diff --git "a/process_0/tokenized_finally.jsonl" "b/process_0/tokenized_finally.jsonl"
new file mode 100644--- /dev/null
+++ "b/process_0/tokenized_finally.jsonl"
@@ -0,0 +1,9182 @@
+{"id": "3993.png", "formula": "\\begin{align*} \\varpi \\bigl ( z ^ { \\left ( t + 1 \\right ) } \\bigr ) = \\sum _ { i = 1 } ^ { n } x _ { i } ^ { \\ : \\left ( t + 1 \\right ) } + \\sum _ { j = 1 } ^ { \\nu } y _ { j } ^ { \\ : \\left ( t + 1 \\right ) } = \\bigl ( \\sum _ { i = 1 } ^ { n } x _ { i } ^ { \\ : \\left ( t \\right ) } \\bigr ) \\bigl ( \\sum _ { j = 1 } ^ { \\nu } y _ { j } ^ { \\ : \\left ( t \\right ) } \\bigr ) \\end{align*}"}
+{"id": "2585.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u = f ( u ) & B , \\\\ u = 0 & \\partial B , \\end{cases} \\end{align*}"}
+{"id": "823.png", "formula": "\\begin{align*} { M _ { i j . m } } = { { \\dot \\partial } _ m } { D _ i } { \\Psi _ j } & = { { \\dot \\partial } _ m } ( ( { \\partial _ i } - G _ i ^ r { { \\dot \\partial } _ r } ) { \\Psi _ j } - { \\Psi _ r } G _ { i j } ^ r ) \\\\ & = { { \\dot \\partial } _ m } ( { \\partial _ i } { \\Psi _ j } - G _ i ^ r { \\Psi _ { j . r } } - { \\Psi _ r } G _ { i j } ^ r ) = - { \\Psi _ r } G _ { i j m } ^ r . \\end{align*}"}
+{"id": "864.png", "formula": "\\begin{align*} \\mathcal { G } ^ i = 0 . \\end{align*}"}
+{"id": "7690.png", "formula": "\\begin{align*} { I _ N } f ( t ) = \\sum \\limits _ { j = 0 } ^ { N - 1 } { { f _ j } { \\C { F } _ j } ( t ) } , \\end{align*}"}
+{"id": "4372.png", "formula": "\\begin{align*} F _ { \\mathcal { Y } , a , b } ( x ) = \\sum _ { ( i , j ) \\in [ m ] \\times [ n ] } \\overline { y } _ { ( i , j ) } g _ { ( i , j ) } ( x ) + \\sum _ { ( q , p ) \\in \\mathcal { Y } } \\Delta y _ { ( q , p ) } g _ { ( q , p ) } ( x ) = F _ { \\mathcal { Y } , 0 , 0 } ( x ) . \\end{align*}"}
+{"id": "8304.png", "formula": "\\begin{align*} L ( \\varepsilon ) y ( t , \\varepsilon ) : = y ^ { ( r ) } ( t , \\varepsilon ) + \\sum \\limits _ { j = 1 } ^ r A _ { r - j } ( t , \\varepsilon ) y ^ { ( r - j ) } ( t , \\varepsilon ) = f ( t , \\varepsilon ) , t \\in ( a , b ) , \\end{align*}"}
+{"id": "3421.png", "formula": "\\begin{align*} \\langle b , \\tau ( \\phi _ 1 , \\phi _ 2 ) \\rangle _ { \\Lambda ^ + } = - \\langle \\sqrt { - 1 } \\rho ( b ) \\phi _ 1 , \\phi _ 2 \\rangle _ { S ^ + } \\end{align*}"}
+{"id": "2217.png", "formula": "\\begin{align*} - \\partial _ \\tau ^ 2 v + 2 \\partial _ \\tau v + 3 v = g ( e ^ { - \\tau } , z ) \\equiv g ' ( \\tau , z ) . \\end{align*}"}
+{"id": "6118.png", "formula": "\\begin{align*} \\boldsymbol u _ { n + 1 } = \\boldsymbol u _ n + \\tau \\varphi _ 1 ( \\tau K ) ( K \\boldsymbol u _ n + \\boldsymbol g ( t _ n , \\boldsymbol u _ n ) ) \\end{align*}"}
+{"id": "8365.png", "formula": "\\begin{align*} \\pi ^ { - 1 } ( D ) = G \\times F _ 0 . \\end{align*}"}
+{"id": "3677.png", "formula": "\\begin{align*} \\left \\{ 1 ^ { \\left ( d n - \\binom { d + 1 } { 2 } - d \\right ) } , ( n - d / 2 ) ^ { ( d - 1 ) } , n ^ { ( 1 ) } \\right \\} \\end{align*}"}
+{"id": "2799.png", "formula": "\\begin{align*} K ^ { ( 1 ) } = \\begin{bmatrix} 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ 1 & 0 & 0 \\end{bmatrix} , \\end{align*}"}
+{"id": "8716.png", "formula": "\\begin{gather*} A n n _ { H } ( M ) = A n n _ { H } ^ { \\mathrm { l e f t } } ( M ) \\cap A n n _ { H } ^ { \\mathrm { r i g h t } } ( M ) = \\\\ \\{ a \\in H | \\ [ a , M ] = \\langle 0 \\rangle = [ M , a ] \\} \\end{gather*}"}
+{"id": "8536.png", "formula": "\\begin{align*} \\begin{array} { l c l } a _ 1 \\mu _ 1 & = & - \\alpha \\log _ 1 ( \\pi _ 2 ) + \\gamma \\log _ 1 ( u ( K ) ) , \\\\ k \\lambda _ 1 + \\mu _ 1 a _ 1 & = & - \\beta , \\\\ a _ 2 \\mu _ 2 & = & \\alpha \\log _ 1 ( \\pi _ 2 ) - \\gamma \\log _ 1 ( u ( K ) ) , \\\\ k \\lambda _ 2 + \\mu _ 2 a _ 2 & = & - \\beta . \\end{array} \\end{align*}"}
+{"id": "7711.png", "formula": "\\begin{align*} E ( x , t ) = \\log \\lambda _ { m a x } ( \\{ b _ { i j } \\} ) - \\tilde { d } \\log h ( x , t ) + \\tilde { l } | \\nabla h | ^ { 2 } , \\end{align*}"}
+{"id": "4552.png", "formula": "\\begin{align*} R m _ h = O _ h ' ( \\rho ^ { - 6 } ) \\ , \\end{align*}"}
+{"id": "8922.png", "formula": "\\begin{align*} k _ { \\lambda w } ^ { \\lambda v } | _ { \\sigma } = k _ { \\lambda w } ^ { \\lambda v } | _ { t _ { 0 } \\sigma } = k _ { \\lambda w } ^ { \\lambda v } | _ { t _ { 0 } \\sigma + 0 \\nu } \\leq k _ { \\lambda w } ^ { \\lambda v } | _ { t _ { 0 } \\sigma + 1 \\nu } = k _ { \\lambda w } ^ { \\lambda v } | _ { \\rho } . \\end{align*}"}
+{"id": "214.png", "formula": "\\begin{align*} N _ T ( X _ 1 , X _ 2 ) = [ T ( X _ 1 ) , T ( X _ 2 ) ] + T ^ 2 ( [ X _ 1 , X _ 2 ] ) - T ( [ T ( X _ 1 ) , X _ 2 ] ) - T ( [ X _ 1 , T ( X _ 2 ) ] ) , \\ \\forall X _ 1 , X _ 2 \\in \\mathfrak { X } ( M ) , \\end{align*}"}
+{"id": "565.png", "formula": "\\begin{align*} g \\phi _ n - \\omega ^ 2 \\phi & = - j \\omega \\frac { p } { \\rho } & \\Gamma _ f , \\end{align*}"}
+{"id": "7440.png", "formula": "\\begin{align*} \\left ( - \\frac { \\partial } { \\partial t } + A \\right ) \\Phi _ n ( t , p ) & = - \\left ( \\sum _ { j = 1 } ^ d \\mu ^ { ( j ) } ( p ) e _ j \\right ) \\otimes \\Phi _ { n - 1 } ( t , p ) \\\\ - \\sum _ { j = 1 } ^ d e _ j & \\otimes \\frac { \\partial \\Phi _ { n - 1 } ( t , p ) } { \\partial p _ j } - \\left ( \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ d e _ { j } \\otimes e _ { j } \\right ) \\otimes \\Phi _ { n - 2 } ( t , p ) . \\end{align*}"}
+{"id": "4035.png", "formula": "\\begin{align*} \\mathbb { X } \\sim _ 0 \\mathbb { Y } \\iff \\mathcal { L } ( X ) = \\mathcal { L } ( Y ) \\iff \\mathbb { E } _ { \\mathbb { P } } [ S ( X ) ] = \\mathbb { E } _ { \\mathbb { Q } } [ S ( Y ) ] . \\end{align*}"}
+{"id": "2082.png", "formula": "\\begin{align*} N _ { i , 1 } ^ s = - \\alpha \\int _ 0 ^ { \\frac { \\lfloor s T \\rfloor } { T } } \\int _ 0 ^ 1 f ( u ) A ( \\frac { i } { d } , y ) \\Theta ( u , y ) d y d u + o ( 1 ) , N _ { i , 3 } ^ s \\to 0 \\ d , T \\to \\infty . \\end{align*}"}
+{"id": "3055.png", "formula": "\\begin{align*} z ^ 2 = \\frac { b _ { 1 1 } } { 4 } x ^ 4 + b _ { 1 2 } x ^ 3 y + b _ { 2 2 } x ^ 2 y ^ 2 + 2 b _ { 2 3 } x y ^ 3 + b _ { 3 3 } y ^ 4 \\end{align*}"}
+{"id": "5043.png", "formula": "\\begin{align*} B ( x , r ) = \\{ y \\in X : d ( y , x ) < r \\} \\end{align*}"}
+{"id": "6605.png", "formula": "\\begin{align*} \\phi _ { \\Phi _ k , \\beta } ( x , y ) = \\frac { 1 } { 2 \\pi i } \\int _ { b - i U } ^ { b + i U } \\left ( \\sum _ { n \\le x } \\left ( \\frac { \\Phi _ k ( n ) } { n ^ \\beta } \\right ) ^ { - z } \\right ) \\frac { y ^ z } { z } \\ , d z + O \\left ( \\frac { y ^ { b } } { U } \\sum _ { n \\leq x } \\Phi _ k ( n ) ^ { - b } n ^ { b \\beta } \\right ) . \\end{align*}"}
+{"id": "4703.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\alpha ( 1 _ { W } ) + \\sum \\limits _ { l + 1 } ^ { r - 1 } \\alpha ( 1 _ { U _ k } ) = \\sum \\limits _ { l } ^ { r - 2 } \\alpha ( 1 _ { V _ k } ) . \\\\ \\beta ( 1 _ { W } ) + \\sum \\limits _ { l + 1 } ^ { r - 1 } \\beta ( 1 _ { U _ k } ) = \\sum \\limits _ { l } ^ { r - 2 } \\beta ( 1 _ { V _ k } ) . \\end{array} \\right . \\end{align*}"}
+{"id": "5086.png", "formula": "\\begin{align*} L _ s \\cdot \\varepsilon _ \\theta & = q ^ { - k } ( E _ { s } + R _ { s } \\Psi _ { s } ) ( \\varepsilon _ \\theta ) \\\\ & = q ^ { - k } ( ( q ^ k - 1 ) \\varepsilon _ \\theta - R _ { s } \\varepsilon _ \\theta ) \\\\ & = ( 1 - q ^ { - k } ) \\varepsilon _ \\theta - q ^ { - k } R _ { s } \\cdot \\varepsilon _ \\theta , \\end{align*}"}
+{"id": "4312.png", "formula": "\\begin{align*} \\| u \\| _ { m b } = \\| u \\| _ { c b } \\end{align*}"}
+{"id": "2427.png", "formula": "\\begin{align*} V _ { t } = ( V ^ { m } ) _ { x x } + \\mu V ^ { m } - \\delta ( 1 - p ) V , t > 0 , x \\in \\mathbb { R } , m = \\dfrac { 1 } { 1 - p } > 1 \\end{align*}"}
+{"id": "7492.png", "formula": "\\begin{align*} { D } ^ { ' } _ { \\lambda _ 0 } = D _ { \\lambda _ 0 } \\cap \\left ( \\left \\{ \\mathbf { r } ( x ) \\leq \\tfrac { 3 } { 2 } \\sqrt { \\gamma t + ( \\Lambda + 1 ) ^ 2 } , - \\lambda _ 0 / 4 \\leq l \\leq \\lambda _ 0 / 4 \\right \\} \\times [ 0 , T ] \\right ) , \\end{align*}"}
+{"id": "9162.png", "formula": "\\begin{align*} G _ { \\beta , \\beta ' } = ( w _ { \\beta , 1 } - w _ { \\beta ' , 1 } ) \\cdot G _ { [ 2 , n , j ] , \\beta ' } , \\end{align*}"}
+{"id": "3056.png", "formula": "\\begin{align*} ( A , B ) = \\left ( \\begin{pmatrix} - a & 0 & 0 \\\\ 0 & 0 & 1 / 2 \\\\ 0 & 1 / 2 & b / 4 a d \\end{pmatrix} , \\begin{pmatrix} 0 & 0 & 1 / 2 \\\\ 0 & d & 0 \\\\ 1 / 2 & 0 & - c / 4 a d \\end{pmatrix} \\right ) \\end{align*}"}
+{"id": "7178.png", "formula": "\\begin{align*} \\alpha ( T ) : = \\frac { \\| v _ 0 ^ 2 - ( v _ 0 + V ) ^ 2 \\| _ Y } { v _ 0 ^ 2 - 1 } + \\frac { M ^ 2 } { v _ 0 ^ 2 - 1 } + \\frac { 2 M \\| v _ 0 + V \\| _ Y } { v _ 0 ^ 2 - 1 } + \\frac { 1 } { \\gamma ( v _ 0 ^ 2 - 1 ) } < 1 , t \\in [ 0 , T ] \\end{align*}"}
+{"id": "4333.png", "formula": "\\begin{align*} \\inf _ { x \\in \\mathcal { X } } & \\ \\left \\{ \\sup _ { \\mathcal { S } \\subseteq [ m ] : | \\mathcal { S } | \\leq \\Gamma } \\left \\{ \\sum _ { i \\in \\mathcal { S } } \\max \\{ 0 , x _ i - \\overline { b } _ i + \\Delta b _ i \\} + \\sum _ { i \\in [ m ] \\setminus \\mathcal { S } } \\max \\{ 0 , x _ i - \\overline { b } _ i \\} \\right \\} \\right \\} . \\end{align*}"}
+{"id": "2678.png", "formula": "\\begin{align*} \\delta \\dot { q } ^ { i } ( t _ { 1 } ) = \\delta \\dot { q } ^ { i } ( t _ { 2 } ) = 0 \\end{align*}"}
+{"id": "3881.png", "formula": "\\begin{align*} \\hat \\gamma ^ n ( \\{ x \\} \\times \\{ y \\} \\times [ 0 , t ] ) & = \\int _ 0 ^ t \\exp ( - s ) \\eta ^ n ( x , y \\mid s ) d s \\\\ \\hat \\theta ^ n ( \\{ x \\} \\times \\{ y \\} \\times [ 0 , t ] ) & = \\int _ 0 ^ t \\exp ( - s ) \\eta ^ n _ { ( 1 ) } ( x \\mid s ) G ( M ^ n ( s ) ) ( x , y ) d s . \\end{align*}"}
+{"id": "1113.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ \\infty | z _ i | \\right ) ^ { \\alpha } \\leq \\sum _ { i = 1 } ^ \\infty | z _ i | ^ { \\alpha } . \\end{align*}"}
+{"id": "3591.png", "formula": "\\begin{align*} \\alpha ^ 3 f - ( 1 + a ) \\alpha ^ 2 f + ( a + b ) \\alpha f - b f = 0 , \\end{align*}"}
+{"id": "6690.png", "formula": "\\begin{align*} K = \\langle c ^ p , a _ 1 , \\ldots , a _ n \\rangle , \\end{align*}"}
+{"id": "1568.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ N V ^ i _ j \\ , ( V ^ j - \\ell _ j ) \\ , n ^ i & = - \\frac { 1 } { | U | } \\sum _ { i , j , k = 1 } ^ N V ^ i _ j \\ , ( V ^ j - \\ell _ j ) \\ , ( V ^ k - \\ell _ k ) \\ , u _ { k i } \\\\ & = - \\frac { 1 } { | U | } \\ , \\sum _ { i , j , k , h = 1 } ^ N F _ { i h } ( D u ) \\ , u _ { h j } \\ , ( V ^ j - \\ell _ j ) \\ , u _ { i k } \\ , ( V ^ k - \\ell _ k ) \\\\ & = - \\frac { 1 } { | U | } \\ , \\big ( D ^ 2 F ( D u ) \\ , U , U \\big ) < 0 , \\end{align*}"}
+{"id": "2405.png", "formula": "\\begin{align*} v _ q \\left ( \\widetilde { A _ n } ^ { ( j - 1 - v ) } ( - k ) \\right ) = v _ q ( ( j - 1 - v ) ! a _ { n , 2 s + 4 - ( j - 1 - v ) , k } ) \\geqslant s + 2 - ( j - 1 - v ) . \\end{align*}"}
+{"id": "8424.png", "formula": "\\begin{align*} [ b ] = \\{ g ^ { - 1 } b \\sigma ( g ) \\mid g \\in G ( \\breve F ) \\} \\subseteq G ( \\breve F ) \\end{align*}"}
+{"id": "7278.png", "formula": "\\begin{align*} [ - u + w , u + w ] = \\left [ \\tfrac { \\widetilde { \\theta } + \\eta \\tfrac { \\alpha _ t } { 1 - q } - 2 \\sqrt { \\tfrac { \\alpha _ t } { 1 - q } } \\sqrt { \\eta \\widetilde { \\theta } + 1 - q } } { 1 - q } , \\tfrac { \\widetilde { \\theta } + \\eta \\tfrac { \\alpha _ t } { 1 - q } + 2 \\sqrt { \\tfrac { \\alpha _ t } { 1 - q } } \\sqrt { \\eta \\widetilde { \\theta } + 1 - q } } { 1 - q } \\right ] . \\end{align*}"}
+{"id": "583.png", "formula": "\\begin{align*} c = \\sum \\limits _ { n = 1 } ^ \\infty \\| x _ n - y _ n \\| \\cdot \\| x _ n ^ \\ast \\| < 1 . \\end{align*}"}
+{"id": "4998.png", "formula": "\\begin{align*} P _ N ( \\{ x \\} ) = \\sum _ { i = 1 } ^ N P _ N ( y _ i , x _ 2 , \\dots , x _ N ) \\prod _ { \\substack { j = 1 \\\\ j \\neq i } } ^ { N } \\frac { x _ 1 - y _ j } { y _ i - y _ j } , \\end{align*}"}
+{"id": "5608.png", "formula": "\\begin{align*} \\hat { a } - \\hat { v } = a ' - v ' \\leq a - v . \\end{align*}"}
+{"id": "348.png", "formula": "\\begin{align*} x = a + \\sqrt { 2 a b } , y = b + \\sqrt { 2 a b } , z = a + b + \\sqrt { 2 a b } . \\end{align*}"}
+{"id": "2438.png", "formula": "\\begin{align*} \\begin{cases} \\phi ' = \\psi , \\\\ \\psi ' = - c \\phi ^ { - 1 } \\psi + \\gamma \\phi ^ { - 1 } \\psi ^ { 2 } - k \\phi + \\delta p , \\end{cases} \\left ( \\ , ' = \\dfrac { d } { d \\xi } \\ , \\right ) . \\end{align*}"}
+{"id": "5969.png", "formula": "\\begin{align*} P ( x ) = \\sqrt { P ( x ) } \\cdot \\sqrt { P ( x ) } \\leq \\sqrt { P ( x ) } \\cdot \\sqrt { L _ 0 \\cdot Q ( x ) } , \\end{align*}"}
+{"id": "7453.png", "formula": "\\begin{align*} t ^ j - ( t - m \\tau _ 0 ) ^ j = \\sum _ { q = 1 } ^ { j } { j \\choose q } ( - m \\tau _ 0 ) ^ q t ^ { j - q } \\leq C ( j , t ) \\cdot m \\end{align*}"}
+{"id": "5618.png", "formula": "\\begin{align*} - \\tilde { H } _ { e f } & = \\left ( M _ { e _ 3 e _ 2 } M _ { f _ 1 e _ 2 } + M _ { e _ 3 f _ 2 } M _ { f _ 1 f _ 2 } + \\mathbf { 1 } \\{ f _ 1 = e _ 3 \\} \\sum _ { u \\in V _ 2 } M _ { e _ 3 u } ^ 2 \\right ) A _ { f _ 1 f _ 3 , f _ 2 } \\\\ & : = M ^ { ( 1 ) } _ { e f } + M ^ { ( 2 ) } _ { e f } + M ^ { ( 3 ) } _ { e f } . \\end{align*}"}
+{"id": "2924.png", "formula": "\\begin{align*} \\mathsf P _ { n , m } : = \\begin{cases} \\displaystyle \\frac { 1 } { 2 + \\sigma } \\frac { n + 2 } { n + 1 } , & \\mbox { i f } m = n + 1 , \\\\ \\\\ \\displaystyle \\frac { \\sigma } { 2 + \\sigma } , & \\mbox { i f } m = n , \\\\ \\\\ \\displaystyle \\frac { 1 } { 2 + \\sigma } \\frac { n } { n + 1 } , & \\mbox { i f } m = n - 1 \\ge 0 , \\\\ \\\\ \\displaystyle 0 , & \\mbox { o t h e r w i s e , } \\end{cases} n \\ge 0 . \\end{align*}"}
+{"id": "4240.png", "formula": "\\begin{align*} E _ 6 ( \\tau ) ^ 2 = 1 - 1 0 0 8 q + 2 2 0 7 5 2 q ^ 2 + \\cdots . \\end{align*}"}
+{"id": "5228.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } D A G I ( p \\| q ) } { \\partial q _ { j } } = \\sum _ { j } p _ { j } \\left [ \\underbrace { \\frac { a - 1 } { a - b } \\left ( \\overline { M G } \\right ) ^ { a - 2 } - \\frac { b - 1 } { a - b } \\left ( \\overline { M G } \\right ) ^ { b - 2 } } _ { Z } \\right ] \\frac { \\partial \\left ( \\overline { M G } \\right ) } { \\partial q _ { j } } \\end{align*}"}
+{"id": "8395.png", "formula": "\\begin{align*} B _ 1 ( a ) B _ 1 ( b ) & = B _ 1 ( B _ 1 ( a ) b B _ 2 ( a ) ) , \\forall a , b \\in G , \\\\ B _ 2 ( b ) B _ 2 ( a ) & = B _ 2 ( B _ 1 ( a ) b B _ 2 ( a ) ) , \\forall a , b \\in G . \\end{align*}"}
+{"id": "4091.png", "formula": "\\begin{align*} R _ 0 ( \\lambda ) f = e ^ { i \\lambda r } r ^ { \\frac { 1 - d } { 2 } } h ( \\theta ) + O _ { L ^ 2 ( \\R ^ d ) } , \\ , \\ , \\ , \\ , h ( \\theta ) = c _ n \\lambda ^ { \\frac { d - 1 } { 2 } } \\widehat { f } ( \\lambda \\theta ) , \\end{align*}"}
+{"id": "8462.png", "formula": "\\begin{align*} \\partial _ t ( u _ h ) _ { \\pm } ^ { ( \\ell ) } = \\dfrac { ( u _ m ( x ) ) _ { \\pm } ^ { ( \\ell ) } - ( u _ { m - 1 } ( x ) ) _ { \\pm } ^ { ( \\ell ) } } { h } \\end{align*}"}
+{"id": "7051.png", "formula": "\\begin{align*} \\frac { 1 } { q } \\partial _ t \\| w \\| _ q ^ q + I _ q + ( q - 2 ) J _ q = \\langle b _ m \\cdot w , \\nabla \\cdot ( w | w | ^ { q - 2 } ) \\rangle \\end{align*}"}
+{"id": "7815.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } \\ ! \\ ! I _ { M } & \\ ! \\ ! \\ ! \\ ! I _ { M } \\ ! \\ ! \\ ! \\\\ \\end{array} \\right ] P _ { \\operatorname { B o x } [ \\boldsymbol { 0 } , \\boldsymbol { 1 } ] } \\biggl ( \\boldsymbol { \\rho } \\left ( i + \\frac { 1 } { 2 } \\right ) - \\begin{bmatrix} \\boldsymbol { \\lambda } ^ { \\star } \\\\ \\boldsymbol { \\lambda } ^ { \\star } \\end{bmatrix} \\biggl ) = \\boldsymbol { 1 } , \\end{align*}"}
+{"id": "6524.png", "formula": "\\begin{align*} \\mathbb { P } ( \\varepsilon _ 1 \\leq - x ) & = ( \\sigma _ 1 + o ( 1 ) ) x ^ { - \\alpha } h ( x ) \\mathbb { P } ( \\varepsilon _ 1 > x ) = ( \\sigma _ 2 + o ( 1 ) ) x ^ { - \\alpha } h ( x ) \\end{align*}"}
+{"id": "714.png", "formula": "\\begin{align*} f ( x ) = \\phi _ 1 ^ 6 ( x ) + ( 9 - 6 x ) \\phi _ 1 ^ 5 ( x ) - ( 5 x + 2 5 ) \\phi _ 1 ^ 4 ( x ) + ( 2 4 x + 1 8 ) \\phi _ 1 ^ 3 ( x ) - 1 8 x \\phi _ 1 ^ 2 ( x ) \\\\ + ( 4 x - 4 ) \\phi _ 1 ( x ) + 1 - m . \\end{align*}"}
+{"id": "673.png", "formula": "\\begin{align*} \\Delta f ^ { i } - \\frac { \\partial } { \\partial x ^ { i } } ( \\delta f ) = 2 \\frac { \\partial } { \\partial x ^ { j } } ( R ^ { 0 } | _ { S ^ { 1 } } f ) _ { i , j } = \\frac { \\partial } { \\partial x ^ { j } } \\left ( \\frac { \\partial f _ { i } } { \\partial x ^ { j } } - \\frac { \\partial f _ { j } } { \\partial x ^ { i } } \\right ) = - \\left ( { \\rm c u r l } ^ { \\intercal } \\ , { \\rm c u r l } \\ , ( f ) \\right ) _ { i } , \\end{align*}"}
+{"id": "5496.png", "formula": "\\begin{align*} A & { } = A _ { j - 1 } [ X _ { j + 1 } ; \\sigma ^ * _ { j + 1 } ] \\ldots [ X _ N ; \\sigma ^ * _ N ] [ X _ j ; \\sigma _ j ' , \\delta _ j ' ] , \\\\ \\hat { A } & { } = A _ { j - 1 } [ X _ { j + 1 } ^ { \\pm 1 } ; \\sigma ^ * _ { j + 1 } ] \\ldots [ X _ N ^ { \\pm 1 } ; \\sigma ^ * _ N ] [ X _ j ; \\sigma _ j ' , \\delta _ j ' ] , \\end{align*}"}
+{"id": "3680.png", "formula": "\\begin{align*} ( \\lambda - 2 ) x ( \\{ i , j \\} ) = \\sum _ { k \\in [ n ] \\setminus ( [ d ] \\cup \\{ j \\} ) } x ( \\{ i , k \\} ) + \\frac { \\sqrt { 2 } } { 2 } \\sum _ { k \\in [ d ] \\setminus \\{ i \\} } x ( \\{ i , k \\} ) . \\end{align*}"}
+{"id": "4826.png", "formula": "\\begin{align*} \\delta ^ { \\mbox { \\tiny ( \\itshape k \\upshape ) } } _ a = \\begin{cases} \\delta \\mbox { w i t h p r o b a b l t y } p _ a , \\\\ 0 \\mbox { w i t h p r o b a b i l i t y } 1 - p _ a . \\end{cases} \\end{align*}"}
+{"id": "3458.png", "formula": "\\begin{align*} \\delta _ R ( L ) : = \\delta _ R ( \\Sigma ( L ) , \\iota , \\mathfrak { t } _ L ) , \\ \\ \\ \\bar { \\delta } _ R ( L ) : = \\overline \\delta _ R ( \\Sigma ( L ) , \\iota , \\mathfrak { t } _ L ) , \\ \\ \\ \\underline { \\delta } _ R ( L ) : = \\underline \\delta _ R ( \\Sigma ( L ) , \\iota , \\mathfrak { t } _ L ) , \\end{align*}"}
+{"id": "943.png", "formula": "\\begin{align*} R f ( x ) = \\mathbb E _ x \\int ^ { \\infty } _ 0 f ( X _ t ) \\ , d t , R ^ V f ( x ) = \\mathbb E _ x \\int ^ { \\tau _ V } _ 0 f ( X _ t ) \\ , d t , x \\in E , \\end{align*}"}
+{"id": "7290.png", "formula": "\\begin{align*} \\tilde { S } _ { m , r } ( \\alpha , n ) : = \\frac { 1 } { m } \\sum _ { j = 0 } ^ { m - 1 } \\sec ^ { n } \\left ( \\frac { 2 ( j + \\alpha ) } { m } \\pi \\right ) e ^ { 2 \\pi i r j / m } . \\end{align*}"}
+{"id": "8530.png", "formula": "\\begin{align*} f _ C & = f _ C ( \\varphi ( 1 , 0 ) ) \\cdot \\log + f _ C ( \\varphi ( 0 , 1 ) ) \\cdot \\phi \\\\ & = f _ C ( 1 , 0 ) \\cdot \\log + s ^ { - 1 } \\log _ 1 ( \\pi _ 2 ) \\cdot f _ C ( 1 , - 1 ) \\cdot \\phi \\\\ & = v _ 1 \\cdot \\log + s ^ { - 1 } \\log _ 1 ( \\pi _ 2 ) ( v _ 1 - v _ 2 ) \\cdot \\phi \\end{align*}"}
+{"id": "990.png", "formula": "\\begin{align*} L = - \\phi ( - \\Delta ) , \\end{align*}"}
+{"id": "4114.png", "formula": "\\begin{align*} Q ^ { ( r ) } ( \\theta ) { { \\psi } } ^ { ( r ) } ( \\theta ) + \\sum _ { i \\in \\mathcal { J } } ( \\mu ^ { ( r ) } _ i - \\lambda _ i ) \\xi _ i ( \\theta , r ) \\bigl ( { \\psi } ^ { ( r ) } ( \\theta ) - { \\psi } _ i ^ { ( r ) } ( \\theta ) \\bigr ) = \\epsilon ( \\theta , r ) , \\theta \\in \\R ^ J _ - , \\end{align*}"}
+{"id": "6753.png", "formula": "\\begin{align*} \\widetilde { \\xi } ^ k = \\frac { 1 } { \\tau ^ k } \\breve { \\xi } ^ { k } - \\frac { 1 - \\tau ^ k } { \\tau ^ k } \\breve { \\xi } ^ { k - 1 } , ~ { \\rm f o r } ~ \\xi = v ~ { \\rm o r } ~ w , \\end{align*}"}
+{"id": "5761.png", "formula": "\\begin{align*} & \\frac { d } { d t } z ^ { ( k , \\ell ) } _ j = \\mathcal { W } ^ { ( k , \\ell ) } _ j , \\ \\frac { d } { d t } \\bar { z } ^ { ( k , \\ell ) } _ j - m \\bar { z } ^ { ( k , \\ell ) } _ j = \\overline { \\mathcal { W } } ^ { ( k , \\ell ) } _ j . \\end{align*}"}
+{"id": "327.png", "formula": "\\begin{align*} v _ t = ( m - 1 ) v \\Delta v + | \\nabla v | ^ 2 + K ( m , p ) v ^ { ( m + p - 2 ) / ( m - 1 ) } . \\end{align*}"}
+{"id": "3117.png", "formula": "\\begin{align*} \\varphi ^ * \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & : = \\left ( \\begin{array} { c c c } a ^ { \\ell _ 1 } & 0 & 0 \\\\ 0 & a ^ { \\ell _ 2 } & 0 \\\\ 0 & 0 & a ^ { \\ell _ 3 } \\end{array} \\right ) = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "7861.png", "formula": "\\begin{align*} \\mathbf { \\Phi } _ { \\mathbf { A } } = [ \\mathbf { s } _ { \\mathbf { A } } ^ { ( 0 ) } , \\mathbf { s } _ { \\mathbf { A } } ^ { ( 1 ) } , \\ldots , \\mathbf { s } _ { \\mathbf { A } } ^ { ( M - 1 ) } ] , \\end{align*}"}
+{"id": "1449.png", "formula": "\\begin{align*} y ^ 2 + x y = x ^ 3 + ( 1 / t ) x ^ 2 + 1 , \\end{align*}"}
+{"id": "2325.png", "formula": "\\begin{align*} \\mathcal { C } _ { q } = 3 ^ { - \\frac { 7 } { 4 } } q ^ { \\frac { 3 } { q } } \\left ( q - 2 \\right ) ^ { \\frac { 3 } { 2 q } } \\left ( \\frac { q } { q - 2 } \\right ) ^ { \\frac { 3 } { 4 } } ( 4 \\pi ( 1 - \\tau ) ) ^ { - \\frac { 3 } { 2 } ( \\frac { 1 } { 3 } - \\frac { 1 } { q } ) } e ^ { - 6 ( \\frac { 1 } { 3 } - \\frac { 1 } { q } ) } . \\end{align*}"}
+{"id": "3818.png", "formula": "\\begin{align*} \\lvert \\omega \\rvert & = ( 2 t ) / 2 \\sum _ { i = 0 } ^ { 2 t - 1 } n _ i ^ 2 + \\sum _ { i = 0 } ^ { 2 t - 1 } i n _ i = \\sum _ { i = 0 } ^ { t - 1 } ( ( 2 t ) n _ i ^ 2 + ( 2 ( i - t ) - 1 ) n _ i ) . \\end{align*}"}
+{"id": "7743.png", "formula": "\\begin{align*} \\nu ' = \\nu - 8 m ^ 2 \\tilde { M } ^ 2 \\epsilon ^ { \\frac { 1 } { m } } , \\ \\ \\zeta ' = \\zeta + 8 m ^ 2 \\tilde { M } ^ 2 \\epsilon ^ { \\frac { 1 } { m } } . \\end{align*}"}
+{"id": "2612.png", "formula": "\\begin{align*} s _ 4 - s _ 3 = s _ 3 - s _ 1 \\end{align*}"}
+{"id": "2301.png", "formula": "\\begin{align*} P _ r ( t ) = \\bigl [ ( 1 - x ^ 2 ) \\bigl ( 1 + t ^ 3 + ( t + t ^ 2 ) ( 1 + x ^ 2 + x ^ { - 2 } ) \\bigr ) ^ { r } \\bigr ] _ 0 = \\displaystyle \\sum _ { k = 0 } ^ r R _ k \\binom { r } { k } ( 1 + t ^ 3 ) ^ { r - k } ( t + t ^ 2 ) ^ k , \\end{align*}"}
+{"id": "1066.png", "formula": "\\begin{align*} \\| f _ { n , s } \\| _ { L } ^ 2 = \\int _ { 0 } ^ { \\infty } \\frac { d \\ell } { \\{ \\ell + ( s + 1 ) / n \\} ^ { 2 + 2 d } } = \\frac { 1 } { 1 + 2 d } \\left ( \\frac { n } { s + 1 } \\right ) ^ { 1 + 2 d } . \\end{align*}"}
+{"id": "1799.png", "formula": "\\begin{align*} \\partial _ t u + \\partial _ x f ( \\zeta , u ) = D _ \\zeta G ( \\zeta , u ) \\ , D \\zeta \\end{align*}"}
+{"id": "3761.png", "formula": "\\begin{align*} | t A + ( 1 - t ) B \\setminus S ^ 1 | & = \\int _ { x \\in \\mathbb { R } ^ { d - 1 } } | S _ x \\setminus S ^ 1 | \\\\ & \\geq t ^ { d - 1 } \\int _ { x \\in \\mathbb { R } ^ { d - 1 } } | S _ { t x } \\setminus S ^ 1 | \\\\ & \\geq t ^ { d - 1 } \\int _ { x \\in \\mathbb { R } ^ { d - 1 } } | t A _ x | \\\\ & = t ^ { d } \\int _ { x \\in \\mathbb { R } ^ { d - 1 } } | A _ x | \\\\ & = t ^ { d } | A | . \\end{align*}"}
+{"id": "5180.png", "formula": "\\begin{align*} A = \\frac { 1 } { \\beta ( \\beta - 1 ) } \\sum _ { i } p _ { i } ^ { \\beta } - \\frac { 1 } { \\beta - 1 } \\sum _ { i } p _ { i } q ^ { \\beta - 1 } _ { i } \\ ; \\ ; ; \\ ; \\ ; B = - \\frac { 1 } { \\beta } q ^ { \\beta } _ { i } \\end{align*}"}
+{"id": "1981.png", "formula": "\\begin{align*} - \\dfrac { 1 } { 2 N ^ 2 } \\left ( \\partial ^ 2 _ t g _ { i i } + 2 \\partial _ t N \\ , h _ { i i } \\right ) - \\frac { k } { 2 N ^ 2 } \\left ( g ^ { i j } \\partial ^ 2 _ t g _ { i j } + 2 \\partial _ t N \\ , H \\right ) & \\geq - ( 1 - q ) \\lambda _ i ^ 2 - ( 1 - q ) k \\sum _ i \\lambda _ i ^ 2 \\\\ & = - ( 1 - q ) \\lambda _ i ^ 2 - ( 1 - q ) k | h | ^ 2 \\ , . \\end{align*}"}
+{"id": "7258.png", "formula": "\\begin{align*} a = q , b = 0 c = \\tfrac { \\eta \\beta _ t } { \\eta \\beta _ t - ( 1 + \\eta x ) ( 1 - q ) ^ 2 } , \\end{align*}"}
+{"id": "8266.png", "formula": "\\begin{align*} & e ^ { C _ 2 s } D ^ 2 = \\exp \\left ( C _ 2 \\left ( \\log \\frac { 1 } { D } \\right ) ^ { \\alpha } \\right ) D ^ 2 \\\\ = & \\exp \\left ( C _ 2 \\left ( \\log \\frac { 1 } { D } \\right ) ^ { \\alpha } \\right ) e ^ { 2 \\log D } = \\exp \\left ( - 2 \\left ( \\log \\frac { 1 } { D } \\right ) + C _ 2 \\left ( \\log \\frac { 1 } { D } \\right ) ^ { \\alpha } \\right ) . \\end{align*}"}
+{"id": "8509.png", "formula": "\\begin{align*} \\pi _ r ( w _ { n _ k } ) = \\pi _ { r + l _ k } ( w _ { n _ k } ) = a , \\end{align*}"}
+{"id": "1903.png", "formula": "\\begin{align*} \\qquad \\mathcal { A } ( u | _ \\Omega ) = h _ { f , g } . \\end{align*}"}
+{"id": "1104.png", "formula": "\\begin{align*} \\log \\left ( 2 + | x _ 0 | + 2 ^ i r \\right ) \\leq \\log ( 2 + | x _ 0 | + r ) ^ { i + 1 } = ( i + 1 ) \\log ( 2 + | x _ 0 | + r ) . \\end{align*}"}
+{"id": "2147.png", "formula": "\\begin{align*} \\vartheta _ J = - 1 9 . 0 0 0 0 , u _ J = ( 1 . 5 0 0 0 , 0 . 0 0 0 0 ) , v _ J = ( 1 . 0 0 0 0 , 0 . 0 0 0 0 , 2 . 0 0 0 0 ) . \\end{align*}"}
+{"id": "8562.png", "formula": "\\begin{align*} F ( n , k ) = \\frac { ( a ) _ k ( b ) _ k } { ( n ) _ k ( 3 n ) _ k } \\end{align*}"}
+{"id": "5140.png", "formula": "\\begin{align*} \\frac { \\partial A } { \\partial q _ { j } } = \\frac { \\partial A ( p _ { j } , q _ { j } ) } { \\partial q _ { j } } \\ : \\ : \\ : \\ : ; \\ : \\ : \\ : \\ : \\frac { \\partial ( X + Y ) } { \\partial q _ { j } } = \\frac { \\partial X ( p _ { j } , q _ { j } ) } { \\partial q _ { j } } + \\frac { \\partial Y ( p _ { j } , q _ { j } ) } { \\partial q _ { j } } \\end{align*}"}
+{"id": "3032.png", "formula": "\\begin{align*} H = p _ s ^ T p _ s + V ( s , p _ x ) \\ , , V ( p _ x , s ) = p _ x ^ T { \\cal V } ^ { - 1 } p _ x \\ , , \\end{align*}"}
+{"id": "3066.png", "formula": "\\begin{align*} \\mu _ F = \\dim _ { \\mathbb { C } } \\frac { \\mathbb { C } \\{ x , y \\} } { \\langle F _ x , F _ y \\rangle } = \\sum _ { l = 1 } ^ { g } ( n _ l - 1 ) v _ l - v _ 0 + 1 . \\end{align*}"}
+{"id": "275.png", "formula": "\\begin{align*} \\ddot x + f ( x ) \\ , \\dot x + g ( x ) = 0 , \\end{align*}"}
+{"id": "2250.png", "formula": "\\begin{align*} \\small \\ddot { r } ( t ) & + [ ( 2 n - 2 p - 1 ) \\cot t - ( 2 p + 1 ) \\tan t ] \\ , \\dot { r } ( t ) \\\\ [ 1 e x ] & + \\left [ \\frac { p } { \\cos ^ 2 t } - \\frac { ( n - p - 1 ) } { \\sin ^ 2 t } \\right ] \\ , \\sin 2 r ( t ) - \\frac { \\sin 4 r ( t ) } { \\sin ^ 2 2 t } = 0 \\end{align*}"}
+{"id": "8361.png", "formula": "\\begin{align*} P ( w ) : = { \\rm S t a b } _ G ( X _ { w B } ) = \\{ g \\in G : \\ g X _ { w B } \\subseteq X _ { w B } \\} , \\end{align*}"}
+{"id": "8735.png", "formula": "\\begin{align*} \\eta _ x = \\left ( 1 - \\frac { p _ - ( x ) + p _ + ( x ) } { \\mu ( \\{ x \\} ) } \\right ) \\delta _ x + \\frac { 1 } { \\mu ( \\{ x \\} ) } \\int _ { F _ \\nu ( x - ) - p _ - ( x ) } ^ { F _ \\nu ( x - ) } \\delta _ { F _ \\nu ^ { - 1 } ( v ) } d v + \\frac { 1 } { \\mu ( \\{ x \\} ) } \\int _ { F _ \\nu ( x ) } ^ { F _ \\nu ( x ) + p _ + ( x ) } \\delta _ { F _ \\nu ^ { - 1 } ( v ) } d v . \\end{align*}"}
+{"id": "3219.png", "formula": "\\begin{align*} u ( T ) = 0 , \\end{align*}"}
+{"id": "9194.png", "formula": "\\begin{align*} e ^ + _ { 1 , j + 1 } ( z ) = ( 1 - q ^ 2 ) ^ { - 1 } \\cdot [ e ^ + _ { 1 j } ( z ) , e ^ { ( 0 ) } _ { j , j + 1 } ] _ q . \\end{align*}"}
+{"id": "417.png", "formula": "\\begin{align*} U _ N = \\max \\{ U _ R , U _ S \\} , ~ ~ U _ M = \\min \\{ U _ R , U _ S \\} . \\end{align*}"}
+{"id": "4926.png", "formula": "\\begin{align*} \\begin{gathered} \\mathcal { C O } ^ 0 \\circ \\mathcal { O C } ^ 0 ( 1 _ { L } ) = 2 p _ L , \\\\ \\mathcal { C O } ^ 0 \\circ \\mathcal { O C } ^ 0 ( p _ L ) = \\beta _ L 1 _ L . \\end{gathered} \\end{align*}"}
+{"id": "8554.png", "formula": "\\begin{align*} F ( n , k ) = \\frac { ( a ) _ { k } ( b ) _ { k } } { ( \\alpha _ { 1 } + \\beta _ { 1 } n ) _ { k } ( \\alpha _ { 2 } + \\beta _ { 2 } n ) _ { k } } \\end{align*}"}
+{"id": "1383.png", "formula": "\\begin{align*} I _ { \\lim } : = ( x _ { 2 } ^ { 4 } , \\ , x _ { 4 } ^ { 2 } , \\ , x _ { 2 } x _ { 4 } , \\ , x _ { 3 } x _ { 4 } , \\ , x _ { 1 } ^ { 2 } x _ { 4 } , \\ , x _ { 2 } x _ { 3 } - x _ { 1 } x _ { 4 } , \\ , x _ { 3 } ^ { 2 } , \\ , x _ { 1 } x _ { 3 } ) , \\end{align*}"}
+{"id": "831.png", "formula": "\\begin{align*} a ( X ) : = ( 1 / 2 ) ( \\nabla _ i X _ { j } - \\nabla _ j X _ { i } ) d x ^ { i } \\wedge d x ^ { j } . \\end{align*}"}
+{"id": "3308.png", "formula": "\\begin{align*} \\begin{cases} g _ 1 ( t , B _ t ) = - g ( t , B _ t ) \\frac { t + B _ t ^ 2 } { 2 t ^ 2 } , \\\\ g _ 2 ( t , B _ t ) = g ( t , B _ t ) \\frac { B _ t } { t } , \\\\ g _ { 2 2 } ( t , B _ t ) = g ( t , B _ t ) \\frac { t + B _ t ^ 2 } { t ^ 2 } . \\end{cases} \\end{align*}"}
+{"id": "6469.png", "formula": "\\begin{align*} m _ { \\lambda _ j } = { { p + j } \\choose { j } } - { { p + j - 2 } \\choose { j - 2 } } , j = 2 , \\ldots \\end{align*}"}
+{"id": "7362.png", "formula": "\\begin{align*} L ( v ) ( x , t ) : = \\frac { 1 } { 2 } \\int _ 0 ^ t d s \\int _ { x - t + s } ^ { x + t - s } v ( y , s ) d y . \\end{align*}"}
+{"id": "830.png", "formula": "\\begin{align*} \\nabla _ k ( \\nabla _ j X ^ { i } + C ^ { i } _ { j h } \\nabla _ 0 X ^ { h } ) + X ^ { h } R ^ { i } _ { j h k } = \\delta ^ { i } _ { j } \\Psi _ { k } + \\delta ^ { i } _ { k } \\Psi _ { j } + C ^ { i } _ { j k } \\Psi , \\end{align*}"}
+{"id": "2572.png", "formula": "\\begin{align*} X ( \\mathbf { a } ) X ( \\mathbf { b } ) = q ^ { \\frac { 1 } { 2 } \\Lambda ( \\mathbf { a } , \\mathbf { b } ) } X ( \\mathbf { a } + \\mathbf { b } ) , \\end{align*}"}
+{"id": "5519.png", "formula": "\\begin{align*} d _ { \\Omega } & \\geq \\deg ( G ) - ( 2 g - 2 ) + s - a \\\\ & = 2 a ( q ^ 2 - q - 1 ) - \\frac { q ^ 2 ( q ^ n - 2 q ^ { n - 1 } - q ^ { n - 2 } - q + 1 ) } { q + 1 } . \\end{align*}"}
+{"id": "8908.png", "formula": "\\begin{align*} \\sinh ( n \\theta ) = \\begin{cases} \\sinh \\theta V _ n ( 2 \\sinh \\theta ) & n : , \\\\ \\sinh 2 \\theta V _ n ( 2 \\sinh \\theta ) & n : . \\end{cases} \\end{align*}"}
+{"id": "2079.png", "formula": "\\begin{align*} & - \\int _ 0 ^ { \\frac { \\lfloor s T \\rfloor } { T } } \\Theta ( u , x ) f ' ( u ) \\mathrm { d } u + f ( s ) { \\Theta } ( \\frac { \\lfloor s T \\rfloor } { T } , x ) - f ( 0 ) { \\Theta } ( 0 , x ) + o ( 1 ) \\\\ & = - \\eta d T \\big ( \\int _ 0 ^ { \\frac { \\lfloor s T \\rfloor } { T } } \\int _ 0 ^ 1 f ( u ) A ( x , y ) \\Theta ( u , y ) \\mathrm { d } y \\mathrm { d } u + o ( 1 ) \\big ) + \\sigma ^ 2 \\eta ^ 2 T \\big ( \\int _ 0 ^ s f ( u ) \\mathrm { d } \\xi _ 1 ( u , x ) + o ( 1 ) \\big ) . \\end{align*}"}
+{"id": "4023.png", "formula": "\\begin{align*} \\lim _ { m \\rightarrow + \\infty } \\frac { x _ { 2 } ^ { \\left ( t _ { 0 } + m \\right ) } } { x _ { 1 } ^ { \\left ( t _ { 0 } + m \\right ) } } = \\begin{cases} 0 & \\mbox { i f } \\gamma _ { 2 } = 0 , \\delta _ { 1 } \\neq 0 , \\\\ \\frac { x _ { 2 } ^ { \\left ( t _ { 0 } \\right ) } } { x _ { 1 } ^ { \\left ( t _ { 0 } \\right ) } } & \\mbox { i f } \\gamma _ { 2 } = \\delta _ { 1 } = 0 , \\\\ + \\infty & \\mbox { i f } \\gamma _ { 2 } \\neq 0 , \\delta _ { 1 } = 0 . \\end{cases} \\end{align*}"}
+{"id": "3485.png", "formula": "\\begin{align*} d f = \\langle \\nabla f ( x ) , d x \\rangle , \\ d ^ 2 f = \\langle \\nabla ^ 2 f ( x ) d x _ 2 , d x _ 1 \\rangle \\end{align*}"}
+{"id": "9177.png", "formula": "\\begin{align*} F = \\frac { f ( \\{ x _ { i , r } \\} _ { i \\in I } ^ { 1 \\leq r \\leq k _ { i } } ) } { \\prod _ { i < j } ^ { a _ { i j } \\neq 0 } \\prod _ { 1 \\leq r \\leq k _ { i } } ^ { 1 \\leq s \\leq k _ { j } } ( x _ { i , r } - x _ { j , s } ) } , \\end{align*}"}
+{"id": "6254.png", "formula": "\\begin{align*} P ( - i D ) E = \\delta . \\end{align*}"}
+{"id": "8311.png", "formula": "\\begin{align*} \\{ X , \\alpha \\wedge \\beta \\} & = \\{ X , \\alpha \\} \\beta + ( - 1 ) ^ { k } \\alpha \\{ X , \\beta \\} \\\\ & = ( \\iota _ X \\alpha ) \\beta + ( - 1 ) ^ { k } \\alpha ( \\iota _ X \\beta ) \\\\ & = \\iota _ X ( \\alpha \\wedge \\beta ) , \\end{align*}"}
+{"id": "7139.png", "formula": "\\begin{align*} \\widehat { A } _ \\mu ( k ) = \\sum _ \\lambda \\frac { \\varepsilon _ \\mu ( k , \\lambda ) } { \\sqrt { 2 \\omega V } } \\Bigl ( a _ \\lambda ( k ) e ^ { i \\ , k x } + a ^ * _ \\lambda ( k ) e ^ { - i \\ , k x } \\Bigr ) \\end{align*}"}
+{"id": "7969.png", "formula": "\\begin{align*} H ^ { \\frac { 1 } { 2 } } \\Lambda ^ { 0 } ( \\partial \\Omega ) = H ^ { \\frac { 1 } { 2 } } \\Lambda ^ { 0 } ( \\Sigma ) \\oplus H _ { 0 0 } ^ { \\frac { 1 } { 2 } } \\Lambda ^ { 0 } ( \\Gamma ) . \\end{align*}"}
+{"id": "3608.png", "formula": "\\begin{align*} \\alpha ^ 3 \\tau _ 0 ^ 2 \\tau _ 1 = ( 1 + a ) \\alpha ^ 2 \\tau _ 0 \\tau _ 1 - ( a + b ) \\alpha \\tau _ 0 + b . \\end{align*}"}
+{"id": "9112.png", "formula": "\\begin{align*} A ( \\xi ) [ V ] = \\ker L ( \\xi ) \\ ; . \\end{align*}"}
+{"id": "6341.png", "formula": "\\begin{align*} \\partial E : = \\{ ( 1 + \\rho ( \\varphi ) ) \\varphi : \\varphi \\in S ^ { n - 1 } \\} , \\end{align*}"}
+{"id": "7507.png", "formula": "\\begin{align*} | \\tilde { \\nabla } l _ s | _ { \\tilde { g } } ( x , t ) = | \\nabla ^ { g _ 0 ( t / s ) } l | _ { g _ 0 ( t / s ) } ( \\phi _ s ( \\Phi _ s ^ { - 1 } ( x ) ) ) = 1 , \\end{align*}"}
+{"id": "6375.png", "formula": "\\begin{align*} \\frac { \\phi ' ( s ) } { ( n \\phi ( s ) ) ^ { ( n - 1 ) / n } } \\leq \\cosh ( s ) ^ { d - 1 + ( n - 1 ) / n } = \\cosh ( s ) ^ { d - 1 / n } . \\end{align*}"}
+{"id": "935.png", "formula": "\\begin{align*} \\mu = f \\cdot m + \\nu \\end{align*}"}
+{"id": "5351.png", "formula": "\\begin{align*} b _ i ^ { S \\cup \\{ j \\} } & = b _ i ^ S + w ^ S _ j \\ , x _ { i j } ^ { 1 , S \\cup \\{ j \\} } , j \\in N ^ { \\{ 0 , 1 \\} } \\setminus S \\\\ b _ i ^ S & = b _ i ^ { S \\setminus \\{ j \\} } + w ^ S _ j \\ , x _ { i j } ^ { 0 , S \\setminus \\{ j \\} } , j \\in S . \\end{align*}"}
+{"id": "8775.png", "formula": "\\begin{align*} \\int _ { \\phi _ \\uparrow ( F _ \\mu ( x - ) ) } ^ { \\phi _ \\uparrow ( F _ \\mu ( x ) ) } \\left ( F _ { \\tilde \\nu _ l } ^ { - 1 } \\left ( \\frac w { \\nu _ l ( \\R ) } \\right ) - x \\right ) d w + \\int _ { F _ \\mu ( x - ) - \\phi _ \\uparrow ( F _ \\mu ( x - ) ) } ^ { F _ \\mu ( x ) - \\phi _ \\uparrow ( F _ \\mu ( x ) ) } \\left ( F _ { \\tilde \\nu _ r } ^ { - 1 } \\left ( \\frac w { \\nu _ r ( \\R ) } \\right ) - x \\right ) d w = 0 . \\end{align*}"}
+{"id": "8744.png", "formula": "\\begin{align*} \\pi \\left ( \\bigcup _ { x \\in A } \\{ x \\} \\times \\left \\{ \\Gamma _ x \\cap \\{ ( - \\infty , x ) \\cup ( x , + \\infty ) \\} \\right \\} \\right ) = \\int _ { x \\in A } \\pi _ x \\left ( \\Gamma _ x \\cap \\{ ( - \\infty , x ) \\cup ( x , + \\infty ) \\} \\right ) \\mu ( d x ) = 0 , \\end{align*}"}
+{"id": "2305.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u + u \\cdot \\nabla u + \\nabla p = 0 , \\\\ \\mathrm { d i v } u = 0 , \\end{cases} \\end{align*}"}
+{"id": "2754.png", "formula": "\\begin{align*} & \\omega _ { \\Xi , \\Psi } ( t ) : = d \\Xi ^ { \\alpha } ( t ) \\wedge d \\Psi _ { \\alpha } ( t ) = d \\Xi ^ { \\alpha } ( t ) \\wedge 0 , \\\\ & \\omega _ { Q , P } ( t ) : = d Q ^ { a } ( t ) \\wedge d P _ { a } ( t ) . \\end{align*}"}
+{"id": "3823.png", "formula": "\\begin{align*} P ( q ) : = \\sum _ { n = 0 } ^ \\infty p ( n ) q ^ n = \\prod _ { n = 0 } ^ \\infty \\frac { 1 } { 1 - q ^ n } = \\frac { 1 } { ( q ; q ) _ \\infty } , \\end{align*}"}
+{"id": "3892.png", "formula": "\\begin{align*} \\phi _ 1 - \\phi _ 2 & = \\begin{dcases} \\kappa \\circ f - \\kappa & f , \\\\ \\left . \\frac { d } { d t } \\right | _ { t = 0 } ( \\kappa \\circ f ^ t ) & f ^ t . \\end{dcases} \\end{align*}"}
+{"id": "9059.png", "formula": "\\begin{align*} f ^ * ( x ) = \\int _ { 0 } ^ { \\infty } \\chi _ { \\{ | f | > t \\} } ^ * ( x ) d t , \\end{align*}"}
+{"id": "1807.png", "formula": "\\begin{align*} ( \\sigma _ 1 , \\ldots , \\sigma _ n ) = E ( z ^ + , z ^ - , u _ r , u _ \\ell ) \\ , . \\end{align*}"}
+{"id": "7266.png", "formula": "\\begin{align*} x w _ n ( x ) = w _ { n + 1 } ( x ) + ( \\widetilde { A } _ n + \\widetilde { C } _ n ) w _ n ( x ) + \\widetilde { A } _ { n - 1 } \\widetilde { C } _ n w _ { n - 1 } ( x ) , n \\geq 0 , \\end{align*}"}
+{"id": "4600.png", "formula": "\\begin{align*} N \\left ( - \\Delta ^ N _ \\Omega + \\lambda V \\right ) = ( 2 \\pi ) ^ { - d } \\left | B _ 1 ( 0 ) \\right | \\lambda ^ { \\frac { d } { 2 } } \\int _ \\Omega | V | ^ { \\frac { d } { 2 } } + o \\left ( \\lambda ^ { \\frac { d } { 2 } } \\right ) \\mathrm { \\ a s \\ } \\lambda \\rightarrow \\infty . \\end{align*}"}
+{"id": "8512.png", "formula": "\\begin{align*} H _ { l _ k } [ a ] = H _ { l _ k } [ G _ { r + l _ k , n _ k } [ b ] ] = G _ { r , n _ k } [ b ] = a , \\end{align*}"}
+{"id": "7786.png", "formula": "\\begin{align*} M T ( c ) = F \\times [ 0 , 1 ] / ( f , 1 ) \\sim ( c ( f ) , 0 ) , ~ \\forall { f } \\in { F } . \\end{align*}"}
+{"id": "8580.png", "formula": "\\begin{align*} C _ t ^ { t _ 1 } = ( J _ k + [ - \\varepsilon _ k ( G ^ { t _ 0 } _ t ) B _ { t _ 0 } ] ^ { k \\cdot } _ + ) C ^ { t _ 0 } _ t , \\end{align*}"}
+{"id": "2552.png", "formula": "\\begin{align*} C ^ { - \\infty } ( M , L ) = C ^ \\infty ( M , L ; \\Omega ) ' \\ ; , C ^ { \\prime \\ , - k } ( M , L ) = C ^ k ( M , L ; \\Omega ) ' \\ ; . \\end{align*}"}
+{"id": "8004.png", "formula": "\\begin{align*} I _ { d y n } ( W ) = \\mathcal { I } _ 1 \\left ( W , F _ W , G _ W , H _ W \\right ) , \\end{align*}"}
+{"id": "8187.png", "formula": "\\begin{align*} I ( { \\bf X } ^ { 2 } ; Y ^ { 2 } | { \\bf S } ) = H ( Y _ 1 | { \\bf S } ) + H ( Y _ 2 | Y _ 1 , { \\bf S } ) . \\end{align*}"}
+{"id": "6981.png", "formula": "\\begin{align*} 1 = z _ { n + 1 } \\overline { \\eta } + \\overline { z _ { n + 1 } } \\eta . \\end{align*}"}
+{"id": "6065.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k + 1 } \\lambda _ i L _ i ^ d + \\sum _ { i = 1 } ^ { k + 1 } \\lambda _ i l _ i g _ i = 0 . \\end{align*}"}
+{"id": "4078.png", "formula": "\\begin{align*} e ^ { - i z _ 0 s } \\tilde { E } _ i ( i z _ 0 s ) = e ^ { i z s } E _ i ( - i z s ) , \\ , \\ , \\ , \\ , \\ e ^ { i z _ 0 s } \\tilde { E } _ i ( - i z _ 0 s ) = e ^ { - i z s } E _ i ( i z s ) + e ^ { - i z s } 2 \\pi i \\end{align*}"}
+{"id": "2790.png", "formula": "\\begin{align*} \\frac { \\partial L _ { T } } { \\partial \\Xi ^ { \\alpha } } = - \\frac { \\partial H _ { T } } { \\partial \\Xi ^ { \\alpha } } = 0 \\end{align*}"}
+{"id": "7758.png", "formula": "\\begin{align*} \\xi _ { n } = \\# \\{ i : s _ { i } \\leq n \\} . \\end{align*}"}
+{"id": "367.png", "formula": "\\begin{align*} K _ p & = - \\sum _ { i = 1 } ^ { p - 1 } \\dfrac { 1 } { i } \\binom { p - 1 } { i - 1 } b ^ { i - 1 } \\bigg ( \\sum _ { j = 1 } ^ { p - i - 1 } ( - 1 ) ^ j \\binom { p - i } { j } x ^ { p - i - j } a ^ { j - 1 } \\bigg ) \\\\ & - \\sum _ { i = 1 } ^ { p - 1 } \\dfrac { ( - 1 ) ^ i } { i } \\binom { p - 1 } { i - 1 } b ^ { p - i - 1 } a ^ { i - 1 } . \\end{align*}"}
+{"id": "5624.png", "formula": "\\begin{align*} \\sigma ( x _ 0 , x _ 1 , x _ 2 , \\ldots ) = ( x _ 1 , x _ 2 , x _ 3 , \\ldots ) . \\end{align*}"}
+{"id": "6225.png", "formula": "\\begin{align*} \\begin{cases} \\dot { w } = h _ 3 - c - D _ 3 g _ 3 / w \\ & \\mbox { i n } \\ ( \\gamma , \\beta ) , \\\\ w < 0 \\ & \\mbox { i n } \\ [ \\gamma , \\beta ) , \\\\ w ( \\beta ) = 0 . \\end{cases} \\end{align*}"}
+{"id": "8261.png", "formula": "\\begin{align*} & H ( Y _ i | Y ^ { i - 1 } , \\underline S ) \\\\ = & \\sum _ { \\underline s } \\sum _ { y ^ { i - 1 } } P _ { Y ^ { i - 1 } , \\underline S } ( y ^ { i - 1 } , \\underline s ) H ( P _ { Y _ i | Y ^ { i - 1 } , \\underline S } ( 1 | y ^ { i - 1 } , \\underline s ) ) . \\end{align*}"}
+{"id": "815.png", "formula": "\\begin{align*} 2 { \\pounds } _ { \\hat { X } } { E _ { j l } } = { \\pounds } _ { \\hat { X } } G _ { j i l } ^ i = { \\Psi _ { i . j . l } } { y ^ i } + { \\Psi _ { i . j } } \\delta _ l ^ i + { \\Psi _ { i . l } } \\delta _ j ^ i + { \\Psi _ { j . l } } \\delta _ i ^ i , \\end{align*}"}
+{"id": "3864.png", "formula": "\\begin{align*} \\hat M ^ 3 ( t ) = M ^ 1 ( T ) + \\int _ 0 ^ t \\hat \\eta _ { ( 1 ) } ^ 3 ( s ) d s - \\int _ 0 ^ t \\hat M ^ { 3 } ( s ) d s , \\ ; t \\in [ 0 , T ] , \\ ; \\ ; \\hat M ^ 3 ( T ) = m _ 3 . \\end{align*}"}
+{"id": "5234.png", "formula": "\\begin{align*} L _ { d } D A H ( p \\| q ) = \\frac { \\left ( M A \\right ) ^ { a - 1 } - \\left ( M A \\right ) ^ { b - 1 } } { a - b } - \\left [ \\frac { \\left ( M H \\right ) ^ { a - 1 } - \\left ( M H \\right ) ^ { b - 1 } } { a - b } \\right ] \\end{align*}"}
+{"id": "804.png", "formula": "\\begin{align*} { R } i c _ { i j } = { \\cal R } i c \\ g _ { i j } . \\end{align*}"}
+{"id": "1808.png", "formula": "\\begin{align*} \\phi ( \\pmb \\alpha , \\pmb \\beta ) = 0 \\mbox { f o r a l l } \\pmb \\alpha , \\pmb \\beta \\mbox { w i t h } \\mathcal { A } _ { \\pmb \\alpha , \\pmb \\beta } = \\emptyset \\ , . \\end{align*}"}
+{"id": "1901.png", "formula": "\\begin{align*} \\sum _ { y \\sim x } \\omega ( x , y ) u ( y ) = m . \\end{align*}"}
+{"id": "4664.png", "formula": "\\begin{align*} \\mathcal H ^ 3 _ g \\left ( \\Omega _ { \\epsilon ^ { \\gamma ^ k } , \\epsilon } \\right ) \\leq \\Lambda _ 0 \\sum _ { i = 1 } ^ { k - 1 } \\epsilon ^ { \\gamma ^ i ( 2 - \\gamma ) } < \\Lambda _ 0 \\epsilon ^ { \\gamma ( 2 - \\gamma ) } \\left ( 1 - \\epsilon ^ { ( \\gamma - 1 ) ( 2 - \\gamma ) } \\right ) ^ { - 1 } , \\end{align*}"}
+{"id": "7434.png", "formula": "\\begin{align*} \\dd X _ t = - \\theta X _ t \\ , \\dd t + \\sigma \\ , \\dd W _ t , \\end{align*}"}
+{"id": "4003.png", "formula": "\\begin{align*} \\bigl | \\varpi \\circ W ^ { t } \\left ( z \\right ) - \\varpi \\left ( z ^ { * } \\right ) \\bigr | \\leq \\sum _ { i = 1 } ^ { n + \\nu } \\bigl | x _ { i } ^ { \\ : \\left ( t \\right ) } - x _ { i } ^ { * } \\bigr | = \\left \\Vert W ^ { t } \\left ( z \\right ) - z ^ { * } \\right \\Vert , \\end{align*}"}
+{"id": "7130.png", "formula": "\\begin{align*} V ( u _ { t } ) = V ( u ) - c ^ { 2 } \\ln t \\rightarrow + \\infty \\ \\ t \\rightarrow 0 . \\end{align*}"}
+{"id": "4236.png", "formula": "\\begin{align*} & Q ( X , \\tau ) = \\left [ \\widehat { A } ( T X ) { \\rm c h } ( \\triangle ( X ) ) + 3 2 \\widehat { A } ( T X ) \\right ] ^ { ( 8 ) } \\\\ & + \\left [ 2 \\widehat { A } ( T X ) { \\rm c h } ( \\triangle ( X ) ) { \\rm c h } ( \\widetilde { T X } ) + 3 2 \\widehat { A } ( T X ) { \\rm c h } ( \\widetilde { T X } + \\wedge ^ 2 \\widetilde { T X } ) \\right ] ^ { ( 8 ) } q + O ( q ^ { \\frac { 3 } { 2 } } ) . \\end{align*}"}
+{"id": "4564.png", "formula": "\\begin{align*} \\nabla _ { \\alpha } R _ { \\beta \\gamma } & = \\nabla _ \\beta R _ { \\alpha \\gamma } + O _ g ' ( r _ q ) \\\\ & = \\nabla _ \\beta R _ { \\mu \\nu } J ^ \\mu _ \\alpha J ^ \\nu _ \\gamma + O _ g ' ( r _ q ) \\\\ & = \\nabla _ { \\mu } R _ { \\beta \\nu } J _ { \\alpha } ^ { \\mu } J _ { \\gamma } ^ { \\nu } + O _ g ' ( r _ q ) , \\end{align*}"}
+{"id": "4355.png", "formula": "\\begin{gather*} [ \\overline { u } _ i , \\overline { u } _ i + \\Delta u _ i ] = \\{ u _ i \\in \\R \\colon ( - 2 \\overline { u } _ i - \\Delta u _ i ) u _ i + u _ i ^ 2 \\leq - \\overline { u } _ i ^ 2 - \\overline { u } _ i \\cdot \\Delta u _ i \\} . \\end{gather*}"}
+{"id": "1893.png", "formula": "\\begin{align*} g ( v ) = \\min \\{ g ( v ) , M \\} + ( \\max \\{ g ( v ) , M \\} - M ) , \\end{align*}"}
+{"id": "8336.png", "formula": "\\begin{align*} \\Pi _ t : = ( \\psi _ t ) _ { * } \\Pi _ 0 = \\Pi + ( W _ t ) ^ { \\gamma } \\end{align*}"}
+{"id": "370.png", "formula": "\\begin{align*} ( x - a ) ^ p = f _ p ( x , b , a ) + g _ p ( b , a ) \\textnormal { a n d } K _ p = \\dfrac { f _ p ( x , b , a ) } { p a b } + \\dfrac { g _ p ( b , a ) } { p a b } . \\end{align*}"}
+{"id": "1359.png", "formula": "\\begin{align*} E ( A _ \\alpha ( G ) ) \\geq \\begin{cases} ( 1 - \\alpha ) n , \\ , \\ , \\ , \\ , s _ n = 0 \\ , \\ , \\ , \\ , G \\cong K _ { \\frac { n } { 2 } , \\frac { n } { 2 } } \\\\ 2 ( 1 - \\alpha ) n r \\dfrac { \\sqrt { s _ n } } { r + s _ n } , \\ , \\ , \\ , \\ , s _ n > 0 . \\end{cases} \\end{align*}"}
+{"id": "1057.png", "formula": "\\begin{align*} S _ { 1 , k } ( n , s , t ) & : = \\sum _ { \\ell = 0 } ^ { \\infty } \\frac { 1 } { ( s + \\ell + 2 ) ^ { 1 + d } ( t + 1 ) ^ d } \\Vert \\tilde { b } ^ k _ { n , 1 , \\ell } \\Vert , \\\\ S _ { 2 , k } ( n , s , t ) & : = \\sum _ { \\ell = 0 } ^ { \\infty } \\sum _ { u = 0 } ^ { t - 1 } \\frac { 1 } { ( s + \\ell + 2 ) ^ { 1 + d } ( u + 1 ) ^ d } \\Vert \\tilde { b } ^ k _ { n , t - u , \\ell } - \\tilde { b } ^ k _ { n , t + 1 - u , \\ell } \\Vert . \\end{align*}"}
+{"id": "169.png", "formula": "\\begin{align*} X _ 2 \\ = \\ X _ 1 \\times \\{ Y _ x \\} \\ = \\ & \\ \\bigcup _ { x \\in X } \\ \\{ x \\} \\times Y _ x , \\\\ R _ 2 \\ = \\ R _ 1 \\ltimes \\{ S _ x \\} \\ & \\ ( x _ 1 , y _ 1 ) , ( x _ 2 , y _ 2 ) \\in X _ 2 , \\\\ ( ( x _ 1 , y _ 1 ) , ( x _ 2 , y _ 2 ) ) \\in R _ 2 \\Longleftrightarrow & \\begin{cases} ( x _ 1 , x _ 2 ) \\in R _ 1 ^ { \\circ } \\\\ x _ 1 = x _ 2 \\ \\ ( y _ 1 , y _ 2 ) \\in S _ { x _ 1 } . \\end{cases} \\end{align*}"}
+{"id": "7224.png", "formula": "\\begin{align*} \\int _ { V _ { \\min } } ^ { V _ { F } } \\left ( \\partial _ { t } p + \\partial _ { v } ( h p ) - a \\partial _ { v v } p \\right ) \\phi d v = 0 . \\end{align*}"}
+{"id": "7911.png", "formula": "\\begin{align*} \\mathrm { t r } ( \\alpha ) ( X _ { 1 } , \\ldots , X _ { k } ) : = \\alpha ( X _ { 1 } ^ { \\parallel } , \\ldots , X _ { k } ^ { \\parallel } ) , \\end{align*}"}
+{"id": "1100.png", "formula": "\\begin{align*} \\int _ 0 ^ r h ( t ) t ^ { a + n - 1 } [ \\log ( 2 + t ) ] ^ b \\ , d t = \\frac { 1 } { a + n } r ^ { a + n } [ \\log ( 2 + r ) ] ^ b , \\end{align*}"}
+{"id": "874.png", "formula": "\\begin{align*} T _ 2 = w ^ 1 C _ { 1 2 } + w ^ 2 C _ { 2 2 } = w ^ 1 ( { w ^ 1 } _ { \\substack { \\\\ x _ 2 } } - { w ^ 2 } _ { \\substack { \\\\ x _ 1 } } ) . \\end{align*}"}
+{"id": "5287.png", "formula": "\\begin{align*} L _ { d } K L ( q \\| p ) = \\frac { 1 } { a - b } \\sum _ { i } \\left [ \\frac { q _ { i } ^ { a } } { p _ { i } ^ { a - 1 } } - \\frac { q _ { i } ^ { b } } { p _ { i } ^ { b - 1 } } \\right ] + \\sum _ { i } p _ { i } - \\sum _ { i } q _ { i } \\end{align*}"}
+{"id": "8139.png", "formula": "\\begin{align*} & c _ { 0 1 } ^ 1 = 1 , c _ { 0 2 } ^ 2 = 1 , c _ { 0 7 } ^ 7 = - 1 , c _ { 0 8 } ^ 8 = - 1 \\ , \\\\ & c _ { 1 3 } ^ 1 = 1 , c _ { 1 5 } ^ 2 = 1 , c _ { 1 7 } ^ 3 = 1 , c _ { 1 7 } ^ 0 = - 1 , c _ { 1 8 } ^ 4 = 1 \\ , \\\\ & c _ { 2 4 } ^ 1 = 1 , c _ { 2 6 } ^ 2 = 1 , c _ { 2 7 } ^ 5 = 1 , c _ { 2 8 } ^ 6 = 1 , c _ { 2 8 } ^ 0 = - 1 \\ , \\\\ & c _ { 3 4 } ^ 4 = - 1 , c _ { 3 5 } ^ 5 = 1 , c _ { 3 7 } ^ 7 = 1 \\ , \\\\ & c _ { 4 5 } ^ 3 = - 1 , c _ { 4 5 } ^ 6 = 1 , c _ { 4 6 } ^ 4 = - 1 , c _ { 4 7 } ^ 8 = 1 \\ , \\\\ & c _ { 5 6 } ^ 5 = 1 , c _ { 5 8 } ^ 7 = 1 \\ , \\\\ & c _ { 6 8 } ^ 8 = 1 \\ . \\end{align*}"}
+{"id": "1946.png", "formula": "\\begin{align*} \\left | B _ { 2 R } \\cap \\left \\{ \\left ( w - 2 ^ { - k + 1 } t \\right ) _ { - } \\leq 2 ^ { - k } t \\right \\} \\right | = \\left | B _ { 2 R } \\cap \\left \\{ w \\geq 2 ^ { - k } t \\right \\} \\right | \\geq \\left | B _ { 2 R } \\cap \\{ w \\geq t \\} \\right | \\geq \\nu \\left | B _ { 2 R } \\right | \\end{align*}"}
+{"id": "4292.png", "formula": "\\begin{align*} N ^ p _ { \\beta } ( \\mathbf { x } ) : = \\max \\{ m \\ , : \\ , \\tau _ m < 1 \\} . \\end{align*}"}
+{"id": "436.png", "formula": "\\begin{align*} \\begin{aligned} & \\min _ { x \\in \\Re ^ n } \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ n ( x _ i - y _ i ) ^ 2 , \\\\ & x _ { 1 } \\le x _ { 2 } \\le \\ldots \\le x _ { n } , \\end{aligned} \\end{align*}"}
+{"id": "6703.png", "formula": "\\begin{align*} \\dim ( G ) = \\log _ p \\lvert \\Omega _ { \\{ 1 \\} } ( P ) \\rvert - s ( G ) = \\dd ( P ) - s ( G ) = \\dd ( G ) - s ( G ) = \\dd ( G ) - \\dd ( T ) . \\end{align*}"}
+{"id": "3330.png", "formula": "\\begin{align*} \\Q _ { z e r o } & = \\lbrace ( x , t ) : | x _ j | \\leq \\eta ^ { \\alpha _ j } , j = 1 , \\ldots , N , - 1 - \\eta ^ 2 < t \\leq - 1 \\rbrace , \\\\ \\Q _ { e x t } & = \\lbrace ( x , t ) : | x _ j | \\leq 2 ^ { \\alpha _ j } R , j = 1 , \\ldots , N , - 1 - \\eta ^ 2 < t \\leq 0 \\rbrace , \\end{align*}"}
+{"id": "8212.png", "formula": "\\begin{align*} a _ i ( { \\bf x } _ i ) = P _ { { \\bf X } _ i | Y ^ i } ( { \\bf x } _ i | y ^ { i } ) . \\end{align*}"}
+{"id": "1347.png", "formula": "\\begin{align*} E ( A _ \\alpha ( G ) ) ^ 2 = \\left ( \\sum _ { i = 1 } ^ { n } s _ i \\right ) ^ 2 = \\sum _ { i = 1 } ^ { n } s _ i ^ 2 + 2 \\sum _ { 1 \\leq i \\leq j \\leq n } s _ i s _ j \\end{align*}"}
+{"id": "9030.png", "formula": "\\begin{align*} \\varphi _ 1 ( a ) = a \\quad & \\varphi _ 1 ( y ) = z \\\\ \\varphi _ 2 ( b ) = b \\quad & \\varphi _ 2 ( y ) = z . \\end{align*}"}
+{"id": "3200.png", "formula": "\\begin{align*} f _ { X Y } \\circ \\pi _ { X ' X } = \\pi _ { Y ' Y } \\circ f _ { X ' Y ' } , \\end{align*}"}
+{"id": "6818.png", "formula": "\\begin{align*} \\sum _ { i , j } \\beta _ { i j } k _ { i j } ^ N + K _ N = 0 , \\qquad \\sum _ { i j } \\beta _ { i j } l _ { i j } ^ { N - 1 } + L _ { N - 1 } = 0 . \\end{align*}"}
+{"id": "8871.png", "formula": "\\begin{align*} \\kappa < \\frac { 4 ( n - 1 ) ( n - 2 ) } { 5 n - 4 + \\sqrt { 9 n ^ 2 + 8 n - 1 6 } } & = \\frac { 4 ( n - 1 ) ( n - 2 ) \\big \\{ 5 n - 4 - \\sqrt { 9 n ^ 2 + 8 n - 1 6 } \\big \\} } { ( 5 n - 4 ) ^ 2 - 9 n ^ 2 - 8 n + 1 6 } \\\\ & = \\frac { 5 n - 4 - \\sqrt { 9 n ^ 2 + 8 n - 1 6 } } { 4 } . \\end{align*}"}
+{"id": "7751.png", "formula": "\\begin{align*} S ^ { - 1 } ( A + G ) S = \\tilde { A } , \\end{align*}"}
+{"id": "8381.png", "formula": "\\begin{align*} S _ { 2 } : = \\{ \\beta \\in R ^ { + } ( w ^ { - 1 } ) : s u p p ( \\beta ) \\nsubseteq J \\} = R ^ { + } ( w ^ { - 1 } ) \\setminus S _ { 1 } = \\{ \\beta _ { k + 1 } , \\ldots , \\beta _ { r } \\} . \\end{align*}"}
+{"id": "4055.png", "formula": "\\begin{align*} R _ 0 ( z ) ( x , y ) = \\sum _ { k = 0 } ^ { \\frac { d - 3 } { 2 } } \\frac { \\tilde { c _ k } } { | x - y | ^ { d - 1 - s _ k } } \\partial _ s ^ { s _ k } \\Big \\{ e ^ { i z s } E _ i ( - i z s ) + e ^ { - i z s } E _ i ( - i z s ) + 2 \\pi i e ^ { i z s } \\Big \\} \\Big | _ { ( s = | x - y | ) } \\end{align*}"}
+{"id": "5144.png", "formula": "\\begin{align*} \\frac { \\partial A ( p , q ) } { \\partial q _ { j } } = \\frac { \\partial A ( p _ { j } , q _ { j } ) } { \\partial q _ { j } } = \\frac { \\partial A _ { j } } { \\partial q _ { j } } \\end{align*}"}
+{"id": "3711.png", "formula": "\\begin{align*} H = \\int _ { \\mathcal { M } } d _ g ^ 2 x : \\phi ^ \\dagger ( - \\nabla ^ 2 _ g + m ^ 2 ) \\phi + \\pi ^ 2 : \\ ; - \\ , \\alpha \\ , \\phi ^ { ( - ) } ( a ) \\phi ^ { ( + ) } ( a ) \\ ; , \\end{align*}"}
+{"id": "7277.png", "formula": "\\begin{align*} y \\widetilde { Q } _ n ( y ; x , t , s ) = \\widetilde { Q } _ { n + 1 } ( y ; x , t , s ) + \\mathcal { \\widetilde { A } } _ { n } ( x , & t , s ) \\widetilde { Q } _ n ( y ; x , t , s ) \\\\ & + \\mathcal { \\widetilde { B } } _ { n } ( x , t , s ) \\widetilde { Q } _ { n - 1 } ( y ; x , t , s ) , n \\geq 0 , \\end{align*}"}
+{"id": "5002.png", "formula": "\\begin{align*} \\tilde a ( x ) = \\prod _ { k = 1 } ^ { N } \\left ( q x - q ^ { - 1 } y _ k \\right ) , \\tilde d ( x ) = \\prod _ { k = 1 } ^ { N } \\left ( x - y _ k \\right ) . \\end{align*}"}
+{"id": "650.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial a } \\cosh ( \\ell ( a , c ) ) & = \\frac { \\sinh ( a - c ) } { 2 \\sinh ^ 2 a } \\left ( 2 \\sinh ^ 2 a - \\sinh ^ 2 c - \\cosh ( a - c ) \\sinh a \\sinh c \\right ) \\\\ & > \\frac { \\sinh ( a - c ) } { 2 \\sinh a } \\left ( \\sinh a - \\cosh ( a - c ) \\sinh c \\right ) \\\\ & = \\frac { \\sinh ^ 2 ( a - c ) \\cosh c } { 2 \\sinh a } > 0 , \\end{align*}"}
+{"id": "1023.png", "formula": "\\begin{align*} \\lambda ( \\gamma ) = \\sum _ { \\beta \\in \\Gamma ^ { t _ \\lambda } ( n ) } \\lambda ( \\beta ) \\ , p _ { t _ \\lambda } ( \\beta , \\gamma ) \\end{align*}"}
+{"id": "8390.png", "formula": "\\begin{align*} \\Delta ' ( \\phi ( a ) ) = \\phi ( a _ 1 ) \\otimes \\phi ( a _ 2 ) , \\epsilon ' ( \\phi ( a ) ) = \\epsilon ( a ) \\end{align*}"}
+{"id": "3952.png", "formula": "\\begin{align*} \\begin{bmatrix} - \\lambda _ { N + 1 } D + \\left ( Q \\right ) + \\frac { \\alpha _ 0 \\ell _ 1 } { 2 } I _ 3 & I _ 3 & I _ 3 \\\\ * & - 2 \\alpha _ 0 I _ 3 & 0 \\\\ * & * & - 2 \\alpha _ 1 I _ 3 \\end{bmatrix} \\prec 0 . \\end{align*}"}
+{"id": "5213.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } D _ { \\alpha \\beta } I ( p \\| q ) } { \\partial q _ { j } } = - \\frac { 1 } { \\alpha } \\left [ \\left ( \\frac { a - 1 } { a - b } \\right ) ( X . Y ) ^ { a - 2 } - \\left ( \\frac { b - 1 } { a - b } \\right ) ( X . Y ) ^ { b - 2 } \\right ] \\left [ M - N \\right ] \\end{align*}"}
+{"id": "5566.png", "formula": "\\begin{align*} \\Phi _ { x y } = ( Q Q ^ * ) _ { x y } = \\sum _ { z } Q _ { x z } Q _ { y z } \\leq \\frac { K ^ 2 \\rho ^ 2 } { n } . \\end{align*}"}
+{"id": "6126.png", "formula": "\\begin{align*} \\| u _ { 1 2 } \\| & = \\| v _ { 1 1 } v _ { 1 2 } + v _ { 1 2 } v _ { 2 2 } \\| \\\\ & \\leq \\| v _ { 1 2 } \\| \\big ( \\| v _ { 1 1 } \\| + \\| v _ { 2 2 } \\| \\big ) \\\\ & \\leq \\| v _ { 1 1 } ^ 2 + v _ { 1 2 } v _ { 1 2 } ^ * \\| ^ { 1 / 2 } \\big ( \\| v _ { 1 1 } ^ 2 + v _ { 1 2 } v _ { 1 2 } ^ * \\| ^ { 1 / 2 } + \\| v _ { 1 2 } ^ * v _ { 1 2 } + v _ { 2 2 } ^ 2 \\| ^ { 1 / 2 } \\big ) \\\\ & = \\| u _ { 1 1 } \\| ^ { 1 / 2 } \\big ( \\| u _ { 1 1 } \\| ^ { 1 / 2 } + \\| u _ { 2 2 } \\| ^ { 1 / 2 } \\big ) , \\end{align*}"}
+{"id": "2308.png", "formula": "\\begin{align*} ( 1 - y ^ { 2 } ) U _ { \\theta } ' + 2 y U _ { \\theta } + \\frac { 1 } { 2 } U ^ { 2 } _ { \\theta } = c _ { 1 } ( 1 - y ) + c _ { 2 } ( 1 + y ) + c _ { 3 } ( 1 - y ^ { 2 } ) , \\ ; \\ , \\mathrm { i n } \\ ; ( - 1 , 1 ) , \\end{align*}"}
+{"id": "8044.png", "formula": "\\begin{align*} P ( \\varepsilon _ 9 ^ N > \\epsilon ) & \\leq \\sum _ { l = 1 } ^ { + \\infty } \\frac { T } { \\delta } \\exp \\left ( \\frac { \\gamma _ N ^ 2 } { N } \\left ( K _ { 1 1 } - \\frac { 1 0 ^ l } { l } \\frac { 2 T \\epsilon } { e ^ { 2 0 T } \\delta } \\right ) \\right ) \\\\ & \\leq \\frac { 2 T } { \\delta } \\exp \\left ( \\frac { \\gamma _ N ^ 2 } { N } \\left ( K _ { 1 1 } - \\frac { 2 0 T \\epsilon } { \\delta e ^ { 2 0 T } } \\right ) \\right ) . \\end{align*}"}
+{"id": "8890.png", "formula": "\\begin{align*} \\mathcal I [ u ( t ) ] & = \\| \\sqrt { \\mathcal K _ \\lambda } u \\| ^ 2 - \\mathcal P [ u ] \\\\ & = p \\mathcal E [ u ] - ( p - 1 ) \\| \\sqrt { \\mathcal K _ \\lambda } u \\| ^ 2 \\\\ & < p \\mathcal E [ \\varphi ] - ( p - 1 ) \\| \\sqrt { \\mathcal K _ \\lambda } \\varphi \\| ^ 2 < 0 , \\end{align*}"}
+{"id": "1610.png", "formula": "\\begin{align*} \\b Q _ { i ( p - 1 ) } ( x \\otimes y ) \\mapsto ( - 1 ) ^ { \\tfrac { n m ( p - 1 ) } { 2 } } \\sum _ { j + k = i } & \\big ( \\b Q _ { j ( p - 1 ) } ( x ) \\otimes Q _ { k ( p - 1 ) } ( y ) \\\\ & + ( - 1 ) ^ n Q _ { j ( p - 1 ) } ( x ) \\otimes \\b Q _ { k ( p - 1 ) } ( y ) \\big ) . \\end{align*}"}
+{"id": "8030.png", "formula": "\\begin{align*} & h _ 1 ( T ) W _ { T , 1 } ( f _ 1 ) - h _ 1 ( 0 ) W _ { 0 , 1 } ( f _ 1 ) - \\int _ 0 ^ T \\partial _ s h _ 1 ( s ) W _ { s , 1 } ( f _ 1 ) d s \\\\ & = - \\int _ 0 ^ T h _ 1 ( s ) W _ { s , 1 } \\bigotimes W _ { s , 3 } ( \\lambda ( \\cdot , \\ast ) f _ 1 ( \\cdot ) ) d s . \\end{align*}"}
+{"id": "2691.png", "formula": "\\begin{align*} \\Phi ^ { ( 1 ) } _ { i } : = P ^ { ( 2 ) } _ { i } - f _ { i } ( Q _ { ( 1 ) } ^ { j } , Q _ { ( 2 ) } ^ { j } ) : \\approx 0 , \\end{align*}"}
+{"id": "7970.png", "formula": "\\begin{align*} \\mathrm { t r } \\big ( \\mathrm { l i } ( \\mu ) \\big ) = \\mu , \\quad \\forall \\mu \\in H ^ { \\frac { 1 } { 2 } } \\Lambda ^ { 0 } ( \\partial \\Omega ) . \\end{align*}"}
+{"id": "3092.png", "formula": "\\begin{align*} v \\in W _ \\ell & \\Longleftrightarrow h ( z ) \\ , v = z ^ { \\ell } \\ , v \\\\ & \\Longleftrightarrow P h ' ( z ) P ^ { - 1 } \\ , v = z ^ { \\ell } \\ , v \\\\ & \\Longleftrightarrow h ' ( z ) P ^ { - 1 } \\ , v = z ^ { \\ell } \\ , P ^ { - 1 } \\ , v \\\\ & \\Longleftrightarrow P ^ { - 1 } \\ , v \\in { W ' } _ \\ell . \\end{align*}"}
+{"id": "2292.png", "formula": "\\begin{align*} A \\mapsto X \\ , , B \\mapsto Y \\ , , C \\mapsto Z : = [ X , Y ] \\ , , \\end{align*}"}
+{"id": "2671.png", "formula": "\\begin{align*} L ^ { ( 2 ) } = \\sum _ { i = 1 } ^ { n } f _ { i } ( \\dot { q } ^ { j } , q ^ { j } ) \\ddot { q } ^ { i } + g ( \\dot { q } ^ { j } , q ^ { j } ) . \\end{align*}"}
+{"id": "3966.png", "formula": "\\begin{align*} \\lambda ( T _ 1 T _ 2 ) ^ * \\rho ( T _ 1 T _ 2 ) & = \\lambda ( T _ 2 ) ^ * \\lambda ( T _ 1 ) ^ * \\rho ( T _ 1 ) \\rho ( T _ 2 ) \\\\ & = \\lambda ( T _ 2 ) ^ * \\int _ X c ( T _ 1 , x ) d \\mu ( x ) \\rho ( T _ 2 ) \\\\ & = \\int _ X c ( T _ 1 , T _ 2 ( x ) ) d \\mu ( x ) \\lambda ( T _ 2 ) ^ * \\rho ( T _ 2 ) \\\\ & = \\int _ X c ( T _ 1 , T _ 2 ( x ) ) c ( T _ 2 , x ) d \\mu ( x ) , \\end{align*}"}
+{"id": "8407.png", "formula": "\\begin{align*} [ B _ 1 ( a ) , B _ 1 ( b ) ] & = B _ 1 ( B _ 1 ( a ) B _ 1 ( b ) - B _ 1 ( b ) B _ 1 ( a ) + B _ 2 ( b ) B _ 2 ( a ) - B _ 2 ( a ) B _ 2 ( b ) ) \\\\ & = B _ 1 ( [ B _ 1 ( a ) , B _ 1 ( b ) ] - [ B _ 2 ( a ) , B _ 2 ( b ) ] ) . \\end{align*}"}
+{"id": "3286.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ^ { d + r } ; q ^ d ) _ k ^ { d - r - 1 } ( q ^ r ; q ^ d ) _ k ^ { r + 1 } q ^ { d k } } { ( q ^ d ; q ^ d ) _ k ^ d } \\equiv \\frac { ( 1 - q ^ r ) ^ { r + 1 } ( q ^ d ; q ^ d ) _ { n - 1 - ( n + r ) / d } } { ( - 1 ) ^ { n - 1 - ( n + r ) / d } ( q ^ d ; q ^ d ) _ { ( n + r ) / d } ^ { d - 1 } } q ^ { A ( d , n , r ) - r } , \\end{align*}"}
+{"id": "868.png", "formula": "\\begin{align*} L _ { i j } = w _ { i : j } + w _ { j : i } , \\ : \\ : S _ { i } = w ^ s L _ { s i } , \\ : \\ : C _ { i j } = w _ { i : j } - w _ { j : i } , \\ : \\ : T _ i = w ^ s C _ { s i } , \\end{align*}"}
+{"id": "2189.png", "formula": "\\begin{align*} \\omega = \\omega _ r ( r , z , t ) \\bar e _ r + \\omega _ \\varphi ( r , z , t ) \\bar e _ \\varphi + \\omega _ z ( r , z , t ) \\bar e _ z . \\end{align*}"}
+{"id": "5088.png", "formula": "\\begin{align*} ( L _ s \\cdot \\varepsilon _ { \\theta } ) ( t ) & = ( 1 - q ^ { - k } ) \\theta ( t ) - 0 , \\end{align*}"}
+{"id": "8906.png", "formula": "\\begin{align*} \\log ( 1 + t ) = 2 \\sinh ^ { - 1 } \\left ( \\frac { t } { 2 \\sqrt { 1 + t } } \\right ) \\end{align*}"}
+{"id": "2111.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\Theta ( s , x ) U ( s , x ) \\mathrm { d } x - \\frac { 1 } { d } \\sum _ { i = 1 } ^ d \\Theta ( \\frac { \\lfloor s T \\rfloor } { T } , \\frac { i } { d } ) U ( \\frac { \\lfloor s T \\rfloor } { T } , \\frac { i } { d } ) & = o ( 1 ) \\\\ \\frac { 2 } { d } \\sum _ { i = 1 } ^ d \\Theta ( \\frac { \\lfloor s T \\rfloor } { T } , \\frac { i } { d } ) \\big ( U ( \\frac { \\lfloor s T \\rfloor } { T } , \\frac { i } { d } ) - U ^ { d , T } ( \\frac { \\lfloor s T \\rfloor } { T } , \\frac { i } { d } ) \\big ) & = o ( 1 ) . \\end{align*}"}
+{"id": "719.png", "formula": "\\begin{align*} ( a = b ) \\Longrightarrow ( \\forall P ) ( P ( a ) \\Leftrightarrow P ( b ) ) . \\end{align*}"}
+{"id": "2125.png", "formula": "\\begin{align*} \\begin{aligned} \\sup _ { N \\in \\mathbb { N } } \\mathbb { P } ( \\bar \\mu ^ N \\in H ^ c _ M ) & = \\sup _ { N \\in \\mathbb { N } } \\mathbb { P } ( \\bar \\mu ^ N ( G _ \\alpha ( f ) ) > M ) \\leq \\frac { \\sup _ { N \\in \\mathbb { N } } \\mathbb { E } [ \\bar \\mu ^ N ( G _ \\alpha ( f ) ) ] } { M } . \\end{aligned} \\end{align*}"}
+{"id": "1973.png", "formula": "\\begin{align*} E q u > & p ^ s + 1 - ( \\tau + 1 ) p ^ t - \\frac { ( p - \\tau + 1 ) p ^ { s } - p ^ { t + 1 } } { p ^ { t - 1 } + 1 } \\\\ = & \\frac { p ^ { s - t - 1 } - ( p - \\tau ) p ^ s - ( \\tau + 1 ) p ^ { 2 t - 1 } + p ^ { t + 1 } - ( \\tau + 1 ) p ^ t } { p ^ { t - 1 } + 1 } + 1 \\\\ > & 0 , \\end{align*}"}
+{"id": "1735.png", "formula": "\\begin{align*} Q _ j = X ^ T \\widehat { Q } _ j X \\quad \\quad \\forall \\ , j , \\end{align*}"}
+{"id": "585.png", "formula": "\\begin{align*} \\tilde h _ j = \\underset { I \\subset \\Delta ^ \\ast _ j } { \\sum \\limits _ { I \\in \\mathcal { D } _ { m _ j } } } \\theta _ j ( I ) h _ I , \\end{align*}"}
+{"id": "7900.png", "formula": "\\begin{align*} \\mathbf { \\Phi } ^ { ' } = [ \\mathbf { \\Phi } _ { \\mathbf { A } _ { 1 } } ^ { ' } , \\cdots , \\mathbf { \\Phi } _ { \\mathbf { A } _ { L } } ^ { ' } ] . \\end{align*}"}
+{"id": "3368.png", "formula": "\\begin{align*} E _ 1 & = - E _ { 0 1 } + E _ { 2 7 } - E _ { 3 6 } + E _ { 4 5 } , \\\\ E _ 2 & = - E _ { 0 2 } - E _ { 1 7 } + E _ { 3 5 } + E _ { 4 6 } , \\\\ E _ 3 & = - E _ { 0 3 } + E _ { 1 6 } - E _ { 2 5 } + E _ { 4 7 } , \\\\ E _ 4 & = - E _ { 0 4 } - E _ { 1 5 } - E _ { 2 6 } - E _ { 3 7 } , \\\\ E _ 5 & = - E _ { 0 5 } + E _ { 1 4 } + E _ { 2 3 } - E _ { 6 7 } , \\\\ E _ 6 & = - E _ { 0 6 } - E _ { 1 3 } + E _ { 2 4 } + E _ { 5 7 } , \\\\ E _ 7 & = - E _ { 0 7 } + E _ { 1 2 } + E _ { 3 4 } - E _ { 5 6 } , \\end{align*}"}
+{"id": "4047.png", "formula": "\\begin{align*} \\mathcal { D } _ { \\mathcal { S } } ^ 2 ( \\mathbb { X } , \\mathbb { Y } ) ^ 2 & = \\mathbb { E } _ { X , X ^ \\prime } [ \\langle S ^ 2 ( \\hat X ^ 1 ) , S ^ 2 ( \\hat X ^ { \\prime , 1 } ) \\rangle _ { \\mathcal { H } ^ 2 } ] \\\\ & + \\mathbb { E } _ { Y , Y ^ \\prime } [ \\langle S ^ 2 ( \\hat Y ^ 1 ) , S ^ 2 ( \\hat Y ^ { \\prime , 1 } ) \\rangle _ { \\mathcal { H } ^ 2 } ] \\\\ & - 2 \\mathbb { E } _ { X , Y } [ \\langle S ^ 2 ( \\hat X ^ 1 ) , S ^ 2 ( \\hat Y ^ 1 ) \\rangle _ { \\mathcal { H } ^ 2 } ] . \\end{align*}"}
+{"id": "7773.png", "formula": "\\begin{align*} u _ { s _ k } \\geq \\nu _ n \\geq \\nu _ { s _ k } - 2 \\epsilon _ { s _ k } ^ { \\frac { 1 } { 2 m } } = u _ { s _ k } - 2 \\epsilon _ { s _ k } ^ { \\frac { 1 } { 2 m } } > \\frac { u _ { s _ k } } { 2 } . \\end{align*}"}
+{"id": "2317.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ { p } _ { \\sigma } ( \\mathbb { R } ^ { 3 } ) } : = \\| u \\| _ { L ^ { p } ( \\mathbb { R } ^ { 3 } ) } , \\quad \\| u \\| _ { \\dot { H } ^ { 1 } _ { \\sigma } ( \\mathbb { R } ^ { 3 } ) } = \\| \\nabla u \\| _ { L ^ { 2 } ( \\mathbb { R } ^ { 3 } ) } . \\end{align*}"}
+{"id": "6848.png", "formula": "\\begin{align*} d g _ K & = e ^ 1 \\wedge \\ldots \\wedge e ^ { 2 N } \\\\ & = \\det ( ( e _ { i j } ) ) \\ , d \\phi _ { N - 1 } \\wedge d \\psi _ { N - 1 } \\wedge \\ldots \\wedge d \\phi _ 1 \\wedge d \\psi _ 1 . \\end{align*}"}
+{"id": "4496.png", "formula": "\\begin{align*} M _ { H , j } ( Z _ 0 , J , \\rho ) = \\| \\tilde F _ 0 \\| ^ 2 _ { \\partial D _ j \\times M _ j , \\rho } . \\end{align*}"}
+{"id": "8982.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c l } F ( D ^ 2 u _ n ) - \\beta { u _ n ( r ^ 2 + 1 / n ) ^ { - \\gamma / 2 } } = ( r ^ 2 + 1 / n ) ^ { - \\gamma / 2 } f ( r ) & \\hbox { i n } \\ B ( 0 , 1 ) \\\\ u _ n = b & \\hbox { o n } \\ \\partial B ( 0 , 1 ) \\end{array} \\right . \\end{align*}"}
+{"id": "8545.png", "formula": "\\begin{align*} \\int _ M ( A / R ) : = A \\otimes _ R M \\simeq P _ A ( M \\wedge \\mathrm { T A Q } ( A / R ) ) . \\end{align*}"}
+{"id": "2833.png", "formula": "\\begin{align*} \\delta \\Xi ^ { 2 } ( t _ { 1 } ) = 0 . \\end{align*}"}
+{"id": "3016.png", "formula": "\\begin{align*} \\begin{array} { l } \\overline { R } _ { 0 1 1 0 } = - \\overline { R } _ { 0 2 0 2 } = ( \\beta - 1 ) , \\\\ \\overline { R } _ { 0 1 0 2 } = - \\alpha \\\\ \\overline { R } _ { 1 2 1 2 } = \\alpha ^ 2 - ( \\beta - 1 ) ^ 2 \\end{array} \\end{align*}"}
+{"id": "6193.png", "formula": "\\begin{align*} \\begin{array} { l } b _ { n - 1 } = - \\alpha _ 1 e _ { n - 1 } + e _ n ~ ~ ~ \\ , { \\rm a n d } ~ ~ ~ \\ , b _ { n + 1 } = - \\alpha _ 1 e _ { n - 3 } + e _ { n - 2 } ~ , ~ ~ ~ \\ , { \\rm s o } \\\\ \\\\ t _ { n + 1 } = - \\alpha _ 1 e _ { n - 1 } + e _ n + \\alpha _ 1 e _ { n - 3 } - e _ { n - 2 } ~ . \\end{array} \\end{align*}"}
+{"id": "529.png", "formula": "\\begin{align*} \\left ( x ^ { \\alpha + \\beta } u _ { x } v \\right ) _ { x } = x ^ { \\alpha + \\beta } u _ { x } v _ { x } + x ^ { \\beta } \\left ( \\left ( x ^ { \\alpha } u _ { x } \\right ) _ x + \\beta x ^ { \\alpha - 1 } u _ { x } + \\frac { \\mu } { x ^ { 2 - \\alpha } } u \\right ) v - \\frac { \\mu } { x ^ { 2 - \\alpha - \\beta } } u v \\in L ^ { 1 } ( 0 , 1 ) . \\end{align*}"}
+{"id": "4032.png", "formula": "\\begin{align*} \\begin{cases} \\begin{array} { c c l l l l } x _ { 1 } ' & = & 0 \\\\ x _ { 2 } ' & = & \\frac { 1 - \\eta } { 2 - \\eta } x _ { 1 } y _ { 1 } & & + \\frac { 1 - \\eta } { 2 - \\eta } x _ { 2 } y _ { 1 } \\\\ y _ { 1 } ' & = & 0 \\\\ y _ { 2 } ' & = & \\frac { 1 } { 2 - \\eta } x _ { 1 } y _ { 1 } & + x _ { 1 } y _ { 2 } & + \\frac { 1 } { 2 - \\eta } x _ { 2 } y _ { 1 } & + x _ { 2 } y _ { 2 } \\end{array} \\end{cases} \\end{align*}"}
+{"id": "5637.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ d \\frac { P _ { j i } } { P _ { 0 i } } B _ { x , i } = 0 . \\end{align*}"}
+{"id": "8601.png", "formula": "\\begin{align*} u _ t + u u _ x + \\Lambda ^ { \\frac { 1 } { 2 } } u + \\mathcal { H } \\Lambda ^ { \\frac { 1 } { 2 } } u = 0 , \\end{align*}"}
+{"id": "8478.png", "formula": "\\begin{align*} \\lim _ { h \\searrow 0 } \\left \\| ( \\bar { u } _ h ) _ - - u _ - \\right \\| _ { L ^ \\gamma ( K _ T ) } = 0 . \\end{align*}"}
+{"id": "1523.png", "formula": "\\begin{align*} G ( z ) = F ( D F ^ { - 1 } ( z ) ) \\end{align*}"}
+{"id": "9048.png", "formula": "\\begin{align*} \\sum _ { k \\in I ^ a } X ^ Q ( T ^ { ( k ) } v ) = - \\sum _ { k \\in I ^ a } \\lambda _ { k } X ^ Q ( v ) ( v \\in V ^ { \\otimes n } ) . \\end{align*}"}
+{"id": "7428.png", "formula": "\\begin{align*} \\frac { \\abs { I } \\abs { J ^ \\prime } } { \\abs { I ^ \\prime } \\abs { J } } & = \\abs { r _ s ( i _ 1 , \\ldots , i _ k ) } \\abs { D ( \\Psi _ { i _ 1 , \\ldots , i _ k } ) ( \\xi ) - D ( \\Psi _ { i _ 1 , \\ldots , i _ k } ) ( \\tilde { \\xi } ) } , \\end{align*}"}
+{"id": "4458.png", "formula": "\\begin{align*} \\int _ { \\partial D _ 1 } f _ 1 \\overline { f } \\rho _ 1 | d z _ 1 | = 0 \\end{align*}"}
+{"id": "4425.png", "formula": "\\begin{gather*} \\omega _ t \\colon 0 = t _ 0 < t _ 1 < \\ldots < t _ M = T , \\intertext { w i t h m e s h i n t e r v a l s } I _ j \\coloneqq ( t _ { j - 1 } , t _ j ) \\tau _ j \\coloneqq t _ j - t _ { j - 1 } , \\ j = 1 , 2 , \\dots , M . \\end{gather*}"}
+{"id": "5650.png", "formula": "\\begin{align*} X \\cap Y = \\bigcup _ { i , j } C _ i \\cap C _ j ' \\end{align*}"}
+{"id": "7055.png", "formula": "\\begin{align*} S _ 1 & = \\langle b _ n \\cdot w , | w | ^ { q - 2 } ( \\partial _ t u + b _ n \\cdot w ) \\rangle \\\\ & = B _ q + \\langle b _ n \\cdot w , | w | ^ { q - 2 } \\partial _ t u \\rangle . \\end{align*}"}
+{"id": "7287.png", "formula": "\\begin{align*} C _ { m , r } ( n ) : = \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m - 1 } \\csc ^ { 2 n } \\left ( \\frac { j } { m } \\pi \\right ) e ^ { 2 \\pi i r j / m } . \\end{align*}"}
+{"id": "7292.png", "formula": "\\begin{align*} \\tilde { C } _ { m , r } ( n ) : = \\frac { 1 } { m } \\sum _ { j \\in \\{ 1 , \\ldots , m - 1 \\} \\setminus \\{ j _ m \\} ^ \\ast } \\csc ^ { n } \\left ( \\frac { 2 j } { m } \\pi \\right ) e ^ { 2 \\pi i r j / m } . \\end{align*}"}
+{"id": "1416.png", "formula": "\\begin{align*} s ^ { \\alpha - 1 } e ^ { - t s ^ \\alpha } & = - \\frac { 1 } { \\alpha t } \\frac { d } { d s } e ^ { - t s ^ \\alpha } = \\frac { 1 } { \\alpha t } \\int _ 0 ^ \\infty e ^ { - s x } x f _ \\alpha ( x \\vert t ) \\ , d x \\end{align*}"}
+{"id": "6128.png", "formula": "\\begin{align*} u = \\begin{pmatrix} u _ { 1 1 } & u _ { 1 2 } \\\\ u _ { 1 2 } ^ * & u _ { 2 2 } \\end{pmatrix} \\coloneqq \\rho _ { m , j } ^ { ( 2 ) } ( d _ j ^ * d _ j ) - \\rho _ { m , n } ^ { ( 2 ) } \\big ( \\rho _ { n , j } ^ { ( 2 ) } ( d _ j ) ^ * \\rho _ { n , j } ^ { ( 2 ) } ( d _ j ) \\big ) , \\end{align*}"}
+{"id": "4358.png", "formula": "\\begin{gather*} \\max \\{ 0 , x _ i - \\overline { b } _ i , \\max \\{ 0 , x _ i - \\overline { b } _ i + \\Delta b _ i \\} - \\theta ^ k ( x ) \\} = \\max \\{ 0 , x _ i - \\overline { b } _ i , x _ i - \\overline { b } _ i + \\Delta b _ i - \\Delta b _ k \\} . \\end{gather*}"}
+{"id": "4522.png", "formula": "\\begin{align*} \\left \\{ \\begin{matrix*} [ l ] \\partial _ t n _ N + \\displaystyle \\sum _ { i = 1 } ^ N \\partial _ { s _ i } n _ N + p _ N ( [ s ] _ N ) n _ N = 0 , \\\\ [ 5 p t ] n _ N ( t , s _ 1 = 0 , s _ 2 , . . . , s _ N ) = \\int _ { u = 0 } ^ \\infty p _ N ( s _ 2 , . . . , s _ N , u ) \\ , n _ N ( t , s _ 2 , . . . , s _ N , u ) \\ , d u . \\end{matrix*} \\right . \\end{align*}"}
+{"id": "1315.png", "formula": "\\begin{align*} \\{ \\mu _ I , \\mu _ J \\} _ \\pm & = \\pm C _ { I J } ^ K \\mu _ K , \\\\ \\{ \\mu _ I , q ^ i \\} _ \\pm & = \\pm \\rho ^ i _ I , \\\\ \\{ q ^ i , q ^ j \\} _ \\pm & = 0 . \\end{align*}"}
+{"id": "2166.png", "formula": "\\begin{align*} | f _ d ( A _ d '' ) + f _ d ( A _ d '' ) - f _ d ( A _ d '' ) | \\leq 2 L K = 8 L | f _ d ( A _ d '' ) | . \\end{align*}"}
+{"id": "7021.png", "formula": "\\begin{align*} ( \\partial _ t - \\Delta + b \\cdot \\nabla ) u = | \\mathsf { h } | f , u ( 0 ) = 0 , \\end{align*}"}
+{"id": "2311.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u ^ { c , \\gamma } + u ^ { c , \\gamma } \\cdot \\nabla u ^ { c , \\gamma } + \\nabla p ^ { c , \\gamma } = f ^ { c , \\gamma } , \\\\ \\mathrm { d i v } u ^ { c , \\gamma } = 0 , \\end{cases} \\end{align*}"}
+{"id": "8317.png", "formula": "\\begin{align*} \\epsilon ( Y ) + Y - X - \\Pi ^ { \\sharp } ( \\delta ) = \\beta + \\delta - \\alpha + \\epsilon ^ { * } ( \\alpha ) \\in T M \\cap T ^ { * } M , \\end{align*}"}
+{"id": "4393.png", "formula": "\\begin{gather*} p ^ * _ i = \\sup \\{ 0 , \\sup _ { u ^ i \\in \\mathcal { U } _ i } f _ i ( x ^ * , u ^ i ) - f _ i ( x ^ * , \\overline { u } ^ i ) ) - \\theta ^ * \\} \\ \\forall i \\in [ m ] \\end{gather*}"}
+{"id": "6863.png", "formula": "\\begin{align*} H _ { { \\varphi } , \\sf E } = \\sum _ { j \\in \\mathbb { J } } E _ j \\langle \\varphi _ j , \\cdot \\rangle \\ , \\varphi _ j \\end{align*}"}
+{"id": "8137.png", "formula": "\\begin{align*} k _ { A _ 2 } = d ^ { ( 3 ) } _ { i j k } p _ { i } p _ { j } p _ { k } + d ^ { ( 2 ) } _ { i j } p _ { i } p _ { j } + d ^ { ( 1 ) } _ { i } p _ { i } + d ^ { ( 0 ) } \\ , \\end{align*}"}
+{"id": "736.png", "formula": "\\begin{align*} t _ { y , z } & \\equiv t _ { y + J _ k ( v ) , z - 1 } \\pmod { a } \\\\ & ( 0 \\le y \\le J _ { k } ( v ) - 1 , 0 \\le z \\le q - 1 ; 0 \\le y \\le r - 1 , z = q ) \\ , , \\\\ t _ { y , 0 } & \\equiv t _ { y + r , q } \\pmod { a } \\quad ( 0 \\le y \\le J _ k ( v ) - r - 1 ) \\ , , \\\\ t _ { J _ k ( v ) - r + y , 0 } & \\equiv t _ { y , q + 1 } \\pmod { a } \\quad ( 0 \\le y \\le r - 1 ) \\ , . \\end{align*}"}
+{"id": "744.png", "formula": "\\begin{align*} \\overset { \\ _ * } { R } ^ { i } _ { m } = R ^ i _ { m } . \\end{align*}"}
+{"id": "8899.png", "formula": "\\begin{align*} ( 1 + t ) ^ x = \\sum _ { n = 0 } ^ \\infty ( x ) _ n \\frac { t ^ n } { n ! } = \\sum _ { n = 0 } ^ \\infty \\binom { x } { n } t ^ n , \\end{align*}"}
+{"id": "2652.png", "formula": "\\begin{align*} w = \\textbf { 1 } _ { I } \\int _ 0 ^ t e ^ { i ( - s ) \\Delta _ { x } } ( P _ { \\leq M } F ( s ) ) \\ , d s , \\end{align*}"}
+{"id": "671.png", "formula": "\\begin{align*} \\begin{aligned} G _ { m - r } = \\sum \\limits _ { \\ell = 0 } ^ { \\lfloor \\frac { m - r } { 2 } \\rfloor } c _ { \\ell , m - r } \\mathfrak { i } _ { ( 2 ) } ^ { \\ell } \\mathfrak { j } _ { ( 2 ) } ^ { \\ell } \\sum \\limits _ { p = 0 } ^ { r } ( - 1 ) ^ { r - p } \\frac { 1 } { p ! } \\binom { r } { p } \\mathfrak { j } _ { x ^ { \\otimes ( r - p ) } } \\delta ^ { p } N ^ { p } f , \\end{aligned} \\end{align*}"}
+{"id": "7463.png", "formula": "\\begin{align*} 1 = \\lim _ { j \\to \\infty } \\int _ 0 ^ t f _ j ( \\tau ) \\ , \\dd \\tau \\neq \\int _ 0 ^ t \\lim _ { j \\to \\infty } f _ j ( \\tau ) \\ , \\dd \\tau = 0 . \\end{align*}"}
+{"id": "6184.png", "formula": "\\begin{align*} \\begin{aligned} & \\| v ^ k - \\widetilde { v } ^ k \\| ^ 2 _ { G ^ k } = \\tau ^ k \\beta ^ k \\| \\bar { x } _ 2 ^ k - \\bar { x } _ 2 ^ { k + 1 } \\| ^ 2 _ { D } + ( 2 - \\gamma ) \\tau ^ k \\beta ^ k \\| A \\bar { x } ^ { k + 1 } - b \\| ^ 2 \\\\ & + 2 \\tau ^ k \\beta ^ k ( \\bar { x } _ 2 ^ k - \\bar { x } _ 2 ^ { k + 1 } ) ^ T A _ 2 ^ T ( A \\bar { x } ^ { k + 1 } - b ) \\geq ( 1 - \\gamma ) \\tau ^ k \\beta ^ k \\| A \\widetilde { x } ^ k - b \\| ^ 2 , \\end{aligned} \\end{align*}"}
+{"id": "6102.png", "formula": "\\begin{align*} & \\sum _ { ( 1 ) } c _ { ( 1 ) } \\otimes d _ { ( 1 ) } \\otimes b _ u - a _ u \\otimes 1 \\otimes \\Delta ( b _ u ) = ( \\Delta ( a _ u ) - 1 \\otimes a _ u ) \\otimes b _ u - a _ u \\otimes 1 \\otimes \\Delta ( b _ u ) . \\end{align*}"}
+{"id": "832.png", "formula": "\\begin{align*} \\delta ( i _ { ( X ) } a ( X ) ) = g ( a ( X ) , a ( X ) ) - [ 2 R _ { i j } X ^ { i } X ^ { j } + ( n - 1 ) X ^ { i } \\Psi _ { i } ] , \\end{align*}"}
+{"id": "2319.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } w - \\Delta w + w \\cdot \\nabla w + w \\cdot \\nabla u ^ { c , \\gamma } + u ^ { c , \\gamma } \\cdot \\nabla w + \\nabla \\pi = 0 , \\\\ \\mathrm { d i v } w = 0 , \\\\ w ( x , 0 ) = w _ { 0 } . \\end{cases} \\end{align*}"}
+{"id": "6430.png", "formula": "\\begin{align*} A ^ n _ i = n ^ { 1 / 2 \\alpha _ 0 } \\frac { \\left | \\Delta _ i ^ n X - \\Delta _ { i - 1 } ^ n X \\right | ^ { 1 / 2 } } { X _ { \\frac { i - 2 } { n } } ^ { 1 / 2 \\alpha _ 0 } } . \\end{align*}"}
+{"id": "4835.png", "formula": "\\begin{align*} \\sigma ' _ i = \\left \\{ \\begin{array} { l l } \\sigma _ i & \\mbox { i f $ 2 \\leq i \\leq c _ 1 $ } , \\\\ \\sigma _ i - 1 & \\mbox { i f $ c _ 1 + 1 \\leq i \\leq n $ . } \\end{array} \\right . \\end{align*}"}
+{"id": "5355.png", "formula": "\\begin{align*} c _ i ^ S = h _ i ^ 0 - h _ i ^ 1 + \\beta \\ , \\sum _ { j \\in N } ( p _ { i j } ^ 0 - p _ { i j } ^ 1 ) \\ , v ^ { S } _ j , i \\in N , \\end{align*}"}
+{"id": "3017.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\overline { \\mathrm { R i c } } _ { 0 0 } = 2 ( \\beta - 1 ) , \\ \\ \\ & \\overline { \\mathrm { R i c } } _ { 1 1 } = \\overline { \\mathrm { R i c } } _ { 2 2 } = \\beta ( \\beta - 1 ) - \\alpha ^ 2 , \\ \\ \\ & \\overline { \\mathrm { R i c } } _ { 1 2 } = \\alpha . \\\\ \\end{array} \\end{align*}"}
+{"id": "6471.png", "formula": "\\begin{align*} | A | ^ 4 - m K | A | ^ 2 - m ^ 2 K H ^ 2 = 0 . \\end{align*}"}
+{"id": "2117.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\mathrm { d } X ( t ) = b ( X ( t ) , \\mbox { L a w } ( X ( t ) ) ) \\mathrm { d } t + \\sigma ( X ( t ) , \\mbox { L a w } ( X ( t ) ) ) \\mathrm { d } W ( t ) - \\mathrm { d } K ( t ) , \\\\ & | K ( t ) | = \\int ^ { t } _ { 0 } \\mathbf { 1 } _ { \\partial \\mathcal { D } } ( X ( s ) ) \\mathrm { d } | K | ( s ) , K ( t ) = \\int ^ { t } _ { 0 } \\mathbf { n } ( X ( s ) ) \\mathrm { d } | K | ( s ) , \\\\ & \\mbox { L a w } ( X ( 0 ) ) = \\nu _ { 0 } , X ( t ) \\in \\overline { \\mathcal { D } } , t \\in [ 0 , T ] , \\end{aligned} \\right . \\end{align*}"}
+{"id": "2569.png", "formula": "\\begin{align*} \\phi _ k ( t ) = h ( t ) \\int _ 0 ^ t \\int _ 0 ^ { t _ { m - 1 } } \\cdots \\int _ 0 ^ { t _ 1 } f _ k ( t _ 0 ) \\ , d t _ 0 \\cdots d t _ { m - 1 } \\ ; , \\end{align*}"}
+{"id": "8146.png", "formula": "\\begin{align*} D _ 4 & \\ = \\ \\frac { 8 } { 3 } \\left ( 3 J _ 4 J _ 3 - 2 J _ 4 J _ 0 \\right ) A _ 8 - 4 J _ 4 J _ 1 A _ 7 - 1 0 J _ 4 J _ 1 A _ 6 + 1 0 J _ 4 J _ 2 A _ 4 \\ , \\end{align*}"}
+{"id": "372.png", "formula": "\\begin{align*} \\begin{aligned} z - y & = 2 ^ { p d } p ^ { p e - 1 } r '^ p , & & \\phi _ p ( z , y ) = p r _ 1 ^ p , & & x = 2 ^ { d } p ^ { e } r ' r _ 1 , \\\\ z - x & = s ^ p , & & \\phi _ p ( z , x ) = s _ 1 ^ p , & & y = s s _ 1 , \\\\ x + y & = t ^ p , & & \\phi _ p ( x , - y ) = t _ 1 ^ p , & & z = t t _ 1 , \\end{aligned} \\end{align*}"}
+{"id": "2744.png", "formula": "\\begin{align*} X _ { \\psi _ { 1 } } \\xi = \\{ \\xi , \\psi _ { 1 } \\} \\approx 1 \\end{align*}"}
+{"id": "1792.png", "formula": "\\begin{align*} 0 = \\Delta ^ 2 \\left ( a x _ 1 \\Re ( z _ 1 ^ { \\mu + 1 } ) - b \\Re ( z _ 1 ^ { \\mu + 2 } ) \\right ) = \\Delta ^ 2 ( c x _ 1 ^ { \\mu + 2 } ) = ( \\mu + 2 ) ( \\mu + 1 ) \\mu ( \\mu - 1 ) c x _ 1 ^ { \\mu - 2 } \\end{align*}"}
+{"id": "3423.png", "formula": "\\begin{align*} H = \\{ 1 , - 1 \\} \\subset G . \\end{align*}"}
+{"id": "7632.png", "formula": "\\begin{align*} \\Psi = \\langle b _ 0 t , \\ , b _ 1 t , \\ , a _ 2 + b _ 2 t , a _ 3 + b _ 3 t , a _ 4 + b _ 4 t , a _ 5 + b _ 5 t , a _ 6 + b _ 6 t , a _ 7 + b _ 7 t . \\rangle \\end{align*}"}
+{"id": "6006.png", "formula": "\\begin{align*} u _ k \\le d ( c - p ) ^ { - 1 } k ^ { - p } + o ( k ^ { - p } ) , & \\mbox { i f $ c > p $ } , \\cr u _ k = O ( k ^ { - c } \\ln k ) , & \\mbox { i f $ c = p $ } , \\cr u _ k = O ( k ^ { - c } ) , & \\mbox { i f $ c < p $ } . \\end{align*}"}
+{"id": "1994.png", "formula": "\\begin{align*} \\psi _ j ^ { n + 1 } = \\sum _ { l \\in \\mathcal { T } _ M } \\widetilde { ( \\psi ^ { n + 1 } ) } _ l e ^ { i \\mu _ l ( x _ j - a ) } , j \\in \\mathcal { T } ^ 0 _ M , \\end{align*}"}
+{"id": "2073.png", "formula": "\\begin{align*} f ( s \\pm T ^ { - 1 } ) = f ( s ) + O ( T ^ { - 1 } ) , T \\left ( f ( s ) - f ( s - T ^ { - 1 } ) \\right ) = f ' ( s ) + O ( T ^ { - 1 } ) . \\end{align*}"}
+{"id": "5568.png", "formula": "\\begin{align*} \\langle v , \\Phi ^ t w \\rangle = \\sum _ { x y } ( \\Phi ^ t ) _ { x y } v ( x ) w ( y ) \\leq \\frac { K ^ 2 \\rho ^ { 2 t } } { n } \\| v \\| _ 1 \\| w \\| _ 1 . \\end{align*}"}
+{"id": "6984.png", "formula": "\\begin{align*} \\lvert \\eta \\rvert ^ 2 e ^ { i \\theta } = \\eta ^ 2 = z _ { n + 1 } ^ 2 - { \\displaystyle \\sum _ { j = 1 } ^ { n } } z _ j ^ 2 = z _ { n + 1 } ^ 2 - { \\displaystyle \\sum _ { j = 1 } ^ { n } } \\lvert z _ j \\rvert ^ 2 e ^ { i \\theta } , \\end{align*}"}
+{"id": "7436.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u - \\mathcal { A } u = f & [ 0 , T ] \\times \\R ^ d , \\\\ u = 0 & t = 0 . \\end{cases} \\end{align*}"}
+{"id": "7316.png", "formula": "\\begin{align*} \\int \\limits _ { 0 } ^ { \\infty } e ^ { - ( s + 1 ) t } I _ { | \\ell + j m | } ( t ) d t = \\frac { 1 } { \\sqrt { s ^ 2 + 2 s } } \\left ( s + 1 - \\sqrt { ( s + 1 ) ^ 2 - 1 } \\right ) ^ { | \\ell + j m | } . \\end{align*}"}
+{"id": "7378.png", "formula": "\\begin{align*} \\| ( u _ { j + 1 } ) _ x - ( u _ j ) _ x \\| _ 1 = O \\left ( \\frac { 1 } { 2 ^ j } \\right ) \\end{align*}"}
+{"id": "5383.png", "formula": "\\begin{align*} c ^ { S _ 2 } _ 0 & = \\frac { \\lambda _ 0 } { \\alpha + \\lambda _ 0 + \\mu _ 1 } \\ , \\Delta h _ 1 \\\\ c ^ { S _ { k + 2 } } _ k & = \\frac { \\lambda _ k } { a _ { k + 1 } } \\ , \\frac { \\displaystyle \\Delta h _ { k + 1 } + \\frac { c ^ { S _ { k + 1 } } _ { k - 1 } } { \\rho _ { k - 1 } } } { \\alpha + \\lambda _ k + \\mu _ { k + 1 } } , 1 \\leq k \\leq n - 1 . \\end{align*}"}
+{"id": "5251.png", "formula": "\\begin{align*} \\overline { Z } _ { i } = \\frac { \\overline { p } _ { i } } { \\alpha \\overline { p } _ { i } + \\left ( 1 - \\alpha \\right ) \\overline { q } _ { i } } \\end{align*}"}
+{"id": "3397.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { k } \\eta _ { i } \\left \\Vert \\nabla f ( x _ { i } ) \\right \\Vert ^ { 2 } + \\Delta _ { k + 1 } & \\le \\left ( \\sqrt { \\Delta _ { 1 } } + 2 \\sqrt { A } C _ { 1 } \\right ) ^ { 2 } \\end{align*}"}
+{"id": "3169.png", "formula": "\\begin{align*} ( a _ s , \\ldots , a _ 1 , a _ 0 , i ) ^ \\psi = \\left ( \\sum \\limits _ { r = 1 } ^ s a _ r k ^ r + a _ 0 \\right ) t + i \\equiv a _ 0 t + i \\pmod { k } . \\end{align*}"}
+{"id": "8733.png", "formula": "\\begin{align*} \\hat \\mu _ I = \\frac 1 I \\sum _ { i = 1 } ^ I \\delta _ { X _ i - \\bar X _ I } \\mbox { w i t h } \\bar X _ I = \\frac 1 I \\sum _ { i = 1 } ^ I X _ i \\quad \\mbox { a n d } \\hat \\nu _ J = \\frac 1 J \\sum _ { j = 1 } ^ J \\delta _ { Y _ j - \\bar Y _ J } \\mbox { w i t h } \\bar Y _ J = \\frac 1 J \\sum _ { j = 1 } ^ J Y _ j , \\end{align*}"}
+{"id": "5473.png", "formula": "\\begin{align*} c & = o ( 1 ) , \\\\ a & = o ( c ) , \\\\ c ^ { - K } & = o ( a ^ { 1 - K } ) . \\end{align*}"}
+{"id": "3011.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\xi } g ( x , y ) = g ( \\nabla _ x \\xi , y ) + g ( x , \\nabla _ y \\xi ) = 2 g ( x , \\phi y ) . \\end{align*}"}
+{"id": "5291.png", "formula": "\\begin{align*} K L I _ { n } ( q \\| p ) = \\sum _ { i } p _ { i } - K _ { 0 } \\sum _ { i } q _ { i } \\end{align*}"}
+{"id": "4072.png", "formula": "\\begin{align*} E _ i ( \\pm i \\sigma ) = \\int _ { - \\infty } ^ { \\pm i \\sigma } \\frac { e ^ { p } } { p } d p \\end{align*}"}
+{"id": "6973.png", "formula": "\\begin{align*} \\eta ^ { 2 } = - \\langle \\mathbf { z } , \\mathbf { \\overline { z } } \\rangle = - \\langle \\mathbf { z } , \\mathbf { z } \\rangle = \\lvert \\langle \\mathbf { z } , \\mathbf { z } \\rangle \\rvert , \\end{align*}"}
+{"id": "7727.png", "formula": "\\begin{align*} ( H _ { \\varepsilon ^ { - 1 } V , \\alpha , x } u ) _ n = u _ { n + 1 } + u _ { n - 1 } + \\varepsilon ^ { - 1 } V ( x + n \\alpha ) u _ n , \\ \\ n \\in \\Z , \\end{align*}"}
+{"id": "6399.png", "formula": "\\begin{align*} u _ n = \\begin{pmatrix} \\frac { 1 } { n ^ { 1 / \\alpha _ 0 - 1 / 2 } } & 0 \\\\ 0 & \\frac { 1 } { \\sqrt { n } } v _ n \\end{pmatrix} \\mbox { w h e r e } v _ n = \\begin{pmatrix} v _ n ^ { 1 1 } & v _ n ^ { 1 2 } \\\\ v _ n ^ { 2 1 } & v _ n ^ { 2 2 } \\\\ \\end{pmatrix} . \\end{align*}"}
+{"id": "5612.png", "formula": "\\begin{align*} & n ( 4 ( k + 1 ) s ) ^ { 2 s + 3 } \\left ( 1 + 4 p \\right ) ^ { 1 6 s ^ 2 } 2 ^ { 6 s } K ^ { 2 2 s } \\theta _ 1 ^ { 4 s k } \\cdot \\left ( \\frac { K } { d } \\right ) ^ { 2 s k } \\sum _ { g = 0 } ^ { \\infty } \\left ( \\frac { 2 K ^ { 1 2 } \\left ( 1 + 4 p \\right ) ^ { 8 s } ( 4 ( k + 1 ) s ) ^ { 6 s } } { n } \\right ) ^ g . \\end{align*}"}
+{"id": "9012.png", "formula": "\\begin{align*} e _ { 1 1 } \\cdot e _ { 1 1 } = - e _ { 1 1 } \\cdot e _ { n n } = e _ { n n } \\cdot e _ { n n } = e _ { 1 n } \\end{align*}"}
+{"id": "1415.png", "formula": "\\begin{align*} M ^ \\nu _ { \\alpha , \\mu } ( 1 \\vert \\lambda ) & = \\int _ 0 ^ \\infty e ^ { - \\lambda t } \\ , d R ^ \\nu _ { \\alpha , \\mu } ( t ) ( \\lambda \\ge 0 ) \\\\ { \\rm w h e r e } d R ^ \\nu _ { \\alpha , \\mu } ( t ) & = \\frac { 1 } { \\Gamma ( \\mu ) } \\ , \\{ I _ + ^ \\nu \\ , f _ \\alpha ( \\cdot \\vert t ) \\} ( 1 ) \\ , t ^ { \\mu - 1 } \\ , d t \\\\ & = \\frac { 1 } { \\Gamma ( \\mu ) } \\ , \\{ I _ + ^ \\nu \\ , f _ \\alpha \\} ( t ^ { - 1 / \\alpha } ) \\ , t ^ { \\mu + ( \\nu - 1 ) / \\alpha - 1 } \\ , d t \\end{align*}"}
+{"id": "6769.png", "formula": "\\begin{align*} \\begin{cases} { x } ^ { k + 1 } & = \\arg \\min \\limits _ { x \\in \\mathcal { X } } \\left \\{ \\Phi ( x , y ^ k ) + \\frac { r } { 2 } \\| x - x ^ k \\| ^ 2 \\right \\} , \\\\ \\widetilde { y } ^ k & = \\arg \\max \\limits _ { y \\in \\mathcal { Y } } \\left \\{ \\Phi ( [ { x } ^ { k + 1 } + \\alpha ( { x } ^ { k + 1 } - x ^ k ) ] , y ) - \\frac { s } { 2 } \\| y - y ^ k \\| ^ 2 \\right \\} , \\\\ y ^ { k + 1 } & = \\widetilde { y } ^ k - ( 1 - \\alpha ) \\frac { 1 } { s } A ( x ^ { k + 1 } - x ^ k ) . \\end{cases} \\end{align*}"}
+{"id": "3171.png", "formula": "\\begin{align*} d X \\left ( t \\right ) = \\Delta \\ln \\left ( X \\left ( t \\right ) \\right ) d t \\end{align*}"}
+{"id": "1957.png", "formula": "\\begin{align*} c ( x ) & = ( x - 1 ) ^ { p ^ s - p ^ { s - t } } g ( x ) \\\\ & = ( x ^ { p ^ { s - t } } - 1 ) ^ { p ^ t - 1 } g ( x ) \\\\ & = \\left [ \\sum _ { j = 0 } ^ { p ^ { t } - 1 } \\binom { p ^ { t } - 1 } { j } ( - 1 ) ^ { ( p ^ { t } - 1 - j ) } x ^ { j p ^ { s - t } } \\right ] g ( x ) . \\end{align*}"}
+{"id": "1825.png", "formula": "\\begin{align*} \\star ( \\phi _ 0 \\wedge \\star ( \\phi _ 0 \\wedge \\omega ) ) = 2 \\omega + \\star ( \\phi _ 0 \\wedge \\omega ) , \\forall \\ ; \\omega \\in \\Lambda ^ 2 V ^ * , \\end{align*}"}
+{"id": "1302.png", "formula": "\\begin{align*} \\partial _ t \\rho = - \\div ( \\mathbf { Q } \\rho ) + \\div ( \\boldsymbol { D } \\nabla \\rho ) , \\end{align*}"}
+{"id": "5787.png", "formula": "\\begin{align*} \\sum _ { i \\in I _ 1 } \\left ( | \\tilde { \\mathcal { E } } _ { i , 1 } ( t ) | ^ 2 + | \\tilde { \\mathcal { E } } _ { i , 2 } ( t ) | ^ 2 \\right ) + \\sum _ { i \\in \\mathbb { Z } \\setminus \\{ 0 \\} } | \\tilde { \\mathcal { E } } _ { i } ( t ) | ^ 2 = \\Vert \\tilde { \\mathcal { E } } \\Vert ^ 2 _ G = 2 m ^ { - 2 } \\| \\tilde { E } ( u ) \\| ^ 2 _ { L ^ 2 } ( t ) \\leq C | z ( t ) | ^ q . \\end{align*}"}
+{"id": "4709.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\beta ( g ) ( t ) \\leq \\beta ( g _ { 2 / 2 ^ n } ) ( t ' ) = \\alpha ( g _ { 2 / 2 ^ n } ) ( t ' ) \\leq \\alpha ( ( g _ { 2 / 2 ^ n } ) _ { 1 / 2 ^ n } ) ( t ) = \\alpha ( g _ { 3 / 2 ^ n } ) ( t ) . \\\\ \\alpha ( g ) ( t ) \\leq \\alpha ( g _ { 1 / 2 ^ n } ) ( t ' ) = \\beta ( g _ { 1 / 2 ^ n } ) ( t ' ) \\leq \\beta ( ( g _ { 1 / 2 ^ n } ) _ { 2 / 2 ^ n } ) ( t ) = \\beta ( g _ { 3 / 2 ^ n } ) ( t ) . \\end{array} \\right . \\end{align*}"}
+{"id": "9035.png", "formula": "\\begin{align*} E _ { 0 2 } & = \\frac { 5 } { 6 } E _ { 0 1 } \\circ E _ { 0 1 } - \\frac { 5 } { 2 } E _ { 0 0 } , \\\\ E _ { 0 3 } & = \\frac { 5 } { 2 } E _ { 0 1 } \\circ E _ { 0 1 } \\circ E _ { 0 1 } - \\frac { 2 7 } { 2 } E _ { 0 1 } - 6 E _ { 1 0 } , \\\\ E _ { 1 1 } & = E _ { 1 0 } \\circ E _ { 0 1 } - \\frac { 4 } { 3 } E _ { 0 1 } \\circ E _ { 0 1 } + 4 E _ { 0 0 } . \\end{align*}"}
+{"id": "5133.png", "formula": "\\begin{align*} L D 2 ( p \\| q ) = \\left [ \\log A \\left ( p , q \\right ) - \\log X \\left ( p , q \\right ) - \\log Y \\left ( p , q \\right ) \\right ] \\end{align*}"}
+{"id": "4446.png", "formula": "\\begin{align*} \\lim _ { j \\rightarrow + \\infty } \\ll f _ j , g \\gg _ { \\partial M , \\rho } = \\ll f , g \\gg _ { \\partial M , \\rho } \\end{align*}"}
+{"id": "4798.png", "formula": "\\begin{align*} u _ \\star ( x ) : = \\sup \\{ t \\leq 0 : \\mu ( t ) < { \\bf b } _ \\theta \\vert x - ( - r _ t \\cos \\theta E _ n ) \\vert ^ n \\} = \\sup \\{ t \\leq 0 : r _ t < \\vert x + r _ t \\cos \\theta E _ n \\vert \\} . \\end{align*}"}
+{"id": "8915.png", "formula": "\\begin{align*} \\sum _ { \\substack { d \\mid n \\\\ d \\equiv j \\ ( 2 ) } } \\mu \\left ( \\frac { n } { d } \\right ) = 0 , \\prod _ { d \\mid n } d ^ { \\mu ( n / d ) } = \\Phi _ n ( 1 ) \\end{align*}"}
+{"id": "7331.png", "formula": "\\begin{align*} \\partial _ s ^ n \\left . F _ { m , r } ( s , \\alpha ) \\right | _ { s = - 1 } & = ( - 1 ) ^ n n ! \\cdot \\frac { 1 } { m } \\sum _ { j = 0 } ^ { m - 1 } \\frac { e ^ { 2 \\pi i \\frac { j r } { m } } } { \\left ( - 1 + 2 \\sin ^ 2 \\left ( \\pi \\frac { ( j + \\alpha ) } { m } \\right ) \\right ) ^ { n + 1 } } \\\\ & = - n ! \\tilde { S } _ { m , r } ( \\alpha , n + 1 ) . \\end{align*}"}
+{"id": "2949.png", "formula": "\\begin{align*} \\frac { k ! ( n - 1 ) ! } { ( k + n - 1 ) ! } \\int _ { \\R ^ n } \\int _ { \\R ^ n } \\varphi _ k ( 2 i y , 2 i v ) p _ { 2 t } ( 2 y , 2 v ) d y \\ , d v = e ^ { 2 ( 2 k + n ) t } . \\end{align*}"}
+{"id": "5409.png", "formula": "\\begin{align*} c ^ { S _ 1 } _ 0 = \\frac { \\lambda _ 0 } { \\alpha + \\mu _ 1 } \\ , \\Delta h _ 1 . \\end{align*}"}
+{"id": "5965.png", "formula": "\\begin{align*} | c | = \\sum _ { u \\in A , v \\in B } f _ { u , v } , \\end{align*}"}
+{"id": "4749.png", "formula": "\\begin{align*} S ( \\vert t \\vert ) S ( \\vert s \\vert ) C ( x ) = _ X S ( \\vert t \\vert ) \\overline { S ( \\vert s \\vert ) C ( x ) } \\end{align*}"}
+{"id": "2323.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } w _ { 2 } - \\Delta w _ { 2 } + w _ { 2 } \\cdot \\nabla w _ { 2 } + w _ { 2 } \\cdot \\nabla ( u ^ { c , \\gamma } + w _ { 1 } ) + ( u ^ { c , \\gamma } + w _ { 1 } ) \\cdot \\nabla w _ { 2 } + \\nabla \\pi _ { 2 } = 0 , \\\\ \\mathrm { d i v } w _ { 2 } = 0 , \\\\ w _ { 2 } ( x , 0 ) = w _ { 0 2 } . \\end{cases} \\end{align*}"}
+{"id": "5014.png", "formula": "\\begin{align*} S ^ * ( \\nu ) = \\left \\{ j \\in N \\colon \\nu _ j ^ * > \\nu \\right \\} , \\nu \\in \\mathbb { R } . \\end{align*}"}
+{"id": "7064.png", "formula": "\\begin{align*} X _ t = x - \\int _ 0 ^ t b ( X _ s ) d s + \\sqrt { 2 } \\int _ 0 ^ t \\sigma ( X _ s ) \\circ d W _ s , \\end{align*}"}
+{"id": "3673.png", "formula": "\\begin{align*} | E ( T \\setminus v ) | < | E | - d = d | V ( T \\setminus v ) | - \\binom { d + 1 } { 2 } , \\end{align*}"}
+{"id": "4823.png", "formula": "\\begin{align*} \\min _ { \\mathbf { y } \\in Y } \\Big ( \\frac { 1 } { K } \\min _ { v \\in \\{ 1 , \\ldots v \\} } R _ v + \\varepsilon _ K \\Big ) y _ 1 + h y _ 2 = \\min \\Big \\{ \\frac { 1 } { K } \\min _ { v \\in \\{ 1 , \\ldots v \\} } R _ v + \\varepsilon _ K ; h \\Big \\} . \\end{align*}"}
+{"id": "5578.png", "formula": "\\begin{align*} \\vec f _ { \\phi , t } ( g , e ) = \\left ( \\frac { m n } { d ^ 2 } \\right ) ^ t \\sum _ { o _ 2 = e _ 3 , \\dots , o _ { 2 t } } \\left ( \\prod _ { s = 1 } ^ { t } M _ { \\iota ( o _ { 2 s } ) , \\iota ( o _ { 2 s + 1 } ) } M _ { \\iota ( o _ { 2 s + 2 } ) , \\iota ( o _ { 2 s + 1 } ) } \\right ) \\phi _ { \\iota ( o _ { 2 t + 2 } ) } , \\end{align*}"}
+{"id": "6686.png", "formula": "\\begin{align*} L / \\Phi _ p ( N ) = \\langle \\overline { x } \\rangle \\ltimes \\langle \\overline { y _ 1 } , \\ldots , \\overline { y _ l } \\rangle \\cong C _ q \\ltimes C _ p ^ { \\ , l } M \\Phi _ p ( N ) / \\Phi _ p ( N ) \\cong M \\cong C _ p ^ { \\ , m } \\end{align*}"}
+{"id": "629.png", "formula": "\\begin{align*} A _ k ( t ) = \\frac { 1 } { n } \\int ( h _ L - h _ K ) ( u ) d S _ { M _ k } ( u ) = \\frac { 1 } { n } ( h _ L - h _ K ) ( u _ k ) S _ { M _ k } ( u _ k ) = \\frac { \\kappa _ { n - 1 } } { n } ( h _ { L } ( u _ k ) - h _ K ( u _ k ) ) . \\end{align*}"}
+{"id": "5108.png", "formula": "\\begin{align*} \\dim \\mathcal { C } _ { A _ 2 } ( v ) & = 1 \\cdot 5 + 2 \\cdot 3 + 1 \\cdot 6 \\\\ & = 2 0 \\\\ \\dim \\mathcal { C } _ { A _ 3 } ( v ) & = 1 \\cdot 2 3 + 4 \\cdot 2 0 + 3 \\cdot 6 + 6 \\cdot 1 2 + 1 \\cdot 2 4 \\\\ & = 2 1 7 \\\\ \\dim \\mathcal { C } _ { A _ 4 } ( v ) & = 1 \\cdot 1 1 9 + 5 \\cdot 1 1 5 + 1 0 \\cdot 5 0 + 1 0 \\cdot 1 0 0 + 1 5 \\cdot 3 0 + 1 0 \\cdot 6 0 + 1 \\cdot 1 2 0 \\\\ & = 3 3 6 4 \\end{align*}"}
+{"id": "373.png", "formula": "\\begin{align*} \\begin{aligned} z - y & = 2 ^ { p d } r _ 0 ^ p , & & \\phi _ p ( z , y ) = r _ 1 ^ p , & & x = 2 ^ { d } r _ 0 r _ 1 , \\\\ z - x & = p ^ { p e - 1 } s '^ p , & & \\phi _ p ( z , x ) = p s _ 1 ^ p , & & y = p ^ e s ' s _ 1 , \\\\ x + y & = t ^ p , & & \\phi _ p ( x , - y ) = t _ 1 ^ p , & & z = t t _ 1 , \\end{aligned} \\end{align*}"}
+{"id": "8067.png", "formula": "\\begin{align*} \\begin{pmatrix} W _ { T , 1 } \\\\ W _ { T , 2 } \\\\ W _ { T , 3 } \\end{pmatrix} \\cdot \\begin{pmatrix} f _ 1 \\\\ f _ 2 \\\\ f _ 3 \\end{pmatrix} = \\mathcal { B } _ { 1 2 } ( ( h _ 1 , h _ 2 , h _ 3 ) , R _ 4 \\Phi _ T ^ { - 1 } \\vec { f } ) + \\mathcal { B } _ { 1 0 } ( ( \\tilde { F } , \\tilde { G } , \\tilde { H } ) , \\vec { f } _ { \\Phi , T } ) . \\end{align*}"}
+{"id": "6408.png", "formula": "\\begin{align*} u _ n ^ { - 1 } ( \\hat { \\theta } _ { 1 , n } - \\hat { \\theta } _ { n } ) = & ( u _ n ^ T J _ n ( \\hat { \\theta } _ { 0 , n } ) u _ n ) ^ { - 1 } \\left [ u _ n ^ T J _ n ( \\hat { \\theta } _ { 0 , n } ) u _ n \\right . \\\\ & \\left . - u _ n ^ T \\int _ 0 ^ 1 J _ n ( \\hat { \\theta } _ { n } + t ( \\hat { \\theta } _ { 0 , n } - \\hat { \\theta } _ { n } ) ) d t \\ u _ n \\right ] u _ n ^ { - 1 } ( \\hat { \\theta } _ { 0 , n } - \\hat { \\theta } _ { n } ) . \\end{align*}"}
+{"id": "4432.png", "formula": "\\begin{gather*} D _ t v ^ j \\coloneqq \\alpha _ j \\delta _ t v ^ j + \\beta _ j \\delta _ t v ^ { j - 1 } , \\alpha _ j = \\frac { 2 \\tau _ j + \\tau _ { j - 1 } } { \\tau _ j + \\tau _ { j - 1 } } , \\beta _ j = - \\frac { \\tau _ j } { \\tau _ j + \\tau _ { j - 1 } } \\ , , j = 2 , \\dots , M . \\end{gather*}"}
+{"id": "3853.png", "formula": "\\begin{align*} \\eta ^ * ( x , y \\mid s ) = \\eta ^ * _ { ( 1 ) } ( x \\mid s ) \\eta ^ * _ { 2 | 1 } ( y \\mid s , x ) , \\ ; \\ ; ( x , y ) \\in \\Delta ^ o \\times \\Delta ^ o . \\end{align*}"}
+{"id": "548.png", "formula": "\\begin{align*} x ^ 2 y '' + x y ' + ( x ^ 2 - \\nu ^ 2 ) y = 0 . \\end{align*}"}
+{"id": "747.png", "formula": "\\begin{align*} B _ { _ F } ( \\varphi ) ( \\hat { X } , Y ) = \\textrm { H e s s } ( \\varphi ) ( \\hat { X } , Y ) - ( d \\varphi \\otimes d \\varphi ) ( \\varrho \\hat { X } , Y ) - \\frac { 1 } { n } ( \\Delta \\varphi - \\| g r a d \\varphi \\| ^ 2 ) g ( \\varrho \\hat { X } , Y ) , \\end{align*}"}
+{"id": "7547.png", "formula": "\\begin{align*} \\begin{dcases} \\partial _ t \\bar \\rho + \\nabla \\cdot ( \\bar \\rho \\bar u ) = 0 \\\\ \\partial _ t \\bar u + \\bar u \\cdot \\nabla \\bar u + \\nabla \\big ( h _ 1 ^ \\prime ( \\bar \\rho ) + h _ 2 ^ \\prime ( \\bar \\rho ) \\big ) = 0 \\end{dcases} \\end{align*}"}
+{"id": "1904.png", "formula": "\\begin{align*} \\mathcal { A } u = h _ { f , g } . \\end{align*}"}
+{"id": "8070.png", "formula": "\\begin{align*} & \\{ | \\tau _ c ^ N - \\tau _ c | \\leq \\delta \\} \\cap \\{ \\frac { N } { \\gamma _ N } ( \\tau _ c ^ N - \\tau _ c ) > x \\} \\\\ & \\subseteq \\left \\{ \\sum _ { k = 1 } ^ 3 \\eta ^ N _ { \\tau _ c ^ N , k } ( f _ k ) < - \\zeta _ 1 ( \\epsilon , x ) \\right \\} \\cap \\{ | \\tau _ c ^ N - \\tau _ c | \\leq \\delta \\} \\end{align*}"}
+{"id": "3639.png", "formula": "\\begin{align*} f ( z _ 0 , w _ 0 \\sigma ( z _ 0 ) ^ 2 ) = L _ 1 ( w _ 0 \\sigma ( z _ 0 ) ^ 2 ) = 1 , f ( z _ 0 , w _ 0 ) = L _ 1 ( w _ 0 ) = 0 , \\end{align*}"}
+{"id": "6872.png", "formula": "\\begin{align*} H _ { { \\varphi } \\sf E } = A _ 0 + A _ 1 , A _ i = \\sum _ { j \\in \\mathbb { J } _ i } E _ j \\langle \\varphi _ j , \\cdot \\rangle \\varphi _ j . \\end{align*}"}
+{"id": "1798.png", "formula": "\\begin{align*} D ^ + _ { v } \\Xi ( z , z , u ) = \\lim _ { t \\to 0 + } \\dfrac { \\Xi ( z + t \\ , v , z , u ) - \\Xi ( z , z , u ) } { t } \\ , . \\end{align*}"}
+{"id": "8258.png", "formula": "\\begin{align*} P _ { Y ^ j , \\underline S } ( y ^ j , \\underline S ) = \\frac { c _ { y ^ { j } } } { M } , \\\\ P _ { Y _ { j + 1 } | Y ^ { j } , \\underline S } ( 1 | y ^ j \\underline S ) = \\frac { b _ { y ^ { j } } } { c _ { y ^ { j } } } , \\end{align*}"}
+{"id": "3049.png", "formula": "\\begin{align*} \\psi _ { \\alpha , q } ( z ) = \\sum _ { j = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ { j } } { j ! \\Gamma ( j + q + \\frac { 3 } { 2 } ) } \\left ( \\frac { \\alpha z } { 2 } \\right ) ^ { 2 j + q + 1 } \\end{align*}"}
+{"id": "2583.png", "formula": "\\begin{align*} q ^ { - 1 } c _ { n - 3 } = & \\Big [ \\sum \\limits _ { i = 1 } ^ { \\frac { n } { 2 } - 2 } b _ i ( q ^ { - ( \\frac { n } { 2 } - i ) } + q ^ { \\frac { n } { 2 } - 2 - i } ) + b _ { \\frac { n } { 2 } - 1 } q ^ { - 1 } \\Big ] 2 h ^ 2 \\\\ & + ( q ^ { - ( \\frac { n } { 2 } - 1 ) } + q ^ { - ( \\frac { n } { 2 } - 3 ) } + \\ldots + q ^ { \\frac { n } { 2 } - 5 } + q ^ { \\frac { n } { 2 } - 3 } ) \\end{align*}"}
+{"id": "8052.png", "formula": "\\begin{align*} \\limsup _ { N \\rightarrow + \\infty } \\frac { N } { \\gamma _ N ^ 2 } \\log P ( | \\varepsilon ^ N _ { 1 2 } | \\geq \\epsilon ) = - \\infty \\end{align*}"}
+{"id": "1207.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd t } \\bigg | _ { t = 0 } \\vert ( u + t \\varphi ) ( x ) \\vert ^ { m } = m \\vert u ( x ) \\vert ^ { m - 2 } u ( x ) \\varphi ( x ) , \\ \\ \\forall \\ , \\varphi \\in L ^ m ( \\Omega ) . \\end{align*}"}
+{"id": "1915.png", "formula": "\\begin{align*} V ( x ) \\rho ( x ) \\geq c _ 0 \\ , \\mathbf { d } ( x ) ^ { - \\alpha } \\rho ( x ) = c _ 0 \\ , \\mathbf { d } ( x ) ^ { - \\alpha } \\cdot \\max \\{ \\mathbf { d } ^ \\alpha ( x ) , r ^ \\alpha \\} \\geq c _ 0 \\forall \\ , \\ , x \\in G ; \\end{align*}"}
+{"id": "6754.png", "formula": "\\begin{align*} 1 / \\tau ^ { k - 1 } = ( 1 - \\tau ^ k ) / \\tau ^ k , ~ \\tau ^ { - 1 } \\in ( 0 , 1 ) . \\end{align*}"}
+{"id": "4388.png", "formula": "\\begin{align*} \\inf _ { x \\in \\mathcal { X } } & \\ \\sum _ { i \\in [ m ] } \\max \\{ 0 , x _ i - \\overline { b } _ i , x _ i - \\overline { b } _ i + \\Delta b _ i - \\Delta b _ k \\} + \\Gamma \\Delta b _ k \\\\ \\mathrm { s . t . } & \\ x _ k \\in [ \\overline { b } _ k - \\Delta b _ k , \\overline { b } _ k ] \\end{align*}"}
+{"id": "8154.png", "formula": "\\begin{align*} P _ { \\hat { \\underline S } | \\underline X ^ L Y ^ L } ( { \\underline s } | \\underline x ^ L , y ^ L ) & = P _ { { \\underline S } | \\underline X ^ L Y ^ L } ( { \\underline s } | \\underline x ^ L , y ^ L ) \\\\ & = \\frac { P _ { { \\underline S } , \\underline X ^ L Y ^ L } ( { \\underline s } , \\underline x ^ L , y ^ L ) } { P _ { \\underline X ^ L Y ^ L } ( \\underline x ^ L , y ^ L ) } . \\end{align*}"}
+{"id": "4550.png", "formula": "\\begin{align*} \\Big | p _ \\infty ( [ s ] _ \\infty ) - p _ \\infty ( [ s ' ] _ \\infty ) \\Big | \\leq \\delta \\ , V _ { \\infty , \\beta , a } ( [ s ] _ \\infty , [ s ' ] _ \\infty ) \\textup { a n d } \\gamma : = \\frac { \\beta a _ - } { 1 + \\beta } - \\frac { a \\delta } { \\beta } > 0 . \\end{align*}"}
+{"id": "912.png", "formula": "\\begin{align*} u = g \\quad \\partial D . \\end{align*}"}
+{"id": "5750.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } e ^ { \\varepsilon t } \\| u \\| _ { C ^ 1 } ( t ) = \\infty . \\end{align*}"}
+{"id": "5976.png", "formula": "\\begin{align*} E _ G ( 0 ) \\geq \\lim _ { n \\rightarrow \\infty } - \\frac { 1 } { n } \\cdot \\frac { n - | V | B } { B } \\cdot \\frac { M - 1 } { M } \\cdot B ( | f | - \\epsilon ) = \\frac { M - 1 } { M } \\cdot ( | f | - \\epsilon ) . \\end{align*}"}
+{"id": "5719.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\ell + 1 } \\Vert \\mathbf { L } ^ j ( v , w ) \\Vert _ { G } \\approx \\Vert ( v , w ) \\Vert _ { H ^ 1 \\times H ^ 0 } + \\Vert \\mathbf { L } ( v , w ) \\Vert _ { H ^ \\ell \\times H ^ { \\ell - 1 } } \\end{align*}"}
+{"id": "4784.png", "formula": "\\begin{align*} v _ x = - C \\left ( \\left ( \\frac { J _ { \\varphi _ 2 ( k ) } x } { \\varphi _ 2 ( k ) } \\right ) _ k \\right ) . \\end{align*}"}
+{"id": "442.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & x ^ { q } _ i ( t ) = x _ i ^ * ( t _ q ) , \\forall \\ , i \\in V _ m \\backslash V _ { B ^ q } , \\\\ [ 2 m m ] & x ^ { q } _ i ( t ) = \\big ( ( \\sum _ { i \\in V _ { B ^ q } } f _ i ) ^ { * } \\big ) ' \\bigg ( t + \\beta ^ { q } \\bigg ) , \\forall ~ i \\in V _ { B ^ q } , \\\\ [ 2 m m ] & x ^ { q } _ { m + 1 } ( t ) = ( f ^ * _ { m + 1 } ) ' ( - t ) , \\\\ \\end{aligned} \\right . \\end{align*}"}
+{"id": "1159.png", "formula": "\\begin{align*} Q ( I , k ) & = I \\times [ \\ell ( I ) k , \\ell ( I ) ( k + 1 ) ) \\subset R \\times [ \\ell ( I ) k , 0 ) \\\\ & \\subset \\widetilde { R } \\times \\left [ - \\ell \\left ( \\widetilde { R } \\right ) , 0 \\right ) = P _ R . \\end{align*}"}
+{"id": "2728.png", "formula": "\\begin{align*} & { _ { * } \\Phi } ^ { ( s + 1 ) } _ { \\alpha } = \\{ { _ { * } \\Phi } ^ { ( s ) } _ { \\alpha } , H _ { T } \\} + \\frac { \\partial { _ { * } \\Phi } ^ { ( s ) } _ { \\alpha } } { \\partial t } : \\approx 0 , \\\\ & { _ { * } \\Phi } ^ { ( m _ { \\alpha } + 1 ) } _ { \\alpha } : = { _ { * } X } _ { T } { _ { * } \\Phi } ^ { ( m _ { \\alpha } ) } _ { \\alpha } = { _ { * } C } _ { \\alpha s } ^ { \\beta } { _ { * } \\Phi } ^ { ( s ) } _ { \\beta } , \\end{align*}"}
+{"id": "2769.png", "formula": "\\begin{align*} \\sigma _ { 3 } : & T ^ { * } M | _ { Q , P } \\rightarrow T ^ { * } M \\\\ & ; ( \\sigma _ { 3 } ^ { * } \\Xi ^ { a } : = \\epsilon ^ { a } , \\sigma _ { 3 } ^ { * } \\Psi _ { a } : = \\epsilon _ { a } , \\sigma _ { 3 } ^ { * } \\Theta ^ { \\alpha } : = \\epsilon ^ { \\alpha } , \\sigma _ { 3 } ^ { * } \\Theta _ { \\alpha } : = \\epsilon _ { \\alpha } , \\sigma _ { 3 } ^ { * } Q ^ { i } , \\sigma _ { 3 } ^ { * } P _ { i } ) \\\\ & \\mapsto ( \\Xi ^ { a } , \\Psi _ { a } , \\Theta ^ { \\alpha } , \\Theta _ { \\alpha } , Q ^ { i } , P _ { i } ) \\end{align*}"}
+{"id": "2855.png", "formula": "\\begin{align*} \\mathcal { M } ( f ) ( x ) : = \\sup _ { B \\ni x } \\frac { 1 } { | B | } \\int _ B \\left | f ( y ) \\right | \\ , d y , \\end{align*}"}
+{"id": "1494.png", "formula": "\\begin{align*} & f _ { 1 } ( m ) f _ 2 ( n ) [ m - n ] - f _ 1 ( m ) f _ 2 ( m + n ) [ m + d - n ] = 0 , \\\\ & f _ 1 ( m ) g _ 2 ( n ) [ m - n ] - g _ 1 ( m ) f _ 2 ( n ) [ n - m ] - f _ 1 ( m ) g _ 2 ( m + n ) [ m + d - n ] = 0 . \\end{align*}"}
+{"id": "4608.png", "formula": "\\begin{align*} & \\quad \\int _ D | \\nabla u | ^ 2 + \\int _ D \\frac { 1 } { 2 } \\tilde { V } | u | ^ 2 \\geq \\int _ D | \\nabla v | ^ 2 + \\int _ D \\frac { 1 } { 2 } \\tilde { V } \\left ( 2 | v | ^ 2 + 2 | u _ D | ^ 2 \\right ) \\\\ & = \\int _ D | \\nabla v | ^ 2 + \\int _ D \\tilde { V } | v | ^ 2 + \\left | \\frac { 1 } { | D | } \\int _ D u \\right | ^ 2 \\int _ D \\tilde { V } \\ge \\frac { 1 } { | D | } \\int _ D \\tilde { V } \\int _ D | u | ^ 2 . \\end{align*}"}
+{"id": "4129.png", "formula": "\\begin{align*} & \\psi ^ { ( r ) } ( \\theta ) - \\frac { 1 } { 1 - d _ j \\eta _ j } \\psi ^ { ( r ) } _ j ( \\theta ) = o ( 1 ) , \\end{align*}"}
+{"id": "8905.png", "formula": "\\begin{align*} F _ k ( x _ 1 , \\dots , x _ k ) = ( x _ 1 ) _ k + 2 \\sum _ { m = 1 } ^ { \\lfloor k / 2 \\rfloor } B _ { 2 m } { k \\choose 2 m } ( x _ 1 - m ) _ { k - 2 m } \\Omega _ m ( x _ 2 , \\dots , x _ { 2 m } ) . \\end{align*}"}
+{"id": "7004.png", "formula": "\\begin{align*} \\widetilde { \\Lambda } _ 0 : = \\{ \\mathbf { w } = ( w _ 1 , \\cdots , w _ { n + 1 } ) \\in \\mathbb { C } ^ { n + 1 } : w _ 1 \\ ! = \\ ! \\cdots \\ ! = \\ ! w _ m \\ ! = \\ ! w _ { n + 1 } \\ ! = \\ ! 0 \\} \\cong \\mathbb { C } ^ { n - m + 1 } , \\end{align*}"}
+{"id": "8038.png", "formula": "\\begin{align*} \\mathcal { Y } _ { \\vec { f } } ^ N ( t , \\xi ^ N ) = \\exp \\left ( \\frac { \\gamma _ N ^ 2 } { N } \\left ( \\eta _ t ^ N ( \\vec { f } ) - \\eta _ 0 ^ N ( \\vec { f } ) - \\int _ 0 ^ t \\eta _ s ^ N ( \\mathcal { A } _ s ^ N \\vec { f } ) d s + \\sum _ { k = 1 } ^ 4 \\varepsilon _ { k , t } ^ N ( \\vec { f } ) \\right ) \\right ) , \\end{align*}"}
+{"id": "4527.png", "formula": "\\begin{align*} n _ N ^ { ( K ) } ( t , [ s ] _ { K } ) = \\int _ { s _ K } ^ \\infty n _ N ^ { ( K + 1 ) } ( t , s _ 1 , . . . , s _ { K + 1 } ) \\ , d s _ { K + 1 } . \\end{align*}"}
+{"id": "3595.png", "formula": "\\begin{align*} \\tau _ 0 ( z ) = ( \\tau ' ( z ) ) ^ { \\frac { 1 } { p } } , \\ ; \\tau _ 1 ( z ) = ( \\tau ' \\circ \\tau ( z ) ) ^ { \\frac { 1 } { p } } , \\tau _ 2 ( z ) = ( \\tau ' \\circ \\tau ^ 2 ( z ) ) ^ { \\frac { 1 } { p } } , \\end{align*}"}
+{"id": "9055.png", "formula": "\\begin{align*} \\{ 1 , \\dotsc , n \\} = \\{ i ^ 1 _ 1 , \\dotsc , i ^ 1 _ { m _ 1 } \\} \\sqcup \\{ i ^ 2 _ 1 , \\dotsc , i ^ 2 _ { m _ 2 } \\} \\sqcup \\dotsb \\sqcup \\{ i ^ p _ 1 , \\dotsc , i ^ p _ { m _ p } \\} \\end{align*}"}
+{"id": "5502.png", "formula": "\\begin{align*} X _ { i , j } X _ { s , t } = \\begin{cases} X _ { s , t } X _ { i , j } & i < s , \\ , j > t ; \\\\ q X _ { s , t } X _ { i , j } & ( i = s , \\ , j < t ) ( i < s , \\ , j = t ) ; \\\\ X _ { s , t } X _ { i , j } + ( q - q ^ { - 1 } ) X _ { i , t } X _ { s , j } & i < s , \\ , j < t . \\end{cases} \\end{align*}"}
+{"id": "5783.png", "formula": "\\begin{align*} \\tilde { E } ( u ) & = \\Pi ^ \\perp \\bigg [ E ( u ) - \\left ( H ( u ^ T ) \\right ) '' + m \\left ( H ( u ^ T ) \\right ) ' - \\mathcal { L } _ { \\Sigma } H ( u ^ T ) \\bigg ] \\\\ & = \\Pi ^ \\perp \\bigg [ - \\mathcal { M } _ { \\Sigma } ( u ) + \\mathcal { L } _ { \\Sigma } u + N _ 1 ( u ) - \\left ( H ( u ^ T ) \\right ) '' + m \\left ( H ( u ^ T ) \\right ) ' - \\mathcal { L } _ { \\Sigma } H ( u ^ T ) \\bigg ] \\\\ & = : \\textsc { I + I I } , \\end{align*}"}
+{"id": "3403.png", "formula": "\\begin{align*} \\eta _ { t } \\lambda _ { t } \\sqrt { 2 L } & = \\frac { \\sqrt { \\Delta _ { 1 } } T ^ { \\frac { 1 - p } { 3 p - 2 } } } { 8 \\sqrt { L } \\gamma } \\sqrt { 2 L } \\le \\frac { \\sqrt { \\Delta _ { 1 } } } { 4 \\sqrt { 2 } \\gamma } = C _ { 1 } \\end{align*}"}
+{"id": "3675.png", "formula": "\\begin{align*} E ( S _ { n , d } ) = \\left \\{ \\{ i , j \\} : \\ , i \\in [ d ] , \\ , j \\in [ n ] \\setminus \\{ i \\} \\right \\} . \\end{align*}"}
+{"id": "5457.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } | X _ t ^ { i , N } - \\bar X _ t ^ i | + | V _ t ^ { i , N } - \\bar V _ t ^ i | = 0 \\mbox { a . s . } \\end{align*}"}
+{"id": "5480.png", "formula": "\\begin{align*} \\mathbb E | E | ^ { K } & = \\mathbb E 1 \\cdot | E | ^ n \\cdot | E | ^ { 1 / a _ 1 } \\cdot | E | ^ { 1 / a _ 2 } \\cdots \\\\ & \\le \\left ( \\mathbb E 1 ^ r \\right ) ^ { 1 / r } \\cdot \\left ( \\mathbb E | E | ^ { 2 n } \\right ) ^ { 1 / 2 } \\cdot \\left ( \\mathbb E | E | ^ 2 \\right ) ^ { 1 / 2 a _ 1 } \\cdots \\\\ & = \\left ( \\mathbb E | E | ^ { 2 n } \\right ) ^ { 1 / 2 } \\cdot \\left ( \\mathbb E | E | ^ 2 \\right ) ^ { ( K - n ) / 2 } \\end{align*}"}
+{"id": "4678.png", "formula": "\\begin{align*} \\mathbf { B } _ { - } ( D ) = \\bigcap _ { E \\equiv D } \\mathrm { S u p p } ( E ) , \\end{align*}"}
+{"id": "8507.png", "formula": "\\begin{align*} G _ k ( a ) = F _ { n _ k } ( a ) = F _ { n _ k } ( a _ { n _ k } ) \\to x \\end{align*}"}
+{"id": "5688.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } t ^ { 1 / ( p - 2 ) } \\Vert u ( t ) \\Vert _ { L ^ 2 } = \\infty . \\end{align*}"}
+{"id": "8437.png", "formula": "\\begin{align*} = \\frac { 1 } { r ^ 2 } \\left ( N \\| c \\| ^ 2 ( r ^ 2 - 1 ) + \\sum _ J \\| g _ J \\| ^ 2 \\right ) . \\end{align*}"}
+{"id": "4982.png", "formula": "\\begin{align*} S _ { N - 1 , i } ( \\{ \\nu \\} \\setminus \\nu _ i ) = \\prod _ { \\substack { j , k = 1 \\\\ j , k \\ne i } } ^ N ( \\nu _ j - \\nu _ k + 1 ) , \\end{align*}"}
+{"id": "2683.png", "formula": "\\begin{align*} S ^ { ( d ) } = \\int ^ { t _ { f } } _ { t _ { i } } L ^ { ( d ) } \\left ( D ^ { d } { q ^ { i } } , \\cdots , D q ^ { i } , q ^ { i } , t \\right ) d t , \\end{align*}"}
+{"id": "1968.png", "formula": "\\begin{align*} E q u > & p ^ s + 1 - ( \\tau + 1 ) p ^ { t } - ( p - \\tau ) p ^ { s - t - 1 } ( 1 + \\frac { \\delta - 1 } { \\frac { p - \\tau - 1 } { \\tau + 1 } \\delta + 1 } ) \\\\ \\ge & p ^ s + 1 - ( \\tau + 1 ) p ^ { t } - ( p - \\tau ) ( \\tau + 2 ) p ^ { s - t - 1 } \\end{align*}"}
+{"id": "5422.png", "formula": "\\begin{align*} S _ { h _ k } f ( x ) - S _ { h _ k } f ( \\tfrac { 1 } { 2 } ) = f ( T ^ { \\ell _ k + i } ( x ) ) = \\left \\{ \\begin{array} { l c r } 1 & & T ^ { \\ell _ k + i } ( x ) < \\tfrac { 1 } { 2 } , \\\\ 0 & & T ^ { \\ell _ k + i } ( x ) = \\tfrac { 1 } { 2 } , \\\\ - 1 & & T ^ { \\ell _ k + i } ( x ) > \\tfrac { 1 } { 2 } . \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "5344.png", "formula": "\\begin{align*} v = \\min \\ , \\left \\{ \\sum _ { k = 1 } ^ m \\sum _ { j _ k \\in J _ k } c _ { j _ k } ^ k \\ , x _ { j _ k } ^ { k , u } : u \\in \\mathcal { U } \\right \\} , \\end{align*}"}
+{"id": "8468.png", "formula": "\\begin{align*} & \\left \\| ( u _ h ) _ + - ( u _ { h ' } ) _ + \\right \\| _ { L ^ \\gamma ( K _ T ) } \\\\ [ 2 m m ] & \\leq \\left \\| ( u _ h ) _ + - ( u _ h ) _ + ^ { ( \\ell ) } \\right \\| _ { L ^ \\gamma ( K _ T ) } + \\left \\| ( u _ h ) _ + ^ { ( \\ell ) } - ( u _ { h ' } ) _ + ^ { ( \\ell ) } \\right \\| _ { L ^ \\gamma ( K _ T ) } + \\left \\| ( u _ { h ' } ) _ + ^ { ( \\ell ) } - ( u _ { h ' } ) _ + \\right \\| _ { L ^ \\gamma ( K _ T ) } \\\\ [ 2 m m ] & : = \\mathbf { I } _ { h , \\ell } + \\mathbf { I } _ { h , h ' , \\ell } + \\mathbf { I } _ { h ' , \\ell } . \\end{align*}"}
+{"id": "6130.png", "formula": "\\begin{align*} \\| \\rho ^ { ( r ) } _ { m , n } \\big ( \\rho ^ { ( r ) } _ { n , k } ( x ) \\rho ^ { ( r ) } _ { n , k } ( x ) ^ * \\big ) \\| & < \\| \\rho ^ { ( r ) } _ { m , k } ( x ) ^ * \\| ^ 2 + \\eta = \\| \\rho ^ { ( r ) } _ { m , k } ( x ) \\| ^ 2 + \\eta \\ \\\\ \\| \\rho ^ { ( r ) } _ { m , n } \\big ( \\rho ^ { ( r ) } _ { n , k } ( y ) ^ * \\rho ^ { ( r ) } _ { n , k } ( y ) \\big ) \\| & < \\| \\rho ^ { ( r ) } _ { m , k } ( y ) \\| ^ 2 + \\eta . \\end{align*}"}
+{"id": "2463.png", "formula": "\\begin{align*} \\phi ' ( s ) = p c ^ { - 1 } \\phi ^ { 2 } - p c ^ { - 3 } ( \\mu c ^ { 2 } + 1 ) \\phi ^ { 3 } + O ( \\phi ^ { 4 } ) . \\end{align*}"}
+{"id": "6070.png", "formula": "\\begin{align*} f ( \\lambda ) = 0 \\ \\Longrightarrow \\ \\lambda \\in \\sigma _ { \\rm p } ( T ) \\ \\Longrightarrow \\ e ^ { \\pm i \\arccos \\lambda } \\in \\sigma _ { \\rm p } ( U ) . \\end{align*}"}
+{"id": "8054.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\limsup _ { N \\rightarrow + \\infty } \\sup _ { \\sigma \\in \\mathcal { T } , t \\leq \\delta } \\widetilde { P } _ { f _ 1 , f _ 2 , f _ 3 } ^ { N , \\tilde { F } , \\tilde { G } , \\tilde { H } } \\left ( \\left | \\eta ^ N _ { t + \\sigma } ( \\vec { f } ) - \\eta ^ N _ { \\sigma } ( \\vec { f } ) \\right | > \\epsilon \\right ) = 0 \\end{align*}"}
+{"id": "3254.png", "formula": "\\begin{align*} & \\big ( S ^ { \\wedge , ( l ) } \\ : V \\ : S ^ { \\vee , ( r ) } \\big ) ( x , y ) \\\\ & = - 2 \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { n ! } \\int _ { - \\infty } ^ 0 \\alpha ^ { l } \\ : ( 1 - \\alpha ) ^ r \\ : ( \\alpha - \\alpha ^ 2 ) ^ n \\ : ( \\Box ^ n V ) \\big | _ { \\alpha y + ( 1 - \\alpha ) x } \\ : d \\alpha \\ : S ^ { \\bowtie , ( n + l + 1 ) } ( x , y ) \\ : . \\end{align*}"}
+{"id": "7823.png", "formula": "\\begin{align*} \\max _ { \\substack { \\boldsymbol { V } } } \\sum _ { k = 1 } ^ { K } R _ { k } \\left ( \\boldsymbol { v } _ { k } \\right ) . \\end{align*}"}
+{"id": "4096.png", "formula": "\\begin{align*} \\lambda = \\alpha + P ' \\lambda , \\end{align*}"}
+{"id": "5972.png", "formula": "\\begin{align*} d _ H ( \\vec { x } ^ { ( j ) } _ { m _ 1 , \\ell _ 1 } , \\vec { x } ^ { ( j ) } _ { m _ 2 , \\ell _ 2 } ) \\geq B / 2 + \\min ( \\ell _ 1 , \\ell _ 2 ) \\geq \\max ( \\ell _ 1 , \\ell _ 2 ) + \\min ( \\ell _ 1 , \\ell _ 2 ) = \\ell _ 1 + \\ell _ 2 , \\end{align*}"}
+{"id": "4229.png", "formula": "\\begin{align*} { \\widehat { A } ( T X , \\nabla ^ { T X } ) } { \\rm c h } ( 2 ^ k \\Theta _ 3 ( T _ { C } X ) ) = \\prod _ { j = 1 } ^ { k } \\frac { 2 x _ j \\theta ' ( 0 , \\tau ) } { \\theta ( x _ j , \\tau ) } \\frac { \\theta _ 3 ( x _ j , \\tau ) } { \\theta _ 3 ( 0 , \\tau ) } , \\end{align*}"}
+{"id": "1739.png", "formula": "\\begin{align*} c _ { \\sigma ( j ) } : = \\begin{cases} 1 & \\mbox { i f } S _ { \\sigma ( j - 1 ) , \\sigma ( j ) } = 0 \\\\ \\frac { 1 } { c _ { \\sigma ( j - 1 ) } S _ { \\sigma ( j - 1 ) , \\sigma ( j ) } } & \\mbox { i f } S _ { \\sigma ( j - 1 ) , \\sigma ( j ) } \\neq 0 . \\end{cases} \\end{align*}"}
+{"id": "6251.png", "formula": "\\begin{align*} r _ i = \\left ( 2 - \\frac { 3 } { \\c d } \\right ) \\lambda _ g . \\end{align*}"}
+{"id": "1810.png", "formula": "\\begin{align*} v = S \\left ( z , \\mathbf q ( z , u , v ) \\right ) ( u ) \\end{align*}"}
+{"id": "4581.png", "formula": "\\begin{align*} c _ 1 ( Y ) [ \\Sigma _ a ] - c _ 1 ( Y ) [ \\Sigma _ { - a } ] = \\sum _ { t _ j \\in ( - a , a ) } - \\frac { 1 } { r _ j s _ j } . \\end{align*}"}
+{"id": "148.png", "formula": "\\begin{align*} 2 \\bigg [ G \\big ( s + \\log ( r _ 1 ) \\big ) \\cdot { r _ 1 } ^ { \\alpha ( \\eta ) } + G \\big ( s + \\log ( r _ 2 ) \\big ) \\cdot { r _ 2 } ^ { \\alpha ( \\eta ) } \\bigg ] = G ( s ) + e ^ { ( 2 n - \\alpha ( \\eta ) ) s } R ( e ^ { - s } ) . \\end{align*}"}
+{"id": "5805.png", "formula": "\\begin{align*} | Y _ + ( t ) | ^ 2 + | Y _ 0 ( t ) | ^ 2 + | Y _ - ( t ) | ^ 2 & \\leq \\sum _ { k + \\ell \\leq s } \\sum _ { i = 1 } ^ \\infty | \\mathcal { E } ^ { k , \\ell } _ i ( t ) | ^ 2 = \\sum _ { k + \\ell \\leq s } \\| \\mathcal { L } _ { \\Sigma } ^ { \\ell } \\partial ^ { k } _ t E _ 2 ( u ) \\| _ { L ^ 2 } ^ 2 ( t ) . \\end{align*}"}
+{"id": "1853.png", "formula": "\\begin{align*} \\hat { \\mathcal { E } } ( A , B _ r ( x ) ) : = r ^ { - 3 } \\int _ { B _ r ( x ) } | F _ A | ^ 2 , \\end{align*}"}
+{"id": "5360.png", "formula": "\\begin{align*} v _ i ^ S & = v _ i ^ { S \\cup \\{ j \\} } + c ^ { S } _ j \\ , x _ { i j } ^ { 1 , S \\cup \\{ j \\} } , j \\in N ^ { \\{ 0 , 1 \\} } \\setminus S \\\\ v _ i ^ { S \\setminus \\{ j \\} } & = v _ i ^ S + c ^ { S } _ j \\ , x _ { i j } ^ { 0 , S \\setminus \\{ j \\} } , j \\in S . \\end{align*}"}
+{"id": "851.png", "formula": "\\begin{align*} W _ { i } : = g ( X , u ) \\dot \\delta _ { i } h - F ^ { - 1 } g ( X , l ) h . l _ { i } , \\end{align*}"}
+{"id": "7685.png", "formula": "\\begin{align*} \\lambda _ 0 = \\sup _ { s \\in ( 0 , 1 ) } \\sup _ { \\mu > 0 } \\inf _ { E \\in I } \\widehat { \\lambda } _ { s , \\mu } ( E ) \\end{align*}"}
+{"id": "7549.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ \\Omega \\tfrac { 1 } { 2 } \\bar \\rho | \\bar u | ^ 2 + h _ 1 ( \\bar \\rho ) + h _ 2 ( \\bar \\rho ) \\ d x = 0 \\ , \\end{align*}"}
+{"id": "2835.png", "formula": "\\begin{align*} K ^ { ( 1 ) } = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{bmatrix} \\end{align*}"}
+{"id": "5571.png", "formula": "\\begin{align*} \\overline { f _ { \\phi _ i , t } } & = \\nu _ i ^ { 2 t } \\phi _ i , \\\\ \\overline { f _ { \\phi _ i , t } f _ { \\phi _ j , t } } & = ( \\nu _ i \\nu _ j ) ^ { 2 t } \\left ( \\sum _ { s = 0 } ^ { t } \\frac { \\Phi ^ s } { ( \\nu _ i \\nu _ j d ) ^ { 2 s } } \\right ) ( \\phi _ i \\circ \\phi _ j ) \\end{align*}"}
+{"id": "8278.png", "formula": "\\begin{align*} S ( t ) x = 0 \\mbox { f o r a l l } t \\geq 0 \\Rightarrow x = 0 \\end{align*}"}
+{"id": "3357.png", "formula": "\\begin{align*} d ( \\sigma ^ 1 , \\sigma ^ 2 ) = | | \\sigma ^ 1 - \\sigma ^ 2 | | _ { L ^ 1 } = \\int _ { \\mathbb { R } ^ n } | \\sigma ^ 1 ( z ) - \\sigma ^ 2 ( z ) | d z . \\end{align*}"}
+{"id": "3342.png", "formula": "\\begin{align*} & \\mathcal { L } _ { A _ f + X _ f } ( \\mathbb { J } + J ) , \\\\ = & \\mathcal { L } _ { A _ f } ( \\mathbb { J } ) + \\mathcal { L } _ { A _ f } ( J ) + \\mathcal { L } _ { X _ f } ( \\mathbb { J } ) + \\mathcal { L } _ { X _ f } ( J ) , \\\\ = & \\mathcal { L } _ { A _ f } ( J ) + \\mathcal { L } _ { X _ f } ( J ) . \\end{align*}"}
+{"id": "4949.png", "formula": "\\begin{align*} \\int _ { X } e ^ { L ' } _ { \\pm } = ( - 1 ) ^ { n ( n - 1 ) / 2 } \\cdot \\frac { 1 } { 4 \\beta } . \\end{align*}"}
+{"id": "3164.png", "formula": "\\begin{align*} y & = \\left ( a _ { s - \\ell } ^ \\tau k ^ s + \\sum _ { r = \\ell } ^ { s - 1 } a _ { r - \\ell } k ^ r \\right ) t + i k ^ \\ell + \\sum _ { r = 0 } ^ { \\ell - 1 } a _ { r + s + 1 - \\ell } k ^ r \\\\ & = a _ { s - \\ell } ^ \\tau n + \\left ( \\sum _ { r = \\ell } ^ { s - 1 } a _ { r - \\ell } k ^ r \\right ) t + i k ^ \\ell + \\sum _ { r = 0 } ^ { \\ell - 1 } a _ { r + s + 1 - \\ell } k ^ r . \\end{align*}"}
+{"id": "254.png", "formula": "\\begin{align*} Q = \\left ( A \\frac { d } { d x } - 3 A ' \\right ) \\left ( A \\frac { d } { d x } + \\gamma A - A ' \\right ) b . \\end{align*}"}
+{"id": "8986.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } F ( D ^ 2 u ) - \\beta { u r ^ { - \\gamma } } = r ^ { - \\gamma } f ( r ) & \\hbox { i n } \\ B ( 0 , 1 ) \\setminus \\{ 0 \\} \\\\ u = b & \\hbox { o n } \\ \\partial B ( 0 , 1 ) \\end{array} \\right . \\end{align*}"}
+{"id": "5474.png", "formula": "\\begin{align*} c = a ^ { \\frac { K - 1 } { 2 K } } . \\end{align*}"}
+{"id": "1007.png", "formula": "\\begin{align*} \\| u \\| ^ 2 _ { V ^ D } = 2 \\int _ { D \\times \\mathbb R ^ d } ( u ( x ) - u ( y ) ) ^ 2 j ( x , y ) \\ , d x \\ , d y < \\infty . \\end{align*}"}
+{"id": "5441.png", "formula": "\\begin{align*} r ^ { S _ 0 \\oplus S _ 1 } _ { ( a ^ - , i ) } = R _ i - ( 1 - a ^ - ) c _ i + \\beta \\sum _ { j \\in N } p _ { i j } f ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , j ) } - \\beta f ^ { S _ 0 \\oplus S _ 1 } _ { ( 0 , i ) } . \\end{align*}"}
+{"id": "559.png", "formula": "\\begin{align*} - \\frac { P } { \\rho } & = g z + \\Phi _ t + \\frac { 1 } { 2 } \\left \\| \\nabla \\Phi \\right \\| ^ 2 & \\Omega . \\end{align*}"}
+{"id": "7703.png", "formula": "\\begin{align*} I _ { q } ( \\Omega ) = \\frac { q } { \\omega _ { n } } \\int _ { \\Omega } \\widetilde { V } _ { q - 1 } ( \\Omega , z ) d z . \\end{align*}"}
+{"id": "2982.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\left | \\tilde { \\mathcal { L } } ( t ) \\right | } { D _ 1 } & = \\left | \\int _ { D _ { X , [ l ] } ( 0 ) } ^ { D _ { X , [ l ] } ( t ) } \\phi ( s ) d s \\right | = \\int ^ { D _ { X , [ l ] } ( 0 ) } _ { D _ { X , [ l ] } ( t ) } \\phi ( s ) d s \\\\ & \\geq \\int ^ { D _ { X , [ l ] } ( 0 ) } _ { B e ^ { - t C } } \\phi ( s ) d s = \\frac { B ^ { 1 - \\alpha } e ^ { t C ( \\alpha - 1 ) } - \\left ( D _ { X , [ l ] } ( 0 ) \\right ) ^ { 1 - \\alpha } } { \\alpha - 1 } . \\end{aligned} \\end{align*}"}
+{"id": "79.png", "formula": "\\begin{align*} \\begin{aligned} \\begin{cases} w _ 1 = p - \\gamma - 2 \\sqrt { ( p - 1 ) ( 1 - \\gamma ) } + \\eta , & w _ 2 = 2 , \\\\ w _ 3 = 0 , & w _ 4 = 0 , \\end{cases} \\end{aligned} \\end{align*}"}
+{"id": "8654.png", "formula": "\\begin{align*} \\sup _ { x \\in K } \\langle x , y _ 0 ^ * \\rangle - F ( x ) & = \\langle x _ 0 , y _ 0 ^ * \\rangle - F ( x _ 0 ) , \\\\ & \\leq F ^ * ( y _ 0 ^ * ) - D ( y _ 0 ^ * , x _ 0 ^ * ) , \\\\ & \\leq F ^ * ( y _ 0 ^ * ) - \\inf _ { x ^ * \\in ( F ^ * ) } D ( y _ 0 ^ * , x ^ * ) , \\\\ & = \\sup _ { x \\in K } \\langle x , y _ 0 ^ * \\rangle - F ( x ) . \\end{align*}"}
+{"id": "5192.png", "formula": "\\begin{align*} T = \\frac { 1 } { ( \\beta - 1 ) ( \\alpha + \\beta - 1 ) } \\end{align*}"}
+{"id": "7005.png", "formula": "\\begin{gather*} \\widetilde { Q } _ m : \\ ; \\mathbb { C } ^ { n , 1 } \\rightarrow \\mathbb { C } ^ { m , 1 } \\\\ \\mathbf { z } = ( z _ 1 , z _ 2 , . . . , z _ { n + 1 } ) \\mapsto \\widetilde { Q } _ m ( \\mathbf { z } ) = ( z _ 1 , \\cdots , z _ m , z _ { n + 1 } ) . \\end{gather*}"}
+{"id": "1412.png", "formula": "\\begin{align*} \\varphi ( x ) & = \\int _ 0 ^ \\infty e ^ { - x t } \\ , d F ( t ) \\end{align*}"}
+{"id": "8955.png", "formula": "\\begin{align*} & 0 < \\alpha : = \\min \\Big \\{ | B _ { \\frac { r _ i } { 2 } } ( x _ i ) \\cap B _ { \\frac { r _ j } { 2 } } ( x _ j ) \\cap \\Omega | \\ , : \\ , i , j \\in \\{ 1 , \\dots , N \\} , B _ { \\frac { r _ i } { 2 } } ( x _ i ) \\cap B _ { \\frac { r _ j } { 2 } } ( x _ j ) \\cap \\Omega \\neq \\emptyset \\Big \\} . \\end{align*}"}
+{"id": "3157.png", "formula": "\\begin{align*} ( u ( t ) - I _ 3 ) \\ , P = P \\ , ( u ^ \\sharp ( t ) - I _ 3 ) . \\end{align*}"}
+{"id": "5079.png", "formula": "\\begin{align*} \\Psi _ s = \\sum _ { r \\in ( \\mathbb { F } _ { q ^ k } ) ^ \\times } \\psi ( r ) H _ { s } ( r ) , \\end{align*}"}
+{"id": "9001.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\left | g _ { \\lambda } ^ n ( t , x _ n , y _ n ) - \\int _ 0 ^ { \\infty } e ^ { - \\lambda t } \\mathbb { E } [ p ^ { \\infty } ( N _ { t } ^ n / n , x _ n , y _ n ) ] d t \\right | = 0 . \\end{align*}"}
+{"id": "5138.png", "formula": "\\begin{align*} L _ { d } D 1 ( p \\| q ) = & \\left [ \\frac { A ^ { a - 1 } \\left ( p , q \\right ) - A ^ { b - 1 } \\left ( p , q \\right ) } { a - b } \\right ] \\\\ & - \\left [ \\frac { ( X + Y ) ^ { a - 1 } \\left ( p , q \\right ) - ( X + Y ) ^ { b - 1 } \\left ( p , q \\right ) } { a - b } \\right ] \\end{align*}"}
+{"id": "76.png", "formula": "\\begin{align*} \\lim _ { y \\to 0 , y \\neq 0 } F \\big ( D g ( y ) , D ^ 2 g ( y ) \\big ) = 0 . \\end{align*}"}
+{"id": "6745.png", "formula": "\\begin{align*} Q ( m , z , p ) & = \\sum _ { j = 0 } ^ { p - 1 } \\frac { ( - 2 ) ^ j } { m ^ { j + 2 } } \\sum _ { k = 1 } ^ \\infty \\frac { ( - 1 ) ^ k \\zeta ( 2 k ) z ^ { 2 k } } { k ^ { p - j } } \\\\ & - 2 \\sum _ { j = 0 } ^ { p - 1 } \\frac { ( - 2 ) ^ j } { m ^ { j + 2 } } P ( m , z , p - 1 - j ) + \\left ( - \\frac { 2 } { m } \\right ) ^ { p } S _ { 3 , m } ( z ) , \\end{align*}"}
+{"id": "8782.png", "formula": "\\begin{align*} \\mu ( \\{ x \\} ) \\int _ \\R ( z - x ) ^ + \\pi _ x ( d z ) & - \\mu ( \\{ x \\} ) \\int _ \\R ( x - z ) ^ + \\pi _ x ( d z ) = 0 \\\\ & = \\int _ { F _ \\nu ( x ) } ^ { F _ \\nu ( x ) + p _ + ( x ) } ( F _ \\nu ^ { - 1 } ( v ) - x ) d v - \\int _ { F _ \\nu ( x - ) - p _ - ( x ) } ^ { F _ \\nu ( x - ) } ( x - F _ \\nu ^ { - 1 } ( v ) ) d v . \\end{align*}"}
+{"id": "8029.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } I _ { i n i } ( W ^ n _ 0 ) + I _ { d y n } ( W ^ n ) = 0 . \\end{align*}"}
+{"id": "973.png", "formula": "\\begin{align*} u ( X _ t ) = g ( X _ { \\tau _ V } ) + A ^ \\mu _ { \\tau _ V } - A ^ \\mu _ t - ( M _ { \\tau _ V } - M _ t ) , t \\le \\tau _ V , P _ x \\end{align*}"}
+{"id": "780.png", "formula": "\\begin{align*} \\omega ( a ^ { - 1 } c , c ^ { - 1 } b ) = \\gamma ( a , c ) \\gamma ( c , b ) \\gamma ( a , b ) ^ { - 1 } \\end{align*}"}
+{"id": "930.png", "formula": "\\begin{align*} \\int _ D | f ( x , u ( y ) ) | G _ D ( x , y ) \\ , d y \\sim \\int _ D e ^ { u ( x ) } \\delta ^ { \\alpha } ( x ) \\ , d x = \\int _ D e ^ { P _ D g ( x ) } e ^ { R ^ D f ( u ( x ) ) } \\delta ^ { \\alpha } ( x ) \\ , d x , \\end{align*}"}
+{"id": "5224.png", "formula": "\\begin{align*} \\sum _ { j } q _ { j } \\frac { \\partial L _ { d } D G H I ( p \\| q ) } { \\partial q _ { j } } = 0 \\end{align*}"}
+{"id": "3779.png", "formula": "\\begin{align*} \\epsilon & = \\tilde { c } _ 0 \\times \\frac { 1 } { \\sqrt { \\sum _ { i \\in { \\mathcal { C } } _ j } N _ i } } + \\tilde { c } _ 1 \\Delta _ { \\max } \\sum _ { i \\in [ M ] } \\sum _ { t = 0 } ^ { T - 1 } \\exp \\left ( - \\tilde { c } _ 2 N _ i n _ x \\left ( \\frac { \\rho ^ { ( i ) } \\| \\Sigma ^ { ( i ) } _ t \\| } { \\rho ^ { ( i ) } \\| \\Sigma ^ { ( i ) } _ t \\| + \\sqrt { n _ x } } \\right ) ^ 2 \\right ) , \\end{align*}"}
+{"id": "4097.png", "formula": "\\begin{align*} \\rho _ j = \\lambda _ j / \\mu _ j . \\end{align*}"}
+{"id": "4503.png", "formula": "\\begin{align*} J _ { \\ell } = J : = \\lceil K \\ell ^ { K } \\log 2 \\rceil \\ll ~ \\ell ^ { K } . \\end{align*}"}
+{"id": "8826.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } = \\Delta u + u ( 1 - u ) , x \\in \\mathbb { R } , \\ , \\ , t \\ge 0 . \\end{align*}"}
+{"id": "3741.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial a } \\mathrm { B } _ { z } \\left ( a , b \\right ) = z ^ { a } \\left \\{ \\ln z \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( 1 - b \\right ) _ { k } \\ , z ^ { z } } { k ! \\left ( a + k \\right ) } - \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( 1 - b \\right ) _ { k } } { k ! \\left ( a + k \\right ) ^ { 2 } } \\right \\} . \\end{align*}"}
+{"id": "1063.png", "formula": "\\begin{align*} \\Vert T \\Vert = \\pi . \\end{align*}"}
+{"id": "8107.png", "formula": "\\begin{align*} \\L _ k : = 2 ^ { - k } \\Z ^ d , \\end{align*}"}
+{"id": "846.png", "formula": "\\begin{align*} { { \\dot \\partial } _ j } ( g ( X , l ) ) = & { { \\dot \\partial } _ j } ( { g _ { k m } } { X ^ k } { l ^ m } ) = 2 { C _ { k m j } } { X ^ k } { l ^ m } + { g _ { k m } } { { \\dot \\partial } _ j } { X ^ k } { l ^ m } + { g _ { k m } } { X ^ k } { { \\dot \\partial } _ j } { l ^ m } \\\\ = & { g _ { k l } } { X ^ k } ( \\frac { { \\delta _ j ^ m - { l ^ m } { l _ j } } } { F } ) = \\frac { { { X _ j } - g ( X , l ) { l _ j } } } { F } . \\end{align*}"}
+{"id": "3039.png", "formula": "\\begin{align*} \\widehat { X } _ { 7 } = - L _ 1 \\ , , \\widehat { X } _ { 5 } = L _ 2 \\ , , \\widehat { X } _ { 3 } = - L _ 3 \\ , , L _ i = \\epsilon _ { i j k } s _ j p _ { k } \\ , . \\end{align*}"}
+{"id": "1902.png", "formula": "\\begin{align*} - ( \\mathrm { D e g } ( x ) + V ( x ) ) u ( x ) + \\sum _ { y \\in \\Omega } P ( x , y ) u ( y ) = f ( x ) - \\sum _ { y \\notin \\Omega } P ( x , y ) g ( y ) \\forall \\ , \\ , x \\in \\Omega . \\end{align*}"}
+{"id": "7879.png", "formula": "\\begin{align*} \\mathbf { D } = \\left ( \\begin{array} { c c c c c c } 2 ( i - j ) & c _ { 1 } & & & & \\\\ c _ { 1 } & 2 ( i - j ) & c _ { 2 } & & & \\\\ & c _ { 2 } & 2 ( i - j ) & c _ { 3 } & & \\\\ & & \\ddots & \\ddots & \\ddots & \\\\ & & & c _ { m - 2 } & 2 ( i - j ) & c _ { m - 1 } \\\\ & & & & c _ { m - 1 } & 2 ( i - j ) \\end{array} \\right ) , \\end{align*}"}
+{"id": "2355.png", "formula": "\\begin{align*} \\mathcal { C } _ { q } = e ^ { - \\frac { 1 } { T } ( \\frac { 1 } { 3 } - \\frac { 1 } { q } ) \\int _ { 0 } ^ { T } ( 3 + \\frac { 3 } { 2 } \\ln \\frac { 4 \\pi ( r ( t ) - 2 ) ( 1 - \\tau ) } { r ( t ) ^ { 2 } } ) d t } . \\end{align*}"}
+{"id": "2123.png", "formula": "\\begin{align*} \\langle \\tilde { M } ^ { \\Theta } _ { f ^ i } , \\tilde { M } ^ { \\Theta } _ { g ^ j } \\rangle ( t ) = \\int _ 0 ^ t \\sigma _ { i j } ( s , \\phi ( s ) , \\nu _ { \\Theta } ( s ) ) \\mathrm { d } s . \\end{align*}"}
+{"id": "7620.png", "formula": "\\begin{align*} W ^ H = \\{ x \\in W : h \\cdot x = x \\mbox { f o r a n y } h \\in H \\} . \\end{align*}"}
+{"id": "1286.png", "formula": "\\begin{align*} m ( t ) - \\bar { m } ( t ) = \\frac { e ^ { - \\lambda _ 2 t } - e ^ { - \\lambda _ 1 t } } { \\lambda _ 1 - \\lambda _ 2 } ( \\lambda _ 2 x _ 0 + v _ 0 ) . \\end{align*}"}
+{"id": "6681.png", "formula": "\\begin{align*} l = \\dd \\ ! \\big ( N / \\big ( ( H \\cap A ) \\Phi _ p ( N ) \\big ) \\big ) = \\dd \\ ! \\big ( N / \\big ( ( H \\cap B ) \\Phi _ p ( N ) \\big ) \\big ) < \\dd ( N / \\Phi _ p ( N ) ) = r _ p . \\end{align*}"}
+{"id": "7042.png", "formula": "\\begin{align*} K ^ g _ t & = \\sum _ { s \\leq t } \\mathbf { 1 } _ A \\left ( \\omega _ { s - } \\right ) g ( \\omega _ { s } ) + \\int _ 0 ^ t \\mathbf { 1 } _ A ( \\omega _ { s - } ) \\bigl ( - \\Delta g + b \\cdot \\nabla g \\bigr ) ( \\omega _ s ) d s \\\\ & = \\sum _ { s \\leq t } \\mathbf { 1 } _ A \\left ( \\omega _ { s - } \\right ) g ( \\omega _ s ) . \\end{align*}"}
+{"id": "5277.png", "formula": "\\begin{align*} \\widetilde { x } ^ { k + 1 } = x ^ { k } \\left ( \\frac { U ^ { k } } { V ^ { k } } \\right ) \\end{align*}"}
+{"id": "4563.png", "formula": "\\begin{align*} \\nabla _ { \\alpha } R _ { \\beta \\gamma } = \\nabla _ { \\alpha } R _ { \\mu \\nu } J ^ \\mu _ { \\beta } J ^ { \\nu } _ { \\gamma } . \\end{align*}"}
+{"id": "3863.png", "formula": "\\begin{align*} \\hat M ^ { 2 , \\kappa } ( t ) = M ^ 1 ( T ) + \\int _ 0 ^ t \\hat \\eta _ { ( 1 ) } ^ { 2 , \\kappa } ( s ) d s - \\int _ 0 ^ t \\hat M ^ { 2 , \\kappa } ( s ) d s , \\ ; t \\ge 0 . \\end{align*}"}
+{"id": "5204.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } D _ { \\alpha \\beta } ( p \\| q ) } { \\partial q _ { j } } = \\underbrace { \\frac { Z _ { A } } { \\alpha } p ^ { \\alpha } _ { j } q ^ { \\beta - 2 } _ { j } } _ { U _ { j } } - \\underbrace { \\frac { Z _ { B } } { \\alpha } q ^ { \\alpha + \\beta - 2 } _ { j } } _ { V _ { j } } \\end{align*}"}
+{"id": "3889.png", "formula": "\\begin{align*} A ^ { - 1 } ( - \\omega _ 1 ) & = ( - 4 / 3 , - 1 , - 5 / 3 , - 2 , - 4 / 3 , - 2 / 3 ) \\\\ A ^ { - 1 } ( - \\omega _ 1 ) + \\alpha _ 1 & = ( - 1 / 3 , - 1 , - 5 / 3 , - 2 , - 4 / 3 , - 2 / 3 ) \\\\ \\vdots \\\\ A ^ { - 1 } ( - \\omega _ 1 ) + \\alpha _ 1 + \\alpha _ 3 + \\alpha _ 4 + \\alpha _ 5 + \\alpha _ 6 & = ( - 1 / 3 , - 1 , - 2 / 3 , - 1 , - 1 / 3 , 1 / 3 ) . \\end{align*}"}
+{"id": "228.png", "formula": "\\begin{align*} \\frac { d ^ 2 x } { d t ^ 2 } = X \\left ( x , \\frac { d x } { d t } \\right ) , \\end{align*}"}
+{"id": "4510.png", "formula": "\\begin{align*} \\mathcal { T } : = \\bigg \\{ \\sup _ { X _ { \\ell - 1 } < n \\leqslant X _ { \\ell } } V ( n ) \\leqslant \\frac { 2 C _ 0 T ( \\ell ) } { \\ell ^ { K / 2 } } \\bigg \\} \\end{align*}"}
+{"id": "7635.png", "formula": "\\begin{align*} ( \\alpha _ n \\pi ( \\widetilde { A } ) ) = ( \\pi ( A ) ) = K . \\end{align*}"}
+{"id": "8873.png", "formula": "\\begin{align*} \\| u ( t ) - e ^ { - i t { \\mathcal K } _ \\lambda } \\phi \\| _ { H ^ 1 _ \\lambda } & = \\Big \\| \\int _ t ^ { \\infty } e ^ { i ( t - s ) \\mathcal { K } _ { \\lambda } } \\mathcal { N } [ u ] d s \\Big \\| _ { H _ { \\lambda } ^ 1 } \\\\ & \\lesssim \\| \\mathcal { N } [ u ] \\| _ { \\mathcal { W } _ { \\lambda } ' ( [ t , \\infty ] ) } \\\\ & \\lesssim \\| u \\| ^ { 2 p - 1 } _ { \\mathcal { W } _ { \\lambda } ( [ t , \\infty ] ) } \\rightarrow 0 \\end{align*}"}
+{"id": "1966.png", "formula": "\\begin{align*} E q u : = n - k + 1 - ( \\left \\lceil \\frac { k } { r } \\right \\rceil - 1 ) ( \\delta - 1 ) - d . \\end{align*}"}
+{"id": "6327.png", "formula": "\\begin{align*} [ q _ x , a _ y ^ * ] = \\delta _ { x , y } - u _ 0 ( y ) = \\delta _ { x , y } - 1 , \\forall x , y \\in \\Lambda . \\end{align*}"}
+{"id": "2149.png", "formula": "\\begin{align*} \\vartheta _ J = 1 . 4 1 9 1 , u _ J = ( 0 . 6 4 1 4 , 0 . 7 6 7 2 ) , v _ J = ( 0 . 0 0 0 0 , 0 . 0 8 3 9 , 0 . 9 0 1 1 ) . \\end{align*}"}
+{"id": "4362.png", "formula": "\\begin{align*} \\inf _ { x , p , \\theta } & \\ \\Gamma \\theta + \\sum _ { i \\in [ m ] } f _ i ( x , \\overline { u } ^ i ) + p _ i , \\\\ \\mathrm { s . t . \\ ; } & x \\in \\mathcal { X } , \\\\ & p _ i + \\theta \\geq \\sup _ { u ^ i \\in \\mathcal { U } _ i } f _ i ( x , u ^ i ) - f _ i ( x , \\overline { u } ^ i ) \\ \\forall i \\in [ m ] , \\\\ & p \\in \\R ^ m _ { \\geq 0 } , \\theta \\in \\R _ { \\geq 0 } . \\end{align*}"}
+{"id": "125.png", "formula": "\\begin{align*} \\beta \\left \\| \\varepsilon \\left ( \\vec { v } - P \\vec { v } \\right ) \\right \\| & \\leq \\sup _ { q \\in Q , ~ \\| q \\| = 1 } \\big ( { \\rm d i v } ( \\vec { v } - P \\vec { v } ) , q \\big ) \\\\ & = \\sup _ { q \\in Q , ~ \\| q \\| = 1 } \\big ( { \\rm d i v } \\vec { v } , q \\big ) = \\left \\| \\operatorname { d i v } \\vec { v } \\right \\| , \\end{align*}"}
+{"id": "5779.png", "formula": "\\begin{align*} & \\tilde { E } ( u ) : = \\left ( \\tilde u ^ \\perp \\right ) '' - m \\left ( \\tilde u ^ \\perp \\right ) ' + \\mathcal { L } _ { \\Sigma } \\tilde u ^ \\perp , \\tilde { \\mathcal { E } } : = ( 0 , \\tilde { E } ( u ) ) . \\end{align*}"}
+{"id": "1480.png", "formula": "\\begin{align*} [ \\alpha ( x ) , [ y , z ] ] = [ [ x , y ] , \\alpha ( z ) ] + [ \\alpha ( y ) , [ x , z ] ] . \\end{align*}"}
+{"id": "5538.png", "formula": "\\begin{align*} M = \\sum _ { i = 1 } ^ n \\nu _ i \\phi _ i \\psi _ i ^ * , \\end{align*}"}
+{"id": "4181.png", "formula": "\\begin{align*} \\varphi ( g _ 1 g _ 2 ) \\varphi ( g _ 1 g ^ { - 1 } _ 2 ) \\overset { ( \\ref { e q : v a r - n o n c o m m } ) } { = } \\varphi ( g _ 1 ) \\varphi ( g _ 2 ) \\varphi ( g ^ { - 1 } _ 2 ) \\varphi ( g _ 1 ) \\overset { { \\bf A 2 } } { = } \\varphi ( g _ 1 ) \\varphi ( g _ 1 ) \\overset { { \\bf A 3 } } { = } \\varphi ( g ^ { 2 } _ 1 ) \\ , . \\end{align*}"}
+{"id": "4517.png", "formula": "\\begin{align*} \\mathcal { T } ^ { ( 2 ) } : = \\bigg \\{ \\sup _ { X _ { \\ell - 1 } < n \\leqslant X _ { \\ell } } V ^ { ( 2 ) } ( n ) \\leqslant \\frac { 2 C _ 0 T ( \\ell ) } { \\ell ^ { K / 2 } } \\bigg \\} \\end{align*}"}
+{"id": "7625.png", "formula": "\\begin{align*} \\begin{pmatrix} g _ 2 ( z ) \\\\ g _ 3 ( z ) \\end{pmatrix} = \\begin{pmatrix} \\cos \\theta & - \\sin \\theta \\\\ \\sin \\theta & \\cos \\theta \\end{pmatrix} \\begin{pmatrix} f _ 2 ( z ) \\\\ f _ 3 ( z ) \\end{pmatrix} = \\begin{pmatrix} f _ 2 ( z ) \\cos \\theta - f _ 3 ( z ) \\sin \\theta \\\\ f _ 2 ( z ) \\sin \\theta + f _ 3 ( z ) \\cos \\theta \\end{pmatrix} . \\end{align*}"}
+{"id": "399.png", "formula": "\\begin{align*} \\begin{aligned} & a _ { S } ( \\kappa _ x , \\kappa _ y , \\bold { s } ) = e ^ { - \\jmath \\boldsymbol { \\kappa } ^ { T } \\bold { s } } = e ^ { - \\jmath ( \\kappa _ x s _ x + \\kappa _ y s _ y + \\kappa _ z s _ z ) } , \\\\ & a _ { R } ( k _ x , k _ y , \\bold { r } ) = e ^ { \\jmath \\bold { k } ^ { T } \\bold { r } } = e ^ { - \\jmath ( k _ x r _ x + k _ y r _ y + k _ z r _ z ) } , \\end{aligned} \\end{align*}"}
+{"id": "1448.png", "formula": "\\begin{align*} \\begin{aligned} J _ v ^ 1 ( E / L ) : = \\underset { w \\mid v } \\prod \\frac { H ^ 1 _ { f l } ( L _ w , E _ { p ^ \\infty } ) } { i m ( \\kappa _ w ) } \\ K _ v ^ 1 ( E / L ) : = \\underset { w \\mid v } \\prod H ^ 1 _ { f l } ( L _ w , E _ { p ^ \\infty } ) . \\end{aligned} \\end{align*}"}
+{"id": "586.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { j - 1 } \\Big | \\Big \\langle { \\frac { T ( \\tilde h _ i ) } { \\| h _ i \\| _ F } } , { \\frac { \\tilde h _ j } { \\| h _ j \\| _ { F ^ \\ast } } } \\Big \\rangle \\Big | < { \\frac { \\eta _ j } { 2 } } , \\end{align*}"}
+{"id": "6873.png", "formula": "\\begin{align*} H _ { { \\varphi } \\sf E } f = S _ 0 ^ { 1 / 2 } H _ { e { \\sf E } _ 0 } S _ 0 ^ { 1 / 2 } f + ( I - S _ 0 ) ^ { 1 / 2 } H _ { e { \\sf E } _ 1 } ( I - S _ 0 ) ^ { 1 / 2 } f \\end{align*}"}
+{"id": "1012.png", "formula": "\\begin{align*} \\mathfrak D _ { [ b ] } ( L ) = \\{ \\eta \\in \\mathfrak D ( L ) : L \\eta \\in L ^ \\infty ( E ; m ) \\} . \\end{align*}"}
+{"id": "6489.png", "formula": "\\begin{align*} \\Sigma _ k & = \\Sigma _ { k + 2 } + g _ k \\\\ & = \\Sigma ' _ { k + 1 } + \\max ( g ' _ k , 0 ) - g ' _ k \\\\ & = \\Sigma ' _ { k + 1 } + \\max ( - g ' _ k , 0 ) \\ge 0 . \\end{align*}"}
+{"id": "7353.png", "formula": "\\begin{align*} D g = - 2 \\theta _ g \\otimes g . \\end{align*}"}
+{"id": "9174.png", "formula": "\\begin{align*} { \\langle 2 \\rangle _ { v } } ^ { | \\beta | - 2 } \\langle 1 \\rangle _ { v } \\cdot \\prod _ { \\ell = j } ^ { n - 1 } \\big \\{ ( v ^ { - 4 ( n - \\ell ) - 2 } - 1 ) ( v ^ { - 4 ( n - \\ell ) + 6 } - 1 ) \\big \\} . \\end{align*}"}
+{"id": "1999.png", "formula": "\\begin{align*} \\xi ^ { n , \\pm } ( x ) = Q ^ { n , \\pm } ( x ) + P ^ { n , \\pm } ( x ) , \\end{align*}"}
+{"id": "4451.png", "formula": "\\begin{align*} & \\int _ { M _ j } \\int _ { \\partial D _ j } \\left | \\frac { f ^ * } { g _ l } \\right | ^ 2 | d z _ j | d \\mu _ j ( \\hat w _ j ) \\\\ \\le & C _ 0 \\int _ { M _ j } \\int _ { \\partial D _ j } \\left | f ^ * \\right | ^ 2 e ^ { - 2 \\log | g _ l | } \\frac { \\rho } { e ^ { - 2 \\log | g _ l | } } | d z _ j | d \\mu _ j ( \\hat w _ j ) \\\\ \\le & C _ 0 \\| f \\| ^ 2 _ { \\partial M , \\rho } \\\\ < & + \\infty \\end{align*}"}
+{"id": "1869.png", "formula": "\\begin{align*} \\beta \\tau _ 2 \\beta = \\tau ^ { - 1 } _ 2 \\alpha \\tau _ 2 \\alpha = \\tau _ 2 . \\end{align*}"}
+{"id": "3478.png", "formula": "\\begin{align*} \\langle b ( x ) , x \\rangle = \\langle \\nabla \\log \\pi ( x ) , x \\rangle = - \\langle \\nabla U ( x ) , x \\rangle \\end{align*}"}
+{"id": "8541.png", "formula": "\\begin{align*} \\begin{array} { l c l } 0 & = & ( \\alpha _ 2 - \\alpha _ 1 ) \\log _ 1 ( \\pi _ 2 ) + \\beta \\log _ 1 ( u ( K ) ) , \\\\ k \\lambda _ 1 + a _ 1 \\mu _ 1 & = & - s \\alpha _ 1 , \\\\ 0 & = & ( \\alpha _ 1 - \\alpha _ 2 ) \\log _ 1 ( \\pi _ 2 ) - \\beta \\log _ 1 ( u ( K ) ) , \\\\ k \\lambda _ 2 + a _ 2 \\mu _ 2 & = & - s \\alpha _ 2 . \\end{array} \\end{align*}"}
+{"id": "7829.png", "formula": "\\begin{align*} L ( \\boldsymbol { \\rho } ; \\boldsymbol { \\lambda } ) = \\left \\| \\boldsymbol { \\rho } - \\boldsymbol { \\rho } \\left ( i + \\frac { 1 } { 2 } \\right ) \\right \\| ^ { 2 } + \\sum _ { m = 1 } ^ { M } \\lambda _ m \\left ( \\boldsymbol { e } _ m ^ { \\mathrm { T } } \\boldsymbol { \\rho } - 1 \\right ) , \\end{align*}"}
+{"id": "2615.png", "formula": "\\begin{align*} & s _ 1 = 0 , \\ , s _ 2 = 1 0 , \\ , s _ 3 = 2 3 , \\ , s _ 4 = 4 0 , \\\\ & s _ i : = a _ { i - 1 } + s _ { i - 2 } - s _ { i - 4 } \\ , \\ , \\ , \\ , i = 5 , \\dots , n . \\end{align*}"}
+{"id": "2391.png", "formula": "\\begin{align*} A _ n ( t ) & : = 2 ^ { ( 6 s + 1 2 ) n } \\cdot ( 4 t + 2 n ) ^ { \\delta } \\cdot \\frac { \\left ( t + \\frac { 1 } { 4 } \\right ) _ { n } ^ { s + 2 } \\left ( t + \\frac { 3 } { 4 } \\right ) _ { n } ^ { s + 2 } } { ( t ) _ { n + 1 } ^ { 2 s + 4 } } , \\\\ B _ n ( t ) & : = 2 ^ { ( 3 s + 6 ) n } \\cdot \\frac { \\left ( t + \\frac { 3 } { 4 } \\right ) _ { n } ^ { s + 2 } } { ( t ) _ { n + 1 } ^ { s + 2 } } . \\end{align*}"}
+{"id": "5598.png", "formula": "\\begin{align*} H ^ { ( 2 ) } = \\sum _ { i = 1 } ^ r \\nu _ i ^ 2 \\chi _ i \\check { \\chi } _ i ^ * + \\tilde { H } . \\end{align*}"}
+{"id": "3430.png", "formula": "\\begin{align*} e _ { \\Z _ 2 } ( \\tilde { \\R } ) = e ( E \\Z _ 2 \\times _ { \\Z _ 2 } \\tilde { \\R } \\to B \\Z _ 2 ) \\in \\tilde { H } ^ 1 _ { \\Z _ 2 } ( S ^ 0 ) . \\end{align*}"}
+{"id": "4008.png", "formula": "\\begin{align*} W : \\begin{cases} x _ { 1 } ' & = \\ ; \\bigl ( \\gamma _ { 1 } x _ { 1 } + \\delta _ { 1 } x _ { 2 } \\bigr ) y \\medskip \\\\ x _ { 2 } ' & = \\ ; \\bigl ( \\gamma _ { 2 } x _ { 1 } + \\delta _ { 2 } x _ { 2 } \\bigr ) y \\medskip \\\\ y ' & = \\ ; \\bigl ( \\gamma x _ { 1 } + \\delta x _ { 2 } \\bigr ) y . \\end{cases} \\end{align*}"}
+{"id": "7745.png", "formula": "\\begin{align*} \\hat { G } _ { i j } ( k , \\lambda ) = L _ { i j } ^ { - 1 } ( k ; \\lambda ) \\hat H _ { i j } ( k , \\lambda ) . \\end{align*}"}
+{"id": "1217.png", "formula": "\\begin{align*} \\limsup _ { p \\to \\infty } F _ { p } ( U _ p , V _ p ) = \\max \\bigg \\{ \\vert u \\vert _ s , \\vert v \\vert _ t \\bigg \\} = F _ { \\infty } ( u , v ) , \\end{align*}"}
+{"id": "7259.png", "formula": "\\begin{align*} p _ n ( y ) : = u ^ n B _ n ( \\tfrac { y - w } { u } ; x , t ) , n \\geq 0 , \\end{align*}"}
+{"id": "1321.png", "formula": "\\begin{align*} \\langle \\mathrm { d } \\Phi _ X , H _ p v \\rangle & = \\bigl \\langle \\mathrm { d } \\Phi _ X , T \\mu ( v ) - V _ p ( \\nabla ^ \\ast _ v \\mu ) \\bigr \\rangle \\\\ & = v \\bigl [ \\langle \\mu , X \\rangle \\bigr ] - \\langle \\nabla _ v ^ \\ast \\mu , X \\rangle \\\\ & = \\langle \\mu , \\nabla _ v X \\rangle , \\end{align*}"}
+{"id": "8259.png", "formula": "\\begin{align*} & C _ ( B _ ) = \\frac { 1 } { L } \\max _ { b _ { y ^ { i - 1 } } , i \\in [ L ] } \\sum _ { i = 1 } ^ L \\sum _ { y ^ { i - 1 } } \\frac { c _ { y ^ { i - 1 } } } { M } H \\left ( \\frac { b _ { y ^ { i - 1 } } } { c _ { y ^ { i - 1 } } } \\right ) , \\end{align*}"}
+{"id": "7939.png", "formula": "\\begin{align*} [ \\alpha , \\beta ] _ { 1 } : = [ \\alpha ^ { \\sharp } , \\beta ^ { \\sharp } ] ^ { \\flat } = ( - 1 ) ^ { n - 1 } \\delta ( \\alpha \\wedge \\beta ) , \\end{align*}"}
+{"id": "5058.png", "formula": "\\begin{align*} \\epsilon _ { i _ 0 } = \\frac { \\tau _ { i _ 0 } q _ { i _ 0 } ^ 2 \\delta } { q _ { i _ 0 } ^ 2 - \\alpha L - q _ { i _ 0 } \\alpha L } , ~ ~ ~ ~ \\forall i _ 0 \\in \\mathcal { V } , \\end{align*}"}
+{"id": "1192.png", "formula": "\\begin{align*} \\left | \\left ( I - \\operatorname { T a y l } _ x ^ { | \\alpha | } \\right ) a ( y ) \\right | & \\leq | a ( y ) | + \\sum _ { \\beta \\in \\mathbb { Z } _ + ^ n , \\ , | \\beta | \\leq | \\alpha | } \\frac { | ( y - x ) ^ \\beta | } { \\beta ! } | \\partial ^ \\beta a ( x ) | \\\\ & \\lesssim 1 + \\sum _ { k = 0 } ^ { | \\alpha | } | y - x | ^ k \\sim 1 + | y - x | ^ { | \\alpha | } . \\end{align*}"}
+{"id": "5314.png", "formula": "\\begin{align*} S ( \\nu ) = \\left \\{ j \\in N ^ { \\{ 0 , 1 \\} } : \\nu \\leq \\nu _ j \\right \\} \\in \\mathcal { F } , \\nu \\in \\mathbb { R } . \\end{align*}"}
+{"id": "6670.png", "formula": "\\begin{align*} T _ j ( \\bar { s } , \\bar { t } ) = A _ j T ( s , t ) , \\end{align*}"}
+{"id": "6241.png", "formula": "\\begin{align*} \\mathcal { R } : = \\left \\{ ( \\omega , \\gamma ) \\colon \\sqrt { 3 } - 1 < \\omega < 1 \\ \\hbox { a n d } \\ \\frac { 2 - \\omega } { 3 } < \\gamma < 1 - \\frac { 1 } { \\sqrt { 3 } } \\right \\} . \\end{align*}"}
+{"id": "2609.png", "formula": "\\begin{align*} D _ 3 ( n ) = n + 2 . \\end{align*}"}
+{"id": "1594.png", "formula": "\\begin{align*} F _ \\lambda ( z ) = F ( z / \\lambda ) . \\end{align*}"}
+{"id": "4071.png", "formula": "\\begin{align*} R _ 0 ( z _ 0 ) = \\sum _ { j = 1 } ^ { \\frac { d - 1 } { 2 } } B _ j ( r , z _ 0 ) \\end{align*}"}
+{"id": "1033.png", "formula": "\\begin{align*} \\C ^ { \\infty \\times q } : = \\left \\{ \\mathbf { y } = ( y _ 1 ^ \\top , y _ 2 ^ \\top , \\dots ) ^ \\top : \\mbox { $ y _ k \\in \\C ^ { q \\times q } $ f o r $ k \\in \\N $ } \\right \\} . \\end{align*}"}
+{"id": "8314.png", "formula": "\\begin{align*} ( \\omega _ { 1 } ^ { \\flat } \\wedge \\omega _ 2 ^ { \\flat } ) \\eta _ 1 \\wedge \\eta _ 2 = \\iota _ { \\eta _ 1 } \\omega _ 1 \\iota _ { \\eta _ 2 } \\omega _ 2 - \\iota _ { \\eta _ 2 } \\omega _ 1 \\iota _ { \\eta _ 1 } \\omega _ 2 , \\end{align*}"}
+{"id": "6740.png", "formula": "\\begin{align*} S _ { 3 , m } ( z ) = \\sum _ { k = 1 } ^ \\infty \\frac { ( - 1 ) ^ k \\zeta ( 2 k ) z ^ { 2 k } } { ( 2 k + m ) ^ 2 } . \\end{align*}"}
+{"id": "152.png", "formula": "\\begin{align*} \\rho _ k ( x _ 1 , \\ldots , x _ k ) = \\det { \\big ( K ( x _ i , x _ j ) \\big ) } _ { 1 \\leq i , j \\leq k } . \\end{align*}"}
+{"id": "7945.png", "formula": "\\begin{align*} \\frac { \\delta _ { \\beta } \\mathcal { F } } { \\delta v } = d w ' , \\end{align*}"}
+{"id": "8793.png", "formula": "\\begin{align*} \\psi _ \\rho ( \\alpha ) = \\alpha + \\alpha ^ { 2 - \\rho } ( 1 - \\alpha ) ^ { \\rho - 1 } + 1 - \\rho , \\end{align*}"}
+{"id": "7037.png", "formula": "\\begin{align*} ( \\mu + \\Lambda _ { C _ \\infty } ( b _ n ) ) ^ { - 1 } \\upharpoonright \\mathcal S = \\Theta _ p ( \\mu , b _ n ) \\upharpoonright \\mathcal S \\end{align*}"}
+{"id": "2565.png", "formula": "\\begin{align*} K ' ( M , L ) = K ( M , L ; \\Omega ) ' \\ ; , J ' ( M , L ) = J ( M , L ; \\Omega ) ' \\ ; . \\end{align*}"}
+{"id": "6711.png", "formula": "\\begin{align*} P ( E ) y ( x ) : = a _ n y ( x + n ) + a _ { n - 1 } y ( x + n - 1 ) + . . . + a _ 0 y ( x ) = 0 , a _ i \\in \\mathbb { R } , \\ , i = 0 , 2 , 3 , . . . , n , \\ , a _ 0 a _ n \\neq 0 , \\end{align*}"}
+{"id": "5679.png", "formula": "\\begin{align*} \\int _ \\Sigma \\left \\langle \\mathcal { M } _ \\Sigma ( u ) , \\zeta \\right \\rangle \\ , d \\mu = - \\frac { d } { d s } \\mathcal { F } _ \\Sigma ( u + s \\zeta ) | _ { s = 0 } . \\end{align*}"}
+{"id": "6421.png", "formula": "\\begin{align*} J _ { n , i , 2 2 } ^ { 2 1 } ( \\theta ) = \\frac { \\ln \\left ( n / X _ { \\frac { i - 1 } { n } } \\right ) X _ { \\frac { i - 1 } { n } } } { \\delta \\alpha ^ 2 X _ { \\frac { i - 1 } { n } } ^ { 1 / \\alpha } } k ' _ \\alpha ( z ^ n _ i ( \\theta ) ) - \\frac { X _ { \\frac { i - 1 } { n } } } { \\delta X _ { \\frac { i - 1 } { n } } ^ { 1 / \\alpha } } f ' _ \\alpha ( z ^ n _ i ( \\theta ) ) \\end{align*}"}
+{"id": "7048.png", "formula": "\\begin{align*} \\langle h w _ \\eta ^ q \\rangle _ { x , \\eta } & = \\langle \\langle h ( \\eta ) | \\eta \\cdot w ( x ) | ^ q \\rangle _ \\eta \\rangle _ x \\\\ & \\geq \\langle \\langle h ( \\eta ) | \\eta \\cdot w ( x ) | ^ q \\mathbf { 1 } _ { A _ { t , x } } ( \\eta ) \\rangle _ \\eta \\rangle _ x \\\\ & \\geq \\langle c ^ q | w ( x ) | ^ q \\langle h ( \\eta ) \\mathbf { 1 } _ { A _ { t , x } } ( \\eta ) \\rangle _ \\eta \\rangle _ x = C \\langle | w | ^ q \\rangle _ x , C > 0 . \\end{align*}"}
+{"id": "4546.png", "formula": "\\begin{align*} \\min \\{ p _ N ( [ s ] _ { N } ) , p _ N ( [ s ' ] _ { N } ) \\} + \\big ( p _ N ( [ s ] _ { N } ) - p _ N ( [ s ' ] _ N ) \\big ) _ { + } = p _ N ( [ s ] _ N ) . \\end{align*}"}
+{"id": "2214.png", "formula": "\\begin{align*} - \\psi _ { 1 , r r r } ^ { ( 1 ) } - \\psi _ { 1 , r z z } ^ { ( 1 ) } - { 3 \\over r } \\psi _ { 1 , r r } ^ { ( 1 ) } + { 3 \\over r ^ 2 } \\psi _ { 1 , r } ^ { ( 1 ) } = g _ { , r } ^ { ( 1 ) } . \\end{align*}"}
+{"id": "4361.png", "formula": "\\begin{align*} \\inf _ { x \\in \\mathcal { X } } & \\ \\sum _ { i \\in [ m ] } \\max \\{ 0 , x _ i - \\overline { b } _ i , x _ i - \\overline { b } _ i + \\Delta b _ i - x _ k + \\overline { b } _ k - \\Delta b _ k \\} + \\Gamma ( x _ k - \\overline { b } _ k + \\Delta b _ k ) \\\\ \\mathrm { s . t . } & \\ x _ k \\geq b _ k . \\end{align*}"}
+{"id": "261.png", "formula": "\\begin{align*} \\frac { d ^ 2 y } { d \\tau ^ 2 } + 1 = 0 \\end{align*}"}
+{"id": "680.png", "formula": "\\begin{align*} \\mathfrak { C } ( u , \\beta , \\psi ) : = \\begin{Bmatrix} \\begin{array} { l | l } x \\in \\mathbb { R } ^ { n } & ( x - u ) \\cdot \\beta = | x - u | \\cos \\psi \\end{array} \\end{Bmatrix} . \\end{align*}"}
+{"id": "768.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { r r r } \\alpha _ 1 ^ { ' } x _ 1 ^ 2 + ( \\alpha _ 2 ^ { ' } + \\alpha _ 3 ^ { ' } ) x _ 1 x _ 2 + \\alpha _ 4 ^ { ' } x _ 2 ^ 2 - \\lambda ( x _ 1 , x _ 2 ) x _ 1 & = & 0 \\\\ \\beta _ 1 ^ { ' } x _ 1 ^ 2 + ( \\beta _ 2 ^ { ' } + \\beta _ 3 ^ { ' } ) x _ 1 x _ 2 + \\beta _ 4 ^ { ' } x _ 2 ^ 2 - \\lambda ( x _ 1 , x _ 2 ) x _ 2 & = & 0 \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "2194.png", "formula": "\\begin{align*} \\Gamma _ { , t } + v \\cdot \\nabla \\Gamma - \\nu \\bigg ( \\Delta + { 2 \\over r } \\partial _ r \\bigg ) \\Gamma + 2 { v _ \\varphi \\over r } \\Phi = F _ \\varphi / r \\equiv \\bar F _ \\varphi . \\end{align*}"}
+{"id": "2162.png", "formula": "\\begin{align*} A _ d ' = \\{ a _ { i _ 1 } , \\dots , a _ { i _ { K / 2 } } \\} . \\end{align*}"}
+{"id": "7617.png", "formula": "\\begin{align*} D _ j ( X ) = F ( \\C , j ) _ { + } \\wedge _ { S _ j } X ^ { \\wedge j } , \\end{align*}"}
+{"id": "3931.png", "formula": "\\begin{align*} \\dot z _ n ( t ) = & \\left ( - \\lambda _ n D + Q \\right ) z _ n ( t ) + F _ n [ z ( t ) ] + B \\sum _ { j = 1 } ^ { N } b _ { j , n } u _ j ( t ) . \\end{align*}"}
+{"id": "4262.png", "formula": "\\begin{align*} f ( t ) = u _ 0 + \\int _ 0 ^ t \\partial _ s f ( s ) \\ , d s , \\end{align*}"}
+{"id": "3031.png", "formula": "\\begin{align*} s ^ \\mu s _ \\mu = 1 \\ , , p _ s ^ T s = 0 \\ , . \\end{align*}"}
+{"id": "9057.png", "formula": "\\begin{align*} X ^ { \\{ 1 \\} \\sqcup \\{ 2 \\} } ( v \\otimes w ) = \\{ v , w \\} , X ^ { \\{ 1 , 2 \\} } ( v \\otimes w ) = v \\cdot w . \\end{align*}"}
+{"id": "4197.png", "formula": "\\begin{align*} & 2 4 0 \\left \\{ \\widehat { A } ( T X ) { \\rm d e t } ^ { \\frac { 1 } { 2 } } { \\rm c o s h } ( \\frac { \\sqrt { - 1 } } { 4 \\pi } R ^ V ) \\right \\} ^ { ( 8 ) } = \\left \\{ \\widehat { A } ( T X ) { \\rm d e t } ^ { \\frac { 1 } { 2 } } { \\rm c o s h } ( \\frac { \\sqrt { - 1 } } { 4 \\pi } R ^ V ) \\right . \\\\ & \\left . \\cdot { \\rm c h } ( \\widetilde { T X } + 2 \\wedge ^ 2 \\widetilde { V _ C } - \\widetilde { V _ C } \\otimes \\widetilde { V _ C } + \\widetilde { V _ C } ) \\right \\} ^ { ( 8 ) } . \\end{align*}"}
+{"id": "742.png", "formula": "\\begin{align*} x ( n ) = k \\iff t _ { ( x \\upharpoonright n ) ^ { \\frown } \\langle k \\rangle } \\subseteq y . \\end{align*}"}
+{"id": "349.png", "formula": "\\begin{align*} \\begin{aligned} x ^ 2 = ( z - y ) ( z + y ) & = a ( 2 x + 2 b - a ) \\\\ x ^ 2 - 2 a x + a ^ 2 & = 2 a b \\\\ ( x - a ) ^ 2 & = 2 a b , \\end{aligned} \\end{align*}"}
+{"id": "1884.png", "formula": "\\begin{align*} \\left . \\nabla _ z \\mathcal { M } _ { p , v + \\lambda u } ^ { \\Re , + , u } ( K , z ) \\right | _ { z = 0 } = 0 . \\end{align*}"}
+{"id": "5120.png", "formula": "\\begin{align*} S _ { a b } ( p ) = \\left [ - \\frac { \\sum _ { i } p ^ { a \\alpha } _ { i } - \\sum _ { i } p ^ { b \\alpha } _ { i } } { a \\alpha - b \\alpha } \\right ] _ { \\alpha = 1 } & = - \\frac { \\sum _ { i } p ^ { a } _ { i } - \\sum _ { i } p ^ { b } _ { i } } { a - b } \\\\ & = - \\sum _ { i } p _ { i } \\left ( \\frac { p ^ { a - 1 } _ { i } - p ^ { b - 1 } _ { i } } { a - b } \\right ) \\end{align*}"}
+{"id": "4713.png", "formula": "\\begin{align*} \\mathcal { R } \\widetilde { \\mathcal { V } } ^ { ( h ) } ( \\psi ^ { ( \\leq h ) } ) \\ ; : = \\ ; ( 1 - \\mathcal { L } ) \\widetilde { \\mathcal { V } } ^ { ( h ) } ( \\psi ^ { ( \\leq h ) } ) \\end{align*}"}
+{"id": "7306.png", "formula": "\\begin{align*} K _ { X _ { m } , \\chi _ { \\beta } } ( x + k m , y ; t ) = \\chi _ \\beta ( k ) K _ { X _ { m } , \\chi _ { \\beta } } ( x , y ; t ) , k \\in \\mathbb { Z } . \\end{align*}"}
+{"id": "296.png", "formula": "\\begin{align*} \\dot { z } _ l = F _ l ( z ) = \\sum _ { k = 1 } ^ r a _ { l , k } \\ , z ^ { \\alpha ( k ) } , l = 1 , \\dots , n . \\end{align*}"}
+{"id": "2950.png", "formula": "\\begin{align*} U _ t ( \\xi + i v ) = \\int _ { \\R ^ n } p _ { 2 t } ( 2 y , 2 v ) e ^ { - 2 y \\cdot \\xi } d y = c _ n ( \\sinh 4 t ) ^ { - n / 2 } e ^ { - ( \\coth 2 t ) | v | ^ 2 + ( \\tanh 2 t ) | \\xi | ^ 2 } \\end{align*}"}
+{"id": "7272.png", "formula": "\\begin{align*} x _ k = - \\tfrac { 1 } { 1 - q } \\left ( \\eta s q ^ k - ( \\theta + \\eta s ) \\right ) , k = 0 , 1 , 2 , \\ldots , \\end{align*}"}
+{"id": "6437.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { k \\in \\Z } f \\left ( k \\frac { \\pi } { b } \\right ) \\frac { 1 } { 2 b } \\widehat { \\gamma } \\left ( x - k \\frac { \\pi } { b } \\right ) , \\end{align*}"}
+{"id": "5921.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { C } ( A ( n , p ) ) ) = p ^ { n } ( p ^ { 2 n } - 1 ) ( p ^ { 2 n } - p ^ { n } - 1 ) ^ { 2 } , M _ { 2 } ( \\mathcal { C } ( A ( n , p ) ) ) = \\dfrac { p ^ { n } ( p ^ { 2 n } - 1 ) ( p ^ { 2 n } - p ^ { n } - 1 ) ^ { 3 } } { 2 } , \\end{align*}"}
+{"id": "5159.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } D _ { \\alpha } ( p \\| q ) } { \\partial q _ { j } } = & \\frac { 1 } { \\alpha } \\left [ \\frac { a - 1 } { a - b } A ^ { a - 2 } - \\frac { b - 1 } { a - b } A ^ { b - 2 } \\right ] p ^ { \\alpha } _ { j } q ^ { - \\alpha } _ { j } \\\\ & - \\frac { 1 } { \\alpha } \\left [ \\frac { a - 1 } { a - b } B ^ { a - 2 } - \\frac { b - 1 } { a - b } B ^ { b - 2 } \\right ] \\end{align*}"}
+{"id": "954.png", "formula": "\\begin{align*} P _ V ( x , B ) = P _ x ( X _ { \\tau _ V } \\in B ) , x \\in E \\setminus N , \\end{align*}"}
+{"id": "8642.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } W _ 1 ( \\mu _ n , \\nu ) = 0 . \\end{align*}"}
+{"id": "2726.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial v ^ { i } } \\rightarrow \\frac { \\partial p _ { j } } { \\partial v ^ { i } } \\frac { \\partial } { \\partial p _ { j } } = K ^ { ( 1 ) } _ { i j } \\frac { \\partial } { \\partial p _ { j } } . \\end{align*}"}
+{"id": "5597.png", "formula": "\\begin{align*} T P S _ { \\Delta } = \\sum _ { i = 1 } ^ n \\nu _ i ^ 2 T \\phi _ i \\phi _ i ^ * S _ { \\Delta } = \\sum _ { i = 1 } ^ n \\nu _ i ^ 2 \\chi _ i \\check { \\chi } _ i ^ * . \\end{align*}"}
+{"id": "5128.png", "formula": "\\begin{align*} D 2 ( p \\| q ) = \\left \\lbrace A \\left ( p , q \\right ) - \\left [ X \\left ( p , q \\right ) . \\ , Y \\left ( p , q \\right ) \\right ] \\right \\rbrace \\end{align*}"}
+{"id": "1924.png", "formula": "\\begin{align*} \\omega _ 0 ( x , y ) = 1 d ( x , y ) = 1 [ d ( y , x _ 0 ) = d ( x , x _ 0 ) \\pm 1 ] \\ , . \\end{align*}"}
+{"id": "2198.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ k \\intop _ { \\R _ + } | \\partial _ r ^ i u | ^ 2 r ^ { 2 ( \\mu - k + i ) } r d r \\sim \\sum _ { i = 0 } ^ k \\intop _ \\R | \\partial _ \\tau ^ i u ' | ^ 2 e ^ { 2 h \\tau } d \\tau \\end{align*}"}
+{"id": "6550.png", "formula": "\\begin{align*} T _ N - U _ N = \\frac { 1 } { 2 \\pi } \\sum ^ { N } _ { n = 1 } \\sum ^ { \\infty } _ { j = 1 } \\int _ { \\R } \\widehat { K } ( u ) \\phi ( u ) ( e ^ { \\iota u a _ j \\varepsilon _ { n - j } } - \\phi _ { \\varepsilon } ( a _ j u ) - \\iota u a _ j \\varepsilon _ { n - j } ) \\ , d u . \\end{align*}"}
+{"id": "8916.png", "formula": "\\begin{align*} 1 + 2 \\sum _ { m = 1 } ^ \\infty \\frac { B _ { 2 m } } { ( 2 m ) ! } \\omega _ m ( n ) x ^ { 2 m } & = \\frac { 1 } { \\Phi _ n ( 1 ) } \\prod _ { d \\mid n } \\left ( \\sinh \\left ( d \\sinh ^ { - 1 } \\left ( \\frac { x } { 2 } \\right ) \\right ) \\right ) ^ { \\mu ( n / d ) } \\\\ & = \\frac { 1 } { \\Phi _ n ( 1 ) } \\prod _ { d \\mid n } V _ d ( x ) ^ { \\mu ( n / d ) } . \\end{align*}"}
+{"id": "8785.png", "formula": "\\begin{align*} & \\theta : = \\frac { 1 } { C } \\int _ { F _ \\nu ( x - ) - q ( x ) } ^ { F _ \\nu ( x - ) } ( 1 - p ( v ) ) \\delta _ { F _ \\nu ^ { - 1 } ( v ) } d v + \\int _ { F _ \\nu ( x ) } ^ { F _ \\nu ( x ) + \\mu ( \\{ x \\} ) - q ( x ) } ( 1 - p ( v ) ) \\delta _ { F _ \\nu ^ { - 1 } ( v ) } d v \\\\ \\mbox { a n d } & \\vartheta : = \\frac { 1 } { C } \\int _ 0 ^ { F _ \\nu ( x - ) - q ( x ) } p ( v ) \\delta _ { F _ \\nu ^ { - 1 } ( v ) } d v + \\int _ { F _ \\nu ( x ) + \\mu ( \\{ x \\} ) - q ( x ) } ^ 1 p ( v ) \\delta _ { F _ \\nu ^ { - 1 } ( v ) } d v \\end{align*}"}
+{"id": "5755.png", "formula": "\\begin{align*} & \\mathcal { E } _ { i } ( t ) : = G ( \\mathcal { E } ( t ) , \\Psi _ { i } ) \\ \\textup { f o r } \\ i \\in \\mathbb { Z } \\setminus \\{ 0 \\} , \\\\ & \\mathcal { W } _ j ( t ) : = G ( \\mathcal { E } ( t ) , \\Upsilon _ j ) , \\ \\overline { \\mathcal { W } } _ j ( t ) : = G ( \\mathcal { E } ( t ) , \\overline { \\Upsilon } _ j ) \\ \\textup { f o r } \\ 1 \\leq j \\leq J . \\end{align*}"}
+{"id": "599.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { \\binom { 2 k } { k } ^ { 2 } } { 3 2 ^ { k } } ( 2 H _ { 2 k } - H _ { k } ) = \\frac { \\ln ( 2 ) \\ , \\Gamma ^ { 2 } \\left ( \\frac { 1 } { 4 } \\right ) } { 4 \\pi \\sqrt { \\pi } } . \\end{align*}"}
+{"id": "247.png", "formula": "\\begin{align*} b _ 2 ( x _ 2 ) = \\frac { b _ 1 ( x _ 1 ) } { | A _ 1 ( x _ 1 ) | } , \\end{align*}"}
+{"id": "4916.png", "formula": "\\begin{align*} ( x _ 1 f _ 1 + x _ 2 f _ 2 ) + y _ 0 ( x _ 0 + a _ 1 x _ 1 + a _ 2 x _ 2 ) = 0 \\end{align*}"}
+{"id": "369.png", "formula": "\\begin{align*} g _ p ( b , a ) = - p a b \\sum _ { i = 1 } ^ { p - 1 } \\dfrac { ( - 1 ) ^ i } { i } \\binom { p - 1 } { i - 1 } b ^ { p - i - 1 } a ^ { i - 1 } , \\end{align*}"}
+{"id": "4872.png", "formula": "\\begin{align*} ( H + l F ) ^ 2 = e + 2 l = g - 2 , \\end{align*}"}
+{"id": "4217.png", "formula": "\\begin{align*} \\theta _ 2 ( v , \\tau ) = \\prod _ { j = 1 } ^ { \\infty } [ ( 1 - q ^ j ) ( 1 - e ^ { 2 \\pi \\sqrt { - 1 } v } q ^ { j - \\frac { 1 } { 2 } } ) ( 1 - e ^ { - 2 \\pi \\sqrt { - 1 } v } q ^ { j - \\frac { 1 } { 2 } } ) ] , \\end{align*}"}
+{"id": "4405.png", "formula": "\\begin{align*} \\min _ { x } & \\ \\sum _ { i , j \\in \\mathcal { I } , \\mathcal { J } } u _ i q _ j x _ { i , j } , \\\\ \\mathrm { s . t . } & \\ x \\in \\mathcal { X } \\subseteq \\left \\{ x \\in \\{ 0 , 1 \\} ^ { | \\mathcal { I } | \\cdot | \\mathcal { J } | } : \\sum _ { j \\in \\mathcal { J } } x _ { i , j } = 1 \\ \\forall i \\in \\mathcal { I } \\right \\} \\end{align*}"}
+{"id": "6468.png", "formula": "\\begin{align*} \\lambda _ j = \\frac { j ( j + p - 1 ) } { R ^ 2 } , j = 0 , 1 , 2 , \\ldots \\end{align*}"}
+{"id": "6499.png", "formula": "\\begin{align*} \\lambda _ t = \\mu + \\int _ { ( 0 , t ) } \\Phi ( t - s ) d H _ s , t \\geq 0 . \\end{align*}"}
+{"id": "5957.png", "formula": "\\begin{align*} \\frac { M _ { 2 } ( \\mathcal { N C } ( G ) ) } { | e ( \\mathcal { N C } ( G ) ) | } = 3 0 6 > \\frac { 3 3 6 0 } { 1 1 } = \\frac { M _ { 1 } ( \\mathcal { N C } ( G ) ) } { | v ( \\mathcal { N C } ( G ) ) | } . \\end{align*}"}
+{"id": "6744.png", "formula": "\\begin{align*} \\frac { 1 } { k ^ p ( 2 k + m ) ( 2 k + n ) } & = \\frac { 1 } { m n k ^ p } + \\frac { 2 } { m ( m - n ) } \\left ( \\sum _ { j = 0 } ^ { p - 2 } \\frac { ( - 2 ) ^ j } { m ^ { j + 1 } k ^ { p - 1 - j } } + \\Big ( \\frac { 2 } { m } \\Big ) ^ { p - 1 } \\frac { ( - 1 ) ^ { p - 1 } } { 2 k + m } \\right ) \\\\ & - \\frac { 2 } { n ( m - n ) } \\left ( \\sum _ { j = 0 } ^ { p - 2 } \\frac { ( - 2 ) ^ j } { n ^ { j + 1 } k ^ { p - 1 - j } } + \\Big ( \\frac { 2 } { n } \\Big ) ^ { p - 1 } \\frac { ( - 1 ) ^ { p - 1 } } { 2 k + n } \\right ) . \\end{align*}"}
+{"id": "8531.png", "formula": "\\begin{align*} \\begin{array} { l c l } - a _ 1 \\mu _ 1 k ^ { - 1 } & = & s ^ { - 1 } \\log _ 1 ( \\pi _ 2 ) ( v _ 1 - v _ 2 ) , \\\\ - ( \\lambda _ 1 + k ^ { - 1 } \\mu _ 1 a _ 1 ) & = & v _ 1 , \\\\ - a _ 2 \\mu _ 2 k ^ { - 1 } & = & s ^ { - 1 } \\log _ 1 ( \\pi _ 2 ) ( v _ 2 - v _ 1 ) , \\\\ - ( \\lambda _ 2 + k ^ { - 1 } \\mu _ 2 a _ 2 ) & = & v _ 2 . \\end{array} \\end{align*}"}
+{"id": "207.png", "formula": "\\begin{align*} d t = f ( x ^ 1 , \\ldots , x ^ n ) \\ d \\tau , f ( x ^ 1 , \\ldots , x ^ n ) > 0 , \\end{align*}"}
+{"id": "3413.png", "formula": "\\begin{align*} I _ \\varphi ( \\rho ( \\xi ) \\phi ) = \\rho ( \\iota ^ * ( \\xi ) ) I _ \\varphi ( \\phi ) \\end{align*}"}
+{"id": "653.png", "formula": "\\begin{align*} \\begin{cases} f \\in L ^ { 1 } ( \\mathbb { R } ^ { n } ; S ^ { m } ) \\cap L _ { \\rm l o c } ^ { 2 } ( \\mathbb { R } ^ { n } ; S ^ { m } ) & \\\\ f \\in L ^ { 2 } ( \\mathbb { R } ^ { n } ; S ^ { m } ) & \\end{cases} \\end{align*}"}
+{"id": "2990.png", "formula": "\\begin{align*} \\begin{array} { l l l } T ( x , y ) & = & \\overline { \\nabla } _ x y - \\overline { \\nabla } _ y x - [ x , y ] \\\\ & = & \\alpha \\{ \\eta ( x ) y - \\eta ( y ) x \\} + \\beta \\{ \\eta ( x ) \\varphi y - \\eta ( y ) \\varphi x \\} , \\end{array} \\end{align*}"}
+{"id": "7476.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { 4 } r ^ j \\left | \\left ( \\nabla ^ G \\right ) ^ j \\left ( ( \\Phi _ s \\circ \\tilde { F } _ s ) ^ * g - G \\right ) \\right | _ G + r ^ j \\left | ( \\nabla ^ G ) ^ j ( \\tilde { F } _ { s } ^ { * } g _ 0 ( s ) - G ) \\right | _ G < \\delta \\end{align*}"}
+{"id": "8489.png", "formula": "\\begin{align*} I _ 2 = - \\dfrac { 1 } { h } \\int _ \\Omega | u _ { m - 1 } | ^ { q - 1 } u _ { m - 1 } \\varphi \\ , d x \\end{align*}"}
+{"id": "6764.png", "formula": "\\begin{align*} \\begin{pmatrix} x _ 2 ^ { k + 1 } \\\\ \\lambda ^ { k + 1 } \\end{pmatrix} = \\begin{pmatrix} x _ 2 ^ { k } \\\\ \\lambda ^ { k } \\end{pmatrix} - \\begin{pmatrix} I _ { n _ 2 } & 0 \\\\ - s \\beta A _ 2 & ( r + s ) I _ l \\end{pmatrix} \\begin{pmatrix} x _ 2 ^ { k } - \\widetilde { x } _ 2 ^ k \\\\ \\lambda ^ { k } - \\widetilde { \\lambda } ^ k \\end{pmatrix} . \\end{align*}"}
+{"id": "312.png", "formula": "\\begin{align*} L : = \\sigma ( m - 1 ) + 2 ( p - 1 ) > 0 . \\end{align*}"}
+{"id": "3033.png", "formula": "\\begin{align*} { \\cal V } _ { \\mu \\nu } = - ( { Y } _ { \\mu } ) _ { \\alpha } ^ \\rho ( { Y } _ { \\nu } ) _ { \\rho } ^ \\beta s ^ \\alpha s _ \\beta \\ , , \\Im \\mathcal { V } = 0 \\ , . \\end{align*}"}
+{"id": "5697.png", "formula": "\\begin{align*} u = u ^ T + H ( u ^ T ) + \\tilde { u } ^ \\perp . \\end{align*}"}
+{"id": "1129.png", "formula": "\\begin{align*} \\left \\| \\left \\{ \\mathbf { 1 } _ P 2 ^ { - j n ( \\tau - \\frac { 1 } { p } ) } \\right \\} _ { j \\geq j _ P } \\right \\| _ { L \\dot A _ { p , q } } & = \\left \\| \\mathbf { 1 } _ P \\right \\| _ { L ^ p } \\left \\| \\left \\{ 2 ^ { - j n ( \\tau - \\frac { 1 } { p } ) } \\right \\} _ { j \\geq j _ P } \\right \\| _ { \\ell ^ q } \\\\ & \\sim | P | ^ { \\frac { 1 } { p } } 2 ^ { - j _ P n ( \\tau - \\frac { 1 } { p } ) } = | P | ^ { \\frac { 1 } { p } } | P | ^ { \\tau - \\frac { 1 } { p } } = | P | ^ { \\tau } \\end{align*}"}
+{"id": "7513.png", "formula": "\\begin{align*} g _ s = \\xi _ 3 \\left ( \\tfrac { r _ s } { s ^ { 1 / 4 } } \\right ) \\left ( \\sum _ { k = 1 } ^ { 2 n _ 0 } f _ k ( \\Phi _ { s , k } ^ { - 1 } ) ^ * g _ { 0 , k } ( s ) \\right ) + \\left ( 1 - \\xi _ 3 \\left ( \\tfrac { r _ s } { s ^ { 1 / 4 } } \\right ) \\right ) ( \\Psi _ s ^ { - 1 } ) ^ * g _ Z , \\end{align*}"}
+{"id": "8810.png", "formula": "\\begin{align*} \\frac 1 \\rho \\partial _ x f ( z _ - , z _ - ) & = - \\frac { z _ + - z _ - } { z _ + - y _ - } \\left ( z _ - - y _ - \\right ) ^ { \\rho - 1 } + \\frac { z _ - - y _ - } { z _ + - y _ - } \\left ( z _ + - z _ - \\right ) ^ { \\rho - 1 } \\\\ & = \\frac { z _ + - z _ - } { z _ + - y _ - } \\left ( z _ - - y _ - \\right ) ^ { \\rho - 1 } \\left ( \\left ( \\frac { z _ - - y _ - } { z _ + - z _ - } \\right ) ^ { 2 - \\rho } - 1 \\right ) . \\end{align*}"}
+{"id": "5827.png", "formula": "\\begin{align*} \\frac { \\partial u ^ A } { \\partial t } = \\bar { g } \\left ( \\vec { H } , \\frac { \\partial } { \\partial y ^ B } \\right ) ( h _ u ) ^ { B A } . \\end{align*}"}
+{"id": "3012.png", "formula": "\\begin{align*} \\overline { \\mathcal { L } } _ { \\xi } g ( x , y ) = \\mathcal { L } _ { \\xi } g ( x , y ) - 2 \\alpha g ( \\varphi x , \\varphi y ) - 2 \\beta g ( \\varphi x , y ) . \\end{align*}"}
+{"id": "3427.png", "formula": "\\begin{align*} \\underline { d } ( \\Sigma ^ V X ) = \\underline { d } ( X ) + \\dim V \\overline { d } ( \\Sigma ^ V X ) = \\overline { d } ( X ) + \\dim V . \\end{align*}"}
+{"id": "1020.png", "formula": "\\begin{align*} \\Gamma ^ { t } ( n ) = \\Gamma ( n ) / \\cong _ { t } \\end{align*}"}
+{"id": "6217.png", "formula": "\\begin{align*} \\Delta ( F , \\phi _ 0 ) ( \\phi ) : = \\frac { 1 } { \\phi - \\phi _ 0 } \\int _ { \\phi _ 0 } ^ \\phi \\delta ( F , \\phi _ 0 ) ( \\psi ) \\ , d \\psi = \\frac { 1 } { \\phi - \\phi _ 0 } \\int _ { \\phi _ 0 } ^ \\phi \\frac { F ( \\psi ) - F ( \\phi _ 0 ) } { \\psi - \\phi _ 0 } \\ , d \\psi . \\end{align*}"}
+{"id": "7747.png", "formula": "\\begin{align*} H ^ { - 1 } ( \\cdot + \\alpha ) ( A ' + f ^ { ( r e ) } ) H ( \\cdot ) = A '' . \\end{align*}"}
+{"id": "722.png", "formula": "\\begin{align*} ( A , a _ 0 ) \\cong ( B , b _ 0 ) \\to \\pi _ 1 ( A , a _ 0 ) = _ { \\mathfrak { U } } \\pi _ 1 ( B , b _ 0 ) . \\end{align*}"}
+{"id": "8900.png", "formula": "\\begin{align*} F _ k ( x _ 1 , \\dots , x _ k ) : = k ! \\sum _ { \\substack { \\lambda _ 1 , \\dots , \\lambda _ k \\geq 0 \\\\ \\lambda _ 1 + 2 \\lambda _ 2 + \\cdots + k \\lambda _ k = k } } \\prod _ { j = 1 } ^ k \\frac { ( - s _ j ( x _ 1 , \\dots , x _ j ) ) ^ { \\lambda _ j } } { \\lambda _ j ! j ^ { \\lambda _ j } } , F _ 0 : = 1 , \\end{align*}"}
+{"id": "1285.png", "formula": "\\begin{align*} W _ 2 ^ 2 ( \\mu _ t , \\bar { \\mu } _ t ) ^ 2 = \\Big ( m _ x ( t ) - \\bar { m } ( t ) \\Big ) ^ 2 + \\Big ( \\sqrt { \\sigma _ { x x } ( t ) } - \\sqrt { \\bar { \\sigma } ( t ) } \\Big ) ^ 2 , \\end{align*}"}
+{"id": "6985.png", "formula": "\\begin{align*} 1 & = ( z _ { n + 1 } \\overline { \\eta } + \\overline { z _ { n + 1 } } \\eta ) ^ 2 = 2 ( \\epsilon + 1 ) \\lvert z _ { n + 1 } \\rvert ^ 2 \\lvert \\eta \\rvert ^ 2 . \\end{align*}"}
+{"id": "2122.png", "formula": "\\begin{align*} \\langle \\tilde { M } ^ { \\Theta } _ { g ^ i } , \\tilde { M } ^ { \\Theta } _ { g ^ j } \\rangle ( t ) = \\langle W _ { i } , W _ { j } \\rangle ( t ) = \\delta _ { i j } t . \\end{align*}"}
+{"id": "3501.png", "formula": "\\begin{align*} \\textstyle { \\bigcup _ n } \\rho _ n ( F _ n ) = \\{ \\Psi \\big ( \\lim _ m \\varphi _ m \\circ \\rho _ { m , n } ( x ) \\big ) \\mid k \\geq 0 , \\ x \\in F _ n \\} \\subset \\Psi ( A ) . \\end{align*}"}
+{"id": "4515.png", "formula": "\\begin{align*} \\begin{cases} g _ { j , n } ( r ) = y _ j r \\leqslant n - y _ j , \\\\ g _ { j , n } ( r ) = n - r n - y _ j < r \\leqslant n - y _ { j - 1 } , \\\\ g _ { j , n } ( r ) = y _ { j - 1 } r > n - y _ { j - 1 } \\ , . \\end{cases} \\end{align*}"}
+{"id": "8513.png", "formula": "\\begin{align*} f ( a _ 1 ) f ( a _ 2 ) \\dotsc = f ( b _ 1 ) f ( b _ 2 ) \\dotsc , \\end{align*}"}
+{"id": "2246.png", "formula": "\\begin{align*} \\frac { \\partial F ^ { \\alpha } } { \\partial v ^ j _ { \\beta } } = g ^ { \\alpha \\beta } _ { i j } X ^ i \\ , , \\frac { \\partial F ^ { \\alpha } } { \\partial s ^ { \\beta } } = \\frac { \\partial ^ 2 L } { \\partial s ^ \\beta \\partial v ^ i _ \\alpha } X ^ i \\ , . \\end{align*}"}
+{"id": "3287.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( a ^ { d - 1 } q ^ { d - 1 } , a ^ { d - 3 } q ^ { d - 1 } , \\cdots , a ^ { 3 - d } q ^ { d - 1 } , a ^ { 1 - d } q ^ { d - 1 } ; q ^ d ) _ k q ^ { d k } } { ( q ^ d , a ^ { d - 2 } q ^ d , a ^ { d - 4 } q ^ d , \\cdots , a ^ { 4 - d } q ^ d , a ^ { 2 - d } q ^ d ; q ^ d ) _ k } \\equiv B _ q ( d , n , d - 1 ) \\end{align*}"}
+{"id": "4427.png", "formula": "\\begin{gather*} \\norm { E ^ j } _ \\infty = \\norm { u _ h ^ j - R ^ j } _ \\infty \\le \\eta _ \\mathrm { e l l } ^ j \\coloneqq \\eta \\left ( u _ h ^ j , f ^ j + \\psi ^ j \\right ) \\ , , \\ \\ j = 0 , \\dots , M . \\end{gather*}"}
+{"id": "3129.png", "formula": "\\begin{align*} \\varphi ^ * \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & = \\left ( \\begin{array} { c c c } a ^ { p ^ { e _ 2 } } & 0 & b ^ { p ^ { e _ 2 } } \\\\ 0 & 1 & 0 \\\\ 0 & 0 & d ^ { p ^ { e _ 2 } } \\end{array} \\right ) ( \\ , e _ 2 \\geq 0 \\ , ) . \\end{align*}"}
+{"id": "6361.png", "formula": "\\begin{align*} \\xi ( r ) : = \\frac { \\abs { \\nabla ^ h \\rho } ^ 2 } { \\omega _ 1 ^ 2 ( r ) } + \\frac { \\abs { \\nabla ^ v \\rho } ^ 2 } { \\omega _ 1 ^ 1 ( r ) } , \\end{align*}"}
+{"id": "2279.png", "formula": "\\begin{align*} H ( G , B ) = _ G ( \\mathbb { C } G / B ) = _ G ( _ B ^ G \\textbf { 1 } _ B ) \\ . \\end{align*}"}
+{"id": "136.png", "formula": "\\begin{align*} X = \\sum _ { i \\in I } \\delta _ { x _ i } , \\end{align*}"}
+{"id": "175.png", "formula": "\\begin{align*} \\widehat { A } \\ = \\ \\{ ( x , y ) : x ^ { - 1 } y & \\in A \\} \\ \\ \\widehat { A } ^ { - 1 } \\ = \\ \\widehat { A ^ { - 1 } } , \\\\ \\ \\ \\widehat { A } ^ { \\circ } \\ & = \\ \\{ ( x , y ) : x ^ { - 1 } y \\in A ^ { \\circ } \\} . \\end{align*}"}
+{"id": "2497.png", "formula": "\\begin{align*} \\langle u ( t ) \\rangle = M : = \\langle u ^ { i n } \\rangle \\ ; \\ ; \\ ; \\ ; \\| v ( t ) \\| _ \\infty \\le V : = \\| v ^ { i n } \\| _ \\infty \\ , . \\end{align*}"}
+{"id": "2935.png", "formula": "\\begin{align*} \\sum _ { T ' : T ' \\in P ( T ) , T \\in M _ j , \\bar { s } \\in T ' } v _ { T ' } = \\sum _ { T ' : T ' \\in P ( T ) , T \\in M _ j , \\bar { s } \\in T ' } g _ { T ' } = 0 . \\end{align*}"}
+{"id": "2192.png", "formula": "\\begin{align*} ( \\Phi , \\Gamma ) = ( \\omega _ r / r , \\omega _ \\varphi / r ) . \\end{align*}"}
+{"id": "7400.png", "formula": "\\begin{align*} u _ { t t } - u _ { x x } = | u _ t | ^ p | u | ^ q . \\end{align*}"}
+{"id": "2661.png", "formula": "\\begin{align*} \\frac { \\partial L ^ { ( 1 ) } } { \\partial q ^ { i } } - \\frac { d } { d t } \\left ( \\frac { \\partial L ^ { ( 1 ) } } { \\partial \\dot { q } ^ { i } } \\right ) = 0 \\end{align*}"}
+{"id": "8071.png", "formula": "\\begin{align*} \\limsup _ { N \\rightarrow + \\infty } \\frac { N } { \\gamma _ N ^ 2 } \\log P ( \\frac { N } { \\gamma _ N } ( \\tau _ c ^ N - \\tau _ c ) > x ) & \\leq - x ^ 2 J _ { c o n t r a , \\tau _ c } ( 1 ) \\left ( \\frac { d } { d t } \\sum _ { k = 1 } ^ 3 \\mu _ { t , k } ( f _ k ) \\Bigg | _ { t = \\tau _ c } \\right ) ^ 2 \\\\ & = - J _ { h i t } ( x ) . \\end{align*}"}
+{"id": "2871.png", "formula": "\\begin{align*} \\mathrm { I _ 2 } & = \\limsup _ { \\lambda \\to 0 ^ + } \\lambda ^ q \\left \\| \\left [ \\int _ { B } \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ { \\frac { 1 } { q } } \\mathbf { 1 } _ { ( 2 B ) ^ \\complement } ( \\cdot ) \\right \\| _ { X } \\\\ & \\leq 2 ^ { \\frac { n - \\gamma } { q } } \\limsup _ { \\lambda \\to 0 ^ + } \\lambda ^ q | B | ^ { \\frac { 1 } { q } } \\left \\| | \\cdot | ^ { \\frac { \\gamma - n } { q } } \\mathbf { 1 } _ { ( 2 B ) ^ \\complement } ( \\cdot ) \\right \\| _ { X } = 0 . \\end{align*}"}
+{"id": "6248.png", "formula": "\\begin{align*} \\frac { \\sqrt { d - 1 } } { E _ g } \\sqrt { A ( \\omega ) B ( \\omega ) } > ( 1 + \\omega ) ^ 2 + \\c d ( 1 - \\omega ^ 2 ) - 3 = : E ( \\omega , \\c d ) . \\end{align*}"}
+{"id": "8859.png", "formula": "\\begin{align*} 0 < \\frac 1 { a _ 1 } , \\frac 1 { b _ 1 } < 1 , \\frac { 1 } { a _ 1 } + \\frac { 1 } { b _ 1 } = \\frac { n + 2 } { 2 n } + \\frac { \\alpha } { n } , \\frac { 1 } { a _ 1 } = \\frac { 1 } { a _ 3 } + \\frac { 1 } { r } , \\end{align*}"}
+{"id": "4976.png", "formula": "\\begin{align*} Z _ N ( \\{ \\nu \\} ) = \\prod _ { j , k = 1 } ^ N ( \\nu _ j - \\nu _ k + 1 ) . \\end{align*}"}
+{"id": "849.png", "formula": "\\begin{align*} - \\dot \\delta { Z } = n { F ^ { - 1 } } g ( X , l ) \\Psi - { X ^ i } { \\Psi _ i } - 2 g ( X , { I ^ * } ) \\Psi , \\end{align*}"}
+{"id": "6154.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau ^ k } [ f ( x ) - f ( x ^ { k + 1 } ) ] - \\frac { 1 } { \\tau ^ { k - 1 } } [ f ( x ) - f ( x ^ { k } ) ] + ( x - \\bar { x } ^ { k + 1 } ) ^ T [ - A ^ T \\lambda ^ k \\\\ & + \\tau ^ k \\beta ^ k A ^ T ( A \\bar { x } ^ { k + 1 } - b ) + \\tau ^ k \\beta ^ k ( D - A ^ T A ) ( \\bar { x } ^ { k + 1 } - \\bar { x } ^ k ) \\\\ & + ( 1 - \\tau ^ k ) \\beta ^ k A ^ T ( A { x } ^ { k } - b ) ] \\geq 0 , ~ \\forall x . \\end{aligned} \\end{align*}"}
+{"id": "7309.png", "formula": "\\begin{align*} K _ { X _ { m } , \\chi _ { \\beta } } ( x + \\ell m , y ; t ) & = \\sum _ { k \\in \\mathbb { Z } } e ^ { - 2 \\pi i \\beta k } e ^ { - t } I _ { x - y + ( k + \\ell ) m } ( t ) \\\\ & = \\sum _ { j \\in \\mathbb { Z } } e ^ { - 2 \\pi i \\beta ( j - \\ell ) } e ^ { - t } I _ { x - y + j m } ( t ) \\\\ & = e ^ { 2 \\pi i \\beta \\ell } K _ { X _ { m } , \\chi _ { \\beta } } ( x , y ; t ) . \\end{align*}"}
+{"id": "3007.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\lambda & = & \\beta ^ 2 - \\beta + ( 1 - 2 n ) \\alpha ^ 2 + a , \\\\ \\mu & = & ( 2 - 2 n ) ( \\alpha \\beta - \\alpha ) + \\alpha + b \\\\ \\nu & = & ( 2 n - 1 ) \\alpha ^ 2 + ( 2 n - 2 ) \\alpha \\beta + ( 1 - 2 n ) \\alpha + ( 2 n + 1 ) \\beta - \\beta ^ 2 + c \\end{array} \\end{align*}"}
+{"id": "8457.png", "formula": "\\begin{align*} \\overline { U } _ h ( x , y , t ) : = | \\bar { u } _ h ( x , t ) - \\bar { u } _ h ( y , t ) | ^ { p - 2 } ( \\bar { u } _ h ( x , t ) - \\bar { u } _ h ( y , t ) ) . \\end{align*}"}
+{"id": "6350.png", "formula": "\\begin{align*} 0 = \\int _ { S ^ { n - 1 } } \\varphi \\psi ( R ( 1 + \\rho ( \\varphi ) ) ) d \\varphi . \\end{align*}"}
+{"id": "8920.png", "formula": "\\begin{align*} \\frac { \\sigma ( n - 2 ) } { \\sigma ( n - 1 ) } = \\frac { t \\sigma ( n - 2 ) } { t \\sigma ( n - 1 ) } = \\frac { \\rho ( n - 2 ) } { \\rho ( n - 1 ) } . \\end{align*}"}
+{"id": "2946.png", "formula": "\\begin{align*} \\big ( \\Delta _ \\C + \\partial _ \\rho ^ 2 + \\frac { 1 - s } { \\rho } \\partial _ \\rho \\big ) U ( g , \\rho ) = 0 , \\ , \\ , U ( g , 0 ) = F ( g ) \\end{align*}"}
+{"id": "1303.png", "formula": "\\begin{align*} W _ 2 ( \\rho _ 1 ( t ) , \\hat { \\rho } _ 1 ( t ) ) ^ 2 = ( \\mu _ 1 ( t ) - \\hat { \\mu } _ 1 ( t ) ) ^ 2 + \\Big ( \\sqrt { \\Sigma _ { 1 1 } } ( t ) - \\sqrt { \\hat { \\Sigma } _ { 1 } } ( t ) \\Big ) ^ 2 \\end{align*}"}
+{"id": "5795.png", "formula": "\\begin{align*} \\xi ^ { ( k , \\ell ) } _ i ( t ) : = \\int _ \\Sigma \\left \\langle \\mathcal { L } _ { \\Sigma } ^ { \\ell } \\partial ^ { k } _ t u , \\varphi _ { i } \\right \\rangle \\ , d \\mu , \\ \\mathcal { E } ^ { ( k , \\ell ) } _ i ( t ) : = \\int _ \\Sigma \\left \\langle \\mathcal { L } _ { \\Sigma } ^ { \\ell } \\partial ^ { k } _ t E _ 2 ( u ) , \\varphi _ { i } \\right \\rangle \\ , d \\mu \\end{align*}"}
+{"id": "5172.png", "formula": "\\begin{align*} A = \\frac { 1 } { 1 - \\beta } \\sum _ { i } p _ { i } q ^ { \\beta - 1 } _ { i } + \\frac { 1 } { \\beta } q ^ { \\beta } _ { i } \\ ; \\ ; ; \\ ; \\ ; B = \\frac { 1 } { \\beta ( 1 - \\beta ) } \\sum _ { i } p _ { i } ^ { \\beta } \\end{align*}"}
+{"id": "8936.png", "formula": "\\begin{align*} \\iota ( b ) = ( - I ( b ) , b ) \\pi ( a , b ) = a + I ( b ) . \\end{align*}"}
+{"id": "1377.png", "formula": "\\begin{align*} \\begin{aligned} \\delta _ \\pi ( f ) = & \\ [ \\pi , f ] _ { \\Omega } \\\\ = & \\ \\Big ( 0 , \\ldots , 0 , \\delta _ \\pi ( f ) ^ A _ { \\{ 1 , \\ldots , n + 1 \\} } ; \\delta _ \\pi ( f ) ^ V _ \\emptyset , \\ldots , \\delta _ \\pi ( f ) ^ V _ { \\{ 2 , \\ldots , n + 1 \\} } , 0 \\Big ) \\in \\mathcal { C } _ { \\rm A s s A c t } ^ { n + 1 } ( A , V ) , \\end{aligned} \\end{align*}"}
+{"id": "4941.png", "formula": "\\begin{align*} [ S ] \\circ [ S _ j ] = \\begin{cases} 1 \\\\ 0 . \\end{cases} \\end{align*}"}
+{"id": "5563.png", "formula": "\\begin{align*} \\lambda _ i = \\nu _ i ^ 2 \\left ( 1 + C _ 5 \\left ( \\frac { \\theta } { \\nu _ i } \\right ) ^ { 2 \\ell } \\right ) , \\end{align*}"}
+{"id": "4899.png", "formula": "\\begin{align*} | - K _ S | = A + | B | \\end{align*}"}
+{"id": "2273.png", "formula": "\\begin{align*} \\check R _ { 1 2 } ( b , c ) \\check R _ { 2 3 } ( a , c ) \\check R _ { 1 2 } ( a , b ) = \\check R _ { 2 3 } ( a , b ) \\check R _ { 1 2 } ( a , c ) \\check R _ { 2 3 } ( b , c ) \\ \\ \\ \\ \\ \\end{align*}"}
+{"id": "7030.png", "formula": "\\begin{align*} ( \\mu - \\Delta + b \\cdot \\nabla ) u = f \\end{align*}"}
+{"id": "9020.png", "formula": "\\begin{align*} e _ { 1 1 } \\cdot e _ { 1 1 } = c e _ { 1 1 } , e _ { 1 1 } \\cdot e _ { 1 2 } = - e _ { 1 2 } \\cdot e _ { 2 2 } = c e _ { 1 2 } , e _ { 1 1 } \\cdot e _ { 2 2 } = - e _ { 2 2 } \\cdot e _ { 2 2 } = c e _ { 2 2 } , \\ c \\ne 0 , \\end{align*}"}
+{"id": "8709.png", "formula": "\\begin{align*} ( \\ , \\sigma _ { 1 , \\frac { d } { 4 } } v \\ , | \\ , u \\ , ) _ { X _ G } = 0 , \\ \\ \\mbox { f o r e v e r y $ v \\in H ^ 0 _ b ( X ) ^ G \\cap C ^ \\infty ( X ) ^ G $ } . \\end{align*}"}
+{"id": "6594.png", "formula": "\\begin{align*} I _ 1 = \\int _ { \\varepsilon \\log ( \\frac { 3 } { 2 } ) } ^ { 1 } \\frac { e ^ w } { w } \\ , d w \\leq \\frac { e } { \\varepsilon \\log \\frac { 3 } { 2 } } \\int _ { \\varepsilon \\log ( \\frac { 3 } { 2 } ) } ^ { 1 } \\ , d w \\leq \\frac { e } { \\varepsilon \\log \\frac { 3 } { 2 } } \\ll \\frac { 1 } { \\varepsilon } . \\end{align*}"}
+{"id": "4999.png", "formula": "\\begin{align*} P _ N ( y _ i , x _ 2 , \\dots , x _ N ) & { } = S _ { N - 1 , i } ( x _ 2 , \\dots , x _ N ) \\\\ & { } \\times ( q - q ^ { - 1 } ) \\ , \\prod _ { k = 2 } ^ N ( q x _ k - q ^ { - 1 } y _ i ) \\ , \\prod _ { \\substack { j = 1 \\\\ j \\neq i } } ^ N ( q y _ i - q ^ { - 1 } y _ j ) , \\end{align*}"}
+{"id": "177.png", "formula": "\\begin{align*} A \\ = \\ \\bigcap _ { i \\in \\N } \\ \\pi _ { i } ^ { - 1 } ( A _ i ) \\ = \\ \\overleftarrow { L i m } \\{ A _ i \\} \\end{align*}"}
+{"id": "7027.png", "formula": "\\begin{align*} b _ n : = c _ n \\eta _ { \\varepsilon _ n } \\ast ( \\mathbf { 1 } _ n b ) , \\end{align*}"}
+{"id": "5726.png", "formula": "\\begin{align*} & \\frac { d } { d t } \\xi _ { i , 3 } - 2 ^ { - 1 } m \\xi _ { i , 3 } = \\mathcal { E } _ { i , 3 } , \\\\ & \\frac { d } { d t } \\xi _ { i , 4 } - 2 ^ { - 1 } m \\xi _ { i , 4 } - \\xi _ { i , 3 } = \\mathcal { E } _ { i , 4 } . \\end{align*}"}
+{"id": "1151.png", "formula": "\\begin{align*} \\begin{cases} \\lfloor r \\rfloor : = \\max \\{ k \\in \\mathbb Z : \\ k \\leq r \\} , \\\\ \\lfloor \\ ! \\lfloor r \\rfloor \\ ! \\rfloor : = \\max \\{ k \\in \\mathbb Z : \\ k < r \\} , \\end{cases} \\begin{cases} \\lceil r \\rceil : = \\min \\{ k \\in \\mathbb Z : \\ k \\geq r \\} , \\\\ \\lceil \\ ! \\lceil r \\rceil \\ ! \\rceil : = \\min \\{ k \\in \\mathbb Z : \\ k > r \\} , \\end{cases} \\end{align*}"}
+{"id": "7497.png", "formula": "\\begin{align*} S _ 1 [ h ] = \\chi ^ 2 R _ 0 [ h ] + \\nabla ^ { g _ 0 ( t ) } ( \\chi ^ 2 R _ 1 [ h ] ) , \\end{align*}"}
+{"id": "8812.png", "formula": "\\begin{align*} g ( a , b , c ) & \\ge ( a + b + c ) \\ , b ^ { \\ , \\rho - 1 } - ( b + c ) ( b ^ { \\ , \\rho - 1 } + ( \\rho - 1 ) b ^ { \\rho - 2 } a ) + a c ^ { \\ , \\rho - 1 } \\\\ & = ( 2 - \\rho ) a b ^ { \\rho - 1 } + a c \\left ( c ^ { \\rho - 2 } - ( ( \\rho - 1 ) ^ { \\frac { 1 } { \\rho - 2 } } b ) ^ { \\rho - 2 } \\right ) . \\end{align*}"}
+{"id": "7891.png", "formula": "\\begin{align*} f _ { 0 } ( \\mathbf { x } ) & = \\frac { q } { p } \\mathbf { x } ^ { T } \\mathbf { A } \\mathbf { x } + \\mathcal { L } _ { c } ( \\mathbf { x } ) , \\\\ f _ { 1 } ( \\mathbf { x } ) & = f _ { 0 } ( \\mathbf { x } ) + \\frac { q } { p } x _ { \\pi ( 1 ) } , \\\\ & \\vdots \\\\ f _ { p - 1 } ( \\mathbf { x } ) & = f _ { 0 } ( \\mathbf { x } ) + \\frac { q } { p } ( p - 1 ) x _ { \\pi ( 1 ) } . \\end{align*}"}
+{"id": "7473.png", "formula": "\\begin{align*} \\mathbf { r } : = 2 \\sqrt { f } = ( F ^ { - 1 } ) ^ * r . \\end{align*}"}
+{"id": "3990.png", "formula": "\\begin{align*} W : \\left \\{ \\begin{aligned} x _ { k } ' & = \\sum _ { i , j = 1 } ^ { n , \\nu } \\gamma _ { i j k } x _ { i } y _ { j } , k = 1 , \\ldots , n \\medskip \\\\ y ' _ { r } & = \\sum _ { i , j = 1 } ^ { n , \\nu } \\widetilde { \\gamma } _ { i j r } x _ { i } y _ { j } , r = 1 , \\ldots , \\nu , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4736.png", "formula": "\\begin{align*} S _ { \\xi \\to \\tau } : = S _ { \\tau } ^ { S _ { \\xi } } . \\end{align*}"}
+{"id": "386.png", "formula": "\\begin{align*} M _ { 2 } ( \\xi ; a ) : = \\bigcap \\{ M _ { 2 } \\left ( M _ { 2 } ( \\xi ; b ) \\cap M _ { 3 } ( \\nu ) \\right ) : \\nu < \\xi , b < a \\} . \\end{align*}"}
+{"id": "7615.png", "formula": "\\begin{align*} C _ j ( X ) = F ( X , j ) / S _ j . \\end{align*}"}
+{"id": "3304.png", "formula": "\\begin{align*} L ^ c _ t = \\sqrt { \\frac { c ^ 2 } { c ^ 2 + t } } \\exp \\left ( \\frac { B _ t ^ 2 } { 2 ( c ^ 2 + t ) } \\right ) = : f ( t , B _ t ) . \\end{align*}"}
+{"id": "1841.png", "formula": "\\begin{align*} \\star ( \\psi \\wedge \\star ( \\psi \\wedge F _ { A } ) ) = F _ { A } + \\star ( F _ { A } \\wedge \\phi ) . \\end{align*}"}
+{"id": "2845.png", "formula": "\\begin{align*} \\delta Q ( t _ { 2 } ) = \\delta Q ( t _ { 1 } ) = 0 , \\end{align*}"}
+{"id": "4626.png", "formula": "\\begin{align*} g _ n ( x ' ) = \\sum _ { \\tilde { k } \\in K _ n } \\psi \\left ( 2 ^ { n m } x ' - \\tilde { k } \\right ) = \\psi \\left ( 2 ^ { n m } x ' - k \\right ) \\geq \\frac { 1 } { 4 } . \\end{align*}"}
+{"id": "8952.png", "formula": "\\begin{align*} \\| \\nabla \\varphi - R \\| _ { L ^ { p ( \\cdot ) } ( Q ' ) } & = \\| \\nabla w - P \\| _ { L ^ { p ( \\cdot ) } ( Q ' ) } \\\\ & \\le \\| \\nabla w - I - S \\| _ { L ^ { p ( \\cdot ) } ( Q ' ) } + \\| d ( I + S , S O ( n ) ) \\| _ { L ^ { p ( \\cdot ) } ( Q ' ) } \\\\ & \\le 2 \\| \\nabla w - I - S \\| _ { L ^ { p ( \\cdot ) } ( Q ' ) } + \\| d ( \\nabla w , S O ( n ) ) \\| _ { L ^ { p ( \\cdot ) } ( Q ) } \\\\ & \\le 2 \\| \\nabla w - I - S \\| _ { L ^ { p ( \\cdot ) } ( Q ' ) } + \\| d ( \\nabla \\varphi , S O ( n ) ) \\| _ { L ^ { p ( \\cdot ) } ( Q ) } \\le C \\| d ( \\nabla v , S O ( n ) ) \\| _ { L ^ { p ( \\cdot ) } ( Q ) } \\end{align*}"}
+{"id": "7756.png", "formula": "\\begin{align*} u _ { n } = b _ { 1 } ^ { - 1 } ( \\frac { 4 \\pi h _ { 1 } } { | \\log \\varepsilon | } ) ^ { \\tau } e ^ { - b _ { 2 } e ^ { 4 \\sqrt { n } } } . \\end{align*}"}
+{"id": "5739.png", "formula": "\\begin{align*} X ^ 2 _ - ( t ) = \\sum _ { i : \\gamma ^ + _ i < \\gamma _ * } | \\xi _ i ^ + ( t ) | ^ 2 + \\sum _ { i : \\gamma ^ - _ i < \\gamma _ * } | \\xi _ i ^ - ( t ) | ^ 2 + \\sum _ { i \\in I _ 1 } | \\xi _ { i , 1 } ( t ) | ^ 2 + | \\xi _ { i , 2 } ( t ) | ^ 2 + \\sum _ { i \\in I _ 2 } | \\xi _ { i , 3 } ( t ) | ^ 2 + \\varepsilon _ 1 ^ 2 | \\xi _ { i , 4 } ( t ) | ^ 2 . \\end{align*}"}
+{"id": "5092.png", "formula": "\\begin{align*} L _ s \\cdot \\varepsilon _ { \\theta } & = - q ^ { - k } R _ { s } \\Psi _ { s } \\cdot \\varepsilon _ { \\theta } , \\end{align*}"}
+{"id": "5710.png", "formula": "\\begin{align*} G ( \\mathbf { L } ( v _ 1 , w _ 1 ) ; ( v _ 2 , w _ 2 ) ) = G ( ( v _ 1 , w _ 1 ) ; \\mathbf { L } ^ \\dagger ( v _ 2 , w _ 2 ) ) . \\end{align*}"}
+{"id": "8975.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c } F ( D ^ 2 u _ \\gamma ) + \\lambda _ \\gamma r ^ { - \\gamma } u _ \\gamma = 0 & \\mbox { i n } \\ \\overline { B ( 0 , 1 ) } \\setminus \\{ 0 \\} \\\\ u _ \\gamma = 0 & \\mbox { o n } \\ \\partial B ( 0 , 1 ) . \\end{array} \\right . \\end{align*}"}
+{"id": "4728.png", "formula": "\\begin{align*} { \\tau } _ 1 ( \\bar { a } ) = & \\{ ( i _ 1 , \\dots , i _ n ) \\mid i _ 1 < \\dots < i _ n \\in \\{ 1 , \\dots , m \\} ( a _ { i _ 1 } , \\dots , a _ { i _ n } ) \\in \\varrho \\} , \\\\ { \\tau } _ 2 ( \\bar { a } ) = & \\{ ( i , j ) \\mid i < j \\in \\{ 1 , \\dots , m \\} , \\ ; a _ i = a _ j \\} . \\end{align*}"}
+{"id": "1871.png", "formula": "\\begin{align*} \\alpha \\tau _ 2 \\alpha = \\tau _ 2 \\beta \\tau _ 2 \\beta = \\tau _ 2 ^ { - 1 } . \\end{align*}"}
+{"id": "6481.png", "formula": "\\begin{align*} | A ( \\nabla \\operatorname { g r a d } & f ) | ^ 2 - | A ( \\nabla \\operatorname { g r a d } f ) | \\big ( | A | | \\Delta f | + | A | ^ 3 f + m | H | | d \\phi | | f | \\big ) \\\\ & \\geq - \\frac { 1 } { 4 } \\big ( | A | | \\Delta f | + | A | ^ 3 | f | + m | H | | d \\phi | | f | \\big ) ^ 2 \\\\ & = - \\frac { 3 } { 4 } | A | ^ 2 | \\Delta f | ^ 2 - \\frac { 3 } { 4 } | A | ^ 6 f ^ 2 - \\frac { 3 } { 4 } m ^ 3 H ^ 2 f ^ 2 , \\end{align*}"}
+{"id": "1261.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ; q ^ 2 ) _ k ( - q ^ 2 ; q ^ 2 ) _ k } { ( q ^ 2 ; q ^ 2 ) _ k } q ^ { n ^ 2 - ( k + 1 ) ^ 2 } \\\\ [ 7 p t ] & = \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ; q ^ 2 ) _ n q ^ { k ^ 2 + k } } { ( q ^ 2 ; q ^ 2 ) _ k ( q ^ 2 ; q ^ 2 ) _ { n - k - 1 } ( 1 - q ^ { 2 n - 2 k - 1 } ) } \\\\ [ 7 p t ] & = \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ; q ^ 2 ) _ n q ^ { k ^ 2 + k } } { ( q ^ 2 ; q ^ 2 ) _ { n - 1 } ( 1 - q ^ { 2 n - 2 k - 1 } ) } { n - 1 \\brack k } _ { q ^ 2 } . \\end{align*}"}
+{"id": "3455.png", "formula": "\\begin{align*} [ S W F _ G ( Y , \\mathfrak { t } , \\iota ) ] _ { \\mathrm { l o c } } = [ ( S ^ 0 , m , n ) ] _ { \\mathrm { l o c } } \\end{align*}"}
+{"id": "5136.png", "formula": "\\begin{align*} \\frac { \\partial L _ { d } D ( p \\| q ) } { \\partial q _ { j } } = & \\left \\{ \\left [ \\frac { a - 1 } { a - b } A ^ { a - 2 } - \\frac { b - 1 } { a - b } A ^ { b - 2 } \\right ] \\frac { \\partial A } { \\partial q _ { j } } \\right \\} \\\\ & - \\left \\{ \\left [ \\frac { a - 1 } { a - b } B ^ { a - 2 } - \\frac { b - 1 } { a - b } B ^ { b - 2 } \\right ] \\frac { \\partial B } { \\partial q _ { j } } \\right \\} \\end{align*}"}
+{"id": "5198.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } D _ { \\alpha \\beta } ( p \\| q ) } { \\partial q _ { j } } = \\underbrace { \\frac { Z _ { B } } { \\alpha } p ^ { \\alpha } _ { j } q ^ { \\beta - 2 } _ { j } } _ { U _ { j } } - \\underbrace { \\frac { Z _ { A } } { \\alpha } q ^ { \\alpha + \\beta - 2 } _ { j } } _ { V _ { j } } \\end{align*}"}
+{"id": "873.png", "formula": "\\begin{align*} T _ 1 = w ^ 1 C _ { 1 1 } + w ^ 2 C _ { 2 1 } = w ^ 2 ( { w ^ 2 } _ { \\substack { \\\\ x _ 1 } } - { w ^ 1 } _ { \\substack { \\\\ x _ 2 } } ) . \\end{align*}"}
+{"id": "6144.png", "formula": "\\begin{align*} \\begin{aligned} & f ( y ) \\leq f ( x ) + \\langle \\nabla f ( x ) , y - x \\rangle + \\frac { L } { 2 } \\| y - x \\| ^ 2 , ~ \\forall x , y \\in \\mathbb { R } ^ n . \\\\ \\Longleftrightarrow & f ( y ) \\geq f ( x ) + \\langle \\nabla f ( x ) , y - x \\rangle + \\frac { 1 } { 2 L } \\| \\nabla f ( x ) - \\nabla f ( y ) \\| ^ 2 , ~ \\forall x , y \\in \\mathbb { R } ^ n . \\end{aligned} \\end{align*}"}
+{"id": "4242.png", "formula": "\\begin{align*} E _ 4 ( \\tau ) ^ 4 = 1 + 9 6 0 q + 3 5 4 2 4 0 q ^ 2 + \\cdots , \\end{align*}"}
+{"id": "12.png", "formula": "\\begin{align*} m ( x ; \\kappa , q ) : = R ( \\kappa , q ) q _ + = R _ 0 ( \\kappa ) \\sum _ { \\ell \\geq 1 } [ C _ + q R _ 0 ( \\kappa ) ] ^ { \\ell - 1 } q _ + \\end{align*}"}
+{"id": "9070.png", "formula": "\\begin{align*} x _ i ( t ) = v _ i ( t ) - \\underline { v } _ i , \\mbox { f o r a l l $ i \\in N $ } . \\end{align*}"}
+{"id": "3422.png", "formula": "\\begin{align*} G = \\{ 1 , j , - 1 , - j \\} . \\end{align*}"}
+{"id": "6528.png", "formula": "\\begin{align*} Z ^ { \\alpha , \\beta } _ t = \\int ^ t _ { - \\infty } [ ( t - s ) ^ { 1 - \\beta } - ( - s ) ^ { 1 - \\beta } 1 _ { \\{ s < 0 \\} } ] d Z ^ { \\alpha } _ s , \\end{align*}"}
+{"id": "7647.png", "formula": "\\begin{align*} \\norm { S _ { \\gamma _ a } } _ { \\infty , \\infty } : = \\sup _ { n \\in \\mathbb { Z } ^ d } \\sum _ { m \\in \\mathbb { Z } ^ d } e ^ { ( \\delta - \\gamma _ a ) d ( m , n ) } < \\infty . \\end{align*}"}
+{"id": "8181.png", "formula": "\\begin{align*} D _ 1 & = \\frac { c } { M } \\frac { 2 [ c - 1 ] ^ + } { c } + \\frac { M - c } { M } \\frac { 2 [ M - c - 1 ] ^ + } { M - c } \\\\ & = \\frac { 2 [ c - 1 ] ^ + + 2 [ M - c - 1 ] ^ + } { M } . \\end{align*}"}
+{"id": "6189.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau ^ k } [ f ( x ) - f ( \\breve { x } ^ { k } ) ] - \\frac { 1 } { \\tau ^ { k - 1 } } [ f ( x ) - f ( \\breve { x } ^ { k - 1 } ) ] + ( u - \\widetilde { u } ^ k ) ^ T F ( \\widetilde { u } ^ k ) \\\\ & + c _ 1 ^ k ( A ( x - \\widetilde { x } ^ k ) ) ^ T ( A \\breve { x } ^ { k - 1 } - b ) \\geq ( v - \\widetilde { v } ^ k ) ^ T Q ^ k ( v ^ k - \\widetilde { v } ^ k ) , ~ \\forall u , \\end{aligned} \\end{align*}"}
+{"id": "8249.png", "formula": "\\begin{align*} I ( { A } ^ { L } ; Y ^ { L } | { S } ) & = H ( Y ^ L | S ) - H ( Y ^ L | S , A ^ L ) \\\\ & = H ( Y ^ L | S ) - H ( Y ^ L | S , X ^ L ) \\\\ & = I ( { X } ^ { L } ; Y ^ { L } | { S } ) . \\end{align*}"}
+{"id": "9015.png", "formula": "\\begin{align*} y _ { 1 n } = z _ { 1 n } ^ 1 = z _ { 1 n } ^ n = 0 . \\end{align*}"}
+{"id": "6596.png", "formula": "\\begin{align*} | I _ 3 | = \\int _ { ( 1 / 2 ) \\varepsilon \\log | \\eta | } ^ { \\varepsilon \\log | \\eta | } \\frac { e ^ w } { w } \\ d w \\leq \\frac { e ^ { \\varepsilon \\log | \\eta | } } { \\varepsilon \\log | \\eta | } + e ^ { ( 1 / 2 ) \\varepsilon \\log | \\eta | } \\frac { 1 } { \\varepsilon \\log | \\eta | } \\ll \\frac { e ^ { \\varepsilon \\log | \\eta | } } { \\varepsilon \\log | \\eta | } . \\end{align*}"}
+{"id": "3065.png", "formula": "\\begin{align*} v _ i = \\sum _ { j = 0 } ^ { i - 2 } \\frac { e _ j - e _ { j + 1 } } { e _ { i - 1 } } \\beta _ { j + 1 } + \\beta _ i \\end{align*}"}
+{"id": "2428.png", "formula": "\\begin{align*} U ( t , x ) = ( 1 - p ) ^ { - \\frac { 1 } { p } } ( V ( t , x ) ) ^ { \\frac { 1 } { 1 - p } } \\end{align*}"}
+{"id": "6129.png", "formula": "\\begin{align*} \\| u _ { 2 2 } \\| & = \\| \\rho _ { m , j } ( x _ j ^ * x _ j + a _ j ^ * a _ j ) - \\rho _ { m , n } \\big ( \\rho _ { n , j } ( x _ j ) ^ * \\rho _ { n , j } ( x _ j ) + \\rho _ { n , j } ( a _ j ) ^ * \\rho _ { n , j } ( a _ j ) \\big ) \\| \\leq 4 . \\end{align*}"}
+{"id": "2490.png", "formula": "\\begin{align*} \\partial _ t u & = \\Delta ( u \\gamma ( v ) ) \\ ; \\ ; \\ ; \\ ; ( 0 , \\infty ) \\times \\Omega \\ , , \\\\ \\partial _ t v & = \\Delta v - v + u \\ ; \\ ; \\ ; \\ ; ( 0 , \\infty ) \\times \\Omega \\ , , \\\\ \\nabla ( u \\gamma ( v ) ) \\cdot \\mathbf { n } & = \\nabla v \\cdot \\mathbf { n } = 0 \\ ; \\ ; \\ ; \\ ; ( 0 , \\infty ) \\times \\partial \\Omega \\ , , \\\\ ( u , v ) ( 0 ) & = ( u ^ { i n } , v ^ { i n } ) \\ ; \\ ; \\ ; \\ ; \\Omega \\ , , \\end{align*}"}
+{"id": "8740.png", "formula": "\\begin{align*} \\int _ { y \\in \\R } \\alpha ( d x , d y ) \\mbox { a . e . } , \\ ; \\int y \\ , \\alpha _ x ( d y ) = \\int y \\ , \\alpha ' _ x ( d y ) . \\end{align*}"}
+{"id": "3771.png", "formula": "\\begin{align*} k ( x ) : = \\inf _ { y \\in ( 0 , 1 ] } m ( y ) + y x , x > 0 . \\end{align*}"}
+{"id": "5558.png", "formula": "\\begin{align*} \\check \\chi = J _ \\Delta \\chi . \\end{align*}"}
+{"id": "4475.png", "formula": "\\begin{align*} F _ 0 = & \\frac { g _ 0 p _ * \\left ( f _ { \\tilde u } \\left ( \\prod _ { 1 \\le j \\le m } f _ { z _ j } \\right ) \\left ( \\sum _ { 1 \\le j \\le m } p _ j \\frac { d f _ { z _ j } } { f _ { z _ j } } \\right ) \\right ) } { c _ 0 d z } \\\\ = & \\frac { p _ * \\left ( f _ { u } \\left ( \\prod _ { 1 \\le j \\le m } f ^ { k _ j + 1 } _ { z _ j } \\right ) \\left ( \\sum _ { 1 \\le j \\le m } p _ j \\frac { d f _ { z _ j } } { f _ { z _ j } } \\right ) \\right ) } { c _ 0 d z } , \\end{align*}"}
+{"id": "3598.png", "formula": "\\begin{align*} T ^ 3 - ( 1 + a ) T ^ 2 + ( a + b ) T - b I = 0 , \\end{align*}"}
+{"id": "4147.png", "formula": "\\begin{align*} & Q ^ * ( \\theta ) \\psi ^ { ( r ) } ( \\theta ) + \\sum _ { i \\ge j } r ^ { i } \\bar { \\xi } _ i ( \\theta ) \\bigl ( \\psi ^ { ( r ) } ( \\theta ) - \\psi ^ { ( r ) } _ i ( \\theta ) \\bigr ) \\\\ & = - \\frac { 1 } { 2 } \\sum _ { i \\in \\mathcal { J } } r ^ i { \\xi } ^ * _ i ( \\theta ) \\bigl ( \\psi ^ { ( r ) } ( \\theta ) - \\psi ^ { ( r ) } _ i ( \\theta ) \\bigr ) + o ( r ^ J \\abs { \\theta } ) + o ( \\abs { \\theta } ^ 2 ) . \\end{align*}"}
+{"id": "197.png", "formula": "\\begin{align*} P [ j , k ] ^ { \\circ } ( z + ) \\ = \\ \\bar D _ k - \\ \\cup \\ [ \\bigcup _ { i \\in \\N } \\ ( z + A _ i ) + ] . \\end{align*}"}
+{"id": "5718.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\ell \\Vert \\mathbf { L } ^ j ( v , w ) \\Vert _ { G } \\approx \\Vert ( v , w ) \\Vert _ { H ^ { \\ell + 1 } \\times H ^ \\ell } . \\end{align*}"}
+{"id": "4584.png", "formula": "\\begin{align*} \\int _ M t d \\mu & = 2 \\pi \\int _ { - a } ^ a t A ( t ) d t \\\\ & = \\frac { \\omega ( F ) ^ 3 } { 9 6 \\pi } \\left [ 6 \\frac { \\omega ( c _ + ) - \\omega ( c _ - ) } { \\omega ( F ) } - \\sum _ j \\frac { 1 } { r _ j s _ j } \\left ( \\frac { \\omega ( r _ j E _ j ' ) - \\omega ( - s _ j E _ j ) } { \\omega ( F ) } \\right ) ^ 3 + c _ 1 ( Y ) [ \\Sigma _ { - a } ] + c _ 1 ( Y ) [ \\Sigma _ a ] \\right ] , \\end{align*}"}
+{"id": "3262.png", "formula": "\\begin{align*} \\Phi _ n ( q ) = \\prod _ { \\substack { 1 \\leq k \\leq n \\\\ \\gcd ( k , n ) = 1 } } ( q - \\zeta ^ k ) , \\end{align*}"}
+{"id": "8493.png", "formula": "\\begin{align*} & U _ m ( x , y ) \\bigl ( \\xi _ \\delta ( u _ m ( x ) ) - \\xi _ \\delta ( u _ m ( y ) ) \\bigr ) \\\\ & = | u _ m ( x ) - u _ m ( y ) | ^ { p - 2 } \\bigl ( u _ m ( x ) - u _ m ( y ) \\bigr ) \\bigl ( \\xi _ \\delta ( u _ m ( x ) ) - \\xi _ \\delta ( u _ m ( y ) ) \\bigr ) \\geq 0 \\end{align*}"}
+{"id": "4215.png", "formula": "\\begin{align*} \\theta ( v , \\tau ) = 2 q ^ { \\frac { 1 } { 8 } } { \\rm s i n } ( \\pi v ) \\prod _ { j = 1 } ^ { \\infty } [ ( 1 - q ^ j ) ( 1 - e ^ { 2 \\pi \\sqrt { - 1 } v } q ^ j ) ( 1 - e ^ { - 2 \\pi \\sqrt { - 1 } v } q ^ j ) ] , \\end{align*}"}
+{"id": "7710.png", "formula": "\\begin{align*} \\frac { \\partial [ \\det ( \\nabla ^ { 2 } h + h I ) ] ^ { - 1 } } { \\partial t } & = \\kappa Q [ ( n - 1 ) - \\varepsilon _ { 0 } H ] \\\\ & \\leq \\kappa Q [ ( n - 1 ) - \\varepsilon _ { 0 } ( n - 1 ) \\kappa ^ { \\frac { 1 } { n - 1 } } ] , \\end{align*}"}
+{"id": "5582.png", "formula": "\\begin{align*} \\langle ( B ^ * ) ^ t \\check \\chi _ i , ( B ^ * ) ^ t \\check \\chi _ j \\rangle = \\langle \\Delta B ^ t \\chi _ i , \\Delta B ^ t \\chi _ j \\rangle . \\end{align*}"}
+{"id": "1030.png", "formula": "\\begin{align*} \\Delta _ n ^ { s , t } : = \\Vert ( T _ n ( w ) ^ { - 1 } ) ^ { s , t } - ( T _ { \\infty } ( w ) ^ { - 1 } ) ^ { s , t } \\Vert , n \\in \\N , \\ \\ s , t \\in \\{ 1 , \\dots , n \\} . \\end{align*}"}
+{"id": "8526.png", "formula": "\\begin{align*} \\rho = \\begin{pmatrix} \\chi _ p ( 1 + \\eta \\log ) & \\eta b \\\\ \\eta c & 1 - \\eta \\log \\end{pmatrix} \\end{align*}"}
+{"id": "5072.png", "formula": "\\begin{align*} \\overline { K ( w ) } & = \\overline { K ( s _ { i _ 1 } ) } * \\dots * \\overline { K ( s _ { i _ k } ) } \\end{align*}"}
+{"id": "8253.png", "formula": "\\begin{align*} & \\frac { 1 } { L } \\sum _ { i = 1 } ^ L H ( Y _ i | { \\underline S } , Y ^ { i - 1 } ) \\\\ & = \\frac { 1 } { L } \\Big ( \\sum _ { i = 1 } ^ { t - 1 } H ( Y _ i | { \\underline S } , Y ^ { i - 1 } ) \\\\ & + \\sum _ { y ^ { t - 1 } } P _ { Y ^ { t - 1 } , { \\underline S } } ( y ^ { t - 1 } , { \\underline s } ) \\sum _ { i = t } ^ L H ( Y _ i | { \\underline s } , y ^ { t - 1 } , Y _ t ^ { i - 1 } ) \\Big ) . \\end{align*}"}
+{"id": "3026.png", "formula": "\\begin{align*} \\begin{array} { l } \\lambda = 3 \\alpha ^ 2 + \\alpha - \\beta ^ 2 + \\beta , \\\\ \\mu = 2 \\alpha \\beta + \\beta - 3 \\alpha - 1 , \\\\ \\nu = \\beta ^ 2 - 3 \\alpha ^ 2 - 2 \\alpha \\beta - 6 \\beta + 2 \\alpha + 5 . \\end{array} \\end{align*}"}
+{"id": "9093.png", "formula": "\\begin{align*} a _ { i j } = \\left \\{ \\begin{array} { l l l } \\alpha & \\mbox { w i t h p r o b . $ \\tilde \\Theta _ i \\in [ 0 , 1 ] $ } \\\\ 0 & \\mbox { w i t h p r o b . $ 1 - \\tilde \\Theta _ i $ } \\end{array} \\right . , \\forall i \\in \\mathcal P ^ { ( r ) } , \\ , j \\in \\mathcal N ^ { ( r ) } . \\end{align*}"}
+{"id": "5627.png", "formula": "\\begin{align*} d ( y , T a ) = d ( T x , T a ) = d ( x , a ) < \\delta . \\end{align*}"}
+{"id": "7349.png", "formula": "\\begin{align*} W _ \\phi \\left ( u z ^ 1 z ^ 2 . . . z ^ \\ell \\right ) = \\emptyset . \\end{align*}"}
+{"id": "7052.png", "formula": "\\begin{align*} I _ q = \\sum _ { i = 1 } ^ d \\langle | \\nabla w _ i | ^ 2 , | w | ^ { q - 2 } \\rangle , J _ q = \\langle | \\nabla | w | | ^ 2 , | w | ^ { q - 2 } \\rangle \\end{align*}"}
+{"id": "3252.png", "formula": "\\begin{align*} \\partial _ a \\partial _ \\nu H _ { a , [ \\nu ] } ( x , y ) & = \\frac { a } { 1 6 \\pi } \\ : \\sum _ { j = 1 } ^ \\infty \\frac { ( - 1 ) ^ j } { j ! \\ : ( j + 1 ) ! } \\ : ( j + 1 ) ( - j ) \\ : \\frac { a ^ { j - 1 } } { 4 ^ j } \\ : \\big ( \\xi ^ 2 - \\nu \\big ) ^ { j - 1 } \\\\ & = \\frac { a } { 1 6 \\pi } \\ : \\sum _ { j = 0 } ^ \\infty \\frac { ( - 1 ) ^ j } { j ! \\ : ( j + 1 ) ! } \\ : \\frac { a ^ { j - 1 } } { 4 \\cdot 4 ^ j } \\ : \\big ( \\xi ^ 2 - \\nu \\big ) ^ { j } = \\frac { 1 } { 4 } \\ : H _ { a , [ \\nu ] } ( x , y ) \\end{align*}"}
+{"id": "8714.png", "formula": "\\begin{align*} & { \\rm d e t \\ , } \\left ( ( b _ { j , \\ell } ) ^ { n - d } _ { j , \\ell = 0 } \\right ) \\neq 0 , \\\\ & b _ { 0 , \\ell } = B _ { G , M _ 1 } ( p , a _ { \\ell } ) , \\ \\ \\ell = 0 , \\ldots , n - d , \\\\ & b _ { j , \\ell } = ( Z _ { j , x } B _ { G , M } ) ( p , a _ \\ell ) , \\ \\ \\ell = 0 , \\ldots , n - d , j = 1 , \\ldots , n - d . \\end{align*}"}
+{"id": "2981.png", "formula": "\\begin{align*} \\tilde { \\mathcal { L } } ( t ) : = D _ 1 \\int _ { D _ { X , [ l ] } ( 0 ) } ^ { D _ { X , [ l ] } ( t ) } \\phi ( r ) d r . \\end{align*}"}
+{"id": "7163.png", "formula": "\\begin{align*} \\omega _ 1 \\gg 1 , \\quad \\omega _ 2 = \\omega _ 1 + \\eta \\ , \\ , \\quad \\ , \\eta > 0 . \\end{align*}"}
+{"id": "7429.png", "formula": "\\begin{align*} F _ { \\rm f r i c t i o n } = - A \\dot { x } , \\end{align*}"}
+{"id": "7878.png", "formula": "\\begin{align*} \\mathbf { A } _ { i } = \\begin{cases} \\psi ( \\pi , \\mathbf { a } , \\mathbf { d } _ { i } ) , & 1 \\leq i \\leq 3 ; \\\\ \\psi ( \\pi , \\mathbf { b } , \\mathbf { d } _ { i - 3 } ) , & 4 \\leq i \\leq 6 . \\end{cases} \\end{align*}"}
+{"id": "4619.png", "formula": "\\begin{align*} \\lim _ { j \\rightarrow \\infty } \\lambda ( j ) ^ { - \\frac { d } { 2 } } M ^ { ( d - 1 ) j } = \\infty . \\end{align*}"}
+{"id": "7903.png", "formula": "\\begin{align*} \\lambda \\wedge \\mu = ( - 1 ) ^ { k j } \\mu \\wedge \\lambda , \\end{align*}"}
+{"id": "1942.png", "formula": "\\begin{align*} ( v - u ) f ( x ) = \\phi ( x ) w _ + ( x ) f ( x ) , \\end{align*}"}
+{"id": "4017.png", "formula": "\\begin{align*} \\begin{cases} x _ { 1 } ^ { \\left ( t + 1 \\right ) } & = \\ ; \\gamma _ { 1 } x _ { 1 } ^ { \\left ( t \\right ) } y ^ { \\left ( t \\right ) } \\medskip \\\\ y ^ { \\left ( t + 1 \\right ) } & = \\ ; \\left ( 1 - \\gamma _ { 1 } \\right ) x _ { 1 } ^ { \\left ( t \\right ) } y ^ { \\left ( t \\right ) } . \\end{cases} \\end{align*}"}
+{"id": "8013.png", "formula": "\\begin{align*} & \\sup _ { f _ 1 , f _ 2 , f _ 3 , F , G , H } \\inf _ { W \\in C } \\left ( \\mathcal { I } _ 2 ( W _ 0 , f _ 1 , f _ 2 , f _ 3 ) + \\mathcal { I } _ 1 ( W , F , G , H ) \\right ) \\\\ & = \\inf _ { W \\in C } \\sup _ { f _ 1 , f _ 2 , f _ 3 , F , G , H } \\left ( \\mathcal { I } _ 2 ( W _ 0 , f _ 1 , f _ 2 , f _ 3 ) + \\mathcal { I } _ 1 ( W , F , G , H ) \\right ) \\end{align*}"}
+{"id": "6187.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau ^ k } [ f _ j ( x _ j ) - f _ j ( \\breve { x } _ j ^ { k } ) ] - \\frac { 1 } { \\tau ^ { k - 1 } } [ f _ j ( x _ j ) - f _ j ( \\breve { x } _ j ^ { k - 1 } ) ] + ( x _ j - \\widetilde { x } ^ k _ j ) ^ T [ - A _ j ^ T \\widetilde { \\lambda } ^ k \\\\ + & \\tau ^ k \\beta ^ k \\sum _ { i = 2 } ^ { j } A _ j ^ T A _ i ( \\widetilde { x } _ i ^ k - \\bar { x } _ i ^ { k } ) + ( 1 - \\gamma ) ( 1 - \\tau ^ k ) \\beta ^ k A _ j ^ T ( A \\breve { x } ^ { k - 1 } - b ) ] \\geq 0 , ~ \\forall x _ j . \\end{aligned} \\end{align*}"}
+{"id": "5792.png", "formula": "\\begin{align*} \\sum _ { i : \\lambda _ i \\neq 0 } | \\xi _ i ( t ) | ^ 2 = o ( 1 ) | x ( t ) | ^ 2 . \\end{align*}"}
+{"id": "2017.png", "formula": "\\begin{align*} \\mathcal { N } _ n = \\dfrac { 1 } { \\rho - 1 } \\left ( d _ 1 \\rho ^ { \\ell + m + k } - ( d _ 1 - d _ 2 ) \\rho ^ { m + k } - ( d _ 2 - d _ 3 ) \\rho ^ k - d _ 3 \\right ) . \\end{align*}"}
+{"id": "4208.png", "formula": "\\begin{align*} & - 1 6 6 3 2 \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 1 4 ) } = \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right . \\\\ & \\left . \\cdot { \\rm c h } ( S ^ 2 \\widetilde { T X } + \\widetilde { T X } + \\wedge ^ 2 \\widetilde { L _ R \\otimes C } - \\widetilde { L _ R \\otimes C } - \\widetilde { T X } \\otimes \\widetilde { L _ R \\otimes C } \\right \\} ^ { ( 1 4 ) } . \\end{align*}"}
+{"id": "3577.png", "formula": "\\begin{align*} \\lambda _ 1 + \\lambda _ 2 + \\lambda _ 1 \\lambda _ 2 = 0 . \\end{align*}"}
+{"id": "2154.png", "formula": "\\begin{align*} k A : = \\{ a _ 1 + \\dots + a _ k : a _ 1 , \\dots , a _ k \\in A \\} , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , A ^ { ( k ) } : = \\{ a _ 1 \\cdots a _ k : a _ 1 , \\dots , a _ k \\in A \\} . \\end{align*}"}
+{"id": "3585.png", "formula": "\\begin{align*} \\alpha ^ 3 ( \\tau _ 0 ^ 2 \\tau _ 1 ) \\tau - ( \\alpha ^ 3 ( \\tau _ 0 ^ 2 \\tau _ 1 ) + ( a + b ) \\alpha \\tau _ 0 - b ) i d + ( a + b ) \\alpha \\tau _ 0 \\tau - b i d = 0 , \\end{align*}"}
+{"id": "7946.png", "formula": "\\begin{align*} \\frac { \\delta _ { \\phi } \\mathcal { F } } { \\delta v } = \\delta ( \\ast w ) = - \\ast d w . \\end{align*}"}
+{"id": "3522.png", "formula": "\\begin{align*} \\phi ^ * ( m \\cdot \\textbf { r } _ 0 ^ * ) & = \\kappa \\cdot m \\cdot \\textbf { r } _ 0 ^ * \\\\ \\phi ^ * ( m \\cdot \\textbf { r } _ i ^ * ) & = \\mu _ i \\cdot m \\cdot \\textbf { r } _ 0 ^ * + k _ i \\cdot m \\cdot \\textbf { r } _ { \\sigma ( i ) } ^ * \\\\ d ^ * ( m \\cdot \\textbf { a } _ i ^ * ) & = m \\cdot \\textbf { r } _ 0 ^ * + p _ i \\cdot m \\cdot \\textbf { r } _ i ^ * \\end{align*}"}
+{"id": "8573.png", "formula": "\\begin{align*} C _ t ^ { t _ 1 } = ( J _ k + [ - \\varepsilon _ k ( G ^ { t _ 0 } _ t ) B _ { t _ 0 } ] ^ { k \\cdot } _ + ) C ^ { t _ 0 } _ t , \\end{align*}"}
+{"id": "8566.png", "formula": "\\begin{align*} F ( n , k ) = \\frac { ( a ) _ k ( b ) _ k } { ( 2 n ) _ k ( a + n ) _ k } \\end{align*}"}
+{"id": "7575.png", "formula": "\\begin{align*} m _ { p , q } ( c ) = \\lim \\limits _ { n \\rightarrow \\infty } E _ { p , q } ( u _ { n } ) = \\lim \\limits _ { n \\rightarrow \\infty } E _ { p , q } ( u _ { n } ( \\cdot + y _ { n } ) ) = \\lim \\limits _ { n \\rightarrow \\infty } E _ { p , q } ( v _ { n } ^ { 1 } ) + E _ { p , q } ( z ^ { 1 } ) . \\end{align*}"}
+{"id": "738.png", "formula": "\\begin{align*} g _ 1 ( A ) & = t _ { J _ k ( v ) - 1 , q } - a \\\\ & = ( a - r - 1 ) b + v a ( J _ k ( v ) - J _ { k - 1 } ( v ) - 1 ) - a \\\\ & + \\frac { v a ( a - r ) J _ { k - 1 } ( v ) } { J _ k ( v ) } + ( v a J _ { k - 1 } ( v ) + b J _ k ( v ) ) \\ , . \\end{align*}"}
+{"id": "6426.png", "formula": "\\begin{align*} \\overline { G } ^ { ( d ) } _ n ( a _ 0 , b _ 0 ) & = \\overline { G } _ n ( a _ 0 , b _ 0 , \\delta _ 0 , \\alpha _ 0 ) \\\\ & + \\int _ 0 ^ 1 \\nabla _ { ( \\delta , \\alpha ) } \\overline { G } _ n ( a _ 0 , b _ 0 , \\delta _ 0 + t ( \\tilde { \\delta } _ n - \\delta _ 0 ) , \\alpha _ 0 + t ( \\tilde { \\alpha } _ n - \\alpha _ 0 ) ) d t \\begin{pmatrix} \\tilde { \\delta } _ n - \\delta _ 0 \\\\ \\tilde { \\alpha } _ n - \\alpha _ 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "259.png", "formula": "\\begin{align*} \\frac { d ^ 2 y } { d \\tau ^ 2 } = 0 \\end{align*}"}
+{"id": "3873.png", "formula": "\\begin{align*} V ^ { n , r } = V ^ { n , 0 } + \\sum \\limits _ { k = 1 } ^ r \\frac { 1 } { m ^ n _ 0 + p ^ n _ 0 + k + 2 } \\left ( \\left ( \\bar v ^ { n , m ^ n _ 0 + p ^ n _ 0 + k + 1 } - \\beta ^ 0 _ { ( 1 ) } \\right ) - V ^ { n , k - 1 } \\right ) . \\end{align*}"}
+{"id": "4118.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\sup _ { 0 < \\abs { y } \\le K } \\frac { c ^ { ( r ) } _ { e , i } ( r y ) } { r ^ 2 y ^ 2 } = 0 , \\end{align*}"}
+{"id": "5135.png", "formula": "\\begin{align*} L _ { d } D ( p \\| q ) = \\left \\{ \\frac { A ^ { a - 1 } \\left ( p , q \\right ) - A ^ { b - 1 } \\left ( p , q \\right ) } { a - b } - \\frac { B ^ { a - 1 } \\left ( p , q \\right ) - B ^ { b - 1 } \\left ( p , q \\right ) } { a - b } \\right \\} \\end{align*}"}
+{"id": "6833.png", "formula": "\\begin{align*} e ^ { i ( \\omega _ { N - 1 } - \\omega _ { N - 1 } ' ) } \\cdot \\left [ \\begin{pmatrix} - \\sin ( \\psi _ { N - 1 } ) & 0 & \\ldots & 0 \\end{pmatrix} B ( B ' ) ^ { - 1 } \\begin{pmatrix} - \\sin ( \\psi _ { N - 1 } ' ) \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{pmatrix} \\right ] + & \\\\ \\cos ( \\psi _ { N - 1 } ) \\cos ( \\psi _ { N - 1 } ' ) e ^ { - i ( N - 1 ) ( \\omega _ { N - 1 } - \\omega _ { N - 1 } ' ) } & = 1 . \\end{align*}"}
+{"id": "1755.png", "formula": "\\begin{align*} r _ { j , n - k } = - r _ { k , n - j } \\quad \\quad \\forall \\ , j , k . \\end{align*}"}
+{"id": "3826.png", "formula": "\\begin{align*} G _ 2 ( q ) = \\frac { \\left ( q ^ 6 ; q ^ 6 \\right ) _ \\infty } { 4 \\left ( q ^ 2 ; q ^ 2 \\right ) _ \\infty \\left ( q ^ 3 ; q ^ 3 \\right ) _ \\infty } f ( q ) + \\frac { 3 \\left ( q ^ 3 ; q ^ 3 \\right ) ^ 3 _ \\infty } { 4 ( q ; q ) _ \\infty \\left ( q ^ 2 ; q ^ 2 \\right ) _ \\infty \\left ( q ^ 6 ; q ^ 6 \\right ) _ \\infty } , \\end{align*}"}
+{"id": "7781.png", "formula": "\\begin{align*} \\hat V _ \\ell f _ E ( z ) = \\sum _ { j = - \\ell } ^ \\ell \\hat V _ j z ^ { j + \\ell } - E z ^ \\ell . \\end{align*}"}
+{"id": "5230.png", "formula": "\\begin{align*} & U _ { j } = ( 1 - \\alpha ) \\frac { \\sum _ { j } p _ { j } } { \\sum _ { j } q _ { j } } Z \\ ; \\frac { \\overline { M G } _ { j } } { \\overline { q } _ { j } } \\\\ & V _ { j } = ( 1 - \\alpha ) \\frac { \\sum _ { j } p _ { j } } { \\sum _ { j } q _ { j } } Z \\ ; \\overline { M G } \\end{align*}"}
+{"id": "2335.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } a _ { k } - \\Delta a _ { k } = - \\mathrm { d i v } ( u ^ { c , \\gamma } \\otimes a _ { k - 1 } + a _ { k - 1 } \\otimes u ^ { c , \\gamma } ) - \\nabla \\pi _ { k - 1 } , \\\\ \\mathrm { d i v } a _ { k } = 0 , \\\\ \\pi _ { k - 1 } = ( - \\Delta ) ^ { - 1 } \\partial _ { i } \\partial _ { j } ( u ^ { c , \\gamma } \\otimes a _ { k - 1 } + a _ { k - 1 } \\otimes u ^ { c , \\gamma } ) , \\\\ a _ { k } ( x , 0 ) = w _ { 0 } . \\end{cases} \\end{align*}"}
+{"id": "1124.png", "formula": "\\begin{align*} \\psi _ j * \\vec f & = \\left ( \\widehat { \\psi _ j } \\widehat { \\vec f } \\right ) ^ \\vee = \\left [ 2 ^ { - j n } \\widehat { \\psi } \\left ( 2 ^ { - j } \\cdot \\right ) \\widehat { \\vec f } \\right ] ^ \\vee \\\\ & = 2 ^ { - j \\sigma } \\left [ 2 ^ { - j n } | \\cdot | ^ \\sigma \\widehat { \\varphi } \\left ( 2 ^ { - j } \\cdot \\right ) \\widehat { \\vec f } \\right ] ^ \\vee = 2 ^ { - j \\sigma } \\left ( | \\cdot | ^ \\sigma \\widehat { \\varphi _ j } \\widehat { \\vec f } \\right ) ^ \\vee , \\end{align*}"}
+{"id": "1107.png", "formula": "\\begin{align*} \\left \\| A _ Q A _ R ^ { - 1 } \\right \\| ^ p & = \\sup _ { \\vec z \\in \\mathbb { C } ^ m , \\ , | \\vec z | = 1 } \\left | A _ Q A _ R ^ { - 1 } \\vec z \\right | ^ p \\\\ & \\sim \\sup _ { \\vec z \\in \\mathbb { C } ^ m , \\ , | \\vec z | = 1 } \\left [ | c _ Q | + \\ell ( Q ) \\right ] ^ { - d } \\left | A _ R ^ { - 1 } \\vec z \\right | ^ p \\\\ & \\sim \\left [ \\frac { | c _ R | + \\ell ( R ) } { | c _ Q | + \\ell ( Q ) } \\right ] ^ d . \\end{align*}"}
+{"id": "4420.png", "formula": "\\begin{align*} \\inf _ { x \\in \\mathcal { X } } & \\ f ( x ) , \\\\ \\mathrm { s . t . } & \\sum _ { i \\in [ m ] } \\overline { g } _ i ( x ) \\leq 0 . \\end{align*}"}
+{"id": "1943.png", "formula": "\\begin{align*} r _ { i } = 2 ^ { - 1 } r + 2 ^ { - i - 1 } r , i = 0 , 1 , 2 , \\ldots . \\end{align*}"}
+{"id": "580.png", "formula": "\\begin{align*} I ^ + = \\Big [ { \\frac { i - 1 } { 2 ^ j } } , { \\frac { 2 i - 1 } { 2 ^ { j + 1 } } } \\Big ) , \\ ; \\ ; I ^ - = \\Big [ { \\frac { 2 i - 1 } { 2 ^ { j + 1 } } } , { \\frac { i } { 2 ^ j } } \\Big ) . \\end{align*}"}
+{"id": "2153.png", "formula": "\\begin{align*} A + A : = \\{ a + b : a , b \\in A \\} , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , A A : = \\{ a b : a , b \\in A \\} . \\end{align*}"}
+{"id": "4183.png", "formula": "\\begin{align*} \\varphi ( g _ 1 g _ 2 ) = \\varphi ( g _ 1 ) \\varphi ( g _ 2 ) \\mbox { a n d } \\varphi ( g _ 1 g _ 3 ) = \\varphi ( g _ 3 ) \\varphi ( g _ 1 ) \\ , . \\end{align*}"}
+{"id": "1289.png", "formula": "\\begin{align*} \\dot { x } _ 1 & = a x _ 1 + k ( x _ 2 - x _ 1 ) + \\sigma _ 1 \\dot { W } _ 1 \\\\ \\dot { x } _ 2 & = - k ( x _ 2 - x _ 1 ) + d x _ 2 + \\sigma _ 2 \\dot { W } _ 2 , \\end{align*}"}
+{"id": "4697.png", "formula": "\\begin{align*} \\xi ( f ) : = \\int _ \\R f \\ , d \\mu _ 0 . \\end{align*}"}
+{"id": "854.png", "formula": "\\begin{align*} - \\dot \\delta { U } = X ^ { i } \\dot \\delta _ { i } h + 2 g ( X , I ^ { * } ) h - n F ^ { - 1 } g ( X , l ) h . \\end{align*}"}
+{"id": "3758.png", "formula": "\\begin{align*} \\left . \\frac { \\partial \\mathrm { E } _ { \\alpha , \\beta } \\left ( z \\right ) } { \\partial \\beta } \\right \\vert _ { \\alpha = n } = - \\frac { 1 } { n \\ , \\Gamma \\left ( \\beta \\right ) } \\sum _ { m = 1 } ^ { n } Q \\left ( \\beta , z ^ { 1 / n } e ^ { i 2 \\pi m / n } \\right ) . \\end{align*}"}
+{"id": "8988.png", "formula": "\\begin{align*} u '' = M _ + \\left ( - \\frac { ( N - 1 ) } { r } K _ + ( u ' ) - \\frac { u } { r ^ \\gamma } \\right ) \\hbox { i n } ( 0 , + \\infty ) \\ , , \\end{align*}"}
+{"id": "412.png", "formula": "\\begin{align*} V _ { M } ( \\zeta ) = - \\log \\det ( \\bold { I } _ { M } - \\bold { B } _ { M } ) , \\end{align*}"}
+{"id": "5950.png", "formula": "\\begin{align*} \\frac { M _ { 2 } ( \\mathcal { C } ( G ) ) } { | e ( \\mathcal { C } ( G ) ) | } = \\dfrac { 7 3 } { 9 } > \\dfrac { 7 5 } { 1 1 } = \\frac { M _ { 1 } ( \\mathcal { C } ( G ) ) } { | v ( \\mathcal { C } ( G ) ) | } . \\end{align*}"}
+{"id": "8225.png", "formula": "\\begin{align*} \\{ { \\bf x } _ i ^ { \\star } & = g ^ { \\star } _ i ( w _ i , y ^ { i - 1 } ) \\\\ & = g ^ { \\star } _ i ( w _ i , { \\bf a } _ { i - 1 } ) : i \\in [ L ] \\} \\in \\mathcal X ( B _ ) ( \\mathcal X ( B _ ) ) . \\end{align*}"}
+{"id": "4721.png", "formula": "\\begin{align*} \\nu _ h \\ ; = \\ ; \\gamma ^ 2 \\nu _ { h + 1 } + \\gamma ^ { - 2 h } \\sum _ { \\substack { T \\\\ h _ T = h + 1 } } W _ T ^ { ( h _ T ) } ( 0 , \\underline { \\nu } ) \\ , . \\end{align*}"}
+{"id": "4702.png", "formula": "\\begin{align*} f ' _ n ( x ) & = 2 x \\cdot \\sum _ { \\nu = 1 } ^ { n } \\alpha _ { \\nu } ^ { - 1 } \\sum _ { \\kappa = 1 } ^ { \\nu } ( - 1 ) ^ { \\nu - \\kappa } h _ { \\kappa - 1 } ^ 2 ( x ) \\\\ & = 2 x \\cdot \\sum _ { \\kappa = 1 } ^ { n } h _ { \\kappa - 1 } ^ 2 ( x ) \\sum _ { \\nu = 0 } ^ { n - \\kappa } ( - 1 ) ^ { \\nu } \\alpha _ { \\nu + \\kappa } ^ { - 1 } . \\end{align*}"}
+{"id": "6560.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { a _ N } { \\left ( \\frac { ( \\ell ^ { \\leftarrow } _ { \\beta } ) ^ { \\alpha } } { h \\circ ( \\ell ^ { \\leftarrow } _ { \\beta } ) } \\right ) ^ { \\leftarrow } ( N ) } = ( \\gamma _ 2 + \\gamma _ 1 ) ^ { \\frac { 1 } { \\alpha \\beta } } . \\end{align*}"}
+{"id": "2672.png", "formula": "\\begin{align*} L ^ { ( 2 ) } \\rightarrow L '^ { ( 1 ) } = L ^ { ( 2 ) } + \\frac { d W } { d t } , \\end{align*}"}
+{"id": "767.png", "formula": "\\begin{align*} A x ^ { \\otimes 2 } - \\lambda ( x ) x = 0 \\end{align*}"}
+{"id": "6747.png", "formula": "\\begin{align*} \\min \\limits _ { x _ i \\in \\mathcal { X } _ i } \\left \\{ f ( x ) = \\sum _ { i = 1 } ^ { m } f _ i ( x _ i ) : ~ ( A x : = ) \\sum _ { i = 1 } ^ { m } A _ i x _ i = b ~ ( { \\rm o r } \\geq b ) \\right \\} , \\end{align*}"}
+{"id": "4473.png", "formula": "\\begin{align*} \\lim _ { z \\rightarrow z _ j } \\frac { g _ 1 p _ * \\left ( f _ { u _ 1 } \\left ( \\prod _ { 1 \\le j \\le m } f _ { z _ j } \\right ) \\left ( \\sum _ { 1 \\le j \\le m } p _ j \\frac { d f _ { z _ j } } { f _ { z _ j } } \\right ) \\right ) } { \\sum _ { 0 \\le l \\le k _ j } a _ { j , l } ( z - z _ j ) ^ { l } d z } = c _ 0 \\end{align*}"}
+{"id": "1174.png", "formula": "\\begin{align*} \\left | x ^ \\alpha \\partial ^ \\beta g _ k ( x ) \\right | = \\left | \\sum _ { \\gamma \\leq \\alpha } \\binom { \\alpha } { \\gamma } \\int _ { Q _ { 0 , k } } ( x - y ) ^ { \\gamma } \\partial ^ \\beta \\theta ( x - y ) y ^ { \\alpha - \\gamma } \\eta ( y ) \\ , d y \\right | \\lesssim ( 1 + | k | ) ^ { - M } , \\end{align*}"}
+{"id": "1756.png", "formula": "\\begin{align*} r _ { j , k } = \\begin{cases} 0 & \\mbox { i f } j \\neq k \\\\ \\frac { 2 j - n } { 2 } a ^ \\prime & \\mbox { i f } j = k \\end{cases} \\end{align*}"}
+{"id": "1245.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ; q ^ 2 ) _ k ( - 1 ; q ^ 2 ) _ k } { ( q ^ 2 ; q ^ 2 ) _ k } q ^ { ( n - 1 ) ^ 2 - k ^ 2 } \\\\ [ 7 p t ] & = \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ; q ^ 2 ) _ n q ^ { k ^ 2 - k } } { ( q ^ 2 ; q ^ 2 ) _ k ( q ^ 2 ; q ^ 2 ) _ { n - k - 1 } ( 1 - q ^ { 2 n - 2 k - 1 } ) } \\\\ [ 7 p t ] & = \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ; q ^ 2 ) _ n q ^ { k ^ 2 - k } } { ( q ^ 2 ; q ^ 2 ) _ { n - 1 } ( 1 - q ^ { 2 n - 2 k - 1 } ) } { n - 1 \\brack k } _ { q ^ 2 } . \\end{align*}"}
+{"id": "3239.png", "formula": "\\begin{align*} T _ { m ^ 2 } ^ { [ 0 ] } ( p ) & = \\delta ( p ^ 2 - m ^ 2 ) \\ : \\Theta ( - p ^ 0 ) \\\\ T _ { m ^ 2 } ^ { [ n ] } ( p ) & = \\frac { ( - 1 ) ^ n } { ( n - 1 ) ! } \\ : \\frac { 1 } { 4 ^ n } \\ : \\big ( p ^ 2 - m ^ 2 \\big ) ^ { n - 1 } \\ : \\Theta ( p ^ 2 - m ^ 2 ) \\ : \\Theta ( - p ^ 0 ) \\ : . \\end{align*}"}
+{"id": "6429.png", "formula": "\\begin{align*} \\tilde { \\delta } _ n ( \\frac { 1 } { 2 } ) = \\frac { 1 } { m _ { \\frac { 1 } { 2 } } ( \\alpha _ 0 ) } \\left ( 1 + \\epsilon ^ 1 ( \\tilde { \\alpha } _ n ) \\right ) \\left ( 1 + \\epsilon _ n ^ 2 ( \\tilde { \\alpha } _ n ) \\right ) \\frac { 1 } { n } \\sum _ { i = 2 } ^ n A ^ n _ i \\left ( 1 + ( X _ { \\frac { i - 2 } { n } } ^ { ( 1 / \\alpha _ 0 - 1 / \\hat { \\alpha } _ n ) / 2 } - 1 ) \\right ) \\end{align*}"}
+{"id": "1426.png", "formula": "\\begin{align*} - \\frac { \\widetilde f \\ , ^ \\prime ( s \\vert t ) } { \\widetilde f ( s \\vert t ) } & = t \\ , \\psi \\ , ^ { \\prime } ( s ) = t \\ , \\widetilde \\rho ( s ) \\\\ \\ ; - \\widetilde f \\ , ^ \\prime ( s \\vert t ) & = t \\ , \\widetilde \\rho ( s ) \\times \\widetilde f ( s \\vert t ) \\ \\\\ \\implies x f ( x \\vert t ) & = t \\ , \\{ \\rho \\star f ( \\cdot \\vert t ) \\} ( x ) \\end{align*}"}
+{"id": "1923.png", "formula": "\\begin{align*} \\underline w ( x ) : = - C \\ , h ( x ) + \\gamma . \\end{align*}"}
+{"id": "1398.png", "formula": "\\begin{align*} ( n a ) * a _ j = \\sum _ { k = 1 } ^ { m - j } \\binom { n } { k } a _ { k + j } . \\end{align*}"}
+{"id": "531.png", "formula": "\\begin{align*} u ( \\tau ) = S ( \\tau ) u _ { 0 } - B ( \\tau ) f , \\tau \\in ( 0 , T ] , \\end{align*}"}
+{"id": "4039.png", "formula": "\\begin{align*} \\langle k _ { S _ { \\mathcal { H } _ { \\mathcal { S } } } } ( \\hat x , \\cdot ) , k _ { S _ { \\mathcal { H } _ { \\mathcal { S } } } } ( \\hat y , \\cdot ) \\rangle _ { \\mathcal { H } _ { \\mathcal { S } } ^ 2 } = \\langle S _ { \\mathcal { H } _ { \\mathcal { S } } } ( \\hat x ) , S _ { \\mathcal { H } _ { \\mathcal { S } } } ( \\hat y ) \\rangle _ { \\widetilde { \\mathbb { R } \\oplus \\mathcal { H } _ { \\mathcal { S } } } } \\end{align*}"}
+{"id": "4606.png", "formula": "\\begin{align*} \\sum _ { n \\in \\N } | \\tilde D _ n | = \\infty \\ , . \\end{align*}"}
+{"id": "8730.png", "formula": "\\begin{align*} F _ \\eta ( x - ) < u \\leq F _ \\eta ( x ) \\implies x = F _ \\eta ^ { - 1 } ( u ) \\quad \\mbox { a n d } F _ \\eta ( F _ \\eta ^ { - 1 } ( u ) - ) \\leq u \\leq F _ \\eta ( F _ \\eta ^ { - 1 } ( u ) ) ; \\end{align*}"}
+{"id": "6200.png", "formula": "\\begin{align*} \\begin{array} { l l c } \\gamma _ i \\leq \\alpha _ 1 ~ , ~ ~ ~ \\ , i = 1 , ~ 2 , ~ 3 ~ , & & \\gamma _ j \\geq \\alpha _ 1 ~ , ~ ~ ~ \\ , j \\geq 4 \\\\ \\\\ c _ { n + 1 } \\leq 0 & { \\rm a n d } & c _ 2 < 0 ~ . \\end{array} \\end{align*}"}
+{"id": "7520.png", "formula": "\\begin{align*} d _ { g ( t ) } ( x , \\{ r _ M = \\delta _ 2 \\} ) < \\varepsilon , \\end{align*}"}
+{"id": "5424.png", "formula": "\\begin{align*} S _ { h _ k } f \\mid _ { \\Xi _ k } = S _ { h _ k } f ( \\tfrac { 1 } { 2 } ) + 1 . \\end{align*}"}
+{"id": "4490.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ { \\partial D } | f _ 2 | ^ 2 | d z | \\le C _ 3 \\sum _ { \\beta \\in I _ 1 , | \\alpha | = k } | a _ { \\beta , \\alpha } - \\partial ^ { \\alpha } f _ 1 ^ { * } ( z _ { \\beta } ) | ^ 2 , \\end{align*}"}
+{"id": "8482.png", "formula": "\\begin{align*} \\omega = \\partial _ t ( | u | ^ { \\frac { q - 1 } { 2 } } u ) \\textrm { i n } \\ , \\ , L ^ 2 ( \\Omega _ \\infty ) . \\end{align*}"}
+{"id": "7658.png", "formula": "\\begin{align*} \\| K \\| _ { \\infty , \\infty } : = \\sup _ { n \\in \\mathbb { G } } \\sum _ { u \\in \\mathbb { G } } K ( n , u ) < 1 . \\end{align*}"}
+{"id": "890.png", "formula": "\\begin{align*} L _ { w w } & = 2 ( w ^ 1 ) ^ 2 w ^ 1 _ { x _ 1 } + 2 w ^ 1 w ^ 2 ( w ^ 1 _ { x _ 2 } + w ^ 2 _ { x _ 1 } ) + 2 ( w ^ 2 ) ^ 2 w ^ 2 _ { x _ 2 } \\\\ & = 2 ( c _ 1 + a _ 1 x _ 1 + b _ 1 x _ 2 ) ^ 2 a _ 1 + 2 ( c _ 1 + a _ 1 x _ 1 + b _ 1 x _ 2 ) ( c _ 2 + a _ 2 x _ 1 + b _ 2 x _ 2 ) ( b _ 1 + a _ 2 ) \\\\ & \\ : \\ : \\ : \\ : + 2 ( c _ 2 + a _ 2 x _ 1 + b _ 2 x _ 2 ) ^ 2 b _ 2 , \\end{align*}"}
+{"id": "3201.png", "formula": "\\begin{align*} ( y _ 1 , y _ 2 ) = f _ { X Y } ( x _ 1 , x _ 2 ) = ( x _ 1 ^ a x _ 2 ^ b f _ 1 ( x _ 1 , x _ 2 ) , x _ 1 ^ c x _ 2 ^ d f _ 2 ( x _ 1 , x _ 2 ) ) \\end{align*}"}
+{"id": "8556.png", "formula": "\\begin{align*} f ( n ) = \\sum _ { j = 0 } ^ \\infty \\left ( \\prod _ { i = 0 } ^ { j - 1 } g _ { 2 } ( n + i ) \\right ) g _ { 1 } ( n + j ) \\end{align*}"}
+{"id": "7150.png", "formula": "\\begin{align*} \\frac { d k _ \\nu } { d \\tau } = \\frac { d } { d \\tau } \\left ( \\frac { d x ^ \\mu } { d \\tau } \\right ) = 0 \\ , \\ , , \\end{align*}"}
+{"id": "4102.png", "formula": "\\begin{align*} [ \\ell : k ] = \\{ \\ell , \\ell + 1 , \\ldots , k \\} , P _ { A , k } = P _ { A , \\{ k \\} } , P _ { \\ell , B } = P _ { \\{ \\ell \\} , B } , P _ { k } = P _ { [ 1 : k ] , [ 1 : k ] } , \\end{align*}"}
+{"id": "7496.png", "formula": "\\begin{align*} ( \\partial _ t - L _ t ) ( \\chi ^ 2 h ) & = \\left ( 2 \\chi \\partial _ t \\chi - 2 \\chi \\Delta _ { g _ 0 ( t ) } \\chi - 2 | \\nabla ^ { g _ 0 ( t ) } \\chi | ^ 2 \\right ) h + \\chi ^ 2 R _ 0 [ h ] \\\\ & + \\nabla ^ { g _ 0 ( t ) } ( \\chi ^ 2 R _ 1 [ h ] ) - 2 \\chi \\nabla ^ { g _ 0 ( t ) } \\chi * \\nabla ^ { g _ 0 ( t ) } h - 2 \\chi \\nabla ^ { g _ 0 ( t ) } \\chi * R _ 1 [ h ] . \\end{align*}"}
+{"id": "5946.png", "formula": "\\begin{align*} & \\dfrac { M _ { 2 } ( \\mathcal { N C } ( G ) ) } { | e ( \\mathcal { N C } ( G ) ) | } \\\\ & ~ ~ ~ ~ = \\dfrac { 2 ^ { 1 2 k } - 7 \\cdot 2 ^ { 1 0 k } - 2 ^ { 9 k } + 2 1 \\cdot 2 ^ { 8 k } - 2 6 \\cdot 2 ^ { 6 k } - 2 \\cdot 2 ^ { 5 k } + 1 5 \\cdot 2 ^ { 4 k } + 3 \\cdot 2 ^ { 3 k } + 6 \\cdot 2 ^ { 2 k } - 8 \\cdot 2 ^ { k } } { 2 ^ { 6 k } - 3 \\cdot 2 ^ { 4 k } - 2 ^ { 3 k } + 3 \\cdot 2 ^ { 2 k } } . \\end{align*}"}
+{"id": "2358.png", "formula": "\\begin{align*} & \\int _ { 3 } ^ { q } \\frac { \\ln 4 \\pi ( 1 - \\tau ) } { r ^ { 2 } } d r = \\left ( \\frac { 1 } { 3 } - \\frac { 1 } { q } \\right ) \\ln 4 \\pi ( 1 - \\tau ) , \\\\ & \\int _ { 3 } ^ { q } \\frac { \\ln ( r - 2 ) } { r ^ { 2 } } d r = \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { q } \\right ) \\ln ( q - 2 ) + \\frac { 1 } { 2 } \\ln \\frac { 3 } { q } , \\\\ & \\int _ { 3 } ^ { q } \\frac { \\ln r ^ { 2 } } { r ^ { 2 } } d r = 2 \\left ( \\frac { 1 + \\ln 3 } { 3 } - \\frac { 1 + \\ln q } { q } \\right ) . \\end{align*}"}
+{"id": "7018.png", "formula": "\\begin{align*} X _ t = - \\sqrt { \\delta } \\frac { d - 2 } { 2 } \\int _ 0 ^ t | X _ r | ^ { - 2 } X _ r d r + \\sqrt { 2 } W _ t , \\end{align*}"}
+{"id": "2258.png", "formula": "\\begin{align*} Z ( S ) = \\{ \\ [ M _ i , N ] = 0 \\ , , \\ \\forall i \\} \\ . \\end{align*}"}
+{"id": "3985.png", "formula": "\\begin{align*} \\Theta _ { j } = \\sum _ { k = 1 } ^ { n } \\gamma _ { i p k } \\gamma _ { k p j } + \\sum _ { r = 1 } ^ { \\nu } \\widetilde { \\gamma } _ { i p r } \\gamma _ { i r j } \\ ; \\mbox { a n d } \\ ; \\widetilde { \\Theta } _ { u } = \\sum _ { k = 1 } ^ { n } \\gamma _ { i p k } \\widetilde { \\gamma } _ { k p u } + \\sum _ { r = 1 } ^ { \\nu } \\widetilde { \\gamma } _ { i p r } \\widetilde { \\gamma } _ { i r u } \\end{align*}"}
+{"id": "5323.png", "formula": "\\begin{align*} v ^ { \\boldsymbol { \\pi } } = \\sum _ { j \\in J } c _ j \\ , x ^ { \\boldsymbol { \\pi } } _ j = \\nu _ { \\pi _ 1 } \\ , b ^ { S _ 1 } + \\sum _ { k = 2 } ^ { n } ( \\nu _ { \\pi _ k } - \\nu _ { \\pi _ { k - 1 } } ) \\ , b ^ { S _ k } . \\end{align*}"}
+{"id": "6611.png", "formula": "\\begin{align*} I _ { ( b , \\tau ) } = R _ { k , \\beta } \\left ( \\frac { 1 } { ( 1 - \\delta ) } \\right ) y ^ { \\frac { 1 } { ( 1 - \\delta ) } } + \\frac { x } { 2 \\pi i } \\left ( \\int _ { d - i \\tau } ^ { d + i \\tau } + \\int _ { d + i \\tau } ^ { b + i \\tau } + \\int _ { b - i \\tau } ^ { d - i \\tau } \\right ) R _ { k , \\beta } ( z ) \\frac { ( \\alpha x ^ \\delta ) ^ { z } } { z ( 1 - z + \\delta z ) } d z . \\end{align*}"}
+{"id": "6936.png", "formula": "\\begin{align*} T ^ c = T ^ c _ 1 + T ^ c _ 2 + T ^ c _ { p r i n c } \\end{align*}"}
+{"id": "5869.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { C } ( G ) ) = ( 2 ^ { n - 1 } - 2 ) ( 2 ^ { n - 1 } - 3 ) ^ { 2 } + 2 ^ { n - 1 } , M _ { 2 } ( \\mathcal { C } ( G ) ) = ( 2 ^ { n - 2 } - 1 ) ( 2 ^ { n - 1 } - 3 ) ^ { 3 } + 2 ^ { n - 2 } , \\end{align*}"}
+{"id": "1106.png", "formula": "\\begin{align*} f ' ( t ) & = \\frac { a t ^ a } { 2 + t } [ \\log ( 2 + t ) ] ^ { b - 1 } \\left [ \\frac { 2 + t } { t } \\log ( 2 + t ) + \\frac { b } { a } \\right ] \\\\ & > \\frac { a t ^ a } { 2 + t } [ \\log ( 2 + t ) ] ^ { b - 1 } \\left ( \\log t + \\frac { b } { a } \\right ) > 0 \\end{align*}"}
+{"id": "6281.png", "formula": "\\begin{align*} p = 2 , \\Psi _ p ( x ) = \\frac { 1 } { 2 } \\| x \\| _ 2 ^ 2 , \\end{align*}"}
+{"id": "4379.png", "formula": "\\begin{gather*} F _ { k _ 1 , k _ 2 , k _ 3 , k _ 4 } ( x ) = \\sum _ { ( i , j , r , s ) \\in [ n ] ^ 4 } ( \\overline { c } _ { i , j } d _ { r , s } + \\max \\{ 0 , \\Delta { c } _ { i , j } d _ { r , s } - \\Delta c _ { k _ 1 , k _ 2 } d _ { k _ 3 , k _ 4 } \\} ) x _ { i , r } x _ { j , s } \\end{gather*}"}
+{"id": "3263.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ^ { d + 1 } ; q ^ d ) _ k ^ { d - 1 } ( q ^ { 1 - d } ; q ^ d ) _ k q ^ { d k } } { ( q ^ d ; q ^ d ) _ k ^ d } \\\\ & \\equiv \\frac { ( 1 - q ) ( 1 - q ^ { d - 1 } ) ( q ^ d ; q ^ d ) _ { n - 1 - ( n + 1 ) / d } } { - ( q ^ d ; q ^ d ) _ { ( n + 1 ) / d } ^ { d - 1 } } q ^ { ( d ( d + n ) ( n + 1 ) - ( n + 1 ) ^ 2 ) / ( 2 d ) - 1 } \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "1991.png", "formula": "\\begin{align*} ( s ) = \\frac { \\sin ( s ) } { s } s \\neq 0 , \\ ( 0 ) = 1 . \\end{align*}"}
+{"id": "9021.png", "formula": "\\begin{align*} x _ { k k } = 0 x _ { i j } = y _ { i j } = 0 i \\ne j 1 \\le k \\le n . \\end{align*}"}
+{"id": "73.png", "formula": "\\begin{align*} \\big ( M + \\lambda ( N - M ) \\big ) _ { 1 1 } = M _ { 1 1 } + \\lambda ( N _ { 1 1 } - M _ { 1 1 } ) \\geq c - \\lambda ( \\| N \\| _ { L ^ { \\infty } ( \\Omega _ T ) } + \\| M \\| _ { L ^ \\infty ( \\Omega _ T ) } ) , \\end{align*}"}
+{"id": "1707.png", "formula": "\\begin{align*} M : = \\left \\{ z \\in \\mathbb { C } ^ { n + 1 } \\ , \\left | \\ , \\Phi \\left ( z _ 1 , \\ldots , z _ n , \\overline { z _ 1 } , \\ldots , \\overline { z _ n } \\right ) = z _ 0 + \\overline { z _ 0 } \\right . \\right \\} , \\end{align*}"}
+{"id": "6810.png", "formula": "\\begin{align*} f = f _ { S U ( N ) } ( g ) & = \\sum _ { j = 1 } ^ M \\sum _ { i = 1 } ^ Q c _ { i j } e ^ { i k _ { i j } ^ 1 \\phi _ 1 } \\sin ^ { m _ { i j } ^ 1 } ( \\psi _ 1 ) \\cos ^ { n _ { i j } ^ 1 } ( \\psi _ 1 ) \\cdots e ^ { i k _ { i j } ^ { N - 1 } \\phi _ { N - 1 } } \\sin ^ { m _ { i j } ^ { N - 1 } } ( \\psi _ { N - 1 } ) \\cos ^ { n _ { i j } ^ { N - 1 } } ( \\psi _ { N - 1 } ) \\\\ & \\qquad \\qquad \\cdot ( f _ { S U ( N - 1 ) } ) _ { i j } ( g _ { S U ( N - 1 ) } ) e ^ { i l ^ { N - 1 } _ { i j } \\omega _ { N - 1 } } \\end{align*}"}
+{"id": "6108.png", "formula": "\\begin{align*} \\overline { g } ( V _ \\phi , X ) = - \\langle e _ 0 \\cdot X \\cdot \\phi , \\phi \\rangle \\end{align*}"}
+{"id": "3806.png", "formula": "\\begin{align*} \\dim \\left ( \\bigcap _ { i = 1 } ^ \\ell \\Im ( W _ { [ k ] , A _ { i } } ) \\right ) = \\max _ { P _ 1 \\sqcup P _ 2 \\sqcup \\dots \\sqcup P _ s = [ \\ell ] } \\left ( \\sum _ { i = 1 } ^ s \\Big | \\bigcap _ { j \\in P _ i } A _ j \\Big | - ( s - 1 ) k \\right ) , \\end{align*}"}
+{"id": "4751.png", "formula": "\\begin{align*} [ S ( \\vert t \\vert ) ] ^ { m + 1 } x = S ( \\vert t \\vert ) \\left ( [ S ( \\vert t \\vert ) ] ^ m x \\right ) \\end{align*}"}
+{"id": "8773.png", "formula": "\\begin{align*} \\int _ { 1 - \\frac { F _ \\mu ( x ' ) - F _ \\mu ( x ) } { \\phi _ \\pi ( F _ \\mu ( x ' ) ) } } ^ 1 F _ \\sigma ^ { - 1 } ( w ) d w & \\ge \\int _ { 1 - \\frac { F _ \\mu ( x ' ) - F _ \\mu ( x ) } { \\phi _ \\pi ( F _ \\mu ( x ' ) ) } } ^ 1 F _ \\vartheta ^ { - 1 } ( w ) d w \\\\ & = \\frac { 1 } { \\phi _ \\pi ( F _ \\mu ( x ' ) ) } \\int _ { \\phi _ \\pi ( F _ \\mu ( x ' ) ) + F _ \\mu ( x ) - F _ \\mu ( x ' ) } ^ { \\phi _ \\pi ( F _ \\mu ( x ' ) ) } F _ { \\tilde \\nu _ l } ^ { - 1 } \\left ( \\frac { w } { \\nu _ l ( \\R ) } \\right ) d w . \\end{align*}"}
+{"id": "2296.png", "formula": "\\begin{align*} F _ n ( t ^ { - 1 } ) = ( - 1 ) ^ { ( n - 1 ) } t ^ { 3 n } F _ n ( t ) . \\end{align*}"}
+{"id": "3470.png", "formula": "\\begin{align*} - \\sigma ( \\Sigma ( S ) ) = \\frac { 1 } { 2 } S \\circ S - \\sigma ( K ) = b _ 1 ( S ) = b _ 2 ( \\Sigma ( S ) ) . \\end{align*}"}
+{"id": "7556.png", "formula": "\\begin{align*} \\int _ \\Omega \\tfrac { 1 } { 2 } \\rho | u | ^ 2 + \\varepsilon \\tfrac { 1 } { 2 } n | v | ^ 2 + h _ 1 ( \\rho ) + h _ 2 ( n ) + \\delta \\tfrac { 1 } { 2 } | \\nabla \\phi | ^ 2 \\ d x \\Big | _ { \\tau = 0 } ^ { \\tau = t } \\leq 0 \\ . \\end{align*}"}
+{"id": "77.png", "formula": "\\begin{align*} \\left | D ( | D u | ^ { \\frac { p - 2 + s } 2 } D u ) \\right | & = \\frac { 1 } { 2 } \\alpha ^ { \\frac { p + s } { 2 } } ( \\alpha - 1 ) ( p + s ) | x _ 1 | ^ { \\frac { ( \\alpha - 1 ) ( p + s ) } { 2 } - 1 } \\\\ & = C ( p , s , \\gamma ) | x _ 1 | ^ { \\frac { p + s } { 2 ( \\gamma + 1 ) } - 1 } . \\end{align*}"}
+{"id": "4578.png", "formula": "\\begin{align*} t - t _ 0 = h ( s _ g - s _ 0 ) . \\end{align*}"}
+{"id": "1804.png", "formula": "\\begin{align*} \\begin{cases} f ( z ^ + , u ^ + ) - f ( z ^ - , u ^ - ) = \\Xi ( z ^ + , z ^ - , u ^ - ) \\\\ z ^ + , z ^ - \\in B ( \\bar z ; \\delta ) \\\\ u ^ + , u ^ - \\in B ( \\bar u ; \\delta ) \\end{cases} \\Longleftrightarrow u ^ + = T ( z ^ + , z ^ - ) ( u ^ - ) \\ , . \\end{align*}"}
+{"id": "4535.png", "formula": "\\begin{align*} E _ \\infty ^ { ( K ) } ( t , [ s ] _ K ) = \\sum _ { i = K + 1 } ^ \\infty \\int _ { s _ { K + 1 } = 0 } ^ \\infty \\ ! . . . \\ ! \\int _ { s _ { i } = 0 } ^ \\infty \\varphi _ i ( [ s ] _ i ) n _ \\infty ^ { ( i ) } ( t , [ s ] _ i ) \\ , d [ s ] _ { K + 1 , i } , \\end{align*}"}
+{"id": "5382.png", "formula": "\\begin{align*} \\rho _ i = \\frac { \\lambda _ i } { \\mu _ { i + 1 } } . \\end{align*}"}
+{"id": "2771.png", "formula": "\\begin{align*} & \\omega _ { \\Xi , \\Psi } : = d \\sigma _ { 3 } ^ { * } \\Xi ^ { a } \\wedge d \\sigma _ { 3 } ^ { * } \\Psi _ { a } = 0 , \\\\ & \\omega _ { \\Theta } : = d \\sigma _ { 3 } ^ { * } \\Theta ^ { \\alpha } \\wedge d \\sigma _ { 3 } ^ { * } \\Theta _ { \\alpha } = 0 , \\\\ & \\omega _ { Q , P } : = d \\sigma _ { 3 } ^ { * } Q ^ { i } \\wedge d \\sigma _ { 3 } ^ { * } P _ { a } . \\end{align*}"}
+{"id": "7932.png", "formula": "\\begin{align*} \\langle \\delta \\beta , \\delta \\gamma \\rangle _ { L ^ { 2 } \\Lambda ^ { 1 } ( \\Omega ) } = \\langle \\omega , \\gamma \\rangle _ { L ^ { 2 } \\Lambda ^ { 2 } ( \\Omega ) } , \\quad \\forall \\gamma \\in \\mathring { V } \\Lambda ^ { 2 } ( \\Omega ) , \\end{align*}"}
+{"id": "7535.png", "formula": "\\begin{align*} \\begin{dcases} \\partial _ t \\rho _ { 0 0 } + \\nabla \\cdot m _ { 0 0 } = 0 \\\\ \\partial _ t m _ { 0 0 } + \\nabla \\cdot \\big ( \\frac { m _ { 0 0 } \\otimes m _ { 0 0 } } { \\rho _ { 0 0 } } \\big ) + \\nabla p _ 1 ( \\rho _ { 0 0 } ) + \\rho _ { 0 0 } \\nabla h _ 2 ^ \\prime ( n _ { 0 0 } ) = 0 \\\\ n _ { 0 0 } = \\rho _ { 0 0 } \\ , \\end{dcases} \\end{align*}"}
+{"id": "1425.png", "formula": "\\begin{align*} t ^ \\gamma \\{ I _ + ^ { \\beta - \\alpha \\gamma } f _ \\alpha ( \\cdot \\vert t ) \\} ( x ) & = t ^ { ( \\beta - 1 ) / \\alpha } \\{ I _ + ^ { \\beta - \\alpha \\gamma } f _ \\alpha \\} ( x t ^ { - 1 / \\alpha } ) \\end{align*}"}
+{"id": "4156.png", "formula": "\\begin{align*} g _ j \\cdot \\ldots \\cdot g _ { \\ell } g _ 1 \\cdot \\ldots \\cdot g _ { j - 1 } = 1 _ G \\ , . \\end{align*}"}
+{"id": "7188.png", "formula": "\\begin{align*} R _ v ( x , t ) & = E _ v ( x , t ) - F _ v ( x , t ) , \\\\ R _ w ( x , t ) & = E _ w ( x , t ) - F _ w ( x , t ) . \\end{align*}"}
+{"id": "5844.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { \\lambda _ { n } ^ { \\alpha } } { \\mu _ { n } ^ { \\beta } } = 1 \\lim _ { n \\rightarrow \\infty } \\frac { \\mu _ { n } } { \\mu _ { n } ^ { \\beta } } = 1 \\end{align*}"}
+{"id": "317.png", "formula": "\\begin{align*} u ( x , t ) = ( T - t ) ^ { - \\alpha } f ( | x | ( T - t ) ^ { \\beta } ) , \\ \\alpha = \\frac { \\sigma + 2 } { L } , \\ \\beta = \\frac { m - p } { L } \\end{align*}"}
+{"id": "3473.png", "formula": "\\begin{align*} f _ \\delta : = \\sum _ i f _ { B _ i } \\varphi _ i \\ , , \\end{align*}"}
+{"id": "5446.png", "formula": "\\begin{align*} \\nu _ { ( a ^ - , i ) } ^ { \\hat { S } ^ k } = \\nu _ { ( a _ k ^ - , i _ k ) } ^ * - \\frac { w _ { ( a ^ - , i ) } ^ { \\hat { S } ^ { k - 1 } } } { w _ { ( a ^ - , i ) } ^ { \\hat { S } ^ { k } } } \\big ( \\nu _ { ( a _ k ^ - , i _ k ) } ^ * - \\nu _ { ( a ^ - , i ) } ^ { \\hat { S } ^ { k - 1 } } \\big ) , ( a ^ - , i ) \\in \\hat { N } . \\end{align*}"}
+{"id": "5436.png", "formula": "\\begin{align*} w ^ { S _ 0 \\oplus S _ 1 } _ { ( a ^ - , i ) } & = 1 + \\beta \\sum _ { j \\in N } p _ { i j } g ^ { S _ 0 \\oplus S _ 1 } ( 1 , j ) - \\beta g ^ { S _ 0 \\oplus S _ 1 } _ { ( 0 , i ) } = 1 + \\beta \\sum _ { j \\in S _ 1 } p _ { i j } g ^ { S _ 1 } _ j = w ^ { S _ 1 } _ i + \\beta g ^ { S _ 1 } _ i = w ^ { S _ 1 } _ i / ( 1 - \\beta ) , \\end{align*}"}
+{"id": "2900.png", "formula": "\\begin{align*} & \\lim _ { \\lambda \\to \\infty } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ { L ^ \\Phi ( \\mathbb { R } ^ n ) } = \\left [ \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { L ^ \\Phi ( \\mathbb { R } ^ n ) } ; \\end{align*}"}
+{"id": "8993.png", "formula": "\\begin{align*} v _ T ( z , t , \\delta ) : = \\sup _ { 0 \\vee ( t - \\delta ) \\leq t _ 1 \\leq t _ 2 \\leq ( t + \\delta ) \\wedge T } | z ( t _ 1 ) - z ( t _ 2 ) | \\end{align*}"}
+{"id": "7889.png", "formula": "\\begin{align*} \\mathcal { L } _ { c } ( \\mathbf { x } ) = \\frac { q } { p } \\sum _ { i = h + 1 } ^ { m } v _ { i } x _ { i } + d \\sum _ { i = 1 } ^ { h } x _ { i } p ^ { i - 1 } , \\end{align*}"}
+{"id": "8933.png", "formula": "\\begin{align*} \\dim ( \\mathcal { M } ( L ) ) \\leq & ~ \\frac { 1 } { 2 } [ ( m + n - r - s ) ( m + n - r - s - 1 ) + 2 ( n - s ) ] + ( r + s ) ( m + n - r - s - 1 ) \\\\ & - \\sum _ { i = 2 } ^ { l } ( m + n - r - s - i ) \\\\ & \\leq \\frac { 1 } { 2 } [ ( m + n - r - s - 1 ) ( m + n + r + s ) + 2 ( n - s ) ] - \\sum _ { i = 2 } ^ { l } ( m + n - r - s - i ) \\\\ & \\leq \\frac { 1 } { 2 } [ ( m + n - r - s ) ( m + n + r + s ) + ( n - m - r - 3 s ) ] - \\sum _ { i = 2 } ^ { l } ( m + n - r - s - i ) . \\end{align*}"}
+{"id": "8109.png", "formula": "\\begin{align*} \\mu \\ast \\nu ( A ) = \\int \\int 1 _ A ( x + y ) \\ ; d \\mu ( x ) \\ ; d \\nu ( y ) , \\end{align*}"}
+{"id": "8496.png", "formula": "\\begin{align*} \\big | f ( x , t ) - f ( x , \\tau + t ) \\big | = \\left | \\int _ 0 ^ 1 \\dfrac { d } { d \\sigma } f ( x , t + \\sigma \\tau ) \\ , d \\sigma \\right | & \\leq | \\tau | \\int _ 0 ^ 1 \\Big | \\partial _ t f ( x , t + \\sigma \\tau ) \\Big | \\ , d \\tau , \\end{align*}"}
+{"id": "2457.png", "formula": "\\begin{align*} d s / d \\xi = \\phi ^ { - 1 } \\end{align*}"}
+{"id": "1362.png", "formula": "\\begin{align*} & l \\Biggl ( \\sqrt { \\frac { \\alpha ^ 2 M _ 1 + ( 1 - \\alpha ) ^ 2 2 m - \\frac { 4 \\alpha ^ 2 m ^ 2 } { n } } { n } } + \\frac { 2 \\alpha m } { n } \\Biggr ) + ( n - l ) \\Biggl ( - \\sqrt { \\frac { \\alpha ^ 2 M _ 1 + ( 1 - \\alpha ) ^ 2 2 m - \\frac { 4 \\alpha ^ 2 m ^ 2 } { n } } { n } } + \\frac { 2 \\alpha m } { n } \\Biggr ) \\\\ & = \\sqrt { \\frac { \\alpha ^ 2 M _ 1 + ( 1 - \\alpha ) ^ 2 2 m - \\frac { 4 \\alpha ^ 2 m ^ 2 } { n } } { n } } ( 2 l - n ) + 2 \\alpha m , \\end{align*}"}
+{"id": "3261.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ^ { d - 1 } ; q ^ d ) _ k ^ d q ^ { d k } } { ( q ^ d ; q ^ d ) _ k ^ d } \\equiv \\frac { ( q ^ d ; q ^ d ) _ { ( d - 1 ) ( n - 1 ) / d } q ^ { ( d - 1 ) ( n - 1 ) ( d + n - 1 ) / ( 2 d ) } } { ( q ^ d ; q ^ d ) _ { ( n - 1 ) / d } ^ { d - 1 } ( - 1 ) ^ { ( d - 1 ) ( n - 1 ) / d } } \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "4843.png", "formula": "\\begin{align*} h ( | x | ) - h ( a ) & = h _ A ( | x | ) - h _ A ( a ) + h _ J ( | x | ) - h _ J ( a ) + h _ c ( | x | ) - h _ c ( a ) \\\\ & \\ge h _ A ( | x | ) - h _ A ( a ) = \\int _ a ^ { | x | } h ' ( t ) d t , \\end{align*}"}
+{"id": "5150.png", "formula": "\\begin{align*} \\frac { \\partial A } { \\partial q _ { j } } = \\frac { \\partial A _ { j } } { \\partial q _ { j } } = \\frac { 1 } { \\alpha } \\ ; \\ ; ; \\ ; \\ ; \\frac { \\partial B } { \\partial q _ { j } } = \\frac { \\partial B _ { j } } { \\partial q _ { j } } = \\frac { 1 } { \\alpha } p ^ { \\alpha } _ { j } q ^ { - \\alpha } _ { j } \\end{align*}"}
+{"id": "1785.png", "formula": "\\begin{align*} ( \\widehat { w } , \\widehat { z } , \\widehat { \\zeta } ) = \\Psi ( w , z , \\zeta ) : = \\left ( w + \\widehat { h } ^ 2 , z _ 1 + \\widehat { f } _ { 1 } ^ 1 , \\ldots , z _ { n - 1 } + \\widehat { f } _ { n - 1 } ^ 1 , \\zeta + \\widehat { g } ^ 0 0 \\right ) \\end{align*}"}
+{"id": "3089.png", "formula": "\\begin{align*} { \\rm C o e f f } ( \\varphi _ { a } ^ * ( \\omega _ { i g } ) , t ^ k ) = q _ { i k } ( a _ { \\beta _ g } , \\ldots , a _ { k - v _ i - v _ g + \\beta _ g } ) + r _ { i k } \\cdot a _ { k - v _ i - v _ g + \\beta _ g + 1 } , \\end{align*}"}
+{"id": "6722.png", "formula": "\\begin{align*} \\mathbb { P } _ { 1 2 } & = \\ker ( \\Phi _ 1 ( E ) ) \\oplus \\ker ( \\Phi _ 2 ( E ) ) \\oplus \\ker ( \\Phi _ 3 ( E ) ) \\oplus \\ker ( \\Phi _ 4 ( E ) ) \\oplus \\ker ( \\Phi _ 6 ( E ) ) \\oplus \\ker ( \\Phi _ { 1 2 } ( E ) ) \\\\ & = \\mathbb { P } _ 1 \\oplus \\mathbb { A P } _ 1 \\oplus \\ker ( \\Phi _ 3 ( E ) ) \\oplus \\mathbb { A P } _ 2 \\oplus \\ker ( \\Phi _ 6 ( E ) ) \\oplus \\ker ( \\Phi _ { 1 2 } ( E ) ) . \\end{align*}"}
+{"id": "6968.png", "formula": "\\begin{align*} \\eta ^ { 2 } ( A \\mathbf { z } ) = - \\langle A \\mathbf { z } , \\overline { A \\mathbf { z } } \\rangle = - \\langle A \\mathbf { z } , A \\mathbf { \\overline { z } } \\rangle = - \\langle \\mathbf { z } , \\mathbf { \\overline { z } } \\rangle = \\eta ^ { 2 } ( \\mathbf { z } ) . \\end{align*}"}
+{"id": "6195.png", "formula": "\\begin{align*} \\tau : = - e _ { n - 1 } + e _ { n - 3 } - G ( e _ { n - 2 } - e _ n ) > 0 ~ . \\end{align*}"}
+{"id": "7412.png", "formula": "\\begin{align*} M = \\frac { r - 1 } { 2 } \\cdot 2 - \\frac { r - 1 } { 2 } + 1 = \\frac { r + 1 } { 2 } > 0 . \\end{align*}"}
+{"id": "267.png", "formula": "\\begin{align*} \\ddot { x } = 2 \\dot { x } ^ 2 \\cot x + \\sin x \\cos x , \\end{align*}"}
+{"id": "3888.png", "formula": "\\begin{align*} A ^ { - 1 } \\omega _ 2 & = ( 1 , 2 , 2 , 3 , 2 , 1 ) \\\\ A ^ { - 1 } \\omega _ 2 - \\alpha _ 2 & = ( 1 , 2 , 1 , 3 , 2 , 1 ) \\\\ A ^ { - 1 } \\omega _ 2 - \\alpha _ 2 - \\alpha _ 4 & = ( 1 , 2 , 1 , 2 , 2 , 1 ) \\\\ A ^ { - 1 } \\omega _ 2 - \\alpha _ 2 - \\alpha _ 4 - \\alpha _ 5 & = ( 1 , 2 , 1 , 2 , 1 , 1 ) \\\\ A ^ { - 1 } \\omega _ 2 - \\alpha _ 2 - \\alpha _ 4 - \\alpha _ 5 - \\alpha _ 6 & = ( 1 , 2 , 1 , 2 , 1 , 0 ) . \\end{align*}"}
+{"id": "2909.png", "formula": "\\begin{align*} \\int _ { 0 } ^ 1 \\big | b _ 0 + \\sum _ { k = 1 } ^ n { \\rm e } ^ { i 2 \\pi k \\vartheta } b _ k \\big | { \\rm d } \\vartheta \\geqslant | b _ 0 | . \\end{align*}"}
+{"id": "773.png", "formula": "\\begin{align*} \\Delta ( x ) = x _ { ( 1 ) } \\otimes x _ { ( 2 ) } \\in \\bigoplus _ { a , b , c } H _ { a , b } \\otimes H _ { b , c } . \\end{align*}"}
+{"id": "8665.png", "formula": "\\begin{align*} \\mbox { $ 0 $ i s a r e g u l a r v a l u e o f $ \\mu $ , $ G $ a c t s f r e e l y o n $ \\mu ^ { - 1 } ( 0 ) $ . } \\end{align*}"}
+{"id": "5435.png", "formula": "\\begin{align*} w ^ { S _ 0 \\oplus S _ 1 } _ { ( a ^ - , i ) } & = 1 + \\beta \\sum _ { j \\in N } p _ { i j } g ^ { S _ 0 \\oplus S _ 1 } ( 1 , j ) - \\beta g ^ { S _ 0 \\oplus S _ 1 } _ { ( 0 , i ) } = 1 + \\beta \\sum _ { j \\in S _ 1 } p _ { i j } g ^ { S _ 1 } _ j - \\beta g ^ { S _ 1 } _ i = w ^ { S _ 1 } _ i , \\end{align*}"}
+{"id": "2959.png", "formula": "\\begin{align*} \\mathcal { E } ( V , c ) = \\left \\{ e : V \\to \\N \\ : \\middle | \\ : \\sum _ { v \\in V } e ( v ) \\leq c \\right \\} . \\end{align*}"}
+{"id": "5557.png", "formula": "\\begin{align*} S _ \\Delta = S \\Delta , T _ \\Delta = \\Delta T , J _ \\Delta = \\Delta J = J \\Delta \\end{align*}"}
+{"id": "1488.png", "formula": "\\begin{align*} f ( m ) \\big ( f ( n ) ( \\{ m \\} - \\{ n \\} ) - f ( m + n ) ( \\{ m \\} - \\{ n - d \\} ) \\big ) = 0 . \\end{align*}"}
+{"id": "5919.png", "formula": "\\begin{align*} \\dfrac { M _ { 2 } ( \\mathcal { N C } ( A ( n , \\nu ) ) ) } { | e ( \\mathcal { N C } ( A ( n , \\nu ) ) ) | } & = \\dfrac { 2 ^ { 7 n } ( 2 ^ { n - 1 } - 3 ) - 2 ^ { 6 n } ( 2 ^ { n - 1 } - 9 ) - 2 ^ { 4 n + 1 } ( 5 \\cdot 2 ^ { n } - 2 ) } { 2 ^ { n } ( 2 ^ { n } - 2 ) ( 2 ^ { 2 n } - 2 ^ { n } ) } \\\\ & = \\dfrac { 2 ^ { 5 n } ( 2 ^ { n } - 5 ) + 2 ^ { 3 n + 2 } ( 2 ^ { n + 1 } - 1 ) } { 2 ^ { 2 n } - 2 ^ { n } } \\\\ & = \\dfrac { M _ { 1 } ( \\mathcal { N C } ( A ( n , \\nu ) ) ) } { | v ( \\mathcal { N C } ( A ( n , \\nu ) ) ) | } . \\end{align*}"}
+{"id": "2053.png", "formula": "\\begin{align*} T = o ( d ^ 2 ) , \\sigma = O ( d ) , \\gamma T ^ { - \\frac { 1 } { 2 } } \\to \\zeta \\in ( 0 , \\infty ) , \\gamma \\sigma d ^ { - 1 } T ^ { - \\frac { 1 } { 2 } } \\to \\beta \\in [ 0 , \\infty ) , \\end{align*}"}
+{"id": "4203.png", "formula": "\\begin{align*} & - 2 6 4 \\left \\{ \\widehat { A } ( T X ) { \\rm d e t } ^ { \\frac { 1 } { 2 } } { \\rm c o s h } ( \\frac { \\sqrt { - 1 } } { 4 \\pi } R ^ V ) \\right \\} ^ { ( 2 0 ) } = \\left \\{ \\widehat { A } ( T X ) { \\rm d e t } ^ { \\frac { 1 } { 2 } } { \\rm c o s h } ( \\frac { \\sqrt { - 1 } } { 4 \\pi } R ^ V ) \\right . \\\\ & \\left . \\cdot { \\rm c h } ( \\widetilde { T X } + 2 \\wedge ^ 2 \\widetilde { V _ C } - \\widetilde { V _ C } \\otimes \\widetilde { V _ C } + \\widetilde { V _ C } ) \\right \\} ^ { ( 2 0 ) } . \\end{align*}"}
+{"id": "749.png", "formula": "\\begin{align*} { } ^ c \\nabla _ k { } ^ c \\nabla _ j \\varphi _ i = 2 \\varphi _ i \\varphi _ j \\varphi _ k + \\Phi ( g _ { i k } \\varphi _ j + g _ { j k } \\varphi _ i ) + \\Phi _ k g _ { i j } , \\end{align*}"}
+{"id": "1835.png", "formula": "\\begin{align*} ( 1 , - 1 ) \\cdot ( x _ 1 , x _ 2 , x _ 3 , x _ 4 , x _ 5 , x _ 6 , x _ 7 ) & = ( x _ 1 , x _ 2 , x _ 3 , - x _ 4 , - x _ 5 , - x _ 6 , - x _ 7 ) \\\\ ( - 1 , 1 ) \\cdot ( x _ 1 , x _ 2 , x _ 3 , x _ 4 , x _ 5 , x _ 6 , x _ 7 ) & = ( x _ 1 , - x _ 2 , - x _ 3 , x _ 4 , x _ 5 , - x _ 6 , - x _ 7 ) . \\end{align*}"}
+{"id": "4599.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to 1 } p _ { d , \\gamma } = \\frac { d } { 2 } . \\end{align*}"}
+{"id": "7817.png", "formula": "\\begin{align*} \\begin{aligned} \\max _ { \\boldsymbol { a } , \\{ \\eta _ k \\} _ { k = 1 } ^ { K } } & \\mathcal { D } \\left ( \\boldsymbol { a } , \\{ \\eta _ k \\} _ { k = 1 } ^ { K } \\right ) , \\\\ & a _ { k } \\in [ 0 , 1 ] , \\forall k \\in \\mathcal { K } , \\end{aligned} \\end{align*}"}
