ( \\beta , 1 ) ( t ^ { \\beta } - s ^ { \\beta } ) \\leq C ( \\beta , 1 ) | t - s | ^ { \\beta } . \\end{align*}"}
-{"id": "2771.png", "formula": "\\begin{align*} \\Delta _ k ^ \\pm ( I , \\eta ( P _ 0 ) ) = \\Delta _ k ^ \\pm ( I , P _ 0 ) - P _ 0 + \\eta ( P _ 0 ) . \\end{align*}"}
-{"id": "3588.png", "formula": "\\begin{align*} D _ { * a } ^ \\alpha f : = D _ a ^ \\alpha [ f - T _ { \\lceil \\alpha \\rceil - 1 } [ f ; a ] ] \\end{align*}"}
-{"id": "871.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { 2 } } \\left ( \\begin{array} { c c } 1 & 1 \\\\ 1 & - 1 \\end{array} \\right ) \\left ( \\frac { 1 } { \\sqrt { 5 } } \\left ( \\begin{array} { c c } 2 & 1 \\\\ 1 & - 2 \\end{array} \\right ) \\right ) ^ \\top = \\frac { 1 } { \\sqrt { 1 0 } } \\left ( \\begin{array} { c c } 3 & - 1 \\\\ 1 & 3 \\end{array} \\right ) . \\end{align*}"}
-{"id": "4104.png", "formula": "\\begin{align*} \\mathrm { h e s s } _ b f ( X , Y ) : = X ( Y f ) - ( \\hat { \\nabla } _ X Y ) f . \\end{align*}"}
-{"id": "3092.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } F } { \\mathrm { d } x } ( t , 0 ) = 0 \\end{align*}"}
-{"id": "7073.png", "formula": "\\begin{align*} \\pi _ 2 ( \\nabla _ u \\Phi ( u _ 1 , u _ 2 , \\lambda ) ) = 0 . \\end{align*}"}
-{"id": "6840.png", "formula": "\\begin{align*} S _ { \\rho } ( w _ { \\lambda } + \\theta \\phi ) = S _ { \\rho } ( w _ { \\lambda } ) + \\theta \\Delta \\phi + O ( \\lambda e ^ { w _ { \\lambda } } ) . \\end{align*}"}
-{"id": "6259.png", "formula": "\\begin{align*} | I _ 3 | \\leq & \\frac \\mu { 8 } \\sum _ { q \\geq - 1 } \\lambda _ q ^ { 2 s + 2 \\alpha } \\| b _ q \\| _ 2 ^ 2 + \\frac \\nu { 8 } \\sum _ { q \\geq - 1 } \\lambda _ q ^ { 2 s + 2 } \\| u _ q \\| _ 2 ^ 2 \\\\ & + C _ { \\nu , \\mu } \\left ( \\sum _ { q \\geq - 1 } \\lambda _ q ^ { 2 s } \\| b _ q \\| _ 2 ^ 2 \\right ) ^ { \\gamma _ 1 } + C _ { \\nu , \\mu } \\left ( \\sum _ { q \\geq - 1 } \\lambda _ q ^ { 2 s } \\| u _ q \\| _ 2 ^ 2 \\right ) ^ { \\gamma _ 2 } , \\end{align*}"}
-{"id": "8443.png", "formula": "\\begin{align*} G _ l ( y , \\omega _ { \\pi } ^ { - 1 } ) = \\int _ { 1 + \\varpi ^ l \\mathcal { O } } \\omega _ { \\pi } ^ { - 1 } ( x ) \\psi ( y x ) d ^ { \\times } x , \\end{align*}"}
-{"id": "7191.png", "formula": "\\begin{align*} \\partial _ { i j } z _ \\infty = 0 \\qquad \\textit { i n } B _ 1 \\qquad \\forall i , j = 1 , \\dots , n - 2 . \\end{align*}"}
-{"id": "8483.png", "formula": "\\begin{align*} \\abs { W _ { \\pi } ( g _ { t , l , v } ) } = \\zeta _ F ( 1 ) ^ { - 1 } q ^ { - \\frac { t } { 2 } } \\abs { G _ l ( - v ^ { - 1 } \\varpi ^ { t + l } , \\chi ^ { - 1 } ) } = q ^ { \\frac { n - l } { 2 } } \\abs { I _ { \\chi } ( l , v ) } . \\end{align*}"}
-{"id": "3272.png", "formula": "\\begin{align*} y = x \\tan ( \\theta _ 0 - \\theta _ k ) , y = x \\tan ( \\theta _ 0 ) , \\ , \\ , \\ , \\ , x > 0 . \\end{align*}"}
-{"id": "862.png", "formula": "\\begin{align*} \\varepsilon = \\sqrt { e ^ { - \\alpha } ( 1 + \\alpha ) } < 1 , \\alpha = \\beta ^ 2 \\delta ^ 2 - 1 > 0 \\end{align*}"}
-{"id": "2213.png", "formula": "\\begin{align*} \\widetilde Q _ i = w _ 1 ^ { m _ { i 1 } } \\cdot \\ldots \\cdot w _ n ^ { m _ { i n } } \\cdot Q _ i \\left ( \\dfrac 1 { w _ 1 } , \\ldots , \\dfrac 1 { w _ n } \\right ) . \\end{align*}"}
-{"id": "7532.png", "formula": "\\begin{gather*} w ^ 1 = - 2 \\frac { \\tilde w ^ 1 _ 2 } { \\tilde w ^ 1 } , w ^ 2 = \\frac { \\tilde w ^ 2 } { \\tilde w ^ 1 } \\end{gather*}"}
-{"id": "8501.png", "formula": "\\begin{align*} b _ 1 - b _ 2 = b _ { \\chi _ 1 \\chi _ 2 ^ { - 1 } } \\varpi ^ { a ( \\chi _ 1 ) - a ( \\chi _ 1 \\chi _ 2 ^ { - 1 } ) } . \\end{align*}"}
-{"id": "2567.png", "formula": "\\begin{align*} \\xi : = \\alpha - \\eta > \\frac { \\pi } { 2 } + \\frac { d _ { 0 } } { 4 } \\ , . \\end{align*}"}
-{"id": "778.png", "formula": "\\begin{align*} ( - 1 ) ^ { k + 1 } B _ { 2 k + 1 } ( x ) > 0 , k = 1 , 2 , . . . 0 < x < \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "6665.png", "formula": "\\begin{align*} S _ 2 ' = \\{ ( z , t ) \\in \\mathbb { R } ^ { n - 1 } \\times \\mathbb { R } : f _ x ( z ) \\leq t \\leq ( 1 + \\varepsilon ) \\Delta \\} \\quad . \\end{align*}"}
-{"id": "8165.png", "formula": "\\begin{align*} a _ 3 ^ { q ^ 3 + 1 } = 1 . \\end{align*}"}
-{"id": "5378.png", "formula": "\\begin{align*} \\textbf { R } = ( G , T , V , \\cdot , ( , ) , \\varphi , \\psi ) \\end{align*}"}
-{"id": "7832.png", "formula": "\\begin{align*} P _ m = P \\otimes \\cdots \\otimes P . \\end{align*}"}
-{"id": "4843.png", "formula": "\\begin{align*} w _ p ( n ) = \\sum _ { i = 0 } ^ r ( h _ i - i ) . \\end{align*}"}
-{"id": "8249.png", "formula": "\\begin{align*} E ( s , g ) = v ^ { \\circ } ( s ) ( g ) + [ M ( s ) v ^ { \\circ } ( s ) ] ( g ) + \\sum _ { q \\in F ^ { \\times } } W _ s ( a ( q ) g ) , \\end{align*}"}
-{"id": "634.png", "formula": "\\begin{align*} \\delta _ { ( a + 1 ) k } < \\delta : = \\frac { a ^ { 1 - p / 2 } - [ \\omega + ( 1 - \\omega ) ( 1 + \\rho - 2 \\alpha \\rho ) ^ { 1 - p / 2 } ] } { a ^ { 1 - p / 2 } + [ \\omega + ( 1 - \\omega ) ( 1 + \\rho - 2 \\alpha \\rho ) ^ { 1 - p / 2 } ] } . \\end{align*}"}
-{"id": "594.png", "formula": "\\begin{align*} \\mathcal { R } \\cap \\left ( s _ { 0 } - \\varepsilon , \\frac { 1 } { 2 } \\right ] = \\{ s _ { 0 } \\} . \\end{align*}"}
-{"id": "1085.png", "formula": "\\begin{align*} \\sigma _ { c , i } ^ 2 = \\frac { P _ t } { 2 N } \\sum \\limits _ { k \\in \\mathcal { N } , k \\neq i } E [ | \\alpha _ { k i } | ^ 2 ] . \\end{align*}"}
-{"id": "6766.png", "formula": "\\begin{align*} \\Delta u + \\rho \\left ( \\frac { h e ^ u } { \\int _ M h e ^ u } - \\frac { 1 } { \\abs { M } } \\right ) = 0 , \\end{align*}"}
-{"id": "7223.png", "formula": "\\begin{align*} F _ { 0 , \\lambda } ( R _ \\lambda ) & = F ( R _ { \\lambda } ) + \\lambda \\mathbb { F } ( R _ { \\lambda } - \\Sigma ) \\\\ & \\leq \\liminf _ { i \\to \\infty } \\{ F ( R _ { \\delta _ i , \\lambda } ) + \\lambda | \\mathbb { F } ( R _ { \\delta _ i , \\lambda } - \\Sigma ) - \\eta _ { \\delta _ i } | \\} \\\\ & = \\liminf _ { i \\to \\infty } F _ { \\delta _ i , \\lambda } ( R _ { \\delta _ i , \\lambda } ) . \\end{align*}"}
-{"id": "2891.png", "formula": "\\begin{align*} O _ i = \\{ \\langle 0 , \\sigma ^ { \\smallfrown } ( k - 1 ) \\rangle : \\sigma \\in ( < \\omega ) ^ { k - 1 } c ( \\sigma ) = i \\} . \\end{align*}"}
-{"id": "3548.png", "formula": "\\begin{align*} I ( S _ { \\Lambda } \\in U ) = I ( S _ { \\Lambda _ 0 } \\in U ) = I ( S _ { B _ { \\boldsymbol { t } _ 0 } } \\in U ) . \\end{align*}"}
-{"id": "2397.png", "formula": "\\begin{align*} a _ 6 = 3 a _ 2 + a _ 5 = R _ 2 = 0 , \\ a _ 2 ^ 2 \\leq 4 a _ 3 ^ 2 + 4 a _ 1 a _ 2 a _ 3 , \\ a _ 2 ( 3 a _ 3 + a _ 4 ) \\neq 0 , \\end{align*}"}
-{"id": "8413.png", "formula": "\\begin{align*} n ( \\psi _ E ) = - \\frac { d } { f } . \\end{align*}"}
-{"id": "8109.png", "formula": "\\begin{align*} \\varphi ( x ) ( t ) = \\sum _ { i \\in I } h _ { i , t } ( x ) \\end{align*}"}
-{"id": "5778.png", "formula": "\\begin{align*} \\| u \\| _ { \\dot { L } ^ { \\alpha , p } ( \\R ^ n ) } = \\| f \\| _ { L ^ { p } ( \\R ^ n ) } . \\end{align*}"}
-{"id": "5814.png", "formula": "\\begin{align*} & \\int _ { \\R ^ n } \\vert \\nabla v \\vert ^ { p - 2 } \\nabla v \\cdot \\nabla \\psi \\ ; d x \\\\ = & \\int _ { \\R ^ n } v ^ { q } \\psi \\ ; d \\sigma + \\int _ { \\R ^ n } \\psi \\ ; d \\mu , \\psi \\in \\dot { W } _ { 0 } ^ { 1 , p } ( \\R ^ n ) , \\end{align*}"}
-{"id": "5601.png", "formula": "\\begin{align*} X _ t = x _ 0 + \\int _ 0 ^ t f ( X _ s ) d s + { B } ^ H _ t . \\end{align*}"}
-{"id": "4035.png", "formula": "\\begin{align*} \\frac { 1 } { \\max _ { X \\in S _ p } ( k _ { M , p } ( X ) ) } + \\frac { 1 } { \\min _ { Y \\in S _ q } ( k _ { M , q } ( Y ) ) } = c , \\end{align*}"}
-{"id": "1685.png", "formula": "\\begin{align*} r ( \\eta ) = v \\Rightarrow \\eta \\equiv ( v , Q _ 1 , \\ldots , Q _ { 2 n - 1 } , v ) r ( \\eta ) \\not = v \\Rightarrow \\eta \\equiv ( Q _ 0 , v , Q _ 2 , \\ldots , v , Q _ { 2 n } ) , \\end{align*}"}
-{"id": "2773.png", "formula": "\\begin{align*} \\det \\Big ( M ( \\vec w , k ) \\Big ) = \\det \\begin{pmatrix} c _ { w _ 1 } & c _ { w _ 1 + p } & \\cdots & c _ { w _ 1 + k p } \\\\ c _ { w _ 1 - 1 } & c _ { w _ 1 + p - 1 } & \\cdots & c _ { w _ 1 + k p - 1 } \\\\ c _ { w _ 1 - 2 } & c _ { w _ 1 + p - 2 } & \\cdots & c _ { w _ 1 + k p - 2 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ c _ { w _ 1 - k } & c _ { w _ 1 + p - k } & \\cdots & c _ { w _ 1 + k p - k } \\end{pmatrix} . \\end{align*}"}
-{"id": "2811.png", "formula": "\\begin{align*} S _ { 0 } = \\frac { K r } { \\delta } , \\alpha ^ { + } = \\frac { 1 } { 2 \\sigma ^ { 2 } } \\left [ \\sigma ^ { 2 } - 2 ( r - \\delta ) + \\sqrt { 4 \\delta ^ { 2 } - 8 \\delta r + 4 \\delta \\sigma ^ { 2 } + 4 r ^ { 2 } + 4 \\sigma ^ { 2 } r + \\sigma ^ { 4 } } \\right ] . \\end{align*}"}
-{"id": "6082.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\tfrac { \\Theta _ n ( t ) } { t } = - \\tfrac { 1 } { 2 } \\sqrt { - ( 4 + 8 k - 4 k ^ 2 - 4 n - 4 k n + n ^ 2 ) } , \\end{align*}"}
-{"id": "4219.png", "formula": "\\begin{align*} \\mathbb { E } [ \\mathrm { t r } ( \\rho _ N ^ k ) ] = \\tau [ C _ { 2 k } ( U _ N , U _ N ^ * ) ] = N ^ { - 2 k - 1 } \\sum _ { j = 1 } ^ { k + 1 } \\mathcal { F } ( 2 k , j ) ( N ) _ j . \\end{align*}"}
-{"id": "8459.png", "formula": "\\begin{align*} x _ 2 = ( b _ 2 ) _ 0 + \\alpha \\varpi ^ { r - v ( b _ 2 ) } \\end{align*}"}
-{"id": "6556.png", "formula": "\\begin{align*} \\mathrm { I . } t \\lambda - \\mu ( 1 + \\beta _ { \\xi _ 1 } \\delta ' ) ^ { - 1 } = 0 \\qquad \\quad \\mathrm { I I . } t ( 1 - \\lambda ) + \\mu = 1 . \\end{align*}"}
-{"id": "3083.png", "formula": "\\begin{align*} g _ \\epsilon ( s ) : = | u _ \\epsilon ^ 1 ( s ) | ^ 2 - \\frac { 1 } { 1 + \\delta } \\| u _ \\epsilon ( s ) \\| ^ 2 , \\end{align*}"}
-{"id": "5463.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } U _ { r ^ n z } ( y ) = y ^ \\rho p ( z y ) = : V _ z ( y ) z y \\in C _ p . \\end{align*}"}
-{"id": "2201.png", "formula": "\\begin{align*} \\begin{cases} | 1 - t a _ { 1 1 } z _ 1 | ^ { m _ { 1 1 } } \\cdot \\ldots \\cdot | 1 - a _ { 1 j _ { 1 } } z _ { j 1 } | ^ { m _ { 1 j _ { 1 } } } \\cdot \\ldots \\cdot | 1 - t a _ { 1 n } z _ n | ^ { m _ { 1 n } } = r _ 1 , \\\\ \\ldots \\\\ | 1 - t a _ { n 1 } z _ 1 | ^ { m _ { n 1 } } \\cdot \\ldots \\cdot | 1 - a _ { n j _ { n } } z _ { j n } | ^ { m _ { n j _ { n } } } \\cdot \\ldots \\cdot | 1 - t a _ { n n } z _ n | ^ { m _ { n n } } = r _ n , \\end{cases} \\end{align*}"}
-{"id": "5047.png", "formula": "\\begin{align*} E _ { _ { H , \\mathcal { U } _ i } } = \\eta _ { _ { \\mathcal { U } _ i } } P _ 0 H _ { \\mathcal { U } _ i } t _ e , \\quad \\forall i \\in \\left \\lbrace 1 , 2 , \\ldots , N \\right \\rbrace , \\end{align*}"}
-{"id": "1633.png", "formula": "\\begin{align*} X : = \\bigsqcup _ { i \\geq 1 } G _ i , \\end{align*}"}
-{"id": "8280.png", "formula": "\\begin{align*} K _ p = G ( \\Q _ p ) \\cap C ( V _ { \\Z _ p } ) ^ \\times \\end{align*}"}
-{"id": "3573.png", "formula": "\\begin{align*} \\overline { n ( p ) } = \\lim _ { x \\to \\infty } \\frac { 1 } { \\pi ( x ) } \\sum _ { p \\leq x } n ( p ) = 2 . 9 2 0 0 5 0 \\ldots , \\end{align*}"}
-{"id": "2092.png", "formula": "\\begin{align*} 0 \\leq w ( p , t ) \\leq e ^ { C _ 5 '' t } \\sup _ M | w ( \\cdot , 0 ) | = 0 \\end{align*}"}
-{"id": "1395.png", "formula": "\\begin{align*} \\deg v _ I + \\deg y _ { J } = 2 i , \\end{align*}"}
-{"id": "10051.png", "formula": "\\begin{align*} E ( \\tau , s ) = E ( \\vec { \\tau } , s ) | _ \\mathcal { H } \\end{align*}"}
-{"id": "8673.png", "formula": "\\begin{align*} P = \\begin{pmatrix} P _ { 1 1 } & P _ { 1 2 } & 0 \\\\ P _ { 2 1 } & P _ { 2 2 } & P _ { 2 3 } \\\\ 0 & P _ { 3 2 } & P _ { 3 3 } \\end{pmatrix} , \\end{align*}"}
-{"id": "6085.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow - \\infty } \\Psi ( t ) = 0 \\quad \\mbox { a n d } \\quad \\lim _ { t \\rightarrow - \\infty } e ^ { - t } \\Psi ' ( t ) = 1 . \\end{align*}"}
-{"id": "3989.png", "formula": "\\begin{align*} p ^ { \\beta _ 3 } ( 3 , t ) = - \\sum _ { k = 3 } ^ { \\infty } ( - \\lambda ) ^ k \\underset { \\Omega ^ { k } _ { 3 } } { \\sum } \\frac { t ^ { \\sum _ { j = 0 } ^ 3 k _ j \\beta _ j } } { \\Gamma \\left ( \\sum _ { j = 0 } ^ 3 k _ j \\beta _ j + 1 \\right ) } , \\end{align*}"}
-{"id": "1321.png", "formula": "\\begin{align*} P _ { H _ \\alpha } \\varphi | _ { A _ \\alpha } ( g ) | _ { H _ \\alpha } = P _ { H _ \\alpha } \\pi ( g ) | _ { H _ \\alpha } = \\pi _ \\alpha ( g ) . \\end{align*}"}
-{"id": "4245.png", "formula": "\\begin{align*} \\det ( A _ p ) = c \\lambda ^ { m ^ 2 } ( \\lambda - 1 ) ^ { m ^ 2 } \\cdot \\det ( B _ { m + 1 } ) , \\end{align*}"}
-{"id": "9472.png", "formula": "\\begin{align*} S _ 0 ( 1 ) & = 1 = ( q ^ 2 ; q ^ 2 ) _ 0 , \\\\ S _ 1 ( 1 ) & = 1 - q ^ 2 = ( q ^ 2 ; q ^ 2 ) _ 1 . \\end{align*}"}
-{"id": "4191.png", "formula": "\\begin{align*} E _ t : y ^ 2 = 4 ( x - t _ 1 ) ^ 3 - t _ 2 ( x - t _ 1 ) - t _ 3 , \\end{align*}"}
-{"id": "9274.png", "formula": "\\begin{align*} \\Big \\| { \\sigma } _ \\varepsilon ^ { ( \\varepsilon ^ { - 2 } s _ { n + 1 } , \\ ; 0 ) } ( \\cap _ { j = 1 } ^ n \\{ \\mathcal { M } _ { s _ j \\varepsilon ^ { - 2 } } = \\nu _ j \\} ) - e ^ { ( s _ { n + 1 } - s _ n ) Q } \\mathcal { P } _ { \\nu _ n } \\dots e ^ { ( s _ 2 - s _ 1 ) Q } \\mathcal { P } _ { \\nu _ 1 } e ^ { s _ 1 Q } \\Big \\| _ { 2 , } & \\leq C n \\varepsilon . \\end{align*}"}
-{"id": "9737.png", "formula": "\\begin{align*} y _ { k , n } = y _ k + ( 2 n + 1 + \\theta _ k ) s , \\end{align*}"}
-{"id": "4133.png", "formula": "\\begin{align*} h ( Z , X ) = X ( \\langle \\eta , \\xi \\rangle ) = \\langle D _ X \\eta , \\xi \\rangle + \\langle \\eta , D _ X \\xi \\rangle = \\langle \\eta , d \\xi _ q X \\rangle . \\end{align*}"}
-{"id": "9566.png", "formula": "\\begin{align*} \\frac { b } { n } = \\frac { 1 } { q } - \\frac { n - s p } { n p } \\quad - b q = \\frac q p ( n - s p ) - n \\ , . \\end{align*}"}
-{"id": "1760.png", "formula": "\\begin{align*} & T ( f \\sqrt { d \\mu } ) = T ( f \\sqrt { d \\mu / d \\lambda } \\sqrt { d \\lambda } ) \\\\ \\Longrightarrow & F _ \\mu f \\sqrt { d \\mu } = F _ \\lambda f \\sqrt { d \\mu / d \\lambda } \\sqrt { d \\lambda } \\end{align*}"}
-{"id": "8046.png", "formula": "\\begin{align*} m ( r ( s ) ) \\cos \\zeta ( s ) = { \\rm c o n s t a n t } = \\nu , \\end{align*}"}
-{"id": "3862.png", "formula": "\\begin{align*} \\sum \\limits _ { i \\in I _ g ( x ^ * ) } \\bar \\lambda _ i \\nabla g _ i ( x ^ * ) + \\sum \\limits _ { i = 1 } ^ p \\bar \\mu _ i \\nabla h _ i ( x ^ * ) + \\sum \\limits _ { i \\in I _ 0 ( x ^ k ) } \\bar \\gamma _ i e _ i = 0 \\end{align*}"}
-{"id": "1429.png", "formula": "\\begin{align*} \\phi _ \\ell ( x _ I ) & = e _ \\ell ( x _ { 4 i _ 1 } , \\dots , x _ { 4 i _ { \\ell } } ) \\not = 0 , \\\\ \\phi _ { \\ell + 1 } ( x _ I ) & = e _ \\ell ( x _ { 4 i _ 1 } , \\dots , x _ { 4 i _ { \\ell } } ) ^ 2 \\not = 0 , \\\\ \\phi _ { \\ell + 2 } ( x _ I ) & = 0 . \\end{align*}"}
-{"id": "5534.png", "formula": "\\begin{align*} \\begin{array} { c c } x _ { 1 } \\left ( t _ { 0 } \\right ) = 1 & x _ { 2 } \\left ( t _ { 0 } \\right ) = 0 \\\\ \\dot { x } _ { 1 } \\left ( t _ { 0 } \\right ) = 0 & \\dot { x } _ { 2 } \\left ( t _ { 0 } \\right ) = 1 \\end{array} \\end{align*}"}
-{"id": "272.png", "formula": "\\begin{align*} \\hat { P } _ M ^ { \\rm o d d } = \\hat { M } _ { \\rm o d d } + \\hat { M } _ { \\rm e v e n } ~ ~ ~ ~ \\hat { P } _ M ^ { \\rm e v e n } = \\hat { M } _ { \\rm o d d } + \\hat { M } _ { \\rm e v e n } + \\hat { M } . \\end{align*}"}
-{"id": "8464.png", "formula": "\\begin{align*} S = \\{ x + \\alpha \\Omega ^ { r - 1 } \\colon x \\in S ' , \\alpha \\in \\mathfrak { O } / \\mathfrak { P } \\} . \\end{align*}"}
-{"id": "4585.png", "formula": "\\begin{align*} [ F _ { 0 , 1 } , F _ { 1 , - 1 } ] _ { q ^ { 2 } } = [ F _ { 1 , 1 } , F _ { 0 , - 1 } ] _ { q ^ { 2 } } - ( q _ 1 ^ { - 1 } + q _ 3 ^ { - 1 } ) [ F _ { 1 , 0 } , F _ { 0 , 0 } ] \\ , , \\end{align*}"}
-{"id": "1412.png", "formula": "\\begin{align*} \\Gamma _ i ( m ) = ( m _ 0 , \\dots , m _ n ) , \\end{align*}"}
-{"id": "9779.png", "formula": "\\begin{align*} & \\bar { Z } = \\bar { Z } _ { 0 } \\exp ( - \\frac { 1 } { \\bar { \\rho } _ { 0 } \\bar { u } _ { 0 } A ( 0 ) } \\int _ 0 ^ x A ( \\tau ) \\bar { \\rho } \\phi ( \\bar { T } ) d \\tau ) + O ( 1 ) \\delta _ * ^ 2 . \\end{align*}"}
-{"id": "6802.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int _ { \\mathbb { S } ^ 2 } | \\nabla w _ { \\lambda } | ^ 2 = \\frac { 1 } { 2 } \\langle - \\Delta _ g w _ { \\lambda } , w _ { \\lambda } \\rangle = \\frac { 1 } { 2 } \\langle - \\Delta _ g w _ { \\lambda } , \\tilde { w } _ { \\lambda } \\rangle \\end{align*}"}
-{"id": "1796.png", "formula": "\\begin{align*} M ^ { 1 / y } = a ^ { a ( 1 - b ) } b ^ { a - 1 } . \\end{align*}"}
-{"id": "5439.png", "formula": "\\begin{align*} \\widetilde { M } _ 4 = \\left ( \\begin{array} { r r r r r } 1 & 5 & 1 0 & 1 0 & 5 \\\\ 1 & 6 & 1 4 & 1 6 & 8 \\\\ 1 & 6 & 1 5 & 1 8 & 9 \\\\ \\end{array} \\right ) \\quad \\rightsquigarrow { M } _ 4 = \\left ( \\begin{array} { r r r r r } 1 & 5 & 1 0 & 1 0 & 5 \\\\ 0 & 1 & 4 & 6 & 3 \\\\ 0 & 0 & 1 & 2 & 1 \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "2542.png", "formula": "\\begin{align*} \\left [ L - i \\eta \\mathrm { P } _ 1 \\xi _ 1 - i \\eta \\rho \\right ] ( \\mathrm { P } _ 1 e ) = i \\eta \\mathrm { P } _ 1 \\xi _ 1 ( \\mathrm { P } _ 0 e ) . \\end{align*}"}
-{"id": "7683.png", "formula": "\\begin{align*} y ( t ) = \\left [ \\Pi ~ | ~ 0 \\right ] x ( t ) = \\Pi x _ 1 ( t ) \\ , , \\end{align*}"}
-{"id": "235.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { l } e ^ { - \\frac { \\mu } { l } | k _ { j - 1 } - k _ j | } \\leq e ^ { - \\frac { \\mu } { 2 l } | a - b | } \\prod _ { j = 1 } ^ { l } e ^ { - \\frac { \\mu } { 2 l } | k _ { j - 1 } - k _ j | } . \\end{align*}"}
-{"id": "1230.png", "formula": "\\begin{align*} ( \\nabla f ) ( \\nabla \\hat G ( X ) ) & = - p \\ , k ^ { p - 1 } ( - \\nabla \\hat G ( X ) ) \\ , \\ , ( \\nabla k ( - \\nabla \\hat G ( X ) ) ) \\\\ & = \\frac { - X } { h ( X ) } \\ , p \\ , k ^ { p - 1 } ( - \\nabla \\hat G ( x ) ) \\\\ & = X p ( - \\beta ) ^ { p - 1 } \\ , h ^ { [ ( \\beta - 1 ) ( p - 1 ) - 1 ] } ( X ) \\ , k ^ { p - 1 } ( \\nabla h ( X ) ) \\\\ & = X \\ , p \\ , \\ , ( - \\beta ) ^ { p - 1 } h ( X ) ^ { - n } . \\end{align*}"}
-{"id": "4352.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { j = m } ^ \\infty M _ { a _ j } A M _ { b _ j } f \\Big \\| ^ p \\to 0 \\end{align*}"}
-{"id": "6114.png", "formula": "\\begin{align*} \\mu _ { f } \\left ( A \\right ) = \\mu \\left ( A _ { f ^ { - 1 } } \\right ) , \\end{align*}"}
-{"id": "6973.png", "formula": "\\begin{align*} ( \\det { D ^ 2 u } ) ^ { 1 / n } = \\inf _ { M \\in \\mathcal { M } } { L _ M u } = \\inf _ { M \\in \\mathcal { M } } { t r a c e ( M D ^ 2 u ) } , \\end{align*}"}
-{"id": "240.png", "formula": "\\begin{align*} T _ { x + i y } ^ { a , b } ( G , G ' ) : = \\chi _ a \\{ R _ { x + i y } ( A _ G ) - R _ { x + i y } ( A _ { G ' } ) \\} \\chi _ b . \\end{align*}"}
-{"id": "5802.png", "formula": "\\begin{align*} - \\Delta _ { p } w _ { j } = \\omega _ { j - 1 } \\ ; \\ ; \\R ^ n . \\end{align*}"}
-{"id": "8113.png", "formula": "\\begin{align*} O _ i = \\varphi ^ { - 1 } ( S _ i ) \\setminus \\big ( \\overline { S _ { i - 1 } \\varphi ^ { - 1 } ( S _ { i - 1 } ) } \\cup \\cdots \\cup \\overline { S _ 1 \\varphi ^ { - 1 } ( S _ 1 ) } \\big ) . \\end{align*}"}
-{"id": "2596.png", "formula": "\\begin{align*} \\pi \\tan ( \\tfrac { a \\pi } 2 ) u - S u = f ( 0 , 1 ) \\end{align*}"}
-{"id": "6387.png", "formula": "\\begin{align*} \\mathcal { S } \\left [ P \\left \\vert P _ { } \\right . \\right ] \\overset { } { = } - \\int d x d \\theta P \\left ( x \\theta \\right ) \\log \\left [ \\frac { P \\left ( x \\theta \\right ) } { P _ { } \\left ( x \\theta \\right ) } \\right ] \\end{align*}"}
-{"id": "3295.png", "formula": "\\begin{align*} \\psi _ 2 ( z , m ) = \\ , A \\sum _ { j = 1 } ^ 2 m _ j ^ 2 + 2 \\sum _ { j = 1 } ^ 2 m _ j ^ 2 \\log m _ j - 8 \\pi m _ 1 m _ 2 G ( z ) . \\end{align*}"}
-{"id": "5386.png", "formula": "\\begin{align*} a & : = ( 1 , 2 ) ( 3 , 4 ) ( 5 , 6 ) \\\\ b & : = ( 5 , 6 ) \\\\ c & : = ( 2 , 3 ) ( 4 , 5 ) \\end{align*}"}
-{"id": "1656.png", "formula": "\\begin{align*} a _ i b _ i = d _ i c _ i , a _ i b _ { 1 - i } = d _ i c _ { 1 - i } , \\quad c _ i d _ i = b _ { 1 - i } a _ { 1 - i } . \\end{align*}"}
-{"id": "66.png", "formula": "\\begin{align*} \\frac { \\partial D D ^ * } { \\partial \\textbf { w } ^ * } = 0 \\end{align*}"}
-{"id": "1010.png", "formula": "\\begin{align*} P _ { i j } = \\mathbb { P } ( e _ i ^ h \\rightarrow e _ j ^ h ) , i , j \\in \\{ 1 , 2 \\} . \\end{align*}"}
-{"id": "9324.png", "formula": "\\begin{align*} J _ { 0 } = 0 , \\ J _ { 1 } = 1 , \\ J _ { n + 1 } = J _ { n } + 2 J _ { n - 1 } , \\ n \\geq 1 . \\end{align*}"}
-{"id": "4984.png", "formula": "\\begin{align*} F : = \\frac { \\left \\Vert f \\right \\Vert _ { \\dot { F } ^ { \\alpha , p } _ { q } } } { \\min \\left ( \\eta _ { \\delta } , \\frac { \\delta } { D _ { \\delta } } \\right ) } \\tilde { F } . \\end{align*}"}
-{"id": "9116.png", "formula": "\\begin{align*} 2 f _ 2 ( x _ 1 x _ 2 ) + x _ 1 f _ 1 ( x _ 2 ) + x _ 2 f _ { 1 } ( x _ 1 ) = 0 . \\end{align*}"}
-{"id": "7060.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l c l } - \\triangle u & = & \\lambda \\nabla F ( u ) & & B ^ N \\\\ \\frac { \\partial u } { \\partial \\nu } & = & 0 & & S ^ { N - 1 } , \\end{array} \\right . \\end{align*}"}
-{"id": "7380.png", "formula": "\\begin{align*} F ( n ^ { - 1 } , \\tau ) = \\begin{cases} 1 - \\mathcal { O } \\big ( e ^ { - j ^ { 3 / 2 } N _ { f } } \\big ) & n ^ { - 1 } > 2 \\sqrt { 1 + 2 \\tau } \\\\ \\mathcal { O } \\big ( e ^ { - | j | ^ 3 N _ { f } ^ { 2 } } ) & n ^ { - 1 } < 2 \\sqrt { 1 + 2 \\tau } \\\\ \\end{cases} . \\end{align*}"}
-{"id": "5469.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } U _ { r ^ n z } ( y ) = u _ z ( y ) , y \\in C _ { u _ z } \\end{align*}"}
-{"id": "9695.png", "formula": "\\begin{align*} \\gamma _ 1 ( \\omega _ { k , k + 1 } ) = \\gamma _ 1 ( 0 ) + \\frac { \\partial \\gamma _ 1 } { \\partial \\omega _ { k , k + 1 } } { \\big | _ { \\{ \\omega _ { k , k + 1 } = 0 \\} } } \\omega _ { k , k + 1 } + O ( 1 ) | \\omega _ { k , k + 1 } | ^ 2 . \\end{align*}"}
-{"id": "3810.png", "formula": "\\begin{align*} E _ { k + 1 } : = \\max \\{ z < E _ k \\colon \\ , N _ 0 ( z ) > 0 \\} , \\ ; \\ ; k \\ge 0 . \\end{align*}"}
-{"id": "9246.png", "formula": "\\begin{align*} [ x \\otimes \\alpha , y \\otimes \\beta ] = ( x \\otimes \\alpha ) ( y \\otimes \\beta ) - ( y \\otimes \\beta ) ( x \\otimes \\alpha ) = x y \\otimes \\alpha \\beta - y x \\otimes \\beta \\alpha . \\end{align*}"}
-{"id": "1374.png", "formula": "\\begin{align*} \\varrho _ 0 : = \\left ( \\overline { \\nu } \\overline { \\sigma } { \\underline { \\sigma } } ^ { - 2 } \\underline { \\eta } ^ { - 2 } \\max \\left \\{ 8 \\overline \\nu _ 1 ^ 2 , 1 6 \\overline \\nu _ 2 ^ 2 , 3 2 \\overline \\nu _ 3 ^ 2 , 6 4 \\overline \\nu _ 4 ^ 2 , 1 2 8 \\overline \\nu _ 5 ^ 2 , 1 2 8 \\overline { \\nu } _ 0 ^ 2 , \\ 4 \\kappa _ 0 ^ 2 \\right \\} \\right ) ^ { 1 / 2 } \\ , . \\end{align*}"}