+{"id": "7899.png", "formula": "\\begin{align*} f _ { k } ^ { ( c ) } ( \\mathbf { x } ) = \\frac { q } { p } \\mathbf { x } ^ { T } \\mathbf { A } _ { k } \\mathbf { x } + \\mathcal { L } _ { c } ( \\mathbf { x } ) . \\end{align*}"}
+{"id": "5018.png", "formula": "\\begin{align*} & \\max \\ , \\mathbf { x } ^ { 1 } ( \\mathbf { R } - \\nu \\mathbf { 1 } ) \\\\ & \\\\ & \\begin{bmatrix} \\mathbf { x } ^ { 0 } & \\mathbf { x } ^ { 1 } \\end{bmatrix} \\begin{bmatrix} ( 1 - \\beta ) \\mathbf { I } \\\\ \\mathbf { I } - \\beta \\mathbf { P } \\end{bmatrix} = \\mathbf { e } _ i \\\\ & \\begin{bmatrix} \\mathbf { x } ^ { 0 } & \\mathbf { x } ^ { 1 } \\end{bmatrix} \\geq \\mathbf { 0 } . \\end{align*}"}
+{"id": "1975.png", "formula": "\\begin{align*} h _ { i j } = - \\dfrac { 1 } { 2 N } \\partial _ t g _ { i j } \\ , , \\end{align*}"}
+{"id": "1697.png", "formula": "\\begin{align*} H : = \\mathrm { s p a n } _ { \\mathbb { C } } \\left \\{ X _ 1 , \\ldots , X _ n \\right \\} . \\end{align*}"}
+{"id": "1520.png", "formula": "\\begin{align*} { \\mathcal K } _ H = \\left \\{ M \\in \\R ^ { N \\times N } : { \\rm T r } \\ , ( M \\ , M ) \\ge \\frac { 1 } { H } \\ , { \\rm T r } \\ , ( M \\ , M ^ t ) \\right \\} , \\end{align*}"}
+{"id": "4900.png", "formula": "\\begin{align*} h ^ 2 ( - A ) = h ^ 0 ( K _ S + A ) = h ^ 0 ( - B ) = 0 . \\end{align*}"}
+{"id": "2566.png", "formula": "\\begin{align*} \\left \\{ \\begin{gathered} K ^ { \\prime \\ , ( s ) } ( M , L ) = K ^ { ( - s ) } ( M , L ; \\Omega ) ' \\ ; , K ^ { \\prime \\ , m } ( M , L ) = K ^ { - m } ( M , L ; \\Omega ) ' \\ ; , \\\\ J ^ { \\prime \\ , ( s ) } ( M , L ) = J ^ { ( - s ) } ( M , L ; \\Omega ) ' \\ ; , J ^ { \\prime \\ , m } ( M , L ) = J ^ { - m } ( M , L ; \\Omega ) ' \\ ; , \\end{gathered} \\right . \\end{align*}"}
+{"id": "5077.png", "formula": "\\begin{align*} E _ s & = \\sum _ { r \\in \\mathbb { F } _ { q ^ k } ^ \\times } H _ s ( r ) . \\end{align*}"}
+{"id": "7877.png", "formula": "\\begin{align*} \\mathcal { N } _ { \\mathbf { \\Phi } } = ( p ! ) ^ { 2 m - 2 } \\end{align*}"}
+{"id": "7659.png", "formula": "\\begin{align*} b _ 1 : = \\sum _ { u \\in \\mathbb { G } } W ( u ) \\psi ( u ) < \\infty \\ , \\ , \\mathrm { a n d } \\ , \\ , b _ 2 : = \\sup _ { m \\in \\mathbb { G } } \\sum _ { u \\in \\mathbb { G } } \\frac { W ( u ) } { W ( m ) } K ( u , m ) < 1 . \\end{align*}"}
+{"id": "1637.png", "formula": "\\begin{align*} K _ { 3 } \\left ( D _ { 3 } \\right ) = \\left \\{ u \\in H _ { 0 } ^ { 2 } \\left ( Q _ { T } \\right ) : \\sup _ { Q _ { T } } \\left \\vert u \\right \\vert , \\sup _ { Q _ { T } } \\left \\vert \\nabla u \\right \\vert , \\sup _ { Q _ { T } } \\left \\vert \\Delta u \\right \\vert \\leq D _ { 3 } \\right \\} , \\end{align*}"}
+{"id": "3248.png", "formula": "\\begin{align*} - \\Box _ x S ^ { ( l ) } ( x , y ) & = - \\frac { 1 } { 2 } \\ : \\frac { \\partial } { \\partial x ^ k } \\left ( ( y - x ) ^ k \\ : S ^ { ( l - 1 ) } ( x , y ) \\right ) \\\\ & = 2 \\ : S ^ { ( l - 1 ) } ( x , y ) + \\frac { 1 } { 4 } \\ : ( y - x ) ^ 2 \\ : S ^ { ( l - 2 ) } ( x , y ) \\ : . \\end{align*}"}
+{"id": "1400.png", "formula": "\\begin{align*} ( - a ) * a _ { m - k } & = - a _ { m - k + 1 } - ( - a ) * a _ { m - k + 1 } \\\\ & = - a _ { m - k + 1 } + \\sum _ { j = 1 } ^ { k - 1 } ( - 1 ) ^ { j + 1 } a _ { m - k + 1 + j } \\\\ & = - a _ { m - k + 1 } + \\sum _ { j = 2 } ^ k ( - 1 ) ^ j a _ { m - k + j } \\\\ & = \\sum _ { j = 1 } ^ k ( - 1 ) ^ j a _ { m - k + j } . \\end{align*}"}
+{"id": "4643.png", "formula": "\\begin{align*} \\mu _ \\mathcal { G } ( n ) = \\begin{cases} ( - 1 ) ^ { \\omega ( n ) } , \\quad n = p _ 1 \\dots p _ k , p _ i \\in \\mathcal { G } \\ , \\ , \\forall i = 1 , \\dotsc , k , p _ i \\neq p _ j i \\neq j \\\\ \\ ; 0 , \\quad \\end{cases} . \\end{align*}"}
+{"id": "8901.png", "formula": "\\begin{align*} s _ 1 ( x _ 1 ) & = - x _ 1 , \\\\ s _ 2 ( x _ 1 , x _ 2 ) & = \\frac { 1 } { 3 } ( 3 x _ 1 - x _ 2 ) , \\\\ s _ 3 ( x _ 1 , x _ 2 , x _ 3 ) & = - \\frac { 1 } { 2 } ( 2 x _ 1 - x _ 2 ) , \\\\ s _ 4 ( x _ 1 , \\dots , x _ 4 ) & = \\frac { 1 } { 9 0 } ( 9 0 x _ 1 - 5 5 x _ 2 + x _ 4 ) , \\\\ s _ 5 ( x _ 1 , \\dots , x _ 5 ) & = - \\frac { 1 } { 3 6 } ( 3 6 x _ 1 - 2 5 x _ 2 + x _ 4 ) , \\end{align*}"}
+{"id": "8126.png", "formula": "\\begin{align*} ( \\wp ' ( x ) ) ^ 2 \\ = \\ 4 \\ \\wp ^ 3 ( x ) - g _ 2 \\ \\wp ( x ) \\ - \\ g _ 3 \\ = \\ 4 ( \\wp ( x ) - e _ 1 ) ( \\wp ( x ) - e _ 2 ) ( \\wp ( x ) - e _ 3 ) \\ , \\end{align*}"}
+{"id": "4733.png", "formula": "\\begin{align*} \\forall \\gamma ^ { \\mathbb { N } ^ \\mathbb { N } } , p ^ X , x ^ X \\left ( \\gamma > _ \\mathbb { R } 0 \\land \\gamma ^ { - 1 } ( x - _ X p ) \\in A p \\rightarrow p = _ X J ^ A _ { \\gamma } x \\right ) . \\end{align*}"}
+{"id": "9023.png", "formula": "\\begin{align*} \\frac { 1 } { g } = \\frac { 1 } { q _ 1 } + . . . + \\frac { 1 } { q _ { k } } - \\frac { b } { a } , \\end{align*}"}
+{"id": "2970.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d D _ { V , [ l ] } } { d t } & \\leq - \\frac { \\kappa _ 1 | [ l ] | \\mathcal { A } _ { [ l ] } ( v ) ( 0 ) } { N T _ M ^ \\infty } \\phi ( D _ { X , [ l ] } ) D _ { V , [ l ] } + \\frac { 4 \\kappa _ 1 ( N - | [ l ] | ) \\phi ( \\delta ) } { N T _ m ^ \\infty } \\\\ & = : - C _ 1 \\phi ( D _ { X , [ l ] } ) D _ { V , [ l ] } + C _ 2 , \\mbox { a . e . } ~ t \\in ( 0 , t _ 0 ) . \\end{aligned} \\end{align*}"}
+{"id": "225.png", "formula": "\\begin{align*} \\frac { d \\bar v } { d \\tau } = \\frac 1 f f ' ( q ) \\ , \\bar v \\ , \\bar v + f ^ 2 \\ , F ( q ) , \\end{align*}"}
+{"id": "5737.png", "formula": "\\begin{align*} X _ + ( t ) Y _ + ( t ) = \\sum _ { i : \\gamma ^ + _ i > \\gamma _ * } \\xi _ i ^ + ( t ) \\mathcal { E } ^ + _ i ( t ) + \\sum _ { i : \\gamma ^ - _ i > \\gamma _ * } \\xi _ i ^ - ( t ) \\mathcal { E } ^ - _ i ( t ) . \\end{align*}"}
+{"id": "6256.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } - c = 0 . \\end{align*}"}
+{"id": "7235.png", "formula": "\\begin{align*} \\{ ( [ n ] , \\star ) : ( [ n ] , \\star ) \\cong ( [ n ] , \\ast ) \\} = \\{ ( [ n ] , \\ast _ \\sigma ) : \\sigma \\in S _ n \\} . \\end{align*}"}
+{"id": "33.png", "formula": "\\begin{align*} P ^ { j , n } : = \\{ I : \\{ 1 , \\dots , j \\} \\to \\{ 1 , \\dots , n \\} ; I \\ , \\ , { \\rm s t r i c t l y \\ , \\ , m o n o t o n e } \\} \\end{align*}"}
+{"id": "7908.png", "formula": "\\begin{align*} i _ { X } \\alpha = \\ast ( X ^ { \\flat } \\wedge \\ast \\alpha ) . \\end{align*}"}
+{"id": "3772.png", "formula": "\\begin{align*} P ( x ) & = \\int _ 0 ^ 1 r ^ x G ( 1 - r ) \\ , d r \\\\ & = \\int _ 0 ^ 1 \\exp \\Big ( x \\log r - m ( 1 - r ) \\Big ) d r \\\\ & \\geq \\int _ 0 ^ 1 \\exp \\Big ( 2 x ( r - 1 ) - m ( 1 - r ) \\Big ) d r \\\\ & = \\int _ { 0 } ^ 1 \\exp \\Big ( - 2 x t - m ( t ) \\Big ) d t . \\end{align*}"}
+{"id": "5097.png", "formula": "\\begin{align*} \\mathsf { c } _ s & = \\frac { \\mathsf { a } _ s ^ { 2 } - 1 } { v - v ^ { 3 } } . \\end{align*}"}
+{"id": "2534.png", "formula": "\\begin{align*} g = a _ 0 \\Big ( \\frac { d x } { x } \\Big ) ^ 2 + 2 \\sum _ { j = 1 } ^ { n - 1 } a _ { 0 j } \\ , \\frac { d x } { x } \\ , d y ^ j + \\sum _ { j , k = 1 } ^ { n - 1 } a _ { j k } \\ , d y ^ j \\ , d y ^ k \\ ; , \\end{align*}"}
+{"id": "2032.png", "formula": "\\begin{align*} \\Delta ^ \\Sigma \\ , w = 2 n = \\lambda \\xi \\geq \\lambda w . \\end{align*}"}
+{"id": "816.png", "formula": "\\begin{align*} { \\pounds } _ { \\hat { X } } { I _ k } = ( \\nabla _ { i } X ^ { i } + I _ { h } \\nabla _ { 0 } X ^ { h } ) _ { . k } + { X ^ h } P _ { i h k } ^ i + \\nabla _ { 0 } X ^ { h } Q _ { i h k } ^ i . \\end{align*}"}
+{"id": "1135.png", "formula": "\\begin{align*} 2 ^ { - ( E - \\frac { n } { 2 } ) k _ - } 2 ^ { - ( F + \\frac { n } { 2 } - \\frac { n } { a } ) k _ + } 2 ^ { - ( D - \\frac { n } { a } ) l } = 2 ^ { - ( E - \\frac { n } { 2 } ) k _ - } 2 ^ { - [ F - ( J - \\frac { n } { 2 } ) ] k _ + } 2 ^ { - ( D - J ) l } , \\end{align*}"}
+{"id": "7935.png", "formula": "\\begin{align*} d H _ { 2 } ( \\Sigma ; \\partial \\Sigma _ { v } \\mathcal { N } , \\frac { \\tau } { \\rho } v _ { \\Sigma } ) = \\frac { \\tau } { \\rho } \\int _ { \\Sigma } i _ { \\partial \\Sigma _ { v } \\mathcal { N } } d v _ { \\Sigma } = \\frac { \\tau } { \\rho } \\int _ { \\Sigma } \\partial \\Sigma _ { v } i _ { \\mathcal { N } } \\mathrm { d i v } ( \\mathcal { N } ) v _ { \\Omega } = \\frac { \\tau } { \\rho } \\int _ { \\Sigma } k \\partial \\Sigma . \\end{align*}"}
+{"id": "765.png", "formula": "\\begin{align*} A ( A \\otimes A ) ( x \\otimes x \\otimes y \\otimes y - x \\otimes y \\otimes x \\otimes y ) = 0 . \\end{align*}"}
+{"id": "778.png", "formula": "\\begin{align*} ( x \\otimes y ) ^ { * _ \\beta \\ , \\otimes _ \\beta \\ , * _ \\beta } & = \\beta ( a ^ { - 1 } b , a ^ { - 1 } ) \\beta ( b ^ { - 1 } c , b ^ { - 1 } ) \\beta ( b ^ { - 1 } c , a ^ { - 1 } b ) \\ , x ^ * \\otimes y ^ * \\\\ & = \\beta ( a ^ { - 1 } c , a ^ { - 1 } ) \\ , x ^ * \\otimes y ^ * . \\end{align*}"}
+{"id": "8544.png", "formula": "\\begin{align*} \\begin{array} { l c l } \\beta & = & 0 , \\\\ \\alpha \\log _ 1 ( \\pi _ 2 ) - \\gamma \\log _ 1 ( u ( K ) ) & = & 0 , \\\\ \\lambda _ 1 + k ^ { - 1 } \\mu _ 1 a _ 1 & = & 0 , \\\\ \\lambda _ 2 + k ^ { - 1 } \\mu _ 2 a _ 2 & = & 0 . \\end{array} \\end{align*}"}
+{"id": "3950.png", "formula": "\\begin{align*} & \\gamma ^ 4 \\sum _ { n = N + 1 } ^ { \\infty } y _ n ^ { \\top } ( t ) B \\mathcal B _ n ^ \\top \\mathcal Q _ { N \\times N , \\gamma } \\bar { y } ^ { N } ( t ) \\leq \\frac { 1 } { 2 \\alpha _ 1 } \\sum _ { n = N + 1 } ^ { \\infty } \\left | y _ n ( t ) \\right | ^ 2 \\\\ & + \\frac { \\alpha _ 1 \\gamma ^ 8 } { 2 } \\sum _ { n = N + 1 } ^ { \\infty } \\left | B \\mathcal B _ n ^ \\top \\mathcal Q _ { N \\times N , \\gamma } \\bar { y } ^ { N } ( t ) \\right | ^ 2 , \\end{align*}"}
+{"id": "659.png", "formula": "\\begin{align*} \\begin{aligned} & ( I ^ { 0 } f , I ^ { 1 } f , \\cdots , I ^ { k } f ) = ( 0 , 0 , \\cdots , 0 ) \\iff ( J ^ { 0 } f , J ^ { 1 } f , \\cdots , J ^ { k } f ) = ( 0 , 0 , \\cdots , 0 ) . \\end{aligned} \\end{align*}"}
+{"id": "3389.png", "formula": "\\begin{align*} C _ { 1 } = \\frac { R _ { 1 } } { 2 4 \\gamma } ; C _ { 2 } = \\frac { \\gamma } { 2 6 \\sigma ^ { p } } ; C _ { 3 } = \\frac { \\gamma } { 2 6 T \\sigma ^ { p } } ; A = 3 \\gamma \\end{align*}"}
+{"id": "2384.png", "formula": "\\begin{align*} x _ { I _ k \\cup J _ k } = x _ { I _ k } + x _ { J _ k } \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ u _ { I _ k , J _ k } = \\frac { x _ { J _ k } - x _ { I _ k } } { x _ { I _ k } + x _ { J _ k } } \\ . \\end{align*}"}
+{"id": "6772.png", "formula": "\\begin{align*} Q = \\begin{pmatrix} r I _ n & A ^ T \\\\ \\alpha A & s I _ m \\end{pmatrix} ~ { \\rm a n d } ~ M = \\begin{pmatrix} I _ n & 0 \\\\ - ( 1 - \\alpha ) \\frac { 1 } { s } A & I _ m \\end{pmatrix} . \\end{align*}"}
+{"id": "4542.png", "formula": "\\begin{align*} \\big ( n _ { \\infty } ^ { ( K + 1 ) } \\big ) ^ { ( K ) } = n _ { \\infty } ^ { ( K ) } , \\end{align*}"}
+{"id": "5638.png", "formula": "\\begin{align*} B _ { x , i } = ( 0 , y _ 5 + \\tfrac { 3 } { 2 } y _ 8 - 1 1 0 , 2 2 , 1 1 0 - y _ 5 - y _ 8 , \\tfrac { 1 } { 2 } y _ 8 , y _ 5 , y _ 8 , y _ 8 , y _ 8 , 3 3 0 - y _ 5 - 4 y _ 8 ) . \\end{align*}"}
+{"id": "468.png", "formula": "\\begin{align*} \\tilde { \\pi } ( A , g ) = \\pi ( g ) \\left ( \\prod _ { h \\in A } \\varepsilon ( h ) \\right ) \\left ( \\prod _ { h \\in Y _ g \\setminus A } ( \\pi ( d ( g ) ) - \\varepsilon ( h ) ) \\right ) . \\end{align*}"}
+{"id": "4516.png", "formula": "\\begin{align*} W ( n ) & : = \\sum _ { y _ 0 < k \\leqslant n / 2 } \\frac { 1 } { 2 k ^ 2 } \\bigg | \\sum _ { \\substack { | \\lambda | = n - 2 k \\\\ \\lambda _ 1 < k } } a ( \\lambda ) \\bigg | ^ 2 \\\\ & \\leqslant \\frac { 1 } { 2 y _ 0 } \\sum _ { y _ 0 < k \\leqslant n } \\frac { 1 } { k } \\bigg | \\sum _ { \\substack { | \\lambda | = n - k \\\\ \\lambda _ 1 < k / 2 } } a ( \\lambda ) \\bigg | ^ 2 . \\end{align*}"}
+{"id": "7601.png", "formula": "\\begin{align*} \\begin{cases} \\| \\Delta U _ { \\epsilon } \\| _ { 2 } ^ { 2 } = \\| U _ { \\epsilon } \\| _ { 4 ^ * } ^ { 4 ^ * } \\\\ \\mathcal { S } \\| U _ { \\epsilon } \\| _ { 4 ^ * } ^ { 2 } = \\| \\Delta U _ { \\epsilon } \\| _ { 2 } ^ { 2 } \\\\ \\end{cases} \\end{align*}"}
+{"id": "9142.png", "formula": "\\begin{align*} \\tilde { E } ^ { - } _ { \\beta } = \\begin{cases} \\hat { E } ^ { - } _ { \\beta } & \\ \\beta = [ 1 ] , [ 2 ] , [ 1 , 2 ] \\\\ [ 2 ] _ { v } ! \\hat { E } ^ { - } _ { \\beta } & \\ \\beta = [ 1 , 2 , 2 ] \\\\ [ 3 ] _ { v } ! \\hat { E } ^ { - } _ { \\beta } & \\ \\beta = [ 1 , 2 , 2 , 2 ] , [ 1 , 2 , 1 , 2 , 2 ] \\end{cases} . \\end{align*}"}
+{"id": "8891.png", "formula": "\\begin{align*} i \\partial _ t u & = - \\Delta u + \\frac \\lambda { | x | ^ 2 } u - | x | ^ { - \\tau } | u | ^ { p - 2 } \\Big ( I _ \\alpha * | \\cdot | ^ { - \\tau } | u | ^ p \\Big ) u \\\\ & = - \\Delta u + \\frac \\lambda { | x | ^ 2 } u - \\mathcal N . \\end{align*}"}
+{"id": "7534.png", "formula": "\\begin{align*} \\rho _ 0 & = \\rho _ { 0 0 } + \\delta \\rho _ { 0 1 } + \\delta ^ 2 \\rho _ { 0 2 } + \\dots \\ , \\\\ m _ 0 & = m _ { 0 0 } + \\delta m _ { 0 1 } + \\delta ^ 2 m _ { 0 2 } + \\dots \\ , \\\\ n _ 0 & = n _ { 0 0 } + \\delta n _ { 0 1 } + \\delta ^ 2 n _ { 0 2 } + \\dots \\ , \\\\ \\end{align*}"}
+{"id": "2878.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to 0 ^ + } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ { L ^ p ( \\mathbb { R } ^ n ) } = \\left [ - \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { L ^ p ( \\mathbb { R } ^ n ) } . \\end{align*}"}
+{"id": "6232.png", "formula": "\\begin{align*} s _ { 0 , \\gamma } : = \\inf _ { [ 0 , \\gamma ] } \\delta ( f , \\alpha ) - 2 \\sup _ { [ 0 , \\gamma ] } \\sqrt { \\Delta ( D g , \\alpha ) } \\quad \\hbox { a n d } \\Sigma _ { \\gamma , 1 } : = \\sup _ { [ \\gamma , 1 ] } \\delta ( f , \\beta ) + 2 \\sup _ { [ \\gamma , 1 ] } \\sqrt { \\Delta ( D g , \\beta ) } . \\end{align*}"}
+{"id": "8213.png", "formula": "\\begin{align*} P _ { { \\bf X } _ i | Y ^ { i - 1 } } ( { \\bf x } _ i | y ^ { i - 1 } ) = P _ { { \\bf X } _ i | { \\bf A } _ { i - 1 } } ( { \\bf x } _ i | { \\bf a } _ { i - 1 } ) . \\end{align*}"}
+{"id": "9190.png", "formula": "\\begin{align*} q \\ell ^ - _ { 1 j } ( z ) \\ell ^ - _ { j j } [ 0 ] = \\ell ^ - _ { j j } [ 0 ] \\ell ^ - _ { 1 j } ( z ) , \\end{align*}"}
+{"id": "794.png", "formula": "\\begin{align*} \\nabla _ { k } \\nabla _ { l } \\Psi ^ { i } - \\nabla _ { l } \\nabla _ { k } \\Psi ^ { i } = \\Psi ^ { r } R ^ { i } _ { r k l } - \\dot { \\nabla } _ { r } \\Psi ^ { i } R ^ { r } _ { 0 k l } - \\nabla _ { r } \\Psi ^ { i } S ^ { r } _ { k l } , \\end{align*}"}
+{"id": "4912.png", "formula": "\\begin{align*} f _ * \\O _ C = \\O _ { \\P ^ 1 } \\oplus V , V = \\O _ { \\P ^ 1 } ( - a ) \\oplus \\O _ { \\P ^ 1 } ( - b ) \\end{align*}"}
+{"id": "1570.png", "formula": "\\begin{align*} I ( R , \\rho ) : = \\inf \\left \\{ \\dfrac { \\displaystyle { \\int _ 0 ^ R | \\varphi ' | ^ 2 \\ , d r } } { \\displaystyle { \\int _ \\rho ^ R \\varphi ^ 2 \\ , d r } } : \\varphi \\in C ^ \\infty ( [ 0 , R ] ) \\setminus \\{ 0 \\} , \\varphi ( R ) = 0 \\right \\} . \\end{align*}"}
+{"id": "4815.png", "formula": "\\begin{align*} \\mathcal { S } ^ { ( b f ) } _ 0 : = \\Big \\{ \\mathbf { c } ' \\in \\mathbb { R } ^ { n } : \\mathbf { 0 } \\leq \\mathbf { c } ' \\leq \\mathbf { 1 } \\Big \\} . \\end{align*}"}
+{"id": "4690.png", "formula": "\\begin{align*} F = \\mathrm { N e g } _ { q _ X } ( D ) ^ { \\perp } \\cap \\overline { \\mathrm { M o v } ( X ) } , \\end{align*}"}
+{"id": "6691.png", "formula": "\\begin{align*} b _ i b _ j = b _ { m ( i , j ) } \\end{align*}"}
+{"id": "8919.png", "formula": "\\begin{align*} F _ { k , n } ( x ) = ( x ) _ k + \\sum _ { m = 1 } ^ { \\lfloor k / 2 \\rfloor } \\frac { k ! } { ( k - 2 m ) ! } \\frac { 2 B _ { 2 m } } { ( 2 m ) ! } \\omega _ m ( n ) ( x - m ) _ { k - 2 m } . \\end{align*}"}
+{"id": "4841.png", "formula": "\\begin{align*} \\int _ E g d x & = \\int _ { E _ 1 } \\Big ( \\int _ { E ^ { x _ 2 , x _ 3 , \\ldots , x _ n } } g d x _ 1 \\Big ) d x _ 2 \\ldots d x _ n \\\\ & \\ge \\int _ { E _ 1 } \\Big ( \\int _ { - \\frac { | E ^ { x _ 2 , x _ 3 , \\ldots , x _ n } | } { 2 } } ^ { \\frac { | E ^ { x _ 2 , x _ 3 , \\ldots , x _ n } | } { 2 } } g d x _ 1 \\Big ) d x _ 2 \\ldots d x _ n \\\\ & = \\int _ { E _ { w _ 1 } } g d x . \\end{align*}"}
+{"id": "2522.png", "formula": "\\begin{align*} I _ k ^ { ( s ) } ( M , L ) = \\{ \\ , u \\in C ^ { - \\infty } ( M ) \\mid P _ j u \\subset H ^ s ( M ) , \\ j = 0 , \\dots , k \\ , \\} \\ ; , \\end{align*}"}
+{"id": "517.png", "formula": "\\begin{align*} & O ( 2 ^ { ( k + 1 ) d } \\cdot 2 ^ { ( k + 1 ) d + H _ i } ( ( k + 1 ) d + H _ i ) \\log ^ { 1 + o ( 1 ) } { q } ) \\\\ = & O ( 2 ^ { 2 ( k + 1 ) d + H _ i } ( ( k + 1 ) d + H _ i ) \\log ^ { 1 + o ( 1 ) } { q } ) \\end{align*}"}
+{"id": "5051.png", "formula": "\\begin{align*} \\epsilon _ S ( \\beta _ S , K _ S ) = \\begin{cases} \\left ( \\dfrac { \\ln ( \\frac { c } { \\beta _ S } ) } { q K _ S } \\right ) ^ { \\frac { 1 } { \\max \\{ p , 2 \\} } } & K _ S \\geq \\dfrac { \\ln ( \\frac { c } { \\beta _ S } ) } { q } \\\\ \\left ( \\dfrac { \\ln ( \\frac { c } { \\beta _ S } ) } { q K _ S } \\right ) ^ { \\frac { 1 } { a } } & K _ S < \\dfrac { \\ln ( \\frac { c } { \\beta _ S } ) } { q } . \\end{cases} \\end{align*}"}
+{"id": "6916.png", "formula": "\\begin{align*} n - q - \\frac { \\ ( p - 1 \\ ) \\ ( n - q \\ ) } { p } + \\varepsilon = \\frac { 2 p - 1 } { p } + \\varepsilon < 2 \\end{align*}"}
+{"id": "4163.png", "formula": "\\begin{align*} \\varphi \\big ( g ^ { i } _ 1 g _ 2 \\big ) = \\varphi \\big ( g ^ { i } _ 1 \\big ) \\varphi ( g _ 2 ) \\mbox { a n d } \\varphi \\big ( g ^ { i } _ 1 g _ 3 \\big ) = \\varphi ( g _ 3 ) \\varphi \\big ( g ^ { i } _ 1 \\big ) \\ , , \\end{align*}"}
+{"id": "2659.png", "formula": "\\begin{align*} S ^ { ( 1 ) } = \\int ^ { t _ { 2 } } _ { t _ { 1 } } L ^ { ( 1 ) } ( \\dot { q } ^ { i } , q ^ { i } , t ) d t \\end{align*}"}
+{"id": "2901.png", "formula": "\\begin{align*} & \\lim _ { \\lambda \\to 0 ^ + } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ { L ^ \\Phi ( \\mathbb { R } ^ n ) } = \\left [ - \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { L ^ \\Phi ( \\mathbb { R } ^ n ) } . \\end{align*}"}
+{"id": "8855.png", "formula": "\\begin{align*} A _ 2 & = \\big \\| | x | ^ { - \\tau } | u | ^ { p - 1 } ( I _ \\alpha \\ast | \\cdot | ^ { - \\tau - 1 } | u | ^ { p } ) \\big \\| _ { L _ x ^ { \\frac { 2 n } { n + 2 } } } \\\\ & \\leq \\| | x | ^ { - \\tau } | u | ^ { p - 1 } \\| _ { L _ x ^ { a _ 2 } } \\| | x | ^ { - \\tau - 1 } | u | ^ p \\| _ { L _ x ^ { b _ 2 } } \\\\ & \\lesssim \\| \\nabla u \\| ^ { 2 p - 1 } _ { L _ x ^ r } \\end{align*}"}
+{"id": "5997.png", "formula": "\\begin{align*} w _ j = \\frac { 2 u _ i u _ { i j } } { u ^ 2 } - \\frac { 2 u _ i ^ 2 u _ j } { u ^ 3 } \\end{align*}"}
+{"id": "637.png", "formula": "\\begin{align*} I _ { r + 1 , r } ( x ) & = - \\sinh ( x ) \\biggl ( \\frac { 1 } { r - 2 } \\tanh ^ { r - 2 } ( x ) + \\frac { r - 1 } { ( r - 2 ) ( r - 4 ) } \\tanh ^ { r - 4 } ( x ) \\\\ & + \\frac { ( r - 1 ) ( r - 3 ) } { ( r - 2 ) ( r - 4 ) ( r - 6 ) } \\tanh ^ { r - 6 } ( x ) + \\dots + \\frac { ( r - 1 ) ! ! } { 3 ( r - 2 ) ! ! } \\tanh ^ 2 ( x ) \\biggr ) \\\\ & + \\frac { ( r - 1 ) ! ! } { ( r - 2 ) ! ! } I _ { 3 , 2 } ( x ) . \\end{align*}"}
+{"id": "3395.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { k } Z _ { t } & \\le \\log \\frac { 1 } { \\delta } \\end{align*}"}
+{"id": "9163.png", "formula": "\\begin{align*} G _ { \\beta , \\beta ' } = ( w _ { \\beta , 1 } - v ^ { 4 n - 4 j + 2 } w _ { \\beta ' , 1 } ) ( w _ { \\beta , 1 } - v ^ { 4 n - 4 j - 4 } w _ { \\beta ' , 1 } ) \\cdot G _ { [ 1 , n , j + 1 ] , \\beta ' } , \\end{align*}"}
+{"id": "1240.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ; q ^ 2 ) _ k } { ( q ; q ) _ k } q ^ k \\equiv ( - 1 ) ^ { \\frac { n - 1 } { 2 } } q ^ { \\frac { n ^ 2 - 1 } { 4 } } \\pmod { \\Phi _ { n } ( q ) } , \\end{align*}"}
+{"id": "3052.png", "formula": "\\begin{align*} A _ { \\mu \\nu } ( x ) = \\frac { \\partial q _ \\mu } { \\partial x _ \\nu } = ( { Z } _ \\nu ) _ { \\mu \\sigma } B _ { \\sigma \\rho } ( x ) s _ \\rho = B _ { \\mu \\rho } ( x ) ( { Z } _ \\nu ) _ { \\rho \\sigma } s _ \\sigma \\ , . \\end{align*}"}
+{"id": "7307.png", "formula": "\\begin{align*} K _ { X _ { m } , \\chi _ { \\beta } } ( x , y ; t ) = \\sum _ { k \\in \\mathbb { Z } } e ^ { - 2 \\pi i \\beta k } e ^ { - t } I _ { x - y + k m } ( t ) . \\end{align*}"}
+{"id": "1680.png", "formula": "\\begin{align*} \\dim \\langle F _ 0 ^ { j _ 0 } , \\ldots , F _ 3 ^ { j _ 3 } \\rangle = j _ 0 + \\cdots + j _ 3 \\end{align*}"}
+{"id": "3732.png", "formula": "\\begin{align*} \\mathrm { B } _ { z } \\left ( a , b \\right ) + \\mathrm { B } _ { 1 - z } \\left ( b , a \\right ) = \\mathrm { B } \\left ( a , b \\right ) . \\end{align*}"}
+{"id": "2284.png", "formula": "\\begin{align*} Z _ n ( \\mathfrak { g } ) = \\left \\{ X \\in U ( \\mathfrak { g } ) ^ { \\otimes n } ~ | ~ [ \\delta ^ { ( n ) } ( g ) , X ] = 0 \\ , , \\forall g \\in U ( \\mathfrak { g } ) \\right \\} \\ . \\end{align*}"}
+{"id": "2260.png", "formula": "\\begin{align*} V ^ { \\otimes n } = \\bigoplus _ { \\substack { \\lambda \\vdash n \\\\ [ 0 . 2 e m ] \\ell ( \\lambda ) \\leq N } } V _ { \\lambda } \\otimes S _ { \\lambda } \\ , . \\end{align*}"}
+{"id": "6418.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ n f _ { k j } ( X _ { \\frac { i - 1 } { n } } ) \\left ( h _ j ( z ^ n _ i ( \\theta _ 0 ) ) - h _ j ( n ^ { 1 / \\alpha _ 0 } \\Delta _ i ^ n L ) \\right ) \\to 0 . \\end{align*}"}
+{"id": "4470.png", "formula": "\\begin{align*} F _ 0 : = f _ 0 - f _ 1 f _ 2 , \\end{align*}"}
+{"id": "106.png", "formula": "\\begin{align*} \\lim _ { G \\to P } \\frac { \\sqrt { E } - \\sqrt { \\frac { G } { n - 1 } + K } } { \\sqrt { G } - \\sqrt { P } } = \\frac { ( n - 2 ) \\sqrt { P } } { ( n - 1 ) \\sqrt { \\frac { P } { n - 1 } + K } } . \\end{align*}"}
+{"id": "4471.png", "formula": "\\begin{align*} M _ M = \\int _ M | f _ 0 | ^ 2 \\lambda _ 1 \\lambda _ 2 = \\int _ M | F _ 2 | ^ 2 \\lambda _ 1 \\lambda _ 2 + \\int _ M | f _ 1 f _ 2 | ^ 2 \\lambda _ 1 \\lambda _ 2 \\ge M _ { D _ 1 } \\times M _ { M _ 1 } . \\end{align*}"}
+{"id": "3419.png", "formula": "\\begin{align*} d _ { A _ 0 } : = \\frac { 1 } { 2 } ( d _ { A ' _ 0 } + ( \\iota ^ * \\otimes I ) \\circ d _ { A ' _ 0 } \\circ ( \\iota ^ * \\otimes I ) ) . \\end{align*}"}
+{"id": "1818.png", "formula": "\\begin{align*} F _ A \\wedge \\star _ { \\phi } \\phi = 0 , \\end{align*}"}
+{"id": "1671.png", "formula": "\\begin{align*} \\begin{array} { l l l l l l l } \\omega ( e _ r , f _ r ) = 1 , & & \\omega ( e _ r , \\phi _ 2 ) = 1 , & & \\omega ( e _ r , \\phi _ 3 ) = \\varepsilon a _ 2 , \\\\ \\omega ( f _ r , \\phi _ 2 ) = \\varepsilon , & & \\omega ( f _ r , \\phi _ 3 ) = \\varepsilon a _ 1 , & & \\omega ( \\phi _ 2 , \\phi _ 3 ) = \\varepsilon . \\end{array} \\end{align*}"}
+{"id": "2401.png", "formula": "\\begin{align*} a _ { n , 2 s + 4 , k } & = \\widetilde { A _ n } ( - k ) \\\\ & = 2 ^ { ( 6 s + 1 2 ) n } ( - 4 k + 2 n ) ^ { \\delta } \\left ( \\frac { ( - k + 1 / 4 ) _ { n } ( - k + 3 / 4 ) _ n } { k ! ^ 2 ( n - k ) ! ^ 2 } \\right ) ^ { s + 2 } \\\\ & = ( - 4 k + 2 n ) ^ { \\delta } \\left ( \\frac { ( 4 k ) ! ( 4 n - 4 k ) ! } { ( 2 k ) ! ( 2 n - 2 k ) ! k ! ^ 2 ( n - k ) ! ^ 2 } \\right ) ^ { s + 2 } . \\end{align*}"}
+{"id": "2077.png", "formula": "\\begin{align*} \\sigma _ 1 ^ 2 ( x , y ) = A ( x , y ) , \\forall x , y \\in [ 0 , 1 ] . \\end{align*}"}
+{"id": "3766.png", "formula": "\\begin{align*} \\log ( | \\tilde { g } | ^ t w ) & = ( t / t _ * ) \\log ( | \\tilde { g } | ^ { t _ * } w ^ { t ^ * / t } ) \\\\ & = ( t / t _ * ) \\log ( | \\tilde { g } | ^ { t _ * } w ) + ( t / t _ * ) \\log ( w ^ { t _ * / t - 1 } ) \\\\ & = ( t / t _ * ) \\log ( | \\tilde { g } | ^ { t _ * } w ) + ( t / t _ * ) ( t _ * / t - 1 ) \\log ( w ) . \\end{align*}"}
+{"id": "4040.png", "formula": "\\begin{align*} [ \\textbf { k } ^ x _ t ] _ i = k _ { S _ t } ( \\bar x ^ i | _ { I _ t } , x | _ { I _ t } ) , [ \\textbf { k } ^ y _ s ] _ j = k _ { S _ s } ( \\bar y ^ j | _ { I _ s } , y | _ { I _ s } ) , i \\le M , j \\le N \\end{align*}"}
+{"id": "6199.png", "formula": "\\begin{align*} 1 / \\alpha _ 1 = \\sum _ { j = 2 } ^ { 2 n - 3 } 1 / \\gamma _ j ~ . \\end{align*}"}
+{"id": "3332.png", "formula": "\\begin{align*} \\begin{cases} \\max \\lbrace \\L u - f , \\psi - u \\rbrace & ( x , t ) \\in \\R ^ { N } \\times [ 0 , T ] \\\\ u ( x , t ) = g & ( x , t ) \\in \\R ^ { N } \\times \\lbrace 0 \\rbrace , \\end{cases} \\end{align*}"}
+{"id": "5125.png", "formula": "\\begin{align*} \\sum _ { j } q _ { j } \\frac { \\partial D ( p \\| K ( p , q ) . q ) } { \\partial q _ { j } } = 0 \\end{align*}"}
+{"id": "597.png", "formula": "\\begin{align*} \\| A \\| \\| B \\| = \\| A \\| \\cdot \\| A ^ { - 1 } P \\| \\le \\| A \\| \\cdot \\| A ^ { - 1 } \\| \\cdot \\| P \\| \\le 1 . \\end{align*}"}
+{"id": "7596.png", "formula": "\\begin{align*} \\mathcal { M } _ { 4 ^ * , q } ( c ) = \\{ v \\in S ( c ) , E _ { 4 ^ * , q } ( v ) < E _ { 4 ^ * , q } ( u ) , Q _ { 4 ^ * , q } ( v ) < 0 \\} . \\end{align*}"}
+{"id": "1744.png", "formula": "\\begin{align*} \\quad ( - 1 ) ^ { n + 1 } \\left ( \\sum \\limits _ { r + s = n - 2 - m } S _ { r , s } x _ r x _ s \\right ) = m ! \\left . \\frac { \\partial ^ m } { \\partial z _ n ^ m } \\right | _ { \\{ z _ n = 0 \\} } \\left ( \\sum _ { j = 2 } ^ { n - 2 } ( - 1 ) ^ r \\left ( H _ { \\mathcal { L } } \\right ) _ { n - j , n } \\left ( H _ { \\mathcal { L } } \\right ) _ { n , j } \\right ) . \\end{align*}"}
+{"id": "6368.png", "formula": "\\begin{align*} \\psi ( \\bar \\rho ) - \\psi ( R ) & = \\psi ' ( R ) R \\rho + \\psi '' ( R ( 1 + \\theta \\rho ) ) \\frac { R ^ 2 \\rho ^ 2 } { 2 } \\\\ & = \\phi ' ( R ) R ^ 2 \\rho + \\bigl ( \\phi ' ( R ( 1 + \\theta \\rho ) ) + R ( 1 + \\theta \\rho ) \\phi '' ( R ( 1 + \\theta \\rho ) ) \\bigr ) \\frac { R ^ 2 \\rho ^ 2 } { 2 } . \\end{align*}"}
+{"id": "9135.png", "formula": "\\begin{align*} \\kappa _ { \\beta } = \\begin{cases} | \\beta | - 1 & \\ \\ \\beta \\neq [ 1 , 2 , 1 , 2 , 2 ] \\\\ | \\beta | + 1 & \\ \\ \\beta = [ 1 , 2 , 1 , 2 , 2 ] \\end{cases} . \\end{align*}"}
+{"id": "3407.png", "formula": "\\begin{align*} b ^ + ( X ) + \\frac { 1 } { 2 } b _ 1 ( S ) - \\frac { 1 } { 4 } S \\circ S - \\frac { 1 } { 2 } \\sigma ( L ) + \\frac { 1 } { 2 } \\sigma ( L ' ) = 0 , \\end{align*}"}
+{"id": "5704.png", "formula": "\\begin{align*} \\psi _ i ^ \\pm : = \\bigg ( \\frac { m } { m - 2 \\gamma ^ \\pm _ i } \\varphi _ i , - \\frac { m } { 2 } \\varphi _ i \\bigg ) . \\end{align*}"}
+{"id": "6156.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau ^ k } [ f ( x ) - f ( \\breve { x } ^ { k } ) ] - \\frac { 1 } { \\tau ^ { k - 1 } } [ f ( x ) - f ( \\breve { x } ^ { k - 1 } ) ] + ( u - \\widetilde { u } ^ k ) ^ T F ( \\widetilde { u } ^ k ) \\\\ & + ( 1 - \\tau ^ k ) \\beta ^ k ( A ( x - \\widetilde { x } ^ k ) ) ^ T ( A \\breve { x } ^ { k - 1 } - b ) \\geq ( v - \\widetilde { v } ^ k ) Q ^ k ( v ^ k - \\widetilde { v } ^ k ) , ~ \\forall u . \\end{aligned} \\end{align*}"}
+{"id": "2734.png", "formula": "\\begin{align*} & { \\textrm { ( i ) . } } \\omega _ { m n } = - \\omega _ { n m } \\\\ & { \\textrm { ( i i ) . } } { \\textrm { d e t } } \\omega _ { m n } \\neq 0 \\\\ & { \\textrm { ( i i i ) . } } d \\omega = 0 . \\end{align*}"}
+{"id": "3497.png", "formula": "\\begin{align*} \\Phi = ( \\varphi _ \\lambda ) _ \\lambda \\colon \\textstyle { \\prod } F _ \\lambda / \\textstyle { \\bigoplus } F _ \\lambda & \\longrightarrow \\ell ^ \\infty ( \\Lambda , A ) / c _ 0 ( \\Lambda , A ) , \\end{align*}"}
+{"id": "7696.png", "formula": "\\begin{align*} I _ { 1 } ( \\Omega ) = V o l ( \\Omega ) , \\ \\ I _ { 0 } ( \\Omega ) = \\frac { \\omega _ { n - 1 } } { n \\omega _ { n } } S ( \\Omega ) , \\ \\ I _ { n + 1 } ( \\Omega ) = \\frac { n + 1 } { \\omega _ { n } } V o l ( \\Omega ) ^ { 2 } , \\end{align*}"}
+{"id": "2346.png", "formula": "\\begin{align*} N ( w _ { 1 } , w _ { 2 } ) = - \\int ^ { t } _ { 0 } e ^ { - ( t - s ) \\mathcal { L } } \\mathbb { P } \\mathrm { d i v } ( w _ { 1 } \\otimes w _ { 2 } ) d s . \\end{align*}"}
+{"id": "2979.png", "formula": "\\begin{align*} v _ 1 ^ 2 ( 0 ) - v _ 2 ^ 2 ( 0 ) = \\frac { \\kappa _ 1 ( 1 + \\mathcal { A } ( v ) ( 0 ) ) } { 2 ( 1 - \\alpha ) } ( x _ 2 ^ 2 ( 0 ) - x _ 1 ^ 2 ( 0 ) ) ^ { 1 - \\alpha } , \\end{align*}"}
+{"id": "5390.png", "formula": "\\begin{align*} w ^ { S _ { j + 1 } } _ { j - 1 } = \\frac { \\lambda } { \\displaystyle ( 1 + \\rho ) ^ j \\ , \\prod _ { i = 1 } ^ j a _ i } = \\frac { \\lambda } { 1 + \\cdots + \\rho ^ j } , 1 \\leq j \\leq n . \\end{align*}"}
+{"id": "3100.png", "formula": "\\begin{align*} \\varphi \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & = \\left ( \\begin{array} { c c c } a ^ { \\ell _ 1 } & 0 & c _ 2 \\ , a ^ { p ^ { e _ 2 } } \\ , b ^ { p ^ { e _ 2 } } \\ , d ^ { - \\ell _ 3 } \\\\ 0 & d ^ { - \\ell _ 2 } & c _ 1 \\ , a ^ { p ^ { e _ 1 } } \\ , b ^ { p ^ { e _ 1 } } \\ , d ^ { - \\ell _ 3 } \\\\ 0 & 0 & d ^ { - \\ell _ 3 } \\end{array} \\right ) . \\end{align*}"}
+{"id": "1560.png", "formula": "\\begin{align*} J _ { n } ( u _ { n } ) = \\inf \\{ J _ { n } ( w ) : w \\in \\psi _ n + W ^ { 1 , p } _ { 0 } ( B ) \\} . \\end{align*}"}
+{"id": "8763.png", "formula": "\\begin{align*} \\forall u \\in [ 0 , 1 ] , \\ ; \\phi _ \\uparrow ( u ) \\in [ ( u - \\nu _ r ( \\R ) ) ^ + , u \\wedge \\nu _ l ( \\R ) ] \\mbox { a n d } \\int _ 0 ^ u F _ \\mu ^ { - 1 } ( w ) d w = G ( u , \\phi _ \\uparrow ( u ) ) . \\end{align*}"}
+{"id": "3574.png", "formula": "\\begin{align*} ( 1 + a ) \\alpha ^ 2 \\tau _ 0 \\tau _ 1 ( i d ^ 2 - ( \\tau ^ 2 ) ^ 2 ) - ( a + b ) \\alpha \\tau _ 0 ( i d ^ 2 - \\tau ^ 2 ) = 0 . \\end{align*}"}
+{"id": "3620.png", "formula": "\\begin{align*} \\sigma \\sigma _ 1 = - 1 , \\sigma \\sigma _ 1 = 1 , \\sigma = 1 . \\end{align*}"}
+{"id": "411.png", "formula": "\\begin{align*} \\bold { B } _ { i } = \\begin{bmatrix} \\bold { \\Pi } _ { i } & \\bold { \\Gamma } _ { i } \\\\ \\bold { \\Xi } _ { i } + \\widetilde { \\bold { \\Lambda } } _ { i } & \\bold { \\Pi } _ i ^ { T } \\end{bmatrix} , \\end{align*}"}
+{"id": "1060.png", "formula": "\\begin{align*} & D _ 2 ( m _ 1 , \\ell ) : = \\frac { 1 } { m _ 1 + \\ell } , \\\\ & D _ k ( m _ 1 , \\ell ) : = \\int _ { 0 } ^ { \\infty } d m _ 2 \\cdots \\int _ { 0 } ^ { \\infty } d m _ { k - 1 } \\frac { 1 } { \\{ \\prod _ { i = 1 } ^ { k - 2 } ( m _ { i } + m _ { i + 1 } ) \\} ( m _ { k - 1 } + \\ell ) } , \\ \\ k \\ge 3 . \\end{align*}"}
+{"id": "7996.png", "formula": "\\begin{align*} 0 & \\geq \\Delta g = W ' ( u ) - \\Delta \\zeta \\geq g \\int _ 0 ^ 1 W '' ( t u + ( 1 - t ) ( \\zeta + 1 ) ) \\ , d t + W ' ( \\zeta + 1 ) - 2 B R ^ { - 2 } . \\end{align*}"}
+{"id": "4171.png", "formula": "\\begin{align*} \\varphi \\big ( g ^ { j } _ 1 g _ 3 \\big ) \\varphi \\big ( g ^ { i } _ 1 \\big ) \\overset { ( \\ref { e q : 8 } ) } { = } \\varphi ( g _ 3 ) \\varphi \\big ( g ^ { j } _ 1 \\big ) \\varphi \\big ( g ^ { i } _ 1 \\big ) \\overset { { \\bf A 3 } } { = } \\varphi ( g _ 3 ) \\varphi ( g ^ { n } _ 1 ) = \\varphi ( g ^ { n } _ 1 ) \\varphi ( g _ 3 ) \\overset { { \\bf A 3 } } { = } \\varphi ( g ^ { n } _ 1 g _ 3 ) \\overset { ( \\ref { e q : 1 1 n 3 } ) } { = } \\varphi \\big ( g _ 3 g ^ { j } _ 1 \\big ) \\varphi \\big ( g ^ { i } _ 1 \\big ) \\ , , \\end{align*}"}
+{"id": "5924.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { C } ( A ( n , p ) ) ) = ( p ^ { n } + 1 ) ( p ^ { 2 n } - p ^ { n } ) ( p ^ { 2 n } - p ^ { n } - 1 ) ^ { 2 } = p ^ { n } ( p ^ { 2 n } - 1 ) ( p ^ { 2 n } - p ^ { n } - 1 ) ^ { 2 } \\end{align*}"}
+{"id": "2523.png", "formula": "\\begin{align*} \\langle u , v \\rangle _ { s , k } = \\sum _ { j = 0 } ^ k \\langle P _ j u , P _ j v \\rangle _ s \\ ; , \\end{align*}"}
+{"id": "3081.png", "formula": "\\begin{align*} a _ { k _ { i j } } = - \\left ( r _ k \\cdot a _ { \\frac { \\beta _ j } { e _ j } } ^ { \\gamma _ { 1 j } } \\right ) ^ { - 1 } \\cdot P _ k \\left ( a _ { \\frac { \\beta _ j } { e _ j } } , \\ldots , a _ { k _ { i j } - 1 } \\right ) . \\end{align*}"}
+{"id": "3682.png", "formula": "\\begin{align*} e _ i - e _ j = \\frac { 1 } { 2 } u - \\frac { 1 } { 2 } v . \\end{align*}"}
+{"id": "1851.png", "formula": "\\begin{align*} F _ A \\wedge \\psi + \\sigma _ { \\pmb { \\pi } } ( A ) = 0 \\end{align*}"}
+{"id": "8574.png", "formula": "\\begin{align*} G _ t ^ { t _ 1 } = ( J _ k + [ \\varepsilon _ k ( G ^ { t _ 0 } _ t ) B _ { t _ 0 } ] ^ { \\cdot k } _ + ) G ^ { t _ 0 } _ t . \\end{align*}"}
+{"id": "3707.png", "formula": "\\begin{align*} | N | & \\ge | \\kappa | + 1 \\\\ & = | G | - | G \\setminus \\kappa | + 1 \\\\ & \\ge | G | - \\frac { | G | } { 2 } + 1 \\\\ & > \\frac { | G | } { 2 } . \\end{align*}"}
+{"id": "1834.png", "formula": "\\begin{align*} d \\phi = 0 d \\star _ { \\phi } \\phi = 0 , \\end{align*}"}
+{"id": "4755.png", "formula": "\\begin{align*} \\delta ( \\varepsilon ) = \\inf \\left \\{ 1 - \\norm { x + y } / 2 \\mid \\norm { x } = \\norm { y } = 1 , \\norm { x - y } = \\varepsilon \\right \\} \\end{align*}"}
+{"id": "2269.png", "formula": "\\begin{align*} \\mathcal { R } \\cdot \\Delta ( a ) = \\Delta ^ { o p } ( a ) \\cdot \\mathcal { R } \\ \\ \\ \\ \\forall a \\in U _ q ( \\mathfrak { g } ) \\ . \\end{align*}"}
+{"id": "7134.png", "formula": "\\begin{align*} \\partial _ \\mu F ^ { \\mu \\nu } = 0 \\ , \\ , , \\end{align*}"}
+{"id": "8632.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } ( \\partial _ x X ) = v _ 1 ( \\partial _ x X ) , \\\\ ( \\partial _ x X ) ( 0 ; x ) = 1 , \\end{cases} \\end{align*}"}
+{"id": "7105.png", "formula": "\\begin{align*} \\sqrt { \\delta _ 1 } & = \\| | b | ( \\lambda - \\Delta ) ^ { - \\frac { 1 } { 2 } } \\| _ { 2 \\rightarrow 2 } \\\\ & \\leq \\| b \\| _ { d , w } \\Omega _ d ^ { - \\frac { 1 } { d } } \\| | x | ^ { - 1 } ( \\lambda - \\Delta ) ^ { - \\frac { 1 } { 2 } } \\| _ { 2 \\rightarrow 2 } \\leq \\| b \\| _ { d , w } \\Omega _ d ^ { - \\frac { 1 } { d } } \\frac { 2 } { d - 2 } , \\end{align*}"}
+{"id": "6603.png", "formula": "\\begin{align*} b = \\begin{cases} & \\varepsilon \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mbox { i f } \\ \\delta \\leq 0 , \\ \\mbox { a n d } \\ 0 < \\alpha < 1 \\\\ & \\frac { 1 } { \\log x } \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mbox { i f } \\ \\delta < 0 \\ \\mbox { a n d } \\ 1 \\leq \\alpha . \\end{cases} \\end{align*}"}
+{"id": "5773.png", "formula": "\\begin{align*} u = u ^ T + H ( u ^ T ) + \\tilde { u } ^ \\perp . \\end{align*}"}
+{"id": "598.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { \\binom { 2 k } { k } ^ { 2 } } { ( - 1 6 ) ^ { k } } ( 2 H _ { 2 k } - H _ { k } ) = - \\frac { \\ln ( 2 ) \\ , \\Gamma ^ { 2 } \\left ( \\frac { 1 } { 4 } \\right ) } { 4 \\pi \\sqrt { 2 \\pi } } \\end{align*}"}
+{"id": "576.png", "formula": "\\begin{align*} \\begin{dcases} \\Delta ( B \\Delta \\eta ) + ( 1 - \\nu ) \\left ( 2 B _ { x y } \\eta _ { x y } - B _ { x x } \\eta _ { y y } - B _ { y y } \\eta _ { x x } \\right ) - f = 0 & \\Gamma _ c \\\\ \\Delta \\eta - ( 1 - \\nu ) \\eta _ { \\tau \\tau } = 0 & \\partial \\Gamma _ c \\\\ ( B \\Delta \\eta ) _ n + ( 1 - \\nu ) ( 2 B _ \\tau \\eta _ { n \\tau } - B _ n \\eta _ { n \\tau } + B \\eta _ { n \\tau \\tau } ) = 0 & \\partial \\Gamma _ c \\end{dcases} , \\end{align*}"}
+{"id": "8656.png", "formula": "\\begin{align*} \\liminf _ i W _ 1 ( \\mu _ { i } , \\nu ) = 0 , \\end{align*}"}
+{"id": "2220.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ 2 \\intop _ { \\R _ + } | \\partial _ r ^ j v | ^ 2 r ^ { 2 ( \\mu + j - 2 ) } r d r \\le c \\intop _ { \\R _ + } | f + v _ { , z z } | ^ 2 r ^ { 2 \\mu } r d r . \\end{align*}"}
+{"id": "3040.png", "formula": "\\begin{align*} L ^ 2 = L _ 1 ^ 2 + L _ 2 ^ 2 + L _ 3 ^ 2 | _ { S ^ 2 } = p _ 1 ^ 2 + p _ 2 ^ 2 + p _ 3 ^ 2 \\ , . \\end{align*}"}
+{"id": "8602.png", "formula": "\\begin{align*} u _ t + u u _ x + \\Lambda ^ { \\frac { 1 } { 2 } } u = 0 \\end{align*}"}
+{"id": "2896.png", "formula": "\\begin{align*} & \\lim _ { \\lambda \\to 0 ^ + } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ { L ^ r _ \\omega ( \\mathbb { R } ^ n ) } = \\left [ - \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { L ^ r _ \\omega ( \\mathbb { R } ^ n ) } . \\end{align*}"}
+{"id": "8105.png", "formula": "\\begin{align*} w ^ i _ { k , \\ell } : = e ^ { - t _ i } ( u _ k - u _ \\ell + s _ k - s _ \\ell ) . \\end{align*}"}
+{"id": "636.png", "formula": "\\begin{align*} I _ { r + 1 , r } ( x ) = - \\frac { \\sinh ^ { r - 1 } ( x ) } { ( r - 2 ) \\cosh ^ { r - 2 } ( x ) } + \\frac { r - 1 } { r - 2 } I _ { r - 1 , r - 2 } ( x ) . \\end{align*}"}
+{"id": "4455.png", "formula": "\\begin{align*} \\int _ { \\{ \\psi < - t \\} } | F | ^ 2 e ^ { - \\varphi } = \\frac { G ( 0 ) } { \\int _ 0 ^ { + \\infty } c ( t ) e ^ { - t } d t } e ^ { - t } \\end{align*}"}
+{"id": "4547.png", "formula": "\\begin{align*} \\mathcal { T } _ { V _ { N , \\beta , a } } \\big ( n _ N ( t ) , m _ N ( t ) \\big ) & \\leq T _ { V _ { N , \\beta , a } } \\big ( \\omega _ N ( t ) \\big ) \\\\ & \\leq e ^ { - \\gamma t } T _ { V _ { N , \\beta , a } } \\big ( \\omega _ N ( 0 ) \\big ) \\\\ & = e ^ { - \\gamma t } \\iint V _ { N , \\beta , a } ( [ s ] _ N , [ s ' ] _ N ) \\omega _ N ( 0 , d [ s ] _ N , d [ s ' ] _ N ) . \\end{align*}"}
+{"id": "1276.png", "formula": "\\begin{align*} m _ { 1 1 } & : = e ^ { ( a + d ) / 2 } \\Big [ \\Delta \\cosh { \\frac { 1 } { 2 } \\Delta } + ( a - d ) \\sinh { \\frac { 1 } { 2 } \\Delta } \\Big ] , \\\\ m _ { 1 2 } & : = 2 b e ^ { ( a + d ) / 2 } \\sinh { \\frac { 1 } { 2 } \\Delta } , \\\\ m _ { 2 1 } & : = 2 c e ^ { ( a + d ) / 2 } \\sinh { \\frac { 1 } { 2 } \\Delta } , \\\\ m _ { 2 2 } & : = e ^ { ( a + d ) / 2 } \\Big [ \\Delta \\cosh { \\frac { 1 } { 2 } \\Delta } + ( d - a ) \\sinh { \\frac { 1 } { 2 } \\Delta } \\Big ] . \\end{align*}"}
+{"id": "6044.png", "formula": "\\begin{align*} \\tau _ + = ( - 1 , 1 ) , \\tau _ - = ( 1 , 1 ) . \\end{align*}"}
+{"id": "1332.png", "formula": "\\begin{align*} L ( a ) = \\langle p , a \\rangle \\pm H ( p ) , \\end{align*}"}
+{"id": "4607.png", "formula": "\\begin{align*} \\frac { 1 } { s } + \\frac { 1 } { s ' } = 1 \\end{align*}"}
+{"id": "7083.png", "formula": "\\begin{align*} b _ n : = c _ n \\eta _ { \\varepsilon _ n } \\ast ( \\mathbf { 1 } _ n b ) , \\end{align*}"}
+{"id": "8341.png", "formula": "\\begin{align*} \\mu _ 3 ^ B ( \\alpha , \\beta , \\gamma ) = { - } ( - 1 ) ^ { | \\beta | } ( \\alpha ^ \\flat \\wedge \\beta ^ \\flat \\wedge \\gamma ^ \\flat ) \\Upsilon _ B . \\end{align*}"}
+{"id": "6602.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { a - i T } ^ { a + i T } F _ { k , \\beta } ( s , z ) \\frac { x ^ s } { s } \\ d s = R _ { k , \\beta } ( z ) \\frac { x ^ { 1 - z + \\delta z } } { 1 - z + \\delta z } + O ( x ^ { ( 1 - ( 1 - \\delta ) b ) } \\exp ( - f _ k ( \\log x \\log \\log x ) ^ { \\frac { 1 } { 2 } } ) ) \\end{align*}"}
+{"id": "1043.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } \\| z _ { k } - z _ { k , n } \\| = \\begin{cases} O ( n ^ { 1 - 2 d - \\rho } ) & ( 1 - 2 d < \\rho < 1 - d ) , \\\\ O ( n ^ { - d } \\log n ) & ( \\rho = 1 - d ) , \\\\ O ( n ^ { - d } ) & ( \\rho > 1 - d ) , \\end{cases} n \\to \\infty . \\end{align*}"}
+{"id": "2519.png", "formula": "\\begin{align*} \\| a \\| ' _ { K , \\alpha , \\beta , m } = \\sup _ { x \\in K } \\limsup _ { | \\xi | \\to \\infty } \\frac { \\big | \\partial _ x ^ \\alpha \\partial _ \\xi ^ \\beta a ( x , \\xi ) \\big | } { | \\xi | ^ { m - | \\beta | } } \\ ; . \\end{align*}"}
+{"id": "5926.png", "formula": "\\begin{align*} \\dfrac { M _ { 1 } ( \\mathcal { C } ( A ( n , p ) ) ) } { | v ( \\mathcal { C } ( A ( n , p ) ) ) | } = ( p ^ { 2 n } - p ^ { n } - 1 ) ^ { 2 } = \\dfrac { M _ { 2 } ( \\mathcal { C } ( A ( n , p ) ) ) } { | e ( \\mathcal { C } ( A ( n , p ) ) ) | } . \\end{align*}"}
+{"id": "4914.png", "formula": "\\begin{align*} f = x _ 1 f _ 1 + x _ 2 f _ 2 . \\end{align*}"}
+{"id": "3927.png", "formula": "\\begin{align*} \\left ( \\sum _ { k = 1 } ^ N \\left [ \\left \\| b _ k \\right \\| ^ 2 _ { L ^ 2 } - \\left | \\mathcal { B } _ k \\right | ^ 2 \\right ] \\right ) \\lvert \\mathcal B _ { N \\times N } ^ { - 1 } \\rvert ^ 2 \\leq \\eta \\lambda _ { N + 1 } ^ { \\beta } \\end{align*}"}
+{"id": "651.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial c } \\cosh ( \\ell ( a , c ) ) & = - \\frac { \\sinh ( a - c ) } { 2 \\sinh a } ( 2 \\sinh a + \\cosh c \\sinh ( a - c ) - 2 \\sinh c \\cosh ( a - c ) ) \\\\ & = - \\frac { 3 \\sinh ^ 2 ( a - c ) \\cosh c } { 2 \\sinh a } < 0 . \\end{align*}"}
+{"id": "330.png", "formula": "\\begin{align*} ( I , x _ n ) ^ { [ k ] } = I ^ { [ k ] } + x _ n I ^ { [ k - 1 ] } \\subset S \\end{align*}"}
+{"id": "6186.png", "formula": "\\begin{align*} \\begin{aligned} & f _ j ( x _ j ) - f _ j ( \\breve { x } _ j ^ k ) + ( x _ j - \\breve { x } ^ k _ j ) ^ T ( - A _ j ^ T \\lambda ^ k + \\beta ^ k A _ j ^ T ( \\sum _ { i = 1 } ^ { j } A _ i \\breve { x } _ i ^ k + \\sum _ { i = j + 1 } ^ { m } A _ i \\hat { x } _ i ^ k - b ) ) \\geq 0 . \\end{aligned} \\end{align*}"}
+{"id": "5555.png", "formula": "\\begin{align*} T = \\sum _ { j = 1 } ^ r \\nu _ j w _ j ^ { ( 1 ) } \\otimes w _ j ^ { ( 2 ) } \\otimes \\cdots \\otimes w _ j ^ { ( k ) } , \\end{align*}"}
+{"id": "4387.png", "formula": "\\begin{align*} & \\max \\{ 0 , x _ i - \\overline { b } _ i , \\max \\{ 0 , x _ i - \\overline { b } _ i + \\Delta b _ i \\} - \\theta ^ k ( x ) \\} = \\\\ & \\max \\{ 0 , x _ i - \\overline { b } _ i , x _ i - \\overline { b } _ i + \\Delta b _ i - x _ k + \\overline { b } _ k - \\Delta b _ k \\} . \\end{align*}"}
+{"id": "5311.png", "formula": "\\begin{align*} v _ i ( \\nu ) = \\min \\ , \\left \\{ v _ i ^ u ( \\nu ) : u \\in \\mathcal { U } \\right \\} . \\end{align*}"}
+{"id": "5860.png", "formula": "\\begin{align*} ( m - 2 ) ( m - 3 ) ^ { 3 } ( 2 m - 2 ) & + m ( 2 m - 2 ) \\\\ & > ( m - 2 ) ^ { 2 } ( m - 3 ) ^ { 3 } + m ( m - 2 ) ( m - 3 ) + m ( m - 2 ) ( m - 3 ) ^ { 2 } + m ^ { 2 } \\\\ & = ( ( m - 2 ) ( m - 3 ) + m ) ( ( m - 2 ) ( m - 3 ) ^ { 2 } + m ) . \\end{align*}"}
+{"id": "1620.png", "formula": "\\begin{align*} Q _ { r ( p - 1 ) } \\b Q _ { s ( p - 1 ) } = q _ { g , r } \\sum _ i & t _ { r , s , i } \\tfrac { ( i - s ) ( p - 1 ) } { p i - s ( p - 1 ) - r + 1 } \\b Q _ { ( r - 1 + p s - p i ) ( p - 1 ) } Q _ { i ( p - 1 ) } \\\\ & - q ^ { \\tfrac { p - 1 } { 2 } } ( - 1 ) ^ { \\tfrac { s ( p - 1 ) } { 2 } } \\sum _ i t _ { r , s , i } Q _ { ( r + p s - p i ) ( p - 1 ) } \\b Q _ { i ( p - 1 ) } , \\end{align*}"}
+{"id": "3460.png", "formula": "\\begin{align*} & \\sigma ( \\Sigma ( S ) ) = 2 \\sigma ( X ) - \\frac { 1 } { 2 } S \\circ S - \\sigma ( L ) + \\sigma ( L ' ) , \\\\ & b _ 2 ( \\Sigma ( S ) ) = 2 b _ 2 ( X ) + b _ 1 ( S ) , \\\\ & b ^ + ( \\Sigma ( S ) ) = 2 b ^ + ( X ) + \\frac { 1 } { 2 } b _ 1 ( S ) - \\frac { 1 } { 4 } S \\circ S - \\frac { 1 } { 2 } \\sigma ( L ) + \\frac { 1 } { 2 } \\sigma ( L ' ) , \\\\ & b _ 1 ( \\Sigma ( S ) ) = b _ 3 ( \\Sigma ( S ) ) = 0 . \\end{align*}"}
+{"id": "4716.png", "formula": "\\begin{align*} q _ T : = \\sum _ { \\tilde { T } \\subseteq T } \\left ( M _ { \\tilde { T } } ^ { I } + M _ { \\tilde { T } } ^ \\nu \\right ) \\ , . \\end{align*}"}
+{"id": "8517.png", "formula": "\\begin{align*} h ( \\sigma ^ k ( y ) ) = \\sigma ^ \\ell ( h ( y ) ) . \\end{align*}"}
+{"id": "498.png", "formula": "\\begin{align*} O \\left ( \\sum _ { t = 0 } ^ { \\infty } { \\log ^ { 3 + o ( 1 ) } \\left ( m ^ { 1 / 2 ^ t } \\right ) } \\right ) = O \\left ( \\sum _ { t = 0 } ^ { \\infty } { \\frac { 1 } { 2 ^ { 3 t } } } \\cdot \\log ^ { 3 + o ( 1 ) } { m } \\right ) = O ( \\log ^ { 3 + o ( 1 ) } { m } ) \\end{align*}"}
+{"id": "6743.png", "formula": "\\begin{align*} P ( n , z , p ) = \\sum _ { j = 0 } ^ { p - 1 } \\frac { ( - 2 ) ^ j } { n ^ { j + 1 } } \\sum _ { k = 1 } ^ \\infty \\frac { ( - 1 ) ^ k \\zeta ( 2 k ) z ^ { 2 k } } { k ^ { p - j } } + \\left ( - \\frac { 2 } { n } \\right ) ^ p S _ { 2 , n } ( z ) . \\end{align*}"}
+{"id": "218.png", "formula": "\\begin{align*} X ^ i ( x , v ) = \\sum _ { j = 1 } ^ n B ^ i \\ , _ j ( x ) \\ , v ^ j , i = 1 , \\ldots , n . \\end{align*}"}
+{"id": "5849.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { C } ( D _ { 2 m } ) ) & = ( m - 1 ) ( m - 1 - 1 ) ^ { 2 } + m ( 1 - 1 ) ^ { 2 } = ( m - 1 ) ( m - 2 ) ^ { 2 } \\end{align*}"}
+{"id": "402.png", "formula": "\\begin{align*} \\begin{aligned} n _ S & = \\lceil \\frac { \\pi L _ { S , x } L _ { S , y } } { \\lambda ^ 2 } \\rceil + o ( \\frac { L _ { S , x } L _ { S , y } } { \\lambda ^ 2 } ) , \\\\ n _ R & = \\lceil \\frac { \\pi L _ { R , x } L _ { R , y } } { \\lambda ^ 2 } \\rceil + o ( \\frac { L _ { R , x } L _ { R , y } } { \\lambda ^ 2 } ) , \\end{aligned} \\end{align*}"}
+{"id": "4437.png", "formula": "\\begin{align*} \\int _ { \\partial D } | f | ^ 2 \\varphi | d z | = \\lim _ { r \\rightarrow 1 - 0 } \\int _ { \\partial D _ r } | f | ^ 2 \\varphi | d z | \\end{align*}"}
+{"id": "2406.png", "formula": "\\begin{align*} U ^ { ( v ) } ( t ) = ( - 1 ) ^ { v } v ! \\left ( \\frac { \\delta } { ( t + n / 2 ) ^ { 1 + v } } + \\sum _ { \\ell = 0 } ^ { n - 1 } \\frac { s + 2 } { ( t + \\ell + 1 / 4 ) ^ { 1 + v } } + \\sum _ { \\ell = 0 } ^ { n - 1 } \\frac { s + 2 } { ( t + \\ell + 3 / 4 ) ^ { 1 + v } } - \\sum _ { \\ell = 0 \\atop \\ell \\neq k } ^ { n } \\frac { 2 s + 4 } { ( t + \\ell ) ^ { 1 + v } } \\right ) . \\end{align*}"}
+{"id": "6450.png", "formula": "\\begin{align*} 0 = \\tau _ 3 ( \\phi ) : = - \\bar \\Delta ^ 2 \\tau ( \\phi ) + R ^ N ( \\bar \\nabla _ { e _ j } \\tau ( \\phi ) , \\tau ( \\phi ) ) d \\phi ( e _ j ) + R ^ N ( \\bar \\Delta \\tau ( \\phi ) , d \\phi ( e _ j ) ) d \\phi ( e _ j ) , \\end{align*}"}
+{"id": "3267.png", "formula": "\\begin{align*} \\frac { { ( n - 1 ) ! } ^ d d ^ { d n - d } } { n ^ 2 } \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( \\frac { d + 1 } { d } ) _ k ^ { d - 2 } ( \\frac { 1 } { d } ) _ k ( \\frac { 1 - d } { d } ) _ k } { ( 1 ) _ k ^ d } \\in \\Z . \\end{align*}"}
+{"id": "5868.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { N C } ( G ) ) = 4 0 n ^ { 3 } - 7 2 n ^ { 2 } + 3 2 n M _ { 2 } ( \\mathcal { N C } ( G ) ) = 6 4 n ^ { 4 } - 1 6 0 n ^ { 3 } + 1 2 8 n ^ { 2 } - 3 2 n . \\end{align*}"}
+{"id": "456.png", "formula": "\\begin{align*} \\Delta ( E _ { B ^ { q } } ) = \\left \\{ \\begin{aligned} & \\min _ { ( i , j ) \\in E _ { B ^ { q } } } \\Delta t _ { i , j } , \\mbox { i f } E _ { B ^ { q } } \\neq \\emptyset , \\\\ [ 2 p t ] & + \\infty , \\mbox { o t h e r w i s e } , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4286.png", "formula": "\\begin{align*} \\| F _ m - F \\| _ { \\mathbf { D } _ { q , r } } = O ( m ^ { - \\gamma } ) \\end{align*}"}
+{"id": "5581.png", "formula": "\\begin{align*} f ( T , x ) = \\partial _ W ( f _ { \\phi _ i , t } f _ { \\phi _ j , t } ) , \\end{align*}"}
+{"id": "387.png", "formula": "\\begin{align*} M _ { 2 } ( \\bar { \\xi } * ( \\xi ) ; \\bar { a } * ( a ) ) = M _ { 2 } ( \\bar { \\xi } ; \\bar { a } ) \\cap \\bigcap \\{ M _ { 2 } \\left ( M _ { 2 } ( \\bar { \\xi } * ( \\xi ) ; \\bar { a } * ( b ) ) \\cap M _ { 3 } ( \\nu ) \\right ) : \\nu < \\xi , b < a \\} \\end{align*}"}
+{"id": "306.png", "formula": "\\begin{align*} Q _ { j k } = \\dfrac { K ^ 2 \\left | \\left \\langle L _ { F ' } e _ k ' , e _ j ' \\right \\rangle \\right | ^ 2 } { \\xi ^ 2 \\ , \\left | \\Re \\left ( \\left \\langle L _ { F ' } e _ j ' , e _ j ' \\right \\rangle \\right ) \\right | \\left | \\Re \\left ( \\left \\langle L _ { F ' } e _ k ' , e _ k ' \\right \\rangle \\right ) \\right | } j < k . \\end{align*}"}
+{"id": "6791.png", "formula": "\\begin{align*} \\dot { x } & = \\kappa _ 1 x ^ 2 + c _ 2 \\kappa _ 2 x ^ { a _ 2 } y ^ { b _ 2 } + c _ 3 \\kappa _ 3 x ^ { a _ 3 } y ^ { b _ 3 } , \\\\ \\dot { y } & = \\phantom { \\kappa _ 1 x ^ 2 + } d _ 2 \\kappa _ 2 x ^ { a _ 2 } y ^ { b _ 2 } + d _ 3 \\kappa _ 3 x ^ { a _ 3 } y ^ { b _ 3 } . \\end{align*}"}
+{"id": "2431.png", "formula": "\\begin{align*} \\tilde { U } _ { \\tau } = \\tilde { U } ^ { p } ( v _ { x x } + \\mu \\tilde { U } - \\tilde { U } ^ { - p + 1 } ) , ( \\tau , x ) \\in ( 0 , + \\infty ) \\times ( - L , L ) \\end{align*}"}
+{"id": "5403.png", "formula": "\\begin{align*} w _ i ^ S = \\begin{cases} \\displaystyle { \\lambda _ i \\ , \\left [ 1 - \\Delta b ^ S _ { i + 1 } \\right ] } & 0 \\leq i \\leq n - 1 \\\\ 0 & i = n . \\end{cases} \\end{align*}"}
+{"id": "1780.png", "formula": "\\begin{align*} f _ { j } ^ 0 ( \\zeta ) = c \\zeta ^ I + O ( \\zeta ^ { I + 1 } ) \\end{align*}"}
+{"id": "3257.png", "formula": "\\begin{align*} \\omega _ 0 & = x d _ x + y d _ y \\\\ \\omega _ 1 & = d _ y \\\\ \\omega _ 2 & = ( x ^ 2 - y ^ 2 ) \\ , d _ x + 2 x y \\ , d _ y . \\end{align*}"}
+{"id": "8210.png", "formula": "\\begin{align*} H ( Y _ i | Y ^ { i - 1 } , { \\bf S } ) & \\leq H ( Y _ i | Y ^ { i - 1 } ) \\\\ & = \\sum _ { y ^ { i - 1 } } \\frac { c ^ { y ^ { i - 1 } } } { M } H ( \\frac { b ^ { y ^ { i - 1 } } } { c ^ { y ^ { i - 1 } } } ) . \\end{align*}"}
+{"id": "5654.png", "formula": "\\begin{align*} \\{ y \\in Y : \\exists ( x , v ) \\in X \\times T _ y Y | f ( x ) = y d _ x f ( v ) \\in T Y \\setminus d f ( T X ) \\} . \\end{align*}"}
+{"id": "7297.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { 3 k - 1 } \\csc ^ 2 \\left ( \\frac { 2 j + 1 } { 6 k } \\pi \\right ) \\omega ^ j = 3 k ^ 2 e ^ { - \\frac { i \\pi } { 3 } } \\end{align*}"}
+{"id": "6889.png", "formula": "\\begin{align*} s h _ { ( I _ i ) } = a l t ( S h _ { ( I _ i ) } ) = \\sum _ { \\pi \\in S h _ { ( I _ i ) } } ( - 1 ) ^ \\pi \\pi , \\end{align*}"}
+{"id": "754.png", "formula": "\\begin{align*} { } ^ c \\nabla _ k \\rho _ l = \\phi g _ { l k } , \\end{align*}"}
+{"id": "6676.png", "formula": "\\begin{align*} \\widetilde { H } = \\langle y _ 1 , \\ldots , y _ l , a _ 1 , \\ldots , a _ m \\rangle \\le G . \\end{align*}"}
+{"id": "6047.png", "formula": "\\begin{align*} G ^ 1 = \\frac { 1 } { 2 } \\begin{pmatrix} 0 & - \\frac { E } { N ^ \\gamma } \\\\ - \\frac { E } { N ^ \\gamma } & 0 \\end{pmatrix} \\textrm { a n d } G ^ 2 = \\begin{pmatrix} 0 & 0 \\\\ 0 & - \\frac { E } { N ^ \\gamma } \\end{pmatrix} \\end{align*}"}
+{"id": "5859.png", "formula": "\\begin{align*} ( m - 3 ) ^ { 3 } + 1 - ( m - 3 ) - ( m - 3 ) ^ { 2 } = ( m - 3 ) ( ( m - 3 ) ( m - 4 ) - 1 ) + 1 > 0 . \\end{align*}"}
+{"id": "1067.png", "formula": "\\begin{align*} A _ { s , n } & : = K _ 1 K _ 2 \\sum _ { k = 1 } ^ { \\infty } \\sum _ { t = 1 } ^ { n } \\{ S _ { 1 , 2 k - 1 } ( n , n - s , t - 1 ) + S _ { 2 , 2 k - 1 } ( n , n - s , t - 1 ) \\} , \\\\ B _ { s , n } & : = K _ 1 K _ 2 \\sum _ { k = 1 } ^ { \\infty } \\sum _ { t = 1 } ^ { n } \\{ S _ { 1 , 2 k } ( n , s - 1 , t - 1 ) + S _ { 2 , 2 k } ( n , s - 1 , t - 1 ) \\} . \\end{align*}"}
+{"id": "5715.png", "formula": "\\begin{align*} G ( ( v , 0 ) ; ( v , 0 ) ) = 2 m ^ { - 2 } \\left ( \\sum _ { i \\notin I _ 2 } \\left | 4 ^ { - 1 } m - \\lambda _ i \\right | a _ i ^ 2 + \\sum _ { i \\in I _ 2 } a _ i ^ 2 \\right ) \\geq c \\sum _ { i = 1 } ^ \\infty a _ i ^ 2 = c \\| v \\| ^ 2 _ { L ^ 2 } , \\end{align*}"}
+{"id": "6573.png", "formula": "\\begin{align*} \\phi _ { \\Phi _ k , \\beta } ( x , y ) = R _ { k , \\beta } \\left ( \\frac { 1 } { ( 1 - \\delta ) } \\right ) y ^ { \\frac { 1 } { ( 1 - \\delta ) } } + O _ \\delta ( x \\alpha ^ { \\frac { 1 } { \\log x } } ( \\log x \\log \\log x ) ^ { \\frac { 1 } { 2 } } ) , \\end{align*}"}
+{"id": "8306.png", "formula": "\\begin{align*} P = \\Pi + Z . \\end{align*}"}
+{"id": "7009.png", "formula": "\\begin{align*} d X _ t = - b ( t , X _ t ) d t + \\sqrt { 2 } d W _ t , \\end{align*}"}
+{"id": "6502.png", "formula": "\\begin{align*} \\Psi : = \\sum _ { n = 1 } ^ { + \\infty } \\Phi _ n \\end{align*}"}
+{"id": "3995.png", "formula": "\\begin{align*} 4 \\Bigl ( \\sum _ { k = 1 } ^ { n } x _ { k } ^ { \\ : \\left ( t - 1 \\right ) } \\Bigr ) \\Bigl ( \\sum _ { r = 1 } ^ { \\nu } y _ { r } ^ { \\ : \\left ( t - 1 \\right ) } \\Bigr ) \\leq \\Bigl ( \\sum _ { k = 1 } ^ { n } x _ { k } ^ { \\ : \\left ( t - 1 \\right ) } + \\sum _ { r = 1 } ^ { \\nu } y _ { r } ^ { \\ : \\left ( t - 1 \\right ) } \\Bigr ) ^ { 2 } \\end{align*}"}
+{"id": "8804.png", "formula": "\\begin{align*} { \\rm s q } \\# \\pi - { \\rm s q } \\# \\pi ^ { \\uparrow } = 2 p s ( \\overline \\gamma _ m - \\underline \\gamma _ m ) . \\end{align*}"}
+{"id": "6211.png", "formula": "\\begin{align*} \\mathcal { T } _ g ( d ) : = \\left \\{ ( r _ i , \\lambda _ g ) \\in \\R ^ + \\times \\R ^ + \\colon \\eqref { e : r e s t r 1 - 3 } \\hbox { a n d } \\eqref { e : a r g e n } \\hbox { h o l d } \\right \\} . \\end{align*}"}
+{"id": "5471.png", "formula": "\\begin{align*} \\int _ { S O ( 2 N + 1 ) } \\left ( \\frac { P ' } { P } \\Big ( 1 - \\frac { a } { N } \\Big ) \\right ) ^ { K } = ( - 1 ) ^ K \\left [ \\left ( \\frac { N } { a } \\right ) ^ K - \\frac { N ^ K } { a ^ { K - 1 } } K \\right ] + O \\left ( \\frac { N ^ { K - 1 } } { a ^ { K - 1 } } + \\frac { N ^ { K } } { a ^ { K - 2 } } \\right ) \\end{align*}"}
+{"id": "505.png", "formula": "\\begin{align*} \\kappa _ { i , p } : = \\nu _ p ^ { ( \\kappa _ p ) } ( \\overline { \\beta } _ i ) = \\nu _ p ^ { ( \\kappa _ p ) } ( \\overline { \\beta } _ i \\bmod { p ^ { \\kappa _ p } } ) , \\end{align*}"}
+{"id": "5326.png", "formula": "\\begin{align*} v ^ { k , \\textrm { L P } } = \\min \\ , \\left \\{ \\sum _ { j _ k \\in J _ k } c _ { j _ k } \\ , x _ { j _ k } : \\mathbf { x } ^ k \\in P _ k ( \\mathcal { F } _ k ) \\right \\} , \\end{align*}"}
+{"id": "7749.png", "formula": "\\begin{align*} k _ { i j } + k _ { j n ( j ) } - k _ { i n ( i ) } = \\left \\{ \\begin{array} { l l l } k _ { i j } + k _ { j i } = 0 , & \\textrm { i f } \\ n ( j ) = n ( i ) = i , \\\\ k _ { i j } - k _ { i j } = 0 , & \\textrm { i f } \\ n ( j ) = n ( i ) = j , \\\\ k _ { i j } + k _ { j n ( j ) } - k _ { i n ( j ) } = 0 , & \\textrm { o t h e r w i s e } , \\end{array} \\right . \\end{align*}"}
+{"id": "5243.png", "formula": "\\begin{align*} F \\left ( p \\| q \\right ) = \\sum _ { i } p _ { i } \\log \\frac { p _ { i } } { \\alpha p _ { i } + \\left ( 1 - \\alpha \\right ) q _ { i } } + \\left ( 1 - \\alpha \\right ) \\left ( q _ { i } - p _ { i } \\right ) \\end{align*}"}
+{"id": "6002.png", "formula": "\\begin{align*} \\sigma _ t = \\begin{cases} & \\frac { a } { t ^ { \\frac { 1 } { 4 } } } , \\mbox { i f $ j = 1 $ , } \\\\ & \\frac { a } { t ^ s } , \\mbox { i f $ j = 2 $ , } \\end{cases} \\end{align*}"}
+{"id": "7555.png", "formula": "\\begin{align*} \\theta ( \\tau ) = \\begin{cases} 1 , \\ \\ 0 \\leq \\tau < t \\\\ ( t - \\tau ) / \\kappa + 1 , \\ \\ t \\leq \\tau < t + \\kappa \\\\ 0 , \\ \\ t + \\kappa \\leq \\tau < T \\ . \\end{cases} \\end{align*}"}
+{"id": "4018.png", "formula": "\\begin{align*} \\begin{cases} x _ { 1 } ^ { \\left ( t + 2 \\right ) } & = \\ ; \\gamma _ { 1 } ^ { \\ , 2 ^ { t } } \\left ( 1 - \\gamma _ { 1 } \\right ) ^ { 2 ^ { t } - 1 } \\left ( x _ { 1 } ^ { \\left ( 1 \\right ) } y ^ { \\left ( 1 \\right ) } \\right ) ^ { 2 ^ { t } } \\medskip \\\\ y ^ { \\left ( t + 2 \\right ) } & = \\ ; \\gamma _ { 1 } ^ { \\ , 2 ^ { t } - 1 } \\left ( 1 - \\gamma _ { 1 } \\right ) ^ { 2 ^ { t } } \\left ( x _ { 1 } ^ { \\left ( 1 \\right ) } y ^ { \\left ( 1 \\right ) } \\right ) ^ { 2 ^ { t } } , t \\geq 0 . \\end{cases} \\end{align*}"}
+{"id": "7677.png", "formula": "\\begin{align*} \\norm { B } _ { \\infty , \\infty } : = \\sup _ { n } \\sum _ { l \\in \\Lambda _ L } e ^ { \\delta d ( n , l ) } \\abs { B ( n , l ) } < \\frac { 1 } { 2 } \\end{align*}"}
+{"id": "4021.png", "formula": "\\begin{align*} \\begin{cases} x _ { 1 } ^ { \\left ( 2 t + 2 \\right ) } & = \\ ; \\delta _ { 1 } x _ { 2 } ^ { \\left ( 2 t + 1 \\right ) } y ^ { \\left ( 2 t + 1 \\right ) } \\medskip \\\\ x _ { 2 } ^ { \\left ( 2 t + 1 \\right ) } & = \\ ; \\gamma _ { 2 } x _ { 1 } ^ { \\left ( 2 t \\right ) } y ^ { \\left ( 2 t \\right ) } \\medskip \\\\ y ^ { \\left ( 2 t + 2 \\right ) } & = \\ ; \\delta x _ { 2 } ^ { \\left ( 2 t + 1 \\right ) } y ^ { \\left ( 2 t + 1 \\right ) } \\\\ y ^ { \\left ( 2 t + 1 \\right ) } & = \\ ; \\gamma x _ { 1 } ^ { \\left ( 2 t \\right ) } y ^ { \\left ( 2 t \\right ) } . \\end{cases} \\end{align*}"}
+{"id": "235.png", "formula": "\\begin{align*} \\frac { d \\varphi } { d x } = c \\ , h . \\end{align*}"}
+{"id": "3875.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum \\limits _ { k = 0 } ^ { m ^ n _ 0 - 1 } R ^ { n , k } & = \\frac { 1 } { n } \\sum \\limits _ { k = 0 } ^ { N _ 1 - 1 } R ^ { n , k } + \\frac { 1 } { n } \\sum \\limits _ { k = N _ 1 } ^ { k _ 2 - 1 } R ^ { n , k } + \\frac { 1 } { n } \\sum \\limits _ { k = k _ 2 } ^ { m ^ n _ 0 - 1 } R ^ { n , k } \\\\ & \\le \\left | \\log \\left ( a _ 1 ^ * \\delta _ 0 ^ A \\right ) \\right | \\frac { k _ 1 } { n } + | \\log \\delta _ 1 | \\frac { m ^ n _ 0 } { n } . \\end{align*}"}
+{"id": "522.png", "formula": "\\begin{align*} N ^ { - 1 } \\sum _ { m \\leq N } { \\tau ( m ) } & \\geq N ^ { - 1 } \\sum _ { m = 1 } ^ { \\lfloor \\delta N \\rfloor } { \\log ^ { 1 + \\epsilon } { m } } \\geq ( 2 N ) ^ { - 1 } \\frac { \\delta } { 2 } N \\log ^ { 1 + \\epsilon } \\left ( \\frac { \\delta } { 2 } N \\right ) = \\frac { \\delta } { 4 } \\log ^ { 1 + \\epsilon } \\left ( \\frac { \\delta } { 2 } N \\right ) \\\\ & \\geq \\frac { \\delta } { 8 } \\log ^ { 1 + \\epsilon } { N } > > \\log { N } , \\end{align*}"}
+{"id": "885.png", "formula": "\\begin{align*} G ^ 1 ( x , \\mathbf { y } ) & = \\left ( F P ^ 1 _ 1 \\right ) y ^ 1 + \\left ( F P ^ 1 _ 2 \\right ) y ^ 2 + \\left ( L ^ 1 _ 1 \\right ) \\left ( y ^ 1 \\right ) ^ 2 + ( L ^ 1 _ 2 ) \\left ( y ^ 2 \\right ) ^ 2 + ( Q _ 0 ^ 1 ) y ^ 1 y ^ 2 \\\\ & + \\left ( \\frac { A ^ 1 _ 0 } { F } \\right ) \\left ( y ^ 1 \\right ) ^ 3 + \\left ( \\frac { B ^ 1 _ 0 } { F } \\right ) \\left ( y ^ 1 \\right ) ^ 2 y ^ 2 + \\left ( \\frac { D ^ 1 _ 0 } { F } \\right ) y ^ 1 \\left ( y ^ 2 \\right ) ^ 2 + F ^ 2 R _ 0 ^ 1 . \\end{align*}"}
+{"id": "7558.png", "formula": "\\begin{align*} \\begin{dcases} \\partial _ t \\bar { u } + \\bar { u } \\cdot \\nabla \\bar { u } = - \\nabla \\big ( h _ 1 ^ \\prime ( \\bar { \\rho } ) + \\bar { \\phi } \\big ) \\\\ \\varepsilon \\big ( \\partial _ t \\bar { v } + \\bar { v } \\cdot \\nabla \\bar { v } \\big ) = - \\nabla \\big ( h _ 2 ^ \\prime ( \\bar { n } ) - \\bar { \\phi } \\big ) + \\varepsilon \\frac { \\bar { e } } { \\bar { n } } \\ , \\end{dcases} \\end{align*}"}
+{"id": "8511.png", "formula": "\\begin{align*} G _ { r , n _ k } [ b ] = G _ { r + l _ k , n _ k } [ b ] = a . \\end{align*}"}
+{"id": "5709.png", "formula": "\\begin{align*} \\mathbf { L } \\psi _ { i , 3 } = 2 ^ { - 1 } m \\psi _ { i , 3 } + \\psi _ { i , 4 } , \\ \\mathbf { L } \\psi _ { i , 4 } = 2 ^ { - 1 } m \\psi _ { i , 4 } . \\end{align*}"}
+{"id": "6364.png", "formula": "\\begin{align*} \\xi ( r ) : = r ^ 2 \\frac { \\abs { \\nabla ^ h \\rho } ^ 2 + \\cosh ^ 2 ( r ) \\abs { \\nabla ^ v \\rho } ^ 2 } { \\sinh ^ 2 ( r ) \\cosh ^ 2 ( r ) } , \\end{align*}"}
+{"id": "321.png", "formula": "\\begin{align*} ( f ^ m ) '' ( \\xi ) + \\frac { N - 1 } { \\xi } ( f ^ m ) ' ( \\xi ) - \\alpha f ( \\xi ) + \\beta \\xi f ' ( \\xi ) + \\xi ^ { \\sigma } f ( \\xi ) ^ p = 0 , \\xi = | x | ^ { \\sigma } ( T - t ) ^ { \\beta } . \\end{align*}"}
+{"id": "6117.png", "formula": "\\begin{align*} e _ { i _ 1 \\ldots i _ d } = \\sum _ { j _ d = 1 } ^ { n _ d } \\cdots \\sum _ { j _ 1 = 1 } ^ { n _ 1 } v _ { j _ 1 \\ldots j _ d } \\prod _ { \\mu = 1 } ^ d e ^ \\mu _ { i _ \\mu j _ \\mu } , 1 \\le i _ \\mu \\le n _ \\mu , \\end{align*}"}
+{"id": "3085.png", "formula": "\\begin{align*} \\begin{array} { c } { \\rm C o e f f } ( ( \\omega \\phi ) ( t ^ { n } ) , t ^ { k n } ) = { \\rm C o e f f } ( ( \\omega \\phi ) ( t ) , t ^ { k } ) \\\\ n t ^ { n - 1 } { \\rm C o e f f } ( ( \\omega \\phi ) ( t ^ n ) , t ^ { k n } ) = { \\rm C o e f f } ( \\omega ( \\phi ( t ^ n ) ) , t ^ { n ( k + 1 ) - 1 } ) . \\end{array} \\end{align*}"}
+{"id": "8827.png", "formula": "\\begin{align*} u _ t = d \\Delta u + f ( x - c t , u ) , x \\in \\mathbb { R } , \\ , \\ , t \\ge 0 , \\end{align*}"}
+{"id": "1025.png", "formula": "\\begin{align*} \\lambda ^ d ( \\gamma ) = \\sum _ { \\beta \\in \\Gamma ^ { t _ \\lambda } ( n _ \\lambda ) } \\lambda ( \\beta ) \\ , p _ { t _ \\lambda } ^ d ( \\beta , \\gamma ) \\end{align*}"}
+{"id": "6579.png", "formula": "\\begin{align*} \\chi _ 1 ( n ) = \\begin{cases} & 1 \\ \\ \\ \\ \\mbox { i f } \\ n \\equiv 1 \\ ( m o d \\ 4 ) , \\\\ & - 1 \\ \\ \\mbox { i f } \\ n \\equiv 3 \\ ( m o d \\ 4 ) , \\\\ & 0 \\ \\ \\ \\ \\mbox { i f } \\ n = 2 . \\end{cases} \\end{align*}"}
+{"id": "4849.png", "formula": "\\begin{align*} a _ { m _ * } = C \\mathcal { H } ^ { n - 1 } ( \\partial E _ m ) , \\end{align*}"}
+{"id": "7934.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Omega } \\frac { \\delta \\mathcal { F } } { \\delta v } \\wedge \\partial v = \\lim _ { \\epsilon \\to 0 } \\frac { 1 } { \\epsilon } \\big ( \\mathcal { F } ( v + \\epsilon \\partial v ) - \\mathcal { F } ( v ) \\big ) , \\ \\forall \\partial v \\in P ^ { \\ast } \\Lambda ^ { 1 } ( \\Omega ) . \\end{aligned} \\end{align*}"}
+{"id": "4474.png", "formula": "\\begin{align*} G ( t ) = \\int _ { \\{ 2 \\psi < - t \\} } | F _ 0 | ^ 2 \\tilde \\rho \\end{align*}"}
+{"id": "8316.png", "formula": "\\begin{align*} \\epsilon ( Y ) + Y + \\alpha - \\epsilon ^ { * } ( \\alpha ) = ( X + \\beta ) + ( \\delta + \\Pi ^ { \\sharp } ( \\delta ) ) , \\end{align*}"}
+{"id": "859.png", "formula": "\\begin{align*} \\begin{cases} \\dot { x } = \\cos ( \\theta ( t ) ) + w ^ 1 . \\\\ \\dot { y } = \\sin ( \\theta ( t ) ) + w ^ 2 . \\end{cases} \\end{align*}"}
+{"id": "9178.png", "formula": "\\begin{align*} f ( \\{ x _ { i , r } \\} _ { i \\in I } ^ { 1 \\leq r \\leq k _ { i } } ) = 0 \\ \\ x _ { i , s _ { 1 } } = x _ { i , s _ { 2 } } + d _ { i } \\hbar = \\cdots = x _ { i , s _ { 1 - a _ { i j } } } - d _ { i } a _ { i j } \\hbar = x _ { j , r } - \\frac { d _ { i } a _ { i j } } { 2 } \\hbar \\end{align*}"}
+{"id": "52.png", "formula": "\\begin{align*} ( \\mathcal { F } H \\mathcal { F } ^ * - z ) ^ { - 1 } ( \\xi ) = \\frac { 1 } { R _ z ( \\xi ) } \\mathcal { F } H \\mathcal { F } ^ * + \\frac { z } { R _ z ( \\xi ) } \\end{align*}"}
+{"id": "449.png", "formula": "\\begin{align*} \\Delta ( \\Omega ^ q _ - ) : = \\left \\{ \\begin{aligned} & \\Delta \\overline { t } \\mbox { s a t i s f y i n g } x _ { B ^ q } ( t _ q + \\Delta \\overline { t } ) = \\max _ { ( i , j ) \\in \\Omega ^ q _ - \\cap { \\cal A } ^ q _ < } x ^ * _ i ( t _ q ) , \\mbox { i f } \\ ; \\Omega ^ q _ - \\cap { \\cal A } ^ q _ < \\neq \\emptyset , \\\\ [ 2 p t ] & - \\infty , \\mbox { o t h e r w i s e . } \\end{aligned} \\right . \\end{align*}"}
+{"id": "6303.png", "formula": "\\begin{align*} f ( \\rho , T ) : = \\lim _ { \\substack { N \\to \\infty \\\\ N L ^ { - 3 } \\to \\rho } } \\frac { F _ L ( N ) } { L ^ 3 } . \\end{align*}"}
+{"id": "59.png", "formula": "\\begin{align*} \\prescript { } { i _ 0 - 1 } { ( \\prescript { } { i _ 1 } { \\hat { s } } ) } ( i ) = & \\begin{cases} \\prescript { } { i _ 1 } { \\hat { s } } ( i ) & i < i _ 0 - 1 \\\\ \\prescript { } { i _ 1 } { \\hat { s } } ( i + 1 ) & i _ 0 - 1 \\leq i \\end{cases} \\end{align*}"}
+{"id": "1553.png", "formula": "\\begin{align*} | D F ( z ) | \\le C \\ , \\inf _ { | w | = 1 } | D F ( w ) | \\ , \\eta _ H ( | z | ) , \\end{align*}"}
+{"id": "1190.png", "formula": "\\begin{align*} \\left | \\left ( I - \\operatorname { T a y l } _ x ^ { | \\alpha | - | \\theta | } \\right ) \\partial ^ { \\theta } a ( x + h ) \\right | \\lesssim | h | ^ { | \\alpha | - | \\theta | + 1 } \\sup _ { \\beta \\in \\mathbb { Z } _ + ^ n , \\ , | \\beta | = | \\alpha | - | \\theta | + 1 } \\left \\| \\partial ^ { \\beta + \\theta } a \\right \\| _ { L ^ \\infty } \\leq | h | ^ { | \\alpha | - | \\theta | + 1 } . \\end{align*}"}
+{"id": "8902.png", "formula": "\\begin{align*} \\frac { \\Phi _ n ^ { ( k ) } ( 1 ) } { \\Phi _ n ( 1 ) } = F _ k \\left ( \\frac { \\phi ( n ) } { 2 } , \\frac { J _ 2 ( n ) } { 4 } , \\dots , \\frac { J _ k ( n ) } { 2 k } \\right ) . \\end{align*}"}
+{"id": "3836.png", "formula": "\\begin{align*} \\int _ { - \\frac { 1 } { k ( k + k _ 1 ) } } ^ { - \\frac { 1 } { k ( k + N ) } } = \\sum _ { \\ell = k + k _ 1 } ^ { k + N - 1 } \\int _ { - \\frac { 1 } { k \\ell } } ^ { - \\frac { 1 } { k ( \\ell + 1 ) } } . \\end{align*}"}
+{"id": "3080.png", "formula": "\\begin{align*} P _ k \\left ( a _ { \\frac { \\beta _ j } { e _ j } } , \\ldots , a _ { k _ { i j } - 1 } \\right ) + r _ k \\cdot a _ { \\frac { \\beta _ j } { e _ j } } ^ { \\gamma _ { 1 j } } \\cdot a _ { k _ { i j } } = 0 \\end{align*}"}
+{"id": "8008.png", "formula": "\\begin{align*} \\begin{cases} & \\frac { - \\frac { d } { d t } w _ { t , 1 } ( u ) } { w _ { t , 1 } ( u ) \\int _ { \\mathbb { T } } \\lambda ( u , v ) w _ { t , 3 } ( v ) d v } = \\exp \\left ( - F _ W ( t , u ) + G _ W ( t , u ) \\right ) , \\\\ & \\frac { - \\frac { d } { d t } \\sum _ { k = 1 } ^ 2 w _ { t , k } ( u ) } { \\psi ( u ) w _ { t , 2 } ( u ) } = \\exp \\left ( - G _ W ( t , u ) + H _ W ( t , u ) \\right ) , \\\\ & \\frac { - \\frac { d } { d t } \\sum _ { k = 1 } ^ 3 w _ { t , k } ( u ) } { \\phi ( u ) w _ { t , 3 } ( u ) } = \\exp \\left ( - H _ W ( t , u ) \\right ) \\end{cases} \\end{align*}"}
+{"id": "1370.png", "formula": "\\begin{align*} \\mathcal { C } ^ 0 _ { \\rm { R B A } _ \\lambda } ( A , T ) = & \\ 0 , \\\\ \\mathcal { C } ^ 1 _ { \\rm { R B A } _ \\lambda } ( A , T ) = & \\ \\mathcal { C } ^ 1 _ { \\rm { A l g } } ( A ) , \\\\ \\mathcal { C } ^ n _ { \\rm { R B A } _ \\lambda } ( A , T ) = & \\ \\mathcal { C } ^ n _ { \\rm { A l g } } ( A ) \\oplus \\mathcal { C } ^ { n - 1 } _ { \\rm { R B O } _ \\lambda } ( T ) , \\ , n \\geqslant 2 , \\end{align*}"}
+{"id": "6511.png", "formula": "\\begin{align*} \\tau \\left ( \\Omega \\right ) = \\frac { 1 } { n + 2 } \\int _ { \\partial K } h _ K ( N _ K ( x ) ) | \\nabla u ( x ) | ^ 2 d \\mathcal { H } ^ { n - 1 } ( x ) . \\end{align*}"}
+{"id": "7993.png", "formula": "\\begin{align*} 0 = \\int _ { \\Omega } \\partial _ { z _ i } \\tilde u \\partial _ { z _ i } \\tilde \\varphi + 4 | z | ^ 2 \\bigg ( \\partial _ { x _ i } \\tilde u \\partial _ { x _ i } \\tilde \\varphi + h b ^ { i j } \\partial _ { y _ i } \\tilde u \\partial _ { y _ j } \\tilde \\varphi + \\hat \\varepsilon ^ { - 2 } h ( | \\tilde u | ^ 2 - 1 ) \\tilde u \\tilde \\varphi \\bigg ) \\ , d z d x . \\end{align*}"}
+{"id": "1944.png", "formula": "\\begin{align*} B _ { i } = B _ { r _ i } ( x _ 0 ) , i = 0 , 1 , 2 , \\ldots . \\end{align*}"}
+{"id": "7214.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } p + \\partial _ { v } ( h p ) - a \\partial _ { v v } p = N ( t ) \\delta ( v - V _ R ) , v \\in ( - \\infty , V _ F ] , \\\\ p ( v , 0 ) = p ^ 0 ( v ) , p ( - \\infty , t ) = p ( V _ F , t ) = 0 , \\\\ \\end{cases} \\end{align*}"}
+{"id": "5818.png", "formula": "\\begin{align*} \\Vert E _ 2 ( u ) \\Vert _ { H ^ { \\ell - 1 } } ( t ) = o ( 1 ) \\Vert u \\Vert _ { H ^ { \\ell + 1 } } ( t ) \\ \\textup { a n d } \\ \\Vert E _ 2 ( u ) \\Vert _ { H ^ { \\ell } } ( t ) = o ( 1 ) \\Vert u \\Vert _ { H ^ { \\ell + 2 } } ( t ) . \\end{align*}"}
+{"id": "6702.png", "formula": "\\begin{align*} \\log _ p \\lvert \\Omega _ { \\{ 1 \\} } ( P ) \\rvert = \\dd ( U ) + s ( G ) = \\dim ( G ) + s ( G ) . \\end{align*}"}
+{"id": "2041.png", "formula": "\\begin{align*} \\mathcal { I } ^ n _ { d , e , m } \\ , \\ , = \\ , \\ , \\bigl ( \\ , I _ { \\rm m a x } ( P _ T ) \\ , : \\ , I _ { \\rm m a x } ( \\hat P _ T ) ^ \\infty \\ , \\bigr ) . \\end{align*}"}
+{"id": "2494.png", "formula": "\\begin{align*} \\langle \\mathcal { K } [ z ] \\rangle = 0 \\ , . \\end{align*}"}
+{"id": "486.png", "formula": "\\begin{align*} \\overline { \\alpha } _ { i , h } = \\prod _ { t = 1 } ^ h { \\alpha _ { i _ { - t } } } \\overline { \\beta } _ { i , h } = \\sum _ { t = 1 } ^ h { \\beta _ { i _ { - t } } \\prod _ { k = 1 } ^ { t - 1 } { \\alpha _ { i _ { - k } } } } . \\end{align*}"}
+{"id": "4320.png", "formula": "\\begin{gather*} \\sup _ { z _ i \\in \\mathcal { U } _ i - \\overline { u } _ i } f _ i ( x , z _ i ) = \\Delta u _ i x _ i \\end{gather*}"}
+{"id": "703.png", "formula": "\\begin{align*} f ( x ) = \\Phi ^ 6 - ( 1 2 x - 7 2 ) \\Phi ^ 5 - ( 4 0 0 x - 1 2 2 0 ) \\Phi ^ 4 - ( 4 5 1 2 x - 8 8 3 2 ) \\Phi ^ 3 - \\\\ ( 2 3 0 4 0 x - 3 1 5 3 6 ) \\Phi ^ 2 - ( 5 4 9 7 6 x - 5 4 6 5 6 ) \\Phi - 4 9 9 2 0 x + 3 6 5 4 4 - m \\end{align*}"}
+{"id": "2417.png", "formula": "\\begin{align*} & f _ { ( 0 , \\ldots , 0 , i _ { 2 ^ { m - 1 } } = s , 0 , \\ldots , 0 ) } ( t ) \\\\ = & ~ \\frac { ( s + 2 ) ! } { 2 } \\cdot ( 2 ^ m - 1 ) ! ^ 2 \\cdot ( 2 ^ { m - 1 } - 1 ) ! ^ { 2 s } \\cdot g ( t ) \\cdot \\binom { t + 2 ^ m - 1 } { 2 ^ m - 1 } ^ 2 \\binom { t + 2 ^ { m - 1 } - 1 } { 2 ^ { m - 1 } - 1 } ^ { s } \\binom { t + 2 ^ m - 1 } { 2 ^ { m - 1 } - 1 } ^ { s } . \\end{align*}"}
+{"id": "6051.png", "formula": "\\begin{align*} ( I : p ) _ l : = \\{ q \\in A _ l \\colon p q \\in I \\} \\subset A _ l \\end{align*}"}
+{"id": "8555.png", "formula": "\\begin{align*} f ( n ) = g _ { 1 } ( n ) + g _ { 2 } ( n ) f ( n + 1 ) \\end{align*}"}
+{"id": "737.png", "formula": "\\begin{align*} g _ 1 ( A ) & = t _ { r - 1 , q + 1 } - a \\\\ & = ( a - 1 ) b + a \\bigl ( v ( r - 1 ) - 1 \\bigr ) + \\dfrac { v a ( a - r ) J _ { k - 1 } ( v ) } { J _ k ( v ) } + \\bigl ( v a J _ { k - 1 } ( v ) + b J _ k ( v ) \\bigr ) \\ , . \\end{align*}"}
+{"id": "2312.png", "formula": "\\begin{align*} b ^ { c , \\gamma } = \\int ^ { 1 } _ { - 1 } \\left ( y [ ( U ^ { c , \\gamma } _ { \\theta } ) ' ] ^ { 2 } - \\frac { 2 - y ^ { 2 } } { 1 - y ^ { 2 } } U ^ { c , \\gamma } _ { \\theta } - \\frac { y } { 1 - y ^ { 2 } } ( U _ { \\theta } ^ { c , \\gamma } ) ^ { 2 } \\right ) \\rightarrow 0 , \\quad | ( c , \\gamma ) | \\rightarrow 0 . \\end{align*}"}
+{"id": "6228.png", "formula": "\\begin{align*} \\phi _ { 1 , \\beta } ( 0 ) = \\frac { 1 + \\beta } { 2 } , \\phi _ { \\beta , \\gamma } = \\frac { \\beta + \\gamma } { 2 } , \\phi _ { \\gamma , \\alpha } ( 0 ) = \\frac { \\gamma + \\alpha } { 2 } , \\phi _ { \\alpha , 0 } ( 0 ) = \\frac { \\alpha } { 2 } . \\end{align*}"}
+{"id": "7172.png", "formula": "\\begin{align*} ( 0 , 1 ) \\cdot H ( ( V , W ) , x , t ) & = \\varepsilon ( V - \\gamma S ) \\\\ & \\leq \\varepsilon ( L - \\gamma S ) . \\end{align*}"}
+{"id": "5724.png", "formula": "\\begin{align*} & \\xi _ { i , 1 } ( t ) : = G ( q ( t ) , \\psi _ { i , 1 } ) , \\ \\xi _ { i , 2 } ( t ) : = G ( q ( t ) , \\psi _ { i , 2 } ) \\ \\textup { f o r } \\ i \\in I _ 1 , \\\\ & \\xi _ { i , 3 } ( t ) : = G ( q ( t ) , \\psi _ { i , 3 } ) , \\ \\xi _ { i , 4 } ( t ) : = G ( q ( t ) , \\psi _ { i , 4 } ) \\ \\textup { f o r } \\ i \\in I _ 2 , \\\\ & \\xi ^ \\pm _ { i } ( t ) : = G ( q ( t ) , \\psi ^ \\pm _ { i } ) \\ \\textup { f o r } \\ i \\in I _ 3 \\cup I _ 4 . \\end{align*}"}
+{"id": "151.png", "formula": "\\begin{align*} \\rho _ k ( x _ 1 , \\ldots , x _ k ) = \\rho _ k ( x _ { \\sigma ( 1 ) } , \\ldots , x _ { \\sigma ( k ) } ) , \\end{align*}"}
+{"id": "8405.png", "formula": "\\begin{align*} & B _ 1 ( a ) B _ 1 ( b ) = B _ 1 ( a ) B _ 1 ( \\sigma ( b ) ) = B _ 1 ( B _ 1 ( a _ 1 ) \\sigma ( b ) S ( B _ 2 ( a _ 2 ) ) ) \\\\ = & B _ 1 ( B _ 1 ( a ) \\sigma ( b ) + \\sigma ( b ) S ( B _ 2 ( a ) ) ) = B _ 1 ( B _ 1 ( a ) \\sigma ( b ) - \\sigma ( b ) B _ 2 ( a ) ) . \\end{align*}"}
+{"id": "4924.png", "formula": "\\begin{align*} [ L ] ^ 3 = 4 \\beta _ { L } [ L ] \\end{align*}"}
+{"id": "5114.png", "formula": "\\begin{align*} S _ { T } ( p ) = - \\frac { \\sum _ { i } p ^ { t } _ { i } - \\sum _ { i } p _ { i } } { t - 1 } = - \\frac { \\sum _ { i } p ^ { t } _ { i } - 1 } { t - 1 } = - \\sum _ { i } p _ { i } \\left ( \\frac { p ^ { t - 1 } _ { i } - 1 } { t - 1 } \\right ) \\end{align*}"}
+{"id": "544.png", "formula": "\\begin{align*} u _ { 0 } ( x ) = \\sum _ { k = 1 } ^ { \\infty } a _ { k } \\Phi _ { k } ( x ) . \\end{align*}"}
+{"id": "6363.png", "formula": "\\begin{align*} \\phi '' ( r ) & = ( n - 1 ) \\sinh ^ { n - 2 } ( r ) \\cosh ^ d ( r ) + ( d - 1 ) \\sinh ^ n ( r ) \\cosh ^ { d - 2 } ( r ) \\\\ & = ( n - 1 ) \\coth ( r ) \\phi ' ( r ) + ( d - 1 ) \\tanh ( r ) \\phi ' ( r ) , \\end{align*}"}
+{"id": "4725.png", "formula": "\\begin{align*} S \\ ; = \\ ; S _ { \\mathrm { f r e e } } + S _ { \\mathrm { i n t } } \\ , , \\end{align*}"}
+{"id": "684.png", "formula": "\\begin{align*} ( - \\nabla \\cdot A \\nabla ) ^ { - s } f : = \\mathcal { A } _ { s } ^ { A } f \\end{align*}"}
+{"id": "355.png", "formula": "\\begin{align*} \\begin{aligned} A _ p ( z , y ) & = \\sum _ { i = 0 } ^ { k - 1 } ( z y ) ^ { i } ( z ^ { 2 ( k - i ) } + y ^ { 2 ( k - i ) } ) , \\\\ D _ p ( z , y ) & = \\sum _ { i = 0 } ^ { k - 1 } ( z y ) ^ { i } ( z ^ { k - i } - y ^ { k - i } ) ^ 2 \\end{aligned} \\end{align*}"}
+{"id": "2752.png", "formula": "\\begin{align*} \\tilde { \\sigma } _ { 1 } ( t ) : & T ^ { * } M \\times \\mathbb { R } \\rightarrow T ^ { * } M \\times \\mathbb { R } \\\\ & ; ( { \\sigma _ { 1 } ^ { * } } ( t ) \\Xi ^ { \\alpha } , { \\sigma _ { 1 } ^ { * } } ( t ) \\Psi _ { \\alpha } : = \\epsilon _ { \\alpha } , { \\sigma _ { 1 } ^ { * } } ( t ) Q ^ { i } , { \\sigma _ { 1 } ^ { * } } ( t ) P _ { i } , { \\sigma _ { 1 } ^ { * } } ( t ) u = t ) \\mapsto ( \\Xi ^ { \\alpha } , \\Psi _ { \\alpha } , Q ^ { i } , P _ { i } , u ) . \\end{align*}"}
+{"id": "6208.png", "formula": "\\begin{align*} \\alpha = \\frac 2 3 - \\frac \\omega 3 \\hbox { a n d } \\beta = \\frac 2 3 + \\frac \\omega 3 , \\hbox { f o r } \\omega : = \\sqrt { \\frac { D _ i - 4 D _ g } { D _ i - D _ g } } . \\end{align*}"}
+{"id": "458.png", "formula": "\\begin{align*} \\Delta ( \\Omega ^ q _ - ) : = \\left \\{ \\begin{aligned} & \\Delta \\overline { t } \\mbox { s a t i s f y i n g } x _ { B ^ q } ( t _ q + \\Delta \\overline { t } ) = \\min _ { ( i , j ) \\in \\Omega ^ q _ - \\cap { \\cal A } ^ q _ > } x ^ * _ i ( t _ q ) , \\mbox { i f } \\ ; \\Omega ^ q _ - \\cap { \\cal A } ^ q _ > \\neq \\emptyset , \\\\ [ 2 p t ] & + \\infty , \\mbox { o t h e r w i s e . } \\end{aligned} \\right . \\end{align*}"}
+{"id": "5467.png", "formula": "\\begin{align*} \\dot { x } & = v \\left ( \\cos { \\psi } - \\sin { \\psi } \\tan { \\beta } \\right ) , \\\\ \\dot { y } & = v \\left ( \\sin { \\psi } + \\cos { \\psi } \\tan { \\beta } \\right ) , \\\\ \\dot { \\psi } & = \\frac { v } { l _ r } \\tan { \\beta } , \\\\ \\dot { \\beta } & = \\omega , \\\\ \\dot { v } & = a , \\end{align*}"}
+{"id": "3442.png", "formula": "\\begin{align*} \\tilde { H } _ G ^ \\ast ( \\mathcal { X } ) = [ \\tilde { H } _ G ^ \\ast ( X , m , n ) ] , \\end{align*}"}
+{"id": "4638.png", "formula": "\\begin{align*} h _ x : = f ( x ' ) - x _ d . \\end{align*}"}
+{"id": "8523.png", "formula": "\\begin{align*} \\tau ( \\chi ^ { - 1 } ) \\cdot \\frac { L ( f , \\chi , 1 ) } { \\Omega _ f ^ + } = - \\sum _ { a \\in ( \\Z / m \\Z ) ^ { \\times } } \\chi ( a ) ^ { - 1 } \\cdot \\varphi _ f ( \\{ 0 , \\frac { a } { m } \\} ) \\end{align*}"}
+{"id": "8229.png", "formula": "\\begin{align*} & \\mathbb { E } [ d ( { \\bf S } _ i , \\hat { \\bf S } ) ] \\\\ & = \\sum _ { { \\bf s } _ i } P _ { \\bf S } ( { \\bf s } _ i ) \\left ( - \\frac { 1 } { M } \\sum _ { j = 1 } ^ M s _ { i , j } \\log P ( \\hat { s } _ { i , j } = s _ { i , j } ) \\right ) \\\\ & = \\sum _ { { \\bf s } _ i } P _ { \\bf S } ( { \\bf s } _ i ) \\left ( - \\frac { 1 } { M } \\sum _ { j = 1 } ^ M s _ { i , j } \\log ( \\sum _ { y ^ i } P ( \\hat { s } _ { i , j } = s _ { i , j } | y ^ i ) P _ { Y ^ i } ( y ^ i ) ) \\right ) . \\end{align*}"}
+{"id": "6982.png", "formula": "\\begin{align*} 0 = z _ j \\eta ( z _ j \\overline { \\eta } + \\overline { z _ j } \\eta ) = z _ j ^ 2 \\lvert \\eta \\rvert ^ 2 + \\lvert z _ j \\rvert ^ 2 \\eta ^ 2 \\ ; \\mathrm { f o r } \\ ; 1 \\leq j \\leq n . \\end{align*}"}
+{"id": "3793.png", "formula": "\\begin{align*} \\left \\| W Z ^ { \\top } \\right \\| \\leq 4 \\sigma _ { w , i } \\sqrt { N _ { i } ( 2 n _ x + n _ u ) \\log \\frac { 9 | \\widehat { \\mathcal { C } } _ 1 | T } { \\delta } } \\sum _ { t = 0 } ^ { T - 1 } \\left \\| ( \\Sigma _ { t } ^ { ( i ) } ) ^ { \\frac { 1 } { 2 } } \\right \\| . \\end{align*}"}
+{"id": "3443.png", "formula": "\\begin{align*} d ( X , m , n ) : = d ( X ) - m - n \\in \\Q . \\end{align*}"}
+{"id": "6817.png", "formula": "\\begin{align*} \\sum _ { i , j } \\beta _ { i j } l _ { i j } ^ { N - 1 } + L _ { N - 1 } = 0 . \\end{align*}"}
+{"id": "8561.png", "formula": "\\begin{align*} F ( n , k ) = \\frac { ( a ) _ k ( b ) _ k } { ( k + n ) ( n ) _ k ^ 2 } , \\end{align*}"}
+{"id": "80.png", "formula": "\\begin{align*} \\begin{aligned} \\begin{cases} w _ 1 = p - \\gamma , & w _ 2 = 2 \\\\ w _ 3 = 4 - p + \\gamma , & w _ 4 = 2 , \\end{cases} \\end{aligned} \\end{align*}"}
+{"id": "4796.png", "formula": "\\begin{align*} F _ \\theta ^ o ( x ) = \\frac { \\vert x \\vert ^ 2 } { \\sqrt { \\cos ^ 2 \\theta \\left < x , E _ n \\right > ^ 2 + \\sin ^ 2 \\theta \\vert x \\vert ^ 2 } - \\cos \\theta \\left < x , E _ n \\right > } . \\end{align*}"}
+{"id": "5841.png", "formula": "\\begin{align*} \\left ( x y z + ( 1 - \\alpha ^ 3 ) ^ { - 1 } x ^ 3 \\right ) ^ { | \\alpha ^ 3 | } = \\alpha ^ { - \\frac { ( | \\alpha ^ 3 | - 1 ) | \\alpha ^ 3 | } { 2 } } x ^ { | \\alpha ^ 3 | } y ^ { | \\alpha ^ 3 | } z ^ { | \\alpha ^ 3 | } + ( 1 - \\alpha ^ 3 ) ^ { - | \\alpha ^ 3 | } x ^ { 3 | \\alpha ^ 3 | } . \\end{align*}"}
+{"id": "1101.png", "formula": "\\begin{align*} & \\int _ 0 ^ r t ^ { a + n - 1 } [ \\log ( 2 + t ) ] ^ b \\ , d t \\\\ & \\quad = \\int _ 0 ^ N t ^ { a + n - 1 } [ \\log ( 2 + t ) ] ^ b \\ , d t + \\int _ N ^ r \\cdots \\\\ & \\quad < \\int _ 0 ^ N t ^ { a + n - 1 } [ \\log ( 2 + t ) ] ^ b \\ , d t + \\frac { 2 } { a + n } r ^ { a + n } [ \\log ( 2 + r ) ] ^ b , \\end{align*}"}
+{"id": "5937.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { C } ( G ) ) = \\dfrac { 2 ^ { 6 k } - 6 \\cdot 2 ^ { 5 k } + 1 4 \\cdot 2 ^ { 4 k } - 9 \\cdot 2 ^ { 3 k } - 1 5 \\cdot 2 ^ { 2 k } + 1 5 \\cdot 2 ^ { k } + 8 } { 2 } , \\end{align*}"}
+{"id": "1563.png", "formula": "\\begin{align*} \\| u _ { n } \\| _ { L ^ { p } ( B ) } & \\le \\| u _ { n } - \\psi _ n \\| _ { L ^ { p } ( B ) } + \\| \\psi _ n \\| _ { L ^ { p } ( B ) } \\\\ & \\le C \\ , \\big ( \\| D ( u _ { n } - \\psi _ n ) \\| _ { L ^ { p } ( B ) } + \\| \\psi _ n \\| _ { L ^ { p } ( B ) } \\big ) \\\\ & \\le C \\ , \\big ( \\| D u _ { n } \\| _ { L ^ { p } ( B ) } + \\| \\psi _ n \\| _ { W ^ { 1 , p } ( B ) } \\big ) , \\end{align*}"}
+{"id": "3510.png", "formula": "\\begin{align*} & \\| \\varphi _ n \\big ( \\rho _ { n , n - 1 } ( 1 _ { F _ { n - 1 } } ) \\psi _ n ( a _ k ) \\big ) - \\varphi _ n ( \\rho _ { n , n - 1 } ( 1 _ { F _ { n - 1 } } ) ) \\ \\varphi _ n ( \\psi _ n ( a _ k ) ) \\| \\\\ & < \\big ( 3 \\varepsilon _ n ^ 2 / 3 \\big ) ^ { 1 / 2 } \\\\ & = \\varepsilon _ n . \\end{align*}"}
+{"id": "3234.png", "formula": "\\begin{align*} \\overline { K ^ { \\alpha \\gamma } | _ { \\beta \\delta } ( x , y ) } & = K ^ { \\beta \\delta } | _ { \\alpha \\gamma } ( y , x ) \\\\ \\overline { K ^ \\alpha _ { \\ ; \\ ; \\ , \\gamma } | _ \\beta ^ { \\ ; \\ ; \\ , \\delta } ( x , y ) } & = K ^ \\beta _ { \\ ; \\ ; \\ , \\delta } | _ \\alpha ^ { \\ ; \\ ; \\ , \\gamma } ( y , x ) \\ : . \\end{align*}"}
+{"id": "5367.png", "formula": "\\begin{align*} \\nu _ j = \\max \\ , \\left \\{ \\frac { - v ^ { S } _ j } { b ^ S _ j } : j \\in S \\in 2 ^ N \\right \\} , j \\in N , \\end{align*}"}
+{"id": "5815.png", "formula": "\\begin{align*} X _ + ( t ) + X _ 0 ( t ) = o ( 1 ) X _ - ( t ) . \\end{align*}"}
+{"id": "2232.png", "formula": "\\begin{align*} I _ 1 = R \\intop _ 0 ^ t \\intop _ { - a } ^ a f _ \\varphi v _ { \\varphi , r } \\bigg | _ { r = R } d z d t ' = \\intop _ 0 ^ t \\intop _ { - a } ^ a f _ \\varphi \\bigg ( u _ { , r } - { 1 \\over R } u \\bigg ) \\bigg | _ { r = R } d z d t ' . \\end{align*}"}
+{"id": "1259.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ; q ^ 2 ) _ k ( - 1 ; q ^ 2 ) _ k } { ( q ^ 2 ; q ^ 2 ) _ k } q ^ { ( n - 1 ) ^ 2 - k ^ 2 } \\equiv - q ^ { 1 - n } + \\frac { ( n - 3 ) q ( 1 - q ^ n ) } { 2 } \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "103.png", "formula": "\\begin{align*} a = - \\frac { 1 } { 4 } ( G - P ) ^ 2 , b = P \\cdot E + \\frac { 1 } { 2 } ( G - P ) \\Big ( \\frac { G } { n - 1 } + K - \\frac { ( n - 2 ) P } { n - 1 } \\Big ) \\end{align*}"}
+{"id": "3921.png", "formula": "\\begin{align*} \\binom { N } { k } \\cos ^ { N - k } z \\sin ^ k z \\approx \\frac { v ^ k } { k ! } . \\end{align*}"}
+{"id": "4317.png", "formula": "\\begin{gather*} \\bar { A } = A \\cup \\{ ( 0 , j ) : \\ j \\in N \\} \\cup \\{ ( i , n + 1 ) : \\ i \\in N \\cup \\{ 0 \\} \\} . \\end{gather*}"}
+{"id": "2185.png", "formula": "\\begin{align*} x _ 1 = r \\cos \\varphi , x _ 2 = r \\sin \\varphi , x _ 3 = z . \\end{align*}"}
+{"id": "6587.png", "formula": "\\begin{align*} \\phi _ { f } ( x , y ) = \\frac { 1 } { 2 \\pi i } \\int _ { b - i U } ^ { b + i U ^ { \\prime } } A _ z ( x ) \\frac { y ^ z } { z } d z + O \\left ( y ^ { b } \\sum _ { n \\leq x } f ( n ) ^ { - b } \\left ( \\frac { 1 } { U } + \\frac { 1 } { U ^ { \\prime } } \\right ) \\right ) . \\end{align*}"}
+{"id": "6178.png", "formula": "\\begin{align*} \\begin{aligned} & f _ 1 ( x _ 1 ) - f _ 1 ( x ^ { k + 1 } _ 1 ) + ( x _ 1 - x _ 1 ^ { k + 1 } ) ^ T [ - A _ 1 ^ T \\lambda ^ k \\\\ & + \\beta ^ k A _ 1 ^ T ( A _ 1 { x } _ 1 ^ { k + 1 } + A _ 2 \\hat { x } _ 2 ^ k - b ) ] \\geq 0 , ~ \\forall x _ 1 . \\end{aligned} \\end{align*}"}
+{"id": "1478.png", "formula": "\\begin{align*} \\overline { D } _ G ( S ) = 1 - \\underline { D } _ { G } \\left ( \\bigcup _ { i \\in I } E _ i M \\setminus S _ i \\right ) \\leq d , \\end{align*}"}
+{"id": "802.png", "formula": "\\begin{align*} { \\cal R } i c & = \\ell ^ i \\ell ^ k R i c _ { i k } . \\end{align*}"}
+{"id": "3933.png", "formula": "\\begin{align*} A : = & \\{ - \\lambda _ 1 D + Q , \\ldots , - \\lambda _ N D + Q \\} , \\end{align*}"}
+{"id": "8675.png", "formula": "\\begin{align*} \\Omega ^ { 0 , q } ( X ) ^ G : = \\{ u \\in \\Omega ^ { 0 , q } ( X ) ; \\ , g ^ * u = u , \\ \\ \\forall g \\in G \\} . \\end{align*}"}
+{"id": "420.png", "formula": "\\begin{align*} \\frac { \\gamma _ j } { \\zeta _ j } \\le \\frac { \\log ( 1 - U _ 2 ) } { U _ 2 } : = U _ 3 . \\end{align*}"}
+{"id": "1428.png", "formula": "\\begin{align*} \\tilde m _ 1 h _ 1 & = \\Big ( \\tilde m _ 1 - \\widetilde { \\left ( \\frac { m _ 1 } { q } \\right ) } \\tilde q \\Big ) h _ 1 + \\widetilde { \\left ( \\frac { m _ 1 } { q } \\right ) } ( \\tilde q h _ 1 ) \\\\ & = \\Big ( \\tilde m _ 1 - \\widetilde { \\left ( \\frac { m _ 1 } { q } \\right ) } \\tilde q \\Big ) h _ 1 + \\sum _ { j = 1 } ^ s c _ j ^ \\prime \\widetilde { \\left ( \\frac { m _ 1 } { q } \\right ) } \\tilde m _ j ^ \\prime h _ j ^ \\prime . \\end{align*}"}
+{"id": "5212.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } D _ { \\alpha \\beta } I ( p \\| q ) } { \\partial q _ { j } } = T \\ ; \\left [ \\left ( \\frac { a - 1 } { a - b } \\right ) ( X . Y ) ^ { a - 2 } - \\left ( \\frac { b - 1 } { a - b } \\right ) ( X . Y ) ^ { b - 2 } \\right ] \\frac { \\partial ( X . Y ) } { \\partial q _ { j } } \\end{align*}"}
+{"id": "2415.png", "formula": "\\begin{align*} f _ { ( i _ 1 , \\ldots , i _ n , j ) } ( t ) = & ~ s ! \\binom { s + 2 } { i _ 1 } \\cdots \\binom { s + 2 } { i _ n } \\cdot \\frac { g ^ { ( j ) } ( t ) } { j ! } \\cdot n ! ^ { 2 + j } \\binom { t + n } { n } ^ { 2 + j } \\\\ & \\qquad \\times \\prod _ { k = 1 } ^ { n } \\left ( ( k - 1 ) ! ( n - k ) ! \\binom { t + k - 1 } { k - 1 } \\binom { t + n } { n - k } \\right ) ^ { i _ k } . \\end{align*}"}
+{"id": "8748.png", "formula": "\\begin{align*} f ( x _ + ) - f ( x _ - ) & = \\frac { z - m } { z - y } \\left ( \\varphi ( x _ + - y ) - \\varphi ( x _ - - y ) - \\varphi ( x _ + - m ) + \\varphi ( x _ - - m ) \\right ) \\\\ & + \\frac { m - y } { z - y } \\left ( \\varphi ( z - x _ + ) - \\varphi ( z - x _ - ) - \\varphi ( x _ + - m ) + \\varphi ( x _ - - m ) \\right ) . \\end{align*}"}
+{"id": "8255.png", "formula": "\\begin{align*} d ( \\underline s , \\hat { \\underline s } ) = \\begin{cases} 0 , & \\underline s = \\hat { \\underline s } \\\\ 2 , & \\underline s , \\hat { \\underline s } \\in \\mathcal S \\underline s \\neq \\hat { \\underline s } . \\end{cases} \\end{align*}"}
+{"id": "8095.png", "formula": "\\begin{align*} \\tilde { \\psi } = \\psi \\cdot \\psi \\circ g _ { - r _ 0 } . \\end{align*}"}
+{"id": "8474.png", "formula": "\\begin{align*} \\mathbf { I I } _ { h , \\ell } = \\left \\| ( \\bar { u } _ h ) _ + ^ { ( \\ell ) } - ( u _ h ) _ + ^ { ( \\ell ) } \\right \\| _ { L ^ 1 ( K _ T ) } \\leq h \\left \\| \\partial _ t ( u _ h ) _ + ^ { ( \\ell ) } \\right \\| _ { L ^ 1 ( K _ T ) } . \\end{align*}"}
+{"id": "6409.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } v _ t ( u ) = - R ( v _ t ( u ) ) , v _ 0 ( u ) = u . \\end{align*}"}
+{"id": "8223.png", "formula": "\\begin{align*} J ( { \\bf a } _ { i } ) & = \\max _ { P _ { { \\bf X } _ i | { \\bf A } _ { i - 1 } } ( { \\bf x } _ i | y ^ { i - 1 } ) } \\{ \\xi ( { \\bf a } _ { i } , \\{ P _ { { \\bf X } _ { i + 1 } | { \\bf A } _ { i } } ( { \\bf x } _ { i + 1 } | { \\bf a } _ { i } ) \\} , P _ { \\bf S } ) \\\\ & + \\mathbb { E } \\left [ J \\left ( { \\bf a } _ { i - 1 } \\right ) \\right ] \\} . \\end{align*}"}
+{"id": "2132.png", "formula": "\\begin{align*} \\lambda _ J ( x , y ) = A _ J ( x ) ^ { - T } \\nabla _ z f ( x , y ) . \\end{align*}"}
+{"id": "8296.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\delta \\rho \\ln w } { 3 2 } & \\leq \\frac { 5 r ^ 2 M ^ 2 \\ln w } { \\rho w ^ 2 } + 2 r ^ 2 + 2 M ^ 2 + \\frac { 4 M r } { w } ; \\end{aligned} \\end{align*}"}
+{"id": "2213.png", "formula": "\\begin{align*} \\psi _ 1 ^ { ( i ) } = \\psi _ 1 \\zeta ^ { ( i ) } , \\omega _ 1 ^ { ( i ) } = \\omega _ 1 \\zeta ^ { ( i ) } , \\ \\ i = 1 , 2 . \\end{align*}"}
+{"id": "1836.png", "formula": "\\begin{align*} \\varphi _ { \\tau ( 1 ) } ( \\tilde { \\gamma } _ 1 ( 0 ) ) = x , \\varphi _ { \\tau ( i ) } ( \\tilde { \\gamma } _ i ( t _ { i - 1 } ) ) = \\varphi _ { \\tau ( i - 1 ) } ( \\tilde { \\gamma } _ { i - 1 } ( t _ { i - 1 } ) ) , \\varphi _ { \\tau ( n ) } ( \\tilde { \\gamma } _ n ( 1 ) ) = y . \\end{align*}"}
+{"id": "8470.png", "formula": "\\begin{align*} \\mathbf { I } _ { h , \\ell } & \\leq \\left ( h \\sum _ { m = 1 } ^ N \\left \\| ( u _ m ) _ + - ( u _ m ) _ + ^ { ( \\ell ) } \\right \\| _ { L ^ \\gamma ( K ) } ^ \\gamma \\right ) ^ { \\frac { 1 } { \\gamma } } + \\left ( h \\sum _ { m = 1 } ^ N \\left \\| ( u _ { m - 1 } ) _ + - ( u _ { m - 1 } ) _ + ^ { ( \\ell ) } \\right \\| _ { L ^ \\gamma ( K ) } ^ \\gamma \\right ) ^ { \\frac { 1 } { \\gamma } } \\\\ [ 2 m m ] & = : I _ 1 + I _ 2 . \\end{align*}"}
+{"id": "2718.png", "formula": "\\begin{align*} & X _ { t } : = \\eta ^ { i } \\frac { \\partial } { \\partial v ^ { i } } + v ^ { i } \\frac { \\partial } { \\partial q ^ { i } } + \\frac { \\partial } { \\partial t } , \\\\ & Z _ { \\alpha } : = - \\tau ^ { i } _ { \\alpha } \\frac { \\partial } { \\partial v ^ { i } } . \\end{align*}"}
+{"id": "5556.png", "formula": "\\begin{align*} \\ell = \\left \\lfloor c _ 1 \\log _ d ( n ) \\right \\rfloor , c _ 1 = \\frac { ( k / 2 - 1 ) \\wedge 1 } { 2 5 } . \\end{align*}"}
+{"id": "47.png", "formula": "\\begin{align*} I ( i ) = & I ' ( i ) , i < l , \\\\ I ( l ) < & I ' ( l ) . \\end{align*}"}
+{"id": "8828.png", "formula": "\\begin{align*} v _ t = d \\Delta v + c v _ z + f ( z , v ) , z \\in \\mathbb { R } , \\ , \\ , t \\ge 0 . \\end{align*}"}
+{"id": "828.png", "formula": "\\begin{align*} { \\bf K } ( P , y ) & = \\frac { { { g _ y } ( { \\bf R } _ y ( \\nabla { \\Psi } } ) , \\nabla { \\Psi } ) } { { { g _ y ( y , y ) } { g _ y } ( \\nabla { \\Psi } } , \\nabla { \\Psi } ) - { g _ y } { { ( \\nabla { \\Psi } } , y ) } ^ 2 } . \\end{align*}"}
+{"id": "8999.png", "formula": "\\begin{align*} p ^ n ( t , x _ n , y _ n ) = \\sum _ { k = 0 } ^ { \\infty } \\mathbb { P } _ { \\lfloor \\sqrt { n } x \\rfloor } ( S _ k = \\lfloor \\sqrt { n } y \\rfloor ) \\frac { ( n t ) ^ { k } } { k ! } e ^ { - n t } , \\end{align*}"}
+{"id": "1175.png", "formula": "\\begin{align*} S _ \\varphi \\circ T = \\widehat T \\circ S _ { \\varphi } \\end{align*}"}
+{"id": "4559.png", "formula": "\\begin{align*} \\nabla _ { \\mathfrak { q } ^ * h } ^ 2 ( \\mathfrak { q } ^ * X - X _ \\infty ) & = \\nabla _ { \\mathfrak { q } ^ * h } ^ 2 \\mathfrak { q } ^ * X - \\nabla _ { h _ E } ^ 2 X _ \\infty + ( \\nabla _ { h _ E } ^ 2 - \\nabla _ { \\mathfrak { q } ^ * h } ^ 2 ) X _ \\infty \\\\ & = \\nabla _ { \\mathfrak { q } ^ * h } ^ 2 \\mathfrak { q } ^ * X - \\nabla _ { h _ E } ^ 2 X _ \\infty + O ' ( \\rho ^ { - \\tau - 1 } ) \\nabla _ { h _ E } X _ \\infty + O ' ( \\rho ^ { - \\tau - 2 } ) X _ \\infty \\\\ & = O ' ( \\rho ^ { - \\tau - 1 } ) . \\end{align*}"}
+{"id": "8872.png", "formula": "\\begin{align*} \\| e ^ { i t _ 2 \\mathcal { K } _ { \\lambda } } u ( t _ 2 ) - e ^ { i t _ 2 \\mathcal { K } _ { \\lambda } } u ( t _ 1 ) \\| _ { H _ { \\lambda } ^ 1 } = \\Big \\| \\int _ { t _ 1 } ^ { t _ 2 } e ^ { i s \\mathcal { K } _ { \\lambda } } \\mathcal { N } [ u ] \\Big \\| _ { H _ { \\lambda } ^ 1 } \\\\ \\lesssim \\| \\mathcal { N } [ u ] \\| _ { \\mathcal { W } _ { \\lambda } { ' } ( [ t _ 1 , t _ 2 ] ) } \\\\ \\lesssim \\| u \\| ^ { 2 p - 1 } _ { \\mathcal { W } _ { \\lambda } ( [ t _ 1 , t _ 2 ] ) } \\rightarrow 0 \\end{align*}"}
+{"id": "7599.png", "formula": "\\begin{align*} Q _ { 4 ^ * , q } ( \\psi ( t ) ) & = 2 N ( q - 2 ) E _ { 4 ^ * , q } ( \\psi _ { 0 } ) - ( N ( q - 2 ) - 8 ) \\| \\Delta \\psi ( t ) \\| _ { 2 } ^ { 2 } - \\frac { 2 N ( 4 ^ * - q ) } { 4 ^ * } \\| \\psi ( t ) \\| _ { 4 ^ * } ^ { 4 ^ * } \\\\ & \\leq \\rho ^ * \\frac { N ( q - 2 ) - 8 } { 2 } - 1 - \\frac { N ( q - 2 ) - 8 } { 2 } \\rho ^ * - \\frac { N ( q - 2 ) - 8 } { 2 } \\| \\Delta \\psi ( t ) \\| _ { 2 } ^ { 2 } . \\\\ \\end{align*}"}
+{"id": "968.png", "formula": "\\begin{align*} P _ { U _ n } ( w ) - P _ { D } ( g ) & = R ^ D f ( \\cdot , w ) - R ^ { U _ n } f ( \\cdot , w ) + R ^ D \\mu - R ^ { U _ n } \\mu \\\\ & = \\mathbb E _ \\cdot \\int _ { \\tau _ { U _ n } } ^ { \\tau _ D } f ( X _ s , w ( X _ s ) ) \\ , d s + \\mathbb E _ \\cdot ( A ^ \\mu _ { \\tau _ D } - A ^ \\mu _ { \\tau _ { U _ n } } ) \\quad \\end{align*}"}
+{"id": "4083.png", "formula": "\\begin{align*} \\chi R _ 0 ( z _ m ) \\chi = \\sum _ { j = 1 } ^ { \\frac { d - 1 } { 2 } } B _ j ( r , z _ 0 ) + i m \\Big ( \\frac { z _ 0 } { 2 \\pi } \\Big ) ^ { d - 1 } \\mathcal { E } ^ * _ { \\chi } ( \\bar { z _ 0 } ) \\mathcal { E } _ { \\chi } ( z _ 0 ) \\end{align*}"}
+{"id": "6962.png", "formula": "\\begin{align*} f ( A \\mathbf { z } ) & = 4 \\langle A \\mathbf { x } , A \\mathbf { y } \\rangle ^ 2 - 4 \\langle A \\mathbf { x } , A \\mathbf { x } \\rangle \\langle A \\mathbf { y } , A \\mathbf { y } \\rangle \\\\ & = 4 \\langle \\mathbf { x } , \\mathbf { y } \\rangle ^ 2 - 4 \\langle \\mathbf { x } , \\mathbf { x } \\rangle \\langle \\mathbf { y } , \\mathbf { y } \\rangle \\\\ & = f ( \\mathbf { z } ) , \\end{align*}"}
+{"id": "6763.png", "formula": "\\begin{align*} Q = \\begin{pmatrix} \\beta A _ 2 ^ T A _ 2 & - r A _ 2 ^ T \\\\ - A _ 2 & \\frac { 1 } { \\beta } I _ l \\end{pmatrix} ~ { \\rm a n d } ~ M = \\begin{pmatrix} I _ { n _ 2 } & 0 \\\\ - s \\beta A _ 2 & ( r + s ) I _ l \\end{pmatrix} . \\end{align*}"}
+{"id": "6760.png", "formula": "\\begin{align*} \\begin{aligned} & \\| v ^ { k + 1 } - v ^ * \\| ^ 2 _ { H } - \\| v ^ k - v ^ * \\| ^ 2 _ { H } \\\\ = & A ^ k - A ^ { k - 1 } + B ^ k - B ^ { k - 1 } + 2 ( C ^ k - C ^ { k - 1 } ) \\\\ = & A ^ k - A ^ { k - 1 } + 2 ( C ^ k - C ^ { k - 1 } ) + \\frac { 1 - \\tau ^ k } { ( \\tau ^ k ) ^ 2 } E ^ k - \\frac { 1 - \\tau ^ { k - 1 } } { ( \\tau ^ { k - 1 } ) ^ 2 } E ^ { k - 1 } \\\\ & + \\frac { 1 } { \\tau ^ k } ( D ^ k - D ^ { k - 1 } ) - \\frac { 1 } { \\tau ^ { k - 1 } } ( D ^ { k - 1 } - D ^ { k - 2 } ) + D ^ { k - 1 } - D ^ { k - 2 } . \\end{aligned} \\end{align*}"}
+{"id": "2611.png", "formula": "\\begin{align*} D _ 3 ^ 2 ( n ) = n + 2 . \\end{align*}"}
+{"id": "1118.png", "formula": "\\begin{align*} \\left \\langle \\varphi _ Q , \\psi _ R \\right \\rangle & = | Q | ^ { \\frac 1 2 } | R | ^ { \\frac 1 2 } \\int _ { \\mathbb { R } ^ n } \\varphi _ j ( x - x _ Q ) \\overline { \\psi _ i ( x - x _ R ) } \\ , d x \\\\ & = | Q | ^ { \\frac 1 2 } | R | ^ { \\frac 1 2 } \\int _ { \\mathbb { R } ^ n } \\varphi _ j ( x ) \\widetilde \\psi _ i ( x _ R - x _ Q - x ) \\ , d x = | Q | ^ { \\frac 1 2 } | R | ^ { \\frac 1 2 } \\left ( \\varphi _ j * \\widetilde \\psi _ i \\right ) ( x _ R - x _ Q ) , \\end{align*}"}
+{"id": "3188.png", "formula": "\\begin{align*} I ^ { ( 2 ) } _ n ( \\mu ) = \\int _ { \\frac { \\pi } { 6 n + 4 } } ^ { \\frac { \\pi } { 2 } } \\theta \\sin \\left ( \\mu \\theta \\right ) \\prod _ { k = 0 } ^ { n } \\cos ( ( 3 k + 1 ) \\theta ) \\cos ( ( 3 k + 2 ) \\theta ) \\mathrm { d } \\theta . \\end{align*}"}
+{"id": "5652.png", "formula": "\\begin{align*} C = \\{ g _ 1 ( x ) = 0 , \\dots , g _ \\ell ( x ) = 0 , g _ { \\ell + 1 } ( x ) > 0 , \\dots , g _ N ( x ) > 0 \\} \\end{align*}"}
+{"id": "5272.png", "formula": "\\begin{align*} x ^ { k + 1 } = x ^ { k } + \\alpha ^ { k } x ^ { k } \\left ( U ^ { k } - V ^ { k } \\right ) \\end{align*}"}
+{"id": "8170.png", "formula": "\\begin{align*} & P _ { Y ^ { j - 1 } , \\underline S } ( 0 ^ { j - 1 } , \\underline S ) H ( P _ { Y _ j | Y ^ { j - 1 } \\underline S } ( 1 | 0 ^ { j - 1 } , \\underline S ) ) \\\\ & = ( 1 - \\frac { \\sum _ { k = 1 } ^ { j - 1 } b _ k ^ 0 } { M } ) H ( \\frac { b _ j ^ 0 } { M - \\sum _ { k = 1 } ^ { j - 1 } b _ k ^ 0 } ) . \\end{align*}"}
+{"id": "5900.png", "formula": "\\begin{align*} 6 0 ( 4 n - 1 ) ^ { 3 } + 6 0 ( 3 n - 1 ) ^ { 3 } = 7 6 ( 4 n - 1 ) ^ { 3 } - 1 6 ( 4 n - 1 ) ^ { 3 } + 2 8 5 ( 3 n - 1 ) ^ { 3 } - 2 2 5 ( 3 n - 1 ) ^ { 3 } \\end{align*}"}
+{"id": "9133.png", "formula": "\\begin{align*} \\begin{aligned} & g _ { 1 } = ( v ^ { 6 } + 1 ) x _ { 1 , 1 } ^ { 2 } x _ { 1 , 2 } ^ { 2 } + ( v ^ { 6 } + 1 ) x _ { 1 , 1 } x _ { 1 , 2 } ( x _ { 2 , 1 } x _ { 2 , 2 } + x _ { 2 , 1 } x _ { 2 , 3 } + x _ { 2 , 2 } x _ { 2 , 3 } ) \\\\ & \\ \\ \\ \\ \\ \\ - v ^ { 3 } ( x _ { 1 , 1 } + x _ { 1 , 2 } ) ( x _ { 1 , 1 } x _ { 1 , 2 } x _ { 2 , 1 } + x _ { 1 , 1 } x _ { 1 , 2 } x _ { 2 , 2 } + x _ { 1 , 1 } x _ { 1 , 2 } x _ { 2 , 3 } + x _ { 2 , 1 } x _ { 2 , 2 } x _ { 2 , 3 } ) . \\end{aligned} \\end{align*}"}
+{"id": "6551.png", "formula": "\\begin{align*} T _ N - U _ N = \\frac { 1 } { 2 \\pi } \\sum ^ { N - 1 } _ { j = - \\infty } \\sum ^ { N - 1 } _ { n = 1 \\vee ( j + 1 ) } \\int _ { \\R } \\widehat { K } ( u ) \\phi ( u ) ( e ^ { \\iota u a _ { n - j } \\varepsilon _ { j } } - \\phi _ { \\varepsilon } ( a _ { n - j } u ) - \\iota u a _ { n - j } \\varepsilon _ { j } ) d u . \\end{align*}"}
+{"id": "5889.png", "formula": "\\begin{align*} \\dfrac { M _ { 2 } ( \\mathcal { N C } ( G ) ) } { | e ( \\mathcal { N C } ( G ) ) | } - \\dfrac { M _ { 1 } ( \\mathcal { N C } ( G ) ) } { | v ( \\mathcal { N C } ( G ) ) | } = \\dfrac { n ^ { 2 } ( m ^ { 3 } ( m - 6 ) + m ( 1 3 m - 1 2 ) + 4 ) ) } { 3 ( m - 1 ) ( 2 m - 1 ) } : = \\dfrac { n ^ 2 h ( m ) } { g ( m ) } . \\end{align*}"}
+{"id": "5123.png", "formula": "\\begin{align*} \\sum _ { i } \\frac { \\partial D ( p _ { i } \\| K q _ { i } ) } { \\partial K } = 0 \\end{align*}"}
+{"id": "3115.png", "formula": "\\begin{align*} \\varphi ^ * \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) : = \\left ( \\begin{array} { c c c } a ^ { \\ell _ 1 } & 0 & 0 \\\\ 0 & d ^ { - \\ell _ 2 } & 0 \\\\ 0 & 0 & d ^ { - \\ell _ 2 } \\end{array} \\right ) . \\end{align*}"}
+{"id": "591.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { j - 1 } \\Big | \\Big \\langle { \\frac { T ^ \\ast ( \\tilde h _ i ) } { \\| h _ i \\| _ { F ^ \\ast } } } , { \\frac { 1 _ { \\Delta _ j ^ \\ast } r ^ { \\gamma _ { j , m _ j } } _ { m _ j } } { \\| h _ j \\| _ F } } \\Big \\rangle \\Big | < { \\frac { \\eta _ j } { 2 } } . \\end{align*}"}
+{"id": "4851.png", "formula": "\\begin{align*} ( \\frac { \\epsilon m } { a _ { m _ * } } ) o _ m ( 1 ) & = ( \\frac { \\epsilon m } { \\bar { C } m ^ { \\frac { n - 1 } { n } } } ) o _ m ( 1 ) \\\\ & = \\frac { o _ m ( \\epsilon ) } { \\bar { C } } m ^ { 1 / n } ; \\end{align*}"}
+{"id": "44.png", "formula": "\\begin{align*} \\tilde d ^ * = ( - 1 ) ^ { n i + 1 } * \\tilde d * . \\end{align*}"}
+{"id": "4319.png", "formula": "\\begin{align*} \\inf _ { x \\in \\mathcal { X } } & \\sum _ { i \\in [ m ] } \\ \\sup _ { u ^ i \\in \\mathcal { U } _ i } f _ i ( x , u ^ i ) . \\end{align*}"}
+{"id": "5886.png", "formula": "\\begin{align*} ( m - 1 ) & ( 2 m - 1 ) ( m n - n - 1 ) ^ 3 + m ( 2 m - 1 ) ( n - 1 ) ^ 3 \\\\ & > ( m - 1 ) ^ 2 ( m n - n - 1 ) ^ 3 + m ^ 2 ( n - 1 ) ^ 2 + m ( m - 1 ) ( m n - n - 1 ) ( n - 1 ) ( m n - 2 ) \\\\ & = ( ( m - 1 ) ( m n - n - 1 ) + m ( n - 1 ) ) ( ( m - 1 ) ( m n - n - 1 ) ^ { 2 } + m ( n - 1 ) ^ { 2 } ) . \\end{align*}"}
+{"id": "1005.png", "formula": "\\begin{align*} Y _ t & = g ( X _ { \\tau _ D } ) + \\int ^ { \\tau _ D } _ { t \\wedge \\tau _ D } f ( X _ s , u ( X _ s ) ) \\ , d s \\\\ & \\quad + A ^ { \\mu } _ { \\tau _ D } - A ^ { \\mu } _ { t \\wedge \\tau _ D } - ( M ^ x _ { \\tau _ D } - M ^ x _ { t \\wedge \\tau _ D } ) , t \\ge 0 , P _ x \\mbox { a . s . } \\end{align*}"}
+{"id": "6925.png", "formula": "\\begin{align*} \\forall x \\in \\R , & H _ { 2 \\mu } ^ \\beta ( x ) : = \\frac { 1 } { 2 \\pi } \\int _ \\R e ^ { i x u } e ^ { - \\beta u ^ { 2 \\mu } } d u , \\\\ \\forall x \\in \\R , & E _ { 2 \\mu } ^ \\beta ( x ) : = \\int _ x ^ { + \\infty } H ^ \\beta _ { 2 \\mu } ( y ) d y . \\end{align*}"}
+{"id": "2856.png", "formula": "\\begin{align*} \\kappa ( q , n ) : = \\int _ { \\mathbb { S } ^ { n - 1 } } \\left | e \\cdot \\omega \\right | ^ q \\ , d \\omega = \\frac { 2 \\Gamma ( \\frac { q + 1 } { 2 } ) \\pi ^ { \\frac { n - 1 } { 2 } } } { \\Gamma ( \\frac { q + n } { 2 } ) } , \\end{align*}"}
+{"id": "6805.png", "formula": "\\begin{align*} d g _ { S U ( N ) } = C _ N \\cos ( \\psi _ { N - 1 } ) \\sin ^ { 2 ( N - 1 ) - 1 } ( \\psi _ { N - 1 } ) & \\left [ \\prod _ { j = 1 } ^ { N - 2 } \\cos ^ { 2 j - 1 } ( \\psi _ { j } ) \\sin ( \\psi _ j ) \\right ] \\cdot \\\\ & d \\phi _ 1 \\ldots d \\phi _ { N - 1 } d \\psi _ 1 \\ldots d \\psi _ { N - 1 } d g _ { S U ( N - 1 ) } d \\omega _ N , \\end{align*}"}
+{"id": "4115.png", "formula": "\\begin{align*} & Q ^ * ( \\theta ) \\psi ^ { ( r ) } ( \\theta ) + \\sum _ { i \\in \\mathcal { J } } ( \\mu ^ { r } _ i - \\lambda _ i ) \\bar { \\xi } _ i ( \\theta ) \\bigl ( \\psi ^ { ( r ) } ( \\theta ) - \\psi ^ { ( r ) } _ i ( \\theta ) \\bigr ) \\\\ & = - \\frac { 1 } { 2 } \\sum _ { i \\in \\mathcal { J } } ( \\mu ^ { ( r ) } _ i - \\lambda _ i ) { \\xi } ^ * _ i ( \\theta ) \\bigl ( \\psi ^ { ( r ) } ( \\theta ) - \\psi ^ { ( r ) } _ i ( \\theta ) \\bigr ) + o ( r ^ J \\abs { \\theta } ) + o ( \\abs { \\theta } ^ 2 ) , \\end{align*}"}
+{"id": "8217.png", "formula": "\\begin{align*} P _ { { \\bf X } _ i | Y ^ { i - 1 } } ( { \\bf x } _ i | y _ 1 ^ { t - 1 } , y _ t ^ { i - 1 } ) = P _ { { \\bf X } _ i | { \\bf A } _ { t - 1 } , Y _ t ^ { i - 1 } } ( { \\bf x } _ i | { \\bf a } _ { t - 1 } , y _ t ^ { i - 1 } ) \\end{align*}"}
+{"id": "6930.png", "formula": "\\begin{align*} l _ 1 = \\frac { a _ p } { \\alpha } \\pi ^ c ( 1 ) e = \\frac { a _ p } { \\alpha } \\begin{pmatrix} 1 & \\hdots & 1 \\end{pmatrix} ^ T . \\end{align*}"}
+{"id": "657.png", "formula": "\\begin{align*} \\langle f , g \\rangle : = f _ { i _ { 1 } \\cdots i _ { m } } g _ { i _ { 1 } \\cdots i _ { m } } . \\end{align*}"}
+{"id": "2118.png", "formula": "\\begin{align*} \\nu _ { \\Theta } ( t ) ( B ) : = \\Theta \\big ( \\{ ( \\phi , \\rho , w ) \\in \\mathcal { Z } : \\phi ( t ) \\in B \\} \\big ) , B \\in \\mathcal { B } ( \\overline { \\mathcal { D } } ) , t \\in [ 0 , T ] , \\end{align*}"}
+{"id": "5569.png", "formula": "\\begin{align*} \\tilde p _ t = 1 - ( 1 - p ) ^ { n _ t } = p n _ t + O ( p ^ 2 n ^ 2 ) = O ( n p ) . \\end{align*}"}
+{"id": "7694.png", "formula": "\\begin{align*} \\psi _ { L , N _ G , j } ^ { \\alpha } ( y ; t ) = \\left ( \\frac { L } { \\alpha - 1 } y ^ { \\frac { \\alpha } { 1 - \\alpha } } \\right ) ^ { N _ G + 1 } \\C { F } _ j ^ { ( N _ G + 2 ) } \\left ( t - L \\ , y ^ { \\frac { 1 } { 1 - \\alpha } } \\right ) . \\end{align*}"}
+{"id": "7144.png", "formula": "\\begin{align*} k ^ 2 \\widehat { A } _ \\mu ( k ) = \\eta ^ { \\alpha \\beta } k _ \\alpha k _ \\beta \\delta _ \\mu ^ { ~ \\nu } \\widehat { A } _ \\nu ( k ) = 0 \\ , \\ , . \\end{align*}"}
+{"id": "4227.png", "formula": "\\begin{align*} { \\widehat { A } ( T X , \\nabla ^ { T X } ) } { \\rm c h } ( \\triangle ( X ) ) { \\rm c h } ( \\Theta _ 1 ( T _ { C } X ) ) = \\prod _ { j = 1 } ^ { k } \\frac { 2 x _ j \\theta ' ( 0 , \\tau ) } { \\theta ( x _ j , \\tau ) } \\frac { \\theta _ 1 ( x _ j , \\tau ) } { \\theta _ 1 ( 0 , \\tau ) } , \\end{align*}"}
+{"id": "4330.png", "formula": "\\begin{gather*} \\mathcal { X } = \\left \\{ x \\in \\{ 0 , 1 \\} ^ { [ n ] ^ 2 } : \\sum _ { i \\in [ n ] } x _ { i , r } = 1 \\ \\forall r \\in [ n ] , \\sum _ { r \\in [ n ] } x _ { i , r } = 1 \\ \\forall i \\in [ n ] \\right \\} . \\end{gather*}"}
+{"id": "8630.png", "formula": "\\begin{align*} v _ 1 ( T _ 2 ; x ) \\geq m ( T _ 2 ) = m ( 0 ) q ^ { - 1 } ( T _ 2 ) > - C _ 1 q ^ { - 1 } ( T _ 2 ) . \\end{align*}"}
+{"id": "1091.png", "formula": "\\begin{align*} A = U \\operatorname { d i a g } \\ , ( \\lambda _ 1 , \\ldots , \\lambda _ m ) U ^ * . \\end{align*}"}
+{"id": "1693.png", "formula": "\\begin{align*} M = \\{ z \\in \\mathbb { C } ^ { n + 1 } \\ , | \\ , z _ 0 + \\overline { z _ 0 } = \\Phi ( z _ 1 , \\ldots , z _ n , \\overline { z _ 1 } , \\ldots , \\overline { z _ n } ) \\} \\end{align*}"}
+{"id": "5821.png", "formula": "\\begin{align*} ( g _ u ) _ { i j } = g _ { i j } + \\frac { \\partial u ^ A } { \\partial x ^ i } \\frac { \\partial u ^ B } { \\partial x ^ j } h _ { A B } + \\frac { \\partial u ^ A } { \\partial x ^ i } c _ { j A } + \\frac { \\partial u ^ B } { \\partial x ^ j } c _ { i B } . \\end{align*}"}
+{"id": "6340.png", "formula": "\\begin{align*} | \\nabla _ 1 g ( p , q ) | & = \\left | \\frac { 2 p ^ 2 p + q p } { \\sqrt { p ^ 4 + q p ^ 2 } } \\frac { 1 } { e ^ { \\sqrt { p ^ 4 + q p ^ 2 } } - 1 } \\right | \\leq \\frac { 2 \\sqrt { p ^ 4 + q p ^ 2 } } { | p | \\Big ( e ^ { \\sqrt { p ^ 4 + q p ^ 2 } } - 1 \\Big ) } \\leq 2 | p | ^ { - 1 } e ^ { - \\tfrac { p ^ 2 } { 2 } } , \\end{align*}"}
+{"id": "7336.png", "formula": "\\begin{align*} a _ n ( k ) = \\frac { 1 } { k ! } \\cdot \\prod _ { j = 0 } ^ { k - 1 } \\frac { n ^ 2 - j ^ 2 } { ( 2 j + 1 ) } \\ , \\ , \\ , \\ , \\ , \\end{align*}"}
+{"id": "2545.png", "formula": "\\begin{align*} \\| u \\| _ { K , k , m } = \\sup _ K | P _ k u | \\ ; . \\end{align*}"}
+{"id": "6525.png", "formula": "\\begin{align*} S _ N = \\sum ^ { N } _ { n = 1 } \\big [ K ( X _ n ) - \\mathbb { E } K ( X _ n ) \\big ] \\end{align*}"}
+{"id": "1435.png", "formula": "\\begin{align*} ( \\varepsilon _ * \\Omega _ { ( X , D ) _ { \\bullet } / Y } ) ^ n = \\bigoplus _ { k \\ge 0 } ( \\varepsilon _ k ) _ * \\Omega ^ { n - k } _ { ( X , D ) _ k / Y } \\end{align*}"}
+{"id": "3174.png", "formula": "\\begin{align*} \\tilde { \\Psi } _ \\lambda ( r ) : = \\Psi _ \\lambda ( r ) - \\Psi _ \\lambda ( 0 ) + \\lambda r , \\end{align*}"}
+{"id": "3495.png", "formula": "\\begin{align*} & \\| e \\rho _ m \\big ( \\rho _ { m , k } ( x ) ^ * \\rho _ { m , k } ( x ) \\big ) - e \\rho _ n \\big ( \\rho _ { n , k } ( x ) ^ * \\rho _ { n , k } ( x ) \\big ) \\| \\\\ & = \\| \\rho _ m \\big ( \\rho _ { m , k } ( x ) ^ * \\rho _ { m , k } ( x ) \\big ) - \\rho _ n \\big ( \\rho _ { n , k } ( x ) ^ * \\rho _ { n , k } ( x ) \\big ) \\| , \\end{align*}"}
+{"id": "7039.png", "formula": "\\begin{align*} ( \\mu + \\Lambda _ { C _ \\infty } ( b ) ) ^ { - 1 } | b _ m | h & = \\Theta _ p ( \\mu , b ) | b _ m | h \\\\ & = ( \\mu - \\Delta ) ^ { - \\frac { 1 } { 2 } - \\frac { 1 } { q } } G _ p ( q , b _ m ) | b _ m | ^ { \\frac { 2 } { p } } h \\\\ & - ( \\mu - \\Delta ) ^ { - \\frac { 1 } { 2 } - \\frac { 1 } { q } } Q _ { p } ( q , b ) ( 1 + T _ p ( b ) ) ^ { - 1 } T _ p ( b , b _ m ) | b _ m | ^ { \\frac { 2 } { p } } h . \\end{align*}"}
+{"id": "4543.png", "formula": "\\begin{align*} V _ { N , \\beta , a } ( [ s ] _ N , [ s ' ] _ N ) : = \\sum _ { i = 1 } ^ { N } \\frac { | s _ i - s _ i ' | \\wedge a } { ( 1 + \\beta ) ^ i } \\leq \\frac a \\beta , \\end{align*}"}
+{"id": "1777.png", "formula": "\\begin{align*} w ^ * = w + h ^ 0 + h ^ 1 + \\cdots , z _ j ^ * = z _ j + f _ { j } ^ k + f _ { j } ^ k + \\cdots , \\quad \\mbox { a n d } \\zeta ^ * = \\zeta + g ^ 0 + g ^ 1 + \\cdots . \\end{align*}"}
+{"id": "2458.png", "formula": "\\begin{align*} \\begin{cases} \\phi ' = \\phi \\psi , \\\\ \\psi ' = - c \\psi + \\gamma \\psi ^ { 2 } - k \\phi ^ { 2 } + \\delta p \\phi , \\end{cases} \\left ( \\ , ' = \\dfrac { d } { d s } \\right ) . \\end{align*}"}
+{"id": "5790.png", "formula": "\\begin{align*} \\xi _ i ( t ) : = \\int _ \\Sigma \\left \\langle u , \\varphi _ { i } \\right \\rangle \\ , d \\mu . \\end{align*}"}
+{"id": "3658.png", "formula": "\\begin{align*} R ( G , p ) = R ( G ' , p ' ) X \\end{align*}"}
+{"id": "7100.png", "formula": "\\begin{align*} X _ t = x - \\int _ 0 ^ t b ( X _ s ) d s + Z _ t - Z _ 0 , t \\geq 0 , x \\in \\mathbb R ^ d , \\end{align*}"}
+{"id": "7458.png", "formula": "\\begin{align*} A ^ n _ 0 = \\prod _ { i = 1 } ^ n \\frac { 1 } { \\sum _ { j = i } ^ n c _ j } . \\end{align*}"}
+{"id": "6381.png", "formula": "\\begin{align*} c ( u _ 1 \\otimes u _ 1 ) & = \\xi ^ { m i + t j } u _ 1 \\otimes u _ 1 , & c ( u _ 1 \\otimes u _ 2 ) & = \\xi ^ { i t + m j } u _ 2 \\otimes u _ 1 , \\\\ c ( u _ 2 \\otimes u _ 1 ) & = \\xi ^ { i t + m j + 2 ( i ^ { 2 } - j ^ { 2 } ) } u _ 1 \\otimes u _ 2 , & c ( u _ 2 \\otimes u _ 2 ) & = \\xi ^ { m i + t j } u _ 2 \\otimes u _ 2 . \\end{align*}"}
+{"id": "9161.png", "formula": "\\begin{align*} G _ { \\beta , \\beta ' } = ( w _ { \\beta , 1 } - v ^ { - 4 } w _ { \\beta ' , 1 } ) \\cdot G _ { \\beta , [ i , n - 1 ] } , \\end{align*}"}
+{"id": "5165.png", "formula": "\\begin{align*} L D _ { \\alpha } I ( p \\| q ) = T \\left ( \\log A - \\log X - \\log Y \\right ) \\end{align*}"}
+{"id": "1418.png", "formula": "\\begin{align*} \\frac { s ^ { \\alpha \\gamma - \\beta } } { \\Gamma ( \\gamma + \\theta / \\alpha ) } \\int _ 0 ^ \\infty \\ , t ^ { \\gamma + \\theta / \\alpha - 1 } e ^ { - ( \\lambda + s ^ \\alpha ) t } \\ , d t & = \\frac { s ^ { \\alpha \\gamma - \\beta } } { ( \\lambda + s ^ \\alpha ) ^ { \\gamma + \\theta / \\alpha } } = \\frac { s ^ { \\alpha ( \\gamma + \\theta / \\alpha ) - ( \\beta + \\theta ) } } { ( \\lambda + s ^ \\alpha ) ^ { \\gamma + \\theta / \\alpha } } \\end{align*}"}
+{"id": "5701.png", "formula": "\\begin{align*} \\mathbf { L } ( v , w ) : = \\left ( 2 ^ { - 1 } m v + w , - \\mathcal { L } _ { \\Sigma } v + 4 ^ { - 1 } m ^ 2 v + 2 ^ { - 1 } m w \\right ) . \\end{align*}"}
+{"id": "2100.png", "formula": "\\begin{align*} - ( i + 1 - d x ) P _ { i , 2 } ^ s - ( d x - i ) P _ { i + 1 , 2 } ^ s = \\alpha \\beta \\int _ 0 ^ { \\frac { \\lfloor s T \\rfloor } { T } } f ( u ) \\mathrm { d } \\xi _ 2 ( u , x ) + o ( 1 ) . \\end{align*}"}
+{"id": "5742.png", "formula": "\\begin{align*} \\hat { q } ( t ) : = q ( t ) - e ^ { \\gamma _ * t } \\bigg ( \\sum _ { i \\in I _ 1 } \\textup { R e } \\big ( w _ i e ^ { \\mathbf { i } \\beta _ i t } \\big ) \\psi _ { i , 1 } + \\textup { I m } \\big ( w _ i e ^ { \\mathbf { i } \\beta _ i t } \\big ) \\psi _ { i , 2 } + \\sum _ { i \\in I _ 2 } c _ { i , 3 } \\psi _ { i , 3 } + ( t c _ { i , 3 } + c _ { i , 4 } ) \\psi _ { i , 4 } \\bigg ) \\end{align*}"}
+{"id": "3047.png", "formula": "\\begin{align*} E _ { n , m } = ( \\ell _ 1 + \\ell _ 2 - \\ell _ 3 + 2 m - 2 n - 1 ) ( \\ell _ 1 + \\ell _ 2 - \\ell _ 3 + 2 m - 2 n ) \\ , . \\end{align*}"}
+{"id": "7808.png", "formula": "\\begin{align*} R _ { k } ^ { \\mathrm { l o c } } \\left ( a _ { k } \\right ) = \\frac { f _ { k } } { C _ { k } } = \\frac { 1 } { C _ { k } } \\sqrt { \\frac { ( 1 - a _ { k } ) E _ { k } } { L \\kappa _ k } } , \\forall k \\in \\mathcal { K } . \\end{align*}"}
+{"id": "5922.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { N C } ( A ( n , p ) ) ) = p ^ { 8 n } ( p ^ { n } - 2 ) + p ^ { 5 n } ( 2 p ^ { n } - 1 ) \\end{align*}"}
+{"id": "49.png", "formula": "\\begin{align*} ( \\tilde { U } _ { j , h } ^ * f ) ( \\mu ) : = h ^ { j } \\sum _ { l = 1 } ^ { \\binom { n } { j } } f _ l ( \\mu ) d x ^ { I ^ j _ l } \\end{align*}"}
+{"id": "6285.png", "formula": "\\begin{align*} f ( \\overline { x } _ T ) - f ( x ^ * ) \\leq \\frac 1 T \\sum \\limits _ { k = 0 } ^ { T - 1 } \\langle \\nabla \\hat { f } _ \\tau ( x _ { k } ) , x _ k - x ^ * \\rangle + 2 M _ 2 \\tau . \\end{align*}"}
+{"id": "1465.png", "formula": "\\begin{align*} ( P _ n \\cap X ) _ { F _ n \\setminus S \\Delta } = X _ { F _ n \\setminus S \\Delta } = C ( p ) _ { F _ n \\setminus S \\Delta } . \\end{align*}"}
+{"id": "5246.png", "formula": "\\begin{align*} \\frac { \\partial L _ { d } F \\left ( p \\| q \\right ) } { \\partial q _ { j } } = \\left [ \\frac { a - 1 } { a - b } p _ { j } \\left ( Z _ { j } \\right ) ^ { a - 2 } - \\frac { b - 1 } { a - b } p _ { j } \\left ( Z _ { j } \\right ) ^ { b - 2 } \\right ] \\frac { \\partial Z _ { j } } { \\partial q _ { j } } + \\left ( 1 - \\alpha \\right ) \\end{align*}"}
+{"id": "1484.png", "formula": "\\begin{align*} [ P ( x ) , P ( y ) ] = P ( [ P ( x ) , y ] ) , \\ ; \\ ; \\forall \\ x , y \\in A . \\end{align*}"}
+{"id": "1732.png", "formula": "\\begin{align*} ( z _ 0 , \\ldots , z _ n ) = ( z _ 0 , z _ { 1 , 1 } , \\ldots , z _ { 1 , n _ 1 } , \\ldots , z _ { \\mu , 1 } , \\ldots , z _ { \\mu , n _ { \\mu } } , z _ n ) , \\end{align*}"}
+{"id": "8391.png", "formula": "\\begin{align*} \\Delta ' ( \\phi ( a ) ) & = \\phi ( a _ 2 ) \\otimes \\phi ( a _ 1 ) , \\epsilon ' ( \\phi ( a ) ) = \\epsilon ( a ) . \\end{align*}"}
+{"id": "1305.png", "formula": "\\begin{align*} \\max _ { t > 0 } t e ^ { a t } = \\frac { 1 } { | a | e } . \\end{align*}"}
+{"id": "4665.png", "formula": "\\begin{align*} \\Delta u _ \\alpha & = u _ { \\alpha } '' ( r ) + u _ \\alpha ' ( r ) \\cdot \\frac { n - 1 } { r } \\\\ & = \\zeta ( r ) \\Delta u _ 1 + ( 1 - \\zeta ( r ) ) \\Delta u _ 2 + \\zeta ' ( r ) \\left ( u _ 1 ' ( r ) - u _ 2 ' ( r ) \\right ) . \\end{align*}"}
+{"id": "2129.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } _ { \\Theta ^ { n } } \\Big [ \\Psi \\cdot & \\big ( M ^ { \\Theta , B } _ f ( t _ 1 ) - M ^ { \\Theta , B } _ f ( t _ 0 ) \\big ) \\Big ] \\\\ & \\stackrel { n \\rightarrow \\infty } { \\longrightarrow } \\mathbb { E } _ { \\Theta } \\Big [ \\Psi \\cdot \\big ( M ^ { \\Theta , B } _ f ( t _ 1 ) - M ^ { \\Theta , B } _ f ( t _ 0 ) \\big ) \\Big ] . \\end{aligned} \\end{align*}"}
+{"id": "3400.png", "formula": "\\begin{align*} z _ { t } & = \\frac { 1 } { 2 C _ { 1 } \\max _ { i \\le t } \\sqrt { \\Delta _ { i } } + 4 C _ { 1 } ^ { 2 } \\sqrt { A } } \\end{align*}"}
+{"id": "1258.png", "formula": "\\begin{align*} q ^ { ( n - 1 ) ^ 2 - { n \\choose 2 } } & = q ^ { 1 - n } q ^ { \\frac { n ( n - 1 ) } { 2 } } \\\\ & \\equiv q ^ { 1 - n } \\left ( 1 - \\frac { ( n - 1 ) ( 1 - q ^ n ) } { 2 } \\right ) \\\\ & \\equiv q ^ { 1 - n } + \\frac { ( 1 - n ) q ( 1 - q ^ n ) } { 2 } \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "7313.png", "formula": "\\begin{align*} K _ { X _ { m } , \\chi _ { \\beta } } ( x , y ; t ) = \\sum _ { j = 0 } ^ { m } e ^ { - \\lambda _ j t } \\psi _ j ( x ) \\overline { \\psi _ j ( y ) } \\ , \\ , \\ , \\ , \\ , \\ , x , y \\in G _ m \\ , \\ , \\ , \\ , \\ , \\ , t \\geq 0 . \\end{align*}"}
+{"id": "1862.png", "formula": "\\begin{align*} n _ { \\phi } ( X ) : = \\sum _ { P ^ { \\dagger } } n _ { \\phi } ( X , P ^ { \\dagger } ) \\in \\Z , \\end{align*}"}
+{"id": "5416.png", "formula": "\\begin{align*} f ( x ) = - f \\circ \\mathcal I ( x ) . \\end{align*}"}
+{"id": "2345.png", "formula": "\\begin{align*} \\lim \\limits _ { j \\rightarrow \\infty } \\| a ( \\cdot , t _ { i _ { j } } ) \\| _ { L ^ { p } } = \\| w _ { 0 } \\| _ { L ^ { p } } . \\end{align*}"}
+{"id": "2928.png", "formula": "\\begin{align*} \\sum \\limits _ { l = 1 , \\ldots , k } u _ { i _ l } \\leq - \\frac { ( n - k + 1 - y ) } { y } . \\end{align*}"}
+{"id": "4440.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow 1 - 0 } \\int _ { \\partial D _ r } | f _ n | ^ 2 \\varphi | d z | = \\int _ { \\partial D } | f _ n | ^ 2 \\varphi | d z | \\end{align*}"}
+{"id": "1074.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { [ n \\delta ] } \\| ( T _ { \\infty } ( w ) ^ { - 1 } ) ^ { s , t } \\| \\leq C _ 1 n ^ { - d } , n \\in \\N , ~ ~ t \\in \\{ [ r n ] + 1 , \\dots , n \\} . \\end{align*}"}
+{"id": "7545.png", "formula": "\\begin{align*} \\bar v = \\frac { \\bar \\rho \\bar u - \\nabla \\partial _ t h _ 2 ^ \\prime ( \\bar n ) } { \\bar n } \\ , \\bar \\phi = h _ 2 ^ \\prime ( \\bar \\rho ) \\ . \\end{align*}"}
+{"id": "8419.png", "formula": "\\begin{align*} f _ 1 \\ast f ^ { \\ast - 1 } _ 2 ( a ) & = \\mathcal { B } _ 1 ( f \\ast g ) ( a _ 1 ) \\mathcal { B } _ 2 ( f \\ast g ) ^ { \\ast - 1 } ( a _ 2 ) = f \\ast g ( B _ 1 ( a _ 1 ) ) f \\ast g ( S ( B _ 2 ( a _ 2 ) ) ) \\\\ & = f \\ast g ( \\sigma ( a ) ) = f \\ast g ( a ) . \\end{align*}"}
+{"id": "5399.png", "formula": "\\begin{align*} \\nu _ k ( j _ k ) = \\frac { c _ { k } } { \\mu _ { k } } \\ , \\left [ \\frac { \\rho _ { k } ^ { - j _ k - 1 } - 1 } { ( 1 - \\rho _ { k } ) ^ 2 } - \\frac { j _ { k } + 1 } { 1 - \\rho _ k } \\right ] - r _ k - s _ k . \\end{align*}"}
+{"id": "6672.png", "formula": "\\begin{align*} \\dim ( G ) = \\dd ( G ) - \\log _ p \\lvert \\Omega _ { \\{ 1 \\} } ( G ) \\rvert = \\dd ( G ) - \\dd ( T ) . \\end{align*}"}
+{"id": "1009.png", "formula": "\\begin{align*} - L u = \\mu \\quad D , u = g \\quad \\partial _ { \\chi } D , \\end{align*}"}
+{"id": "5163.png", "formula": "\\begin{align*} & \\frac { \\partial A } { \\partial q _ { j } } = 0 \\\\ & \\frac { \\partial X } { \\partial q _ { j } } = \\frac { 1 - \\alpha } { \\alpha } \\left ( \\sum _ { i } p ^ { \\alpha } _ { i } q ^ { 1 - \\alpha } _ { i } \\right ) ^ { \\frac { 1 } { \\alpha } - 1 } p ^ { \\alpha } _ { j } q ^ { - \\alpha } _ { j } \\\\ & \\frac { \\partial Y } { \\partial q _ { j } } = \\frac { \\alpha - 1 } { \\alpha } \\left ( \\sum _ { i } q _ { i } \\right ) ^ { - \\frac { 1 } { \\alpha } } . \\ 1 \\end{align*}"}
+{"id": "1552.png", "formula": "\\begin{align*} { \\rm d i v } \\ , ( D F ( D u ) ) = 0 \\end{align*}"}
+{"id": "7847.png", "formula": "\\begin{align*} P A P R ( \\mathbf { s } ) = \\frac { 1 } { \\sum _ { i = 0 } ^ { M - 1 } \\mid s _ { i } \\mid ^ 2 } \\cdot \\max _ { t \\in [ 0 , 1 ) } \\mid \\sum _ { k = 0 } ^ { M - 1 } s _ { k } e ^ { \\sqrt { - 1 } 2 \\pi k t } \\mid \\end{align*}"}
+{"id": "6103.png", "formula": "\\begin{align*} \\sum _ u \\left ( \\Delta ( a _ u ) \\otimes b _ u - 1 \\otimes a _ u \\otimes b _ u - a _ u \\otimes 1 \\otimes \\Delta ( b _ u ) \\right ) = 0 , \\end{align*}"}
+{"id": "7392.png", "formula": "\\begin{align*} \\frac { q r + p - 1 } { q + 1 } - \\frac { r ( r - 1 ) } { r + 1 } = \\frac { r ( p + q - r ) + q r + p - 1 } { ( q + 1 ) ( r + 1 ) } > 0 \\end{align*}"}
+{"id": "3194.png", "formula": "\\begin{align*} ( f ^ * \\alpha \\cdot \\beta ) _ X = ( \\alpha \\cdot f _ * \\beta ) _ Y . \\end{align*}"}
+{"id": "1578.png", "formula": "\\begin{align*} 1 = g _ k ( z ) = ( D g _ k ( z ) , z ) = \\alpha ( z ) \\ , ( D G ( z ) , z ) , \\end{align*}"}
+{"id": "6638.png", "formula": "\\begin{align*} Q _ 1 = Q _ 2 \\coloneqq \\langle 1 \\rangle \\oplus \\langle - 1 \\rangle , Q _ 3 = Q _ 4 = Q _ 5 = Q _ 6 \\coloneqq \\langle - 2 \\rangle \\oplus \\langle - 2 \\rangle . \\end{align*}"}
+{"id": "469.png", "formula": "\\begin{align*} \\tilde { \\pi } ( A , g ) \\tilde { \\pi } ( B , h ) & = \\pi ( g ) \\cdot \\prod _ { k \\in A } \\varepsilon ( k ) \\cdot \\prod _ { k \\in Y _ g \\setminus A } ( \\pi ( d ( g ) ) - \\varepsilon ( k ) ) \\cdot \\pi ( h ) \\\\ & \\cdot \\prod _ { \\ell \\in B } \\varepsilon ( \\ell ) \\cdot \\prod _ { \\ell \\in Y _ h \\setminus B } ( \\pi ( d ( h ) ) - \\varepsilon ( \\ell ) ) . \\end{align*}"}
+{"id": "7088.png", "formula": "\\begin{align*} X _ t = x - \\int _ 0 ^ t b ( X _ r ) d r + \\sqrt { 2 } W _ t , 0 \\leq t \\leq T . \\end{align*}"}
+{"id": "4975.png", "formula": "\\begin{align*} Z _ N ( \\nu _ j , \\nu _ j - 1 , \\lambda _ 3 , \\dots , \\lambda _ N ) = 0 , j = 1 , \\dots , N ; \\end{align*}"}
+{"id": "1558.png", "formula": "\\begin{align*} \\psi _ n = u * \\varphi _ { 1 / n } \\in C ^ \\infty ( B ) \\end{align*}"}
+{"id": "8151.png", "formula": "\\begin{align*} \\Delta ^ { ( \\ell ) } = \\frac { 1 } { \\ell } \\sum _ { i = 1 } ^ { \\ell } \\mathbb { E } [ c ( \\underline x _ i ^ L ) ] . \\end{align*}"}
+{"id": "3630.png", "formula": "\\begin{align*} \\alpha ^ 3 \\tau _ 0 ^ 3 f ( \\tau , w \\sigma ^ 3 ) - ( 1 + a ) \\alpha ^ 2 \\tau ^ 2 _ 0 f ( \\tau , w \\sigma ^ 2 ) + ( a + b ) \\alpha \\tau _ 0 f ( \\tau , w \\sigma ) - b f ( \\tau , w ) = 0 , \\end{align*}"}
+{"id": "3870.png", "formula": "\\begin{align*} \\inf _ { j = 0 , \\dots , l _ 0 } \\inf _ { x \\in \\Delta ^ o } \\beta ^ j _ { ( 1 ) } ( x ) \\ge \\delta , \\ ; \\ ; \\inf _ { j = 0 , \\dots , l _ 0 } \\inf _ { ( x , y ) \\in A _ { + } } \\beta ^ j _ { 2 | 1 } ( y \\mid x ) \\ge \\delta . \\end{align*}"}
+{"id": "5996.png", "formula": "\\begin{align*} w = \\frac { | \\nabla u | ^ 2 } { u ^ 2 } . \\end{align*}"}
+{"id": "2399.png", "formula": "\\begin{align*} \\Phi _ n : = \\prod _ { \\sqrt { 1 0 n } < q \\leqslant n \\atop \\left \\{ \\frac { n } { q } \\right \\} > \\frac { 1 } { 2 } } q . \\end{align*}"}
+{"id": "918.png", "formula": "\\begin{align*} u = g \\quad \\partial _ { \\chi } D , \\hat W _ D ( u ) = 0 , \\end{align*}"}
+{"id": "2775.png", "formula": "\\begin{align*} L _ { T } & : = { \\sigma _ { \\tau } ^ { * } } ( t ) \\left [ P _ { a } \\dot { Q } ^ { a } + \\Theta _ { \\alpha } \\dot { \\Theta } ^ { \\alpha } - H _ { T } ( \\Theta ^ { \\alpha } , \\Theta _ { \\alpha } , Q ^ { a } , P _ { a } ) \\right ] \\\\ & = P _ { a } ( t ) \\dot { Q } ^ { a } ( t ) - H _ { T } ( \\Theta ^ { \\alpha } ( t ) = \\epsilon ^ { \\alpha } , \\Theta _ { \\alpha } ( t ) = \\epsilon _ { \\alpha } , Q ^ { a } ( t ) , P _ { a } ( t ) ) . \\end{align*}"}
+{"id": "5500.png", "formula": "\\begin{align*} A ^ { ( j + 1 ) } \\subset A ^ { ( j + 1 ) } S _ j ^ { - 1 } = A ^ { ( j ) } S _ j ^ { - 1 } \\subset A ^ { ( j ) } \\Sigma _ j ^ { - 1 } . \\end{align*}"}
+{"id": "7804.png", "formula": "\\begin{align*} \\boldsymbol { g } _ { r } = \\boldsymbol { h } _ { \\mathrm { d } , r } + \\left ( \\boldsymbol { G } \\right ) ^ { \\mathrm { H } } \\boldsymbol { \\Omega } _ r \\boldsymbol { \\Theta } \\boldsymbol { h } _ { \\mathrm { s } , r } , \\forall r \\in \\mathcal { R } , \\end{align*}"}
+{"id": "1715.png", "formula": "\\begin{align*} H _ j = \\left ( \\begin{array} { c c } A _ j & B _ j \\\\ B _ j ^ * & C _ j \\end{array} \\right ) \\end{align*}"}
+{"id": "2101.png", "formula": "\\begin{align*} P _ { i , 3 } ^ s = ( \\eta d T ) \\sum _ { t = 0 } ^ { \\lfloor s T \\rfloor - 1 } Z ^ { d , T } _ { \\gamma } ( \\frac { t } { T } , \\frac { i } { d } ) . \\end{align*}"}
+{"id": "4908.png", "formula": "\\begin{align*} \\dim \\bigl ( ( \\P ^ 2 ) ^ { [ 1 6 - d ] } \\bigr ) + ( d - 2 ) - 8 = 2 2 - d . \\end{align*}"}
+{"id": "8582.png", "formula": "\\begin{align*} S ^ 1 = \\begin{pmatrix} 0 & 1 / 2 \\\\ 1 / 2 & 0 \\end{pmatrix} , S ^ 2 = \\begin{pmatrix} 0 & - i / 2 \\\\ i / 2 & 0 \\end{pmatrix} , \\mbox { a n d } S ^ 3 = \\begin{pmatrix} 1 / 2 & 0 \\\\ 0 & - 1 / 2 \\end{pmatrix} \\ : . \\end{align*}"}
+{"id": "4426.png", "formula": "\\begin{gather*} \\delta _ t v ^ j \\coloneqq \\frac { v ^ j - v ^ { j - 1 } } { \\tau _ j } \\ , , \\ \\ j = 1 , 2 , \\dots , M . \\end{gather*}"}
+{"id": "2732.png", "formula": "\\begin{align*} \\tilde { L } = \\tilde { L } ( \\dot { Q } _ { ( d - 1 ) } ^ { i } , \\cdots , \\dot { Q } _ { ( 1 ) } ^ { i } , \\dot { Q } _ { ( 0 ) } ^ { i } ; Q _ { ( d - 1 ) } ^ { i } , \\cdots , Q _ { ( 1 ) } ^ { i } , Q _ { ( 0 ) } ^ { i } ; \\dot { \\lambda } ^ { ( \\alpha ) } _ { i } ; \\lambda ^ { ( \\alpha ) } _ { i } ) . \\end{align*}"}
+{"id": "8382.png", "formula": "\\begin{align*} 5 1 3 6 2 4 = s _ 2 s _ 4 s _ 5 s _ 3 s _ 4 s _ 2 s _ 1 . \\end{align*}"}
+{"id": "3619.png", "formula": "\\begin{align*} ( \\sigma \\sigma _ 1 - 1 ) ( \\sigma \\sigma _ 1 + 1 ) ( \\sigma - 1 ) = 0 . \\end{align*}"}
+{"id": "4860.png", "formula": "\\begin{align*} \\Big | \\int _ { E _ m + y _ 0 } g ( x ) d x - \\int _ { E _ m } g ( x ) d x \\Big | \\\\ & \\le | y _ 0 | ^ 2 \\int _ { E _ m } ( | D ^ 2 g ( x ) | + \\frac { 1 } { 6 } ) d x ; \\\\ & = \\mu _ { W ^ { 2 , 1 } } | y _ 0 | ^ 2 , \\end{align*}"}
+{"id": "8640.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } ( \\partial ^ 3 _ x X ) ( \\cdot ; x ) = u ^ { \\prime \\prime \\prime } _ 0 ( x ) - I _ 3 ( \\cdot ; x ) , \\\\ ( \\partial ^ 3 _ x X ) ( 0 ; x ) = 0 . \\end{cases} \\end{align*}"}
+{"id": "8019.png", "formula": "\\begin{align*} I _ { i n i } ( \\hat { W } _ 0 ) = & \\int _ { \\mathbb { T } } \\Bigg ( \\sum _ { k = 1 } ^ 3 f _ k ( u ) \\log \\left ( \\frac { f _ k ( u ) } { \\rho _ { k - 1 } ( u ) } \\right ) \\\\ & - \\left ( 1 - \\sum _ { k = 1 } ^ 3 f _ k ( u ) \\right ) \\log \\left ( \\frac { 1 - \\sum _ { k = 1 } ^ 3 \\rho _ { k - 1 } ( u ) } { 1 - \\sum _ { k = 1 } ^ 3 f _ k ( u ) } \\right ) \\Bigg ) d u . \\end{align*}"}
+{"id": "6040.png", "formula": "\\begin{align*} \\tau _ { \\pm } = \\begin{pmatrix} - \\frac { 1 } { 2 ( E _ { A } - E _ { C } ) \\rho ^ { B } } [ ( E _ { A } - E _ { C } ) ( 1 - \\rho ^ { A } ) - ( E _ { B } - E _ { C } ) ( 1 - \\rho ^ { B } ) \\pm \\delta ] \\\\ 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "9004.png", "formula": "\\begin{align*} e _ { 1 1 } \\cdot e _ { 1 1 } = c e _ { 1 1 } , e _ { 1 1 } \\cdot e _ { 1 2 } = - e _ { 1 2 } \\cdot e _ { 2 2 } = c e _ { 1 2 } , e _ { 1 1 } \\cdot e _ { 2 2 } = - e _ { 2 2 } \\cdot e _ { 2 2 } = c e _ { 2 2 } , \\ c \\ne 0 , \\end{align*}"}
+{"id": "1291.png", "formula": "\\begin{align*} \\mathbf { Q } = \\mathbf { Q } ( k ) = \\begin{pmatrix} a - k & k \\\\ k & - k + d \\end{pmatrix} \\end{align*}"}
+{"id": "3798.png", "formula": "\\begin{align*} \\left \\| \\sum _ { l = 1 } ^ { N _ i } \\chi ^ { ( i ) } _ { l , t } \\chi ^ { ( i ) , \\top } _ { l , t } \\right \\| \\leq \\frac { 9 N _ i } { 4 } , \\end{align*}"}
+{"id": "4958.png", "formula": "\\begin{align*} a ( \\lambda , \\nu ) = \\lambda - \\nu + 1 , b ( \\lambda , \\nu ) = \\lambda - \\nu , c = 1 \\end{align*}"}
+{"id": "9050.png", "formula": "\\begin{align*} ( X ^ \\sigma ) ^ Q ( v ) = X ^ { \\sigma Q } _ { \\sigma ( \\Lambda _ 1 , \\dotsc , \\Lambda _ n ) } ( \\sigma v ) = 0 . \\end{align*}"}
+{"id": "2784.png", "formula": "\\begin{align*} \\frac { \\partial \\Psi _ { a } ( t ) } { \\partial R ^ { b } ( t ) } = 0 . \\end{align*}"}
+{"id": "5858.png", "formula": "\\begin{align*} \\dfrac { M _ { 2 } ( \\mathcal { C } ( D _ { 2 m } ) ) } { | e ( \\mathcal { C } ( D _ { 2 m } ) ) | } = 1 = \\dfrac { M _ { 1 } ( \\mathcal { C } ( D _ { 2 m } ) ) } { | v ( \\mathcal { C } ( D _ { 2 m } ) ) | } . \\end{align*}"}
+{"id": "1034.png", "formula": "\\begin{align*} \\mathbf { y } _ n : = ( y _ 1 ^ \\top , \\dots , y _ n ^ \\top ) ^ \\top \\in \\C ^ { n q \\times q } , n \\in \\N . \\end{align*}"}
+{"id": "8298.png", "formula": "\\begin{align*} B y = c , \\end{align*}"}
+{"id": "6224.png", "formula": "\\begin{align*} \\begin{cases} \\dot { w } = h _ 2 - c _ 2 - D _ 2 g _ 2 / w \\ & \\mbox { i n } \\ ( \\alpha , \\gamma ) , \\\\ w < 0 \\ & \\mbox { i n } \\ ( \\alpha , \\gamma ] , \\\\ w ( \\alpha ) = 0 . \\end{cases} \\end{align*}"}
+{"id": "620.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 u ^ { n + \\frac { 1 } { 2 } } \\ln ( 1 - u ) \\ , d u = - \\frac { 2 \\left ( \\psi \\left ( n + \\frac { 5 } { 2 } \\right ) + \\gamma \\right ) } { 2 n + 3 } . \\end{align*}"}
+{"id": "8099.png", "formula": "\\begin{align*} \\sum _ { j \\in \\Z } p ( t - j ) = 1 , \\forall t \\in \\R . \\end{align*}"}
+{"id": "1705.png", "formula": "\\begin{align*} 0 = \\sum _ { j = 1 } ^ n v _ j \\left ( H _ { \\mathcal { L } } \\right ) _ { j , k } = \\sum _ { j = 1 } ^ n v _ j \\frac { \\partial ^ 2 \\Phi } { \\partial x _ j \\partial x _ k } \\quad \\quad \\forall k \\in \\{ 1 , \\ldots , n \\} , \\end{align*}"}
+{"id": "2376.png", "formula": "\\begin{align*} \\tilde { w } _ l = \\sum _ { k = 0 } ^ N Q _ { l , k } \\tilde { v } _ k \\ \\ \\ \\ \\ \\ \\tilde { v } _ l = \\Gamma ^ { - 1 } \\sum _ { k = 0 } ^ N \\tilde { Q } _ { k , l } \\tilde { w } _ k \\ , \\end{align*}"}
+{"id": "566.png", "formula": "\\begin{align*} E ( \\phi , \\eta , u ) & = 0 & \\Gamma _ c . \\end{align*}"}
+{"id": "2161.png", "formula": "\\begin{align*} A _ d = \\{ a _ { i _ 1 } , \\dots , a _ { i _ K } \\} \\end{align*}"}
+{"id": "7174.png", "formula": "\\begin{align*} ( 1 , 0 ) \\cdot H & = ( 1 - v _ 0 ^ 2 ) L - v _ 0 L ^ 2 - L ^ 3 / 3 - W - ( v _ 0 + L ) ( A ( x ) ^ 2 / 2 + B ( x ) ^ 2 / 2 + A ( x ) B ( x ) \\cos ( \\eta t ) ) \\\\ & \\leq ( 1 - v _ 0 ^ 2 ) L - v _ 0 L ^ 2 - L ^ 3 / 3 - \\frac { ( v _ 0 + L ) } { 2 } ( | A ( x ) | + | B ( x ) | ) ^ 2 + S \\\\ & = ( | A ( x ) | + | B ( x ) | ) ^ 2 \\frac { ( - v _ 0 - L ) } { 2 } + ( - v _ 0 - L / 3 ) L ^ 2 - \\left ( v _ 0 ^ 2 - 1 - \\frac { S } { L } \\right ) L \\\\ & \\leq \\Delta ^ 2 ( - v _ 0 - L ) + ( - v _ 0 - L / 3 ) L ^ 2 - \\left ( v _ 0 ^ 2 - 1 - \\frac { S } { L } \\right ) L . \\end{align*}"}
+{"id": "1791.png", "formula": "\\begin{align*} x _ 1 \\Re ( f _ { n - 1 } ^ { \\mu + 1 } ) - \\Re ( h ^ { \\mu + 2 } ) = \\Re ( Y ^ { \\prime } ) ( P ( z , \\zeta ) - u ) \\end{align*}"}
+{"id": "8757.png", "formula": "\\begin{align*} f _ { \\tilde { \\rho } } ( x _ + ) - f _ { \\tilde { \\rho } } ( x _ - ) & = \\tilde { \\rho } \\left ( \\frac { z - m } { z - y } ( m - y ) ^ { \\tilde { \\rho } - 1 } - \\frac { m - y } { z - y } ( z - m ) ^ { \\tilde { \\rho } - 1 } \\right ) ( x _ + - x _ - ) + o ( x _ + - x _ - ) \\\\ & \\quad + ( m - x _ - ) ^ { \\tilde { \\rho } } - ( x _ + - m ) ^ { \\tilde { \\rho } } . \\end{align*}"}
+{"id": "7253.png", "formula": "\\begin{align*} \\sigma = 0 \\quad | q | < 1 . \\end{align*}"}
+{"id": "6625.png", "formula": "\\begin{align*} H _ N S _ N ( i , j ) & = \\frac { 1 } { J _ N } ( - J _ 1 ( J _ { N - 2 } + J _ { N - 1 } ) + J _ 0 J _ { N - 2 } + 2 J _ 2 ( J _ { N - 3 } + J _ { N - 2 } ) + 2 J _ 0 J _ { N - 3 } ) \\\\ & = \\frac { 1 } { J _ N } ( - J _ { N - 2 } - J _ { N - 1 } + 2 ( J _ { N - 3 } + J _ { N - 2 } ) ) \\\\ & = \\frac { 1 } { J _ N } ( 2 J _ { N - 3 } + J _ { N - 2 } - J _ { N - 1 } ) \\\\ & = 0 . \\end{align*}"}
+{"id": "2353.png", "formula": "\\begin{align*} \\left ( \\sum ^ { n } _ { i = 1 } a _ { i } \\right ) ^ { 2 } \\leq n \\sum ^ { n } _ { i = 1 } a _ { i } ^ { 2 } ; \\end{align*}"}
+{"id": "51.png", "formula": "\\begin{align*} ( F H _ h F ^ * - z ) ^ { - 1 } ( \\xi ) & = \\frac { 1 } { r _ z ( \\xi ) } F H _ h F ^ * + \\frac { z } { r _ z ( \\xi ) } \\end{align*}"}
+{"id": "4034.png", "formula": "\\begin{align*} S ( x ) = \\left ( 1 , \\int _ { 0 < u _ 1 < T } d x _ { u _ 1 } , . . . , \\int _ { 0 < u _ 1 < . . . < u _ k < T } d x _ { u _ 1 } \\otimes . . . \\otimes d x _ { u _ k } , . . . \\right ) , \\end{align*}"}
+{"id": "2570.png", "formula": "\\begin{gather*} \\phi _ k ^ { ( m ) } ( 0 ) = f _ k ( 0 ) \\to \\infty \\ ; , \\\\ \\int _ { - \\infty } ^ 0 a ( t ) \\phi _ k ^ { ( m ) } ( t ) \\ , d t \\to \\frac { a ( 0 ) } 2 \\ ; , \\int _ 0 ^ \\infty b ( t ) \\phi _ k ^ { ( m ) } ( t ) \\ , d t \\to \\frac { b ( 0 ) } 2 \\ ; , \\end{gather*}"}
+{"id": "3288.png", "formula": "\\begin{align*} n _ j = \\begin{cases} ( n + r ) / d - 1 & ( d - r - 1 ) / 2 \\leq j \\leq ( d + r - 1 ) / 2 \\\\ [ 4 p t ] ( n + r ) / d & 1 \\leq j < ( d - r - 1 ) / 2 \\ : \\ : \\ : \\ : ( d + r - 1 ) / 2 < j \\leq d - 1 \\end{cases} \\end{align*}"}
+{"id": "5387.png", "formula": "\\begin{align*} \\bar { v } ^ u = \\lim _ { T \\to \\infty } \\ , \\frac { 1 } { T } \\ , E ^ u \\left [ \\int _ 0 ^ T h _ { L ( t ) } \\ , d t \\right ] . \\end{align*}"}
+{"id": "1117.png", "formula": "\\begin{align*} \\| \\phi \\| _ { S _ M } : = \\sup _ { \\gamma \\in \\mathbb { Z } _ + ^ n , \\ , | \\gamma | \\leq M } \\sup _ { x \\in \\mathbb { R } ^ n } | \\partial ^ \\gamma \\phi ( x ) | ( 1 + | x | ) ^ { n + M + | \\gamma | } . \\end{align*}"}
+{"id": "1032.png", "formula": "\\begin{align*} R _ { s , n } : = \\sum _ { t = 1 } ^ { n } \\Delta _ n ^ { s , t } , n \\in \\N , \\ \\ s \\in \\{ 1 , \\dots , n \\} . \\end{align*}"}
+{"id": "2160.png", "formula": "\\begin{align*} A ' = \\{ a _ i \\in A : a _ { i + 1 } - a _ i \\in D _ { b i g } ' \\} = \\bigcup _ { d \\in D _ { b i g } ' } A _ d . \\end{align*}"}
+{"id": "3821.png", "formula": "\\begin{align*} w ^ 2 = f _ 3 ( x _ 0 : x _ 1 : x _ 2 ) g _ 3 ( x _ 0 : x _ 1 : x _ 2 ) . \\end{align*}"}
+{"id": "5902.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { N C } ( G ) ) & = 1 9 n ( 1 9 n - 1 ) ^ { 2 } - 4 ( 1 9 n - 1 ) \\dfrac { ( 6 1 n ^ { 2 } - 1 9 n ) } { 2 } + 4 n ( 4 n - 1 ) ^ { 2 } + 1 5 n ( 3 n - 1 ) ^ { 2 } \\\\ & = 6 8 5 9 n ^ { 3 } - 2 3 1 8 n ^ { 3 } + 6 4 n ^ { 3 } + 1 3 5 n ^ { 3 } = 4 7 4 0 n ^ { 3 } \\end{align*}"}
+{"id": "8849.png", "formula": "\\begin{align*} \\mathcal N & : = \\mathcal N [ u ] : = | x | ^ { - \\tau } | u | ^ { p - 2 } ( I _ \\alpha * | \\cdot | ^ { - \\tau } | u | ^ p ) u . \\end{align*}"}
+{"id": "8467.png", "formula": "\\begin{align*} \\iint _ { K \\times K } \\dfrac { 1 } { | x - y | ^ { \\frac { p } { p - 1 } \\left ( n + \\bar { s } - ( n + s p ) \\frac { 1 } { p } \\right ) } } \\ , d x d y & \\leq \\int _ K \\left ( \\ , \\int _ { B _ R ( x ) } \\dfrac { d y } { | x - y | ^ { n + \\frac { p } { p - 1 } ( \\bar { s } - s ) } } \\right ) \\ , d x \\\\ [ 3 m m ] & = C ( n ) | K | \\int _ 0 ^ R \\dfrac { d \\rho } { \\rho ^ { 1 + \\frac { p } { p - 1 } ( \\bar { s } - s ) } } \\\\ [ 3 m m ] & = C ( n ) | K | \\dfrac { p - 1 } { p ( s - \\bar { s } ) } R ^ { \\frac { p } { p - 1 } ( s - \\bar { s } ) } \\end{align*}"}
+{"id": "891.png", "formula": "\\begin{align*} w ^ i = C _ i + A _ i x _ 1 + B _ i x _ 2 , i = 1 , 2 , \\end{align*}"}
+{"id": "1035.png", "formula": "\\begin{align*} T _ { \\infty } ( w ) \\mathbf { z } = \\mathbf { y } . \\end{align*}"}
+{"id": "3883.png", "formula": "\\begin{align*} \\hat \\gamma ( \\cdot \\times d s ) = \\hat \\eta ( \\cdot \\mid s ) \\exp ( - s ) d s , \\end{align*}"}
+{"id": "301.png", "formula": "\\begin{align*} \\widehat { F } ( \\widehat { z } _ 1 , \\widehat { z } _ 2 ) = \\begin{cases} ( a - \\frac { 1 } { c } ) \\widehat { z } _ 1 + \\frac { a ^ 2 b } { 2 } \\left ( \\widehat { z } _ 1 ^ 2 - 2 \\widehat { z } _ 1 \\widehat { z } _ 2 + \\widehat { z } _ 2 ^ 2 \\right ) \\\\ ( a + \\frac { 1 } { c } ) \\widehat { z } _ 2 + \\frac { a ^ 2 b } { 2 } \\left ( \\widehat { z } _ 1 ^ 2 - 2 \\widehat { z } _ 1 \\widehat { z } _ 2 + \\widehat { z } _ 2 ^ 2 \\right ) . \\end{cases} \\end{align*}"}
+{"id": "157.png", "formula": "\\begin{align*} \\eta _ { X } ( x ) & = \\left ( \\prod _ { i } u ^ { * } _ i \\right ) x \\left ( \\prod _ { i } u _ { i } \\right ) \\\\ & = \\left ( \\prod _ { i \\in J _ { F } } u ^ { * } _ { i } \\right ) x \\left ( \\prod _ { i \\in J _ { F } } u ^ { * } _ { i } \\right ) \\\\ & = \\left ( \\prod _ { i \\in J _ { G } } u ^ { * } _ { i } \\right ) x \\left ( \\prod _ { i \\in J _ { G } } u ^ { * } _ { i } \\right ) \\ \\end{align*}"}
+{"id": "7510.png", "formula": "\\begin{align*} M _ { s } ^ 2 = \\frac { \\left ( M _ { s } ^ 1 \\backslash \\phi _ 2 ( ( 0 , s ^ { 1 / 4 } ] \\times X \\times [ - \\eta / 2 , \\eta / 2 ] ) \\right ) \\bigsqcup \\left ( \\{ \\mathbf { r } _ s \\leq 1 \\} \\times I _ { \\eta } ^ { 2 } \\right ) } { \\thicksim _ 2 } , \\end{align*}"}
+{"id": "2141.png", "formula": "\\begin{align*} \\begin{array} { l } h = ( 2 - e ^ T x , \\ , x _ 1 , \\ , x _ 2 ) , \\\\ g = ( - z _ 1 + z _ 2 + 2 x _ 1 - 2 . 5 , \\ , - z _ 2 - x _ 1 + 3 x _ 2 + 2 , \\ , z _ 1 , z _ 2 ) . \\end{array} \\end{align*}"}
+{"id": "4394.png", "formula": "\\begin{gather*} f _ i ( x , u ^ i ) = ( u ^ i ) ^ T l _ i ( x ) \\ \\forall u ^ i \\in \\mathcal { U } _ i , \\ x \\in \\mathcal { X } . \\end{gather*}"}
+{"id": "5006.png", "formula": "\\begin{align*} d _ A ( g _ t ' , g _ { t + 1 } ' ) & = d _ A ( g _ t , g _ { t + 1 } ) \\\\ & \\leq d _ A ( g _ t , \\widehat { W } ( t ) ) + d _ A ( \\widehat { W } ( t ) , \\widehat { W } ( t + 1 ) ) + d _ A ( \\widehat { W } ( t + 1 ) , g _ { t + 1 } ) \\\\ & \\leq 2 \\eta + 1 + 2 \\eta = 4 \\eta + 1 , \\end{align*}"}
+{"id": "7192.png", "formula": "\\begin{align*} \\partial _ t \\bar { v } - \\partial _ { x } ^ 2 \\bar { v } & = ( 1 - v _ 0 ^ 2 ) \\bar { v } - v _ 0 \\bar { v } ^ 2 - \\frac { 1 } { 3 } \\bar { v } ^ 3 - \\bar { w } - \\bar { v } X _ 1 ( x , t ) - Z ( \\bar { v } , x , t ) - v _ 0 J _ 0 ^ 2 = \\psi _ 1 ( \\bar { v } , \\bar { w } , x , t ) , \\\\ \\partial _ t \\bar { w } - \\rho \\partial _ { x } ^ { 2 } \\bar { w } & = \\varepsilon ( \\bar { v } - \\gamma \\bar { w } ) + \\varepsilon J _ 0 = \\psi _ 2 ( \\bar { v } , \\bar { w } , x , t ) , \\end{align*}"}
+{"id": "1337.png", "formula": "\\begin{align*} \\theta ^ 2 - \\left \\lbrace 2 r \\alpha + ( 1 - \\alpha ) ( a - c ) \\right \\rbrace \\theta - \\left \\lbrace ( r - c ) ( 1 - \\alpha ) ^ 2 - r \\alpha ( 1 - \\alpha ) ( a - c ) - r ^ 2 \\alpha ^ 2 \\right \\rbrace = 0 \\end{align*}"}
+{"id": "8755.png", "formula": "\\begin{align*} \\begin{dcases} \\pi _ x ( \\{ y \\} ) + \\pi _ x ( \\{ m \\} ) + \\pi _ x ( \\{ z \\} ) = 1 \\\\ y \\ , \\pi _ x ( \\{ y \\} ) + m \\ , \\pi _ x ( \\{ m \\} ) + z \\ , \\pi _ x ( \\{ z \\} ) = x \\end{dcases} \\quad \\implies \\begin{dcases} \\pi _ x ( \\{ y \\} ) & = \\frac { z - x - ( z - m ) \\pi _ { x } ( \\{ m \\} ) } { z - y } , \\\\ \\pi _ x ( \\{ z \\} ) & = \\frac { x - y - ( m - y ) \\pi _ { x } ( \\{ m \\} ) } { z - y } . \\end{dcases} \\end{align*}"}
+{"id": "7381.png", "formula": "\\begin{align*} D : = \\{ ( x , t ) \\in \\R \\times [ 0 , \\infty ) \\ : \\ t - | x | \\ge R \\} . \\end{align*}"}
+{"id": "6052.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ n U \\to A _ { 2 d } , ( q _ 1 , \\dots , q _ n ) \\mapsto \\sum _ { i = 1 } ^ n p _ i q _ i . \\end{align*}"}
+{"id": "1222.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\max \\bigg \\{ \\mathcal { L } _ { s , \\infty } u , \\mathcal { L } ^ { - } _ { s , \\infty } u + \\Lambda _ { 1 , \\infty } \\vert u ( x ) \\vert ^ { \\theta } \\vert v _ \\infty ( x _ 0 ) \\vert ^ { 1 - \\theta } \\bigg \\} = 0 & { \\rm i n } \\ \\ \\Omega , \\\\ u = 0 & { \\rm i n } \\ \\mathbb { R } ^ N \\setminus \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "2283.png", "formula": "\\begin{align*} \\check R _ i ( u ) = \\Sigma _ i ^ { ( 2 ) } - ( q + q ^ { - 1 } ) \\frac { ( q ^ 2 - q ^ { - 2 } ) } { 1 - u q ^ { - 2 } } \\Sigma _ i ^ { ( 1 ) } + q ^ 2 \\frac { ( 1 - q ^ { - 2 } ) ( 1 - q ^ { - 4 } ) } { ( 1 - u ) ( 1 - u q ^ { - 2 } ) } \\ . \\end{align*}"}
+{"id": "2601.png", "formula": "\\begin{align*} \\varphi ' ( r ) & = - \\dfrac { 2 \\beta R r } { ( r ^ 2 + \\alpha ^ 2 ) ^ { \\beta R + 1 } } \\\\ \\varphi '' ( r ) & = - ( - \\beta R - 1 ) \\dfrac { 4 \\beta R r ^ 2 } { ( r ^ 2 + \\alpha ^ 2 ) ^ { \\beta R + 2 } } - \\dfrac { 2 \\beta R } { ( r ^ 2 + \\alpha ^ 2 ) ^ { \\beta R + 1 } } . \\end{align*}"}
+{"id": "616.png", "formula": "\\begin{align*} & \\frac { 4 } { \\pi } \\int _ 0 ^ 1 \\sqrt { 1 - x ^ 2 } \\ln ( x ) \\sum _ { n = 0 } ^ { \\infty } \\left ( - \\frac { x ^ 2 } { 2 } \\right ) ^ n \\binom { - \\frac { 1 } { 2 } } { n } ( n + 1 ) \\ , d x + \\\\ & \\frac { 1 } { 2 } \\sum _ { n = 0 } ^ { \\infty } \\left ( \\frac { 1 } { 3 2 } \\right ) ^ n \\binom { 2 n } { n } ^ 2 \\frac { 1 + 2 ( n + 1 ) \\ln ( 2 ) } { n + 1 } . \\end{align*}"}
+{"id": "2911.png", "formula": "\\begin{align*} \\begin{aligned} H _ n '' ( t ) & \\leqslant \\mathbb { E } _ { n - 1 } \\Big [ 5 \\lambda ^ 2 | Z _ n | ^ 2 \\cosh { \\big ( \\lambda | g _ { n - 1 } + t Z _ n | } \\big ) \\Big ] \\\\ & \\leqslant 5 \\lambda ^ 2 S _ n ^ 2 \\mathbb { E } _ { n - 1 } \\Big [ \\cosh { \\big ( \\lambda | g _ { n - 1 } + t Z _ n | } \\big ) \\Big ] \\\\ & = 5 \\lambda ^ 2 S _ n ^ 2 H _ n ( t ) . \\end{aligned} \\end{align*}"}
+{"id": "8613.png", "formula": "\\begin{align*} \\Sigma _ \\gamma ( t ) = \\{ x \\in \\mathbb { R } : v _ 1 ( t ; x ) \\leq ( 1 - \\gamma ) m ( t ) \\} . \\end{align*}"}
+{"id": "6500.png", "formula": "\\begin{align*} \\lambda _ t : = \\mu + \\int _ { ( 0 , t ) } \\Phi ( t - s ) d H _ s , t \\geq 0 , \\end{align*}"}
+{"id": "2593.png", "formula": "\\begin{align*} f ( t ) = c _ 1 t \\left [ c _ 2 t ^ { \\frac { 1 } { \\beta R } } + c _ 3 t ^ { \\frac { 2 } { \\beta R } } \\right ] , \\end{align*}"}
+{"id": "8963.png", "formula": "\\begin{align*} g ( q , t ) : = \\begin{cases} \\frac { t ^ 2 } { 2 } & , \\\\ \\frac { t ^ { q } } { q } + \\frac { 1 } { 2 } - \\frac { 1 } { q } & . \\end{cases} \\end{align*}"}
+{"id": "2951.png", "formula": "\\begin{align*} \\big ( \\Delta + \\partial _ { \\rho } ^ 2 + \\frac { ( 1 - s ) } { \\rho } \\partial _ \\rho \\big ) u ( x , \\rho ) = 0 , \\ , \\ , u ( x , 0 ) = f ( x ) . \\end{align*}"}
+{"id": "5284.png", "formula": "\\begin{align*} D ^ { s } _ { r } \\left ( p \\| q \\right ) = \\frac { 1 } { s - 1 } \\left \\{ \\left [ \\sum _ { i } \\left ( M G \\right ) _ { i } ^ { 1 - r } \\left ( M A \\right ) _ { i } ^ { r } \\right ] ^ { \\frac { s - 1 } { r - 1 } } - \\left [ \\sum _ { i } \\left ( M A \\right ) _ { i } \\right ] ^ { \\frac { s - 1 } { r - 1 } } \\right \\} \\end{align*}"}
+{"id": "663.png", "formula": "\\begin{align*} ( \\delta f ) _ { i _ { 1 } \\dots i _ { m - 1 } } = \\frac { \\partial f _ { i _ { 1 } \\dots i _ { m } } } { \\partial x ^ { i _ { m } } } . \\end{align*}"}
+{"id": "5302.png", "formula": "\\begin{align*} b ^ u _ i = E _ i ^ u \\left [ \\int _ 0 ^ \\infty \\lambda _ { L ( t ) } \\ , a ( t ) \\ , e ^ { - \\alpha t } \\ , d t \\right ] \\end{align*}"}
+{"id": "1668.png", "formula": "\\begin{align*} g _ 1 \\ , [ v _ 0 , v _ 1 , v _ 2 ] = [ e _ r , f _ r , e _ r + f _ r + x ] \\end{align*}"}
+{"id": "5676.png", "formula": "\\begin{align*} u '' - m u ' + \\mathcal { M } _ \\Sigma u = N _ 1 ( u ) , \\end{align*}"}
+{"id": "979.png", "formula": "\\begin{align*} f _ k ( x , y ) = f ( x , y ) - f ( x , 0 ) + ( f ( x , 0 ) \\wedge k ) \\vee ( - k ) ) \\frac { k \\varrho } { 1 + k \\varrho } . \\end{align*}"}
+{"id": "8485.png", "formula": "\\begin{align*} \\lim _ { h \\searrow 0 } I _ h ^ { ( 1 ) } = 0 . \\end{align*}"}
+{"id": "1510.png", "formula": "\\begin{align*} \\tilde { F } ( z ) = F ( z + \\bar z ) - F ( \\bar z ) \\end{align*}"}
+{"id": "1220.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l } \\mathcal { L } _ { t , p _ n } v = 0 & { \\rm i n } \\ \\ \\Omega \\setminus \\{ x _ 0 \\} , \\\\ v = 0 & { \\rm i n } \\ \\mathbb { R } ^ N \\setminus \\Omega , \\\\ v ( x _ 0 ) = v _ { p _ n } ( x _ 0 ) . \\end{array} \\right . \\end{align*}"}
+{"id": "2666.png", "formula": "\\begin{align*} \\frac { \\partial L ^ { ( 2 ) } } { \\partial q ^ { i } } - \\frac { d } { d t } \\left ( \\frac { \\partial L ^ { ( 2 ) } } { \\partial \\dot { q } ^ { i } } \\right ) + \\frac { d ^ { 2 } } { d t ^ { 2 } } \\left ( \\frac { \\partial L ^ { ( 2 ) } } { \\partial \\ddot { q } ^ { i } } \\right ) = 0 \\end{align*}"}
+{"id": "1842.png", "formula": "\\begin{align*} \\star ( F _ A \\wedge \\phi ) = - F _ A \\Longrightarrow \\star F _ A = - F _ A \\wedge \\phi . \\end{align*}"}
+{"id": "4218.png", "formula": "\\begin{align*} \\theta _ 3 ( v , \\tau ) = \\prod _ { j = 1 } ^ { \\infty } [ ( 1 - q ^ j ) ( 1 + e ^ { 2 \\pi \\sqrt { - 1 } v } q ^ { j - \\frac { 1 } { 2 } } ) ( 1 + e ^ { - 2 \\pi \\sqrt { - 1 } v } q ^ { j - \\frac { 1 } { 2 } } ) ] , \\end{align*}"}
+{"id": "4423.png", "formula": "\\begin{align*} \\inf _ { x , p , \\theta } & \\ f ( x ) , \\\\ \\mathrm { s . t . } & \\ \\sum _ { i \\in [ m ] } g _ i ( x , \\bar { u } ^ i ) + p _ i + \\theta \\Gamma \\le 0 , \\\\ & p _ i + \\theta \\ge \\sup _ { u ^ i \\in \\mathcal { U } _ i } g _ i ( x , u ^ i ) - g _ i ( x , \\bar { u } ^ i ) \\ \\forall i \\in [ m ] , \\\\ & p , \\theta \\ge 0 , \\\\ & x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "1827.png", "formula": "\\begin{align*} V ^ { \\Gamma } : = \\{ v \\in V : \\gamma v = v \\ ; \\ ; \\forall \\gamma \\in \\Gamma \\} \\end{align*}"}
+{"id": "2363.png", "formula": "\\begin{align*} g _ { \\pm } ( T ) = \\frac { 1 \\pm \\sqrt { 1 - 4 a b ( T ) ^ { 2 } C _ { 1 } ^ { 2 } } } { 2 C _ { 1 } b ( T ) } . \\end{align*}"}
+{"id": "8009.png", "formula": "\\begin{align*} \\mathcal { U } ^ N _ { F , G , H } \\left ( T , \\xi ^ N \\right ) = \\exp \\left ( N \\left ( \\mathcal { I } _ 1 \\left ( \\mu ^ N , F , G , H \\right ) \\right ) \\right ) . \\end{align*}"}
+{"id": "4438.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\int _ { \\partial D } | f _ n - f | ^ 2 \\varphi | d z | = 0 . \\end{align*}"}
+{"id": "3313.png", "formula": "\\begin{align*} K ^ { [ \\nu ] } _ { n - \\nu } = \\frac { 1 } { \\sqrt { n } } \\exp \\left ( \\frac { ( ( n - \\nu ) \\mu _ 0 - S _ { n - \\nu } ) ^ 2 } { 2 n } \\right ) ; \\end{align*}"}
+{"id": "6249.png", "formula": "\\begin{align*} E ( \\omega , \\c d ) \\le E \\left ( \\omega , \\frac 3 2 \\right ) = - \\frac { 1 } { 2 } \\omega ^ 2 + 2 \\omega - \\frac { 1 } { 2 } : = \\phi ( \\omega ) . \\end{align*}"}
+{"id": "1230.png", "formula": "\\begin{align*} D _ k : = \\{ j | ~ \\{ u _ j , u _ k \\} \\mbox { i s l i n e a r l y d e p e n d e n t } \\} \\end{align*}"}
+{"id": "5221.png", "formula": "\\begin{align*} L _ { d } D G H I ( p \\| q ) = & \\frac { \\sum _ { j } p _ { j } } { a - b } \\left [ \\left ( \\overline { M G } \\right ) ^ { a - 1 } - \\left ( \\overline { M G } \\right ) ^ { b - 1 } \\right ] \\\\ & - \\frac { \\sum _ { j } p _ { j } } { a - b } \\left [ \\left ( \\overline { M H } \\right ) ^ { a - 1 } - \\left ( \\overline { M H } \\right ) ^ { b - 1 } \\right ] \\end{align*}"}
+{"id": "953.png", "formula": "\\begin{align*} P _ V ( g ) ( x ) = \\mathbb E _ x g ( X _ { \\tau _ V } ) , x \\in E \\setminus N . \\end{align*}"}
+{"id": "1784.png", "formula": "\\begin{align*} w ^ * = w + h ^ 3 + h ^ 4 + \\cdots , \\ , \\ , z _ j ^ * = z _ j + f _ { j } ^ 2 + f _ { j } ^ 3 + \\cdots , \\ , \\ , \\mbox { a n d } \\ , \\ , \\zeta ^ * = \\zeta + g ^ 1 + g ^ 2 + \\cdots . \\end{align*}"}
+{"id": "1292.png", "formula": "\\begin{align*} \\lambda _ { \\pm } = \\lambda _ { \\pm } ( k ) : = \\frac { ( a + d - 2 k ) \\pm \\sqrt { ( a - d ) ^ 2 + 4 k ^ 2 } } { 2 } \\ ; , \\end{align*}"}
+{"id": "8695.png", "formula": "\\begin{align*} \\Box _ \\beta ^ { ( 0 ) } N + \\hat { S } & \\equiv I \\ \\ D \\times D , \\\\ \\Box _ \\beta ^ { ( 0 ) } \\hat { S } & \\equiv 0 \\ \\ D \\times D . \\end{align*}"}
+{"id": "699.png", "formula": "\\begin{align*} S _ i = \\left \\{ \\frac { \\theta ^ { n - u } q _ { i j } ( \\theta ) } { p ^ { \\lfloor { y _ u + \\frac { Y _ { i j } - j V ( \\Phi _ i ( x ) ) } { e } } \\rfloor } } ~ : ~ n - e \\mu _ i < u \\le n , b _ { i t } - e _ { i t } f _ { i t } < j \\le b _ { i t } , 1 \\le t \\le z _ i \\right \\} , \\end{align*}"}
+{"id": "2387.png", "formula": "\\begin{align*} \\left | D \\cdot \\left ( a _ { n , 0 } + \\sum _ { j = 1 } ^ { m } a _ { n , j } \\alpha _ j \\right ) \\right | _ p & \\geqslant \\frac { 1 } { \\left | D \\cdot \\left ( a _ { n , 0 } + \\sum _ { j = 1 } ^ { m } a _ { n , j } \\alpha _ j \\right ) \\right | } \\\\ & \\geqslant \\frac { 1 } { D \\cdot \\left ( 1 + \\sum _ { j = 1 } ^ { m } | \\alpha _ j | \\right ) \\cdot \\max _ { 0 \\leqslant j \\leqslant m } | a _ { n , j } | } , \\end{align*}"}
+{"id": "4821.png", "formula": "\\begin{align*} \\alpha _ { v , 0 } : = 0 \\leq \\alpha _ { v , 1 } \\leq \\ldots \\leq \\alpha _ { v , r _ v } \\leq \\beta _ v \\leq \\alpha _ { v , r _ v + 1 } \\leq \\ldots \\leq \\alpha _ { v , n _ v } \\leq 1 . \\end{align*}"}
+{"id": "5794.png", "formula": "\\begin{align*} 2 N s + N \\leq \\sum _ { i = 1 } ^ N ( 2 k _ i + \\ell _ i ) = 2 k _ 0 + \\ell _ 0 + m _ 2 + 2 \\leq 2 s + 2 N . \\end{align*}"}
+{"id": "8125.png", "formula": "\\begin{align*} \\dim ( B _ { L _ I } \\cdot x ) & = \\dim B _ { L _ I } - \\dim ( B _ { L _ I } ) _ x \\\\ & = \\ell ( w _ 0 ( I ) ) + \\dim T - ( \\dim T - \\ell ( c ) ) \\\\ & = \\ell ( w _ 0 ( I ) ) + \\ell ( c ) \\\\ & = \\ell ( w _ 0 ( I ) c ) , \\end{align*}"}
+{"id": "1306.png", "formula": "\\begin{align*} \\Sigma _ { 1 1 } ( t ) - \\hat { \\Sigma } _ { 1 } ( t ) & = \\frac { 1 } { 2 } \\Bigg [ \\frac { 1 } { a - 2 k } \\Big ( e ^ { 2 ( a - 2 k ) t } - e ^ { 2 \\lambda _ + t } \\Big ) + \\frac { 1 } { a } \\Big ( e ^ { 2 a t } - e ^ { 2 \\lambda _ + t } \\Big ) \\Bigg ] \\\\ & = \\frac { 1 } { 2 } \\frac { 1 } { a - 2 k } \\Big ( e ^ { 2 ( a - 2 k ) t } - e ^ { 2 a t } \\Big ) \\\\ & = \\frac { 1 } { 2 } \\frac { 1 } { 2 k - a } e ^ { 2 a t } \\Big ( 1 - e ^ { - 4 k t } \\Big ) \\\\ & \\leq \\frac { 2 k } { 2 k - a } e ^ { 2 a t } t \\\\ & \\leq \\frac { k } { a ^ 2 e } , \\end{align*}"}
+{"id": "5911.png", "formula": "\\begin{align*} f ( p , q ) : = q ( q - 1 ) ( p - 1 ) ( q - 2 ) ^ 3 + q ( q - 1 ) ( p - 1 ) ( p - 2 ) ^ 3 > q ( q - 1 ) ( p - 1 ) ( p - 2 ) ( q - 2 ) ( p + q - 4 ) . \\end{align*}"}
+{"id": "6324.png", "formula": "\\begin{align*} & n \\sum _ { z \\in \\mathbb { Z } ^ 3 } \\Delta \\omega _ { \\ell , \\lambda } ( P _ z ( x ) - y ) + \\frac { 1 } { 2 } V _ { \\ell } ( x - y ) \\widetilde K ( x , y ) \\\\ & = \\frac { n } { 2 } \\sum _ { z \\in \\mathbb { Z } ^ 3 } \\Big ( \\epsilon _ { \\ell , \\lambda } - \\big ( V _ \\ell ( 1 - \\omega _ { \\ell } ) \\big ) ( P _ z ( x ) - y ) - V _ \\ell ( x - y ) \\omega _ { \\ell , \\lambda } ( P _ z ( x ) - y ) \\Big ) , \\end{align*}"}
+{"id": "731.png", "formula": "\\begin{align*} g _ p ( a _ 1 , a _ 2 , \\dots , a _ k ) & = \\left ( \\max _ { 0 \\le j \\le a _ 1 - 1 } m _ j ^ { ( p ) } \\right ) - a _ 1 \\ , , \\\\ n _ p ( a _ 1 , a _ 2 , \\dots , a _ k ) & = \\frac { 1 } { a _ 1 } \\sum _ { j = 0 } ^ { a _ 1 - 1 } m _ j ^ { ( p ) } - \\frac { a _ 1 - 1 } { 2 } \\ , , \\\\ s _ p ( a _ 1 , a _ 2 , \\dots , a _ k ) & = \\frac { 1 } { 2 a _ 1 } \\sum _ { j = 0 } ^ { a _ 1 - 1 } \\bigl ( m _ j ^ { ( p ) } \\bigr ) ^ 2 - \\frac { 1 } { 2 } \\sum _ { j = 0 } ^ { a _ 1 - 1 } m _ j ^ { ( p ) } + \\frac { a _ 1 ^ 2 - 1 } { 1 2 } \\ , . \\end{align*}"}
+{"id": "2133.png", "formula": "\\begin{align*} \\begin{array} { l l } \\lambda _ { \\{ 1 , 2 \\} } ( x , y ) = \\frac { ( x , \\ , 9 x ) } { 4 0 } , & \\lambda _ { \\{ 1 , 3 \\} } ( x , y ) = \\frac { ( - x , \\ , 9 x ) } { 4 1 } , \\\\ \\lambda _ { \\{ 1 , 4 \\} } ( x , y ) = ( - x , \\ , x ) , & \\lambda _ { \\{ 2 , 3 \\} } ( x , y ) = \\frac { ( x , \\ , x ) } { 9 } , \\\\ \\lambda _ { \\{ 2 , 4 \\} } ( x , y ) = \\frac { ( x , \\ , 9 x ) } { 4 1 } , & \\lambda _ { \\{ 3 , 4 \\} } ( x , y ) = \\frac { ( 9 x , \\ , - x ) } { 4 0 } . \\end{array} \\end{align*}"}
+{"id": "5154.png", "formula": "\\begin{align*} \\frac { \\partial A } { \\partial q _ { j } } = \\frac { \\partial A _ { j } } { \\partial q _ { j } } = - \\frac { 1 } { \\alpha } p ^ { \\alpha } _ { j } q ^ { - \\alpha } _ { j } + \\frac { 1 } { \\alpha } \\ ; \\ ; ; \\ ; \\ ; \\frac { \\partial B } { \\partial q _ { j } } = \\frac { \\partial B _ { j } } { \\partial q _ { j } } = 0 \\end{align*}"}
+{"id": "2940.png", "formula": "\\begin{align*} \\big ( \\Delta + \\partial _ { \\rho } ^ 2 + \\frac { ( 1 - s ) } { \\rho } \\partial _ \\rho \\big ) u ( x , \\rho ) = 0 , \\ , \\ , u ( x , 0 ) = f ( x ) . \\end{align*}"}
+{"id": "1964.png", "formula": "\\begin{align*} c ( x ) & = ( x - 1 ) ^ { p ^ s - p ^ { s - t } } g ( x ) \\\\ & = ( x ^ { p ^ { s - t } } - 1 ) ^ { p ^ t - 1 } g ( x ) \\\\ & = \\left [ \\sum _ { j = 0 } ^ { p ^ { t } - 1 } \\binom { p ^ { t } - 1 } { j } ( - 1 ) ^ { ( p ^ { t } - 1 - j ) } x ^ { j p ^ { s - t } } \\right ] g ( x ) . \\end{align*}"}
+{"id": "6917.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } \\eta _ R ^ { \\theta - 2 - 2 q } v ^ { - a \\ ( n , p , q , \\varepsilon \\ ) } g ^ { b \\ ( n , p , q , \\varepsilon \\ ) } = C R ^ { n - \\min \\ ( \\frac { p a \\ ( n , p , q , \\varepsilon \\ ) } { p - 1 } , a \\ ( n , p , q , \\varepsilon \\ ) + b \\ ( n , p , q , \\varepsilon \\ ) \\ ) } \\end{align*}"}
+{"id": "7457.png", "formula": "\\begin{align*} \\mathcal { A } ^ { c _ 1 , c _ 2 , \\ldots , c _ n } ( t ) & = A ^ n _ 0 + A ^ n _ 1 e ^ { - c _ n t } + A ^ n _ 2 e ^ { - ( c _ n + c _ { n - 1 } ) t } + \\ldots + A ^ n _ n e ^ { - ( c _ n + \\ldots + c _ 1 ) t } , \\end{align*}"}
+{"id": "5819.png", "formula": "\\begin{align*} \\bar { g } = g _ { i j } d x ^ i d x ^ j + h _ { A B } d y ^ A d y ^ B + c _ { i A } ( d x ^ i d y ^ A + d y ^ A d x ^ i ) . \\end{align*}"}
+{"id": "3616.png", "formula": "\\begin{align*} b ( ( \\sigma ^ 2 \\sigma _ 1 ) ^ 2 - ( \\sigma \\sigma _ 1 ) ^ 2 ) + ( a + b ) \\alpha \\tau _ 0 ( ( \\sigma ^ 2 \\sigma _ 1 ) ^ 2 - \\sigma ^ 2 ) - b ( ( \\sigma ^ 2 \\sigma _ 1 ) ^ 2 - 1 ) = 0 . \\end{align*}"}
+{"id": "8608.png", "formula": "\\begin{align*} m ( t ) < 0 ~ ~ { \\rm f o r ~ a n y } ~ ~ t \\in [ 0 , T ) , ~ ~ q ( 0 ) = 1 ~ ~ { \\rm a n d } ~ ~ q ( t ) > 0 ~ ~ { \\rm f o r ~ a n y } ~ ~ t \\in [ 0 , T ) . \\end{align*}"}
+{"id": "6449.png", "formula": "\\begin{align*} \\frac { \\bar \\nabla } { \\partial t } \\tau ( \\phi _ t ) = & - \\bar \\Delta d \\phi _ t ( \\partial _ t ) + R ^ N ( d \\phi _ t ( \\partial _ t ) , d \\phi _ t ( e _ i ) ) d \\phi _ t ( e _ i ) . \\end{align*}"}
+{"id": "5551.png", "formula": "\\begin{align*} T = w ^ { ( 1 ) } \\otimes \\cdots \\otimes w ^ { ( k ) } , \\end{align*}"}
+{"id": "4410.png", "formula": "\\begin{align*} \\inf _ { x , p , \\theta } & \\ \\Gamma \\theta + \\sum _ { i \\in [ m ] } ( x _ i - \\overline { u } _ i ) ^ 2 + p _ i , \\\\ \\mathrm { s . t . } & \\ x \\in \\mathcal { X } , \\\\ & \\max _ { u _ i \\in \\mathcal { U } _ i } ( x _ i - u _ i ) ^ 2 - ( x _ i - \\overline { u } _ i ) ^ 2 \\leq p _ i + \\theta \\ \\forall i \\in [ m ] . \\end{align*}"}
+{"id": "4284.png", "formula": "\\begin{align*} d y _ t = \\sigma ( y _ t ) d w _ t + b ( y _ t ) d t , y _ 0 = a \\ , \\in \\R ^ e . \\end{align*}"}
+{"id": "4193.png", "formula": "\\begin{align*} \\varphi ( g _ 1 g _ 3 ) \\varphi ( g _ 1 g _ 2 ) = \\varphi ( ( g _ 3 g ^ { 2 } _ 1 ) g _ 2 ) \\mbox { o r } \\varphi ( g _ 1 g _ 3 ) \\varphi ( g _ 1 g _ 2 ) = \\varphi ( g _ 2 ( g _ 3 g ^ { 2 } _ 1 ) ) \\ , . \\end{align*}"}
+{"id": "1752.png", "formula": "\\begin{align*} b _ { j , 1 } = r _ { j , 1 } + a _ { j , n - 1 } , b _ { j , k } = r _ { j , k } + a _ { j , n - k } - a _ { j , n + 1 - k } \\zeta , \\quad \\forall \\ , k = 2 , \\ldots , n - 1 \\end{align*}"}
+{"id": "5984.png", "formula": "\\begin{align*} \\nabla \\phi ( x ) = \\frac { \\nabla ( | \\nabla u ( x ) | ^ 2 ) } { ( b ^ 2 - \\rho ^ 2 ( u ( x ) ) ) ^ 2 } + 2 \\frac { | \\nabla u ( x ) | ^ 2 \\nabla ( \\rho ^ 2 ( u ( x ) ) ) } { ( b ^ 2 - \\rho ^ 2 ( u ( x ) ) ) ^ 3 } , \\end{align*}"}
+{"id": "725.png", "formula": "\\begin{align*} f ( t ) \\leq g ( t ) \\implies \\begin{cases} \\deg f < \\deg g , \\\\ \\deg f = \\deg g ~ ~ ( \\forall k ) \\left ( \\left | [ t ^ k ] f ( t ) \\right | \\leq \\left | [ t ^ k ] g ( t ) \\right | \\right ) . \\end{cases} \\end{align*}"}
+{"id": "4981.png", "formula": "\\begin{align*} S _ { N - 1 , i } ( \\nu _ j , \\nu _ j - 1 , \\lambda _ 4 , \\dots , \\lambda _ N ) = 0 , j = 1 , \\dots , i - 1 , i + 1 , \\ldots , N ; \\end{align*}"}
+{"id": "8349.png", "formula": "\\begin{align*} S ^ { M } _ { K } ( E ) = M ( n _ K ^ { - 1 } ( E ) ) = \\int _ { n _ K ^ { - 1 } ( E ) } \\phi ( x ) \\ , d \\mathcal { H } ^ { n - 1 } ( x ) \\end{align*}"}
+{"id": "137.png", "formula": "\\begin{align*} { | | \\varphi | | } ^ p _ { W ^ { t , p } } = \\int _ A \\int _ A \\frac { { | \\varphi ( x ) - \\varphi ( y ) | } ^ p } { { | x - y | } ^ { n + t p } } \\ d x d y \\end{align*}"}
+{"id": "6003.png", "formula": "\\begin{align*} h _ 0 ( t ) = \\begin{cases} & O \\left ( \\frac { d \\gamma ^ 2 _ t } { \\sigma ^ 2 _ t } \\right ) , \\mbox { i f $ j = 1 $ } , \\\\ & O ( { N d ^ 2 \\gamma ^ 2 _ t } ) , \\mbox { i f $ j = 2 $ } . \\end{cases} \\end{align*}"}
+{"id": "6478.png", "formula": "\\begin{align*} H = \\frac { 1 } { p + q } \\big ( - \\frac { R _ 2 } { R _ 1 } p + \\frac { R _ 1 } { R _ 2 } q \\big ) \\nu , | A | ^ 2 = \\frac { R _ 2 ^ 2 } { R _ 1 ^ 2 } p + \\frac { R _ 1 ^ 2 } { R _ 2 ^ 2 } q . \\end{align*}"}
+{"id": "8702.png", "formula": "\\begin{align*} B _ G = P S _ G ( P ^ \\ast P ) ^ { - 1 } P ^ \\ast B _ G . \\end{align*}"}
+{"id": "6880.png", "formula": "\\begin{align*} \\mathcal { E } _ \\varphi = P _ { \\mathcal { R } ( \\theta _ \\varphi ) } \\mathcal { E } , \\mathcal { D } ( \\mathcal { E } _ \\varphi ) = \\mathcal { D } ( \\mathcal { E } ) \\cap \\mathcal { R } ( \\theta _ \\varphi ) . \\end{align*}"}
+{"id": "9171.png", "formula": "\\begin{align*} \\tilde { \\mathbf { E } } _ { \\beta , s } ^ { \\pm , ( k ) } \\coloneqq \\begin{cases} \\frac { ( \\tilde { E } ^ { \\pm } _ { \\beta , s } ) ^ { k } } { [ k ] _ { v _ { \\beta } } ! } & \\ \\beta = [ i , j ] \\\\ \\frac { ( \\tilde { E } ^ { \\pm } _ { \\beta , s } ) ^ { k } } { ( [ 2 ] _ { v } ! ) ^ { k } [ k ] _ { v _ { \\beta } } ! } & \\ \\beta = [ i , n , j ] \\end{cases} . \\end{align*}"}
+{"id": "7841.png", "formula": "\\begin{align*} \\nabla ^ { \\perp \\psi } _ { E _ a } B ^ j ( E _ a , H _ 2 ) = B ^ j ( \\nabla ^ { \\mathbb { C } P ^ p } _ { E _ a } E _ a , H _ 2 ) - H _ 2 . \\end{align*}"}
+{"id": "2503.png", "formula": "\\begin{align*} u ( t ) - \\Delta A ( t ) = u ^ { i n } \\ ; \\ ; \\ ; \\ ; \\ ; H ^ 1 ( \\Omega ) ' \\ , . \\end{align*}"}
+{"id": "6591.png", "formula": "\\begin{align*} S _ { r } ^ + = \\{ ( \\sigma + i t , \\xi + i \\eta ) ; \\ \\sigma \\geq 0 , \\ ( 1 - \\delta ) \\xi \\geq 0 ; \\ \\sigma + ( 1 - \\delta ) \\xi > r \\} . \\end{align*}"}
+{"id": "8625.png", "formula": "\\begin{align*} \\| v _ 1 ( 0 ; \\cdot ) \\| _ { L ^ \\infty ( \\mathbb { R } ) } = \\| u _ 0 ^ \\prime \\| _ { L ^ \\infty ( \\mathbb { R } ) } < C _ 1 q ^ { - 1 } ( 0 ) , \\end{align*}"}
+{"id": "7684.png", "formula": "\\begin{align*} \\Gamma ( s _ 0 ) = e \\ln \\left ( \\frac { \\lambda } { 2 M _ { \\infty } } \\right ) \\frac { 2 M _ { \\infty } } { \\lambda } . \\end{align*}"}
+{"id": "7342.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } U ^ { \\{ Y \\} } _ 0 & = Y ; \\\\ U ^ { \\{ Y \\} } _ { n + 1 } & = \\left ( U ^ { \\{ Y \\} } _ { n } \\odot _ \\phi ( V ^ 0 \\circ \\theta ^ n , \\Sigma ^ 0 \\circ \\theta ^ n ) \\right ) \\odot _ \\phi ( V ^ 1 \\circ \\theta ^ n , \\Sigma ^ 1 \\circ \\theta ^ n ) , \\ , n \\in \\N . \\end{array} \\right . \\end{align*}"}
+{"id": "6857.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } q _ n x ^ n = G ( x ) < e ^ { G ( x ) } & = \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( \\sum _ { n = 0 } ^ { \\infty } q _ n x ^ n \\right ) ^ k } { k ! } \\end{align*}"}
+{"id": "5562.png", "formula": "\\begin{align*} \\lambda _ i ^ \\ell = \\nu _ i ^ { 2 \\ell } + C _ 4 \\theta ^ { 2 \\ell } = \\nu _ i ^ { 2 \\ell } \\left ( 1 + C _ 4 \\left ( \\frac { \\theta } { \\nu _ i } \\right ) ^ { 2 \\ell } \\right ) , \\end{align*}"}
+{"id": "8158.png", "formula": "\\begin{align*} c _ j = \\min ( \\frac { M - \\sum _ { k = 1 } ^ { j - 1 } c _ k } { 2 } , B _ ) , 1 < j \\leq L . \\end{align*}"}
+{"id": "5310.png", "formula": "\\begin{align*} v _ i ^ u ( \\nu ) = v _ i ^ u + \\nu \\ , b _ i ^ u . \\end{align*}"}
+{"id": "5256.png", "formula": "\\begin{align*} T _ { i } = \\alpha p _ { i } + \\left ( 1 - \\alpha \\right ) q _ { i } \\end{align*}"}
+{"id": "4630.png", "formula": "\\begin{align*} u _ { n , k } ( x ) : = \\begin{cases} \\sin \\left ( 2 ^ { \\gamma m n } \\left ( x _ d - a _ { n , k } \\right ) \\right ) & \\mathrm { \\ f o r \\ } x ' \\in Q ( n , k ) , \\ , x _ d \\geq a _ { n , k } \\\\ 0 & \\mathrm { \\ e l s e } . \\end{cases} \\end{align*}"}
+{"id": "4520.png", "formula": "\\begin{align*} \\widetilde { \\mathbb { P } } ^ { ( 2 ) } _ { \\ell } : = \\mathbb { P } \\bigg [ \\sup _ { \\substack { X _ { \\ell - 1 } < n \\leqslant X _ { \\ell } } } \\widetilde { V } ^ { ( 2 ) } ( n ) > \\frac { T ( \\ell ) } { \\ell ^ { K / 2 } } \\bigg ] . \\end{align*}"}
+{"id": "424.png", "formula": "\\begin{align*} \\bold { p } _ { j } = \\bold { B } _ { j } \\bold { p } _ { j } + \\bold { q } _ { j } + \\bold { v } _ { j } , \\end{align*}"}
+{"id": "7653.png", "formula": "\\begin{align*} G ( m , n ; z ) = \\langle \\delta _ m , ( H _ { \\omega } - z ) ^ { - 1 } \\delta _ n \\rangle \\end{align*}"}
+{"id": "5298.png", "formula": "\\begin{align*} \\bar { K } _ 0 ( u ) = M _ 0 K _ 0 ( - u + 8 \\eta ) | _ { ( c , c _ 1 , c _ 2 , c _ 3 ) \\rightarrow \\ , ( c ' , c ' _ 1 , c ' _ 2 , c ' _ 3 ) } , \\end{align*}"}
+{"id": "1823.png", "formula": "\\begin{align*} \\Lambda ^ 2 V ^ * : = \\Lambda ^ 2 _ 7 \\oplus \\Lambda ^ 2 _ { 1 4 } , \\end{align*}"}
+{"id": "5171.png", "formula": "\\begin{align*} D _ { \\beta } ( p \\| q ) = \\frac { 1 } { \\beta ( \\beta - 1 ) } \\sum _ { i } p ^ { \\beta } _ { i } - \\frac { 1 } { \\beta - 1 } \\sum _ { i } p _ { i } q ^ { \\beta - 1 } _ { i } + \\frac { 1 } { \\beta } q ^ { \\beta } _ { i } \\end{align*}"}
+{"id": "3385.png", "formula": "\\begin{align*} C _ { 1 } = \\frac { c _ { 1 } } { 2 4 } ; C _ { 2 } = \\frac { 1 } { 2 6 c _ { 2 } } ; C _ { 3 } = \\frac { 1 } { 5 2 c _ { 2 } } ; A = \\gamma + \\frac { 2 \\sigma ^ { p } } { c _ { 2 } } \\end{align*}"}
+{"id": "5547.png", "formula": "\\begin{align*} \\delta _ { i , \\ell } = \\min _ { \\nu _ j \\neq \\nu _ i } \\left | 1 - \\left ( \\frac { \\nu _ j } { \\nu _ i } \\right ) ^ { 2 \\ell } \\right | . \\end{align*}"}
+{"id": "7180.png", "formula": "\\begin{align*} F _ v ( x , t ) & = g _ 1 ( t ) * F _ v ( \\cdot , 0 ) + \\int _ 0 ^ t e ^ { ( 1 - v _ 0 ^ 2 ) ( t - \\tau ) } g _ 1 ( t - \\tau ) * \\left ( ( ( v _ 0 ^ 2 - ( v _ 0 + V ) ^ 2 ) + \\varphi _ { 1 } ) F _ v - F _ w + \\varphi _ 3 \\right ) d \\tau . \\end{align*}"}
+{"id": "8017.png", "formula": "\\begin{align*} & \\langle \\mathcal { M } _ { f , k _ 1 } ^ N , \\mathcal { M } _ { g , k _ 2 } ^ N \\rangle ( t ) = \\\\ & \\int _ 0 ^ t \\mathcal { L } ^ N \\left ( \\mu _ { s , k _ 1 } ^ N ( f ) \\mu _ { s , k _ 2 } ^ N ( g ) \\right ) - \\mu _ { s , k _ 1 } ^ N ( f ) \\mathcal { L } ^ N \\left ( \\mu _ { s , k _ 2 } ^ N ( g ) \\right ) - \\mu _ { s , k _ 2 } ^ N ( g ) \\mathcal { L } ^ N \\left ( \\mu _ { s , k _ 1 } ^ N ( f ) \\right ) d s . \\end{align*}"}
+{"id": "8040.png", "formula": "\\begin{align*} \\varepsilon _ 5 ^ N ( \\vec { f } ) = & | \\eta _ 0 ^ N ( \\vec { f } ) | + K _ 6 T \\| \\vec { f } \\| _ \\infty ^ 2 + \\frac { \\gamma _ N T } { N } K _ 7 \\| \\vec { f } \\| _ \\infty ^ 3 \\exp \\left ( 2 \\| \\vec { f } \\| _ \\infty \\frac { \\gamma _ N } { N } \\right ) \\\\ & + \\frac { K _ 8 T } { \\gamma _ N } \\| \\vec { f } \\| _ \\infty + \\max _ { 0 \\leq t \\leq T } \\mathcal { Z } _ t ^ N ( \\vec { f } ) , \\end{align*}"}
+{"id": "2219.png", "formula": "\\begin{align*} \\hat v = { 1 \\over \\lambda ^ 2 + 2 i \\lambda + 3 } \\hat g ' \\equiv R ( \\lambda ) \\hat g ' . \\end{align*}"}
+{"id": "6172.png", "formula": "\\begin{align*} \\begin{aligned} & f ( x ) - f ( x ^ { k + 1 } ) + ( x - x ^ { k + 1 } ) ^ T [ - A ^ T \\hat { \\lambda } ^ k + \\beta ^ k A ^ T ( A \\hat { x } ^ k - b ) \\\\ & + \\beta ^ k D ( x ^ { k + 1 } - \\hat { x } ^ k ) + \\frac { \\sigma ( 1 - \\tau ^ k ) } { \\tau ^ k } ( x ^ { k + 1 } - x ^ k ) ] \\geq \\frac { \\sigma } { 2 } \\| x ^ { k + 1 } - x \\| ^ 2 , ~ \\forall x . \\end{aligned} \\end{align*}"}
+{"id": "4363.png", "formula": "\\begin{align*} \\sup _ { s } & \\ \\sum _ { i \\in [ m ] } f _ i ( x , \\overline { u } ^ i ) + s _ i ( \\sup _ { u ^ i \\in \\mathcal { U } _ i } f _ i ( x , u ^ i ) - f _ i ( x , \\overline { u } ^ i ) ) , \\\\ & \\ \\sum _ { i \\in [ m ] } s _ i \\leq \\Gamma , \\\\ & \\ s \\in \\{ 0 , 1 \\} ^ m . \\end{align*}"}
+{"id": "502.png", "formula": "\\begin{align*} O \\left ( \\log { x } \\cdot \\frac { 2 x ^ { 1 / 2 } } { \\log { x } } \\right ) = O \\left ( x ^ { 1 / 2 } \\right ) \\subseteq o \\left ( \\sum _ { q \\leq x } { 1 } \\right ) . \\end{align*}"}
+{"id": "2397.png", "formula": "\\begin{align*} a _ { n , i , k } = \\frac { 1 } { ( 2 s + 4 - i ) ! } \\left ( ( t + k ) ^ { 2 s + 4 } A _ n ( t ) \\right ) ^ { ( 2 s + 4 - i ) } \\big | _ { t = - k } . \\end{align*}"}
+{"id": "9085.png", "formula": "\\begin{align*} \\begin{array} { l l l } X ^ { * } = ( I - J ^ { ( r ) } \\hat { C } C \\hat { C } ^ { - 1 } J ^ { ( r ) } ) ^ { - 1 } \\cdot \\Big [ J ^ { ( r ) } \\Big ( ( \\hat { C } C \\hat { C } ^ { - 1 } - I ) \\underline { v } + \\hat { C } ( D p - U ( J ^ { ( r ) } X ^ { * } ) - \\mathbf { 1 } \\frac { \\beta } { 2 } ) \\Big ) \\Big ] . \\end{array} \\end{align*}"}
+{"id": "3690.png", "formula": "\\begin{align*} \\left \\{ 2 d - 2 \\sum _ { i = 1 } ^ d \\cos ( 2 \\pi m k / n ) : \\ , m = 0 , \\ldots , n - 1 \\right \\} . \\end{align*}"}
+{"id": "8128.png", "formula": "\\begin{align*} x = \\frac { f ' ( y _ 1 ) - f ' ( y _ 2 ) } { f ( y _ 1 ) f ' ( y _ 2 ) - f ( y _ 2 ) f ' ( y _ 1 ) } \\ , , y = \\frac { 2 ( f ( y _ 1 ) - f ( y _ 2 ) ) } { f ( y _ 1 ) f ' ( y _ 2 ) - f ( y _ 2 ) f ' ( y _ 1 ) } \\ , \\end{align*}"}
+{"id": "8427.png", "formula": "\\begin{align*} x = x _ 1 \\xrightarrow { s _ 1 } \\cdots \\xrightarrow { s _ n } x _ { n + 1 } , \\end{align*}"}
+{"id": "9092.png", "formula": "\\begin{align*} X _ + ( 0 ) \\geq 0 \\Longrightarrow X _ + ( t ) \\geq 0 , \\mbox { f o r a l l $ t = 0 , 1 , 2 , \\ldots $ } . \\end{align*}"}
+{"id": "2033.png", "formula": "\\begin{align*} { \\rm d e t } \\begin{bmatrix} c _ 5 & c _ 4 & c _ 3 \\\\ c _ 4 & c _ 3 & c _ 2 \\\\ c _ 3 & c _ 2 & c _ 1 \\end{bmatrix} \\ , \\ , = \\ , \\ , c _ 1 c _ 3 c _ 5 - c _ 1 c _ 4 ^ 2 - c _ 2 ^ 2 c _ 5 + 2 c _ 2 c _ 3 c _ 4 - c _ 3 ^ 3 = 0 . \\end{align*}"}
+{"id": "2825.png", "formula": "\\begin{align*} \\phi ^ { ( 1 ) } _ { 1 } : = p _ { 1 } : \\approx 0 , \\phi ^ { ( 1 ) } _ { 2 } : = p _ { 2 } - p _ { 3 } + p _ { 4 } : \\approx 0 , \\end{align*}"}
+{"id": "6073.png", "formula": "\\begin{align*} & \\Lambda = 2 | \\phi _ { 2 , 1 } \\phi _ { 2 , 2 } | < | \\lambda _ 0 | \\leq 1 , \\\\ & \\left ( \\frac { \\omega _ { 2 , 2 } ^ 2 } { \\phi _ { 2 , 2 } ^ 2 } - 1 \\right ) | a | = \\frac { \\omega _ { 2 , 2 } ^ 2 } { \\phi _ { 2 , 2 } ^ 2 } \\sqrt { a ^ 2 - 1 } \\ ( > 0 ) . \\end{align*}"}
+{"id": "6359.png", "formula": "\\begin{align*} \\phi ^ { ( k ) } ( R ( 1 + \\tau ) ) & = R ^ { n - k } ( 1 + \\tau ) ^ { n - k } \\omega _ k ( R ( 1 + \\tau ) ) \\\\ & \\geq ( 1 + ( n - k ) \\tau ) R ^ { n - k } \\bigl ( \\omega _ k ( R ) + \\omega _ k ' ( R ) R \\tau \\bigr ) \\\\ & = ( 1 + ( n - k ) \\tau ) \\bigl ( 1 + R \\frac { \\omega _ k ' ( R ) } { \\omega _ k ( R ) } \\tau \\bigr ) \\phi ^ { ( k ) } ( R ) \\\\ & \\geq ( 1 - A _ k \\abs { \\tau } ) \\phi ^ { ( k ) } ( R ) . \\end{align*}"}
+{"id": "5577.png", "formula": "\\begin{align*} \\sum _ { x \\in [ n ] } f ( T , x ) = \\sum _ { x \\in [ n ] } \\phi _ j ( x ) \\nu _ i ^ { 2 t + 2 } \\phi _ i ( x ) = \\nu _ i ^ { 2 t + 2 } \\delta _ { i j } , \\end{align*}"}
+{"id": "195.png", "formula": "\\begin{align*} D _ j \\ = \\ \\{ x \\in \\Z [ 2 ] : x _ j = 0 \\} , \\bar D _ j \\ = \\ \\{ x \\in \\Z [ 2 ] : x _ j = 1 \\} . \\end{align*}"}
+{"id": "610.png", "formula": "\\begin{align*} B = \\frac { 1 } { 4 } \\beta \\left ( \\frac { 1 } { 2 } , \\frac { 3 } { 4 } \\right ) = \\int _ 0 ^ 1 \\frac { t ^ 2 } { \\sqrt { 1 - t ^ 4 } } \\ , d t = \\sum _ { n = 0 } ^ { \\infty } \\left ( \\frac { 1 } { 4 } \\right ) ^ n \\binom { 2 n } { n } \\frac { 1 } { 4 n + 3 } = \\frac { \\sqrt { 2 \\pi ^ 3 } } { \\Gamma ^ 2 \\left ( \\frac { 1 } { 4 } \\right ) } \\end{align*}"}
+{"id": "5305.png", "formula": "\\begin{align*} S ( \\nu ) = \\left \\{ j \\in N ^ { \\{ 0 , 1 \\} } : \\nu \\leq \\nu _ j \\right \\} , \\nu \\in \\mathbb { R } . \\end{align*}"}
+{"id": "7388.png", "formula": "\\begin{align*} \\frac { ( q - 1 ) ( p + q ) + p } { q } = \\frac { ( p - 1 ) ( p + q ) - q } { p } = p + q - 1 . \\end{align*}"}
+{"id": "8834.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } S ( 0 ) [ \\phi ] ( x ) & = & \\phi ( x ) , \\\\ S ( t ) [ \\phi ] ( x ) & = & \\frac { \\exp ( - \\mu t ) } { \\sqrt { 4 d \\pi t } } \\int _ { - \\infty } ^ \\infty \\phi ( y ) \\exp \\left ( - \\frac { ( x + c t - y ) ^ 2 } { 4 d t } \\right ) { \\rm d } y , t > 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "4712.png", "formula": "\\begin{align*} \\mathcal { R } \\widetilde { \\mathcal { V } } ^ { ( 0 ) } ( \\psi ^ { ( \\leq 0 ) } ) \\ ; : = \\ ; ( 1 - \\mathcal { L } ) \\widetilde { \\mathcal { V } } ^ { ( 0 ) } ( \\psi ^ { ( \\leq 0 ) } ) \\ , . \\end{align*}"}
+{"id": "4603.png", "formula": "\\begin{align*} h _ x : = h _ { x , l } : = f _ l ( x ' ) - x _ d > 0 \\ , . \\end{align*}"}
+{"id": "1876.png", "formula": "\\begin{align*} \\lim _ { q \\to 0 } \\int _ 0 ^ \\infty r ^ { q } \\phi ( r ) d r = \\int _ 0 ^ \\infty \\phi ( r ) d r \\end{align*}"}
+{"id": "5826.png", "formula": "\\begin{align*} \\mathcal { M } _ \\Sigma ( u ) ^ A = \\bar { g } \\left ( \\vec { H } , \\frac { \\partial } { \\partial y ^ B } \\right ) ( h _ 0 ) ^ { B A } \\sqrt { \\frac { \\det g _ u } { \\det g _ 0 } } . \\end{align*}"}
+{"id": "5980.png", "formula": "\\begin{align*} \\begin{cases} & \\Delta u + h u = 0 ~ ~ ~ ~ ~ ~ ~ ~ M , \\\\ & u _ \\nu | _ { \\partial M } \\geq 0 \\end{cases} \\end{align*}"}
+{"id": "8359.png", "formula": "\\begin{align*} b \\cdot ( g , x ) = ( g b ^ { - 1 } , \\rho ( b ) \\cdot x ) = ( g b ^ { - 1 } , \\chi ( b ) x ) ( g \\in G , x \\in k , b \\in B ) . \\end{align*}"}
+{"id": "3366.png", "formula": "\\begin{align*} \\begin{aligned} \\left | \\mathbb { E } ^ { u ^ \\circ } \\left [ \\mu \\Phi ( x _ K ) \\right ] - \\mathbb { E } ^ { u } \\left [ \\mu \\Phi ( x _ K ) \\right ] \\right | \\leq e _ \\Phi ( \\hat { \\phi } ^ K d _ 0 + s \\sum _ { i = 1 } ^ { K - 1 } \\hat { \\phi } ^ i ) . \\end{aligned} \\end{align*}"}
+{"id": "3187.png", "formula": "\\begin{align*} I ^ { ( 1 ) } _ n ( \\mu ) : = \\int _ 0 ^ { \\frac { \\pi } { 6 n + 4 } } \\theta \\sin \\left ( \\mu \\theta \\right ) \\prod _ { k = 0 } ^ { n } \\cos ( ( 3 k + 1 ) \\theta ) \\cos ( ( 3 k + 2 ) \\theta ) \\mathrm { d } \\theta . \\end{align*}"}
+{"id": "7761.png", "formula": "\\begin{align*} \\Lambda ^ { ( j + 1 ) } = \\Lambda ^ { ( j ) } \\backslash \\mathcal R _ j ( \\tilde \\Lambda ) , \\ \\ \\mathcal R _ j ( \\tilde \\Lambda ) = \\cup _ { i \\in J _ j ( \\tilde \\Lambda ) } I ^ { ( j ) } _ i , \\end{align*}"}
+{"id": "4272.png", "formula": "\\begin{align*} \\Delta _ f \\hat v _ h + \\frac { \\hat v _ h } { 2 \\tau } = \\div _ f \\div _ f h , \\qquad \\int _ M \\hat v _ h e ^ { - f } \\ , d V = 0 . \\end{align*}"}
+{"id": "4936.png", "formula": "\\begin{align*} \\begin{gathered} \\frac { 1 } { 2 \\beta ^ { 1 / 2 } } [ L ] = e ^ { L } _ { + } - e ^ { L } _ { - } , \\\\ \\frac { 1 } { 2 \\beta ^ { 1 / 2 } } [ L ' ] = e ^ { L ' } _ { + } - e ^ { L ' } _ { - } , \\end{gathered} \\end{align*}"}
+{"id": "8649.png", "formula": "\\begin{align*} x ^ * \\in \\partial H ( x ) \\Leftrightarrow x \\in \\partial H ^ * ( x ^ * ) \\Leftrightarrow H ( x ) + H ^ * ( x ^ * ) = \\langle x , x ^ * \\rangle . \\end{align*}"}
+{"id": "2876.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to 0 ^ + } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ X = \\left [ - \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { X } . \\end{align*}"}
+{"id": "8356.png", "formula": "\\begin{align*} f ( r , \\theta ) = f \\Big ( \\frac { r } { \\rho _ L ( \\theta ) } \\rho _ L ( \\theta ) , \\theta \\Big ) & \\geq \\frac { r } { \\rho _ L ( \\theta ) } f \\Big ( \\rho _ L ( \\theta ) , \\theta \\Big ) + f ( 0 , \\theta ) \\left ( 1 - \\frac { r } { \\rho _ L ( \\theta ) } \\right ) \\\\ & \\geq f ( 0 , \\theta ) \\Big ( 1 - \\frac { r } { \\rho _ L ( \\theta ) } \\Big ) . \\end{align*}"}
+{"id": "1550.png", "formula": "\\begin{align*} \\| v \\| _ { L ^ A ( { \\mathcal O } ) } + \\| | D v | \\| _ { L ^ A ( { \\mathcal O } ) } , \\| g \\| _ { L ^ A ( { \\mathcal O } ) } = \\inf \\left \\{ s > 0 : \\int _ { \\mathcal O } A ( | g | / s ) \\ , d x \\le 1 \\right \\} . \\end{align*}"}
+{"id": "8143.png", "formula": "\\begin{align*} [ h _ { A _ 2 } ( J ) , k _ { A _ 2 } ( J ) ] & \\ = \\ D _ 1 + D _ 2 \\tau + D _ 3 \\mu + D _ 4 \\tau ^ 2 + D _ 5 \\tau \\mu + D _ 6 \\mu ^ 2 + D _ 7 \\tau ^ 2 \\mu \\ + \\\\ & D _ 8 \\tau \\mu ^ 2 + D _ 9 \\mu ^ 3 + D _ { 1 0 } \\tau ^ 3 \\mu + D _ { 1 1 } \\tau ^ 2 \\mu ^ 2 + D _ { 1 2 } \\tau \\mu ^ 3 \\ , \\end{align*}"}
+{"id": "6631.png", "formula": "\\begin{align*} H _ N S _ N ( i , j ) & = \\frac { 1 } { J _ N } ( - J _ { N - 3 } ( J _ 2 + J _ 3 ) + ( - 1 ) ^ { N - 2 } J _ 0 J _ 2 - J _ { N - 2 } ( J _ 1 + J _ 2 ) \\\\ & ~ + ( - 1 ) ^ { N - 1 } J _ 0 J _ 1 + 2 J _ { N - 1 } ) \\\\ & = \\frac { 1 } { J _ N } ( - 4 J _ { N - 3 } - 2 J _ { N - 2 } + 2 J _ { N - 1 } ) \\\\ & = 0 . \\end{align*}"}
+{"id": "4089.png", "formula": "\\begin{align*} R _ 0 ( \\lambda ) = 2 \\lambda R _ { \\Delta } ( \\lambda ) + R _ 0 ( - \\lambda ) \\end{align*}"}
+{"id": "8077.png", "formula": "\\begin{align*} ( n ) _ \\ast \\mu _ { n x } ^ u = \\mu _ x ^ u , \\end{align*}"}
+{"id": "7111.png", "formula": "\\begin{align*} b _ \\varepsilon - b & = I _ 1 + I _ 2 , I _ 1 : = E _ { \\varepsilon } ( 1 - \\mathbf { 1 } _ { B _ R } ) b , I _ 2 : = E _ { \\varepsilon } ( \\mathbf { 1 } _ { B _ R } b ) - \\mathbf { 1 } _ { B _ R } b \\end{align*}"}
+{"id": "9046.png", "formula": "\\begin{align*} \\{ a , b c \\} = \\{ a , b \\} c + ( - 1 ) ^ { ( p ( a ) + q ) p ( b ) } b \\{ a , c \\} \\end{align*}"}
+{"id": "3228.png", "formula": "\\begin{align*} B _ i : = \\{ b \\in B \\vert \\dim _ { k ( b ) } ( L _ b ) = i \\} \\end{align*}"}
+{"id": "1753.png", "formula": "\\begin{align*} a _ { j , k } = 0 \\quad \\quad \\forall \\ , 2 \\leq j , k \\leq n - 1 \\mbox { w i t h } j \\neq n + 1 - k . \\end{align*}"}
+{"id": "8723.png", "formula": "\\begin{gather*} f ( x ) = \\xi _ { 1 } f ( a _ { 1 } ) + \\xi _ { 2 } f ( a _ { 2 } ) + \\xi _ { 3 } f ( a _ { 3 } ) = \\\\ \\xi _ { 1 } a _ { 1 } + \\xi _ { 1 } \\alpha _ { 3 } a _ { 3 } + \\xi _ { 2 } a _ { 2 } + \\xi _ { 2 } \\beta _ { 3 } a _ { 3 } + \\xi _ { 3 } a _ { 3 } = \\\\ \\xi _ { 1 } a _ { 1 } + \\xi _ { 2 } a _ { 2 } + ( \\xi _ { 1 } \\alpha _ { 3 } + \\xi _ { 2 } \\beta _ { 3 } + \\xi _ { 3 } ) a _ { 3 } . \\end{gather*}"}
+{"id": "1555.png", "formula": "\\begin{align*} J _ n ( w , B _ { 2 R } ) = \\int _ { B _ { 2 R } } F _ n ( D w ) \\ , d x \\end{align*}"}
+{"id": "3144.png", "formula": "\\begin{align*} \\psi ^ * \\left ( \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ) = \\left ( \\begin{array} { c c c } a ^ { 2 \\ , p ^ { e _ 1 } } & 0 & b ^ { 2 \\ , p ^ { e _ 1 } } \\\\ a ^ { p ^ { e _ 1 } } \\ , c ^ { p ^ { e _ 1 } } & 1 & b ^ { p ^ { e _ 1 } } \\ , d ^ { p ^ { e _ 1 } } \\\\ c ^ { 2 \\ , p ^ { e _ 1 } } & 0 & d ^ { 2 \\ , p ^ { e _ 1 } } \\end{array} \\right ) . \\end{align*}"}
+{"id": "2961.png", "formula": "\\begin{align*} \\C _ { \\alpha , \\beta } ^ 2 \\left \\| \\frac { f } { r ^ 2 } \\right \\| _ { L _ \\beta ^ 2 } ^ 2 = \\left \\| \\L _ { \\alpha } f \\right \\| _ { L _ \\beta ^ 2 } ^ 2 - \\left \\| \\L _ { \\alpha } f + \\C _ { \\alpha , \\beta } \\frac { f } { r ^ 2 } \\right \\| _ { L _ \\beta ^ 2 } ^ 2 - 2 \\C _ { \\alpha , \\beta } \\left \\| \\frac { f ^ * } { r } \\right \\| _ { L _ \\beta ^ 2 } ^ 2 \\end{align*}"}
+{"id": "3940.png", "formula": "\\begin{align*} & \\operatorname { c o l } \\left \\{ u _ j \\right \\} _ { j = 1 } ^ N = \\mathcal { B } _ { N \\times N } ^ { - 1 } \\left [ - \\mathcal { G } _ N + K \\right ] y ^ { N } ( t ) , \\\\ & K = \\operatorname { d i a g } \\left \\{ K _ j \\right \\} _ { j = 1 } ^ N \\in \\mathbb { R } ^ { N \\times 3 N } , \\end{align*}"}
+{"id": "5091.png", "formula": "\\begin{align*} ( \\overline { \\mathsf { a } } _ s \\cdot \\varepsilon _ { \\theta } ) ( t u s ) & = q ^ { - 1 } \\theta ( t ) \\sum _ { t '' \\in T ( \\mathbb { F } _ { q ^ k } ) } \\theta ( t '' ) \\psi ( t '' ) . \\end{align*}"}
+{"id": "2248.png", "formula": "\\begin{align*} \\Phi ^ * \\eta ^ { \\alpha } _ L = \\eta ^ { \\alpha } _ L \\ , , \\Phi ^ * E _ L = E _ L \\ , . \\end{align*}"}
+{"id": "4963.png", "formula": "\\begin{align*} L _ { j k } ( \\lambda , \\nu ) = \\begin{pmatrix} a ( \\lambda , \\nu ) \\pi _ k ^ { + } + b ( \\lambda , \\nu ) \\pi _ k ^ { - } & c \\sigma _ k ^ { - } \\\\ c \\sigma _ k ^ { + } & b ( \\lambda , \\nu ) \\pi _ k ^ { + } + a ( \\lambda , \\nu ) \\pi _ k ^ { - } \\end{pmatrix} _ { [ \\mathcal { V } _ j ] } . \\end{align*}"}
+{"id": "6206.png", "formula": "\\begin{align*} u _ t + f ( u ) _ x = \\left ( D ( u ) u _ x \\right ) _ x + g ( u ) , \\end{align*}"}
+{"id": "8321.png", "formula": "\\begin{align*} { \\sf N } _ k ( Q _ 1 \\odot \\ldots \\odot Q _ k ) = P \\{ \\{ \\ldots \\{ { \\sf n } Q _ 1 , Q _ 2 \\} , \\ldots \\} , Q _ k \\} , \\end{align*}"}
+{"id": "723.png", "formula": "\\begin{align*} d _ { \\chi } ( \\mathcal { G } , \\mathcal { H } ) = | \\chi ( \\mathcal { G } ) - \\chi ( \\mathcal { H } ) | . \\end{align*}"}
+{"id": "3450.png", "formula": "\\begin{align*} F ( a , \\phi ) = ( d ^ + a , \\ ; D \\phi , \\ ; d ^ { \\ast } a , \\ ; \\Pi ^ - \\circ r _ 0 ( a , \\phi ) , \\ ; \\Pi ^ - \\circ r _ 1 ( a , \\phi ) , \\ ; \\Pi _ 2 \\circ r _ 0 ( a , \\phi ) , \\ ; \\Pi _ 2 \\circ r _ 1 ( a , \\phi ) ) \\end{align*}"}
+{"id": "6937.png", "formula": "\\begin{align*} \\forall \\tau _ p \\in \\left [ - \\eta , \\varepsilon \\right ] , \\forall t \\in [ - \\ell _ { p } , \\ell _ { p } ] , \\Im ( \\gamma _ { p } ( t ) ) = t , \\Re ( \\gamma _ { p } ( t ) ) = h _ { p } ( t ) : = \\Psi ^ { - 1 } \\left ( \\Psi ( \\tau _ p ) - A _ I t ^ { 2 \\mu } \\right ) . \\end{align*}"}
+{"id": "1497.png", "formula": "\\begin{align*} & \\alpha \\circ P _ { d } = P _ { d } \\circ \\alpha \\\\ & \\Longleftrightarrow \\forall m \\in \\mathbb { Z } , ~ q ^ { m } + q ^ { - m } = q ^ { m + d } + q ^ { - m - d } \\Longleftrightarrow \\forall m \\in \\mathbb { Z } , ~ q ^ m ( 1 - q ^ d ) = q ^ { - m - d } ( 1 - q ^ d ) \\\\ & \\begin{array} [ b ] { l } \\\\ \\Longleftrightarrow \\forall m \\in \\mathbb { Z } , ~ q ^ m = q ^ { - m - d } \\end{array} \\Longleftrightarrow \\forall m \\in \\mathbb { Z } , ~ q ^ { 2 m + d } = 1 , \\end{align*}"}
+{"id": "734.png", "formula": "\\begin{align*} g _ 0 ( A ) & = t _ { r - 1 , q } - a \\\\ & = ( a - 1 ) b + a \\bigl ( v ( r - 1 ) - 1 \\bigr ) + \\frac { v a ( a - r ) J _ { k - 1 } ( v ) } { J _ k ( v ) } \\ , . \\end{align*}"}
+{"id": "5865.png", "formula": "\\begin{align*} \\dfrac { M _ { 2 } ( \\mathcal { N C } ( D _ { 2 m } ) ) } { | e ( \\mathcal { N C } ( D _ { 2 m } ) ) | } = \\dfrac { 8 m ^ { 4 } - 4 0 m ^ { 3 } + 6 4 m ^ { 2 } - 3 2 m } { 3 m ( m - 2 ) } . \\end{align*}"}
+{"id": "6680.png", "formula": "\\begin{align*} A = F _ j B = F _ { j + 1 } = \\Phi ( F _ j ) = A ^ p ; \\end{align*}"}
+{"id": "7662.png", "formula": "\\begin{align*} K ( m , u ) = \\frac { 2 ^ { s } M ^ s _ { \\infty } } { \\lambda ^ s } \\delta _ { | m - u | = 1 } , \\ , \\ , \\ , \\psi ( m ) = \\frac { 2 ^ { s } M _ { \\infty } } { \\lambda ^ s } \\delta _ { m , n } . \\end{align*}"}
+{"id": "5553.png", "formula": "\\begin{align*} \\theta _ 1 & = \\sqrt { \\frac { n ^ { k / 2 } \\prod _ { j = 1 } ^ k \\norm * { w ^ { ( j ) } } _ 4 ^ 2 } d } , & \\theta _ 2 & = \\frac { n ^ { k / 2 } \\prod _ { j = 1 } ^ k \\norm * { w ^ { ( j ) } } _ \\infty } d \\\\ K & = n ^ { k / 2 } \\prod _ { j = 1 } ^ k \\frac { \\norm { w ^ { ( j ) } } _ { \\infty } ^ 2 } { \\norm { w ^ { ( j ) } } _ 4 ^ 2 } , & \\kappa ^ { ( i ) } & = \\sqrt { n } \\| w ^ { ( i ) } \\| _ { \\infty } \\leq \\sqrt { n } \\max _ i \\| w ^ { ( i ) } \\| _ { \\infty } \\eqqcolon \\kappa _ { \\mathrm { m a x } } . \\end{align*}"}
+{"id": "4523.png", "formula": "\\begin{align*} \\partial _ t n + \\sum _ { i = 1 } ^ N \\partial _ { s _ i } \\big [ g _ i ( [ s ] _ N ) n \\big ] + p _ N ( [ s ] _ N , n _ N ) n _ N = 0 . \\end{align*}"}
+{"id": "5035.png", "formula": "\\begin{align*} a _ { j j } ^ S = \\frac { 1 - \\beta p _ { j j } + \\beta \\mathbf { A } _ { j S } ^ S \\mathbf { P } _ { S j } } { 1 - \\beta } . \\end{align*}"}
+{"id": "6509.png", "formula": "\\begin{align*} \\frac { 1 } { n } h _ { K ( 0 ) } ( \\xi ) d S _ { K ( 0 ) } ( \\xi ) = \\varphi d \\xi , \\end{align*}"}
+{"id": "3855.png", "formula": "\\begin{align*} \\tau ^ { \\kappa } ( x , y \\mid t ) = \\frac { \\kappa ( 1 - \\kappa ) \\eta ^ 0 ( x , y \\mid t ) + \\kappa \\pi ^ * _ x G ( \\pi ^ * ) _ { x , y } } { 2 \\kappa ( 1 - \\kappa ) + \\kappa ^ 2 } , \\end{align*}"}
+{"id": "9044.png", "formula": "\\begin{align*} T S ^ i - S ^ i T = 0 , S ^ i S ^ j + S ^ j S ^ i = 2 \\delta _ { i , j } T ( i , j \\in [ N ] ) . \\end{align*}"}
+{"id": "7892.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { p - 1 } R _ { \\mathbf { s } _ { n } } ( \\tau ) & = \\sum _ { n = 0 } ^ { p - 1 } \\sum _ { i = 0 } ^ { p ^ m - 1 - \\tau } \\omega _ { q } ^ { f _ { n , i } - f _ { n , i + \\tau } } \\\\ & = \\sum _ { i = 0 } ^ { p ^ m - 1 - \\tau } \\sum _ { n = 0 } ^ { p - 1 } \\omega _ { q } ^ { f _ { n , i } - f _ { n , j } } ( ) , \\end{align*}"}
+{"id": "8140.png", "formula": "\\begin{align*} J _ 1 & = p _ x \\ , & J _ 2 & = p _ y \\ , & J _ 3 & = q _ x p _ x \\ , \\\\ [ 3 p t ] J _ 4 & = q _ y p _ x \\ , & J _ 5 & = q _ x p _ y \\ , & J _ 6 & = q _ y p _ y \\ , \\\\ [ 3 p t ] J _ 7 & = q _ x \\left ( q _ x p _ x + q _ y p _ y + 3 \\nu \\right ) \\ , & J _ 8 & = q _ y \\left ( q _ x p _ x + q _ y p _ y + 3 \\nu \\right ) \\ , & & \\end{align*}"}
+{"id": "4739.png", "formula": "\\begin{align*} [ j ] _ \\mathcal { M } ( u , v , z , w ) = \\begin{cases} ( \\langle w , j ^ A _ { u , v } ( z ) \\rangle ) _ \\circ & \\exists x , y \\in X \\left ( u \\in A x \\land v \\in A y \\land z = _ X x - _ X y \\right ) , \\\\ ( \\langle w , \\tilde { j } ( z ) \\rangle ) _ \\circ & , \\end{cases} \\end{align*}"}
+{"id": "8188.png", "formula": "\\begin{align*} & _ { B _ 1 , B _ 2 ^ 0 , B _ 2 ^ 1 } \\ f ( B _ 1 , B _ 2 ^ 0 , B _ 2 ^ 1 ) \\\\ & \\frac { 1 } { 2 } \\left ( B _ 1 + \\frac { B _ 1 } { M } B _ 2 ^ 1 + ( 1 - \\frac { B _ 1 } { M } ) B _ 2 ^ 0 \\right ) \\leq B _ , \\\\ & \\ \\ 0 \\leq B _ 1 \\leq M , \\ 0 \\leq B _ 2 ^ 1 \\leq B _ 1 , \\ 0 \\leq B _ 2 ^ 0 \\leq M - B _ 1 . \\end{align*}"}
+{"id": "6013.png", "formula": "\\begin{align*} c ^ { A B } + c ^ { B C } + c ^ { C A } = c ^ { B A } + c ^ { C B } + c ^ { A C } . \\end{align*}"}
+{"id": "3114.png", "formula": "\\begin{align*} \\varphi ^ * \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & : = \\left ( \\begin{array} { c c c } a ^ { \\ell _ 1 } & 0 & 0 \\\\ 0 & a ^ { \\ell _ 1 } & a ^ { \\ell _ 1 } \\ , b ^ { p ^ { e _ 1 } } \\ , d ^ { p ^ { e _ 1 } } \\\\ 0 & 0 & d ^ { - \\ell _ 3 } \\end{array} \\right ) . \\end{align*}"}
+{"id": "2573.png", "formula": "\\begin{gather*} b ' _ { k l } = \\begin{cases} - b _ { k l } , & k = i l = i , \\\\ b _ { k l } + \\frac { | b _ { k i } | b _ { i l } + b _ { k i } | b _ { i l } | } { 2 } , & . \\end{cases} \\end{gather*}"}
+{"id": "4031.png", "formula": "\\begin{align*} \\begin{cases} x _ { 1 } ' & = \\ ; x _ { 2 } ' = y _ { 1 } ' = 0 \\\\ y _ { 2 } ' & = \\ ; \\left ( x _ { 1 } + x _ { 2 } \\right ) \\left ( y _ { 1 } + y _ { 2 } \\right ) \\end{cases} \\end{align*}"}
+{"id": "2693.png", "formula": "\\begin{align*} \\Phi ^ { ( 2 ) } _ { i } : = \\{ \\Phi ^ { ( 1 ) } _ { i } , H _ { T } \\} \\approx - P ^ { ( 1 ) } _ { i } - \\frac { \\partial f _ { i } } { \\partial Q _ { ( 1 ) } ^ { j } } Q _ { ( 2 ) } ^ { j } + \\frac { \\partial g } { \\partial Q _ { ( 2 ) } ^ { i } } \\approx 0 , \\end{align*}"}
+{"id": "4809.png", "formula": "\\begin{align*} \\begin{gathered} \\mathcal { Q } ( \\widehat { \\mathbf { C } } ) : = \\Big \\{ \\mathbb { Q } \\in \\mathcal { Q } ( \\mathbb { R } ^ { n } ) : \\ ; \\mathbb { Q } \\{ \\mathbf { c } \\in \\mathcal { S } _ 0 \\} = 1 , \\ ; W _ 1 ( \\widehat { \\mathbb { Q } } _ K , \\mathbb { Q } ) \\leq \\varepsilon _ K \\Big \\} . \\\\ \\end{gathered} \\end{align*}"}
+{"id": "5226.png", "formula": "\\begin{align*} D A G I ( p \\| q ) = \\sum _ { j } p _ { j } \\left [ \\overline { M A } - \\overline { M G } \\right ] = \\sum _ { j } p _ { j } \\left [ 1 - \\overline { M G } \\right ] \\end{align*}"}
+{"id": "7115.png", "formula": "\\begin{align*} - \\Delta u + \\lambda u - \\gamma ( \\Phi \\ast | u | ^ { 2 } ) u = f ( u ) , \\ \\forall x \\in \\mathbb { R } ^ { 2 } . \\end{align*}"}
+{"id": "6824.png", "formula": "\\begin{align*} \\sum _ { i , j } \\beta _ { i j } k _ { i j } ^ { N - m - 1 } + K _ { N - m - 1 } = 0 \\qquad \\forall m \\in \\{ 2 , \\ldots , N - 3 \\} . \\end{align*}"}
+{"id": "8001.png", "formula": "\\begin{align*} I _ { d y n } ( W ) = \\sup _ { F , G , H \\in C ^ { 1 , 0 } ( [ 0 , T ] \\times \\mathbb { T } ) } \\left \\{ \\mathcal { I } _ 1 ( W , F , G , H ) \\right \\} . \\end{align*}"}
+{"id": "1940.png", "formula": "\\begin{align*} Y _ { j + 1 } \\leq K b ^ { j } ( Y _ { j } ^ { 1 + \\delta _ 1 } + Y _ { j } ^ { 1 + \\delta _ 2 } ) , ~ ~ j = 0 , 1 , 2 , \\ldots , \\end{align*}"}
+{"id": "2798.png", "formula": "\\begin{align*} L _ { 1 } = \\dot { q } ^ { 1 } \\dot { q } ^ { 3 } + \\frac { 1 } { 2 } q ^ { 2 } \\left ( q ^ { 3 } \\right ) ^ { 2 } . \\end{align*}"}
+{"id": "4187.png", "formula": "\\begin{align*} \\varphi ( g _ 1 g ^ { - 1 } _ 2 ) \\varphi ( g _ 2 g _ 3 ) \\overset { { \\bf A 6 } } { = } \\varphi ( g _ 1 ) \\varphi ( g ^ { - 1 } _ 2 ) \\varphi ( g _ 2 ) \\varphi ( g _ 3 ) \\overset { { \\bf A 2 } } { = } \\varphi ( g _ 1 ) \\varphi ( g _ 3 ) \\overset { ( \\ref { e q : a n t i 3 } ) } { = } \\varphi ( g _ 3 g _ 1 ) \\ , , \\end{align*}"}
+{"id": "2359.png", "formula": "\\begin{align*} \\mathcal { C } _ { q } = 3 ^ { - \\frac { 7 } { 4 } } q ^ { \\frac { 3 } { q } } \\left ( q - 2 \\right ) ^ { \\frac { 3 } { 2 q } } \\left ( \\frac { q } { q - 2 } \\right ) ^ { \\frac { 3 } { 4 } } ( 4 \\pi ( 1 - \\tau ) ) ^ { - \\frac { 3 } { 2 } ( \\frac { 1 } { 3 } - \\frac { 1 } { q } ) } e ^ { - 6 ( \\frac { 1 } { 3 } - \\frac { 1 } { q } ) } . \\end{align*}"}
+{"id": "826.png", "formula": "\\begin{align*} R _ r ^ m { \\Psi _ r } { \\Psi ^ m } = { k } \\{ { \\Psi _ m } { \\Psi ^ m } { F ^ 2 } - { \\Psi _ i } { y ^ i } { \\Psi ^ m } { y _ m } \\} . \\end{align*}"}
+{"id": "5420.png", "formula": "\\begin{align*} S _ { 2 n } f ( T ^ { - n } ( \\tfrac { 1 } { 2 } ) ) = - f ( T ^ { n } ( \\tfrac { 1 } { 2 } ) ) . \\end{align*}"}
+{"id": "835.png", "formula": "\\begin{align*} { \\pounds } _ { \\hat { X } } I _ { k } = ( \\nabla _ { i } X ^ { i } + I _ { i } \\nabla _ { 0 } X ^ { i } ) _ { . k } ) = f _ { . k } , \\end{align*}"}
+{"id": "248.png", "formula": "\\begin{align*} \\frac { d b _ 2 } { d x _ 2 } = B \\in \\mathbb { R } . \\end{align*}"}
+{"id": "1630.png", "formula": "\\begin{align*} \\partial _ { \\nu } u \\mid _ { S _ { T } } = \\partial _ { \\nu } m \\mid _ { S _ { T } } = 0 , \\end{align*}"}
+{"id": "7530.png", "formula": "\\begin{align*} \\lim \\limits _ { r \\to \\infty } \\frac { h _ i ( r ) } { r ^ { \\gamma _ i } } = \\frac { k _ i } { \\gamma _ i - 1 } \\ , \\end{align*}"}
+{"id": "940.png", "formula": "\\begin{align*} h _ V ( g ) = g \\quad \\mbox { q . e . o n } V ^ c . \\end{align*}"}
+{"id": "6419.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ n F ( X _ { \\frac { i - 1 } { n } } ) H ( n ^ { 1 / \\alpha _ 0 } \\Delta _ i ^ n L ) \\xrightarrow [ n \\to \\infty ] { \\mathcal { L } - s } \\Sigma ^ { 1 / 2 } \\mathcal { N } . \\end{align*}"}
+{"id": "556.png", "formula": "\\begin{align*} P ( t ) = H ' ( t ) / H ( t ) = \\sum p _ r t ^ r \\ ; \\Rightarrow \\ ; H ( t ) = \\exp ( p _ 1 \\cdot \\sum \\wp _ n \\frac { t ^ n } { n } ) \\ ; , \\end{align*}"}
+{"id": "7393.png", "formula": "\\begin{align*} C _ 3 : = \\frac { 1 } { 2 } \\min \\{ 2 C _ 1 , C _ { 3 1 } , C _ { 3 2 } , C _ { 3 3 } , C _ { 3 4 } \\} , \\end{align*}"}
+{"id": "1273.png", "formula": "\\begin{align*} \\rho ( t , x ) = \\frac { 1 } { \\sqrt { ( 2 \\pi ) ^ d \\det \\Sigma ( t ) } } \\exp \\Big [ - \\frac { 1 } { 2 } ( x - \\mu ( t ) ) ^ T \\Sigma ^ { - 1 } ( t ) ( x - \\mu ( t ) ) \\Big ] \\end{align*}"}
+{"id": "7229.png", "formula": "\\begin{align*} - h ( V _ R ) \\phi ( V _ R ) \\left ( p ( V _ R ^ - ) - p ( V _ R ^ + ) \\right ) + a \\phi ( V _ R ) \\left ( \\partial _ v p ( V _ R ^ - ) - \\partial _ v p ( V _ R ^ + ) + \\partial _ v p ( V _ F ) \\right ) = 0 . \\end{align*}"}
+{"id": "6019.png", "formula": "\\begin{align*} \\delta : = \\tfrac 2 3 \\sqrt { ( E _ A - E _ C ) ^ 2 + ( E _ B - E _ C ) ^ 2 - ( E _ A - E _ C ) ( E _ B - E _ C ) } \\ , . \\end{align*}"}
+{"id": "4477.png", "formula": "\\begin{align*} \\int _ { \\{ z \\in D : \\psi ( z ) = r \\} } F _ 0 \\overline { F _ 0 - f } e ^ { - \\varphi } \\left ( \\frac { \\partial \\psi } { \\partial v _ z } \\right ) ^ { - 1 } | d z | = 0 \\end{align*}"}
+{"id": "4385.png", "formula": "\\begin{align*} \\inf _ { k \\in [ m ] _ 0 } \\left \\{ \\inf _ { x \\in \\mathcal { X } } \\left \\{ \\Gamma \\theta ^ k ( x ) + \\sum _ { i \\in [ m ] } \\max \\{ 0 , x _ i - \\overline { b } _ i , \\max \\{ 0 , x _ i - \\overline { b } _ i + \\Delta b _ i \\} - \\theta ^ k ( x ) \\} \\right \\} \\right \\} \\end{align*}"}
+{"id": "6906.png", "formula": "\\begin{align*} - \\Delta _ p u = u ^ { p ^ * - 1 } \\quad \\R ^ n , \\end{align*}"}
+{"id": "6831.png", "formula": "\\begin{align*} \\widetilde { h _ { S O ( n ) } } ( x _ 1 , \\ldots x _ { \\frac { ( n - 1 ) ( n - 2 ) } { 2 } } , z _ 1 , \\ldots z _ { n - 1 } ) : = \\sum _ { i , j } & c _ { i j } z _ 1 ^ { k _ { i j } ^ 1 } x _ 1 ^ { m _ { i j } ^ 1 } ( 1 - x _ 1 ^ 2 ) ^ { \\frac { n _ { i j } ^ 1 } { 2 } } \\cdots x _ { n - 1 } ^ { m _ { i j } ^ { n - 1 } } ( 1 - x ^ 2 _ { n - 1 } ) ^ { \\frac { n _ { i j } ^ { n - 1 } } { 2 } } \\\\ & \\cdot ( \\widetilde { h _ { S O ( n - 1 ) } } ) _ { i j } ( x _ n , \\ldots , x _ { \\frac { ( n - 1 ) ( n - 2 ) } { 2 } } , z _ { n - 2 } , \\ldots , z _ { n - 1 } ) . \\end{align*}"}
+{"id": "777.png", "formula": "\\begin{align*} x ^ { * _ \\beta } = \\beta ( a ^ { - 1 } b , a ^ { - 1 } ) x ^ * , x \\in H _ { a , b } , \\end{align*}"}
+{"id": "500.png", "formula": "\\begin{align*} \\alpha _ i = r _ i , \\beta _ i = \\frac { e _ i + r _ i i - \\overline { f } ( i ) } { d } \\bmod { s } . \\end{align*}"}
+{"id": "3525.png", "formula": "\\begin{align*} \\hat \\Phi \\cdot [ A ] = [ B ] & \\Longleftrightarrow \\big ( \\Phi _ B \\ \\hat \\Phi \\ \\Phi _ A ^ { - 1 } \\big ) \\cdot [ \\Phi _ A \\cdot A ] = [ \\Phi _ B \\cdot B ] \\\\ & \\Longleftrightarrow \\hat \\Phi ' \\cdot [ A ' ] = [ B ' ] , \\end{align*}"}
+{"id": "1411.png", "formula": "\\begin{align*} \\frac { 1 } { \\Gamma ( \\nu ) } \\int ^ x _ 0 ( x - u ) ^ { \\nu - 1 } d F ( u ) \\nu \\ge 0 \\end{align*}"}
+{"id": "5666.png", "formula": "\\begin{align*} \\left ( \\frac { \\| \\boldsymbol { a } _ \\perp ^ { ( 3 ) } \\| ^ 2 \\tan ^ 2 \\theta } { \\| \\boldsymbol { a } ^ { ( 3 ) } \\| ^ 2 } , \\| \\boldsymbol { a } _ \\perp ^ { ( 3 ) } \\| ^ 2 \\tan ^ 2 \\theta \\right ) _ v = 1 \\end{align*}"}
+{"id": "2340.png", "formula": "\\begin{align*} C _ { 0 } K ( c , \\gamma ) < \\begin{cases} 1 , & p = 2 , \\\\ \\min \\Big \\{ \\frac { 1 } { 2 } , \\frac { 2 ( p - 2 ) } { p ^ { 2 } } \\Big \\} , & p > 2 , \\end{cases} \\end{align*}"}
+{"id": "3491.png", "formula": "\\begin{align*} \\theta ( 1 ) \\theta ( y z ) = \\theta ( y ) \\theta ( z ) . \\end{align*}"}
+{"id": "2026.png", "formula": "\\begin{align*} f ( R ) & \\geq f ( \\sigma ^ n A ) \\geq ( \\alpha A ^ \\gamma + \\beta ) ^ n f ( A ) \\\\ & \\geq ( \\alpha A ^ \\gamma + \\beta ) ^ { \\log _ \\sigma \\left ( \\frac { R } { A } \\right ) } \\frac { f ( A ) } { \\alpha A ^ \\gamma + \\beta } \\\\ & = \\left ( \\frac { R } { A } \\right ) ^ { \\log _ \\sigma ( \\alpha A ^ \\gamma + \\beta ) } \\frac { f ( A ) } { \\alpha A ^ \\gamma + \\beta } \\end{align*}"}
+{"id": "7469.png", "formula": "\\begin{align*} H e s s _ { g _ N } f _ N = \\frac { 1 } { 2 } \\mathcal { L } _ { \\nabla f _ N } g _ N = R i c ( g _ N ) + \\lambda { g _ N } , \\end{align*}"}
+{"id": "6153.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau ^ k } [ f ( x ) - f ( x ^ { k + 1 } ) ] - \\frac { 1 } { \\tau ^ { k - 1 } } [ f ( x ) - f ( x ^ { k } ) ] + ( x - \\bar { x } ^ { k + 1 } ) ^ T [ - A ^ T \\lambda ^ k \\\\ & + \\beta ^ k A ^ T ( A \\hat { x } ^ { k } - b ) + \\beta ^ k D ( x ^ { k + 1 } - \\hat { x } ^ k ) ] \\geq 0 , ~ \\forall x . \\end{aligned} \\end{align*}"}
+{"id": "4588.png", "formula": "\\begin{align*} \\mathcal { F } ( \\mathfrak { E } , [ \\omega ] ) = & - \\int _ M ( s _ g - s _ 0 ) ^ 2 d \\mu \\\\ = & - \\frac 1 h \\int _ M t s _ g d \\mu + s _ 0 \\frac 1 h \\int _ M t d \\mu \\\\ = & - \\frac 1 h T _ s + s _ 0 \\frac 1 h T _ 1 . \\end{align*}"}
+{"id": "4108.png", "formula": "\\begin{align*} Q ( \\theta ) \\phi ^ { ( r ) } ( \\theta ) + \\sum _ { i \\in \\mathcal { J } } ( \\mu ^ { ( r ) } _ i - \\lambda _ i ) \\xi _ i ( \\theta ) \\big ( \\phi ^ { ( r ) } ( \\theta ) - \\phi ^ { ( r ) } _ i ( \\theta ) \\big ) = 0 , \\theta \\in \\R ^ J _ - , \\end{align*}"}
+{"id": "5752.png", "formula": "\\begin{align*} \\mathbf { L } \\Psi _ i = \\mathbf { L } ^ \\dagger \\Psi _ i = \\Gamma _ i \\Psi _ i . \\end{align*}"}
+{"id": "2221.png", "formula": "\\begin{align*} \\theta = { 3 - s \\over q } - { 1 \\over 2 } > 0 \\end{align*}"}
+{"id": "8708.png", "formula": "\\begin{align*} S _ G ( P ^ \\ast P ) \\hat \\sigma \\hat \\sigma ^ \\ast = S _ G ( I + R ) S _ G , \\end{align*}"}
+{"id": "6504.png", "formula": "\\begin{align*} L ^ 2 ( \\mathbb X ^ j ) : = \\left \\{ f : \\mathbb { X } ^ j \\to \\real , \\ ; \\int _ { \\mathbb { X } ^ j } | f ( x _ 1 , \\cdots , x _ j ) | ^ 2 d x _ 1 \\cdots d x _ j < + \\infty \\right \\} . \\end{align*}"}
+{"id": "3409.png", "formula": "\\begin{align*} \\delta _ R ( K ( p , q ) ) = \\underline { \\delta } _ R ( K ( p , q ) ) = \\bar { \\delta } _ R ( K ( p , q ) ) = - \\frac { \\sigma ( K ( p , q ) ) } { 1 6 } . \\end{align*}"}
+{"id": "5187.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } D _ { \\beta } I ( p \\| q ) } { \\partial q _ { j } } = T \\left \\{ \\underbrace { \\left [ \\frac { a - 1 } { a - b } ( X . Y ) ^ { a - 2 } - \\frac { b - 1 } { a - b } ( X . Y ) ^ { b - 2 } \\right ] } _ { Z > 0 } \\frac { \\partial ( X . Y ) } { \\partial q _ { j } } \\right \\} \\end{align*}"}
+{"id": "2644.png", "formula": "\\begin{align*} \\left ( \\mathcal { F } e ^ { i t \\Delta } f \\right ) ( \\xi ) = e ^ { - i t | \\xi | ^ 2 } ( \\mathcal { F } f ) ( \\xi ) . \\end{align*}"}
+{"id": "813.png", "formula": "\\begin{align*} { \\cal { L } } _ { \\hat { X } } \\nabla _ { 0 } C ^ { i } _ { j k } = \\nabla _ { 0 } { \\cal { L } } _ { \\hat { X } } C ^ { i } _ { j k } + \\Psi C ^ { i } _ { j k } + y ^ { i } C ^ { s } _ { j k } \\Psi _ { s } , \\end{align*}"}
+{"id": "944.png", "formula": "\\begin{align*} \\tau _ V = \\inf \\{ t > 0 : X _ t \\notin V \\} . \\end{align*}"}
+{"id": "277.png", "formula": "\\begin{align*} Q \\equiv \\Bigl ( A \\frac { d } { d x } - 3 A ' \\Bigr ) \\Bigl ( A \\frac { d } { d x } + \\gamma A - A ' \\Bigr ) b = 0 , \\end{align*}"}
+{"id": "1806.png", "formula": "\\begin{align*} f ( z ^ + , u ^ * ) - f ( z ^ - , u ^ * ) - \\Xi ( z ^ + , z ^ - , u ^ * ) + D _ u f ( z ^ * , u ^ * ) \\left ( T ( z ^ + , z ^ - ) ( u ) - u \\right ) = \\mathcal { E } _ 1 + \\mathcal { E } _ 2 + \\mathcal { E } _ 3 \\end{align*}"}
+{"id": "8039.png", "formula": "\\begin{align*} \\varepsilon _ { 1 , t } ^ N ( \\vec { f } ) = \\int _ 0 ^ t & \\frac { 1 } { N \\gamma _ N } \\sum _ { i = 1 } ^ N \\sum _ { j = 1 } ^ N \\lambda _ 1 \\left ( \\frac { i } { N } \\right ) \\lambda _ 2 \\left ( \\frac { j } { N } \\right ) \\times \\\\ & \\left ( \\mathbb { E } \\left ( S _ s ^ N ( i ) I _ s ^ N ( j ) \\right ) - \\mathbb { E } \\left ( S _ s ^ N ( i ) \\right ) \\mathbb { E } \\left ( I _ s ^ N ( j ) \\right ) \\right ) \\left ( - f _ 1 \\left ( \\frac { i } { N } \\right ) + f _ 2 \\left ( \\frac { i } { N } \\right ) \\right ) d s \\end{align*}"}
+{"id": "7161.png", "formula": "\\begin{align*} I ( x , t ) = A ( x ) \\omega _ 1 \\cos ( \\omega _ 1 t ) + B ( x ) \\omega _ 2 \\cos ( \\omega _ 2 t ) , \\end{align*}"}
+{"id": "3020.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\lambda = \\alpha ^ 2 + \\alpha + \\beta - \\beta ^ 2 , \\ \\ \\ & \\mu = \\beta - \\alpha - 1 , \\ \\ \\ & \\nu = ( \\beta - 1 ) ( \\beta + 3 ) - \\alpha ^ 2 . \\\\ \\end{array} \\end{align*}"}
+{"id": "7359.png", "formula": "\\begin{align*} \\Delta ^ h ( e ^ { ( 2 - n ) \\psi } ) = 0 . \\end{align*}"}
+{"id": "8085.png", "formula": "\\begin{align*} n p _ t ^ - = \\tilde { p } _ t ^ - ( n ) u _ t ^ + ( n ) , \\forall n \\in N _ 1 ^ + . \\end{align*}"}
+{"id": "4910.png", "formula": "\\begin{align*} L ^ \\sharp = L ( p _ 1 + \\cdots + p _ b ) \\end{align*}"}
+{"id": "1087.png", "formula": "\\begin{align*} E _ { 1 , 1 } + E _ { 1 , 2 } \\leq \\sum _ { s = 1 } ^ { n } \\Vert ( T _ n ( w ) ^ { - 1 } ) ^ { s , t } - ( T _ \\infty ( w ) ^ { - 1 } ) ^ { s , t } \\| \\leq M _ 4 n ^ { - d } , n \\in \\N , ~ ~ t \\in H _ \\delta ( n ) . \\end{align*}"}
+{"id": "3102.png", "formula": "\\begin{align*} \\varphi \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) = \\left ( \\begin{array} { c c c } a ^ { \\ell _ 1 } & 0 & 0 \\\\ 0 & a ^ { \\ell _ 1 } & 0 \\\\ 0 & 0 & d ^ { - \\ell _ 3 } \\end{array} \\right ) . \\end{align*}"}
+{"id": "1363.png", "formula": "\\begin{align*} l _ n ^ \\alpha \\big ( x _ 1 \\otimes \\dots \\otimes x _ n \\big ) = \\sum _ { i = 0 } ^ { \\infty } \\frac { 1 } { i ! } l _ { n + i } \\big ( \\alpha ^ { \\otimes i } \\otimes x _ 1 \\otimes \\dots \\otimes x _ n \\big ) , \\forall x _ 1 , \\ldots , x _ n \\in \\mathfrak { L } , \\end{align*}"}
+{"id": "2819.png", "formula": "\\begin{align*} \\delta Q ^ { 1 } ( t _ { 2 } ) = \\delta Q ^ { 1 } ( t _ { 1 } ) = 0 . \\end{align*}"}
+{"id": "2386.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\left ( \\max _ { 0 \\leqslant j \\leqslant m } | a _ { n , j } | \\cdot \\left | a _ { n , 0 } + \\sum _ { j = 1 } ^ { m } a _ { n , j } \\alpha _ j \\right | _ p \\right ) = 0 \\end{align*}"}
+{"id": "8758.png", "formula": "\\begin{align*} f _ { \\rho } ( x _ + ) - f _ { \\rho } ( x _ - ) & = \\left ( 2 \\rho \\left ( \\frac { z - m } { z - y } ( m - y ) ^ { \\rho - 1 } - \\frac { m - y } { z - y } ( z - m ) ^ { \\rho - 1 } \\right ) + \\beta \\right ) \\varepsilon + o ( \\varepsilon ) . \\end{align*}"}
+{"id": "3406.png", "formula": "\\begin{align*} [ S ] \\equiv 0 \\operatorname { m o d } 2 b ^ + ( X ) + \\frac { 1 } { 2 } b _ 1 ( S ) - \\frac { 1 } { 4 } S \\circ S = 0 , \\end{align*}"}
+{"id": "4105.png", "formula": "\\begin{align*} f _ \\theta ( z ) = \\exp ( \\langle \\theta , z \\rangle ) , z \\in \\mathbb { R } _ + ^ J , \\end{align*}"}
+{"id": "3787.png", "formula": "\\begin{align*} \\mathbb { P } \\left \\{ g ^ { ( i ) } _ t V ^ { ( i ) } _ t \\leq \\bar { t } \\right \\} = \\mathbb { P } \\left \\{ \\frac { V ^ { ( i ) } _ t } { N _ i n _ x } - 1 \\leq \\frac { 4 \\alpha \\| \\Sigma ^ { ( i ) } _ t \\| } { ( 1 - 2 \\alpha ) ^ 2 \\rho ^ { ( i ) } \\| \\Sigma ^ { ( i ) } _ t \\| + 4 \\sqrt { n _ x } } \\right \\} , \\end{align*}"}
+{"id": "3844.png", "formula": "\\begin{align*} n ^ { - 1 } \\int _ 0 ^ { t _ n } \\psi _ e ( s ) d s = 1 . \\end{align*}"}
+{"id": "7230.png", "formula": "\\begin{align*} \\partial _ v p ( V _ R ^ - ) - \\partial _ v p ( V _ R ^ + ) + \\partial _ v p ( V _ F ) = 0 . \\end{align*}"}
+{"id": "8798.png", "formula": "\\begin{align*} 1 - \\alpha _ \\rho ^ { 2 - \\rho } ( 1 - \\alpha _ \\rho ) ^ { \\rho - 1 } \\ln { \\frac { 1 - \\alpha _ \\rho } { \\alpha _ \\rho } } & = 1 - \\int _ 0 ^ 1 \\frac { \\rho - 1 - \\alpha _ \\rho } { 1 - \\alpha _ \\rho + ( \\rho - 2 ) u } d u . \\end{align*}"}
+{"id": "2002.png", "formula": "\\begin{align*} \\widetilde { ( \\eta ^ { n , \\pm } ) } _ l = \\begin{cases} & 0 , n = 0 , \\\\ & p _ l ^ { \\pm } ( \\widetilde { ( f ( \\psi ( t _ n ) ) ) } _ l - \\widetilde { ( f ( \\psi ^ n ) ) } _ l ) \\\\ & + q _ l ^ { \\pm } ( \\widetilde { ( \\delta _ t ^ - f ( \\psi ( t _ n ) ) ) } _ l - \\widetilde { ( \\delta _ t ^ - f ( \\psi ^ n ) ) } _ l ) , n \\ge 1 . \\end{cases} \\end{align*}"}
+{"id": "3815.png", "formula": "\\begin{align*} | \\omega | = \\frac { t } { 2 } \\sum _ { i = 0 } ^ { t - 1 } n _ i ^ 2 + \\sum _ { i = 0 } ^ { t - 1 } i n _ i . \\end{align*}"}
+{"id": "5285.png", "formula": "\\begin{align*} K L ( q \\| p ) = \\sum _ { i } q _ { i } \\log \\frac { q _ { i } } { p _ { i } } + p _ { i } - q _ { i } \\end{align*}"}
+{"id": "8042.png", "formula": "\\begin{align*} \\limsup _ { M \\rightarrow + \\infty } \\limsup _ { N \\rightarrow + \\infty } \\frac { N } { \\gamma ^ 2 _ N } \\log P \\left ( \\varepsilon _ 6 ^ N > M \\right ) = - \\infty . \\end{align*}"}
+{"id": "3431.png", "formula": "\\begin{align*} d ( S ^ 0 ) = \\underline { d } ( S ^ 0 ) = \\overline { d } ( S ^ 0 ) = 0 . \\end{align*}"}
+{"id": "5640.png", "formula": "\\begin{align*} 2 x _ 4 = x _ 6 = x _ 7 = x _ 8 , x _ 3 = 1 1 7 - x _ 5 - x _ 8 , x _ 5 = x _ 1 - \\tfrac { 3 } { 2 } x _ 8 + 1 1 7 . \\end{align*}"}
+{"id": "3172.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } d X _ { t } - \\Delta \\ln X _ { t } d t = X _ t \\circ d W _ { t } , & ( t , \\xi ) \\in \\left [ 0 , T \\right ] \\times \\mathbb { R } ^ { d } , \\\\ X _ { 0 } = x . & \\end{array} \\right . \\end{align*}"}
+{"id": "1959.png", "formula": "\\begin{align*} d ( { ( { \\hat { C } } ^ { p ^ t } ) ^ { \\oplus p ^ { s - t - 1 } } | } _ { T } ) = m i n \\{ d ( { { \\hat { C } } ^ { p ^ t } | } _ { T _ j } ) | 0 \\le j \\le p ^ { s - t - 1 } - 1 \\} . \\end{align*}"}
+{"id": "3734.png", "formula": "\\begin{align*} \\frac { d } { d x } \\left [ \\left ( x \\right ) _ { n } \\right ] = \\left ( x \\right ) _ { n } \\left [ \\psi \\left ( x + n \\right ) - \\psi \\left ( x \\right ) \\right ] , \\end{align*}"}
+{"id": "3163.png", "formula": "\\begin{align*} x & = \\left ( a _ { s - \\ell } k ^ s + \\sum _ { r = \\ell } ^ s a _ { r - \\ell } k ^ r \\right ) t + i k ^ \\ell + \\sum _ { r = 0 } ^ { \\ell - 1 } a _ { r + s + 1 - \\ell } k ^ r \\\\ & = a _ { s - \\ell } n + \\left ( \\sum _ { r = \\ell } ^ { s - 1 } a _ { r - \\ell } k ^ r \\right ) t + i k ^ \\ell + \\sum _ { r = 0 } ^ { \\ell - 1 } a _ { r + s + 1 - \\ell } k ^ r \\end{align*}"}
+{"id": "3024.png", "formula": "\\begin{align*} \\begin{array} { l } ( \\overline { \\mathcal { L } } _ { \\xi } g ) _ { 1 1 } = ( \\overline { \\mathcal { L } } _ { \\xi } g ) _ { 2 2 } = ( \\overline { \\mathcal { L } } _ { \\xi } g ) _ { 3 3 } = ( \\overline { \\mathcal { L } } _ \\xi g ) _ { 4 4 } = - 2 \\alpha , \\\\ ( \\overline { \\mathcal { L } } _ \\xi g ) _ { 1 3 } = ( \\overline { \\mathcal { L } } _ \\xi g ) _ { 2 4 } = 2 ( 1 - \\beta ) \\end{array} \\end{align*}"}
+{"id": "3387.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { k } \\frac { z _ { t } \\eta _ { t } } { \\alpha _ { t } } \\left ( f \\left ( y _ { t + 1 } \\right ) - f \\left ( x ^ { * } \\right ) \\right ) - \\frac { z _ { t } \\eta _ { t } \\left ( 1 - \\alpha _ { t } \\right ) } { \\alpha _ { t } } \\left ( f \\left ( y _ { t } \\right ) - f \\left ( x ^ { * } \\right ) \\right ) \\end{align*}"}
+{"id": "5152.png", "formula": "\\begin{align*} & U _ { j } = \\frac { 1 } { \\alpha } \\left [ \\frac { a - 1 } { a - b } B ^ { a - 2 } - \\frac { b - 1 } { a - b } B ^ { b - 2 } \\right ] p ^ { \\alpha } _ { j } q ^ { - \\alpha } _ { j } \\\\ & V _ { j } = \\frac { 1 } { \\alpha } \\left [ \\frac { a - 1 } { a - b } A ^ { a - 2 } - \\frac { b - 1 } { a - b } A ^ { b - 2 } \\right ] \\end{align*}"}
+{"id": "4250.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\| u ( t ) - e ^ { i t \\Delta } S _ a ( u _ 0 ) \\| _ { L ^ 2 } = 0 , \\end{align*}"}
+{"id": "466.png", "formula": "\\begin{align*} \\lambda ( g ^ { - 1 } ) \\lambda ( g h ) & = \\sum _ { \\substack { A \\ni g \\\\ C \\ni h ^ { - 1 } g ^ { - 1 } } } ( A , g ^ { - 1 } ) ( C , g h ) = \\sum _ { \\substack { C \\ni h ^ { - 1 } \\\\ C \\ni h ^ { - 1 } g ^ { - 1 } } } ( h C , g ^ { - 1 } ) ( C , g h ) = \\sum _ { \\substack { C \\ni h ^ { - 1 } \\\\ C \\ni h ^ { - 1 } g ^ { - 1 } } } ( C , h ) . \\end{align*}"}
+{"id": "6109.png", "formula": "\\begin{align*} g ( U _ \\phi , X ) = \\langle e _ 0 \\cdot X \\cdot \\phi , \\phi \\rangle \\end{align*}"}
+{"id": "7420.png", "formula": "\\begin{align*} b ' _ 1 & = b _ 1 c _ 1 \\\\ b ' _ 2 & = b _ 1 c _ 2 + b _ 2 c _ 1 + ( a _ 2 - d _ 2 ) a _ 3 \\\\ b ' _ 3 & = b _ 1 c _ 3 + b _ 2 c _ 2 + b _ 3 c _ 1 + ( a _ 2 - d _ 2 ) a _ 4 - a _ 3 d _ 3 . \\\\ \\end{align*}"}
+{"id": "1275.png", "formula": "\\begin{align*} \\exp \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} = \\frac { 1 } { \\Delta } \\begin{pmatrix} m _ { 1 1 } & m _ { 1 2 } \\\\ m _ { 2 1 } & m _ { 2 2 } \\end{pmatrix} , \\end{align*}"}
+{"id": "2994.png", "formula": "\\begin{align*} g ( T ( x , y ) , z ) = g ( H ( x , y ) , z ) - g ( H ( y , x ) , z ) . \\end{align*}"}
+{"id": "5744.png", "formula": "\\begin{align*} \\frac { d } { d t } \\left ( e ^ { - ( \\gamma _ * + \\mathbf { i } \\beta _ i ) t } \\left ( \\xi _ { i , 1 } ( t ) + \\mathbf { i } \\xi _ { i , 2 } ( t ) \\right ) \\right ) = e ^ { - ( \\gamma _ * + \\mathbf { i } \\beta _ i ) t } \\left ( \\mathcal { E } _ { i , 1 } ( t ) + \\mathbf { i } \\mathcal { E } _ { i , 2 } ( t ) \\right ) . \\end{align*}"}
+{"id": "2568.png", "formula": "\\begin{align*} K ' ( M , L ) \\xrightarrow { \\cong } \\prod _ { m = 0 } ^ \\infty C ^ 2 _ m \\ ; , K ^ { \\prime \\ , ( s ) } ( M , L ) \\xrightarrow { \\cong } \\prod _ { m < s - 1 / 2 } C ^ 2 _ m \\quad ( s > 1 / 2 ) \\ ; . \\end{align*}"}
+{"id": "4641.png", "formula": "\\begin{align*} \\lambda _ { \\mathcal { G } _ 2 } ( n ) = \\begin{cases} 1 , \\quad n = p _ 1 ^ { \\alpha _ 1 } \\dots p _ k ^ { \\alpha _ k } , p _ i \\in \\mathcal { G } _ 2 \\ , \\ , \\forall i = 1 , \\dotsc , k \\\\ 0 , \\quad \\end{cases} \\end{align*}"}
+{"id": "3394.png", "formula": "\\begin{align*} & 6 4 \\left ( \\log \\frac { 1 } { \\delta } + \\frac { 6 0 \\sigma ^ { p } C _ { 3 } } { C _ { 1 } ^ { 2 } } \\right ) ^ { 2 } + \\frac { 4 8 \\sigma ^ { 2 p } C _ { 2 } C _ { 3 } + 1 4 0 \\sigma ^ { p } C _ { 3 } } { C _ { 1 } ^ { 2 } } \\\\ & = 6 4 \\left ( \\log \\frac { 1 } { \\delta } + 6 0 \\log \\frac { 1 } { \\delta } \\frac { 3 2 } { \\Delta _ { 1 } } \\frac { \\Delta _ { 1 } } { 2 0 4 8 } \\right ) ^ { 2 } + \\left ( 4 8 \\frac { \\Delta _ { 1 } } { 2 0 4 8 } + 1 4 0 \\frac { \\Delta _ { 1 } } { 2 0 4 8 } \\right ) \\frac { 3 2 } { \\Delta _ { 1 } } \\\\ & \\le 2 5 6 \\gamma ^ { 2 } = A \\end{align*}"}
+{"id": "6455.png", "formula": "\\begin{align*} \\langle R ^ N ( \\bar \\nabla _ { e _ j } \\big ( R ^ N ( V , d \\phi ( e _ k ) ) d \\phi ( e _ k ) \\big ) , \\tau ( \\phi ) ) d \\phi ( e _ j ) , V \\rangle \\\\ = \\langle R ^ N ( V , d \\phi ( e _ j ) ) \\tau ( \\phi ) , \\bar \\nabla _ { e _ j } \\big ( R ^ N ( V , d \\phi ( e _ k ) ) d \\phi ( e _ k ) \\big ) \\rangle . \\end{align*}"}
+{"id": "8185.png", "formula": "\\begin{align*} B _ 1 ^ { \\star } & = \\min ( \\sqrt { \\frac { 4 M B _ - M ^ 2 } { 2 } } , \\frac { M } { 2 } ) \\\\ B _ 2 ^ { 0 \\star } & = \\frac { M - B _ 1 ^ { \\star } } { 2 } , \\\\ B _ 2 ^ 1 & = \\frac { B _ 1 ^ { \\star } } { 2 } . \\end{align*}"}
+{"id": "6563.png", "formula": "\\begin{align*} J _ k ( n ) & : = | \\{ ( a _ 1 , a _ 2 , \\cdots , a _ k ) ; \\ 1 \\leq a _ i \\leq n ; \\ \\gcd ( a _ 1 , a _ 2 , \\cdots , a _ k , n ) = 1 \\} | \\\\ & = n ^ { k } \\prod _ { p | n } \\left ( 1 - \\frac { 1 } { p ^ k } \\right ) . \\end{align*}"}
+{"id": "8674.png", "formula": "\\begin{align*} ( \\ , u \\ , | \\ , v \\ , ) _ X : = \\int _ X \\langle \\ , u \\ , | \\ , v \\ , \\rangle d v _ X , \\end{align*}"}
+{"id": "6286.png", "formula": "\\begin{align*} \\frac 1 T \\sum \\limits _ { k = 0 } ^ { T - 1 } \\langle \\hat { g } _ { k + 1 } , x _ k - x ^ * \\rangle \\leq \\frac 1 2 \\frac { R _ { 0 } ^ { 2 } } { \\nu T } + \\frac { \\nu } { 2 } \\frac { 1 } { T } \\sum \\limits _ { k = 0 } ^ { T - 1 } \\| \\hat { g } _ { k + 1 } \\| ^ { 2 } _ q . \\end{align*}"}
+{"id": "553.png", "formula": "\\begin{align*} u _ { 1 } ( x ) = \\dfrac { \\sqrt { - \\mu } } { 2 } \\left [ x ^ { \\sqrt { - \\mu } } - x ^ { - \\sqrt { - \\mu } } \\right ] , \\ , \\ , \\ , v _ { 1 } ( x ) = - \\frac { 1 } { 2 } \\left [ x ^ { \\sqrt { - \\mu } } + x ^ { - \\sqrt { - \\mu } } \\right ] \\end{align*}"}
+{"id": "2298.png", "formula": "\\begin{align*} Q _ r ( t ) = \\bigl [ ( 1 - x ^ 2 ) ( 1 + t x ^ { 2 } ) ^ { r } ( 1 + t x ^ { - 2 } ) ^ { r } \\bigr ] _ 0 . \\end{align*}"}
+{"id": "7634.png", "formula": "\\begin{align*} \\begin{aligned} \\mu _ p ( _ { \\alpha _ n p } B ( p , C ) ) & = \\mu _ p ( \\widetilde { M } ( \\infty ) ) - \\mu _ p ( \\widetilde { M } ( \\infty ) - _ { \\alpha _ n p } B ( p , C ) ) \\\\ & \\geq \\mu _ p ( \\widetilde { M } ( \\infty ) ) - \\lambda - \\epsilon . \\end{aligned} \\end{align*}"}
+{"id": "7328.png", "formula": "\\begin{align*} \\tilde { f } _ { m , r } ( z , \\alpha ) = - \\sum _ { n = 0 } ^ { \\infty } \\tilde { S } _ { m , r } ( \\alpha , n + 1 ) z ^ n \\end{align*}"}
+{"id": "3009.png", "formula": "\\begin{align*} ( \\mathcal { L } _ { v } g ) ( x , y ) = g ( \\nabla _ x v , y ) + g ( x , \\nabla _ y v ) . \\end{align*}"}
+{"id": "6768.png", "formula": "\\begin{align*} ( \\lambda - \\breve { \\lambda } ^ k ) ^ T [ ( \\sum _ { j = 1 } ^ { m } A _ j \\widetilde { x } _ j ^ k - b ) + \\frac { 1 } { \\beta } ( \\widetilde { \\lambda } ^ k - \\lambda ^ k ) ] \\geq 0 , ~ \\forall \\lambda \\in \\Lambda . \\end{align*}"}
+{"id": "4444.png", "formula": "\\begin{align*} \\psi _ 1 - \\log r = \\sum _ { 1 \\le k \\le m _ 1 } p _ { 1 , k } G _ { D _ { 1 , r } } ( \\cdot , z _ { 1 , k } ) \\end{align*}"}
+{"id": "1181.png", "formula": "\\begin{align*} p _ \\gamma ( y ) : = \\sum _ { \\alpha \\in \\mathbb { Z } _ + ^ n , \\ , | \\alpha | \\leq N - k } v _ { \\gamma + \\alpha } \\frac { y ^ \\alpha } { \\alpha ! } \\end{align*}"}
+{"id": "84.png", "formula": "\\begin{align*} \\det ( M ) = c _ 3 ( c _ 3 + c _ 4 ) P _ \\theta - \\frac { \\big ( c _ 3 P _ \\theta + ( c _ 3 + c _ 4 ) - ( 2 c _ 1 + c _ 2 ) \\big ) ^ 2 } { 4 } \\geq c \\end{align*}"}
+{"id": "2863.png", "formula": "\\begin{align*} \\mathrm { I } : = \\| f \\| _ { L ^ p ( \\mathbb { R } ^ n ) } + \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\frac { | f ( \\cdot ) - f ( y ) | ^ q } { | \\cdot - y | ^ { n + q } } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ { L ^ p ( \\mathbb { R } ^ n ) } < \\infty , \\end{align*}"}
+{"id": "2009.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l l } \\alpha + \\beta + \\gamma = 1 , \\\\ \\alpha \\beta + \\beta \\gamma + \\alpha \\gamma = 0 , \\\\ \\alpha \\beta \\gamma = 1 . \\end{array} \\right . \\end{align*}"}
+{"id": "4080.png", "formula": "\\begin{align*} [ e ^ { i z _ 1 s } \\tilde { E } _ i ( - i z _ 1 s ) + e ^ { - i z _ 1 s } \\tilde { E } _ i ( i z _ 1 s ) ] - [ e ^ { i z _ 0 s } \\tilde { E } _ i ( i z _ 0 s ) + e ^ { - i z _ 0 s } \\tilde { E } _ i ( - i z _ 0 s ) ] = 2 \\pi i [ e ^ { i z _ 0 s } + e ^ { - i z _ 0 s } ] \\end{align*}"}
+{"id": "1971.png", "formula": "\\begin{align*} E q u > & p ^ s + 1 - ( \\tau + 1 ) p ^ t - \\left ( ( p - \\tau + 1 ) p ^ { s - t - 1 } - 1 \\right ) \\times \\left ( 1 + \\frac { \\tau p ^ t - 1 } { ( p - \\tau ) p ^ { t - 1 } + 1 } \\right ) \\\\ = & p ^ s + 1 - ( \\tau + 1 ) p ^ t - \\left ( ( p - \\tau + 1 ) p ^ { s - 2 } - p ^ { t - 1 } \\right ) \\times \\frac { p ( \\tau - 1 ) - \\tau } { ( p - \\tau ) p ^ { t - 1 } + 1 } . \\end{align*}"}
+{"id": "4381.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathcal { X } \\subseteq \\{ 0 , 1 \\} ^ n } & \\sum _ { i \\in [ n ] } \\sum _ { j \\in [ i ] } p _ { i , j } x _ i x _ j \\end{align*}"}
+{"id": "7701.png", "formula": "\\begin{align*} \\widetilde { V } _ { q } ( \\Omega , z ) = \\widetilde { V } ^ { + } _ { q } ( \\Omega , z ) . \\end{align*}"}
+{"id": "5288.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } K L ( q \\| p ) } { \\partial q _ { j } } = 1 - \\left [ \\frac { a } { \\left ( a - b \\right ) } \\frac { q _ { i } ^ { a - 1 } } { p _ { i } ^ { a - 1 } } - \\frac { b } { \\left ( a - b \\right ) } \\frac { q _ { i } ^ { b - 1 } } { p _ { i } ^ { b - 1 } } \\right ] \\end{align*}"}
+{"id": "4886.png", "formula": "\\begin{align*} h ^ 1 ( \\Sigma ' , \\O _ { \\Sigma ' } ( C ' ) ) = h ^ 1 ( \\Gamma , \\mathcal D ) + h ^ 1 ( \\Gamma , \\mathcal D \\otimes \\L ) + h ^ 1 ( \\Gamma , \\mathcal D \\otimes \\L ^ { \\otimes 2 } ) = h ^ 1 ( \\Gamma , \\mathcal D \\otimes \\L ^ { \\otimes 2 } ) . \\end{align*}"}
+{"id": "4930.png", "formula": "\\begin{align*} \\zeta _ { e ^ { L } _ { + } - e ^ { L } _ { - } } ( H ) = \\max \\zeta _ { e ^ { L } _ { \\pm } } ( H ) \\end{align*}"}
+{"id": "1710.png", "formula": "\\begin{align*} \\left ( T _ k \\right ) _ { r , s } : = \\delta _ { r - s , 1 } = \\begin{cases} 1 & \\mbox { i f } r - s = 1 \\\\ 0 & \\mbox { i f } r - s \\neq 1 . \\end{cases} \\end{align*}"}
+{"id": "6287.png", "formula": "\\begin{align*} f ( \\overline { x } _ T ) - f ( x ^ * ) \\leq \\frac 1 T \\sum \\limits _ { k = 0 } ^ { T - 1 } \\langle \\nabla \\hat { f } _ \\tau ( x _ { k } ) , x _ k - x ^ * \\rangle + 2 M _ 2 \\tau . \\end{align*}"}
+{"id": "8290.png", "formula": "\\begin{align*} \\begin{aligned} G _ { i j } & = \\frac { u _ 1 u _ { 1 i j } } { w ^ 2 \\ln w } + \\frac { \\sum _ k u _ { k i } u _ { k j } } { w ^ 2 \\ln w } - \\frac { 2 u ^ 2 _ 1 u _ { 1 i } u _ { 1 j } } { w ^ 4 \\ln w } - \\frac { 2 u ^ 2 _ 1 u _ { 1 i } u _ { 1 j } } { w ^ 4 \\ln ^ 2 w } \\\\ & + \\varphi ' u _ { i j } + \\frac { \\rho _ { i j } } { \\rho } + \\frac { \\varphi ' } { \\rho } ( u _ i \\rho _ j + u _ j \\rho _ i ) . \\end{aligned} \\end{align*}"}
+{"id": "2705.png", "formula": "\\begin{align*} q ^ { i } = q ^ { i } ( t , c ^ { I } ) , v ^ { i } = v ^ { i } ( t , c ^ { I } ) . \\end{align*}"}
+{"id": "220.png", "formula": "\\begin{align*} \\bar X ^ i \\left ( \\bar x , \\frac { d \\bar x } { d \\tau } \\right ) = { f ^ 2 } \\ , X ^ i \\left ( \\bar x , \\frac 1 f \\frac { d \\bar x } { d \\tau } \\right ) + \\frac d { d \\tau } ( \\log f ) \\ , \\frac { d \\bar x ^ i } { d \\tau } , \\end{align*}"}
+{"id": "8329.png", "formula": "\\begin{align*} [ \\partial _ s , V _ t ] _ { \\widetilde { \\gamma } } = 0 \\end{align*}"}
+{"id": "3611.png", "formula": "\\begin{align*} ( 1 + a ) \\alpha ^ 2 \\tau _ 0 \\tau _ 1 ( \\sigma ^ 2 \\sigma _ 1 - \\sigma \\sigma _ 1 ) - ( a + b ) \\alpha \\tau _ 0 ( \\sigma ^ 2 \\sigma _ 1 - \\sigma ) + b ( \\sigma ^ 2 \\sigma _ 1 - 1 ) = 0 . \\end{align*}"}
+{"id": "423.png", "formula": "\\begin{align*} \\begin{aligned} \\bold { p } _ { j } & = [ \\psi _ { j , 1 } , \\psi _ { j , 2 } , . . . , \\psi _ { j , j } , \\rho ^ 2 \\widetilde { t } _ { 1 , 1 } ^ 2 \\psi _ { j , 1 } + \\Theta _ { j , 1 } , \\\\ & \\rho ^ 2 \\widetilde { t } _ { 2 , 2 } ^ 2 \\psi _ { j , 2 } + \\Theta _ { j , 2 } , . . . , \\rho ^ 2 \\widetilde { t } _ { j , j } ^ 2 \\psi _ { j , j } + \\Theta _ { j , j } ] ^ { T } , \\\\ \\bold { q } _ { j } & = M [ \\Gamma _ { j , 1 } , \\Gamma _ { j , 2 } , . . . , \\Gamma _ { j , j } , \\Pi _ { j , 1 } , \\Pi _ { j , 2 } , . . . , \\Pi _ { j , j } ] ^ { T } . \\end{aligned} \\end{align*}"}
+{"id": "8728.png", "formula": "\\begin{align*} p _ d = \\frac { 1 } { 2 } + \\frac { 1 } { 2 } ( - w _ 1 - \\ldots - w _ { d - 1 } + w _ { d } ) - \\lambda \\end{align*}"}
+{"id": "9186.png", "formula": "\\begin{align*} e _ { i , j + 1 } ( u ) = ( 1 - q ^ 2 ) ^ { i - j } \\cdot [ \\cdots [ [ e _ { i , i + 1 } ( u ) , e ^ { ( 0 ) } _ { i + 1 , i + 2 } ] _ { q } , e ^ { ( 0 ) } _ { i + 2 , i + 3 } ] _ { q } , \\cdots , e ^ { ( 0 ) } _ { j , j + 1 } ] _ { q } \\end{align*}"}
+{"id": "961.png", "formula": "\\begin{align*} | \\Pi _ V ( u ) | = | u - P _ V ( u ) | \\ge | u | - P _ V ( | u | ) . \\end{align*}"}
+{"id": "4754.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\norm { J _ t x - S ( t ) x } } { t } = 0 . \\end{align*}"}
+{"id": "1652.png", "formula": "\\begin{align*} \\lambda \\left ( \\delta \\right ) = \\ln \\left [ \\delta ^ { \\left ( 3 \\left ( T + 1 \\right ) \\right ) ^ { - 1 } } \\right ] , \\end{align*}"}
+{"id": "4832.png", "formula": "\\begin{align*} \\Lambda ( { \\bf S } ) = \\Big \\{ \\mu ( A ) : = \\frac { 1 } { 2 } \\big ( \\sum _ { \\alpha \\not \\in A } \\alpha - \\sum _ { \\alpha \\in A } \\alpha \\big ) \\ ; \\mid \\ ; A \\subset \\Phi ^ + ( { \\mathfrak q } ) \\Big \\} \\subset \\mathfrak { t } _ { \\mathbb R } ^ * \\end{align*}"}
+{"id": "2170.png", "formula": "\\begin{align*} f _ d ( x ) = \\ln \\left ( \\frac { x + d } { x } \\right ) . \\end{align*}"}
+{"id": "960.png", "formula": "\\begin{align*} \\mathfrak D ( \\mathcal E ) = \\Big \\{ u \\in L ^ 2 ( D ; m ) : \\int _ D \\int _ D \\frac { ( u ( x ) - u ( y ) ) ^ 2 } { | x - y | ^ { d + \\alpha } } \\ , d x \\ , d y < \\infty \\Big \\} . \\end{align*}"}
+{"id": "7107.png", "formula": "\\begin{align*} \\tilde { b } _ m = \\mathbf { 1 } _ m b + \\big ( \\tilde { b } _ m - \\mathbf { 1 } _ m b \\big ) . \\end{align*}"}
+{"id": "4960.png", "formula": "\\begin{align*} v _ N ( \\{ \\lambda \\} ) = \\prod _ { 1 \\leq j < k \\leq N } ( \\lambda _ k - \\lambda _ j ) , \\end{align*}"}
+{"id": "7126.png", "formula": "\\begin{align*} B _ { 1 } ( u _ { n } ^ { 2 } , ( u _ { n } - u _ { c } ) ^ { 2 } ) = V _ { 1 } ( u _ { n } ) - 2 B _ { 1 } ( u _ { n } ^ { 2 } , ( u _ { n } - u _ { c } ) u _ { c } ) - B _ { 1 } ( u _ { n } ^ { 2 } , u _ { c } ^ { 2 } ) . \\end{align*}"}
+{"id": "1593.png", "formula": "\\begin{align*} \\frac { M } { 2 } = G \\left ( \\frac { z } { g _ { M / 2 } ( z ) } \\right ) \\le C \\ , \\eta _ { H + 1 , 1 + \\frac { 1 } { H } } \\left ( \\frac { | z | } { g _ { M / 2 } ( z ) } \\right ) , \\end{align*}"}
+{"id": "965.png", "formula": "\\begin{align*} \\Pi _ V ( w ) & = \\Pi _ V ( P _ D ( g ) ) + \\Pi _ V ( R ^ D \\mu ) + \\Pi _ V ( R ^ { D } f ( \\cdot , w ) ) \\\\ & = P _ D g - P _ V ( P _ D ( ( g ) ) + R ^ { V } f ( \\cdot , u ) + R ^ V \\mu = R ^ { V } f ( \\cdot , w ) + R ^ V \\mu \\mbox { q . e . } \\end{align*}"}
+{"id": "3632.png", "formula": "\\begin{align*} \\alpha ^ 3 \\tau _ 0 ^ 3 L ( w _ 0 \\sigma ( z _ 0 ) ^ 3 ) - ( 1 + a ) \\alpha ^ 2 \\tau ^ 2 _ 0 L ( w _ 0 \\sigma ( z _ 0 ) ^ 2 ) + ( a + b ) \\alpha \\tau _ 0 L ( w _ 0 \\sigma ( z _ 0 ) ) - b L ( w _ 0 ) = 0 , \\end{align*}"}
+{"id": "8035.png", "formula": "\\begin{align*} \\limsup _ { \\delta \\rightarrow 0 } \\limsup _ { N \\rightarrow + \\infty } \\frac { N } { \\gamma _ N ^ 2 } \\log \\sup _ { \\sigma \\in \\mathcal { T } } P \\left ( \\sup _ { 0 \\leq t \\leq \\delta } \\left ( \\eta ^ N _ { t + \\sigma } ( \\vec { f } ) - \\eta ^ N _ \\sigma ( \\vec { f } ) \\right ) > \\epsilon \\right ) = - \\infty \\end{align*}"}
+{"id": "5109.png", "formula": "\\begin{align*} S \\left ( p \\right ) = - \\sum _ { i } p _ { i } \\Lambda \\left ( p _ { i } \\right ) \\end{align*}"}
+{"id": "5882.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { C } ( G ) ) & = \\dfrac { ( m n - n ) ( m n - n - 1 ) ^ { 3 } } { 2 } + m \\cdot \\dfrac { n ( n - 1 ) ^ { 3 } } { 2 } \\\\ & = \\dfrac { ( m n - n ) ( m n - n - 1 ) ^ { 3 } + m n ( n - 1 ) ^ { 3 } } { 2 } . \\end{align*}"}
+{"id": "7501.png", "formula": "\\begin{align*} \\chi ^ 2 R _ 1 [ h ] = \\chi ^ 2 \\left [ \\left ( ( g _ 0 ( t ) + h ( t ) ) ^ { p q } - ( g _ 0 ( t ) ) ^ { p q } \\right ) \\nabla _ q ^ { g _ 0 ( t ) } h \\right ] . \\end{align*}"}
+{"id": "7582.png", "formula": "\\begin{align*} N - \\Delta \\varphi _ { R } ( r ) & = N - \\varphi _ { R } '' ( r ) - \\varphi _ { R } ' ( r ) \\frac { N - 1 } { r } \\\\ & = 1 - \\varphi _ { R } '' ( r ) + N - 1 - \\varphi _ { R } ' ( r ) \\frac { N - 1 } { r } \\\\ & = 1 - \\varphi _ { R } '' ( r ) + ( N - 1 ) ( 1 - \\frac { \\varphi _ { R } ' ( r ) } { r } ) \\geq 0 \\\\ \\end{align*}"}
+{"id": "5530.png", "formula": "\\begin{align*} \\big | \\tilde f ( \\rho ) \\big | \\le C _ r n ^ { \\alpha + \\beta - 1 } , \\ ; \\rho \\in \\tilde S _ n ; \\big | \\tilde d _ n \\big | = \\Big | \\lim _ { \\rho \\to z _ n } \\tilde f ( \\rho ) \\Big | \\le C _ r n ^ { \\alpha + \\beta - 1 } . \\end{align*}"}
+{"id": "6348.png", "formula": "\\begin{align*} \\phi ( r ) : = \\int _ 0 ^ r \\tau ^ { n - 1 } \\omega _ g ( \\tau ) \\ , d \\tau , \\end{align*}"}
+{"id": "1250.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ; q ^ 2 ) _ k ( - 1 ; q ^ 2 ) _ k } { ( q ^ 2 ; q ^ 2 ) _ k } q ^ { ( n - 1 ) ^ 2 - k ^ 2 } \\\\ [ 7 p t ] & = \\sum _ { \\substack { 0 \\le k \\le n - 1 \\\\ [ 3 p t ] k \\not = ( n - 1 ) / 2 } } a _ { n , k } + a _ { n , ( n - 1 ) / 2 } \\\\ [ 7 p t ] & \\equiv \\sum _ { \\substack { 0 \\le k \\le n - 1 \\\\ [ 3 p t ] k \\not = ( n - 1 ) / 2 } } b _ { n , k } + a _ { n , ( n - 1 ) / 2 } \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "2605.png", "formula": "\\begin{align*} D _ { \\mathcal { I } } ( S ) : = \\{ s _ x - s _ { y } \\mid x - y \\in \\mathcal { I } \\} . \\end{align*}"}
+{"id": "7959.png", "formula": "\\begin{align*} v = d \\phi + \\delta N _ { \\beta } ( \\omega ) \\quad \\Omega , \\end{align*}"}
+{"id": "7644.png", "formula": "\\begin{align*} \\nu _ { i } \\left ( \\mathcal { F } _ { i } ( x ) \\right ) = \\begin{cases} 1 & \\mathcal { F } _ { i } ( x ) \\geq y _ { i } ^ { 0 } , \\\\ n _ { i } ( \\mathcal { F } _ i ( x ) ) & y _ { i } ^ { 0 } \\geq \\mathcal { F } _ { i } ( x ) \\geq y _ { i } ^ { 1 } , \\\\ 0 & \\mathcal { F } _ { i } ( x ) \\leq y _ { i } ^ { 1 } , \\end{cases} \\end{align*}"}
+{"id": "2231.png", "formula": "\\begin{align*} \\nu \\intop _ \\Omega ( u _ { , r } ^ 2 ) _ { , r } d r d z = \\nu \\intop _ { - a } ^ a u _ { , r } ^ 2 \\bigg | _ { r = 0 } ^ { r = R } d z = \\nu \\intop _ { - a } ^ a u _ { , r } ^ 2 \\bigg | _ { r = R } d z \\end{align*}"}
+{"id": "1787.png", "formula": "\\begin{align*} \\Psi ( M _ { f ^ * ( x ) } ) = M _ { c _ 1 f ^ * ( c _ 2 x ) } , \\end{align*}"}
+{"id": "6439.png", "formula": "\\begin{align*} k ^ \\Theta _ z ( w ) = \\frac { i } { 2 \\pi } \\frac { 1 - \\overline { \\Theta ( z ) } \\Theta ( w ) } { w - \\overline { z } } , z , w \\in \\C ^ + . \\end{align*}"}
+{"id": "8688.png", "formula": "\\begin{align*} T = \\frac { i } { \\sqrt { 2 } } ( L _ n - \\bar L _ n ) . \\end{align*}"}
+{"id": "6358.png", "formula": "\\begin{align*} r ^ { n - 1 } \\omega _ 1 ( r ) & = r ^ { n - 1 } + o ( r ^ { n } ) , \\\\ r ^ { n - 2 } \\omega _ 2 ( r ) & = ( n - 1 ) r ^ { n - 2 } + o ( r ^ { n - 1 } ) , \\\\ r ^ { n - 3 } \\omega _ 3 ( r ) & = ( n - 1 ) ( n - 2 ) r ^ { n - 3 } + ( 2 d n - d - n ) r ^ { n - 1 } + o ( r ^ { n } ) , \\end{align*}"}
+{"id": "6544.png", "formula": "\\begin{align*} S _ { N , 1 } - T _ { N , 1 } & = \\frac { 1 } { 2 \\pi } \\sum ^ N _ { n = 1 } \\sum ^ { \\infty } _ { j = m _ 0 + 1 } \\int _ { \\R } \\widehat { K } ( u ) \\prod ^ { j - 1 } _ { k = 1 } \\phi _ { \\varepsilon } ( a _ k u ) ( e ^ { \\iota u a _ j \\varepsilon _ { n - j } } - \\phi _ { \\varepsilon } ( a _ j u ) ) ( e ^ { \\iota u \\widetilde { X } _ { n , j } } - \\mathbb { E } e ^ { \\iota u \\widetilde { X } _ { n , j } } ) d u , \\end{align*}"}
+{"id": "285.png", "formula": "\\begin{align*} e _ { 2 k } e _ { 2 l } = C _ { k + l - 1 } ^ { l } e _ { 2 k + 2 l } , k \\geq 1 , \\ l \\geq 1 . \\end{align*}"}
+{"id": "1898.png", "formula": "\\begin{align*} \\nabla _ { x y } f : = f ( y ) - f ( x ) \\ , ; \\end{align*}"}
+{"id": "1861.png", "formula": "\\begin{align*} n _ { \\phi } ( X , P ^ { \\dagger } ) : = \\# \\mathcal { M } ^ * _ { \\phi , \\pmb { \\pi } } ( X , P ^ { \\dagger } ) \\in \\Z \\end{align*}"}
+{"id": "3756.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } a _ { n k } = \\sum _ { k = 0 } ^ { \\infty } \\theta _ { n , k } \\ , a _ { k } , \\end{align*}"}
+{"id": "7592.png", "formula": "\\begin{align*} \\mathcal { M } _ { p , q } ( c ) = \\{ v \\in S ( c ) , E _ { p , q } ( v ) < E _ { p , q } ( u ) , Q _ { p , q } ( v ) < 0 \\} . \\end{align*}"}
+{"id": "1119.png", "formula": "\\begin{align*} | \\langle \\psi _ Q , \\phi \\rangle | & = | \\langle \\psi _ Q , \\phi _ { Q _ { 0 , \\mathbf { 0 } } } \\rangle | \\\\ & \\lesssim \\| \\psi \\| _ { S _ { M + 1 } } \\| \\phi \\| _ { S _ { M + 1 } } \\left [ \\min \\left \\{ [ \\ell ( Q ) ] ^ { - 1 } , \\ell ( Q ) \\right \\} \\right ] ^ { M + \\frac { n } { 2 } } \\left [ 1 + \\frac { | x _ Q | } { \\ell ( Q ) \\vee 1 } \\right ] ^ { - ( n + M ) } , \\end{align*}"}
+{"id": "4258.png", "formula": "\\begin{align*} \\mathcal { F } _ { \\sigma } ^ { - 1 } [ e ^ { - i z \\sigma } \\hat h ( \\xi - \\sigma ) ] ( \\sigma ' ) = e ^ { i ( \\xi \\sigma ' - z \\xi ) } h ( z - \\sigma ' ) . \\end{align*}"}
+{"id": "7327.png", "formula": "\\begin{align*} \\partial _ s ^ n \\left . F _ { m , r } ( s ) \\right | _ { s = 0 } = ( - 1 ) ^ n n ! 2 ^ { - ( n + 1 ) } \\cdot C _ { m , r } ( n + 1 ) . \\end{align*}"}
+{"id": "2276.png", "formula": "\\begin{align*} \\Delta ( x ) = x \\otimes e ^ { - \\alpha h / 2 } + e ^ { \\alpha h / 2 } \\otimes x \\ , , \\ \\ \\ \\Delta ( y ) = y \\otimes e ^ { - \\alpha h / 2 } + e ^ { \\alpha h / 2 } \\otimes y \\ , , \\ \\ \\ \\Delta ( h ) = h \\otimes 1 + 1 \\otimes h \\ , , \\end{align*}"}
+{"id": "6053.png", "formula": "\\begin{align*} U = U ' \\oplus V \\oplus W \\end{align*}"}
+{"id": "850.png", "formula": "\\begin{align*} \\dot \\delta { ( Y - Z ) } = 2 g ( X , { I ^ * } ) \\Psi = 2 \\rho \\Psi . \\end{align*}"}
+{"id": "8677.png", "formula": "\\begin{align*} g ^ { T M ' } = d \\omega _ 0 ( \\cdot , J \\cdot ) . \\end{align*}"}
+{"id": "3955.png", "formula": "\\begin{align*} K _ 0 = \\begin{pmatrix} - 3 . 7 0 8 & - 2 6 . 3 2 9 & - 2 . 2 2 2 \\end{pmatrix} \\end{align*}"}
+{"id": "3928.png", "formula": "\\begin{align*} z _ i ( t , \\cdot ) = \\sum _ { n = 1 } ^ \\infty z _ { i , n } ( t ) \\varphi _ n ( \\cdot ) , i = 1 , 2 , 3 \\end{align*}"}
+{"id": "8097.png", "formula": "\\begin{align*} \\norm { f } ^ \\star _ 1 = e ^ \\star _ { 1 , 0 } ( f ) + e ^ \\star _ { 1 , 1 } ( f ) . \\end{align*}"}
+{"id": "6586.png", "formula": "\\begin{align*} \\chi ^ { ( f ) } _ y ( n ) = \\begin{cases} 1 & f ( n ) \\leq y , \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "884.png", "formula": "\\begin{align*} w ^ 1 = c _ 1 + a _ 1 x _ 1 + b _ 1 x _ 2 a n d w ^ 2 = c _ 2 + a _ 2 x _ 1 + b _ 2 x _ 2 . \\end{align*}"}
+{"id": "7280.png", "formula": "\\begin{align*} M i n i m i z e \\ \\ \\ & c ^ T x \\\\ s u b j e c t \\ t o \\ \\ \\ \\ & A x \\leq b \\\\ & x \\in \\mathbb { Z } ^ n \\\\ \\end{align*}"}
+{"id": "5855.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { C } ( D _ { 2 m } ) ) = ( m - 2 ) ( m - 2 - 1 ) ^ { 2 } + \\dfrac { m } { 2 } \\cdot 2 ( 2 - 1 ) ^ { 2 } = ( m - 2 ) ( m - 3 ) ^ { 2 } + m \\end{align*}"}
+{"id": "8176.png", "formula": "\\begin{align*} c _ { y ^ { i - 1 } } b _ { y ^ { i - 1 } } = \\frac { 1 } { 4 } \\left ( ( c _ { y ^ { i - 1 } } + b _ { y ^ { i - 1 } } ) ^ 2 - ( c _ { y ^ { i - 1 } } - b _ { y ^ { i - 1 } } ) ^ 2 \\right ) , \\end{align*}"}
+{"id": "4213.png", "formula": "\\begin{align*} \\Theta _ 3 ( T _ { C } X ) = \\bigotimes _ { n = 1 } ^ { \\infty } S _ { q ^ n } ( \\widetilde { T _ C X } ) \\otimes \\bigotimes _ { m = 1 } ^ { \\infty } \\wedge _ { q ^ { m - \\frac { 1 } { 2 } } } ( \\widetilde { T _ C X } ) . \\end{align*}"}
+{"id": "5689.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } { u ( t ) } / { \\Vert u ( t ) \\Vert _ { L ^ 2 } } = w \\ \\textup { i n } \\ C ^ \\infty ( \\Sigma ; \\mathbf { V } ) . \\end{align*}"}
+{"id": "4652.png", "formula": "\\begin{align*} a b + n = r ^ { 2 } , r < n \\sqrt { N } . \\end{align*}"}
+{"id": "2719.png", "formula": "\\begin{align*} Y ^ { ( 1 ) } _ { \\alpha } : = [ X _ { t } , Z _ { \\alpha } ] = \\tau ^ { i } _ { \\alpha } \\frac { \\partial } { \\partial q ^ { i } } + \\left ( - Z _ { \\alpha } \\eta ^ { i } - X _ { t } \\tau ^ { i } _ { \\alpha } \\right ) \\frac { \\partial } { \\partial v ^ { i } } . \\end{align*}"}
+{"id": "8694.png", "formula": "\\begin{align*} [ \\ , u \\ , | \\ , v \\ , ] = ( \\ , \\tilde P u \\ , | \\ , \\tilde P v ) _ { M } , \\ \\ u , v \\in W ^ { - \\frac { 1 } { 2 } } ( X , T ^ { * 0 , q } M ' ) . \\end{align*}"}
+{"id": "8888.png", "formula": "\\begin{align*} \\mathcal E [ u _ 0 ] < \\mathcal E [ \\varphi ] = f ( t _ 1 ) , \\end{align*}"}
+{"id": "2720.png", "formula": "\\begin{align*} Z _ { \\alpha } \\tau _ { \\beta } ^ { i } - Z _ { \\beta } \\tau _ { \\alpha } ^ { i } = T _ { \\alpha \\beta } ^ { \\gamma } \\tau ^ { i } _ { \\gamma } , \\end{align*}"}
+{"id": "8621.png", "formula": "\\begin{align*} C _ 0 = 2 ( \\| u _ 0 \\| _ { L ^ \\infty ( \\mathbb { R } ) } + \\| u _ 0 ^ \\prime \\| ) _ { L ^ \\infty ( \\mathbb { R } ) } ) , ~ ~ C _ 1 = 2 \\| u _ 0 ^ \\prime \\| ) _ { L ^ \\infty ( \\mathbb { R } ) } , ~ ~ C _ 2 = ( - m ( 0 ) ) ^ { 3 / 4 } \\end{align*}"}
+{"id": "6987.png", "formula": "\\begin{align*} \\mathcal { B } _ x : = \\{ W _ { 1 } = A _ 1 / \\mathrm { c o s h } ( t _ { 1 } ) , W _ { 2 } = A _ 2 , \\cdots , W _ n = A _ n , W _ { n + 1 } = A _ { n + 1 } / \\mathrm { c o s h } ( t _ { 1 } ) \\} \\end{align*}"}
+{"id": "3014.png", "formula": "\\begin{align*} \\begin{array} { l } \\lambda = a + \\alpha b - \\beta ^ 2 + \\beta + ( 2 n - 1 ) \\alpha ^ 2 \\\\ \\mu = b + \\beta b + ( \\alpha - \\alpha \\beta ) ( 2 n - 2 ) \\\\ \\nu = \\alpha b - ( 2 n + 1 ) \\beta + \\beta ^ 2 + ( 1 - 2 n ) \\alpha ^ 2 + c - \\beta b + ( \\alpha \\beta - \\alpha ) ( 2 n - 2 ) \\end{array} \\end{align*}"}
+{"id": "158.png", "formula": "\\begin{align*} f ( \\eta _ { X } ( x ) ) & = f ( \\left ( \\prod _ { i \\in J _ { F } } u ^ { * } _ { i } \\right ) x \\left ( \\prod _ { i \\in J _ { F } } u _ { i } \\right ) ) \\\\ & = \\left ( \\prod _ { i \\in J _ { F } } u ^ { * } _ { i } \\right ) f ( x ) \\left ( \\prod _ { i \\in J _ { F } } u _ { i } \\right ) \\\\ & = \\eta _ { Y } ( f ( x ) ) \\\\ & = \\eta _ { Y } ( \\alpha _ { * } ( f ) ( x ) ) \\\\ \\end{align*}"}
+{"id": "689.png", "formula": "\\begin{align*} A = & f ( 1 ) ^ 2 - g ( 1 ) ^ 2 \\\\ B = & f ( - 1 ) ^ 2 - g ( - 1 ) ^ 2 \\\\ C = & | f ( i ) | ^ 2 - | g ( i ) | ^ 2 \\\\ D = & \\left ( | f ( \\omega ) | ^ 2 + | g ( \\omega ) | ^ 2 \\right ) \\left ( | f ( \\omega ^ 3 ) | ^ 2 + | g ( \\omega ^ 3 ) | ^ 2 \\right ) . \\end{align*}"}
+{"id": "2165.png", "formula": "\\begin{align*} | D ' | = \\sum _ { j = 1 } ^ { 4 \\log | A | } \\sum _ { \\stackrel { d \\in D ' : } { 2 ^ { j - 1 } K \\leq | 2 f _ d ( A _ d '' ) - 2 f _ d ( A _ d '' ) | < 2 ^ { j } K } } 1 . \\end{align*}"}
+{"id": "5962.png", "formula": "\\begin{align*} f _ { \\varphi } & \\ : = \\ : | ( a c + b c ) - ( c - a b ) | \\\\ & \\ : = \\ : \\left ( \\left ( a c + b c \\right ) - \\left ( a ^ { \\prime } c + b ^ { \\prime } c \\right ) \\right ) + \\left ( \\left ( a ^ { \\prime } c + b ^ { \\prime } c \\right ) - \\left ( a c + b c \\right ) \\right ) \\\\ & \\ : = \\ : a b c + a ^ { \\prime } b ^ { \\prime } c \\end{align*}"}
+{"id": "3494.png", "formula": "\\begin{align*} \\rho _ k ( x ) ^ * \\rho _ k ( x ) = \\lim _ n e \\rho _ n \\big ( \\rho _ { n , k } ( x ) ^ * \\rho _ { n , k } ( x ) \\big ) . \\end{align*}"}
+{"id": "429.png", "formula": "\\begin{align*} \\begin{aligned} - \\begin{bmatrix} 1 - F _ { j , j - 1 } & 0 & - \\Gamma _ { j } \\\\ - G _ { j - 1 , j - 1 } - \\widetilde { t } _ { j } \\rho ^ 2 & 0 & 1 - F _ { j , j - 1 } ^ { T } \\\\ x & - M \\rho ^ 2 \\widetilde { t } _ { j } ^ 2 & 1 - f _ { j , j } \\end{bmatrix} \\end{aligned} \\end{align*}"}
+{"id": "1547.png", "formula": "\\begin{align*} A ^ * ( s ) = \\sup _ { t \\ge 0 } \\left ( s \\ , t - A ( t ) \\right ) = \\int _ 0 ^ s a ^ { - 1 } ( \\sigma ) \\ , d \\sigma \\end{align*}"}
+{"id": "256.png", "formula": "\\begin{align*} \\frac { d ^ 2 x } { d t ^ 2 } + \\gamma ( x ) \\left ( \\frac { d x } { d t } \\right ) ^ 2 + A ( x ) \\frac { d x } { d t } + b ( x ) = 0 . \\end{align*}"}
+{"id": "3122.png", "formula": "\\begin{align*} u _ { \\psi ^ * } ^ - ( s ) = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 2 \\ , s ^ { p ^ { e _ 1 } } & 1 & 0 \\\\ 2 \\ , s ^ { 2 \\ , p ^ { e _ 1 } } & 2 \\ , s ^ { p ^ { e _ 1 } } & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "3566.png", "formula": "\\begin{align*} T ^ 2 f = \\alpha ^ 2 \\tau _ 0 \\tau _ 1 f \\circ \\tau ^ 2 T ^ 3 f = \\alpha ^ 3 \\tau _ 0 \\tau _ 1 \\tau _ 2 f \\circ \\tau ^ 3 , \\end{align*}"}
+{"id": "5941.png", "formula": "\\begin{align*} \\dfrac { M _ { 1 } ( \\mathcal { C } ( G ) ) } { | v ( \\mathcal { C } ( G ) ) | } = \\dfrac { 2 ^ { 5 k } - 4 \\cdot 2 ^ { 4 k } + 4 \\cdot 2 ^ { 3 k } + 4 \\cdot 2 ^ { 2 k } - 5 \\cdot 2 ^ { k } - 4 } { 2 ^ { 3 k } - 2 ^ { k } - 1 } \\end{align*}"}
+{"id": "5157.png", "formula": "\\begin{align*} A = \\frac { 1 } { 1 - \\alpha } \\sum _ { i } p _ { i } + \\frac { 1 } { \\alpha \\left ( \\alpha - 1 \\right ) } \\sum _ { i } p ^ { \\alpha } _ { i } q ^ { 1 - \\alpha } _ { i } \\ ; \\ ; ; \\ ; \\ ; B = - \\frac { 1 } { \\alpha } \\sum _ { i } q _ { i } \\end{align*}"}
+{"id": "5013.png", "formula": "\\begin{align*} f _ i ^ \\tau ( \\mathbf { R } , \\mathbf { Q } ) = Q _ i + f _ i ^ \\tau ( \\mathbf { R } - ( \\mathbf { I } - \\beta \\mathbf { P } ) \\mathbf { Q } , \\mathbf { 0 } ) . \\end{align*}"}
+{"id": "1131.png", "formula": "\\begin{align*} J : = \\left \\{ \\begin{aligned} & \\frac { n } { \\min \\{ 1 , p \\} } & & , \\\\ & \\frac { n } { \\min \\{ 1 , p , q \\} } & & . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3859.png", "formula": "\\begin{align*} \\begin{aligned} \\sigma ^ { \\kappa } ( x , y \\mid t ) & \\doteq \\frac { \\kappa ( 1 - \\kappa ) \\eta ^ 0 _ { ( 1 ) } ( x \\mid s ) G ( \\pi ^ * ) _ { x , y } + \\kappa ( 1 - \\kappa ) \\pi ^ * _ x G ( M ^ { 0 } ( s ) ) _ { x , y } + \\kappa ^ 2 \\pi ^ { * } _ x G ( \\pi ^ * ) _ { x , y } } { 2 \\kappa ( 1 - \\kappa ) + \\kappa ^ 2 } . \\end{aligned} \\end{align*}"}
+{"id": "5871.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { N C } ( G ) ) = 4 \\cdot 2 ^ { 4 n - 4 } - 2 0 \\cdot 2 ^ { 3 n - 3 } + 3 2 \\cdot 2 ^ { 2 n - 2 } - 1 6 \\cdot 2 ^ { n - 1 } . \\end{align*}"}
+{"id": "3672.png", "formula": "\\begin{align*} \\frac { 2 | E | } { | V | } = 2 d - \\frac { d ( d + 1 ) } { | V | } > d . \\end{align*}"}
+{"id": "7628.png", "formula": "\\begin{align*} r _ d ( f _ 1 ( z ) , f _ 2 ( z ) , f _ 3 ( z ) ) = \\frac { \\tilde { r } _ d ( f _ 1 ( z ) , f _ 2 ( z ) , f _ 3 ( z ) ) } { \\vert \\tilde { r } _ d ( f _ 1 ( z ) , f _ 2 ( z ) , f _ 3 ( z ) ) \\vert } \\end{align*}"}
+{"id": "5458.png", "formula": "\\begin{align*} \\mathring { B } _ t ^ i : = B _ t ^ i + \\frac { 1 } { \\sqrt { 2 \\sigma } } \\int _ 0 ^ t \\frac { 1 } { N } H ( X ^ { i , N } _ s - X ^ { 1 , N } _ s , V ^ { i , N } _ s - V ^ { 1 , N } _ s ) d s \\end{align*}"}
+{"id": "5932.png", "formula": "\\begin{align*} \\dfrac { M _ { 2 } ( \\mathcal { C } ( G ) ) } { | e ( \\mathcal { C } ( G ) ) | } & - \\dfrac { M _ { 1 } ( \\mathcal { C } ( G ) ) } { | v ( \\mathcal { C } ( G ) ) | } \\\\ & = \\dfrac { 2 q ^ { 8 } ( q - 5 ) + q ^ { 5 } ( 1 4 q ^ { 2 } - 1 3 ) + q ^ { 3 } ( 2 4 q ^ { 3 } - q + 4 ) + q ( 8 q - 7 ) + 1 } { q ^ { 5 } ( q ^ { 2 } - 2 q - 2 ) + q ( 3 q ^ { 3 } - q - 3 ) + ( 4 q ^ { 3 } - 1 ) } : = \\dfrac { f ( q ) } { g ( q ) } . \\end{align*}"}
+{"id": "612.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\left ( \\frac { 1 } { 4 } \\right ) ^ { n } \\binom { 2 n } { n } \\frac { O _ { 2 n } } { 4 n + 3 } = \\frac { \\pi ^ { 3 / 2 } ( 3 \\ln ( 2 ) + 2 ) } { 2 \\sqrt { 2 } \\Gamma ^ { 2 } \\left ( \\frac { 1 } { 4 } \\right ) } \\end{align*}"}
+{"id": "164.png", "formula": "\\begin{align*} u ^ { F , G } _ { X , Y } ( a ( b _ { i } \\boxtimes c _ { j } ) ) & = \\sum _ { k } c _ { k } \\boxtimes b _ { i } \\langle c _ { k } \\ | \\ a c _ { j } \\rangle \\\\ & = \\sum _ { k } c _ { k } \\boxtimes \\langle c _ { k } \\ | \\ a c _ { j } \\rangle b _ { i } \\\\ & = \\sum _ { k } c _ { k } \\langle c _ { k } \\ | \\ a c _ { j } \\rangle \\boxtimes b _ { i } \\\\ & = a c _ { j } \\boxtimes b _ { i } = a ( c _ { j } \\boxtimes b _ i ) \\\\ & = a u ^ { F , G } _ { X , Y } ( b _ { i } \\boxtimes c _ { j } ) . \\end{align*}"}
+{"id": "7040.png", "formula": "\\begin{align*} & \\mathbb E _ x \\int _ 0 ^ t \\bigl | b \\cdot \\nabla g \\bigr | ( \\omega _ s ) d s \\\\ & \\leq \\liminf _ n \\mathbb E _ x \\int _ 0 ^ t \\bigl | b _ n \\cdot \\nabla g \\bigr | ( \\omega _ s ) d s = \\liminf _ n \\int _ 0 ^ t e ^ { - s \\Lambda _ { C _ \\infty } ( b ) } \\bigl | b _ n \\cdot \\nabla g \\bigr | ( x ) d s \\\\ & \\leq e ^ { \\mu T } \\liminf _ n ( \\mu + \\Lambda _ { C _ \\infty } ( b ) ) ^ { - 1 } | b _ n | | \\nabla g | ( x ) \\end{align*}"}
+{"id": "8719.png", "formula": "\\begin{gather*} L _ { 3 } = L e i _ { 3 } ( 3 , F ) = F a _ { 1 } \\oplus F a _ { 2 } \\oplus F a _ { 3 } , \\ \\mbox { w h e r e } [ a _ { 1 } , a _ { 1 } ] = [ a _ { 1 } , a _ { 2 } ] = a _ { 3 } , \\\\ [ a _ { 1 } , a _ { 3 } ] = [ a _ { 2 } , a _ { 1 } ] = [ a _ { 2 } , a _ { 2 } ] = [ a _ { 2 } , a _ { 3 } ] = [ a _ { 3 } , a _ { 1 } ] = [ a _ { 3 } , a _ { 2 } ] = [ a _ { 3 } , a _ { 3 } ] = 0 . \\end{gather*}"}
+{"id": "6488.png", "formula": "\\begin{align*} g _ u = \\begin{cases} - \\sum _ { v \\in \\N ; i ' _ v = i _ 1 } g ' _ v & , \\\\ g ' _ { u - 1 } & . \\\\ \\end{cases} \\end{align*}"}
+{"id": "6696.png", "formula": "\\begin{align*} L = G _ { n + 1 } \\trianglelefteq G _ n \\trianglelefteq \\ldots \\trianglelefteq G _ 1 = G \\end{align*}"}
+{"id": "2972.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\pm } ( D _ X , D _ V ) : = D _ V { \\pm } \\frac { \\kappa _ 1 \\mathcal { A } ( v ) ( 0 ) } { T _ M ^ \\infty } \\Phi ( D _ X ) , \\end{align*}"}
+{"id": "6876.png", "formula": "\\begin{align*} \\psi _ j ^ { ( n ) } = \\frac { 1 } { \\sqrt { n } } , j = 1 , \\ldots { n } , \\psi _ { n + 1 } ^ { ( n ) } = 0 . \\end{align*}"}
+{"id": "2435.png", "formula": "\\begin{align*} u ( t , x ) = ( U ( t , x ) ) ^ { p } , \\gamma = \\dfrac { p - 1 } { p } , k = p \\mu . \\end{align*}"}
+{"id": "8889.png", "formula": "\\begin{align*} f \\big ( \\| \\sqrt { \\mathcal K _ \\lambda } u \\| ^ 2 \\big ) \\leq \\mathcal E [ u ] = \\mathcal E [ u _ 0 ] < f ( t _ 1 ) . \\end{align*}"}
+{"id": "6541.png", "formula": "\\begin{align*} S _ N - S _ { N , 1 } = \\sum ^ N _ { n = 1 } \\sum ^ { m _ 0 } _ { j = 1 } \\left ( \\mathbb { E } \\big [ K ( X _ n ) | \\mathcal { F } _ { n - j } \\big ] - \\mathbb { E } \\big [ K ( X _ n ) | \\mathcal { F } _ { n - j - 1 } \\big ] \\right ) : = \\sum ^ N _ { n = 1 } \\sum ^ { m _ 0 } _ { j = 1 } \\mathcal { P } _ { n , n - j } \\end{align*}"}
+{"id": "730.png", "formula": "\\begin{align*} \\mathcal { I } _ 2 ( H ( f _ 1 ) ) & = ( x _ 1 x _ 2 , \\ x _ 1 x _ 4 , \\ x _ 2 x _ 3 , \\ x _ 3 x _ 4 ) = \\mathfrak { p } _ 1 \\cap \\mathfrak { p } _ 2 . \\end{align*}"}
+{"id": "7128.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 } } | u _ { t } | ^ { 2 } d x = \\int _ { \\mathbb { R } ^ { 2 } } | u | ^ { 2 } d x = c , \\ A ( u _ { t } ) = t ^ { 2 } A ( u ) , \\ V ( u _ { t } ) = V ( u ) - c ^ { 2 } \\ln t , \\end{align*}"}
+{"id": "4651.png", "formula": "\\begin{align*} D _ { 3 , n } = \\frac { 1 } { 2 } \\left ( D _ { 2 , n } ( N ) - N \\cdot [ n ] + O ( 1 ) \\right ) . \\end{align*}"}
+{"id": "5363.png", "formula": "\\begin{align*} \\widehat { \\mathbf { h } } ^ 0 = \\mathbf { c } ^ { J } . \\end{align*}"}
+{"id": "4267.png", "formula": "\\begin{align*} R c + \\nabla ^ 2 f = \\frac 1 { 2 \\tau } g , \\end{align*}"}
+{"id": "36.png", "formula": "\\begin{align*} \\langle d x ^ I ; d x ^ { I ' } \\rangle _ { \\bigwedge \\nolimits ^ j ( \\Z ^ n ) } : = \\begin{cases} 1 & I = I ' \\\\ 0 & \\end{cases} \\ . \\end{align*}"}
+{"id": "1816.png", "formula": "\\begin{align*} d \\phi = 0 d \\star _ { \\phi } \\phi = 0 , \\end{align*}"}
+{"id": "8684.png", "formula": "\\begin{align*} u = U - \\sqrt { - 1 } J _ G U , v = V - \\sqrt { - 1 } J _ G V . \\end{align*}"}
+{"id": "4254.png", "formula": "\\begin{align*} G _ \\xi [ f , g , h ] ( \\eta , \\sigma ) : = \\hat f ( \\xi - \\eta ) \\hat { \\bar { g } } ( \\eta - \\xi + \\sigma ) \\hat h ( \\xi - \\sigma ) . \\end{align*}"}
+{"id": "635.png", "formula": "\\begin{align*} \\ddot { \\lambda } _ { H _ r , d _ r } ( \\rho ) = \\frac { \\cosh ( \\rho ) \\sinh ^ { \\frac { 2 ( n - r ) } { r } - 1 } ( \\rho ) \\left ( n H _ r \\frac { \\sinh ^ n ( \\rho ) } { \\cosh ^ r ( \\rho ) } - ( n - r ) ( n H _ r I _ { n , r } ( \\rho ) + d _ r ) \\right ) } { r ( n H _ r I _ { n , r } ( \\rho ) + d _ r ) ^ { \\frac { r - 1 } { r } } \\left ( \\sinh ^ { \\frac { 2 ( n - r ) } { r } } ( \\rho ) - ( n H _ r I _ { n , r } ( \\rho ) + d _ r ) ^ { \\frac { 2 } { r } } \\right ) ^ { \\frac { 3 } { 2 } } } . \\end{align*}"}
+{"id": "6677.png", "formula": "\\begin{align*} \\lvert G : \\widetilde { H } \\rvert < \\lvert G : H \\rvert \\dd ( \\widetilde { H } ) = r _ p , \\end{align*}"}
+{"id": "8088.png", "formula": "\\begin{align*} \\Phi _ t ( n ) : = ( \\phi _ 1 J _ 1 ) \\circ u ^ + _ t ( n ) \\cdot J _ t ^ { - 1 } ( n ) , \\end{align*}"}
+{"id": "963.png", "formula": "\\begin{align*} u = P _ D ( g ) + R ^ D f ( \\cdot , u ) + R ^ D \\mu \\mbox { q . e . } \\end{align*}"}
+{"id": "1052.png", "formula": "\\begin{align*} h ( z ) = ( 1 - z ) ^ { - d } g ( z ) \\mbox { a n d } h _ { \\sharp } ( z ) = ( 1 - z ) ^ { - d } g _ { \\sharp } ( z ) , \\end{align*}"}
+{"id": "5544.png", "formula": "\\begin{align*} K = \\frac { L ^ 2 } { \\rho } . \\end{align*}"}
+{"id": "5721.png", "formula": "\\begin{align*} \\Vert ( v , w ) \\Vert _ { H ^ { \\ell + 1 } \\times H ^ \\ell } \\leq C \\sum _ { j = 0 } ^ { \\ell + 1 } \\Vert \\mathbf { L } ^ j ( v , w ) \\Vert _ { G } . \\end{align*}"}