-{"id": "5628.png", "formula": "\\begin{align*} \\frac 1 { ( a - 1 ) \\ , ( k + 1 ) ^ { a - 1 } } \\ , \\le \\ , \\sum _ { j = k + 1 } ^ { \\infty } \\frac { 1 } { j ^ a } \\ , \\le \\ , \\frac 1 { ( a - 1 ) \\ , ( k + 1 / 2 ) ^ { a - 1 } } . \\end{align*}"}
-{"id": "5522.png", "formula": "\\begin{align*} h ( x ) & = \\frac { h ( x _ { n + 1 } ) - h ( x _ { n } ) } { x _ { n + 1 } - x _ n } ( x - x _ n ) + h ( x _ n ) \\\\ & = \\frac { \\sqrt { 2 } } { 2 } \\left ( \\frac { \\sqrt { 2 } + 1 } { 3 } \\right ) ^ n x + ( \\sqrt { 2 } + 1 ) ^ n \\left ( \\frac { 2 - \\sqrt { 2 } } { 4 } \\right ) . \\end{align*}"}
-{"id": "2786.png", "formula": "\\begin{align*} u _ { i } = g ( t _ { i } , u _ { i } ) + \\underbrace { \\sum _ { j = 0 } ^ { i } w _ { i , j } k _ { 1 } ( t _ { i } , t _ { j } , { u } _ { i } , { u } _ { j } ) } _ { } + \\underbrace { \\sum _ { j = 0 } ^ { i } \\omega _ { j } k _ { 2 } ( t _ { i } , t _ { j } , { u } _ { i } , { u } _ { j } ) } _ { } , i = 0 , 1 , \\cdots , n , \\end{align*}"}
-{"id": "2362.png", "formula": "\\begin{align*} \\big ( \\chi _ n ( g ) f \\big ) \\left ( \\begin{smallmatrix} x \\\\ y \\end{smallmatrix} \\right ) = f \\big ( g ^ { - 1 } \\left ( \\begin{smallmatrix} x \\\\ y \\end{smallmatrix} \\right ) \\big ) , \\end{align*}"}
-{"id": "186.png", "formula": "\\begin{align*} \\bigwedge _ { i \\in I } \\delta _ { i } ( a ) = \\bigcap \\{ \\delta _ { i } ( a ) : i \\in I \\} \\end{align*}"}
-{"id": "1939.png", "formula": "\\begin{align*} \\alpha _ j = \\beta + ( j - 1 ) - \\frac { n } { 2 } - \\frac { ( n - 1 ) } { 2 } ( j - 2 ) \\max { \\left ( 0 , \\frac { 1 } { 2 } - \\frac { \\beta } { n } \\right ) } . \\end{align*}"}
-{"id": "404.png", "formula": "\\begin{align*} X _ { n j } ^ { ( \\varepsilon x \\sigma _ { n } ) } = X _ { n j } I ( X _ { n j } \\leq \\varepsilon x \\sigma _ { n } ) \\ \\ \\ S _ { n } ^ { ( \\varepsilon x \\sigma _ { n } ) } = \\sum _ { j = 1 } ^ { k _ { n } } X _ { n j } ^ { ( \\varepsilon x \\sigma _ { n } ) } . \\end{align*}"}
-{"id": "2806.png", "formula": "\\begin{align*} \\mathcal { F } _ { s } \\{ V ( t , S ) \\} = & \\int _ { 0 } ^ { \\infty } V ( t , S ) \\sin ( \\omega S ) \\mathrm { d } S , \\\\ \\mathcal { F } _ { c } \\{ V ( t , S ) \\} = & \\int _ { 0 } ^ { \\infty } V ( t , S ) \\cos ( \\omega S ) \\mathrm { d } S , \\end{align*}"}
-{"id": "3286.png", "formula": "\\begin{align*} p _ 1 = \\dfrac { a } { 2 } , p _ 2 = \\dfrac { b } { 2 } , p _ 3 = \\dfrac { a + b } { 2 } . \\end{align*}"}
-{"id": "9044.png", "formula": "\\begin{align*} d ( D _ 3 ) \\le 1 2 < 1 4 = n ^ 2 - 2 \\end{align*}"}
-{"id": "7811.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 v _ m } { \\partial r ^ 2 } - ( - \\Delta r + \\frac { 2 m } { r } + 2 \\rho ^ { \\prime } ) \\frac { \\partial v _ m } { \\partial r } + A _ r v _ m + ( \\frac { m ( m + 1 ) } { r ^ 2 } + \\frac { m } { r } ( 2 \\rho ^ { \\prime } - \\Delta r ) - V _ 0 - V _ 1 - V _ 2 + \\lambda ) v _ m = 0 . \\end{align*}"}
-{"id": "722.png", "formula": "\\begin{align*} \\Theta = \\frac { D - 8 d + 2 \\sqrt { ( 2 D - d ) ^ 2 + 1 5 D } } { 1 5 } > 0 . \\end{align*}"}
-{"id": "6535.png", "formula": "\\begin{align*} g \\left ( ( C - s ( C ) ) ^ { \\circ } \\right ) = 0 = s \\left ( ( C - g ( C ) ) ^ { \\circ } \\right ) . \\end{align*}"}
-{"id": "7662.png", "formula": "\\begin{align*} \\mathrm { E } [ s _ 1 ] \\le O ( \\log r ) \\sum _ { k = 1 } ^ { r ^ { 0 . 1 } } \\Pr [ \\rho \\ge k ] + ( 2 r + 1 ) \\Pr [ \\rho \\ge r ^ { 0 . 1 } ] . \\end{align*}"}
-{"id": "7303.png", "formula": "\\begin{align*} \\max _ j \\abs { s _ { n , j } } = \\frac { \\abs { a _ { n , j } - \\bar { a } _ n } } { \\sqrt [ ] { \\sum _ { j = 1 } ^ n \\left ( a _ { n , j } - \\bar { a } _ n \\right ) ^ 2 } } \\to 0 \\end{align*}"}
-{"id": "277.png", "formula": "\\begin{align*} \\underset { t \\rightarrow 0 ^ { + } } { \\lim } t ^ { 1 - \\alpha } x \\left ( t \\right ) = x _ { 0 } , \\end{align*}"}
-{"id": "4973.png", "formula": "\\begin{align*} \\left \\Vert \\Delta _ j f \\right \\Vert _ { L ^ { \\infty } } \\lesssim 2 ^ { \\frac { j d } p } \\left \\Vert \\Delta _ j f \\right \\Vert _ { L ^ p } = 2 ^ { \\alpha j } \\left \\Vert \\Delta _ j f \\right \\Vert _ { L ^ p } \\leq \\left \\Vert f \\right \\Vert _ { \\dot { F } ^ { \\alpha , p } _ q } , \\end{align*}"}
-{"id": "3879.png", "formula": "\\begin{align*} f ( \\omega ; s ; x ) = \\frac { 2 } { \\pi ^ { 1 / 2 } } \\left ( \\frac { x } { 4 l } \\right ) ^ { s } \\int _ { 0 } ^ { \\infty } \\omega ( y ) \\cos \\left ( \\frac { x y } { 2 l } \\right ) d y , \\end{align*}"}
-{"id": "1369.png", "formula": "\\begin{align*} W _ 1 ( H _ 1 ) = Q \\left ( \\Delta _ { 1 / c } \\circ ( Q ^ T H _ 1 Q ) \\right ) Q ^ T , \\forall \\ , H _ 1 \\in { \\cal S } ^ q \\ , . \\end{align*}"}
-{"id": "6622.png", "formula": "\\begin{align*} M _ t = \\det ( D f ( 0 ) ) ^ { ( t - 1 ) / 2 } K ( \\theta _ t ) A ( \\rho _ t ) N ( n _ t ) t \\in [ 0 , 1 ] \\end{align*}"}
-{"id": "7774.png", "formula": "\\begin{align*} | u _ { R } ( \\ell ) - u _ { R } ( m ) | & \\leq N _ { \\ell , m } ^ { 1 / 2 } \\bigg ( \\sum _ { i = 1 } ^ { N _ { \\ell , m } } | D _ { \\rho _ { i } } u _ { R } ( \\ell _ { i } ) | ^ { 2 } \\bigg ) ^ { 1 / 2 } \\leq C | \\ell - m | ^ { 1 / 2 } \\| D u _ { R } \\| _ { \\ell ^ { 2 } _ { \\mathcal { N } } ( \\L ) } \\\\ & \\leq C C _ { * } \\lambda | \\ell - m | ^ { 1 / 2 } = C _ { 1 } \\lambda | \\ell - m | ^ { 1 / 2 } . \\end{align*}"}
-{"id": "8561.png", "formula": "\\begin{align*} \\psi ( u ) : = \\begin{cases} \\displaystyle { \\frac { 1 } { 2 } \\int _ \\Omega H ( \\nabla u ( x ) ) ^ 2 \\ , d x } \\ & \\mbox { i f } \\ u \\in H ^ 1 _ 0 ( \\Omega ) , \\\\ + \\infty \\ & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "5002.png", "formula": "\\begin{align*} A _ { \\alpha } ( t ) = \\begin{cases} C 2 ^ { ( 1 - \\alpha ) a _ { \\alpha } t } & \\textrm { i f } \\alpha < 1 , \\\\ C a _ { \\alpha } t & \\textrm { i f } \\alpha = 1 , \\\\ C & \\textrm { i f } \\alpha > 1 . \\end{cases} \\end{align*}"}
-{"id": "4791.png", "formula": "\\begin{align*} [ b , \\mathcal R ^ k _ j ] _ 1 ( f , g ) : = \\ , & \\mathcal R ^ k _ j ( b f , g ) - b \\mathcal R ^ k _ j ( f , g ) , \\\\ [ b , \\mathcal R ^ k _ j ] _ 2 ( f , g ) : = \\ , & \\mathcal R ^ k _ j ( f , b g ) - b \\mathcal R ^ k _ j ( f , g ) . \\end{align*}"}
-{"id": "7341.png", "formula": "\\begin{align*} \\begin{aligned} \\| A _ i \\cdot \\nabla u \\| _ { L ^ { s _ i } ( [ 0 , T ] , L ^ { p _ i } ( \\R ^ 3 ) ) } \\ ; \\lesssim \\ ; \\| A \\| _ { L ^ { a _ i } ( [ 0 , T ] , L ^ { b _ i } ( \\R ^ 3 ) ) } \\| u \\| _ { X ^ { ( 4 , 3 ) } [ 0 , T ] } \\ , . \\end{aligned} \\end{align*}"}
-{"id": "7646.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { 2 ^ { n - 2 } } \\big ( E _ j E _ j ^ * + F _ j F _ j ^ * \\big ) = 2 ^ { n - 1 } \\bigg ( y ^ 2 + r z ^ 2 + 2 \\sum _ { j = 1 } ^ { 2 ^ { n - 3 } - 1 } k _ j x _ j ^ 2 \\bigg ) I _ m . \\end{align*}"}
-{"id": "7703.png", "formula": "\\begin{align*} Z _ { i m } = \\left [ \\begin{array} { c c } Q _ { i j } Q _ { m j } & 0 \\\\ 0 & 0 \\end{array} \\right ] = \\left [ \\begin{array} { c c } u _ { i j } u _ { m j } & 0 \\\\ 0 & 0 \\end{array} \\right ] \\ , , \\end{align*}"}
-{"id": "2938.png", "formula": "\\begin{align*} \\theta _ \\epsilon ( e , i ) : = \\frac { \\exp \\left ( - \\frac { l _ i } { \\epsilon } \\right ) } { \\sum _ { j \\in e } \\exp \\left ( - \\frac { l _ j } { \\epsilon } \\right ) } . \\end{align*}"}
-{"id": "6100.png", "formula": "\\begin{align*} \\left ( \\begin{smallmatrix} q ' ( t ) \\\\ p ' ( t ) \\end{smallmatrix} \\right ) = \\left ( \\begin{smallmatrix} p ( t ) \\\\ - ( 2 n - 2 ) p ( t ) - ( ( 2 n - 1 ) + c ^ 2 ) \\sin ( q ) \\end{smallmatrix} \\right ) = : V ( q , p ) . \\end{align*}"}
-{"id": "140.png", "formula": "\\begin{align*} M = \\begin{pmatrix} a & b \\\\ 0 & a \\end{pmatrix} ; N = \\begin{pmatrix} x & y \\\\ z & w \\end{pmatrix} \\end{align*}"}
-{"id": "9847.png", "formula": "\\begin{align*} ( e ^ { - x ^ { 2 } } ) \\hat { \\ , } ( y ) = \\sqrt { \\pi } e ^ { - y ^ { 2 } / 4 } . \\end{align*}"}
-{"id": "5597.png", "formula": "\\begin{align*} A _ z ( n , \\chi _ 0 ) & = q ^ n \\frac { n ^ { z - 1 } } { \\Gamma ( z ) } F _ d ( 1 / q , z ) + O _ { A } ( q ^ n n ^ { \\Re z - 2 } ( 1 + \\log m ) ^ { K _ A } ) \\\\ & = \\left ( \\prod _ { p | d } \\left ( 1 + \\frac { z } { q ^ { \\deg p } } \\right ) ^ { - 1 } \\right ) F ( 1 / q , z ) q ^ n \\frac { n ^ { z - 1 } } { \\Gamma ( z ) } + O _ { A } ( q ^ n n ^ { \\Re z - 2 } ( 1 + \\log m ) ^ { K _ A } ) . \\end{align*}"}
-{"id": "6291.png", "formula": "\\begin{align*} A _ 8 \\lesssim & \\sum _ { p > Q _ 1 + 2 } \\| \\nabla u _ { \\leq Q _ 1 } \\| _ \\infty \\| b _ p \\| _ 2 \\sum _ { | p - q | \\leq 2 } \\lambda _ q ^ { 2 s } \\| b _ q \\| _ 2 \\\\ \\lesssim & Q _ 1 f ( t ) \\sum _ { p > Q _ 1 + 2 } \\| b _ p \\| _ 2 \\sum _ { | p - q | \\leq 2 } \\lambda _ q ^ { 2 s } \\| b _ q \\| _ 2 \\\\ \\lesssim & Q _ 1 f ( t ) \\sum _ { q > Q _ 1 } \\lambda _ q ^ { 2 s } \\| b _ q \\| _ 2 ^ 2 ; \\end{align*}"}
-{"id": "4641.png", "formula": "\\begin{align*} \\sum _ { n = 2 ^ { k - 1 } } ^ { 2 ^ k - 1 } 1 _ { [ - B , B ] } \\left ( t + \\sum _ { i = 0 } ^ { n - 1 } g ( T ^ i x ) \\right ) \\ge & \\sum _ { n = 2 ^ { k - 1 } } ^ { 2 ^ k - 1 } 1 _ { [ - B + C , B - C ] } \\left ( t + \\sum _ { i = 0 } ^ { n - 1 } f ( T ^ i x ) \\right ) \\\\ \\ge & \\beta \\tau \\sum _ { n = 0 } ^ { 2 ^ k - 1 } 1 _ { [ - B - C , B + C ] } \\left ( t + \\sum _ { i = 0 } ^ { n - 1 } f ( T ^ i x ) \\right ) \\\\ \\ge & \\beta \\tau \\sum _ { n = 0 } ^ { 2 ^ k - 1 } 1 _ { [ - B , B ] } \\left ( \\sum _ { i = 0 } ^ { n - 1 } t + g ( T ^ i x ) \\right ) \\end{align*}"}
-{"id": "5535.png", "formula": "\\begin{align*} \\Phi \\left ( t , t _ { 0 } \\right ) = \\left [ \\begin{array} { c c } x _ { 1 } \\left ( t \\right ) & x _ { 2 } \\left ( t \\right ) \\\\ \\dot { x } _ { 1 } \\left ( t \\right ) & \\dot { x } _ { 2 } \\left ( t \\right ) \\end{array} \\right ] t \\in \\left ( - \\infty , \\infty \\right ) \\end{align*}"}
-{"id": "1735.png", "formula": "\\begin{align*} ( Q , d _ Q ) = ( L ^ 2 ( X , \\mathcal { F } , \\mu ) , d _ { L ^ 2 ( X , \\mathcal { F } , \\mu ) } ) , \\tilde { \\Sigma } = \\tilde { \\mathcal { S } } , \\ { \\Sigma } = { \\mathcal { S } } . \\end{align*}"}
-{"id": "2909.png", "formula": "\\begin{align*} Z = \\{ \\langle y , 0 \\rangle : y \\in X \\} \\cup \\{ k : k = \\langle y , c + 1 \\rangle \\ , \\wedge \\ , c < _ { \\mathcal { O } } ^ X b \\ , \\wedge \\ , f ( \\langle n _ \\sigma , k \\rangle ) = 1 \\} . \\end{align*}"}
-{"id": "8477.png", "formula": "\\begin{align*} A _ { \\pm } = - \\frac { 1 } { 2 v } \\pm \\frac { Y } { 2 v } \\varpi ^ { \\delta } \\end{align*}"}
-{"id": "7768.png", "formula": "\\begin{align*} D _ { R } : = \\{ ( x _ { 1 } , x _ { 2 } ) \\in \\mathbb { R } ^ { 2 } \\ , | \\ , ( x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } ) ^ { 1 / 2 } < R \\ , \\} . \\end{align*}"}
-{"id": "6126.png", "formula": "\\begin{align*} L \\left ( f \\right ) \\left ( x , t \\right ) = \\frac { 1 } { 2 } \\nabla _ { x } f ^ { T } \\boldsymbol { \\sigma } _ { \\mu _ { \\left ( \\xi ^ { - 1 } \\right ) ^ { t } } } ^ { 2 } \\nabla _ { x } f + \\frac { \\partial } { \\partial t } f \\end{align*}"}
-{"id": "8839.png", "formula": "\\begin{align*} \\partial _ t \\Psi _ v ( t , x ) = - \\frac { x + v t } { 2 t } \\lambda \\chi ' ( \\lambda ( x - v t ) ) e ^ { i \\phi ( t , x ) } + i \\partial _ t \\phi ( t , x ) \\chi ( \\lambda ( x - v t ) e ^ { i \\phi ( t , x ) } . \\end{align*}"}
-{"id": "7686.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } \\dot { z } _ 1 ( t ) \\\\ \\dot { z } _ 2 ( t ) \\end{array} \\right ] = \\left [ \\begin{array} { c c } 0 & I _ { N - 1 } \\\\ - \\bar { \\Lambda } & - \\bar { \\Lambda } \\end{array} \\right ] \\left [ \\begin{array} { c } z _ 1 ( t ) \\\\ z _ 2 ( t ) \\end{array} \\right ] + \\left [ \\begin{array} { c } 0 \\\\ Q \\end{array} \\right ] w ( t ) \\ , , \\end{align*}"}
-{"id": "4813.png", "formula": "\\begin{align*} \\mathbf { y } = \\frac m n \\textbf { x } = \\bigg ( \\frac { m d } n \\bigg ) \\frac { \\mathbf { x } } { d } \\in \\bigg \\langle \\frac { \\mathbf { x } } { d } \\bigg \\rangle . \\end{align*}"}
-{"id": "6332.png", "formula": "\\begin{align*} \\omega = L ( \\eta ) \\end{align*}"}
-{"id": "6858.png", "formula": "\\begin{align*} I = \\bigcup _ { i = 0 } ^ { 3 } F _ { i } ( I ) . \\end{align*}"}
-{"id": "239.png", "formula": "\\begin{align*} & \\chi _ a \\{ h ( A _ G ) - h ( A _ { G ' } ) \\} \\chi _ b \\\\ & \\quad = \\frac { 1 } { 2 \\pi } \\int _ { \\R ^ 2 } \\mathrm { d } x \\mathrm { d } y \\ , \\omega _ { h , n } ( x , y ) \\chi _ a \\{ R _ { x + i y } ( A _ G ) - R _ { x + i y } ( A _ { G ' } ) \\} \\chi _ b \\\\ & = : \\frac { 1 } { 2 \\pi } \\int _ { \\R ^ 2 } \\mathrm { d } x \\mathrm { d } y \\ , \\omega _ { h , n } ( x , y ) T _ { x + i y } ^ { a , b } ( G , G ' ) , \\end{align*}"}
-{"id": "1148.png", "formula": "\\begin{align*} a _ 1 & = a _ 0 ^ 2 \\\\ a _ j & = \\frac { [ 1 ] _ x a _ { j - 1 } ^ 2 + a _ { j - 2 } ^ 4 } { [ j ] _ x } \\mbox { f o r $ j \\geq 2 $ . } \\end{align*}"}
-{"id": "7235.png", "formula": "\\begin{align*} ( \\mathcal { W } _ { \\psi } f ) ( 1 ) = \\int _ { \\hat { G } } ^ { } \\Phi _ { \\pi } ^ { \\psi } ( f ) d \\mu _ { \\pi } . \\end{align*}"}
-{"id": "2377.png", "formula": "\\begin{align*} \\Phi ( s , \\lambda , w ) = \\lambda ^ k \\Phi ( s , \\lambda , w + k ) + \\sum _ { n = 0 } ^ { k - 1 } \\frac { \\lambda ^ n } { ( n + w ) ^ s } . \\end{align*}"}
-{"id": "3538.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l l } & - d { p } ^ n ( t ) = \\{ b _ x ( \\bar { X } ^ n { ( t ) } , \\bar { u } ^ n ( t ) ) ^ { } p ^ n ( t ) - \\beta ^ { n , 0 } f _ x ( \\bar { X } ^ n { ( t ) } , \\bar { u } ^ n ( t ) ) - \\displaystyle \\frac { \\sum _ { j = i } ^ { n } \\beta ^ { n , j } } { n } b _ x ( \\bar { X } ^ n { ( t ) } , \\bar { u } ^ n ( t ) ) \\} d t , \\\\ & p ^ n ( t _ { i } ) = - \\beta ^ 0 \\Psi _ { x } ( \\bar { X } ^ n ( t _ n ) 1 _ { i = n } ( i ) + p ^ n ( t _ { i } ^ { + } ) , t \\in ( t _ { i - 1 } , t _ { i } ) , \\ i = 1 , 2 , \\ldots , n , \\end{array} \\end{align*}"}
-{"id": "2911.png", "formula": "\\begin{align*} k \\in Z ^ c \\Leftrightarrow f _ { y , c } ( \\lambda ) = f ( \\langle n _ \\sigma , k \\rangle ) = 1 , \\end{align*}"}
-{"id": "6651.png", "formula": "\\begin{align*} | \\mathrm { c o n v } [ K , x ] | _ n = \\frac { 1 } { 2 } \\left ( | K | _ n + \\frac { 1 } { n } \\int _ { \\partial K } | \\langle x - y , u ( y ) \\rangle | \\mathrm { d } \\mu ( y ) \\right ) , \\end{align*}"}
-{"id": "10011.png", "formula": "\\begin{align*} R _ \\Lambda ( m ) ( \\phi ) = \\sum _ { \\substack { x \\in \\Lambda ' \\\\ \\langle x , x \\rangle = m } } \\phi ( x ) \\end{align*}"}
-{"id": "9841.png", "formula": "\\begin{align*} \\hat { f } ( y ) = \\int _ { - \\infty } ^ { \\infty } f ( x ) e ^ { - i x y } d x \\end{align*}"}
-{"id": "3878.png", "formula": "\\begin{align*} Z _ C ( s ) = \\frac { \\zeta ( s ) } { \\pi ^ { 1 / 2 + s } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\sigma _ { 2 i r } ( l ^ 2 ) } { l ^ { 2 i r } } \\frac { \\zeta ( s + 2 i r ) \\zeta ( s - 2 i r ) } { \\left | \\zeta ( 1 + 2 i r \\right ) | ^ 2 } h ( \\omega ; s ; r ) d r , \\end{align*}"}
-{"id": "9238.png", "formula": "\\begin{align*} \\epsilon ( [ \\alpha _ { 1 } , \\alpha _ { 2 } ] , \\alpha _ { 3 } ) + \\epsilon ( [ \\alpha _ { 2 } , \\alpha _ { 3 } ] , \\alpha _ { 1 } ) + \\epsilon ( [ \\alpha _ { 3 } , \\alpha _ { 1 } ] , \\alpha _ { 2 } ) = 0 \\end{align*}"}
-{"id": "9878.png", "formula": "\\begin{align*} A = \\left [ { \\begin{array} { c c } - 1 & - 1 \\\\ 1 & 0 \\\\ \\end{array} } \\right ] . \\end{align*}"}
-{"id": "4731.png", "formula": "\\begin{align*} { L } _ 0 = \\frac { - ( \\beta ^ 2 c _ 0 + 2 \\beta \\gamma f _ 0 + \\gamma ^ 2 d _ 0 ) } { 2 \\ , ( \\alpha + \\beta a _ 0 + \\gamma b _ 0 ) } . \\end{align*}"}
-{"id": "5029.png", "formula": "\\begin{align*} G ^ \\omega _ { n , n } ( z ) = \\left [ P _ n A P _ n + \\omega _ n C _ n - z P _ n - P _ n A ( I - P _ n ) ( \\tilde { A } ^ \\omega - z ) ^ { - 1 } ( I - P _ n ) A P _ n \\right ] ^ { - 1 } , \\end{align*}"}
-{"id": "116.png", "formula": "\\begin{align*} r _ 2 = r _ 0 + \\sqrt { \\sum _ { i = 1 } ^ k ( 1 / \\rho _ i ) ^ 2 } . \\end{align*}"}
-{"id": "7102.png", "formula": "\\begin{align*} \\begin{aligned} k _ t = | \\nabla k | ^ 2 - k k _ { p p } | \\nabla p | ^ 2 + k \\Delta k . \\end{aligned} \\end{align*}"}
-{"id": "9856.png", "formula": "\\begin{align*} { \\displaystyle \\int \\hat { f } g = \\int f \\hat { g } } . \\end{align*}"}
-{"id": "1937.png", "formula": "\\begin{align*} u _ s = R _ k ( q e ^ { i k \\theta \\cdot ( \\cdot ) } ) + R _ k ( q u _ s ( k , \\theta , \\cdot ) ) . \\end{align*}"}
-{"id": "385.png", "formula": "\\begin{align*} S _ { \\Psi } : = \\sum _ { n = m } ^ \\infty \\frac { 1 } { n h ( n ) } \\P \\left ( | S _ n | > ( 1 + \\varepsilon ) \\sigma _ n \\sqrt { 2 \\ln \\Psi ( n ) } \\right ) . \\end{align*}"}
-{"id": "5750.png", "formula": "\\begin{align*} \\ & \\frac { \\psi _ { x , y - 1 , z } ( a , b ) } { \\psi _ { x , y - 1 , z - 1 } ( a , b ) } \\frac { \\psi _ { x , y , z - 1 } ( a , b ) } { \\psi _ { x , y , z } ( a , b ) } + \\frac { \\psi _ { x + 1 , y - 1 , z - 1 } ( a , b ) } { \\psi _ { x , y - 1 , z - 1 } ( a , b ) } \\frac { \\psi _ { x - 1 , y , z } ( a , b ) } { \\psi _ { x , y , z } ( a , b ) } = \\\\ & \\frac { ( y + z + a + b ) ( 2 x + y + z + 2 a + 2 b ) } { ( x + y + z + a + b ) ( x + y + z + 2 a + 2 b ) } + \\frac { x ( x + a + b ) } { ( x + y + z + a + b ) ( x + y + z + 2 a + 2 b ) } = 1 \\end{align*}"}
-{"id": "9838.png", "formula": "\\begin{align*} \\varphi _ 5 ( x , y ) = x ^ 5 + ( - 5 0 + 2 0 \\sqrt { 5 } ) x ^ 3 y ^ 2 + ( 2 2 5 - 1 0 0 \\sqrt { 5 } ) x y ^ 4 \\end{align*}"}
-{"id": "579.png", "formula": "\\begin{align*} { \\displaystyle \\int _ { \\gamma } g ( \\zeta ) d \\zeta } = 0 , \\end{align*}"}
-{"id": "9333.png", "formula": "\\begin{align*} J _ { n } ^ { ( 3 ) } = \\frac { 2 } { 7 } 2 ^ { n } - \\frac { 3 + 2 i \\sqrt { 3 } } { 2 1 } \\omega _ { 1 } ^ { n } - \\frac { 3 - 2 i \\sqrt { 3 } } { 2 1 } \\omega _ { 2 } ^ { n } \\end{align*}"}
-{"id": "3695.png", "formula": "\\begin{align*} W [ n ] \\backslash W [ n ] ^ { s s } = \\bigcup _ { \\# ( I ) \\neq j } D _ { I , j } . \\end{align*}"}
-{"id": "2517.png", "formula": "\\begin{align*} \\int ( \\nabla _ \\xi h , \\mathcal { L } \\nabla _ \\xi h ) \\varrho \\ , m _ { 0 } = - \\int \\left ( \\sigma \\nabla _ \\xi \\partial _ k h , \\nabla _ \\xi \\partial _ k h \\right ) \\varrho \\ , m _ { 0 } - \\int \\Psi ( x , \\xi ) \\left | \\nabla _ \\xi h \\right | ^ 2 \\varrho \\ , m _ { 0 } . \\end{align*}"}
-{"id": "5931.png", "formula": "\\begin{align*} \\forall i , j \\in \\{ 0 , \\ldots , d - 1 \\} & & \\rho _ i \\rho _ j \\neq \\rho _ j \\rho _ i ~ ~ \\Longleftrightarrow ~ ~ | i - j | = 1 \\end{align*}"}
-{"id": "7981.png", "formula": "\\begin{align*} | J _ s ^ N ( t ) | & = | Y _ 1 ^ { N } ( t ) | + | Y _ 2 ^ { N } ( 1 - t ) | \\\\ & \\fallingdotseq t | \\dot \\gamma _ 3 ( t ) ^ N | + ( 1 - t ) | \\dot \\gamma _ 2 ( t ) ^ N | \\\\ & = t \\sin \\beta _ s a _ 3 + ( 1 - t ) \\sin \\alpha _ s a _ 2 , \\end{align*}"}
-{"id": "1478.png", "formula": "\\begin{align*} f \\in \\mathcal { B } \\iff g = 1 / f \\in \\mathcal { A } . \\end{align*}"}
-{"id": "9171.png", "formula": "\\begin{align*} X _ \\omega : = - \\sqrt { - 1 } g ^ { i \\bar j } { \\frac { \\partial e ^ { h _ \\omega } } { \\partial \\bar z ^ j } } { \\frac { \\partial } { \\partial z ^ i } } \\end{align*}"}
-{"id": "1880.png", "formula": "\\begin{align*} f _ j ( x ) : = \\sum _ { i = 1 } ^ { m _ j } f _ j ^ i \\ 1 _ { F _ j ^ i } ( x ) , \\end{align*}"}
-{"id": "1946.png", "formula": "\\begin{align*} P _ { B } ( \\eta ) : & = \\int _ { B } \\frac { F _ r ( \\eta ) } { 1 - r } \\ , d r \\\\ & = \\sum _ { 0 \\le j < N ( \\eta ) } \\int _ { \\{ 2 ^ { - ( j + 1 ) } < | 1 - r | < 2 ^ { - j } \\} } \\chi _ { \\{ | 1 - r | < \\delta _ \\eta \\} } ( r ) \\frac { F _ r ( \\eta ) } { 1 - r } \\ , d r . \\end{align*}"}
-{"id": "7054.png", "formula": "\\begin{align*} z _ n = y _ n , \\end{align*}"}
-{"id": "1618.png", "formula": "\\begin{align*} C ^ * ( \\Lambda ) = \\overline { \\operatorname { s p a n } } \\{ t _ \\alpha t ^ * _ \\beta \\ , : \\ , \\alpha , \\beta \\in \\Lambda , \\ ; s ( \\alpha ) = s ( \\beta ) \\} . \\end{align*}"}
-{"id": "6437.png", "formula": "\\begin{align*} h _ { \\mathcal { M } } ^ { } = \\lim _ { \\tau \\rightarrow \\infty } \\left \\{ \\lim _ { \\Delta \\tau \\rightarrow 0 } \\left [ \\frac { \\mathcal { S } _ { \\mathcal { M } } \\left ( \\tau + \\Delta \\tau \\right ) - \\mathcal { S } _ { \\mathcal { M } } \\left ( \\tau \\right ) } { \\Delta \\tau } \\right ] \\right \\} . \\end{align*}"}
-{"id": "6883.png", "formula": "\\begin{align*} \\frac { d } { d x } \\left [ p ( x ) \\frac { d u } { d x } \\right ] + [ \\lambda \\rho ( x ) - q ( x ) ] u = 0 \\end{align*}"}
-{"id": "3151.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } _ { \\geqslant 0 } } x ^ { n } m _ { \\alpha , \\beta } ( \\mathrm { d } x ) & \\leqslant c _ { 1 } \\frac { e ^ { - \\alpha } } { \\beta ^ { n } } \\left ( \\alpha + \\alpha ^ { 2 } + \\cdots + \\alpha ^ { n - 1 } + \\alpha ^ { n } \\sum _ { m = 0 } ^ { \\infty } \\frac { \\alpha ^ { m } } { m ! } \\right ) \\\\ & \\leqslant c _ { 2 } \\left ( \\frac { 1 } { \\beta ^ { n } } + \\frac { \\alpha ^ { n } } { \\beta ^ { n } } \\right ) , \\quad \\alpha , \\beta > 0 , \\end{align*}"}
-{"id": "2842.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } n ^ { k - 1 } \\mid u _ n ^ { \\alpha } - u _ { n - 1 } ^ { \\alpha } \\mid ^ { k } < \\infty . \\end{align*}"}
-{"id": "9189.png", "formula": "\\begin{align*} R ( h _ 1 ) & = R ( h _ 2 ) + R ( h _ 1 - h _ 2 ) = R ( h _ 2 ) + ( D - C ) \\left [ \\frac { \\beta ( t ) } { 2 ( \\pi t ) ^ 2 } - \\frac { \\cos ( 2 \\pi T t ) } { 2 ( \\pi t ) ^ 2 } \\hat { \\varphi } ( t ) \\right ] \\\\ & = ( C - D ) \\left [ \\frac { \\cos ( 2 \\pi T t ) } { 2 ( \\pi t ) ^ 2 } \\hat { \\varphi } ( t ) \\right ] + B , \\end{align*}"}
-{"id": "674.png", "formula": "\\begin{align*} \\varphi ( 0 , \\mu ) = 1 , \\varphi ' ( 0 , \\mu ) = - \\cot \\alpha . \\end{align*}"}
-{"id": "6897.png", "formula": "\\begin{align*} \\int \\Psi _ m ^ { ( x ) } d \\mu = \\frac { 1 } { 3 } ( \\mu ( A ) + \\mu ( B ) ) = \\frac { 1 } { 3 } ( \\mu _ 0 ^ { i ( A ) } \\mu _ 1 ^ { m - i ( A ) } + \\mu _ 0 ^ { i ( B ) } \\mu _ 1 ^ { m - i ( B ) } ) \\end{align*}"}
-{"id": "2113.png", "formula": "\\begin{align*} \\phi ( p ) = \\gamma _ p ( 0 ) = \\gamma _ q ( \\psi ( q ) - \\psi ( p ) ) . \\end{align*}"}
-{"id": "1189.png", "formula": "\\begin{align*} & v _ { x _ { i } } = ( u ^ { 1 + \\epsilon } ) _ { x _ { i } } = ( 1 + \\epsilon ) u ^ { \\epsilon } u _ { x _ { i } } , \\\\ & v _ { x _ { i } x _ { j } } = ( 1 + \\epsilon ) \\epsilon u ^ { \\epsilon - 1 } u _ { x _ { i } } u _ { x _ j } + ( 1 + \\epsilon ) u ^ { \\epsilon } u _ { x _ { i } x _ { j } } . \\end{align*}"}
-{"id": "1502.png", "formula": "\\begin{align*} \\alpha ( x ) + \\omega _ { 1 } ( x ) + \\omega _ { 2 } ( x ) = x ^ { 2 } \\ \\textrm { a n d } \\ \\alpha ( x ) \\omega _ { 1 } ( x ) \\omega _ { 2 } ( x ) = 1 . \\end{align*}"}