+{"id": "8323.png", "formula": "\\begin{align*} \\Pi _ t = ( \\phi _ t ) _ { * } \\Pi _ { 0 } \\end{align*}"}
+{"id": "1028.png", "formula": "\\begin{align*} T _ { \\infty } ( w ) : = \\left ( \\begin{matrix} \\gamma ( 0 ) & \\gamma ( - 1 ) & \\gamma ( - 2 ) & \\cdots \\cr \\gamma ( 1 ) & \\gamma ( 0 ) & \\gamma ( - 1 ) & \\cdots \\cr \\gamma ( 2 ) & \\gamma ( 1 ) & \\gamma ( 0 ) & \\cdots \\cr \\vdots & \\vdots & \\vdots & \\ddots \\end{matrix} \\right ) . \\end{align*}"}
+{"id": "4202.png", "formula": "\\begin{align*} & 4 8 0 \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 1 6 ) } = \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right . \\\\ & \\left . \\cdot { \\rm c h } ( \\widetilde { T X } + 2 \\wedge ^ 2 \\widetilde { L _ R \\otimes C } - \\widetilde { L _ R \\otimes C } \\otimes \\widetilde { L _ R \\otimes C } + \\widetilde { L _ R \\otimes C } ) \\right \\} ^ { ( 1 6 ) } . \\end{align*}"}
+{"id": "530.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } \\left ( x ^ { \\alpha + \\beta } u _ { x } v _ { x } - \\frac { \\mu } { x ^ { 2 - \\alpha - \\beta } } u v \\right ) \\mathrm { d } x = - \\int _ { 0 } ^ { 1 } x ^ { \\beta } \\left ( \\left ( x ^ { \\alpha } u _ { x } \\right ) _ { x } + \\beta x ^ { \\alpha - 1 } u _ { x } + \\frac { \\mu } { { x ^ { 2 - \\alpha } } } u \\right ) v \\mathrm { d } x \\end{align*}"}
+{"id": "263.png", "formula": "\\begin{align*} \\frac { d ^ 2 y } { d \\tau ^ 2 } + \\alpha \\frac { d y } { d \\tau } + B y + C = 0 , \\end{align*}"}
+{"id": "4580.png", "formula": "\\begin{align*} \\frac { d ^ 2 A } { d t ^ 2 } ( t ) = 2 \\pi \\sum \\frac { 1 } { r _ j s _ j } \\delta _ { t _ j } . \\end{align*}"}
+{"id": "7109.png", "formula": "\\begin{align*} \\| b _ \\varepsilon \\varphi \\| _ 2 ^ 2 & \\leq \\langle E _ \\varepsilon | b | ^ 2 , \\varphi ^ 2 \\rangle = \\| b \\sqrt { E _ \\varepsilon \\varphi ^ 2 } \\| ^ 2 _ 2 \\\\ & \\leq \\delta \\| \\nabla \\sqrt { E _ \\varepsilon \\varphi ^ 2 } \\| _ 2 ^ 2 + c _ \\delta \\| \\varphi \\| _ 2 ^ 2 , \\varphi \\in W ^ { 1 , 2 } , \\end{align*}"}
+{"id": "3880.png", "formula": "\\begin{align*} R ^ { n , m ^ n _ 0 + l ^ { n , 0 } } \\le | \\log ( \\delta _ 1 / 2 ) | . \\end{align*}"}
+{"id": "8745.png", "formula": "\\begin{align*} \\pi \\left ( \\left \\{ ( x , y ) : ( y - x ) ^ 2 > z \\right \\} \\right ) \\le f ( z ) \\mbox { w h e r e } f ( z ) = \\left ( \\mu ( \\{ x : 4 x ^ 2 > z \\} ) + \\nu ( \\{ y : 4 y ^ 2 > z \\} ) \\right ) \\wedge 1 . \\end{align*}"}
+{"id": "575.png", "formula": "\\begin{align*} T & = u , & 1 - \\frac { \\omega ^ 2 m } { g \\rho } & = v . \\end{align*}"}
+{"id": "7124.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\int _ { \\mathbb { R } ^ { 2 } } F ( u _ { n } ) d x = \\int _ { \\mathbb { R } ^ { 2 } } F ( u _ { c } ) d x . \\end{align*}"}
+{"id": "6135.png", "formula": "\\begin{align*} \\| \\varphi _ k ( x ) - \\varphi _ n ( \\psi _ n ( \\varphi _ k ( x ) ) ) \\| & = \\| ( \\Psi ^ { - 1 } \\circ \\rho _ k ) ( x ) - ( \\Psi ^ { - 1 } \\circ \\rho _ n ) ( \\psi _ n ( \\varphi _ k ( x ) ) ) \\| \\\\ & = \\| \\rho _ k ( x ) - \\rho _ n ( \\psi _ n ( \\varphi _ k ( x ) ) ) \\| \\\\ & < \\| \\rho _ k ( x ) - \\rho _ n ( \\rho _ { n , k } ( x ) ) ) \\| + \\varepsilon \\\\ & = \\varepsilon . \\end{align*}"}
+{"id": "1592.png", "formula": "\\begin{align*} G ( y ) \\le C \\ , | y | \\ , \\eta _ H ( | y | ) = C \\ , \\eta _ { H + 1 , 1 + \\frac { 1 } { H } } ( | y | ) \\forall \\ , y \\in \\R ^ N . \\end{align*}"}
+{"id": "8705.png", "formula": "\\begin{align*} & B _ G = B ^ { ( 0 ) } _ { \\leq \\lambda _ 0 } \\circ Q _ G \\ \\ \\mbox { o n $ L ^ 2 ( M ) $ } , \\\\ & B _ G ( x , y ) = \\int _ G B _ { \\leq \\lambda _ 0 } ( x , g y ) d \\mu ( g ) . \\end{align*}"}
+{"id": "6976.png", "formula": "\\begin{align*} \\langle \\widetilde { \\Pi } ( \\mathbf { z } ) , \\widetilde { \\Pi } ( \\mathbf { z } ) \\rangle = 2 \\lvert \\eta ( \\mathbf { z } ) \\rvert ^ 2 ( \\langle \\mathbf { z } , \\mathbf { z } \\rangle - \\lvert \\eta ( \\mathbf { z } ) \\rvert ^ 2 ) < 0 . \\end{align*}"}
+{"id": "6645.png", "formula": "\\begin{align*} \\nu ^ { + } ( E , \\mathrm { i } y ) & = \\omega ^ { + } ( E , \\mathrm { i } y ) , \\\\ \\nu ^ { - } ( E , \\mathrm { i } y ) & = \\omega ^ { - } ( E , \\mathrm { i } y ) . \\end{align*}"}
+{"id": "5286.png", "formula": "\\begin{align*} - \\frac { \\partial K L ( q \\| p ) } { \\partial q _ { j } } = \\log p _ { j } - \\log q _ { j } \\end{align*}"}
+{"id": "318.png", "formula": "\\begin{align*} ( f ^ m ) '' ( \\xi ) - \\alpha f ( \\xi ) + \\beta \\xi f ' ( \\xi ) + \\xi ^ { \\sigma } f ( \\xi ) ^ p = 0 , \\xi = | x | ^ { \\sigma } ( T - t ) ^ { \\beta } . \\end{align*}"}
+{"id": "3809.png", "formula": "\\begin{align*} ( \\phi ( x ) - G x _ j ) | _ { I _ j } = 0 j \\in [ t ] . \\end{align*}"}
+{"id": "1290.png", "formula": "\\begin{align*} \\dot { \\mathbf { z } } = \\mathbf { Q } ~ \\mathbf { z } \\ ; , \\end{align*}"}
+{"id": "6356.png", "formula": "\\begin{align*} A _ k & : = ( n - k ) + \\max _ { r \\in [ 0 , R _ 0 ] } r \\frac { \\omega _ k ' ( r ) } { \\omega _ k ( r ) } , \\\\ B _ k & : = \\max _ { r \\in [ 0 , R _ 0 ] } \\frac { \\omega _ k ( 2 r ) } { \\omega _ k ( r ) } , \\\\ C _ k & : = \\omega _ k ( 2 R _ 0 ) , \\\\ \\end{align*}"}
+{"id": "7271.png", "formula": "\\begin{align*} x _ k = - \\tfrac { 1 } { 1 - q } \\left ( \\theta q ^ k - ( \\theta + \\eta s ) \\right ) , k = 0 , 1 , 2 , \\ldots , \\end{align*}"}
+{"id": "1590.png", "formula": "\\begin{align*} \\int _ { A ( M , R / 2 ) } ( g _ M ( V ) - 1 ) ^ 2 \\ , d x = 0 \\end{align*}"}
+{"id": "5670.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l } - \\Delta _ g u + V ( \\sigma ) u = f ( u ) & \\hbox { o n $ \\mathcal { M } $ } \\\\ u > 0 & \\hbox { o n $ \\mathcal { M } $ } , \\end{array} \\right . \\end{align*}"}
+{"id": "8747.png", "formula": "\\begin{align*} f ( x _ + ) - f ( x _ - ) & = \\frac { z - m } { z - y } \\left ( \\varphi ( x _ + - y ) - \\varphi ( x _ - - y ) - \\varphi ( m - x _ + ) + \\varphi ( m - x _ - ) \\right ) \\\\ & + \\frac { m - y } { z - y } \\left ( \\varphi ( z - x _ + ) - \\varphi ( z - x _ - ) - \\varphi ( m - x _ + ) + \\varphi ( m - x _ - ) \\right ) . \\end{align*}"}
+{"id": "2120.png", "formula": "\\begin{align*} \\langle \\tilde { M } ^ { \\Theta } _ { f ^ i } , \\tilde { M } ^ { \\Theta } _ { f ^ j } \\rangle ( t ) = \\int ^ { t } _ { 0 } \\sum ^ { d _ { 1 } } _ { k = 1 } \\sigma _ { i k } \\sigma _ { k j } ( s , \\phi ( s ) , \\nu _ { \\Theta } ( s ) ) \\mathrm { d } s . \\end{align*}"}
+{"id": "1078.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { n } \\Vert ( T _ n ( \\tilde { w } ) ^ { - 1 } ) ^ { s , t } \\| \\leq B _ 3 , n \\in \\N , \\ \\ t \\in \\{ 1 , \\dots , [ ( n + 1 ) / 2 ] \\} . \\end{align*}"}
+{"id": "1974.png", "formula": "\\begin{align*} R _ { \\alpha \\beta } - \\frac { 1 } { 2 } R \\gamma _ { \\alpha \\beta } = \\mathcal { T } _ { \\alpha \\beta } \\ , , \\end{align*}"}
+{"id": "2341.png", "formula": "\\begin{align*} \\tau _ { p } : = \\begin{cases} C _ { 0 } K ( c , \\gamma ) , & p = 2 , \\\\ \\frac { C _ { 0 } K ( c , \\gamma ) } { \\min \\{ \\frac { 1 } { 2 } , \\frac { 2 ( p - 2 ) } { p } - C _ { 0 } ( p - 1 ) K ( c , \\gamma ) \\} } , & p > 2 . \\end{cases} \\end{align*}"}
+{"id": "5003.png", "formula": "\\begin{align*} \\det [ p _ k ( x _ j ) ] _ { j , k = 1 , \\ldots , N } = Q _ N ( \\{ p \\} ) \\prod _ { 1 \\le j < k \\le N } ( x _ k - x _ j ) . \\end{align*}"}
+{"id": "5838.png", "formula": "\\begin{align*} \\begin{array} { r @ { \\ } l } \\left ( 1 - \\alpha ^ { 3 n - 3 j } \\right ) \\kappa ^ n _ { 3 n - 2 j , j , j } & = \\displaystyle \\alpha ^ { 3 n - 5 j - 3 } \\left ( \\sum _ { \\nu = 0 } ^ { j } \\alpha ^ { 3 \\nu } \\right ) \\kappa ^ n _ { 3 n - 2 j - 2 , j + 1 , j + 1 } \\\\ & = \\displaystyle \\alpha ^ { 3 n - 5 j - 3 } ( 1 - \\alpha ^ 3 ) ^ { - 1 } ( 1 - \\alpha ^ { 3 j + 3 } ) \\kappa ^ n _ { 3 n - 2 j - 2 , j + 1 , j + 1 } , \\quad \\cdots ( \\ast ) \\end{array} \\end{align*}"}
+{"id": "3533.png", "formula": "\\begin{align*} \\psi _ 1 ( z _ 1 ) = \\psi ( z _ 1 , z _ 2 ) ( z _ 1 \\in \\mathbb { T } ) . \\end{align*}"}
+{"id": "8820.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\infty v _ j 2 ^ { - j s } \\varphi ( 2 ^ { - j } ) ^ { - 1 } = \\infty . \\end{align*}"}
+{"id": "662.png", "formula": "\\begin{align*} \\sigma ( i _ { 1 } , \\cdots , i _ { p } ) u _ { i _ { 1 } , \\cdots , i _ { m } } = \\frac { 1 } { p ! } \\sum _ { \\pi \\in \\Pi _ { p } } u _ { i _ { \\pi ( 1 ) } \\cdots i _ { \\pi ( p ) } i _ { p + 1 } \\cdots i _ { m } } , \\end{align*}"}
+{"id": "109.png", "formula": "\\begin{align*} a ( \\vec { u } , \\vec { v } ) & = ( \\varepsilon ( \\vec { u } ) , \\varepsilon ( \\vec { v } ) ) , \\\\ b ( \\vec { v } , q ) & = ( \\operatorname { d i v } \\vec { v } , q ) . \\end{align*}"}
+{"id": "8145.png", "formula": "\\begin{align*} D _ 2 & \\ = \\ \\frac { 2 } { 9 } \\left ( - 6 J _ 6 ^ 2 - 6 J _ 5 J _ 4 + 3 J _ 3 J _ 0 + 4 J _ 0 ^ 2 - 8 J _ 6 + 3 J _ 3 + 1 0 J _ 0 - 1 4 \\right ) A _ 9 \\ + \\\\ & \\frac { 8 } { 9 } \\left ( 3 J _ 6 J _ 5 + 9 J _ 4 J _ 1 + 4 J _ 5 \\right ) A _ 8 - \\frac { 2 } { 9 } \\left ( 1 2 J _ 5 J _ 1 - 1 3 J _ 2 J _ 0 \\right ) A _ 7 \\ - \\\\ & - \\frac { 2 8 } { 9 } J _ 6 J _ 2 A _ 5 + \\frac { 2 8 } { 9 } J _ 6 J _ 1 A _ 3 \\ , \\end{align*}"}
+{"id": "2134.png", "formula": "\\begin{align*} x \\left [ \\nabla _ z f ( x , y ) - A _ { \\{ 1 , 2 \\} } ( x ) ^ T \\lambda _ { \\{ 1 , 2 \\} } ( x , y ) \\right ] = 0 . \\end{align*}"}
+{"id": "6085.png", "formula": "\\begin{align*} h _ { \\lambda } : = [ P - ( P ^ { \\perp } ( T - \\lambda ) P ^ { \\perp } ) ^ { - 1 } P ^ { \\perp } T P ] \\psi _ { \\lambda } . \\end{align*}"}
+{"id": "2813.png", "formula": "\\begin{align*} & H _ { T } = H + \\zeta ^ { \\alpha } \\phi ^ { ( 1 ) } _ { \\alpha } , \\\\ & H = ( q ^ { 1 } ) ^ { 2 } + ( q ^ { 2 } ) ^ { 2 } . \\end{align*}"}
+{"id": "6181.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau ^ k } [ f _ 2 ( x _ 2 ) - f _ 2 ( \\breve { x } _ 2 ^ { k } ) ] - \\frac { 1 } { \\tau ^ { k - 1 } } [ f _ 2 ( x _ 2 ) - f _ 2 ( \\breve { x } _ 2 ^ { k - 1 } ) ] + ( x _ 2 - \\widetilde { x } _ 2 ^ { k } ) ^ T [ - A _ 2 ^ T \\widetilde { \\lambda } ^ k \\\\ & + \\tau ^ k \\beta ^ k D ( \\widetilde { x } _ 2 ^ { k } - \\bar { x } _ 2 ^ k ) + ( 1 - \\gamma ) ( 1 - \\tau ^ k ) \\beta ^ k A _ 2 ^ T ( A \\breve { x } ^ { k - 1 } - b ) ] \\geq 0 , ~ \\forall x _ 2 . \\end{aligned} \\end{align*}"}
+{"id": "3083.png", "formula": "\\begin{align*} \\frac { I ( F , G ) } { v ' _ 0 } = \\frac { n _ q v _ q + v _ 0 \\cdot c ( \\mathcal { C } _ F , \\mathcal { C } _ G ) - \\beta _ q } { n _ 0 \\cdot \\ldots \\cdot n _ q } . \\end{align*}"}
+{"id": "2429.png", "formula": "\\begin{align*} V _ { t } = ( V ^ { m } ) _ { x x } , t > 0 , x \\in \\mathbb { R } , m > 1 . \\end{align*}"}
+{"id": "6948.png", "formula": "\\begin{align*} f : = - \\bigg \\lvert \\begin{array} { c c } \\langle \\mathbf { z _ 1 } , \\mathbf { z _ 1 } \\rangle & \\langle \\mathbf { z _ 1 } , \\mathbf { z _ 2 } \\rangle \\\\ \\langle \\mathbf { z _ 2 } , \\mathbf { z _ 1 } \\rangle & \\langle \\mathbf { z _ 2 } , \\mathbf { z _ 2 } \\rangle \\end{array} \\bigg \\vert \\end{align*}"}
+{"id": "3721.png", "formula": "\\begin{align*} b ^ d _ 1 \\leq - 1 - \\frac { 2 } { d + 2 } = b ^ d _ 2 . \\end{align*}"}
+{"id": "3301.png", "formula": "\\begin{align*} D = \\bigcup _ { \\substack { 0 < a < b < \\infty \\\\ a , b \\in \\mathbb Q } } D ^ { a , b } , \\end{align*}"}
+{"id": "6559.png", "formula": "\\begin{align*} \\gamma _ 2 = \\sigma _ 2 ( C ^ + _ K ) ^ { \\alpha \\beta } 1 _ { \\{ C ^ + _ K > 0 \\} } + \\sigma _ 1 ( C ^ - _ K ) ^ { \\alpha \\beta } 1 _ { \\{ C ^ - _ K > 0 \\} } \\ ; \\ ; \\ ; \\ ; \\gamma _ 1 = \\sigma _ 2 | C ^ + _ K | ^ { \\alpha \\beta } 1 _ { \\{ C ^ + _ K < 0 \\} } + \\sigma _ 1 | C ^ - _ K | ^ { \\alpha \\beta } 1 _ { \\{ C ^ - _ K < 0 \\} } . \\end{align*}"}
+{"id": "6636.png", "formula": "\\begin{align*} q _ Y ( \\varphi v , \\varphi w ) = q _ { X , a } ( a v , a w ) = q _ X ( v , w ) . \\end{align*}"}
+{"id": "4558.png", "formula": "\\begin{align*} g _ 1 ( \\rho ) = f ( \\rho ) , \\ g _ 2 ( \\rho ) = f ( \\rho ) \\int \\frac { 1 } { f ^ 2 ( \\rho ) } , \\end{align*}"}
+{"id": "5933.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { N C } ( G ) ) & = ( q - 1 ) ( q ^ { 3 } - q - 1 ) ( ( q - 1 ) ( q ^ { 3 } - q - 1 ) - 1 ) ^ { 2 } \\\\ & ~ ~ ~ ~ - 4 ( ( q - 1 ) ( q ^ { 3 } - q - 1 ) - 1 ) \\dfrac { q ( q - 1 ) } { 2 } ( q ^ { 4 } - 2 q ^ { 3 } - q ^ { 2 } + 2 q + 1 ) \\\\ & ~ ~ ~ ~ ~ + q ( q - 1 ) ( q ^ { 6 } - 4 q ^ { 5 } + 4 q ^ { 4 } + 2 q ^ { 3 } - 4 q ^ { 2 } + q - 1 ) \\\\ & = ( q - 1 ) ( q ^ { 1 1 } - 2 q ^ { 1 0 } - 4 q ^ { 9 } + 9 q ^ { 8 } + 5 q ^ { 7 } - 1 5 q ^ { 6 } + q ^ { 5 } + 7 q ^ { 4 } - 2 q ^ { 3 } + q ^ { 2 } - q ) \\end{align*}"}
+{"id": "8280.png", "formula": "\\begin{align*} S ( t ) S ( s ) = \\int _ s ^ { t + s } S ( r ) d r - \\int _ 0 ^ t S ( r ) d r . \\end{align*}"}
+{"id": "1199.png", "formula": "\\begin{align*} \\left | \\partial _ x ^ \\alpha \\partial _ y ^ \\beta \\mathcal K ( x , y ) - \\partial _ x ^ \\alpha \\partial _ y ^ \\beta \\mathcal K ( x , y + v ) \\right | & \\leq C | v | ^ \\rho | x - y | ^ { - n - | \\alpha | - | \\beta | - \\rho } \\\\ & \\quad \\quad \\begin{cases} | \\alpha | \\leq \\lfloor s \\rfloor , \\\\ | \\beta | = \\lfloor J - n \\rfloor - | \\alpha | \\end{cases} \\end{align*}"}
+{"id": "8414.png", "formula": "\\begin{align*} & \\Delta _ { H ^ { \\otimes 3 } } ( B _ 1 ( a _ 1 ) \\otimes a _ 2 \\otimes B _ 2 ( a _ 3 ) ) \\\\ = & ( B _ 1 ( a _ 1 ) \\otimes a _ 3 \\otimes B _ 2 ( a _ 5 ) ) \\otimes ( B _ 1 ( a _ 2 ) \\otimes a _ 4 \\otimes B _ 2 ( a _ 6 ) ) \\\\ = & ( B _ 1 ( a _ 1 ) \\otimes a _ 2 \\otimes B _ 2 ( a _ 3 ) ) \\otimes ( B _ 1 ( a _ 4 ) \\otimes a _ 5 \\otimes B _ 2 ( a _ 6 ) ) . \\end{align*}"}
+{"id": "264.png", "formula": "\\begin{align*} y = \\int ^ x A ( \\zeta ) \\exp \\left ( \\int ^ \\zeta \\gamma ( \\eta ) \\ , d \\eta \\right ) \\ , d \\zeta , d \\tau = | A ( x ) | \\ , d t . \\end{align*}"}
+{"id": "4382.png", "formula": "\\begin{align*} \\inf _ { x , p , \\theta } & \\ \\Gamma \\theta + \\sum _ { i \\in [ m ] } ( x _ i - \\overline { u } _ i ) ^ 2 + p _ i , \\\\ \\mathrm { s . t . } & \\ x \\in \\mathcal { X } , \\\\ & \\max _ { u _ i \\in \\mathcal { U } _ i } ( x _ i - u _ i ) ^ 2 - ( x _ i - \\overline { u } _ i ) ^ 2 \\leq p _ i + \\theta \\ \\forall i \\in [ m ] . \\end{align*}"}
+{"id": "5392.png", "formula": "\\begin{align*} \\nu _ j = \\frac { 1 } { \\mu } \\sum _ { i = 1 } ^ { j + 1 } \\Delta h _ i \\ , \\left ( 1 + \\cdots + \\rho ^ { i - 1 } \\right ) = \\begin{cases} \\displaystyle \\frac { 1 } { \\mu } \\sum _ { i = 1 } ^ { j + 1 } \\Delta h _ i \\ , \\frac { \\rho ^ i - 1 } { \\rho - 1 } & \\rho \\neq 1 \\\\ \\\\ \\displaystyle \\frac { 1 } { \\mu } \\sum _ { i = 1 } ^ { j + 1 } i \\ , \\Delta h _ i & \\rho = 1 . \\end{cases} \\end{align*}"}
+{"id": "5153.png", "formula": "\\begin{align*} A = \\frac { 1 } { \\alpha } \\sum _ { i } q _ { i } + \\frac { 1 } { \\alpha \\left ( \\alpha - 1 \\right ) } \\sum _ { i } p ^ { \\alpha } _ { i } q ^ { 1 - \\alpha } _ { i } \\ ; \\ ; ; \\ ; \\ ; B = \\frac { 1 } { \\alpha - 1 } \\sum _ { i } p _ { i } \\end{align*}"}
+{"id": "4907.png", "formula": "\\begin{align*} \\mathbf f ( \\mathbf x ) + t \\mathbf f _ v ( \\mathbf x ) = \\mathbf 0 , t ^ 2 = 0 \\end{align*}"}
+{"id": "5112.png", "formula": "\\begin{align*} \\frac { d f ( \\alpha ) } { d ( \\alpha : t ) } = \\frac { f ( \\alpha t ) - f ( \\alpha ) } { \\alpha t - \\alpha } \\end{align*}"}
+{"id": "1146.png", "formula": "\\begin{align*} \\mathrm { I I } & = \\sum _ { i = j + 1 } ^ { \\infty } 2 ^ { ( j - i ) \\widetilde { F } } \\sum _ { k \\in \\mathbb Z ^ n } \\left ( 1 + 2 ^ j | x _ Q - 2 ^ { - i } k | \\right ) ^ { - \\widetilde { D } } \\\\ & \\sim \\sum _ { i = j + 1 } ^ { \\infty } 2 ^ { ( j - i ) \\widetilde { F } } \\int _ { \\mathbb R ^ n } \\left ( 1 + 2 ^ j | x _ Q - 2 ^ { - i } y | \\right ) ^ { - \\widetilde { D } } \\ , d y \\sim \\sum _ { i = j + 1 } ^ { \\infty } 2 ^ { ( j - i ) ( \\widetilde { F } - n ) } \\sim 1 . \\end{align*}"}
+{"id": "3524.png", "formula": "\\begin{align*} d _ j = \\frac { \\gcd ( p _ { i _ 1 } p _ { i _ 2 } \\ldots p _ { i _ j } \\ \\ 1 \\le i _ 1 < i _ 2 < \\ldots < i _ j \\le m ) } { \\gcd ( p _ { i _ 1 } p _ { i _ 2 } \\ldots p _ { i _ { j - 1 } } \\ \\ 1 \\le i _ 1 < i _ 2 < \\ldots < i _ { j - 1 } \\le m ) } . \\end{align*}"}
+{"id": "204.png", "formula": "\\begin{align*} \\max ( \\bar u ( H x , H y ) , \\bar u ( H y , H z ) ) \\ = \\ & \\max ( u ( x _ 1 , y _ 1 ) , u ( y _ 1 , z _ 1 ) ) \\\\ \\ge \\ u ( x _ 1 , z _ 1 ) \\ \\ge \\ & \\bar u ( H x , H z ) . \\end{align*}"}
+{"id": "8970.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } { \\mathcal F } _ { \\varepsilon _ j } ( u _ j ) = \\mathcal { F } ( u ) . \\end{align*}"}
+{"id": "6865.png", "formula": "\\begin{align*} H _ { { \\varphi } \\sf E } f = \\sum _ { j \\in \\mathbb { J } _ 0 } E _ j \\langle \\varphi _ j , f \\rangle \\ , \\varphi _ j + \\sum _ { j \\in \\mathbb { J } _ 1 } E _ j \\langle \\varphi _ j , f \\rangle \\ , \\varphi _ j , \\end{align*}"}
+{"id": "8037.png", "formula": "\\begin{align*} \\left ( \\mathcal { A } _ s ^ N \\vec { f } \\right ) _ 3 ( u ) = \\frac { 1 } { N } \\sum _ { j = 1 } ^ N \\mathbb { E } S _ s ^ N ( j ) \\lambda \\left ( \\frac { j } { N } , u \\right ) \\left ( - f _ 1 \\left ( \\frac { j } { N } \\right ) + f _ 2 \\left ( \\frac { j } { N } \\right ) \\right ) - \\phi ( u ) f _ 3 ( u ) , \\end{align*}"}
+{"id": "3543.png", "formula": "\\begin{align*} H _ { 1 } ^ { \\infty } ( \\mathbb { D } ) = \\{ f \\in { H } ^ { \\infty } ( \\mathbb { D } ) : f ' ( 0 ) = 0 \\} , \\end{align*}"}
+{"id": "2631.png", "formula": "\\begin{align*} E _ i ( S ) = \\{ s _ x + s _ y : 1 \\leq x - y \\leq i \\} . \\end{align*}"}
+{"id": "3546.png", "formula": "\\begin{align*} \\overline { R a n ( \\tau ^ 2 ) } = \\overline { R a n ( \\tau ^ 3 ) } = \\overline { i d ^ 2 } = \\overline { i d ^ 3 } = \\overline { \\mathbb { D } } . \\end{align*}"}
+{"id": "4360.png", "formula": "\\begin{align*} \\inf _ { x \\in \\mathcal { X } } & \\ \\sum _ { i \\in [ m ] } \\max \\{ 0 , x _ i - \\overline { b } _ i , x _ i - \\overline { b } _ i + \\Delta b _ i - \\Delta b _ k \\} + \\Gamma \\Delta b _ k \\\\ \\mathrm { s . t . } & \\ x _ k \\in [ \\overline { b } _ k - \\Delta b _ k , \\overline { b } _ k ] \\end{align*}"}
+{"id": "7698.png", "formula": "\\begin{align*} h _ { i } = e _ { i } \\cdot F ( x ) . \\end{align*}"}
+{"id": "4291.png", "formula": "\\begin{align*} \\mathbf { x } ^ k _ { s , t } : = \\int _ { s \\le t _ 1 \\le \\cdots \\le t _ k \\le t } d x _ { t _ 1 } \\otimes \\cdots \\otimes d x _ { t _ k } , ( s , t ) \\in \\triangle \\end{align*}"}
+{"id": "5992.png", "formula": "\\begin{align*} ( | \\nabla u | ^ 2 ) _ { \\nu } ( x _ 0 ) = - 2 H ( x _ 0 ) | \\nabla u | ^ 2 ( x _ 0 ) . \\end{align*}"}
+{"id": "7203.png", "formula": "\\begin{align*} T _ 0 ( P ) = \\begin{bmatrix} f _ 1 ( P ) \\\\ f _ 2 ( P ) \\\\ \\vdots \\\\ f _ N ( P ) \\\\ \\end{bmatrix} . \\end{align*}"}
+{"id": "7441.png", "formula": "\\begin{align*} n ! ! = \\begin{cases} \\prod _ { k = 0 } ^ { \\frac { n + 1 } { 2 } } ( 2 k - 1 ) ; \\\\ \\prod _ { k = 1 } ^ { \\frac { n } { 2 } } ( 2 k ) ; \\\\ 1 n \\in \\{ 0 , - 1 \\} . \\end{cases} \\end{align*}"}
+{"id": "9000.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } n p ^ n ( t , x _ n , y _ n ) = p ^ { \\infty } ( t , x , y ) \\end{align*}"}
+{"id": "834.png", "formula": "\\begin{align*} { \\cal { L } } _ { \\hat { X } } C ^ { i } _ { j k } = \\dot { \\nabla } _ { k } ( \\nabla _ { j } X ^ { i } + C ^ { i } _ { j h } \\nabla _ { 0 } X ^ { h } ) + X ^ { h } P ^ { i } _ { \\ j h k } + \\nabla _ { 0 } X ^ { h } Q ^ { i } _ { \\ j h k } . \\end{align*}"}
+{"id": "3715.png", "formula": "\\begin{align*} ( \\sqrt { \\sigma ^ 2 + m ^ 2 } - E ) \\psi _ { \\sigma } - \\alpha \\sum _ { \\nu } \\frac { f _ { \\sigma } ^ { * } ( a ) } { \\sqrt { \\omega _ { \\nu } } } \\frac { f _ { \\nu } ( a ) } { \\sqrt { \\omega _ { \\nu } } } \\psi _ { \\nu } = \\chi _ { \\sigma } \\ ; . \\end{align*}"}
+{"id": "2783.png", "formula": "\\begin{align*} \\frac { \\partial L } { \\partial R ^ { a } ( t ) } = \\Psi _ { a } ( t ) = { c o n s t a n t } , \\end{align*}"}
+{"id": "4750.png", "formula": "\\begin{align*} \\begin{cases} [ S ( \\vert t \\vert ) ] ^ 1 x = \\overline { S ( \\vert t \\vert ) x } , \\\\ [ S ( \\vert t \\vert ) ] ^ { m + 1 } x = \\overline { S ( \\vert t \\vert ) \\left ( [ S ( \\vert t \\vert ) ] ^ m x \\right ) } . \\end{cases} \\end{align*}"}
+{"id": "8863.png", "formula": "\\begin{align*} 0 < \\frac 1 { a _ 2 } , \\frac { 1 } { b _ 2 } < 1 , \\frac 1 { a _ 2 } + \\frac 1 { b _ 2 } = \\frac { n + 2 } { 2 n } + \\frac { \\alpha } { n } , \\frac { 1 } { b _ 2 } = \\frac { 1 } { b _ 4 } + \\frac 1 r \\end{align*}"}
+{"id": "1089.png", "formula": "\\begin{align*} \\| A \\| : = \\sup _ { \\vec z \\in \\mathbb { C } ^ m , \\ , | \\vec z | = 1 } | A \\vec z | . \\end{align*}"}
+{"id": "8184.png", "formula": "\\begin{align*} R = \\frac { 1 } { 2 } ( H ( \\frac { B _ 1 ^ { \\star } } { M } ) + 1 ) , \\end{align*}"}
+{"id": "7952.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Omega } \\ast d w \\wedge [ \\eta , u ] _ { 1 } & = \\int _ { \\Omega } d w \\wedge \\ast \\delta ( \\eta \\wedge u ) = \\int _ { \\partial \\Omega } ( \\langle d w , \\eta \\rangle _ { \\Lambda ^ { 1 } } i _ { \\mathcal { N } } u ) v _ { \\partial \\Omega } . \\end{aligned} \\end{align*}"}
+{"id": "179.png", "formula": "\\begin{align*} ( x _ n ) _ A \\ = \\ \\begin{cases} 1 \\ \\ \\ \\ n \\in A , \\\\ 0 \\ \\ \\ \\ n \\not \\in A . \\end{cases} \\end{align*}"}
+{"id": "7782.png", "formula": "\\begin{align*} f _ E ( z ) = f _ E ' ( z ) = 0 . \\end{align*}"}
+{"id": "343.png", "formula": "\\begin{align*} I _ 0 ^ { [ k ] } : I ^ { [ \\ell ] } = ( I _ 0 ^ { [ k ] } : I _ 0 ^ { [ \\ell ] } ) \\cap \\big ( I _ 0 ^ { [ k ] } : \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\sum _ { \\substack { q \\in [ s ] \\\\ \\nu ( I _ q ) \\ge \\ell - 1 } } \\ ! \\ ! \\ ! \\ ! u _ q I _ q ^ { [ \\ell - 1 ] } \\big ) = I _ 0 ^ { [ k ] } , \\end{align*}"}
+{"id": "6901.png", "formula": "\\begin{align*} F ( 0 ) & = \\left ( \\log \\left ( \\cosh 1 + \\frac { k ( \\eta \\sinh 2 + 2 \\gamma \\sinh 1 ) } { 4 ( \\gamma + \\eta \\cosh 1 ) ^ 2 } \\right ) ^ 2 \\right ) ^ 2 - 4 \\intertext { a n d } F \\left ( \\frac { \\pi } { 2 } \\right ) & = \\left ( \\log \\left ( \\cos 1 - \\frac { k ( \\eta \\sin 2 + 2 \\gamma \\sin 1 ) } { 4 ( \\gamma + \\eta \\cos 1 ) ^ { 2 } } \\right ) ^ 2 \\right ) ^ 2 - 4 \\end{align*}"}
+{"id": "4276.png", "formula": "\\begin{align*} ( \\cdot , \\cdot ) _ f = \\int _ M < \\cdot , \\cdot > e ^ { - f } d V . \\end{align*}"}
+{"id": "3060.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} - a _ { 1 1 } & 0 & a _ { 1 3 } / 2 \\\\ 0 & 0 & a _ { 2 3 } / 2 \\\\ a _ { 1 3 } / 2 & a _ { 2 3 } / 2 & a _ { 3 3 } \\end{pmatrix} , \\begin{pmatrix} 0 & 0 & b _ { 1 3 } / 2 \\\\ 0 & b _ { 2 2 } & b _ { 2 3 } / 2 \\\\ b _ { 1 3 } / 2 & b _ { 2 3 } / 2 & b _ { 3 3 } \\end{pmatrix} \\right ) \\end{align*}"}
+{"id": "6039.png", "formula": "\\begin{align*} \\delta = \\sqrt { [ ( E _ { A } - E _ { C } ) ( 1 - \\rho ^ { A } ) - ( E _ { B } - E _ { C } ) ( 1 - \\rho ^ { B } ) ] ^ 2 + 4 ( E _ { A } - E _ { C } ) ( E _ { B } - E _ { C } ) \\rho ^ { A } \\rho ^ { B } } . \\end{align*}"}
+{"id": "6460.png", "formula": "\\begin{align*} \\bar \\nabla _ { e _ j } \\big ( R ^ N ( V , d \\phi ( e _ j ) ) \\tau ( \\phi ) \\big ) = & K \\big ( | \\tau ( \\phi ) | ^ 2 V + \\langle d \\phi , \\bar \\nabla \\tau ( \\phi ) \\rangle V + \\langle d \\phi ( e _ j ) , \\tau ( \\phi ) \\rangle \\bar \\nabla _ { e _ j } V \\\\ & - \\langle \\bar \\nabla _ { e _ j } V , \\tau ( \\phi ) \\rangle d \\phi ( e _ j ) - \\langle V , \\bar \\nabla _ { e _ j } \\tau ( \\phi ) \\rangle d \\phi ( e _ j ) - \\langle V , \\tau ( \\phi ) \\rangle \\tau ( \\phi ) \\big ) . \\end{align*}"}
+{"id": "2191.png", "formula": "\\begin{align*} u = r v _ \\varphi \\end{align*}"}
+{"id": "8053.png", "formula": "\\begin{align*} \\lim _ { M \\rightarrow + \\infty } \\limsup _ { N \\rightarrow + \\infty } \\widetilde { P } _ { f _ 1 , f _ 2 , f _ 3 } ^ { N , \\tilde { F } , \\tilde { G } , \\tilde { H } } \\left ( | \\eta _ t ^ N ( \\vec { f } ) | \\geq M \\right ) = 0 \\end{align*}"}
+{"id": "3265.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ^ { d + 1 } ; q ^ d ) _ k ^ { d - 1 } ( q ^ { 1 - d } ; q ^ d ) _ k q ^ { d k } } { ( q ^ d ; q ^ d ) _ k ^ d } \\\\ & \\equiv \\frac { ( 1 - q ) ( 1 - q ^ { d - 1 } ) ( q ^ d ; q ^ d ) _ { n - 1 - ( n + 1 ) / d } } { - ( - 1 ) ^ { ( n + 1 ) / d } ( q ^ d ; q ^ d ) _ { ( n + 1 ) / d } ^ { d - 1 } } q ^ { ( d ( d + n ) ( n + 1 ) - ( n + 1 ) ^ 2 ) / ( 2 d ) - 1 } \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "6008.png", "formula": "\\begin{align*} S _ { \\alpha , \\beta } ( t , u ) = \\langle \\phi _ \\alpha ( t , u ) \\phi _ \\beta ( 0 , 0 ) \\rangle _ { \\mu _ \\rho } \\ , . \\end{align*}"}
+{"id": "6556.png", "formula": "\\begin{align*} \\lim \\limits _ { x \\to \\infty } \\frac { 1 } { x ^ { \\frac { 1 } { \\beta } } \\ell ^ { \\frac { 1 } { \\beta } } ( x ^ { \\frac { 1 } { \\beta } } ) } \\eta _ K ( x ) = \\int ^ { \\infty } _ 0 \\big ( K _ { \\infty } ( t ^ { - \\beta } ) - K _ { \\infty } ( 0 ) \\big ) d t . \\end{align*}"}
+{"id": "8459.png", "formula": "\\begin{align*} & \\left ( | u _ { m } | ^ { q - 1 } u _ { m } - | u _ { m - 1 } | ^ { q - 1 } u _ { m - 1 } \\right ) ( u _ { m } - u _ { m - 1 } ) \\\\ [ 2 m m ] & \\geq C ( q ) \\big ( | u _ m | + | u _ { m - 1 } | \\big ) ^ { q - 1 } | u _ m - u _ { m - 1 } | ^ 2 . \\end{align*}"}
+{"id": "7837.png", "formula": "\\begin{align*} H ^ \\psi = \\frac { p } { p + q } H _ 1 + \\frac { q } { p + q } H _ 2 . \\end{align*}"}
+{"id": "305.png", "formula": "\\begin{align*} \\left \\langle L _ { F ' } e _ { k } ' , e _ { j } ' \\right \\rangle = \\begin{cases} \\mu ^ { | \\alpha ( j ) | - | \\alpha ( k ) | } \\sum _ { l = 1 } ^ n \\alpha _ l ( k ) \\ , a _ { l , ( \\alpha ( j ) - \\alpha ( k ) ) _ l } & | \\alpha ( j ) | > | \\alpha ( k ) | \\\\ \\sum _ { l = 1 } ^ n \\alpha _ l ( k ) \\ , a _ { l , l } & j = k \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "5252.png", "formula": "\\begin{align*} L _ { d } F I \\left ( p \\| q \\right ) = \\sum _ { j } p _ { j } \\sum _ { i } \\frac { \\overline { p } _ { i } } { a - b } \\left [ \\left ( \\overline { Z } _ { i } \\right ) ^ { a - 1 } - \\left ( \\overline { Z } _ { i } \\right ) ^ { b - 1 } \\right ] \\end{align*}"}
+{"id": "1618.png", "formula": "\\begin{align*} q ^ { \\tfrac { p - 1 } { 2 } } ( - 1 ) ^ { \\tfrac { s ( p - 1 ) } { 2 } } \\sum _ i ( - 1 ) ^ { \\tfrac { r - i } { 2 } } \\binom { ( i - s ) \\tfrac { ( p - 1 ) } { 2 } - 1 } { \\tfrac { p i - r - s ( p - 1 ) } { 2 } } Q _ { ( r + p s - p i ) ( p - 1 ) } Q _ { i ( p - 1 ) } ( x ) . \\end{align*}"}
+{"id": "7462.png", "formula": "\\begin{align*} \\Lambda = { \\rm D i a g } \\ , \\left ( \\frac { 1 } { 2 } , \\ , \\frac { \\lambda _ 1 } { \\lambda _ 1 + \\lambda _ 2 } , \\ , \\frac { 1 } { 2 } , \\ , \\frac { \\lambda _ 2 } { \\lambda _ 1 + \\lambda _ 2 } \\right ) . \\end{align*}"}
+{"id": "688.png", "formula": "\\begin{align*} m _ 1 & = F ( 1 , 1 , 1 ) F ( - 1 , 1 , 1 ) \\equiv ( \\alpha _ 1 + \\alpha _ 2 + \\gamma _ 1 + \\gamma _ 2 ) ^ 2 , \\\\ m _ 2 & = F ( 1 , - 1 , 1 ) F ( - 1 , - 1 , 1 ) \\equiv ( \\alpha _ 1 + \\alpha _ 2 - \\gamma _ 1 - \\gamma _ 2 ) ^ 2 , \\\\ m _ 3 & = F ( 1 , 1 , - 1 ) F ( - 1 , 1 , - 1 ) \\equiv ( \\alpha _ 1 - \\alpha _ 2 + \\gamma _ 1 - \\gamma _ 2 ) ^ 2 , \\\\ m _ 4 & = F ( 1 , - 1 , - 1 ) F ( - 1 , - 1 , - 1 ) \\equiv ( \\alpha _ 1 - \\alpha _ 2 - \\gamma _ 1 + \\gamma _ 2 ) ^ 2 . \\end{align*}"}
+{"id": "926.png", "formula": "\\begin{align*} \\hat W ^ x _ D ( u ) : = \\lim _ { V \\nearrow D , V \\subset \\subset D } P _ V ( u R ^ D \\kappa _ D ) ( x ) = 0 , \\end{align*}"}
+{"id": "8502.png", "formula": "\\begin{align*} | f ( a _ 1 ) | + | f ( a _ 2 ) | + \\dotsb + | f ( a _ n ) | = | f ( u ) | = | u | = n . \\end{align*}"}
+{"id": "7835.png", "formula": "\\begin{align*} { } H ^ { \\phi } = \\frac { p } { p + 2 q } H . \\end{align*}"}
+{"id": "4348.png", "formula": "\\begin{gather*} G ^ k ( x ) : = \\sum _ { j \\in [ m ] } ( \\overline { u } _ j + \\max \\{ 0 , \\Delta u _ j - \\Delta u _ k \\} ) l _ j ( x ) . \\end{gather*}"}
+{"id": "4545.png", "formula": "\\begin{align*} \\mathcal { T } _ { V _ { N , \\beta , a } } \\big ( n _ N ( t ) , m _ N ( t ) \\big ) \\leq \\int _ { \\mathcal { C } _ N \\times \\mathcal { C } _ N } V _ { N , \\beta , a } ( [ s ] _ N , [ s ' ] _ N ) \\omega _ N ( t , d [ s ] _ N , d [ s ' ] _ N ) = : T _ { V _ { N , \\beta , a } } ( \\omega _ N ) , \\end{align*}"}
+{"id": "8447.png", "formula": "\\begin{align*} \\| v \\| _ { W ^ { s , p } ( K ) } : = \\| v \\| _ { L ^ p ( K ) } + [ v ] _ { W ^ { s , p } ( K ) } \\end{align*}"}
+{"id": "1933.png", "formula": "\\begin{align*} - \\Delta u + ( - \\Delta ) ^ s u = 0 , \\end{align*}"}
+{"id": "3810.png", "formula": "\\begin{align*} \\begin{aligned} H _ { [ n - k ] , A _ j } y _ j & = H _ { [ n - k ] , A _ j } ( \\phi ( x ) - G x _ j ) | _ { A _ j } \\\\ & = H ( \\phi ( x ) - G x _ j ) - H _ { [ n - k ] , I _ j } ( \\phi ( x ) - G x _ j ) | _ { I _ j } \\\\ & = H \\phi ( x ) \\end{aligned} \\end{align*}"}
+{"id": "394.png", "formula": "\\begin{align*} f _ { d _ { 1 } } = h _ { d _ { 1 } } \\ , \\& \\ , f ^ { d _ { 1 } } < ^ { d _ { 1 } } h ^ { \\prime } ( d _ { 1 } ) \\end{align*}"}
+{"id": "4206.png", "formula": "\\begin{align*} & 2 1 6 0 \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 1 0 ) } = \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right . \\\\ & \\left . \\cdot { \\rm c h } ( S ^ 2 \\widetilde { T X } + \\widetilde { T X } + \\wedge ^ 2 \\widetilde { L _ R \\otimes C } - \\widetilde { L _ R \\otimes C } - \\widetilde { T X } \\otimes \\widetilde { L _ R \\otimes C } \\right \\} ^ { ( 1 0 ) } . \\end{align*}"}
+{"id": "1123.png", "formula": "\\begin{align*} 2 ^ { j ( s - \\sigma ) } W ^ { \\frac { 1 } { p } } \\left [ \\varphi _ j * \\left ( \\dot I _ \\sigma \\vec f \\right ) \\right ] & = 2 ^ { j ( s - \\sigma ) } W ^ { \\frac { 1 } { p } } \\left [ \\varphi _ j * \\left ( | \\cdot | ^ \\sigma \\widehat { \\vec f } \\right ) ^ \\vee \\right ] \\\\ & = 2 ^ { j ( s - \\sigma ) } W ^ { \\frac { 1 } { p } } \\left ( \\widehat { \\varphi _ j } | \\cdot | ^ \\sigma \\widehat { \\vec f } \\right ) ^ \\vee . \\end{align*}"}
+{"id": "3683.png", "formula": "\\begin{align*} e _ i - e _ j = \\frac { 1 } { 2 } u - \\frac { 1 } { 2 } v . \\end{align*}"}
+{"id": "6700.png", "formula": "\\begin{align*} \\Omega _ { \\{ 1 \\} } ( G / U ^ { p ^ k } ) = \\Omega _ { \\{ 1 \\} } ( N / U ^ { p ^ k } ) . \\end{align*}"}
+{"id": "8816.png", "formula": "\\begin{align*} \\begin{cases} s > 0 , & \\\\ s = 0 & 0 < q \\le 2 . \\end{cases} \\end{align*}"}
+{"id": "467.png", "formula": "\\begin{align*} \\lambda ( g ) \\lambda ( d ( g ) ) & = \\left ( \\sum _ { A \\ni g } ( A , g ) \\right ) \\left ( \\sum _ { B \\ni d ( g ) } ( B , d ( g ) ) \\right ) = \\sum _ { A \\ni g } ( A , g ) ( A , d ( g ) ) = \\sum _ { A \\ni g } ( A , g ) = \\lambda ( g ) . \\end{align*}"}
+{"id": "7007.png", "formula": "\\begin{align*} a ( T , x ) = x \\bar { a } ( T , x + 1 ) \\end{align*}"}
+{"id": "6723.png", "formula": "\\begin{align*} \\begin{bmatrix} a _ { 0 } & a _ { 1 } & \\cdots & a _ { n - 1 } \\\\ a _ { n - 1 } & a _ { 0 } & \\cdots & a _ { n - 2 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a _ { 1 } & a _ { 2 } & \\cdots & a _ { 0 } \\end{bmatrix} , \\end{align*}"}
+{"id": "4140.png", "formula": "\\begin{align*} u _ i + w _ { i j } = \\begin{cases} 0 & i < j , \\\\ - 1 + w _ { j j } , & i = j , \\\\ w _ { i j } & i > j . \\end{cases} \\end{align*}"}
+{"id": "5296.png", "formula": "\\begin{align*} L _ { d } K L I ( q \\| p ) = \\sum _ { j } p _ { j } \\sum _ { i } \\frac { \\overline { q } _ { i } } { a - b } \\left [ \\left ( \\frac { \\overline { q } _ { i } } { \\overline { p } _ { i } } \\right ) ^ { a - 1 } - \\left ( \\frac { \\overline { q } _ { i } } { \\overline { p } _ { i } } \\right ) ^ { b - 1 } \\right ] \\end{align*}"}
+{"id": "3974.png", "formula": "\\begin{align*} \\mu \\otimes \\eta ( [ \\alpha _ \\pi ] ( T ) A _ n \\bigtriangleup A _ n ) & = \\int _ X \\eta ( \\alpha _ \\pi ( T ^ { - 1 } x , x ) A _ n ^ { T ^ { - 1 } x } \\bigtriangleup A _ n ^ x ) \\ d \\mu ( x ) \\\\ & = \\frac { 1 } { \\pi } \\int _ X \\theta _ { n , T ^ { - 1 } x , x } \\ d \\mu ( x ) \\end{align*}"}
+{"id": "1205.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta _ p u = \\lambda \\alpha ( p ) \\Vert u \\Vert _ r ^ { \\alpha ( p ) - r } \\vert u \\vert ^ r \\Vert v \\Vert _ r ^ { \\beta ( p ) } & { \\rm i n } \\ \\ \\Omega , \\\\ - \\Delta _ p v = \\lambda \\beta ( p ) \\Vert u \\Vert _ r ^ { \\alpha ( p ) } \\Vert v \\Vert _ r ^ { \\beta ( p ) - r } \\vert v \\vert ^ r & { \\rm i n } \\ \\ \\Omega , \\\\ u = v = 0 & { \\rm o n } \\ \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
+{"id": "4170.png", "formula": "\\begin{align*} \\varphi ( g ^ { n } _ 1 g _ 3 ) = \\varphi \\big ( g ^ { j } _ 1 \\big ) \\varphi \\big ( g ^ { i } _ 1 g _ 3 \\big ) \\overset { ( \\ref { e q : 5 } ) } { = } \\varphi \\big ( g ^ { j } _ 1 \\big ) \\varphi ( g _ 3 ) \\varphi \\big ( g ^ { i } _ 1 \\big ) \\overset { ( \\ref { e q : 9 } ) } { = } \\varphi \\big ( g _ 3 g ^ { j } _ 1 \\big ) \\varphi \\big ( g ^ { i } _ 1 \\big ) \\ , . \\end{align*}"}
+{"id": "7964.png", "formula": "\\begin{align*} \\bar { H } ( \\omega , \\phi _ { \\partial } , \\Sigma ) = \\frac { 1 } { 2 } \\int _ { \\Omega } \\beta \\wedge \\ast \\omega + \\int _ { \\Omega } \\Phi v _ { \\Omega } + \\frac { 1 } { 2 } \\int _ { \\partial \\Omega } \\mathrm { t r } ( \\phi ) \\wedge \\ast \\boldsymbol { n } ( d \\phi ) + \\frac { \\tau } { \\rho } \\int _ { \\Sigma } v _ { \\Sigma } . \\end{align*}"}
+{"id": "573.png", "formula": "\\begin{align*} \\nabla \\cdot ( T \\nabla \\eta ) & = ( g \\rho - \\omega ^ 2 m ) \\eta + j \\omega \\rho \\phi & \\Gamma _ c , \\end{align*}"}
+{"id": "6388.png", "formula": "\\begin{align*} d X _ t = ( a - b X _ t ) d t + \\delta X _ { t - } ^ { 1 / \\alpha } d L ^ \\alpha _ t , X _ 0 = x _ 0 t \\in [ 0 , 1 ] \\end{align*}"}
+{"id": "8732.png", "formula": "\\begin{align*} \\int _ \\R f ( x ) \\ , \\eta ( d x ) = \\int _ 0 ^ 1 f ( F _ \\eta ^ { - 1 } ( v ) ) \\ , d v \\end{align*}"}
+{"id": "3415.png", "formula": "\\begin{align*} u I _ { \\varphi } \\circ u I _ { \\varphi } ( \\phi ( x ) ) & = u ( x ) I _ { \\varphi } ( u ( x ) I _ { \\varphi } ( \\phi ( x ) ) ) \\\\ & = u ( x ) \\iota ^ { \\ast } ( \\overline { u ( x ) } \\varphi \\circ c ( \\phi ( x ) ) ) \\\\ & = u ( x ) \\overline { u ( \\iota ^ { - 1 } ( x ) ) } I _ { \\varphi } ^ 2 ( \\phi ( x ) ) \\\\ & = u ( x ) \\overline { u ( \\iota ^ { - 1 } ( x ) ) } u _ { \\varphi } ( x ) . \\end{align*}"}
+{"id": "8247.png", "formula": "\\begin{align*} I ( { A } ^ { L } ; Y ^ { L } | { S } ) = I ( { A } ^ { L } \\to Y ^ { L } | { S } ) , \\end{align*}"}
+{"id": "7776.png", "formula": "\\begin{align*} B ^ { - 1 } ( \\cdot + \\alpha , \\lambda ) ( A ( \\lambda ) + F ( \\cdot , \\lambda ) ) B ( \\cdot , \\lambda ) = A _ { N } ( \\lambda ) + F _ N ( \\cdot , \\lambda ) = : \\tilde A ( \\lambda ) + \\tilde F ( \\cdot , \\lambda ) , \\end{align*}"}
+{"id": "9034.png", "formula": "\\begin{align*} A _ \\alpha = v _ \\alpha ( A _ { \\epsilon _ 1 } , A _ { \\epsilon _ 2 } , \\ldots , A _ { \\epsilon _ \\ell } ) , \\end{align*}"}
+{"id": "3155.png", "formula": "\\begin{align*} \\Lambda ^ \\sharp & : = \\bigl \\{ \\ , { \\rm ( I ) ^ \\sharp } , \\ ; { \\rm ( I V ) ^ \\sharp } , \\ ; { \\rm ( V ) ^ \\sharp } , \\ ; { \\rm ( V I I ) ^ \\sharp } , \\ ; { \\rm ( X I I ) ^ \\sharp } \\ , \\bigr \\} . \\end{align*}"}
+{"id": "6695.png", "formula": "\\begin{align*} \\mathcal { S } = \\mathcal { S } _ { \\pi , r } & = \\{ S \\mid S \\} \\\\ & \\subseteq \\{ S \\mid S \\} \\end{align*}"}
+{"id": "7117.png", "formula": "\\begin{align*} \\gamma _ { c } ^ { \\rho } : = \\inf _ { u \\in S ( c ) \\cap \\mathcal { B } _ { \\rho } } J ( u ) , \\end{align*}"}
+{"id": "8540.png", "formula": "\\begin{align*} \\begin{array} { l c l } 0 & = & s ^ { - 1 } \\log _ 1 ( \\pi _ 2 ) ( v _ 1 - v _ 2 ) , \\\\ - ( \\lambda _ 1 + k ^ { - 1 } \\mu _ 1 a _ 1 ) & = & v _ 1 , \\\\ 0 & = & s ^ { - 1 } \\log _ 1 ( \\pi _ 2 ) ( v _ 2 - v _ 1 ) , \\\\ - ( \\lambda _ 2 + k ^ { - 1 } \\mu _ 2 a _ 2 ) & = & v _ 2 . \\end{array} \\end{align*}"}
+{"id": "1907.png", "formula": "\\begin{align*} \\partial _ t v ( x , t ) = \\partial _ t v _ + ( x , t ) . \\end{align*}"}
+{"id": "2952.png", "formula": "\\begin{align*} \\big ( \\Delta _ \\C + \\partial _ { \\rho } ^ 2 + \\frac { ( 1 - s ) } { \\rho } \\partial _ \\rho \\big ) U ( z , \\rho ) = 0 , \\ , \\ , U ( z , 0 ) = F ( z ) \\end{align*}"}
+{"id": "4802.png", "formula": "\\begin{align*} \\mathbf { c } \\in \\mathcal { S } _ 0 : = \\{ \\mathbf { c } ' \\in \\mathbb { R } ^ { n } : \\mathbf { B } ^ { \\mbox { \\tiny \\upshape ( 0 ) } } \\mathbf { c } ' \\leq \\mathbf { b } ^ { \\mbox { \\tiny \\upshape ( 0 ) } } \\} \\end{align*}"}
+{"id": "2542.png", "formula": "\\begin{align*} x ^ { - m ' } | h - u | = x ^ { m - m ' } | x ^ { - m } \\lambda ( x ) f - x ^ { - m } u | \\le S | f - x ^ { - m } u | < \\epsilon \\ ; . \\end{align*}"}
+{"id": "7444.png", "formula": "\\begin{align*} \\big [ ( A B ) \\otimes ( C D ) \\big ] ^ { i j } _ { k \\ell } = \\sum _ { \\alpha , \\beta = 1 } ^ d A ^ i _ \\alpha B ^ \\alpha _ k C ^ j _ \\beta D ^ \\beta _ \\ell = \\big [ ( A \\otimes C ) ( B \\otimes D ) \\big ] ^ { i j } _ { k \\ell } . \\end{align*}"}
+{"id": "1440.png", "formula": "\\begin{align*} t _ 1 = x _ 1 ^ { a _ 1 } \\cdots x _ l ^ { a _ l } , t _ 2 = x _ { l + 1 } , \\dots , t _ k = x _ { l + k - 1 } \\end{align*}"}
+{"id": "2586.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u = f ( u ) & B \\\\ \\displaystyle { \\frac { \\partial u } { \\partial \\nu } + \\beta u = 0 } & \\partial B \\end{cases} \\end{align*}"}
+{"id": "3817.png", "formula": "\\begin{align*} \\lvert \\omega \\rvert & = ( t + 1 ) \\sum _ { i = 0 } ^ { 2 t + 1 } n _ i ^ 2 + \\sum _ { i = 0 } ^ { 2 t + 1 } i n _ i = 2 \\left ( ( t + 1 ) \\sum _ { i = 0 } ^ { t - 1 } n _ i ^ 2 + \\sum _ { i = 0 } ^ { t - 1 } ( i - t - 1 ) n _ i ) \\right ) \\end{align*}"}
+{"id": "1281.png", "formula": "\\begin{align*} \\lambda _ { 1 } = \\frac { \\gamma + \\sqrt { \\gamma ^ 2 - 4 \\omega ^ 2 } } { 2 } , \\lambda _ { 2 } = \\frac { \\gamma - \\sqrt { \\gamma ^ 2 - 4 \\omega ^ 2 } } { 2 } , \\quad \\lambda _ 1 + \\lambda _ 2 = \\gamma , \\lambda _ 1 \\lambda _ 2 = \\omega ^ 2 , \\lambda _ 1 - \\lambda _ 2 = \\sqrt { \\gamma ^ 2 - 4 \\omega ^ 2 } . \\end{align*}"}
+{"id": "697.png", "formula": "\\begin{align*} T _ g ( Y ) = Y ^ d + \\sum _ { j = 0 } ^ { d - 1 } \\overline { \\Big ( \\frac { a _ { e j } } { p ^ { v _ p ( a _ 0 ) + e j \\lambda } } \\Big ) } Y ^ j \\end{align*}"}
+{"id": "1165.png", "formula": "\\begin{align*} \\vec f : = \\vec t _ { I _ 0 } \\operatorname { E x t } \\left ( \\theta ^ { ( \\lambda ' ) } _ { I _ 0 } \\right ) = \\frac { 1 } { \\varphi ( - k _ 0 ) } \\vec u _ { Q ( I _ 0 , k _ 0 ) } \\left [ \\theta ^ { ( \\lambda ' ) } \\otimes \\varphi \\right ] _ { Q ( I _ 0 , k _ 0 ) } , \\end{align*}"}
+{"id": "1168.png", "formula": "\\begin{align*} \\left | A _ { I _ 0 , V } \\vec z \\right | \\sim \\left \\| \\vec g \\right \\| _ { \\dot A ^ { s - \\frac { 1 } { p } , \\frac { n } { n - 1 } \\tau } _ { p , q } ( V , \\mathbb { R } ^ { n - 1 } ) } = \\left \\| \\operatorname { T r } \\vec f \\right \\| _ { \\dot A ^ { s - \\frac { 1 } { p } , \\frac { n } { n - 1 } \\tau } _ { p , q } ( V , \\mathbb { R } ^ { n - 1 } ) } \\lesssim \\left \\| \\vec f \\right \\| _ { \\dot A ^ { s , \\tau } _ { p , q } ( W , \\mathbb { R } ^ n ) } \\lesssim \\left | A _ { Q ( I _ 0 , k _ 0 ) , W } \\vec z \\right | . \\end{align*}"}
+{"id": "383.png", "formula": "\\begin{align*} f _ 5 ( x , b , a ) & = 5 a b ( 4 x ^ 3 - 6 a x ^ 2 + 4 a ^ 2 x + 6 b x ^ 2 + 4 b ^ 2 x - 6 a b x ) , \\\\ g _ 5 ( b , a ) & = 5 a b ( b ^ 3 - 2 a b ^ 2 + 2 a ^ 2 b - a ^ 3 ) , \\end{align*}"}
+{"id": "1436.png", "formula": "\\begin{gather*} L _ m ( \\varepsilon _ * \\Omega _ { ( X , D ) _ { \\bullet } / Y } ) ^ n = \\bigoplus _ { k \\ge - m } ( \\varepsilon _ k ) _ * \\Omega ^ { n - k } _ { ( X , D ) _ k / Y } \\\\ F ^ p ( \\varepsilon _ * \\Omega _ { ( X , D ) _ { \\bullet } / Y } ) ^ n = \\bigoplus _ { 0 \\le k \\le n - p } ( \\varepsilon _ k ) _ * \\Omega ^ { n - k } _ { ( X , D ) _ k / Y } , \\end{gather*}"}
+{"id": "5169.png", "formula": "\\begin{align*} \\sum _ { j } q _ { j } \\frac { \\partial L D _ { \\alpha } I ( p \\| q ) } { \\partial q _ { j } } = 0 \\end{align*}"}
+{"id": "1837.png", "formula": "\\begin{align*} \\gamma ^ s ( 0 ) = x , \\gamma ^ 0 = \\gamma , \\quad , \\gamma ^ 1 = \\gamma ' , \\gamma ^ s ( 1 ) = y . \\end{align*}"}
+{"id": "4295.png", "formula": "\\begin{align*} d y _ t = \\sigma ( y _ t ) d x _ t , y _ 0 = a . \\end{align*}"}
+{"id": "6404.png", "formula": "\\begin{align*} m _ p ( \\alpha ) = \\frac { 2 ^ { p / \\alpha } 2 ^ p \\Gamma ( \\frac { p + 1 } { 2 } ) \\Gamma ( 1 - \\frac { p } { \\alpha } ) } { \\sqrt { \\pi } \\Gamma ( 1 - \\frac { p } { 2 } ) } \\ ; \\Gamma ( a ) = \\int _ 0 ^ { \\infty } x ^ { a - 1 } e ^ { - x } d x . \\end{align*}"}
+{"id": "8837.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } S _ { \\alpha _ 0 ^ * , k } ( 0 ) [ \\varphi ] & = & \\varphi _ k , \\\\ S _ { \\alpha _ 0 ^ * , k } ( t ) [ \\varphi ] ( x ) & = & \\frac { \\exp ( - \\alpha _ 0 ^ * t ) } { \\sqrt { 4 \\pi d _ k t } } \\int _ { \\mathbb { R } } \\varphi _ k ( y ) \\exp \\left ( - \\frac { ( x - y ) ^ 2 } { 4 d _ k t } \\right ) { \\rm d } y , t > 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "5867.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { C } ( G ) ) = ( 2 n - 2 ) ( 2 n - 3 ) ^ { 2 } + 2 n , M _ { 2 } ( \\mathcal { C } ( G ) ) = ( n - 1 ) ( 2 n - 3 ) ^ { 3 } + n , \\end{align*}"}
+{"id": "614.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\left ( \\frac { 1 } { 3 2 } \\right ) ^ { k } \\binom { 2 k } { k } ^ 2 H _ { k } = \\frac { \\sqrt { \\pi } ( \\pi - 4 \\ln ( 2 ) ) } { 2 \\Gamma ^ { 2 } \\left ( \\frac { 3 } { 4 } \\right ) } . \\end{align*}"}
+{"id": "8007.png", "formula": "\\begin{align*} J _ { i n i } ( \\varpi ) = \\sup _ { f _ 1 , f _ 2 , f _ 3 \\in C ^ \\infty ( \\mathbb { T } ) } \\left \\{ \\mathcal { J } _ 2 ( \\varpi , f _ 1 , f _ 2 , f _ 3 ) \\right \\} . \\end{align*}"}
+{"id": "6392.png", "formula": "\\begin{align*} h _ \\alpha ( x ) = \\frac { \\varphi ' _ \\alpha } { \\varphi _ \\alpha } ( x ) , k _ \\alpha ( x ) = 1 + x h _ \\alpha ( z ) , f _ \\alpha ( x ) = \\frac { \\partial _ \\alpha \\varphi _ \\alpha } { \\varphi _ \\alpha } ( x ) . \\end{align*}"}
+{"id": "8565.png", "formula": "\\begin{align*} F ( n , k ) = \\frac { ( a ) _ k ( b ) _ k } { ( n ) _ k ( a + 2 n ) _ k } \\end{align*}"}
+{"id": "1374.png", "formula": "\\begin{align*} \\begin{aligned} f = \\ & ( f ^ { A } _ { \\emptyset } , f ^ { A } _ { \\{ 1 \\} } , \\dots , f ^ { A } _ { \\{ 1 , \\dots , n \\} } ; f ^ { V } _ { \\emptyset } , f ^ { V } _ { \\{ 1 \\} } , \\dots , f ^ { V } _ { \\{ 1 , \\dots , n \\} } ) , \\end{aligned} \\end{align*}"}
+{"id": "5047.png", "formula": "\\begin{align*} I ( x , y ) = \\frac { 1 } { d ( x , y ) ^ { \\theta p } \\ , \\nu ( B ( x , d ( x , y ) ) ) } . \\end{align*}"}
+{"id": "6741.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty { \\frac { { \\zeta ( 2 k ) } } { { k ( k + 2 ) ( 2 k + 3 ) } } } & = - \\frac { 7 } { { 7 2 } } + \\frac { \\ln ( 2 \\pi ) } { 6 } - \\frac { { \\zeta ( 3 ) } } { { 2 \\pi ^ 2 } } , \\\\ \\sum _ { k = 1 } ^ \\infty { \\frac { { \\zeta ( 2 k ) } } { { k ( k + 1 ) ( 2 k + 5 ) } } } & = - \\frac { 7 } { { 5 0 } } + \\frac { \\ln ( 2 \\pi ) } { 5 } - \\frac { { 2 \\zeta ( 3 ) } } { { 3 \\pi ^ 2 } } + \\frac { { \\zeta ( 5 ) } } { { \\pi ^ 4 } } . \\end{align*}"}
+{"id": "5174.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } D _ { \\beta } ( p \\| q ) } { \\partial q _ { j } } = \\left [ \\frac { a - 1 } { a - b } A ^ { a - 2 } - \\frac { b - 1 } { a - b } A ^ { b - 2 } \\right ] \\left [ p _ { j } q ^ { \\beta - 2 } _ { j } - q ^ { \\beta - 1 } _ { j } \\right ] \\end{align*}"}
+{"id": "2936.png", "formula": "\\begin{align*} \\sum _ { T ' \\in { \\bar S } \\setminus \\{ \\{ i , \\bar { i } , 2 n + 1 \\} , i = 1 , \\ldots , n \\} , i \\in { \\bar S } } v _ { T ' } = \\sum _ { T ' \\in { \\bar S } \\setminus \\{ \\{ i , \\bar { i } , 2 n + 1 \\} , i = 1 , \\ldots , n \\} , \\bar { i } \\in { \\bar S } } v _ { T ' } = \\alpha _ i \\in \\{ \\pm 2 , \\pm 4 \\} . \\end{align*}"}
+{"id": "7695.png", "formula": "\\begin{align*} I _ { q } ( \\Omega ) = \\int _ { \\pounds ^ { n } } | \\Omega \\cap \\ell | ^ { q } d \\ell , { \\rm r e a l } \\ q \\geq 0 , \\end{align*}"}
+{"id": "2175.png", "formula": "\\begin{align*} ( f _ d ^ { - 1 } ) ^ { ( 1 ) } ( x ) & = \\frac { ( e ^ d - 1 ) e ^ x } { ( e ^ x - 1 ) ( e ^ d - e ^ x ) } , \\\\ ( f _ d ^ { - 1 } ) ^ { ( 2 ) } ( x ) & = \\frac { ( e ^ d - 1 ) e ^ x ( e ^ { 2 x } - e ^ d ) } { ( e ^ x - 1 ) ^ 2 ( e ^ d - e ^ x ) ^ 2 } , \\\\ ( f _ d ^ { - 1 } ) ^ { ( 3 ) } ( x ) & = \\frac { ( e ^ d - 1 ) e ^ x [ e ^ { 4 x } + ( e ^ d + 1 ) e ^ { 3 x } - 6 e ^ d e ^ { 2 x } + ( e ^ { 2 d } + e ^ d ) e ^ x + e ^ { 2 d } ] } { ( e ^ x - 1 ) ^ 3 ( e ^ d - e ^ x ) ^ 3 } . \\end{align*}"}
+{"id": "3428.png", "formula": "\\begin{align*} { d } ( \\Sigma ^ V X ) = { d } ( X ) + \\dim V . \\end{align*}"}
+{"id": "3816.png", "formula": "\\begin{align*} n _ k : = \\left \\lfloor \\frac { \\lambda _ i - i } { t } \\right \\rfloor + 1 , i = \\min \\lbrace \\nu \\mid \\lambda _ \\nu \\equiv k \\pmod t \\rbrace . \\end{align*}"}
+{"id": "4331.png", "formula": "\\begin{align*} & \\min _ { x \\in \\mathcal { X } } \\left \\{ \\sum _ { ( i , j , r , s ) \\in [ n ] ^ 4 } \\overline { c } _ { i , j } d _ { r , s } x _ { i , r } x _ { j , s } + \\max _ { \\mathcal { S } \\subseteq [ n ] ^ 2 : | \\mathcal { S } | \\le \\Gamma } \\left \\{ \\sum _ { ( i , j ) \\in \\mathcal { S } } \\sum _ { r , s \\in [ n ] } \\Delta \\overline { c } _ { i , j } d _ { r , s } x _ { i , r } x _ { j , s } \\right \\} \\right \\} . \\end{align*}"}
+{"id": "4416.png", "formula": "\\begin{align*} \\inf _ { x \\in \\mathcal { X } } & \\ \\sum _ { i \\in [ m ] } \\max \\{ 0 , x _ i - \\overline { b } _ i , x _ i - \\overline { b } _ i + \\Delta b _ i - \\Delta b _ k \\} + \\Gamma \\Delta b _ k \\\\ \\mathrm { s . t . } & \\ x _ k \\in [ \\overline { b } _ k - \\Delta b _ k , \\overline { b } _ k ] \\end{align*}"}
+{"id": "8318.png", "formula": "\\begin{align*} \\epsilon ( Y ) - \\Pi ^ { \\sharp } ( \\delta ) = X - Y , \\end{align*}"}
+{"id": "1815.png", "formula": "\\begin{align*} | \\bigcup _ { i = 1 } ^ { k } S _ i | \\leq k \\textrm { f o r e v e r y } 1 \\leq k \\leq n - 1 . \\end{align*}"}
+{"id": "4987.png", "formula": "\\begin{align*} p _ k ( \\lambda _ j ) = \\sum _ { i = 1 } ^ N p _ { k i } \\lambda _ j ^ { i - 1 } \\end{align*}"}
+{"id": "6266.png", "formula": "\\begin{align*} w ( [ J ] _ k , \\lambda ) & = - \\sum _ { i = 1 } ^ k i \\left ( a \\binom { n + i - 1 } { i } + b \\left ( \\frac { d } { r ! } i ^ r + O ( i ^ { r - 1 } ) \\right ) \\right ) + O ( 1 ) \\\\ & = - \\sum _ { i = 1 } ^ k i a \\binom { n + i - 1 } { i } - b \\frac { d } { r ! ( r + 2 ) } k ^ { r + 2 } + O ( k ^ { r + 1 } ) \\\\ & = \\frac { - b d ( r + 1 ) } { ( r + 2 ) ! } k ^ { r + 2 } - \\sum _ { i = 1 } ^ k i a \\binom { n + i - 1 } { i } + O ( k ^ { r + 1 } ) , \\end{align*}"}
+{"id": "521.png", "formula": "\\begin{align*} \\sum _ { z = 0 } ^ { L _ { i _ t } - 1 } { \\overline { \\alpha } _ { i _ t } ^ z } = \\begin{cases} \\frac { \\overline { \\alpha } _ { i _ t } ^ { L _ { i _ t } } - 1 } { \\overline { \\alpha } _ { i _ t } - 1 } , & \\overline { \\alpha } _ { i _ t } \\not = 1 , \\\\ L _ { i _ t } \\overline { \\alpha } _ { i _ t } , & \\overline { \\alpha } _ { i _ t } = 1 , \\end{cases} \\end{align*}"}
+{"id": "3141.png", "formula": "\\begin{align*} u ^ - _ { \\psi ^ * } ( s ) = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ v _ { 2 , \\ , 1 } ( s ) & 1 & 0 \\\\ v _ { 3 , \\ , 1 } ( s ) & v _ { 3 , \\ , 2 } ( s ) & 1 \\end{array} \\right ) ( \\ v _ { 2 , \\ , 1 } ( S ) , v _ { 3 , \\ , 1 } ( S ) , v _ { 3 , \\ , 2 } ( S ) \\in k [ S ] \\ , ) . \\end{align*}"}
+{"id": "6119.png", "formula": "\\begin{align*} \\begin{aligned} \\boldsymbol u _ { n 2 } & = \\boldsymbol u _ n + \\tau \\varphi _ 1 ( \\tau K ) ( K \\boldsymbol u _ n + \\boldsymbol g ( t _ n , \\boldsymbol u _ n ) ) , \\\\ \\boldsymbol u _ { n + 1 } & = \\boldsymbol u _ { n 2 } + \\tau \\varphi _ 2 ( \\tau K ) ( \\boldsymbol g ( t _ { n + 1 } , \\boldsymbol u _ { n 2 } ) - \\boldsymbol g ( t _ n , \\boldsymbol u _ n ) ) . \\end{aligned} \\end{align*}"}
+{"id": "8192.png", "formula": "\\begin{align*} R & = \\frac { 1 } { 2 } \\left ( R _ 1 + R _ 2 \\right ) \\\\ & = \\frac { 1 } { 2 } \\left ( H ( \\frac { B _ 1 } { M } ) + \\frac { B _ 1 } { M } H ( \\frac { B _ 2 ^ 1 } { B _ 1 } ) + ( 1 - \\frac { B _ 1 } { M } ) H ( \\frac { B _ 2 ^ 0 } { M - B _ 1 } ) \\right ) , \\end{align*}"}
+{"id": "2811.png", "formula": "\\begin{align*} K ^ { ( 1 ) } = \\begin{bmatrix} 0 & 0 \\\\ 0 & 0 \\end{bmatrix} , \\end{align*}"}
+{"id": "4103.png", "formula": "\\begin{align*} d _ j = & \\frac { \\sigma ^ 2 _ j } { 2 ( 1 - w _ { j j } ) } , \\\\ \\sigma _ j ^ 2 = & \\sum _ { i < j } \\alpha _ i \\bigl ( w _ { i j } ^ 2 c _ { e , i } ^ 2 + w _ { i j } ( 1 - w _ { i j } ) \\bigr ) + \\alpha _ j c _ { e , j } ^ 2 + \\sum _ { i > j } \\lambda _ i \\bigl ( w _ { i j } ^ 2 c _ { s , i } ^ 2 + w _ { i j } ( 1 - w _ { i j } ) ) \\\\ & + \\lambda _ j \\bigl ( c _ { s , j } ^ 2 ( 1 - w _ { j j } ) ^ 2 + w _ { j j } ( 1 - w _ { j j } ) \\bigr ) . \\end{align*}"}
+{"id": "655.png", "formula": "\\begin{align*} ( \\xi ^ { ( 1 ) } \\otimes \\cdots \\otimes \\xi ^ { ( m ) } ) _ { i _ { 1 } \\cdots i _ { m } } : = \\xi _ { i _ { 1 } } ^ { ( 1 ) } \\cdots \\xi _ { i _ { m } } ^ { ( m ) } . \\end{align*}"}
+{"id": "2410.png", "formula": "\\begin{align*} \\sup _ { | t + k | = \\frac { 1 } { 8 } } \\left | \\left ( t + \\frac { 1 } { 4 } \\right ) _ { n } \\right | & \\leqslant k ! ( n - k ) ! , \\\\ \\sup _ { | t + k | = \\frac { 1 } { 8 } } \\left | \\left ( t + \\frac { 3 } { 4 } \\right ) _ { n } \\right | & \\leqslant k ! ( n - k ) ! , \\\\ \\sup _ { | t + k | = \\frac { 1 } { 8 } } \\left | \\left ( t \\right ) _ { n + 1 } ^ { - 1 } \\right | & \\leqslant \\frac { 1 0 0 n ^ 2 } { k ! ( n - k ) ! } . \\end{align*}"}
+{"id": "6230.png", "formula": "\\begin{align*} \\lim _ { \\xi \\to \\{ \\xi _ \\beta ^ 2 \\} ^ + } \\phi _ { \\beta , \\gamma } ' ( \\xi ) = \\lim _ { \\xi \\to \\{ \\xi _ \\beta ^ 2 \\} ^ + } \\frac { z \\left ( \\phi _ { \\beta , \\gamma } ( \\xi ) \\right ) } { D \\left ( \\phi _ { \\beta , \\gamma } ( \\xi ) \\right ) } = \\lim _ { s \\to \\beta ^ - } \\frac { z ( s ) } { D ( s ) } = \\lim _ { t \\to \\gamma ^ + } \\frac { w ( t ) } { D _ 3 ( t ) } , \\end{align*}"}
+{"id": "2462.png", "formula": "\\begin{align*} \\left \\{ ( \\phi , \\psi ) \\mid \\psi ( s ) = p c ^ { - 1 } \\phi - p c ^ { - 3 } ( \\mu c ^ { 2 } + 1 ) \\phi ^ { 2 } + O ( \\phi ^ { 3 } ) \\right \\} \\end{align*}"}
+{"id": "5315.png", "formula": "\\begin{align*} \\partial _ { \\mathcal { F } } ^ - S = \\left \\{ j \\in S : \\ , S \\setminus \\{ j \\} \\in \\mathcal { F } \\right \\} \\partial _ { \\mathcal { F } } ^ + S = \\left \\{ j \\in J \\setminus S : \\ , S \\cup \\{ j \\} \\in \\mathcal { F } \\right \\} , \\end{align*}"}
+{"id": "5320.png", "formula": "\\begin{align*} v ^ { \\textrm { L P } } = \\min \\ , \\left \\{ \\sum _ { j \\in J } c _ j \\ , x ^ { \\boldsymbol { \\pi } } _ j : \\boldsymbol { \\pi } \\in \\Pi ( \\mathcal { F } ) \\right \\} . \\end{align*}"}
+{"id": "1700.png", "formula": "\\begin{align*} K _ q : = \\{ v \\in H _ q \\ , | \\ , \\mathcal { L } ( v , w ) = 0 \\quad \\forall \\ , w \\in H _ q \\} . \\end{align*}"}
+{"id": "5060.png", "formula": "\\begin{align*} C _ 1 ( x ) : = \\sum _ { i _ s = 1 } ^ k x _ { ( i _ s ) } e _ { i _ s } , \\end{align*}"}
+{"id": "5415.png", "formula": "\\begin{align*} S _ n f ( x ) : = \\begin{cases} \\sum _ { i = 0 } ^ { n - 1 } f ( T ^ { i } x ) & n \\ge 1 , \\\\ 0 & n = 0 , \\\\ - \\sum _ { i = - n } ^ { - 1 } f ( T ^ { i } x ) & n \\le 1 , \\end{cases} \\end{align*}"}
+{"id": "4836.png", "formula": "\\begin{align*} N _ \\lambda \\not = 0 \\quad \\Leftrightarrow \\lambda \\in \\mathcal { P } ( n ) . \\end{align*}"}
+{"id": "7319.png", "formula": "\\begin{align*} G _ { X _ { m } , \\chi _ { \\beta } } ( x , y ; s ) = \\frac { 1 } { m } \\sum _ { j = 0 } ^ { m - 1 } \\frac { 1 } { s + 2 \\sin ^ 2 \\left ( \\pi \\frac { j + \\beta } { m } \\right ) } \\exp \\left ( 2 \\pi i \\frac { j + \\beta } { m } ( x - y ) \\right ) . \\end{align*}"}
+{"id": "5303.png", "formula": "\\begin{align*} v ^ u _ i ( \\nu ) = v _ i ^ u + \\nu \\ , b _ i ^ u . \\end{align*}"}
+{"id": "6689.png", "formula": "\\begin{align*} H = \\langle c , a _ 1 ^ { \\ , p ^ { n - 1 } } , a _ 2 ^ { \\ , p ^ { n - 2 } } , \\ldots , a _ { n - 1 } ^ { \\ , p } , a _ n \\rangle \\le _ \\mathrm { o } G \\end{align*}"}
+{"id": "2902.png", "formula": "\\begin{align*} & \\lim _ { \\lambda \\to \\infty } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ { ( E _ \\Phi ^ r ) _ t ( \\mathbb { R } ^ n ) } = \\left [ \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { ( E _ \\Phi ^ r ) _ t ( \\mathbb { R } ^ n ) } ; \\end{align*}"}
+{"id": "7149.png", "formula": "\\begin{align*} k _ \\nu = \\frac { 1 } { 2 } \\eta _ { \\nu \\mu } \\frac { d x ^ \\mu } { d \\tau } \\ , \\ , , \\end{align*}"}
+{"id": "3322.png", "formula": "\\begin{align*} \\nabla _ { v } \\cdot \\left ( u ( v , x , t ) \\ , v + \\nabla _ v u ( v , x , t ) \\right ) + v \\cdot \\nabla _ { x } u ( v , x , t ) - \\partial _ t u ( v , x , t ) = f ( v , x , t ) , \\end{align*}"}
+{"id": "3558.png", "formula": "\\begin{align*} T f ( z ) = f ( e ^ { i \\theta } z ) ( f \\in { H } ^ { \\infty } _ { 0 , 1 , 2 , \\ldots , n } ( \\mathbb { D } ) ) , \\end{align*}"}
+{"id": "5554.png", "formula": "\\begin{align*} \\ell = \\left \\lfloor c _ 1 \\log _ d ( n ) \\right \\rfloor , c _ 1 = \\frac { ( k / 2 - 1 ) \\wedge 1 } { 2 5 } . \\end{align*}"}
+{"id": "3045.png", "formula": "\\begin{align*} - \\left ( \\frac { \\partial ^ 2 \\psi } { \\partial \\chi ^ 2 } + \\cot \\chi \\frac { \\partial \\psi } { \\partial \\chi } + \\frac { 1 } { ^ 2 \\chi } \\frac { \\partial ^ 2 \\psi } { \\partial \\xi ^ 2 } \\right ) + \\frac { 1 } { 1 - 2 \\lambda ^ 2 } \\left [ \\frac { 1 } { ^ 2 \\chi } \\left ( \\frac { k _ 1 ^ 2 } { ^ 2 \\xi } + \\frac { k _ 2 ^ 2 } { ^ 2 \\xi } \\right ) + \\frac { k _ 3 ^ 2 } { \\cos ^ 2 \\chi } \\right ] \\psi = E \\psi \\ , \\end{align*}"}
+{"id": "3303.png", "formula": "\\begin{align*} M _ t = M _ 0 + \\int _ { 0 } ^ t ( \\sigma _ s \\d B _ s + b _ s \\d s ) , \\end{align*}"}
+{"id": "2974.png", "formula": "\\begin{align*} { \\theta } _ 1 ( 0 ) + { \\theta } _ 2 ( 0 ) = 0 \\mbox { a n d t h u s , } \\quad { \\theta } _ 1 ( t ) + { \\theta } _ 2 ( t ) = 0 , t \\in [ 0 , \\infty ) . \\end{align*}"}
+{"id": "6452.png", "formula": "\\begin{align*} & \\langle R ^ N ( \\bar \\nabla _ { e _ j } \\tau ( \\phi ) , R ^ N ( V , d \\phi ( e _ i ) ) d \\phi ( e _ i ) ) d \\phi ( e _ j ) , V \\rangle - \\langle R ^ N ( R ^ N ( V , d \\phi ( e _ j ) ) \\bar \\nabla _ { e _ j } \\tau ( \\phi ) , d \\phi ( e _ k ) ) d \\phi ( e _ k ) , V \\rangle \\\\ & = 2 \\langle R ^ N ( d \\phi ( e _ j ) , V ) \\bar \\nabla _ { e _ j } \\tau ( \\phi ) , R ^ N ( V , d \\phi ( e _ k ) ) d \\phi ( e _ k ) \\rangle . \\end{align*}"}
+{"id": "3592.png", "formula": "\\begin{align*} X f = \\alpha ( f \\circ \\tau ) ( f \\in H ^ { \\infty } ( \\mathbb { D } ) ) . \\end{align*}"}
+{"id": "2076.png", "formula": "\\begin{align*} N _ { i , 1 } ^ s & = - \\eta d T \\big ( \\frac { 1 } { d T } \\sum _ { t = 0 } ^ { \\lfloor s T \\rfloor - 1 } \\sum _ { j = 1 } ^ d f ( \\frac { t } { T } ) A ( \\frac { i } { d } , \\frac { j } { d } ) \\Theta ( \\frac { t } { T } , \\frac { j } { d } ) + o ( 1 ) \\big ) \\\\ & = - \\eta d T \\big ( \\int _ 0 ^ { \\frac { \\lfloor s T \\rfloor } { T } } \\int _ 0 ^ 1 f ( u ) A ( \\frac { i } { d } , y ) \\Theta ( u , y ) \\mathrm { d } y \\mathrm { d } u + o ( 1 ) \\big ) . \\end{align*}"}
+{"id": "1403.png", "formula": "\\begin{align*} ( n ( t + u ) ) _ i & = n _ i + \\sum _ { k = 0 } ^ { i - 1 } \\binom { n _ 1 } { k } ( t + u ) _ { i - k } \\\\ & = n _ i + \\sum _ { k = 0 } ^ { i - 1 } \\binom { n _ 1 } { k } ( t _ { i - k } + u _ { i - k } ) \\\\ & = n _ i + \\sum _ { k = 0 } ^ { i - 1 } \\binom { n _ 1 } { k } t _ { i - k } - n _ i + n _ i + \\sum _ { k = 0 } ^ { i - 1 } \\binom { n _ 1 } { k } u _ { i - k } \\\\ & = ( n t ) _ i - n _ i + ( n u ) _ i = ( n t - n + n u ) _ i \\end{align*}"}
+{"id": "431.png", "formula": "\\begin{align*} \\begin{aligned} \\begin{bmatrix} 1 - F _ { j , j } & - \\Gamma _ { j - 1 } & 0 \\\\ - G _ { j - 1 , j - 1 } - \\widetilde { t } _ { j } \\rho ^ 2 & 1 - F _ { j , j - 1 } ^ { T } & 0 \\\\ 0 & 0 & 1 \\end{bmatrix} - \\begin{bmatrix} 1 - F _ { j , j - 1 } & 1 _ { i - j } - f _ { j , j } & - \\Gamma _ { j } \\\\ - G _ { j - 1 , j - 1 } - \\widetilde { t } _ { j } \\rho ^ 2 & - G _ { j - 1 , j - 1 } - \\widetilde { t } _ { j } \\rho ^ 2 & 1 - F _ { j , j - 1 } ^ { T } \\\\ x & - g _ { j , j } & - f _ { j , j } \\end{bmatrix} \\end{aligned} \\end{align*}"}
+{"id": "7700.png", "formula": "\\begin{align*} \\overline { \\nabla } h _ { \\Omega } ( x ) = \\sum _ { i } h _ { i } e _ { i } + h x , F _ { i } ( x ) = \\sum _ { j } ( h _ { i j } + h \\delta _ { i j } ) e _ { j } , F _ { i j } ( x ) = \\sum _ { k } ( h _ { i j k } + h _ { k } \\delta _ { i j } ) e _ { k } - ( h _ { i j } + h \\delta _ { i j } ) x . \\end{align*}"}
+{"id": "1472.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\sup _ { g \\in G } \\frac { | \\Delta M \\cap g F _ n | } { | F _ n | } = \\overline { D } _ G ( \\Delta M ) = 0 . \\end{align*}"}
+{"id": "160.png", "formula": "\\begin{align*} ( 1 _ { Y } \\boxtimes u _ { X , Z } ) \\circ ( u _ { X , Y } \\boxtimes 1 _ { Z } ) ( b _ { i } \\boxtimes c _ { j } \\boxtimes d _ { k } ) & = c _ { j } \\boxtimes d _ { k } \\boxtimes b _ { i } \\\\ & = u ^ { F , K } _ { X , Y \\boxtimes Z } ( b _ { i } \\boxtimes c _ { j } \\boxtimes d _ { k } ) \\\\ & = u _ { X , Y \\boxtimes Z } ( b _ { i } \\boxtimes c _ { j } \\boxtimes d _ { k } ) , \\end{align*}"}
+{"id": "9024.png", "formula": "\\begin{align*} | G | = | \\Phi | + | G | \\sum _ { i = 1 } ^ { k _ 1 } \\frac { 1 } { \\lambda _ { i } } + | G | \\sum _ { i = 1 } ^ { k _ 2 } \\frac { \\nu _ i } { \\mu _ i } + | G | \\sum _ { i = 1 } ^ { k _ 3 } \\left ( \\frac { 1 } { \\omega _ i } - \\frac { 1 } { q _ i } \\right ) , \\end{align*}"}
+{"id": "1609.png", "formula": "\\begin{align*} Q _ { i ( p - 1 ) } ( x \\otimes y ) \\mapsto ( - 1 ) ^ { \\tfrac { n m ( p - 1 ) } { 2 } } \\sum _ { j + k = i } Q _ { j ( p - 1 ) } ( x ) \\otimes Q _ { k ( p - 1 ) } ( y ) \\end{align*}"}
+{"id": "2695.png", "formula": "\\begin{align*} q ( t ) = \\left \\{ \\begin{array} { l l } \\frac { \\alpha } { \\beta ^ { 2 } } \\exp { [ \\beta ( t - t _ { 0 } ) ] } + q _ { 0 } & ( t < t _ { 0 } ) \\\\ \\frac { \\alpha } { \\beta } ( t - t _ { 0 } ) + \\frac { \\alpha } { \\beta ^ { 2 } } + q _ { 0 } & ( t _ { 0 } < t ) \\end{array} \\right . \\end{align*}"}
+{"id": "4486.png", "formula": "\\begin{align*} M ( Z _ 0 , J , \\tilde \\rho ) = \\int _ M | F _ 0 | ^ 2 \\tilde \\rho \\end{align*}"}
+{"id": "6343.png", "formula": "\\begin{align*} t V _ e ( t ) = Y ( t ) = \\frac { \\sinh ( \\sqrt { - \\kappa } t ) } { \\sqrt { - \\kappa } } V ( t ) , \\end{align*}"}
+{"id": "1684.png", "formula": "\\begin{align*} l _ a ^ + & = \\langle e _ 1 - a \\ , f _ 2 , \\ e _ 2 + a \\ , f _ 1 \\rangle , \\\\ l _ b ^ - & = \\langle e _ 1 + b \\ , e _ 2 , \\ - f _ 2 + b \\ , f _ 1 \\rangle , \\\\ l _ a ^ + \\cap l _ b ^ - & = \\langle \\phi _ { a b } \\rangle , \\ \\ \\mbox { f o r } \\phi _ { a b } \\coloneqq e _ 1 + b \\ , e _ 2 - a \\ , f _ 2 + a b \\ , f _ 1 , \\end{align*}"}
+{"id": "8829.png", "formula": "\\begin{align*} v _ t = d \\Delta v + c v _ z + f _ { \\pm } ^ { \\infty } ( v ) , z \\in \\mathbb { R } , \\ , \\ , t \\ge 0 . \\end{align*}"}
+{"id": "1531.png", "formula": "\\begin{align*} \\alpha & = \\inf \\left \\{ \\lambda _ { \\rm m i n } ( D ^ 2 d ( z ) ) : | z | \\ge r / 2 \\right \\} , & \\beta & = \\sup \\left \\{ | D ^ 2 \\varphi ( z ) | _ 2 : | z - w | \\ge r / 4 \\right \\} , \\\\ \\gamma & = \\sup \\left \\{ \\lambda _ { \\rm m a x } ( D ^ 2 d ( z ) ) : | z - w | \\le r / 4 \\right \\} , & \\delta & = \\inf \\left \\{ \\lambda _ { \\rm m i n } ( D ^ 2 s ( z ) ) : | z - w | \\ge r / 4 \\right \\} . \\end{align*}"}
+{"id": "5188.png", "formula": "\\begin{align*} \\frac { \\partial ( X . Y ) } { \\partial q _ { j } } = \\beta ( 1 - \\beta ) \\left [ \\left ( \\frac { \\sum _ { i } p _ { i } q ^ { \\beta - 1 } _ { i } } { \\sum _ { i } q ^ { \\beta } _ { i } } \\right ) ^ { \\beta } q ^ { \\beta - 1 } _ { j } - \\left ( \\frac { \\sum _ { i } p _ { i } q ^ { \\beta - 1 } _ { i } } { \\sum _ { i } q ^ { \\beta } _ { i } } \\right ) ^ { \\beta - 1 } p _ { j } q ^ { \\beta - 2 } _ { j } \\right ] \\end{align*}"}
+{"id": "5861.png", "formula": "\\begin{align*} \\dfrac { M _ { 2 } ( \\mathcal { C } ( D _ { 2 m } ) ) } { | e ( \\mathcal { C } ( D _ { 2 m } ) ) | } = \\dfrac { ( m - 2 ) ( m - 3 ) ^ { 3 } + m } { ( m - 2 ) ( m - 3 ) + m } > \\dfrac { ( m - 2 ) ( m - 3 ) ^ { 2 } + m } { 2 m - 2 } = \\dfrac { M _ { 1 } ( \\mathcal { C } ( D _ { 2 m } ) ) } { | v ( \\mathcal { C } ( D _ { 2 m } ) ) | } . \\end{align*}"}
+{"id": "1369.png", "formula": "\\begin{align*} l ' ( \\iota ( s ^ { - 1 } f ) , \\iota ( \\theta _ { 1 } ) , \\dots \\iota ( \\theta _ { i } ) ) & = l ' ( s ^ { - 1 } \\widetilde { f } , \\theta _ { 1 } , \\dots \\theta _ { i } ) = \\iota \\left ( l ' ( s ^ { - 1 } \\widetilde { f } , \\theta _ { 1 } , \\dots \\theta _ { i } ) \\right ) . \\end{align*}"}
+{"id": "3004.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\overline { \\mathrm { R i c } } ( x , y ) & = & \\mathrm { R i c } ( x , y ) + [ \\beta ^ 2 - \\beta + ( 1 - 2 n ) \\alpha ^ 2 ] g ( x , y ) \\\\ & & + [ ( 2 n + 1 ) \\beta - \\beta ^ 2 + ( 2 n - 1 ) \\alpha ^ 2 ] \\eta ( x ) \\eta ( y ) \\\\ & & + [ ( \\alpha \\beta - \\alpha ) ( 2 - 2 n ) + \\alpha ] g ( x , \\varphi y ) \\end{array} \\end{align*}"}
+{"id": "3624.png", "formula": "\\begin{align*} 0 = \\big ( \\frac { 2 } { \\lambda _ { 2 } ^ { 2 } - 1 } + 1 \\big ) ( f ( z , w ) - f ( z , - w ) ) = \\frac { \\lambda _ { 2 } ^ { 2 } + 1 } { \\lambda _ { 2 } ^ { 2 } - 1 } ( f ( z , w ) - f ( z , - w ) ) , \\end{align*}"}
+{"id": "4615.png", "formula": "\\begin{align*} V \\left ( x ' , x _ d \\right ) : = - c \\left ( f ( x ' ) - x _ d \\right ) ^ { \\frac { 2 } { d } ( - 1 + \\epsilon ) } \\mathrm { \\ f o r \\ } \\left ( x ' , x _ d \\right ) \\in \\Omega \\subset \\R ^ { d - 1 } \\times \\R \\end{align*}"}
+{"id": "255.png", "formula": "\\begin{align*} \\left ( A J \\frac { d } { d x } - 3 J A ' \\right ) P & = J \\left ( A \\frac { d } { d x } - 3 A ' \\right ) P . \\end{align*}"}
+{"id": "7984.png", "formula": "\\begin{align*} \\begin{aligned} \\langle d N _ { \\phi } ( e _ { \\phi } ^ { i } ) , d \\big ( \\mathrm { l i } _ { \\phi } ( f _ { \\phi } ^ { j } ) \\big ) \\rangle _ { L ^ { 2 } \\Lambda ^ { 1 } ( \\Omega ) } = \\int _ { \\partial \\Omega } e _ { \\phi } ^ { i } \\wedge f _ { \\phi } ^ { j } . \\end{aligned} \\end{align*}"}
+{"id": "721.png", "formula": "\\begin{align*} Y _ { \\leq } : = \\prod _ { a , b : A } \\biggl ( ( b \\leq _ A a ) \\times ( a \\leq _ A b ) \\to ( a = _ { A } b ) \\biggr ) . \\end{align*}"}
+{"id": "7527.png", "formula": "\\begin{align*} \\begin{dcases} \\partial _ t \\rho + \\nabla \\cdot ( \\rho u ) = 0 \\\\ \\partial _ t ( \\rho u ) + \\nabla \\cdot \\big ( \\rho u \\otimes u \\big ) + \\nabla p _ 1 ( \\rho ) + \\rho \\nabla h _ 2 ^ \\prime ( n ) = 0 \\\\ - \\delta \\Delta h _ 2 ^ \\prime ( n ) + n = \\rho \\ , \\end{dcases} \\end{align*}"}
+{"id": "2476.png", "formula": "\\begin{align*} H _ { + } = \\{ y \\in \\mathbb { S } ^ { 2 } \\ , | \\ , y _ { 3 } > 0 \\} , H _ { - } = \\{ y \\in \\mathbb { S } ^ { 2 } \\ , | \\ , y _ { 3 } < 0 \\} \\end{align*}"}
+{"id": "8394.png", "formula": "\\begin{align*} B ( a ) B ( b ) = B ( a _ 1 B ( a _ 2 ) b S ( B ( a _ 3 ) ) ) . \\end{align*}"}
+{"id": "4113.png", "formula": "\\begin{align*} & f _ { ( \\theta , r ) } ( x ) = \\exp \\Bigl ( \\langle \\theta , z \\rangle - \\sum _ { j } \\gamma _ j ( \\theta _ j , r ) ( \\alpha _ j u _ j \\wedge r ^ { - 1 } ) - \\sum _ { j } \\xi _ j ( \\theta , r ) ( \\mu _ j ^ { ( r ) } v _ j \\wedge r ^ { - 1 } ) \\Bigr ) . \\end{align*}"}
+{"id": "2280.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } L \\\\ p \\end{array} \\right ] _ q : = \\frac { [ L ] _ q ! } { [ L - p ] _ q ! [ p ] _ q ! } \\ , \\ \\ \\ \\ \\ \\ ( a \\ , ; \\ , q ) _ p = \\prod _ { r = 0 } ^ { p - 1 } ( 1 - a q ^ r ) \\ . \\end{align*}"}
+{"id": "7175.png", "formula": "\\begin{align*} \\| f ( x , t ) \\| _ Y = \\sup _ { 0 < t < T } \\sup _ { x \\in \\R } | f ( x , t ) | . \\end{align*}"}
+{"id": "6856.png", "formula": "\\begin{align*} G ( x ) = \\sum _ { n = 0 } ^ { \\infty } q _ n x ^ n \\end{align*}"}
+{"id": "6411.png", "formula": "\\begin{align*} z _ 1 ^ n ( \\theta ) = \\frac { n ^ { 1 / \\alpha } } { \\delta x ^ { 1 / \\alpha } } \\left ( \\frac { 1 } { n } ( a _ 0 - a ) - \\int _ 0 ^ \\frac { 1 } { n } ( b _ 0 X _ s - b x ) d s + \\delta _ 0 \\int _ 0 ^ \\frac { 1 } { n } X _ { s - } ^ { 1 / \\alpha _ 0 } d L _ s \\right ) . \\end{align*}"}
+{"id": "4151.png", "formula": "\\begin{align*} & Q ^ * ( \\tilde { \\theta } ) \\psi ^ { ( r ) } ( \\tilde { \\theta } ) + r ^ j \\bar { \\xi } _ j ( \\tilde { \\theta } ) \\bigl ( \\psi ^ { ( r ) } ( \\tilde { \\theta } ) - \\psi ^ { ( r ) } _ i ( \\tilde { \\theta } ) \\bigr ) = o ( r ^ { 2 j + 1 } ) . \\end{align*}"}
+{"id": "1027.png", "formula": "\\begin{align*} T _ n ( w ) : = \\left ( \\begin{matrix} \\gamma ( 0 ) & \\gamma ( - 1 ) & \\cdots & \\gamma ( - n + 1 ) \\cr \\gamma ( 1 ) & \\gamma ( 0 ) & \\cdots & \\gamma ( - n + 2 ) \\cr \\vdots & \\vdots & \\ddots & \\vdots \\cr \\gamma ( n - 1 ) & \\gamma ( n - 2 ) & \\cdots & \\gamma ( 0 ) \\end{matrix} \\right ) \\in \\C ^ { q n \\times q n } \\end{align*}"}
+{"id": "2403.png", "formula": "\\begin{align*} U ( t ) : = & ~ \\frac { \\widetilde { A _ n } ^ { \\prime } ( t ) } { \\widetilde { A _ n } ( t ) } \\\\ = & ~ \\frac { \\delta } { t + n / 2 } + ( s + 2 ) \\sum _ { \\ell = 0 } ^ { n - 1 } \\left ( \\frac { 1 } { t + \\ell + 1 / 4 } + \\frac { 1 } { t + \\ell + 3 / 4 } \\right ) - ( 2 s + 4 ) \\sum _ { \\ell = 0 \\atop \\ell \\neq k } ^ { n } \\frac { 1 } { t + \\ell } . \\end{align*}"}
+{"id": "676.png", "formula": "\\begin{align*} \\mathcal { A } _ { s } : L ^ { p } ( \\mathbb { R } ^ { n } ) \\rightarrow L ^ { q } ( \\mathbb { R } ^ { n } ) \\frac { 1 } { q } = \\frac { 1 } { p } - \\frac { 2 s } { n } , \\end{align*}"}
+{"id": "5017.png", "formula": "\\begin{align*} 2 \\sum _ { k = 2 } ^ { n } \\left \\{ ( 2 / 3 ) k ^ 3 + O ( k ^ 2 ) \\right \\} = ( 1 / 3 ) n ^ 4 + O ( n ^ 3 ) . \\end{align*}"}
+{"id": "1064.png", "formula": "\\begin{align*} T ^ { k - 1 } f ( x ) = \\int _ 0 ^ { \\infty } D _ k ( x , y ) f ( y ) d y , k \\ge 2 . \\end{align*}"}
+{"id": "2776.png", "formula": "\\begin{align*} \\delta L _ { T } = \\left [ \\dot { Q } ^ { a } - \\frac { \\partial H _ { T } } { \\partial P _ { a } } \\right ] \\delta P _ { a } + \\left [ - \\dot { P } _ { a } - \\frac { \\partial H _ { T } } { \\partial Q ^ { a } } \\right ] \\delta Q ^ { a } + \\frac { d } { d t } \\left [ P _ { a } \\delta Q ^ { a } \\right ] \\end{align*}"}
+{"id": "4013.png", "formula": "\\begin{align*} \\begin{cases} \\left ( \\gamma _ { 1 } - \\delta _ { 2 } \\right ) x _ { 1 } + \\delta _ { 1 } x _ { 2 } & = \\ ; 0 \\\\ \\left ( 1 - \\gamma _ { 1 } \\right ) x _ { 1 } + \\left ( 1 - \\delta _ { 1 } - \\delta _ { 2 } \\right ) x _ { 2 } & = \\ ; 1 \\end{cases} \\end{align*}"}
+{"id": "5505.png", "formula": "\\begin{align*} d _ { j , n } ( X _ l ) & { } = \\begin{cases} X _ l & n = 0 ; \\\\ \\delta _ j ( X _ l ) & n = 1 ; \\\\ 0 & n > 1 , \\end{cases} \\end{align*}"}
+{"id": "4655.png", "formula": "\\begin{align*} c = a + b + \\frac { e } { n } + \\frac { 2 } { n ^ 2 } ( a b e + r x y ) , \\end{align*}"}
+{"id": "4059.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } \\frac { e ^ { - i r | x | - \\epsilon r } } { r - z } ( - r ) ^ { \\frac { d - 1 } { 2 } - k } d r = - \\int _ { - \\infty } ^ { 0 } \\frac { e ^ { r | x | } e ^ { - i \\epsilon r } } { r + i z } ( - i r ) ^ { \\frac { d - 1 } { 2 } - k } d r . \\end{align*}"}
+{"id": "1569.png", "formula": "\\begin{align*} \\zeta ( r ) = \\int _ { A ( \\ell , r ) } | D V | _ 2 \\ , | V - D \\ell | \\ , d x 0 < r < R , \\end{align*}"}
+{"id": "8094.png", "formula": "\\begin{align*} e ' _ { k , \\ell } ( \\psi _ \\varpi f ) = e ' _ { k , \\ell } ( \\psi _ \\varpi \\Psi _ r f ) \\ll _ k \\norm { \\Psi _ r f } _ k ' . \\end{align*}"}
+{"id": "1819.png", "formula": "\\begin{align*} H ^ 1 ( X , \\rho _ A ) = 0 , \\end{align*}"}
+{"id": "1584.png", "formula": "\\begin{align*} \\tilde { y } = \\frac { \\tilde x } { g _ h ( \\tilde x ) } . \\end{align*}"}
+{"id": "4054.png", "formula": "\\begin{align*} R _ 0 ( z ) : = ( \\sqrt { - \\Delta } - z ) ^ { - 1 } , \\ , \\ , \\ , \\ , \\ , \\ , R _ V ( z ) : = ( \\sqrt { - \\Delta } + V - z ) ^ { - 1 } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ \\Im z > 0 , \\ , \\ , \\ , \\Re z > 0 . \\end{align*}"}
+{"id": "8456.png", "formula": "\\begin{align*} & \\iint _ { \\Omega _ T } \\partial _ t v _ h ( x , t ) \\ , \\varphi ( x , t ) \\ , d x d t + \\frac { 1 } { 2 } \\int _ 0 ^ T \\iint _ { \\mathbb { R } ^ n \\times \\mathbb { R } ^ n } \\dfrac { \\overline { U } _ h ( x , y , t ) } { | x - y | ^ { n + s p } } ( \\varphi ( x , t ) - \\varphi ( y , t ) ) \\ , d x d y d t = 0 \\end{align*}"}
+{"id": "5290.png", "formula": "\\begin{align*} U _ { j } = 1 + \\frac { a } { b - a } \\left ( \\frac { q _ { j } } { p _ { j } } \\right ) ^ { a - 1 } \\ ; \\ ; \\ ; ; \\ ; \\ ; \\ ; V _ { j } = \\frac { b } { b - a } \\left ( \\frac { q _ { j } } { p _ { j } } \\right ) ^ { b - 1 } \\end{align*}"}
+{"id": "1659.png", "formula": "\\begin{align*} \\xi = \\xi \\left ( T \\right ) = \\frac { T + c } { c ^ { 2 } } = \\frac { T + 2 + \\sqrt { T + 1 / 4 } } { \\left ( 2 + \\sqrt { T + 1 / 4 } \\right ) ^ { 2 } } \\in \\left ( 0 , 1 \\right ) . \\end{align*}"}
+{"id": "4829.png", "formula": "\\begin{align*} ( V [ \\lambda ] ) = \\prod _ { \\alpha \\in \\Phi ^ + ( { \\mathfrak h } ) } \\frac { \\langle \\lambda + \\rho ( { \\mathfrak h } ) , \\alpha \\rangle } { \\langle \\rho ( { \\mathfrak h } ) , \\alpha \\rangle } . \\end{align*}"}
+{"id": "7563.png", "formula": "\\begin{align*} i \\partial \\psi - \\gamma \\Delta ^ 2 \\psi + \\epsilon \\Delta \\psi + | \\psi | ^ { 2 \\sigma } \\psi = 0 , \\gamma > 0 , \\ \\epsilon \\in \\mathbb { R } , \\ 4 \\leq \\sigma \\leq \\frac { 4 } { N - 4 } , \\end{align*}"}
+{"id": "8418.png", "formula": "\\begin{align*} \\Psi ( f ) ( \\sigma ( a ) \\circ \\sigma ( b ) ) & = f ( B _ 1 ( a _ 1 ) ) f ( S ( B _ 2 ( a _ 2 ) ) ) f ( \\sigma ( b ) ) \\\\ & = f ( B _ 1 ( a _ 1 ) S ( B _ 2 ( a _ 2 ) ) ) f ( \\sigma ( b ) ) \\\\ & = \\Psi ( f ) ( \\sigma ( a ) ) \\Psi ( f ) ( \\sigma ( b ) ) . \\end{align*}"}
+{"id": "7874.png", "formula": "\\begin{align*} \\mathbf { A } _ { i } - \\mathbf { A } _ { j } & = \\psi ( \\pi , \\mathbf { a } , \\mathbf { d } _ { i } ) - \\psi ( \\pi , \\mathbf { b } , \\mathbf { d } _ { j } ) \\\\ & = \\psi ( \\pi , \\mathbf { a } - \\mathbf { b } , \\mathbf { d } _ { i } - \\mathbf { d } _ { j } ) . \\end{align*}"}
+{"id": "1137.png", "formula": "\\begin{align*} \\left \\| B \\vec t ^ { ( N ) } \\right \\| _ { \\dot b _ { p , q } ^ s } & \\gtrsim \\left [ \\sum _ { j = 0 } ^ { N - 1 } \\left ( \\sum _ { k \\in \\mathbb { Z } ^ n , \\ , | k | < 2 ^ { j - 1 } } 2 ^ { - j n } 2 ^ { ( j - N ) ( F + s - \\frac { n } { 2 } ) p } \\right ) ^ { \\frac { q } { p } } \\right ] ^ { \\frac { 1 } { q } } \\sim \\left [ \\sum _ { j = 0 } ^ { N - 1 } 2 ^ { ( j - N ) ( F + s - \\frac { n } { 2 } ) q } \\right ] ^ { \\frac { 1 } { q } } . \\end{align*}"}
+{"id": "2910.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ \\mathbb { E } \\big [ X _ { T \\wedge ( n , K ) } \\ , | \\ , \\mathcal { F } _ 0 \\big ] \\ , \\big | \\ , \\mathcal { S } _ 0 \\big ] = \\mathbb { E } \\big [ X _ { T \\wedge ( n , K ) } \\ , | \\ , \\mathcal { S } _ 0 \\big ] \\leqslant 2 \\mathbb { E } \\big [ X _ 0 \\ , | \\ , \\mathcal { S } _ 0 \\big ] . \\end{align*}"}
+{"id": "8959.png", "formula": "\\begin{align*} & \\| F \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } = \\| \\tilde F \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } \\le C \\| f \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } , \\\\ & \\| G \\| _ { L ^ { q ( \\cdot ) } ( \\Omega ) } \\le \\| \\tilde { G } \\| _ { L ^ { q ( \\cdot ) } ( \\Omega ) } + \\| I + S - P \\| _ { L ^ { q ( \\cdot ) } ( \\Omega ) } \\le C \\| g \\| _ { L ^ { q ( \\cdot ) } ( \\Omega ) } . \\end{align*}"}
+{"id": "2424.png", "formula": "\\begin{align*} f ^ { ( s ) } ( t ) = & \\sum _ { ( i _ 1 , \\ldots , i _ n , j ) \\in I } s ! \\binom { s + 2 } { i _ 1 } \\cdots \\binom { s + 2 } { i _ n } \\cdot ( t + 1 ) ^ { s + 2 - i _ 1 } \\cdots ( t + n ) ^ { s + 2 - i _ n } \\cdot \\frac { g ^ { ( j ) } ( t ) } { j ! } \\\\ = & \\sum _ { ( i _ 1 , \\ldots , i _ n , j ) \\in I } f _ { ( i _ 1 , \\ldots , i _ n , j ) } ( t ) , \\end{align*}"}
+{"id": "6996.png", "formula": "\\begin{align*} \\widetilde { \\Lambda } _ 0 = \\{ ( z _ 1 , z _ 2 , . . . , z _ n , z _ { n + 1 } ) \\in \\mathbb { C } ^ { n , 1 } : z _ 1 = z _ 2 = z _ { n + 1 } = 0 \\} \\cong \\mathbb { C } ^ { n - 2 } . \\end{align*}"}
+{"id": "3730.png", "formula": "\\begin{align*} \\mathrm { B } \\left ( x , y \\right ) = \\frac { \\Gamma \\left ( x \\right ) \\Gamma \\left ( y \\right ) } { \\Gamma \\left ( x + y \\right ) } . \\end{align*}"}
+{"id": "7716.png", "formula": "\\begin{align*} \\frac { h _ { j } - h _ { j t } } { h - h _ { t } } & = - \\sum _ { i , k } b ^ { i k } \\nabla _ { j } b _ { i k } + \\nabla _ { j } \\psi \\\\ & = - \\sum _ { i } b ^ { i i } ( h _ { j i i } + h _ { i } \\delta _ { i j } ) + \\nabla _ { j } \\psi , \\end{align*}"}
+{"id": "7322.png", "formula": "\\begin{align*} F _ { m , r } ( s , \\beta ) = \\frac { 1 } { m } \\sum _ { j = 0 } ^ { m - 1 } \\frac { e ^ { 2 \\pi i \\frac { j r } { m } } } { s + 2 \\sin ^ 2 \\left ( \\pi \\frac { ( j + \\beta ) } { m } \\right ) } \\end{align*}"}
+{"id": "6175.png", "formula": "\\begin{align*} \\varphi ^ k ( x _ 1 , x _ 2 , \\lambda ) : = - \\lambda ^ T ( A _ 1 x _ 1 + A _ 2 x _ 2 - b ) + \\frac { \\beta ^ k } { 2 } \\| A _ 1 x _ 1 + A _ 2 x _ 2 - b \\| ^ 2 , ~ \\beta ^ k > 0 , \\end{align*}"}
+{"id": "8501.png", "formula": "\\begin{align*} v _ n = x _ 1 x _ 2 \\dotsc x _ n w _ n , n = 1 , 2 , \\dotsc , \\end{align*}"}
+{"id": "6821.png", "formula": "\\begin{align*} \\left ( \\sum _ { i , j } \\beta _ { i j } k _ { i j } ^ { N - 1 } + K _ { N - 1 } \\right ) + \\left ( \\sum _ { i j } \\beta _ { i j } k _ { i j } ^ N + K _ N \\right ) + \\left ( \\sum _ { i , j } \\beta _ { i j } l _ { i j } ^ { N - 1 } + L _ { N - 1 } \\right ) = 0 . \\end{align*}"}
+{"id": "1949.png", "formula": "\\begin{align*} Y _ { i + 1 } \\leq C 2 ^ { i ( N + 2 q ) \\kappa } Y _ i ^ { \\frac { \\kappa ( \\gamma - 1 ) } { \\gamma } } , i = 1 , 2 , 3 , \\ldots . \\end{align*}"}
+{"id": "1889.png", "formula": "\\begin{align*} \\langle \\nabla F ( u ) , u \\rangle = F ( u ) \\langle \\nabla F ( u ) , i u \\rangle = 0 , \\end{align*}"}
+{"id": "4321.png", "formula": "\\begin{gather*} \\mathcal { X } : = \\left \\{ x \\in \\{ 0 , 1 \\} ^ { [ m ] \\times | \\mathcal { J } | } : \\sum _ { i \\in [ m ] } x _ { i , j } = 1 \\ \\forall j \\in \\mathcal { J } , \\ \\sum _ { j \\in \\mathcal { J } } x _ { i , j } = 1 \\ \\forall i \\in [ m ] \\right \\} . \\end{gather*}"}
+{"id": "4488.png", "formula": "\\begin{align*} M _ H ( Z _ 0 , \\mathcal { I } ( 2 \\psi ) , \\rho ) = \\sum _ { 1 \\le j \\le m } \\frac { 2 | a _ j | ^ 2 t _ j } { ( k _ j + 1 ) c _ { \\beta } ( z _ j ) ^ { 2 ( k _ j + 1 ) } } \\rho ( z _ j ) . \\end{align*}"}
+{"id": "8396.png", "formula": "\\begin{align*} & [ B _ 1 ( a ) , B _ 1 ( b ) ] = B _ 1 ( [ B _ 1 ( a ) , B _ 1 ( b ) ] - [ B _ 2 ( a ) , B _ 2 ( b ) ] ) , \\quad \\forall a , b \\in \\mathfrak { g } , \\\\ & [ B _ 2 ( b ) , B _ 2 ( a ) ] = B _ 2 ( [ B _ 1 ( a ) , B _ 1 ( b ) ] - [ B _ 2 ( a ) , B _ 2 ( b ) ] ) , \\quad \\forall a , b \\in \\mathfrak { g } . \\end{align*}"}
+{"id": "8227.png", "formula": "\\begin{align*} \\mathcal { P } ( { \\bf X } ^ L ) ^ { \\star } = \\left \\{ P _ { { \\bf X } _ i | Y ^ { i - 1 } } ( { \\bf x } _ i | y ^ { i - 1 } ) , i \\in [ L ] \\right \\} . \\end{align*}"}
+{"id": "4087.png", "formula": "\\begin{align*} \\psi _ 0 ( x ) = \\frac { c _ d } { ( \\lambda r ) ^ { \\frac { d - 1 } { 2 } } } ( e ^ { i \\lambda r } \\phi ( \\theta ) + e ^ { - i \\lambda r } i ^ { 1 - d } \\phi ( - \\theta ) ) + O ( r ^ { - \\frac { d + 1 } { 2 } } ) . \\end{align*}"}
+{"id": "1367.png", "formula": "\\begin{align*} \\delta _ \\pi ( f ) = & \\ [ \\pi , f ] _ { \\Omega } , \\\\ d _ T ( \\theta ) = & \\ \\lambda P [ \\pi , \\theta ] _ { \\Omega } + P \\big [ [ \\pi , T ] _ { \\Omega } , \\theta \\big ] _ { \\Omega } \\\\ = & \\ \\lambda [ \\mu _ V , \\theta ] _ { \\Omega } + \\big [ [ \\pi , T ] _ { \\Omega } , \\theta \\big ] _ { \\Omega } , \\\\ h _ T ( f ) = & \\ \\sum _ { k = 1 } ^ n \\frac { 1 } { k ! } \\lambda ^ { n - k } P \\big [ \\ldots [ f , \\underbrace { T ] _ { \\Omega } , \\ldots , T } _ k \\big ] _ { \\Omega } . \\end{align*}"}
+{"id": "2654.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { K } \\langle \\phi _ { k - 1 } , \\tilde { w } ( s _ k ) - \\tilde { w } ( s _ { k - 1 } ) \\rangle _ { L ^ 2 _ x } . \\end{align*}"}
+{"id": "2711.png", "formula": "\\begin{align*} { _ { * } a } ^ { i } = \\frac { \\partial H } { \\partial p _ { i } } , { _ { * } b } _ { i } = - \\frac { \\partial H } { \\partial q ^ { i } } . \\end{align*}"}
+{"id": "6305.png", "formula": "\\begin{align*} \\widetilde { K } ( x , y ) = - n \\omega _ { \\ell , \\lambda } ( x - y ) , \\forall x , y \\in \\{ z \\in \\Lambda : { \\rm d i s t } ( z , \\partial \\Lambda ) > \\lambda \\} \\end{align*}"}
+{"id": "1227.png", "formula": "\\begin{align*} \\eta = \\begin{cases} \\sqrt 2 \\left ( 1 + \\dfrac { 2 \\ , \\| D \\| _ 2 } { \\sigma _ k + \\tilde \\sigma _ k } \\right ) , & \\mbox { i f $ r = k $ } , \\\\ [ 1 e m ] \\sqrt 2 \\left ( 1 + \\dfrac { 2 \\ , \\| D \\| _ 2 } { \\sigma _ r + \\tilde \\sigma _ r } \\right ) + \\dfrac { ( 2 \\sqrt 2 + 4 ) \\ , \\| D \\| _ 2 } { \\max \\{ \\sigma _ r , \\tilde \\sigma _ r \\} } , & \\mbox { i f $ r < k $ } . \\end{cases} \\end{align*}"}
+{"id": "6333.png", "formula": "\\begin{align*} \\mathbb { H } _ { n , \\ell } = f _ { M _ 0 } \\mathbb { H } _ { n , \\ell } f _ { M _ 0 } + g _ { M _ 0 } \\mathbb { H } _ { n , \\ell } g _ { M _ 0 } + \\mathcal { E } _ { M _ 0 } , \\end{align*}"}
+{"id": "2350.png", "formula": "\\begin{align*} \\bar { \\tau } _ { p } : = \\frac { C _ { 0 } K ( c , \\gamma ) } { \\min \\{ \\frac { 1 } { 2 } , \\frac { 2 ( p - 2 ) } { 3 p } - C _ { 0 } ( p - 1 ) K ( c , \\gamma ) \\} } < 1 . \\end{align*}"}
+{"id": "4968.png", "formula": "\\begin{align*} f ( \\lambda , \\mu ) = \\frac { a ( \\lambda , \\mu ) } { b ( \\lambda , \\mu ) } , g ( \\lambda , \\mu ) = \\frac { c } { b ( \\lambda , \\mu ) } . \\end{align*}"}
+{"id": "9188.png", "formula": "\\begin{align*} q \\ell ^ - _ { 1 j } ( z ) \\ell ^ - _ { j , j + 1 } [ 0 ] = q \\ell ^ - _ { j , j + 1 } [ 0 ] \\ell ^ - _ { 1 j } ( z ) + ( 1 - q ^ 2 ) \\ell ^ - _ { j j } [ 0 ] \\ell ^ - _ { 1 , j + 1 } ( z ) . \\end{align*}"}
+{"id": "2224.png", "formula": "\\begin{align*} { 3 \\over 2 } - { 3 ( d - 2 \\varepsilon ) \\over 2 d } + 1 - \\varepsilon _ 1 = 1 + { 3 \\over d } \\varepsilon - \\varepsilon _ 1 = 1 - \\bigg ( 1 - { 3 \\over d } \\bigg ) \\varepsilon _ 1 + { 3 \\over d } \\varepsilon _ 2 . \\end{align*}"}
+{"id": "1142.png", "formula": "\\begin{align*} & 2 ^ { - ( E - \\frac { n } { 2 } ) k _ - } 2 ^ { - ( F + \\frac { n } { 2 } - \\frac { n } { a } ) k _ + } 2 ^ { - ( D - \\frac { n } { a } ) l } \\\\ & \\quad = 2 ^ { - ( E - \\frac { n } { 2 } ) k _ - } 2 ^ { - [ F + \\frac { n } { 2 } - J - ( \\frac { n } { a } - J ) ] k _ + } 2 ^ { - [ D - J - ( \\frac { n } { a } - J ) ] l } , \\end{align*}"}
+{"id": "1741.png", "formula": "\\begin{align*} \\Phi \\left ( z _ 1 , \\ldots , z _ n , \\overline { z _ 1 } , \\ldots , \\overline { z _ { n } } \\right ) : = \\sum _ { 0 < j < j + k \\leq n } S _ { j , k } x _ j x _ k x _ n ^ { n - j - k } , \\end{align*}"}
+{"id": "2344.png", "formula": "\\begin{align*} \\lim \\limits _ { i \\rightarrow \\infty } \\| a ( \\cdot , t _ { i } ) - w _ { 0 } \\| _ { L ^ { p } } = 0 . \\end{align*}"}
+{"id": "7049.png", "formula": "\\begin{align*} \\varphi = - \\nabla \\cdot ( \\frac { w } { | w | } | w | ^ { q - 1 } ) \\end{align*}"}
+{"id": "3236.png", "formula": "\\begin{align*} \\delta _ q \\hat { P } ( p ) & = \\frac { i } { ( 2 \\pi ) ^ 4 } \\ : \\big ( \\hat { \\Lambda } _ q * \\hat { P } ) \\Big ( p - \\frac { q } { 2 } \\Big ) \\\\ \\delta _ { - q } ^ * \\hat { P } ( p ) & = - \\frac { i } { ( 2 \\pi ) ^ 4 } \\ : \\big ( \\hat { \\Lambda } _ { - q } ^ * * \\hat { P } ) \\Big ( p + \\frac { q } { 2 } \\Big ) \\ : . \\end{align*}"}
+{"id": "371.png", "formula": "\\begin{align*} K _ p & \\equiv 2 \\sum _ { i = 1 } ^ { k } \\dfrac { 1 } { 2 i - 1 } \\binom { 2 k } { 2 i - 2 } b ^ { 2 ( i - 1 ) } a ^ { 2 ( k - i ) + 1 } \\\\ & - \\sum _ { j = 1 } ^ { 2 k } \\dfrac { ( - 1 ) ^ j } { j } \\binom { 2 k } { j - 1 } b ^ { 2 k - j } a ^ { j - 1 } \\pmod { p } , \\end{align*}"}
+{"id": "3819.png", "formula": "\\begin{align*} y ^ 2 = x ( x ^ 2 + a ( t ) x + b ( t ) ) . \\end{align*}"}
+{"id": "4396.png", "formula": "\\begin{align*} \\inf _ { k \\in [ m ] _ 0 } & \\left \\{ \\inf _ { \\mathcal { Q } \\subseteq [ m ] } \\left \\{ \\inf _ { x \\in \\mathcal { X _ \\mathcal { Q } } } \\left \\{ \\Gamma \\Delta u _ k ^ T l _ k ( x ) + \\sum _ { i \\in [ m ] } \\overline { u } _ i ^ T l _ i ( x ) + \\sum _ { q \\in \\mathcal { Q } } \\Delta u _ q ^ T l _ q ( x ) - \\Delta u _ k ^ T l _ k ( x ) \\right \\} \\right \\} \\right \\} , \\end{align*}"}
+{"id": "2505.png", "formula": "\\begin{align*} \\| \\nabla P \\| _ 2 \\le \\| \\nabla P ^ { i n } \\| _ 2 + \\| \\nabla A \\| _ 2 \\le c _ 1 : = \\| u ^ { i n } \\| _ 2 + \\sqrt { \\| u ^ { i n } \\| _ \\infty \\| v ^ { i n } \\| _ 1 \\| \\gamma ' \\| _ { L ^ \\infty ( 0 , V ) } } \\ , . \\end{align*}"}
+{"id": "4275.png", "formula": "\\begin{align*} \\int _ M e ^ { - f } < \\div _ f ^ { \\dagger } \\omega , h > d V = \\int _ M e ^ { - f } < \\omega , \\div _ f h > d V . \\end{align*}"}
+{"id": "2850.png", "formula": "\\begin{align*} \\lim _ { s \\to 1 ^ - } ( 1 - s ) \\| f \\| _ { \\dot { W } ^ { s , p } ( \\mathbb { R } ^ n ) } ^ p = C _ { ( p , n ) } \\| \\ , | \\nabla f | \\ , \\| _ { L ^ p ( \\mathbb { R } ^ n ) } ^ p , \\end{align*}"}
+{"id": "5074.png", "formula": "\\begin{align*} \\left ( \\mathrm { t r } _ { q ^ k } \\left ( f \\right ) * \\mathrm { t r } _ { q ^ k } \\left ( \\overline { K ( s ) } \\right ) \\right ) ( x ) & = \\frac { 1 } { q ^ k } \\sum _ { y \\in x Q _ s ( \\mathbb { F } _ { q ^ k } ) } \\psi ( \\langle x , y \\rangle ) f ( y ) . \\end{align*}"}
+{"id": "7186.png", "formula": "\\begin{align*} \\| F _ w - F _ w ^ { ( 0 ) } \\| _ Y & \\leq \\| F _ w - F _ w ^ { ( N + 1 ) } \\| _ { Y } + \\sum _ { k = 0 } ^ { N } \\left \\| F _ w ^ { ( k + 1 ) } - F _ w ^ { ( k ) } \\right \\| _ { Y } \\\\ & \\leq \\| F _ w - F _ w ^ { ( N + 1 ) } \\| _ { Y } + \\sum _ { k = 0 } ^ N \\frac { 1 } { \\gamma } \\left \\| F _ v ^ { ( k + 1 ) } - F _ v ^ { ( k ) } \\right \\| _ { Y } \\\\ & \\leq \\| F _ w - F _ w ^ { ( N + 1 ) } \\| _ { Y } + \\frac { 1 } { \\gamma } \\frac { \\alpha } { 1 - \\alpha } \\frac { C } { \\omega _ 1 } . \\end{align*}"}
+{"id": "8401.png", "formula": "\\begin{align*} \\Delta ( a \\circ b ) & = \\Delta ( B _ 1 ( a _ 1 ) b S ( B _ 2 ( a _ 2 ) ) ) = B _ 1 ( a _ 1 ) b _ 1 S ( B _ 2 ( a _ 4 ) ) \\otimes B _ 1 ( a _ 2 ) b _ 2 S ( B _ 2 ( a _ 3 ) ) \\\\ & = B _ 1 ( a _ 1 ) b _ 1 S ( B _ 2 ( a _ 2 ) ) \\otimes B _ 1 ( a _ 3 ) b _ 2 S ( B _ 2 ( a _ 4 ) ) = a _ 1 \\circ b _ 1 \\otimes a _ 2 \\circ b _ 2 \\end{align*}"}
+{"id": "4653.png", "formula": "\\begin{align*} b s ^ { 2 } - a t ^ { 2 } = n ( b - a ) . \\end{align*}"}
+{"id": "8358.png", "formula": "\\begin{align*} & \\int _ 0 ^ { \\rho _ L ( \\theta ) } h ( f ( r , \\theta ) ) G ( r , \\theta ) \\ , d r \\\\ & \\quad \\geq \\left ( \\int _ 0 ^ { \\rho _ L ( \\theta ) } G ( r , \\theta ) \\ , d r \\right ) \\cdot \\left ( ( \\alpha + 1 ) \\int _ 0 ^ { 1 } h \\Big ( f ( 0 , \\theta ) \\tau \\Big ) ( 1 - \\tau ) ^ { \\alpha } d \\tau \\right ) \\\\ & \\quad = \\beta _ { \\alpha } ( \\theta ) \\left ( \\int _ 0 ^ { \\rho _ L ( \\theta ) } G ( r , \\theta ) \\ , d r \\right ) = \\beta _ { \\alpha } ( \\theta ) \\mathbb { G } ( \\rho _ L ( \\theta ) , \\theta ) ; \\end{align*}"}
+{"id": "7814.png", "formula": "\\begin{align*} \\boldsymbol { \\rho } ^ { \\star } = P _ { \\operatorname { B o x } [ \\boldsymbol { 0 } , \\boldsymbol { 1 } ] } \\left ( \\boldsymbol { \\rho } \\left ( i + \\frac { 1 } { 2 } \\right ) - \\begin{bmatrix} \\boldsymbol { \\lambda } ^ { \\star } \\\\ \\boldsymbol { \\lambda } ^ { \\star } \\end{bmatrix} \\right ) , \\end{align*}"}
+{"id": "5141.png", "formula": "\\begin{align*} L _ { d } D 2 ( p \\| q ) = \\left \\{ \\log _ { d } A \\left ( p , q \\right ) - \\log _ { d } \\left [ X \\left ( p , q \\right ) . Y \\left ( p , q \\right ) \\right ] \\right \\} \\end{align*}"}
+{"id": "1434.png", "formula": "\\begin{align*} ( X , D ) _ n : = ( D \\cap X ) _ n \\setminus D _ n = \\coprod _ { \\substack { k + l + 1 = n \\\\ l \\ge 0 } } D _ k \\cap X _ l , \\end{align*}"}
+{"id": "2894.png", "formula": "\\begin{align*} & \\lim _ { \\lambda \\to 0 ^ + } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ { L ^ { { r } ( \\cdot ) } ( \\mathbb { R } ^ n ) } = \\left [ - \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { L ^ { { r } ( \\cdot ) } ( \\mathbb { R } ^ n ) } . \\end{align*}"}
+{"id": "2011.png", "formula": "\\begin{align*} b = \\dfrac { \\beta ^ 2 } { \\beta ^ 3 + 2 } c = \\dfrac { \\gamma ^ 2 } { \\gamma ^ 3 + 2 } \\end{align*}"}
+{"id": "7778.png", "formula": "\\begin{align*} \\tilde B _ { \\lambda } ^ { - 1 } ( \\cdot + \\alpha ) ( A _ { m _ * } ( \\lambda ) + F _ { m _ * } ( \\cdot , \\lambda ) ) \\tilde B _ \\lambda ( \\cdot ) = \\tilde A ( \\lambda ) , \\end{align*}"}
+{"id": "877.png", "formula": "\\begin{align*} L _ { 0 0 } & = L _ { 1 1 } y ^ 1 y ^ 1 + L _ { 2 2 } y ^ 2 y ^ 2 + 2 L _ { 1 2 } y ^ 1 y ^ 2 , \\\\ & = 2 { w ^ 1 } _ { \\substack { \\\\ x _ 1 } } ( y ^ 1 ) ^ 2 + 2 ( { w ^ 1 } _ { \\substack { \\\\ x _ 2 } } + { w ^ 2 } _ { \\substack { \\\\ x _ 1 } } ) y ^ 1 y ^ 2 + 2 { w ^ 2 } _ { \\substack { \\\\ x _ 2 } } ( y ^ 2 ) ^ 2 . \\end{align*}"}
+{"id": "6069.png", "formula": "\\begin{align*} \\{ e ^ { \\pm i \\arccos \\lambda } \\mid \\lambda \\in \\sigma _ { \\rm p } ( T ) \\} \\subset \\sigma _ { \\rm p } ( U ) , \\{ e ^ { \\pm i \\arccos \\lambda } \\mid \\lambda \\in \\sigma _ { \\rm e s s } ( T ) \\} = \\sigma _ { \\rm e s s } ( U ) . \\end{align*}"}
+{"id": "1978.png", "formula": "\\begin{align*} \\dfrac { \\partial _ t h _ { i i } } { N } = - \\dfrac { \\partial ^ 2 _ t g _ { i i } } { 2 N ^ 2 } - \\frac { \\partial _ t N h _ { i i } } { N ^ 2 } \\ , . \\end{align*}"}
+{"id": "6882.png", "formula": "\\begin{align*} \\mathcal { E } _ \\varphi = P _ { \\mathcal { R } ( \\theta _ \\varphi ) } \\mathcal { E } , \\mathcal { D } ( \\mathcal { E } _ \\varphi ) = \\mathcal { D } ( \\mathcal { E } ) \\cap \\mathcal { R } ( \\theta _ \\varphi ) . \\end{align*}"}
+{"id": "3361.png", "formula": "\\begin{align*} \\begin{aligned} \\begin{cases} Z ( \\sigma _ k , k ) = \\inf _ { u _ k \\in U _ K } \\mathbb { E } ^ \\dagger \\left [ Z ( \\Sigma ^ * ( u _ k , y _ { k + 1 } ) \\sigma _ k , k + 1 ) \\right ] , \\\\ Z ( \\sigma _ K , K ) = \\int _ { \\mathbb { R } ^ n } \\sigma _ K ( z ) \\left ( \\mu \\Phi ( z ) \\right ) d z . \\end{cases} \\end{aligned} \\end{align*}"}
+{"id": "7738.png", "formula": "\\begin{align*} S ^ { - 1 } ( \\lambda ) A ( \\lambda ) S ( \\lambda ) = \\mathrm { d i a g } \\{ A _ { 1 1 } ( \\lambda ) , \\cdots , A _ { l l } ( \\lambda ) \\} , \\end{align*}"}
+{"id": "7114.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta u + \\lambda u - \\gamma w u = f ( u ) \\ & \\ \\mathbb { R } ^ { 2 } , \\\\ - \\Delta w = u ^ { 2 } \\ & \\ \\mathbb { R } ^ { 2 } . \\end{array} \\right . \\end{align*}"}
+{"id": "5195.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } D _ { \\alpha \\beta } ( p \\| q ) } { \\partial q _ { j } } = & - \\left [ \\left ( \\underbrace { \\frac { a - 1 } { a - b } A ^ { a - 2 } - \\frac { b - 1 } { a - b } A ^ { b - 2 } } _ { Z _ { A } > 0 } \\right ) \\frac { \\partial A } { \\partial q _ { j } } \\right ] \\\\ & + \\left [ \\left ( \\underbrace { \\frac { a - 1 } { a - b } B ^ { a - 2 } - \\frac { b - 1 } { a - b } B ^ { b - 2 } } _ { Z _ { B } > 0 } \\right ) \\frac { \\partial B } { \\partial q _ { j } } \\right ] \\end{align*}"}
+{"id": "5359.png", "formula": "\\begin{align*} v _ i ^ S + \\sum _ { j \\in S } c ^ { S } _ j \\ , x _ { i j } ^ { 0 , u } = v _ i ^ u + \\sum _ { j \\in N ^ { \\{ 0 , 1 \\} } \\setminus S } c ^ { S } _ j \\ , x _ { i j } ^ { 1 , u } . \\end{align*}"}
+{"id": "3067.png", "formula": "\\begin{align*} \\begin{array} { c c c l } \\varphi ^ * : & \\Omega ^ 1 & \\rightarrow & \\mathbb { C } \\{ t \\} \\\\ & \\omega = A d x + B d y & \\mapsto & \\varphi ^ * ( A ) x ' ( t ) + \\varphi ^ * ( B ) y ' ( t ) \\end{array} \\end{align*}"}
+{"id": "6057.png", "formula": "\\begin{align*} M = S T = ( x _ i S ) \\frac { T } { x _ i } \\in V ^ 2 , \\end{align*}"}
+{"id": "3512.png", "formula": "\\begin{align*} & ( \\psi _ m \\circ \\varphi _ { m - 1 } \\circ \\psi _ { m - 1 } \\circ \\hdots \\varphi _ { n + 1 } \\circ \\psi _ { n + 1 } ) ( \\varphi _ n ( x ) ) \\\\ & = \\psi _ m ( \\varphi _ n ( x ) ) \\\\ & = \\psi _ m ( \\varphi _ m ( \\varphi _ { m , n } ( x ) ) ) \\\\ & = \\varphi _ { m , n } ( x ) . \\end{align*}"}
+{"id": "4482.png", "formula": "\\begin{align*} G ( t ) = \\int _ { \\{ 2 \\psi < - t \\} } | F _ 0 | ^ 2 \\tilde \\rho \\end{align*}"}
+{"id": "6510.png", "formula": "\\begin{align*} \\begin{cases} \\Delta u ( x ) = - 1 & x \\in \\Omega , \\\\ u ( x ) = 0 & x \\in \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "2626.png", "formula": "\\begin{align*} ( s _ { j + 3 } - s _ { j + 2 } ) - ( s _ { j + 1 } - s _ j ) & \\leq ( s _ { j ' + 1 } - s _ { j ' } ) - ( s _ { j + 1 } - s _ j ) \\\\ & = ( s _ { i + 3 } - s _ { i + 1 } ) - ( s _ { i + 2 } - s _ { i } ) \\\\ & = ( s _ { i + 3 } - s _ { i + 2 } ) - ( s _ { i + 1 } - s _ { i } ) . \\end{align*}"}
+{"id": "3010.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\mathcal { L } _ { v } g + \\mathrm { R i c } + \\lambda g + \\mu \\widetilde { g } + \\nu \\eta \\otimes \\eta = 0 . \\end{align*}"}
+{"id": "127.png", "formula": "\\begin{align*} \\vec { u } _ { \\lambda } = \\frac { \\lambda } { \\lambda + 1 } \\vec { u } _ { \\infty } + \\frac { 1 } { \\lambda + 1 } \\vec { u } _ { 0 } . \\end{align*}"}
+{"id": "6932.png", "formula": "\\begin{align*} \\forall \\tau \\in \\C , \\varphi ( \\tau ) : = - \\frac { \\tau } { \\alpha } + ( - 1 ) ^ { \\mu + 1 } \\frac { \\beta } { \\alpha ^ { 2 \\mu + 1 } } \\tau ^ { 2 \\mu } \\end{align*}"}
+{"id": "4307.png", "formula": "\\begin{align*} d J _ t & = \\sum _ { j = 1 } ^ d \\nabla V _ j ( y _ t ) J _ t d w _ t ^ j + \\nabla V _ 0 ( y _ t ) J _ t d t , J _ 0 = \\mathrm { I d } _ e , \\\\ d K _ t & = - \\sum _ { j = 1 } ^ d K _ t \\nabla V _ j ( y _ t ) d w _ t ^ j - K _ t \\nabla V _ 0 ( y _ t ) d t , K _ 0 = \\mathrm { I d } _ e . \\end{align*}"}
+{"id": "4590.png", "formula": "\\begin{align*} h = & \\left . \\left ( - T _ 2 + \\frac { 1 } { T _ 0 } T _ 1 ^ 2 \\right ) \\right / \\left ( - T _ s + s _ 0 T _ 1 \\right ) . \\end{align*}"}
+{"id": "6814.png", "formula": "\\begin{align*} \\int _ { S U ( N - 1 ) } f _ { S U ( N - 1 ) } & ( g _ { S U ( N - 1 ) } ) _ { i j } ^ { \\beta _ { i j } } h _ { S U ( N - 1 ) } ( g _ { S U ( N - 1 ) } ) \\ , d g _ { S U ( N - 1 ) } \\\\ & = \\int _ { S U ( N - 1 ) } f _ { S U ( N - 1 ) } ( u ' g _ { S U ( N - 1 ) } ) _ { i j } ^ { \\beta _ { i j } } h _ { S U ( N - 1 ) } ( u ' g _ { S U ( N - 1 ) } ) \\ , d g _ { S U ( N - 1 ) } \\\\ & = \\int _ { S U ( N - 1 ) } f _ { S U ( N - 1 ) } ( g _ { S U ( N - 1 ) } u ' ) _ { i j } ^ { \\beta _ { i j } } h _ { S U ( N - 1 ) } ( g _ { S U ( N - 1 ) } u ' ) \\ , d g _ { S U ( N - 1 ) } \\end{align*}"}
+{"id": "1138.png", "formula": "\\begin{align*} \\vec t _ { Q } : = \\begin{cases} | Q | ^ { \\frac { s } { n } + \\frac 1 2 } \\vec e & \\ell ( Q ) \\leq 1 | x _ Q | < 1 , \\\\ \\vec { \\mathbf { 0 } } & . \\end{cases} \\end{align*}"}
+{"id": "6540.png", "formula": "\\begin{align*} S _ { N , 1 } = \\sum ^ N _ { n = 1 } \\sum ^ { \\infty } _ { j = m _ 0 + 1 } \\Big ( \\mathbb { E } \\big [ K ( X _ n ) | \\mathcal { F } _ { n - j } \\big ] - \\mathbb { E } \\big [ K ( X _ n ) | \\mathcal { F } _ { n - j - 1 } \\big ] \\Big ) . \\end{align*}"}
+{"id": "1061.png", "formula": "\\begin{align*} L : = L ^ 2 ( ( 0 , \\infty ) , d x ) \\end{align*}"}
+{"id": "2230.png", "formula": "\\begin{align*} v _ { r , r } + v _ { z , z } + { v _ r \\over r } = 0 . \\end{align*}"}
+{"id": "8031.png", "formula": "\\begin{align*} J _ { d y n } ( W ) = \\frac { 1 } { 2 } \\mathcal { B } _ { 1 0 } \\left ( ( \\tilde { F } _ W , \\tilde { G } _ W , \\tilde { H } _ W ) , ( \\tilde { F } _ W , \\tilde { G } _ W , \\tilde { H } _ W ) \\right ) = \\mathcal { J } _ 1 ( W , \\tilde { F } _ W , \\tilde { G } _ W , \\tilde { H } _ W ) \\end{align*}"}
+{"id": "8578.png", "formula": "\\begin{align*} \\tilde { C } _ t ^ { t _ 1 } = ( J _ k + [ - \\varepsilon _ k ( \\tilde { G } ^ { t _ 0 } _ t ) \\tilde { B } _ { t _ 0 } ] ^ { k \\cdot } _ + ) \\tilde { C } ^ { t _ 0 } _ t . \\end{align*}"}
+{"id": "762.png", "formula": "\\begin{align*} C ^ h _ { i j } \\varphi _ h - \\dot { \\Phi } _ k g _ { i j } y ^ k = 0 . \\end{align*}"}
+{"id": "7046.png", "formula": "\\begin{align*} \\frac { 1 } { q } \\partial _ t \\langle h w _ \\eta ^ q \\rangle _ { x , \\eta } & + \\frac { 4 ( q - 1 ) } { q ^ 2 } \\langle h | \\nabla w ^ { \\frac { q } { 2 } } _ \\eta | ^ 2 \\rangle _ { x , \\eta } \\\\ & - \\frac { 2 } { q } \\langle b \\cdot \\nabla w _ \\eta ^ { \\frac { q } { 2 } } , h w _ \\eta ^ { \\frac { q } { 2 } } \\rangle _ { x , \\eta } - \\sum _ { i = 1 } ^ d \\langle ( \\eta \\cdot \\nabla b _ i ) \\partial _ { \\eta _ i } w _ \\eta , h w _ \\eta ^ { q - 1 } \\rangle _ { x , \\eta } = 0 , \\end{align*}"}
+{"id": "4380.png", "formula": "\\begin{gather*} \\mathcal { M } : = \\{ ( k _ 1 , k _ 2 , k _ 3 , k _ 4 ) \\in [ n ] ^ 4 : \\ k _ 1 < k _ 2 , k _ 3 < k _ 4 \\} \\cup \\{ ( 0 , 0 , 0 , 0 ) \\} . \\end{gather*}"}
+{"id": "6453.png", "formula": "\\begin{align*} \\langle R ^ N ( R ^ N ( V , d \\phi ( e _ j ) ) \\tau ( \\phi ) , \\tau ( \\phi ) ) d \\phi ( e _ j ) , V \\rangle = | R ^ N ( V , d \\phi ( e _ j ) ) \\tau ( \\phi ) | ^ 2 . \\end{align*}"}
+{"id": "3222.png", "formula": "\\begin{align*} \\mathfrak { h } ( X / B ) = \\bigoplus _ n \\mathfrak { h } ^ n ( X / B ) \\ , \\end{align*}"}
+{"id": "5005.png", "formula": "\\begin{align*} p _ k ( x ) = x ^ { k - 1 } , k = 1 , \\ldots , N , \\end{align*}"}
+{"id": "3674.png", "formula": "\\begin{align*} a _ d ( T ) \\le a _ d ( T \\setminus v ) + 1 = 1 , \\end{align*}"}
+{"id": "8528.png", "formula": "\\begin{align*} \\kappa _ { \\Sigma } ( g ) & = g ( e _ 1 ) - e _ 1 \\\\ & = \\chi _ p ( g ) ( 1 + \\eta \\log ( g ) ) \\cdot e _ 1 + \\eta c ( g ) \\cdot e _ 2 - e _ 1 \\\\ & = ( \\phi ( g ) + \\log ( g ) ) \\cdot \\eta e _ 1 + c ( g ) \\cdot \\eta e _ 2 \\end{align*}"}
+{"id": "1011.png", "formula": "\\begin{align*} \\| u - P _ D g \\| ^ 2 _ { F ( D ) } = \\langle \\mu , u - P _ D g \\rangle \\le \\| u - P _ D g \\| _ { F ( D ) } \\| \\mu \\| _ { F ^ * ( D ) } . \\end{align*}"}
+{"id": "2870.png", "formula": "\\begin{align*} \\rho ( t ) : = \\sup _ { x \\in \\mathbb { R } ^ n } \\sup _ { | h | \\leq t } \\left | \\nabla f ( x + h ) - \\nabla f ( x ) \\right | . \\end{align*}"}
+{"id": "7854.png", "formula": "\\begin{align*} f ( \\mathbf { x } ) & = \\sum _ { \\mathbf { b } \\in \\mathbb { F } _ { 3 } ^ { 2 } } \\lambda _ { \\mathbf { b } } ( \\prod _ { i = 1 } ^ { m } x _ { i } ^ { b _ { i } } ) \\\\ & = 1 + 2 x _ 1 x _ 2 + x _ 1 ^ 2 , \\end{align*}"}
+{"id": "7752.png", "formula": "\\begin{align*} A Y - Y A = H \\in \\mathcal B _ \\delta ^ 1 , \\end{align*}"}
+{"id": "3528.png", "formula": "\\begin{align*} ( T f ) ( z ) = f ( e ^ { i \\theta } z ) , \\end{align*}"}
+{"id": "1031.png", "formula": "\\begin{align*} C _ { t , n } \\le \\begin{cases} M n ^ { - d } t ^ { - d } , & n \\in \\N , ~ ~ t \\in \\{ 1 , \\dots , [ \\delta n ] \\} , \\\\ M n ^ { ( 1 / 2 ) - d } ( n + 1 - t ) ^ { - ( 1 / 2 ) - d } , & n \\in \\N , ~ ~ t \\in \\{ [ \\delta n ] + 1 , \\dots , n \\} . \\end{cases} \\end{align*}"}
+{"id": "7061.png", "formula": "\\begin{align*} ( \\nabla _ { r } a _ { i j } ) _ { i = 1 } ^ d \\in \\mathbf { F } _ { \\delta _ { r j } } \\end{align*}"}
+{"id": "4448.png", "formula": "\\begin{align*} \\ll g , g _ z \\gg _ { \\partial M , \\rho } = g ( z ) \\end{align*}"}
+{"id": "1535.png", "formula": "\\begin{align*} I \\ge \\frac { 2 \\ , \\sqrt { \\lambda _ 1 \\ , \\lambda _ N } } { \\lambda _ 1 + \\lambda _ N } = \\frac { 2 \\ , \\sqrt { \\lambda _ N / \\lambda _ 1 } } { \\lambda _ N / \\lambda _ 1 + 1 } \\ge \\frac { 2 \\ , \\sqrt { H } } { H + 1 } \\end{align*}"}
+{"id": "692.png", "formula": "\\begin{align*} a _ 0 = & \\frac { 1 } { 2 } ( \\alpha _ 1 - \\alpha _ 2 ) , a _ 1 = \\beta _ 1 , a _ 2 = \\frac { 1 } { 2 } ( \\alpha _ 1 + \\alpha _ 2 ) , a _ 3 = \\beta _ 2 , \\\\ b _ 0 = & \\frac { 1 } { 2 } ( \\alpha _ 3 - \\alpha _ 4 ) , b _ 1 = \\beta _ 3 , b _ 2 = \\frac { 1 } { 2 } ( \\alpha _ 3 + \\alpha _ 4 ) , b _ 3 = \\beta _ 4 . \\end{align*}"}
+{"id": "7249.png", "formula": "\\begin{align*} \\alpha _ t : = \\tau + ( 1 - q ) t , \\beta _ t : = \\eta \\alpha _ t , \\gamma _ t : = \\theta - \\eta t . \\end{align*}"}
+{"id": "8101.png", "formula": "\\begin{align*} \\int _ \\R e ^ { - z t } \\int _ { N _ 1 ^ + } p _ j ( t ) & \\phi ( n ) f _ \\varpi ( g _ { t + j T _ 0 } n x ) \\ ; d \\mu _ x ^ u ( n ) d t \\\\ & = \\sum _ { w \\in \\Z } e ^ { - z w } \\int _ \\R e ^ { - z t } \\int _ { N _ 1 ^ + } p _ { j , w } ( t ) \\phi ( n ) f _ \\varpi ( g _ { t + w + j T _ 0 } n x ) \\ ; d \\mu _ x ^ u ( n ) d t . \\end{align*}"}
+{"id": "8648.png", "formula": "\\begin{align*} \\partial \\left ( F - \\langle \\cdot , y _ 0 ^ * \\rangle + 1 _ K \\right ) = \\partial F - \\{ y _ 0 ^ * \\} + \\partial 1 _ K . \\end{align*}"}
+{"id": "7159.png", "formula": "\\begin{align*} \\partial _ t u + \\mathfrak { L } u = F ( u , t , \\omega , x ) , x \\in \\R , t \\geq 0 , \\end{align*}"}
+{"id": "1956.png", "formula": "\\begin{align*} \\langle I ( x ) \\rangle = & \\{ f ( x ) I ( x ) | f ( x ) \\in { \\mathbb { F } _ { q } [ x ] } / { \\langle x ^ n - 1 \\rangle } \\} \\\\ = & \\{ f ( x ) I ( x ) | f ( x ) \\in \\mathbb { F } _ { q } [ x ] , f ( x ) = 0 0 \\leq d e g ( f ( x ) ) < n - d e g ( I ( x ) ) \\} , \\end{align*}"}
+{"id": "4764.png", "formula": "\\begin{align*} \\varphi _ 2 ( \\varepsilon , b , n , \\omega , \\varphi ) : = \\psi ( \\varepsilon ^ 2 \\min \\{ \\varphi _ 1 ( \\varepsilon / 2 , b , n , \\varphi ) , \\lambda _ 0 / 2 \\} / 4 , b + 2 n + 3 n ^ 2 , \\omega ) . \\end{align*}"}
+{"id": "7746.png", "formula": "\\begin{align*} \\mathcal T _ { a N } F _ { i i } ^ { ( r e ) } ( \\theta ) = \\hat { F } _ { i i } ^ { ( r e ) } ( 0 ) , \\end{align*}"}
+{"id": "280.png", "formula": "\\begin{align*} e _ 1 e _ i = e _ { i + 1 } . \\end{align*}"}
+{"id": "1926.png", "formula": "\\begin{align*} \\Delta h ( x ) & = \\frac { \\hat C } { \\mu _ c } \\sum _ { y \\in G } \\omega _ 0 ( x , y ) [ \\mathbf { d } ^ { - \\beta } ( y ) - \\mathbf { d } ^ { - \\beta } ( x ) ] \\\\ & = - \\beta \\frac { \\hat C } { \\mu _ c } \\sum _ { y \\in G } \\omega _ 0 ( x , y ) \\xi ^ { - \\beta - 1 } [ \\mathbf { d } ( y ) - \\mathbf { d } ( x ) ] , \\end{align*}"}
+{"id": "6021.png", "formula": "\\begin{align*} \\begin{aligned} j ^ A _ { x , x + 1 } = & \\xi ^ { A } _ { x } - \\xi ^ { A } _ { x + 1 } + \\frac { E _ { A } - E _ { B } } { 2 N ^ \\gamma } ( \\xi ^ { A } _ { x } \\xi ^ { B } _ { x + 1 } + \\xi ^ { B } _ { x } \\xi ^ { A } _ { x + 1 } ) \\\\ & + \\frac { E _ { A } - E _ { C } } { 2 N ^ \\gamma } ( \\xi ^ { A } _ { x + 1 } + \\xi ^ { A } _ { x } - 2 \\xi ^ { A } _ { x } \\xi ^ { A } _ { x + 1 } - \\xi ^ { A } _ { x } \\xi ^ { B } _ { x + 1 } - \\xi ^ { B } _ { x } \\xi ^ { A } _ { x + 1 } ) \\ , , \\end{aligned} \\end{align*}"}
+{"id": "4137.png", "formula": "\\begin{align*} u = ( w _ { 1 j } , \\ldots , w _ { j - 1 , j } , 1 , 0 , \\ldots , 0 ) ' \\in \\R ^ J , \\end{align*}"}
+{"id": "6292.png", "formula": "\\begin{align*} T = \\tilde { O } \\left ( \\left [ \\frac { 2 ^ \\frac { r ^ 2 + 1 } { r } \\sigma _ q } { \\mu _ r ^ { 1 / r } } \\cdot \\frac { 1 } { \\varepsilon ^ { \\frac { ( r - 1 ) } { r } } } \\right ] ^ \\frac { 1 + \\kappa } { \\kappa } \\right ) , T _ k = \\tilde { O } \\left ( \\left [ \\frac { \\sigma _ q 2 ^ { ( 1 + r ) } } { \\mu _ r R _ 0 ^ { r - 1 } } 2 ^ { k ( r - 1 ) } \\right ] ^ \\frac { 1 + \\kappa } { \\kappa } \\right ) . \\end{align*}"}
+{"id": "7714.png", "formula": "\\begin{align*} 0 \\geq \\nabla _ { i i } \\widetilde { E } = b ^ { 1 1 } \\nabla _ { i i } b _ { 1 1 } - ( b ^ { 1 1 } ) ^ { 2 } ( \\nabla _ { i } b _ { 1 1 } ) ^ { 2 } - \\tilde { d } \\left ( \\frac { h _ { i i } } { h } - \\frac { h ^ { 2 } _ { i } } { h ^ { 2 } } \\right ) + 2 \\tilde { l } \\left [ \\sum _ { j } h _ { j } h _ { j i i } + h ^ { 2 } _ { i i } \\right ] . \\end{align*}"}
+{"id": "2699.png", "formula": "\\begin{align*} Q ( t ) = \\left \\{ \\begin{array} { l l } q _ { 0 } & ( t < t _ { 0 } ) \\\\ \\frac { \\alpha } { \\beta } ( t - t _ { 0 } ) + q _ { 0 } & ( t > t _ { 0 } ) \\end{array} \\right . \\end{align*}"}
+{"id": "1022.png", "formula": "\\begin{align*} p _ t ( \\delta _ 1 , \\ldots , \\delta _ m ; \\gamma ) = \\sum _ { \\beta \\in \\Gamma ^ t ( n ' ) } p _ t ( \\delta _ 1 , \\ldots , \\delta _ m ; \\beta ) \\ , p _ t ( \\beta ; \\gamma ) . \\end{align*}"}
+{"id": "8234.png", "formula": "\\begin{align*} \\sum _ { { \\bf x } _ { i + 1 } } P ( { \\bf x } _ { i + 1 } \\otimes { \\bf s } = 1 | Y ^ i = 0 ^ i ) = \\frac { 1 } { 2 } \\end{align*}"}
+{"id": "7195.png", "formula": "\\begin{align*} \\frac { \\omega _ 1 } { 2 \\pi } \\int _ { t - \\pi / \\omega _ 1 } ^ { t + \\pi / \\omega _ 1 } v _ 0 J _ 0 ^ 2 ( s ) d s = v _ 0 \\left ( \\frac { A ( x ) ^ 2 } { 2 } + \\frac { B ( x ) ^ 2 } { 2 } + A ( x ) B ( x ) \\cos ( \\eta t ) \\right ) + O \\left ( \\frac { \\left ( | A ( x ) | ^ 2 + | B ( x ) | ^ 2 \\right ) \\eta } { \\omega _ 1 } \\right ) . \\end{align*}"}
+{"id": "7602.png", "formula": "\\begin{align*} \\Delta u _ { \\epsilon } = \\Delta U _ { \\epsilon } \\varphi + U _ { \\epsilon } \\Delta \\varphi + 2 \\nabla U _ { \\epsilon } \\cdot \\nabla \\varphi . \\end{align*}"}
+{"id": "7430.png", "formula": "\\begin{align*} F _ { \\rm L o r e n t z } = q \\dot { x } \\wedge H . \\end{align*}"}
+{"id": "902.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c c c c } 0 & 0 & \\ldots & 0 & a _ { _ { n - 1 } } \\\\ a _ { _ 0 } & 0 & \\ldots & 0 & 0 \\\\ 0 & a _ { _ 1 } & \\ddots & \\vdots & \\vdots \\\\ \\vdots & \\ddots & \\ddots & 0 & \\vdots \\\\ 0 & \\ldots & 0 & a _ { _ { n - 2 } } & 0 \\end{array} \\right ) \\end{align*}"}
+{"id": "401.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { E } _ { S } & = \\{ ( m _ x , m _ y ) \\in \\mathbb { Z } ^ { 2 } | ( \\frac { m _ x } { L _ { S , x } } ) ^ 2 + ( \\frac { m _ y } { L _ { S , y } } ) ^ 2 \\le 1 \\} , \\\\ \\mathcal { E } _ { R } & = \\{ ( l _ x , l _ y ) \\in \\mathbb { Z } ^ { 2 } | ( \\frac { l _ x } { L _ { r R x } } ) ^ 2 + ( \\frac { l _ y } { L _ { R , y } } ) ^ 2 \\le 1 \\} . \\end{aligned} \\end{align*}"}
+{"id": "3786.png", "formula": "\\begin{align*} \\mathbb { P } \\left \\{ f ^ { ( i ) } _ t V ^ { ( i ) } _ t \\leq \\bar { t } \\right \\} = \\mathbb { P } \\left \\{ \\frac { V ^ { ( i ) } _ t } { N _ i n _ x } - 1 \\leq \\frac { - 4 \\alpha \\| \\Sigma ^ { ( i ) } _ t \\| } { ( 1 + 2 \\alpha ) ^ 2 \\rho ^ { ( i ) } \\| \\Sigma ^ { ( i ) } _ t \\| + 4 \\sqrt { n _ x } } \\right \\} , \\end{align*}"}
+{"id": "4186.png", "formula": "\\begin{align*} \\varphi ( g _ 2 g _ 3 ) = \\varphi ( g _ 2 ) \\varphi ( g _ 3 ) \\ , . \\end{align*}"}
+{"id": "670.png", "formula": "\\begin{align*} \\left ( R ^ { 0 } | _ { S ^ { 1 } } f \\right ) _ { p , q } = \\alpha ( p , q ) \\frac { \\partial f _ { p } } { \\partial x ^ { q } } = \\frac { 1 } { 2 } \\left ( \\frac { \\partial f _ { p } } { \\partial x ^ { q } } - \\frac { \\partial f _ { q } } { \\partial x ^ { p } } \\right ) \\equiv \\frac { 1 } { \\sqrt { 2 } } \\left ( { \\rm c u r l } \\ , ( f ) \\right ) _ { p , q } \\end{align*}"}
+{"id": "1677.png", "formula": "\\begin{align*} g \\cdot v _ 4 = x _ 0 + \\omega ( e ' _ r , v _ 4 ) \\ , e _ r + \\omega ( e ' _ { r - 1 } , v _ 4 ) \\ , e _ { r - 1 } + \\omega ( f ' _ { r - 1 } , v _ 4 ) \\ , f _ { r - 1 } + \\omega ( f ' _ r , v _ 4 ) \\ , f _ r . \\\\ \\end{align*}"}
+{"id": "9008.png", "formula": "\\begin{align*} a * b = a \\cdot _ 1 b + a \\cdot _ 2 b \\end{align*}"}
+{"id": "5560.png", "formula": "\\begin{align*} J _ \\Delta B = B ^ * J _ \\Delta . \\end{align*}"}
+{"id": "5459.png", "formula": "\\begin{align*} L _ + : = \\begin{bmatrix} L _ 1 & C ( \\alpha , \\beta ) A \\\\ C ( \\alpha , \\beta ) A & L _ 3 \\end{bmatrix} , L _ - : = \\begin{bmatrix} L _ 2 & C ( \\alpha , \\beta ) B \\\\ C ( \\alpha , \\beta ) B & L _ 4 \\end{bmatrix} \\end{align*}"}
+{"id": "1457.png", "formula": "\\begin{align*} X = X ( \\mathrm { S U } ( h ) , K ) : = \\mathrm { S U } ( h ) ( K ) \\times \\mathbb { A } / \\backsim , \\end{align*}"}
+{"id": "4913.png", "formula": "\\begin{align*} a + b = g + 2 0 < a \\leq b \\leq 2 a . \\end{align*}"}
+{"id": "8521.png", "formula": "\\begin{align*} \\det ( M ) & \\sim \\det ( a _ { n _ a } ( F _ b ) ) _ { 1 \\leq a , b \\leq g } ^ { - 1 } \\cdot \\prod _ { b = 1 } ^ g \\Omega _ { F _ b } ^ + \\cdot \\det ( a _ { n _ a } ( F _ b ) ) _ { 1 \\leq a , b \\leq g } \\\\ & \\sim \\prod _ { b = 1 } ^ g \\Omega _ { F _ b } ^ + \\end{align*}"}
+{"id": "8357.png", "formula": "\\begin{align*} \\psi ( y ) = \\psi ( a ) + \\int _ a ^ y \\psi ' ( s ) \\ , d s . \\end{align*}"}
+{"id": "8776.png", "formula": "\\begin{align*} & \\{ ( ( x _ - , y _ - ) , ( x _ + , y _ + ) ) \\in \\{ \\R ^ 2 \\} ^ 2 : y _ + < y _ - < x _ - < x _ + \\} \\cap \\hat \\Gamma ^ 2 = \\emptyset \\\\ \\mbox { a n d } & \\{ ( ( x _ - , z _ - ) , ( x _ + , z _ + ) ) \\in \\{ \\R ^ 2 \\} ^ 2 : x _ - < x _ + < z _ + < z _ - \\} \\cap \\hat \\Gamma ^ 2 = \\emptyset . \\end{align*}"}
+{"id": "7326.png", "formula": "\\begin{align*} \\lim _ { \\beta \\downarrow 0 } \\left ( F _ { m , r } ( s , \\beta ) - \\frac { 1 } { m \\left ( s + 2 \\sin ^ { 2 } \\left ( \\frac { \\beta } { m } \\pi \\right ) \\right ) } \\right ) = \\frac { U _ { m - \\ell - 1 } ( s + 1 ) + U _ { \\ell - 1 } ( s + 1 ) } { T _ { m } ( s + 1 ) - 1 } - \\frac { 1 } { m s } = F _ { m , r } ( s ) . \\end{align*}"}
+{"id": "8373.png", "formula": "\\begin{align*} R ^ { + } ( w ^ { - 1 } ) : = \\{ \\beta \\in R ^ { + } : w ^ { - 1 } ( \\beta ) < 0 \\} . \\end{align*}"}
+{"id": "8108.png", "formula": "\\begin{align*} \\norm { \\mu } _ q : = \\left ( \\sum _ { \\l \\in \\L _ k } \\mu ( \\l ) ^ q \\right ) ^ { 1 / q } . \\end{align*}"}
+{"id": "4738.png", "formula": "\\begin{align*} & \\forall x ^ X , y ^ X \\Big ( j x x = _ \\mathbb { R } \\norm { x } _ X ^ 2 \\land \\vert j x y \\vert \\leq _ \\mathbb { R } \\norm { x } _ X \\norm { y } _ X \\\\ & \\qquad \\qquad \\land \\forall \\alpha ^ { \\mathbb { N } ^ \\mathbb { N } } , \\beta ^ { \\mathbb { N } ^ \\mathbb { N } } , u ^ X , v ^ X \\left ( j x ( \\alpha u + _ X \\beta v ) = _ \\mathbb { R } \\alpha j x u + _ \\mathbb { R } \\beta j x v \\right ) \\Big ) , \\end{align*}"}
+{"id": "4814.png", "formula": "\\begin{align*} \\underline { \\mathbf { s } } = \\frac { 1 } { K } \\sum _ { k = 1 } ^ K \\mathbf { l } ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } } \\leq \\mathbf { s } \\leq \\overline { \\mathbf { s } } = \\frac { 1 } { K } \\sum _ { k = 1 } ^ K \\mathbf { u } ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } } . \\end{align*}"}
+{"id": "128.png", "formula": "\\begin{align*} \\widehat { \\vec { u } } _ { \\lambda } = 2 | \\xi | ^ { - 2 } \\left ( I - \\frac { 2 \\lambda + 1 } { 2 ( \\lambda + 1 ) } \\Pi _ \\xi \\right ) \\widehat { \\vec { f } } , \\widehat { \\vec { u } } _ { 0 } = 2 | \\xi | ^ { - 2 } \\left ( I - \\frac { 1 } { 2 } \\Pi _ \\xi \\right ) \\widehat { \\vec { f } } . \\end{align*}"}
+{"id": "7275.png", "formula": "\\begin{align*} \\widetilde { Q } _ { n } ( y ; x , t , s ) = \\tfrac { 1 } { u ^ n } \\overline { w } _ n \\left ( u y + w ; a , b , c , d \\right ) , \\end{align*}"}
+{"id": "5611.png", "formula": "\\begin{align*} k \\leq \\lfloor \\log n \\rfloor , s = \\left \\lfloor \\frac { \\log \\left ( \\frac { n } { 8 ( d \\vee K ) ^ 3 K ^ 9 } \\right ) } { 8 \\log \\log n } \\right \\rfloor \\end{align*}"}
+{"id": "3813.png", "formula": "\\begin{align*} \\Big | \\bigcap _ { j \\in [ t ] . } A _ { j } \\Big | = n ^ { \\prime } - \\Big | \\bigcup _ { j \\in [ t ] } I ' _ j \\Big | = 0 . \\end{align*}"}
+{"id": "3411.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\underline { \\delta } _ R ( \\# _ n K ) } { n } = \\delta ( K ) \\lim _ { n \\to \\infty } \\frac { \\bar { \\delta } _ R ( \\# _ n K ) } { n } = \\delta ( K ) . \\end{align*}"}
+{"id": "3146.png", "formula": "\\begin{align*} u ^ - _ { \\psi ^ * } ( s ) = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ v _ { 2 , \\ , 1 } ( s ) & 1 & 0 \\\\ v _ { 3 , \\ , 1 } ( s ) & v _ { 3 , \\ , 2 } ( s ) & 1 \\end{array} \\right ) ( \\ v _ { 2 , \\ , 1 } ( S ) , v _ { 3 , \\ , 1 } ( S ) , v _ { 3 , \\ , 2 } ( S ) \\in k [ S ] \\ , ) . \\end{align*}"}
+{"id": "1373.png", "formula": "\\begin{align*} P ^ { n } _ { 0 } : = & \\left \\{ \\emptyset \\right \\} , \\\\ P ^ { n } _ { i } : = & \\Big \\{ [ n ] \\mbox { \\ o f c a r d i n a l i t y $ i $ } \\Big \\} , 1 \\leqslant i \\leqslant n , \\\\ P ( [ n ] ) : = & \\bigsqcup _ { i = 0 } ^ { n } P _ { i } ^ { n } . \\end{align*}"}
+{"id": "9026.png", "formula": "\\begin{align*} \\frac { | G | } { | \\Phi | } \\left ( \\sum _ { i = 1 } ^ { k _ 1 } \\frac { 1 } { \\lambda _ { i } } + \\sum _ { i = 1 } ^ { k _ 2 } \\frac { \\nu _ i } { \\mu _ i } + \\sum _ { i = 1 } ^ { k _ 3 } \\frac { 1 } { \\omega _ i } - 1 \\right ) = \\sum _ { i = 1 } ^ { k _ 3 } \\frac { | G | } { | \\Phi | } \\frac { 1 } { q _ i } - 1 . \\end{align*}"}
+{"id": "1417.png", "formula": "\\begin{align*} t ^ { \\gamma + \\theta / \\alpha } \\{ I _ + ^ { \\beta - \\alpha \\gamma } f _ \\alpha ( \\cdot \\vert t ) \\} ( x ) = t ^ { ( \\beta + \\theta - 1 ) / \\alpha } \\{ I _ + ^ { \\beta - \\alpha \\gamma } f _ \\alpha \\} ( x t ^ { - 1 / \\alpha } ) \\end{align*}"}
+{"id": "4932.png", "formula": "\\begin{align*} \\begin{gathered} e ^ { S } _ { \\pm } = \\pm \\frac { 1 } { 4 \\sqrt { \\beta } } [ S ] + \\frac { 1 } { 8 \\beta } [ S ] ^ 2 , \\\\ e ^ { S _ j } _ { \\pm } = \\pm \\frac { 1 } { 4 \\sqrt { \\beta } } [ S _ j ] + \\frac { 1 } { 8 \\beta } [ S _ j ] ^ 2 . \\end{gathered} \\end{align*}"}
+{"id": "3960.png", "formula": "\\begin{align*} \\iota ^ { \\odot n } ( U ) \\pi _ n ( x _ 1 , \\ldots , x _ n ) & = \\pi _ n ( \\alpha ( \\gamma _ 1 ) x _ 1 , \\ldots , \\alpha ( \\gamma _ n ) x _ n ) \\\\ & = \\pi _ n ( \\alpha ^ n ( g ) ( x _ 1 , \\ldots , x _ n ) ) \\\\ & = T \\pi _ n ( x _ 1 , \\ldots , x _ n ) . \\end{align*}"}
+{"id": "5799.png", "formula": "\\begin{align*} X ^ 2 _ 0 ( t ) = & \\sum _ { k + \\ell \\leq s } \\sum _ { 1 \\leq j \\leq J } | x ^ { ( k , \\ell ) } _ j ( t ) | ^ 2 , \\ X ^ 2 _ \\pm ( t ) = \\sum _ { k + \\ell \\leq s } \\sum _ { i : \\pm \\lambda _ i > 0 } | \\xi ^ { ( k , \\ell ) } _ i ( t ) | ^ 2 . \\end{align*}"}
+{"id": "5725.png", "formula": "\\begin{align*} & \\frac { d } { d t } \\xi _ { i , 1 } - 2 ^ { - 1 } m \\xi _ { i , 1 } + \\beta _ i \\xi _ { i , 2 } = \\mathcal { E } _ { i , 1 } , \\\\ & \\frac { d } { d t } \\xi _ { i , 2 } - 2 ^ { - 1 } m \\xi _ { i , 2 } - \\beta _ i \\xi _ { i , 1 } = \\mathcal { E } _ { i , 2 } . \\end{align*}"}
+{"id": "1778.png", "formula": "\\begin{align*} P ( z , \\zeta ) + f \\left ( \\Re ( z _ 1 ) \\right ) + \\Re \\left ( h ( F + i v , z , \\zeta ) \\right ) = P ( z ^ * , \\zeta ^ * ) + f ^ * \\left ( \\Re ( z _ 1 ^ * ) \\right ) \\ , \\ , \\forall \\ , ( w , z , \\zeta ) \\in M _ f \\end{align*}"}
+{"id": "3946.png", "formula": "\\begin{align*} \\mathcal V [ y ] : = \\frac { 1 } { 2 } ( \\bar { y } ^ N ) ^ \\top \\bar P \\bar { y } ^ N + \\frac { \\rho } { 2 } \\sum _ { n = N + 1 } ^ { + \\infty } \\vert y _ n \\vert ^ 2 , \\end{align*}"}
+{"id": "101.png", "formula": "\\begin{align*} G : = p - 1 + s - \\gamma \\quad E : = s + 1 + \\frac { p - 1 } { n - 1 } . \\end{align*}"}
+{"id": "7399.png", "formula": "\\begin{align*} J _ + ( - x , t ) = J _ - ( x , t ) . \\end{align*}"}
+{"id": "4625.png", "formula": "\\begin{align*} | f ( x ' ) - f ( y ' ) | & = \\left \\lvert \\sum _ { j = 0 } ^ \\infty 2 ^ { - \\gamma m j } g _ j ( x ' ) - \\sum _ { j = 0 } ^ \\infty 2 ^ { - \\gamma m j } g _ j ( y ' ) \\right \\rvert \\\\ & \\leq \\sum _ { j = 0 } ^ { n ' } 2 ^ { - \\gamma m j } | g _ j ( x ' ) - g _ j ( y ' ) | + \\sum _ { j = n ' + 1 } ^ \\infty 2 ^ { - \\gamma m j } | g _ j ( x ' ) - g _ j ( y ' ) | \\end{align*}"}
+{"id": "924.png", "formula": "\\begin{align*} u ( x ) & = \\int _ { \\Omega } g ( \\omega _ { \\tau _ D ( \\omega ) } ) \\ , d P _ x ( \\omega ) \\\\ & \\quad + \\int _ \\Omega \\int _ 0 ^ { \\tau _ D ( \\omega ) } f ( \\omega _ t , u ( \\omega _ t ) ) \\ , d t \\ , d P _ x ( \\omega ) + \\int _ \\Omega A ^ \\mu _ { \\tau _ D ( \\omega ) } ( \\omega ) \\ , d P _ x ( \\omega ) , \\end{align*}"}
+{"id": "5011.png", "formula": "\\begin{align*} ( 1 - \\beta ) \\mathbf { x } _ i ^ { 0 , \\tau } + \\mathbf { x } _ i ^ { 1 , \\tau } ( \\mathbf { I } - \\beta \\mathbf { P } ) = \\mathbf { e } _ i . \\end{align*}"}
+{"id": "3778.png", "formula": "\\begin{align*} \\mathbb { P } \\left \\{ \\mathcal { M } _ i ^ { j , j ^ { \\prime } } \\right \\} \\leq c _ 1 \\sum _ { t = 0 } ^ { T - 1 } \\exp \\left ( - c _ 2 N _ i n _ x \\left ( \\frac { \\alpha \\rho ^ { ( i ) } \\| \\Sigma ^ { ( i ) } _ t \\| } { \\rho ^ { ( i ) } \\| \\Sigma ^ { ( i ) } _ t \\| + \\sqrt { n _ x } } \\right ) ^ 2 \\right ) , \\end{align*}"}
+{"id": "8189.png", "formula": "\\begin{align*} & _ { B _ 1 , B _ 2 ^ 0 , B _ 2 ^ 1 } \\ f ( B _ 1 , B _ 2 ^ 0 , B _ 2 ^ 1 ) \\\\ & B _ 1 \\leq B _ , \\ B _ 2 ^ 1 \\leq B _ , \\ B _ 2 ^ 0 \\leq B _ , \\\\ & \\ \\ 0 \\leq B _ 1 \\leq M , \\ 0 \\leq B _ 2 ^ 1 \\leq B _ 1 , \\ 0 \\leq B _ 2 ^ 0 \\leq M - B _ 1 . \\end{align*}"}
+{"id": "9066.png", "formula": "\\begin{align*} \\mu ( F _ 2 ) = 2 ( b + b _ { 2 1 } ) . \\end{align*}"}
+{"id": "5084.png", "formula": "\\begin{align*} \\sum _ { t \\in T _ { s } ( \\mathbb { F } _ { q ^ k } ) } \\theta ( t ) & = \\begin{cases} q ^ k - 1 & s \\in W _ { \\theta } ^ \\circ \\\\ 0 & s \\not \\in W _ { \\theta } ^ \\circ . \\end{cases} \\end{align*}"}
+{"id": "5960.png", "formula": "\\begin{align*} \\left \\vert \\left \\vert a - b \\right \\vert - \\left \\vert a - c \\right \\vert \\right \\vert & \\ , \\ , = \\ , \\ , \\left \\vert \\left \\vert \\left \\vert b - a \\right \\vert - a \\right \\vert - c \\right \\vert \\\\ & \\ , \\ , = \\ , \\ , \\left \\vert \\left \\vert b - \\left \\vert a - a \\right \\vert \\right \\vert - c \\right \\vert \\\\ & \\ , \\ , = \\ , \\ , \\left \\vert \\left \\vert b - 0 \\right \\vert - c \\right \\vert \\\\ & \\ , \\ , = \\ , \\ , \\left \\vert b - c \\right \\vert \\end{align*}"}
+{"id": "8481.png", "formula": "\\begin{align*} \\iint _ { \\Omega _ \\infty } \\partial _ t w _ h \\cdot \\varphi \\ , d x d t = - \\iint _ { \\Omega _ \\infty } w _ h \\ , \\partial _ t \\varphi \\ , d x d t \\end{align*}"}
+{"id": "3260.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { p - 1 } \\frac { ( \\frac { d - 1 } { d } ) _ k ^ d } { k ! ^ d } \\equiv - \\Gamma _ p \\big ( \\tfrac 1 d \\big ) ^ d \\pmod { p ^ 2 } , \\end{align*}"}
+{"id": "4116.png", "formula": "\\begin{align*} & \\gamma _ i ( \\theta _ i , r ) = \\theta _ i + \\frac { 1 } { 2 } \\gamma _ i ^ * ( \\theta _ i ) + o ( r ^ { J } \\theta _ i ) + o ( \\theta _ i ^ 2 ) , i \\in \\mathcal { E } \\\\ & \\xi _ i ( \\theta , r ) = \\bar { \\xi } _ i ( \\theta ) + \\frac { 1 } { 2 } \\xi _ i ^ * ( \\theta ) + o ( r ^ { J } \\abs { \\theta } ) + o ( \\abs { \\theta } ^ 2 ) i \\in \\mathcal { J } . \\end{align*}"}
+{"id": "8096.png", "formula": "\\begin{align*} \\psi ( n ) = \\psi ( n ^ { - 1 } ) , \\psi ( m n m ^ { - 1 } ) = \\psi ( n ) , \\forall n \\in N ^ + , m \\in M . \\end{align*}"}
+{"id": "6239.png", "formula": "\\begin{align*} \\inf _ { [ 0 , \\gamma ] } \\delta ( f , \\alpha ) - \\sup _ { [ \\gamma , 1 ] } \\delta ( f , \\beta ) = \\frac 2 3 \\omega \\left [ p \\left ( \\gamma - \\frac 2 3 \\right ) - q \\right ] . \\end{align*}"}
+{"id": "7991.png", "formula": "\\begin{align*} \\varepsilon ^ 2 \\Delta _ z v = 4 | z | ^ 2 ( | v | ^ 2 - 1 ) v , \\end{align*}"}
+{"id": "1167.png", "formula": "\\begin{align*} \\left \\| \\vec f \\right \\| _ { \\dot A ^ { s , \\tau } _ { p , q } ( W , \\mathbb { R } ^ n ) } & \\lesssim \\left \\| \\vec u \\right \\| _ { \\dot a ^ { s , \\tau } _ { p , q } ( W , \\mathbb { R } ^ n ) } \\sim \\left \\| \\vec u \\right \\| _ { \\dot a ^ { s , \\tau } _ { p , q } ( \\mathbb { A } ( W ) , \\mathbb { R } ^ n ) } \\\\ & = [ \\ell ( Q ( I , k _ 0 ) ) ] ^ { - n \\tau - s + n ( \\frac { 1 } { p } - \\frac { 1 } { 2 } ) } \\left | A _ { Q ( I , k _ 0 ) , W } \\vec u _ { Q ( I , k _ 0 ) } \\right | = \\left | A _ { Q ( I , k _ 0 ) , W } \\vec z \\right | , \\end{align*}"}
+{"id": "8697.png", "formula": "\\begin{align*} & \\mbox { $ a ( x , y , t ) \\sim \\sum ^ { + \\infty } _ { j = 0 } a _ j ( x , y ) t ^ { n - 1 - \\frac { d } { 2 } - j } $ i n $ S ^ { n - 1 - \\frac { d } { 2 } } _ { 1 , 0 } ( D _ 0 \\times D _ 0 \\times \\mathbb R _ + ) $ } , \\\\ & \\mbox { $ a _ j ( x , y ) \\in C ^ \\infty ( D _ 0 \\times D _ 0 ) $ , $ j = 0 , 1 , \\ldots $ } , \\\\ & \\mbox { $ a _ 0 ( x , x ) $ i s g i v e n b y ~ \\cite [ T h e o r e m 1 . 6 ] { H H } } . \\end{align*}"}
+{"id": "5413.png", "formula": "\\begin{align*} T ^ { - n } \\circ \\mathcal I = \\mathcal I \\circ T ^ n , \\end{align*}"}
+{"id": "759.png", "formula": "\\begin{align*} \\overset { \\ _ * } R ^ h _ { j k } = \\kappa F ( l _ k \\delta ^ h _ j - l _ j \\delta ^ h _ k ) , \\end{align*}"}
+{"id": "1581.png", "formula": "\\begin{align*} | \\bar x | = \\sup _ { \\{ G = k \\} } | z | | \\bar y | = \\inf _ { \\{ G = k \\} } | z | . \\end{align*}"}
+{"id": "5371.png", "formula": "\\begin{align*} \\bar { x } ^ { a , u } _ j = \\lim _ { T \\to \\infty } \\ , \\frac { 1 } { T } \\ , E _ i ^ u \\left [ \\sum _ { t = 0 } ^ T 1 \\{ X ( t ) = j , a ( t ) = a \\} \\right ] = \\lim _ { \\beta \\nearrow 1 } \\ , ( 1 - \\beta ) \\ , x _ { i j } ^ { a , u } ( \\beta ) . \\end{align*}"}
+{"id": "5942.png", "formula": "\\begin{align*} \\dfrac { M _ { 2 } ( \\mathcal { C } ( G ) ) } { | e ( \\mathcal { C } ( G ) ) | } = \\dfrac { 2 ^ { 6 k } - 6 \\cdot 2 ^ { 5 k } + 1 4 \\cdot 2 ^ { 4 k } - 9 \\cdot 2 ^ { 3 k } - 1 5 \\cdot 2 ^ { 2 k } + 1 5 \\cdot 2 ^ { k } + 8 } { 2 ^ { 4 k } - 2 \\cdot 2 ^ { 3 k } - 2 \\cdot 2 ^ { 2 k } + 3 \\cdot 2 ^ { k } + 2 } . \\end{align*}"}
+{"id": "4280.png", "formula": "\\begin{align*} \\Delta _ f \\omega = ( 2 \\lambda - \\frac { 1 } { 2 } ) \\omega . \\end{align*}"}
+{"id": "1512.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ N ( D \\partial _ \\alpha u ) \\ , \\partial _ \\alpha u = D \\frac { | D u | ^ 2 } { 2 } \\end{align*}"}
+{"id": "4852.png", "formula": "\\begin{align*} g _ w ( s ) & = \\int _ { \\partial ( A - s w ) \\cap A } \\displaystyle \\langle \\nu _ { A } ( y + s w ) , w \\rangle d \\mathcal { H } ^ { n - 1 } ( y ) \\\\ & = \\int _ { \\partial ( A - s w ) \\cap A + ( s - t ) w } \\displaystyle \\langle \\nu _ { A } ( x + t w ) , w \\rangle d \\mathcal { H } ^ { n - 1 } ( x ) \\\\ & \\leq g _ w ( t ) , \\end{align*}"}
+{"id": "167.png", "formula": "\\begin{align*} R \\cap R ^ { - 1 } \\ = \\ 1 _ X , R \\cup R ^ { - 1 } \\ = \\ X \\times X . \\end{align*}"}
+{"id": "2673.png", "formula": "\\begin{align*} f _ { i } + \\frac { \\partial W } { \\partial \\dot { q } ^ { i } } = 0 \\end{align*}"}
+{"id": "8079.png", "formula": "\\begin{align*} \\norm { ( u , s ) } ' : = \\left ( \\norm { u } ^ 4 + | s | ^ 2 \\right ) ^ { 1 / 4 } . \\end{align*}"}
+{"id": "7680.png", "formula": "\\begin{align*} C _ { \\gamma } ( n - m ) : = \\sum ^ { \\infty } _ { N = 0 } \\gamma ^ N \\# S _ { N } ( n , m ) \\end{align*}"}
+{"id": "3195.png", "formula": "\\begin{align*} \\{ | x _ 1 | = | x _ 2 | \\leq 1 \\} \\cup \\{ | x _ 1 | \\geq | x _ 2 | = 1 \\} \\cup \\{ | x _ 2 | \\geq | x _ 1 | = 1 \\} \\end{align*}"}
+{"id": "6862.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathbb { J } } c _ j \\varphi _ j = 0 \\iff \\{ c _ j \\} \\in \\ker \\theta _ \\varphi ^ * = \\ell _ 2 ( \\mathbb { J } ) \\ominus { \\mathcal R } ( \\theta _ \\varphi ) = { \\mathcal R } ( \\theta _ \\psi ) . \\end{align*}"}
+{"id": "3152.png", "formula": "\\begin{align*} \\varphi ^ \\sharp \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) = \\left ( \\begin{array} { c c c } a ^ { 2 \\ , p ^ { e _ 1 } } & a ^ { p ^ { e _ 1 } } \\ , b ^ { p ^ { e _ 1 } } & b ^ { 2 \\ , p ^ { e _ 1 } } \\\\ 0 & 1 & 0 \\\\ 0 & 0 & d ^ { 2 \\ , p ^ { e _ 1 } } \\end{array} \\right ) ( \\ , e _ 1 \\geq 0 \\ , ) . \\end{align*}"}
+{"id": "7270.png", "formula": "\\begin{align*} x _ k = - \\tfrac { 1 } { 1 - q } \\left ( \\eta s q ^ k + \\tfrac { \\eta \\theta + 1 - q } { \\eta q ^ k } - ( \\theta + \\eta s ) \\right ) , \\end{align*}"}
+{"id": "6196.png", "formula": "\\begin{align*} \\begin{array} { l l l } S & : = & - S _ 1 ( J ) / ( n - 2 ) + S _ 3 ( J ) / { n \\choose 3 } \\\\ \\\\ & \\leq & - S _ 1 ( J ) ( 1 / ( n - 2 ) - S _ 2 ( J ) { n + 1 \\choose 3 } / ( n + 1 ) { n \\choose 3 } { n + 1 \\choose 2 } ) ~ . \\end{array} \\end{align*}"}
+{"id": "5506.png", "formula": "\\begin{align*} T _ 4 & { } : = X _ 4 , & T _ 3 & { } : = X _ 3 , & T _ 2 & { } : = X _ 2 - \\frac { q ^ 4 } { ( q ^ 2 + 1 ) ( q + q ^ { - 1 } ) } X _ 3 ^ 2 X _ 4 ^ { - 1 } , & T _ 1 & { } : = X _ 1 - \\frac { q ^ 2 ( q + q ^ { - 1 } ) } { q ^ 2 - 1 } X _ 2 X _ 3 ^ { - 1 } . \\end{align*}"}
+{"id": "1793.png", "formula": "\\begin{align*} \\hat { f } ^ { ( 4 ) } = { f } ^ { ( 4 ) } \\quad \\mbox { a n d } ( \\hat { f } ^ * ) ^ { ( 4 ) } = ( { f } ^ * ) ^ { ( 4 ) } , \\end{align*}"}
+{"id": "937.png", "formula": "\\begin{align*} F ( V ) = \\{ u \\in F : u = 0 \\mbox { q . e . o n } V ^ c : = E \\setminus V \\} . \\end{align*}"}
+{"id": "3997.png", "formula": "\\begin{align*} \\mathcal { O } ^ { \\ : n , \\nu } = \\left \\{ \\left ( x _ { 1 } , \\ldots , x _ { n } , y _ { 1 } , \\ldots , y _ { \\nu } \\right ) \\in \\mathbb { R } ^ { n + \\nu } : x _ { 1 } = \\cdots = x _ { n } = 0 \\mbox { o r } y _ { 1 } = \\cdots = y _ { \\nu } = 0 \\right \\} . \\end{align*}"}
+{"id": "5089.png", "formula": "\\begin{align*} ( L _ s \\cdot \\varepsilon _ { \\theta } ) ( t u s ) & = 0 - q ^ { - k } \\theta ( t ) , \\end{align*}"}
+{"id": "6620.png", "formula": "\\begin{align*} J _ n & = \\frac { 1 } { 3 } ( 2 ^ n - ( - 1 ) ^ n ) \\\\ J _ { n + 1 } + J _ n & = 2 ^ n \\\\ J _ { n + 1 } - 2 J _ n & = ( - 1 ) ^ n \\\\ J _ m ( J _ { n + 1 } + 2 J _ { n - 1 } ) + J _ n ( J _ { m + 1 } + 2 J _ { m - 1 } ) & = 2 J _ { m + n } \\end{align*}"}
+{"id": "6485.png", "formula": "\\begin{align*} b _ { u , v } = | \\{ \\} | - | \\{ \\} | . \\end{align*}"}
+{"id": "5238.png", "formula": "\\begin{align*} D A H I ( p \\| q ) = \\sum _ { j } p _ { j } \\left ( \\overline { M A } - \\overline { M H } \\right ) = \\sum _ { j } p _ { j } \\left ( 1 - \\overline { M H } \\right ) \\end{align*}"}
+{"id": "6656.png", "formula": "\\begin{align*} \\liminf _ { n \\rightarrow \\infty } \\nu _ { n } ( E , \\mathrm { i } ( y + 2 \\epsilon ) ) \\geq \\frac { 1 } { \\epsilon } ( L ( E , \\mathrm { i } ( y + 2 \\epsilon ) ) - L ( E , \\mathrm { i } ( y + \\epsilon ) ) ) = \\omega ^ { + } ( E , \\mathrm { i } y ) . \\end{align*}"}
+{"id": "1178.png", "formula": "\\begin{align*} \\begin{aligned} \\left | \\partial _ x ^ \\alpha \\partial _ y ^ \\beta \\mathcal K ( x , y ) - \\partial _ x ^ \\alpha \\partial _ y ^ \\beta \\mathcal K ( x , y + v ) \\right | \\leq C | v | ^ { ( F - | \\alpha | ) ^ { * * } } | x - y | ^ { - n - | \\alpha | - ( F - | \\alpha | ) } , \\end{aligned} \\end{align*}"}
+{"id": "3799.png", "formula": "\\begin{align*} \\| \\bar { Z } \\bar { Z } ^ \\top \\| \\leq \\frac { 9 } { 4 } \\sum _ { i \\in \\widehat { \\mathcal { C } } _ 1 \\cap \\overline { \\mathcal { C } _ 1 } } \\sum _ { t = 0 } ^ { T - 1 } N _ i \\| \\Sigma ^ { ( i ) } _ t \\| . \\end{align*}"}
+{"id": "8172.png", "formula": "\\begin{align*} H ( Y _ j | y ^ { j - 1 } \\in \\beta _ k ^ { j - 1 } , \\underline S ) = 1 , \\forall k \\leq j - 1 . \\end{align*}"}
+{"id": "5357.png", "formula": "\\begin{align*} c ^ S _ j = 0 , j \\in N ^ { \\{ 1 \\} } . \\end{align*}"}
+{"id": "173.png", "formula": "\\begin{align*} q ( x ) _ { i } \\ = \\ g _ { i + 1 } ( \\pi _ { i + 1 } ( x ) ) , \\ \\ \\ \\ i \\in \\Z _ + \\end{align*}"}
+{"id": "3638.png", "formula": "\\begin{align*} \\alpha ^ 3 \\tau _ 0 ^ { 3 } f ( \\tau , w ) - ( 1 + a ) \\alpha ^ 2 \\tau _ 0 ^ { 2 } f ( \\tau , w \\sigma ^ 2 ) + ( a + b ) \\alpha \\tau _ 0 f ( \\tau , w \\sigma ) - b f ( \\tau , w ) = 0 . \\end{align*}"}
+{"id": "4173.png", "formula": "\\begin{align*} \\varphi \\big ( g ^ { j } _ 1 g _ 3 \\big ) \\varphi \\big ( g ^ { i } _ 1 g _ 2 \\big ) = \\varphi ( ( g ^ { n } _ 1 g _ 3 ) g _ 2 ) \\mbox { o r } \\varphi \\big ( g ^ { j } _ 1 g _ 3 \\big ) \\varphi \\big ( g ^ { i } _ 1 g _ 2 \\big ) = \\varphi ( g _ 2 ( g ^ { n } _ 1 g _ 3 ) ) \\ , . \\end{align*}"}
+{"id": "6202.png", "formula": "\\begin{align*} \\phi ( - \\infty ) = 1 \\hbox { a n d } \\phi ( \\infty ) = 0 . \\end{align*}"}
+{"id": "6912.png", "formula": "\\begin{align*} v \\ ( x \\ ) : = c _ 1 + c _ 2 \\left | x - x _ 0 \\right | ^ { \\frac { p } { p - 1 } } \\quad \\forall x \\in \\R ^ n \\end{align*}"}
+{"id": "7992.png", "formula": "\\begin{align*} 0 & = \\int _ { N \\mathcal { B } ( 1 ) } \\nabla u \\cdot \\nabla \\varphi + \\hat \\varepsilon ^ { - 2 } ( | u | ^ 2 - 1 ) u \\cdot \\varphi \\\\ & = \\int _ { \\Omega } \\Big ( g ^ { i j } \\partial _ { y _ i } \\tilde u \\partial _ { y _ j } \\tilde \\varphi + \\hat \\varepsilon ^ { - 2 } ( | \\tilde u | ^ 2 - 1 ) \\tilde u \\tilde \\varphi \\Big ) \\sqrt { | \\det ( g ) | } \\ , d y . \\end{align*}"}
+{"id": "1591.png", "formula": "\\begin{align*} G ( y ) \\le C \\ , | y | \\ , \\inf _ { | z | = 1 } \\eta _ H \\left ( \\frac { | y | } { | D F ( z ) | } \\right ) . \\end{align*}"}
+{"id": "4400.png", "formula": "\\begin{align*} F _ { \\mathcal { Y } , a , b } ( x ) = \\sum _ { ( i , j ) \\in [ m ] \\times [ n ] } \\overline { y } _ { ( i , j ) } g _ { ( i , j ) } ( x ) + \\sum _ { ( q , p ) \\in \\mathcal { Y } } \\Delta y _ { ( q , p ) } g _ { ( q , p ) } ( x ) = F _ { \\mathcal { Y } , 0 , 0 } ( x ) . \\end{align*}"}
+{"id": "5347.png", "formula": "\\begin{align*} \\widehat { v } _ i ( \\nu ) = \\min \\ , \\left \\{ \\sum _ { j \\in N ^ { \\{ 0 , 1 \\} } } \\widehat { h } _ j ^ 0 \\ , x _ { i j } ^ { 0 , u } + \\nu \\ , b _ i ^ u : u \\in \\mathcal { U } \\right \\} , \\end{align*}"}
+{"id": "1274.png", "formula": "\\begin{align*} \\mu ( t ) : = e ^ { t C } \\mu ( 0 ) , \\Sigma ( t ) : = e ^ { t C } \\Sigma ( 0 ) e ^ { t C ^ T } + 2 \\int _ 0 ^ t e ^ { s C } D e ^ { s C ^ T } \\ , d s . \\end{align*}"}
+{"id": "3176.png", "formula": "\\begin{align*} R _ y ( u ) = D _ \\eta N ( y , u ) - N ( y , N ( y , u ) ) + N ( y , [ y , u ] ) - [ y , N ( y , u ) ] . \\end{align*}"}
+{"id": "7790.png", "formula": "\\begin{align*} \\begin{aligned} \\dot \\varphi _ t ( \\tau ) \\leq - \\mu \\varphi _ t ( \\tau ) + L \\psi _ t ( \\tau ) , \\end{aligned} \\end{align*}"}
+{"id": "5567.png", "formula": "\\begin{align*} ( \\Phi ^ t ) _ { x y } & = [ ( Q Q ^ * ) ^ t ] _ { x y } = \\sum _ { k } \\mu _ k ^ { 2 t } v _ k ( x ) v _ k ( y ) \\\\ & \\leq \\rho ^ { 2 t - 2 } \\sum _ { k } \\mu _ k ^ { 2 } | v _ k ( x ) | | v _ k ( y ) | \\\\ & \\leq \\rho ^ { 2 t - 2 } \\sqrt { ( \\Phi ) _ { x x } } \\sqrt { ( \\Phi ) _ { y y } } \\leq \\frac { K ^ 2 \\rho ^ { 2 t } } { n } , \\end{align*}"}
+{"id": "2222.png", "formula": "\\begin{align*} 1 - \\theta = { 3 \\over 2 } - { 3 - s \\over q } > 0 . \\end{align*}"}
+{"id": "771.png", "formula": "\\begin{align*} \\Delta ( h \\# a ) = ( h _ { ( 1 ) } \\# { a _ { ( 1 ) } } ^ { ( 1 ) } ) \\otimes ( h _ { ( 2 ) } { a _ { ( 1 ) } } ^ { ( 2 ) } \\# a _ { ( 2 ) } ) , \\end{align*}"}
+{"id": "8631.png", "formula": "\\begin{align*} v _ 1 ( T _ { 3 , n } ; x ) = m ( T _ { 3 , n } ) ~ ~ { \\rm a n d } ~ ~ m ( t ) \\leq v _ 1 ( t ; x ) \\leq \\frac { 1 } { ( 1 + \\epsilon ) ^ { 1 / ( 2 + ( n - 1 ) \\sigma ) } } m ( t ) \\end{align*}"}
+{"id": "1249.png", "formula": "\\begin{align*} b _ { n , k } = q ( 1 - q ^ n ) \\frac { ( - 1 ) ^ { k + 1 } } { 1 - q ^ { 2 k + 1 } } . \\end{align*}"}
+{"id": "2042.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & - C _ 1 & C _ 2 \\end{bmatrix} ^ t \\ , \\ , = \\ , \\ , \\bigl [ \\ , 0 \\ , , \\ , - c _ { 0 0 1 } , - c _ { 0 1 0 } , - c _ { 1 0 0 } , \\ , \\ , c _ { 0 0 2 } , \\ , c _ { 0 1 1 } , \\ , c _ { 0 2 0 } , \\ , c _ { 1 0 1 } , \\ , c _ { 1 1 0 } , \\ , c _ { 2 0 0 } \\ , \\bigr ] ^ t . \\end{align*}"}
+{"id": "8368.png", "formula": "\\begin{align*} \\dim G \\times _ B X _ { \\underline { w } } & = \\dim G + \\dim X _ { \\underline { w } } - \\dim B \\\\ & = \\dim U ^ - + \\dim X _ { \\underline { w } } . \\end{align*}"}
+{"id": "433.png", "formula": "\\begin{align*} | F _ { 2 j , k } ^ { ( c ) } | = | \\sum _ { p = 1 , p \\neq k } ^ { 2 j - 1 } [ \\bold { F } ] _ { j , k } F _ { 2 j , k } ^ { j , p } + [ \\bold { F } ] _ { j , 2 j } F _ { 2 j , k } ^ { j , 2 j } + [ \\bold { F } ] _ { j , j } F _ { 2 j , k } ^ { j , j } | \\le \\| \\bold { F } ^ { ( i ) } \\| _ 2 ( \\sum _ { p = 1 , p \\neq k } ^ { 2 j - 1 } | F _ { 2 j , k } ^ { j , p } | ^ 2 ) ^ { \\frac { 1 } { 2 } } \\frac { j ^ { \\frac { 3 } { 2 } } } { M ^ 2 } , \\end{align*}"}
+{"id": "5915.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { C } ( A ( n , \\nu ) ) ) = ( 2 ^ { n } - 1 ) 2 ^ { n } ( 2 ^ { n } - 1 ) ^ { 2 } = 2 ^ { n } ( 2 ^ { n } - 1 ) ^ { 3 } \\end{align*}"}
+{"id": "8325.png", "formula": "\\begin{align*} [ \\Pi , W ] _ { S N } + \\frac { 1 } { 2 } [ W , W ] _ { \\gamma } { - } \\frac { 1 } { 6 } ( W ^ \\sharp \\wedge W ^ \\sharp \\wedge W ^ \\sharp ) \\Upsilon ^ G _ { T M } = 0 . \\end{align*}"}
+{"id": "239.png", "formula": "\\begin{align*} \\bar { v } = \\frac { d \\bar { x } } { d \\bar { t } } = \\frac { 1 } { h } \\frac { d \\varphi } { d x } \\frac { d x } { d t } = \\frac { 1 } { h } \\frac { d \\varphi } { d x } v , \\end{align*}"}
+{"id": "6400.png", "formula": "\\begin{align*} r _ n ( \\theta _ 0 ) v _ n \\to \\overline { v } = \\begin{pmatrix} \\overline { v } _ { 1 1 } & \\overline { v } _ { 1 2 } \\\\ \\overline { v } _ { 2 1 } & \\overline { v } _ { 2 2 } \\\\ \\end{pmatrix} \\ ; \\ ; \\det ( \\overline { v } ) > 0 \\ ; \\ ; r _ n ( \\theta ) = \\begin{pmatrix} \\frac { 1 } { \\delta } & \\frac { \\ln ( n ) } { \\alpha ^ 2 } \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "6568.png", "formula": "\\begin{align*} \\phi _ { \\Phi _ k , \\beta } ( x , y ) = \\begin{cases} | \\{ n \\leq x ; \\ \\ \\frac { \\phi ( n ) } { n ^ { \\delta } } \\leq y \\} | & k , \\\\ | \\{ n \\leq x ; \\ \\ \\frac { \\phi ( n ) \\alpha _ { k } ( n ) } { n ^ { \\delta } } \\leq y \\} | & k . \\end{cases} \\end{align*}"}
+{"id": "6699.png", "formula": "\\begin{align*} N = U \\times T \\trianglelefteq _ \\mathrm { o } G \\dd ( G ) = \\dd ( G / U ) . \\end{align*}"}
+{"id": "6401.png", "formula": "\\begin{align*} I ( \\theta ) = \\begin{pmatrix} I ^ { 1 1 } ( \\theta ) & \\\\ I ^ { 2 1 } ( \\theta ) & I ^ { 2 2 } ( \\theta ) \\end{pmatrix} \\end{align*}"}
+{"id": "269.png", "formula": "\\begin{align*} \\ddot { x } + \\frac { 1 } { x } \\dot { x } ^ 2 + x \\ , \\dot { x } + \\frac { 1 } { 2 } = 0 , \\end{align*}"}
+{"id": "2498.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } t } \\| v ( t ) \\| _ 1 + \\int _ 0 ^ t \\| ( u v ) ( s ) \\| _ 1 \\ \\mathrm { d } s = \\| v ^ { i n } \\| _ 1 \\ , , t \\ge 0 \\ , , \\end{align*}"}
+{"id": "8338.png", "formula": "\\begin{align*} \\mu _ k ^ B ( \\alpha _ 1 , \\ldots , \\alpha _ k ) : = P \\{ \\{ \\ldots \\{ \\Theta _ B , \\alpha _ 1 \\} , \\ldots , \\} , \\alpha _ k \\} . \\end{align*}"}
+{"id": "4825.png", "formula": "\\begin{align*} \\begin{gathered} \\alpha _ a = \\frac { m _ a ^ 2 ( 1 - m _ a ) } { \\sigma ^ 2 _ a } - m _ a , \\beta _ a = \\alpha _ a ( \\frac { 1 } { m _ a } - 1 ) , \\end{gathered} \\end{align*}"}
+{"id": "6609.png", "formula": "\\begin{align*} \\frac { x } { 2 \\pi i } \\int _ { b - i \\tau } ^ { b + i \\tau } R _ { k , \\beta } ( z ) \\frac { ( \\alpha x ^ \\delta ) ^ { z } } { z ( 1 - z + \\delta z ) } d z = R _ { k , \\beta } \\left ( \\frac { 1 } { ( 1 - \\delta ) } \\right ) y ^ { \\frac { 1 } { ( 1 - \\delta ) } } + E _ 2 , \\end{align*}"}
+{"id": "363.png", "formula": "\\begin{align*} ( u \\pm v ) ^ p = u ^ p \\pm \\binom { p } { 1 } u ^ { p - 1 } v + \\binom { p } { 2 } u ^ { p - 2 } v ^ 2 \\pm \\cdots + \\binom { p } { p - 1 } u v ^ { p - 1 } \\pm v ^ p , \\end{align*}"}
+{"id": "2451.png", "formula": "\\begin{align*} \\begin{cases} U ( \\xi ) \\sim A _ { 3 } ( \\xi _ { + } - \\xi ) , \\\\ U ' ( \\xi ) \\sim - A _ { 4 } \\end{cases} { \\rm { a s } } \\xi \\nearrow \\xi _ { + } - 0 , \\end{align*}"}
+{"id": "1021.png", "formula": "\\begin{align*} p _ t ^ d ( \\delta ; \\gamma ) = p _ t ( \\delta ; \\gamma ) \\ , ( 1 + o _ n ( 1 ) ) . \\end{align*}"}
+{"id": "8324.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Pi _ { t } = d _ { \\Pi _ { t } } Y _ { t } . \\end{align*}"}
+{"id": "7766.png", "formula": "\\begin{align*} | g _ j ( \\lambda , u ) - g _ j ( \\lambda ' , u ) | \\leq | \\frac { \\partial } { \\partial \\lambda } g _ j ( \\lambda , u ) | _ { \\tilde \\delta _ { j + 1 } } | \\lambda - \\lambda ' | < \\frac { ( 2 \\tilde R _ j ) ^ { m ^ 2 } } { \\tilde \\delta _ j / 2 } \\tilde \\delta _ { j + 1 } = \\frac { 1 } { 2 K _ j } , \\end{align*}"}
+{"id": "91.png", "formula": "\\begin{align*} \\begin{cases} w _ 1 = p - \\gamma , & w _ 2 = 2 , \\\\ w _ 3 = 4 - p + \\gamma , & w _ 4 = 2 , \\end{cases} \\end{align*}"}
+{"id": "6600.png", "formula": "\\begin{align*} I = & R _ { k , \\beta } ( z ) \\frac { x ^ { 1 - z + \\delta z } } { 1 - z + \\delta z } + \\frac { 1 } { 2 \\pi i } \\left \\{ \\int _ { a - i T } ^ { c - i T } + \\int _ { c - i T } ^ { c + i T } + \\int _ { c + i T } ^ { a + i T } \\right \\} F _ { k , \\beta } ( s , z ) \\frac { x ^ s } { s } \\ d s \\\\ = & R _ { k , \\beta } ( z ) \\frac { x ^ { 1 - z + \\delta z } } { 1 - z + \\delta z } + J _ 1 + J _ 2 + J _ 3 , \\end{align*}"}
+{"id": "1877.png", "formula": "\\begin{align*} \\lim _ { q \\to - 1 ^ + } \\frac { 1 } { \\Gamma ( q + 1 ) } \\int _ 0 ^ \\infty r ^ { q } \\phi ( r ) d r = \\phi ( 0 ) , \\end{align*}"}
+{"id": "2290.png", "formula": "\\begin{align*} w _ { i j k } = 0 \\qquad x _ { i j k \\ell } = 0 , \\qquad \\forall 1 \\leq i < j < k < \\ell \\leq n \\ . \\end{align*}"}
+{"id": "6036.png", "formula": "\\begin{align*} \\begin{aligned} j ( \\rho ) & = \\begin{pmatrix} - \\nabla \\rho ^ { A } + \\frac { 1 } { N ^ \\gamma } \\Big ( \\rho ^ { A } ( 1 - \\rho ^ { A } ) ( E _ { A } - E _ { C } ) - \\rho ^ { A } \\rho ^ { B } ( E _ { B } - E _ { C } ) \\Big ) \\\\ - \\nabla \\rho ^ { B } + \\frac { 1 } { N ^ \\gamma } \\Big ( \\rho ^ { B } ( 1 - \\rho ^ { B } ) ( E _ { B } - E _ { C } ) - \\rho ^ { A } \\rho ^ { B } ( E _ { A } - E _ { C } ) \\Big ) \\end{pmatrix} . \\end{aligned} \\end{align*}"}
+{"id": "4351.png", "formula": "\\begin{gather*} F _ { k _ 1 , k _ 2 , k _ 3 , k _ 4 } ( x ) = \\sum _ { ( i , j , r , s ) \\in [ n ] ^ 4 } ( \\overline { c } _ { i , j } d _ { r , s } + \\max \\{ 0 , \\Delta { c } _ { i , j } d _ { r , s } - \\Delta c _ { k _ 1 , k _ 2 } d _ { k _ 3 , k _ 4 } \\} ) x _ { i , r } x _ { j , s } \\end{gather*}"}
+{"id": "6995.png", "formula": "\\begin{gather*} \\widetilde { Q } : \\ ; \\mathbb { C } ^ { n , 1 } \\rightarrow \\mathbb { C } ^ { 2 , 1 } \\\\ \\mathbf { z } = ( z _ 1 , z _ 2 , . . . , z _ { n + 1 } ) \\mapsto \\widetilde { Q } ( \\mathbf { z } ) = ( z _ 1 , z _ 2 , z _ { n + 1 } ) . \\end{gather*}"}
+{"id": "1328.png", "formula": "\\begin{align*} \\dot { q } & = \\rho ( a ) , \\\\ \\nabla _ a ^ \\ast p & = - \\rho ^ \\ast \\bigl ( \\mathrm { d } U ( q ) \\bigr ) . \\end{align*}"}
+{"id": "6059.png", "formula": "\\begin{align*} U ^ { ( 1 ) } : = x _ { n + 1 } A ( n + 1 ) _ { d - 1 } \\oplus U \\subset A ( n + 1 ) _ d \\end{align*}"}
+{"id": "7198.png", "formula": "\\begin{align*} X - Y = \\sum _ { j = 1 } ^ m \\alpha _ j ( X _ j - Y _ j ) . \\end{align*}"}
+{"id": "1701.png", "formula": "\\begin{align*} \\mathrm { a d } _ v ( X _ q + K _ q ) : = \\left [ V , \\overline { X } \\right ] _ q \\pmod { \\overline { H } \\oplus K } \\quad \\quad \\forall X \\in \\Gamma ( H ) , \\end{align*}"}
+{"id": "3694.png", "formula": "\\begin{align*} \\lambda - s a _ { \\ell - 1 } ( \\lambda ) / a _ \\ell ( \\lambda ) = \\mu _ 1 . \\end{align*}"}
+{"id": "6366.png", "formula": "\\begin{align*} f _ { 1 , k } ^ R ( x ) : = \\sqrt { n } \\frac { x ^ k } { R } , k = 1 , \\dots , n , \\end{align*}"}
+{"id": "4119.png", "formula": "\\begin{align*} c ^ { ( r ) } _ { e , i } ( \\theta _ i ( r ) ) = o ( ( \\theta _ i ( r ) ) ^ 2 ) r \\to 0 . \\end{align*}"}
+{"id": "8928.png", "formula": "\\begin{align*} \\dim ( \\mathcal { M } ( L ) ) = \\dim ( \\mathcal { M } ( L / \\gamma _ c ( L ) ) ) + \\dim ( \\gamma _ c ( L ) ) ( \\dim ( \\frac { L } { \\gamma _ 2 ( L ) } ) - 1 ) - \\dim ( k e r ( \\lambda _ c ) ) . \\end{align*}"}
+{"id": "8731.png", "formula": "\\begin{align*} F _ \\eta ( x ) > 0 , \\ , F _ \\eta ( x - ) < 1 , \\quad \\mbox { a n d } F _ \\eta ^ { - 1 } ( F _ \\eta ( x ) ) = x ; \\end{align*}"}
+{"id": "2907.png", "formula": "\\begin{align*} \\widetilde { V } ( y _ j , x _ i ) : = \\sum _ { y _ { j - 1 } < p \\leqslant y _ j } \\bigg | \\sum _ { \\substack { n \\leqslant x _ i / p \\\\ P ( n ) < p } } f ( n ) \\bigg | ^ 2 . \\end{align*}"}
+{"id": "4172.png", "formula": "\\begin{align*} \\varphi \\big ( g ^ { j } _ 1 g _ 3 \\big ) \\varphi \\big ( g ^ { i } _ 1 g _ 2 \\big ) = \\varphi ( g _ 3 ) \\varphi ( g ^ { n } _ 1 ) \\varphi ( g _ 2 ) \\overset { ( \\beta _ 2 ) } { = } \\varphi ( g ^ { n } _ 1 g _ 3 ) \\varphi ( g _ 2 ) \\ , , \\end{align*}"}
+{"id": "5450.png", "formula": "\\begin{align*} d = 0 , \\beta = 0 . 9 5 , \\mathbf { R } ^ 0 = \\mathbf { 0 } , \\mathbf { R } ^ 1 = \\begin{bmatrix} 0 . 7 2 2 1 \\\\ 0 . 9 6 8 5 \\\\ 0 . 1 5 5 7 \\end{bmatrix} , \\mathbf { P } = \\begin{bmatrix} 0 . 8 0 6 1 & 0 . 1 5 7 4 & 0 . 0 3 6 5 \\\\ 0 . 1 9 5 7 & 0 . 0 0 6 7 & 0 . 7 9 7 6 \\\\ 0 . 1 3 7 8 & 0 . 5 9 5 9 & 0 . 2 6 6 3 \\end{bmatrix} . \\end{align*}"}
+{"id": "3249.png", "formula": "\\begin{align*} \\big ( & S ^ { \\wedge , ( l ) } \\ : V \\ : S ^ \\vee _ 0 \\big ) ( x , y ) \\\\ & = - 2 \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { n ! } \\int _ { - \\infty } ^ 0 \\alpha ^ { l } \\ : ( \\alpha - \\alpha ^ 2 ) ^ n \\ : ( \\Box ^ n V ) \\big | _ { \\alpha y + ( 1 - \\alpha ) x } \\ : d \\alpha \\ ; S ^ { \\bowtie , ( n + l + 1 ) } ( x , y ) \\ : . \\end{align*}"}
+{"id": "1479.png", "formula": "\\begin{align*} [ x , y ] & = - [ y , x ] & \\\\ [ \\alpha ( x ) , [ y , z ] ] & + [ \\alpha ( y ) , [ z , x ] ] + [ \\alpha ( z ) , [ x , y ] ] = 0 . & \\end{align*}"}
+{"id": "983.png", "formula": "\\begin{align*} F ( x , y ) : = f ( x , y + u _ 2 ( x ) ) - f ( x , u _ 2 ( x ) ) , x \\in E , \\ , y \\in \\mathbb R . \\end{align*}"}
+{"id": "1248.png", "formula": "\\begin{align*} \\frac { ( q ; q ^ 2 ) _ { n - 1 } } { ( q ^ 2 ; q ^ 2 ) _ { n - 1 } } = \\frac { ( q ; q ^ 2 ) _ { n - 1 } } { ( q ; q ) _ { n - 1 } ( - q ; q ) _ { n - 1 } } \\equiv \\frac { - q [ n ] } { ( - q ; q ) _ { n - 1 } } \\equiv - q [ n ] \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "7286.png", "formula": "\\begin{align*} C _ { m , r } ( \\beta , n ) : = \\frac { 1 } { m } \\sum _ { j = 0 } ^ { m - 1 } \\csc ^ { 2 n } \\left ( \\frac { j + \\beta } { m } \\pi \\right ) e ^ { 2 \\pi i r j / m } . \\end{align*}"}
+{"id": "1295.png", "formula": "\\begin{align*} \\alpha ( a - k ) + \\alpha ^ 2 k = k + \\alpha ( d - k ) \\Longleftrightarrow k \\alpha ^ 2 + ( a - d ) \\alpha + k = 0 \\ ; , \\end{align*}"}
+{"id": "1667.png", "formula": "\\begin{align*} S ' _ { r } \\coloneqq \\begin{cases} G _ { r - 1 , 0 } & \\mbox { i f } ( \\varepsilon , d ) = ( + 1 , 1 ) , \\\\ [ - 3 p t ] G _ { r - 2 } & \\mbox { i f } ( \\varepsilon , d ) = ( - 1 , 0 ) , \\\\ [ - 3 p t ] G _ { r - 2 , 1 } & \\mbox { i f } ( \\varepsilon , d ) = ( + 1 , 0 ) . \\end{cases} \\end{align*}"}
+{"id": "4973.png", "formula": "\\begin{align*} \\Lambda ( \\mu ; \\nu _ { j _ 1 } ) = a ( \\mu ) f ( \\nu _ { j _ 1 } - 1 , \\mu ) , \\end{align*}"}
+{"id": "5385.png", "formula": "\\begin{align*} \\nu _ j = \\frac { c ^ { S _ { j + 2 } } _ j } { w ^ { S _ { j + 2 } } _ j } , 0 \\leq j \\leq n - 1 . \\end{align*}"}
+{"id": "7856.png", "formula": "\\begin{align*} f ( \\mathbf { x } ) = \\sum _ { i = 1 } ^ { m - 1 } a _ { i } x _ { \\pi ( i ) } x _ { \\pi ( i + 1 ) } + \\sum _ { k = 1 } ^ { m } d _ { k } x _ { k } ^ { 2 } + \\sum _ { k = 1 } ^ { m } c _ { k } x _ { k } = \\mathbf { x } \\mathbf { A } \\mathbf { x } ^ { T } + \\mathbf { c } \\mathbf { x } ^ { T } , \\end{align*}"}
+{"id": "2956.png", "formula": "\\begin{align*} c ( \\lambda , \\mu ) = \\begin{cases} \\max \\{ 2 | \\lambda | + | \\mu | , | \\lambda | + 2 | \\mu | \\} & \\lambda \\neq \\mu \\\\ 2 | \\lambda | & \\lambda = \\mu . \\end{cases} \\end{align*}"}
+{"id": "1813.png", "formula": "\\begin{align*} \\displaystyle \\check { \\mathcal { D } } _ \\delta ^ \\zeta = \\bigcap _ { h > 0 } \\mathcal { D } _ \\delta ^ { \\zeta ^ h } \\qquad \\qquad \\hat { \\mathcal { D } } _ \\delta ^ \\zeta = \\overline { \\bigcup _ { h > 0 } \\mathcal { D } _ \\delta ^ { \\zeta ^ h } } \\end{align*}"}
+{"id": "1647.png", "formula": "\\begin{align*} v \\left ( x , T \\right ) = v _ { T } \\left ( x \\right ) , p \\left ( x , T \\right ) = p _ { T } \\left ( x \\right ) , x \\in \\Omega . \\end{align*}"}
+{"id": "7683.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( \\abs { G ^ { \\Lambda ' } _ { L } ( m , n ; z ) } ^ s \\right ) \\leq \\sum ^ { \\infty } _ { j = 0 } \\Gamma ( s ) ^ { j + 1 } \\# S ^ { \\Lambda ' } _ { j } ( n , m ) . \\end{align*}"}
+{"id": "7432.png", "formula": "\\begin{align*} \\mathcal { F } [ P ] : = \\int _ s ^ t \\int _ s ^ r \\dd P \\otimes \\dd P . \\end{align*}"}
+{"id": "2075.png", "formula": "\\begin{align*} \\eqref { e q : t e s t f u n c t i o n a p p l i e d s u m } = - \\int _ 0 ^ { \\frac { \\lfloor s T \\rfloor } { T } } \\Theta ( u , x ) f ' ( u ) \\mathrm { d } u + f ( s ) { \\Theta } ( \\frac { \\lfloor s T \\rfloor } { T } , x ) - f ( 0 ) { \\Theta } ( 0 , x ) + o ( 1 ) . \\end{align*}"}
+{"id": "8729.png", "formula": "\\begin{align*} \\overline { \\mathcal M } ^ \\rho _ \\rho ( \\mu , \\nu ) = \\sup _ { \\pi \\in \\Pi _ { \\mathrm { M } } ( \\mu , \\nu ) } \\int _ { \\R \\times \\R } \\vert x - y \\vert ^ \\rho \\ , \\pi ( d x , d y ) \\mbox { a n d } \\underline { \\mathcal M } ^ \\rho _ \\rho ( \\mu , \\nu ) = \\inf _ { \\pi \\in \\Pi _ { \\mathrm { M } } ( \\mu , \\nu ) } \\int _ { \\R \\times \\R } \\vert x - y \\vert ^ \\rho \\ , \\pi ( d x , d y ) . \\end{align*}"}
+{"id": "132.png", "formula": "\\begin{align*} \\langle A _ \\lambda ^ h \\vec { u } _ h , \\vec { v } _ h \\rangle = a ( \\vec { u } _ h , \\vec { v } _ h ) + \\lambda b ( \\vec { v } _ h , \\Pi _ h { \\rm d i v } \\vec { u } _ h ) = \\langle \\vec { f } , \\vec { v } _ h \\rangle \\quad \\forall \\vec { v } \\in V _ h , \\end{align*}"}
+{"id": "4316.png", "formula": "\\begin{gather*} \\bar { A } = A \\cup \\{ ( 0 , j ) : \\ j \\in N \\} \\cup \\{ ( i , n + 1 ) : \\ i \\in N \\cup \\{ 0 \\} \\} . \\end{gather*}"}
+{"id": "7010.png", "formula": "\\begin{align*} d X _ t = - b ( t , X _ t ) d t + \\sigma ( t , X _ t ) d W _ t \\end{align*}"}
+{"id": "8089.png", "formula": "\\begin{align*} \\int _ { N _ 1 ^ + } \\phi ( n ) \\psi ( g _ t n x ) \\ ; d \\mu _ x ^ u ( n ) = e ^ { - \\d t } \\sum _ { i \\in I ( \\L ) } \\int _ { N _ 1 ^ + } \\rho _ i ( n n _ i ) \\phi ( g _ { - t } n n _ i g _ t ) \\psi ( n x _ i ) \\ ; d \\mu _ { x _ i } ^ u ( n ) . \\end{align*}"}
+{"id": "3230.png", "formula": "\\begin{align*} n ^ * \\circ \\mbox { N m } _ { p _ s } = \\mbox { N m } _ { \\epsilon } \\circ \\tilde { n } ^ * \\ . \\end{align*}"}
+{"id": "3625.png", "formula": "\\begin{align*} Q f ( z , w ) = \\frac { 1 } { \\lambda _ { 1 } ^ { 2 } - 1 } \\big ( f ( z , - w ) - f ( z , w ) \\big ) \\end{align*}"}
+{"id": "8141.png", "formula": "\\begin{align*} p _ x = { \\cal D } _ { \\delta } \\ , \\ q _ x = X _ { \\delta } \\ , \\end{align*}"}
+{"id": "4766.png", "formula": "\\begin{align*} \\varphi _ 3 ' ( \\varepsilon , b , c , \\eta , n , \\varphi ) & : = \\min \\{ \\varphi _ 1 ( \\varepsilon / 3 , b , n , \\varphi ) , \\varphi _ 1 ( \\eta ( \\min \\{ \\varepsilon / 3 b , 2 \\} ) c / 2 , b , n , \\varphi ) , \\\\ & \\qquad \\qquad \\qquad \\varphi _ 1 ( c / 2 , b , n , \\varphi ) , \\lambda _ 0 / 2 \\} \\end{align*}"}
+{"id": "366.png", "formula": "\\begin{align*} ( x - a ) ^ p = ( y - b ) ^ p = \\big ( z - ( a + b ) \\big ) ^ p = p a b K _ p , \\end{align*}"}
+{"id": "7849.png", "formula": "\\begin{align*} R _ { \\mathbf { a } , \\mathbf { b } } ( \\tau ) = \\begin{cases} \\sum _ { k = 0 } ^ { M - 1 - \\tau } a _ { k } b _ { k + \\tau } ^ { * } , & 0 \\leq \\tau \\leq M - 1 , \\\\ \\sum _ { k = 0 } ^ { M - 1 + \\tau } a _ { k - \\tau } b _ { k } ^ { * } , & 1 - M \\leq \\tau \\leq - 1 , \\\\ 0 , & \\mid \\tau \\mid \\geq M . \\end{cases} \\end{align*}"}
+{"id": "4063.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } R ^ { \\epsilon } _ 0 ( x , 0 ) ( z ) = \\frac { 1 } { ( 2 \\pi ) ^ d } \\sum _ { k = 0 } ^ { \\frac { d - 3 } { 2 } } \\frac { c _ k ( - i ) ^ { s _ k } } { | x | ^ { d - 1 - s _ k } } \\partial _ s ^ { s _ k } \\Big \\{ e ^ { i z s } E _ i ( - i z s ) + e ^ { - i z s } E _ i ( i z s ) + 2 \\pi i e ^ { i z s } \\Big \\} \\Big | _ { ( s = | x | ) } . \\end{align*}"}
+{"id": "2278.png", "formula": "\\begin{align*} [ L ] _ q : = \\frac { q ^ L - q ^ { - L } } { q - q ^ { - 1 } } = q ^ { L - 1 } + q ^ { L - 3 } + \\dots + q ^ { - ( L - 1 ) } \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ [ L ] _ q ! : = [ 1 ] _ q [ 2 ] _ q \\dots [ L ] _ q \\ . \\end{align*}"}
+{"id": "4614.png", "formula": "\\begin{align*} \\int _ \\Omega | \\nabla u | ^ 2 + \\int _ \\Omega \\lambda W | u | ^ 2 = \\sum _ { j \\in J _ m } \\left ( \\int _ { Q ^ m _ j } | \\nabla u _ j | ^ 2 + \\int _ { Q ^ m _ j } \\lambda W | u _ j | ^ 2 \\right ) < 0 . \\end{align*}"}
+{"id": "4554.png", "formula": "\\begin{align*} h = \\frac { 1 } { 1 - \\left ( \\frac { a } { r } \\right ) ^ 4 } d r ^ 2 + r ^ 2 \\left ( \\sigma _ x ^ 2 + \\sigma _ y ^ 2 + \\left ( 1 - \\left ( \\frac { a } { r } \\right ) ^ 4 \\right ) \\sigma _ z ^ 2 \\right ) \\end{align*}"}
+{"id": "8435.png", "formula": "\\begin{align*} \\psi ( t ) : = \\frac { 2 \\sqrt { n } } { n + 1 } \\Bigl ( t ( n + 1 - t ) \\Bigr ) ^ { 1 / 2 } + \\left | 1 - \\frac { 2 t } { n + 1 } \\right | , 0 \\leq t \\leq n + 1 , \\end{align*}"}
+{"id": "3399.png", "formula": "\\begin{align*} P _ { t } & = \\frac { C _ { 1 } } { \\lambda _ { t } \\eta _ { t } \\sqrt { 2 L } } \\ge 1 \\\\ Q _ { t } & = \\frac { C _ { 1 } ^ { 2 } \\sqrt { A } } { 2 L \\eta _ { t } ^ { 2 } \\lambda _ { t } ^ { 2 } } \\ge 1 \\end{align*}"}
+{"id": "2172.png", "formula": "\\begin{align*} ( f _ d ^ { - 1 } ) ^ { ( 1 ) } ( x ) & = \\frac { - d e ^ x } { ( e ^ x - 1 ) ^ 2 } , \\\\ ( f _ d ^ { - 1 } ) ^ { ( 2 ) } ( x ) & = \\frac { d e ^ x ( e ^ x + 1 ) } { ( e ^ x - 1 ) ^ 3 } , \\\\ ( f _ d ^ { - 1 } ) ^ { ( 3 ) } ( x ) & = \\frac { - d e ^ x ( e ^ { 2 x } + 4 e ^ x + 1 ) } { ( e ^ x - 1 ) ^ 4 } . \\end{align*}"}
+{"id": "6396.png", "formula": "\\begin{align*} G _ n ^ 4 ( \\theta ) = - \\partial _ { \\alpha } L _ n ( \\theta ) = \\sum _ { i = 1 } ^ n \\frac { \\ln \\left ( n / X _ { \\frac { i - 1 } { n } } \\right ) } { \\alpha ^ 2 } k _ \\alpha ( z ^ n _ i ( \\theta ) ) - f _ \\alpha ( z ^ n _ i ( \\theta ) ) . \\end{align*}"}
+{"id": "3133.png", "formula": "\\begin{align*} u ^ - _ { \\psi ^ * } ( s ) = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ v _ { 2 , \\ , 1 } ( s ) & 1 & 0 \\\\ v _ { 3 , \\ , 1 } ( s ) & v _ { 3 , \\ , 2 } ( s ) & 1 \\end{array} \\right ) ( \\ v _ { 2 , \\ , 1 } ( S ) , v _ { 3 , \\ , 1 } ( S ) , v _ { 3 , \\ , 2 } ( S ) \\in k [ S ] \\ , ) . \\end{align*}"}
+{"id": "2113.png", "formula": "\\begin{align*} B ( x _ 1 , x _ 2 , x _ 3 , x _ 4 ) = \\sum _ { p \\in P _ 4 ^ 2 } \\prod _ { ( i , j ) \\in p } A ( x _ i , x _ j ) \\end{align*}"}
+{"id": "4642.png", "formula": "\\begin{align*} \\displaystyle B _ 2 ( N ) = \\sum _ { 1 \\leq n \\leq N } 2 ^ { \\omega ( n ) } \\cdot \\lambda _ { \\mathcal { G } _ 2 } ( n ) = \\sum _ { 1 \\leq n \\leq N } b _ 2 ( n ) . \\end{align*}"}
+{"id": "3440.png", "formula": "\\begin{align*} ( X , m , n ) = \\begin{cases} \\Sigma ^ { - m \\R } \\Sigma ^ { - n \\tilde { \\R } } X G = \\Z _ 2 , \\\\ \\Sigma ^ { - m \\tilde { \\R } } \\Sigma ^ { - n \\C } X G = \\Z _ 4 . \\end{cases} \\end{align*}"}
+{"id": "7391.png", "formula": "\\begin{align*} \\frac { q ( p + q ) + p - 1 } { q + 1 } = \\frac { p ( p + q ) + q - 1 } { p + 1 } = p + q - 1 . \\end{align*}"}
+{"id": "1747.png", "formula": "\\begin{align*} e _ j : = X _ { - j } + i Y _ { - j } , e _ { n - 1 + j } : = X _ { - j } - i Y _ { - j } \\quad \\forall \\ , j = 1 , \\ldots , n - 1 , \\end{align*}"}
+{"id": "1905.png", "formula": "\\begin{align*} S : = G \\times [ 0 , + \\infty ) ; \\end{align*}"}
+{"id": "8661.png", "formula": "\\begin{align*} \\underline { \\mathfrak { g } } _ x = { \\rm S p a n \\ , } \\left \\{ \\xi _ { M ' } ( x ) ; \\ , \\xi \\in \\mathfrak { g } \\ , \\right \\} . \\end{align*}"}
+{"id": "3807.png", "formula": "\\begin{align*} \\dim \\left ( \\bigcap _ { i = 1 } ^ \\ell \\Im ( ( G ^ \\top ) _ { [ k ] , A _ i } ) \\right ) = \\dim \\left ( \\bigcap _ { i = 1 } ^ \\ell \\Im ( W _ { [ k ] , A _ i } ) \\right ) . \\end{align*}"}
+{"id": "2702.png", "formula": "\\begin{align*} X _ { t } : = a ^ { i } \\frac { \\partial } { \\partial v ^ { i } } + b ^ { i } \\frac { \\partial } { \\partial q ^ { i } } + \\frac { \\partial } { \\partial t } , \\end{align*}"}
+{"id": "2504.png", "formula": "\\begin{align*} \\| \\nabla A ( t ) \\| _ 2 ^ 2 & = \\langle - \\Delta A ( t ) , A ( t ) \\rangle _ { ( H ^ 1 ) ' , H ^ 1 } \\\\ & \\le \\langle u ( t ) - \\Delta A ( t ) , A ( t ) \\rangle _ { ( H ^ 1 ) ' , H ^ 1 } = \\int _ \\Omega u ^ { i n } A ( t ) \\ \\mathrm { d } x \\\\ & \\le \\| u ^ { i n } \\| _ \\infty \\| A ( t ) \\| _ 1 \\ , . \\end{align*}"}
+{"id": "5843.png", "formula": "\\begin{align*} \\lambda _ { i _ 0 , j _ 0 , k _ 0 } x ^ { i _ 0 } y ^ { j _ 0 } z ^ { k _ 0 } = \\gamma \\left ( x ^ { | \\alpha | } \\right ) ^ { i _ 1 } \\left ( y ^ { | \\alpha | } \\right ) ^ { j _ 1 } \\left ( z ^ { | \\alpha | } \\right ) ^ { k _ 1 } \\left ( x ^ { | \\alpha ^ 3 | } y ^ { | \\alpha ^ 3 | } z ^ { | \\alpha ^ 3 | } \\right ) ^ { i _ 2 } , \\end{align*}"}
+{"id": "5261.png", "formula": "\\begin{align*} U _ { j } = ( 1 - \\alpha ) \\left [ 1 + \\left ( \\frac { a } { b - a } \\right ) \\left ( \\frac { T _ { j } } { p _ { j } } \\right ) ^ { a - 1 } \\right ] \\ ; \\ ; ; \\ ; \\ ; V _ { j } = \\left ( \\frac { b } { b - a } \\right ) \\left ( \\frac { T _ { j } } { p _ { j } } \\right ) ^ { b - 1 } \\end{align*}"}
+{"id": "2757.png", "formula": "\\begin{align*} \\Theta _ { i } = C _ { i } ^ { \\ j } \\theta _ { j } , \\end{align*}"}
+{"id": "6921.png", "formula": "\\begin{align*} F ( e ^ { i t } ) \\underset { t \\rightarrow 0 } = \\exp ( - i \\alpha t - \\beta t ^ { 2 \\mu } + o ( t ^ { 2 \\mu } ) ) . \\end{align*}"}
+{"id": "6310.png", "formula": "\\begin{align*} - \\Delta \\omega = \\frac { 1 } { 2 } V ( 1 - \\omega ) \\R ^ 3 , \\lim _ { | x | \\to \\infty } \\omega ( x ) = 0 \\end{align*}"}
+{"id": "3391.png", "formula": "\\begin{align*} S _ { t } & = \\sum _ { i = 1 } ^ { t } Z _ { i } \\end{align*}"}
+{"id": "5425.png", "formula": "\\begin{align*} S _ 0 ^ * ( \\nu ) = \\big \\{ ( 0 , i ) \\colon \\nu ^ * _ { ( 0 , i ) } > \\nu \\big \\} S _ 1 ^ * ( \\nu ) = \\big \\{ ( 1 , i ) \\colon \\nu ^ * _ { ( 1 , i ) } > \\nu \\big \\} , \\nu \\in \\mathbb { R } . \\end{align*}"}
+{"id": "5698.png", "formula": "\\begin{align*} u ^ T = \\sum _ { j = 1 } ^ J x _ j \\varphi _ { \\iota + j } . \\end{align*}"}
+{"id": "8157.png", "formula": "\\begin{align*} & D ( B _ ) = \\sum _ { j = 1 } ^ L \\frac { 2 [ c _ j - 2 ^ { L - j } ] ^ + } { M } + \\frac { 2 [ M - \\sum _ { j = 1 } ^ { L } c _ { j } - 1 ] ^ + } { M } , \\end{align*}"}
+{"id": "2978.png", "formula": "\\begin{align*} \\frac { d } { d t } ( v _ 2 ^ 2 - v _ 1 ^ 2 ) = \\underbrace { - \\frac { \\kappa _ 1 ( 1 + \\langle v _ 1 , v _ 2 \\rangle ) } { 2 ( x _ 2 ^ 2 - x _ 1 ^ 2 ) ^ { \\alpha } } } _ { < 0 } ( v _ 2 ^ 2 - v _ 1 ^ 2 ) . \\end{align*}"}
+{"id": "7649.png", "formula": "\\begin{align*} \\lambda _ { \\mathrm { H F } } = 2 M _ { \\infty } \\mu _ d e \\ln \\left ( \\frac { \\lambda _ { \\mathrm { H F } } } { 2 M _ { \\infty } } \\right ) \\end{align*}"}
+{"id": "4663.png", "formula": "\\begin{align*} m = \\sqrt { \\frac { A _ g } { 1 6 \\pi } } . \\end{align*}"}
+{"id": "4885.png", "formula": "\\begin{align*} \\deg ( \\mathcal D \\otimes \\L ) = g - 1 , \\deg ( \\mathcal D \\otimes \\L ^ { \\otimes 2 } ) = g - 1 - e . \\end{align*}"}
+{"id": "8068.png", "formula": "\\begin{align*} \\limsup _ { N \\rightarrow + \\infty } \\frac { N } { \\gamma _ N ^ 2 } \\log P ( | \\tau _ c ^ N - \\tau _ c | \\geq \\delta ) = - \\infty . \\end{align*}"}
+{"id": "7202.png", "formula": "\\begin{align*} T ( P ) = \\begin{bmatrix} f _ 1 ( P _ 1 ) \\\\ f _ 2 ( P _ 2 ) \\\\ \\vdots \\\\ f _ N ( P _ N ) \\\\ \\end{bmatrix} , \\end{align*}"}
+{"id": "46.png", "formula": "\\begin{align*} \\langle \\omega , \\eta \\rangle _ { \\ell ^ 2 ( h \\Z ^ n ; \\bigwedge ^ { j } ( h \\Z ^ n ) ) } & = \\frac 1 { h ^ { 2 j } } \\sum _ { \\mu \\in h \\Z ^ n ; I \\in P ^ { j , n } _ + } \\omega _ I ( \\mu ) \\overline { \\eta } _ I ( \\mu ) , \\\\ \\langle f , g \\rangle _ { \\ell ^ 2 \\Big ( h \\Z ^ n ; \\C ^ { \\binom { n } { j } } \\Big ) } & = \\sum _ { \\mu \\in h \\Z ^ n ; 1 \\leq l \\leq \\binom { n } { j } } f _ l ( \\mu ) \\overline { g } _ l ( \\mu ) . \\end{align*}"}
+{"id": "3471.png", "formula": "\\begin{align*} H ^ 2 ( \\Sigma ( S ) ; \\Z ) = H ^ 2 ( \\Sigma ( S ) ; \\Z ) ^ { \\iota ^ * } \\oplus H ^ 2 ( \\Sigma ( S ) ; \\Z ) ^ { - \\iota ^ * } \\end{align*}"}
+{"id": "3916.png", "formula": "\\begin{align*} \\begin{aligned} \\cos n z = \\cos ^ n z - \\binom { n } { 2 } \\cos ^ { n - 2 } z \\sin ^ 2 z + \\binom { n } { 4 } \\cos ^ { n - 4 } z \\sin ^ 4 z - \\\\ - \\binom { n } { 6 } \\cos ^ { n - 6 } z \\sin ^ 6 z + \\& c . \\end{aligned} \\end{align*}"}
+{"id": "7512.png", "formula": "\\begin{align*} M _ { s } = \\frac { \\left ( M _ { s } ^ { n _ 0 - 1 } \\backslash \\phi _ { n _ 0 } ( ( 0 , s ^ { 1 / 4 } ] \\times X \\times [ - \\eta / 2 , \\eta / 2 ] ) \\right ) \\bigsqcup \\left ( \\{ \\mathbf { r } _ s \\leq 1 \\} \\times [ L / 2 - \\eta , L / 2 ] \\right ) } { \\thicksim _ { n _ 0 } } , \\end{align*}"}
+{"id": "3544.png", "formula": "\\begin{align*} \\overline { R a n ( f ) } = \\overline { R a n ( T f ) } . \\end{align*}"}
+{"id": "7329.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n - 1 } \\binom { n } { j } t _ m ( n - j ) \\tilde { c } _ { m , r } ( \\alpha , j ) + ( t _ m ( 0 ) - \\cos 2 \\pi \\alpha ) \\tilde { c } _ { m , r } ( \\alpha , n ) = u _ { m - \\ell - 1 } ( n ) + e ^ { 2 \\pi i \\alpha } u _ { \\ell - 1 } ( n ) , \\end{align*}"}
+{"id": "5241.png", "formula": "\\begin{align*} \\sum _ { j } q _ { j } \\frac { \\partial L _ { d } D A H I ( p \\| q ) } { \\partial q _ { j } } = 0 \\end{align*}"}
+{"id": "8770.png", "formula": "\\begin{align*} \\mu & \\left ( \\left \\{ y \\in \\R : \\forall x \\in { \\mathbb Q } , \\ ; \\hat \\pi ^ \\uparrow _ y ( ( - \\infty , x ] ) = \\pi ^ \\uparrow _ y ( ( - \\infty , x ] ) \\right \\} \\right ) = 1 = \\mu \\left ( \\left \\{ y \\in \\R : \\hat \\pi ^ \\uparrow _ y = \\pi ^ \\uparrow _ y \\right \\} \\right ) . \\end{align*}"}
+{"id": "81.png", "formula": "\\begin{align*} \\begin{cases} c _ 1 = w _ 1 + w _ 3 \\kappa , & c _ 2 = - 2 w _ 3 \\theta \\kappa , \\\\ c _ 3 = w _ 2 + w _ 4 \\kappa , & c _ 4 = - ( w _ 2 \\gamma + w _ 4 \\kappa ( 2 + \\gamma ) ) \\theta , \\end{cases} \\end{align*}"}
+{"id": "791.png", "formula": "\\begin{align*} \\mathcal { X } _ T = L ^ \\infty V \\cap L ^ 2 D ( A ) \\cap L ^ 3 W ^ { 2 , 3 } \\cap W ^ { 1 , \\infty } L ^ 2 \\cap H ^ 1 V \\cap H ^ 2 V ' \\end{align*}"}
+{"id": "2947.png", "formula": "\\begin{align*} U ( g , \\rho ) = \\rho ^ s \\int _ K \\varphi _ { s , \\rho } ( k ) F ( k g ) d k . \\end{align*}"}
+{"id": "1579.png", "formula": "\\begin{align*} D g _ k ( z ) = \\frac { D G ( z ) } { \\big ( D G ( z ) , z \\big ) } . \\end{align*}"}
+{"id": "1365.png", "formula": "\\begin{align*} ( a \\cdot _ { \\alpha , \\ , \\beta } b ) \\cdot _ { \\alpha \\beta , \\gamma } c = a \\cdot _ { \\alpha , \\ , \\beta \\gamma } ( b \\cdot _ { \\beta , \\gamma } c ) \\end{align*}"}
+{"id": "8866.png", "formula": "\\begin{align*} \\| \\mathcal N [ u ] - \\mathcal N [ v ] \\| _ { L _ x ^ { \\frac { 2 n } { n + 2 } } } & \\leq \\| | x | ^ { - \\tau } ( | u | ^ { p - 2 } + | v | ^ { p - 2 } ) | u - v | ( I _ \\alpha \\ast | \\cdot | ^ { - \\tau } | u | ^ p ) \\| _ { L _ x ^ { \\frac { 2 n } { n + 2 } } } \\\\ & \\qquad + \\| | x | ^ { - \\tau } | v | ^ { p - 1 } \\big ( I _ \\alpha \\ast | \\cdot | ^ { - \\tau } ( | u | ^ { p - 1 } + | v | ^ { p - 1 } ) | u - v | \\big ) \\| _ { L _ x ^ { \\frac { 2 n } { n + 2 } } } \\\\ & : = B _ 1 + B _ 2 . \\end{align*}"}
+{"id": "7673.png", "formula": "\\begin{align*} C _ 2 = \\frac { 4 8 \\norm { F } _ { \\infty } C _ { a } } { \\eta ^ 2 } S _ { \\delta - \\nu } S _ { \\delta - \\frac { \\gamma _ a } { 2 } } ( \\lambda | g | + | g | ^ 2 C _ 1 ) \\end{align*}"}
+{"id": "5731.png", "formula": "\\begin{align*} X _ + ( t ) Y _ + ( t ) = & \\sum \\xi ^ + _ i ( t ) \\mathcal { E } ^ + _ i ( t ) + \\sum _ { i : \\gamma ^ - _ i > \\gamma } \\xi ^ + _ - ( t ) \\mathcal { E } ^ + _ - ( t ) + \\sum _ { i \\in I _ 1 } \\xi _ { i , 1 } ( t ) \\mathcal { E } _ { i , 1 } ( t ) + \\xi _ { i , 1 } ( t ) \\mathcal { E } _ { i , 1 } ( t ) \\\\ & + \\sum _ { i \\in I _ 2 } \\xi _ { i , 3 } ( t ) \\mathcal { E } _ { i , 3 } ( t ) + \\xi _ { i , 4 } ( t ) \\mathcal { E } _ { i , 4 } ( t ) . \\end{align*}"}
+{"id": "161.png", "formula": "\\begin{align*} & \\sum _ { i } | 1 _ { X ^ { k } } \\otimes b _ { i } \\otimes 1 _ { X ^ { k } } \\rangle _ { A _ { [ a - k , b + k ] } } \\ \\langle 1 _ { X ^ { k } } \\otimes b _ { i } \\otimes 1 _ { X ^ { k } } | \\\\ & = \\sum _ { i } ( 1 _ { X ^ { k } } \\otimes b _ { i } \\otimes 1 _ { X ^ { k } } ) \\circ ( 1 _ { X ^ { k } } \\otimes b ^ { * } _ { i } \\otimes 1 _ { X ^ { k } } ) \\\\ & = 1 _ { X ^ { 2 k + b - a + 1 } \\otimes Z } = i d _ { F ^ { k } _ { [ a , b ] } ( Z , c ) } \\end{align*}"}
+{"id": "2655.png", "formula": "\\begin{align*} \\frac { 1 } { q } + \\frac { 1 } { q ^ { ' } } = 1 \\end{align*}"}
+{"id": "9176.png", "formula": "\\begin{align*} \\zeta _ { i j } ( z ) = 1 + \\frac { ( \\alpha _ { i } , \\alpha _ { j } ) \\cdot \\hbar } { 2 z } . \\end{align*}"}
+{"id": "4442.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ { \\partial D } | \\frac { g _ 1 + g _ 2 } { 2 } | ^ 2 \\rho | d z | + \\frac { 1 } { 2 \\pi } \\int _ { \\partial D } | \\frac { g _ 1 - g _ 2 } { 2 } | ^ 2 \\rho | d z | = M _ H ( Z _ 0 , \\mathfrak { a } , \\rho ) , \\end{align*}"}
+{"id": "4093.png", "formula": "\\begin{align*} R _ V ( z ) f + R _ 0 ( z ) V R ( z ) f = R _ 0 ( z ) f \\Rightarrow ( I + T ( z ) ) u = R _ 0 ( z ) f \\end{align*}"}
+{"id": "2460.png", "formula": "\\begin{align*} \\{ ( \\phi , \\psi ) \\mid \\psi ( s ) = - k c ^ { - 1 } \\phi ^ { 2 } + O ( \\phi ^ { 4 } ) \\} . \\end{align*}"}
+{"id": "4002.png", "formula": "\\begin{align*} V ^ { t + 1 } \\left ( z \\right ) = V \\left ( V ^ { t } \\left ( z \\right ) \\right ) = \\frac { 1 } { \\varpi \\circ W \\left ( V ^ { t } \\left ( z \\right ) \\right ) } W \\left ( V ^ { t } \\left ( z \\right ) \\right ) \\end{align*}"}
+{"id": "332.png", "formula": "\\begin{align*} J = x _ n x _ { n - 1 } \\big [ I ( G _ 3 ) ^ { [ k ] } + x _ { n - 2 } I ( G _ 3 ) ^ { [ k - 1 ] } + \\sum _ { j = 1 } ^ { t } x _ { i _ j } I ( G _ 2 ) ^ { [ k - 1 ] } \\big ] . \\end{align*}"}
+{"id": "5782.png", "formula": "\\begin{align*} \\frac { d } { d t } \\tilde \\xi _ { i } - \\Gamma _ i \\tilde \\xi _ { i } = \\tilde { \\mathcal { E } } _ { i } . \\end{align*}"}
+{"id": "7282.png", "formula": "\\begin{align*} C _ { m } ^ { \\mathrm { a l t } } ( \\beta , n ) : = \\sum _ { j = \\delta ( \\beta ) } ^ { m - 1 } ( - 1 ) ^ j \\csc ^ { n } \\left ( \\frac { ( j + \\beta ) } { m } \\pi \\right ) . \\end{align*}"}
+{"id": "1451.png", "formula": "\\begin{align*} H _ i ( H H _ { \\sigma } , M ) : = \\mathrm { T o r } _ { i - 1 } ^ { \\mathbb { Z } [ H ] } ( R , M ) = H _ { i - 1 } ( H , R \\otimes M ) . \\end{align*}"}
+{"id": "388.png", "formula": "\\begin{align*} M _ { k } ( ( \\nu _ { k } , \\nu _ { k + 1 } , \\ldots , \\nu _ { N - 1 } ) ) : = \\bigcap _ { i \\geq k } M _ { i } ( \\nu _ { i } ) . \\end{align*}"}
+{"id": "3556.png", "formula": "\\begin{align*} f ' ( \\tau ( 0 ) ) \\tau ' ( 0 ) = 0 . \\end{align*}"}
+{"id": "5955.png", "formula": "\\begin{align*} \\frac { M _ { 2 } ( \\mathcal { N C } ( G ) ) } { | e ( \\mathcal { N C } ( G ) ) | } = 1 2 0 > 1 1 8 . 8 = \\frac { M _ { 1 } ( \\mathcal { N C } ( G ) ) } { | v ( \\mathcal { N C } ( G ) ) | } . \\end{align*}"}
+{"id": "1076.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { n } \\| ( T _ n ( w ) ^ { - 1 } ) ^ { s , t } - ( T _ { \\infty } ( w ) ^ { - 1 } ) ^ { s , t } \\| \\le B _ 2 , n \\in \\N , \\ t \\in \\{ 1 , \\dots , [ ( n + 1 ) / 2 ] \\} . \\end{align*}"}
+{"id": "8270.png", "formula": "\\begin{align*} N ( \\alpha ) : = \\sum _ { P \\subset X } \\nu ( \\alpha , P ) P , \\end{align*}"}
+{"id": "1355.png", "formula": "\\begin{align*} \\left ( \\frac { 2 m \\alpha } { n } + h \\right ) + ( l - 1 ) \\left ( \\frac { 2 m \\alpha } { n } - h \\right ) + a \\left ( \\frac { 2 m \\alpha } { n } + k ) + ( n - a - l \\right ) \\left ( \\frac { 2 m \\alpha } { n } - k \\right ) = 2 m \\alpha . \\end{align*}"}
+{"id": "5253.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } F I \\left ( p \\| q \\right ) } { \\partial q _ { j } } = - \\sum _ { j } p _ { j } \\left [ \\frac { a - 1 } { a - b } \\sum _ { i } \\overline { p } _ { i } \\left ( \\overline { Z } _ { i } \\right ) ^ { a - 2 } - \\frac { b - 1 } { a - b } \\sum _ { i } \\overline { p } _ { i } \\left ( \\overline { Z } _ { i } \\right ) ^ { b - 2 } \\right ] \\frac { \\partial \\overline { Z } _ { i } } { \\partial q _ { j } } \\end{align*}"}
+{"id": "5602.png", "formula": "\\begin{align*} \\Gamma _ { i i } ^ { ( t ) } \\leq \\sum _ { s = 0 } ^ t \\frac { K ^ 2 \\rho ^ { 2 s } } { ( \\nu _ i ^ 2 d ) ^ { 2 s } } \\leq K ^ 2 \\log ( n ) \\theta _ 1 ^ { 4 t } \\nu _ i ^ { - 4 t } , \\end{align*}"}
+{"id": "2706.png", "formula": "\\begin{align*} { _ { * } L } : = q ^ { i } d p _ { i } - H ( q ^ { i } , p _ { i } ) d t , \\end{align*}"}
+{"id": "2422.png", "formula": "\\begin{align*} v _ 2 \\left ( s ! \\binom { s + 2 } { i _ { * } } n ! ^ { 2 + j } \\prod _ { k = 1 } ^ { n } \\left ( ( k - 1 ) ! ( n - k ) ! \\right ) ^ { i _ k } \\right ) \\geqslant ( s + 2 ) n - ( 2 s + 2 ) m + s + v _ 2 ( ( s + 2 ) ! ) , \\end{align*}"}
+{"id": "9104.png", "formula": "\\begin{align*} X _ i ( 0 ) \\geq 0 \\Longrightarrow X _ i ( t ) \\geq 0 , \\mbox { f o r a l l $ t = 0 , 1 , 2 , \\ldots $ } . \\end{align*}"}
+{"id": "4178.png", "formula": "\\begin{align*} \\varphi \\big ( g ^ { j } _ 1 g _ 3 \\big ) \\varphi \\big ( g ^ { i } _ 1 g _ 2 \\big ) = \\varphi ( g _ 3 ) \\varphi ( g ^ { n } _ 1 g _ 2 ) = \\varphi ( g _ 3 ( g ^ { n } _ 1 g _ 2 ) ) \\ , , \\end{align*}"}
+{"id": "4909.png", "formula": "\\begin{align*} b _ { \\max } = 2 g + 1 . \\end{align*}"}
+{"id": "6886.png", "formula": "\\begin{align*} ( a \\otimes b ) \\cdot x = a x \\sigma ( b ) . \\end{align*}"}
+{"id": "4340.png", "formula": "\\begin{align*} \\inf _ { k \\in [ m ] _ 0 } & \\left \\{ \\inf _ { \\mathcal { Q } \\subseteq [ m ] } \\left \\{ \\inf _ { x \\in \\mathcal { X _ \\mathcal { Q } } } \\left \\{ \\Gamma \\Delta u _ k ^ T l _ k ( x ) + \\sum _ { i \\in [ m ] } \\overline { u } _ i ^ T l _ i ( x ) + \\sum _ { q \\in \\mathcal { Q } } \\Delta u _ q ^ T l _ q ( x ) - \\Delta u _ k ^ T l _ k ( x ) \\right \\} \\right \\} \\right \\} , \\end{align*}"}
+{"id": "6300.png", "formula": "\\begin{align*} \\begin{aligned} \\binom { n - t _ 1 } { 3 } - \\binom { n - t _ 1 - t _ 2 } { 3 } & = \\binom { t _ 2 } { 3 } + \\binom { t _ 2 } { 2 } ( n - t _ 1 - t _ 2 ) + t _ 2 \\binom { n - t _ 1 - t _ 2 } { 2 } \\\\ & = \\binom { t _ 2 } { 3 } + \\binom { t _ 2 } { 2 } ( | W | + 3 t _ 2 ) + t _ 2 \\binom { | W | + 3 t _ 2 } { 2 } \\\\ & = \\binom { t _ 2 } { 3 } + \\binom { t _ 2 } { 2 } | W | + 3 \\binom { t _ 2 } { 2 } t _ 2 + t _ 2 \\binom { | W | } { 2 } + t _ 2 \\binom { 3 t _ 2 } { 2 } + 3 t _ 2 ^ 2 | W | \\\\ & \\ge 3 7 \\binom { t _ 2 } { 3 } + 8 \\binom { t _ 2 } { 2 } | W | , \\end{aligned} \\end{align*}"}
+{"id": "8473.png", "formula": "\\begin{align*} & \\left \\| ( \\bar { u } _ h ) _ + - ( u _ h ) _ + \\right \\| _ { L ^ 1 ( K _ T ) } \\\\ [ 3 m m ] & \\leq \\left \\| ( \\bar { u } _ h ) _ + - ( \\bar { u } _ h ) _ + ^ { ( \\ell ) } \\right \\| _ { L ^ 1 ( K _ T ) } + \\left \\| ( \\bar { u } _ h ) _ + ^ { ( \\ell ) } - ( u _ h ) _ + ^ { ( \\ell ) } \\right \\| _ { L ^ 1 ( K _ T ) } + \\left \\| ( u _ h ) _ + ^ { ( \\ell ) } - ( u _ h ) _ + \\right \\| _ { L ^ 1 ( K _ T ) } \\\\ [ 3 m m ] & = : \\mathbf { I } _ { h , \\ell } + \\mathbf { I I } _ { h , \\ell } + \\mathbf { I I I } _ { h , \\ell } . \\end{align*}"}
+{"id": "7291.png", "formula": "\\begin{align*} \\tilde { C } _ { m , r } ( \\beta , n ) : = \\frac { 1 } { m } \\sum _ { j = 0 } ^ { m - 1 } \\csc ^ { n } \\left ( \\frac { 2 ( j + \\beta ) } { m } \\pi \\right ) e ^ { 2 \\pi i r j / m } . \\end{align*}"}
+{"id": "7516.png", "formula": "\\begin{align*} l _ { s / \\eta _ 1 } = \\pi _ 2 \\circ \\tilde { \\phi } _ { 1 / \\eta _ 1 } ^ { - 1 } \\circ \\Phi _ s ^ { - 1 } = \\pi _ 2 \\circ \\tilde { \\phi } _ { \\eta _ 1 } \\circ \\Phi _ { s } ^ { - 1 } = \\frac { 1 } { \\sqrt { \\eta _ 1 } } l _ s , \\end{align*}"}
+{"id": "7191.png", "formula": "\\begin{align*} v = \\bar { v } + A ( x ) \\sin ( \\omega _ 1 t ) + B ( x ) \\sin ( \\omega _ 2 t ) + O \\left ( \\frac { 1 } { \\omega _ 1 } \\right ) , w = \\bar { w } + O \\left ( \\frac { 1 } { \\omega _ 1 } \\right ) , \\end{align*}"}
+{"id": "7873.png", "formula": "\\begin{align*} \\mathbf { A } _ { i } = \\begin{cases} \\psi ( \\pi , \\mathbf { a } , \\mathbf { d } _ { i } ) , & 1 \\leq i \\leq p , \\\\ \\psi ( \\pi , \\mathbf { b } , \\mathbf { d } _ { i } ) , & p + 1 \\leq i \\leq 2 p , \\end{cases} \\end{align*}"}
+{"id": "4493.png", "formula": "\\begin{align*} M _ { D _ j } = \\pi M _ { \\partial D _ j } \\end{align*}"}
+{"id": "5393.png", "formula": "\\begin{align*} \\nu _ j = \\frac { h } { \\mu } \\sum _ { i = 1 } ^ { j + 1 } \\left ( 1 + \\cdots + \\rho ^ { i - 1 } \\right ) = \\begin{cases} \\displaystyle \\frac { h } { \\mu } \\ , \\left [ \\frac { \\rho ^ { j + 2 } - 1 } { ( \\rho - 1 ) ^ 2 } - \\frac { j + 2 } { \\rho - 1 } \\right ] & \\rho \\neq 1 \\\\ \\\\ \\displaystyle \\frac { h } { \\mu } \\ , \\frac { ( j + 1 ) \\ , ( j + 2 ) } { 2 } & \\rho = 1 . \\end{cases} \\end{align*}"}
+{"id": "5339.png", "formula": "\\begin{align*} \\nu _ { \\pi _ k } = \\nu ^ { S _ k } _ { \\pi _ k } \\leq \\nu ^ { S _ k } _ { \\pi _ l } \\leq \\nu ^ { S _ { k + 1 } } _ { \\pi _ l } \\leq \\cdots \\leq \\nu ^ { S _ { l - 1 } } _ { \\pi _ { l } } \\leq \\nu ^ { S _ l } _ { \\pi _ l } = \\nu _ { \\pi _ l } . \\end{align*}"}
+{"id": "3356.png", "formula": "\\begin{align*} \\min _ { u ^ \\circ } \\ \\ \\mathcal { J } ^ \\circ ( u ^ \\circ ) : = \\mathbb { E } ^ \\dagger \\left [ \\int _ { \\mathbb { R } ^ n } \\sigma ^ \\circ _ K ( z ) \\left ( \\mu \\Phi ( z ) \\right ) d z \\right ] , \\end{align*}"}
+{"id": "3035.png", "formula": "\\begin{align*} { T } _ \\ell = \\sum _ { i , j } c _ { i j } ^ \\ell { X } _ { i } { X } _ j \\ , , [ { T } _ \\ell , Z _ \\rho ] = 0 \\ , , c _ { j k } ^ \\ell \\in \\mathbb { C } \\ , , \\end{align*}"}
+{"id": "2115.png", "formula": "\\begin{align*} & \\big | B ( x _ 1 , x _ 1 , x _ 1 , x _ 1 ) + B ( x _ 2 , x _ 2 , x _ 2 , x _ 2 ) + 6 B ( x _ 1 , x _ 1 , x _ 2 , x _ 2 ) - 4 B ( x _ 1 , x _ 1 , x _ 1 , x _ 2 ) - 4 B ( x _ 1 , x _ 2 , x _ 2 , x _ 2 ) \\big | \\\\ = & 3 \\big | A ( x _ 1 , x _ 1 ) + A ( x _ 2 , x _ 2 ) - 2 A ( x _ 1 , x _ 2 ) \\big | ^ 2 \\le C | x _ 1 - x _ 2 | ^ 4 , \\end{align*}"}
+{"id": "6346.png", "formula": "\\begin{align*} \\partial E = \\{ \\exp _ o ( R ( 1 + \\rho ( \\varphi ) ) ) : \\varphi \\in S ^ { n - 1 } \\} = \\{ ( R ( 1 + \\rho ( \\varphi ) ) , \\varphi ) : \\varphi \\in S ^ { n - 1 } \\} , \\end{align*}"}
+{"id": "4428.png", "formula": "\\begin{gather*} \\norm { \\delta _ t E ^ j } _ \\infty = \\norm { \\delta _ t \\left ( u _ h - R \\right ) ^ j } _ \\infty \\le \\eta _ \\mathrm { e l l , \\delta } ^ j \\coloneqq \\eta \\left ( \\delta _ t u _ h ^ j , \\delta _ t \\left ( f + \\psi \\right ) ^ j \\right ) \\ , , \\ \\ j = 1 , \\dots , M . \\end{gather*}"}
+{"id": "2046.png", "formula": "\\begin{align*} H _ T \\ , \\ , : = \\ , \\ , \\begin{small} \\begin{bmatrix} T _ m & T _ { m - 1 } & \\cdots & T _ { m - e } \\\\ T _ { m - 1 } & T _ { m - 2 } & \\cdots & T _ { m - e - 1 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ T _ { d + 2 } & T _ { d + 1 } & \\cdots & T _ { d - e + 2 } \\\\ T _ { d + 1 } & T _ { d } & \\cdots & T _ { d - e + 1 } \\end{bmatrix} . \\end{small} \\end{align*}"}
+{"id": "7607.png", "formula": "\\begin{align*} \\| u _ { \\epsilon } \\| _ { 4 ^ * } ^ { 4 ^ * } & = R ^ { 4 ^ * } \\omega \\epsilon ^ { N } \\int _ { 0 } ^ { 2 } \\frac { ( \\varphi ^ { 4 ^ * } ( r ) - 1 ) r ^ { N - 1 } } { ( \\epsilon ^ { 2 } + r ^ { 2 } ) ^ { N } } d r + R ^ { 4 ^ * } \\omega \\epsilon ^ { N } \\int _ { 0 } ^ { 2 } \\frac { r ^ { N - 1 } } { ( \\epsilon ^ { 2 } + r ^ { 2 } ) ^ { N } } d r \\\\ & = I _ { 1 } ( \\epsilon ) + I _ { 2 } ( \\epsilon ) . \\\\ \\end{align*}"}
+{"id": "3087.png", "formula": "\\begin{align*} \\varphi ^ * _ a ( H ) = \\varphi ^ * _ a ( F _ 0 ^ { \\gamma _ 0 } ) \\cdot \\varphi ^ * _ a ( F _ 1 ^ { \\gamma _ 1 } ) \\cdot \\ldots \\cdot \\underbrace { \\varphi ^ * _ a ( F _ l ) \\cdot \\varphi ^ * _ a ( F _ l ^ { \\gamma _ l - 1 } ) } \\cdot \\ldots \\cdot \\varphi ^ * _ a ( F _ j ^ { \\gamma _ j } ) . \\end{align*}"}
+{"id": "6257.png", "formula": "\\begin{align*} E _ { n } ( x ) \\sim \\begin{cases} ( 2 / \\pi ) \\log ( | x | ) , \\quad & n = 2 , \\\\ [ 5 p t ] - ( \\Gamma ( n / 2 - 1 ) / \\pi ) ( 2 / c _ { 1 } ) ^ { n / 2 - 1 } | x | ^ { 2 - n } , & n > 2 . \\end{cases} \\end{align*}"}
+{"id": "537.png", "formula": "\\begin{align*} f ( t ) : = \\sum _ { k = 1 } ^ { \\infty } \\frac { a _ { k } \\mathrm { e } ^ { - \\lambda _ { k } T } } { \\mathcal { O } _ { \\alpha + \\beta + \\gamma } \\left ( \\Phi _ { k } \\right ) } \\psi _ { k } ( t ) . \\end{align*}"}
+{"id": "8458.png", "formula": "\\begin{align*} \\sum \\limits _ { m = 1 } ^ k \\int _ \\Omega \\left ( | u _ { m } | ^ { q - 1 } u _ { m } - | u _ { m - 1 } | ^ { q - 1 } u _ { m - 1 } \\right ) u _ m \\ , d x & \\geq \\sum _ { m = 1 } ^ k \\frac { q } { q + 1 } \\int _ \\Omega \\left ( | u _ m | ^ { q + 1 } - | u _ { m - 1 } | ^ { q + 1 } \\right ) \\ , d x \\\\ [ 2 m m ] & = \\frac { q } { q + 1 } \\int _ \\Omega \\left ( | u _ k | ^ { q + 1 } - | u _ { 0 } | ^ { q + 1 } \\right ) \\ , d x . \\end{align*}"}
+{"id": "8196.png", "formula": "\\begin{align*} R = f ( B _ 1 ^ { \\star } , B _ 2 ^ { 0 \\star } , B _ 2 ^ { 1 \\star } ) , \\end{align*}"}
+{"id": "1153.png", "formula": "\\begin{align*} \\langle m _ Q , m _ P \\rangle & = \\int _ { \\mathbb R ^ n } g ( x _ Q - y ) h ( y - x _ P ) \\ , d y \\\\ & = \\int _ { \\mathbb R ^ n } g ( x ) h ( x _ Q - x _ P - x ) \\ , d x = ( g * h ) ( x _ Q - x _ P ) . \\end{align*}"}
+{"id": "5407.png", "formula": "\\begin{align*} \\Delta b ^ { S _ { k + 1 } } _ k - \\Delta b ^ { S _ { k + 2 } } _ k = \\frac { 1 } { a _ k } \\ , \\frac { \\lambda _ k \\ , ( 1 - \\Delta b ^ { S _ { k + 2 } } _ { k + 1 } ) } { \\alpha + \\lambda _ { k - 1 } + \\mu _ k } . \\end{align*}"}
+{"id": "6317.png", "formula": "\\begin{align*} [ a ( g _ 1 ) , a ( g _ 2 ) ] = [ a ^ * ( g _ 1 ) , a ^ * ( g _ 2 ) ] = 0 , [ a ( g _ 1 ) , a ^ * ( g _ 2 ) ] = \\langle g _ 1 , g _ 2 \\rangle \\end{align*}"}
+{"id": "8241.png", "formula": "\\begin{align*} \\omega _ i ( { \\bf x } ^ i , y ^ i ) = \\omega _ { i - 1 } ( { \\bf x } ^ { i - 1 } , y ^ { i - 1 } ) \\cap \\eta _ i ^ { y _ i } . \\end{align*}"}
+{"id": "1203.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } ( - \\Delta _ p ) ^ r u ( x ) = \\lambda \\alpha ( p ) \\vert u ( x ) \\vert ^ { \\alpha ( p ) - 2 } u ( x ) \\vert v ( x ) \\vert ^ { \\beta ( p ) } & { \\rm i n } \\ \\ \\Omega , \\\\ ( - \\Delta _ p ) ^ s v ( x ) = \\lambda \\beta ( p ) \\vert u ( x ) \\vert ^ { \\alpha ( p ) } \\vert v ( x ) \\vert ^ { \\beta ( p ) - 2 } v ( x ) & { \\rm i n } \\ \\ \\Omega , \\\\ u = v = 0 & { \\rm i n } \\ \\mathbb { R } ^ N \\setminus \\Omega , \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "5510.png", "formula": "\\begin{align*} ( \\ell - 1 ) ^ { 2 } \\mid \\ell ^ { n } - 1 \\iff \\ell - 1 \\mid w _ { n } ( \\ell ) \\iff \\ell - 1 \\mid n \\cdot 1 _ { S } \\iff n \\cdot 1 _ { S } = 0 _ { S } , \\end{align*}"}
+{"id": "5522.png", "formula": "\\begin{align*} \\ell y : = - y '' ( x ) + q ( x ) y ( a ) = \\lambda y ( x ) , x \\in ( 0 , \\pi ) , \\end{align*}"}
+{"id": "7819.png", "formula": "\\begin{align*} y _ k ^ { \\star } = \\frac { \\sqrt { \\left ( 1 + \\eta _ k \\right ) \\left | \\left ( \\boldsymbol { v } _ { k } \\right ) ^ { \\mathrm { H } } \\ ! \\boldsymbol { g } _ { k } \\right | ^ { 2 } a _ { k } { E } _ { k } } } { \\sum _ { l \\in \\mathcal { K } } \\left | \\left ( \\boldsymbol { v } _ { k } \\right ) ^ { \\mathrm { H } } \\ ! \\boldsymbol { g } _ { l } \\right | ^ { 2 } a _ { l } { E } _ { l } + L \\sigma ^ 2 \\left \\| \\boldsymbol { v } _ k \\right \\| ^ 2 } , \\forall k \\in \\mathcal { K } , \\end{align*}"}
+{"id": "7494.png", "formula": "\\begin{align*} ( \\partial _ t - L _ t ) h = R _ 0 [ h ] + \\nabla R _ 1 [ h ] , \\end{align*}"}
+{"id": "4463.png", "formula": "\\begin{align*} \\int _ { \\partial D _ 1 } f _ 1 \\overline { f } \\rho _ 1 | d z _ 1 | = 0 \\end{align*}"}
+{"id": "4248.png", "formula": "\\begin{align*} \\begin{cases} ( i \\partial _ t + \\Delta ) u = [ 1 + a ] | u | ^ 2 u , \\\\ u | _ { t = 0 } = u _ 0 , \\end{cases} \\end{align*}"}
+{"id": "221.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\dfrac { d \\bar x ^ i } { d \\tau } & = & f \\ , v ^ i = \\bar v ^ i \\\\ \\dfrac { d \\bar v ^ i } { d \\tau } & = & { f ^ 2 } \\ , X ^ i \\left ( \\bar x , \\dfrac 1 f \\ , \\bar v \\right ) + \\dfrac d { d \\tau } ( \\log f ) \\ , \\bar v ^ i \\end{array} \\right . . \\end{align*}"}
+{"id": "8954.png", "formula": "\\begin{align*} B _ { R _ x } ( x ) \\cap \\Omega = \\left \\{ x + \\sum _ { i = 1 } ^ n \\xi _ i v _ i \\in \\R ^ n \\ , : \\ , \\xi \\in B _ { R _ x } ( 0 ) , \\ , \\xi _ n < \\phi _ x ( \\xi _ 1 , \\dots , \\xi _ { n - 1 } ) \\right \\} . \\end{align*}"}
+{"id": "1717.png", "formula": "\\begin{align*} v _ j = \\left ( \\begin{array} { c } v ^ \\prime _ j \\\\ 1 \\end{array} \\right ) \\end{align*}"}
+{"id": "3755.png", "formula": "\\begin{align*} \\left . \\frac { \\partial \\mathrm { E } _ { \\alpha , \\beta } \\left ( z \\right ) } { \\partial \\beta } \\right \\vert _ { \\alpha = 1 / q } = - \\sum _ { h = 0 } ^ { q - 1 } \\frac { \\psi \\left ( \\frac { h } { q } + \\beta \\right ) + \\tilde { Q } \\left ( \\frac { h } { q } + \\beta , z ^ { q } \\right ) } { \\Gamma \\left ( \\frac { h } { q } + \\beta \\right ) } z ^ { h } , \\end{align*}"}
+{"id": "1702.png", "formula": "\\begin{align*} V : = \\sum _ { j = 1 } ^ n v _ j X _ j \\end{align*}"}
+{"id": "9073.png", "formula": "\\begin{align*} X ( 0 ) = J ^ { ( r ) } x ( 0 ) \\in \\mathbb R _ + ^ n . \\end{align*}"}
+{"id": "7876.png", "formula": "\\begin{align*} c _ { k } = \\begin{cases} a _ { \\tau } , & [ k + t + \\tau ] _ { m } = \\tau ; \\\\ - b _ { m - \\tau } , & [ k + t + \\tau ] _ { m } = m ; \\\\ a _ { [ k + t + \\tau ] _ { m } } - b _ { [ k + t ] _ { m } } , & \\end{cases} \\end{align*}"}
+{"id": "1421.png", "formula": "\\begin{align*} E _ \\alpha ( - x ) & = \\frac { 1 } { 2 \\pi i } \\oint _ { C } \\frac { s ^ { \\alpha - 1 } e ^ s } { x + s ^ { \\alpha } } \\ , d s = \\frac { 1 } { 2 \\pi i \\alpha } \\oint _ { C ^ \\prime } \\frac { e ^ { z ^ { \\frac { 1 } { \\alpha } } } } { x + z } \\ , d z \\end{align*}"}
+{"id": "82.png", "formula": "\\begin{align*} 2 c _ 1 + c _ 2 = 2 ( w _ 1 + w _ 3 \\kappa ) - 2 w _ 3 \\theta \\kappa \\geq c \\end{align*}"}
+{"id": "3628.png", "formula": "\\begin{align*} P ^ 2 f ( z , w ) = 2 \\bigg ( \\frac { \\lambda _ { 1 } ^ { 2 } } { 1 - \\lambda _ { 1 } ^ { 2 } } \\bigg ) ^ 2 \\big ( f ( z , w ) - f ( z , - w ) \\big ) , \\end{align*}"}
+{"id": "6521.png", "formula": "\\begin{align*} \\frac { 1 } { n - p } \\frac { d } { d t } C _ p ( K + t L ) \\bigg | _ { t = 0 ^ + } \\geq C _ { p } \\left ( K \\right ) ^ { 1 - \\frac { 1 } { n - p } } C _ { p } \\left ( L \\right ) ^ \\frac { 1 } { n - p } , \\end{align*}"}
+{"id": "1474.png", "formula": "\\begin{align*} E _ i = T _ i ^ o \\cap g _ i F _ { k _ i } ^ { - M ^ 2 } = g _ i F _ { k _ i } ^ { - M } \\cap g _ i F _ { k _ i } ^ { - M ^ 2 } = g _ i F _ { k _ i } ^ { - M ^ 2 } = T _ i ^ { - M ^ 2 } . \\end{align*}"}
+{"id": "9157.png", "formula": "\\begin{align*} G _ { \\beta , \\beta ' } = ( w _ { \\beta , 1 } - v ^ { ( 4 \\ell - 4 n - 2 ) } w _ { \\beta ' , 1 } ) ( w _ { \\beta , 1 } - v ^ { ( 4 \\ell - 4 n + 6 ) } w _ { \\beta ' , 1 } ) \\cdot G _ { \\beta , [ i , n , k + 1 ] } . \\end{align*}"}
+{"id": "1554.png", "formula": "\\begin{align*} t \\ , \\eta _ { \\frac { 1 } { H + 1 } , \\frac { H } { H + 1 } } ( 1 / t ) = \\eta _ { \\frac { 1 } { H + 1 } , \\frac { H } { H + 1 } } ( t ) \\end{align*}"}
+{"id": "7210.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } p + \\partial _ { v } ( h p ) - a \\partial _ { v v } p = 0 , v \\in ( - \\infty , V _ F ] / \\{ V _ R \\} , \\\\ p ( v , 0 ) = p ^ 0 ( v ) , p ( - \\infty , t ) = p ( V _ F , t ) = 0 , \\\\ p ( V ^ - _ R , t ) = p ( V ^ + _ R , t ) , \\partial _ v p ( V ^ - _ R , t ) = \\partial _ v p ( V ^ + _ R , t ) + \\frac { N ( t ) } { a } , \\\\ \\end{cases} \\end{align*}"}
+{"id": "638.png", "formula": "\\begin{align*} \\frac { ( r - 1 ) ! ! } { ( r - 2 ) ! ! } & = 1 + \\frac { 1 } { r - 2 } + \\frac { r - 1 } { ( r - 2 ) ( r - 4 ) } + \\frac { ( r - 1 ) ( r - 3 ) } { ( r - 2 ) ( r - 4 ) ( r - 6 ) } + \\\\ & + \\dots + \\frac { ( r - 1 ) ! ! } { 3 ( r - 2 ) ! ! } , \\end{align*}"}
+{"id": "4986.png", "formula": "\\begin{align*} \\det \\left [ p _ k ( \\lambda _ j ) \\right ] _ { j , k = 1 , \\ldots , N } = Q _ N ( \\{ p \\} ) \\prod _ { 1 \\leq j < k \\leq N } ( \\lambda _ k - \\lambda _ j ) , \\end{align*}"}
+{"id": "3290.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ^ { d + r } ; q ^ d ) _ k ^ { d - r } ( q ^ r ; q ^ d ) _ k ^ { r - 1 } ( q ^ { r - d } ; q ^ d ) _ k q ^ { d k } } { ( q ^ d ; q ^ d ) _ k ^ d } \\\\ & = \\frac { [ d ] } { [ r ] } \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ^ { d + r } ; q ^ d ) _ k ^ { d - r - 1 } ( q ^ r ; q ^ d ) _ k ^ r ( q ^ { r - d } ; q ^ d ) _ k q ^ { d k } } { ( q ^ d ; q ^ d ) _ k ^ d } - \\frac { [ d - r ] } { q ^ { - r } [ r ] } \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ^ { d + r } ; q ^ d ) _ k ^ { d - r - 1 } ( q ^ r ; q ^ d ) _ k ^ { r + 1 } q ^ { d k } } { ( q ^ d ; q ^ d ) _ k ^ d } . \\end{align*}"}
+{"id": "735.png", "formula": "\\begin{align*} g _ 0 ( A ) & = t _ { J _ k ( v ) - 1 , q - 1 } - a \\\\ & = ( a - r - 1 ) b + v a ( J _ k ( v ) - J _ { k - 1 } ( v ) - 1 ) - a + \\frac { v a ( a - r ) J _ { k - 1 } ( v ) } { J _ k ( v ) } \\ , . \\end{align*}"}
+{"id": "2862.png", "formula": "\\begin{align*} R _ { X } g ( x ) : = \\sum _ { k = 0 } ^ \\infty \\frac { \\mathcal { M } ^ k g ( x ) } { 2 ^ k \\| \\mathcal { M } \\| ^ k _ { X \\to X } } , \\end{align*}"}
+{"id": "8780.png", "formula": "\\begin{align*} \\int _ \\R ( z - x ) ^ + \\nu ( d z ) - \\int _ \\R ( y - x ) ^ + \\mu ( d y ) = \\int _ \\R ( x - z ) ^ + \\nu ( d z ) - \\int _ \\R ( x - y ) ^ + \\mu ( d y ) = \\frac 1 2 \\left ( u _ \\nu ( x ) - u _ \\mu ( x ) \\right ) . \\end{align*}"}
+{"id": "4383.png", "formula": "\\begin{gather*} [ \\overline { u } _ i , \\overline { u } _ i + \\Delta u _ i ] = \\{ u _ i \\in \\R \\colon ( - 2 \\overline { u } _ i - \\Delta u _ i ) u _ i + u _ i ^ 2 \\leq - \\overline { u } _ i ^ 2 - \\overline { u } _ i \\cdot \\Delta u _ i \\} . \\end{gather*}"}
+{"id": "6038.png", "formula": "\\begin{align*} v _ { \\pm } = \\frac { 1 } { 2 N ^ \\gamma } [ ( E _ { A } - E _ { C } ) ( 1 - 3 \\rho ^ { A } ) + ( E _ { B } - E _ { C } ) ( 1 - 3 \\rho ^ { B } ) \\pm \\delta ] , \\\\ \\end{align*}"}
+{"id": "6725.png", "formula": "\\begin{align*} a _ { n - 1 } y + a _ 1 E ^ 2 y + . . . + a _ { n - 2 } E ^ { n - 1 } y = 0 . \\end{align*}"}
+{"id": "6089.png", "formula": "\\begin{align*} S ( x ) = \\begin{cases} x & x \\in H _ 0 , \\\\ - x - \\sum m ( S ( x _ { ( 1 ) } ) \\otimes x _ { ( 2 ) } ) & x \\in H _ n n \\geq 1 . \\end{cases} \\end{align*}"}
+{"id": "8538.png", "formula": "\\begin{align*} \\kappa _ { \\Sigma } ( g ) & = g ( e _ 1 ) - e _ 1 \\\\ & = ( 1 + \\eta \\log ( g ) ) \\cdot e _ 1 + \\eta c ( g ) \\cdot e _ 2 - e _ 1 \\\\ & = \\log ( g ) \\cdot \\eta e _ 1 + c ( g ) \\cdot \\eta e _ 2 \\end{align*}"}
+{"id": "8490.png", "formula": "\\begin{align*} U _ m ^ { ( \\theta ) } ( x , y ) : = \\Big | ( u _ m + \\theta \\varepsilon \\varphi ) ( x ) - ( u _ m + \\theta \\varepsilon \\varphi ) ( y ) \\Big | ^ { p - 2 } \\Big ( ( u _ m + \\theta \\varepsilon \\varphi ) ( x ) - ( u _ m + \\theta \\varepsilon \\varphi ) ( y ) \\Big ) . \\end{align*}"}
+{"id": "4249.png", "formula": "\\begin{align*} ( i \\partial _ t + \\Delta ) u = a | u | ^ 2 u , u | _ { t = 0 } = u _ 0 . \\end{align*}"}
+{"id": "1679.png", "formula": "\\begin{align*} [ V , ( v ^ j ) ] \\sim [ W , ( w ^ j ) ] \\Leftrightarrow \\exists \\ \\phi : V \\xrightarrow { \\cong \\ } W \\mbox { w i t h } \\ \\phi ( v ^ j ) = w ^ j \\ \\ \\mbox { f o r a l l } j \\in \\{ 0 , 1 , 2 , 3 \\} . \\end{align*}"}
+{"id": "6974.png", "formula": "\\begin{align*} \\mathbf { z } \\overline { \\eta } ( \\mathbf { z } ) + \\overline { \\mathbf { z } } \\eta ( \\mathbf { z } ) = 2 \\eta ( \\mathbf { z } ) \\mathbf { z } . \\end{align*}"}
+{"id": "6207.png", "formula": "\\begin{align*} f ( u ) & = - \\left ( C _ i D _ i + C _ g D _ g \\right ) u ( 1 - u ) ^ 2 - C _ g D _ g u ( 1 - u ) , \\\\ D ( u ) & = D _ i \\left ( 1 - 4 u + 3 u ^ 2 \\right ) + D _ g \\left ( 4 u - 3 u ^ 2 \\right ) , \\\\ g ( u ) & = \\lambda _ g u ( 1 - u ) + \\left [ \\lambda _ i - \\lambda _ g - \\left ( k _ i - k _ g \\right ) \\right ] u ( 1 - u ) ^ 2 - k _ g u . \\end{align*}"}
+{"id": "8929.png", "formula": "\\begin{align*} \\dim ( \\mathcal { M } ( L ) ) = \\dim ( \\mathcal { M } ( L / \\gamma _ 2 ( L ) ) ) + \\dim ( \\gamma _ 2 ( L ) ) ( \\dim ( \\frac { L } { \\gamma _ 2 ( L ) } ) - 1 ) - \\sum _ { i = 2 } ^ { c } \\dim ( k e r ( \\lambda _ i ) ) . \\end{align*}"}
+{"id": "5971.png", "formula": "\\begin{align*} d _ H ( \\vec { x } ^ { ( j ) } _ { m _ 1 , \\ell _ 1 } , \\vec { x } ^ { ( j ) } _ { m _ 2 , \\ell _ 2 } ) = | \\ell _ 1 - \\ell _ 2 | , \\end{align*}"}
+{"id": "1122.png", "formula": "\\begin{align*} \\sum _ { i = j - 1 } ^ { j + 1 } I _ i ( x ) \\lesssim \\left ( u _ { p \\wedge q , \\widetilde { \\lambda } } ^ * \\right ) _ { Q ^ * } + \\left ( u _ { p \\wedge q , \\widetilde { \\lambda } } ^ * \\right ) _ Q + \\left ( u _ { p \\wedge q , \\widetilde { \\lambda } } ^ * \\right ) _ { Q ^ { * * } } , \\end{align*}"}
+{"id": "1977.png", "formula": "\\begin{align*} \\tilde { R } _ { 1 1 } = u ^ { - 2 k } \\left \\lbrace R _ { 1 1 } - k ( n - 2 ) ( \\ln u ) _ { s s } - k \\frac { \\Delta u } { u } + k \\frac { \\vert \\nabla u \\vert ^ 2 } { u ^ 2 } \\right \\rbrace \\ , . \\end{align*}"}
+{"id": "3708.png", "formula": "\\begin{align*} \\psi _ { k } ^ { ( 0 ) } ( x ) & = e ^ { - i \\theta _ k + i \\pi } \\phi _ { k } ( x ) \\ ; , \\\\ \\psi _ { k } ^ { ( 1 ) } ( x ) & = \\alpha _ R \\phi _ k ( 0 ) e ^ { - i \\theta _ k + i \\pi } \\left ( \\sum _ { n \\neq k } \\frac { \\phi _ n ( x ) \\overline { \\phi _ n ( 0 ) } } { E _ n - E _ k } + \\int _ { \\sigma _ c ( H _ 0 ) } \\frac { \\chi _ { \\lambda } ( x ) \\overline { \\chi _ { \\lambda } ( 0 ) } } { \\lambda - E _ k } d \\mu ( \\lambda ) \\right ) \\ ; , \\end{align*}"}
+{"id": "1972.png", "formula": "\\begin{align*} E q u > & p ^ s + 1 - ( \\tau + 1 ) p - ( \\tau - 1 ) p ^ { s - 1 } + \\tau p ^ { s - 2 } + \\frac { p ( \\tau - 1 ) - \\tau } { p - \\tau + 1 } \\\\ > & p ^ s + 1 - p ^ 2 - ( p - 2 ) p ^ { s - 1 } - \\tau \\\\ = & p ^ { s - 1 } - p ^ 2 + p ^ { s - 1 } - \\tau + 1 \\\\ > & 1 , \\end{align*}"}
+{"id": "238.png", "formula": "\\begin{align*} \\bar { A } ( \\bar { x } ) = \\frac { 1 } { h ( x ) } A ( x ) \\qquad \\bar { b } ( \\bar { x } ) = \\frac { c } { h ( x ) } b ( x ) . \\end{align*}"}
+{"id": "4146.png", "formula": "\\begin{align*} & \\tilde { \\theta } _ \\ell = w _ { \\ell j } r ^ { j + 1 / 2 } \\eta _ j + o ( r ^ { j + 1 / 2 } ) , \\ell = 1 , \\ldots , j - 1 , \\\\ & \\tilde { \\theta } _ j = r ^ { j + 1 / 2 } \\eta _ j + o ( r ^ { j + 1 / 2 } ) , \\\\ & \\tilde { \\theta } _ k = o ( r ^ { j + 1 / 2 } ) , k = j + 1 , \\ldots , J . \\end{align*}"}
+{"id": "8921.png", "formula": "\\begin{align*} J : = \\{ j \\in \\Sigma : t _ { 0 } \\sigma ( j ) = \\rho ( j ) \\} . \\end{align*}"}
+{"id": "5085.png", "formula": "\\begin{align*} \\overline { \\mathsf { a } } _ { s } \\cdot f & = L _ s \\cdot f \\\\ \\overline { \\mathsf { e } _ s } \\cdot f & = E _ s \\cdot f \\end{align*}"}
+{"id": "4169.png", "formula": "\\begin{align*} \\varphi ( g ^ { n } _ 1 g _ 3 ) = \\varphi \\big ( g ^ { i } _ 1 g _ 3 \\big ) \\varphi \\big ( g ^ { j } _ 1 \\big ) \\overset { ( \\ref { e q : 5 } ) } { = } \\varphi ( g _ 3 ) \\varphi \\big ( g ^ { i } _ 1 \\big ) \\varphi \\big ( g ^ { j } _ 1 \\big ) \\overset { { \\bf A 3 } } { = } \\varphi ( g _ 3 ) \\varphi ( g ^ { n } _ 1 ) \\ , , \\end{align*}"}
+{"id": "8385.png", "formula": "\\begin{align*} H ^ 1 ( X _ { \\underline { w } } , T _ { \\underline { w } } ) = H ^ 1 ( X _ { \\underline { w } } , \\mathcal { L } _ { \\alpha _ { 1 } } ) . \\end{align*}"}
+{"id": "6025.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\mathbb { E } _ { \\nu _ { \\rho } } \\Big [ \\sup _ { 0 \\leq s \\leq T } \\Big | M _ s ^ N - M _ { s ^ - } ^ N \\Big | \\Big ] = 0 , \\end{align*}"}
+{"id": "4409.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathcal { X } \\subseteq \\{ 0 , 1 \\} ^ n } & \\sum _ { i \\in [ n ] } \\sum _ { j \\in [ i ] } p _ { i , j } x _ i x _ j \\end{align*}"}
+{"id": "3297.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\frac { - W _ { - 1 } ( - \\delta ) } { \\log ( 1 / \\delta ) } = 1 . \\end{align*}"}
+{"id": "8619.png", "formula": "\\begin{align*} \\| v _ 1 ( t ; \\cdot ) \\| _ { L ^ \\infty ( \\mathbb { R } ) } = \\| ( \\partial _ x u ) ( \\cdot , t ) \\| _ { L ^ \\infty ( \\mathbb { R } ) } < C _ 1 q ^ { - 1 } ( t ) , \\end{align*}"}
+{"id": "4854.png", "formula": "\\begin{align*} u _ j ( \\hat { z } ) & \\leq u _ j ( \\hat { z } - s \\hat { w } ) + \\langle \\nabla ^ + u _ j ( \\hat { z } - s \\hat { w } ) , s \\hat { w } \\rangle \\\\ & = u _ j ( \\hat { z } - s \\hat { w } ) - \\frac { s } { \\sqrt { 1 + | \\nabla ^ + u _ j ( \\hat { x } ^ * ) | ^ 2 } } \\langle \\nabla ^ + u _ j ( \\hat { z } - s \\hat { w } ) , \\nabla ^ + u _ j ( \\hat { x } ^ * ) \\rangle . \\end{align*}"}
+{"id": "359.png", "formula": "\\begin{align*} \\gcd ( z - y , \\phi _ p ( z , y ) ) = 1 . \\end{align*}"}
+{"id": "6896.png", "formula": "\\begin{align*} M ( z ) = p ( z ) + \\frac { z p ' ( z ) } { \\eta p ( z ) + \\gamma } p ( z ) = \\cosh \\sqrt { w ( z ) } \\end{align*}"}
+{"id": "5828.png", "formula": "\\begin{align*} \\frac { \\partial u ^ A } { \\partial t } - \\mathcal { M } _ \\Sigma ( u ) ^ A = \\left ( \\sqrt { \\frac { \\det g _ 0 } { \\det g _ u } } ( h _ u ) ^ { A B } ( h _ 0 ) _ { B C } - \\delta _ C ^ A \\right ) \\cdot \\mathcal { M } _ \\Sigma ( u ) ^ C . \\end{align*}"}
+{"id": "6.png", "formula": "\\begin{align*} \\limsup _ { \\delta \\to 0 } \\ \\sup _ { q \\in Q } \\ \\sup _ { | y | < \\delta } \\| q ( \\cdot + y ) - q ( \\cdot ) \\| _ { H ^ \\sigma } = 0 . \\end{align*}"}
+{"id": "4811.png", "formula": "\\begin{align*} \\mathcal { S } ^ { ( i n t ) } _ k : = \\Big \\{ \\mathbf { c } ' \\in \\mathbb { R } ^ { n } : \\mathbf { l } ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } } \\leq \\mathbf { c } ' \\leq \\mathbf { u } ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } } \\Big \\} , \\end{align*}"}
+{"id": "5660.png", "formula": "\\begin{align*} \\boldsymbol { b } ^ { ( 4 ) } & = u \\boldsymbol { a } ^ { ( 4 ) } + r ( a _ 2 , - a _ 1 , a _ 4 , - a _ 3 ) + s ( a _ 3 , - a _ 4 , - a _ 1 , a _ 2 ) + t ( a _ 4 , a _ 3 , - a _ 2 , - a _ 1 ) \\\\ & = ( u a _ 1 + r a _ 2 + s a _ 3 + t a _ 4 , u a _ 2 - r a _ 1 - s a _ 4 + t a _ 3 , u a _ 3 + r a _ 4 - s a _ 1 - t a _ 2 , u a _ 4 - r a _ 3 + s a _ 2 - t a _ 1 ) . \\end{align*}"}
+{"id": "3003.png", "formula": "\\begin{align*} \\overline { \\mathrm { s c a l } } = \\sum _ { i = 0 } ^ { 2 n } \\overline { \\mathrm { R i c } } ( e _ i , e _ i ) , \\end{align*}"}
+{"id": "8808.png", "formula": "\\begin{align*} f ( x , m ) = \\frac { z _ - - m } { z _ - - y _ + } ( x - y _ + ) ^ \\rho + \\frac { m - y _ + } { z _ - - y _ + } ( z _ - - x ) ^ \\rho - \\frac { z _ + - m } { z _ + - y _ - } ( x - y _ - ) ^ \\rho - \\frac { m - y _ - } { z _ + - y _ - } ( z _ + - x ) ^ \\rho . \\end{align*}"}
+{"id": "4799.png", "formula": "\\begin{align*} u _ \\star ( x ) = u _ \\ast \\left ( { \\bf b } _ \\theta ( F ^ o _ \\theta ( x ) ) ^ n \\right ) , \\end{align*}"}
+{"id": "3325.png", "formula": "\\begin{align*} { \\rm r a n k \\ L i e } \\left ( \\partial _ { x _ 1 } , \\dots , \\partial _ { x _ { m _ 0 } } , Y \\right ) ( x , t ) = N + 1 , \\forall \\ , ( x , t ) \\in \\R ^ { N + 1 } , \\end{align*}"}
+{"id": "5244.png", "formula": "\\begin{align*} Z _ { i } = \\frac { p _ { i } } { \\alpha p _ { i } + \\left ( 1 - \\alpha \\right ) q _ { i } } \\end{align*}"}
+{"id": "8881.png", "formula": "\\begin{align*} B _ 1 + B _ 2 & = \\frac { 2 N ( 2 - p ) - 4 \\tau } { p } \\int _ { | x | < R } | x | ^ { - \\tau } | u | ^ p ( I _ \\alpha * | \\cdot | ^ { - \\tau } | u | ^ { p } ) d x \\\\ & + O \\bigg ( \\int _ { | x | > R } | x | ^ { - \\tau } | u | ^ { p } ( I _ \\alpha * | \\cdot | ^ { - \\tau } | u | ^ p ) d x \\bigg ) \\\\ & = \\frac { 2 ( N ( 2 - p ) - 2 \\tau ) } p \\mathcal { P } [ u ] + O \\bigg ( \\int _ { | x | > R } | x | ^ { - \\tau } | u | ^ { p } ( I _ \\alpha * | \\cdot | ^ { - \\tau } | u | ^ p ) d x \\bigg ) . \\end{align*}"}
+{"id": "4646.png", "formula": "\\begin{align*} \\left ( \\frac { q } { p } \\right ) & = \\left ( \\frac { - 1 } { p } \\right ) \\left ( \\frac { | q | } { p } \\right ) = ( - 1 ) ^ { \\frac { p - 1 } { 2 } } \\left ( \\frac { p } { | q | } \\right ) \\cdot ( - 1 ) ^ { \\frac { p - 1 } { 2 } \\frac { | q | - 1 } { 2 } } = \\left ( \\frac { p } { | q | } \\right ) \\cdot ( - 1 ) ^ { \\frac { p - 1 } { 2 } \\frac { | q | + 1 } { 2 } } \\\\ & = \\left ( \\frac { p } { | q | } \\right ) \\cdot ( - 1 ) ^ { \\frac { p - 1 } { 2 } \\frac { - q + 1 } { 2 } } = \\left ( \\frac { p } { | q | } \\right ) \\cdot ( - 1 ) ^ { \\frac { p - 1 } { 2 } \\frac { q - 1 } { 2 } } \\end{align*}"}
+{"id": "2378.png", "formula": "\\begin{align*} R _ { k , l } = R _ k ( \\lambda ( l ) ) \\ \\ \\ \\ \\ \\ \\tilde { R } _ { k , l } = \\frac { B ( 0 ) \\dots B ( l - 1 ) } { D ( 1 ) \\dots D ( l ) } \\frac { A _ 0 \\dots A _ { k - 1 } } { C _ 1 \\dots C _ k } R _ { k , l } \\ \\ \\ \\ \\ \\forall k , l = 0 , \\dots , N \\ . \\end{align*}"}
+{"id": "3159.png", "formula": "\\begin{align*} ( i + j n ) ^ { \\sigma } = i k + j \\ \\ i \\in [ n ] \\ , \\ , j \\in [ k ] . \\end{align*}"}
+{"id": "8962.png", "formula": "\\begin{align*} \\nabla u _ j - R _ j ^ \\eta = F _ j ^ \\eta + G _ j ^ \\eta \\quad , \\end{align*}"}
+{"id": "2005.png", "formula": "\\begin{align*} \\left | \\widetilde { ( e _ M ^ { q + 1 } ) } _ l \\right | ^ 2 & \\le 6 \\biggl ( \\left | \\widetilde { ( e _ M ^ { 0 , + } ) } _ l \\right | ^ 2 + \\left | \\widetilde { ( e _ M ^ { 0 , - } ) } _ l \\right | ^ 2 \\\\ & \\ + ( q + 1 ) \\sum _ { k = 0 } ^ q \\left ( \\left | \\widetilde { ( \\xi ^ { k , + } ) } _ l \\right | ^ 2 + \\left | \\widetilde { ( \\xi ^ { k , - } ) } _ l \\right | ^ 2 + \\left | \\widetilde { ( \\eta ^ { k , + } ) } _ l \\right | ^ 2 + \\left | \\widetilde { ( \\eta ^ { k , - } ) } _ l \\right | ^ 2 \\right ) \\biggr ) . \\end{align*}"}
+{"id": "6841.png", "formula": "\\begin{align*} \\mathbf { 1 } _ N = \\begin{pmatrix} e ^ { - i \\omega _ { N - 1 } } & & & \\\\ & \\ddots & & \\\\ & & e ^ { - i \\omega _ { N - 1 } } & \\\\ & & & e ^ { i ( N - 1 ) \\omega _ { N - 1 } } \\end{pmatrix} \\begin{pmatrix} [ F _ { N - 1 } ( \\phi _ N , \\ldots , \\omega _ { N - 2 } ) ] ^ { - 1 } & 0 \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} X & \\\\ & u ^ { ( N - 1 ) } _ { N N } \\end{pmatrix} . \\end{align*}"}
+{"id": "2069.png", "formula": "\\begin{align*} - \\int _ 0 ^ s U ( u , x ) f ' ( u ) \\mathrm { d } u + f ( s ) U ( s , x ) - f ( 0 ) U ( 0 , x ) = - \\alpha \\int _ 0 ^ s \\int _ 0 ^ 1 f ( u ) A ( x , y ) U ( u , y ) \\mathrm { d } y \\mathrm { d } u . \\end{align*}"}
+{"id": "5264.png", "formula": "\\begin{align*} L _ { d } G I \\left ( p \\| q \\right ) = \\frac { \\sum _ { j } p _ { j } } { a - b } \\sum _ { i } \\left [ \\frac { \\overline { T } ^ { a } _ { i } } { \\overline { p } ^ { a - 1 } _ { i } } - \\frac { \\overline { T } ^ { b } _ { i } } { \\overline { p } ^ { b - 1 } _ { i } } \\right ] \\end{align*}"}
+{"id": "1635.png", "formula": "\\begin{align*} \\lambda _ { 0 } = \\lambda _ { 0 } \\left ( T , c \\right ) = 1 6 \\left ( T + c \\right ) ^ { 2 } > 1 6 c ^ { 2 } > 6 4 . \\end{align*}"}
+{"id": "8233.png", "formula": "\\begin{align*} P ( { \\bf s } \\otimes { \\bf x } _ 2 = 1 | Y _ 1 = 0 ) = \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "5749.png", "formula": "\\begin{align*} \\left \\Vert u ( t ) - e ^ { \\gamma _ * t } \\sum _ { i \\in I _ 1 } \\textup { R e } \\left ( w _ i e ^ { \\mathbf { i } \\beta _ i t } \\right ) \\varphi _ i - e ^ { \\gamma _ * t } \\sum _ { i \\in I _ 2 } ( t c _ { i , 3 } + c _ { i , 4 } ) \\varphi _ i \\right \\Vert _ { H ^ 1 } = O ( e ^ { ( \\gamma _ * - \\varepsilon ) t } ) . \\end{align*}"}
+{"id": "6501.png", "formula": "\\begin{align*} \\Phi _ 1 : = \\Phi , \\Phi _ n ( t ) : = \\int _ 0 ^ t \\Phi ( t - s ) \\Phi _ { n - 1 } ( s ) d s , t \\in \\real _ + , \\ ; n \\in \\mathbb { N } ^ * . \\end{align*}"}
+{"id": "7902.png", "formula": "\\begin{align*} H ( \\vec { v } , \\Sigma ) = \\frac { 1 } { 2 } \\int _ { \\Omega ( t ) } | \\vec { v } | ^ { 2 } d ^ { 3 } x + \\tau \\int _ { \\Sigma ( t ) } d s , \\end{align*}"}
+{"id": "3170.png", "formula": "\\begin{align*} ( x , a _ s , \\ldots , a _ 1 , y ) ^ { \\sigma } = \\left ( a _ s , \\ldots , a _ 1 , \\left \\lfloor \\frac { y k + x } { t } \\right \\rfloor , ( y k + x ) \\bmod t \\right ) . \\end{align*}"}
+{"id": "4673.png", "formula": "\\begin{align*} ( 1 - \\| p \\| _ \\infty ) \\cdot p _ i & \\leq \\pi _ S ( i ) \\leq 2 \\cdot p _ i , \\textstyle { \\sum _ { k } } \\ , p _ k = S , \\\\ \\pi _ { S - 1 } ( i ) & \\leq \\pi _ S ( i ) , \\\\ \\P _ S ( \\{ i , j \\} \\subseteq I ) & \\leq \\pi _ S ( i ) \\cdot \\pi _ S ( j ) , i \\neq j . \\end{align*}"}
+{"id": "3074.png", "formula": "\\begin{align*} \\psi _ { u } ( t ) = \\left ( t ^ { \\beta _ 0 } , \\displaystyle \\sum _ { \\beta _ 1 \\leq j < \\beta _ { g } } c _ j t ^ { j } + u t ^ { \\beta _ { g } } + \\displaystyle \\sum _ { j > \\beta _ { g } } s _ j ( u ) t ^ { j } \\right ) , \\end{align*}"}
+{"id": "4045.png", "formula": "\\begin{align*} \\Gamma : = k _ { \\mathcal { Z } } ( \\cdot , \\cdot ) _ { \\mathcal { H } } \\end{align*}"}
+{"id": "2539.png", "formula": "\\begin{align*} \\partial _ x ( u _ { } ) - ( \\partial _ x u ) _ { } = u ( x _ 1 ) \\ , \\delta _ { x _ 1 } - u ( x _ 0 ) \\ , \\delta _ { x _ 0 } \\end{align*}"}
+{"id": "2227.png", "formula": "\\begin{align*} \\delta = { 1 6 \\over 3 } + \\varepsilon _ * , \\delta _ 0 = { 8 \\over 3 } + \\varepsilon _ { 0 * } . \\end{align*}"}
+{"id": "5117.png", "formula": "\\begin{align*} S ( p ) = \\left [ - \\frac { \\sum _ { i } p ^ { \\alpha t } _ { i } - \\sum _ { i } p ^ { \\alpha t ^ { - 1 } } _ { i } } { \\alpha t - \\alpha t ^ { - 1 } } \\right ] _ { \\alpha = 1 } = - \\frac { \\sum _ { i } p ^ { t } _ { i } - \\sum _ { i } p ^ { t ^ { - 1 } } _ { i } } { t - t ^ { - 1 } } \\end{align*}"}
+{"id": "9003.png", "formula": "\\begin{align*} e _ { 1 1 } \\cdot e _ { 1 1 } = - e _ { 1 1 } \\cdot e _ { n n } = e _ { n n } \\cdot e _ { n n } = e _ { 1 n } \\end{align*}"}
+{"id": "6323.png", "formula": "\\begin{align*} \\widetilde { K } ( x , y ) = - \\sum _ { p \\in \\pi \\mathbb { N } _ 0 ^ 3 } n \\widehat { \\omega } _ { \\ell , \\lambda } ( p ) u _ p ( x ) u _ p ( y ) \\ , K ( x , y ) = - \\sum _ { p \\in \\pi \\mathbb { N } _ 0 ^ 3 \\setminus \\{ 0 \\} } n \\widehat { \\omega } _ { \\ell , \\lambda } ( p ) u _ p ( x ) u _ p ( y ) . \\end{align*}"}
+{"id": "3235.png", "formula": "\\begin{align*} K ^ { \\alpha \\gamma } | _ { \\beta \\delta } ( x , y ) & = K ^ { \\gamma \\alpha } | _ { \\delta \\beta } ( x , y ) \\\\ K ^ { \\beta \\delta } | _ { \\alpha \\gamma } ( y , x ) & = K ^ { \\delta \\beta } | _ { \\gamma \\alpha } ( y , x ) \\\\ K ^ \\alpha _ { \\ ; \\ ; \\ , \\gamma } | _ \\beta ^ { \\ ; \\ ; \\ , \\delta } ( x , y ) & = K ^ \\delta _ { \\ ; \\ ; \\ , \\beta } | _ \\gamma ^ { \\ ; \\ ; \\ , \\alpha } ( y , x ) = \\overline { K ^ \\gamma _ { \\ ; \\ ; \\ , \\alpha } | _ \\delta ^ { \\ ; \\ ; \\ , \\beta } ( x , y ) } \\ : . \\end{align*}"}
+{"id": "7532.png", "formula": "\\begin{align*} \\begin{dcases} \\partial _ t \\rho _ 0 + \\nabla \\cdot m _ 0 = 0 \\\\ \\partial _ t m _ 0 + \\nabla \\cdot \\big ( \\frac { m _ 0 \\otimes m _ 0 } { \\rho _ 0 } \\big ) + \\nabla p _ 1 ( \\rho _ 0 ) + \\rho _ 0 \\nabla h _ 2 ^ \\prime ( n _ 0 ) = 0 \\\\ - \\delta \\Delta h _ 2 ^ \\prime ( n _ 0 ) + n _ 0 = \\rho _ 0 \\ , \\end{dcases} \\end{align*}"}
+{"id": "1953.png", "formula": "\\begin{align*} L : = 2 \\cdot 4 ^ { \\frac { s j _ 0 } { 2 } } \\| u \\| _ { L ^ { \\infty } ( B _ { 4 R ( x _ 0 ) } ) } + \\operatorname { T a i l } ( u ; x _ 0 , 4 R ) + \\| f \\| ^ { \\frac { 1 } { p - 1 } } _ { L ^ \\gamma ( B _ { 4 R ( x _ 0 ) } ) } . \\end{align*}"}
+{"id": "7597.png", "formula": "\\begin{align*} \\frac { d } { d t } M _ { \\varphi _ { R } } [ \\psi ( t ) ] & \\leq 8 \\| \\Delta \\psi ( t ) \\| _ { 2 } ^ { 2 } - 8 \\mu \\gamma _ { q } \\| \\psi ( t ) \\| _ { q } ^ { q } - 8 \\| \\psi ( t ) \\| _ { 4 ^ * } ^ { 4 ^ * } \\\\ & + ( C \\xi + R ^ { - 1 6 } ) \\| \\Delta \\psi ( t ) \\| ^ { 2 } _ { 2 } + O ( R ^ { - 4 } + \\xi ^ { - \\frac { q - 2 } { 1 0 - q } } R ^ { - \\frac { 4 ( q - 2 ) ( N - 1 ) } { 1 0 - q } } ) ( N = 5 ) , \\end{align*}"}
+{"id": "829.png", "formula": "\\begin{align*} \\nabla _ 0 t _ { i j } = 2 \\Psi g _ { i j } + y _ i \\Psi _ { j } + y _ { j } \\Psi _ { i } , \\end{align*}"}
+{"id": "5667.png", "formula": "\\begin{align*} ( q , q ) _ q ( q , 2 ^ b p _ 1 \\cdots p _ s ) _ q = 1 \\quad ( ) , ( q , p _ i ) _ { p _ i } ( p _ i , p _ i ) _ { p _ i } = 1 \\quad ( ) . \\end{align*}"}
+{"id": "2288.png", "formula": "\\begin{align*} \\det A = \\frac { 1 } { n ! } \\sum _ { \\rho , \\sigma \\in S _ { n } } s g n ( \\rho ) s g n ( \\sigma ) A _ { \\rho ( 1 ) , \\sigma ( 1 ) } A _ { \\rho ( 2 ) , \\sigma ( 2 ) } \\dots A _ { \\rho ( n ) , \\sigma ( n ) } , \\end{align*}"}
+{"id": "8797.png", "formula": "\\begin{align*} ( \\rho - 2 ) \\ln { \\frac { 1 - \\alpha _ \\rho } { \\alpha _ \\rho } } = \\ln { \\frac { \\rho - 1 - \\alpha _ \\rho } { 1 - \\alpha _ \\rho } } = ( \\rho - 2 ) \\int _ 0 ^ 1 \\frac { d u } { 1 - \\alpha _ \\rho + ( \\rho - 2 ) u } . \\end{align*}"}
+{"id": "7811.png", "formula": "\\begin{align*} \\max _ { \\boldsymbol { x } , \\mu } \\ \\mathcal { F } ( \\boldsymbol { x } , \\mu ) = f ( \\boldsymbol { x } ) - \\mu \\Phi ( \\boldsymbol { x } ) , \\end{align*}"}
+{"id": "1392.png", "formula": "\\begin{align*} \\lim _ { Z \\to \\infty } \\frac { P ( Z ) } { Z ^ { \\frac { 5 } { 3 } } } = p _ \\infty > 0 . \\end{align*}"}
+{"id": "8539.png", "formula": "\\begin{align*} \\kappa _ { C } ( g ) & = g ( e _ 2 ) - e _ 2 \\\\ & = b ( g ) \\cdot \\eta e _ 1 - \\log ( g ) \\cdot \\eta e _ 2 . \\end{align*}"}
+{"id": "1976.png", "formula": "\\begin{align*} \\gamma = - d t ^ 2 + a ( t ) ^ 2 g ^ { K } \\end{align*}"}
+{"id": "8273.png", "formula": "\\begin{align*} ( A + B ) ' = A ' + B ' = \\{ ( y ' , x ' + B ' y ' ) : ( y ' , x ' ) \\in A ' \\} , \\end{align*}"}
+{"id": "6782.png", "formula": "\\begin{align*} \\Gamma = \\widetilde { \\Gamma } \\begin{bmatrix} c _ 1 & c _ 2 & c _ 3 \\\\ d _ 1 & d _ 2 & d _ 3 \\end{bmatrix} , \\end{align*}"}
+{"id": "9084.png", "formula": "\\begin{align*} X _ i ( 0 ) \\geq 0 \\Longrightarrow X _ i ( t ) \\geq 0 , \\mbox { f o r a l l $ t = 0 , 1 , 2 , \\ldots $ } \\end{align*}"}
+{"id": "6458.png", "formula": "\\begin{align*} \\operatorname { H e s s } E _ 3 ( \\phi ) ( V , V ) : = \\frac { d ^ 2 } { d t ^ 2 } \\big | _ { t = 0 } E _ 3 ( \\phi _ t ) \\end{align*}"}
+{"id": "3691.png", "formula": "\\begin{align*} a _ d ( P _ { n , d } ) \\leq a ( C _ { n , d } ) = 2 d - 2 \\sum _ { i = 1 } ^ d \\cos ( 2 \\pi k / n ) . \\end{align*}"}
+{"id": "1325.png", "formula": "\\begin{align*} \\dot { q } & = \\rho ( a ) , \\\\ \\overline { \\nabla } ^ \\ast _ a p & = \\pm \\rho ^ \\ast \\bigl ( \\mathrm { d } H ^ { \\mathrm { h o r } } ( p ) \\bigr ) , \\end{align*}"}
+{"id": "6294.png", "formula": "\\begin{align*} N = \\tilde { O } \\left ( \\frac { 1 } { r } \\log _ 2 \\left ( \\frac { \\mu _ r R _ 0 ^ r } { 2 \\varepsilon } \\right ) \\right ) , \\end{align*}"}
+{"id": "246.png", "formula": "\\begin{align*} \\frac { d ^ 2 x _ 2 } { d t _ 2 ^ 2 } + \\alpha \\frac { d x _ 2 } { d t _ 2 } + b _ 2 ( x _ 2 ) = 0 , \\end{align*}"}
+{"id": "2527.png", "formula": "\\begin{align*} I ( M , L ) = \\bigcup _ m I ^ m ( M , L ) \\ ; , I ^ { ( \\infty ) } ( M , L ) = I ^ { - \\infty } ( M , L ) : = \\bigcap _ m I ^ m ( M , L ) \\ ; . \\end{align*}"}
+{"id": "6806.png", "formula": "\\begin{align*} h ( g ) = \\sum _ { j = 1 } ^ M \\sum _ { i = 1 } ^ Q c _ { i j } & e ^ { i k _ { i j } ^ 1 \\phi _ 1 } \\sin ^ { m _ { i j } ^ 1 } ( \\psi _ 1 ) \\cos ^ { n _ { i j } ^ 1 } ( \\psi _ 1 ) \\cdots e ^ { i k _ { i j } ^ { N - 1 } \\phi _ { N - 1 } } \\sin ^ { m _ { i j } ^ { N - 1 } } ( \\psi _ { N - 1 } ) \\cos ^ { n _ { i j } ^ { N - 1 } } ( \\psi _ { N - 1 } ) \\\\ & \\cdot ( h _ { S U ( N - 1 ) } ) _ { i j } ( g _ { S U ( N - 1 ) } ) e ^ { i l ^ N _ { i j } \\omega _ { N - 1 } } , \\end{align*}"}
+{"id": "7456.png", "formula": "\\begin{align*} \\mathcal { A } ^ { c _ 1 , c _ 2 , \\dots , c _ n } ( t ) : = \\idotsint \\limits _ { 0 \\leq t _ 1 < \\dots < t _ n \\leq t } e ^ { - c _ 1 t _ 1 } \\ , \\dd t _ 1 \\dots e ^ { - c _ n t _ n } \\ , \\dd t _ n , \\end{align*}"}
+{"id": "4696.png", "formula": "\\begin{align*} \\lim _ { \\nu \\to \\infty } \\frac { \\alpha _ { \\nu , i } } { \\alpha _ { \\nu , j } } = 0 . \\end{align*}"}
+{"id": "8422.png", "formula": "\\begin{align*} A _ j = ( A _ 0 , u _ { n - t } , \\ldots , u _ { n - t - j + 1 } ) = ( I _ t ( L _ n ) ^ s , u _ { n - t + 1 } , \\ldots , u _ { n - t - j + 1 } ) . \\end{align*}"}
+{"id": "3199.png", "formula": "\\begin{align*} \\{ \\phi \\in L ^ 1 ( X ) \\ , : \\ , d d ^ c _ X \\phi \\geq - S \\mbox { a n d } \\sup \\phi = 0 \\} \\end{align*}"}
+{"id": "6884.png", "formula": "\\begin{align*} \\Lambda ( R ) _ g = \\left \\{ \\sum a _ d t ^ d \\in \\Lambda ( R ) \\ , | \\ , \\forall n \\in \\N , \\ , a _ d \\in R _ { d g } \\right \\} . \\end{align*}"}
+{"id": "1763.png", "formula": "\\begin{align*} \\left . \\tfrac { d } { d t } \\mathrm { F l } ^ { X } _ { t } ( z ) \\right | _ { t = 0 } = \\Re ( X ) ( z ) \\quad \\quad \\forall \\ , z \\in \\mathbb { C } ^ { n + 1 } . \\end{align*}"}
+{"id": "5550.png", "formula": "\\begin{align*} T = \\sum _ { i = 1 } ^ r \\nu _ i \\ , w _ i ^ { ( 1 ) } \\otimes w _ i ^ { ( 2 ) } \\otimes \\cdots \\otimes w _ i ^ { ( k ) } , \\end{align*}"}
+{"id": "5735.png", "formula": "\\begin{align*} X ^ 2 _ + ( t ) = \\sum _ { i : \\gamma ^ + _ i > \\gamma _ * } | \\xi _ i ^ + ( t ) | ^ 2 + \\sum _ { i : \\gamma ^ - _ i > \\gamma _ * } | \\xi _ i ^ - ( t ) | ^ 2 . \\end{align*}"}
+{"id": "3341.png", "formula": "\\begin{align*} \\Q _ r ( z _ 0 ) : = B _ r ( v _ 0 ) \\times U _ r ( x _ 0 , t _ 0 ) : = B _ r ( v _ 0 ) \\times B _ { r ^ { 1 + s p } } ( x _ 0 ) \\times ( t _ 0 - r ^ { s p } , t _ 0 ) . \\end{align*}"}
+{"id": "604.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { \\infty } \\left ( \\frac { 1 } { 3 2 } \\right ) ^ { k } \\binom { 2 k } { k } ^ 2 \\frac { H _ { 2 k } } { 2 k - 1 } = \\\\ & \\frac { \\sqrt { \\pi } \\left ( \\pi + 3 \\ln ( 2 ) - 4 \\right ) } { 2 \\Gamma ^ { 2 } \\left ( \\frac { 1 } { 4 } \\right ) } - \\frac { \\Gamma ^ { 2 } \\left ( \\frac { 1 } { 4 } \\right ) \\left ( \\pi - 3 \\ln ( 2 ) - 2 \\right ) } { 1 6 \\pi ^ { 3 / 2 } } \\end{align*}"}
+{"id": "7190.png", "formula": "\\begin{align*} X ( ( R _ v , R _ w ) , x , t ) = \\left ( \\begin{array} { c } ( 1 - ( v _ 0 + V ) ^ 2 + \\varphi _ 1 ) R _ v - \\frac { R _ v ^ { 3 } } { 3 } - R _ w - R _ v ^ 2 F _ v - R _ v F _ v ^ 2 - \\frac { F _ v ^ 3 } { 3 } + \\varphi _ 2 ( R _ v + F _ v ) ^ 2 \\\\ \\varepsilon ( R _ v - \\gamma R _ w ) \\end{array} \\right ) \\end{align*}"}
+{"id": "2668.png", "formula": "\\begin{align*} K ^ { ( 2 ) } _ { i j } \\ddddot { q } ^ { j } + E ^ { ( 2 ) } _ { i j } \\dddot { q } ^ { j } + { \\textrm { ( u p \\ t o \\ 2 n d - o r d e r \\ t e r m s ) } } = 0 \\end{align*}"}
+{"id": "69.png", "formula": "\\begin{align*} \\begin{cases} c _ 1 = w _ 1 + w _ 3 \\kappa , & c _ 2 = \\big ( w _ 1 ( p - 2 + s ) + w _ 3 ( p - 4 + s ) \\kappa \\big ) \\theta , \\\\ c _ 3 = w _ 2 + w _ 4 \\kappa , & c _ 4 = \\big ( w _ 2 ( p - 2 + s - \\gamma ) + w _ 4 ( p - 4 + s - \\gamma ) \\kappa \\big ) \\theta . \\end{cases} \\end{align*}"}
+{"id": "5595.png", "formula": "\\begin{align*} B ^ { ( \\ell ) } = \\underline { B } ^ { ( \\ell ) } + H B ^ { ( \\ell - 1 ) } + \\sum _ { t = 1 } ^ { \\ell - 1 } \\underline { B } ^ { ( t - 1 ) } H ^ { ( 2 ) } B ^ { ( \\ell - t - 1 ) } + \\underline { B } ^ { ( \\ell - 1 ) } H ^ { ( 1 ) } - \\sum _ { t = 0 } ^ { \\ell } R _ t ^ { ( \\ell ) } , \\end{align*}"}
+{"id": "439.png", "formula": "\\begin{align*} \\min _ { x \\in \\Re ^ { | V _ { B _ k } | } } \\left \\{ \\sum _ { i \\in V _ { B _ k } } \\hat f _ i ( x _ i ) \\mid x _ i = x _ j , \\ , \\forall \\ , ( i , j ) \\in E _ { B _ k } \\right \\} , 1 \\le k \\le K , \\end{align*}"}
+{"id": "3820.png", "formula": "\\begin{align*} y ^ 2 = x ( x ^ 2 - 2 a ( t ) x + ( a ( t ) ^ 2 - 4 b ( t ) ) , \\end{align*}"}
+{"id": "5824.png", "formula": "\\begin{align*} \\frac { d } { d s } \\mathcal { F } _ \\Sigma ( u + s \\xi ) \\big | _ { s = 0 } = - \\int \\bar { g } \\left ( \\vec { H } , \\xi ^ A \\frac { \\partial } { \\partial y ^ A } \\right ) \\sqrt { \\det g _ u } \\ , d x ^ 1 \\wedge \\dots \\wedge d x ^ n . \\end{align*}"}
+{"id": "5988.png", "formula": "\\begin{align*} \\frac { \\nabla \\phi } { \\phi } = \\frac { 4 r _ { \\partial M } \\nabla r _ { \\partial M } } { R ^ 2 - r _ { \\partial M } ^ 2 } \\end{align*}"}
+{"id": "6302.png", "formula": "\\begin{align*} H _ { N } = \\sum _ { i = 1 } ^ N - \\Delta _ { x _ i } + \\sum _ { 1 \\leq i < j \\leq N } V ( x _ i - x _ j ) \\end{align*}"}
+{"id": "8726.png", "formula": "\\begin{align*} p _ 1 + \\ldots + p _ d = 1 \\end{align*}"}
+{"id": "7610.png", "formula": "\\begin{align*} \\| u _ { \\epsilon } \\| _ { q } ^ { q } & = R ^ { q } \\omega \\epsilon ^ { \\frac { ( N - 4 ) q } { 2 } } \\int _ { 0 } ^ { 2 } \\frac { \\varphi ^ { q } ( r ) r ^ { N - 1 } } { ( \\epsilon ^ { 2 } + r ^ { 2 } ) ^ { \\frac { ( N - 4 ) q } { 2 } } } d r \\\\ & = \\epsilon ^ { \\frac { ( N - 4 ) q } { 2 } } R ^ { q } \\omega \\int _ { 0 } ^ { 2 } \\frac { \\varphi ^ { q } ( r ) } { r ^ { ( N - 4 ) q - ( N - 1 ) } } d r + o ( \\epsilon ^ { \\frac { ( N - 4 ) q } { 2 } } ) . \\\\ \\end{align*}"}
+{"id": "2277.png", "formula": "\\begin{align*} \\mathcal { R } = e ^ { \\alpha ( h \\otimes h ) / 2 } \\sum _ { n \\geq 0 } \\frac { ( q - q ^ { - 1 } ) ^ n } { [ n ] _ q ! } q ^ { n ( n - 1 ) / 2 } \\Bigl ( e ^ { - \\alpha h / 2 } y \\otimes x e ^ { \\alpha h / 2 } \\Bigr ) ^ n \\ , , \\end{align*}"}
+{"id": "1580.png", "formula": "\\begin{align*} \\sup _ { \\S ^ { N - 1 } } g _ k = \\sup _ { \\{ G = k \\} } | z | \\inf _ { \\S ^ { N - 1 } } g _ k = \\inf _ { \\{ G = k \\} } | z | . \\end{align*}"}
+{"id": "900.png", "formula": "\\begin{align*} & L _ 2 ^ 1 = \\frac { 1 } { 4 } ( A _ 1 K x _ 1 + B _ 1 K x _ 2 + C _ 1 K ) , \\\\ & R _ 0 ^ 1 = \\frac { 1 } { 4 } ( A _ 1 x _ 1 \\sum _ { l + k \\leq 2 } M _ { l k } x _ 1 ^ { l } x _ 2 ^ k + B _ 1 x _ 2 \\sum _ { l + k \\leq 2 } M _ { l k } x _ 1 ^ { l } x _ 2 ^ k + C _ 1 \\sum _ { l + k \\leq 2 } M _ { l k } x _ 1 ^ { l } x _ 2 ^ k \\\\ & \\quad - ( A _ 3 x _ 1 + B _ 3 x _ 2 + C _ 3 ) ) , \\\\ & A _ 0 ^ 1 = - \\frac { E } { 4 } , \\\\ & B _ 0 ^ 1 = - \\frac { J } { 4 } , \\\\ & D _ 0 ^ 1 = - \\frac { K } { 4 } , \\end{align*}"}
+{"id": "4972.png", "formula": "\\begin{align*} \\Lambda ( \\mu ; \\nu _ { j _ 1 } , \\dots , \\nu _ { j _ k } ) = a ( \\mu ) \\prod _ { l = 1 } ^ k f ( \\nu _ { j _ l } - 1 , \\mu ) , \\end{align*}"}
+{"id": "8624.png", "formula": "\\begin{align*} \\| v _ 0 ( 0 ; \\cdot ) \\| _ { L ^ \\infty ( \\mathbb { R } ) } = \\| u _ 0 \\| _ { L ^ \\infty ( \\mathbb { R } ) } < C _ 0 , \\end{align*}"}
+{"id": "174.png", "formula": "\\begin{align*} A \\cap A ^ { - 1 } \\ = \\ \\{ e \\} , A \\cup A ^ { - 1 } \\ = \\ G . \\end{align*}"}
+{"id": "4038.png", "formula": "\\begin{align*} \\mu _ { \\hat X ^ 1 _ t } = \\int k _ S ( x , \\cdot ) \\hat X ^ 1 _ t ( d x ) = \\mathbb { E } _ { \\mathbb { P } } [ k _ S ( X , \\cdot ) | \\mathcal { F } _ t ] . \\end{align*}"}
+{"id": "6395.png", "formula": "\\begin{align*} G _ n ^ 3 ( \\theta ) = - \\partial _ { \\delta } L _ n ( \\theta ) = \\sum _ { i = 1 } ^ n \\frac { 1 } { \\delta } k _ \\alpha ( z ^ n _ i ( \\theta ) ) \\end{align*}"}
+{"id": "7479.png", "formula": "\\begin{align*} R _ 0 [ \\Bar { h } ] : = \\Bar { h } ^ { - 1 } * \\Bar { h } * R m ( g _ 0 ( t ) ) + ( g ( t ) ) ^ { - 1 } * ( g ( t ) ) ^ { - 1 } * \\nabla ^ { g _ 0 ( t ) } \\Bar { h } ( t ) * \\nabla ^ { g _ 0 ( t ) } \\Bar { h } ( t ) , \\end{align*}"}
+{"id": "8877.png", "formula": "\\begin{align*} \\frac { \\alpha + n } { 2 n } = \\frac { 1 } { a _ 1 } + \\frac { 1 } { a _ 2 } , \\frac { n - 2 } { 2 n } \\leq \\frac { 1 } { ( p - 1 ) a _ 1 } \\leq \\frac 1 2 . \\end{align*}"}
+{"id": "7377.png", "formula": "\\begin{align*} \\begin{array} { l } \\| L ( | w _ j | ^ { p - 1 } | w _ j - w _ { j - 1 } | ) | u _ j | ^ q \\| _ 1 \\\\ = \\| L \\left ( \\left | | w _ j | ^ { ( p - 1 ) / p } | w _ j - w _ { j - 1 } | ^ { 1 / p } \\right | ^ p | u _ j | ^ q \\right ) \\| _ 1 \\\\ \\le C ( T + R ) \\| | w _ j | ^ { ( p - 1 ) / p } | w _ j - w _ { j - 1 } | ^ { 1 / p } \\| _ 2 ^ p \\| u _ j \\| _ 1 ^ q \\\\ \\le C ( T + R ) \\| w _ j \\| _ 2 ^ { p - 1 } \\| w _ j - w _ { j - 1 } \\| _ 2 \\| u _ j \\| _ 1 ^ q \\end{array} \\end{align*}"}
+{"id": "1873.png", "formula": "\\begin{align*} [ b , \\rho ( \\tau _ i ) ] = [ \\rho ( \\tau _ 1 ) , \\rho ( \\tau _ i ) ] = 1 = [ \\rho ( \\alpha ) , \\rho ( \\tau _ i ) ] = [ a , \\rho ( \\tau _ i ) ] . \\end{align*}"}
+{"id": "74.png", "formula": "\\begin{align*} S = w _ 1 G D _ 1 ( p - 2 + s ) + w _ 2 G D _ 2 ( p - 2 + s ) \\end{align*}"}
+{"id": "7250.png", "formula": "\\begin{align*} [ n ] _ q = 1 + q + \\ldots + q ^ { n - 1 } n \\geq 1 [ 0 ] _ q = 0 . \\end{align*}"}
+{"id": "3571.png", "formula": "\\begin{align*} \\alpha ^ 3 \\tau _ 0 \\tau _ 1 \\tau _ 2 = ( 1 + a ) \\alpha ^ 2 \\tau _ 0 \\tau _ 1 - ( a + b ) \\alpha \\tau _ 0 + b , \\end{align*}"}
+{"id": "1357.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n s _ i \\geq \\frac { \\alpha ^ 2 M _ 1 + ( 1 - \\alpha ) ^ 2 2 m - \\frac { 4 \\alpha ^ 2 m ^ 2 } { n } } { s _ 1 } \\ , \\ , \\ , \\ , \\ , ( \\ , s _ n = 0 ) \\end{align*}"}
+{"id": "5307.png", "formula": "\\begin{align*} \\mathcal { F } = \\left \\{ S _ 1 , \\ldots , S _ { n + 1 } \\right \\} , \\end{align*}"}
+{"id": "8488.png", "formula": "\\begin{align*} \\limsup _ { h \\searrow 0 } I _ h \\leq \\limsup _ { h \\searrow 0 } I _ h ^ { ( 1 ) } + 2 \\limsup _ { h \\searrow 0 } I ^ { ( 2 ) } _ h & = 0 , \\end{align*}"}
+{"id": "8821.png", "formula": "\\begin{align*} \\widetilde { \\psi } ( x ) = e ^ { - 1 / ( 1 - | x | ^ 2 ) } | x | < 1 \\widetilde { \\psi } ( x ) = 0 | x | \\geq 1 . \\end{align*}"}
+{"id": "3162.png", "formula": "\\begin{align*} ( a _ s , \\ldots , a _ 0 , i ) ^ { \\sigma } = \\left ( a _ { s - 1 } , \\ldots , a _ 0 , \\left \\lfloor \\frac { i k + a _ s } { t } \\right \\rfloor , ( i k + a _ s ) \\bmod { t } \\right ) . \\end{align*}"}
+{"id": "6393.png", "formula": "\\begin{align*} G _ n ^ 1 ( \\theta ) = - \\partial _ a L _ n ( \\theta ) = \\frac { n ^ { 1 / \\alpha } } { n } \\sum _ { i = 1 } ^ n \\frac { 1 } { \\delta X _ { \\frac { i - 1 } { n } } ^ { 1 / \\alpha } } h _ \\alpha ( z ^ n _ i ( \\theta ) ) \\end{align*}"}
+{"id": "8480.png", "formula": "\\begin{align*} | v _ h - { \\bar v } _ h | & \\leq C \\left ( | \\bar { u } _ h ( t ) | + | \\bar { u } _ h ( t - h ) | \\right ) ^ { q - 1 } \\left | \\bar { u } _ h ( t ) - \\bar { u } _ h ( t - h ) \\right | \\\\ [ 2 m m ] & \\leq C \\left | \\bar { u } _ h ( t ) - \\bar { u } _ h ( t - h ) \\right | ^ q \\\\ [ 2 m m ] & = C h ^ q \\left ( | \\bar { u } _ h ( t ) | + | \\bar { u } _ h ( t - h ) | \\right ) ^ { \\frac { q ( 1 - q ) } { 2 } } \\left ( | \\bar { u } _ h ( t ) | + | \\bar { u } _ h ( t - h ) | \\right ) ^ { \\frac { q ( q - 1 ) } { 2 } } | \\partial _ t u _ h | ^ q \\end{align*}"}
+{"id": "4964.png", "formula": "\\begin{align*} \\sigma ^ + = \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} , \\sigma ^ - = \\begin{pmatrix} 0 & 0 \\\\ 1 & 0 \\end{pmatrix} , \\pi ^ + = \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} , \\pi ^ - = \\begin{pmatrix} 0 & 0 \\\\ 0 & 1 \\end{pmatrix} , \\end{align*}"}
+{"id": "7160.png", "formula": "\\begin{align*} \\partial _ s v + \\frac { 1 } { \\omega } \\mathfrak { L } v = \\frac { 1 } { \\omega } \\tilde { F } ( v , s , \\omega , x ) , \\end{align*}"}
+{"id": "5129.png", "formula": "\\begin{align*} L D ( p \\| q ) = \\left [ \\log A \\left ( p , q \\right ) - \\log B \\left ( p , q \\right ) \\right ] \\end{align*}"}
+{"id": "757.png", "formula": "\\begin{align*} ( \\Phi _ k - \\varphi _ k \\Phi ) = F ^ { - 2 } ( y _ k ( \\Phi _ 0 - \\varphi _ 0 \\Phi ) - R ^ h _ { \\ k } \\varphi _ h ) , \\end{align*}"}
+{"id": "4398.png", "formula": "\\begin{gather*} B _ { q , . } g ( x ) = \\sum _ { i \\in [ m ] , j \\in [ n ] } B _ { q , ( i , j ) } g _ { i , j } ( x ) = \\sum _ { i \\in [ m ] } B _ { q , ( i , j ( i ) ) } g _ { i , j ( i ) } ( x ) = B _ { q , ( q , j ( q ) ) } g _ { q , j ( q ) } ( x ) . \\end{gather*}"}
+{"id": "2897.png", "formula": "\\begin{align*} \\left ( \\left [ L ^ r _ \\omega ( \\mathbb { R } ^ n ) \\right ] ^ \\frac { 1 } { p } \\right ) ' = L ^ { ( \\frac { r } { p } ) ' } _ { \\omega ^ { 1 - ( \\frac { r } { p } ) ' } } ( \\mathbb { R } ^ n ) . \\end{align*}"}
+{"id": "4504.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } \\big [ | A _ 0 ( n ) | ^ 2 \\big ] = \\sum _ { \\substack { | \\lambda | = n \\\\ \\lambda _ 1 \\leqslant y _ 0 } } \\mathbb { E } \\big [ | a ( \\lambda ) | ^ 2 \\big ] = \\sum _ { \\substack { | \\lambda | = n \\\\ \\lambda _ 1 \\leqslant y _ 0 } } \\prod _ k \\frac { 1 } { k ^ { m _ k } m _ k ! } . \\end{aligned} \\end{align*}"}
+{"id": "4966.png", "formula": "\\begin{align*} a ( \\lambda ) = \\prod _ { j = 1 } ^ { N } a ( \\lambda , \\nu _ j ) , d ( \\lambda ) = \\prod _ { j = 1 } ^ { N } b ( \\lambda , \\nu _ j ) . \\end{align*}"}
+{"id": "3284.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( a ^ 2 q , q ^ 4 , q / a ^ 2 ; q ^ 3 ) _ k q ^ { 3 k } } { ( a q ^ 3 , q ^ 3 / a , q ^ 3 ; q ^ 3 ) _ k } \\equiv \\frac { ( 1 - a ^ 2 q ) ( 1 - q / a ^ 2 ) ( q ^ 3 ; q ^ 3 ) _ { n - 1 - ( n + 1 ) / 3 } } { ( a q ^ 3 , q ^ 3 / a ; q ^ 3 ) _ { ( n + 1 ) / 3 } } q ^ { ( n ^ 2 + 5 n - 2 ) / 3 } . \\end{align*}"}
+{"id": "3177.png", "formula": "\\begin{align*} \\langle [ u , y ] , v \\rangle _ y - \\langle [ v , y ] , u \\rangle _ y = \\langle [ u , y ] , v \\rangle _ y + \\langle [ v , u ] , y \\rangle _ y + \\langle [ y , v ] , u \\rangle _ y = 0 , \\end{align*}"}
+{"id": "3493.png", "formula": "\\begin{align*} \\rho _ n ( x ) = [ ( \\rho _ { m , n } ( x ) ) _ { m > n } ] \\end{align*}"}
+{"id": "288.png", "formula": "\\begin{align*} \\psi ( u _ i ) ( a _ 1 , \\ldots , a _ r ) & = \\psi ( \\pi ( t _ i ) ) ( p ( x _ 1 ) , \\ldots , p ( x _ r ) ) \\\\ & = \\psi ( \\pi ( t _ i ) ) ( ( p \\circ \\pi ) ( x _ 1 ) , \\ldots , ( p \\circ \\pi ) ( x _ r ) ) \\\\ & = ( p \\circ \\pi ) ( t _ i ) \\\\ & = p ( u _ i ) , \\end{align*}"}
+{"id": "2253.png", "formula": "\\begin{align*} \\small N _ 1 & = \\{ E _ { 1 , j } : 2 \\leq j \\leq p + 1 \\} , \\ , & & N _ 2 = \\{ F _ { 1 , j } : 2 \\leq j \\leq p + 1 \\} , \\\\ N _ 3 & = \\{ E _ { p + 2 , j } : p + 3 \\leq j \\leq n + 1 \\} , \\ , & & N _ 4 = \\{ F _ { p + 2 , j } : p + 3 \\leq j \\leq n + 1 \\} . \\end{align*}"}
+{"id": "6446.png", "formula": "\\begin{align*} \\operatorname { I n d } ^ { \\rm { n o r } } ( M \\to N ) : = \\max \\{ \\dim L , L \\subset C ^ \\infty _ 0 ( M ) \\mid \\operatorname { H e s s } E _ 3 ( \\phi ) ( f \\nu , f \\nu ) < 0 , ~ ~ \\forall f \\in L \\} , \\end{align*}"}
+{"id": "8319.png", "formula": "\\begin{align*} \\Pi ^ { \\sharp } ( \\delta ) = \\epsilon ( X ) \\end{align*}"}
+{"id": "3189.png", "formula": "\\begin{align*} E ( n ) & \\leq \\exp \\left ( - \\frac { 4 2 n + 4 3 } { 6 4 } + \\frac { 3 3 ( 6 n + 4 ) } { 1 2 8 \\pi \\left ( 1 - \\frac { 3 \\pi ^ 2 } { 2 0 4 8 2 8 8 } \\right ) } \\right ) \\\\ [ 5 p t ] & = \\exp \\left ( \\left ( - \\frac { 2 1 } { 3 2 } + \\frac { 9 9 } { 6 4 \\pi \\left ( 1 - \\frac { 3 \\pi ^ 2 } { 2 0 4 8 2 8 8 } \\right ) } \\right ) n - \\frac { 4 3 } { 6 4 } + \\frac { 3 3 } { 3 2 \\pi \\left ( 1 - \\frac { 3 \\pi ^ 2 } { 2 0 4 8 2 8 8 } \\right ) } \\right ) \\\\ [ 5 p t ] & < \\exp \\left ( - 0 . 1 6 3 n - 0 . 3 4 3 \\right ) . \\end{align*}"}
+{"id": "5217.png", "formula": "\\begin{align*} D G H ( p \\| q ) = M G - M H \\end{align*}"}
+{"id": "8351.png", "formula": "\\begin{align*} \\mu \\in \\Lambda \\iff \\frac { d \\mu ( x ) } { d x } = \\phi ( x ) , \\phi \\colon \\R ^ n \\to \\R ^ + , \\phi \\in L ^ 1 _ { } ( \\R ^ n ) . \\end{align*}"}
+{"id": "6049.png", "formula": "\\begin{align*} \\phi _ x ^ + = \\frac { 1 } { 2 } \\bar \\xi ^ { A } _ x + \\bar \\xi ^ { B } _ x , \\phi _ x ^ - = \\frac { 1 } { 2 } \\bar \\xi ^ { A } _ x . \\end{align*}"}
+{"id": "8550.png", "formula": "\\begin{align*} p _ { 1 } ( n ) F ( n + r , k ) + p _ { 2 } ( n ) F ( n , k ) = G ( n , k + 1 ) - G ( n , k ) \\end{align*}"}
+{"id": "6011.png", "formula": "\\begin{align*} L _ { N } f ( \\eta ) = N ^ a \\sum _ { x \\in \\mathbb { T } _ N } c _ x ( \\eta ) [ f ( \\eta ^ { x , x + 1 } ) - f ( \\eta ) ] . \\end{align*}"}
+{"id": "2295.png", "formula": "\\begin{align*} F _ n ( t ) = \\frac { P _ r ( t ) } { ( 1 - t ^ 2 ) ^ { 3 ( r + 1 ) } } , \\end{align*}"}
+{"id": "4469.png", "formula": "\\begin{align*} \\int _ { M _ 1 } f _ 2 \\overline { g } \\lambda _ 2 = 0 \\end{align*}"}
+{"id": "8293.png", "formula": "\\begin{align*} u _ { 1 1 } = \\frac { w ^ 2 \\ln w } { \\rho u _ 1 } \\left ( - \\varphi ' \\rho u _ 1 + { 2 x _ 1 } \\right ) \\leq - \\frac { \\varphi ' } { 2 } w ^ 2 \\ln w = - \\frac { 1 } { 2 u } w ^ 2 \\ln w , \\end{align*}"}
+{"id": "4967.png", "formula": "\\begin{align*} R _ { j k } ( \\lambda , \\mu ) \\left [ L _ { j l } ( \\lambda , \\nu ) \\otimes L _ { k l } ( \\mu , \\nu ) \\right ] = \\left [ L _ { k l } ( \\mu , \\nu ) \\otimes L _ { j l } ( \\lambda , \\nu ) \\right ] R _ { j k } ( \\lambda , \\mu ) . \\end{align*}"}
+{"id": "1184.png", "formula": "\\begin{align*} & f ( y ) - \\sum _ { \\alpha \\in \\mathbb { Z } _ + ^ n , \\ , | \\alpha | \\leq N } \\frac { ( y - z ) ^ \\alpha } { \\alpha ! } \\partial ^ \\alpha f ( z ) \\\\ & \\quad = \\sum _ { \\alpha \\in \\mathbb { Z } _ + ^ n , \\ , | \\alpha | = N } \\frac { ( y - z ) ^ \\alpha } { \\alpha ! } \\int _ 0 ^ 1 [ \\partial ^ \\alpha f ( z + t ( y - z ) ) - \\partial ^ \\alpha f ( z ) ] N ( 1 - t ) ^ { N - 1 } \\ , d t , \\end{align*}"}
+{"id": "2533.png", "formula": "\\begin{align*} u = a ( x , y ) \\ , \\Big | \\frac { d x } { x } d y \\Big | ^ s \\mapsto u | _ { \\partial M } = a ( 0 , y ) \\ , | d y | ^ s \\ ; . \\end{align*}"}
+{"id": "1709.png", "formula": "\\begin{align*} A = \\sum _ { j = 1 } ^ { n - 2 } x _ n ^ { j - 1 } T ^ j _ { n - 1 } \\end{align*}"}
+{"id": "6226.png", "formula": "\\begin{align*} \\begin{cases} \\dot { z } = \\dot { f } - c - D g / z \\ & \\mbox { i n } \\ ( \\beta , 1 ) , \\\\ z < 0 \\ & \\mbox { i n } \\ ( \\beta , 1 ) , \\\\ z ( \\beta ) = z ( 1 ) = 0 , \\end{cases} \\end{align*}"}
+{"id": "2084.png", "formula": "\\begin{align*} \\eqref { e q : t e s t f u n c t i o n a p p l i e d s u m } & = - \\int _ 0 ^ { \\frac { \\lfloor s T \\rfloor } { T } } \\Theta ( u , x ) f ' ( u ) d u + f ( s ) { \\Theta } ( \\frac { \\lfloor s T \\rfloor } { T } , x ) - f ( 0 ) { \\Theta } ( 0 , x ) + o ( 1 ) , \\\\ \\eqref { e q : t e s t f u n c t i o n a p p l i e d s u m } & = - ( i + 1 - d x ) ( N _ { i , 1 } ^ s + N _ { i , 2 } ^ s + N _ { i , 3 } ^ s ) - ( d x - i ) ( N _ { i + 1 , 1 } ^ s + N _ { i + 1 , 2 } ^ s + N _ { i + 1 , 3 } ^ s ) , \\end{align*}"}
+{"id": "11.png", "formula": "\\begin{align*} m ' = - i \\kappa m + i C _ + [ q ( m + 1 ) ] , \\end{align*}"}
+{"id": "8328.png", "formula": "\\begin{align*} [ \\widetilde { \\Pi } , \\widetilde { W } _ t ] _ { S N } = [ \\Pi , W _ t ] _ { S N } + [ \\Pi , X _ t ] _ { S N } \\wedge \\partial _ s . \\end{align*}"}
+{"id": "7944.png", "formula": "\\begin{align*} w = - N _ { \\phi } \\big ( \\mathrm { t r } ( \\frac { \\delta \\mathcal { F } } { \\delta v } ) \\big ) \\in H \\Lambda ^ { 0 } ( \\Omega ) , \\end{align*}"}
+{"id": "5570.png", "formula": "\\begin{align*} f _ { \\phi , t } ( g , o ) = \\left ( \\frac { m n } { d ^ 2 } \\right ) ^ t \\sum _ { o _ 1 , \\dots , o _ { 2 t } } \\left ( \\prod _ { s = 0 } ^ { t - 1 } M _ { \\iota ( o _ { 2 s } ) , \\iota ( o _ { 2 s + 1 } ) } M _ { \\iota ( o _ { 2 s + 2 } ) , \\iota ( o _ { 2 s + 1 } ) } \\right ) \\phi _ { \\iota ( o _ { 2 t } ) } \\end{align*}"}
+{"id": "4806.png", "formula": "\\begin{align*} \\ell ( \\mathbf { x } , \\mathbf { c } ) = \\mathbf { c } ^ \\top \\mathbf { T } \\mathbf { x } + \\mathbf { t } _ 1 ^ \\top \\mathbf { x } + \\mathbf { t } _ 2 ^ \\top \\mathbf { c } + t _ 0 , \\end{align*}"}
+{"id": "4507.png", "formula": "\\begin{align*} V ( n ) : = \\sum _ { y _ 0 < k \\leqslant n } \\frac { 1 } { k } \\bigg | \\sum _ { \\substack { | \\lambda | = n - k \\\\ \\lambda _ 1 < k } } a ( \\lambda ) \\bigg | ^ 2 . \\end{align*}"}
+{"id": "2861.png", "formula": "\\begin{align*} [ \\mu ] _ { A _ { p ' } ( \\mathbb { R } ^ n ) } ^ { p - 1 } = [ \\omega ] _ { A _ p ( \\mathbb { R } ^ n ) } \\leq [ \\omega ] _ { A _ 1 ( \\mathbb { R } ^ n ) } , \\ \\ \\left [ L _ \\omega ^ p ( \\mathbb { R } ^ n ) \\right ] ' = L _ \\mu ^ { p ' } ( \\mathbb { R } ^ n ) ; \\end{align*}"}
+{"id": "5677.png", "formula": "\\begin{align*} u ' - \\mathcal { M } _ \\Sigma u = N _ 2 ( u ) . \\end{align*}"}
+{"id": "2211.png", "formula": "\\begin{align*} - \\intop _ \\Omega \\bigg ( u _ { , r r } + { 1 \\over r } u _ { , r } + u _ { , z z } \\bigg ) u r ^ { - 2 } d x - 2 \\intop _ \\Omega { 1 \\over r } u _ { , r } u r ^ { - 2 } d x = \\intop _ \\Omega \\omega _ { 1 , z } u r ^ { - 2 } d x . \\end{align*}"}
+{"id": "2933.png", "formula": "\\begin{align*} \\sum _ { i \\in \\{ 1 , \\ldots , n \\} } \\alpha _ i = 0 . \\end{align*}"}
+{"id": "8452.png", "formula": "\\begin{align*} \\| v \\| _ { W ^ { s , p } ( K _ T ) } : = \\| v \\| _ { L ^ p ( K _ T ) } + [ v ] _ { W ^ { s , p } ( K _ T ) } . \\end{align*}"}
+{"id": "4997.png", "formula": "\\begin{align*} P _ N ( \\{ y \\} ) = ( q - q ^ { - 1 } ) ^ N \\prod _ { \\substack { j , k = 1 \\\\ j \\neq k } } ^ N ( q y _ j - q ^ { - 1 } y _ k ) . \\end{align*}"}
+{"id": "6311.png", "formula": "\\begin{align*} - \\Delta \\omega _ \\ell = \\frac { 1 } { 2 } V _ \\ell ( 1 - \\omega _ \\ell ) . \\end{align*}"}
+{"id": "9110.png", "formula": "\\begin{align*} \\bigcap _ { \\xi \\in \\mathbb { R } ^ n \\setminus \\{ 0 \\} } A ( \\xi ) [ V ] = \\{ 0 \\} \\ ; . \\end{align*}"}
+{"id": "5662.png", "formula": "\\begin{align*} \\frac { \\| \\boldsymbol { a } ^ { ( 3 ) } \\| ^ 2 x ^ 2 } { \\| \\boldsymbol { a } _ \\perp ^ { ( 3 ) } \\| ^ 2 \\tan ^ 2 \\theta } + \\frac { y ^ 2 } { \\| \\boldsymbol { a } _ \\perp ^ { ( 3 ) } \\| ^ 2 \\tan ^ 2 \\theta } = 1 \\end{align*}"}
+{"id": "898.png", "formula": "\\begin{align*} & P ^ 1 _ 1 = - \\frac { 1 } { 2 } ( A _ 1 A _ 5 x _ 1 ^ 2 + A _ 1 B _ 5 x _ 1 x _ 2 + B _ 1 A _ 5 x _ 1 x _ 2 + B _ 1 B _ 5 x _ 2 ^ 2 + ( A _ 5 C _ 1 \\\\ & + A _ 1 C _ 5 ) x _ 1 + ( C _ 1 B _ 5 + B _ 1 C _ 5 ) x _ 2 + C _ 1 C _ 5 ) - \\frac { 1 } { 4 } \\left ( \\sum _ { l + k \\leq 2 } M _ { l k } x _ 1 ^ { l } x _ 2 ^ k \\right ) , \\\\ & L _ 1 ^ 1 = \\frac { 1 } { 2 } ( A _ 5 x _ 1 + B _ 5 x _ 2 + C _ 5 ) + \\frac { 1 } { 4 } ( A _ 1 E x _ 1 + B _ 1 E x _ 2 + C _ 1 E ) , \\end{align*}"}
+{"id": "6402.png", "formula": "\\begin{align*} G _ n ^ { ( d ) } ( a , b ) = - \\nabla _ { ( a , b ) } L _ n ( a , b , \\tilde { \\delta } _ n , \\tilde { \\alpha } _ n ) = - ( \\partial _ a L _ n ( a , b , \\tilde { \\delta } _ n , \\tilde { \\alpha } _ n ) , \\partial _ b L _ n ( a , b , \\tilde { \\delta } _ n , \\tilde { \\alpha } _ n ) ) ^ T . \\end{align*}"}
+{"id": "6956.png", "formula": "\\begin{align*} z _ 1 \\overline { w _ 1 } + \\cdots z _ n \\overline { w _ n } - z _ { n + 1 } \\overline { w _ { n + 1 } } = 0 , \\end{align*}"}
+{"id": "4086.png", "formula": "\\begin{align*} H _ 1 ( z ) : = \\begin{cases} \\frac { H ( z ) } { c _ { 1 , p } z ^ { p } } & \\ , \\ , c _ { 1 , p } \\neq 0 \\\\ \\frac { H ( z ) } { c _ { 2 , p } z ^ { p } \\log ( z ) } & \\ , \\ , \\ c _ { 2 , p / 2 } \\neq 0 \\\\ \\frac { H ( z ) } { z ^ { p } ( c _ { 1 , p } + c _ { 2 , p } \\log ( z ) ) } & \\ , \\ , \\ c _ { 1 , p } , c _ { 2 , p / 2 } \\neq 0 \\\\ \\end{cases} \\end{align*}"}
+{"id": "7777.png", "formula": "\\begin{align*} \\mathcal R _ j = \\cup _ { \\tilde \\Lambda \\in \\Pi _ { m _ * } } R _ j ( \\tilde \\Lambda ) , \\ \\ \\mathcal R = \\cup _ { j = 1 } ^ \\infty \\mathcal R _ j . \\end{align*}"}
+{"id": "8180.png", "formula": "\\begin{align*} P _ { { \\underline X } _ 1 } ( { \\underline x } _ 1 ) = \\begin{cases} r _ 0 \\ & { \\underline x _ 1 } = \\underline { 0 } \\\\ r _ { c } \\alpha _ { c , \\ell } & { { \\underline x } _ 1 } = { \\underline c } _ { \\ell } , \\forall \\ell \\in \\left [ { M \\choose c } \\right ] , c \\in [ B _ ] , \\end{cases} \\end{align*}"}
+{"id": "454.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { M } ( E _ { B ^ { q } } ^ + ) \\ ; = \\ ; \\{ ( i , j ) \\in E _ { B ^ { q } } \\mid \\Delta t _ { i , j } = \\Delta t _ q , \\ ; \\mbox { a n d } \\ ; i \\triangleleft j \\} , \\\\ \\mathcal { M } ( E _ { B ^ { q } } ^ - ) \\ ; = \\ ; \\{ ( i , j ) \\in E _ { B ^ { q } } \\mid \\Delta t _ { i , j } = \\Delta t _ q , \\ ; \\mbox { a n d } \\ ; j \\triangleleft i \\} , \\\\ \\end{cases} \\end{align*}"}
+{"id": "4509.png", "formula": "\\begin{align*} V ( n , y _ { j } ) : = \\frac { 1 } { y _ { j } } \\sum _ { y _ { j - 1 } < k \\leqslant y _ j } \\bigg | \\sum _ { \\substack { | \\lambda | = n - k \\\\ \\lambda _ 1 < k } } a ( \\lambda ) \\bigg | ^ 2 . \\end{align*}"}
+{"id": "7966.png", "formula": "\\begin{align*} \\ast \\boldsymbol { n } ( d \\phi ) = \\Sigma _ { t } = ( i _ { \\mathcal { N } } u ' ) v _ { \\Sigma } \\Sigma . \\end{align*}"}
+{"id": "5317.png", "formula": "\\begin{align*} \\sum _ { j \\in S } w _ j ^ { S } \\ , x _ j & \\geq b ^ S , \\ , S \\in \\mathcal { F } \\setminus \\{ J \\} \\\\ \\sum _ { j \\in J } w ^ { J } _ j \\ , x _ j & = b ^ J \\\\ x _ j & \\geq 0 , j \\in J . \\end{align*}"}
+{"id": "6277.png", "formula": "\\begin{align*} \\phi ( x , \\xi ) = f ( x , \\xi ) + \\delta ( x ) , \\end{align*}"}
+{"id": "3563.png", "formula": "\\begin{align*} T ^ 3 - ( 1 + a ) T ^ 2 + ( a + b ) T - b I = 0 , \\end{align*}"}
+{"id": "4735.png", "formula": "\\begin{align*} \\widehat { \\mathbb { N } } : = \\mathbb { N } , \\ ; \\widehat { X } : = \\mathbb { N } , \\ ; \\widehat { \\xi \\to \\tau } : = \\widehat { \\xi } \\to \\widehat { \\tau } . \\end{align*}"}
+{"id": "5489.png", "formula": "\\begin{align*} \\int _ { | M | \\le 3 | E | } | M | ^ K = o \\left ( \\mathbb E | M | ^ { K } \\right ) , \\end{align*}"}
+{"id": "8678.png", "formula": "\\begin{align*} T Y ' = T ^ H Y ' \\oplus \\underline { \\mathfrak { g } } _ { Y ' } . \\end{align*}"}
+{"id": "7539.png", "formula": "\\begin{align*} - \\int _ 0 ^ T \\int _ \\Omega \\psi _ t n \\ d x d t - \\int _ 0 ^ T \\int _ \\Omega \\nabla \\psi \\cdot ( n v ) \\ d x d t - \\int _ \\Omega \\psi n \\big | _ { t = 0 } \\ d x = 0 \\ , \\end{align*}"}
+{"id": "568.png", "formula": "\\begin{align*} \\omega ^ 2 = g k \\tanh { k h _ 0 } . \\end{align*}"}
+{"id": "6124.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ t u ( t , x _ 1 , x _ 2 , x _ 3 ) & = \\varepsilon \\Delta u ( t , x _ 1 , x _ 2 , x _ 3 ) + \\alpha ( \\partial _ { x _ 1 } + \\partial _ { x _ 2 } + \\partial _ { x _ 3 } ) u ( t , x _ 1 , x _ 2 , x _ 3 ) \\\\ & + g ( t , x _ 1 , x _ 2 , x _ 3 , u ( t , x _ 1 , x _ 2 , x _ 3 ) ) , \\\\ u _ 0 ( x _ 1 , x _ 2 , x _ 3 ) & = 6 4 x _ 1 ( 1 - x _ 1 ) x _ 2 ( 1 - x _ 2 ) x _ 3 ( 1 - x _ 3 ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3420.png", "formula": "\\begin{align*} I ( \\sqrt { - 1 } \\rho ( a ) \\phi ) = - \\sqrt { - 1 } I ( \\rho ( a ) \\phi ) = - \\sqrt { - 1 } \\rho ( \\iota ^ * a ) I ( \\phi ) \\end{align*}"}
+{"id": "6444.png", "formula": "\\begin{align*} \\varphi ( x _ n ) = \\gamma + 2 \\pi n , n \\in \\Z . \\end{align*}"}
+{"id": "7043.png", "formula": "\\begin{align*} \\bigl ( \\partial _ t - \\Delta + b _ n \\cdot \\nabla \\bigr ) u _ n = 0 , u _ n ( s , \\cdot ) = f ( \\cdot ) \\in C _ c ^ \\infty . \\end{align*}"}
+{"id": "8823.png", "formula": "\\begin{align*} f _ j ^ * ( c \\varphi ( 2 ^ { - j } ) ^ p ) \\ge C \\varphi ( 2 ^ { - j } ) ^ { - 1 } , j = 0 , 1 , 2 , \\ldots , \\end{align*}"}
+{"id": "7800.png", "formula": "\\begin{align*} K ^ * = \\begin{bmatrix} - 0 . 3 5 7 5 & 0 . 0 2 2 8 & 0 . 1 6 8 2 & - 0 . 1 1 0 2 & - 0 . 1 3 0 0 \\\\ 0 . 2 1 8 8 & 0 . 6 0 7 2 & 0 . 6 7 9 3 & - 0 . 7 7 1 8 & - 0 . 3 1 5 8 \\end{bmatrix} , \\end{align*}"}
+{"id": "9082.png", "formula": "\\begin{align*} c _ { i l } = 0 , \\mbox { f o r a l l $ i , l \\in N $ : $ i \\in \\mathcal P ^ { ( r ) } $ , $ l \\in \\mathcal N ^ { ( r ) } $ } . \\end{align*}"}
+{"id": "1333.png", "formula": "\\begin{align*} p = \\mathrm { d } L ^ { \\mathrm { v e r } } ( a ) . \\end{align*}"}
+{"id": "1980.png", "formula": "\\begin{align*} \\dfrac { \\partial _ t H } { N } = 2 \\vert h \\vert ^ 2 - \\dfrac { g ^ { i j } \\partial _ t ^ 2 g _ { i j } } { 2 N ^ 2 } - \\dfrac { \\partial _ t N H } { N ^ 2 } \\ , . \\end{align*}"}
+{"id": "7656.png", "formula": "\\begin{align*} \\lambda _ 0 = \\sup _ { s \\in ( 0 , 1 ) } \\sup _ { \\mu > 0 } \\inf _ { E \\in I } \\widehat { \\lambda } _ { s , \\mu } ( E ) \\end{align*}"}
+{"id": "7468.png", "formula": "\\begin{align*} \\left ( C ( \\mathbb { S } ^ { n - 1 } ) \\times \\mathbb { R } , G = d r ^ 2 + r ^ 2 g + d l ^ 2 \\right ) , \\end{align*}"}
+{"id": "7212.png", "formula": "\\begin{align*} h ( v , N ) = - v + b N , a ( N ) = a _ 0 + a _ 1 N , \\end{align*}"}
+{"id": "7639.png", "formula": "\\begin{align*} \\Tilde { \\mathcal { A } } = \\{ \\left < x , \\mu _ { \\Tilde { \\mathcal { A } } } ( x ) , \\nu _ { \\Tilde { \\mathcal { A } } } ( x ) \\right > \\mid x \\in \\mathbb X \\} , \\end{align*}"}
+{"id": "7143.png", "formula": "\\begin{align*} \\begin{cases} \\dot { k } = - \\displaystyle { \\frac { \\partial p ( x , k ) } { \\partial x } } \\ , \\ , , \\\\ [ 3 m m ] \\dot { x } = \\displaystyle { \\frac { \\partial p ( x , k ) } { \\partial k } } \\ , \\ , . \\end{cases} \\end{align*}"}
+{"id": "3193.png", "formula": "\\begin{align*} a _ n ( m ) = a _ { n - 1 } ( m ) + a _ { n - 1 } ( m - 3 n - 1 ) + a _ { n - 1 } ( m - 3 n - 2 ) + a _ { n - 1 } ( m - 6 n - 3 ) . \\end{align*}"}
+{"id": "8749.png", "formula": "\\begin{align*} \\beta ( d x , d y ) & = \\delta _ { ( x _ - , y _ - ) } + \\frac { z - y _ - } { z - y _ + } \\delta _ { ( x _ + , y _ + ) } + \\frac { y _ - - y _ + } { z - y _ + } \\delta _ { ( x _ + , z ) } , \\\\ \\gamma ( d x , d y ) & = \\delta _ { ( x _ + , y _ - ) } + \\frac { z - y _ - } { z - y _ + } \\delta _ { ( x _ - , y _ + ) } + \\frac { y _ - - y _ + } { z - y _ + } \\delta _ { ( x _ - , z ) } , \\end{align*}"}
+{"id": "7687.png", "formula": "\\begin{align*} \\Phi ( V ) = \\sum _ { m \\in \\mathbb { Z } ^ d } a ( n , m ) \\langle { \\delta _ m } , F ( A + \\lambda V _ { \\omega } + g V ) \\delta _ m \\rangle \\end{align*}"}
+{"id": "5488.png", "formula": "\\begin{align*} \\int _ { | M | > 3 | E | } | M + E | ^ K d U & = \\int _ { | M | > 3 | E | } | M | ^ K d U + O _ K \\left ( \\int | M | ^ { K - 1 } | E | + | M | ^ { K - 2 } | E | ^ 2 d U \\right ) . \\end{align*}"}
+{"id": "4867.png", "formula": "\\begin{align*} 2 \\alpha H ^ { \\alpha - 1 } \\nabla _ \\Sigma H \\cdot \\langle A , \\nabla _ \\Sigma A \\rangle = \\frac \\alpha H | \\nabla _ \\Sigma H | ^ 2 \\left ( 2 H ^ { \\alpha + 1 } + \\alpha \\frac v { H } \\right ) . \\end{align*}"}
+{"id": "4861.png", "formula": "\\begin{align*} w _ m ( \\epsilon ) & \\le \\Big | \\int _ { E _ m + y _ 0 } g ( x ) d x - \\int _ { E _ m } g ( x ) d x \\Big | \\\\ & \\le \\mu | y _ 0 | ^ 2 = \\Big ( | | D ^ 2 g | | _ { L ^ 1 ( E _ m ) } + m \\frac { 1 } { 6 } \\Big ) ( \\frac { \\epsilon m } { a _ { m _ * } } ) ^ 2 . \\end{align*}"}
+{"id": "4877.png", "formula": "\\begin{align*} g = 2 q - 1 + b , \\deg ( B ) = 2 b . \\end{align*}"}
+{"id": "6970.png", "formula": "\\begin{align*} \\eta ^ { 2 } ( \\alpha \\mathbf { z } ) = - \\langle \\alpha \\mathbf { z } , \\overline { \\alpha \\mathbf { z } } \\rangle = - \\lvert \\alpha \\rvert ^ 2 \\langle \\mathbf { z } , \\mathbf { \\overline { z } } \\rangle = \\lvert \\alpha \\rvert ^ 2 \\eta ^ { 2 } ( \\mathbf { z } ) , \\end{align*}"}
+{"id": "6934.png", "formula": "\\begin{align*} l _ { e x t r } : = \\left ( \\frac { \\Psi ( \\varepsilon _ \\# ) - \\Psi ( - \\eta ) } { A _ I } \\right ) ^ \\frac { 1 } { 2 \\mu } , \\end{align*}"}
+{"id": "4518.png", "formula": "\\begin{align*} \\mathcal { T } ^ { ( 2 ) } _ n : = \\bigg \\{ V ^ { ( 2 ) } ( n ) \\leqslant \\frac { 2 C _ 0 T ( \\ell ) } { \\ell ^ { K / 2 } } \\bigg \\} . \\end{align*}"}
+{"id": "1721.png", "formula": "\\begin{align*} M : = \\{ z \\in \\mathbb { C } ^ 2 \\oplus \\mathbb { C } ^ { n _ 1 } \\oplus \\cdots \\oplus \\mathbb { C } ^ { n _ \\mu } \\ , | \\ , \\Re ( z _ 0 ) = \\Phi \\} \\end{align*}"}
+{"id": "750.png", "formula": "\\begin{align*} { } ^ c \\nabla _ k { } ^ c \\nabla _ j \\varphi _ i - { } ^ c \\nabla _ j { } ^ c \\nabla _ k \\varphi _ i = - \\overset { \\ _ * } { R } ^ h _ { i j k } \\varphi _ h - R ^ h _ { j k } \\varphi _ { h ; i } , \\end{align*}"}