-{"id": "8284.png", "formula": "\\begin{align*} \\bigoplus _ { p + q = k } V ^ { ( p , q ) } = \\{ v \\in V _ \\C : \\mathrm { w t } ( z ) \\cdot v = z ^ k \\cdot v , \\ , \\forall z \\in \\C ^ \\times \\} . \\end{align*}"}
-{"id": "3595.png", "formula": "\\begin{align*} f ( x ) - T _ { \\lceil \\alpha \\rceil - 1 } [ f ; a ] ( x ) = J _ a ^ \\alpha D _ { * a } ^ \\alpha f ( x ) . \\end{align*}"}
-{"id": "4025.png", "formula": "\\begin{align*} h ( d \\eta _ p X , d \\eta _ p Y ) = c ( p ) \\cdot h ( X , Y ) , \\end{align*}"}
-{"id": "8480.png", "formula": "\\begin{align*} W _ { \\pi } ( g _ { t , l , v } ) = q ^ { \\frac { \\delta } { 2 } } \\sum _ { \\pm } \\gamma _ F ( - 1 \\pm \\sqrt { \\Delta } , \\rho ) \\gamma _ F ( \\Delta \\pm \\sqrt { \\Delta } , \\abs { 2 \\{ \\frac { \\delta - \\rho } { 2 } \\} } ) \\chi ^ 2 ( - \\frac { 1 } { 2 v } \\pm \\frac { \\sqrt { \\Delta } } { 2 v } ) \\psi ( \\varpi ^ { - \\frac { n } { 2 } } ( \\frac { \\Delta - 3 } { 4 v } ) ) . \\end{align*}"}
-{"id": "4792.png", "formula": "\\begin{align*} [ b - C , \\mathcal R ^ k _ j ] _ 1 ( f , g ) = [ b , \\mathcal R ^ k _ j ] _ 1 ( f , g ) , \\\\ [ b - C , \\mathcal R ^ k _ j ] _ 2 ( f , g ) = [ b , \\mathcal R ^ k _ j ] _ 2 ( f , g ) , \\end{align*}"}
-{"id": "8970.png", "formula": "\\begin{gather*} { \\cal Z } _ w w { \\cal H } _ { \\langle w ^ { - 1 } r w \\rangle ; \\gamma } ( X ) W _ I = { \\cal Z } _ w w W _ I , \\end{gather*}"}
-{"id": "4702.png", "formula": "\\begin{align*} c _ \\psi : = \\int _ { \\R ^ d } \\frac { \\abs { \\hat { \\psi } ( \\omega ) } ^ 2 } { \\abs { \\omega _ 1 } ^ d } \\ , \\mathrm { d } \\omega < \\infty . \\end{align*}"}
-{"id": "249.png", "formula": "\\begin{align*} \\vec { K } ^ d _ { n , l } : = \\big \\{ \\vec { k } = ( k _ 1 , . . . , k _ { n - 1 } ) : \\ , k _ i \\in \\{ 1 , . . . , d \\} \\setminus \\{ l \\} , \\ , k _ i \\neq k _ j ( i \\neq j ) \\big \\} , \\end{align*}"}
-{"id": "9269.png", "formula": "\\begin{align*} \\mathcal { P } X = \\sum _ { \\nu \\in \\sigma ( \\mathcal { N } ) } \\mathcal { P } _ \\nu X , \\ , \\ , \\ , \\ , \\mathcal { P } _ \\nu X : = P _ { \\nu } X P _ { \\nu } . \\end{align*}"}
-{"id": "6385.png", "formula": "\\begin{align*} Y _ j & = \\max ( X _ { 1 j } , \\ldots , X _ { n j } ) . \\end{align*}"}
-{"id": "257.png", "formula": "\\begin{align*} \\eqref { p f : F i n a l P a r t 2 } & \\leq C _ 4 \\left ( \\sum _ { a _ n = L - 1 } ^ \\infty a _ n ^ { m - 1 - \\widetilde { q } / 2 } \\right ) ^ 2 \\leq C _ 5 L ^ { 2 m - \\widetilde { q } } . \\end{align*}"}
-{"id": "3496.png", "formula": "\\begin{align*} \\int _ { \\gamma _ i } \\omega _ j \\omega _ i & = \\int _ { \\phi ( \\gamma _ i ) } \\phi ( \\omega _ j ) \\phi ( \\omega _ i ) \\\\ & = ( - 1 ) ^ { \\sigma _ j } \\int _ { \\alpha \\gamma _ i \\alpha ^ { - 1 } } \\omega _ j \\omega _ i + ( - 1 ) ^ { \\sigma _ i + \\sigma _ j } a _ { i j i } \\end{align*}"}
-{"id": "5574.png", "formula": "\\begin{align*} \\Delta \\left ( \\alpha , \\beta \\right ) = S _ { 2 ^ { k - 1 } } + Z _ { 2 ^ { k - 1 } } \\end{align*}"}
-{"id": "8080.png", "formula": "\\begin{align*} x _ j ^ \\ell m = \\left ( \\prod _ { i = 1 } ^ { r } ( x _ j y _ i ) ^ { p _ i } \\right ) ( x _ j y _ { r + 1 } ) ^ { \\ell - q _ r } y _ { r + 1 } ^ { p _ { r + 1 } - ( l - q _ r ) } \\left ( \\prod _ { i = r + 2 } ^ k y _ i ^ { p _ i } \\right ) . \\end{align*}"}
-{"id": "1555.png", "formula": "\\begin{align*} \\psi ( z ) = z + \\sum _ { m = 0 } ^ { \\infty } { b _ { m } z ^ { - m } } \\end{align*}"}
-{"id": "798.png", "formula": "\\begin{align*} ( \\alpha _ 1 , \\alpha _ 2 , b _ 1 , b _ 2 ) = \\begin{cases} \\left ( \\frac { 2 x } { y ^ B } , y , k , r \\right ) & , \\\\ \\left ( \\frac { y ^ B } { 2 x } , y , k , | r | \\right ) & . \\end{cases} \\end{align*}"}
-{"id": "9923.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\Phi \\left ( S ^ 2 ( Z ) \\right ) & = & e _ 1 \\widetilde C Z + e _ 0 \\widetilde C W \\\\ \\Phi \\left ( S ^ 2 ( Z , W ) \\right ) & = & - e _ 2 \\widetilde C Z - e _ 1 \\widetilde C W \\\\ \\Phi \\left ( S ^ 2 ( W ) \\right ) & = & e _ 3 \\widetilde C Z + e _ 2 \\widetilde C W , \\\\ \\end{array} \\end{align*}"}
-{"id": "9047.png", "formula": "\\begin{align*} \\left ( \\begin{array} { l l l } \\lambda _ 1 & 0 & 0 \\\\ 0 & \\lambda _ 2 & 0 \\\\ 0 & 0 & \\lambda _ 3 \\end{array} \\right ) , \\quad \\lambda _ j > 0 , \\ , \\ , j = 1 , 2 , 3 , \\end{align*}"}
-{"id": "3023.png", "formula": "\\begin{align*} x ( t ) : = \\int _ 0 ^ \\infty e ^ { - s \\Lambda ^ { - 1 } } \\Lambda ^ { - 1 } y ( s + t ) \\ , d s , t \\ge 0 , \\end{align*}"}
-{"id": "6797.png", "formula": "\\begin{align*} e ^ { w _ { \\lambda } ( y ) } = O ( \\lambda ^ 2 ) . \\end{align*}"}
-{"id": "7244.png", "formula": "\\begin{align*} = \\int _ { \\hat { G } } ^ { } \\int _ { G } ^ { } f ( x ) ( \\int _ { U } ^ { } \\theta _ { \\pi } ( u x ) \\psi _ { n } ( u ) d u ) d x d \\mu _ { \\pi } . \\end{align*}"}
-{"id": "8849.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac 1 { N ^ 2 } \\sum _ { i , j = 1 } ^ N P _ n ^ { ( d ) } ( \\langle \\mathbf { x } _ i , \\mathbf { x } _ j \\rangle ) = 0 \\quad n \\geq 1 , \\end{align*}"}
-{"id": "8822.png", "formula": "\\begin{align*} Z ( p , \\chi , s ) = p ^ { - n } \\sum _ { { \\substack { I \\subset T , \\\\ \\forall i \\in I : d \\mid N _ { i } } } } c _ { I , \\chi } ^ { 0 } \\prod _ { i \\in I } \\frac { ( p - 1 ) t ^ { N _ { i } } p ^ { - \\nu _ { i } } } { 1 - t ^ { N _ { i } } p ^ { - \\nu _ { i } } } , \\end{align*}"}
-{"id": "2537.png", "formula": "\\begin{align*} & \\left \\| \\bar { x } ^ N \\int _ { | \\bar { \\eta } | < \\delta \\sqrt { t } } e ^ { i \\bar { \\eta } \\bar { x } - A _ j | \\bar { \\eta } | ^ 2 + O ( 1 ) | \\bar { \\eta } | ^ 3 / \\sqrt { t } } \\left ( \\vert e _ j ( 0 ) \\rangle \\langle e _ j ( 0 ) \\vert + \\varepsilon _ j ( \\bar { \\eta } / \\sqrt { t } ) \\right ) d \\bar { \\eta } \\right \\| _ { L ^ 2 _ \\xi } \\\\ & \\leq C _ N + C \\sum _ { k = 1 } ^ { N } \\bar { x } ^ { N - k } e ^ { - c \\delta ^ 2 t } . \\end{align*}"}
-{"id": "8256.png", "formula": "\\begin{align*} \\frac { L _ { \\nu } ( 2 s , \\chi _ 1 \\chi _ 2 ^ { - 1 } ) } { L _ { \\nu } ( 2 s + 1 , \\chi _ 1 \\chi _ 2 ^ { - 1 } ) } = \\frac { 4 \\pi } { 4 s + 4 i t _ { \\nu } } \\ll _ { \\sigma } ( 1 + \\abs { t + t _ { \\nu } } _ { \\nu } ) ^ { - \\frac { 1 } { 2 } } \\ll \\abs { T _ { s , \\nu } } _ { \\nu } ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "6492.png", "formula": "\\begin{align*} \\vec { B } _ { } = B _ { \\perp } \\hat { B } _ { \\perp } + B _ { \\parallel } \\hat { B } _ { \\parallel } , \\end{align*}"}
-{"id": "1237.png", "formula": "\\begin{align*} I & = n \\int _ { D _ 1 } f ( \\nabla v ) d x + \\sum _ { k , j = 1 } ^ n \\int _ { D _ 1 } x _ k f _ { \\eta _ j } ( \\nabla v ) v _ { x _ j x _ k } d x \\\\ & = n \\int _ { D _ 1 } f ( \\nabla v ) d x + I _ 1 . \\end{align*}"}
-{"id": "2937.png", "formula": "\\begin{align*} E ^ \\Delta : = \\{ e \\in E ( H ) : | e | \\leq \\Delta , \\deg _ H ( i ) \\leq \\Delta \\forall i \\in e \\} . \\end{align*}"}
-{"id": "941.png", "formula": "\\begin{align*} E \\left [ \\exp \\left \\{ \\frac { c _ q } { 2 } \\left ( \\frac { | \\Delta _ { i , j } | } { \\sigma _ { i , j } } \\right ) ^ { 2 / q } \\right \\} - 1 \\right ] & = \\int _ 0 ^ \\infty P \\left ( \\frac { | \\Delta _ { i , j } | } { \\sigma _ { i , j } } > \\{ \\log ( ( 1 + u ) ^ { 2 / c _ q } ) \\} ^ { q / 2 } \\right ) d u \\\\ & \\leq \\int _ 0 ^ \\infty ( 1 + u ) ^ { - 2 } d u = 1 . \\end{align*}"}
-{"id": "3327.png", "formula": "\\begin{align*} L ( s ) = \\binom { K - 2 } { s - 1 } + \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K - 1 } { s + i } ( N - 1 ) ^ i N \\end{align*}"}
-{"id": "5184.png", "formula": "\\begin{align*} \\rho _ { 0 } = | T ( \\ell _ { * } , \\ell _ { * } , r _ { * } ) | < 1 , \\end{align*}"}
-{"id": "5013.png", "formula": "\\begin{align*} k _ j ( x ) : = \\varphi _ j ( x + 2 ^ { - j } r ) . \\end{align*}"}
-{"id": "6391.png", "formula": "\\begin{align*} \\int d \\theta P \\left ( x \\theta \\right ) = P \\left ( x \\right ) = \\delta \\left ( x - x ^ { \\prime } \\right ) \\end{align*}"}
-{"id": "6901.png", "formula": "\\begin{align*} \\lambda _ { m } x _ { 0 } = \\frac { 3 } { 2 } ( ( 2 x _ { 0 } - x _ { 2 } - x _ { 1 } ) + ( 2 x _ { 0 } - x _ { 2 } ' - x _ { 1 } ' ) ) \\end{align*}"}
-{"id": "1358.png", "formula": "\\begin{align*} [ { \\cal P } _ { \\tau } \\theta ] ' ( X ; H ) = Q [ p _ { \\tau } ^ { [ 1 ] } ( \\Lambda ( X ) , Q ^ T H Q ) ] Q ^ T . \\end{align*}"}
-{"id": "9174.png", "formula": "\\begin{align*} \\Lambda _ { 1 , X } = \\{ u \\in C ^ \\infty ( M ) | \\triangle _ { \\omega _ 0 } u - \\frac { X } { 1 - \\theta _ X ( \\omega _ 0 ) } u = - u \\} . \\end{align*}"}
-{"id": "2243.png", "formula": "\\begin{align*} J _ \\gamma = ( - 1 ) ^ n \\sum _ { j = 1 } ^ s w _ { j 1 } ^ { \\gamma _ 1 + 1 } \\cdot w _ { j 2 } ^ { \\gamma _ 2 + 1 } \\cdots w _ { j n } ^ { \\gamma _ n + 1 } . \\end{align*}"}
-{"id": "9803.png", "formula": "\\begin{align*} 3 \\sum _ { i = 1 } ^ m \\deg _ { x _ i } ( u ) = \\deg ( u ) = \\deg ( v ) = 3 \\sum _ { i = 1 } ^ m \\deg _ { x _ i } ( v ) , \\end{align*}"}
-{"id": "8104.png", "formula": "\\begin{align*} g _ i = \\max _ { t \\in S _ i } \\alpha _ { t ^ { - 1 } } ( \\hat { g } _ { i , t } ) , \\end{align*}"}
-{"id": "9653.png", "formula": "\\begin{align*} N _ { \\beta , \\rm { s } } ( t ) = \\omega ^ 2 \\Lambda _ \\beta ^ 2 ( t ) \\left ( \\log d - \\log \\delta \\right ) / 8 , \\end{align*}"}
-{"id": "2530.png", "formula": "\\begin{align*} \\begin{aligned} \\left | T _ 2 \\right | & \\lesssim \\begin{cases} \\frac { 1 } { D } \\int h ^ 2 \\left < \\xi \\right > ^ { \\gamma + 1 } \\varrho \\ , m _ 0 , & \\gamma \\in [ - 1 , 1 ] , \\\\ \\alpha \\delta \\int h ^ 2 \\left < \\xi \\right > ^ { 2 ( 1 + \\gamma ) } \\varrho \\ , m _ 0 , & \\gamma \\in [ - 2 , - 1 ) \\end{cases} \\\\ & \\lesssim \\int h ^ 2 \\left < \\xi \\right > ^ { \\gamma + 1 } \\varrho \\ , m _ 0 . \\end{aligned} \\end{align*}"}
-{"id": "5043.png", "formula": "\\begin{align*} f ^ { + } ( \\tau ) & = q - 6 4 \\ , q ^ 2 - 1 8 3 6 \\ , q ^ 3 + 4 0 9 6 \\ , q ^ 4 + 3 9 9 0 \\ , q ^ 5 + 1 1 7 5 0 4 \\ , q ^ 6 + \\cdots \\\\ f ^ { - } ( \\tau ) & = q + 6 4 \\ , q ^ 2 + 1 2 3 6 \\ , q ^ 3 + 4 0 9 6 \\ , q ^ 4 - 5 7 4 5 0 \\ , q ^ 5 + 7 9 1 0 4 \\ , q ^ 6 + \\cdots \\ , , \\end{align*}"}
-{"id": "7349.png", "formula": "\\begin{align*} b _ i * b _ j * { \\textstyle \\frac { 1 } { 2 } } M _ { i j } \\ ; = \\ ; 1 i , j \\in \\{ 1 , 2 \\} \\ , , \\end{align*}"}
-{"id": "4273.png", "formula": "\\begin{align*} \\mathrm { p r ' } \\left ( f \\cdot \\Big ( x ( x - 1 ) ( x - \\lambda ) \\Big ) ^ { m } \\right ) = \\mathrm { p r ' } ( x ^ m , x ^ { m + 1 } , \\cdots , x ^ { 2 m } ) \\cdot ( B \\cdot \\beta ) \\end{align*}"}
-{"id": "6813.png", "formula": "\\begin{align*} - \\Delta \\tilde { V } = \\sum \\limits _ { j = 1 } ^ 4 \\frac { 8 a ^ 2 ( a ^ 2 | z _ { \\xi _ j } | ^ 2 - 1 ) } { ( 1 + a ^ 2 | z _ { \\xi _ j } | ^ 2 ) ^ 3 } \\cdot \\frac { ( 1 + | x _ { \\xi _ j } | ^ 2 ) ^ 2 } { 4 } . \\end{align*}"}
-{"id": "2193.png", "formula": "\\begin{align*} \\begin{cases} f _ 1 ( z ) = q _ 1 ( z ) + Q _ 1 ( z ) , \\\\ \\ldots \\\\ f _ n ( z ) = q _ n ( z ) + Q _ n ( z ) , \\end{cases} \\end{align*}"}
-{"id": "5960.png", "formula": "\\begin{align*} \\sigma ' ( t ) = a _ 1 ( t ) \\vect { e } ( t ) + a _ 2 ( t ) \\vect { e } _ 2 ( t ) + \\cdots + a _ n ( t ) \\vect { e } _ n ( t ) . \\end{align*}"}
-{"id": "6805.png", "formula": "\\begin{align*} L ( u ) = \\lambda ^ 2 \\mathcal { L } ( u ) . \\end{align*}"}
-{"id": "6614.png", "formula": "\\begin{align*} \\liminf _ { J ' \\ni n \\to \\infty } \\int _ { \\Omega _ 1 } \\left | D f _ 0 ^ { - 1 } - D g _ { n , 0 } ^ { - 1 } \\right | ^ { p ^ * } \\ , d \\mu = 0 \\ . \\end{align*}"}
-{"id": "4481.png", "formula": "\\begin{align*} B ( x , \\delta ) = \\{ y \\in X : \\rho ( x , y ) < \\delta \\} . \\end{align*}"}
-{"id": "8274.png", "formula": "\\begin{align*} F ( s , g ) = c ( s ) \\hat { F } ( - s , g ) . \\end{align*}"}
-{"id": "8943.png", "formula": "\\begin{gather*} \\sum _ { w \\in D _ n / D _ { n - 1 } } \\ ! \\ ! \\ ! w \\cdot \\frac { \\prod _ { 1 \\le i \\le 2 n - 3 } \\vartheta ( z _ n - y _ i ) \\prod _ { 1 \\le i \\le n } \\vartheta ( Y + z _ i ) \\prod _ { 1 \\le i < n } \\vartheta ( Y - z _ i ) } { \\prod _ { 1 \\le i < n } \\vartheta ( z _ n + z _ i ) \\vartheta ( z _ n - z _ i ) } \\\\ \\hphantom { \\sum _ { w \\in D _ n / D _ { n - 1 } } \\ ! \\ ! \\ ! } { } = \\vartheta ( 2 Y ) \\prod _ { 1 \\le i \\le 2 n - 3 } \\vartheta ( Y - y _ i ) , \\end{gather*}"}
-{"id": "7915.png", "formula": "\\begin{align*} 2 f ( z ) \\leq | x _ 1 + y _ 1 | + | x _ 2 + y _ 2 | \\leq | x _ 1 | + | y _ 1 | + | x _ 2 | + | y _ 2 | = f ( x ) + f ( y ) . \\end{align*}"}
-{"id": "4566.png", "formula": "\\begin{align*} & e _ j v = 0 ( 1 \\le j \\le n - 1 ) \\ , , e _ 0 ^ { ( l ) } v = 0 \\ , , Z _ r v = 0 ( r \\ge 1 ) , \\\\ & K _ i v = \\kappa _ i v ( 1 \\le i \\le n - 1 ) , C v = \\prod _ { i = 0 } ^ { n - 1 } \\kappa _ i \\cdot v \\ , , \\end{align*}"}
-{"id": "4205.png", "formula": "\\begin{align*} \\frac { 2 } { a } \\frac { d a } { d \\rho } = b ^ 2 + c ^ 2 - a ^ 2 - 2 b c , \\end{align*}"}
-{"id": "10010.png", "formula": "\\begin{align*} \\tilde { g } ( \\tau ) = \\sum _ { m > 0 } \\tilde { c } ( m ) q ^ m \\end{align*}"}
-{"id": "9355.png", "formula": "\\begin{align*} j O _ { n + 3 } ^ { ( 3 ) } - 3 J O _ { n + 3 } ^ { ( 3 ) } = \\sum _ { s = 0 } ^ { 7 } ( j _ { n + s + 3 } ^ { ( 3 ) } - 3 J _ { n + s + 3 } ^ { ( 3 ) } ) e _ { s } \\end{align*}"}
-{"id": "2293.png", "formula": "\\begin{align*} u = A ^ s u = 0 \\mbox { o n } \\partial \\Omega . \\end{align*}"}
-{"id": "2299.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\| u ^ N \\| _ { L ^ 2 } ^ 2 + \\nu \\| A ^ { s / 2 } u ^ N \\| _ { L ^ 2 } ^ 2 = - \\langle u ^ N , \\mathcal { U } ^ \\alpha ( u ^ N , u ^ N ) \\rangle . \\end{align*}"}
-{"id": "2368.png", "formula": "\\begin{align*} \\lambda \\mu \\big ( \\lambda ^ { - 1 } - \\lambda + \\mu - \\mu ^ { - 1 } \\big ) = \\frac 1 2 \\big ( i + \\sqrt { 3 } \\big ) \\not = 0 , \\end{align*}"}
-{"id": "3609.png", "formula": "\\begin{align*} \\int _ \\Omega | \\nabla u | ^ { p - 2 } ( \\nabla u , \\nabla \\varphi ) \\ , d x \\ , = \\ , \\int _ \\Omega f ( u ) \\varphi \\ , d x \\qquad \\forall \\varphi \\in C ^ 1 _ c ( \\Omega \\setminus \\Gamma ) \\ , . \\end{align*}"}
-{"id": "6862.png", "formula": "\\begin{align*} \\mu _ { k } ^ { ( m ) } = \\frac { \\mu ( I _ { k } ^ { ( m ) } ) + \\mu ( I _ { k - 1 } ^ { ( m ) } ) } { 2 } . \\end{align*}"}
-{"id": "9457.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ n } { ( q ^ { n } ; q ) _ { n + 1 } ( q ^ { 2 n + 2 } ; q ^ 2 ) _ { \\infty } } & = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { 2 n ^ 2 + 2 n + 1 } } { ( q ; q ^ 2 ) _ { n + 1 } } , \\\\ \\sum _ { n = 0 } ^ { \\infty } q ^ n ( - q ^ { n + 1 } ; q ) _ n ( - q ^ { 2 n + 2 } ; q ^ 2 ) _ { \\infty } & = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ 2 + n } } { ( q ; q ^ 2 ) _ { n + 1 } } . \\end{align*}"}
-{"id": "7157.png", "formula": "\\begin{align*} & ~ ~ Z ( a ^ * ) \\beta K - Z ( n ) \\beta K + 2 c _ { k , s _ n } \\\\ & = Z ( a ^ * ) \\beta K - Z ( n ) \\beta K + 2 \\sqrt { 2 \\ln k / s _ n } \\\\ & \\leq Z ( a ^ * ) \\beta K - Z ( n ) \\beta K + \\delta _ n = 0 . \\end{align*}"}
-{"id": "46.png", "formula": "\\begin{align*} G _ { \\sigma } ( x ) = \\frac { 1 } { \\sqrt { 2 \\pi } \\sigma } e x p \\left ( - \\frac { x ^ 2 } { 2 \\sigma ^ 2 } \\right ) \\end{align*}"}
-{"id": "9643.png", "formula": "\\begin{align*} \\bar { \\Phi } _ { \\rm { d n } } | ^ { \\tau _ { i } } _ { \\tau _ { i - 1 } } = \\frac { \\int _ { \\tau _ { i - 1 } } ^ { \\tau _ { i } } \\sum _ { \\beta = 1 } ^ { \\kappa } \\Phi _ { \\rm { s t } , \\beta } ( t ) d t + \\Omega _ { \\rm { m } } ( \\tau _ { i } ) } { \\tau _ { i } - \\tau _ { i - 1 } } , ~ i = 1 , \\cdots , N _ \\tau . \\end{align*}"}
-{"id": "4215.png", "formula": "\\begin{align*} Z _ g ( A _ 1 \\otimes \\cdots \\otimes A _ K ) ( i , j ) : = \\sum _ { \\substack { \\kappa : V \\to [ N ] \\\\ \\kappa ( v _ { \\mathrm { i n } } ) = j , \\kappa ( v _ { \\mathrm { o u t } } ) = i } } \\prod _ { r = 1 } ^ K A _ r ( \\kappa ( w _ r ) , \\kappa ( v _ r ) ) , \\end{align*}"}
-{"id": "1299.png", "formula": "\\begin{align*} \\textstyle u _ k \\ = \\ \\frac { a } { c } - \\frac { 1 } { 3 c } \\left ( \\left ( \\frac { 3 + B } { 2 } \\right ) \\ + \\ \\sqrt { 2 7 + B ^ 2 } \\ \\cos \\left ( \\frac { k \\pi } { 3 } + \\frac { 1 } { 3 } \\arctan \\frac { 3 \\sqrt { 3 } } { B } \\right ) \\right ) \\end{align*}"}
-{"id": "4854.png", "formula": "\\begin{align*} s = & \\ ; 6 \\cdot ( 2 \\cdot 5 + 5 \\cdot 0 - 5 ) - 1 7 - 1 3 \\\\ = & \\ ; 0 . \\end{align*}"}
-{"id": "3451.png", "formula": "\\begin{align*} \\begin{cases} \\dot { u } ( t ) = \\lambda ( u ( t ) - g ( t ) ) + \\dot { g } ( t ) , t \\in ( 0 , T ] , \\\\ u ( 0 ) = g ( 0 ) , \\end{cases} \\end{align*}"}
-{"id": "334.png", "formula": "\\begin{align*} \\gamma ( n ) : = \\sum _ { i \\in I } \\alpha _ i ( n ) - \\sum _ { j \\in J } \\beta _ j ( n ) + \\frac { | J | - | I | } { 2 } . \\end{align*}"}
-{"id": "1484.png", "formula": "\\begin{align*} \\alpha = \\cfrac { 2 \\beta } { \\tan \\beta } - 1 . \\end{align*}"}
-{"id": "244.png", "formula": "\\begin{align*} U _ { \\sigma } \\chi _ { \\Lambda _ { L / 2 } ( \\sigma ) } \\{ h ( g ( H ) _ { \\Lambda _ L } ) - f _ 0 \\} U _ { \\sigma } & = \\chi _ { [ - L , 0 ] ^ d } U _ { \\sigma } \\{ h ( g ( H ) _ { \\Lambda _ L } ) - f _ 0 \\} U _ { \\sigma } \\\\ & = \\chi _ { [ - L , 0 ] ^ d } \\{ h ( g ( H ^ { R _ { \\sigma } } ) _ { \\Lambda _ L } ) - f ^ { R _ { \\sigma } } _ 0 \\} . \\end{align*}"}
-{"id": "3164.png", "formula": "\\begin{align*} \\mu _ { X _ { t } ^ { x } } = \\mu _ { Y _ { t } ^ { y } } \\ast \\mu _ { Z _ { t } } , \\end{align*}"}
-{"id": "2697.png", "formula": "\\begin{align*} p ^ { n } { x } \\beta + p ^ { 2 n } z & = y \\beta ^ 2 + p ^ { n } { w } \\beta \\\\ & = - a _ 2 ( a ' _ 1 ) ^ { - 1 } y \\beta - p ^ { 2 n } a ' _ 3 ( a ' _ 1 ) ^ { - 1 } y + p ^ { n } { w } \\beta . \\end{align*}"}
-{"id": "9455.png", "formula": "\\begin{align*} \\omega ( z ; q ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ n q ^ { n } } { ( q ; q ^ 2 ) _ { n + 1 } } , \\nu ( z ; q ) = \\sum _ { n = 0 } ^ { \\infty } ( q / z ; q ^ 2 ) _ n ( - z q ) ^ n . \\end{align*}"}
-{"id": "7502.png", "formula": "\\begin{align*} \\| W _ N ^ k \\| _ { H _ 2 ^ j } & \\leq C \\| W _ N \\| _ { H _ 2 ^ 2 } \\ ; \\ ; j \\leq 2 , \\ ; \\ ; \\\\ \\| W _ N ^ k \\| _ { H _ 2 ^ j } & \\leq C k ^ { \\frac { j - 2 } { 2 } } \\| W _ N \\| _ { H _ 2 ^ 2 } \\ ; \\ ; j = 3 , \\dots , 6 . \\end{align*}"}
-{"id": "3450.png", "formula": "\\begin{align*} C _ 2 = 2 \\big ( 1 + \\| u \\| _ { C ( [ 0 , T ] ; \\R ^ d ) } \\big ) ^ 2 \\big \\| L _ { K _ u } + g \\big \\| _ { L ^ 2 ( 0 , T ; \\R ) } ^ 2 . \\end{align*}"}
-{"id": "4552.png", "formula": "\\begin{align*} & , q ^ { h } q ^ { h ' } = q ^ { h + h ' } , q ^ 0 = 1 \\ , , \\\\ & q ^ { h } E _ i ( z ) q ^ { - h } = q ^ { ( h , \\alpha _ i ) } E _ i ( z ) \\ , , q ^ { h } F _ i ( z ) q ^ { - h } = q ^ { - ( h , \\alpha _ i ) } F _ i ( z ) \\ , , q ^ h K ^ \\pm _ i ( z ) = K ^ \\pm _ i ( z ) q ^ h \\ , . \\end{align*}"}
-{"id": "809.png", "formula": "\\begin{align*} Q _ { k } ( x ) = x ^ { k } + ( x + 1 ) ^ k + ( x + 2 ) ^ k + . . . + 2 ^ { k } ( x - 1 ) ^ k \\end{align*}"}
-{"id": "1651.png", "formula": "\\begin{align*} R _ g \\subseteq D _ f \\quad . \\end{align*}"}
-{"id": "5863.png", "formula": "\\begin{align*} A _ k ( n , d , w ) = \\max \\left \\{ \\sum _ { \\omega \\in \\Omega _ k ^ d } b _ { \\omega } y _ { \\omega } \\ , \\ , \\big { | } \\ , \\ , M = F _ { \\emptyset } - \\sum _ { \\omega \\in \\Omega _ k ^ d } F _ { \\omega } y _ { \\omega } \\succeq 0 \\right \\} . \\end{align*}"}
-{"id": "6350.png", "formula": "\\begin{align*} \\Sigma _ k = ( D \\varphi _ k ^ { E } ( \\vec { y } ^ * ) ) ^ t \\operatorname { d i a g } ( W _ 1 '' ( y _ 1 ^ * ) , W _ 2 '' ( y _ 2 ^ * ) , W _ 1 '' ( y _ 1 ^ * ) , \\ldots ) ^ { - 1 } ( D \\varphi _ k ^ { E } ( \\vec { y } ^ * ) ) . \\end{align*}"}
-{"id": "6953.png", "formula": "\\begin{align*} \\Gamma _ n = \\{ \\lambda \\in \\mathbb { R } ^ n , \\lambda _ i > 0 , i = 1 , 2 , . . . , n \\} . \\end{align*}"}
-{"id": "5307.png", "formula": "\\begin{align*} Z _ M ( t ) : = \\left \\{ \\begin{array} { l l } B ( \\theta ^ { 0 } ) & \\mbox { f o r } t = 0 \\\\ Z ^ { m } & \\mbox { f o r } t \\in ] t _ { m - 1 , M } , t _ { m , M } ] \\end{array} \\right . \\mbox { i n } \\Omega , \\end{align*}"}
-{"id": "7529.png", "formula": "\\begin{gather*} w ^ 1 _ 1 + w ^ 1 w ^ 1 _ 2 - w ^ 1 _ { 2 2 } = 0 , \\\\ w ^ 2 _ 1 + w ^ 1 w ^ 2 _ 2 - w ^ 2 _ { 2 2 } = 0 . \\end{gather*}"}
-{"id": "4345.png", "formula": "\\begin{align*} \\Omega _ k ( B _ j ) : = \\{ z \\in \\mathbb { C } ^ n ; \\operatorname { d i s t } _ \\infty ( z , B _ j ) \\leq k \\} \\end{align*}"}
-{"id": "7117.png", "formula": "\\begin{align*} Q _ 0 v ^ i v ^ j \\geq & ~ ( \\alpha + 1 ) C F ^ { - \\alpha - 1 } \\left ( \\sum _ k \\dot { f } ^ k \\kappa _ k ^ 2 - F \\kappa _ 1 \\right ) \\\\ = & ~ ( \\alpha + 1 ) C F ^ { - \\alpha - 1 } \\sum _ k \\dot { f } ^ k \\kappa _ k \\left ( \\kappa _ k - \\kappa _ 1 \\right ) ~ \\geq ~ 0 \\end{align*}"}
-{"id": "8955.png", "formula": "\\begin{gather*} h ^ { - 1 } ( f _ 0 + ( s + 1 ) f _ 1 { } ^ s h ) h = f _ 0 + h ^ { - 1 } ( s + 1 ) f _ 1 ( { } ^ s h h ) = f _ 0 + { } ^ s h ( s + 1 ) f _ 1 , \\end{gather*}"}
-{"id": "1104.png", "formula": "\\begin{align*} \\sigma _ { Z _ { s , j } ^ { ( i ) } } ^ 2 & = 1 + 2 m _ { c , j } ^ { ( i ) } , \\\\ \\sigma _ { Z _ { 0 , j } ^ { ( i ) } } ^ 2 & = m _ { c , j } ^ { ( i ) } + \\left ( m _ { c , j } ^ { ( i ) } \\right ) ^ 2 + \\sigma _ { t h } ^ 2 , \\end{align*}"}
-{"id": "9247.png", "formula": "\\begin{align*} [ x _ { 1 } ^ { + } \\otimes a _ { 1 } ^ { - } , x _ { 2 } ^ { + } \\otimes a _ { 2 } ^ { - } ] = [ \\left [ \\begin{array} { c c } x _ { 1 } ^ { + } \\otimes a _ { 1 } ^ { - } & 0 \\\\ 0 & 0 \\end{array} \\right ] , \\left [ \\begin{array} { c c } x _ { 2 } ^ { + } \\otimes a _ { 2 } ^ { - } & 0 \\\\ 0 & 0 \\end{array} \\right ] ] = \\left [ \\begin{array} { c c } x _ { 1 } ^ { + } x _ { 2 } ^ { + } \\otimes a _ { 1 } ^ { - } a _ { 2 } ^ { - } - x _ { 2 } ^ { + } x _ { 1 } ^ { + } \\otimes a _ { 2 } ^ { - } a _ { 1 } ^ { - } & 0 \\\\ 0 & 0 \\end{array} \\right ] \\end{align*}"}
-{"id": "8128.png", "formula": "\\begin{align*} S _ { j , 1 } & = \\{ s \\in S _ j : s U _ j \\cap A \\neq \\emptyset \\} , \\\\ S _ { j , 2 } & = \\{ s \\in S _ j : s U _ j \\cap B _ - \\neq \\emptyset \\} \\end{align*}"}
-{"id": "1567.png", "formula": "\\begin{align*} c ( x , y ) = s ( x ) + s ( y ) - s ( x + y ) . \\end{align*}"}
-{"id": "6686.png", "formula": "\\begin{align*} | H _ p ^ n ( \\Delta ) | _ n = \\frac { p - 1 } { n ( n - 1 + p ) } | B _ p ^ { n - 1 } | _ { n - 1 } \\left ( \\frac { p } { p - 1 } \\Delta \\right ) ^ { \\frac { n - 1 + p } { p } } ( 1 + \\psi _ p ^ n ( \\Delta ) ) \\quad , \\end{align*}"}
-{"id": "733.png", "formula": "\\begin{align*} m ( a _ 1 , a _ 2 ) : = \\inf _ { ( u _ 1 , u _ 2 ) \\in S ( a _ 1 , a _ 2 ) } J ( u _ 1 , u _ 2 ) . \\end{align*}"}
-{"id": "4922.png", "formula": "\\begin{align*} 1 = \\pi ( I ) \\ne \\pi ( A ) \\pi ( A ) ^ * . \\end{align*}"}