+{"id": "7303.png", "formula": "\\begin{align*} L _ { X _ m } ( n , \\chi ) = ( - 1 ) ^ { n + 1 } 2 ^ { n } \\frac { m } { \\overline { \\tau ( \\chi ) } } \\sum _ { r = 0 } ^ { m - 1 } \\overline { \\chi ( r ) } c _ { m , r } ( n - 1 ) , n \\in \\mathbb { N } , \\end{align*}"}
+{"id": "1920.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { V } \\Delta h = - 1 \\quad & \\ , \\ , \\ , G \\setminus B _ { \\hat R } , \\\\ h > 0 & \\ , \\ , \\ , G \\setminus B _ { \\hat R } , \\\\ \\lim _ { x \\to \\infty } h ( x ) = 0 \\ , , & \\end{aligned} \\end{align*}"}
+{"id": "8884.png", "formula": "\\begin{align*} C _ { N , \\tau , \\alpha , \\lambda } = \\frac { \\mathcal P ( \\varphi ) } { \\| \\sqrt { \\mathcal K _ \\lambda } \\varphi \\| ^ { 2 p } } = ( { \\mathcal P [ \\varphi ] } ) ^ { 1 - p } \\end{align*}"}
+{"id": "7486.png", "formula": "\\begin{align*} & \\frac { \\partial } { \\partial s } K _ L ( x , t , \\cdot , \\cdot ) = - L _ s K _ L ( x , t , \\cdot , \\cdot ) + R ( g _ 0 ( s ) ) K _ L ( x , t , \\cdot , \\cdot ) , \\\\ & \\frac { \\partial } { \\partial s } g = 2 R i c ( g _ 0 ( \\tau ) ) , \\\\ & \\lim _ { \\tau \\to 0 } K _ L ( x , t , \\cdot , \\tau ) = \\delta _ x . \\end{align*}"}
+{"id": "1247.png", "formula": "\\begin{align*} { n - 1 \\brack k } _ { q ^ 2 } & = \\frac { ( 1 - q ^ { 2 n - 2 } ) ( 1 - q ^ { 2 n - 4 } ) \\cdots ( 1 - q ^ { 2 n - 2 k } ) } { ( 1 - q ^ 2 ) ( 1 - q ^ 4 ) \\cdots ( 1 - q ^ { 2 k } ) } \\\\ [ 7 p t ] & \\equiv \\frac { ( 1 - q ^ { - 2 } ) ( 1 - q ^ { - 4 } ) \\cdots ( 1 - q ^ { - 2 k } ) } { ( 1 - q ^ 2 ) ( 1 - q ^ 4 ) \\cdots ( 1 - q ^ { 2 k } ) } \\pmod { \\Phi _ n ( q ) } \\\\ [ 7 p t ] & = ( - 1 ) ^ k q ^ { - k ^ 2 - k } . \\end{align*}"}
+{"id": "7997.png", "formula": "\\begin{align*} \\bigg | \\varepsilon _ k \\frac { | \\nabla u _ k | ^ 2 } { 2 } + \\frac { W ( u _ k ) } { \\varepsilon _ k } - 2 | \\nabla w _ k | \\bigg | & = \\Bigg ( \\sqrt { \\frac { \\varepsilon _ k } { 2 } | \\nabla u _ k | } - \\sqrt { \\frac { W ( u _ k ) } { \\varepsilon _ k } } \\Bigg ) ^ 2 = | \\xi _ k | , \\end{align*}"}
+{"id": "279.png", "formula": "\\begin{align*} \\frac { d ^ 2 x } { d \\tau ^ 2 } + \\tilde g ( x ) \\frac { d x } { d \\tau } + \\tilde h ( x ) = 0 , \\end{align*}"}
+{"id": "2701.png", "formula": "\\begin{align*} & K ^ { ( 1 ) } _ { i j } : = \\partial ^ { 2 } L / \\partial \\dot { q } ^ { i } \\partial \\dot { q } ^ { j } , \\\\ & S _ { i } : = - v ^ { j } \\frac { \\partial ^ { 2 } L } { \\partial q ^ { i } \\partial v ^ { j } } - \\frac { \\partial ^ { 2 } L } { \\partial t \\partial v ^ { i } } + \\frac { \\partial L } { \\partial q ^ { i } } . \\end{align*}"}
+{"id": "5993.png", "formula": "\\begin{align*} 0 \\leq & \\phi _ { \\nu } = \\frac { ( | \\nabla u | ^ 2 ) _ { \\nu } } { ( b ^ 2 - \\rho ^ 2 ) ^ 2 } + \\frac { 2 | \\nabla u | ^ 2 ( \\rho ^ 2 \\circ u ) _ { \\nu } } { ( b ^ 2 - \\rho ^ 2 ) ^ 3 } \\\\ = & - 2 H \\frac { | \\nabla u | ^ 2 } { ( b ^ 2 - \\rho ^ 2 ) ^ 2 } + \\frac { 4 \\rho | \\nabla u | ^ 2 \\langle \\nabla ^ N \\rho , d u ( \\nu ) \\rangle } { ( b ^ 2 - \\rho ^ 2 ) ^ 3 } . \\end{align*}"}
+{"id": "138.png", "formula": "\\begin{align*} J ( z ) = J ( | z | ) = \\frac { 1 } { 2 } \\frac { d ^ 2 } { d r ^ 2 } \\bigg | _ { r = | z | } r ^ 2 \\big [ \\coth { ( r ) } - 1 \\big ] . \\end{align*}"}
+{"id": "6467.png", "formula": "\\begin{align*} 0 = \\lambda _ 0 \\leq \\lambda _ 1 \\leq \\lambda _ 2 \\leq \\ldots \\to \\infty . \\end{align*}"}
+{"id": "1085.png", "formula": "\\begin{align*} I _ n & = O ( n ^ { - d + \\kappa ( 1 - d ) } ) + O \\left ( \\sum _ { t = [ n ^ { \\kappa } ] + 1 } ^ { n } \\| y _ t \\| \\right ) , n \\to \\infty , \\\\ J _ n & = O \\left ( \\sum _ { t = n + 1 } ^ { \\infty } \\| y _ t \\| \\right ) , n \\to \\infty . \\end{align*}"}
+{"id": "4429.png", "formula": "\\begin{gather*} \\hat { \\phi } ( \\cdot , t ) \\coloneqq \\frac { t _ { j } - t } { \\tau _ j } \\ , \\phi ^ { j - 1 } + \\frac { t - t _ { j - 1 } } { \\tau _ j } \\ , \\phi ^ { j } \\quad \\ \\ t \\in [ t _ { j - 1 } , t _ j ] , j = 1 , \\dots , M . \\end{gather*}"}
+{"id": "5529.png", "formula": "\\begin{align*} \\hat \\Delta ( z _ n ^ 2 ) = - \\tilde d _ n \\hat \\rho _ n + \\hat d _ n ( z _ n - \\rho _ n ) , \\tilde d _ n : = \\lim _ { \\rho \\to z _ n } \\tilde f ( \\rho ) , \\tilde f ( \\rho ) : = \\frac { \\tilde \\Delta ( \\rho ^ 2 ) } { \\rho - \\tilde \\rho _ n } , \\end{align*}"}
+{"id": "2555.png", "formula": "\\begin{align*} \\bigoplus _ { m = 0 } ^ \\infty C ^ 0 _ m \\to C ^ { - \\infty } _ L ( M ) \\ ; , \\end{align*}"}
+{"id": "104.png", "formula": "\\begin{align*} b ^ 2 - 4 a c = G \\cdot P \\cdot E \\Big ( \\frac { G } { n - 1 } + K \\Big ) . \\end{align*}"}
+{"id": "5388.png", "formula": "\\begin{align*} a _ j = \\frac { 1 } { 1 + \\rho } \\ , \\frac { 1 + \\cdots + \\rho ^ j } { 1 + \\cdots + \\rho ^ { j - 1 } } = \\begin{cases} \\displaystyle \\frac { 1 } { 1 + \\rho } \\ , \\frac { \\rho ^ { j + 1 } - 1 } { \\rho ^ j - 1 } & \\rho \\neq 1 \\\\ \\\\ \\displaystyle \\frac { 1 } { 2 } \\ , \\frac { j + 1 } { j } & \\rho = 1 . \\end{cases} \\end{align*}"}
+{"id": "1036.png", "formula": "\\begin{align*} T _ n ( w ) \\mathbf { z } _ n = \\mathbf { y } _ n . \\end{align*}"}
+{"id": "3828.png", "formula": "\\begin{align*} \\xi ( q ) : = \\frac { \\left ( - q ^ 3 ; q ^ 3 \\right ) _ \\infty } { \\left ( q ^ 2 ; q ^ 2 \\right ) _ \\infty } = : \\sum _ { n = 0 } ^ \\infty r ( n ) q ^ n . \\end{align*}"}
+{"id": "5633.png", "formula": "\\begin{align*} 1 + | s _ 3 | + \\cdots + | s _ { 2 k - 1 } | = | b _ 2 | + | b _ 4 | + \\cdots + | b _ { 2 k } | , \\end{align*}"}
+{"id": "5546.png", "formula": "\\begin{align*} \\kappa = \\sqrt { n } \\max _ i \\norm { \\phi _ i } _ \\infty . \\end{align*}"}
+{"id": "1000.png", "formula": "\\begin{align*} R ^ D \\beta ( x ) = \\int _ { ( D \\cup \\partial _ m D ) ^ c } P _ D ( x , y ) \\ , \\gamma ( d y ) \\quad x \\in D \\end{align*}"}
+{"id": "7839.png", "formula": "\\begin{align*} \\nabla _ { \\bar E _ { \\alpha } } ^ { \\perp \\psi } H ^ { \\psi } = \\frac { q } { p + q } \\bar \\nabla ^ { \\perp } _ { \\bar E _ { \\alpha } } H _ 2 + \\frac { p } { p + q } B ^ j ( \\bar E _ { \\alpha } , H _ 1 ) , A ^ { \\psi } _ { H ^ { \\psi } } \\bar E _ { \\alpha } = \\frac { q } { p + q } \\bar A _ { H _ 2 } \\bar E _ { \\alpha } . \\end{align*}"}
+{"id": "7495.png", "formula": "\\begin{align*} & L _ t h _ { i j } = \\Delta _ { g _ 0 ( t ) } h _ { i j } + 2 R m ( g _ 0 ( t ) ) _ { i j k l } h _ { k l } - R i c ( g _ 0 ( t ) ) _ { i k } h _ { k j } - R i c ( g _ 0 ( t ) ) _ { j k } h _ { k i } , \\\\ & R _ 0 [ h ] = R m ( g _ 0 ( t ) ) * h * h + O ( h ^ 3 ) * R m ( g _ 0 ( t ) ) + g ^ { - 1 } * g ^ { - 1 } * \\nabla ^ { g _ 0 ( t ) } h * \\nabla ^ { g _ 0 ( t ) } h , \\\\ & \\nabla R _ 1 [ h ] = \\nabla _ p ^ { g _ 0 ( t ) } \\left ( \\left ( ( g _ 0 ( t ) + h ( t ) ) ^ { p q } - ( g _ { 0 } ( t ) ) ^ { p q } \\right ) \\nabla _ q ^ { g _ 0 ( t ) } h \\right ) , \\end{align*}"}
+{"id": "524.png", "formula": "\\begin{align*} \\nu _ { l , l ' } : = \\begin{cases} \\emptyset , & l \\mid l ' , \\\\ \\neg , & , \\end{cases} \\end{align*}"}
+{"id": "2688.png", "formula": "\\begin{align*} Q _ { ( \\alpha ) } ^ { i } : = D ^ { \\alpha - 1 } q ^ { i } , \\ \\ \\ P ^ { ( \\alpha ) } _ { i } : = \\sum ^ { d } _ { \\beta = \\alpha \\geq 1 } ( - D ) ^ { \\beta - \\alpha } \\frac { \\partial L } { \\partial ( D ^ { \\beta } q ^ { i } ) } , \\end{align*}"}
+{"id": "8464.png", "formula": "\\begin{align*} | u _ m | + | u _ { m - 1 } | = ( u _ { m - 1 } - u _ m ) + 2 u _ m & \\leq ( u _ { m - 1 } - u _ m ) + \\frac { 2 } { \\ell } \\\\ & \\leq ( u _ { m - 1 } - u _ m ) + 2 ( u _ { m - 1 } - u _ m ) \\\\ & = 3 ( u _ { m - 1 } - u _ m ) \\\\ & \\leq 3 | u _ m - u _ { m - 1 } | . \\end{align*}"}
+{"id": "2136.png", "formula": "\\begin{align*} f ( \\hat { x } , q ( \\hat { x } , \\hat { y } ) ) = f ( \\hat { x } , \\hat { z } ) = \\inf \\limits _ { z \\in Z ( \\hat { x } ) } f ( \\hat { x } , z ) < f ( \\hat { x } , \\hat { y } ) . \\end{align*}"}
+{"id": "7559.png", "formula": "\\begin{align*} \\Sigma _ i ( t ) & \\leq C \\int _ 0 ^ t \\Psi ( \\tau ) \\ d \\tau \\ , \\ \\ \\ i = 1 , 2 \\ , \\\\ \\Sigma _ 3 ( t ) & \\leq C \\varepsilon t + C \\int _ 0 ^ t \\Psi ( \\tau ) \\ d \\tau \\ , \\\\ \\Sigma _ 4 ( t ) & \\leq C \\delta t \\ , \\\\ \\Sigma _ j ( t ) & \\leq C \\delta t + C \\int _ 0 ^ t \\Psi ( \\tau ) \\ d \\tau \\ , \\ \\ \\ j = 5 , 6 \\ . \\end{align*}"}
+{"id": "7079.png", "formula": "\\begin{align*} H : = ( \\zeta - \\Delta ) ^ { - \\frac { 1 } { 4 } } | b | ^ { \\frac { 1 } { 2 } } , S : = b ^ { \\frac { 1 } { 2 } } \\cdot \\nabla ( \\zeta - \\Delta ) ^ { - \\frac { 3 } { 4 } } \\end{align*}"}
+{"id": "5268.png", "formula": "\\begin{align*} x ^ { k + 1 } _ { j } = x ^ { k } _ { j } + \\alpha ^ { k } _ { j } x ^ { k } _ { j } \\left ( - \\frac { \\partial D } { \\partial x } \\right ) ^ { k } _ { j } \\end{align*}"}
+{"id": "4848.png", "formula": "\\begin{align*} \\Big | \\int _ { E _ m + y } g ( x ) d x - \\int _ { E _ m } g ( x ) d x \\Big | \\\\ & = \\int _ { E _ m } g ( x + y ) - g ( x ) d x \\\\ & = y \\cdot \\int _ { E _ m } \\nabla g ( x ) d x + \\int _ { E _ m } o _ x ( | y | ) d x \\\\ & = \\int _ { E _ m } o _ x ( | y | ) d x . \\end{align*}"}
+{"id": "1650.png", "formula": "\\begin{align*} \\max _ { \\overline { Q } _ { T } } \\varphi _ { \\lambda , k } ^ { 2 } \\left ( t \\right ) = e ^ { 2 \\lambda \\left ( T + 1 \\right ) ^ { k } } . \\end{align*}"}
+{"id": "4953.png", "formula": "\\begin{align*} ( n - 1 ) r ^ 2 \\lambda ^ 2 = \\mbox { t r } U ^ 2 _ x ( 0 ) = ( \\mbox { t r } U _ x ( 0 ) ) ^ 2 \\ \\mbox { a n d } \\ ( N - n + 1 ) r \\lambda = - \\langle \\nabla V ( p ) , \\tilde K ( p ) \\rangle . \\end{align*}"}
+{"id": "2051.png", "formula": "\\begin{align*} \\mathrm { d } \\Theta ( s , x ) = - \\alpha \\int _ 0 ^ 1 A ( x , y ) \\Theta ( s , y ) \\mathrm { d } y \\mathrm { d } s + \\alpha \\beta \\mathrm { d } \\xi _ 1 ( s , x ) . \\end{align*}"}
+{"id": "3355.png", "formula": "\\begin{align*} \\min _ { \\phi ^ \\circ } \\ \\ \\mathcal { I } ( \\phi ^ \\circ ) : = \\mathbb { E } ^ { u ^ \\circ } \\left [ ( \\mu \\Gamma ( x _ K ) ) \\right ] - \\gamma + \\sum _ { i = 1 } ^ K H ( \\phi ^ \\circ _ i ) , \\end{align*}"}
+{"id": "1860.png", "formula": "\\begin{align*} n ^ c _ { \\phi } ( X ) : = \\sum _ { P ^ { \\dagger } } n _ { \\phi } ( X , P ^ { \\dagger } ) \\in \\Z , \\end{align*}"}
+{"id": "2449.png", "formula": "\\begin{align*} \\begin{cases} U ( \\xi ) \\sim A _ { 1 } ( \\xi - \\xi _ { - } ) , \\\\ U ' ( \\xi ) \\sim A _ { 2 } \\end{cases} { \\rm { a s } } \\xi \\searrow \\xi _ { - } + 0 , \\end{align*}"}
+{"id": "6668.png", "formula": "\\begin{align*} Z _ 0 & = \\abs { T _ s } ^ 2 T , \\\\ Z _ 1 & = 2 \\langle T _ { s s } , T _ s \\rangle T + 2 \\abs { T _ s } ^ 2 T _ s , \\end{align*}"}
+{"id": "5182.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } D _ { \\beta } ( p \\| q ) } { \\partial q _ { j } } = \\left [ \\frac { a - 1 } { a - b } A ^ { a - 2 } - \\frac { b - 1 } { a - b } A ^ { b - 2 } \\right ] p _ { j } q ^ { \\beta - 2 } _ { j } - \\left [ \\frac { a - 1 } { a - b } B ^ { a - 2 } - \\frac { b - 1 } { a - b } B ^ { b - 2 } \\right ] q ^ { \\beta - 1 } _ { j } \\end{align*}"}
+{"id": "7975.png", "formula": "\\begin{align*} \\dot { \\mathcal { F } } ( v , \\Sigma ) = \\{ \\mathcal { F } , H \\} _ { D } ( v , \\Sigma ) - \\int _ { \\Gamma } E ( \\frac { \\delta \\mathcal { F } } { \\delta \\Sigma } ) \\wedge \\ast \\boldsymbol { n } ( v ) , \\end{align*}"}
+{"id": "4235.png", "formula": "\\begin{align*} \\Theta _ 3 ( T _ { C } X ) = 1 + q ^ { \\frac { 1 } { 2 } } \\widetilde { T _ C X } + q ( \\widetilde { T _ C X } + \\wedge ^ 2 \\widetilde { T _ C X } ) + O ( q ^ { \\frac { 3 } { 2 } } ) , \\end{align*}"}
+{"id": "8529.png", "formula": "\\begin{align*} \\kappa _ { C } ( g ) & = g ( e _ 2 ) - e _ 2 \\\\ & = b ( g ) \\cdot \\eta e _ 1 - \\log ( g ) \\cdot \\eta e _ 2 . \\end{align*}"}
+{"id": "1661.png", "formula": "\\begin{align*} p \\left ( x , 0 \\right ) = p _ { 0 } \\left ( x \\right ) = m _ { 0 } ^ { \\left ( 1 \\right ) } \\left ( x \\right ) - m _ { 0 } ^ { \\left ( 2 \\right ) } \\left ( x \\right ) , x \\in \\Omega . \\end{align*}"}
+{"id": "2018.png", "formula": "\\begin{align*} \\left | ( \\rho - 1 ) a \\alpha ^ n - d _ 1 \\cdot \\rho ^ { \\ell + m + k } \\right | & = \\left | - ( \\rho - 1 ) \\Pi ( n ) - ( d _ 1 - d _ 2 ) \\cdot \\rho ^ { m + k } - ( d _ 2 - d _ 3 ) \\cdot \\rho ^ k - d _ 3 \\right | \\\\ & \\leqslant ( \\rho - 1 ) \\cdot \\alpha ^ { - n / 2 } + ( \\rho - 1 ) \\cdot \\rho ^ { m + k } + ( \\rho - 1 ) \\cdot \\rho ^ k + ( \\rho - 1 ) \\\\ & < 3 ( \\rho - 1 ) \\cdot \\rho ^ { m + k } , \\end{align*}"}
+{"id": "1612.png", "formula": "\\begin{align*} Q _ { r ( p - 1 ) } \\b Q _ { s ( p - 1 ) } ( x ) = q _ { g , r } & \\sum _ j \\tfrac { ( j - s ) ( p - 1 ) } { p j - s ( p - 1 ) - r + 1 } t _ { r , s , j } \\ , \\b Q _ { ( r - 1 + p s - p j ) ( p - 1 ) } Q _ { j ( p - 1 ) } ( x ) \\\\ & - q _ { g , n , s } \\sum _ j t _ { r , s , j } \\ , Q _ { ( r + p s - p j ) ( p - 1 ) } \\b Q _ { j ( p - 1 ) } ( x ) \\end{align*}"}
+{"id": "7063.png", "formula": "\\begin{align*} X _ t = x - \\int _ 0 ^ t b ( X _ s ) d s + \\sqrt { 2 } \\int _ 0 ^ t \\sigma ( X _ s ) d W _ s . \\end{align*}"}
+{"id": "8685.png", "formula": "\\begin{align*} u ^ H = U ^ H - \\sqrt { - 1 } J U ^ H , v = V ^ H - \\sqrt { - 1 } J V ^ H \\in T ^ { 1 , 0 } X \\cap \\mathbb { C } T Y ' . \\end{align*}"}
+{"id": "4676.png", "formula": "\\begin{align*} \\mathbf { B } _ { + } ( D ) : = \\bigcap _ { D = A + E } \\mathrm { S u p p } ( E ) , \\end{align*}"}
+{"id": "4356.png", "formula": "\\begin{gather*} \\alpha : = - 2 \\overline { u } _ i - \\Delta u _ i , \\ \\beta : = 1 , \\ \\gamma = - \\overline { u } _ i ^ 2 - \\overline { u } _ i \\Delta u _ i . \\ \\end{gather*}"}
+{"id": "6784.png", "formula": "\\begin{align*} \\det J _ { \\mathrm { r e d } } = \\mu | \\mu | \\left ( \\frac { 2 } { \\bar { x } \\bar { y } } + \\frac { 1 } { \\bar { x } \\bar { z } } + \\frac { 1 } { \\bar { x } \\bar { w } } \\right ) , \\end{align*}"}
+{"id": "2782.png", "formula": "\\begin{align*} \\delta \\left ( { \\sigma _ { \\tau } ^ { * } } ( t ) I \\right ) : = 0 \\end{align*}"}
+{"id": "6983.png", "formula": "\\begin{align*} z _ j ^ 2 = - \\lvert z _ j \\rvert ^ 2 e ^ { i \\theta } \\ ; \\ ; 1 \\leq j \\leq n . \\end{align*}"}
+{"id": "803.png", "formula": "\\begin{align*} \\tilde R _ { i j } & = c ( x ) g _ { i j } , \\\\ { R } i c _ { i j } & = ( n - 1 ) k ( x ) g _ { i j } , \\end{align*}"}
+{"id": "6750.png", "formula": "\\begin{align*} v ^ { k + 1 } = v ^ k - M ( v ^ k - \\widetilde { v } ^ k ) . \\end{align*}"}
+{"id": "8650.png", "formula": "\\begin{align*} F ^ * ( x ^ * ) + 1 _ K ^ * ( x ^ * - y _ 0 ^ * ) & \\geq \\langle x , x ^ * \\rangle - F ( x ) + \\langle - x , x ^ * - y _ 0 ^ * \\rangle , \\\\ & = \\langle x , y _ 0 ^ * \\rangle - F ( x ) . \\end{align*}"}
+{"id": "6582.png", "formula": "\\begin{align*} h ( y ) = \\begin{cases} 1 & y > 1 , \\\\ \\frac { 1 } { 2 } & y = 1 , \\\\ 0 & 0 < y < 1 . \\end{cases} \\end{align*}"}
+{"id": "2938.png", "formula": "\\begin{align*} \\sum \\limits _ { \\{ i , \\bar { i } , 2 n + 1 \\} , l = 1 , \\ldots , n } v _ { \\{ i , \\bar { i } , 2 n + 1 \\} } = \\sum \\limits _ { i = 1 , \\ldots , n } \\alpha _ i = 0 . \\end{align*}"}
+{"id": "2792.png", "formula": "\\begin{align*} \\delta \\Xi ^ { \\alpha } ( t _ { 2 } ) - \\delta \\Xi ^ { \\alpha } ( t _ { 1 } ) = \\delta \\int ^ { t _ { 2 } } _ { t _ { 1 } } \\left [ \\zeta ^ { \\alpha } + f ( Q ^ { i } , P _ { i } ) \\right ] d t = \\delta \\int ^ { t _ { 2 } } _ { t _ { 1 } } \\zeta ^ { \\alpha } d t , \\end{align*}"}
+{"id": "7228.png", "formula": "\\begin{align*} \\partial _ v p ( V _ { \\min } ) = 0 . \\end{align*}"}
+{"id": "3348.png", "formula": "\\begin{align*} \\frac { d \\mathcal { P } ^ u } { d \\mathcal { P } ^ \\dagger } = Z _ k = \\prod _ { i = 1 } ^ k \\Psi ( x _ { i - 1 } , y _ i ) , \\end{align*}"}
+{"id": "1282.png", "formula": "\\begin{align*} \\frac { 1 } { \\lambda _ 1 } e ^ { - 2 \\lambda _ 1 t } + \\frac { 1 } { \\lambda _ 2 } e ^ { - 2 \\lambda _ 2 t } & = \\frac { ( \\lambda _ 1 + \\lambda _ 2 ) e ^ { - 2 \\lambda _ 2 t } } { \\lambda _ 1 \\lambda _ 2 } + \\frac { ( e ^ { - 2 \\lambda _ 1 t } - e ^ { - 2 \\lambda _ 2 t } ) } { \\lambda _ 1 } \\\\ & = \\frac { \\gamma e ^ { - 2 \\lambda _ 2 t } } { \\omega ^ 2 } + \\frac { ( e ^ { - 2 \\lambda _ 1 t } - e ^ { - 2 \\lambda _ 2 t } ) } { \\lambda _ 1 } \\end{align*}"}
+{"id": "3216.png", "formula": "\\begin{align*} I _ { 1 , n } \\leq C \\sum \\limits _ { k = 1 } ^ n \\frac { \\lambda _ k ^ 2 | \\Psi _ k | ^ 2 } { \\left ( T ^ { \\alpha + \\mu } \\frac { 1 } { T ^ { \\alpha + \\beta } } \\right ) ^ 2 } \\leq C T ^ { 2 ( \\beta - \\mu ) } \\sum \\limits _ { k = 1 } ^ n \\lambda _ k ^ 2 | \\Psi _ k | ^ 2 . \\end{align*}"}
+{"id": "627.png", "formula": "\\begin{align*} b ( K ) = \\sup _ { K _ 1 , . . . , K _ n } \\frac { V ( K _ 1 , . . . , K _ n ) V ( K ) } { V ( K _ 1 , K [ n - 1 ] ) V ( K _ 2 , . . . , K _ n , K ) } , \\end{align*}"}
+{"id": "7411.png", "formula": "\\begin{align*} p = r , \\ q = r - 1 , \\ a = 2 , \\end{align*}"}
+{"id": "8841.png", "formula": "\\begin{align*} \\begin{cases} i \\partial _ t u - \\mathcal K _ \\lambda u = \\epsilon | x | ^ { - \\tau } | u | ^ { p - 2 } ( I _ \\alpha * | \\cdot | ^ { - \\tau } | u | ^ p ) u , \\\\ u ( x , 0 ) = u _ 0 ( x ) , ( x , t ) \\in \\mathbb { R } ^ n \\times \\mathbb { R } , \\end{cases} \\end{align*}"}
+{"id": "29.png", "formula": "\\begin{align*} d _ j ^ * g ( r ) = \\sum _ { r \\subset s } \\frac { m ( s ) } { m ( r ) } g ( s ) \\end{align*}"}
+{"id": "478.png", "formula": "\\begin{align*} \\Gamma _ { \\bigotimes _ { j \\in I } { g _ j } } = \\bigotimes _ { j \\in I } { \\Gamma _ { g _ j } } . \\end{align*}"}
+{"id": "5536.png", "formula": "\\begin{align*} A : = \\{ i < \\theta \\mid p _ i \\Vdash _ { \\P } ` ` \\dot { b } _ i \" \\} \\end{align*}"}
+{"id": "8411.png", "formula": "\\begin{align*} B _ 1 ( T ( \\sigma ( a _ 1 ) ) ) B _ 1 ( a _ 2 ) & = S ( B _ 1 ( a _ 1 ) ) B _ 1 ( \\sigma ( a _ 2 ) ) B _ 1 ( T ( \\sigma ( a _ 3 ) ) ) B _ 1 ( a _ 4 ) \\\\ & = S ( B _ 1 ( a _ 1 ) ) B _ 1 ( \\sigma ( a _ 2 ) \\circ T ( \\sigma ( a _ 3 ) ) ) B _ 1 ( a _ 4 ) \\\\ & = S ( B _ 1 ( a _ 1 ) ) \\epsilon ( a _ 2 ) B _ 1 ( a _ 3 ) = \\epsilon ( a ) 1 . \\end{align*}"}
+{"id": "6042.png", "formula": "\\begin{align*} \\delta = \\frac 2 3 E . \\end{align*}"}
+{"id": "6491.png", "formula": "\\begin{align*} \\chi _ q ( L ( m ) ) \\chi _ q ( L ( m ' ) ) = \\chi _ q ( L ( m _ 1 ) ) + \\chi _ q ( L ( m _ 2 ) ) \\end{align*}"}
+{"id": "978.png", "formula": "\\begin{align*} w = R ^ D f _ h ( \\cdot , w ) + R ^ D \\mu \\quad \\mbox { q . e . } , \\end{align*}"}
+{"id": "7396.png", "formula": "\\begin{align*} \\| ( U _ { j + 1 } ) _ x - ( U _ j ) _ x \\| _ 3 = O \\left ( \\frac { 1 } { 2 ^ j } \\right ) \\quad \\mbox { a s } \\ j \\rightarrow \\infty . \\end{align*}"}
+{"id": "1489.png", "formula": "\\begin{align*} & \\alpha \\circ P _ d ( L _ m ) = \\alpha ( f ( m + d ) L _ { m + d } ) = f ( m + d ) \\alpha ( L _ { m + d } ) = f ( m + d ) ( 1 + q ^ { m + d } ) L _ { m + d } , \\\\ & P _ d \\circ \\alpha ( L _ m ) = P _ d ( \\alpha ( L _ m ) ) = P _ d ( ( 1 + q ^ { m } ) L _ m ) = ( 1 + q ^ { m } ) P _ d ( L _ m ) = ( 1 + q ^ { m } ) f ( m + d ) L _ { m + d } , \\\\ & \\alpha \\circ P _ d ( L _ m ) = P _ d \\circ \\alpha ( L _ m ) \\ \\Leftrightarrow \\forall \\ m \\in \\mathbb { Z } : \\ f ( m + d ) ( q ^ { d } - 1 ) = 0 , \\end{align*}"}
+{"id": "2049.png", "formula": "\\begin{align*} \\Theta ^ { d , T } ( \\cdot , x ) = ( \\lfloor { d x } \\rfloor + 1 - d x ) \\Theta ^ { d , T } \\Big ( \\cdot , \\frac { \\lfloor { d x } \\rfloor } { d } \\Big ) + ( d x - \\lfloor { d x } \\rfloor ) \\Theta \\Big ( \\cdot , \\frac { \\lfloor { d x } \\rfloor + 1 } { d } \\Big ) . \\end{align*}"}
+{"id": "1047.png", "formula": "\\begin{align*} \\| v _ { n + 1 } ^ { - 1 } - v _ { \\infty } ^ { - 1 } \\| + \\sum _ { k = 1 } ^ { n } \\| v _ { n + 1 } ^ { - 1 } \\phi _ { n , k } - v _ { \\infty } ^ { - 1 } \\phi _ { k } \\| = O ( n ^ { - d } ) , n \\to \\infty . \\end{align*}"}
+{"id": "6897.png", "formula": "\\begin{align*} | M ( z _ { 0 } ) - 1 | ^ { 2 } & = \\left | \\cosh \\sqrt { w ( z _ { 0 } ) } - 1 + \\frac { z _ { 0 } w ' ( z _ { 0 } ) \\sinh \\sqrt { w ( z _ { 0 } ) } } { 2 \\sqrt { w ( z _ { 0 } ) } ( \\eta \\cosh \\sqrt { w ( z _ { 0 } ) } + \\gamma ) } \\right | ^ { 2 } \\\\ & = \\left | \\cosh e ^ { i t / 2 } - 1 + \\frac { k e ^ { i t / 2 } \\sinh e ^ { i t / 2 } } { 2 ( \\eta \\cosh e ^ { i t / 2 } + \\gamma ) } \\right | ^ { 2 } \\\\ & = \\frac { A _ { N } ( \\tau ) } { A _ { D } ( \\tau ) } ( \\tau = t / 2 , - \\pi \\leq \\tau \\leq \\pi ) , \\end{align*}"}
+{"id": "9089.png", "formula": "\\begin{align*} \\begin{array} { l l l } ( I - J ^ { ( r ) } \\hat { C } C \\hat { C } ^ { - 1 } J ^ { ( r ) } ) X ^ { * } = J ^ { ( r ) } \\Big [ ( \\hat { C } C \\hat { C } ^ { - 1 } - I ) \\underline { v } + \\hat { C } \\Big ( D p - U ( J ^ { ( r ) } X ^ { * } ) - \\mathbf { 1 } \\frac { \\beta } { 2 } \\Big ) \\Big ] . \\end{array} \\end{align*}"}
+{"id": "390.png", "formula": "\\begin{align*} M h _ { k } ^ { a } ( \\vec { \\nu } ) = \\bigcap _ { i \\geq k } M h _ { i } ^ { a } ( \\nu _ { i } ) . \\end{align*}"}
+{"id": "3076.png", "formula": "\\begin{align*} P _ k ( a _ { \\beta _ g } , \\ldots , a _ { k _ { i g } - 1 } ) + r _ k \\cdot a _ { \\beta _ g } ^ { \\gamma _ { 1 g } } \\cdot a _ { k _ { i g } } = 0 \\end{align*}"}
+{"id": "1141.png", "formula": "\\begin{align*} \\mathrm { I I } = \\left \\| \\left \\{ \\gamma _ j E _ j \\left ( g _ j \\right ) \\right \\} _ { j \\in \\mathbb Z } \\right \\| _ { L ^ { ( \\frac { p } { a } ) ' } \\ell ^ { ( \\frac { q } { a } ) ' } } , \\end{align*}"}
+{"id": "3272.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 - ( n + r ) / d } \\frac { ( q ^ { d + r } ; q ^ d ) _ k ^ { d - r - 1 } ( q ^ { d + r } ; q ^ d ) _ { k - 2 } ^ { r + 1 } ( 1 - q ^ { d k - d + r } ) ^ r q ^ { d k } } { ( q ^ d ; q ^ d ) _ k ^ d } \\equiv 0 \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "6301.png", "formula": "\\begin{align*} f ( x _ T ) = e ( Q ' _ { T } ) + | E _ { h _ 1 , k _ 1 } | \\le 2 4 + 2 8 = 5 2 . \\end{align*}"}
+{"id": "1265.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ; q ^ 2 ) _ k ( - q ^ 2 ; q ^ 2 ) _ k } { ( q ^ 2 ; q ^ 2 ) _ k } q ^ { n ^ 2 - ( k + 1 ) ^ 2 } \\equiv 1 + \\frac { ( 1 - n ) ( 1 - q ^ n ) } { 2 } \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "3134.png", "formula": "\\begin{align*} \\varphi ^ * \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) = \\left ( \\begin{array} { c c c } a ^ { 2 \\ , p ^ { e _ 1 } } & a ^ { p ^ { e _ 1 } } \\ , b ^ { p ^ { e _ 1 } } & b ^ { 2 \\ , p ^ { e _ 1 } } \\\\ 0 & 1 & 0 \\\\ 0 & 0 & d ^ { 2 \\ , p ^ { e _ 1 } } \\end{array} \\right ) ( \\ , e _ 1 \\geq 0 \\ , ) . \\end{align*}"}
+{"id": "8618.png", "formula": "\\begin{align*} \\| v _ 0 ( t ; \\cdot ) \\| _ { L ^ \\infty ( \\mathbb { R } ) } = \\| u ( \\cdot , t ) \\| _ { L ^ \\infty ( \\mathbb { R } ) } < C _ 0 , \\end{align*}"}
+{"id": "2377.png", "formula": "\\begin{align*} ( t , v ) = ( x + y , \\frac { y - x } { x + y } ) \\ \\ \\ \\quad \\Leftrightarrow \\ \\ \\ ( x , y ) = ( \\frac { t ( 1 - v ) } { 2 } , \\frac { t ( 1 + v ) } { 2 } ) \\ . \\end{align*}"}
+{"id": "270.png", "formula": "\\begin{align*} \\left ( A \\frac { d } { d x } + \\gamma A - A ' \\right ) b = \\left ( x \\frac { d } { d x } \\right ) ( 1 / 2 ) = 0 , \\end{align*}"}
+{"id": "751.png", "formula": "\\begin{align*} g _ { i j } ( \\Phi _ k - \\varphi _ k \\Phi ) - g _ { i k } ( \\Phi _ j - \\varphi _ j \\Phi ) = - \\overset { \\ _ * } R ^ h _ { i j k } \\varphi _ h - R ^ h _ { j k } \\varphi _ { h ; i } . \\end{align*}"}
+{"id": "3183.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n \\sin ^ 2 ( k x ) & \\overset { \\eqref { e q - 2 . 2 - 2 } } { = } \\frac { n } { 2 } - \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ n \\cos ( 2 k x ) \\\\ [ 5 p t ] & \\overset { \\eqref { e q - 2 . 1 0 } } { = } \\frac { n } { 2 } - \\frac { 1 } { 2 } \\left ( \\frac { \\sin ( ( 2 n + 1 ) x ) } { 2 \\sin ( x ) } - \\frac { 1 } { 2 } \\right ) \\\\ [ 5 p t ] & = \\frac { n } { 2 } - \\frac { \\sin ( ( 2 n + 1 ) x ) } { 4 \\sin ( x ) } + \\frac { 1 } { 4 } , \\end{align*}"}
+{"id": "99.png", "formula": "\\begin{align*} \\begin{aligned} & ( \\lambda - 2 \\eta ) \\int _ { Q _ { 2 r } } ( | D u | ^ 2 + \\epsilon ) ^ { \\frac { p - 2 + s } { 2 } } | D ^ 2 u | ^ 2 \\phi ^ 2 d x d t \\\\ & \\leq \\frac { C } { \\eta } \\int _ { Q _ { 2 r } } ( | D u | ^ 2 + \\epsilon ) ^ { \\frac { p - 2 + s } { 2 } } | D u | ^ 2 | D \\phi | ^ 2 d x d t + C \\int _ { Q _ { 2 r } } ( | D u | ^ 2 + \\epsilon ) ^ { \\frac { p + s - \\gamma } { 2 } } | \\phi _ t | \\phi d x d t , \\end{aligned} \\end{align*}"}
+{"id": "6698.png", "formula": "\\begin{align*} H = G ^ { f ( \\pi , r ) } = \\langle g ^ { f ( \\pi , r ) } \\mid g \\in G \\rangle \\trianglelefteq _ \\mathrm { o } G H \\subseteq K ; \\end{align*}"}
+{"id": "6998.png", "formula": "\\begin{align*} z _ 1 \\overline { w _ 1 } + z _ 2 \\overline { w _ 2 } - z _ { n + 1 } \\overline { w _ { n + 1 } } = 0 \\end{align*}"}
+{"id": "4419.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathcal { X } } & \\ \\overline { c } ^ T x . \\\\ \\end{align*}"}
+{"id": "2043.png", "formula": "\\begin{align*} 0 \\ , \\ , = \\ , \\ , \\frac { \\partial f } { \\partial \\lambda ^ j _ \\alpha } \\ , \\ , = \\ , \\ , \\frac { \\partial f ' } { \\partial \\lambda ^ j _ \\alpha } \\ , \\ , = \\ ! \\sum _ { c _ \\beta \\in C _ { d - e + 1 } } \\ ! \\ ! \\ ! c _ \\beta \\cdot f _ { \\alpha + \\beta } d - e + 2 \\leq j \\leq d + 1 \\alpha \\in M _ { d _ j , d _ j } . \\end{align*}"}
+{"id": "1221.png", "formula": "\\begin{align*} \\left ( A _ { p _ n , t } ( \\varphi ( z _ n ) ) \\right ) ^ { p _ n - 1 } - \\left ( B _ { p _ n , t } ( \\varphi ( z _ n ) ) \\right ) ^ { p _ n - 1 } & = 2 \\int _ { \\mathbb { R } ^ N } \\frac { \\vert \\varphi ( z _ n ) - \\varphi ( y ) \\vert ^ { p _ n - 2 } ( \\varphi ( z _ n ) - \\varphi ( y ) ) } { \\vert z _ n - y \\vert ^ { N + s p _ n } } \\dd y \\\\ & \\leq 0 , \\forall n \\geq n _ 0 . \\end{align*}"}
+{"id": "2639.png", "formula": "\\begin{align*} \\frac { 2 } { p } + \\frac { d } { q } = \\frac { d } { 2 } , 2 \\leq p , q \\leq \\infty ( p , q , d ) \\neq ( 2 , \\infty , 2 ) . \\end{align*}"}
+{"id": "41.png", "formula": "\\begin{align*} [ ( U _ 1 \\circ d _ 0 ) f ] ( \\mu ) = \\sum _ { i = 1 } ^ j ( d _ 0 f ) ( \\mu ; \\delta _ j ) d x ^ { i } = \\sum _ { i = 1 } ^ j ( d _ 0 f ) ( \\mu , \\mu + \\delta _ i ) d x ^ { i } = & \\sum _ { i = 1 } ^ j ( f ( \\mu + \\delta _ i ) - f ( \\mu ) ) d x ^ { i } \\\\ = & \\sum _ { i = 1 } ^ j \\mathcal { D } _ i f ( \\mu ) d x ^ { i } \\end{align*}"}
+{"id": "319.png", "formula": "\\begin{align*} z ( x , t ) = ( T - t ) ^ { - \\alpha } f ( ( 1 + | x | ) ( T - t ) ^ { \\beta } ) , \\end{align*}"}
+{"id": "7950.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Omega } \\frac { \\delta \\mathcal { F } } { \\delta v } \\wedge d ( \\partial \\phi ) = & \\big \\langle \\ast \\frac { \\delta \\mathcal { F } } { \\delta v } , d ( \\partial \\phi ) \\big \\rangle _ { L ^ { 2 } \\Lambda ^ { 1 } ( \\Omega ) } = ( - 1 ) ^ { n - 1 } \\int _ { \\partial \\Omega } \\mathrm { t r } ( \\frac { \\delta \\mathcal { F } } { \\delta v } ) \\wedge \\partial \\phi _ { \\partial } . \\\\ \\end{aligned} \\end{align*}"}
+{"id": "6907.png", "formula": "\\begin{align*} u _ { \\mu , x _ 0 } \\ ( x \\ ) : = \\ ( \\frac { n ^ { \\frac { 1 } { p } } \\ ( \\frac { n - p } { p - 1 } \\ ) ^ { \\frac { p - 1 } { p } } \\mu ^ { \\frac { 1 } { p - 1 } } } { \\mu ^ { \\frac { p } { p - 1 } } + \\left | x - x _ 0 \\right | ^ { \\frac { p } { p - 1 } } } \\ ) ^ { \\frac { n - p } { p } } \\quad \\forall x \\in \\R ^ n , \\end{align*}"}
+{"id": "7789.png", "formula": "\\begin{align*} V _ n ( C \\setminus M _ j ) = 1 \\end{align*}"}
+{"id": "6555.png", "formula": "\\begin{align*} = _ 1 + _ 2 + _ 3 , \\end{align*}"}
+{"id": "8142.png", "formula": "\\begin{align*} p _ x = { \\cal D } _ { q } \\ , \\ q _ x = X _ { q } \\ \\end{align*}"}
+{"id": "4441.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ { \\partial D } | g | ^ 2 \\rho | d z | \\le \\lim _ { s \\rightarrow + \\infty } \\frac { 1 } { 2 \\pi } \\int _ { \\partial D } | f _ s | ^ 2 \\rho | d z | = M _ H ( Z _ 0 , \\mathfrak { a } , \\rho ) . \\end{align*}"}
+{"id": "8073.png", "formula": "\\begin{align*} \\liminf _ { N \\rightarrow + \\infty } \\frac { N } { \\gamma _ N ^ 2 } \\log P ( \\frac { N } { \\gamma _ N } ( \\tau _ c ^ N - \\tau _ c ) > x ) & \\geq - \\left ( \\frac { d } { d t } \\sum _ { k = 1 } ^ 3 \\mu _ { t , k } ( f _ k ) \\Bigg | _ { t = \\tau _ c } \\right ) ^ 2 x ^ 2 J _ { c o n t r a , \\tau _ c } ( 1 ) \\\\ & = - J _ { h i t } ( x ) . \\end{align*}"}
+{"id": "7415.png", "formula": "\\begin{align*} D ' : = \\{ ( x , t ) \\in \\R \\times [ 0 , T ] \\ : \\ x \\ge R \\quad \\mbox { a n d } t - x \\ge R \\} . \\end{align*}"}
+{"id": "2496.png", "formula": "\\begin{align*} \\int _ \\Omega ( v ( t ) \\vartheta ( t ) - v ^ { i n } \\vartheta ( 0 ) ) \\ \\mathrm { d } x + \\int _ 0 ^ t \\int _ \\Omega \\nabla v ( s ) \\cdot \\nabla \\vartheta ( s ) \\ \\mathrm { d } x \\mathrm { d } s & + \\int _ 0 ^ t \\int _ \\Omega ( u v ) ( s ) \\vartheta ( s ) \\ \\mathrm { d } x \\mathrm { d } s \\\\ & = \\int _ 0 ^ t \\int _ \\Omega v ( s ) \\partial _ t \\vartheta ( s ) \\ \\mathrm { d } x \\mathrm { d } s \\end{align*}"}
+{"id": "4923.png", "formula": "\\begin{align*} \\begin{gathered} T _ { p , q , r } = x _ 1 ^ 2 + x _ 2 ^ 2 + \\cdots + x _ { n - 2 } ^ 2 + h ( x , y , z ) , \\\\ h ( x , y , z ) = x ^ p + y ^ q + z ^ r + a x y z , \\ a \\neq 0 \\end{gathered} \\end{align*}"}
+{"id": "8012.png", "formula": "\\begin{align*} & \\limsup _ { N \\rightarrow + \\infty } \\frac { 1 } { N } \\log P ( \\mu ^ N \\in C ) \\\\ & \\leq - \\inf _ { W \\in C } \\left ( \\sum _ { k = 1 } ^ 3 W _ { 0 , k } ( f _ k ) + \\mathcal { I } _ 1 ( W , F , G , H ) \\right ) \\\\ & + \\int _ \\mathbb { T } \\log \\left ( 1 + \\sum _ { k = 1 } ^ 3 \\rho _ { k - 1 } ( u ) \\left ( \\exp \\left ( f _ k ( u ) \\right ) - 1 \\right ) \\right ) d u \\\\ & = - \\inf _ { W \\in C } \\left ( \\mathcal { I } _ 2 ( W _ 0 , f _ 1 , f _ 2 , f _ 3 ) + \\mathcal { I } _ 1 ( W , F , G , H ) \\right ) . \\end{align*}"}
+{"id": "4940.png", "formula": "\\begin{align*} \\begin{gathered} [ S _ 1 ] = E _ 2 - E _ 3 , \\\\ [ S _ 2 ] = E _ 3 - E _ 4 , \\\\ [ S _ 3 ] = H - E _ 2 - E _ 3 - E _ 4 . \\end{gathered} \\end{align*}"}
+{"id": "7200.png", "formula": "\\begin{align*} L ( L R ) ( L L R R ) R L R & = L L L R R ( L R ) ( R L ) R \\\\ & = L ( L L R R ) ( R L ) L R R \\\\ & = L R L ( L L R R ) ( L R ) R \\\\ & = L R L L R L L R R R . \\end{align*}"}
+{"id": "1413.png", "formula": "\\begin{align*} \\{ I _ + ^ \\nu \\ , f _ { \\alpha = 1 } ( \\cdot \\vert t ) \\} ( x ) & = \\begin{cases} \\frac { 1 } { \\Gamma ( \\nu ) } ( x - t ) ^ { \\nu - 1 } & t \\le x \\\\ 0 & t > x \\end{cases} \\end{align*}"}
+{"id": "4933.png", "formula": "\\begin{align*} e ^ { S } _ { - } = e ^ { S _ 1 } _ { - } \\end{align*}"}
+{"id": "5714.png", "formula": "\\begin{align*} \\mathbf { L } ^ \\dagger \\psi _ { i , 3 } = 2 ^ { - 1 } m \\psi _ { i , 3 } , \\ \\mathbf { L } ^ \\dagger \\psi _ { i , 4 } = 2 ^ { - 1 } m \\psi _ { i , 4 } + \\psi _ { i , 3 } . \\end{align*}"}
+{"id": "6522.png", "formula": "\\begin{align*} \\Sigma _ { \\mathrm { N i s } } ^ { \\mathrm { h p } } : = \\{ Q ^ { \\mathrm { h p } } \\rightarrow X \\} _ Q \\cup \\{ \\emptyset \\rightarrow h _ { \\emptyset } \\} \\end{align*}"}
+{"id": "416.png", "formula": "\\begin{align*} \\begin{aligned} & V _ { i i d } ( \\zeta ) \\approx \\widetilde { V } ( U _ R , U _ S , \\Delta ) = \\\\ & \\begin{cases} - \\log ( 1 - \\frac { U _ M } { U _ N } ) , ~ U _ M \\neq U _ N , \\\\ \\frac { 1 } { 2 } [ \\log ( \\frac { U _ S U _ R } { 4 \\sigma ^ 2 \\Delta ^ 4 } ) + 2 ( \\frac { U _ { S } U _ { R } } { \\sigma ^ 2 \\Delta ^ 4 } ) ^ { - \\frac { 1 } { 2 } } ] , ~ U _ M = U _ N , \\end{cases} \\end{aligned} \\end{align*}"}
+{"id": "3282.png", "formula": "\\begin{align*} [ n ] = \\Phi _ n ( q ) \\prod _ { \\substack { 1 < m < n \\\\ m \\mid n } } \\Phi _ m ( q ) \\end{align*}"}
+{"id": "5741.png", "formula": "\\begin{align*} X _ - ( t ) \\leq e ^ { ( \\gamma _ * - \\varepsilon _ 1 ) t } \\left ( X _ - ( 0 ) + \\int _ 0 ^ \\infty e ^ { - ( \\gamma _ * - \\varepsilon _ 1 ) \\tau } | Y _ - ( \\tau ) | d \\tau \\right ) = O ( e ^ { ( \\gamma _ * - \\varepsilon _ 1 ) t } ) . \\end{align*}"}
+{"id": "5080.png", "formula": "\\begin{align*} L _ s ^ 2 & = 1 - q ^ { - k } ( E _ s - L _ s E _ s ) \\\\ \\underbrace { L _ s L _ t L _ s L _ t \\dots } _ { m _ { s t } } & = \\underbrace { L _ t L _ s L _ t L _ s \\dots } _ { m _ { s t } } \\\\ R _ t L _ s & = L _ s R _ { t ' } , t ' = s t s ^ { - 1 } ( t \\in T ( \\mathbb { F } _ { q ^ k } ) ) \\\\ R _ { t _ 1 } R _ { t _ 2 } & = R _ { t _ 1 t _ 2 } ( t _ 1 , t _ 2 \\in T ( \\mathbb { F } _ { q ^ k } ) ) . \\end{align*}"}
+{"id": "5748.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } u ( t ) / \\Vert u ( t ) \\Vert _ { L ^ 2 } = w \\ \\textup { i n } \\ H ^ 1 ( \\Sigma , \\mathbf { V } ) . \\end{align*}"}
+{"id": "5949.png", "formula": "\\begin{align*} \\frac { M _ { 2 } ( \\mathcal { N C } ( G ) ) } { | e ( \\mathcal { N C } ( G ) ) | } = 7 6 . 5 > \\frac { 8 4 0 } { 1 1 } = \\frac { M _ { 1 } ( \\mathcal { N C } ( G ) ) } { | v ( \\mathcal { N C } ( G ) ) | } . \\end{align*}"}
+{"id": "7787.png", "formula": "\\begin{align*} M ( m , p , q ) = \\{ ( u , v , w ) \\in { S ^ 5 \\subset \\mathbb { C } ^ 3 } \\mid { u ^ m + v ^ p + w ^ q = 0 } \\} , \\end{align*}"}
+{"id": "6924.png", "formula": "\\begin{align*} E ^ { s } ( z ) = E ^ { s s } ( z ) \\quad E ^ u ( z ) = E ^ c ( z ) \\oplus E ^ { s u } ( z ) , \\end{align*}"}
+{"id": "4541.png", "formula": "\\begin{align*} \\frac { \\alpha _ N ( t ^ * _ N ) } { t ^ * _ N } & = a _ - \\left ( \\frac { a _ - } { a _ + } \\right ) ^ { N - 1 } \\frac { 1 } { N } \\int _ 0 ^ { N } \\frac { 1 } { ( N - 1 ) ! } \\tilde { s } ^ { N - 1 } e ^ { - \\tilde { s } } d \\tilde { s } \\geq \\frac { a _ - } { 2 N } \\left ( \\frac { a _ - } { a _ + } \\right ) ^ { N - 1 } , \\end{align*}"}
+{"id": "3847.png", "formula": "\\begin{align*} \\gamma + \\frac { 1 } { 2 ( n + 1 ) } < \\sum \\limits _ { k = 1 } ^ n k ^ { - 1 } - \\log n < \\gamma + \\frac { 1 } { 2 ( n - 1 ) } , \\end{align*}"}
+{"id": "499.png", "formula": "\\begin{align*} O ( \\log \\log { m } \\cdot \\log ^ { 2 + o ( 1 ) } { m } ) = O ( \\log ^ { 2 + o ( 1 ) } { m } ) \\end{align*}"}
+{"id": "5854.png", "formula": "\\begin{align*} \\dfrac { M _ { 2 } ( \\mathcal { N C } ( D _ { 2 m } ) ) } { | e ( \\mathcal { N C } ( D _ { 2 m } ) ) | } - \\dfrac { M _ { 1 } ( \\mathcal { N C } ( D _ { 2 m } ) ) } { | v ( \\mathcal { N C } ( D _ { 2 m } ) ) | } = \\dfrac { m ^ { 2 } ( m - 5 ) + 4 ( 2 m - 1 ) } { 3 m ( m - 1 ) ( 2 m - 1 ) } : = \\dfrac { f ( m ) } { g ( m ) } . \\end{align*}"}
+{"id": "8743.png", "formula": "\\begin{align*} \\mu ( d x ) \\mbox { a . e . } , \\ ; x \\in A \\Rightarrow \\pi _ x \\left ( \\Gamma _ x \\cap \\{ ( - \\infty , x ) \\cup ( x , + \\infty ) \\} \\right ) = 0 . \\end{align*}"}
+{"id": "8256.png", "formula": "\\begin{align*} \\frac { 1 } { L } \\max _ { P _ { { X } ^ L | | Y ^ { L - 1 } } } \\sum _ { j = 1 } ^ L H ( { Y } _ { j } | Y ^ { j - 1 } , { S } ) , \\end{align*}"}
+{"id": "4772.png", "formula": "\\begin{align*} d : = d ( 0 , \\mathrm { r a n } A ) = \\inf \\{ \\norm { y } \\mid y \\in \\mathrm { r a n } A \\} \\end{align*}"}
+{"id": "8362.png", "formula": "\\begin{align*} X _ { w B } = P _ { s _ { i _ 1 } } P _ { s _ { i _ 2 } } \\cdots P _ { s _ { i _ m } } / B . \\end{align*}"}
+{"id": "1177.png", "formula": "\\begin{align*} \\mathcal O ^ { \\lfloor \\ ! \\lfloor \\ell \\rfloor \\ ! \\rfloor } & : = \\left \\{ f \\in C ^ \\infty : \\ C \\right . \\\\ & \\qquad \\qquad \\quad \\left . | f ( x ) | \\leq C ( 1 + | x | ) ^ { \\lfloor \\ ! \\lfloor \\ell \\rfloor \\ ! \\rfloor } x \\in \\mathbb { R } ^ n \\right \\} , \\end{align*}"}
+{"id": "3757.png", "formula": "\\begin{align*} \\theta _ { n , k } = \\frac { 1 } { n } \\sum _ { m = 1 } ^ { n } \\exp \\left ( \\frac { 2 \\pi \\ , i \\ , m \\ , k } { n } \\right ) . \\end{align*}"}
+{"id": "4412.png", "formula": "\\begin{gather*} \\alpha : = - 2 \\overline { u } _ i - \\Delta u _ i , \\ \\beta : = 1 , \\ \\gamma = - \\overline { u } _ i ^ 2 - \\overline { u } _ i \\Delta u _ i . \\ \\end{gather*}"}
+{"id": "45.png", "formula": "\\begin{align*} \\langle \\sum _ I ( \\Delta \\omega _ I ) d x ^ I ; \\sum _ I \\omega _ I d x ^ I \\rangle = & \\sum _ { \\mu \\in \\Z ^ n } \\sum _ I ( \\Delta \\omega _ I ( \\mu ) ) \\overline { \\omega _ I ( \\mu ) } \\\\ = & \\sum _ { \\mu \\in \\Z ^ n } \\sum _ I \\sum _ { \\alpha = 1 } ^ n ( ( \\mathcal { D } _ { \\alpha } ^ 2 \\omega _ I ) ( \\mu - e _ \\alpha ) ) \\overline { \\omega _ I ( \\mu ) } \\\\ = & \\sum _ { \\mu \\in \\Z ^ n } \\sum _ I \\sum _ { \\alpha = 1 } ^ n | \\mathcal { D } _ { \\alpha } \\omega _ I ( \\mu ) | ^ 2 \\ . \\end{align*}"}
+{"id": "7603.png", "formula": "\\begin{align*} I _ { 4 } ( \\epsilon ) & = \\epsilon ^ { N - 4 } R ^ { 2 } ( 4 - N ) ^ { 2 } \\int _ { \\mathbb { R } ^ { N } } \\frac { 4 | \\varphi | ^ { 2 } } { ( \\epsilon ^ 2 + | x | ^ { 2 } ) ^ { N - 2 } } + ( 2 - N ) ^ { 2 } \\frac { | x | ^ { 4 } | \\varphi | ^ { 2 } } { ( \\epsilon ^ 2 + | x | ^ { 2 } ) ^ { N } } + 4 ( 2 - N ) \\frac { | x | ^ { 2 } | \\varphi | ^ { 2 } } { ( \\epsilon ^ 2 + | x | ^ { 2 } ) ^ { N - 1 } } d x \\\\ & = K _ { 1 } ( \\epsilon ) + K _ { 2 } ( \\epsilon ) + K _ { 3 } ( \\epsilon ) . \\end{align*}"}
+{"id": "4893.png", "formula": "\\begin{align*} D \\cdot C = m p + q + R , \\end{align*}"}
+{"id": "1698.png", "formula": "\\begin{align*} \\left [ X _ j , \\overline { X _ k } \\right ] = \\frac { \\partial ^ 2 \\Phi } { \\partial z _ j \\partial \\overline { z _ k } } \\left ( \\frac { \\partial } { \\partial \\overline { z _ 0 } } - \\frac { \\partial } { \\partial z _ 0 } \\right ) = 2 i \\frac { \\partial ^ 2 \\Phi } { \\partial x _ j \\partial x _ k } \\frac { \\partial } { \\partial y _ 0 } \\quad \\forall \\ , j , k \\in \\{ 1 , \\ldots , n \\} , \\end{align*}"}
+{"id": "4160.png", "formula": "\\begin{align*} \\varphi ( g _ 1 g _ 2 ) = \\varphi ( g _ 1 ) \\varphi ( g _ 2 ) \\varphi ( g _ 1 g _ 2 ) = \\varphi ( g _ 2 ) \\varphi ( g _ 1 ) \\end{align*}"}
+{"id": "8286.png", "formula": "\\begin{align*} G ( x ) : = \\ln \\ln w + \\varphi ( u ) + \\ln \\rho ( x ) , \\end{align*}"}
+{"id": "4390.png", "formula": "\\begin{align*} \\inf _ { x , p , \\theta } & \\ \\Gamma \\theta + \\sum _ { i \\in [ m ] } f _ i ( x , \\overline { u } ^ i ) + p _ i , \\\\ \\mathrm { s . t . \\ ; } & x \\in \\mathcal { X } , \\\\ & p _ i + \\theta \\geq \\sup _ { u ^ i \\in \\mathcal { U } _ i } f _ i ( x , u ^ i ) - f _ i ( x , \\overline { u } ^ i ) \\ \\forall i \\in [ m ] , \\\\ & p \\in \\R ^ m _ { \\geq 0 } , \\theta \\in \\R _ { \\geq 0 } . \\end{align*}"}
+{"id": "1852.png", "formula": "\\begin{align*} \\pi _ 7 ( F _ A ) = - \\star ( \\psi \\wedge \\star \\sigma _ { \\pmb { \\pi } } ( A ) ) . \\end{align*}"}
+{"id": "5930.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { C } ( G ) ) & = \\dfrac { q ( q + 1 ) } { 2 } ( q ^ { 2 } - 3 q + 2 ) ( q ^ { 2 } - 3 q + 1 ) ^ { 2 } + \\dfrac { q ( q - 1 ) } { 2 } ( q ^ { 2 } - q ) ( q ^ { 2 } - q - 1 ) ^ { 2 } \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ + ( q + 1 ) ( q ^ { 2 } - 2 q + 1 ) ( q ^ { 2 } - 2 q ) ^ { 2 } \\\\ & = q ( q - 1 ) ( q ^ { 6 } - 4 q ^ { 5 } + 4 q ^ { 4 } + 2 q ^ { 3 } - 4 q ^ { 2 } + q - 1 ) \\end{align*}"}
+{"id": "5312.png", "formula": "\\begin{align*} v _ i ( \\nu ) = \\begin{cases} \\displaystyle \\min _ { a \\in \\{ 0 , 1 \\} } \\ , h _ i ^ a + \\nu \\ , \\theta _ i ^ 1 \\ , a + \\beta \\sum _ { j \\in N } p _ { i j } ^ a \\ , v _ j ( \\nu ) & i \\in N ^ { \\{ 0 , 1 \\} } \\\\ \\displaystyle h _ i ^ 1 + \\nu \\ , \\theta _ i ^ 1 + \\beta \\sum _ { j \\in N } p _ { i j } ^ 1 \\ , v _ j ( \\nu ) & i \\in N ^ { \\{ 1 \\} } . \\end{cases} \\end{align*}"}
+{"id": "3702.png", "formula": "\\begin{align*} \\bar { \\alpha } \\cdot \\bar { \\theta } _ { l } \\cdot \\overline { B _ 2 ( \\alpha ) } = \\lambda \\beta \\otimes \\phi _ l + \\lambda \\gamma \\otimes \\theta _ l + \\theta _ l \\otimes \\lambda \\gamma + \\phi _ l \\otimes \\lambda \\beta . \\end{align*}"}
+{"id": "7162.png", "formula": "\\begin{align*} A ( x ) = \\frac { a } { d _ 1 ^ 2 + x ^ 2 } , B ( x ) = \\frac { b } { d _ 2 ^ 2 + ( x - x _ 0 ) ^ 2 } , \\end{align*}"}
+{"id": "6812.png", "formula": "\\begin{align*} h = h _ { S U ( N ) } ( g ) & = e ^ { i K _ 1 \\phi _ 1 } \\sin ^ { R _ 1 } ( \\psi _ 1 ) \\cos ^ { S _ 1 } ( \\psi _ 1 ) \\cdots e ^ { i K _ { N - 1 } \\phi _ { N - 1 } } \\sin ^ { R _ { N - 1 } } ( \\psi _ { N - 1 } ) \\cos ^ { S _ { N - 1 } } ( \\psi _ { N - 1 } ) \\\\ & \\quad \\cdot h _ { S U ( N - 1 ) } ( g _ { S U ( N - 1 ) } ) e ^ { i L _ { N - 1 } \\omega _ { N - 1 } } . \\end{align*}"}
+{"id": "4257.png", "formula": "\\begin{align*} G _ \\xi [ f , g , h ] ( \\eta , \\sigma ) : = \\hat f ( \\xi - \\eta ) \\hat { \\bar { g } } ( \\eta - \\xi + \\sigma ) \\hat h ( \\xi - \\sigma ) . \\end{align*}"}
+{"id": "5777.png", "formula": "\\begin{align*} \\mathcal { W } _ j = - m ^ { - 1 } \\int _ \\Sigma ( N _ 1 ( u ) - \\mathcal { M } _ \\Sigma ( u ) ) \\varphi _ { \\iota + j } \\ , d \\mu , \\end{align*}"}
+{"id": "7356.png", "formula": "\\begin{align*} f ^ G ( x ) : = \\int _ G ( \\overline \\gamma ^ * f ) ( x ) \\ , \\mathrm { v o l } ^ G , x \\in \\widetilde M . \\end{align*}"}
+{"id": "4395.png", "formula": "\\begin{gather*} f _ { i , * } ( x , v ^ i ) \\neq - \\infty \\Leftrightarrow v ^ i = l _ i ( x ) , \\end{gather*}"}
+{"id": "9109.png", "formula": "\\begin{align*} \\begin{array} { l l l } d : = v e c ( D ) = [ D _ { \\bullet 1 } ^ T \\ldots D _ { \\bullet m } ^ T ] ^ T , \\\\ \\mathbf { \\hat e _ i } : = e _ i \\otimes \\mathbf 1 _ m = [ \\mathbf 0 ^ T \\ldots \\mathbf 0 ^ T \\ , \\mathbf 1 ^ T _ m \\ , \\mathbf 0 ^ T \\ldots \\mathbf 0 ^ T ] ^ T , \\\\ \\mathbf { \\tilde e _ i } : = \\mathbf 1 _ m \\otimes e _ i = [ e _ i ^ T \\ldots e _ i ^ T ] ^ T , \\\\ \\mathbf { \\theta _ i } : = [ p _ 1 e _ i ^ T \\ldots p _ m e _ i ^ T ] ^ T . \\end{array} \\end{align*}"}
+{"id": "1338.png", "formula": "\\begin{align*} x ^ 2 - \\left \\lbrace 2 r \\alpha + ( 1 - \\alpha ) ( a - c ) \\right \\rbrace x - \\left \\lbrace ( r - c ) ( 1 - \\alpha ) ^ 2 - r \\alpha ( 1 - \\alpha ) ( a - c ) - r ^ 2 \\alpha ^ 2 \\right \\rbrace = 0 \\end{align*}"}
+{"id": "6138.png", "formula": "\\begin{align*} \\psi _ n ( \\lambda _ s ) = P _ n \\lambda _ s P _ n = \\sum _ { r \\in \\mathcal { F } _ n \\cap s \\mathcal { F } _ n } e _ { r , s ^ { - 1 } r } , \\end{align*}"}
+{"id": "4223.png", "formula": "\\begin{align*} \\theta _ 2 ( v , \\tau + 1 ) = \\theta _ 3 ( v , \\tau ) , ~ ~ \\theta _ 2 ( v , - \\frac { 1 } { \\tau } ) = \\left ( \\frac { \\tau } { \\sqrt { - 1 } } \\right ) ^ { \\frac { 1 } { 2 } } e ^ { \\pi \\sqrt { - 1 } \\tau v ^ 2 } \\theta _ 1 ( \\tau v , \\tau ) ; \\end{align*}"}
+{"id": "326.png", "formula": "\\begin{align*} \\Delta v _ k ( x , t ) \\geq - \\frac { K } { t } , v _ k = \\frac { m } { m - 1 } u _ k ^ { m - 1 } . \\end{align*}"}
+{"id": "6955.png", "formula": "\\begin{align*} \\{ w \\in \\mathbb { P } _ { \\mathbb { C } } ^ n : p _ 1 \\overline { w _ 1 } + \\cdots + p _ n \\overline { w _ n } - p _ { n + 1 } \\overline { w _ { n + 1 } } = 0 \\} . \\end{align*}"}
+{"id": "7324.png", "formula": "\\begin{align*} F _ { m , r } ( s , \\beta ) - \\frac { 1 } { m \\left ( s + 2 \\sin ^ 2 \\left ( \\pi \\frac { \\beta } { m } \\right ) \\right ) } = \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m - 1 } \\frac { e ^ { 2 \\pi i \\frac { j r } { m } } } { s + 2 \\sin ^ 2 \\left ( \\pi \\frac { j + \\beta } { m } \\right ) } . \\end{align*}"}
+{"id": "4979.png", "formula": "\\begin{align*} P _ N ( \\{ \\lambda \\} ) = \\sum _ { i = 1 } ^ N P _ N ( \\nu _ i , \\lambda _ 2 , \\dots , \\lambda _ N ) \\prod _ { \\substack { j = 1 \\\\ j \\neq i } } ^ { N } \\frac { \\lambda _ 1 - \\nu _ j } { \\nu _ i - \\nu _ j } . \\end{align*}"}
+{"id": "1737.png", "formula": "\\begin{align*} \\widehat { F } \\big ( \\Psi ( z ) \\big ) = \\sum _ { j = 1 } ^ { n - 1 } \\left [ ( x _ 1 , \\ldots , x _ { n - 1 } ) X ^ { T } \\widehat { Q } _ { j } X \\left ( \\begin{array} { c } x _ 1 \\\\ \\vdots \\\\ x _ { n - 1 } \\end{array} \\right ) \\right ] x _ n ^ { j - 1 } = F ( z ) . \\end{align*}"}
+{"id": "5232.png", "formula": "\\begin{align*} & M A = \\sum _ { i } \\alpha p _ { i } + \\left ( 1 - \\alpha \\right ) q _ { i } = \\sum _ { i } M A _ { i } \\\\ & M H = \\sum _ { i } \\frac { p _ { i } q _ { i } } { \\left ( 1 - \\alpha \\right ) p _ { i } + \\alpha q _ { i } } = \\sum _ { i } M H _ { i } \\end{align*}"}
+{"id": "5200.png", "formula": "\\begin{align*} & \\frac { \\partial A } { \\partial q _ { j } } = \\frac { 1 } { \\alpha } q ^ { \\alpha + \\beta - 2 } _ { j } - \\frac { 1 } { \\alpha } p ^ { \\alpha } _ { j } q ^ { \\beta - 2 } _ { j } \\\\ & \\frac { \\partial B } { \\partial q _ { j } } = 0 \\end{align*}"}
+{"id": "2860.png", "formula": "\\begin{align*} \\| f \\| _ { \\dot { W } ^ { 1 , p } _ \\omega ( \\mathbb { R } ^ n ) } : = \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { L ^ p _ \\omega ( \\mathbb { R } ^ n ) } . \\end{align*}"}
+{"id": "3936.png", "formula": "\\begin{align*} \\kappa : = \\frac { d _ 3 - d _ 2 } { q _ { 2 , 1 } } . \\end{align*}"}
+{"id": "440.png", "formula": "\\begin{align*} t ^ * \\in \\begin{cases} \\{ - \\lambda _ { i _ m , m + 1 } \\} , & { \\rm i f ~ } x ^ * _ { i _ m } ( t ^ * ) > x ^ * _ { m + 1 } ( t ^ * ) , \\\\ [ - \\lambda _ { i _ m , m + 1 } , \\mu _ { i _ m , m + 1 } ] , & { \\rm i f ~ } x ^ * _ { i _ m } ( t ^ * ) = x ^ * _ { m + 1 } ( t ^ * ) , \\\\ \\{ \\mu _ { i _ m , m + 1 } \\} , & { \\rm i f ~ } x ^ * _ { i _ m } ( t ^ * ) < x ^ * _ { m + 1 } ( t ^ * ) . \\end{cases} \\end{align*}"}
+{"id": "6378.png", "formula": "\\begin{align*} x - \\sigma \\frac { S } { \\| \\nu ^ \\Sigma \\| ^ 2 } \\nu ^ \\Sigma = x ^ { M \\cap \\Sigma } . \\end{align*}"}
+{"id": "1120.png", "formula": "\\begin{align*} \\left | A _ Q \\left ( S _ { \\varphi } \\vec f \\right ) _ Q \\right | \\leq \\left | A _ Q \\left \\langle \\vec f , \\varphi _ Q \\right \\rangle \\right | = | Q | ^ { \\frac 1 2 } \\left | A _ Q \\left ( \\widetilde { \\varphi } _ { j _ Q } * \\vec f \\right ) ( x _ Q ) \\right | \\leq \\sup _ { \\mathbb { A } , \\widetilde { \\varphi } , Q } \\left ( \\vec f \\right ) , \\end{align*}"}
+{"id": "6342.png", "formula": "\\begin{align*} \\partial E = \\{ \\exp _ o ( R ( 1 + \\rho ( \\varphi ) ) ) : \\varphi \\in S ^ { n - 1 } \\} , \\end{align*}"}
+{"id": "966.png", "formula": "\\begin{align*} \\Pi _ V ( w ) = R ^ { V } f ( \\cdot , w ) + R ^ V \\mu = R ^ { V } ( \\mathbf 1 _ V f ( \\cdot , w ) ) + R ^ V ( \\mathbf 1 _ V \\cdot \\mu ) \\in F ( V ) . \\end{align*}"}
+{"id": "309.png", "formula": "\\begin{align*} u _ t = \\Delta u ^ m + u ^ p , \\end{align*}"}
+{"id": "4839.png", "formula": "\\begin{align*} \\lambda _ j = n - j \\quad \\mbox { f o r a l l $ j \\in \\{ 2 , \\ldots , n \\} \\setminus \\{ j _ 0 \\} $ , } \\lambda _ { j _ 0 } = n - j _ 0 + 1 . \\end{align*}"}
+{"id": "455.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { M } ( \\Omega ^ q _ + ) = { } \\{ ( i , j ) \\in \\Omega ^ q _ + \\cap { \\cal A } ( t _ q ) _ > \\mid x ^ q _ { i } ( t _ q + \\Delta t _ q ) = x ^ * _ j ( t _ q ) \\} , \\\\ \\mathcal { M } ( \\Omega ^ q _ - ) = { } \\{ ( i , j ) \\in \\Omega ^ q _ - \\cap { \\cal A } ( t _ q ) _ < \\mid x ^ q _ { j } ( t _ q + \\Delta t _ q ) = x ^ * _ i ( t _ q ) \\} . \\\\ \\end{cases} \\end{align*}"}
+{"id": "5754.png", "formula": "\\begin{align*} & \\xi _ { i } ( t ) : = G ( q ( t ) , \\Psi _ { i } ) \\ \\textup { f o r } \\ i \\in \\mathbb { Z } \\setminus \\{ 0 \\} , \\\\ & z _ j ( t ) : = G ( q ( t ) , \\Upsilon _ j ) , \\ \\bar { z } _ j ( t ) : = G ( q ( t ) , \\overline { \\Upsilon } _ j ) \\ \\textup { f o r } \\ 1 \\leq j \\leq J , \\end{align*}"}
+{"id": "5663.png", "formula": "\\begin{align*} \\begin{pmatrix} w _ 1 & w _ 2 & w _ 3 \\end{pmatrix} \\begin{pmatrix} \\sin ^ 2 \\theta \\\\ & - \\cos ^ 2 \\theta \\\\ & & - \\cos ^ 2 \\theta \\end{pmatrix} \\begin{pmatrix} w _ 1 \\\\ w _ 2 \\\\ w _ 3 \\end{pmatrix} = 0 , \\end{align*}"}
+{"id": "6827.png", "formula": "\\begin{align*} \\Phi _ N ( \\phi _ 1 , \\ldots \\phi _ { \\frac { N ( N - 1 ) } { 2 } } ) : = \\left ( \\prod _ { 1 \\leq k \\leq N - 1 } e ^ { \\phi _ k \\lambda _ { ( k + 1 ) ^ 2 - 2 } } \\right ) \\cdot \\begin{pmatrix} \\Phi _ { N - 1 } ( \\phi _ { N } , \\ldots , \\phi _ { \\frac { N ( N - 1 ) } { 2 } } ) & 0 \\\\ 0 & 1 \\end{pmatrix} , \\end{align*}"}
+{"id": "1143.png", "formula": "\\begin{align*} \\left \\| B \\vec { t } \\right \\| _ { \\dot a ^ { 0 , \\tau } _ { p , q } ( W ) } ^ r & \\lesssim \\sum _ { k \\in \\mathbb { Z } } \\sum _ { l = 0 } ^ \\infty ( 1 + k _ - + l ) \\\\ & \\quad \\times \\left [ 2 ^ { - ( E - \\frac { n } { 2 } ) k _ - } 2 ^ { - k _ + ( F + \\frac { n } { 2 } - \\frac { n } { a } ) } 2 ^ { - ( D - \\frac { n } { a } ) l } 2 ^ { ( k _ - + l ) n \\widehat \\tau } \\left \\| \\vec { t } \\right \\| _ { \\dot a ^ { 0 , \\tau } _ { p , q } ( W ) } \\right ] ^ r , \\end{align*}"}
+{"id": "4513.png", "formula": "\\begin{align*} \\widetilde { \\mathbb { P } } ^ { ( 1 ) } _ { \\ell } : = \\mathbb { P } \\bigg [ \\sup _ { \\substack { X _ { \\ell - 1 } < n \\leqslant X _ { \\ell } } } \\widetilde { V } ( n ) > \\frac { T ( \\ell ) } { \\ell ^ { K / 2 } } \\bigg ] . \\end{align*}"}
+{"id": "7584.png", "formula": "\\begin{align*} \\frac { d } { d t } M _ { \\varphi _ { R } } [ u ( t ) ] = A _ { R } ^ { 1 } [ u ( t ) ] + B _ { R } ^ { 2 } [ u ( t ) ] + B _ { R } ^ { 3 } [ u ( t ) ] \\end{align*}"}
+{"id": "3372.png", "formula": "\\begin{align*} v _ i = \\abs { v } \\sin \\theta \\sin \\varphi , v _ j = \\abs { v } \\sin \\theta \\cos \\varphi , \\end{align*}"}
+{"id": "1699.png", "formula": "\\begin{align*} H _ { \\mathcal { L } } : = \\left ( \\begin{array} { c c c } \\frac { \\partial ^ 2 \\Phi } { \\partial x _ 1 ^ 2 } & \\cdots & \\frac { \\partial ^ 2 \\Phi } { \\partial x _ 1 \\partial x _ n } \\\\ \\vdots & \\ddots & \\vdots \\\\ \\frac { \\partial ^ 2 \\Phi } { \\partial x _ n \\partial x _ 1 } & \\cdots & \\frac { \\partial ^ 2 \\Phi } { \\partial x _ n ^ 2 } \\end{array} \\right ) . \\end{align*}"}
+{"id": "1847.png", "formula": "\\begin{align*} & I _ 1 ( P ^ { \\dagger } ) = \\frac { 1 } { 2 4 } \\int _ Y c _ 2 \\cdot p _ 1 + \\frac { 1 } { 1 2 } \\int _ Y - 2 c _ 2 c _ 1 ^ 2 + 2 c _ 3 c _ 1 + c ^ 2 _ 2 - 2 c _ 4 \\\\ & I _ 2 ( P ^ { \\dagger } ) + I _ 3 ( P ^ { \\dagger } ) = \\pm \\frac { 1 } { 2 } \\int _ { \\Sigma ^ { ( 4 ) } _ Y } c _ 2 . \\end{align*}"}
+{"id": "3279.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q , q ^ { - 1 } ; q ^ 2 ) _ k q ^ { 2 k } } { ( q ^ 2 ; q ^ 2 ) _ k ^ 2 } & \\equiv 0 ~ n \\geq 3 ~ ~ n \\equiv 1 \\pmod 2 , \\\\ \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ^ 4 , q , q ^ { - 2 } , ; q ^ 3 ) _ k q ^ { 3 k } } { ( q ^ 3 ; q ^ 3 ) _ k ^ 3 } & \\equiv 0 \\quad ~ n \\geq 5 ~ ~ n \\equiv 2 \\pmod 3 . \\end{align*}"}
+{"id": "2820.png", "formula": "\\begin{align*} - \\dot { P } _ { 1 } - Q ^ { 1 } = 0 , \\dot { Q } ^ { 1 } - P _ { 1 } = 0 . \\end{align*}"}
+{"id": "3917.png", "formula": "\\begin{align*} \\normalsize \\begin{aligned} \\sin n z = \\binom { n } { 1 } \\cos ^ { n - 1 } z \\sin z - \\binom { n } { 3 } \\cos ^ { n - 3 } z \\sin ^ 3 z + \\binom { n } { 5 } \\cos ^ { n - 5 } z \\sin ^ 5 z - \\& c . \\end{aligned} \\end{align*}"}
+{"id": "2212.png", "formula": "\\begin{align*} \\intop _ \\Omega { u _ { , z } ^ 2 \\over r ^ 2 } d x = \\intop _ \\Omega \\bigg ( u _ { , r r } + { 3 \\over r } u _ { , r } \\bigg ) u r ^ { - 2 } d x + \\intop _ \\Omega \\omega _ { 1 , z } u r ^ { - 2 } d x . \\end{align*}"}
+{"id": "4595.png", "formula": "\\begin{align*} \\min _ { \\mathfrak { E } } s _ g = & 8 \\pi \\left ( 1 + \\sum a \\right ) \\left ( \\sum a + \\sum a ^ 3 + 3 \\sum a b + 6 \\sum a ^ 2 b + \\sum a ^ 3 b \\right ) \\Big / \\\\ [ 0 . 1 c m ] & \\left ( \\sum a ^ 2 + \\sum a ^ 4 + \\sum a b + 4 \\sum a ^ 2 b + 4 \\sum a ^ 3 b + 2 \\sum a ^ 4 b + 3 \\sum a ^ 2 b ^ 2 + 4 \\sum a ^ 3 b ^ 2 + \\sum a ^ 4 b ^ 2 \\right ) . \\end{align*}"}
+{"id": "2520.png", "formula": "\\begin{align*} \\frac { | \\partial _ x ^ \\alpha \\partial _ \\xi ^ \\beta \\phi ( a ) ( x , \\xi ) | } { ( 1 + | \\xi | ) ^ { m - | \\beta | } } \\le \\begin{cases} C ' & \\\\ C ' ( 1 + R ) ^ { | \\beta | - m } & \\ ; . \\end{cases} \\end{align*}"}
+{"id": "7641.png", "formula": "\\begin{align*} \\begin{array} { l l } \\operatorname { M i n } & \\mathcal { F } ( x ) = ( \\mathcal { F } _ { 1 } ( x ) , . . . , \\mathcal { F } _ { k } ( x ) ) ^ T \\\\ & x \\in \\mathcal { X } \\end{array} \\end{align*}"}
+{"id": "4561.png", "formula": "\\begin{align*} 2 \\nabla _ { \\alpha } W ^ { \\alpha } _ { \\beta \\gamma \\eta } = ( \\nabla _ { \\eta } R _ { \\beta \\gamma } - \\nabla _ { \\gamma } R _ { \\beta \\eta } ) - \\frac 1 6 ( g _ { \\beta \\gamma } \\nabla _ { \\eta } s _ g - g _ { \\beta \\eta } \\nabla _ { \\gamma } s _ g ) . \\end{align*}"}
+{"id": "3243.png", "formula": "\\begin{align*} \\beta x & = - p u = - \\varepsilon p ^ 0 \\\\ \\alpha x & = \\sqrt { \\varepsilon ^ 2 + \\beta ^ 2 } \\ : x = \\sqrt { - \\varepsilon ^ 2 p ^ 2 + ( p u ) ^ 2 } \\\\ & = \\sqrt { - \\varepsilon ^ 2 \\ , \\big ( ( p ^ 0 ) ^ 2 - | \\vec { p } \\ , | ^ 2 \\big ) + \\varepsilon ^ 2 ( p ^ 0 ) ^ 2 } = \\sqrt { \\varepsilon ^ 2 \\ , | \\vec { p } \\ , | ^ 2 } = \\varepsilon \\ , | \\vec { p } \\ , | \\ : . \\end{align*}"}
+{"id": "3724.png", "formula": "\\begin{align*} - \\frac { 2 } { 5 } = b _ 2 < \\mu _ H ( E _ 1 ) = \\frac { c _ 1 } { r _ 1 } < 0 . \\end{align*}"}
+{"id": "2606.png", "formula": "\\begin{align*} | D _ 2 ( S ) | = | D _ { \\{ 1 \\} } ( S ) | + | D _ { \\{ 2 \\} } ( S ) \\setminus D _ { \\{ 1 \\} } ( S ) | \\geq n - 1 + 1 = n . \\end{align*}"}
+{"id": "7199.png", "formula": "\\begin{align*} F G - G F & = F _ 1 F _ 2 G - G F _ 1 F _ 2 \\\\ & = F _ 1 F _ 2 G - F _ 1 G F _ 2 + F _ 1 G F _ 2 - G F _ 1 F _ 2 \\\\ & = F _ 1 ( F _ 2 G - G F _ 2 ) + ( F _ 1 G - G F _ 1 ) F _ 2 . \\end{align*}"}
+{"id": "5402.png", "formula": "\\begin{align*} \\mathbf { P } ^ 1 - \\mathbf { P } ^ 0 = \\frac { 1 } { \\Lambda } \\ , \\left [ \\begin{array} { c c c c c c } \\lambda _ 0 & - \\lambda _ 0 & & & & \\\\ & \\lambda _ 1 & - \\lambda _ 1 & & & \\\\ & & \\ddots & \\ddots & & \\\\ & & & \\lambda _ { n - 1 } & - \\lambda _ { n - 1 } & \\\\ & & & & 0 & 0 \\end{array} \\right ] , \\end{align*}"}
+{"id": "3038.png", "formula": "\\begin{align*} V _ { a , b } ( \\xi ) = \\frac { g _ - ( g _ { - } - 1 ) } { \\sin ^ 2 \\xi } + \\frac { g _ + ( g _ { + } - 1 ) } { \\cos ^ 2 \\xi } \\ , , \\end{align*}"}
+{"id": "7940.png", "formula": "\\begin{align*} \\eta _ { t } = [ \\eta , u ' ] _ { 1 } , \\end{align*}"}
+{"id": "3763.png", "formula": "\\begin{align*} G ( x ) = \\exp ( - c / x ^ \\alpha ) \\end{align*}"}
+{"id": "3091.png", "formula": "\\begin{align*} W _ \\ell & : = \\{ \\ , v \\in V \\mid h ( z ) \\ , v = z ^ \\ell \\ , v \\ , \\} , \\\\ { W ' } _ \\ell & : = \\{ \\ , v \\in V \\mid h ' ( z ) \\ , v = z ^ \\ell \\ , v \\ , \\} . \\end{align*}"}
+{"id": "2408.png", "formula": "\\begin{align*} \\sum _ { q \\leqslant Y } \\log q = ( 1 + o ( 1 ) ) Y , Y \\rightarrow \\infty . \\end{align*}"}
+{"id": "3070.png", "formula": "\\begin{align*} F _ i ( x , y ) = \\prod _ { \\alpha \\in U _ { m _ i } } ( y - \\eta _ i ( \\alpha \\cdot x ^ { \\frac { 1 } { m _ i } } ) ) \\end{align*}"}
+{"id": "1910.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ T \\sum _ { x \\in G } \\ , V ( x ) \\ , & \\partial _ t v _ + ( x , t ) v _ + ( x , t ) \\ , \\eta ^ 2 ( x ) e ^ { \\xi ( x , t ) } \\mu ( x ) \\ , d t \\\\ & \\le \\int _ 0 ^ T \\sum _ { x \\in G } \\Delta ( v _ + ( x , t ) ) \\ , v _ + ( x , t ) \\ , \\eta ^ 2 ( x ) e ^ { \\xi ( x , t ) } \\mu ( x ) \\ , d t . \\end{aligned} \\end{align*}"}
+{"id": "147.png", "formula": "\\begin{align*} 2 \\bigg [ F \\bigg ( \\frac { t } { ( 1 - ( 1 / \\eta ) ) / 2 } \\bigg ) \\cdot { \\bigg ( \\frac { 1 - ( 1 / \\eta ) } { 2 } \\bigg ) } ^ { 2 n } + F \\bigg ( \\frac { t } { ( 1 / \\eta ) } \\bigg ) \\cdot { \\bigg ( \\frac { 1 } { \\eta } \\bigg ) } ^ { 2 n } \\bigg ] = F ( t ) + R ( t ) , \\end{align*}"}
+{"id": "4574.png", "formula": "\\begin{align*} & \\theta = h _ 1 \\tau - a _ 1 , \\end{align*}"}
+{"id": "1208.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd t } \\bigg | _ { t = 0 } \\left ( \\int _ { \\Omega } \\vert ( u + t \\varphi ) ( x ) \\vert ^ { \\alpha } \\dd x \\right ) \\vert v ( x _ 0 ) \\vert ^ { \\beta } = \\alpha \\left ( \\int _ { \\Omega } \\vert u ( x ) \\vert ^ { \\alpha - 2 } u ( x ) \\varphi ( x ) \\dd x \\right ) \\vert v ( x _ 0 ) \\vert ^ { \\beta } . \\end{align*}"}
+{"id": "4639.png", "formula": "\\begin{align*} f ( x ' ) = \\sum _ { j = 0 } ^ \\infty 2 ^ { - { \\gamma } j m } g _ j ( x ' ) \\ge g _ 0 ( x ' ) \\geq \\frac { 1 } { 4 } \\end{align*}"}
+{"id": "6233.png", "formula": "\\begin{align*} \\begin{aligned} c _ 0 & = \\min \\{ c _ { 0 , \\alpha } ^ * , c _ { \\alpha , \\gamma } ^ * \\} \\\\ & \\ge \\min \\big \\{ \\inf _ { [ 0 , \\alpha ) } \\delta ( f , \\alpha ) - 2 \\sup _ { [ 0 , \\alpha ) } \\sqrt { \\Delta ( D g , \\alpha ) } , \\inf _ { ( \\alpha , \\gamma ] } \\delta ( f , \\alpha ) - 2 \\sup _ { ( \\alpha , \\gamma ] } \\sqrt { \\Delta ( D g , \\alpha ) } \\big \\} \\\\ & \\ge \\inf _ { [ 0 , \\gamma ] } \\delta ( f , \\alpha ) - 2 \\sup _ { [ 0 , \\gamma ] } \\sqrt { \\Delta ( D g , \\alpha ) } = s _ { 0 , \\gamma } . \\end{aligned} \\end{align*}"}
+{"id": "2587.png", "formula": "\\begin{align*} \\begin{cases} - \\varepsilon ^ 2 \\Delta u = f ( u ) - u & B \\\\ \\displaystyle { \\varepsilon \\frac { \\partial u } { \\partial \\nu } + \\beta u = 0 } & \\partial B \\end{cases} \\end{align*}"}
+{"id": "7075.png", "formula": "\\begin{align*} & \\langle u ( t ) , \\varphi \\rangle - \\langle f , \\varphi \\rangle \\\\ & + \\mu \\int _ 0 ^ t \\langle u , \\varphi \\rangle d s + \\int _ 0 ^ t \\bigl \\langle b \\cdot \\nabla u , \\varphi \\bigr \\rangle d s - \\sigma \\int _ 0 ^ t \\langle u , \\nabla \\varphi \\rangle d B _ s + \\frac { \\sigma ^ 2 } { 2 } \\int _ 0 ^ t \\bigl \\langle u , \\Delta \\varphi \\bigr \\rangle d s = 0 . \\end{align*}"}
+{"id": "1934.png", "formula": "\\begin{align*} - \\Delta _ p u + ( - \\Delta ) _ { p } ^ { s } u = 0 , \\end{align*}"}
+{"id": "3337.png", "formula": "\\begin{align*} \\cos \\theta : = \\frac { v + v _ { * } } { | v - v _ { * | } } \\sigma . \\end{align*}"}
+{"id": "3390.png", "formula": "\\begin{align*} f \\left ( y _ { T + 1 } \\right ) - f \\left ( x ^ { * } \\right ) & \\le \\frac { 2 R _ { 1 } ^ { 2 } \\alpha _ { T } } { \\eta _ { T } } = 6 R _ { 1 } ^ { 2 } c \\gamma ^ { 2 } L \\alpha _ { T } ^ { 2 } \\\\ & = 6 \\max \\left \\{ 1 0 ^ { 4 } L \\gamma ^ { 2 } R _ { 1 } ^ { 2 } ( T + 1 ) ^ { - 2 } ; 6 R _ { 1 } \\left ( T + 1 \\right ) ^ { - 1 } \\left ( 2 6 T \\right ) ^ { \\frac { 1 } { p } } \\gamma ^ { \\frac { p - 1 } { p } } \\sigma \\right \\} . \\end{align*}"}
+{"id": "1576.png", "formula": "\\begin{align*} \\frac { d } { d t } G ( t \\ , z ) = \\left ( D G ( t \\ , z ) , z \\right ) = t \\left ( ( D ^ 2 F ) ^ { - 1 } ( D F ^ { - 1 } ( t \\ , z ) ) \\ , z , z \\right ) \\ge 0 , \\end{align*}"}
+{"id": "6185.png", "formula": "\\begin{align*} \\begin{aligned} \\lambda ^ { k + 1 } = \\lambda ^ k = \\dots = \\lambda ^ 0 = 0 , ~ \\forall k . \\end{aligned} \\end{align*}"}
+{"id": "1990.png", "formula": "\\begin{align*} \\kappa _ l ( s ) = e ^ { i \\beta _ l ^ + s } - e ^ { i \\beta _ l ^ - s } . \\end{align*}"}
+{"id": "2257.png", "formula": "\\begin{align*} \\small \\ddot { \\xi } ( x ) - ( n - 1 ) \\tanh x \\ , \\dot { \\xi } ( x ) - n \\tanh ^ 2 x \\ , \\xi ( x ) + ( \\tfrac { \\lambda } { 4 } + 1 ) \\tfrac { 1 } { \\cosh ^ 2 x } \\ , \\xi ( x ) = 0 \\end{align*}"}
+{"id": "2971.png", "formula": "\\begin{align*} D _ { X } ( t ) & = D _ { X } ( 0 ) + \\int _ { 0 } ^ { t } \\frac { d D _ { X } ( s ) } { d s } d s \\leq D _ { X } ( 0 ) + \\int _ { 0 } ^ { t } D _ { V } ( s ) d s \\\\ & \\leq D _ { X } ( 0 ) + \\int _ { 0 } ^ { t } D _ { V } ( 0 ) \\exp \\left ( - \\frac { \\kappa _ 1 \\mathcal { A } ( v ) ( 0 ) \\phi ( D _ X ^ \\infty ) } { T _ M ^ \\infty } s \\right ) d s \\\\ & \\leq D _ { X } ( 0 ) + \\frac { T _ M ^ \\infty } { \\kappa _ 1 \\mathcal { A } ( v ) ( 0 ) { \\phi ( D _ X ^ \\infty ) } } \\cdot D _ { V } ( 0 ) < D _ { X } ^ \\infty , \\forall t \\in [ 0 , t ^ * ] , \\end{align*}"}
+{"id": "2436.png", "formula": "\\begin{align*} \\phi ( \\xi ) = u ( t , x ) , \\xi = x - c t , 0 < c \\in \\mathbb { R } . \\end{align*}"}
+{"id": "7636.png", "formula": "\\begin{align*} \\mathcal { E } ( x ) = E ( \\mathcal { R } ^ { T } x ) = \\sum _ { k = 1 } ^ { n } \\mathcal { L } _ { k } x _ { k } . \\end{align*}"}
+{"id": "7260.png", "formula": "\\begin{align*} \\mathbf { A } _ t ( g ) ( x ) = \\left \\{ \\begin{array} { l l } \\tfrac { 1 + \\eta x } { 2 } g '' ( x ) & \\textrm { w h e n $ \\theta + \\eta ( t + \\tau ) = 2 x $ , } \\\\ \\tfrac { 1 + \\eta x } { \\theta + \\eta ( t + \\tau ) - 2 x } \\left ( \\tfrac { g ( \\theta + \\eta ( t + \\tau ) - x ) - g ( x ) } { \\theta + \\eta ( t + \\tau ) - 2 x } - g ' ( x ) \\right ) & \\textrm { w h e n $ \\theta + \\eta ( t + \\tau ) \\ne 2 x $ . } \\end{array} \\right . \\end{align*}"}
+{"id": "3425.png", "formula": "\\begin{align*} \\mathcal { J } = ( U ^ i , Q U ^ j ) \\end{align*}"}
+{"id": "4100.png", "formula": "\\begin{align*} X ^ { ( r ) } = \\big ( Z ^ { ( r ) } , R _ { e } ^ { ( r ) } , R _ { s } ^ { ( r ) } \\big ) \\end{align*}"}
+{"id": "920.png", "formula": "\\begin{align*} - \\int _ D \\eta \\ , L u \\ , d m = \\int _ D a \\nabla u \\cdot \\nabla \\eta \\ , d m = 0 , \\eta \\in C _ c ^ \\infty ( D ) . \\end{align*}"}
+{"id": "2898.png", "formula": "\\begin{align*} & \\lim _ { \\lambda \\to \\infty } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ { L ^ { r , \\tau } ( \\mathbb { R } ^ n ) } = \\left [ \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { L ^ { r , \\tau } ( \\mathbb { R } ^ n ) } ; \\end{align*}"}
+{"id": "3706.png", "formula": "\\begin{align*} M _ { ( H , S ( X ) , \\varphi ) } M _ { ( G , S ( X ) , \\rho _ 2 \\varphi ) } & = \\sum _ { \\substack { \\psi : F _ { K , S ( X ^ 2 ) } \\to H \\\\ \\psi p \\not \\in S ( X ) \\\\ \\psi | _ { S ( X ) } = \\varphi } } M _ { ( H , S ( X ^ 2 ) , \\psi ) } \\end{align*}"}
+{"id": "3651.png", "formula": "\\begin{align*} \\lambda _ { r _ i } ( M _ i ) = \\lambda _ 2 ( L ( G [ A _ i ] , q _ i ) ) = a ( G [ A _ i ] ) . \\end{align*}"}
+{"id": "7871.png", "formula": "\\begin{align*} \\mathcal { N } _ { \\mathbf { \\Phi } } = ( p ! ) ^ { m - 1 } . \\end{align*}"}
+{"id": "887.png", "formula": "\\begin{align*} S _ 1 + T _ 1 = 2 a _ 1 ( c _ 1 + a _ 1 x _ 1 + b _ 1 x _ 2 ) + 2 a _ 2 ( c _ 2 + a _ 2 x _ 1 + b _ 2 x _ 2 ) . \\end{align*}"}
+{"id": "2925.png", "formula": "\\begin{align*} x p _ n ( x ) = a _ n p _ { n + 1 } ( x ) + b _ n p _ n ( x ) + c _ n p _ { n - 1 } ( x ) , n = 0 , 1 , 2 , \\dots \\end{align*}"}
+{"id": "8967.png", "formula": "\\begin{align*} & A ( x ' ) : = \\Big \\{ ( x _ n , \\nu ) \\in \\R ^ 2 \\ , : \\ , x _ n \\in ( \\varphi ( x ' ) , r ) , \\ , \\nu \\in \\big ( x _ n - 2 c _ 1 ( x _ n - \\varphi ( x ' ) ) , x _ n - 2 ( x _ n - \\varphi ( x ' ) ) \\big ) \\Big \\} , \\\\ & B ( x ' ) : = \\Big \\{ ( x _ n , \\nu ) \\in \\R ^ 2 \\ , : \\ , \\nu \\in \\big ( 2 c _ 1 \\varphi ( x ' ) - r ( 2 c _ 1 - 1 ) , \\varphi ( x ' ) \\big ) , \\ , x _ n \\in \\big ( \\frac { - \\nu + 2 c _ 1 \\varphi ( x ' ) } { 2 c _ 1 - 1 } , - \\nu + 2 \\varphi ( x ' ) \\big ) \\Big \\} . \\end{align*}"}
+{"id": "4533.png", "formula": "\\begin{align*} 0 \\leq E _ N ^ { ( K ) } ( t , [ s ] _ K ) \\leq \\left ( \\sum _ { i = K + 1 } ^ N \\| \\varphi _ i \\| _ \\infty \\right ) n _ N ^ { ( K ) } ( t , [ s ] _ K ) , \\end{align*}"}
+{"id": "6628.png", "formula": "\\begin{align*} H _ N S _ N ( i , j ) & = \\frac { 1 } { J _ N } ( - J _ 1 J _ { N - j } ( J _ { j - 2 } + J _ { j - 1 } ) + 4 J _ 2 J _ { N - j } ( J _ { j - 4 } + J _ { j - 3 } ) ) \\\\ & = \\frac { 1 } { J _ N } ( - 2 J _ { N - j } ( J _ { j - 3 } + J _ { j - 2 } ) + 4 J _ { N - j } ( J _ { j - 4 } + J _ { j - 3 } ) ) \\\\ & = \\frac { 2 J _ { N - j } } { J _ N } ( 2 J _ { j - 4 } + J _ { j - 3 } - J _ { j - 2 } ) \\\\ & = 0 . \\end{align*}"}
+{"id": "1673.png", "formula": "\\begin{align*} g \\cdot v _ i & = \\omega ( e ' _ r , v _ i ) \\ , e _ r + \\omega ( e ' _ { r - 1 } , v _ i ) \\ , e _ { r - 1 } + \\omega ( f ' _ { r - 1 } , v _ i ) \\ , f _ { r - 1 } + \\omega ( f ' _ r , v _ i ) \\ , f _ r . \\end{align*}"}
+{"id": "7780.png", "formula": "\\begin{align*} f _ E ( z ) = \\textrm { d e t } ( z I - A ( E ) ) = \\prod _ { \\sigma _ i \\in \\Sigma ( E ) } ( z - \\sigma _ i ) . \\end{align*}"}
+{"id": "4637.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ K \\sum _ { D \\in \\mathcal { F } _ k } \\int _ D | V | \\frac { 1 } { | D | } < \\infty , \\end{align*}"}
+{"id": "2513.png", "formula": "\\begin{align*} C ^ \\infty ( M ) = \\bigcap _ s H ^ s ( M ) \\ ; C ^ { - \\infty } ( M ) = \\bigcup _ s H ^ s ( M ) \\ ; . \\end{align*}"}
+{"id": "2610.png", "formula": "\\begin{align*} D _ i ^ 2 ( n ) : = \\min \\{ | D _ i ( S ) | : S \\in \\mathcal { S } ^ 2 ( n ) \\} . \\end{align*}"}
+{"id": "5445.png", "formula": "\\begin{align*} r _ { ( a ^ - , i ) } ^ { \\hat { S } ^ k } - r _ { ( a ^ - , i ) } ^ { \\hat { S } ^ { k - 1 } } = \\frac { r _ { ( a _ k ^ - , i _ k ) } ^ { \\hat { S } ^ { k - 1 } } } { w _ { ( a _ k ^ - , i _ k ) } ^ { \\hat { S } ^ { k - 1 } } } \\big ( w _ { ( a ^ - , i ) } ^ { \\hat { S } ^ k } - w _ { ( a ^ - , i ) } ^ { \\hat { S } ^ { k - 1 } } \\big ) , ( a ^ - , i ) \\in \\hat { N } , \\end{align*}"}
+{"id": "6789.png", "formula": "\\begin{align*} u _ 1 & = c _ 2 d _ 3 - c _ 3 d _ 2 , \\\\ u _ 2 & = c _ 3 d _ 1 - c _ 1 d _ 3 , \\\\ u _ 3 & = c _ 1 d _ 2 - c _ 2 d _ 1 . \\end{align*}"}
+{"id": "2064.png", "formula": "\\begin{align*} & - \\int _ 0 ^ s \\Theta ( u , x ) f ' ( u ) \\mathrm { d } u + f ( s ) { \\Theta } ( s , x ) - f ( 0 ) { \\Theta } ( 0 , x ) \\\\ & = - \\eta d T \\big ( \\int _ 0 ^ { s } \\int _ 0 ^ 1 f ( u ) A ( x , y ) \\Theta ( u , y ) \\mathrm { d } y \\mathrm { d } u + o ( 1 ) \\big ) + \\sigma ^ 2 \\eta ^ 2 T \\big ( \\int _ 0 ^ s f ( u ) \\mathrm { d } \\xi _ 1 ( u , x ) + o ( 1 ) \\big ) . \\end{align*}"}
+{"id": "1935.png", "formula": "\\begin{align*} A ^ - ( k , x _ 0 , r ) : = B _ { r } ( x _ 0 ) \\cap \\big \\{ x \\in \\Bbb { R } ^ N | ~ u < k \\big \\} A ^ + ( k , x _ 0 , r ) : = B _ { r } ( x _ 0 ) \\cap \\big \\{ x \\in \\Bbb { R } ^ N | ~ u > k \\big \\} , \\end{align*}"}
+{"id": "6299.png", "formula": "\\begin{align*} \\begin{aligned} w ( h ) & : = \\sum _ { ( v , E ) \\in V ( R ) \\times E ( R ) } h ( v , E ) \\\\ & = \\sum _ { e \\in \\mathcal { F ' } } \\sum _ { v \\in V ( R ) } \\sum _ { v ' : g ( v ' ) = v } \\sum _ { E \\in E ( R ) } w ( e , E , v ' ) \\\\ & \\ge 2 ( \\alpha + \\varepsilon ' ) 4 ^ { j } t \\times \\frac { 2 } { 4 ^ { j } } \\ge 4 ( \\alpha + \\varepsilon ' ) t , \\end{aligned} \\end{align*}"}
+{"id": "2810.png", "formula": "\\begin{align*} L _ { 2 } = q ^ { 1 } \\dot { q } ^ { 2 } - q ^ { 2 } \\dot { q } ^ { 1 } - \\left ( q ^ { 1 } \\right ) ^ { 2 } - \\left ( q ^ { 2 } \\right ) ^ { 2 } . \\end{align*}"}
+{"id": "3569.png", "formula": "\\begin{align*} 0 & = \\tau _ 0 ( z _ 0 ) \\{ \\alpha ^ 3 \\tau _ 1 ( z _ 0 ) \\tau _ 2 ( z _ 0 ) L ( \\tau ( z _ 0 ) ^ 3 ) - ( 1 + a ) \\tau _ 1 ( z _ 0 ) L ( \\tau ( z _ 0 ) ^ 2 ) + ( a + b ) \\alpha L ( \\tau ( z _ 0 ) ) \\\\ & + ( a + b ) \\alpha L ( \\tau ( z _ 0 ) ) \\} - b L ( z _ 0 ) \\\\ & = - b , \\end{align*}"}
+{"id": "7177.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } \\partial _ t F _ v - \\partial _ { x } ^ 2 F _ v & = ( 1 - ( v _ 0 + V ) ^ 2 + \\varphi _ 1 ) F _ v - F _ w + \\varphi _ 3 , \\\\ \\partial _ t F _ w - \\rho \\partial _ { x } ^ 2 F _ w & = \\varepsilon F _ v - \\varepsilon \\gamma F _ w + \\varepsilon J _ 0 , \\\\ F _ v ( x , 0 ) = 0 , & F _ w ( x , 0 ) = 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "7863.png", "formula": "\\begin{align*} \\mu ( \\mathbf { \\Phi } ) = \\max \\limits _ { \\substack { 1 \\leq i , j \\leq L \\\\ 0 \\leq c _ { 1 } , c _ { 2 } \\leq M - 1 } } \\frac { \\mid \\langle \\mathbf { s } _ { \\mathbf { A } _ { i } } ^ { ( c _ { 1 } ) } , \\mathbf { s } _ { \\mathbf { A } _ { j } } ^ { ( c _ { 2 } ) } \\rangle \\mid } { M } , \\end{align*}"}
+{"id": "3535.png", "formula": "\\begin{align*} \\sum _ { n , m = 0 } ^ { \\infty } | a _ { n m } z _ 2 ^ m | ^ 2 \\leq \\sum _ { n , m = 0 } ^ { \\infty } | a _ { n m } | ^ { 2 } | z _ 2 ^ m | ^ { 2 } = \\sum _ { n , m = 0 } ^ { \\infty } | a _ { n m } | ^ { 2 } < \\infty , \\end{align*}"}
+{"id": "5716.png", "formula": "\\begin{align*} 0 = 2 ^ { - 1 } m ^ { 2 } G ( ( v , w ) ; ( 0 , \\varphi _ j ) ) = \\int _ { \\Sigma } \\left \\langle w , \\varphi _ j \\right \\rangle \\ , d \\mu , \\end{align*}"}
+{"id": "8167.png", "formula": "\\begin{align*} & C ^ { \\prime } _ ( B _ ) = \\frac { 1 } { L } \\sum _ { j = 1 } ^ L ( 1 - \\frac { \\sum _ { k = 1 } ^ { j - 1 } b _ { k } ^ 0 } { M } ) H ( \\frac { b _ { j } ^ 0 } { M - \\sum _ { k = 1 } ^ { j - 1 } b _ k } ) \\\\ & + \\frac { \\sum _ { k = 1 } ^ { j - 1 } b _ { k } ^ 0 } { M } , \\forall k \\in [ 2 : L ] , \\end{align*}"}
+{"id": "396.png", "formula": "\\begin{align*} [ { \\tt Q } ] _ { A ^ { ( \\rho ) } } J = [ { \\tt Q } ] _ { \\lnot A ^ { ( \\rho ) } } J = [ { \\tt Q } ] _ { A } J \\cap [ \\partial { \\tt Q } ] J \\cap [ \\rho ] J \\end{align*}"}
+{"id": "4123.png", "formula": "\\begin{align*} \\epsilon ( \\theta ( r ) , r ) = o ( \\abs { \\theta ( r ) } r ^ J ) , \\end{align*}"}
+{"id": "669.png", "formula": "\\begin{align*} W ^ { k } f = 0 \\iff R ^ { k } f = 0 \\end{align*}"}
+{"id": "8417.png", "formula": "\\begin{align*} \\Psi ( f ) ( \\sigma ( a ) \\circ \\sigma ( b ) ) & = f ( B _ 1 ( a _ 1 ) \\sigma ( b ) S ( B _ 2 ) ( a _ 2 ) ) \\\\ & = f ( B _ 1 ( a _ 1 ) ) f ( \\sigma ( b ) ) f ( S ( B _ 2 ( a _ 2 ) ) ) . \\end{align*}"}
+{"id": "4740.png", "formula": "\\begin{align*} \\norm { x } ^ 2 + t \\langle y , j _ { u , v } x \\rangle = \\langle x + t y , j _ { u , v } x \\rangle \\leq \\norm { x } \\norm { x + t y } \\end{align*}"}
+{"id": "2806.png", "formula": "\\begin{align*} & L _ { T } : = \\tilde { \\sigma } _ { 1 } ^ { * } ( t ) \\left [ \\Psi _ { i } \\dot { \\Xi } ^ { i } - H _ { T } \\right ] \\\\ & \\therefore L _ { T } = \\Psi _ { 1 } \\Xi ^ { 3 } + \\frac { 1 } { 2 } \\Xi ^ { 2 } ( \\Psi _ { 3 } ) ^ { 2 } + \\frac { d } { d t } \\left ( \\Psi _ { a ' } \\Xi ^ { a ' } \\right ) , \\end{align*}"}
+{"id": "1749.png", "formula": "\\begin{align*} [ e _ j , e _ { 2 n - 1 - j } ] = e _ 0 \\quad \\forall \\ , j = 1 , \\ldots , n - 1 . \\end{align*}"}
+{"id": "5066.png", "formula": "\\begin{align*} \\dim \\mathcal { C } ( v ) = | W | \\sum _ { I \\subset S } D _ I / N _ I \\end{align*}"}
+{"id": "168.png", "formula": "\\begin{align*} x \\ \\ & \\Longleftrightarrow \\overline { R ^ { \\circ } ( x ) } = R ( x ) \\\\ x \\ \\ & \\Longleftrightarrow \\overline { R ^ { \\circ - 1 } ( x ) } = R ^ { - 1 } ( x ) \\\\ x \\ \\ & \\Longleftrightarrow x \\ \\ . \\end{align*}"}
+{"id": "96.png", "formula": "\\begin{align*} c _ 3 + c _ 4 = & 2 ( 1 - \\gamma ) - 2 \\kappa + 2 ( 2 + \\gamma ) \\kappa ^ 2 . \\end{align*}"}
+{"id": "9115.png", "formula": "\\begin{align*} A _ { n } \\ ( n \\geq 1 ) \\colon & d _ { i } = 1 \\ ( 1 \\leq i \\leq n ) , \\\\ B _ { n } \\ ( n \\geq 2 ) \\colon & d _ { i } = 2 \\ ( 1 \\leq i \\leq n - 1 ) , \\ d _ { n } = 1 , \\\\ G _ 2 \\colon & d _ { 1 } = 3 , \\ d _ { 2 } = 1 . \\end{align*}"}
+{"id": "8182.png", "formula": "\\begin{align*} \\Delta ^ n : = \\mathbb { E } [ d ( S ^ n , \\hat { S } ^ n ) ] = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\mathbb { E } [ d ( S _ i , \\hat { S } _ i ) ] \\end{align*}"}
+{"id": "398.png", "formula": "\\begin{align*} \\begin{aligned} & h _ { s } ( \\bold { r } , \\bold { s } ) = \\frac { 1 } { 4 \\pi ^ 2 } \\iiiint _ { \\mathcal { D } _ { k } \\times \\mathcal { D } _ { \\kappa } } a _ { R } ( k _ x , k _ y , \\bold { r } ) \\\\ & \\times H _ { a } ( k _ x , k _ y , \\kappa _ x , \\kappa _ y ) a _ { S } ( \\kappa _ x , \\kappa _ y , \\bold { r } ) \\mathrm { d } k _ x \\mathrm { d } k _ y \\mathrm { d } \\kappa _ x \\mathrm { d } \\kappa _ y , \\end{aligned} \\end{align*}"}
+{"id": "1134.png", "formula": "\\begin{align*} \\vec { v } = \\sum _ i O ( 1 ) \\vec { v } _ i \\end{align*}"}
+{"id": "7820.png", "formula": "\\begin{align*} z _ k ^ { \\star } = \\sqrt [ 4 ] { \\frac { ( 1 - a _ { k } ) E _ { k } } { L \\kappa _ k } } , \\forall k \\in \\mathcal { K } . \\end{align*}"}
+{"id": "1603.png", "formula": "\\begin{align*} \\epsilon ( i _ 1 \\dots i _ { r - 1 } n i _ { r + 1 } \\dots i _ { n - 1 } ) = ( - 1 ) ^ { n - 1 - r } \\epsilon ( i _ 1 \\dots i _ { n - 1 } ) , \\end{align*}"}
+{"id": "304.png", "formula": "\\begin{align*} F ' _ l ( z ' ) = \\sum _ { k = 1 } ^ { r } \\mu ^ { | \\alpha ( k ) | - 1 } a _ { l , k } z '^ { \\alpha ( k ) } . \\end{align*}"}
+{"id": "4980.png", "formula": "\\begin{align*} P _ N ( \\nu _ i , \\lambda _ 2 , \\dots , \\lambda _ N ) = S _ { N - 1 , i } ( \\lambda _ 2 , \\dots , \\lambda _ N ) \\prod _ { k = 2 } ^ N ( \\lambda _ k - \\nu _ i + 1 ) \\ , \\prod _ { j = 1 } ^ N ( \\nu _ i - \\nu _ j + 1 ) , \\end{align*}"}
+{"id": "2184.png", "formula": "\\begin{align*} f _ { } ( \\mathfrak { S } ^ { ( 2 ) } \\otimes _ { p _ 1 , \\mathfrak { S } } \\mathfrak { M } ) \\subset \\bigcap _ { n = 1 } ^ { \\infty } ( \\mathfrak { S } ^ { ( 2 ) } \\otimes _ { p _ 2 , \\mathfrak { S } } \\mathfrak { M } + p ^ n \\mathfrak { S } ^ { ( 2 ) } [ E ^ { - 1 } ] ^ { \\wedge } _ p \\otimes _ { p _ 2 , \\mathfrak { S } } \\mathfrak { M } ) \\end{align*}"}
+{"id": "6566.png", "formula": "\\begin{align*} \\phi _ { \\Phi _ k , \\beta } ( x , y ) = | \\{ n \\leq x ; \\ \\ \\frac { \\Phi _ k ( n ) } { n ^ { \\beta } } \\leq y \\} | . \\end{align*}"}
+{"id": "6960.png", "formula": "\\begin{align*} 1 ) & \\ ; 0 \\leq \\langle \\mathbf { z } , \\mathbf { z } \\rangle = x _ 1 ^ 2 + y _ 1 ^ 2 + \\cdots + x _ n ^ 2 + y _ n ^ 2 - x _ { n + 1 } ^ 2 - y _ { n + 1 } ^ 2 \\\\ 2 ) & \\ ; 0 \\geq f ( \\mathbf { z } ) = { \\displaystyle \\sum _ { j = 1 } ^ { n } } 4 ( x _ j y _ { n + 1 } - x _ { n + 1 } y _ { j } ) ^ { 2 } - \\sum _ { 1 \\leq j < k \\leq n } 4 ( x _ j y _ { k } - y _ { k } x _ { j } ) ^ { 2 } . \\end{align*}"}
+{"id": "6308.png", "formula": "\\begin{align*} \\widetilde K ( x , y ) = - \\sum _ { p \\in \\pi \\mathbb { N } _ 0 ^ 3 } n \\widehat \\omega _ { \\ell , \\lambda } ( p ) u _ p ( x ) u _ p ( y ) , \\end{align*}"}
+{"id": "672.png", "formula": "\\begin{align*} c _ { \\ell , s } = \\left ( \\prod _ { p = 0 } ^ { s - \\ell - 1 } ( n - 1 + 2 p ) \\right ) \\frac { ( - 1 ) ^ { \\ell } s ! } { 2 ^ { \\ell } \\ell ! ( s - 2 \\ell ) ! } , \\end{align*}"}
+{"id": "3657.png", "formula": "\\begin{align*} \\lambda & \\geq \\frac { \\lambda _ 2 ( \\mathcal { L } ( s ^ { k - 1 } ( G ) ) ) } { 2 ( 2 - \\lambda ) } \\geq \\frac { \\lambda _ 2 ( \\mathcal { L } ( s ^ { k - 1 } ( G ) ) ) } { 4 } \\\\ & \\geq \\frac { 1 } { 4 } \\frac { \\min \\{ \\lambda _ 2 ( \\mathcal { L } ( G ) ) , 4 \\} } { 4 ^ { k - 1 } } = \\frac { \\min \\{ \\lambda _ 2 ( \\mathcal { L } ( G ) ) , 4 \\} } { 4 ^ k } . \\end{align*}"}
+{"id": "6168.png", "formula": "\\begin{align*} 1 / ( \\tau ^ { k - 1 } ) ^ 2 = ( 1 - \\tau ^ k ) / ( \\tau ^ k ) ^ 2 , ~ \\tau ^ { - 1 } \\in ( 0 , 1 ) . \\end{align*}"}
+{"id": "1393.png", "formula": "\\begin{align*} \\mathcal { S } ' ( Z ) = - \\frac { 3 } { 2 } \\frac { \\frac { 5 } { 3 } P ( Z ) - P ' ( Z ) Z } { Z ^ 2 } . \\end{align*}"}
+{"id": "7552.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\bar \\varphi f \\ d x = \\int _ \\Omega \\delta \\nabla \\bar \\varphi \\cdot \\nabla \\phi \\ d x \\ . \\end{align*}"}
+{"id": "7343.png", "formula": "\\begin{align*} U \\circ \\theta = \\left ( U \\odot _ \\phi ( V ^ 0 , \\Sigma ^ 0 ) \\right ) \\odot _ \\phi ( V ^ 1 , \\Sigma ^ 1 ) , \\end{align*}"}
+{"id": "7295.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n } \\binom { n } { j } a _ m ( j + 1 ) c _ { m , r } ( n - j ) = b _ { m - \\ell - 1 } ( n + 1 ) + b _ { \\ell - 1 } ( n + 1 ) - \\frac { a _ m ( n + 2 ) } { m } . \\end{align*}"}
+{"id": "5250.png", "formula": "\\begin{align*} F I \\left ( p \\| q \\right ) = \\sum _ { j } p _ { j } \\sum _ { i } \\overline { p } _ { i } \\log \\frac { \\overline { p } _ { i } } { \\alpha \\overline { p } _ { i } + \\left ( 1 - \\alpha \\right ) \\overline { q } _ { i } } \\end{align*}"}
+{"id": "1947.png", "formula": "\\begin{align*} \\left | B _ { 2 R } \\cap \\left \\{ \\left ( w - 2 ^ { - k + 1 } t \\right ) _ { - } \\geq 3 \\cdot 2 ^ { - k - 1 } t \\right \\} \\right | = \\left | B _ { 2 R } \\cap \\left \\{ w \\leq 2 ^ { - k - 1 } t \\right \\} \\right | . \\end{align*}"}
+{"id": "8402.png", "formula": "\\begin{align*} \\epsilon ( a \\circ b ) & = \\epsilon ( B _ 1 ( a _ 1 ) b S ( B _ 2 ( a _ 2 ) ) ) = \\epsilon ( B _ 1 ( a _ 1 ) ) \\epsilon ( b ) \\epsilon ( S ( B _ 2 ( a _ 2 ) ) ) \\\\ & = \\epsilon ( a _ 1 ) \\epsilon ( b ) \\epsilon ( a _ 2 ) = \\epsilon ( a ) \\epsilon ( b ) . \\end{align*}"}
+{"id": "3901.png", "formula": "\\begin{align*} S _ { 0 } ( T ) \\coloneqq \\sum _ { n = 1 } ^ \\infty \\sum _ { e ^ { n \\ell ( \\gamma ) } \\leq T } \\ell ( \\gamma ) e ^ { n \\ell _ \\psi ( \\gamma ) } , \\ \\ \\hat { S } _ { 0 } ( T ) \\coloneqq \\sum _ { e ^ { \\ell ( \\gamma ) } \\leq T } \\ell ( \\gamma ) e ^ { \\ell _ \\psi ( \\gamma ) } , \\end{align*}"}
+{"id": "3743.png", "formula": "\\begin{align*} _ { 2 } F _ { 1 } \\left ( \\left . \\begin{array} { c } \\alpha , \\beta \\\\ \\beta + 1 \\end{array} \\right \\vert z \\right ) = \\beta \\ , z ^ { - \\beta } \\ , \\mathrm { B } _ { z } \\left ( \\beta , 1 - \\alpha \\right ) , \\end{align*}"}
+{"id": "4681.png", "formula": "\\begin{align*} \\mathbf { B } _ { - } ( D ) = \\left ( \\bigcap _ { E \\equiv P ( D ) } \\mathrm { S u p p } ( E ) \\right ) \\cup \\mathrm { S u p p } ( N ( D ) ) . \\end{align*}"}
+{"id": "4027.png", "formula": "\\begin{align*} \\lim _ { m \\rightarrow + \\infty } \\frac { x _ { 2 } ^ { \\left ( t _ { 0 } + m \\right ) } } { \\varpi \\circ W \\left ( z ^ { \\left ( t _ { 0 } + m \\right ) } \\right ) } = \\begin{cases} \\qquad 0 & \\mbox { i f } \\gamma _ { 2 } = 0 , \\delta _ { 1 } \\neq 0 , \\\\ \\frac { \\delta _ 2 x _ { 2 } ^ { ( t _ { 0 } ) } } { x _ { 1 } ^ { \\left ( t _ { 0 } \\right ) } + x _ { 2 } ^ { \\left ( t _ { 0 } \\right ) } } & \\mbox { i f } \\gamma _ { 2 } = \\delta _ { 1 } = 0 , \\\\ \\qquad \\delta _ 2 & \\mbox { i f } \\gamma _ { 2 } \\neq 0 , \\delta _ { 1 } = 0 , \\end{cases} \\end{align*}"}
+{"id": "5764.png", "formula": "\\begin{align*} X ^ 2 _ + ( t ) = \\sum _ { k + \\ell \\leq s } \\left ( \\sum _ { i \\in I _ 1 } \\left ( | \\xi ^ { ( k , \\ell ) } _ { i , 1 } ( t ) | ^ 2 + | \\xi ^ { ( k , \\ell ) } _ { i , 2 } ( t ) | ^ 2 \\right ) + \\sum _ { i \\in \\mathbb { N } } | \\xi ^ { ( k , \\ell ) } _ i ( t ) | ^ 2 + \\sum _ { 1 \\leq j \\leq J } | \\bar { z } ^ { ( k , \\ell ) } _ j ( t ) | ^ 2 \\right ) . \\end{align*}"}
+{"id": "2698.png", "formula": "\\begin{align*} \\ddot { Q } = \\frac { \\alpha } { \\beta } \\delta ( t - t _ { 0 } ) \\end{align*}"}
+{"id": "769.png", "formula": "\\begin{align*} m \\triangleleft h = R ( m ^ { ( 2 ) } , h ) m ^ { ( 1 ) } , h \\in H , \\ m \\in M . \\end{align*}"}
+{"id": "3459.png", "formula": "\\begin{align*} \\kappa ( L ) : = \\kappa ( \\Sigma ( L ) , \\iota , \\mathfrak { t } _ L ) . \\end{align*}"}
+{"id": "7951.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Omega } \\ast d w \\wedge \\eta ' = ( - 1 ) ^ { n - 1 } \\langle w , \\delta \\eta ' \\rangle _ { L ^ { 2 } \\Lambda ^ { 0 } ( \\Omega ) } + ( - 1 ) ^ { n - 1 } \\int _ { \\partial \\Omega } \\mathrm { t r } ( w ) \\wedge \\ast \\boldsymbol { n } ( \\eta ' ) = 0 , \\end{aligned} \\end{align*}"}
+{"id": "8568.png", "formula": "\\begin{align*} F ( n , k ) = \\frac { ( a ) _ k ^ 3 } { ( n ) _ k ^ 3 } \\end{align*}"}
+{"id": "4780.png", "formula": "\\begin{align*} \\norm { \\frac { d } { \\norm { z } } z + \\frac { J _ t x } { \\norm { J _ t x } } \\frac { \\norm { J _ t x } } { t } } & \\leq \\norm { \\frac { d } { \\norm { z } } z - \\frac { z } { \\norm { z } } \\frac { \\norm { J _ t x } } { t } } + \\norm { \\frac { z } { \\norm { z } } \\frac { \\norm { J _ t x } } { t } + \\frac { J _ t x } { \\norm { J _ t x } } \\frac { \\norm { J _ t x } } { t } } \\\\ & = \\left \\vert d - \\frac { \\norm { J _ t x } } { t } \\right \\vert + \\frac { \\norm { J _ t x } } { t } \\norm { \\frac { z } { \\norm { z } } + \\frac { J _ t x } { \\norm { J _ t x } } } . \\end{align*}"}
+{"id": "7506.png", "formula": "\\begin{align*} | h ( t ) - g _ 0 ( t ) | _ { g _ 0 ( t ) } + \\sqrt { t } | \\nabla ^ { g _ 0 ( t ) } h ( t ) | _ { g _ 0 ( t ) } & = | s ^ { - 1 } Q ^ * \\hat { g } ( s t ) - s ^ { - 1 } Q ^ * \\tilde { g } ( s t ) | _ { g _ 0 ( t ) } \\\\ & + \\sqrt { t } | \\nabla ^ { g _ 0 ( t ) } s ^ { - 1 } Q ^ * \\hat { g } ( s t ) | _ { g _ 0 ( t ) } \\\\ & \\leq Q ^ * \\left ( | \\hat { g } - \\tilde { g } | _ { \\tilde { g } } + \\sqrt { t } | \\nabla ^ { \\tilde { g } } \\hat { g } | _ { \\tilde { g } } \\right ) ( s t ) \\\\ & \\leq 1 \\end{align*}"}
+{"id": "2809.png", "formula": "\\begin{align*} L _ { T } = \\sigma _ { 1 } ^ { * } ( t ) \\left [ \\Psi _ { i } \\dot { \\Xi } ^ { i } - H _ { T } \\right ] = { c o n s t a n t } , \\end{align*}"}
+{"id": "1318.png", "formula": "\\begin{align*} \\{ F , G \\} _ \\pm = \\pm \\biggl ( \\frac { \\partial F } { \\partial p _ i } \\frac { \\partial G } { \\partial q ^ i } - \\frac { \\partial F } { \\partial q ^ i } \\frac { \\partial G } { \\partial p _ i } \\biggr ) , \\end{align*}"}
+{"id": "2259.png", "formula": "\\begin{align*} E = \\bigoplus _ i A _ i ^ { \\oplus \\dim ( B _ i ) } \\ , , \\quad \\ \\ \\ E = \\bigoplus _ i B _ i ^ { \\oplus \\dim ( A _ i ) } \\ , , \\quad \\ \\ \\ E = \\bigoplus _ i A _ i \\otimes B _ i \\ , . \\end{align*}"}
+{"id": "8850.png", "formula": "\\begin{align*} \\frac 1 q = \\frac { 1 } { 2 ( 2 p - 1 ) } , \\frac { n - 2 } { 2 n } \\leq \\frac 1 r \\leq \\frac 1 2 , \\frac { 2 } q + \\frac { n } { r } = \\frac { n } { 2 } , 0 < \\alpha < n . \\end{align*}"}
+{"id": "2738.png", "formula": "\\begin{align*} \\phi ^ { * } \\omega _ { Q , P } = \\omega _ { q , p } \\end{align*}"}
+{"id": "4762.png", "formula": "\\begin{align*} \\langle y _ 0 , x _ 0 - x \\rangle _ s = - \\lambda \\norm { A _ \\lambda x } ^ 2 \\end{align*}"}
+{"id": "5429.png", "formula": "\\begin{align*} S _ 1 ^ * \\left ( \\nu _ { ( 0 , i ) } ^ * \\right ) = \\big \\{ j \\in N \\colon \\nu _ { ( 1 , j ) } ^ * > \\nu _ { ( 0 , i ) } ^ * \\big \\} . \\end{align*}"}
+{"id": "3447.png", "formula": "\\begin{align*} I ( a ) = - \\iota ^ { \\ast } a , I ( \\phi ) = \\tilde { \\iota } ( \\phi ) \\cdot j , \\end{align*}"}
+{"id": "8371.png", "formula": "\\begin{align*} \\underline { w } ^ j : = ( s _ { i _ 1 } , \\dots , s _ { i _ { j - 1 } } , s _ { i _ { j + 1 } } , \\dots , s _ { i _ l } ) . \\end{align*}"}
+{"id": "6347.png", "formula": "\\begin{align*} \\nu = \\Bigl ( \\varphi - \\frac { \\nabla \\rho } { 1 + \\rho } \\Bigr ) \\Bigl ( 1 + \\frac { \\abs { \\nabla \\rho } ^ 2 } { ( 1 + \\rho ) ^ 2 } \\Bigr ) ^ { - 1 / 2 } , \\end{align*}"}
+{"id": "7726.png", "formula": "\\begin{align*} \\norm { f } _ { \\mathcal L ^ { p , \\lambda } _ k } = [ f ] _ { \\mathcal L ^ { p , \\lambda } _ k } + \\norm { f } _ { L ^ p } . \\end{align*}"}
+{"id": "2146.png", "formula": "\\begin{align*} F _ { \\min } = 1 . 5 6 4 1 , x ^ * = 1 . 9 1 1 1 , y ^ * = ( 2 . 9 7 8 4 , 2 . 2 3 1 5 ) . \\end{align*}"}
+{"id": "6864.png", "formula": "\\begin{align*} \\mathcal { E } \\{ c _ j \\} = \\{ E _ j c _ j \\} , \\mathcal { D } ( \\mathcal { E } ) = \\{ \\{ c _ j \\} \\in \\ell ^ 2 ( \\mathbb { J } ) : \\{ E _ j c _ j \\} \\in \\ell ^ 2 ( \\mathbb { J } ) \\} . \\end{align*}"}
+{"id": "7431.png", "formula": "\\begin{align*} F _ { \\rm L o r e n t z } = q B \\dot { x } B \\in \\mathfrak { s o } ( 3 ; \\R ) . \\end{align*}"}
+{"id": "4649.png", "formula": "\\begin{align*} C _ q ( 2 N ) - C _ q ( N ) = & B _ q ( 2 N ) + B _ q \\left ( \\left \\lfloor \\frac { N } { 2 } \\right \\rfloor \\right ) + 2 B _ q \\left ( \\left \\lfloor \\frac { N } { 4 } \\right \\rfloor \\right ) + \\\\ + & B _ q \\left ( \\left \\lfloor \\frac { 2 N } { | q | } \\right \\rfloor \\right ) + B _ q \\left ( \\left \\lfloor \\frac { N } { 2 | q | } \\right \\rfloor \\right ) + 2 B _ q \\left ( \\left \\lfloor \\frac { N } { 4 | q | } \\right \\rfloor \\right ) . \\end{align*}"}
+{"id": "1689.png", "formula": "\\begin{align*} w = u + i v = z _ 0 \\quad \\mbox { a n d } \\zeta = s + i t = z _ n , \\end{align*}"}
+{"id": "3773.png", "formula": "\\begin{align*} P ( x ) & \\geq \\int _ { \\zeta _ { 2 x } } ^ 1 \\exp \\Big ( - 2 x t - m ( t ) \\Big ) d t \\\\ & \\geq \\exp \\big ( - m ( \\zeta _ { 2 x } ) \\big ) \\int _ { \\zeta _ { 2 x } } ^ 1 \\exp ( - 2 x t ) d t \\\\ & = \\exp \\Big ( - m ( \\zeta _ { 2 x } ) \\Big ) \\frac { \\exp ( - 2 x \\zeta _ { 2 x } ) - \\exp ( - 2 x ) } { 2 x } \\\\ & = \\frac { \\exp \\big ( - k ( 2 x ) \\big ) } { 2 x } \\Big ( 1 - \\exp ( - 2 x + 2 x \\zeta _ { 2 x } ) \\Big ) \\\\ & \\geq \\frac { \\exp \\big ( - k ( 2 x ) \\big ) } { 4 x } . \\end{align*}"}
+{"id": "7764.png", "formula": "\\begin{align*} \\Lambda ^ { ( j + 1 ) } = \\Lambda ^ { ( j ) } \\backslash \\mathcal R _ j ( \\tilde \\Lambda ) , \\ \\ \\mathcal R _ j ( \\tilde \\Lambda ) = \\cup _ { i \\in J _ j ( \\tilde \\Lambda ) } I ^ { ( j ) } _ i , \\end{align*}"}
+{"id": "3603.png", "formula": "\\begin{align*} T ^ 3 f = \\alpha ^ 3 \\tau _ 0 \\tau _ 1 \\tau _ 2 f ( \\tau ^ 3 , w \\sigma \\sigma _ 1 \\sigma _ 2 ) , \\end{align*}"}
+{"id": "3530.png", "formula": "\\begin{align*} P ^ 2 = P . \\end{align*}"}
+{"id": "7976.png", "formula": "\\begin{align*} \\dot { \\mathcal { F } } ( v , \\Sigma ) = \\int _ { \\Omega } \\frac { \\delta \\mathcal { F } } { \\delta v } \\wedge v _ { t } + \\int _ { \\Sigma } \\frac { \\delta \\mathcal { F } } { \\delta \\Sigma } \\wedge \\Sigma _ { t } , \\end{align*}"}
+{"id": "1987.png", "formula": "\\begin{align*} i \\frac { d } { d s } \\widehat { \\psi } _ l ( t _ n + s ) - \\alpha \\frac { d ^ 2 } { d s ^ 2 } \\widehat { \\psi } _ l ( t _ n + s ) - | \\mu _ l | ^ 2 \\widehat { \\psi } _ l ( t _ n + s ) - \\varepsilon ^ { 2 } f _ l ^ n ( s ) = 0 , \\ s \\in \\mathbb { R } , \\end{align*}"}
+{"id": "7364.png", "formula": "\\begin{align*} L ' ( v ) ( x , t ) : = \\frac { 1 } { 2 } \\int _ 0 ^ t \\{ v ( x + t - s , s ) + v ( x - t + s , s ) \\} d s . \\end{align*}"}
+{"id": "8367.png", "formula": "\\begin{align*} ( 1 \\times \\mathbf { m } ) ( t ) = g \\cdot ( 1 \\times \\mathbf { m } ) ( z ) = g \\cdot y \\in D ' . \\end{align*}"}
+{"id": "6630.png", "formula": "\\begin{align*} H _ N S _ N ( i , j ) & = \\frac { 1 } { J _ N } ( - J _ { i - 2 } J _ { N - j } ( J _ { j - i + 1 } + J _ { j - i + 2 } ) - J _ { i - 1 } J _ { N - j } ( J _ { j - i } + J _ { j - i + 1 } ) \\\\ & ~ + 2 J _ i J _ { N - j } ( J _ { j - i - 1 } + J _ { j - i } ) ) \\\\ & = \\frac { J _ { N - j } } { J _ N } ( - 2 J _ { i - 2 } ( J _ { j - i } + J _ { j - i + 1 } ) - J _ { i - 1 } ( J _ { j - i } + J _ { j - i + 1 } ) \\\\ & ~ + J _ i ( J _ { j - i } + J _ { j - i + 1 } ) ) \\\\ & = \\frac { J _ { N - j } } { J _ N } ( - 2 J _ { i - 2 } - J _ { i - 1 } + J _ i ) ( J _ { j - i - 1 } + J _ { j - i } ) \\\\ & = 0 . \\end{align*}"}
+{"id": "196.png", "formula": "\\begin{align*} P [ j , k ] \\ = \\ \\widehat { A } + \\ & \\cup \\ \\widehat { A } - \\ \\cup \\\\ [ ( D _ j - \\times D _ j + ) \\cup ( \\bar D _ j - \\times \\bar D _ j + ) ] \\ & \\cup \\ [ ( D _ j + \\times \\bar D _ j - ) \\cup ( \\bar D _ j + \\times D _ j - ) ] . \\end{align*}"}
+{"id": "1068.png", "formula": "\\begin{align*} \\phi ( x ) : = \\int _ { 0 } ^ { 1 } \\frac { 1 } { ( x + s ) ^ { 1 + d } } d s , x > 0 . \\end{align*}"}
+{"id": "8444.png", "formula": "\\begin{align*} \\nabla \\mathcal { E } ( u ) = ( - \\Delta ) ^ s _ p u \\end{align*}"}
+{"id": "1251.png", "formula": "\\begin{align*} a _ { n , ( n - 1 ) / 2 } & = \\frac { q ^ { \\frac { n ^ 2 - 4 n + 3 } { 4 } } ( q ; q ^ 2 ) _ n } { ( 1 - q ^ { n } ) ( q ^ 2 ; q ^ 2 ) _ { n - 1 } } { n - 1 \\brack \\frac { n - 1 } { 2 } } _ { q ^ 2 } \\\\ [ 7 p t ] & \\equiv \\frac { ( - 1 ) ^ { \\frac { n - 1 } { 2 } } q ^ { - n + 1 } ( q ; q ^ 2 ) _ n ( - q ; q ) _ { n - 1 } ^ 2 } { ( 1 - q ^ { n } ) ( q ^ 2 ; q ^ 2 ) _ { n - 1 } } \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "7252.png", "formula": "\\begin{align*} \\mathbf A _ t f ( x ) = ( 1 + \\eta x ) \\mathcal { L } _ { x , t , \\theta , \\tau , \\eta , q } \\left ( \\tfrac { \\partial } { \\partial x } \\ , \\tfrac { f ( y ) - f ( x ) } { y - x } \\right ) . \\end{align*}"}
+{"id": "3996.png", "formula": "\\begin{align*} \\varpi \\bigl ( z ^ { \\left ( t \\right ) } \\bigr ) \\leq \\max _ { i , j , p , q } \\left \\{ \\gamma _ { i j } \\widetilde { \\gamma } _ { p q } \\right \\} \\Bigl ( \\sum _ { i = 1 } ^ { n } x _ { i } ^ { \\ : \\left ( t - 2 \\right ) } \\Bigr ) ^ { 2 } \\Bigl ( \\sum _ { j = 1 } ^ { \\nu } y _ { j } ^ { \\ ; \\left ( t - 2 \\right ) } \\Bigr ) ^ { 2 } \\end{align*}"}
+{"id": "3768.png", "formula": "\\begin{align*} f _ { N , j } = N \\frac { | I _ j \\cap F | } { | 1 _ { I _ j \\setminus E } | } 1 _ { I _ j \\setminus E } - N 1 _ { I _ j \\cap F } . \\end{align*}"}
+{"id": "8737.png", "formula": "\\begin{align*} & \\int _ { \\R ^ 2 } \\varphi ( \\vert y - x \\vert ^ 2 ) \\pi ^ \\uparrow ( d x , d y ) = \\inf _ { \\pi \\in \\Pi _ M ( \\mu , \\underline \\nu ) } \\int _ { \\R ^ 2 } \\varphi ( \\vert y - x \\vert ^ 2 ) \\pi ( d x , d y ) \\mbox { a n d } \\\\ & \\int _ { \\R ^ 2 } \\varphi ( \\vert y - x \\vert ^ 2 ) \\pi ^ \\downarrow ( d x , d y ) = \\sup _ { \\pi \\in \\Pi _ M ( \\mu , \\overline \\nu ) } \\int _ { \\R ^ 2 } \\varphi ( \\vert y - x \\vert ^ 2 ) \\pi ( d x , d y ) . \\end{align*}"}
+{"id": "350.png", "formula": "\\begin{align*} \\phi _ n ( z , y ) = \\dfrac { z ^ n - y ^ n } { z - y } = \\sum _ { i = 0 } ^ { n - 1 } z ^ { n - i - 1 } y ^ i , \\end{align*}"}
+{"id": "8657.png", "formula": "\\begin{align*} & \\mbox { $ \\langle \\ , d \\rho ( x ) \\ , | \\ , d \\rho ( x ) \\ , \\rangle = 1 $ o n $ X $ } , \\\\ & \\rho ( g \\circ x ) = \\rho ( x ) , \\ \\ \\forall x \\in M ' , \\ \\ \\forall g \\in G . \\end{align*}"}
+{"id": "6773.png", "formula": "\\begin{align*} \\begin{pmatrix} x ^ { k + 1 } \\\\ y ^ { k + 1 } \\end{pmatrix} = \\begin{pmatrix} x ^ { k } \\\\ y ^ { k } \\end{pmatrix} - \\begin{pmatrix} I _ n & 0 \\\\ - ( 1 - \\alpha ) \\frac { 1 } { s } A & I _ m \\end{pmatrix} \\begin{pmatrix} x ^ { k } - \\widetilde { x } ^ k \\\\ y ^ { k } - \\widetilde { y } ^ k \\end{pmatrix} . \\end{align*}"}
+{"id": "6194.png", "formula": "\\begin{align*} 1 / \\alpha _ 1 - 1 / \\gamma _ 1 - \\cdots - 1 / \\gamma _ { 2 n - 2 } = 0 ~ . \\end{align*}"}
+{"id": "7712.png", "formula": "\\begin{align*} \\widetilde { E } ( x , t ) = { \\rm l o g } b _ { 1 1 } - \\tilde { d } \\log h ( x , t ) + \\tilde { l } | \\nabla h | ^ { 2 } . \\end{align*}"}
+{"id": "3854.png", "formula": "\\begin{align*} \\sum _ { x \\in \\Delta ^ o } \\pi ^ * _ x G ( \\pi ^ * ) _ { x , y } = \\pi ^ * _ y , \\ ; y \\in \\Delta ^ o . \\end{align*}"}
+{"id": "4683.png", "formula": "\\begin{align*} \\mathbf { B } _ { + } ( D ) : = \\bigcap _ { \\substack { D = A + N \\\\ A \\\\ N } } \\mathrm { S u p p } ( N ) , \\end{align*}"}
+{"id": "8811.png", "formula": "\\begin{align*} \\frac { ( z _ + - y _ - ) } { \\rho } \\partial _ x f ( z _ - , y _ + ) & = ( z _ + - y _ - ) ( z _ - - y _ + ) ^ { \\rho - 1 } - ( z _ + - y _ + ) ( z _ - - y _ - ) ^ { \\rho - 1 } + ( y _ + - y _ - ) ( z _ + - z _ - ) ^ { \\rho - 1 } \\\\ & = g \\left ( y _ + - y _ - , z _ - - y _ + , z _ + - z _ - \\right ) , \\end{align*}"}
+{"id": "6204.png", "formula": "\\begin{align*} a ^ i + b ^ i = a ^ g + b ^ g = 2 , \\end{align*}"}
+{"id": "4742.png", "formula": "\\begin{align*} \\omega ^ { + , M } ( b , k ) = \\max \\{ \\omega ^ + ( a , j ) \\mid a \\leq b , j \\leq k \\} \\end{align*}"}
+{"id": "5259.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } G \\left ( p \\| q \\right ) } { \\partial q _ { j } } = \\left ( 1 - \\alpha \\right ) \\left \\lbrace 1 - \\left [ \\left ( \\frac { a } { a - b } \\right ) \\left ( \\frac { T _ { j } } { p _ { j } } \\right ) ^ { a - 1 } - \\left ( \\frac { b } { a - b } \\right ) \\left ( \\frac { T _ { j } } { p _ { j } } \\right ) ^ { b - 1 } \\right ] \\right \\rbrace \\end{align*}"}
+{"id": "6060.png", "formula": "\\begin{align*} U ^ { ( l ) } : = ( U ^ { ( l - 1 ) } ) ^ { ( 1 ) } \\subset A ( n + l ) _ d \\end{align*}"}
+{"id": "4947.png", "formula": "\\begin{align*} \\begin{gathered} [ L ] = 2 \\beta ^ { 1 / 2 } ( e ^ { L } _ { + } - e ^ { L } _ { - } ) , \\\\ [ L ' ] = 2 \\beta ^ { 1 / 2 } ( e ^ { L ' } _ { + } - e ^ { L ' } _ { - } ) , \\end{gathered} \\end{align*}"}
+{"id": "2690.png", "formula": "\\begin{align*} P ^ { ( 2 ) } _ { i } = f _ { i } ( Q _ { ( 1 ) } ^ { j } , Q _ { ( 2 ) } ^ { j } ) . \\end{align*}"}
+{"id": "4536.png", "formula": "\\begin{align*} \\left \\{ \\begin{matrix*} [ l ] \\partial _ t m + \\displaystyle \\sum _ { i = 1 } ^ { N _ 1 } \\partial _ { s _ i } m + p _ { N _ 1 } ( [ s ] _ { N _ 1 } ) m = E _ { N _ 2 } ^ { ( N _ 1 ) } , \\\\ [ 5 p t ] m ( t , s _ 1 = 0 , [ s ] _ { 2 , N _ 1 } ) = \\int _ { u = 0 } ^ \\infty \\big [ p _ { N _ 1 } m - E _ { N _ 2 } ^ { ( N _ 1 ) } \\big ] ( t , [ s ] _ { 2 , N _ 1 } , u ) d u . \\end{matrix*} \\right . \\end{align*}"}
+{"id": "632.png", "formula": "\\begin{align*} n \\frac { \\sinh ^ { n - 1 } ( \\rho ) } { \\cosh ^ { r - 1 } ( \\rho ) } H _ r = \\frac { d } { d \\rho } \\left ( \\sinh ^ { n - r } ( \\rho ) \\frac { \\dot { \\lambda } ^ r } { ( 1 + \\dot { \\lambda } ^ 2 ) ^ { \\frac { r } { 2 } } } \\right ) , r = 1 , \\dots , n . \\end{align*}"}
+{"id": "4195.png", "formula": "\\begin{align*} \\begin{aligned} g _ 1 ( g _ 1 g _ 2 ) \\neq ( g _ 1 g _ 2 ) g _ 1 & \\mbox { a n d } g _ 1 g _ 3 \\neq g _ 3 g _ 1 \\ , , \\\\ \\varphi ( g _ 1 ( g _ 1 g _ 2 ) ) = \\varphi ( g _ 1 ) \\varphi ( g _ 1 g _ 2 ) & \\mbox { a n d } \\varphi ( g _ 1 g _ 3 ) = \\varphi ( g _ 3 ) \\varphi ( g _ 1 ) \\ , , \\end{aligned} \\end{align*}"}
+{"id": "6731.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 x ^ w e ^ { - \\mu x } d x = \\frac { w ! } { \\mu ^ { w + 1 } } - e ^ { - \\mu } \\frac { w ! } { \\mu ^ { w + 1 } } \\sum _ { j = 0 } ^ w \\frac { \\mu ^ j } { j ! } , \\end{align*}"}
+{"id": "8228.png", "formula": "\\begin{align*} \\mathbb { E } [ b ( { \\bf x } _ i ) ] & = \\mathbb { E } \\left [ \\mathbb { E } [ b ( { \\bf x } _ i ) | Y ^ { i - 1 } ] \\right ] \\\\ & = \\sum _ { \\bf s } \\sum _ { y ^ { i - 1 } } P _ { Y ^ { i - 1 } , { \\bf S } } ( y ^ { i - 1 } , { \\bf s } ) \\sum _ { { \\bf x } _ i } P _ { { \\bf X } _ i | Y ^ { i - 1 } } b ( { \\bf x } _ i ) , \\end{align*}"}
+{"id": "526.png", "formula": "\\begin{align*} & C _ 1 = \\langle x \\rangle y = y \\langle x \\rangle = \\{ y , y x , y x ^ 2 , \\ldots , y x ^ { p - 1 } \\} , C _ 2 = \\langle y \\rangle = \\{ 1 , y , y ^ 2 , \\ldots , y ^ { q - 1 } \\} , \\\\ & C _ 3 = \\langle y x \\rangle = \\{ 1 , y x , y ^ 2 x ^ { 1 + r } , y ^ 3 x ^ { 1 + r + r ^ 2 } , \\ldots , y ^ { q - 1 } x ^ { 1 + r + r ^ 2 + \\cdots + r ^ { q - 2 } } \\} . \\end{align*}"}
+{"id": "5989.png", "formula": "\\begin{align*} \\Delta r ^ 2 _ { \\partial M } = 2 r _ { \\partial M } \\Delta r _ { \\partial M } + 2 \\leq 2 ( n - 1 ) \\sqrt { K } r _ { \\partial M } + 2 \\leq C ( 1 + \\sqrt { K } r _ { \\partial M } ) . \\end{align*}"}
+{"id": "5656.png", "formula": "\\begin{align*} T _ { \\tilde { x } } Z \\oplus T _ { \\tilde { x } } \\varphi ^ { - 1 } ( R _ x ) = T _ { \\tilde { x } } E . \\end{align*}"}
+{"id": "5682.png", "formula": "\\begin{align*} I _ 1 : = & \\{ i \\in \\mathbb { N } \\ , : \\ , \\lambda _ i > 4 ^ { - 1 } m ^ 2 \\} , \\ I _ 2 : = \\{ i \\in \\mathbb { N } \\ , : \\ , \\lambda _ i = 4 ^ { - 1 } m ^ 2 \\} , \\\\ I _ 3 : = & \\{ i \\in \\mathbb { N } \\ , : \\ , \\lambda _ i = 0 \\} , \\ I _ 4 : = \\mathbb { N } \\setminus ( I _ 1 \\cup I _ 2 \\cup I _ 3 ) . \\end{align*}"}
+{"id": "7092.png", "formula": "\\begin{align*} ( \\lambda + \\partial _ t - \\Delta ) ^ { - \\frac { \\alpha } { 2 } } h ( t , x ) : = \\int _ { - \\infty } ^ t \\int _ { \\mathbb R ^ d } e ^ { - \\lambda ( t - s ) } \\frac { 1 } { ( 4 \\pi ( t - s ) ) ^ { \\frac { d } { 2 } } } \\frac { 1 } { ( t - s ) ^ { \\frac { 2 - \\alpha } { 2 } } } e ^ { - \\frac { | x - y | ^ 2 } { 4 ( t - s ) } } h ( s , y ) d s d y , \\end{align*}"}
+{"id": "3564.png", "formula": "\\begin{align*} a = \\lambda _ 1 + \\lambda _ 2 b = \\lambda _ 1 \\lambda _ 2 . \\end{align*}"}
+{"id": "7646.png", "formula": "\\begin{align*} \\lambda _ { \\mathrm { A n d } } = 2 \\norm { \\rho } _ { \\infty } \\mu _ d e \\ln \\left ( \\frac { \\lambda _ { \\mathrm { A n d } } } { 2 \\norm { \\rho } _ { \\infty } } \\right ) \\end{align*}"}
+{"id": "3739.png", "formula": "\\begin{align*} \\frac { 1 } { \\left ( 1 - t \\right ) ^ { \\nu } } = \\sum _ { k = 0 } ^ { \\infty } \\left ( \\nu \\right ) _ { k } \\frac { t ^ { k } } { k ! } , \\end{align*}"}
+{"id": "5052.png", "formula": "\\begin{align*} L _ S d _ W ( \\mathbb { \\hat { Q } } _ { K _ S } , \\mathbb { P } ) \\geq L _ S \\left ( \\mathbb { E } _ { \\mathbb { \\hat { Q } } _ { K _ S } } \\left [ \\frac { 1 } { L _ S } u _ S ( \\xi ) \\right ] - \\mathbb { E } _ { \\mathbb { P } } \\left [ \\frac { 1 } { L _ S } u _ S ( \\xi ) \\right ] \\right ) , \\end{align*}"}
+{"id": "3057.png", "formula": "\\begin{align*} \\sum _ { g \\in G ( \\Z ) / \\langle \\Gamma , H _ { v ' } ( \\Z ) \\rangle } \\phi ( g v ' ) [ H _ { g v ' } ( \\Z ) : \\Gamma _ { H _ { g v ' } } ] = \\sum _ { g \\in G ( \\Z / m \\Z ) / H _ { v ' } ( \\Z / m \\Z ) } \\phi ( g v ' ) \\times \\# H _ { g v ' } ( \\Z / m \\Z ) . \\end{align*}"}
+{"id": "6151.png", "formula": "\\begin{align*} 1 / \\tau ^ { k - 1 } = ( 1 - \\tau ^ k ) / \\tau ^ k , ~ \\tau ^ { - 1 } \\in ( 0 , 1 ) , \\end{align*}"}
+{"id": "2021.png", "formula": "\\begin{align*} \\left | ( \\rho - 1 ) a \\alpha ^ n - ( d _ 1 \\cdot \\rho ^ \\ell - ( d _ 1 - d _ 2 ) ) \\rho ^ { m + k } \\right | & = \\left | - ( \\rho - 1 ) \\Pi ( n ) - ( d _ 2 - d _ 3 ) \\cdot \\rho ^ k - d _ 3 \\right | \\\\ & \\leqslant ( \\rho - 1 ) \\cdot \\alpha ^ { - n / 2 } + ( \\rho - 1 ) \\cdot \\rho ^ k + \\rho - 1 \\\\ & < 2 \\cdot \\rho ^ { k + 1 } . \\end{align*}"}
+{"id": "6639.png", "formula": "\\begin{align*} M _ 1 = M _ 2 = \\begin{pmatrix} \\frac { 3 } { 2 } & - \\frac { 1 } { 2 } \\\\ \\frac { 1 } { 2 } & - \\frac { 3 } { 2 } \\end{pmatrix} , M _ 3 = M _ 4 = M _ 5 = M _ 6 = \\begin{pmatrix} 1 & 1 \\\\ 1 & - 1 \\end{pmatrix} , \\end{align*}"}
+{"id": "5893.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { C } ( G ) ) = ( p + 1 ) ( p n - n ) ( p n - n - 1 ) ^ { 2 } M _ { 2 } ( \\mathcal { C } ( G ) ) = \\dfrac { ( p + 1 ) ( p n - n ) ( p n - n - 1 ) ^ { 3 } } { 2 } . \\end{align*}"}
+{"id": "2595.png", "formula": "\\begin{align*} f ( t ) = c _ 1 t \\left [ c _ 2 t ^ { \\frac { 1 } { \\beta R } } + c _ 3 t ^ { \\frac { 2 } { \\beta R } } \\right ] , \\end{align*}"}
+{"id": "7526.png", "formula": "\\begin{align*} \\begin{dcases} \\partial _ t \\rho + \\nabla \\cdot ( \\rho u ) = 0 \\\\ \\partial _ t ( \\rho u ) + \\nabla \\cdot ( \\rho u \\otimes u ) + \\nabla p _ 1 ( \\rho ) = - \\rho \\nabla \\phi \\\\ \\partial _ t n + \\nabla \\cdot ( n v ) = 0 \\\\ \\varepsilon \\big ( \\partial _ t ( n v ) + \\nabla \\cdot ( n v \\otimes v ) \\big ) + \\nabla p _ 2 ( n ) = n \\nabla \\phi \\\\ - \\delta \\Delta \\phi = \\rho - n \\ \\end{dcases} \\end{align*}"}
+{"id": "8822.png", "formula": "\\begin{align*} f _ j ( x ) = \\sum _ { k = 1 } ^ { k _ j } \\lambda _ { j , m _ k } \\psi _ { j , m _ k } ( x ) . \\end{align*}"}
+{"id": "4306.png", "formula": "\\begin{align*} d y _ t = \\sum _ { j = 1 } ^ d V _ j ( y _ t ) d w _ t ^ j + V _ 0 ( y _ t ) d t , y _ 0 = 0 . \\end{align*}"}
+{"id": "4927.png", "formula": "\\begin{align*} [ L ] = 2 \\sqrt { \\beta _ { L } } ( e ^ { L } _ { + } - e ^ { L } _ { - } ) . \\end{align*}"}
+{"id": "7179.png", "formula": "\\begin{align*} \\partial _ t e ^ { - ( 1 - v _ 0 ^ 2 ) t } F _ v - \\partial _ x ^ 2 e ^ { - ( 1 - v _ 0 ^ 2 ) t } F _ v & = ( ( v _ 0 ^ 2 - ( v _ 0 + V ) ^ 2 ) + \\varphi _ { 1 } ) e ^ { - ( 1 - v _ 0 ^ 2 ) t } F _ v - e ^ { - ( 1 - v _ 0 ^ 2 ) t } F _ w + e ^ { - ( 1 - v _ 0 ^ 2 ) t } \\varphi _ 3 . \\end{align*}"}
+{"id": "5617.png", "formula": "\\begin{align*} a \\leq \\frac { 4 k s - 6 ( g + s ) } { 2 } + 6 ( g + s ) + 2 s ( 2 \\ell - 2 k ) = 2 s ( 2 \\ell - k ) + 3 ( g + s ) . \\end{align*}"}
+{"id": "8271.png", "formula": "\\begin{align*} f _ * ( N ( f ^ * \\alpha ) ) = g _ * ( N ( g ^ * \\alpha ) ) . \\end{align*}"}
+{"id": "4101.png", "formula": "\\begin{align*} & ( w _ { 1 j } , \\ldots , w _ { j - 1 , j } ) ' = ( I - P _ { j - 1 } ) ^ { - 1 } P _ { [ 1 : j - 1 ] , j } , \\\\ & ( w _ { j j } , \\ldots , w _ { J , j } ) ' = P _ { [ j : J ] , j } + P _ { [ j : J ] , [ 1 : j - 1 ] } ( I - P _ { j - 1 } ) ^ { - 1 } P _ { [ 1 : j - 1 ] , j } , \\end{align*}"}
+{"id": "9159.png", "formula": "\\begin{align*} G _ { \\beta , \\beta ' } = ( w _ { \\beta , 1 } - v ^ { ( 4 \\ell - 4 n - 2 ) } w _ { \\beta ' , 1 } ) \\cdot G _ { \\beta , [ i , n , \\ell + 1 ] } , \\end{align*}"}
+{"id": "1644.png", "formula": "\\begin{align*} v _ { T } \\left ( x \\right ) = u _ { T } ^ { \\left ( 1 \\right ) } \\left ( x \\right ) - u _ { T } ^ { \\left ( 2 \\right ) } \\left ( x \\right ) , p _ { T } \\left ( x \\right ) = m _ { T } ^ { \\left ( 1 \\right ) } \\left ( x \\right ) - m _ { T } ^ { \\left ( 2 \\right ) } \\left ( x \\right ) , x \\in \\Omega , \\end{align*}"}
+{"id": "6271.png", "formula": "\\begin{align*} f ( \\overline { x } _ T ) - f ( x ^ * ) \\leq \\frac 1 T \\sum \\limits _ { k = 0 } ^ { T - 1 } \\langle \\nabla \\hat { f } _ \\tau ( x _ { k } ) , x _ k - x ^ * \\rangle + 2 M _ 2 \\tau . \\end{align*}"}
+{"id": "7489.png", "formula": "\\begin{align*} \\mathcal { D } _ { \\gamma , \\Lambda , s } ^ { c o n e } = \\left \\{ ( p , t ) \\in \\{ r _ s \\leq \\tfrac { 3 } { 4 } \\} \\times [ 0 , ( 3 2 \\gamma ) ^ { - 1 } ] , r _ s ( p ) \\geq \\sqrt { \\gamma t + s \\Lambda ^ 2 } \\right \\} \\end{align*}"}
+{"id": "9119.png", "formula": "\\begin{align*} \\zeta _ { i j } ( z ) = \\frac { z - v ^ { - ( \\alpha _ { i } , \\alpha _ { j } ) } } { z - 1 } . \\end{align*}"}
+{"id": "5443.png", "formula": "\\begin{align*} r ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , i ) } = R _ i + \\beta \\sum _ { j \\in N } p _ { i j } f ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , j ) } - \\beta f ^ { S _ 0 \\oplus S _ 1 } _ { ( 0 , i ) } = R _ i + \\beta \\sum _ { j \\in S _ 1 } p _ { i j } f ^ { S _ 1 } _ j - \\beta ( f ^ { S _ 1 } _ i - c _ i ) = r ^ { S _ 1 } _ i + \\beta c _ i , \\end{align*}"}
+{"id": "1658.png", "formula": "\\begin{align*} c = c \\left ( T \\right ) = 2 + \\sqrt { T + \\frac { 1 } { 4 } } . \\end{align*}"}
+{"id": "6171.png", "formula": "\\begin{align*} \\begin{cases} \\hat { \\lambda } ^ k = \\lambda ^ k - ( 1 - \\tau ^ k ) \\beta ^ k ( A { x } ^ { k } - b ) , \\\\ \\hat { x } ^ k = x ^ k + \\frac { \\tau ^ k ( 1 - \\tau ^ { k - 1 } ) } { \\tau ^ { k - 1 } } ( x ^ k - x ^ { k - 1 } ) , \\\\ x ^ { k + 1 } \\in \\arg \\min \\limits _ x \\{ f ( x ) + x ^ T \\nabla _ x \\varphi ^ k ( \\hat { x } ^ k , \\hat { \\lambda } ^ k ) + \\frac { \\beta ^ k } { 2 } \\| x - \\hat { x } ^ k \\| _ D ^ 2 + \\frac { \\sigma ( 1 - \\tau ^ k ) } { 2 \\tau ^ k } \\| x - x ^ k \\| ^ 2 \\} . \\end{cases} \\end{align*}"}
+{"id": "3852.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\exp ( - s ) \\check \\Lambda ( e _ x \\mid s ) d s = \\gamma ^ * ( \\{ e _ x \\} \\times [ 0 , t ] ) = \\beta ^ * _ { ( 1 , 3 ) } ( \\{ e _ x \\} \\times [ 0 , t ] ) = \\int _ 0 ^ t \\exp ( - s ) \\eta ^ * _ { ( 1 ) } ( x \\mid s ) d s , \\end{align*}"}
+{"id": "7663.png", "formula": "\\begin{align*} K ( m , u ) = D _ { s , 1 } \\abs { \\lambda } ^ s \\abs { G _ 0 ( m , u ; z ) } ^ s , \\ , \\ , \\ , \\psi ( m ) = \\abs { G _ 0 ( m , n ; z ) } ^ s \\end{align*}"}
+{"id": "3869.png", "formula": "\\begin{align*} \\beta ^ j ( x , y ) = \\beta ^ j _ { ( 1 ) } ( x ) \\beta ^ j _ { 2 | 1 } ( y \\mid x ) , \\ ; x , y \\in \\Delta ^ o . \\end{align*}"}
+{"id": "3679.png", "formula": "\\begin{align*} L ^ { - } _ { e , e } = 2 . \\end{align*}"}
+{"id": "2054.png", "formula": "\\begin{align*} \\mathrm { d } U ( s , x ) = - \\alpha \\int _ 0 ^ 1 A ( x , y ) U ( s , y ) \\mathrm { d } y \\mathrm { d } s + \\alpha \\beta \\mathrm { d } \\xi _ 2 ( s , x ) + \\alpha \\zeta \\mathrm { d } \\xi _ 3 ( s , x ) , \\end{align*}"}
+{"id": "1857.png", "formula": "\\begin{align*} r ( x ) = \\sup \\{ r < D ( x ) : \\| F _ A \\| _ { L ^ { 7 / 2 } ( B _ r ( x ) ) } \\leq \\epsilon _ 4 \\} , \\end{align*}"}
+{"id": "6846.png", "formula": "\\begin{align*} \\mu : = e ^ 1 \\wedge \\ldots \\wedge e ^ { 2 ( N - 1 ) } . \\end{align*}"}
+{"id": "4801.png", "formula": "\\begin{align*} \\min _ { \\mathbf { x } \\in X } \\max _ { \\mathbf { c } \\in \\mathcal { C } } \\ell ( \\mathbf { x } , \\mathbf { c } ) = & \\min _ { \\mathbf { x } , t } t \\\\ \\mbox { s . t . } & \\max _ { \\mathbf { c } \\in \\mathcal { C } } \\Big ( \\ell ( \\mathbf { x } , \\mathbf { c } ) - t \\Big ) \\leq 0 \\\\ & \\mathbf { x } \\in X , \\end{align*}"}
+{"id": "581.png", "formula": "\\begin{align*} \\Psi : Z \\to X ^ \\ast , \\ ; \\ ; g \\to x _ g ^ \\ast , \\ ; \\ ; \\mbox { w i t h } \\ ; \\ ; x _ g ^ \\ast ( f ) = \\int \\limits _ 0 ^ 1 f ( x ) g ( x ) \\mathrm { d } x , \\ ; \\ ; \\mbox { f o r } \\ ; \\ ; f \\in X , \\end{align*}"}
+{"id": "1881.png", "formula": "\\begin{align*} \\alpha _ { k , l } ^ { ( n , p ) } = \\pi ^ n \\frac { \\Gamma \\left ( \\frac { p + k - l } { 2 } + 1 \\right ) \\Gamma \\left ( \\frac { p - k + l } { 2 } + 1 \\right ) } { \\Gamma \\left ( \\frac { p + k + l } { 2 } + n \\right ) \\Gamma \\left ( \\frac { p - k - l } { 2 } + 1 \\right ) } . \\end{align*}"}
+{"id": "6927.png", "formula": "\\begin{align*} u _ J = \\sum _ { j _ 0 = 1 } ^ J \\delta _ { j _ 0 } \\end{align*}"}
+{"id": "3382.png", "formula": "\\begin{align*} C _ { 1 } = \\frac { R _ { 1 } } { 2 4 \\gamma } ; C _ { 2 } = \\frac { \\gamma } { 2 6 \\sigma ^ { p } } ; C _ { 3 } = \\frac { \\gamma } { 2 6 T \\sigma ^ { p } } ; A = 3 \\gamma \\end{align*}"}
+{"id": "5247.png", "formula": "\\begin{align*} \\frac { \\partial Z _ { j } } { \\partial q _ { j } } = - \\left ( 1 - \\alpha \\right ) \\frac { p _ { j } } { \\left [ \\alpha p _ { j } + \\left ( 1 - \\alpha \\right ) q _ { j } \\right ] ^ { 2 } } \\end{align*}"}
+{"id": "3019.png", "formula": "\\begin{align*} \\begin{array} { l l l } ( \\overline { \\mathcal { L } } _ { \\xi } g ) _ { 1 1 } = ( \\overline { \\mathcal { L } } _ { \\xi } g ) _ { 2 2 } = - 2 \\alpha , \\ \\ & ( \\overline { \\mathcal { L } } _ { \\xi } g ) _ { 1 2 } = 2 ( 1 - \\beta ) . \\\\ \\end{array} \\end{align*}"}
+{"id": "274.png", "formula": "\\begin{align*} \\left ( A \\frac { d } { d x } - 3 A ' \\right ) \\left ( A \\frac { d } { d x } + \\gamma \\ , A - A ' \\right ) b = ( \\Delta _ \\R - 3 ) \\Delta _ \\R b = 0 , \\end{align*}"}
+{"id": "1546.png", "formula": "\\begin{align*} a ^ { - 1 } ( s ) = \\inf _ { | z | \\ge s } | D F ^ { - 1 } ( z ) | \\end{align*}"}
+{"id": "5651.png", "formula": "\\begin{align*} f ( Z ) = \\pi _ 2 ( \\pi _ 1 ^ { - 1 } ( Z ) \\cap \\Gamma _ f ) \\end{align*}"}
+{"id": "7219.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { V _ { \\min } } ^ { V _ { F } } \\left ( \\partial _ t p \\phi - h p \\partial _ v \\phi + a \\partial _ v p \\partial _ v \\phi \\right ) d v \\\\ + & \\left ( h p \\phi | _ { V _ { \\min } } ^ { V _ R ^ - } + h p \\phi | _ { V _ R ^ + } ^ { V _ F } \\right ) - \\left ( a \\partial _ v p \\phi | _ { V _ { \\min } } ^ { V _ R ^ - } + a \\partial _ v p \\phi | _ { V _ { V _ R ^ + } } ^ { V _ F } \\right ) = 0 . \\end{aligned} \\end{align*}"}
+{"id": "507.png", "formula": "\\begin{align*} \\gcd \\left ( \\alpha _ i ^ { L _ i } , s \\right ) = \\prod _ { p \\mid \\gcd ( \\alpha _ i , s ) } { p ^ { \\kappa _ p } } , \\end{align*}"}
+{"id": "2174.png", "formula": "\\begin{align*} f _ d ^ { - 1 } : ( 0 , d ) \\rightarrow \\mathbb R , \\ , \\ , \\ , \\ , \\ , \\ , f _ d ^ { - 1 } ( x ) = \\ln \\left ( \\frac { \\lambda ( e ^ { x } - 1 ) } { e ^ d - e ^ x } \\right ) . \\end{align*}"}
+{"id": "4042.png", "formula": "\\begin{align*} \\hat v ( \\cdot ) = \\sum _ { i = 1 } ^ M a _ i \\hat K ^ 2 _ { \\mathcal { S } } ( \\cdot , \\mathbb { Y } ^ i ) , \\end{align*}"}
+{"id": "2777.png", "formula": "\\begin{align*} \\delta \\left ( { \\sigma _ { \\tau } ^ { * } } ( t ) I \\right ) = \\int ^ { t _ { 2 } } _ { t _ { 1 } } \\left [ \\dot { Q } ^ { a } - \\frac { \\partial H _ { T } } { \\partial P _ { a } } \\right ] \\delta P _ { a } d t + \\left [ - \\dot { P } _ { a } - \\frac { \\partial H _ { T } } { \\partial Q ^ { a } } \\right ] \\delta Q ^ { a } d t + \\left [ P _ { a } \\delta Q ^ { a } \\right ] ^ { t _ { 2 } } _ { t _ { 1 } } . \\end{align*}"}
+{"id": "5430.png", "formula": "\\begin{align*} g ^ S _ i = \\begin{cases} 1 + \\beta \\sum _ { j \\in S } p _ { i j } g ^ S _ j & i \\in S \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "5732.png", "formula": "\\begin{align*} e ^ { - \\gamma ( t _ 2 - t _ 1 ) } X _ - ( t _ 2 ) \\leq & X _ - ( t _ 1 ) + \\int _ { t _ 1 } ^ { t _ 2 } e ^ { - \\gamma ( s - t _ 1 ) } Y _ - ( s ) \\ , d s \\\\ \\leq & X _ - ( t _ 1 ) + \\int _ { t _ 1 } ^ { \\infty } e ^ { - \\gamma ( s - t _ 1 ) } | Y _ - ( s ) | \\ , d s . \\end{align*}"}
+{"id": "1518.png", "formula": "\\begin{align*} { \\rm d i v } \\ , ( A ( x ) \\ , D v ) = 0 \\end{align*}"}
+{"id": "5785.png", "formula": "\\begin{align*} \\textsc { I I } = - D ^ 2 H ( z - \\bar { z } ) [ ( z - \\bar { z } ) ' , ( z - \\bar { z } ) ' ] - D H ( z - \\bar { z } ) [ ( z - \\bar { z } ) '' ] + m D H ( z - \\bar { z } ) [ ( z - \\bar { z } ) ' ] . \\end{align*}"}
+{"id": "1297.png", "formula": "\\begin{align*} \\boldsymbol { D } = \\begin{pmatrix} \\sigma _ 1 & 0 \\\\ 0 & \\sigma _ 2 \\end{pmatrix} \\ ; , \\end{align*}"}
+{"id": "8200.png", "formula": "\\begin{align*} & \\mathbb { E } [ b ( { \\bf X } _ { i } ) ] \\\\ & = \\mathbb { E } [ \\mathbb { E } [ b ( { \\bf X } _ { i } ) | Y ^ { i - 1 } ] ] \\\\ & = \\sum _ { y ^ { i } } b ( { \\bf x } _ { i } ) \\sum _ { { \\bf x } _ { i } } P _ { { \\bf X } _ { i } | Y ^ { i } } ( { \\bf x } _ { i } | y ^ { i - 1 } ) P _ { Y ^ { i - 1 } } ( y ^ { i - 1 } ) , \\end{align*}"}
+{"id": "2773.png", "formula": "\\begin{align*} \\sigma _ { 3 } ^ { ( 2 ) } : & T ^ { * } M | _ { \\Xi , \\Psi } \\times T ^ { * } M | _ { Q , P } \\rightarrow T ^ { * } M \\\\ ; & ( \\sigma _ { 3 } ^ { * } \\Xi ^ { a } , \\sigma _ { 3 } ^ { * } \\Psi _ { a } , \\sigma _ { 3 } ^ { * } \\Theta ^ { \\alpha } : = \\epsilon ^ { \\alpha } , \\sigma _ { 3 } ^ { * } \\Theta _ { \\alpha } : = \\epsilon _ { \\alpha } , \\sigma _ { 3 } ^ { * } Q ^ { i } , \\sigma _ { 3 } ^ { * } P _ { i } ) \\mapsto ( \\Xi ^ { a } , \\Psi _ { a } , \\Theta ^ { \\alpha } , \\Theta _ { \\alpha } , Q ^ { i } , P _ { i } ) . \\end{align*}"}
+{"id": "5432.png", "formula": "\\begin{align*} g ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , i ) } & = 1 + \\beta \\sum _ { j \\in S _ 1 } p _ { i j } g ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , j ) } + \\beta \\sum _ { j \\in N \\setminus S _ 1 } p _ { i j } g ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , j ) } = 1 + \\beta \\sum _ { j \\in S _ 1 } p _ { i j } g ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , j ) } , \\end{align*}"}
+{"id": "473.png", "formula": "\\begin{align*} \\sum _ { A \\supseteq B } ( A , h ) - \\sum _ { A \\supseteq B \\cup \\{ x _ 1 \\} } ( A , h ) = \\sum _ { \\substack { A \\supseteq B \\\\ A \\not \\ni x _ 1 } } ( A , h ) , \\end{align*}"}
+{"id": "5102.png", "formula": "\\begin{align*} \\mathcal { C } ( v ) & = \\bigoplus _ { R \\subset W } J _ { R } e _ { R } . \\end{align*}"}
+{"id": "2173.png", "formula": "\\begin{align*} f _ d ( x ) = \\ln \\left ( \\frac { e ^ { x + d } + \\lambda } { e ^ x + \\lambda } \\right ) . \\end{align*}"}
+{"id": "1617.png", "formula": "\\begin{align*} \\binom { ( i - \\tfrac { s } { 2 } ) ( p - 1 ) - 1 } { p i - r - \\tfrac { s ( p - 1 ) } { 2 } } = \\binom { i ( p - 1 ) - p ^ { t + 1 } } { r - i - 1 } \\equiv \\binom { i ( p - 1 ) } { r - i + p ^ { t + 1 } - 1 } \\bmod { p } . \\end{align*}"}
+{"id": "5465.png", "formula": "\\begin{align*} \\dot x = x ( e ^ { { k x } ^ 2 } - 1 ) + ( 4 - x ^ 2 ) u , x ( 0 ) = 0 , \\end{align*}"}
+{"id": "4984.png", "formula": "\\begin{align*} p _ i ( \\lambda ) = \\sum _ { j = 1 } ^ N p _ { i j } \\lambda ^ { j - 1 } , i = 1 , \\ldots , N . \\end{align*}"}
+{"id": "518.png", "formula": "\\begin{align*} O \\left ( \\sum _ { k = 0 } ^ { h - 1 } { 2 ^ { 2 ( k + 1 ) d + H _ i } ( ( k + 1 ) d + H _ i ) \\log ^ { 1 + o ( 1 ) } { q } } \\right ) \\subseteq O ( 2 ^ { 2 h d + H _ i } ( h d + H _ i ) \\log ^ { 1 + o ( 1 ) } { q } ) , \\end{align*}"}
+{"id": "991.png", "formula": "\\begin{align*} \\phi ( \\lambda ) = \\int _ 0 ^ \\infty ( 1 - e ^ { - \\lambda t } ) \\hat \\mu ( t ) \\ , d t , \\lambda > 0 , \\end{align*}"}
+{"id": "6441.png", "formula": "\\begin{align*} \\| f \\| ^ 2 = \\sum _ { n \\in \\Z } \\frac { | f ( x _ n ) | ^ 2 } { \\| k ^ \\Theta _ { x _ n } \\| ^ 2 } . \\end{align*}"}
+{"id": "8715.png", "formula": "\\begin{gather*} A n n _ { H } ^ { \\mathrm { l e f t } } ( M ) = \\{ a \\in H | \\ [ a , M ] = \\langle 0 \\rangle \\} , \\\\ A n n _ { H } ^ { \\mathrm { r i g h t } } ( M ) = \\{ a \\in H | \\ [ M , a ] = \\langle 0 \\rangle \\} . \\end{gather*}"}
+{"id": "176.png", "formula": "\\begin{align*} h _ x ( y ) \\ = \\ & x y ^ { - 1 } x \\ \\ h _ x ( x ) \\ = \\ x , \\\\ & x ^ { - 1 } h _ x ( y ) \\ = \\ y ^ { - 1 } x , \\\\ & h _ x \\circ h _ x = 1 _ G . \\end{align*}"}
+{"id": "8238.png", "formula": "\\begin{align*} C _ = \\frac { 1 } { L } \\sup _ { P _ { { \\bf X } ^ L \\| Y ^ { L - 1 } } } I ( { \\bf X } ^ L \\to Y ^ L ) , \\end{align*}"}
+{"id": "2849.png", "formula": "\\begin{align*} \\| f \\| _ { \\dot { W } ^ { s , p } ( \\mathbb { R } ^ n ) } : & = \\left [ \\int _ { \\mathbb { R } ^ n } \\int _ { \\mathbb { R } ^ n } \\frac { | f ( x ) - f ( y ) | ^ p } { | x - y | ^ { s p + n } } \\ , d x \\ , d y \\right ] ^ { \\frac { 1 } { p } } \\\\ & = : \\left \\| \\frac { f ( x ) - f ( y ) } { | x - y | ^ { s + \\frac { n } { p } } } \\right \\| _ { L ^ p ( \\mathbb { R } ^ n \\times \\mathbb { R } ^ n ) } . \\end{align*}"}
+{"id": "5910.png", "formula": "\\begin{align*} \\dfrac { M _ { 1 } ( \\mathcal { C } ( G ) ) } { | v ( \\mathcal { C } ( G ) ) | } = \\dfrac { ( q - 1 ) ( q - 2 ) ^ { 2 } + q ( p - 1 ) ( p - 2 ) ^ { 2 } } { p q - 1 } \\end{align*}"}
+{"id": "590.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { j - 1 } \\Big | \\Big \\langle Y _ i , { \\frac { 1 _ { \\Delta ^ \\ast _ j } r ^ \\theta _ m } { \\| h _ j \\| _ { F ^ \\ast } } } \\Big \\rangle \\Big | = 0 , \\end{align*}"}
+{"id": "1095.png", "formula": "\\begin{align*} d _ p ( w ) : = \\limsup _ { i \\to \\infty } \\frac { 1 } { i } \\log _ 2 a _ i , \\end{align*}"}
+{"id": "22.png", "formula": "\\begin{align*} \\lim _ { \\kappa \\to \\infty } \\ , \\sup _ { q \\in Q _ * } \\ , \\sup _ { | t | \\leq T } \\norm { n ( t ) - n ( 0 ) } _ { H ^ { - 2 } } = 0 . \\end{align*}"}
+{"id": "8762.png", "formula": "\\begin{align*} 1 - F _ { \\tilde \\eta } ( F _ \\eta ^ { - 1 } ( 1 - \\hat \\eta ( \\R ) u ) ) & = \\tilde \\eta ( ( F _ \\eta ^ { - 1 } ( 1 - \\hat \\eta ( \\R ) u ) , + \\infty ) ) \\le \\frac { \\eta ( ( F _ \\eta ^ { - 1 } ( 1 - \\hat \\eta ( \\R ) u ) , + \\infty ) ) } { \\hat \\eta ( \\R ) } \\\\ & = \\frac { 1 - F _ \\eta ( F _ \\eta ^ { - 1 } ( 1 - \\hat \\eta ( \\R ) u ) ) } { \\hat \\eta ( \\R ) } \\le u \\end{align*}"}
+{"id": "250.png", "formula": "\\begin{align*} \\frac { d } { d x } ( A _ 0 ^ { - 3 } z ) = A _ 0 ^ { - 3 } z ' - 3 A _ 0 ^ { - 4 } A ' _ 0 z = A _ 0 ^ { - 4 } ( A _ 0 z ' - 3 A ' _ 0 z ) , \\end{align*}"}
+{"id": "1571.png", "formula": "\\begin{align*} \\varphi ( r ) = \\begin{cases} 1 & \\\\ [ 5 p t ] \\cos \\left ( \\dfrac { \\pi } { 2 } \\ , \\dfrac { r - \\rho } { R - \\rho } \\right ) & , \\end{cases} \\end{align*}"}
+{"id": "5353.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathcal { F } ^ * } w ^ { S ^ * } _ j \\ , x _ { i j } ^ u & = b _ i ^ u + \\sum _ { j \\in S } w ^ S _ j \\ , x _ { i j } ^ { 0 , u } \\\\ & = b _ i ^ S + \\sum _ { j \\in N ^ { \\{ 0 , 1 \\} } \\setminus S } w ^ { S } _ j \\ , x _ { i j } ^ { 1 , u } \\\\ & \\geq b _ i ^ S = b _ i ^ { S ^ * } , \\end{align*}"}
+{"id": "341.png", "formula": "\\begin{align*} \\begin{aligned} I ^ { [ k ] } \\ & = \\ I _ 0 ^ { [ k ] } + \\sum _ { \\substack { p \\in [ s ] \\\\ \\nu ( I _ p ) \\ge k - 1 } } u _ p I _ p ^ { [ k - 1 ] } , & & & I ^ { [ \\ell ] } \\ & = \\ I _ 0 ^ { [ \\ell ] } + \\sum _ { \\substack { q \\in [ s ] \\\\ \\nu ( I _ q ) \\ge \\ell - 1 } } u _ q I _ q ^ { [ \\ell - 1 ] } . \\end{aligned} \\end{align*}"}
+{"id": "6667.png", "formula": "\\begin{align*} \\abs { T _ s } ^ 2 _ t & = 2 \\langle ( T _ s ) _ t , T _ s \\rangle \\\\ & = 2 \\langle ( T _ t ) _ s + \\abs { T _ s } ^ 2 T _ s , T _ s \\rangle \\\\ & = 2 \\langle ( T _ { s s } + \\abs { T _ s } ^ 2 T ) _ s , T _ s \\rangle + 2 \\abs { T _ s } ^ 4 \\\\ & = 2 \\langle T _ { s s s } , T _ s \\rangle + 2 \\abs { T _ s } ^ 2 _ s \\langle T , T _ s \\rangle + 4 \\abs { T _ s } ^ 4 \\\\ & = \\abs { T _ s } ^ 2 _ { s s } - 2 \\abs { T _ { s s } } ^ 2 + 4 \\abs { T _ s } ^ 4 , \\end{align*}"}
+{"id": "555.png", "formula": "\\begin{align*} E ( t ) = \\prod ( 1 + x _ i t ) , \\ ; H ( t ) = E ( - t ) ^ { - 1 } \\end{align*}"}
+{"id": "8942.png", "formula": "\\begin{align*} \\Vert F \\Vert _ { L ^ p ( \\Omega ) } \\le C \\Vert f \\Vert _ { L ^ p ( \\Omega ) } , \\ \\ \\Vert G \\Vert _ { L ^ q ( \\Omega ) } \\le C \\Vert g \\Vert _ { L ^ q ( \\Omega ) } . \\end{align*}"}
+{"id": "947.png", "formula": "\\begin{align*} R \\mu ( x ) = \\mathbb E _ x A ^ { \\mu } _ \\infty , R ^ V \\mu ( x ) = \\mathbb E _ x A ^ { \\mu } _ { \\tau _ V } , x \\in E \\setminus N , \\end{align*}"}
+{"id": "1766.png", "formula": "\\begin{align*} \\pi _ r \\circ \\mathrm { F l } ^ { Y _ { - 1 } ^ \\prime } _ { t } ( z ) = z _ r + \\delta _ { r , n } i t , \\pi _ r \\circ \\mathrm { F l } ^ { Y _ { - j } } _ { t } ( z ) = z _ r + \\delta _ { r , j } i t \\quad \\quad \\forall \\ , j \\in \\{ 1 , \\ldots , n \\} , \\end{align*}"}
+{"id": "5747.png", "formula": "\\begin{align*} \\Vert q ( t ) - e ^ { \\gamma _ * t } \\sum _ { j = 1 } ^ N a _ j \\psi ^ + _ { i + j - 1 } \\Vert _ G = O ( e ^ { ( \\gamma _ * - \\varepsilon ) t } ) . \\end{align*}"}
+{"id": "5400.png", "formula": "\\begin{align*} h _ k ( j _ k ) = c _ k \\ , j _ k ^ 2 + s _ k \\ , \\lambda _ k \\ , 1 \\{ j _ k = 0 \\} - r _ k \\ , \\lambda _ k \\ , 1 \\{ j _ k > 0 \\} . \\end{align*}"}
+{"id": "6307.png", "formula": "\\begin{align*} \\widetilde { K } ( x , y ) = - \\sum _ { z \\in \\mathbb { Z } ^ 3 } n \\omega _ { \\ell , \\lambda } ( P _ z ( x ) - y ) . \\end{align*}"}
+{"id": "7402.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } a _ { n + 1 } = ( p + q ) a _ n + 1 , & a _ 1 = 0 , \\\\ b _ { n + 1 } = q b _ n + p c _ n + 1 , & b _ 1 = 0 , \\\\ c _ { n + 1 } = q b _ n + p c _ n , & c _ 1 = 0 \\end{array} \\right . \\end{align*}"}
+{"id": "2530.png", "formula": "\\begin{align*} \\langle A \\delta _ L ^ u , v \\rangle = \\langle \\delta _ L ^ u , A ^ t v \\rangle = \\langle u , ( A ^ t v ) | _ L \\rangle = \\langle u , ( A ^ t ) | _ L ( v | _ L ) \\rangle = \\langle A ' u , v | _ L \\rangle = \\langle \\delta _ L ^ { A ' u } , v \\rangle \\ ; . \\end{align*}"}
+{"id": "6529.png", "formula": "\\begin{align*} g ^ { \\leftarrow } ( x ) = \\inf \\{ s > A : g ( s ) \\geq x \\} \\end{align*}"}
+{"id": "58.png", "formula": "\\begin{align*} \\prescript { } { i _ 1 } { ( \\prescript { } { i _ 0 } { \\hat { s } } ) } ( i ) = & \\begin{cases} \\prescript { } { i _ 0 } { \\hat { s } } ( i ) & i < i _ 1 \\\\ \\prescript { } { i _ 0 } { \\hat { s } } ( i + 1 ) & i _ 1 \\leq i \\end{cases} \\\\ = & \\begin{cases} \\hat { s } ( i ) & i < i _ 1 \\\\ \\hat { s } ( i + 1 ) & i _ 1 \\leq i \\leq i _ 0 - 2 \\\\ \\hat { s } ( i + 2 ) & i _ 0 - 1 \\leq i \\end{cases} \\end{align*}"}
+{"id": "2243.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 L } { \\partial v ^ j _ \\beta \\partial v ^ i _ \\alpha } \\Gamma _ { \\alpha \\beta } ^ j + \\frac { \\partial ^ 2 L } { \\partial q ^ j \\partial v ^ i _ \\alpha } v _ \\alpha ^ j + \\frac { \\partial ^ 2 L } { \\partial s ^ \\beta \\partial v ^ i _ \\alpha } \\Gamma _ \\alpha ^ \\beta \\displaystyle - \\frac { \\partial L } { \\partial q ^ i } = \\frac { \\partial L } { \\partial s ^ \\alpha } \\frac { \\partial L } { \\partial v ^ i _ \\alpha } \\ , . \\end{align*}"}
+{"id": "7897.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { p - 1 } R _ { \\mathbf { s } _ { n } } ( \\tau ) = 0 . \\end{align*}"}
+{"id": "3583.png", "formula": "\\begin{align*} ( 1 + a ) \\alpha ^ 2 \\tau _ 0 \\tau _ 1 ( \\tau - i d ) + b ( \\tau - i d ) = 0 . \\end{align*}"}
+{"id": "3323.png", "formula": "\\begin{align*} \\begin{cases} & d V _ { t } = \\sqrt { 2 } d W _ { t } - V _ t d t , \\\\ & d X _ { t } = V _ t d t , \\end{cases} \\end{align*}"}
+{"id": "1931.png", "formula": "\\begin{align*} K _ { s q } ( x , y ) = \\frac { a ( x , y ) } { | x - y | ^ { N + s q } } \\end{align*}"}
+{"id": "7616.png", "formula": "\\begin{align*} X ^ { \\wedge j } = X \\wedge \\cdots \\wedge X \\mbox { ( $ j $ - t i m e s ) } \\end{align*}"}
+{"id": "2547.png", "formula": "\\begin{align*} \\dot \\AA ^ m ( M ) = I ^ m _ M ( \\breve M , \\partial M ) \\subset I ^ m ( \\breve M , \\partial M ) \\quad ( m \\in \\R ) \\ ; , \\end{align*}"}
+{"id": "242.png", "formula": "\\begin{align*} \\bar { A } ( \\bar { x } ) \\bar { v } + \\bar { b } ( \\bar { x } ) = \\frac { c } { h ( x ) } \\left ( A ( x ) v + b ( x ) \\right ) . \\end{align*}"}
+{"id": "3862.png", "formula": "\\begin{align*} \\hat M ^ 2 ( t ) = M _ 1 ( T ) + \\int _ 0 ^ t \\hat \\eta _ { ( 1 ) } ^ 2 ( s ) d s - \\int _ 0 ^ t \\hat M ^ { 2 } ( s ) d s , \\ ; t \\in [ 0 , T ] , \\ ; \\ ; \\hat M ^ 2 ( T ) = m _ 2 . \\end{align*}"}
+{"id": "7218.png", "formula": "\\begin{align*} \\int _ { V _ { \\min } } ^ { V _ { F } } \\left ( \\partial _ { t } p + \\partial _ { v } ( h p ) - a \\partial _ { v v } p \\right ) \\phi d v = 0 . \\end{align*}"}
+{"id": "3276.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } ( - 1 ) ^ k { n \\brack k } q ^ { \\binom { n - k } { 2 } + j k } = 0 \\quad 0 \\leq j \\leq n - 1 , \\end{align*}"}
+{"id": "4629.png", "formula": "\\begin{align*} \\Omega _ { n , k } : = \\left \\{ x \\in \\Omega \\bigm | x ' \\in Q ( n , k ) , \\ x _ d \\in \\left ( f _ { n - 1 } ( x ' ) , f ( x ' ) \\right ) \\right \\} . \\end{align*}"}
+{"id": "6173.png", "formula": "\\begin{align*} & \\frac { 1 - \\tau ^ k } { ( \\tau ^ k ) ^ 2 } \\| \\breve { x } ^ { k } - \\breve { x } ^ { k - 1 } \\| ^ 2 + \\frac { 1 } { \\tau ^ k } \\| \\breve { x } ^ { k } - x \\| ^ 2 = \\| \\widetilde { x } ^ { k } - x \\| ^ 2 + \\frac { 1 - \\tau ^ k } { \\tau ^ k } \\| \\breve { x } ^ { k - 1 } - x \\| ^ 2 , ~ \\\\ & 2 ( \\widetilde { x } ^ k - x ) ^ T \\frac { 1 } { \\tau ^ k } ( \\breve { x } ^ { k } - \\breve { x } ^ { k - 1 } ) = \\| \\widetilde { x } ^ k - x \\| ^ 2 + \\frac { 1 } { ( \\tau ^ k ) ^ 2 } \\| \\breve { x } ^ { k } - \\breve { x } ^ { k - 1 } \\| ^ 2 - \\| \\breve { x } ^ { k - 1 } - x \\| ^ 2 . \\end{align*}"}
+{"id": "8720.png", "formula": "\\begin{gather*} f ( a _ { 1 } ) = \\alpha _ { 1 } a _ { 1 } + \\alpha _ { 2 } a _ { 2 } + \\alpha _ { 3 } a _ { 3 } , \\\\ f ( a _ { 2 } ) = \\beta _ { 2 } a _ { 2 } + \\beta _ { 3 } a _ { 3 } . \\end{gather*}"}
+{"id": "690.png", "formula": "\\begin{align*} F ( \\omega ) & = ( c _ 0 + 2 c _ 2 + 4 c _ 4 + \\ldots ) + \\sqrt { 2 } ( c _ 1 + 2 c _ 3 + 4 c _ 5 + \\cdots ) , \\\\ G ( \\omega ) & = ( d _ 0 + 2 d _ 2 + 4 d _ 4 + \\ldots ) + \\sqrt { 2 } ( d _ 1 + 2 d _ 3 + 4 d _ 5 + \\cdots ) , \\end{align*}"}
+{"id": "2332.png", "formula": "\\begin{align*} x = a + N ( x , x ) . \\end{align*}"}
+{"id": "1471.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { | \\partial _ { M ^ 2 } F _ n | } { | F _ n | } = 0 \\lim _ { n \\to \\infty } | F _ n | = \\infty \\end{align*}"}
+{"id": "4939.png", "formula": "\\begin{align*} e ^ { S _ j } _ { + } = e ^ { S _ { j + 1 } } _ { - } . \\end{align*}"}
+{"id": "3663.png", "formula": "\\begin{align*} x _ z ^ T L ( G , p ) x _ z = \\sum _ { i j \\in E } ( y _ i - y _ j ) ^ 2 \\left \\langle z , d _ { i j } \\right \\rangle ^ 2 . \\end{align*}"}
+{"id": "8840.png", "formula": "\\begin{align*} \\chi _ a ( x ) = \\left \\{ \\begin{array} { l l } 1 & x > a , \\\\ x - a + 1 & a - 1 \\leq x \\leq a , \\\\ 0 , & x < a - 1 . \\end{array} \\right . \\end{align*}"}
+{"id": "7427.png", "formula": "\\begin{align*} \\abs { x - y } & = \\sum _ { \\ell = j } ^ { j ^ \\prime } \\abs { I _ { i _ 1 , \\ldots , i _ k , \\ell } } = \\sum _ { \\ell = j } ^ { j ^ \\prime } \\frac { 1 } { \\abs { i _ 1 } ^ { p _ 1 } + \\cdots + \\abs { i _ k } ^ { p _ k } + \\abs { \\ell } ^ r } \\\\ & \\succ \\sum _ { \\ell = j } ^ { j ^ \\prime } \\frac { 1 } { \\abs { \\ell } ^ r } \\succ \\int _ { \\ell = j } ^ { j ^ \\prime } \\frac { 1 } { x ^ r } d x \\succ \\frac { 1 } { \\abs { j } ^ { r - 1 } } , \\end{align*}"}
+{"id": "2407.png", "formula": "\\begin{align*} d _ n = e ^ { ( 1 + o ( 1 ) ) n } \\quad n \\rightarrow \\infty . \\end{align*}"}
+{"id": "6807.png", "formula": "\\begin{align*} h _ { S U ( 2 ) } ( e ^ { \\phi _ 1 \\lambda _ 3 } e ^ { \\psi _ 1 \\lambda _ 1 } e ^ { \\omega _ 1 \\lambda _ 3 } ) = \\sum _ { j = 1 } ^ M c _ j e ^ { i k _ j \\phi _ 1 } \\sin ^ { m _ j } ( \\psi _ 1 ) e ^ { i l _ j \\omega _ 1 } + c _ j ' e ^ { i k _ j ' \\phi _ 1 } \\sin ^ { m _ j ' } ( \\psi _ 1 ) \\cos ( \\psi _ 1 ) e ^ { i l _ j ' \\omega _ 1 } . \\end{align*}"}
+{"id": "3622.png", "formula": "\\begin{align*} R f ( z , w ) = \\frac { 1 } { \\lambda _ { 2 } ^ { 2 } - 1 } ( f ( z , - w ) - f ( z , w ) ) , \\end{align*}"}
+{"id": "5412.png", "formula": "\\begin{align*} \\pi _ 1 \\circ \\pi _ 0 ^ { - 1 } ( k ) = d + 1 - k , \\quad k \\in \\{ 1 , \\ldots , d \\} . \\end{align*}"}
+{"id": "1890.png", "formula": "\\begin{align*} d ^ 2 F ( u ) u = 0 d ^ 2 F ( u ) i u = i \\nabla F ( u ) . \\end{align*}"}
+{"id": "6091.png", "formula": "\\begin{align*} \\Delta ( 1 ) : = 1 \\otimes 1 , \\quad \\quad \\Delta ( x _ 1 ) : = 1 \\otimes x _ 1 + x _ 1 \\otimes 1 . \\end{align*}"}
+{"id": "3069.png", "formula": "\\begin{align*} \\varphi _ { i } ( t ) = \\left ( t ^ { \\frac { \\beta _ 0 } { e _ { i - 1 } } } , \\eta _ i ( t ) \\right ) : = \\left ( t ^ { \\frac { \\beta _ 0 } { e _ { i - 1 } } } , \\sum _ { \\beta _ 1 \\leq k < \\beta _ i } c _ k t ^ { \\frac { k } { e _ { i - 1 } } } \\right ) , \\end{align*}"}
+{"id": "9080.png", "formula": "\\begin{align*} \\begin{array} { c c } \\Big ( \\hat { C } ^ { - 1 } ( \\hat { C } C \\hat { C } ^ { - 1 } - I ) \\underline { v } + \\hat { C } ^ { - 1 } \\hat { C } \\Big ( D p - U ( J ^ { ( r ) } X ) - \\mathbf { 1 } \\frac { \\beta } { 2 } \\Big ) \\Big ) _ i \\\\ = \\Big ( ( C \\hat { C } ^ { - 1 } - \\hat { C } ^ { - 1 } ) \\underline { v } + D p - U ( J ^ { ( r ) } X ) - \\mathbf { 1 } \\frac { \\beta } { 2 } \\Big ) _ i \\geq 0 . \\end{array} \\end{align*}"}
+{"id": "1340.png", "formula": "\\begin{align*} p _ 1 x _ k & = \\alpha d _ k x _ k + ( 1 - \\alpha ) \\sum _ { k \\sim j } x _ j \\\\ & \\geq \\alpha \\delta x _ k + ( 1 - \\alpha ) \\delta x _ k = \\delta x _ k . \\end{align*}"}
+{"id": "8742.png", "formula": "\\begin{align*} \\widetilde \\Gamma = \\Gamma \\setminus \\bigcup _ { x \\in A } \\{ x \\} \\times \\left \\{ \\Gamma _ x \\cap \\{ ( - \\infty , x ) \\cup ( x , + \\infty ) \\} \\right \\} . \\end{align*}"}
+{"id": "1524.png", "formula": "\\begin{align*} \\{ F ( D u ) > k \\} = \\{ G ( V ) > k \\} = \\{ g _ k ( V ) > 1 \\} \\end{align*}"}
+{"id": "3181.png", "formula": "\\begin{align*} & 2 \\sin ( x ) \\left ( \\frac { 1 } { 2 } + \\sum _ { k = 1 } ^ n \\cos ( 2 k x ) \\right ) \\\\ [ 5 p t ] & = \\sin ( x ) + 2 \\sin ( x ) \\cos ( 2 x ) + 2 \\sin ( x ) \\cos ( 4 x ) + \\cdots + 2 \\sin ( x ) \\cos ( 2 n x ) \\\\ [ 5 p t ] & \\overset { \\eqref { e q - 2 . 3 } } { = } \\sin ( x ) + \\left ( \\sin ( 3 x ) - \\sin ( x ) \\right ) + \\left ( \\sin ( 5 x ) - \\sin ( 3 x ) \\right ) \\\\ [ 5 p t ] & \\quad + \\cdots + \\left ( \\sin ( ( 2 n + 1 ) x ) - \\sin ( ( 2 n - 1 ) x ) \\right ) \\\\ [ 5 p t ] & = \\sin ( ( 2 n + 1 ) x ) . \\end{align*}"}
+{"id": "2550.png", "formula": "\\begin{align*} \\dot \\AA ( M ) \\cap \\AA ' ( M ) = C ^ \\infty ( M ) \\ ; . \\end{align*}"}
+{"id": "8548.png", "formula": "\\begin{align*} F ( n , k ) = \\frac { ( a ) _ k ( b ) _ k } { ( n ) _ k ^ 2 } \\end{align*}"}
+{"id": "9169.png", "formula": "\\begin{align*} \\prod _ { \\beta = [ i , n , j ] \\in \\Delta ^ { + } } \\prod _ { \\ell = j } ^ { n - 1 } \\big \\{ ( v ^ { - 4 n + 4 \\ell - 2 } - 1 ) ^ { d _ { \\beta } } ( v ^ { - 4 n + 4 \\ell + 6 } - 1 ) ^ { d _ { \\beta } } \\big \\} . \\end{align*}"}
+{"id": "5878.png", "formula": "\\begin{align*} \\dfrac { M _ { 1 } ( \\mathcal { N C } ( G ) ) } { | v ( \\mathcal { N C } ( G ) ) | } = \\dfrac { 8 n ( 4 0 n ^ { 2 } - 7 2 n + 3 2 ) ) } { 8 n - 4 } \\dfrac { M _ { 2 } ( \\mathcal { N C } ( G ) ) } { | e ( \\mathcal { N C } ( G ) ) | } = \\dfrac { 2 n ( 5 1 2 n ^ { 3 } - 1 2 8 0 n ^ { 2 } + 1 0 2 4 n - 2 5 6 ) } { 2 4 n ( n - 1 ) } . \\end{align*}"}
+{"id": "5116.png", "formula": "\\begin{align*} \\frac { d f ( \\alpha ) } { d ( \\alpha : t ) } = \\frac { f ( \\alpha t ) - f ( \\alpha t ^ { - 1 } ) } { \\alpha t - \\alpha t ^ { - 1 } } \\end{align*}"}
+{"id": "7621.png", "formula": "\\begin{align*} f ( z ) = z ^ d + \\sum _ { k = 1 } ^ d a _ k z ^ { d - k } \\mapsto ( a _ 1 , \\cdots , a _ d ) . \\end{align*}"}
+{"id": "3987.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } \\gamma _ { i p k } = \\sum _ { r = 1 } ^ { \\nu } \\widetilde { \\gamma } _ { i p r } = \\tfrac { 1 } { 2 } , \\quad \\left ( 1 \\leq i \\leq n , 1 \\leq p \\leq \\nu \\right ) . \\end{align*}"}
+{"id": "365.png", "formula": "\\begin{align*} x - a = y - b = z - ( a + b ) = x + y - z . \\end{align*}"}
+{"id": "1226.png", "formula": "\\begin{align*} \\mbox { i t e r a t i v e l y s o l v e $ H ( X _ { i - 1 } ) X _ i = X _ i \\Omega _ i $ f o r $ X _ i $ } , \\end{align*}"}
+{"id": "4878.png", "formula": "\\begin{align*} & h ^ 0 ( \\Sigma ' , \\O _ { \\Sigma ' } ( K _ { \\Sigma ' } ) ) = h ^ 1 ( \\Sigma ' , \\O _ { \\Sigma ' } ( K _ { \\Sigma ' } + C ' ) ) = 0 , \\\\ & h ^ 1 ( \\Sigma ' , \\O _ { \\Sigma ' } ( K _ { \\Sigma ' } ) ) = q , \\\\ & h ^ 0 ( C ' , \\O _ { C ' } ( K _ { C ' } ) ) = g , \\end{align*}"}
+{"id": "4278.png", "formula": "\\begin{align*} \\Delta _ f : = e ^ f \\div ( e ^ { - f } \\d ) = \\Delta - \\nabla f \\cdot \\nabla , \\end{align*}"}
+{"id": "3984.png", "formula": "\\begin{align*} x ^ { 3 } = 2 \\sum _ { j = 1 } ^ { n } \\Theta _ { j } e _ { j } + 2 \\sum _ { u = 1 } ^ { \\nu } \\widetilde { \\Theta } _ { u } \\widetilde { e } _ { u } \\end{align*}"}
+{"id": "7550.png", "formula": "\\begin{align*} - \\int _ 0 ^ T \\int _ \\Omega \\big ( \\tfrac { 1 } { 2 } \\bar \\rho | \\bar u | ^ 2 + h _ 1 ( \\bar \\rho ) + h _ 2 ( \\bar \\rho ) \\big ) \\dot { \\theta } ( t ) \\ d x d t = \\int _ \\Omega \\big ( \\tfrac { 1 } { 2 } \\bar \\rho | \\bar u | ^ 2 + h _ 1 ( \\bar \\rho ) + h _ 2 ( \\bar \\rho ) \\big ) \\big | _ { t = 0 } \\theta ( 0 ) \\ d x \\ , \\end{align*}"}
+{"id": "3196.png", "formula": "\\begin{align*} \\{ | x _ 1 | = 1 \\} \\cup \\{ | x _ 2 | = 1 \\} . \\end{align*}"}
+{"id": "3667.png", "formula": "\\begin{align*} \\mathbb { E } _ z [ x _ z ^ T L ( K _ n , p ) x _ z ] = \\frac 1 d \\sum _ { i < j } ( y _ i - y _ j ) ^ 2 = n / d . \\end{align*}"}
+{"id": "6306.png", "formula": "\\begin{align*} P _ z : \\Lambda \\to \\Lambda + z , ( P _ z ( x ) ) _ i = ( - 1 ) ^ { z _ i } x _ i + z _ i , \\end{align*}"}
+{"id": "1094.png", "formula": "\\begin{align*} d _ p ( W ) : = \\limsup _ { i \\to \\infty } \\frac { 1 } { i } \\log _ 2 a _ i , \\end{align*}"}
+{"id": "3542.png", "formula": "\\begin{align*} T f ( z ) e ^ { 3 i \\theta } z = f ( e ^ { i \\theta } z ) e ^ { 3 i \\theta } z , \\end{align*}"}
+{"id": "1899.png", "formula": "\\begin{align*} \\begin{aligned} & V \\in \\mathfrak F , \\\\ & V ( x ) \\geq 0 x \\in G , \\\\ & V ( x ) \\geq c _ 0 \\ , d ^ { - \\alpha } ( x , x _ 0 ) \\textrm { f o r a l l } \\ ; \\ ; x \\in G \\setminus B _ { R _ 0 } ( x _ 0 ) \\ , . \\end{aligned} \\end{align*}"}
+{"id": "1720.png", "formula": "\\begin{align*} P _ { f , n } ( x _ 1 , \\ldots , x _ n ) : = f ( x _ 1 ) + \\sum _ { j + k \\leq n + 1 } x _ j x _ k \\left ( \\Re \\xi \\right ) ^ { n + 1 - j - k } . \\end{align*}"}
+{"id": "3949.png", "formula": "\\begin{align*} & \\mathcal { M } _ { N , \\gamma } = \\operatorname { d i a g } \\left \\{ \\Gamma T _ j ^ { - \\top } T _ { j } ^ { - 1 } \\Gamma \\right \\} _ { j = 1 } ^ N \\prec \\gamma ^ 6 \\sigma _ N ^ 2 I _ { 3 N } , \\end{align*}"}
+{"id": "4853.png", "formula": "\\begin{align*} A \\cap B _ { r _ j } ( y _ j ) = \\{ x \\in B _ { r _ j } ( y ) : x _ n < u ( x _ 1 , \\ldots , x _ { n - 1 } ) \\} . \\end{align*}"}
+{"id": "324.png", "formula": "\\begin{align*} u _ { \\epsilon , 0 } ( x ) = \\left \\{ \\begin{array} { l l } u _ 0 ( x ) , & { \\rm f o r } \\ x \\in ( - \\infty , R _ { \\epsilon } ) , \\\\ \\frac { \\epsilon ( R _ 0 + r - x ) } { R _ 0 + r - R _ { \\epsilon } } , & \\rm { f o r } \\ x \\in [ R _ { \\epsilon } , R _ 0 + r ] , \\\\ 0 , & { \\rm f o r } \\ x \\in ( R _ 0 + r , \\infty ) , \\end{array} \\right . \\end{align*}"}
+{"id": "6272.png", "formula": "\\begin{align*} \\frac 1 T \\sum \\limits _ { k = 0 } ^ { T - 1 } \\langle \\hat { g } _ { k + 1 } , x _ k - x ^ * \\rangle \\leq \\frac 1 2 \\frac { R _ 0 ^ { 2 } } { \\nu T } + \\frac { \\nu } { 2 } \\frac { 1 } { T } \\sum \\limits _ { k = 0 } ^ { T - 1 } | | \\hat { g } _ { k + 1 } | | ^ { 2 } _ q . \\end{align*}"}
+{"id": "9147.png", "formula": "\\begin{align*} \\begin{aligned} F ( \\{ x _ { i , r } \\} _ { 1 \\leq i \\leq n } ^ { 1 \\leq r \\leq k _ { i } } ) = 0 & x _ { i , 1 } = v ^ { 4 } x _ { i , 2 } = v ^ { 2 } x _ { i + 1 , 1 } 1 \\leq i \\leq n - 1 , \\\\ & x _ { i , 1 } = v ^ { 4 } x _ { i , 2 } = v ^ { 2 } x _ { i - 1 , 1 } 2 \\leq i \\leq n - 1 , \\\\ & x _ { n , 1 } = v ^ { 2 } x _ { n , 2 } = v ^ { 4 } x _ { n , 3 } = v ^ { 2 } x _ { n - 1 , 1 } . \\end{aligned} \\end{align*}"}
+{"id": "8056.png", "formula": "\\begin{align*} \\eta _ { t , 1 } ^ N ( f ) = & \\eta _ { 0 , 1 } ^ N ( f ) + \\widetilde { \\mathfrak { M } } _ { f , k } ^ N ( t ) - \\int _ 0 ^ t K _ { 1 5 , s } ^ N ( f ) d s \\\\ & - \\int _ 0 ^ t \\mu _ { s , 1 } ^ N \\bigotimes \\mu _ { s , 3 } ^ N \\left ( \\lambda ( \\cdot , \\ast ) ( \\tilde { G } ( \\cdot ) - \\tilde { F } ( \\cdot ) ) \\right ) d s + o ( 1 ) , \\end{align*}"}
+{"id": "624.png", "formula": "\\begin{align*} V ( L , P [ n - 1 ] ) = \\frac { 1 } { n } \\sum _ { u \\in E ( P ) } h _ L ( u ) V _ { n - 1 } ( P ^ u ) . \\end{align*}"}
+{"id": "838.png", "formula": "\\begin{align*} { \\pounds } _ { \\hat { X } } { J _ k } = { \\pounds } _ { \\hat { X } } \\lambda ( x , y ) I _ { k } = ( { \\pounds } _ { \\hat { X } } \\lambda ( x , y ) ) I _ { k } + \\lambda ( x , y ) { \\pounds } _ { \\hat { X } } I _ k . \\end{align*}"}
+{"id": "6459.png", "formula": "\\begin{align*} | R ^ N ( V , d \\phi ( e _ j ) ) \\tau ( \\phi ) | ^ 2 = K ^ 2 \\big ( & | \\langle d \\phi , \\tau ( \\phi ) \\rangle | ^ 2 | V | ^ 2 + | d \\phi | ^ 2 | \\langle V , \\tau ( \\phi ) \\rangle | ^ 2 \\\\ & - 2 \\langle d \\phi ( e _ i ) , \\tau ( \\phi ) \\rangle \\langle V , \\tau ( \\phi ) \\rangle \\langle V , d \\phi ( e _ i ) \\rangle \\big ) . \\end{align*}"}
+{"id": "1824.png", "formula": "\\begin{align*} \\Lambda ^ 2 _ 7 = \\left \\{ \\omega \\in \\Lambda ^ 2 V ^ * : \\star ( \\phi _ 0 \\wedge \\omega ) = 2 \\omega ) \\right \\} \\Lambda ^ 2 _ { 1 4 } = \\left \\{ \\omega \\in \\Lambda ^ 2 V ^ * : \\star ( \\phi _ 0 \\wedge \\omega ) = - \\omega \\right \\} . \\end{align*}"}
+{"id": "1218.png", "formula": "\\begin{align*} \\frac { 1 } { R ^ { s \\theta + ( 1 - \\theta ) t } } = \\max \\big \\{ \\vert u _ \\infty \\vert _ s , \\vert v _ \\infty \\vert _ s \\big \\} & \\leq \\liminf _ { n \\to \\infty } \\left ( p _ n ^ { \\frac { 1 } { p _ n } } \\sqrt [ p _ n ] { \\Lambda _ 1 ( p _ n ) } \\right ) \\\\ & \\leq \\limsup _ { n \\to \\infty } \\left ( p _ n ^ { \\frac { 1 } { p _ n } } \\sqrt [ p _ n ] { \\Lambda _ 1 ( p _ n ) } \\right ) \\leq \\frac { 1 } { R ^ { s \\theta + ( 1 - \\theta ) t } } , \\end{align*}"}
+{"id": "6397.png", "formula": "\\begin{align*} W _ n ^ { ( \\eta ) } = \\{ ( \\delta , \\alpha ) ; \\left | \\left | w _ n ^ { - 1 } \\begin{pmatrix} \\delta - \\delta _ 0 \\\\ \\alpha - \\alpha _ 0 \\end{pmatrix} \\right | \\right | \\leq \\eta \\} \\ln ( n ) ^ q w _ n \\rightarrow 0 \\ ; \\forall q > 0 . \\end{align*}"}
+{"id": "2908.png", "formula": "\\begin{align*} a _ f = \\begin{cases} 1 & f \\\\ - 1 & f . \\end{cases} \\end{align*}"}
+{"id": "7925.png", "formula": "\\begin{align*} \\begin{aligned} \\mathfrak { H } ^ { k } & : = \\{ \\mu \\in H \\Lambda ^ { k } ( \\Omega ) \\cap \\mathring { H } ^ { \\ast } \\Lambda ^ { k } ( { \\Omega } ) \\mid d \\mu = 0 , \\ \\delta \\mu = 0 \\} , \\\\ \\mathring { \\mathfrak { H } } ^ { k } & : = \\{ \\mu \\in \\mathring { H } \\Lambda ^ { k } ( \\Omega ) \\cap H ^ { \\ast } \\Lambda ^ { k } ( { \\Omega } ) \\mid d \\mu = 0 , \\ \\delta \\mu = 0 \\} . \\end{aligned} \\end{align*}"}
+{"id": "1116.png", "formula": "\\begin{align*} \\left \\| \\left \\{ g _ j \\right \\} _ { j \\in \\mathbb Z } \\right \\| _ { L ^ p \\ell ^ q ( \\widehat P ) } & \\leq \\left \\| \\left \\{ \\gamma _ j E _ j \\left ( f _ j \\right ) \\right \\} _ { j \\in \\mathbb Z } \\right \\| _ { L ^ p \\ell ^ q ( \\widehat P ) } \\\\ & \\lesssim \\left \\| \\left \\{ E _ j \\left ( f _ j \\right ) \\right \\} _ { j \\in \\mathbb Z } \\right \\| _ { L ^ p \\ell ^ q ( \\widehat P ) } = \\left \\| \\left \\{ f _ j \\right \\} _ { j \\in \\mathbb Z } \\right \\| _ { L ^ p \\ell ^ q ( \\widehat P ) } , \\end{align*}"}
+{"id": "6243.png", "formula": "\\begin{align*} \\begin{aligned} \\tau ( \\omega , \\gamma , 0 ) & = \\omega ^ 2 + 9 \\ , \\gamma ^ 2 - 3 \\ , \\omega \\gamma + 5 \\omega - 2 1 \\gamma + 4 = \\left ( \\omega - \\frac { 3 } { 2 } \\gamma + \\frac 5 2 \\right ) ^ 2 + \\frac { 2 7 } { 4 } ( 1 - \\gamma ) ^ 2 - 9 . \\end{aligned} \\end{align*}"}
+{"id": "4022.png", "formula": "\\begin{align*} \\frac { x _ { 1 } ^ { \\left ( t + 1 \\right ) } } { x _ { 2 } ^ { \\left ( t + 1 \\right ) } } = \\frac { \\gamma _ { 1 } x _ { 1 } ^ { \\left ( t \\right ) } + \\delta _ { 1 } x _ { 2 } ^ { \\left ( t \\right ) } } { \\gamma _ { 2 } x _ { 1 } ^ { \\left ( t \\right ) } + \\delta _ { 2 } x _ { 2 } ^ { \\left ( t \\right ) } } , \\quad \\forall t \\geq t _ { 0 } , \\end{align*}"}
+{"id": "7006.png", "formula": "\\begin{gather*} Q _ m : \\ ; \\mathbb { P } _ \\mathbb { C } ^ { n } \\setminus [ \\widetilde { \\Lambda } _ 0 ] \\rightarrow \\mathbb { P } _ \\mathbb { C } ^ { m } \\\\ z = [ z _ 1 , z _ 2 , . . . , z _ { n + 1 } ] \\mapsto Q _ m ( z ) = [ z _ 1 , \\cdots , z _ m , z _ { n + 1 } ] . \\end{gather*}"}
+{"id": "7267.png", "formula": "\\begin{align*} x p _ n ( x ) = p _ { n + 1 } ( x ) + ( a + 1 ) q ^ n p _ n ( x ) - a q ^ { n - 1 } ( 1 - q ^ n ) p _ { n - 1 } ( x ) , n \\geq 0 , \\end{align*}"}
+{"id": "6845.png", "formula": "\\begin{align*} \\omega _ g = g ^ { - 1 } d g = \\sum _ { j = 1 } ^ { N ^ 2 - 1 } g ^ { - 1 } \\frac { \\partial g } { \\partial x _ j } d x _ j , \\end{align*}"}
+{"id": "9111.png", "formula": "\\begin{align*} A ( \\Deriv ) u = e \\delta _ 0 \\end{align*}"}
+{"id": "1130.png", "formula": "\\begin{align*} b _ { Q , R } ^ { D E F } : = \\left [ 1 + \\frac { | x _ Q - x _ R | } { \\ell ( Q ) \\vee \\ell ( R ) } \\right ] ^ { - D } \\times \\left \\{ \\begin{aligned} & \\left [ \\frac { \\ell ( Q ) } { \\ell ( R ) } \\right ] ^ E & & \\ell ( Q ) \\leq \\ell ( R ) , \\\\ & \\left [ \\frac { \\ell ( R ) } { \\ell ( Q ) } \\right ] ^ F & & \\ell ( R ) < \\ell ( Q ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "4657.png", "formula": "\\begin{align*} \\chi _ { 4 q , 4 q - 1 } ( a ) = \\left ( \\frac { q } { a } \\right ) , \\end{align*}"}
+{"id": "322.png", "formula": "\\begin{align*} z ( x , t ) = ( T - t ) ^ { - \\alpha } f ( ( 1 + | x | ) ( T - t ) ^ { \\beta } ) , f ( \\xi ) = \\left \\{ \\begin{array} { l l } f _ 1 ( \\xi ; A ) , & \\xi \\in [ 0 , \\overline { \\xi } ] , \\\\ f _ 2 ( \\xi ) , & \\xi \\geq \\overline { \\xi } , \\end{array} \\right . \\end{align*}"}
+{"id": "2028.png", "formula": "\\begin{align*} G ( t ) = \\alpha t \\prod _ { 0 \\leq j \\leq k } \\ln ^ { [ j ] } ( t ) \\end{align*}"}
+{"id": "5532.png", "formula": "\\begin{align*} ( y \\cap D ( i , \\gamma _ y ) ) \\cap x = D ( i , \\gamma _ y ) \\cap x = D ( i , \\gamma _ x ) \\cap x \\in S _ x , \\end{align*}"}
+{"id": "8163.png", "formula": "\\begin{align*} P _ { { \\underline X } ^ { i } , Y ^ { i } } ( { \\underline X } ^ { i } , y ^ { i } ) = \\frac { c _ { y ^ i } } { M } , \\\\ | \\mathcal B _ { y ^ { i } } ( { \\underline X } ^ { i } ) | = c _ { y ^ i } , \\end{align*}"}
+{"id": "4112.png", "formula": "\\begin{align*} Q ( \\theta ) { \\tilde { \\psi } } ^ { ( r ) } ( \\theta ) + \\sum _ { i \\in \\mathcal { J } } ( \\mu ^ { ( r ) } _ i - \\lambda _ i ) \\xi _ i ( \\theta ) \\bigl ( \\tilde { \\psi } ^ { ( r ) } ( \\theta ) - \\tilde { \\psi } _ i ^ { ( r ) } ( \\theta ) \\bigr ) = 0 , \\theta \\in \\R ^ J _ - , \\end{align*}"}
+{"id": "6941.png", "formula": "\\begin{align*} h _ 1 W _ { g \\omega _ I } \\sim h _ 2 W _ { g \\omega _ I } W _ { g \\omega _ I } h _ 1 W _ { g \\omega _ I } = W _ { g \\omega _ I } h _ 2 W _ { g \\omega _ I } . \\end{align*}"}
+{"id": "1687.png", "formula": "\\begin{align*} x _ 0 = \\sum _ { 0 < j < j + k \\leq n } S _ { j , k } x _ j x _ k x _ n ^ { n - j - k } , \\end{align*}"}
+{"id": "5807.png", "formula": "\\begin{align*} | X _ + ( t ) | ^ 2 + | X _ 0 ( t ) | ^ 2 + | X _ - ( t ) | ^ 2 = ( 1 + o ( 1 ) ) | X _ 0 ( t ) | ^ 2 . \\end{align*}"}
+{"id": "3839.png", "formula": "\\begin{align*} L ^ { n + 2 } \\doteq \\frac { 1 } { n + 2 } \\sum \\limits _ { i = 0 } ^ { n + 1 } \\delta _ { X _ i } . \\end{align*}"}
+{"id": "7460.png", "formula": "\\begin{align*} \\Lambda ^ I _ J & = \\prod _ { \\ell = 1 } ^ n \\ , \\ , \\left \\{ \\ , \\ , \\idotsint \\limits _ { 0 \\leq t _ 1 < \\dots < t _ n < \\infty } \\left ( - \\lambda _ { i _ \\ell } e ^ { - \\lambda _ { i _ \\ell } t _ \\ell } \\delta _ { i _ \\ell , j _ \\ell } \\right ) \\ , \\dd t _ 1 \\ldots \\dd t _ n \\right \\} , \\end{align*}"}
+{"id": "2682.png", "formula": "\\begin{align*} \\frac { \\partial L ^ { ( 2 ) } } { \\partial \\ddot { q } ^ { i } } = 0 . \\end{align*}"}
+{"id": "7831.png", "formula": "\\begin{align*} P _ { \\operatorname { B o x } [ \\boldsymbol { 0 } , \\boldsymbol { 1 } ] } \\left ( \\boldsymbol { \\rho } \\left ( i + \\frac { 1 } { 2 } \\right ) - \\begin{bmatrix} \\boldsymbol { \\lambda } ^ { \\star } \\\\ \\boldsymbol { \\lambda } ^ { \\star } \\end{bmatrix} \\right ) . \\end{align*}"}
+{"id": "4812.png", "formula": "\\begin{align*} & \\min \\Big ( \\lambda \\varepsilon _ K + \\frac { 1 } { K } \\sum _ { k = 1 } ^ K s _ k \\Big ) \\\\ \\mbox { s . t . } & \\hat { \\mathbf { c } } ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } \\top } \\mathbf { x } \\leq s _ k \\forall k \\in \\mathcal { K } \\\\ & \\lambda \\geq 1 . \\end{align*}"}
+{"id": "5093.png", "formula": "\\begin{align*} \\mathrm { t r } ( \\mathsf { a } _ s ) = i ( - g _ s ) , \\\\ \\mathrm { t r } ( \\mathsf { e } _ s ) = i ( e _ s ) . \\end{align*}"}
+{"id": "3046.png", "formula": "\\begin{align*} \\mathcal { P T } \\ , \\Phi _ m ( \\xi ) = \\varepsilon \\ , \\Phi _ m ( \\xi ) \\ , , \\mathcal { P T } \\ , \\Psi _ { m , n } ( \\chi ) = \\varepsilon \\ , \\Psi _ { m , n } ( \\chi ) , \\varepsilon ^ 2 = 1 \\ , . \\end{align*}"}
+{"id": "3306.png", "formula": "\\begin{align*} \\d L _ t ^ c = \\left ( f _ 1 + \\frac { 1 } { 2 } f _ { 2 2 } \\right ) \\d t + f _ 2 \\ , \\d B _ t = 0 \\ , \\d t + \\frac { L _ t ^ c B _ t } { c ^ 2 + t } \\ , \\d B _ t . \\end{align*}"}
+{"id": "1390.png", "formula": "\\begin{align*} A [ < 0 ] ^ { \\perp _ { f } } & = \\{ Y \\in \\mathcal { R } ( A ) \\mid \\mathrm { H o m } _ { \\mathrm { D } ( A ) } ( A [ < 0 ] , X ) = 0 \\} \\\\ & = \\{ Y \\in \\mathcal { R } ( A ) \\mid \\mathrm { H } ^ { i } \\mathrm { R H o m } _ { A } ( A , X ) = 0 ~ \\textrm { f o r } ~ i > 0 \\} \\\\ & = \\{ Y \\in \\mathcal { R } ( A ) \\mid \\mathrm { H } ^ { i } X = 0 ~ \\mathrm { f o r } ~ i > 0 \\} . \\end{align*}"}
+{"id": "512.png", "formula": "\\begin{align*} O \\left ( \\sum _ { t = 1 } ^ K { ( 2 ^ { m _ t } \\cdot 2 ^ { m _ { j _ t } } m _ { j _ t } \\log ^ { 1 + o ( 1 ) } { q } ) } \\right ) \\subseteq O ( d 4 ^ d \\log ^ { 1 + o ( 1 ) } { q } ) \\end{align*}"}
+{"id": "3353.png", "formula": "\\begin{align*} J ( u ) = \\mathbb { E } ^ \\dagger \\left [ \\prod _ { i = 1 } ^ K \\Psi ( x _ { i - 1 } , y _ i ) \\cdot \\mu \\left ( \\sum _ { i = 0 } ^ { K - 1 } L ( x _ i , u _ i ) + \\Phi ( x _ K ) \\right ) \\right ] , \\end{align*}"}
+{"id": "7334.png", "formula": "\\begin{align*} \\sum _ { r = 0 } ^ { m - 1 } \\chi ( r ) C _ { m , r } ( n ) & = \\frac { 1 } { m } \\sum _ { r = 0 } ^ { m - 1 } \\chi ( r ) \\sum _ { j = 1 } ^ { m - 1 } \\csc ^ { 2 n } \\left ( \\frac { j \\pi } { m } \\right ) e ^ { \\frac { 2 \\pi i r } { m } j } \\\\ & = \\frac { \\tau ( \\chi ) } { m } \\sum _ { j = 0 } ^ { m - 1 } \\overline { \\chi ( j ) } \\csc ^ { 2 n } \\left ( \\frac { j \\pi } { m } \\right ) \\\\ & = \\frac { \\tau ( \\chi ) } { m } \\overline { L _ { X _ m } ( n , \\chi ) } . \\end{align*}"}
+{"id": "4819.png", "formula": "\\begin{align*} \\sum _ { a \\in \\mathcal { A } } \\max \\{ 0 , x _ a - x ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } } _ a \\} = \\begin{cases} 0 , \\mbox { i f } \\mathbf { x } = \\mathbf { x } ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } } , \\\\ h , \\mbox { i f } \\mathbf { x } = \\mathbf { x } ^ { \\mbox { \\tiny \\upshape ( \\itshape j \\upshape ) } } , \\ ; j \\neq k \\end{cases} \\end{align*}"}
+{"id": "4479.png", "formula": "\\begin{align*} M _ H ( Z _ 0 , \\mathfrak { a } , \\rho ) = \\frac { 1 } { 2 \\pi } \\int _ { \\partial D } | F _ 0 | ^ 2 e ^ { - \\varphi } \\left ( \\frac { \\partial \\psi } { \\partial v _ z } \\right ) ^ { - 1 } | d z | . \\end{align*}"}
+{"id": "311.png", "formula": "\\begin{align*} u _ t = \\Delta u ^ m + | x | ^ { \\sigma } u ^ p , \\end{align*}"}
+{"id": "5378.png", "formula": "\\begin{align*} \\bar { v } _ t = \\min \\ , \\left \\{ \\bar { v } ^ u : \\bar { b } ^ u = t , u \\in \\mathcal { U } \\right \\} . \\end{align*}"}
+{"id": "6234.png", "formula": "\\begin{align*} p : = C _ i D _ i + C _ g D _ g q : = C _ g D _ g . \\end{align*}"}
+{"id": "7941.png", "formula": "\\begin{align*} v = d \\phi + \\eta \\quad \\Omega . \\end{align*}"}
+{"id": "8308.png", "formula": "\\begin{align*} { \\sf M } _ k ( \\omega _ 1 \\odot \\ldots \\odot \\omega _ k ) = P \\{ \\{ \\ldots \\{ { \\sf m } \\omega _ 1 , \\omega _ 2 \\} , \\ldots \\} , \\omega _ k \\} \\end{align*}"}
+{"id": "7590.png", "formula": "\\begin{align*} B _ { R } ^ { 2 } [ u ] & = \\mu \\frac { - 2 N ( q - 2 ) } { q } \\int _ { \\mathbb { R } ^ { N } } | u | ^ { q } - \\mu \\frac { 2 ( q - 2 ) } { q } \\int _ { | x | \\geq R } ( \\Delta \\varphi _ { R } - N ) | u | ^ { q } \\\\ & = \\mu \\frac { - 2 N ( q - 2 ) } { q } \\int _ { \\mathbb { R } ^ { N } } | u | ^ { q } + O ( R ^ { - \\frac { ( q - 2 ) ( N - 1 ) } { 2 } } \\| \\Delta u \\| _ { 2 } ^ { \\frac { q - 2 } { 4 } } ) , \\end{align*}"}
+{"id": "6730.png", "formula": "\\begin{align*} P ( n , z ) & = - \\pi z \\int _ 0 ^ 1 { y ^ n d y } + { \\ln ( 2 \\pi z ) } \\int _ 0 ^ 1 { y ^ { n - 1 } d y } + \\int _ 0 ^ 1 { y ^ { n - 1 } \\ln y } \\ , d y \\\\ & \\ , - \\int _ 0 ^ 1 y ^ { n - 1 } \\ln \\ ! \\big ( { 1 - e ^ { - 2 \\pi y z } } \\big ) d y \\\\ & = - \\frac { \\pi z } { { n + 1 } } + \\frac { { \\ln ( 2 \\pi z ) } } n - \\frac 1 { n ^ 2 } - \\int _ 0 ^ 1 { y ^ { n - 1 } \\ln \\ ! \\big ( { 1 - e ^ { - 2 \\pi y z } } \\big ) d y } . \\end{align*}"}
+{"id": "1913.png", "formula": "\\begin{align*} | \\rho ( x ) - \\rho ( y ) | & \\leq | \\mathbf { d } ^ \\beta ( x ) - \\mathbf { d } ^ \\beta ( y ) | \\leq \\beta \\sigma ^ { \\beta - 1 } | \\mathbf { d } ^ \\beta ( x ) - \\mathbf { d } ^ \\beta ( y ) | \\\\ & \\leq \\beta \\sigma ^ { \\beta - 1 } d ( x , y ) \\qquad \\forall \\ , \\ , y \\in G , \\end{align*}"}
+{"id": "7446.png", "formula": "\\begin{align*} \\Upsilon = \\left \\{ \\Upsilon ^ I _ J \\right \\} : = \\left \\{ \\Upsilon ^ { i _ 1 , \\ldots , i _ n } _ { j _ 1 , \\ldots , j _ n } \\right \\} . \\end{align*}"}
+{"id": "2030.png", "formula": "\\begin{align*} \\Delta ^ { \\Sigma } \\ , | x | ^ 2 = 2 n . \\end{align*}"}
+{"id": "942.png", "formula": "\\begin{align*} P _ t f ( x ) : = \\mathbb E _ x f ( X _ t ) , x \\in E , \\ , t \\ge 0 . \\end{align*}"}
+{"id": "5222.png", "formula": "\\begin{align*} \\frac { \\partial \\overline { M G } } { \\partial q _ { j } } = \\frac { 1 - \\alpha } { \\sum _ { j } q _ { j } } \\left [ \\overline { p } ^ { \\alpha } _ { j } \\overline { q } ^ { - \\alpha } _ { j } - \\sum _ { i } \\overline { p } ^ { \\alpha } _ { i } \\overline { q } ^ { 1 - \\alpha } _ { i } \\right ] = \\frac { 1 - \\alpha } { \\sum _ { j } q _ { j } } \\left [ \\frac { \\overline { M G } _ { j } } { \\overline { q } _ { j } } - \\sum _ { i } \\overline { M G } _ { i } \\right ] \\end{align*}"}
+{"id": "8736.png", "formula": "\\begin{align*} \\eta _ x = \\frac { 1 } { \\mu ( \\{ x \\} ) } \\int _ { F _ \\nu ( x - ) - q ( x ) } ^ { F _ \\nu ( x - ) } \\delta _ { F _ \\nu ^ { - 1 } ( v ) } d v + \\frac { 1 } { \\mu ( \\{ x \\} ) } \\int _ { F _ \\nu ( x ) } ^ { F _ \\nu ( x ) + \\mu ( \\{ x \\} ) - q ( x ) } \\delta _ { F _ \\nu ^ { - 1 } ( v ) } d v . \\end{align*}"}
+{"id": "1840.png", "formula": "\\begin{align*} \\Lambda ^ 2 T ^ * X = \\Lambda ^ 2 _ 7 \\oplus \\Lambda ^ 2 _ { 1 4 } . \\end{align*}"}
+{"id": "4954.png", "formula": "\\begin{align*} \\Gamma = \\{ \\tilde \\gamma \\in \\mathcal G ( \\mathcal H ) : \\tilde \\gamma ( t ) \\in B _ { \\delta } ( x ) \\subset S _ \\tau \\} \\end{align*}"}
+{"id": "2848.png", "formula": "\\begin{align*} \\delta Q ( t _ { 2 } ) = \\delta Q ( t _ { 1 } ) = 0 , \\end{align*}"}
+{"id": "4771.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\norm { J _ t x } } { t } = d ( 0 , \\mathrm { r a n } A ) \\end{align*}"}
+{"id": "3488.png", "formula": "\\begin{align*} \\hat { f } - P \\hat { f } = f - \\pi ( f ) , \\end{align*}"}
+{"id": "4148.png", "formula": "\\begin{align*} & Q ^ * ( \\theta ) \\psi ^ { ( r ) } ( \\theta ) + r ^ j \\bar { \\xi } _ j ( \\theta ) \\bigl ( \\psi ^ { ( r ) } ( \\theta ) - \\psi ^ { ( r ) } _ i ( \\theta ) \\bigr ) = o ( r ^ { 2 j } ) . \\end{align*}"}
+{"id": "4212.png", "formula": "\\begin{align*} \\Theta _ 2 ( T _ { C } X ) = \\bigotimes _ { n = 1 } ^ { \\infty } S _ { q ^ n } ( \\widetilde { T _ C X } ) \\otimes \\bigotimes _ { m = 1 } ^ { \\infty } \\wedge _ { - q ^ { m - \\frac { 1 } { 2 } } } ( \\widetilde { T _ C X } ) , \\end{align*}"}
+{"id": "7256.png", "formula": "\\begin{align*} 0 \\leq 1 + \\tfrac { \\eta \\gamma _ t } { 1 - q } + \\tfrac { \\eta \\beta _ t } { ( 1 - q ) ^ 2 } = 1 + \\tfrac { \\eta \\widetilde { \\theta } } { 1 - q } , \\end{align*}"}
+{"id": "7454.png", "formula": "\\begin{align*} \\Psi ^ 1 _ n ( t , p ) & = \\idotsint \\limits _ { 0 \\leq t _ 1 < \\cdots < t _ n \\leq t } \\left ( - \\frac { M } { m } e ^ { - \\frac { M } { m } t _ 1 } \\boldsymbol p \\right ) \\otimes \\cdots \\otimes \\left ( - \\frac { M } { m } e ^ { - \\frac { M } { m } t _ n } \\boldsymbol p \\right ) d t _ 1 \\dots d t _ n . \\end{align*}"}
+{"id": "4164.png", "formula": "\\begin{align*} \\varphi \\big ( g _ 2 g ^ { i } _ 1 \\big ) = \\varphi ( g _ 2 ) \\varphi \\big ( g ^ { i } _ 1 \\big ) \\mbox { a n d } \\varphi \\big ( g _ 3 g ^ { i } _ 1 \\big ) = \\varphi \\big ( g ^ { i } _ 1 \\big ) \\varphi ( g _ 3 ) \\ , , \\end{align*}"}
+{"id": "8285.png", "formula": "\\begin{align*} \\sum _ { i , j } \\frac { \\partial F } { \\partial a _ { i j } } a _ { i j } = \\psi , \\end{align*}"}
+{"id": "6407.png", "formula": "\\begin{align*} \\hat { \\theta } _ { 1 , n } - \\hat { \\theta } _ { n } = J _ n ( \\hat { \\theta } _ { 0 , n } ) ^ { - 1 } \\left ( J _ n ( \\hat { \\theta } _ { 0 , n } ) - \\int _ 0 ^ 1 J _ n ( \\hat { \\theta } _ { n } + t ( \\hat { \\theta } _ { 0 , n } - \\hat { \\theta } _ { n } ) ) d t \\right ) ( \\hat { \\theta } _ { 0 , n } - \\hat { \\theta } _ { n } ) , \\end{align*}"}
+{"id": "7158.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } \\partial _ t f - \\partial _ x ^ 2 f & = & f - f ^ 3 / 3 - g + I ( x , t ) & , \\forall x \\in \\R , t \\geq 0 , \\\\ \\partial _ t g & = & \\varepsilon ( f - \\gamma g + \\beta ) & , \\forall x \\in \\R , t \\geq 0 , \\\\ f ( x , 0 ) = f _ 0 ( x ) , & & g ( x , 0 ) = g _ 0 ( x ) & , \\forall x \\in \\R , \\end{array} \\right . \\end{align*}"}
+{"id": "1919.png", "formula": "\\begin{align*} \\sum _ { x \\in B _ r ( x _ 0 ) } \\ ! \\ ! \\ ! V ( x ) \\ , v ^ 2 _ + ( x , T ) \\mu ( x ) & \\leq \\frac { c ( T , \\lambda , r ) } { ( r _ 1 ) ^ 2 } \\sum _ { x \\in G } e ^ { - \\Lambda \\mathbf { d } ^ \\alpha ( x ) } \\chi _ { \\{ r < \\mathbf { d } ( x ) \\le r _ 1 \\} } ( x ) \\mu ( x ) \\\\ & = \\frac { c ( T , \\lambda , r ) } { ( r _ 1 ) ^ 2 } \\sum _ { x \\in G } e ^ { - \\Lambda d ^ \\alpha ( x , x _ 0 ) } \\chi _ { \\{ r < \\mathbf { d } ( x ) \\le r _ 1 \\} } ( x ) \\mu ( x ) . \\end{align*}"}
+{"id": "410.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { N } C _ { M } ( \\zeta ) \\xrightarrow { N \\xlongrightarrow { c } \\infty } \\frac { 1 } { N } \\overline { C } _ { M } ( \\zeta ) , \\end{aligned} \\end{align*}"}
+{"id": "574.png", "formula": "\\begin{align*} \\eta _ n & = 0 & \\partial \\Gamma _ c . \\end{align*}"}
+{"id": "3213.png", "formula": "\\begin{align*} \\partial _ t ^ \\alpha u _ k ( t ) = ( \\partial _ t ^ \\alpha u ( t ) , v _ k ) = - t ^ { \\beta } ( A u ( t ) , v _ k ) = - t ^ { \\beta } ( u ( t ) , A v _ k ) \\end{align*}"}
+{"id": "8971.png", "formula": "\\begin{align*} ( D W ( v ) , u ) : = \\int _ { \\mathbb R ^ N } \\nabla v \\cdot \\nabla u + | v | ^ { r - 2 } v u , \\ , v \\in H , \\ ; \\ ; u \\in H . \\end{align*}"}
+{"id": "6054.png", "formula": "\\begin{align*} U ^ 2 = ( U ' ) ^ 2 + ( V + W ) ^ 2 + U ' ( V + W ) . \\end{align*}"}
+{"id": "4895.png", "formula": "\\begin{align*} ( - K _ { S ' } ) \\cdot C = ( C - M - F ) \\cdot C = 4 g - 4 + \\varepsilon - ( 2 g - 2 ) = 2 g - 2 + \\varepsilon . \\end{align*}"}
+{"id": "1996.png", "formula": "\\begin{align*} \\xi ^ { n } ( x ) = \\sum _ { l \\in \\mathcal { T } _ M } \\widetilde { ( \\xi ^ n ) } _ l e ^ { i \\mu _ l ( x - a ) } , \\end{align*}"}
+{"id": "2518.png", "formula": "\\begin{align*} \\| a \\| _ { Q , C ^ k } = \\sup _ { ( x , \\xi ) \\in Q , \\ | \\alpha | + | \\beta | \\le k } | \\partial _ x ^ \\alpha \\partial _ \\xi ^ \\beta a ( x , \\xi ) | \\ ; . \\end{align*}"}
+{"id": "541.png", "formula": "\\begin{align*} \\left \\langle u ( \\tau ) , z ^ { \\tau } \\right \\rangle _ { \\mathcal { H } ^ { - s } , \\mathcal { H } ^ { s } } = \\left \\langle u _ { 0 } , S ( \\tau ) z ^ \\tau \\right \\rangle _ { \\mathcal { H } ^ { - s } , \\mathcal { H } ^ { s } } - \\int _ { 0 } ^ { \\tau } f ( t ) \\lim _ { x \\rightarrow 1 ^ - } S ( \\tau - t ) z ^ { \\tau } _ { x } ( x ) \\mathrm { d } t , \\end{align*}"}
+{"id": "5863.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { N C } ( D _ { 2 m } ) ) & = \\dfrac { ( 2 m - 2 ) ( 2 m - 3 ) ^ { 3 } } { 2 } + 2 \\dfrac { ( m - 2 ) ^ { 2 } ( m - 3 ) ^ { 2 } + 2 m ( m - 2 ) ( m - 3 ) + m ^ { 2 } } { 4 } \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - 3 \\dfrac { ( m - 2 ) ( m - 3 ) + m } { 2 } ( 2 m - 3 ) ^ { 2 } + ( 2 m - 2 - \\dfrac { 3 } { 2 } ) ( ( m - 2 ) ( m - 3 ) ^ { 2 } + m ) \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - \\dfrac { ( m - 2 ) ( m - 3 ) ^ { 3 } + m } { 2 } \\\\ & = \\dfrac { 1 } { 2 } ( 8 m ^ { 4 } - 4 0 m ^ { 3 } + 6 4 m ^ { 2 } - 3 2 m ) \\\\ & = 4 m ^ { 4 } - 2 0 m ^ { 3 } + 3 2 m ^ { 2 } - 1 6 m . \\end{align*}"}
+{"id": "8190.png", "formula": "\\begin{align*} R _ 1 = H ( \\frac { B _ 1 } { M } ) . \\end{align*}"}
+{"id": "741.png", "formula": "\\begin{align*} H _ p ( A ; x ) = \\frac { 1 } { 1 - x ^ a } \\sum _ { w \\in { \\rm A p } _ p ( A ; a ) } x ^ w \\ , , \\end{align*}"}
+{"id": "5321.png", "formula": "\\begin{align*} y ^ { \\boldsymbol { \\pi } , S } = \\begin{cases} \\nu _ { \\pi _ k } - \\nu _ { \\pi _ { k - 1 } } & \\\\ \\nu _ { \\pi _ 1 } & \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "4672.png", "formula": "\\begin{align*} \\P \\left ( \\| \\frac { 1 } { N } \\sum _ { n = 1 } ^ N Y _ n \\| \\geq m + t \\right ) \\leq \\kappa \\exp \\left ( \\frac { - N t ^ 2 } { 2 r ^ 2 + ( r + m ) t } \\right ) , \\end{align*}"}
+{"id": "1406.png", "formula": "\\begin{align*} d G ( x \\vert \\mu , \\lambda ) & = \\frac { 1 } { \\Gamma ( \\mu ) } \\ , x ^ { \\mu - 1 } e ^ { - \\lambda x } \\ , d x \\end{align*}"}
+{"id": "7827.png", "formula": "\\begin{align*} \\boldsymbol { y } _ { \\mathrm { t } } = \\left ( \\boldsymbol { G } _ { \\mathrm { t } } \\boldsymbol { \\Theta } _ { \\mathrm { t } } \\boldsymbol { H } _ { \\mathrm { s , t } } + \\boldsymbol { H } _ { \\mathrm { d , t } } \\right ) \\boldsymbol { s } _ { \\mathrm { t } } + \\boldsymbol { z } , \\end{align*}"}
+{"id": "8611.png", "formula": "\\begin{align*} \\frac { d v _ 0 } { d t } + K _ 0 ( t ; x ) + \\phi _ 0 ( t ; x ) = 0 , \\end{align*}"}
+{"id": "6874.png", "formula": "\\begin{align*} \\varphi _ j ^ { ( n ) } = { e } _ j ^ { ( n ) } - \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } { e } _ i ^ { ( n ) } , 1 \\leq { j } \\leq { n } , \\varphi _ { n + 1 } ^ { ( n ) } = \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ { n } { e } _ i ^ { ( n ) } , \\end{align*}"}
+{"id": "3490.png", "formula": "\\begin{align*} \\rho _ k ( x ) \\rho _ k ( y ) = \\lim _ n \\rho _ n \\big ( \\rho _ { n , k } ( x ) \\rho _ { n , k } ( y ) \\big ) \\end{align*}"}
+{"id": "1147.png", "formula": "\\begin{align*} \\dot a ^ { s , \\tau } _ { p , q } ( W ) = \\dot f ^ { s + n ( \\tau - \\frac 1 p ) } _ { \\infty , \\infty } ( \\mathbb A ) = \\dot f ^ { s + n ( \\tau - \\frac 1 p ) , 1 / p } _ { p , \\infty } ( W ) , \\quad \\begin{cases} \\tau > \\frac 1 p \\quad \\\\ ( \\tau , q ) = ( \\frac 1 p , \\infty ) , \\end{cases} \\end{align*}"}
+{"id": "1492.png", "formula": "\\begin{align*} f ( m ) \\big ( f ( n ) ( \\{ m \\} - \\{ n \\} ) - f ( m + n ) ( \\{ m \\} - \\{ n \\} ) \\big ) = 0 . \\end{align*}"}
+{"id": "7565.png", "formula": "\\begin{align*} E _ { p , q } ( u _ { t } ) = \\frac { 1 } { 2 } t ^ { 2 } \\| \\Delta u \\| _ { 2 } ^ { 2 } - \\frac { \\mu } { q } t ^ { \\frac { N ( q - 2 ) } { 4 } } \\| u \\| _ { q } ^ { q } - \\frac { 1 } { p } t ^ { \\frac { N ( p - 2 ) } { 4 } } \\| u \\| _ { p } ^ { p } \\end{align*}"}
+{"id": "3576.png", "formula": "\\begin{align*} ( a + b ) ( i d - \\tau ) ( \\tau ^ 2 - \\tau ) = 0 . \\end{align*}"}
+{"id": "8124.png", "formula": "\\begin{align*} \\dim ( H \\cdot x ) = \\dim H - \\dim H _ x . \\end{align*}"}
+{"id": "6293.png", "formula": "\\begin{align*} \\Delta _ k = \\tilde { O } \\left ( \\frac { \\mu _ r ^ 2 R _ 0 ^ { ( 2 r - 1 ) } } { M _ 2 \\sqrt { d } } \\frac { 1 } { 2 ^ { k ( 2 r - 1 ) } } \\right ) . \\end{align*}"}
+{"id": "2469.png", "formula": "\\begin{align*} \\lim _ { \\xi \\to + \\infty } u ( \\xi ) = \\lim _ { \\xi \\to + \\infty } u ' ( \\xi ) = 0 \\end{align*}"}
+{"id": "3434.png", "formula": "\\begin{align*} ( f ^ H ) ^ \\ast \\tau _ G ' = e _ G ( \\tilde { \\R } ) \\tau _ G , \\end{align*}"}
+{"id": "7341.png", "formula": "\\begin{align*} \\mathbb W _ 2 & = \\left \\{ w \\in \\mathbb W \\ , : \\ , | w | \\mbox { i s e v e n } \\right \\} ; \\\\ \\mathbb W _ 2 ( r ) & = \\left \\{ w \\in \\mathbb W _ 2 \\ , : \\ , | w | \\le 2 r \\right \\} , \\ , r \\in \\N _ + . \\end{align*}"}
+{"id": "8710.png", "formula": "\\begin{align*} & \\sigma ^ * _ 2 \\sigma _ 2 = S _ { \\widehat X _ G } P ^ * _ { M _ G } B _ { M _ G } P _ { M _ G } S _ { \\widehat X _ G } \\\\ & = S _ { \\widehat X _ G } P ^ * _ { M _ G } \\Bigr ( P _ { M _ G } \\hat S _ { \\widehat X _ G } ( P ^ * _ { M _ G } P _ { M _ G } ) ^ { - 1 } P ^ * _ { M _ G } + F ) P _ { M _ G } S _ { \\widehat X _ G } \\\\ & = S _ { \\widehat X _ G } P ^ * _ { M _ G } P _ { M _ G } \\hat S _ { \\widehat X _ G } S _ { \\widehat X _ G } + S _ { \\widehat X _ G } \\hat F S _ { \\widehat X _ G } , \\end{align*}"}
+{"id": "8075.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow + \\infty } \\frac { N } { \\gamma _ N ^ 2 } \\log P ( \\frac { N } { \\gamma _ N } ( \\tau _ c ^ N - \\tau _ c ) < - x ) = - J _ { h i t } ( x ) \\end{align*}"}
+{"id": "1316.png", "formula": "\\begin{align*} \\{ F , G \\} _ \\pm = \\pm C _ { I J } ^ K \\frac { \\partial F } { \\partial \\mu _ I } \\frac { \\partial G } { \\partial \\mu _ J } \\mu _ K \\pm \\rho ^ i _ I \\biggl ( \\frac { \\partial F } { \\partial \\mu _ I } \\frac { \\partial G } { \\partial q ^ i } - \\frac { \\partial F } { \\partial q ^ i } \\frac { \\partial G } { \\partial \\mu _ I } \\biggr ) , \\end{align*}"}
+{"id": "756.png", "formula": "\\begin{align*} y _ j ( \\Phi _ k - \\varphi _ k \\Phi ) - y _ k ( \\Phi _ j - \\varphi _ j \\Phi ) = - R ^ h _ { j k } \\varphi _ h , \\end{align*}"}
+{"id": "2753.png", "formula": "\\begin{align*} { \\sigma _ { 1 } ^ { * } } ( t ) \\omega = \\omega _ { Q , P } ( t ) + \\omega _ { \\Xi , \\Psi } ( t ) , \\end{align*}"}
+{"id": "8043.png", "formula": "\\begin{align*} \\limsup _ { M \\rightarrow + \\infty } \\limsup _ { N \\rightarrow + \\infty } \\frac { N } { \\gamma ^ 2 _ N } \\log P \\left ( \\varepsilon _ 7 ^ N + \\varepsilon _ 8 ^ N > M \\right ) = - \\infty . \\end{align*}"}
+{"id": "1573.png", "formula": "\\begin{align*} \\int _ { B _ { 1 } } | U | ^ 2 \\ , d x & \\le \\int _ { B _ 2 } | E ( U ) | ^ 2 \\ , d x \\le C \\ , \\left ( \\int _ { B _ 2 } | E ( U ) | \\ , d x \\right ) ^ { { \\frac { 4 } { N + 2 } } } \\left ( \\int _ { B _ 2 } | D E ( U ) | ^ 2 \\ , d x \\right ) ^ { { \\frac { N } { N + 2 } } } \\\\ & \\le C \\ , \\left ( \\int _ { B _ 1 } | U | \\ , d x \\right ) ^ { \\frac { 4 } { N + 2 } } \\left ( \\int _ { B _ 1 } | D U | ^ 2 \\ , d x \\right ) ^ { { \\frac { N } { N + 2 } } } \\end{align*}"}
+{"id": "2991.png", "formula": "\\begin{align*} ( \\overline { \\nabla } _ x g ) ( y , z ) = 0 , \\end{align*}"}
+{"id": "8260.png", "formula": "\\begin{align*} c _ { y ^ { i } } = \\begin{cases} c _ { y ^ { i - 1 } } - b _ { y ^ { i - 1 } } , & y _ { i } = 0 \\\\ b _ { y ^ { i - 1 } } , & y _ { i } = 1 \\end{cases} , \\forall 1 \\leq i \\leq L . \\end{align*}"}
+{"id": "1688.png", "formula": "\\begin{align*} z _ j = x _ j + i y _ j . \\end{align*}"}
+{"id": "3752.png", "formula": "\\begin{align*} \\frac { \\partial \\mathrm { W } _ { \\alpha , \\beta } \\left ( z \\right ) } { \\partial \\alpha } = z \\frac { \\partial } { \\partial z } \\left ( \\frac { \\partial \\mathrm { W } _ { \\alpha , \\beta } \\left ( z \\right ) } { \\partial \\beta } \\right ) . \\end{align*}"}
+{"id": "1017.png", "formula": "\\begin{align*} \\int _ E u ( - L \\eta ) \\ , d m = \\int _ E P _ D g ( - L \\eta ) \\ , d m + \\int _ D \\eta \\ , d \\mu . \\end{align*}"}
+{"id": "3785.png", "formula": "\\begin{align*} \\mathbb { P } \\left \\{ f ^ { ( i ) } _ t V ^ { ( i ) } _ t \\leq \\bar { t } \\right \\} = \\mathbb { P } \\left \\{ V ^ { ( i ) } _ t \\leq \\frac { 4 \\bar { t } } { \\sigma ^ 2 _ { w , i } \\sqrt { n _ x } \\left ( ( 1 + 2 \\alpha ) ^ 2 \\rho ^ { ( i ) } \\frac { \\| \\Sigma ^ { ( i ) } _ t \\| } { \\sqrt { n _ x } } + 4 \\right ) } \\right \\} , \\end{align*}"}
+{"id": "1606.png", "formula": "\\begin{align*} \\Phi _ { r } | _ { \\Gamma _ t } = ( - 1 ) ^ { n - 1 } \\sigma _ { r } ( \\kappa ) \\ , d \\textup { v o l } _ { \\Gamma _ t } . \\end{align*}"}
+{"id": "2966.png", "formula": "\\begin{align*} \\left \\| \\frac { f ' } { r } + \\frac { \\beta - 3 } { 2 r ^ 2 } f \\right \\| _ { L _ \\beta ^ 2 } ^ 2 = \\frac { 1 } { ( \\beta - \\alpha - 2 ) ^ 2 } \\left \\| \\L _ { \\alpha } f + \\C _ { \\alpha , \\beta } \\frac { f } { r ^ 2 } \\right \\| _ { L _ \\beta ^ 2 } ^ 2 - \\frac { 1 } { ( \\beta - \\alpha - 2 ) ^ 2 } \\left \\| f ^ \\# \\right \\| _ { L _ \\beta ^ 2 } ^ 2 , \\end{align*}"}
+{"id": "3105.png", "formula": "\\begin{align*} \\varphi \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & = \\left ( \\begin{array} { c c c } a ^ { 2 \\ , p ^ { e _ 1 - 1 } } & 0 & c _ 2 \\ , a ^ { 2 \\ , p ^ { e _ 1 - 1 } } \\ , b ^ { p ^ { e _ 1 } } \\ , d ^ { p ^ { e _ 1 } } \\\\ 0 & a ^ { 2 \\ , p ^ { e _ 1 - 1 } } & c _ 1 \\ , a ^ { 2 \\ , p ^ { e _ 1 - 1 } } \\ , b ^ { p ^ { e _ 1 } } \\ , d ^ { p ^ { e _ 1 } } \\\\ 0 & 0 & d ^ { 4 \\ , p ^ { e _ 1 - 1 } } \\end{array} \\right ) . \\end{align*}"}
+{"id": "3745.png", "formula": "\\begin{align*} \\mathrm { B } _ { z ^ { 2 } } \\left ( \\lambda , 0 \\right ) = 2 \\int _ { 0 } ^ { z } \\frac { u ^ { 2 \\lambda - 1 } } { 1 - u ^ { 2 } } \\ , d u . \\end{align*}"}
+{"id": "7810.png", "formula": "\\begin{align*} & \\rho _ { m } ^ { \\mathrm { x } } - \\left ( \\rho _ { m } ^ { \\mathrm { x } } \\right ) ^ { 2 } = 0 , \\forall \\mathrm { x } \\in \\{ \\mathrm { t } , \\mathrm { r } \\} , m \\in \\mathcal { M } , \\\\ & 0 \\leq \\rho _ { m } ^ { \\mathrm { t } } , \\rho _ { m } ^ { \\mathrm { r } } \\leq 1 , m \\in \\mathcal { M } . \\end{align*}"}
+{"id": "1409.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty e ^ { - s x } E _ \\alpha ( - \\lambda x ^ \\alpha ) \\ , d x & = \\frac { s ^ { \\alpha - 1 } } { \\lambda + s ^ \\alpha } { \\rm R e } ( s ) \\ge 0 \\end{align*}"}
+{"id": "2889.png", "formula": "\\begin{align*} & \\lim _ { \\lambda \\to \\infty } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ { M _ r ^ \\alpha ( \\mathbb { R } ^ n ) } = \\left [ \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { M _ r ^ \\alpha ( \\mathbb { R } ^ n ) } ; \\end{align*}"}
+{"id": "2003.png", "formula": "\\begin{align*} \\delta _ x ^ + ( f ( \\psi ( x _ j , t _ n ) ) - f ( \\psi ^ n _ j ) ) = & \\int _ 0 ^ 1 | \\phi _ { 1 , j } ( \\theta ) | ^ 2 \\delta _ x ^ + \\psi ( x _ j , t _ n ) - | \\phi _ { 2 , j } ( \\theta ) | ^ 2 \\delta _ x ^ + \\psi ^ n _ j d \\theta \\\\ & + \\int _ 0 ^ 1 ( \\phi _ { 1 , j } ( \\theta ) ) ^ 2 \\delta _ x ^ + \\overline { \\psi ( x _ j , t _ n ) } - ( \\phi _ { 2 , j } ( \\theta ) ) ^ 2 \\delta _ x ^ + \\overline { \\psi ^ n _ j } d \\theta , \\end{align*}"}
+{"id": "5176.png", "formula": "\\begin{align*} A = \\frac { 1 } { \\beta ( \\beta - 1 ) } \\sum _ { i } p _ { i } ^ { \\beta } + \\frac { 1 } { \\beta } q ^ { \\beta } _ { i } \\ ; \\ ; ; \\ ; \\ ; B = \\frac { 1 } { \\beta - 1 } \\sum _ { i } p _ { i } q ^ { \\beta - 1 } _ { i } \\end{align*}"}
+{"id": "7963.png", "formula": "\\begin{align*} \\frac { \\delta \\tilde { \\mathcal { F } } } { \\delta \\eta } = ( - 1 ) ^ { n - 1 } d \\big ( \\frac { \\delta \\bar { \\mathcal { F } } } { \\delta \\omega } \\big ) . \\end{align*}"}
+{"id": "8376.png", "formula": "\\begin{align*} w = w _ { 0 , J ( w ) } c , \\end{align*}"}
+{"id": "4253.png", "formula": "\\begin{align*} J ( t ) = x + 2 i t \\partial _ x = e ^ { i t \\Delta } x e ^ { - i t \\Delta } . \\end{align*}"}
+{"id": "6926.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow - \\infty } E _ { 2 \\mu } ^ \\beta ( x ) = \\int _ { - \\infty } ^ { + \\infty } H ^ \\beta _ { 2 \\mu } ( y ) d y = 1 . \\end{align*}"}
+{"id": "2685.png", "formula": "\\begin{align*} \\sum ^ { d } _ { \\beta = \\alpha } \\left [ ( - D ) ^ { \\beta - \\alpha } \\frac { \\partial L ^ { ( d ) } } { \\partial ( D ^ { \\beta } q ^ { i } ) } + \\frac { \\partial W } { \\partial ( D ^ { \\alpha - 1 } q ^ { i } ) } \\right ] = 0 \\ \\ \\ \\ \\ ( \\alpha \\geq 2 ) \\end{align*}"}
+{"id": "2993.png", "formula": "\\begin{align*} T ( x , y ) = H ( x , y ) - H ( y , x ) , \\end{align*}"}
+{"id": "3269.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ^ { d + r } ; q ^ d ) _ k ^ { d - r - 1 } ( q ^ r ; q ^ d ) _ k ^ { r } ( q ^ { r - d } ; q ^ d ) _ k q ^ { d k } } { ( q ^ d ; q ^ d ) _ k ^ d } \\equiv 0 \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "7072.png", "formula": "\\begin{align*} d u + \\mu u d t + b \\cdot \\nabla u d t + \\sigma \\nabla u d W _ t - \\frac { \\sigma ^ 2 } { 2 } \\Delta u d t = 0 \\end{align*}"}
+{"id": "3536.png", "formula": "\\begin{align*} \\| T ( \\varphi g ) \\| _ { \\infty } = \\| T ( \\varphi ) T ( g ) \\| _ { \\infty } = \\| ( T \\varphi ) \\psi _ 1 \\| _ { \\infty } \\leq 1 - \\epsilon , \\end{align*}"}
+{"id": "6123.png", "formula": "\\begin{align*} A ^ { \\mathsf T } \\boldsymbol U ( t ) + \\boldsymbol U ( t ) A + C + \\boldsymbol U ( t ) B \\boldsymbol U ( t ) = 0 . \\end{align*}"}
+{"id": "1875.png", "formula": "\\begin{align*} \\phi \\mapsto \\langle r ^ q _ + , \\phi \\rangle = \\int _ 0 ^ \\infty r ^ q \\phi ( r ) d r \\end{align*}"}
+{"id": "3476.png", "formula": "\\begin{align*} \\sigma ^ { \\mathsf { R e Q U } } ( x ) = ( x \\vee 0 ) ^ 2 , \\sigma ^ { \\mathsf { R e L U } } ( x ) = ( x \\vee 0 ) \\end{align*}"}
+{"id": "563.png", "formula": "\\begin{align*} \\phi _ n & = j \\omega \\eta & \\Gamma _ f \\cup \\Gamma _ c \\end{align*}"}
+{"id": "3979.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ { \\nu } \\widetilde { \\gamma } _ { j p r } = 0 , \\quad \\left ( 1 \\leq j \\leq n , 1 \\leq p \\leq \\nu \\right ) . \\end{align*}"}
+{"id": "3233.png", "formula": "\\begin{align*} K ^ { \\alpha \\gamma } | _ { \\beta \\delta } ( x , y ) & = K ^ { \\gamma \\alpha } | _ { \\delta \\beta } ( x , y ) = \\overline { K ^ { \\beta \\delta } | _ { \\alpha \\gamma } ( y , x ) } \\\\ K ^ \\alpha _ { \\ ; \\ ; \\ , \\gamma } | _ \\beta ^ { \\ ; \\ ; \\ , \\delta } ( x , y ) & = K ^ \\delta _ { \\ ; \\ ; \\ , \\beta } | _ \\gamma ^ { \\ ; \\ ; \\ , \\alpha } ( y , x ) = \\overline { K ^ \\gamma _ { \\ ; \\ ; \\ , \\alpha } | _ \\delta ^ { \\ ; \\ ; \\ , \\beta } ( x , y ) } \\ : . \\end{align*}"}
+{"id": "3808.png", "formula": "\\begin{align*} M = \\begin{pmatrix} H _ { [ n - k ] , A _ 1 } & H _ { [ n - k ] , A _ 2 } & & & \\\\ H _ { [ n - k ] , A _ 1 } & & H _ { [ n - k ] , A _ 3 } & & \\\\ \\vdots & & & \\ddots & \\\\ H _ { [ n - k ] , A _ 1 } & & & & H _ { [ n - k ] , A _ t } \\end{pmatrix} . \\end{align*}"}
+{"id": "2761.png", "formula": "\\begin{align*} \\omega = d \\Theta ^ { \\alpha } \\wedge d \\Theta _ { \\alpha } + d Q ^ { i } \\wedge d P _ { i } \\end{align*}"}
+{"id": "6963.png", "formula": "\\begin{align*} f ( \\mathbf { z } ) = \\displaystyle - \\sum _ { j = 1 } ^ { n } ( z _ j \\overline { z _ { n + 1 } } - z _ { n + 1 } \\overline { z _ { j } } ) ^ { 2 } + \\sum _ { 1 \\leq j < k \\leq n } ( z _ j \\overline { z _ { k } } - z _ { k } \\overline { z _ { j } } ) ^ 2 \\end{align*}"}
+{"id": "5733.png", "formula": "\\begin{align*} \\Lambda : = \\{ \\gamma < 0 \\ , : \\ , \\| u \\| _ { C ^ 1 } ( t ) = O ( e ^ { \\gamma t } ) \\} \\end{align*}"}
+{"id": "8245.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ { \\ell } \\sum _ { j = 1 } ^ L \\mathbb { E } [ b ( X _ { i , j } ) | W = w ] \\leq B _ , \\end{align*}"}
+{"id": "6581.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { a - i T } ^ { a + i T ^ { \\prime } } \\frac { y ^ s } { s } d s = h ( y ) + O \\left ( \\frac { y ^ a } { | \\log y | } \\left ( \\frac { 1 } { T } + \\frac { 1 } { T ^ { \\prime } } \\right ) \\right ) \\quad ( y \\neq 1 ) , \\end{align*}"}
+{"id": "3433.png", "formula": "\\begin{align*} \\tilde { H } _ { \\Z _ 2 } ^ { 2 } ( \\Sigma ^ { \\R } G _ + ) \\cong \\tilde { H } _ { \\Z _ 2 } ^ { 1 } ( G _ + ) \\cong H ^ 1 ( \\mathrm { p t } \\sqcup \\mathrm { p t } ) = 0 , \\end{align*}"}
+{"id": "5994.png", "formula": "\\begin{align*} F ^ { \\frac { 1 } { 2 } } ( x _ 0 ) = R ^ 2 \\phi ^ { \\frac { 1 } { 2 } } ( x _ 0 ) \\leq \\frac { C \\sqrt { K } R ^ 2 } { b } \\leq \\frac { C R ( 1 + \\sqrt { K } R ) } { b } . \\end{align*}"}
+{"id": "271.png", "formula": "\\begin{align*} y = \\frac { 1 } { 3 } x ^ 3 , d \\tau = x \\ , d t \\end{align*}"}
+{"id": "8415.png", "formula": "\\begin{align*} & ( B _ 1 ( a _ 1 ) \\otimes a _ 2 \\otimes B _ 2 ( a _ 3 ) ) \\cdot _ { H ^ { \\otimes 3 } } ( B _ 1 ( b _ 1 ) \\otimes b _ 2 \\otimes B _ 2 ( b _ 3 ) ) \\\\ = & B _ 1 ( a _ 1 ) B _ 1 ( b _ 1 ) \\otimes B _ 1 ( a _ 2 ) b _ 2 S ( B _ 2 ( a _ 3 ) ) \\otimes B _ 2 ( a _ 4 ) B _ 2 ( b _ 3 ) , \\end{align*}"}
+{"id": "1718.png", "formula": "\\begin{align*} A _ j v ^ \\prime _ j + B _ j = 0 \\quad \\mbox { a n d } B _ j ^ * v ^ \\prime _ j + C _ j = 0 \\end{align*}"}
+{"id": "9043.png", "formula": "\\begin{align*} \\lambda _ \\alpha \\lambda _ \\beta - \\lambda _ \\beta \\lambda _ \\alpha = 0 , \\lambda _ \\alpha \\theta _ \\beta ^ i - \\theta _ \\beta ^ i \\lambda _ \\alpha = 0 , \\theta _ \\alpha ^ i \\theta _ \\beta ^ j + \\theta _ \\beta ^ j \\theta _ \\alpha ^ i = - 2 \\delta _ { \\alpha , \\beta } \\delta _ { i , j } \\lambda _ \\alpha , \\end{align*}"}
+{"id": "1477.png", "formula": "\\begin{align*} \\underline { D } _ { G } \\left ( \\bigcup _ { i \\in I } E _ i M \\setminus S _ i \\right ) & \\geq \\left ( 1 - d + \\frac { \\varepsilon } { 8 } \\right ) \\left ( 1 - \\frac { \\varepsilon } { 1 0 0 } \\right ) \\\\ & = 1 - d + \\varepsilon \\left ( \\frac { 1 } { 8 } - \\frac { 1 - d + \\frac { \\varepsilon } { 8 } } { 1 0 0 } \\right ) \\\\ & \\geq 1 - d . \\end{align*}"}
+{"id": "8844.png", "formula": "\\begin{align*} \\| u \\| _ { \\dot W ^ { 1 , r } _ \\lambda } & : = \\| \\sqrt { \\mathcal K _ \\lambda } u \\| _ { L ^ r } \\textnormal { a n d } \\| u \\| _ { W ^ { 1 , r } _ \\lambda } : = \\| \\langle \\sqrt { \\mathcal K _ \\lambda } \\rangle u \\| _ { L ^ r } , \\end{align*}"}
+{"id": "8833.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } \\frac { \\partial u } { \\partial t } & = & d \\Delta u ( t , x ) + c u _ x - \\mu u , & t > 0 , \\\\ u ( 0 , x ) & = & \\phi ( x ) , & x \\in \\mathbb { R } , \\end{array} \\right . \\end{align*}"}
+{"id": "1948.png", "formula": "\\begin{align*} \\left [ ( w - 2 ^ { - k + 1 } t ) _ { - } \\right ] _ { W ^ { 1 , p } \\left ( B _ { 2 R } \\right ) } ^ p & \\leq C H _ { 4 R } ( 2 ^ { - k + 1 } t ) \\left | B _ { 4 R } \\right | \\\\ & \\leq C \\left ( \\frac { ( 2 ^ { - k + 1 } t ) ^ { p } } { R ^ p } + \\frac { ( 2 ^ { - k + 1 } t ) ^ q } { R ^ { s q } } \\right ) \\left | B _ { 4 R } \\right | , \\\\ & \\leq C R ^ { N - p } ( 2 ^ { - k + 1 } t ) ^ { p } , k = k _ 0 , \\ldots , m - 2 . \\end{align*}"}
+{"id": "1422.png", "formula": "\\begin{align*} E _ \\alpha ( - x ) & = \\int _ { 0 } ^ \\infty e ^ { - x t } d P _ \\alpha ( t ) \\end{align*}"}
+{"id": "8248.png", "formula": "\\begin{align*} I ( { A } ^ L \\to Y ^ L | S ) = \\sum _ { i = 1 } ^ L I ( { A } ^ i ; Y _ i | Y ^ { i - 1 } , S ) . \\end{align*}"}
+{"id": "375.png", "formula": "\\begin{align*} 2 x = c - b + a , 2 y = c + b - a , 2 z = c + b + a , \\end{align*}"}
+{"id": "3380.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { k } Z _ { t } & \\le \\log \\frac { 1 } { \\delta } \\end{align*}"}
+{"id": "5543.png", "formula": "\\begin{align*} \\tau _ i = \\frac { \\theta } { \\nu _ i } \\in ( 0 , 1 ) . \\end{align*}"}
+{"id": "4300.png", "formula": "\\begin{align*} d y _ t & = \\sigma ( y _ t ) d x _ t , y _ 0 = a , \\\\ d J _ t & = \\nabla \\sigma ( y _ t ) \\langle J _ t , d x _ t \\rangle , J _ 0 = A , \\\\ d K _ t & = - K _ t \\cdot \\nabla \\sigma ( y _ t ) \\langle \\bullet , d x _ t \\rangle , K _ 0 = B . \\end{align*}"}
+{"id": "8766.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 g \\left ( \\frac { \\phi _ \\uparrow ( u ) } { \\nu _ l ( \\R ) } \\right ) \\frac { \\phi ' _ \\uparrow ( u ) } { \\nu _ l ( \\R ) } d u = \\int _ 0 ^ 1 g ( v ) d v . \\end{align*}"}
+{"id": "5953.png", "formula": "\\begin{align*} \\frac { M _ { 2 } ( \\mathcal { N C } ( G ) ) } { | e ( \\mathcal { N C } ( G ) ) | } = \\frac { 9 7 3 2 0 } { 2 3 4 } > \\frac { 9 5 4 6 } { 2 3 } = \\frac { M _ { 1 } ( \\mathcal { N C } ( G ) ) } { | v ( \\mathcal { N C } ( G ) ) | } . \\end{align*}"}
+{"id": "6809.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\pi / 2 } \\sin ^ { k + p } ( \\phi ) \\cos ^ { l + q } ( \\phi ) \\ ; d \\phi & = \\int _ 0 ^ 1 x ^ { k + p } ( 1 - x ^ 2 ) ^ { \\frac { l + q - 1 } { 2 } } \\ ; d x . \\end{align*}"}
+{"id": "7936.png", "formula": "\\begin{align*} \\dot { \\mathcal { F } } ( v , \\Sigma ) = \\{ \\mathcal { F } , H \\} ( v , \\Sigma ) . \\end{align*}"}
+{"id": "4869.png", "formula": "\\begin{align*} J _ 2 + J _ 3 = - H ^ \\alpha | \\nabla _ \\Sigma H | ^ 2 q ( \\epsilon , \\alpha ) + O ( H ^ { \\alpha + 4 } ) , \\end{align*}"}
+{"id": "4951.png", "formula": "\\begin{align*} U _ x ' ( t ) = A _ x '' ( t ) A ^ { - 1 } _ x ( t ) - A _ x ' ( t ) A ^ { - 1 } _ x ( t ) A _ x ' ( t ) A _ x ^ { - 1 } ( t ) = - { R } ( \\cdot , \\gamma _ x ' ) \\gamma _ x ' - U _ x ^ 2 ( t ) . \\end{align*}"}
+{"id": "6438.png", "formula": "\\begin{align*} D _ \\mu ( \\delta ) = \\sup \\left \\{ \\frac { \\mu ( I ) } { | I | } \\ , : \\ I \\mbox { c l o s e d i n t e r v a l , } | I | = \\delta \\right \\} = \\sup _ { x \\in \\R } \\frac { \\mu ( [ x , x + \\delta ] ) } { \\delta } . \\end{align*}"}
+{"id": "2756.png", "formula": "\\begin{align*} \\sigma _ { 1 } ( t ) : & T ^ { * } M | _ { Q , P } \\times \\mathbb { R } \\rightarrow T ^ { * } M \\times \\mathbb { R } \\\\ & ; ( { \\sigma _ { 1 } ^ { * } } ( t ) \\Xi ^ { \\alpha } : = \\epsilon ^ { \\alpha } , { \\sigma _ { 1 } ^ { * } } ( t ) \\Psi _ { \\alpha } : = \\epsilon _ { \\alpha } , { \\sigma _ { 1 } ^ { * } } ( t ) Q ^ { i } , { \\sigma _ { 1 } ^ { * } } ( t ) P _ { i } , { \\sigma _ { 1 } ^ { * } } ( t ) u = t ) \\\\ & \\mapsto ( \\Xi ^ { \\alpha } , \\Psi _ { \\alpha } , Q ^ { i } , P _ { i } , u ) . \\end{align*}"}
+{"id": "4620.png", "formula": "\\begin{align*} \\limsup _ { \\lambda \\rightarrow \\infty } \\lambda ^ { - \\frac { d } { 2 } } N \\left ( - \\Delta ^ N _ \\Omega + \\lambda V \\right ) = \\infty . \\end{align*}"}
+{"id": "5048.png", "formula": "\\begin{align*} \\mathbb { E } _ \\mathbb { P } [ u _ S ( \\xi ) ] = \\int _ { \\Xi } u _ S ( \\xi ) \\mathbb { P } ( d \\xi ) . \\end{align*}"}
+{"id": "8289.png", "formula": "\\begin{align*} G _ i = \\frac { w _ i } { w \\ln w } + \\varphi ' u _ i + \\frac { \\rho _ i } { \\rho } = \\frac { u _ 1 u _ { 1 i } } { w ^ 2 \\ln w } + \\varphi ' u _ i + \\frac { \\rho _ i } { \\rho } , \\end{align*}"}
+{"id": "9005.png", "formula": "\\begin{align*} 2 z \\cdot [ x , y ] = [ z \\cdot x , y ] + [ x , z \\cdot y ] . \\end{align*}"}
+{"id": "1263.png", "formula": "\\begin{align*} c _ { n , ( n - 1 ) / 2 } & = q ^ { n - 1 } a _ { n , ( n - 1 ) / 2 } \\\\ [ 7 p t ] & \\equiv ( - 1 ) ^ { \\frac { n - 1 } { 2 } } \\left ( 1 + \\frac { ( 1 - n ) ( 1 - q ^ n ) } { 2 } \\right ) \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "1345.png", "formula": "\\begin{align*} \\alpha d ( v ) = \\dfrac { \\alpha ( d ( v ) + d ( u ' ) ) - \\sqrt { \\alpha ^ 2 \\{ d ( v ) + d ( u ' ) \\} ^ 2 - 4 \\alpha ^ 2 d ( v ) d ( u ' ) } } { 2 } , \\end{align*}"}
+{"id": "8064.png", "formula": "\\begin{align*} R _ { s , 3 } ^ { g _ 2 , g _ 3 } ( f ) = \\int _ { \\mathbb { T } } \\psi ( u ) \\theta _ s ^ E ( u ) f ( u ) ( g _ 3 ( u ) - g _ 2 ( u ) ) d u + \\int _ \\mathbb { T } \\phi ( u ) \\theta _ s ^ I ( u ) f ( u ) g _ 3 ( u ) \\end{align*}"}
+{"id": "3590.png", "formula": "\\begin{align*} \\alpha ^ 2 \\tau _ 0 \\tau _ 1 = - ( a + b ) = - ( \\lambda _ 1 + \\lambda _ 2 + \\lambda _ 1 \\lambda _ 2 ) = 1 , \\end{align*}"}
+{"id": "6331.png", "formula": "\\begin{align*} e _ p = \\sqrt { A _ p ^ 2 - B _ p ^ 2 } , e _ { \\rm B o g } = \\frac { 1 } { 2 } \\sum _ { p \\neq 0 } \\left [ \\sqrt { A _ p ^ 2 - B _ p ^ 2 } - A _ p \\right ] + \\frac { 1 } { 2 } \\sum _ { p \\neq 0 } \\frac { ( n \\widehat { \\epsilon } _ { \\ell , \\lambda } ( p ) ) ^ 2 } { 2 p ^ 2 } , \\end{align*}"}
+{"id": "1252.png", "formula": "\\begin{align*} \\frac { ( q ; q ^ 2 ) _ n ( - q ; q ) _ { n - 1 } ^ 2 } { ( 1 - q ^ { n } ) ( q ^ 2 ; q ^ 2 ) _ { n - 1 } } = { 2 n \\brack n } \\frac { 1 } { 1 + q ^ n } . \\end{align*}"}
+{"id": "1232.png", "formula": "\\begin{align*} \\Phi = S _ { A } ^ c \\circ \\mathrm { A d } _ { U } \\end{align*}"}
+{"id": "5834.png", "formula": "\\begin{align*} f _ D ( H ) = \\textrm { c r } _ D ( H ) + \\textrm { c r } _ D ( H , G \\setminus E ( H ) ) / 2 . \\end{align*}"}
+{"id": "8772.png", "formula": "\\begin{align*} \\frac { 1 } { F _ \\mu ( x ' ) } \\int _ { F _ \\mu ( x ) } ^ { F _ \\mu ( x ' ) } F _ \\mu ^ { - 1 } ( w ) d w \\ge \\frac { 1 } { F _ \\mu ( x ' ) } \\int _ u ^ { F _ \\mu ( x ' ) } \\pi _ { F _ { \\mu } ^ { - 1 } ( w ) } ( ( - \\infty , F _ { \\mu } ^ { - 1 } ( w ) ) ) { F _ \\mu ^ { - 1 } ( w ) } d w . \\end{align*}"}
+{"id": "2560.png", "formula": "\\begin{align*} ( - \\partial _ x ^ 2 + B ) B _ 0 u _ 0 & = - B _ 0 \\partial _ x u - B _ 1 u _ 0 - B _ 2 u - B _ 3 \\partial _ x u + \\partial _ x ^ { m - 1 } \\delta _ L ^ { B _ 0 ' A v } + [ B , B _ 0 ] u _ 0 \\\\ & \\in K ^ { ( - \\frac 3 2 - m ) } ( M , L ) + K ^ { ( \\frac 1 2 - m - \\epsilon ) } ( M , L ) = K ^ { ( - \\frac 3 2 - m ) } ( M , L ) \\ ; . \\end{align*}"}
+{"id": "8246.png", "formula": "\\begin{align*} \\frac { 1 } { \\ell } \\sum _ { i = 1 } ^ { \\ell } \\mathbb { E } [ b ^ { ( L ) } _ ( X _ { i } ^ L ) | W = w ] \\leq B _ . \\end{align*}"}
+{"id": "8842.png", "formula": "\\begin{align*} \\mathcal M [ u ( t ) ] : = \\int _ { \\mathbb { R } ^ n } | u ( x , t ) | ^ 2 d x = \\mathcal M [ u _ 0 ] , \\end{align*}"}
+{"id": "6866.png", "formula": "\\begin{align*} A _ 0 = S _ 0 ^ { 1 / 2 } H _ { e { \\sf E } _ 0 } S _ 0 ^ { 1 / 2 } , H _ { e { \\sf E } _ 0 } = \\sum _ { j \\in \\mathbb { J } _ 0 } E _ j \\langle e _ j , \\cdot \\rangle \\ , e _ j , { \\sf E } _ 0 = \\{ E _ j , j \\in \\mathbb { J } _ 0 \\} . \\end{align*}"}
+{"id": "4583.png", "formula": "\\begin{align*} \\int _ M d \\mu & = 2 \\pi \\int _ { - a } ^ a A ( t ) d t \\\\ & = \\frac { \\omega ( F ) ^ 2 } { 8 } \\left [ 4 \\frac { \\omega ( c _ + ) + \\omega ( c _ - ) } { \\omega ( F ) } + \\sum _ j \\frac { 1 } { r _ j s _ j } \\left ( \\frac { \\omega ( r _ j E _ j ' ) - \\omega ( - s _ j E _ j ) } { \\omega ( F ) } \\right ) ^ 2 + c _ 1 ( Y ) [ \\Sigma _ a ] - c _ 1 ( Y ) [ \\Sigma _ { - a } ] \\right ] , \\end{align*}"}
+{"id": "9049.png", "formula": "\\begin{align*} X ^ Q ( ( S ^ i ) ^ { ( k ) } v ) = - ( - 1 ) ^ { p ( X ) } \\theta _ k ^ i X ^ Q ( v ) ( v \\in V ^ { \\otimes n } ) . \\end{align*}"}
+{"id": "3648.png", "formula": "\\begin{align*} a _ d ( G ) \\ge \\lim _ { c \\to \\infty } \\lambda _ { \\binom { d + 1 } { 2 } + 1 } ( L ( G , p _ c ) ) = \\lim _ { c \\to \\infty } \\lambda _ { m } ( L ^ - ( G , p _ c ) ) = \\lambda _ { m } ( L ^ - ) . \\end{align*}"}
+{"id": "5717.png", "formula": "\\begin{align*} 0 = 2 ^ { - 1 } m ^ { 2 } G ( ( v , w ) ; ( \\varphi _ j , 0 ) ) = C _ j \\int _ { \\Sigma } \\left \\langle v , \\varphi _ j \\right \\rangle \\ , d \\mu . \\end{align*}"}
+{"id": "3316.png", "formula": "\\begin{align*} \\Pr [ \\exists n \\ge 1 , \\ Q _ n \\ge 1 / c ] \\le 2 c + c ( 1 + 2 \\log ( 2 / 1 . 7 9 c ) ) = 3 c + 2 c \\log ( 2 / 1 . 7 9 c ) . \\end{align*}"}
+{"id": "451.png", "formula": "\\begin{align*} t _ { q + 1 } = \\max \\{ t _ q + \\Delta t _ q , t _ q + \\Delta \\widetilde { t } _ q , - \\lambda _ { i _ m , m + 1 } \\} . \\end{align*}"}
+{"id": "4769.png", "formula": "\\begin{align*} \\varphi _ 4 ' ( \\varepsilon , b , c , \\eta , n , \\omega , \\varphi ) & : = \\min \\{ \\varphi _ 1 ( \\varepsilon / 3 , b , n , \\varphi ) , \\varphi _ 2 ( \\varepsilon / 3 , b , n , \\omega , \\varphi ) , \\\\ & \\qquad \\varphi _ 1 ( \\eta ( \\min \\{ \\varepsilon , 2 \\} ) c / 4 , b , n , \\varphi ) , \\sqrt { \\varphi _ 2 ( c / 2 , b , n , \\omega , \\varphi ) } , \\\\ & \\qquad \\qquad \\eta ( \\min \\{ \\varepsilon , 2 \\} ) c / 8 b , \\sqrt { \\varphi _ 2 ( c / 2 , b , n , \\omega , \\varphi ) } , 1 , \\lambda _ 0 / 2 \\} \\end{align*}"}
+{"id": "2121.png", "formula": "\\begin{align*} \\tilde { M } ^ { \\Theta } _ { g ^ { i , j } } ( t , ( \\phi , r , w ) ) : = w _ { i } ( t ) w _ { j } ( t ) - w _ { i } ( 0 ) w _ { j } ( 0 ) - \\frac { 1 } { 2 } \\delta _ { i j } t \\end{align*}"}
+{"id": "6589.png", "formula": "\\begin{align*} \\phi _ { \\Phi _ k , \\ \\beta } ( x , y ) = \\begin{cases} 0 \\ \\ \\ \\mbox { i f } \\ \\ y \\leq 1 \\\\ [ x ] \\ \\ \\ \\mbox { i f } \\ \\ y \\geq x \\end{cases} \\end{align*}"}
+{"id": "6385.png", "formula": "\\begin{align*} \\varphi ( f _ { y } ( x ) , y ) F _ { x , y } = \\varphi ( f _ { x } ( g _ { x } ( y ) ) , x ) G _ { x , y } \\qquad x , y \\in X . \\end{align*}"}
+{"id": "6082.png", "formula": "\\begin{align*} J _ { \\pm } ( x ) = \\begin{cases} { \\rm s g n } ( x ) ( \\mp 1 ) ^ { x + 1 } \\pi , & x \\neq 0 , \\\\ 0 , & x = 0 . \\end{cases} \\end{align*}"}
+{"id": "1166.png", "formula": "\\begin{align*} \\left ( \\operatorname { T r } \\vec f \\right ) ( x ' ) = \\vec t _ { I _ 0 } \\theta ^ { ( \\lambda ' ) } _ { I _ 0 } ( x ' ) = \\vec g ( x ' ) . \\end{align*}"}
+{"id": "1632.png", "formula": "\\begin{align*} u \\left ( x , 0 \\right ) = u _ { 0 } \\left ( x \\right ) , m \\left ( x , 0 \\right ) = m _ { 0 } \\left ( x \\right ) , x \\in \\Omega . \\end{align*}"}
+{"id": "3099.png", "formula": "\\begin{align*} u _ \\varphi ( t ) & = \\left ( \\begin{array} { c c c } 1 & 0 & c _ 2 \\ , t ^ { p ^ { e _ 2 } } \\\\ 0 & 1 & c _ 1 \\ , t ^ { p ^ { e _ 1 } } \\\\ 0 & 0 & 1 \\end{array} \\right ) ( \\ , c _ 1 , c _ 2 \\in k , e _ 1 , e _ 2 \\geq 0 \\ , ) , \\end{align*}"}
+{"id": "1176.png", "formula": "\\begin{align*} \\langle T ( f ) , g \\rangle : = \\lim _ { j \\to \\infty } \\langle T ( \\phi _ j f ) , g \\rangle \\end{align*}"}
+{"id": "6766.png", "formula": "\\begin{align*} \\begin{aligned} f _ 2 ( x _ 2 ) & - f _ 2 ( \\breve { x } _ 2 ^ k ) + ( x _ 2 - \\breve { x } _ 2 ^ k ) ^ T ( - A _ 2 ^ T \\widetilde { \\lambda } ^ k \\\\ & + \\beta A _ 2 ^ T A _ 2 [ \\widetilde { x } _ 2 ^ k - x _ 2 ^ k ] - r A _ 2 ^ T [ \\widetilde { \\lambda } ^ k - \\lambda ^ k ] ) \\geq 0 , ~ \\forall x _ 2 \\in \\mathcal { X } _ 2 . \\end{aligned} \\end{align*}"}
+{"id": "2039.png", "formula": "\\begin{align*} \\begin{bmatrix} \\ , \\ , 0 \\ , & \\ , 0 \\ , & T _ 2 T _ 0 - T _ 1 ^ 2 & \\ , T _ 2 T _ 1 - T _ 3 T _ 0 & \\ , T _ 3 T _ 1 - T _ 2 ^ 2 \\ , \\end{bmatrix} ^ t . \\end{align*}"}
+{"id": "4791.png", "formula": "\\begin{align*} u _ \\ast ( s ) = \\sup \\{ t \\leq 0 : \\mu ( t ) < s \\} . \\end{align*}"}
+{"id": "8589.png", "formula": "\\begin{align*} f ( R , \\mu ) : = \\sum _ { X \\subset \\mathcal { V } _ \\mathbb { Z } : | X | = N } e ^ { - \\mu d _ N ( X , R ) } \\leq \\left ( 1 + \\frac { 2 M } { \\mu } \\right ) e ^ { \\frac { 4 M } { \\mu } } \\ : . \\end{align*}"}
+{"id": "8680.png", "formula": "\\begin{align*} J \\underline { \\mathfrak { g } } | _ { Y ' } \\oplus T Y ' = T M ' | _ { Y ' } . \\end{align*}"}
+{"id": "4715.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { V } } ^ { ( h - 1 ) } ( \\psi ^ { ( \\leq h - 1 ) } ) + E _ { h - 1 } \\ ; & = \\ ; \\log \\left ( \\int P _ { g ^ { ( \\leq h - 1 ) } } e ^ { \\mathcal { V } ^ { ( h ) } ( \\psi ^ { ( \\leq h - 1 ) } + \\psi ^ { ( h ) } ) } \\right ) \\\\ & = \\sum _ { r = 1 } ^ { + \\infty } \\frac { 1 } { r ! } \\mathbb { E } ^ T _ { h } ( \\mathcal { V } ^ { ( h ) } ( \\psi ^ { ( { \\leq } h - 1 ) } + \\cdot ) ; r ) \\end{align*}"}
+{"id": "8475.png", "formula": "\\begin{align*} \\lim _ { h \\searrow 0 } \\mathbf { I I } _ { h , \\ell } = 0 . \\end{align*}"}
+{"id": "3444.png", "formula": "\\begin{align*} & \\overline { d } ( X , m , n ) : = \\overline { d } ( X ) - m - 2 n \\in \\Q , \\\\ & \\underline { d } ( X , m , n ) : = \\underline { d } ( X ) - m - 2 n \\in \\Q . \\end{align*}"}
+{"id": "8838.png", "formula": "\\begin{align*} c ^ * ( \\pm \\infty ) : = \\inf \\limits _ { \\mu > 0 } \\frac { 1 } { \\mu } \\log \\lambda _ \\pm ( \\mu ) . \\end{align*}"}
+{"id": "4698.png", "formula": "\\begin{align*} \\sum _ { \\nu , j \\in \\N } \\gamma _ j \\alpha _ { \\nu , 1 } ^ { - 1 } \\cdot h _ \\nu ^ 2 ( x _ j ) & \\geq \\sum _ { j \\in \\N } \\gamma _ j \\cdot \\alpha _ { 2 , 1 } ^ { - 1 } \\cdot h _ 2 ^ 2 ( x _ j ) \\\\ & \\geq 2 ^ { - 7 } \\cdot \\sum _ { j \\in \\N } ( j - 1 ) ^ 2 \\cdot ( j + 1 ) ^ { - 3 } = \\infty . \\end{align*}"}
+{"id": "8398.png", "formula": "\\begin{align*} \\Delta ( \\sigma ( a ) ) & = \\Delta ( B _ 1 ( a _ 1 ) S ( B _ 2 ( a _ 2 ) ) ) = B _ 1 ( a _ 1 ) S ( B _ 2 ( a _ 4 ) ) \\otimes B _ 1 ( a _ 2 ) S ( B _ 2 ( a _ 3 ) ) . \\end{align*}"}
+{"id": "6648.png", "formula": "\\begin{align*} L ( E , \\mathrm { i } y ) = \\int _ { \\mathbb { C } } \\log | \\tilde { E } - E | \\mathrm { d } \\mathcal { N } ^ { \\mathrm { i } y } ( \\tilde { E } ) . \\end{align*}"}
+{"id": "3452.png", "formula": "\\begin{align*} \\hat { W } = \\left ( ( - \\infty , 0 ] \\times Y _ 0 \\right ) \\cup _ { Y _ 0 } W \\cup _ { Y _ 1 } \\left ( [ 0 , \\infty ) \\times Y _ 1 \\right ) \\end{align*}"}
+{"id": "7770.png", "formula": "\\begin{align*} R _ n & < 2 R _ 1 = : R , \\\\ \\tilde { M } _ { n } & \\leq ( \\tilde { M } _ { 1 } \\frac { | \\ln \\epsilon _ { 1 } | } { 2 \\pi h _ { 1 } } ) ^ { b _ { 3 } \\ln n } e ^ { b _ { 3 } e ^ { 4 \\sqrt { n } } } , \\\\ \\delta _ { n } & > \\delta _ { 1 } ( \\tilde M _ 1 \\frac { | \\ln \\epsilon _ { 1 } | } { 2 \\pi h _ { 1 } } ) ^ { - b _ { 4 } \\ln n } e ^ { - b _ { 4 } e ^ { 4 \\sqrt { n } } } , \\\\ c _ { n } & > ( c _ { 1 } \\frac { 2 \\pi h _ { 1 } } { | \\log \\epsilon _ { 1 } | } ) ^ { n ^ { b _ { 5 } } } e ^ { - b _ { 6 } e ^ { 5 \\sqrt { n } } } , \\end{align*}"}
+{"id": "7182.png", "formula": "\\begin{align*} F _ v ^ { ( k + 1 ) } ( x , t ) & = g _ 1 ( t ) * F _ v ( \\cdot , 0 ) + \\int _ 0 ^ t e ^ { ( 1 - v _ 0 ^ 2 ) ( t - \\tau ) } g _ 1 ( t - \\tau ) * \\left ( ( ( v _ 0 ^ 2 - ( v _ 0 + V ) ^ 2 ) + \\varphi _ { 1 } ) F _ v ^ { ( k ) } - F _ w ^ { ( k ) } + \\varphi _ 3 \\right ) d \\tau \\\\ F _ w ^ { ( k + 1 ) } ( x , t ) & = g _ \\rho ( t ) * F _ w ( \\cdot , 0 ) + \\varepsilon \\int _ 0 ^ { t } e ^ { - \\varepsilon \\gamma ( t - \\tau ) } g _ \\rho ( t - \\tau ) * \\left ( F _ v ^ { ( k + 1 ) } + J _ 0 \\right ) d \\tau , \\end{align*}"}
+{"id": "772.png", "formula": "\\begin{align*} \\mathrm { a d } _ \\pi : = ( \\iota \\otimes \\pi ) \\mathrm { a d } \\colon H \\to H \\otimes K . \\end{align*}"}
+{"id": "4029.png", "formula": "\\begin{align*} \\begin{cases} \\begin{array} { c c l } x _ { 1 } & = & 0 \\\\ x _ { 2 } & = & \\frac { 1 - \\mu } { 3 - \\mu } x _ { 2 } \\left ( y _ { 1 } + y _ { 2 } \\right ) \\\\ y _ { 1 } & = & \\frac { 1 - \\mu } { 3 - \\mu } x _ { 2 } \\left ( y _ { 1 } + y _ { 2 } \\right ) \\\\ y _ { 2 } & = & \\frac { 1 + \\mu } { 3 - \\mu } x _ { 2 } \\left ( y _ { 1 } + y _ { 2 } \\right ) , \\end{array} \\end{cases} \\end{align*}"}
+{"id": "241.png", "formula": "\\begin{align*} \\frac { d ^ 2 \\bar { x } } { d \\bar { t } ^ 2 } = \\frac { c } { h ( x ) } \\frac { d ^ 2 x } { d t ^ 2 } , \\end{align*}"}
+{"id": "6623.png", "formula": "\\begin{align*} H _ N S _ N ( i , j ) & = \\frac { 1 } { J _ N } ( 2 J _ 1 J _ { N - 2 } - J _ { N - 2 } ( J _ 1 + J _ 2 ) + ( - 1 ) ^ { N + 1 } J _ 1 J _ 0 ) \\\\ & = \\frac { 1 } { J _ N } ( 2 J _ { N - 2 } - 2 J _ { N - 2 } ) \\\\ & = 0 . \\end{align*}"}
+{"id": "3907.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 2 \\\\ 0 & 1 & 2 & 2 & 1 & 0 & 0 & 1 & 2 \\end{bmatrix} . \\end{align*}"}
+{"id": "3251.png", "formula": "\\begin{align*} H _ { a , [ \\nu ] } ^ { ( p ) } & : = \\partial _ a ^ p H _ { a , [ \\nu ] } \\\\ H _ { a , [ \\nu ] } ^ { ( - p ) } & : = \\big ( 4 \\partial _ \\nu \\big ) ^ p H _ { a , [ \\nu ] } \\ : , \\end{align*}"}
+{"id": "3000.png", "formula": "\\begin{align*} \\overline { R } ( x , y ) z = \\overline { \\nabla } _ x \\overline { \\nabla } _ y z - \\overline { \\nabla } _ y \\overline { \\nabla } _ x z - \\overline { \\nabla } _ { [ x , y ] } z . \\end{align*}"}
+{"id": "958.png", "formula": "\\begin{align*} P _ V ( g ) = \\int _ { \\partial V } g ( y ) P _ V ( x , y ) \\ , \\sigma ( d y ) , x \\in D . \\end{align*}"}
+{"id": "6914.png", "formula": "\\begin{align*} & n - \\min \\ ( \\frac { n p - 3 p + 2 } { p - 1 } , \\frac { n p - 2 p + 1 } { p } + \\varepsilon \\ ) \\\\ & \\qquad = \\max \\ ( \\frac { 3 p - n - 2 } { p - 1 } , \\frac { 2 p - 1 } { p } + \\varepsilon \\ ) < 2 \\end{align*}"}
+{"id": "626.png", "formula": "\\begin{align*} V ( L _ 1 , L _ 2 , K { [ n - 2 ] } ) = \\frac { 1 } { n ( n - 1 ) } V _ { n - 2 } ( \\pi _ { U ^ { \\perp } } K ) | d e t _ 2 ( u , v ) | = \\frac { 2 } { n ( n - 1 ) } V _ { n - 2 } ( \\pi _ { U ^ { \\perp } } K ) V _ 2 ( L _ 1 , L _ 2 ) \\ , \\ , \\end{align*}"}
+{"id": "5366.png", "formula": "\\begin{align*} \\nu _ j = \\max \\ , \\left \\{ \\frac { - v ^ { S } _ j } { b ^ S _ j } : j \\in S \\in \\{ S _ 1 , \\ldots , S _ n \\} \\right \\} , j \\in N ^ { \\{ 0 , 1 \\} } , \\end{align*}"}
+{"id": "8161.png", "formula": "\\begin{align*} & | \\mathcal B _ { y ^ { i + 1 } } ( \\underline x ^ { i + 1 } ) | \\\\ & = y _ { i + 1 } | \\mathcal B _ { y ^ { i } } ( \\underline x ^ { i } ) \\cap \\mathcal B _ i ^ e | + ( 1 - y _ { i + 1 } ) | \\mathcal B _ { y ^ { i } } ( \\underline x ^ { i } ) \\cap \\mathcal B _ i ^ { e , c } | \\\\ & \\geq y _ { i + 1 } | \\mathcal B _ i ^ e | + ( 1 - y _ { i + 1 } ) ( | \\mathcal B _ { y ^ { i } } ( \\underline x ^ { i } ) | - | \\mathcal B _ i ^ { e } | ) \\end{align*}"}
+{"id": "5607.png", "formula": "\\begin{align*} \\hat { a } - \\hat { v } & = { a } ' - { v } ' = a - h - v ' \\geq a - v - h . \\end{align*}"}
+{"id": "3534.png", "formula": "\\begin{align*} \\psi _ 1 ( z _ 1 ) = \\psi ( z _ 1 , z _ 2 ) = \\sum _ { n , m = 0 } ^ { \\infty } a _ { n m } z _ 1 ^ n z _ 2 ^ m . \\end{align*}"}
+{"id": "6785.png", "formula": "\\begin{align*} \\dot { x } _ j = x _ j \\left ( r _ j + \\sum _ { k \\neq j } b _ { j k } x _ k \\right ) j = 1 , \\ldots , n . \\end{align*}"}
+{"id": "3507.png", "formula": "\\begin{align*} \\| \\bar { h } \\Psi ( a ) \\| = \\| h \\Psi ( a ) \\| = \\| \\Psi ( a ) \\| = \\| a \\| . \\end{align*}"}
+{"id": "4251.png", "formula": "\\begin{align*} S _ a ( u _ 0 ) = u _ 0 - i \\int _ 0 ^ \\infty e ^ { - i t \\Delta } [ a | u ( t ) | ^ 2 u ( t ) ] \\ , d t . \\end{align*}"}
+{"id": "3405.png", "formula": "\\begin{align*} \\frac { 1 } { L \\eta _ { t } } \\left ( \\frac { 1 } { \\lambda _ { t } } \\right ) ^ { p } & \\le \\frac { 1 } { \\sigma ^ { p } } = C _ { 2 } . \\end{align*}"}
+{"id": "1201.png", "formula": "\\begin{align*} ( - \\Delta _ p ) ^ { r } \\phi ( x ) = \\lim _ { \\varepsilon \\to 0 } \\int _ { \\mathbb { R } ^ { N } \\setminus B _ { \\varepsilon } ( x ) } \\frac { \\vert \\phi ( x ) - \\phi ( y ) \\vert ^ { p - 2 } ( \\phi ( x ) - \\phi ( y ) ) } { \\vert x - y \\vert ^ { N + r p } } \\dd x \\dd y \\end{align*}"}
+{"id": "3698.png", "formula": "\\begin{align*} \\mu _ t ( 1 - \\epsilon _ t ) - \\epsilon _ 0 ^ 2 \\sqrt { d - 1 } - 2 k d \\sqrt { \\epsilon _ { t - 1 } } \\leq c _ 1 ^ 2 \\mu _ t + c _ 2 ^ 2 \\mu = \\mu + ( \\mu _ t - \\mu ) c _ 1 ^ 2 . \\end{align*}"}
+{"id": "7140.png", "formula": "\\begin{align*} k ^ \\mu \\varepsilon _ \\mu ( k , \\lambda ) = 0 \\ , \\ , . \\end{align*}"}
+{"id": "3139.png", "formula": "\\begin{align*} \\varphi ^ * \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & = \\left ( \\begin{array} { c c c } a ^ { 2 \\ , p ^ { e _ 1 } } & 0 & 0 \\\\ 0 & 1 & b ^ { p ^ { e _ 1 } } \\ , d ^ { p ^ { e _ 1 } } \\\\ 0 & 0 & d ^ { 2 \\ , p ^ { e _ 1 } } \\end{array} \\right ) ( \\ , e _ 1 \\geq 0 \\ , ) . \\end{align*}"}
+{"id": "2264.png", "formula": "\\begin{align*} \\check R _ { 1 } ( u _ 1 , u _ 2 ) \\check R _ { 2 } ( u _ 1 , u _ 3 ) \\check R _ { 1 } ( u _ 2 , u _ 3 ) = \\check R _ { 2 } ( u _ 2 , u _ 3 ) \\check R _ { 1 } ( u _ 1 , u _ 3 ) \\check R _ { 2 } ( u _ 1 , u _ 2 ) \\ \\ \\ \\ \\ \\end{align*}"}
+{"id": "1182.png", "formula": "\\begin{align*} \\partial _ i p _ \\gamma & = \\sum _ { \\alpha \\in \\mathbb { Z } _ + ^ n , \\ , | \\alpha | \\leq N - k , \\ , \\alpha _ i > 0 } v _ { \\gamma + \\alpha } \\frac { y ^ { \\alpha - e _ i } } { ( \\alpha - e _ i ) ! } \\\\ & = \\sum _ { \\beta \\in \\mathbb { Z } _ + ^ n , \\ , | \\beta | \\leq N - k - 1 } v _ { \\gamma + e _ i + \\beta } \\frac { y ^ { \\beta } } { \\beta ! } = p _ { \\gamma + e _ i } . \\end{align*}"}
+{"id": "9136.png", "formula": "\\begin{align*} X _ { i } = \\big \\{ x ^ { ( \\beta , s ) } _ { i , t } \\ , \\big | \\ , \\beta \\in \\Delta ^ { + } , 1 \\leq s \\leq d _ { \\beta } , 1 \\leq t \\leq \\nu _ { \\beta , i } \\big \\} , \\end{align*}"}
+{"id": "5806.png", "formula": "\\begin{align*} \\sum _ { k + \\ell \\leq s } \\| \\mathcal { L } _ { \\Sigma } ^ { \\ell } \\partial ^ { k } _ t E _ 2 ( u ) \\| _ { L ^ 2 } ^ 2 ( t ) = o ( 1 ) \\sum _ { k + \\ell \\leq s } \\| \\mathcal { L } _ { \\Sigma } ^ { \\ell } \\partial ^ { k } _ t u \\| _ { L ^ 2 } ^ 2 ( t ) . \\end{align*}"}
+{"id": "6553.png", "formula": "\\begin{align*} \\mathbb { E } | R _ { N , 1 } | ^ { \\alpha ' } & \\leq 2 \\sum ^ N _ { n = 1 } \\mathbb { E } \\Big | \\sum ^ { \\infty } _ { j = N - n + 1 } K ^ { \\infty } _ { j , n } \\Big | ^ { \\alpha ' } \\\\ & \\leq c _ 9 \\ , \\left ( 1 + \\sum ^ { N - 1 } _ { n = 1 } \\mathbb { E } \\left | ( N - n ) ^ { 1 - \\beta ' } | \\varepsilon _ n | + ( N - n ) ^ { 1 - \\alpha ' \\beta ' } \\right | ^ { \\alpha ' } \\right ) \\\\ & \\leq c _ { 1 0 } \\ , N ^ { 1 + \\alpha ' ( 1 - \\beta ' ) } \\end{align*}"}
+{"id": "6967.png", "formula": "\\begin{align*} \\widetilde { \\Pi } ( \\mathbf { z } ) = \\mathbf { z } \\overline { \\eta } ( \\mathbf { z } ) + \\mathbf { \\overline { z } } \\eta ( \\mathbf { z } ) . \\end{align*}"}
+{"id": "9182.png", "formula": "\\begin{align*} ( \\bar { 1 } , \\ldots , \\bar { N } ) : = ( n - \\tfrac { 1 } { 2 } , \\ldots , \\tfrac { 1 } { 2 } , 0 , - \\tfrac { 1 } { 2 } , \\ldots , - n + \\tfrac { 1 } { 2 } ) . \\end{align*}"}
+{"id": "4062.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { 0 } \\frac { e ^ { r s } } { r \\mp i z } d r = e ^ { \\pm i z s } E _ i ( \\mp i z s ) . \\end{align*}"}
+{"id": "4053.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s / 2 } \\psi ( x ) = \\frac { 2 ^ s \\Gamma ( \\frac { d + s } { 2 } ) } { \\pi ^ { d / 2 } | \\Gamma ( - \\frac { s } { 2 } ) | } \\int _ { \\R ^ d } \\frac { \\psi ( x ) - \\psi ( y ) } { | x - y | ^ { d + s } } d y . \\end{align*}"}
+{"id": "6492.png", "formula": "\\begin{align*} \\xi _ { n - s } = \\begin{cases} \\xi _ { n } + 1 & n - s \\equiv 0 \\pmod 2 , \\\\ \\xi _ { n } + 3 & n - s \\equiv 1 \\pmod 2 , \\end{cases} \\xi _ { 2 n - s } = \\xi _ s - 2 \\ 1 \\le s \\le n - 1 , \\end{align*}"}
+{"id": "621.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 u ^ { n + \\frac { 1 } { 2 } } \\ln ( 1 - u ) \\ , d u = \\frac { 4 \\ln ( 2 ) - 4 O _ { n + 1 } - \\frac { 4 } { 2 n + 3 } } { 2 n + 3 } . \\end{align*}"}
+{"id": "955.png", "formula": "\\begin{align*} P _ V ( g ) ( x ) = g ( x ) \\mbox { q . e . } x \\in E \\setminus V . \\end{align*}"}
+{"id": "3499.png", "formula": "\\begin{align*} \\| a - a \\varphi _ n ( 1 _ { F _ n } ) \\| & \\leq 2 \\eta + \\| \\varphi _ n ( \\psi _ n ( a ) ) - \\varphi _ n ( \\psi _ n ( a ) ) \\varphi _ n ( 1 _ { F _ n } ) \\| \\\\ & = 2 \\eta + \\| \\varphi _ n ( \\psi _ n ( a ) 1 _ { F _ n } ) - \\varphi _ n ( \\psi _ n ( a ) ) \\varphi _ n ( 1 _ { F _ n } ) \\| \\\\ & < 2 \\eta + \\sqrt { 3 \\eta } \\\\ & < \\varepsilon . \\end{align*}"}
+{"id": "140.png", "formula": "\\begin{align*} { [ f ] } _ { W ^ { 1 , p } ( A ) } = { \\bigg ( \\int _ A { | \\nabla f | } ^ p \\ d x \\bigg ) } ^ { \\frac { 1 } { p } } , \\end{align*}"}
+{"id": "5756.png", "formula": "\\begin{align*} \\frac { d } { d t } \\xi _ { i } - \\Gamma _ i \\xi _ { i } = \\mathcal { E } _ { i } , \\end{align*}"}
+{"id": "2242.png", "formula": "\\begin{align*} \\Gamma _ \\alpha ^ \\alpha = L \\ , , \\end{align*}"}
+{"id": "3148.png", "formula": "\\begin{align*} u _ { \\psi ^ * } ^ - ( s ) = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "6777.png", "formula": "\\begin{gather*} \\left | \\bigcap _ { i \\in I } A _ i \\right | = \\left | \\bigcap _ { j \\in I } \\overline { B _ j } \\right | = \\sum _ { J \\subseteq I } ( - 1 ) ^ { | J | } \\left | \\bigcap _ { j \\in J } B _ j \\right | = \\sum _ { J \\subseteq I } ( - 1 ) ^ { | J | } \\frac { n } { \\prod _ { i \\in I } i \\prod _ { j \\in J } j ^ { n _ j - 1 } } . \\end{gather*}"}
+{"id": "4991.png", "formula": "\\begin{align*} a ( \\nu _ j - 1 ) = d ( \\nu _ j ) = 0 , j = 1 , \\dots , N , \\end{align*}"}
+{"id": "4092.png", "formula": "\\begin{align*} \\hat { u } ( \\xi ) = \\frac { \\widehat { V u } ( \\xi ) } { | \\xi | - \\lambda } = \\chi ( | \\xi | < \\lambda / 2 ) \\frac { \\widehat { V u } ( \\xi ) } { | \\xi | - \\lambda } + \\chi ( | \\xi | \\geq \\lambda / 2 ) \\frac { \\widehat { V u } ( \\xi ) } { | \\xi | - \\lambda } \\end{align*}"}
+{"id": "8010.png", "formula": "\\begin{align*} \\mathbb { E } \\exp \\left ( N \\sum _ { k = 1 } ^ 3 \\mu ^ N _ { 0 , k } ( f _ k ) \\right ) & = \\mathbb { E } \\left ( \\exp \\left ( N \\sum _ { k = 1 } ^ 3 \\mu ^ N _ { 0 , k } ( f _ k ) \\right ) \\mathcal { U } ^ N _ { F , G , H } \\left ( T , \\xi ^ N \\right ) \\right ) \\\\ & \\geq \\mathbb { E } \\left ( \\exp \\left ( N \\sum _ { k = 1 } ^ 3 \\mu ^ N _ { 0 , k } ( f _ k ) \\right ) \\mathcal { U } ^ N _ { F , G , H } \\left ( T , \\xi ^ N \\right ) 1 _ { \\{ \\mu ^ N \\in C \\} } \\right ) . \\end{align*}"}
+{"id": "3840.png", "formula": "\\begin{align*} \\eta ( x , y \\mid s ) = \\eta _ { ( 1 ) } ( x \\mid s ) \\eta _ { 2 | 1 } ( y \\mid s , x ) , \\end{align*}"}
+{"id": "283.png", "formula": "\\begin{align*} e _ { 2 k + 1 } e _ { 2 l } = C _ { k + l } ^ { k } e _ { 2 k + 2 l + 1 } . \\end{align*}"}
+{"id": "2892.png", "formula": "\\begin{align*} & \\lim _ { \\lambda \\to 0 ^ + } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ { L ^ { \\vec { r } } ( \\mathbb { R } ^ n ) } = \\left [ - \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { L ^ { \\vec { r } } ( \\mathbb { R } ^ n ) } . \\end{align*}"}
+{"id": "6284.png", "formula": "\\begin{align*} \\frac 1 T \\sum \\limits _ { k = 0 } ^ { T - 1 } \\langle g _ { k + 1 } , x _ k - x ^ * \\rangle \\leq \\frac { \\kappa } { \\kappa + 1 } \\frac { R _ { 0 } ^ { \\frac { 1 + \\kappa } { \\kappa } } } { \\nu T } + \\frac { \\nu ^ { \\kappa } } { 1 + \\kappa } \\frac { 1 } { T } \\sum \\limits _ { k = 0 } ^ { T - 1 } \\| g _ { k + 1 } \\| ^ { 1 + \\kappa } _ q . \\end{align*}"}
+{"id": "6825.png", "formula": "\\begin{align*} \\sum _ { i , j } \\beta _ { i j } k _ { i j } ^ { m } + K _ { m } = 0 \\qquad \\forall m \\in \\{ 1 , \\ldots , N - 1 \\} . \\end{align*}"}
+{"id": "2767.png", "formula": "\\begin{align*} \\omega = d \\Xi ^ { a } \\wedge d \\Psi _ { a } + d \\Theta ^ { \\alpha } \\wedge d \\Theta _ { \\alpha } + d Q ^ { i } \\wedge d P _ { i } \\end{align*}"}
+{"id": "4805.png", "formula": "\\begin{align*} X : = \\Big \\{ \\mathbf { x } \\in \\mathbb { R } ^ { n _ 1 } _ + \\times \\mathbb { Z } ^ { n _ 2 } _ + : \\mathbf { G } _ 1 \\mathbf { x } ^ { ( 1 ) } + \\mathbf { G } _ 2 \\mathbf { x } ^ { ( 2 ) } \\leq \\mathbf { g } \\Big \\} \\end{align*}"}
+{"id": "1639.png", "formula": "\\begin{align*} D = \\max \\left ( D _ { 1 } , D _ { 2 } , D _ { 3 } , D _ { 4 } \\right ) . \\end{align*}"}
+{"id": "4935.png", "formula": "\\begin{align*} e ^ { S } _ { + } = e ^ { S _ 2 } _ { + } . \\end{align*}"}
+{"id": "8831.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\rightarrow \\infty } \\sup \\left \\{ | | Q _ n [ \\varphi ] ( \\cdot , x ) | | : x \\geq n ( c ^ * _ 1 + \\frac { \\varepsilon } { 3 } ) \\mbox { o r } x \\leq - n ( c ^ * _ 2 + \\frac { \\varepsilon } { 3 } ) \\right \\} = 0 . \\end{align*}"}
+{"id": "9087.png", "formula": "\\begin{align*} X ( t + 1 ) = X ( t ) = X ^ { * } . \\end{align*}"}
+{"id": "870.png", "formula": "\\begin{align*} C _ { 1 1 } = C _ { 2 2 } = 0 , C _ { 1 2 } = { w _ 1 } _ { \\substack { \\\\ x _ 2 } } - { w _ 2 } _ { \\substack { \\\\ x _ 1 } } , C _ { 2 1 } = { w _ 2 } _ { \\substack { \\\\ x _ 1 } } - { w _ 1 } _ { \\substack { \\\\ x _ 2 } } \\end{align*}"}
+{"id": "2116.png", "formula": "\\begin{align*} E ( x _ 1 , \\cdots , x _ 8 ) = \\sum _ { p \\in P _ 8 ^ 2 } \\prod _ { ( i , j ) \\in p } A ( x _ i , x _ j ) \\end{align*}"}
+{"id": "234.png", "formula": "\\begin{align*} \\frac { d ^ 2 x _ 1 } { d t _ 1 ^ 2 } = 0 . \\end{align*}"}
+{"id": "5236.png", "formula": "\\begin{align*} & U _ { j } = \\frac { \\left ( 1 - \\alpha \\right ) p ^ { 2 } _ { j } } { \\left [ \\left ( 1 - \\alpha \\right ) p _ { j } + \\alpha q _ { j } \\right ] ^ { 2 } } \\ ; Y \\\\ & V _ { j } = \\left ( 1 - \\alpha \\right ) Z \\end{align*}"}
+{"id": "2922.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } ( | N _ { i j } ( f ) | ^ { 2 m } \\ , | \\ , \\mathcal { F } _ { j - 1 } ) & \\leqslant \\bigg ( \\sum _ { \\substack { d \\leqslant x _ i \\\\ v _ { P ( d ) } ( d ) \\geqslant 2 } } ^ { y _ { j - 1 } , y _ j } \\tau _ { 2 m - 1 } ( d ) | \\Psi _ f ( x _ i / d , y _ { j - 1 } ) | ^ 2 \\bigg ) ^ m . \\end{aligned} \\end{align*}"}
+{"id": "2495.png", "formula": "\\begin{align*} \\langle u ( t ) , \\vartheta ( t ) \\rangle _ { ( H ^ 1 ) ' , H ^ 1 } - \\int _ \\Omega u ^ { i n } \\vartheta ( 0 ) \\ \\mathrm { d } x & = \\int _ 0 ^ t \\int _ \\Omega u ( s ) \\gamma ( v ( s ) ) \\Delta \\vartheta ( s ) \\ \\mathrm { d } x \\mathrm { d } s \\\\ & + \\int _ 0 ^ t \\langle u ( s ) , \\partial _ t \\vartheta ( s ) \\rangle _ { ( H ^ 1 ) ' , H ^ 1 } \\ \\mathrm { d } s \\end{align*}"}
+{"id": "3165.png", "formula": "\\begin{align*} 3 \\cdot 2 ^ r = 2 ^ d + 2 . \\end{align*}"}
+{"id": "3113.png", "formula": "\\begin{align*} \\varphi ^ * \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) : = \\left ( \\begin{array} { c c c } a ^ { \\ell _ 1 } & 0 & 0 \\\\ 0 & a ^ { \\ell _ 1 } & 0 \\\\ 0 & 0 & d ^ { - \\ell _ 3 } \\end{array} \\right ) . \\end{align*}"}
+{"id": "3751.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( a \\right ) _ { k } \\left ( b \\right ) _ { k } } { k ! \\left ( b + 1 \\right ) _ { k } } \\left [ \\psi \\left ( a + k \\right ) - \\psi \\left ( a \\right ) \\right ] z ^ { k } = b \\ , z ^ { - b } \\frac { \\partial } { \\partial a } \\mathrm { B } _ { z } \\left ( b , 1 - a \\right ) . \\end{align*}"}
+{"id": "8883.png", "formula": "\\begin{align*} \\mathcal P [ \\varphi ] : = \\int _ { \\R ^ n } | x | ^ { - \\tau } | \\varphi | ^ p ( I _ \\alpha * | \\cdot | ^ { - \\tau } | \\varphi | ^ p ) d x = \\| \\sqrt { \\mathcal K _ \\lambda } \\varphi \\| ^ 2 . \\end{align*}"}
+{"id": "9181.png", "formula": "\\begin{align*} i ' : = N + 1 - i , \\end{align*}"}
+{"id": "5237.png", "formula": "\\begin{align*} \\overline { M H } = \\sum _ { i } \\frac { \\overline { p } _ { i } \\overline { q } _ { i } } { \\left ( 1 - \\alpha \\right ) \\overline { p } _ { i } + \\alpha \\overline { q } _ { i } } \\ : \\ : ; \\ : \\overline { M A } = \\sum _ { i } \\alpha \\overline { p } _ { i } + \\left ( 1 - \\alpha \\right ) \\overline { q } _ { i } = 1 \\end{align*}"}
+{"id": "2641.png", "formula": "\\begin{align*} \\mathcal { R } _ { \\alpha \\beta } ( f ( x _ 1 , \\ldots , x _ { \\alpha - 1 } , \\frac { x _ { \\alpha } - x _ { \\beta } } { \\sqrt { 2 } } , x _ { \\alpha + 1 } , \\ldots , x _ { \\beta - 1 } , \\frac { x _ { \\alpha } + x _ { \\beta } } { \\sqrt { 2 } } , x _ { \\beta + 1 } , \\ldots , x _ N ) ) = f ( x _ 1 , \\ldots , x _ N ) . \\end{align*}"}
+{"id": "2484.png", "formula": "\\begin{align*} \\tau = \\left ( 4 H ^ 2 \\left [ 1 + \\lambda R + ( R \\| A \\| + \\| b \\| ) ( 4 R \\| A \\| + \\| b \\| ) \\right ] \\right ) ^ { - 1 } \\end{align*}"}
+{"id": "2089.png", "formula": "\\begin{align*} n ^ d _ t \\le C _ { d , T } ( t ) + 2 0 0 0 \\big ( C _ 2 ^ 4 t ^ 3 + 3 ( C _ 8 + C _ { 5 } ^ 2 ) \\big ) \\exp \\left ( C _ { \\tau , 1 } C _ { s , 1 } ^ 4 \\sigma ^ { - 4 } \\tau ^ 4 \\right ) \\eta ^ 4 d ^ 4 t ^ 3 \\sum _ { k = 0 } ^ { t - 1 } C _ { d , T } ( k ) \\le C _ { \\tau , 2 } C _ { d , T } ( t ) . \\end{align*}"}
+{"id": "1805.png", "formula": "\\begin{align*} T ( z ^ + , z ^ - ) ( u ^ { - } ) = \\phi \\left ( z ^ + , f ( z ^ { - } , u ^ { - } ) + \\Xi ( z ^ { + } , z ^ { - } , u ^ { - } ) \\right ) . \\end{align*}"}
+{"id": "3059.png", "formula": "\\begin{align*} g ( x , y ) = \\frac { ( 8 d ' + 1 ) } { 4 } x ^ 4 - 4 b ' x ^ 2 y ^ 2 + 2 x y ^ 3 + \\left ( \\frac { 4 { b ' } ^ 2 } { 8 d ' + 1 } - \\frac { 8 c ' + 1 } { 4 ( 8 d ' + 1 ) } \\right ) y ^ 4 . \\end{align*}"}
+{"id": "1551.png", "formula": "\\begin{align*} { \\rm d i v } \\ , ( D F ( D u ) ) = 0 \\end{align*}"}
+{"id": "3103.png", "formula": "\\begin{align*} \\varphi \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & = \\left ( \\begin{array} { c c c } a ^ { \\ell _ 1 } & 0 & 0 \\\\ 0 & a ^ { \\ell _ 1 } & c _ 1 \\ , a ^ { \\ell _ 1 } \\ , b ^ { p ^ { e _ 1 } } \\ , d ^ { p ^ { e _ 1 } } \\\\ 0 & 0 & d ^ { - \\ell _ 3 } \\end{array} \\right ) . \\end{align*}"}
+{"id": "1194.png", "formula": "\\begin{align*} \\begin{cases} E \\geq N \\notin \\mathbb Z , \\\\ E > 0 , \\end{cases} \\begin{cases} F \\geq ( K \\vee M ) - n , \\\\ F > \\lfloor L \\rfloor , \\end{cases} G \\geq \\lfloor \\ ! \\lfloor N \\rfloor \\ ! \\rfloor _ + , H \\geq \\lfloor L \\rfloor . \\end{align*}"}
+{"id": "5370.png", "formula": "\\begin{align*} \\bar { v } ^ u = \\lim _ { T \\to \\infty } \\ , \\frac { 1 } { T } \\ , E _ i ^ u \\left [ \\sum _ { t = 0 } ^ T h _ { X ( t ) } ^ { a ( t ) } \\right ] = \\lim _ { \\beta \\nearrow 1 } \\ , ( 1 - \\beta ) \\ , v _ i ^ u ( \\beta ) , \\end{align*}"}
+{"id": "999.png", "formula": "\\begin{align*} \\beta ( x ) = \\int _ { D ^ c } j ( x , y ) \\gamma ( d y ) . \\end{align*}"}
+{"id": "8230.png", "formula": "\\begin{align*} P _ { { \\bf X } _ { i + 1 } , Y ^ { i + 1 } , { \\bf S } } = P _ { Y ^ i , { \\bf S } } P _ { { \\bf X } _ { i + 1 } | Y ^ i } P _ { Y _ { i + 1 } | { \\bf X } _ { i + 1 } , { \\bf S } } . \\end{align*}"}
+{"id": "2646.png", "formula": "\\begin{align*} \\eta ^ d ( \\xi ) = \\eta _ 1 ( \\xi _ 1 ) \\eta _ 1 ( \\xi _ 2 ) \\eta _ 1 ( \\xi _ 3 ) \\cdots \\eta _ 1 ( \\xi _ d ) . \\end{align*}"}
+{"id": "83.png", "formula": "\\begin{align*} c _ 3 = w _ 2 + w _ 4 \\kappa \\geq c , \\end{align*}"}
+{"id": "5757.png", "formula": "\\begin{align*} & \\frac { d } { d t } z _ j = \\mathcal { W } _ j , \\ \\frac { d } { d t } \\bar { z } _ j - m \\bar { z } _ j = \\overline { \\mathcal { W } } _ j . \\end{align*}"}
+{"id": "5061.png", "formula": "\\begin{align*} C _ 2 ( x ) : = \\frac { \\Vert x \\Vert } { \\xi } \\cdot ( x ) \\cdot 2 ^ { - ( b - 1 ) } \\circ \\left \\lfloor \\frac { 2 ^ { ( b - 1 ) \\vert x \\vert } } { \\Vert x \\Vert } + u \\right \\rfloor , \\end{align*}"}
+{"id": "8048.png", "formula": "\\begin{align*} \\limsup _ { \\delta \\rightarrow 0 } \\limsup _ { N \\rightarrow + \\infty } \\frac { N } { \\gamma _ N ^ 2 } \\log \\sup _ { \\sigma \\in \\mathcal { T } } P \\left ( \\sup _ { 0 \\leq t \\leq \\delta } \\left ( \\eta ^ N _ { t + \\sigma } ( \\vec { f } ) - \\eta ^ N _ \\sigma ( \\vec { f } ) \\right ) > \\epsilon , ( D _ 2 ^ N ) ^ c \\right ) = - \\infty . \\end{align*}"}
+{"id": "6959.png", "formula": "\\begin{align*} f ( \\mathbf { z } ) = { \\displaystyle \\sum _ { j = 1 } ^ { n } } 4 ( x _ j y _ { n + 1 } - x _ { n + 1 } y _ { j } ) ^ { 2 } - \\sum _ { 1 \\leq j < k \\leq n } 4 ( x _ j y _ { k } - y _ { k } x _ { j } ) ^ { 2 } . \\end{align*}"}
+{"id": "7060.png", "formula": "\\begin{align*} X _ t = x - \\int _ 0 ^ t b ( s , X _ s ) d s + \\sqrt { 2 } \\int _ 0 ^ t \\sigma ( s , X _ s ) \\circ d W _ s , \\end{align*}"}
+{"id": "4192.png", "formula": "\\begin{align*} \\varphi ( g _ 1 g _ 3 ) \\varphi ( g _ 1 g _ 2 ) \\overset { ( \\ref { e q : a n t i } ) } { = } \\varphi ( g _ 3 ) \\varphi ( g _ 1 ) \\varphi ( g _ 1 ) \\varphi ( g _ 2 ) \\overset { { \\bf A 3 } } { = } \\varphi ( g _ 3 ) \\varphi ( g ^ { 2 } _ 1 ) \\varphi ( g _ 2 ) \\overset { ( \\ref { e q : c o m m } ) } { = } \\varphi ( g _ 3 g ^ { 2 } _ 1 ) \\varphi ( g _ 2 ) \\ , , \\end{align*}"}
+{"id": "3643.png", "formula": "\\begin{align*} d _ { u v } = \\begin{cases} \\frac { p ( u ) - p ( v ) } { \\| p ( u ) - p ( v ) \\| } & p ( u ) \\neq p ( v ) , \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "1660.png", "formula": "\\begin{align*} v \\left ( x , 0 \\right ) = v _ { 0 } \\left ( x \\right ) = u _ { 0 } ^ { \\left ( 1 \\right ) } \\left ( x \\right ) - u _ { 0 } ^ { \\left ( 2 \\right ) } \\left ( x \\right ) , x \\in \\Omega , \\end{align*}"}
+{"id": "2095.png", "formula": "\\begin{align*} & 1 2 5 C _ { \\tau } \\eta ^ 4 d ^ 4 ( t _ 2 - t _ 1 ) \\big ( C _ 2 ^ 4 ( t _ 2 - t _ 1 ) ^ 2 + ( C _ 8 ' + C _ { 5 } '^ 2 ) \\big ) \\sum _ { t = t _ 1 } ^ { t _ 2 - 1 } C _ { d , T } ( t ) \\\\ & \\le C ' T ^ { - 4 } ( t _ 2 - t _ 1 ) ^ 3 ( t _ 2 - t _ 1 ) C C _ { s , 2 } ^ 2 \\big ( \\frac { t _ 2 - 1 } { T } \\big ) ^ 2 = O \\bigg ( \\big ( \\frac { t _ 2 - t _ 1 } { T } \\big ) ^ 2 \\bigg ) . \\end{align*}"}
+{"id": "6030.png", "formula": "\\begin{align*} \\mathbb E _ { \\nu _ \\rho } [ ( \\bar \\xi ^ A _ x + D _ 2 ^ + \\bar \\xi ^ B _ x ) ( \\bar \\xi ^ A _ y + D _ 2 ^ - \\bar \\xi ^ B _ y ) ] = 0 \\end{align*}"}
+{"id": "5929.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { N C } ( G ) ) & = \\dfrac { q ( q - 1 ) } { 2 } ( q ^ { 1 4 } - 3 q ^ { 1 3 } - 4 q ^ { 1 2 } + 1 9 q ^ { 1 1 } - 4 7 q ^ { 9 } + 2 8 q ^ { 8 } + 4 3 q ^ { 7 } - 5 0 q ^ { 6 } + 1 1 q ^ { 5 } + 4 q ^ { 4 } \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - 1 2 q ^ { 3 } + 1 9 q ^ { 2 } - 1 1 q + 2 ) . \\end{align*}"}
+{"id": "3508.png", "formula": "\\begin{align*} \\| \\Psi ( b ) \\Psi ( a ) - \\bar { h } \\Psi ( a ) \\| & \\overset { \\eqref { e q : h 3 } } { = } \\| \\bar { h } \\Psi ( b a ) - \\bar { h } \\Psi ( a ) \\| \\\\ & \\overset { \\phantom { \\eqref { e q : h 3 } } } { = } \\| \\bar { h } \\Psi ( b a - a ) \\| \\\\ & \\overset { \\eqref { e q : h 2 } } { = } \\| b a - a \\| \\end{align*}"}
+{"id": "3367.png", "formula": "\\begin{align*} \\left | \\mathbb { E } ^ { u ^ \\circ } \\left [ \\mu \\Gamma ( x _ K ) \\right ] - \\mathbb { E } ^ { u } \\left [ \\mu \\Gamma ( x _ K ) \\right ] \\right | \\leq e _ \\Gamma ( \\hat { \\phi } ^ K d _ 0 + s \\sum _ { i = 1 } ^ { K - 1 } \\hat { \\phi } ^ i ) . \\end{align*}"}
+{"id": "5584.png", "formula": "\\begin{align*} \\overline f = \\Phi ( \\nu _ i \\nu _ j ) ^ { 2 t } \\left ( \\sum _ { s = 0 } ^ { t } \\frac { \\Phi ^ s } { ( \\nu _ i \\nu _ j d ) ^ { 2 s } } \\right ) ( \\phi _ i \\circ \\phi _ j ) = ( \\nu _ i \\nu _ j ) ^ { 2 t + 2 } \\left ( \\Gamma _ { i j } ^ { ( t + 1 ) } - \\delta _ { i j } \\right ) . \\end{align*}"}
+{"id": "4778.png", "formula": "\\begin{align*} \\varphi _ 2 ' ( \\varepsilon , b , D , \\eta , E , f ) : = \\max \\{ \\varphi ( \\varepsilon / 3 , b , f ) , \\varphi _ 1 ( \\varepsilon / 6 E , b , D , f ( 2 D \\eta ( \\min \\{ \\varepsilon / 1 2 E , 2 \\} ) ) , \\eta , f ) \\} \\end{align*}"}
+{"id": "6692.png", "formula": "\\begin{align*} N = \\{ [ x _ 1 , y _ 1 ] \\cdots [ x _ r , y _ r ] \\mid x _ 1 , y _ 1 , \\dots , x _ r , y _ r \\in G \\} \\subseteq _ \\mathrm { c } G , \\end{align*}"}
+{"id": "3532.png", "formula": "\\begin{align*} K _ 2 = \\{ z _ 1 \\in \\mathbb { T } : ( z _ 1 , z _ 2 ) \\in K _ 1 z _ 2 \\in \\mathbb { T } \\} , \\end{align*}"}
+{"id": "5031.png", "formula": "\\begin{align*} \\mathbf { w } ^ \\emptyset = \\mathbf { 1 } \\mathbf { r } ^ \\emptyset = \\mathbf { R } . \\end{align*}"}
+{"id": "482.png", "formula": "\\begin{align*} A ' = ( x \\mapsto - x + b ) = ( x \\mapsto b ^ { - 1 } x ) \\cdot ( x \\mapsto - x + 1 ) \\cdot ( x \\mapsto b x ) , \\end{align*}"}
+{"id": "1706.png", "formula": "\\begin{align*} \\Phi \\left ( z _ 1 , \\ldots , z _ n , \\overline { z _ 1 } , \\ldots , \\overline { z _ n } \\right ) : = f ( x _ 1 ) + \\sum _ { 0 < j < j + k \\leq n } x _ j x _ k x _ n ^ { n - j - k } , \\end{align*}"}
+{"id": "798.png", "formula": "\\begin{align*} { D _ m } { D _ k } \\Psi - { D _ k } { D _ m } \\Psi & = ( { \\delta _ m } { \\delta _ k } \\Psi - { \\delta _ r } \\Psi G _ { m k } ^ r ) - ( { \\delta _ k } { \\delta _ m } \\Psi - { \\delta _ r } \\Psi G _ { k m } ^ r ) \\\\ & = { \\delta _ m } { \\delta _ k } \\Psi - { \\delta _ k } { \\delta _ m } \\Psi = ( { \\delta _ m } { \\delta _ k } - { \\delta _ k } { \\delta _ m } ) \\Psi = \\left [ { { \\delta _ m } , { \\delta _ k } } \\right ] \\Psi . \\end{align*}"}
+{"id": "7612.png", "formula": "\\begin{align*} D _ j = D _ j ( S ^ 1 ) = F ( \\C , j ) _ + \\wedge _ { S _ j } ( S ^ 1 ) ^ { \\wedge j } \\end{align*}"}
+{"id": "716.png", "formula": "\\begin{align*} w \\left ( \\frac { z } { r } \\right ) & = e ^ { \\frac { - \\beta } { r ^ 2 } ( | z _ 1 | ^ 2 + \\cdots + | z _ n | ^ 2 ) } \\\\ & \\le e ^ { - \\beta ( | z _ 1 | ^ 2 + \\cdots + | z _ n | ^ 2 ) } \\\\ & = w ( z ) . \\end{align*}"}
+{"id": "8380.png", "formula": "\\begin{align*} t \\in { \\rm S t a b } _ { T } ( \\eta ) & \\iff w _ { 0 , J ( w ) } t w _ { 0 , J ( w ) } \\xi = \\xi \\\\ & \\iff w _ { 0 , J ( w ) } t w _ { 0 , J ( w ) } \\in { \\rm S t a b } _ { T } ( \\xi ) \\\\ & \\iff t \\in w _ { 0 , J ( w ) } { \\rm S t a b } _ { T } ( \\xi ) w _ { 0 , J ( w ) } . \\end{align*}"}
+{"id": "3947.png", "formula": "\\begin{align*} \\mathcal Q _ { N \\times N , \\gamma } : = \\mathcal B _ { N \\times N } ^ { - 1 } \\left ( - \\gamma ^ { - 4 } \\overline { \\mathcal { G } } _ N + I _ N \\otimes K _ 0 \\right ) . \\end{align*}"}
+{"id": "7511.png", "formula": "\\begin{align*} M _ { s } ^ k = \\frac { \\left ( M _ { s } ^ { k - 1 } \\backslash \\phi _ k ( ( 0 , s ^ { 1 / 4 } ] \\times X \\times [ - \\eta / 2 , \\eta / 2 ] ) \\right ) \\bigsqcup \\left ( \\{ \\mathbf { r } _ s \\leq 1 \\} \\times I _ { \\eta } ^ { k } \\right ) } { \\thicksim _ k } , \\end{align*}"}
+{"id": "4402.png", "formula": "\\begin{align*} F _ { \\overline { \\mathcal { Y } } , a } ( x ) & : = \\sum _ { ( i , j ) \\in [ m ] \\times [ n ] } \\overline { y } _ { ( i , j ) } g _ { ( i , j ) } ( x ) + \\sum _ { ( q , p ) \\in \\overline { \\mathcal { Y } } } \\Delta y _ { ( q , p ( q ) ) } g _ { ( q , p ( q ) ) } ( x ) - \\Delta y _ { ( a , b ( a ) ) } g _ { ( a , b ( a ) ) } ( x ) . \\end{align*}"}
+{"id": "7769.png", "formula": "\\begin{align*} r _ { n + 1 } & = m ^ { 2 } r _ { n } , \\ \\ \\nu _ { n + 1 } = \\nu _ n - b \\tilde M _ n ^ 4 \\epsilon _ n ^ { \\frac { 1 } { m } } , \\\\ \\delta _ { n + 1 } & = { b } ^ { - 1 } R _ n ^ { - ( 2 m - 1 ) } \\nu _ { n } ^ { m } \\delta _ { n } , \\ \\ \\tilde { M } _ { n + 1 } = \\tilde { M } _ { n } + 2 0 m \\tilde M _ n ^ 2 \\epsilon _ n , \\\\ c _ { n + 1 } & = ( { b } ^ { - 1 } R _ { n } ^ { - 1 } r _ { n } ^ { - 1 } \\delta _ { n } c _ { n } ) ^ { \\kappa ^ { r _ { n } } } - ( { b } R _ n \\tilde { M } _ { n } \\delta _ { n } ^ { - 1 } r _ { n } ) ^ { \\kappa r _ { n } } \\epsilon _ { n } , \\end{align*}"}
+{"id": "4265.png", "formula": "\\begin{align*} K _ \\varphi ( x ) = \\int _ 0 ^ \\infty \\tfrac { 1 } { 1 + t ^ 2 } \\exp \\bigl \\{ - \\tfrac { x ^ 2 } { 1 + t ^ 2 } \\bigr \\} \\ , d t . \\end{align*}"}
+{"id": "3185.png", "formula": "\\begin{align*} 0 \\leq \\mu = d _ n - 2 m \\leq 6 n + 3 . \\end{align*}"}
+{"id": "2623.png", "formula": "\\begin{align*} | D _ 4 ( S ) | \\geq | D _ 2 ( S ) | + 2 = | D _ { \\{ 1 \\} } ( S ) \\cup [ D _ { \\{ 2 \\} } ( S ) \\setminus D _ { \\{ 1 \\} } ( S ) ] | + 2 \\geq ( n - 1 ) + \\left ( \\frac { n } { 3 } - \\frac { 7 } { 3 } \\right ) + 2 = \\frac { 4 } { 3 } n - \\frac { 4 } { 3 } . \\end{align*}"}
+{"id": "3586.png", "formula": "\\begin{align*} \\alpha ^ 3 ( \\tau _ 0 ^ 2 \\tau _ 1 ) ( \\tau - i d ) + ( a + b ) \\alpha \\tau _ 0 ( \\tau - i d ) = 0 . \\end{align*}"}
+{"id": "3760.png", "formula": "\\begin{align*} | S | & \\geq | S \\cap f ^ { - 1 } ( Z ) | + | S \\setminus f ^ { - 1 } ( Z ) | \\\\ & \\geq | ( \\{ 0 , 1 \\} + Z ) \\cap f ^ { - 1 } ( Z ) | + | ( X + Y ) \\setminus f ^ { - 1 } ( Z ) | \\\\ & \\geq 2 | Z | + | f ( X + Y ) \\setminus Z | \\\\ & \\geq 2 | Z | + \\big ( \\min \\{ 1 , | X | + | Y | \\} - | Z | \\big ) \\\\ & = \\min \\{ 1 , | X | + | Y | \\} + | Z | . \\end{align*}"}
+{"id": "3457.png", "formula": "\\begin{align*} S W F ( L ) : = S W F ( \\Sigma ( L ) , \\iota , \\mathfrak { t } _ L ) . \\end{align*}"}
+{"id": "5337.png", "formula": "\\begin{align*} \\nu _ { \\pi _ m } = \\min \\ , \\left \\{ \\nu ^ { S _ m } _ j : j \\in S _ m \\right \\} . \\end{align*}"}
+{"id": "8584.png", "formula": "\\begin{align*} \\mathfrak { N } : = \\{ k M : k \\in \\{ M , M + 1 , M + 2 , \\dots \\} \\} \\ : . \\end{align*}"}
+{"id": "2805.png", "formula": "\\begin{align*} H _ { T } = - \\Psi _ { 1 } \\Xi ^ { 3 } - \\frac { 1 } { 2 } \\Xi ^ { 2 } ( \\Psi _ { 3 } ) ^ { 2 } + \\zeta \\Psi _ { 2 } . \\end{align*}"}
+{"id": "1621.png", "formula": "\\begin{align*} \\b Q ^ s _ { g \\oplus h } ( x y ) \\mapsto \\sum _ { j + k = s } ( - 1 ) ^ { 2 j k ( p - 1 ) } \\left ( \\b Q ^ j _ g ( x ) Q ^ k _ h ( y ) + ( - 1 ) ^ n Q ^ j _ g ( x ) \\b Q ^ k _ h ( y ) \\right ) . \\end{align*}"}
+{"id": "1886.png", "formula": "\\begin{align*} \\mathcal { M } _ { 2 , w } ^ { | \\cdot | , u } ( K , z ) = \\mathcal { M } _ { 2 , w } ^ { \\Re , + , u } ( K , z ) + \\mathcal { M } _ { 2 , w } ^ { \\Re , + , u } ( K , - z ) + \\mathcal { M } _ { 2 , w } ^ { \\Re , + , u } ( K , i z ) + \\mathcal { M } _ { 2 , w } ^ { \\Re , + , u } ( K , - i z ) , \\end{align*}"}
+{"id": "647.png", "formula": "\\begin{align*} h ^ * \\coloneqq \\int _ { \\rho _ - } ^ { 2 \\rho _ - } d _ n ^ { \\frac 1 n } = d _ n ^ { \\frac 3 n } . \\end{align*}"}
+{"id": "4196.png", "formula": "\\begin{align*} & \\left \\{ \\widehat { A } ( T X ) { \\rm c h } ( \\triangle ( X ) ) { \\rm c h } ( \\widetilde { T X } ) + 1 6 \\widehat { A } ( T X ) { \\rm c h } ( \\widetilde { T X } + \\wedge ^ 2 \\widetilde { T X } ) \\right \\} ^ { ( 8 ) } \\\\ & = 2 4 0 \\left \\{ \\widehat { A } ( T X ) { \\rm c h } ( \\triangle ( X ) ) + 3 2 \\widehat { A } ( T X ) \\right \\} ^ { ( 8 ) } . \\end{align*}"}
+{"id": "4540.png", "formula": "\\begin{align*} \\frac { \\alpha _ N ( t ^ * ) } { t ^ * } & = \\frac { ( a _ - ) ^ N } { a _ + t ^ * } \\frac { 1 } { ( a _ + ) ^ { N - 1 } } \\int _ 0 ^ { t ^ * } \\frac { 1 } { ( N - 1 ) ! } ( a _ + s ) ^ { N - 1 } e ^ { - s a _ + } d ( a _ + s ) \\\\ & = a _ - \\left ( \\frac { a _ - } { a _ + } \\right ) ^ { N - 1 } \\frac { 1 } { \\tau } \\int _ 0 ^ { \\tau } \\frac { 1 } { ( N - 1 ) ! } \\tilde { s } ^ { N - 1 } e ^ { - \\tilde { s } } d \\tilde { s } , \\end{align*}"}
+{"id": "4719.png", "formula": "\\begin{align*} \\prod _ { \\tilde { T } \\subseteq T } \\gamma ^ { - h _ { \\tilde { T } } } e ^ { - \\eta C \\tilde { \\gamma } ^ { - h _ { \\tilde { T } } } M _ { \\tilde { T } } } & \\leq e ^ { - \\frac { \\eta } { 2 } C \\tilde { \\gamma } ^ { - h _ T } } \\prod _ { \\tilde { T } \\subseteq T } \\left ( \\gamma ^ { - h _ { \\tilde { T } } } e ^ { - \\frac { \\eta } { 2 } C \\tilde { \\gamma } ^ { - h _ { \\tilde { T } } / \\tau } } \\right ) ^ { M _ { \\tilde { T } } } \\ , . \\end{align*}"}
+{"id": "1065.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { n - 1 } \\{ S _ { 1 , k } ( n , s , t ) + S _ { 2 , k } ( n , s , t ) \\} \\\\ \\leq \\frac { \\pi K _ 3 \\{ r \\sin ( \\pi d ) \\} ^ { k - 1 } n ^ { ( 1 / 2 ) - d } } { ( 1 - 4 d ^ 2 ) ^ { 1 / 2 } ( s + 1 ) ^ { ( 1 / 2 ) + d } } . \\end{align*}"}
+{"id": "156.png", "formula": "\\begin{align*} \\alpha ^ { - 1 } ( Z _ { B } ( B _ { F ^ { c } } ) ) & = Z _ { A } ( \\alpha ^ { - 1 } ( B _ { F ^ { c } } ) ) \\\\ & \\subseteq Z _ { A } ( A _ { ( F ^ { \\prime } ) ^ { c } } ) \\\\ & \\subseteq A _ { F ^ { \\prime \\prime } } \\end{align*}"}
+{"id": "5801.png", "formula": "\\begin{align*} X _ 0 ( t ) Y _ 0 ( t ) = \\sum _ { k + \\ell \\leq s } \\sum _ { 1 \\leq j \\leq J } x ^ { ( k , \\ell ) } _ j ( t ) \\mathcal { W } ^ { ( k , \\ell ) } _ j ( t ) , \\end{align*}"}
+{"id": "8332.png", "formula": "\\begin{align*} \\begin{cases} [ \\Pi , W _ { t } ] _ { S N } + \\frac { 1 } { 2 } [ W _ t , W _ t ] _ { S N } = 0 \\\\ \\frac { d } { d t } W _ { t } = [ \\Pi + W _ t , X _ { t } ] _ { S N } \\end{cases} \\end{align*}"}
+{"id": "2148.png", "formula": "\\begin{align*} F _ { \\min } = - 2 9 . 2 0 0 0 , x ^ * = ( 0 . 0 0 0 0 , 0 . 9 0 0 0 ) , y ^ * = ( 0 . 0 0 0 0 , 0 . 6 0 0 0 , 0 . 4 0 0 0 0 ) . \\end{align*}"}
+{"id": "117.png", "formula": "\\begin{align*} \\begin{aligned} \\inf _ { q \\in Q } \\sup _ { \\vec { v } \\in V } \\frac { ( \\operatorname { d i v } \\vec { v } , q ) } { \\| \\varepsilon ( \\vec { v } ) \\| \\| q \\| } = \\inf _ { \\vec { v } \\in W ^ { \\perp } } \\sup _ { q \\in Q } \\frac { ( \\operatorname { d i v } \\vec { v } , q ) } { \\| \\varepsilon ( \\vec { v } ) \\| \\| q \\| } \\ge \\beta > 0 . \\end{aligned} \\end{align*}"}
+{"id": "6946.png", "formula": "\\begin{gather*} \\iota : \\mathrm { S O } _ + ( m , 1 ) \\rightarrow \\mathrm { S U } ( n , 1 ) \\\\ G = \\left ( \\begin{matrix} A & \\mathbf { b } \\\\ \\mathbf { c } & d \\end{matrix} \\right ) \\mapsto \\left ( \\begin{matrix} A & 0 & \\mathbf { b } \\\\ 0 & I & 0 \\\\ \\mathbf { c } & 0 & d \\end{matrix} \\right ) \\end{gather*}"}
+{"id": "4460.png", "formula": "\\begin{align*} F _ 0 : = f _ 0 - f _ 1 f _ 2 , \\end{align*}"}
+{"id": "7067.png", "formula": "\\begin{align*} X _ t = x - \\int _ 0 ^ t b ( X _ r ) d r + \\sigma W _ t , \\end{align*}"}
+{"id": "970.png", "formula": "\\begin{align*} w _ i = R ^ D \\hat f _ i ( \\cdot , w _ i ) + R ^ D \\mu _ i D , i = 1 , 2 . \\end{align*}"}
+{"id": "6482.png", "formula": "\\begin{align*} - 2 m & | H | | d \\phi | | A | | f | | \\Delta f | - 2 m | H | | A | ^ 3 | d \\phi | f ^ 2 \\\\ & = - 2 \\sqrt { m } | A | ^ 2 \\sqrt { | A | ^ 2 - m } | f | | \\Delta f | - 2 \\sqrt { m } | A | ^ 4 \\sqrt { | A | ^ 2 - m } f ^ 2 . \\end{align*}"}
+{"id": "5055.png", "formula": "\\begin{align*} \\mathbb { E } _ C \\begin{bmatrix} \\Vert C ( x ) - x \\Vert \\end{bmatrix} \\leq \\sqrt { \\varphi } \\Vert x \\Vert , \\forall x \\in \\mathbb { R } ^ d . \\end{align*}"}
+{"id": "1797.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t u + \\partial _ x f ( \\zeta ^ h , u ) = \\sum \\limits _ { \\bar x \\in \\mathcal { I } ( \\zeta ^ h ) } \\Xi \\left ( \\zeta ^ h ( \\bar x + ) , \\zeta ^ h ( \\bar x - ) , u ( \\cdot , \\bar x - ) \\right ) \\ ; \\delta _ { \\bar x } \\\\ u ( 0 , x ) = u _ o ( x ) , \\end{array} \\right . \\end{align*}"}
+{"id": "6738.png", "formula": "\\begin{align*} F _ n = \\frac { \\alpha ^ n - \\beta ^ n } { \\alpha - \\beta } , L _ n = \\alpha ^ n + \\beta ^ n , n \\geq 0 , \\end{align*}"}
+{"id": "448.png", "formula": "\\begin{align*} \\Delta ( \\Omega ^ q _ + ) : = \\left \\{ \\begin{aligned} & \\Delta \\overline { t } \\mbox { s a t i s f y i n g } x _ { B ^ q } ( t _ q + \\Delta \\overline { t } ) = \\max _ { ( i , j ) \\in \\Omega ^ q _ + \\cap { \\cal A } ^ q _ > } x ^ * _ j ( t _ q ) , \\mbox { i f } \\ ; \\Omega ^ q _ + \\cap { \\cal A } ^ q _ > \\neq \\emptyset , \\\\ [ 2 p t ] & - \\infty , \\mbox { o t h e r w i s e , } \\end{aligned} \\right . \\end{align*}"}
+{"id": "141.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 ^ { + } } \\frac { 1 } { t ^ { n + p } } \\int _ A \\int _ A { | f ( x ) - f ( y ) | } ^ p \\cdot J \\bigg ( \\frac { x - y } { t } \\bigg ) \\ d x d y = K ( n , p ) \\cdot { \\big ( { [ f ] } _ { W ^ { 1 , p } ( A ) } \\big ) } ^ p , \\end{align*}"}
+{"id": "609.png", "formula": "\\begin{align*} A = \\frac { 1 } { 4 } \\beta \\left ( \\frac { 1 } { 2 } , \\frac { 1 } { 4 } \\right ) = \\int _ 0 ^ 1 \\frac { 1 } { \\sqrt { 1 - t ^ 4 } } \\ , d t = \\sum _ { n = 0 } ^ { \\infty } \\left ( \\frac { 1 } { 4 } \\right ) ^ n \\binom { 2 n } { n } \\frac { 1 } { 4 n + 1 } = \\frac { \\Gamma ^ 2 \\left ( \\frac { 1 } { 4 } \\right ) } { 4 \\sqrt { 2 \\pi } } \\end{align*}"}
+{"id": "4759.png", "formula": "\\begin{align*} \\psi ( \\varepsilon , b , \\omega ) : = \\frac { \\omega ( 2 b , \\varepsilon ) } { 2 b } . \\end{align*}"}
+{"id": "1160.png", "formula": "\\begin{align*} \\left [ \\operatorname { E x t } \\theta ^ { ( \\lambda ' ) } _ I \\right ] ( x ) : = & \\ , \\frac { [ \\ell ( I ) ] ^ { \\frac 1 2 } } { \\varphi ( - k _ 0 ) } \\left [ \\theta ^ { ( \\lambda ' ) } \\otimes \\varphi \\right ] _ { Q ( I , k _ 0 ) } ( x ) \\\\ = & \\ , \\frac { [ \\ell ( I ) ] ^ { \\frac 1 2 } } { \\varphi ( - k _ 0 ) } \\theta ^ { ( ( \\lambda ' , 0 ) ) } _ { Q ( I , k _ 0 ) } ( x ) \\\\ = & \\ , \\frac { 1 } { \\varphi ( - k _ 0 ) } \\theta ^ { ( \\lambda ' ) } _ I ( x ' ) \\varphi \\left ( \\frac { x _ n } { \\ell ( I ) } - k _ 0 \\right ) , \\end{align*}"}
+{"id": "2868.png", "formula": "\\begin{align*} \\mathrm { Z } _ 1 & = | I _ i | ^ \\gamma \\omega \\left ( I _ i \\right ) \\leq | I _ i | ^ { \\gamma + 1 } \\inf _ { t \\in I _ i } \\mathcal { M } ( \\omega ) ( t ) \\\\ & \\lesssim [ \\omega ] _ { A _ 1 ( \\mathbb { R } ) } \\inf _ { t \\in I _ i } \\omega ( t ) \\int _ { I _ i } f ( s ) \\ , d s \\leq [ \\omega ] _ { A _ 1 ( \\mathbb { R } ) } \\int _ { I _ i } f ( s ) \\omega ( s ) \\ , d s . \\end{align*}"}
+{"id": "545.png", "formula": "\\begin{align*} f ( t ) : = \\sum _ { k = 1 } ^ { \\infty } \\frac { a _ { k } \\mathrm { e } ^ { - \\lambda _ { k } T } } { \\Phi ' _ { k } ( 1 ) } \\psi _ { k } ( t ) . \\end{align*}"}
+{"id": "2167.png", "formula": "\\begin{align*} I _ 1 = f _ d ^ { - 1 } ( J _ 1 ) , \\dots , I _ { m + 1 } = f _ d ^ { - 1 } ( J _ { m + 1 } ) . \\end{align*}"}
+{"id": "842.png", "formula": "\\begin{align*} ( n + 1 ) X ^ { i } \\Psi _ { i } = \\nabla _ { i } \\left ( X ^ { i } f \\right ) - f ( \\nabla _ { i } X ^ { i } + I _ { i } \\nabla _ { 0 } X ^ { i } ) + f \\nabla _ { 0 } \\rho + h \\rho - \\rho \\Psi . \\end{align*}"}
+{"id": "9125.png", "formula": "\\begin{align*} \\{ d _ \\beta \\} _ { \\beta \\in \\Delta ^ + } < \\{ d ' _ \\beta \\} _ { \\beta \\in \\Delta ^ + } \\Longleftrightarrow \\exists \\ \\gamma \\in \\Delta ^ + \\ \\mathrm { s . t . } \\ d _ \\gamma < d ' _ \\gamma \\ \\mathrm { a n d } \\ d _ \\beta = d ' _ \\beta \\ \\mathrm { f o r \\ a l l } \\ \\beta < \\gamma . \\end{align*}"}
+{"id": "4662.png", "formula": "\\begin{align*} 2 h _ { \\epsilon , \\beta } ' + \\frac { 3 } { 2 } h _ { \\epsilon , \\beta } ^ 2 = \\frac { 3 } { 2 } \\epsilon ^ 2 > 0 . \\end{align*}"}
+{"id": "1256.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ; q ^ 2 ) _ k ( - 1 ; q ^ 2 ) _ k } { ( q ^ 2 ; q ^ 2 ) _ k } q ^ { ( n - 1 ) ^ 2 - k ^ 2 } \\equiv q ^ { 1 - n } + \\frac { ( 1 - n ) q ( 1 - q ^ n ) } { 2 } \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "2195.png", "formula": "\\begin{align*} \\Phi | _ { t = 0 } = \\Phi ( 0 ) , \\ \\ \\Gamma | _ { t = 0 } = \\Gamma ( 0 ) . \\end{align*}"}
+{"id": "8086.png", "formula": "\\begin{align*} \\psi ( b ^ - _ 1 ( n ) n z _ 0 ) - \\psi ( n z _ 0 ) ) = \\int _ 0 ^ 1 \\frac { \\partial } { \\partial t } \\tilde { \\psi } _ t ( b ^ - _ t ( n ) n z _ 0 ) \\ ; d t = \\int _ 0 ^ 1 L _ { \\tilde { w } _ t } ( \\tilde { \\psi } _ t ) ( p ^ - _ t ( n ) n z _ 0 ) \\ ; d t . \\end{align*}"}
+{"id": "5364.png", "formula": "\\begin{align*} \\nu _ { \\pi _ k } & = \\frac { v ^ { S _ { k + 1 } } - v ^ { S _ k } } { b ^ { S _ { k } } - b ^ { S _ { k + 1 } } } \\\\ & = \\min \\ , \\left \\{ \\frac { v ^ { S _ { k } \\setminus \\{ j \\} } - v ^ { S _ k } } { b ^ { S _ { k } } - b ^ { S _ { k } \\setminus \\{ j \\} } } : j \\in S _ k \\right \\} \\\\ & = \\max \\ , \\left \\{ \\frac { v ^ { S _ k } - v ^ { S _ k \\cup \\{ j \\} } } { b ^ { S _ k \\cup \\{ j \\} } - b ^ { S _ k } } : j \\in N ^ { \\{ 0 , 1 \\} } \\setminus S _ k \\right \\} . \\end{align*}"}
+{"id": "2036.png", "formula": "\\begin{align*} P _ T \\ , \\ , = \\ , \\ , \\begin{bmatrix} c _ m & c _ { m - 1 } & \\cdots & c _ { m - e } \\\\ c _ { m - 1 } & c _ { m - 2 } & \\cdots & c _ { m - e - 1 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ c _ { d + 2 } & c _ { d + 1 } & \\cdots & c _ { d - e + 2 } \\\\ c _ { d + 1 } & c _ { d } & \\cdots & c _ { d - e + 1 } \\end{bmatrix} . \\end{align*}"}
+{"id": "5610.png", "formula": "\\begin{align*} - | \\hat { E } _ 3 | - 2 | \\hat { E } _ 2 | + 2 \\hat { v } _ 2 = \\sum _ { v \\in \\hat { V } _ { \\tilde \\gamma } ^ 2 } ( 2 - \\deg ( v ) ) \\leq 0 . \\end{align*}"}
+{"id": "4152.png", "formula": "\\begin{align*} & Q ^ * ( \\tilde { \\theta } ) \\psi ^ { ( r ) } ( \\tilde { \\theta } ) + ( 1 - w _ { j j } ) r ^ { 2 j + 1 / 2 } \\eta _ j \\bigl ( \\psi ^ { ( r ) } ( \\tilde { \\theta } ) - \\psi ^ { ( r ) } _ i ( \\tilde { \\theta } ) \\bigr ) = o ( r ^ { 2 j + 1 } ) . \\end{align*}"}
+{"id": "2050.png", "formula": "\\begin{align*} \\partial _ s \\Theta ( s , x ) = - \\alpha \\int _ 0 ^ 1 A ( x , y ) \\Theta ( s , y ) \\mathrm { d } y . \\end{align*}"}
+{"id": "4481.png", "formula": "\\begin{align*} M _ H ( Z _ 0 , J , \\rho ) = \\| \\tilde F _ 0 \\| _ { \\partial M , \\rho } ^ 2 . \\end{align*}"}
+{"id": "5803.png", "formula": "\\begin{align*} X _ 0 ' = Y _ 0 , X _ - ' + b X _ - \\leq Y _ - . \\end{align*}"}
+{"id": "5281.png", "formula": "\\begin{align*} L S _ { d } \\left ( x . y \\right ) = \\frac { \\exp \\epsilon \\log ( x . y ) - \\exp ( - \\epsilon ) \\log ( x . y ) } { 2 \\ ; \\epsilon } \\end{align*}"}
+{"id": "2791.png", "formula": "\\begin{align*} \\frac { \\partial L _ { T } } { \\partial \\zeta ^ { \\alpha } } = - \\frac { \\partial H _ { T } } { \\partial \\zeta ^ { \\alpha } } = \\Psi _ { \\alpha } ( t ) = 0 . \\end{align*}"}
+{"id": "2318.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } u - \\Delta u + u \\cdot \\nabla u + \\nabla p = f ^ { c , \\gamma } , \\\\ \\mathrm { d i v } u = 0 , \\\\ u ( x , 0 ) = u ^ { c , \\gamma } + w _ { 0 } , \\end{cases} \\end{align*}"}
+{"id": "5928.png", "formula": "\\begin{align*} \\dfrac { M _ { 2 } ( \\mathcal { N C } ( A ( n , p ) ) ) } { | e ( \\mathcal { N C } ( A ( n , p ) ) ) | } & = \\dfrac { ( p ^ { 3 n } - p ^ { n } ) ( p ^ { 8 n } ( p ^ { n } - 3 ) + p ^ { 6 n } ( 3 p ^ { n } - 1 ) ) } { p ^ { 2 n } ( p ^ { n } - 1 ) ( p ^ { 3 n } - p ^ { n } ) } \\\\ & = \\dfrac { p ^ { 8 n } ( p ^ { n } - 2 ) + p ^ { 5 n } ( 2 p ^ { n } - 1 ) } { p ^ { 3 n } - p ^ { n } } \\\\ & = \\dfrac { M _ { 1 } ( \\mathcal { N C } ( A ( n , p ) ) ) } { | v ( \\mathcal { N C } ( A ( n , p ) ) ) | } . \\end{align*}"}
+{"id": "6787.png", "formula": "\\begin{align*} \\dot { x } & = c _ 1 \\kappa _ 1 x ^ { a _ 1 } y ^ { b _ 1 } + c _ 2 \\kappa _ 2 x ^ { a _ 2 } y ^ { b _ 2 } + c _ 3 \\kappa _ 3 x ^ { a _ 3 } y ^ { b _ 3 } , \\\\ \\dot { y } & = d _ 1 \\kappa _ 1 x ^ { a _ 1 } y ^ { b _ 1 } + d _ 2 \\kappa _ 2 x ^ { a _ 2 } y ^ { b _ 2 } + d _ 3 \\kappa _ 3 x ^ { a _ 3 } y ^ { b _ 3 } \\end{align*}"}
+{"id": "3436.png", "formula": "\\begin{align*} \\underline { d } ( X ) = \\dim V - \\overline { d } ( X ' ) . \\end{align*}"}
+{"id": "5143.png", "formula": "\\begin{align*} \\frac { \\partial L _ { d } D 2 ( p \\| q ) } { \\partial q _ { j } } = & \\left \\{ \\left [ \\frac { a - 1 } { a - b } A ^ { a - 2 } - \\frac { b - 1 } { a - b } A ^ { b - 2 } \\right ] \\frac { \\partial A } { \\partial q _ { j } } \\right \\} \\\\ & - \\left \\{ \\left [ \\frac { a - 1 } { a - b } ( X . Y ) ^ { a - 2 } - \\frac { b - 1 } { a - b } ( X . Y ) ^ { b - 2 } \\right ] \\frac { \\partial ( X . Y ) } { \\partial q _ { j } } \\right \\} \\end{align*}"}
+{"id": "3237.png", "formula": "\\begin{align*} & \\delta _ { - q } ^ * \\hat { P } ( p ) \\\\ & = - \\sum _ { \\alpha , \\beta = 1 } ^ 3 \\lim _ { \\mu \\searrow 0 } \\ : \\hat { P } _ { m _ \\beta } \\Big ( p + \\frac { q } { 2 } \\Big ) \\ : \\big ( g _ { - q } ^ { \\alpha , \\beta } \\big ) ^ * \\Big ( p + \\frac { q } { 2 } \\Big ) \\ : \\frac { 1 } { 2 p q + m _ \\alpha ^ 2 - m _ \\beta ^ 2 + i \\mu \\ : \\Big ( p ^ 0 - \\frac { q ^ 0 } { 2 } \\Big ) } \\ : . \\end{align*}"}
+{"id": "7017.png", "formula": "\\begin{align*} e ^ { - t \\Lambda _ p } : = \\biggl [ e ^ { - t \\Lambda } \\upharpoonright L ^ 2 \\cap L ^ p \\biggr ] _ { L ^ p \\rightarrow L ^ p } ^ { \\rm c l o s } . \\end{align*}"}
+{"id": "5142.png", "formula": "\\begin{align*} L _ { d } D 2 ( p \\| q ) = \\left \\{ \\frac { A ^ { a - 1 } - A ^ { b - 1 } } { a - b } - \\left [ \\frac { ( X . Y ) ^ { a - 1 } - ( X . Y ) ^ { b - 1 } } { a - b } \\right ] \\right \\} \\end{align*}"}
+{"id": "1745.png", "formula": "\\begin{align*} \\sum _ { j + k = n - 1 } S _ { j , k } \\neq 0 . \\end{align*}"}
+{"id": "2629.png", "formula": "\\begin{align*} L = \\{ \\ell _ { j , s _ h } : 1 \\leq j \\leq n , 1 \\leq h \\leq n / 2 \\} . \\end{align*}"}
+{"id": "8940.png", "formula": "\\begin{align*} S _ i = | R _ i | + | V _ i | + | T _ i | = \\left \\{ \\begin{array} { l r } \\# \\Sigma ( d - i + 1 ) + 2 \\# \\Sigma ( d - i ) , & i < d ; \\\\ \\# \\Sigma ( 1 ) + \\# P , & i = d ; \\\\ 0 , & i > d ; \\end{array} \\right . \\end{align*}"}
+{"id": "6227.png", "formula": "\\begin{align*} \\phi ' = \\frac { z ( \\phi ) } { D ( \\phi ) } , \\end{align*}"}
+{"id": "6245.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { f } ( \\alpha ) & = - C _ g D _ g \\left ( 3 ( \\alpha - 1 ) ^ 2 - 1 \\right ) + C _ i D _ i ( 1 - \\alpha ) ( 3 \\alpha - 1 ) \\\\ & = \\frac { C _ g D _ g \\left ( 3 - ( 1 + \\omega ) ^ 2 \\right ) + C _ i D _ i ( 1 - \\omega ^ 2 ) } { 3 } . \\end{aligned} \\end{align*}"}