-{"id": "8668.png", "formula": "\\begin{align*} \\int _ { \\Omega } e ^ { I ( x , y ) } \\Psi ( y ) \\pi ( d y ) = \\rho \\Psi ( x ) . \\end{align*}"}
-{"id": "1748.png", "formula": "\\begin{align*} \\sum _ { \\nu \\in v \\Lambda ^ n } S _ \\nu S _ \\nu ^ * ( f \\sqrt { d \\mu } ) & = \\sum _ { \\nu \\in v \\Lambda ^ n } S _ \\nu ( f \\circ \\sigma _ \\nu ) \\sqrt { d ( \\mu \\circ \\sigma _ \\nu ) } \\\\ & = \\sum _ { \\nu \\in v \\Lambda ^ n } f \\circ \\sigma _ \\nu \\circ \\sigma ^ n \\sqrt { d ( \\mu \\circ \\sigma _ \\nu \\circ \\sigma _ \\nu ^ { - 1 } ) } \\\\ & = \\sum _ { \\nu \\in v \\Lambda ^ n } \\chi _ { Z ( \\nu ) } \\cdot f \\sqrt { d \\mu } , \\end{align*}"}
-{"id": "8540.png", "formula": "\\begin{align*} \\hat f _ \\phi ( t , m , v ) = [ \\hat f _ \\phi ( t , m , v ) - ( \\hat f _ \\phi ( t , m , \\cdot ) \\ast _ v \\chi _ { \\eta } ) ( v ) ] + ( \\hat f _ \\phi ( t , m , \\cdot ) \\ast _ v \\chi _ { \\eta } ) ( v ) . \\end{align*}"}
-{"id": "880.png", "formula": "\\begin{align*} R _ { n , 1 } & = \\sum _ { i = 1 } ^ { N _ n } E \\left [ \\max _ { 1 \\leq k \\leq d _ n } \\left | \\sum _ { j = 1 } ^ { N _ n } \\gamma _ { n , k } ( i , j ) W ^ { ( i ) } _ j \\right | ^ 3 \\right ] ( E [ | Y _ i | ^ { 3 } ] + E [ | G _ i | ^ 3 ] ) , \\\\ R _ { n , 2 } & = \\max _ { 1 \\leq k , l \\leq d _ n } \\left | \\mathfrak { C } _ n ( k , l ) - E [ Q _ { n , k } ( G ) Q _ { n , l } ( G ) ] \\right | , \\\\ R _ { n , 3 } & = \\max _ { 1 \\leq k \\leq d _ n } \\sqrt { E [ Q _ { n , k } ( G ) ^ 4 ] - 3 E [ Q _ { n , k } ( G ) ^ 2 ] ^ 2 } . \\end{align*}"}
-{"id": "3688.png", "formula": "\\begin{align*} \\frac { \\xi _ i } { \\xi _ { s ( i ) } } = 1 \\mbox { f o r a l l $ i $ w i t h $ s ( i ) \\neq i $ } \\end{align*}"}
-{"id": "3669.png", "formula": "\\begin{align*} l ^ { W [ n ] } _ { G H H } = \\alpha _ W [ n ] + b ^ { W [ n ] } _ { G H H } : M _ W [ n ] _ { \\mathbb Q } \\to M _ G [ n ] _ { \\mathbb Q } \\end{align*}"}
-{"id": "9184.png", "formula": "\\begin{align*} C _ { } = \\left \\{ \\begin{array} { r c l } 1 , & & \\mbox { i f } ~ ~ M = K - 1 , \\\\ 1 - \\frac { T } { N } , & & \\mbox { i f } ~ ~ M < K - 1 ~ ~ \\mbox { a n d } ~ ~ \\rho \\geq \\frac { T } { N - T } , \\\\ 0 , & & \\mbox { o t h e r w i s e } , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "5510.png", "formula": "\\begin{align*} W \\stackrel { \\mathcal { D } } { = } \\sum _ { i \\geq 1 } T _ i ^ \\alpha W _ i , \\end{align*}"}
-{"id": "5907.png", "formula": "\\begin{align*} P ( \\frac { S _ n } n \\ge r , \\ n = 1 , 2 , \\cdots ) > 0 . \\end{align*}"}
-{"id": "6372.png", "formula": "\\begin{align*} H _ { \\mu } ( t ) : = \\int \\frac { d \\mu ( s ) } { t - s } \\end{align*}"}
-{"id": "9386.png", "formula": "\\begin{align*} \\| \\mathfrak { a } \\| _ { L ^ \\infty ( \\mathbb { R } ) } : = \\sup _ { t \\in \\mathbb R } | a _ t | \\le \\| \\mathfrak { a } \\| _ { V _ \\rho } \\quad \\ \\ \\rho \\ge 1 . \\end{align*}"}
-{"id": "3798.png", "formula": "\\begin{align*} \\chi ^ g _ \\sigma ( \\omega , U ) : = \\frac { 1 } { L } \\sum _ { i = n } ^ { n + L - 1 } g ( \\theta _ { ( \\sigma ( i ) , i ) } ( \\omega , U ) ) , \\end{align*}"}
-{"id": "3491.png", "formula": "\\begin{align*} \\int _ { \\alpha ^ { - 1 } } \\omega _ 1 \\cdots \\omega _ k = ( - 1 ) ^ k \\int _ \\alpha \\omega _ k \\cdots \\omega _ 1 . \\end{align*}"}
-{"id": "6125.png", "formula": "\\begin{align*} L _ { t } \\left ( f \\right ) \\left ( x \\right ) = \\frac { 1 } { 2 } \\nabla _ { x } f ^ { T } \\boldsymbol { \\sigma } _ { \\mu _ { \\left ( \\xi ^ { - 1 } \\right ) ^ { t } } } ^ { 2 } \\nabla _ { x } f , \\end{align*}"}
-{"id": "6962.png", "formula": "\\begin{align*} \\delta ( u , x , y ) = u ( x + y ) + u ( x - y ) - 2 u ( x ) , \\end{align*}"}
-{"id": "8119.png", "formula": "\\begin{align*} \\dim ( A _ 2 ) \\leq \\dim ( F A _ 1 ' ) = \\dim ( A _ 1 ' ) < \\dim ( A _ 1 ) . \\end{align*}"}
-{"id": "7893.png", "formula": "\\begin{align*} d ( 0 , m ( x , 1 ) ) ^ 2 - d ( 0 , m ( 0 , 1 ) ) ^ 2 = ( 1 + x ) ^ 2 / 4 - 1 / 4 \\ge 0 \\end{align*}"}
-{"id": "7500.png", "formula": "\\begin{align*} i \\varepsilon \\ , \\partial _ t \\omega _ { N , t } = \\left [ \\sqrt { 1 - \\varepsilon ^ 2 \\Delta } + \\left ( V * \\rho _ t \\right ) , \\omega _ { N , t } \\right ] \\end{align*}"}
-{"id": "6661.png", "formula": "\\begin{align*} S _ 2 = S \\cap \\{ \\xi \\in \\mathbb { R } ^ n : \\langle \\xi - x , u ( x ) \\rangle \\geq - ( 1 + \\varepsilon ) \\Delta \\} \\quad . \\end{align*}"}
-{"id": "6696.png", "formula": "\\begin{align*} a _ { \\delta } \\geq & \\frac { 1 - c _ 1 \\delta ^ { \\frac { p } { n - 1 + p } } ( 1 + s _ 1 ( \\delta ) ) } { 1 - c _ 3 \\delta ^ { \\alpha \\frac { p ' } { n - 1 + p ' } } ( 1 + s _ 3 ( \\delta ' ) ) } = 1 + c _ 3 \\delta ^ { \\alpha \\frac { p ' } { n - 1 + p ' } } ( 1 + o _ { \\delta } ( 1 ) + o _ { \\delta ' } ( 1 ) ) \\\\ \\geq & 1 + c _ 2 \\delta ^ { \\frac { 2 } { n + 1 } } ( 1 + o _ { \\delta } ( 1 ) + o _ { \\delta ' } ( 1 ) ) \\quad . \\end{align*}"}
-{"id": "7379.png", "formula": "\\begin{align*} \\Z _ { S e l b } ^ d = \\det \\begin{bmatrix} \\frac { 1 } { | d | } \\sum _ { z \\in d } z ^ { - j + l } f _ { S e l b } ( z ) \\end{bmatrix} _ { j , l = 0 } ^ { N _ { f } - 1 } = D ^ { d } _ { N _ f } ( f _ { S e l b } , | d | ) . \\end{align*}"}
-{"id": "854.png", "formula": "\\begin{align*} d X _ s = f ( s , X _ s , u _ s ) d E _ s + \\sqrt { 2 \\nu } d B _ { E _ s } s \\in ( t , T ] , \\end{align*}"}
-{"id": "8788.png", "formula": "\\begin{align*} \\textnormal { C o n t } _ { 0 } ^ { \\geq m } ( f ) : = \\{ x \\in ( t K [ [ t ] ] ) ^ { n } \\mid f ( x ) \\equiv 0 \\textnormal { m o d } t ^ { m } \\} . \\end{align*}"}
-{"id": "6650.png", "formula": "\\begin{align*} K ^ { \\delta } = \\{ x \\in \\mathbb { R } ^ n : | \\mathrm { c o n v } [ K , x ] | _ n \\leq ( 1 + \\delta ) | K | _ n \\} \\quad . \\end{align*}"}
-{"id": "6589.png", "formula": "\\begin{align*} \\phi _ n = \\varphi _ n \\circ \\varphi _ 0 ^ { - 1 } \\circ \\phi \\circ \\varphi _ 0 \\circ \\varphi _ n ^ { - 1 } \\ . \\end{align*}"}
-{"id": "9060.png", "formula": "\\begin{align*} a _ { 1 1 } ^ 1 = 0 , \\ , \\ , a _ { 2 2 } ^ 2 = 0 , \\ , \\ , b _ { 2 1 } ^ 2 = 0 , \\ , \\ , b _ { 2 1 } ^ 3 = 0 , \\end{align*}"}
-{"id": "2907.png", "formula": "\\begin{align*} f ( \\langle n _ \\sigma , k \\rangle ) = 1 \\Leftrightarrow y \\in T J ( H _ c ^ X ) \\Leftrightarrow k \\in H _ b ^ X \\Leftrightarrow \\sigma ( k ) = H _ b ^ X ( k ) . \\end{align*}"}
-{"id": "246.png", "formula": "\\begin{align*} [ 0 , L ] ^ d = \\bigcup _ { \\pi \\in \\mathcal { S } ^ d } \\{ x \\in [ 0 , L ] ^ d : \\ , x _ { \\pi ( 1 ) } \\leq . . . \\leq x _ { \\pi ( d ) } \\} , \\end{align*}"}
-{"id": "9172.png", "formula": "\\begin{align*} \\mathcal E ^ D ( \\omega ) = \\int _ M ( e ^ { h _ \\omega } - 1 ) ^ 2 \\omega ^ n . \\end{align*}"}
-{"id": "825.png", "formula": "\\begin{align*} \\left \\langle \\mathcal { L } u , u \\right \\rangle = 0 , \\quad \\mbox { f o r } u ( x , 0 ) = u _ 0 ( x ) , \\end{align*}"}
-{"id": "4149.png", "formula": "\\begin{align*} \\sum _ { \\tau = t + h } ^ { n } Z _ { N , J } ^ { \\tau } & + \\sum _ { \\tau = t } ^ { t + h - 1 } n ^ { t + h - \\tau } Z _ { i _ { \\tau } , J } ^ { \\tau } \\leq 1 + \\sum _ { \\tau = 1 } ^ { h - 1 } n ^ { \\tau } , \\\\ & \\forall \\ J \\subseteq V , \\ \\lvert J \\rvert = h \\in [ n - 1 ] , \\ t \\in [ n - h ] , \\ i _ t , \\dotsc , i _ { t + h - 1 } \\in N . \\end{align*}"}
-{"id": "5632.png", "formula": "\\begin{align*} Y _ { \\infty } ^ { { \\rm a b s } } = \\sum _ { j = 1 } ^ { \\infty } | x _ j \\xi _ j | . \\end{align*}"}
-{"id": "1165.png", "formula": "\\begin{align*} S _ { n , r } ( x _ 1 , x _ 2 , \\cdots , x _ n ) = \\sum _ { n \\geq i _ 1 > i _ 2 > \\cdots > i _ r \\geq 1 } \\prod _ { j = 1 } ^ r x _ { i _ j } ^ { 2 ^ { n - j + 1 - i _ j } } . \\end{align*}"}
-{"id": "3345.png", "formula": "\\begin{align*} \\frac { \\binom { K - 2 } { s - 1 } } { \\binom { K - 2 } { s - 1 } + \\sum _ { i = 0 } ^ { K - 1 - s } \\binom { K - 1 } { s + i } ( N - 1 ) ^ i N } & \\geq \\frac { \\binom { K - 2 } { s - 1 } + \\binom { K - 2 } { s - 2 } } { \\binom { K - 2 } { s - 1 } + \\binom { K - 2 } { s - 2 } + \\sum _ { i = 0 } ^ { K - s } \\left [ \\binom { K - 1 } { s + i } + \\binom { K - 1 } { s + i - 1 } \\right ] ( N - 1 ) ^ i N } , \\end{align*}"}
-{"id": "9487.png", "formula": "\\begin{align*} p _ { n , d } = p _ { n - 1 , d + r - 2 } + ( d - 1 ) \\cdot s _ { n - 1 , d - 2 } + 2 \\sum \\limits _ { i = 0 } ^ { n - 1 } \\sum \\limits _ { j = 0 } ^ { d - 2 } s _ { n - i - 1 , d - j - 2 } \\cdot p _ { i , j } , \\end{align*}"}
-{"id": "9784.png", "formula": "\\begin{align*} & ( s ^ { - 1 } d s - g ^ { - 1 } d g ) \\omega _ 1 + ( s - g ) \\omega _ 2 \\\\ & = ( s ^ { - 1 } d s - g ^ { - 1 } d g ) \\big { ( } ( s - g ) \\tau _ 1 + g ^ { - 1 } d g \\tau _ 2 \\big { ) } + ( s - g ) g ^ { - 1 } d g \\tau _ 1 \\\\ & = s ^ { - 1 } d s ( s - g ) \\tau _ 1 + { ( } s ^ { - 1 } d s - g ^ { - 1 } d g { ) } g ^ { - 1 } d g \\tau _ 2 \\\\ & = s ^ { - 1 } d s \\big { ( } ( s - g ) \\tau _ 1 + g ^ { - 1 } d g \\tau _ 2 \\big { ) } . \\end{align*}"}
-{"id": "7046.png", "formula": "\\begin{align*} 1 = \\sum \\{ \\tilde { \\sigma } _ i \\tilde { \\sigma } _ j \\} = \\sum _ { 1 \\leq i < j \\leq n - 1 } { t _ i t _ j \\sigma _ i \\sigma _ j } + g ( t ) \\sigma _ n ( t _ 1 \\sigma _ 1 + t _ 2 \\sigma _ 2 + . . . + t _ { n - 1 } \\sigma _ { n - 1 } ) . \\end{align*}"}
-{"id": "8054.png", "formula": "\\begin{align*} \\lambda ^ { 1 / 2 } ( A ^ { 1 / 2 } J ^ T A J A ^ { 1 / 2 } ) & = \\lambda ( | A ^ { 1 / 2 } ( i J ) A ^ { 1 / 2 } | ) \\\\ & = ( d _ n , d _ n , d _ { n - 1 } , d _ { n - 1 } , \\dots , d _ 1 , d _ 1 ) = \\d ( A ) . \\end{align*}"}
-{"id": "701.png", "formula": "\\begin{gather*} \\mu _ n ( q , \\alpha _ 0 , \\beta ) = \\mu _ n ( q _ 0 , \\alpha _ 0 , \\beta _ 0 ) , \\\\ | \\kappa _ n ( q , \\alpha _ 0 , \\beta ) | \\geq | \\kappa _ n ( q _ 0 , \\alpha _ 0 , \\beta _ 0 ) | , \\end{gather*}"}
-{"id": "7134.png", "formula": "\\begin{align*} d _ c ( m , n ) = \\frac { \\lambda _ 0 \\gamma _ m } { f _ { m , n } } . \\end{align*}"}
-{"id": "7885.png", "formula": "\\begin{align*} C _ k = - \\inf _ { 0 \\le p \\le k } H ( p ) , \\end{align*}"}
-{"id": "6737.png", "formula": "\\begin{align*} & \\Big | \\mathbb { P } ( R _ { N } > \\beta _ { N } t ) - \\mathbb { P } ( \\sigma ^ { + } ( u ) \\notin V , \\forall u \\in \\{ N ^ { 1 + \\delta } , \\dots , \\beta _ { N } t \\} ) \\Big | \\leq \\\\ \\leq & \\Big | \\mathbb { P } ( R _ { N } \\leq \\beta _ { N } t ) - \\mathbb { P } ( N ^ { 1 + \\delta } \\leq R _ { N } \\leq \\beta _ { N } t ) \\Big | \\\\ = & \\mathbb { P } ( R _ { N } < N ^ { 1 + \\delta } ) \\xrightarrow [ ] { N \\rightarrow \\infty } 0 \\end{align*}"}
-{"id": "1704.png", "formula": "\\begin{align*} U ^ { - 1 } ( h ) ( x ) \\ ; = \\ ; \\sqrt { g _ 1 ( x ) } \\cdot h ( x ) , \\ ; h \\in \\ ; L ^ 2 ( X , \\mu ' ) . \\end{align*}"}
-{"id": "5366.png", "formula": "\\begin{align*} \\theta _ { X \\boxtimes Y } = c _ { Y , X } \\circ c _ { X , Y } \\circ \\left ( \\theta _ X \\boxtimes \\theta _ Y \\right ) . \\end{align*}"}
-{"id": "435.png", "formula": "\\begin{align*} p _ { 1 , k _ 1 , k _ 2 } ( x , t ) & = \\frac { ( - 1 ) ^ { k _ 2 } \\pi ^ { k _ 1 + k _ 2 } } { 4 ^ n ( n + k _ 1 - 1 ) ! } \\abs * { t } ^ { n + k _ 1 - 1 } e ^ { - \\frac { 1 } { 4 } d ( x , t ) ^ 2 } \\left [ 1 + O \\left ( \\frac { 1 } { \\abs { t } } + \\kappa \\right ) \\right ] . \\end{align*}"}