+{"id": "7408.png", "formula": "\\begin{align*} C _ { 4 1 } = \\frac { 2 } { \\sqrt { R } } \\cdot \\frac { ( p + q ) ^ { 2 ( p + q - 1 ) S _ { p + q } } } { A C _ 5 ( 2 ^ { - 1 } f _ 0 ) ^ { p + q - 1 } } \\end{align*}"}
+{"id": "7611.png", "formula": "\\begin{align*} D ( d ; m , n ) = ( m n - 2 ) ( \\lfloor d / n \\rfloor + 1 ) - 1 . \\end{align*}"}
+{"id": "1277.png", "formula": "\\begin{align*} d x ( t ) & = v ( t ) \\ , d t \\\\ d v ( t ) & = - \\omega ^ 2 x ( t ) \\ , d t - \\gamma v ( t ) \\ , d t + \\sqrt { 2 \\gamma \\beta ^ { - 1 } } \\ , d W ( t ) , \\\\ ( x ( 0 ) , v ( 0 ) ) & = ( x _ 0 , v _ 0 ) \\end{align*}"}
+{"id": "2262.png", "formula": "\\begin{align*} R _ 1 R _ { 2 } R _ 1 = R _ { 2 } R _ 1 R _ { 2 } \\ \\ \\ \\ \\ \\end{align*}"}
+{"id": "2916.png", "formula": "\\begin{align*} L ^ { ( 2 ) } _ { \\ell } ( x _ i ; f ) : = \\frac { 1 } { x _ i } \\sum _ { \\substack { j = 1 \\\\ x _ i \\geqslant y _ { j - 1 } } } ^ J \\int _ { y _ { j } } ^ { y _ j ( 1 + 1 / X ) } X \\sum _ { \\max \\{ t / ( 1 + 1 / X ) , y _ { j - 1 } \\} < p \\leqslant y _ j } \\frac { 1 } { p } \\big | \\Psi _ f ^ { \\prime } ( x _ i / t , p ) \\big | ^ 2 { \\rm d } t . \\end{align*}"}
+{"id": "4834.png", "formula": "\\begin{align*} \\sigma = ( ( n - c _ 1 ) ^ { c _ 1 } , \\ldots , ( n - c _ k ) ^ { c _ k } ) \\end{align*}"}
+{"id": "1215.png", "formula": "\\begin{align*} U _ n ( x ) & = \\frac { u _ n ( x ) } { \\left ( \\displaystyle \\int _ { \\Omega } \\vert u _ n \\vert ^ { \\alpha ( p ) } \\dd x \\right ) ^ { \\frac { 1 } { p } } \\vert v _ n ( x _ 0 ) \\vert ^ { \\frac { \\beta ( p ) } { p } } } \\intertext { a n d } V _ n ( x ) & = \\frac { v _ n ( x ) } { \\left ( \\displaystyle \\int _ { \\Omega } \\vert u _ n \\vert ^ { \\alpha ( p ) } \\dd x \\right ) ^ { \\frac { 1 } { p } } \\vert v _ n ( x _ 0 ) \\vert ^ { \\frac { \\beta ( p ) } { p } } } , \\end{align*}"}
+{"id": "8305.png", "formula": "\\begin{align*} B ( \\varepsilon ) y ( \\cdot ; \\varepsilon ) = c ( \\varepsilon ) , \\end{align*}"}
+{"id": "6859.png", "formula": "\\begin{align*} H _ { p h } f = \\sum { E } _ j \\langle \\varphi _ j , f \\rangle \\ , \\varphi _ j , \\end{align*}"}
+{"id": "6201.png", "formula": "\\begin{align*} u _ t + f ( u ) _ x = \\left ( D ( u ) u _ x \\right ) _ x + g ( u ) , t \\ge 0 , \\ , x \\in \\R , \\end{align*}"}
+{"id": "7576.png", "formula": "\\begin{align*} m _ { p , q } ( c ) = E _ { p , q } ( z ^ { 1 } ) = m _ { p , q } ( c _ { 1 } ) \\end{align*}"}
+{"id": "7482.png", "formula": "\\begin{align*} Y : = Y _ 0 + \\nabla Y _ 1 : = \\{ & ( R _ 0 ( t ) + \\nabla ^ { { g } _ 0 ( t ) } _ i R ^ i _ 1 ( t ) ) _ { t \\geq 0 } | ( R _ 0 ( t ) ) _ { t \\geq 0 } \\subset S ^ 2 T ^ * ( N \\times \\mathbb { R } ) ; \\\\ & ( R _ 1 ( t ) ) _ { t \\geq 0 } \\subset S ^ 2 T ^ * ( N \\times \\mathbb { R } ) \\otimes \\Gamma ( T ( N \\times \\mathbb { R } ) ) \\} , \\end{align*}"}
+{"id": "1018.png", "formula": "\\begin{align*} \\int _ E u ( - L \\eta ) \\ , d m = \\int _ D \\eta \\ , d \\mu , \\end{align*}"}
+{"id": "6045.png", "formula": "\\begin{align*} R = \\frac 1 2 \\begin{pmatrix} - 1 & 1 \\\\ 1 & 1 \\end{pmatrix} , R ^ { - 1 } = \\begin{pmatrix} - 1 & 1 \\\\ 1 & 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "6152.png", "formula": "\\begin{align*} \\begin{aligned} & f ( x ) - f ( x ^ { k + 1 } ) + ( x - x ^ { k + 1 } ) ^ T [ - A ^ T \\lambda ^ k \\\\ + & \\beta ^ k A ^ T ( A \\hat { x } ^ { k } - b ) + \\beta ^ k D ( x ^ { k + 1 } - \\hat { x } ^ k ) ] \\geq 0 , ~ \\forall x . \\end{aligned} \\end{align*}"}
+{"id": "1557.png", "formula": "\\begin{align*} F _ n ( w ) \\ge \\frac { | w | ^ { p } } { C } \\frac { 1 } { C } \\le \\inf _ { | z | = 1 } | D F _ n ( z ) | \\le C , \\end{align*}"}
+{"id": "1736.png", "formula": "\\begin{align*} \\Psi ( z _ 0 , \\ldots , z _ n ) : = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & X & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) \\left ( \\begin{array} { c } z _ 0 \\\\ \\vdots \\\\ z _ n \\end{array} \\right ) \\end{align*}"}
+{"id": "1254.png", "formula": "\\begin{align*} \\sum _ { \\substack { 0 \\le k \\le n - 1 \\\\ [ 3 p t ] k \\not = ( n - 1 ) / 2 } } b _ { n , k } \\equiv - \\frac { 1 + ( - 1 ) ^ { \\frac { n - 3 } { 2 } } } { 2 } q ( 1 - q ^ n ) \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "9140.png", "formula": "\\begin{gather*} Z ^ { \\beta } _ { i } = \\big \\{ x ^ { ( \\beta , s ) } _ { i , t } \\ , \\big | \\ , 1 \\leq s \\leq d _ { \\beta } , 1 \\leq t \\leq \\nu _ { \\beta , i } \\big \\} \\forall \\ \\beta \\in \\Delta ^ { + } , i \\in \\beta , \\\\ Z ^ { > \\beta } _ { i } = \\big \\{ x ^ { ( \\alpha , s ) } _ { i , t } \\ , \\big | \\ , \\beta < \\alpha , 1 \\leq s \\leq d _ { \\alpha } , i \\in \\alpha , 1 \\leq t \\leq \\nu _ { \\alpha , i } \\} \\forall \\ \\beta \\in \\Delta ^ { + } , 1 \\leq i \\leq 2 . \\end{gather*}"}
+{"id": "7619.png", "formula": "\\begin{align*} f ^ H = f \\vert X ^ H : X ^ H \\to Y ^ H \\end{align*}"}
+{"id": "3942.png", "formula": "\\begin{align*} & K _ j = \\gamma ^ 4 K _ 0 \\Gamma ^ { - 1 } ; \\Gamma = \\operatorname { d i a g } \\left \\{ \\gamma ^ { 3 } , \\gamma ^ { 2 } , \\gamma \\right \\} \\\\ & \\Rightarrow K = \\gamma ^ 4 \\left ( I _ N \\otimes \\left [ K _ 0 \\Gamma ^ { - 1 } \\right ] \\right ) . \\end{align*}"}
+{"id": "3954.png", "formula": "\\begin{align*} & D = \\left \\lbrace 2 , 2 . 5 , 3 \\right \\rbrace , q _ { 2 , 1 } = q _ { 3 , 2 } = 1 , { Q _ 1 = 0 } . \\end{align*}"}
+{"id": "9179.png", "formula": "\\begin{align*} \\begin{aligned} & x ^ { ( \\beta , s ) } _ { 1 , t } \\mapsto w _ { \\beta , s } + t \\hbar , 1 \\leq t \\leq 2 , \\\\ & x ^ { ( \\beta , s ) } _ { 2 , t } \\mapsto w _ { \\beta , s } - \\tfrac { 3 } { 2 } \\hbar + t \\hbar , 1 \\leq t \\leq 3 . \\end{aligned} \\end{align*}"}
+{"id": "2658.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta u & = u \\cdot [ - k \\cdot | H | \\cdot r \\cdot \\cos \\alpha - k \\cdot n + k ^ 2 \\cdot r ^ 2 \\cdot \\sin ^ 2 \\beta ] . \\end{aligned} \\end{align*}"}
+{"id": "2256.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow 0 } r ( t ) = 0 \\lim _ { t \\rightarrow \\frac { \\pi } { 2 } } r ( t ) = k \\tfrac { \\pi } { 2 } , \\end{align*}"}
+{"id": "543.png", "formula": "\\begin{align*} z ^ { \\tau } = \\sum _ { k = 1 } ^ \\infty a _ { k } \\Phi _ { k } , \\end{align*}"}
+{"id": "5178.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } D _ { \\beta } ( p \\| q ) } { \\partial q _ { j } } = - \\left [ \\frac { a - 1 } { a - b } A ^ { a - 2 } - \\frac { b - 1 } { a - b } A ^ { b - 2 } \\right ] q ^ { \\beta - 1 } _ { j } + \\left [ \\frac { a - 1 } { a - b } B ^ { a - 2 } - \\frac { b - 1 } { a - b } B ^ { b - 2 } \\right ] p _ { j } q ^ { \\beta - 2 } _ { j } \\end{align*}"}
+{"id": "5485.png", "formula": "\\begin{align*} A = | M | ^ 2 , B = | E | ^ 2 + M \\overline { E } + \\overline { M } E , \\end{align*}"}
+{"id": "5132.png", "formula": "\\begin{align*} L _ { d } D 1 ( p \\| q ) = \\left \\lbrace \\log _ { d } A \\left ( p , q \\right ) - \\log _ { d } \\left [ X \\left ( p , q \\right ) + Y \\left ( p , q \\right ) \\right ] \\right \\rbrace \\end{align*}"}
+{"id": "5263.png", "formula": "\\begin{align*} \\overline { T } _ { i } = \\alpha \\overline { p } _ { i } + \\left ( 1 - \\alpha \\right ) \\overline { q } _ { i } \\end{align*}"}
+{"id": "4803.png", "formula": "\\begin{align*} \\widehat { \\mathbf { C } } = \\Big \\{ ( \\hat { c } ^ { \\mbox { \\tiny ( \\itshape k \\upshape ) } } _ 1 , \\ldots , \\hat { c } ^ { \\mbox { \\tiny ( \\itshape k \\upshape ) } } _ n ) ^ \\top , \\ ; k \\in \\mathcal { K } : = \\{ 1 , \\ldots , K \\} \\Big \\} , \\end{align*}"}
+{"id": "8164.png", "formula": "\\begin{align*} \\frac { 1 } { L } \\max _ { P _ { { X } ^ L | | Y ^ { L - 1 } } } \\sum _ { j = 1 } ^ L H ( { Y } _ { j } | Y ^ { j - 1 } , { S } ) , \\end{align*}"}
+{"id": "8875.png", "formula": "\\begin{align*} 0 < \\frac { \\alpha + n } { 2 n p } \\leq \\frac 1 2 < 1 , - \\frac { \\alpha + n } { 2 p } < - \\frac { \\tau } { p } \\leq 0 , \\frac { \\tau } { p } - 1 = \\frac { \\alpha + n } { 2 p } - \\frac { n } 2 . \\end{align*}"}
+{"id": "7384.png", "formula": "\\begin{align*} \\| U ^ 0 \\| _ \\infty : = \\sup _ { ( x , t ) \\in \\R \\times [ 0 , T ] } | U ^ 0 ( x , t ) | . \\end{align*}"}
+{"id": "2071.png", "formula": "\\begin{align*} \\Bar { \\Theta } ^ { d , T } ( \\frac { t + 1 } { T } , \\frac { i } { d } ) - \\Bar { \\Theta } ^ { d , T } ( \\frac { t } { T } , \\frac { i } { d } ) = - \\eta \\sum _ { j = 1 } ^ d W ( \\frac { t } { T } , \\frac { i } { d } ) W ( \\frac { t } { T } , \\frac { j } { d } ) \\Bar { \\Theta } ^ { d , T } ( \\frac { t } { T } , \\frac { j } { d } ) + \\eta W ( \\frac { t } { T } , \\frac { i } { d } ) \\varepsilon ^ t . \\end{align*}"}
+{"id": "3956.png", "formula": "\\begin{align*} \\gamma = 5 , \\rho = 1 . 2 4 \\cdot 1 0 ^ { - 4 } . \\end{align*}"}
+{"id": "2663.png", "formula": "\\begin{align*} \\delta S _ { 1 } = \\int ^ { t _ { 2 } } _ { t _ { 1 } } \\sum _ { i = 1 } \\frac { \\partial L ^ { ( 1 ) } } { \\partial \\dot { q } ^ { i } } \\delta \\dot { q } ^ { i } . \\end{align*}"}
+{"id": "8583.png", "formula": "\\begin{align*} \\Lambda _ \\ell : = \\{ 1 , 2 , \\dots , \\ell \\} \\times \\{ 1 , 2 , \\dots , M \\} \\ : . \\end{align*}"}
+{"id": "3912.png", "formula": "\\begin{align*} ( \\cos z + i \\sin z ) ( \\cos z - i \\sin z ) = 1 . \\end{align*}"}
+{"id": "5205.png", "formula": "\\begin{align*} U _ { j } = - \\frac { Z _ { B } } { \\alpha } q ^ { \\alpha + \\beta - 2 } _ { j } \\ ; \\ ; ; \\ ; \\ ; V _ { j } = - \\frac { Z _ { A } } { \\alpha } p ^ { \\alpha } _ { j } q ^ { \\beta - 2 } _ { j } \\end{align*}"}
+{"id": "2559.png", "formula": "\\begin{align*} \\partial _ x ^ m C ^ \\infty ( L ; \\Omega ^ { - 1 } N L ) \\cap K ^ { ( - \\frac 1 2 - m ) } ( M , L ) = 0 \\ ; . \\end{align*}"}
+{"id": "6145.png", "formula": "\\begin{align*} u = v = \\begin{pmatrix} x \\\\ \\lambda \\end{pmatrix} , u ' = v ' = \\begin{pmatrix} x ' \\\\ \\lambda \\end{pmatrix} , u ^ * = v ^ * = \\begin{pmatrix} x ^ * \\\\ \\lambda ^ * \\end{pmatrix} , F ( u ) = \\begin{pmatrix} - A ^ T \\lambda \\\\ A x - b \\end{pmatrix} . \\end{align*}"}
+{"id": "4956.png", "formula": "\\begin{align*} m _ 2 ( H ) : = \\max \\left \\{ \\tfrac { e ( F ) - 1 } { v ( F ) - 2 } : F \\subseteq H , \\ , v ( F ) > 2 \\right \\} . \\end{align*}"}
+{"id": "6242.png", "formula": "\\begin{align*} \\mathcal { T } : = \\left \\{ ( \\omega , \\gamma ) \\in \\mathbb { R } ^ 2 : \\ , 0 < \\omega < 1 \\ \\mbox { a n d } \\ \\frac { 2 - \\omega } { 3 } < \\gamma < \\frac { 2 + \\omega } { 3 } \\right \\} , \\end{align*}"}
+{"id": "1443.png", "formula": "\\begin{align*} G = \\bigoplus _ { i = 1 } ^ k \\mathbb { Z } / c _ i \\mathbb { Z } \\ni ( b _ 1 , \\dots , b _ k ) \\mapsto \\sum _ { i = 1 } ^ { k } d _ i b _ i \\in \\mathbb { Z } / e \\mathbb { Z } \\end{align*}"}
+{"id": "6733.png", "formula": "\\begin{align*} S _ 2 ( z ) & = \\frac { 1 } { 2 n } - \\frac { \\pi z } { 2 } \\int _ 0 ^ 1 x ^ { n } \\coth ( \\pi z x ) d x \\\\ & = \\frac { 1 } { 2 n } - \\frac { \\pi z } { 2 } \\int _ 0 ^ 1 x ^ { n } \\Big ( 1 + \\frac { 2 e ^ { - 2 \\pi z x } } { 1 - e ^ { - 2 \\pi z x } } \\Big ) d x \\\\ & = \\frac { 1 } { 2 n } - \\frac { \\pi z } { 2 } \\Big ( \\frac { 1 } { n + 1 } + 2 \\sum _ { m = 1 } ^ \\infty \\int _ 0 ^ 1 x ^ { n } e ^ { - 2 \\pi z m x } d x \\Big ) . \\end{align*}"}
+{"id": "4454.png", "formula": "\\begin{align*} & \\int _ { \\{ \\psi < - t \\} } | F _ { t } | ^ { 2 } e ^ { - \\varphi } c ( - \\psi ) + \\int _ { \\{ \\psi < - t \\} } | \\hat { F } - F _ { t } | ^ { 2 } e ^ { - \\varphi } c ( - \\psi ) \\\\ = & \\int _ { \\{ \\psi < - t \\} } | \\hat { F } | ^ { 2 } e ^ { - \\varphi } c ( - \\psi ) . \\end{align*}"}
+{"id": "7120.png", "formula": "\\begin{align*} g ( t , v ) : = t ^ { - 2 } F ( t v ) - F ( v ) + \\frac { 1 - t ^ { p - 2 } } { p - 2 } [ f ( v ) v - 2 F ( v ) ] \\geq 0 , \\ \\ t > 0 \\ v \\in \\mathbb { R } . \\end{align*}"}
+{"id": "4903.png", "formula": "\\begin{align*} d = \\deg ( H \\cap S ) \\geq m \\delta \\geq m ( d - 3 ) , \\end{align*}"}
+{"id": "821.png", "formula": "\\begin{align*} ( 1 - n ) ( { D _ i } { \\Psi _ j } ) = k ( n - 1 ) { t _ { i j } } , \\end{align*}"}
+{"id": "131.png", "formula": "\\begin{align*} \\langle A _ \\lambda ^ h \\vec { u } _ h , \\vec { v } _ h \\rangle = a _ \\lambda ( \\vec { u } _ h , \\vec { v } _ h ) = \\langle \\vec { f } , \\vec { v } _ h \\rangle \\quad \\forall \\vec { v } \\in V _ h . \\end{align*}"}
+{"id": "7337.png", "formula": "\\begin{align*} b _ n ( k ) = \\frac { 1 } { ( n + 1 ) k ! } \\cdot \\prod _ { j = 0 } ^ { k } \\frac { ( n + 1 ) ^ 2 - j ^ 2 } { ( 2 j + 1 ) } \\ , \\ , \\ , \\ , \\ , \\end{align*}"}
+{"id": "8756.png", "formula": "\\begin{align*} \\int \\vert x - y \\vert ^ { \\tilde \\rho } \\pi ( d x , d y ) = \\int \\vert x - y \\vert ^ { \\tilde \\rho } \\gamma ( d x , d y ) + \\mu _ \\varepsilon ( \\{ x _ - \\} ) \\pi _ { x _ - } ( \\{ m \\} ) & ( f _ \\rho ( x _ + ) - f _ \\rho ( x _ - ) ) . \\end{align*}"}
+{"id": "4161.png", "formula": "\\begin{align*} \\varphi ( g _ 2 g _ 1 ) = \\varphi ( g _ 1 ) \\varphi ( g _ 2 ) \\varphi ( g _ 2 g _ 1 ) = \\varphi ( g _ 2 ) \\varphi ( g _ 1 ) \\ , . \\end{align*}"}
+{"id": "5954.png", "formula": "\\begin{align*} \\frac { M _ { 2 } ( \\mathcal { C } ( G ) ) } { | e ( \\mathcal { C } ( G ) ) | } = 1 7 . 1 2 5 > 1 2 . 4 = \\frac { M _ { 1 } ( \\mathcal { C } ( G ) ) } { | v ( \\mathcal { C } ( G ) ) | } . \\end{align*}"}
+{"id": "2171.png", "formula": "\\begin{align*} f _ d ^ { - 1 } : ( 0 , \\infty ) \\rightarrow ( 0 , \\infty ) , \\ , \\ , \\ , \\ , \\ , \\ , f _ d ^ { - 1 } ( x ) = \\frac { d } { e ^ x - 1 } . \\end{align*}"}
+{"id": "7937.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Sigma } \\frac { \\tau k } { \\rho } \\wedge \\mathrm { t r } ( \\frac { \\delta \\mathcal { F } } { \\delta v } ) = \\int _ { \\partial \\Omega } \\mathrm { t r } ( \\frac { \\tilde { p } } { \\rho } ) \\wedge \\mathrm { t r } ( \\frac { \\delta \\mathcal { F } } { \\delta v } ) = ( - 1 ) ^ { n - 1 } \\int _ { \\Omega } \\frac { \\delta \\mathcal { F } } { \\delta v } \\wedge d \\big ( \\frac { \\tilde { p } } { \\rho } \\big ) . \\end{aligned} \\end{align*}"}
+{"id": "7302.png", "formula": "\\begin{align*} \\tilde { L } _ { X _ m } ( s , \\chi ) = \\sum _ { j = 1 } ^ { m - 1 } \\chi ( j ) \\csc ^ { 2 s } \\left ( \\frac { j \\pi } { m } \\right ) \\cot \\left ( \\frac { j \\pi } { m } \\right ) . \\end{align*}"}
+{"id": "5376.png", "formula": "\\begin{align*} \\bar { v } ^ S + f _ i ^ S = \\begin{cases} \\displaystyle h _ i ^ 1 + \\sum _ { j \\in N } p _ { i j } ^ 1 \\ , f ^ S _ j & i \\in S \\\\ \\\\ \\displaystyle h _ i ^ 0 + \\sum _ { j \\in N } p _ { i j } ^ 0 \\ , f ^ S _ j & i \\in N \\setminus S . \\end{cases} \\end{align*}"}
+{"id": "4060.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } \\int _ { - \\infty } ^ { 0 } \\frac { e ^ { r | x | } e ^ { \\pm i \\epsilon r } } { r \\mp i z } ( - i r ) ^ { s _ k } d r = \\Big ( ( - i \\partial _ s ) ^ { s _ k } \\int _ { - \\infty } ^ { 0 } \\frac { e ^ { r s } } { r \\mp i z } d r \\Big ) \\Big | _ { s = | x | } \\end{align*}"}
+{"id": "5658.png", "formula": "\\begin{align*} K ^ { T } _ { i } ( G / B ) = \\bigoplus _ { w \\in W } K ^ { T } _ { i } ( S ) . \\end{align*}"}
+{"id": "6121.png", "formula": "\\begin{align*} \\varphi _ \\ell ( \\tau K ) = ( \\ell ! ) ^ { d - 1 } \\left ( \\varphi _ \\ell ( \\tau A _ d ) \\otimes \\varphi _ \\ell ( \\tau A _ { d - 1 } ) \\otimes \\cdots \\otimes \\varphi _ \\ell ( \\tau A _ 1 ) ) \\right ) + \\mathcal { O } ( \\tau ^ 2 ) . \\end{align*}"}
+{"id": "1945.png", "formula": "\\begin{align*} & k _ { i } = \\left ( 1 - 2 ^ { - i } \\right ) \\bar { k } \\bar { k } > 0 , w _ { i } = \\left ( u - k _ { i } \\right ) _ { + } , ~ ~ i = 0 , 1 , 2 , \\ldots . \\end{align*}"}
+{"id": "8448.png", "formula": "\\begin{align*} [ v ] _ { W ^ { s , p } ( K ) } : = \\left ( \\ , \\ , \\iint _ { K \\times K } \\frac { | v ( x ) - v ( y ) | ^ p } { | x - y | ^ { n + s p } } \\ , d x d y \\right ) ^ \\frac { 1 } { p } . \\end{align*}"}
+{"id": "8127.png", "formula": "\\begin{align*} Y = \\sum _ 1 ^ 3 x _ i \\ , \\ y _ i = x _ i - \\frac { 1 } { 3 } Y \\ , \\end{align*}"}
+{"id": "796.png", "formula": "\\begin{align*} { { \\bf K } } ( p , y ) : = { \\textbf { g } _ y ( { \\bf R } _ y ( v ) , v ) \\over \\textbf { g } _ y ( y , y ) \\textbf { g } _ y ( v , v ) - \\textbf { g } _ y ( v , y ) \\textbf { g } _ y ( v , y ) } . \\end{align*}"}
+{"id": "8880.png", "formula": "\\begin{align*} \\partial _ k = \\frac { x _ k } r \\partial _ r , \\quad \\partial _ l \\partial _ k = \\Big ( \\frac { \\delta _ { l k } } r - \\frac { x _ l x _ k } { r ^ 3 } \\Big ) \\partial _ r + \\frac { x _ l x _ k } { r ^ 2 } \\partial _ r ^ 2 \\end{align*}"}
+{"id": "579.png", "formula": "\\begin{align*} \\mathcal { D } = \\Big \\{ \\Big [ { \\frac { i - 1 } { 2 ^ j } } , { \\frac { i } { 2 ^ j } } \\Big ) : j \\in \\mathbb { N } \\cup \\{ 0 \\} , 1 \\le i \\le 2 ^ j \\Big \\} . \\end{align*}"}
+{"id": "4199.png", "formula": "\\begin{align*} & - 5 0 4 \\left \\{ \\widehat { A } ( T X ) { \\rm d e t } ^ { \\frac { 1 } { 2 } } { \\rm c o s h } ( \\frac { \\sqrt { - 1 } } { 4 \\pi } R ^ V ) \\right \\} ^ { ( 1 2 ) } = \\left \\{ \\widehat { A } ( T X ) { \\rm d e t } ^ { \\frac { 1 } { 2 } } { \\rm c o s h } ( \\frac { \\sqrt { - 1 } } { 4 \\pi } R ^ V ) \\right . \\\\ & \\left . \\cdot { \\rm c h } ( \\widetilde { T X } + 2 \\wedge ^ 2 \\widetilde { V _ C } - \\widetilde { V _ C } \\otimes \\widetilde { V _ C } + \\widetilde { V _ C } ) \\right \\} ^ { ( 1 2 ) } . \\end{align*}"}
+{"id": "5331.png", "formula": "\\begin{align*} \\nu _ { \\pi _ k } = \\nu ^ { S _ { k } } _ { \\pi _ k } . \\end{align*}"}
+{"id": "4288.png", "formula": "\\begin{align*} \\mathbf { x } _ { s , t } = \\mathbf { x } _ { s , u } \\otimes \\mathbf { x } _ { u , t } , s \\le u \\le t . \\\\ \\max _ { 1 \\le i \\le [ p ] } \\| \\mathbf { x } ^ i \\| _ { p / i \\textrm { - } \\mathrm { v a r } } < \\infty . \\end{align*}"}
+{"id": "3600.png", "formula": "\\begin{align*} T f ( z , w ) = \\alpha ( \\tau ' ( z ) ) ^ { \\frac { 1 } { p } } f ( \\tau ( z ) , w \\sigma ( z ) ) , \\end{align*}"}
+{"id": "4287.png", "formula": "\\begin{align*} \\sup _ { y \\in \\R ^ e } \\bigl \\| [ ( 1 - \\Delta ) ^ { \\beta / 2 } \\varphi _ { m ^ { - \\delta } } ] ( F _ m - y ) - [ ( 1 - \\Delta ) ^ { \\beta / 2 } \\delta _ y ] ( F ) \\bigr \\| _ { \\mathbf { D } _ { q , - r } } = O ( m ^ { - \\gamma \\wedge \\delta } ) \\end{align*}"}
+{"id": "4233.png", "formula": "\\begin{align*} \\Theta _ 1 ( T _ { C } X ) = 1 + 2 q \\widetilde { T _ C X } + O ( q ^ 2 ) , \\end{align*}"}
+{"id": "8195.png", "formula": "\\begin{align*} B _ 1 ^ { \\star } = \\min ( B _ , \\frac { 2 ( M - B _ ) } { 3 } ) . \\end{align*}"}
+{"id": "8634.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } ( \\partial ^ 3 _ x X ) = v _ 3 ( \\partial _ x X ) ^ 3 + 3 v _ 2 ( \\partial _ x X ) ( \\partial ^ 2 _ x X ) + v _ 1 ( \\partial ^ 3 _ x X ) , \\\\ ( \\partial ^ 3 _ x X ) ( 0 ; x ) = 0 \\end{cases} \\end{align*}"}
+{"id": "1600.png", "formula": "\\begin{align*} d \\theta ^ i = \\sum _ { j = 1 } ^ n \\theta ^ j \\wedge \\omega ^ i _ j \\quad \\quad \\ ; \\ ; \\quad \\ ; \\ ; d \\omega _ j ^ i = \\Omega _ j ^ i - \\sum _ { k = 1 } ^ n \\omega ^ k _ j \\wedge \\omega ^ i _ k , \\end{align*}"}
+{"id": "5872.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { C } ( G ) ) = ( 4 n - 4 ) ( 4 n - 4 - 1 ) ^ { 2 } + n \\cdot 4 ( 4 - 1 ) ^ { 2 } = ( 4 n - 4 ) ( 4 n - 5 ) ^ { 2 } + 3 6 n \\end{align*}"}
+{"id": "8221.png", "formula": "\\begin{align*} P _ { { \\bf A } _ { i } | { \\bf A } ^ { i - 1 } } ( { \\bf a } _ i | { \\bf a } ^ { i - 1 } ) = P _ { { \\bf A } _ i | { \\bf A } _ { i - 1 } } ( { \\bf x } _ i | { \\bf a } _ { i - 1 } ) . \\end{align*}"}
+{"id": "5262.png", "formula": "\\begin{align*} G I \\left ( p \\| q \\right ) = \\sum _ { j } p _ { j } \\sum _ { i } \\left [ \\alpha \\overline { p } _ { i } + \\left ( 1 - \\alpha \\right ) \\overline { q } _ { i } \\right ] \\log \\frac { \\alpha \\overline { p } _ { i } + \\left ( 1 - \\alpha \\right ) \\overline { q } _ { i } } { \\overline { p } _ { i } } \\end{align*}"}
+{"id": "295.png", "formula": "\\begin{align*} \\epsilon _ j > \\max _ { \\substack { k = 1 , \\dots , j - 1 } } \\epsilon _ k \\ , Q _ { j k } , \\end{align*}"}
+{"id": "5131.png", "formula": "\\begin{align*} L D 1 ( p \\| q ) = \\left \\lbrace \\log A \\left ( p , q \\right ) - \\log \\left [ X \\left ( p , q \\right ) + Y \\left ( p , q \\right ) \\right ] \\right \\rbrace \\end{align*}"}
+{"id": "8707.png", "formula": "\\begin{align*} { \\rm C o k e r \\ , } \\hat \\sigma _ s = { \\rm C o k e r \\ , } \\hat \\sigma : = \\{ u \\in H ^ 0 _ b ( X ) ^ G _ { s - \\frac { 1 } { 2 } } ; \\ , ( \\ , u \\ , | \\ , \\hat \\sigma v ) _ X = 0 , \\forall v \\in H ^ 0 ( \\overline M ) ^ G _ s \\cap C ^ \\infty ( \\overline M ) \\} . \\end{align*}"}
+{"id": "710.png", "formula": "\\begin{align*} f ( x ) = \\Phi ^ 3 ( x ) - 2 7 \\Phi ^ 2 ( x ) + 2 4 3 \\Phi ( x ) - 7 2 9 - m \\end{align*}"}
+{"id": "2135.png", "formula": "\\begin{align*} \\lambda _ { \\{ 1 \\} } ( x , y ) = 2 ( y - x ) , \\lambda _ { \\{ j \\} } ( x , y ) = - 2 ( y - x ) \\ , \\ , \\ , \\ , j = 2 , 3 , 4 . \\end{align*}"}
+{"id": "2207.png", "formula": "\\begin{align*} \\intop _ \\Omega \\ ! \\psi _ { 1 , r r z } \\psi _ { 1 , z z z } d x + \\ ! \\intop _ \\Omega \\ ! \\psi _ { 1 , z z z } ^ 2 d x + 3 \\ ! \\intop _ \\Omega { 1 \\over r } \\ ! \\psi _ { 1 , r z } \\psi _ { 1 , z z z } d x = - \\ ! \\intop _ \\Omega \\omega _ { 1 , z } \\psi _ { 1 , z z z } d x . \\end{align*}"}
+{"id": "2599.png", "formula": "\\begin{align*} \\dfrac { \\varphi ' ( r ) } { \\varphi ( r ) } = - \\left ( 2 \\beta R \\right ) \\dfrac { r } { r ^ 2 + \\alpha ^ 2 } . \\end{align*}"}
+{"id": "889.png", "formula": "\\begin{align*} L _ { 0 0 } = 2 a _ 1 ( y ^ 1 ) ^ 2 + 2 ( a _ 2 + b _ 1 ) y ^ 1 y ^ 2 + 2 b _ 2 ( y ^ 2 ) ^ 2 , \\end{align*}"}
+{"id": "4260.png", "formula": "\\begin{align*} x f ( s ) = x e ^ { - i s \\Delta } u ( s ) = e ^ { - i s \\Delta } J ( s ) u ( s ) \\end{align*}"}
+{"id": "1543.png", "formula": "\\begin{align*} F ( z ) \\ge \\frac { 1 } { C } \\sup _ { | w | = 1 } F ( w ) \\ , | z | ^ { 1 + \\frac { 1 } { H } } . \\end{align*}"}
+{"id": "6263.png", "formula": "\\begin{align*} g ( s _ { 0 } ( x ) ) = - g ( x ) , g ( s _ { \\rho } ( x ) ) = - g ( x ) , a . e . \\ x \\in \\Gamma , \\end{align*}"}
+{"id": "8946.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\Delta ( | \\nabla \\varphi | ^ 2 - n ) = \\nabla \\varphi \\cdot \\Delta \\nabla \\varphi + | \\nabla ^ 2 \\varphi | ^ 2 = | \\nabla ^ 2 \\varphi | ^ 2 \\quad , \\end{align*}"}
+{"id": "5814.png", "formula": "\\begin{align*} X _ + ( t ) + X _ - ( t ) = o ( 1 ) X _ 0 ( t ) , \\end{align*}"}
+{"id": "8240.png", "formula": "\\begin{align*} \\hat { \\bf S } _ i \\in \\Omega _ i ( { \\bf X } ^ i , Y ^ i ) , \\ \\Omega _ i ( { \\bf X } ^ i , Y ^ i ) = S ( { \\bf X } ^ i , Y ^ i ) . \\end{align*}"}
+{"id": "5183.png", "formula": "\\begin{align*} & U _ { j } = \\left [ \\frac { a - 1 } { a - b } A ^ { a - 2 } - \\frac { b - 1 } { a - b } A ^ { b - 2 } \\right ] p _ { j } q ^ { \\beta - 2 } _ { j } \\\\ & V _ { j } = \\left [ \\frac { a - 1 } { a - b } B ^ { a - 2 } - \\frac { b - 1 } { a - b } B ^ { b - 2 } \\right ] q ^ { \\beta - 1 } _ { j } \\end{align*}"}
+{"id": "5194.png", "formula": "\\begin{align*} L _ { d } D _ { \\alpha \\beta } ( p \\| q ) = \\frac { 1 } { a - b } \\left [ A ^ { a - 1 } - A ^ { b - 1 } - \\left ( B ^ { a - 1 } - B ^ { b - 1 } \\right ) \\right ] \\end{align*}"}
+{"id": "8166.png", "formula": "\\begin{align*} \\sum _ { y ^ j \\in \\beta _ k ^ j } P _ { Y ^ j , \\underline S } ( y ^ j , \\underline S ) = \\frac { b _ k } { M } . \\end{align*}"}
+{"id": "432.png", "formula": "\\begin{align*} | C _ 2 | = | \\sum _ { k = 1 } ^ { 2 j - 1 } [ \\bold { F } ] _ { 2 j , k } F _ { 2 j , k } ^ { ( c ) } + [ \\bold { F } ] _ { 2 j , 2 j } F _ { 2 j , 2 j } ^ { ( c ) } | \\le \\| \\bold { F } ^ { ( 2 j ) } \\| _ { 2 } ( \\sum _ { k = 1 } ^ { 2 j } | F _ { 2 j , k } ^ { ( c ) } | ^ 2 ) ^ { \\frac { 1 } { 2 } } + | [ \\bold { F } ] _ { 2 j , 2 j } F _ { 2 j , 2 j } ^ { ( c ) } | \\le \\frac { j ^ { \\frac { 1 } { 2 } } } { M } \\frac { j ^ { \\frac { 3 } { 2 } } } { M ^ 2 } \\end{align*}"}
+{"id": "6131.png", "formula": "\\begin{align*} \\| \\rho ^ { ( r ) } _ k ( x ) \\| ^ 2 & = \\| \\rho ^ { ( r ) } _ k ( x ) ^ * \\rho ^ { ( r ) } _ k ( x ) \\| \\\\ & = \\| \\rho ^ { ( r ) } _ n ( \\rho ^ { ( r ) } _ { n , k } ( x ) ^ * ) \\rho ^ { ( r ) } _ n ( \\rho ^ { ( r ) } _ { n , k } ( x ) ) \\| \\\\ & \\leq \\| \\rho ^ { ( r ) } _ n ( \\rho ^ { ( r ) } _ { n , k } ( x ) ^ * \\rho ^ { ( r ) } _ { n , k } ( x ) ) \\| , \\end{align*}"}
+{"id": "3992.png", "formula": "\\begin{align*} \\varpi : A \\rightarrow \\mathbb { R } , \\quad \\varpi \\left ( e _ { i } \\right ) = \\varpi \\left ( \\widetilde { e } _ { j } \\right ) = 1 . \\end{align*}"}
+{"id": "8330.png", "formula": "\\begin{align*} [ \\widetilde { W } _ t , \\widetilde { W } _ t ] _ { \\widetilde { \\gamma } } = [ W _ t , W _ t ] _ { \\gamma } - 2 \\left ( \\frac { d } { d t } W _ t + [ X _ t , W _ t ] _ { \\gamma } \\right ) \\wedge \\partial _ s . \\end{align*}"}
+{"id": "133.png", "formula": "\\begin{align*} \\left ( A ^ h _ { \\lambda } \\right ) ^ { - 1 } \\eqsim M ^ h _ { \\lambda } : = \\frac { \\lambda } { 1 + \\lambda } P _ h A _ h ^ { - 1 } + \\frac { 1 } { \\lambda + 1 } A _ h ^ { - 1 } . \\end{align*}"}
+{"id": "4902.png", "formula": "\\begin{align*} C \\cdot F = \\bar C \\cdot F = 1 \\end{align*}"}
+{"id": "6915.png", "formula": "\\begin{align*} q & > 1 , \\allowdisplaybreaks \\\\ \\frac { n - 2 p + 1 - q } { p } + 2 \\varepsilon & = 0 \\allowdisplaybreaks \\\\ \\frac { \\ ( p - 1 \\ ) \\ ( n - q \\ ) } { p } - \\varepsilon & = \\frac { \\ ( p - 1 \\ ) \\ ( 2 p - 1 \\ ) } { p } - \\varepsilon \\ ( 2 p - 1 \\ ) \\\\ & \\in \\ ( 0 , \\frac { n p - n + p } { p } \\ ) \\end{align*}"}
+{"id": "8944.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\nabla u - \\nabla v \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } & \\le 2 \\| d ( \\nabla u , S O ( n ) ) \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } + 2 ( C + 1 ) \\| M _ \\Omega ( d ( \\nabla u , S O ( n ) ) ) \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } \\\\ & \\le C \\| d ( \\nabla u , S O ( n ) ) \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } \\end{aligned} \\end{align*}"}
+{"id": "861.png", "formula": "\\begin{align*} \\rho = 1 - \\Vert w \\Vert ^ 2 & = 1 - ( w ^ 1 ) ^ 2 - ( w ^ 2 ) ^ 2 . \\end{align*}"}
+{"id": "2199.png", "formula": "\\begin{align*} \\intop _ { - \\infty + i h } ^ { + \\infty + i h } \\sum _ { j = 0 } ^ k | \\lambda | ^ { 2 j } | \\hat u ( \\lambda ) | ^ 2 d \\lambda = \\intop _ \\R \\sum _ { j = 0 } ^ k | \\partial _ \\tau ^ j u | ^ 2 e ^ { 2 h \\tau } d \\tau . \\end{align*}"}
+{"id": "2080.png", "formula": "\\begin{align*} & - \\int _ 0 ^ s \\Theta ( u , x ) f ' ( u ) \\mathrm { d } u + f ( s ) { \\Theta } ( s , x ) - f ( 0 ) { \\Theta } ( 0 , x ) \\\\ & = - \\eta d T \\big ( \\int _ 0 ^ { s } \\int _ 0 ^ 1 f ( u ) A ( x , y ) \\Theta ( u , y ) \\mathrm { d } y \\mathrm { d } u + o ( 1 ) \\big ) + \\sigma ^ 2 \\eta ^ 2 T \\big ( \\int _ 0 ^ s f ( u ) \\mathrm { d } \\xi _ 1 ( u , x ) + o ( 1 ) \\big ) . \\end{align*}"}
+{"id": "4551.png", "formula": "\\begin{align*} \\mu _ { \\infty } ^ { ( K ) } = \\mu _ { K } . \\end{align*}"}
+{"id": "7843.png", "formula": "\\begin{align*} \\nabla ^ { \\perp \\psi } _ { \\bar E _ { \\alpha } } \\nabla ^ { \\perp \\psi } _ { \\bar E _ { \\alpha } } H ^ { \\psi } = \\frac { q } { p + q } \\bar \\nabla ^ { \\perp } _ { \\bar E _ { \\alpha } } \\bar \\nabla ^ { \\perp } _ { \\bar E _ { \\alpha } } H _ 2 + \\frac { p } { p + q } B ^ j ( \\nabla ^ { \\mathbb { C } P ^ q } _ { \\bar E _ { \\alpha } } \\bar E _ { \\alpha } , H _ 1 ) - \\frac { p } { p + q } H _ 1 . \\end{align*}"}
+{"id": "4247.png", "formula": "\\begin{align*} ( i \\partial _ t + \\Delta ) u = [ 1 + a ] | u | ^ 2 u . \\end{align*}"}
+{"id": "7244.png", "formula": "\\begin{align*} Z _ n \\coloneqq \\frac { 1 } { n ! } \\sum _ { \\sigma \\in S _ n } \\prod _ { i = 1 } ^ n t _ i ^ { j _ i ( \\sigma ) } = \\sum _ { j \\in J _ n } \\frac { 1 } { \\prod _ { i = 1 } ^ n i ^ { j _ i } j _ i ! } \\prod _ { i = 1 } ^ n t _ i ^ { j _ i } \\in \\mathbb Q [ t _ 1 , \\dots , t _ n ] \\end{align*}"}
+{"id": "212.png", "formula": "\\begin{align*} i ( \\Gamma ) \\omega _ L = d E _ L . \\end{align*}"}
+{"id": "4296.png", "formula": "\\begin{align*} \\mathbf { z } = \\int \\tilde { \\sigma } _ a ( \\mathbf { z } ) d \\mathbf { z } \\mbox { w i t h } \\pi _ 1 \\mathbf { z } = \\mathbf { x } , \\end{align*}"}
+{"id": "4770.png", "formula": "\\begin{align*} \\Phi ( \\varepsilon , b , \\eta , \\omega , \\varphi , n ) : = \\left ( \\min \\{ \\varphi _ 3 ( \\varepsilon / 2 , b , \\eta , n , \\varphi ) , \\varphi _ 4 ( \\varepsilon / 2 , b , \\eta , n , \\omega , \\varphi ) \\} \\right ) ^ 2 \\end{align*}"}
+{"id": "6373.png", "formula": "\\begin{align*} 0 < s \\leq s _ 1 = \\min \\Bigl \\{ 1 , \\phi ^ { - 1 } \\Bigl ( \\frac { v } { 2 n \\omega _ n } \\Bigr ) \\Bigr \\} , \\end{align*}"}
+{"id": "3084.png", "formula": "\\begin{align*} \\psi _ { \\star } ( t ) : = \\psi _ { c _ { \\beta _ j } } ( t ) \\ \\ \\ \\mbox { a n d } \\ \\ \\ F _ { \\star } : = F _ { c _ { \\beta _ j } } . \\end{align*}"}
+{"id": "4804.png", "formula": "\\begin{align*} \\hat { \\mathbf { c } } ^ { \\mbox { \\tiny ( \\itshape k \\upshape ) } } \\in \\mathcal { S } _ k : = \\Big \\{ \\mathbf { c } ' \\in \\mathbb { R } ^ { n } : \\mathbf { B } ^ { \\mbox { \\tiny ( \\itshape k \\upshape ) } } \\mathbf { c } ' \\leq \\mathbf { b } ^ { \\mbox { \\tiny ( \\itshape k \\upshape ) } } \\Big \\} \\subseteq \\mathcal { S } _ 0 \\end{align*}"}
+{"id": "4201.png", "formula": "\\begin{align*} & 4 8 0 \\left \\{ \\widehat { A } ( T X ) { \\rm d e t } ^ { \\frac { 1 } { 2 } } { \\rm c o s h } ( \\frac { \\sqrt { - 1 } } { 4 \\pi } R ^ V ) \\right \\} ^ { ( 1 2 ) } = \\left \\{ \\widehat { A } ( T X ) { \\rm d e t } ^ { \\frac { 1 } { 2 } } { \\rm c o s h } ( \\frac { \\sqrt { - 1 } } { 4 \\pi } R ^ V ) \\right . \\\\ & \\left . \\cdot { \\rm c h } ( \\widetilde { T X } + 2 \\wedge ^ 2 \\widetilde { V _ C } - \\widetilde { V _ C } \\otimes \\widetilde { V _ C } + \\widetilde { V _ C } ) \\right \\} ^ { ( 1 2 ) } . \\end{align*}"}
+{"id": "1796.png", "formula": "\\begin{align*} f \\left ( \\zeta ( \\bar x + ) , u ( t , \\bar x + ) \\right ) - f \\left ( \\zeta ( \\bar x - ) , u ( t , \\bar x - ) \\right ) = \\Xi \\big ( \\zeta ( \\bar x + ) , \\zeta ( \\bar x - ) , u ( t , \\bar x - ) \\big ) \\mbox { f o r a . e . } t > 0 \\end{align*}"}
+{"id": "6477.png", "formula": "\\begin{align*} A ( X _ 1 ) = - \\frac { R _ 2 } { R _ 1 } X _ 1 , A ( X _ 2 ) = \\frac { R _ 1 } { R _ 2 } X _ 2 \\end{align*}"}
+{"id": "654.png", "formula": "\\begin{align*} R ^ { k } f | _ U = 0 \\end{align*}"}
+{"id": "4944.png", "formula": "\\begin{align*} ( \\tau _ { L } ) _ \\ast [ L ' ] = [ L ' ] - ( - 1 ) ^ { n ( n - 1 ) / 2 } ( [ L ] \\circ [ L ' ] ) [ L ] , \\end{align*}"}
+{"id": "1872.png", "formula": "\\begin{align*} \\alpha \\tau _ 1 \\alpha = \\tau _ 1 \\beta \\tau _ 1 \\beta = \\tau _ 1 \\gamma \\tau _ 1 \\gamma = \\tau ^ { - 1 } _ 1 . \\end{align*}"}
+{"id": "1605.png", "formula": "\\begin{align*} d \\textup { v o l } _ { \\Gamma _ t } ( e _ 1 , \\dots , e _ { n - 1 } ) : = d \\textup { v o l } _ M ( e _ n , e _ 1 , \\dots , e _ { n - 1 } ) = \\epsilon ( n 1 \\dots n - 1 ) = ( - 1 ) ^ { n - 1 } . \\end{align*}"}
+{"id": "8819.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\infty 2 ^ { - j s q ' } \\varphi ( 2 ^ { - j } ) ^ { - q ' } = \\infty \\end{align*}"}
+{"id": "19.png", "formula": "\\begin{align*} \\beta ( \\lambda \\kappa ; q _ \\lambda ) = \\beta ( \\kappa ; q ) \\quad \\end{align*}"}
+{"id": "7598.png", "formula": "\\begin{align*} \\rho ^ * = \\frac { 4 N ( q - 2 ) | E _ { 4 ^ * , q } ( \\psi _ { 0 } ) | + 2 } { N ( q - 2 ) - 8 } . \\end{align*}"}
+{"id": "1081.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { [ n \\delta ] } \\Delta _ n ^ { s , t } \\leq \\{ M _ 4 ( \\tilde { w } , 1 - \\delta ) + C _ 1 + C _ 2 \\} n ^ { - d } , n \\in \\N , ~ ~ t \\in \\{ [ r n ] + 1 , \\dots , n \\} . \\end{align*}"}
+{"id": "3221.png", "formula": "\\begin{align*} H ^ n ( X ) = \\bigoplus _ { a + b = n } H ^ a ( B , ^ p \\ ! \\ ! R ^ { b } f _ * \\mathcal { C } ) , \\end{align*}"}
+{"id": "5887.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { N C } ( G ) ) & = ( 2 m n - n ) ( 2 m n - n - 1 ) ^ { 2 } - 4 ( 2 m n - n - 1 ) \\dfrac { ( m n - n ) ( m n - n - 1 ) + m n ( n - 1 ) } { 2 } \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ + n ( m - 1 ) ( m n - n - 1 ) ^ { 2 } + m n ( n - 1 ) ^ { 2 } \\\\ & = 5 m ^ { 3 } n ^ { 3 } - 9 m ^ { 2 } n ^ { 3 } + 4 m n ^ { 3 } \\\\ & = n ^ { 3 } ( 5 m ^ { 3 } - 9 m ^ { 2 } + 4 m ) \\end{align*}"}
+{"id": "6693.png", "formula": "\\begin{align*} b \\in B ( H ) = \\{ [ x _ 1 , y _ 1 ] \\cdots [ x _ r , y _ r ] z ^ q \\mid x _ 1 , y _ 1 , \\dots , x _ r , y _ r , z \\in H \\} \\subseteq _ \\mathrm { c } \\Phi ( H ) . \\end{align*}"}
+{"id": "1751.png", "formula": "\\begin{align*} a _ { j , k } = a _ { k , j } . \\end{align*}"}
+{"id": "8750.png", "formula": "\\begin{align*} \\int \\varphi ( \\vert x - y \\vert ) \\ , \\beta ( d x , d y ) - \\int \\varphi ( \\vert x - y \\vert ) \\ , \\gamma ( d x , d y ) = f ( x _ + ) - f ( x _ - ) < 0 . \\end{align*}"}
+{"id": "203.png", "formula": "\\begin{align*} A + \\ = \\ \\{ i + : i & \\leq n \\} , A - \\ = \\ \\{ i - : i \\leq n \\} , \\\\ B + \\ = \\ \\{ i + : n < i & \\leq n + m \\} , B - \\ = \\ \\{ i - : n < i \\leq n + m \\} , \\\\ C \\ = \\ K \\setminus ( A + \\cup \\ A - \\cup \\ & B + \\cup B - ) \\ = \\ \\{ i + , i - : n + m < i \\} \\cup \\{ \\infty \\} . \\end{align*}"}
+{"id": "6935.png", "formula": "\\begin{align*} T ^ c = T ^ c _ 1 + T ^ c _ 2 \\end{align*}"}
+{"id": "1604.png", "formula": "\\begin{gather*} d ( \\theta ^ 1 \\wedge \\dots \\wedge \\theta ^ k ) = \\sum \\epsilon ( i _ 1 \\dots i _ k ) \\ , d \\theta ^ { i _ 1 } \\wedge \\theta ^ { i _ 2 } \\wedge \\dots \\wedge \\theta ^ { i _ k } \\\\ = ( - 1 ) ^ { k - 1 } \\sum \\epsilon ( i _ 1 \\dots i _ k ) \\ , \\theta ^ { i _ 1 } \\wedge \\dots \\wedge \\theta ^ { i _ { k - 1 } } \\wedge d \\theta ^ { i _ k } , \\end{gather*}"}
+{"id": "2394.png", "formula": "\\begin{align*} S _ n & = \\rho _ { n , 0 } + \\sum _ { 2 \\leqslant i \\leqslant 2 s + 4 \\atop i \\equiv \\delta \\pmod { 2 } } \\rho _ { n , i } ~ \\zeta _ 2 \\left ( i + s + 1 , \\frac { 1 } { 4 } \\right ) , \\\\ T _ n & = \\sigma _ { n , 0 } + \\sum _ { i = 2 } ^ { s + 2 } \\sigma _ { n , i } ~ \\zeta _ 2 \\left ( i + s + 1 , \\frac { 1 } { 4 } \\right ) , \\end{align*}"}
+{"id": "639.png", "formula": "\\begin{align*} f ( \\rho ) & = \\sinh ( \\rho ) - I _ { r + 1 , r } ( \\rho ) - d _ r \\\\ & = \\sinh ( \\rho ) \\biggl ( 1 + \\frac { 1 } { r - 2 } \\tanh ^ { r - 2 } ( \\rho ) + \\frac { r - 1 } { ( r - 2 ) ( r - 4 ) } \\tanh ^ { r - 4 } ( \\rho ) \\\\ & + \\dots + \\frac { ( r - 1 ) ( r - 3 ) \\cdots 5 } { ( r - 2 ) ( r - 4 ) \\cdots 2 } \\tanh ^ 2 ( \\rho ) - \\frac { ( r - 1 ) ! ! } { ( r - 2 ) ! ! } \\biggr ) \\\\ & + \\frac { ( r - 1 ) ! ! } { ( r - 2 ) ! ! } \\arctan ( \\sinh ( \\rho ) ) - d _ r . \\end{align*}"}
+{"id": "786.png", "formula": "\\begin{align*} U = ( u _ { i j } ) _ { i , j } \\quad \\mbox { i s u n i t a r y a n d } U = F \\bar { U } _ Z F ^ { - 1 } , \\end{align*}"}
+{"id": "4371.png", "formula": "\\begin{align*} F _ { \\mathcal { Y } , a , b } ( x ) & : = \\sum _ { ( i , j ) \\in [ m ] \\times [ n ] } \\overline { y } _ { ( i , j ) } g _ { ( i , j ) } x + \\sum _ { ( q , p ) \\in \\mathcal { Y } } \\Delta y _ { ( q , p ) } g _ { ( q , p ) } ( x ) - \\Delta y _ { ( a , b ) } g _ { ( a , b ) } ( x ) . \\end{align*}"}
+{"id": "5635.png", "formula": "\\begin{align*} \\bigg \\| S ^ { n _ k } \\bigg ( \\sum _ { i = 0 } ^ { s _ { 2 k + 1 } - 1 } x _ i e _ i \\bigg ) \\bigg \\| \\leq \\bigg \\| \\sum _ { i = 0 } ^ { s _ { 2 k + 1 } - 1 } \\frac { x _ i } { 2 ^ k } e _ i \\bigg \\| < \\frac { 1 } { 2 ^ k } \\| x \\| < \\frac { \\epsilon } { 2 } , \\end{align*}"}
+{"id": "6398.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ n F ( X _ { \\frac { i - 1 } { n } } ) H ( z ^ n _ i ( \\theta _ 0 ) ) \\xrightarrow [ n \\to \\infty ] { \\mathcal { L } - s } \\Sigma ^ { 1 / 2 } \\mathcal { N } \\end{align*}"}
+{"id": "4464.png", "formula": "\\begin{align*} \\ll f _ 2 , g \\gg _ { S _ 1 , \\rho _ 2 } = 0 \\end{align*}"}
+{"id": "2182.png", "formula": "\\begin{align*} f ( \\mathfrak { M } _ 1 ) \\subset ( \\mathfrak { M } _ 2 \\otimes _ { \\mathfrak { S } } \\mathcal { O } _ { \\mathcal { E } } ) \\cap ( \\mathfrak { M } _ 2 \\otimes _ { \\mathfrak { S } } \\mathfrak { S } _ g ) = \\mathfrak { M } _ 2 \\otimes _ { \\mathfrak { S } } ( \\mathcal { O } _ { \\mathcal { E } } \\cap \\mathfrak { S } _ g ) = \\mathfrak { M } _ 2 . \\end{align*}"}
+{"id": "1638.png", "formula": "\\begin{align*} K _ { 4 } \\left ( D _ { 4 } \\right ) = \\left \\{ u \\in H _ { 0 } ^ { 2 } \\left ( Q _ { T } \\right ) : \\sup _ { Q _ { T } } \\left \\vert u \\right \\vert , \\sup _ { Q _ { T } } \\left \\vert \\nabla u \\right \\vert \\leq D _ { 4 } \\right \\} . \\end{align*}"}
+{"id": "525.png", "formula": "\\begin{align*} & C _ 1 : = \\langle i \\rangle j = j \\langle i \\rangle = \\{ \\pm j , \\pm k \\} , C _ 2 : = \\langle j \\rangle i = i \\langle j \\rangle = \\{ \\pm i , \\pm k \\} , \\\\ & C _ 3 : = \\langle k \\rangle j = j \\langle k \\rangle = \\{ \\pm i , \\pm j \\} . \\end{align*}"}
+{"id": "8839.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } \\frac { \\partial { \\bf u } } { \\partial t } & = & D \\frac { \\partial ^ 2 { \\bf u } } { \\partial x ^ 2 } + \\zeta ( x ) { \\bf u } , & t > 0 , \\\\ { \\bf u } ( 0 , x ) & = & \\varphi ( x ) , & x \\in \\mathbb { R } , \\end{array} \\right . \\end{align*}"}
+{"id": "7234.png", "formula": "\\begin{align*} \\begin{array} { c | c c c } \\ast _ \\sigma & \\sigma ( 1 ) & \\cdots & \\sigma ( n ) \\\\ \\hline \\sigma ( 1 ) & \\sigma ( 1 ) \\ast _ \\sigma \\sigma ( 1 ) = \\sigma ( 1 \\ast 1 ) & \\cdots & \\sigma ( 1 ) \\ast _ \\sigma \\sigma ( n ) = \\sigma ( 1 \\ast n ) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\sigma ( n ) & \\sigma ( n ) \\ast _ \\sigma \\sigma ( 1 ) = \\sigma ( n \\ast 1 ) & \\cdots & \\sigma ( n ) \\ast _ \\sigma \\sigma ( n ) = \\sigma ( n \\ast n ) \\\\ \\end{array} . \\end{align*}"}
+{"id": "3106.png", "formula": "\\begin{align*} u _ \\varphi ( t ) & = \\left ( \\begin{array} { c c c } 1 & c _ 1 \\ , t ^ { p ^ { e _ 1 } } & c _ 2 \\ , t ^ { p ^ { e _ 2 } } \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) ( \\ , c _ 1 , c _ 2 \\in k , e _ 1 , e _ 2 \\geq 0 \\ , ) , \\end{align*}"}
+{"id": "3609.png", "formula": "\\begin{align*} \\lambda _ 1 = - 1 , \\lambda _ 2 = - 1 , \\lambda _ 1 = - \\lambda _ 2 . \\end{align*}"}
+{"id": "8500.png", "formula": "\\begin{align*} f [ a ] = \\pi _ 1 ( f ( a ) ) . \\end{align*}"}
+{"id": "5675.png", "formula": "\\begin{align*} X = \\bigcup \\limits _ { i \\in I } U _ i \\end{align*}"}
+{"id": "1536.png", "formula": "\\begin{align*} F _ { \\delta } ( z ) = \\inf _ { w \\in \\R ^ { N } } \\Big \\{ F ( w ) + \\frac { 1 } { 2 \\ , \\delta } | w - z | ^ { 2 } \\Big \\} , \\end{align*}"}
+{"id": "5168.png", "formula": "\\begin{align*} - \\frac { \\partial L D _ { \\alpha } I ( p \\| q ) } { \\partial q _ { j } } = - \\frac { 1 } { \\alpha } \\left [ \\left ( \\sum _ { i } q _ { i } \\right ) ^ { - 1 } - \\left ( \\sum _ { i } p ^ { \\alpha } _ { i } q ^ { 1 - \\alpha } _ { i } \\right ) ^ { - 1 } p ^ { \\alpha } _ { j } q ^ { - \\alpha } _ { j } \\right ] \\end{align*}"}
+{"id": "4724.png", "formula": "\\begin{align*} \\nu _ h \\ ; = \\ ; B _ h ( \\underline { \\nu } ) \\ , . \\end{align*}"}
+{"id": "5460.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\sup _ k \\int _ { | x | > R } | u _ k ^ * | ^ 4 + | v _ k ^ * | ^ 4 d x = 0 . \\end{align*}"}
+{"id": "3846.png", "formula": "\\begin{align*} \\check \\Lambda ^ n ( A \\times [ 0 , t ] ) \\doteq \\int _ { t _ n - t } ^ { t _ n } \\bar \\Lambda ^ n ( A \\mid s ) d s = \\int _ { 0 } ^ { t } \\check \\Lambda ^ n ( A \\mid s ) d s , \\end{align*}"}
+{"id": "6278.png", "formula": "\\begin{align*} y _ { k + 1 } = \\nabla ( \\Psi ^ * ) ( \\nabla \\Psi ( x _ k ) - \\nu g _ { k + 1 } ) , x _ { k + 1 } = \\arg \\min \\limits _ { x \\in \\mathcal { } } D _ { \\Psi } ( x , y _ { k + 1 } ) . \\end{align*}"}
+{"id": "5299.png", "formula": "\\begin{align*} t ( u ) = t r _ 0 \\{ \\bar { K } _ 0 ( u ) T _ 0 ( u ) K _ 0 ( u ) \\hat { T } _ 0 ( u ) \\} , \\end{align*}"}
+{"id": "536.png", "formula": "\\begin{align*} u _ { 0 } ( x ) = \\sum _ { k = 1 } ^ { \\infty } a _ { k } \\Phi _ { k } ( x ) . \\end{align*}"}
+{"id": "2541.png", "formula": "\\begin{align*} \\bigcap _ s \\dot \\AA ^ { ( s ) } ( M ) = \\dot C ^ \\infty ( M ) \\ ; , \\quad \\bigcap _ s \\AA ^ { ( s ) } ( M ) = C ^ \\infty ( M ) \\ ; , \\end{align*}"}
+{"id": "4138.png", "formula": "\\begin{align*} & \\sum _ { k \\in \\mathcal { J } } P _ { i k } u _ k = w _ { i j } , i \\in \\mathcal { J } . \\end{align*}"}
+{"id": "1589.png", "formula": "\\begin{align*} k _ n \\le k _ { n + 1 } \\le 2 \\ , k _ n , \\frac { k _ { n + 1 } } { k _ n } - 1 \\ge \\frac { 1 } { 2 ^ { n + 2 } } , ( r _ { n } - r _ { n + 1 } ) ^ 2 = \\frac { R ^ 2 } { 4 ^ { n + 2 } } , \\end{align*}"}
+{"id": "7168.png", "formula": "\\begin{align*} g _ \\sigma ( x , t ) = \\frac { 1 } { \\sqrt { 4 \\pi \\sigma t } } \\exp { \\left ( \\frac { - x ^ 2 } { 4 \\pi \\sigma t } \\right ) } , t > 0 , g _ 0 ( x , t ) = \\delta ( x ) . \\end{align*}"}
+{"id": "6547.png", "formula": "\\begin{align*} T _ { N } - T _ { N , 2 } & = \\sum ^ N _ { n = 1 } \\sum ^ { m _ 0 } _ { j = 1 } \\Big ( \\mathbb { E } \\big [ K ( X _ n + a _ j \\widetilde { \\varepsilon } _ { n - j } ) | \\varepsilon _ { n - j } \\big ] - \\mathbb { E } \\big [ K ( X _ n + a _ j \\widetilde { \\varepsilon } _ { n - j } ) \\big ] \\Big ) \\end{align*}"}
+{"id": "6562.png", "formula": "\\begin{align*} \\Phi _ k ( n ) = | \\{ ( x _ 1 , x _ 2 , \\cdots , x _ k ) \\in \\left ( \\mathbb { Z } / n \\mathbb { Z } \\right ) ^ k ; \\ \\gcd ( x _ 1 ^ 2 + x _ 2 ^ 2 + \\cdots + x _ k ^ 2 , n ) = 1 \\} | . \\end{align*}"}
+{"id": "5586.png", "formula": "\\begin{align*} \\langle \\hat { u } _ i , B ^ { \\ell } u _ j \\rangle = \\frac { \\langle B ^ { 3 \\ell } \\chi _ j , \\check \\chi _ i \\rangle } { \\nu _ i ^ { 2 \\ell + 2 } \\nu _ j ^ { 2 \\ell } } . \\end{align*}"}
+{"id": "2836.png", "formula": "\\begin{align*} L _ { T } : = & \\sigma ^ { * } _ { 3 } ( t ) \\left [ P _ { 1 } \\dot { Q } ^ { 1 } + \\Psi _ { a } \\dot { \\Xi } ^ { a } + \\Theta _ { 1 } \\dot { \\Theta } ^ { 1 } - H _ { T } \\right ] \\\\ = & P _ { 1 } \\dot { Q } ^ { 1 } - H _ { T } + { c o n s t a n t } \\end{align*}"}
+{"id": "3247.png", "formula": "\\begin{align*} & \\frac { \\partial } { \\partial x ^ k } \\frac { d } { d a } \\sum _ { j = 0 } ^ \\infty \\frac { ( - 1 ) ^ j } { j ! \\ : ( j + 1 ) ! } \\ : \\frac { a ^ { j + 1 } \\ : \\xi ^ { 2 j } } { 4 ^ j } \\\\ & = \\sum _ { j = 0 } ^ \\infty \\frac { ( - 1 ) ^ j } { j ! \\ : ( j + 1 ) ! } \\ : \\ : ( j + 1 ) \\ : \\big ( - 2 j \\ , \\xi _ k \\big ) \\ : \\frac { a ^ { j } \\ : \\xi ^ { 2 j - 2 } } { 4 ^ j } = \\frac { 1 } { 2 } \\ : \\xi _ k \\sum _ { j = 0 } ^ \\infty \\frac { ( - 1 ) ^ j } { j ! \\ : ( j + 1 ) ! } \\ : \\frac { a ^ { j + 1 } \\ : \\xi ^ { 2 j } } { 4 ^ j } \\ : . \\end{align*}"}
+{"id": "356.png", "formula": "\\begin{align*} \\phi _ p ( z , y ) = \\dfrac { z ^ p - y ^ p } { z - y } = A _ p ( z , y ) + ( z y ) ^ k = D _ p ( z , y ) + p ( z y ) ^ k . \\end{align*}"}
+{"id": "1128.png", "formula": "\\begin{align*} \\left \\| \\vec { t } \\right \\| _ { \\dot f ^ { s + n ( \\tau - \\frac { 1 } { p } ) } _ { \\infty , \\infty } ( \\mathbb A ) } = \\left \\| \\left \\{ 2 ^ { j [ s + n ( \\tau - \\frac { 1 } { p } ) ] } \\left | A _ j \\vec { t } _ j \\right | \\right \\} _ { j \\in \\mathbb Z } \\right \\| _ { L \\dot A _ { \\infty , \\infty } } = \\sup _ { j \\in \\mathbb Z } 2 ^ { j [ s + n ( \\tau - \\frac { 1 } { p } ) ] } \\left \\| \\ , \\left | A _ j \\vec { t } _ j \\right | \\ , \\right \\| _ { L ^ \\infty } . \\end{align*}"}
+{"id": "7606.png", "formula": "\\begin{align*} \\| u _ { \\epsilon } \\| _ { 4 ^ * } ^ { 4 ^ * } & = R ^ { 4 ^ * } \\int _ { \\mathbb { R } ^ { N } } \\varphi ^ { 4 ^ * } ( x ) ( \\frac { \\epsilon } { \\epsilon ^ { 2 } + | x | ^ { 2 } } ) ^ { N } d x \\\\ & = R ^ { 4 ^ * } \\int _ { \\mathbb { R } ^ { N } } \\varphi ^ { 4 ^ * } ( x ) \\frac { \\epsilon ^ { N } } { ( \\epsilon ^ { 2 } + | x | ^ { 2 } ) ^ { N } } d x . \\end{align*}"}
+{"id": "6933.png", "formula": "\\begin{align*} \\forall \\tau \\in B _ { \\varepsilon ^ \\star _ 0 } ( 0 ) , \\varpi ( \\tau ) = \\varphi ( \\tau ) + \\xi ( \\tau ) \\tau ^ { 2 \\mu + 1 } . \\end{align*}"}
+{"id": "6771.png", "formula": "\\begin{align*} \\begin{pmatrix} x ^ { k + 1 } \\\\ y ^ { k + 1 } \\end{pmatrix} = \\begin{pmatrix} x ^ { k } \\\\ y ^ { k } \\end{pmatrix} - \\begin{pmatrix} I _ n & 0 \\\\ - ( 1 - \\alpha ) \\frac { 1 } { s } A & I _ m \\end{pmatrix} \\begin{pmatrix} x ^ { k } - \\widetilde { x } ^ k \\\\ y ^ { k } - \\widetilde { y } ^ k \\end{pmatrix} . \\end{align*}"}
+{"id": "8534.png", "formula": "\\begin{align*} \\begin{array} { l c l } - a _ 1 \\mu _ 1 k ^ { - 1 } & = & 0 , \\\\ \\lambda _ 1 + k ^ { - 1 } \\mu _ 1 a _ 1 & = & 0 , \\\\ - a _ 2 \\mu _ 2 k ^ { - 1 } & = & 0 , \\\\ \\lambda _ 2 + k ^ { - 1 } \\mu _ 2 a _ 2 & = & 0 . \\end{array} \\end{align*}"}
+{"id": "186.png", "formula": "\\begin{align*} ( a _ 1 , b _ 1 ) , \\ \\ ( b _ 2 , a _ 2 ) , ( a _ 1 , b _ 2 ) , & \\ \\ ( a _ 2 , b _ 1 ) , ( a _ 1 , a _ 2 ) , \\ \\ ( b _ 1 , b _ 2 ) , \\\\ ( c , a _ 1 ) , \\ \\ ( c , a _ 2 ) , & ( b _ 1 , c ) , \\ \\ ( b _ 2 , c ) . \\end{align*}"}
+{"id": "8759.png", "formula": "\\begin{align*} f _ 1 ( x _ + ) - f _ 1 ( x _ - ) = \\frac { 2 \\varepsilon ( \\beta ( z - m ) - ( 2 m - y - z ) ) } { z - y } . \\end{align*}"}
+{"id": "4036.png", "formula": "\\begin{align*} k _ S ( x , y ) : = \\langle S ( x ) , S ( y ) \\rangle _ { \\mathcal { H } ^ 1 } , x , y \\in ( \\mathbb { R } ^ d ) ^ I . \\end{align*}"}
+{"id": "3104.png", "formula": "\\begin{align*} \\varphi \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & = \\left ( \\begin{array} { c c c } a ^ { \\ell _ 1 } & 0 & c _ 2 \\ , a ^ { \\ell _ 1 } \\ , b ^ { p ^ { e _ 2 } } \\ , d ^ { p ^ { e _ 2 } } \\\\ 0 & a ^ { \\ell _ 1 } & 0 \\\\ 0 & 0 & d ^ { - \\ell _ 3 } \\end{array} \\right ) . \\end{align*}"}
+{"id": "1219.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\mathcal { L } _ { t , \\infty } v = 0 & { \\rm i n } \\ \\ \\Omega \\setminus \\{ x _ 0 \\} , \\\\ v = 0 & { \\rm i n } \\ \\mathbb { R } ^ N \\setminus \\Omega , \\\\ v ( x _ 0 ) = v _ \\infty ( x _ 0 ) . \\end{array} \\right . \\end{align*}"}
+{"id": "2472.png", "formula": "\\begin{align*} \\xi _ { - } = \\lim _ { \\tau \\to - \\infty } \\xi ( \\tau ) , \\end{align*}"}
+{"id": "4015.png", "formula": "\\begin{align*} a _ { m + 1 } = c _ { m } \\left ( \\gamma _ { 1 } a _ { m } + \\delta _ { 1 } \\gamma _ { 2 } b _ { m } \\right ) , b _ { m + 1 } = c _ { m } \\left ( a _ { m } + \\delta _ { 2 } b _ { m } \\right ) , c _ { m + 1 } = c _ { m } \\left ( \\gamma a _ { m } + \\delta \\gamma _ { 2 } b _ { m } \\right ) . \\end{align*}"}
+{"id": "7688.png", "formula": "\\begin{align*} \\partial \\Lambda _ L = \\{ u \\in \\mathbb { Z } ^ d : \\ , \\ , \\mathrm { d i s t } ( u , \\Lambda _ L ) = 1 \\ , \\ , \\ , \\ , \\mathrm { d i s t } ( u , \\Lambda ^ { c } _ L ) = 1 \\} \\end{align*}"}
+{"id": "5789.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } e ^ { \\varepsilon t } \\| u \\| _ { C ^ 1 } ( t ) = \\infty . \\end{align*}"}
+{"id": "4404.png", "formula": "\\begin{gather*} G ^ k ( x ) : = \\sum _ { j \\in [ m ] } ( \\overline { u } _ j + \\max \\{ 0 , \\Delta u _ j - \\Delta u _ k \\} ) l _ j ( x ) . \\end{gather*}"}
+{"id": "2354.png", "formula": "\\begin{align*} \\sum ^ { m } _ { i = 1 } x _ { i } ^ { c } y _ { i } ^ { d } \\leq \\left ( \\sum ^ { m } _ { i = 1 } x _ { i } \\right ) ^ { c } \\left ( \\sum ^ { m } _ { i = 1 } y _ { i } \\right ) ^ { d } . \\end{align*}"}
+{"id": "1382.png", "formula": "\\begin{align*} I ( \\Gamma _ \\lambda ) = ( x _ 3 x _ 1 - \\lambda ^ 2 x _ 2 ^ 2 , \\ , \\lambda ( x _ 2 x _ 3 - x _ 1 x _ 4 ) , \\ , \\lambda ^ 2 x _ 2 x _ 4 , \\ , \\lambda x _ 3 x _ 4 , \\ , \\lambda ^ 2 x _ 4 ^ 2 , \\ , x _ 3 ^ 2 ) . \\end{align*}"}
+{"id": "3906.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow \\infty } \\frac { \\sum _ { \\gamma \\in P _ * ( T , \\Delta ) } e ^ { \\ell _ \\psi ( \\gamma ) + i t \\ell _ \\phi ( \\gamma ) / \\sqrt { T } } } { \\sum _ { \\gamma \\in P _ * ( T , \\Delta ) } e ^ { \\ell _ \\psi ( \\gamma ) } } = e ^ { - \\sigma _ { \\phi , \\psi } ^ 2 t ^ 2 / 2 } . \\end{align*}"}
+{"id": "2329.png", "formula": "\\begin{align*} \\Omega _ { \\rho } : = \\{ x \\in \\mathbb { R } ^ { 3 } \\ , | \\ , | x ' | \\leq \\rho | x | \\} . \\end{align*}"}
+{"id": "8686.png", "formula": "\\begin{align*} [ u ^ H , v ^ H ] = W - \\sqrt { - 1 } J W . \\end{align*}"}
+{"id": "4122.png", "formula": "\\begin{align*} \\gamma _ i ( \\theta _ i ( r ) , r ) = O ( \\theta _ i ( r ) ) , \\xi _ i ( \\theta ( r ) , r ) = O ( \\abs { \\theta ( r ) } ) r \\to 0 . \\end{align*}"}
+{"id": "3281.png", "formula": "\\begin{align*} \\frac { ( q ^ d ; q ^ d ) _ { n - 1 } ^ d } { ( 1 - q ) ^ { d n - d } } = \\prod _ { 1 \\leq m < n } \\bigg ( \\frac { 1 - q ^ { m d } } { 1 - q } \\bigg ) ^ d \\equiv 0 \\pmod { \\prod _ { \\substack { 1 < m < n \\\\ m \\mid n } } \\Phi _ m ( q ) ^ 2 } . \\end{align*}"}
+{"id": "5410.png", "formula": "\\begin{align*} f = \\chi _ { ( 0 , 1 / 2 ) } - \\chi _ { ( 1 / 2 , 1 ) } . \\end{align*}"}
+{"id": "508.png", "formula": "\\begin{align*} O \\left ( \\log { q } \\cdot \\left ( \\log ^ { 1 + o ( 1 ) } { q } + \\sum _ { p \\mid \\gcd ( \\overline { \\alpha } _ i , s ) } { \\left ( \\log \\log { q } \\cdot \\log ^ { 2 + o ( 1 ) } { p ^ { \\nu _ p ( s ) } } \\right ) } \\right ) \\right ) = O ( \\log ^ { 2 + o ( 1 ) } { q } ) \\end{align*}"}
+{"id": "5044.png", "formula": "\\begin{align*} \\| u \\| _ { B ^ \\theta _ p ( Y ) } ^ p : = [ u ] _ { \\theta , p } ^ p + \\| u \\| _ { L ^ p ( Y ) } ^ p < \\infty . \\end{align*}"}
+{"id": "2385.png", "formula": "\\begin{align*} S _ n & : = \\int _ { \\mathbb { Z } _ 2 } A _ n ^ { ( s ) } \\left ( t + \\frac { 1 } { 4 } \\right ) \\mathrm { d } t \\\\ & = \\rho _ { n , 0 } + \\sum _ { 2 \\leqslant i \\leqslant 2 s + 4 \\atop i \\equiv \\delta \\pmod { 2 } } \\rho _ { n , i } ~ \\zeta _ 2 \\left ( i + s + 1 , \\frac { 1 } { 4 } \\right ) . \\end{align*}"}
+{"id": "4592.png", "formula": "\\begin{align*} \\min s _ g & = \\min \\left \\{ \\frac { 1 } { h } ( t - t _ 0 ) \\right \\} + s _ 0 \\\\ & = \\frac 1 h \\min t - \\frac { 2 } { h [ \\omega ] ^ 2 } \\int _ M t d \\mu + \\frac { 8 \\pi c _ 1 [ \\omega ] } { [ \\omega ] ^ 2 } \\\\ & = - \\frac { \\omega ( F ) } { 4 \\pi h } - \\frac { 1 } { h T _ 0 } T _ 1 + \\frac { 4 \\pi c _ 1 [ \\omega ] } { T _ 0 } . \\end{align*}"}
+{"id": "8543.png", "formula": "\\begin{align*} \\begin{array} { l c l } 0 & = & - \\alpha \\log _ 1 ( \\pi _ 2 ) + \\gamma \\log _ 1 ( u ( K ) ) , \\\\ k \\lambda _ 1 + \\mu _ 1 a _ 1 & = & - \\beta , \\\\ 0 & = & \\alpha \\log _ 1 ( \\pi _ 2 ) - \\gamma \\log _ 1 ( u ( K ) ) , \\\\ k \\lambda _ 2 + \\mu _ 2 a _ 2 & = & - \\beta . \\end{array} \\end{align*}"}
+{"id": "3903.png", "formula": "\\begin{align*} S _ k ( T ) = \\frac { 1 } { 2 \\pi i } \\int _ { d - i \\infty } ^ { d + i \\infty } \\frac { \\eta ( s , 0 ) T ^ { s + k } } { s ( s + 1 ) \\cdots ( s + k ) } \\ , d s . \\end{align*}"}
+{"id": "3138.png", "formula": "\\begin{align*} u ^ - _ { \\psi ^ * } ( s ) = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ v _ { 2 , \\ , 1 } ( s ) & 1 & 0 \\\\ v _ { 3 , \\ , 1 } ( s ) & v _ { 3 , \\ , 2 } ( s ) & 1 \\end{array} \\right ) ( \\ v _ { 2 , \\ , 1 } ( S ) , v _ { 3 , \\ , 1 } ( S ) , v _ { 3 , \\ , 2 } ( S ) \\in k [ S ] \\ , ) . \\end{align*}"}
+{"id": "1103.png", "formula": "\\begin{align*} \\sup _ { x _ 0 \\in \\mathbb { R } ^ n , \\ , r \\in ( 0 , \\infty ) } \\left ( \\frac { | x _ 0 | + r } { | x _ 0 | + 2 ^ i r } \\right ) ^ a = 2 ^ { - i a } . \\end{align*}"}
+{"id": "693.png", "formula": "\\begin{align*} x _ i ^ 2 + y _ i ^ 2 + z _ i ^ 2 & = x _ i ^ 2 + \\left ( \\frac { b w _ i + e x _ i } { a } \\right ) ^ 2 + w _ i ^ 2 c ^ 2 \\\\ & = \\frac { ( a ^ 2 + e ^ 2 ) } { a ^ 2 } \\left ( x _ i + \\frac { b e w _ i } { a ^ 2 + e ^ 2 } \\right ) ^ 2 + \\frac { w _ i ^ 2 | | \\vec { a } | | ^ 2 } { ( a ^ 2 + e ^ 2 ) } = \\frac { w _ i ^ 2 | | \\vec { a } | | ^ 2 } { ( a ^ 2 + e ^ 2 ) } \\left ( U _ i ^ 2 + 1 \\right ) . \\end{align*}"}
+{"id": "7206.png", "formula": "\\begin{gather*} a _ 1 \\wedge m _ * ( b ) = a _ 2 \\wedge m _ * ( b ) \\implies \\\\ m ( a _ 1 ) \\wedge b = m ( a _ 1 ) \\wedge m \\circ m _ * ( b ) = m ( a _ 2 ) \\wedge m \\circ m _ * ( b ) = m ( a _ 2 ) \\wedge b \\\\ \\iff ( m ( a _ 1 ) , m ( a _ 2 ) ) \\in \\Phi _ b \\iff ( a _ 1 , a _ 2 ) \\in m ^ { - 1 } ( \\Phi _ b ) . \\end{gather*}"}
+{"id": "704.png", "formula": "\\begin{align*} f ( x ) = \\Phi ^ 3 ( x ) - ( 6 x ^ 3 - 1 8 x ^ 2 + 6 8 x - 2 2 4 ) \\Phi ^ 2 ( x ) - ( 8 0 0 x ^ 3 - 1 1 5 2 x ^ 2 + 2 1 4 4 x - 3 5 2 0 ) \\Phi ( x ) \\\\ - ( 1 3 1 8 4 x ^ 3 - 1 2 8 0 0 x ^ 2 + 1 1 7 7 6 x + 1 7 7 2 8 + m ) \\end{align*}"}
+{"id": "7544.png", "formula": "\\begin{align*} & - \\int _ 0 ^ T \\int _ \\Omega \\big ( \\tfrac { 1 } { 2 } \\bar \\rho | \\bar u | ^ 2 + h _ 1 ( \\bar \\rho ) + h _ 2 ( \\bar n ) + \\tfrac { 1 } { 2 } | \\nabla h _ 2 ^ \\prime ( \\bar n ) | \\big ) \\dot { \\theta } ( t ) \\ d x d t \\\\ & = \\int _ \\Omega \\big ( \\tfrac { 1 } { 2 } \\bar \\rho | \\bar u | ^ 2 + h _ 1 ( \\bar \\rho ) + h _ 2 ( \\bar n ) + \\tfrac { 1 } { 2 } | \\nabla h _ 2 ^ \\prime ( \\bar n ) | \\big ) \\big | _ { t = 0 } \\theta ( 0 ) \\ d x \\ , \\end{align*}"}
+{"id": "4074.png", "formula": "\\begin{align*} \\tilde { E } _ i ( \\pm i z _ 0 s ) = E _ i ( \\pm i z _ 0 s ) = h ( | z _ 0 | s ) \\log | z _ 0 s | + e ^ { \\pm i z _ 0 s } ( 1 - h ( | z _ 0 | s ) ) a ( \\pm i z _ 0 s ) \\end{align*}"}
+{"id": "2955.png", "formula": "\\begin{align*} \\Xi ( \\Gamma ) = \\min \\left \\{ c ( \\lambda , \\mu ) \\ : \\middle | \\ : \\lambda , \\mu \\in \\Lambda , \\ , \\{ \\lambda , \\mu \\} \\in \\overline { E } \\right \\} \\end{align*}"}
+{"id": "8218.png", "formula": "\\begin{align*} P _ { { \\bf X } _ { t - 1 } | Y ^ { t - 1 } } ( { \\bf x } _ { t - 1 } | y ^ { t - 1 } ) = P _ { { \\bf X } _ { t - 1 } | Y ^ { t - 1 } } ( { \\bf x } _ { t - 1 } | \\tilde { y } ^ { t - 1 } ) . \\end{align*}"}
+{"id": "8964.png", "formula": "\\begin{align*} B _ j : = \\Big \\{ x \\in \\Omega \\ , : \\ , \\varepsilon _ j ^ \\frac { 1 } { 2 } | \\nabla u _ j ( x ) | \\le 1 \\Big \\} . \\end{align*}"}
+{"id": "5908.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { C } ( G ) ) & = ( q - 1 ) ( q - 1 - 1 ) ^ { 2 } + q ( p - 1 ) ( p - 1 - 1 ) ^ { 2 } \\\\ & = ( q - 1 ) ( q - 2 ) ^ { 2 } + q ( p - 1 ) ( p - 2 ) ^ { 2 } \\end{align*}"}
+{"id": "2932.png", "formula": "\\begin{align*} \\sum _ { T \\in B _ 0 \\cup T } w _ T = 0 . \\end{align*}"}
+{"id": "2893.png", "formula": "\\begin{align*} & \\lim _ { \\lambda \\to \\infty } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ { L ^ { { r } ( \\cdot ) } ( \\mathbb { R } ^ n ) } = \\left [ \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { L ^ { { r } ( \\cdot ) } ( \\mathbb { R } ^ n ) } ; \\end{align*}"}
+{"id": "6659.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi \\mathrm { i } n } \\oint _ { | z | = 1 } \\frac { f ' _ { n } ( z ) } { f _ { n } ( z ) } \\mathrm { d } z \\leq \\frac { 1 } { 2 \\pi n \\epsilon } \\int _ { 0 } ^ { 2 \\pi } \\bigg ( \\log | f _ { n } ( \\mathrm { e } ^ { \\mathrm { i } ( x - \\mathrm { i } \\epsilon ) } ) | - \\log | f _ { n } ( \\mathrm { e } ^ { \\mathrm { i } x } ) | \\bigg ) \\mathrm { d } x . \\end{align*}"}
+{"id": "5278.png", "formula": "\\begin{align*} x ^ { k + 1 } = \\frac { \\widetilde { x } ^ { k + 1 } } { \\sum _ { l } \\tilde { x } ^ { k + 1 } _ { l } } C \\end{align*}"}
+{"id": "1928.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta h ( x ) \\leq \\frac { \\hat C \\beta } { \\mu _ c } \\left \\{ - b [ \\mathbf { d } ( x ) + 1 ] ^ { - \\beta - 1 } + \\mathbf { d } ^ { - \\beta - 1 } ( x ) \\right \\} . \\end{aligned} \\end{align*}"}
+{"id": "4891.png", "formula": "\\begin{align*} | C + K _ { S ' } | = F + | M | , \\end{align*}"}
+{"id": "6931.png", "formula": "\\begin{align*} \\forall \\tau \\in B _ { \\varepsilon ^ \\star _ 0 } ( 0 ) , F ( e ^ { \\varpi ( \\tau ) } ) = e ^ \\tau . \\end{align*}"}
+{"id": "7888.png", "formula": "\\begin{align*} \\mathbf { D } _ { i , j } = \\left ( \\begin{array} { c c c c c c c } 2 ( i - j ^ { ' } ) & c _ { 1 } & & & & & \\\\ c _ { 1 } & \\ddots & \\ddots & & & & \\\\ & \\ddots & 2 ( i - j ^ { ' } ) & c _ { s - 1 } & & & \\\\ & & c _ { s - 1 } & 2 ( i - j ^ { ' } ) - 2 e & c _ { s } & & \\\\ & & & c _ { s } & 2 ( i - j ^ { ' } ) & \\ddots & \\\\ & & & & \\ddots & \\ddots & c _ { m - 1 } \\\\ & & & & & c _ { m - 1 } & 2 ( i - j ^ { ' } ) \\end{array} \\right ) . \\end{align*}"}
+{"id": "6570.png", "formula": "\\begin{align*} R _ { k , \\beta } ( z ) = \\begin{cases} \\prod _ { p } \\left ( 1 - \\frac { 1 } { p } + \\frac { 1 } { p } \\left ( \\frac { p } { p - 1 } \\right ) ^ z \\right ) & k , \\\\ \\prod _ { p } \\left ( 1 - \\frac { 1 } { p } + \\frac { 1 } { p } \\left ( \\frac { p } { p - 1 } \\right ) ^ z \\alpha _ k ( p ) ^ { - z } \\right ) & k . \\end{cases} \\end{align*}"}
+{"id": "996.png", "formula": "\\begin{align*} \\mathbb E _ { z } ( \\mathbf 1 _ D u ( X _ { \\tau _ V } ) \\mathbf 1 _ A ( X _ { \\tau _ V - } ) ) = \\int _ { A \\cap V } \\int _ { D \\setminus \\bar V } G _ V ( z , x ) \\ , u ( y ) j ( | x - y | ) \\ , d y \\ , d x . \\end{align*}"}
+{"id": "7980.png", "formula": "\\begin{align*} \\mathrm { t r } ( \\tilde { f } ) = \\mathrm { t r } \\big ( \\mathrm { l i } _ { \\phi } ( f _ { \\phi } ) - \\mathrm { l i } ' _ { \\phi } ( f _ { \\phi } ) \\big ) = 0 . \\end{align*}"}
+{"id": "2437.png", "formula": "\\begin{align*} \\phi \\phi '' = - c \\phi ' + \\gamma ( \\phi ' ) ^ { 2 } - k \\phi ^ { 2 } + \\delta p \\phi , \\left ( \\ , ' = \\dfrac { d } { d \\xi } , '' = \\dfrac { d ^ { 2 } } { d \\xi ^ { 2 } } \\ , \\right ) . \\end{align*}"}
+{"id": "1304.png", "formula": "\\begin{align*} | \\mu _ 1 ( t ) - \\hat { \\mu } _ 1 ( t ) | & = \\frac { 1 } { 2 } \\Big | \\Big ( e ^ { ( a - 2 k ) t } + e ^ { a t } ) x _ 1 + ( e ^ { a t } - e ^ { ( a - 2 k ) t } ) x _ 2 \\Big ) - e ^ { a t } x _ 1 \\Big | \\\\ & = \\frac { 1 } { 2 } | ( e ^ { a t } - e ^ { ( a - 2 k ) t } ) ( x _ 2 - x _ 1 ) | \\\\ & = \\frac { 1 } { 2 } e ^ { a t } | x _ 2 - x _ 1 | ( 1 - e ^ { - 2 k t } ) \\\\ & \\leq | x _ 2 - x _ 1 | k e ^ { a t } t \\\\ & \\leq k \\ , | x _ 2 - x _ 1 | \\frac { 1 } { | a | e } , \\end{align*}"}
+{"id": "6669.png", "formula": "\\begin{align*} \\gamma _ j ( p _ j , 0 ) & = 0 \\in \\R ^ n , \\\\ T _ j ( p _ j , 0 ) & = ( 1 , 0 , \\dots , 0 ) = e _ 1 , \\\\ N _ j ( p _ j , 0 ) & = ( 0 , 1 , 0 , \\dots , 0 ) = e _ 2 , \\\\ ( B _ i ) _ j ( p _ j , 0 ) & = ( 0 , \\dots , 0 , 1 , 0 , \\dots , 0 ) = e _ { i + 2 } , i = 1 , \\dots , n - 2 . \\end{align*}"}
+{"id": "2375.png", "formula": "\\begin{align*} h ( \\mu | \\nu ) = \\int \\log \\Big ( \\frac { \\dd \\mu } { \\dd \\nu } \\Big ) \\dd \\mu \\ , , \\end{align*}"}
+{"id": "4065.png", "formula": "\\begin{align*} [ \\chi R _ 0 ( z ) \\chi ] ( x , y ) = \\chi ( x ) \\frac { \\alpha _ 1 } { | x - y | ^ { d - 1 } } \\chi ( y ) + O _ { C } ( | z | ) . \\end{align*}"}
+{"id": "7077.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\langle ( \\lambda - \\Delta ) ^ { \\frac { 1 } { 4 } } u , \\partial _ t ( \\lambda - \\Delta ) ^ { \\frac { 1 } { 4 } } \\varphi \\rangle d t & = \\int _ { 0 } ^ { \\infty } \\langle ( \\lambda - \\Delta ) ^ { \\frac { 3 } { 4 } } u , ( \\lambda - \\Delta ) ^ { \\frac { 3 } { 4 } } \\varphi \\rangle d t \\\\ & + \\int _ { 0 } ^ { \\infty } \\langle b \\cdot \\nabla u , ( \\lambda - \\Delta ) ^ { \\frac { 1 } { 2 } } \\varphi \\rangle \\end{align*}"}
+{"id": "7082.png", "formula": "\\begin{align*} c _ p ^ { - 1 } \\| A ^ \\frac { 1 } { 2 } u ^ \\frac { p } { 2 } \\| _ 2 ^ 2 \\leqslant \\langle A _ p u , u ^ { p - 1 } \\rangle , c _ p : = \\frac { p p ' } { 4 } , p ' = \\frac { p } { p - 1 } \\end{align*}"}
+{"id": "4763.png", "formula": "\\begin{align*} \\left \\langle \\frac { S ( t ) x - x } { t } , j _ { v , y _ 0 } ( x - x _ 0 ) \\right \\rangle & \\geq - \\norm { \\frac { S ( t ) x - x } { t } } \\norm { x - J _ \\lambda x } \\\\ & = - \\norm { \\frac { S ( t ) x - x } { t } } \\lambda \\norm { A _ \\lambda x } . \\end{align*}"}
+{"id": "1385.png", "formula": "\\begin{align*} t \\circ [ y _ 1 : y _ 2 : y _ 3 : y _ 4 : y _ 5 ] = [ y _ 1 : t y _ 2 : y _ 3 : t ^ { - 1 } y _ 4 : y _ 5 ] . \\end{align*}"}
+{"id": "7248.png", "formula": "\\begin{align*} \\mathbf A _ t \\ , f \\ , ( x ) = \\int _ \\R \\ , \\tfrac { \\partial } { \\partial x } \\left ( \\tfrac { f ( y ) - f ( x ) } { y - x } \\right ) \\ , \\nu _ { x , t } ( d y ) , \\end{align*}"}
+{"id": "7304.png", "formula": "\\begin{align*} ( \\partial _ { t } + \\Delta _ X ) K _ X ( x , y ; t ) = 0 \\ , \\ , \\ , \\ , \\ , \\end{align*}"}
+{"id": "7146.png", "formula": "\\begin{align*} \\bigl ( \\widetilde { p } ( k ) \\bigr ) _ \\nu ^ { ~ \\mu } = \\delta _ \\nu ^ { ~ \\mu } \\ , \\ , , q ( k ) = \\eta ^ { \\alpha \\beta } k _ \\alpha k _ \\beta \\ , \\ , , \\end{align*}"}
+{"id": "180.png", "formula": "\\begin{align*} ( ( x , t ) , ( y , s ) ) \\in R _ 2 ^ { \\circ } \\Longleftrightarrow \\begin{cases} ( x , y ) \\in L _ 1 ^ { \\circ } , \\\\ x = y \\ \\ \\ \\ ( t , s ) \\in S _ x ^ { \\circ } . \\end{cases} \\end{align*}"}
+{"id": "4654.png", "formula": "\\begin{align*} O \\left ( \\sqrt { N } \\log ^ { 2 } N \\cdot N ^ { 0 . 0 1 } \\cdot \\log N \\right ) = O \\left ( N ^ { 0 . 5 2 } \\right ) . \\end{align*}"}
+{"id": "437.png", "formula": "\\begin{align*} \\partial h _ { i , j } ( x ) = \\begin{cases} \\{ - \\lambda _ { i , j } \\} , & { \\rm i f ~ } x < 0 , \\\\ [ - \\lambda _ { i , j } , \\mu _ { i , j } ] , & { \\rm i f ~ } x = 0 , \\\\ \\{ \\mu _ { i , j } \\} , & { \\rm i f ~ } x > 0 . \\end{cases} \\end{align*}"}
+{"id": "1938.png", "formula": "\\begin{align*} \\operatorname { T a i l } _ a ( v ; x _ { 0 } , r ) & : = \\left ( r ^ { s q } \\underset { x \\in B _ { 2 r } ( x _ 0 ) } { { \\rm e s s } \\sup } \\int _ { \\mathbb { R } ^ { N } \\backslash B _ { r } \\left ( x _ { 0 } \\right ) } a ( x , y ) \\frac { | v ( y ) | ^ { q - 1 } } { \\left | y - x _ { 0 } \\right | ^ { N + s q } } \\ , d y \\right ) ^ { \\frac { 1 } { q - 1 } } . \\end{align*}"}
+{"id": "1774.png", "formula": "\\begin{align*} \\widehat X _ { - 1 } : = \\left ( 2 z _ { n - 1 } - 2 a z _ 1 ^ 3 \\right ) \\frac { \\partial } { \\partial z _ { 0 } } + \\frac { \\partial } { \\partial z _ { 1 } } - z _ n \\frac { \\partial } { \\partial z _ { 2 } } - 3 a z _ 1 ^ 2 \\frac { \\partial } { \\partial z _ { n - 1 } } . \\end{align*}"}
+{"id": "1854.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial r } \\hat { \\mathcal { E } } ( A , B _ r ( 0 ) ) = - 3 r ^ { - 4 } \\int _ { B _ r } | F _ A | ^ 2 + r ^ { - 3 } \\int _ { \\partial B _ r } | F _ A | ^ 2 . \\end{align*}"}
+{"id": "7929.png", "formula": "\\begin{align*} \\begin{aligned} & \\boldsymbol { n } ( \\delta \\beta ) = ( - 1 ) ^ { n - 1 } \\boldsymbol { n } ( \\ast d \\ast \\beta ) = ( - 1 ) ^ { n - 1 } \\ast \\mathrm { t r } ( d \\ast \\beta ) = ( - 1 ) ^ { n - 1 } \\ast d \\mathrm { t r } ( \\ast \\beta ) = 0 , \\\\ & \\boldsymbol { n } ( \\alpha ) = ( - 1 ) ^ { n - 1 } \\ast \\mathrm { t r } ( \\ast \\alpha ) = 0 . \\end{aligned} \\end{align*}"}
+{"id": "1380.png", "formula": "\\begin{align*} \\frac { | K \\cap L | } { | I | } = \\frac { | K | } { | I | } - \\frac { | K \\setminus L | } { | I | } > \\frac { | K | } { | I | } - u ( G ) v ( G ) . \\end{align*}"}
+{"id": "4684.png", "formula": "\\begin{align*} q _ X ( f _ { * } P ( D ) , E ' ) \\int ( f _ { * } P ( D ) ) ^ { 2 n } = q _ X ( f _ { * } P ( D ) ) \\int ( f _ { * } P ( D ) ) ^ { 2 n - 1 } \\cdot E ' = 0 , \\end{align*}"}
+{"id": "6518.png", "formula": "\\begin{align*} \\frac { 1 } { n + 2 } h _ { K ( 0 ) } ( \\xi ) d S _ { K ( 0 ) } ^ { \\mu _ 0 } ( \\xi ) = \\varphi ( \\xi ) d \\xi , \\end{align*}"}
+{"id": "8283.png", "formula": "\\begin{align*} a _ { i j } = \\frac { 1 } { w } \\sum _ { k , l } \\gamma ^ { i k } ( \\delta _ { k l } + u _ k u _ l + u u _ { k l } ) \\gamma ^ { l j } , \\end{align*}"}
+{"id": "2210.png", "formula": "\\begin{align*} u = \\psi _ { 1 , z } . \\end{align*}"}
+{"id": "9108.png", "formula": "\\begin{align*} & \\sum _ { l = 1 } ^ { n } \\Tilde { k } _ { i l } < \\frac { 1 - \\sum _ { l = 1 } ^ { n } a _ { i l } } { \\hat { c } _ { i i } } , \\sum _ { l = 1 } ^ { n } \\Tilde { k } _ { i l } \\geq \\frac { - \\sum _ { l = 1 } ^ { n } a _ { i l } } { \\hat { c } _ { i i } } , \\end{align*}"}
+{"id": "974.png", "formula": "\\begin{align*} w ( X _ t ) = A ^ \\mu _ { \\tau _ V } - A ^ \\mu _ t - ( N _ { \\tau _ V } - N _ t ) , t \\le \\tau _ V , P _ x \\end{align*}"}
+{"id": "3900.png", "formula": "\\begin{align*} \\mu _ \\psi ( \\phi ) = \\lim _ { n \\to \\infty } \\frac { \\sum _ { \\gamma \\in P ( n ) } \\exp ( \\ell _ { \\psi } ( \\gamma ) ) \\delta _ \\gamma ( \\phi ) } { \\sum _ { \\gamma \\in P ( n ) } \\exp ( \\ell _ { \\psi } ( \\gamma ) ) } \\leq \\lim _ { n \\to \\infty } | \\phi | _ \\infty \\frac { \\sum _ { \\gamma \\in Q ' ( n ) } \\exp ( \\ell _ { \\psi } ( \\gamma ) ) } { \\sum _ { \\gamma \\in P ( n ) } \\exp ( \\ell _ { \\psi } ( \\gamma ) ) } . \\end{align*}"}
+{"id": "675.png", "formula": "\\begin{align*} ( ( - \\Delta ) ^ s + q ) u = 0 \\mbox { i n } \\Omega . \\end{align*}"}
+{"id": "5103.png", "formula": "\\begin{align*} ( 1 + g _ t ) e _ t g _ s ^ { - 1 } & = g _ s ^ { - 1 } ( 1 + g _ s g _ t g _ s ^ { - 1 } ) e _ { t s t } . \\end{align*}"}
+{"id": "888.png", "formula": "\\begin{align*} S _ 2 + T _ 2 = 2 b _ 1 ( c _ 1 + a _ 1 x _ 1 + b _ 1 x _ 2 ) + 2 b _ 2 ( c _ 2 + a _ 2 x _ 1 + b _ 2 x _ 2 ) . \\end{align*}"}
+{"id": "7123.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } V _ { 2 } ( u _ { n } ) = V _ { 2 } ( u _ { c } ) V _ { 1 } ( u _ { c } ) \\leq \\liminf _ { n \\rightarrow \\infty } V _ { 1 } ( u _ { n } ) , \\end{align*}"}
+{"id": "3205.png", "formula": "\\begin{align*} { c _ 0 } = 1 , { c _ k } = \\prod _ { j = 0 } ^ { k - 1 } \\frac { \\Gamma ( \\alpha ( j m + l ) + 1 ) } { \\Gamma ( \\alpha ( j m + l + 1 ) + 1 ) } . \\end{align*}"}
+{"id": "4439.png", "formula": "\\begin{align*} \\limsup _ { r \\rightarrow 1 - 0 } \\int _ { \\partial D _ r } | f _ n - f | ^ 2 \\varphi | d z | & \\le L _ 1 ^ 2 \\limsup _ { r \\rightarrow 1 - 0 } \\int _ { \\partial D _ r } | f _ n - f | ^ 2 \\frac { \\partial \\psi } { \\partial v _ z } | d z | \\\\ & \\le L _ 1 ^ 2 \\int _ { \\partial D } | f _ n - f | ^ 2 \\frac { \\partial \\psi } { \\partial v _ z } | d z | \\\\ & \\le L _ 1 ^ 4 \\int _ { \\partial D } | f _ n - f | ^ 2 \\varphi | d z | . \\end{align*}"}
+{"id": "8335.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Pi _ { t } = [ \\Pi _ t , \\widehat { X _ { t } } ] _ { S N } . \\end{align*}"}
+{"id": "2592.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta \\varphi = f ( \\varphi ) & B _ R , \\\\ \\dfrac { \\partial \\varphi } { \\partial \\nu } + \\beta \\varphi = 0 & \\partial B _ R , \\end{cases} \\end{align*}"}
+{"id": "4090.png", "formula": "\\begin{align*} R _ { \\Delta } ( \\lambda ) f = e ^ { i \\lambda r } r ^ { \\frac { 1 - d } { 2 } } h ( \\theta ) + O _ { L ^ 2 ( \\R ^ d ) } , \\ , \\ , \\ , \\ , h ( \\theta ) = c _ n \\lambda ^ { \\frac { d - 3 } { 2 } } \\widehat { f } ( \\lambda \\theta ) , \\end{align*}"}
+{"id": "8423.png", "formula": "\\begin{align*} X _ x ( b ) = \\{ g \\in G ( \\breve F ) / I \\mid g ^ { - 1 } b \\sigma ( g ) \\in I x I \\} . \\end{align*}"}
+{"id": "7284.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } C _ { m } ^ { \\mathrm { a l t } } ( 0 , 2 n ) y ^ { 2 n } = 1 - \\frac { m y } { \\sqrt { 1 - y ^ 2 } } \\csc ( m \\arcsin ( y ) ) . \\end{align*}"}
+{"id": "5837.png", "formula": "\\begin{align*} \\textbf { ( i i ) : } \\displaystyle \\left ( 1 - \\alpha ^ { i - j } \\right ) \\lambda _ { i , j , k } & = \\alpha ^ { i - 3 j - 3 } \\left ( \\sum _ { \\nu = 0 } ^ { j } \\alpha ^ { 3 \\nu } \\right ) \\lambda _ { i - 2 , j + 1 , k + 1 } , \\\\ \\displaystyle \\left ( 1 - \\alpha ^ { i - k } \\right ) \\lambda _ { i , j , k } & = \\alpha ^ { i - 2 j - k - 3 } \\left ( \\sum _ { \\nu = 0 } ^ { k } \\alpha ^ { 3 \\nu } \\right ) \\lambda _ { i - 2 , j + 1 , k + 1 } \\end{align*}"}
+{"id": "389.png", "formula": "\\begin{align*} M _ { 2 } ( \\bar { \\nu } ) \\cap M _ { 2 } ( \\bar { \\mu } ) = M _ { 2 } ( \\max \\{ \\bar { \\nu } , \\bar { \\mu } \\} ) . \\end{align*}"}
+{"id": "6379.png", "formula": "\\begin{align*} \\beta _ x ( a ^ i b ^ j , a ^ k b ^ \\ell ) = \\xi ^ { i \\ell - j k } . \\end{align*}"}
+{"id": "788.png", "formula": "\\begin{align*} x _ i x _ { m - i + 1 } = w ^ 2 \\qquad \\beta ( x _ i ^ { - 1 } x _ { m - i + 1 } , w ) = \\tau \\lambda _ i . \\end{align*}"}
+{"id": "3822.png", "formula": "\\begin{align*} w ^ 2 = c ( \\tau ) b ( x _ 0 : x _ 2 , \\tau ) , \\end{align*}"}
+{"id": "3668.png", "formula": "\\begin{align*} Z ( z ) = n x _ z ^ T L ( G , p ) x _ z - a ( G ) x _ z ^ T L ( K _ n , p ) x _ z , \\end{align*}"}
+{"id": "1073.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { \\infty } \\Vert ( T _ { \\infty } ( w ) ^ { - 1 } ) ^ { s , t } \\| \\le B _ 1 , t \\in \\N . \\end{align*}"}
+{"id": "2477.png", "formula": "\\begin{align*} \\overline { U } _ { k } = \\{ y \\in \\mathbb { R } ^ { 3 } \\ , | \\ , y _ { k } = 1 \\} , \\overline { V } _ { k } = \\{ y \\in \\mathbb { R } ^ { 3 } \\ , | \\ , y _ { k } = - 1 \\} \\end{align*}"}
+{"id": "5968.png", "formula": "\\begin{align*} S = \\{ x \\ , | \\ , L ( x ) \\leq L _ 0 \\} . \\end{align*}"}
+{"id": "8815.png", "formula": "\\begin{align*} m _ 1 = & \\ ; 0 , \\\\ m _ 2 \\ge & \\ ; 2 ( n _ 2 + 1 ) , \\\\ \\vdots & \\ ; \\\\ m _ k \\ge & \\ ; 2 \\max \\big ( m _ { k - 1 } + n _ 2 + 1 , m _ { k - 2 } + n _ 3 + 1 , \\ldots , m _ 2 + n _ { k - 1 } + 1 , n _ k + 1 \\big ) . \\end{align*}"}
+{"id": "4670.png", "formula": "\\begin{align*} \\mathcal { H } : = \\left \\{ ( I , \\sigma , c ) \\ ; | \\ ; \\hat { I } = I \\right \\} \\end{align*}"}
+{"id": "1625.png", "formula": "\\begin{align*} z ^ r _ { p , q } & \\begin{cases} q & 1 \\leq p \\leq k - r \\\\ q \\leq M + k - p & k - r < p \\leq k \\end{cases} \\\\ & \\begin{cases} q & r + 1 \\leq p < k \\\\ q \\leq M + 1 & p = k . \\end{cases} \\end{align*}"}
+{"id": "6971.png", "formula": "\\begin{align*} \\widetilde { \\Pi } ( \\alpha \\mathbf { z } ) = \\alpha \\mathbf { z } \\overline { \\eta } ( \\alpha \\mathbf { z } ) + \\overline { \\alpha \\mathbf { z } } \\eta ( \\alpha \\mathbf { z } ) = \\alpha \\mathbf { z } \\overline { \\alpha } \\ , \\overline { \\eta } ( \\mathbf { z } ) + \\overline { \\alpha } \\ , \\mathbf { \\overline { z } } \\alpha \\eta ( \\mathbf { z } ) = \\lvert \\alpha \\rvert ^ 2 \\widetilde { \\Pi } ( \\mathbf { z } ) . \\end{align*}"}
+{"id": "1769.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\beta _ { n , j } = ( 2 j - 2 - n ) ( z _ { j - 1 } + \\beta _ { n , j - 1 } ) \\quad \\mbox { a n d } \\beta _ { n , j } ( 0 ) = 0 \\quad \\quad \\forall j \\in \\{ 3 , \\ldots , n - 1 \\} . \\end{align*}"}
+{"id": "2987.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\nabla _ x \\xi = \\phi x , & & ( \\nabla _ x \\eta ) y = g ( x , \\phi y ) , \\\\ R ( x , y ) \\xi = - \\eta ( y ) x + \\eta ( x ) y , & & ( x , \\xi ) = - 2 n \\ \\eta ( x ) , \\\\ R ( \\xi , y ) \\xi = \\phi ^ 2 y , & & ( x , \\xi ) ( \\xi , \\xi ) = - 2 n , \\end{array} \\end{align*}"}
+{"id": "532.png", "formula": "\\begin{align*} z ^ { \\tau } = \\sum _ { k = 1 } ^ \\infty a _ { k } \\Phi _ { k } , \\end{align*}"}
+{"id": "7572.png", "formula": "\\begin{align*} A _ { n } : = \\eta ( 1 , D _ { n } ) = \\{ u _ { t _ { u } } : u \\in D _ { n } \\} \\in \\mathcal { G } . \\end{align*}"}
+{"id": "7113.png", "formula": "\\begin{align*} \\mathcal { H } : = \\left \\{ \\psi \\in H ^ { 1 } ( \\mathbb { R } ^ { 2 } ) | \\int _ { \\mathbb { R } ^ { 2 } } \\ln ( \\sqrt { 1 + | x | ^ { 2 } } ) \\psi ^ { 2 } d x < \\infty \\right \\} . \\end{align*}"}
+{"id": "2662.png", "formula": "\\begin{align*} \\delta q ^ { i } ( t _ { 1 } ) = \\delta q ^ { i } ( t _ { 2 } ) = 0 . \\end{align*}"}
+{"id": "3068.png", "formula": "\\begin{align*} \\omega = H _ 1 \\cdot ( n x d y - m y d x ) + d H _ 2 . \\end{align*}"}
+{"id": "4679.png", "formula": "\\begin{align*} \\mathbf { B } _ { - } ( D ) = \\bigcap _ { E \\equiv D } \\mathrm { S u p p } ( P ( E ) + N ( E ) ) , \\end{align*}"}
+{"id": "5063.png", "formula": "\\begin{align*} & x _ i ^ { a - c , ( 1 ) } ( k ) = x _ i ^ { a , ( 1 ) } ( k ) - x _ i ^ { c , ( 1 ) } ( k - 1 ) , \\\\ & y _ i ^ { a - c , ( 1 ) } ( k ) = y _ i ^ { a , ( 1 ) } ( k ) - y _ i ^ { c , ( 1 ) } ( k - 1 ) . \\end{align*}"}
+{"id": "6748.png", "formula": "\\begin{align*} \\begin{aligned} & \\theta ( u ) = \\sum _ { i = 1 } ^ { m } f _ i ( x _ i ) , ~ u = \\begin{pmatrix} x _ 1 \\\\ \\vdots \\\\ x _ m \\end{pmatrix} , ~ w = \\begin{pmatrix} x _ 1 \\\\ \\vdots \\\\ x _ m \\\\ \\lambda \\end{pmatrix} , ~ F ( w ) = \\begin{pmatrix} - A _ 1 ^ T \\lambda \\\\ \\vdots \\\\ - A _ m ^ T \\lambda \\\\ A x - b \\end{pmatrix} , \\end{aligned} \\end{align*}"}
+{"id": "8204.png", "formula": "\\begin{align*} & C ( L = i + 1 ) \\\\ & = \\max _ { P _ { { \\bf X } _ j | Y ^ { j - 1 } } \\in \\mathcal X ( B ^ { \\prime } _ { a , j } ) , j \\in [ i ] } \\left ( \\sum _ { j = 1 } ^ i H ( Y _ j | Y ^ { j - 1 } , { \\bf S } ) \\right . \\\\ & \\left . + \\max _ { P _ { { \\bf X } _ { i + 1 } | Y ^ i } \\in \\mathcal X ( B ^ { \\prime } _ { a , i + 1 } ) } H ( Y _ { i + 1 } | Y ^ { i } , { \\bf S } ) \\right ) , \\end{align*}"}
+{"id": "5010.png", "formula": "\\begin{align*} = 1 0 \\log _ { 1 0 } \\frac { } { } \\end{align*}"}
+{"id": "7422.png", "formula": "\\begin{align*} b ' _ 0 & = b _ 0 c _ 0 \\\\ b ' _ 1 & = b _ 0 c _ 1 + b _ 1 c _ 0 + ( a _ 1 - d _ 1 ) a _ 2 \\\\ b ' _ 2 & = b _ 0 c _ 2 + b _ 1 c _ 1 + b _ 2 c _ 0 + ( a _ 1 - d _ 1 ) a _ 3 - a _ 2 d _ 2 \\\\ b ' _ 3 & = b _ 0 c _ 3 + b _ 1 c _ 2 + b _ 2 c _ 1 + b _ 3 c _ 0 + ( a _ 1 - d _ 1 ) a _ 4 - a _ 2 d _ 3 - a _ 3 d _ 2 . \\\\ \\end{align*}"}
+{"id": "4177.png", "formula": "\\begin{align*} \\varphi ( g _ 2 g ^ { n } _ 1 g _ 3 ) \\overset { ( \\ref { e q : c a s e 1 - 6 } ) } { = } \\varphi \\big ( g ^ { j } _ 1 g _ 3 \\big ) \\varphi \\big ( g ^ { i } _ 1 g _ 2 \\big ) \\overset { ( \\ref { e q : c a s e 1 - 4 } ) } { = } \\varphi ( g _ 3 ) \\varphi ( g ^ { n } _ 1 ) \\varphi ( g _ 2 ) \\overset { ( \\beta _ 1 ) } { = } \\varphi ( g _ 3 ) \\varphi ( g ^ { n } _ 1 g _ 2 ) \\ , . \\end{align*}"}
+{"id": "8433.png", "formula": "\\begin{align*} \\| P \\| _ K = \\max _ { x \\in K } \\sum _ { j = 1 } ^ { n + 1 } | \\lambda _ j ( x ) | . \\end{align*}"}
+{"id": "1602.png", "formula": "\\begin{align*} \\lambda \\wedge \\phi ( e _ 1 , \\dots , e _ { k + \\ell } ) = \\sum \\epsilon ( i _ 1 \\dots i _ { k + \\ell } ) \\ , \\lambda ( e _ { i _ 1 } , \\dots , e _ { i _ k } ) \\ , \\phi ( e _ { i _ { k + 1 } } , \\dots , e _ { i _ { k + \\ell } } ) \\end{align*}"}
+{"id": "6296.png", "formula": "\\begin{align*} \\Delta _ k = \\tilde { O } \\left ( \\frac { \\mu _ r ^ 2 R _ 0 ^ { ( 2 r - 1 ) } } { M _ 2 \\sqrt { d } } \\frac { 1 } { 2 ^ { k ( 2 r - 1 ) } } \\right ) . \\end{align*}"}
+{"id": "8604.png", "formula": "\\begin{align*} \\| u _ 0 ^ { ( n ) } \\| _ { L ^ \\infty ( \\mathbb { R } ) } \\leq ( ( n - 1 ) g ) ^ { 2 ( n - 1 ) } ~ ~ f o r ~ ~ n = 2 , 3 , . . . \\end{align*}"}
+{"id": "6410.png", "formula": "\\begin{align*} \\ ; b \\neq 0 & v _ t ( u ) = \\frac { u e ^ { - b t } } { \\left ( 1 + \\frac { \\delta ^ \\alpha } { \\alpha b } u ^ { \\alpha - 1 } \\left ( 1 - e ^ { - ( \\alpha - 1 ) b t } \\right ) \\right ) ^ { \\frac { 1 } { \\alpha - 1 } } } \\\\ \\ ; b = 0 & v _ t ( u ) = u \\left ( \\frac { \\alpha } { \\alpha + ( \\alpha - 1 ) \\delta ^ \\alpha u ^ { \\alpha - 1 } t } \\right ) ^ \\frac { 1 } { \\alpha - 1 } . \\end{align*}"}
+{"id": "1925.png", "formula": "\\begin{align*} \\begin{aligned} \\operatorname { c a r d i n a l i t y } \\{ y \\in G \\ , : d ( x , y ) = 1 , d ( y , x _ 0 ) = d ( x , x _ 0 ) + 1 \\} = b , \\\\ \\operatorname { c a r d i n a l i t y } \\{ y \\in G \\ , : d ( x , y ) = 1 , d ( y , x _ 0 ) = d ( x , x _ 0 ) - 1 \\} = 1 \\ , . \\end{aligned} \\end{align*}"}
+{"id": "2549.png", "formula": "\\begin{align*} \\bigcap _ s \\dot \\AA ^ { \\prime \\ , ( s ) } ( M ) = \\bigcap _ m \\dot \\AA ^ { \\prime \\ , m } ( M ) \\ ; . \\end{align*}"}
+{"id": "364.png", "formula": "\\begin{align*} x - a = x + y - z , 2 x + b - a = x + y , y = x + b - a , \\end{align*}"}
+{"id": "4757.png", "formula": "\\begin{align*} \\varphi _ 1 ( \\varepsilon , b , n , \\varphi ) : = \\min \\{ \\varphi ( \\varepsilon , b + 2 n + 3 n ^ 2 ) / b , \\lambda _ 0 / 2 \\} . \\end{align*}"}
+{"id": "143.png", "formula": "\\begin{align*} \\int _ { \\Omega \\times \\Omega ^ c } J \\bigg ( \\frac { x - y } { t } \\bigg ) \\ d x d y = \\sum _ { i = 1 } ^ { 1 2 } \\int _ { U _ i \\times V _ i } J \\bigg ( \\frac { x - y } { t } \\bigg ) \\ d x d y + \\sum _ { i \\neq j } \\int _ { U _ i \\times V _ j } J \\bigg ( \\frac { x - y } { t } \\bigg ) \\ d x d y . \\end{align*}"}
+{"id": "1242.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ^ { 2 d + 1 } ; q ^ 2 ) _ k } { ( q ; q ) _ k } q ^ k \\equiv ( - 1 ) ^ { \\frac { n - 1 } { 2 } + d } q ^ { \\frac { n ^ 2 - ( 2 d + 1 ) ^ 2 } { 4 } } \\pmod { \\Phi _ { n } ( q ) } , \\end{align*}"}
+{"id": "5199.png", "formula": "\\begin{align*} & A = \\frac { 1 } { \\alpha ( \\alpha + \\beta - 1 ) } \\sum _ { i } q ^ { \\alpha + \\beta - 1 } _ { i } + \\frac { 1 } { \\alpha ( 1 - \\beta ) } \\sum _ { i } p ^ { \\alpha } _ { i } q ^ { \\beta - 1 } _ { i } \\\\ & B = \\frac { 1 } { ( 1 - \\beta ) ( \\alpha + \\beta - 1 ) } \\sum _ { i } p ^ { \\alpha + \\beta - 1 } _ { i } \\end{align*}"}
+{"id": "1206.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd t } \\bigg | _ { t = 0 } \\left ( \\frac { 1 } { p } [ u + t \\varphi ] _ { r , p } ^ { p } \\right ) = \\big \\langle ( - \\Delta _ p ) ^ r u , \\varphi \\big \\rangle , \\ \\ \\forall \\varphi \\in W _ { 0 } ^ { r , p } ( \\Omega ) . \\end{align*}"}
+{"id": "3268.png", "formula": "\\begin{align*} \\frac { ( q ^ d ; q ^ d ) _ { n - 1 } ^ d } { ( 1 - q ) ^ { d n - d } } \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ^ { d + 1 } ; q ^ d ) _ k ^ { d - 2 } ( q , q ^ { 1 - d } ; q ^ d ) _ k q ^ { d k } } { ( q ^ d ; q ^ d ) _ k ^ d } \\equiv 0 \\pmod { [ n ] ^ 2 } . \\end{align*}"}
+{"id": "7035.png", "formula": "\\begin{align*} X _ t = x - \\int _ 0 ^ t b ( X _ r ) d r + \\sqrt { 2 } W _ t , 0 \\leq t \\leq T \\end{align*}"}
+{"id": "5455.png", "formula": "\\begin{align*} | U _ t | = | U _ { t ^ - } | . \\end{align*}"}
+{"id": "945.png", "formula": "\\begin{align*} P _ x ( \\tau _ V = 0 ) = 1 \\mbox { q . e . } x \\in V ^ c . \\end{align*}"}
+{"id": "3474.png", "formula": "\\begin{align*} g _ { \\varphi ^ * } ( x ) = f ( x ) - \\pi ( f ) , \\end{align*}"}
+{"id": "4526.png", "formula": "\\begin{align*} n _ N ^ { ( K ) } ( t , [ s ] _ { K } ) = \\int _ { s _ { K } \\leq s _ { K + 1 } \\leq . . . \\leq s _ N } n _ N ( t , s _ 1 , . . . , s _ { N } ) \\ , d [ s ] _ { K + 1 , N } . \\end{align*}"}
+{"id": "2112.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 U ( s , x ) ^ 2 \\mathrm { d } x - \\frac { 1 } { d } \\sum _ { i = 1 } ^ d U ( \\frac { \\lfloor s T \\rfloor } { T } , \\frac { i } { d } ) ^ 2 & = o ( 1 ) \\\\ \\frac { 1 } { d } \\sum _ { i = 1 } ^ d \\big ( U ^ { d , T } ( \\frac { \\lfloor s T \\rfloor } { T } , \\frac { i } { d } ) ^ 2 - U ( \\frac { \\lfloor s T \\rfloor } { T } , \\frac { i } { d } ) ^ 2 \\big ) & = o ( 1 ) . \\end{align*}"}
+{"id": "414.png", "formula": "\\begin{align*} V _ { i i d } ( \\zeta ) = - \\log ( 1 - \\frac { v ( \\eta , \\zeta ) ^ 2 } { \\eta } ) , \\end{align*}"}
+{"id": "2840.png", "formula": "\\begin{align*} L _ { 4 } = - \\frac { 1 } { 2 } q \\ddot { q } - \\frac { 1 } { 2 } q ^ { 2 } . \\end{align*}"}
+{"id": "6275.png", "formula": "\\begin{align*} T = \\tilde { O } \\left ( \\frac { B ^ { \\frac { ( 1 + \\kappa ) } { \\kappa } } } { R _ 0 ^ { \\frac { ( r - 1 ) ( 1 + \\kappa ) } { \\kappa } } } \\cdot \\left [ \\frac { \\mu _ r R _ 0 } { 2 \\varepsilon } \\right ] ^ { \\left ( \\frac { ( r - 1 ) ( 1 + \\kappa ) } { r \\kappa } \\right ) } \\right ) , B = \\frac { 2 ^ { ( 1 + r ) } \\left ( 2 M _ 2 + \\sqrt { d } a _ { 2 , q ^ * } \\mathcal { D } + 2 \\sigma _ { q , \\kappa } \\right ) } { \\mu _ r } \\end{align*}"}
+{"id": "2800.png", "formula": "\\begin{align*} \\phi ^ { ( 1 ) } : = p _ { 2 } : \\approx 0 . \\end{align*}"}
+{"id": "6236.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ( f , \\phi ) ( \\phi _ 0 ) & = - ( p + q ) + ( 2 p + q ) ( \\phi + \\phi _ 0 ) - p ( \\phi ^ 2 + \\phi \\phi _ 0 + \\phi _ 0 ^ 2 ) . \\end{aligned} \\end{align*}"}
+{"id": "572.png", "formula": "\\begin{align*} \\nabla \\cdot ( T \\nabla \\zeta ) & = g \\rho \\zeta + \\rho \\Phi _ t + m \\zeta _ { t t } + m g & \\Gamma _ c , \\end{align*}"}
+{"id": "1235.png", "formula": "\\begin{align*} \\Phi ( T ) = \\lim _ { n \\to \\infty } \\sum _ { i , j = 1 } ^ n t _ { i j } \\Phi ( E _ { i j } ) = \\lim _ { n \\to \\infty } \\sum _ { i , j = 1 } ^ n t _ { i j } \\Psi ( E _ { i j } ) = \\Psi ( T ) . \\end{align*}"}
+{"id": "5210.png", "formula": "\\begin{align*} T = \\frac { 1 } { ( \\beta - 1 ) ( \\alpha + \\beta - 1 ) } \\end{align*}"}
+{"id": "7724.png", "formula": "\\begin{align*} \\mathcal L _ k ^ { p , \\lambda } \\big ( \\Omega \\big ) : = \\Bigg \\{ f \\in L ^ p \\big ( \\Omega \\big ) : \\sup _ { z \\in \\Omega , r > 0 } r ^ { - \\lambda } \\inf _ { P \\in \\mathcal P _ k } \\int _ { Q _ r ( z ) \\cap \\Omega } \\abs { f - P } ^ p \\dd z < + \\infty \\Bigg \\} \\end{align*}"}
+{"id": "165.png", "formula": "\\begin{align*} u ^ { F , G } _ { X , Y } ( a ( b _ { i } \\boxtimes c _ { j } ) ) & = \\sum _ { l } c _ { j } \\boxtimes b _ { l } \\langle b _ { l } \\ | \\ a b _ { i } \\rangle \\\\ & = c _ { j } \\boxtimes a b _ { i } = c _ { j } a \\boxtimes b _ { i } \\\\ & = a c _ { j } \\boxtimes b _ { i } = a ( c _ { j } \\boxtimes b _ { i } ) \\\\ & = a u ^ { F , G } _ { X , Y } ( b _ { i } \\boxtimes c _ { j } ) . \\end{align*}"}
+{"id": "2814.png", "formula": "\\begin{align*} \\omega = d q ^ { i } \\wedge d p _ { i } = d \\Theta ^ { 1 } \\wedge d \\Theta _ { 1 } + d Q ^ { 1 } \\wedge d P _ { 1 } , \\end{align*}"}
+{"id": "6980.png", "formula": "\\begin{align*} 0 = z _ j \\overline { \\eta } + \\overline { z _ j } \\eta \\mathrm { , } \\ ; \\ ; 1 \\leq j \\leq n , \\end{align*}"}
+{"id": "7859.png", "formula": "\\begin{align*} f ( \\mathbf { x } ) = \\mathbf { x } \\mathbf { A } \\mathbf { x } ^ { T } , \\end{align*}"}
+{"id": "1329.png", "formula": "\\begin{align*} \\overline { \\nabla } _ a b = \\bigl ( \\overline { \\nabla } _ { \\dot { x } } ( \\delta x ) , \\widetilde { \\nabla } _ { \\dot { x } } \\overline { \\eta } + [ \\overline { v } , \\overline { \\eta } ] - \\widetilde { R } ( \\dot { x } , \\delta x ) \\bigr ) . \\end{align*}"}
+{"id": "5038.png", "formula": "\\begin{align*} \\mathbf { r } _ { S _ k ^ c } ^ { S _ k } : = \\mathbf { r } _ { S _ k ^ c } ^ { S _ { k - 1 } } - r _ { i _ k } ^ { S _ { k - 1 } } \\mathbf { A } _ { S _ k ^ c i _ k } ^ { S _ { k } } , \\end{align*}"}
+{"id": "2709.png", "formula": "\\begin{align*} \\dot { q } ^ { i } = \\frac { \\partial H } { \\partial p _ { i } } , \\dot { p } ^ { i } = - \\frac { \\partial H } { \\partial q _ { i } } . \\end{align*}"}
+{"id": "7907.png", "formula": "\\begin{align*} \\delta : = ( - 1 ) ^ { n ( k + 1 ) + 1 } \\ast d \\ast . \\end{align*}"}
+{"id": "1467.png", "formula": "\\begin{align*} y \\in X , y \\vert _ S = p y \\vert _ { F _ n \\Delta ^ 2 \\setminus S \\Delta } = q = x \\vert _ { F _ n \\Delta ^ 2 \\setminus S \\Delta } . \\end{align*}"}
+{"id": "4222.png", "formula": "\\begin{align*} \\theta _ 1 ( v , \\tau + 1 ) = e ^ { \\frac { \\pi \\sqrt { - 1 } } { 4 } } \\theta _ 1 ( v , \\tau ) , ~ ~ \\theta _ 1 ( v , - \\frac { 1 } { \\tau } ) = \\left ( \\frac { \\tau } { \\sqrt { - 1 } } \\right ) ^ { \\frac { 1 } { 2 } } e ^ { \\pi \\sqrt { - 1 } \\tau v ^ 2 } \\theta _ 2 ( \\tau v , \\tau ) ; \\end{align*}"}
+{"id": "3475.png", "formula": "\\begin{align*} f : \\R ^ { p _ 0 } \\rightarrow \\R ^ { p _ { L + 1 } } \\ , , f ( x ) = W _ L \\circ \\sigma _ { v _ { L } } \\circ W _ { L - 1 } \\circ \\sigma _ { v _ { L - 1 } } \\circ \\dots \\circ W _ 1 \\circ \\sigma _ { v _ 1 } \\circ W _ 0 \\circ x , \\end{align*}"}
+{"id": "2923.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { P } \\big [ \\mathcal { B } _ { \\ell } ^ { ( 2 ) } \\big ] & \\leqslant \\mathbb { P } \\big [ \\mathcal { B } _ { \\ell } ^ { ( 2 ) } \\cap \\mathcal { X } _ { \\ell } \\big ] + \\mathbb { P } \\big [ \\ , \\overline { \\mathcal { X } _ { \\ell } } \\ , \\big ] . \\end{aligned} \\end{align*}"}
+{"id": "1433.png", "formula": "\\begin{align*} & D _ k = D _ k \\cap X _ { - 1 } = \\coprod _ { \\lambda _ 0 < \\lambda _ 1 < \\cdots < \\lambda _ k } D _ { \\lambda _ 0 } \\cap D _ { \\lambda _ 1 } \\cap \\cdots \\cap D _ { \\lambda _ k } \\\\ & X _ l = D _ { - 1 } \\cap X _ l = \\coprod _ { i _ 0 < i _ 1 < \\cdots < i _ l } X _ { i _ 0 } \\cap X _ { i _ 1 } \\cap \\cdots \\cap X _ { i _ l } \\end{align*}"}
+{"id": "8652.png", "formula": "\\begin{align*} \\langle x , y _ 0 ^ * \\rangle - F ( x ) & = \\langle x , y _ 0 ^ * - x ^ * \\rangle + F ^ * ( x ^ * ) , \\\\ & = F ^ * ( y _ 0 ^ * ) - \\left ( F ^ * ( y _ 0 ^ * ) - F ^ * ( x ^ * ) - \\langle x , y ^ * _ 0 - x ^ * \\rangle \\right ) , \\\\ & \\leq F ^ * ( y _ 0 ^ * ) - ( F ^ * ( y _ 0 ^ * ) - F ^ * ( x ^ * ) - \\sup _ { z \\in \\partial F ^ * ( x ^ * ) \\cap K } \\langle z , y ^ * _ 0 - x ^ * \\rangle . \\end{align*}"}
+{"id": "6558.png", "formula": "\\begin{align*} \\lim \\limits _ { x \\to \\infty } \\frac { ( \\ell ^ { \\leftarrow } _ { \\beta } ( x ) ) ^ { \\alpha } } { h ( \\ell ^ { \\leftarrow } _ { \\beta } ( x ) ) } \\P \\big ( \\eta _ K ( \\varepsilon _ 1 ) > x \\big ) = \\gamma _ 2 \\quad \\lim \\limits _ { x \\to \\infty } \\frac { ( \\ell ^ { \\leftarrow } _ { \\beta } ( x ) ) ^ { \\alpha } } { h ( \\ell ^ { \\leftarrow } _ { \\beta } ( x ) ) } \\P \\big ( \\eta _ K ( \\varepsilon _ 1 ) < - x \\big ) = \\gamma _ 1 , \\end{align*}"}
+{"id": "6966.png", "formula": "\\begin{align*} \\eta ^ 2 ( \\mathbf { z } ) = - \\langle \\mathbf { z } , \\mathbf { \\overline { z } } \\rangle = z _ { n + 1 } ^ 2 - z _ 1 ^ 2 - \\cdots - z _ n ^ 2 . \\end{align*}"}
+{"id": "6572.png", "formula": "\\begin{align*} \\phi _ { \\Phi _ k , k - 1 } ( x , y ) = R _ { k , k - 1 } ( 1 ) y + O \\left ( y \\exp \\left ( - \\sqrt { r _ k \\log y \\log \\log y } \\right ) \\right ) , \\end{align*}"}
+{"id": "4232.png", "formula": "\\begin{align*} E _ 6 ( \\tau ) = 1 - 5 0 4 q - 1 6 6 3 2 q ^ 2 - 1 2 2 9 7 6 q ^ 3 + \\cdots . \\end{align*}"}
+{"id": "3861.png", "formula": "\\begin{align*} \\hat { M } ^ 1 ( t ) = M ^ 1 ( T ) + \\int _ 0 ^ t \\hat \\eta _ { ( 1 ) } ^ 1 ( s ) d s - \\int _ 0 ^ t \\hat M ^ { 1 } ( s ) d s , \\ ; t \\in [ 0 , T ] , \\end{align*}"}
+{"id": "2985.png", "formula": "\\begin{align*} \\begin{array} { l l l } g ( x , \\xi ) = \\eta ( x ) , & & g ( x , \\phi y ) = g ( \\phi x , y ) , \\\\ g ( \\xi , \\xi ) = 1 , & & \\eta ( \\nabla _ x \\xi ) = 0 , \\end{array} \\end{align*}"}
+{"id": "2551.png", "formula": "\\begin{align*} \\tilde \\pi _ * \\tilde \\pi ^ * = 2 \\ ; , \\quad \\tilde \\pi ^ * \\tilde \\pi _ * = A _ \\sigma \\ ; , \\end{align*}"}
+{"id": "9144.png", "formula": "\\begin{align*} \\underbrace { x _ { i , 1 } ^ { r } \\star \\cdots \\star x _ { i , 1 } ^ { r } } _ { \\ell \\rm \\ t i m e s } = v _ { i } ^ { - \\frac { \\ell ( \\ell - 1 ) } { 2 } } [ \\ell ] _ { v _ { i } } ! \\cdot ( x _ { i , 1 } \\cdots x _ { i , \\ell } ) ^ { r } . \\end{align*}"}
+{"id": "5684.png", "formula": "\\begin{align*} f = f ( 0 ) + \\sum _ { j \\geq p } f _ j , \\end{align*}"}
+{"id": "6252.png", "formula": "\\begin{align*} \\Phi g ( x ) : = \\int _ { \\Gamma } g ( y ) E _ { P } ( x - y ) d \\sigma ( y ) , x \\in \\R ^ { n } , \\end{align*}"}
+{"id": "5203.png", "formula": "\\begin{align*} & \\frac { \\partial A } { \\partial q _ { j } } = - \\frac { 1 } { \\alpha } p ^ { \\alpha } _ { j } q ^ { \\beta - 2 } _ { j } \\\\ & \\frac { \\partial B } { \\partial q _ { j } } = - \\frac { 1 } { \\alpha } q ^ { \\alpha + \\beta - 2 } _ { j } \\end{align*}"}
+{"id": "1789.png", "formula": "\\begin{align*} \\Re ( Y ) ( P ( z , \\zeta ) - u ) = c x _ 1 ^ { \\mu + 2 } \\quad \\quad \\forall \\ , ( w , z , \\zeta ) \\in M _ 0 . \\end{align*}"}
+{"id": "6344.png", "formula": "\\begin{align*} \\omega _ g ( r , \\varphi ) = \\omega _ g ( r ) = \\frac { \\sinh ^ { n - 1 } ( r ) \\cosh ^ { d - 1 } ( r ) } { r ^ { n - 1 } } . \\end{align*}"}
+{"id": "4866.png", "formula": "\\begin{align*} 2 H ^ \\alpha \\langle A , \\nabla _ \\Sigma A \\rangle = \\nabla _ \\Sigma H \\left ( 2 H ^ { \\alpha + 1 } + \\alpha H ^ { \\alpha - 1 } ( H ^ 2 - | A | ^ 2 ) \\right ) = \\nabla _ \\Sigma H \\left ( 2 H ^ { \\alpha + 1 } + \\alpha \\frac v { H } \\right ) \\end{align*}"}
+{"id": "2851.png", "formula": "\\begin{align*} \\nu _ \\gamma ( E ) : = \\iint _ { \\{ ( x , y ) \\in E : \\ x \\neq y \\} } \\left | x - y \\right | ^ { \\gamma - n } \\ , d x \\ , d y . \\end{align*}"}
+{"id": "862.png", "formula": "\\begin{align*} b _ 1 = - \\frac { w _ 1 } { \\rho } = - \\frac { w _ 1 } { 1 - ( w ^ 1 ) ^ 2 - ( w ^ 2 ) ^ 2 } , b _ 2 = - \\frac { w _ 2 } { \\rho } = - \\frac { w _ 2 } { 1 - ( w ^ 1 ) ^ 2 - ( w ^ 2 ) ^ 2 } \\end{align*}"}
+{"id": "4906.png", "formula": "\\begin{align*} \\dim ( R _ { C , L } ) & = h ^ 0 ( K _ C ) h ^ 0 ( L ) - h ^ 0 ( K _ C + L ) . \\end{align*}"}
+{"id": "7254.png", "formula": "\\begin{align*} \\prod \\limits _ { n = 1 } ^ N ( 1 + \\eta \\gamma _ t [ n ] _ q + \\eta \\beta _ t [ n ] _ q ^ 2 + x \\eta q ^ n ) \\geq 0 N \\geq 1 ; \\end{align*}"}
+{"id": "443.png", "formula": "\\begin{align*} \\tilde { z } _ { i , j } ^ { q } ( t ) = \\left \\{ \\begin{aligned} & z _ { i , j } ^ { q } ( t ) , \\ , \\mbox { i f } ( i , j ) \\in { \\cal A } ^ q _ = , \\\\ [ 2 p t ] & z _ { i , j } ^ * ( t _ q ) , \\ , \\mbox { o t h e r w i s e } , \\end{aligned} \\right . \\forall \\ , ( i , j ) \\in E _ m . \\end{align*}"}
+{"id": "7851.png", "formula": "\\begin{align*} \\mu ( \\mathbf { S } ) = \\max \\limits _ { 1 \\leq i \\neq j \\leq N } \\frac { \\mid \\langle \\mathbf { s } _ { i } , \\mathbf { s } _ { j } \\rangle \\mid } { \\| \\mathbf { s } _ { i } \\| _ { 2 } \\| \\mathbf { s } _ { j } \\| _ { 2 } } , \\end{align*}"}
+{"id": "3646.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k ( n _ i - r _ i + 1 ) = n + k - \\sum _ { i = 1 } ^ k r _ i = n - m + 1 \\end{align*}"}
+{"id": "4874.png", "formula": "\\begin{align*} m = C \\cdot \\theta . \\end{align*}"}
+{"id": "3340.png", "formula": "\\begin{align*} Q _ r ( z _ 0 ) : = \\big \\{ z : ~ | v - v _ 0 | < r , ~ | x - x _ 0 - ( t - t _ 0 ) v _ 0 | < r ^ { 1 + s p } , ~ t _ 0 - r ^ { s p } < t < t _ 0 \\big \\} , \\end{align*}"}
+{"id": "3742.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial a } \\mathrm { B } _ { z } \\left ( a , b \\right ) = z ^ { a } \\left \\{ \\frac { \\ln z } { a } \\ , _ { 2 } F _ { 1 } \\left ( \\left . \\begin{array} { c } 1 - b , a \\\\ a + 1 \\end{array} \\right \\vert z \\right ) - \\frac { 1 } { a ^ { 2 } } \\ , _ { 3 } F _ { 2 } \\left ( \\left . \\begin{array} { c } 1 - b , a , a \\\\ a + 1 , a + 1 \\end{array} \\right \\vert z \\right ) \\right \\} . \\end{align*}"}
+{"id": "8350.png", "formula": "\\begin{align*} R _ { p } K \\subseteq R _ { q } K \\subseteq R _ { \\infty } K = D K . \\end{align*}"}
+{"id": "8231.png", "formula": "\\begin{align*} C ( L = 2 ) & = \\frac { 1 } { 2 } \\max _ { P _ { { \\bf X } _ 1 } , P _ { { \\bf X } _ 2 | Y _ 1 } \\in \\mathcal X ( B _ ) } ( H ( Y _ 1 | { \\bf S } ) + H ( Y _ 2 | Y _ 1 , { \\bf S } ) ) . \\end{align*}"}
+{"id": "3887.png", "formula": "\\begin{align*} X ^ s ( e _ 1 \\wedge \\cdots \\wedge e _ n ) = s ! \\sum _ { \\substack { I \\subseteq \\{ 1 , \\ldots , n \\} \\\\ | I | = 2 s } } \\pm P _ I \\Big ( \\bigwedge _ { \\substack { 1 \\leq r \\leq k \\\\ r \\notin I } } e _ r \\Big ) \\in \\bigwedge ^ { n - 2 s } F _ 1 \\end{align*}"}
+{"id": "1642.png", "formula": "\\begin{align*} y _ { 1 } z _ { 1 } - y _ { 2 } z _ { 2 } = \\widetilde { y } z _ { 1 } + \\widetilde { z } y _ { 2 } . \\end{align*}"}
+{"id": "6487.png", "formula": "\\begin{align*} e _ { u , v } = \\begin{cases} \\delta _ { u , v } & , \\\\ \\delta _ { u , v ^ + } - \\delta _ { u , 1 } & , \\end{cases} \\end{align*}"}
+{"id": "6794.png", "formula": "\\begin{align*} c _ 1 > 0 , \\ c _ 2 = - 1 , \\ c _ 3 > 0 , \\ \\ d _ 1 > 0 , \\ d _ 2 = - 1 , \\ d _ 3 \\geq 0 , \\ \\ \\frac { 1 } { 2 } \\left ( \\frac { d _ 3 } { c _ 3 } + \\frac { d _ 1 } { c _ 1 } \\right ) < \\frac { d _ 2 } { c _ 2 } < \\frac { d _ 1 } { c _ 1 } . \\end{align*}"}
+{"id": "8198.png", "formula": "\\begin{align*} & P _ { Y _ 3 | Y ^ 2 , { \\bf S } } ( 1 | 0 0 , { \\bf s } ) = \\frac { B _ 3 ^ { 0 0 } } { M - B _ 1 - B _ 2 ^ 0 } , \\ P _ { Y _ 3 | Y ^ 2 , { \\bf S } } ( 1 | 0 1 , { \\bf s } ) = \\frac { B _ 3 ^ { 0 1 } } { B _ 2 ^ 0 } , \\\\ & P _ { Y _ 3 | Y ^ 2 , { \\bf S } } ( 1 | 1 0 , { \\bf s } ) = \\frac { B _ 3 ^ { 1 0 } } { B _ 1 - B _ 2 ^ 1 } , \\ P _ { Y _ 3 | Y ^ 2 , { \\bf S } } ( 1 | 1 1 , { \\bf s } ) = \\frac { B _ 3 ^ { 1 1 } } { B _ 2 ^ 1 } . \\end{align*}"}
+{"id": "551.png", "formula": "\\begin{align*} \\begin{aligned} & j _ { \\nu , k } \\geq \\left ( k - \\frac { 1 } { 4 } \\right ) \\pi \\nu \\in \\left [ 0 , 1 / 2 \\right ] , \\\\ & j _ { \\nu , k } \\geq \\left ( k - \\frac { 1 } { 8 } \\right ) \\pi \\nu \\in \\left [ 1 / 2 , \\infty \\right ] . \\end{aligned} \\end{align*}"}
+{"id": "5518.png", "formula": "\\begin{align*} k _ { \\Omega } & = \\deg ( D ) + g - 1 - \\deg ( G ) \\\\ & = q ^ { 2 n + 1 } - q ^ { n + 2 } + \\frac { 5 q ^ { n + 2 } + q ^ n - q ^ 3 + q ^ 2 - 2 q + 2 } { 2 ( q + 1 ) } - a ( 2 q ^ 2 - 2 q - 1 ) . \\end{align*}"}
+{"id": "825.png", "formula": "\\begin{align*} { D _ 0 } { D _ m } \\Psi - { D _ m } { D _ 0 } \\Psi = { k } ( { \\Psi _ m } { F ^ 2 } - \\Psi { y _ m } ) . \\end{align*}"}
+{"id": "6415.png", "formula": "\\begin{align*} \\sup _ { \\theta \\in A \\times W _ n ^ { ( \\eta ) } } \\ln ( n ) ^ q \\left | \\frac { 1 } { n } \\sum _ { i = 1 } ^ n f ( X _ { \\frac { i - 1 } { n } } , \\delta _ 0 , \\alpha _ 0 ) \\left ( h ( n ^ { 1 / \\alpha _ 0 } \\Delta _ i ^ n L , \\alpha ) - h ( n ^ { 1 / \\alpha _ 0 } \\Delta _ i ^ n L , \\alpha _ 0 ) \\right ) \\right | \\to 0 , \\end{align*}"}
+{"id": "6767.png", "formula": "\\begin{align*} \\begin{aligned} f _ i ( x _ i ) & - f _ i ( \\breve { x } _ i ^ k ) + ( x _ i - \\breve { x } _ i ^ k ) ^ T [ - A _ i ^ T \\widetilde { \\lambda } ^ k \\\\ & + \\beta A _ i ^ T \\sum _ { j = 1 } ^ { i } A _ j ( \\widetilde { x } _ j ^ { k } - x _ j ^ k ) + A _ i ^ T ( \\widetilde { \\lambda } ^ k - \\lambda ^ k ) ] \\geq 0 , ~ \\forall x _ i \\in \\mathcal { X } _ i . \\end{aligned} \\end{align*}"}
+{"id": "4971.png", "formula": "\\begin{align*} A ( \\mu ) B ( \\lambda ) = f ( \\lambda , \\mu ) B ( \\lambda ) A ( \\mu ) + g ( \\mu , \\lambda ) B ( \\mu ) A ( \\lambda ) . \\end{align*}"}
+{"id": "7483.png", "formula": "\\begin{align*} \\psi _ { \\tilde { x } , R } ( \\Tilde { y } , t ) : = \\xi \\left ( \\frac { d _ { g _ N ( t ) } ( x , y ) } { R } \\right ) \\xi \\left ( \\tfrac { l } { R } \\right ) , \\end{align*}"}
+{"id": "5411.png", "formula": "\\begin{align*} \\chi _ { ( 0 , 1 / 2 ) } - \\chi _ { ( 1 / 2 , 1 ) } = \\chi _ { ( 0 , 1 / 2 ) } - \\chi _ { ( 0 , 1 / 2 ) } \\circ T , \\end{align*}"}
+{"id": "2828.png", "formula": "\\begin{align*} H _ { T } = \\frac { 1 } { 2 } ( P _ { 1 } ) ^ { 2 } + \\frac { 1 } { 2 } ( Q ^ { 1 } ) ^ { 2 } + \\Psi _ { 1 } P _ { 1 } + \\zeta ^ { 1 } \\Psi _ { 1 } + f ( \\Xi ^ { 1 } , \\Psi _ { 2 } , \\Theta ^ { 1 } ) + g ( \\Psi _ { 1 } , \\Psi _ { 2 } , \\Theta ^ { 1 } , \\Theta _ { 1 } ) , \\end{align*}"}
+{"id": "4680.png", "formula": "\\begin{align*} \\mathbf { B } _ { - } ( D ) = \\left ( \\bigcap _ { E \\equiv D } \\mathrm { S u p p } ( P ( E ) ) \\right ) \\cup \\mathrm { S u p p } ( N ( D ) ) . \\end{align*}"}
+{"id": "2085.png", "formula": "\\begin{align*} N _ { i , 1 } ^ s & = - ( \\eta d T ) \\int _ 0 ^ { \\frac { \\lfloor s T \\rfloor } { T } } \\int _ 0 ^ 1 f ( u ) A ( \\frac { i } { d } , y ) \\Theta ( u , y ) d y d u + o ( 1 ) \\to 0 \\ d , T \\to \\infty . \\end{align*}"}
+{"id": "4560.png", "formula": "\\begin{align*} \\nabla _ { h } \\lambda ^ { - 1 / 3 } = J K _ \\infty + O ( \\rho ^ { - 3 } ) = O ( \\rho ^ { - 3 } ) . \\end{align*}"}
+{"id": "1759.png", "formula": "\\begin{align*} f _ 1 ( 0 ) = 1 \\quad \\mbox { a n d } f _ 2 ( 0 ) = . . . = f _ { n - 1 } ( 0 ) = g ( 0 ) = h ( 0 ) = 0 . \\end{align*}"}
+{"id": "6312.png", "formula": "\\begin{align*} - \\Delta \\omega _ { \\ell , \\lambda } = \\frac { 1 } { 2 } V _ \\ell ( 1 - \\omega _ { \\ell } ) - \\frac { 1 } { 2 } \\epsilon _ { \\ell , \\lambda } , \\end{align*}"}
+{"id": "7084.png", "formula": "\\begin{align*} b _ n \\in \\mathbf { F } ^ { \\scriptscriptstyle 1 / 2 } _ { \\delta } \\end{align*}"}
+{"id": "8078.png", "formula": "\\begin{align*} \\norm { u } ^ 2 : = \\sum _ i | u _ i | ^ 2 , \\end{align*}"}
+{"id": "4346.png", "formula": "\\begin{align*} F _ { \\overline { \\mathcal { Y } } , a } ( x ) & : = \\sum _ { ( i , j ) \\in [ m ] \\times [ n ] } \\overline { y } _ { ( i , j ) } g _ { ( i , j ) } ( x ) + \\sum _ { ( q , p ) \\in \\overline { \\mathcal { Y } } } \\Delta y _ { ( q , p ( q ) ) } g _ { ( q , p ( q ) ) } ( x ) - \\Delta y _ { ( a , b ( a ) ) } g _ { ( a , b ( a ) ) } ( x ) . \\end{align*}"}
+{"id": "1498.png", "formula": "\\begin{align*} m f _ 2 ( m ) g _ 1 ( 0 ) = & - m g _ 2 ( m ) f _ 1 ( 0 ) + m f _ 2 ( m ) g _ 1 ( m ) + m g _ 2 ^ { 2 } ( m ) . \\end{align*}"}
+{"id": "2749.png", "formula": "\\begin{align*} \\sigma _ { 1 } ^ { * } \\omega = \\omega _ { Q , P } , \\end{align*}"}
+{"id": "5559.png", "formula": "\\begin{align*} S & = T ^ * J , & T & = S ^ * J , \\\\ S _ \\Delta T & = S T _ \\Delta = A A ^ * , & B & = T S _ \\Delta - J _ \\Delta . \\end{align*}"}
+{"id": "3446.png", "formula": "\\begin{align*} & [ S W F _ { \\Z _ 4 } ( \\Sigma ( 2 , 3 , 1 2 k - 1 ) , \\iota ) ] _ { \\mathrm { l o c } } = [ ( \\tilde { G } , 0 , 0 ) ] _ { \\mathrm { l o c } } , \\\\ & [ S W F _ { \\Z _ 4 } ( \\Sigma ( 2 , 3 , 1 2 k - 5 ) , \\iota ) ] _ { \\mathrm { l o c } } = [ ( \\tilde { G } , 0 , 1 / 4 ) ] _ { \\mathrm { l o c } } , \\\\ & [ S W F _ { \\Z _ 4 } ( \\Sigma ( 2 , 3 , 1 2 k + 1 ) , \\iota ) ] _ { \\mathrm { l o c } } = [ ( S ^ 0 , 0 , 0 ) ] _ { \\mathrm { l o c } } , \\\\ & [ S W F _ { \\Z _ 4 } ( \\Sigma ( 2 , 3 , 1 2 k + 5 ) , \\iota ) ] _ { \\mathrm { l o c } } = [ ( S ^ 0 , 0 , - 1 / 4 ) ] _ { \\mathrm { l o c } } \\end{align*}"}
+{"id": "7734.png", "formula": "\\begin{align*} \\chi _ { \\Sigma ( \\lambda ) } ( X ) = X ^ { m } + a _ { 1 } ( \\lambda ) X ^ { m - 1 } + \\cdots + a _ { m } ( \\lambda ) , \\\\ \\chi _ { \\Sigma ' ( \\lambda ) } ( X ) = X ^ { m } + b _ { 1 } ( \\lambda ) X ^ { m - 1 } + \\cdots + b _ { m } ( \\lambda ) . \\end{align*}"}
+{"id": "1195.png", "formula": "\\begin{align*} \\begin{cases} E \\geq N , \\\\ E > \\lfloor N \\rfloor _ + , \\end{cases} \\begin{cases} F \\geq ( K \\vee M ) - n , \\\\ F > \\lfloor L \\rfloor , \\end{cases} G \\geq \\lfloor N \\rfloor _ + , H \\geq \\lfloor L \\rfloor . \\end{align*}"}
+{"id": "2475.png", "formula": "\\begin{align*} \\begin{cases} \\dot { \\phi } = P ( \\phi , \\psi ) , \\\\ \\dot { \\psi } = Q ( \\phi , \\psi ) , \\end{cases} \\end{align*}"}
+{"id": "4353.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathcal { X } \\subseteq \\{ 0 , 1 \\} ^ n } & \\sum _ { i \\in [ n ] } \\sum _ { j \\in [ i ] } p _ { i , j } x _ i x _ j \\end{align*}"}
+{"id": "8563.png", "formula": "\\begin{align*} F ( n , k ) = \\frac { ( a ) _ k ( b ) _ k } { ( 2 n ) _ k ( 3 n ) _ k } \\end{align*}"}
+{"id": "1549.png", "formula": "\\begin{align*} u \\in W ^ { 1 , A } ( { \\mathcal O } ) : = \\left \\{ v \\in W ^ { 1 , 1 } ( { \\mathcal O } ) : A ( | v | ) , A ( | D v | ) \\in L ^ 1 ( { \\mathcal O } ) \\right \\} , \\end{align*}"}
+{"id": "5840.png", "formula": "\\begin{align*} \\begin{array} { r @ { \\ } l } \\left ( 1 - \\alpha ^ { 3 n - 3 j } \\right ) \\kappa ^ n _ { 3 n - 2 j , j , j } & = \\displaystyle \\alpha ^ { 3 n - 5 j - 3 } \\left ( \\sum _ { \\nu = 0 } ^ { j } \\alpha ^ { 3 \\nu } \\right ) \\kappa ^ n _ { 3 n - 2 j - 2 , j + 1 , j + 1 } \\\\ & = \\displaystyle \\alpha ^ { 3 n - 5 j - 3 } ( 1 - \\alpha ^ 3 ) ^ { - 1 } ( 1 - \\alpha ^ { 3 j + 3 } ) \\kappa ^ n _ { 3 n - 2 j - 2 , j + 1 , j + 1 } , \\end{array} \\end{align*}"}
+{"id": "9074.png", "formula": "\\begin{align*} \\begin{array} { l l } X ( t + 1 ) = J ^ { ( r ) } \\hat { C } C \\hat { C } ^ { - 1 } J ^ { ( r ) } X ( t ) + J ^ { ( r ) } \\Big [ ( \\hat { C } C \\hat { C } ^ { - 1 } - I ) \\underline { v } + \\hat { C } \\Big ( D p - U ( J ^ { ( r ) } X ( t ) ) - \\mathbf { 1 } \\frac { \\beta } { 2 } \\Big ) \\Big ] . \\end{array} \\end{align*}"}
+{"id": "3748.png", "formula": "\\begin{align*} 2 \\frac { \\partial } { \\partial \\lambda } \\int _ { 0 } ^ { z } \\frac { u ^ { 2 \\lambda - 1 } } { 1 - u ^ { 2 } } \\ , d u = 4 \\int _ { 0 } ^ { z } \\frac { u ^ { 2 \\lambda - 1 } } { 1 - u ^ { 2 } } \\ , \\ln u \\ , d u . \\end{align*}"}
+{"id": "5269.png", "formula": "\\begin{align*} \\left ( - \\frac { \\partial D } { \\partial x } \\right ) ^ { k } _ { j } = U ^ { k } _ { j } - V ^ { k } _ { j } \\ \\ ; \\ \\ \\ \\ U ^ { k } _ { j } > 0 \\ \\ ; \\ \\ \\ \\ V ^ { k } _ { j } > 0 \\end{align*}"}
+{"id": "916.png", "formula": "\\begin{align*} L = - ( - \\Delta ) ^ { \\alpha / 2 } _ { D } \\end{align*}"}
+{"id": "5507.png", "formula": "\\begin{align*} \\phi ' ( t _ 1 ) & { } = \\varphi ( x _ 1 ^ 1 y _ 1 ^ 0 z _ 1 ^ 1 ) = \\varphi ( x _ 1 ) \\varphi ( z _ 1 ) = \\xi \\varphi ( x _ 1 ) , \\\\ \\phi ' ( t _ 3 ) & { } = \\varphi ( x _ 1 ^ { - 1 } y _ 1 ^ 0 z _ 1 ^ 0 ) = \\varphi ( x _ 1 ) ^ { - 1 } = \\lambda ^ { - \\ell } \\varphi ( x _ 1 ^ { \\ell - 1 } ) , \\\\ \\phi ' ( t _ 4 ) & { } = \\varphi ( x _ 1 ^ 0 y _ 1 ^ { - 1 } z _ 1 ^ 0 ) = \\varphi ( y _ 1 ) ^ { - 1 } = \\varphi ( y _ 1 ^ { \\ell - 1 } ) , \\end{align*}"}
+{"id": "5379.png", "formula": "\\begin{align*} p = \\frac { t - \\bar { b } ^ { S _ { k + 1 } } } { \\bar { b } ^ { S _ k } - \\bar { b } ^ { S _ { k + 1 } } } , q = 1 - p . \\end{align*}"}
+{"id": "5772.png", "formula": "\\begin{align*} z ^ { ( k , \\ell ) } _ j = \\frac { d } { d t } z ^ { ( k - 1 , \\ell ) } _ j = \\mathcal { W } ^ { ( k - 1 , \\ell ) } _ j . \\end{align*}"}
+{"id": "2821.png", "formula": "\\begin{align*} L _ { T } = \\frac { 1 } { 2 } ( \\dot { Q } ^ { 1 } ) ^ { 2 } - \\frac { 1 } { 2 } ( Q ^ { 1 } ) ^ { 2 } + { c o n s t a n t } , \\end{align*}"}
+{"id": "7924.png", "formula": "\\begin{align*} \\mathfrak { B } ^ { \\ast k } : = \\delta H ^ { \\ast } \\Lambda ^ { k + 1 } ( \\Omega ) , \\quad \\mathring { \\mathfrak { B } } ^ { \\ast k } : = \\delta \\mathring { H } ^ { \\ast } \\Lambda ^ { k + 1 } ( \\Omega ) . \\end{align*}"}
+{"id": "6616.png", "formula": "\\begin{align*} \\Phi _ k ( n _ s ) & = \\left ( \\frac { 2 ^ { \\frac { k } { 2 } } } { 2 ^ { \\frac { k } { 2 } } - 1 } \\right ) n _ s ^ { k - 1 } \\phi ( n _ s ) \\prod _ { p \\leq n _ s } \\left ( 1 - \\frac { 1 } { p ^ { \\frac { k } { 2 } } } \\right ) \\\\ & = \\left ( \\frac { 2 ^ { \\frac { k } { 2 } } } { 2 ^ { \\frac { k } { 2 } } - 1 } \\right ) \\left ( \\zeta \\left ( \\frac { k } { 2 } \\right ) ^ { - 1 } + O \\left ( \\frac { 1 } { n _ s ^ { \\frac { k } { 2 } - 1 } } \\right ) \\right ) \\frac { e ^ { - \\gamma } n _ s ^ k } { \\log \\log n _ s } . \\end{align*}"}
+{"id": "1058.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta _ n ^ { s , t } & \\leq K _ 1 K _ 2 \\sum _ { k = 1 } ^ { \\infty } \\{ S _ { 1 , 2 k - 1 } ( n , n - s , t - 1 ) + S _ { 2 , 2 k - 1 } ( n , n - s , t - 1 ) \\} \\\\ & \\qquad + K _ 1 K _ 2 \\sum _ { k = 1 } ^ { \\infty } \\{ S _ { 1 , 2 k } ( n , s - 1 , t - 1 ) + S _ { 2 , 2 k } ( n , s - 1 , t - 1 ) \\} . \\end{aligned} \\end{align*}"}
+{"id": "7797.png", "formula": "\\begin{align*} \\begin{aligned} & ~ ~ ~ K ^ { ( i ) \\top } R K ^ { ( i ) } + S ^ { \\top } K ^ { ( i ) } + K ^ { ( i ) \\top } S + Q \\\\ & = Q - S ^ { \\top } R ^ { - 1 } S + ( R K ^ { ( i ) } + S ) ^ { \\top } R ^ { - 1 } ( R K ^ { ( i ) } + S ) > 0 . \\end{aligned} \\end{align*}"}
+{"id": "6024.png", "formula": "\\begin{align*} \\bar \\pi ^ { N , \\alpha } _ s ( d u ) = \\frac { 1 } { N } \\sum _ { x \\in \\mathbb { T } _ N } [ \\xi ^ \\alpha _ x ( s ) - \\rho _ \\alpha ] \\delta _ { \\frac x N } ( d u ) . \\end{align*}"}
+{"id": "216.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } { \\displaystyle \\frac { d x ^ i } { d t } } & = & v ^ i \\\\ { \\displaystyle \\frac { d v ^ i } { d t } } & = & X ^ i ( x ^ 1 , \\ldots , x ^ n , v ^ 1 , \\ldots , v ^ n ) \\end{array} \\right . i = 1 , \\ldots , n , \\end{align*}"}
+{"id": "3801.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ | \\widehat { \\mathcal { C } } _ 1 \\cap \\overline { \\mathcal { C } _ 1 } | \\right ] & \\leq c _ 6 \\sum _ { i \\in [ M ] } \\sum _ { t = 0 } ^ { T - 1 } \\exp \\left ( - c _ 7 N _ i n _ x \\left ( \\frac { \\alpha \\rho ^ { ( i ) } \\| \\Sigma ^ { ( i ) } _ t \\| } { \\rho ^ { ( i ) } \\| \\Sigma ^ { ( i ) } _ t \\| + \\sqrt { n _ x } } \\right ) ^ 2 \\right ) , \\\\ \\end{align*}"}
+{"id": "7622.png", "formula": "\\begin{align*} F _ n ( f ) ( z ) = ( f ( z ) , f ( z ) + f ^ { \\prime } ( z ) , f ( z ) + f ^ { \\prime \\prime } ( z ) , \\cdots , f ( z ) + f ^ { ( n - 1 ) } ( z ) ) . \\end{align*}"}
+{"id": "5454.png", "formula": "\\begin{align*} \\Delta U _ t = - 2 ( U _ { t ^ - } \\cdot n ( Y _ t ) ) n ( Y _ t ) \\textbf { 1 } _ { \\{ Y _ t \\in \\partial D \\} } = \\Delta \\tilde K _ t \\ , . \\end{align*}"}
+{"id": "5104.png", "formula": "\\begin{align*} ( 1 + g _ s g _ t g _ s ^ { - 1 } ) e _ { s t s } & = ( 1 + g _ { s t s } ) e _ { s t s } + ( u ^ { - 1 } - 1 ) g _ s g _ t ( 1 + g _ s ) e _ { \\langle s , t \\rangle } , \\end{align*}"}
+{"id": "2025.png", "formula": "\\begin{align*} f ( R ) & \\geq f ( A ) \\left ( ( p - 1 ) C h ^ 2 + 1 \\right ) ^ { \\frac { R - A } { h } - 1 } \\\\ & = f ( A ) e ^ { - \\theta A - 1 } e ^ { \\theta R } . \\end{align*}"}
+{"id": "4776.png", "formula": "\\begin{align*} \\varphi _ 1 ( \\varepsilon , b , D , c , \\eta , f ) & : = \\max \\Bigg \\{ \\psi ( c + 1 , b , D ) , \\psi \\left ( \\left ( \\frac { 4 } { \\varepsilon } + 1 \\right ) c , b , D \\right ) , \\\\ & \\qquad \\qquad \\qquad \\left ( \\frac { D \\left ( 2 \\eta ( \\min \\{ \\varepsilon / 2 , 2 \\} ) \\right ) ^ 2 } { 1 8 } \\right ) ^ { - 1 } ( c + b ) , \\\\ & \\qquad \\qquad \\qquad \\qquad \\varphi \\left ( \\frac { D \\left ( 2 \\eta ( \\min \\{ \\varepsilon / 2 , 2 \\} ) \\right ) ^ 2 } { 1 8 } , b , f \\right ) \\Bigg \\} \\end{align*}"}
+{"id": "8342.png", "formula": "\\begin{align*} Z _ { \\mathcal { A } } ( \\mathcal { H } ) = \\left \\{ x \\in \\mathcal { H } | \\alpha ( x ) \\bullet \\mathcal { H } = \\mathcal { H } \\bullet \\alpha ( x ) = 0 \\right \\} , \\end{align*}"}
+{"id": "2843.png", "formula": "\\begin{align*} \\delta I ' = \\int ^ { t _ { 2 } } _ { t _ { 1 } } ( - \\ddot { q } - q ) \\delta q d t + \\frac { d } { d t } \\left [ \\left ( \\dot { q } + \\frac { \\partial C } { \\partial q } \\right ) \\delta q \\right ] ^ { t _ { 2 } } _ { t _ { 1 } } \\end{align*}"}
+{"id": "8254.png", "formula": "\\begin{align*} Y _ { i , j } = { \\underline S } _ i ^ T { \\underline X } _ { i , j } , \\end{align*}"}
+{"id": "2787.png", "formula": "\\begin{align*} \\delta \\Xi ^ { a ' } ( t _ { 1 } ) : = 0 , \\end{align*}"}
+{"id": "8084.png", "formula": "\\begin{align*} \\norm { \\rho f } _ k = \\norm { f } _ k , \\norm { \\rho f } ' _ k = \\norm { f } ' _ k . \\end{align*}"}
+{"id": "1656.png", "formula": "\\begin{align*} u _ { 1 } \\left ( x , 0 \\right ) = u _ { 0 } ^ { \\left ( 1 \\right ) } \\left ( x \\right ) , m _ { 1 } \\left ( x , 0 \\right ) = m _ { 0 } ^ { \\left ( 1 \\right ) } \\left ( x \\right ) , x \\in \\Omega . \\end{align*}"}
+{"id": "5918.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { N C } ( A ( n , \\nu ) ) ) & = ( 2 ^ { 2 n } - 2 ^ { n } ) ( 2 ^ { 2 n } - 2 ^ { n } - 1 ) ^ { 2 } - 4 ( 2 ^ { 2 n } - 2 ^ { n } - 1 ) ( 2 ^ { n - 1 } ( 2 ^ { n } - 1 ) ^ { 2 } ) + 2 ^ { n } ( 2 ^ { n } - 1 ) ^ { 3 } \\\\ & = 2 ^ { 6 n } - 5 \\cdot 2 ^ { 5 n } + 8 \\cdot 2 ^ { 4 n } - 4 \\cdot 2 ^ { 3 n } \\\\ & = 2 ^ { 5 n } ( 2 ^ { n } - 5 ) + 2 ^ { 3 n + 2 } ( 2 ^ { n + 1 } - 1 ) \\end{align*}"}
+{"id": "8051.png", "formula": "\\begin{align*} \\varepsilon _ { 1 6 } ^ N = \\int _ 0 ^ T & \\frac { 1 } { N \\gamma _ N } \\sum _ { i = 1 } ^ N \\sum _ { j = 1 } ^ N \\lambda _ 1 \\left ( \\frac { i } { N } \\right ) \\lambda _ 2 \\left ( \\frac { j } { N } \\right ) \\times \\\\ & \\left ( \\mathbb { E } \\left ( S _ s ^ N ( i ) I _ s ^ N ( j ) \\right ) - \\mathbb { E } \\left ( S _ s ^ N ( i ) \\right ) \\mathbb { E } \\left ( I _ s ^ N ( j ) \\right ) \\right ) \\left ( G _ s \\left ( \\frac { i } { N } \\right ) - F _ s \\left ( \\frac { i } { N } \\right ) \\right ) d s , \\end{align*}"}
+{"id": "8016.png", "formula": "\\begin{align*} & P \\left ( \\sup _ { 0 \\leq t \\leq \\delta } \\left | \\mu _ { t + \\sigma , k } ^ N ( f ) - \\mu _ { \\sigma , k } ^ N ( f ) \\right | > \\epsilon \\right ) \\\\ & \\leq P \\left ( \\| f \\| _ \\infty Y ( N K _ 2 \\delta ) > N \\epsilon \\right ) \\leq e ^ { - K N \\epsilon } \\mathbb { E } e ^ { K \\| f \\| _ \\infty Y ( N K _ 2 \\delta ) } \\\\ & = e ^ { - K N \\epsilon } e ^ { N K _ 2 \\delta \\left ( e ^ { K \\| f \\| _ \\infty } - 1 \\right ) } . \\end{align*}"}
+{"id": "1611.png", "formula": "\\begin{align*} \\b Q _ { i ( p - 1 ) } ( x y ) \\mapsto ( - 1 ) ^ { \\tfrac { n m ( p - 1 ) } { 2 } } \\sum _ { j + k = i } & \\big ( \\b Q _ { j ( p - 1 ) } ( x ) Q _ { k ( p - 1 ) } ( y ) \\\\ & + ( - 1 ) ^ n Q _ { j ( p - 1 ) } ( x ) \\b Q _ { k ( p - 1 ) } ( y ) \\big ) . \\end{align*}"}
+{"id": "7011.png", "formula": "\\begin{align*} b ( x ) = \\sqrt { \\delta } \\frac { d - 2 } { 2 } \\mathbf { 1 } _ { | x | < 1 } | x | ^ { - 2 } x . \\end{align*}"}
+{"id": "4338.png", "formula": "\\begin{gather*} f _ i ( x , u ^ i ) = ( u ^ i ) ^ T l _ i ( x ) \\ \\forall u ^ i \\in \\mathcal { U } _ i , \\ x \\in \\mathcal { X } . \\end{gather*}"}
+{"id": "1420.png", "formula": "\\begin{align*} P _ \\alpha ( t ) & = \\ \\frac { 1 } { \\alpha } \\int _ 0 ^ t f _ \\alpha ( u ^ { - 1 / \\alpha } ) \\ , u ^ { - 1 / \\alpha - 1 } \\ , d u = \\int _ { t ^ { - 1 / \\alpha } } ^ \\infty f _ \\alpha ( y ) \\ , d y \\\\ & = 1 - \\int _ 0 ^ { t ^ { - 1 / \\alpha } } f _ \\alpha ( y ) \\ , d y \\equiv 1 - F _ \\alpha ( t ^ { - 1 / \\alpha } ) = 1 - F _ \\alpha ( 1 \\vert t ) \\end{align*}"}
+{"id": "4669.png", "formula": "\\begin{align*} \\P _ S ( I ) : = \\P _ B ( I \\ ; | \\ ; | I | = S ) . \\end{align*}"}
+{"id": "6125.png", "formula": "\\begin{align*} u = v ^ 2 = \\begin{pmatrix} v _ { 1 1 } ^ 2 + v _ { 1 2 } v _ { 1 2 } ^ * & v _ { 1 1 } v _ { 1 2 } + v _ { 1 2 } v _ { 2 2 } \\\\ v _ { 1 2 } ^ * v _ { 1 1 } + v _ { 2 2 } v _ { 1 2 } ^ * & v _ { 1 2 } ^ * v _ { 1 2 } + v _ { 2 2 } ^ 2 \\end{pmatrix} . \\end{align*}"}
+{"id": "1529.png", "formula": "\\begin{align*} \\lambda _ { \\rm m i n } ( D ^ 2 d ( z ) ) & = | z | ^ { p - 2 } , & \\lambda _ { \\rm m a x } ( D ^ 2 d ( z ) ) & = ( p - 1 ) \\ , | z | ^ { p - 2 } , \\\\ \\lambda _ { \\rm m i n } ( D ^ 2 s ( z ) ) & = ( q - 1 ) \\ , | z | ^ { q - 2 } , & \\lambda _ { \\rm m a x } ( D ^ 2 s ( z ) ) & = | z | ^ { q - 2 } . \\end{align*}"}
+{"id": "1051.png", "formula": "\\begin{align*} g ( e ^ { i \\theta } ) g ( e ^ { i \\theta } ) ^ * = g _ { \\sharp } ( e ^ { i \\theta } ) ^ * g _ { \\sharp } ( e ^ { i \\theta } ) , \\theta \\in [ - \\pi , \\pi ) \\end{align*}"}
+{"id": "3258.png", "formula": "\\begin{align*} k _ 0 & = x / y \\\\ k _ 1 & = 1 / y \\\\ k _ 2 & = ( x ^ 2 + y ^ 2 ) / y . \\end{align*}"}
+{"id": "7439.png", "formula": "\\begin{align*} - \\sum _ { j = 1 } ^ d \\mu ^ { ( j ) } ( p ) e _ j & = \\left ( - \\frac { \\partial } { \\partial t } + A \\right ) \\Phi _ 1 ( t , p ) \\\\ & = \\left ( - \\frac { \\partial } { \\partial t } + \\sum ^ d _ { j = 1 } \\mu ^ { ( j ) } ( p ) \\frac { \\partial } { \\partial p _ j } + \\frac { 1 } { 2 } \\sum ^ d _ { j = 1 } \\frac { \\partial ^ 2 } { \\partial p _ j ^ 2 } \\right ) \\Phi _ 1 ( t , p ) , \\end{align*}"}
+{"id": "1743.png", "formula": "\\begin{align*} ( - 1 ) ^ { n + 1 } \\left ( H _ { \\mathcal { L } } \\right ) _ { n , n } = \\sum _ { j = 2 } ^ { n - 2 } ( - 1 ) ^ r \\left ( H _ { \\mathcal { L } } \\right ) _ { n - j , n } \\left ( H _ { \\mathcal { L } } \\right ) _ { n , j } \\end{align*}"}
+{"id": "2577.png", "formula": "\\begin{align*} q ^ { \\frac { 1 } { 2 } } ( q ^ { \\frac { 1 } { 2 } } X _ 2 + h ) ( q ^ { \\frac { 1 } { 2 } } X _ 3 + h ) = & q ^ 2 X ( - 1 , 3 ) + ( q + q ^ 2 ) h X ( - 1 , 2 ) + q h X ( 0 , 1 ) \\\\ & + ( q h ^ 2 + q ^ 2 ) X ( - 1 , 1 ) + q h X ( - 1 , 0 ) + q ^ { \\frac { 1 } { 2 } } h ^ 2 . \\end{align*}"}
+{"id": "8927.png", "formula": "\\begin{align*} \\dim ( \\gamma _ c ( L ) ) + \\dim ( \\mathcal { M } ( L ) ) = \\dim ( \\mathcal { M } ( L / \\gamma _ c ( L ) ) ) + \\dim \\frac { [ \\gamma _ c ( F ) + R , F ] } { [ R , F ] } \\end{align*}"}
+{"id": "8057.png", "formula": "\\begin{align*} \\widetilde { \\eta } _ { t , 1 } ( f ) = & \\widetilde { \\eta } _ { 0 , 1 } ( f ) - \\int _ 0 ^ t \\int _ \\mathbb { T } \\widetilde { \\eta } _ { s , 3 } ( \\lambda _ 2 ) \\lambda _ 1 ( v ) \\theta _ s ^ S ( v ) f ( v ) d v d s \\\\ & - \\int _ 0 ^ t \\int _ { \\mathbb { T } } \\widetilde { \\eta } _ { s , 1 } ( \\lambda _ 1 f ) \\lambda _ 2 ( v ) \\theta _ s ^ I ( v ) d v d s \\\\ & - \\int _ 0 ^ t \\mu _ { s , 1 } \\bigotimes \\mu _ { s , 3 } \\left ( \\lambda ( \\cdot , \\ast ) ( \\tilde { G } ( \\cdot ) - \\tilde { F } ( \\cdot ) ) \\right ) d s \\end{align*}"}
+{"id": "3885.png", "formula": "\\begin{align*} & \\int _ 0 ^ { \\infty } \\exp ( - s ) R ( \\eta ( \\cdot \\mid s ) \\| \\eta _ { ( 1 ) } ( \\cdot \\mid s ) \\otimes P ^ o ( \\cdot , \\cdot ) ) d s \\\\ & \\ge R \\left ( \\int _ 0 ^ { \\infty } \\exp ( - s ) \\eta ( \\cdot \\mid s ) d s \\| \\int _ 0 ^ { \\infty } \\exp ( - s ) \\eta _ { ( 1 ) } ( \\cdot \\mid s ) \\otimes P ^ o ( \\cdot , \\cdot ) d s \\right ) \\\\ & = R ( \\gamma \\| m \\otimes P ^ o ) \\end{align*}"}
+{"id": "9148.png", "formula": "\\begin{align*} \\begin{aligned} & g _ { 1 } = x _ { j - 1 , 1 } ^ { s _ { j - 1 } + 2 } ( x _ { j , 1 } x _ { j , 2 } ) ^ { s _ { j } } \\prod _ { \\ell = i } ^ { j - 2 } x _ { \\ell , 1 } ^ { s _ { \\ell } + 1 } \\prod _ { \\ell = j + 1 } ^ { n } ( x _ { \\ell , 1 } x _ { \\ell , 2 } ) ^ { s _ { \\ell } + 1 } , \\\\ & g _ { 2 } = x _ { i , 1 } ^ { s _ { i } } \\prod _ { \\ell = i + 1 } ^ { j - 1 } x _ { \\ell , 1 } ^ { s _ { \\ell } + 1 } \\prod _ { \\ell = j } ^ { n } ( x _ { \\ell , 1 } x _ { \\ell , 2 } ) ^ { s _ { \\ell } + 1 } . \\end{aligned} \\end{align*}"}
+{"id": "495.png", "formula": "\\begin{align*} R _ K R _ { K - 1 } \\cdots R _ 1 \\cdot \\begin{pmatrix} P ( T ) \\\\ Q ( T ) \\end{pmatrix} = \\begin{pmatrix} P _ { N - 1 } ( T ) \\\\ P _ N ( T ) \\end{pmatrix} = \\begin{pmatrix} P _ { N - 1 } ( T ) \\\\ 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "5229.png", "formula": "\\begin{align*} \\frac { \\partial \\overline { M G } } { \\partial q _ { j } } = \\frac { \\left ( 1 - \\alpha \\right ) } { \\sum _ { j } q _ { j } } \\left [ \\overline { p } ^ { \\alpha } _ { j } \\overline { q } ^ { - \\alpha } _ { j } - \\sum _ { i } \\overline { p } ^ { \\alpha } _ { i } \\overline { q } ^ { 1 - \\alpha } _ { i } \\right ] = \\frac { \\left ( 1 - \\alpha \\right ) } { \\sum _ { j } q _ { j } } \\left [ \\frac { \\overline { M G } _ { j } } { \\overline { q } _ { j } } - \\overline { M G } \\right ] \\end{align*}"}
+{"id": "2388.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } f ( t ) \\mathrm { d } t \\equiv \\frac { 1 } { p ^ m } \\sum _ { k = 0 } ^ { p ^ m - 1 } f ( k ) \\pmod { p ^ { \\triangle _ m ( f ) - 1 } \\mathbb { Z } _ p } . \\end{align*}"}
+{"id": "6840.png", "formula": "\\begin{align*} U ^ { ( N - 1 ) } = \\begin{pmatrix} u _ { 1 1 } ^ { ( N - 1 ) } & \\ldots & u ^ { ( N - 1 ) } _ { 1 ( N - 1 ) } & 0 \\\\ u _ { 2 1 } ^ { ( N - 1 ) } & \\ldots & u ^ { ( N - 1 ) } _ { 2 ( N - 1 ) } & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots \\\\ u _ { ( N - 1 ) 1 } ^ { ( N - 1 ) } & \\ldots & u ^ { ( N - 1 ) } _ { ( N - 1 ) ( N - 1 ) } & 0 \\\\ u _ { N 1 } ^ { ( N - 1 ) } & \\ldots & u ^ { ( N - 1 ) } _ { N ( N - 1 ) } & u ^ { ( N - 1 ) } _ { N N } \\end{pmatrix} . \\end{align*}"}
+{"id": "9052.png", "formula": "\\begin{align*} \\sigma ( ( S ^ i ) ^ { ( k ) } v ) = ( S ^ i ) ^ { ( \\sigma ( k ) ) } ( \\sigma v ) ( v \\in V ^ { \\otimes n } ) , \\end{align*}"}
+{"id": "4528.png", "formula": "\\begin{align*} p _ N = \\sum _ { i = 1 } ^ N \\varphi _ i ( s _ 1 , . . . , s _ i ) , \\varphi _ i > 0 , \\sum _ { i = 1 } ^ \\infty \\| \\varphi _ i \\| _ { \\infty } \\leq a _ + < \\infty , \\end{align*}"}
+{"id": "5397.png", "formula": "\\begin{align*} h _ k ( j _ k ) = c _ k ( j _ k ) + s _ k \\ , \\lambda _ k ( 0 ) \\ , 1 \\{ j _ k = 0 \\} - r _ k ( j _ k ) \\ , \\lambda _ k ( j _ k ) \\ , 1 \\{ j _ k > 0 \\} . \\end{align*}"}
+{"id": "7305.png", "formula": "\\begin{align*} \\lim _ { t \\downarrow 0 } K _ X ( x , y ; t ) = \\delta _ { x } ( y ) . \\end{align*}"}
+{"id": "2176.png", "formula": "\\begin{align*} | \\{ ( a , b ) \\in A \\times A : a - b = t \\} | \\lesssim K ^ { \\frac { 4 0 5 } { 4 1 } } | A | ^ { \\frac { 2 } { 3 } - \\frac { 1 } { 1 2 3 } } . \\end{align*}"}
+{"id": "8058.png", "formula": "\\begin{align*} & P ( \\eta ^ N \\in O ) \\\\ & = \\mathbb { E } _ { \\widetilde { P } _ { h _ 1 ^ m , h _ 2 ^ m , h _ 3 ^ m } ^ { N , F ^ m , G ^ m , H ^ m } } \\left ( \\left ( \\mathcal { Y } _ { F ^ m , G ^ m , H ^ m } ^ N ( t , \\xi ^ N ) \\right ) ^ { - 1 } \\frac { d P } { d \\tilde { P } ^ N _ { h _ 1 ^ m , h _ 2 ^ m , h _ 3 ^ m } } 1 _ { \\{ \\eta ^ N \\in O \\} } \\right ) . \\end{align*}"}
+{"id": "2247.png", "formula": "\\begin{align*} F ^ { \\alpha } = - i _ X \\eta ^ \\alpha _ L \\end{align*}"}
+{"id": "6037.png", "formula": "\\begin{align*} J = \\frac { 1 } { N ^ \\gamma } \\begin{pmatrix} ( 1 - 2 \\rho ^ { A } ) ( E _ { A } - E _ { C } ) - \\rho ^ { B } ( E _ { B } - E _ { C } ) & - \\rho ^ { A } ( E _ { B } - E _ { C } ) \\\\ - \\rho ^ { B } ( E _ { A } - E _ { C } ) & - \\rho ^ { A } ( E _ { A } - E _ { C } ) + ( 1 - 2 \\rho ^ { B } ) ( E _ { B } - E _ { C } ) \\end{pmatrix} \\end{align*}"}
+{"id": "3868.png", "formula": "\\begin{align*} m ^ n _ j \\doteq m ^ n _ { j - 1 } + l ^ { n , j - 1 } + 1 , \\ ; \\ ; j = 1 , \\dots , l _ 0 . \\end{align*}"}
+{"id": "7354.png", "formula": "\\begin{align*} D _ X = \\nabla ^ g _ X + \\theta _ g ( X ) \\mathrm { I d } + \\theta _ g \\wedge X , \\forall X \\in \\mathrm { T } M , \\end{align*}"}
+{"id": "6177.png", "formula": "\\begin{align*} \\frac { 1 } { \\beta ( \\tau ^ k ) ^ 2 } + \\frac { \\sigma '' } { \\tau ^ k } \\geq \\frac { 1 } { \\beta ( \\tau ^ { k + 1 } ) ^ 2 } + \\frac { \\sigma '' ( 1 - \\gamma ) } { \\tau ^ { k + 1 } } , ~ \\sigma '' = \\sigma ' - \\frac { \\sigma '^ 2 } { \\sigma ' + 1 } , ~ \\sigma ' = \\frac { \\sigma _ { \\rm m i n } ( A _ m A _ m ^ T ) } { L } , \\end{align*}"}
+{"id": "2338.png", "formula": "\\begin{align*} & \\frac { 1 } { p } \\frac { d } { d t } \\| a _ { k } \\| _ { L ^ { p } } ^ { p } + \\frac { 4 ( p - 2 ) } { p ^ { 2 } } \\| \\nabla | a _ { k } | ^ { \\frac { p } { 2 } } \\| _ { L ^ { 2 } } ^ { 2 } + \\| | \\nabla a _ { k } | | a _ { k } | ^ { \\frac { p - 2 } { 2 } } \\| _ { L ^ { 2 } } ^ { 2 } \\\\ & = \\int _ { \\mathbb { R } ^ { 3 } } ( u ^ { c , \\gamma } \\otimes a _ { k - 1 } + a _ { k - 1 } \\otimes u ^ { c , \\gamma } ) \\cdot \\nabla ( | a _ { k } | ^ { p - 2 } a _ { k } ) + \\int _ { \\mathbb { R } ^ { 3 } } \\pi _ { k - 1 } \\mathrm { d i v } ( | a _ { k } | ^ { p - 2 } a _ { k } ) . \\end{align*}"}
+{"id": "7311.png", "formula": "\\begin{align*} \\psi _ j ( x ) = \\frac { 1 } { \\sqrt { m } } \\exp \\left ( 2 \\pi i \\frac { j + \\beta } { m } x \\right ) \\ , \\ , \\ , \\ , \\ , \\ , x \\in G _ m \\ , \\ , \\ , \\ , \\ , \\ , j = 0 , \\ldots , m - 1 . \\end{align*}"}
+{"id": "4566.png", "formula": "\\begin{align*} R i c _ { g , a b } = - 2 s _ g ^ { - 1 } \\nabla _ { a } \\nabla _ { b } s _ g + \\left ( - s _ g ^ { - 1 } \\Delta s _ g + 3 | d \\log s _ g | ^ 2 \\right ) g _ { a b } . \\end{align*}"}
+{"id": "4741.png", "formula": "\\begin{align*} \\langle y , x \\rangle _ s : = \\mathrm { s u p } \\left \\{ \\langle y , j \\rangle \\mid j \\in J ( x ) \\right \\} . \\end{align*}"}
+{"id": "8292.png", "formula": "\\begin{align*} \\frac { u _ 1 u _ { 1 i } } { w ^ 2 \\ln w } = \\frac { 2 x _ i } { \\rho } , i \\geq 2 . \\end{align*}"}
+{"id": "7613.png", "formula": "\\begin{align*} D ( d ; m , n ) = \\begin{cases} d & \\mbox { i f } ( m , n ) = ( 3 , 1 ) , \\\\ \\lfloor d / 3 \\rfloor & \\mbox { i f } ( m , n ) = ( 1 , 3 ) . \\end{cases} \\end{align*}"}
+{"id": "5804.png", "formula": "\\begin{align*} & | X _ + ( t ) | ^ 2 + | X _ 0 ( t ) | ^ 2 + | X _ - ( t ) | ^ 2 = \\sum _ { k + \\ell \\leq s } \\| \\mathcal { L } _ { \\Sigma } ^ { \\ell } \\partial ^ { k } _ t u \\| _ { L ^ 2 } ^ 2 ( t ) . \\end{align*}"}
+{"id": "2803.png", "formula": "\\begin{align*} \\omega = d q ^ { i } \\wedge d p _ { i } = d \\Xi ^ { i } \\wedge d \\Psi _ { i } , \\end{align*}"}
+{"id": "5023.png", "formula": "\\begin{align*} \\max \\ , \\left \\{ \\mathbf { x } ^ 1 \\mathbf { R } \\colon \\begin{bmatrix} \\mathbf { x } ^ { 0 } & \\mathbf { x } ^ { 1 } \\end{bmatrix} \\begin{bmatrix} ( 1 - \\beta ) \\mathbf { I } \\\\ \\mathbf { I } - \\beta \\mathbf { P } \\end{bmatrix} = \\mathbf { e } _ i , \\begin{bmatrix} \\mathbf { x } ^ { 0 } & \\mathbf { x } ^ { 1 } \\end{bmatrix} \\geq \\mathbf { 0 } \\right \\} , \\end{align*}"}
+{"id": "6276.png", "formula": "\\begin{align*} \\Delta _ k = \\tilde { O } \\left ( \\frac { R _ k ^ 2 } { T _ k ^ { \\frac { 2 \\kappa } { 1 + \\kappa } } } \\right ) = \\tilde { O } \\left ( \\frac { R _ 0 ^ 4 } { B ^ 2 2 ^ { 2 k r } } \\right ) . \\end{align*}"}
+{"id": "2047.png", "formula": "\\begin{align*} \\theta ^ { t + 1 } & = \\theta ^ { t } + \\eta \\left ( y ^ t - \\langle x ^ t , \\theta ^ { t } \\rangle \\right ) x ^ t , \\end{align*}"}
+{"id": "7478.png", "formula": "\\begin{align*} ( \\partial _ t - L _ t ) \\Bar { h } = R _ 0 [ \\Bar { h } ] + \\nabla ^ { g _ 0 ( t ) } R _ 1 [ \\Bar { h } ] , \\end{align*}"}
+{"id": "2763.png", "formula": "\\begin{align*} \\sigma _ { 2 } : T ^ { * } M | _ { Q , P } \\rightarrow T ^ { * } M ; ( \\sigma _ { 2 } ^ { * } \\Theta ^ { \\alpha } : = \\epsilon ^ { \\alpha } , \\sigma _ { 2 } ^ { * } \\Theta _ { \\alpha } : = \\epsilon _ { \\alpha } , \\sigma _ { 2 } ^ { * } Q ^ { i } , \\sigma _ { 2 } ^ { * } P _ { i } ) \\mapsto ( \\Theta ^ { \\alpha } , \\Theta _ { \\alpha } , Q ^ { i } , P _ { i } ) , \\end{align*}"}
+{"id": "2713.png", "formula": "\\begin{align*} q ^ { i } = { _ { * } q } ^ { i } ( t , { _ { * } c } ^ { I } ) , p _ { i } = p _ { i } ( t , { _ { * } c } ^ { I } ) . \\end{align*}"}
+{"id": "3736.png", "formula": "\\begin{align*} _ { p } F _ { q } \\left ( \\left . \\begin{array} { c } a _ { 1 } , \\ldots , a _ { p } \\\\ b _ { 1 } , \\ldots b _ { q } \\end{array} \\right \\vert z \\right ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( a _ { 1 } \\right ) _ { k } \\cdots \\left ( a _ { p } \\right ) _ { k } } { \\left ( b _ { 1 } \\right ) _ { k } \\cdots \\left ( b _ { q } \\right ) _ { k } } \\frac { z ^ { k } } { k ! } , \\end{align*}"}
+{"id": "2620.png", "formula": "\\begin{align*} | D _ 4 ( S ) | \\geq 1 + | D _ { \\{ 2 , 4 \\} } ( S ) | = 1 + | D _ { \\{ 4 \\} } ( S ) \\cup [ D _ { \\{ 2 \\} } ( S ) \\setminus D _ { \\{ 4 \\} } ( S ) ] | \\geq 1 + n - 4 + \\frac { n } { 4 } + 2 = \\frac { 5 n } { 4 } - 1 . \\end{align*}"}
+{"id": "3175.png", "formula": "\\begin{align*} \\left \\langle X \\left ( t \\right ) , e _ { j } \\right \\rangle _ { 2 } = \\left \\langle x , e _ { j } \\right \\rangle _ { 2 } - \\int _ { 0 } ^ { t } \\int _ { \\mathbb { R } ^ { d } } \\left \\langle \\nabla \\eta \\left ( X \\right ) , \\nabla e _ { j } \\right \\rangle _ { \\mathbb { R } ^ { d } } d \\xi d s + \\left \\langle \\int _ { 0 } ^ { t } X \\left ( s \\right ) \\circ d W _ { s } , e _ { j } \\right \\rangle _ { 2 } . \\end{align*}"}
+{"id": "1314.png", "formula": "\\begin{align*} \\{ \\Phi _ X , g \\circ \\pi \\} _ \\pm & = \\pm \\rho ( X ) [ g ] \\circ \\pi , \\\\ \\{ f \\circ \\pi , g \\circ \\pi \\} _ \\pm & = 0 . \\end{align*}"}
+{"id": "7529.png", "formula": "\\begin{align*} r h _ i ^ { \\prime \\prime } ( r ) = p _ i ^ \\prime ( r ) \\ , r h _ i ^ \\prime ( r ) = p _ i ( r ) + h _ i ( r ) \\ , \\end{align*}"}
+{"id": "4636.png", "formula": "\\begin{align*} \\int _ \\Omega | \\nabla u _ { n , k } | ^ 2 + \\int _ \\Omega \\lambda V | u _ { n , k } | ^ 2 \\leq b _ \\nabla 2 ^ { - ( d - 1 ) m n } \\cdot 2 ^ { \\gamma m n } \\left ( 1 - 2 \\lambda 2 ^ { - \\gamma m n \\tfrac { 2 } { d } ( - 1 + \\epsilon ) } \\cdot 2 ^ { - 2 \\gamma m n } \\right ) < 0 \\end{align*}"}
+{"id": "4205.png", "formula": "\\begin{align*} & 2 4 0 \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 1 0 ) } = \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } ( \\widetilde { T X } - \\widetilde { L _ R \\otimes C } ) \\right \\} ^ { ( 1 0 ) } . \\end{align*}"}
+{"id": "560.png", "formula": "\\begin{align*} - \\frac { P } { \\rho } & = g z + \\Phi _ t & \\Omega . \\end{align*}"}
+{"id": "7916.png", "formula": "\\begin{align*} \\begin{aligned} \\langle d \\lambda , \\mu \\rangle _ { L ^ { 2 } \\Lambda ^ { k } ( \\Omega ) } & = \\langle \\lambda , \\delta \\mu \\rangle _ { L ^ { 2 } \\Lambda ^ { k - 1 } ( \\Omega ) } + \\int _ { \\partial \\Omega } ( \\lambda ) \\wedge ( \\ast \\mu ) \\\\ & = \\langle \\lambda , \\delta \\mu \\rangle _ { L ^ { 2 } \\Lambda ^ { k - 1 } ( \\Omega ) } + \\int _ { \\partial \\Omega } \\mathrm { t r } ( \\lambda ) \\wedge \\ast \\boldsymbol { n } ( \\mu ) , \\end{aligned} \\end{align*}"}
+{"id": "6457.png", "formula": "\\begin{align*} R ^ N ( X , Y ) Z = K ( \\langle Y , Z \\rangle X - \\langle X , Z \\rangle Y ) , \\end{align*}"}
+{"id": "8867.png", "formula": "\\begin{align*} \\frac { n } { a _ 1 } = \\frac { ( p - 1 ) n } { r } + \\tau - p + 2 , \\frac { n } { b _ 1 } = \\frac { p n } { r } + \\tau - p . \\end{align*}"}
+{"id": "5606.png", "formula": "\\begin{align*} v ' = t _ 1 + t _ 2 + t _ { \\geq 3 } & \\geq v - h , \\\\ t _ 1 + 2 t _ 2 + 3 t _ { \\geq 3 } & \\leq \\sum _ { k \\geq 1 } k t _ k = 2 ( a - h ) . \\end{align*}"}
+{"id": "993.png", "formula": "\\begin{align*} \\eta _ V [ u ] ( A ) = \\mathbb E _ { x _ 0 } ( \\mathbf 1 _ D u ( X _ { \\tau _ V } ) \\mathbf 1 _ A ( X _ { \\tau _ V - } ) ) , \\end{align*}"}
+{"id": "5189.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } D _ { \\beta } I ( p \\| q ) } { \\partial q _ { j } } = Z \\ ; \\left [ \\left ( \\frac { \\sum _ { i } p _ { i } q ^ { \\beta - 1 } _ { i } } { \\sum _ { i } q ^ { \\beta } _ { i } } \\right ) ^ { \\beta - 1 } p _ { j } q ^ { \\beta - 2 } _ { j } - \\left ( \\frac { \\sum _ { i } p _ { i } q ^ { \\beta - 1 } _ { i } } { \\sum _ { i } q ^ { \\beta } _ { i } } \\right ) ^ { \\beta } q ^ { \\beta - 1 } _ { j } \\right ] \\end{align*}"}
+{"id": "4634.png", "formula": "\\begin{align*} & \\int _ \\Omega | \\nabla u _ { n , k } | ^ 2 + \\int _ \\Omega \\lambda V | u _ { n , k } | ^ 2 \\leq b _ \\nabla 2 ^ { - ( d - 1 ) m n } \\cdot 2 ^ { \\gamma m n } - \\lambda b _ V 2 ^ { - \\gamma m n \\frac { 2 } { d } ( - 1 + \\epsilon ) } b _ 2 2 ^ { - ( d - 1 ) m n } 2 ^ { - \\gamma m n } \\\\ & = b _ \\nabla 2 ^ { - ( d - 1 ) m n } \\cdot 2 ^ { \\gamma m n } \\left ( 1 - 2 \\lambda 2 ^ { - \\gamma m n \\frac { 2 } { d } ( - 1 + \\epsilon ) } \\cdot 2 ^ { - 2 \\gamma m n } \\right ) . \\end{align*}"}
+{"id": "1054.png", "formula": "\\begin{align*} \\tilde { A } _ k : = - \\sum _ { u = k } ^ { \\infty } \\tilde { a } _ u , k \\in \\N \\cup \\{ 0 \\} . \\end{align*}"}
+{"id": "5334.png", "formula": "\\begin{align*} \\nu _ { \\pi _ k } = \\min \\ , \\left \\{ \\nu ^ { S _ k } _ j : j \\in \\partial _ { \\mathcal { F } } ^ - S _ k \\right \\} . \\end{align*}"}
+{"id": "7131.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 } } \\nabla u _ { n } \\nabla \\varphi d x + \\lambda _ { n } \\int _ { \\mathbb { R } ^ { 2 } } u _ { n } \\varphi d x + \\left [ V _ { 1 } ^ { \\prime } ( u _ { n } ) - V _ { 2 } ^ { \\prime } ( u _ { n } ) \\right ] \\varphi - \\int _ { \\mathbb { R } ^ { 2 } } f ( u _ { n } ) \\varphi d x = o ( 1 ) \\Vert \\varphi \\Vert . \\end{align*}"}
+{"id": "6010.png", "formula": "\\begin{align*} \\eta ^ { x , y } ( z ) = \\begin{cases} \\eta ( y ) & z = x \\\\ \\eta ( x ) & z = y \\\\ \\eta ( z ) & \\textrm { o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "4375.png", "formula": "\\begin{gather*} F _ { i , j , k , l } ( x ) : = \\max \\{ 0 , \\Delta u _ i B _ { i , ( i , j ) } - \\Delta u _ k B _ { k , ( k , l ) } \\} x _ { i , j } . \\end{gather*}"}
+{"id": "122.png", "formula": "\\begin{align*} \\begin{aligned} M _ { \\lambda } ^ { - 1 } & = \\left ( \\frac { 1 } { \\lambda + 1 } \\left ( I + \\lambda P \\right ) A ^ { - 1 } \\right ) ^ { - 1 } \\\\ & = A \\left ( ( \\lambda + 1 ) I - \\lambda P \\right ) \\\\ & = A + \\lambda A ( I - P ) . \\end{aligned} \\end{align*}"}
+{"id": "3015.png", "formula": "\\begin{align*} \\overline { \\mathcal { L } } _ { v } g ( x , y ) = \\mathcal { L } _ { v } g ( x , y ) + 2 \\alpha k \\eta ( x ) \\eta ( y ) - 2 \\alpha k g ( x , y ) - 2 \\beta g ( x , \\varphi y ) . \\end{align*}"}
+{"id": "5690.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } { u ( t ) } / { \\Vert u ( t ) \\Vert _ { L ^ 2 } } = w \\ \\textup { i n } \\ C ^ \\infty ( \\Sigma ; \\mathbf { V } ) . \\end{align*}"}
+{"id": "7357.png", "formula": "\\begin{align*} L _ { \\varphi _ \\alpha ( \\overline \\gamma ) } = \\varphi _ \\alpha L _ { \\overline \\gamma } \\varphi _ \\alpha ^ { - 1 } , \\quad \\forall \\overline \\gamma \\in G , \\alpha \\in \\pi _ 1 ( M ) . \\end{align*}"}
+{"id": "5486.png", "formula": "\\begin{align*} \\int _ { | M | > 3 | E | } | M + E | ^ K d U & = \\int _ { | M | > 3 | E | } ( | M | ^ 2 + | E | ^ 2 + M \\overline { E } + \\overline { M } E ) ^ { K / 2 } d U \\\\ & = \\int _ { | M | > 3 | E | } ( A + B ) ^ { K / 2 } d U \\\\ & = \\int _ { | M | > 3 | E | } | M | ^ K \\left ( 1 + \\frac { B } { A } \\right ) ^ { K / 2 } d U \\\\ & = \\int _ { | M | > 3 | E | } | M | ^ K \\left ( 1 + \\frac { K } { 2 } \\frac { B } { A } + O _ K \\left ( \\frac { B ^ 2 } { A ^ 2 } \\right ) \\right ) d U , \\end{align*}"}
+{"id": "2714.png", "formula": "\\begin{align*} q ^ { i } = { q } ^ { i } ( t , c ^ { I } ) , p _ { i } = p _ { i } ( t , c ^ { I } ) . \\end{align*}"}
+{"id": "8938.png", "formula": "\\begin{align*} J _ i ^ T = \\begin{pmatrix} e _ i & \\cdots & e _ i \\end{pmatrix} \\end{align*}"}
+{"id": "3516.png", "formula": "\\begin{align*} \\begin{aligned} \\langle x _ 1 , \\ldots , y _ g , a _ 1 , \\ldots , a _ m \\ | \\ & \\Pi [ x _ i , y _ i ] \\cdot a _ 1 \\ldots a _ m = 1 , \\ \\\\ & a _ i ^ { p _ i } = 1 i = 1 , \\ldots , m \\rangle , \\end{aligned} \\end{align*}"}
+{"id": "7242.png", "formula": "\\begin{align*} \\ast _ { \\sigma \\circ \\tau } & = \\sigma \\circ \\tau \\circ \\ast \\circ ( ( \\sigma \\circ \\tau ) ^ { - 1 } ) ^ { \\times k } \\\\ & = \\sigma \\circ \\tau \\circ \\ast \\circ ( \\tau ^ { - 1 } ) ^ { \\times k } \\circ ( \\sigma ^ { - 1 } ) ^ { \\times k } \\\\ & = \\sigma \\circ \\ast _ { \\tau } \\circ ( \\sigma ^ { - 1 } ) ^ { \\times k } \\\\ & = { ( \\ast _ { \\tau } ) } _ { \\sigma } \\sigma , \\tau \\in S _ n , \\end{align*}"}
+{"id": "7721.png", "formula": "\\begin{align*} & - 2 \\tilde { l } \\sum _ { j } h _ { j } \\nabla _ { j } \\psi - b ^ { 1 1 } \\nabla _ { 1 1 } \\psi \\\\ & \\leq C _ { 1 } \\tilde { l } + C _ { 2 } b _ { 1 1 } + C _ { 3 } b ^ { 1 1 } + C _ { 4 } \\tilde { d } + C _ { 5 } . \\end{align*}"}
+{"id": "5067.png", "formula": "\\begin{align*} n _ s & = \\rho _ s \\begin{pmatrix} 0 & 1 \\\\ - 1 & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "7652.png", "formula": "\\begin{align*} D _ { s , 1 } = \\sup _ { L } \\sup _ { \\Lambda ' \\subset \\Lambda _ L } \\sup _ { \\gamma \\in \\mathbb { C } , n _ 0 \\in \\Lambda ' } \\frac { \\psi ^ { n _ 0 } _ s ( \\gamma ) } { \\phi ^ { n _ 0 } _ s ( \\gamma ) } . \\end{align*}"}
+{"id": "2065.png", "formula": "\\begin{align*} & - \\int _ 0 ^ s \\Theta ( u , x ) f ' ( u ) \\mathrm { d } u + f ( s ) { \\Theta } ( s , x ) - f ( 0 ) { \\Theta } ( 0 , x ) \\\\ & = - \\alpha \\int _ 0 ^ { s } \\int _ 0 ^ 1 f ( u ) A ( x , y ) \\Theta ( u , y ) \\mathrm { d } y \\mathrm { d } u + \\alpha ^ 2 \\beta ^ 2 \\int _ 0 ^ s f ( u ) \\mathrm { d } \\xi _ 1 ( u , x ) . \\end{align*}"}
+{"id": "4443.png", "formula": "\\begin{align*} \\| f \\| _ { \\partial D _ 1 \\times M _ 1 , \\rho } = \\frac { 1 } { 2 \\pi } \\lim _ { r \\rightarrow 1 - 0 } \\int _ { M _ { 1 , r } } \\int _ { \\partial D _ { 1 , r } } | f ^ * ( w _ 1 , \\hat w _ 1 ) | ^ 2 \\rho _ 1 ( w _ 1 ) | d w _ 1 | \\lambda _ 1 ( \\hat w _ 1 ) d \\mu _ 1 ( \\hat w _ 1 ) \\end{align*}"}
+{"id": "4450.png", "formula": "\\begin{align*} & \\sum _ { 1 \\le j \\le n } \\int _ { M _ j } \\int _ { \\partial D _ j } | f ( w _ j , \\hat w _ j ) | ^ 2 \\rho | d w _ j | d \\mu _ j ( \\hat w _ j ) \\\\ \\le & 2 \\liminf _ { r \\rightarrow 1 - 0 } \\frac { \\int _ { \\{ z \\in M : 2 \\psi ( z ) \\ge \\log r \\} } | g | ^ 2 \\hat \\rho } { - \\log r } \\\\ = & 2 \\liminf _ { r \\rightarrow 1 - 0 } \\frac { \\int _ { \\{ z \\in M : 2 \\psi ( z ) \\ge \\log r \\} } | g | ^ 2 \\hat \\rho } { 1 - r } \\\\ < & + \\infty . \\end{align*}"}
+{"id": "31.png", "formula": "\\begin{align*} d _ j ^ * d _ j f = - d _ { j - 1 } d _ { j - 1 } ^ * f \\ . \\end{align*}"}
+{"id": "1121.png", "formula": "\\begin{align*} \\left | \\left ( \\varphi _ j * \\psi _ R \\right ) ( x ) \\right | & = \\left | \\int _ { \\mathbb { R } ^ n } \\varphi _ j ( x - y ) \\psi _ R ( y ) \\ , d y \\right | = | R | ^ { \\frac 1 2 } \\left | \\left ( \\varphi _ j * \\psi _ i \\right ) ( x - x _ R ) \\right | \\\\ & \\lesssim | R | ^ { \\frac 1 2 } 2 ^ { - | i - j | M } \\frac { 2 ^ { - ( i \\wedge j ) M } } { [ 2 ^ { - ( i \\wedge j ) } + | x - x _ R | ] ^ { n + M } } \\\\ & \\sim | R | ^ { - \\frac 1 2 } \\frac { 1 } { \\{ 1 + [ \\ell ( R ) ] ^ { - 1 } | x - x _ R | \\} ^ { n + M } } . \\end{align*}"}
+{"id": "6650.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } \\log | f _ { n } ( E , x + \\mathrm { i } y ) | = \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { ( 2 \\pi ) ^ { d } } \\int _ { \\mathbb { T } ^ { d } } \\frac { 1 } { n } \\log | f _ { n } ( E , x + \\mathrm { i } y ) | \\mathrm { d } x = L ( E , \\mathrm { i } y ) . \\end{align*}"}
+{"id": "8174.png", "formula": "\\begin{align*} R = \\frac { 1 } { L } \\sum _ { j = 1 } ^ L R _ j . \\end{align*}"}
+{"id": "4155.png", "formula": "\\begin{align*} \\mathcal B ( G _ 0 ) = \\{ S \\in \\mathcal F ( G _ 0 ) \\colon 1 _ G \\in \\pi ( S ) \\} \\subset \\mathcal F ( G _ 0 ) \\end{align*}"}
+{"id": "558.png", "formula": "\\begin{align*} \\Delta \\Phi = 0 . \\end{align*}"}
+{"id": "6993.png", "formula": "\\begin{align*} \\mathcal { H } _ \\mathbb { R } ^ 2 : = \\{ w \\in \\mathcal { P } _ \\mathbb { R } ^ 2 : w _ 1 ^ 2 + w _ 2 ^ 2 - w _ { n + 1 } ^ 2 < 0 \\} . \\end{align*}"}
+{"id": "5609.png", "formula": "\\begin{align*} \\hat { a } = \\hat { a } - \\hat { v } + \\hat { v } = a ' - v ' + \\hat { v } \\leq a - v + \\hat { v } \\leq 3 ( a - v ) + 6 s . \\end{align*}"}
+{"id": "5896.png", "formula": "\\begin{align*} \\dfrac { M _ { 1 } ( \\mathcal { N C } ( G ) ) } { | v ( \\mathcal { N C } ( G ) ) | } = \\dfrac { ( p + 1 ) ( p n - n ) ( p ^ { 4 } n ^ { 2 } - 2 p ^ { 3 } n ^ { 2 } + p ^ { 2 } n ^ { 2 } ) } { p ^ { 2 } n - n } \\end{align*}"}
+{"id": "7268.png", "formula": "\\begin{align*} y Q _ n ( y ; x , t , s ) & = Q _ { n + 1 } ( y ; x , t , s ) + \\mathcal { A } _ { n } ( x , t , s ) Q _ n ( y ; x , t , s ) \\\\ & + \\mathcal { B } _ { n } ( x , t , s ) Q _ { n - 1 } ( y ; x , t , s ) , n \\ge 0 \\end{align*}"}
+{"id": "7421.png", "formula": "\\begin{align*} c ' _ 2 & = b _ 1 c _ 2 + ( a _ 2 - d _ 2 ) a _ 3 \\\\ c ' _ 3 & = b _ 1 c _ 3 + b _ 2 c _ 2 + ( a _ 2 - d _ 2 ) a _ 4 - a _ 3 d _ 3 . \\\\ \\end{align*}"}
+{"id": "1155.png", "formula": "\\begin{align*} \\widetilde s - \\widetilde J & = ( s + n \\widehat \\tau ) - \\left [ J _ \\tau + \\left ( n \\widehat \\tau \\wedge \\frac { d } { p } \\right ) \\right ] = s - J _ \\tau - n + \\left ( n \\widehat \\tau - \\frac { d } { p } \\right ) _ + , \\\\ \\left ( n \\widehat \\tau - \\frac { d } { p } \\right ) _ + & = \\left ( \\left [ n \\tau - \\frac { n } { p } + \\frac { d } { p } \\right ] _ + - \\frac { d } { p } \\right ) _ + = n \\left ( \\tau - \\frac { 1 } { p } \\right ) _ + , \\end{align*}"}
+{"id": "8524.png", "formula": "\\begin{align*} L ( \\tilde { J } ^ { ( p ) } _ { D } , s ) = L ( \\tilde { J } ^ { ( p ) } , \\chi _ D , s ) = \\prod _ { \\sigma } L ( f ^ { \\sigma } , \\chi _ D , s ) , \\end{align*}"}
+{"id": "1795.png", "formula": "\\begin{align*} \\Psi \\left ( \\zeta ( \\bar x _ i + ) , u ( t , \\bar x _ i + ) , \\zeta ( \\bar x _ i - ) , u ( t , \\bar x _ i - ) \\right ) = 0 \\mbox { f o r a . e . } t > 0 \\mbox { a n d } i = 1 , \\ldots , N \\end{align*}"}
+{"id": "226.png", "formula": "\\begin{align*} \\frac { d \\bar v } { d \\tau } = f f ' ( q ) \\ , 2 ( E - \\mathcal { V } ) - f ^ 2 \\ , \\mathcal { V } ' ( q ) = \\frac d { d q } \\left ( f ^ 2 ( E - \\mathcal { V } ) \\right ) , \\end{align*}"}
+{"id": "2600.png", "formula": "\\begin{align*} \\varphi ( r ) = \\dfrac { c } { ( r ^ 2 + \\alpha ^ 2 ) ^ { \\beta R } } . \\end{align*}"}
+{"id": "6652.png", "formula": "\\begin{align*} u ( z ) = \\int _ { \\Omega _ { 1 } } \\log | z - \\zeta | \\mathrm { d } \\mu ( \\zeta ) + g ( z ) , \\end{align*}"}
+{"id": "4938.png", "formula": "\\begin{align*} e ^ { S _ j } _ { \\pm } = \\pm \\frac { 1 } { 4 \\sqrt { \\beta } } [ S _ j ] + \\frac { 1 } { 8 \\beta } [ S _ j ] ^ 2 . \\end{align*}"}
+{"id": "3976.png", "formula": "\\begin{align*} a _ { i } = \\sum _ { j = 1 } ^ { n } \\alpha _ { j i } e _ { j } \\mbox { a n d } \\widetilde { a } _ { p } = \\sum _ { q = 1 } ^ { \\nu } \\widetilde { \\alpha } _ { q p } \\widetilde { e } _ { p } \\end{align*}"}
+{"id": "2045.png", "formula": "\\begin{align*} \\binom { n + d } { n - 1 } - \\sum _ { i = 1 } ^ e \\binom { n - 1 + i } { n - 1 } \\ , \\ , = \\ , \\ , \\binom { n + d } { n - 1 } - \\binom { n + e } { n } + 1 \\ , \\ , = \\ , \\ , W ( d , e ) . \\end{align*}"}
+{"id": "980.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\mathbb E _ x \\sup _ { t \\le \\tau _ D } | Y ^ { x , k } _ t - Y ^ { x } _ t | ^ { 1 / 2 } = 0 . \\end{align*}"}
+{"id": "7846.png", "formula": "\\begin{align*} - \\Delta ^ { \\perp \\psi } H ^ { \\psi } = - \\frac { p } { p + q } \\Delta ^ { \\perp } H _ 1 - \\frac { q } { p + q } \\bar \\Delta ^ { \\perp } H _ 2 + \\frac { p q } { p + q } \\left ( 2 B ^ j ( H _ 1 , H _ 2 ) - H _ 1 - H _ 2 \\right ) . \\end{align*}"}
+{"id": "4413.png", "formula": "\\begin{align*} \\inf _ { k \\in [ m ] _ 0 } \\left \\{ \\inf _ { x \\in \\mathcal { X } } \\left \\{ \\Gamma \\theta ^ k ( x ) + \\sum _ { i \\in [ m ] } \\max \\{ 0 , x _ i - \\overline { b } _ i , \\max \\{ 0 , x _ i - \\overline { b } _ i + \\Delta b _ i \\} - \\theta ^ k ( x ) \\} \\right \\} \\right \\} \\end{align*}"}
+{"id": "3553.png", "formula": "\\begin{align*} g = T ^ { - 1 } ( T g ) = T ^ { - 1 } ( g \\circ \\tau ) = g \\circ ( \\tau \\circ \\psi ) ( g \\in { H } ^ { \\infty } _ { 1 } ( \\mathbb { D } ) ) . \\end{align*}"}
+{"id": "2944.png", "formula": "\\begin{align*} \\big ( \\Delta + \\partial _ \\rho ^ 2 + \\frac { 1 - s } { \\rho } \\partial _ \\rho \\big ) \\big ( \\rho ^ s \\varphi _ { s , \\rho } \\big ) ( k ) = 0 . \\end{align*}"}
+{"id": "7919.png", "formula": "\\begin{align*} \\begin{aligned} \\mathrm { t r } ( d f _ { 1 } ) \\wedge \\ast \\boldsymbol { n } ( \\mu \\wedge d f _ { 2 } ) & = \\langle d f _ { 1 } , i _ { \\mathcal { N } } ( \\mu \\wedge d f _ { 2 } ) \\rangle _ { \\Lambda ^ { 1 } } v _ { \\partial \\Omega } \\\\ & = \\langle d f _ { 1 } , i _ { \\mathcal { N } } \\mu \\wedge d f _ { 2 } - \\mu \\wedge i _ { \\mathcal { N } } d f _ { 2 } \\rangle _ { \\Lambda ^ { 1 } } v _ { \\partial \\Omega } . \\end{aligned} \\end{align*}"}
+{"id": "2098.png", "formula": "\\begin{align*} - ( i + 1 - d x ) P _ { i , 1 } ^ s - ( d x - i ) P _ { i + 1 , 1 } ^ s = - \\alpha \\int _ 0 ^ { \\frac { \\lfloor s T \\rfloor } { T } } \\int _ 0 ^ 1 f ( u ) A ( x , y ) U ( u , y ) \\mathrm { d } y \\mathrm { d } u + o ( 1 ) . \\end{align*}"}
+{"id": "5000.png", "formula": "\\begin{align*} S _ { N - 1 , i } ( y _ j , q ^ { - 2 } y _ j , x _ 4 , \\dots , x _ N ) = 0 , j = 1 , \\dots , i - 1 , i + 1 , \\ldots , N ; \\end{align*}"}
+{"id": "1678.png", "formula": "\\begin{align*} q ( x ' _ 0 ) = - q \\big ( \\tilde \\phi _ 4 \\ , ( \\pi _ 4 ( \\mathbf { p } ) \\big ) . \\end{align*}"}
+{"id": "8024.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\limsup _ { N \\rightarrow + \\infty } \\sup _ { \\sigma \\in \\mathcal { T } , t \\leq \\delta } \\hat { P } ^ { N , F , G , H } _ { f _ 1 , f _ 2 , f _ 3 } \\left ( \\left | \\mu ^ N _ { t + \\sigma , k } ( f ) - \\mu ^ N _ { \\sigma , k } ( f ) \\right | > \\epsilon \\right ) = 0 \\end{align*}"}
+{"id": "1506.png", "formula": "\\begin{align*} \\underline { u } : = \\delta ^ { \\frac { 1 } { p } } z _ { \\delta } ^ { \\omega } - \\delta \\overline { u } : = \\delta ^ { - p } z ^ { \\overline { \\omega } } - \\delta \\end{align*}"}
+{"id": "5523.png", "formula": "\\begin{align*} y ^ { ( \\alpha ) } ( 0 ) = y ^ { ( \\beta ) } ( \\pi ) = 0 , \\end{align*}"}
+{"id": "4472.png", "formula": "\\begin{align*} M _ H ( Z _ 0 , \\mathfrak { a } , \\rho ) = \\frac { M ( Z _ 0 , \\mathfrak { a } , \\tilde \\rho ) } { \\pi \\int _ { 0 } ^ { + \\infty } c ( t ) e ^ { - t } d t } \\end{align*}"}
+{"id": "5279.png", "formula": "\\begin{align*} L _ { d } \\left ( x . y \\right ) = \\frac { ( x . y ) ^ { a - 1 } - ( x . y ) ^ { b - 1 } } { a - b } \\end{align*}"}
+{"id": "748.png", "formula": "\\begin{align*} \\big ( B _ F ( \\varphi ) \\big ) _ { i j } = { } ^ c \\nabla _ i \\varphi _ j - \\varphi _ i \\varphi _ j - \\frac { 1 } { n } ( \\Delta \\varphi - \\| g r a d \\varphi \\| ^ 2 ) g _ { i j } , \\end{align*}"}
+{"id": "1521.png", "formula": "\\begin{align*} D V = D ^ 2 F ( D u ) \\ , D ^ 2 u , \\end{align*}"}
+{"id": "631.png", "formula": "\\begin{align*} V _ 2 ( R _ A , R _ B ) & = V _ 2 ( L _ 1 ( A ) + L _ 2 ( A ) , L _ 1 ( B ) + L _ 2 ( B ) ) = V _ 2 ( L _ 1 ( A ) , L _ 2 ( B ) ) + V _ 2 ( L _ 2 ( A ) , L _ 1 ( B ) ) \\\\ & = \\frac { 1 } { 2 } \\left ( | \\pi _ 1 ( A ) | | \\pi _ 2 ( B ) | + | \\pi _ 2 ( A ) | | \\pi _ 1 ( B ) | \\right ) . \\end{align*}"}
+{"id": "3345.png", "formula": "\\begin{align*} \\left ( \\int _ { X / B } \\beta \\wedge \\gamma ^ n \\right ) ( v ) & = \\left ( \\int _ { X / B } \\beta ( v ) \\wedge \\gamma ( v ) ^ n \\right ) , \\\\ & = \\int _ { X / B } ( \\rho + \\Delta _ V \\langle \\mu , v \\rangle ) \\wedge ( \\omega + \\langle \\mu , v \\rangle ) ^ n . \\end{align*}"}
+{"id": "4831.png", "formula": "\\begin{align*} _ { V [ \\lambda ] } ( \\mu ) = \\sum _ { w \\in W ( { \\mathfrak h } , { \\mathfrak t } ) } ( - 1 ) ^ { \\ell ( w ) } { \\mathcal P } ( w ( \\lambda + \\rho ( { \\mathfrak h } ) ) - ( \\mu + \\rho ( { \\mathfrak h } ) ) ) . \\end{align*}"}
+{"id": "1539.png", "formula": "\\begin{align*} { \\rm C } _ H = \\{ F : \\R ^ N \\to \\R \\} \\end{align*}"}
+{"id": "6433.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 2 } ^ n \\left ( A _ i - \\delta _ 0 ^ { 1 / 2 } n ^ { 1 / 2 \\alpha _ 0 } | \\Delta _ i ^ n L - \\Delta _ { i - 1 } ^ n L | ^ { 1 / 2 } \\right ) = o _ P ( \\frac { 1 } { \\sqrt { n } } ) . \\end{align*}"}
+{"id": "3604.png", "formula": "\\begin{align*} \\alpha ^ 3 \\tau _ 0 \\tau _ 1 \\tau _ 2 f ( \\tau ^ 3 , w \\sigma \\sigma _ 1 \\sigma _ 2 ) - ( 1 + a ) \\alpha ^ 2 \\tau _ 0 \\tau _ 1 f ( \\tau ^ 2 , w \\sigma \\sigma _ 1 ) + ( a + b ) \\alpha \\tau _ 0 f ( \\tau , w \\sigma ) - b f = 0 , \\end{align*}"}
+{"id": "3.png", "formula": "\\begin{align*} q ( \\phi ; q _ 0 ) = \\bigl ( e ^ { J \\nabla H _ \\phi } q _ 0 \\bigr ) ( x = 0 ) . \\end{align*}"}
+{"id": "38.png", "formula": "\\begin{align*} \\langle \\omega , \\eta \\rangle _ { \\Omega _ c ^ j ( \\Z ^ n ) } = \\sum _ { \\mu \\in \\Z ^ n } \\sum _ { I \\in P ^ { j , n } _ + } \\omega _ { I } ( \\mu ) \\overline { \\eta _ { I } ( \\mu ) } \\ . \\end{align*}"}
+{"id": "3246.png", "formula": "\\begin{align*} & \\big ( S ^ { \\wedge , ( l ) } \\ : V \\ : S ^ { \\wedge , ( r ) } \\big ) ( x , y ) \\\\ & = \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { n ! } \\int _ 0 ^ 1 \\alpha ^ { l } \\ : ( 1 - \\alpha ) ^ r \\ : ( \\alpha - \\alpha ^ 2 ) ^ n \\ : ( \\Box ^ n V ) \\big | _ { \\alpha y + ( 1 - \\alpha ) x } \\ : d \\alpha \\ ; S ^ { \\wedge , ( n + l + r + 1 ) } ( x , y ) \\ : . \\end{align*}"}
+{"id": "198.png", "formula": "\\begin{align*} Z _ { z + } \\ = \\ \\{ z + \\} \\ \\cup \\ \\{ \\bar d _ k - \\} \\ \\cup \\ \\{ i + : i \\ge j - 1 \\} \\ \\cup \\ [ \\bigcup _ { i = 1 } ^ { j - 2 } \\ ( z + A _ i ) + ] . \\end{align*}"}
+{"id": "7000.png", "formula": "\\begin{align*} \\ ! & \\{ [ 0 \\ ! : \\ ! z _ 2 \\ ! : \\ ! \\cdots \\ ! : \\ ! z _ { n + 1 } ] : z _ 2 , \\ ! \\cdots \\ ! z _ { n + 1 } \\ ! \\in \\ ! \\mathbb { C } \\} \\setminus \\ ! \\{ [ 0 \\ ! : \\ ! 1 \\ ! : \\ ! z _ 3 \\ ! : \\ ! \\cdots \\ ! : \\ ! z _ { n + 1 } ] : z _ 3 , \\ ! \\cdots \\ ! , \\ ! z _ { n + 1 } \\ ! \\in \\ ! \\mathbb { C } \\} = \\\\ & \\{ [ 0 \\ ! : \\ ! 0 \\ ! : \\ ! z _ 3 \\ ! : \\ ! \\cdots \\ ! : \\ ! z _ { n + 1 } ] : z _ 3 , \\ ! \\cdots \\ ! , \\ ! z _ { n + 1 } \\ ! \\in \\ ! \\mathbb { C } \\} \\end{align*}"}
+{"id": "126.png", "formula": "\\begin{align*} A _ \\lambda \\vec { u } _ \\lambda = \\vec { f } , A _ \\infty \\vec { u } _ \\infty = \\vec { f } , A \\vec { u } _ 0 = \\vec { f } , \\end{align*}"}
+{"id": "7437.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ d b _ { i , j } ( x ) \\xi _ i \\xi _ j \\geq c _ 0 | \\xi | ^ 2 \\xi \\in \\R ^ d x \\in \\R ^ d . \\end{align*}"}
+{"id": "5588.png", "formula": "\\begin{align*} B _ { e f } = \\begin{cases} A _ { f _ 1 f _ 2 } A _ { f _ 3 f _ 2 } \\mathbf { 1 } \\{ e \\to f \\} & e , f \\in \\vec { E } _ 2 , \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "8537.png", "formula": "\\begin{align*} \\begin{array} { l c l } \\beta & = & 0 , \\\\ \\alpha \\log _ 1 ( \\pi _ 2 ) - \\gamma \\log _ 1 ( u ( K ) ) & = & 0 . \\end{array} \\end{align*}"}
+{"id": "7424.png", "formula": "\\begin{align*} G / H _ s = \\{ f _ 1 ^ { i _ 1 } \\cdots f _ k ^ { i _ k } g _ s ^ j H _ s \\ ; : \\ ; i _ 1 , . . . , i _ k , j \\in \\mathbb { Z } \\} . \\end{align*}"}
+{"id": "3746.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\lambda } \\mathrm { B } _ { z ^ { 2 } } \\left ( \\lambda , 0 \\right ) = 2 \\ln z \\ , \\mathrm { B } _ { z ^ { 2 } } \\left ( \\lambda , 0 \\right ) - \\frac { z ^ { 2 \\lambda } } { \\lambda ^ { 2 } } \\ , _ { 3 } F _ { 2 } \\left ( \\left . \\begin{array} { c } 1 , \\lambda , \\lambda \\\\ \\lambda + 1 , \\lambda + 1 \\end{array} \\right \\vert z ^ { 2 } \\right ) . \\end{align*}"}
+{"id": "1850.png", "formula": "\\begin{align*} \\mathcal { T } _ j : = \\mathcal { J } _ j \\oplus \\mathcal { K } _ j , \\end{align*}"}
+{"id": "5987.png", "formula": "\\begin{align*} \\nabla F ( x _ 0 ) = 0 \\end{align*}"}
+{"id": "4983.png", "formula": "\\begin{align*} S _ { N - 1 , i } ( \\lambda _ 2 , \\dots , \\lambda _ N ) = Z _ { N - 1 } ( \\lambda _ 2 , \\dots , \\lambda _ N ; \\{ \\nu \\} \\setminus \\nu _ i ) , \\end{align*}"}
+{"id": "4303.png", "formula": "\\begin{align*} ( y _ t , J _ t , K _ t ) : = ( y _ 0 + \\mathbf { y } ^ 1 _ { 0 , t } , J _ 0 + \\mathbf { J } ^ 1 _ { 0 , t } , K _ 0 + \\mathbf { K } ^ 1 _ { 0 , t } ) . \\end{align*}"}
+{"id": "1331.png", "formula": "\\begin{align*} \\overline { \\nabla } _ { \\dot { x } } ^ \\ast y & = \\pm \\frac { \\partial H } { \\partial x } - \\bigl \\langle \\overline { \\mu } , \\widetilde { R } ( \\dot { x } , \\cdot ) \\bigr \\rangle , \\\\ \\widetilde { \\nabla } _ { \\dot { x } } ^ \\ast \\overline { \\mu } & = \\operatorname { a d } ^ \\ast _ { \\overline { v } } \\overline { \\mu } . \\end{align*}"}
+{"id": "2366.png", "formula": "\\begin{align*} & ( w ( s _ { 2 } ) , \\varphi ( s _ { 2 } ) ) + \\int _ { s _ { 1 } } ^ { s _ { 2 } } [ ( \\nabla w , \\nabla \\varphi ) + ( w \\cdot \\nabla w + w \\cdot \\nabla u ^ { c , \\gamma } + u ^ { c , \\gamma } \\cdot \\nabla w , \\varphi ) ] \\\\ & = ( w ( s _ { 1 } ) , \\varphi ( s _ { 1 } ) ) + \\int ^ { s _ { 2 } } _ { s _ { 1 } } ( w , \\partial _ { t } \\varphi ) d t \\end{align*}"}
+{"id": "7227.png", "formula": "\\begin{align*} a \\partial _ v p ( V _ { \\min } ) \\phi ( V _ { \\min } ) = 0 . \\end{align*}"}
+{"id": "7668.png", "formula": "\\begin{align*} r _ L ( u , v ; t ) = & \\left ( G _ L ( u , v ; t - i \\eta ) - G _ L ( u , v ; t + i \\eta ) \\right ) G _ L ( v , u ; t - i \\eta ) \\\\ & + G _ L ( u , v ; t + i \\eta ) \\left ( G _ L ( v , u ; t - i \\eta ) - G _ L ( v , u ; t + i \\eta ) \\right ) . \\end{align*}"}
+{"id": "1714.png", "formula": "\\begin{align*} M _ 2 : = \\left \\{ z \\in \\mathbb { C } ^ { n _ 2 + 1 } \\ , \\left | \\ , \\Re ( z _ 0 ) = \\Phi _ 2 \\left ( z _ 1 , \\ldots , z _ { { n _ 2 } } , \\overline { z _ 1 } , \\ldots , \\overline { z _ { { n _ 2 } } } \\right ) \\right . \\right \\} \\end{align*}"}
+{"id": "8925.png", "formula": "\\begin{align*} \\dim ( \\mathcal { M } ( L ) ) \\leq \\frac { 1 } { 2 } [ ( n + m ) ( n - m - 1 ) - \\sum _ { i = 2 } ^ { \\min \\{ c , n - m \\} } ( n - m - i ) . \\end{align*}"}
+{"id": "7571.png", "formula": "\\begin{align*} d \\Psi ( u ) [ \\varphi ] & = t _ { u } ^ { 2 } \\int _ { \\mathbb { R } ^ { N } } \\Delta u \\cdot \\Delta \\varphi d x - t _ { u } ^ { \\frac { N ( q - 2 ) } { 4 } } \\mu \\int _ { \\mathbb { R } ^ { N } } | u | ^ { q - 2 } u \\cdot \\varphi d x - t _ { u } ^ { \\frac { N ( p - 2 ) } { 4 } } \\int _ { \\mathbb { R } ^ { N } } | u | ^ { p - 2 } u \\cdot \\varphi d x \\\\ & = d E _ { p , q } ( u _ { s _ { u } } ) [ \\varphi _ { s _ { u } } ] \\\\ \\end{align*}"}
+{"id": "5121.png", "formula": "\\begin{align*} L o g _ { a b } ( x ) \\equiv L o g _ { d } ( x ) = \\frac { x ^ { a - 1 } - x ^ { b - 1 } } { a - b } \\end{align*}"}
+{"id": "1634.png", "formula": "\\begin{align*} \\varphi _ { \\lambda } \\left ( t \\right ) = \\exp \\left ( \\left ( T - t + c \\right ) ^ { \\lambda } \\right ) , t \\in \\left ( 0 , T \\right ) . \\end{align*}"}
+{"id": "5233.png", "formula": "\\begin{align*} D A H ( p \\| q ) = M A - M H \\end{align*}"}
+{"id": "2379.png", "formula": "\\begin{align*} \\tilde { w } _ l = \\sum _ { k = 0 } ^ N R _ { l , k } \\tilde { v } _ k \\ \\ \\ \\ \\ \\ \\tilde { v } _ l = \\Gamma ^ { - 1 } \\sum _ { k = 0 } ^ N \\tilde { R } _ { k , l } \\tilde { w } _ k \\ , \\end{align*}"}
+{"id": "3913.png", "formula": "\\begin{align*} ( \\cos z \\pm i \\sin z ) ^ n = ( \\cos n z \\pm i \\sin n z ) . \\end{align*}"}
+{"id": "6295.png", "formula": "\\begin{align*} T = \\tilde { O } \\left ( \\left [ \\frac { 2 ^ \\frac { r ^ 2 + 1 } { r } \\sigma _ q \\beta } { \\mu _ r ^ { 1 / r } } \\cdot \\frac { 1 } { \\varepsilon ^ { \\frac { ( r - 1 ) } { r } } } \\right ] ^ \\frac { 1 + \\kappa } { \\kappa } \\right ) , T _ k = \\tilde { O } \\left ( \\left [ \\frac { \\sigma _ q \\beta 2 ^ { ( 1 + r ) } } { \\mu _ r R _ 0 ^ { r - 1 } } 2 ^ { k ( r - 1 ) } \\right ] ^ \\frac { 1 + \\kappa } { \\kappa } \\right ) . \\end{align*}"}
+{"id": "3053.png", "formula": "\\begin{align*} \\mathcal { V } = - A ^ T ( B ^ { - 1 } ) ^ 2 A = - ( B ^ { - 1 } A ) ^ T ( B ^ { - 1 } A ) = - ( A ( x = 0 ) ) ^ T ( A ( x = 0 ) \\ , . \\end{align*}"}
+{"id": "2414.png", "formula": "\\begin{align*} v _ 2 \\left ( \\int _ { \\mathbb { Z } _ 2 } f ^ { ( s ) } ( t ) \\mathrm { d } t \\right ) = ( s + 2 ) n - ( 2 s + 3 ) m + s + v _ 2 ( ( s + 2 ) ! ) - 1 . \\end{align*}"}
+{"id": "7164.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l l } \\partial _ t f - \\partial _ x ^ 2 f & = f - f ^ 3 / 3 - g + I ( x , t ) & , \\forall x \\in \\R , t \\geq 0 , \\\\ \\partial _ t g - \\rho \\partial _ x ^ 2 g & = \\varepsilon ( f - \\gamma g + \\beta ) & , \\forall x \\in \\R , t \\geq 0 , \\\\ f ( x , 0 ) = f _ 0 ( x ) , & g ( x , 0 ) = g _ 0 ( x ) & , \\forall x \\in \\R , \\end{array} \\right . \\end{align*}"}
+{"id": "6191.png", "formula": "\\begin{align*} \\begin{aligned} & ( Q ^ k ) ^ T + Q ^ k = \\begin{pmatrix} \\beta ^ k ( J ^ T + J ) & - \\widetilde { I } ^ T \\\\ - \\widetilde { I } & \\frac { 2 } { \\beta ^ k } I _ l \\end{pmatrix} \\succeq \\begin{pmatrix} \\beta ^ k ( J ^ T + J ) & - \\widetilde { I } ^ T \\\\ - \\widetilde { I } & \\frac { 1 } { \\beta ^ k } I _ l \\end{pmatrix} = \\frac { 1 } { \\gamma ^ 2 } ( N ^ k ) ^ T N ^ k , \\\\ & ( M ^ k ) ^ T H ^ k M ^ k = \\frac { 1 } { \\gamma } ( N ^ k ) ^ T N ^ k . \\end{aligned} \\end{align*}"}
+{"id": "222.png", "formula": "\\begin{align*} \\bar { \\Gamma } ( \\bar x , \\bar v ) = { \\displaystyle \\sum _ { j = 1 } ^ n \\bar { v } ^ { j } \\frac { \\partial } { \\partial \\bar { x } ^ { j } } } + { \\displaystyle \\sum _ { i = 1 } ^ n } \\left ( { \\displaystyle \\sum _ { j = 1 } ^ n } \\frac { 1 } { f } \\bar { v } ^ { j } \\frac { \\partial f } { \\partial \\bar x ^ { j } } \\bar { v } ^ { i } + f ^ { 2 } X ^ { i } \\left ( \\bar x , \\dfrac 1 f \\ , \\bar v \\right ) \\right ) \\frac { \\partial } { \\partial \\bar { v } ^ { i } } , \\end{align*}"}
+{"id": "5864.png", "formula": "\\begin{align*} \\dfrac { M _ { 1 } ( \\mathcal { N C } ( D _ { 2 m } ) ) } { | v ( \\mathcal { N C } ( D _ { 2 m } ) ) | } = \\dfrac { 5 m ^ { 3 } - 1 8 m ^ { 2 } + 1 6 m } { 2 m - 2 } \\end{align*}"}
+{"id": "3054.png", "formula": "\\begin{align*} \\begin{array} { l l l } H & = 4 p _ q ^ T p _ r = ( p _ q ^ T , p _ r ^ T ) ( 2 \\mathcal { I } ) \\left ( \\begin{array} { c } p _ q \\\\ p _ r \\end{array} \\right ) = ( p _ x ^ T , p _ s ^ T ) J ^ { - 1 } ( 2 \\mathcal { I } ) ( J ^ { - 1 } ) ^ T \\left ( \\begin{array} { c } p _ x \\\\ p _ s \\end{array} \\right ) \\\\ & = p _ s ^ T p _ s + p _ x ^ T \\mathcal { V } ^ { - 1 } p _ x \\ , . \\end{array} \\end{align*}"}
+{"id": "8032.png", "formula": "\\begin{align*} J _ { i n i } ( W _ 0 ) = \\frac { 1 } { 2 } \\int _ { \\mathbb { T } } \\sum _ { k = 1 } ^ 3 \\frac { ( h ^ W _ k ( u ) ) ^ 2 } { \\rho _ { k - 1 } ( u ) } + \\frac { ( \\sum _ { k = 1 } ^ 3 h ^ W _ k ( u ) ) ^ 2 } { 1 - \\sum _ { k = 1 } ^ 3 \\rho _ { k - 1 } ( u ) } d u . \\end{align*}"}
+{"id": "2016.png", "formula": "\\begin{align*} \\Lambda = \\eta _ 1 ^ { b _ 1 } \\cdots \\eta _ s ^ { b _ s } - 1 . \\end{align*}"}
+{"id": "7339.png", "formula": "\\begin{align*} w ' = w \\odot _ { \\phi } ( v , \\sigma ) . \\end{align*}"}
+{"id": "1496.png", "formula": "\\begin{align*} \\alpha \\circ P _ { d } ( L _ { m } ) & = \\alpha ( P _ { d } ( L _ { m } ) ) = \\alpha ( f _ 1 ( m + d ) L _ { m + d } + g _ 1 ( m + d ) W _ { m + d } ) \\\\ & = f _ 1 ( m + d ) ( q ^ { m + d } + q ^ { - ( m + d ) } ) L _ { m + d } + g _ 1 ( m + d ) ( q ^ { m + d } + q ^ { - ( m + d ) } ) W _ { m + d } , \\\\ P _ { d } \\circ \\alpha ( L _ { m } ) & = P _ { d } ( \\alpha ( L _ { m } ) ) = P _ { d } ( ( q ^ { m } + q ^ { - m } ) L _ { m } ) \\\\ & = f _ 1 ( m + d ) ( q ^ { m } + q ^ { - m } ) L _ { m } + g _ 1 ( m + d ) ( q ^ { m } + q ^ { - m } ) W _ { m } . \\end{align*}"}
+{"id": "7153.png", "formula": "\\begin{align*} \\sum _ { \\lambda , \\lambda ' } \\varepsilon _ \\mu ( k , \\lambda ) \\varepsilon _ \\nu ( k , \\lambda ' ) \\eta _ { \\lambda \\lambda ' } = \\eta _ { \\mu \\nu } \\ , \\ , . \\end{align*}"}
+{"id": "120.png", "formula": "\\begin{align*} M _ { \\lambda } = \\frac { \\lambda } { 1 + \\lambda } P A ^ { - 1 } + \\frac { 1 } { \\lambda + 1 } A ^ { - 1 } , \\end{align*}"}
+{"id": "2781.png", "formula": "\\begin{align*} \\delta Q ^ { a } ( t _ { 1 } ) = \\delta Q ^ { a } ( t _ { 2 } ) = 0 . \\end{align*}"}
+{"id": "4501.png", "formula": "\\begin{align*} a ( \\lambda ) : = \\prod _ { k \\geqslant 0 } \\bigg ( \\frac { X ( k ) } { \\sqrt { k } } \\bigg ) ^ { m _ k } \\frac { 1 } { m _ k ! } , \\end{align*}"}
+{"id": "4268.png", "formula": "\\begin{align*} \\div _ f h = e ^ { f } \\div ( e ^ { - f } h ) = \\div h - h ( \\nabla f , \\cdot ) , \\end{align*}"}
+{"id": "4076.png", "formula": "\\begin{align*} e ^ { i z _ 0 s } \\tilde { E } _ i ( - i z _ 0 s ) = e ^ { i z _ 0 s } h ( | z _ 0 | s ) \\log | z _ 0 s | + ( 1 - h ( | z _ 0 | s ) ) b ( \\pm i z _ 0 s ) . \\end{align*}"}
+{"id": "488.png", "formula": "\\begin{align*} f ( x ) = \\begin{cases} 0 , & x = 0 , \\\\ \\omega ^ 5 x ^ 9 , & x \\in C _ 0 , \\\\ x ^ 3 , & x \\in C _ 1 , \\\\ x ^ { 1 7 } , & x \\in C _ 2 , \\\\ \\omega ^ 3 x ^ { 3 4 } , & x \\in C _ 3 , \\\\ \\omega ^ 4 x ^ 9 , & x \\in C _ 4 , \\end{cases} \\end{align*}"}
+{"id": "1690.png", "formula": "\\begin{align*} x _ 0 = f ( x _ 1 ) + \\sum _ { 0 < j < j + k \\leq n } x _ j x _ k x _ n ^ { n - j - k } \\end{align*}"}
+{"id": "4792.png", "formula": "\\begin{align*} ( u _ \\star ) _ { F , \\S } ( x ) : = u _ \\ast ( \\kappa _ { F , \\S } ( F ^ o ( x ) ) ^ n ) , \\end{align*}"}
+{"id": "4120.png", "formula": "\\begin{align*} & a ^ { ( r ) } \\theta _ i - \\theta _ i = \\theta _ i o ( r ^ { J } ) = o ( r ^ { J } \\theta _ i ) , \\\\ & ( a ^ { ( r ) } ) ^ 3 \\sigma ^ 2 _ { ( r ) } \\theta ^ 2 _ i - c ^ 2 _ { e , i } \\theta ^ 2 _ i = \\theta ^ 2 _ i o ( r ^ { J - 1 } ) = o ( r ^ { J - 1 } \\theta ^ 2 _ i ) = o ( \\theta _ i ^ 2 ) . \\end{align*}"}
+{"id": "4461.png", "formula": "\\begin{align*} M _ { H , 1 } ( Z _ 0 , J , \\rho _ 1 \\lambda _ 2 ) & = \\| f _ 0 \\| ^ 2 _ { \\partial D _ 1 \\times M _ 1 , \\rho _ 1 \\lambda _ 2 } \\\\ & = \\| F _ 2 \\| ^ 2 _ { \\partial D _ 1 \\times M _ 1 , \\rho _ 1 \\lambda _ 2 } + \\| f _ 1 f _ 2 \\| ^ 2 _ { \\partial D _ 1 \\times M _ 1 , \\rho _ 1 \\lambda _ 2 } \\\\ & \\ge M _ { \\partial D _ 1 } \\times M _ { M _ 1 } . \\end{align*}"}
+{"id": "87.png", "formula": "\\begin{align*} \\sup _ \\theta X _ 1 ( \\theta ) & = \\max \\{ X _ 1 ( 0 ) , X _ 1 ( 1 ) \\} \\\\ & = \\max \\Big \\{ 0 , \\big ( \\sqrt { p - 1 } - \\sqrt { 1 - \\gamma } \\big ) ^ 2 \\Big \\} \\\\ & = \\big ( \\sqrt { p - 1 } - \\sqrt { 1 - \\gamma } \\big ) ^ 2 . \\end{align*}"}
+{"id": "1665.png", "formula": "\\begin{align*} \\lambda \\left ( \\delta \\right ) = \\ln \\left [ \\delta ^ { - \\left ( 3 \\left ( T + c \\right ) ^ { - 1 } \\right ) } \\right ] . \\end{align*}"}
+{"id": "8926.png", "formula": "\\begin{align*} [ x , [ y , z ] ] = [ [ x , y ] , z ] + ( - 1 ) ^ { | z | | x + y | } [ [ z , x ] , y ] . \\end{align*}"}
+{"id": "7452.png", "formula": "\\begin{align*} \\Xi _ 1 \\leq t ^ j \\left ( \\frac { \\Lambda K ^ 2 d ^ 2 } { \\lambda } \\right ) ^ { j - 1 } \\frac { \\Lambda K ^ 2 d ^ 2 } { \\lambda } \\cdot m = C \\left ( K , \\Lambda , \\lambda ^ { - 1 } , j , d , t \\right ) \\cdot m . \\end{align*}"}
+{"id": "8670.png", "formula": "\\begin{align*} \\omega _ 0 = J ^ t ( d \\rho ) , \\end{align*}"}
+{"id": "5206.png", "formula": "\\begin{align*} U _ { j } = - \\frac { Z _ { A } } { \\alpha } q ^ { \\alpha + \\beta - 2 } _ { j } \\ ; \\ ; ; \\ ; \\ ; V _ { j } = - \\frac { Z _ { A } } { \\alpha } q ^ { \\alpha + \\beta - 2 } _ { j } \\end{align*}"}
+{"id": "1438.png", "formula": "\\begin{align*} f ^ * t _ 1 = x _ 1 ^ { a _ 1 } \\cdots x _ l ^ { a _ l } \\end{align*}"}
+{"id": "8817.png", "formula": "\\begin{align*} \\begin{cases} s > 0 , & \\\\ s = 0 & 0 < q \\le \\min ( p , 2 ) . \\end{cases} \\end{align*}"}
+{"id": "1663.png", "formula": "\\begin{align*} v \\left ( x , 0 \\right ) = v _ { 0 } \\left ( x \\right ) , p \\left ( x , 0 \\right ) = p _ { 0 } \\left ( x \\right ) . \\end{align*}"}
+{"id": "2090.png", "formula": "\\begin{align*} C _ { d , T } ( t ) = \\big ( \\frac { t } { T } \\big ) ^ 2 O ( \\gamma ^ 4 T ^ { - 2 } + \\gamma ^ 4 d ^ { - 4 } + { \\gamma ^ 4 \\sigma ^ 4 d ^ { - 4 } T ^ { - 2 } } ) . \\end{align*}"}
+{"id": "5369.png", "formula": "\\begin{align*} \\bar { b } ^ u = \\lim _ { T \\to \\infty } \\ , \\frac { 1 } { T } \\ , E _ i ^ u \\left [ \\sum _ { t = 0 } ^ T \\theta _ { X ( t ) } ^ 1 \\ , a ( t ) \\right ] = \\lim _ { \\beta \\nearrow 1 } \\ , ( 1 - \\beta ) \\ , b _ i ^ u ( \\beta ) , \\end{align*}"}
+{"id": "4491.png", "formula": "\\begin{align*} M _ H ( Z _ 0 , J , \\rho ) = \\| F _ 0 \\| _ { \\partial M , \\rho } ^ 2 \\ge \\sum _ { 1 \\le j \\le n } M _ { H , j } ( Z _ 0 , J , \\rho ) . \\end{align*}"}
+{"id": "2196.png", "formula": "\\begin{align*} \\omega _ 1 = \\omega _ \\varphi / r . \\end{align*}"}
+{"id": "5053.png", "formula": "\\begin{align*} \\lim _ { K _ S \\rightarrow \\infty } d _ W ( \\mathbb { P } , \\mathbb { \\hat { Q } } _ { K _ S } ) = 0 , \\ \\mathbb { P } ^ \\infty . \\end{align*}"}
+{"id": "3204.png", "formula": "\\begin{align*} E _ { \\alpha , m , l } ( z ) = \\sum \\limits _ { k = 0 } ^ \\infty { c _ k } { z ^ k } , \\end{align*}"}
+{"id": "1626.png", "formula": "\\begin{align*} \\Psi ^ { r + 1 } _ { p , q } & \\begin{cases} q & 1 \\leq p < k - r \\\\ q \\leq M + k - p & k - r \\leq p \\leq k \\end{cases} \\\\ & \\begin{cases} q & r + 1 \\leq p < k \\\\ q \\leq M + 1 & p = k . \\end{cases} \\end{align*}"}
+{"id": "3072.png", "formula": "\\begin{align*} \\omega _ { i j } = v _ { i } F _ i d F _ j - v _ { j } F _ j d F _ i . \\end{align*}"}
+{"id": "7068.png", "formula": "\\begin{align*} u ( t ) - f + \\int _ 0 ^ t b \\cdot \\nabla u d s + \\sigma \\int _ 0 ^ t \\nabla u \\circ d W _ s = 0 , t \\in [ 0 , T ] , \\end{align*}"}
+{"id": "3676.png", "formula": "\\begin{align*} \\left \\{ 0 ^ { \\left ( \\binom { d + 1 } { 2 } \\right ) } , 1 ^ { \\left ( d n - \\binom { d + 1 } { 2 } - d \\right ) } , ( n - d / 2 ) ^ { ( d - 1 ) } , n ^ { ( 1 ) } \\right \\} \\end{align*}"}
+{"id": "4631.png", "formula": "\\begin{align*} V ( x ) : = \\begin{cases} - b _ V \\left ( f ( x ) - x _ d \\right ) ^ { \\frac { 2 } { d } ( - 1 + \\epsilon ) } & \\mathrm { \\ f o r \\ } x ' \\in Q ^ { ( d - 1 ) } , \\ , 0 \\leq x _ d < f ( x ' ) \\\\ 0 & \\ \\mathrm { e l s e } . \\end{cases} \\end{align*}"}
+{"id": "5812.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } | z | ^ { - p } f ( z ) = \\hat f _ p ( \\theta ^ * ) \\ge 0 . \\end{align*}"}
+{"id": "65.png", "formula": "\\begin{align*} f ( 0 ) = f ' ( 0 ) = f '' ( 0 ) = 0 , \\ f '' ( r ) > 0 r > 0 , \\end{align*}"}
+{"id": "6451.png", "formula": "\\begin{align*} \\frac { \\bar \\nabla } { \\partial t } \\bar \\nabla _ { X } \\tau ( \\phi _ t ) = R ^ N ( d \\phi _ t ( \\partial _ t ) , d \\phi _ t ( X ) ) \\tau ( \\phi _ t ) + \\bar \\nabla _ { X } \\frac { \\bar \\nabla } { \\partial t } \\tau ( \\phi _ t ) , \\end{align*}"}
+{"id": "7380.png", "formula": "\\begin{align*} \\begin{array} { l l } C _ 0 : = \\max & \\{ 3 ^ { p + q } A C M ^ { p + q - 1 } , 3 ^ r B C M ^ { r - 1 } , \\\\ & \\quad 2 ^ { p + q + 3 } 3 ^ { p + q - 1 } ( p + q ) A C M ^ { p + q - 1 } , 2 ^ { r + 3 } 3 ^ { r - 1 } r B C M ^ { r - 1 } , \\\\ & 3 ^ { p + q } ( p + q ) A C M ^ { p + q - 1 } , 3 ^ r r B C M ^ { r - 1 } , \\\\ & \\quad 2 \\cdot 3 ^ { p + q - 1 } p A C M ^ { p + q - 1 } \\} . \\end{array} \\end{align*}"}
+{"id": "6734.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty { \\frac { { \\zeta ( 2 k ) } } { { k ( k + 1 ) } } } & = - \\frac { 1 } { 2 } + \\ln ( 2 \\pi ) , \\\\ \\sum _ { k = 1 } ^ \\infty { \\frac { \\zeta ( 2 k ) } { k ( k + 2 ) } } & = - \\frac { 1 } { 8 } + \\frac { \\ln ( 2 \\pi ) } { 2 } + \\frac { 3 \\ , \\zeta ( 3 ) } { 2 \\pi ^ 2 } , \\end{align*}"}
+{"id": "2647.png", "formula": "\\begin{align*} \\mathcal { F } ( P _ { \\leq M } f ) ( \\xi ) : = \\eta ^ d \\left ( \\frac { \\xi } { M } \\right ) \\mathcal { F } ( f ) ( \\xi ) , \\xi \\in \\mathbb { Z } ^ d , \\end{align*}"}
+{"id": "2596.png", "formula": "\\begin{align*} \\dfrac { \\partial \\varphi } { \\partial \\nu } + \\beta \\varphi = 0 \\ ; \\ ; \\partial B _ R \\end{align*}"}
+{"id": "2035.png", "formula": "\\begin{align*} \\begin{matrix} { \\rm d i m } \\bigl ( \\mathcal { T } ^ n _ { d , e , m } \\bigr ) \\ , \\ , \\leq \\ , \\ , { \\rm m i n } \\bigl \\{ \\ , \\binom { d + n } { n } + \\binom { e + n } { n } - 2 \\ , , \\ , \\binom { m + n } { n } - 1 \\ , \\bigr \\} . \\end{matrix} \\end{align*}"}
+{"id": "708.png", "formula": "\\begin{align*} f ( x ) = \\Phi ^ 3 ( x ) + 9 \\delta \\Phi ^ 2 ( x ) + 2 7 \\Phi ( x ) + 2 7 \\delta - m \\end{align*}"}
+{"id": "40.png", "formula": "\\begin{align*} ( { U } _ j f ) ( \\mu ) : = \\sum _ { I \\in P ^ { j , i } _ + } f ( \\mu ; \\delta _ { I ( 1 ) } , \\hdots , \\delta _ { I ( j ) } ) \\ , d x ^ I \\ . \\end{align*}"}
+{"id": "5481.png", "formula": "\\begin{align*} S _ N ( x ) = \\dfrac { \\sin \\frac { N x } { 2 } } { \\sin \\frac { x } { 2 } } . \\end{align*}"}
+{"id": "3271.png", "formula": "\\begin{align*} ( q ^ r ; q ^ d ) _ k ^ r ( q ^ { r - d } ; q ^ d ) _ k = ( 1 - q ^ { r - d } ) ( 1 - q ^ r ) ^ { r + 1 } ( q ^ { d + r } ; q ^ d ) _ { k - 2 } ^ { r + 1 } ( 1 - q ^ { d k - d + r } ) ^ r . \\end{align*}"}
+{"id": "1613.png", "formula": "\\begin{align*} \\b Q _ { r ( p - 1 ) } \\b Q _ { s ( p - 1 ) } ( x ) = - q _ { g , n , s } \\sum _ j t _ { r , s , j } \\ , \\b Q _ { ( r + p s - p j ) ( p - 1 ) } \\b Q _ { j ( p - 1 ) } ( x ) ) , \\end{align*}"}
+{"id": "3926.png", "formula": "\\begin{align*} \\mathcal B _ { N \\times N } : = \\operatorname { c o l } \\left \\{ \\mathcal B _ { j } ^ { \\top } \\right \\} _ { j = 1 } ^ { N } , \\end{align*}"}
+{"id": "2020.png", "formula": "\\begin{align*} h ( \\eta _ 1 ) & = h \\left ( \\dfrac { ( \\rho - 1 ) a } { d _ 1 } \\right ) \\leqslant h ( \\rho - 1 ) + h ( a ) + h ( d _ 1 ) \\\\ & \\leqslant 2 \\log \\rho + \\dfrac { 1 } { 3 } \\log 3 1 \\leqslant 4 \\log \\rho , \\end{align*}"}
+{"id": "3703.png", "formula": "\\begin{align*} \\pi ( \\gamma ) = \\bigg ( \\gamma ( 0 ) , \\gamma ( \\frac { 1 } { n - 1 } ) , \\dots , \\gamma ( \\frac { n - 1 } { n - 1 } ) \\bigg ) . \\end{align*}"}
+{"id": "5497.png", "formula": "\\begin{align*} y _ l = \\begin{cases} x _ l & l \\geq j ; \\\\ \\sum _ { n = 0 } ^ { \\infty } q _ j ^ { \\frac { n ( n + 1 ) } { 2 } } ( q _ j - 1 ) ^ { - n } d _ { j , n } \\circ \\sigma _ j ^ { - n } ( x _ l ) x _ j ^ { - n } & l < j . \\end{cases} \\end{align*}"}
+{"id": "1526.png", "formula": "\\begin{align*} C = C ( K ) , \\eta = \\eta _ { K ^ { 1 / ( N - 1 ) } } . \\end{align*}"}
+{"id": "7171.png", "formula": "\\begin{align*} H ( ( V , W ) , x , t ) = \\left ( \\begin{array} { c } ( 1 - v _ 0 ^ 2 ) V - ( \\frac { A ( x ) ^ 2 } { 2 } + \\frac { B ( x ) ^ 2 } { 2 } + A ( x ) B ( x ) \\cos ( \\eta t ) ) ( V + v _ 0 ) - v _ 0 V ^ 2 - \\frac { V ^ 3 } { 3 } - W \\\\ \\varepsilon ( V - \\gamma W ) \\end{array} \\right ) , \\end{align*}"}
+{"id": "7310.png", "formula": "\\begin{align*} K _ { X _ { m } , \\chi _ { \\beta } } ( x , y ; t ) = e ^ { - 2 \\pi i \\beta \\frac { \\ell - ( x - y ) } { m } } \\sum _ { j = - \\infty } ^ { \\infty } e ^ { - 2 \\pi i \\beta j } e ^ { - t } I _ { \\ell + j m } ( t ) . \\end{align*}"}
+{"id": "4671.png", "formula": "\\begin{align*} \\P \\left ( \\| \\sum _ { n = 1 } ^ N X _ n \\| \\geq t \\right ) \\leq K \\left ( \\frac { e \\mu _ { m a x } } { t } \\right ) ^ { \\frac { t } { \\eta } } . \\end{align*}"}
+{"id": "4322.png", "formula": "\\begin{gather*} \\mathcal { X } = \\left \\{ x \\in \\{ 0 , 1 \\} ^ { [ n ] ^ 2 } : \\sum _ { i \\in [ n ] } x _ { i , r } = 1 \\ \\forall r \\in [ n ] , \\sum _ { r \\in [ n ] } x _ { i , r } = 1 \\ \\forall i \\in [ n ] \\right \\} . \\end{gather*}"}
+{"id": "3932.png", "formula": "\\begin{align*} \\dot z ^ N ( t ) = A z ^ N ( t ) + F ^ N [ z ( t ) ] + \\tilde B u ( t ) , \\end{align*}"}
+{"id": "2205.png", "formula": "\\begin{align*} 3 \\intop _ \\Omega \\bigg | { 1 \\over r } \\psi _ { 1 , r } \\bigg | ^ 2 d x = - \\intop _ \\Omega \\psi _ { 1 , r r } { 1 \\over r } \\psi _ { 1 , r } d x - \\intop _ \\Omega \\psi _ { 1 , z z } { 1 \\over r } \\psi _ { 1 , r } d x - \\intop _ \\Omega \\omega _ 1 { 1 \\over r } \\psi _ { 1 , r } d x . \\end{align*}"}
+{"id": "9165.png", "formula": "\\begin{align*} G _ { \\beta , \\beta ' } = ( w _ { \\beta , 1 } - v ^ { - 4 } w _ { \\beta ' , 1 } ) \\cdot G _ { \\beta ' , \\beta ' } , \\end{align*}"}
+{"id": "3788.png", "formula": "\\begin{align*} \\mathbb { P } \\left \\{ \\mathcal { M } _ i ^ { 1 , j } \\right \\} \\leq c _ 1 \\sum _ { t = 0 } ^ { T - 1 } \\exp \\left ( - c _ 2 N _ i n _ x \\left ( \\frac { \\alpha \\rho ^ { ( i ) } \\| \\Sigma ^ { ( i ) } _ t \\| } { \\rho ^ { ( i ) } \\| \\Sigma ^ { ( i ) } _ t \\| + \\sqrt { n _ x } } \\right ) ^ 2 \\right ) . \\end{align*}"}
+{"id": "5475.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ N \\frac { 1 } { | z _ 0 - z _ j | ^ 2 } \\ll N ( N + | X _ 1 | ) . \\end{align*}"}
+{"id": "5645.png", "formula": "\\begin{align*} a & = \\left ( 1 , x _ 1 , 0 , \\tfrac { 3 5 1 } { 8 } - x _ 1 , \\tfrac { 3 5 1 } { 8 } , x _ 1 - \\tfrac { 1 1 7 } { 8 } , \\tfrac { 3 5 1 } { 4 } , \\tfrac { 3 5 1 } { 4 } , \\tfrac { 3 5 1 } { 4 } , \\tfrac { 1 1 7 } { 8 } - x _ 1 \\right ) , \\\\ a Q & = \\left ( 3 5 2 , 0 , 3 7 4 4 , 0 , 0 , 3 2 ( 4 x _ 1 - \\tfrac { 1 1 7 } { 2 } ) , 0 , 0 , 0 , 3 2 ( \\tfrac { 1 1 7 } { 2 } - 4 x _ 1 ) \\right ) . \\end{align*}"}
+{"id": "1003.png", "formula": "\\begin{align*} \\bigcup _ { n \\ge 1 } U _ n = V \\quad \\end{align*}"}
+{"id": "9094.png", "formula": "\\begin{align*} \\hat F _ n ( \\tau ) = \\sum _ { i = 1 } ^ n \\mathbf 1 _ { \\{ \\hat k _ i < \\tilde \\Theta _ i \\tau \\} } , \\forall \\tau = 0 , \\ldots , n \\end{align*}"}
+{"id": "7730.png", "formula": "\\begin{align*} \\Sigma _ { k = - \\ell } ^ { \\ell } \\hat V _ { k } u _ { n - k } + \\varepsilon W ( \\theta + n \\alpha ) u _ { n } = E u _ { n } , n \\in \\mathbb { Z } . \\end{align*}"}
+{"id": "8093.png", "formula": "\\begin{align*} \\norm { f } _ { k , B , s } = \\sum _ { \\ell = 0 } ^ k \\frac { 1 } { B ^ \\ell } e _ { k , \\ell } ( \\psi f ) \\leq \\norm { \\psi } _ { C ^ { 2 k } } \\sum _ { \\ell = 0 } ^ k \\frac { 1 } { B ^ \\ell } \\sum _ { i = 0 } ^ \\ell e _ { k , i , s } ( f ) \\leq \\norm { \\psi } _ { C ^ { 2 k } } \\sum _ { \\ell = 0 } ^ k \\frac { k - \\ell } { B } \\frac { e _ { k , \\ell , s } ( f ) } { B ^ \\ell } . \\end{align*}"}
+{"id": "8343.png", "formula": "\\begin{align*} \\left [ d , \\varphi \\right ] ( x \\bullet y ) = ( \\left [ d , \\varphi \\right ] ( x ) ) \\bullet \\alpha ( y ) = \\alpha ( x ) \\bullet ( \\left [ d , \\varphi \\right ] ( y ) ) \\end{align*}"}
+{"id": "3218.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow + 0 } J _ t ^ { \\alpha - 1 } u ( t ) = 0 \\end{align*}"}
+{"id": "8644.png", "formula": "\\begin{align*} \\partial _ t \\rho ( t ) - \\nabla \\cdot ( \\rho ( t ) \\nabla u _ 0 ( t ) ) = 0 , \\end{align*}"}
+{"id": "4961.png", "formula": "\\begin{align*} v _ N ( \\{ \\lambda \\} ) = \\prod _ { 1 \\leq j < k \\leq N } \\sin \\gamma ( \\lambda _ k - \\lambda _ j ) . \\end{align*}"}
+{"id": "867.png", "formula": "\\begin{align*} \\zeta ^ i & = \\frac { 1 } { 4 } \\left ( \\frac { y ^ i } { F } - w ^ i \\right ) \\left ( 2 F S _ 0 - L _ { 0 0 } - F ^ 2 L _ { w w } \\right ) - \\frac { 1 } { 4 } F ^ 2 \\left ( S ^ i + T ^ i \\right ) - \\frac { 1 } { 2 } F C ^ i _ 0 . \\end{align*}"}
+{"id": "2102.png", "formula": "\\begin{align*} & ( i + 1 - d z _ 1 ) ( j + 1 - d z _ 2 ) \\Tilde { B } ( \\frac { i } { d } , x , \\frac { j } { d } , y ) + ( i + 1 - d z _ 1 ) ( d z _ 2 - j ) \\Tilde { B } ( \\frac { i } { d } , x , \\frac { j + 1 } { d } , y ) \\\\ & + ( d z _ 1 - i ) ( j + 1 - d z _ 2 ) \\Tilde { B } ( \\frac { i + 1 } { d } , x , \\frac { j + 1 } { d } , y ) + ( d z _ 1 - i ) ( d z _ 2 - j ) \\Tilde { B } ( \\frac { i + 1 } { d } , x , \\frac { j + 1 } { d } , y ) \\\\ & = \\Tilde { B } ( z _ 1 , x , z _ 2 , y ) + o ( 1 ) \\end{align*}"}
+{"id": "289.png", "formula": "\\begin{align*} \\dot { z } = F ( z ) , z \\in \\mathbb { D } ^ n ( \\rho ) , \\end{align*}"}
+{"id": "3108.png", "formula": "\\begin{align*} \\varphi \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & = \\left ( \\begin{array} { c c c } a ^ { \\ell _ 1 } & c _ 1 \\ , a ^ { \\ell _ 1 } \\ , b ^ { p ^ { e _ 1 } } \\ , d ^ { p ^ { e _ 1 } } & 0 \\\\ 0 & d ^ { - \\ell _ 2 } & 0 \\\\ 0 & 0 & d ^ { - \\ell _ 2 } \\end{array} \\right ) . \\end{align*}"}
+{"id": "3902.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { d - i \\infty } ^ { d + i \\infty } \\frac { x ^ s } { s ( s + 1 ) \\cdots ( s + k ) } \\ , d s = \\begin{dcases} 0 & 0 < x < 1 , \\\\ \\frac { 1 } { k ! } \\left ( 1 - \\frac { 1 } { x } \\right ) ^ k & x \\geq 1 . \\end{dcases} \\end{align*}"}
+{"id": "2108.png", "formula": "\\begin{align*} U ^ N ( s , x ) & = U ( 0 , x ) - \\alpha \\int _ 0 ^ { s } \\int _ 0 ^ 1 A ( x , y ) U ( u \\wedge \\tau _ N , y ) 1 _ { ( 0 , \\tau _ N ) } ( u ) \\mathrm { d } y \\mathrm { d } u \\\\ & + \\alpha \\beta \\int _ 0 ^ s 1 _ { ( 0 , \\tau _ N ) } ( u ) \\mathrm { d } \\xi _ 2 ( u \\wedge \\tau _ N , x ) + \\alpha \\zeta \\int _ 0 ^ s 1 _ { ( 0 , \\tau _ N ) } ( u ) \\mathrm { d } \\xi _ 3 ( u \\wedge \\tau _ N , x ) . \\end{align*}"}
+{"id": "917.png", "formula": "\\begin{align*} L = \\Delta + \\Delta ^ { \\alpha / 2 } \\end{align*}"}
+{"id": "7016.png", "formula": "\\begin{align*} b ( x ) = \\pm \\sqrt { \\delta } \\frac { d - 2 } { 2 } \\frac { x } { | x | ^ 2 } , \\end{align*}"}
+{"id": "6898.png", "formula": "\\begin{align*} A _ { D } ( \\tau ) & = 1 6 ( ( \\gamma + \\eta \\cos ( \\sin \\tau ) \\cosh ( \\cos \\tau ) ) ^ { 2 } + \\eta ^ 2 \\sin ^ 2 ( \\sin \\tau ) \\sinh ^ 2 ( \\cos \\tau ) ) ^ 2 \\end{align*}"}
+{"id": "3697.png", "formula": "\\begin{align*} h ( \\lambda ) : = \\lambda - s a _ { \\ell - 1 } / a _ \\ell = \\mu . \\end{align*}"}
+{"id": "5643.png", "formula": "\\begin{align*} a _ \\mathcal { F } Q _ \\mathcal { F } = \\left ( 3 5 2 , 6 4 ( 2 x _ 8 - 1 1 7 ) , 3 2 ( 3 5 1 - 4 x _ 8 ) \\right ) . \\end{align*}"}
+{"id": "7481.png", "formula": "\\begin{align*} & \\partial _ t \\Bar { g } ( t ) = - 2 R i c ( \\Bar { g } ( t ) ) + \\mathcal { L } _ { V ( \\Bar { g } ( t ) , { g } _ 0 ( t ) ) } ( \\Bar { g } ( t ) ) , \\\\ & \\Bar { g } ( t ) : = { g } _ 0 ( t ) + \\Bar { h } ( t ) , \\end{align*}"}
+{"id": "6253.png", "formula": "\\begin{align*} P ( - i D ) u ( x ) & = 0 , x \\in \\Omega , \\\\ [ 5 p t ] u ( x ) & = 0 , x \\in \\Gamma , \\end{align*}"}
+{"id": "8144.png", "formula": "\\begin{align*} D _ { 1 } & \\ = \\ - \\frac { 2 } { 9 } \\left ( 8 J _ 4 J _ 2 + 3 J _ 3 J _ 1 \\right ) A _ 9 - \\frac { 2 } { 9 } \\left ( 8 J _ 5 J _ 1 - 8 J _ 3 J _ 2 - 1 1 J _ 2 J _ 0 \\right ) A _ 8 \\ - \\\\ & \\frac { 4 } { 3 } J _ 2 J _ 1 A _ 7 - \\frac { 2 2 } { 9 } J _ 2 J _ 1 A _ 6 + \\frac { 4 } { 9 } J _ 2 J _ 1 A _ 5 + \\frac { 2 2 } { 9 } J _ 2 ^ 2 A _ 4 - \\frac { 4 } { 9 } J _ 1 ^ 2 A _ 3 \\ , \\end{align*}"}
+{"id": "6999.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\mathbb { C } } ^ n \\ ! \\setminus \\ ! \\{ [ 1 \\ ! : \\ ! z _ 2 \\ ! : \\ ! \\cdots \\ ! : \\ ! z _ { n + 1 } ] : z _ 2 , \\ ! \\cdots \\ ! , \\ ! z _ { n + 1 } \\ ! \\in \\ ! \\mathbb { C } \\} \\ ! = \\ ! \\{ [ 0 \\ ! : \\ ! z _ 2 \\ ! : \\ ! \\cdots \\ ! : \\ ! z _ { n + 1 } ] : z _ 2 , \\ ! \\cdots \\ ! , \\ ! z _ { n + 1 } \\ ! \\in \\ ! \\mathbb { C } \\} . \\end{align*}"}
+{"id": "8209.png", "formula": "\\begin{align*} c ^ { y ^ { i - 1 } 0 } = c ^ { y ^ { i - 1 } } - b ^ { y ^ { i - 1 } } , Y _ i = 0 . \\end{align*}"}
+{"id": "4327.png", "formula": "\\begin{align*} \\inf _ { x \\in \\mathcal { X } } & \\sum _ { i \\in [ m ] } \\ \\sup _ { u ^ i \\in \\mathcal { U } _ i } f _ i ( x , u ^ i ) . \\end{align*}"}
+{"id": "866.png", "formula": "\\begin{align*} G ^ i = \\mathcal { G } ^ i + \\zeta ^ i , \\end{align*}"}
+{"id": "2367.png", "formula": "\\begin{align*} \\| w ( t ) \\| _ { L ^ { 2 } } ^ { 2 } + \\int ^ { t } _ { s } \\| \\nabla \\otimes w ( \\tau ) \\| _ { L ^ { 2 } } ^ { 2 } d \\tau \\leq \\| w ( s ) \\| _ { L ^ { 2 } } ^ { 2 } , \\quad \\lim \\limits _ { t \\rightarrow \\infty } \\| w ( t ) \\| _ { L ^ { 2 } } = 0 , \\end{align*}"}
+{"id": "24.png", "formula": "\\begin{align*} q _ + ( t , z ) = \\tfrac { 1 } { 2 \\pi i } I _ + \\Big ( \\big ( X - t \\kappa R ( \\kappa ; q ^ 0 ) ^ 2 - z \\big ) ^ { - 1 } q ^ 0 _ + \\Big ) \\end{align*}"}
+{"id": "6077.png", "formula": "\\begin{align*} \\Psi _ { m _ { \\pm } ^ \\# } \\in { \\mathcal H } _ { \\infty } \\backslash { \\mathcal H } , \\ U _ { \\infty } \\Psi _ { m _ { \\pm } ^ \\# } = m _ { \\pm } ^ \\# \\Psi _ { m _ { \\pm } ^ \\# } . \\end{align*}"}
+{"id": "4730.png", "formula": "\\begin{align*} u _ x ( t ) = \\lim _ { n \\to \\infty } \\left ( \\mathrm { I d } + \\frac { t } { n } A \\right ) ^ { - n } x . \\end{align*}"}
+{"id": "7917.png", "formula": "\\begin{align*} \\gamma = \\langle \\lambda , i _ { \\mathcal { N } } \\mu \\rangle _ { \\Lambda ^ { k - 1 } } v _ { \\partial \\Omega } , \\end{align*}"}
+{"id": "8384.png", "formula": "\\begin{align*} J ( \\underline { w } ) : = \\{ \\alpha _ { i _ { j } } : s _ { i _ { j } } s _ { i _ { k } } = s _ { i _ { k } } s _ { i _ { j } } \\} . \\end{align*}"}
+{"id": "4445.png", "formula": "\\begin{align*} f = \\sum _ { l = 1 } ^ { + \\infty } f _ l g _ l . \\end{align*}"}
+{"id": "3296.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } Q _ n \\ge \\liminf _ { n \\to \\infty } \\frac { 1 } { \\sqrt { 4 n } } V ( \\sqrt { n } \\Delta _ \\mu ) = \\infty . \\end{align*}"}
+{"id": "193.png", "formula": "\\begin{align*} C _ 1 \\ = \\ E _ 1 \\ = \\ F _ 2 \\ = \\ E , C _ 2 \\ = \\ F _ 1 \\ = \\ E _ 2 \\ = \\ F . \\end{align*}"}
+{"id": "32.png", "formula": "\\begin{align*} \\tau f = \\begin{cases} f & f \\in \\ell ^ 2 ( X ) _ { } \\\\ - f & f \\in \\ell ^ 2 ( X ) _ { } \\\\ \\end{cases} \\end{align*}"}
+{"id": "231.png", "formula": "\\begin{align*} \\gamma _ 1 = \\gamma _ 0 + \\frac { h ' } { h } , A _ 1 = \\frac { A _ 0 } { h } \\qquad b _ 1 = \\frac { b _ 0 } { h ^ 2 } . \\end{align*}"}
+{"id": "7704.png", "formula": "\\begin{align*} \\int _ { S _ { 1 } } ( \\frac { h _ { E _ { t _ { j } } } } { n } ) ^ { p } f ( x ) d x & \\leq c _ { 0 } \\left ( \\int _ { S _ { 1 } } ( \\frac { h _ { E _ { t _ { j } } } } { n } ) ^ { - n } d x \\right ) ^ { - \\frac { p } { n } } | S _ { 1 } | ^ { \\frac { p + n } { n } } \\\\ & \\leq c _ { 1 } | S _ { 1 } | ^ { \\frac { p + n } { n } } \\rightarrow 0 . \\end{align*}"}
+{"id": "6417.png", "formula": "\\begin{align*} T _ n \\leq C _ K \\frac { \\ln ( n ) ^ q w _ n } { n } \\sum _ { i = 1 } ^ n \\ \\left ( 1 + \\frac { 1 } { X _ { \\frac { i - 1 } { n } } ^ q } \\right ) \\sup _ { ( a , b ) \\in A , \\ ; ( \\delta , \\alpha ) \\in W _ n ^ { ( \\eta ) } } \\left | h ( z ^ n _ i ( \\theta ) , \\alpha ) \\right | { \\bf 1 } _ { \\Omega _ K } , & \\end{align*}"}
+{"id": "7995.png", "formula": "\\begin{align*} \\frac { d } { d r } \\bigg ( \\frac { e ^ { C n r } } { r ^ { n - 1 } } E ( u , B _ r ) \\bigg ) & = - ( n - 1 - C n r ) \\frac { e ^ { C n r } } { r ^ n } E ( u , B _ r ) + \\frac { e ^ { C n r } } { r ^ { n - 1 } } \\int _ { \\partial B _ r } e _ \\varepsilon ( u ) , \\end{align*}"}
+{"id": "3829.png", "formula": "\\begin{align*} A : = - \\frac { 5 h ^ 2 h ' k } { 7 2 } + \\frac { 5 k + 1 8 } { 7 2 } h + \\frac { 5 h ' k } { 7 2 } = \\frac { 5 \\frac k 6 \\left ( 1 - h ^ 2 \\right ) h ' } { 1 2 } + \\frac { 5 k + 1 8 } { 7 2 } h . \\end{align*}"}
+{"id": "9126.png", "formula": "\\begin{align*} H _ { \\underline { k } } \\coloneqq \\big \\{ h \\in H \\ , \\big | \\ , ( h ) = \\underline { k } \\big \\} , H _ { \\underline { k } , \\underline { d } } \\coloneqq \\big \\{ h \\in H \\ , \\big | \\ , ( h ) = \\underline { d } \\big \\} . \\end{align*}"}
+{"id": "7410.png", "formula": "\\begin{align*} F '' ( t ) = \\int _ { \\R } \\{ A | u _ t ( x , t ) | ^ p | u ( x , t ) | ^ q + B | u ( x , t ) | ^ r \\} d x \\end{align*}"}
+{"id": "1266.png", "formula": "\\begin{align*} q ^ { n ^ 2 - { n + 1 \\choose 2 } } = q ^ { \\frac { n ( n - 1 ) } { 2 } } \\equiv 1 + \\frac { ( 1 - n ) ( 1 - q ^ n ) } { 2 } \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "8616.png", "formula": "\\begin{align*} \\frac { d v _ 1 } { d t } ( \\cdot ; x _ 1 ) = - v _ 1 ^ 2 ( \\cdot ; x _ 1 ) - K _ 1 ( \\cdot ; x _ 1 ) - \\phi _ 1 ( \\cdot ; x _ 1 ) \\geq ( - 1 - \\frac { \\gamma } { 2 } ) v _ 1 ^ 2 ( \\cdot ; x _ 1 ) \\end{align*}"}
+{"id": "5585.png", "formula": "\\begin{align*} \\langle u _ i , \\hat { u } _ j \\rangle = \\frac { \\langle B ^ { \\ell } \\chi _ i , ( B ^ * ) ^ { \\ell } \\check { \\chi } _ j \\rangle } { \\nu _ i ^ { 2 \\ell } \\nu _ j ^ { 2 \\ell + 2 } } = \\frac { \\langle B ^ { 2 \\ell } \\chi _ i , \\check { \\chi } _ j \\rangle } { \\nu _ i ^ { 2 \\ell } \\nu _ j ^ { 2 \\ell + 2 } } . \\end{align*}"}
+{"id": "6795.png", "formula": "\\begin{align*} \\dot { x } & = c _ 1 \\kappa _ 1 x + c _ 2 \\kappa _ 2 x y + c _ 3 \\kappa _ 3 y , \\\\ \\dot { y } & = d _ 1 \\kappa _ 1 x + d _ 2 \\kappa _ 2 x y + d _ 3 \\kappa _ 3 y \\end{align*}"}
+{"id": "7586.png", "formula": "\\begin{align*} [ \\Delta , i \\Gamma _ { \\varphi _ { R } } ] = [ \\Delta , \\nabla \\varphi _ { R } \\cdot \\nabla + \\nabla \\cdot \\nabla \\varphi _ { R } ] = 4 \\partial _ { k } ( \\partial ^ { 2 } _ { k l } \\varphi _ { R } ) \\partial _ { l } + \\Delta ^ { 2 } \\varphi _ { R } . \\end{align*}"}
+{"id": "413.png", "formula": "\\begin{align*} \\begin{aligned} & \\overline { C } _ { i i d } ( \\zeta ) = \\frac { \\pi } { \\lambda ^ 2 } [ U _ R \\log ( 1 + \\zeta ^ { - 1 } - \\rho v ( \\eta , \\zeta ) ) \\\\ & + U _ S \\log ( 1 + \\eta \\zeta ^ { - 1 } - \\rho v ( \\eta , \\zeta ) ) - U _ S v ( \\eta , \\zeta ) ] , \\end{aligned} \\end{align*}"}
+{"id": "7488.png", "formula": "\\begin{align*} \\tilde { g } ( t ) = \\xi _ 1 ( r _ s ) ( \\Phi _ { s } ^ { - 1 } ) ^ * g _ 0 ( t + s ) + ( 1 - \\xi _ 1 ( r _ s ) ) \\hat { g } ( 0 ) , \\end{align*}"}
+{"id": "5641.png", "formula": "\\begin{align*} a & = \\left ( 1 , x _ 1 , 0 , \\frac { x _ 8 } { 2 } - x _ 1 , \\frac { x _ 8 } { 2 } , x _ 1 - \\frac { 3 x _ 8 } { 2 } + 1 1 7 , x _ 8 , x _ 8 , x _ 8 , - x _ 1 - \\frac { 5 x _ 8 } { 2 } + 2 3 4 \\right ) , \\\\ a Q & = \\left ( 3 5 2 , 0 , 6 4 ( 2 x _ 8 - 1 1 7 ) , 0 , 0 , 3 2 ( 1 1 7 + 4 x _ 1 - 2 x _ 8 ) , 0 , 0 , 0 , 6 4 ( 1 1 7 - 2 x _ 1 - x _ 8 ) \\right ) . \\end{align*}"}
+{"id": "6918.png", "formula": "\\begin{align*} \\forall t \\geq 0 , \\forall x \\geq 0 , \\quad & \\partial _ t u + v \\partial _ x u = 0 , \\\\ \\forall x \\geq 0 , & u ( 0 , x ) = u _ 0 ( x ) \\in \\R . \\end{align*}"}
+{"id": "2716.png", "formula": "\\begin{align*} K ^ { ( 1 ) } _ { i j } \\tau ^ { j } _ { \\alpha } = 0 , \\end{align*}"}
+{"id": "2524.png", "formula": "\\begin{align*} I ( M , L ) = \\bigcup _ s I ^ { ( s ) } ( M , L ) \\ ; , I ^ { ( \\infty ) } ( M , L ) = \\bigcap _ s I ^ { ( s ) } ( M , L ) \\ ; . \\end{align*}"}
+{"id": "6328.png", "formula": "\\begin{align*} \\theta _ M ( \\mathcal N ) : = \\theta ( \\mathcal N / M ) , \\end{align*}"}
+{"id": "3128.png", "formula": "\\begin{align*} u ^ - _ { \\psi ^ * } ( s ) = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ v _ { 2 , \\ , 1 } ( s ) & 1 & 0 \\\\ v _ { 3 , \\ , 1 } ( s ) & v _ { 3 , \\ , 2 } ( s ) & 1 \\end{array} \\right ) ( \\ v _ { 2 , \\ , 1 } ( S ) , v _ { 3 , \\ , 1 } ( S ) , v _ { 3 , \\ , 2 } ( S ) \\in k [ S ] \\ , ) . \\end{align*}"}
+{"id": "5771.png", "formula": "\\begin{align*} z ^ { ( k , \\ell ) } _ j = G ( \\mathbf { L } q ^ { k , \\ell - 1 } , \\Upsilon _ j ) = G ( q ^ { k , \\ell - 1 } , \\mathbf { L } ^ \\dagger \\Upsilon _ j ) = 0 . \\end{align*}"}
+{"id": "3645.png", "formula": "\\begin{align*} L ^ { - } ( G , p ) _ { e , e ' } = \\begin{cases} 2 & e = e ' = \\{ u , v \\} p ( u ) \\neq p ( v ) , \\\\ d _ { u v } \\cdot d _ { u w } & e = \\{ u , v \\} , ~ e ' = \\{ u , w \\} \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "8972.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { 0 } ^ { + \\infty } e ^ { - t } \\| u ( t , \\cdot ) \\| _ { L ^ 2 ( \\R ^ N ) } ^ 2 \\ , d t & \\le 2 \\| w _ 0 \\| _ { L ^ 2 ( \\mathbb R ^ N ) } ^ 2 \\\\ & \\quad + 8 \\| w _ 1 \\| _ { L ^ 2 ( \\mathbb R ^ N ) } ^ 2 + 1 6 \\int _ { 0 } ^ { + \\infty } e ^ { - t } \\| u '' ( t , \\cdot ) \\| _ { L ^ 2 ( \\R ^ N ) } ^ 2 \\ , d t . \\end{aligned} \\end{align*}"}
+{"id": "303.png", "formula": "\\begin{align*} L _ \\mu = \\left ( 1 0 + 2 \\sum _ { k = 0 } ^ \\infty \\dfrac { 1 0 ^ { k + 2 } } { 3 0 ^ { k + 1 } } \\right ) + \\left ( 1 0 + 4 \\sum _ { k = 0 } ^ \\infty \\dfrac { ( k + 1 ) 1 0 ^ { k + 2 } } { 2 0 ^ { k + 2 } } \\right ) = 7 3 / 3 , \\end{align*}"}
+{"id": "8596.png", "formula": "\\begin{align*} \\sinh { d ( A _ 1 A _ 2 , l ) } \\sinh { \\frac { \\ell ( c _ i ) } 2 } & = \\frac { b _ k + b _ j } { 2 a _ j } \\frac { 2 a _ j } { b _ k + b _ j } = 1 \\end{align*}"}
+{"id": "6913.png", "formula": "\\begin{align*} \\frac { n p - 3 p + 2 } { p } + \\frac { p - 1 } { p } + \\varepsilon & = \\frac { n p - 2 p + 1 } { p } + \\varepsilon < \\frac { n p - n + p } { p } , \\allowdisplaybreaks \\\\ 0 < \\frac { p - 1 } { p } + \\varepsilon & < 1 \\end{align*}"}
+{"id": "2251.png", "formula": "\\begin{align*} \\small h _ { F S } = \\sum _ { j , k = 1 } ^ n \\frac { ( 1 + | z | ^ 2 ) \\ , \\delta _ { j k } - \\bar { z _ j } \\ , z _ k } { ( 1 + | z | ^ 2 ) ^ 2 } \\ , d z _ j \\ , d \\bar { z } _ k . \\end{align*}"}
+{"id": "1039.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } \\Vert \\phi _ { n , k } - \\phi _ k \\Vert \\le K \\sum _ { k = n + 1 } ^ { \\infty } \\Vert \\phi _ k \\Vert , n \\in \\N . \\end{align*}"}
+{"id": "6280.png", "formula": "\\begin{align*} f ( \\overline { x } _ T ) - f ( x ^ * ) = \\tilde { O } \\left ( \\sqrt { 8 M _ 2 \\sqrt { d } \\Delta \\mathcal { D } _ \\Psi } + \\sqrt { \\frac { 3 2 M _ 2 \\mathcal { D } _ { \\Psi } d a _ { q } \\Delta } { T ^ { \\frac { \\kappa } { ( 1 + \\kappa ) } } } } + \\frac { 2 \\sqrt { d } a _ { q } M _ 2 \\mathcal { D } _ { \\Psi } } { T ^ { \\frac { \\kappa } { 1 + \\kappa } } } \\right ) . \\end{align*}"}
+{"id": "2939.png", "formula": "\\begin{align*} \\sigma ( z ) = ( 2 \\pi ) ^ { - n / 2 } \\ , \\int _ { \\R ^ n } e ^ { - i x \\cdot \\xi } \\psi ( \\xi + \\frac { 1 } { 2 } y , \\xi - \\frac { 1 } { 2 } y ) d \\xi . \\end{align*}"}
+{"id": "3971.png", "formula": "\\begin{align*} \\Vert \\rho _ { t _ n } ( x , y ) \\xi _ n ( x ) - \\xi _ n ( y ) \\Vert ^ 2 & = \\Vert ( \\rho ( x , y ) ( 0 ) ) _ { t _ n } - 0 _ { t _ n } \\Vert ^ 2 & \\\\ & = \\Vert ( \\rho ( x , y ) ( 0 ) ) _ { t _ n } \\Vert ^ 2 + \\Vert 0 _ { t _ n } \\Vert ^ 2 - 2 \\langle ( \\rho ( x , y ) ( 0 ) ) _ { t _ n } , 0 _ { t _ n } \\rangle & \\\\ & = 2 \\left ( 1 - \\exp ( - t _ n \\Vert \\rho ( x , y ) ( 0 ) \\Vert ^ 2 ) \\right ) & \\end{align*}"}
+{"id": "3523.png", "formula": "\\begin{align*} \\widehat H ^ 2 ( \\widehat \\Delta ; M ) = \\bigoplus _ { i = 1 } ^ { m } M \\cdot \\mathbf { r } _ i ^ * \\ / ( \\mathbf { r } _ 0 ^ * + p _ i \\cdot \\mathbf { r } _ i ^ * \\ \\ i = 1 , \\ldots , m ) . \\end{align*}"}
+{"id": "2244.png", "formula": "\\begin{align*} \\Gamma _ \\alpha ^ \\alpha \\circ \\phi _ s ^ { ( 1 ) } = L \\circ \\phi _ s ^ { ( 1 ) } . \\end{align*}"}
+{"id": "380.png", "formula": "\\begin{align*} ( x - a ) ^ 3 = 6 a b x + 3 a b ^ 2 - 3 a ^ 2 b = 3 a b ( 2 x + b - a ) . \\end{align*}"}
+{"id": "3381.png", "formula": "\\begin{align*} z _ { t } & \\le \\frac { 1 } { 2 C _ { 1 } \\left ( R _ { 1 } + 8 A C _ { 1 } \\right ) } \\\\ z _ { k } & \\ge \\frac { 1 } { 2 \\eta _ { k } \\lambda _ { k } \\left ( R _ { 1 } + 8 A C _ { 1 } \\right ) + 1 6 A \\eta _ { k } ^ { 2 } \\lambda _ { k } ^ { 2 } } \\\\ & = \\frac { 1 } { 2 C _ { 1 } \\left ( R _ { 1 } + 1 6 A C _ { 1 } \\right ) } \\end{align*}"}
+{"id": "858.png", "formula": "\\begin{align*} \\dot { \\gamma } = \\left ( x , y ^ i \\right ) , \\ddot { \\gamma } = \\left ( \\dot { \\gamma } , y ^ i , - 2 G ^ i \\right ) \\end{align*}"}
+{"id": "2186.png", "formula": "\\begin{align*} \\bar e _ r = ( \\cos \\varphi , \\sin \\varphi , 0 ) , \\ \\ \\bar e _ \\varphi = ( - \\sin \\varphi , \\cos \\varphi , 0 ) , \\ \\ \\bar e _ z = ( 0 , 0 , 1 ) \\end{align*}"}
+{"id": "3951.png", "formula": "\\begin{align*} \\dot { { V } } ( t ) + 2 \\delta { V } ( t ) & \\leq \\eta ( t ) ^ { \\top } \\Phi \\eta ( t ) \\\\ & + \\rho \\sum _ { n = N + 1 } ^ { \\infty } y _ n ^ { \\top } ( t ) W _ n y _ n ( t ) \\leq 0 \\end{align*}"}
+{"id": "2396.png", "formula": "\\begin{align*} & d _ n ^ { \\ell } \\cdot \\frac { 1 } { \\ell ! } F _ { 1 / 4 } ^ { ( \\ell ) } ( t ) \\big | _ { t = - k } \\in \\mathbb { Z } , \\\\ & d _ n ^ { \\ell } \\cdot \\frac { 1 } { \\ell ! } F _ { 3 / 4 } ^ { ( \\ell ) } ( t ) \\big | _ { t = - k } \\in \\mathbb { Z } , \\\\ & d _ n ^ { \\ell } \\cdot \\frac { 1 } { \\ell ! } \\left ( ( t + k ) G ( t ) \\right ) ^ { ( \\ell ) } \\big | _ { t = - k } \\in \\mathbb { Z } . \\end{align*}"}
+{"id": "1998.png", "formula": "\\begin{align*} \\widetilde { ( Q ^ { n , \\pm } ) } _ l = \\frac { i \\varepsilon ^ 2 } { \\alpha \\beta _ l } \\int _ 0 ^ \\tau e ^ { - i \\beta _ l ^ \\pm w } f _ l ^ n ( w ) d w \\mp ( p _ l ^ \\pm \\widehat { ( f ( \\psi ( t _ n ) ) ) } _ l + q _ l ^ \\pm \\widehat { ( g ( \\psi ( t _ n ) ) ) } _ l ) , \\end{align*}"}
+{"id": "3712.png", "formula": "\\begin{align*} - \\nabla _ { g } ^ 2 f _ { \\sigma } ( x ) = \\lambda ( \\sigma ) f _ { \\sigma } ( x ) \\ ; , \\end{align*}"}
+{"id": "5045.png", "formula": "\\begin{align*} | u ( x ) - u ( y ) | ^ p \\le \\sup _ i | u _ i ( x ) - u _ i ( y ) | ^ p \\le \\sum _ { i = 1 } ^ \\infty | u _ i ( x ) - u _ i ( y ) | ^ p \\end{align*}"}
+{"id": "3938.png", "formula": "\\begin{align*} \\max _ { n \\in \\mathbb N } \\vert T _ n ^ { - 1 } \\vert , \\max _ { n \\in \\mathbb N } \\vert T _ n \\vert \\leq \\sigma _ N : = 1 + \\vert \\kappa \\vert \\lambda _ N . \\end{align*}"}
+{"id": "2866.png", "formula": "\\begin{align*} ( a _ { i + 1 } - a _ i ) ^ { - ( \\gamma + 1 ) } \\int _ { a _ i } ^ { a _ { i + 1 } } f ( s ) \\ , d s = \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "7793.png", "formula": "\\begin{align*} \\begin{aligned} & ( A + B K ) ^ { \\top } P + P ( A + B K ) + ( C + D K ) ^ { \\top } P ( C + D K ) \\\\ & \\qquad \\qquad \\qquad + K ^ { \\top } R K + S ^ { \\top } K + K ^ { \\top } S + Q = 0 . \\end{aligned} \\end{align*}"}
+{"id": "5722.png", "formula": "\\begin{align*} \\sum _ { k + \\ell \\le s } \\Vert \\mathcal { E } ^ { ( k , \\ell ) } ( u ) \\Vert _ { G } ( t ) = o ( 1 ) \\sum _ { k + \\ell \\le s } \\Vert q ^ { ( k , \\ell ) } ( u ) \\Vert _ { G } ( t ) \\end{align*}"}
+{"id": "2304.png", "formula": "\\begin{align*} g _ i ( u ) = g _ i + ( q - q ^ { - 1 } ) \\frac { e _ i } { u - 1 } \\ . \\end{align*}"}
+{"id": "2459.png", "formula": "\\begin{align*} E _ { 0 } : ( \\phi , \\psi ) = ( 0 , 0 ) , E _ { 2 } : ( \\phi , \\psi ) = \\left ( 0 , c \\gamma ^ { - 1 } \\right ) . \\end{align*}"}
+{"id": "1115.png", "formula": "\\begin{align*} g _ j : = \\left | A _ j \\left ( \\varphi _ j * \\vec f \\right ) \\right | h _ j : = \\left | W ^ { \\frac { 1 } { p } } \\left ( \\varphi _ j * \\vec { f } \\right ) \\right | . \\end{align*}"}
+{"id": "6098.png", "formula": "\\begin{align*} \\Delta ( x _ j ) & = 1 \\otimes x _ j + \\sum a _ j \\otimes b _ j . \\end{align*}"}
+{"id": "4774.png", "formula": "\\begin{align*} \\psi ( K , b , D ) : = \\frac { b + K } { D } . \\end{align*}"}
+{"id": "7915.png", "formula": "\\begin{align*} \\int _ { \\Omega } d \\alpha = \\int _ { \\partial \\Omega } \\mathrm { t r } ( \\alpha ) . \\end{align*}"}
+{"id": "5699.png", "formula": "\\begin{align*} \\| u ^ T \\| _ { C ^ s } ( t ) \\leq C \\sum _ { k = 0 } ^ s \\left | \\frac { d ^ k } { d t ^ k } x ( t ) \\right | \\leq C \\| u \\| _ { C ^ s } ( t ) . \\end{align*}"}
+{"id": "3531.png", "formula": "\\begin{align*} | \\psi ( z _ 1 , z _ 2 ) | = \\begin{cases} 1 & \\mbox { i f } ( z _ 1 , z _ 2 ) \\in K _ 1 \\\\ 1 - \\epsilon & \\mbox { i f } ( z _ 1 , z _ 2 ) \\in \\mathbb { T } ^ 2 \\setminus K _ 1 . \\end{cases} \\end{align*}"}
+{"id": "6728.png", "formula": "\\begin{align*} S _ 1 ( z ) & = - \\ln \\Big ( \\frac { \\sinh ( \\pi z ) } { \\pi z } \\Big ) = \\pi z + \\ln \\Big ( \\frac { 2 \\pi z } { e ^ { 2 \\pi z } - 1 } \\Big ) \\\\ & = - \\pi z + \\ln ( 2 \\pi z ) - \\ln \\left ( { 1 - e ^ { - 2 \\pi z } } \\right ) ; \\end{align*}"}
+{"id": "7857.png", "formula": "\\begin{align*} A ( i , j ) = \\begin{cases} d _ { i } , & i = j ; \\\\ a _ { k } , & \\pi ( k ) = i , \\pi ( k + 1 ) = j ; \\\\ 0 , & \\end{cases} \\end{align*}"}
+{"id": "4752.png", "formula": "\\begin{align*} \\omega _ { K , b } ( m ) = 2 ^ { - ( m + 3 ) } / \\max \\{ 1 , A ^ * ( b + 1 ) \\} . \\end{align*}"}
+{"id": "2608.png", "formula": "\\begin{align*} D _ i ( n ) : = \\min \\{ | D _ i ( S ) | : S \\in \\mathcal { S } ( n ) \\} . \\end{align*}"}
+{"id": "5743.png", "formula": "\\begin{align*} X ^ 2 _ + ( t ) & = \\sum _ { i : \\gamma ^ + _ i > \\gamma _ * } ( \\xi _ i ^ + ( t ) ) ^ 2 + \\sum _ { i : \\gamma ^ - _ i > \\gamma _ * } ( \\xi _ i ^ - ( t ) ) ^ 2 , \\\\ X ^ 2 _ - ( t ) & = \\sum _ { i : \\gamma ^ + _ i < \\gamma _ * } ( \\xi _ i ^ + ( t ) ) ^ 2 + \\sum _ { i : \\gamma ^ - _ i < \\gamma _ * } ( \\xi _ i ^ - ( t ) ) ^ 2 . \\end{align*}"}
+{"id": "8792.png", "formula": "\\begin{align*} f _ \\rho ( x ) = \\frac { z - m } { z - y } \\vert x - y \\vert ^ \\rho + \\frac { m - y } { z - y } \\vert z - x \\vert ^ \\rho - \\vert x - m \\vert ^ \\rho . \\end{align*}"}
+{"id": "100.png", "formula": "\\begin{align*} P : = p - 1 \\quad K : = \\gamma + 1 , \\end{align*}"}
+{"id": "1983.png", "formula": "\\begin{align*} & X _ M : = \\left \\{ u = ( u _ 0 , u _ 1 , \\ldots , u _ M ) ^ T \\in \\mathbb { C } ^ { M + 1 } ~ | ~ u _ 0 = u _ M \\right \\} , \\\\ & Y _ M : = \\left \\{ e ^ { i \\mu _ l ( x - a ) } , \\ x \\in \\overline { \\Omega } , \\ l \\in \\mathcal { T } _ M \\right \\} . \\end{align*}"}
+{"id": "3273.png", "formula": "\\begin{align*} \\frac { ( q ^ { d + r - ( d - 1 ) n } ; q ^ d ) _ k q ^ { d k } } { ( q ^ d ; q ^ d ) _ k } & = ( - 1 ) ^ k { n - 1 - ( n + r ) / d \\brack k } _ { q ^ d } q ^ { d \\binom { k } { 2 } + ( n + 2 d + r - d n ) k } , \\\\ d \\binom { k } { 2 } + ( n + 2 d + r - d n ) k & = d \\binom { n - 1 - ( n + r ) / d - k } { 2 } - d \\binom { n - 1 - ( n + r ) / d } { 2 } , \\end{align*}"}
+{"id": "7145.png", "formula": "\\begin{align*} p ( k ) = \\eta ^ { \\alpha \\beta } k _ \\alpha k _ \\beta \\delta _ \\mu ^ { ~ \\nu } \\ , \\ , . \\end{align*}"}
+{"id": "3982.png", "formula": "\\begin{align*} x ^ { 2 } = 2 \\sum _ { k = 1 } ^ { n } \\gamma _ { i p k } e _ { k } + 2 \\sum _ { r = 1 } ^ { \\nu } \\widetilde { \\gamma } _ { i p r } \\widetilde { e } _ { r } . \\end{align*}"}
+{"id": "2326.png", "formula": "\\begin{align*} \\lim _ { q \\rightarrow 3 ^ { + } } \\mathcal { C } _ { q } = 1 , \\quad \\lim _ { q \\rightarrow + \\infty } \\mathcal { C } _ { q } = \\frac { 1 } { 3 ^ { \\frac { 7 } { 4 } } e ^ { 2 } \\sqrt { 4 \\pi ( 1 - \\tau ) } } . \\end{align*}"}
+{"id": "701.png", "formula": "\\begin{align*} f ( x ) = \\Phi ^ 6 ( x ) + 1 2 \\Phi ^ 5 ( x ) + 6 0 \\Phi ^ 4 ( x ) + 1 6 0 \\Phi ^ 3 ( x ) + 2 4 0 \\Phi ^ 2 ( x ) + 1 9 2 \\Phi ( x ) + 6 4 - m \\end{align*}"}
+{"id": "3021.png", "formula": "\\begin{align*} \\begin{array} { l } \\xi = e _ 0 , \\ \\ \\ \\varphi e _ 1 = e _ 3 , \\ \\ \\ \\varphi e _ 2 = e _ 4 , \\ \\ \\ \\varphi e _ 3 = e _ 1 , \\ \\ \\ \\varphi _ 4 = e _ 2 , \\\\ g ( e _ i , e _ i ) = 1 , \\ \\ \\ g ( e _ i , e _ j ) = 0 , \\ \\ \\ i , j \\in \\{ 0 , 1 , \\ldots , 4 \\} , \\ i \\neq j . \\end{array} \\end{align*}"}
+{"id": "6533.png", "formula": "\\begin{align*} C ^ { \\pm } _ K = \\int ^ { \\infty } _ 0 ( K _ { \\infty } ( \\pm t ^ { - \\beta } ) - K _ { \\infty } ( 0 ) ) d t , \\end{align*}"}
+{"id": "3660.png", "formula": "\\begin{align*} a ( G ) = \\lambda _ 2 ( L ( G , p ) ) = \\lambda _ i ( L ^ { - } ( G , p ) ) \\ge \\lambda _ i ( L ^ { - } ( G ' , p ' ) ) = \\lambda _ i ( L ^ { - } ( G ' , p ' ) ) = a ( G ' ) , \\end{align*}"}
+{"id": "6780.png", "formula": "\\begin{align*} \\dot { x } _ i = \\sum _ { j = 1 } ^ m \\beta _ { i j } x ^ { \\alpha _ j } ( i = 1 , \\ldots , n ) , \\end{align*}"}
+{"id": "4622.png", "formula": "\\begin{align*} Q ( j , k ) : = \\left \\{ x ' \\in \\R ^ { d - 1 } \\Bigm | 2 ^ { j m } x ' - k \\in Q ^ { ( d - 1 ) } \\right \\} \\subset Q ^ { ( d - 1 ) } . \\end{align*}"}
+{"id": "724.png", "formula": "\\begin{align*} d ( G , H ) & = \\sum _ { p } | \\nu _ p ( | G | ) - \\nu _ p ( | H | ) | , & n & = \\prod _ p p ^ { \\nu _ p ( n ) } . \\end{align*}"}
+{"id": "15.png", "formula": "\\begin{align*} d m | _ 0 ( g ) = R _ 0 ( \\kappa ) C _ + g . \\end{align*}"}
+{"id": "4449.png", "formula": "\\begin{align*} \\| \\frac { g _ 1 + g _ 2 } { 2 } \\| _ { \\partial M , \\rho } ^ 2 + \\| \\frac { g _ 1 - g _ 2 } { 2 } \\| _ { \\partial M , \\rho } ^ 2 = \\frac { \\| g _ 1 \\| _ { \\partial M , \\rho } ^ 2 + \\| g _ 2 \\| _ { \\partial M , \\rho } ^ 2 } { 2 } = M _ H ( Z _ 0 , J , \\rho ) , \\end{align*}"}
+{"id": "622.png", "formula": "\\begin{align*} & \\int _ 0 ^ 1 \\frac { \\sqrt { 1 - x ^ 2 } \\ln ( x ) } { \\left ( 2 - x ^ 2 \\right ) ^ { 3 / 2 } } \\ , d x = \\\\ & \\frac { 1 } { 4 } \\sum _ { n = 0 } ^ { \\infty } \\left ( - \\frac { 1 } { 2 } \\right ) ^ n \\binom { n } { \\left \\lfloor \\frac { n } { 2 } \\right \\rfloor } \\frac { ( n + 1 ) \\left ( 4 \\ln ( 2 ) - 4 O _ { n + 1 } - \\frac { 4 } { 2 n + 3 } \\right ) } { 2 n + 3 } . \\end{align*}"}
+{"id": "4026.png", "formula": "\\begin{align*} \\lim _ { m \\rightarrow + \\infty } \\frac { x _ { 1 } ^ { \\left ( t _ { 0 } + m \\right ) } } { \\varpi \\circ W \\left ( z ^ { \\left ( t _ { 0 } + m \\right ) } \\right ) } = \\begin{cases} \\gamma _ 1 & \\mbox { i f } \\gamma _ { 2 } = 0 , \\delta _ { 1 } \\neq 0 , \\\\ \\frac { \\gamma _ 1 x _ { 1 } ^ { ( t _ { 0 } ) } } { x _ { 1 } ^ { \\left ( t _ { 0 } \\right ) } + x _ { 2 } ^ { \\left ( t _ { 0 } \\right ) } } & \\mbox { i f } \\gamma _ { 2 } = \\delta _ { 1 } = 0 , \\\\ \\qquad 0 & \\mbox { i f } \\gamma _ { 2 } \\neq 0 , \\delta _ { 1 } = 0 , \\end{cases} \\end{align*}"}
+{"id": "7447.png", "formula": "\\begin{align*} \\left [ \\Upsilon \\Theta \\right ] ^ I _ J = \\sum _ { K \\in \\{ 1 , \\ldots , d \\} ^ n } \\Upsilon ^ I _ K \\Theta ^ K _ J = \\sum _ { k _ 1 = 1 } ^ d \\cdots \\sum _ { k _ n = 1 } ^ d \\Upsilon ^ { i _ 1 , \\ldots , i _ n } _ { k _ 1 , \\ldots , k _ n } \\Upsilon ^ { k _ 1 , \\ldots , k _ n } _ { j _ 1 , \\ldots , j _ n } . \\end{align*}"}
+{"id": "7379.png", "formula": "\\begin{align*} \\begin{array} { l } | w _ j | ^ { p - 2 } w _ j ( w _ j ) _ x | u _ j | ^ q - | w _ { j - 1 } | ^ { p - 2 } w _ { j - 1 } ( w _ { j - 1 } ) _ x | u _ { j - 1 } | ^ q \\\\ = ( | w _ j | ^ { p - 2 } w _ j - | w _ { j - 1 } | ^ { p - 2 } w _ { j - 1 } ) ( w _ j ) _ x | u _ j | ^ q \\\\ \\quad + | w _ { j - 1 } | ^ { p - 2 } w _ { j - 1 } ( ( w _ j ) _ x - ( w _ { j - 1 } ) _ x ) | u _ j | ^ q \\\\ \\quad + | w _ { j - 1 } | ^ { p - 2 } w _ { j - 1 } ( w _ { j - 1 } ) _ x ( | u _ j | ^ q - | u _ { j - 1 } | ^ q ) . \\end{array} \\end{align*}"}
+{"id": "7152.png", "formula": "\\begin{align*} \\varepsilon _ \\mu ( k , \\lambda ) \\varepsilon ^ \\mu ( k , \\lambda ' ) = \\eta _ { \\lambda \\lambda ' } \\ , \\ , . \\end{align*}"}
+{"id": "6940.png", "formula": "\\begin{align*} \\sum _ { j = i _ k + 1 } ^ { i _ { k + 1 } } \\big ( x _ { i } ( j ) - \\omega _ { i } ( j ) \\big ) = \\sum _ { j = i _ k + 1 } ^ { i _ { k + 1 } } \\big ( y _ { i } ( j ) - \\omega _ { i } ( j ) \\big ) , \\end{align*}"}
+{"id": "4640.png", "formula": "\\begin{align*} f ( \\mu ) = \\frac { 1 } { d ^ 2 } \\mu ^ 2 \\left ( \\mu - ( d + 1 ) \\right ) - \\frac { 1 } { d + 1 } \\mu \\left [ \\frac { 1 } { d } \\mu ^ 2 - d \\right ] \\le - 1 \\end{align*}"}
+{"id": "6043.png", "formula": "\\begin{align*} v _ { \\pm } = \\pm \\frac { E } { 3 N ^ \\gamma } , \\end{align*}"}
+{"id": "8572.png", "formula": "\\begin{align*} G _ t ^ { t _ 0 } = ( \\bar { C } _ { t _ 0 } ^ t ) ^ T . \\end{align*}"}
+{"id": "66.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 , x \\neq 0 } F ( D g ( x ) , D ^ 2 g ( x ) ) = 0 . \\end{align*}"}
+{"id": "824.png", "formula": "\\begin{align*} { D _ i } { D _ m } { \\Psi _ j } - { D _ m } { D _ i } { \\Psi _ j } = { k } ( { \\Psi _ m } { g _ { i j } } - { \\Psi _ i } { g _ { j m } } ) . \\end{align*}"}
+{"id": "3232.png", "formula": "\\begin{align*} H ^ 0 \\left ( R \\Gamma _ c ( Z _ b , F _ b ( \\delta ) ) [ 2 \\delta ] \\right ) = H _ c ^ { 2 \\delta } ( Z _ b , F _ b ) ( \\delta ) \\rightarrow F _ b \\end{align*}"}
+{"id": "9118.png", "formula": "\\begin{align*} F = \\frac { f ( \\{ x _ { i , r } \\} _ { i \\in I } ^ { 1 \\leq r \\leq k _ { i } } ) } { \\prod _ { i < j } ^ { a _ { i j } \\neq 0 } \\prod _ { 1 \\leq r \\leq k _ { i } } ^ { 1 \\leq s \\leq k _ { j } } ( x _ { i , r } - x _ { j , s } ) } , \\end{align*}"}
+{"id": "8169.png", "formula": "\\begin{align*} & D _ { \\max } ( B _ ) = \\sum _ { j = 1 } ^ L \\frac { 2 [ b _ j ^ 0 - 2 ^ { L - j } ] ^ + } { M } + \\frac { 2 [ M - \\sum _ { j = 1 } ^ { L } b _ { j } ^ 0 - 1 ] ^ + } { M } . \\end{align*}"}
+{"id": "9146.png", "formula": "\\begin{align*} \\tilde { \\mathbf { E } } ^ { + } _ { h } \\ = \\prod _ { ( \\beta , s ) \\in \\Delta ^ { + } \\times \\mathbb { Z } } \\limits ^ { \\rightarrow } \\tilde { \\mathbf { E } } _ { \\beta , s } ^ { + , ( h ( \\beta , s ) ) } , \\tilde { \\mathbf { E } } ^ { - } _ { h } \\ = \\prod _ { ( \\beta , s ) \\in \\Delta ^ { + } \\times \\mathbb { Z } } \\limits ^ { \\rightarrow } \\tilde { \\mathbf { E } } _ { \\beta , s } ^ { - , ( h ( \\beta , s ) ) } . \\end{align*}"}
+{"id": "144.png", "formula": "\\begin{align*} \\int _ { 3 U _ 1 \\times 3 V _ 1 } J \\bigg ( \\frac { x - y } { t } \\bigg ) \\ d x d y = 3 ^ 4 \\cdot \\int _ { U _ 1 \\times V _ 1 } J \\bigg ( \\frac { x - y } { ( t / 3 ) } \\bigg ) \\ d x d y . \\end{align*}"}
+{"id": "8973.png", "formula": "\\begin{align*} \\langle w ' ( t , \\cdot ) , h ( \\cdot ) \\rangle = \\int _ { \\R ^ N } w _ 1 ( x ) h ( x ) \\ , d x + \\int _ 0 ^ t \\langle w '' ( \\tau , \\cdot ) , h ( \\cdot ) \\rangle \\ , d \\tau \\qquad \\mbox { f o r e v e r y $ t \\in [ 0 , T ] $ } , \\end{align*}"}
+{"id": "3833.png", "formula": "\\begin{align*} K _ k ^ { [ 6 ] } ( \\nu ; n ) : = K _ k ^ { [ 6 ] } ( \\nu ; n , 0 ) . \\end{align*}"}
+{"id": "1865.png", "formula": "\\begin{align*} n _ { \\phi } ( T ^ 3 \\times K 3 , P ^ { \\dagger } ) = 2 . \\end{align*}"}
+{"id": "7127.png", "formula": "\\begin{align*} Q ( v _ { c } ) = A ( v _ { c } ) - \\frac { 1 } { 4 } \\Vert v _ { c } \\Vert _ { L ^ { 2 } } ^ { 4 } + \\int _ { \\mathbb { R } ^ { 2 } } ( 2 F ( v _ { c } ) - f ( v _ { c } ) v _ { c } ) d x = 0 . \\end{align*}"}
+{"id": "5673.png", "formula": "\\begin{gather*} c = \\inf _ { \\gamma \\in \\Gamma } \\max _ { 0 \\leq t \\leq 1 } J ( \\gamma ( t ) ) \\leq d \\end{gather*}"}
+{"id": "2334.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 3 } } w _ { 0 } \\varphi d x + \\int ^ { \\infty } _ { 0 } \\int _ { \\mathbb { R } ^ { 3 } } \\left ( a ( \\partial _ { t } \\varphi + \\Delta \\varphi + u ^ { c , \\gamma } \\cdot \\nabla \\varphi ) - ( a \\cdot \\nabla ) u ^ { c , \\gamma } \\varphi \\right ) d x d t = 0 . \\end{align*}"}
+{"id": "6027.png", "formula": "\\begin{align*} \\Big ( D _ 1 \\bar { \\xi } ^ { A } _ { x + 1 } ( s ) + D _ 2 \\bar { \\xi } ^ { B } _ { x + 1 } ( s ) \\Big ) - \\Big ( D _ 1 \\bar { \\xi } ^ { A } _ { x + 1 } ( s ^ - ) + D _ 2 \\bar { \\xi } ^ { B } _ { x + 1 } ( s ^ - ) \\Big ) = D _ 1 - D _ 2 \\ , . \\end{align*}"}
+{"id": "7121.png", "formula": "\\begin{align*} \\Phi ( t ) = \\frac { t ^ { 2 } } { 2 } A ( u ) - t ^ { - 2 } \\int _ { \\mathbb { R } ^ { 2 } } F ( t u ) d x . \\end{align*}"}
+{"id": "2237.png", "formula": "\\begin{align*} J ^ \\alpha = \\displaystyle \\frac { \\displaystyle \\partial } { \\displaystyle \\partial v ^ i _ \\alpha } \\otimes \\d q ^ i \\ , . \\end{align*}"}
+{"id": "4211.png", "formula": "\\begin{align*} \\Theta _ 1 ( T _ { C } X ) = \\bigotimes _ { n = 1 } ^ { \\infty } S _ { q ^ n } ( \\widetilde { T _ C X } ) \\otimes \\bigotimes _ { m = 1 } ^ { \\infty } \\wedge _ { q ^ m } ( \\widetilde { T _ C X } ) , \\end{align*}"}
+{"id": "6367.png", "formula": "\\begin{align*} \\abs * { \\int _ { S ^ { n - 1 } } \\rho \\phi ' ( R ) d \\varphi } & = \\abs * { \\int _ { S ^ { n - 1 } } \\frac { R \\rho ^ 2 } { 2 } \\phi '' ( R ) + \\frac { R ^ 2 \\rho ^ 3 } { 6 } \\phi ''' ( R ( 1 + \\rho \\eta ) ) \\ , d \\varphi } \\\\ & \\leq ( C _ 2 + C _ 3 ) \\int _ { S ^ { n - 1 } } \\frac { \\rho ^ 2 } { 2 } \\ , \\phi ' ( R ) \\ , d \\varphi , \\end{align*}"}
+{"id": "3197.png", "formula": "\\begin{align*} D _ 1 = ( \\tilde { x } _ 1 = 0 ) + ( \\tilde { x } _ 2 = 0 ) + ( \\tilde { x } _ 0 = 0 ) \\mbox { a n d } D _ 2 = ( \\tilde { x } _ 1 = 0 ) + ( \\tilde { x } _ 2 = 0 ) + \\sqrt { 2 } ( \\tilde { x } _ 0 = 0 ) . \\end{align*}"}
+{"id": "8803.png", "formula": "\\begin{align*} \\hat \\tau = \\frac { z _ + - m } { z _ + - y _ + } \\delta _ { y _ + } + \\frac { m - y _ + } { z _ + - y _ + } \\delta _ { z _ + } , \\check \\tau = \\frac { z _ - - m } { z _ - - y _ - } \\delta _ { y _ - } + \\frac { m - y _ - } { z _ - - y _ - } \\delta _ { z _ - } . \\end{align*}"}
+{"id": "1223.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } C _ { p _ n } ( \\varphi ( z _ n ) ) & = \\lim _ { n \\to \\infty } \\left ( \\sqrt [ p _ n ] { \\Lambda _ 1 ( p _ n ) } \\sqrt [ p _ n ] { \\alpha ( p _ n ) } \\vert \\varphi ( z _ n ) + \\sigma _ n \\vert ^ { \\frac { \\alpha ( p _ n ) } { p _ n - 1 } } v _ { p _ n } ( x _ 0 ) ^ { \\frac { \\beta ( p _ n ) } { p _ n - 1 } } \\right ) \\\\ & = \\Lambda _ { 1 , \\infty } \\vert \\varphi ( z _ 0 ) \\vert ^ { \\theta } v _ { \\infty } ( x _ 0 ) ^ { 1 - \\theta } \\end{align*}"}
+{"id": "3851.png", "formula": "\\begin{align*} \\beta ^ * ( \\{ e _ x \\} \\times \\{ e _ y \\} \\times [ 0 , t ] ) = \\int _ 0 ^ t \\exp ( - s ) \\check \\eta ^ { * } ( \\{ e _ x \\} \\times \\{ e _ y \\} , s ) d s , \\end{align*}"}
+{"id": "3802.png", "formula": "\\begin{align*} \\| \\mathcal { H } _ 2 \\| \\leq \\bar { c } _ 1 \\Delta _ { \\max } \\sum _ { i \\in [ M ] } \\sum _ { t = 0 } ^ { T - 1 } \\exp \\left ( - \\bar { c } _ 2 N _ i n _ x \\left ( \\frac { \\alpha \\rho ^ { ( i ) } \\| \\Sigma ^ { ( i ) } _ t \\| } { \\rho ^ { ( i ) } \\| \\Sigma ^ { ( i ) } _ t \\| + \\sqrt { n _ x } } \\right ) ^ 2 \\right ) , \\end{align*}"}
+{"id": "2310.png", "formula": "\\begin{align*} u ^ { c , \\gamma } = & u _ { r } ^ { c , \\gamma } e _ { r } + u _ { \\theta } ^ { c , \\gamma } e _ { \\theta } = \\frac { ( U _ { \\theta } ^ { c , \\gamma } ) ' } { r } e _ { r } + \\frac { U _ { \\theta } ^ { c , \\gamma } } { r \\sin \\theta } e _ { \\theta } , \\\\ p ^ { c , \\gamma } = & u _ { r } ^ { c , \\gamma } - \\frac { 1 } { 2 } ( u _ { \\theta } ^ { c , \\gamma } ) ^ { 2 } = \\frac { 1 } { r } \\left ( ( U _ { \\theta } ^ { c , \\gamma } ) ' - \\frac { ( U _ { \\theta } ^ { c , \\gamma } ) ^ { 2 } } { 2 r \\sin ^ { 2 } \\theta } \\right ) . \\end{align*}"}
+{"id": "6577.png", "formula": "\\begin{align*} \\prod _ { p \\leq x } \\left ( 1 - \\frac { 1 } { p ^ j } \\right ) = \\zeta ( j ) ^ { - 1 } + O \\left ( \\frac { 1 } { x ^ { j - 1 } } \\right ) . \\end{align*}"}
+{"id": "1015.png", "formula": "\\begin{align*} \\mathcal H = \\{ f \\in \\mathcal B _ b ( E ) : \\eqref { e q 8 . 3 } \\ , \\ , \\ , \\ , \\eta \\in \\mathfrak D _ { [ b ] } ( L ) \\cap F ( D ) \\} . \\end{align*}"}
+{"id": "3215.png", "formula": "\\begin{align*} I _ { 1 , n } \\leq \\sum \\limits _ { k = 1 } ^ n { \\left | \\frac { \\Psi _ k \\Gamma ( \\mu + \\alpha + 1 ) } { \\Gamma ( \\mu + 1 ) T ^ { \\alpha + \\mu } E _ { \\alpha , 1 + \\frac { \\beta } { \\alpha } , 1 + \\frac { \\beta + \\mu } { \\alpha } } ( - \\lambda _ { k } T ^ { \\alpha + \\beta } ) } \\right | ^ 2 } . \\end{align*}"}
+{"id": "4787.png", "formula": "\\begin{align*} d = _ \\mathbb { R } 0 \\to C \\vert A x \\vert = _ \\mathbb { R } 0 \\to C \\end{align*}"}
+{"id": "108.png", "formula": "\\begin{align*} \\begin{aligned} & a _ { \\lambda } ( \\vec { u } , \\vec { v } ) = a ( \\vec { u } , \\vec { v } ) + \\lambda b ( \\vec { v } , \\operatorname { d i v } \\vec { u } ) = \\langle \\vec { f } , \\vec { v } \\rangle , \\end{aligned} \\end{align*}"}
+{"id": "7023.png", "formula": "\\begin{align*} ( \\mu + \\partial _ t - \\Delta + b \\cdot \\nabla ) ^ { - 1 } f & : = ( \\mu + \\partial _ t - \\Delta ) ^ { - 1 } \\\\ & - ( \\mu + \\partial _ t - \\Delta ) ^ { - \\frac { 1 } { 2 } - \\frac { 1 } { q } } Q _ { p } ( 1 + T _ p ) ^ { - 1 } G _ { p } ( \\mu + \\partial _ t - \\Delta ) ^ { - \\frac { 1 } { 2 } + \\frac { 1 } { r } } f , \\end{align*}"}
+{"id": "8835.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } S ( 0 ) [ \\varphi ( 0 , \\cdot ) ] ( x ) & = & \\varphi ( 0 , x ) , \\\\ S ( t ) [ \\varphi ( 0 , \\cdot ) ] ( x ) & = & \\frac { \\exp ( - \\mu t ) } { \\sqrt { 4 d \\pi t } } \\int _ { - \\infty } ^ \\infty \\varphi ( 0 , y ) \\exp \\left ( - \\frac { ( x + c t - y ) ^ 2 } { 4 d t } \\right ) { \\rm d } y , t > 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "6803.png", "formula": "\\begin{align*} \\begin{cases} \\frac { q ^ 2 - 1 } 4 & q \\equiv 1 \\pmod 4 \\\\ \\frac { ( q - 1 ) ( q + 3 ) } 4 & q \\equiv - 1 \\pmod 4 \\end{cases} \\end{align*}"}
+{"id": "3812.png", "formula": "\\begin{align*} \\dim \\left ( \\bigcap _ { i = 1 } ^ t \\Im ( H _ { [ n ' - k ] , A _ { i } } ) \\right ) = \\max _ { P _ 1 \\sqcup P _ 2 \\sqcup \\dots \\sqcup P _ s = [ t ] } \\left ( \\sum _ { i = 1 } ^ s \\Big | \\bigcap _ { j \\in P _ i } A _ j \\Big | - ( s - 1 ) ( n ' - k ) \\right ) . \\end{align*}"}
+{"id": "6721.png", "formula": "\\begin{align*} x ^ { 1 2 } - 1 & = ( x - 1 ) ( x + 1 ) ( x ^ 2 + x + 1 ) ( x ^ 2 + 1 ) ( x ^ 2 - x + 1 ) ( x ^ 4 - x ^ 2 + 1 ) \\\\ & = \\Phi _ 1 ( x ) \\Phi _ 2 ( x ) \\Phi _ 3 ( x ) \\Phi _ 4 ( x ) \\Phi _ 6 ( x ) \\Phi _ { 1 2 } ( x ) \\end{align*}"}
+{"id": "4604.png", "formula": "\\begin{align*} K : = \\left \\lbrace ( x ' , x _ d ) \\in \\R ^ { d - 1 } \\times \\R \\ , \\middle \\vert \\ , 0 < x _ d \\ , , \\ 0 \\le | x ' | < \\psi ( x _ d ) \\right \\rbrace \\ , . \\end{align*}"}
+{"id": "5892.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { C } ( G ) ) = \\begin{cases} \\dfrac { ( m n - n ) ( m n - n - 1 ) ^ { 3 } + m n ( n - 1 ) ^ { 3 } } { 2 } , & \\\\ \\dfrac { ( m n - 2 n ) ( m n - 2 n - 1 ) ^ { 3 } + m n ( 2 n - 1 ) ^ { 3 } } { 2 } , & \\end{cases} \\end{align*}"}
+{"id": "2066.png", "formula": "\\begin{align*} & - \\int _ 0 ^ s \\Theta ( u , x ) f ' ( u ) \\mathrm { d } u + f ( s ) { \\Theta } ( s , x ) - f ( 0 ) { \\Theta } ( 0 , x ) = \\alpha ^ 2 \\int _ 0 ^ s f ( u ) \\mathrm { d } \\xi _ 1 ( u , x ) . \\end{align*}"}
+{"id": "904.png", "formula": "\\begin{align*} f _ i ( A ) x = 0 , x \\in \\mathbb { F } _ q ^ n \\end{align*}"}
+{"id": "1511.png", "formula": "\\begin{align*} J ( u , { \\mathcal O } ) = \\int _ { \\mathcal O } F ( D u ) + f \\ , u \\ , d x , { \\mathcal O } \\Subset \\Omega , \\end{align*}"}
+{"id": "2957.png", "formula": "\\begin{align*} \\Omega ' ( v ) = \\{ w \\in V \\mid \\{ w , v \\} \\in E \\} \\Omega ( v ) = \\Omega ' ( v ) \\cup \\{ v \\} , \\end{align*}"}
+{"id": "1812.png", "formula": "\\begin{align*} \\zeta ^ h ( x _ i ) = \\zeta ( x _ { i - 1 } + ) \\mbox { a n d } \\zeta ^ h ( x _ i + ) = \\zeta ( x _ i + ) \\ , . \\end{align*}"}
+{"id": "8683.png", "formula": "\\begin{align*} ( J _ G V ) ^ H = J ( V ^ H ) . \\end{align*}"}
+{"id": "4159.png", "formula": "\\begin{align*} \\varphi ( g _ 1 g _ 2 ) = \\varphi ( g _ 1 ) \\varphi ( g _ 2 ) \\mbox { o r } \\varphi ( g _ 1 g _ 2 ) = \\varphi ( g _ 2 ) \\varphi ( g _ 1 ) \\ , . \\end{align*}"}
+{"id": "379.png", "formula": "\\begin{align*} ( x + y - z ) ^ p - p a b K _ p = x ^ p + y ^ p - z ^ p = 0 . \\end{align*}"}
+{"id": "8522.png", "formula": "\\begin{align*} \\tau ( \\chi ^ { - 1 } ) \\cdot L ( f , \\chi , 1 ) = \\sum _ { a \\in ( \\Z / m \\Z ) ^ { \\times } } \\chi ( a ) ^ { - 1 } \\cdot \\int _ { \\frac { a } { m } } ^ { 0 } 2 i \\pi f ( z ) d z \\end{align*}"}
+{"id": "8354.png", "formula": "\\begin{align*} \\mu ( ( 1 - \\lambda ) K + \\lambda K ^ 1 ( 0 , \\bar x ) ) ^ s = \\lambda \\mu ( K ) ^ s + ( 1 - \\lambda ) \\mu ( K ^ 1 ( 0 , \\bar x ) ) ^ s , \\end{align*}"}
+{"id": "3095.png", "formula": "\\begin{align*} u _ \\varphi ( t ) & = \\left ( \\begin{array} { c c c } 1 & c _ 1 \\ , t ^ { p ^ { e _ 1 } } & \\frac { 1 } { 2 } \\ , \\lambda \\ , c _ 1 ^ 2 \\ , t ^ { 2 \\ , p ^ { e _ 1 } } \\\\ 0 & 1 & \\lambda \\ , c _ 1 \\ , t ^ { p ^ { e _ 1 } } \\\\ 0 & 0 & 1 \\end{array} \\right ) ( \\ , c _ 1 \\in k \\backslash \\{ 0 \\} , \\lambda \\in k \\backslash \\{ 0 \\} , e _ 1 \\geq 0 \\ , ) , \\end{align*}"}
+{"id": "6321.png", "formula": "\\begin{align*} K = Q ^ { \\otimes 2 } \\widetilde { K } , Q = 1 - | u _ 0 \\rangle \\langle u _ 0 | , \\widetilde { K } ( x , y ) = - \\sum _ { z \\in \\mathbb { Z } ^ 3 } n \\omega _ { \\ell , \\lambda } ( P _ z ( x ) - y ) . \\end{align*}"}
+{"id": "2190.png", "formula": "\\begin{align*} F = F _ r ( r , z , t ) \\bar e _ r + F _ \\varphi ( r , z , t ) \\bar e _ \\varphi + F _ z ( r , z , t ) \\bar e _ z . \\end{align*}"}
+{"id": "2445.png", "formula": "\\begin{align*} \\begin{cases} u ( \\xi ) \\sim c k ^ { - 1 } \\xi ^ { - 1 } , \\\\ u ' ( \\xi ) \\sim - c k ^ { - 1 } \\xi ^ { - 2 } , \\end{cases} { \\rm { a s } } \\xi \\to + \\infty . \\end{align*}"}
+{"id": "181.png", "formula": "\\begin{align*} Q \\ = \\ \\bigcup _ { n = 2 } ^ { \\infty } \\ ( \\{ t \\} \\times A _ n ) \\times ( \\{ t _ n \\} \\times A _ 1 ) . \\end{align*}"}
+{"id": "5729.png", "formula": "\\begin{align*} M _ \\gamma ( t _ 1 ) \\leq \\limsup _ { \\gamma ' \\to - \\infty } M _ { \\gamma ' } ( t _ 1 ) \\leq 2 \\limsup _ { \\gamma ' \\to - \\infty } \\Vert \\Pi _ { \\gamma ' } q ( t _ 1 ) \\Vert _ G = 0 . \\end{align*}"}
+{"id": "1424.png", "formula": "\\begin{align*} \\{ I _ + ^ { \\beta - \\alpha \\gamma } f _ \\alpha \\} ( x ) & = \\frac { 1 } { \\Gamma ( \\beta - \\alpha \\gamma ) } \\int _ 0 ^ x ( x - u ) ^ { \\beta - \\alpha \\gamma - 1 } f _ \\alpha ( u ) \\ , d u \\end{align*}"}
+{"id": "6424.png", "formula": "\\begin{align*} I _ { n , i , 2 2 } ^ { 2 1 } ( \\theta ) = - \\frac { \\ln \\left ( X _ { \\frac { i - 1 } { n } } \\right ) X _ { \\frac { i - 1 } { n } } } { \\delta \\alpha ^ 2 X _ { \\frac { i - 1 } { n } } ^ { 1 / \\alpha } } k ' _ \\alpha ( z ^ n _ i ( \\theta ) ) - \\frac { X _ { \\frac { i - 1 } { n } } } { \\delta X _ { \\frac { i - 1 } { n } } ^ { 1 / \\alpha } } f ' _ \\alpha ( z ^ n _ i ( \\theta ) ) \\end{align*}"}
+{"id": "5007.png", "formula": "\\begin{align*} E ( \\theta , \\phi ) = E \\ : \\Gamma \\ : f ^ 2 ( \\theta ) \\sum _ { m = 1 } ^ { M } \\sum _ { n = 1 } ^ { N } e ^ { j \\Phi _ { m n } } e ^ { j k . \\hat { r } _ { m n } } . \\end{align*}"}
+{"id": "9201.png", "formula": "\\begin{gather*} K K ^ { - 1 } = K ^ { - 1 } K = 1 , \\ \\ K E = q ^ 2 E K , \\ \\ K F = q ^ { - 2 } F K , \\ \\ E F - F E = \\kappa ( E F - F E ) . \\end{gather*}"}
+{"id": "3030.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } \\overline { P } \\\\ P \\end{array} \\right ) = ( J ^ { - 1 } ) ^ { T } \\left ( \\begin{array} { c } p _ { x } \\\\ p _ { s } \\end{array} \\right ) \\ , , \\end{align*}"}
+{"id": "2156.png", "formula": "\\begin{align*} f _ d : I \\rightarrow J , \\ , \\ , \\ , \\ , \\ , f _ d ( x ) : = f ( x + d ) - f ( x ) . \\end{align*}"}
+{"id": "2456.png", "formula": "\\begin{align*} \\begin{cases} V ( \\xi ) \\sim A _ { 5 } ( \\xi _ { + } - \\xi ) ^ { \\frac { 1 } { m - 1 } } , \\\\ V ' ( \\xi ) \\sim - A _ { 6 } ( \\xi _ { + } - \\xi ) ^ { - \\frac { m - 2 } { m - 1 } } \\end{cases} { \\rm { a s } } \\xi \\nearrow \\xi _ { + } - 0 , \\end{align*}"}
+{"id": "2630.png", "formula": "\\begin{align*} I ( P , L ) : = \\{ ( p , \\ell ) \\in P \\times L : p \\in \\ell \\} . \\end{align*}"}
+{"id": "3573.png", "formula": "\\begin{align*} ( 1 + a ) \\alpha ^ 2 \\tau _ 0 \\tau _ 1 ( i d - \\tau ^ 2 ) = ( a + b ) \\alpha \\tau _ 0 ( i d - \\tau ) , \\end{align*}"}
+{"id": "4056.png", "formula": "\\begin{align*} E _ i ( \\sigma ) = \\gamma + \\ln ( - \\sigma ) + \\sum _ { k = 1 } ^ { \\infty } \\frac { \\sigma ^ k } { k k ! } . \\end{align*}"}
+{"id": "7618.png", "formula": "\\begin{align*} D _ j = D _ j ( S ^ 1 ) . \\end{align*}"}
+{"id": "9022.png", "formula": "\\begin{align*} y _ { i i } = y _ { j j } 1 \\le i , j \\le n . \\end{align*}"}
+{"id": "8947.png", "formula": "\\begin{align*} \\begin{aligned} | \\nabla \\varphi ( x ) - O | & \\le | \\nabla \\varphi ( x ) - B | + d ( B , S O ( n ) ) \\le 2 | \\nabla \\varphi ( x ) - B | + d ( \\nabla \\varphi ( x ) , S O ( n ) ) \\\\ & \\le C \\| d ( \\nabla v , S O ( n ) ) \\| _ { L ^ { p ( \\cdot ) } ( Q ) } ^ \\frac { 1 } { 2 } + d ( \\nabla \\varphi ( x ) , S O ( n ) ) \\end{aligned} \\end{align*}"}
+{"id": "1029.png", "formula": "\\begin{align*} C _ { t , n } : = \\sum _ { s = 1 } ^ { n } \\Delta _ n ^ { s , t } , n \\in \\N , \\ \\ t \\in \\{ 1 , \\dots , n \\} , \\end{align*}"}
+{"id": "75.png", "formula": "\\begin{align*} D \\varphi ( x _ 0 , t _ 0 ) = D u ( x _ 0 , t _ 0 ) \\neq 0 , \\phi _ t ( x _ 0 , t _ 0 ) = u _ t ( x _ 0 , t _ 0 ) = 0 \\end{align*}"}
+{"id": "8617.png", "formula": "\\begin{align*} v _ 1 ( t ) = \\frac { v _ 1 ( 0 ) } { 1 + v _ 1 ( 0 ) \\int _ { 0 } ^ { t } ( 1 + ( v _ 1 ^ { - 2 } ( K _ 1 + \\phi _ 1 ) ) ( \\tau ) ) d \\tau } = : m ( 0 ) r ^ { - 1 } ( t ) . \\end{align*}"}
+{"id": "7763.png", "formula": "\\begin{align*} d _ H ( \\hat \\Sigma _ i ( \\lambda ) , \\hat \\Sigma _ i ( \\lambda _ 0 ) ) = d _ H ( \\tilde \\Sigma _ i ( \\lambda ) , \\tilde \\Sigma _ i ( \\lambda _ 0 ) ) < 2 \\nu _ { m _ 1 } , \\end{align*}"}
+{"id": "8535.png", "formula": "\\begin{align*} \\lambda _ 1 = \\lambda _ 2 = \\mu _ 1 = \\mu _ 2 = 0 \\end{align*}"}
+{"id": "5708.png", "formula": "\\begin{align*} \\psi _ { i , 3 } : = ( 0 , 2 ^ { - 1 / 2 } m \\varphi _ i ) , \\ \\psi _ { i , 4 } : = ( 2 ^ { - 1 / 2 } m \\varphi _ i , 0 ) . \\end{align*}"}
+{"id": "1757.png", "formula": "\\begin{align*} X = a ^ \\prime U _ 0 - \\frac { n a ^ \\prime } { 2 } V _ 0 \\in \\mathfrak { g } , \\end{align*}"}
+{"id": "2413.png", "formula": "\\begin{align*} S _ n = 2 ^ { ( 9 s + 1 8 ) n + 4 s + 8 } \\cdot \\int _ { \\mathbb { Z } _ 2 } f ^ { ( s ) } ( t ) \\mathrm { d } t \\end{align*}"}
+{"id": "3310.png", "formula": "\\begin{align*} K _ { n - 1 } ' = \\frac { 1 } { \\sqrt { n } } \\exp \\left ( \\frac { 1 } { 2 } n \\left ( \\mu _ 0 - \\frac { \\mu _ 0 + X _ 2 + \\dots + X _ n } { n } \\right ) ^ 2 \\right ) = \\frac { 1 } { \\sqrt { n } } \\exp \\left ( \\frac { ( ( n - 1 ) \\mu _ 0 - ( S _ n - S _ 1 ) ) ^ 2 } { 2 n } \\right ) . \\end{align*}"}
+{"id": "7087.png", "formula": "\\begin{align*} b = \\mathsf { f } d x + \\mathsf { h } , \\end{align*}"}
+{"id": "1353.png", "formula": "\\begin{align*} t ( h + k ) = n k . \\end{align*}"}
+{"id": "6574.png", "formula": "\\begin{align*} \\sum _ { p \\leq x } \\frac { \\log p } { p } = \\log x + R ( x ) , \\end{align*}"}
+{"id": "9072.png", "formula": "\\begin{align*} \\begin{array} { l l l } x ( t + 1 ) & = v ( t + 1 ) - \\underline v = \\hat C V ( t + 1 ) - \\underline v = \\hat C \\Big [ C V ( t ) + D p - \\phi ( v ) \\Big ] - \\underline v \\\\ & = \\hat { C } C \\hat { C } ^ { - 1 } x ( t ) + ( \\hat { C } C \\hat { C } ^ { - 1 } - I ) \\underline { v } + \\hat { C } ( D p - U ( x ( t ) ) - \\mathbf { 1 } \\frac { \\beta } { 2 } ) . \\end{array} \\end{align*}"}
+{"id": "3925.png", "formula": "\\begin{align*} \\begin{aligned} d _ i \\varphi ^ { \\prime \\prime } ( x ) + \\bar \\lambda \\varphi ( x ) = & 0 , 0 < x < L , \\\\ \\gamma _ { 1 1 } \\varphi ( 0 ) + \\gamma _ { 1 2 } \\varphi ^ \\prime ( 0 ) = & \\gamma _ { 2 1 } \\varphi ( L ) + \\gamma _ { 2 2 } \\varphi ^ \\prime ( L ) = 0 , \\end{aligned} \\end{align*}"}
+{"id": "5391.png", "formula": "\\begin{align*} \\nu _ 0 & = \\frac { \\Delta h _ 1 } { \\mu } \\\\ \\nu _ j & = \\nu _ { j - 1 } + \\Delta h _ { j + 1 } \\ , \\frac { 1 + \\cdots + \\rho ^ j } { \\mu } , 1 \\leq j \\leq n - 1 , \\end{align*}"}
+{"id": "7344.png", "formula": "\\begin{align*} W _ \\phi \\left ( w z ^ 1 \\cdots z ^ { C + 1 } \\right ) = W _ \\phi \\left ( W _ \\phi ( w z ^ 1 \\cdots z ^ { C } ) z ^ { C + 1 } \\right ) = W _ \\phi \\left ( W _ \\phi \\left ( w ^ 1 w ^ 2 \\cdots w ^ { C + 1 } z ^ 1 \\cdots z ^ { C } \\right ) z ^ { C + 1 } \\right ) . \\end{align*}"}
+{"id": "6001.png", "formula": "\\begin{align*} \\sup _ { B _ { R / 2 } ( \\partial M ) } \\frac { | \\nabla u | ^ 2 } { u ^ 2 } = \\sup _ { B _ { R / 2 } ( \\partial M ) } w \\leq C ( n ) \\left ( \\frac { 1 } { R ^ 2 } + K \\right ) . \\end{align*}"}
+{"id": "3465.png", "formula": "\\begin{align*} b ^ { + } _ { \\iota } ( W ) = m , b ^ { - } _ { \\iota } ( W ) = n + 8 . \\end{align*}"}
+{"id": "4727.png", "formula": "\\begin{align*} S _ { \\mathrm { f r e e } } \\ ; = \\ ; S ^ { ( \\xi ) } _ { \\mathrm { f r e e } } + S ^ { ( \\psi ) } _ { \\mathrm { f r e e } } \\ , , S _ { \\mathrm { i n t } } \\ ; = \\ ; S ^ { ( \\xi ) } _ { \\mathrm { i n t } } + S ^ { ( \\psi ) } _ { \\mathrm { i n t } } + Q _ { \\mathrm { i n t } } ^ { ( \\psi , \\xi ) } \\ , . \\end{align*}"}
+{"id": "6716.png", "formula": "\\begin{align*} y ( x + n ) - y ( x ) = 0 , \\end{align*}"}
+{"id": "3934.png", "formula": "\\begin{align*} y _ n = T _ n z _ n , n \\geq 1 \\end{align*}"}
+{"id": "1654.png", "formula": "\\begin{align*} \\lambda \\left ( \\delta _ { 0 } \\right ) = \\ln \\left [ \\delta ^ { \\left ( 3 \\left ( T + 1 \\right ) \\right ) ^ { - 1 } } \\right ] \\geq \\lambda _ { 1 } . \\end{align*}"}
+{"id": "6646.png", "formula": "\\begin{align*} ( H _ { 2 \\lambda \\cos , y } \\psi ) _ { n } = \\psi _ { n + 1 } + \\psi _ { n - 1 } + 2 \\lambda \\cos ( x + \\mathrm { i } y + n \\alpha ) \\psi _ { n } , \\end{align*}"}
+{"id": "9132.png", "formula": "\\begin{align*} \\begin{aligned} & x ^ { ( \\beta , s ) } _ { 1 , t } \\mapsto v ^ { 2 t } w _ { \\beta , s } , 1 \\leq t \\leq 2 , \\\\ & x ^ { ( \\beta , s ) } _ { 2 , t } \\mapsto v ^ { - 3 + 2 t } w _ { \\beta , s } , 1 \\leq t \\leq 3 . \\end{aligned} \\end{align*}"}
+{"id": "3448.png", "formula": "\\begin{align*} U _ i = \\mathrm { I m } ( d : \\Omega ^ 0 ( Y _ i ) \\to \\Omega ^ 1 ( Y _ i ) ) ^ I . \\end{align*}"}
+{"id": "5373.png", "formula": "\\begin{align*} \\bar { b } ^ S + a _ i ^ S = \\begin{cases} \\displaystyle \\theta _ i ^ 1 + \\sum _ { j \\in N } p _ { i j } ^ 1 \\ , a ^ S _ j & i \\in S \\cup N ^ { \\{ 1 \\} } \\\\ \\\\ \\displaystyle \\sum _ { j \\in N } p _ { i j } ^ 0 \\ , a ^ S _ j & i \\in N ^ { \\{ 0 , 1 \\} } \\setminus S . \\end{cases} \\end{align*}"}
+{"id": "7176.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } \\partial _ t E _ v - \\partial _ x ^ 2 E _ v & = ( 1 - ( v _ 0 + V ) ^ 2 + \\varphi _ 1 ) E _ v + \\varphi _ 2 E _ v ^ 2 - \\frac { 1 } { 3 } E _ v ^ 3 - E _ w + \\varphi _ 3 , \\\\ \\partial _ t E _ w - \\rho \\partial _ { x } ^ 2 E _ w & = \\varepsilon \\left ( E _ v - \\gamma E _ w \\right ) + \\varepsilon J _ 0 , \\\\ E _ v ( 0 ) = 0 , & E _ w ( 0 ) = 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "5056.png", "formula": "\\begin{align*} f _ i ^ { ( 1 ) } = f _ i ^ { ( 2 ) } ~ \\forall i \\neq i _ 0 , ~ ~ D ( f _ { i _ 0 } ^ { ( 1 ) } , f _ { i _ 0 } ^ { ( 2 ) } ) \\leq \\delta , \\end{align*}"}
+{"id": "3607.png", "formula": "\\begin{align*} \\alpha ^ 3 \\tau _ 0 ^ 2 \\tau _ 1 \\tau ^ m - ( 1 + a ) \\alpha ^ 2 \\tau _ 0 \\tau _ 1 z ^ m + ( a + b ) \\alpha \\tau _ 0 \\tau ^ m - b z ^ m = 0 , \\end{align*}"}
+{"id": "644.png", "formula": "\\begin{align*} \\mu _ { H _ r , \\epsilon } ( \\rho ) = \\int _ { \\epsilon } ^ { \\rho } \\frac { ( n H _ r J _ { n , r , \\epsilon } ( \\xi ) ) ^ { \\frac { 1 } { r } } } { \\sqrt { \\cosh ^ { \\frac { 2 ( n - r ) } { r } } ( \\xi ) - ( n H _ r J _ { n , r , \\epsilon } ( \\xi ) ) ^ { \\frac { 2 } { r } } } } \\ , d \\xi , \\rho \\geq \\epsilon . \\end{align*}"}
+{"id": "1070.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { s = 0 } ^ { n - 1 } S _ { 1 , k } ( n , s , t ) & \\leq n ^ { - d } ( t + 1 ) ^ { - d } K _ 3 \\left ( \\frac { r \\sin ( \\pi d ) } { \\pi } \\right ) ^ { k - 1 } \\| \\zeta _ { } \\| _ L \\| T \\| ^ { k - 1 } \\| \\phi _ { } \\| _ L \\\\ & = n ^ { - d } ( t + 1 ) ^ { - d } \\| \\phi _ { } \\| _ L K _ 3 \\{ r \\sin ( \\pi d ) \\} ^ { k - 1 } . \\end{aligned} \\end{align*}"}
+{"id": "9045.png", "formula": "\\begin{align*} T ( a b ) = ( T a ) b + a ( T b ) , S ^ i ( a b ) = ( S ^ i a ) b + ( - 1 ) ^ { p ( a ) } a ( S ^ i b ) ( i \\in [ N ] ) . \\end{align*}"}
+{"id": "7519.png", "formula": "\\begin{align*} d _ Z ( z _ 1 , z _ 2 ) & < d _ { Z , { \\delta _ 1 } } ( z _ 1 , z _ 2 ) \\leq d _ Z ( z _ 1 , q _ 1 ) + d _ Z ( z _ 2 , q _ 2 ) + d _ { Z , \\delta _ 1 } ( q _ 1 , q _ 2 ) \\\\ & \\leq d _ Z ( z _ 1 , q _ 1 ) + d _ Z ( z _ 2 , q _ 2 ) + d _ { Z } ( q _ 1 , q _ 2 ) + \\varepsilon \\\\ & = d _ Z ( z _ 1 , z _ 2 ) + \\varepsilon , \\end{align*}"}
+{"id": "8045.png", "formula": "\\begin{align*} \\limsup _ { \\delta \\rightarrow 0 } \\limsup _ { N \\rightarrow + \\infty } \\frac { N } { \\gamma _ N ^ 2 } \\log \\sup _ { \\sigma \\in \\mathcal { T } } P \\left ( \\varepsilon _ 9 ^ N > \\epsilon \\right ) = - \\infty . \\end{align*}"}
+{"id": "2500.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\| A ( t ) - A _ \\infty \\| _ 2 = \\lim _ { t \\to \\infty } \\int _ \\Omega \\nabla \\vartheta \\cdot \\nabla ( A ( t ) - A _ \\infty ) \\ \\mathrm { d } x = 0 \\ , . \\end{align*}"}
+{"id": "6235.png", "formula": "\\begin{align*} f ( u ) & = - p u ( 1 - u ) ^ 2 - q u ( 1 - u ) , \\\\ D ( u ) & = 3 ( D _ i - D _ g ) ( u - \\alpha ) ( u - \\beta ) , \\\\ g ( u ) & = ( r _ i + \\lambda _ g ) \\cdot u ( 1 - u ) ( u - \\gamma ) . \\end{align*}"}
+{"id": "2664.png", "formula": "\\begin{align*} \\frac { \\partial L ^ { ( 1 ) } } { \\partial \\dot { q } ^ { i } } = 0 . \\end{align*}"}
+{"id": "6842.png", "formula": "\\begin{align*} \\mathbf { 1 } _ N = \\begin{pmatrix} [ F _ { N - 1 } ( \\phi _ N , \\ldots , \\omega _ { N - 2 } ) ] ^ { - 1 } & 0 \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} e ^ { - i \\omega _ { N - 1 } } & & & \\\\ & \\ddots & & \\\\ & & e ^ { - i \\omega _ { N - 1 } } & \\\\ & & & e ^ { i ( N - 1 ) \\omega _ { N - 1 } } \\end{pmatrix} \\begin{pmatrix} X & \\\\ & u ^ { ( N - 1 ) } _ { N N } \\end{pmatrix} . \\end{align*}"}
+{"id": "7086.png", "formula": "\\begin{align*} ( \\lambda - \\Delta + b \\cdot \\nabla ) u = f , \\end{align*}"}
+{"id": "4628.png", "formula": "\\begin{align*} f ( x ' ) - a _ { n , k } \\leq f ( x ' ) - f _ { n - 1 } ( x ' ) = \\sum _ { j = n } ^ \\infty 2 ^ { - \\gamma m j } g _ j ( x ' ) \\leq \\frac { 1 } { 2 } 2 ^ { - \\gamma m n } \\sum _ { j = 0 } ^ \\infty 2 ^ { - \\gamma m j } \\le 2 ^ { - \\gamma m n } . \\end{align*}"}
+{"id": "709.png", "formula": "\\begin{align*} f ( x ) = \\Phi ^ 6 ( x ) - 1 8 \\delta \\Phi ^ 5 ( x ) + 1 3 5 \\Phi ^ 4 ( x ) - 5 4 0 \\delta \\Phi ^ 3 ( x ) + 1 2 1 5 \\Phi ^ 2 ( x ) - 1 4 5 8 \\delta \\Phi ( x ) + 7 2 9 - m . \\end{align*}"}
+{"id": "8232.png", "formula": "\\begin{align*} P _ { { \\bf X _ 2 } | Y _ 1 } ( { \\bf x } _ 2 | 0 ) & = \\begin{cases} r _ 0 ^ { y _ 1 = 0 } \\ & { \\bf x } _ 2 = 0 ^ M \\\\ r _ b ^ { y _ 1 = 0 } \\alpha _ { b , \\ell } ^ { y _ 1 = 0 } & { \\bf x } _ 2 = { \\bf b } _ { \\ell } \\end{cases} , \\\\ & \\forall \\ell \\in \\left [ { M \\choose b } \\right ] , b \\in [ B _ ] . \\end{align*}"}
+{"id": "3025.png", "formula": "\\begin{align*} \\begin{array} { l } \\lambda = \\beta ^ 2 - \\beta - 3 \\alpha ^ 2 , \\\\ \\mu = - 2 \\alpha \\beta + 3 \\alpha , \\\\ \\nu = 3 \\alpha ^ 2 + 2 \\alpha \\beta - 3 \\alpha + 5 \\beta - \\beta ^ 2 - 4 . \\end{array} \\end{align*}"}
+{"id": "6094.png", "formula": "\\begin{align*} S _ d \\begin{pmatrix} \\alpha \\\\ r \\end{pmatrix} S _ d \\begin{pmatrix} \\beta \\\\ s \\end{pmatrix} = S _ d \\begin{pmatrix} \\alpha \\beta \\\\ r + s \\end{pmatrix} + \\sum \\limits _ { i + j = r + s } \\Delta _ { r , s } ^ { j } S _ d \\begin{pmatrix} \\alpha \\beta & 1 \\\\ i & j \\end{pmatrix} , \\end{align*}"}
+{"id": "1939.png", "formula": "\\begin{align*} \\gamma > \\max \\left \\{ \\frac { q } { q - p } , \\frac { p } { p - 1 } , \\frac { N } { p } \\right \\} \\ p \\le N , \\ \\gamma = 1 \\ p > N . \\end{align*}"}
+{"id": "3240.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty ( \\alpha - \\alpha ^ 2 ) ^ n \\ : \\Box ^ n V | _ { \\alpha p } \\ : d \\alpha & = \\int _ 0 ^ \\infty \\alpha ^ n \\ : \\Box ^ n V | _ { \\alpha p } \\ : d \\alpha + \\O \\Big ( \\frac { 1 } { \\varepsilon \\underline { \\omega } } \\Big ) \\\\ & = \\frac { 1 } { \\underline { \\omega } ^ { n + 1 } } \\int _ 0 ^ \\infty \\alpha ^ n \\ : \\Box ^ n V | _ { \\alpha \\hat { p } } \\ : d \\alpha + \\O \\Big ( \\frac { 1 } { \\varepsilon \\underline { \\omega } } \\Big ) \\ : , \\end{align*}"}
+{"id": "1801.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t a + \\partial _ x q = 0 \\\\ \\partial _ t q + \\partial _ x \\left ( \\dfrac { q ^ 2 } { a } + \\dfrac 1 2 \\ , g \\ , \\zeta _ 1 \\ , a ^ 2 \\right ) = - \\dfrac 1 2 \\ , g \\ , a ^ 2 \\ , \\partial _ x \\zeta _ 1 - g \\ , a \\ , \\partial _ x \\zeta _ 2 \\end{array} \\right . \\end{align*}"}
+{"id": "5649.png", "formula": "\\begin{align*} X \\cup Y = C _ 1 \\cup \\dots \\cup C _ \\ell \\cup C _ 1 ' \\cup \\dots \\cup C _ k ' \\end{align*}"}
+{"id": "3336.png", "formula": "\\begin{align*} v ' = \\frac { v + v _ * } { 2 } + \\frac { | v - v _ * | } { 2 } \\sigma v ' _ * = \\frac { v + v _ * } { 2 } - \\frac { | v - v _ * | } { 2 } \\sigma , \\end{align*}"}
+{"id": "2327.png", "formula": "\\begin{align*} \\begin{cases} \\frac { 1 } { s _ { i } } + \\frac { \\alpha _ { i } } { n - 1 } > 0 , \\\\ \\beta _ { i } \\geq 0 , \\end{cases} \\quad \\mathrm { o r } \\begin{cases} \\frac { 1 } { s _ { i } } + \\frac { \\alpha _ { i } } { n - 1 } > 0 , \\\\ \\beta _ { i } < 0 , \\\\ \\frac { 1 } { s _ { i } } + \\frac { \\alpha _ { i } + \\beta _ { i } } { n } > 0 , \\end{cases} i = 1 , 2 , 3 , \\end{align*}"}
+{"id": "6016.png", "formula": "\\begin{align*} \\partial _ t \\begin{pmatrix} \\rho ^ A \\\\ \\rho ^ B \\end{pmatrix} = \\Delta \\begin{pmatrix} \\rho ^ A \\\\ \\rho ^ B \\end{pmatrix} - \\nabla \\cdot \\left ( \\chi ( \\rho ) \\cdot g _ E \\right ) , \\end{align*}"}
+{"id": "4830.png", "formula": "\\begin{align*} \\big ( 2 \\langle \\lambda - \\mu , \\mu + \\rho ( { \\mathfrak h } ) \\rangle + \\mid \\mid \\lambda - \\mu \\mid \\mid ^ 2 \\big ) _ { V [ \\lambda ] } ( \\mu ) = 2 \\sum _ { \\alpha \\in \\Phi ^ + ( { \\mathfrak h } ) } \\sum _ { k \\geq 1 } \\langle \\mu + k \\alpha , \\alpha \\rangle _ { V [ \\lambda ] } ( \\mu + k \\alpha ) . \\end{align*}"}
+{"id": "2087.png", "formula": "\\begin{align*} - \\int _ 0 ^ s \\Theta ( u , x ) f ' ( u ) d u + f ( s ) { \\Theta } ( s , x ) - f ( 0 ) { \\Theta } ( 0 , x ) = \\alpha \\int _ 0 ^ s f ( u ) d \\xi _ { 1 } ( u , x ) . \\end{align*}"}
+{"id": "969.png", "formula": "\\begin{align*} u _ i = P _ D ( g _ i ) + R ^ D f _ i ( \\cdot , u _ i ) + R ^ D \\mu _ i D , i = 1 , 2 . \\end{align*}"}
+{"id": "7289.png", "formula": "\\begin{align*} S _ { m , r } ( n ) : = \\frac { 1 } { m } \\sum _ { j \\in \\{ 0 , \\ldots , m - 1 \\} \\setminus \\{ j _ m \\} ^ \\ast } \\sec ^ { 2 n } \\left ( \\frac { j } { m } \\pi \\right ) e ^ { 2 \\pi i r j / m } , \\end{align*}"}
+{"id": "4350.png", "formula": "\\begin{align*} \\min _ { x } & \\ \\sum _ { ( i , j , r , s ) \\in [ n ] ^ 4 } c _ { i , j } d _ { r , s } x _ { i , r } x _ { j , s } \\\\ \\mathrm { s . t . } & \\ x \\in \\mathcal { X } = \\{ x \\in \\{ 0 , 1 \\} ^ { [ n ] ^ 2 } : \\sum _ { i \\in [ n ] } x _ { i , r } = 1 \\ \\forall r \\in [ n ] , \\ \\sum _ { r \\in [ n ] } x _ { i , r } = 1 \\ \\forall i \\in [ n ] \\} \\end{align*}"}
+{"id": "7913.png", "formula": "\\begin{align*} \\mathrm { t r } ( \\alpha \\wedge \\beta ) = \\mathrm { t r } ( \\alpha ) \\wedge \\mathrm { t r } ( \\beta ) , \\mathrm { t r } ( d \\alpha ) = d ( \\mathrm { t r } ( \\alpha ) ) , \\quad \\boldsymbol { n } ( \\delta \\alpha ) = \\delta \\boldsymbol { n } ( \\alpha ) , \\end{align*}"}
+{"id": "856.png", "formula": "\\begin{align*} \\delta ( i _ { ( X ) } a ( X ) ) = g ( a ( X ) , a ( X ) ) + \\frac { n - 1 } { n + 1 } \\delta ( ( X + \\rho V ) f ) ^ { 2 } - 2 R _ { i j } X ^ { i } X ^ { j } + D i v , \\end{align*}"}
+{"id": "4531.png", "formula": "\\begin{align*} E _ N ^ { ( K ) } ( t , [ s ] _ K ) = \\sum _ { i = K + 1 } ^ N \\int _ { s _ { K + 1 } = 0 } ^ \\infty \\ ! . . . \\ ! \\int _ { s _ { N } = 0 } ^ \\infty \\varphi _ i ( [ s ] _ i ) n _ N ( t , [ s ] _ N ) \\ , d [ s ] _ { K + 1 , N } . \\end{align*}"}
+{"id": "2762.png", "formula": "\\begin{align*} T ^ { * } M = T ^ { * } M | _ { \\Theta } \\times T ^ { * } M | _ { Q , P } , \\end{align*}"}
+{"id": "2874.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\left \\| f - f _ k \\right \\| _ { \\dot { W } ^ { 1 , X } ( \\mathbb { R } ^ n ) } = 0 \\quad \\lim _ { k \\to \\infty } \\left \\| ( f - f _ k ) \\mathbf { 1 } _ { B ( { \\bf 0 } , R ) } \\right \\| _ X = 0 . \\end{align*}"}
+{"id": "8930.png", "formula": "\\begin{align*} \\phi _ { i } ( x _ 1 , x _ 2 , \\ldots , x _ { i + 1 } ) & = \\\\ + \\{ ( - 1 ) ^ { ( | x _ 1 | + \\ldots + | x _ { i - 2 } | ) | x _ { i } | } - ( - 1 ) ^ { | x _ { i - 1 } | | x _ { i + 1 } | } \\} & ( - 1 ) ^ { ( | x _ 1 | + \\ldots + | x _ { i - 2 } | ) | x _ { i + 1 } | } \\overline { [ [ x _ 1 , \\ldots } \\overline { \\ldots , x _ { i - 1 } ] _ l , [ x _ { i } , x _ { i + 1 } ] ] } \\\\ & \\in \\ker ( \\lambda _ { i } ) . \\end{align*}"}
+{"id": "1350.png", "formula": "\\begin{align*} 2 \\sum _ { 1 \\leq i \\leq j \\leq n } s _ i s _ j \\geq \\bigg | \\sum _ { i = 1 } ^ n p _ i ^ 2 - \\frac { 4 \\alpha ^ 2 m ^ 2 } { n } \\bigg | = s _ i ^ 2 \\ , \\ , ( \\ref { l e m m a 2 } ( i i i ) ) \\end{align*}"}
+{"id": "6462.png", "formula": "\\begin{align*} \\bar \\nabla _ { e _ i } ( k \\nu ) = ( \\nabla _ { e _ i } k ) \\nu + k \\nabla _ { e _ i } \\nu = ( \\nabla _ { e _ i } k ) \\nu - k A ( e _ i ) \\end{align*}"}
+{"id": "2677.png", "formula": "\\begin{align*} \\delta S ^ { ( 2 ) } = \\int ^ { t _ { 2 } } _ { t _ { 1 } } \\sum _ { i = 1 } ^ { n } \\left [ \\frac { \\partial L ^ { ( 2 ) } } { \\partial \\dot { q } ^ { i } } - \\frac { d } { d t } \\left ( \\frac { \\partial L ^ { ( 2 ) } } { \\partial \\ddot { q } ^ { i } } \\right ) \\right ] \\delta \\dot { q } ^ { i } d t + \\left [ \\sum _ { i = 1 } ^ { n } \\left ( \\frac { \\partial L ^ { ( 2 ) } } { \\partial \\dot { q } ^ { i } } \\right ) \\delta \\dot { q } ^ { i } \\right ] ^ { t _ { 2 } } _ { t _ { 1 } } . \\end{align*}"}
+{"id": "7633.png", "formula": "\\begin{align*} \\mu _ p ( _ p ( B ( \\alpha q , R ) ) ) = \\int _ { _ p B ( \\alpha q , R ) } \\mathrm { e } ^ { - \\delta _ \\Gamma \\beta _ \\xi ( p , \\alpha q ) } \\ , \\mathrm { d } \\mu _ { \\alpha q } ( \\xi ) . \\end{align*}"}
+{"id": "1349.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ n p _ i p _ j = \\sum _ { i = 1 } ^ n p _ i ^ 2 + 2 \\sum _ { 1 \\leq i \\leq j \\leq n } p _ i p _ j \\end{align*}"}
+{"id": "8549.png", "formula": "\\begin{align*} F ( n , k ) = \\frac { ( a ) _ k ( b ) _ k } { ( n ) _ k ( 2 n ) _ k } \\end{align*}"}
+{"id": "1917.png", "formula": "\\begin{align*} & \\sum _ { x \\in G } V ( x ) \\eta ^ 2 ( x ) v ^ 2 _ + ( x , T ) e ^ { \\xi ( x . T ) } \\mu ( x ) - \\sum _ { x \\in G } V ( x ) \\eta ^ 2 ( x ) v ^ 2 _ + ( x , 0 ) e ^ { \\xi ( x . 0 ) } \\mu ( x ) \\\\ & \\leq 2 \\int _ 0 ^ T \\sum _ { x , y \\in G } v ^ 2 _ + ( x , t ) e ^ { \\xi ( y , t ) } [ \\eta ( y ) - \\eta ( x ) ] ^ 2 \\omega ( x , y ) \\ , d t . \\end{align*}"}
+{"id": "4200.png", "formula": "\\begin{align*} & - 5 0 4 \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 1 2 ) } = \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right . \\\\ & \\left . \\cdot { \\rm c h } ( \\widetilde { T X } + 2 \\wedge ^ 2 \\widetilde { L _ R \\otimes C } - \\widetilde { L _ R \\otimes C } \\otimes \\widetilde { L _ R \\otimes C } + \\widetilde { L _ R \\otimes C } ) \\right \\} ^ { ( 1 2 ) } . \\end{align*}"}
+{"id": "8416.png", "formula": "\\begin{align*} \\Theta ( \\overline { \\mathcal { B } _ 1 ( f ) \\ast \\mathcal { B } _ 1 ( g ) ) } & = \\Theta ( \\overline { \\mathcal { B } _ 1 ( f \\ast g ) } ) \\\\ & = \\overline { \\mathcal { B } _ 2 ( f \\ast g ) } = \\overline { \\mathcal { B } _ 2 ( f ) } \\ast \\overline { \\mathcal { B } _ 2 ( g ) } \\\\ & = \\overline { \\mathcal { B } _ 2 ( f ) } = \\Theta ( \\overline { \\mathcal { B } _ 2 ( f ) } ) . \\end{align*}"}
+{"id": "219.png", "formula": "\\begin{align*} \\frac { d } { d t } = \\frac 1 f \\frac { d } { d \\tau } , \\frac { d ^ 2 } { d t ^ 2 } = \\frac 1 f \\frac { d } { d \\tau } \\left ( \\left ( \\frac 1 f \\right ) \\frac { d } { d \\tau } \\right ) = \\frac 1 { f ^ 2 } \\frac { d ^ 2 } { d \\tau ^ 2 } - \\frac 1 { f ^ 3 } \\frac { d f } { d \\tau } \\frac { d } { d \\tau } , \\end{align*}"}
+{"id": "5914.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { N C } ( G ) ) & = ( p q - 1 ) ( p q - 2 ) ^ { 2 } - 4 ( p q - 2 ) \\dfrac { ( q - 1 ) ( q - 2 ) + q ( p - 1 ) ( p - 2 ) } { 2 } \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ + ( q - 1 ) ( q - 2 ) ^ { 2 } + q ( p - 1 ) ( p - 2 ) ^ { 2 } \\\\ & = p ^ { 3 } q ^ { 3 } - 2 p ^ { 2 } q ^ { 2 } - p q ^ { 3 } - p ^ { 3 } q ^ { 2 } + p q ^ { 2 } - 3 q ^ { 2 } - 3 q p ^ { 2 } + 2 q + p ^ { 3 } q + q ^ { 3 } - 4 \\end{align*}"}
+{"id": "6203.png", "formula": "\\begin{align*} \\left ( D ( \\phi ) \\phi ^ { \\prime } \\right ) ^ { \\prime } + \\left ( c - \\dot f ( \\phi ) \\right ) \\phi ' + g ( \\phi ) = 0 . \\end{align*}"}
+{"id": "1441.png", "formula": "\\begin{align*} E _ r ^ { p , q } ( R f _ * \\Omega _ { X / Y } ( \\log E ) , F ) \\Rightarrow E ^ { p + q } ( R f _ * \\Omega _ { X / Y } ( \\log E ) ) = R ^ { p + q } f _ * \\Omega _ { X / Y } ( \\log E ) \\end{align*}"}
+{"id": "8609.png", "formula": "\\begin{align*} \\frac { d v _ n } { d t } + \\sum ^ n _ { j = 1 } \\binom { n } { j } v _ j v _ { n + 1 - j } + K _ n ( t ; x ) + \\phi _ n ( t ; x ) = 0 ~ ~ f o r ~ ~ n = 2 , 3 , . . . , \\end{align*}"}
+{"id": "8668.png", "formula": "\\begin{align*} \\underline { \\mathfrak { h } } _ { x } = { \\rm S p a n \\ , } \\left \\{ \\xi _ { M ' , H } ( x ) ; \\ , \\xi \\in \\mathfrak { h } \\ , \\right \\} . \\end{align*}"}
+{"id": "4274.png", "formula": "\\begin{align*} \\div _ f h = \\div h - h ( \\nabla f , \\cdot ) = \\nabla _ j h _ { i j } - h _ { i j } \\nabla _ j f . \\end{align*}"}
+{"id": "5521.png", "formula": "\\begin{align*} d \\geq \\deg ( G ) - ( 2 g - 2 ) + 1 + b + 1 0 M - c _ 1 - c _ 2 = \\frac { u ^ 2 } { 2 } + 1 2 . \\end{align*}"}
+{"id": "1586.png", "formula": "\\begin{align*} h = F ( y ) \\ge | y | \\ , | D F ( y ) | / C . \\end{align*}"}
+{"id": "1988.png", "formula": "\\begin{align*} f _ l ^ n ( s ) = \\widehat { ( f ( \\psi _ M ( t _ n + s ) ) ) } _ l . \\end{align*}"}
+{"id": "5034.png", "formula": "\\begin{align*} \\nu _ i ^ { S \\cup \\{ j \\} } & = \\frac { \\nu _ i ^ S w _ i ^ S - \\nu _ j ^ S \\frac { w _ j ^ S } { a _ { j j } ^ S } a _ { i j } ^ S } { w _ i ^ S - \\frac { w _ j ^ S } { a _ { j j } ^ S } a _ { i j } ^ S } = \\nu _ j ^ S - \\frac { w _ i ^ S } { w _ i ^ S - \\frac { w _ j ^ S } { a _ { j j } ^ S } a _ { i j } ^ S } ( \\nu _ j ^ S - \\nu _ i ^ S ) = \\nu _ j ^ S - \\frac { w _ i ^ S } { w _ i ^ { S \\cup \\{ j \\} } } ( \\nu _ j ^ S - \\nu _ i ^ S ) , \\end{align*}"}
+{"id": "1307.png", "formula": "\\begin{align*} W _ 2 ( \\rho _ 1 ( t ) , \\rho _ \\infty ) ^ 2 = \\mu _ 1 ( t ) ^ 2 + \\Big ( \\sqrt { \\Sigma _ { 1 1 } } ( t ) - \\sqrt { \\overline { \\Sigma } } \\Big ) ^ 2 \\leq \\mu _ 1 ( t ) ^ 2 + \\Big | \\Sigma _ { 1 1 } ( t ) - \\overline { \\Sigma } _ { 1 1 } \\Big | . \\end{align*}"}
+{"id": "7896.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { p - 1 } \\omega _ { q } ^ { f _ { 0 , i ^ { ( n ) } } - f _ { 0 , j ^ { ( n ) } } } = \\omega _ { q } ^ { f _ { 0 , i } - f _ { 0 , j } } \\sum _ { n = 0 } ^ { p - 1 } \\omega _ { q } ^ { \\frac { q } { p } n a _ { v - 1 } ( i _ { \\pi ( v ) } - j _ { \\pi ( v ) } ) } = 0 . \\end{align*}"}
+{"id": "3701.png", "formula": "\\begin{align*} \\bar { \\alpha } _ i \\cdot \\bar { B } _ 2 ( \\alpha _ i ) \\cdot \\bar { B } _ 2 ( \\alpha _ k ) & = \\alpha _ k ^ 2 \\otimes \\bigg [ \\binom { a _ 1 } { 2 } + \\binom { a _ i } { 2 } \\bigg ] \\gamma + \\alpha _ i ^ 2 \\otimes \\binom { a _ 1 } { 2 } \\gamma \\\\ & + \\binom { a _ 1 } { 2 } \\gamma \\otimes \\alpha _ i ^ 2 + \\bigg [ \\binom { a _ 1 } { 2 } + \\binom { a _ i } { 2 } \\bigg ] \\gamma \\otimes \\alpha _ k ^ 2 \\\\ \\end{align*}"}
+{"id": "8504.png", "formula": "\\begin{align*} | ( g _ 1 \\circ g _ 2 \\circ \\dotsb \\circ g _ m ) ^ n ( a ) | = | g ^ n ( a ) | = | a | = 1 , \\end{align*}"}
+{"id": "2412.png", "formula": "\\begin{align*} A _ n \\left ( t + \\frac { 1 } { 4 } \\right ) = 2 ^ { ( 9 s + 1 8 ) n + 4 s + 8 } \\cdot f ( t ) , \\end{align*}"}
+{"id": "4289.png", "formula": "\\begin{align*} d _ { p \\textrm { - } \\mathrm { v a r } } ( \\mathbf { x } , \\hat { \\mathbf { x } } ) : = \\max _ { 1 \\le i \\le [ p ] } \\| \\mathbf { x } ^ i - \\hat { \\mathbf { x } } ^ i \\| _ { p / i \\textrm { - } \\mathrm { v a r } } . \\end{align*}"}
+{"id": "5829.png", "formula": "\\begin{align*} d \\mu = e ^ { - | X ( 0 ) | ^ 2 / 4 } \\sqrt { \\det g _ 0 } d x ^ 1 \\wedge \\dots \\wedge d x ^ n . \\end{align*}"}
+{"id": "6291.png", "formula": "\\begin{align*} N = \\tilde { O } \\left ( \\frac { 1 } { r } \\log _ 2 \\left ( \\frac { \\mu _ r R _ 0 ^ r } { 2 \\varepsilon } \\right ) \\right ) , \\end{align*}"}
+{"id": "7300.png", "formula": "\\begin{align*} \\frac { 1 } { 3 k } \\sum _ { j = 0 } ^ { 3 k - 1 } \\sec ^ 2 \\left ( \\frac { 4 j } { 3 k } \\pi \\right ) \\omega ^ j = - k \\end{align*}"}
+{"id": "7660.png", "formula": "\\begin{align*} \\varphi _ 2 ( n ) = \\abs { \\omega ( n ) - \\omega _ { \\alpha } ( n ) } \\delta _ { n \\neq n _ 0 } , \\ , \\ , \\ , W ( n ) = e ^ { \\delta | n - l | } . \\end{align*}"}
+{"id": "7419.png", "formula": "\\begin{align*} c ' _ 3 & = b _ 2 c _ 3 + ( a _ 3 - d _ 3 ) a _ 4 . \\\\ \\end{align*}"}
+{"id": "5740.png", "formula": "\\begin{align*} X _ - ( t ) Y _ - ( t ) = & \\sum _ { i : \\gamma ^ + _ i < \\gamma _ * } \\xi _ i ^ + ( t ) \\mathcal { E } ^ + _ i ( t ) + \\sum _ { i : \\gamma ^ - _ i < \\gamma _ * } \\xi _ i ^ - ( t ) \\mathcal { E } ^ - _ i ( t ) + \\sum _ { i \\in I _ 1 } \\xi _ { i , 1 } ( t ) \\mathcal { E } _ { i , 1 } ( t ) + \\xi _ { i , 2 } ( t ) \\mathcal { E } _ { i , 2 } ( t ) \\\\ & + \\sum _ { i \\in I _ 2 } \\xi _ { i , 3 } ( t ) \\mathcal { E } _ { i , 3 } ( t ) + \\varepsilon _ 1 ^ 2 \\xi _ { i , 4 } ( t ) \\mathcal { E } _ { i , 4 } ( t ) . \\end{align*}"}
+{"id": "1102.png", "formula": "\\begin{align*} 2 ^ { - i a } & = \\sup _ { r \\in ( 0 , \\infty ) } \\left ( \\frac { r } { 2 ^ i r } \\right ) ^ a \\leq \\sup _ { x _ 0 \\in \\mathbb { R } ^ n , \\ , r \\in ( 0 , \\infty ) } \\left ( \\frac { | x _ 0 | + r } { | x _ 0 | + 2 ^ i r } \\right ) ^ a \\\\ & \\leq \\sup _ { x _ 0 \\in \\mathbb { R } ^ n , \\ , r \\in ( 0 , \\infty ) } \\left ( \\frac { | x _ 0 | + r } { 2 ^ i | x _ 0 | + 2 ^ i r } \\right ) ^ a = 2 ^ { - i a } , \\end{align*}"}
+{"id": "4084.png", "formula": "\\begin{align*} \\chi R _ 0 ( z _ m ) \\chi = \\sum _ { j = 1 } ^ { \\frac { d - 1 } { 2 } } \\tilde { B } _ j ( r , - z _ 0 ) + 2 i \\Big ( \\frac { z _ 0 } { 2 \\pi } \\Big ) ^ { d - 1 } \\mathcal { E } ^ * _ { \\chi } ( i \\bar { z _ 0 } ) \\mathcal { E } _ { \\chi } ( - i \\bar { z _ 0 } ) + i m \\Big ( \\frac { z _ 0 } { 2 \\pi } \\Big ) ^ { d - 1 } \\mathcal { E } ^ * _ { \\chi } ( z _ 0 ) \\mathcal { E } _ { \\chi } ( \\bar { z _ 0 } ) \\end{align*}"}
+{"id": "7351.png", "formula": "\\begin{align*} \\lambda _ { s t a g e 1 , \\beta } ( \\delta ) = \\ln \\frac { \\sum _ { \\mu = 0 } ^ { M _ { \\beta } - 1 } P ( x _ { \\beta } ( \\delta ) = q _ { \\mu } ^ { \\beta } \\mid y _ { \\beta } ( \\delta ) ) } { P ( x _ { \\beta } ( \\delta ) = 0 \\mid y _ { \\beta } ( \\delta ) ) } . \\end{align*}"}
+{"id": "9028.png", "formula": "\\begin{align*} 1 = \\frac { 1 } { | G | } + \\sum _ { i = 1 } ^ r ( 1 - \\frac { 1 } { g _ i } ) + \\frac { 1 } { 2 } \\sum _ { i = r + 1 } ^ { r + s } ( 1 - \\frac { 1 } { g _ i } ) \\end{align*}"}
+{"id": "452.png", "formula": "\\begin{align*} \\begin{aligned} x _ { i _ m } ^ * ( t _ { q + 1 } ) - x _ { m + 1 } ^ * ( t _ { q + 1 } ) = { } & x _ { i _ m } ^ q ( t _ { q + 1 } ) - x _ { m + 1 } ^ q ( t _ { q + 1 } ) \\\\ [ 2 p t ] \\ge { } & x _ { i _ m } ^ q ( t _ { q } + \\Delta \\widetilde { t } _ q ) - x _ { m + 1 } ^ q ( t _ { q } + \\Delta \\widetilde { t } _ q ) = 0 . \\end{aligned} \\end{align*}"}
+{"id": "1585.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 | D F ( y + t \\ , ( x - y ) ) - D F ( y ) | \\ , d t & \\le C \\ , | D F ( y ) | \\ , \\int _ 0 ^ 1 \\eta _ H \\left ( t \\ , \\frac { | x - y | } { | y | } \\right ) \\ , d x \\\\ & \\le C \\ , | D F ( y ) | \\ , \\eta _ H \\left ( \\frac { | x - y | } { | y | } \\right ) \\ , \\int _ 0 ^ 1 \\eta _ H ( t ) \\ , d t \\\\ & \\le \\frac { C H } { H + 1 } \\ , | D F ( y ) | \\ , \\eta _ H \\left ( \\frac { | x - y | } { | y | } \\right ) . \\end{align*}"}
+{"id": "3717.png", "formula": "\\begin{align*} G ( x , y | E ) = G _ 0 ( x , y ) + \\tilde { G } _ 0 ( x , a | E ) \\Phi ^ { - 1 } ( E ) \\tilde { G } _ 0 ( a , y | E ) . \\end{align*}"}
+{"id": "6209.png", "formula": "\\begin{align*} \\gamma = \\frac { r _ i } { r _ i + \\lambda _ g } , \\end{align*}"}
+{"id": "1429.png", "formula": "\\begin{align*} m _ 1 e _ i < m _ 2 e _ j \\Longleftrightarrow i > j ( i = j m _ 1 < m _ 2 ) . \\end{align*}"}
+{"id": "1486.png", "formula": "\\begin{align*} P _ { d } ( L _ { m } ) = f ( m + d ) L _ { m + d } , \\forall \\ m \\in \\mathbb { Z } , \\end{align*}"}
+{"id": "2638.png", "formula": "\\begin{align*} \\hat { x } _ { \\alpha } = ( x _ 1 , \\ldots , x _ { \\alpha - 1 } , x _ { \\alpha + 1 } , \\ldots , x _ N ) \\in \\mathbb { T } ^ { d ( N - 1 ) } . \\end{align*}"}
+{"id": "5419.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } f \\big ( T ^ { i } \\circ \\mathcal I ( \\tfrac { 1 } { 2 } ) \\big ) = \\sum _ { i = - n } ^ { - 1 } f \\big ( \\mathcal I \\circ T ^ i ( \\tfrac { 1 } { 2 } ) \\big ) = - \\sum _ { i = - n } ^ { - 1 } f ( T ^ i ( \\tfrac { 1 } { 2 } ) ) = S _ { - n } f ( \\tfrac { 1 } { 2 } ) , \\end{align*}"}
+{"id": "7626.png", "formula": "\\begin{align*} \\tilde { r } _ d ( f _ 1 ( z ) , f _ 2 ( z ) , f _ 3 ( z ) ) = \\prod _ { j = 1 } ^ l \\big ( f _ 2 ( x _ j ) + \\sqrt { - 1 } f _ 3 ( x _ j ) \\big ) ^ { \\epsilon ( j ) } \\end{align*}"}
+{"id": "1559.png", "formula": "\\begin{align*} \\inf \\{ J _ { n } ( w ) : w \\in { \\rm L i p } ( \\overline { B } ) , w = \\psi _ n \\ \\} . \\end{align*}"}
+{"id": "2637.png", "formula": "\\begin{align*} P _ { \\leq M } ( V u ) = P _ { \\leq M } ( P _ { \\leq M _ 0 } V u ) \\sim \\sum _ { i + j \\leq M , i \\leq M _ 0 } P _ i V P _ j u \\sim \\sum _ { i \\leq M _ 0 } P _ i V \\sum _ { j = 0 } ^ { M - i } P _ j u \\\\ \\sim \\sum _ { i \\leq M _ 0 } P _ i V P _ { \\leq M } u \\sim P _ { \\leq M _ 0 } V P _ { \\leq M } u = V P _ { \\leq M } u , \\end{align*}"}
+{"id": "6473.png", "formula": "\\begin{align*} A ( X ) = - \\sqrt { 2 } X \\end{align*}"}
+{"id": "6657.png", "formula": "\\begin{align*} L ( E , \\mathrm { i } y ) = \\max \\{ \\log \\lambda + y , L ( E , \\mathrm { i } 0 ) \\} . \\end{align*}"}
+{"id": "2297.png", "formula": "\\begin{align*} Q _ r ( 1 ) = c _ r = \\binom { 2 r } { r } - \\binom { 2 r } { r + 1 } . \\end{align*}"}
+{"id": "1662.png", "formula": "\\begin{align*} \\partial _ { \\nu } v \\mid _ { S _ { T } } = 0 , \\partial _ { \\nu } p \\mid _ { S _ { T } } = 0 , \\end{align*}"}
+{"id": "3043.png", "formula": "\\begin{align*} \\mathcal { P T } \\ , V _ \\lambda ( s _ 1 , s _ 2 , s _ 3 ) = \\overline { V _ \\lambda } ( s _ 1 , s _ 2 , - s _ 3 ) = V _ \\lambda ( s _ 1 , s _ 2 , s _ 3 ) \\ , . \\end{align*}"}
+{"id": "619.png", "formula": "\\begin{align*} & \\int _ 0 ^ 1 \\frac { \\sqrt { 1 - x ^ 2 } \\ln ( x ) } { \\left ( 2 - x ^ 2 \\right ) ^ { 3 / 2 } } \\ , d x = \\\\ & \\frac { 1 } { 4 } \\int _ 0 ^ 1 \\left ( \\sum _ { n = 0 } ^ { \\infty } \\left ( - \\frac { 1 } { 2 } \\right ) ^ n \\binom { n } { \\left \\lfloor \\frac { n } { 2 } \\right \\rfloor } ( n + 1 ) u ^ { n + \\frac { 1 } { 2 } } \\ln ( 1 - u ) \\right ) \\ , d u . \\end{align*}"}
+{"id": "844.png", "formula": "\\begin{align*} { Y _ i } = g ( X , l ) { \\Psi _ i } - { F ^ { - 1 } } g ( X , l ) \\Psi { l _ i } , \\end{align*}"}
+{"id": "8909.png", "formula": "\\begin{align*} \\sinh ( n \\theta ) & = \\begin{cases} ( - 1 ) ^ { \\frac { n + 1 } { 2 } } i T _ n ( i \\sinh \\theta ) & n : , \\\\ ( - 1 ) ^ { \\frac { n } { 2 } } i \\cosh \\theta U _ { n - 1 } ( i \\sinh \\theta ) & n : . \\end{cases} \\end{align*}"}
+{"id": "3930.png", "formula": "\\begin{align*} & \\dot z _ n ( t ) = \\int _ 0 ^ L z _ t ( t , x ) \\varphi _ n ( x ) d x \\\\ & = \\left [ D z _ x ( \\cdot ) \\varphi _ n ( \\cdot ) - D z ( \\cdot ) \\varphi _ n ^ { \\prime } ( \\cdot ) \\right ] _ 0 ^ L \\\\ & + \\left ( - \\lambda _ n D + Q \\right ) z _ n ( t ) + F _ n [ z ( t ) ] \\\\ & + B \\sum _ { j = 1 } ^ N u _ j ( t ) \\int _ 0 ^ L \\varphi _ n ( x ) b _ j ( x ) d x , \\end{align*}"}
+{"id": "4271.png", "formula": "\\begin{align*} \\delta ^ 2 _ g \\nu ( h , h ) = \\left . \\frac { d ^ 2 } { d s ^ 2 } \\right | _ { s = 0 } \\nu ( g ( s ) ) = \\frac { 1 } { ( 4 \\pi \\tau ) ^ { n / 2 } } \\int _ M < N _ f h , h > e ^ { - f } d V , \\\\ \\end{align*}"}
+{"id": "7085.png", "formula": "\\begin{align*} u = ( \\mu - \\Delta ) ^ { - 1 } f - ( \\mu - \\Delta ) ^ { - \\frac { 1 } { 2 } - \\frac { 1 } { 2 q } } Q _ { p } ( q ) ( 1 + T _ p ) ^ { - 1 } G _ { p } ( r ) ( \\mu - \\Delta ) ^ { - \\frac { 1 } { 2 } + \\frac { 1 } { 2 r } } f , f \\in L ^ p \\end{align*}"}
+{"id": "2963.png", "formula": "\\begin{align*} \\left \\| \\frac { f ' } { r } \\right \\| _ { L _ { \\beta } ^ 2 } ^ 2 = \\left \\| \\left ( \\frac { f } { r } \\right ) ' + \\frac { f } { r ^ 2 } \\right \\| _ { L _ { \\beta } ^ 2 } ^ 2 = \\left \\| \\left ( \\frac { f } { r } \\right ) ' \\right \\| _ { L _ { \\beta } ^ 2 } ^ 2 + 2 \\Re \\left \\langle \\left ( \\frac { f } { r } \\right ) ' , \\frac { f } { r ^ 2 } \\right \\rangle + \\left \\| \\frac { f } { r ^ 2 } \\right \\| _ { L _ { \\beta } ^ 2 } ^ 2 . \\end{align*}"}
+{"id": "7019.png", "formula": "\\begin{align*} b ( x ) = \\sqrt { \\delta } \\frac { d - 2 } { 2 } \\frac { x } { | x | ^ 2 } \\in \\mathbf { F } _ \\delta \\end{align*}"}
+{"id": "7768.png", "formula": "\\begin{align*} r _ { n + 1 } & = r _ { n } , \\ \\ \\nu _ { n + 1 } = \\nu _ n - b \\tilde M _ n ^ 4 \\epsilon _ n ^ { \\frac { 1 } { m } } , \\\\ \\delta _ { n + 1 } & = \\delta _ { n } , \\ \\ \\tilde { M } _ { n + 1 } = \\tilde { M } _ { n } + 2 0 m \\tilde M _ n ^ 2 \\epsilon _ { n } , \\\\ c _ { n + 1 } & = c _ { n } - { b } R _ { n } ^ { 3 m ^ 2 } \\tilde { M } _ { n } ^ { m + 2 } ( 2 \\delta _ { n } ^ { - 1 } r _ { n } ) ^ { r _ { n } } \\epsilon _ { n } . \\end{align*}"}
+{"id": "2980.png", "formula": "\\begin{align*} \\frac { d ( x _ 2 ^ 2 - x _ 1 ^ 2 ) } { d t } \\leq v _ 2 ^ 2 - v _ 1 ^ 2 \\leq - \\frac { \\kappa _ 1 ( 1 + \\mathcal { A } ( v ) ( 0 ) ) } { 2 ( 1 - \\alpha ) } ( x _ 2 ^ 2 - x _ 1 ^ 2 ) ^ { 1 - \\alpha } = : - a ( x _ 2 ^ 2 - x _ 1 ^ 2 ) ^ { 1 - \\alpha } . \\end{align*}"}
+{"id": "3329.png", "formula": "\\begin{align*} & \\left ( L ^ { 2 } ( \\O _ { N - m _ { 0 } + 1 } ; H ^ { 1 } _ { c , x ^ { ( 0 ) } } ) \\right ) ^ { * } = L ^ { 2 } ( \\O _ { N - m _ { 0 } + 1 } ; H ^ { - 1 } _ { c , x ^ { ( 0 ) } } ) \\\\ & \\left ( L ^ { 2 } ( \\O _ { N - m _ { 0 } + 1 } ; H ^ { - 1 } _ { c , x ^ { ( 0 ) } } ) \\right ) ^ { * } = L ^ { 2 } ( \\O _ { N - m _ { 0 } + 1 } ; H ^ { 1 } _ { c , x ^ { ( 0 ) } } ) . \\end{align*}"}
+{"id": "1493.png", "formula": "\\begin{align*} & P _ d ^ { 1 } ( L _ m ) = \\beta + \\nu \\delta _ { m + 2 d , 0 } L _ { m + d } , \\quad q = 1 d \\in \\mathbb { Z } , \\\\ & P _ d ^ { 2 } ( L _ m ) = \\mu \\frac { m + d } { m + 2 d } \\gamma \\delta _ { m + d , \\mathbb { Z } \\setminus \\{ - d \\} } L _ { m + d } , \\quad q = 1 d \\in \\mathbb { Z } , \\\\ & P _ d ^ { 3 } ( L _ m ) = \\beta + \\nu \\delta _ { q ^ { m } , 1 } L _ { m + d } , \\quad q \\neq 1 q ^ d = 1 , \\end{align*}"}
+{"id": "2034.png", "formula": "\\begin{align*} \\frac { P ( x ) } { Q ( x ) } = \\sum _ { | \\gamma | \\leq m } c _ { \\gamma } \\ , x ^ { \\gamma } + \\hbox { t e r m s o f o r d e r $ \\ , \\geq m + 1 $ } . \\end{align*}"}
+{"id": "211.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\dfrac { d q ^ i } { d t } & = & v ^ i \\\\ \\dfrac { d v ^ i } { d t } & = & f ^ i ( q , v ) \\end{array} \\right . i = 1 , \\ldots , n , \\end{align*}"}
+{"id": "4506.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ | A _ 3 ( n ) | ^ 2 \\big ] & = \\sum _ { \\substack { | \\lambda | = n \\\\ m _ { \\lambda _ 1 } ( \\lambda ) \\geqslant 3 \\\\ \\lambda _ 1 > y _ 0 } } \\mathbb { E } \\big [ | a ( \\lambda ) | ^ 2 \\big ] \\\\ & \\leqslant \\sum _ { y _ 0 < k \\leqslant n / 3 } \\frac { 1 } { k ^ 3 } \\sum _ { \\substack { | \\lambda | = n - 3 k \\\\ \\lambda _ 1 \\leqslant k } } \\mathbb { E } \\big [ | a ( \\lambda ) | ^ 2 \\big ] . \\end{align*}"}
+{"id": "5319.png", "formula": "\\begin{align*} v ^ { \\textrm { L P } } = \\min \\ , \\left \\{ \\sum _ { j \\in J } c _ j \\ , x _ j : \\mathbf { x } \\in P ( \\mathcal { F } ) \\right \\} . \\end{align*}"}
+{"id": "2380.png", "formula": "\\begin{align*} ( t , u _ 1 , u _ 2 ) = ( x + y + z , \\ \\frac { y - x } { x + y } , \\ \\frac { z - ( x + y ) } { x + y + z } ) \\ . \\end{align*}"}
+{"id": "6523.png", "formula": "\\begin{align*} X _ n = \\sum ^ { \\infty } _ { i = 1 } a _ i \\varepsilon _ { n - i } , \\end{align*}"}
+{"id": "8291.png", "formula": "\\begin{align*} \\frac { u _ 1 u _ { 1 1 } } { w ^ 2 \\ln w } = - \\varphi ' u _ 1 + \\frac { 2 x _ 1 } { \\rho } , \\end{align*}"}
+{"id": "308.png", "formula": "\\begin{align*} u ( x , 0 ) = u _ 0 ( x ) , x \\in \\real ^ N . \\end{align*}"}
+{"id": "6992.png", "formula": "\\begin{align*} \\mathcal { P } _ \\mathbb { R } ^ 2 : = \\{ [ w _ 1 : w _ 2 : 0 : \\cdots : 0 : w _ { n + 1 } ] \\in \\mathbb { P } _ { \\mathbb { C } } ^ { n } : w _ 1 , w _ 2 , w _ { n + 1 } \\in \\mathbb { R } \\} \\end{align*}"}
+{"id": "3959.png", "formula": "\\begin{align*} T _ A ( x ) = T ^ { \\tau _ A ( x ) } ( x ) . \\end{align*}"}
+{"id": "88.png", "formula": "\\begin{align*} \\inf _ \\theta X _ 2 ( \\theta ) & = \\min \\Big \\{ X _ 2 ( 0 ) , X _ 2 \\Big ( \\frac { p - 2 - \\gamma } { ( p - 2 ) \\gamma } \\Big ) , X _ 2 ( 1 ) \\Big \\} \\\\ & = \\min \\Big \\{ 4 , \\frac { p - 2 } { \\gamma } + \\frac { \\gamma } { p - 2 } + 2 , \\big ( \\sqrt { p - 1 } + \\sqrt { 1 - \\gamma } \\big ) ^ 2 \\Big \\} \\\\ & = \\min \\Big \\{ 4 , \\big ( \\sqrt { p - 1 } + \\sqrt { 1 - \\gamma } \\big ) ^ 2 \\Big \\} . \\end{align*}"}
+{"id": "291.png", "formula": "\\begin{align*} \\left \\langle L _ F e _ { k } , e _ { j } \\right \\rangle = \\begin{cases} \\sum _ { l = 1 } ^ n \\alpha _ l ( k ) \\ , a _ { l , ( \\alpha ( j ) - \\alpha ( k ) ) _ l } & \\textrm { i f } | \\alpha ( j ) | \\geq | \\alpha ( k ) | \\\\ 0 & \\textrm { i f } | \\alpha ( j ) | < | \\alpha ( k ) | . \\end{cases} \\end{align*}"}
+{"id": "3654.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { d } ( | A _ i | - 1 ) + \\sum _ { 1 \\le i < j \\le d } ( | A _ i | + | A _ j | - 1 ) = d | V | - \\binom { d + 1 } { 2 } \\end{align*}"}
+{"id": "2872.png", "formula": "\\begin{align*} \\limsup _ { \\lambda \\to \\infty } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ X = \\left [ \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { X } , \\end{align*}"}
+{"id": "4246.png", "formula": "\\begin{align*} Q ( X , L , \\tau ) = \\left \\{ \\left ( \\prod _ { j = 1 } ^ { 2 k + 1 } \\frac { x _ j \\theta ' ( 0 , \\tau ) } { \\theta ( x _ j , \\tau ) } \\right ) \\frac { \\sqrt { - 1 } \\theta ( u , \\tau ) } { \\theta _ 1 ( 0 , \\tau ) \\theta _ 2 ( 0 , \\tau ) \\theta _ 3 ( 0 , \\tau ) } \\right \\} ^ { ( 4 k + 2 ) } . \\end{align*}"}
+{"id": "901.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c c c c } 0 & 0 & \\ldots & 0 & a _ { _ { n - 1 } } \\\\ a _ { _ 0 } & 0 & \\ldots & 0 & 0 \\\\ 0 & a _ { _ 1 } & \\ddots & \\vdots & \\vdots \\\\ \\vdots & \\ddots & \\ddots & 0 & \\vdots \\\\ 0 & \\ldots & 0 & a _ { _ { n - 2 } } & 0 \\end{array} \\right ) \\end{align*}"}
+{"id": "1313.png", "formula": "\\begin{align*} \\{ \\Phi _ X , \\Phi _ Y \\} _ \\pm = \\pm \\Phi _ { [ X , Y ] } , \\end{align*}"}
+{"id": "93.png", "formula": "\\begin{align*} 2 c _ 1 + c _ 2 = & 2 ( p - \\gamma ) + 2 ( 4 - p + \\gamma ) \\kappa ^ 2 . \\end{align*}"}
+{"id": "463.png", "formula": "\\begin{align*} P _ S = \\prod _ { h \\in S } \\varepsilon ( h ) \\prod _ { h \\in X _ e \\setminus S } ( \\pi ( e ) - \\varepsilon ( h ) ) . \\end{align*}"}
+{"id": "5008.png", "formula": "\\begin{align*} = \\frac { D _ r - D _ a } { D _ r } . \\end{align*}"}
+{"id": "5207.png", "formula": "\\begin{align*} U _ { j } = - \\frac { Z _ { A } } { \\alpha } q ^ { \\alpha + \\beta - 2 } _ { j } \\ ; \\ ; ; \\ ; \\ ; V _ { j } = - \\frac { Z _ { B } } { \\alpha } p ^ { \\alpha } _ { j } q ^ { \\beta - 2 } _ { j } \\end{align*}"}
+{"id": "661.png", "formula": "\\begin{align*} N ^ { 0 } | _ { S ^ { 0 } } : W ^ { \\ell , p } ( \\mathbb { R } ^ { n } ) \\rightarrow W ^ { \\ell , q } ( \\mathbb { R } ^ { n } ) \\frac { 1 } { q } = \\frac { 1 } { p } - \\frac { 1 } { n } , \\end{align*}"}
+{"id": "6957.png", "formula": "\\begin{align*} z _ 1 w _ 1 + \\cdots z _ n w _ n - z _ { n + 1 } w _ { n + 1 } = 0 , \\end{align*}"}
+{"id": "3644.png", "formula": "\\begin{align*} \\lambda _ { k } ( L ( G , p ) ) = \\lambda _ { | E | - d | V | + k } ( L ^ { - } ( G , p ) ) , \\end{align*}"}
+{"id": "5119.png", "formula": "\\begin{align*} \\frac { d _ { a b } f ( \\alpha ) } { d _ { a b } ( \\alpha ) } = \\frac { f ( a \\alpha ) - f ( b \\alpha ) } { a \\alpha - b \\alpha } \\end{align*}"}
+{"id": "4050.png", "formula": "\\begin{align*} \\langle \\mu _ { \\hat X ^ 1 _ t } ( \\omega ) , \\mu _ { \\hat Y ^ 1 _ s } ( \\omega ^ \\prime ) \\rangle _ { \\mathcal { H } _ { \\mathcal { S } } } = \\langle \\mathbb { E } _ { \\mathbb { P } } [ S ( X ) | \\mathcal { F } _ t ] ( \\omega ) , \\mathbb { E } _ { \\mathbb { Q } } [ S ( Y ) | \\mathcal { G } _ s ] ( \\omega ^ \\prime ) \\rangle _ { \\mathcal { H } ^ 1 } , \\end{align*}"}
+{"id": "2694.png", "formula": "\\begin{align*} \\ddot { q } = \\frac { 1 } { \\beta } \\dddot { q } + \\frac { \\alpha } { \\beta } \\delta ( t - t _ { 0 } ) \\end{align*}"}
+{"id": "4354.png", "formula": "\\begin{align*} \\inf _ { x , p , \\theta } & \\ \\Gamma \\theta + \\sum _ { i \\in [ m ] } ( x _ i - \\overline { u } _ i ) ^ 2 + p _ i , \\\\ \\mathrm { s . t . } & \\ x \\in \\mathcal { X } , \\\\ & \\max _ { u _ i \\in \\mathcal { U } _ i } ( x _ i - u _ i ) ^ 2 - ( x _ i - \\overline { u } _ i ) ^ 2 \\leq p _ i + \\theta \\ \\forall i \\in [ m ] . \\end{align*}"}
+{"id": "5594.png", "formula": "\\begin{align*} H _ { e f } & = \\frac { d ^ 2 } { m n } \\mathbf { 1 } \\{ e \\to f \\} \\\\ H _ { e f } ^ { ( 1 ) } & = \\mathbf { 1 } \\{ e \\to f \\} M _ { f _ 1 f _ 3 , f _ 2 } \\\\ H ^ { ( 2 ) } _ { e f } & = \\sum _ { e \\to g \\to f } M _ { e _ 3 f _ 1 , g _ 2 } A _ { f _ 1 f _ 3 , f _ 2 } , \\end{align*}"}
+{"id": "2322.png", "formula": "\\begin{align*} w ( x , t ) = e ^ { - t \\mathcal { L } } w _ { 0 } - \\int ^ { t } _ { 0 } e ^ { - ( t - s ) \\mathcal { L } } \\mathbb { P } \\mathrm { d i v } ( w \\otimes w ) d s . \\end{align*}"}
+{"id": "8344.png", "formula": "\\begin{align*} \\left [ d , \\varphi \\right ] ( x \\bullet y ) = ( d ( \\varphi \\circ ( x ) ) \\bullet \\alpha ( y ) + \\alpha ( x ) \\bullet ( d \\circ \\varphi ( y ) ) - ( \\varphi \\circ d ( x ) ) \\bullet \\alpha ( y ) - \\alpha ( x ) \\bullet ( \\varphi \\circ d ( y ) ) . \\end{align*}"}
+{"id": "2004.png", "formula": "\\begin{align*} & \\phi _ { 1 , j } ( \\theta ) = \\theta \\psi ( x _ { j + 1 } , t _ n ) + ( 1 - \\theta ) \\psi ( x _ j , t _ n ) , \\phi _ { 2 , j } ( \\theta ) = \\theta \\psi ^ n _ { j + 1 } + ( 1 - \\theta ) \\psi ^ n _ { j } . \\end{align*}"}
+{"id": "1883.png", "formula": "\\begin{align*} \\mathcal { M } _ { p , v } ^ { \\Re , + , u } ( K , z ) = \\int _ { v ^ + \\cap ( K \\cap H _ { u , z } ) } \\Re ( x \\cdot v ) ^ p d x , \\end{align*}"}
+{"id": "506.png", "formula": "\\begin{align*} \\gcd ( \\overline { \\alpha } _ i - 1 , p ^ { \\kappa _ p } ) = \\gcd ( ( \\overline { \\alpha } _ i \\bmod { p ^ { \\kappa _ p } } ) - 1 , p ^ { \\kappa _ p } ) \\mid \\overline { \\beta _ i } \\bmod { p ^ { \\kappa _ p } } , \\end{align*}"}
+{"id": "7546.png", "formula": "\\begin{align*} \\Sigma _ i ( t ) & \\leq C \\int _ 0 ^ t \\Phi ( \\tau ) \\ d \\tau \\ , i = 1 , 2 , * \\ , \\\\ \\Sigma _ 3 ( t ) & \\leq C \\varepsilon t + C \\int _ 0 ^ t \\Phi ( \\tau ) \\ d \\tau \\ . \\\\ \\end{align*}"}
+{"id": "8183.png", "formula": "\\begin{align*} \\mathbb { E } [ b ( X ^ n ) ] = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\mathbb { E } [ b ( X _ i ) ] \\end{align*}"}
+{"id": "3127.png", "formula": "\\begin{align*} u _ { \\psi ^ * } ( t ) & = \\left ( \\begin{array} { c c c } 1 & t ^ { p ^ { e _ 1 } } & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "112.png", "formula": "\\begin{align*} \\langle A \\vec { u } , \\vec { v } \\rangle = a ( \\vec { u } , \\vec { v } ) : = \\left ( \\varepsilon \\left ( \\vec { u } \\right ) , \\varepsilon \\left ( \\vec { v } \\right ) \\right ) . \\end{align*}"}
+{"id": "2532.png", "formula": "\\begin{align*} I ^ { \\prime \\ , ( s ) } ( M , L ) = I ^ { ( - s ) } ( M , L ; \\Omega ) ' \\ ; , I ^ { \\prime \\ , m } ( M , L ) = I ^ { - m } ( M , L ; \\Omega ) ' \\ ; . \\end{align*}"}
+{"id": "7374.png", "formula": "\\begin{align*} \\| ( u , w ) \\| _ X : = \\| u \\| _ 1 + \\| u _ x \\| _ 1 + \\| w \\| _ 2 + \\| w _ x \\| _ 2 , \\end{align*}"}
+{"id": "8560.png", "formula": "\\begin{align*} & - ( a - n ) ^ 2 ( b - n ) ^ 2 \\sum _ { k = 0 } ^ { m } F ( n + 1 , k ) + n ^ 2 ( - 1 - a - b + 2 n ) ( - a - b + 2 n ) \\sum _ { k = 0 } ^ { m } F ( n , k ) \\\\ & = G ( n , m + 1 ) - G ( n , 0 ) , \\end{align*}"}
+{"id": "587.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { j - 1 } \\Big | \\Big \\langle { \\frac { T ^ \\ast ( \\tilde h _ i ) } { { \\| h _ i \\| } _ { F ^ \\ast } } } , { \\frac { \\tilde h _ j } { \\| h _ j \\| _ F } } \\Big \\rangle \\Big | = \\sum \\limits _ { i = 1 } ^ { j - 1 } \\Big | \\Big \\langle { \\frac { \\tilde h _ i } { \\| h _ i \\| _ { F ^ \\ast } } } , { \\frac { T ( \\tilde h _ j ) } { \\| h _ j \\| _ F } } \\Big \\rangle \\Big | < { \\frac { \\eta _ j } { 2 } } , \\end{align*}"}
+{"id": "4125.png", "formula": "\\begin{align*} \\sum _ { i \\in \\mathcal { J } } \\bigl ( \\alpha _ i \\bar { \\gamma } _ i ( \\theta _ i ) + \\lambda _ i \\bar { \\xi } _ i ( \\theta ) \\bigr ) = 0 \\theta \\in \\R ^ J , \\end{align*}"}
+{"id": "3093.png", "formula": "\\begin{align*} \\det h _ \\psi ( z ) = \\prod _ { i = 1 } ^ r z ^ { m _ i \\ , \\dim _ k V _ { m _ i } } = z ^ { d } , \\end{align*}"}
+{"id": "8669.png", "formula": "\\begin{align*} T ^ { 1 , 0 } _ x \\widehat X \\oplus T ^ { 0 , 1 } _ x \\widehat X \\oplus \\underline { \\mathfrak { h } } _ { x } = \\mathbb C T _ x \\widehat X , \\ \\ \\mbox { f o r e v e r y $ x \\in \\widehat X $ } , \\end{align*}"}
+{"id": "6739.png", "formula": "\\begin{align*} Q ( m , z ) = \\frac { 1 } { m ^ 2 } S _ 1 ( z ) - \\frac { 2 } { m ^ 2 } S _ { 2 , m } ( z ) - \\frac { 2 } { m } S _ { 3 , m } ( z ) \\end{align*}"}
+{"id": "5483.png", "formula": "\\begin{align*} \\mathbb E | M + E | ^ { K } = \\int _ { | M | > 3 | E | } | M + E | ^ K d U + \\int _ { | M | \\le 3 | E | } | M + E | ^ K d U . \\end{align*}"}
+{"id": "8599.png", "formula": "\\begin{align*} H ' ( 0 ) & = - v \\sum _ { k = s - u } ^ { s - 1 } ( 2 m + 1 - 2 k ) \\cosh ( x _ k ' ) + u \\sum _ { k = s } ^ { s + v } ( 2 m + 1 - 2 k ) \\cosh ( x _ k ' ) \\\\ & > - v \\sum _ { k = s - u } ^ { s - 1 } ( 2 m + 1 - 2 k ) \\cosh ( x _ s ' ) + u \\sum _ { k = s } ^ { s + v } ( 2 m + 1 - 2 k ) \\cosh ( x _ s ' ) \\\\ & \\geqslant ( - u v ( 2 m + 1 - 2 s ) + u v ( 2 m + 1 - 2 s ) ) \\cosh ( x _ s ' ) = 0 \\end{align*}"}
+{"id": "2680.png", "formula": "\\begin{align*} K ^ { ( 2 ) } _ { i j } \\dddot { q } ^ { j } + { \\textrm { ( u p t o 2 n d - o r d e r t e r m s ) } } = 0 . \\end{align*}"}
+{"id": "8664.png", "formula": "\\begin{align*} { \\rm C o k e r \\ , } \\tilde \\sigma _ { G , s } = { \\rm C o k e r \\ , } \\tilde \\sigma _ G : = \\{ u \\in H ^ 0 _ b ( \\widehat X _ G ) _ { s - \\frac { d } { 4 } - \\frac { 1 } { 2 } } ; \\ , ( \\ , u \\ , | \\ , \\tilde \\sigma _ { G , s } v ) _ { \\widehat X _ G } = 0 , \\forall v \\in H ^ 0 ( \\overline M ) ^ G _ s \\cap C ^ \\infty ( \\overline M ) \\} . \\end{align*}"}
+{"id": "4565.png", "formula": "\\begin{align*} \\nabla _ { \\alpha } R _ { \\beta \\gamma } = \\nabla _ { \\mu } R _ { \\nu \\gamma } J _ { \\alpha } ^ \\mu J _ { \\beta } ^ \\nu + O _ g ' ( r _ q ) . \\end{align*}"}
+{"id": "7531.png", "formula": "\\begin{align*} \\partial _ t \\big ( \\tfrac { 1 } { 2 } \\rho | u | ^ 2 + \\varepsilon \\tfrac { 1 } { 2 } n | v | ^ 2 + h _ 1 ( \\rho ) + h _ 2 ( n ) + \\delta \\tfrac { 1 } { 2 } | \\nabla \\phi | ^ 2 \\big ) + \\nabla \\cdot \\mathcal { F } = 0 \\ , \\end{align*}"}
+{"id": "8156.png", "formula": "\\begin{align*} P _ { Y ^ L } ( 0 ^ L ) & = 1 - \\sum _ { k = 1 } ^ { L } \\sum _ { y ^ L \\in \\beta _ k ^ L } P _ { Y ^ L } ( y ^ L ) = 1 - \\sum _ { k = 1 } ^ { L } \\frac { c _ k } { M } . \\end{align*}"}
+{"id": "1575.png", "formula": "\\begin{align*} G ( z ) \\ge \\frac { \\sup _ { | v | = 1 } | D F ^ { - 1 } ( v ) | } { C } \\ , | z | ^ { 1 + 1 / H } | z | > 1 , \\end{align*}"}
+{"id": "1682.png", "formula": "\\begin{align*} l _ i ^ + \\cap l _ i ^ - \\cap ( l _ j ^ + + l _ j ^ - ) \\subset ( l _ i ^ + \\cap l _ i ^ - \\cap l _ j ^ + ) + ( l _ i ^ + \\cap l _ i ^ - \\cap l _ j ^ - ) = \\{ 0 \\} . \\end{align*}"}
+{"id": "1401.png", "formula": "\\begin{align*} ( n t ) _ i & = n _ i + \\sum _ { k = 0 } ^ { i - 1 } \\binom { n _ 1 } { k } t _ { i - k } , \\\\ ( t u ) _ i & = t _ i + \\sum _ { l = 0 } ^ { i - 1 } \\binom { n _ 1 } { l } u _ { i - l } . \\end{align*}"}
+{"id": "5527.png", "formula": "\\begin{align*} \\Delta ( \\rho ^ 2 ) = \\frac { \\sin \\rho \\pi } { \\rho } + \\int _ 0 ^ \\pi \\frac { \\cos \\rho t } { \\rho ^ 2 } W ( t ) \\ , d t , \\int _ 0 ^ \\pi W ( t ) \\ , d t = 0 , \\end{align*}"}
+{"id": "6604.png", "formula": "\\begin{align*} \\phi _ { \\Phi _ k , \\beta } ( x , y ) = & \\frac { x } { 2 \\pi i } \\int _ { b - i \\tau } ^ { b + i \\tau } R _ { k , \\beta } ( z ) \\frac { ( \\alpha x ^ { \\delta } ) ^ z } { z ( 1 - ( 1 - \\delta ) z ) } \\ , d z + E _ 1 , \\end{align*}"}
+{"id": "1430.png", "formula": "\\begin{align*} K _ { X ' } + D ' = g ^ * ( K _ X + D ) + E \\end{align*}"}
+{"id": "1627.png", "formula": "\\begin{align*} \\Psi ^ { r } _ { p , q } & \\begin{cases} q & 1 \\leq p \\leq k - r \\\\ q \\leq M + k - p & k - r < p \\leq k \\end{cases} \\\\ & \\begin{cases} q & r \\leq p < k \\\\ q \\leq M + 1 & p = k . \\end{cases} \\end{align*}"}
+{"id": "7413.png", "formula": "\\begin{align*} C _ { 4 3 } = 2 ^ { 4 / ( r + 1 ) } C _ 0 ' \\left ( \\frac { A R f _ 0 ^ { p + q } } { 2 ^ { p + q + 4 } } \\right ) ^ { - ( r - 1 ) / ( r + 1 ) } > 0 . \\end{align*}"}
+{"id": "5541.png", "formula": "\\begin{align*} L = \\sqrt { m n } \\max _ { x \\in [ n ] , y \\in [ m ] } | M _ { x y } | . \\end{align*}"}
+{"id": "4779.png", "formula": "\\begin{align*} \\varphi _ 2 ( \\varepsilon , b , \\eta , E , f ) & : = \\max \\{ \\varphi ( \\varepsilon / 4 , b , f ) , \\varphi ( \\varepsilon / 3 , b , f ) , \\\\ & \\qquad \\varphi _ 1 ( \\varepsilon / 6 E , b , \\varepsilon / 4 , f ( \\varepsilon \\eta ( \\min \\{ \\varepsilon / 1 2 E , 2 \\} ) / 2 ) , \\eta , f ) \\} \\end{align*}"}
+{"id": "6920.png", "formula": "\\begin{align*} \\forall t \\geq 0 , \\forall x \\in \\R , \\quad & \\partial _ t u + v \\partial _ x u = 0 , \\\\ \\forall x \\in \\R , & u ( 0 , x ) = u _ 0 ( x ) . \\end{align*}"}
+{"id": "5644.png", "formula": "\\begin{align*} a & = \\left ( 1 , x _ 1 , 0 , \\tfrac { 1 1 7 } { 4 } - x _ 1 , \\tfrac { 1 1 7 } { 4 } , \\tfrac { 1 1 7 } { 4 } + x _ 1 , \\tfrac { 1 1 7 } { 2 } , \\tfrac { 1 1 7 } { 2 } , \\tfrac { 1 1 7 } { 2 } , \\tfrac { 3 5 1 } { 4 } - x _ 1 \\right ) , \\\\ a Q & = ( 3 5 2 , 0 , 0 , 0 , 0 , 1 2 8 x _ 1 , 0 , 0 , 0 , 3 2 ( 1 1 7 - 4 x _ 1 ) ) . \\end{align*}"}
+{"id": "8699.png", "formula": "\\begin{align*} \\gamma B _ G = ( N _ b \\Box ^ { ( 0 ) } _ { b , G } + S _ G ) \\gamma B _ G = S _ G \\gamma B _ G , \\end{align*}"}
+{"id": "6545.png", "formula": "\\begin{align*} T _ { N , 2 } = \\sum ^ N _ { n = 1 } \\sum ^ { \\infty } _ { j = m _ 0 + 1 } \\Big ( \\mathbb { E } \\big [ K ( X _ n + a _ j \\widetilde { \\varepsilon } _ { n - j } ) | \\varepsilon _ { n - j } \\big ] - \\mathbb { E } \\big [ K ( X _ n + a _ j \\widetilde { \\varepsilon } _ { n - j } ) \\big ] \\Big ) . \\end{align*}"}
+{"id": "5493.png", "formula": "\\begin{align*} s _ 1 \\log ( \\gamma s _ 1 ) \\le \\sum _ { i = 1 } ^ \\ell \\log \\left ( ( m _ i k _ i ) ^ { m _ i } ) \\right ) , \\end{align*}"}
+{"id": "8643.png", "formula": "\\begin{align*} f = \\sum _ { n = 1 } ^ \\infty u _ n , \\end{align*}"}
+{"id": "6947.png", "formula": "\\begin{align*} \\Lambda _ { K u l } ( \\iota ( \\Gamma ) ) = \\underset { z \\in \\mathcal { S } } { \\bigcup } H _ { z } . \\end{align*}"}
+{"id": "2575.png", "formula": "\\begin{align*} X _ 4 = & X ( - 2 , 3 ) + ( q ^ { - \\frac { 1 } { 2 } } + q ^ { \\frac { 1 } { 2 } } ) h X ( - 2 , 2 ) + ( q ^ { - 1 } + h ^ 2 + q ) X ( - 2 , 1 ) + ( q ^ { - \\frac { 1 } { 2 } } + q ^ { \\frac { 1 } { 2 } } ) h X ( - 2 , 0 ) \\\\ & + X ( - 2 , - 1 ) + h X ( - 1 , 1 ) + h ^ 2 X ( - 1 , 0 ) + h X ( - 1 , - 1 ) + X ( 0 , - 1 ) . \\end{align*}"}
+{"id": "3467.png", "formula": "\\begin{align*} b ^ { + } _ { \\varphi } ( W ) = b ^ { + } ( W ) , b ^ { - } _ { \\varphi } ( W ) = b ^ { - } ( W ) - 1 , \\sigma _ { \\varphi } ( W ) = \\sigma ( W ) + 1 . \\end{align*}"}
+{"id": "2748.png", "formula": "\\begin{align*} \\sigma _ { 1 } : T ^ { * } M | _ { Q , P } \\rightarrow T ^ { * } M ; ( \\sigma _ { 1 } ^ { * } \\Xi ^ { \\alpha } : = \\epsilon ^ { \\alpha } , \\sigma _ { 1 } ^ { * } \\Psi _ { \\alpha } : = \\epsilon _ { \\alpha } , \\sigma _ { 1 } ^ { * } Q ^ { i } , \\sigma _ { 1 } ^ { * } P _ { i } ) \\mapsto ( \\Xi ^ { \\alpha } , \\Psi _ { \\alpha } , Q ^ { i } , P _ { i } ) \\end{align*}"}
+{"id": "1110.png", "formula": "\\begin{align*} \\| \\{ f _ j \\} _ { j \\in \\mathbb Z } \\| _ { L \\dot B _ { p q } ( E \\times J ) } : = \\| \\{ f _ j \\} _ { j \\in \\mathbb Z } \\| _ { \\ell ^ q L ^ p ( E \\times J ) } : = \\| \\{ f _ j \\} _ { j \\in \\mathbb Z } \\| _ { \\ell ^ q ( J ; L ^ p ( E ) ) } : = \\left [ \\sum _ { j \\in J } \\| f _ j \\| _ { L ^ p ( E ) } ^ q \\right ] ^ { \\frac { 1 } { q } } \\end{align*}"}
+{"id": "2905.png", "formula": "\\begin{align*} M _ f ^ { ( 2 ) } ( x _ i ) : = \\sum _ { j = 1 } ^ J \\sum _ { \\substack { y _ { j - 1 } ^ 2 < d \\leqslant x _ i \\\\ v _ { P ( d ) } ( d ) \\geqslant 2 } } ^ { y _ { j - 1 } , y _ j } f ( d ) \\sum _ { \\substack { n \\leqslant x _ i / d \\\\ P ( n ) \\leqslant y _ { j - 1 } } } f ( n ) . \\end{align*}"}
+{"id": "4618.png", "formula": "\\begin{align*} \\lambda ( j ) : = M ^ { 2 \\gamma j \\left ( 1 + \\frac { 1 } { d } ( - 1 + \\epsilon ) \\right ) } \\end{align*}"}
+{"id": "7032.png", "formula": "\\begin{align*} e ^ { - t \\Lambda _ { C _ \\infty } ( b ) } \\upharpoonright L ^ 2 \\cap C _ \\infty = e ^ { - t \\Lambda _ { p } ( b ) } \\upharpoonright L ^ 2 \\cap C _ \\infty = e ^ { - t \\Lambda ( b ) } \\upharpoonright L ^ 2 \\cap C _ \\infty , \\end{align*}"}
+{"id": "6431.png", "formula": "\\begin{align*} \\tilde { \\delta } _ n ( \\frac { 1 } { 2 } ) - \\delta _ 0 ^ { 1 / 2 } = \\frac { 1 } { m _ { \\frac { 1 } { 2 } } ( \\alpha _ 0 ) } \\frac { 1 } { n } \\sum _ { i = 2 } ^ n A _ i - \\delta _ 0 ^ { 1 / 2 } + \\frac { 1 } { m _ { \\frac { 1 } { 2 } } ( \\alpha _ 0 ) } \\epsilon _ n ^ 2 ( \\tilde { \\alpha } _ n ) \\frac { 1 } { n } \\sum _ { i = 2 } ^ n A _ i + R _ n \\end{align*}"}
+{"id": "6352.png", "formula": "\\begin{align*} \\norm { \\rho } _ { C ^ 1 } : = \\norm { \\rho } _ { C ^ 1 ( S ^ { n - 1 } ) } = \\sup _ { \\varphi \\in S ^ { n - 1 } } \\Bigl ( \\abs { \\rho ( \\varphi ) } + \\abs { \\nabla \\rho ( \\varphi ) } \\Bigr ) \\leq \\varepsilon , \\end{align*}"}
+{"id": "4142.png", "formula": "\\begin{align*} & \\sum _ { i = 1 } ^ J \\lambda _ i \\xi ^ * _ i ( u ) = \\lambda _ j c _ { s , j } ^ 2 ( 1 - w _ { j j } ) ^ 2 + \\sum _ { i > j } \\lambda _ i c _ { s , i } ^ 2 w _ { i j } ^ 2 + \\sum _ { i \\in \\mathcal { J } } \\lambda _ i b _ i , \\end{align*}"}
+{"id": "6527.png", "formula": "\\begin{align*} \\ell _ { \\beta } ( x ) : = x ^ { \\frac { 1 } { \\beta } } \\ell ^ { \\frac { 1 } { \\beta } } ( x ^ { \\frac { 1 } { \\beta } } ) \\end{align*}"}
+{"id": "5190.png", "formula": "\\begin{align*} \\sum _ { j } q _ { j } \\frac { \\partial L _ { d } B E I ( p \\| q ) } { \\partial q _ { j } } = 0 \\end{align*}"}
+{"id": "6362.png", "formula": "\\begin{align*} \\phi ''' ( r ) = \\phi ' ( r ) \\Bigl ( \\frac { \\phi '' ( r ) ^ 2 } { \\phi ' ( r ) ^ 2 } - \\lambda _ 1 ^ r \\Bigr ) , \\end{align*}"}
+{"id": "6218.png", "formula": "\\begin{align*} \\inf _ { [ 0 , \\gamma ] } \\delta ( f , \\alpha ) = \\delta ( f , \\alpha ) ( \\gamma ) = \\frac { f ( \\gamma ) - f ( \\alpha ) } { \\gamma - \\alpha } , \\sup _ { [ \\gamma , 1 ] } \\delta ( f , \\beta ) = \\delta ( f , \\beta ) ( \\gamma ) = \\frac { f ( \\gamma ) - f ( \\beta ) } { \\gamma - \\beta } , \\end{align*}"}
+{"id": "4693.png", "formula": "\\begin{align*} \\mathcal { C } _ i = g _ i ^ { * } ( \\mathrm { N e f } ( Y _ i ) ) \\cap \\mathrm { E f f } ( X ) + \\sum _ j \\mathbf { R } ^ { \\geq 0 } E _ i ^ j , \\end{align*}"}
+{"id": "1888.png", "formula": "\\begin{align*} \\left . \\Delta \\right | _ { z = 0 } F ( u + z w ) ^ p = p ^ 2 | \\lambda | ^ 2 F ( u ) ^ p + p F ( u ) ^ { p - 1 } \\left . \\Delta \\right | _ { z = 0 } F ( u + z v ) , \\end{align*}"}
+{"id": "4457.png", "formula": "\\begin{align*} M _ { H , 1 } ( Z _ 0 , J , \\rho _ 1 \\lambda _ 2 ) = \\| f _ 0 \\| ^ 2 _ { \\partial D _ 1 \\times M _ 1 , \\rho _ 1 \\lambda _ 2 } . \\end{align*}"}
+{"id": "7057.png", "formula": "\\begin{align*} ( \\partial _ t - \\Delta + b _ m \\cdot \\nabla ) u = | \\mathsf { f } _ k | h , u ( s , \\cdot ) = g . \\end{align*}"}
+{"id": "7503.png", "formula": "\\begin{align*} Q ^ * r _ s = 2 \\sqrt { s f } . \\end{align*}"}
+{"id": "7500.png", "formula": "\\begin{align*} \\chi ^ 2 g ^ { - 1 } * g ^ { - 1 } * \\nabla ^ { g _ 0 ( t ) } h * \\nabla ^ { g _ 0 ( t ) } h & = \\chi ^ 2 ( 1 - \\chi ^ 2 ) g ^ { - 1 } * g ^ { - 1 } * \\nabla ^ { g _ 0 ( t ) } h * \\nabla ^ { g _ 0 ( t ) } h \\\\ & + g ^ { - 1 } * g ^ { - 1 } * \\nabla ^ { g _ 0 ( t ) } ( \\chi ^ 2 h ) * \\nabla ^ { g _ 0 ( t ) } ( \\chi ^ 2 h ) \\\\ & + g ^ { - 1 } * g ^ { - 1 } * \\chi ^ 2 * \\nabla ^ { g _ 0 ( t ) } \\chi * \\nabla ^ { g _ 0 ( t ) } \\chi * h * h \\\\ & + g ^ { - 1 } * g ^ { - 1 } * \\chi ^ 3 * \\nabla ^ { g _ 0 ( t ) } \\chi * \\nabla ^ { g _ 0 ( t ) } h * h . \\end{align*}"}
+{"id": "1629.png", "formula": "\\begin{align*} [ H , Q ] \\equiv H Q - Q H = 0 . \\end{align*}"}
+{"id": "9199.png", "formula": "\\begin{align*} { \\ell ^ - _ { n + 1 , n + 1 } [ 0 ] } ^ { - 1 } \\ell ^ \\pm _ { 1 , n + 1 } ( z ) = \\ell ^ \\pm _ { 1 , n + 1 } ( z ) { \\ell ^ - _ { n + 1 , n + 1 } [ 0 ] } ^ { - 1 } . \\end{align*}"}
+{"id": "9129.png", "formula": "\\begin{align*} | \\beta | : = \\sum _ { i \\in I } \\nu _ { \\beta , i } . \\end{align*}"}
+{"id": "2764.png", "formula": "\\begin{align*} & \\omega _ { \\Theta } : = d \\sigma ^ { * } \\Theta ^ { \\alpha } \\wedge d \\sigma ^ { * } \\Theta _ { \\alpha } = 0 , \\\\ & \\omega _ { Q , P } : = d \\sigma ^ { * } Q ^ { i } \\wedge d \\sigma ^ { * } P _ { i } . \\end{align*}"}
+{"id": "8046.png", "formula": "\\begin{align*} \\limsup _ { \\delta \\rightarrow 0 } \\limsup _ { N \\rightarrow + \\infty } \\frac { N } { \\gamma _ N ^ 2 } \\log \\sup _ { \\sigma \\in \\mathcal { T } } P \\left ( \\varepsilon _ { 1 0 } ^ N > \\epsilon \\right ) = - \\infty . \\end{align*}"}
+{"id": "4790.png", "formula": "\\begin{align*} F ( \\nabla F ^ o ( x ) ) = 1 , \\nabla _ \\xi F ( \\nabla _ x F ^ o _ \\theta ( x ) ) = \\frac { x } { F ^ o ( x ) } . \\end{align*}"}
+{"id": "1255.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ; q ^ 2 ) _ k ( - 1 ; q ^ 2 ) _ k } { ( q ^ 2 ; q ^ 2 ) _ k } q ^ { ( n - 1 ) ^ 2 - k ^ 2 } \\\\ [ 7 p t ] & \\equiv - \\frac { \\left ( 1 + ( - 1 ) ^ { \\frac { n - 3 } { 2 } } \\right ) q ( 1 - q ^ n ) } { 2 } \\\\ [ 7 p t ] & + ( - 1 ) ^ { \\frac { n - 1 } { 2 } } \\left ( q ^ { - n + 1 } + \\frac { ( 1 - n ) q ( 1 - q ^ n ) } { 2 } \\right ) \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "3941.png", "formula": "\\begin{align*} & I _ N \\otimes Q _ 0 + \\left ( I _ N \\otimes B \\right ) K = \\operatorname { d i a g } \\left \\{ Q _ 0 + B K _ j \\right \\} _ { j = 1 } ^ N \\end{align*}"}
+{"id": "4111.png", "formula": "\\begin{align*} \\tilde { f } _ { \\theta } ( x ) = \\exp \\Big ( \\langle \\theta , z \\rangle - \\sum _ { i } \\alpha _ i \\gamma _ i ( \\theta _ i ) u _ i - \\sum _ { i } \\mu ^ { ( r ) } _ i \\xi _ i ( \\theta ) v _ i \\Big ) , \\end{align*}"}
+{"id": "3346.png", "formula": "\\begin{align*} ( J _ b + i ) \\left ( \\frac { \\partial } { \\partial \\bar z _ i } + \\sum _ { i = 1 } ^ n h _ { i , j } ( z , b ) \\frac { \\partial } { \\partial z _ j } \\right ) = 0 . \\end{align*}"}
+{"id": "9187.png", "formula": "\\begin{align*} & \\frac { z - w } { q z - q ^ { - 1 } w } \\ell ^ - _ { 1 j } ( z ) \\ell ^ - _ { j , j + 1 } ( w ) + \\frac { ( q - q ^ { - 1 } ) z } { q z - q ^ { - 1 } w } \\ell ^ - _ { j j } ( z ) \\ell ^ - _ { 1 , j + 1 } ( w ) = \\\\ & \\frac { z - w } { q z - q ^ { - 1 } w } \\ell ^ - _ { j , j + 1 } ( w ) \\ell ^ - _ { 1 j } ( z ) + \\frac { ( q - q ^ { - 1 } ) w } { q z - q ^ { - 1 } w } \\ell ^ - _ { j j } ( w ) \\ell ^ - _ { 1 , j + 1 } ( z ) . \\end{align*}"}
+{"id": "7233.png", "formula": "\\begin{align*} b \\star b ' = \\sigma ( \\sigma ^ { - 1 } ( b ) \\ast \\sigma ^ { - 1 } ( b ' ) ) b , b ' \\in [ m ] . \\end{align*}"}
+{"id": "1838.png", "formula": "\\begin{align*} u \\cdot A : = u A u ^ { - 1 } = A - ( d _ A u ) u ^ { - 1 } . \\end{align*}"}
+{"id": "307.png", "formula": "\\begin{align*} \\partial _ t u = \\Delta u ^ m + ( 1 + | x | ) ^ { \\sigma } u ^ p , ( x , t ) \\in \\real ^ N \\times ( 0 , \\infty ) , \\ N \\geq 1 , \\end{align*}"}
+{"id": "1540.png", "formula": "\\begin{align*} F _ n ( z ) = n \\big ( G ( w + z / n ) - G ( w ) \\big ) + c , D F _ n ( z ) = D G ( w + z / n ) , D ^ 2 F _ n = \\frac { { \\rm I d } } { n } . \\end{align*}"}
+{"id": "8646.png", "formula": "\\begin{align*} D ( y ^ * , x ^ * ) : = F ^ * ( y ^ * ) - F ^ * ( x ^ * ) - \\sup _ { x \\in \\partial F ^ * ( x ^ * ) \\cap K } \\langle x , y ^ * - x ^ * \\rangle . \\end{align*}"}
+{"id": "1937.png", "formula": "\\begin{align*} H _ r ( t ) : = \\frac { t ^ { p } } { r ^ p } + \\frac { t ^ { q } } { r ^ { s q } } , t \\ge 0 . \\end{align*}"}
+{"id": "1294.png", "formula": "\\begin{align*} \\partial _ t ^ { m i c r o } \\langle x _ 2 \\rangle & : = k \\langle x _ 1 \\rangle + ( d - k ) \\langle x _ 2 \\rangle \\\\ & = ( k + \\alpha ( d - k ) ) \\langle x _ 1 \\rangle \\ ; . \\end{align*}"}
+{"id": "2088.png", "formula": "\\begin{align*} n ^ d _ t & \\le C _ { d , T } ( t ) + 2 0 0 0 \\big ( C _ 2 ^ 4 + 3 ( C _ 8 + C _ { 5 } ^ 2 ) \\big ) \\eta ^ 4 d ^ 4 t ^ 3 \\sum _ { k = 0 } ^ { t - 1 } C _ { d , T } ( k ) \\exp \\big ( 2 0 0 0 \\eta ^ 4 d ^ 4 \\big ( C _ 2 ^ 4 t ^ 3 + 3 ( C _ 8 + C _ { 5 } ^ 2 ) \\big ) t ^ 3 ( t - k - 1 ) \\big ) . \\end{align*}"}
+{"id": "17.png", "formula": "\\begin{align*} \\int w ( x ; \\kappa , q ) \\ , d x = 2 \\beta ( \\kappa ) - \\kappa \\frac { \\partial \\beta } { \\partial \\kappa } + 2 \\int q \\ , d x . \\end{align*}"}
+{"id": "4191.png", "formula": "\\begin{align*} \\begin{aligned} g _ 3 g _ 2 \\neq g _ 2 g _ 3 & \\mbox { a n d } g _ 3 g ^ { 2 } _ 1 \\neq g ^ { 2 } _ 1 g _ 3 \\ , , \\\\ \\varphi ( g _ 3 g _ 2 ) = \\varphi ( g _ 3 ) \\varphi ( g _ 2 ) & \\mbox { a n d } \\varphi ( g _ 3 g ^ { 2 } _ 1 ) = \\varphi ( g ^ { 2 } _ 1 ) \\varphi ( g _ 3 ) \\ , . \\end{aligned} \\end{align*}"}
+{"id": "146.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ \\infty { ( 3 ^ { 2 n - \\alpha ( 3 ) } ) } ^ i \\cdot | R ( t / 3 ^ i ) | \\leq \\sum _ { i = 0 } ^ \\infty { ( 3 ^ { 2 n - \\alpha ( 3 ) } ) } ^ i \\cdot C _ J \\cdot D ( \\Omega ) \\cdot t ^ { 2 n } \\cdot { ( 3 ^ { 2 n } ) } ^ { - i } < \\infty . \\end{align*}"}
+{"id": "260.png", "formula": "\\begin{align*} y = x , d \\tau = \\exp \\left ( - \\int ^ x \\gamma ( \\zeta ) \\ , d \\zeta \\right ) \\ , d t . \\end{align*}"}
+{"id": "4139.png", "formula": "\\begin{align*} \\xi ^ * _ i ( u ) & = c ^ 2 _ { s , i } \\Bigl ( - u _ i + \\sum _ { k \\in \\mathcal { J } } P _ { i , k } u _ k \\Bigr ) ^ 2 + \\sum _ { k \\in \\mathcal { J } } P _ { i k } u _ k ^ 2 - \\big ( \\sum _ { k \\in \\mathcal { J } } P _ { i k } u _ k \\big ) ^ 2 \\\\ & = c _ { s , i } ^ 2 ( - u _ i + w _ { i j } ) ^ 2 + \\sum _ { k \\in \\mathcal { J } } P _ { i k } ( u _ k ^ 2 - u _ k ) + w _ { i j } - w _ { i j } ^ 2 . \\end{align*}"}
+{"id": "2240.png", "formula": "\\begin{align*} L ( q , v _ t , v _ x , s ^ t , s ^ x ) = L _ \\circ - \\gamma s ^ t = \\frac { 1 } { 2 } \\rho v _ t ^ 2 - \\frac { 1 } { 2 } \\tau v _ x ^ 2 - \\gamma s ^ t \\ , . \\end{align*}"}
+{"id": "4020.png", "formula": "\\begin{align*} \\begin{cases} x _ { 1 } ^ { \\left ( 2 t + 1 \\right ) } & = \\ ; \\delta _ { 1 } x _ { 2 } ^ { \\left ( 2 t \\right ) } y ^ { \\left ( 2 t \\right ) } \\medskip \\\\ x _ { 2 } ^ { \\left ( 2 t + 2 \\right ) } & = \\ ; \\gamma _ { 2 } x _ { 1 } ^ { \\left ( 2 t + 1 \\right ) } y ^ { \\left ( 2 t + 1 \\right ) } \\medskip \\\\ y ^ { \\left ( 2 t + 1 \\right ) } & = \\ ; \\delta x _ { 2 } ^ { \\left ( 2 t \\right ) } y ^ { \\left ( 2 t \\right ) } \\\\ y ^ { \\left ( 2 t + 2 \\right ) } & = \\ ; \\gamma x _ { 1 } ^ { \\left ( 2 t + 1 \\right ) } y ^ { \\left ( 2 t + 1 \\right ) } . \\end{cases} \\end{align*}"}
+{"id": "3656.png", "formula": "\\begin{align*} \\lambda _ { i } ( X ^ { T } A X ) = \\theta _ i \\mu _ i , \\end{align*}"}
+{"id": "1965.png", "formula": "\\begin{align*} g ( x ) = & ( x - 1 ) ^ { ( \\tau - 1 ) \\rho + i ' } f ( x ) \\\\ = & ( x ^ { \\rho } - 1 ) ^ { \\tau - 1 } \\left ( \\sum _ { \\theta = 0 } ^ { i ' } \\binom { i ' } { \\theta } ( - 1 ) ^ { i ' - \\theta } x ^ { \\theta } \\right ) \\left ( \\sum _ { j = 0 } ^ { \\rho - 1 } \\bar { f _ j } ( x ^ { \\rho } ) x ^ j \\right ) \\end{align*}"}
+{"id": "2145.png", "formula": "\\begin{align*} \\vartheta _ J = 1 . 6 2 2 1 , u _ J = 1 . 9 9 0 1 , v _ J = ( 2 . 9 7 0 9 , 2 . 1 9 9 1 ) . \\end{align*}"}
+{"id": "2564.png", "formula": "\\begin{align*} \\{ \\ , u \\in C ^ r ( L _ { \\pm 1 } ; \\Omega ^ { - 1 } N L _ { \\pm 1 } ) \\mid \\check \\sigma ^ * u = \\pm u \\ , \\} \\ ; . \\end{align*}"}
+{"id": "6515.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u ( x ) = \\lambda _ 1 ( K ) u ( x ) & x \\in ( K ) , \\\\ \\ ; u ( x ) = 0 & x \\in \\partial K . \\end{cases} \\end{align*}"}
+{"id": "2921.png", "formula": "\\begin{align*} I _ { 0 } : = \\frac { 1 } { \\log y _ { 0 } } \\int _ { - \\infty } ^ { + \\infty } \\bigg | \\frac { F _ { 0 } ( 1 / 2 + i t ) } { 1 / 2 + i t } \\bigg | ^ 2 { \\rm d } t \\end{align*}"}
+{"id": "2879.png", "formula": "\\begin{align*} \\mathcal { B } _ r ( f ) ( x ) : = \\frac { 1 } { | B ( x , r ) | } \\int _ { B ( x , r ) } \\left | f ( y ) \\right | \\ , d y . \\end{align*}"}
+{"id": "1788.png", "formula": "\\begin{align*} \\Re \\left ( g ^ { \\mu } P _ \\zeta - h ^ { \\mu + 2 } ( P + i v , z , \\zeta ) + \\sum _ { j = 1 } ^ { n - 1 } f _ j ^ { \\mu + 1 } P _ { z _ j } \\right ) = c x _ 1 ^ { \\mu + 2 } \\end{align*}"}
+{"id": "1209.png", "formula": "\\begin{align*} \\left \\langle ( - \\Delta _ p ) ^ s u , \\varphi \\right \\rangle + \\left \\langle ( - \\Delta _ p ) ^ t v , \\psi \\right \\rangle = & \\lambda \\left [ \\alpha ( p ) \\vert u \\vert ^ { \\alpha ( p ) - 2 } u ( x ) \\vert v ( x _ 0 ) \\vert ^ { \\beta ( p ) } \\varphi ( x ) \\right . \\\\ & \\left . + \\ \\beta ( p ) \\left ( \\int _ { \\Omega } \\vert u ( x ) \\vert ^ { \\alpha ( p ) } \\dd x \\right ) \\vert v ( x _ 0 ) \\vert ^ { \\beta ( p ) - 2 } v ( x _ 0 ) \\psi ( x _ 0 ) \\right ] \\end{align*}"}
+{"id": "5877.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { N C } ( G ) ) & = \\dfrac { ( 8 n - 4 ) ( 8 n - 5 ) ^ { 3 } } { 2 } + 2 ( ( 2 n - 2 ) ^ { 2 } ( 4 n - 5 ) ^ { 2 } + 1 2 n ( 2 n - 2 ) ( 4 n - 5 ) + 3 6 n ^ { 2 } ) \\\\ & ~ ~ ~ ~ ~ ~ - 3 ( ( 2 n - 2 ) ( 4 n - 5 ) + 6 n ) ( 8 n - 5 ) ^ { 2 } + ( 8 n - 4 - \\dfrac { 3 } { 2 } ) ( ( 4 n - 4 ) ( 4 n - 5 ) ^ { 2 } + 3 6 n ) \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - ( 2 n - 2 ) ( 4 n - 5 ) ^ { 3 } - 5 4 n \\\\ & = 1 0 2 4 n ^ { 4 } - 2 5 6 0 n ^ { 3 } + 2 0 4 8 n ^ { 2 } - 5 1 2 n \\\\ & = 2 n ( 5 1 2 n ^ { 3 } - 1 2 8 0 n ^ { 2 } + 1 0 2 4 n - 2 5 6 ) . \\end{align*}"}
+{"id": "2535.png", "formula": "\\begin{align*} \\dot C ^ \\infty ( M ) = \\bigcap _ { m \\ge 0 } x ^ m C ^ \\infty ( M ) \\subset C ^ \\infty ( M ) \\ ; , \\end{align*}"}
+{"id": "2369.png", "formula": "\\begin{align*} \\begin{cases} \\| a \\| _ { C _ { T } L _ { x } ^ { 2 } } + \\| \\nabla a \\| _ { L _ { T } ^ { 2 } L _ { x } ^ { 2 } } \\leq C ( c , \\gamma ) \\| w _ { 0 } \\| _ { L ^ { 2 } } , \\\\ \\| a \\| _ { C _ { T } L _ { x } ^ { p } \\cap L _ { T } ^ { \\frac { 4 p } { 3 } } L _ { x } ^ { 2 p } } \\leq C ( p , c , \\gamma ) \\| w _ { 0 } \\| _ { L ^ { p } } . \\end{cases} \\end{align*}"}
+{"id": "5646.png", "formula": "\\begin{align*} a Q = ( 3 5 2 , 0 , 3 7 4 4 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ) \\end{align*}"}
+{"id": "896.png", "formula": "\\begin{align*} C _ 0 ^ 1 = N y ^ 2 , \\ : \\ : C _ 0 ^ 2 = - N y ^ 1 . \\end{align*}"}
+{"id": "2057.png", "formula": "\\begin{align*} \\mathrm { d } U ( s , x ) = - \\alpha \\int _ 0 ^ 1 A ( x , y ) U ( s , y ) \\mathrm { d } y \\mathrm { d } s . \\end{align*}"}
+{"id": "4648.png", "formula": "\\begin{align*} \\beta _ { - 1 } ( s ) = \\frac { \\zeta ( s ) } { \\zeta ( 2 s ) } \\cdot \\frac { L ( s , \\chi _ { 4 , 3 } ) } { ( 1 + 2 ^ { - s } ) } . \\end{align*}"}
+{"id": "3270.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ^ { d + 1 } ; q ^ d ) _ k ^ { d - 2 } ( q , q ^ { 1 - d } ; q ^ d ) _ k q ^ { d k } } { ( q ^ d ; q ^ d ) _ k ^ d } \\equiv 0 \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "2927.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 , \\ldots , n , u _ i > 0 } u _ i \\geq \\frac { x } { k } \\geq \\frac { n - k + 1 - y } { k } \\geq \\frac { n - k + 1 - ( m + 2 ) ( k - 1 ) } { k } . \\end{align*}"}
+{"id": "2735.png", "formula": "\\begin{align*} \\omega : = \\frac { 1 } { 2 } J _ { m n } d z ^ { m } \\wedge d z ^ { n } = d q ^ { i } \\wedge d p _ { i } \\end{align*}"}
+{"id": "2470.png", "formula": "\\begin{align*} \\begin{cases} \\lambda ( \\tau ) = A _ { 1 } e ^ { - \\gamma \\tau } ( 1 + o ( 1 ) ) , \\\\ x ( \\tau ) = A _ { 2 } e ^ { p ^ { - 1 } \\tau } ( 1 + o ( 1 ) ) \\end{cases} \\end{align*}"}
+{"id": "658.png", "formula": "\\begin{align*} T \\mathbb { S } ^ { n - 1 } : = \\begin{Bmatrix} \\begin{array} { l | l } ( x , \\xi ) \\in \\mathbb { R } ^ { n } \\times \\mathbb { S } ^ { n - 1 } & \\langle x , \\xi \\rangle = 0 \\end{array} \\end{Bmatrix} . \\end{align*}"}
+{"id": "1628.png", "formula": "\\begin{gather*} r = ( x ^ 2 _ 1 + x _ 2 ^ 2 + x _ 3 ^ 2 ) ^ \\frac 1 2 , \\tilde r = ( x ^ 2 _ 1 + x _ 2 ^ 2 ) ^ \\frac 1 2 , \\varphi = \\arctan \\left ( \\frac { x _ 2 } { x _ 1 } \\right ) , \\ \\ \\theta = \\arctan \\left ( \\frac { \\tilde r } { x _ 3 } \\right ) . \\end{gather*}"}
+{"id": "7053.png", "formula": "\\begin{align*} \\langle b _ m \\cdot w , \\nabla \\cdot ( w | w | ^ { q - 2 } ) \\rangle & = \\langle b _ m \\cdot w , \\Delta u | w | ^ { q - 2 } \\rangle + ( q - 2 ) \\langle b _ m \\cdot w , | w | ^ { q - 3 } w \\cdot \\nabla | w | \\rangle \\\\ & : = S _ 1 + S _ 2 . \\end{align*}"}
+{"id": "6561.png", "formula": "\\begin{align*} \\int _ { \\R } \\widehat { K } ( u ) \\phi ( u ) u \\ , d u = 0 \\end{align*}"}
+{"id": "814.png", "formula": "\\begin{align*} { \\cal { L } } _ { \\hat { X } } C ^ { i } _ { j k } = \\dot { \\nabla } _ { k } ( \\nabla _ { j } X ^ { i } + C ^ { i } _ { j h } \\nabla _ { 0 } X ^ { h } ) + X ^ { h } P ^ { i } _ { \\ j h k } + \\nabla _ { 0 } X ^ { h } Q ^ { i } _ { \\ j h k } . \\end{align*}"}
+{"id": "4917.png", "formula": "\\begin{align*} g = x _ 0 g _ 0 + x _ 1 g _ 1 + x _ 2 g _ 2 + x _ 3 g _ 3 , \\end{align*}"}
+{"id": "691.png", "formula": "\\begin{align*} X + \\sqrt { 2 } Y & = \\left ( \\frac { \\alpha _ 1 + \\alpha _ 2 } { 2 } + \\frac { \\sqrt { 2 } } { 2 } ( \\beta _ 1 + \\beta _ 2 ) \\right ) ^ 2 + \\left ( \\frac { \\alpha _ 1 - \\alpha _ 2 } { 2 } + \\frac { \\sqrt { 2 } } { 2 } ( \\beta _ 1 - \\beta _ 2 ) \\right ) ^ 2 \\\\ & + \\left ( \\frac { \\alpha _ 3 + \\alpha _ 4 } { 2 } + \\frac { \\sqrt { 2 } } { 2 } ( \\beta _ 3 + \\beta _ 4 ) \\right ) ^ 2 + \\left ( \\frac { \\alpha _ 3 - \\alpha _ 4 } { 2 } + \\frac { \\sqrt { 2 } } { 2 } ( \\beta _ 3 - \\beta _ 4 ) \\right ) ^ 2 . \\end{align*}"}
+{"id": "5348.png", "formula": "\\begin{align*} b _ i ^ { S } = \\begin{cases} \\displaystyle \\theta _ i ^ 1 + \\beta \\ , \\sum _ { j \\in N } p _ { i j } ^ 1 \\ , b _ j ^ S & i \\in S \\cup N ^ { \\{ 1 \\} } \\\\ \\displaystyle \\beta \\ , \\sum _ { j \\in N } p _ { i j } ^ 0 \\ , b ^ S _ j , & i \\in N ^ { \\{ 0 , 1 \\} } \\setminus S ; \\end{cases} \\end{align*}"}
+{"id": "26.png", "formula": "\\begin{align*} \\big ( X - t \\kappa R ( \\kappa ; q ^ 0 ) ^ 2 - z \\big ) ^ { - 1 } q ^ 0 _ + = U ( t ) ^ * ( X - z ) ^ { - 1 } U ( t ) q ^ 0 _ + \\end{align*}"}
+{"id": "1172.png", "formula": "\\begin{align*} \\psi ( x ) = \\int _ { \\mathbb R ^ n } \\theta ( x - y ) \\eta ( y ) \\ , d y = \\sum _ { k \\in \\mathbb Z ^ n } \\int _ { Q _ { 0 , k } } \\theta ( x - y ) \\eta ( y ) \\ , d y = : \\sum _ { k \\in \\mathbb Z ^ n } g _ k ( x ) . \\end{align*}"}
+{"id": "9097.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } X ( t + 1 ) = ( A ^ { ( r ) } + J ^ { ( r ) } \\hat { C } K ^ { ( r ) } ) X ( t ) + J ^ { ( r ) } ( ( \\hat { C } C \\hat { C } ^ { - 1 } - I ) \\underline { v } + \\hat { C } ( \\underline { u } ^ { ( r ) } - U ( J ^ { ( r ) } X ) - \\mathbf { 1 } \\frac { \\beta } { 2 } ) , \\\\ v ( t ) = J ^ { ( r ) } X ( t ) + \\underline { v } . \\end{array} \\right . \\end{align*}"}
+{"id": "2027.png", "formula": "\\begin{align*} f ( R ) \\geq f ( A \\sigma ^ n ) \\geq \\alpha ^ n f ( A ) \\geq \\alpha ^ { \\log _ { \\sigma } \\left ( \\frac { R } { A \\sigma } \\right ) } f ( A ) = f ( A ) \\left ( \\frac { R } { A \\sigma } \\right ) ^ { \\frac { \\ln ( \\alpha ) } { \\ln ( \\sigma ) } } , \\end{align*}"}
+{"id": "5075.png", "formula": "\\begin{align*} \\left ( \\mathrm { t r } _ { q ^ k } \\left ( f \\right ) * \\mathrm { t r } _ { q ^ k } \\left ( E ( s ) \\right ) \\right ) ( x ) & = \\sum _ { x ^ { - 1 } y \\in T _ s ( \\mathbb { F } _ { q ^ k } ) \\subset X ( \\mathbb { F } _ { q ^ k } ) } f ( y ) . \\end{align*}"}
+{"id": "2152.png", "formula": "\\begin{align*} | A A | \\leq K | A | \\implies \\ , \\forall d \\in \\mathbb R \\setminus \\{ 0 \\} , \\ , \\ , | \\{ ( a , b ) \\in A \\times A : a - b = d \\} | \\ll K ^ C | A | ^ { \\frac { 2 } { 3 } - c ' } , \\end{align*}"}
+{"id": "3896.png", "formula": "\\begin{align*} | Q ' _ * ( n ) | \\leq | Q ' ( n ) | \\leq \\sum _ { k = 1 } ^ n | Q ' _ * ( k ) | . \\end{align*}"}
+{"id": "3358.png", "formula": "\\begin{align*} \\begin{aligned} \\ \\ & d ( \\sigma _ { k + 1 } ^ \\phi ( \\Bar { \\sigma } _ k ) , \\sigma _ { k + 1 } ^ \\phi ( \\hat { \\sigma } _ k ) ) \\leq \\phi ^ { - 1 } ( y ) \\cdot \\\\ & \\left ( \\int _ { \\mathbb { R } ^ n } | \\Bar { \\sigma } _ k ( \\xi ) - \\hat { \\sigma } _ k ( \\xi ) | ^ p d \\xi \\right ) ^ { 1 / p } \\cdot \\left ( \\int _ { \\mathbb { R } ^ n } | \\Tilde { T } ( u , y , \\xi ) | ^ q d \\xi \\right ) ^ { 1 / q } , \\end{aligned} \\end{align*}"}
+{"id": "6143.png", "formula": "\\begin{align*} \\min \\left \\{ f ( x ) = \\sum _ { i = 1 } ^ { m } f _ i ( x _ i ) : ~ ( A x : = ) \\sum _ { i = 1 } ^ { m } A _ i x _ i = b \\right \\} , \\end{align*}"}
+{"id": "3521.png", "formula": "\\begin{align*} ( \\phi _ * \\circ d ) \\ ( \\textbf { r } _ 0 ) & = \\phi _ * ( \\textbf { a } _ 1 + \\ldots + \\textbf { a } _ m ) = \\sum _ { i = 1 } ^ m k _ i \\cdot \\textbf { a } _ { \\sigma ( i ) } \\\\ ( d \\circ \\phi _ * ) \\ ( \\textbf { r } _ 0 ) & = d ( \\kappa \\cdot \\textbf { r } _ 0 + \\sum _ { i = 1 } ^ m \\mu _ i \\cdot \\textbf { r } _ { \\sigma ( i ) } ) = \\sum _ { i = 1 } ^ m ( \\kappa + p _ i \\mu _ i ) \\cdot \\textbf { a } _ { \\sigma ( i ) } \\end{align*}"}
+{"id": "7404.png", "formula": "\\begin{align*} C _ 5 : = \\frac { ( p + q - 1 ) ^ 2 } { 4 } > 0 , \\end{align*}"}
+{"id": "7822.png", "formula": "\\begin{align*} w _ k & = \\left ( \\sum _ { k \\in \\mathcal { K } } y _ k ^ 2 \\right ) | ( \\boldsymbol { v } _ { k } ) ^ { \\mathrm { H } } \\boldsymbol { g } _ { k } | ^ { 2 } { E } _ { k } > 0 , \\\\ o _ k & = y _ k \\sqrt { ( 1 + \\eta _ k ) \\left | \\left ( \\boldsymbol { v } _ { k } \\right ) ^ { \\mathrm { H } } \\ ! \\boldsymbol { g } _ { k } \\right | ^ { 2 } { E } _ { k } } > 0 , \\\\ s _ k & = - \\sqrt { \\frac { z _ k ^ 2 } { 4 C _ k } } ( \\frac { E _ k } { L \\kappa _ k } ) ^ { \\frac { 1 } { 4 } } < 0 . \\end{align*}"}
+{"id": "282.png", "formula": "\\begin{align*} e _ { 2 k } e _ 2 = k e _ { 2 k + 2 } , \\ e _ { 2 k + 1 } e _ 2 = ( k + 1 ) e _ { 2 k + 3 } , \\ e _ 2 e _ { 2 k } = e _ { 2 k + 2 } , \\ e _ 2 e _ { 2 k + 1 } = 0 . \\end{align*}"}
+{"id": "1572.png", "formula": "\\begin{align*} I ( R , \\rho ) = \\frac { \\pi ^ 2 } { 4 \\ , ( R - \\rho ) ^ 2 } , \\end{align*}"}
+{"id": "617.png", "formula": "\\begin{align*} & \\frac { 4 } { \\pi } \\int _ 0 ^ 1 \\sqrt { 1 - x ^ 2 } \\ln ( x ) \\sum _ { n = 0 } ^ { \\infty } \\left ( - \\frac { x ^ 2 } { 2 } \\right ) ^ n \\binom { - \\frac { 1 } { 2 } } { n } ( n + 1 ) \\ , d x + \\\\ & \\frac { 4 \\sqrt { \\pi } } { \\Gamma ^ 2 \\left ( \\frac { 1 } { 4 } \\right ) } + \\frac { \\ln ( 2 ) \\Gamma ^ 2 \\left ( \\frac { 1 } { 4 } \\right ) } { 2 \\pi ^ { 3 / 2 } } . \\end{align*}"}
+{"id": "14.png", "formula": "\\begin{align*} d m | _ q ( g ) = \\frac { d } { d \\theta } m ( x ; \\kappa , q + \\theta g ) \\bigg | _ { \\theta = 0 } = R ( \\kappa , q ) \\bigl [ ( m + 1 ) C _ + g \\bigr ] , \\end{align*}"}
+{"id": "5632.png", "formula": "\\begin{align*} \\sup _ { n \\geq 1 } M _ 1 ^ n = \\infty \\inf _ { n , k \\geq 1 } M _ n ^ k > 0 . \\end{align*}"}
+{"id": "5161.png", "formula": "\\begin{align*} K _ { 0 , \\alpha } = \\left [ \\frac { \\sum _ { i } p ^ { \\alpha } _ { i } q ^ { 1 - \\alpha } _ { i } } { \\sum _ { i } q _ { i } } \\right ] ^ { \\frac { 1 } { \\alpha } } \\end{align*}"}
+{"id": "8307.png", "formula": "\\begin{align*} { \\sf m } : = \\{ \\eta , - \\} : C ^ \\infty ( T ^ \\ast [ 2 ] A [ 1 ] ) [ 2 ] \\to C ^ \\infty ( T ^ \\ast [ 2 ] A [ 1 ] ) [ 2 ] . \\end{align*}"}
+{"id": "2514.png", "formula": "\\begin{align*} \\langle u , v \\rangle ' _ s = \\sum _ k \\sum _ { | \\alpha | \\le s } \\int _ B f _ k ^ 2 ( x ) \\cdot \\partial ^ \\alpha ( u y _ { p _ k } ^ { - 1 } ) ( x ) \\cdot \\overline { \\partial ^ \\alpha ( v y _ { p _ k } ^ { - 1 } ) ( x ) } \\ , d x \\ ; , \\end{align*}"}
+{"id": "775.png", "formula": "\\begin{align*} x \\cdot _ \\beta y = \\beta ( a ^ { - 1 } b , c ^ { - 1 } ) x y , x \\in H _ { a , b } , \\ y \\in H _ { c , d } , \\end{align*}"}
+{"id": "4996.png", "formula": "\\begin{align*} P _ N ( y _ j , q ^ { - 2 } y _ j , x _ 3 , \\dots , x _ N ) = 0 , j = 1 , \\dots , N ; \\end{align*}"}
+{"id": "5134.png", "formula": "\\begin{align*} L _ { d } D 2 ( p \\| q ) = \\left \\lbrace \\log _ { d } A \\left ( p , q \\right ) - \\log _ { d } \\left [ X \\left ( p , q \\right ) . Y \\left ( p , q \\right ) \\right ] \\right \\rbrace \\end{align*}"}
+{"id": "695.png", "formula": "\\begin{align*} ( x _ 1 ^ 2 + y _ 1 ^ 2 + z _ 1 ^ 2 ) ( x _ 2 ^ 2 + y _ 2 ^ 2 + z _ 2 ^ 2 ) & = \\frac { w _ 1 ^ 2 w _ 2 ^ 2 | | \\vec { a } | | ^ 4 } { ( a ^ 2 + e ^ 2 ) ^ 2 } ( U _ 1 ^ 2 + 1 ) \\ ; \\left ( \\frac { 1 } { U _ 1 ^ 2 } + 1 \\right ) \\\\ & = \\frac { w _ 1 ^ 2 w _ 2 ^ 2 | | \\vec { a } | | ^ 4 } { ( a ^ 2 + e ^ 2 ) ^ 2 } \\left ( 2 + U _ 1 ^ 2 + \\frac { 1 } { U _ 1 ^ 2 } \\right ) \\geq \\frac { 4 w _ 1 ^ 2 w _ 2 ^ 2 | | \\vec { a } | | ^ 4 } { ( a ^ 2 + e ^ 2 ) ^ 2 } . \\end{align*}"}
+{"id": "790.png", "formula": "\\begin{align*} W = ( w _ { i j } ) _ { i , j } \\mbox { i s u n i t a r y a n d } W = B \\bar { W } B ^ { - 1 } d , \\end{align*}"}
+{"id": "6565.png", "formula": "\\begin{align*} \\Phi _ k ( n ) = \\begin{cases} n ^ { k - 1 } \\phi ( n ) & k , \\\\ n ^ { k - 1 } \\phi ( n ) \\prod _ { \\substack { p | n \\\\ p > 2 } } \\ \\left ( 1 - \\frac { ( - 1 ) ^ { \\frac { k ( p - 1 ) } { 4 } } } { p ^ { \\frac { k } { 2 } } } \\right ) & k . \\end{cases} \\end{align*}"}
+{"id": "5223.png", "formula": "\\begin{align*} & U _ { j } = ( 1 - \\alpha ) \\frac { \\sum _ { j } p _ { j } } { \\sum _ { j } q _ { j } } \\left [ Y \\frac { \\overline { M H } ^ { 2 } _ { j } } { \\overline { q } ^ { 2 } _ { j } } + Z \\sum _ { i } \\overline { M G } _ { i } \\right ] \\\\ & V _ { j } = ( 1 - \\alpha ) \\frac { \\sum _ { j } p _ { j } } { \\sum _ { j } q _ { j } } \\left [ Z \\frac { \\overline { M G } _ { j } } { \\overline { q } _ { j } } + Y \\sum _ { i } \\frac { \\overline { M H } ^ { 2 } _ { i } } { \\overline { q } _ { i } } \\right ] \\end{align*}"}
+{"id": "8791.png", "formula": "\\begin{align*} f _ \\rho ( x ) = \\frac { z - y _ - } { z - y _ + } \\vert x - y _ + \\vert ^ \\rho + \\frac { y _ - - y _ + } { z - y _ + } \\vert z - x \\vert ^ \\rho - \\vert x - y _ - \\vert ^ \\rho . \\end{align*}"}
+{"id": "5346.png", "formula": "\\begin{align*} \\widehat { \\mathbf { h } } ^ 0 = \\mathbf { h } ^ { 0 } - ( \\mathbf { I } - \\beta \\ , \\mathbf { P } ^ { 0 } ) \\ , ( \\mathbf { I } - \\beta \\ , \\mathbf { P } ^ { 1 } ) ^ { - 1 } \\ , \\mathbf { h } ^ { 1 } . \\end{align*}"}
+{"id": "6192.png", "formula": "\\begin{align*} \\begin{aligned} & \\left \\| \\left ( 1 - \\frac { 1 } { \\gamma } \\right ) \\bar { \\lambda } ^ k + \\frac { 1 } { \\gamma } \\bar { \\lambda } ^ { k + 1 } - \\lambda ^ * \\right \\| ^ 2 \\\\ = & \\frac { 1 } { \\gamma } \\| \\bar { \\lambda } ^ { k + 1 } - \\lambda ^ * \\| ^ 2 - \\left ( \\frac { 1 } { \\gamma } - 1 \\Big ) \\| \\bar { \\lambda } ^ k - \\lambda ^ * \\| ^ 2 + \\frac { 1 } { \\gamma } \\Big ( \\frac { 1 } { \\gamma } - 1 \\right ) \\| \\bar { \\lambda } ^ { k + 1 } - \\bar { \\lambda } ^ k \\| ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "2906.png", "formula": "\\begin{align*} V ( x _ i ) = \\sum _ { y _ { 0 } < p \\leqslant y _ J } \\bigg | \\sum _ { \\substack { n \\leqslant x _ i / p \\\\ P ( n ) < p } } f ( n ) \\bigg | ^ 2 . \\end{align*}"}
+{"id": "7225.png", "formula": "\\begin{align*} \\partial _ { t } p + \\partial _ { v } ( h p ) - a \\partial _ { v v } p = 0 , \\forall v \\in ( V _ { \\min } , V _ R ) \\cup ( V _ R , V _ F ) . \\end{align*}"}
+{"id": "2425.png", "formula": "\\begin{align*} f _ { ( i _ 1 , \\ldots , i _ n , j ) } ( t ) = & ~ s ! \\binom { s + 2 } { i _ 1 } \\cdots \\binom { s + 2 } { i _ n } \\cdot \\frac { g ^ { ( j ) } ( t ) } { j ! } \\cdot n ! ^ { 2 + j } \\binom { t + n } { n } ^ { 2 + j } \\\\ & \\qquad \\times \\prod _ { k = 1 } ^ { n } \\left ( ( k - 1 ) ! ( n - k ) ! \\binom { t + k - 1 } { k - 1 } \\binom { t + n } { n - k } \\right ) ^ { i _ k } . \\end{align*}"}
+{"id": "3096.png", "formula": "\\begin{align*} \\varphi \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & = \\left ( \\begin{array} { c c c } a ^ { 2 \\ , p ^ { e _ 1 } } & c _ 1 \\ , a ^ { p ^ { e _ 1 } } \\ , b ^ { p ^ { e _ 1 } } & \\frac { 1 } { 2 } \\ , \\lambda \\ , c _ 1 ^ 2 \\ , b ^ { 2 \\ , p ^ { e _ 1 } } \\\\ 0 & 1 & \\lambda \\ , c _ 1 \\ , b ^ { p ^ { e _ 1 } } \\ , d ^ { p ^ { e _ 1 } } \\\\ 0 & 0 & d ^ { 2 \\ , p ^ { e _ 1 } } \\end{array} \\right ) . \\end{align*}"}
+{"id": "30.png", "formula": "\\begin{align*} \\Delta _ j ( X ) = d _ j ^ * d _ j + d _ { j - 1 } d _ { j - 1 } ^ * \\ . \\end{align*}"}
+{"id": "7318.png", "formula": "\\begin{align*} \\sinh \\left ( j \\cosh ^ { - 1 } ( s + 1 ) \\right ) & = \\frac { 1 } { 2 } \\left ( ( s + 1 + \\sqrt { s ^ 2 + 2 s } ) ^ j - ( s + 1 - \\sqrt { s ^ 2 + 2 s } ) ^ j \\right ) \\\\ & = j \\sqrt { s ^ 2 + 2 s } + O ( s ) \\ , \\ , \\ , \\ , \\ , \\end{align*}"}
+{"id": "4069.png", "formula": "\\begin{align*} \\chi R _ V \\chi = \\chi R _ 0 \\chi ( I + V R _ 0 \\chi ) ^ { - 1 } . \\end{align*}"}
+{"id": "7269.png", "formula": "\\begin{align*} x _ k = - \\tfrac { 1 } { 1 - q } \\left ( \\theta q ^ k + s \\tfrac { \\eta \\theta + 1 - q } { \\theta q ^ k } - ( \\theta + \\eta s ) \\right ) , \\end{align*}"}
+{"id": "7893.png", "formula": "\\begin{align*} f _ { n , i } - f _ { n , j } = f _ { 0 , i } - f _ { 0 , j } + \\frac { q } { p } n ( i _ { \\pi ( 1 ) } - j _ { \\pi ( 1 ) } ) , \\end{align*}"}
+{"id": "5939.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { N C } ( G ) ) = \\frac { 1 } { 2 } ( 2 ^ { 1 2 k } - 7 \\cdot 2 ^ { 1 0 k } - 2 ^ { 9 k } + 2 1 \\cdot 2 ^ { 8 k } - 2 6 \\cdot 2 ^ { 6 k } - 2 \\cdot 2 ^ { 5 k } + 1 5 \\cdot 2 ^ { 4 k } + 3 \\cdot 2 ^ { 3 k } + 6 \\cdot 2 ^ { 2 k } - 8 \\cdot 2 ^ { k } ) . \\end{align*}"}
+{"id": "7968.png", "formula": "\\begin{align*} E ( \\mu _ { \\Sigma } ) | _ { \\Sigma } = \\mu _ { \\Sigma } , \\quad \\forall \\mu _ { \\Sigma } \\in H ^ { \\frac { 1 } { 2 } } \\Lambda ^ { 0 } ( \\Sigma ) , \\end{align*}"}
+{"id": "6640.png", "formula": "\\begin{align*} M _ 1 = M _ 2 \\coloneqq \\begin{pmatrix} 2 & - 1 \\\\ 1 & - 2 \\end{pmatrix} M _ 3 = M _ 4 \\coloneqq \\begin{pmatrix} 0 & \\frac { 3 } { 2 } \\\\ 2 & 0 \\end{pmatrix} , \\end{align*}"}
+{"id": "5306.png", "formula": "\\begin{align*} v _ i ( \\nu ) = \\min \\ , \\left \\{ v _ i ^ S ( \\nu ) : S \\in 2 ^ { N ^ { \\{ 0 , 1 \\} } } \\right \\} . \\end{align*}"}
+{"id": "268.png", "formula": "\\begin{align*} y = \\frac { 1 } { 2 \\sin ^ 2 ( x ) } , d \\tau = | \\cot ( x ) | \\ d t , \\end{align*}"}
+{"id": "4674.png", "formula": "\\begin{align*} \\overline { \\mathrm { A m p } _ k ( X ) } = \\mathrm { b M o b } _ k ( X ) ^ { \\vee } , \\end{align*}"}
+{"id": "1238.png", "formula": "\\begin{align*} { n \\brack k } = { n \\brack k } _ q = \\begin{cases} \\displaystyle \\frac { ( q ; q ) _ n } { ( q ; q ) _ k ( q ; q ) _ { n - k } } & , \\\\ [ 1 0 p t ] 0 & \\end{cases} \\end{align*}"}
+{"id": "2967.png", "formula": "\\begin{align*} 1 + \\frac { 2 \\C _ { \\alpha , \\beta } } { \\left ( \\beta - \\alpha - 2 \\right ) ^ 2 } & = \\frac { 1 } { 2 } + \\frac { 2 \\alpha \\beta - \\beta ^ 2 + 4 \\beta - 6 \\alpha - 3 + ( \\beta - ( \\alpha + 2 ) ) ^ 2 } { 2 \\left ( \\beta - \\alpha - 2 \\right ) ^ 2 } \\\\ & = \\frac { 1 } { 2 } + \\frac { ( \\alpha - 1 ) ^ 2 } { 2 \\left ( \\beta - \\alpha - 2 \\right ) ^ 2 } > 0 , \\end{align*}"}
+{"id": "4650.png", "formula": "\\begin{align*} c = a + b + \\frac { e } { n } + \\frac { 2 } { n ^ 2 } ( a b e + r x y ) , \\end{align*}"}
+{"id": "6746.png", "formula": "\\begin{align*} ( w - w ' ) ^ T ( F ( w ) - F ( w ' ) ) = 0 , ~ \\forall w , w ' . \\end{align*}"}
+{"id": "8113.png", "formula": "\\begin{align*} p ^ + _ \\ell x = u _ \\ell ^ - \\cdot p ^ + _ { \\ell _ 0 } x \\in ( Z ^ { ( e ^ { t _ \\star } \\epsilon ) } \\cap N ^ - _ { e ^ { t _ \\star } \\iota _ j } ) \\cdot p ^ + _ { \\ell _ 0 } x , \\forall \\ell \\in C _ { \\rho , j , i } ( \\ell _ 0 ) . \\end{align*}"}
+{"id": "6160.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau ^ k } S ^ { k + 1 } - \\frac { 1 } { \\tau ^ { k - 1 } } S ^ k + ( 1 - \\tau ^ k ) \\beta ^ k ( b - A \\widetilde { x } ^ k ) ^ T ( A \\breve { x } ^ { k - 1 } - b ) \\\\ \\geq & \\frac { 1 } { 2 } \\Big ( \\| v ^ { k + 1 } - v ' \\| ^ 2 _ { H ^ k } - \\| v ^ k - v ' \\| ^ 2 _ { H ^ k } + \\| v ^ k - \\widetilde { v } ^ k \\| ^ 2 _ { G ^ k } \\Big ) \\\\ \\geq & \\frac { 1 } { 2 } \\Big ( \\| v ^ { k + 1 } - v ' \\| ^ 2 _ { H ^ { k + 1 } _ 0 } - \\| v ^ k - v ' \\| ^ 2 _ { H ^ k _ 0 } + \\| v ^ k - \\widetilde { v } ^ k \\| ^ 2 _ { G ^ k } \\Big ) . \\end{aligned} \\end{align*}"}
+{"id": "6374.png", "formula": "\\begin{align*} \\phi ( s ) = \\int _ 0 ^ s \\phi ' ( \\tau ) \\ , d \\tau & = \\int _ 0 ^ s \\sinh ^ { n - 1 } ( \\tau ) \\cosh ^ { d - 1 } ( \\tau ) \\ , d \\tau \\geq \\frac { 1 } { \\cosh ( s ) } \\int _ 0 ^ s \\sinh ^ { n - 1 } ( \\tau ) \\cosh ( \\tau ) \\ , d \\tau \\\\ & = \\frac { 1 } { n \\cosh ( s ) } \\sinh ^ n ( s ) , \\end{align*}"}
+{"id": "7136.png", "formula": "\\begin{align*} \\partial ^ \\mu A _ { \\mu } = 0 \\Longrightarrow \\partial _ t A _ 0 + { \\boldsymbol { \\nabla } } \\cdot { \\boldsymbol { A } } = 0 \\ , \\ , . \\end{align*}"}
+{"id": "4106.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\phi ^ { r } ( r \\eta _ 1 , \\ldots , r ^ J \\eta _ J ) = \\prod _ { j \\in \\mathcal { J } } \\frac { 1 } { 1 - d _ j \\eta _ j } , \\eta = ( \\eta _ 1 , \\ldots , \\eta _ J ) ' \\in \\R ^ J _ - . \\end{align*}"}
+{"id": "4921.png", "formula": "\\begin{align*} x _ 1 ^ 2 + x _ 2 ^ 2 + \\cdots + x _ { n - 1 } ^ 2 + p ( y , z ) = 0 \\end{align*}"}
+{"id": "3731.png", "formula": "\\begin{align*} \\mathrm { B } _ { z } \\left ( x , y \\right ) = \\int _ { 0 } ^ { z } t ^ { x - 1 } \\left ( 1 - t \\right ) ^ { y - 1 } d t , \\end{align*}"}
+{"id": "5636.png", "formula": "\\begin{align*} E _ i = \\sum _ { j = 0 } ^ d Q _ { j i } A _ j , \\end{align*}"}
+{"id": "3642.png", "formula": "\\begin{align*} P = \\frac { I + T + T ^ 2 } { 3 } , Q = \\frac { I + \\lambda ^ 2 T + \\lambda T ^ 2 } { 3 } , R = \\frac { I + \\lambda T + \\lambda ^ 2 T ^ 2 } { 3 } , \\end{align*}"}
+{"id": "5700.png", "formula": "\\begin{align*} \\frac { \\partial ^ k } { \\partial t ^ k } H ( x ( t ) ) = \\sum _ { i = 1 } ^ k \\sum _ { \\substack { k _ 1 + \\dots + k _ i = k \\\\ k _ j \\geq 1 } } D ^ i H ( x ( t ) ) \\left [ \\frac { d ^ { k _ 1 } x ( t ) } { d t ^ { k _ 1 } } , \\dots , \\frac { d ^ { k _ i } x ( t ) } { d t ^ { k _ i } } \\right ] . \\end{align*}"}
+{"id": "4377.png", "formula": "\\begin{align*} \\min _ { x } & \\ \\sum _ { i , j \\in \\mathcal { I } , \\mathcal { J } } u _ i q _ j x _ { i , j } , \\\\ \\mathrm { s . t . } & \\ x \\in \\mathcal { X } \\subseteq \\left \\{ x \\in \\{ 0 , 1 \\} ^ { | \\mathcal { I } | \\cdot | \\mathcal { J } | } : \\sum _ { j \\in \\mathcal { J } } x _ { i , j } = 1 \\ \\forall i \\in \\mathcal { I } \\right \\} \\end{align*}"}
+{"id": "6796.png", "formula": "\\begin{align*} \\dot { x } & = c _ 1 \\kappa _ 1 x + c _ 2 \\kappa _ 2 y + c _ 3 \\kappa _ 3 , \\\\ \\dot { y } & = d _ 1 \\kappa _ 1 x + d _ 2 \\kappa _ 2 y + d _ 3 \\kappa _ 3 \\end{align*}"}
+{"id": "9184.png", "formula": "\\begin{align*} & x ^ + _ i ( u ) \\mapsto \\frac { e ^ + _ { i , i + 1 } ( u q ^ i ) - e ^ - _ { i , i + 1 } ( u q ^ i ) } { q - q ^ { - 1 } } , x ^ - _ i ( u ) \\mapsto \\frac { f ^ + _ { i + 1 , i } ( u q ^ i ) - f ^ - _ { i + 1 , i } ( u q ^ i ) } { q ^ { 1 - \\delta _ { i n } / 2 } - q ^ { - 1 + \\delta _ { i n } / 2 } } , \\\\ & \\psi _ i ( u ) \\mapsto h ^ - _ { i + 1 } ( u q ^ i ) h ^ - _ { i } ( u q ^ i ) ^ { - 1 } , \\varphi _ i ( u ) \\mapsto h ^ + _ { i + 1 } ( u q ^ i ) h ^ + _ { i } ( u q ^ i ) ^ { - 1 } , \\end{align*}"}
+{"id": "7226.png", "formula": "\\begin{align*} \\begin{aligned} & p ( V _ { \\min } , t ) = p ( V _ F , t ) = 0 . \\\\ \\end{aligned} \\end{align*}"}
+{"id": "3210.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ t ^ \\alpha T _ k ( t ) + \\lambda _ k t ^ { \\beta } T _ k ( t ) = t ^ { \\mu } f _ k , t > 0 ; \\\\ & \\lim \\limits _ { t \\rightarrow + 0 } J _ t ^ { \\alpha - 1 } T _ k ( t ) = \\varphi _ k . \\end{aligned} \\right . \\end{align*}"}
+{"id": "7840.png", "formula": "\\begin{align*} \\nabla ^ { \\perp \\psi } _ { E _ a } \\nabla ^ { \\perp \\psi } _ { E _ a } H ^ { \\psi } = \\frac { p } { p + q } \\nabla ^ { \\perp \\psi } _ { E _ a } \\nabla ^ { \\perp } _ { E _ a } H _ 1 + \\frac { q } { p + q } \\nabla ^ { \\perp \\psi } _ { E _ a } B ^ j ( E _ a , H _ 2 ) . \\end{align*}"}
+{"id": "2274.png", "formula": "\\begin{align*} h = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & - 1 \\end{array} \\right ) \\ , , \\ \\ \\ x = \\left ( \\begin{array} { c c } 0 & 1 \\\\ 0 & 0 \\end{array} \\right ) \\ , , \\ \\ \\ y = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 1 & 0 \\end{array} \\right ) \\ \\ \\ \\ \\ \\ \\ \\ [ h , x ] = 2 x \\ , , \\ \\ [ h , y ] = - 2 y \\ , , \\ \\ [ x , y ] = h \\ . \\end{align*}"}
+{"id": "8893.png", "formula": "\\begin{align*} \\Re \\big ( \\bar u \\partial _ k ( \\frac { u } { | x | ^ 2 } ) - \\frac { u } { | x | ^ 2 } \\partial _ k \\bar u \\big ) = - 2 \\frac { x _ k } { | x | ^ 4 } { | u | ^ 2 } . \\end{align*}"}
+{"id": "5434.png", "formula": "\\begin{align*} w ^ { S _ 0 \\oplus S _ 1 } _ { ( 0 , i ) } = w ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , i ) } = 1 + \\beta \\sum _ { j \\in N } p _ { i j } g ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , j ) } - \\beta g ^ { S _ 0 \\oplus S _ 1 } _ { ( 0 , i ) } . \\end{align*}"}
+{"id": "7333.png", "formula": "\\begin{align*} F _ { m , \\chi } ( s ) : = \\frac { m } { \\tau ( \\chi ) } \\sum _ { r = 0 } ^ { m - 1 } \\chi ( r ) \\left ( \\frac { U _ { m - r - 1 } ( s + 1 ) + U _ { r - 1 } ( s + 1 ) } { T _ { m } ( s + 1 ) - 1 } - \\frac { 1 } { m s } \\right ) , \\end{align*}"}
+{"id": "2055.png", "formula": "\\begin{align*} \\mathrm { d } U ( s , x ) = - \\alpha \\int _ 0 ^ 1 A ( x , y ) U ( s , y ) \\mathrm { d } y \\mathrm { d } s + \\alpha \\beta \\mathrm { d } \\xi _ 2 ( s , x ) , \\end{align*}"}
+{"id": "7537.png", "formula": "\\begin{align*} \\partial _ t \\hat { \\mathcal { H } } + \\nabla \\cdot \\hat { \\mathcal { F } } = \\mathcal { J } _ 1 + \\mathcal { J } _ 2 + \\mathcal { J } _ 3 \\ , \\end{align*}"}
+{"id": "7750.png", "formula": "\\begin{align*} A '' : = H ^ { - 1 } ( \\theta + \\alpha ) ( A ' + f ^ { ( r e ) } ( \\theta ) ) H ( \\theta ) . \\end{align*}"}
+{"id": "554.png", "formula": "\\begin{align*} - ( p u ' ) ' + q u = \\lambda w u , \\quad ( 0 , 1 ) , \\lambda \\in \\R . \\end{align*}"}
+{"id": "8663.png", "formula": "\\begin{align*} \\widehat X _ G : = \\widehat X / G , ~ \\widetilde X _ G = \\widetilde X / G . \\end{align*}"}
+{"id": "8807.png", "formula": "\\begin{align*} & \\int \\vert x - y \\vert ^ \\rho \\pi ( d x , d y ) - \\int \\vert x - y \\vert ^ \\rho \\overline \\pi ^ \\star ( d x , d y ) = p s \\left ( f ( x _ - , m ) - f ( x _ + , m ) \\right ) , \\end{align*}"}
+{"id": "5903.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { N C } ( G ) ) & = \\dfrac { 1 9 n ( 1 9 n - 1 ) ^ { 3 } } { 2 } + 2 \\times \\dfrac { ( 6 1 n ^ { 2 } - 1 9 n ) ^ { 2 } } { 4 } - 3 \\times \\dfrac { ( 6 1 n ^ { 2 } - 1 9 n ) } { 2 } ( 1 9 n - 1 ) ^ { 2 } \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ + ( 1 9 n - \\frac { 3 } { 2 } ) ( 4 n ( 4 n - 1 ) ^ { 2 } + 1 5 n ( 3 n - 1 ) ^ { 2 } ) - \\dfrac { 4 n ( 4 n - 1 ) ^ { 3 } + 1 5 n ( 3 n - 1 ) ^ { 3 } } { 2 } \\\\ & = \\frac { 1 } { 2 } \\times 7 4 8 8 0 n ^ { 4 } = 3 7 4 4 0 n ^ { 4 } . \\end{align*}"}
+{"id": "426.png", "formula": "\\begin{align*} \\begin{aligned} & [ \\bold { S } _ { j - 1 } ^ { - 1 } ] ^ { ( 1 , 2 ) } = [ \\bold { I } _ { j - 1 } - \\bold { \\Pi } _ { j - 1 } - \\bold { \\Gamma } _ { j - 1 } \\times \\\\ & ( \\bold { I } _ { j - 1 } - \\bold { \\Pi } _ { j - 1 } ^ { T } ) ^ { - 1 } \\bold { \\Xi } _ { j - 1 } ] ^ { - 1 } \\bold { \\Gamma } _ { j - 1 } ( \\bold { I } _ { j - 1 } - \\bold { \\Pi } _ { j - 1 } ^ { T } ) ^ { - 1 } , \\end{aligned} \\end{align*}"}
+{"id": "8506.png", "formula": "\\begin{align*} \\Lambda ( A ) = \\displaystyle \\bigcup _ { a \\in A } \\Lambda ( a ) , \\end{align*}"}
+{"id": "3842.png", "formula": "\\begin{align*} \\bar \\xi ^ n ( e _ x , e _ y \\mid t ) = 0 , \\ ; \\ ; \\bar \\Xi ^ n ( e _ x , e _ y \\mid t ) = 0 , \\ ; \\mbox { a . s . } \\end{align*}"}
+{"id": "8738.png", "formula": "\\begin{align*} \\forall \\rho > 0 , \\ ; ( \\rho - 2 ) \\int _ { \\R ^ 2 } \\vert y - x \\vert ^ \\rho \\pi ^ \\uparrow ( d x , d y ) & = \\inf _ { \\pi \\in \\Pi _ M ( \\mu , \\underline \\nu ) } ( \\rho - 2 ) \\int _ { \\R ^ 2 } \\vert y - x \\vert ^ \\rho \\pi ( d x , d y ) \\mbox { a n d } \\\\ ( \\rho - 2 ) \\int _ { \\R ^ 2 } \\vert y - x \\vert ^ \\rho \\pi ^ \\downarrow ( d x , d y ) & = \\sup _ { \\pi \\in \\Pi _ M ( \\mu , \\overline \\nu ) } ( \\rho - 2 ) \\int _ { \\R ^ 2 } \\vert y - x \\vert ^ \\rho \\pi ( d x , d y ) . \\end{align*}"}
+{"id": "919.png", "formula": "\\begin{align*} - L u = f ( \\cdot , u ) + \\mu \\quad D , u = g \\quad \\partial _ { \\chi } D , \\hat W _ D ( u ) = 0 \\quad \\partial D . \\end{align*}"}
+{"id": "3589.png", "formula": "\\begin{align*} ( 1 + \\lambda _ { 1 } ) ( 1 + \\lambda _ { 2 } ) ( \\lambda _ 1 + \\lambda _ 2 ) = 0 . \\end{align*}"}
+{"id": "1269.png", "formula": "\\begin{align*} W ^ 2 _ 2 ( \\mu , \\nu ) : = \\inf _ { \\gamma \\in \\Gamma ( \\mu , \\nu ) } \\int _ { \\R ^ d \\times \\R ^ d } | x - y | ^ 2 \\ , \\gamma ( d x , d y ) , \\end{align*}"}
+{"id": "7860.png", "formula": "\\begin{align*} \\mathbf { A } = \\psi ( \\pi , \\mathbf { a } , \\mathbf { d } ) = \\left ( \\begin{array} { c c c c } 2 & 2 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 1 & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "8348.png", "formula": "\\begin{align*} g _ { \\mu , K } ( x ) = \\mu ( K \\cap ( K + x ) ) . \\end{align*}"}
+{"id": "9131.png", "formula": "\\begin{align*} \\begin{aligned} F ( \\{ x _ { i , r } \\} _ { 1 \\leq i \\leq 2 } ^ { 1 \\leq r \\leq k _ { i } } ) = 0 & x _ { 1 , 1 } = v ^ { 6 } x _ { 1 , 2 } = v ^ { 3 } x _ { 2 , 1 } , \\\\ & x _ { 2 , 1 } = v ^ { 2 } x _ { 2 , 2 } = v ^ { 4 } x _ { 2 , 3 } = v ^ { 6 } x _ { 2 , 4 } = v ^ { 3 } x _ { 1 , 1 } . \\end{aligned} \\end{align*}"}
+{"id": "2733.png", "formula": "\\begin{align*} \\omega : = \\frac { 1 } { 2 } \\omega _ { m n } d z ^ { m } \\wedge d z ^ { n } \\end{align*}"}
+{"id": "5069.png", "formula": "\\begin{align*} ( \\tilde { A } _ s - v ) ( \\tilde { A } _ s + v ^ { - 1 } ) & = 0 . \\end{align*}"}
+{"id": "1545.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 ^ + } \\frac { A ( t ) } { t } = 0 , \\lim _ { t \\to \\infty } \\frac { A ( t ) } { t } = \\infty , \\end{align*}"}
+{"id": "4283.png", "formula": "\\begin{align*} d y ( m ) _ t = \\sigma ( y ( m ) _ t ) d w ( m ) _ t + b ( y ( m ) _ t ) d t , y ( m ) _ 0 = a . \\end{align*}"}
+{"id": "3462.png", "formula": "\\begin{align*} b ^ + ( X ) + g ( S ) - \\frac { 1 } { 4 } S \\circ S - \\frac { 1 } { 2 } \\sigma ( L ) + \\frac { 1 } { 2 } \\sigma ( L ' ) = 0 \\end{align*}"}
+{"id": "3037.png", "formula": "\\begin{align*} \\widehat { H } _ { a , b } = p _ 1 ^ 2 + p _ 2 ^ 2 + V _ { a , b } ( s _ 1 , s _ 2 ) \\ , , \\end{align*}"}
+{"id": "5282.png", "formula": "\\begin{align*} L S _ { d } \\left ( x . y \\right ) = \\frac { 1 + \\epsilon \\log ( x . y ) - ( 1 - \\epsilon \\log ( x . y ) ) } { 2 \\ ; \\epsilon } \\end{align*}"}
+{"id": "7689.png", "formula": "\\begin{align*} \\tau = t - L \\ , y ^ { \\frac { 1 } { m - \\alpha } } , \\end{align*}"}
+{"id": "3999.png", "formula": "\\begin{align*} S ^ { \\ : n , \\nu } = S ^ { \\ : n + \\nu - 1 } \\setminus \\mathcal { O } ^ { \\ : n , \\nu } \\end{align*}"}
+{"id": "2371.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\Lambda ( g _ N , \\alpha ) = \\Lambda ( f , \\alpha ) ^ 2 \\ , . \\end{align*}"}
+{"id": "1941.png", "formula": "\\begin{align*} & 0 \\leq \\int _ { \\Omega } \\big ( F ( x , D v ) - F ( x , D u ) \\big ) d x + \\int _ { B _ { r _ 1 } } \\int _ { B _ { r _ 1 } } a ( x , y ) \\big ( | v ( x ) - v ( y ) | ^ q - | u ( x ) - u ( y ) | ^ q \\big ) d \\mu \\\\ & \\quad + 2 \\int _ { \\mathbb { R } ^ { N } \\backslash B _ { r _ 1 } } \\int _ { B _ { r _ 1 } } a ( x , y ) \\big ( | v ( x ) - v ( y ) | ^ q - | u ( x ) - u ( y ) | ^ q \\big ) d \\mu + \\int _ { B _ { r _ 1 } } ( v - u ) f ( x ) d x \\\\ & = : I _ { 1 } + I _ { 2 } + I _ { 3 } + I _ { 4 } . \\end{align*}"}
+{"id": "7640.png", "formula": "\\begin{align*} A = \\{ \\left < x , \\mu _ A ( x ) , \\nu _ A ( x ) \\right > \\mid x \\in X \\} \\end{align*}"}
+{"id": "7705.png", "formula": "\\begin{align*} \\int _ { S _ { 2 } } ( \\frac { h _ { E _ { t _ { j } } } } { n } ) ^ { p } f ( x ) d x \\rightarrow 0 . \\end{align*}"}
+{"id": "3560.png", "formula": "\\begin{align*} T f = \\alpha ( \\tau ' ) ^ { \\frac { 1 } { p } } f \\circ \\tau ( f \\in H ^ p ( { \\mathbb { D } } ) ) . \\end{align*}"}
+{"id": "3911.png", "formula": "\\begin{align*} \\sin ^ 2 z + \\cos ^ 2 z = 1 , \\end{align*}"}
+{"id": "809.png", "formula": "\\begin{align*} \\delta { \\pi _ 2 } = - F { g ^ { i j } } \\frac { \\partial { b _ i } } { { \\partial { y ^ j } } } . \\end{align*}"}
+{"id": "9075.png", "formula": "\\begin{align*} b ^ { ( r ) } ( X ) : = J ^ { ( r ) } \\Big [ ( \\hat { C } C \\hat { C } ^ { - 1 } - I ) \\underline { v } + \\hat { C } \\Big ( D p - U ( J ^ { ( r ) } X ) - \\mathbf { 1 } \\frac { \\beta } { 2 } \\Big ) \\Big ] \\end{align*}"}
+{"id": "5368.png", "formula": "\\begin{align*} \\nu _ j = \\max \\ , \\left \\{ \\frac { v ^ { S \\setminus \\{ j \\} } _ j - v ^ { S } _ j } { b ^ S _ j - b ^ { S \\setminus \\{ j \\} } _ j } : j \\in S \\in \\mathcal { F } \\right \\} , j \\in N ^ { \\{ 0 , 1 \\} } . \\end{align*}"}
+{"id": "3670.png", "formula": "\\begin{align*} \\lambda _ { \\binom { d + 1 } { 2 } + 1 } ( L ( G , p ) ) \\leq \\frac { q ^ T L ( G , p ) q } { \\| q \\| ^ 2 } = a ( G ) . \\end{align*}"}
+{"id": "1335.png", "formula": "\\begin{align*} a = \\mp \\mathrm { d } H ^ { \\mathrm { v e r } } ( p ) . \\end{align*}"}
+{"id": "1509.png", "formula": "\\begin{align*} { \\rm e } ( z ) = \\frac { \\lambda _ { \\rm m a x } ( D ^ 2 F ( z ) ) } { \\lambda _ { \\rm m i n } ( D ^ 2 F ( z ) ) } . \\end{align*}"}
+{"id": "3312.png", "formula": "\\begin{align*} K _ { n - \\nu } ^ { ( \\nu ) } = \\frac { 1 } { \\sqrt { n } } \\exp \\left ( \\frac { 1 } { 2 } n \\left ( \\mu _ 0 - \\frac { \\nu \\mu _ 0 + X _ { \\nu + 1 } + \\dots + X _ n } { n } \\right ) ^ 2 \\right ) = \\frac { 1 } { \\sqrt { n } } \\exp \\left ( \\frac { ( ( n - \\nu ) \\mu _ 0 - ( S _ n - S _ { \\nu } ) ) ^ 2 } { 2 n } \\right ) , \\end{align*}"}
+{"id": "4131.png", "formula": "\\begin{align*} \\phi ^ { ( r ) } ( 0 , \\ldots , 0 , r ^ j \\eta _ j , \\ldots , r ^ J \\eta _ J ) - \\frac { 1 } { 1 - d _ j \\eta _ j } \\phi ^ { ( r ) } ( 0 , \\ldots , 0 , r ^ { j + 1 } \\eta _ { j + 1 } , \\ldots , r ^ J \\eta _ J ) = o ( 1 ) \\end{align*}"}
+{"id": "9063.png", "formula": "\\begin{align*} \\mu ( F _ i ) = \\sum _ { I \\in { \\mathcal { A } _ i } } \\mu ( I ) < \\frac { 1 } { \\lambda } \\sum _ { I \\in \\mathcal { A } _ i } \\int _ { I } f d \\mu = \\frac { 1 } { \\lambda } \\int _ { F _ i } f d \\mu . \\end{align*}"}
+{"id": "6798.png", "formula": "\\begin{align*} \\dot { v } & = v ( ( d \\kappa _ 2 - \\kappa _ 1 ) + ( \\kappa _ 2 - \\kappa _ 3 ) v - \\kappa _ 3 C w ) , \\\\ \\dot { w } & = w ( - \\kappa _ 1 + \\kappa _ 2 v ) . \\end{align*}"}
+{"id": "3505.png", "formula": "\\begin{align*} \\bar { h } \\Psi ( a b ) & = h \\Psi ( a b ) = \\Psi ( a ) \\Psi ( b ) , \\ \\\\ \\bar { h } \\Psi ( a ) & = h \\Psi ( a ) = \\Psi ( a ) h = \\Psi ( a ) \\bar { h } . \\end{align*}"}
+{"id": "7122.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } i \\partial _ { t } \\psi + ( \\Delta - m \\ln ( 1 + | x | ) ) \\psi - \\frac { \\gamma } { 2 \\pi } \\psi \\int _ { \\mathbb { R } ^ { 2 } } \\ln \\left ( \\frac { | x - y | } { 1 + | x | } \\right ) | \\psi ( y ) | ^ { 2 } d y + f ( \\psi ) = 0 , & \\\\ \\psi ( 0 , x ) = \\psi _ { 0 } ( x ) , & \\end{array} \\right . \\end{align*}"}
+{"id": "3905.png", "formula": "\\begin{align*} S _ { 0 , t } ( T ) \\coloneqq \\sum _ { n = 1 } ^ \\infty \\sum _ { e ^ { n \\ell ( \\gamma ) } \\leq T } \\ell ( \\gamma ) e ^ { n \\ell _ \\psi ( \\gamma ) + i t n \\ell _ \\phi ( \\gamma ) } , & & \\hat { S } _ { 0 , t } ( T ) = \\sum _ { e ^ { \\ell ( \\gamma ) } \\leq T } \\ell ( \\gamma ) e ^ { \\ell _ \\psi ( \\gamma ) + i t \\ell _ \\phi ( \\gamma ) } , \\end{align*}"}
+{"id": "4781.png", "formula": "\\begin{align*} t = \\max \\{ \\varphi ( \\varepsilon / 8 , b , f ) , \\varphi _ 1 ( \\varepsilon / 8 ( E + 1 ) , b , D , c , \\eta , f ) , \\varphi _ 2 ( \\varepsilon / 2 , b , \\eta , E , f ) \\} \\end{align*}"}
+{"id": "7989.png", "formula": "\\begin{align*} \\begin{aligned} \\begin{pmatrix} f _ { b } \\\\ e _ { b } \\end{pmatrix} = \\begin{pmatrix} - 1 & 0 \\\\ 0 & E ( \\cdot ) \\end{pmatrix} \\begin{pmatrix} \\frac { \\delta \\bar { H } } { \\delta \\phi _ { \\partial } } \\\\ \\frac { \\delta \\bar { H } } { \\delta \\Sigma } \\end{pmatrix} . \\end{aligned} \\end{align*}"}
+{"id": "8885.png", "formula": "\\begin{align*} \\mathcal E [ u ] & = \\| \\sqrt { \\mathcal K _ \\lambda } u \\| ^ 2 - \\frac 1 p \\mathcal P [ u ] \\\\ & \\geq \\Big ( 1 - \\frac { 1 } p \\Big ( \\frac { \\mathcal P [ u ] } { \\mathcal P [ \\varphi ] } \\Big ) ^ \\frac { p - 1 } p \\Big ) \\| \\sqrt { \\mathcal K _ \\lambda } u \\| ^ 2 \\\\ & \\geq \\Big ( 1 - \\frac { c ^ { ( p - 1 ) / p } } p \\Big ) \\| \\sqrt { \\mathcal K _ \\lambda } u \\| ^ 2 . \\end{align*}"}
+{"id": "3245.png", "formula": "\\begin{align*} S ^ \\wedge _ a ( x , y ) & = - \\frac { 1 } { 2 \\pi } \\ : \\delta ( \\xi ^ 2 ) \\ : \\Theta ( - \\xi ^ 0 ) + 2 \\ , \\ : \\Theta ( \\xi ^ 2 ) \\ : \\Theta ( - \\xi ^ 0 ) \\ : H _ a ( x , y ) \\\\ & = - \\frac { 1 } { 2 \\pi } \\ : \\delta ( \\xi ^ 2 ) \\ : \\Theta ( - \\xi ^ 0 ) + \\frac { a } { 8 \\pi } \\ : \\Theta ( \\xi ^ 2 ) \\ : \\Theta ( - \\xi ^ 0 ) + \\O \\big ( a ^ 2 \\big ) \\ : , \\end{align*}"}
+{"id": "1482.png", "formula": "\\begin{align*} & \\left [ L _ m , L _ n \\right ] = ( \\{ m \\} - \\{ n \\} ) L _ { m + n } , \\\\ & \\alpha ( L _ n ) = ( 1 + q ^ n ) L _ n . \\end{align*}"}
+{"id": "4000.png", "formula": "\\begin{align*} V : \\left \\{ \\begin{aligned} x _ { k } ' & = \\frac { \\sum _ { i , j = 1 } ^ { n , \\nu } \\gamma _ { i j k } x _ { i } y _ { j } } { \\left ( \\sum _ { i = 1 } ^ { n } x _ { i } \\right ) \\bigl ( \\sum _ { j = 1 } ^ { \\nu } y _ { j } \\bigr ) } , k = 1 , \\ldots , n \\medskip \\\\ y ' _ { r } & = \\frac { \\sum _ { i , j = 1 } ^ { n , \\nu } \\widetilde { \\gamma } _ { i j r } x _ { i } y _ { j } } { \\left ( \\sum _ { i = 1 } ^ { n } x _ { i } \\right ) \\bigl ( \\sum _ { j = 1 } ^ { \\nu } y _ { j } \\bigr ) } , r = 1 , \\ldots , \\nu . \\end{aligned} \\right . \\end{align*}"}
+{"id": "9180.png", "formula": "\\begin{align*} \\hat { P } _ { \\lambda _ { h , \\beta } } = { \\mathop { S y m } } _ { \\mathfrak { S } _ { d _ { \\beta } } } \\left ( \\prod _ { s = 1 } ^ { d _ { \\beta } } p _ { \\beta , r _ { \\beta } ( h , s ) } ( w _ { \\beta , s } ) \\prod _ { 1 \\leq s < r \\leq d _ { \\beta } } \\Big ( 1 + \\frac { ( \\beta , \\beta ) \\cdot \\hbar } { 2 ( w _ { \\beta , s } - w _ { \\beta , r } ) } \\Big ) \\right ) . \\end{align*}"}
+{"id": "55.png", "formula": "\\begin{align*} \\left ( Q _ h ^ * \\dfrac { a _ l } { r _ z } Q _ h - Q _ h ^ * Q _ h \\dfrac { A _ l } { R _ z } \\right ) \\psi = \\sum _ { \\mu \\in \\{ 0 , \\pm 1 \\} ^ n } \\hat { \\varphi } ( h \\xi ) \\overline { \\hat { \\varphi } ( h \\xi + \\mu ) } \\mathcal B _ h ( \\xi + h ^ { - 1 } \\mu ) \\psi ( \\xi + h ^ { - 1 } \\mu ) , \\end{align*}"}
+{"id": "5574.png", "formula": "\\begin{align*} \\tilde f _ { \\phi , t } ( g , o ) = \\sum _ { o _ 1 , \\dots , o _ { t } } \\left ( \\prod _ { s = 0 } ^ { t - 1 } \\tilde W _ { \\iota ( o _ { s } ) , \\iota ( o _ { s + 1 } ) } \\right ) \\phi _ { \\iota ( o _ { t } ) } . \\end{align*}"}
+{"id": "7074.png", "formula": "\\begin{align*} & \\langle u ( t ) , \\varphi \\rangle - \\langle f , \\varphi \\rangle \\\\ & + \\mu \\langle \\int _ 0 ^ t u d s , \\varphi \\rangle + \\bigl \\langle b \\cdot \\nabla \\int _ 0 ^ t u d s , \\varphi \\bigr \\rangle - \\sigma \\bigl \\langle \\int _ 0 ^ t u d B _ s , \\nabla \\varphi \\bigr \\rangle + \\frac { \\sigma ^ 2 } { 2 } \\bigl \\langle \\nabla \\int _ 0 ^ t u d s , \\nabla \\varphi \\bigr \\rangle = 0 . \\end{align*}"}
+{"id": "7442.png", "formula": "\\begin{align*} \\big [ ( A \\otimes B ) ( v \\otimes w ) \\big ] ^ { i j } = \\big [ ( A v ) \\otimes ( B w ) \\big ] ^ { i j } = ( A v ) ^ i ( B w ) ^ j = \\sum _ { \\alpha , \\beta = 1 } ^ { d } A ^ i _ \\alpha v ^ \\alpha B ^ j _ \\beta w ^ \\beta . \\end{align*}"}
+{"id": "2525.png", "formula": "\\begin{align*} u ( x ) & = ( 2 \\pi ) ^ { - n ' } \\int _ { \\R ^ { n ' } } e ^ { i \\langle x ' , \\xi \\rangle } a ( x '' , \\xi ) \\ , d \\xi \\ ; , \\\\ a ( x '' , \\xi ) & = \\int _ { \\R ^ { n ' } } e ^ { - i \\langle x ' , \\xi \\rangle } u ( x ' , x '' ) \\ , d x ' \\ ; . \\end{align*}"}
+{"id": "7116.png", "formula": "\\begin{align*} J ( u ) = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ { 2 } } | \\nabla u | ^ { 2 } d x + \\frac { 1 } { 4 } \\int _ { \\mathbb { R } ^ { 2 } } \\int _ { \\mathbb { R } ^ { 2 } } \\ln | x - y | u ^ { 2 } ( x ) u ^ { 2 } ( y ) d x d y - \\int _ { \\mathbb { R } ^ { 2 } } F ( u ) d x \\end{align*}"}
+{"id": "3637.png", "formula": "\\begin{align*} ( 1 + a ) ( a + b ) - b = ( 1 + \\lambda _ 1 + \\lambda _ 2 ) ( \\lambda _ 1 + \\lambda _ 2 + \\lambda _ 1 \\lambda _ 2 ) - \\lambda _ 1 \\lambda _ 2 = 0 . \\end{align*}"}
+{"id": "5766.png", "formula": "\\begin{align*} X _ - ( t ) Y _ - ( t ) = \\sum _ { k + \\ell \\leq s } \\sum _ { i \\in - \\mathbb { N } } \\xi ^ { ( k , \\ell ) } _ i ( t ) \\mathcal { E } ^ { ( k , \\ell ) } _ i ( t ) . \\end{align*}"}
+{"id": "2302.png", "formula": "\\begin{align*} R _ n = \\bigl [ ( 1 - x ^ 2 ) ( 1 + x ^ 2 + x ^ { - 2 } ) ^ n \\bigr ] _ 0 = \\sum _ { i = 0 } ^ n ( - 1 ) ^ { n - i } \\binom { n } { i } c _ i . \\end{align*}"}
+{"id": "7523.png", "formula": "\\begin{align*} d _ { g ( t _ k ) } \\left ( \\{ r _ M = \\sqrt { \\hat { \\gamma } t _ k } \\} , \\{ r _ M = 0 \\} \\right ) \\leq \\hat { C } \\sqrt { t _ k } . \\end{align*}"}
+{"id": "3034.png", "formula": "\\begin{align*} \\widehat { X } = \\frac { 1 } { 2 } p _ s ^ T ( X + \\overline { X } ) s + \\frac { 1 } { 2 } k ^ T \\mathcal { A } ^ { - 1 } ( X - \\overline { X } ) s \\ , , \\mathcal { A } _ { \\mu \\nu } = ( Z _ \\nu ) _ { \\mu \\sigma } s _ \\sigma \\ , . \\end{align*}"}
+{"id": "6762.png", "formula": "\\begin{align*} \\min \\limits _ { x _ 1 \\in \\mathcal { X } _ 1 , x _ 2 \\in \\mathcal { X } _ 2 } \\left \\{ f _ 1 ( x _ 1 ) + f _ 2 ( x _ 2 ) : ~ A _ 1 x _ 1 + A _ 2 x _ 2 = b \\right \\} . \\end{align*}"}
+{"id": "9002.png", "formula": "\\begin{align*} \\int _ E f d \\mu = \\lim _ { t \\searrow 0 } \\frac { 1 } { t } \\int _ E \\mathbb { E } _ x [ \\int _ 0 ^ t f ( Z _ s ) d A _ s ] d m ( x ) \\end{align*}"}
+{"id": "9025.png", "formula": "\\begin{align*} \\frac { 1 } { g } = \\frac { 1 } { q _ 1 } + . . . + \\frac { 1 } { q _ { k _ 3 } } - \\frac { b } { a } , \\end{align*}"}
+{"id": "6041.png", "formula": "\\begin{align*} \\begin{aligned} \\phi _ x ^ + & = - \\frac { E _ A - E _ { C } } { 3 \\delta } \\bar { \\xi } ^ { A } _ x - \\frac { c _ - } { 3 \\delta } \\bar { \\xi } ^ { B } _ x , \\\\ \\phi _ x ^ - & = \\frac { E _ A - E _ { C } } { 3 \\delta } \\bar { \\xi } ^ { A } _ x + \\frac { c _ + } { 3 \\delta } \\bar { \\xi } ^ { B } _ x , \\end{aligned} \\end{align*}"}
+{"id": "8533.png", "formula": "\\begin{align*} \\begin{array} { l c l } a _ 1 \\mu _ 1 & = & ( \\alpha _ 2 - \\alpha _ 1 ) \\log _ 1 ( \\pi _ 2 ) + \\beta \\log _ 1 ( u ( K ) ) , \\\\ k \\lambda _ 1 + a _ 1 \\mu _ 1 & = & - s \\alpha _ 1 , \\\\ a _ 2 \\mu _ 2 & = & ( \\alpha _ 1 - \\alpha _ 2 ) \\log _ 1 ( \\pi _ 2 ) - \\beta \\log _ 1 ( u ( K ) ) , \\\\ k \\lambda _ 2 + a _ 2 \\mu _ 2 & = & - s \\alpha _ 2 . \\end{array} \\end{align*}"}
+{"id": "1236.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { p ^ r - 1 } \\frac { 1 } { 2 ^ k } { 2 k \\choose k } \\equiv ( - 1 ) ^ { \\frac { p ^ r - 1 } { 2 } } \\pmod { p } , \\end{align*}"}
+{"id": "3922.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { N } a ^ j & = \\Big [ \\Big ( \\frac { 1 - a ^ { n _ 1 + 1 } } { 1 - a } , \\frac { 1 - a ^ { n _ 2 + 1 } } { 1 - a } , \\frac { 1 - a ^ { n _ 3 + 1 } } { 1 - a } , . . . \\Big ) \\Big ] \\\\ & = \\frac { 1 - a ^ { N + 1 } } { 1 - a } . \\end{align*}"}
+{"id": "3740.png", "formula": "\\begin{align*} \\int x ^ { p } \\ln x \\ , d x = x ^ { p + 1 } \\left [ \\frac { \\ln x } { p + 1 } - \\frac { 1 } { \\left ( p + 1 \\right ) ^ { 2 } } \\right ] , \\end{align*}"}
+{"id": "7927.png", "formula": "\\begin{align*} \\begin{aligned} d I ( V ; X , \\alpha ) & = \\lim _ { t \\to 0 } \\frac { 1 } { t } \\Big ( \\int _ { V ( t ) } \\alpha ( \\xi _ { t } x ) - \\int _ { V } \\alpha ( x ) \\Big ) \\\\ & = \\lim _ { t \\to 0 } \\frac { 1 } { t } \\Big ( \\int _ { V } \\xi _ { t } ^ { \\ast } \\alpha ( \\xi _ { t } x ) - \\int _ { V } \\alpha ( x ) \\Big ) \\\\ & = \\int _ { V } \\lim _ { t \\to 0 } \\frac { 1 } { t } \\Big ( \\xi _ { t } ^ { \\ast } \\alpha ( \\xi _ { t } x ) - \\alpha ( x ) \\Big ) \\\\ & = \\int _ { V } \\mathfrak { L } _ { X } \\alpha , \\end{aligned} \\end{align*}"}
+{"id": "1681.png", "formula": "\\begin{align*} \\Vert J \\Vert _ 1 = j _ 0 + \\ldots + j _ 3 = n - 2 \\end{align*}"}
+{"id": "9088.png", "formula": "\\begin{align*} \\begin{array} { l l l } X ^ * = J ^ { ( r ) } \\hat { C } C \\hat { C } ^ { - 1 } J ^ { ( r ) } X ^ * + J ^ { ( r ) } \\Big [ ( \\hat { C } C \\hat { C } ^ { - 1 } - I ) \\underline { v } + \\hat { C } ( D p - U ( J ^ { ( r ) } X ^ * ) - \\mathbf { 1 } \\frac { \\beta } { 2 } ) \\Big ] . \\end{array} \\end{align*}"}
+{"id": "1878.png", "formula": "\\begin{align*} \\lim _ { p \\to - 2 ^ + } \\frac { 1 } { \\Gamma ( p + 2 ) } \\int _ 0 ^ \\infty r ^ { p + 1 } g _ { u , v } ( r ) d r = g _ { u , v } ( 0 ) \\end{align*}"}
+{"id": "430.png", "formula": "\\begin{align*} \\begin{aligned} - M \\begin{bmatrix} 1 - F _ { j , j } & 1 _ { i - j } - f _ { j , j } & - \\Gamma _ { j } \\\\ - G _ { j - 1 , j - 1 } - \\widetilde { t } _ { j } \\rho ^ 2 & - G _ { j - 1 , j - 1 } - \\widetilde { t } _ { j } \\rho ^ 2 & 1 - F _ { j , j - 1 } ^ { T } \\\\ x & - g _ { j , j } & - f _ { j , j } \\end{bmatrix} \\end{aligned} \\end{align*}"}
+{"id": "1346.png", "formula": "\\begin{align*} \\alpha ^ 2 ( d ( v ) - d ( u ' ) ) ^ 2 + 4 ( 1 - \\alpha ) ^ 2 = \\alpha ^ 2 ( d ( u ' ) - d ( v ) ) ^ 2 , \\end{align*}"}
+{"id": "2531.png", "formula": "\\begin{align*} I ' ( M , L ) = I ( M , L ; \\Omega ) ' \\ ; . \\end{align*}"}
+{"id": "9060.png", "formula": "\\begin{align*} | \\{ x : - a < x < 0 \\ , \\ , \\mbox { a n d } \\ , \\ , f ^ * ( x ) > t \\} | = \\left \\{ \\begin{array} { l l } a , & \\mbox { { \\rm i f } $ t < f ^ * ( a ) $ , } \\\\ \\frac 1 2 | \\{ f ^ * > t \\} | , \\quad & \\mbox { { \\rm i f } $ t \\geq f ^ * ( a ) $ } . \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "4325.png", "formula": "\\begin{align*} \\inf _ { x \\in \\mathcal { X } } & \\ \\left \\{ \\sup _ { \\mathcal { S } \\subseteq [ m ] : | \\mathcal { S } | \\leq \\Gamma } \\left \\{ \\sum _ { i \\in \\mathcal { S } } \\max \\{ 0 , x _ i - \\overline { b } _ i + \\Delta b _ i \\} + \\sum _ { i \\in [ m ] \\setminus \\mathcal { S } } \\max \\{ 0 , x _ i - \\overline { b } _ i \\} \\right \\} \\right \\} . \\end{align*}"}
+{"id": "4857.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } \\operatorname { d i v } \\psi ( y ) \\mathbf { 1 } _ { ( A - x ) } ( y ) d y & = \\int _ { \\mathbb { R } ^ n } \\langle \\psi ( y ) , \\nu _ { ( A - x ) } ( y ) \\rangle d \\mathcal { H } ^ { n - 1 } \\lfloor \\partial ( A - x ) ( y ) \\\\ & = \\int _ { \\mathbb { R } ^ n } \\langle \\psi ( z - x ) , \\nu _ { A } ( z ) \\rangle d \\mathcal { H } ^ { n - 1 } \\lfloor \\partial A ( z ) . \\end{align*}"}
+{"id": "984.png", "formula": "\\begin{align*} \\kappa _ D ( d x ) = \\mathbf 1 _ D \\cdot J ( d x , D ^ c ) + \\mathbf 1 _ D \\cdot \\kappa ( d x ) . \\end{align*}"}
+{"id": "8638.png", "formula": "\\begin{align*} I _ 2 ( t ; x ) = \\int _ { 0 } ^ { t } ( ( K _ 2 + \\phi _ 2 ) ( \\partial _ x X ) ^ 2 + ( K _ 1 + \\phi _ 1 ) ( \\partial ^ 2 _ x X ) ) ( \\tau ; x ) d \\tau , \\end{align*}"}
+{"id": "1829.png", "formula": "\\begin{align*} \\lambda _ { \\beta \\alpha } \\circ g _ { \\alpha } = \\lambda _ { \\beta \\alpha } ( g _ { \\alpha } ) \\circ \\lambda _ { \\beta \\alpha } . \\end{align*}"}
+{"id": "63.png", "formula": "\\begin{align*} \\Delta u : = \\sum _ { i = 1 } ^ n u _ { x _ i x _ i } \\end{align*}"}
+{"id": "6988.png", "formula": "\\begin{align*} x = [ \\mathrm { t a n h } ( t _ 1 ) \\mathrm { c o s } ( t _ { n } ) \\mathrm { s i n } ( t _ { n - 1 } ) \\cdots \\mathrm { s i n } ( t _ { 2 } ) : \\cdots : \\mathrm { t a n h } ( t _ 1 ) \\mathrm { c o s } ( t _ 2 ) : 1 ] , \\end{align*}"}
+{"id": "5274.png", "formula": "\\begin{align*} x ^ { k + 1 } = x ^ { k } \\left ( \\frac { U ^ { k } } { V ^ { k } } \\right ) \\end{align*}"}
+{"id": "2226.png", "formula": "\\begin{align*} \\delta = { 4 \\varepsilon \\over \\theta } = { 4 \\varepsilon \\over ( 1 - { 3 \\over d } ) \\varepsilon _ 1 - { 3 \\over d } \\varepsilon _ 2 } , \\delta _ 0 = { 2 \\varepsilon \\over ( 1 - { 3 \\over d } ) \\varepsilon _ 1 - { 3 \\over d } \\varepsilon _ 2 } . \\end{align*}"}
+{"id": "1691.png", "formula": "\\begin{align*} f ^ { ( 4 ) } ( x ) = c _ 1 ( f ^ * ) ^ { ( 4 ) } ( c _ 2 x ) \\end{align*}"}
+{"id": "1894.png", "formula": "\\begin{align*} \\Delta u - V u = 0 \\ ; G , \\end{align*}"}
+{"id": "1169.png", "formula": "\\begin{align*} \\vec t ^ { ( \\lambda ) } _ Q : = [ \\ell ( Q ) ] ^ { - \\frac 1 2 } \\left \\langle \\vec f , \\theta ^ { ( \\lambda ) } _ Q \\right \\rangle . \\end{align*}"}
+{"id": "1001.png", "formula": "\\begin{align*} \\Xi _ W = \\{ \\mathcal S : \\mathcal S \\mbox { i s a $ W $ - t o t a l f a m i l y } \\} , \\end{align*}"}
+{"id": "3750.png", "formula": "\\begin{align*} _ { 3 } F _ { 2 } \\left ( \\left . \\begin{array} { c } 1 , 1 , a \\\\ 2 , 2 \\end{array} \\right \\vert z \\right ) = \\frac { \\psi \\left ( 2 - a \\right ) + \\gamma + \\ln z + \\mathrm { B } _ { 1 - z } \\left ( 2 - a , 0 \\right ) } { \\left ( 1 - a \\right ) z } . \\end{align*}"}
+{"id": "6536.png", "formula": "\\begin{align*} | 1 - \\phi _ { \\varepsilon } ( \\lambda ) | = | \\mathbb { E } ( 1 - \\cos ( \\lambda \\varepsilon ) ) - \\iota \\mathbb { E } \\sin ( \\lambda \\varepsilon ) | & \\leq c _ 1 \\ , ( | \\lambda | ^ { \\alpha ' } \\mathbb { E } | \\varepsilon | ^ { \\alpha ' } ) \\wedge 1 \\leq c _ 2 ( | \\lambda | ^ { \\alpha ' } \\wedge 1 ) , \\end{align*}"}
+{"id": "3250.png", "formula": "\\begin{align*} \\big ( S ^ { \\wedge , ( l ) } & \\ : V \\ : S ^ { \\bowtie , ( r ) } \\big ) ( x , y ) = - \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { n ! } \\ : S ^ { \\bowtie , ( n + l + r + 1 ) } ( x , y ) \\\\ & \\times \\int _ { - \\infty } ^ 0 \\alpha ^ { l } \\ : ( 1 - \\alpha ) ^ r \\ : ( \\alpha - \\alpha ^ 2 ) ^ n \\ : ( \\Box ^ n V ) \\big | _ { \\alpha y + ( 1 - \\alpha ) x } \\ : d \\alpha \\ : . \\end{align*}"}
+{"id": "729.png", "formula": "\\begin{align*} M & = \\begin{bmatrix} - x _ 3 & 0 & x _ 1 & 0 \\\\ 0 & - x _ 4 & 0 & x _ 2 \\end{bmatrix} . \\end{align*}"}
+{"id": "459.png", "formula": "\\begin{align*} \\pi ( h ^ { - 1 } ) \\varepsilon ( g ) & = \\pi ( h ^ { - 1 } ) \\pi ( g ) \\pi ( g ^ { - 1 } ) = \\pi ( h ^ { - 1 } g ) \\pi ( g ^ { - 1 } ) = \\pi ( h ^ { - 1 } g ) \\pi ( g ^ { - 1 } h ) \\pi ( h ^ { - 1 } g ) \\pi ( g ^ { - 1 } ) \\\\ & = \\pi ( h ^ { - 1 } g ) \\pi ( g ^ { - 1 } h ) \\pi ( h ^ { - 1 } g g ^ { - 1 } ) = \\varepsilon ( h ^ { - 1 } g ) \\pi ( h ^ { - 1 } r ( g ) ) = \\varepsilon ( h ^ { - 1 } g ) \\pi ( h ^ { - 1 } r ( h ) ) \\\\ & = \\varepsilon ( h ^ { - 1 } g ) \\pi ( h ^ { - 1 } ) , \\end{align*}"}
+{"id": "3396.png", "formula": "\\begin{align*} \\mbox { L H S } & = \\frac { 1 } { 2 } \\sum _ { t = 1 } ^ { k } z _ { t } \\eta _ { t } \\left \\Vert \\nabla f ( x _ { t } ) \\right \\Vert ^ { 2 } + z _ { k } \\Delta _ { k + 1 } - z _ { 1 } \\Delta _ { 1 } + \\sum _ { t = 2 } ^ { k } \\left ( z _ { k - 1 } - z _ { k } \\right ) \\Delta _ { k } \\\\ & \\ge \\frac { 1 } { 2 } \\sum _ { t = 1 } ^ { k } z _ { t } \\eta _ { t } \\left \\Vert \\nabla f ( x _ { t } ) \\right \\Vert ^ { 2 } + z _ { k } \\Delta _ { k + 1 } - z _ { 1 } \\Delta _ { 1 } \\end{align*}"}
+{"id": "4207.png", "formula": "\\begin{align*} & - 5 0 4 \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 1 4 ) } = \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } ( \\widetilde { T X } - \\widetilde { L _ R \\otimes C } ) \\right \\} ^ { ( 1 4 ) } . \\end{align*}"}
+{"id": "7148.png", "formula": "\\begin{align*} \\frac { d x ^ \\mu } { d \\tau } = 2 \\eta ^ { \\mu \\nu } k _ \\nu \\quad { \\mbox { a n d } } \\frac { d k _ \\nu } { d \\tau } = 0 \\ , \\ , . \\end{align*}"}
+{"id": "7678.png", "formula": "\\begin{align*} G ^ { \\Lambda ' } _ { L } ( m , n ; z ) = - G ^ { \\Lambda ' } _ { L } ( m , m ; z ) \\sum _ { \\substack { m ' \\in \\Lambda ' \\\\ | m ' - m | = 1 } } G ^ { \\Lambda ' \\setminus \\{ m \\} } _ { L } ( m ' , n ; z ) . \\end{align*}"}
+{"id": "163.png", "formula": "\\begin{align*} & \\sum _ { l } \\langle j _ { a , b } ( f _ s ) \\ | \\kappa _ { 0 } ( f ^ { \\prime } _ { l } ) \\rangle j _ { a , b } ( e _ { i } ) \\langle \\kappa _ { 0 } ( f ^ { \\prime } _ { l } ) \\ | j _ { a , b } ( f _ { j } ) ) \\rangle \\\\ & = j _ { a , b } \\left ( ( f ^ { * } _ { s } \\otimes 1 _ { Z } ) \\circ 1 _ { X ^ { b - a } } \\otimes c _ { Z , W } ) \\circ ( e _ { i } \\otimes 1 _ { W } ) \\circ f _ { j } \\right ) \\end{align*}"}
+{"id": "434.png", "formula": "\\begin{align*} | F _ { 2 j , k } ^ { ( c ) } | \\le \\| \\bold { F } ^ { ( i ) } \\| _ 2 ( \\sum _ { p = 1 , p \\neq k } ^ { 2 j - 1 } | F _ { 2 j , k } ^ { j , p } | ^ 2 ) ^ { \\frac { 1 } { 2 } } = \\frac { j ^ { \\frac { 1 } { 2 } } } { M } \\frac { j ^ { \\frac { 3 } { 2 } } } { M ^ 2 } + \\frac { j ^ { \\frac { 1 } { 2 } } } { M ^ 2 } \\le \\frac { K _ f j } { M ^ 2 } , \\end{align*}"}
+{"id": "3538.png", "formula": "\\begin{align*} X ( \\alpha + \\beta f ) = \\alpha + \\beta T ^ { - 1 } f , \\end{align*}"}
+{"id": "5875.png", "formula": "\\begin{align*} \\dfrac { M _ { 2 } ( \\mathcal { C } ( G ) ) } { | e ( \\mathcal { C } ( G ) ) | } - \\dfrac { M _ { 1 } ( \\mathcal { C } ( G ) ) } { | v ( \\mathcal { C } ( G ) ) | } = \\dfrac { 3 2 n ^ { 4 } ( 2 n - 1 1 ) + 3 2 n ^ { 2 } ( 2 1 n - 1 6 ) + 1 2 8 n } { 8 n ^ { 2 } ( n - 2 ) + 1 6 n - 5 } : = \\frac { f ( n ) } { g ( n ) } . \\end{align*}"}
+{"id": "7729.png", "formula": "\\begin{align*} \\mathcal { N } ( E ) = \\hat { \\mathcal { N } } ( E ) . \\end{align*}"}
+{"id": "6834.png", "formula": "\\begin{align*} 1 & \\leq | \\sin ( \\psi _ { N - 1 } ) \\sin ( \\psi _ { N - 1 } ' ) | \\cdot \\left | \\left ( B ( B ' ) ^ { - 1 } \\right ) _ { 1 1 } \\right | + | \\cos ( \\psi _ { N - 1 } ) \\cos ( \\psi _ { N - 1 } ' ) | \\\\ & \\leq \\sin ( \\psi _ { N - 1 } ) \\sin ( \\psi _ { N - 1 } ' ) + \\cos ( \\psi _ { N - 1 } ) \\cos ( \\psi _ { N - 1 } ' ) \\\\ & = \\cos ( \\psi _ { N - 1 } - \\psi _ { N - 1 } ' ) \\leq 1 . \\end{align*}"}
+{"id": "5712.png", "formula": "\\begin{align*} \\mathbf { L } ^ \\dagger \\psi ^ \\pm _ i = \\gamma ^ \\pm _ i \\psi ^ \\pm _ i . \\end{align*}"}
+{"id": "240.png", "formula": "\\begin{align*} \\frac { d ^ 2 \\bar { x } } { d \\bar { t } ^ 2 } = \\frac { 1 } { h } \\left [ \\frac { d } { d x } \\left ( \\frac { 1 } { h } \\frac { d \\varphi } { d x } \\right ) v ^ 2 + \\left ( \\frac { 1 } { h } \\frac { d \\varphi } { d x } \\right ) \\frac { d ^ 2 x } { d t ^ 2 } \\right ] . \\end{align*}"}
+{"id": "3605.png", "formula": "\\begin{align*} P = \\frac { 1 } { 3 } ( I + T + T ^ 2 ) , Q = \\frac { 1 } { 3 } ( I + \\lambda ^ 2 T + \\lambda T ^ 2 ) , R = \\frac { 1 } { 3 } ( I + \\lambda T + \\lambda ^ 2 T ^ 2 ) . \\end{align*}"}
+{"id": "1024.png", "formula": "\\begin{align*} \\lambda ^ \\star = \\lim _ { n \\to \\infty } \\min _ { \\gamma \\in \\Gamma ^ { t _ \\lambda } ( n ) } \\lambda ( \\gamma ) . \\end{align*}"}
+{"id": "1239.png", "formula": "\\begin{align*} \\Phi _ n ( q ) = \\prod _ { \\substack { 1 \\le k \\le n \\\\ [ 3 p t ] ( n , k ) = 1 } } ( q - \\zeta ^ k ) , \\end{align*}"}
+{"id": "5705.png", "formula": "\\begin{align*} \\mathbf { L } \\psi ^ { \\pm } _ i = \\gamma ^ \\pm _ i \\psi ^ \\pm _ i . \\end{align*}"}
+{"id": "6176.png", "formula": "\\begin{align*} \\varphi ^ k ( x _ 1 , \\cdots , x _ m , \\lambda ) : = - \\lambda ^ T \\Big ( \\sum _ { i = 1 } ^ { m } A _ i x _ i - b \\Big ) + \\frac { \\beta ^ k } { 2 } \\Big \\| \\sum _ { i = 1 } ^ { m } A _ i x _ i - b \\Big \\| ^ 2 , \\beta ^ k > 0 , \\end{align*}"}
+{"id": "6298.png", "formula": "\\begin{align*} e ( R ) \\ge \\max \\left \\{ \\binom { 4 \\alpha t } { 3 } , \\binom { t } { 3 } - \\binom { t - \\alpha t } { 3 } \\right \\} + \\frac { \\gamma t ^ 3 } { 4 } . \\end{align*}"}
+{"id": "7470.png", "formula": "\\begin{align*} ( C ( X ) , g _ c = d r ^ 2 + r ^ 2 g _ X , o ) \\end{align*}"}
+{"id": "2010.png", "formula": "\\begin{align*} \\mathcal { N } _ 0 & = a + b + c = 0 , \\\\ \\mathcal { N } _ 1 & = a \\alpha + b \\beta + c \\gamma = 1 , \\\\ \\mathcal { N } _ 2 & = a \\alpha ^ 2 + b \\beta ^ 2 + c \\gamma ^ 2 = 1 , \\end{align*}"}
+{"id": "4734.png", "formula": "\\begin{align*} & \\forall \\gamma ^ { \\mathbb { N } ^ \\mathbb { N } } , \\lambda ^ { \\mathbb { N } ^ \\mathbb { N } } , x ^ X \\Big ( \\gamma > _ \\mathbb { R } 0 \\land \\lambda > _ \\mathbb { R } 0 \\land x \\in \\mathrm { d o m } ( J ^ A _ \\lambda ) \\\\ & \\qquad \\qquad \\qquad \\rightarrow J ^ A _ \\lambda x = _ X J ^ A _ { \\gamma } \\left ( \\frac { \\gamma } { \\lambda } x + _ X \\left ( 1 - \\frac { \\gamma } { \\lambda } \\right ) J ^ A _ \\gamma x \\right ) \\Big ) . \\end{align*}"}
+{"id": "6951.png", "formula": "\\begin{align*} W ^ { \\bot } = \\{ \\mathbf { z } \\in \\mathbb { C } ^ { n + 1 } : \\langle \\mathbf { z } , \\mathbf { w } \\rangle = 0 \\ ; \\mathrm { f o r \\ ; e v e r y \\ ; \\mathbf { w } \\in W } \\} \\end{align*}"}
+{"id": "7880.png", "formula": "\\begin{align*} U = \\begin{cases} U _ { 0 } \\cup U _ { 1 } , & \\\\ U _ { 0 } \\cup U _ { 2 } , & \\\\ U _ { 1 } \\cup U _ { 2 } , & \\\\ \\end{cases} \\end{align*}"}
+{"id": "7288.png", "formula": "\\begin{align*} S _ { m , r } ( \\alpha , n ) : = \\frac { 1 } { m } \\sum _ { j = 0 } ^ { m - 1 } \\sec ^ { 2 n } \\left ( \\frac { j + \\alpha } { m } \\pi \\right ) e ^ { 2 \\pi i r j / m } . \\end{align*}"}
+{"id": "8148.png", "formula": "\\begin{align*} & P _ { W { \\underline X } ^ n { \\underline S } ^ { \\ell } Y ^ n } ( w , { \\underline x } ^ n , { \\underline s } ^ { \\ell } , y ^ n ) = P _ W ( w ) \\\\ & \\times \\prod _ { i = 1 } ^ { \\ell } \\left ( P _ { \\underline S } ( { \\underline s } _ i ) \\prod _ { j = 1 } ^ L P _ { Y | { \\underline X \\underline S } } ( y _ { i , j } | { \\underline x } _ { i , j } { \\underline s } _ i ) P ( { \\underline x } _ { i , j } | w , { y } _ { i } ^ { j - 1 } ) \\right ) \\end{align*}"}
+{"id": "4729.png", "formula": "\\begin{align*} \\begin{cases} u ' ( t ) \\in - A u ( t ) , \\ , 0 < t < \\infty \\\\ u ( 0 ) = x \\end{cases} \\end{align*}"}
+{"id": "5857.png", "formula": "\\begin{align*} \\dfrac { M _ { 2 } ( \\mathcal { C } ( D _ { 2 m } ) ) } { | e ( \\mathcal { C } ( D _ { 2 m } ) ) | } = \\dfrac { ( m - 2 ) ( m - 3 ) ^ { 3 } + m } { ( m - 2 ) ( m - 3 ) + m } \\dfrac { M _ { 1 } ( \\mathcal { C } ( D _ { 2 m } ) ) } { | v ( \\mathcal { C } ( D _ { 2 m } ) ) | } = \\dfrac { ( m - 2 ) ( m - 3 ) ^ { 2 } + m } { 2 m - 2 } . \\end{align*}"}
+{"id": "6537.png", "formula": "\\begin{align*} S _ N = \\sum ^ { N } _ { n = 1 } \\big [ K ( X _ n ) - \\mathbb { E } K ( X _ n ) \\big ] \\end{align*}"}
+{"id": "8753.png", "formula": "\\begin{align*} \\int \\vert x - y \\vert \\ , \\beta ( d x , d y ) - \\int \\vert x - y \\vert \\ , \\gamma ( d x , d y ) = g ( x _ - ) - g ( x _ + ) < 0 . \\end{align*}"}
+{"id": "4896.png", "formula": "\\begin{align*} \\dim ( | - K _ { S ' } - C | ) = r - \\gamma \\geq 0 \\end{align*}"}
+{"id": "739.png", "formula": "\\begin{align*} & n _ p \\bigl ( a , a + b , a J _ { k - 1 } ( v ) + b J _ k ( v ) \\bigr ) \\\\ & = \\frac { 1 } { 2 } \\biggl ( - v ( a - r ^ 2 ) + ( a - 1 ) ( b - 1 ) + \\frac { v ( a + r ) ( a - r ) J _ { k - 1 } ( v ) } { J _ k ( v ) } \\\\ & + v ( a - r ) \\bigl ( J _ k ( v ) - J _ { k - 1 } ( v ) \\bigr ) \\biggr ) \\\\ & + \\frac { p } { 2 } a J _ k ( v ) \\bigl ( 2 ( v a + b ) - v ( J _ k ( v ) - J _ { k - 1 } ( v ) \\bigr ) - \\frac { p ^ 2 } { 2 } a v J _ k ( v ) \\bigl ( J _ k ( v ) - J _ { k - 1 } ( v ) \\bigr ) \\ , , \\end{align*}"}
+{"id": "7795.png", "formula": "\\begin{align*} \\begin{aligned} & ( A + B K ^ { ( i ) } ) ^ { \\top } P ^ { ( i + 1 ) } + P ^ { ( i + 1 ) } ( A + B K ^ { ( i ) } ) + ( C + D K ^ { ( i ) } ) ^ { \\top } P ^ { ( i + 1 ) } ( C + D K ^ { ( i ) } ) \\\\ & ~ ~ ~ \\qquad \\qquad \\qquad + K ^ { ( i ) \\top } R K ^ { ( i ) } + S ^ { \\top } K ^ { ( i ) } + K ^ { ( i ) \\top } S + Q = 0 , \\end{aligned} \\end{align*}"}
+{"id": "292.png", "formula": "\\begin{align*} \\left \\langle L _ F e _ { k } , e _ { j } \\right \\rangle = \\begin{cases} \\sum _ { l = 1 } ^ { n } \\ , \\alpha _ l ( j ) \\ , a _ { l , \\alpha ( l ) } & \\textrm { i f } j = k \\\\ \\alpha _ { l } ( k ) \\ , a _ { l , \\alpha ( r ) } & \\textrm { i f } \\alpha ( j ) = ( \\alpha _ 1 ( k ) , \\cdots , \\alpha _ { l } ( k ) - 1 , \\cdots , \\\\ & \\alpha _ { r } ( k ) + 1 , \\cdots , \\alpha _ n ( k ) ) , \\\\ 0 & \\textrm { o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "1646.png", "formula": "\\begin{align*} \\partial _ { \\nu } v \\mid _ { S _ { T } } = 0 , \\partial _ { \\nu } p \\mid _ { S _ { T } } = 0 , \\end{align*}"}
+{"id": "8983.png", "formula": "\\begin{align*} u '' = M _ + \\left ( - \\frac { ( N - 1 ) } { r } K _ + ( u ' ) - \\frac { u } { r ^ \\gamma } \\right ) \\hbox { i n } ( 0 , + \\infty ) \\ , , \\end{align*}"}
+{"id": "6267.png", "formula": "\\begin{align*} [ \\bar { \\rho } \\bar { U } ] \\Big | _ { r = r _ s } = 0 , [ \\bar { \\rho } \\bar { U } ^ 2 + \\bar { P } ] \\Big | _ { r = r _ s } = 0 , [ \\bar { B } ] \\Big | _ { r = r _ s } = 0 , \\bar { K } ^ + > \\bar { K } ^ - , \\end{align*}"}
+{"id": "2995.png", "formula": "\\begin{align*} g ( H ( x , y ) , z ) = - g ( H ( x , z ) , y ) . \\end{align*}"}
+{"id": "1040.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sum _ { k = 1 } ^ { n } \\Vert \\phi _ { n , k } - \\phi _ k \\Vert = 0 . \\end{align*}"}
+{"id": "7240.png", "formula": "\\begin{align*} \\phi ( \\ast ( a _ 1 , \\dots , a _ k ) ) = \\star ( \\phi ( a _ 1 ) , \\dots , \\phi ( a _ k ) ) a _ 1 , \\dots , a _ k \\in M . \\end{align*}"}
+{"id": "6653.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi \\mathrm { i } } \\oint _ { | z | = r _ { 1 } } \\frac { f ' ( z ) } { f ( z ) } \\mathrm { d } z - \\frac { 1 } { 2 \\pi \\mathrm { i } } \\oint _ { | z | = r _ { 1 } } \\frac { g ' ( z ) } { g ( z ) } \\mathrm { d } z = \\sum _ { | z _ { i } | = r _ { 1 } } \\frac { 1 } { 2 \\pi \\mathrm { i } } \\oint _ { | z | = r _ { 1 } } \\frac { 1 } { z - z _ { i } } \\mathrm { d } z = \\frac { 1 } { 2 } \\# \\{ | z _ { i } | = r _ { 1 } \\} . \\end{align*}"}
+{"id": "8620.png", "formula": "\\begin{align*} \\| v _ n ( t ; \\cdot ) \\| _ { L ^ \\infty ( \\mathbb { R } ) } = \\| ( \\partial _ x ^ n u ) ( \\cdot , t ) \\| _ { L ^ \\infty ( \\mathbb { R } ) } < C _ 2 ( ( n - 1 ) g ) ^ { 2 ( n - 1 ) } q ^ { - 1 - ( n - 1 ) \\sigma } ( t ) , \\end{align*}"}
+{"id": "6384.png", "formula": "\\begin{align*} q _ { x , y \\triangleright z } q _ { y , z } = q _ { x \\triangleright y , x \\triangleright z } q _ { x , z } \\qquad x , y , z \\in X \\end{align*}"}
+{"id": "7816.png", "formula": "\\begin{align*} \\max _ { \\substack { \\boldsymbol { a } } } & \\sum _ { k = 1 } ^ { K } R _ { k } ( \\boldsymbol { a } ) + \\sum _ { k = 1 } ^ { K } R _ { k } ^ { \\mathrm { l o c } } \\left ( a _ { k } \\right ) , \\\\ & a _ { k } \\in [ 0 , 1 ] , \\forall k \\in \\mathcal { K } . \\end{align*}"}
+{"id": "6569.png", "formula": "\\begin{align*} \\alpha _ { k } ( p ^ m ) = \\begin{cases} 1 & p = 2 , \\\\ \\left ( 1 - \\frac { ( - 1 ) ^ { \\frac { k ( p - 1 ) } { 4 } } } { p ^ { \\frac { k } { 2 } } } \\right ) & \\end{cases} \\end{align*}"}
+{"id": "3086.png", "formula": "\\begin{align*} { \\rm C o e f f } ( \\varphi ^ * ( F _ i ) , t ^ k ) = P _ { i k } ( c _ { \\beta _ 1 } , \\ldots , c _ { k - v _ i + \\beta _ i - 1 } ) + \\left ( \\prod _ { l = 0 } ^ { i - 2 } b _ l \\right ) c _ { k - v _ i + \\beta _ i } \\end{align*}"}
+{"id": "4667.png", "formula": "\\begin{align*} | \\{ b ^ n \\} _ { n = 0 } ^ \\infty \\pmod p | > 2 \\sqrt { p } \\end{align*}"}
+{"id": "4048.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 u } { \\partial t \\partial s } ( t , s ) = \\langle ( 1 , \\frac { \\partial } { \\partial t } \\mu _ { \\hat X ^ 1 _ t } ( \\omega ) ) , ( 1 , \\frac { \\partial } { \\partial s } \\mu _ { \\hat Y ^ 1 _ s } ( \\omega ^ \\prime ) ) \\rangle _ { \\mathbb { R } \\oplus \\mathcal { H } _ { \\mathcal { S } } } u ( t , s ) . \\end{align*}"}
+{"id": "7855.png", "formula": "\\begin{align*} f _ n ( \\mathbf { x } ) = \\sum _ { i = 1 } ^ { m - 1 } a _ { i } x _ { \\pi ( i ) } x _ { \\pi ( i + 1 ) } + \\sum _ { t = 1 } ^ { p - 1 } \\sum _ { k = 1 } ^ { m } c _ { t , k } x _ { k } ^ { t } + n x _ { \\pi ( 1 ) } , \\end{align*}"}
+{"id": "7471.png", "formula": "\\begin{align*} f _ N ( F ( r , x ) ) = \\frac { r ^ 2 } { 4 } , \\end{align*}"}
+{"id": "3111.png", "formula": "\\begin{align*} \\varphi ^ * \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & : = \\left ( \\begin{array} { c c c } a ^ { \\ell _ 1 } & a ^ { \\ell _ 1 } \\ , b ^ { p ^ { e _ 1 } } \\ , d ^ { p ^ { e _ 1 } } & a ^ { \\ell _ 1 } \\ , b ^ { p ^ { e _ 2 } } \\ , d ^ { p ^ { e _ 2 } } \\\\ 0 & a ^ { \\ell _ 2 } & 0 \\\\ 0 & 0 & d ^ { - \\ell _ 3 } \\end{array} \\right ) . \\end{align*}"}
+{"id": "5728.png", "formula": "\\begin{align*} ( \\Pi _ \\gamma q ) ( t ) = \\sum _ { i : \\gamma ^ - _ i \\le \\gamma } \\xi _ i ^ - ( t ) \\psi _ i ^ - . \\end{align*}"}
+{"id": "1954.png", "formula": "\\begin{align*} u & \\ge \\frac { M _ j + m _ j } { 2 } + \\frac { L } { 2 \\cdot 4 ^ { \\alpha j } } \\delta - \\frac { L } { 2 \\cdot 4 ^ { \\alpha j } } \\\\ & = M _ j - \\frac { M _ j - m _ j } { 2 } - \\frac { L } { 2 \\cdot 4 ^ { \\alpha j } } ( 1 - \\delta ) \\\\ & = M _ j - \\frac { L } { 2 \\cdot 4 ^ { \\alpha j } } ( 2 - \\delta ) B _ { 4 ^ { - j } R } . \\end{align*}"}
+{"id": "5816.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } e ^ { ( b - \\varepsilon ) t } ( X _ + ( t ) + X _ 0 ( t ) + X _ - ( t ) ) = 0 . \\end{align*}"}
+{"id": "7650.png", "formula": "\\begin{align*} \\lambda _ { \\mathrm { A n d } } = 2 \\norm { \\rho } _ { \\infty } \\mu _ d e \\ln \\left ( \\frac { \\lambda _ { \\mathrm { A n d } } } { 2 \\norm { \\rho } _ { \\infty } } \\right ) . \\end{align*}"}
+{"id": "7335.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } ( - 1 ) ^ n 2 ^ { - ( n + 1 ) } \\overline { L _ { X _ m } ( n + 1 , \\chi ) } s ^ n = \\frac { m } { \\tau ( \\chi ) } \\sum _ { r = 0 } ^ { m - 1 } \\chi ( r ) F _ { m , r } ( s ) . \\end{align*}"}
+{"id": "8980.png", "formula": "\\begin{align*} \\bar \\lambda _ \\gamma ^ { \\prime } : = \\sup \\{ \\mu \\ , : \\ \\exists \\ , u \\in C ^ 2 ( B ( 0 , 1 ) \\setminus \\{ 0 \\} ) \\ , , \\ u > 0 \\hbox { i n } B ( 0 , 1 ) \\setminus \\{ 0 \\} , \\ u \\hbox { r a d i a l } , \\ \\ F ( D ^ 2 u ) + \\mu \\frac { u } { r ^ \\gamma } \\leq 0 \\} \\ , , \\end{align*}"}
+{"id": "7842.png", "formula": "\\begin{align*} \\nabla ^ { \\perp \\psi } _ { E _ a } \\nabla ^ { \\perp \\psi } _ { E _ a } H ^ { \\psi } = \\frac { p } { p + q } \\nabla ^ { \\perp } _ { E _ a } \\nabla ^ { \\perp } _ { E _ a } H _ 1 + \\frac { q } { p + q } B ^ j ( \\nabla ^ { \\mathbb { C } P ^ p } _ { E _ a } E _ a , H _ 2 ) - \\frac { q } { p + q } H _ 2 . \\end{align*}"}
+{"id": "2225.png", "formula": "\\begin{align*} X ( t ) = \\| \\Phi \\| _ { V ( \\Omega ^ t ) } + \\| \\Gamma \\| _ { V ( \\Omega ^ t ) } . \\end{align*}"}
+{"id": "7802.png", "formula": "\\begin{align*} \\boldsymbol { y } = \\sum _ { t \\in \\mathcal { T } } \\boldsymbol { g } _ { t } \\sqrt { p _ { t } } s _ { t } + \\sum _ { r \\in \\mathcal { R } } \\boldsymbol { g } _ { r } \\sqrt { p _ { r } } s _ { r } + \\boldsymbol { z } , \\end{align*}"}
+{"id": "6617.png", "formula": "\\begin{align*} \\int _ { a - i \\infty } ^ { a + i \\infty } \\frac { y ^ s } { s ( s + 1 ) \\cdots ( s + N ) } d s = J ( y ) , \\end{align*}"}
+{"id": "2553.png", "formula": "\\begin{align*} C ^ \\infty ( M , L ) = \\bigcap _ k C ^ k ( M , L ) \\ ; , C ^ { - \\infty } ( M , L ) = \\bigcup _ k C ^ { \\prime \\ , - k } ( M , L ) \\ ; . \\end{align*}"}
+{"id": "3238.png", "formula": "\\begin{align*} \\delta _ q P ( \\xi ) & = - \\frac { 1 } { 6 } \\sum _ { \\alpha , \\beta = 1 } ^ 3 \\Phi _ q ^ { \\beta , \\alpha } \\Big ( - \\epsilon ( \\xi ^ 0 ) \\ , \\hat { \\vec { \\xi } } \\ , \\Big ) \\ : \\xi ^ 0 \\ : \\frac { e ^ { \\frac { i } { 2 } q \\xi } } { q \\xi } \\ : P ( \\xi ) \\Big ( 1 + \\O \\Big ( \\frac { \\varepsilon } { t } \\Big ) \\bigg ) + ( \\deg < 2 ) \\ : . \\end{align*}"}
+{"id": "6213.png", "formula": "\\begin{align*} \\tilde { \\mathcal { S } } : = \\left \\{ ( \\omega , \\c d ) \\colon - \\frac { 1 6 \\omega } { ( \\omega + 2 ) ^ 2 } < \\c d < 1 - \\frac { 1 } { \\omega } \\right \\} . \\end{align*}"}
+{"id": "3209.png", "formula": "\\begin{align*} u ( t ) = \\sum \\limits _ { k = 1 } ^ \\infty T _ k ( t ) v _ k , \\end{align*}"}
+{"id": "8651.png", "formula": "\\begin{align*} F ^ * ( x _ 0 ^ * ) + 1 _ K ^ * ( x _ 0 ^ * - y _ 0 ^ * ) & = \\langle x _ 0 , x _ 0 ^ * \\rangle - F ( x _ 0 ) + \\langle - x _ 0 , x _ 0 ^ * - y _ 0 ^ * \\rangle , \\\\ & = \\langle x _ 0 , y _ 0 ^ * \\rangle - F ( x _ 0 ) . \\end{align*}"}
+{"id": "2360.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\| w _ { 0 , k } - w _ { 0 } \\| _ { L ^ { 3 } } = 0 . \\end{align*}"}
+{"id": "5115.png", "formula": "\\begin{align*} L o g _ { T } ( x ) = \\frac { x ^ { t - 1 } - 1 } { t - 1 } \\end{align*}"}
+{"id": "797.png", "formula": "\\begin{align*} & { D _ m } { D _ k } \\Psi - { D _ k } { D _ m } \\Psi = K _ { 0 m k } ^ r { \\dot \\partial _ r } \\Psi , \\end{align*}"}
+{"id": "2888.png", "formula": "\\begin{align*} \\| f \\| _ { M _ r ^ \\alpha ( \\mathbb { R } ^ n ) } : = \\sup _ { B \\in \\mathbb { B } } \\left | B \\right | ^ { \\frac { 1 } { \\alpha } - \\frac { 1 } { r } } \\| f \\| _ { L ^ r ( B ) } < \\infty , \\end{align*}"}
+{"id": "3537.png", "formula": "\\begin{align*} H ^ { \\infty } ( \\mathbb { D } ) = \\mathbb { C } \\dotplus H ^ { \\infty } _ { 0 } ( \\mathbb { D } ) . \\end{align*}"}
+{"id": "3028.png", "formula": "\\begin{align*} y ^ \\mu \\overline { y } _ \\mu = 1 \\ , , y _ \\mu \\in \\C \\ , , \\mu = 1 , \\ldots , n \\ , , \\end{align*}"}
+{"id": "642.png", "formula": "\\begin{align*} \\int _ { \\rho _ - } ^ { \\rho _ + } \\frac { s ( \\xi ) ^ { \\frac 1 n } } { \\sqrt { 1 - s ( \\xi ) ^ { \\frac 2 n } } } \\ , d \\xi = \\frac { \\rho _ + - \\rho _ - } { 2 } \\int _ { - 1 } ^ 1 \\frac { u ^ { \\frac 1 n } } { \\sqrt { 1 - u ^ { \\frac 2 n } } } \\ , d u = 0 . \\end{align*}"}
+{"id": "7825.png", "formula": "\\begin{align*} \\max _ { \\{ \\theta _ m \\} _ { m = 1 } ^ { M } } \\ ! \\ ! \\ ! \\ ! \\mathcal { H } \\ ! \\left ( \\ ! \\{ \\theta _ m \\} _ { m = 1 } ^ { M } \\ ! \\right ) \\ ! \\ ! : = \\ ! \\ ! \\sum _ { t } \\ ! R _ { t } \\ ! \\left ( \\ ! \\{ \\theta _ m \\ ! \\} _ { m = 1 } ^ { M } \\ ! \\right ) \\ ! + \\ ! \\sum _ { r } \\ ! R _ { r } \\ ! \\left ( \\ ! \\{ \\theta _ m \\} _ { m = 1 } ^ { M } \\ ! \\right ) \\ ! , \\end{align*}"}
+{"id": "8532.png", "formula": "\\begin{align*} f _ { \\Sigma } & = - s \\alpha _ 2 \\cdot \\log + ( ( \\alpha _ 1 - \\alpha _ 2 ) \\log _ 2 ( \\pi _ 1 ) + \\beta \\log _ 2 ( u ( K ) ) ) \\cdot \\phi \\\\ & = - s \\alpha _ 2 \\cdot \\log + ( ( \\alpha _ 1 - \\alpha _ 2 ) \\log _ 1 ( \\pi _ 2 ) - \\beta \\log _ 1 ( u ( K ) ) ) \\cdot \\phi \\end{align*}"}
+{"id": "7979.png", "formula": "\\begin{align*} \\mathrm { t r } \\big ( \\mathrm { l i } _ { \\phi } ( \\mu ) \\big ) = \\mu , \\ \\forall \\mu \\in H ^ { \\frac { 1 } { 2 } } \\Lambda ^ { 0 } ( \\partial \\Omega ) , \\end{align*}"}
+{"id": "4329.png", "formula": "\\begin{gather*} \\mathcal { X } : = \\left \\{ x \\in \\{ 0 , 1 \\} ^ { [ m ] \\times | \\mathcal { J } | } : \\sum _ { i \\in [ m ] } x _ { i , j } = 1 \\ \\forall j \\in \\mathcal { J } , \\ \\sum _ { j \\in \\mathcal { J } } x _ { i , j } = 1 \\ \\forall i \\in [ m ] \\right \\} . \\end{gather*}"}
+{"id": "2052.png", "formula": "\\begin{align*} \\mathrm { d } \\Theta ( s , x ) = \\alpha \\mathrm { d } \\xi _ 1 ( s , x ) . \\end{align*}"}
+{"id": "7050.png", "formula": "\\begin{align*} \\langle \\partial _ t u , \\varphi \\rangle + \\langle - \\Delta u , \\varphi \\rangle + \\langle b _ m \\cdot w , \\varphi \\rangle = 0 , \\end{align*}"}
+{"id": "7574.png", "formula": "\\begin{align*} \\| \\Delta u _ { n } \\| _ { 2 } ^ { 2 } = \\| \\Delta v _ { n } ^ { 1 } ( \\cdot + y _ { n } ) + \\Delta z ^ { 1 } \\| _ { 2 } ^ { 2 } = \\| \\Delta v _ { n } ^ { 1 } \\| _ { 2 } ^ { 2 } + \\| \\Delta z ^ { 1 } \\| _ { 2 } ^ { 2 } + o _ { n } ( 1 ) . \\end{align*}"}
+{"id": "3121.png", "formula": "\\begin{align*} \\varphi ^ * \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & = \\left ( \\begin{array} { c c c } a ^ { 2 \\ , p ^ { e _ 1 } } & a ^ { p ^ { e _ 1 } } \\ , b ^ { p ^ { e _ 1 } } & \\frac { 1 } { 2 } \\ , b ^ { 2 \\ , p ^ { e _ 1 } } \\\\ 0 & 1 & b ^ { p ^ { e _ 1 } } \\ , d ^ { p ^ { e _ 1 } } \\\\ 0 & 0 & d ^ { 2 \\ , p ^ { e _ 1 } } \\end{array} \\right ) ( \\ , e _ 1 \\geq 0 \\ , ) . \\end{align*}"}
+{"id": "1296.png", "formula": "\\begin{align*} \\langle \\dot { x } _ 1 \\rangle = ( a - k + k \\hat { \\alpha } ) \\langle x _ 1 \\rangle \\ ; , \\end{align*}"}
+{"id": "6847.png", "formula": "\\begin{align*} ( d g _ K ) _ { x K } = \\mu _ { x 1 } = ( e ^ 1 ) _ { x 1 } \\wedge \\ldots \\wedge ( e ^ { 2 N - 1 } ) _ { x 1 } . \\end{align*}"}
+{"id": "6183.png", "formula": "\\begin{align*} \\begin{pmatrix} \\bar { x } _ 2 ^ { k + 1 } \\\\ \\bar { \\lambda } ^ { k + 1 } \\end{pmatrix} = \\begin{pmatrix} \\bar { x } ^ { k } _ 2 \\\\ \\bar { \\lambda } ^ { k } \\end{pmatrix} - \\begin{pmatrix} I _ { n _ 2 } & 0 \\\\ - \\gamma \\tau ^ k \\beta ^ k A _ 2 & \\gamma I _ l \\end{pmatrix} \\begin{pmatrix} \\bar { x } _ 2 ^ { k } - \\widetilde { x } _ 2 ^ k \\\\ \\bar { \\lambda } ^ k - \\widetilde { \\lambda } ^ k \\end{pmatrix} . \\end{align*}"}
+{"id": "4132.png", "formula": "\\begin{align*} w _ { i j } = P _ { i j } + \\sum _ { k < j } P _ { i k } w _ { k j } , i \\in \\cal { J } \\end{align*}"}
+{"id": "5533.png", "formula": "\\begin{align*} ( x , y ) \\in R x = y . \\end{align*}"}
+{"id": "57.png", "formula": "\\begin{align*} ( \\prescript { } { j - i _ 0 + 1 } { \\hat { s } } ) ^ * ( i ) = & \\prescript { } { j - i _ 0 + 1 } { \\hat { s } } ( j - i ) \\\\ = & \\begin{cases} \\hat { s } ( j - i ) & j - i < j - i _ 0 + 1 \\\\ \\hat { s } ( j - i + 1 ) & j - i _ 0 + 1 \\leq j - i \\end{cases} \\\\ = & \\begin{cases} \\hat { s } ( j - i ) & i _ 0 \\leq i \\\\ \\hat { s } ( j - i + 1 ) & i < i _ 0 \\end{cases} \\end{align*}"}
+{"id": "4707.png", "formula": "\\begin{align*} u ( t ) = q _ { l , \\hat { l } } \\circ j _ x ( t ) t \\in \\overline { D _ x \\cap Q _ x } \\end{align*}"}
+{"id": "4717.png", "formula": "\\begin{align*} f _ { e x t } : = \\left \\{ \\begin{array} { l c l } e ^ { - \\frac { \\zeta } { 1 6 } \\tilde { \\gamma } ^ { - h _ T ^ { e x t } } } & & \\\\ 1 & & \\end{array} \\right . \\ , . \\end{align*}"}
+{"id": "8404.png", "formula": "\\begin{align*} ( a _ 1 \\circ b ) S ( \\sigma ( a _ 2 ) ) ( a _ 3 \\circ c ) = & ( B _ 1 ( a _ 1 ) b S ( B _ 2 ( a _ 2 ) ) ) S ( \\sigma ( a _ 3 ) ) ( B _ 1 ( a _ 4 ) c S ( B _ 2 ( a _ 5 ) ) ) \\\\ = & ( B _ 1 ( a _ 1 ) b S ( B _ 2 ( a _ 2 ) ) ) B _ 2 ( a _ 4 ) S ( B _ 1 ( a _ 3 ) ) ( B _ 1 ( a _ 5 ) c S ( B _ 2 ( a _ 6 ) ) ) \\\\ = & ( B _ 1 ( a _ 1 ) b S ( B _ 2 ( a _ 2 ) ) ) B _ 2 ( a _ 3 ) S ( B _ 1 ( a _ 4 ) ) ( B _ 1 ( a _ 5 ) c S ( B _ 2 ( a _ 6 ) ) ) \\\\ = & B _ 1 ( a _ 1 ) b c S ( B _ 2 ( a _ 2 ) ) = a \\circ ( b c ) , \\end{align*}"}
+{"id": "5888.png", "formula": "\\begin{align*} \\dfrac { M _ { 1 } ( \\mathcal { N C } ( G ) ) } { | v ( \\mathcal { N C } ( G ) ) | } = \\dfrac { m n ^ { 2 } ( 5 m ^ { 2 } - 9 m + 4 ) } { 2 m - 1 } \\dfrac { M _ { 2 } ( \\mathcal { N C } ( G ) ) } { | e ( \\mathcal { N C } ( G ) ) | } = \\dfrac { 4 n ^ { 2 } ( 2 m ^ { 3 } - 5 m ^ { 2 } + 4 m - 1 ) } { 3 ( m - 1 ) } . \\end{align*}"}
+{"id": "6313.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\epsilon _ { \\ell , \\lambda } & = \\Delta ( \\omega _ { \\ell , \\lambda } - \\omega _ \\ell ) = 2 \\nabla \\omega _ \\ell \\cdot \\nabla \\chi _ \\lambda + \\omega _ \\ell \\Delta \\chi _ \\lambda . \\end{align*}"}
+{"id": "3118.png", "formula": "\\begin{align*} d _ { ( 1 ) ^ * } ( \\varphi ) & : = ( \\ , \\dim _ k \\ , ( V ) ^ \\varphi _ { \\ell _ 1 } , \\ ; \\dim _ k \\ , ( V ) ^ \\varphi _ { \\ell _ 2 } , \\ ; \\dim _ k \\ , ( V ) ^ \\varphi _ { \\ell _ 3 } \\ , ) , \\\\ d ' _ { ( 1 ) ^ * } ( \\varphi ) & : = ( \\ , \\dim _ k \\ , ( V ' ) ^ \\varphi _ { \\ell _ 1 } , \\ ; \\dim _ k \\ , ( V ' ) ^ \\varphi _ { \\ell _ 2 } , \\ ; \\dim _ k \\ , ( V ' ) ^ \\varphi _ { \\ell _ 3 } \\ , ) . \\end{align*}"}
+{"id": "3612.png", "formula": "\\begin{align*} - b ( \\sigma ^ 2 \\sigma _ 1 - \\sigma \\sigma _ 1 ) - ( a + b ) \\alpha \\tau _ 0 ( \\sigma ^ 2 \\sigma _ 1 - \\sigma ) + b ( \\sigma ^ 2 \\sigma _ 1 - 1 ) = 0 . \\end{align*}"}
+{"id": "2390.png", "formula": "\\begin{align*} \\zeta _ p ( s , x ) = \\zeta _ p ( s , 1 - x ) . \\end{align*}"}
+{"id": "7681.png", "formula": "\\begin{align*} \\chi ( \\gamma ) : = \\sum _ { m \\in \\mathbb { Z } ^ d } C _ { \\gamma } ( m ) = \\sum ^ { \\infty } _ { N = 0 } C _ N \\gamma ^ N \\end{align*}"}
+{"id": "1828.png", "formula": "\\begin{align*} \\varphi _ { \\gamma } = \\varphi _ { \\alpha } \\circ \\lambda _ { \\alpha \\gamma } = \\varphi _ { \\beta } \\circ \\lambda _ { \\beta \\gamma } \\end{align*}"}
+{"id": "405.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { E } _ { l _ x , l _ y } & = [ \\frac { 2 \\pi l _ x } { L _ { R , x } } , \\frac { 2 \\pi ( l _ x + 1 ) } { L _ { R , x } } ] \\times [ \\frac { 2 \\pi l _ y } { L _ { R , y } } , \\frac { 2 \\pi ( l _ y + 1 ) } { L _ { R , y } } ] \\\\ \\mathcal { E } _ { m _ x , m _ y } & = [ \\frac { 2 \\pi m _ x } { L _ { S , x } } , \\frac { 2 \\pi ( m _ x + 1 ) } { L _ { S , x } } ] \\times [ \\frac { 2 \\pi m _ y } { L _ { S , y } } , \\frac { 2 \\pi ( m _ y + 1 ) } { L _ { S , y } } ] . \\end{aligned} \\end{align*}"}
+{"id": "6122.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\boldsymbol U ' ( t ) = A ^ { \\mathsf T } \\boldsymbol U ( t ) + \\boldsymbol U ( t ) A + C + \\boldsymbol U ( t ) B \\boldsymbol U ( t ) , \\\\ & \\boldsymbol U ( 0 ) = \\boldsymbol Z , \\end{aligned} \\right . \\end{align*}"}
+{"id": "561.png", "formula": "\\begin{align*} \\Delta \\phi & = 0 , \\\\ - \\frac { p } { \\rho } & = g z + j \\omega \\phi . \\end{align*}"}
+{"id": "8206.png", "formula": "\\begin{align*} c ^ { y ^ { i - 1 } } = M P _ { Y ^ { i - 1 } } ( y ^ { i - 1 } ) , \\ b ^ { y ^ { i - 1 } } = M P _ { Y ^ i } ( y ^ { i - 1 } 1 ) . \\end{align*}"}
+{"id": "7314.png", "formula": "\\begin{align*} G _ { X _ { m } , \\chi _ { \\beta } } ( x , y ; s ) & = \\frac { e ^ { - 2 \\pi i \\beta \\frac { \\ell - ( x - y ) } { m } } } { \\sqrt { s ^ 2 + 2 s } } \\\\ & \\cdot \\frac { \\sinh \\left ( m - \\ell \\right ) \\cosh ^ { - 1 } ( s + 1 ) + e ^ { 2 \\pi i \\beta } \\sinh \\left ( \\ell \\cosh ^ { - 1 } ( s + 1 ) \\right ) } { \\cosh \\left ( m \\cosh ^ { - 1 } ( s + 1 ) \\right ) - \\cos 2 \\pi \\beta } . \\end{align*}"}
+{"id": "1897.png", "formula": "\\begin{align*} | \\nabla d ( x , x _ 0 ) | ^ 2 & = \\frac 1 { \\mu ( x ) } \\sum _ { y \\in G } \\omega ( x , y ) [ d ( y , x _ 0 ) - d ( x , x _ 0 ) ] ^ 2 \\\\ & \\leq \\frac 1 { \\mu ( x ) } \\sum _ { y \\in G } \\omega ( x , y ) d ^ 2 ( x , y ) \\ ; x \\in G . \\end{align*}"}
+{"id": "4610.png", "formula": "\\begin{align*} Y : = \\frac { 1 } { d } \\left ( \\frac { d - 1 } { \\gamma } + 1 \\right ) \\leq \\frac { 1 } { d } \\left ( \\frac { 2 d - 1 } { 2 } + 1 \\right ) = 1 + \\frac { 1 } { 2 d } , \\end{align*}"}
+{"id": "4430.png", "formula": "\\begin{gather*} u _ h ^ j \\coloneqq 2 w _ h ^ j - v _ h ^ j , j = 1 , \\dots , M . \\end{gather*}"}
+{"id": "8149.png", "formula": "\\begin{align*} q ^ { \\star } ( \\underline x ^ n , y ^ n ) = ( \\hat { \\underline s } ^ { \\star } ( \\underline x _ 1 ^ L , y _ 1 ^ L ) , \\hat { \\underline s } ^ { \\star } ( \\underline x _ 2 ^ L , y _ 2 ^ L ) \\cdots , \\hat { \\underline s } ^ { \\star } ( \\underline x _ { \\ell } ^ L , y _ { \\ell } ^ L ) ) \\end{align*}"}
+{"id": "5526.png", "formula": "\\begin{align*} \\Delta ( \\lambda ) = ( - 1 ) ^ { \\alpha ( 1 - \\beta ) } \\frac { \\varphi _ { | \\beta - \\alpha | } ( \\pi \\rho ) } { \\rho ^ { 1 - \\alpha - \\beta } } + \\frac { 1 } { \\rho ^ { 2 - \\alpha - \\beta } } \\int _ 0 ^ \\pi \\varphi _ { 1 - | \\beta - \\alpha | } ( t \\rho ) \\ , W ( t ) \\ , d t , W \\in L _ 2 ( 0 , \\pi ) , \\end{align*}"}
+{"id": "5127.png", "formula": "\\begin{align*} D 1 ( p \\| q ) = \\left \\lbrace A \\left ( p , q \\right ) - \\left [ X \\left ( p , q \\right ) + Y \\left ( p , q \\right ) \\right ] \\right \\rbrace \\end{align*}"}
+{"id": "8202.png", "formula": "\\begin{align*} & P _ { Y _ { i - 1 } , { \\bf S } } ( 1 y ^ { i - 2 } , { \\bf s } ) \\leq \\frac { B _ { a , i - 1 } } { | \\mathcal S ( y ^ { i - 2 } ) | } P _ { Y _ { i - 2 } , { \\bf S } } ( y ^ { i - 2 } , { \\bf s } ) , \\\\ & P _ { Y _ { i - 1 } , { \\bf S } } ( 0 y ^ { i - 2 } , { \\bf s } ) = 1 - P _ { Y _ { i - 1 } , { \\bf S } } ( 1 y ^ { i - 2 } , { \\bf s } ) . \\end{align*}"}
+{"id": "3496.png", "formula": "\\begin{align*} \\| \\rho _ k ( x ) \\| ^ 2 = \\| \\rho _ k ( x ) ^ * \\rho _ k ( x ) \\| = \\| e \\bar { z } \\| = \\| \\bar { z } \\| . \\end{align*}"}
+{"id": "1161.png", "formula": "\\begin{align*} \\left ( \\operatorname { T r } \\circ \\operatorname { E x t } \\right ) \\left [ \\theta ^ { ( \\lambda ' ) } _ I \\right ] ( x ' ) = \\frac { [ \\ell ( I ) ] ^ { \\frac 1 2 } } { \\varphi ( - k _ 0 ) } \\left [ \\operatorname { T r } \\theta ^ { ( ( \\lambda ' , 0 ) ) } _ { Q ( I , k _ 0 ) } \\right ] ( x ' ) = \\theta ^ { ( \\lambda ' ) } _ { I } ( x ' ) . \\end{align*}"}
+{"id": "1271.png", "formula": "\\begin{align*} W _ 2 ( \\N ( u _ 1 , \\sigma _ 1 ^ 2 ) , \\N ( u _ 2 , \\sigma _ 2 ^ 2 ) ^ 2 = ( u _ 1 - u _ 2 ) ^ 2 + ( \\sigma _ 1 - \\sigma _ 2 ) ^ 2 . \\end{align*}"}
+{"id": "404.png", "formula": "\\begin{align*} \\begin{aligned} & { A } ( l _ x , l _ y , m _ x , m _ y ) = \\frac { 1 } { 4 \\pi ^ 2 } \\iiiint _ { \\mathcal { E } _ { l _ x , l _ y } \\times \\mathcal { E } _ { m _ x , m _ y } } \\\\ & \\frac { \\delta ( \\kappa _ x - k _ x ) \\delta ( k _ y - \\kappa _ y ) } { \\gamma _ { S } ( \\kappa _ x , \\kappa _ y ) } \\mathrm { d } k _ x \\mathrm { d } k _ y \\mathrm { d } \\kappa _ x \\mathrm { d } \\kappa _ y \\end{aligned} \\end{align*}"}
+{"id": "9101.png", "formula": "\\begin{align*} \\underline { u } ^ { ( r ) } = ( \\hat { C } ^ { - 1 } - C \\hat { C } ^ { - 1 } ) \\underline { v } + U ( J ^ { ( r ) } X ) + \\mathbf { 1 } \\frac { \\beta } { 2 } + J ^ { ( r ) } \\xi , \\xi > 0 , \\end{align*}"}
+{"id": "3048.png", "formula": "\\begin{align*} \\left ( - \\frac { d ^ { 2 } } { d z ^ 2 } + \\frac { E } { z ^ 2 } \\right ) \\psi ( z ) = \\alpha ^ 2 \\psi ( z ) \\ , . \\end{align*}"}
+{"id": "6015.png", "formula": "\\begin{align*} \\mu _ N ( \\eta ) = \\begin{pmatrix} N \\\\ n _ A , \\ , n _ B , \\ , n _ C \\end{pmatrix} ^ { - 1 } = \\frac { n _ A ! \\ , n _ B ! \\ , n _ C ! } { N ! } . \\end{align*}"}
+{"id": "5327.png", "formula": "\\begin{align*} c ^ { S _ 1 } _ j & = c _ j , j \\in S _ 1 = J \\\\ c ^ { S _ { k } } _ j & = c ^ { S _ { k - 1 } } _ j - \\frac { c ^ { S _ { k - 1 } } _ { \\pi _ { k - 1 } } } { w ^ { S _ { k - 1 } } _ { \\pi _ { k - 1 } } } \\ , \\left [ w ^ { S _ { k - 1 } } _ j - w ^ { S _ { k } } _ j \\right ] , j \\in S _ { k } , 2 \\leq k \\leq n , \\end{align*}"}
+{"id": "8173.png", "formula": "\\begin{align*} R _ j & = P _ { Y ^ { j - 1 } , \\underline S } ( 0 ^ { j - 1 } , \\underline S ) H ( P _ { Y _ j | Y ^ { j - 1 } \\underline S } ( 1 | 0 ^ { j - 1 } , \\underline S ) ) \\\\ & + ( 1 - P _ { Y ^ { j - 1 } , \\underline S } ( 0 ^ { j - 1 } , \\underline S ) ) H ( \\frac { 1 } { 2 } ) \\\\ & = ( 1 - \\frac { \\sum _ { k = 1 } ^ { j - 1 } b _ { k } ^ 0 } { M } ) H ( \\frac { b _ j ^ 0 } { M - \\sum _ { k = 1 } ^ { j - 1 } b _ k ^ 0 } ) ) + \\frac { \\sum _ { k = 1 } ^ { j - 1 } b _ { k } ^ 0 } { M } . \\end{align*}"}
+{"id": "5059.png", "formula": "\\begin{align*} \\min _ { x } f ( x ) = \\frac { 1 } { 6 } \\sum _ { i = 1 } ^ 6 \\Vert A _ i x - b _ i \\Vert ^ 2 . \\end{align*}"}
+{"id": "4977.png", "formula": "\\begin{align*} P _ N ( \\nu _ j , \\nu _ j - 1 , \\lambda _ 3 , \\dots , \\lambda _ N ) = 0 , j = 1 , \\dots , N ; \\end{align*}"}
+{"id": "2887.png", "formula": "\\begin{align*} s = ( 1 - \\eta ) s _ 0 + \\eta \\quad \\frac { 1 } { q } = \\frac { 1 - \\eta } { q _ 0 } + \\eta . \\end{align*}"}
+{"id": "3466.png", "formula": "\\begin{align*} c ' = ( 0 _ { S } , E _ { 1 } , \\ldots , E _ { k } , 0 _ { E S } ) , \\end{align*}"}
+{"id": "6132.png", "formula": "\\begin{align*} \\Phi = ( \\varphi _ \\lambda ) _ \\lambda \\colon \\textstyle { \\prod } F _ \\lambda / \\textstyle { \\bigoplus } F _ \\lambda & \\longrightarrow \\ell ^ \\infty ( \\Lambda , A ) / c _ 0 ( \\Lambda , A ) , \\end{align*}"}
+{"id": "8586.png", "formula": "\\begin{align*} \\chi _ \\mathcal { A } : = \\sum _ { X \\in \\mathcal { A } } \\chi _ { \\{ X \\} } \\ : . \\end{align*}"}
+{"id": "1772.png", "formula": "\\begin{align*} F ( z ) = x _ 0 - \\sum _ { 0 < j < j + k \\leq n } x _ j x _ k x _ n ^ { n - j - k } , \\end{align*}"}
+{"id": "2737.png", "formula": "\\begin{align*} \\{ f , g \\} : = \\omega ( X _ { f } , X _ { g } ) , \\end{align*}"}
+{"id": "5986.png", "formula": "\\begin{align*} F ( x ) : = ( R ^ 2 - r ^ 2 _ { \\partial M } ( x ) ) ^ 2 \\phi ( x ) . \\end{align*}"}
+{"id": "6063.png", "formula": "\\begin{align*} V ^ 2 = y ^ 2 A _ { 2 d - 2 } ' + y A ' _ { d - 1 } U + U ^ 2 = \\bigoplus _ { i = 2 } ^ { 2 d } y ^ i A _ { 2 d - i } \\oplus y ( A _ { d - 1 } U ) \\oplus U ^ 2 \\end{align*}"}
+{"id": "3352.png", "formula": "\\begin{align*} \\begin{cases} \\sigma _ k = \\Sigma ^ * ( u _ { k - 1 } , y _ k ) \\sigma _ { k - 1 } , \\\\ \\sigma _ 0 = \\rho . \\end{cases} \\end{align*}"}
+{"id": "8934.png", "formula": "\\begin{align*} \\sigma _ I = \\sigma _ { i _ 1 , \\dotsc , i _ r } = \\delta _ 1 \\times \\dotsb \\times \\delta _ d ; \\end{align*}"}
+{"id": "6120.png", "formula": "\\begin{align*} \\boldsymbol u _ { n + 1 } = \\boldsymbol u _ n + \\tau \\varphi _ 1 ( \\tau K _ n ) ( K \\boldsymbol u _ n + \\boldsymbol g ( \\boldsymbol u _ n ) ) . \\end{align*}"}
+{"id": "5770.png", "formula": "\\begin{align*} | X _ 0 ( t ) | ^ 2 = ( 1 + o ( 1 ) ) | z ( t ) | ^ 2 . \\end{align*}"}
+{"id": "4121.png", "formula": "\\begin{align*} & \\sup _ { r \\in ( 0 , 1 ) } \\sup _ { x = ( z , u , v ) } f _ { ( \\theta ( r ) , r ) } ( x ) \\le M . \\end{align*}"}
+{"id": "8047.png", "formula": "\\begin{align*} \\limsup _ { \\delta \\rightarrow 0 } \\limsup _ { N \\rightarrow + \\infty } \\frac { N } { \\gamma _ N ^ 2 } \\log \\sup _ { \\sigma \\in \\mathcal { T } } P \\left ( \\varepsilon _ { 1 1 } ^ N > \\epsilon \\right ) = - \\infty . \\end{align*}"}
+{"id": "5600.png", "formula": "\\begin{align*} \\| B ^ { ( \\ell ) } x \\| \\leq & \\| \\underline { B } ^ { ( \\ell ) } \\| + \\| H B ^ { ( \\ell - 1 ) } \\| + \\sum _ { t = 1 } ^ { \\ell - 1 } \\sum _ { i = 1 } ^ r \\nu _ i ^ 2 \\| \\underline { B } ^ { ( t - 1 ) } \\chi _ i \\| \\left | \\langle \\check { \\chi } _ i , B ^ { ( \\ell - t - 1 ) } x \\rangle \\right | \\\\ & + \\sum _ { t = 1 } ^ { \\ell - 1 } \\| \\underline { B } ^ { ( t - 1 ) } \\tilde { H } B ^ { ( \\ell - t - 1 ) } \\| + L ^ 2 \\| \\underline { B } ^ { ( \\ell - 1 ) } \\| + \\sum _ { t = 0 } ^ { \\ell } \\| R _ t ^ { ( \\ell ) } \\| . \\end{align*}"}
+{"id": "8645.png", "formula": "\\begin{align*} \\int _ \\Omega \\varphi _ 0 ^ { c _ 2 } d \\mu + \\int _ \\Omega \\varphi _ 0 d \\rho _ 0 & = \\frac { 1 } { 2 } W _ 2 ^ 2 ( \\mu , \\rho _ 0 ) , \\end{align*}"}
+{"id": "8243.png", "formula": "\\begin{align*} D ( B _ ) & = \\frac { 1 } { L } \\sum _ { i = 1 } ^ L d _ i , \\\\ d _ i & = P _ { Y ^ i } ( 0 ^ i ) \\log ( M - \\sum _ { j = 1 } ^ { i } b _ j ^ 0 ) \\\\ & + \\sum _ { j = 1 } ^ i P _ { Y ^ i } ( \\beta _ j ^ i ) \\log ( \\max ( 1 , \\frac { b _ j ^ 0 } { 2 ^ { i - j } } ) ) . \\end{align*}"}
+{"id": "6654.png", "formula": "\\begin{align*} \\omega ^ { + } ( E , \\mathrm { i } y ) = \\frac { L ( E , \\mathrm { i } y _ { 2 } ) - L ( E , \\mathrm { i } y _ { 1 } ) } { y _ { 2 } - y _ { 1 } } , \\ \\ [ y _ { 1 } , y _ { 2 } ] \\subset ( y , y + h ' ) . \\end{align*}"}
+{"id": "7875.png", "formula": "\\begin{align*} [ k ] _ { m } = \\begin{cases} m , & ; \\\\ v , & k = s m + v , v \\not = 0 . \\end{cases} \\end{align*}"}
+{"id": "1197.png", "formula": "\\begin{align*} \\begin{cases} N : = 0 \\in ( s , E ) & s < 0 , \\\\ N \\in ( s , \\lceil \\ ! \\lceil s \\rceil \\ ! \\rceil \\wedge E ) & s \\geq 0 , \\end{cases} \\end{align*}"}
+{"id": "4945.png", "formula": "\\begin{align*} \\begin{gathered} [ L ] = 2 \\sqrt { \\beta } ( e ^ { L } _ + - e ^ { L } _ - ) , \\\\ [ L ' ] = 2 \\sqrt { \\beta } ( e ^ { L ' } _ + - e ^ { L ' } _ - ) . \\end{gathered} \\end{align*}"}
+{"id": "4104.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { ( A _ j ( t ) ) } { t } = \\bigl ( c ^ 2 p _ j ^ 2 + p _ j ( 1 - p _ j ) \\bigr ) \\nu . \\end{align*}"}
+{"id": "2604.png", "formula": "\\begin{align*} D _ i ( S ) : = \\{ s _ x - s _ { y } : 1 \\leq x - y \\leq i \\} . \\end{align*}"}
+{"id": "1738.png", "formula": "\\begin{align*} \\sigma \\left ( j \\right ) : = \\left \\lfloor \\tfrac { n } { 2 } \\right \\rfloor - ( - 1 ) ^ { n + j } \\left \\lfloor \\tfrac { j } { 2 } \\right \\rfloor \\quad \\quad \\forall \\ , j \\in \\{ 1 , \\ldots , n \\} . \\end{align*}"}
+{"id": "5945.png", "formula": "\\begin{align*} \\dfrac { M _ { 1 } ( \\mathcal { N C } ( G ) ) } { | v ( \\mathcal { N C } ( G ) ) | } = \\dfrac { 2 ^ { 9 k } - 5 \\cdot 2 ^ { 7 k } - 2 ^ { 6 k } + 9 \\cdot 2 ^ { 5 k } - 5 \\cdot 2 ^ { 3 k } - 3 \\cdot 2 ^ { 2 k } + 3 \\cdot 2 ^ { k } } { 2 ^ { 3 k } - 2 ^ { k } - 1 } \\end{align*}"}
+{"id": "6727.png", "formula": "\\begin{align*} P ( n , z ) = \\int _ 0 ^ 1 { y ^ { n - 1 } S _ 1 ( z y ) d y } . \\end{align*}"}
+{"id": "1469.png", "formula": "\\begin{align*} \\lim _ { i \\in I } \\frac { | { \\partial _ M F _ i } | } { | F _ i | } = 0 . \\end{align*}"}
+{"id": "8966.png", "formula": "\\begin{align*} ( x ' , x _ n - \\lambda \\delta ( x ) ) \\in Q _ R \\cap \\Omega \\quad . \\end{align*}"}
+{"id": "1870.png", "formula": "\\begin{align*} \\alpha \\tau _ 1 \\alpha = \\tau _ 1 \\gamma \\tau _ 1 \\gamma = \\tau _ 1 ^ { - 1 } . \\end{align*}"}
+{"id": "4243.png", "formula": "\\begin{align*} E _ 4 ( \\tau ) E _ 6 ( \\tau ) ^ 2 = 1 - 7 6 8 q - 1 9 0 0 8 q ^ 2 + \\cdots . \\end{align*}"}
+{"id": "62.png", "formula": "\\begin{align*} u _ t - \\Delta _ p u = 0 \\end{align*}"}
+{"id": "1455.png", "formula": "\\begin{align*} \\mathrm { S t a b } _ { G } ( \\xi ) = \\bigcup \\left \\lbrace \\mathrm { S t a b } _ { G } ( w ) : w \\in \\mathrm { V } ( \\mathfrak { r } _ \\xi ) \\right \\rbrace . \\end{align*}"}
+{"id": "8387.png", "formula": "\\begin{align*} A _ 1 & = M _ 4 M _ 2 ^ { - 1 } + M _ 2 ^ { - 1 } X ^ { - 1 } Y ^ { - 1 } ( I - X ) - M _ 2 ^ { - 1 } X ^ { - 1 } , \\\\ A _ 2 & = X M _ 2 , \\\\ A _ 3 & = M _ 2 ^ { - 1 } X ^ { - 1 } ( Y - I ) , \\\\ A _ 4 & = Y ^ { - 1 } ( I - X ) M _ 2 , \\\\ A _ 5 & = M _ 2 ^ { - 1 } ( M _ 1 - Y ) , \\end{align*}"}
+{"id": "3166.png", "formula": "\\begin{align*} ( j , i ) ^ \\psi = j n + i \\ \\ ( j , i ) ^ \\sigma = \\left ( \\left \\lfloor \\frac { i k + j } { n } \\right \\rfloor , ( i k + j ) \\bmod { n } \\right ) , \\end{align*}"}
+{"id": "6774.png", "formula": "\\begin{align*} F _ { \\cdot , { c } } ( \\kappa , { x } ) = F ( \\kappa , { c } , { x } ) . \\end{align*}"}
+{"id": "5879.png", "formula": "\\begin{align*} \\dfrac { M _ { 2 } ( \\mathcal { N C } ( G ) ) } { | e ( \\mathcal { N C } ( G ) ) | } - \\dfrac { M _ { 1 } ( \\mathcal { N C } ( G ) ) } { | v ( \\mathcal { N C } ( G ) ) | } = \\dfrac { 6 4 n ^ { 3 } ( n - 6 ) + 6 4 n ( 1 3 n - 1 2 ) + 2 5 6 } { 2 4 ( n - 1 ) ( 2 n - 1 ) } : = \\dfrac { f ( n ) } { g ( n ) } . \\end{align*}"}
+{"id": "8299.png", "formula": "\\begin{align*} L ( k ) y ( t , k ) : = y ^ { ( r ) } ( t , k ) + \\sum _ { j = 1 } ^ { r } A _ { r - j } ( t , k ) y ^ { ( r - j ) } ( t , k ) = f ( t , k ) , t \\in ( a , b ) , \\end{align*}"}
+{"id": "6534.png", "formula": "\\begin{align*} \\lim \\limits _ { x \\to \\infty } \\frac { \\ell ( x \\ell ^ { \\frac { 1 } { \\beta } } ( x ) ) } { \\ell ( x ) } = 1 , \\end{align*}"}
+{"id": "2526.png", "formula": "\\begin{align*} \\bar m = m + n / 4 - n ' / 2 \\ ; , \\end{align*}"}
+{"id": "2348.png", "formula": "\\begin{align*} \\int ^ { T } _ { 0 } \\int _ { \\mathbb { R } ^ { 3 } } \\left ( z ( \\partial _ { t } \\varphi + \\Delta \\varphi + u ^ { c , \\gamma } \\cdot \\nabla \\varphi ) + ( w _ { 1 } \\otimes w _ { 2 } ) \\cdot \\nabla \\varphi - ( z \\cdot \\nabla ) u ^ { c , \\gamma } \\varphi \\right ) d x d t = 0 . \\end{align*}"}
+{"id": "606.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\left ( \\frac { 1 } { 1 6 } \\right ) ^ { n } H _ { 2 n } ' \\binom { 2 n } { n } ^ { 2 } \\frac { f _ n } { n + 1 } \\end{align*}"}
+{"id": "6943.png", "formula": "\\begin{align*} \\ker f _ { v , w } | _ { U _ v \\oplus f _ { u , v } ( V _ u ) } = \\ker f _ { v , w } | _ { U _ v } \\oplus \\ker f _ { v , w } | _ { f _ { u , v } ( V _ u ) } . \\end{align*}"}
+{"id": "1951.png", "formula": "\\begin{align*} m _ i \\leq u \\leq M _ i B _ { 4 ^ { 1 - i } R } , i = 0 , 1 , 2 , 3 , \\ldots \\end{align*}"}
+{"id": "8924.png", "formula": "\\begin{align*} \\sigma ' ( s ) : = \\begin{cases} \\sigma ( s ) & s < k - 1 \\\\ \\sigma ( n - 1 ) & s = k - 1 \\end{cases} \\qquad ( s \\in \\Sigma _ { k } ) . \\end{align*}"}
+{"id": "4824.png", "formula": "\\begin{align*} X = \\Big \\{ \\mathbf { x } \\in \\{ 0 , 1 \\} ^ { n } : \\Vert \\mathbf { x } \\Vert _ 1 = h \\Big \\} . \\end{align*}"}
+{"id": "9122.png", "formula": "\\begin{align*} ( \\alpha , s ) < ( \\beta , t ) \\alpha < \\beta \\ \\ \\alpha = \\beta , s < t . \\end{align*}"}
+{"id": "1468.png", "formula": "\\begin{align*} \\vert \\Gamma _ { F _ n \\Delta ^ 2 } \\vert \\geq \\vert \\tau ( Q _ n ) \\vert = | Q _ n | \\geq | ( Q _ n ) _ { F _ n \\setminus S \\Delta } | = | X _ { { F _ n \\setminus S \\Delta } } | \\geq \\frac { | X _ { F _ n } | } { | A | ^ { | S \\Delta | } } . \\end{align*}"}
+{"id": "9047.png", "formula": "\\begin{align*} a ( \\Lambda _ 1 , \\dotsc , \\Lambda _ n ) \\cdot T = a ( \\Lambda _ 1 , \\dotsc , \\Lambda _ n ) \\Bigl ( - \\sum _ { k = 1 } ^ n \\Lambda _ k \\Bigr ) , a ( \\Lambda _ 1 , \\dotsc , \\Lambda _ n ) \\cdot S ^ i = a ( \\Lambda _ 1 , \\dotsc , \\Lambda _ n ) \\Bigl ( - \\sum _ { k = 1 } ^ n \\theta _ k ^ i \\Bigr ) \\end{align*}"}
+{"id": "3078.png", "formula": "\\begin{align*} \\psi _ { u } ( t ) = \\left ( t ^ { \\frac { \\beta _ 0 } { e _ j } } , \\displaystyle \\sum _ { \\beta _ 1 \\leq l < \\beta _ { j } } c _ l t ^ { \\frac { l } { e _ j } } + u t ^ { \\frac { \\beta _ { j } } { e _ j } } + \\displaystyle \\sum _ { l > \\frac { \\beta _ { j } } { e _ j } } s _ l ( u ) t ^ { l } \\right ) , \\end{align*}"}
+{"id": "8118.png", "formula": "\\begin{align*} w U _ { \\alpha } w ^ { - 1 } = U _ { w ( \\alpha ) } , \\end{align*}"}
+{"id": "199.png", "formula": "\\begin{align*} P [ j ] \\ = \\ \\widehat { A } + \\ & \\cup \\ \\widehat { A } - \\ \\cup \\\\ ( D _ j - \\times D _ j + ) \\ \\cup \\ ( \\bar D _ j - \\times \\bar D _ j + ) \\ & \\cup \\ ( D _ j + \\times \\bar D _ j - ) \\ \\cup \\ ( \\bar D _ j + \\times D _ j - ) . \\end{align*}"}
+{"id": "4837.png", "formula": "\\begin{align*} N _ \\lambda = N _ { ( 2 , 2 , 1 , 1 ) } + N _ { ( 2 , 2 , 2 ) } + 2 N _ { ( 3 , 2 , 1 ) } . \\end{align*}"}
+{"id": "5004.png", "formula": "\\begin{align*} p _ k ( x ) = \\prod _ { \\substack { l = 1 \\\\ l \\neq k } } ^ N \\ , ( x - y _ l ) , k = 1 , \\ldots , N . \\end{align*}"}
+{"id": "5974.png", "formula": "\\begin{align*} B _ i = \\Bigg \\lceil \\frac { f _ i } { \\sum _ { j \\in S _ e } f _ j } \\cdot B \\Bigg \\rceil . \\end{align*}"}
+{"id": "6575.png", "formula": "\\begin{align*} \\sum _ { p \\leq x } \\frac { 1 } { p } = \\log \\log x + b _ 0 + O \\left ( \\frac { 1 } { \\log x } \\right ) , \\end{align*}"}
+{"id": "5113.png", "formula": "\\begin{align*} S ( p ) = \\left [ - \\frac { \\sum _ { i } p ^ { \\alpha t } _ { i } - \\sum _ { i } p ^ { \\alpha } _ { i } } { \\alpha t - \\alpha } \\right ] _ { \\alpha = 1 } \\end{align*}"}
+{"id": "9098.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\tilde { A } ^ { ( r ) } : = [ \\tilde a _ { i j } ] _ { i , j = 1 , \\ldots , n } = ( A ^ { ( r ) } + \\hat { C } \\Tilde { K } ^ { ( r ) } ) = \\begin{bmatrix} \\hat { c } _ { 1 1 } \\Tilde { k } _ { 1 1 } & \\ldots & a _ { 1 n } + \\hat { c } _ { 1 1 } \\Tilde { k } _ { 1 n } \\\\ \\vdots & \\ddots & \\vdots \\\\ a _ { n 1 } + \\hat { c } _ { n n } \\Tilde { k } _ { n 1 } & \\ldots & \\hat { c } _ { n n } \\Tilde { k } _ { n n } \\\\ \\end{bmatrix} . \\end{array} \\end{align*}"}
+{"id": "8734.png", "formula": "\\begin{align*} & \\int _ \\R ( z - x ) ^ + \\nu ( d z ) = \\int _ \\R ( y - x ) ^ + \\mu ( d y ) + \\int _ { F _ \\nu ( x ) } ^ { F _ \\nu ( x ) + p _ + ( x ) } ( F _ \\nu ^ { - 1 } ( v ) - x ) d v . \\\\ \\mbox { a n d } & \\int _ \\R ( x - z ) ^ + \\nu ( d z ) = \\int _ \\R ( x - y ) ^ + \\mu ( d y ) + \\int _ { F _ \\nu ( x - ) - p _ - ( x ) } ^ { F _ \\nu ( x - ) } ( x - F _ \\nu ^ { - 1 } ( v ) ) d v \\end{align*}"}
+{"id": "8994.png", "formula": "\\begin{align*} w _ T ( z , \\delta ) : = \\sup _ { 0 \\leq t \\leq T } \\sup _ { \\substack { 0 \\vee ( t - \\delta ) \\leq t _ 1 \\leq t _ 2 \\leq t _ 3 \\leq ( t + \\delta ) \\wedge T } } \\inf _ { 0 \\leq \\theta \\leq 1 } | z ( t _ 2 ) - \\left ( \\theta z ( t _ 1 ) + ( 1 - \\theta ) z ( t _ 3 ) \\right ) | \\end{align*}"}
+{"id": "3684.png", "formula": "\\begin{align*} \\Phi _ { i , j , k } ( e ) = \\begin{cases} 1 & e = \\{ i , j \\} , \\\\ - 1 & e = \\{ i , k \\} , \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "5944.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { N C } ( G ) ) & = ( 2 ^ { 3 k } - 2 ^ { k } - 1 ) ( 2 ^ { 3 k } - 2 ^ { k } - 2 ) ^ { 2 } \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - 4 \\cdot ( 2 ^ { 3 k } - 2 ^ { k } - 2 ) \\dfrac { ( 2 ^ { 4 k } - 2 \\cdot 2 ^ { 3 k } - 2 \\cdot 2 ^ { 2 k } + 3 \\cdot 2 ^ { k } + 2 ) } { 2 } \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ + ( 2 ^ { 5 k } - 4 \\cdot 2 ^ { 4 k } + 4 \\cdot 2 ^ { 3 k } + 4 \\cdot 2 ^ { 2 k } - 5 \\cdot 2 ^ { k } - 4 ) \\\\ & = 2 ^ { 9 k } - 5 \\cdot 2 ^ { 7 k } - 2 ^ { 6 k } + 9 \\cdot 2 ^ { 5 k } - 5 \\cdot 2 ^ { 3 k } - 3 \\cdot 2 ^ { 2 k } + 3 \\cdot 2 ^ { k } \\end{align*}"}
+{"id": "5938.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { N C } ( G ) ) = 2 ^ { 9 k } - 5 \\cdot 2 ^ { 7 k } - 2 ^ { 6 k } + 9 \\cdot 2 ^ { 5 k } - 5 \\cdot 2 ^ { 3 k } - 3 \\cdot 2 ^ { 2 k } + 3 \\cdot 2 ^ { k } \\end{align*}"}
+{"id": "35.png", "formula": "\\begin{align*} ( d _ h f ) ( s ) = \\sum _ { r \\subset s } f ( r ) ; \\quad ( d _ h ^ * f ) ( s ) = \\frac 1 { h ^ 2 } \\sum _ { s \\subset r } f ( r ) \\ . \\end{align*}"}
+{"id": "3320.png", "formula": "\\begin{align*} A ( x , t ) = & \\left ( a _ { i j } ( x , t ) \\right ) _ { 1 \\leq i , j \\leq N } , \\ , \\ , \\ , b ( x , t ) : = \\left ( b _ 1 ( x , t ) , \\ldots , b _ { m _ 0 } ( x , t ) , 0 , \\ldots , 0 \\right ) , \\\\ & Y : = \\sum _ { i , j = 1 } ^ N b _ { i j } x _ j \\partial _ { x _ i } u ( x , t ) - \\partial _ t u ( x , t ) . \\end{align*}"}
+{"id": "444.png", "formula": "\\begin{align*} \\min _ { x \\in \\Re ^ { | V _ { B ^ q } | } } \\left \\{ \\sum _ { i \\in V _ { B ^ q } } \\hat f _ i ( x _ i ) - t x _ { i _ m } \\mid x _ i = x _ j , \\ , \\forall \\ , ( i , j ) \\in E _ { B ^ q } \\right \\} , \\end{align*}"}
+{"id": "8945.png", "formula": "\\begin{align*} \\begin{aligned} \\| d ( \\nabla v , S O ( n ) ) \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } & \\le \\| d ( \\nabla u , S O ( n ) ) \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } + \\| \\nabla u - \\nabla v \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } \\\\ & \\le C \\| d ( \\nabla u , S O ( n ) ) \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } \\end{aligned} \\end{align*}"}
+{"id": "4889.png", "formula": "\\begin{align*} \\O _ { \\Sigma ' } ( C ' ) = \\O _ { \\Sigma ' } ( 2 E ) \\otimes \\varphi '^ * ( { ( \\L ^ \\vee ) } ^ { \\otimes 2 } ) = \\mathcal A ^ { \\otimes 2 } \\end{align*}"}
+{"id": "210.png", "formula": "\\begin{align*} \\mathcal { L } _ \\Delta S = - S . \\end{align*}"}
+{"id": "5648.png", "formula": "\\begin{align*} \\sum _ i ( \\mu _ i - 1 4 ) ^ 2 ( 2 2 - \\mu _ i ) & = \\sum _ i \\left ( - \\mu _ i ( \\mu _ i - 1 ) ( \\mu _ i - 2 ) + 4 7 \\mu _ i ( \\mu _ i - 1 ) - 7 6 3 \\mu _ i + 4 3 1 2 \\right ) \\\\ & = 7 8 \\left ( - 1 2 2 8 9 2 + 4 7 \\times 6 5 4 5 - 7 6 3 \\times 3 4 1 + 7 5 4 6 0 \\right ) \\\\ & = 0 . \\end{align*}"}
+{"id": "4882.png", "formula": "\\begin{align*} a = g - 1 + e . \\end{align*}"}
+{"id": "4117.png", "formula": "\\begin{align*} c ^ { ( r ) } _ { e , i } ( y ) = \\gamma _ i ( y , r ) - y a ^ { ( r ) } - \\frac { 1 } { 2 } y ^ 2 ( a ^ { ( r ) } ) ^ 3 \\sigma ^ 2 _ { ( r ) } . \\end{align*}"}
+{"id": "6839.png", "formula": "\\begin{align*} u ^ { ( N - 1 ) } _ { 1 N } & = u ^ { ( N - 2 ) } _ { 1 N } e ^ { - i \\phi _ { N - 1 } } \\cos ( \\psi _ { N - 1 } ) - u ^ { ( N - 2 ) } _ { N N } \\sin ( \\psi _ { N - 1 } ) , \\\\ u ^ { ( N - 1 ) } _ { N N } & = u ^ { ( N - 2 ) } _ { 1 N } e ^ { - i \\phi _ { N - 1 } } \\sin ( \\psi _ { N - 1 } ) + u ^ { ( N - 2 ) } _ { N N } \\cos ( \\psi _ { N - 1 } ) . \\end{align*}"}
+{"id": "951.png", "formula": "\\begin{align*} P _ V g ( x ) = \\int _ { V ^ c } g ( y ) \\ , P _ V ( x , d y ) , x \\in E \\setminus N . \\end{align*}"}
+{"id": "5395.png", "formula": "\\begin{align*} \\nu _ j & = \\frac { h } { \\mu } \\ , \\frac { \\left ( j + 1 \\right ) \\ , \\left ( j + 2 \\right ) \\ , \\left ( 4 j + 3 \\right ) } { 6 } . \\end{align*}"}
+{"id": "5147.png", "formula": "\\begin{align*} D _ { \\alpha } \\left ( p \\| q \\right ) = \\frac { 1 } { \\alpha \\left ( \\alpha - 1 \\right ) } \\left \\lbrace \\sum _ { i } p ^ { \\alpha } _ { i } q ^ { 1 - \\alpha } _ { i } - \\left ( \\alpha \\sum _ { i } p _ { i } + \\left ( 1 - \\alpha \\right ) \\sum _ { i } q _ { i } \\right ) \\right \\rbrace \\end{align*}"}
+{"id": "9039.png", "formula": "\\begin{align*} ( V , \\nabla ) = ( V , T , S ^ 1 , \\ldots , S ^ N ) , \\end{align*}"}
+{"id": "892.png", "formula": "\\begin{align*} S _ { ( i - 2 ) } + T _ { ( i - 2 ) } = A _ i x _ 1 + B _ i x _ 2 + C _ i , i = 3 , 4 , \\end{align*}"}
+{"id": "9091.png", "formula": "\\begin{align*} X _ + ( t + 1 ) = A ^ { ( r ) } _ + X _ + ( t ) + A ^ { ( r ) } _ - X _ - ( t ) + b ^ { ( r ) } _ + ( X ( t ) ) . \\end{align*}"}
+{"id": "7135.png", "formula": "\\begin{align*} A _ \\mu ( x ) \\longrightarrow A '^ { \\ , \\mu } ( x ) = A ^ \\mu ( x ) + \\partial ^ \\mu \\chi ( x ) \\ , \\ , , \\end{align*}"}
+{"id": "8520.png", "formula": "\\begin{align*} \\det ( M ) & = \\det ( P ) ^ { - 1 } \\cdot \\det ( \\int _ { T _ { n _ a } c } 2 i \\pi F _ b ( z ) d z ) _ { 1 \\leq a , b \\leq g } \\\\ & = \\det ( P ) ^ { - 1 } \\cdot \\det ( \\int _ { c } 2 i \\pi a _ { n _ a } ( F _ b ) \\cdot F _ b ( z ) d z ) _ { 1 \\leq a , b \\leq g } \\\\ & \\sim \\det ( a _ { n _ a } ( F _ b ) ) _ { 1 \\leq a , b \\leq g } ^ { - 1 } \\cdot \\det ( \\int _ { c } 2 i \\pi a _ { n _ a } ( F _ b ) \\cdot F _ b ( z ) d z ) _ { 1 \\leq a , b \\leq g } \\end{align*}"}
+{"id": "3229.png", "formula": "\\begin{align*} \\gamma ^ { - 1 } V ^ { i , j } = V ^ { j , i } . \\end{align*}"}
+{"id": "1516.png", "formula": "\\begin{align*} { \\rm d i v } \\ , \\big ( D ^ 2 F ( D u ) D ( F ( D u ) ) \\big ) = { \\rm T r } \\ , \\big ( D ^ 2 F ( D u ) \\ , D ^ 2 u \\ , D ^ 2 F ( D u ) \\ , D ^ 2 u ) . \\end{align*}"}
+{"id": "3514.png", "formula": "\\begin{align*} \\varphi _ { m , k } ( \\hat { x } ) \\varphi _ { m , k } ( \\hat { y } ) & = \\big ( \\hat { x } \\oplus \\big ( \\oplus _ { j = k + 1 } ^ m \\rho _ { j , k } ( x ) \\big ) \\big ) \\big ( \\hat { y } \\oplus \\big ( \\oplus _ { j = k + 1 } ^ m \\rho _ { j , k } ( y ) \\big ) \\big ) \\\\ & = \\hat { x } \\hat { y } \\oplus \\big ( \\oplus _ { j = k + 1 } ^ m \\rho _ { j , k } ( x ) \\rho _ { j , k } ( y ) \\big ) , \\\\ \\varphi _ { m , l } ( 1 _ { B _ l } ) & = 1 _ { B _ l } \\oplus \\big ( \\oplus _ { j = l + 1 } ^ m \\rho _ { j , l } ( 1 _ { F _ l } ) \\big ) , \\end{align*}"}
+{"id": "4876.png", "formula": "\\begin{align*} \\bar C ^ 2 = C ^ 2 + 4 h + \\nu , p _ a ( \\bar C ) = g + h , \\end{align*}"}
+{"id": "7736.png", "formula": "\\begin{align*} \\mathrm { R e s } ( \\chi , \\chi ; u ) = \\mathrm { R e s } ( \\chi , \\acute \\chi _ u ) . \\end{align*}"}
+{"id": "8764.png", "formula": "\\begin{align*} F _ \\mu ^ { - 1 } ( u ) = \\phi _ \\uparrow ' ( u ) F _ { \\tilde \\nu _ l } ^ { - 1 } \\left ( \\frac { \\phi _ \\uparrow ( u ) } { \\nu _ l ( \\R ) } \\right ) + ( 1 - \\phi _ \\uparrow ' ( u ) ) F _ { \\tilde \\nu _ r } ^ { - 1 } \\left ( \\frac { u - \\phi _ \\uparrow ( u ) } { \\nu _ r ( \\R ) } \\right ) , \\ ; d u \\mbox { a . e . o n } ( 0 , 1 ) , \\end{align*}"}
+{"id": "4785.png", "formula": "\\begin{align*} d = 0 d > 0 \\end{align*}"}
+{"id": "3626.png", "formula": "\\begin{align*} Q ^ 2 f ( z , w ) = \\frac { 2 } { ( \\lambda _ { 1 } ^ { 2 } - 1 ) ^ 2 } \\big ( f ( z , w ) - f ( z , - w ) \\big ) , \\end{align*}"}
+{"id": "4911.png", "formula": "\\begin{align*} \\delta _ { \\mu , b } = b + \\min ( g , \\mu ) - \\min ( g , \\mu + b ) = \\begin{cases} 0 & \\mu + b \\leq g \\\\ b + \\mu - g & g - b \\leq \\mu \\leq g \\\\ b & \\mu = g + 1 , \\\\ \\end{cases} \\end{align*}"}
+{"id": "1414.png", "formula": "\\begin{align*} \\{ I _ + ^ \\nu \\ , f _ \\alpha ( \\cdot \\vert t ) \\} ( x ) & = t ^ { ( \\nu - 1 ) / \\alpha } \\{ I _ + ^ \\nu \\ , f _ \\alpha \\} ( x t ^ { - 1 / \\alpha } ) \\end{align*}"}
+{"id": "6159.png", "formula": "\\begin{align*} S ^ { k + 1 } : = f ( x ' ) - f ( \\breve { x } ^ { k } ) + \\lambda ^ T ( A \\breve { x } ^ { k } - b ) , ~ x ' \\in \\Omega . \\end{align*}"}
+{"id": "8334.png", "formula": "\\begin{align*} \\begin{cases} [ \\Pi _ t , \\Pi _ t ] _ { S N } = 0 \\\\ \\frac { d } { d t } \\Pi _ { t } = [ \\Pi _ t , \\widehat { X _ { t } } ] _ { S N } \\end{cases} . \\end{align*}"}
+{"id": "4403.png", "formula": "\\begin{gather*} F _ { i , j , k , l } ( x ) : = \\max \\{ 0 , \\Delta u _ i B _ { i , ( i , j ) } - \\Delta u _ k B _ { k , ( k , l ) } \\} x _ { i , j } . \\end{gather*}"}
+{"id": "6092.png", "formula": "\\begin{align*} \\Delta ( \\mathfrak { u } ) = 1 \\otimes \\mathfrak { u } + \\sum \\mathfrak { u } _ { ( 1 ) } \\otimes \\mathfrak { u } _ { ( 2 ) } , \\Delta ( \\mathfrak { v } ) = 1 \\otimes \\mathfrak { v } + \\sum \\mathfrak { v } _ { ( 1 ) } \\otimes \\mathfrak { v } _ { ( 2 ) } . \\end{align*}"}
+{"id": "2613.png", "formula": "\\begin{align*} s _ 4 - s _ 1 = s _ 5 - s _ 3 . \\end{align*}"}
+{"id": "5620.png", "formula": "\\begin{align*} X \\setminus \\{ 0 \\} \\subseteq \\bigcup _ { n = 0 } ^ \\infty T ^ n ( U ) . \\end{align*}"}
+{"id": "3796.png", "formula": "\\begin{align*} \\| \\bar { Z } \\bar { Z } ^ \\top \\| & = \\left \\| \\sum _ { i \\in \\widehat { \\mathcal { C } } _ 1 \\cap \\overline { \\mathcal { C } _ 1 } } \\sum _ { l = 1 } ^ { N _ i } \\sum _ { t = 0 } ^ { T - 1 } z ^ { ( i ) } _ { l , t } z ^ { ( i ) , \\top } _ { l , t } \\right \\| \\\\ & \\leq \\sum _ { i \\in \\widehat { \\mathcal { C } } _ 1 \\cap \\overline { \\mathcal { C } _ 1 } } \\left \\| \\sum _ { l = 1 } ^ { N _ i } \\sum _ { t = 0 } ^ { T - 1 } z ^ { ( i ) } _ { l , t } z ^ { ( i ) , \\top } _ { l , t } \\right \\| \\end{align*}"}
+{"id": "470.png", "formula": "\\begin{align*} \\tilde { \\pi } \\circ \\lambda ( g ) & = \\tilde { \\pi } \\left ( \\sum _ { A \\ni g ^ { - 1 } } ( A , g ) \\right ) = \\sum _ { A \\ni g ^ { - 1 } } \\tilde { \\pi } ( A , g ) = \\pi ( g ) \\cdot \\sum _ { A \\ni g ^ { - 1 } } \\sum _ { h \\in A } \\varepsilon ( h ) \\cdot \\prod _ { h \\in Y _ g \\setminus A } ( \\pi ( d ( g ) ) - \\varepsilon ( h ) ) \\end{align*}"}
+{"id": "7518.png", "formula": "\\begin{align*} d _ { g ( t ) } \\left ( \\left \\{ r _ M = \\sqrt { \\gamma t } \\right \\} , \\left \\{ r _ M = 0 \\right \\} \\right ) \\leq C _ { \\zeta } \\sqrt { t } , \\end{align*}"}
+{"id": "3029.png", "formula": "\\begin{align*} H = \\frac { 2 } { m } { P } _ \\mu \\overline { P } ^ \\mu \\ , , \\end{align*}"}
+{"id": "6514.png", "formula": "\\begin{align*} \\mu ^ + ( \\partial K ) \\coloneqq \\liminf _ { \\epsilon \\to 0 } \\frac { \\mu \\left ( K + \\epsilon B _ 2 ^ n \\right ) - \\mu ( K ) } { \\epsilon } = \\int _ { \\partial K } \\phi ( x ) d \\mathcal { H } ^ { n - 1 } ( x ) , \\end{align*}"}
+{"id": "3432.png", "formula": "\\begin{align*} \\underline { d } ( \\tilde { G } ) = 0 , d ( \\tilde { G } ) = 1 , \\overline { d } ( \\tilde { G } ) = 2 . \\end{align*}"}
+{"id": "8063.png", "formula": "\\begin{align*} R _ { s , 2 } ^ { g _ 1 , g _ 2 , g _ 3 } ( f ) = & \\int _ { \\mathbb { T } ^ 2 } \\lambda ( u , v ) \\theta _ s ^ S ( u ) \\theta _ s ^ I ( v ) f ( u ) ( g _ 2 ( u ) - g _ 1 ( u ) ) d u d v \\\\ & - \\int _ { \\mathbb { T } } \\psi ( u ) \\theta _ s ^ E ( u ) f ( u ) ( g _ 3 ( u ) - g _ 2 ( u ) ) d u \\end{align*}"}
+{"id": "8250.png", "formula": "\\begin{align*} \\frac { 1 } { L } \\max _ { \\prod _ { i = 1 } ^ L P _ { { X } _ i | { X } ^ { i - 1 } , Y ^ { i - 1 } } } I ( { X } ^ { L } ; Y ^ { L } | { S } ) , \\end{align*}"}
+{"id": "8977.png", "formula": "\\begin{align*} \\bar \\lambda _ \\gamma = \\inf _ { u \\in H _ 0 ^ 1 ( B ( 0 , 1 ) ) , \\int _ { B ( 0 , 1 ) } \\frac { | u ( x ) | ^ 2 } { | x | ^ \\gamma } = 1 } \\int _ { B ( 0 , 1 ) } | \\nabla u | ^ 2 \\end{align*}"}
+{"id": "2643.png", "formula": "\\begin{align*} f ( z ) = c \\sum _ { ( \\xi _ { 1 } , \\ldots , \\xi _ { d } ) \\in \\mathbb { Z } ^ d } ( \\mathcal { F } f ) ( \\xi ) e ^ { i z \\cdot \\xi } . \\end{align*}"}
+{"id": "1754.png", "formula": "\\begin{align*} r _ { j , k } = r _ { j + 1 , k + 1 } - \\delta _ { j , k } a ^ \\prime \\quad \\quad \\forall \\ , j < k < n - 1 , \\end{align*}"}
+{"id": "897.png", "formula": "\\begin{align*} \\zeta ^ 1 = & \\frac { 1 } { 4 } \\left ( \\frac { y ^ 1 } { F } - ( C _ 1 + A _ 1 x _ 1 + B _ 1 x _ 2 ) \\right ) ( 2 F ( ( A _ 5 x _ 1 + B _ 5 x _ 2 + C _ 5 ) y ^ 1 + ( A _ 6 x _ 1 \\\\ & + B _ 6 x _ 2 + C _ 6 ) y ^ 2 ) - ( E ( y ^ 1 ) ^ 2 + J y ^ 1 y ^ 2 + K ( y ^ 2 ) ^ 2 ) - F ^ 2 \\left ( \\sum _ { l + k \\leq 2 } M _ { l k } x _ 1 ^ { l } x _ 2 ^ k \\right ) ) \\\\ & - \\frac { 1 } { 4 } F ^ 2 ( A _ 3 x _ 1 + B _ 3 x _ 2 + C _ 3 ) - \\frac { 1 } { 2 } F N y ^ 2 . \\end{align*}"}
+{"id": "5179.png", "formula": "\\begin{align*} & U _ { j } = \\left [ \\frac { a - 1 } { a - b } B ^ { a - 2 } - \\frac { b - 1 } { a - b } B ^ { b - 2 } \\right ] p _ { j } q ^ { \\beta - 2 } _ { j } \\\\ & V _ { j } = \\left [ \\frac { a - 1 } { a - b } A ^ { a - 2 } - \\frac { b - 1 } { a - b } A ^ { b - 2 } \\right ] q ^ { \\beta - 1 } _ { j } \\end{align*}"}
+{"id": "4568.png", "formula": "\\begin{align*} \\rho _ { g , J , a b } = \\frac { s _ g } { 6 } g _ { a b } + s _ g ^ { - 2 } \\left ( | d s _ g | ^ 2 g _ { a b } - ( d s _ g ) _ a ( d s _ g ) _ b - ( J d s _ g ) _ a ( J d s _ g ) _ b \\right ) . \\end{align*}"}
+{"id": "706.png", "formula": "\\begin{align*} f ( x ) = \\Phi ^ 3 ( x ) - ( 6 x ^ 3 - 6 x ^ 2 + 2 0 x - 3 2 ) \\Phi ^ 2 ( x ) - ( 6 4 x ^ 3 + 3 2 x ^ 2 + 9 6 x + 1 2 8 ) \\Phi ( x ) - 6 4 - m \\end{align*}"}
+{"id": "5322.png", "formula": "\\begin{align*} w _ { \\pi _ { k } } ^ { S _ 1 } \\ , y ^ { S _ 1 } + \\cdots + w _ { \\pi _ { k } } ^ { S _ k } \\ , y ^ { S _ k } = c _ { \\pi _ { k } } , 1 \\leq k \\leq n . \\end{align*}"}
+{"id": "3716.png", "formula": "\\begin{align*} \\psi _ { \\sigma } = \\frac { 1 } { ( \\omega _ { \\sigma } - E ) } \\chi _ { \\sigma } + \\frac { 1 } { ( \\omega _ { \\sigma } - E ) } f ^ * _ { \\sigma } ( a ) \\bigg [ \\frac 1 { \\alpha } - \\sum _ { \\tau } \\frac { | f _ { \\tau } ( a ) | ^ 2 } { \\omega _ { \\tau } ( \\omega _ { \\tau } - E ) } \\bigg ] ^ { - 1 } \\sum _ { \\nu } \\frac { f _ { \\nu } ( a ) } { \\omega _ { \\nu } } \\frac { 1 } { ( \\omega _ { \\nu } - E ) } \\chi _ { \\nu } \\ ; . \\end{align*}"}
+{"id": "8846.png", "formula": "\\begin{align*} - \\Delta \\varphi + \\frac { \\lambda } { | x | ^ 2 } \\varphi = | x | ^ { - \\tau } | \\varphi | ^ { p - 2 } ( I _ \\alpha * | \\cdot | ^ { - \\tau } | \\varphi | ^ p ) \\varphi , 0 \\neq \\varphi \\in { H ^ 1 _ \\lambda } . \\end{align*}"}
+{"id": "6479.png", "formula": "\\begin{align*} P ( x ) = 3 ( p + q ) x ^ 3 - ( 2 q + 7 p ) x ^ 2 + 5 p x - p . \\end{align*}"}
+{"id": "5707.png", "formula": "\\begin{align*} \\mathbf { L } \\psi _ { i , 1 } = 2 ^ { - 1 } m \\psi _ { i , 1 } + \\beta _ i \\psi _ { i , 2 } , \\ \\mathbf { L } \\psi _ { i , 2 } = 2 ^ { - 1 } m \\psi _ { i , 2 } - \\beta _ i \\psi _ { i , 1 } . \\end{align*}"}
+{"id": "407.png", "formula": "\\begin{align*} \\sigma ^ 2 ( l _ x , l _ y , m _ x , m _ y ) = \\sigma ^ 2 _ { S } ( m _ x , m _ y ) \\sigma ^ 2 _ { R } ( l _ x , l _ y ) , \\end{align*}"}
+{"id": "6240.png", "formula": "\\begin{align*} p \\left ( \\gamma - \\frac 2 3 \\right ) - q & = C _ g D _ g \\left ( \\frac { r _ i } { r _ i + \\lambda _ g } - \\frac 5 3 \\right ) + C _ i D _ i \\left ( \\frac { r _ i } { r _ i + \\lambda _ g } - \\frac 2 3 \\right ) \\\\ & = \\frac { 1 } { 3 ( r _ i + \\lambda _ g ) } \\left ( C _ g D _ g ( - 2 r _ i - 5 \\lambda _ g ) + C _ i D _ i ( r _ i - 2 \\lambda _ g ) \\right ) . \\end{align*}"}
+{"id": "4995.png", "formula": "\\begin{align*} p _ k ( \\lambda ) = \\lambda ^ { k - 1 } , k = 1 , \\ldots , N , \\end{align*}"}
+{"id": "7914.png", "formula": "\\begin{align*} \\ast \\boldsymbol { n } ( \\alpha ) = \\mathrm { t r } ( \\ast \\alpha ) , \\ast \\mathrm { t r } ( \\alpha ) = \\boldsymbol { n } ( \\ast \\alpha ) . \\end{align*}"}
+{"id": "7755.png", "formula": "\\begin{align*} N _ { n , p } = a _ { n } ^ { p } N _ { n } , \\ \\ K _ { n } = 1 4 4 m R \\gamma ^ { - 1 } ( 3 m N _ { n , m + 1 } ) ^ { \\tau } . \\end{align*}"}
+{"id": "7321.png", "formula": "\\begin{align*} F _ { m , r } ( s , \\beta ) = e ^ { - 2 \\pi i \\beta \\ell / m } \\cdot \\frac { U _ { m - \\ell - 1 } ( s + 1 ) + e ^ { 2 \\pi i \\beta } U _ { \\ell - 1 } ( s + 1 ) } { T _ { m } ( s + 1 ) - \\cos 2 \\pi \\beta } . \\end{align*}"}
+{"id": "8050.png", "formula": "\\begin{align*} \\varepsilon _ { 1 4 } ^ N = & \\int _ 0 ^ T \\eta _ { s , 3 } ^ N ( \\lambda _ 2 ) \\Bigg ( \\int _ \\mathbb { T } \\theta _ s ^ S ( v ) \\lambda _ 1 ( v ) \\left ( G _ s ( v ) - F _ s ( v ) \\right ) d v \\\\ & - \\frac { 1 } { N } \\sum _ { j = 0 } ^ { N - 1 } \\mathbb { E } S _ s ^ N ( j ) \\lambda _ 1 \\left ( \\frac { j } { N } \\right ) \\left ( G _ s \\left ( \\frac { j } { N } \\right ) - F _ s \\left ( \\frac { j } { N } \\right ) \\right ) \\Bigg ) d s , \\end{align*}"}
+{"id": "665.png", "formula": "\\begin{align*} \\alpha ( i _ { 1 } i _ { 2 } ) u _ { i _ { 1 } i _ { 2 } j _ { 1 } \\cdots j _ { p } } : = \\frac { 1 } { 2 } ( u _ { i _ { 1 } i _ { 2 } j _ { 1 } \\cdots j _ { p } } - u _ { i _ { 2 } i _ { 1 } j _ { 1 } \\cdots j _ { p } } ) . \\end{align*}"}
+{"id": "4743.png", "formula": "\\begin{align*} \\hat { x } _ n = _ X \\begin{cases} x _ n & \\forall k < _ \\mathbb { N } n \\left ( [ \\norm { x _ k - _ X x _ { k + 1 } } _ X ] ( k + 1 ) < _ \\mathbb { Q } 6 \\cdot 2 ^ { - k - 1 } \\right ) , \\\\ x _ k & \\min k < _ \\mathbb { N } n : [ \\norm { x _ k - _ X x _ { k + 1 } } _ X ] ( k + 1 ) \\geq _ \\mathbb { Q } 6 \\cdot 2 ^ { - k - 1 } . \\end{cases} \\end{align*}"}
+{"id": "6475.png", "formula": "\\begin{align*} \\langle A ( \\operatorname { g r a d } f ) , d \\phi ( e _ j ) \\rangle \\nabla _ { e _ j } f = - \\sqrt { 2 } | \\nabla f | ^ 2 . \\end{align*}"}
+{"id": "8667.png", "formula": "\\begin{align*} \\sigma _ G : H ^ 0 ( \\overline { M } ) ^ G \\cap C ^ \\infty ( \\overline { M } ) \\to H ^ 0 ( \\overline { M } _ G ) , \\sigma _ G = \\iota _ G \\circ \\iota ^ * . \\end{align*}"}
+{"id": "1050.png", "formula": "\\begin{align*} H _ { \\delta } ( n ) : = \\{ 1 , \\dots , [ \\delta n ] \\} \\mbox { a n d } T _ { \\delta } ( n ) : = \\{ n - [ \\delta n ] + 1 , \\dots , n \\} . \\end{align*}"}
+{"id": "4467.png", "formula": "\\begin{align*} M _ M = \\int _ M | f _ 0 | ^ 2 \\lambda _ 1 \\lambda _ 2 . \\end{align*}"}
+{"id": "3002.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\overline { R } ( \\xi , y ) \\xi & = & ( 1 - \\beta ) \\varphi ^ 2 y - \\alpha \\varphi ( y ) . \\\\ \\end{array} \\end{align*}"}
+{"id": "9031.png", "formula": "\\begin{align*} B _ \\varepsilon ( g ( x ) ) \\cap [ g ( y ) , g ( z ) ] & = \\varnothing \\\\ \\end{align*}"}
+{"id": "3599.png", "formula": "\\begin{align*} a = \\lambda _ 1 + \\lambda _ 2 b = \\lambda _ 1 \\lambda _ 2 . \\end{align*}"}
+{"id": "6133.png", "formula": "\\begin{align*} \\Phi ( \\rho _ k ( x ) ) & = \\Phi ( [ ( \\rho _ { m , k } ( x ) ) _ { m > k } ] ) \\\\ & = [ ( \\varphi _ m ( \\rho _ { m , k } ( x ) ) ) _ { m > k } ] \\\\ & = [ ( ( \\Psi ^ { - 1 } \\circ \\rho _ m ) ( \\rho _ { m , k } ( x ) ) _ { m > k } ] \\\\ & = [ ( \\Psi ^ { - 1 } \\circ \\rho _ k ( x ) ) _ { m > k } ] \\\\ & = \\iota \\circ \\Psi ^ { - 1 } \\circ \\rho _ k ( x ) \\end{align*}"}
+{"id": "6634.png", "formula": "\\begin{align*} q _ X ( e v , w ) = q _ X ( v , e ' w ) , \\forall v , w \\in T ( X ) . \\end{align*}"}
+{"id": "1882.png", "formula": "\\begin{align*} \\Delta _ z \\rho ( u + z v ) | _ { z = 0 } \\geq 0 , \\end{align*}"}
+{"id": "8465.png", "formula": "\\begin{align*} | u _ m | + | u _ { m - 1 } | = u _ m + u _ { m - 1 } \\leq 2 \\ell . \\end{align*}"}
+{"id": "189.png", "formula": "\\begin{align*} ( Y _ { i + } , S _ { i + } ) \\ = \\ & ( Z + , P + ) , \\\\ ( Y _ { i - } , S _ { i - } ) \\ = \\ & ( Z - , P - ) \\end{align*}"}
+{"id": "3664.png", "formula": "\\begin{align*} \\mathbb { E } _ z [ \\left \\langle z , y \\right \\rangle ^ 2 ] = \\frac { 1 } { d } \\mathbb { E } _ z \\left [ \\sum _ { i = 1 } ^ { d } \\left \\langle z , u _ i \\right \\rangle ^ 2 \\right ] = \\frac { \\mathbb E [ \\| z \\| ^ 2 ] } d = \\frac 1 d , \\end{align*}"}
+{"id": "2252.png", "formula": "\\begin{align*} \\small P _ t X _ j = \\sum _ { k = 1 } ^ { 2 n - 1 } Q ( P _ t X _ j , X _ k ) X _ k = \\sum _ { k = 1 } ^ { 2 n - 1 } g _ { F S } ( X _ j ^ { \\ast } , X _ k ^ { \\ast } ) _ { \\vert \\gamma ( t ) } X _ k . \\end{align*}"}
+{"id": "8150.png", "formula": "\\begin{align*} \\Delta ^ { ( \\ell ) } & = \\frac { 1 } { \\ell } \\sum _ { i = 1 } ^ { \\ell } \\mathbb { E } [ d ( \\underline S _ i , \\hat { \\underline S } _ i ) ] \\geq \\frac { 1 } { \\ell } \\sum _ { i = 1 } ^ { \\ell } \\mathbb { E } [ d ( \\underline S _ i , \\hat { \\underline s } ^ { \\star } ( \\underline X _ i ^ L , Y _ i ^ L ) ) ] , \\end{align*}"}
+{"id": "660.png", "formula": "\\begin{align*} \\begin{aligned} & ( N ^ { k } f ) _ { i _ { 1 } \\cdots i _ { m } } ( x ) \\\\ & = 2 \\sum \\limits _ { \\ell = 0 } ^ k \\binom { k } { \\ell } ( - 1 ) ^ { \\ell } x _ { p _ { 1 } \\cdots p _ { 2 k - \\ell } } ^ { \\otimes ( 2 k - \\ell ) } \\left ( f _ { j _ { 1 } \\dots j _ { m } } * \\Xi _ { p _ { 1 } \\cdots p _ { 2 k - \\ell } i _ { 1 } \\cdots i _ { m } j _ { 1 } \\dots j _ { m } } \\right ) ( x ) , \\end{aligned} \\end{align*}"}
+{"id": "2362.png", "formula": "\\begin{align*} g ( T ) : = & \\| v _ { k } \\| _ { C _ { T } L _ { x } ^ { 3 } \\cap L _ { T } ^ { 4 } L _ { x } ^ { 6 } } + \\big \\| \\nabla | v _ { k } | ^ { \\frac { 3 } { 2 } } \\big \\| ^ { \\frac { 2 } { 3 } } _ { L _ { T } ^ { 2 } L _ { x } ^ { 2 } } , \\\\ a : = & \\| v _ { k } ( \\cdot , 0 ) \\| _ { L ^ { 3 } } , \\ ; \\ , b ( T ) : = e ^ { C _ { 2 } \\int _ { 0 } ^ { T } \\| w \\| _ { L ^ { 6 } } ^ { 4 } d t } . \\end{align*}"}
+{"id": "3635.png", "formula": "\\begin{align*} \\alpha ^ 3 \\tau _ 0 ( z _ 0 ) ^ 3 + ( a + b ) \\alpha \\tau _ 0 ( z _ 0 ) = 0 , \\end{align*}"}
+{"id": "1922.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta \\overline v ( x ) - V ( x ) \\overline v ( x ) & = \\frac { 1 } { \\mu ( x ) } \\sum _ { y \\in G } \\omega ( x , y ) [ \\overline v ( y ) - \\overline v ( x ) ] - V ( x ) \\overline v ( x ) \\\\ & = - \\gamma \\ , V ( x ) \\le 0 \\ , \\ , x \\in G . \\end{aligned} \\end{align*}"}
+{"id": "5516.png", "formula": "\\begin{align*} \\tilde { M } = \\begin{cases} 2 a \\frac { ( q ^ { n - 1 } - 1 ) } { q ^ 2 - 1 } , & 1 \\leq a \\leq \\frac { q } { 2 } , \\\\ \\frac { q ^ { n - 1 } - q } { q - 1 } , & \\frac { q } { 2 } + 1 \\leq a \\leq q + 1 . \\end{cases} \\end{align*}"}
+{"id": "1108.png", "formula": "\\begin{align*} w ( x ) : = w _ 1 ( x ) w _ 2 ( x ) : = | x | ^ { - d } | x - x _ 0 | ^ { ( p - 1 ) \\widetilde d } . \\end{align*}"}
+{"id": "5216.png", "formula": "\\begin{align*} & M G = \\sum _ { i } p ^ { \\alpha } _ { i } q ^ { 1 - \\alpha } _ { i } = \\sum _ { i } M G _ { i } \\\\ & M H = \\sum _ { i } \\frac { p _ { i } q _ { i } } { \\left ( 1 - \\alpha \\right ) p _ { i } + \\alpha q _ { i } } = \\sum _ { i } M H _ { i } \\end{align*}"}
+{"id": "941.png", "formula": "\\begin{align*} R _ { \\alpha } f ( x ) = \\mathbb E _ x \\int ^ { \\infty } _ 0 e ^ { - \\alpha t } f ( X _ t ) \\ , d t , x \\in E , \\alpha > 0 , \\end{align*}"}
+{"id": "5881.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { N C } ( G ) ) = n ^ { 3 } ( 5 m ^ { 3 } - 9 m ^ { 2 } + 4 m ) M _ { 2 } ( \\mathcal { N C } ( G ) ) = n ^ { 4 } ( 4 m ^ { 4 } - 1 0 m ^ { 3 } + 8 m ^ { 2 } - 2 m ) , \\end{align*}"}
+{"id": "8713.png", "formula": "\\begin{align*} & { \\rm d e t \\ , } \\left ( ( a _ { j , \\ell } ) ^ { n - d } _ { j , \\ell = 0 } \\right ) \\neq 0 , \\\\ & a _ { 0 , \\ell } = f _ { \\ell } ( p ) , \\ \\ \\ell = 0 , \\ldots , n - d , \\\\ & a _ { j , \\ell } = ( Z _ { j } f _ { \\ell } ) ( p ) , \\ \\ \\ell = 0 , \\ldots , n - d , j = 1 , \\ldots , n - d . \\end{align*}"}
+{"id": "9027.png", "formula": "\\begin{align*} | G | = 1 + \\sum _ { \\in X } \\frac { | G | } { | S t a b _ x | } ( | G _ x | - 1 ) , \\end{align*}"}
+{"id": "7137.png", "formula": "\\begin{align*} \\partial ^ \\nu \\partial _ \\nu A _ { \\mu } = 0 \\ , \\ , . \\end{align*}"}
+{"id": "6182.png", "formula": "\\begin{align*} \\begin{aligned} \\lambda ^ { k + 1 } = & \\lambda ^ k - \\gamma [ ( 1 - \\tau ^ { k } ) \\beta ^ k ( A x ^ { k } - b ) - ( 1 - \\tau ^ { k + 1 } ) \\beta ^ { k + 1 } ( A x ^ { k + 1 } - b ) ] \\\\ & - \\gamma \\tau ^ k \\beta ^ k ( A \\bar { x } ^ { k + 1 } - b ) , ~ \\gamma \\in ( 0 , 1 ] . \\end{aligned} \\end{align*}"}
+{"id": "3509.png", "formula": "\\begin{align*} & \\| \\rho _ { n , n - 1 } ( 1 _ { F _ { n - 1 } } ) \\psi _ n ( a _ k ) - h _ n \\psi _ n ( a _ k ) \\| \\\\ & \\overset { \\phantom { \\eqref { e q : s e c o n d a p p r o x 1 } } } { = } \\| \\psi _ n ( \\varphi _ { n - 1 } ( 1 _ { F _ { n - 1 } } ) ) \\psi _ n ( a _ k ) - h _ n \\psi _ n ( a _ k ) \\| \\\\ & \\overset { \\eqref { e q : s e c o n d a p p r o x 1 } } { < } \\varepsilon _ { n - 1 } \\end{align*}"}
+{"id": "8199.png", "formula": "\\begin{align*} R & = \\frac { 1 } { 3 } \\left ( R _ 1 + R _ 2 + R _ 3 \\right ) = : g ( { \\bf B } ) , \\end{align*}"}
+{"id": "8949.png", "formula": "\\begin{align*} \\begin{aligned} \\| d ( \\nabla \\varphi , S O ( n ) ) \\| _ { L ^ { p ( \\cdot ) } ( Q ' ) } & \\le \\| d ( \\nabla v , S O ( n ) ) \\| _ { L ^ { p ( \\cdot ) } ( Q ' ) } + \\| \\nabla \\psi \\| _ { L ^ { p ( \\cdot ) } ( Q ' ) } \\\\ & \\le C \\| d ( \\nabla v , S O ( n ) ) \\| _ { L ^ { p ( \\cdot ) } ( Q ) } \\end{aligned} \\end{align*}"}
+{"id": "337.png", "formula": "\\begin{align*} g _ { I ( G ) } ( k ) = \\min \\{ g _ { I ( G _ 1 ) } ( k ) + 1 , g _ { I ( G _ 2 ) } ( k - 1 ) , g _ { I ( G _ 3 ) } ( k - 1 ) , g _ { I ( G _ 3 ) } ( k ) + 1 \\} . \\end{align*}"}
+{"id": "7373.png", "formula": "\\begin{align*} \\begin{array} { l l } X : = & \\{ ( u , w ) \\in \\{ C ^ 1 ( \\R \\times [ 0 , T ] ) \\} ^ 2 \\ : \\ \\| ( u , w ) \\| _ X < \\infty , \\\\ & \\quad \\mbox { s u p p } \\ ( u , w ) \\subset \\{ ( x , t ) \\in \\R \\times [ 0 , T ] \\ : \\ | x | \\le t + R \\} \\} , \\end{array} \\end{align*}"}
+{"id": "3376.png", "formula": "\\begin{align*} I _ 5 : = I _ { 5 , 1 } + I _ { 5 , 2 } + I _ { 5 , 3 } . \\end{align*}"}
+{"id": "3729.png", "formula": "\\begin{align*} \\left ( x \\right ) _ { n } = \\frac { \\Gamma \\left ( x + n \\right ) } { \\Gamma \\left ( x \\right ) } , \\end{align*}"}
+{"id": "9173.png", "formula": "\\begin{align*} \\tilde { E } ^ { - } _ { \\beta } = \\begin{cases} \\hat { E } ^ { - } _ { \\beta } & \\ \\beta = [ i , j ] \\\\ [ 2 ] _ { v } ! \\hat { E } ^ { - } _ { \\beta } & \\ \\beta = [ i , n , j ] \\end{cases} . \\end{align*}"}
+{"id": "6994.png", "formula": "\\begin{align*} \\Lambda = \\bigcup _ { p \\in \\partial \\mathcal { H } _ \\mathbb { R } ^ 2 } H _ p . \\end{align*}"}
+{"id": "484.png", "formula": "\\begin{align*} \\gcd ( \\prod _ { j = 0 } ^ { k - 1 } { \\alpha _ { i _ { t - k + j } } } , s ) = \\gcd ( \\prod _ { j = 0 } ^ { k - 2 } { \\alpha _ { i _ { t - k + j } } } , s ) , \\end{align*}"}
+{"id": "7581.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\int _ { \\mathbb { R } ^ { N } } | u ( t , x ) | ^ { \\frac { 2 ( N + 4 ) } { N - 4 } } d x d t = + \\infty . \\end{align*}"}
+{"id": "5027.png", "formula": "\\begin{align*} \\mathbf { x } ^ 1 \\mathbf { R } = f _ i ^ S - \\sum _ { j \\in S } r _ j ^ S x _ j ^ 0 + \\sum _ { j \\in S ^ c } r _ j ^ S x _ j ^ 1 , \\end{align*}"}
+{"id": "2373.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int V ( u _ 1 , v _ 1 , u _ 2 , v _ 2 ) \\alpha _ n ( \\dd u _ 1 , \\dd v _ 1 ) \\beta _ n ( \\dd u _ 2 , \\dd v _ 2 ) = 0 \\ , . \\end{align*}"}
+{"id": "878.png", "formula": "\\begin{align*} L _ { w w } & = w _ 1 ^ 2 L _ { 1 1 } + 2 w _ 1 w _ 2 L _ { 1 2 } + w _ 2 ^ 2 L _ { 2 2 } , \\\\ & = 2 ( w ^ 1 ) ^ 2 { w ^ 1 } _ { \\substack { \\\\ x _ 1 } } + 2 w ^ 1 w ^ 2 ( { w ^ 1 } _ { \\substack { \\\\ x _ 2 } } + { w ^ 2 } _ { \\substack { \\\\ x _ 1 } } ) + 2 ( w ^ 2 ) ^ 2 { w ^ 2 } _ { \\substack { \\\\ x _ 2 } } . \\end{align*}"}
+{"id": "3192.png", "formula": "\\begin{align*} B _ n ( q ^ { - 1 } ) & = \\prod _ { k = 0 } ^ { n } ( 1 + q ^ { - ( 3 k + 1 ) } ) ( 1 + q ^ { - ( 3 k + 2 ) } ) \\\\ [ 5 p t ] & = q ^ { - d _ n } \\prod _ { k = 0 } ^ { n } ( 1 + q ^ { ( 3 k + 1 ) } ) ( 1 + q ^ { ( 3 k + 2 ) } ) \\\\ [ 5 p t ] & = q ^ { - d _ n } B _ n ( q ) . \\end{align*}"}
+{"id": "1648.png", "formula": "\\begin{align*} b = 1 \\end{align*}"}
+{"id": "2804.png", "formula": "\\begin{align*} & \\Xi ^ { 1 } : = q ^ { 1 } , \\Xi ^ { 2 } : = q ^ { 2 } , \\Xi ^ { 3 } : = - p _ { 3 } \\\\ & \\Psi _ { 1 } : = \\phi ^ { ( 3 ) } , \\Psi _ { 2 } : = \\phi ^ { ( 1 ) } , \\Psi _ { 3 } : = \\phi ^ { ( 2 ) } . \\end{align*}"}
+{"id": "6751.png", "formula": "\\begin{align*} Q = H M , \\end{align*}"}
+{"id": "7047.png", "formula": "\\begin{align*} \\partial _ t \\langle h w _ \\eta ^ q \\rangle _ { x , \\eta } & + \\frac { 4 ( q - 1 ) } { q } \\langle h | \\nabla w ^ { \\frac { q } { 2 } } _ \\eta | ^ 2 \\rangle _ { x , \\eta } \\\\ & - 4 \\langle b \\cdot \\nabla w _ \\eta ^ { \\frac { q } { 2 } } , h w _ \\eta ^ { \\frac { q } { 2 } } \\rangle _ { x , \\eta } + 4 \\theta \\sum _ { i = 1 } ^ d \\langle b _ i \\nabla w _ \\eta ^ { \\frac { q } { 2 } } , \\frac { \\kappa \\eta _ i \\eta } { 1 + \\kappa | \\eta | ^ 2 } h w _ \\eta ^ { \\frac { q } { 2 } } \\rangle _ { x , \\eta } = 0 . \\end{align*}"}
+{"id": "2589.png", "formula": "\\begin{align*} \\dfrac { d u } { d \\nu } ( - R ) = - \\dfrac { d u } { d x } ( - R ) = \\dfrac { d u } { d x } ( \\lambda ) < 0 . \\end{align*}"}
+{"id": "948.png", "formula": "\\begin{align*} R ^ V \\mu = 0 \\quad \\mbox { q . e . i n } V ^ c . \\end{align*}"}
+{"id": "8363.png", "formula": "\\begin{align*} ( p _ 1 , \\dots , p _ m ) \\cdot ( b _ 1 , \\dots , b _ m ) = ( p _ 1 b _ 1 , b _ 1 ^ { - 1 } p _ 2 b _ 2 , \\dots , b _ { m - 1 } ^ { - 1 } p _ m b _ m ) , \\end{align*}"}
+{"id": "7237.png", "formula": "\\begin{align*} \\sigma ^ k ( x _ { p , q } ) = \\sigma ^ k ( \\sigma ^ p ( a ) \\ast \\sigma ^ q ( b ) ) = \\sigma ^ { k + p } ( a ) \\ast \\sigma ^ { k + q } ( b ) = x _ { ( k + p ) \\bmod r , \\ , ( k + q ) \\bmod s } \\end{align*}"}
+{"id": "2830.png", "formula": "\\begin{align*} L _ { T } : = & \\tilde { \\sigma } ^ { * } _ { 3 } ( t ) \\left [ P _ { 1 } \\dot { Q } ^ { 1 } + \\Psi _ { a } \\dot { \\Xi } ^ { a } + \\Theta _ { 1 } \\dot { \\Theta } ^ { 1 } - H _ { T } \\right ] \\\\ = & \\frac { d } { d t } \\left [ \\Psi _ { 2 } \\Xi ^ { 2 } \\right ] + P _ { 1 } \\dot { Q } ^ { 1 } - H _ { T } + { c o n s t a n t } , \\end{align*}"}
+{"id": "8491.png", "formula": "\\begin{align*} \\sup _ { t > 0 } \\| \\bar { u } _ h ( t ) \\| _ { L ^ \\infty ( \\Omega ) } & \\leq \\| u _ 0 \\| _ { L ^ \\infty ( \\Omega ) } , \\\\ \\sup _ { t > 0 } \\| u _ h ( t ) \\| _ { L ^ \\infty ( \\Omega ) } & \\leq \\| u _ 0 \\| _ { L ^ \\infty ( \\Omega ) } ; \\end{align*}"}
+{"id": "3283.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ N \\frac { ( q ^ { - N } , b _ 1 q ^ { n _ 1 } , \\cdots , b _ m q ^ { n _ m } ; q ) _ k } { ( q , b _ 1 , \\cdots , b _ m ; q ) _ k } q ^ k = ( - 1 ) ^ N \\frac { ( q ; q ) _ N b _ 1 ^ { n _ 1 } \\cdots b _ m ^ { n _ m } } { ( b _ 1 ; q ) _ { n _ 1 } \\cdots ( b _ m ; q ) _ { n _ m } } q ^ { \\binom { n _ 1 } { 2 } + \\cdots + \\binom { n _ m } { 2 } } . \\end{align*}"}
+{"id": "5185.png", "formula": "\\begin{align*} D _ { \\beta } I ( p \\| q ) = \\underbrace { \\frac { 1 } { \\beta ( \\beta - 1 ) } } _ { T } \\left [ \\underbrace { \\sum _ { i } p ^ { \\beta } _ { i } } _ { A } - \\underbrace { \\left ( \\sum _ { i } p _ { i } q ^ { \\beta - 1 } _ { i } \\right ) ^ { \\beta } } _ { X } \\times \\underbrace { \\left ( \\sum _ { i } q ^ { \\beta } _ { i } \\right ) ^ { 1 - \\beta } } _ { Y } \\right ] \\end{align*}"}
+{"id": "4773.png", "formula": "\\begin{align*} \\varphi ( \\varepsilon , b , f ) : = \\frac { 8 ( b + f ( \\varepsilon / 2 ) ) } { \\varepsilon } . \\end{align*}"}
+{"id": "6635.png", "formula": "\\begin{align*} E = \\mathrm { E n d } _ { \\mathrm { H d g } } ( T ( X ) ) \\simeq \\left ( T ( X ) \\otimes T ( X ) \\right ) ^ { 2 , 2 } . \\end{align*}"}
+{"id": "7732.png", "formula": "\\begin{align*} H _ \\eta ^ \\varrho ( \\Omega ) : = \\inf \\{ \\sum _ { i = 1 } ^ \\infty ( \\mathrm { d i a m } \\Omega _ i ) ^ \\varrho \\ : \\ \\Omega \\subseteq \\cup _ i \\Omega _ i , \\ \\mathrm { a n d } \\ \\mathrm { d i a m } ( \\Omega _ i ) < \\eta \\} . \\end{align*}"}
+{"id": "2779.png", "formula": "\\begin{align*} { \\sigma _ { \\tau } ^ { * } } ( t ) H _ { T } : = P _ { a } \\dot { Q } ^ { a } - L _ { T } ( Q ^ { a } , \\dot { Q } ^ { a } ) \\end{align*}"}
+{"id": "7089.png", "formula": "\\begin{align*} ( 1 - c _ p m _ d \\delta ) \\| | | b | _ 1 ^ { - \\frac { 1 } { p ' } } ( \\mu - \\Delta ) u \\| _ p \\leq \\| | b | _ 1 ^ { - \\frac { 1 } { p ' } } f \\| _ p , | b | _ 1 : = | b | + 1 \\end{align*}"}
+{"id": "6641.png", "formula": "\\begin{align*} \\textstyle \\mathrm { t r } ( x ) = \\Sigma _ i b ^ i ( x \\cdot b _ i ) \\end{align*}"}
+{"id": "5796.png", "formula": "\\begin{align*} \\frac { d } { d t } \\xi ^ { ( k , \\ell ) } _ i ( t ) - \\lambda _ i \\xi ^ { ( k , \\ell ) } _ i ( t ) = \\mathcal { E } ^ { ( k , \\ell ) } _ i ( t ) . \\end{align*}"}
+{"id": "1891.png", "formula": "\\begin{align*} 2 \\pi p \\langle r ^ { p - 1 } _ + , \\mathcal { M } _ { 2 , w } ^ { | \\cdot | , u } ( K , \\cdot ) \\rangle . \\end{align*}"}
+{"id": "3804.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { L + 1 } | I _ j | = n ( L + 1 ) - n \\cdot \\sum _ { j = 1 } ^ { L + 1 } \\delta ( c _ j , y ) \\geq n ( L + 1 ) ( 1 - \\rho ) . \\end{align*}"}
+{"id": "8518.png", "formula": "\\begin{align*} \\sigma ^ \\ell ( h ( y ) ) = h ( z ) = h ( \\sigma ^ k ( y ) ) , \\end{align*}"}
+{"id": "3220.png", "formula": "\\begin{align*} = - t ^ { \\beta } ( u ( t ) , \\lambda _ k v _ k ) + t ^ { \\mu } f _ k = - \\lambda _ k t ^ { \\beta } ( u ( t ) , v _ k ) + t ^ { \\mu } f _ k = - \\lambda _ k t ^ { \\beta } u _ k ( t ) + t ^ { \\mu } f _ k , t > 0 . \\end{align*}"}
+{"id": "4795.png", "formula": "\\begin{align*} F _ \\theta ( \\xi ) = \\vert \\xi \\vert - \\cos \\theta \\left < \\xi , E _ n \\right > . \\end{align*}"}
+{"id": "384.png", "formula": "\\begin{align*} K _ 5 = 4 x ^ 3 + 6 ( b - a ) x ^ 2 + 2 ( 2 b ^ 2 - a b + 2 a ^ 2 ) x - 2 a b c + b ^ 3 - a ^ 3 , \\end{align*}"}
+{"id": "4904.png", "formula": "\\begin{align*} | K _ { S ' } + C | = F + | M | , \\end{align*}"}
+{"id": "8026.png", "formula": "\\begin{align*} & d \\langle \\mathcal { M } ^ N _ { f , k } , \\Lambda ^ N _ { F , G , H } \\rangle _ t = \\Big ( \\mathcal { L } ^ N \\left ( \\mu _ { t , k } ^ N ( f ) V _ { F , G , H } ^ N ( t , \\xi ^ N ) \\right ) - V _ { F , G , H } ^ N ( t , \\xi ^ N ) \\mathcal { L } ^ N \\mu _ { t , k } ^ N ( f ) \\\\ & - \\mu _ { t , k } ^ N ( f ) \\mathcal { L } ^ N V _ { F , G , H } ^ N ( t , \\xi ^ N ) \\Big ) d t . \\end{align*}"}
+{"id": "4068.png", "formula": "\\begin{align*} \\chi R _ V \\chi + \\chi R _ V V R _ 0 \\chi = \\chi R _ 0 \\chi \\Rightarrow \\chi R _ V \\chi ( I + V R _ 0 \\chi ) = \\chi R _ 0 \\chi . \\end{align*}"}
+{"id": "3454.png", "formula": "\\begin{align*} \\delta _ R ( Y , \\iota , \\mathfrak { t } ) + \\delta _ R ( Y ' , \\iota ' , \\mathfrak { t } ' ) = \\delta _ R ( Y \\# Y ' , \\iota \\# \\iota ' , \\mathfrak { t } \\# \\mathfrak { t } ' ) . \\end{align*}"}
+{"id": "7.png", "formula": "\\begin{align*} R ( \\kappa ) - R _ 0 ( \\kappa ) = R _ 0 ( \\kappa ) C _ + q \\sqrt { R _ 0 ( \\kappa ) } \\cdot \\sqrt { R _ 0 ( \\kappa ) } \\bigl [ 1 + C _ + q R ( \\kappa ) \\bigr ] . \\end{align*}"}
+{"id": "3484.png", "formula": "\\begin{align*} V = \\prod _ { \\ell = 0 } ^ { L } ( p _ { \\ell } + 1 ) \\ , . \\end{align*}"}
+{"id": "1234.png", "formula": "\\begin{align*} \\lim _ n \\rho _ n & = \\lim _ { n \\to \\infty } \\sum _ { i , j = 1 } ^ n \\sqrt { \\lambda _ i \\lambda _ j } E _ { i j } \\otimes E _ { i j } \\\\ & = \\lim _ { n , m \\to \\infty } \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ m \\sqrt { \\lambda _ i \\lambda _ j } E _ { i j } \\otimes E _ { i j } \\\\ & = \\sum _ { i , j } \\sqrt { \\lambda _ i \\lambda _ j } E _ { i j } \\otimes E _ { i j } \\end{align*}"}
+{"id": "8615.png", "formula": "\\begin{align*} | K _ 1 ( t ; x _ j ) + \\phi _ 1 ( t ; x _ j ) | & \\leq \\epsilon ^ 2 m ^ 2 ( t ) \\leq 4 \\epsilon ^ 2 v _ 1 ^ 2 ( t ; x _ j ) \\\\ & < \\frac { \\gamma } { 2 } v _ 1 ^ 2 ( t ; x _ j ) ~ ~ { \\rm f o r ~ a n y } ~ ~ t \\in [ t _ 1 , t _ 2 ] ~ ~ { \\rm a n d } ~ ~ j = 1 , 2 . \\end{align*}"}
+{"id": "4224.png", "formula": "\\begin{align*} \\theta _ 3 ( v , \\tau + 1 ) = \\theta _ 2 ( v , \\tau ) , ~ ~ \\theta _ 3 ( v , - \\frac { 1 } { \\tau } ) = \\left ( \\frac { \\tau } { \\sqrt { - 1 } } \\right ) ^ { \\frac { 1 } { 2 } } e ^ { \\pi \\sqrt { - 1 } \\tau v ^ 2 } \\theta _ 3 ( \\tau v , \\tau ) , \\end{align*}"}
+{"id": "5405.png", "formula": "\\begin{align*} \\det ( \\mathbf { I } - \\mathbf { B } ^ { k } ) & = \\det ( \\mathbf { I } - \\mathbf { B } ^ { k - 1 } ) - \\frac { \\lambda _ { k - 1 } \\ , \\mu _ { k - 1 } } { ( \\alpha + \\lambda _ { k - 2 } + \\mu _ { k - 1 } ) \\ , ( \\alpha + \\lambda _ { k - 1 } + \\mu _ k ) } \\ , \\det ( \\mathbf { I } - \\mathbf { B } ^ { k - 2 } ) \\end{align*}"}
+{"id": "5492.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ \\ell \\frac { \\gamma } { k _ i } m _ i k _ i \\right ) \\log \\left ( \\sum _ { i = 1 } ^ \\ell \\frac { \\gamma } { k _ i } m _ i k _ i \\right ) \\le \\sum _ { i = 1 } ^ \\ell \\frac { \\gamma } { k _ i } m _ i k _ i \\log ( m _ i k _ i ) . \\end{align*}"}
+{"id": "8224.png", "formula": "\\begin{align*} \\{ P _ { { \\bf X } _ i | { \\bf A } _ { i - 1 } } ( { \\bf x } _ i | { \\bf a } _ { i - 1 } ) = P _ { { \\bf X } | { \\bf A } } ( { \\bf x } _ i | { \\bf a } _ { i - 1 } ) \\} \\end{align*}"}
+{"id": "7012.png", "formula": "\\begin{align*} \\| b \\| _ { M _ { 2 + \\varepsilon } } : = \\sup _ { r > 0 , x \\in \\mathbb R ^ d } r \\biggl ( \\frac { 1 } { | B _ r | } \\int _ { B _ r ( x ) } | b | ^ { 2 + \\varepsilon } d x \\biggr ) ^ { \\frac { 1 } { 2 + \\varepsilon } } < \\infty \\end{align*}"}
+{"id": "2271.png", "formula": "\\begin{align*} ( \\Delta \\otimes ) ( \\mathcal { R } ) = \\mathcal { R } _ { 1 3 } \\mathcal { R } _ { 2 3 } \\ \\ \\ \\ \\ \\ ( \\otimes \\Delta ) ( \\mathcal { R } ) = \\mathcal { R } _ { 1 3 } \\mathcal { R } _ { 1 2 } \\ . \\end{align*}"}
+{"id": "1112.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathbb { Z } } \\overline { \\widehat { \\varphi } \\left ( 2 ^ j \\xi \\right ) } \\widehat { \\psi } \\left ( 2 ^ j \\xi \\right ) = 1 \\xi \\in \\mathbb { R } ^ n \\setminus \\{ \\mathbf { 0 } \\} . \\end{align*}"}
+{"id": "4901.png", "formula": "\\begin{align*} \\phi = ( \\phi _ { \\mathfrak l } , \\phi _ { \\mathfrak m } ) : C \\to \\P ^ 1 \\times \\P ^ 1 \\end{align*}"}
+{"id": "4230.png", "formula": "\\begin{align*} Q ( X , \\tau ) = \\prod _ { j = 1 } ^ { 2 k } \\left ( \\frac { 2 x _ j \\theta ' ( 0 , \\tau ) } { \\theta ( x _ j , \\tau ) } \\left ( \\frac { \\theta _ 1 ( x _ j , \\tau ) } { \\theta _ 1 ( 0 , \\tau ) } + \\frac { \\theta _ 2 ( x _ j , \\tau ) } { \\theta _ 2 ( 0 , \\tau ) } + \\frac { \\theta _ 3 ( x _ j , \\tau ) } { \\theta _ 3 ( 0 , \\tau ) } \\right ) \\right ) . \\end{align*}"}
+{"id": "2554.png", "formula": "\\begin{align*} C ^ \\infty ( M , L ) = C ^ { - \\infty } ( M , L ; \\Omega ) ' \\ ; . \\end{align*}"}
+{"id": "5273.png", "formula": "\\begin{align*} x ^ { k + 1 } = x ^ { k } + \\alpha ^ { k } x ^ { k } \\left ( \\frac { U ^ { k } } { V ^ { k } } - 1 \\right ) \\end{align*}"}
+{"id": "5901.png", "formula": "\\begin{align*} 7 6 ( 4 n - 1 ) ^ { 3 } + 2 8 5 ( 3 n - 1 ) ^ { 3 } & > 1 6 ( 4 n - 1 ) ^ { 3 } + 2 2 5 ( 3 n - 1 ) ^ { 3 } + 6 0 ( 4 n - 1 ) ( 3 n - 1 ) ( 7 n - 2 ) \\\\ & = ( 4 ( 4 n - 1 ) + 1 5 ( 3 n - 1 ) ) ( 4 ( 4 n - 1 ) ^ { 2 } + 1 5 ( 3 n - 1 ) ^ { 2 } ) . \\end{align*}"}
+{"id": "8681.png", "formula": "\\begin{align*} T M ' | _ { Y ' } = \\underline { \\mathfrak { g } } | _ { Y ' } \\oplus J \\underline { \\mathfrak { g } } | _ { Y ' } \\oplus T ^ H Y ' . \\end{align*}"}
+{"id": "2342.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow \\infty } \\| a _ { k } - a \\| _ { L _ { t } ^ { \\infty } L _ { x } ^ { p } } + \\| \\nabla | a _ { k } - a | ^ { \\frac { p } { 2 } } \\| ^ { \\frac { 2 } { p } } _ { L ^ { 2 } _ { t } L ^ { 2 } _ { x } } = 0 . \\end{align*}"}
+{"id": "4623.png", "formula": "\\begin{align*} f _ n : Q ^ { ( d - 1 ) } \\rightarrow \\left [ 0 , \\infty \\right ) \\ , , \\ x ' \\mapsto \\sum _ { j = 0 } ^ n 2 ^ { - { \\gamma } j m } g _ j ( x ' ) . \\end{align*}"}
+{"id": "715.png", "formula": "\\begin{align*} \\lim _ { r \\to 1 ^ { - } } \\int _ { E } | f _ r | ^ p d \\mu _ \\alpha ( z ) = \\int _ { E } | f | ^ p d \\mu _ \\alpha ( z ) . \\end{align*}"}
+{"id": "8865.png", "formula": "\\begin{align*} 0 < \\frac 1 { ( p - 1 ) b _ 4 } \\leq \\frac 1 r \\leq 1 , 0 \\leq \\frac { \\tau } { p - 1 } < \\frac { n } { ( p - 1 ) b _ 4 } , \\frac { \\tau } { p - 1 } - 1 = \\frac { n } { ( p - 1 ) b _ 4 } - \\frac { n } { r } . \\end{align*}"}
+{"id": "4378.png", "formula": "\\begin{align*} \\min _ { x } & \\ \\sum _ { ( i , j , r , s ) \\in [ n ] ^ 4 } c _ { i , j } d _ { r , s } x _ { i , r } x _ { j , s } \\\\ \\mathrm { s . t . } & \\ x \\in \\mathcal { X } = \\{ x \\in \\{ 0 , 1 \\} ^ { [ n ] ^ 2 } : \\sum _ { i \\in [ n ] } x _ { i , r } = 1 \\ \\forall r \\in [ n ] , \\ \\sum _ { r \\in [ n ] } x _ { i , r } = 1 \\ \\forall i \\in [ n ] \\} \\end{align*}"}
+{"id": "8028.png", "formula": "\\begin{align*} \\langle \\mathcal { M } ^ N _ { h , 3 } , \\hat { \\Lambda } ^ N _ { F , G , H } \\rangle _ t = & \\int _ 0 ^ t \\mu _ { s , 2 } ^ N \\left ( \\psi ( \\cdot ) h ( \\cdot ) \\left ( e ^ { - G _ s ( \\cdot ) + H _ s ( \\cdot ) } - 1 \\right ) \\right ) d s \\\\ & - \\int _ 0 ^ t \\mu _ { s , 3 } ^ N \\left ( \\phi ( \\cdot ) h ( \\cdot ) \\left ( e ^ { - H _ s ( \\cdot ) } - 1 \\right ) \\right ) d s \\end{align*}"}
+{"id": "946.png", "formula": "\\begin{align*} u ( x ) = \\mathbb E _ x g ( X _ { \\tau _ D } ) + \\mathbb E _ x \\int _ 0 ^ { \\tau _ D } f ( X _ t , u ( X _ t ) ) \\ , d t + \\mathbb E _ x A ^ \\mu _ { \\tau _ D } . \\end{align*}"}
+{"id": "5456.png", "formula": "\\begin{align*} \\frac { \\rm { d } \\mathbb { Q } } { \\rm { d } \\mathbb { P } } = Z _ T : = \\exp \\left ( - \\frac { 1 } { \\sqrt { 2 \\sigma } } \\int _ 0 ^ T H \\ast f _ s ^ N ( X _ s ^ { 1 , N } , V _ s ^ { 1 , N } ) \\rm { d } B ^ 1 _ s - \\frac { 1 } { 4 \\sigma } \\int _ 0 ^ T \\big | H \\ast f _ s ^ N ( X _ s ^ { 1 , N } , V _ s ^ { 1 , N } ) \\big | ^ 2 \\rm { d } s \\right ) . \\end{align*}"}
+{"id": "1097.png", "formula": "\\begin{align*} \\int _ 0 ^ r t ^ { a + n - 1 } [ \\log ( 2 + t ) ] ^ b \\ , d t & > \\left [ \\log \\left ( 2 + \\frac { r } { 2 } \\right ) \\right ] ^ b \\int _ { \\frac { r } { 2 } } ^ r t ^ { a + n - 1 } \\ , d t \\\\ & = \\frac { 1 - 2 ^ { - ( a + n ) } } { a + n } r ^ { a + n } \\left [ \\log \\left ( 2 + \\frac { r } { 2 } \\right ) \\right ] ^ b \\\\ & > \\frac { 1 - 2 ^ { - ( a + n ) } } { 2 ^ b ( a + n ) } r ^ { a + n } [ \\log ( 2 + r ) ] ^ b . \\end{align*}"}
+{"id": "3255.png", "formula": "\\begin{align*} & \\big ( S ^ { \\wedge , ( l ) } \\ : V \\ : S ^ { \\wedge , ( r ) } \\big ) ( x , y ) \\\\ & = \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { n ! } \\int _ 0 ^ 1 \\alpha ^ { l } \\ : ( 1 - \\alpha ) ^ r \\ : ( \\alpha - \\alpha ^ 2 ) ^ n \\ : ( \\Box ^ n V ) \\big | _ { \\alpha y + ( 1 - \\alpha ) x } \\ : d \\alpha \\ ; S ^ { \\wedge , ( n + l + r + 1 ) } ( x , y ) \\ : . \\end{align*}"}
+{"id": "6724.png", "formula": "\\begin{align*} a _ 0 y + a _ 1 E y + a _ 2 E ^ 2 y + . . . + a _ { n - 1 } E ^ { n - 1 } y = 0 , \\end{align*}"}
+{"id": "3562.png", "formula": "\\begin{align*} P = \\frac { ( T - \\lambda _ { 1 } I ) ( T - \\lambda _ 2 I ) } { ( 1 - \\lambda _ 1 ) ( 1 - \\lambda _ 2 ) } , Q = \\frac { ( T - I ) ( T - \\lambda _ 2 I ) } { ( \\lambda _ 1 - 1 ) ( \\lambda _ 1 - \\lambda _ 2 ) } , R = \\frac { ( T - I ) ( T - \\lambda _ 1 I ) } { ( \\lambda _ 2 - 1 ) ( \\lambda _ 2 - \\lambda _ 1 ) } \\end{align*}"}
+{"id": "5524.png", "formula": "\\begin{align*} \\lambda _ n = \\rho _ n ^ 2 , \\rho _ n = z _ n + \\frac { \\nu _ n } { n } \\varphi _ \\alpha ( a z _ n ) , n \\in { \\mathbb N } , \\{ \\nu _ n \\} _ { n \\in { \\mathbb N } } \\in \\ell _ 2 , \\end{align*}"}
+{"id": "817.png", "formula": "\\begin{align*} { \\pounds } _ { \\hat { X } } \\nabla _ { 0 } I _ { k } = \\nabla _ { 0 } ( { \\pounds } _ { \\hat { X } } { I _ k } ) + \\Psi { I _ k } . \\end{align*}"}
+{"id": "6902.png", "formula": "\\begin{align*} \\eta _ { 3 } \\leq \\eta - \\eta _ { 2 } \\leq 0 \\begin{cases} \\eta \\geq \\eta _ { 0 } & 0 < s \\leq - 1 + \\sqrt { \\cosh 1 } , \\\\ 0 \\leq \\eta - \\eta _ { 0 } \\leq \\eta _ { 1 } & - 1 + \\sqrt { \\cosh 1 } < s \\leq 1 / \\sqrt { 2 } , \\end{cases} \\end{align*}"}
+{"id": "8939.png", "formula": "\\begin{align*} \\alpha ( a ) = ( a , J ( a ) ) \\rho ( a , b ) = b - J ( a ) . \\end{align*}"}
+{"id": "3803.png", "formula": "\\begin{align*} \\epsilon _ r & : = \\Bar { c } _ 0 \\times \\frac { 1 } { \\sqrt { \\sum _ { i \\in \\widehat { \\mathcal { C } } ^ { ( r ) } _ j } N _ i } } + \\bar { c } _ 1 \\Delta _ { \\max } \\sum _ { i \\in [ M ] } \\sum _ { t = 0 } ^ { T - 1 } \\exp \\left ( - \\bar { c } _ 2 N _ i n _ x \\left ( \\frac { \\alpha ^ { ( r ) } \\rho ^ { ( i ) } \\| \\Sigma ^ { ( i ) } _ t \\| } { \\rho ^ { ( i ) } \\| \\Sigma ^ { ( i ) } _ t \\| + \\sqrt { n _ x } } \\right ) ^ 2 \\right ) , \\end{align*}"}
+{"id": "8132.png", "formula": "\\begin{align*} h ^ { ( q _ 1 , q _ 2 ) } \\ = \\ h _ { A _ 2 } ( D _ { q _ 1 , x } , X _ { q _ 1 , x } , D _ { q _ 2 , y } , X _ { q _ 2 , y } ) \\ , \\end{align*}"}
+{"id": "6788.png", "formula": "\\begin{align*} \\Gamma = \\begin{bmatrix} c _ 1 & c _ 2 & c _ 3 \\\\ d _ 1 & d _ 2 & d _ 3 \\end{bmatrix} \\ , . \\end{align*}"}
+{"id": "3293.png", "formula": "\\begin{align*} \\ell _ n ( \\mu ; \\mu _ 0 ) : = \\exp \\left \\{ \\sum _ { i = 1 } ^ n \\frac { ( X _ i - \\mu _ 0 ) ^ 2 - ( X _ i - \\mu ) ^ 2 } { 2 } \\right \\} . \\end{align*}"}
+{"id": "2144.png", "formula": "\\begin{align*} F _ { \\min } = - 1 . 2 0 9 9 , x ^ * = 1 . 8 8 8 9 , y ^ * = ( 0 . 8 8 8 9 , 0 . 0 0 0 0 ) . \\end{align*}"}
+{"id": "503.png", "formula": "\\begin{align*} O ( \\pi ( x ) ) = O \\left ( \\frac { x } { \\log { x } } \\right ) = O \\left ( \\frac { x } { 2 \\log { x } } \\right ) . \\end{align*}"}
+{"id": "6714.png", "formula": "\\begin{align*} P ( E ) y ( x ) = P ( E ) m ^ x = P ( m ) y ( x ) . \\end{align*}"}
+{"id": "528.png", "formula": "\\begin{align*} u _ { x } ( 1 ) : = \\lim _ { x \\rightarrow 1 ^ { - } } x ^ { \\delta } u _ { x } ( x ) , \\end{align*}"}
+{"id": "2074.png", "formula": "\\begin{align*} \\frac { 1 } { T } \\sum _ { t = 1 } ^ { \\lfloor s T \\rfloor - 1 } \\Bar { \\Theta } ^ { d , T } ( \\frac { t } { T } , x ) \\left ( f ' ( \\frac { t } { T } ) + O ( T ^ { - 1 } ) \\right ) = \\int _ 0 ^ { \\frac { \\lfloor s T \\rfloor } { T } } \\Theta ( u , x ) f ' ( u ) \\mathrm { d } u + o ( 1 ) . \\end{align*}"}
+{"id": "3256.png", "formula": "\\begin{align*} - \\int _ 0 ^ 1 S ^ { \\wedge , ( 1 ) } _ { [ \\nu / \\alpha ] } ( 0 , y ) \\ : d \\alpha & = - 2 \\int _ 0 ^ 1 \\Theta \\Big ( \\xi ^ 2 - \\frac { \\nu } { \\alpha } \\Big ) \\ : H ^ { ( 1 ) } _ { [ \\nu / \\alpha ] } ( 0 , y ) \\ : d \\alpha \\\\ & = - \\frac { 1 } { 8 \\pi } \\int _ 0 ^ 1 \\Theta \\Big ( \\xi ^ 2 - \\frac { \\nu } { \\alpha } \\Big ) \\ : d \\alpha = - \\frac { 1 } { 8 \\pi } \\Big ( 1 - \\frac { \\nu } { \\xi ^ 2 } \\Big ) \\end{align*}"}
+{"id": "8571.png", "formula": "\\begin{align*} C _ t ^ { t _ 0 } = ( \\bar { G } _ { t _ 0 } ^ t ) ^ T , \\end{align*}"}
+{"id": "1473.png", "formula": "\\begin{align*} T _ i = g _ i F _ { k _ i } . \\end{align*}"}
+{"id": "1475.png", "formula": "\\begin{align*} S = G \\ , \\setminus \\bigcup _ { i \\in I } \\left ( E _ i M \\setminus S _ i \\right ) . \\end{align*}"}
+{"id": "5664.png", "formula": "\\begin{align*} x = \\frac { \\boldsymbol { a } _ \\perp ^ { ( 3 ) } \\cdot \\boldsymbol { v } ^ { ( 3 ) } } { \\boldsymbol { a } ^ { ( 3 ) } \\cdot \\boldsymbol { v } ^ { ( 3 ) } } , y = \\frac { ( \\boldsymbol { a } ^ { ( 3 ) } \\times \\boldsymbol { a } _ \\perp ^ { ( 3 ) } ) \\cdot \\boldsymbol { v } ^ { ( 3 ) } } { \\boldsymbol { a } ^ { ( 3 ) } \\cdot \\boldsymbol { v } ^ { ( 3 ) } } . \\end{align*}"}
+{"id": "8378.png", "formula": "\\begin{align*} \\dim ( U _ { w } ) = \\ell ( w _ { 0 , J ( w ) } ) = \\# R _ { 1 } , \\end{align*}"}
+{"id": "3354.png", "formula": "\\begin{align*} \\mathcal { J } ( u ) : = \\mathbb { E } ^ \\dagger \\left [ \\int _ { \\mathbb { R } ^ n } \\sigma _ K ( z ) \\left ( \\mu \\Phi ( z ) \\right ) d z \\right ] . \\end{align*}"}
+{"id": "1514.png", "formula": "\\begin{align*} G ( z ) = \\int _ 0 ^ { | z | } \\lambda _ { \\rm m i n } ( s ) \\ , s \\ , d s \\end{align*}"}
+{"id": "1126.png", "formula": "\\begin{align*} \\left | A _ Q \\left ( S _ { \\varphi } \\vec f \\right ) _ Q \\right | = \\left | A _ Q \\left \\langle \\vec f , \\varphi _ Q \\right \\rangle \\right | = | Q | ^ { \\frac 1 2 } \\left | A _ Q \\left ( \\widetilde { \\varphi } _ { j _ Q } * \\vec f \\right ) ( x _ Q ) \\right | \\leq \\sup _ { \\mathbb { A } , \\widetilde { \\varphi } , Q } \\left ( \\vec f \\right ) . \\end{align*}"}
+{"id": "8484.png", "formula": "\\begin{align*} \\left \\{ ( x , y , t ) \\in B \\times B \\times ( 0 , T ) : x = y \\right \\} \\end{align*}"}
+{"id": "8614.png", "formula": "\\begin{align*} v _ 1 ( t _ 1 ; x _ 2 ) = m ( t _ 1 ) < \\frac { 1 } { 2 } m ( t _ 1 ) . \\end{align*}"}
+{"id": "6964.png", "formula": "\\begin{align*} f ( \\alpha \\mathbf { z } ) & = \\lvert \\langle \\alpha \\mathbf { z } , \\overline { \\alpha } \\mathbf { \\overline { z } } \\rangle \\rvert ^ 2 - \\langle \\alpha \\mathbf { z } , \\alpha \\mathbf { z } \\rangle ^ 2 \\\\ & = \\lvert \\alpha \\rvert ^ 4 ( \\lvert \\langle \\mathbf { z } , \\mathbf { \\overline { z } } \\rangle \\rvert ^ 2 - \\langle \\mathbf { z } , \\mathbf { z } \\rangle ^ 2 ) \\\\ & = \\lvert \\alpha \\rvert ^ 4 f ( \\mathbf { z } ) . \\end{align*}"}
+{"id": "2692.png", "formula": "\\begin{align*} H _ { T } = P ^ { ( 1 ) } _ { i } Q _ { ( 2 ) } ^ { i } - g + v ^ { s } \\Phi ^ { ( 1 ) } _ { s } , \\end{align*}"}
+{"id": "2289.png", "formula": "\\begin{align*} y _ { i j k \\ell m } = 0 \\qquad z _ { i j k \\ell m p } = 0 \\ . \\end{align*}"}
+{"id": "72.png", "formula": "\\begin{align*} S = w _ 1 G D _ 1 ( p - 2 + s ) + w _ 2 G D _ 2 ( p - 2 + s ) + \\epsilon w _ 3 G D _ 1 ( p - 4 + s ) + \\epsilon w _ 4 G D _ 2 ( p - 4 + s ) \\end{align*}"}
+{"id": "230.png", "formula": "\\begin{align*} \\frac { d ^ 2 x _ 1 } { d t _ 1 ^ 2 } + \\left ( \\gamma _ 0 + \\frac { h ' } { h } \\right ) \\left ( \\frac { d x _ 1 } { d t _ 1 } \\right ) ^ 2 + \\frac { 1 } { h } A _ 0 \\frac { d x _ 1 } { d t _ 1 } + \\frac { 1 } { h ^ 2 } b _ 0 = 0 , \\end{align*}"}
+{"id": "7731.png", "formula": "\\begin{align*} \\hat \\gamma ( E ) = \\int _ { \\R } \\ln | E - E ' | d \\hat { \\mathcal N } ( E ' ) - \\ln | \\hat V _ { \\ell } | . \\end{align*}"}
+{"id": "1779.png", "formula": "\\begin{align*} h ^ 0 = h ^ 1 = f _ { 1 } ^ 0 = \\cdots = f _ { n - 1 } ^ 0 = 0 . \\end{align*}"}
+{"id": "6017.png", "formula": "\\begin{align*} T _ { v t } f \\left ( \\frac { x } { N } \\right ) = f \\left ( \\frac { x - v t } { N } \\right ) . \\end{align*}"}
+{"id": "313.png", "formula": "\\begin{align*} \\int _ { \\real ^ N } u ( x , t ) \\eta ( x , t ) \\ , d x & + \\int _ 0 ^ t \\int _ { \\real ^ N } \\left ( - u ( x , \\tau ) \\eta _ t ( x , \\tau ) - u ^ m ( x , \\tau ) \\Delta \\eta ( x , \\tau ) \\right ) d x \\ , d \\tau \\\\ & = \\int _ 0 ^ t \\int _ { \\real ^ N } ( 1 + | x | ) ^ { \\sigma } u ^ p ( x , \\tau ) \\eta ( x , \\tau ) d x \\ , d \\tau . \\end{align*}"}
+{"id": "7538.png", "formula": "\\begin{align*} - \\int _ 0 ^ T \\int _ \\Omega \\varphi _ t \\rho \\ d x d t - \\int _ 0 ^ T \\int _ \\Omega \\nabla \\varphi \\cdot ( \\rho u ) \\ d x d t - \\int _ \\Omega \\varphi \\rho \\big | _ { t = 0 } \\ d x = 0 \\ , \\end{align*}"}
+{"id": "8055.png", "formula": "\\begin{align*} & \\langle \\mathfrak { M } _ { f , 1 } ^ N , \\widetilde { \\Theta } ^ N _ { \\tilde { F } , \\tilde { G } , \\tilde { H } } \\rangle _ t = \\\\ & - \\int _ 0 ^ t \\frac { 1 } { N \\gamma _ N } \\sum _ { i = 1 } ^ N \\sum _ { j = 1 } ^ N S _ t ^ N ( i ) I _ t ^ N ( j ) f ( i / N ) \\lambda ( i / N , j / N ) \\left ( e ^ { \\frac { \\gamma _ N } { N } ( - \\tilde { F } ( i / N ) + \\tilde { G } ( i / N ) ) } - 1 \\right ) d s . \\end{align*}"}
+{"id": "5374.png", "formula": "\\begin{align*} \\bar { w } _ i ^ S = \\lim _ { \\beta \\nearrow 1 } w _ i ^ S ( \\beta ) = \\theta _ i ^ 1 \\ , 1 \\{ i \\in N ^ { \\{ 0 , 1 \\} } \\} + \\sum _ { j \\in N } ( p _ { i j } ^ 1 - p _ { i j } ^ 0 ) \\ , a ^ S _ j , i \\in N . \\end{align*}"}
+{"id": "342.png", "formula": "\\begin{align*} \\begin{aligned} I ^ { [ k ] } : I ^ { [ \\ell ] } \\ & = \\ ( I _ 0 ^ { [ k ] } : I ^ { [ \\ell ] } ) + \\ ! \\ ! \\ ! \\ ! \\ ! \\sum _ { \\substack { p \\in [ s ] \\\\ \\nu ( I _ p ) \\ge k - 1 } } \\ ! \\ ! \\ ! \\ ! ( u _ p I _ p ^ { [ k - 1 ] } : I ^ { [ \\ell ] } ) . \\end{aligned} \\end{align*}"}
+{"id": "8724.png", "formula": "\\begin{gather*} \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ \\lambda & 1 + \\lambda & 0 \\\\ 0 & 0 & 1 + \\lambda \\end{array} \\right ) \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ \\mu & 1 + \\mu & 0 \\\\ 0 & 0 & 1 + \\mu \\end{array} \\right ) = \\\\ \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ ( 1 + \\lambda ) ( 1 + \\mu ) - 1 & ( 1 + \\lambda ) ( 1 + \\mu ) & 0 \\\\ 0 & 0 & ( 1 + \\lambda ) ( 1 + \\mu ) \\end{array} \\right ) \\end{gather*}"}
+{"id": "1843.png", "formula": "\\begin{align*} \\left ( F _ { \\theta _ 1 } - \\frac { r _ 1 } { r } F _ { \\theta } \\right ) \\wedge \\psi = 0 . \\end{align*}"}
+{"id": "114.png", "formula": "\\begin{align*} \\| \\vec { u } \\| \\lesssim \\left ( \\| \\vec { u } \\| _ { H ^ { - 1 } ( \\Omega ) } ^ 2 + \\sum _ { j = 1 } ^ d \\left \\| \\frac { \\partial \\vec { u } } { \\partial x _ j } \\right \\| _ { H ^ { - 1 } ( \\Omega ) } ^ 2 \\right ) ^ { 1 / 2 } \\end{align*}"}
+{"id": "7472.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { 4 } \\sup _ { \\partial B ( o , r ) } r ^ j | ( \\nabla ^ { g _ c } ) ^ j ( F ^ * g _ N - g _ c ) | _ { g _ c } = k _ { e } ( r ) , \\end{align*}"}
+{"id": "2439.png", "formula": "\\begin{align*} & \\lim _ { \\xi \\nearrow \\xi _ { + } - 0 } \\phi ( \\xi ) = 0 , \\lim _ { \\xi \\nearrow \\xi _ { + } - 0 } | \\phi ' ( \\xi ) | = \\infty , \\\\ & \\lim _ { \\xi \\searrow \\xi _ { - } + 0 } \\phi ( \\xi ) = 0 , \\lim _ { \\xi \\searrow \\xi _ { - } + 0 } | \\phi ' ( \\xi ) | = \\infty \\end{align*}"}
+{"id": "8486.png", "formula": "\\begin{align*} \\int _ { B ^ c } \\dfrac { d y } { | x - y | ^ { n + s p } } & \\leq \\int _ { B _ d ( x ) ^ c } \\dfrac { d y } { | x - y | ^ { n + s p } } \\\\ [ 3 m m ] & = C ( n ) \\int _ d ^ \\infty \\dfrac { d \\rho } { \\rho ^ { 1 + s p } } = C ( n ) \\dfrac { d ^ { - s p } } { s p } . \\end{align*}"}
+{"id": "2106.png", "formula": "\\begin{align*} \\Bar { U } ( s , x ) - U ( s , x ) = - \\alpha \\int _ 0 ^ s \\int _ 0 ^ 1 A ( x , y ) \\left ( \\Bar { U } ( u , y ) - U ( u , y ) \\right ) \\mathrm { d } y \\mathrm { d } u , \\end{align*}"}
+{"id": "4579.png", "formula": "\\begin{align*} - 2 \\pi c _ 1 ( Y ) [ \\Sigma _ c ] + 2 \\pi c _ 1 ( Y ) [ \\Sigma _ b ] = - 2 \\pi \\sum _ { t _ j \\in [ b , c ] } c _ 1 ( Y ) [ S _ j ] . \\end{align*}"}
+{"id": "761.png", "formula": "\\begin{align*} - \\Gamma ^ h _ { i j k } \\varphi _ h y ^ k - C ^ h _ { i j k } \\varphi _ h y ^ k - \\dot { \\Phi } _ { k } g _ { i j } y ^ k = 0 . \\end{align*}"}
+{"id": "8551.png", "formula": "\\begin{align*} p ( n + 1 , k ) F ( n + 1 , k ) - p ( n , k ) F ( n , k ) = G ( n , k + 1 ) - G ( n , k ) . \\end{align*}"}
+{"id": "6420.png", "formula": "\\begin{align*} J _ n ( \\theta ) = \\begin{pmatrix} \\nabla _ \\theta G _ n ^ 1 ( \\theta ) ^ T \\\\ \\nabla _ \\theta G _ n ^ 2 ( \\theta ) ^ T \\\\ \\nabla _ \\theta G _ n ^ 3 ( \\theta ) ^ T \\\\ \\nabla _ \\theta G _ n ^ 4 ( \\theta ) ^ T \\end{pmatrix} = \\begin{pmatrix} J _ n ^ { 1 1 } ( \\theta ) & \\\\ J _ n ^ { 2 1 } ( \\theta ) & J _ n ^ { 2 2 } ( \\theta ) \\end{pmatrix} \\end{align*}"}
+{"id": "6804.png", "formula": "\\begin{align*} \\begin{cases} ( q + 1 ) ^ 2 + 4 & q \\equiv 1 \\pmod 4 \\\\ ( q + 1 ) ^ 2 & q \\equiv 3 \\pmod 4 . \\end{cases} \\end{align*}"}
+{"id": "7719.png", "formula": "\\begin{align*} \\nabla _ { j } \\widetilde { V } _ { q - 1 } ( \\Omega _ { t } , \\overline { \\nabla } h ) = \\frac { ( q - 1 ) ( n - q + 1 ) } { n } \\int _ { \\Omega _ { t } } | y - z | ^ { q - n - 3 } [ ( y - z ) \\cdot e _ { j } ] b _ { j j } d y , \\end{align*}"}
+{"id": "8676.png", "formula": "\\begin{align*} \\langle \\mu ( x ) , \\xi \\rangle = \\omega _ 0 ( \\xi _ { M ' } ( x ) ) , x \\in M ' , \\xi \\in \\mathfrak { g } . \\end{align*}"}
+{"id": "9071.png", "formula": "\\begin{align*} \\phi ( v ) = U ( x ) + U ^ { ( o f f s e t ) } = U ( x ) + \\mathbf { 1 } \\frac { \\beta } { 2 } . \\end{align*}"}
+{"id": "7197.png", "formula": "\\begin{align*} h \\left ( - \\left ( 1 + \\frac { 1 } { \\delta \\gamma } \\right ) ^ { 1 / 2 } \\right ) & = - \\left ( 1 + \\frac { 1 } { \\delta \\gamma } \\right ) ^ { 3 / 2 } + 3 \\left ( 1 - \\frac { 1 } { \\gamma } \\right ) \\left ( 1 + \\frac { 1 } { \\delta \\gamma } \\right ) ^ { 1 / 2 } + 3 \\frac { \\beta } { \\gamma } , \\\\ & = \\left ( 1 + \\frac { 1 } { \\delta \\gamma } \\right ) ^ { 1 / 2 } \\left ( 2 - \\frac { 3 + 1 / \\delta } { \\gamma } \\right ) + 3 \\frac { \\beta } { \\gamma } , \\end{align*}"}
+{"id": "8263.png", "formula": "\\begin{align*} c _ { y ^ { i - 1 } 0 } & = M P _ { Y ^ i } ( y ^ { i - 1 } 0 ) = M ( P _ { Y ^ { i - 1 } } ( y ^ { i - 1 } ) - P _ { Y ^ i } ( y ^ { i - 1 } 1 ) ) \\\\ & = c _ { y ^ { i - 1 } } - b _ { y ^ { i - 1 } } , \\ Y _ i = 0 , \\end{align*}"}
+{"id": "4842.png", "formula": "\\begin{align*} \\int _ E h d x & = \\int _ { E \\cap B _ a } h d x + \\int _ { E \\setminus B _ a } h ( | x | ) d x \\\\ & \\ge \\int _ { E \\cap B _ a } h d x + \\int _ { E \\setminus B _ a } h ( | T ( x ) | ) d x \\\\ & = \\int _ { E \\cap B _ a } h d x + \\int _ { B _ a \\setminus E } h ( | x | ) d x \\\\ & = \\int _ { B _ a } h d x . \\end{align*}"}
+{"id": "2124.png", "formula": "\\begin{align*} \\sup _ { N \\in \\mathbb { N } } \\mathbb { E } \\left [ \\frac { 1 } { N } \\sum ^ { N } _ { i = 1 } \\int ^ { T } _ { 0 } | h ^ { N } _ { i } ( t ) | ^ { 2 } \\mathrm { d } t \\right ] < \\infty . \\end{align*}"}
+{"id": "4335.png", "formula": "\\begin{align*} \\sup _ { s } & \\ \\sum _ { i \\in [ m ] } f _ i ( x , \\overline { u } ^ i ) + s _ i ( \\sup _ { u ^ i \\in \\mathcal { U } _ i } f _ i ( x , u ^ i ) - f _ i ( x , \\overline { u } ^ i ) ) , \\\\ & \\ \\sum _ { i \\in [ m ] } s _ i \\leq \\Gamma , \\\\ & \\ s \\in \\{ 0 , 1 \\} ^ m . \\end{align*}"}
+{"id": "5823.png", "formula": "\\begin{align*} \\mathcal { F } _ \\Sigma ( u ) = \\int \\sqrt { \\det g _ u } \\ , d x ^ 1 \\wedge \\dots \\wedge d x ^ n . \\end{align*}"}
+{"id": "293.png", "formula": "\\begin{align*} Q _ { j k } = \\dfrac { \\left | \\left \\langle L _ { F } e _ k , e _ j \\right \\rangle \\right | ^ 2 } { 4 \\left | \\Re \\left ( \\left \\langle L _ { F } e _ j , e _ j \\right \\rangle \\right ) \\right | \\left | \\Re \\left ( \\left \\langle L _ { F } e _ k , e _ k \\right \\rangle \\right ) \\right | b _ { j k } b _ { k j } } \\end{align*}"}
+{"id": "8432.png", "formula": "\\begin{align*} P f ( x ) = \\sum \\limits _ { j = 1 } ^ { n + 1 } f \\left ( x ^ { ( j ) } \\right ) \\lambda _ j ( x ) . \\end{align*}"}
+{"id": "8991.png", "formula": "\\begin{align*} { \\cal M } ^ + ( D ^ 2 u ) + \\bar \\lambda _ \\gamma { u \\over r ^ \\gamma } = 0 \\end{align*}"}
+{"id": "5340.png", "formula": "\\begin{align*} \\nu _ j = \\max \\ , \\left \\{ \\nu ^ { S } _ j : j \\in S \\in \\{ S _ 1 , \\ldots , S _ n \\} \\right \\} , j \\in J . \\end{align*}"}
+{"id": "4822.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ K \\min \\big \\{ S ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } } + \\sum _ { a \\in A } \\max \\{ 0 , x _ a - x ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } } _ a \\} ; h \\big \\} \\geq \\sum _ { v = 1 } ^ V \\widetilde { \\beta } _ v \\Big ( n _ v \\overline { S } _ v + ( K - n _ v ) h \\Big ) = \\sum _ { v = 1 } ^ V \\widetilde { \\beta } _ v R _ v \\geq \\min _ { v \\in \\{ 1 , \\ldots v \\} } R _ v , \\end{align*}"}
+{"id": "2931.png", "formula": "\\begin{align*} \\sum _ { T ' \\in P ( \\{ a _ 1 , a _ 2 , a _ 3 \\} ) , s \\in T ' } { g _ { T ' } } = 0 \\mbox { f o r } s \\in \\{ a _ 2 , a _ 3 , \\bar { a } _ 2 , \\bar { a } _ 3 \\} . \\end{align*}"}
+{"id": "7850.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { N - 1 } R _ { \\mathbf { a } _ i } ( \\tau ) = 0 , ~ ~ 1 \\leq \\tau \\leq M - 1 , \\end{align*}"}
+{"id": "4858.png", "formula": "\\begin{align*} f _ w ( t ) & = \\int _ 0 ^ t f ' _ w ( \\xi ) d \\xi = \\int _ 0 ^ t 2 g _ w ( \\xi ) d \\xi \\geq \\displaystyle 2 \\inf _ { w \\in \\mathbb { R } ^ n } g _ w ( s ) t \\\\ & = \\displaystyle 2 \\inf _ { w \\in \\mathbb { R } ^ n } g _ w ( s ) | y | . \\\\ \\end{align*}"}
+{"id": "9056.png", "formula": "\\begin{align*} i ^ 1 _ 1 = 1 < i ^ 2 _ 1 < \\dotsb < i ^ p _ 1 , i ^ l _ 1 = \\min \\{ i ^ l _ 1 , \\dotsc , i ^ l _ { m _ l } \\} ( l = 1 , 2 , \\dotsc , p ) . \\end{align*}"}
+{"id": "726.png", "formula": "\\begin{align*} d ( \\mathcal { G } _ 1 , \\mathcal { G } _ 2 ) & = \\left | n _ 1 - n _ 2 \\right | + \\sum _ { k \\geq 0 } \\left | [ t ^ k ] \\left ( ( - t ) ^ { n _ 1 } \\chi _ { \\mathcal { G } _ 1 } ( - t ^ { - 1 } ) - ( - t ) ^ { n _ 2 } \\chi _ { \\mathcal { G } _ 2 } ( - t ^ { - 1 } ) \\right ) \\right | . \\end{align*}"}
+{"id": "8746.png", "formula": "\\begin{align*} f ( x ) = \\frac { z - m } { z - y } \\varphi ( \\vert x - y \\vert ) + \\frac { m - y } { z - y } \\varphi ( \\vert z - x \\vert ) - \\varphi ( \\vert x - m \\vert ) . \\end{align*}"}
+{"id": "6155.png", "formula": "\\begin{align*} \\breve { x } ^ k : = x ^ { k + 1 } , ~ \\widetilde { x } ^ k : = \\bar { x } ^ { k + 1 } , ~ { \\rm a n d } ~ \\widetilde { \\lambda } ^ k : = \\lambda ^ k - \\tau ^ k \\beta ^ k ( A \\widetilde { x } ^ k - b ) . \\end{align*}"}
+{"id": "1243.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ; q ^ 2 ) _ k ( - 1 ; q ^ 2 ) _ k } { ( q ^ 2 ; q ^ 2 ) _ k } q ^ { 2 k } & \\equiv ( - 1 ) ^ { \\frac { n - 1 } { 2 } } \\pmod { \\Phi _ { n } ( q ) } , \\\\ [ 7 p t ] \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ; q ^ 2 ) _ k ( - q ^ 2 ; q ^ 2 ) _ k } { ( q ^ 2 ; q ^ 2 ) _ k } q ^ { 2 k + 1 } & \\equiv ( - 1 ) ^ { \\frac { n - 1 } { 2 } } \\pmod { \\Phi _ { n } ( q ) } , \\end{align*}"}
+{"id": "4184.png", "formula": "\\begin{align*} g _ 1 g _ 2 = g _ 2 g _ 1 g _ 1 g _ 3 = g _ 3 g _ 1 \\ , . \\end{align*}"}
+{"id": "7389.png", "formula": "\\begin{align*} \\frac { p + q - 1 } { p + q } - \\frac { r - 1 } { r + 1 } = \\frac { 2 ( p + q ) - ( r + 1 ) } { ( p + q ) ( r + 1 ) } \\ge 0 \\end{align*}"}
+{"id": "7898.png", "formula": "\\begin{align*} \\mathbf { \\Phi } _ { \\mathbf { A } _ { k } } ^ { ' } = [ \\mathbf { s } _ { k } ^ { ( 0 ) } , \\mathbf { s } _ { k } ^ { ( 1 ) } , \\cdots , \\mathbf { s } _ { k } ^ { ( p ^ { m } - 1 ) } ] . \\end{align*}"}
+{"id": "7223.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { V _ { \\min } } ^ { V _ { F } } \\left ( \\partial _ { t } p + \\partial _ { v } ( h p ) - a \\partial _ { v v } p \\right ) \\phi d v - h ( V _ R ) \\phi ( V _ R ) \\left ( p ( V _ R ^ - ) - p ( V _ R ^ + ) \\right ) \\\\ + & a \\phi ( V _ R ) \\left ( \\partial _ v p ( V _ R ^ - ) - \\partial _ v p ( V _ R ^ + ) + \\partial _ v p ( V _ F ) \\right ) - a \\partial _ v p ( V _ { \\min } ) \\phi ( V _ { \\min } ) = 0 , \\end{aligned} \\end{align*}"}
+{"id": "3167.png", "formula": "\\begin{align*} \\left ( k - 1 , \\frac { ( k - 1 ) ( n - 1 ) + i } { k } \\right ) ^ { \\sigma } = \\left ( \\left \\lfloor \\frac { ( k - 1 ) n + i } { n } \\right \\rfloor , ( ( k - 1 ) n + i ) \\bmod { n } \\right ) = ( k - 1 , i ) . \\end{align*}"}
+{"id": "8279.png", "formula": "\\begin{align*} \\dot { u } ( t ) \\in A u ( t ) + x , \\ , \\ , t > 0 , \\ , \\ , u ( 0 ) = 0 . \\end{align*}"}
+{"id": "5580.png", "formula": "\\begin{align*} \\sum _ { x \\in V _ 1 } f ( G , x ) = \\langle B ^ t \\chi _ i , B ^ t \\chi _ j \\rangle , \\end{align*}"}
+{"id": "1675.png", "formula": "\\begin{align*} \\alpha _ 1 \\big ( \\Phi _ 4 ( \\mathbf { a } , \\mathbf { b } , c _ 1 ) \\big ) & = a _ 1 , \\beta _ 1 \\big ( \\Phi _ 4 ( \\mathbf { a } , \\mathbf { b } , c _ 1 ) \\big ) = b _ 1 , \\gamma _ 1 \\big ( \\Phi _ 4 ( \\mathbf { a } , \\mathbf { b } , c _ 1 ) \\big ) = c _ 1 , \\\\ \\alpha _ 2 \\big ( \\Phi _ 4 ( \\mathbf { a } , \\mathbf { b } , c _ 1 ) \\big ) & = a _ 2 , \\beta _ 2 \\big ( \\Phi _ 4 ( \\mathbf { a } , \\mathbf { b } , c _ 1 ) \\big ) = b _ 2 . \\end{align*}"}
+{"id": "800.png", "formula": "\\begin{align*} { \\textbf S } ( x , y ) : = \\frac { d } { { d t } } \\left [ { \\tau ( \\sigma ( t ) , \\dot \\sigma ( t ) ) } \\right ] \\left | _ { t = 0 } . \\right . \\end{align*}"}
+{"id": "3207.png", "formula": "\\begin{align*} f = \\sum \\limits _ { k = 1 } ^ \\infty f _ k v _ k . \\end{align*}"}
+{"id": "8853.png", "formula": "\\begin{align*} 0 < \\frac { 1 } { ( p - 1 ) a _ 1 } \\leq \\frac 1 r \\leq 1 , 0 \\leq \\frac { \\tau + 1 } { p - 1 } < \\frac { n } { ( p - 1 ) a _ 1 } , \\frac { \\tau + 1 } { p - 1 } - 1 = \\frac { n } { ( p - 1 ) a _ 1 } - \\frac { n } { r } \\end{align*}"}
+{"id": "320.png", "formula": "\\begin{align*} z ( x , 0 ) = T _ 0 ^ { - \\alpha } f ( ( 1 + | x | ) T _ 0 ^ { \\beta } ) \\geq \\| u _ 0 \\| _ { \\infty } \\geq u _ 0 ( x ) \\end{align*}"}
+{"id": "1462.png", "formula": "\\begin{align*} L _ i = \\{ ( x , y ) \\in \\Z ^ 2 \\colon a _ i x + b _ i y + c _ i = 0 \\} . \\end{align*}"}
+{"id": "8741.png", "formula": "\\begin{align*} A = \\left \\{ x \\in \\R : \\pi _ x \\left ( \\Gamma _ x \\cap ( - \\infty , x ) \\right ) \\times \\pi _ x \\left ( \\Gamma _ x \\cap ( x , + \\infty ) \\right ) = 0 \\right \\} , \\end{align*}"}
+{"id": "8397.png", "formula": "\\begin{align*} B _ 1 ( a ) B _ 1 ( b ) = B _ 1 ( B _ 1 ( a _ 1 ) b S ( B _ 2 ( a _ 2 ) ) ) , \\\\ B _ 2 ( a ) B _ 2 ( b ) = B _ 2 ( B _ 1 ( a _ 1 ) b S ( B _ 2 ( a _ 2 ) ) ) . \\end{align*}"}
+{"id": "5703.png", "formula": "\\begin{align*} q ' = \\mathbf { L } q + \\mathcal { E } . \\end{align*}"}
+{"id": "3636.png", "formula": "\\begin{align*} ( 1 + a ) \\alpha ^ 2 \\tau _ 0 ( z _ 0 ) ^ 2 + b = 0 . \\end{align*}"}
+{"id": "3915.png", "formula": "\\begin{align*} \\begin{aligned} \\cos n z = \\frac { ( \\cos z + i \\sin z ) ^ n + ( \\cos z - i \\sin z ) ^ n } { 2 } , \\\\ \\sin n z = \\frac { ( \\cos z + i \\sin z ) ^ n - ( \\cos z - i \\sin z ) ^ n } { 2 i } . \\end{aligned} \\end{align*}"}
+{"id": "3963.png", "formula": "\\begin{align*} [ \\alpha ] ( T ) ( x , z ) = ( T ( x ) , \\alpha ( x , T ( x ) ) z ) . \\end{align*}"}
+{"id": "3837.png", "formula": "\\begin{align*} S _ { 6 1 } ^ { [ 1 ] } = \\sum _ { \\substack { 1 \\le k \\le N \\\\ 6 \\mid k } } ( - 1 ) ^ { \\frac k 2 + 1 } \\sum _ { m \\ge 0 } a ( m ) K _ k ^ { [ 1 ] } ( n , m ) \\int _ { - \\frac { 1 } { k ( k + N ) } } ^ \\frac { 1 } { k ( k + N ) } e ^ { \\frac { 2 \\pi } { k } \\left ( n z - \\frac m z \\right ) } d \\Phi . \\end{align*}"}
+{"id": "871.png", "formula": "\\begin{align*} S _ 1 = w ^ 1 L _ { 1 1 } + w ^ 2 L _ { 2 1 } = 2 w ^ 1 { w ^ 1 } _ { \\substack { \\\\ x _ 1 } } + w ^ 2 ( { w ^ 1 } _ { \\substack { \\\\ x _ 2 } } + { w ^ 2 } _ { \\substack { \\\\ x _ 1 } } ) . \\end{align*}"}
+{"id": "6977.png", "formula": "\\begin{align*} 0 = f ( \\mathbf { z } ) = \\lvert \\eta ( \\mathbf { z } ) \\rvert ^ 4 - \\langle \\mathbf { z } , \\mathbf { z } \\rangle ^ 2 , \\end{align*}"}
+{"id": "3429.png", "formula": "\\begin{align*} e _ G ( E ) = e ( E G \\times _ G E \\to E G \\times _ G B ) \\in H _ G ^ \\ast ( B ) . \\end{align*}"}
+{"id": "4720.png", "formula": "\\begin{align*} \\nu _ h \\ ; = \\ ; \\gamma ^ { - 2 h } \\widehat { \\mathcal { V } } _ 0 ^ { ( h ) } ( 0 , \\underline { \\nu } ) \\end{align*}"}
+{"id": "6337.png", "formula": "\\begin{align*} | \\partial _ q g ( p , q ) | = \\left | \\frac { 1 } { e ^ { \\sqrt { p ^ 4 + q p ^ 2 } } - 1 } \\frac { p ^ 2 } { 2 \\sqrt { p ^ 4 + q p ^ 2 } } \\right | \\leq \\frac { 1 } { 2 } \\frac { 1 } { e ^ { p ^ 2 } - 1 } . \\end{align*}"}
+{"id": "1822.png", "formula": "\\begin{align*} g _ 0 ( u \\times v , u ) = 0 | u \\times v | ^ 2 = | u | ^ 2 | v | ^ 2 - g _ 0 ( u , v ) ^ 2 , \\forall \\ ; u , v \\in V . \\end{align*}"}
+{"id": "4838.png", "formula": "\\begin{align*} a = \\begin{pmatrix} a ^ { ( 1 ) } & \\mathbf { 1 } \\\\ \\mathbf { 0 } & a ^ { ( 2 ) } \\end{pmatrix} \\end{align*}"}
+{"id": "6828.png", "formula": "\\begin{align*} d g _ { S O ( N ) } = C _ N \\prod _ { j = 1 } ^ { N - 1 } \\sin ^ { j - 1 } ( \\phi _ j ) d \\phi _ 1 \\ldots d \\phi _ { N - 1 } d g _ { S O ( N - 1 ) } , \\end{align*}"}
+{"id": "7661.png", "formula": "\\begin{align*} \\varphi ( m ) = \\sup _ { \\Lambda \\subset \\mathbb { Z } ^ d } \\mathbb { E } \\left ( \\abs { G ^ { \\Lambda } ( m , n ; z ) } ^ s \\right ) , \\ , \\ , \\ , W ( m ) = e ^ { \\nu ' | m - n | } , \\end{align*}"}
+{"id": "2865.png", "formula": "\\begin{align*} E ( f , \\gamma ) & = \\left \\{ ( x , y ) \\in E ( f , \\gamma ) : \\ x > y \\right \\} \\cup \\left \\{ ( x , y ) \\in E ( f , \\gamma ) : \\ x < y \\right \\} \\\\ & = : E _ + ( f , \\gamma ) \\cup E _ - ( f , \\gamma ) . \\end{align*}"}
+{"id": "5535.png", "formula": "\\begin{align*} Y : = \\{ u \\in \\Lambda \\mid ( j ( u ) , k ( u ) ) = ( j ^ * , k ^ * ) \\} \\end{align*}"}
+{"id": "7183.png", "formula": "\\begin{align*} F _ v ^ { ( 1 ) } - F _ v ^ { ( 0 ) } = \\int _ 0 ^ t e ^ { ( 1 - v _ 0 ^ 2 ) ( t - \\tau ) } g _ 1 ( t - \\tau ) * \\varphi _ 3 d \\tau , \\end{align*}"}
+{"id": "9081.png", "formula": "\\begin{align*} c _ { i l } = 0 , \\mbox { f o r a l l $ i , l \\in N $ : $ j ^ { ( r ) } _ { i i } \\neq j ^ { ( r ) } _ { l l } $ } . \\end{align*}"}
+{"id": "5271.png", "formula": "\\begin{align*} x ^ { k + 1 } _ { j } = x ^ { k } _ { j } + \\alpha ^ { k } _ { j } x ^ { k } _ { j } \\left ( \\frac { U ^ { k } _ { j } } { V ^ { k } _ { j } } - 1 \\right ) \\end{align*}"}
+{"id": "155.png", "formula": "\\begin{align*} X _ N ( \\varphi ) \\equiv \\sum _ { i = 1 } ^ N \\varphi ( z _ i ) \\end{align*}"}
+{"id": "3700.png", "formula": "\\begin{align*} \\gamma : = \\min \\left ( \\beta _ { k } ^ - , \\ \\ \\min _ { 1 \\leq i < k - 1 } ( \\beta _ i ^ - - \\beta _ { i + 1 } ^ + ) \\right ) \\sqrt { d - 1 } \\geq 4 k d \\epsilon _ 0 ^ { 1 / 4 ^ k } . \\end{align*}"}
+{"id": "7099.png", "formula": "\\begin{align*} \\mathbb E _ x ^ i u _ n ( T \\wedge \\tau _ R , \\omega _ { T \\wedge \\tau _ R } ) & = u _ n ( 0 , x ) + \\mathbb E _ x ^ i \\int _ 0 ^ { T \\wedge \\tau _ R } f ( t , \\omega _ t ) d t \\\\ & + \\mathbb E _ x ^ i \\int _ 0 ^ { T \\wedge \\tau _ R } \\big [ ( b - b _ n ) \\cdot \\nabla u _ n \\big ] ( t , \\omega _ t ) d t \\end{align*}"}
+{"id": "1716.png", "formula": "\\begin{align*} H = \\left ( \\begin{array} { c c c } A _ 1 & 0 & B _ 1 \\\\ 0 & A _ 2 & B _ 2 \\\\ B _ 1 ^ * & B _ 2 ^ * & C _ 1 + C _ 2 \\end{array} \\right ) . \\end{align*}"}
+{"id": "6629.png", "formula": "\\begin{align*} H _ N S _ N ( i , j ) & = \\frac { 1 } { J _ N } ( - J _ { i - 2 } J _ { N - j } ( J _ { j - i + 1 } + J _ { j - i + 2 } ) - J _ { i - 1 } J _ { N - j } ( J _ { j - i } + J _ { j - i + 1 } ) \\\\ & ~ + 2 J _ i J _ { N - i - 1 } ) \\\\ & = \\frac { 1 } { J _ N } ( - J _ { j - 3 } J _ { N - j } ( J _ 2 + J _ 3 ) - J _ { j - 2 } J _ { N - j } ( J _ 1 + J _ 2 ) + 2 J _ { j - 1 } J _ { N - j } ) \\\\ & = \\frac { 1 } { J _ N } ( - 4 J _ { j - 3 } J _ { N - j } - 2 J _ { j - 2 } J _ { N - j } + 2 J _ { j - 1 } J _ { N - j } ) \\\\ & = \\frac { 2 J _ { N - j } } { J _ N } ( - 2 J _ { j - 3 } - J _ { j - 2 } + J _ { j - 1 } ) \\\\ & = 0 . \\end{align*}"}
+{"id": "8347.png", "formula": "\\begin{align*} h _ { \\Pi _ { \\mu } K } ( \\theta ) & = \\frac { 1 } { 2 } \\int _ { \\partial K } | \\langle \\theta , n _ K ( y ) \\rangle | \\phi ( y ) \\ , d y - \\frac { 1 } { 2 } \\int _ { \\partial K } \\langle n _ K ( y ) , \\theta \\rangle \\phi ( y ) \\ , d y \\\\ & = \\int _ { \\partial K } \\langle n _ K ( y ) , \\theta \\rangle _ { - } \\phi ( y ) \\ , d y , \\end{align*}"}
+{"id": "178.png", "formula": "\\begin{align*} a _ i , b _ i \\in & U _ { i - 1 } , \\overline { U _ { i } } \\subset U _ { i - 1 } , \\\\ \\{ a _ i , z , b _ i \\} \\ \\ & 3 - \\ z \\in \\overline { U _ i } . \\end{align*}"}
+{"id": "6867.png", "formula": "\\begin{align*} \\langle A _ 0 f , g \\rangle = \\langle H _ { e { \\sf E } _ 0 } S _ 0 ^ { 1 / 2 } f , S _ 0 ^ { 1 / 2 } g \\rangle , f , g \\in \\mathcal { D } ( A _ 0 ) . \\end{align*}"}
+{"id": "1069.png", "formula": "\\begin{align*} \\sum _ { s = 0 } ^ { n - 1 } \\{ S _ { 1 , k } ( n , s , t ) + S _ { 2 , k } ( n , s , t ) \\} \\\\ \\leq \\frac { 2 K _ 3 \\| \\phi \\| _ L \\{ r \\sin ( \\pi d ) \\} ^ { k - 1 } n ^ { 1 - d } } { ( 1 - 2 d ) ^ { 1 / 2 } ( t + 1 ) ^ { d } ( n - t ) } . \\end{align*}"}
+{"id": "4970.png", "formula": "\\begin{align*} \\left [ B ( \\lambda ) , B ( \\mu ) \\right ] = 0 , \\end{align*}"}
+{"id": "7551.png", "formula": "\\begin{align*} \\bar n = \\bar \\rho \\ , \\bar v = \\bar u \\ , \\bar \\phi = h _ 2 ^ \\prime ( \\bar \\rho ) \\ . \\end{align*}"}
+{"id": "2151.png", "formula": "\\begin{align*} \\begin{array} { l l } \\vartheta _ J = - 6 . 8 9 9 5 , & u _ J = ( 1 . 0 0 0 0 , 1 . 0 0 0 0 , 0 . 3 6 6 0 , - 1 . 3 6 6 0 ) , \\\\ & v _ J = ( 0 . 5 0 0 0 , - 0 . 5 0 0 0 , 0 . 0 0 0 0 , 3 . 0 0 0 0 ) . \\end{array} \\end{align*}"}
+{"id": "7954.png", "formula": "\\begin{align*} \\boldsymbol { n } ( u ) = ( - 1 ) ^ { n - 1 } \\ast \\partial \\Sigma = ( - 1 ) ^ { n - 1 } \\partial \\Sigma _ { v } ( \\ast v _ { \\Sigma } ) = ( \\partial \\Sigma _ { v } ) ^ { \\flat } \\quad \\Sigma . \\end{align*}"}
+{"id": "2834.png", "formula": "\\begin{align*} & - \\dot { P } _ { 1 } - Q ^ { 1 } = 0 , \\\\ & - P _ { 1 } + \\dot { Q } ^ { 1 } = 0 . \\end{align*}"}
+{"id": "1213.png", "formula": "\\begin{align*} F _ { \\infty } ( u , v ) \\leq \\lim _ { p \\to \\infty } \\inf \\Lambda _ 1 ( p ) ^ { \\frac { 1 } { p } } = \\frac { 1 } { R ^ { s \\theta + ( 1 - \\theta ) t } } = \\max \\{ \\vert u _ \\infty \\vert _ s , \\vert v _ \\infty \\vert _ t \\} , \\end{align*}"}
+{"id": "7983.png", "formula": "\\begin{align*} 0 = \\delta d \\tilde { f } = \\delta d \\hat { f } , \\end{align*}"}
+{"id": "5549.png", "formula": "\\begin{align*} \\langle \\zeta _ i ^ R , \\phi _ i \\rangle = \\sqrt { 1 - \\tau _ i ^ 4 } + o ( 1 ) , \\langle \\zeta _ i ^ L , \\phi _ i \\rangle = \\sqrt { 1 - \\tau _ i ^ 4 } + o ( 1 ) . \\end{align*}"}
+{"id": "3333.png", "formula": "\\begin{align*} 0 = \\iiint \\limits _ { \\O _ { v } \\times \\O _ { x t } } A ( v , x , t ) \\nabla _ v u \\cdot \\nabla _ v \\phi \\ , \\dd v \\ , \\dd x \\ , \\dd t + \\iint \\limits _ { \\O _ { x t } } \\langle f ( \\cdot , x , t ) - Y u ( \\cdot , x , t ) | \\phi ( \\cdot , x , t ) \\rangle \\ , \\dd x \\ , \\dd t \\end{align*}"}
+{"id": "8704.png", "formula": "\\begin{align*} B _ G \\tau & = B _ G P ( P ^ \\ast P ) ^ { - 1 } S _ G P ^ \\ast \\tau \\\\ & = B _ G P ( P ^ \\ast P ) ^ { - 1 } S _ G \\tilde { \\tau } P ^ \\ast \\tau + B _ G P ( P ^ \\ast P ) ^ { - 1 } S _ G ( 1 - \\tilde { \\tau } ) P ^ \\ast \\tau . \\end{align*}"}
+{"id": "2103.png", "formula": "\\begin{align*} P _ 4 ^ s & = \\gamma \\sum _ { t = 1 } ^ { \\lfloor s T \\rfloor - 1 } f ( \\frac { t } { T } ) O ( \\eta + d ^ { - 1 } T ^ { - 1 } + T ^ { - 2 } ) = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { \\lfloor s T \\rfloor - 1 } f ( \\frac { t } { T } ) O ( d ^ { - 1 } \\gamma + T ^ { - 1 } \\gamma ) . \\end{align*}"}
+{"id": "2665.png", "formula": "\\begin{align*} S ^ { ( 2 ) } = \\int ^ { t _ { 2 } } _ { t _ { 1 } } L ^ { ( 2 ) } ( \\ddot { q } ^ { i } , \\dot { q } ^ { i } , q ^ { i } , t ) d t , \\end{align*}"}
+{"id": "8215.png", "formula": "\\begin{align*} \\sum _ { i = t } ^ L H ( Y _ i | { \\bf S } , y ^ { t - 1 } , Y _ t ^ { i - 1 } ) . \\end{align*}"}
+{"id": "381.png", "formula": "\\begin{align*} z = K _ 3 - ( x - a ) , \\end{align*}"}
+{"id": "3737.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial a } \\mathrm { B } _ { z } \\left ( a , b \\right ) = \\ln z \\ , \\mathrm { B } _ { z } \\left ( a , b \\right ) - \\frac { z ^ { a } } { a ^ { 2 } } \\ , _ { 3 } F _ { 2 } \\left ( \\left . \\begin{array} { c } 1 - b , a , a \\\\ a + 1 , a + 1 \\end{array} \\right \\vert z \\right ) . \\end{align*}"}
+{"id": "5791.png", "formula": "\\begin{align*} x _ j ( t ) : = \\xi _ { \\iota + j } ( t ) . \\end{align*}"}
+{"id": "2110.png", "formula": "\\begin{align*} \\Bar { U } ( s , x ) - U ( s , x ) = - \\alpha \\int _ 0 ^ s \\int _ 0 ^ 1 A ( x , y ) \\left ( \\Bar { U } ( u , y ) - U ( u , y ) \\right ) \\mathrm { d } y \\mathrm { d } u , \\end{align*}"}
+{"id": "6390.png", "formula": "\\begin{align*} L _ n ( \\theta ) = \\sum _ { i = 1 } ^ n \\log \\left ( \\frac { n ^ { 1 / \\alpha } } { \\delta X _ { \\frac { i - 1 } { n } } ^ { 1 / \\alpha } } \\varphi _ \\alpha \\left ( n ^ { 1 / \\alpha } \\frac { X _ { \\frac { i } { n } } - X _ { \\frac { i - 1 } { n } } - \\frac { a } { n } + \\frac { b } { n } X _ { \\frac { i - 1 } { n } } } { \\delta X _ { \\frac { i - 1 } { n } } ^ { 1 / \\alpha } } \\right ) \\right ) , \\end{align*}"}
+{"id": "5377.png", "formula": "\\begin{align*} \\bar { c } _ i ^ S = \\lim _ { \\beta \\nearrow 1 } c _ i ^ S ( \\beta ) = h _ i ^ 0 - h _ i ^ 1 + \\sum _ { j \\in N } ( p _ { i j } ^ 0 - p _ { i j } ^ 1 ) \\ , f ^ S _ j , i \\in N . \\end{align*}"}
+{"id": "4176.png", "formula": "\\begin{align*} \\varphi ( g ^ { n } _ 1 g _ 2 ) \\overset { { \\bf A 3 } } { = } \\varphi ( g _ 2 ) \\varphi ( g ^ { n } _ 1 ) \\overset { { \\bf A 3 } } { = } \\varphi ( g _ 2 ) \\varphi \\big ( g ^ { j } _ 1 \\big ) \\varphi \\big ( g ^ { i } _ 1 \\big ) \\overset { ( \\ref { e q : 9 } ) } { = } \\varphi \\big ( g _ 2 g ^ { j } _ 1 \\big ) \\varphi \\big ( g ^ { i } _ 1 \\big ) \\ , , \\end{align*}"}
+{"id": "6835.png", "formula": "\\begin{align*} \\cos ( \\omega _ { N - 1 } - \\omega _ { N - 1 } ' + \\xi ) \\sin ^ 2 ( \\psi _ { N - 1 } ) + \\cos ( ( N - 1 ) ( \\omega _ { N - 1 } - \\omega _ { N - 1 } ' ) ) \\cos ^ 2 ( \\psi _ { N - 1 } ) & = 1 , \\\\ \\sin ( \\omega _ { N - 1 } - \\omega _ { N - 1 } ' + \\xi ) \\sin ^ 2 ( \\psi _ { N - 1 } ) - \\sin ( ( N - 1 ) ( \\omega _ { N - 1 } - \\omega _ { N - 1 } ' ) ) \\cos ^ 2 ( \\psi _ { N - 1 } ) & = 0 . \\end{align*}"}
+{"id": "3794.png", "formula": "\\begin{align*} Z Z ^ \\top \\succeq \\frac { 1 } { 4 } \\sum _ { i \\in \\widehat { \\mathcal { C } } _ 1 } N _ i \\sum _ { t = 0 } ^ { T - 1 } { \\Sigma } _ { t } ^ { ( i ) } , \\end{align*}"}
+{"id": "8219.png", "formula": "\\begin{align*} P _ { { \\bf X } _ i | Y ^ { i - 1 } } ( { \\bf x } _ i | y ^ { t - 1 } , y _ t ^ { i - 1 } ) & = P _ { { \\bf X } _ i | Y ^ { i - 1 } } ( { \\bf x } _ i | { \\bf a } _ { t - 1 } , y _ t ^ { i - 1 } ) \\\\ & = P _ { { \\bf X } _ i | Y ^ { i - 1 } } ( { \\bf x } _ i | \\tilde { { \\bf a } } _ { t - 1 } , y _ t ^ { i - 1 } ) \\\\ & = P _ { { \\bf X } _ i | Y ^ { i - 1 } } ( { \\bf x } _ i | \\tilde { y } ^ { t - 1 } , y _ t ^ { i - 1 } ) . \\end{align*}"}
+{"id": "6642.png", "formula": "\\begin{align*} [ \\Gamma ] _ * = ( h _ X ^ { 2 n - 2 } \\cup \\bullet ) \\circ \\varphi \\circ \\psi \\colon T ( X ' ) \\rightarrow H ^ { 4 n - 2 } ( X , \\Q ) . \\end{align*}"}
+{"id": "5802.png", "formula": "\\begin{align*} X _ - ( t ) Y _ - ( t ) = \\sum _ { k + \\ell \\leq s } \\sum _ { i : \\lambda _ i < 0 } \\xi ^ { ( k , \\ell ) } _ i ( t ) \\mathcal { E } ^ { ( k , \\ell ) } _ i ( t ) . \\end{align*}"}
+{"id": "2635.png", "formula": "\\begin{align*} ( i \\partial _ t + H _ N ) u ( t , x _ 1 , \\ldots , x _ N ) = 0 , u ( 0 , x _ 1 , \\ldots , x _ N ) = u _ 0 ( x _ 1 , \\ldots , x _ N ) \\in L ^ 2 _ { x _ 1 , \\ldots , x _ N } , \\end{align*}"}
+{"id": "1185.png", "formula": "\\begin{align*} \\left | f ( y ) - \\operatorname { T a y l } _ z ^ N f ( y ) \\right | & \\leq \\sum _ { \\alpha \\in \\mathbb { Z } _ + ^ n , \\ , | \\alpha | = N } \\frac { | ( y - z ) ^ \\alpha | } { \\alpha ! } \\int _ 0 ^ 1 S | t ( y - z ) | ^ s N ( 1 - t ) ^ { N - 1 } \\ , d t \\\\ & \\lesssim S | y - z | ^ { N + s } , \\end{align*}"}
+{"id": "1885.png", "formula": "\\begin{align*} \\mathcal { M } _ { 2 , v } ^ { | \\cdot | , u } ( K , z ) = \\int _ { K \\cap H _ { u , z } } | x \\cdot v | ^ 2 d x , \\end{align*}"}
+{"id": "1336.png", "formula": "\\begin{align*} & A _ \\alpha ^ 2 z - \\left \\lbrace 2 r \\alpha + ( 1 - \\alpha ) ( a - c ) \\right \\rbrace A _ \\alpha z - \\left \\lbrace ( r - c ) ( 1 - \\alpha ) ^ 2 - r \\alpha ( 1 - \\alpha ) ( a - c ) - r ^ 2 \\alpha ^ 2 \\right \\rbrace I z \\\\ & = c ( 1 - \\alpha ) ^ 2 J z = 0 \\end{align*}"}
+{"id": "7445.png", "formula": "\\begin{align*} \\big [ \\xi \\theta \\big ] ^ { i j } _ { k \\ell } = \\sum _ { \\alpha , \\beta = 1 } ^ d \\xi ^ { i j } _ { \\alpha \\beta } \\theta ^ { \\alpha \\beta } _ { k \\ell } . \\end{align*}"}
+{"id": "971.png", "formula": "\\begin{align*} \\mathbb E _ x ( A ^ \\mu _ { \\tau _ D } ) ^ 2 = 2 \\mathbb E _ x \\int _ 0 ^ { \\tau _ D } R ^ D \\mu ( X _ t ) \\ , d A ^ \\mu _ t \\le 2 \\| R ^ D \\mu \\| _ \\infty \\mathbb E _ x A ^ \\mu _ { \\tau _ D } \\le 2 \\| R ^ D \\mu \\| ^ 2 _ \\infty , \\end{align*}"}
+{"id": "8809.png", "formula": "\\begin{align*} \\pi & = \\underline \\pi ^ \\star + p s \\delta _ { x _ - } \\otimes \\left ( \\check \\tau - \\hat \\tau \\right ) + p s \\delta _ { x _ + } \\otimes \\left ( \\hat \\tau - \\check \\tau \\right ) , \\end{align*}"}
+{"id": "7834.png", "formula": "\\begin{align*} \\langle R ^ N ( X , Y ) Z , W \\rangle = \\langle R ( X , Y ) Z , W \\rangle + \\langle B ( X , Z ) , B ( Y , W ) \\rangle - \\langle B ( X , W ) , B ( Y , Z ) \\rangle , \\end{align*}"}
+{"id": "6627.png", "formula": "\\begin{align*} H _ N S _ N ( i , j ) & = \\frac { 1 } { J _ N } ( - J _ 1 J _ { N - 3 } ( J _ 1 + J _ 2 ) + 2 J _ 2 J _ { N - 3 } ) \\\\ & = \\frac { 1 } { J _ N } ( - 2 J _ { N - 3 } + 2 J _ { N - 3 } ) \\\\ & = 0 . \\end{align*}"}
+{"id": "9013.png", "formula": "\\begin{align*} x _ { k k } = 0 x _ { i j } = z _ { i j } ^ k = 0 1 \\le i < j \\le n 1 < k < n . \\end{align*}"}
+{"id": "1152.png", "formula": "\\begin{align*} \\begin{cases} r ^ * : = r - \\lfloor r \\rfloor \\in [ 0 , 1 ) , \\\\ r ^ { * * } : = r - \\lfloor \\ ! \\lfloor r \\rfloor \\ ! \\rfloor \\in ( 0 , 1 ] . \\end{cases} \\end{align*}"}
+{"id": "3720.png", "formula": "\\begin{align*} \\begin{pmatrix} - 1 & - 1 \\\\ 1 - d & - d \\end{pmatrix} . \\end{align*}"}
+{"id": "7918.png", "formula": "\\begin{align*} \\int _ { \\Omega } d f _ { 1 } \\wedge \\ast \\delta ( \\mu \\wedge d f _ { 2 } ) = \\int _ { \\partial \\Omega } \\big ( \\langle d f _ { 1 } , \\mu \\rangle _ { \\Lambda ^ { 1 } } i _ { \\mathcal { N } } d f _ { 2 } \\big ) v _ { \\partial \\Omega } . \\end{align*}"}
+{"id": "7296.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { 3 k - 1 } \\csc ^ { 4 } \\left ( \\frac { j \\pi } { 3 k } \\right ) \\cos \\left ( \\frac { 2 \\pi j } { 3 } \\right ) = - \\frac { 1 } { 4 5 } \\left ( 3 9 k ^ 4 + 3 0 k ^ 2 + 1 1 \\right ) \\end{align*}"}
+{"id": "284.png", "formula": "\\begin{align*} e _ { 2 k + 1 } e _ { 2 l + 1 } = C _ { k + l } ^ { l } e _ { 2 k + 2 l + 2 } . \\end{align*}"}
+{"id": "6608.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { b - i \\tau } ^ { b + i \\tau } \\left ( \\sum _ { n \\le x } \\left ( \\frac { \\Phi _ k ( n ) } { n ^ \\beta } \\right ) ^ { - z } \\right ) \\frac { y ^ z } { z } \\ , d z & = \\frac { x } { 2 \\pi i } \\int _ { b - i \\tau } ^ { b + i \\tau } R _ { k , \\beta } ( z ) \\frac { ( \\alpha x ^ \\delta ) ^ { z } } { z ( 1 - z + \\delta z ) } d z \\\\ & + O \\left ( x ^ { 1 + \\delta b } \\alpha ^ { b } \\exp { ( - \\sqrt { f _ k ^ { \\prime \\prime \\prime } \\log x \\log \\log x } } ) \\right ) . \\end{align*}"}
+{"id": "8442.png", "formula": "\\begin{align*} \\begin{cases} \\ , \\ , \\partial _ t ( | u | ^ { q - 1 } u ) + ( - \\Delta ) _ p ^ s u = 0 & \\textrm { i n } \\Omega _ \\infty : = \\Omega \\times ( 0 , \\infty ) , \\\\ \\ , \\ , u = 0 & \\textrm { o n } \\ ! \\ ! ( \\mathbb { R } ^ n \\setminus \\Omega ) \\times ( 0 , \\infty ) , \\\\ \\ , \\ , u ( \\cdot , 0 ) = u _ 0 ( \\cdot ) & \\textrm { i n } \\Omega , \\end{cases} \\end{align*}"}
+{"id": "2029.png", "formula": "\\begin{align*} \\Delta ^ { \\Sigma } \\ , x _ i = 0 \\forall i = 1 , . . . , m . \\end{align*}"}
+{"id": "3623.png", "formula": "\\begin{align*} R ^ 2 f ( z , w ) = \\frac { 2 } { ( \\lambda _ { 2 } ^ { 2 } - 1 ) ^ 2 } ( f ( z , w ) - f ( z , - w ) ) , \\end{align*}"}
+{"id": "7713.png", "formula": "\\begin{align*} 0 = \\nabla _ { i } \\widetilde { E } & = b ^ { 1 1 } \\nabla _ { i } b _ { 1 1 } - \\tilde { d } \\frac { h _ { i } } { h } + 2 \\tilde { l } \\sum _ { j } h _ { j } h _ { j i } \\\\ & = b ^ { 1 1 } ( b _ { 1 i 1 } ) - \\tilde { d } \\frac { h _ { i } } { h } + 2 \\tilde { l } h _ { i } h _ { i i } , \\end{align*}"}
+{"id": "8800.png", "formula": "\\begin{align*} \\psi _ \\rho ( \\alpha ) & = \\alpha + ( 1 - \\alpha ) \\left ( 1 + ( 2 - \\rho ) \\ln { \\frac { \\alpha } { 1 - \\alpha } } \\right ) + 1 - \\rho + { \\mathcal O } \\left ( ( 2 - \\rho ) ^ 2 \\right ) = ( 2 - \\rho ) \\left ( h ( \\alpha ) + { \\mathcal O } ( 2 - \\rho ) \\right ) . \\end{align*}"}
+{"id": "1216.png", "formula": "\\begin{align*} \\liminf _ { p \\to \\infty } F _ { p } ( u _ p , v _ p ) & \\geq \\max \\left \\{ \\vert u \\vert _ s , \\vert v \\vert _ t \\right \\} = F _ { \\infty } ( u , v ) , \\end{align*}"}
+{"id": "2493.png", "formula": "\\begin{align*} - \\Delta \\mathcal { K } [ z ] = z \\ ; \\ ; \\ ; \\ ; \\Omega \\ , , \\nabla \\mathcal { K } [ z ] \\cdot \\mathbf { n } = 0 \\ ; \\ ; \\ ; \\ ; \\partial \\Omega \\ , , \\end{align*}"}
+{"id": "1633.png", "formula": "\\begin{align*} \\varphi _ { \\lambda , k } \\left ( t \\right ) = \\exp \\left ( \\lambda \\left ( t + b \\right ) ^ { k } \\right ) , t \\in \\left ( 0 , T \\right ) . \\end{align*}"}
+{"id": "8894.png", "formula": "\\begin{align*} \\Phi _ n ( x ) : = \\prod _ { \\substack { 0 < k \\leq n \\\\ ( k , n ) = 1 } } \\left ( x - e ^ { 2 \\pi i k / n } \\right ) \\end{align*}"}
+{"id": "6329.png", "formula": "\\begin{align*} \\langle g , B _ c f \\rangle = n ^ { - 1 / 2 } \\int _ { \\Lambda ^ 2 } \\overline { ( Q ^ { \\otimes 2 } g ) } ( x _ 1 , x _ 2 ) K ( x _ 1 , x _ 2 ) ( Q f ) ( x _ 2 ) \\dd x _ 1 \\dd x _ 2 , \\end{align*}"}
+{"id": "557.png", "formula": "\\begin{align*} H ^ * _ \\Q | [ X / / G ] | = H ^ 0 ( G , H ^ * _ \\Q ( X ) ) \\end{align*}"}
+{"id": "4624.png", "formula": "\\begin{align*} a _ { n , k } : = \\underset { x ' \\in Q \\left ( n , k \\right ) } { \\sup } f _ { n - 1 } ( x ' ) . \\end{align*}"}
+{"id": "3527.png", "formula": "\\begin{align*} { H } _ { 1 } ^ { \\infty } ( \\mathbb { D } ) = \\{ f \\in { H } ^ { \\infty } ( \\mathbb { D } ) : f ' ( 0 ) = 0 \\} . \\end{align*}"}
+{"id": "7609.png", "formula": "\\begin{align*} \\| u _ { \\epsilon } \\| _ { q } ^ { q } & = R ^ { q } \\omega \\epsilon ^ { \\frac { N } { 2 } } \\int _ { 0 } ^ { 2 } \\frac { ( \\varphi ^ { q } ( r ) - 1 ) r ^ { N - 1 } } { ( \\epsilon ^ { 2 } + r ^ { 2 } ) ^ { \\frac { N } { 2 } } } d r + R ^ { q } \\omega \\epsilon ^ { \\frac { N } { 2 } } \\int _ { 0 } ^ { 2 } \\frac { r ^ { N - 1 } } { ( \\epsilon ^ { 2 } + r ^ { 2 } ) ^ { \\frac { N } { 2 } } } d r \\\\ & = I _ { 1 } ( \\epsilon ) + I _ { 2 } ( \\epsilon ) . \\\\ \\end{align*}"}
+{"id": "5890.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { C } ( G ) ) = 2 n ( 2 n - 1 ) ^ { 2 } + 3 n ( n - 1 ) ^ { 2 } , M _ { 2 } ( \\mathcal { C } ( G ) ) = \\frac { 2 n ( 2 n - 1 ) ^ { 3 } + 3 n ( n - 1 ) ^ { 3 } } { 2 } , M _ { 1 } ( \\mathcal { N C } ( G ) ) = 6 6 n ^ { 3 } \\end{align*}"}
+{"id": "3500.png", "formula": "\\begin{align*} \\rho _ k ( x ) & = [ ( \\rho _ { m , k } ( x ) ) _ { m > k } ] \\\\ & = [ ( \\psi _ m ( \\varphi _ { m - 1 } ( \\rho _ { m - 1 , k } ( x ) ) ) ) _ { m > k } ] \\\\ & = [ ( \\psi _ m ( a ) ) _ m ] \\\\ & = \\Psi ( a ) , \\end{align*}"}
+{"id": "7438.png", "formula": "\\begin{align*} \\dd Z ( s ) = \\big \\{ - \\partial _ t u ( t - s , X _ s ) + \\mathcal { A } u ( t - s , X _ s ) \\big \\} \\ , \\dd s + V \\cdot \\nabla u ( t - s , X _ s ) \\cdot \\dd W _ s , \\end{align*}"}
+{"id": "1246.png", "formula": "\\begin{align*} a _ { n , k } = \\frac { ( q ; q ^ 2 ) _ n q ^ { k ^ 2 - k } } { ( q ^ 2 ; q ^ 2 ) _ { n - 1 } ( 1 - q ^ { 2 n - 2 k - 1 } ) } { n - 1 \\brack k } _ { q ^ 2 } . \\end{align*}"}
+{"id": "3886.png", "formula": "\\begin{align*} P ( L ^ { n K , X } ( x ) = 0 ) = P ( L ^ { ( n - 1 ) K , X } ( x ) = 0 , L ^ { n K , X } ( x ) = 0 ) \\le ( 1 - \\gamma ) P ( L ^ { ( n - 1 ) K , X } ( x ) = 0 ) . \\end{align*}"}
+{"id": "7933.png", "formula": "\\begin{align*} \\beta = N _ { \\beta } ( \\omega ) , \\end{align*}"}
+{"id": "1419.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty d Q ^ { \\gamma } _ { \\alpha , \\beta , \\theta } ( t ) & = E ^ { \\gamma + \\theta / \\alpha } _ { \\alpha , \\beta + \\theta } ( 0 ) = \\frac { 1 } { \\Gamma ( \\beta + \\theta ) } \\end{align*}"}
+{"id": "7708.png", "formula": "\\begin{align*} 0 \\geq \\nabla _ { i i } Q & = \\frac { - h _ { t i i } } { h - \\varepsilon _ { 0 } } + \\frac { 2 h _ { t i } h _ { i } + h _ { t } h _ { i i } } { ( h - \\varepsilon _ { 0 } ) ^ { 2 } } - \\frac { 2 h _ { t } h ^ { 2 } _ { i } } { ( h - \\varepsilon _ { 0 } ) ^ { 3 } } \\\\ & = \\frac { - h _ { t i i } } { h - \\varepsilon _ { 0 } } + \\frac { h _ { t } h _ { i i } } { ( h - \\varepsilon _ { 0 } ) ^ { 2 } } . \\end{align*}"}
+{"id": "2656.png", "formula": "\\begin{align*} \\frac { 1 } { q } + \\frac { 1 } { \\frac { q } { q - 2 } } = \\frac { 1 } { q ' } . \\end{align*}"}
+{"id": "9016.png", "formula": "\\begin{align*} y _ { i i } = 0 , \\ z _ { i i } ^ 1 = z _ { 1 1 } ^ i z _ { i i } ^ n = z _ { n n } ^ i 1 < i < n . \\end{align*}"}
+{"id": "6649.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { y } \\nu ^ { + } ( E , \\mathrm { i } \\tilde { y } ) \\mathrm { d } \\tilde { y } + L ( E , \\mathrm { i } 0 ) = \\int _ { \\mathbb { C } } \\log | \\tilde { E } - E | \\mathrm { d } \\mathcal { N } ^ { \\mathrm { i } y } ( \\tilde { E } ) . \\end{align*}"}
+{"id": "5575.png", "formula": "\\begin{align*} f _ { \\phi , t } ( T , x ) \\stackrel { d } { = } \\tilde f _ { \\phi , t } ( \\tilde T , x ) . \\end{align*}"}
+{"id": "1299.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\mathbb { E } [ z _ 1 ( t ) ^ 2 ] \\equiv \\overline { \\Sigma } _ { 1 1 } = - \\frac { 1 } { 2 } \\Big ( \\frac { 1 } { a - 2 k } + \\frac { 1 } { a } \\Big ) . \\end{align*}"}
+{"id": "1846.png", "formula": "\\begin{align*} \\int _ Y c _ 1 ^ 4 = 0 \\int _ Y c _ 1 ^ 2 \\cdot p _ 1 = 0 \\int _ { \\Sigma ^ { ( 4 ) } _ Y } c _ 1 ^ 2 = 0 \\int _ { \\Sigma ^ { ( 2 ) } _ Y } c _ 1 = 0 . \\end{align*}"}
+{"id": "7796.png", "formula": "\\begin{align*} \\begin{aligned} ( A + B K ^ { ( i ) } ) ^ { \\top } P ^ { ( i ) } + P ^ { ( i ) } ( A + B K ^ { ( i ) } ) + ( C + D K ^ { ( i ) } ) ^ { \\top } P ^ { ( i ) } ( C + D K ^ { ( i ) } ) < 0 . \\end{aligned} \\end{align*}"}
+{"id": "5350.png", "formula": "\\begin{align*} \\mathbf { w } _ { N ^ { \\{ 0 , 1 \\} } } ^ S & = \\boldsymbol { \\theta } _ { N ^ { \\{ 0 , 1 \\} } } ^ 1 + \\beta \\ , \\left ( \\mathbf { P } _ { N ^ { \\{ 0 , 1 \\} } , N } ^ 1 - \\mathbf { P } _ { N ^ { \\{ 0 , 1 \\} } , N } ^ 0 \\right ) \\ , \\mathbf { b } ^ S \\\\ \\mathbf { w } _ { N ^ { \\{ 1 \\} } } ^ S & = \\beta \\ , \\left ( \\mathbf { P } _ { N ^ { \\{ 1 \\} } , N } ^ 1 - \\mathbf { P } _ { N ^ { \\{ 1 \\} } , N } ^ 0 \\right ) \\ , \\mathbf { b } ^ S = \\mathbf { 0 } , \\end{align*}"}