-{"id": "7156.png", "formula": "\\begin{align*} & \\mathbb { P } \\{ \\bar { z } ' _ { a ^ * , s } \\geq Z ( a ^ * ) / \\beta K + c _ { k , s } \\} \\leq e ^ { - 4 \\ln k } = k ^ { - 4 } , \\\\ & \\mathbb { P } \\{ \\bar { z } ' _ { n , s _ n } \\leq Z ( n ) / \\beta K - c _ { k , s _ n } \\} \\leq k ^ { - 4 } . \\end{align*}"}
-{"id": "2373.png", "formula": "\\begin{align*} Z ( s , \\Gamma , \\chi ) = \\prod _ { j = 1 } ^ m Z ( s , \\Gamma , \\chi _ j ) . \\end{align*}"}
-{"id": "7083.png", "formula": "\\begin{align*} ( a ; q ) _ { \\infty } & : = \\prod _ { n = 0 } ^ { \\infty } ( 1 - a q ^ { n } ) . \\end{align*}"}
-{"id": "6647.png", "formula": "\\begin{align*} \\int _ \\Omega \\varphi _ 1 \\varphi _ 2 \\cdots \\varphi _ n \\ , d \\mu \\ ; \\leq \\ ; \\prod _ { i = 1 } ^ { n } \\left ( \\int _ { \\Omega } \\varphi _ i ^ n \\ , d \\mu \\right ) ^ { \\frac { 1 } { n } } \\ , \\end{align*}"}
-{"id": "10153.png", "formula": "\\begin{align*} { d _ k ( i ) } = { \\boldsymbol x _ k ^ H ( i ) \\boldsymbol \\omega _ 0 + n _ k ( i ) } , ~ ~ ~ k = 1 , 2 , \\ldots , 1 4 , \\end{align*}"}
-{"id": "7354.png", "formula": "\\begin{align*} \\displaystyle { \\not } D = e ^ a . \\nabla _ { X _ a } . \\end{align*}"}
-{"id": "4243.png", "formula": "\\begin{align*} \\frac { \\deg ( E _ 0 ) } { \\mathrm { r a n k } ( E _ 0 ) } = \\frac { \\deg ( L ) } { 1 - p ^ f } . \\end{align*}"}
-{"id": "892.png", "formula": "\\begin{align*} H _ 0 : \\rho _ m = 0 m = 1 , \\dots , M H _ 1 : \\rho _ m \\neq 0 m = 1 , \\dots , M . \\end{align*}"}
-{"id": "6933.png", "formula": "\\begin{align*} \\langle \\Gamma _ { f } g , h \\rangle _ { L ^ 2 ( \\Xi ) } = ( f , g * \\bar { h } ) = \\mu ( \\mathcal { F } ^ { - 1 } g \\cdot \\mathcal { F } ^ { - 1 } \\bar { h } ) . \\end{align*}"}
-{"id": "4752.png", "formula": "\\begin{align*} { \\varepsilon } = F ( x ^ 1 , Y , Z ) \\ , { \\Delta \\sqrt { - \\det g ^ { i j } } } / { L _ 0 ( x ^ 0 ) } , \\end{align*}"}
-{"id": "9965.png", "formula": "\\begin{align*} \\bar { \\upsilon } _ { k i } = \\dfrac { \\rho _ { k i i } } { \\rho _ { k i i } + { \\displaystyle \\sum _ { j \\neq i } ^ { L } } \\kappa _ { j } \\mathbb { E } \\left [ \\mathsf { \\Gamma } _ { j i } \\right ] + \\sigma ^ { 2 } } \\end{align*}"}
-{"id": "331.png", "formula": "\\begin{align*} \\delta _ { \\nu } ( n ) : = \\min \\Big \\{ \\ , 1 , \\ , \\prod _ { i \\in I _ { \\nu } } \\mathrm { d i s t } ( \\alpha _ i ^ { ( \\nu ) } ( n ) , \\ , \\Z _ { \\le 0 } + | \\sigma _ i | \\Z _ { \\le - \\nu } ) \\ , \\Big \\} > 0 , \\nu = 0 , 1 , \\ , \\ , n \\in \\N , \\end{align*}"}
-{"id": "4520.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | \\phi _ 1 ( \\Delta _ n ) | = 1 = \\lim _ { n \\to \\infty } | \\phi _ 2 ( \\Delta _ n ) | . \\end{align*}"}
-{"id": "7531.png", "formula": "\\begin{gather*} w ^ j _ 1 + w ^ 1 w ^ j _ 2 - w ^ j _ { 2 2 } + F ^ j ( z _ 2 , w ^ 1 , w ^ 2 ) = 0 , j = 1 , 2 , \\end{gather*}"}
-{"id": "13.png", "formula": "\\begin{align*} V _ { \\sigma } ^ C ( C _ 1 , C _ 2 ) = \\frac { 1 } { 2 \\pi \\sigma } - \\frac { 1 } { 2 \\pi \\sigma } E [ ( C _ 1 - C _ 2 ) ( C _ 1 - C _ 2 ) ^ * ] + h _ { \\sigma ^ 4 } \\end{align*}"}
-{"id": "5422.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } - \\Delta u + q u = 0 \\ \\ \\textrm { i n } \\Omega , \\\\ u = 0 , \\ \\textrm { o n } \\partial \\omega _ { + , \\epsilon / 2 , - \\theta } \\times \\R , \\end{array} \\right . \\end{align*}"}
-{"id": "489.png", "formula": "\\begin{align*} \\phi _ \\delta ' ( s ) = \\pi \\frac { s } { \\sigma ( s ) } \\left [ 1 - \\delta ( s ) \\rho \\left ( \\delta ( s ) \\right ) + \\frac { \\delta ^ 2 } { \\sigma ( s ) ^ { \\frac { 3 } { 2 } } } \\rho ' \\left ( \\delta ( s ) \\right ) \\right ] , \\end{align*}"}
-{"id": "1125.png", "formula": "\\begin{align*} \\widetilde { \\Phi ^ c } : = \\sum _ { \\sigma \\in \\Phi ^ c } \\sigma ^ { - 1 } . \\end{align*}"}
-{"id": "260.png", "formula": "\\begin{align*} f ( t ) = A ( t ) \\oplus h ( t ) , \\rho _ A = \\rho | _ { [ - \\frac 1 { N } , \\frac 1 { N } ] } \\rho _ h = \\rho | _ { \\R \\setminus [ - \\frac 1 { N } , \\frac 1 { N } ] } . \\end{align*}"}
-{"id": "474.png", "formula": "\\begin{align*} \\partial _ 1 ^ { k _ 1 + 3 } \\left [ ( \\psi _ { \\pi / 2 } - P _ { 2 , 0 } \\psi _ { \\pi / 2 } ) a _ { k _ 1 , k _ 2 , \\pi / 2 } \\right ] ( 0 ) = \\frac { ( k _ 1 + 3 ) ! } { 3 ! } ( - 1 ) ^ { k _ 1 } i ^ { k _ 2 - n } \\pi \\left ( i \\frac { \\pi } { 2 } \\right ) ^ { n + k _ 1 + k _ 2 } , \\end{align*}"}
-{"id": "6503.png", "formula": "\\begin{align*} \\mathcal { P } = { \\displaystyle \\iiiint } \\psi \\left ( k _ { 1 } , k _ { 2 } , t \\right ) \\psi \\left ( k _ { 3 } , k _ { 4 } , t \\right ) \\psi ^ { \\ast } \\left ( k _ { 1 } , k _ { 4 } , t \\right ) \\psi ^ { \\ast } \\left ( k _ { 3 } , k _ { 2 } , t \\right ) d k _ { 1 } d k _ { 2 } d k _ { 3 } d k _ { 4 } , \\end{align*}"}
-{"id": "201.png", "formula": "\\begin{align*} \\mu \\ast v = \\mu \\ast f ( 1 ) = f ( \\mu \\ast 1 ) = f ( \\mu ) = f ( g ( v ) ) = v . \\end{align*}"}
-{"id": "2256.png", "formula": "\\begin{align*} \\begin{cases} \\widetilde f _ 1 = w _ 1 ( w _ 2 - a _ 2 ) ^ 2 + a _ 3 = 0 , \\\\ \\widetilde f _ 2 = ( w _ 1 - b _ 1 ) ^ 2 ( w _ 2 - b _ 2 ) + b _ 3 = 0 . \\end{cases} \\end{align*}"}
-{"id": "8641.png", "formula": "\\begin{align*} B = \\{ B _ s : s \\in [ 0 , T ] \\} = ( X _ 0 , \\ldots , X _ { N + 1 } ) . \\end{align*}"}
-{"id": "9608.png", "formula": "\\begin{align*} & | D _ k ^ \\# H _ j D _ { k ' } ^ \\# ( x , y ) | \\\\ & = \\bigg | \\int _ { S ( x , 9 ( A _ 0 ) ^ 2 2 ^ j ) } \\int D _ k ^ \\# ( x , u ) [ k _ j ( u , v ) - k _ j ( x , v ) ] [ D _ { k ' } ^ \\# ( v , y ) - D _ { k ' } ^ \\# ( u , y ) ] d \\mu ( u ) d \\mu ( v ) \\bigg | \\\\ & \\lesssim \\frac { 2 ^ { - | j - k ' | \\varepsilon } } { V _ { - k ' } ( y ) } \\int _ { S ( x , 9 ( A _ 0 ) ^ 2 2 ^ j ) } \\bigg ( \\int | D _ k ^ \\# ( x , u ) | | k _ j ( u , v ) - k _ j ( x , v ) | d \\mu ( u ) \\bigg ) d \\mu ( v ) \\end{align*}"}
-{"id": "5568.png", "formula": "\\begin{align*} Q = \\left [ \\begin{array} { c c c c } 0 & 1 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & 1 \\\\ 0 & 0 & \\cdots & 0 \\end{array} \\right ] \\in \\mathbb { R } ^ { 2 ^ { k } \\times 2 ^ { k } } \\end{align*}"}
-{"id": "6392.png", "formula": "\\begin{align*} \\int d x d \\theta \\left [ - \\log P \\left ( x \\theta \\right ) - 1 + \\log P _ { } \\left ( x \\theta \\right ) + \\alpha + \\beta f \\left ( \\theta \\right ) + \\gamma \\left ( x \\right ) \\right ] \\delta P \\left ( x \\theta \\right ) = 0 \\end{align*}"}
-{"id": "8450.png", "formula": "\\begin{align*} \\gamma _ F ( A , \\rho ) = \\begin{cases} \\chi _ F ( A ) \\epsilon ( \\frac { 1 } { 2 } , \\chi _ F ) & \\rho > 0 , \\\\ 1 & \\rho = 0 \\rho . \\end{cases} \\end{align*}"}
-{"id": "5401.png", "formula": "\\begin{align*} a = ( 1 , 4 ) ( 2 , 6 ) ( 3 , 5 ) ( 8 , 9 ) , b = ( 1 , 4 ) ( 2 , 8 ) ( 6 , 9 ) ( 1 0 , 1 1 ) , c = ( 2 , 7 ) ( 3 , 4 ) ( 5 , 9 ) . \\end{align*}"}
-{"id": "383.png", "formula": "\\begin{align*} | b _ { n , r , s } | & \\le C \\sum _ { j , k = 1 } ^ n ( | j + r | + | k + s | ) ^ { - \\beta } L ( | j + r | + | k + s | ) \\\\ & \\le 4 C \\sum _ { j , k = 1 } ^ { 2 n } ( j + k ) ^ { - \\beta } L ( j + k ) \\\\ & \\ll \\sum _ { j = 1 } ^ { 2 n } j ^ { 1 - \\beta } \\max _ { 1 \\le k \\le 2 n } L ( j + k ) \\\\ & \\propto n ^ { 2 - \\beta } \\max _ { 1 \\le k \\le 2 n } L ( 2 n + k ) \\propto n ^ { 2 - \\beta } L ( n ) . \\end{align*}"}
-{"id": "6560.png", "formula": "\\begin{align*} | ( F _ { \\xi } - s ( F _ { \\xi } ) ) ^ { \\circ } | _ { n - 1 } = | B _ 1 ^ { n - 1 } | _ { n - 1 } = \\frac { 2 ^ { n - 1 } } { ( n - 1 ) ! } . \\end{align*}"}
-{"id": "9556.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in W ( \\Phi ( B _ n ) ) } { ( - 1 ) ^ { \\ell ( \\sigma ) } x ^ { L _ { \\Phi ( B _ n ) } ( \\sigma ) } } = \\prod _ { i = 1 } ^ { n } ( 1 - x ^ { i } ) . \\end{align*}"}
-{"id": "6627.png", "formula": "\\begin{align*} \\| f _ n - g _ n \\| _ { \\mathbb { W } ^ { 1 , p } ( \\varphi _ n ( K _ n ) , \\mathbb { R } ^ d ) } = \\| f _ n - g _ n \\| _ { C ^ 0 ( \\varphi _ n ( K _ n ) ) } + \\left ( \\int _ { \\varphi _ n ( K _ n ) } | D f _ n - D g _ n | ^ { p } \\ , d \\mu \\right ) ^ { \\frac { 1 } { p } } \\ . \\end{align*}"}
-{"id": "6090.png", "formula": "\\begin{align*} \\Phi ^ { '' } ( r ) + ( n - 1 ) \\tfrac { \\Phi ' ( r ) } { r } - \\frak { e } _ k \\tfrac { \\sin ( 2 \\Phi ( r ) ) } { r ^ 2 } - \\tfrac { \\sin ( 2 \\Phi ( r ) ) } { 2 } g ' ( r ) ^ 2 = 0 . \\end{align*}"}
-{"id": "5875.png", "formula": "\\begin{align*} \\begin{aligned} & I _ i ( r ) < \\infty \\ \\ r \\le \\mu _ i \\ \\ P _ i ^ { ( ) } ( - \\infty , r ] ) > 0 , \\ \\\\ & r > \\mu _ i \\ \\ P _ i ^ { ( ) } ( [ r , \\infty ) ) > 0 . \\end{aligned} \\end{align*}"}
-{"id": "9533.png", "formula": "\\begin{align*} x ^ n + a _ 1 x ^ { n - 1 } + \\dots + a _ n = 0 , \\ \\ \\ a _ i \\in I ^ i . \\end{align*}"}
-{"id": "1723.png", "formula": "\\begin{align*} t _ \\lambda \\pi ( f ) = \\pi ( f \\circ \\sigma ^ n ) t _ \\lambda \\end{align*}"}
-{"id": "2232.png", "formula": "\\begin{align*} \\frac 1 { \\widetilde F _ j ( w ) } = \\sum _ { p = 0 } ^ \\infty \\frac { ( - 1 ) ^ p t ^ p \\widetilde Q _ j ^ p ( w ) } { \\widetilde q _ j ^ { ( p + 1 ) } } . \\end{align*}"}
-{"id": "6695.png", "formula": "\\begin{align*} \\left ( 1 + ( 1 + \\psi _ 2 ( \\delta ) ) c _ 4 \\delta ^ { \\frac { 2 } { n + 1 } } \\right ) \\frac { 1 } { \\sqrt [ p ] { n } } \\sum _ { i = 1 } ^ n \\pm e _ i \\in \\partial \\left [ ( B _ p ^ n ) ^ { \\delta } \\right ] \\quad , \\end{align*}"}
-{"id": "8260.png", "formula": "\\begin{align*} \\tilde { w } ( 1 ) = \\int _ { - \\infty } ^ { \\infty } w ( r ) d r > 0 . \\end{align*}"}
-{"id": "1929.png", "formula": "\\begin{align*} \\Theta _ { p , q } & = \\underset { n \\rightarrow \\infty } \\limsup \\frac { 1 } { n } \\log d ( \\mathcal { F } _ { i } ^ { n } ( q ) , \\mathcal { F } _ { i } ^ { n } ( p ) ) \\Delta _ { p , q } = \\underset { n \\rightarrow \\infty } \\limsup \\frac { 1 } { n } \\log d ( \\mathcal { F } _ { i } ^ { - n } ( q ) , \\mathcal { F } _ { i } ^ { - n } ( p ) ) . \\end{align*}"}
-{"id": "2330.png", "formula": "\\begin{align*} \\textbf { d } ( x , y ) = \\begin{cases} \\min \\{ 1 , d ( x , y ) \\} & \\mbox { i f } x , y \\in M _ { i } \\\\ 1 & \\mbox { i f } x \\in M _ { i } , y \\in M _ { j } \\mbox { a n d } i \\neq j . \\\\ \\end{cases} \\end{align*}"}
-{"id": "4233.png", "formula": "\\begin{align*} y ( n ) & = \\sum _ { l = 0 } ^ { L } x ( n - D - l ) h _ l + z ( n ) , \\end{align*}"}
-{"id": "2549.png", "formula": "\\begin{align*} P ( E ; \\Omega ) : = \\sup \\left \\{ \\int _ { \\Omega } \\chi _ { E } ( x ) \\ , \\div g ( x ) \\ , d x \\ , : \\ g \\in C ^ 1 _ c ( \\Omega ; \\ , \\R ^ n ) \\ , , \\| g \\| _ { \\infty } \\leq 1 \\right \\} \\ , . \\end{align*}"}
-{"id": "6440.png", "formula": "\\begin{align*} p _ { } \\left ( x _ { 1 } , \\ldots , x _ { l } \\right ) = { \\displaystyle \\prod \\limits _ { j = 1 } ^ { l } } p _ { j } \\left ( x _ { j } \\right ) \\overset { } { \\longrightarrow } p _ { } ^ { \\prime } \\left ( x _ { 1 } , \\ldots , x _ { l } \\right ) \\neq { \\displaystyle \\prod \\limits _ { j = 1 } ^ { l } } p _ { j } \\left ( x _ { j } \\right ) \\end{align*}"}
-{"id": "9825.png", "formula": "\\begin{align*} W _ { 1 2 } ( x , y ) = x ^ { 1 2 } - 3 3 x ^ 8 y ^ 4 - 3 3 x ^ 4 y ^ 8 + y ^ { 1 2 } . \\end{align*}"}
-{"id": "2880.png", "formula": "\\begin{align*} R _ { 0 } + R _ { 1 } + R _ { \\frac { N - 2 h + 1 } { N } } = P ( x ) , \\end{align*}"}
-{"id": "3868.png", "formula": "\\begin{align*} 0 = \\nabla \\ell ( x ^ * , \\lambda ^ * , \\mu ^ * , \\gamma ^ * ) d _ x . \\end{align*}"}
-{"id": "3616.png", "formula": "\\begin{align*} \\operatorname { C a p } _ p ^ { D } ( R _ { \\lambda } ( \\Gamma ) ) = 0 , \\end{align*}"}