+{"id": "3878.png", "formula": "\\begin{align*} h _ x ( \\bar L ^ { n , k + 1 } ) & = h _ x ( \\bar L ^ { n } ( t _ k ) ) = h _ x ( \\bar L ^ { n } ( t _ n - T + ( t _ k - t _ n + T ) ) ) \\\\ & = h _ x ( \\hat M ^ n ( t _ k - ( t _ n - T ) ) ) \\le h _ x ( \\hat M ( t _ k - ( t _ n - T ) ) ) + \\delta _ 1 ^ { - 1 } \\sup _ { t \\in [ 0 , T ] } \\| \\hat M ^ n ( t ) - \\hat M ( t ) \\| , \\end{align*}"}
+{"id": "8297.png", "formula": "\\begin{align*} ( L y ) ( t ) : = y ^ { ( r ) } ( t ) + \\sum \\limits _ { j = 1 } ^ r A _ { r - j } ( t ) y ^ { ( r - j ) } ( t ) = f ( t ) , t \\in ( a , b ) , \\end{align*}"}
+{"id": "6758.png", "formula": "\\begin{align*} \\begin{aligned} & \\| v ^ { k + 1 } - v ^ * \\| ^ 2 _ { H } \\\\ = & \\| ( I - M ) ( v ^ k - v ^ * ) + M ( \\widetilde { v } ^ k - v ^ * ) \\| ^ 2 _ H \\\\ = & \\underbrace { \\| ( I - M ) ( v ^ k - v ^ * ) \\| ^ 2 _ H } _ { : = A ^ k } + \\underbrace { \\| M ( \\widetilde { v } ^ k - v ^ * ) \\| ^ 2 _ H } _ { : = B ^ k } \\\\ & + 2 \\underbrace { ( v ^ k - v ^ * ) ^ T ( I - M ) ^ T H M ( \\widetilde { v } ^ k - v ^ * ) } _ { : = C ^ k } . \\end{aligned} \\end{align*}"}
+{"id": "8011.png", "formula": "\\begin{align*} & \\exp \\left ( N \\left ( \\int _ \\mathbb { T } \\log \\left ( 1 + \\sum _ { k = 1 } ^ 3 \\rho _ { k - 1 } ( u ) \\left ( \\exp \\left ( f _ k ( u ) \\right ) - 1 \\right ) \\right ) d u + o ( 1 ) \\right ) \\right ) \\\\ & \\geq P ( \\mu ^ N \\in C ) \\times \\\\ & \\exp \\left ( N \\left ( \\inf _ { W \\in C } \\left ( \\sum _ { k = 1 } ^ 3 W _ { 0 , k } ( f _ k ) + \\mathcal { I } _ 1 ( W , F , G , H ) \\right ) \\right ) \\right ) . \\end{align*}"}
+{"id": "6986.png", "formula": "\\begin{align*} x & = [ x _ 1 : \\cdots : x _ n : 1 ] \\\\ & = [ \\mathrm { t a n h } ( t _ 1 ) \\mathrm { c o s } ( t _ { n } ) { \\displaystyle \\prod _ { k = 2 } ^ { n - 1 } } { \\mathrm { s i n } ( t _ { k } ) } : \\cdots : \\mathrm { t a n h } ( t _ 1 ) \\mathrm { c o s } ( t _ 2 ) : 1 ] , \\end{align*}"}
+{"id": "1253.png", "formula": "\\begin{align*} \\sum _ { \\substack { 0 \\le k \\le n - 1 \\\\ [ 3 p t ] k \\not = ( n - 1 ) / 2 } } b _ { n , k } = - q ( 1 - q ^ n ) \\sum _ { \\substack { 0 \\le k \\le n - 1 \\\\ [ 3 p t ] k \\not = ( n - 1 ) / 2 } } \\frac { ( - 1 ) ^ { k } } { 1 - q ^ { 2 k + 1 } } . \\end{align*}"}
+{"id": "4524.png", "formula": "\\begin{align*} \\mathcal { C } _ N = \\{ 0 \\leq s _ 1 \\leq . . . \\leq s _ N \\} , \\mathcal { C } _ \\infty = \\{ 0 \\leq s _ 1 . . . \\leq s _ N \\leq . . . \\} . \\end{align*}"}
+{"id": "5196.png", "formula": "\\begin{align*} & A = \\frac { 1 } { ( \\beta - 1 ) ( \\alpha + \\beta - 1 ) } \\sum _ { i } p ^ { \\alpha + \\beta - 1 } _ { i } + \\frac { 1 } { \\alpha ( \\alpha + \\beta - 1 ) } \\sum _ { i } q ^ { \\alpha + \\beta - 1 } _ { i } \\\\ & B = \\frac { 1 } { \\alpha ( \\beta - 1 ) } \\sum _ { i } p ^ { \\alpha } _ { i } q ^ { \\beta - 1 } _ { i } \\end{align*}"}
+{"id": "7487.png", "formula": "\\begin{align*} h ( x , t ) = \\int _ { 0 } ^ t \\int _ { N \\times \\mathbb { R } } < K _ L ( x , t , y , s ) , R ( y , s ) > d \\mu _ { g _ 0 ( s ) } ( y ) d s , \\end{align*}"}
+{"id": "1309.png", "formula": "\\begin{align*} | \\Sigma _ { 1 1 } ( t ) - \\overline { \\Sigma } _ { 1 1 } | = \\frac { 1 } { 2 } \\Big | \\frac { e ^ { 2 ( a - 2 k ) t } } { a - 2 k } + \\frac { e ^ { 2 a t } } { a } \\Big | = \\frac { 1 } { 2 } e ^ { 2 a t } \\Big | \\frac { 1 } { a } + \\frac { e ^ { - 4 k t } } { a - 2 k } \\Big | \\leq C e ^ { 2 a t } . \\end{align*}"}
+{"id": "4627.png", "formula": "\\begin{align*} f ( x ' ) - a _ { n , k } & \\geq f _ n ( x ' ) - a _ { n , k } = f _ n ( x ' ) - f _ { n - 1 } ( x ' ) + f _ { n - 1 } ( x ' ) - a _ { n , k } \\\\ & \\geq 2 ^ { - \\gamma m n } g _ n ( x ' ) - | f _ { n - 1 } ( x ' ) - a _ { n , k } | \\geq \\frac { 1 } { 4 } 2 ^ { - \\gamma m n } - \\frac { 1 } { 8 } 2 ^ { - \\gamma m n } = \\frac { 1 } { 8 } 2 ^ { - \\gamma m n } , \\end{align*}"}
+{"id": "7455.png", "formula": "\\begin{align*} P _ t = e ^ { - t \\frac { M } { m } } \\boldsymbol p + \\int _ 0 ^ t e ^ { - ( t - s ) \\frac { M } { m } } d B _ s , \\end{align*}"}
+{"id": "124.png", "formula": "\\begin{align*} \\| \\operatorname { d i v } \\vec { v } \\| & = \\sup _ { q \\in Q , ~ \\| q \\| = 1 } ( { \\rm d i v } \\vec { v } , q ) = \\sup _ { q \\in Q , ~ \\| q \\| = 1 } ( { \\rm d i v } ( \\vec { v } - P \\vec { v } ) , q ) \\\\ & \\leq \\left \\| { \\rm t r } [ \\varepsilon ( \\vec { v } - P \\vec { v } ) ] \\right \\| \\leq \\sqrt { d } \\left \\| \\varepsilon ( \\vec { v } - P \\vec { v } ) \\right \\| . \\end{align*}"}
+{"id": "5503.png", "formula": "\\begin{align*} E _ 1 ^ 3 E _ 2 - ( q ^ 2 + 1 + q ^ { - 2 } ) E _ 1 ^ 2 E _ 2 E _ 1 + ( q ^ 2 + 1 + q ^ { - 2 } ) E _ 1 E _ 2 E _ 1 ^ 2 - E _ 2 E _ 1 ^ 3 & { } = 0 \\\\ E _ 2 ^ 2 E _ 1 - ( q ^ 2 + q ^ { - 2 } ) E _ 2 E _ 1 E _ 2 + E _ 1 E _ 2 ^ 2 & { } = 0 . \\end{align*}"}
+{"id": "2389.png", "formula": "\\begin{align*} v _ p \\left ( \\frac { f ( k ) ^ p - f ( k _ { - } ) ^ p } { k - k _ { - } } \\right ) & = v _ p \\left ( \\frac { f ( k ) - f ( k _ { - } ) } { k - k _ { - } } \\cdot \\sum _ { i = 0 } ^ { p - 1 } f ( k ) ^ { i } f ( k _ { - } ) ^ { p - 1 - i } \\right ) \\\\ & \\geqslant - \\left \\lfloor \\frac { \\log n } { \\log p } \\right \\rfloor + 1 , \\end{align*}"}
+{"id": "6637.png", "formula": "\\begin{align*} q ( \\psi v , w ) = q ( v , \\psi w ) . \\end{align*}"}
+{"id": "3339.png", "formula": "\\begin{align*} z _ 0 \\circ z = ( v _ 0 + v , x _ 0 + x + t v _ 0 , t _ 0 + t ) , \\forall z _ 0 = ( v _ 0 , x _ 0 , t _ 0 ) \\in \\R ^ { 2 n + 1 } , \\end{align*}"}
+{"id": "8861.png", "formula": "\\begin{align*} 0 < \\frac 1 { p b _ 1 } \\leq \\frac 1 r \\leq 1 , 0 \\leq \\frac { \\tau } { p } < \\frac { n } { p b _ 1 } , \\frac { \\tau } { p } - 1 = \\frac { n } { p b _ 1 } - \\frac { n } { r } . \\end{align*}"}
+{"id": "4711.png", "formula": "\\begin{align*} \\psi \\ ; = \\ ; \\psi ^ { ( \\leq 0 ) } + \\psi ^ { ( 1 ) } \\end{align*}"}
+{"id": "4613.png", "formula": "\\begin{align*} Q ^ m _ j : = 2 ^ { - m } \\left ( j + ( 0 , 1 ) ^ d \\right ) \\end{align*}"}
+{"id": "8331.png", "formula": "\\begin{align*} \\begin{cases} [ \\Pi , W _ t ] _ { S N } + \\frac { 1 } { 2 } [ W _ t , W _ t ] _ { \\gamma } { - } \\frac { 1 } { 6 } ( W _ t ^ { \\sharp } \\wedge W _ t ^ { \\sharp } \\wedge W _ t ^ { \\sharp } ) \\Upsilon ^ G _ { T M } = 0 \\\\ [ \\Pi , X _ t ] _ { S N } - \\left ( \\frac { d } { d t } W _ t + [ X _ t , W _ t ] _ { \\gamma } \\right ) { - } \\frac { 1 } { 2 } ( X _ t ^ { \\sharp } \\wedge W _ t ^ { \\sharp } \\wedge W _ t ^ { \\sharp } ) \\Upsilon ^ G _ { T M } = 0 \\end{cases} . \\end{align*}"}
+{"id": "533.png", "formula": "\\begin{align*} \\Psi _ { k } ( z ) : = \\dfrac { \\Lambda ( z ) } { \\Lambda ' ( i \\lambda _ { k } ) ( z - i \\lambda _ { k } ) } , k \\geq 1 , \\end{align*}"}
+{"id": "8431.png", "formula": "\\begin{align*} \\langle \\nu _ G ( b _ x ) - \\nu _ G ( b ) , 2 \\rho - 2 \\rho _ J \\rangle = 0 . \\end{align*}"}
+{"id": "39.png", "formula": "\\begin{align*} \\tilde d _ { 0 } \\omega = \\sum _ { l = 1 } ^ n ( \\mathcal { D } _ { l } \\omega ) d x _ l . \\end{align*}"}
+{"id": "8869.png", "formula": "\\begin{align*} \\frac { n } { a _ 2 } = \\frac { ( p - 1 ) n } { r } + \\tau - p + 1 , \\frac { n } { b _ 2 } = \\frac { p n } { r } + \\tau - p + 1 \\end{align*}"}
+{"id": "2263.png", "formula": "\\begin{align*} R _ { 1 2 } ( u _ 1 , u _ 2 ) R _ { 1 3 } ( u _ 1 , u _ 3 ) R _ { 2 3 } ( u _ 2 , u _ 3 ) = R _ { 2 3 } ( u _ 2 , u _ 3 ) R _ { 1 3 } ( u _ 1 , u _ 3 ) R _ { 1 2 } ( u _ 1 , u _ 2 ) \\ \\ \\ \\ \\ \\end{align*}"}
+{"id": "5276.png", "formula": "\\begin{align*} \\widetilde { x } ^ { k + 1 } = x ^ { k } + \\alpha ^ { k } x ^ { k } \\left ( \\frac { U ^ { k } } { V ^ { k } } - 1 \\right ) \\end{align*}"}
+{"id": "9121.png", "formula": "\\begin{align*} \\begin{aligned} & [ i , j ] \\coloneqq [ i , i + 1 , \\dots , j ] , \\ \\ 1 \\leq i \\leq j \\leq n , \\\\ & [ i , n , j ] \\coloneqq [ i , \\dots , n , n , n - 1 , \\dots , j ] , \\ \\ 1 \\leq i < j \\leq n . \\end{aligned} \\end{align*}"}
+{"id": "6944.png", "formula": "\\begin{align*} \\ker f _ { v , w } | _ { U _ v } = \\ker f _ { v , w } | _ { V _ v } \\oplus \\ker f _ { v , w } | _ { f _ { u , v } ( U _ u ) } \\end{align*}"}
+{"id": "4098.png", "formula": "\\begin{align*} \\begin{aligned} & \\mu ^ { ( r ) } _ j - \\lambda _ j = r ^ j , j \\in \\mathcal { J } . \\end{aligned} \\end{align*}"}
+{"id": "3481.png", "formula": "\\begin{align*} B _ j ^ { m , K } ( x ) = \\begin{cases} \\frac { ( x - a _ j ) B _ j ^ { m - 1 , K } ( x ) + ( a _ { j + m + 1 } - x ) B _ { j + 1 } ^ { m - 1 , K } ( x ) } { a _ { j + m + 1 } - a _ j } , \\ a _ j < a _ { j + m + 1 } , a _ j \\leq x < a _ { j + m + 1 } , \\\\ 0 , \\ \\ , . \\end{cases} \\end{align*}"}
+{"id": "2704.png", "formula": "\\begin{align*} f ^ { I } ( q ^ { i } , v ^ { i } , t ) = c ^ { I } , \\end{align*}"}
+{"id": "2048.png", "formula": "\\begin{align*} \\theta ^ { t + 1 } _ i & = \\theta ^ { t } _ i + \\eta x ^ t _ i y ^ t - \\eta \\sum _ { j = 1 } ^ { d } x ^ t _ i x ^ t _ j \\theta ^ { t } _ j \\\\ & = \\theta ^ t _ i + \\eta x ^ t _ i \\Big ( \\sum _ { j = 1 } ^ d x ^ t _ j \\theta ^ * _ j + \\varepsilon { ^ t } - \\sum _ { j = 1 } ^ d x ^ t _ j \\theta ^ t _ j \\Big ) \\\\ & = \\theta ^ t _ i - \\eta \\sum _ { j = 1 } ^ d x ^ t _ i x ^ t _ j ( \\theta ^ t _ j - \\theta ^ * _ j ) + \\eta x ^ t _ i \\varepsilon ^ t . \\end{align*}"}
+{"id": "3259.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { p - 1 } \\frac { ( \\frac { 1 } { 2 } ) _ k ^ 2 } { k ! ^ 2 } \\equiv ( - 1 ) ^ { ( p - 1 ) / 2 } \\pmod { p ^ 2 } . \\end{align*}"}
+{"id": "3094.png", "formula": "\\begin{align*} 2 \\ , d & = \\sum _ { i = 1 } ^ r m _ i \\ , \\dim _ k V _ { m _ i } + \\sum _ { i = 1 } ^ r ( - m _ { r - i + 1 } ) \\ , \\dim _ k V _ { m _ { r - i + 1 } } \\\\ & = \\sum _ { i = 1 } ^ r m _ i \\ , \\dim _ k V _ { m _ i } - \\sum _ { i = 1 } ^ r m _ i \\ , \\dim _ k V _ { m _ i } \\\\ & = 0 , \\end{align*}"}
+{"id": "185.png", "formula": "\\begin{align*} h _ { n + 1 } ( x ) \\ = \\ ( h _ n ( x ) , \\pi _ { n h _ n ( x ) } ( x ) ) \\ \\ x \\in X . \\end{align*}"}
+{"id": "9105.png", "formula": "\\begin{align*} \\Tilde { k } : = v e c ( \\Tilde { K } ) = [ \\Tilde { K } _ { \\bullet 1 } ^ T \\ldots \\Tilde { K } _ { \\bullet n } ^ T ] ^ T , \\delta : = \\Big [ \\frac { a _ { \\bullet 1 } } { \\hat { c } _ { 1 1 } } ^ T \\ldots \\frac { a _ { \\bullet n } } { \\hat { c } _ { n n } } ^ T \\Big ] ^ T . \\end{align*}"}
+{"id": "3023.png", "formula": "\\begin{align*} \\begin{array} { l } \\overline { R i c } _ { 0 0 } = 4 ( \\beta - 1 ) , \\\\ \\overline { R i c } _ { 1 1 } = \\overline { R i c } _ { 2 2 } = \\overline { R i c } _ { 3 3 } = \\overline { R i c } _ { 4 4 } = \\beta ^ 2 - \\beta - 3 \\alpha ^ 2 , \\\\ \\overline { R i c } _ { 1 3 } = \\overline { R i c } _ { 2 4 } = 3 \\alpha - 2 \\alpha \\beta . \\end{array} \\end{align*}"}
+{"id": "1669.png", "formula": "\\begin{align*} \\Pi & \\coloneqq \\omega ( v _ 0 , v _ 1 ) \\ , \\omega ( v _ 1 , v _ 2 ) \\ , \\omega ( v _ 2 , v _ 0 ) , \\\\ \\lambda & \\coloneqq \\frac { \\sqrt { \\Pi } } { \\omega ( v _ 1 , v _ 0 ) } = \\frac { \\omega ( v _ 1 , v _ 2 ) \\ , \\omega ( v _ 0 , v _ 2 ) } { \\sqrt { \\Pi } } , \\\\ \\mu & \\coloneqq \\lambda ^ { - 1 } \\cdot \\omega ( v _ 3 , v _ 2 ) . \\end{align*}"}
+{"id": "5913.png", "formula": "\\begin{align*} ( q - 1 ) ( q - 2 ) ^ 3 & ( p q - 1 ) + ( p - 1 ) ( p - 2 ) ^ 3 q ( p q - 1 ) \\\\ > & ( q - 1 ) ^ 2 ( q - 2 ) ^ 3 + ( p - 1 ) ^ 2 ( p - 2 ) ^ 3 q ^ 2 + q ( q - 1 ) ( p - 1 ) ( p - 2 ) ( q - 2 ) ( p + q - 4 ) \\\\ = & \\Big { ( } ( q - 1 ) ( q - 2 ) + q ( p - 1 ) ( p - 2 ) \\Big { ) } \\Big { ( } ( q - 1 ) ( q - 2 ) ^ { 2 } + q ( p - 1 ) ( p - 2 ) ^ { 2 } \\Big { ) } \\end{align*}"}
+{"id": "5225.png", "formula": "\\begin{align*} \\overline { M G } = \\sum _ { i } \\overline { p } ^ { \\alpha } _ { i } \\overline { q } ^ { 1 - \\alpha } _ { i } \\ : \\ : ; \\ : \\overline { M A } = \\sum _ { i } \\alpha \\overline { p } _ { i } + \\left ( 1 - \\alpha \\right ) \\overline { q } _ { i } = 1 \\end{align*}"}
+{"id": "6274.png", "formula": "\\begin{align*} N = \\tilde { O } \\left ( \\frac { 1 } { r } \\log _ 2 \\left ( \\frac { \\mu _ r R _ 0 } { 2 \\varepsilon } \\right ) \\right ) , \\end{align*}"}
+{"id": "1086.png", "formula": "\\begin{align*} z _ { s , n } & = \\sum _ { t = 1 } ^ { n } ( T _ { n } ( w ) ^ { - 1 } ) ^ { s , t } y _ t , s \\in \\{ 1 , \\dots , n \\} , \\\\ z _ s & = \\sum _ { t = 1 } ^ { \\infty } ( T _ \\infty ( w ) ^ { - 1 } ) ^ { s , t } y _ t , s \\in \\N . \\end{align*}"}
+{"id": "7832.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } \\ ! \\ ! I _ { M } , & \\ ! \\ ! \\ ! \\ ! I _ { M } \\ ! \\ ! \\ ! \\\\ \\end{array} \\right ] P _ { \\operatorname { B o x } [ \\boldsymbol { 0 } , \\boldsymbol { 1 } ] } \\biggl ( \\boldsymbol { \\rho } \\left ( i + \\frac { 1 } { 2 } \\right ) - \\begin{bmatrix} \\boldsymbol { \\lambda } ^ { \\star } \\\\ \\boldsymbol { \\lambda } ^ { \\star } \\end{bmatrix} \\biggl ) = \\boldsymbol { 1 } . \\end{align*}"}
+{"id": "4810.png", "formula": "\\begin{align*} \\mathcal { S } ^ { ( i n t ) } _ 0 : = \\Big \\{ \\mathbf { c } ' \\in \\mathbb { R } ^ { n } : \\mathbf { l } \\leq \\mathbf { c } ' \\leq \\mathbf { u } \\Big \\} , \\end{align*}"}
+{"id": "5537.png", "formula": "\\begin{align*} b : = \\{ x \\in S \\mid \\exists q \\leq _ { \\P } p ~ ( q \\Vdash _ { \\P } ` ` x \\in \\dot { b } _ i ) \" \\} . \\end{align*}"}
+{"id": "1344.png", "formula": "\\begin{align*} \\lambda = \\dfrac { \\alpha ( d ( v ) + d ( u ' ) ) \\pm \\sqrt { \\alpha ^ 2 \\{ d ( v ) + d ( u ' ) \\} ^ 2 - 4 \\alpha ^ 2 d ( v ) d ( u ' ) } } { 2 } . \\end{align*}"}
+{"id": "78.png", "formula": "\\begin{align*} S & = w _ 1 G D _ 1 ( p - 2 + s ) + w _ 2 G D _ 2 ( p - 2 + s ) + \\epsilon w _ 3 G D _ 1 ( p - 4 + s ) + \\epsilon w _ 4 G D _ 2 ( p - 4 + s ) \\\\ & = w _ 1 G D _ 1 ( 0 ) + w _ 2 G D _ 2 ( 0 ) + w _ 3 \\epsilon G D _ 1 ( - 2 ) + w _ 4 \\epsilon G D _ 2 ( - 2 ) , \\end{align*}"}
+{"id": "4700.png", "formula": "\\begin{align*} \\| T ^ { \\otimes } \\| = \\prod _ { j \\in \\N } \\| T _ j \\| , \\end{align*}"}
+{"id": "7982.png", "formula": "\\begin{align*} \\mathrm { t r } ( \\hat { f } ) = 0 \\quad \\partial \\Omega . \\end{align*}"}
+{"id": "8603.png", "formula": "\\begin{align*} u _ t + u u _ x + \\mathcal { H } \\Lambda ^ { \\frac { 1 } { 2 } } u = 0 . \\end{align*}"}
+{"id": "7059.png", "formula": "\\begin{align*} X _ t = x - \\int _ 0 ^ t b ( s , X _ s ) d s + \\sqrt { 2 } \\int _ 0 ^ t \\sigma ( s , X _ s ) d W _ s , \\in \\mathbb R ^ d \\end{align*}"}
+{"id": "3143.png", "formula": "\\begin{align*} u _ { \\psi ^ * } ^ - ( s ) = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ s ^ { p ^ { e _ 1 } } & 1 & 0 \\\\ s ^ { 2 \\ , p ^ { e _ 1 } } & 0 & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "233.png", "formula": "\\begin{align*} \\frac { d ^ 2 x _ 1 } { d t _ 1 ^ 2 } + A _ 1 ( x _ 1 ) \\frac { d x _ 1 } { d t _ 1 } + b _ 1 ( x _ 1 ) = 0 , \\end{align*}"}
+{"id": "807.png", "formula": "\\begin{align*} & \\int \\limits _ { S M } ( \\delta X ) \\eta = - \\int \\limits _ { S M } { ( { g ^ { i j } } } { D _ i } { X _ j } ) \\eta = 0 , \\ \\textrm { a n d } & \\int \\limits _ { S M } ( \\dot \\delta X ) \\eta = - \\int \\limits _ { S M } { F { g ^ { i j } } } \\frac { { \\partial { X _ i } } } { { \\partial { y ^ j } } } \\eta = 0 , \\end{align*}"}
+{"id": "6580.png", "formula": "\\begin{align*} \\prod _ { p \\leq x } \\left ( 1 - \\frac { \\chi _ 1 ( p ) } { p ^ j } \\right ) = L ( j , \\chi _ 1 ) ^ { - 1 } + \\begin{cases} & O \\left ( \\frac { 1 } { x ^ { j - 1 } } \\right ) \\ \\ \\mbox { i f } \\ j > 1 , \\\\ & O \\left ( \\frac { 1 } { \\log x } \\right ) \\ \\ \\mbox { i f } \\ j = 1 . \\end{cases} \\end{align*}"}
+{"id": "217.png", "formula": "\\begin{align*} X ^ i ( x , v ) = \\sum _ { j = 1 } ^ n ( A ^ i \\ , _ j \\ , x ^ j + B ^ i \\ , _ j \\ , v ^ j ) , i = 1 , \\ldots , n . \\end{align*}"}
+{"id": "1588.png", "formula": "\\begin{align*} k _ n = M \\ , ( 1 - 2 ^ { - ( n + 1 ) } ) , r _ n = \\frac { R } { 2 } + \\frac { R } { 2 ^ { n + 1 } } , X _ n = \\int _ { A ( k _ n , r _ n ) } ( g _ { k _ n } ( V ) - 1 ) ^ 2 \\ , d x , \\end{align*}"}
+{"id": "382.png", "formula": "\\begin{align*} ( x + y - z ) ^ 3 = 3 ( z - y ) ( z - x ) ( x + y ) , \\end{align*}"}
+{"id": "2492.png", "formula": "\\begin{align*} & u ( t ) \\rightharpoonup u _ \\infty : = \\Delta A _ \\infty + u ^ { i n } \\ ; \\ ; \\ ; \\ ; \\ ; H ^ 1 ( \\Omega ) ' \\ ; \\ ; \\ ; t \\to \\infty \\ , , \\\\ & \\lim _ { t \\to \\infty } \\| v ( t ) \\| _ p = 0 \\ , , p \\in [ 1 , \\infty ) \\ , . \\end{align*}"}
+{"id": "2092.png", "formula": "\\begin{align*} n ^ d _ t & \\le C _ { d , T } ( t ) + 2 5 0 \\eta ^ 4 d ^ 4 \\left ( C _ 2 ^ 4 t ^ 3 + \\left ( C _ 8 + C _ { 5 } ^ 2 \\right ) t \\right ) \\sum _ { k = 0 } ^ { t - 1 } C _ { d , T } ( k ) \\exp ( 2 5 0 \\eta ^ 4 d ^ 4 \\left ( C _ 2 ^ 4 t ^ 3 + \\left ( C _ 8 + C _ { 5 } ^ 2 \\right ) t \\right ) ( t - k - 1 ) ) \\end{align*}"}
+{"id": "8998.png", "formula": "\\begin{align*} g _ { \\lambda } ^ { H } ( 0 , 0 ) = \\frac { 1 } { 2 \\pi } \\Gamma ( 1 - \\frac { 1 } { \\alpha } ) \\Gamma ( \\frac { 1 } { \\alpha } ) = \\frac { 1 } { 2 \\alpha \\sin ( \\pi / \\alpha ) } , \\end{align*}"}
+{"id": "7156.png", "formula": "\\begin{align*} D _ p \\omega = 2 \\eta ^ { \\mu \\nu } k _ \\mu \\cdot \\frac { \\partial \\omega } { \\partial x ^ \\nu } \\ , \\ , . \\end{align*}"}
+{"id": "1212.png", "formula": "\\begin{align*} F _ { p } ( u , v ) = \\left \\{ \\begin{array} { l l } G _ { p } ( u , v ) + \\chi _ { S _ { p } } ( u , v ) , & ( u , v ) \\in X ^ { * } _ { s , t , p } ( \\Omega ) ; \\\\ \\infty , & , \\end{array} \\right . \\end{align*}"}
+{"id": "6590.png", "formula": "\\begin{align*} \\alpha _ { k } ( p ) \\geq \\begin{cases} 1 & k \\equiv 2 \\pmod 4 p \\equiv 3 \\pmod { 4 } , \\\\ \\left ( 1 - \\frac { 1 } { 2 ^ { k / 2 } } \\right ) & . \\end{cases} \\end{align*}"}
+{"id": "462.png", "formula": "\\begin{align*} \\pi ( g ) \\pi ( h ) \\pi ( h ^ { - 1 } ) & = \\phi ( [ g ] ) \\phi ( [ h ] ) \\phi ( [ h ^ { - 1 } ] ) = \\phi ( [ g ] [ h ] [ h ^ { - 1 } ] ) \\\\ & = \\phi ( [ g h ] [ h ^ { - 1 } ] ) = \\phi ( [ g h ] ) \\phi ( [ h ^ { - 1 } ] ) = \\pi ( g h ) \\pi ( h ^ { - 1 } ) . \\end{align*}"}
+{"id": "7273.png", "formula": "\\begin{align*} A = 0 , B = - \\tfrac { \\eta } { \\sqrt { 1 - q + \\eta \\widetilde { \\theta } } } , C = - \\tfrac { \\widetilde { \\theta } + \\tfrac { \\eta \\tau } { 1 - q } - \\sqrt { \\theta ^ 2 - 4 \\tau } } { 2 \\sqrt { 1 - q + \\eta \\widetilde { \\theta } } } D = - \\tfrac { \\widetilde { \\theta } + \\tfrac { \\eta \\tau } { 1 - q } + \\sqrt { \\theta ^ 2 - 4 \\tau } } { 2 \\sqrt { 1 - q + \\eta \\widetilde { \\theta } } } \\end{align*}"}
+{"id": "1817.png", "formula": "\\begin{align*} \\phi _ 0 : = d x _ { 1 2 3 } + d x _ { 1 4 5 } + d x _ { 1 6 7 } + d x _ { 2 4 6 } - d x _ { 2 5 7 } - d x _ { 3 4 7 } - d x _ { 3 5 6 } , \\end{align*}"}
+{"id": "4365.png", "formula": "\\begin{gather*} p ^ * _ i = \\sup \\{ 0 , \\sup _ { u ^ i \\in \\mathcal { U } _ i } f _ i ( x ^ * , u ^ i ) - f _ i ( x ^ * , \\overline { u } ^ i ) ) - \\theta ^ * \\} \\ \\forall i \\in [ m ] \\end{gather*}"}
+{"id": "1892.png", "formula": "\\begin{align*} \\rho _ { E _ { a , b } } ( u _ 1 \\sin ( t ) , u _ 2 \\cos ( t ) ) = \\left ( \\frac { \\sin ( t ) ^ 2 } { a ^ 2 } + \\frac { \\cos ( t ) ^ 2 } { b ^ 2 } \\right ) ^ { - 1 / 2 } , t \\in [ 0 , \\pi / 2 ] . \\end{align*}"}
+{"id": "2014.png", "formula": "\\begin{align*} h ( \\eta ) = \\frac { 1 } { d } \\left ( \\log | a _ 0 | + \\sum _ { j = 1 } ^ { d } \\log \\max \\left ( 1 , \\left | \\eta ^ { ( j ) } \\right | \\right ) \\right ) \\end{align*}"}
+{"id": "3453.png", "formula": "\\begin{align*} [ S W F _ G ( Y _ 0 , \\mathfrak { t } _ 0 , \\iota _ 0 ) \\wedge S W F _ G ( Y _ 1 , \\mathfrak { t } _ 1 , \\iota _ 1 ) ] _ { \\rm { l o c } } = [ S W F _ G ( Y ^ \\# , \\mathfrak { t } ^ \\# , \\iota ^ \\# ) ] _ { \\rm { l o c } } , \\end{align*}"}
+{"id": "7207.png", "formula": "\\begin{gather*} ( m ( a _ 1 ) , m ( a _ 2 ) ) \\in \\Phi _ b \\iff m ( a _ 1 ) \\wedge b = m ( a _ 2 ) \\wedge b \\implies \\\\ m _ * \\circ m ( a _ 1 ) \\wedge m _ * ( b ) = m _ * \\circ m ( a _ 2 ) \\wedge m _ * ( b ) \\iff \\\\ ( m _ * \\circ m ( a _ 1 ) , m _ * \\circ m ( a _ 2 ) ) \\in \\Phi _ { m _ * ( b ) } . \\end{gather*}"}
+{"id": "6174.png", "formula": "\\begin{align*} \\begin{cases} \\hat { \\lambda } ^ k = \\lambda ^ k - ( 1 - \\gamma ) ( 1 - \\tau ^ k ) \\beta ^ k ( A { x } ^ { k } - b ) , \\\\ \\hat { x } ^ k = x ^ k + \\frac { \\tau ^ k ( 1 - \\tau ^ { k - 1 } ) } { \\tau ^ { k - 1 } } ( x ^ k - x ^ { k - 1 } ) , \\\\ x ^ { k + 1 } \\in \\arg \\min \\limits _ x \\{ f ( x ) + x ^ T \\nabla _ x \\varphi ^ k ( \\hat { x } ^ k , \\hat { \\lambda } ^ k ) + \\frac { \\beta ^ k } { 2 } \\| x - \\hat { x } ^ k \\| _ D ^ 2 + \\frac { \\sigma ( 1 - \\tau ^ k ) } { 2 \\tau ^ k } \\| x - x ^ k \\| ^ 2 \\} . \\end{cases} \\end{align*}"}
+{"id": "910.png", "formula": "\\begin{align*} L u ( x ) = \\mathcal I u ( x ) : = \\mathrm { P } . \\mathrm { V } . \\int _ { \\mathbb { R } ^ d } ( u ( y ) - u ( x ) ) j ( | x - y | ) \\ , d y \\end{align*}"}
+{"id": "0.png", "formula": "\\begin{align*} q ( t , x ) \\mapsto q _ \\lambda ( t , x ) = \\lambda q ( \\lambda ^ 2 t , \\lambda x ) \\qquad \\end{align*}"}
+{"id": "1992.png", "formula": "\\begin{align*} g ( \\phi ( 0 ) ) = \\frac { d } { d t } f ( \\phi ( t ) ) | _ { t = 0 } , g ( \\phi ( t _ n ) ) = \\delta ^ - _ t f ( \\phi ( t _ n ) ) , \\ n \\ge 1 , \\end{align*}"}
+{"id": "1533.png", "formula": "\\begin{align*} \\delta = \\frac { 2 \\ , \\sqrt { H } } { H + 1 } , H = \\frac { 1 + \\sqrt { 1 - \\delta ^ 2 } } { 1 - \\sqrt { 1 - \\delta ^ 2 } } , \\end{align*}"}
+{"id": "3791.png", "formula": "\\begin{align*} \\| \\mathcal { H } _ 1 \\| & = \\| \\widehat { \\Theta } _ 1 - \\Theta _ 1 \\| \\left \\| I - \\frac { 2 \\eta _ 1 } { | \\widehat { \\mathcal { C } } _ 1 | } Z Z ^ { \\top } \\right \\| + \\frac { 2 \\eta _ 1 } { | \\widehat { \\mathcal { C } } _ 1 | } \\sum _ { i \\in \\widehat { \\mathcal { C } } _ 1 } \\| W Z ^ { \\top } \\| , \\end{align*}"}
+{"id": "6892.png", "formula": "\\begin{align*} f _ B ( v ) w = v f _ A ( w ) , \\end{align*}"}
+{"id": "2536.png", "formula": "\\begin{align*} x ^ m \\dot C ^ \\infty ( M ) = \\dot C ^ \\infty ( M ) \\quad ( m \\in \\R ) \\ ; , \\end{align*}"}
+{"id": "894.png", "formula": "\\begin{align*} L _ { 0 0 } = E ( y ^ 1 ) ^ 2 + J y ^ 1 y ^ 2 + K ( y ^ 2 ) ^ 2 , \\end{align*}"}
+{"id": "6991.png", "formula": "\\begin{align*} \\{ [ w _ 1 : w _ 2 : 0 : \\cdots : 0 : w _ { n + 1 } ] \\in \\mathbb { P } _ { \\mathbb { C } } ^ { n } : w _ 1 , w _ 2 , w _ { n + 1 } \\in \\mathbb { R } , w _ 1 ^ 2 + w _ 2 ^ 2 - w _ { n + 1 } ^ 2 = 0 \\} . \\end{align*}"}
+{"id": "5258.png", "formula": "\\begin{align*} L _ { d } G \\left ( p \\| q \\right ) = \\sum _ { i } \\frac { T _ { i } } { a - b } \\left [ \\left ( \\frac { T _ { i } } { p _ { i } } \\right ) ^ { a - 1 } - \\left ( \\frac { T _ { i } } { p _ { i } } \\right ) ^ { b - 1 } \\right ] + \\left ( 1 - \\alpha \\right ) \\left ( p _ { i } - q _ { i } \\right ) \\end{align*}"}
+{"id": "5427.png", "formula": "\\begin{align*} v _ { ( a ^ - , i ) } ^ * ( \\nu ) = \\max \\big \\{ \\beta v _ { ( 0 , i ) } ^ * ( \\nu ) , R _ i - ( 1 - a ^ - ) c _ i - \\nu + \\beta \\sum _ { j \\in N } p _ { i j } v _ { ( 1 , j ) } ^ * ( \\nu ) \\big \\} . \\end{align*}"}
+{"id": "733.png", "formula": "\\begin{align*} & g _ p ( a , a + b , a F _ { 2 } + b F _ 3 ) \\\\ & = \\begin{cases} ( a - 1 ) b + \\dfrac { a ( a - 3 ) } { 2 } + p ( a + 2 b ) & ; \\\\ ( a - 1 ) b + \\dfrac { a ( a - 2 ) } { 2 } + p ( a + 2 b ) & \\ , . \\end{cases} \\end{align*}"}
+{"id": "1499.png", "formula": "\\begin{align*} \\int _ { \\Omega } | \\nabla u | ^ { p - 2 } \\nabla u \\nabla \\varphi d x + \\int _ { \\Omega } \\frac { | \\nabla u | ^ { p } } { u + \\delta } \\varphi d x = \\int _ { \\Omega } f ( x , u , \\nabla u ) \\varphi d x , \\end{align*}"}
+{"id": "7361.png", "formula": "\\begin{align*} u ^ 0 ( x , t ) : = \\frac { 1 } { 2 } \\{ f ( x + t ) + f ( x - t ) \\} + \\frac { 1 } { 2 } \\int _ { x - t } ^ { x + t } g ( y ) d y , \\end{align*}"}
+{"id": "6405.png", "formula": "\\begin{align*} \\tilde { \\delta } _ { n } ( p ) = \\frac { 1 } { m _ p ( \\tilde { \\alpha } _ { n } ( p ) ) } \\frac { 1 } { n } \\sum _ { i = 2 } ^ n n ^ { p / \\tilde { \\alpha } _ { n } ( p ) } \\frac { \\left | \\Delta _ i ^ n X - \\Delta _ { i - 1 } ^ n X \\right | ^ { p } } { X _ { \\frac { i - 2 } { n } } ^ { p / \\tilde { \\alpha } _ { n } ( p ) } } . \\end{align*}"}
+{"id": "3568.png", "formula": "\\begin{align*} L ( \\tau ( z _ 0 ) ) = L ( \\tau ^ 2 ( z _ 0 ) ) = L ( \\tau ^ 3 ( z _ 0 ) ) = 0 L ( z _ 0 ) = 1 . \\end{align*}"}
+{"id": "7066.png", "formula": "\\begin{align*} - a _ n \\cdot \\nabla ^ 2 + b _ n \\cdot \\nabla = - \\nabla \\cdot a _ n \\cdot \\nabla + ( \\nabla a _ n + b _ n ) \\cdot \\nabla . \\end{align*}"}
+{"id": "4922.png", "formula": "\\begin{align*} \\# \\left ( S _ i \\cap S _ j \\right ) = \\begin{cases} 1 \\\\ 0 . \\end{cases} \\end{align*}"}
+{"id": "925.png", "formula": "\\begin{align*} u ( x ) = \\int _ { D ^ c } g ( y ) \\ , P _ D ( x , d y ) + \\int _ D f ( y , u ( y ) ) G _ D ( x , y ) \\ , d y + \\int _ D G _ D ( x , y ) \\ , \\mu ( d y ) . \\end{align*}"}
+{"id": "4775.png", "formula": "\\begin{align*} 1 / 2 + 1 / 2 \\langle y , j \\rangle = \\langle ( x + y ) / 2 , j \\rangle \\leq 1 - \\eta ( \\varepsilon ) \\end{align*}"}
+{"id": "1262.png", "formula": "\\begin{align*} c _ { n , k } = \\frac { ( q ; q ^ 2 ) _ n q ^ { k ^ 2 + k } } { ( q ^ 2 ; q ^ 2 ) _ { n - 1 } ( 1 - q ^ { 2 n - 2 k - 1 } ) } { n - 1 \\brack k } _ { q ^ 2 } = q ^ { 2 k } a _ { n , k } . \\end{align*}"}
+{"id": "3629.png", "formula": "\\begin{align*} 2 \\bigg ( \\frac { \\lambda _ { 1 } ^ { 2 } } { 1 - \\lambda _ { 1 } ^ { 2 } } \\bigg ) ^ 2 \\big ( f ( z , w ) - f ( z , - w ) \\big ) = \\frac { \\lambda _ { 1 } ^ { 2 } } { 1 - \\lambda _ { 1 } ^ { 2 } } \\big ( f ( z , - w ) - f ( z , w ) \\big ) , \\end{align*}"}
+{"id": "1038.png", "formula": "\\begin{align*} \\mathcal { A } _ { \\rho } ^ { \\infty \\times q } & : = \\left \\{ \\mathbf { y } = ( y _ 1 ^ \\top , y _ 2 ^ \\top , \\dots ) ^ \\top \\in \\C ^ { \\infty \\times q } : \\sup _ { k \\in \\N } k ^ { \\rho } \\| y _ k \\| < \\infty \\right \\} , \\\\ \\mathcal { B } ^ { \\infty \\times q } & : = \\left \\{ \\mathbf { y } = ( y _ 1 ^ \\top , y _ 2 ^ \\top , \\dots ) ^ \\top \\in \\C ^ { \\infty \\times q } : \\sum _ { k = 1 } ^ { \\infty } \\| y _ k \\| < \\infty \\right \\} . \\end{align*}"}
+{"id": "1202.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } ( - \\Delta _ p ) ^ { s } u = \\lambda _ 1 ( s , p ) \\vert u \\vert ^ { p - 2 } u ( x ) & { \\rm i n } \\ \\ \\Omega , \\\\ u = 0 & { \\rm i n } \\ \\mathbb { R } ^ N \\setminus \\Omega . \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "1761.png", "formula": "\\begin{align*} X = X _ { - 1 } + p U _ 0 + q V _ 0 + r V _ { 2 - n } + X _ 1 , \\end{align*}"}
+{"id": "4794.png", "formula": "\\begin{align*} v _ h : = \\begin{cases} - h \\quad & u < t - h , \\\\ u - t \\quad & t - h < u < t , \\\\ 0 \\quad & u \\geq t , \\end{cases} \\end{align*}"}
+{"id": "34.png", "formula": "\\begin{align*} \\partial ( s ) : = \\cup _ { i = 1 } ^ j \\{ ( - 1 ) ^ { j - i } ( \\lfloor s \\rfloor ; \\prescript { } { i } { \\hat { s } } ) \\} \\bigcup \\cup _ { i = 1 } ^ j \\{ ( - 1 ) ^ { i } ( \\lceil s \\rceil ; ( \\prescript { } { i } { \\hat { s } } ) ^ * ) \\} \\ . \\end{align*}"}
+{"id": "2178.png", "formula": "\\begin{align*} X = A ( t ) , \\ , \\ , \\ , \\ , \\lambda = t . \\end{align*}"}
+{"id": "5462.png", "formula": "\\begin{align*} V '' ( t ) = 1 6 E ( u ( t ) , v ( t ) ) - 8 M ( u ( t ) , v ( t ) ) = 1 6 E ( u _ 0 , v _ 0 ) - 8 M ( u _ 0 , v _ 0 ) . \\end{align*}"}
+{"id": "1301.png", "formula": "\\begin{align*} \\hat { D } = - \\lambda _ + \\overline { \\Sigma } _ { 1 1 } \\ ; . \\end{align*}"}
+{"id": "5470.png", "formula": "\\begin{align*} \\mathcal J _ K \\sim \\dfrac { N ^ { 2 K } } { ( 2 a ) ^ { 2 K - 1 } } \\cdot \\binom { 2 K - 2 } { K - 1 } \\end{align*}"}
+{"id": "998.png", "formula": "\\begin{align*} W _ D [ u ] = \\nu . \\end{align*}"}
+{"id": "7031.png", "formula": "\\begin{align*} \\langle e ^ { - t \\Lambda _ { C _ \\infty } ( b ) } ( x , \\cdot ) \\rangle = 1 x \\in \\mathbb R ^ d , t > 0 . \\end{align*}"}
+{"id": "5352.png", "formula": "\\begin{align*} b ^ S _ j & < b ^ { S \\cup \\{ j \\} } _ j , j \\in N ^ { \\{ 0 , 1 \\} } \\setminus S \\\\ b ^ S _ j & > b ^ { S \\setminus \\{ j \\} } _ j , j \\in S . \\end{align*}"}
+{"id": "1343.png", "formula": "\\begin{align*} \\lambda ^ 2 - \\lambda \\alpha \\{ d ( v ) + d ( u ' ) \\} + \\alpha ^ 2 d ( v ) d ( u ' ) - ( 1 - \\alpha ) ^ 2 = 0 \\end{align*}"}
+{"id": "162.png", "formula": "\\begin{align*} \\mu ^ { [ a , b ] } _ { ( Z , c ) , ( W , d ) } \\circ u _ { F _ { a , b } ( Z , c ) , F _ { a , b } ( W , d ) } ( j _ { a , b } ( e _ { i } ) \\boxtimes j _ { a , b } ( f _ { j } ) ) = j _ { a , b } \\left ( 1 _ { X ^ { b - a } } \\otimes c _ { Z , W } ) \\circ ( e _ { j } \\otimes 1 _ { W } ) \\circ f _ { j } \\right ) \\end{align*}"}
+{"id": "5798.png", "formula": "\\begin{align*} \\sum _ { k + \\ell \\leq s } \\sum _ { i = 1 } ^ \\infty | \\xi ^ { ( k , \\ell ) } _ i ( t ) | ^ 2 = ( 1 + o ( 1 ) ) | x ( t ) | ^ 2 . \\end{align*}"}
+{"id": "6685.png", "formula": "\\begin{align*} H = L \\ltimes M . \\end{align*}"}
+{"id": "5124.png", "formula": "\\begin{align*} K ( p , q ) + \\sum _ { j } q _ { j } \\frac { \\partial K ( p , q ) } { \\partial q _ { j } } = 0 \\end{align*}"}
+{"id": "3713.png", "formula": "\\begin{align*} | \\psi \\rangle = \\sum _ \\sigma \\psi _ \\sigma { a ^ \\dagger _ \\sigma \\over \\sqrt { \\omega _ \\sigma } } | 0 \\rangle , \\end{align*}"}
+{"id": "6826.png", "formula": "\\begin{align*} \\sum _ { i , j } \\beta _ { i j } k _ { i j } ^ m + K _ m = 0 , \\sum _ { i , j } \\beta _ { i j } l _ { i j } ^ { n } + L _ n = 0 , \\qquad \\forall m \\in \\left \\{ 1 , 2 , \\ldots , \\frac { N ( N - 1 ) } { 2 } \\right \\} , \\ , n \\in \\{ 1 \\ldots , N \\} . \\end{align*}"}
+{"id": "7394.png", "formula": "\\begin{align*} \\begin{array} { l l } C _ { 3 1 } : = \\min & \\left [ \\{ 2 ^ { p + q - 1 } p A C ( 5 N ) ^ { p + q } N ^ { - 1 } \\} ^ { - 1 / ( p + q ) } , \\right . \\\\ & \\quad \\{ 2 ^ { p + q - 1 } q A C ( 5 N ) ^ { p + q } N ^ { - 1 } \\} ^ { - 1 / ( p + q ) } , \\\\ & \\quad \\left . \\{ 2 ^ { r - 1 } r B C ( 5 N ) ^ r N ^ { - 1 } \\} ^ { - 1 / ( r + 1 ) } \\right ] , \\end{array} \\end{align*}"}
+{"id": "4572.png", "formula": "\\begin{align*} \\eta \\left ( S ^ 3 / L ( q , p ) \\right ) = \\frac 1 3 \\left ( \\sum _ { i = 1 } ^ k e _ i + \\frac { q + q ^ { - 1 , p } } { p } \\right ) - k , \\end{align*}"}
+{"id": "8155.png", "formula": "\\begin{align*} P _ { \\hat { \\underline S } | \\underline X ^ L Y ^ L } ( { \\underline s } | \\underline x ^ L , y ^ L ) & = P _ { { \\underline S } | \\underline X ^ L Y ^ L } ( { \\underline s } | \\underline x ^ L , y ^ L ) \\\\ & = \\begin{cases} \\frac { 1 } { | { \\mathcal B } _ { y ^ L } ( \\underline x ^ L ) | } , & \\forall \\underline s \\in { \\mathcal B } _ { y ^ L } ( \\underline x ^ L ) \\\\ 0 , & \\end{cases} , \\end{align*}"}
+{"id": "121.png", "formula": "\\begin{align*} ( I + t P ) ^ { - 1 } = I - \\frac { t } { t + 1 } P , \\quad \\forall t \\in \\mathbb { R } \\backslash \\{ - 1 \\} . \\end{align*}"}
+{"id": "97.png", "formula": "\\begin{align*} c _ 3 + c _ 4 = 2 ( 1 - \\gamma ) - \\frac { 1 } { 2 ( 2 + \\gamma ) } > 0 \\end{align*}"}
+{"id": "3126.png", "formula": "\\begin{align*} \\varphi ^ * \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & = \\left ( \\begin{array} { c c c } a ^ { 2 \\ , p ^ { e _ 1 } } & a ^ { p ^ { e _ 1 } } \\ , b ^ { p ^ { e _ 1 } } & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & d ^ { 2 \\ , p ^ { e _ 1 } } \\end{array} \\right ) ( \\ , e _ 1 \\geq 0 \\ , ) . \\end{align*}"}
+{"id": "7294.png", "formula": "\\begin{align*} c _ { m , r } ( n ) : = ( - 1 ) ^ { n } 2 ^ { - ( n + 1 ) } C _ { m , r } ( n + 1 ) . \\end{align*}"}
+{"id": "1189.png", "formula": "\\begin{align*} & | \\partial ^ \\alpha T a ( x ) - \\partial ^ \\alpha T a ( x + h ) | \\\\ & \\quad = \\left | \\int _ 0 ^ 1 h \\cdot \\nabla \\partial ^ \\alpha T a ( x + t h ) \\ , d t \\right | \\\\ & \\quad \\lesssim | h | \\int _ 0 ^ 1 \\max _ { i \\in \\{ 1 , \\ldots , n \\} } | \\partial ^ { \\alpha + e _ i } T a ( x + t h ) | \\ , d t \\\\ & \\quad \\lesssim | h | \\int _ 0 ^ 1 | x + t h | ^ { - M } \\ , d t \\lesssim | h | ^ { N ^ { * * } } | x | ^ { - M } . \\end{align*}"}
+{"id": "9033.png", "formula": "\\begin{align*} R _ i = \\{ ( g x _ 0 , g x ) \\mid x \\in \\Lambda _ i , \\ g \\in G \\} . \\end{align*}"}
+{"id": "7345.png", "formula": "\\begin{align*} \\mathbf M ^ { * , 0 } _ { \\infty } ( \\phi ) : = \\mathbf M ^ { \\{ U ^ * _ 0 , 0 \\} } _ { \\infty } ( \\phi ) = \\mathbf M ^ { \\{ U ^ * _ 0 , 0 \\} } _ { c ^ * _ { 0 } } ( \\phi ) \\cup \\ , \\bigcup _ { i = 0 } ^ { \\infty } \\mathbf M ^ { \\{ \\emptyset , c ^ * _ i \\} } _ { c ^ * _ { i + 1 } } ( \\phi ) , \\end{align*}"}
+{"id": "4126.png", "formula": "\\begin{align*} & \\theta _ k = \\eta _ k r ^ k , k = j + 1 , \\ldots , J , \\\\ & \\theta _ j = \\eta _ j r ^ j + \\frac { 1 } { 1 - w _ { j j } } \\sum _ { i = j + 1 } ^ J w _ { j i } \\theta _ i , \\\\ & \\theta _ \\ell = w _ { \\ell j } \\theta _ j + \\sum _ { i = j + 1 } ^ { J } w _ { \\ell i } \\theta _ i , \\ell = 1 , \\ldots , j - 1 . \\end{align*}"}
+{"id": "2307.png", "formula": "\\begin{gather*} e _ { r } = \\begin{pmatrix} \\sin \\theta \\cos \\phi \\\\ \\sin \\theta \\sin \\phi \\\\ \\cos \\theta \\end{pmatrix} , e _ { \\theta } = \\begin{pmatrix} \\cos \\theta \\cos \\phi \\\\ \\cos \\theta \\sin \\phi \\\\ - \\sin \\theta \\end{pmatrix} , e _ { \\phi } = \\begin{pmatrix} - \\sin \\phi \\\\ \\cos \\phi \\\\ 0 \\end{pmatrix} . \\end{gather*}"}
+{"id": "7262.png", "formula": "\\begin{align*} m _ 1 & : = \\sharp \\big ( \\{ a b , a c , a d , b c , b d , c d \\} \\cap [ 1 , \\infty ) \\big ) , \\\\ m _ 2 & : = \\sharp \\big ( \\{ q a b , q a c , q a d , q b c , q b d , q c d \\} \\cap [ 1 , \\infty ) \\big ) . \\end{align*}"}
+{"id": "827.png", "formula": "\\begin{align*} { { g _ y } ( { \\bf R } _ y ( \\nabla { \\Psi } } ) , \\nabla { \\Psi } ) = { k } \\{ { { F ^ 2 } ( y ) { g _ y } ( \\nabla { \\Psi } } , \\nabla { \\Psi } ) - { g _ y } { { ( \\nabla { \\Psi } } , y ) } ^ 2 \\} , \\end{align*}"}
+{"id": "1114.png", "formula": "\\begin{align*} \\left | 2 ^ { j s } g _ j ( x ) \\right | ^ r & \\lesssim \\int _ { \\mathbb { R } ^ n } \\frac { 2 ^ { j n } } { ( 1 + 2 ^ j | x - z | ) ^ M } \\left | 2 ^ { j s } h _ j ( z ) \\right | ^ r \\ , d z \\\\ & = \\int _ { 3 P } \\frac { 2 ^ { j n } } { ( 1 + 2 ^ j | x - z | ) ^ M } \\left | 2 ^ { j s } h _ j ( z ) \\right | ^ r \\ , d z + \\sum _ { k \\in \\mathbb { Z } ^ n , \\ , \\| k \\| _ \\infty \\geq 2 } \\int _ { P + k \\ell ( P ) } \\cdots \\\\ & = : I _ j ( x ) + J _ j ( x ) , \\end{align*}"}
+{"id": "5990.png", "formula": "\\begin{align*} \\phi _ { \\nu } ( x _ 0 ) = \\frac { F _ { \\nu } ( x _ 0 ) } { R ^ 4 } \\geq 0 . \\end{align*}"}
+{"id": "776.png", "formula": "\\begin{align*} S _ \\beta ( x ) = \\beta ( a ^ { - 1 } b , b ) S ( x ) , x \\in H _ { a , b } . \\end{align*}"}
+{"id": "6259.png", "formula": "\\begin{align*} \\Phi g ( t , x ) = \\int _ { \\Gamma } g ( u , y ) \\frac { e ^ { i ( x - y ) ^ { 2 } / ( 4 ( t - u ) ) } } { \\sqrt { t - u } } d \\sigma ( u , y ) , \\end{align*}"}
+{"id": "4915.png", "formula": "\\begin{align*} x _ 0 + a _ 1 x _ 1 + a _ 2 x _ 2 = 0 \\end{align*}"}
+{"id": "1503.png", "formula": "\\begin{align*} \\begin{array} { l } \\int _ { \\Omega } | \\nabla u _ { n } | ^ { p - 2 } \\nabla u _ { n } \\nabla \\varphi \\mathrm { d } x + \\int _ { \\Omega } \\frac { | \\nabla u _ { n } | ^ { p } } { u _ { n } + \\delta + \\frac { 1 } { n } } \\varphi \\mathrm { d } x = \\int _ { \\Omega } f ( x , u _ { n } , \\nabla u _ { n } ) \\varphi \\mathrm { d } x , \\end{array} \\end{align*}"}
+{"id": "8722.png", "formula": "\\begin{gather*} T = \\{ f | \\ f \\in E n d ( L ) , f ( a _ { 1 } ) \\in a _ { 1 } + [ L , L ] , f ( a _ { 2 } ) \\in a _ { 2 } + [ L , L ] \\} = \\\\ C _ { E n d ( L ) } ( L / [ L , L ] ) . \\end{gather*}"}
+{"id": "2960.png", "formula": "\\begin{align*} \\R _ n ^ 2 \\left \\| \\frac { f } { | x | ^ 2 } \\right \\| _ 2 ^ 2 & = \\| \\Delta _ r f \\| _ 2 ^ 2 - \\left \\| \\Delta _ r f + \\R _ n \\frac { f } { | x | ^ 2 } \\right \\| _ 2 ^ 2 - 2 \\R _ n \\left \\| \\frac { f ^ * } { | x | } \\right \\| _ 2 ^ 2 \\\\ & = \\| \\Delta _ r f \\| _ 2 ^ 2 - \\left ( 1 + \\frac { 1 } { 2 } \\R _ n \\right ) \\left \\| \\Delta _ r f + \\R _ n \\frac { f } { | x | ^ 2 } \\right \\| _ 2 ^ 2 + \\frac { 1 } { 2 } \\R _ n \\left \\| f ^ \\# \\right \\| _ 2 ^ 2 , \\end{align*}"}
+{"id": "5848.png", "formula": "\\begin{align*} M _ { 1 } ( \\overline { \\Gamma } ) = | v ( \\Gamma ) | ( | v ( \\Gamma ) | - 1 ) ^ { 2 } - 4 | e ( \\Gamma ) | ( | v ( \\Gamma ) | - 1 ) + M _ { 1 } ( \\Gamma ) \\end{align*}"}
+{"id": "4587.png", "formula": "\\begin{align*} s _ 0 = \\frac { 8 \\pi c _ 1 [ \\omega ] } { [ \\omega ] ^ 2 } = \\frac { 4 \\pi c _ 1 [ \\omega ] } { T _ 0 } . \\end{align*}"}
+{"id": "483.png", "formula": "\\begin{align*} b \\cdot \\{ 0 , 2 , 3 , 4 , \\ldots , 2 ^ { v - t - 1 } \\} = \\{ 0 , 2 b , 3 b , 4 b , \\ldots , 2 ^ { v - t - 1 } b \\} \\end{align*}"}
+{"id": "2474.png", "formula": "\\begin{align*} \\lambda _ { \\tau } = ( c \\lambda - \\gamma ) \\lambda , ( \\gamma = ( p - 1 ) / p , \\ , c > 0 ) . \\end{align*}"}
+{"id": "5020.png", "formula": "\\begin{align*} \\begin{bmatrix} \\mathbf { x } _ { S } ^ { 1 } & \\mathbf { x } _ { S ^ c } ^ { 0 } \\end{bmatrix} \\mathbf { B } ^ S = \\mathbf { e } _ i \\Longrightarrow \\begin{bmatrix} \\mathbf { x } _ { i S } ^ { 1 , S } & \\mathbf { x } _ { i S ^ c } ^ { 0 , S } \\end{bmatrix} = \\mathbf { e } _ i \\mathbf { H } ^ S . \\end{align*}"}
+{"id": "5266.png", "formula": "\\begin{align*} \\frac { \\partial \\overline { T } _ { i } } { \\partial q _ { j } } = \\frac { \\partial \\overline { T } _ { i } } { \\partial \\overline { q } _ { i } } \\frac { \\partial \\overline { q } _ { i } } { \\partial q _ { j } } = \\left ( 1 - \\alpha \\right ) \\frac { \\delta _ { i j } - \\overline { q } _ { i } } { \\sum _ { j } q _ { j } } \\end{align*}"}
+{"id": "8443.png", "formula": "\\begin{align*} ( - \\Delta ) _ p ^ s u ( x , t ) & : = \\mathrm { P . V . } \\int _ { \\mathbb { R } ^ n } \\frac { | u ( x , t ) - u ( y , t ) | ^ { p - 2 } ( u ( x , t ) - u ( y , t ) ) } { | x - y | ^ { n + s p } } \\ , d y \\\\ & = \\lim _ { \\varepsilon \\searrow 0 } \\int _ { \\mathbb { R } ^ n \\setminus B _ \\varepsilon ( x ) } \\frac { | u ( x , t ) - u ( y , t ) | ^ { p - 2 } ( u ( x , t ) - u ( y , t ) ) } { | x - y | ^ { n + s p } } \\ , d y , \\end{align*}"}
+{"id": "2099.png", "formula": "\\begin{align*} \\sigma _ 2 ^ 2 ( x , y ) = A ( x , y ) , \\forall x , y \\in [ 0 , 1 ] . \\end{align*}"}
+{"id": "6770.png", "formula": "\\begin{align*} \\begin{cases} \\widetilde { x } ^ k = \\arg \\min \\limits _ { x \\in \\mathcal { X } } \\left \\{ \\Phi ( x , y ^ k ) + \\frac { r } { 2 } \\| x - x ^ k \\| ^ 2 \\right \\} , \\\\ \\widetilde { y } ^ k = \\arg \\max \\limits _ { y \\in \\mathcal { Y } } \\left \\{ \\Phi ( [ \\widetilde { x } ^ k + \\alpha ( \\widetilde { x } ^ k - x ^ k ) ] , y ) - \\frac { s } { 2 } \\| y - y ^ k \\| ^ 2 \\right \\} . \\end{cases} \\end{align*}"}
+{"id": "1407.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\tfrac { n } { \\mu } \\ , d G ( x \\vert \\tfrac { \\mu } { n } , \\lambda ) & = x ^ { - 1 } e ^ { - \\lambda x } \\ , d x \\end{align*}"}
+{"id": "5245.png", "formula": "\\begin{align*} L _ { d } F \\left ( p \\| q \\right ) = \\sum _ { i } \\frac { p _ { i } } { a - b } \\left [ \\left ( Z _ { i } \\right ) ^ { a - 1 } - \\left ( Z _ { i } \\right ) ^ { b - 1 } \\right ] + \\left ( 1 - \\alpha \\right ) \\left ( q _ { i } - p _ { i } \\right ) \\end{align*}"}
+{"id": "7448.png", "formula": "\\begin{align*} \\int _ { \\blacktriangle } F \\left ( t _ 1 , \\ldots , t _ n \\right ) \\ , \\dd t _ 1 \\ldots \\ , \\dd t _ n = \\frac { 1 } { n ! } \\left [ \\int _ 0 ^ L f ( t ) \\ , \\dd t \\right ] ^ { \\otimes n } . \\end{align*}"}
+{"id": "8912.png", "formula": "\\begin{align*} W _ n ( x ) & : = \\prod _ { \\substack { 1 \\leq k < n / 2 \\\\ ( k , n ) = 1 } } \\left ( x ^ 2 + 4 \\sin ^ 2 \\left ( \\frac { \\pi k } { n } \\right ) \\right ) \\end{align*}"}
+{"id": "3520.png", "formula": "\\begin{align*} \\phi _ * ( \\textbf { a } _ i ) & = k _ i \\cdot \\textbf { a } _ { \\sigma ( i ) } \\\\ \\phi _ * ( \\textbf { r } _ i ) & = p _ i \\cdot \\textbf { r } _ { \\sigma ( i ) } i = 1 , 2 , 3 \\\\ \\phi _ * ( \\textbf { r } _ 0 ) & = \\kappa \\cdot \\textbf { r } _ 0 + \\sum _ { i = 1 } ^ m \\mu _ i \\cdot \\textbf { r } _ { \\sigma ( i ) } \\kappa , \\mu _ i \\in \\Z . \\end{align*}"}
+{"id": "7578.png", "formula": "\\begin{align*} \\| \\Delta v _ \\epsilon \\| _ { 2 } ^ { 2 } = \\mu \\gamma _ { q } t _ \\epsilon ^ { \\frac { N ( q - 2 ) } { 4 } - 2 } \\| v _ \\epsilon \\| _ { q } ^ { q } + \\gamma _ { 4 ^ * } t _ \\epsilon ^ { \\frac { N ( 4 ^ * - 2 ) } { 4 } - 2 } \\| v _ \\epsilon \\| _ { 4 ^ * } ^ { 4 ^ * } . \\end{align*}"}
+{"id": "6163.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau ^ k } [ f ( x ) - f ( \\breve { x } ^ { k } ) ] - \\frac { 1 } { \\tau ^ { k - 1 } } [ f ( x ) - f ( \\breve { x } ^ { k - 1 } ) ] + ( u - \\widetilde { u } ^ k ) ^ T F ( \\widetilde { u } ^ k ) \\\\ & + c ^ k ( A ( x - \\widetilde { x } ^ k ) ) ^ T ( A \\breve { x } ^ { k - 1 } - b ) \\geq ( v - \\widetilde { v } ^ k ) ^ T Q ^ k ( v ^ k - \\widetilde { v } ^ k ) , ~ \\forall u , \\end{aligned} \\end{align*}"}
+{"id": "1764.png", "formula": "\\begin{align*} \\pi _ r \\circ \\mathrm { F l } ^ { X _ { - j } } _ { t } ( z ) = z _ r + \\delta _ { r , j } t - \\delta _ { r , j + 1 } t z _ n + \\delta _ { r , 0 } ( 2 t z _ { n - j } + \\alpha _ { n , j , t } ) , \\end{align*}"}
+{"id": "6020.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\nu _ \\rho } \\big [ c _ x ( \\eta ) [ ( D _ 1 \\xi ^ { A } _ { x + 1 } + D _ 2 \\xi ^ { B } _ { x + 1 } ) - ( D _ 1 \\xi ^ { A } _ { x } + D _ 2 \\xi ^ { B } _ { x } ) ] ^ 2 \\big ] = \\frac { 4 } { 9 } ( D _ 1 ^ 2 + D _ 2 ^ 2 - D _ 1 D _ 2 ) \\ , . \\end{align*}"}
+{"id": "863.png", "formula": "\\begin{align*} ( a ) _ { 2 \\times 2 } & = \\frac { 1 } { \\rho ^ 2 } \\begin{pmatrix} \\rho + ( w _ 1 ) ^ 2 & w _ 1 w _ 2 \\\\ w _ 1 w _ 2 & \\rho + ( w _ 2 ) ^ 2 \\end{pmatrix} = \\frac { 1 } { \\rho ^ 2 } \\begin{pmatrix} 1 - ( w _ 2 ) ^ 2 & w _ 1 w _ 2 \\\\ w _ 1 w _ 2 & 1 - ( w _ 1 ) ^ 2 \\end{pmatrix} . \\end{align*}"}
+{"id": "7923.png", "formula": "\\begin{align*} \\mathfrak { Z } ^ { \\ast k } : = \\{ \\mu \\in H ^ { \\ast } \\Lambda ^ { k } ( \\Omega ) \\mid \\delta \\mu = 0 \\} , \\mathring { \\mathfrak { Z } } ^ { \\ast k } : = \\{ \\mu \\in \\mathring { H } ^ { \\ast } \\Lambda ^ { k } ( \\Omega ) \\mid \\delta \\mu = 0 \\} , \\end{align*}"}
+{"id": "6790.png", "formula": "\\begin{align*} J = \\mu \\begin{bmatrix} \\sum _ { i = 1 } ^ 3 a _ i c _ i u _ i & \\sum _ { i = 1 } ^ 3 b _ i c _ i u _ i \\\\ \\sum _ { i = 1 } ^ 3 a _ i d _ i u _ i & \\sum _ { i = 1 } ^ 3 b _ i d _ i u _ i \\end{bmatrix} \\begin{bmatrix} 1 / \\bar { x } & 0 \\\\ 0 & 1 / \\bar { y } \\end{bmatrix} . \\end{align*}"}
+{"id": "8802.png", "formula": "\\begin{align*} \\int y \\ , \\pi _ { x _ - } ( d y ) = x _ - = \\int y \\ , \\pi _ { x _ - } ^ { \\uparrow } ( d y ) , \\end{align*}"}
+{"id": "3371.png", "formula": "\\begin{align*} \\chi ( y ) = \\left \\{ \\begin{array} { l l } 1 & \\ \\ \\ \\ \\abs { y } \\leq 1 , \\\\ 0 & \\ \\ \\ \\ \\abs { y } \\geq 2 , \\end{array} \\right . \\end{align*}"}
+{"id": "799.png", "formula": "\\begin{align*} & { D _ m } { D _ k } \\Psi - { D _ k } { D _ m } \\Psi = K _ { 0 m k } ^ r { \\dot \\partial _ r } \\Psi . \\end{align*}"}
+{"id": "8267.png", "formula": "\\begin{align*} \\beta _ \\lambda ^ m : = ( 1 - \\lambda ) \\beta _ 0 ^ m + \\lambda \\beta ^ m _ 1 . \\end{align*}"}
+{"id": "7406.png", "formula": "\\begin{align*} S _ r : = \\sum _ { j = 0 } ^ \\infty \\frac { j + 1 } { r ^ { j + 1 } } < \\infty . \\end{align*}"}
+{"id": "3441.png", "formula": "\\begin{align*} \\tilde { H } _ G ^ \\ast ( X , m , n ) : = \\tilde { H } _ G ^ { \\ast + m + k _ { G } \\cdot n } ( X ) . \\end{align*}"}
+{"id": "5461.png", "formula": "\\begin{align*} F ( u , v ) : = \\frac { 1 } { 3 6 } | u | ^ 4 + \\frac { 9 } { 4 } | v | ^ 4 + | u | ^ 2 | v | ^ 2 + \\frac { 1 } { 9 } \\Re ( \\bar { u } ^ 3 v ) . \\end{align*}"}
+{"id": "6601.png", "formula": "\\begin{align*} R _ { k , \\beta } ( z ) = \\begin{cases} G _ { k , \\beta } ( 1 - ( 1 - \\delta ) z , z ) & k , \\\\ I _ { k , \\beta } ( 1 - ( 1 - \\delta ) z , z ) & k . \\end{cases} \\end{align*}"}
+{"id": "3480.png", "formula": "\\begin{align*} B _ j ^ { 0 , K } = \\begin{cases} \\frac { 1 } { a _ { j + 1 } - a _ j } , \\ a _ j < a _ { j + 1 } , a _ j \\leq x < a _ { j + 1 } , \\\\ 0 , \\ \\ , . \\end{cases} \\end{align*}"}
+{"id": "7567.png", "formula": "\\begin{align*} f ( t ) t - 2 F ( t ) > \\frac { 8 } { N } F ( t ) , \\quad \\mathrm { f o r } \\ \\mathrm { a l l } \\ t \\not = 0 . \\end{align*}"}
+{"id": "1655.png", "formula": "\\begin{align*} P _ { \\varepsilon , T } = \\Omega \\times \\left ( 0 , T - \\varepsilon \\right ) \\subset Q _ { T } . \\end{align*}"}
+{"id": "2164.png", "formula": "\\begin{align*} A _ d '' : = \\{ a _ { i _ 1 } , \\dots , a _ { i _ { K / 4 } } \\} , \\end{align*}"}
+{"id": "6382.png", "formula": "\\begin{align*} \\delta _ { i , m } ( w _ { r } ) = \\sum _ k \\chi _ { m , m - 2 i } \\ , e _ { r k } \\otimes w _ { k } . \\end{align*}"}
+{"id": "3160.png", "formula": "\\begin{align*} ( i + j n ) ^ { \\rho _ \\tau } = i + j ^ \\tau n \\ \\ i \\in [ n ] \\ , \\ , j \\in [ k ] . \\end{align*}"}
+{"id": "6339.png", "formula": "\\begin{align*} S ( \\Gamma ) \\leq \\sum _ { j = 1 } ^ { M _ B } S ( \\Gamma _ { j } ) = \\sum _ { j = 1 } ^ { M _ B } \\sum _ { n = 0 } ^ N S ( \\Gamma _ { j , n } ) \\end{align*}"}
+{"id": "5975.png", "formula": "\\begin{align*} \\sum _ { i \\in S _ e } B _ i = \\sum _ { i \\in S _ e } \\Bigg \\lceil \\frac { f _ i } { \\sum _ { j \\in S _ e } f _ j } \\cdot B \\Bigg \\rceil \\leq \\sum _ { i \\in S _ e } \\Bigg ( \\frac { f _ i } { \\sum _ { j \\in S _ e } f _ j } \\cdot B + 1 \\Bigg ) = B + | S _ e | \\leq B + k . \\end{align*}"}
+{"id": "2337.png", "formula": "\\begin{align*} \\| a _ { 1 } \\| _ { L _ { t } ^ { \\infty } L _ { x } ^ { p } } + \\big \\| \\nabla | a _ { 1 } | ^ { \\frac { p } { 2 } } \\big \\| ^ { \\frac { 2 } { p } } _ { L _ { t } ^ { 2 } L _ { x } ^ { 2 } } \\leq \\| w _ { 0 } \\| _ { L ^ { p } } \\begin{cases} 1 + \\frac { 1 } { \\sqrt { 2 } } , & p = 2 , \\\\ 1 + \\sqrt [ p ] { \\frac { p - 2 } { 4 p } } , & p > 2 . \\end{cases} \\end{align*}"}
+{"id": "325.png", "formula": "\\begin{align*} & v _ t = ( m - 1 ) v \\Delta v + | \\nabla v | ^ 2 + K ( m , p ) v ^ { ( m + p - 2 ) / ( m - 1 ) } , \\\\ & K ( m , p ) = m \\left ( \\frac { m - 1 } { m } \\right ) ^ { ( m + p - 2 ) / ( m - 1 ) } . \\end{align*}"}
+{"id": "4957.png", "formula": "\\begin{align*} e _ { \\Gamma ' } ( S ) \\geq e _ { \\Gamma } ( S ) - \\gamma e ( \\Gamma ) \\geq d \\binom { | S | } { 2 } - \\frac { \\gamma } { 2 } n ^ 2 = d \\binom { | S | } { 2 } - \\frac { \\gamma } { \\rho ^ 2 } \\frac { | S | ^ 2 } { 2 } \\geq d ' \\ , \\binom { | S | } { 2 } , \\end{align*}"}
+{"id": "7346.png", "formula": "\\begin{align*} \\mathbf M ^ { * , - \\infty } _ { \\infty } ( \\phi ) = \\bigcup _ { i \\in \\Z } \\mathbf M ^ { \\{ \\emptyset , c ^ * _ i \\} } _ { c ^ * _ { i + 1 } } ( \\phi ) , \\end{align*}"}
+{"id": "3765.png", "formula": "\\begin{align*} | g _ N ( z ) | & = \\exp \\Big ( P _ { f _ N } ( z ) \\Big ) \\\\ & \\leq \\exp \\Big ( \\frac { C _ N } { 1 - | z | } \\Big ) \\\\ & \\leq \\exp \\Big ( \\frac { C _ N } { d } \\log ( 1 / G ( 1 - | z | ) ) \\Big ) \\\\ & = \\Big ( G ( 1 - | z | ) \\Big ) ^ { - C _ N / d } . \\end{align*}"}
+{"id": "4124.png", "formula": "\\begin{align*} & Q ( \\theta ( r ) ) = Q ^ * ( \\theta ( r ) ) + o ( \\abs { \\theta ( r ) } ^ 2 ) ) \\end{align*}"}
+{"id": "921.png", "formula": "\\begin{align*} P _ V ( u ) ( x ) : = \\int _ E u ( y ) \\ , P _ V ( x , d y ) = u ( x ) - \\pi _ V ( u ) ( x ) , u \\in F \\cap \\mathcal B _ b ( E ) , \\ , \\ , m \\end{align*}"}
+{"id": "3709.png", "formula": "\\begin{align*} G _ N ( x , y | E ) = G _ { N - 1 } ( x , y | E ) + G _ { N - 1 } ( x , a _ N | E ) \\Phi ^ { - 1 } ( E ) G _ { N - 1 } ( a _ N , y | E ) , \\end{align*}"}
+{"id": "630.png", "formula": "\\begin{align*} F ( L _ k ( t ) , M _ k ) = A _ k ( t ) V ( K ) - B _ k ( t ) V ( K , M _ k [ n - 1 ] ) \\geq \\left ( c _ n V ( K ) - \\frac { \\kappa _ n w ( K ) } { n } S _ K ( 2 U _ k ) \\right ) t > 0 . \\end{align*}"}
+{"id": "205.png", "formula": "\\begin{align*} d t = r \\ , d \\tau \\Longleftrightarrow \\frac { d \\tau } { d t } = \\frac 1 r , \\end{align*}"}
+{"id": "3614.png", "formula": "\\begin{align*} \\alpha ^ 3 \\tau _ 0 ^ 2 \\tau _ 1 ( \\sigma ^ 2 \\sigma _ 1 ) ^ 2 - ( 1 + a ) \\alpha ^ 2 \\tau _ 0 \\tau _ 1 ( \\sigma \\sigma _ 1 ) ^ 2 + ( a + b ) \\alpha \\tau _ 0 \\sigma ^ 2 - b = 0 . \\end{align*}"}
+{"id": "166.png", "formula": "\\begin{align*} R _ 1 \\times R _ 2 \\ = \\ \\{ ( x _ 1 , x _ 2 ) , ( y _ 1 , y _ 2 ) ) : ( x _ 1 , y _ 1 ) \\in R _ 1 \\ \\ \\ \\ ( x _ 2 , y _ 2 ) \\in R _ 2 \\} . \\end{align*}"}
+{"id": "7852.png", "formula": "\\begin{align*} f ( \\mathbf { x } ) = \\sum _ { \\mathbf { b } \\in \\mathbb { F } _ { p } ^ { m } } \\lambda _ { \\mathbf { b } } ( \\prod _ { i = 1 } ^ { m } x _ { i } ^ { b _ { i } } ) , \\end{align*}"}
+{"id": "2797.png", "formula": "\\begin{align*} L _ { T } = P _ { i } \\dot { Q } ^ { i } + \\Theta _ { \\alpha } \\dot { \\Theta } ^ { \\alpha } - H _ { T } ( \\Xi ^ { a } = \\epsilon ^ { a } , \\Psi _ { a } = \\epsilon _ { a } , \\Theta ^ { \\alpha } , \\Theta _ { \\alpha } , Q ^ { i } , P _ { i } ) . \\end{align*}"}
+{"id": "7608.png", "formula": "\\begin{align*} \\| u _ { \\epsilon } \\| _ { q } ^ { q } & = R ^ { q } \\omega \\epsilon ^ { \\frac { ( N - 4 ) q } { 2 } } \\int _ { 0 } ^ { 2 } \\frac { ( \\varphi ^ { q } ( r ) - 1 ) r ^ { N - 1 } } { ( \\epsilon ^ { 2 } + r ^ { 2 } ) ^ { \\frac { ( N - 4 ) q } { 2 } } } d r + R ^ { q } \\omega \\epsilon ^ { \\frac { ( N - 4 ) q } { 2 } } \\int _ { 0 } ^ { 2 } \\frac { r ^ { N - 1 } } { ( \\epsilon ^ { 2 } + r ^ { 2 } ) ^ { \\frac { ( N - 4 ) q } { 2 } } } d r \\\\ & = I _ { 1 } ( \\epsilon ) + I _ { 2 } ( \\epsilon ) . \\\\ \\end{align*}"}
+{"id": "7818.png", "formula": "\\begin{align*} \\eta _ k ^ { \\star } = \\frac { a _ { k } { E } _ { k } \\left | \\left ( \\boldsymbol { v } _ { k } \\right ) ^ { \\mathrm { H } } \\ ! \\boldsymbol { g } _ { k } \\right | ^ { 2 } } { \\sum _ { l \\neq k } a _ { l } { E } _ { l } \\left | \\left ( \\boldsymbol { v } _ { k } \\right ) ^ { \\mathrm { H } } \\boldsymbol { g } _ { l } \\right | ^ { 2 } \\ ! + \\ ! L \\sigma ^ { 2 } \\left \\| \\boldsymbol { v } _ { k } \\right \\| ^ { 2 } } , \\forall k \\in \\mathcal { K } . \\end{align*}"}
+{"id": "2293.png", "formula": "\\begin{align*} \\ell = \\frac { 1 } { 3 } ( m _ 1 + 2 m _ 2 + m ' _ 1 + 2 m ' _ 2 - m '' _ 1 - 2 m '' _ 2 ) \\ \\ \\ \\ \\ \\ \\ \\ n = \\frac { 1 } { 3 } ( 2 m _ 1 + m _ 2 + 2 m ' _ 1 + m ' _ 2 - 2 m '' _ 1 - m '' _ 2 ) \\ . \\end{align*}"}
+{"id": "5811.png", "formula": "\\begin{align*} - 2 ^ { - 1 } p r ^ { p - 1 } ( t ) \\hat { f } _ p ( \\theta ^ * ) \\leq - 8 ^ { - 1 } A ^ { - 1 } p \\hat { f } _ p ( \\theta ^ * ) r ^ { \\varepsilon } ( t ) | \\theta ' ( t ) | = - 4 c _ 3 r ^ { \\varepsilon } ( t ) | \\theta ' ( t ) | . \\end{align*}"}
+{"id": "8856.png", "formula": "\\begin{align*} 0 < \\frac 1 { a _ 2 } , \\frac { 1 } { b _ 2 } < 1 , \\frac { 1 } { a _ 2 } + \\frac { 1 } { b _ 2 } = \\frac { n + 2 } { 2 n } + \\frac { \\alpha } { n } , \\end{align*}"}
+{"id": "135.png", "formula": "\\begin{align*} \\mathfrak { R } = \\left \\{ \\vec { c } + \\mathfrak { m } \\vec { x } \\ ; | \\ ; \\vec { c } \\in \\mathbb { R } ^ d , \\mathfrak { m } \\in \\mathfrak { s o } ( d ) \\right \\} , \\end{align*}"}
+{"id": "7103.png", "formula": "\\begin{align*} \\mu \\| w \\| _ p ^ p + I _ p + ( p - 2 ) J _ p = \\langle | b | ^ { 1 - \\frac { 2 } { p } } f , - \\nabla \\cdot ( w | w | ^ { p - 2 } ) \\rangle , \\end{align*}"}
+{"id": "6904.png", "formula": "\\begin{align*} A _ { D } ( \\tau ) & = 1 6 ( ( \\gamma + \\eta \\cos ( \\sin \\tau ) \\cosh ( \\cos \\tau ) ) ^ 2 + \\eta ^ 2 \\sin ^ 2 ( \\sin \\tau ) \\sinh ^ 2 ( \\cos \\tau ) ) ^ 2 . \\end{align*}"}
+{"id": "1019.png", "formula": "\\begin{align*} m _ { q , c } ( L ) = \\lim _ { n \\to \\infty } \\ , \\min _ { \\gamma \\in \\Gamma _ { q , c } ( n ) } s _ L ( \\gamma ^ { ( 1 ) } ) + \\ldots + s _ L ( \\gamma ^ { ( c ) } ) . \\end{align*}"}
+{"id": "8854.png", "formula": "\\begin{align*} 0 < \\frac { 1 } { p b _ 1 } \\leq \\frac 1 r \\leq 1 , 0 \\leq \\frac { \\tau } { p } < \\frac { n } { p b _ 1 } , \\frac { \\tau } { p } - 1 = \\frac { n } { p b _ 1 } - \\frac { n } { r } . \\end{align*}"}
+{"id": "2114.png", "formula": "\\begin{align*} & \\big | B ( x _ 1 , x _ 3 , x _ 1 , x _ 3 ) + B ( x _ 2 , x _ 3 , x _ 2 , x _ 3 ) - 2 B ( x _ 1 , x _ 3 , x _ 2 , x _ 3 ) \\big | \\\\ = & \\big | A ( x _ 1 , x _ 3 ) \\left ( A ( x _ 1 , x _ 1 ) + A ( x _ 2 , x _ 2 ) - 2 A ( x _ 1 , x _ 2 ) \\right ) - 2 \\left ( A ( x _ 1 , x _ 3 ) - A ( x _ 2 , x _ 3 ) \\right ) ^ 2 \\big | \\le C | x _ 1 - x _ 2 | ^ 2 , \\end{align*}"}
+{"id": "9130.png", "formula": "\\begin{align*} P _ { \\lambda _ { h , \\beta } } = _ { \\mathfrak { S } _ { d _ { \\beta } } } \\left ( w _ { \\beta , 1 } ^ { r _ { \\beta } ( h , 1 ) } \\cdots w _ { \\beta , d _ { \\beta } } ^ { r _ { \\beta } ( h , d _ { \\beta } ) } \\prod _ { 1 \\leq i < j \\leq d _ { \\beta } } \\frac { w _ { \\beta , i } - v _ { \\beta } ^ { - 2 } w _ { \\beta , j } } { w _ { \\beta , i } - w _ { \\beta , j } } \\right ) . \\end{align*}"}
+{"id": "2766.png", "formula": "\\begin{align*} \\sigma _ { 2 } ( t ) : & T ^ { * } M | _ { Q , P } \\times \\mathbb { R } \\rightarrow T ^ { * } M \\times \\mathbb { R } \\\\ & ; ( { \\sigma _ { 2 } ^ { * } } ( t ) \\Theta ^ { \\alpha } : = \\epsilon ^ { \\alpha } , { \\sigma _ { 2 } ^ { * } } ( t ) \\Theta _ { \\alpha } : = \\epsilon _ { \\alpha } , { \\sigma _ { 2 } ^ { * } } ( t ) Q ^ { i } , { \\sigma _ { 2 } ^ { * } } ( t ) P _ { i } , { \\sigma _ { 2 } ^ { * } } ( t ) u = t ) \\\\ & \\mapsto ( \\Theta ^ { \\alpha } , \\Theta _ { \\alpha } , Q ^ { i } , P _ { i } , u ) , \\end{align*}"}
+{"id": "5870.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { N C } ( G ) ) = 5 \\cdot 2 ^ { 3 n - 3 } - 1 8 \\cdot 2 ^ { 2 n - 2 } + 1 6 \\cdot 2 ^ { n - 1 } \\end{align*}"}
+{"id": "6316.png", "formula": "\\begin{align*} a ^ * ( g ) = \\int _ { \\Lambda } g ( x ) a _ x ^ * d x , a ( g ) = \\int _ { \\Lambda } \\overline { g ( x ) } a _ x d x \\end{align*}"}
+{"id": "3581.png", "formula": "\\begin{align*} \\alpha ^ 3 ( \\tau _ 0 ^ 2 \\tau _ 1 ) = ( 1 + a ) \\alpha ^ 2 \\tau _ 0 \\tau _ 1 - ( a + b ) \\alpha \\tau _ 0 + b , \\end{align*}"}
+{"id": "8698.png", "formula": "\\begin{align*} \\mbox { $ N _ b \\Box ^ { ( 0 ) } _ { b , G } + S _ G = I $ o n $ { \\rm D o m \\ , } \\Box ^ { ( 0 ) } _ { b , G } $ } , \\\\ \\mbox { $ \\Box ^ { ( 0 ) } _ { b , G } N _ b + S _ G = I $ o n $ L ^ 2 ( X ) ^ G $ } . \\end{align*}"}
+{"id": "4602.png", "formula": "\\begin{align*} \\frac { \\tilde { p } } { s } - \\frac { 1 } { 2 s ' } = \\frac { d } { 2 } , \\end{align*}"}
+{"id": "8059.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow + \\infty } \\widetilde { P } _ { h _ 1 ^ m , h _ 2 ^ m , h _ 3 ^ m } ^ { N , F ^ m , G ^ m , H ^ m } \\left ( D _ { \\epsilon , 3 } \\cap \\{ \\eta ^ N \\in O \\} \\right ) = 1 . \\end{align*}"}
+{"id": "3426.png", "formula": "\\begin{align*} \\underline { d } ( X ) = i + n \\overline { d } ( X ) = j + n . \\end{align*}"}
+{"id": "5484.png", "formula": "\\begin{align*} \\int _ { | M | \\le 3 | E | } | M + E | ^ K & \\le \\int _ { | M | \\le 3 | E | } ( | M | + | E | ) ^ K d U \\\\ & \\le \\int _ { | M | \\le 3 | E | } ( 4 | E | ) ^ K d U \\\\ & \\ll _ K \\int _ { U ( N ) } | E | ^ K d U \\\\ & = \\mathbb E | E | ^ { K } \\\\ & = o \\left ( \\mathbb E | M | ^ { K } \\right ) \\end{align*}"}
+{"id": "1387.png", "formula": "\\begin{align*} \\begin{pmatrix} a _ { 0 0 } & 0 & a _ { 1 0 } & 0 \\\\ 0 & a _ { 0 0 } & 0 & - a _ { 1 0 } \\\\ a _ { 0 1 } & 0 & a _ { 1 1 } & 0 \\\\ 0 & - a _ { 0 1 } & 0 & a _ { 1 1 } \\end{pmatrix} \\end{align*}"}
+{"id": "6838.png", "formula": "\\begin{align*} u _ { 1 N } '' & = u _ { 1 N } ' e ^ { - i \\phi _ 2 } \\cos ( \\psi _ 2 ) - u _ { 3 N } ' \\sin ( \\psi _ 2 ) , \\\\ u _ { 3 N } '' & = u _ { 1 N } ' e ^ { - i \\phi _ 2 } \\sin ( \\psi _ 2 ) + u _ { 3 N } ' \\cos ( \\psi _ 2 ) . \\end{align*}"}
+{"id": "8765.png", "formula": "\\begin{align*} \\pi ^ \\uparrow ( d x , d y ) = \\int _ 0 ^ 1 \\bigg ( & \\phi _ \\uparrow ' ( u ) \\delta _ { \\left ( F _ \\mu ^ { - 1 } ( u ) , F _ { \\tilde \\nu _ l } ^ { - 1 } \\left ( \\frac { \\phi _ \\uparrow ( u ) } { \\nu _ l ( \\R ) } \\right ) \\right ) } ( d x , d y ) \\\\ & + ( 1 - \\phi _ \\uparrow ' ( u ) ) \\delta _ { \\left ( F _ \\mu ^ { - 1 } ( u ) , F _ { \\tilde \\nu _ r } ^ { - 1 } \\left ( \\frac { u - \\phi _ \\uparrow ( u ) } { \\nu _ r ( \\R ) } \\right ) \\right ) } ( d x , d y ) \\bigg ) d u . \\end{align*}"}
+{"id": "8281.png", "formula": "\\begin{align*} \\dot { u } ( t ) = \\Delta u ( t ) \\mbox { i n } { \\mathcal D } ( \\Omega ) ' \\mbox { f o r a l l } t > 0 \\end{align*}"}
+{"id": "4167.png", "formula": "\\begin{align*} \\varphi \\big ( g ^ { j } _ 1 g _ 3 \\big ) \\varphi \\big ( g ^ { i } _ 1 g _ 2 \\big ) \\overset { ( \\ref { e q : 8 } ) } { = } \\varphi ( g _ 3 ) \\varphi \\big ( g ^ { j } _ 1 \\big ) \\varphi \\big ( g ^ { i } _ 1 g _ 2 \\big ) \\overset { ( \\ref { e q : 5 } ) } { = } \\varphi ( g _ 3 ) \\varphi \\big ( g ^ { j } _ 1 \\big ) \\varphi \\big ( g ^ { i } _ 1 \\big ) \\varphi ( g _ 2 ) \\overset { { \\bf A 3 } } { = } \\varphi ( g _ 3 ) \\varphi ( g ^ { n } _ 1 ) \\varphi ( g _ 2 ) \\ , . \\end{align*}"}
+{"id": "1574.png", "formula": "\\begin{align*} | D F ^ { - 1 } ( z ) | \\ge \\frac { \\sup _ { | v | = 1 } | D F ^ { - 1 } ( v ) | } { C \\ , \\eta _ H ( 1 / | z | ) } . \\end{align*}"}
+{"id": "276.png", "formula": "\\begin{align*} \\frac { d ^ 2 x } { d t ^ 2 } + \\gamma ( x ) \\Bigl ( \\frac { d x } { d t } \\Bigr ) ^ 2 + A ( x ) \\frac { d x } { d t } + b ( x ) = 0 , \\end{align*}"}
+{"id": "8990.png", "formula": "\\begin{align*} u ( r ) = r ^ { - \\frac { \\tilde { N } _ + - 2 } { 2 } } ( c _ 1 ( - \\ln r ) + c _ 2 ) . \\end{align*}"}
+{"id": "3559.png", "formula": "\\begin{align*} P = \\frac { ( T - \\lambda I ) ( T - \\mu I ) } { ( \\lambda - 1 ) ( \\mu - 1 ) } , Q = \\frac { ( T - I ) ( T - \\mu I ) } { ( \\lambda - 1 ) ( \\lambda - \\mu ) } , R = \\frac { ( T - I ) ( T - \\lambda I ) } { ( \\mu - 1 ) ( \\mu - \\lambda ) } . \\end{align*}"}
+{"id": "5702.png", "formula": "\\begin{align*} q ( u ) : = ( u , u ' - 2 ^ { - 1 } m u ) , \\ \\mathcal { E } ( u ) : = ( 0 , E _ 1 ( u ) ) . \\end{align*}"}
+{"id": "5587.png", "formula": "\\begin{align*} f ( g , o ) = \\sum _ { e : e _ 1 = o } \\left ( \\vec f _ { \\phi _ i , t + 1 } ( g , e ) - \\nu _ i ^ 2 \\vec f _ { \\phi _ i , t } ( g , e ) \\right ) ^ 2 \\end{align*}"}
+{"id": "346.png", "formula": "\\begin{align*} z ^ n - y ^ n = ( z - y ) ( z ^ { n - 1 } + y ^ { n - 1 } ) + z y ( z ^ { n - 2 } - y ^ { n - 2 } ) \\end{align*}"}
+{"id": "2717.png", "formula": "\\begin{align*} \\Phi ^ { ( 2 ) } _ { \\alpha } : = \\tau ^ { i } _ { \\alpha } S _ { i } : \\approx 0 . \\end{align*}"}
+{"id": "2788.png", "formula": "\\begin{align*} \\delta \\Xi ^ { a ' } ( t _ { 2 } ) : = 0 \\end{align*}"}
+{"id": "2446.png", "formula": "\\begin{align*} \\begin{cases} u ( \\xi ) \\sim A _ { 3 } ( \\xi _ { + } - \\xi ) ^ { p } , \\\\ u ' ( \\xi ) \\sim - A _ { 4 } ( \\xi _ { + } - \\xi ) ^ { p - 1 } \\end{cases} { \\rm { a s } } \\xi \\nearrow \\xi _ { + } - 0 , \\end{align*}"}
+{"id": "338.png", "formula": "\\begin{align*} g _ { I ( G ) } ( k + 1 ) & = \\min \\{ g _ { I ( G _ 1 ) } ( k + 1 ) + 1 , g _ { I ( G _ 2 ) } ( k ) , g _ { I ( G _ 3 ) } ( k ) , g _ { I ( G _ 3 ) } ( k + 1 ) + 1 \\} , \\\\ g _ { I ( G ) } ( k ) & = \\min \\{ g _ { I ( G _ 1 ) } ( k ) + 1 , g _ { I ( G _ 2 ) } ( k - 1 ) , g _ { I ( G _ 3 ) } ( k - 1 ) , g _ { I ( G _ 3 ) } ( k ) + 1 \\} . \\end{align*}"}
+{"id": "1044.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } \\| z _ { k } - z _ { k , n } \\| = O ( n ^ { - d + \\kappa ( 1 - d ) } ) + O \\bigg ( \\sum _ { t = [ n ^ { \\kappa } ] + 1 } ^ { \\infty } \\| y _ { t } \\| \\bigg ) , n \\to \\infty . \\end{align*}"}
+{"id": "9041.png", "formula": "\\begin{align*} a b - ( - 1 ) ^ { p ( a ) p ( b ) } b a = \\int _ { - T } ^ 0 d \\Lambda [ a _ \\Lambda b ] . \\end{align*}"}
+{"id": "8595.png", "formula": "\\begin{align*} M _ k = 2 \\cosh ^ { - 1 } ( 1 + 2 k ) = 2 \\log ( 1 + 2 k + 2 \\sqrt { k ^ 2 + k } ) \\end{align*}"}
+{"id": "5592.png", "formula": "\\begin{align*} \\underline { B } ^ { ( k ) } _ { e f } = \\sum _ { \\gamma \\in F _ { e f } ^ { 2 k + 2 } } \\underline X _ { e _ 1 e _ 3 , e _ 2 } \\prod _ { s = 1 } ^ { k } \\underline A _ { \\gamma _ { 2 s } \\gamma _ { 2 s + 2 } , \\gamma _ { 2 s + 1 } } . \\end{align*}"}
+{"id": "2675.png", "formula": "\\begin{align*} \\delta S '^ { ( 1 ) } = { \\textrm { t h e s a m e t e r m s t o } } \\delta S ^ { ( 2 ) } + \\left [ \\sum ^ { n } _ { i = 1 } \\left \\{ \\frac { \\partial L ^ { ( 2 ) } } { \\partial \\dot { q } ^ { i } } - \\frac { d } { d t } \\left ( \\frac { \\partial L ^ { ( 2 ) } } { \\partial \\ddot { q } ^ { i } } \\right ) + \\frac { \\partial W } { \\partial q ^ { i } } \\right \\} \\delta q ^ { i } \\right ] ^ { t _ { 2 } } _ { t _ { 1 } } \\end{align*}"}
+{"id": "5693.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } { u ( t ) } / { \\Vert u ( t ) \\Vert _ { L ^ 2 } } = w \\ \\textup { i n } \\ C ^ \\infty ( \\Sigma ; \\mathbf { V } ) , \\end{align*}"}
+{"id": "4141.png", "formula": "\\begin{align*} b _ i = \\sum _ { k \\in \\mathcal { J } } P _ { i k } ( u _ k ^ 2 - u _ k ) + w _ { i j } - w _ { i j } ^ 2 , \\end{align*}"}
+{"id": "618.png", "formula": "\\begin{align*} \\left ( \\left ( \\frac { d } { d u } \\right ) ^ { n } \\frac { 1 } { ( u + 1 ) \\sqrt { 1 - u ^ 2 } } \\right ) \\ , \\Bigg | _ { u = 0 } = \\left ( - \\frac { 1 } { 2 } \\right ) ^ n ( n + 1 ) ! \\binom { n } { \\left \\lfloor \\frac { n } { 2 } \\right \\rfloor } . \\end{align*}"}
+{"id": "3058.png", "formula": "\\begin{align*} - k = - 4 \\det ( A ) = f ( b ^ 2 - 4 a c ) + ( a e ^ 2 - b e d + c d ^ 2 ) . \\end{align*}"}
+{"id": "8917.png", "formula": "\\begin{align*} \\prod _ { d \\mid n } V _ d ( x ) ^ { \\mu ( n / d ) } = W _ n ( x ) , \\end{align*}"}
+{"id": "5083.png", "formula": "\\begin{align*} E _ s \\cdot f = ( q ^ k - 1 ) \\pi _ s ( f ) . \\end{align*}"}
+{"id": "7566.png", "formula": "\\begin{align*} m _ { p , q } ( c _ { 2 } ) \\leq \\inf _ { u \\in S ( c _ { 1 } ) } \\max _ { t \\in ( 0 , + \\infty ) } E _ { p , q } ( u _ { t } ) = m _ { p , q } ( c _ { 1 } ) . \\end{align*}"}
+{"id": "1244.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n \\frac { ( a ; q ) _ k ( b ; q ) _ k } { ( q ; q ) _ k } ( - a b ) ^ { n - k } q ^ { \\frac { ( n - k ) ( n + k - 1 ) } { 2 } } = \\sum _ { k = 0 } ^ n \\frac { ( a ; q ) _ { n + 1 } ( - b ) ^ k q ^ { k \\choose 2 } } { ( q ; q ) _ k ( q ; q ) _ { n - k } ( 1 - a q ^ { n - k } ) } . \\end{align*}"}
+{"id": "6078.png", "formula": "\\begin{align*} I _ { \\pm } ( x ) & : = \\lim _ { \\lambda \\to \\pm \\Lambda \\pm 0 } \\int _ { - \\pi } ^ { \\pi } \\frac { - i \\sin k } { 2 \\phi _ { 2 , 1 } \\phi _ { 2 , 2 } \\cos k - \\lambda } e ^ { i k x } \\frac { d k } { 2 \\pi } \\\\ & = 2 \\lim _ { \\lambda \\to \\pm \\Lambda \\pm 0 } \\int _ 0 ^ { \\pi } \\frac { \\sin k \\sin x k } { 2 \\phi _ { 2 , 1 } \\phi _ { 2 , 2 } \\cos k - \\lambda } \\frac { d k } { 2 \\pi } . \\end{align*}"}
+{"id": "8288.png", "formula": "\\begin{align*} \\rho ( x ) = ( r ^ 2 - | x | ^ 2 ) ^ + . \\end{align*}"}
+{"id": "6732.png", "formula": "\\begin{align*} I ( z ) = \\frac { { ( n - 1 ) ! } } { { ( 2 \\pi z ) ^ { n } } } \\sum _ { m = 1 } ^ \\infty { \\frac { 1 } { { m ^ { n + 1 } } } } - \\frac { ( n - 1 ) ! } { ( 2 \\pi z ) ^ n } \\sum _ { j = 0 } ^ { n - 1 } { \\frac { ( 2 \\pi z ) ^ { j } } { j ! } \\sum _ { m = 1 } ^ \\infty { \\frac { e ^ { - 2 \\pi m z } } { m ^ { n - j + 1 } } } } . \\end{align*}"}
+{"id": "8300.png", "formula": "\\begin{align*} B ( k ) y ( \\cdot , k ) = c ( k ) , k \\in \\mathbb { N } , \\end{align*}"}
+{"id": "5967.png", "formula": "\\begin{align*} P ^ k ( \\vec { y } | \\vec { x } ) = \\prod _ { i = 1 } ^ k P ( y _ i | x _ i ) . \\end{align*}"}
+{"id": "3529.png", "formula": "\\begin{align*} T f ( z ) = \\alpha f ( e ^ { i \\theta } z ) , \\end{align*}"}
+{"id": "2746.png", "formula": "\\begin{align*} \\omega = d q ^ { i } \\wedge d p _ { i } = d \\Xi ^ { \\alpha } \\wedge d \\Psi _ { \\alpha } + d Q ^ { i } \\wedge d P _ { i } , \\end{align*}"}
+{"id": "4929.png", "formula": "\\begin{align*} \\zeta _ { e ^ { L } _ { + } + e ^ { L } _ { - } } ( H ) = \\max \\zeta _ { e ^ { L } _ { \\pm } } ( H ) \\end{align*}"}
+{"id": "905.png", "formula": "\\begin{align*} \\Phi _ a ^ m ( c ) = c _ { k - 1 } \\Phi _ a ^ { m - 1 } ( c ) + \\ldots + c _ { 2 } \\Phi _ a ^ { 2 } ( c ) + c _ 1 \\Phi _ a ( c ) + c _ 0 c . \\end{align*}"}
+{"id": "5380.png", "formula": "\\begin{align*} \\bar { v } _ t = ( 1 - p ) \\ , \\bar { v } ^ { S _ { k + 1 } } + p \\ , \\bar { v } ^ { S _ k } . \\end{align*}"}
+{"id": "6055.png", "formula": "\\begin{align*} x ^ \\alpha \\succeq _ { g r l e x } x ^ \\beta : | \\alpha | > | \\beta | | \\alpha | = | \\beta | x ^ \\alpha \\succeq _ { l e x } x ^ \\beta \\end{align*}"}
+{"id": "1080.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { [ n \\delta ] } \\Delta _ n ^ { s , t } \\leq C _ { t , n } \\leq M _ 4 ( w , r ) n ^ { - d } , n \\in \\N , ~ ~ t \\in \\{ 1 , \\dots , [ r n ] \\} . \\end{align*}"}
+{"id": "223.png", "formula": "\\begin{align*} \\bar { \\Gamma } ( \\bar x , \\bar v ) = f ( \\bar x ) \\Gamma ( \\bar x , \\bar v ) , \\end{align*}"}
+{"id": "7824.png", "formula": "\\begin{align*} \\max _ { \\boldsymbol { \\Theta } } & \\sum _ { t } R _ { t } \\left ( \\boldsymbol { \\Theta } \\right ) + \\sum _ { r } R _ { r } \\left ( \\boldsymbol { \\Theta } \\right ) , \\\\ & \\left | ( \\boldsymbol { \\Theta } ) _ { m , m } \\right | = 1 , \\forall m \\in \\mathcal { M } . \\end{align*}"}
+{"id": "4950.png", "formula": "\\begin{align*} A '' _ x + R ( A _ x , \\tilde \\gamma ' _ x ) \\tilde \\gamma ' _ x = 0 \\end{align*}"}
+{"id": "7365.png", "formula": "\\begin{align*} u _ x ^ 0 ( x , t ) = \\frac { 1 } { 2 } \\{ f ' ( x + t ) + f ' ( x - t ) + g ( x + t ) - g ( x - t ) \\} , \\end{align*}"}
+{"id": "7654.png", "formula": "\\begin{align*} G _ L ( m , n ; z ) = \\langle \\delta _ m , ( H _ { \\omega , L } - z ) ^ { - 1 } \\delta _ n \\rangle \\ , \\ , \\ , \\ , G ^ { \\Lambda ' } _ L ( m , n ; z ) = \\langle \\delta _ m , ( H ^ { \\Lambda ' } _ { \\omega , L } - z ) ^ { - 1 } \\delta _ n \\rangle , \\end{align*}"}
+{"id": "1724.png", "formula": "\\begin{align*} F ( z ) = f ( x _ 1 ) + \\sum _ { 0 < j < j + k \\leq n } x _ j x _ k x _ n ^ { n - j - k } . \\end{align*}"}
+{"id": "7184.png", "formula": "\\begin{align*} I ( x , t ) & = \\int _ 0 ^ t \\int _ \\R \\frac { e ^ { - \\frac { | x - y | ^ 2 } { 4 ( t - s ) } + ( 1 - v _ 0 ^ 2 ) ( t - s ) } } { ( 4 \\pi ( t - s ) ) ^ { 1 / 2 } } h ( y , s ) e ^ { i \\omega s } d y d s & = e ^ { i \\omega t } \\int _ 0 ^ t e ^ { - ( i \\omega + ( v _ 0 ^ 2 - 1 ) ) \\tau } \\frac { 1 } { \\sqrt { \\pi } } \\int _ \\R e ^ { - z ^ 2 } h ( x - 2 \\tau ^ { 1 / 2 } z , \\tau ) d z d \\tau . \\end{align*}"}
+{"id": "2368.png", "formula": "\\begin{align*} \\| w \\| _ { L ^ { p } } \\leq C \\| \\nabla w \\| _ { L ^ { 2 } } ^ { \\alpha } \\| w \\| _ { L ^ { 2 } } ^ { 1 - \\alpha } , \\quad \\mathrm { w i t h } \\ ; \\frac { 1 } { p } = \\frac { 1 } { 2 } - \\frac { \\alpha } { 3 } . \\end{align*}"}
+{"id": "6114.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\boldsymbol U ' ( t ) & = A _ 1 \\boldsymbol U ( t ) + \\boldsymbol U ( t ) A _ 2 ^ { \\mathsf T } + C + \\boldsymbol U ( t ) B \\boldsymbol U ( t ) , \\\\ \\boldsymbol U ( 0 ) & = \\boldsymbol U _ 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "7215.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } p + \\partial _ { v } ( h p ) - a \\partial _ { v v } p = 0 , v \\in [ V _ { \\min } , V _ F ] / \\{ V _ R \\} , \\\\ p ( v , 0 ) = p ^ 0 ( v ) , p ( V _ { \\min } , t ) = p ( V _ F , t ) = 0 , \\\\ p ( V ^ - _ R , t ) = p ( V ^ + _ R , t ) , \\partial _ v p ( V ^ - _ R , t ) = \\partial _ v p ( V ^ + _ R , t ) + \\frac { N ( t ) } { a } . \\\\ \\end{cases} \\end{align*}"}
+{"id": "4162.png", "formula": "\\begin{align*} \\varphi ( g _ 2 g _ 1 ) = \\varphi ( g _ 2 ) \\varphi ( g _ 1 ) \\mbox { a n d } \\varphi ( g _ 3 g _ 1 ) = \\varphi ( g _ 1 ) \\varphi ( g _ 3 ) \\ , . \\end{align*}"}
+{"id": "7298.png", "formula": "\\begin{align*} h _ { m , r } ( s ) = \\sum _ { n = 0 } ^ { \\infty } S _ { m , r } ( n + 1 ) s ^ n \\end{align*}"}
+{"id": "4185.png", "formula": "\\begin{align*} \\varphi ( g _ 2 g _ 1 ) = \\varphi ( g _ 2 ) \\varphi ( g _ 1 ) \\mbox { a n d } \\varphi ( g _ 3 g _ 1 ) = \\varphi ( g _ 1 ) \\varphi ( g _ 3 ) \\ , . \\end{align*}"}
+{"id": "7594.png", "formula": "\\begin{align*} Q _ { p , q } ( \\psi ( t ) ) & = 2 N ( q - 2 ) E _ { p , q } ( \\psi ( t ) ) - ( N ( q - 2 ) - 8 ) \\| \\Delta \\psi ( t ) \\| _ { 2 } ^ { 2 } - \\frac { 2 N ( p - q ) } { p } \\| \\psi ( t ) \\| _ { p } ^ { p } \\\\ & \\leq \\rho \\frac { N ( q - 2 ) - 8 } { 2 } - 1 - \\frac { N ( q - 2 ) - 8 } { 2 } \\rho - \\frac { N ( q - 2 ) - 8 } { 2 } \\| \\Delta \\psi ( t ) \\| _ { 2 } ^ { 2 } . \\\\ \\end{align*}"}
+{"id": "845.png", "formula": "\\begin{align*} - \\dot \\delta { Y } = { X ^ i } { \\Psi _ i } - n { F ^ { - 1 } } g ( X , l ) \\Psi . \\end{align*}"}
+{"id": "7548.png", "formula": "\\begin{align*} \\int _ \\Omega \\bar \\rho _ 0 \\ d x = \\bar M < \\infty \\ , \\end{align*}"}
+{"id": "2793.png", "formula": "\\begin{align*} \\delta \\left ( { \\tilde { \\sigma } _ { 3 } ^ { * } } ( t ) I \\right ) = \\int ^ { t _ { 2 } } _ { t _ { 1 } } \\frac { \\partial L _ { T } } { \\partial \\Xi ^ { \\alpha } } \\delta \\Xi ^ { \\alpha } d t . \\end{align*}"}
+{"id": "3744.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { z } \\frac { u ^ { \\alpha } } { 1 - u ^ { 2 } } \\ln u \\ , d u = \\frac { 1 } { 2 } \\ln z \\ , \\mathrm { B } _ { z ^ { 2 } } \\left ( \\frac { 1 + \\alpha } { 2 } , 0 \\right ) - \\frac { z ^ { \\alpha + 1 } } { 4 } \\ , \\Phi \\left ( z ^ { 2 } , 2 , \\frac { 1 + \\alpha } { 2 } \\right ) . \\end{align*}"}
+{"id": "4782.png", "formula": "\\begin{align*} \\eta ( \\varepsilon ) : = 2 ^ { - \\eta ( \\min k [ 2 ^ { - k } \\leq \\varepsilon ] ) } \\end{align*}"}
+{"id": "2992.png", "formula": "\\begin{align*} \\overline { \\nabla } _ x y = \\nabla _ x y + H ( x , y ) , \\end{align*}"}
+{"id": "6539.png", "formula": "\\begin{align*} T _ { N } = \\sum ^ N _ { n = 1 } \\sum ^ { \\infty } _ { j = 1 } \\Big ( \\mathbb { E } \\big [ K ( X _ n + a _ j \\widetilde { \\varepsilon } _ { n - j } ) | \\varepsilon _ { n - j } \\big ] - \\mathbb { E } \\big [ K ( X _ n + a _ j \\widetilde { \\varepsilon } _ { n - j } ) \\big ] \\Big ) , \\end{align*}"}
+{"id": "4304.png", "formula": "\\begin{align*} K _ t = K _ { \\tau _ { j - 1 } } K ( j ) _ t = \\{ K ( 1 ) _ { \\tau _ { 1 } } \\cdots K ( j - 1 ) _ { \\tau _ { j - 1 } } \\} K ( j ) _ t , t \\in [ \\tau _ { j - 1 } , \\tau _ { j } ] . \\end{align*}"}
+{"id": "4612.png", "formula": "\\begin{align*} N \\left ( - \\Delta ^ N _ \\Omega + \\lambda W \\right ) = ( 2 \\pi ) ^ { - d } \\left | B _ 1 ( 0 ) \\right | \\lambda ^ \\frac { d } { 2 } \\int _ \\Omega | W | ^ \\frac { d } { 2 } + o \\left ( \\lambda ^ \\frac { d } { 2 } \\right ) \\mathrm { \\ a s \\ } \\lambda \\rightarrow \\infty . \\end{align*}"}
+{"id": "3412.png", "formula": "\\begin{align*} \\iota ( z _ 1 , z _ 2 , \\ldots , z _ n ) = ( - z _ 1 , z _ 2 , \\ldots , z _ n ) . \\end{align*}"}
+{"id": "4544.png", "formula": "\\begin{align*} \\gamma : = \\frac { \\beta a _ - } { 1 + \\beta } - \\frac { a \\delta } { \\beta } > 0 . \\end{align*}"}
+{"id": "3718.png", "formula": "\\begin{align*} G ( x , y ) = \\sum _ { \\sigma } \\frac { f _ { \\sigma } ( x ) f ^ * _ { \\sigma } ( y ) } { ( \\omega _ { \\sigma } - E ) } + \\sum _ { \\sigma } \\frac { f _ { \\sigma } ( x ) } { ( \\omega _ { \\sigma } - E ) } \\frac { f ^ * _ { \\sigma } ( a ) } { \\sqrt { \\omega _ { \\sigma } } } \\Phi ^ { - 1 } ( E ) \\sum _ { \\nu } \\frac { f _ { \\nu } ( a ) } { \\sqrt { \\omega _ { \\nu } } } \\frac { f ^ * _ { \\nu } ( y ) } { ( \\omega _ { \\nu } - E ) } \\end{align*}"}
+{"id": "8847.png", "formula": "\\begin{align*} \\frac 1 { C _ { N , \\tau , \\alpha , \\lambda } } = \\inf \\Big \\{ \\frac { \\| \\sqrt { \\mathcal K _ \\lambda } u \\| ^ { 2 p } } { \\mathcal P [ u ] } , \\quad 0 \\neq u \\in H ^ 1 _ \\lambda \\Big \\} . \\end{align*}"}
+{"id": "8152.png", "formula": "\\begin{align*} \\min _ { P _ { { \\underline X } ^ L } } \\frac { 1 } { \\ell } \\sum _ { i = 1 } ^ { \\ell } \\sum _ { { \\underline x } ^ L } P _ { { \\underline X } ^ L } ( { \\underline x } ^ L ) c ( \\underline x ^ L ) , \\end{align*}"}
+{"id": "8497.png", "formula": "\\begin{align*} I & \\leq C ( n , s ^ \\prime , \\bar { s } , T ) \\int _ K \\| \\partial _ t f ( x , \\cdot ) \\| _ { L ^ 1 ( I ) } \\ , d x \\\\ & = C ( n , s ^ \\prime , \\bar { s } , T ) \\| \\partial _ t f \\| _ { L ^ 1 ( K \\times I ) } . \\end{align*}"}
+{"id": "2187.png", "formula": "\\begin{align*} u = u _ r ( r , z , t ) \\bar e _ r + u _ \\varphi ( r , z , t ) \\bar e _ \\varphi + u _ z ( r , z , t ) \\bar e _ z , \\end{align*}"}
+{"id": "6694.png", "formula": "\\begin{align*} \\Phi ^ { 2 R + 1 } ( F ) = \\big \\{ x ^ { q ^ { 2 R + 1 } } \\mid x \\in F \\big \\} \\end{align*}"}
+{"id": "4314.png", "formula": "\\begin{align*} \\| u \\| _ { m M } = \\| u \\| _ { \\C } = \\| [ I d _ \\C \\otimes u ] ( t _ E ) \\| _ { \\max } . \\end{align*}"}
+{"id": "6470.png", "formula": "\\begin{align*} \\Delta | A | ^ 2 + | A | ^ 4 - m K | A | ^ 2 - m ^ 2 K H ^ 2 = 0 , A ( \\operatorname { g r a d } | A | ^ 2 ) = 0 . \\end{align*}"}
+{"id": "1288.png", "formula": "\\begin{align*} W _ 2 ( \\bar { \\mu } _ t , \\bar { \\mu } _ \\infty ) ^ 2 & = ( \\bar { m } ( t ) - \\bar { m } _ \\infty ) ^ 2 + ( \\sqrt { \\bar { \\sigma } ( t ) } - \\sqrt { \\bar { \\sigma } _ \\infty } ) ^ 2 \\\\ & \\leq \\bar { m } ( t ) ^ 2 + | \\overline { \\sigma } ( t ) - \\overline { \\sigma } _ \\infty | \\\\ & = e ^ { - 2 \\lambda _ 2 t } \\Big [ x _ 0 ^ 2 + \\frac { 1 } { \\beta \\omega ^ 2 } \\Big ] . \\end{align*}"}
+{"id": "2934.png", "formula": "\\begin{align*} \\sum _ { T ' : T ' \\in P ( T ) , T \\in M _ j , j \\in T ' } v _ { T ' } = \\sum _ { T ' : T ' \\in P ( T ) , T \\in M _ j , \\bar { j } \\in T ' } v _ { T ' } = 0 . \\end{align*}"}
+{"id": "2465.png", "formula": "\\begin{align*} \\begin{cases} \\lambda ' = c \\lambda - \\gamma + k x ^ { 2 } - \\delta p \\lambda x , \\\\ x ' = p ^ { - 1 } \\lambda ^ { - 1 } x + c x + k \\lambda ^ { - 1 } x ^ { 3 } - \\delta p x ^ { 2 } . \\end{cases} \\end{align*}"}
+{"id": "2571.png", "formula": "\\begin{align*} \\sum ^ { m } _ { k = 1 } \\lambda _ { k i } b _ { k j } = \\left \\{ \\begin{aligned} d _ { j } & , ~ ~ i = j ; \\\\ 0 & , ~ ; \\end{aligned} \\right . \\end{align*}"}
+{"id": "1508.png", "formula": "\\begin{align*} \\frac { \\partial w } { \\partial \\eta _ { p } } : = | \\nabla w | ^ { p - 2 } \\frac { \\partial w } { \\partial \\eta } , \\forall \\ , w \\in W ^ { 1 , p } ( \\Omega ) \\cap C ^ { 1 } ( \\overline { \\Omega } ) , \\end{align*}"}
+{"id": "5070.png", "formula": "\\begin{align*} C _ w & = \\sum _ { x \\leq w } ( - 1 ) ^ { l ( w ) - l ( x ) } v ^ { l ( x ) - l ( w ) } P _ { x , w } \\tilde { A } _ { x ^ { - 1 } } ^ { - 1 } \\end{align*}"}
+{"id": "3649.png", "formula": "\\begin{align*} Q _ { e , e ' } = \\begin{cases} | e \\cap e ' | & e , e ' \\in E ' , \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "5634.png", "formula": "\\begin{align*} \\bigg \\| S ^ { n _ k } \\bigg ( \\sum _ { i = s _ { 2 k + 1 } } ^ \\infty x _ i e _ i \\bigg ) \\bigg \\| \\leq \\bigg \\| \\sum _ { i = s _ { 2 k + 1 } } ^ \\infty 2 x _ i e _ i \\bigg \\| \\leq \\bigg \\| \\sum _ { i = N } ^ \\infty 2 x _ i e _ i \\bigg \\| < \\frac { \\epsilon } { 2 } . \\end{align*}"}
+{"id": "2473.png", "formula": "\\begin{align*} \\phi ( \\xi ) & = \\lambda ^ { - 1 } x \\sim A _ { 8 } e ^ { \\tau } \\sim A _ { 9 } ( \\xi - \\xi _ { - } ) ^ { p } { \\rm { a s } } \\xi \\searrow \\xi _ { - } + 0 , \\\\ \\psi ( \\xi ) & = \\lambda ^ { - 1 } \\sim A _ { 1 0 } e ^ { \\gamma \\tau } \\sim A _ { 1 1 } ( \\xi - \\xi _ { - } ) ^ { p - 1 } { \\rm { a s } } \\xi \\searrow \\xi _ { - } + 0 \\end{align*}"}
+{"id": "1564.png", "formula": "\\begin{align*} \\lambda _ j \\ge \\frac { 1 } { \\beta } \\ , \\lambda _ i , \\beta = \\frac { \\lambda _ { \\rm m a x } ( P ) } { \\lambda _ { \\rm m i n } ( P ) } , \\end{align*}"}
+{"id": "5661.png", "formula": "\\begin{align*} m _ { i j } = a _ i a _ j - \\delta _ { i j } \\| \\boldsymbol { a } ^ { ( n ) } \\| ^ 2 \\cos ^ 2 \\theta , \\end{align*}"}
+{"id": "6715.png", "formula": "\\begin{align*} y ( x ) = \\sum _ { j = 1 } ^ { n } \\mu _ j ( x ) m _ j ^ x , \\end{align*}"}
+{"id": "9065.png", "formula": "\\begin{align*} \\mu ( F _ 1 ) = 2 ( b + b _ { 1 2 } ) , \\end{align*}"}
+{"id": "841.png", "formula": "\\begin{align*} ( n + 1 ) \\Psi _ { i } = \\nabla _ { i } f + h I _ { i } - \\Psi I _ i . \\end{align*}"}
+{"id": "534.png", "formula": "\\begin{align*} F _ { k } ( z ) : = \\Psi _ { k } ( z ) \\dfrac { H _ { a , \\theta } ( z ) } { H _ { a , \\theta } ( i \\lambda _ { k } ) } , z \\in \\C . \\end{align*}"}
+{"id": "7417.png", "formula": "\\begin{align*} \\int _ { y - R } ^ { y + R } \\left ( | u ( y , s ) | ^ { ( p + q ) / p } \\right ) _ s d s = | u ( y , y + R ) | ^ { ( p + q ) / p } . \\end{align*}"}
+{"id": "7397.png", "formula": "\\begin{align*} \\alpha = s + y , \\ \\beta = s - y . \\end{align*}"}
+{"id": "4529.png", "formula": "\\begin{align*} \\mathcal { N } _ N ( t , [ s ] _ { N - 1 } ) = \\int _ { s _ { N - 1 } } ^ { + \\infty } n _ { N } ( t , [ s ] _ N ) p _ { N } ( [ s ] _ N ) d s _ N , \\mathcal { N } _ { \\infty } ( [ s ] _ { \\infty } ) = p _ { \\infty } ( [ s ] _ { \\infty } ) n _ { \\infty } ( [ s ] _ { \\infty } ) . \\end{align*}"}
+{"id": "447.png", "formula": "\\begin{align*} \\Omega ^ q _ + = \\left \\{ ( i , j ) \\in \\Omega ^ q \\mid i \\in V _ { B ^ { q } } , \\ , j \\in V _ m \\backslash V _ { B ^ { q } } \\right \\} \\mbox { a n d } \\Omega ^ q _ - = \\left \\{ ( i , j ) \\in \\Omega ^ q \\mid i \\in V _ m \\backslash V _ { B ^ { q } } , \\ , j \\in V _ { B ^ { q } } \\right \\} , \\end{align*}"}
+{"id": "5316.png", "formula": "\\begin{align*} S _ k = \\{ \\pi _ k , \\ldots , \\pi _ n \\} , 1 \\leq k \\leq n . \\end{align*}"}
+{"id": "6205.png", "formula": "\\begin{align*} \\frac { P _ m ^ { i , g } l ^ 2 } { 2 \\tau } \\sim D _ { i , g } , \\frac { P _ b ^ { i , g } } { 2 \\tau } \\sim \\lambda _ { i , g } , \\frac { P _ d ^ { i , g } } { 2 \\tau } \\sim k _ { i , g } , P _ b ^ { i , g } , P _ d ^ { i , g } = O ( \\tau ) , \\hbox { f o r } l \\to 0 , \\tau \\to 0 . \\end{align*}"}
+{"id": "3557.png", "formula": "\\begin{align*} \\Gamma f = \\hat f , \\end{align*}"}
+{"id": "8795.png", "formula": "\\begin{align*} \\psi _ \\rho ' ( \\alpha ) = 1 + ( 2 - \\rho ) \\left ( \\frac { 1 - \\alpha } { \\alpha } \\right ) ^ { \\rho - 1 } - ( \\rho - 1 ) \\left ( \\frac { \\alpha } { 1 - \\alpha } \\right ) ^ { 2 - \\rho } . \\end{align*}"}
+{"id": "1622.png", "formula": "\\begin{align*} \\b Q ^ s _ { g \\oplus h } ( x \\otimes y ) \\mapsto \\sum _ { j + k = s } ( - 1 ) ^ { 2 j k ( p - 1 ) } \\left ( \\b Q ^ j _ g ( x ) \\otimes Q ^ k _ h ( y ) + ( - 1 ) ^ n Q ^ j _ g ( x ) \\otimes \\b Q ^ k _ h ( y ) \\right ) . \\end{align*}"}
+{"id": "1228.png", "formula": "\\begin{align*} \\rho = \\sum _ { n = 1 } ^ { N } p _ n \\rho _ n ^ { ( 1 ) } \\otimes \\rho _ n ^ { ( 2 ) } , \\end{align*}"}
+{"id": "2294.png", "formula": "\\begin{align*} F _ n ( t ) = \\sum _ { k \\geq 0 } \\dim \\bigl ( Z _ n ( s l _ 2 ) _ { \\leq k } / Z _ n ( s l _ 2 ) _ { < k } \\bigr ) t ^ k , \\end{align*}"}
+{"id": "6779.png", "formula": "\\begin{align*} \\dot { x } = \\Gamma ( \\kappa \\circ x ^ { \\Gamma _ l ^ \\top } ) , \\end{align*}"}
+{"id": "2883.png", "formula": "\\begin{align*} D _ { \\lambda , \\frac { \\gamma } { q } , s } [ f ] : = \\left \\{ ( x , y ) \\in \\mathbb { R } ^ n \\times \\mathbb { R } ^ n : \\ x \\neq y , \\ \\frac { | f ( x ) - f ( y ) | } { | x - y | ^ { s + \\frac { \\gamma } { q } } } > \\lambda \\right \\} . \\end{align*}"}
+{"id": "3374.png", "formula": "\\begin{align*} I _ 5 : = I _ { 5 , 1 } + I _ { 5 , 2 } + I _ { 5 , 3 } . \\end{align*}"}
+{"id": "3834.png", "formula": "\\begin{align*} C : = \\frac { 1 } { 7 2 k } \\left ( \\left ( - 8 h ^ 2 h ' + 5 h ' - 1 6 h - 1 8 \\right ) k ^ 2 + 1 8 k - 9 h ' \\right ) . \\end{align*}"}
+{"id": "3278.png", "formula": "\\begin{align*} ( q ^ { d + 1 } , q ^ { 1 - d } ; q ^ d ) _ k = - q [ d - 1 ] \\left ( 1 + \\frac { 1 - q ^ d } { q ^ d - q ^ { d k + 1 } } \\right ) ( q ; q ^ d ) _ k ^ 2 . \\end{align*}"}
+{"id": "5406.png", "formula": "\\begin{align*} \\Delta \\mathbf { b } ^ k - \\Delta \\widehat { \\mathbf { b } } ^ k & = ( \\mathbf { I } - \\mathbf { B } ^ k ) ^ { - 1 } \\ , ( \\mathbf { b } ^ k - \\widehat { \\mathbf { b } } ^ k ) \\\\ & = \\frac { \\lambda _ k \\ , \\left [ 1 - \\Delta b ^ { S _ { k + 2 } } _ { k + 1 } \\right ] } { \\alpha + \\lambda _ { k - 1 } + \\mu _ k } \\ , ( \\mathbf { I } - \\mathbf { B } ^ k ) ^ { - 1 } \\ , \\mathbf { e } ^ k . \\end{align*}"}
+{"id": "3587.png", "formula": "\\begin{align*} \\alpha ^ 2 \\tau _ 0 \\tau _ 1 = - ( a + b ) . \\end{align*}"}
+{"id": "4436.png", "formula": "\\begin{align*} f ( w ) = \\frac { 1 } { 2 \\pi \\sqrt { - 1 } } \\int _ { \\partial D } \\frac { f _ * ( z ) } { z - w } d z \\end{align*}"}
+{"id": "7509.png", "formula": "\\begin{align*} Z _ s : = Z \\backslash \\bigcup _ { k = 1 } ^ { n _ 0 } \\phi _ k \\left ( ( 0 , s ^ { 1 / 4 } ] \\times X \\times [ - \\eta / 2 , \\eta / 2 ] \\right ) . \\end{align*}"}
+{"id": "7029.png", "formula": "\\begin{align*} u = ( \\mu - \\Delta ) ^ { - 1 } f - ( \\mu - \\Delta ) ^ { - \\frac { 1 } { 2 } - \\frac { 1 } { q } } Q _ { p } ( q ) ( 1 + T _ p ) ^ { - 1 } G _ { p } ( r ) ( \\mu - \\Delta ) ^ { - \\frac { 1 } { 2 } + \\frac { 1 } { r } } f , f \\in L ^ p \\end{align*}"}
+{"id": "3343.png", "formula": "\\begin{align*} \\iota _ v R & = \\iota _ v ( d J d \\log ( g _ { \\alpha \\bar { \\beta } } ) ) , \\\\ & = \\mathcal { L } _ v ( J d \\log ( g _ { \\alpha \\bar { \\beta } } ) ) - d ( \\mathcal { L } _ { J v } \\log ( g _ { \\alpha \\bar { \\beta } } ) ) . \\end{align*}"}
+{"id": "1278.png", "formula": "\\begin{align*} \\mu _ t = \\mathcal { N } ( m ( t ) , \\sigma ( t ) ) , \\bar { \\mu } _ t = \\mathcal { N } ( \\bar { m } _ t , \\bar { \\sigma } ( t ) ) , \\end{align*}"}
+{"id": "549.png", "formula": "\\begin{align*} x J _ { \\nu } ^ { \\prime } ( x ) - \\nu J _ { \\nu } ( x ) = - x J _ { \\nu + 1 } ( x ) . \\end{align*}"}
+{"id": "2829.png", "formula": "\\begin{align*} & f ( \\Xi ^ { 1 } , \\Psi _ { 2 } , \\Theta ^ { 1 } ) : = - \\frac { 1 } { 1 8 } ( \\Xi ^ { 1 } ) ^ { 2 } - \\frac { 1 } { 9 } \\Xi ^ { 1 } ( - 5 \\Theta ^ { 1 } + 4 \\Psi _ { 2 } ) , \\\\ & g ( \\Psi _ { 1 } , \\Psi _ { 2 } , \\Theta ^ { 1 } , \\Theta _ { 1 } ) : = - \\frac { 1 } { 1 8 } ( 5 \\Theta ^ { 1 } - \\Psi _ { 2 } ) ^ { 2 } + \\frac { 1 } { 2 } ( 3 \\Theta _ { 1 } - \\Psi _ { 1 } ) ( \\Theta _ { 1 } - \\Psi _ { 1 } ) - \\frac { 1 } { 3 } \\Psi _ { 2 } ( \\Theta ^ { 1 } - \\Psi _ { 2 } ) . \\end{align*}"}
+{"id": "5723.png", "formula": "\\begin{align*} \\sum _ { k + \\ell \\le m } \\Vert \\mathcal { E } ^ { ( k , \\ell ) } ( u ) \\Vert _ { G } ( t ) & \\approx \\sum _ { k = 0 } ^ s \\Vert \\partial _ t ^ k E _ 1 ( u ) \\Vert _ { H ^ { s - k } } ( t ) . \\end{align*}"}
+{"id": "8636.png", "formula": "\\begin{align*} ( v _ 2 ( \\partial _ x X ) ^ 2 + v _ 1 ( \\partial _ x ^ 2 X ) ) ( t ; x ) = u ^ { \\prime \\prime } _ 0 ( x ) - I _ 2 ( t ; x ) , \\end{align*}"}
+{"id": "2594.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta \\varphi = f ( \\varphi ) & B _ R , \\\\ \\dfrac { \\partial \\varphi } { \\partial \\nu } + \\beta \\varphi = 0 & \\partial B _ R , \\end{cases} \\end{align*}"}
+{"id": "7514.png", "formula": "\\begin{align*} \\max _ { M _ s } \\sum _ { j = 0 } ^ { 2 } r _ s ^ { j + 2 } | ( \\nabla ^ { g _ s ( t ) } ) ^ j R m ( g _ s ( t ) ) | _ { g _ s ( t ) } \\leq C _ M , \\end{align*}"}
+{"id": "2914.png", "formula": "\\begin{align*} W _ { \\ell } ( x _ i ; f ) : = \\sum _ { y _ { 0 } < p \\leqslant y _ J } \\frac { X } { p } \\int _ { p } ^ { p ( 1 + 1 / X ) } \\big | \\Psi ^ { \\prime } _ f ( x _ i / p , x _ i / t , p ) \\big | ^ 2 { \\rm d } t . \\end{align*}"}
+{"id": "4633.png", "formula": "\\begin{align*} | V ( x ) | = b _ V \\left ( f ( x ' ) - x _ d \\right ) ^ { \\frac { 2 } { d } ( - 1 + \\epsilon ) } \\geq b _ V 2 ^ { - \\gamma m n \\frac { 2 } { d } ( - 1 + \\epsilon ) } . \\end{align*}"}
+{"id": "1158.png", "formula": "\\begin{align*} Q ( I , k ) & = I \\times [ \\ell ( I ) k , \\ell ( I ) ( k + 1 ) ) \\subset R \\times [ 0 , \\ell ( R ) ( k + 1 ) ) \\\\ & \\subset \\widetilde { R } \\times \\left [ 0 , \\ell \\left ( \\widetilde { R } \\right ) \\right ) = P _ R . \\end{align*}"}
+{"id": "5603.png", "formula": "\\begin{align*} ( \\gamma _ { 2 i , 2 k } , \\gamma _ { 2 i , 2 k + 1 } , \\gamma _ { 2 i , 2 k + 2 } ) & = ( \\gamma _ { 2 i - 1 , 2 k } , \\gamma _ { 2 i - 1 , 2 k + 1 } , \\gamma _ { 2 i - 1 , 2 k + 2 } ) , \\\\ ( \\gamma _ { 2 i + 1 , 0 } , \\gamma _ { 2 i + 1 , 1 } , \\gamma _ { 2 i + 1 , 2 } ) & = ( \\gamma _ { 2 i , 0 } , \\gamma _ { 2 i , 1 } , \\gamma _ { 2 i , 2 } ) \\end{align*}"}
+{"id": "3978.png", "formula": "\\begin{align*} \\sum _ { k , r = 1 } ^ { n , \\nu } \\gamma _ { i r k } \\widetilde { \\gamma } _ { j p r } + \\sum _ { s , r = 1 } ^ { \\nu } \\widetilde { \\gamma } _ { i r s } \\widetilde { \\gamma } _ { j p r } = \\sum _ { r = 1 } ^ { \\nu } \\Bigl ( \\sum _ { k = 1 } ^ { n } \\gamma _ { i r k } + \\sum _ { s = 1 } ^ { \\nu } \\widetilde { \\gamma } _ { i r s } \\Bigr ) \\widetilde { \\gamma } _ { j p r } = \\sum _ { r = 1 } ^ { \\nu } \\widetilde { \\gamma } _ { j p r } \\end{align*}"}
+{"id": "7949.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Omega } \\frac { \\delta \\mathcal { F } } { \\delta v } \\wedge \\partial v = \\int _ { \\Omega } \\Big ( \\frac { \\delta \\mathcal { F } } { \\delta v } \\wedge d ( \\partial \\phi ) + \\frac { \\delta _ { \\beta } \\mathcal { F } } { \\delta v } \\wedge \\partial \\eta - \\ast d w \\wedge \\eta ' - \\ast d w \\wedge [ \\eta , u ] _ { 1 } \\Big ) . \\\\ \\end{aligned} \\end{align*}"}
+{"id": "5158.png", "formula": "\\begin{align*} \\frac { \\partial A } { \\partial q _ { j } } = \\frac { \\partial A _ { j } } { \\partial q _ { j } } = - \\frac { 1 } { \\alpha } p ^ { \\alpha } _ { j } q ^ { - \\alpha } _ { j } \\ ; \\ ; ; \\ ; \\ ; \\frac { \\partial B } { \\partial q _ { j } } = \\frac { \\partial B _ { j } } { \\partial q _ { j } } = - \\frac { 1 } { \\alpha } \\end{align*}"}
+{"id": "3375.png", "formula": "\\begin{align*} I : = I _ 1 + I _ 2 + I _ 3 + I _ 4 + I _ 5 . \\end{align*}"}
+{"id": "2107.png", "formula": "\\begin{align*} U ( s , x ) = U ( 0 , x ) - \\alpha \\int _ 0 ^ s \\int _ 0 ^ 1 A ( x , y ) U ( u , y ) \\mathrm { d } y \\mathrm { d } u + \\alpha \\beta \\int _ 0 ^ s \\mathrm { d } \\xi _ 2 ( u , x ) + \\alpha \\zeta \\int _ 0 ^ s \\mathrm { d } \\xi _ 3 ( u , x ) . \\end{align*}"}
+{"id": "2059.png", "formula": "\\begin{align*} A ( x , y ) = a _ 0 + \\sum _ { k = 1 } ^ \\infty b _ k \\cos \\left ( 2 \\pi k \\left ( x - y \\right ) \\right ) , \\forall \\ x , y \\in [ 0 , 1 ] . \\end{align*}"}
+{"id": "6583.png", "formula": "\\begin{align*} A ( x ) = \\sum ^ { \\prime } _ { n \\leq x } f ( n ) , \\end{align*}"}
+{"id": "8454.png", "formula": "\\begin{align*} U _ m ( x , y ) : = | u _ m ( x ) - u _ m ( y ) | ^ { p - 2 } ( u _ m ( x ) - u _ m ( y ) ) . \\end{align*}"}
+{"id": "784.png", "formula": "\\begin{align*} \\delta ( u _ { i j } ) = \\sum _ { s , t } u _ { s t } \\otimes S ( z _ { i s } ) z _ { t j } , \\end{align*}"}
+{"id": "6598.png", "formula": "\\begin{align*} \\left \\{ z = b + i \\eta : 0 < ( 1 - \\delta ) b \\leq \\frac { 1 } { 2 } , | \\eta | \\leq \\exp \\left ( \\left ( \\frac { 1 } { 2 } \\log x \\log \\log x \\right ) ^ { \\frac { 1 } { 2 } } \\right ) \\right \\} . \\end{align*}"}
+{"id": "2352.png", "formula": "\\begin{align*} \\begin{cases} \\| a \\| _ { L _ { T } ^ { \\frac { 4 p } { 3 } } L _ { x } ^ { 2 p } } \\leq C _ { 1 } \\| w _ { 0 } \\| _ { L ^ { p } } , \\\\ \\| N ( w _ { 1 } , w _ { 2 } ) \\| _ { L _ { T } ^ { \\frac { 4 p } { 3 } } L _ { x } ^ { 2 p } } \\leq C _ { 2 } T ^ { \\frac { p - 3 } { 2 p } } \\| w _ { 1 } \\| _ { L _ { T } ^ { \\frac { 4 p } { 3 } } L _ { x } ^ { 2 p } } \\| w _ { 2 } \\| _ { L _ { T } ^ { \\frac { 4 p } { 3 } } L _ { x } ^ { 2 p } } . \\end{cases} \\end{align*}"}
+{"id": "2061.png", "formula": "\\begin{align*} \\Sigma _ d ( i , j ) = a _ 0 + \\sum _ { k = 1 } ^ \\infty b _ k \\cos \\left ( 2 \\pi k ( i - j ) / d \\right ) , \\forall \\ i , j \\in [ d ] . \\end{align*}"}
+{"id": "6576.png", "formula": "\\begin{align*} \\prod _ { p \\leq x } \\left ( 1 - \\frac { 1 } { p } \\right ) = \\frac { e ^ { - \\gamma } } { \\log x } \\left ( 1 + O \\left ( \\frac { 1 } { \\log x } \\right ) \\right ) , \\end{align*}"}
+{"id": "7717.png", "formula": "\\begin{align*} & \\frac { h _ { 1 1 } - h _ { 1 1 t } } { h - h _ { t } } - \\frac { ( h _ { 1 } - h _ { 1 t } ) ^ { 2 } } { ( h - h _ { t } ) ^ { 2 } } \\\\ & = - \\sum _ { i } b ^ { i i } \\nabla _ { 1 1 } b _ { i i } + \\sum _ { i , k } b ^ { i i } b ^ { k k } ( \\nabla _ { 1 } b _ { i k } ) ^ { 2 } + \\nabla _ { 1 1 } \\psi . \\end{align*}"}
+{"id": "8989.png", "formula": "\\begin{align*} - ( u ^ \\prime _ \\gamma r ^ { \\tilde { N } _ + - 1 } ) ^ \\prime = \\bar \\lambda _ { \\gamma , v a r } u _ \\gamma r ^ { \\tilde { N } _ + - 1 - \\gamma } \\end{align*}"}
+{"id": "5267.png", "formula": "\\begin{align*} \\frac { \\partial D } { \\partial x } = H ^ { T } \\frac { \\partial D } { \\partial q } \\end{align*}"}
+{"id": "2730.png", "formula": "\\begin{align*} { _ { * } \\rho } _ { i } : = K ^ { ( 1 ) } _ { i j } \\left ( d q ^ { j } - \\frac { \\partial H } { \\partial p _ { j } } d t \\right ) , { _ { * } \\theta } _ { i } : = d p _ { i } + \\frac { \\partial H } { \\partial q ^ { i } } d t . \\end{align*}"}
+{"id": "520.png", "formula": "\\begin{align*} \\gcd ( \\overline { \\alpha } _ { i _ t } ^ { L _ { i _ t } } , s ) = \\gcd ( \\overline { \\alpha } _ { i _ t } ^ { L _ { i _ t } + 1 } , s ) = : s '' _ { i _ t } . \\end{align*}"}
+{"id": "374.png", "formula": "\\begin{align*} K _ p \\equiv \\begin{cases} b & \\textnormal { f o r F L T 2 C a s e I } , \\\\ a & \\textnormal { f o r F L T 2 C a s e I I } , \\end{cases} \\pmod { p } , \\end{align*}"}
+{"id": "6425.png", "formula": "\\begin{align*} Q ( X ) = \\int _ 0 ^ 1 \\int _ { \\R } d s \\varphi _ { \\alpha } ( u ) d u \\left ( \\frac { 1 } { \\delta _ 0 } \\frac { 1 } { X _ s ^ { 1 / \\alpha _ 0 } } h _ { \\alpha _ 0 } ( u ) x - \\frac { 1 } { \\delta _ 0 } \\frac { X _ s } { X _ s ^ { 1 / \\alpha _ 0 } } h _ { \\alpha _ 0 } ( u ) y \\right . \\\\ \\left . + k _ { \\alpha _ 0 } ( u ) z - [ f _ { \\alpha _ 0 } ( u ) + \\frac { \\ln ( X _ s ) } { \\alpha _ 0 } k _ { \\alpha _ 0 } ( u ) ] w \\right ) ^ 2 , \\end{align*}"}
+{"id": "6349.png", "formula": "\\begin{align*} \\psi ( r ) : = \\int _ 0 ^ r \\tau ^ n \\omega _ g ( \\tau ) \\ , d \\tau , \\end{align*}"}
+{"id": "3856.png", "formula": "\\begin{align*} \\begin{aligned} \\eta ^ { \\kappa } ( x , y \\mid t ) & = ( 1 - \\kappa ) \\eta ^ { 0 } ( x , y \\mid t ) + \\kappa \\pi ^ { * } _ x G ( \\pi ^ * ) _ { x , y } \\\\ & = ( 1 - \\kappa ) ^ 2 \\eta ^ { 0 } ( x , y \\mid t ) + ( 2 \\kappa ( 1 - \\kappa ) + \\kappa ^ 2 ) \\tau ^ { \\kappa } ( x , y \\mid t ) . \\end{aligned} \\end{align*}"}
+{"id": "3540.png", "formula": "\\begin{align*} T ^ { - 1 } ( \\tilde { g } ) = g - g ( 0 ) . \\end{align*}"}
+{"id": "3923.png", "formula": "\\begin{align*} Q & = Q _ 0 + Q _ 1 ; \\\\ Q _ 0 & = { \\begin{pmatrix} 0 & 0 & 0 \\cr q _ { 2 , 1 } & 0 & 0 \\\\ 0 & q _ { 3 , 2 } & 0 \\end{pmatrix} , \\ Q _ 1 = \\begin{pmatrix} q _ { 1 , 1 } & q _ { 1 , 2 } & q _ { 1 , 3 } \\cr 0 & q _ { 2 , 2 } & q _ { 2 , 3 } \\\\ 0 & 0 & q _ { 3 , 3 } \\end{pmatrix} } . \\end{align*}"}
+{"id": "7904.png", "formula": "\\begin{align*} \\langle \\lambda , \\mu \\rangle _ { \\Lambda ^ { k } } : = \\sum _ { \\sigma \\in S ( k , n ) } \\lambda ( E _ { \\sigma ( 1 ) } , \\ldots , E _ { \\sigma ( k ) } ) \\mu ( E _ { \\sigma ( 1 ) } , \\ldots , E _ { \\sigma ( k ) } ) , \\end{align*}"}
+{"id": "7971.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { F } _ { 1 } & : = P ^ { \\ast } \\Lambda ^ { 1 } ( \\Omega ) \\times H ^ { - \\frac { 1 } { 2 } } \\Lambda ^ { n - 1 } ( \\Sigma ) \\times H ^ { - \\frac { 1 } { 2 } } \\Lambda ^ { n - 1 } ( \\Gamma ) , \\\\ \\mathcal { E } _ { 1 } & : = P \\Lambda ^ { n - 1 } ( \\Omega ) \\times H ^ { \\frac { 1 } { 2 } } \\Lambda ^ { 0 } ( \\Sigma ) \\times H ^ { \\frac { 1 } { 2 } } \\Lambda ^ { 0 } ( \\Gamma ) , \\end{aligned} \\end{align*}"}
+{"id": "2598.png", "formula": "\\begin{align*} \\cos ( \\theta ) = \\dfrac { R ^ 2 + a ^ 2 - r ^ 2 } { 2 a R } . \\end{align*}"}
+{"id": "2163.png", "formula": "\\begin{align*} t _ d : = f _ d ( a _ { i _ { K / 2 } } ) . \\end{align*}"}
+{"id": "8483.png", "formula": "\\begin{align*} \\overline { U } _ h ( x , y , t ) & : = | \\bar { u } _ h ( x , t ) - \\bar { u } _ h ( y , t ) | ^ { p - 2 } ( \\bar { u } _ h ( x , t ) - \\bar { u } _ h ( y , t ) ) , \\\\ [ 2 m m ] U ( x , y , t ) & : = | u ( x , t ) - u ( y , t ) | ^ { p - 2 } ( u ( x , t ) - u ( y , t ) ) . \\end{align*}"}
+{"id": "7166.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l l } \\partial _ t v - \\partial _ x ^ 2 v & = ( 1 - v _ 0 ^ 2 ) v - v _ 0 v ^ 2 - \\frac { 1 } { 3 } v ^ 3 - w + I ( x , t ) , & \\forall x \\in \\R , t \\geq 0 , \\\\ \\partial _ t w - \\rho \\partial _ { x } ^ { 2 } w & = \\varepsilon ( v - \\gamma w ) , & \\forall x \\in \\R , t \\geq 0 , \\\\ v ( x , 0 ) = \\bar { f } _ 0 ( x ) , & w ( x , 0 ) = \\bar { g } _ 0 ( x ) , & \\forall x \\in \\R , \\end{array} \\right . \\end{align*}"}
+{"id": "5605.png", "formula": "\\begin{align*} \\sum _ { \\gamma ' : \\gamma ' \\sim \\gamma } \\prod _ { e \\in E _ { \\tilde \\gamma } } | M _ e | ^ { \\tilde m _ e } & \\leq \\left ( \\sqrt \\frac { K \\rho } { m n } \\right ) ^ { 4 s k - 2 a + 2 h } \\sum _ { \\gamma ' : \\gamma ' \\sim \\gamma } \\prod _ { e \\in E _ { \\tilde \\gamma } \\setminus H } \\frac { Q _ e } { \\sqrt { m n } } , \\\\ & = ( K \\rho ) ^ { 2 s k + h - a } ( \\sqrt { m n } ) ^ { a - h - 4 s k } \\sum _ { \\gamma ' : \\gamma ' \\sim \\gamma } \\prod _ { e \\in E _ { \\tilde \\gamma } \\setminus H } { Q _ e } , \\end{align*}"}
+{"id": "5830.png", "formula": "\\begin{align*} \\mathcal { F } _ \\Sigma ( u ) = \\int e ^ { - | X ( u ) | ^ 2 / 4 } \\sqrt { \\det g _ u } d x ^ 1 \\wedge \\dots \\wedge d x ^ n . \\end{align*}"}
+{"id": "8858.png", "formula": "\\begin{align*} 0 < \\frac { 1 } { p b _ 2 } \\leq \\frac 1 r \\leq 1 , 0 \\leq \\frac { \\tau + 1 } { p } < \\frac { n } { p b _ 2 } , \\frac { \\tau + 1 } { p } - 1 = \\frac { n } { p b _ 2 } - \\frac { n } { r } . \\end{align*}"}
+{"id": "2742.png", "formula": "\\begin{align*} S ^ { m } _ { n } : = \\frac { \\partial Z ^ { m } } { \\partial z ^ { n } } \\end{align*}"}
+{"id": "3223.png", "formula": "\\begin{align*} \\mathfrak { h } ( X / B ) = \\mathfrak { h } ^ { - N \\geq n \\geq N } ( X / B ) \\oplus \\bigoplus _ { n < - N \\ , \\textrm { o r } \\ , n > N } \\mathfrak { h } ^ n ( X / B ) \\ , \\end{align*}"}
+{"id": "8287.png", "formula": "\\begin{align*} \\varphi ( u ) = \\ln u \\end{align*}"}
+{"id": "855.png", "formula": "\\begin{align*} \\dot \\delta ( W - U ) = 2 g ( X , I ^ { * } ) h = 2 \\rho h , \\end{align*}"}
+{"id": "5057.png", "formula": "\\begin{align*} \\mathbb { E } [ \\Vert \\Theta ( k + 1 ) \\Vert ] & \\leq h ^ k \\mathbb { E } [ \\Vert \\Theta ( 0 ) \\Vert ] + \\vartheta \\bar { d _ \\eta } \\sum _ { t = 0 } ^ { k - 1 } h ^ { k - 1 - t } \\bar { q } ^ t \\\\ & \\leq h ^ k \\mathbb { E } [ \\Vert \\Theta ( 0 ) \\Vert ] + \\vartheta \\bar { d _ \\eta } h ^ { k - 1 } \\sum _ { t = 0 } ^ { k - 1 } \\left ( \\frac { \\bar { q } } { h } \\right ) ^ t \\\\ & \\leq ( \\mathbb { E } [ \\Vert \\Theta ( 0 ) \\Vert ] + \\frac { \\vartheta \\bar { d _ \\eta } } { h - \\bar { q } } ) h ^ k . \\end{align*}"}
+{"id": "129.png", "formula": "\\begin{align*} \\begin{pmatrix} \\frac 1 2 | \\xi | ^ { 2 } ( I + \\Pi _ \\xi ) & \\xi \\\\ \\xi ^ * & 0 \\end{pmatrix} \\begin{pmatrix} \\widehat { \\vec { u } } _ \\infty \\\\ \\widehat { p } \\end{pmatrix} = \\begin{pmatrix} \\widehat { \\vec { f } } \\\\ 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "8072.png", "formula": "\\begin{align*} \\liminf _ { N \\rightarrow + \\infty } \\frac { N } { \\gamma _ N ^ 2 } \\log P ( \\frac { N } { \\gamma _ N } ( \\tau _ c ^ N - \\tau _ c ) > x ) & \\geq - ( J _ { i n i } ( W ^ { \\tau _ c + \\delta , b } _ 0 ) + J _ { d y n , \\tau _ c + \\delta } ( W ^ { \\tau _ c + \\delta , b } ) ) \\\\ & = - b ^ 2 J _ { c o n t r a , \\tau _ c + \\delta } ( 1 ) \\end{align*}"}
+{"id": "6033.png", "formula": "\\begin{align*} D _ n ( f ) = \\sum _ { x \\in \\mathbb { T } _ N } I _ { x , x + 1 } ^ n ( f ) , \\end{align*}"}
+{"id": "8193.png", "formula": "\\begin{align*} \\frac { \\partial f ( B _ 1 , B _ 2 ^ { 0 \\star } , B _ 2 ^ { 1 \\star } ) } { \\partial { B _ 1 } } = & \\frac { 1 } { M } \\left ( \\log ( 1 - \\frac { B _ 1 } { M } ) - \\log ( \\frac { B _ 1 } { M } ) \\right ) . \\end{align*}"}
+{"id": "4777.png", "formula": "\\begin{align*} \\frac { 1 } { 1 + 2 \\eta ( \\min \\{ \\varepsilon / 2 , 2 \\} ) } & = 1 - 2 \\eta ( \\min \\{ \\varepsilon / 2 , 2 \\} ) + \\frac { \\left ( 2 \\eta ( \\min \\{ \\varepsilon / 2 , 2 \\} ) \\right ) ^ 2 } { 1 + 2 \\eta ( \\min \\{ \\varepsilon / 2 , 2 \\} ) } \\\\ & \\geq 1 - 2 \\eta ( \\min \\{ \\varepsilon / 2 , 2 \\} ) + \\frac { \\left ( 2 \\eta ( \\min \\{ \\varepsilon / 2 , 2 \\} ) \\right ) ^ 2 } { 3 } . \\end{align*}"}
+{"id": "6885.png", "formula": "\\begin{align*} \\Lambda _ M ( R ) = \\bigoplus _ { g \\in M } \\Lambda ( R ) _ g . \\end{align*}"}
+{"id": "5775.png", "formula": "\\begin{align*} u ^ T ( t ) = \\sum _ { j = 1 } ^ { J } ( z _ j ( t ) - \\bar { z } _ j ( t ) ) \\varphi _ { \\iota + j } . \\end{align*}"}
+{"id": "3975.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ { n , \\nu } \\gamma _ { i j k } x _ { i } ^ { \\ : \\left ( t \\right ) } y _ { j } ^ { \\ : \\left ( t \\right ) } \\quad \\Bigl ( \\mbox { r e s p . } \\sum _ { i , j = 1 } ^ { n , \\nu } \\widetilde { \\gamma } _ { i j r } x _ { i } ^ { \\ : \\left ( t \\right ) } y _ { j } ^ { \\ : \\left ( t \\right ) } \\Bigr ) . \\end{align*}"}
+{"id": "1267.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ; q ^ 2 ) _ k ( - q ^ 2 ; q ^ 2 ) _ k } { ( q ^ 2 ; q ^ 2 ) _ k } q ^ { n ^ 2 - ( k + 1 ) ^ 2 } \\equiv - 1 + \\frac { ( n + 1 ) ( 1 - q ^ n ) } { 2 } \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "4447.png", "formula": "\\begin{align*} \\| f \\| _ { \\partial M , \\rho } ^ 2 \\le \\lim _ { j \\rightarrow + \\infty } \\| f _ j \\| ^ 2 _ { \\partial M , \\rho } = M _ H ( Z _ 0 , J , \\rho ) . \\end{align*}"}
+{"id": "95.png", "formula": "\\begin{align*} c _ 3 = 2 + 2 \\kappa \\geq 2 > 0 . \\end{align*}"}
+{"id": "9017.png", "formula": "\\begin{align*} z _ { i i } ^ j = z _ { j j } ^ i 1 < i , j < n . \\end{align*}"}
+{"id": "5696.png", "formula": "\\begin{align*} - u '' - m u ' + \\mathcal { M } _ \\Sigma u = N _ 1 ( u ) . \\end{align*}"}
+{"id": "1139.png", "formula": "\\begin{align*} t _ Q : = \\begin{cases} 1 & \\ell ( Q ) = 1 , \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "6837.png", "formula": "\\begin{align*} u _ { 1 N } ' & = u _ { 1 N } e ^ { - i \\phi _ 1 } \\cos ( \\psi _ 1 ) - u _ { 2 N } e ^ { i \\phi _ 1 } \\sin ( \\psi _ 1 ) , \\\\ u _ { 2 N } ' & = u _ { 1 N } e ^ { - i \\phi _ 1 } \\sin ( \\psi _ 1 ) + u _ { 2 N } e ^ { i \\phi _ 1 } \\cos ( \\psi _ 1 ) . \\end{align*}"}
+{"id": "7643.png", "formula": "\\begin{align*} \\mu _ { i } \\left ( \\mathcal { F } _ { i } ( x ) \\right ) = \\begin{cases} 0 & \\mathcal { F } _ { i } ( x ) \\geq y _ { i } ^ { 0 } , \\\\ m _ { i } ( \\mathcal { F } _ i ( x ) ) & y _ { i } ^ { 0 } \\geq \\mathcal { F } _ { i } ( x ) \\geq y _ { i } ^ { 1 } , \\\\ 1 & \\mathcal { F } _ { i } ( x ) \\leq y _ { i } ^ { 1 } , \\end{cases} \\end{align*}"}
+{"id": "8207.png", "formula": "\\begin{align*} c ^ { y ^ { i - 1 } 1 } = M P _ { Y ^ i } ( y ^ { i - 1 } 1 ) = b ^ { y ^ { i - 1 } } , Y _ i = 1 . \\end{align*}"}
+{"id": "7002.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\mathbb { C } } ^ n \\setminus \\Lambda _ 0 = & \\\\ & \\{ [ 1 \\ ! : \\ ! z _ 2 \\ ! : \\ ! \\cdots \\ ! : \\ ! z _ { n + 1 } ] : z _ 2 , \\ ! \\cdots \\ ! , \\ ! z _ { n + 1 } \\ ! \\in \\ ! \\mathbb { C } \\} \\cup \\\\ & \\{ [ 0 \\ ! : \\ ! 1 \\ ! : \\ ! z _ 3 \\ ! : \\ ! \\cdots \\ ! : \\ ! z _ { n + 1 } ] : z _ 3 , \\ ! \\cdots \\ ! , \\ ! z _ { n + 1 } \\ ! \\in \\ ! \\mathbb { C } \\} \\cup \\\\ & \\{ [ 0 \\ ! : \\ ! 0 \\ ! : \\ ! z _ 3 \\ ! : \\ ! \\cdots \\ ! : \\ ! z _ { n } \\ ! : \\ ! 1 ] : z _ 3 , \\ ! \\cdots \\ ! , \\ ! z _ { n } \\ ! \\in \\ ! \\mathbb { C } \\} . \\end{align*}"}
+{"id": "421.png", "formula": "\\begin{align*} T \\xlongrightarrow [ M \\rightarrow \\infty ] { \\mathcal { P } } \\sum _ { j = 1 } ^ { M } ( \\frac { \\rho ^ 2 \\widetilde { t } _ { j , j } ^ 2 \\psi _ { j , j } } { M } + \\frac { 2 \\Theta _ { j , j } } { M } ) : = K _ { M } , \\end{align*}"}
+{"id": "5514.png", "formula": "\\begin{align*} 0 & = c _ { d + m - r } \\\\ [ 2 m m ] & = f _ { d } g _ { m - r } + \\Biggl ( \\ , \\sum _ { t = 1 } ^ { \\min \\{ d , r \\} } f _ { d - t } g _ { m - r + t } \\Biggr ) \\\\ [ 2 m m ] \\shortintertext { ( r e c a l l t h a t w e m u s t h a v e $ d - t \\geq 0 $ a n d $ m - r + t \\leq m $ ) } \\\\ [ - 3 m m ] & = f _ { d } g _ { m - r } + A . \\end{align*}"}
+{"id": "6034.png", "formula": "\\begin{align*} I ^ n _ { x , x + 1 } ( f ) = \\frac { N ^ 2 } { 2 } \\int _ { \\Omega _ N } c _ x ^ { \\alpha \\beta } ( \\eta ) ( f ( \\eta ^ { x , x + 1 } ) - f ( \\eta ) ) ^ 2 \\nu _ { \\rho } ( \\dd \\eta ) . \\end{align*}"}
+{"id": "4166.png", "formula": "\\begin{align*} \\varphi \\big ( g _ 2 g ^ { j } _ 1 \\big ) = \\varphi ( g _ 2 ) \\varphi \\big ( g ^ { j } _ 1 \\big ) \\mbox { a n d } \\varphi \\big ( g _ 3 g ^ { j } _ 1 \\big ) = \\varphi \\big ( g ^ { j } _ 1 \\big ) \\varphi ( g _ 3 ) \\ , . \\end{align*}"}
+{"id": "6662.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi \\mathrm { i } n } \\oint _ { | z | = 1 } \\frac { f _ { n } ' ( z ) } { f _ { n } ( z ) } \\mathrm { d } z \\geq \\frac { 1 } { 2 \\pi n \\epsilon } \\int _ { 0 } ^ { 2 \\pi } \\bigg ( \\log | f _ { n } ( \\mathrm { e } ^ { \\mathrm { i } x } ) | - \\log | f _ { n } ( \\mathrm { e } ^ { \\mathrm { i } ( x - \\mathrm { i } \\epsilon ) } ) | \\bigg ) \\mathrm { d } x . \\end{align*}"}
+{"id": "1450.png", "formula": "\\begin{align*} \\chi ( G ) = \\sum _ { v } 1 / \\sharp \\mathrm { S t a b } _ G ( v ) - \\sum _ { e } 1 / \\sharp \\mathrm { S t a b } _ G ( e ) , \\end{align*}"}
+{"id": "154.png", "formula": "\\begin{align*} \\bigg | e ^ { N s } - \\sum _ { m = 0 } ^ N \\frac { N ^ m } { m ! } { s } ^ m \\bigg | \\leq \\frac { e ^ { N s } } { C \\sqrt { N } } \\cdot \\delta ^ N . \\end{align*}"}
+{"id": "8598.png", "formula": "\\begin{align*} \\cosh { \\frac { \\ell ( \\gamma _ p ' ) } 2 } = \\cosh { d ( P _ 1 ' , \\widetilde { x ' } ) } = \\frac { \\sinh { d ( P _ 1 ' , Q _ 1 ' ) } } { \\sinh { d ( O , \\widetilde { x ' } ) } } > \\frac { \\sinh { \\log \\frac 4 { \\ell ( c _ i ) } } } { \\sinh { \\log \\frac 2 { \\ell ( c _ i ) } } } = \\frac { 1 6 - \\ell ( c _ i ) ^ 2 } { 8 - 2 \\ell ( c _ i ) ^ 2 } > 2 \\end{align*}"}
+{"id": "9095.png", "formula": "\\begin{align*} \\mathbf 1 _ { \\{ \\hat k _ i < \\tilde \\Theta _ i \\tau \\} } = \\left \\{ \\begin{array} { l l } 1 & \\mbox { i f $ \\hat k _ i < \\tilde \\Theta _ i \\tau $ } , \\\\ 0 & \\mbox { o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
+{"id": "3456.png", "formula": "\\begin{align*} h ( o ) ( l ) : = \\frac { 1 } { 2 } \\operatorname { l k } ( p ( l ) , L ) : H _ 1 ( \\Sigma ( L ) \\setminus N _ L ; \\Z _ 2 ) \\to \\Z _ 2 , \\end{align*}"}
+{"id": "1453.png", "formula": "\\begin{align*} X = X ( \\mathcal { G } , k , P ) : = \\mathcal { G } ( k ) \\times \\mathbb { A } _ 0 / \\backsim , \\end{align*}"}
+{"id": "9107.png", "formula": "\\begin{align*} & \\sum _ { l = 1 } ^ { n } a _ { i l } + \\sum _ { l = 1 } ^ { n } \\hat { c } _ { i i } \\Tilde { k } _ { i l } < 1 , \\sum _ { l = 1 } ^ { n } a _ { i l } + \\sum _ { l = 1 } ^ { n } \\hat { c } _ { i i } \\Tilde { k } _ { i l } \\geq 0 . \\end{align*}"}
+{"id": "2538.png", "formula": "\\begin{align*} \\dot H ^ s ( M ) = H ^ s _ M ( \\breve M ) \\subset H ^ s ( \\breve M ) \\ ; . \\end{align*}"}
+{"id": "4110.png", "formula": "\\begin{align*} \\theta = ( r \\eta _ 1 , \\ldots , r ^ J \\eta _ J ) \\to 0 r \\to 0 \\end{align*}"}
+{"id": "7867.png", "formula": "\\begin{align*} r _ { m i n } = \\min _ { 1 \\leq i \\not = j \\leq L } r a n k _ { p } ( \\mathbf { Q } _ { i , j } ) . \\end{align*}"}
+{"id": "4392.png", "formula": "\\begin{align*} \\inf _ { k \\in [ m ] _ 0 } & \\left \\{ \\inf _ { x \\in \\mathcal { X } } \\left \\{ \\Gamma \\theta ^ k ( x ) + \\sum _ { i \\in [ m ] } f _ i ( x , \\overline { u } ^ i ) + \\sup \\{ 0 , \\theta ^ i ( x ) - \\theta ^ k ( x ) \\} \\right \\} \\right \\} , \\\\ \\end{align*}"}
+{"id": "3944.png", "formula": "\\begin{align*} \\overline { \\mathcal { G } } _ N = \\mathcal { G } _ N ( I _ N \\otimes \\Gamma ) = \\operatorname { d i a g } \\left \\{ G _ j \\Gamma \\right \\} _ { j = 1 } ^ N \\in \\mathbb { R } ^ { N \\times 3 N } . \\end{align*}"}
+{"id": "2858.png", "formula": "\\begin{align*} [ \\omega ] _ { A _ p ( \\mathbb { R } ^ n ) } : = \\sup _ { Q \\subset \\mathbb { R } ^ n } \\left [ \\frac { 1 } { | Q | } \\int _ Q \\omega ( x ) \\ , d x \\right ] \\left \\{ \\frac { 1 } { | Q | } \\int _ Q \\left [ \\omega ( x ) \\right ] ^ { 1 - p ' } \\ , d x \\right \\} ^ { p - 1 } < \\infty , \\end{align*}"}
+{"id": "8779.png", "formula": "\\begin{align*} \\int _ { \\R ^ 2 } \\varphi ( | v - w | ) \\alpha ' _ p ( d v , d w ) - \\int _ { \\R ^ 2 } \\varphi ( | v - w | ) \\alpha _ p ( d v , d w ) = \\frac { 1 } { 3 } \\left ( f ( x _ - ) - f ( x _ + ) \\right ) > 0 , \\end{align*}"}
+{"id": "5333.png", "formula": "\\begin{align*} v ^ { \\rm L P } = \\sum _ { k = 1 } ^ m \\nu _ { \\pi _ k } \\ , \\left [ b ^ { S _ k } - b ^ { S _ { k + 1 } } \\right ] + \\sum _ { j \\in S _ { m + 1 } } c ^ { S _ { m + 1 } } _ j \\ , x ^ { \\boldsymbol { \\pi } } _ j . \\end{align*}"}
+{"id": "4666.png", "formula": "\\begin{align*} | \\{ F _ n \\} _ { n = 0 } ^ \\infty \\pmod p | > 3 \\sqrt { p } , \\end{align*}"}
+{"id": "5239.png", "formula": "\\begin{align*} L _ { d } D A H I ( p \\| q ) = - \\sum _ { j } p _ { j } \\left [ \\frac { \\left ( \\overline { M H } \\right ) ^ { a - 1 } - \\left ( \\overline { M H } \\right ) ^ { b - 1 } } { a - b } \\right ] \\end{align*}"}
+{"id": "5148.png", "formula": "\\begin{align*} D _ { \\alpha } \\left ( p \\| q \\right ) = \\frac { 1 } { \\alpha \\left ( \\alpha - 1 \\right ) } \\sum _ { i } p ^ { \\alpha } _ { i } q ^ { 1 - \\alpha } _ { i } - \\frac { 1 } { \\alpha - 1 } \\sum _ { i } p _ { i } + \\frac { 1 } { \\alpha } \\sum _ { i } q _ { i } \\end{align*}"}
+{"id": "7783.png", "formula": "\\begin{align*} \\frac { \\partial ^ { l } g ( E _ { 0 } , u _ { 0 } ) } { \\partial E ^ { l } } = 0 . \\end{align*}"}
+{"id": "7070.png", "formula": "\\begin{align*} u ( t ) - f + \\mu \\int _ 0 ^ t u d s + \\int _ 0 ^ t b \\cdot \\nabla u d s + \\sigma \\int _ 0 ^ t \\nabla u \\circ d W _ s = 0 , t \\in [ 0 , T ] . \\end{align*}"}
+{"id": "6647.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 ^ { + } } \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } N _ { n } ( \\mathrm { e } ^ { - ( y _ { 2 } + \\epsilon ) } , \\mathrm { e } ^ { - ( y _ { 1 } - \\epsilon ) } ) = \\nu ^ { + } ( E , \\mathrm { i } y _ { 2 } ) - \\nu ^ { - } ( E , \\mathrm { i } y _ { 1 } ) . \\end{align*}"}
+{"id": "8898.png", "formula": "\\begin{align*} \\log \\left ( \\frac { \\sinh ( t / 2 ) } { t / 2 } \\right ) = \\sum _ { n = 2 } ^ \\infty \\frac { B _ n } { n } \\frac { t ^ n } { n ! } \\end{align*}"}
+{"id": "8948.png", "formula": "\\begin{align*} | e w ( x ) - I | & \\le d ( \\nabla w ( x ) , S O ( n ) ) + C | \\nabla w ( x ) - I | ^ 2 \\\\ & = d ( \\nabla \\varphi ( x ) , S O ( n ) ) + C | \\nabla \\varphi ( x ) - O | ^ 2 \\\\ & \\le d ( \\nabla \\varphi ( x ) , S O ( n ) ) + 2 C d ( \\nabla \\varphi ( x ) , S O ( n ) ) ^ 2 + 2 C \\| d ( \\nabla v , S O ( n ) ) \\| _ { L ^ { p ( \\cdot ) } ( Q ) } . \\end{align*}"}
+{"id": "7739.png", "formula": "\\begin{align*} \\begin{matrix} \\begin{array} { l } M ' = ( 2 R ) ^ { m ^ 2 } , \\\\ c ' = ( \\acute b R \\nu '^ { - 1 } \\delta ^ { - 1 } \\tilde M ) ^ { - r m ^ 3 ( m + 6 ) } c , \\\\ \\zeta ' = 1 0 m \\nu ' , \\end{array} & \\begin{array} { l } \\delta ' = \\acute b ^ { - 1 } ( ( R ^ 2 \\tilde M ) ^ { - 1 } \\nu ' ) ^ { 3 m } \\delta , \\\\ r ' = r , \\\\ \\tilde { M } ' = \\acute b ( \\nu '^ { - 1 } \\tilde { M } ) ^ { m ^ 2 ( m + 2 ) } , \\end{array} \\end{matrix} \\end{align*}"}
+{"id": "589.png", "formula": "\\begin{align*} \\bigg \\| { \\frac { T ( \\tilde h _ i ) } { \\| h _ i \\| _ F } } - Y _ i \\bigg \\| _ { F ^ \\ast } < { \\frac { \\eta _ j } { 2 ( j - 1 ) } } , \\ ; \\ ; i = 2 , \\ldots , j - 1 . \\end{align*}"}
+{"id": "2723.png", "formula": "\\begin{align*} \\dot { \\Phi } ^ { ( 2 ) } _ { \\alpha } = d \\Phi ^ { ( 2 ) } _ { \\alpha } ( X _ { T } ) : \\approx 0 . \\end{align*}"}
+{"id": "5164.png", "formula": "\\begin{align*} \\sum _ { j } q _ { j } \\frac { \\partial L _ { d } D _ { \\alpha } I ( p \\| q ) } { \\partial q _ { j } } = 0 \\end{align*}"}
+{"id": "792.png", "formula": "\\begin{align*} \\mathcal { Y } _ T = L ^ \\infty H ^ 1 \\cap H ^ 1 L ^ 2 , \\end{align*}"}
+{"id": "7323.png", "formula": "\\begin{align*} F _ { m , r } ( s , \\beta ) & = \\sum \\limits _ { n = 0 } ^ { \\infty } \\partial _ s ^ n \\left . F _ { m , r } ( s , \\beta ) \\right | _ { s = 0 } \\frac { s ^ { n } } { n ! } \\\\ & = \\sum \\limits _ { n = 0 } ^ { \\infty } \\left ( ( - 1 ) ^ n 2 ^ { - ( n + 1 ) } \\cdot C _ { m , r } ( \\beta , n + 1 ) \\right ) s ^ { n } \\end{align*}"}
+{"id": "4107.png", "formula": "\\begin{align*} P _ { i 0 } = 1 - \\sum _ { j \\in \\mathcal { J } } P _ { i j } , i \\in \\mathcal { J } . \\end{align*}"}
+{"id": "6783.png", "formula": "\\begin{align*} J _ { \\mathrm { r e d } } = \\mu \\begin{bmatrix} c ^ \\top \\\\ d ^ \\top \\end{bmatrix} \\Delta _ u \\Gamma _ l ^ \\top \\Delta _ { 1 / \\bar { x } } \\widetilde { \\Gamma } , \\end{align*}"}
+{"id": "6557.png", "formula": "\\begin{align*} \\lim \\limits _ { x \\to - \\infty } \\frac { 1 } { | x | ^ { \\frac { 1 } { \\beta } } \\ell ^ { \\frac { 1 } { \\beta } } ( | x | ^ { \\frac { 1 } { \\beta } } ) } \\eta _ K ( x ) = \\int ^ { \\infty } _ 0 \\big ( K _ { \\infty } ( - t ^ { - \\beta } ) - K _ { \\infty } ( 0 ) \\big ) d t . \\end{align*}"}
+{"id": "4368.png", "formula": "\\begin{align*} \\inf _ { k \\in [ m ] _ 0 } & \\left \\{ \\inf _ { \\mathcal { Q } \\subseteq [ m ] } \\left \\{ \\inf _ { x \\in \\mathcal { X _ \\mathcal { Q } } } \\left \\{ \\Gamma \\Delta u _ k ^ T l _ k ( x ) + \\sum _ { i \\in [ m ] } \\overline { u } _ i ^ T l _ i ( x ) + \\sum _ { q \\in \\mathcal { Q } } \\Delta u _ q ^ T l _ q ( x ) - \\Delta u _ k ^ T l _ k ( x ) \\right \\} \\right \\} \\right \\} , \\end{align*}"}
+{"id": "357.png", "formula": "\\begin{align*} \\begin{aligned} \\phi _ p ( x , - y ) & = \\dfrac { x ^ p + y ^ p } { x + y } = \\sum _ { i = 0 } ^ { p - 1 } x ^ { p - i - 1 } ( - y ) ^ i \\\\ & = A _ p ( x , - y ) + ( - 1 ) ^ k ( x y ) ^ k = D _ p ( x , - y ) + ( x y ) ^ k , \\\\ \\end{aligned} \\end{align*}"}
+{"id": "7133.png", "formula": "\\begin{align*} Q ( \\bar { u } _ { c } ) = A ( \\bar { u } _ { c } ) - \\frac { 1 } { 4 } \\Vert \\bar { u } _ { c } \\Vert _ { L ^ { 2 } } ^ { 4 } + \\int _ { \\mathbb { R } ^ { 2 } } ( 2 F ( \\bar { u } _ { c } ) - f ( \\bar { u } _ { c } ) \\bar { u } _ { c } ) d x = 0 \\end{align*}"}
+{"id": "4282.png", "formula": "\\begin{align*} d y _ t = \\sigma ( y _ t ) \\circ d w _ t + b ( y _ t ) d t , y _ 0 = a . \\end{align*}"}
+{"id": "8020.png", "formula": "\\begin{align*} P \\left ( \\mu ^ N \\in O \\right ) = \\mathbb { E } _ { \\hat { P } ^ { N , F , G , H } _ { f _ 1 , f _ 2 , f _ 3 } } \\left ( \\frac { d P } { d P ^ N _ { f _ 1 , f _ 2 , f _ 3 } } \\frac { 1 } { \\mathcal { U } ^ N _ { F , G , H } \\left ( T , \\xi ^ N \\right ) } 1 _ { \\{ \\mu ^ N \\in O \\} } \\right ) . \\end{align*}"}
+{"id": "491.png", "formula": "\\begin{align*} \\gcd ( r _ i , s ) = \\gcd ( r _ i , \\frac { q - 1 } { d } ) = 1 . \\end{align*}"}
+{"id": "3107.png", "formula": "\\begin{align*} \\varphi \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) = \\left ( \\begin{array} { c c c } a ^ { \\ell _ 1 } & 0 & 0 \\\\ 0 & d ^ { - \\ell _ 2 } & 0 \\\\ 0 & 0 & d ^ { - \\ell _ 2 } \\end{array} \\right ) . \\end{align*}"}
+{"id": "602.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { 3 2 } \\right ) ^ { k } \\binom { 2 k } { k } ^ { 2 } O _ { k } ^ { ( 2 ) } \\end{align*}"}
+{"id": "3370.png", "formula": "\\begin{align*} & \\omega : = e ^ { 1 5 } + e ^ { 2 6 } + e ^ { 3 7 } , \\\\ & \\omega _ + = ( e ^ 1 + e ^ 5 ) ( e ^ 2 + e ^ 6 ) ( e ^ 3 + e ^ 7 ) , \\omega _ - = ( e ^ 1 - e ^ 5 ) ( e ^ 2 - e ^ 6 ) ( e ^ 3 - e ^ 7 ) . \\end{align*}"}
+{"id": "5.png", "formula": "\\begin{align*} C _ + + C _ - = 1 C _ + f + C _ - f = f + \\textstyle \\int \\ ! f . \\end{align*}"}
+{"id": "208.png", "formula": "\\begin{align*} \\frac { d x ^ i } { d \\tau } = f ( x ^ 1 , \\ldots , x ^ n ) \\ X ^ i ( x ^ 1 , \\ldots , x ^ n ) , i = 1 , \\ldots , n . \\end{align*}"}
+{"id": "8515.png", "formula": "\\begin{align*} \\pi _ k ( \\sigma ( w ) ) = \\pi _ { k + 1 } ( w ) , \\end{align*}"}
+{"id": "2423.png", "formula": "\\begin{align*} & v _ 2 \\left ( n ! ^ { 2 + j } \\prod _ { k = 1 } ^ { n } \\left ( ( k - 1 ) ! ( n - k ) ! \\right ) ^ { i _ k } \\right ) \\\\ \\geqslant & ( 2 + j ) ( n - m ) + ( i _ 1 + \\cdots + i _ n ) ( n - 2 m + 2 ) - i _ { * } \\\\ = & ( 2 + j ) ( m - 2 ) + ( i _ 1 + \\cdots + i _ n + j + 2 ) ( n - 2 m + 2 ) - i _ { * } \\\\ \\geqslant & 2 ( m - 2 ) + ( s + 2 ) ( n - 2 m + 2 ) - i _ { * } . \\end{align*}"}
+{"id": "4252.png", "formula": "\\begin{align*} \\| u \\| _ { H ^ { k , \\ell } } = \\| \\langle \\partial _ x \\rangle ^ k \\langle x \\rangle ^ \\ell u \\| _ { L ^ 2 } , \\end{align*}"}
+{"id": "3994.png", "formula": "\\begin{align*} S ^ { \\ : n + \\nu - 1 } = \\left \\{ \\left ( x _ { 1 } , \\ldots , x _ { n } , y _ { 1 } , \\ldots , y _ { \\nu } \\right ) \\in \\mathbb { R } ^ { n + \\nu } : x _ { i } \\geq 0 , y _ { i } \\geq 0 , \\sum _ { i = 1 } ^ { n } x _ { i } + \\sum _ { i = 1 } ^ { \\nu } y _ { i } = 1 \\right \\} \\end{align*}"}
+{"id": "7651.png", "formula": "\\begin{align*} \\phi ^ { n _ 0 } _ s ( \\gamma ) = \\int ^ { \\infty } _ { \\infty } \\frac { \\rho ^ { \\mathrm { e f f } , \\Lambda ' } _ { n _ 0 , L } ( v ) } { \\abs { v - \\gamma } ^ s } \\ , d v \\end{align*}"}
+{"id": "7858.png", "formula": "\\begin{align*} f ( \\mathbf { x } ) & = \\sum _ { i = 1 } ^ { m - 1 } a _ { i } x _ { \\pi ( i ) } x _ { \\pi ( i + 1 ) } + \\sum _ { k = 1 } ^ { m } d _ { k } x _ { k } ^ { 2 } \\\\ & = 2 x _ { 1 } x _ { 2 } + x _ { 2 } x _ { 4 } + x _ { 4 } x _ { 3 } + 2 x _ { 1 } ^ { 2 } + x _ { 3 } ^ { 2 } + x _ { 4 } ^ { 2 } \\end{align*}"}
+{"id": "2012.png", "formula": "\\begin{align*} \\mathcal { N } _ { - n } = \\dfrac { a } { \\alpha ^ n } + \\dfrac { b } { \\beta ^ n } + \\dfrac { c } { \\gamma ^ n } . \\end{align*}"}
+{"id": "5981.png", "formula": "\\begin{align*} r _ { \\partial M } : = d ( \\cdot , \\partial M ) , \\end{align*}"}
+{"id": "1765.png", "formula": "\\begin{align*} \\alpha _ { n , j , t } : = \\begin{cases} t ^ 2 & \\mbox { i f } n = 2 j \\\\ - t ^ 2 z _ n & \\mbox { i f } n = 2 j + 1 \\\\ 0 & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
+{"id": "4033.png", "formula": "\\begin{align*} \\begin{cases} \\begin{array} { c c l } x _ { 1 } ' & = & 0 \\\\ x _ { 2 } ' & = & \\left ( \\frac { 1 - \\mu } { 2 - \\mu } x _ { 1 } + \\frac { 1 - \\mu } { 3 - \\mu } x _ { 2 } \\right ) \\left ( y _ { 1 } + y _ { 2 } \\right ) \\\\ y _ { 1 } ' & = & \\left ( \\frac { 1 - \\mu } { 2 - \\mu } x _ { 1 } + \\frac { 1 - \\mu } { 3 - \\mu } x _ { 2 } \\right ) \\left ( y _ { 1 } + y _ { 2 } \\right ) \\\\ y _ { 2 } ' & = & \\left ( \\frac { \\mu } { 2 - \\mu } x _ { 1 } + \\frac { 1 + \\mu } { 3 - \\mu } x _ { 2 } \\right ) \\left ( y _ { 1 } + y _ { 2 } \\right ) . \\end{array} \\end{cases} \\end{align*}"}
+{"id": "1615.png", "formula": "\\begin{align*} \\sum _ j ( - 1 ) ^ { \\tfrac { s - j } { 2 } } & \\binom { j \\tfrac { ( p - 1 ) } { 2 } } { \\tfrac { s - j } { 2 } } Q _ { ( r + p s - p j ) ( p - 1 ) } Q _ { j ( p - 1 ) } \\\\ & = q ^ { \\tfrac { p - 1 } { 2 } } \\sum _ i ( - 1 ) ^ { \\tfrac { r + s ( p - 1 ) - i } { 2 } } \\binom { \\tfrac { i ( p - 1 ) } { 2 } } { \\tfrac { r + s ( p - 1 ) - i } { 2 } } Q _ { ( r + p s - p i ) ( p - 1 ) } Q _ { i ( p - 1 ) } , \\end{align*}"}
+{"id": "1995.png", "formula": "\\begin{align*} g ( \\psi _ 0 ) = 2 | \\psi _ 0 | ^ 2 \\psi _ 1 + ( \\psi _ 0 ) ^ 2 \\overline { \\psi _ 1 } , g ( \\psi ^ n ) = \\delta _ t ^ - f ( \\psi ^ n ) , n \\ge 1 . \\end{align*}"}
+{"id": "6111.png", "formula": "\\begin{align*} g ( \\nabla _ A U , B ) & = g ( \\nabla _ B U , A ) \\end{align*}"}
+{"id": "2824.png", "formula": "\\begin{align*} K ^ { ( 1 ) } = \\begin{bmatrix} 0 & 0 & 0 & 0 \\\\ 0 & 2 & 1 & - 1 \\\\ 0 & 1 & 1 & 0 \\\\ 0 & - 1 & 0 & 1 \\end{bmatrix} , \\end{align*}"}
+{"id": "6369.png", "formula": "\\begin{align*} \\abs * { \\int _ { S ^ { n - 1 } } \\frac { x ^ k } { R } \\rho \\phi ' ( R ) \\ , d x } & = \\abs * { \\int _ { S ^ { n - 1 } } \\frac { x ^ k } { R } \\frac { \\rho ^ 2 } { 2 } \\Bigl ( \\phi ' ( R ( 1 + \\theta \\rho ) ) + R ( 1 + \\theta \\rho ) \\phi '' ( R ( 1 + \\theta \\rho ) ) \\Bigr ) \\ , d x } \\\\ & \\leq ( B _ 1 + 2 C _ 2 ) \\int _ { S ^ { n - 1 } } \\frac { \\rho ^ 2 } { 2 } \\phi ' ( R ) \\ , d x , \\end{align*}"}
+{"id": "1723.png", "formula": "\\begin{align*} A = T _ { n _ 1 } \\oplus \\cdots \\oplus T _ { n _ \\mu } \\end{align*}"}
+{"id": "427.png", "formula": "\\begin{align*} \\begin{aligned} & | [ \\bold { S } _ { j - 1 } ^ { - 1 } ] ^ { ( 1 , 2 ) } _ { m , n } | \\le \\max _ { a , b } | [ \\bold { \\Gamma } _ { j - 1 } ] _ { a , b } | \\| ( \\bold { I } _ { j - 1 } - \\bold { \\Pi } _ { j - 1 } ) ^ { - 1 } _ { ( n ) } \\| _ 1 \\\\ & \\| ( \\bold { I } _ { j - 1 } - \\bold { \\Pi } _ { j - 1 } - \\bold { \\Xi } _ { j - 1 } ( \\bold { I } _ { j - 1 } - \\bold { \\Pi } _ { j - 1 } ^ { T } ) ^ { - 1 } \\bold { \\Gamma } _ { j - 1 } ) ^ { - 1 } _ { ( m ) } \\| _ { 1 } \\\\ & \\le \\frac { K } { M } , \\end{aligned} \\end{align*}"}
+{"id": "6048.png", "formula": "\\begin{align*} R = \\frac 1 2 \\begin{pmatrix} 1 & 2 \\\\ 1 & 0 \\end{pmatrix} , R ^ { - 1 } = \\begin{pmatrix} 0 & 2 \\\\ 1 & - 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "2997.png", "formula": "\\begin{align*} \\begin{array} { l l l } 2 g ( H ( x , y ) , z ) & = & g ( T ( x , y ) , z ) + g ( T ( z , x ) , y ) + g ( T ( z , y ) , x ) \\\\ & = & 2 g ( \\alpha ( g ( x , y ) \\xi - \\eta ( y ) x ) + \\beta ( g ( \\varphi x , y ) \\xi - \\eta ( y ) \\varphi x ) , z ) . \\end{array} \\end{align*}"}
+{"id": "5491.png", "formula": "\\begin{align*} \\left ( \\frac { \\Delta } { e } \\right ) ^ n e ^ { - \\frac { 2 0 n } { \\log ^ 2 \\Delta } } \\le p e r ( A ) = \\sum _ { s = 1 } ^ { n / 2 } f ( G , s ) 2 ^ s . \\end{align*}"}
+{"id": "3547.png", "formula": "\\begin{align*} f - f ( \\tau ( z _ 0 ) ) = ( i d - \\tau ( z _ 0 ) ) g , \\end{align*}"}
+{"id": "6688.png", "formula": "\\begin{align*} a _ i ^ { \\ , c } = a _ i a _ { i + 1 } , a _ n ^ { \\ , c } = a _ n . \\end{align*}"}
+{"id": "393.png", "formula": "\\begin{align*} M h ^ { a } _ { c } ( f ) : = \\bigcap \\{ M h ^ { a } _ { d } ( f ( d ) ) : d \\in { \\rm s u p p } ( f ^ { c } ) \\} = \\bigcap \\{ M h ^ { a } _ { d } ( f ( d ) ) : c \\leq d \\in { \\rm s u p p } ( f ) \\} . \\end{align*}"}
+{"id": "4864.png", "formula": "\\begin{align*} v = H ^ \\alpha ( H ^ 2 - | A | ^ 2 ) , \\end{align*}"}
+{"id": "7360.png", "formula": "\\begin{align*} | u _ t | ^ p | u | ^ q = ( p / ( p + q ) ) ^ p \\left | \\left ( | u | ^ { ( p + q ) / p } \\right ) _ t \\right | ^ p \\end{align*}"}
+{"id": "2736.png", "formula": "\\begin{align*} J = \\begin{bmatrix} 0 & I _ { n \\times n } \\\\ - I _ { n \\times n } & 0 \\end{bmatrix} . \\end{align*}"}
+{"id": "1519.png", "formula": "\\begin{align*} \\Delta V = { \\rm D i v } \\ , ( D V ) , \\end{align*}"}
+{"id": "6619.png", "formula": "\\begin{align*} \\frac { 1 } { x ^ { N } } \\sum _ { n \\leq x } ( x - n ) ^ { N } f ( n ) = \\frac { 1 } { 2 \\pi i } \\int _ { a - i \\infty } ^ { a + i \\infty } F ( s ) \\ y ^ s \\frac { d s } { s ( s + 1 ) \\cdots ( s + N ) } , \\end{align*}"}
+{"id": "8769.png", "formula": "\\begin{align*} \\mu & \\left ( \\left \\{ y \\in \\R : \\hat \\pi ^ \\uparrow _ y ( B ) > \\pi ^ \\uparrow _ y ( B ) \\right \\} \\right ) = 0 = \\mu \\left ( \\left \\{ y \\in \\R : \\hat \\pi ^ \\uparrow _ y ( B ) < \\pi ^ \\uparrow _ y ( B ) \\right \\} \\right ) . \\end{align*}"}
+{"id": "8784.png", "formula": "\\begin{align*} G ( \\mu ( \\{ x \\} ) - p _ + ( x ) ) = \\int _ { F _ \\nu ( x - ) - p _ - ( x ) } ^ { F _ \\nu ( x - ) + p _ + ( x ) - \\mu ( \\{ x \\} ) } ( F _ \\nu ^ { - 1 } ( v ) - x ) d v \\le 0 \\end{align*}"}
+{"id": "182.png", "formula": "\\begin{align*} Q \\ = \\ \\{ ( x , y , z ) : & ( x , z ) , ( z , y ) \\in R \\} \\ \\cup \\\\ & \\{ ( x , y , z ) : ( y , z ) , ( z , x ) \\in R \\} , \\\\ Q ^ { \\circ } \\ = \\ \\{ ( x , y , z ) : & ( x , z ) , ( z , y ) \\in R ^ { \\circ } \\} \\ \\cup \\\\ & \\{ ( x , y , z ) : ( y , z ) , ( z , x ) \\in R ^ { \\circ } \\} \\end{align*}"}
+{"id": "6023.png", "formula": "\\begin{align*} \\begin{cases} \\frac { 1 } { N ^ \\gamma } \\Big ( D _ 1 \\frac { E _ { B } - E _ A } { 3 } + D _ 2 \\frac { E _ A - E _ { C } } { 3 } \\Big ) + D _ 1 v = 0 \\\\ \\frac { 1 } { N ^ \\gamma } \\Big ( D _ 1 \\frac { E _ { B } - E _ { C } } { 3 } - D _ 2 \\frac { E _ { B } - E _ A } { 3 } \\Big ) + D _ 2 v = 0 . \\end{cases} \\end{align*}"}
+{"id": "5669.png", "formula": "\\begin{align*} ( \\| \\boldsymbol { a } \\| ^ 2 , A ) _ p = ( \\tan ^ 2 \\theta , A ) _ p \\end{align*}"}
+{"id": "1049.png", "formula": "\\begin{align*} ( T _ n ( w ) ^ { - 1 } ) ^ { n + 1 - s , n + 1 - t } = ( T _ n ( \\tilde { w } ) ^ { - 1 } ) ^ { s , t } , n \\in \\N , ~ ~ s , t \\in \\{ 1 , \\dots , n \\} . \\end{align*}"}
+{"id": "4793.png", "formula": "\\begin{align*} f \\le 0 , f \\in L ^ { \\frac { 2 n } { n + 2 } } \\hbox { i f } n \\ge 3 \\hbox { a n d } f \\in L ^ p ( p > 1 ) \\hbox { i f } n = 2 . \\end{align*}"}
+{"id": "6975.png", "formula": "\\begin{align*} \\Pi ( z ) = z , \\ ; \\ ; \\mathrm { i f } \\ ; \\ ; z \\in \\mathbb { H } _ { \\mathbb { R } } ^ n . \\end{align*}"}
+{"id": "9090.png", "formula": "\\begin{align*} X _ i ( 0 ) \\geq 0 \\Longrightarrow X _ i ( t ) \\geq 0 , \\mbox { f o r a l l $ t = 0 , 1 , 2 , \\ldots $ } . \\end{align*}"}
+{"id": "4942.png", "formula": "\\begin{align*} \\{ \\zeta _ j \\} _ { 1 \\leq j \\leq 4 } : = \\{ \\zeta _ { e ^ L } , \\zeta _ { e ^ { S _ 1 } _ + } , \\zeta _ { e ^ { S _ 2 } _ + } , \\zeta _ { e ^ { S } _ + } = \\zeta _ { e ^ { S _ 3 } _ - } \\} \\end{align*}"}
+{"id": "515.png", "formula": "\\begin{align*} O ( \\log \\log { s } \\log ^ { 1 + o ( 1 ) } { s } + \\ell \\log ^ { 1 + o ( 1 ) } { s } ) = O ( \\ell \\log ^ { 1 + o ( 1 ) } { s } ) \\subseteq O ( d \\log ^ { 1 + o ( 1 ) } { q } ) \\end{align*}"}
+{"id": "2617.png", "formula": "\\begin{align*} | D _ 4 ( S ) | \\geq | D _ 2 ( S ) | + 2 = | D _ { \\{ 1 \\} } ( S ) \\cup [ D _ { \\{ 2 \\} } ( S ) \\setminus D _ { \\{ 1 \\} } ( S ) ] | + 2 \\geq ( n - 1 ) + ( n / 4 - 2 ) + 2 = \\frac { 5 n } { 4 } - 1 . \\end{align*}"}
+{"id": "2254.png", "formula": "\\begin{align*} \\small \\Lambda = N _ 1 \\cup N _ 2 \\cup N _ 3 \\cup N _ 4 \\cup D \\end{align*}"}
+{"id": "4066.png", "formula": "\\begin{align*} R _ 0 ( z ) ( x , y ) = \\frac { \\alpha _ 1 } { r ^ { d - 1 } } + O \\Big ( | z | r ^ { 2 - d } \\Big ) \\end{align*}"}
+{"id": "5622.png", "formula": "\\begin{align*} X = \\bigcup _ { i = 0 } ^ \\infty \\bigcap _ { n = i } ^ \\infty T ^ n ( U ) . \\end{align*}"}
+{"id": "4611.png", "formula": "\\begin{align*} \\beta = \\frac { d ^ 2 } { d + 1 } Y \\left [ Y ^ 2 - 1 \\right ] \\leq \\frac { d ^ 2 } { d + 1 } \\left ( 1 + \\frac { 1 } { 2 d } \\right ) \\frac { 1 } { d } \\left ( 1 + \\frac { 1 } { 4 d } \\right ) < 1 \\ , . \\end{align*}"}
+{"id": "8953.png", "formula": "\\begin{align*} \\nabla u ( x ) - S = F ( x ) + G ( x ) \\quad , \\end{align*}"}
+{"id": "5466.png", "formula": "\\begin{align*} h _ 1 ( x ) & = 2 - x , \\\\ h _ 2 ( x ) & = x + 2 . \\end{align*}"}
+{"id": "3186.png", "formula": "\\begin{align*} I _ n ( \\mu ) & = \\left \\{ \\int _ 0 ^ { \\frac { \\pi } { 6 n + 4 } } + \\int _ { \\frac { \\pi } { 6 n + 4 } } ^ { \\frac { \\pi } { 2 } } \\right \\} \\theta \\sin \\left ( \\mu \\theta \\right ) \\prod _ { k = 0 } ^ { n } \\cos ( ( 3 k + 1 ) \\theta ) \\cos ( ( 3 k + 2 ) \\theta ) \\mathrm { d } \\theta \\\\ [ 5 p t ] & = I ^ { ( 1 ) } _ n ( \\mu ) + I ^ { ( 2 ) } _ n ( \\mu ) . \\end{align*}"}
+{"id": "4687.png", "formula": "\\begin{align*} \\mathrm { S C } ( D ) : = \\left \\{ D ' \\in \\mathrm { B i g } ( X ) \\ ; | \\ ; \\mathbf { B } _ { + } ( D ' ) = \\mathbf { B } _ { + } ( D ) \\right \\} . \\end{align*}"}
+{"id": "8641.png", "formula": "\\begin{align*} v _ 3 ( T _ 2 ; x _ 2 ) ( \\partial _ x X ) ( T _ 2 ; x _ 2 ) = 0 . \\end{align*}"}
+{"id": "6371.png", "formula": "\\begin{align*} \\norm { \\rho } _ { C ^ 1 } < \\varepsilon : = \\min \\Bigl \\{ \\frac 1 2 , \\frac { 1 } { 3 K _ 1 } , \\frac { R ^ 2 \\lambda ^ R _ 2 } { 2 4 K _ 2 } , \\frac { 1 } { 1 6 K _ 3 } \\Bigr \\} , \\end{align*}"}
+{"id": "3580.png", "formula": "\\begin{align*} \\alpha ^ 3 ( \\tau _ 0 ^ 2 \\tau _ 1 ) f \\circ \\tau - ( 1 + a ) \\alpha ^ 2 \\tau _ 0 \\tau _ 1 f \\circ i d + ( a + b ) \\alpha \\tau _ 0 f \\circ \\tau - b f = 0 , \\end{align*}"}
+{"id": "5738.png", "formula": "\\begin{align*} X _ + ( t ) \\leq e ^ { ( \\gamma _ * + \\varepsilon _ 1 ) t } \\int _ t ^ \\infty e ^ { - ( \\gamma _ * + \\varepsilon _ 1 ) \\tau } | Y _ + ( \\tau ) | \\ , d \\tau = O ( e ^ { ( \\gamma _ * - \\varepsilon _ 0 ) t } ) . \\end{align*}"}
+{"id": "5448.png", "formula": "\\begin{align*} \\nu ^ { ( 0 , k _ 1 ) } _ { ( 0 , j ) } : = \\nu ^ { ( k _ 1 ) } _ j - \\hat { c } _ j / w ^ { ( k _ 1 ) } _ j , j \\in S _ 1 ^ { k _ 1 } \\setminus S _ 0 ^ { k _ 0 } , \\end{align*}"}
+{"id": "209.png", "formula": "\\begin{align*} \\frac { d \\tau } { d t } = \\frac 1 { f ( \\gamma ( t ) ) } , \\end{align*}"}
+{"id": "8252.png", "formula": "\\begin{align*} \\sum _ { i = t } ^ L H ( Y _ i | { \\underline S } , y ^ { t - 1 } , Y _ t ^ { i - 1 } ) . \\end{align*}"}
+{"id": "8587.png", "formula": "\\begin{align*} R _ i ^ N = \\{ i , i + 1 , \\dots , i + k - 1 \\} \\times \\{ 1 , 2 , \\dots , M \\} \\ : . \\end{align*}"}
+{"id": "8320.png", "formula": "\\begin{align*} \\delta ( V ) = \\delta ( X - \\Pi ^ { \\sharp } ( \\beta ) ) . \\end{align*}"}
+{"id": "8315.png", "formula": "\\begin{align*} \\eta ^ { \\sharp } ( \\psi ( X + \\beta ) ) = - \\omega ^ { \\flat } ( \\epsilon ( X ) ) + \\epsilon ^ { * } \\big ( \\omega ^ { \\flat } ( \\epsilon ( X ) ) - \\beta \\big ) + \\epsilon ( X ) , \\end{align*}"}
+{"id": "1895.png", "formula": "\\begin{align*} \\begin{aligned} & \\ , \\ , \\omega ( x , y ) = \\omega ( y , x ) & \\ , \\ , \\ , ( x , y ) \\in G \\times G ; \\\\ & \\ , \\ , \\omega ( x , x ) = 0 \\quad \\quad \\quad \\ , \\ , & \\ , \\ , \\ , x \\in G ; \\\\ & \\ , \\ , \\displaystyle \\sum _ { y \\in G } \\omega ( x , y ) < \\infty & \\ , \\ , \\ , x \\in G \\ , . \\end{aligned} \\end{align*}"}
+{"id": "4990.png", "formula": "\\begin{align*} a ( \\lambda ) = \\prod _ { k = 1 } ^ { N } ( \\lambda - \\nu _ k + 1 ) , d ( \\lambda ) = \\prod _ { k = 1 } ^ { N } ( \\lambda - \\nu _ k ) . \\end{align*}"}
+{"id": "5227.png", "formula": "\\begin{align*} L _ { d } D A G I ( p \\| q ) = - \\sum _ { j } p _ { j } \\left [ \\frac { \\left ( \\overline { M G } \\right ) ^ { a - 1 } - \\left ( \\overline { M G } \\right ) ^ { b - 1 } } { a - b } \\right ] \\end{align*}"}
+{"id": "5325.png", "formula": "\\begin{align*} \\mathcal { F } = \\left \\{ S = \\bigcup _ { k = 1 } ^ m S _ k : S _ k \\in \\mathcal { F } _ k , 1 \\leq k \\leq m \\right \\} . \\end{align*}"}
+{"id": "3082.png", "formula": "\\begin{align*} \\Lambda _ F = \\{ \\nu ( A d x + B d y ) ; \\ A , B \\in \\mathbb { C } \\{ x \\} [ y ] \\ \\mbox { w i t h } \\ d e g _ y ( A ) < v _ 0 \\ \\mbox { a n d } \\ d e g _ y ( B ) < v _ 0 - 1 \\} . \\end{align*}"}
+{"id": "5847.png", "formula": "\\begin{align*} M _ { 1 } ( \\Gamma _ i ) = l _ i M _ { 1 } ( K _ { m _ i } ) M _ { 2 } ( \\Gamma ) = l _ i M _ { 2 } ( K _ { m _ i } ) . \\end{align*}"}
+{"id": "8346.png", "formula": "\\begin{align*} \\mu ^ + ( \\partial K ) : = \\liminf _ { \\epsilon \\to 0 } \\frac { \\mu \\left ( K + \\epsilon B _ 2 ^ n \\right ) - \\mu ( K ) } { \\epsilon } = \\int _ { \\partial K } \\phi ( y ) \\ , d y . \\end{align*}"}
+{"id": "6495.png", "formula": "\\begin{align*} ( \\xi ' _ 1 , \\xi ' _ 2 , \\xi ' _ 3 ) = ( - 3 , 0 , - 1 ) , ( \\xi _ 1 , \\xi _ 2 , \\xi _ 3 ) = ( - 1 , 0 , - 1 ) . \\end{align*}"}
+{"id": "153.png", "formula": "\\begin{align*} & e ^ { N s } - \\sum _ { m = 0 } ^ { N - 1 } \\frac { N ^ m } { m ! } { s } ^ m = \\sum _ { m \\geq N } \\frac { N ^ m } { m ! } { s } ^ m \\\\ & = \\frac { { ( N s ) } ^ { N } } { N ! } \\cdot \\bigg ( 1 + \\sum _ { m \\geq 1 } \\frac { { ( N s ) } ^ { m } } { ( N + 1 ) \\cdots ( N + m ) } \\bigg ) \\leq C ( \\lambda ) \\cdot \\frac { { ( N s ) } ^ { N } } { N ! } , \\end{align*}"}
+{"id": "8049.png", "formula": "\\begin{align*} \\mathcal { Y } _ { F , G , H } ^ N ( t , \\xi ^ N ) = \\exp \\left ( \\frac { \\gamma _ N ^ 2 } { N } \\left ( \\mathcal { J } _ 1 ( \\eta ^ N , F , G , H ) + \\varepsilon ^ N _ { 1 2 } + o ( 1 ) \\right ) \\right ) , \\end{align*}"}
+{"id": "5561.png", "formula": "\\begin{align*} \\chi _ i = T \\phi _ i \\end{align*}"}
+{"id": "8369.png", "formula": "\\begin{align*} ( G : G \\times _ B X _ { \\underline { w } } ) = r \\iff ( B : X _ { \\underline { w } } ) = r . \\end{align*}"}
+{"id": "6022.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta _ N f \\left ( \\frac { x } { N } \\right ) & = N ^ 2 \\left \\{ f \\left ( \\frac { x + 1 } { N } \\right ) - 2 f \\left ( \\frac { x } { N } \\right ) + f \\left ( \\frac { x - 1 } { N } \\right ) \\right \\} \\\\ \\nabla _ N f \\left ( \\frac { x } { N } \\right ) & = N \\left \\{ f \\left ( \\frac { x + 1 } { N } \\right ) - f \\left ( \\frac { x } { N } \\right ) \\right \\} . \\end{aligned} \\end{align*}"}
+{"id": "8832.png", "formula": "\\begin{align*} f _ * : = \\inf f ( [ \\mathfrak { s } , \\infty ) \\times [ \\delta _ 1 , u ^ { * * } ] ) > 0 , \\ , \\ , \\partial _ u f ( s , u ) > \\frac { { \\rm d } f _ + ^ { \\infty } ( 0 ) } { { \\rm d } u } - \\frac { 2 \\gamma } { 3 } , \\ , \\ , \\forall ( s , u ) \\in [ \\mathfrak { s } , \\infty ) \\times [ 0 , \\delta _ 1 ] . \\end{align*}"}
+{"id": "645.png", "formula": "\\begin{align*} \\ddot { \\mu } _ { H _ r , \\epsilon } ( \\rho ) = \\frac { \\sinh ( \\rho ) \\cosh ^ { \\frac { 2 ( n - r ) } { r } - 1 } ( \\rho ) \\left ( n H _ r \\frac { \\cosh ^ n ( \\rho ) } { \\sinh ^ r ( \\rho ) } - ( n - r ) ( n H _ r J _ { n , r , \\epsilon } ( \\rho ) ) \\right ) } { r ( n H _ r J _ { n , r , \\epsilon } ( \\rho ) ) ^ { \\frac { r - 1 } { r } } \\left ( \\cosh ^ { \\frac { 2 ( n - r ) } { r } } ( \\rho ) - ( n H _ r J _ { n , r , \\epsilon } ( \\rho ) ) ^ { \\frac { 2 } { r } } \\right ) ^ { \\frac { 3 } { 2 } } } . \\end{align*}"}
+{"id": "8567.png", "formula": "\\begin{align*} F ( n , k ) = \\frac { ( - 1 ) ^ k ( a ) _ k ( b ) _ k } { ( a + n + 1 ) _ k ( b + n + 1 ) _ k } , \\end{align*}"}
+{"id": "1936.png", "formula": "\\begin{align*} h _ r ( t ) : = \\frac { t ^ { p - 1 } } { r ^ p } + \\frac { t ^ { q - 1 } } { r ^ { s q } } , g _ r ( t ) : = \\frac { t ^ { \\bar { q } - 1 } } { r ^ { \\bar { q } } } , t \\ge 0 \\end{align*}"}
+{"id": "8466.png", "formula": "\\begin{align*} | u _ m | + | u _ { m - 1 } | = ( u _ { m - 1 } - u _ m ) + 2 u _ m & \\leq ( u _ { m - 1 } - u _ m ) + 2 \\ell \\\\ & \\leq ( u _ { m - 1 } - u _ m ) + 2 ( u _ { m - 1 } - u _ m ) \\\\ & = 3 ( u _ { m - 1 } - u _ m ) \\\\ & = 3 | u _ m - u _ { m - 1 } | . \\end{align*}"}
+{"id": "7499.png", "formula": "\\begin{align*} | R m ( g _ 0 ( t ) ) | _ { g _ 0 ( t ) } = | R m ( g _ N ( t ) ) | _ { g _ N ( t ) } \\leq \\frac { C ( g _ N ) } { \\mathbf { r } ^ 2 } \\leq \\frac { C ( g _ N ) } { \\gamma t + ( \\Lambda + 1 ) ^ 2 } \\end{align*}"}
+{"id": "6246.png", "formula": "\\begin{align*} \\begin{aligned} g ( \\alpha ) & = ( r _ i + \\lambda _ g ) \\alpha ( 1 - \\alpha ) ( \\alpha - \\gamma ) \\\\ & = \\frac { ( r _ i + \\lambda _ g ) ( 2 - \\omega ) ( \\omega + 1 ) ( 2 - \\omega - 3 \\gamma ) } { 2 7 } \\\\ & = \\frac { ( 2 - \\omega ) ( \\omega + 1 ) \\left ( - ( \\omega + 1 ) r _ i + ( 2 - \\omega ) \\lambda _ g \\right ) } { 2 7 } . \\end{aligned} \\end{align*}"}
+{"id": "354.png", "formula": "\\begin{align*} \\begin{aligned} \\phi _ p ( z , y ) & = ( z y ) ^ k + \\sum _ { i = 0 } ^ { k - 1 } ( z y ) ^ { i } ( z ^ { 2 ( k - i ) } + y ^ { 2 ( k - i ) } ) \\\\ & = p ( z y ) ^ k + \\sum _ { i = 0 } ^ { k - 1 } ( z y ) ^ { i } ( z ^ { k - i } - y ^ { k - i } ) ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "5876.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { N C } ( G ) ) & = ( 8 n - 4 ) ( 8 n - 5 ) ^ { 2 } - 4 ( 8 n - 5 ) ( ( 2 n - 2 ) ( 4 n - 5 ) + 6 n ) \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ + ( 4 n - 4 ) ( 4 n - 5 ) ^ { 2 } + 3 6 n \\\\ & = 3 2 0 n ^ { 3 } - 5 7 6 n ^ { 2 } + 2 5 6 n \\\\ & = 8 n ( 4 0 n ^ { 2 } - 7 2 n + 3 2 ) \\end{align*}"}
+{"id": "1485.png", "formula": "\\begin{align*} P ( P ^ { - 1 } ( [ P ^ { - 1 } ( x ) , y ] ) ) & = [ P ^ { - 1 } ( x ) , y ] = [ P ^ { - 1 } ( x ) , P ( P ^ { - 1 } ( y ) ] \\\\ & = [ P ( x ' ) , P ( P ^ { - 1 } ( y ) ] = P ( [ P ( x ' ) , P ^ { - 1 } ( y ) ] ) = P ( [ P ^ { - 1 } ( x ) , P ^ { - 1 } ( y ) ] ) . \\end{align*}"}
+{"id": "3073.png", "formula": "\\begin{align*} \\varphi _ a ( t ) = \\left ( t ^ { \\beta _ 0 } , \\sum _ { \\beta _ 1 \\leq l < \\beta _ g } c _ l t ^ l + \\sum _ { l \\geq \\beta _ g } a _ l t ^ l \\right ) \\end{align*}"}
+{"id": "3492.png", "formula": "\\begin{align*} \\| a \\| + \\| a _ - \\| & = \\| b - a \\| \\\\ & = \\| \\theta ( b ) - \\theta ( a ) \\| \\\\ & \\leq \\| \\| \\theta ( b ) \\| 1 _ { B ^ \\sim } - \\theta ( a ) \\| \\\\ & = \\| \\| \\theta ( a ) \\| 1 _ { B ^ \\sim } - \\theta ( a ) \\| \\\\ & \\leq \\| \\theta ( a ) \\| \\\\ & = \\| a \\| , \\end{align*}"}
+{"id": "7299.png", "formula": "\\begin{align*} \\frac { 1 } { 3 k } \\sum _ { j = 0 } ^ { 3 k - 1 } \\sec \\left ( \\frac { 4 j } { 3 k } \\pi \\right ) \\omega ^ j = ( - 1 ) ^ { \\frac { k - 1 } { 2 } } \\end{align*}"}
+{"id": "253.png", "formula": "\\begin{align*} \\frac { d ^ 2 x } { d t ^ 2 } + \\gamma ( x ) \\left ( \\frac { d x } { d t } \\right ) ^ 2 + A ( x ) \\frac { d x } { d t } + b ( x ) = 0 , \\end{align*}"}
+{"id": "3805.png", "formula": "\\begin{align*} r \\cdot v & = G _ { i , [ k ] } x _ { \\ell , t } - G _ { i , [ k ] } x _ { \\ell ' , t } = G _ { i , [ k ] } ( x _ { \\ell } - x _ t ) - G _ { i , [ k ] } ( x _ { \\ell ' , t } - x _ t ) \\\\ & = G _ { i , [ k ] } x _ { \\ell } - G _ { i , [ k ] } x _ { \\ell ' } = c _ { \\ell , i } - c _ { \\ell ' , i } = 0 \\end{align*}"}
+{"id": "5468.png", "formula": "\\begin{align*} \\mathcal I _ K ( \\sigma , T ) = \\int _ { T } ^ { 2 T } \\left | \\frac { \\zeta ' } { \\zeta } \\left ( \\sigma + i t \\right ) \\right | ^ { 2 K } d t . \\end{align*}"}
+{"id": "3151.png", "formula": "\\begin{align*} \\varphi ^ \\sharp \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & = \\left ( \\begin{array} { c c c } a ^ { p ^ { e _ 2 } } & 0 & b ^ { p ^ { e _ 2 } } \\\\ 0 & 1 & 0 \\\\ 0 & 0 & d ^ { p ^ { e _ 2 } } \\end{array} \\right ) ( \\ , e _ 2 \\geq 0 \\ , ) . \\end{align*}"}
+{"id": "7884.png", "formula": "\\begin{align*} \\mathbf { D } _ { i , j } = \\left ( \\begin{array} { c c c c c c } 2 d _ { 1 } ^ { ' } & c _ { 1 } & & & & \\\\ c _ { 1 } & 2 d _ { 2 } ^ { ' } & c _ { 2 } & & & \\\\ & c _ { 2 } & 2 d _ { 3 } ^ { ' } & c _ { 3 } & & \\\\ & & \\ddots & \\ddots & \\ddots & \\\\ & & & c _ { m - 2 } & 2 d _ { m - 1 } ^ { ' } & c _ { m - 1 } \\\\ & & & & c _ { m - 1 } & 2 d _ { m } ^ { ' } \\end{array} \\right ) . \\end{align*}"}
+{"id": "7564.png", "formula": "\\begin{align*} u _ { t } ( x ) = t ^ { \\frac { N } { 4 } } u ( \\sqrt { t } x ) . \\end{align*}"}
+{"id": "2576.png", "formula": "\\begin{align*} q ^ { \\frac { 1 } { 2 } } X _ 1 X _ 4 = & q ^ 2 X ( - 1 , 3 ) + ( q + q ^ 2 ) h X ( - 1 , 2 ) + ( 1 + q h ^ 2 + q ^ 2 ) X ( - 1 , 1 ) + ( 1 + q ) h X ( - 1 , 0 ) \\\\ & + X ( - 1 , - 1 ) + q h X ( 0 , 1 ) + q ^ { \\frac { 1 } { 2 } } h ^ 2 + h X ( 0 , - 1 ) + X ( 1 , - 1 ) \\end{align*}"}
+{"id": "1229.png", "formula": "\\begin{align*} \\Psi ( T ) = V \\Phi ^ c _ { \\min } ( T ) V ^ * \\quad \\mbox { a n d } \\Phi ^ c _ { \\min } ( T ) = V ^ * \\Psi ( T ) V . \\end{align*}"}
+{"id": "7505.png", "formula": "\\begin{align*} | h ( 0 ) - g _ 0 ( 0 ) | _ { g _ 0 ( 0 ) } = Q ^ * ( | g ( 0 ) - \\tilde { g } ( 0 ) | ) < \\delta \\end{align*}"}
+{"id": "1657.png", "formula": "\\begin{align*} u _ { 2 } \\left ( x , 0 \\right ) = u _ { 0 } ^ { \\left ( 2 \\right ) } \\left ( x \\right ) , m _ { 2 } \\left ( x , 0 \\right ) = m _ { 0 } ^ { \\left ( 2 \\right ) } \\left ( x \\right ) , x \\in \\Omega . \\end{align*}"}
+{"id": "5073.png", "formula": "\\begin{align*} e _ J & = e _ { J ' } , p ( J ) = p ( J ' ) \\end{align*}"}
+{"id": "4888.png", "formula": "\\begin{align*} \\O _ { \\Sigma ' } ( C ' ) = \\O _ { \\Sigma ' } ( 2 E ) \\otimes \\varphi '^ * \\mathcal { M } \\end{align*}"}
+{"id": "1105.png", "formula": "\\begin{align*} 2 ^ { - i a } & = \\sup _ { x _ 0 \\in \\mathbb { R } ^ n , \\ , r \\in ( 0 , \\infty ) } \\left ( \\frac { | x _ 0 | + r } { | x _ 0 | + 2 ^ i r } \\right ) ^ a \\lesssim \\sup _ { x _ 0 \\in \\mathbb { R } ^ n , \\ , r \\in ( 0 , \\infty ) } I ( B ( x _ 0 , r ) , i ) \\\\ & \\lesssim \\sup _ { x _ 0 \\in \\mathbb { R } ^ n , \\ , r \\in ( 0 , \\infty ) } \\left ( \\frac { | x _ 0 | + r } { | x _ 0 | + 2 ^ i r } \\right ) ^ a ( i + 1 ) ^ { - b } = 2 ^ { - i a } ( i + 1 ) ^ { - b } , \\end{align*}"}
+{"id": "2281.png", "formula": "\\begin{align*} \\check R _ i ( u ) = \\sum _ { p = 0 } ^ k ( - q ) ^ { k - p } \\left [ \\begin{array} { c } k \\\\ p \\end{array} \\right ] ^ 2 _ q \\ \\frac { ( q ^ { - 2 } \\ , ; \\ , q ^ { - 2 } ) _ { k - p } } { ( u q ^ { - 2 p } \\ , ; \\ , q ^ { - 2 } ) _ { k - p } } \\ \\Sigma _ i ^ { ( p ) } \\ , \\end{align*}"}
+{"id": "6071.png", "formula": "\\begin{align*} \\Lambda = 2 | \\phi _ { 2 , 1 } \\phi _ { 2 , 2 } | \\leq | \\phi _ { 2 , 1 } | ^ 2 + | \\phi _ { 2 , 2 } | ^ 2 < 1 . \\end{align*}"}
+{"id": "6564.png", "formula": "\\begin{align*} \\Phi _ k ( n ) = | \\{ ( x _ 1 , x _ 2 , \\cdots , x _ k ) \\in \\left ( \\mathbb { Z } / n \\mathbb { Z } \\right ) ^ k ; \\ \\gcd ( x _ 1 ^ 2 + x _ 2 ^ 2 + \\cdots + x _ k ^ 2 , n ) = 1 \\} | . \\end{align*}"}
+{"id": "2228.png", "formula": "\\begin{align*} v _ { \\varphi , t } + v \\cdot \\nabla v _ \\varphi - \\nu \\bigg ( \\Delta v _ \\varphi - { 1 \\over r ^ 2 } v _ \\varphi \\bigg ) = \\psi _ { 1 , z } v _ \\varphi + f _ \\varphi , \\end{align*}"}
+{"id": "8421.png", "formula": "\\begin{align*} f ( a ) & = f ( \\sigma ( a ) ) = f ( B _ 1 ( a _ 1 ) S ( B _ 2 ( a _ 2 ) ) ) \\\\ & = f ( B _ 1 ( a _ 1 ) ) f ( S ( B _ 2 ( a _ 2 ) ) ) = \\mathcal { B } _ 1 ( f ) ( a _ 1 ) \\mathcal { B } _ 2 ( f ) ^ { \\ast - 1 } ( a _ 2 ) = e ' ( a ) , \\end{align*}"}
+{"id": "7091.png", "formula": "\\begin{align*} ( \\lambda - \\partial _ t - \\Delta ) ^ { - \\frac { \\alpha } { 2 } } h ( t , x ) : = \\int _ { t } ^ \\infty \\int _ { \\mathbb R ^ d } e ^ { - \\lambda ( s - t ) } \\frac { 1 } { ( 4 \\pi ( s - t ) ) ^ { \\frac { d } { 2 } } } \\frac { 1 } { ( s - t ) ^ { \\frac { 2 - \\alpha } { 2 } } } e ^ { - \\frac { | x - y | ^ 2 } { 4 ( s - t ) } } h ( s , y ) d s d y , \\end{align*}"}
+{"id": "2193.png", "formula": "\\begin{align*} \\Phi _ { , t } + v \\cdot \\nabla \\Phi - \\nu \\bigg ( \\Delta + { 2 \\over r } \\partial _ r \\bigg ) \\Phi - ( \\omega _ r \\partial _ r + \\omega _ z \\partial _ z ) { v _ r \\over r } = F _ r / r \\equiv \\bar F _ r \\end{align*}"}
+{"id": "8002.png", "formula": "\\begin{align*} I _ { i n i } ( \\varpi ) = \\sup _ { f _ 1 , f _ 2 , f _ 3 \\in C ( \\mathbb { T } ) } \\{ \\mathcal { I } _ 2 ( \\varpi , f _ 1 , f _ 2 , f _ 3 ) \\} . \\end{align*}"}
+{"id": "2977.png", "formula": "\\begin{align*} x _ 2 ( t ) - x _ 1 ( t ) = \\left ( 0 , ~ x ^ 2 _ 2 ( 0 ) - x ^ 2 _ 1 ( 0 ) + 2 \\int _ 0 ^ t \\sin ( \\theta _ 2 ) d s \\right ) . \\end{align*}"}
+{"id": "493.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { N - 1 } { \\deg { Q _ t ( T ) } } = \\sum _ { t = 0 } ^ { N - 1 } { ( \\deg { P _ t ( T ) } - \\deg { P _ { t + 1 } ( T ) } ) } = \\deg { P ( T ) } - \\deg { 1 } = \\deg { P ( T ) } . \\end{align*}"}
+{"id": "4585.png", "formula": "\\begin{align*} \\int _ M t ^ 2 d \\mu & = 2 \\pi \\int _ { - a } ^ a t ^ 2 A ( t ) d t \\\\ & = \\frac { \\omega ( F ) ^ 4 } { 7 6 8 \\pi ^ 2 } \\left [ 8 \\frac { \\omega ( c _ + ) + \\omega ( c _ - ) } { \\omega ( F ) } + \\sum _ j \\frac { 1 } { r _ j s _ j } \\left ( \\frac { \\omega ( r _ j E _ j ' ) - \\omega ( - s _ j E _ j ) } { \\omega ( F ) } \\right ) ^ 4 + c _ 1 ( Y ) [ \\Sigma _ a ] - c _ 1 ( Y ) [ \\Sigma _ { - a } ] \\right ] . \\end{align*}"}
+{"id": "3123.png", "formula": "\\begin{align*} u _ { \\psi ^ * } ( t ) = \\left ( \\begin{array} { c c c } 1 & t ^ { p ^ { e _ 1 } } & \\frac { 1 } { 2 } \\ , t ^ { 2 \\ , p ^ { e _ 1 } } \\\\ 0 & 1 & t ^ { p ^ { e _ 1 } } \\\\ 0 & 0 & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "5846.png", "formula": "\\begin{align*} M _ { 1 } ( \\Gamma ) = \\sum _ { i = 1 } ^ { k } M _ { 1 } ( \\Gamma _ i ) M _ { 2 } ( \\Gamma ) = \\sum _ { i = 1 } ^ { k } M _ { 2 } ( \\Gamma _ i ) . \\end{align*}"}
+{"id": "4189.png", "formula": "\\begin{align*} g _ 1 g _ 2 g ^ { - 1 } _ 1 = g ^ { - 1 } _ 3 g _ 2 g _ 3 = g ^ { - 1 } _ 1 g _ 2 g _ 1 \\ , , \\mbox { w h e n c e } \\ , \\ , g ^ { 2 } _ 1 g _ 2 = g _ 2 g ^ { 2 } _ 1 \\ , . \\end{align*}"}
+{"id": "2070.png", "formula": "\\begin{align*} m _ { t } ^ d & \\le 8 m ^ d _ 0 + 2 1 6 \\eta ^ 4 d ^ 4 \\big [ C _ 2 ^ 4 t ^ 3 + 3 \\big ( C _ 8 ' + C _ { 5 } '^ 2 \\big ) t \\big ] \\sum _ { k = 0 } ^ { t - 1 } m ^ d _ k + 2 1 6 \\big ( C _ { 5 } C _ 1 + C _ 2 ^ 2 \\big ) \\eta ^ 4 \\sigma ^ 4 t ^ 2 \\\\ & \\le 8 m ^ d _ 0 + 2 1 6 \\big ( C _ 2 ^ 4 + 3 C _ 8 ' + 3 C _ { 5 } '^ 2 \\big ) \\eta ^ 4 d ^ 4 t ^ 3 \\sum _ { k = 0 } ^ { t - 1 } m ^ d _ k + 2 1 6 \\big ( C _ { 5 } C _ 1 + C _ 2 ^ 2 \\big ) \\eta ^ 4 \\sigma ^ 4 t ^ 2 . \\end{align*}"}
+{"id": "5842.png", "formula": "\\begin{align*} x ^ { | \\alpha ^ 3 | } y ^ { | \\alpha ^ 3 | } z ^ { | \\alpha ^ 3 | } = \\alpha ^ { \\frac { ( | \\alpha ^ 3 | - 1 ) | \\alpha ^ 3 | } { 2 } } \\left ( x y z + ( 1 - \\alpha ^ 3 ) ^ { - 1 } x ^ 3 \\right ) ^ { | \\alpha ^ 3 | } - \\alpha ^ { \\frac { ( | \\alpha ^ 3 | - 1 ) | \\alpha ^ 3 | } { 2 } } ( 1 - \\alpha ^ 3 ) ^ { - | \\alpha ^ 3 | } x ^ { | \\alpha | } . \\end{align*}"}
+{"id": "5629.png", "formula": "\\begin{align*} d ( T u _ 2 , u _ 1 ) = d ( T ^ { n + 1 } u _ 2 , T ^ n u _ 1 ) \\leq d ( T ^ { n + 1 } u _ 2 , a ) + d ( a , T ^ n u _ 1 ) < \\frac { \\delta } { 4 } + \\frac { \\delta } { 4 } = \\frac { \\delta } { 2 } , \\end{align*}"}
+{"id": "8569.png", "formula": "\\begin{align*} f ( n ) = g _ { 1 } ( n ) + g _ { 2 } ( n ) f ( n + 2 ) , \\end{align*}"}
+{"id": "8061.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow + \\infty } \\widetilde { P } _ { h _ 1 ^ m , h _ 2 ^ m , h _ 3 ^ m } ^ { N , F ^ m , G ^ m , H ^ m } \\left ( D _ { 4 , \\epsilon } \\cap \\{ \\eta ^ N \\in O \\} \\right ) = 1 . \\end{align*}"}
+{"id": "6736.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty { \\frac { { B _ { 2 k } z ^ { 2 k } } } { k ( 2 k ) ! } } = 2 \\ln \\left ( \\frac { 2 } { z } \\sinh \\left ( \\frac { z } { 2 } \\right ) \\right ) . \\end{align*}"}
+{"id": "4374.png", "formula": "\\begin{align*} F _ { \\overline { \\mathcal { Y } } , a } ( x ) & : = \\sum _ { ( i , j ) \\in [ m ] \\times [ n ] } \\overline { y } _ { ( i , j ) } g _ { ( i , j ) } ( x ) + \\sum _ { ( q , p ) \\in \\overline { \\mathcal { Y } } } \\Delta y _ { ( q , p ( q ) ) } g _ { ( q , p ( q ) ) } ( x ) - \\Delta y _ { ( a , b ( a ) ) } g _ { ( a , b ( a ) ) } ( x ) . \\end{align*}"}
+{"id": "5107.png", "formula": "\\begin{align*} \\dim \\mathcal { C } ( v ) & = \\sum _ { I \\subset S } R _ I \\cdot D _ I . \\end{align*}"}
+{"id": "782.png", "formula": "\\begin{align*} \\Delta ( u _ { i j } ) = \\sum _ k u _ { i k } \\otimes u _ { k j } , \\varepsilon ( u _ { i j } ) = \\delta _ { i j } . \\end{align*}"}
+{"id": "8018.png", "formula": "\\begin{align*} I _ { d y n } ( \\hat { W } ) = \\mathcal { I } _ 1 ( \\hat { W } , F , G , H ) \\end{align*}"}
+{"id": "4723.png", "formula": "\\begin{align*} \\nu _ h \\ ; = \\ ; - \\gamma ^ { - 2 h } \\sum _ { j = - \\infty } ^ { h - 1 } \\gamma ^ { 2 j } \\beta _ j ( \\underline { \\nu } ) \\ , . \\end{align*}"}
+{"id": "2864.png", "formula": "\\begin{align*} E ( f , \\gamma ) : = \\left \\{ ( x , y ) \\in \\mathbb { R } \\times \\mathbb { R } : \\ x \\neq y , \\ \\left | \\int _ x ^ y | f ( s ) | \\ , d s \\right | > | x - y | ^ { \\gamma + 1 } \\right \\} \\end{align*}"}
+{"id": "2097.png", "formula": "\\begin{align*} \\eqref { e q : C L T t e s t f u n c t i o n a p p l i e d s u m } = - \\int _ 0 ^ { \\frac { \\lfloor s T \\rfloor } { T } } U ( u , x ) f ' ( u ) \\mathrm { d } u + f ( s ) U ( s , x ) - f ( 0 ) U ( 0 , x ) + o ( 1 ) . \\end{align*}"}
+{"id": "3890.png", "formula": "\\begin{gather*} \\omega _ 1 + ( \\omega _ 2 - \\omega _ 1 ) + ( \\omega _ 3 - \\omega _ 2 ) + \\cdots + ( \\omega _ { n - 2 } - \\omega _ { n - 3 } ) \\\\ + ( \\omega _ n + \\omega _ { n - 1 } - \\omega _ { n - 2 } ) + ( \\omega _ n - \\omega _ { n - 1 } ) = 2 \\omega _ n . \\end{gather*}"}
+{"id": "6497.png", "formula": "\\begin{align*} G ^ \\theta & = \\left \\{ g \\in G : \\theta ( g ) = g \\right \\} , \\\\ G _ \\theta & = \\left \\{ g \\in G : g \\theta ( g ) ^ { - 1 } \\in Z _ G \\right \\} , \\end{align*}"}
+{"id": "8760.png", "formula": "\\begin{align*} \\nu _ 0 ^ \\pi ( \\R ) = \\pi ( \\{ ( x , x ) : x \\in \\R \\} ) = \\pi ( \\{ ( x , x ) : x \\in \\R \\} \\cap \\Gamma ) = \\nu _ 0 ^ \\pi ( \\{ x \\in \\R : ( x , x ) \\in \\Gamma \\} ) . \\end{align*}"}
+{"id": "8761.png", "formula": "\\begin{align*} u _ \\nu ( x ) & = \\int _ { \\R ^ 2 } | z - x | \\pi ( d y , d z ) = \\int _ { ( y , z ) \\in ( - \\infty , x ) \\times \\R } ( x - z ) \\pi _ y ( d z ) \\mu ( d y ) + \\int _ { ( y , z ) \\in ( x , + \\infty ) \\times \\R } ( z - x ) \\pi _ y ( d z ) \\mu ( d y ) \\\\ & = \\int _ { ( - \\infty , x ) } ( x - y ) \\mu ( d y ) + \\int _ { ( x , + \\infty ) } ( y - x ) \\mu ( d y ) = u _ \\mu ( x ) . \\end{align*}"}
+{"id": "3463.png", "formula": "\\begin{align*} b ^ + ( X ) + g ( S ) - \\frac { 1 } { 4 } S \\circ S - \\frac { 1 } { 2 } \\sigma ( L ) + \\frac { 1 } { 2 } \\sigma ( L ' ) = 1 , \\end{align*}"}
+{"id": "7669.png", "formula": "\\begin{align*} \\psi ( n ) = \\frac { C _ a 7 2 \\sqrt { 2 } \\norm { F } _ { \\infty } } { \\eta } \\lambda \\sum _ { m \\in \\Lambda _ L } e ^ { - \\gamma _ a d ( n , m ) - 2 \\nu d ( m , l ) } \\end{align*}"}
+{"id": "3858.png", "formula": "\\begin{align*} \\begin{aligned} G ( M ^ { \\kappa } ( s ) ) _ { x , y } = G ( ( 1 - \\kappa ) M ^ { 0 } ( s ) + \\kappa \\pi ^ * ) _ { x , y } = ( 1 - \\kappa ) G ( M ^ { 0 } ( s ) ) _ { x , y } + \\kappa G ( \\pi ^ * ) _ { x , y } . \\end{aligned} \\end{align*}"}
+{"id": "1989.png", "formula": "\\begin{align*} & \\beta ^ + _ l = \\frac { 1 + \\sqrt { 1 + 4 \\alpha | \\mu _ l | ^ 2 } } { 2 \\alpha } , \\\\ & \\beta ^ - _ l = \\frac { 1 - \\sqrt { 1 + 4 \\alpha | \\mu _ l | ^ 2 } } { 2 \\alpha } = \\frac { - 2 | \\mu _ l | ^ 2 } { 1 + \\sqrt { 1 + 4 \\alpha | \\mu _ l | ^ 2 } } , \\\\ & \\beta _ l = \\beta ^ + _ l - \\beta ^ - _ l = \\frac { \\sqrt { 1 + 4 \\alpha | \\mu _ l | ^ 2 } } { \\alpha } , \\\\ \\end{align*}"}
+{"id": "5248.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } F \\left ( p \\| q \\right ) } { \\partial q _ { j } } = \\left ( 1 - \\alpha \\right ) \\left \\lbrace \\left [ \\frac { a - 1 } { a - b } \\left ( Z _ { j } \\right ) ^ { a } - \\frac { b - 1 } { a - b } \\left ( Z _ { j } \\right ) ^ { b } \\right ] - 1 \\right \\rbrace \\end{align*}"}
+{"id": "1530.png", "formula": "\\begin{align*} \\varphi ( z ) = \\begin{cases} s ( z ) & \\\\ 1 & \\end{cases} \\end{align*}"}
+{"id": "2282.png", "formula": "\\begin{align*} \\check R _ i ( u ) \\check R _ { i + 1 } ( u v ) \\check R _ i ( v ) = \\check R _ { i + 1 } ( v ) \\check R _ i ( u v ) \\check R _ { i + 1 } ( v ) \\ . \\end{align*}"}
+{"id": "1830.png", "formula": "\\begin{align*} S _ X : = \\{ x \\in X : G _ x \\neq 1 \\} \\end{align*}"}
+{"id": "8115.png", "formula": "\\begin{align*} \\frac { \\mu ^ s _ { p ^ + _ { \\ell _ 0 } x } ( Q \\cdot u ^ - _ \\ell ) } { \\mu ^ s _ { p ^ + _ { \\ell _ 0 } x } ( N ^ - _ { r _ j } \\cdot u ^ - _ \\ell ) } = \\frac { \\mu ^ s _ { p ^ + _ { \\ell } x } ( Q ) } { \\mu ^ s _ { p ^ + _ { \\ell } x } ( N ^ - _ { r _ j } ) } = \\frac { \\mu ^ s _ { y _ \\ell } ( Q ^ \\prime ) } { \\mu ^ s _ { y _ \\ell } ( N ^ - _ { e ^ { - s _ { \\rho , \\ell } } r _ j } ) } . \\end{align*}"}
+{"id": "7744.png", "formula": "\\begin{align*} A G - G ^ + A = H \\in \\mathcal B _ { h , \\delta } ^ 1 . \\end{align*}"}
+{"id": "5565.png", "formula": "\\begin{align*} \\left \\langle \\frac { \\zeta _ i } { \\norm { \\zeta _ i } } , \\phi _ i \\right \\rangle = \\left \\langle \\frac { S _ \\Delta \\xi _ i } { \\norm { S _ { \\Delta } \\xi _ i } } , \\phi _ i \\right \\rangle = \\frac { 1 } { \\sqrt { \\gamma _ i } } + O \\left ( \\frac { 2 K ^ 2 d \\sqrt { \\log n } ~ \\sigma } { \\delta _ i } \\right ) , \\end{align*}"}
+{"id": "8801.png", "formula": "\\begin{align*} \\pi _ { x _ - } ( \\{ y _ + \\} ) \\geq 0 = \\pi _ { x _ - } ^ { \\uparrow } ( \\{ y _ + \\} ) , & \\pi _ { x _ - } ( \\{ z _ + \\} ) \\geq 0 = \\pi _ { x _ - } ^ { \\uparrow } ( \\{ z _ + \\} ) , \\\\ \\pi _ { x _ - } ^ { \\uparrow } ( \\{ y _ - \\} ) = \\frac { \\nu ( \\{ y _ - \\} ) } { \\mu ( \\{ x _ - \\} ) } \\geq \\pi _ { x _ - } ( \\{ y _ - \\} ) , & \\pi _ { x _ - } ^ { \\uparrow } ( \\{ z _ - \\} ) = \\frac { \\nu ( \\{ z _ - \\} ) } { \\mu ( \\{ x _ - \\} ) } \\geq \\pi _ { x _ - } ( \\{ z _ - \\} ) . \\end{align*}"}
+{"id": "8102.png", "formula": "\\begin{align*} \\chi _ { N _ 1 ^ + } ( n ) \\phi ( n ) = \\chi _ { N _ 1 ^ + ( j ) } ( n ) \\phi ( n ) , \\forall n \\in N ^ + . \\end{align*}"}
+{"id": "1746.png", "formula": "\\begin{align*} M _ 0 : = \\left \\{ u = \\sum _ { 0 < j < j + k \\leq n } x _ j x _ k s ^ { n - j - k } \\right \\} . \\end{align*}"}
+{"id": "6428.png", "formula": "\\begin{align*} - \\overline { G } ^ { ( d ) } _ n ( a _ 0 , b _ 0 ) = \\int _ 0 ^ 1 \\nabla _ { ( a , b ) } \\overline { G } ^ { ( d ) } _ n ( a _ 0 + t ( \\hat { a } _ n - a _ 0 ) , b _ 0 + t ( \\hat { b } _ n - b _ 0 ) ) d t \\begin{pmatrix} \\hat { a } _ n - a _ 0 \\\\ \\hat { b } _ n - b _ 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "7928.png", "formula": "\\begin{align*} v = d \\phi + \\delta \\beta + \\alpha , \\ \\mathrm { w h e r e } \\end{align*}"}
+{"id": "3897.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\frac { \\sum _ { \\gamma \\in Q ( n ) } \\exp \\left ( \\ell _ \\psi ( \\gamma ) \\right ) } { \\sum _ { \\gamma \\in P ( n ) } \\exp \\left ( \\ell _ \\psi ( \\gamma ) \\right ) } = \\limsup _ { n \\rightarrow \\infty } \\frac { \\sum _ { \\gamma \\in Q _ * ( n ) } \\exp \\left ( \\ell _ \\psi ( \\gamma ) \\right ) } { \\sum _ { \\gamma \\in P _ * ( n ) } \\exp \\left ( \\ell _ \\psi ( \\gamma ) \\right ) } , \\end{align*}"}
+{"id": "3578.png", "formula": "\\begin{align*} ( 1 + a ) ( i d - \\tau ^ 2 ) ( \\tau ^ 2 - \\tau ) = 0 , \\end{align*}"}
+{"id": "6090.png", "formula": "\\begin{align*} \\Delta _ { r , s } ^ { i } = \\begin{cases} ( - 1 ) ^ { r - 1 } \\binom { i - 1 } { r - 1 } + ( - 1 ) ^ { s - 1 } \\binom { i - 1 } { s - 1 } & ( q - 1 ) \\mid i , \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "7193.png", "formula": "\\begin{align*} Z ( \\bar { v } , x , t ) = & - A ( x ) \\sin ( \\omega _ 1 t ) - B ( x ) \\sin ( \\omega _ 2 t ) + \\frac { A ( x ) ^ 3 } { 3 } \\sin ^ 3 ( \\omega _ 1 t ) + \\frac { B ( x ) ^ 3 } { 3 } \\sin ^ 3 ( \\omega _ 2 t ) \\\\ & + A ( x ) \\bar { v } ^ 2 \\sin ( \\omega _ 1 t ) + B ( x ) \\bar { v } ^ 2 \\sin ( \\omega _ 2 t ) + A ( x ) ^ 2 B ( x ) \\sin ^ 2 ( \\omega _ 1 t ) \\sin ( \\omega _ 2 t ) \\\\ & + A ( x ) B ( x ) ^ 2 \\sin ( \\omega _ 1 t ) \\sin ^ 2 ( \\omega _ 2 t ) - \\partial _ x ^ 2 J _ 0 ( x , t ) + v _ 0 ^ 2 J _ 0 . \\end{align*}"}
+{"id": "4326.png", "formula": "\\begin{align*} \\inf _ { x \\in \\mathcal { X } } & \\sum _ { i \\in [ m ] } \\ \\overline { f } _ i ( x ) , \\end{align*}"}
+{"id": "7733.png", "formula": "\\begin{align*} \\mathrm { R e s } ( \\chi , \\chi ; u ) = \\prod _ { 1 \\leq i , j \\leq l } \\mathrm { R e s } ( \\chi _ { i } , \\chi _ { j } ; u ) . \\end{align*}"}
+{"id": "1541.png", "formula": "\\begin{align*} \\eta _ { a , b } ( t ) = \\max \\bigl \\{ t ^ a , t ^ b \\bigr \\} , t \\ge 0 , \\end{align*}"}
+{"id": "2096.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { \\lfloor s T \\rfloor - 1 } f ( \\frac { t } { T } ) \\big ( U ^ { d , T } ( \\frac { t + 1 } { T } , x ) - U ^ { d , T } ( \\frac { t } { T } , x ) \\big ) = & - ( i + 1 - d x ) \\sum _ { k = 1 } ^ 3 P _ { i , k } ^ s - ( d x - i ) \\sum _ { k = 1 } ^ 3 P _ { i + 1 , k } ^ s - P _ 4 ^ s , \\end{align*}"}
+{"id": "4828.png", "formula": "\\begin{align*} E = L \\oplus F , \\ , \\ , \\Phi = \\begin{pmatrix} 0 & \\beta \\\\ \\gamma & 0 \\end{pmatrix} \\end{align*}"}
+{"id": "3226.png", "formula": "\\begin{align*} H ( X ) = \\bigoplus _ n H ( B , { } ^ p \\ ! R ^ { n } f _ * C _ X ) \\end{align*}"}
+{"id": "4959.png", "formula": "\\begin{align*} a ( \\lambda , \\nu ) = \\sin \\gamma ( \\lambda - \\nu + 1 ) , b ( \\lambda , \\nu ) = \\sin \\gamma ( \\lambda - \\nu ) , c = \\sin \\gamma . \\end{align*}"}
+{"id": "2364.png", "formula": "\\begin{align*} g ( 0 ^ { + } ) = & \\| v _ { k } ( \\cdot , 0 ) \\| _ { L ^ { 3 } } \\geq g _ { + } ( 0 ^ { + } ) = \\frac { 1 + \\sqrt { 1 - 4 a b ( 0 ) ^ { 2 } C _ { 1 } ^ { 2 } } } { 2 C _ { 1 } b ( 0 ) } \\\\ \\geq & \\frac { 2 - 4 C _ { 1 } ^ { 2 } \\| v _ { k } ( \\cdot , 0 ) \\| _ { L ^ { 3 } } } { 2 C _ { 1 } } = \\frac { 1 } { C _ { 1 } } - 2 C _ { 1 } \\| v _ { k } ( \\cdot , 0 ) \\| _ { L ^ { 3 } } , \\end{align*}"}
+{"id": "4800.png", "formula": "\\begin{align*} u ( x ) = \\frac { F _ \\theta ^ o ( x ) ^ 2 - r ^ 2 } { 2 } \\end{align*}"}
+{"id": "4483.png", "formula": "\\begin{align*} \\frac { 1 } { \\pi } \\liminf _ { r \\rightarrow 1 - 0 } \\frac { \\int _ { \\{ z \\in D : 2 \\psi \\ge \\log r \\} } | F _ 0 | ^ 2 \\tilde \\rho } { 1 - r } = \\frac { M ( Z _ 0 , J , \\tilde \\rho ) } { \\pi \\int _ { 0 } ^ { + \\infty } c ( t ) e ^ { - t } d t } . \\end{align*}"}
+{"id": "8023.png", "formula": "\\begin{align*} P \\left ( \\sum _ { 0 \\leq t \\leq T } \\left ( \\mu ^ N _ { t , k } ( f ) - \\mu ^ N _ { t - , k } ( f ) \\right ) ^ 2 \\geq \\epsilon \\right ) & \\leq e ^ { - N ^ 2 \\epsilon } \\mathbb { E } \\exp \\left ( \\| f \\| ^ 2 _ \\infty Y ( N K _ 2 T ) \\right ) \\\\ & = e ^ { - N ^ 2 \\epsilon } e ^ { N K _ 2 T ( e ^ { \\| f \\| ^ 2 _ \\infty } - 1 ) } \\end{align*}"}
+{"id": "3893.png", "formula": "\\begin{align*} \\ell _ \\phi ( \\gamma ) \\coloneqq \\begin{dcases} \\sum _ { k = 0 } ^ { \\ell ( \\gamma ) - 1 } \\phi ( f ^ k ( x _ \\gamma ) ) & f , \\\\ \\int _ 0 ^ { \\ell ( \\gamma ) } \\phi ( f ^ s ( x _ \\gamma ) ) \\ , d s & f ^ t . \\end{dcases} \\end{align*}"}
+{"id": "6552.png", "formula": "\\begin{align*} 0 < x ^ { 1 - \\beta ' \\delta } - ( 1 + x ) ^ { 1 - \\beta ' \\delta } = ( \\beta ' \\delta - 1 ) \\int ^ { x + 1 } _ x t ^ { - \\beta ' \\delta } d t < \\int ^ { x + 1 } _ x t ^ { - \\beta ' \\delta } d t < x ^ { - \\beta ' \\delta } \\end{align*}"}
+{"id": "8864.png", "formula": "\\begin{align*} 0 < \\frac { 1 } { ( p - 1 ) a _ 2 } \\leq \\frac 1 r \\leq 1 , 0 \\leq \\frac { \\tau } { p - 1 } \\leq \\frac { n } { ( p - 1 ) a _ 2 } , \\frac { \\tau } { p - 1 } - 1 = \\frac { n } { ( p - 1 ) a _ 2 } - \\frac { n } { r } , \\end{align*}"}
+{"id": "1145.png", "formula": "\\begin{align*} \\widetilde D : = D - \\frac { d } { p } > J , \\widetilde F : = F + s + \\frac { n } { 2 } > J , J : = \\frac { n } { \\min ( 1 , q ) } \\geq n . \\end{align*}"}
+{"id": "5439.png", "formula": "\\begin{align*} f ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , i ) } & = R _ i + \\beta \\sum _ { j \\in S _ 1 } p _ { i j } f ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , j ) } + \\beta \\sum _ { j \\in N \\setminus S _ 1 } p _ { i j } f ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , j ) } = R _ i + \\beta \\sum _ { j \\in S _ 1 } p _ { i j } f ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , j ) } , \\end{align*}"}
+{"id": "972.png", "formula": "\\begin{align*} X _ { \\tau _ V } \\circ \\theta _ { \\tau _ V } = X _ { \\tau _ V } P _ x \\quad x \\in V . \\end{align*}"}
+{"id": "4609.png", "formula": "\\begin{align*} \\beta = \\beta ( d , \\gamma ) = \\frac { 1 } { d + 1 } \\left ( \\frac { d - 1 } { \\gamma } + 1 \\right ) \\left [ \\frac { 1 } { d } \\left ( \\frac { d - 1 } { \\gamma } + 1 \\right ) ^ 2 - d \\right ] < 1 . \\end{align*}"}
+{"id": "7026.png", "formula": "\\begin{align*} G _ p ( r ) & : = b ^ { \\frac { 2 } { p } } \\cdot \\nabla ( \\mu - \\Delta ) ^ { - \\frac { 1 } { 2 } - \\frac { 1 } { r } } , \\\\ Q _ p ( q ) \\upharpoonright { \\mathcal E } & : = ( \\mu - \\Delta ) ^ { - \\frac { 1 } { 2 } + \\frac { 1 } { q } } | b | ^ { 1 - \\frac { 2 } { p } } , \\\\ T _ p \\upharpoonright { \\mathcal E } & : = b ^ { \\frac { 2 } { p } } \\cdot \\nabla ( \\mu - \\Delta ) ^ { - 1 } | b | ^ { 1 - \\frac { 2 } { p } } . \\end{align*}"}
+{"id": "2521.png", "formula": "\\begin{gather*} x = ( x ^ 1 , \\dots , x ^ n ) \\equiv ( x ' , x '' ) : U \\to U ' \\times U '' \\ ; , \\\\ x ' = ( x '^ 1 , \\dots , x '^ { n ' } ) \\ ; , x '' = ( x ''^ { 1 } , \\dots , x ''^ { n '' } ) \\ ; , L _ 0 : = L \\cap U = \\{ x ' = 0 \\} \\ ; . \\end{gather*}"}
+{"id": "7238.png", "formula": "\\begin{align*} \\frac { n ! } { 1 ^ { j _ 1 } \\cdots n ^ { j _ n } j _ 1 ! \\cdots j _ n ! } = \\frac { n ! } { \\prod _ { i = 1 } ^ n i ^ { j _ i } j _ i ! } \\end{align*}"}
+{"id": "5293.png", "formula": "\\begin{align*} K L I ( q \\| p ) = \\sum _ { j } p _ { j } \\sum _ { i } \\overline { q } _ { i } \\log \\frac { \\overline { q } _ { i } } { \\overline { p } _ { i } } \\end{align*}"}
+{"id": "3969.png", "formula": "\\begin{align*} \\pi ( [ x _ 1 , \\dots , x _ n ] , [ y _ 1 , \\dots , y _ n ] ) \\tilde \\eta ( x _ 1 , \\dots , x _ n ) & = \\frac 1 { n ! } \\sum _ { \\sigma \\in \\mathfrak S _ n } \\hat \\pi ( ( x _ 1 , \\dots , x _ n ) , ( y _ 1 , \\dots , y _ n ) ) \\sigma \\eta ( x _ 1 , \\dots , x _ n ) \\\\ & = \\frac 1 { n ! } \\sum _ { \\sigma \\in \\mathfrak S _ n } \\sigma \\eta ( y _ 1 , \\dots , y _ n ) \\\\ & = \\tilde \\eta ( y _ 1 , \\dots , y _ n ) . \\end{align*}"}
+{"id": "361.png", "formula": "\\begin{align*} ( x - a ) ^ n = f _ n ( x , b , a ) + g _ n ( b , a ) , \\end{align*}"}
+{"id": "8083.png", "formula": "\\begin{align*} \\Omega _ r ^ - : = N _ r ^ - \\Omega . \\end{align*}"}
+{"id": "7706.png", "formula": "\\begin{align*} \\int _ { S _ { 3 } } ( \\frac { h _ { E _ { t _ { j } } } } { n } ) ^ { p } f ( x ) d x \\leq c _ { 2 } \\int _ { S _ { 3 } } ( \\frac { 1 } { n \\delta } ) ^ { p } d x = c _ { 2 } ( \\frac { 1 } { n \\delta } ) ^ { p } | S _ { 3 } | \\leq c _ { 3 } \\delta ^ { - p } . \\end{align*}"}
+{"id": "4134.png", "formula": "\\begin{align*} & \\bar { \\xi } _ \\ell ( \\theta ) = 0 , \\ell = 1 , \\ldots , j - 1 , \\\\ & \\bar { \\xi } _ j ( \\theta ) = ( 1 - w _ { j j } ) r ^ j \\eta _ j \\end{align*}"}
+{"id": "6929.png", "formula": "\\begin{align*} L \\cdot \\begin{pmatrix} 1 & \\hdots & 1 \\end{pmatrix} ^ T = 1 \\end{align*}"}
+{"id": "8673.png", "formula": "\\begin{align*} & ( \\ , u \\ , | \\ , v \\ , ) _ { M ' } : = \\int _ { M ' } \\langle \\ , u \\ , | \\ , v \\ , \\rangle d v _ { M ' } , \\ \\ u , v \\in \\Omega ^ { 0 , q } _ c ( M ' ) , \\\\ & ( \\ , u \\ , | \\ , v \\ , ) _ { M } : = \\int _ M \\langle \\ , u \\ , | \\ , v \\ , \\rangle d v _ { M ' } , \\ \\ u , v \\in \\Omega ^ { 0 , q } _ c ( M ) , \\end{align*}"}
+{"id": "3552.png", "formula": "\\begin{align*} T ^ { - 1 } g = g \\circ \\psi ( g \\in { H } ^ { \\infty } _ { 1 } ( \\mathbb { D } ) ) , \\end{align*}"}
+{"id": "7243.png", "formula": "\\begin{align*} \\frac 1 { \\lvert G \\rvert } \\sum _ { g \\in G } \\prod _ { i = 1 } ^ { \\lvert X \\rvert } t _ i ^ { j _ i ( g . ) } \\in \\mathbb Q [ t _ 1 , \\dots , t _ { \\lvert X \\rvert } ] . \\end{align*}"}
+{"id": "886.png", "formula": "\\begin{align*} G ^ 2 ( x , \\mathbf { y } ) & = \\left ( F P ^ 2 _ 1 \\right ) y ^ 1 + ( F P ^ 2 _ 2 ) y ^ 2 + \\left ( L ^ 2 _ 1 \\right ) \\left ( y ^ 1 \\right ) ^ 2 + L ^ 2 _ 2 \\left ( y ^ 2 \\right ) ^ 2 + ( Q _ 0 ^ 2 ) y ^ 1 y ^ 2 \\\\ & + \\left ( \\frac { A ^ 2 _ 0 } { F } \\right ) \\left ( y ^ 1 \\right ) ^ 3 + \\left ( \\frac { B ^ 2 _ 0 } { F } \\right ) \\left ( y ^ 1 \\right ) ^ 2 y ^ 2 + \\left ( \\frac { D ^ 2 _ 0 } { F } \\right ) y ^ 1 \\left ( y ^ 2 \\right ) ^ 2 + F ^ 2 R _ 0 ^ 2 , \\end{align*}"}
+{"id": "5784.png", "formula": "\\begin{align*} \\textsc { I } & = - \\Pi ^ \\perp \\mathcal { M } _ \\Sigma ( u ) + \\mathcal { L } _ { \\Sigma } \\tilde { u } ^ \\perp + \\Pi ^ \\perp N _ 1 ( u ) , \\ \\textsc { I I } = - \\left ( H ( u ^ T ) \\right ) '' + m \\left ( H ( u ^ T ) \\right ) ' . \\end{align*}"}
+{"id": "4928.png", "formula": "\\begin{align*} \\begin{aligned} \\ell ( H , 1 _ L ) & \\geq c ( H , [ L ] ) \\\\ & = c ( H , 2 \\sqrt { \\beta _ { L } } ( e ^ { L } _ { + } - e ^ { L } _ { - } ) ) \\\\ & = c ( H , e ^ { L } _ { + } - e ^ { L } _ { - } ) + \\nu ( 2 \\sqrt { \\beta _ { L } } ) . \\end{aligned} \\end{align*}"}
+{"id": "7467.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { 4 } r ^ j \\left | ( \\nabla ^ G ) ^ j ( \\phi _ { p } ^ * g _ Z - G ) \\right | { \\big | _ { B _ r ( p ) } } < \\kappa ( r ) , \\end{align*}"}
+{"id": "4367.png", "formula": "\\begin{gather*} f _ { i , * } ( x , v ^ i ) \\neq - \\infty \\Leftrightarrow v ^ i = l _ i ( x ) , \\end{gather*}"}
+{"id": "6332.png", "formula": "\\begin{align*} M _ 0 = n ^ { 1 - 8 0 \\kappa } , M = n ^ { 1 - 6 8 \\kappa } , \\lambda = Y ^ { 1 0 \\kappa } , \\delta = Y ^ { 3 \\kappa } \\end{align*}"}
+{"id": "8133.png", "formula": "\\begin{align*} k ^ { ( q _ 1 , q _ 2 ) } \\ = \\ k _ { A _ 2 } ( D _ { q _ 1 , x } , X _ { q _ 1 , x } , D _ { q _ 2 , y } , X _ { q _ 2 , y } ) \\ , \\end{align*}"}
+{"id": "8659.png", "formula": "\\begin{align*} H ^ 0 ( \\overline { M } ) ^ G : = \\left \\{ u \\in H ^ 0 ( \\overline { M } ) ; \\ , h ^ * u = u , ~ ~ h \\in G \\right \\} . \\end{align*}"}
+{"id": "6170.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { ( \\tau ^ k ) ^ 2 } [ S ^ { k + 1 } - \\frac { c \\beta } { 2 } \\| A \\breve { x } ^ { k } - b \\| ^ 2 ] - \\frac { 1 } { ( \\tau ^ { k - 1 } ) ^ 2 } [ S ^ { k } - \\frac { c \\beta } { 2 } \\| A \\breve { x } ^ { k - 1 } - b \\| ^ 2 ] \\\\ \\geq & \\frac { 1 } { 2 } \\Big ( \\| v ^ { k + 1 } - v ' \\| ^ 2 _ { H ^ { k + 1 } _ 0 } - \\| v ^ k - v ' \\| ^ 2 _ { H ^ k _ 0 } \\Big ) , \\end{aligned} \\end{align*}"}
+{"id": "8771.png", "formula": "\\begin{align*} \\exists u \\in [ 0 , F _ \\mu ( x ' ) ] \\mbox { s . t . } \\int _ u ^ { F _ \\mu ( x ' ) } \\pi _ { F _ { \\mu } ^ { - 1 } ( w ) } ( ( - \\infty , F _ { \\mu } ^ { - 1 } ( w ) ) ) d w = F _ \\mu ( x ' ) - F _ \\mu ( x ) . \\end{align*}"}
+{"id": "4270.png", "formula": "\\begin{align*} \\Delta _ f h : = \\Delta h - \\nabla f \\cdot \\nabla h , \\end{align*}"}
+{"id": "5884.png", "formula": "\\begin{align*} f ( m , n ) : = m ( m - 1 ) ( m n - n - 1 ) ^ 3 + m ( m - 1 ) ( n - 1 ) ^ 3 > m ( m - 1 ) ( m n - n - 1 ) ( n - 1 ) ( m n - 2 ) . \\end{align*}"}
+{"id": "8355.png", "formula": "\\begin{align*} \\mathcal { M } _ \\psi : p \\mapsto \\begin{cases} \\int _ 0 ^ \\infty t ^ { p - 1 } ( \\psi ( t ) - \\psi ( 0 ) ) \\ , d t , & p \\in ( - 1 , 0 ) , \\\\ \\int _ 0 ^ \\infty t ^ { p - 1 } \\psi ( t ) \\ , d t , & p > 0 t ^ { p - 1 } \\psi ( t ) \\in L ^ 1 ( \\R ^ + ) , \\end{cases} \\end{align*}"}
+{"id": "5813.png", "formula": "\\begin{align*} | Y _ + ( t ) | ^ 2 + | Y _ 0 ( t ) | ^ 2 + | Y _ - ( t ) | ^ 2 = o ( 1 ) ( | X _ + ( t ) | ^ 2 + | X _ 0 ( t ) | ^ 2 + | X _ - ( t ) | ^ 2 ) \\end{align*}"}
+{"id": "4695.png", "formula": "\\begin{align*} b _ 1 = \\inf _ { j \\in \\N } b _ j . \\end{align*}"}
+{"id": "4760.png", "formula": "\\begin{align*} \\norm { x - S ( t ) x } = \\norm { S ( 0 ) x - S ( t ) x } \\leq 2 \\norm { v } t \\leq 2 b \\frac { \\varepsilon } { 2 b } \\leq \\varepsilon \\end{align*}"}
+{"id": "3198.png", "formula": "\\begin{align*} \\| T \\wedge \\omega \\| = \\| T _ { a v e } \\wedge \\omega \\| . \\end{align*}"}
+{"id": "5691.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\left ( e ^ { - 2 ^ { - 1 } m t } u ( t ) - \\sum _ { i \\in I _ 1 } \\textup { R e } \\left ( w _ i e ^ { \\mathbf { i } \\beta _ i t } \\right ) \\varphi _ i - \\sum _ { i \\in I _ 2 } c _ i \\varphi _ i \\right ) = 0 \\ \\textup { i n } \\ C ^ \\infty ( \\Sigma ; \\mathbf { V } ) . \\end{align*}"}
+{"id": "3661.png", "formula": "\\begin{align*} a ( G ' ) \\geq a ( s ^ k ( G ) ) \\geq \\frac { \\min \\left \\{ \\frac { 2 } { \\Delta } a ( G ) , 8 \\right \\} } { 4 ^ k } \\geq \\frac { \\min \\left \\{ \\frac { 2 } { \\Delta } a ( G ) , 8 \\right \\} } { 4 ^ { \\log _ 2 ( m + 1 ) + 1 } } = \\frac { \\min \\left \\{ \\frac { 1 } { \\Delta } a ( G ) , 4 \\right \\} } { 2 ( m + 1 ) ^ 2 } . \\end{align*}"}
+{"id": "1802.png", "formula": "\\begin{align*} H ( z , \\pmb \\sigma ) = H _ n ( z , \\sigma _ n ) \\circ H _ { n - 1 } ( z , \\sigma _ { n - 1 } ) \\circ \\cdots \\circ H _ 2 ( z , \\sigma _ 2 ) \\circ H _ 1 ( z , \\sigma _ 1 ) \\ , ( u ) \\ , . \\end{align*}"}
+{"id": "8914.png", "formula": "\\begin{align*} 1 + 2 \\sum _ { m = 1 } ^ \\infty \\frac { B _ { 2 m } } { ( 2 m ) ! } \\omega _ m ( n ) x ^ { 2 m } & = \\exp \\left ( \\sum _ { m = 2 } ^ \\infty \\frac { B _ m } { m ! } \\left ( 2 \\sinh ^ { - 1 } \\left ( \\frac { x } { 2 } \\right ) \\right ) ^ m \\frac { J _ m ( n ) } { m } \\right ) \\\\ & = \\prod _ { d \\mid n } \\exp \\left ( \\sum _ { m = 2 } ^ \\infty \\frac { B _ m } { m \\cdot m ! } \\left ( 2 d \\sinh ^ { - 1 } \\left ( \\frac { x } { 2 } \\right ) \\right ) ^ m \\right ) ^ { \\mu ( n / d ) } . \\end{align*}"}
+{"id": "6231.png", "formula": "\\begin{align*} z ( \\phi ) : = D ( \\phi ) \\phi _ { 1 , \\beta } ' \\left ( \\zeta ( \\phi ) \\right ) , \\ \\mbox { f o r } \\ \\phi \\in ( \\beta , 1 ) . \\end{align*}"}
+{"id": "6000.png", "formula": "\\begin{align*} 0 \\leq & w _ \\nu \\leq ( | \\nabla \\log u | ^ 2 ) _ { \\nu } = 2 ( \\log u ) _ { \\nu } ( \\Delta u - ( \\log u ) _ \\nu H ) \\\\ = & 2 \\frac { u _ \\nu } { u } \\left ( \\frac { \\Delta u } { u } - \\frac { | \\nabla u | ^ 2 } { u ^ 2 } - \\frac { u _ \\nu } { u } H \\right ) \\\\ = & 2 \\frac { u _ \\nu } { u } \\left ( - h - \\frac { | \\nabla u | ^ 2 } { u ^ 2 } - \\frac { u _ \\nu } { u } H \\right ) \\\\ \\leq & 2 \\frac { u _ \\nu } { u } \\left ( - w - \\frac { u _ \\nu } { u } H \\right ) . \\end{align*}"}
+{"id": "8594.png", "formula": "\\begin{align*} ( I I ) \\leq ( A ) + ( B ) \\leq \\frac { 1 } { ( j - 1 ) ! } \\left ( \\frac { 2 M } { \\mu } \\right ) ^ { j - 1 } + \\frac { 1 } { j ! } \\left ( \\frac { 2 M } { \\mu } \\right ) ^ { j } = \\frac { 1 } { j ! } \\left ( 1 + \\frac { j \\mu } { 2 M } \\right ) \\left ( \\frac { 2 M } { \\mu } \\right ) ^ j . \\end{align*}"}
+{"id": "5255.png", "formula": "\\begin{align*} G \\left ( p \\| q \\right ) = \\sum _ { i } \\left [ \\alpha p _ { i } + \\left ( 1 - \\alpha \\right ) q _ { i } \\right ] \\log \\frac { \\alpha p _ { i } + \\left ( 1 - \\alpha \\right ) q _ { i } } { p _ { i } } + \\left ( 1 - \\alpha \\right ) \\left ( p _ { i } - q _ { i } \\right ) \\end{align*}"}
+{"id": "8239.png", "formula": "\\begin{align*} I ( { \\bf X } ^ L \\to Y ^ L ) : = \\sum _ { i = 1 } ^ L I ( { \\bf X } ^ i ; Y _ i | Y ^ { i - 1 } ) . \\end{align*}"}
+{"id": "7836.png", "formula": "\\begin{align*} \\nabla ^ { \\perp \\phi } _ { \\bar E _ { \\alpha } } B ^ j ( \\bar E _ { \\alpha } , H ) = B ^ j ( \\nabla ^ { \\mathbb { C } P ^ q } _ { \\bar E _ { \\alpha } } \\bar E _ { \\alpha } , H ) - H . \\end{align*}"}
+{"id": "1636.png", "formula": "\\begin{align*} Q _ { \\varepsilon , T } = \\Omega \\times \\left ( \\varepsilon , T \\right ) \\subset Q _ { T } . \\end{align*}"}
+{"id": "8593.png", "formula": "\\begin{align*} ( I I ) & \\leq \\sum _ { 0 < y _ 1 < \\dots < y _ j \\leq N } e ^ { - \\frac { \\mu } { 2 M } \\sum _ k ( N - y _ k ) } = \\sum _ { 0 \\leq z _ 1 < \\dots < z _ j < N } e ^ { - \\frac { \\mu } { 2 M } \\sum _ k z _ k } . \\end{align*}"}
+{"id": "3770.png", "formula": "\\begin{align*} \\alpha _ n = 2 \\int _ 0 ^ 1 r ^ { 2 n + 1 } G ( 1 - r ) d r \\end{align*}"}
+{"id": "6682.png", "formula": "\\begin{align*} G = H A . \\end{align*}"}
+{"id": "7044.png", "formula": "\\begin{align*} \\partial _ t h - \\Delta h + b _ m \\cdot \\nabla h + ( b _ m - b _ n ) \\cdot \\nabla u _ n = 0 , h ( s , \\cdot ) = 0 \\end{align*}"}
+{"id": "5338.png", "formula": "\\begin{align*} \\nu ^ { S _ { m } } _ { \\pi _ k } & = \\nu _ { \\pi _ m } + \\frac { 1 } { w ^ { S _ m } _ { \\pi _ k } } \\sum _ { l = m + 1 } ^ k ( \\nu _ { \\pi _ { l } } - \\nu _ { \\pi _ { l - 1 } } ) \\ , w ^ { S _ { l } } _ { \\pi _ k } \\geq \\nu _ { \\pi _ m } , \\end{align*}"}
+{"id": "6319.png", "formula": "\\begin{align*} \\mathcal N = \\dd \\Gamma ( \\ 1 ) = \\sum _ { p \\in \\pi \\mathbb { N } _ 0 ^ 3 } a _ p ^ * a _ p , \\mathcal N _ + = \\dd \\Gamma ( Q ) = \\sum _ { p \\neq 0 } a _ p ^ * a _ p , \\end{align*}"}
+{"id": "8098.png", "formula": "\\begin{align*} \\norm { f } ^ \\star _ { B } : = e ^ \\star _ { 1 , 0 } ( f ) + \\frac { e ^ \\star _ { 1 , 1 } ( f ) } { B } . \\end{align*}"}
+{"id": "4214.png", "formula": "\\begin{align*} Q ( X , \\tau ) & = { \\rm I n d } ( ( D \\otimes [ \\triangle ( X ) \\otimes \\Theta _ 1 ( T _ { C } X ) + 2 ^ k \\Theta _ 2 ( T _ { C } X ) + 2 ^ k \\Theta _ 3 ( T _ { C } X ) ] ) _ + ) \\\\ & = \\int _ X \\widehat { A } ( T X ) [ { \\rm c h } ( \\triangle ( X ) ) { \\rm c h } ( \\Theta _ 1 ( T _ { C } X ) ) + 2 ^ k { \\rm c h } ( \\Theta _ 2 ( T _ { C } X ) ) + 2 ^ k { \\rm c h } ( \\Theta _ 3 ( T _ { C } X ) ) ] . \\end{align*}"}
+{"id": "4389.png", "formula": "\\begin{align*} \\inf _ { x \\in \\mathcal { X } } & \\ \\sum _ { i \\in [ m ] } \\max \\{ 0 , x _ i - \\overline { b } _ i , x _ i - \\overline { b } _ i + \\Delta b _ i - x _ k + \\overline { b } _ k - \\Delta b _ k \\} + \\Gamma ( x _ k - \\overline { b } _ k + \\Delta b _ k ) \\\\ \\mathrm { s . t . } & \\ x _ k \\geq b _ k . \\end{align*}"}
+{"id": "3291.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { p - 1 } \\frac { ( \\frac { d + r } { d } ) _ k ^ { d - r } ( \\frac { r } { d } ) _ k ^ { r - 1 } ( \\frac { r - d } { d } ) _ k } { k ! ^ d } \\equiv \\tfrac { d - r } { d } \\big ( \\tfrac { r } { d } \\big ) ^ { r } \\Gamma _ p \\big ( - \\tfrac { r } { d } \\big ) ^ d \\pmod { p ^ 2 } . \\end{align*}"}
+{"id": "6028.png", "formula": "\\begin{align*} \\textrm { V a r } ( D _ 1 \\xi _ x ^ A + D _ 2 \\xi _ x ^ { B } ) = \\frac 2 9 ( D _ 1 ^ 2 + D _ 2 ^ 2 - D _ 1 D _ 2 ) \\ , , \\end{align*}"}
+{"id": "3948.png", "formula": "\\begin{align*} & \\rho \\sum _ { n = N + 1 } ^ { \\infty } y _ n ^ { \\top } ( t ) F _ n ( t ) \\leq \\frac { \\rho } { 2 \\alpha _ 0 } \\sum _ { n = N + 1 } ^ { \\infty } \\left | y _ n ( t ) \\right | ^ 2 \\\\ & - \\frac { \\alpha _ 0 \\rho } { 2 } \\left | F ^ N ( t ) \\right | ^ 2 + \\frac { \\alpha _ 0 \\rho } { 2 } \\sum _ { n = 1 } ^ { \\infty } \\left | F _ n ( t ) \\right | ^ 2 \\end{align*}"}
+{"id": "2741.png", "formula": "\\begin{align*} \\phi ^ { * } d Z ^ { m } = S ^ { m } _ { n } d z ^ { n } , \\end{align*}"}
+{"id": "2578.png", "formula": "\\begin{align*} F _ { m } ( X _ \\delta ) F _ { n } ( X _ \\delta ) = F _ { m + n } ( X _ \\delta ) + F _ { m - n } ( X _ \\delta ) , \\ F _ { n } ( X _ \\delta ) F _ { n } ( X _ \\delta ) = F _ { 2 n } ( X _ \\delta ) + 2 . \\end{align*}"}
+{"id": "1287.png", "formula": "\\begin{align*} W _ 2 ^ 2 ( \\mu , \\bar { \\mu } ) & = \\big ( m ( t ) - \\bar { m } ( t ) \\big ) ^ 2 + \\Big ( \\sqrt { \\sigma ( t ) } - \\sqrt { \\bar { \\sigma } ( t ) } \\Big ) ^ 2 \\\\ & \\leq \\big ( m _ x ( t ) - \\bar { m } ( t ) \\big ) ^ 2 + | \\sigma _ { x x } ( t ) - \\bar { \\sigma } ( t ) | \\\\ & \\leq \\frac { 4 } { \\gamma ^ 2 - 4 \\omega ^ 2 } ( \\omega | x _ 0 | + | v _ 0 | ) ^ 2 + \\frac { 1 6 } { \\beta ( \\gamma ^ 2 - 4 \\omega ^ 2 ) } = \\frac { 4 } { \\gamma ^ 2 - 4 \\omega ^ 2 } \\Big [ ( \\omega | x _ 0 | + | v _ 0 | ) ^ 2 + \\frac { 4 } { \\beta } \\Big ] , \\end{align*}"}
+{"id": "2313.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { R } ^ { 3 } } ( \\nabla u _ { j } ^ { c , \\gamma } \\nabla \\varphi - u ^ { c , \\gamma } _ { j } u ^ { c , \\gamma } \\cdot \\nabla \\varphi - p ^ { c , \\gamma } \\partial _ { x _ { j } } \\varphi ) \\\\ & = \\begin{cases} 0 , & j = 1 , 2 , \\\\ 4 \\pi c _ { 3 } \\int ^ { \\infty } _ { - \\infty } \\ln | x _ { 3 } | \\partial _ { x _ { 3 } } \\varphi ( 0 , 0 , x _ { 3 } ) d x _ { 3 } - b ^ { c , \\gamma } \\varphi ( 0 ) , & j = 3 , \\end{cases} \\end{align*}"}
+{"id": "7838.png", "formula": "\\begin{align*} \\nabla _ { E _ a } ^ { \\perp \\psi } H ^ { \\psi } = \\frac { p } { p + q } \\nabla ^ { \\perp } _ { E _ a } H _ 1 + \\frac { q } { p + q } B ^ j ( E _ a , H _ 2 ) , A ^ { \\psi } _ { H ^ { \\psi } } E _ a = \\frac { p } { p + q } A _ { H _ 1 } E _ a \\end{align*}"}
+{"id": "4993.png", "formula": "\\begin{align*} P _ N ( \\{ \\nu \\} ) = \\prod _ { j , k = 1 } ^ N ( \\nu _ j - \\nu _ k + 1 ) , \\end{align*}"}
+{"id": "4051.png", "formula": "\\begin{align*} f ^ \\star ( x , d ) = \\mathop { \\lim \\sup } _ { y \\to x , t \\downarrow 0 } ~ \\frac { f ( y + t d ) - f ( y ) } { t } . \\end{align*}"}
+{"id": "8752.png", "formula": "\\begin{align*} \\beta ( d x , d y ) & = \\delta _ { ( x _ + , z _ + ) } + \\frac { z _ - - z _ + } { z _ - - y } \\delta _ { ( x _ - , y ) } + \\frac { z _ + - y } { z _ - - y } \\delta _ { ( x _ - , z _ - ) } , \\\\ \\gamma ( d x , d y ) & = \\delta _ { ( x _ - , z _ + ) } + \\frac { z _ - - z _ + } { z _ - - y } \\delta _ { ( x _ + , y ) } + \\frac { z _ + - y } { z _ - - y } \\delta _ { ( x _ + , z _ - ) } , \\end{align*}"}
+{"id": "5308.png", "formula": "\\begin{align*} S _ { k } = \\{ k - 1 , \\ldots , n - 1 \\} , 1 \\leq k \\leq n . \\end{align*}"}
+{"id": "3843.png", "formula": "\\begin{align*} \\begin{aligned} \\bar \\zeta ^ n ( A \\times C \\times B ) \\doteq \\int _ B \\bar \\zeta ^ n \\left ( A \\times C \\mid t , \\bar L ^ n ( a ( t ) ) \\right ) d t , \\end{aligned} \\end{align*}"}
+{"id": "981.png", "formula": "\\begin{align*} u ( x ) = h ( x ) + \\mathbb E _ x g ( X _ { \\tau _ D } ) + \\mathbb E _ x \\int _ 0 ^ { \\tau _ D } f ( X _ t , u ( X _ t ) ) \\ , d t + \\mathbb E _ x A ^ \\mu _ { \\tau _ D } \\quad x \\in E \\end{align*}"}
+{"id": "2336.png", "formula": "\\begin{align*} \\begin{cases} \\frac { 1 } { 2 } \\frac { d } { d t } \\| a _ { 1 } \\| _ { L ^ { 2 } } ^ { 2 } + \\| \\nabla a _ { 1 } \\| _ { L ^ { 2 } } ^ { 2 } = 0 , & p = 2 , \\\\ \\frac { 1 } { p } \\frac { d } { d t } \\| a _ { 1 } \\| _ { L ^ { p } } ^ { p } + \\frac { 4 ( p - 2 ) } { p ^ { 2 } } \\| \\nabla | a _ { 1 } | ^ { \\frac { p } { 2 } } \\| _ { L ^ { 2 } } ^ { 2 } + \\| | \\nabla a _ { 1 } | | a _ { 1 } | ^ { \\frac { p - 2 } { 2 } } \\| _ { L ^ { 2 } } ^ { 2 } = 0 , & p > 2 , \\end{cases} \\end{align*}"}
+{"id": "9183.png", "formula": "\\begin{align*} q : = v ^ 2 \\ \\ ( \\mathrm { s o \\ t h a t } \\ v = q ^ { 1 / 2 } ) , \\xi : = q ^ { 1 - N } . \\end{align*}"}
+{"id": "1311.png", "formula": "\\begin{align*} & | \\mu _ 1 ( t ) - \\hat { \\mu } _ 1 ( t ) | = \\frac { 1 } { 2 } e ^ { a t } | x _ 2 - x _ 1 | ( 1 - e ^ { - 2 k t } ) \\leq C e ^ { a t } , \\\\ & | \\Sigma _ { 1 1 } ( t ) - \\hat { \\Sigma } _ { 1 } ( t ) | = \\frac { 1 } { 2 } \\frac { 1 } { | 2 k - a | } e ^ { 2 a t } \\Big ( 1 - e ^ { - 4 k t } \\Big ) \\leq C e ^ { 2 a t } . \\end{align*}"}
+{"id": "6212.png", "formula": "\\begin{align*} \\mathcal { S } : = \\left \\{ ( \\omega , \\c d ) \\colon 1 + \\frac { 1 } { \\omega } < \\c d < \\frac { 1 2 ( 2 + 3 \\omega ) } { ( 4 + \\omega ) ^ 2 } \\right \\} . \\end{align*}"}
+{"id": "7910.png", "formula": "\\begin{align*} H \\Lambda ^ { k } ( \\Omega ) : = \\{ \\lambda \\in L ^ { 2 } \\Lambda ^ { k } ( \\Omega ) \\mid d \\lambda \\in L ^ { 2 } \\Lambda ^ { k + 1 } ( \\Omega ) \\} , \\end{align*}"}
+{"id": "2062.png", "formula": "\\begin{align*} A ( x , y ) = 1 + \\frac { 1 } { 2 } \\big ( \\sqrt { 2 } \\cos ( 2 \\pi x ) \\big ) \\big ( \\sqrt { 2 } \\cos ( 2 \\pi y ) \\big ) + \\frac { 1 } { 2 } \\big ( \\sqrt { 2 } \\sin ( 2 \\pi x ) \\big ) \\big ( \\sqrt { 2 } \\sin ( 2 \\pi y ) \\big ) . \\end{align*}"}
+{"id": "6445.png", "formula": "\\begin{align*} \\langle \\Theta _ { c _ 1 } B _ { \\Lambda _ 1 } f , S _ { c _ 1 } B _ { \\Lambda _ 1 } \\Theta g \\rangle & = \\int _ \\R \\Theta _ { c _ 1 } ( x ) B _ { \\Lambda _ 1 } ( x ) f ( x ) \\overline { \\Theta _ { c _ 1 } ( x ) B _ { \\Lambda _ 1 } ( x ) \\Theta ( x ) g ( x ) } \\ , \\mbox { d } x \\\\ & = \\int _ \\R f ( x ) \\overline { \\Theta ( x ) g ( x ) } \\ , \\mbox { d } x = 0 , \\end{align*}"}
+{"id": "102.png", "formula": "\\begin{align*} \\operatorname { d e t } ( M ) = a \\Big ( w _ 2 - \\frac { n - 2 } { n - 1 } \\Big ) ^ 2 + b \\Big ( w _ 2 - \\frac { n - 2 } { n - 1 } \\Big ) + c \\end{align*}"}
+{"id": "623.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\left ( \\frac { 1 } { 3 2 } \\right ) ^ k \\binom { 2 k } { k } ^ 2 H _ { 2 k } = \\frac { ( \\pi - 3 \\ln ( 2 ) ) \\Gamma ^ 2 \\left ( \\frac { 1 } { 4 } \\right ) } { 8 \\pi ^ { 3 / 2 } } . \\end{align*}"}
+{"id": "7013.png", "formula": "\\begin{align*} v ( t , x ) : = \\mathbf E _ { X _ 0 = x } [ f ( X _ t ) ] , \\end{align*}"}
+{"id": "8527.png", "formula": "\\begin{align*} \\kappa _ { C } ( g ) = g ( e _ 2 ) - e _ 2 \\end{align*}"}
+{"id": "7015.png", "formula": "\\begin{align*} d X _ t = - b ( t , X _ t ) d t + \\sqrt { 2 } d W _ t , X _ 0 = x \\end{align*}"}
+{"id": "9067.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { p , \\infty } ( \\mathbb { R } , d \\mu ) } = \\sup _ { t \\in \\mathbb { R } } | 2 t | ^ { \\frac { 1 } { p } } f ^ * ( t ) . \\end{align*}"}
+{"id": "9166.png", "formula": "\\begin{align*} G _ { \\beta , \\beta ' } = ( w _ { \\beta , 1 } - v ^ { - 4 } w _ { \\beta ' , 1 } ) ( w _ { \\beta , 1 } - v ^ { 4 } w _ { \\beta ' , 1 } ) \\cdot G _ { [ 2 , n , j ] , [ 2 , n , \\ell ] } , \\end{align*}"}
+{"id": "1364.png", "formula": "\\begin{align*} [ f , g ] _ { \\Omega } : = \\sum _ { i = 1 } ^ m ( - 1 ) ^ { ( i - 1 ) ( n - 1 ) } f \\circ _ i g - ( - 1 ) ^ { ( m - 1 ) ( n - 1 ) } \\sum _ { i = 1 } ^ n ( - 1 ) ^ { ( i - 1 ) ( m - 1 ) } g \\circ _ i f , \\end{align*}"}
+{"id": "7036.png", "formula": "\\begin{align*} \\Theta _ p ( \\mu , b _ n ) - \\Theta _ p ( \\eta , b _ n ) = ( \\nu - \\mu ) \\Theta _ p ( \\mu , b _ n ) \\Theta _ p ( \\nu , b _ n ) , \\mu , \\nu \\geq \\mu _ 0 , \\end{align*}"}
+{"id": "258.png", "formula": "\\begin{align*} Q \\equiv \\left ( A \\frac { d } { d x } - 3 A ' \\right ) \\left ( A \\frac { d } { d x } + \\gamma A - A ' \\right ) b = 0 . \\end{align*}"}
+{"id": "4511.png", "formula": "\\begin{align*} \\mathcal { T } _ n : = \\bigg \\{ V ( n ) \\leqslant \\frac { 2 C _ 0 T ( \\ell ) } { \\ell ^ { K / 2 } } \\bigg \\} . \\end{align*}"}
+{"id": "8087.png", "formula": "\\begin{align*} \\int \\phi _ 1 ( n ) L _ { \\tilde { w } _ t } ( \\tilde { \\psi } _ t ) ( p ^ - _ t ( n ) n z _ 0 ) J _ 1 ( n ) \\ ; d \\mu _ { z _ 0 } ^ u & = \\int ( \\phi _ 1 J _ 1 ) \\circ u ^ + _ t ( n ) L _ { w _ t } ( F _ t ) ( \\tilde { p } ^ - _ t ( n ) u ^ + _ t ( n ) z _ 0 ) J _ t ^ { - 1 } ( n ) \\ ; d \\mu _ { z _ t } ^ u \\\\ & = \\int ( \\phi _ 1 J _ 1 ) \\circ u ^ + _ t ( n ) \\cdot L _ { w _ t } ( F _ t ) ( n z _ t ) \\cdot J _ t ^ { - 1 } ( n ) \\ ; d \\mu _ { z _ t } ^ u ( n ) , \\end{align*}"}
+{"id": "7142.png", "formula": "\\begin{align*} { \\boldsymbol { k } } \\cdot { \\boldsymbol { \\varepsilon } } ( \\boldsymbol { k } ) = 0 \\ , \\ , . \\end{align*}"}
+{"id": "2489.png", "formula": "\\begin{align*} \\partial _ t u & = \\Delta ( u \\gamma ( v ) ) \\ ; \\ ; \\ ; \\ ; ( 0 , \\infty ) \\times \\Omega \\ , , \\\\ \\partial _ t v & = \\Delta v - u v \\ ; \\ ; \\ ; \\ ; ( 0 , \\infty ) \\times \\Omega \\ , , \\\\ \\nabla ( u \\gamma ( v ) ) \\cdot \\mathbf { n } & = \\nabla v \\cdot \\mathbf { n } = 0 \\ ; \\ ; \\ ; \\ ; ( 0 , \\infty ) \\times \\partial \\Omega \\ , , \\\\ ( u , v ) ( 0 ) & = ( u ^ { i n } , v ^ { i n } ) \\ ; \\ ; \\ ; \\ ; \\Omega \\ , , \\end{align*}"}
+{"id": "2188.png", "formula": "\\begin{align*} v = v _ r ( r , z , t ) \\bar e _ r + v _ \\varphi ( r , z , t ) \\bar e _ \\varphi + v _ z ( r , z , t ) \\bar e _ z \\end{align*}"}
+{"id": "7588.png", "formula": "\\begin{align*} \\partial ^ { 2 } _ { k l } f = ( \\delta _ { k l } - \\frac { x _ { k } x _ { l } } { r ^ 2 } ) \\frac { \\partial _ { r } f } { r } + \\frac { x _ { k } x _ { l } } { r ^ 2 } \\partial ^ { 2 } _ { r } f . \\end{align*}"}
+{"id": "7449.png", "formula": "\\begin{align*} \\int _ { \\blacktriangle } F = \\frac { 1 } { \\# \\Sigma _ n } \\int _ { \\bigcup _ { \\sigma \\in \\Sigma _ n } \\blacktriangle ^ \\sigma } F = \\frac { 1 } { n ! } \\int _ { [ 0 , L ] ^ n } F = \\frac { 1 } { n ! } \\left [ \\int _ 0 ^ L f ( t ) \\ , \\dd t \\right ] ^ { \\otimes n } . \\end{align*}"}
+{"id": "8123.png", "formula": "\\begin{align*} U _ { w _ 0 ( I ) d } = U _ { w _ 0 ( I ) } V _ d = V _ d U _ { w _ 0 ( I ) } . \\end{align*}"}
+{"id": "257.png", "formula": "\\begin{align*} \\frac { d ^ 2 y } { d \\tau ^ 2 } + \\alpha \\frac { d y } { d \\tau } + \\beta ( y ) = 0 , \\end{align*}"}
+{"id": "3929.png", "formula": "\\begin{gather*} z _ { i , n } = \\left < z _ i , \\varphi _ n \\right > . \\end{gather*}"}
+{"id": "2807.png", "formula": "\\begin{align*} \\delta _ { { } } ( \\tilde { \\sigma } _ { 1 } ^ { * } ( t ) I ) = \\Psi _ { 1 } \\int ^ { t _ { 2 } } _ { t _ { 1 } } d t \\delta \\Xi ^ { 3 } + \\left [ \\Psi _ { 1 } \\delta \\Xi ^ { 1 } + \\Psi _ { 3 } \\delta \\Xi ^ { 3 } \\right ] ^ { t _ { 2 } } _ { t _ { 1 } } . \\end{align*}"}
+{"id": "5345.png", "formula": "\\begin{align*} \\mathbf { e } _ i \\ , ( \\mathbf { I } - \\beta \\ , \\mathbf { P } ^ { 1 } ) ^ { - 1 } \\ , \\mathbf { h } ^ { 1 } + \\mathbf { x } ^ 0 \\ , \\widehat { \\mathbf { h } } ^ 0 + \\nu \\ , \\mathbf { x } ^ 1 \\ , \\boldsymbol { \\theta } ^ 1 = v _ i ^ { N ^ { \\{ 0 , 1 \\} } } + \\mathbf { x } ^ 0 \\ , \\widehat { \\mathbf { h } } ^ 0 + \\nu \\ , \\mathbf { x } ^ 1 \\ , \\boldsymbol { \\theta } ^ 1 , \\end{align*}"}
+{"id": "2674.png", "formula": "\\begin{align*} W = - \\sum _ { i = 1 } ^ { n } \\int f _ { i } d \\dot { q } ^ { i } + C ( q ^ { j } ) , \\end{align*}"}
+{"id": "7881.png", "formula": "\\begin{align*} U _ { i } = \\begin{cases} \\{ l _ 1 \\mid 1 \\leq l _ { 1 } \\leq m , l _ 1 \\equiv i \\bmod 3 \\} , & ; \\\\ \\{ l _ 1 \\mid 1 \\leq l _ { 1 } \\leq m , l _ 1 \\equiv i \\bmod 3 , l _ 1 \\equiv 1 \\bmod 2 \\} , & . \\end{cases} \\end{align*}"}
+{"id": "3307.png", "formula": "\\begin{align*} K _ t = t ^ { - 1 / 2 } \\exp \\left ( \\frac { B _ t ^ 2 } { 2 t } \\right ) = : g ( t , B _ t ) . \\end{align*}"}
+{"id": "265.png", "formula": "\\begin{align*} \\ddot { x } + \\frac { 2 } { x } \\dot { x } ^ 2 + \\frac { \\omega ^ 2 } { x ^ 3 } = 0 , \\omega \\in \\mathbb { R } , \\end{align*}"}
+{"id": "4466.png", "formula": "\\begin{align*} M _ { S } ( Z _ 0 , J , \\rho _ 1 \\rho _ 2 ) = \\| f _ 0 \\| ^ 2 _ { S , \\rho _ 1 \\rho _ 2 } = \\| F _ 2 \\| ^ 2 _ { S , \\rho _ 1 \\rho _ 2 } + \\| f _ 1 f _ 2 \\| ^ 2 _ { S , \\rho _ 1 \\rho _ 2 } \\ge M _ { \\partial D _ 1 } \\times M _ { S _ 1 } . \\end{align*}"}
+{"id": "2891.png", "formula": "\\begin{align*} & \\lim _ { \\lambda \\to \\infty } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ { L ^ { \\vec { r } } ( \\mathbb { R } ^ n ) } = \\left [ \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { L ^ { \\vec { r } } ( \\mathbb { R } ^ n ) } ; \\end{align*}"}
+{"id": "1932.png", "formula": "\\begin{align*} \\operatorname { T a i l } ( v ; x _ { 0 } , r ) & : = \\left ( r ^ { s q } \\int _ { \\mathbb { R } ^ { N } \\backslash B _ { r } \\left ( x _ { 0 } \\right ) } \\frac { | v ( x ) | ^ { q - 1 } } { \\left | x - x _ { 0 } \\right | ^ { N + s q } } d x \\right ) ^ { \\frac { 1 } { q - 1 } } r > 0 , x _ 0 \\in \\mathbb { R } ^ N . \\end{align*}"}
+{"id": "7973.png", "formula": "\\begin{align*} \\dot { H } = \\int _ { \\Gamma } e _ { b } \\wedge f _ { b } . \\end{align*}"}
+{"id": "2999.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\overline { \\nabla } _ x \\varphi & = & ( \\beta - 1 ) \\{ g ( \\varphi x , \\varphi y ) \\xi + \\eta ( y ) \\varphi ^ 2 ( x ) \\} + \\alpha \\{ g ( x , \\varphi y ) \\xi + \\eta ( y ) \\varphi x \\} , \\\\ \\overline { \\nabla } _ x \\xi & = & ( 1 - \\beta ) \\varphi x - \\alpha \\varphi ^ 2 x , \\\\ ( \\overline { \\nabla } _ x \\eta ) y & = & ( 1 - \\beta ) g ( \\varphi x , y ) - \\alpha g ( x , y ) + \\eta ( x ) \\eta ( y ) , \\\\ \\end{array} \\end{align*}"}
+{"id": "4341.png", "formula": "\\begin{gather*} \\mathcal { X } _ \\mathcal { Q } : = \\{ x \\in \\mathcal { X } : \\Delta u _ q ^ T l _ q ( x ) - \\Delta u _ k ^ T l _ k ( x ) \\geq 0 \\ \\forall q \\in \\mathcal { Q } , \\ \\Delta u _ q ^ T l _ q ( x ) - \\Delta u _ k ^ T l _ k ( x ) \\leq 0 \\ \\forall q \\in [ m ] \\setminus \\mathcal { Q } \\} . \\end{gather*}"}
+{"id": "8487.png", "formula": "\\begin{align*} \\lim _ { h \\searrow 0 } I _ h ^ { ( 2 ) } = 0 . \\end{align*}"}
+{"id": "1961.png", "formula": "\\begin{align*} d ( { { \\hat { C } } ^ { p ^ t } | } _ { T _ 0 } ) \\ ! = \\ ! \\min \\{ \\sum _ { \\theta = 1 } ^ { \\tau + 1 - | N _ 0 | } | T _ { 0 , i _ { \\theta } } | \\mid i _ \\theta \\in \\bar { N _ { 0 } } \\} . \\end{align*}"}
+{"id": "5515.png", "formula": "\\begin{align*} \\Gamma ( Q _ \\infty , Q ) = \\Bigg \\{ ( m j _ 1 - i ( q ^ 2 - q ) , i + m j _ 2 ) & \\mid 1 \\leq i \\leq m - 1 , \\ ; j _ 1 \\geq \\left \\lceil \\frac { i ( q ^ 2 - q ) } { m } \\right \\rceil , \\ ; j _ 2 \\geq 0 , \\\\ & \\mbox { a n d } j _ 1 + j _ 2 = ( q - 2 ) \\left \\lceil \\frac { i } { s } \\right \\rceil + 1 \\Bigg \\} , \\end{align*}"}
+{"id": "6626.png", "formula": "\\begin{align*} H _ N S _ N ( i , j ) & = \\frac { 1 } { J _ N } ( - J _ 1 J _ { N - 2 } + 2 J _ 1 ( J _ { N - 2 } + J _ { N - 1 } ) - 2 J _ 1 J _ { N - 3 } ) \\\\ & = \\frac { 1 } { J _ N } ( - J _ { N - 2 } + 2 ( J _ { N - 2 } + J _ { N - 1 } ) - 2 J _ { N - 3 } ) \\\\ & = \\frac { 1 } { J _ N } ( - 2 J _ { N - 3 } + J _ { N - 2 } + 2 J _ { N - 1 } ) \\\\ & = \\frac { 1 } { J _ N } ( 2 J _ { N - 2 } + J _ { N - 1 } ) \\\\ & = 1 . \\end{align*}"}
+{"id": "682.png", "formula": "\\begin{align*} c _ { 1 } t ^ { - \\frac { n } { 2 } } \\exp \\bigg ( - b _ { 1 } \\frac { | x - y | ^ { 2 } } { t } \\bigg ) \\le k _ { t } ^ { A } ( x , y ) = k _ { t } ^ { A } ( y , x ) \\le c _ { 2 } t ^ { - \\frac { n } { 2 } } \\exp \\bigg ( - b _ { 2 } \\frac { | x - y | ^ { 2 } } { t } \\bigg ) , \\end{align*}"}
+{"id": "7459.png", "formula": "\\begin{align*} A ^ { n + 1 } _ i = - \\frac { A ^ { n } _ { i - 1 } } { c _ { n + 1 } + \\cdots + c _ { n - i + 2 } } i \\in \\{ 1 , \\ldots , n + 1 \\} . \\end{align*}"}
+{"id": "2206.png", "formula": "\\begin{align*} \\intop _ \\Omega \\bigg | { 1 \\over r } \\psi _ { 1 , r } \\bigg | ^ 2 d x + { 1 \\over 2 } \\intop _ { - a } ^ a \\psi _ { 1 , r } ^ 2 \\bigg | _ { r = R } d z \\le c ( | \\psi _ { 1 , z z } | _ { 2 , \\Omega } ^ 2 + | \\omega _ 1 | _ { 2 , \\Omega } ^ 2 ) . \\end{align*}"}
+{"id": "4127.png", "formula": "\\begin{align*} & \\tilde { \\theta } _ k = \\eta _ k r ^ k , k = j + 1 , \\ldots , J , \\\\ & \\tilde { \\theta } _ j = \\eta _ j r ^ { j + 1 / 2 } + \\frac { 1 } { 1 - w _ { j j } } \\sum _ { i = j + 1 } ^ J w _ { j i } \\theta _ i , \\\\ & \\tilde { \\theta } _ \\ell = w _ { \\ell j } \\tilde { \\theta } _ j + \\sum _ { i = j + 1 } ^ { J } w _ { \\ell i } \\theta _ i , \\ell = 1 , \\ldots , j - 1 . \\end{align*}"}
+{"id": "740.png", "formula": "\\begin{align*} & n _ p \\bigl ( a , a + b , a j _ { k - 1 } ( v ) + b j _ k ( v ) \\bigr ) \\\\ & = \\frac { 1 } { 2 } \\biggl ( - v ( a - r ^ 2 ) + ( a - 1 ) ( b - 1 ) + \\frac { v ( a + r ) ( a - r ) j _ { k - 1 } ( v ) } { j _ k ( v ) } \\\\ & + v ( a - r ) \\bigl ( j _ k ( v ) - j _ { k - 1 } ( v ) \\bigr ) \\biggr ) \\\\ & + \\frac { p } { 2 } a j _ k ( v ) \\bigl ( 2 ( v a + b ) - v ( j _ k ( v ) - j _ { k - 1 } ( v ) \\bigr ) - \\frac { p ^ 2 } { 2 } a v j _ k ( v ) \\bigl ( j _ k ( v ) - j _ { k - 1 } ( v ) \\bigr ) \\ , , \\end{align*}"}
+{"id": "758.png", "formula": "\\begin{align*} \\overset { \\ _ * } R ^ h _ j = \\kappa F ^ 2 ( \\delta ^ h _ j - l ^ h l _ j ) , \\end{align*}"}
+{"id": "8274.png", "formula": "\\begin{align*} R ( \\lambda , A ) = \\sum _ { n = 0 } ^ \\infty ( \\lambda _ 0 - \\lambda ) ^ n R ( \\lambda _ 0 , A ) ^ { n + 1 } . \\end{align*}"}
+{"id": "9058.png", "formula": "\\begin{align*} \\int _ { ( \\mathbb { R } ^ n ) ^ p } \\prod _ { j = 1 } ^ { k } f _ j \\left ( \\sum _ { m = 1 } ^ { p } a _ { j m } x _ m \\right ) d x \\leq \\int _ { ( \\mathbb { R } ^ n ) ^ p } \\prod _ { j = 1 } ^ { k } f ^ * _ j \\left ( \\sum _ { m = 1 } ^ { p } a _ { j m } x _ m \\right ) d x , \\end{align*}"}
+{"id": "6516.png", "formula": "\\begin{align*} \\Phi _ { \\nu } ( B _ m ) = - \\frac { 1 } { \\alpha } \\log { m _ n } . \\end{align*}"}
+{"id": "9200.png", "formula": "\\begin{gather*} \\Delta ( K ) = K \\otimes K , \\ \\ \\Delta ( E ) = E \\otimes K + 1 \\otimes E , \\ \\ \\Delta ( F ) = F \\otimes 1 + K ^ { - 1 } \\otimes F , \\\\ \\varepsilon ( K ) = 1 , \\ \\ \\varepsilon ( E ) = 0 , \\ \\ \\varepsilon ( F ) = 0 , \\\\ S ( K ) = K ^ { - 1 } , \\ \\ S ( E ) = - E K ^ { - 1 } , \\ \\ S ( F ) = - K F . \\end{gather*}"}
+{"id": "3884.png", "formula": "\\begin{align*} \\hat \\theta ^ n ( \\{ x \\} \\times \\{ y \\} \\times [ 0 , t ] ) & = \\int _ 0 ^ t \\exp ( - s ) \\eta ^ n _ { ( 1 ) } ( x \\mid s ) G ( M ^ n ( s ) ) ( x , y ) d s \\\\ & \\to \\int _ 0 ^ t \\exp ( - s ) \\eta _ { ( 1 ) } ( x \\mid s ) G ( M ( s ) ) ( x , y ) d s . \\end{align*}"}
+{"id": "7905.png", "formula": "\\begin{align*} \\lambda \\wedge \\ast \\mu = \\langle \\lambda , \\mu \\rangle _ { \\Lambda ^ { k } } v _ { \\Omega } , \\quad \\forall \\lambda , \\mu \\in \\Lambda ^ { k } ( \\Omega ) . \\end{align*}"}
+{"id": "6029.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb E _ { \\nu _ \\rho } \\Big [ \\big ( \\xi ^ { A } _ { x + 1 } - \\xi ^ { A } _ { x } + D _ 2 ^ + ( \\xi ^ { B } _ { x + 1 } - \\xi ^ { B } _ { x } ) \\big ) \\big ( \\xi ^ { A } _ { x + 1 } - \\xi ^ { A } _ { x } + D _ 2 ^ - ( \\xi ^ { B } _ { x + 1 } - \\xi ^ { B } _ { x } ) \\big ) \\Big ] = \\frac { 2 } { 9 } ( 2 - ( D _ 2 ^ + + D _ 2 ^ - ) + 2 D _ 2 ^ + D _ 2 ^ - ) = 0 \\end{aligned} \\end{align*}"}
+{"id": "1111.png", "formula": "\\begin{align*} \\| \\{ f _ j \\} _ { j \\in \\mathbb Z } \\| _ { L \\dot F _ { p q } ( E \\times J ) } : = \\| \\{ f _ j \\} _ { j \\in \\mathbb Z } \\| _ { L ^ p \\ell ^ q ( E \\times J ) } : = \\| \\{ f _ j \\} _ { j \\in \\mathbb Z } \\| _ { L ^ p ( E ; \\ell ^ q ( J ) ) } : = \\left \\| \\left ( \\sum _ { j \\in J } | f _ j | ^ q \\right ) ^ { \\frac { 1 } { q } } \\right \\| _ { L ^ p ( E ) } \\end{align*}"}
+{"id": "5071.png", "formula": "\\begin{align*} \\overline { K ( s ) } = ( - \\overline { \\mathrm { p r } } _ s ) ^ * \\mathcal { L } _ \\psi \\end{align*}"}
+{"id": "516.png", "formula": "\\begin{align*} O ( d \\cdot ( \\log { d } + \\log \\log { q } ) \\cdot d \\log ^ { 1 + o ( 1 ) } { q } ) = O ( d ^ 2 \\log { d } \\log ^ { 1 + o ( 1 ) } { q } ) . \\end{align*}"}
+{"id": "3973.png", "formula": "\\begin{align*} \\Psi _ t ( \\rho \\oplus \\rho ( x , y ) ( \\xi \\oplus \\eta ) ) & = \\Psi _ t ( \\rho ( x , y ) \\xi \\oplus \\rho ( x , y ) \\eta ) & \\\\ & = ( \\rho ( x , y ) \\xi ) _ t \\otimes ( \\rho ( x , y ) \\eta ) _ t & \\\\ & = \\rho _ t ( x , y ) \\xi _ t \\otimes \\rho _ t ( x , y ) \\eta _ t & \\\\ & = ( \\rho _ t \\otimes \\rho _ t ) ( x , y ) ( \\xi _ t \\otimes \\eta _ t ) & \\\\ & = ( \\rho _ t \\otimes \\rho _ t ) ( x , y ) \\Psi _ t ( \\xi \\oplus \\eta ) . & \\end{align*}"}
+{"id": "5111.png", "formula": "\\begin{align*} S ( p ) = \\left [ - \\frac { d } { d \\alpha } \\sum _ { i } p ^ { \\alpha } _ { i } \\right ] _ { \\alpha = 1 } \\equiv \\left [ - \\frac { d } { d \\alpha } f ( \\alpha ) \\right ] _ { \\alpha = 1 } \\end{align*}"}
+{"id": "5979.png", "formula": "\\begin{gather*} { \\rm s u p p \\ ; } \\omega _ j = D _ j , \\ ; \\mu : = \\int _ { D } \\omega _ j > 0 , \\ ; | D _ j | = \\pi r _ j ^ 2 , \\ ; { \\rm d i s t } ( D _ j ) \\leq 2 . 5 r _ j , \\ ; j = 1 , 2 , \\end{gather*}"}
+{"id": "8579.png", "formula": "\\begin{align*} g ' _ j = \\begin{cases} - g _ k & j = k ; \\\\ g _ j + [ b _ { j k } ] _ + g _ k - b _ { j k } m i n ( g _ k , 0 ) & j \\neq k . \\end{cases} \\end{align*}"}
+{"id": "1601.png", "formula": "\\begin{align*} K _ { i j } : = K ( e _ i , e _ j ) = R _ { i j i j } . \\end{align*}"}
+{"id": "7960.png", "formula": "\\begin{align*} \\frac { \\delta \\bar { \\mathcal { F } } } { \\delta \\phi _ { \\partial } } = \\frac { \\delta \\tilde { \\mathcal { F } } } { \\delta \\phi _ { \\partial } } , \\frac { \\delta \\bar { \\mathcal { F } } } { \\delta \\Sigma } = \\frac { \\delta \\tilde { \\mathcal { F } } } { \\delta \\Sigma } = \\frac { \\delta \\mathcal { F } } { \\delta \\Sigma } + ( - 1 ) ^ { n - 1 } \\langle d N _ { \\phi } ( \\frac { \\delta \\tilde { \\mathcal { F } } } { \\delta \\phi _ { \\partial } } ) , \\eta \\rangle _ { \\Lambda ^ { 1 } } . \\end{align*}"}
+{"id": "6660.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi \\mathrm { i } n } \\oint _ { | z | = 1 } \\frac { f ' _ { n } ( z ) } { f _ { n } ( z ) } \\mathrm { d } z = \\frac { 1 } { n } \\sum _ { j = 1 } ^ { n } \\frac { - 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } \\frac { \\partial } { \\partial x } \\arg ( E _ { j } ( x ) - E ) \\mathrm { d } x = 0 . \\end{align*}"}
+{"id": "3653.png", "formula": "\\begin{align*} 1 - \\binom { d + 1 } { 2 } + \\sum _ { i = 1 } ^ { d } r _ i + \\sum _ { 1 \\le i < j \\le d } r _ { i , j } = | E | - d | V | + \\binom { d + 1 } { 2 } + 1 = m . \\end{align*}"}
+{"id": "471.png", "formula": "\\begin{align*} g _ 1 = b _ 1 , g _ 1 g _ 2 = b _ 2 , g _ 1 g _ 2 g _ 3 = b _ 3 , \\ldots g _ { 1 } g _ { 2 } \\cdots g _ { k - 1 } & = b _ { k - 1 } , g _ 1 \\cdots g _ k = h . \\end{align*}"}
+{"id": "2703.png", "formula": "\\begin{align*} & a ^ { i } = ( { K ^ { ( 1 ) } } ^ { - 1 } ) ^ { i j } S _ { j } , \\\\ & b ^ { i } = v ^ { i } . \\end{align*}"}
+{"id": "4220.png", "formula": "\\begin{align*} \\theta ' ( 0 , \\tau ) = \\pi \\theta _ 1 ( 0 , \\tau ) \\theta _ 2 ( 0 , \\tau ) \\theta _ 3 ( 0 , \\tau ) . \\end{align*}"}
+{"id": "3311.png", "formula": "\\begin{align*} K _ { n - 1 } '' = \\frac { 1 } { \\sqrt { n } } \\exp \\left ( \\frac { ( ( n - 1 ) \\mu _ 0 - S _ { n - 1 } ) ^ 2 } { 2 n } \\right ) , \\end{align*}"}
+{"id": "490.png", "formula": "\\begin{align*} \\{ ( 3 ^ 0 1 7 ^ 0 + 2 , 8 ) , ( 3 ^ 1 1 7 ^ 0 + 2 , 8 ) , ( 3 ^ 0 1 7 ^ 1 + 2 , 1 ) \\} = \\{ ( 3 , 8 ) , ( 5 , 8 ) , ( 2 , 1 ) \\} \\end{align*}"}
+{"id": "20.png", "formula": "\\begin{align*} \\beta ( \\kappa ; q + c ) + { \\textstyle \\int } ( q + c ) \\ , d x = \\tfrac { \\kappa ^ 2 } { ( \\kappa - c ) ^ 2 } \\Bigl [ \\beta ( \\kappa - c ; q ) + { \\textstyle \\int } q \\ , d x \\Bigr ] + \\tfrac { c \\kappa } { \\kappa - c } . \\end{align*}"}
+{"id": "1820.png", "formula": "\\begin{align*} \\phi _ 0 : = d x _ { 1 2 3 } + d x _ { 1 4 5 } + d x _ { 1 6 7 } + d x _ { 2 4 6 } - d x _ { 2 5 7 } - d x _ { 3 4 7 } - d x _ { 3 5 6 } , \\end{align*}"}
+{"id": "5977.png", "formula": "\\begin{align*} E _ { \\hat { P } } ( 0 ) = \\sum _ { i } E _ { P ^ { ( i ) } } ( 0 ) . \\end{align*}"}
+{"id": "3792.png", "formula": "\\begin{align*} \\| \\mathcal { H } _ 2 \\| & = \\| \\Theta _ j - \\Theta _ 1 \\| \\frac { 2 \\eta _ 1 } { | \\widehat { \\mathcal { C } } _ 1 | } | \\widehat { \\mathcal { C } } _ 1 \\cap \\overline { \\mathcal { C } _ 1 } | \\| \\bar { Z } \\bar { Z } ^ { \\top } \\| . \\end{align*}"}
+{"id": "7110.png", "formula": "\\begin{align*} \\| \\nabla \\sqrt { E _ \\varepsilon \\varphi ^ 2 } \\| _ 2 & = \\| \\frac { E _ \\varepsilon ( | \\varphi | | \\nabla | \\varphi | ) } { \\sqrt { E _ \\varepsilon \\varphi ^ 2 } } \\| _ 2 \\\\ & \\leq \\| \\sqrt { E _ \\varepsilon | \\nabla | \\varphi | | ^ 2 } \\| _ 2 = \\| E _ \\varepsilon | \\nabla | \\varphi | | ^ 2 \\| _ 1 ^ \\frac { 1 } { 2 } \\\\ & \\leq \\| \\nabla | \\varphi | \\| _ 2 \\leq \\| \\nabla \\varphi \\| _ 2 , \\end{align*}"}
+{"id": "6134.png", "formula": "\\begin{align*} [ ( \\psi _ n ( \\varphi _ k ( x ) ) ) _ n ] = \\Psi ( \\varphi _ k ( x ) ) = \\rho _ k ( x ) = [ ( \\rho _ { n , k } ( x ) ) _ { n > k } ] \\end{align*}"}
+{"id": "1396.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\leq \\alpha \\leq 1 , \\ \\beta > 6 , \\ \\lim _ { Z \\to \\infty } \\mathcal { S } ( Z ) = 0 . \\end{align*}"}
+{"id": "1532.png", "formula": "\\begin{align*} \\lambda _ { \\rm m i n } ( D ^ 2 F ( z ) ) = \\lambda _ { \\rm m i n } ( D ^ 2 d ( z ) ) , \\lambda _ { \\rm m a x } ( D ^ 2 ( F ( z ) ) = \\lambda _ { \\rm m a x } ( D ^ 2 d ( z ) ) , \\end{align*}"}
+{"id": "2072.png", "formula": "\\begin{align*} \\eqref { e q : t e s t f u n c t i o n a p p l i e d s u m } = \\sum _ { t = 1 } ^ { \\lfloor s T \\rfloor - 1 } \\Bar { \\Theta } ^ { d , T } ( \\frac { t } { T } , x ) \\big ( f ( \\frac { t - 1 } { T } ) - f ( \\frac { t } { T } ) \\big ) + f ( \\frac { \\lfloor s T \\rfloor - 1 } { T } ) \\Bar { \\Theta } ^ { d , T } ( \\frac { \\lfloor s T \\rfloor } { T } , x ) - f ( 0 ) \\Bar { \\Theta } ^ { d , T } ( 0 , x ) . \\end{align*}"}
+{"id": "2515.png", "formula": "\\begin{align*} \\| a \\| _ { K , \\alpha , \\beta , m } : = \\sup _ { x \\in K , \\ \\xi \\in \\R ^ l } \\frac { | \\partial _ x ^ \\alpha \\partial _ \\xi ^ \\beta a ( x , \\xi ) | } { ( 1 + | \\xi | ) ^ { m - | \\beta | } } < \\infty \\ ; . \\end{align*}"}
+{"id": "3449.png", "formula": "\\begin{align*} F \\colon & \\Omega ^ 1 ( W ) ^ I \\times \\Gamma ( \\mathbb { S } ^ + ) ^ I \\to \\\\ & \\Omega ^ + ( W ) ^ I \\times \\Gamma ( \\mathbb { S } ^ - ) ^ I \\times \\Omega ^ 0 ( W ) ^ I \\times V ( - Y _ 0 ) ^ { 0 } _ { - \\infty } \\times V ( Y _ 1 ) ^ { 0 } _ { - \\infty } \\times U _ 0 \\times U _ 1 \\end{align*}"}
+{"id": "8134.png", "formula": "\\begin{align*} [ h _ { A _ 2 } ( J ) , k _ { \\rm A _ 2 } ( J ) ] \\ = \\ \\sum _ { i = 1 } ^ 9 \\ c _ i ( J ) \\ A _ i \\ , \\end{align*}"}
+{"id": "1308.png", "formula": "\\begin{align*} \\mu _ 1 ( t ) & = \\frac { 1 } { 2 } \\Big ( ( e ^ { ( a - 2 k ) t } + e ^ { a t } ) x _ 1 + ( e ^ { a t } - e ^ { ( a - 2 k ) t } ) x _ 2 \\Big ) \\\\ & = \\frac { 1 } { 2 } e ^ { a t } \\Big ( ( 1 + e ^ { - 2 k t } ) x _ 1 + ( 1 - e ^ { - 2 k t } ) x _ 2 \\Big ) \\\\ & \\leq C e ^ { a t } . \\end{align*}"}
+{"id": "7958.png", "formula": "\\begin{align*} \\omega _ { t } + d \\ast ( v \\wedge \\ast \\omega ) = 0 . \\end{align*}"}
+{"id": "7293.png", "formula": "\\begin{align*} f _ { m , r } ( s ) = 2 \\frac { U _ { m - \\ell - 1 } ( 1 - 2 s ) + U _ { \\ell - 1 } ( 1 - 2 s ) } { T _ { m } ( 1 - 2 s ) - 1 } + \\frac { 1 } { m s } . \\end{align*}"}
+{"id": "5982.png", "formula": "\\begin{align*} \\left ( | \\nabla u | ^ 2 \\right ) _ { \\nu } = & 2 \\langle d u ( \\nu ) , \\tau ^ { \\partial M } ( u ) - \\tau ( u ) \\rangle - 2 H | d u ( \\nu ) | ^ 2 + 2 \\langle \\nabla ( d u ( \\nu ) ) , d u \\rangle _ { \\partial M } \\\\ & - 2 \\langle d u \\circ A , d u \\rangle _ { \\partial M } , \\end{align*}"}
+{"id": "7375.png", "formula": "\\begin{align*} M : = \\sum _ { \\alpha = 0 } ^ 2 \\| f ^ { ( \\alpha ) } \\| _ { L ^ \\infty ( \\R ) } + \\| g \\| _ { L ^ 1 ( \\R ) } + \\sum _ { \\beta = 0 } ^ 1 \\| g ^ { ( \\beta ) } \\| _ { L ^ \\infty ( \\R ) } . \\end{align*}"}
+{"id": "1993.png", "formula": "\\begin{align*} \\psi _ M ^ { n + 1 } ( x ) = \\sum _ { l \\in \\mathcal { T } _ M } \\widehat { ( \\psi _ M ^ { n + 1 } ) } _ l e ^ { i \\mu _ l ( x - a ) } , x \\in \\overline { \\Omega } , \\end{align*}"}
+{"id": "8786.png", "formula": "\\begin{align*} \\theta ( [ F _ \\nu ^ { - 1 } ( F _ \\nu ( x - ) - q ( x ) ) & , F _ \\nu ^ { - 1 } ( F _ \\nu ( x ) + \\mu ( \\{ x \\} ) - q ( x ) ) ] ) = 1 \\\\ & = \\vartheta ( ( - \\infty , F _ \\nu ^ { - 1 } ( F _ \\nu ( x - ) - q ( x ) ) ] \\cup [ F _ \\nu ^ { - 1 } ( F _ \\nu ( x ) + \\mu ( \\{ x \\} ) - q ( x ) ) , + \\infty ) ) . \\end{align*}"}
+{"id": "7151.png", "formula": "\\begin{align*} \\varepsilon _ \\mu ( k , 0 ) = \\left ( \\begin{array} { c } 1 \\\\ 0 \\\\ 0 \\\\ 0 \\end{array} \\right ) \\ , \\ , , \\ , \\ , \\varepsilon _ \\mu ( k , 1 ) = \\left ( \\begin{array} { c } 0 \\\\ 1 \\\\ 0 \\\\ 0 \\end{array} \\right ) \\ , \\ , , \\ , \\ , \\varepsilon _ \\mu ( k , 2 ) = \\left ( \\begin{array} { c } 0 \\\\ 0 \\\\ 1 \\\\ 0 \\end{array} \\right ) \\ , \\ , , \\ , \\ , \\varepsilon _ \\mu ( k , 3 ) = \\left ( \\begin{array} { c } 0 \\\\ 0 \\\\ 0 \\\\ 1 \\end{array} \\right ) \\ , \\ , . \\end{align*}"}
+{"id": "7475.png", "formula": "\\begin{align*} l _ s = \\pi _ 2 \\circ \\Phi _ s ^ { - 1 } , \\end{align*}"}
+{"id": "5106.png", "formula": "\\begin{align*} D _ I = \\sum _ { T \\subset \\{ \\lambda _ i ^ I \\} _ { i } } ( - 1 ) ^ { | T | } \\frac { ( n + 1 ) ! } { \\prod _ { \\lambda _ i ^ I \\in T } ( \\lambda _ i ^ I + 1 ) ! } \\end{align*}"}
+{"id": "4534.png", "formula": "\\begin{align*} n _ \\infty ^ { ( K ) } ( t , [ s ] _ { K } ) = \\lim _ { N \\to \\infty } n _ { N } ^ { ( K ) } ( t , s _ 1 , s _ 2 , . . . s _ { K } ) , K = 1 , \\ , 2 , . . . \\end{align*}"}
+{"id": "8074.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow + \\infty } \\frac { N } { \\gamma _ N ^ 2 } \\log P ( \\frac { N } { \\gamma _ N } ( \\tau _ c ^ N - \\tau _ c ) > x ) = - J _ { h i t } ( x ) \\end{align*}"}
+{"id": "8981.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } F ( D ^ 2 u ) - \\beta { u r ^ { - \\gamma } } = r ^ { - \\gamma } f ( r ) & \\hbox { i n } \\ B ( 0 , 1 ) \\setminus \\{ 0 \\} \\\\ u = b & \\hbox { o n } \\ \\partial B ( 0 , 1 ) \\end{array} \\right . \\end{align*}"}
+{"id": "9014.png", "formula": "\\begin{align*} x _ { 1 1 } = x _ { n n } = 0 y _ { i j } = z _ { i j } ^ 1 = z _ { i j } ^ n = 0 1 \\le i < j \\le n ( i , j ) \\ne ( 1 , n ) . \\end{align*}"}
+{"id": "7631.png", "formula": "\\begin{align*} f ( z ) = \\Big ( \\prod _ { i = 1 } ^ { d - 2 j } ( z - x _ i ) \\Big ) \\Big ( \\prod _ { k = 1 } ^ j ( z - \\alpha _ k ) ( z - \\overline { \\alpha } _ k ) \\Big ) \\end{align*}"}
+{"id": "419.png", "formula": "\\begin{align*} \\begin{aligned} & \\sigma ^ 2 ( l _ x , l _ y , m _ x , m _ y ) = \\sigma ^ 2 _ { R } ( l _ x , l _ y ) \\sigma ^ 2 _ { S } ( m _ x , m _ y ) \\\\ & \\times \\exp ( - \\frac { ( l _ x - m _ x ) ^ 2 + ( l _ y - m _ y ) ^ 2 } { a } ) , \\end{aligned} \\end{align*}"}
+{"id": "3849.png", "formula": "\\begin{align*} & \\gamma ^ n ( \\{ e _ x \\} \\times [ 0 , t ] ) - \\beta ^ n _ { ( 2 , 3 ) } ( \\{ e _ x \\} \\times [ 0 , t ] ) \\\\ & = n ^ { - 1 } \\int _ 0 ^ { t } \\psi _ e ( t _ n - s ) \\left ( \\bar \\Lambda ^ n ( e _ x \\mid t _ n - s ) - \\bar \\xi ^ n _ { ( 2 ) } ( e _ x \\mid t _ n - s ) \\right ) d s \\\\ & = n ^ { - 1 } \\int _ { t _ n - t } ^ { t _ n } \\psi _ e ( s ) \\left ( \\bar \\Lambda ^ n ( e _ x \\mid s ) - \\bar \\xi ^ n _ { ( 2 ) } ( e _ x \\mid s ) \\right ) d s . \\end{align*}"}
+{"id": "5423.png", "formula": "\\begin{align*} S _ { h _ k } f \\mid _ { \\Xi _ k } = 1 . \\end{align*}"}
+{"id": "115.png", "formula": "\\begin{align*} \\mathfrak { R } = \\left \\{ \\vec { c } + \\mathfrak { m } \\vec { x } \\ ; | \\ ; \\vec { c } \\in \\mathbb { R } ^ d , \\mathfrak { m } \\in \\mathfrak { s o } ( d ) \\right \\} , \\end{align*}"}
+{"id": "3678.png", "formula": "\\begin{align*} L ^ { - } _ { e , e ' } = \\begin{cases} \\frac { 1 } { 2 } & i , j , k \\in [ d ] , \\\\ \\frac { \\sqrt { 2 } } { 2 } & i , j \\in [ d ] , k \\in [ n ] \\setminus [ d ] , \\\\ 1 & i \\in [ d ] , j , k \\notin [ d ] , \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "5952.png", "formula": "\\begin{align*} \\frac { M _ { 2 } ( \\mathcal { C } ( G ) ) } { | e ( \\mathcal { C } ( G ) ) | } = \\frac { 1 1 5 } { 1 9 } > \\frac { 8 6 } { 2 3 } = \\frac { M _ { 1 } ( \\mathcal { C } ( G ) ) } { | v ( \\mathcal { C } ( G ) ) | } . \\end{align*}"}
+{"id": "5936.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { C } ( G ) ) = 2 ^ { 5 k } - 4 \\cdot 2 ^ { 4 k } + 4 \\cdot 2 ^ { 3 k } + 4 \\cdot 2 ^ { 2 k } - 5 \\cdot 2 ^ { k } - 4 , \\end{align*}"}
+{"id": "3872.png", "formula": "\\begin{align*} \\hat M ^ n ( t ) \\doteq \\bar L ^ n ( \\sigma _ { n , t } ) = \\bar L ^ n ( t _ n - T + t ) . \\end{align*}"}
+{"id": "5851.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { N C } ( D _ { 2 m } ) ) & = ( 2 m - 1 ) ( 2 m - 2 ) ^ { 2 } - 4 ( 2 m - 2 ) \\dfrac { ( m - 1 ) ( m - 2 ) } { 2 } + ( m - 1 ) ( m - 2 ) ^ { 2 } \\\\ & = ( m - 1 ) [ ( 8 m - 4 ) ( m - 1 ) - 4 ( m - 1 ) ( m - 2 ) + ( m - 2 ^ { 2 } ) ] \\\\ & = m ( m - 1 ) ( 5 m - 4 ) \\end{align*}"}
+{"id": "7081.png", "formula": "\\begin{align*} G _ p ( r ) & : = b ^ { \\frac { 1 } { p } } \\cdot \\nabla ( \\mu - \\Delta ) ^ { - \\frac { 1 } { 2 } - \\frac { 1 } { 2 r } } , \\\\ Q _ p ( q ) \\upharpoonright { \\mathcal E } & : = ( \\mu - \\Delta ) ^ { - \\frac { 1 } { 2 } + \\frac { 1 } { 2 q } } | b | ^ { 1 - \\frac { 1 } { p } } , \\\\ T _ p \\upharpoonright { \\mathcal E } & : = b ^ { \\frac { 1 } { p } } \\cdot \\nabla ( \\mu - \\Delta ) ^ { - 1 } | b | ^ { 1 - \\frac { 1 } { p } } . \\end{align*}"}
+{"id": "6661.png", "formula": "\\begin{align*} 0 & \\leq \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { 2 \\pi n \\epsilon } \\int _ { 0 } ^ { 2 \\pi } \\bigg ( \\log | f _ { n } ( \\mathrm { e } ^ { \\mathrm { i } ( x - \\mathrm { i } \\epsilon ) } ) | - \\log | f _ { n } ( \\mathrm { e } ^ { \\mathrm { i } x } ) | \\bigg ) \\mathrm { d } x \\\\ & = \\frac { 1 } { \\epsilon } ( L ( E , - \\mathrm { i } \\epsilon ) - L ( E , \\mathrm { i } 0 ) ) = - \\omega ^ { - } ( E , \\mathrm { i } 0 ) . \\end{align*}"}
+{"id": "9128.png", "formula": "\\begin{align*} x ^ { ( \\beta , s ) } _ { i , 1 } \\mapsto v ^ { - i } w _ { \\beta , s } \\ , , \\ , \\dots \\ , , \\ , x ^ { ( \\beta , s ) } _ { j , 1 } \\mapsto v ^ { - j } w _ { \\beta , s } \\forall \\ \\beta = [ i , j ] , \\ 1 \\leq s \\leq d _ { \\beta } . \\end{align*}"}
+{"id": "2750.png", "formula": "\\begin{align*} & \\omega _ { \\Xi , \\Psi } : = d \\sigma _ { 1 } ^ { * } \\Xi ^ { \\alpha } \\wedge d \\sigma _ { 1 } ^ { * } \\Psi _ { \\alpha } = 0 , \\\\ & \\omega _ { Q , P } : = d \\sigma _ { 1 } ^ { * } Q ^ { a } \\wedge d \\sigma _ { 1 } ^ { * } P _ { a } . \\end{align*}"}
+{"id": "4244.png", "formula": "\\begin{align*} Q ( X , V , \\tau ) = \\left ( \\prod _ { j = 1 } ^ { 2 k } \\frac { x _ j \\theta ' ( 0 , \\tau ) } { \\theta ( x _ j , \\tau ) } \\prod _ { r = 1 } ^ { m _ 0 } \\left ( \\frac { \\theta _ 1 ( u _ r , \\tau ) } { \\theta _ 1 ( 0 , \\tau ) } \\frac { \\theta _ 2 ( u _ r , \\tau ) } { \\theta _ 2 ( 0 , \\tau ) } \\frac { \\theta _ 3 ( u _ r , \\tau ) } { \\theta _ 3 ( 0 , \\tau ) } \\right ) \\right ) ^ { ( 4 k ) } . \\end{align*}"}
+{"id": "6595.png", "formula": "\\begin{align*} I _ 2 = \\int _ 1 ^ { ( 1 / 2 ) \\varepsilon \\log | \\eta | } \\frac { e ^ w } { w } \\ , d w \\leq e ^ { ( 1 / 2 ) \\varepsilon \\log | \\eta | } . \\end{align*}"}
+{"id": "3398.png", "formula": "\\begin{align*} z _ { t } & = \\frac { 1 } { 2 P _ { t } \\eta _ { t } \\lambda _ { t } \\max _ { i \\le t } \\sqrt { 2 L \\Delta _ { i } } + 8 Q _ { t } L \\eta _ { t } ^ { 2 } \\lambda _ { t } ^ { 2 } } \\end{align*}"}
+{"id": "297.png", "formula": "\\begin{align*} K = \\sum _ { l = 1 } ^ n \\# \\left \\{ k \\neq l : a _ { l , k } \\neq 0 \\right \\} \\end{align*}"}
+{"id": "4596.png", "formula": "\\begin{align*} N \\left ( - \\Delta _ { \\Omega } - \\lambda \\right ) \\approx \\frac { 1 } { ( 2 \\pi ) ^ { d } } \\left | \\left \\{ ( p , x ) \\in \\R ^ d \\times \\Omega \\bigm | | p | ^ 2 - \\lambda < 0 \\right \\} \\right | = \\frac { | B _ 1 ( 0 ) | } { ( 2 \\pi ) ^ d } | \\Omega | \\lambda ^ \\frac { d } { 2 } \\end{align*}"}
+{"id": "4150.png", "formula": "\\begin{align*} ( \\sigma ^ 2 _ j / 2 ) \\eta _ j ^ 2 \\psi ^ { ( r ) } ( \\theta ) + ( 1 - w _ { j j } ) \\eta _ j \\bigl ( \\psi ^ { ( r ) } ( \\theta ) - \\psi ^ { ( r ) } _ i ( \\theta ) \\bigr ) = o ( 1 ) , \\end{align*}"}
+{"id": "2562.png", "formula": "\\begin{align*} B u = B \\partial _ x ^ m \\delta _ L ^ v = \\partial _ x ^ m B \\delta _ L ^ v = \\partial _ x ^ m \\delta _ L ^ { B ' v } = \\partial _ x ^ m \\delta _ L ^ { A v } \\ ; . \\end{align*}"}
+{"id": "6592.png", "formula": "\\begin{align*} | \\alpha _ { k } ( p ) ^ { - z } ( p - 1 ) ^ { - z } - p ^ { - z } | = & \\left | z \\int _ { \\alpha _ { k } ( p ) ( p - 1 ) } ^ p { u ^ { - z - 1 } } \\ , d u \\right | \\ll _ k | z | ( p - 1 ) ^ { - \\xi - 1 } . \\end{align*}"}
+{"id": "5398.png", "formula": "\\begin{align*} h _ k ( j _ k ) = c _ k \\ , j _ k + s _ k \\ , \\lambda _ k \\ , 1 \\{ j _ k = 0 \\} - r _ k \\ , \\lambda _ k \\ , 1 \\{ j _ k > 0 \\} . \\end{align*}"}
+{"id": "1849.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\left | \\int ^ s _ 0 \\nabla ^ i _ x \\beta ( t ) d t \\right | ^ 2 d s & \\leq \\int _ 0 ^ 1 s \\int ^ s _ 0 | \\nabla ^ i _ x \\beta ( t ) | ^ 2 d t d s \\\\ & \\leq \\frac { 1 } { 2 } \\left ( \\int ^ 1 _ 0 | \\nabla ^ i _ x \\beta ( t ) | ^ 2 d t - \\int ^ 1 _ 0 s | \\nabla ^ i _ x \\beta ( s ) | ^ 2 d s \\right ) \\\\ & \\leq \\int ^ 1 _ 0 | \\nabla ^ i _ x \\beta ( t ) | ^ 2 d t . \\end{align*}"}
+{"id": "336.png", "formula": "\\begin{align*} J _ 1 \\cap J _ 2 = ( x _ { i _ 1 } , \\dots , x _ { i _ t } ) J _ 1 \\end{align*}"}
+{"id": "2755.png", "formula": "\\begin{align*} & \\Xi ^ { \\alpha } : \\approx 0 , \\\\ & \\dot { \\Xi } ^ { \\alpha } = \\{ \\Xi ^ { \\alpha } , H _ { T } \\} : \\approx 0 , \\end{align*}"}
+{"id": "2499.png", "formula": "\\begin{align*} A _ \\infty ( x ) : = \\sup _ { t \\ge 0 } \\{ A ( t , x ) \\} = \\int _ 0 ^ \\infty ( u \\gamma ( v ) ) ( s , x ) \\ \\mathrm { d } s \\ , , x \\in \\Omega \\ , , \\end{align*}"}
+{"id": "1544.png", "formula": "\\begin{align*} a ( t ) = \\sup _ { | z | \\le t } | D F ( z ) | , A ( t ) = \\int _ 0 ^ t a ( \\tau ) \\ , d \\tau . \\end{align*}"}
+{"id": "134.png", "formula": "\\begin{align*} \\begin{aligned} a ( P _ h \\vec { v } _ h , \\vec { w } _ h ) + b ( \\vec { w } _ h , p _ h ) & = a ( \\vec { v } _ h , \\vec { w } _ h ) , & & \\forall \\vec { w } _ h \\in V _ h , \\\\ b ( P _ h \\vec { v } _ h , q _ h ) & = 0 , & & \\forall q _ h \\in Q _ h . \\end{aligned} \\end{align*}"}
+{"id": "2918.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\ell } ^ { \\lambda , k } : = \\mathbb { P } \\bigg [ \\sup _ { X _ { \\ell - 1 } < x _ i \\leqslant X _ { \\ell } } \\sup _ { \\substack { 0 \\leqslant j \\leqslant J } } \\lambda _ { \\ell } ^ { ( k ) } ( x _ i , y _ j , f ) > \\frac { T ( \\ell ) } { \\ell ^ { K / 2 } \\ell \\log \\ell } \\bigg ] \\end{align*}"}
+{"id": "7614.png", "formula": "\\begin{align*} F ( X , j ) = \\{ ( x _ 1 , \\cdots , x _ j ) \\in X ^ j : x _ i \\not = x _ k \\mbox { i f } i \\not = k \\} . \\end{align*}"}
+{"id": "2357.png", "formula": "\\begin{align*} & \\int _ { 3 } ^ { q } \\frac { 1 } { r ^ { 2 } } \\ln \\frac { 4 \\pi ( r - 2 ) ( 1 - \\tau ) } { r ^ { 2 } } d r \\\\ & = \\int _ { 3 } ^ { q } \\frac { \\ln 4 \\pi ( 1 - \\tau ) } { r ^ { 2 } } d r + \\int _ { 3 } ^ { q } \\frac { \\ln ( r - 2 ) } { r ^ { 2 } } d r - \\int _ { 3 } ^ { q } \\frac { \\ln r ^ { 2 } } { r ^ { 2 } } d r . \\end{align*}"}
+{"id": "8505.png", "formula": "\\begin{align*} | F _ n ( a ) | \\leq | F ( w _ n ) ( F _ n ( a ) ) | = | F ( v _ n ) ( a ) | \\leq N . \\end{align*}"}
+{"id": "5512.png", "formula": "\\begin{align*} ( c e ) ^ { k } [ 1 - ( c e ) ^ { k } ] = c ^ { k } e ^ { k } [ ( 1 - e ) ^ { k } d ] = c ^ { k } d [ ( 1 - e ) ^ { k } e ^ { k } ] = 0 . \\end{align*}"}
+{"id": "1713.png", "formula": "\\begin{align*} M _ 1 : = \\left \\{ z \\in \\mathbb { C } ^ { n _ 1 + 1 } \\ , \\left | \\ , \\Re ( z _ 0 ) = \\Phi _ 1 \\left ( z _ 1 , \\ldots , z _ { n _ 1 } , \\overline { z _ 1 } , \\ldots , \\overline { z _ { n _ 1 } } \\right ) \\right . \\right \\} \\end{align*}"}
+{"id": "2482.png", "formula": "\\begin{align*} \\left \\| a ^ { \\frac { \\beta } { \\beta - \\alpha } } \\right \\| _ 2 = \\left \\| a \\right \\| _ { { \\frac { 2 \\beta } { \\beta - \\alpha } } } ^ { { \\frac { \\beta } { \\beta - \\alpha } } } \\leq \\left ( \\frac { 2 \\beta } { \\beta - \\alpha } \\| a \\| _ { \\psi _ 1 } \\right ) ^ { \\frac { \\beta } { \\beta - \\alpha } } . \\end{align*}"}
+{"id": "5137.png", "formula": "\\begin{align*} & \\frac { \\partial A } { \\partial q _ { j } } = \\frac { \\partial A ( p _ { j } , q _ { j } ) } { \\partial q _ { j } } \\equiv \\frac { \\partial A j } { \\partial q _ { j } } \\\\ & \\frac { \\partial B } { \\partial q _ { j } } = \\frac { \\partial B ( p _ { j } , q _ { j } ) } { \\partial q _ { j } } \\equiv \\frac { \\partial B j } { \\partial q _ { j } } \\end{align*}"}
+{"id": "6606.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { b - i \\tau } ^ { b + i \\tau } \\left ( \\sum _ { n \\le x } \\left ( \\frac { \\Phi _ k ( n ) } { n ^ \\beta } \\right ) ^ { - z } \\right ) \\frac { y ^ z } { z } \\ , d z & = \\frac { x } { 2 \\pi i } \\int _ { b - i \\tau } ^ { b + i \\tau } R _ { k , \\beta } ( z ) \\frac { ( \\alpha x ^ \\delta ) ^ { z } } { z ( 1 - z + \\delta z ) } d z \\\\ & + O \\left ( x ^ { 1 - \\varepsilon + \\delta \\varepsilon } y ^ { \\varepsilon } \\exp { ( - \\sqrt { f _ k ^ \\prime \\log x \\log \\log x } } ) \\right ) . \\end{align*}"}
+{"id": "5999.png", "formula": "\\begin{align*} ( w + h ) \\psi \\Delta \\psi \\geq & - \\frac { 1 } { 4 n } ( w + h ) ^ 2 \\psi ^ 2 - C ( n ) ( \\Delta \\psi ) ^ 2 \\\\ = & - \\frac { 1 } { 4 n } ( w + h ) ^ 2 \\psi ^ 2 - C ( n ) [ \\psi '' + \\psi ' \\Delta r _ { \\partial M } ] ^ 2 \\\\ \\geq & - \\frac { 1 } { 4 n } ( w + h ) ^ 2 \\psi ^ 2 - C ( n ) [ | \\psi '' | + | \\psi ' | \\frac { n - 1 } { r _ { \\partial M } } ( 1 + \\sqrt { K } ) r _ { \\partial M } ] ^ 2 \\\\ \\geq & - \\frac { 1 } { 4 n } ( w + h ) ^ 2 \\psi ^ 2 - C ( n ) \\left ( \\frac { 1 } { R ^ 4 } + K ^ 2 \\right ) . \\end{align*}"}
+{"id": "453.png", "formula": "\\begin{align*} t _ { q + 1 } = t _ q + \\Delta t _ q , \\mbox { a n d } x _ { i _ m } ^ * ( t _ { q + 1 } ) - x _ { m + 1 } ^ * ( t _ { q + 1 } ) > 0 . \\end{align*}"}
+{"id": "480.png", "formula": "\\begin{align*} ( x _ 1 ^ { e _ 1 } \\cdots x _ N ^ { e _ N } ) \\divideontimes ( x _ 1 ^ { e ' _ 1 } \\cdots x _ { N ' } ^ { e ' _ { N ' } } ) = \\prod _ { 1 \\leq n \\leq N , 1 \\leq n ' \\leq N ' } { x _ n ^ { e _ n } \\divideontimes x _ { n ' } ^ { e _ { n ' } } } \\end{align*}"}
+{"id": "461.png", "formula": "\\begin{align*} \\phi ( [ g ] [ h ] ) & = \\phi ( [ r ( g ) ] [ g ] [ h ] ) = \\phi ( [ g g ^ { - 1 } ] [ g ] [ h ] ) = \\phi ( [ g ] [ g ^ { - 1 } ] [ g ] [ h ] ) = \\phi ( [ g ] [ g ^ { - 1 } ] [ g h ] ) \\\\ & = \\pi ( g ) \\pi ( g ^ { - 1 } ) \\pi ( g h ) = \\pi ( g ) \\pi ( g ^ { - 1 } ) \\pi ( g ) \\pi ( h ) = \\pi ( g ) \\pi ( h ) = \\phi ( [ g ] ) \\phi ( [ h ] ) , \\end{align*}"}
+{"id": "4616.png", "formula": "\\begin{align*} \\Omega _ { j , k } : = \\left \\{ \\left ( x ' , x _ d \\right ) \\in Q ( j , k ) \\times \\R \\bigm | c ( j ) < x _ d < f ( x ' ) \\right \\} , \\end{align*}"}
+{"id": "3005.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\overline { \\mathrm { s c a l } } & = & \\mathrm { s c a l } + 2 n [ \\beta ^ 2 + ( 1 - 2 n ) \\alpha ^ 2 ] . \\\\ \\end{array} \\end{align*}"}
+{"id": "5758.png", "formula": "\\begin{align*} | \\bar { z } ( t ) | ^ 2 + \\sum _ { i \\in I _ 1 } \\left ( | \\xi _ { i , 1 } ( t ) | ^ 2 + | \\xi _ { i , 2 } ( t ) | ^ 2 \\right ) + \\sum _ { i \\neq 0 } | \\xi _ i ( t ) | ^ 2 = o ( 1 ) | z ( t ) | ^ 2 . \\end{align*}"}
+{"id": "3935.png", "formula": "\\begin{align*} T _ n = \\left \\{ \\begin{array} { l l } I _ 3 + \\lambda _ n \\begin{pmatrix} 0 & \\kappa & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} , & 1 \\leq n \\leq N , \\\\ I _ 3 , & n \\geq N + 1 \\end{array} \\right . , \\end{align*}"}
+{"id": "8830.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\rightarrow \\infty } \\sup \\left \\{ | | Q ^ n [ \\varphi ] ( \\cdot , x ) | | : x \\geq n ( c ^ * _ { L _ { k _ 0 } } + \\frac { \\varepsilon } { 3 } ) \\mbox { o r } x \\leq - n ( c ^ * _ { \\mathcal { S } \\circ \\hat { L } _ { k _ 0 } \\circ \\mathcal { S } } + \\frac { \\varepsilon } { 3 } ) \\right \\} = 0 . \\end{align*}"}
+{"id": "7604.png", "formula": "\\begin{align*} K _ { 1 } ( \\epsilon ) & = \\epsilon ^ { N - 4 } R ^ { 2 } ( 4 - N ) ^ { 2 } \\omega \\int _ { 0 } ^ { 2 } \\big [ \\frac { 4 r ^ { N - 1 } ( | \\varphi ( r ) | ^ { 2 } - 1 ) } { ( \\epsilon ^ 2 + | r | ^ { 2 } ) ^ { N - 2 } } + \\frac { 4 r ^ { N - 1 } } { ( \\epsilon ^ 2 + | r | ^ { 2 } ) ^ { N - 2 } } \\big ] d r \\\\ & = L _ { 1 } ( \\epsilon ) + L _ { 2 } ( \\epsilon ) . \\\\ \\end{align*}"}
+{"id": "4747.png", "formula": "\\begin{align*} \\omega ^ S ( k , b , T ) = 2 ^ { 2 k + 2 } T ^ 2 b ^ 2 \\end{align*}"}
+{"id": "6336.png", "formula": "\\begin{align*} g ( p , q ) = \\log \\left ( 1 - e ^ { - \\sqrt { p ^ 4 + q p ^ 2 } } \\right ) . \\end{align*}"}
+{"id": "3202.png", "formula": "\\begin{align*} \\partial _ t ^ \\alpha h ( t ) = \\frac { d } { d t } J ^ { \\alpha - 1 } h ( t ) , \\ , \\ , J ^ \\sigma h ( t ) = \\frac { 1 } { \\Gamma ( - \\sigma ) } \\int \\limits _ 0 ^ t \\frac { h ( \\xi ) } { ( t - \\xi ) ^ { \\sigma + 1 } } d \\xi , t > 0 , \\end{align*}"}
+{"id": "2037.png", "formula": "\\begin{align*} P _ T \\ , \\ , = \\ , \\ , \\begin{bmatrix} C _ 5 & C _ 4 ' & C _ 3 '' \\\\ C _ 4 & C _ 3 ' & C _ 2 ' \\\\ C _ 3 & C _ 2 & C _ 1 \\end{bmatrix} . \\end{align*}"}
+{"id": "213.png", "formula": "\\begin{align*} \\mathcal { L } _ \\Gamma \\theta _ L = d L , \\end{align*}"}
+{"id": "4846.png", "formula": "\\begin{align*} \\infty & = \\int _ { B _ a } \\liminf g ( y + k ) d y \\\\ & \\le \\liminf \\int _ { B _ a } [ g ( y + k ) - g ( y ) ] d y + \\int _ { B _ a } g ( y ) d y \\\\ & \\le w _ 1 \\Big [ \\mathcal { E } ( E ) - \\mathcal { E } ( B _ a ) \\Big ] ^ { 1 / 2 } + | | g | | _ { L ^ 1 ( B _ a ) } \\\\ & \\le w _ 1 e ^ { 1 / 2 } + | | g | | _ { L ^ 1 ( B _ a ) } \\end{align*}"}
+{"id": "2488.png", "formula": "\\begin{align*} \\begin{aligned} \\lambda _ { | I | } ( C ( \\widehat { A } , \\lambda ) _ I ^ T C ( \\widehat { A } , \\lambda ) _ I ) & \\geq \\lambda _ { | I | } ( C ( A , \\lambda ) _ I ^ T C ( A , \\lambda ) _ I ) - \\frac { 1 } { \\epsilon H } \\\\ & \\geq \\left ( 1 - \\frac { 1 } { \\epsilon } \\right ) \\lambda _ { | I | } ( C ( A , \\lambda ) _ I ^ T C ( A , \\lambda ) _ I ) , \\end{aligned} \\end{align*}"}
+{"id": "2203.png", "formula": "\\begin{align*} \\intop _ \\Omega ( \\psi _ { 1 , z r } ^ 2 + \\psi _ { 1 , z z } ^ 2 ) d x - \\intop _ \\Omega ( \\psi _ { 1 , z } ^ 2 ) _ { , r } d r d z = - \\intop _ \\Omega \\omega _ 1 \\psi _ { 1 , z z } d x , \\end{align*}"}
+{"id": "4682.png", "formula": "\\begin{align*} \\mathbf { B } _ + ( D ) = \\mathbf { B } _ + ( P ( D ) ) \\end{align*}"}
+{"id": "6644.png", "formula": "\\begin{align*} ( H _ { v , y } \\psi ) _ { n } = \\psi _ { n + 1 } + \\psi _ { n - 1 } + v ( x + \\mathrm { i } y + n \\alpha ) \\psi _ { n } , \\end{align*}"}
+{"id": "1686.png", "formula": "\\begin{align*} s ( [ \\varphi ] ) ( \\mathbf { p } ) = \\varphi ( \\mathbf { p } ) ; \\end{align*}"}
+{"id": "3208.png", "formula": "\\begin{align*} | | h | | ^ 2 _ \\tau = \\sum \\limits _ { k = 1 } ^ \\infty \\lambda _ k ^ { 2 \\tau } | h _ k | ^ 2 = | | A ^ \\tau h | | ^ 2 , \\end{align*}"}
+{"id": "6260.png", "formula": "\\begin{align*} \\lim _ { t \\to T _ { - } } \\Phi _ { 2 } g ( t , x ) & = \\lim _ { t \\to T _ { - } } \\int _ { y = 0 } ^ { L } g ( T , y ) \\frac { e ^ { - i ( x - y ) ^ { 2 } / ( 4 ( T - t ) ) } } { - i \\sqrt { T - t } } d y \\\\ [ 1 0 p t ] & = i \\lim _ { t \\to T _ { - } } \\int _ { y = 0 } ^ { L } g ( T , y ) \\frac { \\bar { e ^ { i ( x - y ) ^ { 2 } / ( 4 ( T - t ) ) } } } { \\sqrt { T - t } } d y = i \\frac { \\sqrt { \\pi } } { 2 } e ^ { - i \\pi / 4 } g ( T , x ) = \\lim _ { t \\to T _ { + } } \\Phi _ { 2 } g ( t , x ) , \\end{align*}"}
+{"id": "640.png", "formula": "\\begin{align*} I _ { r + 1 , r } ( x ) & = - \\sinh ( x ) \\biggl ( \\frac { 1 } { r - 2 } \\tanh ^ { r - 2 } ( x ) + \\frac { r - 1 } { ( r - 2 ) ( r - 4 ) } \\tanh ^ { r - 4 } ( x ) \\\\ & + \\frac { ( r - 1 ) ( r - 3 ) } { ( r - 2 ) ( r - 4 ) ( r - 6 ) } \\tanh ^ { r - 6 } ( x ) + \\dots + \\frac { ( r - 1 ) ! ! } { 2 ( r - 2 ) ! ! } \\tanh ( x ) \\biggr ) \\\\ & + \\frac { ( r - 1 ) ! ! } { ( r - 2 ) ! ! } I _ { 2 , 1 } ( x ) . \\end{align*}"}
+{"id": "9198.png", "formula": "\\begin{align*} q \\ell ^ \\pm _ { 1 , n + 1 } ( z ) \\ell ^ - _ { n + 1 , n + 1 } [ 0 ] = q \\ell ^ - _ { n + 1 , n + 1 } [ 0 ] \\ell ^ \\pm _ { 1 , n + 1 } ( z ) , \\end{align*}"}
+{"id": "8106.png", "formula": "\\begin{align*} X = u _ k + s _ k . \\end{align*}"}
+{"id": "7803.png", "formula": "\\begin{align*} \\boldsymbol { g } _ { t } = \\boldsymbol { h } _ { \\mathrm { d } , t } + \\left ( \\boldsymbol { G } \\right ) ^ { \\mathrm { H } } \\boldsymbol { \\Omega } _ t \\boldsymbol { \\Theta } \\boldsymbol { h } _ { \\mathrm { s } , t } , \\forall t \\in \\mathcal { T } , \\end{align*}"}
+{"id": "3565.png", "formula": "\\begin{align*} T f = \\alpha \\tau _ 0 f \\circ \\tau , \\end{align*}"}
+{"id": "687.png", "formula": "\\begin{align*} \\ell _ 1 + \\ell _ 3 & = 4 \\left ( ( \\alpha _ 1 + \\gamma _ 1 ) ^ 2 + ( \\alpha _ 2 + \\gamma _ 2 ) ^ 2 + ( \\beta _ 1 + \\delta _ 1 ) ^ 2 + ( \\beta _ 2 + \\delta _ 2 ) ^ 2 \\right ) , \\\\ \\ell _ 2 + \\ell _ 4 & = 4 \\left ( ( \\alpha _ 1 - \\gamma _ 1 ) ^ 2 + ( \\alpha _ 2 - \\gamma _ 2 ) ^ 2 + ( \\beta _ 1 - \\delta _ 1 ) ^ 2 + ( \\beta _ 2 - \\delta _ 2 ) ^ 2 \\right ) . \\end{align*}"}
+{"id": "7502.png", "formula": "\\begin{align*} C ( ( x _ 0 , 0 ) , t _ 0 , r ) : = \\displaystyle { \\bigcup _ { t \\in ( t _ 0 - r ^ 2 , t _ 0 ) } } \\left ( B _ { g _ N ( t ) } ( x _ 0 , r ) \\times ( - r _ 0 , r _ 0 ) \\right ) \\times \\{ t \\} , \\end{align*}"}
+{"id": "3124.png", "formula": "\\begin{align*} u ^ - _ { \\psi ^ * } ( s ) = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ v _ { 2 , \\ , 1 } ( s ) & 1 & 0 \\\\ v _ { 3 , \\ , 1 } ( s ) & v _ { 3 , \\ , 2 } ( s ) & 1 \\end{array} \\right ) ( \\ v _ { 2 , \\ , 1 } ( S ) , v _ { 3 , \\ , 1 } ( S ) , v _ { 3 , \\ , 2 } ( S ) \\in k [ S ] \\ , ) . \\end{align*}"}
+{"id": "2721.png", "formula": "\\begin{align*} X _ { T } : = X _ { t } + \\zeta ^ { \\alpha } Y ^ { ( 1 ) } _ { \\alpha } + \\xi ^ { \\alpha } Z _ { \\alpha } , \\end{align*}"}
+{"id": "6542.png", "formula": "\\begin{align*} \\mathbb { E } | S _ N - S _ { N , 1 } | ^ 2 & = \\sum ^ N _ { n = 1 } \\sum ^ { m _ 0 } _ { j = 1 } \\mathbb { E } [ \\mathcal { P } _ { n , n - j } ^ 2 ] + \\sum ^ N _ { n = 1 } \\sum ^ { m _ 0 } _ { i , j = 1 } \\mathbb { E } [ \\mathcal { P } _ { n , n - j } \\mathcal { P } _ { n - j + i , n - j } ] \\\\ & \\leq \\sum ^ N _ { n = 1 } \\mathbb { E } [ K ^ 2 ( X _ n ) ] + \\sum ^ N _ { n = 1 } \\sum ^ { m _ 0 } _ { i , j = 1 } \\left ( \\mathbb { E } [ \\mathcal { P } ^ 2 _ { n , n - j } ] + \\mathbb { E } [ \\mathcal { P } ^ 2 _ { n - j + i , n - j } ] \\right ) \\\\ & \\leq c _ 1 N . \\end{align*}"}
+{"id": "2314.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 3 } } u ^ { c , \\gamma } \\cdot \\nabla \\varphi = 0 . \\end{align*}"}
+{"id": "9076.png", "formula": "\\begin{align*} X ( t + 1 ) = A ^ { ( r ) } X ( t ) + b ^ { ( r ) } ( X ( t ) ) . \\end{align*}"}
+{"id": "6494.png", "formula": "\\begin{align*} \\begin{aligned} ( 2 , 1 - 6 m ) & \\mapsto ( 1 , - 6 m ) , \\ \\qquad ( 1 , - 6 m ) \\mapsto ( 1 , - 2 - 6 m ) , \\allowdisplaybreaks \\\\ ( 2 , - 1 - 6 m ) & \\mapsto ( 2 , - 1 - 6 m ) , ( 1 , - 2 - 6 m ) \\mapsto ( 2 , - 3 - 6 m ) , \\allowdisplaybreaks \\\\ ( 2 , - 3 - 6 m ) & \\mapsto ( 2 , - 5 - 6 m ) , ( 1 , - 4 - 6 m ) \\mapsto ( 1 , - 4 - 6 m ) . \\end{aligned} \\end{align*}"}
+{"id": "130.png", "formula": "\\begin{align*} \\widehat { \\vec { u } } _ { \\infty } = 2 | \\xi | ^ { - 2 } ( I - \\Pi _ \\xi ) \\widehat { \\vec { f } } , \\widehat { p } = | \\xi | ^ { - 2 } \\xi ^ * \\widehat { \\vec { f } } . \\end{align*}"}
+{"id": "5599.png", "formula": "\\begin{align*} B ^ { ( \\ell ) } = & \\underline { B } ^ { ( \\ell ) } + H B ^ { ( \\ell - 1 ) } + \\sum _ { t = 1 } ^ { \\ell - 1 } \\sum _ { k = 1 } ^ r \\nu _ k ^ 2 \\underline { B } ^ { ( t - 1 ) } \\chi _ k \\check { \\chi } _ k ^ * B ^ { ( \\ell - t - 1 ) } \\\\ & + \\sum _ { t = 1 } ^ { \\ell - 1 } \\underline { B } ^ { ( t - 1 ) } \\tilde { H } B ^ { ( \\ell - t - 1 ) } + \\underline { B } ^ { ( \\ell - 1 ) } H ^ { ( 1 ) } - \\sum _ { t = 0 } ^ { \\ell } R _ t ^ { ( \\ell ) } . \\end{align*}"}
+{"id": "237.png", "formula": "\\begin{align*} \\frac { d ^ 2 \\bar { x } } { d \\bar { t } ^ 2 } + \\bar { A } ( \\bar { x } ) \\frac { d \\bar x } { d t } + \\bar { b } ( \\bar { x } ) = 0 , \\end{align*}"}
+{"id": "7957.png", "formula": "\\begin{align*} \\dot { \\tilde { \\mathcal { F } } } ( \\eta , \\phi _ { \\partial } , \\Sigma ) = \\{ \\tilde { \\mathcal { F } } , \\tilde { H } \\} ( \\eta , \\phi _ { \\partial } , \\Sigma ) . \\end{align*}"}
+{"id": "3349.png", "formula": "\\begin{align*} \\Psi ( x , y ) : = \\frac { \\phi ( y - h ( x ) ) } { \\phi ( y ) } . \\end{align*}"}
+{"id": "8451.png", "formula": "\\begin{align*} [ v ] _ { W ^ { s , p } ( K _ T ) } : = \\left ( \\ , \\ , \\iint _ { K _ T \\times K _ T } \\frac { | v ( x , t ) - v ( x ^ \\prime , t ^ \\prime ) | ^ p } { \\left ( \\sqrt { | x - x ^ \\prime | ^ 2 + ( t - t ^ \\prime ) ^ 2 } \\right ) ^ { n + 1 + s p } } \\ , d x d t d x ' d t ' \\right ) ^ \\frac { 1 } { p } , \\end{align*}"}
+{"id": "8978.png", "formula": "\\begin{align*} F ( O ^ t M O ) = F ( M ) \\end{align*}"}
+{"id": "1180.png", "formula": "\\begin{align*} \\left | \\partial ^ \\beta f ( y ) - \\partial ^ \\beta f ( z ) \\right | & \\leq \\sum _ { k = 1 } ^ 7 \\left | \\partial ^ \\beta f \\left ( r e ^ { i \\phi \\frac { k } { 7 } } \\right ) - \\partial ^ \\beta f \\left ( r e ^ { i \\phi \\frac { k - 1 } { 7 } } \\right ) \\right | \\\\ & \\lesssim \\sum _ { k = 1 } ^ 7 \\left ( \\frac { r } { 2 } \\right ) ^ { \\varepsilon } r ^ { - n - N - \\varepsilon } \\sim | y | ^ { - n - N } , \\end{align*}"}
+{"id": "2466.png", "formula": "\\begin{align*} \\begin{cases} \\lambda _ { \\tau } = c \\lambda ^ { 2 } - \\gamma \\lambda + k \\lambda x ^ { 2 } - \\delta p \\lambda ^ { 2 } x , \\\\ x _ { \\tau } = p ^ { - 1 } x + c \\lambda x + k x ^ { 3 } - \\delta p \\lambda x ^ { 2 } , \\end{cases} \\end{align*}"}
+{"id": "7670.png", "formula": "\\begin{align*} K ( n , u ) = \\frac { C _ a 7 2 \\sqrt { 2 } \\norm { F } _ { \\infty } } { \\eta } | g | \\sum _ { m \\in \\in \\Lambda _ L } e ^ { - \\gamma _ a d ( n , m ) - 2 \\nu d ( m , u ) } \\end{align*}"}
+{"id": "2838.png", "formula": "\\begin{align*} \\delta Q ^ { 1 } ( t _ { 2 } ) = \\delta Q ^ { 1 } ( t _ { 1 } ) = 0 . \\end{align*}"}
+{"id": "1006.png", "formula": "\\begin{align*} | Y _ { t \\wedge \\tau _ D } | - | Y _ 0 | & \\ge \\int ^ { t \\wedge \\tau _ D } _ 0 \\mbox { s g n } ( Y _ { s - } ) \\ , d Y _ s = - \\int ^ { t \\wedge \\tau _ D } _ 0 \\mbox { s g n } ( Y _ s ) ( f ( X _ s , Y _ s ) - f ( X _ s , 0 ) ) \\ , d s \\\\ & \\quad + \\int ^ { t \\wedge \\tau _ D } _ 0 \\mbox { s g n } ( Y _ s ) ( - f ( X _ s , 0 ) \\ , d s + d A ^ { \\mu } _ s ) + \\int ^ { t \\wedge \\tau _ D } _ 0 \\mbox { s g n } ( Y _ { s - } ) \\ , d M ^ x _ s , \\end{align*}"}
+{"id": "2411.png", "formula": "\\begin{align*} v _ 2 ( ( k - 1 ) ! ( n - k ) ! ) & = k - 1 - \\operatorname { s o d } _ 2 ( k - 1 ) + n - k - \\operatorname { s o d } _ 2 ( n - k ) \\\\ & = n - 1 - \\operatorname { s o d } _ 2 ( k - 1 ) - \\operatorname { s o d } _ 2 ( n - k ) . \\end{align*}"}
+{"id": "4992.png", "formula": "\\begin{align*} \\begin{pmatrix} p _ 1 ( \\nu _ j ) a ( \\nu _ j ) & p _ 2 ( \\nu _ j ) a ( \\nu _ j ) & \\dots & p _ N ( \\nu _ j ) a ( \\nu _ j ) \\\\ p _ 1 ( \\nu _ j ) d ( \\nu _ j - 1 ) & p _ 2 ( \\nu _ j ) d ( \\nu _ j - 1 ) & \\dots & p _ N ( \\nu _ j ) d ( \\nu _ j - 1 ) \\\\ \\hdotsfor { 4 } \\\\ \\hdotsfor { 4 } \\end{pmatrix} , \\end{align*}"}
+{"id": "3051.png", "formula": "\\begin{align*} J = \\frac { \\partial ( q , r ) } { \\partial ( x , s ) } = \\left ( \\begin{array} { c c } A & B \\\\ - ( B ^ { - 1 } ) ^ 2 A & B ^ { - 1 } \\end{array} \\right ) \\ , , J ^ { - 1 } = \\frac { 1 } { 2 } \\left ( \\begin{array} { c c } A ^ { - 1 } & - A ^ { - 1 } B ^ 2 \\\\ B ^ { - 1 } & B \\end{array} \\right ) \\ , , \\end{align*}"}
+{"id": "8203.png", "formula": "\\begin{align*} & H ( Y _ i | Y ^ { i - 1 } , { \\bf S } ) \\\\ \\leq & \\sum _ { y ^ { i - 1 } } \\prod _ { j = 1 } ^ i \\min ( \\frac { 1 } { 2 } , \\frac { B _ { a , j } } { | \\mathcal S ( y ^ { j - 1 } ) | } ) H ( \\min ( \\frac { 1 } { 2 } , \\frac { B _ { a , i - 1 } } { | \\mathcal S ( y ^ { i - 2 } ) | } ) ) . \\end{align*}"}
+{"id": "6074.png", "formula": "\\begin{align*} \\exists \\lambda _ 0 \\in \\mathbb { T } _ - \\cup \\mathbb { T } _ + , \\ f ( \\lambda _ 0 ) = 0 \\ \\Longrightarrow \\ | \\phi _ { 2 , 2 } | < | \\omega _ { 2 , 2 } | . \\end{align*}"}
+{"id": "2109.png", "formula": "\\begin{align*} U _ k ( s , x ) = U ( 0 , x ) - \\alpha \\int _ 0 ^ s \\int _ 0 ^ 1 A ( x , y ) U _ { k - 1 } ( u , y ) \\mathrm { d } y \\mathrm { d } u + \\alpha \\beta \\int _ 0 ^ s \\mathrm { d } \\xi _ 2 ( u , x ) + \\alpha \\zeta \\int _ 0 ^ s \\mathrm { d } \\xi _ 3 ( u , x ) . \\end{align*}"}
+{"id": "7798.png", "formula": "\\begin{align*} \\begin{aligned} & ( A + B K ^ { ( i ) } ) ^ { \\top } P ^ { ( i + 1 ) } + P ^ { ( i + 1 ) } ( A + B K ^ { ( i ) } ) + ( C + D K ^ { ( i ) } ) ^ { \\top } P ^ { ( i + 1 ) } ( C + D K ^ { ( i ) } ) \\\\ & ~ ~ ~ = - [ K ^ { ( i ) \\top } R K ^ { ( i ) } + S ^ { \\top } K ^ { ( i ) } + K ^ { ( i ) \\top } S + Q ] . \\end{aligned} \\end{align*}"}
+{"id": "6829.png", "formula": "\\begin{align*} h _ { S O ( N ) } ( g ) = \\sum _ { j = 1 } ^ M \\sum _ { i = 1 } ^ Q c _ { i j } & e ^ { i k _ { i j } ^ 1 \\phi _ 1 } \\cos ^ { m _ { i j } ^ 1 } ( \\phi _ 2 ) \\sin ^ { n _ { i j } ^ 1 } ( \\phi _ 2 ) \\cdots \\cos ^ { m _ { i j } ^ { N - 1 } } ( \\phi _ { N - 1 } ) \\sin ^ { n _ { i j } ^ { N - 1 } } ( \\phi _ { N - 1 } ) \\\\ & \\cdot ( h _ { S O ( N - 1 ) } ) _ { i j } ( g _ { S O ( N - 1 ) } ) , \\end{align*}"}
+{"id": "3914.png", "formula": "\\begin{align*} ( \\cos z + i \\sin z ) ^ n = \\cos n z + i \\sin n z , \\ ( \\cos z - i \\sin z ) ^ n = \\cos n z - i \\sin n z . \\end{align*}"}
+{"id": "2155.png", "formula": "\\begin{align*} | A A | \\leq K | A | \\implies | \\{ ( a , b ) \\in A \\times A : a - b = t \\} | \\ll K ^ { C } | A | ^ { \\frac { 2 } { 3 } - c } . \\end{align*}"}
+{"id": "2937.png", "formula": "\\begin{align*} v _ { \\{ i , \\bar { i } , 2 n + 1 \\} } = - \\alpha _ i i \\in \\{ 1 , \\ldots , n \\} . \\end{align*}"}
+{"id": "2068.png", "formula": "\\begin{align*} & - \\alpha ^ { - 1 } \\big ( \\int _ 0 ^ s U ( u , x ) f ' ( u ) \\mathrm { d } u + f ( s ) U ( s , x ) - f ( 0 ) U ( 0 , x ) \\big ) \\\\ & = - \\int _ 0 ^ s \\int _ 0 ^ 1 f ( u ) A ( x , y ) U ( u , y ) \\mathrm { d } y \\mathrm { d } u + \\beta \\int _ 0 ^ s f ( u ) \\mathrm { d } \\xi _ 2 ( u , x ) . \\end{align*}"}
+{"id": "1856.png", "formula": "\\begin{align*} r ^ { - 3 } \\| | F _ A \\| ^ 2 _ { L ^ 2 ( B _ r ( x ) ) } & \\leq ( r + 1 / 2 ) ^ { - 3 } \\| F _ A \\| ^ 2 _ { L ^ 2 ( B _ { r + 1 / 2 } ( x ) ) } + c o n s t . \\| \\pi _ 7 ( F _ A ) \\| ^ 2 _ { L ^ { 7 / 2 } ( B _ { r + 1 / 2 } ( x ) ) } \\\\ & \\leq c o n s t . \\left ( \\| F _ A \\| _ { L ^ 2 ( \\mathbb { B } ) } + \\| \\pi _ 7 ( F _ A ) \\| _ { L ^ { 7 / 2 } ( \\mathbb { B } ) } \\right ) ^ 2 , \\end{align*}"}
+{"id": "4019.png", "formula": "\\begin{align*} \\begin{cases} x _ { 1 } ^ { \\left ( t + 1 \\right ) } & = \\ ; \\delta _ { 1 } x _ { 2 } ^ { \\left ( t \\right ) } y ^ { \\left ( t \\right ) } \\medskip \\\\ x _ { 2 } ^ { \\left ( t + 1 \\right ) } & = \\ ; \\gamma _ { 2 } x _ { 1 } ^ { \\left ( t \\right ) } y ^ { \\left ( t \\right ) } \\medskip \\\\ y ^ { \\left ( t + 1 \\right ) } & = \\ ; \\bigl ( \\gamma x _ { 1 } ^ { \\left ( t \\right ) } + \\delta x _ { 2 } ^ { \\left ( t \\right ) } \\bigr ) y ^ { \\left ( t \\right ) } . \\end{cases} \\end{align*}"}
+{"id": "6919.png", "formula": "\\begin{align*} \\forall n \\in \\N , \\forall j \\in \\Z , u ^ { n + 1 } _ j = \\sum _ { k = - r } ^ p a _ k u ^ n _ { j + k } . \\end{align*}"}
+{"id": "1260.png", "formula": "\\begin{align*} - q ^ { ( n - 1 ) ^ 2 - { n + 1 \\choose 2 } } & = - q ^ { 1 - n } q ^ { \\frac { n ( n - 3 ) } { 2 } } \\\\ & \\equiv - q ^ { 1 - n } \\left ( 1 - \\frac { ( n - 3 ) ( 1 - q ^ n ) } { 2 } \\right ) \\\\ & \\equiv - q ^ { 1 - n } + \\frac { ( n - 3 ) q ( 1 - q ^ n ) } { 2 } \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "4644.png", "formula": "\\begin{align*} B ( N ) & = \\sum _ { 1 \\leq n \\leq N } 2 ^ { \\omega ( n ) } \\cdot \\lambda _ \\mathcal { G } ( n ) = \\sum _ { 1 \\leq n \\leq N } \\sum _ { d | n } \\mu ^ 2 _ { \\mathcal { G } } ( d ) \\cdot \\lambda _ \\mathcal { G } ( n ) \\\\ & \\stackrel { ( * * ) } { = } \\sum _ { 1 \\leq n \\leq N } \\sum _ { d | n } \\mu ^ 2 _ { \\mathcal { G } } ( d ) \\cdot \\lambda _ \\mathcal { G } \\left ( \\frac { n } { d } \\right ) = \\sum _ { 1 \\leq n \\leq N } \\left ( \\mu ^ 2 _ { \\mathcal { G } } * \\lambda _ \\mathcal { G } \\right ) ( n ) \\end{align*}"}
+{"id": "4918.png", "formula": "\\begin{align*} \\begin{gathered} \\mathcal { C O } ^ 0 : Q H ( X , \\omega ) \\to H F ( L ) , \\\\ \\mathcal { O C } ^ 0 : H F ( L ) \\to Q H ( X , \\omega ) , \\end{gathered} \\end{align*}"}
+{"id": "7947.png", "formula": "\\begin{align*} \\tilde { \\mathcal { F } } ( \\eta , \\phi _ { \\partial } , \\Sigma ) = \\mathcal { F } ( v , \\Sigma ) , \\end{align*}"}
+{"id": "4569.png", "formula": "\\begin{align*} \\nu ( T , x ) = \\left \\{ \\begin{aligned} 0 , \\ \\\\ 2 , \\ \\end{aligned} \\right . \\end{align*}"}
+{"id": "7674.png", "formula": "\\begin{align*} r _ L ( u , v ) = \\frac { 1 } { 2 \\pi i } \\int ^ { \\infty } _ { - \\infty } r _ { L } ( u , v ; t ) f ( t ) \\ , d t \\end{align*}"}
+{"id": "7828.png", "formula": "\\begin{align*} \\boldsymbol { y } = \\underbrace { [ \\boldsymbol { H } _ 1 \\ \\boldsymbol { H } _ 2 ] } _ { \\boldsymbol { H } } \\boldsymbol { s } + \\boldsymbol { z } , \\end{align*}"}
+{"id": "6318.png", "formula": "\\begin{align*} [ a ^ * _ x , a ^ * _ y ] = [ a _ x , a _ y ] = 0 , [ a _ x , a ^ * _ y ] = \\delta _ { x , y } , \\end{align*}"}
+{"id": "4962.png", "formula": "\\begin{align*} Z _ N = \\frac { \\prod _ { j , k = 1 } ^ { N } a ( \\lambda _ j , \\nu _ k ) } { v _ N ( \\{ \\lambda \\} ) } \\det \\left [ \\lambda _ j ^ { k - 1 } - ( \\lambda _ j + 1 ) ^ { k - 1 } \\prod _ { l = 1 } ^ N \\frac { b ( \\lambda _ j , \\nu _ l ) } { a ( \\lambda _ j , \\nu _ l ) } \\right ] _ { j , k = 1 , \\ldots , N } . \\end{align*}"}
+{"id": "6506.png", "formula": "\\begin{align*} L ^ 2 _ s ( \\mathbb X ^ j ) : = \\left \\{ f \\in L ^ 2 ( \\mathbb X ^ j ) \\textrm { a n d } f \\textrm { i s s y m m e t r i c } \\right \\} \\end{align*}"}
+{"id": "3723.png", "formula": "\\begin{align*} \\frac { - 2 b } { 1 - 2 b ^ 2 + 2 w } = \\mu ^ { 0 } _ { b , w } ( E ) \\ > \\ \\mu ^ { 0 } _ { b , w } ( B ) = \\frac { ( b x - y ) } { ( 2 b ^ 2 - 2 w - 1 ) \\frac { x } { 2 } - ( b + 1 ) y } \\ , . \\end{align*}"}
+{"id": "1014.png", "formula": "\\begin{align*} \\int _ E P _ D ( f ) \\ , L \\eta \\ , d m = 0 . \\end{align*}"}
+{"id": "4401.png", "formula": "\\begin{align*} \\mathcal { X } _ { \\overline { \\mathcal { Y } } } : = \\{ x \\in \\mathcal { X } \\colon & \\Delta y _ { q , p ( q ) } g _ { q , p ( q ) } ( x ) \\geq \\Delta y _ { a , b ( a ) } g _ { a , b ( a ) } ( x ) \\ \\forall q \\in \\overline { \\mathcal { Y } } , \\\\ & \\Delta y _ { q , p ( q ) } g _ { q , p ( q ) } ( x ) \\leq \\Delta y _ { a , b ( a ) } g _ { a , b ( a ) } ( x ) \\ \\forall q \\in [ m ] \\setminus \\overline { \\mathcal { Y } } \\} , \\end{align*}"}
+{"id": "8932.png", "formula": "\\begin{align*} \\dim ( \\mathcal { M } ( L / \\gamma _ 2 ( L ) ) ) & = \\frac { 1 } { 2 } [ ( m + n - r - s ) ^ 2 + n - s - m + r ] \\\\ & = \\frac { 1 } { 2 } [ ( m + n - r - s ) ^ 2 + 2 ( n - s ) - ( m + n - r - s ) ] \\\\ & = \\frac { 1 } { 2 } [ ( m + n - r - s ) ( m + n - r - s - 1 ) + 2 ( n - s ) ] . \\end{align*}"}
+{"id": "5948.png", "formula": "\\begin{align*} \\frac { M _ { 2 } ( \\mathcal { C } ( G ) ) } { | e ( \\mathcal { C } ( G ) ) | } = \\dfrac { 1 6 } { 7 } > \\dfrac { 2 0 } { 1 1 } = \\frac { M _ { 1 } ( \\mathcal { C } ( G ) ) } { | v ( \\mathcal { C } ( G ) ) | } . \\end{align*}"}
+{"id": "8508.png", "formula": "\\begin{align*} F ( w _ { n _ k } ) ( a ) = g \\circ F ( v _ { n _ k } ) ( a ) \\to g ( x ) , \\end{align*}"}
+{"id": "6718.png", "formula": "\\begin{align*} y ( x ) = a _ r ( x ) r ^ x , \\end{align*}"}
+{"id": "3841.png", "formula": "\\begin{align*} \\begin{aligned} & \\bar \\Lambda ^ n ( A \\times B ) \\doteq \\int _ B \\bar \\Lambda ^ n ( A \\mid t ) d t , \\\\ & \\bar \\xi ^ n ( A \\times C \\times B ) \\doteq \\int _ B \\bar \\xi ^ n ( A \\times C \\mid t ) d t , \\ ; \\ ; \\bar \\Xi ^ n ( A \\times C \\times B ) \\doteq \\int _ B \\bar \\Xi ^ n ( A \\times C \\mid t ) d t \\end{aligned} \\end{align*}"}
+{"id": "1148.png", "formula": "\\begin{align*} \\widehat \\tau : = \\left [ \\left ( \\tau - \\frac { 1 } { p } \\right ) + \\frac { d } { n p } \\right ] _ + , \\ \\widetilde { J } : = J _ { \\tau } + \\left [ \\left ( n \\widehat \\tau \\right ) \\wedge \\frac { d } { p } \\right ] , \\widetilde { s } : = s + n \\widehat \\tau . \\end{align*}"}
+{"id": "4892.png", "formula": "\\begin{align*} D = \\overline { \\bigcup _ { C \\in \\mathcal { P } ^ 0 } \\{ q : p + q \\in g ^ 1 _ 2 ( C ) \\} , } \\end{align*}"}
+{"id": "7371.png", "formula": "\\begin{align*} F ( \\lambda ) = O ( | \\lambda | ^ { 1 + \\alpha } ) \\quad \\mbox { w i t h $ \\alpha \\in \\N $ n e a r $ \\lambda = 0 $ } . \\end{align*}"}
+{"id": "1800.png", "formula": "\\begin{align*} \\zeta ( x ) = \\left [ \\begin{array} { c } \\left . 1 \\middle / \\sigma ( x ) \\right . \\\\ b ( x ) \\end{array} \\right ] \\mbox { a n d } G \\left ( z , ( a , q ) \\right ) = \\left [ \\begin{array} { c } 0 \\\\ - \\frac 1 2 \\ , g \\ , a ^ 2 \\ , z _ 1 - g \\ , a \\ , z _ 2 \\end{array} \\right ] \\end{align*}"}
+{"id": "5270.png", "formula": "\\begin{align*} x ^ { k + 1 } _ { j } = x ^ { k } _ { j } + \\alpha ^ { k } _ { j } x ^ { k } _ { j } \\left ( U ^ { k } _ { j } - V ^ { k } _ { j } \\right ) \\end{align*}"}
+{"id": "7648.png", "formula": "\\begin{align*} Q _ { I } ( m , n ) : = \\sup _ { | \\varphi | \\leq 1 } \\abs { \\langle \\delta _ m , \\varphi ( H ) \\delta _ n \\rangle } \\end{align*}"}
+{"id": "4198.png", "formula": "\\begin{align*} & 2 4 0 \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 8 ) } = \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right . \\\\ & \\left . \\cdot { \\rm c h } ( \\widetilde { T X } + 2 \\wedge ^ 2 \\widetilde { L _ R \\otimes C } - \\widetilde { L _ R \\otimes C } \\otimes \\widetilde { L _ R \\otimes C } + \\widetilde { L _ R \\otimes C } ) \\right \\} ^ { ( 8 ) } . \\end{align*}"}
+{"id": "8406.png", "formula": "\\begin{align*} [ B _ 1 ( a ) , B _ 1 ( b ) ] & = B _ 1 ( a ) B _ 1 ( b ) - B _ 1 ( b ) B _ 1 ( a ) \\\\ & = B _ 1 ( B _ 1 ( a ) \\sigma ( b ) - \\sigma ( b ) B _ 2 ( a ) - B _ 1 ( b ) \\sigma ( a ) + \\sigma ( a ) B _ 2 ( b ) ) . \\end{align*}"}
+{"id": "2447.png", "formula": "\\begin{align*} u ( \\xi ) \\sim \\dfrac { 1 } { \\mu } \\sim \\begin{cases} B _ { 1 } e ^ { \\omega _ { 1 } \\xi } + B _ { 2 } e ^ { \\omega _ { 2 } \\xi } + \\dfrac { 1 } { \\mu } , ( D > 0 ) , \\\\ ( B _ { 3 } \\xi + B _ { 4 } ) e ^ { \\omega \\xi } + \\dfrac { 1 } { \\mu } , ( D = 0 ) , \\\\ e ^ { - \\frac { \\mu c } { 2 } \\xi } \\bar { Z } ( \\xi ) + \\dfrac { 1 } { \\mu } , ( D < 0 ) , \\end{cases} \\end{align*}"}
+{"id": "7382.png", "formula": "\\begin{align*} \\begin{array} { l l } Y : = & \\{ ( U , W ) \\in \\{ C ^ 1 ( \\R \\times [ 0 , T ] ) \\} ^ 2 \\ : \\ \\| ( U , W ) \\| _ Y < \\infty , \\\\ & \\mbox { s u p p } \\ ( U , W ) \\subset \\{ | x | \\le t + R \\} \\} \\end{array} \\end{align*}"}
+{"id": "3350.png", "formula": "\\begin{align*} J ( u ) = \\mathbb { E } ^ u \\left [ \\mu \\left ( \\sum _ { i = 0 } ^ { K - 1 } L ( x _ i , u _ i ) + \\Phi ( x _ K ) \\right ) \\right ] , \\end{align*}"}
+{"id": "2684.png", "formula": "\\begin{align*} \\delta S ^ { ( d ) } = \\int ^ { t _ { 2 } } _ { t _ { 1 } } d t \\sum _ { i = 1 } ^ { n } \\left [ \\sum ^ { d } _ { \\alpha = 0 } ( - D ) ^ { \\alpha } \\frac { \\partial L ^ { ( d ) } } { \\partial ( D ^ { \\alpha } q ^ { i } ) } \\right ] \\delta q ^ { i } + \\left [ \\sum _ { i = 1 } ^ { n } \\sum ^ { d } _ { \\beta = \\alpha \\geq 1 } ( - D ) ^ { \\beta - \\alpha } \\frac { \\partial L ^ { ( d ) } } { \\partial ( D ^ { \\beta } q ^ { i } ) } \\delta ( D ^ { \\alpha - 1 } q ^ { i } ) \\right ] ^ { t _ { 2 } } _ { t _ { 1 } } . \\end{align*}"}
+{"id": "4708.png", "formula": "\\begin{align*} \\alpha ( g ) ( t ) = \\beta ( g ) ( t ) t \\in \\partial { C _ { x } } . \\end{align*}"}
+{"id": "1803.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t U + A \\ ; \\partial _ x U = g \\\\ U ( 0 , x ) = u ( \\tau , x ) \\end{array} \\right . \\end{align*}"}
+{"id": "2780.png", "formula": "\\begin{align*} \\delta \\left ( { \\sigma _ { \\tau } ^ { * } } ( t ) I \\right ) = \\int ^ { t _ { 2 } } _ { t _ { 1 } } \\left [ - \\frac { d } { d t } \\left ( \\frac { \\partial L _ { T } } { \\partial \\dot { Q } ^ { a } } \\right ) + \\frac { \\partial L _ { T } } { \\partial Q ^ { a } } \\right ] \\delta Q ^ { a } d t + \\left [ \\frac { \\partial L _ { T } } { \\partial \\dot { Q } ^ { a } } \\delta Q ^ { a } \\right ] ^ { t _ { 2 } } _ { t _ { 1 } } , \\end{align*}"}
+{"id": "3965.png", "formula": "\\begin{align*} \\lambda ( T ) ^ * \\rho ( T ) = \\int _ X c ( T , x ) d \\mu ( x ) , \\end{align*}"}
+{"id": "8911.png", "formula": "\\begin{align*} \\widetilde { V } _ n ( x ) = \\begin{cases} \\displaystyle { \\sum _ { m = 0 } ^ { \\frac { n - 1 } { 2 } } \\frac { n ( n - m - 1 ) ! } { m ! ( n - 2 m ) ! } x ^ { n - 2 m - 1 } } & n : , \\\\ \\displaystyle { \\sum _ { m = 0 } ^ { \\frac { n } { 2 } - 1 } { n - m - 1 \\choose m } x ^ { n - 2 m - 2 } } & n : . \\end{cases} \\end{align*}"}
+{"id": "6338.png", "formula": "\\begin{align*} \\frac { M } { \\ell } \\leq C Y ^ { 3 2 \\kappa } , \\frac { M _ 0 } { \\ell } \\leq C Y ^ { 3 8 \\kappa } , \\frac { n } { \\ell } \\frac { M _ 0 } { M } \\leq C Y ^ { 4 \\kappa } , \\end{align*}"}
+{"id": "2880.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to \\infty } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ X = \\left [ \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { X } . \\end{align*}"}
+{"id": "8787.png", "formula": "\\begin{align*} \\mu ( d x ) \\mbox { a . e . } , \\ ; \\int _ { \\Gamma _ x } y \\pi _ x ( d y ) = x \\mbox { s o t h a t } \\pi _ x ( \\Gamma _ x \\cap ( - \\infty , x ] ) \\wedge \\pi _ x ( \\Gamma _ x \\cap [ x , + \\infty ) ) > 0 . \\end{align*}"}
+{"id": "5797.png", "formula": "\\begin{align*} x ^ { ( k , \\ell ) } _ { j } ( t ) : = \\xi ^ { ( k , \\ell ) } _ { \\iota + j } ( t ) , \\ \\mathcal { W } ^ { ( k , \\ell ) } _ { j } ( t ) : = \\mathcal { E } ^ { ( k , \\ell ) } _ { \\iota + j } ( t ) \\end{align*}"}
+{"id": "5036.png", "formula": "\\begin{align*} \\mathbf { A } _ { S ^ c \\setminus \\{ j \\} , j } ^ S & = - \\frac { \\beta } { 1 - \\beta } \\left \\{ \\mathbf { P } _ { S ^ c \\setminus \\{ j \\} , j } - \\mathbf { A } _ { S ^ c \\setminus \\{ j \\} , S } ^ S \\mathbf { P } _ { S j } \\right \\} . \\end{align*}"}
+{"id": "9019.png", "formula": "\\begin{align*} e _ { 1 1 } \\cdot e _ { 1 1 } = - e _ { 1 1 } \\cdot e _ { 2 2 } = e _ { 2 2 } \\cdot e _ { 2 2 } = e _ { 1 2 } \\end{align*}"}
+{"id": "1150.png", "formula": "\\begin{align*} \\begin{cases} \\widetilde p = \\widetilde r & \\dot a = \\dot b , \\\\ \\widetilde p \\wedge \\widetilde q = \\widetilde r & \\dot a = \\dot f . \\end{cases} \\end{align*}"}
+{"id": "445.png", "formula": "\\begin{align*} \\Delta ( E _ { B ^ { q } } ) : = \\left \\{ \\begin{aligned} & \\max _ { ( i , j ) \\in E _ { B ^ { q } } } \\Delta t _ { i , j } , \\mbox { i f } E _ { B ^ { q } } \\neq \\emptyset , \\\\ [ 2 p t ] & - \\infty , \\mbox { o t h e r w i s e } . \\end{aligned} \\right . \\end{align*}"}
+{"id": "8777.png", "formula": "\\begin{align*} & \\{ ( ( x _ - , y _ - ) , ( x _ + , y _ + ) ) \\in \\{ \\R ^ 2 \\} ^ 2 : y _ + < y _ - \\le x _ - < x _ + \\} \\cap \\Gamma ^ 2 = \\emptyset \\\\ \\mbox { a n d } & \\{ ( ( x _ - , z _ - ) , ( x _ + , z _ + ) ) \\in \\{ \\R ^ 2 \\} ^ 2 : x _ - < x _ + \\le z _ + < z _ - \\} \\cap \\Gamma ^ 2 = \\emptyset . \\end{align*}"}
+{"id": "9143.png", "formula": "\\begin{align*} \\tilde { c } _ { \\beta } = \\begin{cases} c _ { \\beta } & \\ \\beta = [ 1 ] , [ 2 ] , [ 1 , 2 ] \\\\ \\frac { c _ { \\beta } } { [ 2 ] _ { v } ! } & \\ \\beta = [ 1 , 2 , 2 ] \\\\ \\frac { c _ { \\beta } } { [ 3 ] _ { v } ! } & \\ \\beta = [ 1 , 2 , 2 , 2 ] , [ 1 , 2 , 1 , 2 , 2 ] \\end{cases} . \\end{align*}"}
+{"id": "1154.png", "formula": "\\begin{align*} \\begin{aligned} & K _ b \\wedge M _ b \\wedge K _ m \\wedge M _ m > \\widetilde J , \\\\ & [ N _ b \\wedge \\lceil \\ ! \\lceil L _ m \\rceil \\ ! \\rceil \\wedge ( K _ m - n - \\alpha ) ] _ + > \\widetilde s , \\\\ & [ N _ m \\wedge \\lceil \\ ! \\lceil L _ b \\rceil \\ ! \\rceil \\wedge ( K _ b - n - \\alpha ) ] _ + > \\widetilde J - n - \\widetilde s . \\end{aligned} \\end{align*}"}
+{"id": "5173.png", "formula": "\\begin{align*} & \\frac { \\partial A } { \\partial q _ { j } } = \\frac { \\partial A _ { j } } { \\partial q _ { j } } = - p _ { j } q ^ { \\beta - 2 } _ { j } + q ^ { \\beta - 1 } _ { j } \\\\ & \\frac { \\partial B } { \\partial q _ { j } } = \\frac { \\partial B _ { j } } { \\partial q _ { j } } = 0 \\end{align*}"}
+{"id": "1770.png", "formula": "\\begin{align*} ( v _ 0 ^ * , \\ldots , v _ n ^ * ) : = v ( z ^ * ) , \\end{align*}"}
+{"id": "4088.png", "formula": "\\begin{align*} \\psi ( x , \\lambda ) : = \\int _ { \\mathbb { S } ^ { d - 1 } } w ( \\lambda , \\omega , x ) \\phi ( \\omega ) d \\omega \\end{align*}"}
+{"id": "8860.png", "formula": "\\begin{align*} 0 < \\frac { 1 } { ( p - 2 ) a _ 3 } \\leq \\frac 1 r \\leq 1 , 0 \\leq \\frac { \\tau } { p - 2 } < \\frac { n } { ( p - 2 ) a _ 3 } , \\frac { \\tau } { p - 2 } - 1 = \\frac { n } { ( p - 2 ) a _ 3 } - \\frac { n } { r } , \\end{align*}"}
+{"id": "1608.png", "formula": "\\begin{align*} ( \\S ^ { g , n } X ) ( h ) = \\S ^ n X ( h \\oplus g ^ { - 1 } ) , \\end{align*}"}
+{"id": "6490.png", "formula": "\\begin{align*} x x ' = t ^ { a _ 1 } y _ 1 + t ^ { a _ 2 } y _ 2 , \\end{align*}"}
+{"id": "378.png", "formula": "\\begin{align*} x ^ p + y ^ p = z ^ p \\Longrightarrow ( x - a ) ^ p - p a b K _ p = 0 . \\end{align*}"}
+{"id": "7167.png", "formula": "\\begin{align*} \\partial _ t U - A \\partial _ { x } ^ 2 U = F ( U , x , t ) , x \\in \\R , t \\geq t _ 0 , \\end{align*}"}
+{"id": "652.png", "formula": "\\begin{align*} \\ell ( 4 x , 2 x ) & = \\cosh ^ { - 1 } \\left ( \\cosh ( 2 x ) - \\frac { \\sinh ^ 2 ( 2 x ) } { 4 \\cosh ( 2 x ) } \\right ) \\\\ & = \\cosh ^ { - 1 } \\left ( \\frac { 3 ( 2 \\cosh ^ 2 x - 1 ) ^ 2 + 1 } { 4 ( 2 \\cosh ^ 2 x - 1 ) } \\right ) . \\end{align*}"}
+{"id": "5508.png", "formula": "\\begin{align*} t _ 4 : = \\bar { X } _ 4 , t _ 3 = \\bar { X } _ 3 , t _ 2 = \\bar { X } _ 2 - \\frac { q ^ 4 } { ( q ^ 2 + 1 ) ( q + q ^ { - 1 } ) } \\bar { X } _ 3 ^ 2 \\bar { X } _ 4 ^ { - 1 } , t _ 1 = \\bar { X } _ 1 - \\frac { q ^ 2 ( q + q ^ { - 1 } ) } { q ^ 2 - 1 } \\bar { X } _ 2 \\bar { X } _ 3 ^ { - 1 } . \\end{align*}"}
+{"id": "8679.png", "formula": "\\begin{align*} J T ^ H Y ' = T ^ H Y ' = J T Y ' \\cap T Y ' . \\end{align*}"}
+{"id": "7001.png", "formula": "\\begin{align*} \\ ! & \\{ [ 0 \\ ! : \\ ! 0 \\ ! : \\ ! z _ 3 \\ ! : \\ ! \\cdots \\ ! : \\ ! z _ { n + 1 } ] : z _ 3 , \\ ! \\cdots \\ ! , \\ ! z _ { n + 1 } \\ ! \\in \\ ! \\mathbb { C } \\} \\setminus \\ ! \\{ [ 0 \\ ! : \\ ! 0 \\ ! : \\ ! z _ 3 \\ ! : \\ ! \\cdots \\ ! : \\ ! z _ { n } \\ ! : \\ ! 1 ] : z _ 3 , \\ ! \\cdots \\ ! , \\ ! z _ { n } \\ ! \\in \\ ! \\mathbb { C } \\} = \\\\ & \\{ [ 0 \\ ! : \\ ! 0 \\ ! : \\ ! z _ 3 \\ ! : \\ ! \\cdots \\ ! : \\ ! z _ { n } \\ ! : \\ ! 0 ] : z _ 3 , \\ ! \\cdots \\ ! , \\ ! z _ { n } \\ ! \\in \\ ! \\mathbb { C } \\} = \\Lambda _ 0 . \\end{align*}"}
+{"id": "4041.png", "formula": "\\begin{align*} & [ \\textbf { K } ^ X _ { t , t } ] _ { i , j } = k _ { S } ( \\bar x ^ i | _ { I _ t } , \\bar x ^ j | _ { I _ t } ) , [ \\textbf { K } ^ { X , Y } _ { T , T } ] _ { i , j } = k _ S ( \\bar x ^ i , \\bar y ^ j ) , \\\\ & [ \\textbf { K } ^ Y _ { s , s } ] _ { i , j } = k _ S ( \\bar y ^ i | _ { I _ s } , \\bar y ^ j | _ { I _ s } ) . \\end{align*}"}
+{"id": "2081.png", "formula": "\\begin{align*} \\eqref { e q : t e s t f u n c t i o n a p p l i e d s u m } & = - \\int _ 0 ^ { \\frac { \\lfloor s T \\rfloor } { T } } \\Theta ( u , x ) f ' ( u ) d u + f ( s ) { \\Theta } ( \\frac { \\lfloor s T \\rfloor } { T } , x ) - f ( 0 ) { \\Theta } ( 0 , x ) + o ( 1 ) , \\\\ \\eqref { e q : t e s t f u n c t i o n a p p l i e d s u m } & = - ( i + 1 - d x ) ( N _ { i , 1 } ^ s + N _ { i , 2 } ^ s + N _ { i , 3 } ^ s ) - ( d x - i ) ( N _ { i + 1 , 1 } ^ s + N _ { i + 1 , 2 } ^ s + N _ { i + 1 , 3 } ^ s ) , \\end{align*}"}
+{"id": "2404.png", "formula": "\\begin{align*} \\widetilde { A _ n } ^ { ( j ) } ( - k ) & = \\left ( \\widetilde { A _ n } ( t ) \\cdot U ( t ) \\right ) ^ { ( j - 1 ) } \\big | _ { t = - k } \\\\ & = \\sum _ { v = 0 } ^ { j - 1 } \\binom { j - 1 } { v } \\widetilde { A _ n } ^ { ( j - 1 - v ) } ( - k ) \\cdot U ^ { ( v ) } ( - k ) . \\end{align*}"}
+{"id": "7241.png", "formula": "\\begin{align*} \\phi \\circ \\ast = \\star \\circ \\phi ^ { \\times k } . \\end{align*}"}
+{"id": "1534.png", "formula": "\\begin{align*} I : = \\inf _ { v \\ne 0 } \\frac { \\sum _ { i } \\lambda _ i \\ , v _ i ^ 2 } { \\Big ( { \\sum _ i \\lambda _ i ^ 2 \\ , v _ i ^ 2 } \\Big ) ^ { 1 / 2 } \\Big ( { \\sum _ i v _ i ^ 2 } \\Big ) ^ { 1 / 2 } } . \\end{align*}"}
+{"id": "7185.png", "formula": "\\begin{align*} \\| F _ v - F _ v ^ { ( 0 ) } \\| _ Y & \\leq \\| F _ v - F _ v ^ { ( N + 1 ) } \\| _ { Y } + \\sum _ { k = 0 } ^ { N } \\left \\| F _ v ^ { ( k + 1 ) } - F _ v ^ { ( k ) } \\right \\| _ { Y } \\\\ & \\leq \\| F _ v - F _ v ^ { ( N + 1 ) } \\| _ { Y } + \\sum _ { k = 0 } ^ N \\alpha ^ k \\| F _ v ^ { ( 1 ) } - F _ v ^ { ( 0 ) } \\| _ { Y } \\\\ & \\leq \\| F _ v - F _ v ^ { ( N + 1 ) } \\| _ { Y } + \\frac { \\alpha } { 1 - \\alpha } \\frac { C } { \\omega _ 1 } . \\end{align*}"}
+{"id": "8129.png", "formula": "\\begin{align*} [ p _ x , p _ y ] = 0 \\ , \\ [ q _ x , q _ y ] = 0 \\ , \\ [ p _ { x , y } , I ] = 0 \\ , \\ [ q _ { x , y } , I ] = 0 \\ . \\end{align*}"}
+{"id": "601.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\left ( \\frac { 1 } { 3 2 } \\right ) ^ { k } \\binom { 2 k } { k } ^ 2 \\frac { H _ { k } } { k + 1 } = 8 - \\frac { 2 \\Gamma ^ { 2 } \\left ( \\frac { 1 } { 4 } \\right ) } { \\pi ^ { 3 / 2 } } - \\frac { 4 \\pi ^ { 3 / 2 } + 1 6 \\sqrt { \\pi } \\ln ( 2 ) } { \\Gamma ^ { 2 } \\left ( \\frac { 1 } { 4 } \\right ) } \\end{align*}"}
+{"id": "5528.png", "formula": "\\begin{align*} \\big \\| \\{ \\hat \\xi _ n \\} _ { n \\in \\overline \\Omega } \\big \\| = \\big \\| \\{ \\hat \\Delta ( z _ n ^ 2 ) z _ n ^ { 2 - \\alpha - \\beta } \\} _ { n \\in \\overline \\Omega } \\big \\| _ { \\bf a } . \\end{align*}"}
+{"id": "2128.png", "formula": "\\begin{align*} \\mathcal { P } ( \\mathcal { Z } ) \\ni \\Theta \\mapsto \\Phi ( \\Theta ) : = \\bigg ( \\mathbb { E } _ { \\Theta } [ \\Psi \\cdot ( M ^ \\Theta _ f ( t _ 1 ) - M ^ \\Theta _ f ( t _ 0 ) ) ] \\bigg ) ^ { - } , \\end{align*}"}
+{"id": "7096.png", "formula": "\\begin{align*} X _ t = x - \\int _ 0 ^ b b ( s , X _ s ) d s + \\sqrt { 2 } W _ t , t \\geq 0 , \\end{align*}"}
+{"id": "7985.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Omega } e _ { \\eta } ^ { i } \\wedge [ \\eta , d N _ { \\phi } ( e _ { \\phi } ^ { j } ) ] _ { 1 } = - \\int _ { \\Omega } e _ { \\eta } ^ { i } \\wedge \\ast \\Big ( ( \\ast e _ { \\eta } ^ { j } + d N _ { \\phi } ( e _ { \\phi } ^ { j } ) ) \\wedge ( \\ast d \\eta ) \\Big ) . \\end{aligned} \\end{align*}"}
+{"id": "8789.png", "formula": "\\begin{align*} 0 \\le F _ \\mu ( x ) - \\psi _ { \\pi ^ \\downarrow } ( x ) & = { \\pi ^ \\downarrow } ( ( - \\infty , x ] \\times \\R ) - { \\pi ^ \\downarrow } ( ( - \\infty , x ] \\times ( - \\infty , a ] ) \\\\ & = { \\pi ^ \\downarrow } ( ( - \\infty , x ] \\times [ b , + \\infty ) ) \\le \\nu ( [ b , + \\infty ) ) = 1 - F _ \\nu ( a ) , \\end{align*}"}
+{"id": "6633.png", "formula": "\\begin{align*} \\dim _ \\C V ^ { 2 , 0 } = 1 , V ^ { p , q } = 0 | p - q | > 2 . \\end{align*}"}
+{"id": "3472.png", "formula": "\\begin{align*} ( e , h ) = ( 8 / 3 ( 1 - n ) + 2 + 2 m , 1 + m ) , \\end{align*}"}
+{"id": "1454.png", "formula": "\\begin{align*} \\mathcal { T } ( k ) \\cap h \\mathcal { G } ( A ) h ^ { - 1 } = \\left \\lbrace \\mathrm { d i a g } ( t , \\overline { t } t ^ { - 1 } , \\overline { t } ^ { - 1 } ) : t \\in \\mathrm { R } _ { \\ell / k } ( \\mathbb { G } _ { m , \\ell } ) \\right \\rbrace \\cap h \\mathcal { G } ( A ) h ^ { - 1 } . \\end{align*}"}
+{"id": "6167.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau ^ k } [ f ( x ) - f ( \\breve { x } ^ { k } ) ] - \\frac { 1 } { \\tau ^ { k - 1 } } [ f ( x ) - f ( \\breve { x } ^ { k - 1 } ) ] + ( x - \\widetilde { x } ^ { k } ) ^ T [ - A ^ T \\widetilde { \\lambda } ^ k \\\\ & + \\tau ^ k \\beta ^ k ( D - A ^ T A ) ( \\widetilde { x } ^ { k } - \\bar { x } ^ k ) + ( 2 - \\gamma ) ( 1 - \\tau ^ k ) \\beta ^ k A ^ T ( A \\breve { x } ^ { k - 1 } - b ) ] \\geq 0 , ~ \\forall x . \\end{aligned} \\end{align*}"}
+{"id": "8220.png", "formula": "\\begin{align*} & P _ { Y ^ { i - 1 } , { \\bf S } } ( y ^ { i - 1 } , { \\bf s } ) = P _ { Y ^ { i - 1 } , { \\bf S } } ( \\tilde { y } ^ { i - 1 } , { \\bf s } ) , \\\\ & \\sum _ { i = t } ^ L H ( Y _ i | { \\bf S } , y ^ { t - 1 } , Y _ t ^ { i - 1 } ) = \\sum _ { i = t } ^ L H ( Y _ i | { \\bf S } , \\tilde { y } ^ { t - 1 } , Y _ t ^ { i - 1 } ) \\end{align*}"}
+{"id": "6484.png", "formula": "\\begin{align*} x _ { u , v } = ( y _ u , \\varpi _ { i _ v } - w _ v \\varpi _ { i _ v } ) = 0 . \\end{align*}"}
+{"id": "2261.png", "formula": "\\begin{align*} P ' _ { n } = \\frac { 1 } { n ! } \\sum _ { w \\in S _ { n } } s g n ( w ) w . \\end{align*}"}
+{"id": "8886.png", "formula": "\\begin{align*} f ( t ) = t - \\frac { C _ { N , \\tau , \\alpha , \\lambda } } { p } t ^ { p } . \\end{align*}"}
+{"id": "7524.png", "formula": "\\begin{align*} h _ P ( y ) : = 2 \\sum _ { \\gamma < 1 4 0 0 0 } \\alpha \\exp ( - 1 . 5 \\cdot 1 0 ^ { - 6 } \\gamma ^ 2 ) \\cos ( \\gamma y - \\psi ) . \\end{align*}"}
+{"id": "6387.png", "formula": "\\begin{align*} \\begin{aligned} \\xi _ { g , q } = \\sqrt { \\frac { \\epsilon _ { g , q } ^ 2 } { 2 } F _ { 2 M } ^ { - 1 } ( 1 - \\rho _ q ) } , \\\\ \\xi _ { h , q } = \\sqrt { \\frac { \\epsilon _ { h , q } ^ 2 } { 2 } F _ { 2 N } ^ { - 1 } ( 1 - \\rho _ q ) } . \\end{aligned} \\end{align*}"}
+{"id": "8896.png", "formula": "\\begin{align*} J _ k ( n ) = n ^ k \\prod _ { p \\mid n } ( 1 - p ^ { - k } ) = \\sum _ { d \\mid n } \\mu ( n / d ) d ^ k , \\end{align*}"}
+{"id": "10.png", "formula": "\\begin{align*} f ( z ) = \\tfrac { 1 } { 2 \\pi i } I _ + \\bigl ( ( X - z ) ^ { - 1 } f \\bigr ) = \\lim _ { y \\to \\infty } \\tfrac { 1 } { 2 \\pi i } \\bigl \\langle \\chi _ y , ( X - z ) ^ { - 1 } f \\bigr \\rangle \\end{align*}"}
+{"id": "6567.png", "formula": "\\begin{align*} \\phi _ { \\Phi _ k , \\beta } ( x , y ) = \\begin{dcases} | \\{ n \\leq x ; \\ \\ \\frac { \\phi ( n ) } { n ^ { \\delta } } \\leq y \\} | , & k \\\\ | \\{ n \\leq x ; \\ \\ \\frac { \\phi ( n ) \\alpha _ { k } ( n ) } { n ^ { \\delta } } \\leq y \\} | , & k \\end{dcases} \\end{align*}"}
+{"id": "215.png", "formula": "\\begin{align*} \\frac { d ^ 2 x ^ i } { d t ^ 2 } = X ^ i ( x ^ 1 , \\ldots , x ^ n , \\dot x ^ 1 , \\ldots , \\dot x ^ n ) , i = 1 , \\ldots , n , \\end{align*}"}
+{"id": "5122.png", "formula": "\\begin{align*} L o g ( x . y ) = L o g ( x ) + L o g ( y ) \\end{align*}"}
+{"id": "700.png", "formula": "\\begin{align*} f ( x ) = ( \\Phi ( x ) + 2 ) ^ 4 = \\Phi ^ 4 ( x ) + 8 \\Phi ^ 3 ( x ) + 2 4 \\Phi ^ 2 ( x ) + 3 2 \\Phi ( x ) + 1 6 - m \\end{align*}"}
+{"id": "588.png", "formula": "\\begin{align*} \\Big | \\Big \\langle Y _ 1 , \\frac { r ^ \\theta _ m } { \\| h _ 2 \\| _ { F ^ \\ast } } \\Big \\rangle \\Big | = 0 . \\end{align*}"}
+{"id": "8033.png", "formula": "\\begin{align*} \\Xi ( s ) = & - W _ { s , 1 } ( \\mathcal { P } _ { 1 , s } f ) - W _ { s , 3 } ( \\mathcal { P } _ { 2 , s } f ) \\\\ & - \\int _ { \\mathbb { T } ^ 2 } \\lambda ( u , v ) \\theta _ s ^ S ( u ) \\theta _ s ^ I ( v ) f ( u ) \\left ( \\tilde { G } _ s ( u ) - \\tilde { F } _ s ( u ) \\right ) d u d v . \\end{align*}"}
+{"id": "5711.png", "formula": "\\begin{align*} \\mathcal { B } : = \\{ \\psi _ { i , 1 } , \\psi _ { i , 2 } \\} _ { i \\in I _ 1 } \\cup \\{ \\psi _ { i , 3 } , \\psi _ { i , 4 } \\} _ { i \\in I _ 2 } \\cup \\{ \\psi ^ + _ i , \\psi ^ - _ i \\} _ { i \\in I _ 3 \\cup I _ 4 } . \\end{align*}"}
+{"id": "6905.png", "formula": "\\begin{align*} F ( 0 ) & = ( 4 ( \\cosh 1 - 1 ) ( \\eta \\cosh 1 + \\gamma ) ^ 2 + k ( \\eta \\sinh 2 + 2 \\gamma \\sinh 1 ) ) ^ 2 \\\\ & - 1 6 ( ( s + 1 ) ^ 2 - 1 ) ^ 2 ( \\eta \\cosh 1 + \\gamma ) ^ 4 \\intertext { a n d } F \\left ( \\frac { \\pi } { 2 } \\right ) & = ( 4 ( \\cos 1 - 1 ) ( \\eta \\cos 1 + \\gamma ) ^ 2 - k ( \\eta \\sin 2 + 2 \\gamma \\sin 1 ) ) ^ 2 \\\\ & - 1 6 ( ( s + 1 ) ^ 2 - 1 ) ^ 2 ( \\eta \\cos 1 + \\gamma ) ^ 4 . \\end{align*}"}
+{"id": "2846.png", "formula": "\\begin{align*} L '' _ { 4 } = - \\frac { 1 } { 2 } q \\dot { x } - \\frac { 1 } { 2 } q ^ { 2 } + \\lambda ( x - \\dot { q } ) . \\end{align*}"}
+{"id": "3776.png", "formula": "\\begin{align*} x ^ { ( i ) } _ { l , t + 1 } = \\Theta _ j z ^ { ( i ) } _ { l , t } + w ^ { ( i ) } _ { l , t } \\forall \\ 1 \\le l \\le { { N _ i } } 0 \\le t \\le T - 1 , \\end{align*}"}
+{"id": "1729.png", "formula": "\\begin{align*} \\left ( Q _ { j ; k } \\right ) _ { r , s } = \\delta _ { r + s , j + 2 - k } . \\end{align*}"}
+{"id": "6282.png", "formula": "\\begin{align*} p = 1 , \\Psi _ p ( x ) = ( 1 + \\gamma ) \\sum _ { i = 1 } ^ d ( x _ i + \\gamma / d ) \\log ( x _ i + \\gamma / d ) , \\gamma > 0 . \\end{align*}"}
+{"id": "3871.png", "formula": "\\begin{align*} \\sum _ { x \\in \\Delta ^ o } \\beta ^ j _ { ( 1 ) } ( x ) \\beta ^ j _ { 2 | 1 } ( y \\mid x ) = \\beta ^ j _ { ( 1 ) } ( y ) , \\ ; \\ ; y \\in \\Delta ^ o . \\end{align*}"}
+{"id": "6062.png", "formula": "\\begin{align*} V A _ { s } ' & = y A _ { d - 1 } ' A _ { s } ' + U A _ { s } ' = \\left ( \\bigoplus _ { i = 1 } ^ { d + s } y ^ i A _ { d + s - i } \\right ) + A _ { s } U + y A _ { s - 1 } U + \\dots + y ^ s U \\\\ & = \\bigoplus _ { i = 1 } ^ { d + s } y ^ i A _ { d + s - i } \\oplus U A _ { s } \\end{align*}"}
+{"id": "4273.png", "formula": "\\begin{align*} \\div _ f \\omega = \\div \\omega - \\omega ( \\nabla f ) = \\nabla _ i \\omega _ i - \\omega _ i \\nabla _ i f , \\end{align*}"}
+{"id": "683.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathcal { A } _ { s } ^ { A } : L ^ { 2 } ( \\mathbb { R } ^ { n } ) \\rightarrow L ^ { \\frac { 2 n } { n - 4 s } } ( \\mathbb { R } ^ { n } ) \\\\ & \\mathcal { A } _ { s } ^ { A } : L ^ { \\frac { 2 n } { n + 4 s } } ( \\mathbb { R } ^ { n } ) \\rightarrow L ^ { 2 } ( \\mathbb { R } ^ { n } ) \\end{aligned} \\end{align*}"}
+{"id": "1726.png", "formula": "\\begin{align*} x _ 0 = f \\left ( f ^ { ( 4 ) } ( 0 ) x _ 1 \\right ) \\left ( f ^ { ( 4 ) } ( 0 ) \\right ) ^ { - 5 } + \\sum _ { 0 < j < j + k \\leq 4 } x _ j x _ k x _ n ^ { n - j - k } , \\end{align*}"}
+{"id": "1144.png", "formula": "\\begin{align*} \\widetilde D : = D - \\frac { d } { p } > \\frac { n } { \\min ( 1 , q ) } \\geq n , \\widetilde E : = E - s - \\frac { n } { 2 } - \\frac { d } { p } > 0 . \\end{align*}"}
+{"id": "5324.png", "formula": "\\begin{align*} v ^ { { \\rm L P } } = \\nu _ { \\pi _ 1 } \\ , b ^ { S _ 1 } + \\sum _ { k = 2 } ^ { n } ( \\nu _ { \\pi _ k } - \\nu _ { \\pi _ { k - 1 } } ) \\ , b ^ { S _ k } . \\end{align*}"}
+{"id": "3671.png", "formula": "\\begin{align*} a _ d ( K _ { d + 1 } ) = 1 . \\end{align*}"}
+{"id": "7257.png", "formula": "\\begin{align*} \\widetilde { \\theta } : = \\theta + \\tfrac { \\eta \\tau } { 1 - q } . \\end{align*}"}
+{"id": "5094.png", "formula": "\\begin{align*} \\pi _ { \\mathcal { L } } ( \\mathsf { a } _ s ) = \\begin{cases} - v \\tilde { A } _ { s } ^ { - 1 } 1 _ { \\mathcal { L } } & s \\in W _ { \\mathcal { L } } ^ \\circ \\\\ - \\tilde { A } _ s 1 _ { \\mathcal { L } } & s \\not \\in W _ { \\mathcal { L } } ^ \\circ \\end{cases} \\end{align*}"}
+{"id": "7435.png", "formula": "\\begin{align*} \\Phi ( 0 , x ) = \\mathbf { 1 } \\quad \\Phi _ { 0 } ( t , x ) = 1 ( t , x ) \\in [ 0 , T ] \\times \\R ^ d . \\end{align*}"}
+{"id": "2139.png", "formula": "\\begin{align*} \\begin{array} { r c l } S _ 1 & = & \\Big ( \\omega ( \\hat { x } ) - \\omega ( x ^ { ( k ) } ) \\Big ) + \\Big ( \\omega ( x ^ { ( k ) } ) - f ( \\hat { x } , q ^ { ( k ) } ( \\hat { x } , \\hat { y } ) \\Big ) \\\\ & = & \\Big ( \\omega ( \\hat { x } ) - \\omega ( x ^ { ( k ) } ) \\Big ) + \\Big ( f ( x ^ { ( k ) } , q ^ { ( k ) } ( x ^ { ( k ) } , y ^ { ( k ) } ) ) - f ( \\hat { x } , q ^ { ( k ) } ( \\hat { x } , \\hat { y } ) ) \\Big ) \\rightarrow 0 . \\end{array} \\end{align*}"}
+{"id": "345.png", "formula": "\\begin{align*} x ^ n + y ^ n = z ^ n \\end{align*}"}
+{"id": "4555.png", "formula": "\\begin{align*} W ^ + = \\left ( \\begin{matrix} - \\frac { 2 a ^ 4 } { r ^ 6 } & & \\\\ & - \\frac { 2 a ^ 4 } { r ^ 6 } & \\\\ & & \\frac { 4 a ^ 4 } { r ^ 6 } \\end{matrix} \\right ) . \\end{align*}"}
+{"id": "8178.png", "formula": "\\begin{align*} P _ { { \\underline X } _ 1 } ( 1 1 0 0 0 ) = P _ { { \\underline X } _ 1 } ( 0 0 0 1 0 ) = \\frac { 2 } { 1 0 } , \\ P _ { { \\underline X } _ 1 } ( 0 0 0 0 1 ) = \\frac { 3 } { 1 0 } , \\\\ P _ { { \\underline X } _ 1 } ( 1 0 1 0 0 ) = P _ { { \\underline X } _ 1 } ( 0 1 1 0 0 ) = P _ { { \\underline X } _ 1 } ( 0 0 1 1 0 ) = \\frac { 1 } { 1 0 } . \\end{align*}"}
+{"id": "7921.png", "formula": "\\begin{align*} \\mathfrak { Z } ^ { k } : = \\{ \\mu \\in H \\Lambda ^ { k } ( \\Omega ) \\mid d \\mu = 0 \\} , \\mathring { \\mathfrak { Z } } ^ { k } : = \\{ \\mu \\in \\mathring { H } \\Lambda ^ { k } ( \\Omega ) \\mid d \\mu = 0 \\} , \\end{align*}"}
+{"id": "8244.png", "formula": "\\begin{align*} D ( B _ ) & = \\frac { 1 } { L } \\sum _ { i = 1 } ^ L d _ i , \\\\ d _ i & = \\frac { 1 } { M } \\left ( \\sum _ { y ^ i } \\frac { c _ { y ^ i } } { M } \\log c _ { y ^ i } \\right ) \\end{align*}"}
+{"id": "4573.png", "formula": "\\begin{align*} \\eta \\left ( S ^ 3 / \\Gamma \\right ) = - \\frac { n ( n - 1 ) } { 3 ( n + 1 ) } , - \\frac { 2 n ^ 2 - 8 n + 9 } { 6 ( n - 2 ) } , - \\frac { 4 9 } { 3 6 } , - \\frac { 1 2 1 } { 7 2 } , - \\frac { 3 6 1 } { 1 8 0 } , \\end{align*}"}
+{"id": "4095.png", "formula": "\\begin{align*} h = \\frac { c _ d } { ( \\lambda r ) ^ { \\frac { d - 1 } { 2 } } } ( e ^ { i \\lambda r } \\phi ( \\theta ) + e ^ { - i \\lambda r } i ^ { 1 - d } \\phi ( - \\theta ) ) + O ( r ^ { - \\frac { d + 1 } { 2 } } ) . \\end{align*}"}
+{"id": "3384.png", "formula": "\\begin{align*} \\frac { 1 } { T } \\sum _ { t = 2 } ^ { T + 1 } \\Delta _ { t } & \\le \\frac { 8 } { T c _ { 1 } } \\left ( R _ { 1 } + \\frac { c _ { 1 } } { 3 } \\left ( \\gamma + \\frac { 2 \\sigma ^ { p } } { c _ { 2 } } \\right ) \\right ) ^ { 2 } \\max \\left \\{ \\left ( 5 2 T ( 1 + \\log T ) ^ { 2 } c _ { 2 } \\right ) ^ { 1 / p } ; 4 R _ { 1 } L + \\frac { 2 c _ { 1 } } { 3 } L \\left ( \\gamma + \\frac { 2 \\sigma ^ { p } } { c _ { 2 } } \\right ) + 2 \\nabla _ { 1 } ; \\frac { L c _ { 1 } } { 6 } \\right \\} \\\\ & = \\widetilde { O } \\left ( T ^ { \\frac { 1 - p } { p } } \\right ) . \\end{align*}"}
+{"id": "1967.png", "formula": "\\begin{align*} E q u = p ^ s \\ ! - \\ ! ( p \\ ! - \\ ! \\tau ) p ^ { s - t - 1 } \\ ! + \\ ! 1 \\ ! - \\ ! ( \\left \\lceil \\frac { ( p \\ ! - \\ ! \\tau ) p ^ { s - t - 1 } } { ( p \\ ! - \\ ! 1 \\ ! - \\ ! \\tau ) \\lceil \\frac { \\delta } { \\tau + 1 } \\rceil \\ ! + \\ ! 1 } \\right \\rceil \\ ! - \\ ! 1 ) ( \\delta \\ ! - \\ ! 1 ) \\ ! - \\ ! ( \\tau \\ ! + \\ ! 1 ) p ^ { t } . \\end{align*}"}
+{"id": "7741.png", "formula": "\\begin{align*} \\Sigma _ { i } ( \\lambda ) : = W _ { \\nu ' } ( \\Sigma _ { i } ( \\lambda _ { 0 } ) ) \\cap \\Sigma ( \\lambda ) \\subseteq \\Sigma ( \\lambda ) . \\end{align*}"}
+{"id": "810.png", "formula": "\\begin{align*} \\tilde { R } ( X , X ) = \\frac { \\int \\limits _ { S M } R _ { i j } X ^ { i } X ^ { j } \\eta ( g ) } { \\underset { S M } \\int \\eta ( g ) } , \\end{align*}"}
+{"id": "667.png", "formula": "\\begin{align*} \\begin{aligned} & ( W ^ { k } f ) _ { p _ { 1 } \\cdots p _ { m - k } q _ { 1 } \\cdots q _ { m - k } } ^ { i _ { 1 } \\cdots i _ { k } } \\\\ & = 2 ^ { m - k } \\sigma ( q _ { 1 } \\cdots q _ { m - k } i _ { 1 } \\cdots i _ { k } ) \\sigma ( p _ { 1 } \\cdots p _ { m - k } ) ( R ^ { k } f ) _ { p _ { 1 } q _ { 1 } \\cdots p _ { m - k } q _ { m - k } } ^ { i _ { 1 } \\cdots i _ { k } } , \\end{aligned} \\end{align*}"}
+{"id": "7414.png", "formula": "\\begin{align*} G : = \\frac { 1 } { 2 } \\int _ { \\R } g ( x ) d x > 0 . \\end{align*}"}
+{"id": "8519.png", "formula": "\\begin{align*} h ( \\sigma ^ k ( y ) ) = \\sigma ^ { \\ell } ( h ( y ) ) . \\end{align*}"}
+{"id": "2854.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ p ( \\mathbb { R } ^ n ) } : = \\left [ \\int _ { \\mathbb { R } ^ n } | f ( x ) | ^ p \\ , d x \\right ] ^ \\frac { 1 } { p } < \\infty . \\end{align*}"}
+{"id": "1310.png", "formula": "\\begin{align*} W _ 2 ( \\hat { \\rho } _ 1 ( t ) , \\rho _ \\infty ) ^ 2 & = \\hat { \\rho } _ 1 ( t ) ^ 2 + \\Big ( \\sqrt { \\hat { \\Sigma } _ 1 } ( t ) - \\sqrt { \\overline { \\Sigma } } _ { 1 1 } \\Big ) ^ 2 \\\\ & \\leq \\hat { \\rho } _ 1 ( t ) ^ 2 + \\Big | \\hat { \\Sigma } _ 1 ( t ) - \\overline { \\Sigma } _ { 1 1 } \\Big | \\\\ & = ( x _ 1 ^ 2 + \\overline { \\Sigma } _ { 1 1 } ) e ^ { 2 a t } . \\end{align*}"}
+{"id": "7312.png", "formula": "\\begin{align*} \\lambda _ j = 1 - \\frac { 1 } { 2 } \\left ( \\exp \\left ( 2 \\pi i \\frac { j + \\beta } { m } \\right ) + \\exp \\left ( - 2 \\pi i \\frac { j + \\beta } { m } \\right ) \\right ) = 2 \\sin ^ 2 \\left ( \\pi \\frac { ( j + \\beta ) } { m } \\right ) \\end{align*}"}
+{"id": "8392.png", "formula": "\\begin{align*} a \\circ ( b c ) = ( a _ 1 \\circ b ) S ( a _ 2 ) ( a _ 3 \\circ c ) , \\forall a , b , c \\in H . \\end{align*}"}
+{"id": "4557.png", "formula": "\\begin{align*} g '' ( \\rho ) - \\left ( A ( \\tau ^ 2 - \\tau ) ( \\rho + 1 ) ^ { - \\tau - 2 } + A ^ 2 \\tau ^ 2 ( \\rho + 1 ) ^ { - 2 \\tau - 2 } \\right ) g ( \\rho ) = 0 . \\end{align*}"}
+{"id": "6220.png", "formula": "\\begin{align*} \\begin{cases} \\dot { z } ( \\phi ) = \\dot { f } ( \\phi ) - c - \\frac { D ( \\phi ) g ( \\phi ) } { z ( \\phi ) } \\ & \\mbox { i n } \\ ( 0 , \\alpha ) \\cup ( \\alpha , \\beta ) \\cup ( \\beta , 1 ) , \\\\ z < 0 \\ & \\mbox { i n } \\ ( 0 , \\alpha ) \\cup ( \\beta , 1 ) , \\\\ z > 0 \\ & \\mbox { i n } \\ ( \\alpha , \\beta ) , \\\\ z ( 0 ) = z ( \\alpha ) = z ( \\beta ) = z ( 1 ) = 0 . \\end{cases} \\end{align*}"}
+{"id": "2758.png", "formula": "\\begin{align*} \\{ \\Theta _ { \\alpha } , \\Theta _ { s + \\beta } \\} = \\delta _ { \\alpha \\beta } , \\end{align*}"}
+{"id": "7999.png", "formula": "\\begin{align*} & \\xi _ t ( i ) \\\\ & \\begin{cases} \\phi \\left ( \\frac { i } { N } \\right ) & k = 2 l = 3 , \\\\ \\psi \\left ( \\frac { i } { N } \\right ) & k = 1 l = 2 , \\\\ \\frac { 1 } { N } \\sum _ { j \\neq i } \\lambda \\left ( \\frac { i } { N } , \\frac { j } { N } \\right ) 1 _ { \\{ \\xi _ t ( j ) = 2 \\} } & k = 0 l = 1 , \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "2954.png", "formula": "\\begin{align*} \\xi ( \\Gamma ) = \\min \\left \\{ | \\lambda | + | \\mu | \\ : \\middle | \\ : \\lambda , \\mu \\in \\Lambda , \\ , \\{ \\lambda , \\mu \\} \\in \\overline { E } \\right \\} \\end{align*}"}
+{"id": "3362.png", "formula": "\\begin{align*} \\begin{aligned} \\begin{cases} & W ( x _ k , \\phi ^ \\circ _ k , u ^ \\circ _ k , k ) = H ( \\phi ^ \\circ _ k ) \\\\ & \\ \\ \\ \\ + \\mathbb { E } ^ { u ^ \\circ } \\left [ W ( b ( x _ k , u ^ \\circ _ k ) + w _ k , \\phi ^ \\circ _ { k + 1 } , u ^ \\circ _ { k + 1 } , k + 1 ) \\right ] , \\\\ & W ( x _ K , \\phi ^ \\circ _ K , u ^ \\circ _ K , K ) = ( \\mu \\Gamma ( x _ K ) ) . \\end{cases} \\end{aligned} \\end{align*}"}
+{"id": "7930.png", "formula": "\\begin{align*} \\langle d \\phi , d \\psi \\rangle _ { L ^ { 2 } \\Lambda ^ { 1 } ( \\Omega ) } = \\int _ { \\partial \\Omega } \\mathrm { t r } ( \\psi ) \\wedge g _ { \\partial } , \\quad \\forall \\psi \\in H \\Lambda ^ { 0 } ( \\Omega ) , \\end{align*}"}
+{"id": "3132.png", "formula": "\\begin{align*} u _ { \\psi ^ * } ( t ) = \\left ( \\begin{array} { c c c } 1 & 0 & t ^ { p ^ { e _ 2 } } \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "2420.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { Z } _ 2 } g ( t ) \\binom { t + 2 ^ m - 1 } { 2 ^ m - 1 } ^ 2 \\binom { t + 2 ^ { m - 1 } - 1 } { 2 ^ { m - 1 } - 1 } ^ { s } \\binom { t + 2 ^ m - 1 } { 2 ^ { m - 1 } - 1 } ^ { s } \\mathrm { d } t \\\\ \\equiv & ~ \\frac { 1 } { 2 ^ { m } } \\sum _ { k = 0 } ^ { 2 ^ m - 1 } g ( k ) \\binom { k + 2 ^ m - 1 } { 2 ^ m - 1 } ^ 2 \\binom { k + 2 ^ { m - 1 } - 1 } { 2 ^ { m - 1 } - 1 } ^ { s } \\binom { k + 2 ^ m - 1 } { 2 ^ { m - 1 } - 1 } ^ { s } \\pmod { 2 ^ { - m + 1 } \\mathbb { Z } _ 2 } . \\end{align*}"}
+{"id": "7801.png", "formula": "\\begin{align*} x _ j ^ 2 = 1 \\end{align*}"}
+{"id": "8805.png", "formula": "\\begin{align*} \\underline \\gamma _ { y _ + } & = \\frac 1 2 \\left ( \\delta _ { ( x _ + - y _ + ) ^ 2 } + \\frac { z _ - - y _ + } { z _ - - y _ - } \\delta _ { ( x _ - - y _ - ) ^ 2 } + \\frac { y _ + - y _ - } { z _ - - y _ - } \\delta _ { ( z _ - - x _ - ) ^ 2 } \\right ) \\\\ & \\le _ { c x } \\frac 1 2 \\left ( \\delta _ { ( x _ - - y _ + ) ^ 2 } + \\frac { z _ - - y _ + } { z _ - - y _ - } \\delta _ { ( x _ + - y _ - ) ^ 2 } + \\frac { y _ + - y _ - } { z _ - - y _ - } \\delta _ { ( z _ - - x _ + ) ^ 2 } \\right ) = \\overline \\gamma _ { y _ + } . \\end{align*}"}
+{"id": "812.png", "formula": "\\begin{align*} \\nabla _ { 0 } ( { \\pounds } _ { \\hat { X } } g _ { i j } ) = 2 \\Psi g _ { i j } + \\Psi _ i y _ j + \\Psi _ j y _ i , \\end{align*}"}
+{"id": "5751.png", "formula": "\\begin{align*} & \\{ \\Psi _ i \\} _ { i \\in \\mathbb { N } } = \\{ \\psi ^ + _ i \\ , | \\ , i \\in I _ 4 \\ \\textup { a n d } \\ \\gamma ^ { + } _ i > 0 \\} \\cup \\{ \\psi ^ - _ i \\ , | \\ , i \\in I _ 4 \\ \\textup { a n d } \\ \\gamma ^ { - } _ i > 0 \\} , \\\\ & \\{ \\Psi _ i \\} _ { i \\in - \\mathbb { N } } = \\{ \\psi ^ + _ i \\ , | \\ , i \\in I _ 4 \\ \\textup { a n d } \\ \\gamma ^ { + } _ i < 0 \\} \\cup \\{ \\psi ^ - _ i \\ , | \\ , i \\in I _ 4 \\ \\textup { a n d } \\ \\gamma ^ { - } _ i < 0 \\} , \\end{align*}"}
+{"id": "4281.png", "formula": "\\begin{align*} \\Delta _ f ( \\div _ f \\sigma ) = \\div _ f ( \\Delta _ f \\sigma ) - \\frac { 1 } { 2 } \\div _ f \\sigma = - \\frac { 1 } { 2 } \\div _ f \\sigma . \\end{align*}"}
+{"id": "2142.png", "formula": "\\begin{align*} w _ 0 = 0 _ { 2 \\times 1 } , W _ 1 = 0 _ { 2 \\times 2 } , \\xi = 0 _ { 4 \\times 1 } , Y _ 1 = 0 _ { 4 \\times 3 } , \\end{align*}"}
+{"id": "85.png", "formula": "\\begin{align*} \\begin{cases} c _ 1 = w _ 1 , & c _ 2 = 0 , \\\\ c _ 3 = w _ 2 , & c _ 4 = - w _ 2 \\gamma \\theta . \\end{cases} \\end{align*}"}
+{"id": "4238.png", "formula": "\\begin{align*} Q ( X , \\tau ) = \\lambda _ 1 E _ 4 ( \\tau ) ^ 3 + \\lambda _ 2 E _ 6 ( \\tau ) ^ 2 , \\end{align*}"}
+{"id": "3006.png", "formula": "\\begin{align*} \\mathrm { R i c } = a \\ g + b \\ \\widetilde { g } + c \\ \\eta \\otimes \\eta \\end{align*}"}
+{"id": "5396.png", "formula": "\\begin{align*} \\nu _ k ( j _ k ) = \\frac { h _ k } { \\mu _ k } \\ , \\left [ \\frac { \\rho _ k ^ { j _ k + 2 } - 1 } { ( \\rho _ k - 1 ) ^ 2 } - \\frac { j _ k + 2 } { \\rho _ k - 1 } \\right ] , \\end{align*}"}
+{"id": "8957.png", "formula": "\\begin{align*} | e w ( x ) - I | \\le d ( \\nabla w ( x ) , S O ( n ) ) + C | \\nabla w ( x ) - I | ^ 2 = d ( \\nabla v ( x ) , S O ( n ) ) + C | \\nabla v ( x ) - O | ^ 2 \\end{align*}"}
+{"id": "2837.png", "formula": "\\begin{align*} \\delta ( \\sigma ^ { * } _ { 3 } ( t ) I ) = \\int ^ { t _ { 2 } } _ { t _ { 1 } } \\left [ - \\dot { P } _ { 1 } - Q ^ { 1 } \\right ] \\delta Q ^ { 1 } d t + \\left [ \\dot { Q } ^ { 1 } - P _ { 1 } \\right ] \\delta P _ { 1 } d t + \\left [ P _ { 1 } \\delta Q ^ { 1 } \\right ] ^ { t _ { 2 } } _ { t _ { 1 } } \\end{align*}"}
+{"id": "2953.png", "formula": "\\begin{align*} U ( z , \\rho ) = \\rho ^ s \\int _ { \\R ^ n } F ( z - \\xi ) \\varphi _ { s , \\rho } ( \\xi ) d \\xi \\end{align*}"}
+{"id": "2618.png", "formula": "\\begin{align*} ( n - 3 ) - 2 ( n / 4 - 2 ) = \\frac { n } { 2 } + 1 \\end{align*}"}
+{"id": "7664.png", "formula": "\\begin{align*} K _ L ( n , \\omega ; t ) = \\sum _ { m \\in \\mathbb { Z } ^ d } a ( n , m ) \\left ( G _ L ( m , m ; t - i \\eta ) - G _ L ( m , m ; t + i \\eta ) \\right ) \\end{align*}"}
+{"id": "5734.png", "formula": "\\begin{align*} \\Vert ( 0 , E _ 1 ( u ) ) \\Vert ^ 2 _ { G } ( t ) = \\sum _ { i \\in I _ 1 } ( \\mathcal { E } _ { i , 1 } ( t ) ) ^ 2 + ( \\mathcal { E } _ { i , 2 } ( t ) ) ^ 2 + \\sum _ { i \\in I _ 2 } ( \\mathcal { E } _ { i , 3 } ( t ) ) ^ 2 + ( \\mathcal { E } _ { i , 4 } ( t ) ) ^ 2 + \\sum _ { i \\in I _ 3 \\cup I _ 4 } ( \\mathcal { E } _ { i } ^ + ( t ) ) ^ 2 + ( \\mathcal { E } _ { i } ^ - ( t ) ) ^ 2 \\\\ = O ( e ^ { 2 ( \\gamma _ * - \\varepsilon _ 0 ) t } ) . \\end{align*}"}
+{"id": "4489.png", "formula": "\\begin{align*} \\sum _ { \\beta \\in I _ 1 | \\alpha | = k } | \\partial ^ { \\alpha } f _ 1 ^ * ( z _ \\beta ) | ^ 2 \\le C _ 2 \\| f _ 1 ^ * \\| _ { \\partial M , \\rho } ^ 2 \\le C _ 1 C _ 2 \\sum _ { \\beta \\in I _ 1 , \\alpha \\in L _ { k - 1 } } | a _ { \\beta , \\alpha } | ^ 2 , \\end{align*}"}
+{"id": "1093.png", "formula": "\\begin{align*} \\left \\| A _ Q A _ R ^ { - 1 } \\right \\| \\lesssim \\begin{cases} \\displaystyle \\left [ \\frac { \\ell ( R ) } { \\ell ( Q ) } \\right ] ^ { d / p } & Q \\subset R , \\\\ \\displaystyle \\left [ \\frac { \\ell ( Q ) } { \\ell ( R ) } \\right ] ^ { \\widetilde d / p ' } & R \\subset Q . \\end{cases} \\end{align*}"}
+{"id": "8882.png", "formula": "\\begin{align*} 0 < \\frac { \\alpha + n } { 2 n p } \\leq \\frac 1 2 < 1 , - \\frac { \\alpha + n } { 2 p } < - \\frac { \\tau } { p } \\leq 0 , \\frac { \\tau } { p } - 1 = \\frac { \\alpha + n } { 2 p } - \\frac { n } { 2 } . \\end{align*}"}
+{"id": "6360.png", "formula": "\\begin{align*} \\omega _ 1 ^ 1 ( r ) & : = \\frac { \\sinh ^ 2 ( r ) } { r ^ 2 } , \\\\ \\omega _ 1 ^ 2 ( r ) & : = \\frac { \\sinh ^ 2 ( r ) \\cosh ^ 2 ( r ) } { r ^ 2 } , \\end{align*}"}
+{"id": "4868.png", "formula": "\\begin{align*} \\textrm { T r a c e } ( A ^ 3 ) = \\kappa _ 1 ^ 3 + \\kappa _ 2 ^ 3 = ( \\kappa _ 1 + \\kappa _ 2 ) ( \\kappa _ 1 ^ 2 - \\kappa _ 1 \\kappa _ 2 + \\kappa _ 2 ^ 2 ) = H ( | A | ^ 2 - K ) . \\end{align*}"}
+{"id": "5973.png", "formula": "\\begin{align*} S _ e = \\{ i : e \\in p _ i \\} \\end{align*}"}
+{"id": "5022.png", "formula": "\\begin{align*} \\mathbf { r } _ S ^ S & = \\mathbf { f } _ S ^ S - \\beta \\mathbf { f } _ S ^ S = ( 1 - \\beta ) \\mathbf { f } _ S ^ S \\\\ \\mathbf { r } _ { S ^ c } ^ S & = \\mathbf { R } _ { S ^ c } + \\beta \\mathbf { P } _ { S ^ c N } \\mathbf { f } ^ S - \\mathbf { f } _ { S ^ c } ^ S = \\mathbf { R } _ { S ^ c } + \\beta \\mathbf { P } _ { S ^ c N } \\mathbf { f } ^ S . \\end{align*}"}
+{"id": "7246.png", "formula": "\\begin{align*} \\frac { ( n - 1 ) ! } { ( n - i ) ! } t _ i ( n - i ) ! Z _ { n - i } = ( n - 1 ) ! t _ i Z _ { n - i } , \\end{align*}"}
+{"id": "8399.png", "formula": "\\begin{align*} \\Delta ( \\sigma ( a ) ) = B _ 1 ( a _ 1 ) S ( B _ 2 ( a _ 2 ) ) \\otimes B _ 1 ( a _ 3 ) S ( B _ 2 ( a _ 4 ) ) = \\sigma ( a _ 1 ) \\otimes \\sigma ( a _ 2 ) . \\end{align*}"}
+{"id": "5504.png", "formula": "\\begin{align*} X _ 2 X _ 1 & = q ^ { - 2 } X _ 1 X _ 2 , \\\\ X _ 3 X _ 1 & { } = X _ 1 X _ 3 - ( q + q ^ { - 1 } ) X _ 2 , & & X _ 3 X _ 2 = q ^ { - 2 } X _ 2 X _ 3 , \\\\ X _ 4 X _ 1 & { } = q ^ 2 X _ 1 X _ 4 - q ^ 2 X _ 3 , & & X _ 4 X _ 2 = X _ 2 X _ 4 - \\frac { q ^ 2 - 1 } { q + q ^ { - 1 } } X _ 3 ^ 2 , ~ ~ X _ 4 X _ 3 = q ^ { - 2 } X _ 3 X _ 4 . \\end{align*}"}
+{"id": "702.png", "formula": "\\begin{align*} f ( x ) = \\Phi ^ 3 ( x ) - 6 x ^ 2 \\Phi ^ 2 ( x ) + ( 1 6 x ^ 2 - 3 2 ) \\Phi ( x ) + 6 4 - m \\end{align*}"}
+{"id": "3727.png", "formula": "\\begin{align*} \\left | A + B + ( \\{ 0 , 1 \\} ^ d \\times \\{ 0 \\} ^ k ) \\right | ^ { 1 / ( k + d ) } & = \\left | ( A + ( [ 0 , 1 ] ^ d \\times \\{ 0 \\} ^ k ) ) + ( B + ( [ 0 , 1 ] ^ d \\times \\{ 0 \\} ^ k ) ) \\right | ^ { 1 / ( k + d ) } \\\\ & \\geq \\left | A + ( [ 0 , 1 ] ^ d \\times \\{ 0 \\} ^ k ) \\right | ^ { 1 / ( k + d ) } + \\left | B + ( [ 0 , 1 ] ^ d \\times \\{ 0 \\} ^ k ) \\right | ^ { 1 / ( k + d ) } \\\\ & = \\left | A \\right | ^ { 1 / ( k + d ) } + \\left | B \\right | ^ { 1 / ( k + d ) } , \\end{align*}"}
+{"id": "9018.png", "formula": "\\begin{align*} y _ { n n } = - y _ { 1 1 } z _ { 1 1 } ^ n = z _ { n n } ^ 1 . \\end{align*}"}
+{"id": "2452.png", "formula": "\\begin{align*} U ( \\xi ) \\sim \\mu ^ { - \\frac { 1 } { p } } \\sim \\begin{cases} \\left ( B _ { 1 } e ^ { \\omega _ { 1 } \\xi } + B _ { 2 } e ^ { \\omega _ { 2 } \\xi } + \\dfrac { 1 } { \\mu } \\right ) ^ { \\frac { 1 } { p } } , ( D > 0 ) , \\\\ \\left ( ( B _ { 3 } \\xi + B _ { 4 } ) e ^ { \\omega \\xi } + \\dfrac { 1 } { \\mu } \\right ) ^ { \\frac { 1 } { p } } , ( D = 0 ) , \\\\ \\left ( e ^ { - \\frac { \\mu c } { 2 } \\xi } \\bar { Z } ( \\xi ) + \\dfrac { 1 } { \\mu } \\right ) ^ { \\frac { 1 } { p } } , ( D < 0 ) , \\end{cases} \\end{align*}"}
+{"id": "6269.png", "formula": "\\begin{align*} g ( x , \\xi , \\mathbf { e } ) = \\frac { d } { 2 \\tau } ( \\phi ( x + \\tau \\mathbf { e } , \\xi ) - \\phi ( x - \\tau \\mathbf { e } , \\xi ) ) \\mathbf { e } \\end{align*}"}
+{"id": "3295.png", "formula": "\\begin{align*} \\Pr [ K _ 1 ^ d \\ge x ] = \\Pr \\left [ | Z | \\ge \\sqrt { 2 \\log ( x / d ) } \\right ] \\le 2 \\exp \\left ( - \\sqrt { 2 \\log ( x / d ) } ^ 2 / 2 \\right ) = \\frac { 2 d } { x } \\end{align*}"}
+{"id": "2419.png", "formula": "\\begin{align*} \\triangle _ m \\left ( \\binom { t + 2 ^ m - 1 } { 2 ^ m - 1 } ^ 2 \\right ) & \\geqslant - ( m - 1 ) + 1 = - m + 2 , \\\\ \\triangle _ m \\left ( \\binom { t + 2 ^ { m - 1 } - 1 } { 2 ^ { m - 1 } - 1 } \\right ) & \\geqslant - ( m - 2 ) = - m + 2 , \\\\ \\triangle _ m \\left ( \\binom { t + 2 ^ m - 1 } { 2 ^ { m - 1 } - 1 } \\right ) & \\geqslant - ( m - 2 ) = - m + 2 . \\end{align*}"}
+{"id": "9164.png", "formula": "\\begin{align*} G _ { \\beta , \\beta ' } = ( w _ { \\beta , 1 } - v ^ { 4 n - 4 j + 2 } w _ { \\beta ' , 1 } ) ( w _ { \\beta , 1 } - v ^ { 4 n - 4 j - 4 } w _ { \\beta ' , 1 } ) ( w _ { \\beta , 1 } - v ^ { 4 } w _ { \\beta ' , 1 } ) \\cdot G _ { [ 1 , n , j + 1 ] , \\beta ' } , \\end{align*}"}
+{"id": "1722.png", "formula": "\\begin{align*} \\Phi \\left ( z _ { 1 , 1 } , \\ldots , z _ { \\mu , n _ \\mu } , \\xi , \\overline { z _ { 1 , 1 } } , \\ldots , \\overline { z _ { \\mu , n _ \\mu } } , \\overline { \\xi } \\right ) : = \\sum _ { j = 1 } ^ \\mu P _ { f _ j , n _ j } ( \\Re z _ { j , 1 } , \\ldots , \\Re z _ { j , n _ j } ) \\end{align*}"}
+{"id": "7464.png", "formula": "\\begin{align*} \\partial _ t g ( t ) = - 2 R i c ( g ( t ) ) , \\end{align*}"}
+{"id": "2689.png", "formula": "\\begin{align*} H = \\sum _ { i = 1 } ^ { n } \\sum ^ { d - 1 } _ { \\alpha = 1 } P ^ { ( \\alpha ) } _ { i } Q ^ { i } _ { ( \\alpha + 1 ) } + \\sum _ { i = 1 } ^ { n } P ^ { ( d ) } _ { i } F ^ { i } - L , \\end{align*}"}
+{"id": "1.png", "formula": "\\begin{align*} q ( t , x ) = \\widetilde q ( t , x - 2 c t ) + c . \\end{align*}"}
+{"id": "5545.png", "formula": "\\begin{align*} \\eta = \\frac { \\log ( m ) } { \\log ( n ) } - 1 . \\end{align*}"}
+{"id": "3891.png", "formula": "\\begin{gather*} \\omega _ 1 + ( \\omega _ 3 - \\omega _ 1 ) + ( \\omega _ 4 - \\omega _ 3 ) + ( \\omega _ 2 + \\omega _ 5 - \\omega _ 4 ) + ( \\omega _ 2 + \\omega _ 6 - \\omega _ 5 ) \\\\ + ( \\omega _ 2 + \\omega _ 7 - \\omega _ 6 ) + ( \\omega _ 2 + \\omega _ 8 - \\omega _ 7 ) + ( \\omega _ 2 - \\omega _ 8 ) = 5 \\omega _ 2 \\end{gather*}"}
+{"id": "8370.png", "formula": "\\begin{align*} { } ^ G k ( G \\times _ B Z ) = { } ^ G ( k ( G \\times Z ) ^ B ) = ( { } ^ G k ( G \\times Z ) ) ^ B = k ( Z ) ^ B . \\end{align*}"}
+{"id": "3989.png", "formula": "\\begin{align*} z ^ { \\left ( t \\right ) } = W ^ { t } \\left ( z \\right ) = \\sum _ { i = 1 } ^ { n } x _ { i } ^ { \\ : \\left ( t \\right ) } e _ { i } + \\sum _ { p = 1 } ^ { \\nu } y _ { p } ^ { \\ : \\left ( t \\right ) } \\widetilde { e } _ { p } \\end{align*}"}
+{"id": "6614.png", "formula": "\\begin{align*} \\phi _ { \\Phi _ k , \\beta } ( x , y ) & = R _ { ( k , \\beta ) } \\left ( \\frac { 1 } { ( 1 - \\delta ) } \\right ) y ^ { \\frac { 1 } { ( 1 - \\delta ) } } + O _ \\delta ( x \\alpha ^ { \\frac { 1 } { \\log x } } ( \\log x \\log \\log x ) ^ { \\frac { 1 } { 2 } } ) . \\end{align*}"}
+{"id": "923.png", "formula": "\\begin{align*} T _ t f ( x ) = \\int _ { \\Omega } f ( \\omega _ t ) \\ , d P _ x ( \\omega ) m x \\in E . \\end{align*}"}
+{"id": "4761.png", "formula": "\\begin{align*} \\varphi _ 2 ' ( \\varepsilon , b , c , n , \\varphi , \\omega ) : = \\psi ( \\varepsilon c \\min \\{ \\varphi _ 1 ( \\min \\{ \\varepsilon / 2 , c / 2 \\} , b , n , \\varphi ) , \\lambda _ 0 / 2 \\} / 4 , b + 2 n + 3 n ^ 2 , \\omega ) \\end{align*}"}
+{"id": "3125.png", "formula": "\\begin{align*} \\varphi ^ * \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & = \\left ( \\begin{array} { c c c } a ^ { \\ell _ 1 } & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & d ^ { - \\ell _ 3 } \\end{array} \\right ) ( \\ , \\ell _ 1 > 0 > \\ell _ 3 \\ , ) . \\end{align*}"}
+{"id": "8062.png", "formula": "\\begin{align*} P ( \\eta ^ N \\in O ) & \\geq \\exp \\left ( - \\frac { \\gamma _ N ^ 2 } { N } ( J _ { i n i } ( W ^ m _ 0 ) + \\mathcal { J } _ 1 ( W ^ m , F ^ m , G ^ m , H ^ m ) + 4 \\epsilon ) \\right ) \\\\ & \\times P ( D _ { 5 , \\epsilon } \\cap D _ { 4 , \\epsilon } \\cap \\{ \\eta ^ N \\in O \\} ) \\\\ & = \\exp \\left ( - \\frac { \\gamma _ N ^ 2 } { N } ( J _ { i n i } ( W ^ m _ 0 ) + J _ { d y n } ( W ^ m ) + 4 \\epsilon ) \\right ) ( 1 + o ( 1 ) ) . \\end{align*}"}
+{"id": "1431.png", "formula": "\\begin{align*} R ^ p \\pi _ * \\left ( \\mathcal M \\otimes R ^ q f _ * \\omega _ X \\right ) = 0 \\end{align*}"}
+{"id": "4850.png", "formula": "\\begin{align*} a _ { m _ * } & = C \\mathcal { H } ^ { n - 1 } ( \\partial E _ m ) \\\\ & \\approx \\mathcal { H } ^ { n - 1 } ( \\partial \\mu _ m K ) \\\\ & = \\mu _ m ^ { n - 1 } \\mathcal { H } ^ { n - 1 } ( \\partial K ) = ( \\frac { m } { | K | } ) ^ { \\frac { n - 1 } { n } } \\mathcal { H } ^ { n - 1 } ( \\partial K ) \\end{align*}"}
+{"id": "2104.png", "formula": "\\begin{align*} U _ k ( s , x ) = U ( 0 , x ) - \\alpha \\int _ 0 ^ s \\int _ 0 ^ 1 A ( x , y ) U _ { k - 1 } ( u , y ) \\mathrm { d } y \\mathrm { d } u + \\alpha \\beta \\int _ 0 ^ s \\mathrm { d } \\xi _ 2 ( u , x ) + \\alpha \\zeta \\int _ 0 ^ s \\mathrm { d } \\xi _ 3 ( u , x ) . \\end{align*}"}
+{"id": "5800.png", "formula": "\\begin{align*} X _ + X _ + ' = & \\sum _ { k + \\ell \\leq s } \\sum _ { i : \\lambda _ i > 0 } \\left ( \\lambda _ i | \\xi ^ { ( k , \\ell ) } _ i ( t ) | ^ 2 + \\xi ^ { ( k , \\ell ) } _ i ( t ) \\mathcal { E } ^ { ( k , \\ell ) } _ i ( t ) \\right ) \\\\ = & \\sum _ { k + \\ell \\leq s } \\sum _ { i : \\lambda _ i > 0 } \\left ( \\lambda _ i | \\xi ^ { ( k , \\ell ) } _ i ( t ) | ^ 2 \\right ) + X _ + Y _ + , \\end{align*}"}
+{"id": "8265.png", "formula": "\\begin{align*} H ( Y _ { i + 1 } | Y ^ { i } , \\underline S ) = \\sum _ { y ^ { i } } \\frac { c _ { y ^ { i } } } { M } H \\left ( \\frac { b _ { y ^ { i } } } { c _ { y ^ { i } } } \\right ) . \\end{align*}"}
+{"id": "188.png", "formula": "\\begin{align*} \\tilde \\pi ( 0 , x ) \\ = \\ x , \\tilde \\pi ( 1 , x ) \\ = \\ \\infty . \\end{align*}"}
+{"id": "2245.png", "formula": "\\begin{align*} v ^ i _ \\alpha ( \\phi _ s ^ { ( 1 ) } ( t ) ) = \\frac { \\partial \\phi ^ i } { \\partial t ^ \\alpha } \\Big \\vert _ { t } \\ , , \\Gamma ^ i _ { \\alpha \\beta } ( \\phi _ s ^ { ( 1 ) } ( t ) ) = \\frac { \\partial ^ 2 \\phi ^ i } { \\partial t ^ { \\alpha } \\partial t ^ { \\beta } } \\Big \\vert _ { t } \\ , , \\Gamma _ \\alpha ^ \\beta ( \\psi ( t ) ) = \\frac { \\partial s ^ \\alpha } { \\partial t ^ \\beta } \\Big \\vert _ t \\ , . \\end{align*}"}
+{"id": "3962.png", "formula": "\\begin{align*} \\rho _ { \\restriction Y _ n } = \\rho _ \\pi \\circ \\iota _ n , \\end{align*}"}
+{"id": "8201.png", "formula": "\\begin{align*} P _ { Y ^ { i - 1 } , { \\bf S } } ( y ^ { i - 1 } , { \\bf s } ) & = P _ { Y _ { i - 1 } | Y ^ { i - 2 } , { \\bf S } } ( y _ { i - 1 } | y ^ { i - 2 } , { \\bf s } ) P _ { Y ^ { i - 2 } , { \\bf S } } ( y ^ { i - 2 } , { \\bf s } ) \\\\ & = \\prod _ { j = 1 } ^ { i - 1 } \\beta ( y _ { j } , y ^ { j - 1 } ) . \\end{align*}"}
+{"id": "3061.png", "formula": "\\begin{align*} a _ { 3 3 } & = \\frac { - a _ { 1 3 } ^ 2 b _ { 2 2 } + 2 a _ { 1 1 } a _ { 2 3 } b _ { 2 3 } + b } { 4 a _ { 1 1 } b _ { 2 2 } } , \\\\ b _ { 3 3 } & = \\frac { a _ { 1 1 } b _ { 2 3 } ^ 2 - 2 a _ { 1 3 } b _ { 1 3 } b _ { 2 2 } - c } { 4 a _ { 1 1 } b _ { 2 2 } } . \\end{align*}"}
+{"id": "1695.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial z _ j } : = \\frac { 1 } { 2 } \\left ( \\frac { \\partial } { \\partial x _ j } - i \\frac { \\partial } { \\partial y _ j } \\right ) \\quad \\forall \\ , j \\in \\{ 1 , \\ldots , n \\} . \\end{align*}"}
+{"id": "8918.png", "formula": "\\begin{align*} \\Phi _ n ( 1 ) = \\begin{cases} p & n = p ^ r , \\ p , \\\\ 1 & . \\end{cases} \\end{align*}"}
+{"id": "3373.png", "formula": "\\begin{align*} I : = I _ 1 + I _ 2 + I _ 3 + I _ 4 + I _ 5 . \\end{align*}"}
+{"id": "8430.png", "formula": "\\begin{align*} B ( M ) _ { \\tilde x _ { n } } = B ( M ) _ { \\tilde x _ { n + 1 } } \\cup B ( M ) _ { \\tilde s \\tilde x _ { n } } . \\end{align*}"}
+{"id": "2760.png", "formula": "\\begin{align*} \\{ \\Theta _ { i } , \\Theta _ { j } \\} = f _ { i j } ( \\Theta _ { k } ) \\end{align*}"}
+{"id": "8476.png", "formula": "\\begin{align*} \\lim _ { h \\searrow 0 } \\left \\| ( \\bar { u } _ h ) _ + - ( u _ h ) _ + \\right \\| _ { L ^ 1 ( K _ T ) } = 0 . \\end{align*}"}
+{"id": "1004.png", "formula": "\\begin{align*} u = \\xi + h . \\end{align*}"}
+{"id": "2440.png", "formula": "\\begin{align*} \\lim _ { \\xi \\nearrow \\xi _ { + } - 0 } \\phi ( \\xi ) = 0 { \\rm a n d } \\lim _ { \\xi \\nearrow \\xi _ { + } - 0 } \\phi ' ( \\xi ) = - C \\end{align*}"}
+{"id": "3328.png", "formula": "\\begin{align*} x = \\big ( x ^ { ( 0 ) } , x ^ { ( 1 ) } , \\ldots , x ^ { ( \\kappa ) } \\big ) , x ^ { ( 0 ) } \\ ! \\in \\R ^ { m _ 0 } , x ^ { ( j ) } \\ ! \\in \\R ^ { m _ j } , j \\in \\{ 1 , \\ldots , \\kappa \\} , \\end{align*}"}
+{"id": "6813.png", "formula": "\\begin{align*} G / K = \\left \\{ \\left . \\prod _ { 2 \\leq k \\leq N } A ( k ) ( \\phi _ { k - 1 } , \\psi _ { k - 1 } ) \\ , \\right | \\ , \\phi _ k \\in [ 0 , 2 \\pi ] , \\psi _ k \\in \\left [ 0 , \\frac { \\pi } { 2 } \\right ] \\right \\} . \\end{align*}"}
+{"id": "7385.png", "formula": "\\begin{align*} \\begin{array} { l l } N : = & \\d 2 ^ { p + q - 1 } \\sum _ { \\gamma = 0 } ^ 1 A E ( \\| u _ t ^ 0 \\| _ \\infty ^ { p - \\gamma } \\| u _ { t x } ^ 0 \\| _ \\infty ^ \\gamma \\| u _ 0 \\| _ \\infty ^ q + \\| u _ t ^ 0 \\| _ \\infty ^ p \\| u ^ 0 \\| _ \\infty ^ { q - \\gamma } \\| u _ x ^ 0 \\| _ \\infty ^ \\gamma ) \\\\ & \\d + \\sum _ { \\gamma = 0 } ^ 1 2 ^ { r - \\gamma } B E \\| u ^ 0 \\| _ \\infty ^ { r - \\gamma } \\| u _ x ^ 0 \\| ^ \\gamma , \\end{array} \\end{align*}"}
+{"id": "2920.png", "formula": "\\begin{align*} T _ 1 ( \\ell ) = \\frac { T ( \\ell ) } { \\ell \\log \\ell } . \\end{align*}"}
+{"id": "7147.png", "formula": "\\begin{align*} \\frac { d x ^ \\mu } { d \\tau } = \\frac { \\partial q } { \\partial k _ \\mu } \\quad { \\mbox { a n d } } \\frac { d k _ \\nu } { d \\tau } = - \\frac { \\partial q } { \\partial x ^ \\nu } \\ , \\ , . \\end{align*}"}
+{"id": "3699.png", "formula": "\\begin{align*} \\frac { n _ 2 } { \\log n _ 2 + 2 } - \\frac { 1 } { 4 2 0 } \\frac { n _ 2 } { \\log ^ 2 n _ 2 } - \\frac { n _ 1 } { \\log n _ 1 - 4 } - \\frac { 1 } { 4 2 0 } \\frac { n _ 1 } { \\log ^ 2 n _ 1 } = \\Omega ( \\sqrt { d - 1 } / \\log ( d - 1 ) ) . \\end{align*}"}
+{"id": "2321.png", "formula": "\\begin{align*} w ( x , t ) = e ^ { - t \\mathcal { L } } w _ { 0 } - \\int ^ { t } _ { 0 } e ^ { - ( t - s ) \\mathcal { L } } \\mathbb { P } \\mathrm { d i v } ( w \\otimes w ) d s : = a + N ( w , w ) . \\end{align*}"}
+{"id": "7895.png", "formula": "\\begin{align*} i ^ { ( n ) } _ { \\pi ( k ) } = \\begin{cases} i _ { \\pi ( k ) } , & \\\\ i _ { \\pi ( k ) } + n , & \\end{cases} , j ^ { ( n ) } _ { \\pi ( k ) } = \\begin{cases} j _ { \\pi ( k ) } , & \\\\ j _ { \\pi ( k ) } + n , & \\end{cases} \\end{align*}"}
+{"id": "5064.png", "formula": "\\begin{align*} x _ i ^ { c , ( 1 ) } ( k ) ) = x _ i ^ { c , ( 2 ) } ( k ) ) , y _ i ^ { c , ( 1 ) } ( k ) ) = y _ i ^ { c , ( 2 ) } ( k ) ) . \\end{align*}"}
+{"id": "8552.png", "formula": "\\begin{align*} p _ { 1 } ( n ) f ( n + r ) + p _ { 2 } ( n ) f ( n ) = - G ( n , 0 ) , \\end{align*}"}
+{"id": "770.png", "formula": "\\begin{align*} ( a \\otimes b ) ^ * = ( 1 \\otimes b ^ * ) \\cdot ( a ^ * \\otimes 1 ) = R ( b ^ { ( 2 ) * } , a ^ { ( 2 ) * } ) a ^ { ( 1 ) * } \\otimes b ^ { ( 1 ) * } . \\end{align*}"}
+{"id": "3335.png", "formula": "\\begin{align*} \\begin{cases} & P _ { s } = p _ { 0 } + \\sqrt { 2 } \\int \\limits _ 0 ^ s \\sqrt { P _ \\tau ^ 2 + 1 } \\ , d W _ { \\tau } , \\\\ & X _ { s } = x _ { 0 } + \\int \\limits _ { 0 } ^ { s } P _ { \\tau } \\dd \\tau , \\\\ & T _ { s } = t _ { 0 } + \\int \\limits _ { 0 } ^ { s } \\sqrt { P _ \\tau ^ 2 + 1 } \\ , \\dd \\tau , \\end{cases} \\end{align*}"}
+{"id": "1399.png", "formula": "\\begin{align*} 0 = ( ( - a ) + a ) * a _ { m - k } = ( - a ) * a _ { m - k + 1 } + ( - a ) * a _ { m - k } + a _ { m - k + 1 } \\end{align*}"}
+{"id": "3606.png", "formula": "\\begin{align*} \\alpha ^ 3 \\tau _ 0 ^ 2 \\tau _ 1 f ( \\tau , w \\sigma ^ 2 \\sigma _ 1 ) - ( 1 + a ) \\alpha ^ 2 \\tau _ 0 \\tau _ 1 f ( i d , w \\sigma \\sigma _ 1 ) + ( a + b ) \\alpha \\tau _ 0 f ( \\tau , w \\sigma ) - b f = 0 , \\end{align*}"}
+{"id": "4985.png", "formula": "\\begin{align*} Q _ N ( \\{ p \\} ) = \\det [ p _ { i j } ] _ { i , j = 1 , \\ldots , N } , \\end{align*}"}
+{"id": "3631.png", "formula": "\\begin{align*} \\alpha ^ 3 \\tau _ 0 ^ 3 - ( 1 + a ) ( \\alpha \\tau _ 0 ) ^ 2 + ( a + b ) ( \\alpha \\tau _ 0 ) - b = 0 , \\end{align*}"}
+{"id": "7931.png", "formula": "\\begin{align*} \\phi ( x ) = N _ { \\phi } ( g _ { \\partial } ) , \\end{align*}"}
+{"id": "2839.png", "formula": "\\begin{align*} & - \\dot { P } _ { 1 } - Q ^ { 1 } = 0 , \\\\ & \\dot { Q } ^ { 1 } - P _ { 1 } = 0 . \\end{align*}"}
+{"id": "7740.png", "formula": "\\begin{align*} S ^ { - 1 } ( \\lambda ) A ( \\lambda ) S ( \\lambda ) = \\mathrm { d i a g } \\{ A _ { 1 1 } ( \\lambda ) , \\cdots , A _ { l l } ( \\lambda ) \\} , \\end{align*}"}
+{"id": "4462.png", "formula": "\\begin{align*} M _ { S } ( Z _ 0 , J , \\rho _ 1 \\rho _ 2 ) = \\| f _ 0 \\| ^ 2 _ { S , \\rho _ 1 \\rho _ 2 } . \\end{align*}"}
+{"id": "7315.png", "formula": "\\begin{align*} G _ { X _ { m } , \\chi _ { \\beta } } ( x , y ; s ) = e ^ { - 2 \\pi i \\beta \\frac { \\ell - ( x - y ) } { m } } \\sum _ { j = - \\infty } ^ { \\infty } e ^ { - 2 \\pi i \\beta j } \\int \\limits _ { 0 } ^ { \\infty } e ^ { - ( s + 1 ) t } I _ { | \\ell + j m | } ( t ) d t , \\end{align*}"}
+{"id": "4883.png", "formula": "\\begin{align*} \\mathcal E = \\O _ { \\Gamma } \\oplus \\L \\end{align*}"}
+{"id": "21.png", "formula": "\\begin{align*} \\lim _ { \\kappa \\to \\infty } \\ , \\sup _ { q \\in Q _ * } \\ , \\sup _ { | t | \\leq T } \\norm { n ( t ) - n ( 0 ) } _ { H ^ { s + 1 } } = 0 . \\end{align*}"}
+{"id": "5039.png", "formula": "\\begin{align*} \\tilde { \\mathbf { P } } _ { S _ k ^ c S _ { k } ^ c } ^ { ( k ) } & = \\mathbf { I } - ( 1 - \\beta ) \\mathbf { A } _ { S _ k ^ c S _ { k } ^ c } ^ { ( k ) } \\\\ \\tilde { r } _ { S _ k ^ c } ^ { ( k ) } & = ( 1 - \\beta ) \\mathbf { r } _ { S _ k ^ c } ^ { ( k ) } \\\\ \\boldsymbol { \\beta } _ { S _ k ^ c } ^ { ( k ) } & = \\mathbf { 1 } _ { S _ k ^ c } - ( 1 - \\beta ) \\mathbf { w } _ { S _ k ^ c } ^ { ( k ) } , \\end{align*}"}
+{"id": "5912.png", "formula": "\\begin{align*} f ( p , q ) = ( q - 1 ) ( q - 2 ) ^ 3 ( p q - 1 ) - ( q - 1 ) ^ 2 ( q - 2 ) ^ 3 + ( p - 1 ) ( p - 2 ) ^ 3 q ( p q - 1 ) - ( p - 1 ) ^ 2 ( p - 2 ) ^ 3 q ^ 2 . \\end{align*}"}
+{"id": "5681.png", "formula": "\\begin{align*} N _ 1 ( u ) = & a _ 1 \\cdot D u ' + a _ 2 \\cdot u ' + a _ 3 \\cdot \\mathcal { M } _ \\Sigma ( u ) , \\\\ N _ 2 ( u ) = & b _ 1 \\cdot \\mathcal { M } _ \\Sigma ( u ) . \\end{align*}"}
+{"id": "6664.png", "formula": "\\begin{align*} L ( E , \\mathrm { i } y ) = \\int _ { 0 } ^ { y } \\nu ^ { + } ( E , \\mathrm { i } \\tilde { y } ) \\mathrm { d } \\tilde { y } + L ( E , \\mathrm { i } 0 ) . \\end{align*}"}
+{"id": "8868.png", "formula": "\\begin{align*} \\frac { ( 2 p - 1 ) n } { r } = \\alpha + 2 p - 2 \\tau - 1 + \\frac { n } { 2 } . \\end{align*}"}
+{"id": "8605.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d X } { d t } ( t , x ) = u ( X ( t ; x ) , t ) , \\\\ X ( 0 ; t ) = x . \\end{cases} \\end{align*}"}
+{"id": "392.png", "formula": "\\begin{align*} M _ { c } ( f ) : = \\bigcap \\{ M _ { d } ( f ( d ) ) : d \\in { \\rm s u p p } ( f ^ { c } ) \\} = \\bigcap \\{ M _ { d } ( f ( d ) ) : c \\leq d \\in { \\rm s u p p } ( f ) \\} . \\end{align*}"}
+{"id": "1548.png", "formula": "\\begin{align*} F ( z ) = \\int _ 0 ^ { | z | } ( D F ( s \\ , z / | z | ) , z / | z | ) \\ , d s \\le \\int _ 0 ^ { | z | } a ( s ) \\ , d s = A ( | z | ) \\end{align*}"}
+{"id": "4797.png", "formula": "\\begin{align*} \\left < \\frac { z - ( - r \\cos \\theta E _ n ) } { r } , E _ n \\right > = \\cos \\theta . \\end{align*}"}
+{"id": "2374.png", "formula": "\\begin{align*} \\mu _ n = \\sum _ { j = 1 } ^ k \\alpha _ { n , j } * \\delta _ { ( a _ { n , j } , a _ { n , j } ) } + \\beta _ n \\ , , \\end{align*}"}
+{"id": "1077.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { n } \\Vert ( T _ n ( w ) ^ { - 1 } ) ^ { s , t } \\| \\leq B _ 1 + B _ 2 , n \\in \\N , \\ \\ t \\in \\{ 1 , \\dots , [ ( n + 1 ) / 2 ] \\} . \\end{align*}"}
+{"id": "6900.png", "formula": "\\begin{align*} F ( \\tau ) & : = 4 | \\log p ( z _ { 0 } ) | ^ { 2 } - 4 = \\log ^ { 2 } ( A _ { N } ^ { 2 } ( \\tau ) + A _ { D } ^ { 2 } ( \\tau ) ) + 4 \\left ( \\tan ^ { - 1 } \\frac { A _ { N } ( \\tau ) } { A _ { D } ( \\tau ) } \\right ) ^ { 2 } - 4 \\geq 0 . \\end{align*}"}
+{"id": "244.png", "formula": "\\begin{align*} \\frac { d ^ 2 x _ 2 } { d t _ 2 ^ 2 } + \\frac { 1 } { h ( x _ 1 ) } A _ 1 ( x _ 1 ) \\frac { d x _ 2 } { d t _ 2 } + \\frac { 1 } { h ( x _ 1 ) } b _ 1 ( x _ 1 ) = 0 , \\end{align*}"}
+{"id": "2633.png", "formula": "\\begin{align*} s _ n = s _ { n - 2 } + s _ { n - 3 } - s _ { n - 6 } . \\end{align*}"}
+{"id": "5381.png", "formula": "\\begin{align*} \\frac { d } { d t } \\bar { v } _ t = \\nu _ { \\pi _ k } , \\bar { b } ^ { S _ { k + 1 } } < t < \\bar { b } ^ { S _ k } . \\end{align*}"}
+{"id": "4239.png", "formula": "\\begin{align*} E _ 4 ( \\tau ) ^ 3 = 1 + 7 2 0 q + 1 7 9 2 8 0 q ^ 2 + \\cdots , \\end{align*}"}
+{"id": "2007.png", "formula": "\\begin{align*} \\mathcal { N } _ n = \\overline { \\underbrace { d _ 1 \\ldots d _ 1 } _ { \\ell ~ t i m e s } } \\overline { \\underbrace { d _ 2 \\ldots d _ 2 } _ { m ~ t i m e s } } \\overline { \\underbrace { d _ 3 \\ldots d _ 3 } _ { k ~ t i m e s } } \\end{align*}"}
+{"id": "1712.png", "formula": "\\begin{align*} M ^ \\prime = \\left \\{ z \\in \\mathbb { C } ^ { n + 1 } \\ , \\left | \\ , \\Re ( z _ 0 ) = f \\left ( z _ 1 , \\ldots , z _ n , \\overline { z _ 1 } , \\ldots , \\overline { z _ n } \\right ) + \\epsilon \\left ( \\Re z _ { n + 1 } \\right ) ^ 2 \\right . \\right \\} \\end{align*}"}
+{"id": "3112.png", "formula": "\\begin{align*} \\varphi ^ * \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & : = \\left ( \\begin{array} { c c c } a ^ { \\ell _ 1 } & 0 & a ^ { p ^ { e _ 2 } } \\ , b ^ { p ^ { e _ 2 } } \\ , d ^ { - \\ell _ 3 } \\\\ 0 & d ^ { - \\ell _ 2 } & a ^ { p ^ { e _ 1 } } \\ , b ^ { p ^ { e _ 1 } } \\ , d ^ { - \\ell _ 3 } \\\\ 0 & 0 & d ^ { - \\ell _ 3 } \\end{array} \\right ) . \\end{align*}"}
+{"id": "950.png", "formula": "\\begin{align*} h _ V ( g ) ( x ) = \\int _ { V ^ c } g ( y ) \\ , P _ V ( x , d y ) \\mbox { q . e . } x \\in E . \\end{align*}"}
+{"id": "6079.png", "formula": "\\begin{align*} \\lim _ { k \\to 0 + 0 } \\frac { | \\sin k \\sin x k | } { 1 - \\cos k } & = \\lim _ { k \\to 0 + 0 } \\frac { \\sin x k } { \\sin k } \\frac { k } { x k } ( 1 + \\cos k ) x = 2 x , x \\in { \\mathbb Z } \\backslash \\{ 0 \\} . \\end{align*}"}
+{"id": "5275.png", "formula": "\\begin{align*} \\sum _ { l } x ^ { k + 1 } _ { l } = \\sum _ { l } x ^ { k } _ { l } \\end{align*}"}
+{"id": "5479.png", "formula": "\\begin{align*} \\frac { 1 } { r } + \\frac { 1 } { 2 } + \\sum _ { l = 1 } ^ \\infty \\frac { 1 } { 2 a _ l } = 1 . \\end{align*}"}
+{"id": "5019.png", "formula": "\\begin{align*} \\begin{bmatrix} \\begin{bmatrix} \\mathbf { x } _ { S } ^ { 1 } & \\mathbf { x } _ { S ^ c } ^ { 0 } \\end{bmatrix} & \\begin{bmatrix} \\mathbf { x } _ { S } ^ { 0 } & \\mathbf { x } _ { S ^ c } ^ { 1 } \\end{bmatrix} \\end{bmatrix} \\begin{bmatrix} \\mathbf { B } ^ S \\\\ \\mathbf { N } ^ S \\end{bmatrix} = \\mathbf { e } _ i . \\end{align*}"}
+{"id": "656.png", "formula": "\\begin{align*} \\xi ^ { \\otimes m } : = \\xi ^ { ( 1 ) } \\otimes \\cdots \\otimes \\xi ^ { ( m ) } \\in S ^ { m } \\xi _ { i _ { 1 } \\cdots i _ { m } } ^ { \\otimes m } : = ( \\xi ^ { \\otimes m } ) _ { i _ { 1 } \\cdots i _ { m } } . \\end{align*}"}
+{"id": "3549.png", "formula": "\\begin{align*} T f ( z _ 0 ) \\tau ^ 4 ( z _ 0 ) - f ( \\tau ( z _ 0 ) ) \\tau ^ 4 ( z _ 0 ) = ( \\tau ^ 3 ( z _ 0 ) - \\tau ( z _ 0 ) \\tau ^ 2 ( z _ 0 ) ) T ( g \\ , i d ^ 2 ) ( z _ 0 ) , \\end{align*}"}
+{"id": "3075.png", "formula": "\\begin{align*} { \\rm C o e f f } ( \\varphi ^ * _ a ( H _ 1 \\omega _ { i g } ) , t ^ k ) = p _ { k } ( a _ { \\beta _ g } , \\ldots , a _ { k _ { i g } - 1 } ) + r _ k \\cdot a _ { \\beta _ g } ^ { \\gamma _ { 1 g } } \\cdot a _ { k _ { i g } } \\end{align*}"}
+{"id": "4043.png", "formula": "\\begin{align*} \\tilde { \\mathcal { H } } : = \\Big \\{ \\textbf { h } \\in \\textbf { T } ( ( \\mathcal { H } ) ) : \\sum _ { n = 0 } ^ \\infty \\langle \\textbf { h } _ n , \\textbf { h } _ n \\rangle _ { \\mathcal { H } ^ { \\hat \\otimes n } } < \\infty \\Big \\} , \\end{align*}"}
+{"id": "7141.png", "formula": "\\begin{align*} \\varepsilon _ \\mu ( k , \\lambda ) \\longrightarrow \\varepsilon ' _ \\mu ( k , \\lambda ) = \\varepsilon _ \\mu ( k , \\lambda ) + i k _ \\mu \\widehat { \\chi } ( k ) \\ , \\ , , k ^ 2 = k _ \\mu k ^ \\mu = 0 \\omega = k _ 0 = | { \\boldsymbol { k } } | \\ , \\ , . \\end{align*}"}
+{"id": "932.png", "formula": "\\begin{align*} L = - \\phi ( - \\Delta ) \\end{align*}"}
+{"id": "5442.png", "formula": "\\begin{align*} r ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , i ) } & = R _ i + \\beta \\sum _ { j \\in N } p _ { i j } f ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , j ) } - f ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , i ) } = R _ i + \\beta \\sum _ { j \\in S _ 1 } p _ { i j } f ^ { S _ 1 } _ j - f ^ { S _ 1 } _ i = r ^ { S _ 1 } _ i , \\end{align*}"}
+{"id": "3561.png", "formula": "\\begin{align*} \\tau _ 0 = ( \\tau ' ) ^ { \\frac { 1 } { p } } , \\ ; \\tau _ 1 = ( \\tau ' \\circ \\tau ) ^ { \\frac { 1 } { p } } , \\tau _ 2 = ( \\tau ' \\circ \\tau ^ 2 ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
+{"id": "7589.png", "formula": "\\begin{align*} B _ { R } ^ { 2 } [ u ] & = : - \\big < u , [ \\mu | u | ^ { q - 2 } u , \\nabla \\varphi _ { R } \\cdot \\nabla + \\nabla \\cdot \\nabla \\varphi _ { R } ] u \\big > = 2 \\mu \\int _ { \\mathbb { R } ^ { N } } | u | ^ { 2 } \\nabla \\varphi _ { R } \\nabla ( | u | ^ { q - 2 } ) \\\\ & = - \\mu \\frac { 2 ( q - 2 ) } { q } \\int _ { \\mathbb { R } ^ { N } } \\Delta \\varphi _ { R } | u | ^ { q } . \\\\ \\end{align*}"}
+{"id": "5534.png", "formula": "\\begin{align*} X _ u : = \\{ v \\in u ^ \\uparrow \\mid ( j ( u , v ) , k ( u , v ) ) = ( j ( u ) , k ( u ) ) \\} \\end{align*}"}
+{"id": "7542.png", "formula": "\\begin{align*} \\int _ \\Omega \\bar \\rho _ 0 \\ d x = \\int _ \\Omega \\bar n _ 0 \\ d x = \\bar M < \\infty \\ , \\end{align*}"}
+{"id": "1279.png", "formula": "\\begin{align*} \\lambda _ 1 = \\frac { \\gamma + \\sqrt { \\gamma ^ 2 - 4 \\omega ^ 2 } } { 2 } , \\lambda _ 2 = \\frac { \\gamma - \\sqrt { \\gamma ^ 2 - 4 \\omega ^ 2 } } { 2 } = \\frac { 2 \\omega ^ 2 } { \\gamma + \\sqrt { \\gamma ^ 2 - 4 \\omega ^ 2 } } . \\end{align*}"}
+{"id": "8711.png", "formula": "\\begin{align*} \\sigma _ 2 ( P ^ \\ast _ { M _ G } P _ { M _ G } ) ^ { - 1 } \\sigma _ 2 ^ \\ast = B _ { M _ G } ( I + R _ { M _ G } ) B _ { M _ G } , \\end{align*}"}
+{"id": "3461.png", "formula": "\\begin{align*} \\sigma ( L ) = \\sigma ( N _ L ) . \\end{align*}"}
+{"id": "18.png", "formula": "\\begin{align*} \\alpha ( \\kappa ; q ) = \\tfrac 1 { 2 \\pi } \\int _ 0 ^ \\infty \\tfrac { \\beta ( \\kappa + \\xi ; q ) } { \\kappa + \\xi } \\ , d \\xi \\quad \\qquad \\alpha ( \\kappa ; q ) = \\sum _ { \\xi \\in 2 \\pi \\Z _ + } \\tfrac { \\beta ( \\kappa + \\xi ; q ) } { \\kappa + \\xi } \\quad , \\end{align*}"}
+{"id": "6376.png", "formula": "\\begin{align*} \\partial E _ \\varepsilon = \\Bigl \\{ r ( 1 + \\rho _ \\varepsilon ( \\varphi ) ) : \\varphi \\in S ^ { n - 1 } \\Bigr \\} , \\end{align*}"}
+{"id": "3551.png", "formula": "\\begin{align*} T f = f \\circ \\tau ( f \\in { H } ^ { \\infty } _ { 1 } ( \\mathbb { D } ) ) . \\end{align*}"}
+{"id": "6879.png", "formula": "\\begin{align*} \\theta _ h H _ { h { \\sf E } } g = \\{ \\langle h _ j , H _ { h { \\sf E } } g \\rangle \\} _ { j \\in \\mathbb { J } } = \\{ \\langle H _ { h { \\sf E } } h _ j , g \\rangle \\} _ { j \\in \\mathbb { J } } = \\{ E _ j \\langle h _ j , g \\rangle \\} _ { j \\in \\mathbb { J } } = \\mathcal { E } \\theta _ h { g } , \\end{align*}"}
+{"id": "5531.png", "formula": "\\begin{align*} S _ x : = \\{ x \\cap D ( i , \\gamma _ x ) \\mid i < \\theta \\sup ( x \\cap D ( i , \\gamma _ x ) ) = \\sup ( x ) \\} . \\end{align*}"}
+{"id": "9053.png", "formula": "\\begin{align*} & \\sum _ { k \\in I ^ a } ( X \\circ ( Y _ 1 \\odot \\cdots \\odot Y _ m ) ) ^ Q ( T ^ { ( k ) } v ) = - \\sum _ { k \\in I ^ a } \\lambda _ k ( X \\circ ( Y _ 1 \\odot \\cdots \\odot Y _ m ) ) ^ Q ( v ) . \\end{align*}"}
+{"id": "906.png", "formula": "\\begin{align*} A ' = \\left ( \\begin{array} { c c c c c } A _ 1 & & & & \\\\ & A _ 2 & & & \\\\ & & \\ddots & & \\\\ & & & A _ { r - 1 } & \\\\ & & & & A _ { r } \\\\ \\end{array} \\right ) \\end{align*}"}
+{"id": "7686.png", "formula": "\\begin{align*} \\widehat { \\lambda } _ { s , \\mu } ( E ) = \\left ( D _ { s , 1 } \\sup _ { \\delta \\neq 0 } \\sup _ { u \\in \\mathbb { Z } ^ d } \\sum _ { v \\in \\mathbb { Z } ^ d } | G _ 0 ( u , v ; E + i \\delta ) | ^ s e ^ { \\mu | u - v | } \\right ) ^ \\frac { - 1 } { s } . \\end{align*}"}
+{"id": "6950.png", "formula": "\\begin{align*} \\langle \\mathbf { z } , \\mathbf { w } \\rangle = z _ 1 \\overline { w _ 1 } + z _ n \\overline { w _ 2 } + \\cdots z _ n \\overline { w _ n } - z _ { n + 1 } \\overline { w _ { n + 1 } } , \\end{align*}"}
+{"id": "7097.png", "formula": "\\begin{align*} \\mathbb E _ x ^ 1 [ \\int _ 0 ^ T f ( t , \\omega _ t ) d t ] = \\mathbb E _ x ^ 2 [ \\int _ 0 ^ T f ( t , \\omega _ t ) d t ] , \\end{align*}"}
+{"id": "6250.png", "formula": "\\begin{align*} \\gamma = \\frac 2 3 + \\frac { C _ g D _ g } { 3 ( C _ g D _ g + C _ i D _ i ) } = \\frac { 3 - 2 \\c d } { 3 ( 1 - \\c d ) } . \\end{align*}"}
+{"id": "4182.png", "formula": "\\begin{align*} \\varphi ( g _ 1 g ^ { - 1 } _ 3 ) \\varphi ( g _ 1 g _ 3 ) \\overset { ( \\ref { e q : v a r - n o n c o m m } ) } { = } \\varphi ( g _ 1 ) \\varphi ( g ^ { - 1 } _ 3 ) \\varphi ( g _ 3 ) \\varphi ( g _ 1 ) \\overset { { \\bf A 2 } } { = } \\varphi ( g _ 1 ) \\varphi ( g _ 1 ) \\overset { { \\bf A 3 } } { = } \\varphi ( g ^ { 2 } _ 1 ) \\ , . \\end{align*}"}
+{"id": "6162.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau ^ k } [ S ^ { k + 1 } - \\frac { \\beta } { 2 \\tau ^ k } \\| A \\breve { x } ^ { k } - b \\| ^ 2 ] - \\frac { 1 } { \\tau ^ { k - 1 } } [ S ^ { k } - \\frac { \\beta } { 2 \\tau ^ { k - 1 } } \\| A \\breve { x } ^ { k - 1 } - b \\| ^ 2 ] \\\\ \\geq & \\frac { 1 } { 2 } \\Big ( \\| v ^ { k + 1 } - v ' \\| ^ 2 _ { H ^ { k + 1 } _ 0 } - \\| v ^ k - v ' \\| ^ 2 _ { H ^ k _ 0 } \\Big ) . \\end{aligned} \\end{align*}"}
+{"id": "6507.png", "formula": "\\begin{align*} \\int _ s ^ t \\int _ s ^ u \\Phi _ { n - 1 } ( u - r ) f ( r ) d r d u = \\int _ s ^ { t } \\int _ s ^ { v _ n } \\int _ s ^ { v _ { n - 1 } } \\cdots \\int _ s ^ { v _ 2 } \\prod _ { i = 2 } ^ { n } \\Phi ( v _ { i } - v _ { i - 1 } ) f ( { v _ 1 } ) d v _ 1 \\cdots d v _ { n } . \\end{align*}"}
+{"id": "5201.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } D _ { \\alpha \\beta } ( p \\| q ) } { \\partial q _ { j } } = \\underbrace { \\frac { Z _ { A } } { \\alpha } p ^ { \\alpha } _ { j } q ^ { \\beta - 2 } _ { j } } _ { U _ { j } } - \\underbrace { \\frac { Z _ { A } } { \\alpha } q ^ { \\alpha + \\beta - 2 } _ { j } } _ { V _ { j } } \\end{align*}"}
+{"id": "8295.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { u ^ 2 _ 1 \\psi _ u } { w ^ 2 \\ln w } - \\frac { w ^ 2 + 1 } { u w ^ 2 \\ln w } \\sum _ { i , j } \\frac { \\partial F } { \\partial a _ { i j } } a _ { i j } + \\frac { 1 } { u w ^ 2 } \\sum _ { i , j } \\frac { \\partial F } { \\partial a _ { i j } } a _ { i j } \\\\ \\geq \\left ( 1 - \\frac { 2 } { \\ln w } \\right ) \\frac { 1 } { u w ^ 2 } \\sum _ { i , j } \\frac { \\partial F } { \\partial a _ { i j } } a _ { i j } \\geq 0 . \\end{aligned} \\end{align*}"}
+{"id": "6597.png", "formula": "\\begin{align*} R _ { k , \\beta } ( z ) = \\begin{cases} G _ { k , \\beta } ( 1 - ( 1 - \\delta ) z , z ) & k , \\\\ I _ { k , \\beta } ( 1 - ( 1 - \\delta ) z , z ) & k . \\end{cases} \\end{align*}"}
+{"id": "8282.png", "formula": "\\begin{align*} \\dot { u _ n } ( t ) = \\Delta u _ n ( t ) \\mbox { i n } { \\mathcal D } ( \\Omega _ n ) ' \\mbox { f o r a l l } t > 0 \\end{align*}"}
+{"id": "3554.png", "formula": "\\begin{align*} ( z - \\tau ( \\psi ( z ) ) ) ( z + \\tau ( \\psi ( z ) ) ) = 0 . \\end{align*}"}
+{"id": "1293.png", "formula": "\\begin{align*} \\partial _ t ^ { m a c r o } \\langle x _ 2 \\rangle & : = \\frac { \\partial \\langle x _ 2 \\rangle } { \\partial \\langle x _ 1 \\rangle } \\langle \\dot { x } _ 1 \\rangle \\\\ & = ( \\alpha ( a - k ) + \\alpha ^ 2 k ) \\langle x _ 1 \\rangle \\ ; , \\end{align*}"}
+{"id": "6663.png", "formula": "\\begin{align*} 0 & \\geq \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { 2 \\pi n \\epsilon } \\int _ { 0 } ^ { 2 \\pi } \\bigg ( \\log | f _ { n } ( \\mathrm { e } ^ { \\mathrm { i } x } ) | - \\log | f _ { n } ( \\mathrm { e } ^ { \\mathrm { i } ( x + \\mathrm { i } \\epsilon ) } ) | \\bigg ) \\mathrm { d } x \\\\ & = \\frac { 1 } { \\epsilon } ( L ( E , \\mathrm { i } 0 ) - L ( E , \\mathrm { i } \\epsilon ) ) = - \\omega ^ { + } ( E , \\mathrm { i } 0 ) . \\end{align*}"}
+{"id": "60.png", "formula": "\\begin{align*} Z _ X ( S ) : = \\bigcup _ { v \\in S } Z _ X ( v ) . \\end{align*}"}
+{"id": "3957.png", "formula": "\\begin{align*} T ( x ) = g _ n ( x ) x \\in A _ n n \\in \\N \\end{align*}"}
+{"id": "1758.png", "formula": "\\begin{align*} \\mathfrak { g } = \\mbox { s p a n } \\{ Y _ { - n } , X _ { - j } , Y _ { - j } , X ^ \\prime _ { - 1 } , Y ^ \\prime _ { - 1 } , U _ 0 , V _ 0 , V _ { 2 - n } \\ , | \\ , j = 1 , \\ldots , n - 1 \\} . \\end{align*}"}
+{"id": "2786.png", "formula": "\\begin{align*} \\delta \\Xi ^ { a ' } ( t _ { 2 } ) = \\delta \\Xi ^ { a ' } ( t _ { 1 } ) . \\end{align*}"}
+{"id": "441.png", "formula": "\\begin{align*} \\mathcal { A } ^ 0 = \\{ ( i , j , \\# ) \\mid ( i , j ) \\in E _ m , \\ ; x _ i ^ * ( t _ 0 ) \\ , \\# \\ , x _ j ^ * ( t _ 0 ) \\} , \\ , \\mbox { w h e r e } \\# \\in \\{ < , = , > \\} . \\end{align*}"}
+{"id": "2965.png", "formula": "\\begin{align*} \\frac { ( \\beta - 1 - a ) ^ 2 } { 4 } \\left \\| \\frac { g } { r ^ { 1 + \\frac { a } { 2 } } } \\right \\| _ { L _ \\beta ^ 2 } ^ 2 = \\left \\| \\frac { g ' } { r ^ { \\frac { a } { 2 } } } \\right \\| _ { L _ \\beta ^ 2 } ^ 2 - \\left \\| \\frac { \\beta - 1 - a } { 2 } \\frac { g } { r ^ { 1 + \\frac { a } { 2 } } } + \\frac { g ' } { r ^ { \\frac { a } { 2 } } } \\right \\| _ { L _ \\beta ^ 2 } ^ 2 \\end{align*}"}
+{"id": "1183.png", "formula": "\\begin{align*} f ( y ) & = \\sum _ { \\alpha \\in \\mathbb { Z } _ + ^ n , \\ , | \\alpha | < N } \\frac { ( y - z ) ^ \\alpha } { \\alpha ! } \\partial ^ \\alpha f ( z ) \\\\ & \\quad + \\sum _ { \\alpha \\in \\mathbb { Z } _ + ^ n , \\ , | \\alpha | = N } \\frac { ( y - z ) ^ \\alpha } { \\alpha ! } \\int _ 0 ^ 1 \\partial ^ \\alpha f ( z + t ( y - z ) ) N ( 1 - t ) ^ { N - 1 } \\ , d t \\end{align*}"}
+{"id": "6355.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\omega _ k ( r ) = \\begin{cases} 1 , & k = 1 , \\\\ ( n - 1 ) , & k = 2 , \\\\ ( n - 1 ) ( n - 2 ) , & k = 3 . \\end{cases} \\end{align*}"}
+{"id": "8557.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\prod _ { i = 0 } ^ { m } g _ { 2 } ( n + i ) f ( n + m + 1 ) \\end{align*}"}
+{"id": "4765.png", "formula": "\\begin{align*} \\alpha ( a , b ) = \\norm { \\frac { a } { \\norm { a } } - \\frac { b } { \\norm { b } } } \\leq 2 \\end{align*}"}
+{"id": "934.png", "formula": "\\begin{align*} - \\int _ E u \\ , L \\eta \\ , d m = \\int _ D \\eta f ( \\cdot , u ) \\ , d m + \\int _ D \\eta \\ , d \\mu , \\eta \\in \\mathcal C , u = g \\quad \\partial _ { \\chi } D \\end{align*}"}
+{"id": "3705.png", "formula": "\\begin{align*} M _ { ( H , S ( X ) , \\varphi ) } & = | H ^ { \\rm a b } [ | \\mu ( K ) | ] | ^ { - 1 } | H | ^ { - | S ( X ) | + 1 } \\\\ & = | H | ^ { | S ( X ^ { j } ) | - | S ( X ) | } | H ^ { \\rm a b } [ | \\mu ( K ) | ] | ^ { - 1 } | H | ^ { - | S ( X ^ { j } ) | + 1 } \\\\ & = \\sum _ { \\substack { \\psi : F _ { K , S ( X ^ { j } ) } \\to H \\\\ \\psi p \\not \\in S ( X ) \\\\ \\psi | _ { S ( X ) } = \\varphi } } M _ { ( H , S ( X ^ { j } ) , \\psi ) } . \\end{align*}"}
+{"id": "6219.png", "formula": "\\begin{align*} \\inf _ { [ 0 , \\gamma ] } \\delta ( f , \\alpha ) - \\sup _ { [ \\gamma , 1 ] } \\delta ( f , \\beta ) = \\frac { f ( \\gamma ) - f ( \\alpha ) } { \\gamma - \\alpha } - \\frac { f ( \\gamma ) - f ( \\beta ) } { \\gamma - \\beta } > 0 . \\end{align*}"}
+{"id": "2078.png", "formula": "\\begin{align*} N _ { i , 3 } ^ s = - ( \\eta d T ) \\sum _ { j = 1 } ^ d \\sum _ { t = 0 } ^ { \\lfloor s T \\rfloor - 1 } Z _ { t , j } , \\end{align*}"}
+{"id": "7403.png", "formula": "\\begin{align*} a _ n = b _ n + c _ n = \\frac { ( p + q ) ^ { n - 1 } - 1 } { p + q - 1 } \\quad ( n \\in \\N ) , \\end{align*}"}
+{"id": "1084.png", "formula": "\\begin{align*} J _ { 1 , n } & : = \\left ( \\sum _ { s = 1 } ^ n \\Vert \\tilde { A } _ s \\Vert \\right ) \\sum _ { t = n + 1 } ^ { \\infty } \\Vert \\tilde { a } _ { t - 1 } \\Vert \\cdot \\Vert y _ t \\Vert , \\\\ J _ { 2 , n } & : = \\sum _ { s = 1 } ^ n \\sum _ { t = n + 1 } ^ { \\infty } \\sum _ { u = 0 } ^ { s - 2 } \\Vert \\tilde { A } _ { u + 1 } \\Vert \\cdot \\Vert \\tilde { a } _ { t - s + u + 1 } - \\tilde { a } _ { t - s + u } \\Vert \\cdot \\Vert y _ t \\Vert . \\end{align*}"}
+{"id": "4067.png", "formula": "\\begin{align*} R _ V ( z ) - R _ 0 ( z ) = - R _ V ( z ) V R _ 0 ( z ) \\end{align*}"}
+{"id": "1513.png", "formula": "\\begin{align*} A ( z ) = \\frac { D ^ 2 F ( z ) } { \\lambda _ { \\rm m i n } ( z ) } , \\end{align*}"}
+{"id": "4969.png", "formula": "\\begin{align*} R _ { j k } ( \\lambda , \\mu ) \\left [ T _ j ( \\lambda ) \\otimes T _ k ( \\mu ) \\right ] = \\left [ T _ k ( \\mu ) \\otimes T _ j ( \\lambda ) \\right ] R _ { j k } ( \\lambda , \\mu ) . \\end{align*}"}
+{"id": "5623.png", "formula": "\\begin{align*} X = \\bigcap _ { n = j } ^ \\infty T ^ n ( U ) \\supseteq \\bigcup _ { i = 0 } ^ \\infty \\bigcap _ { n = i } ^ \\infty T ^ n ( U ) , \\end{align*}"}
+{"id": "4487.png", "formula": "\\begin{align*} G ( t ) = \\int _ { \\{ 2 \\psi < - t \\} } | F _ 0 | ^ 2 \\tilde \\rho \\end{align*}"}
+{"id": "1985.png", "formula": "\\begin{align*} \\left ( i \\partial _ t - \\alpha \\partial _ { t t } \\right ) \\psi _ M ( x , t ) + \\partial _ { x x } \\psi _ M ( x , t ) - \\varepsilon ^ { 2 } P _ M ( f ( \\psi _ M ( x , t ) ) ) = 0 , \\end{align*}"}
+{"id": "9170.png", "formula": "\\begin{align*} d _ { \\beta } = \\sum _ { r = 1 } ^ { \\ell _ { \\beta } } t _ { \\beta , r } \\forall \\ \\beta \\in \\Delta ^ { + } . \\end{align*}"}
+{"id": "8060.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow + \\infty } \\widetilde { P } _ { h _ 1 ^ m , h _ 2 ^ m , h _ 3 ^ m } ^ { N , F ^ m , G ^ m , H ^ m } \\left ( | \\varepsilon _ { 1 2 } | \\leq \\epsilon \\right ) = 1 . \\end{align*}"}
+{"id": "5231.png", "formula": "\\begin{align*} \\sum _ { j } q _ { j } \\frac { \\partial L _ { d } D A G I ( p \\| q ) } { \\partial q _ { j } } = 0 \\end{align*}"}
+{"id": "6147.png", "formula": "\\begin{align*} Q ^ k = H ^ k M ^ k , \\end{align*}"}
+{"id": "8179.png", "formula": "\\begin{align*} | { \\mathcal B } _ { y ^ j } ( \\underline x ^ j ) | & \\geq y _ j \\min ( | { \\mathcal B } _ { y ^ { j - 1 } } ( \\underline x ^ { j - 1 } ) | , b ( \\underline x _ j ) ) \\\\ & + ( 1 - y _ j ) [ | { \\mathcal B } _ { y ^ { j - 1 } } ( \\underline x ^ { j - 1 } ) | - b ( \\underline x _ j ) ] ^ + , \\forall j \\leq L , \\\\ | { \\mathcal B } _ { y ^ 0 } ( \\underline x ^ 0 ) | & = M . \\end{align*}"}
+{"id": "1848.png", "formula": "\\begin{align*} \\rho _ i ( s , x ) = \\rho _ j ( s , x ) , s \\in [ - \\epsilon , \\epsilon ] , \\ ; x \\in D ^ 6 \\end{align*}"}
+{"id": "7595.png", "formula": "\\begin{align*} \\rho _ { 0 } = \\frac { 8 \\beta _ { 2 } \\sigma | E _ { p , q } ( \\psi _ { 0 } ) | + C _ { \\eta } \\frac { 8 \\mu \\sigma ( \\beta _ { 2 } - \\beta _ { 1 } ) } { q } \\| \\psi _ { 0 } \\| _ { 2 } ^ { 2 } + 2 } { \\beta _ { 2 } \\sigma - 4 } , \\end{align*}"}
+{"id": "8870.png", "formula": "\\begin{align*} \\frac { n } { r } = \\frac { n } { 2 } - \\frac 1 { 2 p - 1 } . \\end{align*}"}
+{"id": "1762.png", "formula": "\\begin{align*} ( n p + 2 q ) f ( x _ 1 ) = \\bigl ( 1 + ( p + q ) x _ 1 \\bigr ) f ' ( x _ 1 ) - 4 x _ 1 ^ 3 . \\end{align*}"}
+{"id": "1742.png", "formula": "\\begin{align*} | H _ { \\mathcal { L } } | & = ( - 1 ) ^ n \\left ( H _ { \\mathcal { L } } \\right ) _ { n , n } + \\sum _ { j = 2 } ^ { n - 2 } ( - 1 ) ^ r \\left ( H _ { \\mathcal { L } } \\right ) _ { n - j , n } \\left ( H _ { \\mathcal { L } } \\right ) _ { n , j } , \\end{align*}"}
+{"id": "8377.png", "formula": "\\begin{align*} \\dim ( X _ { w B } ) = \\# R _ { 1 } + \\# R _ { 2 } = \\dim ( B _ { L ( w ) } ) = \\dim ( T ) + \\dim ( U _ { w } ) , \\end{align*}"}
+{"id": "4484.png", "formula": "\\begin{align*} \\frac { 1 } { \\pi } \\liminf _ { r \\rightarrow 1 - 0 } \\frac { \\int _ { \\{ z \\in D : 2 \\psi \\ge \\log r \\} } | F _ 0 | ^ 2 \\tilde \\rho } { 1 - r } = \\| \\tilde F _ 0 \\| _ { \\partial M , \\rho } ^ 2 . \\end{align*}"}
+{"id": "1204.png", "formula": "\\begin{align*} \\lambda _ { s , p } ^ { \\frac { 1 } { p } } \\to \\Lambda _ { 1 , \\infty } = \\left [ \\frac { 1 } { R } \\right ] ^ { \\theta r + ( 1 - \\theta ) s } \\ \\ \\ \\ p \\to \\infty . \\end{align*}"}
+{"id": "300.png", "formula": "\\begin{align*} F ( z _ 1 , z _ 2 ) = \\begin{cases} a \\left ( z _ 1 - \\frac { 1 } { a c } z _ 2 \\right ) \\\\ a \\left ( z _ 2 - \\frac { 1 } { a c } z _ 1 + b z _ 1 ^ 2 \\right ) , \\end{cases} \\end{align*}"}
+{"id": "2543.png", "formula": "\\begin{align*} \\AA ( M ) = \\bigcup _ m \\AA ^ m ( M ) \\ ; . \\end{align*}"}
+{"id": "3775.png", "formula": "\\begin{align*} x ^ { ( i ) } _ { t + 1 } = A ^ { ( i ) } x ^ { ( i ) } _ t + B ^ { ( i ) } u ^ { ( i ) } _ t + w ^ { ( i ) } _ t , \\ ; \\ t = 0 , 1 , \\ldots , T - 1 \\end{align*}"}
+{"id": "4840.png", "formula": "\\begin{align*} H ( \\frac { 1 } { 2 } ( | S ^ - | + | S ^ + | ) ) & \\le \\frac { 1 } { 2 } ( H ( | S ^ - | ) + H ( | S ^ + | ) ) \\\\ & = \\frac { 1 } { 2 } \\Big ( \\int _ 0 ^ { | S ^ + | } h + \\int _ 0 ^ { | S ^ - | } h \\Big ) . \\end{align*}"}
+{"id": "2852.png", "formula": "\\begin{align*} E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] : = \\left \\{ ( x , y ) \\in \\mathbb { R } ^ n \\times \\mathbb { R } ^ n : \\ x \\neq y , \\ \\frac { | f ( x ) - f ( y ) | } { | x - y | ^ { 1 + \\frac { \\gamma } { q } } } > \\lambda \\right \\} \\end{align*}"}
+{"id": "7525.png", "formula": "\\begin{align*} 2 \\sum _ { i = 1 } ^ n \\alpha ^ * _ i \\left ( 1 - \\left ( \\gamma ^ * _ i y - \\psi ^ * _ i \\mbox { m o d $ 2 \\pi $ } \\right ) ^ 2 \\right ) + \\mbox { ( ` ` e r r o r '' ) } \\end{align*}"}
+{"id": "2964.png", "formula": "\\begin{align*} \\Re \\left \\langle \\frac { f } { r ^ 2 } , { f '' } \\right \\rangle = \\Re \\left \\langle \\frac { f } { r ^ 2 } , { f '' + \\frac { \\alpha } { r } f ' } \\right \\rangle - \\alpha \\Re \\left \\langle \\frac { f } { r ^ 3 } , { f ' } \\right \\rangle , \\end{align*}"}
+{"id": "5249.png", "formula": "\\begin{align*} & U _ { j } = \\left ( 1 - \\alpha \\right ) \\left [ \\frac { a - 1 } { a - b } \\left ( Z _ { j } \\right ) ^ { a } - \\frac { b - 1 } { a - b } \\left ( Z _ { j } \\right ) ^ { b } \\right ] \\\\ & V _ { j } = \\left ( 1 - \\alpha \\right ) \\end{align*}"}
+{"id": "6014.png", "formula": "\\begin{align*} \\mu _ N ( \\eta ^ { x , x + 1 } ) = \\mu _ N ( \\eta ) , \\ , \\ , \\eta \\in \\Omega _ N \\ , \\ , x \\in \\mathbb { T } _ N , \\end{align*}"}
+{"id": "2561.png", "formula": "\\begin{align*} \\bigoplus _ { m = 0 } ^ \\infty C ^ 1 _ m \\to K ( M , L ) \\ ; , \\end{align*}"}
+{"id": "8965.png", "formula": "\\begin{align*} h _ j = h _ j \\chi _ { B _ j } + h _ j \\chi _ { B _ j ^ c } \\quad . \\end{align*}"}
+{"id": "3044.png", "formula": "\\begin{align*} \\widehat { T } _ 1 + \\widehat { T } _ 2 + \\widehat { T } _ 3 = ( 1 - 2 \\lambda ^ 2 ) \\widehat { H } _ \\lambda - ( k _ 1 - k _ 2 - k _ 3 ) ^ 2 \\ , , \\end{align*}"}
+{"id": "4722.png", "formula": "\\begin{align*} \\beta _ h ( \\underline { \\nu } ) : = \\gamma ^ { - 2 h } \\sum _ { \\substack { T \\\\ h _ T = h + 1 } } W _ T ^ { ( h _ T ) } ( 0 , \\underline { \\nu } ) \\end{align*}"}
+{"id": "7325.png", "formula": "\\begin{align*} \\lim _ { \\beta \\downarrow 0 } \\left ( F _ { m , r } ( s , \\beta ) - \\frac { 1 } { m \\left ( s + 2 \\sin ^ { 2 } \\left ( \\frac { \\beta } { m } \\pi \\right ) \\right ) } \\right ) = \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m - 1 } \\frac { e ^ { 2 \\pi i \\frac { j r } { m } } } { s + 2 \\sin ^ 2 \\left ( \\pi \\frac { j } { m } \\right ) } . \\end{align*}"}
+{"id": "3334.png", "formula": "\\begin{align*} ( p _ 0 , x _ 0 , t _ 0 ) \\circ _ { \\mathcal { L } } ( p , x , t ) = \\Big ( & p \\sqrt { | p _ 0 | ^ 2 + 1 } + p _ 0 \\sqrt { | p | ^ 2 + 1 } , \\\\ & x _ 0 + x \\sqrt { | p _ 0 | ^ 2 + 1 } + p _ 0 t , t _ 0 + t \\sqrt { | p _ 0 | ^ 2 + 1 } + p _ 0 \\cdot x \\Big ) . \\end{align*}"}
+{"id": "7332.png", "formula": "\\begin{align*} \\frac { 1 } { m } \\sum _ { j = 0 } ^ { m - 1 } \\frac { e ^ { \\frac { 2 \\pi i r } { m } j } } { z + \\sin \\left ( \\frac { 2 \\pi j } { m } \\right ) } = e ^ { - i \\pi \\ell / 2 } \\cdot \\frac { U _ { m - \\ell - 1 } ( z ) + e ^ { i \\pi m / 2 } U _ { \\ell - 1 } ( z ) } { T _ m ( z ) - \\cos ( m \\pi / 2 ) } . \\end{align*}"}
+{"id": "2842.png", "formula": "\\begin{align*} L ' _ { 4 } = L _ { 4 } + \\frac { d W } { d t } = \\frac { 1 } { 2 } ( \\dot { q } ) ^ { 2 } - \\frac { 1 } { 2 } q ^ { 2 } + \\frac { \\partial C } { \\partial q } \\dot { q } . \\end{align*}"}
+{"id": "6050.png", "formula": "\\begin{align*} ( I : p ) : = \\bigoplus _ { l \\ge 0 } ( I : p ) _ l \\end{align*}"}
+{"id": "8345.png", "formula": "\\begin{align*} D K = \\{ x : K \\cap ( K + x ) \\neq \\emptyset \\} = K + ( - K ) , \\end{align*}"}
+{"id": "8262.png", "formula": "\\begin{align*} c _ { y ^ { i - 1 } 1 } = M P _ { Y ^ i } ( y ^ { i - 1 } 1 ) = b _ { y ^ { i - 1 } } , Y _ i = 1 , \\end{align*}"}
+{"id": "6273.png", "formula": "\\begin{align*} f ( \\overline { x } _ T ) - f ( x ^ * ) \\leq \\frac 1 T \\sum \\limits _ { k = 0 } ^ { T - 1 } \\langle \\nabla \\hat { f } _ \\tau ( x _ { k } ) , x _ k - x ^ * \\rangle + 2 M _ 2 \\tau . \\end{align*}"}
+{"id": "2808.png", "formula": "\\begin{align*} \\delta \\Xi ^ { a ' } ( t _ { 1 } ) = 0 , \\end{align*}"}
+{"id": "5451.png", "formula": "\\begin{align*} \\partial D = \\left \\{ x \\in \\R ^ d : ~ \\ell ( x ) = 0 \\right \\} \\ , . \\end{align*}"}
+{"id": "8471.png", "formula": "\\begin{align*} \\lim \\limits _ { h , h ' \\searrow 0 } \\left \\| ( u _ h ) _ + - ( u _ { h ' } ) _ + \\right \\| _ { L ^ \\gamma ( K _ T ) } = 0 . \\end{align*}"}
+{"id": "8091.png", "formula": "\\begin{align*} V _ 0 = e ^ { 3 \\b T _ 0 } . \\end{align*}"}
+{"id": "116.png", "formula": "\\begin{align*} W : = \\operatorname { K e r } ( B ) = \\left \\{ \\vec { v } \\in V \\ ; \\big | \\ ; B \\vec { v } = \\operatorname { d i v } \\vec { v } = 0 \\right \\} . \\end{align*}"}
+{"id": "6954.png", "formula": "\\begin{align*} \\partial \\mathbb { H } _ { \\mathbb { R } } ^ n = \\partial \\mathbb { H } _ { \\mathbb { C } } ^ n \\cap \\mathbb { P } _ { \\mathbb { R } } ^ n , \\end{align*}"}
+{"id": "4591.png", "formula": "\\begin{align*} \\mathcal { A } ( [ \\omega ] ) = & \\frac { ( c _ 1 [ \\omega ] ) ^ 2 } { 2 T _ 0 } - \\frac { 1 } { 3 2 \\pi ^ 2 } \\left ( - \\frac 1 h T _ s + s _ 0 \\frac 1 h T _ 1 \\right ) . \\end{align*}"}
+{"id": "2356.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\ln \\frac { 4 \\pi ( r ( t ) - 2 ) ( 1 - \\tau ) } { r ( t ) ^ { 2 } } d t = \\frac { T } { \\frac { 1 } { 3 } - \\frac { 1 } { q } } \\int _ { 3 } ^ { q } \\frac { 1 } { r ^ { 2 } } \\ln \\frac { 4 \\pi ( r - 2 ) ( 1 - \\tau ) } { r ^ { 2 } } d r , \\end{align*}"}
+{"id": "8692.png", "formula": "\\begin{align*} \\tilde F ^ { ( q ) } : W ^ 2 ( \\overline M , T ^ { * 0 , q } M ' ) & \\rightarrow L ^ { 2 } _ { ( 0 , q ) } ( M ) \\oplus W ^ { \\frac { 3 } { 2 } } ( X , T ^ { * 0 , q } M ' ) \\\\ u & \\mapsto ( \\tilde \\Box _ f ^ { ( q ) } u , \\gamma u ) . \\end{align*}"}
+{"id": "2044.png", "formula": "\\begin{align*} \\left [ \\frac { \\prod _ { i = 1 } ^ e ( 1 - t ^ { d + 1 - i } ) } { ( 1 - t ) ^ n } \\right ] _ { > 0 } . \\end{align*}"}
+{"id": "8845.png", "formula": "\\begin{align*} \\| u \\| _ { \\dot H ^ 1 _ \\lambda } & : = \\| \\sqrt { \\mathcal K _ \\lambda } u \\| = \\big ( \\| \\nabla u \\| ^ 2 + \\lambda \\| | x | ^ { - 1 } u \\| ^ 2 \\big ) ^ \\frac 1 2 \\simeq \\| u \\| _ { \\dot H ^ 1 } , \\end{align*}"}
+{"id": "3650.png", "formula": "\\begin{align*} N _ { v , e } = \\begin{cases} 1 & v \\in e \\in E ' , \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "8309.png", "formula": "\\begin{align*} \\sf M _ k ( \\omega _ 1 \\odot \\cdots \\odot \\omega _ k ) = P \\{ \\{ \\ldots \\{ { \\sf m } \\omega _ 1 , \\omega _ 2 \\} , \\ldots \\} , \\omega _ k \\} . \\end{align*}"}
+{"id": "2400.png", "formula": "\\begin{align*} a _ { n , 2 s + 4 - j , k } = \\frac { 1 } { j ! } \\widetilde { A _ n } ^ { ( j ) } ( - k ) . \\end{align*}"}
+{"id": "6012.png", "formula": "\\begin{align*} c _ x ( \\eta ) = \\sum _ { \\alpha , \\beta } c ^ { \\alpha , \\beta } \\xi ^ \\alpha _ x \\xi ^ \\beta _ { x + 1 } \\end{align*}"}
+{"id": "2501.png", "formula": "\\begin{align*} \\| A ( t ) \\| _ 1 & = \\int _ 0 ^ t \\int _ \\Omega ( u \\gamma ( v ) ) ( s , x ) \\ \\mathrm { d } x \\mathrm { d } s \\le \\| \\gamma ' \\| _ { L ^ \\infty ( 0 , V ) } \\int _ 0 ^ t \\int _ \\Omega ( u v ) ( s , x ) \\ \\mathrm { d } x \\mathrm { d } s \\\\ & \\le \\| v ^ { i n } \\| _ 1 \\| \\gamma ' \\| _ { L ^ \\infty ( 0 , V ) } \\ , , \\end{align*}"}
+{"id": "5810.png", "formula": "\\begin{align*} x ^ { ( k , \\ell ) } _ j = \\frac { d } { d t } z ^ { ( k - 1 , \\ell ) } _ j = \\mathcal { W } ^ { ( k - 1 , \\ell ) } _ j . \\end{align*}"}
+{"id": "2158.png", "formula": "\\begin{align*} A _ d : = \\{ a _ i \\in A : a _ { i + 1 } - a _ i = d \\} . \\end{align*}"}
+{"id": "7901.png", "formula": "\\begin{align*} \\dot { \\mathcal { F } } ( \\vec { v } , \\Sigma ) = \\{ \\mathcal { F } , H \\} ( \\vec { v } , \\Sigma ) , \\end{align*}"}
+{"id": "4571.png", "formula": "\\begin{align*} \\frac { q } { p } = \\frac { 1 } { \\displaystyle e _ 1 - \\frac { 1 } { \\displaystyle e _ 2 - \\cdots \\frac { 1 } { e _ k } } } . \\end{align*}"}
+{"id": "4415.png", "formula": "\\begin{align*} & \\max \\{ 0 , x _ i - \\overline { b } _ i , \\max \\{ 0 , x _ i - \\overline { b } _ i + \\Delta b _ i \\} - \\theta ^ k ( x ) \\} = \\\\ & \\max \\{ 0 , x _ i - \\overline { b } _ i , x _ i - \\overline { b } _ i + \\Delta b _ i - x _ k + \\overline { b } _ k - \\Delta b _ k \\} . \\end{align*}"}
+{"id": "8558.png", "formula": "\\begin{align*} R ( n , k ) = - n ^ 2 \\left ( - 2 n ( a + b - k + 1 ) + a b - a k + a - b k + b + 3 n ^ 2 \\right ) . \\end{align*}"}
+{"id": "9175.png", "formula": "\\begin{align*} \\tilde { \\mathbf { E } } ^ { + } _ { h } \\ = \\prod _ { ( \\beta , s ) \\in \\Delta ^ { + } \\times \\mathbb { Z } } \\limits ^ { \\rightarrow } \\tilde { \\mathbf { E } } _ { \\beta , s } ^ { + , ( h ( \\beta , s ) ) } , \\tilde { \\mathbf { E } } ^ { - } _ { h } \\ = \\prod _ { ( \\beta , s ) \\in \\Delta ^ { + } \\times \\mathbb { Z } } \\limits ^ { \\rightarrow } \\tilde { \\mathbf { E } } _ { \\beta , s } ^ { - , ( h ( \\beta , s ) ) } . \\end{align*}"}
+{"id": "5336.png", "formula": "\\begin{align*} \\nu ^ { S _ m } _ { \\pi _ k } = \\nu _ { \\pi _ m } + \\frac { 1 } { w ^ { S _ m } _ { \\pi _ k } } \\sum _ { l = m + 1 } ^ k ( \\nu _ { \\pi _ { l } } - \\nu _ { \\pi _ { l - 1 } } ) \\ , w ^ { S _ { l } } _ { \\pi _ k } , m \\leq k \\leq n . \\end{align*}"}
+{"id": "3317.png", "formula": "\\begin{align*} \\ell _ n ( \\mu ; \\mu _ 0 ) = \\exp \\left \\{ \\sum _ { i = 1 } ^ n \\frac { ( X _ i - \\mu _ 0 ) ^ 2 - ( X _ i - \\mu ) ^ 2 } { 2 } \\right \\} \\end{align*}"}
+{"id": "494.png", "formula": "\\begin{align*} R _ k R _ { k - 1 } \\cdots R _ 1 \\cdot \\begin{pmatrix} P ( T ) \\\\ Q ( T ) \\end{pmatrix} = \\begin{pmatrix} P _ j ( T ) \\\\ P _ { j + 1 } ( T ) \\end{pmatrix} \\end{align*}"}
+{"id": "3835.png", "formula": "\\begin{align*} K _ k ^ { [ 8 ] } ( \\nu ; n ) = K _ k ( \\nu ; n , 0 ) . \\end{align*}"}
+{"id": "2826.png", "formula": "\\begin{align*} & \\dot { \\Phi } ^ { ( 1 ) } _ { 1 } = \\{ \\Phi ^ { ( 1 ) } _ { 1 } , H _ { T } \\} \\approx p _ { 3 } \\\\ & \\therefore \\Phi ^ { ( 2 ) } _ { 1 } : = p _ { 3 } : \\approx 0 , \\\\ & \\dot { \\Phi } ^ { ( 1 ) } _ { 2 } = \\{ \\Phi ^ { ( 1 ) } _ { 2 } , H _ { T } \\} \\approx \\frac { 1 } { 3 } p _ { 3 } + q ^ { 1 } + q ^ { 2 } + q ^ { 4 } \\\\ & \\therefore \\Phi ^ { ( 2 ) } _ { 2 } : = \\frac { 1 } { 3 } p _ { 3 } + q ^ { 1 } + q ^ { 2 } + q ^ { 4 } : \\approx 0 . \\end{align*}"}
+{"id": "1930.png", "formula": "\\begin{align*} | E ( M ) | & \\leq \\frac { ( k - 1 ) { r \\choose k - 1 } } { { r - 2 \\choose k - 3 } } \\\\ & = \\frac { r ( r - 1 ) } { k - 2 } \\end{align*}"}
+{"id": "5956.png", "formula": "\\begin{align*} \\frac { M _ { 2 } ( \\mathcal { C } ( G ) ) } { | e ( \\mathcal { C } ( G ) ) | } = \\frac { 1 9 7 } { 1 3 } > \\frac { 1 4 7 } { 1 1 } = \\frac { M _ { 1 } ( \\mathcal { C } ( G ) ) } { | v ( \\mathcal { C } ( G ) ) | } . \\end{align*}"}
+{"id": "4011.png", "formula": "\\begin{align*} \\left ( \\gamma _ { 1 } \\delta _ { 2 } - \\gamma _ { 2 } \\delta _ { 1 } \\right ) y ^ { 2 } - \\left ( \\gamma _ { 1 } + \\delta _ { 2 } \\right ) y + 1 = 0 . \\end{align*}"}
+{"id": "2023.png", "formula": "\\begin{align*} u ^ + : = \\max \\{ u , 0 \\} \\textnormal { a n d } u ^ - : = \\max \\{ - u , 0 \\} . \\end{align*}"}
+{"id": "6101.png", "formula": "\\begin{align*} \\sum \\left ( \\Delta ( a _ { t , \\varepsilon } ) - 1 \\otimes a _ { t , \\varepsilon } \\right ) \\otimes b _ t = \\sum a _ { t , \\varepsilon } \\otimes \\Delta ( b _ t ) . \\end{align*}"}
+{"id": "1376.png", "formula": "\\begin{align*} [ f , g ] _ \\Omega = \\ \\Big ( 0 , \\ldots , 0 , { [ f , g ] _ { \\Omega } } ^ A _ { \\{ 1 , \\ldots , n + m - 1 \\} } , { [ f , g ] _ { \\Omega } } ^ V _ { \\emptyset } , { [ f , g ] _ { \\Omega } } ^ V _ { \\{ 1 \\} } , \\ldots , { [ f , g ] _ { \\Omega } } ^ V _ { \\{ 2 , \\ldots , n + m - 1 \\} } , 0 \\Big ) \\in \\frak { L } ' ( n + m - 1 ) , \\end{align*}"}
+{"id": "4364.png", "formula": "\\begin{align*} \\inf _ { k \\in [ m ] _ 0 } & \\left \\{ \\inf _ { x \\in \\mathcal { X } } \\left \\{ \\Gamma \\theta ^ k ( x ) + \\sum _ { i \\in [ m ] } f _ i ( x , \\overline { u } ^ i ) + \\sup \\{ 0 , \\theta ^ i ( x ) - \\theta ^ k ( x ) \\} \\right \\} \\right \\} , \\\\ \\end{align*}"}
+{"id": "1083.png", "formula": "\\begin{align*} I _ n & = \\begin{cases} O ( n ^ { 1 - 2 d - \\rho } ) & ( 1 - 2 d < \\rho < 1 - d ) , \\\\ O ( n ^ { - d } \\log n ) & ( \\rho = 1 - d ) , \\\\ O ( n ^ { - d } ) & ( \\rho > 1 - d ) , \\end{cases} n \\to \\infty , \\\\ J _ n & = O ( n ^ { 1 - 2 d - \\rho } ) , n \\to \\infty . \\end{align*}"}
+{"id": "395.png", "formula": "\\begin{align*} [ { \\tt Q } ] _ { A ^ { ( \\rho ) } } J = [ { \\tt Q } ] _ { \\lnot A ^ { ( \\rho ) } } J = [ { \\tt Q } ] _ { A } J \\cap [ \\partial { \\tt Q } ] J \\cap [ \\rho ] J \\end{align*}"}
+{"id": "8796.png", "formula": "\\begin{align*} \\psi _ \\rho ' ( \\alpha _ \\rho ) & = 1 + ( 2 - \\rho ) \\frac { \\rho - 1 - \\alpha _ \\rho } { \\alpha _ \\rho } - ( \\rho - 1 ) \\frac { \\rho - 1 - \\alpha _ \\rho } { 1 - \\alpha _ \\rho } = \\frac { ( \\rho - 1 ) ( 2 - \\rho ) } { \\alpha _ \\rho ( 1 - \\alpha _ \\rho ) } , \\end{align*}"}
+{"id": "2645.png", "formula": "\\begin{align*} \\eta _ 1 ( \\xi ) = \\begin{cases} 1 , \\ | \\xi | \\le 1 , \\\\ 0 , \\ | \\xi | \\ge 2 , \\end{cases} \\end{align*}"}
+{"id": "1598.png", "formula": "\\begin{align*} \\omega _ n ^ i ( e _ n ) = \\frac { \\langle \\nabla _ { e _ n } \\nabla u , e _ i \\rangle } { | \\nabla u | } = \\frac { \\langle \\nabla _ { e _ i } \\nabla u , e _ n \\rangle } { | \\nabla u | } = \\frac { \\langle \\nabla _ { e _ i } \\nabla u , \\nabla u \\rangle } { | \\nabla u | ^ 2 } = \\frac { | \\nabla u | _ i } { | \\nabla u | } , \\quad , \\end{align*}"}
+{"id": "8022.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow + \\infty } \\hat { P } ^ { N , F , G , H } _ { f _ 1 , f _ 2 , f _ 3 } \\left ( \\{ \\mu ^ N \\in O \\} \\cap D _ { \\epsilon , 1 } \\cap D _ { \\epsilon , 2 } \\right ) = 1 . \\end{align*}"}
+{"id": "5414.png", "formula": "\\begin{align*} T ^ { - 1 } ( 1 - x ) & = 1 - x - \\sum _ { \\overline \\pi _ 0 ( \\beta ) < \\overline \\pi _ 0 ( \\alpha ) } \\lambda _ \\beta + \\sum _ { \\overline \\pi _ 1 ( \\beta ) < \\overline \\pi _ 1 ( \\alpha ) } \\lambda _ \\beta \\\\ & = 1 - x - \\sum _ { \\pi _ 1 ( \\beta ) < \\pi _ 1 ( \\alpha ) } \\lambda _ \\beta + \\sum _ { \\pi _ 0 ( \\beta ) < \\pi _ 0 ( \\alpha ) } \\lambda _ \\beta \\\\ & = 1 - T ( x ) . \\end{align*}"}
+{"id": "5920.png", "formula": "\\begin{align*} A ( n , p ) = \\Bigg \\lbrace v ( a , b , c ) = \\begin{bmatrix} 1 & 0 & 0 \\\\ a & 1 & 0 \\\\ b & c & 1 \\end{bmatrix} : a , b , c \\in F \\Bigg \\rbrace \\end{align*}"}
+{"id": "720.png", "formula": "\\begin{align*} Z _ { \\preccurlyeq } : = \\prod _ { a , b , c : A } \\biggl ( ( b \\preccurlyeq _ A c ) \\times ( a \\preccurlyeq _ A b ) \\to ( a \\preccurlyeq _ A c ) \\biggr ) . \\end{align*}"}
+{"id": "5329.png", "formula": "\\begin{align*} \\nu ^ { S _ 1 } _ j & = \\frac { c _ j } { w ^ { S _ 1 } _ j } , j \\in S _ 1 = J \\\\ \\nu ^ { S _ { k } } _ j & = \\nu ^ { S _ { k - 1 } } _ j + \\left ( \\frac { w ^ { S _ { k - 1 } } _ j } { w ^ { S _ k } _ j } - 1 \\right ) \\ , \\left [ \\nu ^ { S _ { k - 1 } } _ j - \\nu ^ { S _ { k - 1 } } _ { \\pi _ { k - 1 } } \\right ] , \\ , j \\in S _ { k } , 2 \\leq k \\leq n , \\end{align*}"}
+{"id": "5850.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { C } ( D _ { 2 m } ) ) = \\dfrac { ( m - 1 ) ( m - 1 - 1 ) ^ { 3 } } { 2 } + m \\cdot \\dfrac { 1 ( 1 - 1 ) ^ { 3 } } { 2 } = \\dfrac { ( m - 1 ) ( m - 2 ) ^ { 3 } } { 2 } . \\end{align*}"}
+{"id": "7078.png", "formula": "\\begin{align*} \\| e ^ { - t \\Lambda _ p ( b ) } f \\| _ { q } \\leq c e ^ { t \\omega _ p } t ^ { - \\frac { d } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\| f \\| _ p , t > 0 , \\omega _ p : = \\frac { c _ \\delta } { 2 ( p - 1 ) } , \\end{align*}"}
+{"id": "6532.png", "formula": "\\begin{align*} \\gamma _ 2 & = \\sigma _ 2 ( C ^ + _ K ) ^ { \\alpha \\beta } 1 _ { \\{ C ^ + _ K > 0 \\} } + \\sigma _ 1 ( C ^ - _ K ) ^ { \\alpha \\beta } 1 _ { \\{ C ^ - _ K > 0 \\} } \\ ; \\ ; \\ ; \\ ; \\gamma _ 1 = \\sigma _ 2 | C ^ + _ K | ^ { \\alpha \\beta } 1 _ { \\{ C ^ + _ K < 0 \\} } + \\sigma _ 1 | C ^ - _ K | ^ { \\alpha \\beta } 1 _ { \\{ C ^ - _ K < 0 \\} } \\end{align*}"}
+{"id": "8460.png", "formula": "\\begin{align*} ( u _ h ) _ { \\pm } ^ { ( \\ell ) } ( x , t ) : = \\dfrac { t - t _ { m - 1 } } { h } ( u _ m ( x ) ) _ { \\pm } ^ { ( \\ell ) } + \\dfrac { t _ m - t } { h } ( u _ { m - 1 } ( x ) ) _ { \\pm } ^ { ( \\ell ) } \\end{align*}"}
+{"id": "8441.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n + 1 } | \\lambda _ j ( y ) | = p _ n \\end{align*}"}
+{"id": "8027.png", "formula": "\\begin{align*} \\langle \\mathcal { M } ^ N _ { g , 2 } , \\hat { \\Lambda } ^ N _ { F , G , H } \\rangle _ t = & \\int _ 0 ^ t \\mu _ { s , 1 } ^ N \\bigotimes \\mu _ { s , 3 } ^ N \\left ( \\lambda ( \\cdot , \\ast ) g ( \\cdot ) \\left ( e ^ { - F _ s ( \\cdot ) + G _ s ( \\cdot ) } - 1 \\right ) \\right ) d s \\\\ & - \\int _ 0 ^ t \\mu _ { s , 2 } ^ N \\left ( \\psi ( \\cdot ) g ( \\cdot ) \\left ( e ^ { - G _ s ( \\cdot ) + H _ s ( \\cdot ) } - 1 \\right ) \\right ) d s \\end{align*}"}
+{"id": "8516.png", "formula": "\\begin{align*} \\Omega ( a ) = \\overline { \\bigcup _ { n = 0 } ^ { \\infty } \\sigma ^ n ( \\Lambda ( a ) ) } \\ ; . \\end{align*}"}
+{"id": "3211.png", "formula": "\\begin{align*} T _ k ( t ) = \\frac { \\varphi _ { k } } { \\Gamma ( \\alpha ) } t ^ { \\alpha - 1 } E _ { \\alpha , 1 + \\frac { \\beta } { \\alpha } , 1 + \\frac { \\beta - 1 } { \\alpha } } ( - \\lambda _ k t ^ { \\alpha + \\beta } ) + \\frac { \\Gamma ( \\mu + 1 ) { f _ { k } } } { \\Gamma ( \\mu + \\alpha + 1 ) } t ^ { \\alpha + \\mu } E _ { \\alpha , 1 + \\frac { \\beta } { \\alpha } , 1 + \\frac { \\beta + \\mu } { \\alpha } } ( - \\lambda _ { k } t ^ { \\alpha + \\beta } ) . \\end{align*}"}
+{"id": "957.png", "formula": "\\begin{align*} \\nu ^ V _ m ( A ) = \\int _ V P _ V ( x , A ) \\ , m ( d x ) . \\end{align*}"}
+{"id": "3895.png", "formula": "\\begin{align*} \\mu _ \\psi = \\lim _ { n \\to \\infty } \\frac { \\sum _ { \\gamma \\in P ( n ) } \\exp ( \\ell _ \\psi ( \\gamma ) ) \\delta _ \\gamma } { \\sum _ { \\gamma \\in P ( n ) } \\exp ( \\ell _ \\psi ( \\gamma ) ) } \\end{align*}"}
+{"id": "376.png", "formula": "\\begin{align*} ( x - a ) ^ 2 = - ( b - a ) ^ 2 + b ^ 2 + a ^ 2 = 2 a b , \\end{align*}"}
+{"id": "2915.png", "formula": "\\begin{align*} L ^ { ( 1 ) } _ { \\ell } ( x _ i ; f ) : = \\frac { 1 } { x _ i } \\sum _ { \\substack { j = 1 \\\\ x _ i \\geqslant y _ { j - 1 } } } ^ J \\int _ { y _ { j - 1 } } ^ { y _ j } X \\sum _ { t / ( 1 + 1 / X ) < p \\leqslant t } \\frac { 1 } { p } \\big | \\Psi _ f ^ { \\prime } ( x _ i / t , p ) \\big | ^ 2 { \\rm d } t \\end{align*}"}
+{"id": "408.png", "formula": "\\begin{align*} \\bold { \\Sigma } = \\bold { d } \\widetilde { \\bold { d } } ^ { T } , \\end{align*}"}
+{"id": "5193.png", "formula": "\\begin{align*} D _ { \\alpha \\beta } ( p \\| q ) = & \\frac { 1 } { ( \\beta - 1 ) ( \\alpha + \\beta - 1 ) } \\sum _ { i } p ^ { \\alpha + \\beta - 1 } _ { i } \\\\ & + \\frac { 1 } { \\alpha ( \\alpha + \\beta - 1 ) } \\sum _ { i } q ^ { \\alpha + \\beta - 1 } _ { i } \\\\ & - \\frac { 1 } { \\alpha ( \\beta - 1 ) } \\sum _ { i } p ^ { \\alpha } _ { i } q ^ { \\beta - 1 } _ { i } \\end{align*}"}
+{"id": "2006.png", "formula": "\\begin{align*} \\psi _ 0 ( x ) = \\frac { 1 } { 2 + \\cos ^ 2 ( x ) + \\sin ( x ) } , \\psi _ 1 ( x ) = \\frac { 1 } { 2 + \\sin ^ 2 ( x ) + \\cos ( x ) } . \\end{align*}"}
+{"id": "1664.png", "formula": "\\begin{align*} e ^ { 3 \\left ( T + c \\right ) ^ { \\lambda } } \\delta ^ { 2 } = \\delta . \\end{align*}"}
+{"id": "6110.png", "formula": "\\begin{align*} 2 g ( \\nabla _ { \\widetilde { Y } } U , U ) & = g ( [ \\widetilde { Y } , U ] , U ) . \\end{align*}"}
+{"id": "5769.png", "formula": "\\begin{align*} | X _ + ( t ) | ^ 2 + | X _ 0 ( t ) | ^ 2 + | X _ - ( t ) | ^ 2 = ( 1 + o ( 1 ) ) | X _ 0 ( t ) | ^ 2 . \\end{align*}"}
+{"id": "8479.png", "formula": "\\begin{align*} \\left | | \\bar { u } _ h | ^ { q - 1 } \\bar { u } _ h - v _ h \\right | = \\left | \\bar { v } _ h - v _ h \\right | \\leq h \\left | \\partial _ t v _ h \\right | \\end{align*}"}
+{"id": "5476.png", "formula": "\\begin{align*} \\frac { P ' } { P } ( z ) = M + E , \\end{align*}"}
+{"id": "7062.png", "formula": "\\begin{align*} a ( x ) = I + c \\frac { x \\otimes x } { | x | ^ 2 } , c > - 1 . \\end{align*}"}
+{"id": "1483.png", "formula": "\\begin{align*} & \\alpha \\circ P = P \\circ \\alpha , & \\\\ & [ P ( x ) , P ( y ) ] = P ( [ P ( x ) , y ] ) = P ( [ x , P ( y ) ] ) , & \\end{align*}"}
+{"id": "8130.png", "formula": "\\begin{align*} D _ { q } \\ = \\ x ^ { - 1 } \\frac { T _ { q } - 1 } { q - 1 } \\ , \\ X _ { q } \\ = \\ \\frac { A ( q - 1 ) } { T _ { q } - 1 } \\ , x \\ , \\end{align*}"}
+{"id": "582.png", "formula": "\\begin{align*} P ( f ) = \\sum \\limits _ { j = 1 } ^ \\infty \\Big \\langle f , { \\dfrac { \\Tilde { h } _ j } { \\| h _ j \\| _ { F ^ \\ast } } } \\Big \\rangle { \\dfrac { \\Tilde { h } _ j } { \\| h _ j \\| _ F } } , \\ ; f \\in F , \\end{align*}"}
+{"id": "3980.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { n } \\gamma _ { i p l } \\gamma _ { l q k } = \\sum _ { l = 1 } ^ { n } \\gamma _ { i p l } \\widetilde { \\gamma } _ { l q r } = 0 , \\quad \\left ( 1 \\leq i , k \\leq n , 1 \\leq p , q , r \\leq \\nu \\right ) , \\end{align*}"}
+{"id": "6705.png", "formula": "\\begin{align*} \\mathbb { P } _ p : = \\{ f : \\mathbb { R } \\rightarrow \\mathbb { R } , f ( x + p ) = f ( x ) \\} . \\end{align*}"}
+{"id": "4731.png", "formula": "\\begin{align*} S ( t ) x : = \\lim _ { n \\to \\infty } \\left ( \\mathrm { I d } + \\frac { t } { n } A \\right ) ^ { - n } x \\end{align*}"}
+{"id": "6775.png", "formula": "\\begin{align*} J _ { \\widetilde { \\varphi } } ( u , z ) = J _ { \\varphi } ( u , \\Phi ( z ) ) \\left [ \\begin{array} { c c } { \\rm I d } _ { ( r - s ) \\times ( r - s ) } & 0 _ { ( r - s ) \\times s } \\\\ 0 _ { n \\times ( r - s ) } & J _ { \\Phi } ( z ) \\end{array} \\right ] . \\end{align*}"}
+{"id": "4576.png", "formula": "\\begin{align*} t \\curvearrowright Q _ { i + 1 } = t ^ { - \\theta ( p v _ { i + 1 , V } + q v _ { i + 1 , U } ) + \\tau v _ { i + 1 , U } } Q _ { i + 1 } . \\end{align*}"}
+{"id": "7173.png", "formula": "\\begin{align*} ( 0 , - 1 ) \\cdot H ( ( V , W ) , x , t ) & = - \\varepsilon ( V + \\gamma S ) \\\\ & \\leq \\varepsilon ( L - \\gamma S ) . \\end{align*}"}
+{"id": "5408.png", "formula": "\\begin{align*} \\Delta v ^ { S _ 1 } _ 1 = \\frac { \\Delta h _ 1 } { \\alpha + \\mu _ 1 } , \\end{align*}"}
+{"id": "1525.png", "formula": "\\begin{align*} F ( z ) = G ( h ( z ) ) \\end{align*}"}
+{"id": "8783.png", "formula": "\\begin{align*} \\int _ \\R z \\check \\pi _ y ( d z ) & = \\int _ \\R z ( 1 - f ( z ) ) \\bar \\pi _ y ( d z ) + \\int _ \\R f ( v ) \\int _ \\R z \\theta _ v ( d z ) \\bar \\pi _ y ( d v ) \\\\ & = \\int _ \\R z ( 1 - f ( z ) ) \\bar \\pi _ y ( d z ) + \\int _ \\R v f ( v ) \\bar \\pi _ y ( d v ) = \\int _ \\R z \\bar \\pi _ y ( d z ) = y . \\end{align*}"}
+{"id": "3588.png", "formula": "\\begin{align*} ( 1 + a ) ( a + b ) - b = 0 . \\end{align*}"}
+{"id": "914.png", "formula": "\\begin{align*} n \\cdot a \\nabla u + \\gamma u = 0 \\quad \\quad \\Sigma \\end{align*}"}
+{"id": "8303.png", "formula": "\\begin{gather*} Y ( t ) = \\operatorname { e x p } \\big ( - A ( t - a ) \\big ) , Y ( a ) = I _ { m } ; \\\\ Y ^ { ( k ) } ( t ) = ( - A ) ^ k \\operatorname { e x p } \\big ( - A ( t - a ) \\big ) , Y ^ { ( k ) } ( a ) = ( - A ) ^ k , k \\in \\mathbb { N } . \\end{gather*}"}
+{"id": "8691.png", "formula": "\\begin{align*} \\xi _ M \\mathcal T ^ { k - 1 } ( \\chi f ) & = [ \\xi _ M , \\mathcal T ^ { k - 1 } ] ( \\chi f ) + \\mathcal T ^ { k - 1 } \\xi _ M ( \\chi f ) \\\\ & = [ \\xi _ M , \\mathcal T ^ { k - 1 } ] ( \\chi f ) + \\mathcal T ^ { k - 1 } [ \\xi _ M , \\chi ] f + \\mathcal T ^ { k - 1 } \\chi \\xi _ M f . \\end{align*}"}
+{"id": "2884.png", "formula": "\\begin{align*} D _ { \\lambda , \\frac { \\gamma } { q } , s } [ f ] & \\subset D _ { A ^ { - s } \\lambda , \\frac { \\gamma } { p } , 0 } [ f ] \\cup D _ { A ^ { 1 - s } \\lambda , \\gamma , 1 } [ f ] \\\\ & = D _ { A ^ { - s } \\lambda , \\frac { \\gamma } { p } , 0 } [ f ] \\cup E _ { A ^ { 1 - s } \\lambda , \\gamma } [ f ] , \\end{align*}"}
+{"id": "3190.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } n } { f ( n ) } = \\frac { \\mathrm { d } } { \\mathrm { d } n } e ^ { \\ln f ( n ) } = f ( n ) \\frac { \\mathrm { d } } { \\mathrm { d } n } \\ln { f ( n ) } . \\end{align*}"}
+{"id": "1157.png", "formula": "\\begin{align*} \\varphi ( t ) = \\sqrt { 2 } \\sum _ { k \\in \\mathbb { Z } } h _ k \\varphi ( 2 t - k ) , \\end{align*}"}
+{"id": "5958.png", "formula": "\\begin{align*} a = b & \\ , \\ , \\Longleftrightarrow \\ , \\ , \\left ( a - b = 0 \\ , \\wedge \\ , b - a = 0 \\right ) \\\\ a \\neq b & \\ , \\ , \\Longleftrightarrow \\ , \\ , \\left ( a - b \\neq 0 \\ , \\vee \\ , b - a \\neq 0 \\right ) \\end{align*}"}
+{"id": "7760.png", "formula": "\\begin{align*} Z _ { \\tilde m } ^ { - 1 } ( \\cdot + \\alpha ) ( A _ { \\tilde m } + F _ { \\tilde m } ( \\cdot ) ) Z _ { \\tilde m } ( \\cdot ) = A _ { \\tilde m + 1 } + F _ { \\tilde m + 1 } ( \\cdot ) , \\end{align*}"}
+{"id": "6244.png", "formula": "\\begin{align*} \\tilde { \\mathcal { R } } ( d ) = \\left \\{ ( r _ i , \\lambda _ g ) \\in \\R ^ + \\times \\R ^ + \\colon \\frac { 1 } { \\sqrt { 3 } - 1 } < \\frac { \\lambda _ g } { r _ i } < \\frac { 1 + \\omega } { 2 - \\omega } \\right \\} . \\end{align*}"}
+{"id": "7222.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { V _ { \\min } } ^ { V _ { F } } \\left ( \\partial _ { t } p + \\partial _ { v } ( h p ) - a \\partial _ { v v } p \\right ) \\phi d v \\\\ - & \\left ( h p \\phi | _ { V _ { \\min } } ^ { V _ R ^ - } + h p \\phi | _ { V _ R ^ + } ^ { V _ F } \\right ) + \\left ( a \\partial _ v p \\phi | _ { V _ { \\min } } ^ { V _ R ^ - } + a \\partial _ v p \\phi | _ { V _ { V _ R ^ + } } ^ { V _ F } ) + a \\partial _ v p ( V _ F ) ( \\phi ( V _ R ) - \\phi ( V _ F ) \\right ) = 0 . \\end{aligned} \\end{align*}"}
+{"id": "8647.png", "formula": "\\begin{align*} \\langle x _ 0 , y _ 0 ^ * - x _ 0 ^ * \\rangle = 1 _ K ^ * ( x _ 0 ^ * - y _ 0 ^ * ) . \\end{align*}"}
+{"id": "8941.png", "formula": "\\begin{align*} S _ i ^ \\prime = \\left \\{ \\begin{array} { c c } R _ { N d - d - i } ^ \\prime = \\Sigma _ T ( N d - d - i ) , & i \\leq N d - d ; \\\\ 0 , & i > N d - d . \\end{array} \\right . \\end{align*}"}
+{"id": "2093.png", "formula": "\\begin{align*} n ^ d _ t \\le C _ { d , T } ( t ) + C \\exp \\left ( C \\alpha ^ 4 \\sigma ^ { - 4 } \\tau ^ 4 \\right ) \\alpha ^ 4 t ^ 3 T ^ { - 4 } \\sum _ { k = 0 } ^ { t - 1 } C _ { d , T } ( k ) \\le C _ { \\tau } C _ { d , T } ( t ) . \\end{align*}"}
+{"id": "1432.png", "formula": "\\begin{align*} R ^ p \\pi _ * R ^ q f _ * \\omega _ X = 0 \\end{align*}"}
+{"id": "5354.png", "formula": "\\begin{align*} \\mathbf { v } _ { S } ^ { S } & = \\mathbf { h } _ { S } ^ 1 + \\beta \\ , \\mathbf { P } _ { S N } ^ { 1 } \\ , \\mathbf { v } ^ { S } \\\\ \\mathbf { v } _ { N \\setminus S } ^ { S } & = \\mathbf { h } _ { N \\setminus S } ^ 0 + \\beta \\ , \\mathbf { P } _ { N \\setminus S , N } ^ { 0 } \\ , \\mathbf { v } ^ { S } . \\end{align*}"}
+{"id": "7845.png", "formula": "\\begin{align*} \\nabla ^ { \\perp \\psi } _ { \\nabla ^ { \\Sigma } _ { \\bar E _ { \\alpha } } \\bar E _ { \\alpha } } H ^ { \\psi } = \\frac { q } { p + q } \\nabla ^ { \\perp } _ { \\bar \\nabla _ { \\bar E _ { \\alpha } } \\bar E _ { \\alpha } } H _ 2 + \\frac { p } { p + q } B ^ j ( \\bar \\nabla _ { \\bar E _ { \\alpha } } \\bar E _ { \\alpha } , H _ 1 ) . \\end{align*}"}
+{"id": "5760.png", "formula": "\\begin{align*} \\frac { d } { d t } \\xi ^ { ( k , \\ell ) } _ { i } - \\Gamma _ i \\xi ^ { ( k , \\ell ) } _ { i } = \\mathcal { E } ^ { ( k , \\ell ) } _ { i } , \\end{align*}"}
+{"id": "4988.png", "formula": "\\begin{align*} \\left [ p _ k ( \\lambda _ j ) \\right ] _ { j , k = 1 , \\ldots , N } = [ p _ { k i } ] _ { i , k = 1 , \\ldots , N } \\ ; [ \\lambda _ j ^ { i - 1 } ] _ { i , j = 1 , \\ldots , N } , \\end{align*}"}
+{"id": "5509.png", "formula": "\\begin{align*} w _ { n } ( \\ell ) = \\begin{cases} 1 _ { S } , & \\textnormal { i f } \\ n = 1 ; \\\\ n \\cdot 1 _ { S } + ( \\ell - 1 ) \\sum _ { k = 0 } ^ { n - 2 } ( n - 1 - k ) \\ell ^ { k } , & \\textnormal { o t h e r w i s e } , \\end{cases} \\end{align*}"}
+{"id": "1237.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { p ^ r - 1 } \\frac { 1 } { 2 ^ k } { 2 k \\choose k } \\equiv ( - 1 ) ^ { \\frac { p ^ r - 1 } { 2 } } \\pmod { p ^ 2 } . \\end{align*}"}
+{"id": "4324.png", "formula": "\\begin{align*} \\inf _ { x \\in \\mathcal { X } } & \\ \\sum _ { i \\in [ m ] } \\max \\{ 0 , x _ i - b _ i \\} . \\end{align*}"}
+{"id": "3515.png", "formula": "\\begin{align*} & \\varphi _ { m , n } ( \\varphi _ { n , k } ( \\hat { x } ) \\varphi _ { n , k } ( \\hat { y } ) ) \\\\ & = \\varphi _ { m , n } \\big ( \\hat { x } \\hat { y } \\oplus \\big ( \\oplus _ { j = k + 1 } ^ n \\rho _ { j , k } ( x ) \\rho _ { j , k } ( y ) \\big ) \\big ) \\\\ & = \\hat { x } \\hat { y } \\oplus \\big ( \\oplus _ { j = k + 1 } ^ n \\rho _ { j , k } ( x ) \\rho _ { j , k } ( y ) \\big ) \\oplus \\big ( \\oplus _ { j = n + 1 } ^ m \\rho _ { j , k } \\big ( \\rho _ { n , k } ( x ) \\rho _ { n , k } ( y ) \\big ) \\big ) . \\end{align*}"}
+{"id": "6875.png", "formula": "\\begin{align*} { H _ { { \\varphi _ n } { \\sf E } _ n } } = \\sum _ { j \\in \\mathbb { J } _ n } E _ j ^ { ( n ) } \\langle \\varphi _ j ^ { ( n ) } , \\cdot \\rangle \\ , \\varphi _ j ^ { ( n ) } = \\sum _ { j = 1 } ^ { n + 1 } E _ j ^ { ( n ) } \\langle \\varphi _ j ^ { ( n ) } , \\cdot \\rangle \\ , \\varphi _ j ^ { ( n ) } , { \\sf E } _ n = \\{ E _ j ^ { ( n ) } , j \\in \\mathbb { J } _ n \\} . \\end{align*}"}
+{"id": "9029.png", "formula": "\\begin{align*} \\frac { 1 } { | G | } = \\frac { 1 } { n _ 1 } + . . . + \\frac { 1 } { n _ { r + s } } - \\frac { b } { a } , \\end{align*}"}
+{"id": "4308.png", "formula": "\\begin{align*} \\sup _ { \\xi \\in \\R ^ e } \\bigl \\| [ ( 1 - \\Delta ) ^ { \\beta / 2 } \\varphi _ { m ^ { - \\delta } } ] ( y ( m ) _ t - \\xi ) - [ ( 1 - \\Delta ) ^ { \\beta / 2 } \\delta _ \\xi ] ( y _ t ) \\bigr \\| _ { \\mathbf { D } _ { q , - r } } = O ( m ^ { - \\kappa \\wedge \\delta } ) \\end{align*}"}
+{"id": "2708.png", "formula": "\\begin{align*} & { _ { * } \\rho } ^ { i } : = d q ^ { i } - \\frac { \\partial H } { \\partial p _ { i } } d t , \\\\ & { _ { * } \\theta } _ { i } : = d p _ { i } + \\frac { \\partial H } { \\partial q ^ { i } } d t . \\end{align*}"}
+{"id": "1864.png", "formula": "\\begin{align*} d _ A a \\wedge \\psi = 0 \\Longleftrightarrow d _ A a = 0 . \\end{align*}"}
+{"id": "2137.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow \\bar x } { \\rm d i s t } ( \\bar y , { Z } ( x ) ) = 0 , \\end{align*}"}
+{"id": "4605.png", "formula": "\\begin{align*} \\bigcup _ { m = 1 } ^ { n - 1 } \\tilde D _ m \\supset A _ b \\ , , \\end{align*}"}
+{"id": "8907.png", "formula": "\\begin{align*} V _ n ( x ) : = \\prod _ { 1 \\leq k < n / 2 } \\left ( x ^ 2 + 4 \\sin ^ 2 \\left ( \\frac { \\pi k } { n } \\right ) \\right ) \\end{align*}"}
+{"id": "3866.png", "formula": "\\begin{align*} P \\left ( Z _ { n + 1 } = x \\mid \\sigma \\{ Z _ 0 , \\ldots Z _ n \\} \\right ) \\doteq G ( L ^ { n + 1 , Z } ) ( Z _ n , x ) , \\ ; x \\in \\Delta ^ o , \\end{align*}"}
+{"id": "8985.png", "formula": "\\begin{align*} u ( r ) = r ^ { - \\frac { \\tilde { N } _ + - 2 } { 2 } } ( c _ 1 ( - \\ln r ) + c _ 2 ) . \\end{align*}"}
+{"id": "1037.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sum _ { k = 1 } ^ { n } \\| z _ { k } - z _ { k , n } \\| = 0 \\end{align*}"}
+{"id": "1173.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ n } x ^ \\gamma g _ k ( x ) \\ , d x = \\sum _ { \\beta \\in \\mathbb { Z } _ + ^ n , \\ , \\beta \\leq \\gamma } \\binom { \\gamma } { \\beta } \\int _ { Q _ { 0 , k } } \\left [ \\int _ { \\mathbb R ^ n } ( x - y ) ^ { \\beta } \\theta ( x - y ) \\ , d x \\right ] y ^ { \\gamma - \\beta } \\eta ( y ) \\ , d y = 0 . \\end{align*}"}
+{"id": "977.png", "formula": "\\begin{align*} u ( x ) = \\mathbb E _ x u ( X _ 0 ) = \\mathbb E _ x Y ^ x _ 0 & = \\mathbb E _ x g ( X _ { \\tau _ D } ) + \\mathbb E _ x \\int _ 0 ^ { \\tau _ D } f ( X _ t , u ( X _ t ) ) \\ , d t + \\mathbb E _ x A ^ \\mu _ { \\tau _ D } \\\\ & = P _ D ( g ) ( x ) + R ^ D f ( \\cdot , u ) ( x ) + R ^ D \\mu ( x ) \\end{align*}"}
+{"id": "6972.png", "formula": "\\begin{align*} \\Pi : [ U _ + ] & \\rightarrow \\mathbb { P } _ { \\mathbb { R } } ^ n \\\\ z & \\mapsto \\Pi ( z ) = [ \\widetilde { \\Pi } ( \\mathbf { z } ) ] . \\end{align*}"}
+{"id": "4539.png", "formula": "\\begin{align*} \\frac { \\alpha _ N ( t ^ * ) } { t ^ * } = \\frac { ( a _ - ) ^ N } { t ^ * } \\int _ 0 ^ { t ^ * } \\frac { 1 } { ( N - 1 ) ! } s ^ { N - 1 } e ^ { - s a _ + } d s . \\end{align*}"}
+{"id": "9156.png", "formula": "\\begin{align*} G _ { \\beta , \\beta ' } = ( w _ { \\beta , 1 } - w _ { \\beta ' , 1 } ) \\cdot G _ { \\beta , \\gamma } . \\end{align*}"}
+{"id": "5026.png", "formula": "\\begin{align*} \\begin{bmatrix} \\mathbf { r } _ S ^ S - \\nu \\mathbf { w } _ S ^ S \\\\ - \\mathbf { r } _ { S ^ c } ^ S + \\nu \\mathbf { w } _ { S ^ c } ^ S \\end{bmatrix} . \\end{align*}"}
+{"id": "3285.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ^ { d + r } ; q ^ d ) _ k ^ { d - r } ( q ^ r ; q ^ d ) _ k ^ { r - 1 } ( q ^ { r - d } ; q ^ d ) _ k q ^ { d k } } { ( q ^ d ; q ^ d ) _ k ^ d } \\equiv \\frac { ( 1 - q ^ r ) ^ { r } ( 1 - q ^ { d - r } ) ( q ^ d ; q ^ d ) _ { n - 1 - ( n + r ) / d } } { - ( - 1 ) ^ { n - 1 - ( n + r ) / d } ( q ^ d ; q ^ d ) _ { ( n + r ) / d } ^ { d - 1 } } q ^ { A ( d , n , r ) } , \\end{align*}"}
+{"id": "8191.png", "formula": "\\begin{align*} R _ 2 & = H ( Y _ 2 | Y _ 1 , { \\bf S } ) \\\\ & = P _ { Y _ 1 } ( 1 ) H ( Y _ 2 | Y _ 1 = 1 , { \\bf S } ) + P _ { Y _ 1 } ( 0 ) H ( Y _ 2 | Y _ 1 = 0 , { \\bf S } ) \\\\ & = \\frac { B _ 1 } { M } H ( \\frac { B _ 2 ^ 1 } { B _ 1 } ) + ( 1 - \\frac { B _ 1 } { M } ) H ( \\frac { B _ 2 ^ 0 } { M - B _ 1 } ) . \\end{align*}"}
+{"id": "314.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\to 0 } u ( t ) = u _ 0 , { \\rm w i t h \\ c o n v e r g e n c e \\ i n } \\ L ^ 1 _ { \\rm l o c } ( \\real ^ N ) \\end{align*}"}
+{"id": "4298.png", "formula": "\\begin{align*} \\mathbf { w } ^ k _ { s , t } = \\mathbf { w } ^ k _ { s , u } + \\sum _ { i = 1 } ^ { k - 1 } \\mathbf { w } ^ { k - i } _ { s , u } \\otimes \\mathbf { w } ^ { i } _ { u , t } + \\mathbf { w } ^ k _ { u , t } , 1 \\le k \\le [ p ] , \\ , \\ , s \\le u \\le t . \\end{align*}"}
+{"id": "7330.png", "formula": "\\begin{align*} \\frac { 1 } { m } \\sum _ { j = 0 } ^ { m - 1 } \\frac { e ^ { 2 \\pi i \\frac { j r } { m } } } { s + 2 \\sin ^ 2 \\left ( \\pi \\frac { ( j + \\alpha ) } { m } \\right ) } = e ^ { - 2 \\pi i \\alpha \\ell / m } \\cdot \\frac { U _ { m - \\ell - 1 } ( s + 1 ) + e ^ { 2 \\pi i \\alpha } U _ { \\ell - 1 } ( s + 1 ) } { T _ { m } ( s + 1 ) - \\cos 2 \\pi \\alpha } . \\end{align*}"}
+{"id": "1599.png", "formula": "\\begin{align*} \\omega _ i ^ j = 0 , 1 \\leq i , j \\leq n - 1 . \\end{align*}"}
+{"id": "7104.png", "formula": "\\begin{align*} | b ( t , x ) | = \\frac { | b ( t , x ) | } { \\langle | b ( t , \\cdot ) | ^ q \\rangle ^ { \\frac { 1 } { q } } } \\langle | b ( t , \\cdot ) | ^ q \\rangle ^ { \\frac { 1 } { q } } \\leq \\frac { d } { q } \\biggl ( \\frac { | b ( t , x ) | ^ q } { \\langle | b ( t , \\cdot ) | ^ q \\rangle } \\biggr ) ^ { \\frac { 1 } { d } } + \\frac { 2 } { p } \\bigl ( \\langle | b ( t , \\cdot ) | ^ q \\rangle ^ { \\frac { 1 } { q } } \\bigr ) ^ { \\frac { p } { 2 } } , \\end{align*}"}
+{"id": "3036.png", "formula": "\\begin{align*} [ X _ 3 , X _ 2 ] = 2 X _ 1 \\ , , [ X _ 2 , X _ 1 ] = 2 X _ 3 \\ , , [ X _ 1 , X _ 3 ] = 2 X _ 2 \\ , . \\end{align*}"}
+{"id": "909.png", "formula": "\\begin{align*} - L u = f ( \\cdot , u ) + \\mu \\quad D . \\end{align*}"}
+{"id": "1323.png", "formula": "\\begin{align*} \\Pi _ \\pm \\bigl ( \\mathrm { d } \\Phi _ X ( p ) , \\mathrm { d } \\Phi _ Y ( p ) \\bigr ) & = \\pm \\Bigl [ \\langle p , \\nabla _ X Y \\rangle - \\langle p , \\nabla _ Y X \\rangle - \\bigl \\langle p , T ( X , Y ) \\bigr \\rangle \\Bigr ] \\\\ & = \\pm \\bigl \\langle p , [ X , Y ] \\bigr \\rangle , \\end{align*}"}
+{"id": "7401.png", "formula": "\\begin{align*} \\Sigma : = \\left \\{ ( x , t ) \\in \\R \\times [ 0 , T ] \\ : \\ x \\ge 0 , \\ t + x \\ge R , \\ 0 < t - x < \\frac { R } { 2 } \\right \\} . \\end{align*}"}
+{"id": "7990.png", "formula": "\\begin{align*} \\dot { \\bar { \\mathcal { F } } } ( \\omega , \\phi _ { \\partial } , \\Sigma ) = \\{ \\bar { \\mathcal { F } } , \\bar { H } \\} _ { D } ( \\omega , \\phi _ { \\partial } , \\Sigma ) - \\int _ { \\Gamma } E ( \\frac { \\delta \\bar { \\mathcal { F } } } { \\delta \\Sigma } ) \\wedge \\ast \\boldsymbol { n } ( d \\phi ) , \\end{align*}"}
+{"id": "9042.png", "formula": "\\begin{align*} ( a b ) c - a ( b c ) = \\Bigl ( \\int _ 0 ^ T d \\Lambda a \\Bigr ) [ b _ \\Lambda c ] + ( - 1 ) ^ { p ( a ) p ( b ) } \\Bigl ( \\int _ 0 ^ T d \\Lambda b \\Bigr ) [ a _ \\Lambda c ] . \\end{align*}"}
+{"id": "2614.png", "formula": "\\begin{align*} s _ 5 - s _ 4 = s _ 3 - s _ 1 = s _ 4 - s _ 3 . \\end{align*}"}
+{"id": "1645.png", "formula": "\\begin{align*} \\widetilde { G } _ { 1 } \\left ( x , t \\right ) = \\left ( G _ { 1 , 1 } - G _ { 1 , 2 } \\right ) \\left ( x , t \\right ) , \\widetilde { G } _ { 2 } \\left ( x , t \\right ) = \\left ( G _ { 2 , 1 } - G _ { 2 , 2 } \\right ) \\left ( x , t \\right ) , \\left ( x , t \\right ) \\in Q _ { T } . \\end{align*}"}
+{"id": "2177.png", "formula": "\\begin{align*} r _ { A - A } ( t ) : = | \\{ ( a , b ) \\in A \\times A : a - b = t \\} | . \\end{align*}"}
+{"id": "8197.png", "formula": "\\begin{align*} B _ 1 ^ { \\star } & = \\min ( B _ , \\frac { M } { 2 } ) , \\\\ B _ 2 ^ { 0 \\star } & = \\min ( B _ , \\frac { M - B _ 1 ^ { \\star } } { 2 } ) , \\\\ B _ 2 ^ { 1 \\star } & = \\min ( B _ , \\frac { B _ 1 ^ { \\star } } { 2 } ) . \\end{align*}"}
+{"id": "2642.png", "formula": "\\begin{align*} ( \\mathcal { F } f ) ( \\xi ) = \\int _ { \\mathbb { T } ^ d } f ( z ) e ^ { - i z \\cdot \\xi } \\ , d z , \\end{align*}"}
+{"id": "7022.png", "formula": "\\begin{align*} ( \\mu - \\Delta + b \\cdot \\nabla ) ^ { - 1 } f & : = ( \\mu - \\Delta ) ^ { - 1 } f \\\\ & - ( \\mu - \\Delta ) ^ { - \\frac { 1 } { 2 } - \\frac { 1 } { q } } Q _ { p } ( 1 + T _ p ) ^ { - 1 } G _ { p } ( \\mu - \\Delta ) ^ { - \\frac { 1 } { 2 } + \\frac { 1 } { r } } f , \\end{align*}"}
+{"id": "4856.png", "formula": "\\begin{align*} 0 & \\geq \\langle \\nabla ^ { + } u _ j ( \\hat { z } ) , \\hat { z } - \\hat { x } ^ * \\rangle = \\langle \\nabla ^ { + } u _ j ( \\hat { z } ) , \\gamma _ 1 e _ 1 + \\sum _ { i = 2 } ^ { n - 1 } \\gamma _ i e _ i \\rangle \\\\ & \\geq \\frac { \\gamma _ 1 } { | \\nabla ^ { + } u _ j ( \\hat { x } ^ * ) | } \\langle \\nabla ^ { + } u _ j ( \\hat { z } ) , \\nabla ^ { + } u _ j ( \\hat { x } ^ * ) \\rangle - C _ 0 \\epsilon . \\end{align*}"}
+{"id": "6113.png", "formula": "\\begin{align*} K = A _ d \\oplus A _ { d - 1 } \\oplus \\cdots \\oplus A _ 1 = \\sum _ { \\mu = 1 } ^ d A _ { \\otimes \\mu } , A _ { \\otimes \\mu } = I _ d \\otimes \\cdots \\otimes I _ { \\mu + 1 } \\otimes A _ \\mu \\otimes I _ { \\mu - 1 } \\otimes \\cdots \\otimes I _ 1 , \\end{align*}"}
+{"id": "6756.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau ^ k } [ \\theta ( u ) - \\theta ( \\breve { u } ^ k ) ] - \\frac { 1 } { \\tau ^ { k - 1 } } [ \\theta ( u ) - \\theta ( \\breve { u } ^ { k - 1 } ) ] + ( w - \\widetilde { w } ^ k ) ^ T F ( \\widetilde { w } ^ k ) \\\\ \\geq & \\frac { 1 } { 2 } ( \\| v ^ { k + 1 } - v \\| ^ 2 _ { H } - \\| v ^ k - v \\| ^ 2 _ { H } ) + \\frac { 1 } { 2 } \\| v ^ k - \\widetilde { v } ^ k \\| ^ 2 _ { G } , ~ \\forall w \\in \\Omega . \\end{aligned} \\end{align*}"}
+{"id": "1466.png", "formula": "\\begin{align*} ( Q _ n ) _ { F _ n \\setminus S \\Delta } = ( P _ n \\cap C ( p ) ) _ { F _ n \\setminus S \\Delta } = X _ { F _ n \\setminus S \\Delta } = C ( p ) _ { F _ n \\setminus S \\Delta } . \\end{align*}"}
+{"id": "460.png", "formula": "\\begin{align*} \\varepsilon ( h ) \\varepsilon ( g ) = \\pi ( h ) \\pi ( h ^ { - 1 } ) \\varepsilon ( g ) = \\pi ( h ) \\varepsilon ( h ^ { - 1 } g ) \\pi ( h ^ { - 1 } ) = \\varepsilon ( r ( h ) g ) \\pi ( h ) \\pi ( h ^ { - 1 } ) = \\varepsilon ( g ) \\varepsilon ( h ) . \\end{align*}"}
+{"id": "4586.png", "formula": "\\begin{align*} T = ( T _ s , T _ 0 , T _ 1 , T _ 2 ) \\triangleq \\left ( \\int _ M t s _ g d \\mu , \\int _ M d \\mu , \\int _ M t d \\mu , \\int _ M t ^ 2 d \\mu \\right ) . \\end{align*}"}
+{"id": "7205.png", "formula": "\\begin{align*} m ( a _ 1 ) = m ( a _ 2 ) & \\implies m ( a _ 1 ) \\wedge b = m ( a _ 2 ) \\wedge b \\iff ( m ( a _ 1 ) , m ( a _ 2 ) ) \\in \\Phi _ b \\\\ & \\iff ( a _ 1 , a _ 2 ) \\in m ^ { - 1 } ( \\Phi _ b ) . \\end{align*}"}
+{"id": "2759.png", "formula": "\\begin{align*} \\{ \\Theta _ { i } , \\Theta _ { j } \\} = C _ { i j } ^ { \\ \\ k } \\Theta _ { k } , \\end{align*}"}
+{"id": "3953.png", "formula": "\\begin{align*} S : = \\left ( - 1 + \\frac { \\eta \\left | K _ 0 \\right | ^ 2 } { \\lambda _ { N + 1 } ^ { \\beta _ 1 - \\beta } } \\right ) I _ { 3 N } + \\frac { 2 } { \\lambda _ { N + 1 } ^ { \\beta _ 1 } } \\bar { P } \\bar { P } ^ { \\top } \\prec 0 . \\end{align*}"}
+{"id": "5309.png", "formula": "\\begin{align*} b _ i ^ u = E _ i ^ u \\left [ \\sum _ { t = 0 } ^ { \\infty } \\theta _ { X ( t ) } ^ 1 \\ , a ( t ) \\ , \\beta ^ { t } \\right ] , \\end{align*}"}
+{"id": "3101.png", "formula": "\\begin{align*} u _ \\varphi ( t ) & = \\left ( \\begin{array} { c c c } 1 & 0 & c _ 2 \\ , t ^ { p ^ { e _ 2 } } \\\\ 0 & 1 & c _ 1 \\ , t ^ { p ^ { e _ 1 } } \\\\ 0 & 0 & 1 \\end{array} \\right ) ( \\ , c _ 1 , c _ 2 \\in k , e _ 1 , e _ 2 \\geq 0 \\ , ) , \\end{align*}"}
+{"id": "2822.png", "formula": "\\begin{align*} - \\ddot { Q } ^ { 1 } - Q ^ { 1 } = 0 . \\end{align*}"}
+{"id": "8168.png", "formula": "\\begin{align*} & b _ 1 ^ 0 = \\min ( \\frac { M } { 2 } , B _ ) , \\ b _ j ^ 0 = \\min ( \\frac { M - \\sum _ { k = 1 } ^ { j - 1 } b _ k ^ 0 } { 2 } , B _ ) . \\end{align*}"}
+{"id": "8015.png", "formula": "\\begin{align*} \\limsup _ { \\delta \\rightarrow 0 } \\limsup _ { N \\rightarrow + \\infty } \\frac { 1 } { N } \\log \\sup _ { \\sigma \\in \\mathcal { T } } P \\left ( \\sup _ { 0 \\leq t \\leq \\delta } \\left | \\mu _ { t + \\sigma , k } ^ N ( f ) - \\mu _ { \\sigma , k } ^ N ( f ) \\right | > \\epsilon \\right ) = - \\infty \\end{align*}"}
+{"id": "6270.png", "formula": "\\begin{align*} y _ { k + 1 } = \\nabla ( \\Psi ^ * ) ( \\nabla \\Psi ( x _ k ) - \\nu g _ { k + 1 } ) x _ { k + 1 } = \\arg \\min \\limits _ { x \\in \\mathcal { S } } D _ { \\Psi } ( x , y _ { k + 1 } ) . \\end{align*}"}
+{"id": "5866.png", "formula": "\\begin{align*} \\dfrac { M _ { 2 } ( \\mathcal { N C } ( D _ { 2 m } ) ) } { | e ( \\mathcal { N C } ( D _ { 2 m } ) ) | } - \\dfrac { M _ { 1 } ( \\mathcal { N C } ( D _ { 2 m } ) ) } { | v ( \\mathcal { N C } ( D _ { 2 m } ) ) | } = \\dfrac { m ^ { 3 } ( m ^ { 2 } - 1 2 m + 2 8 ) + m ^ { 2 } ( 2 4 m - 9 6 ) + 6 4 m } { 3 m ( m - 2 ) ( 2 m - 2 ) } : = \\frac { f ( m ) } { g ( m ) } . \\end{align*}"}
+{"id": "8412.png", "formula": "\\begin{align*} T ( \\sigma ( a _ 1 ) ) \\circ \\sigma ( a _ 2 ) & = B _ 1 ( T ( \\sigma ( a _ 1 ) ) ) \\sigma ( a _ 3 ) S ( B _ 2 ( T ( \\sigma ( a _ 2 ) ) ) ) \\\\ & = B _ 1 ( T ( \\sigma ( a _ 1 ) ) ) \\sigma ( a _ 2 ) S ( B _ 2 ( T ( \\sigma ( a _ 3 ) ) ) ) \\\\ & = B _ 1 ( T ( \\sigma ( a _ 1 ) ) ) B _ 1 ( a _ 2 ) S ( B _ 2 ( a _ 3 ) ) S ( B _ 2 ( T ( \\sigma ( a _ 4 ) ) ) ) \\\\ & = \\epsilon ( a ) 1 , \\end{align*}"}
+{"id": "6832.png", "formula": "\\begin{align*} \\begin{pmatrix} A & 0 \\\\ 0 & 1 \\end{pmatrix} : = \\left ( \\prod _ { 2 \\leq k \\leq N - 1 } A ( k ) ( \\phi _ k , \\psi _ k ) \\right ) e ^ { \\lambda _ 3 \\phi _ { N - 1 } } , B : = F _ { N - 1 } ( \\phi _ { N } , \\ldots , \\phi _ { \\frac { N ( N - 1 ) } { 2 } } , \\psi _ { N } , \\ldots , \\psi _ { \\frac { N ( N - 1 ) } { 2 } } , \\omega _ 1 , \\ldots , \\omega _ { N - 2 } ) . \\end{align*}"}
+{"id": "6697.png", "formula": "\\begin{align*} K \\cap L = K \\cap G _ { n + 1 } \\trianglelefteq K \\cap G _ n \\trianglelefteq \\ldots \\trianglelefteq K \\cap G _ 1 = K \\end{align*}"}
+{"id": "3876.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum \\limits _ { k = m ^ n _ 0 + k ^ * } ^ { m ^ n _ 0 + \\tau ^ { n , 0 } - 1 } R ^ { n , k } \\le | \\log \\delta _ 1 | , R ^ { n , m ^ n _ 0 + \\tau ^ { n , 0 } } \\le \\left | \\log \\left ( \\frac { \\delta _ 1 } { 2 } \\right ) \\right | \\end{align*}"}
+{"id": "2398.png", "formula": "\\begin{align*} & d _ n ^ { 2 s + 4 - i } a _ { n , i , k } \\\\ = & \\sum ~ ~ \\frac { d _ n ^ { h } H ^ { ( h ) } ( t ) } { h ! } \\cdot \\prod _ { u = 1 } ^ { s + 2 } \\frac { d _ n ^ { f _ { u } } F _ { 1 / 4 } ^ { ( f _ { u } ) } ( t ) } { f _ { u } ! } \\cdot \\prod _ { u = 1 } ^ { s + 2 } \\frac { d _ n ^ { f _ { u } ^ { \\prime } } F _ { 3 / 4 } ^ { ( f _ { u } ^ { \\prime } ) } ( t ) } { f _ { u } ^ { \\prime } ! } \\cdot \\prod _ { u = 1 } ^ { 2 s + 4 } \\frac { d _ n ^ { g _ u } \\left ( ( t + k ) G ( t ) \\right ) ^ { ( g _ u ) } } { g _ u ! } \\big | _ { t = - k } \\in \\mathbb { Z } , \\end{align*}"}
+{"id": "3314.png", "formula": "\\begin{align*} K ^ { [ \\nu ] } _ { n } = \\frac { 1 } { \\sqrt { n + \\nu } } \\exp \\left ( \\frac { ( n \\mu _ 0 - S _ { n } ) ^ 2 } { 2 ( n + \\nu ) } \\right ) . \\end{align*}"}
+{"id": "1748.png", "formula": "\\begin{align*} e _ 0 : = 4 i X _ { - n } , e _ { 2 n - 1 } : = - X ^ \\prime _ { - 1 } - 2 i Y ^ \\prime _ { - 1 } \\quad \\mbox { a n d } e _ { 2 n } : = - X ^ \\prime _ { - 1 } + 2 i Y ^ \\prime _ { - 1 } , \\end{align*}"}
+{"id": "8353.png", "formula": "\\begin{align*} \\mu ( K ^ \\lambda ( 0 , \\bar x ) ) ^ s = \\lambda \\mu ( K ) ^ s + ( 1 - \\lambda ) \\mu ( K ^ 1 ( 0 , \\bar x ) ) ^ s , \\end{align*}"}
+{"id": "9127.png", "formula": "\\begin{align*} \\lambda _ { h , \\beta } = \\big \\{ r _ { \\beta } ( h , 1 ) \\leq \\dots \\leq r _ { \\beta } ( h , d _ { \\beta } ) \\big \\} \\end{align*}"}
+{"id": "6161.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { ( \\tau ^ k ) ^ 2 } \\| A \\breve { x } ^ { k } - b \\| ^ 2 = & \\| A \\widetilde { x } ^ { k } - b \\| ^ 2 + \\frac { 1 } { ( \\tau ^ { k - 1 } ) ^ 2 } \\| A \\breve { x } ^ { k - 1 } - b \\| ^ 2 \\\\ & + 2 \\frac { 1 - \\tau ^ k } { \\tau ^ { k } } ( A \\widetilde { x } ^ { k } - b ) ^ T ( A \\breve { x } ^ { k - 1 } - b ) . \\end{aligned} \\end{align*}"}
+{"id": "4512.png", "formula": "\\begin{align*} \\mathbb { P } ^ { ( 1 ) } _ { \\ell } : = \\mathbb { P } \\bigg [ \\sup _ { \\substack { X _ { \\ell - 1 } < n \\leqslant X _ { \\ell } \\\\ 1 \\leqslant j \\leqslant J } } V ( n , y _ j ) > \\frac { T _ 1 ( \\ell ) } { \\ell ^ { K / 2 } } \\bigg ] \\end{align*}"}
+{"id": "7728.png", "formula": "\\begin{align*} ( L _ { \\varepsilon W , \\alpha , \\theta } ^ { V } u ) _ { n } = \\Sigma _ { k = - \\ell } ^ \\ell \\hat V _ { k } u _ { n - k } + \\varepsilon W ( \\theta + n \\alpha ) u _ { n } , \\ \\ n \\in \\Z . \\end{align*}"}
+{"id": "5304.png", "formula": "\\begin{align*} v _ i ( \\nu ) = \\min \\ , \\left \\{ v _ i ^ u ( \\nu ) : u \\in \\mathcal { U } \\right \\} . \\end{align*}"}
+{"id": "5961.png", "formula": "\\begin{align*} f _ { \\phi } & \\ : = \\ : \\left ( a c - b d \\right ) + \\left ( a - \\left ( b + c \\right ) \\right ) + \\left ( c - \\left ( a + d \\right ) \\right ) \\\\ & \\ : = \\ : a b ^ { \\prime } c + a c d ^ { \\prime } + a b ^ { \\prime } c ^ { \\prime } + a ^ { \\prime } c d ^ { \\prime } \\\\ & \\ : = \\ : \\left ( a - b \\right ) + \\left ( c - d \\right ) \\end{align*}"}
+{"id": "4502.png", "formula": "\\begin{align*} F _ R ( z ) : = \\exp \\bigg ( \\sum _ { k \\leqslant R } \\frac { X ( k ) } { \\sqrt { k } } z ^ k \\bigg ) . \\end{align*}"}
+{"id": "7869.png", "formula": "\\begin{align*} \\det ( \\mathbf { Q } _ { i , j } ) & = \\begin{vmatrix} 2 ( d _ { i , 1 } - d _ { j , 1 } ) & & & \\\\ & 2 ( d _ { i , 2 } - d _ { j , 2 } ) & & \\\\ & & \\ddots & \\\\ & & & 2 ( d _ { i , m } - d _ { j , m } ) \\end{vmatrix} \\\\ & = 2 ^ m \\prod _ { k = 1 } ^ { m } ( d _ { i , k } - d _ { j , k } ) . \\end{align*}"}
+{"id": "8510.png", "formula": "\\begin{align*} w _ n = G _ { n } [ b ] G _ { 2 , n } [ b ] \\dotsc G _ { n , n } [ b ] , \\end{align*}"}
+{"id": "608.png", "formula": "\\begin{align*} \\psi ( z ) = \\frac { d } { d z } \\ln \\Gamma ( z ) = \\frac { \\Gamma ' ( z ) } { \\Gamma ( z ) } = - \\gamma + \\sum _ { n = 0 } ^ { \\infty } \\frac { z - 1 } { ( n + 1 ) ( n + z ) } . \\end{align*}"}
+{"id": "8687.png", "formula": "\\begin{align*} [ u , v ] = \\pi _ * [ u ^ H , v ^ H ] = \\pi _ * ( X ^ H - \\sqrt { - 1 } J X ^ H ) = X - \\sqrt { - 1 } J _ G X \\in T ^ { 1 , 0 } Y ' _ G , \\end{align*}"}
+{"id": "5498.png", "formula": "\\begin{align*} A ^ { ( j + 1 ) } = A ^ { ( j + 1 ) } _ { j - 1 } [ x _ { j + 1 } ; \\sigma _ { j + 1 } ^ { ( j + 1 ) * } ] \\ldots [ x _ N ; \\sigma _ N ^ { ( j + 1 ) * } ] [ x _ j ; \\sigma ' _ j , \\delta ' _ j ] , \\end{align*}"}
+{"id": "8961.png", "formula": "\\begin{align*} \\tilde f : = f + ( 2 C + 1 ) M _ \\Omega ( f ) \\quad \\tilde g : = g + ( 2 C + 1 ) M _ \\Omega ( g ) \\quad , \\end{align*}"}
+{"id": "145.png", "formula": "\\begin{align*} { ( 3 ^ { 2 n - \\alpha ( 3 ) } ) } ^ i F ( t / 3 ^ i ) = { ( 3 ^ { 2 n - \\alpha ( 3 ) } ) } ^ { i + 1 } F ( t / 3 ^ { i + 1 } ) + { ( 3 ^ { 2 n - \\alpha ( 3 ) } ) } ^ i \\cdot R ( t / 3 ^ i ) . \\end{align*}"}
+{"id": "527.png", "formula": "\\begin{align*} \\mu ( \\delta ) : = \\frac { ( \\delta - 1 ) ^ 2 } { 4 } , \\delta \\in \\R , \\quad \\gamma = \\gamma ( \\alpha , \\beta , \\mu ) : = - ( 1 + \\alpha + \\beta ) / 2 - \\sqrt { \\mu ( \\alpha + \\beta ) - \\mu } . \\end{align*}"}
+{"id": "7221.png", "formula": "\\begin{align*} \\mathbb { H } ^ { 1 } _ 0 ( V _ { \\min } , V _ F ) = \\{ p \\in \\mathbb { H } ^ { 1 } ( V _ { \\min } , V _ F ) : p | _ { V _ { \\min } } = p | _ { V _ F } = 0 \\} . \\end{align*}"}
+{"id": "6068.png", "formula": "\\begin{align*} T \\in \\left ( \\bigcup _ { M \\in S } M ^ + \\right ) \\cap V = \\left ( \\bigcup _ { M \\in S } M ^ + \\right ) \\setminus V ^ \\perp \\end{align*}"}
+{"id": "8313.png", "formula": "\\begin{align*} \\{ \\eta _ 1 \\wedge \\eta _ 2 , \\omega _ 1 \\} & = - \\big ( \\{ \\omega _ 1 , \\eta _ 1 \\} \\eta _ 2 + ( - 1 ) ^ { | \\omega _ 1 | } \\eta _ 1 \\{ \\omega _ 1 , \\eta _ 2 \\} \\big ) \\\\ & = ( - 1 ) ^ { | \\omega _ 1 | } \\{ \\eta _ 1 , \\omega _ 1 \\} \\eta _ 2 + \\eta _ 1 \\{ \\eta _ 2 , \\omega _ 1 \\} \\\\ & = ( - 1 ) ^ { | \\omega _ 1 | } ( \\iota _ { \\eta _ 1 } \\omega _ 1 ) \\eta _ 2 + \\eta _ 1 ( \\iota _ { \\eta _ 2 } \\omega _ 1 ) , \\end{align*}"}
+{"id": "4216.png", "formula": "\\begin{align*} \\theta _ 1 ( v , \\tau ) = 2 q ^ { \\frac { 1 } { 8 } } { \\rm c o s } ( \\pi v ) \\prod _ { j = 1 } ^ { \\infty } [ ( 1 - q ^ j ) ( 1 + e ^ { 2 \\pi \\sqrt { - 1 } v } q ^ j ) ( 1 + e ^ { - 2 \\pi \\sqrt { - 1 } v } q ^ j ) ] , \\end{align*}"}
+{"id": "1405.png", "formula": "\\begin{align*} e ^ { - t s ^ \\alpha } & = \\int _ 0 ^ \\infty e ^ { - s x } \\ , d F _ \\alpha ( x \\vert t ) = \\int _ 0 ^ \\infty e ^ { - s x } f _ \\alpha ( x \\vert t ) \\ , d x \\end{align*}"}
+{"id": "5342.png", "formula": "\\begin{align*} w ^ { S \\setminus \\{ i _ 1 , i _ 2 \\} } _ j = \\frac { \\displaystyle \\frac { w ^ S _ { i _ 1 } } { w ^ { S \\setminus \\{ i _ 2 \\} } _ { i _ 1 } } \\ , w ^ { S \\setminus \\{ i _ 2 \\} } _ j + \\frac { w ^ S _ { i _ 2 } } { w ^ { S \\setminus \\{ i _ 1 \\} } _ { i _ 2 } } \\ , w ^ { S \\setminus \\{ i _ 1 \\} } _ j - w ^ S _ j } { \\displaystyle \\frac { w ^ S _ { i _ 1 } } { w ^ { S \\setminus \\{ i _ 2 \\} } _ { i _ 1 } } + \\frac { w ^ S _ { i _ 2 } } { w ^ { S \\setminus \\{ i _ 1 \\} } _ { i _ 2 } } - 1 } . \\end{align*}"}
+{"id": "9149.png", "formula": "\\begin{align*} \\kappa _ { \\beta } = \\begin{cases} | \\beta | - 1 & \\quad \\ \\beta = [ i , j ] \\\\ | \\beta | + 2 ( n - j ) - 1 & \\quad \\ \\beta = [ i , n , j ] \\end{cases} \\end{align*}"}
+{"id": "989.png", "formula": "\\begin{align*} M _ D \\nu ( x ) = \\int _ { \\partial _ m D } M _ D ( x , y ) \\ , \\nu ( d y ) , x \\in D . \\end{align*}"}
+{"id": "5736.png", "formula": "\\begin{align*} X _ + ( t ) X _ + ' ( t ) = & \\sum _ { i : \\gamma ^ + _ i > \\gamma _ * } \\gamma ^ + _ i | \\xi _ i ^ + ( t ) | ^ 2 + \\sum _ { i : \\gamma ^ - _ i > \\gamma _ * } \\gamma ^ - _ i | \\xi _ i ^ - ( t ) | ^ 2 + \\sum _ { i : \\gamma ^ + _ i > \\gamma _ * } \\xi _ i ^ + ( t ) \\mathcal { E } ^ + _ i ( t ) + \\sum _ { i : \\gamma ^ - _ i > \\gamma _ * } \\xi _ i ^ - ( t ) \\mathcal { E } ^ - _ i ( t ) \\\\ \\geq & ( \\gamma _ * + \\varepsilon _ 1 ) X ^ 2 _ + ( t ) + X _ + ( t ) Y _ + ( t ) . \\end{align*}"}
+{"id": "1200.png", "formula": "\\begin{align*} \\left | \\partial _ x ^ \\alpha \\partial _ y ^ \\beta \\mathcal K ( x , y ) - \\partial _ x ^ \\alpha \\partial _ y ^ \\beta \\mathcal K ( x + u , y ) \\right | & \\leq C | u | ^ \\delta | x - y | ^ { - n - | \\alpha | - | \\beta | - \\delta } \\\\ & \\quad \\quad \\begin{cases} | \\alpha | = \\lfloor s \\rfloor , \\\\ | \\beta | = \\lfloor J - n \\rfloor - \\lfloor s \\rfloor . \\end{cases} \\end{align*}"}
+{"id": "1008.png", "formula": "\\begin{align*} V ^ D ( u , v ) = 2 \\int _ { D \\times \\mathbb R ^ d } ( u ( x ) - u ( y ) ) ( v ( x ) - v ( y ) ) j ( x , y ) \\ , d x \\ , d y , u , v \\in V ^ D . \\end{align*}"}
+{"id": "8718.png", "formula": "\\begin{gather*} L _ { 2 } = L e i _ { 2 } ( 3 , F ) = F a _ { 1 } \\oplus F a _ { 2 } \\oplus F a _ { 3 } , \\ \\mbox { w h e r e } [ a _ { 1 } , a _ { 1 } ] = a _ { 3 } , [ a _ { 1 } , a _ { 2 } ] = \\\\ [ a _ { 1 } , a _ { 3 } ] = [ a _ { 2 } , a _ { 1 } ] = [ a _ { 2 } , a _ { 2 } ] = [ a _ { 2 } , a _ { 3 } ] = [ a _ { 3 } , a _ { 1 } ] = [ a _ { 3 } , a _ { 2 } ] = [ a _ { 3 } , a _ { 3 } ] = 0 . \\end{gather*}"}
+{"id": "848.png", "formula": "\\begin{align*} { g _ { i j } } = { ( \\frac { 1 } { 2 } { F ^ 2 } ) _ { { . i } { . j } } } = { F _ { { . i } } } { F _ { { . j } } } + F { F _ { { . i } { . j } } } = { l _ i } { l _ j } + F { { \\dot \\partial } _ j } { l _ i } . \\end{align*}"}
+{"id": "3482.png", "formula": "\\begin{align*} x \\xrightarrow { 2 } ( ( x + 1 ) ^ 3 , ( x - 1 ) ^ 3 , x ^ 3 ) \\xrightarrow { 1 } \\frac { ( x + 1 ) ^ 3 + ( x - 1 ) ^ 3 - 2 x ^ 3 } { 6 } = x , \\end{align*}"}
+{"id": "2491.png", "formula": "\\begin{align*} \\gamma \\in C ^ 1 ( [ 0 , \\infty ) ) \\ , , \\gamma ( 0 ) = 0 , \\gamma > 0 \\ ; \\ ; \\ ; \\ ; ( 0 , \\infty ) \\ , , \\end{align*}"}
+{"id": "8660.png", "formula": "\\begin{align*} \\mbox { $ ( \\xi _ { M ' } u ) ( x ) = \\frac { \\partial } { \\partial t } \\left ( u ( \\exp ( t \\xi ) \\circ x ) \\right ) | _ { t = 0 } $ , f o r a n y $ u \\in C ^ \\infty ( M ' ) $ } . \\end{align*}"}
+{"id": "464.png", "formula": "\\begin{align*} \\pi ( e ) = \\sum _ { S \\subseteq X _ e } P _ S , \\end{align*}"}
+{"id": "6888.png", "formula": "\\begin{align*} s _ X = \\sum _ { \\pi \\in \\mathfrak { S } _ X } ( - 1 ) ^ \\pi \\pi . \\end{align*}"}
+{"id": "4061.png", "formula": "\\begin{align*} E _ i ( \\sigma ) : = \\int _ { r ( - \\infty ) } ^ { r ( 0 ) } \\frac { e ^ { t } } { { t } } d t = \\gamma + \\ln ( - \\sigma ) + \\sum _ { k = 1 } ^ { \\infty } \\frac { \\sigma ^ k } { k k ! } \\end{align*}"}
+{"id": "4946.png", "formula": "\\begin{align*} \\begin{aligned} ( \\tau _ { L } ) _ \\ast [ L ' ] & = [ L ' ] + [ L ] \\\\ & = 2 \\sqrt { \\beta } \\left ( ( e ^ { L ' } _ + - e ^ { L ' } _ - ) + ( e ^ { L } _ + - e ^ { L } _ - ) \\right ) \\\\ & = 2 \\sqrt { \\beta } ( e ^ { L ' } _ + - e ^ { L } _ - ) . \\end{aligned} \\end{align*}"}
+{"id": "8712.png", "formula": "\\begin{align*} ( \\tau F _ { j } ) ( z ) = \\tau ( z ) ( B _ { G , M _ 1 } F _ { j } ) ( z ) = ( ( \\tau B _ { G , M _ 1 } ) F _ { j } ) ( z ) , \\end{align*}"}
+{"id": "2875.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to \\infty } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ X = \\left [ \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { X } . \\end{align*}"}
+{"id": "3486.png", "formula": "\\begin{align*} J \\bigl ( \\sigma ^ { \\mathsf { R e Q U } } ( g ( x ) ) \\bigr ) = \\operatorname { d i a g } \\bigl ( 2 \\sigma ^ { \\mathsf { R e L U } } ( g ( x ) ) \\bigr ) \\cdot J ( g ( x ) ) , \\end{align*}"}
+{"id": "1046.png", "formula": "\\begin{align*} \\mathbf { z } ^ \\top = ( v _ { \\infty } ^ { - 1 } , v _ { \\infty } ^ { - 1 } \\phi _ { 1 } , v _ { \\infty } ^ { - 1 } \\phi _ { 2 } , \\dots ) \\mathbf { z } _ { n + 1 } ^ \\top = ( v _ { n + 1 } ^ { - 1 } , v _ { n + 1 } ^ { - 1 } \\phi _ { n , 1 } , \\dots , v _ { n + 1 } ^ { - 1 } \\phi _ { n , n } ) , \\end{align*}"}
+{"id": "2067.png", "formula": "\\begin{align*} & - \\alpha ^ { - 1 } \\big ( \\int _ 0 ^ s U ( u , x ) f ' ( u ) \\mathrm { d } u + f ( s ) U ( s , x ) - f ( 0 ) U ( 0 , x ) \\big ) \\\\ & = - \\int _ 0 ^ s \\int _ 0 ^ 1 f ( u ) A ( x , y ) U ( u , y ) \\mathrm { d } y \\mathrm { d } u + \\beta \\int _ 0 ^ s f ( u ) \\mathrm { d } \\xi _ 2 ( u , x ) 1 _ { \\{ \\gamma \\sigma d ^ { - 1 } T ^ { - \\frac { 1 } { 2 } } \\to \\beta \\} } + \\zeta \\int _ 0 ^ s f ( u ) \\mathrm { d } \\xi _ 3 ( u , x ) . \\end{align*}"}
+{"id": "4808.png", "formula": "\\begin{align*} \\begin{gathered} W _ p ( \\mathbb { Q } , \\mathbb { Q } ' ) : = \\Big ( \\inf _ { \\pi \\in \\Pi ( \\mathbb { Q } , \\mathbb { Q } ' ) } \\int _ { \\mathcal { S } _ 0 \\times \\mathcal { S } _ 0 } \\Vert \\mathbf { c } - \\mathbf { c } ' \\Vert _ p \\pi ( \\mbox { \\upshape d } \\mathbf { c } , \\mbox { \\upshape d } \\mathbf { c } ' ) \\Big ) ^ { \\frac { 1 } { p } } , \\end{gathered} \\end{align*}"}
+{"id": "6061.png", "formula": "\\begin{align*} V : = U ^ { ( 1 ) } = y A ' _ { d - 1 } \\oplus U \\subset A ' _ d . \\end{align*}"}
+{"id": "5028.png", "formula": "\\begin{align*} \\mathbf { x } ^ 1 \\mathbf { 1 } = g _ i ^ S - \\sum _ { j \\in S } w _ j ^ S x _ j ^ 0 + \\sum _ { j \\in S ^ c } w _ j ^ S x _ j ^ 1 \\end{align*}"}
+{"id": "7666.png", "formula": "\\begin{align*} S _ { \\beta } : = \\sup _ { u \\in \\mathbb { Z } ^ d } \\sum _ { v \\in \\mathbb { Z } ^ d } e ^ { \\beta d ( u , v ) } . \\end{align*}"}
+{"id": "2467.png", "formula": "\\begin{align*} E _ { 3 } : ( \\lambda , x ) = ( 0 , 0 ) . \\end{align*}"}
+{"id": "4332.png", "formula": "\\begin{align*} \\inf _ { x \\in \\mathcal { X } } & \\ \\sum _ { i \\in [ m ] } \\max \\{ 0 , x _ i - b _ i \\} . \\end{align*}"}
+{"id": "6032.png", "formula": "\\begin{align*} D _ n ( f ) = - \\int _ { \\Omega _ N } f ( \\eta ) L _ N f ( \\eta ) \\nu _ \\rho ( \\dd \\eta ) , \\end{align*}"}
+{"id": "9051.png", "formula": "\\begin{align*} \\sigma ( T ^ { ( k ) } v ) = T ^ { ( \\sigma ( k ) ) } ( \\sigma v ) ( v \\in V ^ { \\otimes n } ) . \\end{align*}"}
+{"id": "2751.png", "formula": "\\begin{align*} \\dot { \\Psi } _ { \\alpha } = \\{ \\Psi _ { \\alpha } , H _ { T } \\} \\approx 0 . \\end{align*}"}
+{"id": "8269.png", "formula": "\\begin{align*} \\beta _ { \\lambda , i } = ( 1 - \\mu ) \\beta _ { 0 , i } + \\mu \\beta _ { \\bar \\lambda , i } . \\end{align*}"}
+{"id": "6797.png", "formula": "\\begin{align*} \\dot { x } & = x ( \\kappa _ 1 x - \\kappa _ 2 y ) , \\\\ \\dot { y } & = y ( d \\kappa _ 2 x - \\kappa _ 3 ( y + C ) ) . \\end{align*}"}
+{"id": "8498.png", "formula": "\\begin{align*} \\Lambda ( A ) = \\overline { S ( A ) } \\setminus S ( A ) . \\end{align*}"}
+{"id": "5156.png", "formula": "\\begin{align*} & U _ { j } = \\frac { 1 } { \\alpha } \\left [ \\frac { a - 1 } { a - b } A ^ { a - 2 } - \\frac { b - 1 } { a - b } A ^ { b - 2 } \\right ] p ^ { \\alpha } _ { j } q ^ { - \\alpha } _ { j } \\\\ & V _ { j } = \\frac { 1 } { \\alpha } \\left [ \\frac { a - 1 } { a - b } A ^ { a - 2 } - \\frac { b - 1 } { a - b } A ^ { b - 2 } \\right ] \\end{align*}"}
+{"id": "8153.png", "formula": "\\begin{align*} d ( \\underline s , \\hat { \\underline s } ) = \\begin{cases} 0 , & \\underline s = \\hat { \\underline s } \\\\ 2 , & \\underline s , \\hat { \\underline s } \\in \\mathcal S \\underline s \\neq \\hat { \\underline s } \\end{cases} , \\end{align*}"}
+{"id": "5417.png", "formula": "\\begin{align*} S _ { n + 1 } f ( \\tfrac { 1 } { 2 } ) = S _ { - n } f ( \\tfrac { 1 } { 2 } ) . \\end{align*}"}
+{"id": "2912.png", "formula": "\\begin{align*} V _ { \\ell } ( x _ i ; f ) : = \\sum _ { y _ { 0 } < p \\leqslant y _ J } \\big | \\Psi _ f ^ { \\prime } ( x _ i / p , p ) \\big | ^ 2 . \\end{align*}"}
+{"id": "2983.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\searrow 1 } \\int _ { D _ X ( 0 ) } ^ \\infty x ^ { - \\alpha } = + \\infty , \\end{align*}"}
+{"id": "1505.png", "formula": "\\begin{align*} - \\Delta _ { p } u = \\left \\{ \\begin{array} { l l } 1 & x \\in \\Omega \\backslash \\overline { \\Omega } _ { \\delta } , \\\\ - 1 & , \\end{array} \\right . u = 0 \\partial \\Omega , \\end{align*}"}
+{"id": "6513.png", "formula": "\\begin{align*} \\mu \\in \\Lambda \\iff \\frac { d \\mu ( x ) } { d x } = \\phi ( x ) , \\phi \\colon \\R ^ n \\to \\R ^ + , \\phi \\in L ^ 1 _ { } ( \\R ^ n ) . \\end{align*}"}
+{"id": "5375.png", "formula": "\\begin{align*} v _ i ^ S ( \\beta ) = \\frac { \\bar { v } ^ S } { 1 - \\beta } + f _ i ^ S + O ( 1 - \\beta ) , \\beta \\nearrow 1 , \\end{align*}"}
+{"id": "562.png", "formula": "\\begin{align*} \\phi _ n & = 0 & \\Gamma _ r , \\end{align*}"}
+{"id": "5372.png", "formula": "\\begin{align*} b _ i ^ S ( \\beta ) = \\frac { \\bar { b } ^ S } { 1 - \\beta } + a _ i ^ S + O ( 1 - \\beta ) , \\beta \\nearrow 1 , \\end{align*}"}
+{"id": "2948.png", "formula": "\\begin{align*} \\int _ G | F ( g ) | ^ 2 w _ { t , \\gamma } ( g ) d g = \\sum _ { \\pi \\in \\widehat { K } } d _ \\pi \\| \\pi ( f ) \\| ^ 2 e ^ { - t \\lambda _ \\pi ^ 2 } \\sigma _ { t , \\gamma } ( \\pi ) . \\end{align*}"}
+{"id": "7536.png", "formula": "\\begin{align*} \\partial _ t \\Big ( \\frac { 1 } { 2 } \\frac { | m _ { 0 0 } | ^ 2 } { \\rho _ 0 } + h _ 1 ( \\rho _ { 0 0 } ) + h _ 2 ( n _ { 0 0 } ) \\Big ) + \\nabla \\cdot \\mathcal { F } _ { 0 0 } = 0 \\ , \\end{align*}"}
+{"id": "3439.png", "formula": "\\begin{align*} \\mathbb { K } _ G = \\begin{cases} \\tilde { \\R } G = \\Z _ 2 \\\\ \\C G = \\Z _ 4 , \\end{cases} k _ G = \\dim _ { \\R } \\mathbb { K } _ G = \\begin{cases} 1 G = \\Z _ 2 \\\\ 2 G = \\Z _ 4 \\end{cases} \\end{align*}"}
+{"id": "4925.png", "formula": "\\begin{align*} e ^ { L } _ { \\pm } = \\pm \\frac { 1 } { 4 \\sqrt { \\beta _ { L } } } [ L ] + \\frac { 1 } { 8 \\beta _ { L } } [ L ] ^ 2 \\end{align*}"}
+{"id": "450.png", "formula": "\\begin{align*} \\Delta t _ q = \\max \\{ \\Delta ( E _ { B ^ { q } } ) , \\Delta ( \\Omega ^ q ) \\} . \\end{align*}"}
+{"id": "150.png", "formula": "\\begin{align*} \\lim _ { s \\to + \\infty } G ( s ) = \\frac { 1 } { 2 r _ 1 ^ { \\alpha ( \\eta ) } \\log ( r _ 1 ) + 2 r _ 2 ^ { \\alpha ( \\eta ) } \\log ( r _ 2 ) } \\cdot \\int _ { 0 } ^ { + \\infty } e ^ { ( 2 n - \\alpha ( \\eta ) ) s } R ( e ^ { - s } ) \\ d s . \\end{align*}"}
+{"id": "3519.png", "formula": "\\begin{align*} \\phi _ * ( \\textsc { \\textbf { a } } _ i ) & = k _ i g _ i \\cdot \\textsc { \\textbf { a } } _ { \\sigma ( i ) } \\\\ \\phi _ * ( \\textsc { \\textbf { r } } _ i ) & = k _ i g _ i \\cdot \\textsc { \\textbf { r } } _ { \\sigma ( i ) } . \\end{align*}"}
+{"id": "962.png", "formula": "\\begin{align*} P _ U ( | u | ) \\le P _ U ( | \\Pi _ V ( u ) | ) + P _ U ( P _ V ( | u | ) ) = P _ U ( | \\Pi _ V ( u ) | ) + P _ V ( | u | ) . \\end{align*}"}
+{"id": "1090.png", "formula": "\\begin{align*} I _ m : = \\left [ \\begin{matrix} 1 & 0 & \\cdots & 0 & 0 \\\\ 0 & 1 & \\cdots & 0 & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & \\cdots & 1 & 0 \\\\ 0 & 0 & \\cdots & 0 & 1 \\end{matrix} \\right ] \\in M _ m ( \\mathbb { C } ) . \\end{align*}"}
+{"id": "7493.png", "formula": "\\begin{align*} \\chi ( x , l , t ) = \\xi _ 1 \\left ( \\tfrac { l } { \\lambda _ 0 } \\right ) \\xi _ 2 \\left ( \\frac { \\mathbf { r } ( x ) } { \\sqrt { \\gamma t + ( \\Lambda + 1 ) ^ 2 } } \\right ) . \\end{align*}"}
+{"id": "1927.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { y \\in G } \\omega _ 0 ( x , y ) \\xi ^ { - \\beta - 1 } [ \\mathbf { d } ( y ) - \\mathbf { d } ( x ) ] & = \\sum _ { \\mathbf { d } ( y ) > \\mathbf { d } ( x ) } \\omega _ 0 ( x , y ) \\xi ^ { - \\beta - 1 } [ \\mathbf { d } ( y ) - \\mathbf { d } ( x ) ] \\\\ & + \\sum _ { \\mathbf { d } ( y ) < \\mathbf { d } ( x ) } \\omega _ 0 ( x , y ) \\xi ^ { - \\beta - 1 } [ \\mathbf { d } ( y ) - \\mathbf { d } ( x ) ] \\\\ & = : \\mathrm { S } ^ + + \\mathrm { S } ^ - . \\end{aligned} \\end{align*}"}
+{"id": "9113.png", "formula": "\\begin{align*} \\partial _ j ( \\partial _ i u ) = \\partial _ i ( \\partial _ j u ) \\end{align*}"}
+{"id": "9117.png", "formula": "\\begin{align*} \\mathop { S y m } _ { z _ { 1 } , \\dots , z _ { 1 - a _ { i j } } } \\sum _ { k = 0 } ^ { 1 - a _ { i j } } ( - 1 ) ^ { k } \\left [ \\begin{matrix} 1 - a _ { i j } \\\\ k \\end{matrix} \\right ] _ { v _ { i } } e _ { i } ( z _ { 1 } ) \\cdots e _ { i } ( z _ { k } ) e _ { j } ( w ) e _ { i } ( z _ { k + 1 } ) \\cdots e _ { i } ( z _ { 1 - a _ { i j } } ) = 0 \\forall \\ i \\neq j . \\end{align*}"}
+{"id": "4732.png", "formula": "\\begin{align*} A _ \\gamma : = \\frac { 1 } { \\gamma } ( I d - J ^ A _ \\gamma ) \\end{align*}"}
+{"id": "783.png", "formula": "\\begin{align*} \\delta _ U ( e _ j ) = \\sum _ i e _ i \\otimes u _ { i j } \\end{align*}"}
+{"id": "1351.png", "formula": "\\begin{align*} & E ( A _ \\alpha ( G ) ) ^ 2 \\geq 2 \\sum _ { i = 1 } ^ n s _ i ^ 2 \\\\ & \\ , E ( A _ \\alpha ( G ) ) \\geq \\sqrt { 2 \\left ( \\alpha ^ 2 M _ 1 + ( 1 - \\alpha ) ^ 2 2 m - \\frac { 4 \\alpha ^ 2 m ^ 2 } { n } \\right ) } \\end{align*}"}
+{"id": "4570.png", "formula": "\\begin{align*} \\gamma = \\left \\{ \\begin{aligned} N \\widehat { \\omega } , \\ , \\\\ \\epsilon \\widehat { \\omega } , \\ \\end{aligned} \\right . \\end{align*}"}
+{"id": "5839.png", "formula": "\\begin{align*} \\left ( 1 - \\alpha ^ { i _ 0 - j _ 0 - 3 j } \\right ) \\lambda _ { i _ 0 - 2 j , j _ 0 + j , k _ 0 + j } = \\alpha ^ { i _ 0 - 3 j _ 0 - 3 j - 3 } \\left ( \\sum _ { \\nu = 0 } ^ { j _ 0 + j } \\alpha ^ { 3 \\nu } \\right ) \\lambda _ { i _ 0 - 2 j - 2 , j _ 0 + j + 1 , k _ 0 + j + 1 } . \\end{align*}"}
+{"id": "1071.png", "formula": "\\begin{align*} \\sum _ { s = 0 } ^ { n - 1 } S _ { 2 , k } ( n , s , t ) \\leq \\frac { ( t + 1 ) ^ { 1 - d } K _ 3 \\| \\phi \\| _ L \\{ r \\sin ( \\pi d ) \\} ^ { k - 1 } } { n ^ d ( n - t ) ( 1 - 2 d ) ^ { 1 / 2 } } . \\end{align*}"}
+{"id": "5098.png", "formula": "\\begin{align*} \\pi ( \\mathsf { c } _ s ) & = \\frac { v ^ { 2 } \\tilde { A } _ { s } ^ { - 2 } - 1 } { v - v ^ { 3 } } \\\\ & = \\frac { v ^ { 2 } ( ( v ^ { - 1 } - v ) \\tilde { A } _ s + ( v ^ { - 2 } - 1 + v ^ 2 ) ) - 1 } { v - v ^ { 3 } } \\\\ & = \\tilde { A } _ s - v = c _ s . \\end{align*}"}
+{"id": "2996.png", "formula": "\\begin{align*} g ( T ( x , y ) , z ) + g ( T ( z , x ) , y ) + g ( T ( z , y ) , x ) = 2 g ( H ( x , y ) , z ) . \\end{align*}"}
+{"id": "2765.png", "formula": "\\begin{align*} \\sigma _ { 2 } ^ { * } \\omega = \\omega _ { Q , P } . \\end{align*}"}
+{"id": "3988.png", "formula": "\\begin{align*} \\sum _ { r , s = 1 } ^ { \\nu } \\widetilde { \\gamma } _ { i r s } \\widetilde { \\gamma } _ { i p r } = \\sum _ { r = 1 } ^ { \\nu } \\widetilde { \\gamma } _ { i p r } \\sum _ { s = 1 } ^ { \\nu } \\widetilde { \\gamma } _ { i r s } = \\tfrac { 1 } { 2 } \\sum _ { r = 1 } ^ { \\nu } \\widetilde { \\gamma } _ { i p r } = \\tfrac { 1 } { 4 } \\end{align*}"}
+{"id": "5685.png", "formula": "\\begin{align*} \\mathbf { C } : = \\left \\{ w \\in \\ker \\mathcal { L } _ { \\Sigma } \\ , : \\ , \\Vert w \\Vert _ { L ^ 2 } = 1 \\ \\textup { a n d } \\ w \\ \\textup { i s a c r i t i c a l p o i n t o f } \\ \\hat { f } _ p \\right \\} . \\end{align*}"}
+{"id": "8222.png", "formula": "\\begin{align*} \\xi ( { \\bf a } _ { i - 1 } , \\{ P _ { { \\bf X } _ i | Y ^ { i - 1 } } ( { \\bf x } _ i | y ^ { i - 1 } ) \\} , P _ { \\bf S } ) = H ( Y _ i | { \\bf S } , Y ^ { i - 1 } ) \\end{align*}"}
+{"id": "7024.png", "formula": "\\begin{align*} \\| g \\| _ { L ^ \\infty ( [ 0 , T ] \\times \\mathbb R ^ d ) } = \\| u _ n - u _ m \\| _ { L ^ \\infty ( [ 0 , T ] \\times \\mathbb R ^ d ) } \\rightarrow 0 n , m \\rightarrow \\infty \\end{align*}"}
+{"id": "5513.png", "formula": "\\begin{align*} c _ { k } = \\sum _ { \\substack { 0 \\leq i \\leq d \\\\ 0 \\leq j \\leq m \\\\ i + j = k } } f _ { i } g _ { j } . \\end{align*}"}
+{"id": "8272.png", "formula": "\\begin{align*} p ^ * \\alpha = q ^ * \\alpha ' + E , \\end{align*}"}
+{"id": "764.png", "formula": "\\begin{align*} \\mathrm { x } \\cdot \\mathrm { y } = \\mathbf { e } A ( x \\otimes y ) \\end{align*}"}
+{"id": "3633.png", "formula": "\\begin{align*} \\alpha ^ 3 \\tau _ 0 ^ 3 f ( \\tau , w \\sigma ) - ( 1 + a ) \\alpha ^ 2 \\tau _ 0 ^ 2 f ( \\tau , w ) + ( a + b ) \\alpha \\tau _ 0 f ( \\tau , w \\sigma ) - b f ( \\tau , w ) = 0 . \\end{align*}"}
+{"id": "1501.png", "formula": "\\begin{align*} b ( x , u ) = 0 u \\in \\lbrack \\underline { u } , \\overline { u } ] , \\end{align*}"}
+{"id": "4209.png", "formula": "\\begin{align*} & 4 8 0 \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 1 8 ) } = \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } ( \\widetilde { T X } - \\widetilde { L _ R \\otimes C } ) \\right \\} ^ { ( 1 8 ) } . \\end{align*}"}
+{"id": "8081.png", "formula": "\\begin{align*} a \\cdot v _ i = e ^ { \\chi _ i ( a ) } v _ i , \\qquad \\forall a \\in A _ i . \\end{align*}"}
+{"id": "1556.png", "formula": "\\begin{align*} z = \\int ( z - y ) \\ , \\varphi ( y ) \\ , d y \\end{align*}"}
+{"id": "9007.png", "formula": "\\begin{align*} ( a _ 1 + a _ 2 ) \\cdot ( b _ 1 + b _ 2 ) = a _ 1 * b _ 1 , \\end{align*}"}
+{"id": "4870.png", "formula": "\\begin{align*} g = \\frac { ( \\delta - 1 ) ( \\delta - 2 ) } 2 . \\end{align*}"}
+{"id": "995.png", "formula": "\\begin{align*} \\mathbb E _ { h } ( \\mathbf 1 _ { D \\setminus \\bar V } u ( X _ { \\tau _ V } ) \\mathbf 1 _ { A \\cap V } ( X _ { \\tau _ V - } ) ) & = 2 \\int _ { \\mathbb R ^ d \\times \\mathbb R ^ d } R ^ V h ( x ) \\mathbf 1 _ { A \\cap V } ( x ) \\mathbf 1 _ { D \\setminus \\bar V } u ( y ) \\ , J ( d x , d y ) \\\\ & = \\int _ { A \\cap V } \\int _ { D \\setminus \\bar V } R ^ V h ( x ) u ( y ) j ( | x - y | ) \\ , d y \\ , d x \\\\ & = \\int _ { A \\cap V } \\int _ { D \\setminus \\bar V } \\int _ V G _ V ( x , z ) h ( z ) \\ , d z \\ , u ( y ) j ( | x - y | ) \\ , d y \\ , d x . \\end{align*}"}
+{"id": "8592.png", "formula": "\\begin{align*} \\partial ^ { i n } R ^ { k M } _ i = \\{ i \\} \\times \\{ 1 , 2 , \\dots , M \\} \\cup \\{ i + k - 1 \\} \\times \\{ 1 , 2 , \\dots , M \\} \\ : . \\end{align*}"}
+{"id": "1072.png", "formula": "\\begin{align*} ( T _ { \\infty } ( w ) ^ { - 1 } ) ^ { s , t } & = \\sum _ { \\ell = 1 } ^ { s \\wedge t } \\tilde { a } _ { s - \\ell } ^ * \\tilde { a } _ { t - \\ell } , s , t \\in \\N , \\\\ ( T _ { \\infty } ( \\tilde { w } ) ^ { - 1 } ) ^ { s , t } & = \\sum _ { \\ell = 1 } ^ { s \\wedge t } a _ { s - \\ell } ^ * a _ { t - \\ell } , s , t \\in \\N , \\end{align*}"}
+{"id": "7461.png", "formula": "\\begin{align*} \\Lambda ^ I _ J = \\prod _ { \\ell = 1 } ^ n \\left ( \\frac { - \\lambda _ { i _ \\ell } } { \\lambda _ { i _ { \\ell } } + \\cdots + \\lambda _ { i _ n } } \\right ) \\ , \\delta _ { i _ \\ell , j _ \\ell } . \\end{align*}"}
+{"id": "1616.png", "formula": "\\begin{align*} \\sum _ j & ( - 1 ) ^ { 1 + \\dots + p ^ t - j } \\binom { j ( p - 1 ) } { 1 + \\dots + p ^ t - j } Q _ { ( 2 r + p s - 2 p j ) ( p - 1 ) } Q _ { 2 j ( p - 1 ) } \\\\ & = q ^ { \\tfrac { p - 1 } { 2 } } \\sum _ i ( - 1 ) ^ { r + p ^ { t + 1 } - 1 - i } \\binom { i ( p - 1 ) } { r + p ^ { t + 1 } - 1 - i } Q _ { ( 2 r + p s - 2 p i ) ( p - 1 ) } Q _ { 2 i ( p - 1 ) } . \\end{align*}"}
+{"id": "7245.png", "formula": "\\begin{align*} Z _ n = \\frac 1 n \\sum _ { i = 1 } ^ n t _ i Z _ { n - i } n > 0 . \\end{align*}"}
+{"id": "2202.png", "formula": "\\begin{align*} \\intop _ \\Omega \\psi _ { 1 , r } \\psi _ { 1 , z z r } d x - \\intop _ \\Omega \\psi _ { 1 , z z } ^ 2 d x + 2 \\intop _ \\Omega \\psi _ { 1 , r z } \\psi _ { 1 , z } d r d z = \\intop _ \\Omega \\omega _ 1 \\psi _ { 1 , z z } d x . \\end{align*}"}
+{"id": "2509.png", "formula": "\\begin{align*} \\| u \\| _ { K , C ^ k } = \\sup _ { x \\in K , \\ | \\alpha | \\le k } | \\partial ^ \\alpha u ( x ) | \\ ; , \\end{align*}"}
+{"id": "5891.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { C } ( G ) ) = \\begin{cases} n ( m - 1 ) ( m n - n - 1 ) ^ { 2 } + m n ( n - 1 ) ^ { 2 } , & \\\\ n ( m - 2 ) ( m n - 2 n - 1 ) ^ { 2 } + m n ( 2 n - 1 ) ^ { 2 } , & \\end{cases} \\end{align*}"}
+{"id": "3617.png", "formula": "\\begin{align*} b ( ( \\sigma \\sigma _ 1 ) ^ 2 - 1 ) = ( a + b ) \\alpha \\tau _ 0 ( ( \\sigma ^ 2 \\sigma _ 1 ) ^ 2 - \\sigma ^ 2 ) . \\end{align*}"}
+{"id": "7553.png", "formula": "\\begin{align*} \\int _ \\Omega f \\nabla \\phi \\cdot \\bar u \\ d x = \\int _ \\Omega \\delta \\nabla \\bar u : \\nabla \\phi \\otimes \\nabla \\phi \\ d x - \\int _ \\Omega \\delta ( \\nabla \\cdot \\bar u ) \\tfrac { 1 } { 2 } | \\nabla \\phi | ^ 2 \\ d x \\ . \\end{align*}"}
+{"id": "6326.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\epsilon _ { \\ell , \\lambda } = \\Delta \\left ( \\omega _ { \\ell , \\lambda } - \\omega _ \\ell \\right ) \\R ^ 3 , \\end{align*}"}
+{"id": "1845.png", "formula": "\\begin{align*} c _ 1 ( P ^ { \\dagger } _ u ) = c _ 1 ( \\det P ^ { \\dagger } _ u ) = \\pi ^ * c _ 1 ( \\det P ^ { \\dagger } ) = \\pi ^ * c _ 1 ( P ^ { \\dagger } ) , \\end{align*}"}
+{"id": "5318.png", "formula": "\\begin{align*} w ^ { S _ k } _ { \\pi _ { k } } \\ , x _ { \\pi _ { k } } + \\cdots + w ^ { S _ k } _ { \\pi _ { n } } \\ , x _ { \\pi _ { n } } = b ^ { S _ k } , 1 \\leq k \\leq n , \\end{align*}"}
+{"id": "3681.png", "formula": "\\begin{align*} \\psi ( e ) = \\begin{cases} \\sqrt { 2 } & i , j \\in [ d ] , \\\\ 1 & i \\in [ d ] , j \\in [ n ] \\setminus [ d ] \\end{cases} \\end{align*}"}
+{"id": "5628.png", "formula": "\\begin{align*} d ( x , a ) = d ( T x , T a ) = d ( y , T a ) < \\delta , \\end{align*}"}
+{"id": "2574.png", "formula": "\\begin{gather*} [ x ] _ + = \\begin{cases} x , & x \\geq 0 ; \\\\ 0 , & x \\leq 0 . \\end{cases} \\end{gather*}"}
+{"id": "5856.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { C } ( D _ { 2 m } ) ) = \\dfrac { ( m - 2 ) ( m - 2 - 1 ) ^ { 3 } } { 2 } + \\dfrac { m } { 2 } \\cdot \\dfrac { 2 ( 2 - 1 ) ^ { 3 } } { 2 } = \\dfrac { ( m - 2 ) ( m - 3 ) ^ { 3 } + m } { 2 } . \\end{align*}"}
+{"id": "7779.png", "formula": "\\begin{align*} B _ \\lambda ^ { - 1 } ( \\cdot + \\alpha ) ( A ( \\lambda ) + F ( \\cdot , \\lambda ) ) B _ \\lambda ( \\cdot ) = \\tilde { A } ( \\lambda ) . \\end{align*}"}
+{"id": "833.png", "formula": "\\begin{align*} { \\cal { L } } _ { \\hat { X } } \\nabla _ 0 C ^ { i } _ { j k } = \\nabla _ 0 ( { \\cal { L } } _ { \\hat { X } } C ^ { i } _ { j k } ) + \\Psi C ^ { i } _ { j k } + y ^ { i } C ^ { s } _ { j k } \\Psi _ { s } , \\end{align*}"}
+{"id": "6223.png", "formula": "\\begin{align*} \\begin{cases} \\dot { w } = h _ 1 - \\tilde c _ 1 - D _ 1 g _ 1 / w \\ & \\mbox { i n } \\ ( 0 , \\alpha ) , \\\\ w < 0 \\ & \\mbox { i n } \\ ( 0 , \\alpha ) , \\\\ w ( 0 ) = w ( \\alpha ) = 0 . \\end{cases} \\end{align*}"}
+{"id": "6622.png", "formula": "\\begin{align*} H _ N S _ N ( i , j ) & = \\frac { 1 } { J _ N } ( 2 J _ 1 ( J _ { N - 2 } + J _ { N - 1 } ) + 2 J _ 0 J _ { N - 2 } - J _ { N - 1 } ) \\\\ & = \\frac { 1 } { J _ N } ( 2 ( J _ { N - 2 } + J _ { N - 1 } ) - J _ { N - 1 } ) \\\\ & = \\frac { 1 } { J _ N } ( 2 J _ { N - 2 } + J _ { N - 1 } ) \\\\ & = 1 . \\end{align*}"}
+{"id": "5235.png", "formula": "\\begin{align*} & \\frac { \\partial \\left ( M A \\right ) } { \\partial q _ { j } } = \\frac { \\partial \\left ( M A _ { j } \\right ) } { \\partial q _ { j } } = \\left ( 1 - \\alpha \\right ) > 0 \\\\ & \\frac { \\partial \\left ( M H \\right ) } { \\partial q _ { j } } = \\frac { \\partial \\left ( M H _ { j } \\right ) } { \\partial q _ { j } } = \\frac { \\left ( 1 - \\alpha \\right ) p ^ { 2 } _ { j } } { \\left [ \\left ( 1 - \\alpha \\right ) p _ { j } + \\alpha q _ { j } \\right ] ^ { 2 } } > 0 \\end{align*}"}
+{"id": "7112.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } i \\partial _ { t } \\psi + \\Delta \\psi + \\gamma w \\psi + f ( \\psi ) = 0 , & \\forall ( t , x ) \\in \\mathbb { R } ^ { 1 + 2 } , \\\\ - \\Delta w = | \\psi | ^ { 2 } , & \\\\ \\psi ( 0 , x ) = \\psi _ { 0 } ( x ) , & \\end{array} \\right . \\end{align*}"}
+{"id": "5788.png", "formula": "\\begin{align*} \\int _ 0 ^ t e ^ { - b ( t - \\tau ) } | z ( \\tau ) | ^ q \\ , d \\tau = & \\int _ 0 ^ { t _ 0 } e ^ { - b ( t - \\tau ) } | z ( \\tau ) | ^ q \\ , d \\tau + \\int _ { t _ 0 } ^ t e ^ { - 2 ^ { - 1 } b ( t - \\tau ) } \\left ( e ^ { - 2 ^ { - 1 } q ^ { - 1 } b ( t - \\tau ) } | z ( \\tau ) | \\right ) ^ q \\ , d \\tau \\\\ \\leq C & e ^ { - b t } \\max _ { \\tau \\in [ 0 , \\infty ) } | z ( \\tau ) | ^ q + C | z ( t ) | ^ q \\leq C | z ( t ) | ^ q . \\end{align*}"}
+{"id": "2471.png", "formula": "\\begin{align*} \\dfrac { d \\tau } { d \\xi } & = \\dfrac { d \\tau } { d s } \\dfrac { d s } { d \\xi } = \\lambda ^ { - 1 } \\phi ^ { - 1 } = x ^ { - 1 } \\sim A _ { 3 } e ^ { - p ^ { - 1 } \\tau } { \\rm { a s } } \\tau \\to - \\infty . \\end{align*}"}
+{"id": "915.png", "formula": "\\begin{align*} L = - ( - \\Delta ) ^ { \\alpha / 2 } \\end{align*}"}
+{"id": "1867.png", "formula": "\\begin{align*} a = \\begin{pmatrix} 1 & & \\\\ & - 1 & \\\\ & & - 1 \\end{pmatrix} , b = \\begin{pmatrix} - 1 & & \\\\ & 1 & \\\\ & & - 1 \\end{pmatrix} , c = \\begin{pmatrix} - 1 & & \\\\ & - 1 & \\\\ & & 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "8434.png", "formula": "\\begin{align*} \\| P \\| _ B = \\max \\limits _ { f _ j = \\pm 1 } \\left [ R \\left ( \\sum _ { i = 1 } ^ n \\left ( \\sum _ { j = 1 } ^ { n + 1 } f _ j l _ { i j } \\right ) ^ 2 \\right ) ^ { 1 / 2 } + \\left | \\sum _ { j = 1 } ^ { n + 1 } f _ j \\lambda _ j ( x ^ { ( 0 ) } ) \\right | \\right ] . \\end{align*}"}
+{"id": "5487.png", "formula": "\\begin{align*} ( 1 + x ) ^ { \\alpha } = 1 + \\alpha x + O _ \\alpha ( x ^ 2 ) , \\end{align*}"}
+{"id": "2627.png", "formula": "\\begin{align*} | D _ 4 ( S ) | \\geq 1 + | D _ { \\{ 2 , 4 \\} } ( S ) | = 1 + | D _ { \\{ 4 \\} } ( S ) \\cup [ D _ { \\{ 2 \\} } ( S ) \\setminus D _ { \\{ 4 \\} } ( S ) ] | \\geq 1 + n - 4 + \\frac { n } { 3 } + \\frac { 5 } { 3 } = \\frac { 4 } { 3 } n - \\frac { 4 } { 3 } . \\end{align*}"}
+{"id": "6735.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty { \\frac { { \\zeta ( 2 k ) } } { { k ( 2 k + 1 ) } } } & = - 1 + \\ln ( 2 \\pi ) , \\\\ \\sum _ { k = 1 } ^ \\infty { \\frac { { \\zeta ( 2 k ) } } { { k ( 2 k + 3 ) } } } & = - \\frac { 1 } { 9 } + \\frac { \\ln ( 2 \\pi ) } { 3 } + \\frac { \\zeta ( 3 ) } { 2 \\pi ^ 2 } . \\end{align*}"}
+{"id": "1775.png", "formula": "\\begin{align*} \\Phi \\circ \\Psi ( M _ { f ( x ) } ) = M _ { { f ( x - x _ 1 ^ * ) } - f ( x _ 1 ^ * ) } , \\pi _ 1 \\circ \\Phi \\circ \\Psi ( z ) = z _ 1 - z _ 1 ^ * , \\quad \\mbox { a n d } \\Phi \\circ \\Psi ( z ^ * ) = 0 . \\end{align*}"}
+{"id": "6391.png", "formula": "\\begin{align*} G _ n ( \\theta ) = - \\nabla _ \\theta L _ n ( \\theta ) = - ( \\partial _ a L _ n ( \\theta ) , \\partial _ b L _ n ( \\theta ) , \\partial _ { \\delta } L _ n ( \\theta ) , \\partial _ { \\alpha } L _ n ( \\theta ) ) ^ T \\end{align*}"}
+{"id": "2881.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to 0 ^ + } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ X = \\left [ - \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { X } . \\end{align*}"}
+{"id": "2989.png", "formula": "\\begin{align*} \\overline { \\nabla } _ x y = \\nabla _ x y + \\alpha \\{ g ( x , y ) \\xi - \\eta ( y ) x \\} + \\beta \\{ g ( \\varphi x , y ) \\xi - \\eta ( y ) \\varphi x \\} , \\end{align*}"}
+{"id": "7562.png", "formula": "\\begin{align*} \\Delta ^ { 2 } u + \\omega u - \\mu | u | ^ { q - 2 } u - | u | ^ { p - 2 } u = 0 , \\quad \\mathrm { i n } \\ \\mathbb { R } ^ { N } . \\end{align*}"}
+{"id": "5220.png", "formula": "\\begin{align*} & \\overline { M G } = \\sum _ { i } \\overline { M G } _ { i } = \\sum _ { i } \\overline { p } ^ { \\alpha } _ { i } \\overline { q } ^ { 1 - \\alpha } _ { i } \\\\ & \\overline { M H } = \\sum _ { i } \\overline { M H } _ { i } = \\sum _ { i } \\frac { \\overline { p } _ { i } \\overline { q } _ { i } } { \\left ( 1 - \\alpha \\right ) \\overline { p } _ { i } + \\alpha \\overline { q } _ { i } } \\end{align*}"}
+{"id": "7395.png", "formula": "\\begin{align*} \\begin{array} { l l } C _ { 3 4 } : = \\min & \\left [ \\{ 2 ^ { r - 1 } 3 r B E \\| u _ x ^ 0 \\| _ \\infty ( 5 N ) ^ { r - 1 } N ^ { - 1 } \\} ^ { - 1 / ( r - 1 ) } , \\right . \\\\ & \\quad \\left . \\{ 2 ^ { r - 1 } 3 r B E \\| u ^ 0 \\| _ \\infty ^ { r - 1 } 5 \\} ^ { - 1 } \\right ] . \\end{array} \\end{align*}"}
+{"id": "4905.png", "formula": "\\begin{align*} 0 \\leq C \\cdot E = g + 1 - e . \\end{align*}"}
+{"id": "4158.png", "formula": "\\begin{align*} g _ 1 g _ 2 = g _ 2 g _ 1 \\mbox { i f a n d o n l y i f } \\varphi ( g _ 1 ) \\varphi ( g _ 2 ) = \\varphi ( g _ 2 ) \\varphi ( g _ 1 ) \\ , . \\end{align*}"}
+{"id": "3610.png", "formula": "\\begin{align*} \\alpha ^ 3 \\tau _ 0 ^ 2 \\tau _ 1 ( \\sigma ^ 2 \\sigma _ 1 ) ^ m - ( 1 + a ) \\alpha ^ 2 \\tau _ 0 \\tau _ 1 ( \\sigma \\sigma _ 1 ) ^ m + ( a + b ) \\alpha \\tau _ 0 \\sigma ^ m - b = 0 . \\end{align*}"}
+{"id": "7466.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\phi _ p ( r , x , 0 ) = p \\in \\Gamma , \\end{align*}"}
+{"id": "9106.png", "formula": "\\begin{align*} \\tilde { a } _ { i l } = a _ { i l } + \\hat { c } _ { i i } \\Tilde { k } _ { i l } \\geq 0 . \\end{align*}"}
+{"id": "8186.png", "formula": "\\begin{align*} f ( B _ 1 , B _ 2 ^ 0 , B _ 2 ^ 1 ) & = \\frac { 1 } { 2 M } \\left ( - B _ 2 ^ 0 \\log B _ 2 ^ 0 - B _ 2 ^ 1 \\log B _ 2 ^ 1 \\right . \\\\ & \\left . - ( B _ 1 - B _ 2 ^ 1 ) \\log ( B _ 1 - B _ 2 ^ 1 ) \\right . \\\\ & \\left . - ( M - B _ 1 - B _ 2 ^ 0 ) \\log ( M - B _ 1 - B _ 2 ^ 0 ) \\right . \\\\ & \\left . + B _ 1 \\log B _ 1 + ( M - B _ 1 ) \\log ( M - B _ 1 ) \\right ) . \\end{align*}"}
+{"id": "6095.png", "formula": "\\begin{align*} S _ { < d } ( x _ { s _ 1 , \\varepsilon _ 1 } \\dots x _ { s _ r , \\varepsilon _ r } ) & = S _ { < d } \\begin{pmatrix} \\varepsilon _ 1 & \\dotsb & \\varepsilon _ n \\\\ s _ 1 & \\dotsb & s _ n \\end{pmatrix} , \\\\ \\zeta _ A ( x _ { s _ 1 , \\varepsilon _ 1 } \\dots x _ { s _ r , \\varepsilon _ r } ) & = \\zeta _ A \\begin{pmatrix} \\varepsilon _ 1 & \\dotsb & \\varepsilon _ n \\\\ s _ 1 & \\dotsb & s _ n \\end{pmatrix} . \\end{align*}"}
+{"id": "8499.png", "formula": "\\begin{align*} f ( a _ 1 a _ 2 \\dotsc ) = f ( a _ 1 ) f ( a _ 2 ) \\dotsc . \\end{align*}"}
+{"id": "2370.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } \\frac { d } { d t } \\| z ( t ) \\| _ { L ^ { 2 } } ^ { 2 } + \\| \\nabla z \\| _ { L ^ { 2 } } ^ { 2 } \\\\ & = \\int _ { \\mathbb { R } ^ { 3 } } ( z \\otimes u ^ { c , \\gamma } + u ^ { c , \\gamma } \\otimes z ) \\cdot \\nabla z d x + \\int _ { \\mathbb { R } ^ { 3 } } ( w \\otimes w ) \\cdot \\nabla z d x . \\end{align*}"}
+{"id": "5280.png", "formula": "\\begin{align*} L S _ { d } \\left ( x . y \\right ) = \\frac { ( x . y ) ^ { \\epsilon } - ( x . y ) ^ { - \\epsilon } } { 2 \\ ; \\epsilon } \\end{align*}"}
+{"id": "2710.png", "formula": "\\begin{align*} { _ { * } X _ { t } } = { _ { * } a } ^ { i } \\frac { \\partial } { \\partial q ^ { i } } + { _ { * } b } ^ { i } \\frac { \\partial } { \\partial p _ { i } } + \\frac { \\partial } { \\partial t } . \\end{align*}"}
+{"id": "5257.png", "formula": "\\begin{align*} \\frac { \\partial T _ { j } } { \\partial q _ { j } } = 1 - \\alpha \\end{align*}"}
+{"id": "2455.png", "formula": "\\begin{align*} \\begin{cases} V ( \\xi ) \\sim A _ { 3 } ( \\xi _ { + } - \\xi ) ^ { \\frac { 1 } { m } } , \\\\ V ' ( \\xi ) \\sim - A _ { 4 } ( \\xi _ { + } - \\xi ) ^ { - \\frac { m - 1 } { m } } \\end{cases} { \\rm { a s } } \\xi \\nearrow \\xi _ { + } - 0 , \\end{align*}"}
+{"id": "8264.png", "formula": "\\begin{align*} H ( Y _ i | Y ^ { i - 1 } , \\underline S ) & \\leq H ( Y _ i | Y ^ { i - 1 } ) \\\\ & = \\sum _ { y ^ { i - 1 } } \\frac { c _ { y ^ { i - 1 } } } { M } H ( \\frac { b _ { y ^ { i - 1 } } } { c _ { y ^ { i - 1 } } } ) , \\end{align*}"}
+{"id": "266.png", "formula": "\\begin{align*} y = \\omega ^ 2 x ^ 2 , d \\tau = \\frac { \\omega ^ 2 } { x } \\ , d t . \\end{align*}"}
+{"id": "6508.png", "formula": "\\begin{align*} \\frac { 1 } { n } h _ { K ( 0 ) } ( \\xi ) \\det ( D ^ 2 h _ { K ( 0 ) } ( \\xi ) + h _ { K ( 0 ) } ( \\xi ) _ { n - 1 } ) = \\varphi . \\end{align*}"}
+{"id": "929.png", "formula": "\\begin{align*} V ^ D ( u , \\eta ) = \\int _ D f ( \\cdot , u ) \\eta \\ , d m + \\int _ D \\eta \\ , d \\mu , \\eta \\in F ( D ) . \\end{align*}"}
+{"id": "5490.png", "formula": "\\begin{align*} \\frac { 1 } { z - z _ j } + \\frac { 1 } { z - \\overline { z _ j } } = 2 \\Re \\frac { 1 } { z - z _ j } , \\end{align*}"}
+{"id": "8425.png", "formula": "\\begin{align*} B ( G ) _ x = \\{ [ b ] \\in B ( G ) \\mid [ b ] \\cap I x I \\neq \\emptyset \\} . \\end{align*}"}
+{"id": "6127.png", "formula": "\\begin{align*} \\rho _ { m , n } ^ { ( 2 ) } \\big ( \\rho _ { n , j } ^ { ( 2 ) } ( d _ j ^ * d _ j ) - \\rho _ { n , j } ^ { ( 2 ) } ( d _ j ) ^ * \\rho _ { n , j } ^ { ( 2 ) } ( d _ j ) \\big ) = \\rho _ { m , j } ^ { ( 2 ) } ( d _ j ^ * d _ j ) - \\rho _ { m , n } ^ { ( 2 ) } \\big ( \\rho _ { n , j } ^ { ( 2 ) } ( d _ j ) ^ * \\rho _ { n , j } ^ { ( 2 ) } ( d _ j ) \\big ) \\geq 0 . \\end{align*}"}
+{"id": "6357.png", "formula": "\\begin{align*} r ^ { n - 1 } \\omega _ 1 ( r ) & = \\phi ' ( r ) = \\sinh ^ { n - 1 } ( r ) \\cosh ^ { d - 1 } ( r ) , \\\\ r ^ { n - 2 } \\omega _ 2 ( r ) & = \\phi '' ( r ) = ( n - 1 ) \\sinh ^ { n - 2 } ( r ) \\cosh ^ { d } ( r ) + ( d - 1 ) \\sinh ^ n ( r ) \\cosh ^ { d - 2 } ( r ) , \\\\ r ^ { n - 3 } \\omega _ 3 ( r ) & = \\phi ''' ( r ) = ( n - 1 ) ( n - 2 ) \\sinh ^ { n - 3 } ( r ) \\cosh ^ { d + 1 } ( r ) \\\\ & \\quad + ( 2 d n - d - n ) \\sinh ^ { n - 1 } ( r ) \\cosh ^ { d - 1 } ( r ) + ( d - 1 ) ( d - 2 ) \\sinh ^ { n + 1 } ( r ) \\cosh ^ { d - 3 } ( r ) . \\end{align*}"}
+{"id": "8066.png", "formula": "\\begin{align*} \\mathcal { B } _ { 1 0 } ( ( \\tilde { F } , \\tilde { G } , \\tilde { H } ) , ( \\hat { F } , \\hat { G } , \\hat { H } ) ) = \\int _ 0 ^ T \\begin{pmatrix} R ^ { \\tilde { F } _ s , \\tilde { G } _ s } _ { s , 1 } \\\\ R ^ { \\tilde { F } _ s , \\tilde { G } _ s , \\tilde { F } _ s } _ { s , 2 } \\\\ R ^ { \\tilde { G } _ s , \\tilde { F } _ s } _ { s , 3 } \\end{pmatrix} \\cdot \\begin{pmatrix} \\hat { F } _ s \\\\ \\hat { G } _ s \\\\ \\hat { H } _ s \\end{pmatrix} d s . \\end{align*}"}
+{"id": "7792.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & A ^ { \\top } P + P A + C ^ { \\top } P C + Q - \\big ( P B + C ^ { \\top } P D + S ^ { \\top } \\big ) \\big ( D ^ { \\top } P D + R \\big ) ^ { - 1 } \\big ( B ^ { \\top } P + D ^ { \\top } P C + S \\big ) = 0 , \\\\ & D ^ { \\top } P D + R > 0 \\end{aligned} \\right . \\end{align*}"}
+{"id": "5767.png", "formula": "\\begin{align*} X _ 0 ' = Y _ 0 , X _ - ' + b X _ - \\leq Y _ - . \\end{align*}"}
+{"id": "2722.png", "formula": "\\begin{align*} & Y ^ { ( s + 1 ) } _ { \\alpha } : = [ X _ { T } , Y ^ { ( s ) } _ { \\alpha } ] \\ \\ \\ ( s = 1 , 2 , \\cdots , y _ { \\alpha } - 1 ) , \\\\ & Y ^ { ( y _ { \\alpha } + 1 ) } _ { \\alpha } : = [ X _ { T } , Y ^ { ( y _ { \\alpha } ) } _ { \\alpha } ] = D _ { \\alpha s } ^ { \\beta } Y ^ { ( s ) } _ { \\beta } , \\end{align*}"}
+{"id": "2945.png", "formula": "\\begin{align*} \\lim _ { \\rho \\rightarrow 0 } \\rho ^ { 1 - s } \\partial _ \\rho \\big ( \\chi _ \\pi \\ast \\rho ^ s \\varphi _ { s , \\rho } ) ( k ) = c _ s \\lambda _ \\pi ^ s \\chi _ \\pi ( k ) \\end{align*}"}
+{"id": "9009.png", "formula": "\\begin{align*} x * y = \\phi \\big ( \\phi ^ { - 1 } ( x ) \\cdot \\phi ^ { - 1 } ( x ) \\big ) , \\ \\ x , y \\in { \\mathfrak L } . \\end{align*}"}
+{"id": "8857.png", "formula": "\\begin{align*} 0 < \\frac 1 { ( p - 1 ) a _ 2 } \\leq \\frac 1 r \\leq 1 , 0 \\leq \\frac { \\tau } { p - 1 } < \\frac { n } { ( p - 1 ) a _ 2 } , \\frac { \\tau } { p - 1 } - 1 = \\frac { n } { ( p - 1 ) a _ 2 } - \\frac { n } { r } , \\end{align*}"}
+{"id": "6142.png", "formula": "\\begin{align*} \\min \\left \\{ f ( x ) = f _ 1 ( x _ 1 ) + f _ 2 ( x _ 2 ) : ~ ( A x : = ) A _ 1 x _ 1 + A _ 2 x _ 2 = b \\right \\} , \\end{align*}"}
+{"id": "4006.png", "formula": "\\begin{align*} \\begin{cases} x ^ { \\left ( t + 1 \\right ) } & = \\ ; \\gamma x ^ { \\left ( t \\right ) } y ^ { \\left ( t \\right ) } \\\\ y ^ { \\left ( t + 1 \\right ) } & = \\left ( 1 - \\gamma \\right ) x ^ { \\left ( t \\right ) } y ^ { \\left ( t \\right ) } \\end{cases} \\end{align*}"}
+{"id": "3498.png", "formula": "\\begin{align*} \\Phi ( \\rho _ k ( x ) ) & = \\Phi ( [ ( \\rho _ { m , k } ( x ) ) _ { m > k } ] ) \\\\ & = [ ( \\varphi _ m ( \\rho _ { m , k } ( x ) ) ) _ { m > k } ] \\\\ & = [ ( ( \\Theta ^ { - 1 } \\circ \\rho _ m ) ( \\rho _ { m , k } ( x ) ) _ { m > k } ] \\\\ & = [ ( \\Theta ^ { - 1 } \\circ \\rho _ k ( x ) ) _ { m > k } ] \\\\ & = \\iota \\circ \\Theta ^ { - 1 } \\circ \\rho _ k ( x ) \\\\ & = \\iota \\circ \\varphi _ k ( x ) . \\end{align*}"}
+{"id": "2899.png", "formula": "\\begin{align*} & \\lim _ { \\lambda \\to 0 ^ + } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ { L ^ { r , \\tau } ( \\mathbb { R } ^ n ) } = \\left [ - \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { L ^ { r , \\tau } ( \\mathbb { R } ^ n ) } . \\end{align*}"}
+{"id": "422.png", "formula": "\\begin{align*} V _ M ( \\rho ) > \\sum _ { j = 1 } ^ { M } \\frac { \\rho ^ 2 \\widetilde { t } _ { j , j } ^ 2 \\sum _ { i = 1 } ^ { N } \\sigma _ { i j } ^ 2 t _ { i } ^ 2 } { M ^ 2 } > \\frac { K \\sum _ { i , j } \\sigma _ { i , j } ^ 2 } { M ^ 2 } . \\end{align*}"}
+{"id": "4478.png", "formula": "\\begin{align*} \\int _ { \\{ z \\in D : \\psi ( z ) = r \\} } | f | ^ 2 e ^ { - \\varphi } \\left ( \\frac { \\partial \\psi } { \\partial v _ z } \\right ) ^ { - 1 } | d z | \\ge \\int _ { \\{ z \\in D : \\psi ( z ) = r \\} } | F _ 0 | ^ 2 e ^ { - \\varphi } \\left ( \\frac { \\partial \\psi } { \\partial v _ z } \\right ) ^ { - 1 } | d z | . \\end{align*}"}
+{"id": "6726.png", "formula": "\\begin{align*} S _ 1 ( z ) = \\sum _ { k = 1 } ^ \\infty \\frac { ( - 1 ) ^ k \\zeta ( 2 k ) } { k } z ^ { 2 k } . \\end{align*}"}
+{"id": "8014.png", "formula": "\\begin{align*} & \\sup _ { f _ 1 , f _ 2 , f _ 3 , F , G , H } \\left ( \\mathcal { I } _ 2 ( W _ 0 , f _ 1 , f _ 2 , f _ 3 ) + \\mathcal { I } _ 1 ( W , F , G , H ) \\right ) \\\\ & = \\sup _ { f _ 1 , f _ 2 , f _ 3 } \\left ( \\mathcal { I } _ 2 ( W _ 0 , f _ 1 , f _ 2 , f _ 3 ) \\right ) + \\sup _ { F , G , H } \\left ( \\mathcal { I } _ 1 ( W , F , G , H ) \\right ) \\\\ & = I _ { i n i } ( W _ 0 ) + I _ { d y n } ( W ) \\end{align*}"}
+{"id": "3212.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ t ^ \\alpha u ( t ) + t ^ { \\beta } A u ( t ) = 0 , t > 0 ; \\\\ & \\lim \\limits _ { t \\rightarrow + 0 } J _ t ^ { \\alpha - 1 } u ( t ) = 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "8138.png", "formula": "\\begin{align*} [ J _ 7 , J _ 8 ] \\ = \\ 0 \\ . \\end{align*}"}
+{"id": "323.png", "formula": "\\begin{align*} E ( x , t ) = t ^ { 1 / ( 1 - p ) } \\varphi ( | x | t ^ { - \\gamma } ) , \\gamma = \\frac { m - p } { 2 ( 1 - p ) } \\end{align*}"}
+{"id": "3981.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { n } \\gamma _ { i p l } = 0 \\quad \\left ( 1 \\leq i \\leq n , 1 \\leq p \\leq \\nu \\right ) . \\end{align*}"}
+{"id": "504.png", "formula": "\\begin{align*} ( z \\mapsto \\alpha z + \\beta ) ( z \\mapsto \\alpha ' z + \\beta ' ) = ( z \\mapsto \\alpha \\alpha ' z + \\alpha ' \\beta + \\beta ' ) . \\end{align*}"}
+{"id": "5431.png", "formula": "\\begin{align*} w ^ S _ i & \\triangleq g ^ { \\langle 1 , S \\rangle } _ i - g ^ { \\langle 0 , S \\rangle } _ i = 1 + \\beta \\sum _ { j \\in N } p _ { i j } g ^ S _ j - \\beta g ^ S _ i = \\begin{cases} ( 1 - \\beta ) g ^ S _ i & i \\in S \\\\ 1 + \\beta \\sum _ { j \\in S } p _ { i j } g ^ S _ j & \\textrm { o t h e r w i s e . } \\end{cases} \\end{align*}"}
+{"id": "5653.png", "formula": "\\begin{align*} \\pi ^ { - 1 } ( C ) = \\{ g _ 1 ( f ( x ) ) = 0 , \\dots , g _ \\ell ( f ( x ) ) = 0 , g _ { \\ell + 1 } ( f ( x ) ) > 0 , \\dots , g _ N ( f ( x ) ) > 0 \\} \\end{align*}"}
+{"id": "822.png", "formula": "\\begin{align*} ( 1 - n ) { D _ 0 } { D _ i } { \\Psi _ j } = k ( n - 1 ) ( 2 \\Psi g _ { i j } + \\Psi _ i y _ j + \\Psi _ j y _ i ) . \\end{align*}"}
+{"id": "5151.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } D _ { \\alpha } ( p \\| q ) } { \\partial q _ { j } } = & - \\frac { 1 } { \\alpha } \\left [ \\frac { a - 1 } { a - b } A ^ { a - 2 } - \\frac { b - 1 } { a - b } A ^ { b - 2 } \\right ] \\\\ & + \\frac { 1 } { \\alpha } \\left [ \\frac { a - 1 } { a - b } B ^ { a - 2 } - \\frac { b - 1 } { a - b } B ^ { b - 2 } \\right ] p ^ { \\alpha } _ { j } q ^ { - \\alpha } _ { j } \\end{align*}"}
+{"id": "1916.png", "formula": "\\begin{align*} \\eta ( x ) : = \\min \\left \\{ \\frac { 2 \\left [ r _ 1 - s - \\mathbf { d } ( x ) \\right ] _ + } { r _ 1 } , \\ , \\ , 1 \\right \\} \\ , \\ ; x \\in G \\ , . \\end{align*}"}
+{"id": "6084.png", "formula": "\\begin{align*} \\lim _ { \\stackrel { \\mu \\to m _ { \\pm } ^ \\# } { \\mu \\in \\Sigma _ { \\pm } ^ \\# } } \\| ( U - \\mu ) \\Psi _ { \\mu } \\| _ 2 = 0 . \\end{align*}"}
+{"id": "9150.png", "formula": "\\begin{align*} c _ { \\beta } = \\begin{cases} { \\langle 2 \\rangle _ { v } } ^ { | \\beta | - 1 } & \\quad \\ \\beta = [ i , j ] \\\\ { \\langle 2 \\rangle _ { v } } ^ { | \\beta | - 1 } \\cdot \\prod _ { \\ell = j } ^ { n - 1 } \\big \\{ ( v ^ { - 4 n + 4 \\ell - 2 } - 1 ) ( v ^ { - 4 n + 4 \\ell + 6 } - 1 ) \\big \\} & \\quad \\ \\beta = [ i , n , j ] \\end{cases} . \\end{align*}"}
+{"id": "7106.png", "formula": "\\begin{align*} b _ m : = c _ m E _ \\varepsilon ( \\mathbf 1 _ m b ) , \\end{align*}"}
+{"id": "5341.png", "formula": "\\begin{align*} c ^ J _ j & = c _ j , j \\in J \\\\ c ^ { S \\setminus \\{ i \\} } _ j & = c ^ S _ j - \\frac { c _ i ^ S } { w _ i ^ S } \\ , \\left [ w ^ S _ j - w ^ { S \\setminus \\{ i \\} } _ j \\right ] \\end{align*}"}
+{"id": "6683.png", "formula": "\\begin{align*} M = H \\cap A = H \\cap B = \\langle b _ 1 , \\ldots , b _ m \\rangle \\cong C _ p ^ { \\ , m } \\end{align*}"}
+{"id": "1583.png", "formula": "\\begin{align*} G ( \\tilde x ) = k g _ h ( \\tilde x ) = \\inf _ { G \\ge k } g _ h , \\end{align*}"}
+{"id": "4070.png", "formula": "\\begin{align*} \\int _ { | z | = \\epsilon } B ( z ) ( I - K ( z ) ) ^ { - 1 } C ( z ) d z \\leq \\ell . \\end{align*}"}
+{"id": "1053.png", "formula": "\\begin{align*} w ( e ^ { i \\theta } ) = h ( e ^ { i \\theta } ) h ( e ^ { i \\theta } ) ^ * = h _ { \\sharp } ( e ^ { i \\theta } ) ^ * h _ { \\sharp } ( e ^ { i \\theta } ) , \\end{align*}"}
+{"id": "5191.png", "formula": "\\begin{align*} D _ { \\alpha \\beta } ( p \\| q ) = T \\left [ \\sum _ { i } p ^ { \\alpha + \\beta - 1 } _ { i } + \\frac { \\beta - 1 } { \\alpha } \\sum _ { i } q ^ { \\alpha + \\beta - 1 } _ { i } - \\frac { \\alpha + \\beta - 1 } { \\alpha } \\sum _ { i } p ^ { \\alpha } _ { i } q ^ { \\beta - 1 } _ { i } \\right ] \\end{align*}"}
+{"id": "641.png", "formula": "\\begin{align*} \\frac { ( r - 1 ) ! ! } { ( r - 2 ) ! ! } & = 1 + \\frac { 1 } { r - 2 } + \\frac { r - 1 } { ( r - 2 ) ( r - 4 ) } + \\frac { ( r - 1 ) ( r - 3 ) } { ( r - 2 ) ( r - 4 ) ( r - 6 ) } + \\\\ & + \\dots + \\frac { ( r - 1 ) ( r - 3 ) \\cdots 4 } { ( r - 2 ) ( r - 4 ) \\cdots 3 } , \\end{align*}"}
+{"id": "139.png", "formula": "\\begin{align*} K ( n ) = \\frac { \\Gamma ( \\frac { n } { 2 } ) } { \\sqrt { \\pi } \\Gamma ( \\frac { n + 1 } { 2 } ) } . \\end{align*}"}
+{"id": "4370.png", "formula": "\\begin{gather*} B _ { q , . } g ( x ) = \\sum _ { i \\in [ m ] , j \\in [ n ] } B _ { q , ( i , j ) } g _ { i , j } ( x ) = \\sum _ { i \\in [ m ] } B _ { q , ( i , j ( i ) ) } g _ { i , j ( i ) } ( x ) = B _ { q , ( q , j ( q ) ) } g _ { q , j ( q ) } ( x ) . \\end{gather*}"}
+{"id": "3464.png", "formula": "\\begin{align*} - 2 \\kappa ( L ) = \\mu ( \\Sigma ( L ) , \\mathfrak { t } _ L ) = \\frac { 1 } { 8 } \\sigma ( \\Sigma ( S ) ) . \\end{align*}"}
+{"id": "3435.png", "formula": "\\begin{align*} d ( X ) = \\dim V - d ( X ' ) \\end{align*}"}
+{"id": "2454.png", "formula": "\\begin{align*} \\begin{cases} V ( \\xi ) \\sim m ^ { - \\frac { 1 } { m - 1 } } \\left ( \\dfrac { k } { c } \\right ) ^ { - \\frac { 1 } { m - 1 } } \\xi ^ { - \\frac { 1 } { m - 1 } } , \\\\ V ' ( \\xi ) \\sim - ( m - 1 ) ^ { - 1 } m ^ { - \\frac { 1 } { m - 1 } } \\left ( \\dfrac { k } { c } \\right ) ^ { - \\frac { 1 } { m - 1 } } \\xi ^ { - \\frac { m } { m - 1 } } , \\end{cases} { \\rm { a s } } \\xi \\to + \\infty . \\end{align*}"}
+{"id": "251.png", "formula": "\\begin{align*} \\frac { d B } { d x } = \\frac { 1 } { A _ 0 ^ 4 } \\left ( A _ 0 \\frac { d } { d x } - 3 A ' _ 0 \\right ) \\left ( A _ 0 \\frac { d } { d x } + \\gamma _ 0 A _ 0 - A ' _ 0 \\right ) b _ 0 = 0 . \\end{align*}"}
+{"id": "1171.png", "formula": "\\begin{align*} E _ Q : = I \\times \\left [ \\ell ( I ) \\left ( k + \\frac 1 3 \\right ) , \\ell ( I ) \\left ( k + \\frac 2 3 \\right ) \\right ) . \\end{align*}"}
+{"id": "4656.png", "formula": "\\begin{align*} \\chi _ { q , q - 1 } ( a ) = \\left ( \\frac { q } { a } \\right ) \\end{align*}"}
+{"id": "27.png", "formula": "\\begin{align*} q _ + ( t , z ) = \\lim _ { y \\to \\infty } \\tfrac { 1 } { 2 \\pi i } \\bigl \\langle U ( t ) ^ * \\chi _ y , \\big ( X - t \\kappa R ( \\kappa ; q ^ 0 ) ^ 2 - z \\big ) ^ { - 1 } q ^ 0 _ + \\bigr \\rangle . \\end{align*}"}
+{"id": "7093.png", "formula": "\\begin{align*} b _ n : = \\mathbf { 1 } _ n b , \\end{align*}"}
+{"id": "2801.png", "formula": "\\begin{align*} & H _ { T } = H + \\zeta \\phi ^ { ( 1 ) } , \\\\ & H = p _ { 1 } p _ { 3 } - \\frac { 1 } { 2 } q ^ { 2 } ( q ^ { 3 } ) ^ { 2 } . \\end{align*}"}
+{"id": "5935.png", "formula": "\\begin{align*} & \\dfrac { M _ { 2 } ( \\mathcal { N C } ( G ) ) } { | e ( \\mathcal { N C } ( G ) ) | } - \\dfrac { M _ { 1 } ( \\mathcal { N C } ( G ) ) } { | v ( \\mathcal { N C } ( G ) ) | } \\\\ & ~ ~ ~ ~ ~ = \\dfrac { q ^ { 1 1 } ( q - 5 ) + q ^ { 8 } ( 1 4 q - 3 5 ) + q ^ { 5 } ( 1 5 q - 1 2 ) + q ^ { 4 } ( 5 q ^ { 6 } - 4 ) + q ^ { 3 } ( 1 8 q ^ { 4 } - 5 ) + 1 6 q ^ { 2 } - 1 0 q + 2 } { ( q ^ { 5 } ( q ^ { 2 } - 2 q - 2 ) + q ^ { 2 } ( 5 q ^ { 2 } - 4 ) + q ^ { 3 } + 1 ) ( q ^ { 3 } - q - 1 ) } \\\\ & ~ ~ ~ ~ ~ : = \\dfrac { f ( q ) } { g ( q ) } . \\end{align*}"}
+{"id": "2285.png", "formula": "\\begin{align*} f ( M _ 1 , \\dots , M _ n ) = f ( g ^ { - 1 } M _ 1 g , \\dots , g ^ { - 1 } M _ n g ) , \\ \\ \\ \\end{align*}"}
+{"id": "679.png", "formula": "\\begin{align*} g : = \\mathcal { A } _ { s } f \\in L ^ { 2 } ( \\mathbb { R } ^ { n } ) , g = ( - \\Delta ) ^ { s } g = 0 \\end{align*}"}
+{"id": "465.png", "formula": "\\begin{align*} \\pi ( e ) & = \\prod _ { h \\in X _ e } \\pi ( e ) = \\prod _ { h \\in X _ e } ( \\pi ( e ) - \\varepsilon ( h ) + \\varepsilon ( h ) ) \\\\ & = \\sum _ { S \\subseteq X _ e } \\left ( \\left ( \\prod _ { h \\in S } \\varepsilon ( h ) \\right ) \\cdot \\left ( \\prod _ { h \\in X _ g \\setminus S } ( \\pi ( e ) - \\varepsilon ( h ) ) \\right ) \\right ) . \\end{align*}"}
+{"id": "2063.png", "formula": "\\begin{align*} A ( x , y ) = a _ 0 + \\sum _ { k = 1 } ^ \\infty \\frac { b _ k } { 2 } \\big ( \\sqrt { 2 } \\cos ( 2 \\pi x ) \\big ) \\big ( \\sqrt { 2 } \\cos ( 2 \\pi y ) \\big ) + \\sum _ { k = 1 } ^ \\infty \\frac { b _ k } { 2 } \\big ( \\sqrt { 2 } \\sin ( 2 \\pi x ) \\big ) \\big ( \\sqrt { 2 } \\sin ( 2 \\pi y ) \\big ) , \\end{align*}"}
+{"id": "3315.png", "formula": "\\begin{align*} \\Pr [ Q _ 1 \\ge 1 / c ] = \\Pr [ V ( Z ) \\ge 2 / c ] \\le \\Pr [ 2 \\exp ( Z ^ 2 / 2 ) \\ge 2 / c ] = \\Pr [ K _ 1 ^ 1 \\ge 1 / c ] \\le 2 c , \\end{align*}"}
+{"id": "5149.png", "formula": "\\begin{align*} A = \\frac { 1 } { 1 - \\alpha } \\sum _ { i } p _ { i } + \\frac { 1 } { \\alpha } \\sum _ { i } q _ { i } \\ ; \\ ; ; \\ ; \\ ; B = \\frac { 1 } { \\alpha \\left ( 1 - \\alpha \\right ) } \\sum _ { i } p ^ { \\alpha } _ { i } q ^ { 1 - \\alpha } _ { i } \\end{align*}"}
+{"id": "4153.png", "formula": "\\begin{align*} \\sqrt { r } ( \\sigma ^ 2 _ j / 2 ) \\eta _ j ^ 2 \\psi ^ { ( r ) } ( \\theta ) + ( 1 - w _ { j j } ) \\eta _ j \\bigl ( \\psi ^ { ( r ) } ( \\tilde { \\theta } ) - \\psi ^ { ( r ) } _ i ( \\tilde { \\theta } ) \\bigr ) = o ( \\sqrt { r } ) , \\end{align*}"}
+{"id": "6383.png", "formula": "\\begin{align*} F _ { x , y } F _ { f _ { y } ( x ) , z } F _ { g _ { x } ( y ) , g _ { f _ { y } ( x ) } ( z ) } = F _ { y , z } F _ { x , g _ { y } ( z ) } F _ { f _ { g _ { y } ( z ) } ( x ) , f _ { z } ( y ) } \\qquad x , y , z \\in X . \\end{align*}"}
+{"id": "2127.png", "formula": "\\begin{align*} Q ^ N _ \\omega ( A \\times R \\times B ) : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\delta _ { \\bar X ^ { i , N } ( \\cdot , \\omega ) } ( A ) \\cdot \\delta _ { \\rho ^ { i , N } _ \\omega } ( R ) \\cdot \\delta _ { W ^ { i } ( \\cdot , \\omega ) } ( B ) \\end{align*}"}
+{"id": "755.png", "formula": "\\begin{align*} Z ^ h _ { j k } = R ^ h _ { j k } - F ^ { - 2 } ( y _ j R ^ h _ k - y _ k R ^ h _ j ) . \\end{align*}"}
+{"id": "908.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 0 & 0 & 0 & 1 \\\\ 3 & 0 & 0 & 0 \\\\ 0 & 4 & 0 & 0 \\\\ 0 & 0 & 3 & 0 \\\\ \\end{pmatrix} \\end{align*}"}
+{"id": "1384.png", "formula": "\\begin{align*} t \\circ [ y _ 1 : y _ 2 : y _ 3 : y _ 4 : y _ 5 ] = [ t y _ 1 : t y _ 2 : t y _ 3 : t y _ 4 : t ^ 2 y _ 5 ] . \\end{align*}"}
+{"id": "1360.png", "formula": "\\begin{align*} E ( A _ \\alpha ( G ) ) & \\leq f ( \\eta _ 1 ) \\leq f ( \\sqrt { \\dfrac { y } { n } } ) \\\\ & = f ( \\sqrt { \\dfrac { y } { n } } + \\sqrt { ( n - 1 ) ( y - \\dfrac { y } { n } ) } ) \\end{align*}"}
+{"id": "227.png", "formula": "\\begin{align*} y = \\varphi ( x ) , d \\tau = h ( x ) \\ d t , \\end{align*}"}
+{"id": "6531.png", "formula": "\\begin{align*} \\lim \\limits _ { x \\to \\infty } \\frac { \\ell ( x \\ell ^ { \\frac { 1 } { \\beta } } ( x ) ) } { \\ell ( x ) } = 1 , \\end{align*}"}
+{"id": "249.png", "formula": "\\begin{align*} B = \\frac { d x _ 1 } { d x _ 2 } \\frac { d } { d x _ 1 } \\left ( \\frac { b _ 1 } { A _ 1 } \\right ) = \\frac { 1 } { A _ 1 } \\frac { d } { d x } \\left ( \\frac { b _ 0 } { A _ 0 h _ 0 } \\right ) = \\frac { 1 } { A _ 0 ^ 3 } \\left ( A _ 0 \\frac { d } { d x } + \\gamma A _ 0 - A ' _ 0 \\right ) b _ 0 . \\end{align*}"}
+{"id": "6072.png", "formula": "\\begin{align*} \\lambda _ 0 ^ { \\pm } = \\pm \\frac { 2 \\omega _ { 2 , 1 } \\omega _ { 2 , 2 } } { \\sqrt { 2 \\omega _ { 2 , 2 } ^ 2 / \\phi _ { 2 , 2 } ^ 2 - 1 } } . \\end{align*}"}
+{"id": "949.png", "formula": "\\begin{align*} \\varphi ( x ) = \\mathbb E _ x \\int _ 0 ^ \\infty e ^ { - A ^ \\mu _ t } f ( X _ t ) \\ , d t , x \\in E . \\end{align*}"}
+{"id": "7723.png", "formula": "\\begin{align*} \\frac { d J ( t ) } { d t } \\Bigg | _ { t = t _ { k } } \\rightarrow 0 a s \\ t _ { k } \\rightarrow + \\infty . \\end{align*}"}
+{"id": "1619.png", "formula": "\\begin{align*} Q _ { r , s , g , n } \\coloneqq Q _ { r ( p - 1 ) } & Q _ { s ( p - 1 ) } \\\\ - q ^ { \\tfrac { p - 1 } { 2 } } & ( - 1 ) ^ { \\tfrac { s ( p - 1 ) } { 2 } } \\sum _ i ( - 1 ) ^ { \\tfrac { r - i } { 2 } } \\binom { ( i - s ) \\tfrac { p - 1 } { 2 } - 1 } { \\tfrac { r - i } { 2 } - 1 } Q _ { ( r + p s - p i ) ( p - 1 ) } Q _ { i ( p - 1 ) } \\end{align*}"}
+{"id": "8904.png", "formula": "\\begin{align*} \\Omega _ 1 ( x _ 2 ) & = x _ 2 , \\\\ \\Omega _ 2 ( x _ 2 , x _ 4 ) & = x _ 4 - 5 x _ 2 ( x _ 2 - 1 ) , \\\\ \\Omega _ 3 ( x _ 2 , x _ 4 , x _ 6 ) & = x _ 6 - 7 x _ 4 ( x _ 2 - 1 ) + \\frac { 3 5 } { 3 } x _ 2 ( x _ 2 - 1 ) ( x _ 2 - 2 ) + \\frac { 1 4 } { 3 } x _ 2 , \\end{align*}"}
+{"id": "513.png", "formula": "\\begin{align*} \\gcd \\left ( \\prod _ { t = 0 } ^ { h ' - 1 } { \\alpha _ { i _ { k - h ' + t } } } , s \\right ) = \\gcd \\left ( \\prod _ { t = 0 } ^ { h ' - 2 } { \\alpha _ { i _ { k - h ' + t } } } , s \\right ) , \\end{align*}"}
+{"id": "4285.png", "formula": "\\begin{align*} \\varphi _ \\rho ( x ) = ( 2 \\pi \\rho ^ 2 ) ^ { - e / 2 } e ^ { - | x | ^ 2 / 2 \\rho ^ 2 } , ( x , \\rho ) \\in \\R ^ e \\times ( 0 , \\infty ) . \\end{align*}"}
+{"id": "4144.png", "formula": "\\begin{align*} & \\lim _ { r \\to 0 } \\frac { Q ^ * ( \\theta ) } { r ^ { 2 j } } = \\eta _ j ^ 2 Q ^ * ( w _ { 1 j } , \\ldots , w _ { j - 1 , j } , 1 , 0 , \\ldots , 0 ) , \\\\ & \\lim _ { r \\to 0 } \\frac { Q ^ * ( \\tilde { \\theta } ) } { r ^ { 2 j + 1 } } = \\eta _ j ^ 2 Q ^ * ( w _ { 1 j } , \\ldots , w _ { j - 1 , j } , 1 , 0 , \\ldots , 0 ) . \\end{align*}"}
+{"id": "907.png", "formula": "\\begin{align*} A ' = \\left ( \\begin{array} { c c c c c } A _ { i _ 1 } & & & & \\\\ & A _ { i _ 2 } & & & \\\\ & & \\ddots & & \\\\ & & & A _ { i _ { s - 1 } } & \\\\ & & & & A _ { i _ s } \\\\ \\end{array} \\right ) \\end{align*}"}
+{"id": "7485.png", "formula": "\\begin{align*} & \\frac { \\partial } { \\partial s } K ( x , t , \\cdot , \\cdot ) = \\Delta _ { g ( \\tau ) } K ( x , t , \\cdot , \\cdot ) , \\\\ & \\frac { \\partial } { \\partial s } g = 2 R i c ( g ( \\tau ) ) , \\\\ & \\lim _ { \\tau \\to 0 } K ( x , t , \\cdot , \\tau ) = \\delta _ x . \\end{align*}"}
+{"id": "4937.png", "formula": "\\begin{align*} 1 _ X = \\sum _ { 1 \\leq j \\leq l } e _ j . \\end{align*}"}
+{"id": "4323.png", "formula": "\\begin{align*} & \\min _ { x \\in \\mathcal { X } } \\left \\{ \\sum _ { ( i , j , r , s ) \\in [ n ] ^ 4 } \\overline { c } _ { i , j } d _ { r , s } x _ { i , r } x _ { j , s } + \\max _ { \\mathcal { S } \\subseteq [ n ] ^ 2 : | \\mathcal { S } | \\le \\Gamma } \\left \\{ \\sum _ { ( i , j ) \\in \\mathcal { S } } \\sum _ { r , s \\in [ n ] } \\Delta \\overline { c } _ { i , j } d _ { r , s } x _ { i , r } x _ { j , s } \\right \\} \\right \\} . \\end{align*}"}
+{"id": "6687.png", "formula": "\\begin{align*} \\widetilde { H } / K \\cong L K / K \\ltimes \\widetilde { M } / K \\cong L \\ltimes M = H , \\end{align*}"}
+{"id": "5898.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { C } ( G ) ) = 4 n ( 4 n - 1 ) ^ { 2 } + 5 \\cdot 3 n ( 3 n - 1 ) ^ { 2 } = 4 n ( 4 n - 1 ) ^ { 2 } + 1 5 n ( 3 n - 1 ) ^ { 2 } \\end{align*}"}
+{"id": "3539.png", "formula": "\\begin{align*} X ( ( \\alpha _ 1 + \\beta _ 1 f _ 1 ) + \\gamma ( \\alpha _ 2 + \\beta _ 2 f _ 2 ) ) & = X ( \\alpha _ 1 + \\gamma \\alpha _ 2 + \\beta _ 1 f _ 1 + \\gamma \\beta _ 2 f _ 2 ) \\\\ & = \\alpha _ 1 + \\gamma \\alpha _ 2 + T ^ { - 1 } ( \\beta _ 1 f _ 1 + \\gamma \\beta _ 2 f _ 2 ) \\\\ & = \\alpha _ 1 + T ^ { - 1 } ( \\beta _ 1 f _ 1 ) + \\gamma ( \\alpha _ 2 + T ^ { - 1 } ( \\beta _ 2 f _ 2 ) ) \\\\ & = X ( \\alpha _ 1 + \\beta _ 1 f _ 1 ) + \\gamma X ( \\alpha _ 2 + \\beta _ 2 f _ 2 ) . \\end{align*}"}
+{"id": "4145.png", "formula": "\\begin{align*} & \\theta _ \\ell = w _ { \\ell j } r ^ j \\eta _ j + o ( r ^ j ) , \\ell = 1 , \\ldots , j - 1 , \\\\ & \\theta _ j = r ^ j \\eta _ j + o ( r ^ j ) , \\\\ & \\theta _ k = o ( r ^ j ) , k = j + 1 , \\ldots , J . \\end{align*}"}
+{"id": "2300.png", "formula": "\\begin{align*} P _ r ( t ) = \\bigl [ ( 1 - x ^ 2 ) ( 1 + t ) ^ r ( 1 + t x ^ { 2 } ) ^ { r } ( 1 + t x ^ { - 2 } ) ^ { r } \\bigr ] _ 0 . \\end{align*}"}
+{"id": "7451.png", "formula": "\\begin{align*} A ^ { \\otimes j } - B ^ { \\otimes j } & = \\sum _ { \\{ q _ 1 , q _ 2 \\in \\mathbb { N } : \\ , q _ 1 + q _ 2 = j - 1 \\} } B ^ { \\otimes q _ 1 } \\otimes ( A - B ) \\otimes A ^ { \\otimes q _ 2 } . \\end{align*}"}
+{"id": "6097.png", "formula": "\\begin{align*} \\Delta ( x _ { u , \\alpha } ) & = 1 \\otimes x _ { u , \\alpha } + \\sum a _ { u , \\alpha } \\otimes b _ u , \\\\ \\Delta ( x _ { v , \\beta } ) & = 1 \\otimes x _ { v , \\beta } + \\sum a _ { v , \\beta } \\otimes b _ v , \\end{align*}"}
+{"id": "1437.png", "formula": "\\begin{align*} E _ r ^ { p , q } ( K , L ) = & ( E _ r ^ { p , q } ( K _ { \\mathbb { R } } , L ) , ( E _ r ^ { p , q } ( K _ { \\mathcal { O } } , L ) , F ) , E _ r ^ { p , q } ( \\alpha ) ) \\\\ & \\Rightarrow E _ { \\infty } ^ { p , q } ( K , L ) = ( E _ { \\infty } ^ { p , q } ( K _ { \\mathbb { R } } , L ) , ( E _ { \\infty } ^ { p , q } ( K _ { \\mathcal { O } } , L ) , F ) , E _ { \\infty } ^ { p , q } ( \\alpha ) ) , \\end{align*}"}
+{"id": "9040.png", "formula": "\\begin{align*} T ( a b ) = ( T a ) b + a ( T b ) , S ^ i ( a b ) = ( S ^ i a ) b + ( - 1 ) ^ { p ( a ) } a ( S ^ i b ) \\quad ( i \\in [ N ] ) . \\end{align*}"}
+{"id": "487.png", "formula": "\\begin{align*} \\Theta _ { i , h } ( x ) : = \\begin{cases} \\emptyset , & H _ i = ( h = ) 0 , \\\\ \\neg \\theta _ { i , 1 } ( x ) , & h = 0 < H _ i , \\\\ \\theta _ { i , h } ( x ) \\wedge ( \\neg \\theta _ { i , h + 1 } ( x ) ) , & 0 < h < H _ i , \\\\ \\theta _ { i , H _ i } ( x ) , & h = H _ i > 0 . \\end{cases} \\end{align*}"}
+{"id": "13.png", "formula": "\\begin{align*} m ( x + h ; \\kappa , q ) = m ( x ; \\kappa , q ( \\cdot + h ) ) \\quad h \\in \\R . \\end{align*}"}
+{"id": "2877.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to \\infty } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ { L ^ p ( \\mathbb { R } ^ n ) } = \\left [ \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { L ^ p ( \\mathbb { R } ^ n ) } . \\end{align*}"}
+{"id": "5931.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { C } ( G ) ) & = \\dfrac { q ( q + 1 ) } { 2 } ( q ^ { 2 } - 3 q + 2 ) \\dfrac { ( q ^ { 2 } - 3 q + 1 ) ^ { 3 } } { 2 } + \\dfrac { q ( q - 1 ) } { 2 } ( q ^ { 2 } - q ) \\dfrac { ( q ^ { 2 } - q - 1 ) ^ { 3 } } { 2 } \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ + ( q + 1 ) ( q ^ { 2 } - 2 q + 1 ) \\dfrac { ( q ^ { 2 } - 2 q ) ^ { 3 } } { 2 } \\\\ & = \\dfrac { q ( q - 1 ) } { 2 } ( q ^ { 8 } - 6 q ^ { 7 } + 1 4 q ^ { 6 } - 1 5 q ^ { 5 } + 3 q ^ { 4 } + 1 2 q ^ { 3 } - 1 6 q ^ { 2 } + 9 q - 1 ) . \\end{align*}"}
+{"id": "5418.png", "formula": "\\begin{align*} S _ { n + 1 } f ( \\tfrac { 1 } { 2 } ) = \\sum _ { i = 0 } ^ { n } f \\big ( T ^ { i } ( \\tfrac { 1 } { 2 } ) \\big ) = \\sum _ { i = 1 } ^ { n } f \\big ( T ^ { i } ( \\tfrac { 1 } { 2 } ) \\big ) = \\sum _ { i = 1 } ^ { n } f \\big ( T ^ { i } \\circ \\mathcal I ( \\tfrac { 1 } { 2 } ) \\big ) . \\end{align*}"}
+{"id": "6461.png", "formula": "\\begin{align*} \\nabla _ { X } \\nu = - A ( X ) , \\end{align*}"}
+{"id": "3182.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n \\cos ( 2 k x ) = \\frac { \\sin ( ( 2 n + 1 ) x ) } { 2 \\sin ( x ) } - \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "5155.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } D _ { \\alpha } ( p \\| q ) } { \\partial q _ { j } } = \\frac { 1 } { \\alpha } \\left [ \\frac { a - 1 } { a - b } A ^ { a - 2 } - \\frac { b - 1 } { a - b } A ^ { b - 2 } \\right ] \\left ( p ^ { \\alpha } _ { j } q ^ { - \\alpha } _ { j } - 1 \\right ) \\end{align*}"}
+{"id": "3597.png", "formula": "\\begin{align*} P = \\frac { ( T - \\lambda _ { 1 } I ) ( T - \\lambda _ 2 I ) } { ( 1 - \\lambda _ 1 ) ( 1 - \\lambda _ 2 ) } , Q = \\frac { ( T - I ) ( T - \\lambda _ 2 I ) } { ( \\lambda _ 1 - 1 ) ( \\lambda _ 1 - \\lambda _ 2 ) } , R = \\frac { ( T - I ) ( T - \\lambda _ 1 I ) } { ( \\lambda _ 2 - 1 ) ( \\lambda _ 2 - \\lambda _ 1 ) } , \\end{align*}"}
+{"id": "200.png", "formula": "\\begin{align*} K ' _ 1 \\ & = \\ ( \\bigcup _ { k > i + 1 } \\ \\{ k - , k + \\} \\times \\Z [ 2 ] ) \\cup \\{ \\infty \\} , \\\\ K ' _ 2 \\ & = \\ \\{ 1 - , 2 - , \\dots , ( i - 1 ) - , ( i - , \\bar d ) , ( i + 1 ) - , \\\\ & ( ( i + 2 ) - , a ) , ( ( i + 3 ) - , a ) , \\dots , \\infty \\} . \\\\ K ' _ 3 \\ & = \\ \\{ ( i - , \\bar d ) \\} \\cup \\{ ( i + , k ) : j - 1 \\le k \\le \\infty \\} \\cup ( \\bigcup _ { k < j - 1 } \\ \\{ i + \\} \\times \\{ z + A _ k \\} ) . \\end{align*}"}
+{"id": "4399.png", "formula": "\\begin{align*} F _ { \\mathcal { Y } , a , b } ( x ) & : = \\sum _ { ( i , j ) \\in [ m ] \\times [ n ] } \\overline { y } _ { ( i , j ) } g _ { ( i , j ) } x + \\sum _ { ( q , p ) \\in \\mathcal { Y } } \\Delta y _ { ( q , p ) } g _ { ( q , p ) } ( x ) - \\Delta y _ { ( a , b ) } g _ { ( a , b ) } ( x ) . \\end{align*}"}
+{"id": "2818.png", "formula": "\\begin{align*} \\delta \\left ( \\sigma ^ { * } _ { 2 } ( t ) I \\right ) = \\int ^ { t _ { 2 } } _ { t _ { 1 } } \\left [ - \\dot { P } _ { 1 } - Q ^ { 1 } \\right ] \\delta Q ^ { 1 } d t + \\left [ \\dot { Q } ^ { 1 } - P _ { 1 } \\right ] \\delta P _ { 1 } d t + \\left [ P _ { 1 } \\delta Q ^ { 1 } \\right ] ^ { t _ { 2 } } _ { t _ { 1 } } . \\end{align*}"}
+{"id": "9168.png", "formula": "\\begin{align*} \\tilde { \\mathcal { E } } ^ { \\epsilon } _ { h } \\ = \\prod _ { ( \\beta , s ) \\in \\Delta ^ { + } \\times \\mathbb { Z } } \\limits ^ { \\rightarrow } ( \\tilde { \\mathcal { E } } ^ { \\epsilon } _ { \\beta , s } ) ^ { h ( \\beta , s ) } \\forall \\ h \\in H , \\end{align*}"}
+{"id": "8041.png", "formula": "\\begin{align*} | \\eta _ t ^ N ( \\vec { f } ) | \\leq \\varepsilon _ 5 ^ N ( \\vec { f } ) + \\sum _ { l = 1 } ^ { + \\infty } \\int _ 0 ^ t \\ldots \\int _ 0 ^ { t _ { l - 1 } } \\sum _ { \\mathcal { C } _ { t _ j } ^ N \\in \\{ A _ { t _ j } ^ N , \\mathcal { B } _ { 1 1 } \\} \\atop 1 \\leq j \\leq l } \\varepsilon _ 5 ^ N \\left ( \\mathcal { C } _ { t _ { l } } ^ N \\ldots \\mathcal { C } _ { t _ 1 } ^ N \\vec { f } \\right ) d t _ 1 d t _ 2 \\ldots d t _ { l } \\end{align*}"}
+{"id": "8689.png", "formula": "\\begin{align*} \\langle \\omega _ 0 , \\xi _ M \\rangle T v _ J = - \\xi _ M v _ J + \\sum _ { j = 1 } ^ { n - 1 } a _ j L _ j v _ J + \\sum _ { j = 1 } ^ { n - 1 } b _ j \\bar L _ j v _ J + O ( | z | ) D v _ J . \\end{align*}"}
+{"id": "2197.png", "formula": "\\begin{align*} \\hat f ( \\lambda ) = { 1 \\over \\sqrt { 2 \\pi } } \\intop _ \\R e ^ { - i \\lambda \\tau } f ( \\tau ) d \\tau , \\check { \\hat f } _ { ( \\tau ) } = { 1 \\over \\sqrt { 2 \\pi } } \\intop _ \\R e ^ { i \\lambda \\tau } \\hat f ( \\lambda ) d \\lambda \\end{align*}"}
+{"id": "3327.png", "formula": "\\begin{align*} ( v , x , t ) \\circ ( v _ { 0 } , x _ { 0 } , t _ { 0 } ) = ( v _ { 0 } + v , x _ { 0 } + x + t v _ { 0 } , t _ { 0 } + t ) . \\end{align*}"}
+{"id": "6673.png", "formula": "\\begin{align*} A = F _ j B = F _ { j + 1 } = \\Phi ( F _ j ) = A ^ p . \\end{align*}"}
+{"id": "7799.png", "formula": "\\begin{align*} P ^ * = \\begin{bmatrix} 0 . 4 1 5 2 & 0 . 3 8 9 0 & 0 . 2 0 6 8 & 0 . 0 1 6 1 & - 0 . 4 0 6 0 \\\\ 0 . 3 8 9 0 & 2 . 7 2 1 1 & 1 . 9 1 0 0 & - 2 . 6 0 7 7 & - 0 . 7 7 5 7 \\\\ 0 . 2 0 6 8 & 1 . 9 1 0 0 & 1 . 8 5 3 7 & - 1 . 8 3 3 3 & - 0 . 8 9 8 0 \\\\ 0 . 0 1 6 1 & - 2 . 6 0 7 7 & - 1 . 8 3 3 3 & 4 . 2 4 0 9 & - 0 . 2 6 6 4 \\\\ - 0 . 4 0 6 0 & - 0 . 7 7 5 7 & - 0 . 8 9 8 0 & - 0 . 2 6 6 4 & 2 . 1 5 4 0 \\end{bmatrix} , \\end{align*}"}
+{"id": "9153.png", "formula": "\\begin{align*} G _ { \\beta } = ( w _ { \\beta , 1 } - v ^ { \\pm ( 4 n - 6 ) } w _ { \\beta , 2 } ) ( w _ { \\beta , 1 } - v ^ { \\pm ( 4 n - 1 4 ) } w _ { \\beta , 2 } ) ( w _ { \\beta , 1 } - v ^ { \\pm 4 } w _ { \\beta , 2 } ) \\cdot G _ { \\gamma } . \\end{align*}"}
+{"id": "5885.png", "formula": "\\begin{align*} f ( m , n ) = ( m - 1 ) ( 2 m - 1 ) ( m n - n - 1 ) ^ 3 - ( m - 1 ) ^ 2 ( m n - n - 1 ) ^ 3 + m ( 2 m - 1 ) ( n - 1 ) ^ 3 - m ^ 2 ( n - 1 ) ^ 2 \\end{align*}"}
+{"id": "6436.png", "formula": "\\begin{align*} \\left | | \\xi ^ 1 _ i + \\xi ^ 2 _ i | ^ { 1 / 2 } - | \\xi ^ 1 _ i | ^ { 1 / 2 } \\right | & = \\frac { | | \\xi ^ 1 _ i + \\xi ^ 2 _ i | - | \\xi ^ 1 _ i | | } { | \\xi ^ 1 _ i + \\xi ^ 2 _ i | ^ { 1 / 2 } + | \\xi ^ 1 _ i | ^ { 1 / 2 } } \\\\ & \\leq C \\left ( | \\xi ^ 1 _ i | ^ { - 1 / 2 } | \\xi ^ 2 _ i | { \\bf 1 } _ { | \\xi ^ 1 _ i | > \\epsilon _ n } + | \\xi ^ 2 _ i | ^ { 1 / 2 } { \\bf 1 } _ { | \\xi ^ 1 _ i | \\leq \\epsilon _ n } \\right ) , \\end{align*}"}
+{"id": "3319.png", "formula": "\\begin{align*} E _ 1 ^ { i j } = R ^ j f _ * ( E \\otimes \\Omega _ { Y / X } ^ i ) \\quad \\Rightarrow R ^ { i + j } f _ * ( E \\otimes \\Omega ^ \\bullet _ { Y / X } ) , \\end{align*}"}
+{"id": "7069.png", "formula": "\\begin{align*} u ( t ) : = f ( \\Psi _ { t } ^ { - 1 } ) , t \\geqslant 0 , \\end{align*}"}
+{"id": "922.png", "formula": "\\begin{align*} - L [ \\Pi _ { V _ n } ( u ) ] = \\mathbf 1 _ { V _ n } \\cdot f ( \\cdot , u ) + \\mathbf 1 _ { V _ n } \\cdot \\mu , n \\ge 1 , \\end{align*}"}
+{"id": "8992.png", "formula": "\\begin{align*} A _ t ^ n = \\int _ 0 ^ { t } \\exp { ( \\gamma X ^ n ( Z ^ n _ s ) - \\frac { \\gamma ^ 2 } { 2 } \\mathbb { E } ^ { X ^ n } [ X ^ n ( Z ^ n _ s ) ^ 2 ] ) } d s , \\end{align*}"}
+{"id": "7870.png", "formula": "\\begin{align*} \\mu ( \\mathbf { \\Phi } ) = \\frac { 1 } { \\sqrt { p ^ { r _ { m i n } } } } = \\frac { 1 } { \\sqrt { p ^ { m } } } = \\sqrt { \\frac { 1 } { M } } , \\end{align*}"}
+{"id": "1446.png", "formula": "\\begin{align*} E ^ { ( k ) } = \\coprod _ { 1 \\le i _ 1 < \\dots < i _ k \\le l } E _ { i _ 1 } \\cap \\cdots \\cap E _ { i _ k } \\end{align*}"}
+{"id": "4366.png", "formula": "\\begin{gather*} f _ i ( x , u ^ i ) = ( u ^ i ) ^ T l _ i ( x ) \\ \\forall u ^ i \\in \\mathcal { U } _ i , \\ x \\in \\mathcal { X } . \\end{gather*}"}
+{"id": "1447.png", "formula": "\\begin{align*} F ^ p A ^ n _ { \\mathbb { C } } : = \\bigoplus _ { 0 \\le q \\le n - p } \\widetilde { \\Omega } _ { \\widetilde { X } } ^ { n + 1 } ( \\log E ) / W _ q \\widetilde { \\Omega } _ { \\widetilde { X } } ^ { n + 1 } ( \\log E ) \\end{align*}"}
+{"id": "9068.png", "formula": "\\begin{align*} \\frac { 1 } { s _ 0 + t } \\int _ { - s _ 0 } ^ { t } a ( u ) d u = a ( s _ 0 ) . \\end{align*}"}
+{"id": "4562.png", "formula": "\\begin{align*} \\nabla _ { \\eta } R _ { \\beta \\gamma } - \\nabla _ { \\gamma } R _ { \\beta \\eta } & = 2 \\nabla _ { \\alpha } W ^ { \\alpha } _ { \\beta \\gamma \\eta } + \\frac 1 6 ( g _ { \\beta \\gamma } \\nabla _ { \\eta } s _ g - g _ { \\beta \\eta } \\nabla _ { \\gamma } s _ g ) \\\\ & = O _ g ' ( r _ q ) , \\end{align*}"}
+{"id": "1577.png", "formula": "\\begin{align*} \\big ( D g _ k ( z ) , z \\big ) & = g _ k ( z ) , \\forall \\ , z \\ne 0 , \\\\ D g _ k ( \\lambda \\ , z ) & = D g _ k ( z ) , \\forall \\ , \\lambda > 0 , z \\ne 0 , \\end{align*}"}
+{"id": "5845.png", "formula": "\\begin{align*} M _ { 1 } ( \\Gamma ) = \\sum \\limits _ { v \\in v ( \\Gamma ) } \\deg ( v ) ^ { 2 } M _ { 2 } ( \\Gamma ) = \\sum \\limits _ { u v \\in e ( \\Gamma ) } \\deg ( u ) \\deg ( v ) , \\end{align*}"}
+{"id": "2083.png", "formula": "\\begin{align*} - \\int _ 0 ^ s \\Theta ( u , x ) f ' ( u ) d u + f ( s ) { \\Theta } ( s , x ) - f ( 0 ) { \\Theta } ( 0 , x ) = & - \\alpha \\int _ 0 ^ { s } \\int _ 0 ^ 1 f ( u ) A ( x , y ) \\Theta ( u , y ) d y d u \\\\ & + \\alpha \\beta \\int _ 0 ^ s f ( u ) d \\xi _ 1 ( s , x ) . \\end{align*}"}
+{"id": "6289.png", "formula": "\\begin{align*} \\frac { 1 } { T } \\sum \\limits _ { k = 0 } ^ { T - 1 } \\| \\hat { g } _ { k + 1 } \\| ^ { 2 } _ q \\leq 1 2 \\sigma _ { q , \\kappa } ^ { 1 + \\kappa } c ^ { 1 - \\kappa } \\log \\left ( \\frac 4 \\delta \\left [ \\log \\left ( \\sqrt { T } \\right ) + 2 \\right ] ^ 2 \\right ) + \\frac { 2 0 } { T } c ^ 2 \\log \\left ( \\frac { 1 2 } { \\delta } \\right ) . \\end{align*}"}
+{"id": "6335.png", "formula": "\\begin{align*} 2 \\hbar ^ { 5 } \\left | E _ { n , \\ell } - 4 \\pi \\mathfrak { a } n ^ 2 \\ell ^ { - 1 } - 4 \\pi \\frac { 1 2 8 } { 1 5 \\sqrt { \\pi } } \\left ( \\mathfrak { a } \\frac { n } { \\ell } \\right ) ^ { 5 / 2 } \\right | = \\left | \\hbar ^ 3 \\sum _ { p \\in \\pi \\mathbb { N } _ 0 ^ 3 \\setminus \\{ 0 \\} } g ( p \\hbar ) - \\pi ^ { - 3 } \\int _ { \\R ^ 3 _ { \\geq 0 } } g ( z ) \\dd ^ 3 z \\right | . \\end{align*}"}
+{"id": "418.png", "formula": "\\begin{align*} \\widetilde { { C } } ( L ^ 2 , L ^ 2 , \\sigma ^ 2 ) = \\frac { \\pi L ^ 2 } { \\lambda ^ 2 } \\log ( \\frac { L ^ { 4 } } { e \\sigma ^ 2 \\Delta ^ 4 } ) + \\frac { 2 \\pi \\sigma \\Delta ^ 2 } { \\lambda ^ 2 } . \\end{align*}"}
+{"id": "8216.png", "formula": "\\begin{align*} { \\bf a } _ i = F _ { B C J R } ( { \\bf a } _ { i - 1 } , \\{ P _ { { \\bf X } _ i | Y ^ { i - 1 } } ( { \\bf x } _ i | y ^ { i - 1 } ) \\} , y _ i , P _ { \\bf S } ) . \\end{align*}"}
+{"id": "6759.png", "formula": "\\begin{align*} \\begin{aligned} & \\| M ( \\widetilde { v } ^ { k } - v ^ * ) \\| ^ 2 _ H + \\frac { 1 - \\tau ^ k } { \\tau ^ k } \\underbrace { \\| M ( \\breve { v } ^ { k - 1 } - v ^ * ) \\| ^ 2 _ H } _ { : = D ^ { k - 1 } } \\\\ = & \\frac { 1 - \\tau ^ k } { ( \\tau ^ k ) ^ 2 } \\underbrace { \\| M ( \\breve { v } ^ { k } - \\breve { v } ^ { k - 1 } ) \\| ^ 2 _ H } _ { : = E ^ k } + \\frac { 1 } { \\tau ^ k } \\underbrace { \\| M ( \\breve { v } ^ { k } - v ^ * ) \\| ^ 2 _ H } _ { D ^ k } . \\end{aligned} \\end{align*}"}
+{"id": "1214.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\mathcal { L } _ { \\sigma , p } u = 0 & { \\rm i n } \\ \\ \\Omega , \\\\ u = 0 & { \\rm i n } \\ \\mathbb { R } ^ N \\setminus \\Omega , \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "3898.png", "formula": "\\begin{align*} X _ n = \\frac { \\ell _ { \\phi ' } } { \\sqrt { n } } = \\frac { \\ell _ { \\phi } - n \\mu _ \\psi ( \\phi ) } { \\sqrt { n } } . \\end{align*}"}
+{"id": "5146.png", "formula": "\\begin{align*} \\frac { \\partial L D 2 ( p \\| q ) } { \\partial q _ { j } } = \\left \\lbrace \\frac { 1 } { A } \\frac { \\partial A } { \\partial q _ { j } } - \\left [ \\frac { 1 } { X } \\frac { \\partial X } { \\partial q _ { j } } + \\frac { 1 } { Y } \\frac { \\partial Y } { \\partial q _ { j } } \\right ] \\right \\rbrace \\end{align*}"}
+{"id": "7038.png", "formula": "\\begin{align*} \\Theta _ { p } ( \\mu , b _ n ) f & - ( \\mu - \\Delta ) ^ { - 1 } f = \\\\ & - ( \\mu - \\Delta ) ^ { - \\frac { 1 } { 2 } - \\frac { 1 } { q } } Q _ { p } ( q ) \\bigl ( 1 + T _ { p } \\bigr ) ^ { - 1 } b _ n ^ { \\frac { 2 } { p } } \\cdot \\nabla ( \\lambda - \\Delta ) ^ { - 1 } ( \\mu - \\Delta ) ^ { - 1 } ( \\lambda - \\Delta ) f , \\end{align*}"}
+{"id": "8825.png", "formula": "\\begin{align*} f _ j ( x ) & = 2 ^ { - j s } \\varphi ( 2 ^ { - j } ) ^ { - 1 } \\tilde { \\psi } ( 2 ^ j x ) , \\\\ g _ j ( x ) & = \\sum _ { k = 1 } ^ { k _ j } f _ j ( x - m _ k ) , \\end{align*}"}
+{"id": "5021.png", "formula": "\\begin{align*} \\mathbf { w } _ S ^ S & = \\mathbf { g } _ S ^ S - \\beta \\mathbf { g } _ S ^ S = ( 1 - \\beta ) \\mathbf { g } _ S ^ S \\\\ \\mathbf { w } _ { S ^ c } ^ S & = \\mathbf { 1 } _ { S ^ c } + \\beta \\mathbf { P } _ { S ^ c N } \\mathbf { g } ^ S - \\mathbf { g } _ { S ^ c } ^ S = \\mathbf { 1 } _ { S ^ c } + \\beta \\mathbf { P } _ { S ^ c S } \\mathbf { g } _ S ^ S . \\end{align*}"}
+{"id": "3735.png", "formula": "\\begin{align*} \\frac { d } { d x } \\left [ \\frac { 1 } { \\left ( x \\right ) _ { n } } \\right ] = \\frac { 1 } { \\left ( x \\right ) _ { n } } \\left [ \\psi \\left ( x \\right ) - \\psi \\left ( x + n \\right ) \\right ] , \\end{align*}"}
+{"id": "8721.png", "formula": "\\begin{gather*} f ( x ) = \\xi _ { 1 } f ( a _ { 1 } ) + \\xi _ { 2 } f ( a _ { 2 } ) + \\xi _ { 3 } f ( a _ { 3 } ) = \\\\ \\xi _ { 1 } a _ { 1 } + \\xi _ { 1 } \\alpha _ { 2 } a _ { 2 } + \\xi _ { 1 } \\alpha _ { 3 } a _ { 3 } + \\xi _ { 2 } ( ( 1 + \\alpha _ { 2 } ) a _ { 2 } + \\beta _ { 3 } a _ { 3 } ) + \\xi _ { 3 } ( 1 + \\alpha _ { 2 } ) a _ { 3 } = \\\\ \\xi _ { 1 } a _ { 1 } + ( \\xi _ { 1 } \\alpha _ { 2 } + \\xi _ { 2 } + \\xi _ { 2 } \\alpha _ { 2 } ) a _ { 2 } + ( \\xi _ { 1 } \\alpha _ { 3 } + \\xi _ { 2 } \\beta _ { 3 } + \\xi _ { 3 } ( 1 + \\alpha _ { 2 } ) ) a _ { 3 } . \\end{gather*}"}
+{"id": "4318.png", "formula": "\\begin{align*} \\inf _ { x \\in \\mathcal { X } } & \\sum _ { i \\in [ m ] } \\ \\overline { f } _ i ( x ) , \\end{align*}"}
+{"id": "5208.png", "formula": "\\begin{align*} K _ { 0 , \\alpha \\beta } = \\left [ \\frac { \\sum _ { i } p ^ { \\alpha } _ { i } q ^ { \\beta - 1 } _ { i } } { \\sum _ { i } q ^ { \\alpha + \\beta - 1 } _ { i } } \\right ] ^ { \\frac { 1 } { \\alpha } } \\end{align*}"}
+{"id": "4046.png", "formula": "\\begin{align*} \\mathbb { P } _ X = \\mathbb { Q } _ Y & \\iff \\mathbb { E } _ { \\mathbb { P } } [ \\langle \\ell , S ( X ) \\rangle _ { \\mathcal { H } ^ 1 } ] = \\mathbb { E } _ { \\mathbb { Q } } [ \\langle \\ell , S ( Y ) \\rangle _ { \\mathcal { H } ^ 1 } ] , \\forall \\ell \\in T ( \\mathbb { R } ^ { d + 1 } ) \\\\ & \\iff \\mathbb { E } _ { \\mathbb { P } } [ S ( X ) ] = \\mathbb { E } _ { \\mathbb { Q } } [ S ( Y ) ] . \\end{align*}"}
+{"id": "7569.png", "formula": "\\begin{align*} \\Delta ^ { 2 } u + \\omega u = \\mu | u | ^ { q - 2 } u + u | ^ { p - 2 } u , \\end{align*}"}
+{"id": "837.png", "formula": "\\begin{align*} ( n + 1 ) \\Psi _ { i } = \\nabla _ { i } f + \\nabla _ { 0 } f _ { . i } . \\end{align*}"}
+{"id": "1504.png", "formula": "\\begin{align*} - \\Delta _ { p } z = 1 , \\Omega , z = 0 \\partial \\Omega \\end{align*}"}
+{"id": "1979.png", "formula": "\\begin{align*} g ^ { i j } \\partial _ t g _ { i j } = - 2 N g ^ { i j } h _ { i j } = - 2 N H \\ , , \\end{align*}"}
+{"id": "7926.png", "formula": "\\begin{align*} d I ( V ; X , \\alpha ) : = \\lim _ { t \\to 0 } \\frac { 1 } { t } \\Big ( I ( V ( t ) ; \\alpha ) - I ( V ( 0 ) ; \\alpha ) \\Big ) . \\end{align*}"}
+{"id": "6701.png", "formula": "\\begin{align*} \\Omega _ { \\{ 1 \\} } ( U / U ^ { p ^ k } ) \\times \\Omega _ { \\{ 1 \\} } ( T ) = U ^ { p ^ { k - 1 } } / U ^ { p ^ k } \\times \\Omega _ { \\{ 1 \\} } ( G ) \\end{align*}"}
+{"id": "5068.png", "formula": "\\begin{align*} A _ s ^ 2 & = ( v ^ 2 - 1 ) A _ s + v ^ 2 . \\end{align*}"}
+{"id": "4827.png", "formula": "\\begin{align*} \\delta ^ { \\mbox { \\tiny ( \\itshape k \\upshape ) } } _ a = \\begin{cases} \\frac { k - 1 } { K _ { m a x } } \\mbox { w i t h p r o b a b l t y } p _ a , \\\\ 0 \\mbox { w i t h p r o b a b i l i t y } 1 - p _ a , \\end{cases} \\end{align*}"}
+{"id": "2272.png", "formula": "\\begin{align*} \\check R _ 1 \\check R _ { 2 } \\check R _ 1 = \\check R _ { 2 } \\check R _ 1 \\check R _ { 2 } \\ \\ \\ \\ \\ \\end{align*}"}
+{"id": "9191.png", "formula": "\\begin{align*} { \\ell ^ - _ { j j } [ 0 ] } ^ { - 1 } \\ell ^ - _ { 1 j } ( z ) = q ^ { - 1 } \\ell ^ - _ { 1 j } ( z ) { \\ell ^ - _ { j j } [ 0 ] } ^ { - 1 } . \\end{align*}"}
+{"id": "6712.png", "formula": "\\begin{align*} P ( E ) = a _ n E ^ n + a _ { n - 1 } E ^ { n - 1 } + . . . + a _ 1 E + a _ 0 + I \\end{align*}"}
+{"id": "3018.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\lambda = \\beta ( \\beta - 1 ) - \\alpha ^ 2 , \\ \\ \\ & \\mu = \\alpha , \\ \\ \\ & \\nu = \\alpha ( \\alpha - 1 ) - ( \\beta - 1 ) ( \\beta - 2 ) . \\\\ \\end{array} \\end{align*}"}
+{"id": "6908.png", "formula": "\\begin{align*} v : = u ^ { - \\frac { p } { n - p } } . \\end{align*}"}
+{"id": "3550.png", "formula": "\\begin{align*} ( T f - f \\circ \\tau ) \\tau ^ 4 = 0 . \\end{align*}"}
+{"id": "1445.png", "formula": "\\begin{align*} \\mathcal { M } : = \\mathcal { O } _ { \\widetilde { X } } \\cap j _ * \\mathcal { O } ^ * _ { \\widetilde { X } \\setminus E } \\end{align*}"}
+{"id": "9197.png", "formula": "\\begin{align*} a _ { i , n + 1 } ( z / w ) = \\begin{cases} q ^ { - 1 } ( z / w - \\xi ) ( z / w - 1 ) + ( \\xi - 1 ) ( q ^ { - 2 } - 1 ) z / w & \\ \\ i = n + 1 \\\\ ( q ^ { - 2 } - 1 ) q ^ { \\bar { i } - \\overline { n + 1 } } \\xi \\cdot ( z / w - 1 ) & \\ \\ i < n + 1 \\\\ ( q ^ { - 2 } - 1 ) q ^ { \\bar { i } - \\overline { n + 1 } } \\cdot ( z / w - 1 ) z / w & \\ \\ i > n + 1 \\end{cases} . \\end{align*}"}
+{"id": "5692.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } t ^ { 1 / ( p - 2 ) } \\Vert u ( t ) \\Vert _ { L ^ 2 } = \\beta \\in ( 0 , \\infty ) . \\end{align*}"}
+{"id": "2657.png", "formula": "\\begin{align*} \\big \\| \\| \\textbf { 1 } _ { [ - s _ k , - s _ { k - 1 } ] } ( P _ { \\leq M } u ) \\| _ { V ^ p _ { \\Delta _ x } } \\big \\| _ { l ^ { p } } = \\big \\{ \\sum _ { k = 1 } ^ { K } \\| \\mathbf { 1 } _ { [ - s _ k , - s _ { k - 1 } ) } P _ { \\leq M } u \\| ^ { p } _ { V ^ p _ { \\Delta _ x } } \\big \\} ^ { \\frac { 1 } { p } } \\leq \\| P _ { \\leq M } u \\| _ { V ^ p _ { \\Delta _ x } } . \\end{align*}"}
+{"id": "6895.png", "formula": "\\begin{align*} & \\begin{cases} \\eta B ( \\cosh 1 - 1 ) \\leq ( A - B ) \\log ( 1 + B ) - 1 < B \\leq B _ { 0 } , \\\\ \\eta B ( \\cos 1 - 1 ) \\geq ( A - B ) \\log ( 1 - B ) B _ { 0 } < B < 0 , \\\\ \\eta B ( \\cos 1 - 1 ) \\leq ( A - B ) \\log ( 1 - B ) 0 < B < 1 , \\end{cases} \\end{align*}"}
+{"id": "56.png", "formula": "\\begin{align*} \\prescript { } { i _ 0 } { ( \\hat { s } ^ * } ) ( i ) = & \\begin{cases} \\hat { s } ^ * ( i ) & i < i _ 0 \\\\ \\hat { s } ^ * ( i + 1 ) & i _ 0 \\leq i \\end{cases} \\\\ = & \\begin{cases} \\hat { s } ( j - i + 1 ) & i < i _ 0 \\\\ \\hat { s } ( j - i ) & i _ 0 \\leq i \\end{cases} \\end{align*}"}
+{"id": "852.png", "formula": "\\begin{align*} - \\dot \\delta { W } = X ^ { i } \\dot \\delta _ { i } h - n F ^ { - 1 } g ( X , l ) h . \\end{align*}"}
+{"id": "5040.png", "formula": "\\begin{align*} n ^ { 2 } + \\sum _ { k = 1 } ^ { n - 1 } \\left \\{ 3 ( n - k ) + ( n - k ) ^ { 2 } \\right \\} + O ( n ) = ( 1 / 3 ) n ^ { 3 } + 2 n ^ { 2 } + O ( n ) . \\end{align*}"}
+{"id": "5759.png", "formula": "\\begin{align*} & \\frac { d } { d t } \\xi ^ { ( k , \\ell ) } _ { i , 1 } - 2 ^ { - 1 } m \\xi ^ { ( k , \\ell ) } _ { i , 1 } + \\beta _ i \\xi ^ { ( k , \\ell ) } _ { i , 2 } = \\mathcal { E } ^ { ( k , \\ell ) } _ { i , 1 } , \\\\ & \\frac { d } { d t } \\xi ^ { ( k , \\ell ) } _ { i , 2 } - 2 ^ { - 1 } m \\xi ^ { ( k , \\ell ) } _ { i , 2 } - \\beta _ i \\xi ^ { ( k , \\ell ) } _ { i , 1 } = \\mathcal { E } ^ { ( k , \\ell ) } _ { i , 2 } , \\end{align*}"}
+{"id": "8768.png", "formula": "\\begin{align*} \\mathcal { M } : = \\left \\{ A \\in \\mathcal { B ( \\R ) } \\mid \\int _ { A } \\pi ^ \\uparrow _ y \\mu ( d y ) = \\int _ { A } \\hat \\pi ^ \\uparrow _ y \\mu ( d y ) \\right \\} \\end{align*}"}
+{"id": "8862.png", "formula": "\\begin{align*} A _ 4 & = \\| | x | ^ { - \\tau } | u | ^ { p - 1 } ( I _ \\alpha \\ast | \\cdot | ^ { - \\tau } | u | ^ { p - 1 } \\nabla u ) \\| _ { L _ x ^ { \\frac { 2 n } { n + 2 } } } \\\\ & \\lesssim \\| | x | ^ { - \\tau } | u | ^ { p - 1 } \\| _ { L _ x ^ { a _ 2 } } \\| | x | ^ { - \\tau } | u | ^ { p - 1 } \\nabla u \\| _ { L _ x ^ { b _ 2 } } \\\\ & \\leq \\| | x | ^ { - \\frac { \\tau } { p - 1 } } | u | \\| ^ { p - 1 } _ { L _ x ^ { ( p - 1 ) a _ 2 } } \\| | x | ^ { - \\frac { \\tau } { p - 1 } } u \\| ^ { p - 1 } _ { L _ x ^ { ( p - 1 ) b _ 4 } } \\| \\nabla u \\| _ { L _ x ^ r } \\\\ & \\lesssim \\| \\nabla u \\| _ { L _ x ^ r } ^ { 2 p - 1 } \\end{align*}"}
+{"id": "550.png", "formula": "\\begin{align*} j _ { \\nu , k } = \\left ( k + \\frac { \\nu } { 2 } - \\frac { 1 } { 4 } \\right ) \\pi - \\frac { 4 \\nu ^ { 2 } - 1 } { 8 \\left ( k + \\frac { \\nu } { 2 } - \\frac { 1 } { 4 } \\right ) \\pi } + O \\left ( \\frac { 1 } { k ^ { 3 } } \\right ) , k \\rightarrow \\infty \\end{align*}"}
+{"id": "8553.png", "formula": "\\begin{align*} f ( n ) = \\frac { - G ( n , 0 ) } { p _ { 2 } ( n ) } - \\frac { p _ { 1 } ( n ) } { p _ { 2 } ( n ) } f ( n + r ) . \\end{align*}"}
+{"id": "3253.png", "formula": "\\begin{align*} & \\partial _ a \\partial _ \\nu S _ { a , [ \\nu ] } ^ \\times ( x , y ) = \\partial _ a \\partial _ \\nu \\big ( \\Theta ( \\xi ^ 2 - \\nu ) \\ : H _ { a , [ \\nu ] } ( x , y ) \\big ) \\\\ & = \\Theta ( \\xi ^ 2 - \\nu ) \\ : \\partial _ a \\partial _ \\nu H _ { a , [ \\nu ] } ( x , y ) + \\big ( \\partial _ \\nu \\Theta ( \\xi ^ 2 - \\nu ) \\big ) \\ : \\partial _ a H _ { a , [ \\nu ] } ( x , y ) \\ : . \\end{align*}"}
+{"id": "4266.png", "formula": "\\begin{align*} \\hat K _ \\varphi ( \\xi ) & = \\int _ 0 ^ \\infty ( 1 + t ^ 2 ) ^ { - 1 } \\int _ \\R \\exp \\bigl \\{ - i x \\xi - \\tfrac { x ^ 2 } { 1 + t ^ 2 } \\bigr \\} \\ , d x \\ , d t \\\\ & = \\sqrt { \\pi } \\int _ 0 ^ \\infty ( 1 + t ^ 2 ) ^ { - \\frac 1 2 } \\exp \\bigl \\{ - \\tfrac { \\xi ^ 2 ( 1 + t ^ 2 ) } { 4 } \\bigr \\} \\ , d t . \\end{align*}"}
+{"id": "5540.png", "formula": "\\begin{align*} \\rho = \\norm { Q } , Q = \\sqrt { m n } ( M \\circ M ) . \\end{align*}"}
+{"id": "3503.png", "formula": "\\begin{align*} \\bar { h } \\Psi ( a ) & = \\Psi ( a ) \\bar { h } , \\\\ \\| \\bar { h } \\Psi ( a ) \\| & = \\| \\Psi ( a ) \\| = \\| a \\| , \\ \\\\ \\bar { h } \\Psi ( a b ) & = \\Psi ( a ) \\Psi ( b ) . \\end{align*}"}
+{"id": "789.png", "formula": "\\begin{align*} A \\bar A = \\tau \\beta ( w , w ) ^ { 2 } X ( t _ w ) ^ { - 2 } . \\end{align*}"}
+{"id": "2453.png", "formula": "\\begin{align*} \\begin{cases} V ( \\xi ) \\sim A _ { 1 } ( \\xi - \\xi _ { - } ) ^ { \\frac { 1 } { m } } , \\\\ V ' ( \\xi ) \\sim A _ { 2 } ( \\xi - \\xi _ { - } ) ^ { - \\frac { m - 1 } { m } } \\end{cases} { \\rm { a s } } \\xi \\searrow \\xi _ { - } + 0 , \\end{align*}"}
+{"id": "2700.png", "formula": "\\begin{align*} \\rho ^ { i } : = & d q ^ { i } - v ^ { i } d t = 0 , \\\\ \\theta _ { i } : = & K ^ { ( 1 ) } _ { i j } d v ^ { j } - S _ { i } d t \\end{align*}"}
+{"id": "7521.png", "formula": "\\begin{align*} | R m ( g _ N ) | _ { g _ N } ( q _ { \\max } ) = \\max _ { N } | R m ( g _ N ) | _ { g _ N } . \\end{align*}"}
+{"id": "3825.png", "formula": "\\begin{align*} \\chi ( q ) : = \\sum _ { n = 0 } ^ \\infty \\frac { ( - q ; q ) _ n q ^ { n ^ 2 } } { \\left ( - q ^ 3 ; q ^ 3 \\right ) _ n } . \\end{align*}"}
+{"id": "4818.png", "formula": "\\begin{align*} \\max _ { \\hat { \\mathbf { c } } ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } } \\in \\mathcal { S } ^ { ( b f ) } _ k } \\hat { \\mathbf { c } } ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } \\top } \\mathbf { x } = \\min \\Big \\{ S ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } } + \\sum _ { a \\in \\mathcal { A } } \\max \\{ 0 , x _ a - x ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } } _ a \\} ; h \\Big \\} \\end{align*}"}
+{"id": "71.png", "formula": "\\begin{align*} \\| N \\| _ { L ^ \\infty ( \\Omega _ T ) } : = \\sup \\{ | N ( x , t ) | : ( x , t ) \\in \\Omega _ T \\} \\end{align*}"}
+{"id": "1794.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t a + \\partial _ x q = 0 \\\\ \\partial _ t q + \\partial _ x \\left ( \\dfrac { q ^ 2 } { a } + \\dfrac 1 2 \\ , g \\ , \\dfrac { a ^ 2 } \\sigma \\right ) = \\dfrac 1 2 \\ , g \\ , \\dfrac { a ^ 2 } { \\sigma ^ 2 } \\ , \\partial _ x \\sigma - g \\ , a \\ , \\partial _ x b \\ , , \\end{array} \\right . \\end{align*}"}
+{"id": "2216.png", "formula": "\\begin{align*} - r \\partial _ r ( r \\partial _ r v ) - 2 r \\partial _ r v + 3 v = g ( r , z ) . \\end{align*}"}
+{"id": "2179.png", "formula": "\\begin{align*} | 8 A - 7 A | \\leq | 8 A - 8 A | \\leq \\frac { | 8 A + 8 A | ^ 2 } { | 8 A | } = \\frac { | 1 6 A | ^ 2 } { | 8 A | } \\end{align*}"}
+{"id": "2486.png", "formula": "\\begin{align*} \\widetilde { \\tau } = \\left ( 4 \\widehat { H } ^ 2 \\left [ 1 + \\lambda R + \\left ( R \\| \\widehat { A } \\| + \\sqrt { \\widehat { Z } } \\right ) \\left ( 4 R \\| \\widehat { A } \\| + \\sqrt { \\widehat { Z } } \\right ) \\right ] \\right ) ^ { - 1 } , \\end{align*}"}
+{"id": "5130.png", "formula": "\\begin{align*} L _ { d } D ( p \\| q ) = \\left [ \\log _ { d } A \\left ( p , q \\right ) - \\log _ { d } B \\left ( p , q \\right ) \\right ] \\end{align*}"}
+{"id": "8408.png", "formula": "\\begin{align*} \\widetilde { B _ 1 } \\left ( \\sum \\alpha _ i g _ i \\right ) = \\sum \\alpha _ i B _ 1 ( g _ i ) , \\\\ \\widetilde { B _ 2 } \\left ( \\sum \\alpha _ i g _ i \\right ) = \\sum \\alpha _ i B _ 2 ( g _ i ) ^ { - 1 } \\end{align*}"}
+{"id": "2507.png", "formula": "\\begin{align*} \\langle u ( t ) - u _ \\infty , \\vartheta \\rangle _ { ( H ^ 1 ) ' , H ^ 1 } & = \\langle u ( t ) - u ^ { i n } + u ^ { i n } - u _ \\infty , \\vartheta \\rangle _ { ( H ^ 1 ) ' , H ^ 1 } \\\\ & = \\langle \\Delta ( A ( t ) - A _ \\infty ) , \\vartheta \\rangle _ { ( H ^ 1 ) ' , H ^ 1 } \\\\ & = - \\int _ \\Omega \\nabla ( A ( t ) - A _ \\infty ) \\cdot \\nabla \\vartheta \\ \\mathrm { d } x \\ , , \\end{align*}"}
+{"id": "2789.png", "formula": "\\begin{align*} \\delta \\left ( { \\tilde { \\sigma } _ { 3 } ^ { * } } ( t ) I \\right ) = \\int ^ { t _ { 2 } } _ { t _ { 1 } } \\frac { \\partial L _ { T } } { \\partial \\Xi ^ { \\alpha } } \\delta \\Xi ^ { \\alpha } d t + \\frac { \\partial L _ { T } } { \\partial \\zeta ^ { \\alpha } } \\delta \\zeta ^ { \\alpha } d t + \\left [ \\Psi _ { \\alpha } \\delta \\Xi ^ { \\alpha } \\right ] ^ { t _ { 2 } } _ { t _ { 1 } } . \\end{align*}"}
+{"id": "8352.png", "formula": "\\begin{align*} \\rho _ { D K } ( \\theta ) \\le \\frac { F ( \\mu ( K ) ) } { F ' ( \\mu ( K ) } h _ { \\Pi _ \\mu K } ( \\theta ) ^ { - 1 } = \\frac { F ( \\mu ( K ) ) } { F ' ( \\mu ( K ) } \\rho _ { \\Pi _ \\mu ^ \\circ K } ( \\theta ) \\end{align*}"}
+{"id": "8968.png", "formula": "\\begin{align*} \\begin{aligned} g ( p ( x ) , | \\nabla u ( x ) - R | ) & \\le g \\big ( p ( x ) , | \\nabla u ( x ) - \\nabla v ( x ) | + | \\nabla v ( x ) - R | \\big ) \\\\ & \\le 2 \\left ( | \\nabla u ( x ) - \\nabla v ( x ) | ^ { p ( x ) } + | \\nabla v ( x ) - R | ^ 2 \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "6665.png", "formula": "\\begin{align*} L ( E , \\mathrm { i } y ) = \\int _ { \\mathbb { C } } \\log | \\tilde { E } - E | \\mathrm { d } \\mathcal { N } ^ { \\mathrm { i } y } ( \\tilde { E } ) . \\end{align*}"}
+{"id": "7285.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } C _ m ( \\beta , 2 n ) y ^ { 2 n } = \\frac { m y } { \\sqrt { 1 - y ^ 2 } } \\frac { \\sin ( 2 m \\arcsin ( y ) ) } { \\cos ( 2 m \\arcsin ( y ) ) - \\cos 2 m \\beta } . \\end{align*}"}
+{"id": "5831.png", "formula": "\\begin{align*} \\frac { d } { d s } \\mathcal { F } _ \\Sigma ( u + s \\xi ) \\big | _ { s = 0 } = - \\int \\left \\langle \\vec { H } + \\frac { X ^ \\perp ( u ) } { 2 } , \\xi ^ A \\frac { \\partial } { \\partial y ^ A } \\right \\rangle e ^ { - | X ( u ) | ^ 2 / 4 } \\sqrt { \\det g _ u } \\ , d x ^ 1 \\wedge \\dots \\wedge d x ^ n , \\end{align*}"}
+{"id": "2056.png", "formula": "\\begin{align*} d = O ( T ^ { \\frac { 1 } { 2 } } ) , \\sigma = O ( T ^ { \\frac { 1 } { 2 } } ) \\ , \\qquad \\gamma d ^ { - 1 } \\to 0 . \\end{align*}"}
+{"id": "6238.png", "formula": "\\begin{align*} \\begin{cases} 2 p + q \\le 0 , \\\\ - p + q \\le 0 . \\end{cases} \\end{align*}"}
+{"id": "385.png", "formula": "\\begin{align*} x ^ p + y ^ p - z ^ p = R ^ p + \\sum _ { i = 1 } ^ { p - 1 } \\binom { p } { i } \\bigg ( ( z - y ) ^ { p - i } R ^ i + ( - z ) ^ { p - i } y ^ i \\bigg ) , \\end{align*}"}
+{"id": "7974.png", "formula": "\\begin{align*} \\begin{aligned} \\begin{pmatrix} f _ { b } \\\\ e _ { b } \\end{pmatrix} = \\begin{pmatrix} ( - 1 ) ^ { n } \\mathrm { t r } _ { \\Gamma } ( \\cdot ) & 0 \\\\ 0 & E ( \\cdot ) | _ { \\Gamma } \\end{pmatrix} \\begin{pmatrix} \\frac { \\delta H } { \\delta v } \\\\ \\frac { \\delta H } { \\delta \\Sigma } \\end{pmatrix} . \\end{aligned} \\end{align*}"}
+{"id": "5793.png", "formula": "\\begin{align*} u ' - \\mathcal { L } _ { \\Sigma } u = E _ 2 ( u ) . \\end{align*}"}
+{"id": "1950.png", "formula": "\\begin{align*} j _ 0 : = \\max \\Bigg \\{ \\frac { 2 } { s q } \\log _ 4 \\left ( \\frac { 3 2 ^ { q + 1 } N ( 1 + \\left | B _ 1 \\right | ) } { s \\delta ^ { q - 1 } } \\right ) , \\frac { 2 \\gamma ( p - 1 ) } { p \\gamma - N } \\log _ 4 \\left ( \\frac { 4 } { \\delta | B _ 1 | ^ { \\frac { 1 } { \\gamma ( p - 1 ) } } } \\right ) \\Bigg \\} . \\end{align*}"}
+{"id": "2287.png", "formula": "\\begin{align*} \\Gamma = 8 ( k _ 1 - k _ 2 + k _ 3 - k _ 4 ) ( k _ 1 k _ 3 - k _ 2 k _ 4 ) - 3 2 ( k _ 1 k _ 3 + k _ 2 k _ 4 ) \\ . \\end{align*}"}
+{"id": "4919.png", "formula": "\\begin{align*} \\begin{gathered} \\zeta _ { e } : C ^ { \\infty } ( X ) \\longrightarrow \\R \\\\ \\zeta _ { e } ( H ) : = \\lim _ { k \\to + \\infty } \\frac { c ( H ^ { \\# k } , e ) } { k } . \\end{gathered} \\end{align*}"}
+{"id": "6610.png", "formula": "\\begin{align*} E _ 2 \\ll _ { \\delta } \\begin{cases} x \\exp \\left ( - 4 \\exp \\left ( \\kappa ( c _ k \\alpha x ^ \\delta ) ^ { \\frac { 1 } { 4 } } \\right ) ^ { - 1 } \\right ) & \\ k \\ , \\\\ x \\exp \\left ( - 2 \\exp \\left ( \\kappa ( \\alpha x ^ \\delta ) ^ { \\frac { 1 } { 2 } } \\right ) ^ { - 1 } \\right ) & \\ k \\ . \\end{cases} \\end{align*}"}
+{"id": "1224.png", "formula": "\\begin{align*} \\left \\langle ( - \\Delta _ p ) ^ s u , \\varphi \\right \\rangle + \\left \\langle ( - \\Delta _ p ) ^ s v , \\psi \\right \\rangle = \\lambda & \\left [ \\alpha ( p ) \\vert u ( x _ 1 ) \\vert ^ { \\alpha ( p ) - 2 } u ( x _ 1 ) \\vert v ( x _ 2 ) \\vert ^ { \\beta ( p ) } \\varphi ( x _ 1 ) \\right . \\\\ & \\left . \\ + \\beta ( p ) \\vert u ( x _ 1 ) \\vert ^ { \\alpha ( p ) } \\vert v ( x _ 2 ) \\vert ^ { \\beta ( p ) - 2 } v ( x _ 2 ) \\psi ( x _ 2 ) \\right ] \\end{align*}"}
+{"id": "4647.png", "formula": "\\begin{align*} \\lambda _ \\mathcal { G } ( n ) = \\begin{cases} 1 , \\quad n = p _ 1 ^ { \\alpha _ 1 } \\dots p _ k ^ { \\alpha _ k } , p _ i \\in \\mathcal { G } \\\\ 0 , \\quad \\end{cases} , \\end{align*}"}
+{"id": "5126.png", "formula": "\\begin{align*} D ( p \\| q ) = \\left [ A \\left ( p , q \\right ) - B \\left ( p , q \\right ) \\right ] \\ ; \\ ; ; \\ ; \\ ; A \\left ( p , q \\right ) > 0 \\ ; \\ ; ; \\ ; \\ ; B \\left ( p , q \\right ) > 0 \\end{align*}"}
+{"id": "8818.png", "formula": "\\begin{align*} \\widetilde { \\varphi } ( t ) = \\min ( \\varphi ( t ) , 1 ) = \\begin{cases} \\varphi ( t ) , & 0 < t \\leq 1 , \\\\ 1 , & t \\geq 1 . \\end{cases} \\end{align*}"}
+{"id": "1395.png", "formula": "\\begin{align*} \\lim _ { Z \\to \\infty } \\mathcal { S } ( Z ) = 0 , \\end{align*}"}
+{"id": "8739.png", "formula": "\\begin{align*} x _ + - y _ + & = z _ - - y _ + + \\alpha _ \\rho ( y _ - - z _ - ) = z _ - - y _ + - \\alpha _ \\rho ( z _ + + z _ - - y _ + - y _ - ) + \\alpha _ \\rho ( z _ + - y _ + ) \\\\ & > z _ - - y _ + - \\beta ( z _ + + z _ - - y _ + - y _ - ) + \\alpha _ \\rho ( z _ + - y _ + ) = \\alpha _ \\rho ( z _ + - y _ + ) . \\end{align*}"}
+{"id": "1241.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ; q ^ 2 ) _ k } { ( q ; q ) _ k } q ^ k \\equiv ( - 1 ) ^ { \\frac { n - 1 } { 2 } } q ^ { \\frac { n ^ 2 - 1 } { 4 } } \\pmod { \\Phi _ { n } ( q ) ^ 2 } , \\end{align*}"}
+{"id": "1041.png", "formula": "\\begin{align*} ( T _ { \\infty } ( w ) ^ { - 1 } ) ^ { s , t } = ( ( T _ { \\infty } ( w ) ^ { - 1 } ) ^ { t , s } ) ^ * , s , t \\in \\{ 1 , \\dots , n \\} . \\end{align*}"}
+{"id": "1832.png", "formula": "\\begin{align*} \\pi ^ { - 1 } ( x ) = T _ { \\tilde { x } } \\tilde { U } / G _ { \\tilde { x } } , \\end{align*}"}
+{"id": "7568.png", "formula": "\\begin{align*} m _ { p , q } ( c ) & = \\frac { c - \\alpha } { c } m _ { p , q } ( c ) + \\frac { \\alpha } { c } _ { p , q } m _ { p , q } ( c ) \\\\ & = \\frac { c - \\alpha } { c } m _ { p , q } ( \\frac { c } { c - \\alpha } ( c - \\alpha ) ) + \\frac { \\alpha } { c } m _ { p , q } ( \\frac { c } { \\alpha } \\alpha ) \\\\ & \\leq m _ { p , q } ( c - \\alpha ) + m _ { p , q } ( \\alpha ) , \\\\ \\end{align*}"}
+{"id": "5590.png", "formula": "\\begin{align*} B ^ { ( k ) } _ { e f } = \\sum _ { \\gamma \\in F _ { e f } ^ { 2 k + 2 } } X _ { e _ 1 e _ 2 } X _ { e _ 3 e _ 2 } \\prod _ { s = 1 } ^ { k } A _ { \\gamma _ { 2 s } \\gamma _ { 2 s + 2 } , \\gamma _ { 2 s + 1 } } . \\end{align*}"}
+{"id": "7573.png", "formula": "\\begin{align*} \\Delta ^ { 2 } z ^ { 1 } + \\omega z ^ { 1 } - \\mu _ { n } | z ^ { 1 } | ^ { q - 2 } z ^ { 1 } - | z ^ { 1 } | ^ { p - 2 } z ^ { 1 } = 0 . \\end{align*}"}
+{"id": "1284.png", "formula": "\\begin{align*} \\bar { \\sigma } ( t ) = \\frac { D _ r } { \\alpha } ( 1 - e ^ { - 2 \\alpha t } ) = \\frac { 1 } { \\omega ^ 2 \\beta } ( 1 - e ^ { - 2 \\lambda _ 2 t } ) . \\end{align*}"}
+{"id": "7138.png", "formula": "\\begin{align*} { \\boldsymbol { k } } \\cdot { \\boldsymbol { \\widehat { A } } } ( \\boldsymbol { k } ) = 0 \\ , \\ , , \\end{align*}"}
+{"id": "8607.png", "formula": "\\begin{align*} m ( t ) = \\inf _ { x \\in \\mathbb { R } } v _ 1 ( t ; x ) = \\inf _ { x \\in \\mathbb { R } } ( \\partial _ x u ) ( X ( t ; x ) , t ) = : m ( 0 ) q ^ { - 1 } ( t ) . \\end{align*}"}
+{"id": "5907.png", "formula": "\\begin{align*} | e ( \\Gamma _ { G } \\mathcal { C } ( G ) ) | = \\binom { q - 1 } { 2 } + q \\cdot \\binom { p - 1 } { 2 } = \\dfrac { ( q - 1 ) ( q - 2 ) + q ( p - 1 ) ( p - 2 ) } { 2 } . \\end{align*}"}
+{"id": "2255.png", "formula": "\\begin{align*} \\ddot { r } ( t ) & + [ ( 2 n - 2 p - 1 ) \\cot t - ( 2 p + 1 ) \\tan t ] \\ , \\dot { r } ( t ) \\\\ & + \\left [ \\frac { p } { \\cos ^ 2 t } - \\frac { ( n - p - 1 ) } { \\sin ^ 2 t } \\right ] \\ , \\sin 2 r ( t ) - \\frac { \\sin 4 r ( t ) } { \\sin ^ 2 2 t } = 0 \\end{align*}"}
+{"id": "8629.png", "formula": "\\begin{align*} \\frac { d v _ 1 } { d t } ( t ; x ) = - v _ 1 ^ 2 ( t ; x ) - K _ 1 ( t ; x ) - \\phi _ 1 ( t ; x ) \\leq | K _ 1 ( t ; x ) + \\phi _ 1 ( t ; x ) | \\end{align*}"}
+{"id": "4485.png", "formula": "\\begin{align*} M _ H ( Z _ 0 , J , \\rho ) = \\| \\tilde F _ 0 \\| ^ 2 _ { \\partial M , \\rho } , \\end{align*}"}
+{"id": "4746.png", "formula": "\\begin{align*} S ( t ) x = \\lim _ { n \\to \\infty } \\left ( \\mathrm { I d } + \\frac { t } { n } A \\right ) ^ { - n } x \\end{align*}"}
+{"id": "8690.png", "formula": "\\begin{align*} \\langle \\omega _ 0 , \\xi _ M \\rangle T \\mathcal T ^ { k - 1 } ( \\chi f ) = - \\xi _ M \\mathcal T ^ { k - 1 } ( \\chi f ) + \\sum _ { j = 1 } ^ { n - 1 } a _ j L _ j \\mathcal T ^ { k - 1 } ( \\chi f ) + \\sum _ { j = 1 } ^ { n - 1 } b _ j \\overline L _ j \\mathcal T ^ { k - 1 } ( \\chi f ) + O ( | z | ) D \\mathcal T ^ { k - 1 } ( \\chi f ) . \\end{align*}"}
+{"id": "6585.png", "formula": "\\begin{align*} \\phi _ { f } ( x , y ) : = \\sum _ { \\substack { n \\leq x \\\\ f ( n ) \\leq y } } 1 = | \\{ n \\leq x ; \\ \\ f ( n ) \\leq y \\} | , \\end{align*}"}
+{"id": "4417.png", "formula": "\\begin{align*} \\inf _ { x \\in \\mathcal { X } } & \\ \\sum _ { i \\in [ m ] } \\max \\{ 0 , x _ i - \\overline { b } _ i , x _ i - \\overline { b } _ i + \\Delta b _ i - x _ k + \\overline { b } _ k - \\Delta b _ k \\} + \\Gamma ( x _ k - \\overline { b } _ k + \\Delta b _ k ) \\\\ \\mathrm { s . t . } & \\ x _ k \\geq b _ k . \\end{align*}"}
+{"id": "7194.png", "formula": "\\begin{align*} \\frac { \\omega _ 1 } { 2 \\pi } \\int _ { t - \\pi / \\omega _ 1 } ^ { t + \\pi / \\omega _ 1 } \\left ( \\frac { A ( x ) ^ 2 } { 2 } + \\frac { B ( x ) ^ 2 } { 2 } + A ( x ) B ( x ) \\cos ( \\eta s ) \\right ) d s = \\frac { A ( x ) ^ 2 } { 2 } + \\frac { B ( x ) ^ 2 } { 2 } + A ( x ) B ( x ) \\cos ( \\eta t ) + O \\left ( \\frac { \\vert A ( x ) B ( x ) \\eta \\vert } { \\omega _ 1 } \\right ) , \\end{align*}"}
+{"id": "3318.png", "formula": "\\begin{align*} U _ \\delta & = \\{ x \\in X \\ , : \\ , \\} , \\\\ U ' _ \\delta & = U _ \\delta \\cap X ' \\end{align*}"}
+{"id": "3338.png", "formula": "\\begin{align*} \\partial _ t u ( v , x , t ) + v \\cdot \\nabla _ x u ( v , x , t ) + \\mathcal { L } _ { K } ( u ) = f ( v , x , t , u ) , \\mbox { i n } ~ \\R ^ { 2 n + 1 } . \\end{align*}"}
+{"id": "3116.png", "formula": "\\begin{align*} \\varphi ^ * \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & : = \\left ( \\begin{array} { c c c } a ^ { \\ell _ 1 } & a ^ { \\ell _ 1 } \\ , b ^ { p ^ { e _ 1 } } \\ , d ^ { p ^ { e _ 1 } } & 0 \\\\ 0 & d ^ { - \\ell _ 2 } & 0 \\\\ 0 & 0 & d ^ { - \\ell _ 2 } \\end{array} \\right ) . \\end{align*}"}
+{"id": "1156.png", "formula": "\\begin{align*} a ( x , D ) ^ { \\# } \\left ( x ^ \\beta \\right ) = 0 \\in \\mathcal { S } _ \\infty ' , \\end{align*}"}
+{"id": "6099.png", "formula": "\\begin{align*} \\Delta ( x _ u ) = 1 \\otimes x _ u + \\sum a _ u \\otimes b _ u , \\Delta ( x _ v ) = 1 \\otimes x _ v + \\sum a _ v \\otimes b _ v . \\end{align*}"}
+{"id": "5478.png", "formula": "\\begin{align*} K = n + \\sum _ { l = 1 } ^ \\infty \\frac { 1 } { a _ l } \\end{align*}"}
+{"id": "4668.png", "formula": "\\begin{align*} \\P _ B ( I ) = \\prod _ { i \\in I } p _ i \\prod _ { j \\notin I } ( 1 - p _ j ) . \\end{align*}"}
+{"id": "1811.png", "formula": "\\begin{align*} \\tilde \\zeta ( x ) = \\left \\{ \\begin{array} { l l } \\zeta ( \\xi ) & x \\leq \\xi , \\\\ \\zeta ( \\xi + ) & x > \\xi . \\end{array} \\right . \\end{align*}"}
+{"id": "7239.png", "formula": "\\begin{align*} \\ast _ { \\sigma } = \\sigma \\circ \\ast \\circ ( \\sigma ^ { - 1 } \\times \\sigma ^ { - 1 } ) , \\end{align*}"}
+{"id": "6713.png", "formula": "\\begin{align*} E y ( x ) = E m ^ x = m ^ { x + 1 } = m m ^ x = m y ( x ) . \\end{align*}"}
+{"id": "2299.png", "formula": "\\begin{align*} \\bigl [ ( 1 - x ^ 2 ) ( 1 + x ^ { 2 } ) ^ { r } ( 1 + x ^ { - 2 } ) ^ { r } \\bigr ] _ 0 = \\bigl [ ( 1 - x ^ 2 ) ( x + x ^ { - 1 } ) ^ { 2 r } \\bigr ] _ 0 = c _ r \\ . \\end{align*}"}
+{"id": "4519.png", "formula": "\\begin{align*} \\mathbb { P } ^ { ( 2 ) } _ { \\ell } : = \\mathbb { P } \\bigg [ \\sup _ { \\substack { X _ { \\ell - 1 } < n \\leqslant X _ { \\ell } \\\\ 1 \\leqslant j \\leqslant J } } V ^ { ( 2 ) } ( n , y _ j ) > \\frac { T _ 1 ( \\ell ) } { \\ell ^ { K / 2 } } \\bigg ] \\end{align*}"}
+{"id": "3877.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum \\limits _ { k = 0 } ^ { n - 1 } R ^ { n , k } \\le \\left | \\log \\left ( a _ 1 ^ * \\delta _ 0 ^ A \\right ) \\right | \\frac { k _ 1 } { n } + | \\log \\delta _ 1 | \\frac { m ^ n _ 0 } { n } + \\left | \\log \\left ( \\frac { \\delta _ 1 } { k ^ * + 2 } \\right ) \\right | \\frac { k ^ * } { n } + | \\log \\delta _ 1 | + \\frac { \\left | \\log \\left ( \\frac { \\delta _ 1 } { 2 } \\right ) \\right | } { n } . \\end{align*}"}
+{"id": "7922.png", "formula": "\\begin{align*} \\mathfrak { B } ^ { k } : = d H \\Lambda ^ { k - 1 } ( \\Omega ) , \\quad \\mathring { \\mathfrak { B } } ^ { k } : = d \\mathring { H } \\Lambda ^ { k - 1 } ( \\Omega ) . \\end{align*}"}
+{"id": "7491.png", "formula": "\\begin{align*} 0 < H : = \\max \\bigg \\{ \\sup _ { D _ { \\lambda _ 0 } \\cap \\{ t = 0 \\} } | g - g _ 0 | _ { g _ 0 } , & \\sup _ { A _ { \\lambda _ 0 } } ( | g - g _ 0 | _ { g _ 0 } + \\mathbf { r } | \\nabla ^ { g _ 0 } g | _ { g _ 0 } ) \\bigg \\} \\leq \\delta _ 1 . \\end{align*}"}
+{"id": "5671.png", "formula": "\\begin{align*} \\Delta _ g = \\frac { \\partial ^ 2 } { \\partial r ^ 2 } + ( N - 1 ) \\frac { h ' ( r ) } { h ( r ) } \\frac { \\partial } { \\partial r } + \\frac { 1 } { h ( r ) ^ 2 } \\Delta _ { \\mathbb { S } ^ { N - 1 } } . \\end{align*}"}
+{"id": "6720.png", "formula": "\\begin{align*} u _ { t t } - c ^ 2 u _ { x x } = 0 . \\end{align*}"}
+{"id": "4269.png", "formula": "\\begin{align*} ( \\cdot , \\cdot ) _ f = \\int _ M < \\cdot , \\cdot > e ^ { - f } d V , \\end{align*}"}
+{"id": "4714.png", "formula": "\\begin{align*} \\mathbf { b } _ h = \\mathbf { b } _ { h + 1 } + \\mathbf { c } _ h \\ , . \\end{align*}"}
+{"id": "7554.png", "formula": "\\begin{align*} \\int _ \\Omega f _ n \\nabla \\phi _ n \\cdot \\bar u \\ d x = \\int _ \\Omega \\delta \\nabla \\bar u : \\nabla \\phi _ n \\otimes \\nabla \\phi _ n \\ d x - \\int _ \\Omega \\delta ( \\nabla \\cdot \\bar u ) \\tfrac { 1 } { 2 } | \\nabla \\phi _ n | ^ 2 \\ d x \\ . \\end{align*}"}
+{"id": "4005.png", "formula": "\\begin{align*} \\begin{cases} \\left ( 1 - \\gamma y \\right ) x & = \\ ; 0 \\\\ \\left ( 1 - \\left ( 1 - \\gamma \\right ) x \\right ) y & = \\ ; 0 . \\end{cases} \\end{align*}"}
+{"id": "4844.png", "formula": "\\begin{align*} | E \\setminus B _ a | & = \\int _ { E _ * } \\sum _ j \\int _ { r _ { - , j } ( \\sigma ) } ^ { r _ { + , j } ( \\sigma ) } r ^ { n - 1 } d r d \\sigma . \\end{align*}"}
+{"id": "1582.png", "formula": "\\begin{align*} 1 = \\frac { F ( D F ^ { - 1 } ( \\bar y ) ) } { F ( D F ^ { - 1 } ( \\bar x ) ) } \\le C \\ , \\frac { | D F ^ { - 1 } ( \\bar y ) | } { | D F ^ { - 1 } ( \\bar x ) | } \\ , \\eta _ H \\left ( \\frac { | D F ^ { - 1 } ( \\bar y ) | } { | D F ^ { - 1 } ( \\bar x ) | } \\right ) = C \\ , \\eta _ { 1 + H , 1 + 1 / H } \\left ( \\frac { | D F ^ { - 1 } ( \\bar y ) | } { | D F ^ { - 1 } ( \\bar x ) | } \\right ) \\end{align*}"}
+{"id": "7627.png", "formula": "\\begin{align*} f _ 1 ( z ) = ( z - x _ 1 ) ( z - x _ 2 ) \\cdots ( z - x _ l ) g ( z ) ( x _ 1 \\leq x _ 2 \\leq \\cdots \\leq x _ l ) \\end{align*}"}
+{"id": "4264.png", "formula": "\\begin{align*} K _ \\varphi ( x ) = \\int _ 0 ^ \\infty | e ^ { i t \\Delta } \\varphi ( x ) | ^ 4 \\ , d t \\end{align*}"}
+{"id": "5895.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { N C } ( G ) ) & = ( p ^ { 2 } n - n ) ( p ^ { 2 } n - n - 1 ) ^ { 2 } - 4 ( p ^ { 2 } n - n - 1 ) \\dfrac { ( p ^ { 2 } n - n ) ( p n - n - 1 ) } { 2 } \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ + ( p ^ { 2 } n - n ) ( p n - n - 1 ) ^ { 2 } \\\\ & = ( p ^ { 2 } n - n ) ( p ^ { 4 } n ^ { 2 } - 2 p ^ { 3 } n ^ { 2 } + p ^ { 2 } n ^ { 2 } ) \\\\ & = ( p + 1 ) ( p n - n ) ( p ^ { 4 } n ^ { 2 } - 2 p ^ { 3 } n ^ { 2 } + p ^ { 2 } n ^ { 2 } ) \\end{align*}"}
+{"id": "4748.png", "formula": "\\begin{align*} S ( \\vert t \\vert ) \\overline { S ( \\vert s \\vert ) C ( x ) } = _ X \\overline { S ( \\vert t \\vert ) \\overline { S ( \\vert s \\vert ) C ( x ) } } . \\end{align*}"}
+{"id": "8655.png", "formula": "\\begin{align*} \\partial F ( \\varphi ) = \\{ \\rho \\in \\mathcal { P } ( \\Omega ) \\mid \\int _ \\Omega \\varphi ^ { c _ 2 } d \\mu + \\int _ \\Omega \\varphi d \\rho = \\frac { 1 } { 2 } W _ 2 ^ 2 ( \\rho , \\mu ) \\} . \\end{align*}"}
+{"id": "8570.png", "formula": "\\begin{align*} & \\frac { \\sqrt { 2 } } { 2 } = \\sum _ { n = 0 } ^ \\infty \\frac { 3 + 1 8 4 n - 3 3 6 n ^ 2 } { ( 4 n - 1 ) ( 4 n - 3 ) } \\frac { \\binom { 4 n } { 2 n } } { 2 ^ { 1 0 n } } , \\\\ & \\frac { \\pi ^ 2 } { 2 } = \\sum _ { n = 1 } ^ \\infty \\frac { ( 1 4 n - 3 ) ( 3 n - 1 ) } { n ^ 3 ( 2 n - 1 ) } \\frac { 1 6 ^ n } { \\binom { 4 n } { 2 n } ^ 2 \\binom { 2 n } { n } } . \\end{align*}"}
+{"id": "4016.png", "formula": "\\begin{align*} \\begin{cases} x _ { 1 } ^ { \\left ( t + 1 \\right ) } & = \\ ; \\delta _ { 1 } x _ { 2 } ^ { \\left ( t \\right ) } y ^ { \\left ( t \\right ) } \\medskip \\\\ x _ { 2 } ^ { \\left ( t + 1 \\right ) } & = \\ ; \\gamma _ { 2 } x _ { 1 } ^ { \\left ( t \\right ) } y ^ { \\left ( t \\right ) } \\medskip \\\\ y ^ { \\left ( t + 1 \\right ) } & = \\ ; \\bigl ( \\left ( 1 - \\gamma _ { 2 } \\right ) x _ { 1 } ^ { \\left ( t \\right ) } + \\left ( 1 - \\delta _ { 1 } \\right ) x _ { 2 } ^ { \\left ( t \\right ) } \\bigr ) y ^ { \\left ( t \\right ) } . \\end{cases} \\end{align*}"}
+{"id": "2926.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } \\frac { \\int _ { - 2 } ^ 2 F ( x ) ( x + \\sigma ) ^ L \\sqrt { 4 - x ^ 2 } d x } { \\int _ { - 2 } ^ 2 ( x + \\sigma ) ^ L \\sqrt { 4 - x ^ 2 } d x } = F ( 2 ) . \\end{align*}"}
+{"id": "7358.png", "formula": "\\begin{align*} \\varphi _ \\alpha ^ * \\mathrm { v o l } ^ G = \\mathrm { v o l } ^ G . \\end{align*}"}
+{"id": "4817.png", "formula": "\\begin{align*} & \\min _ { \\mathbf { y } \\in Y } \\Big \\{ \\Big ( \\frac { 1 } { K } \\sum _ { k = 1 } ^ K \\max _ { \\hat { \\mathbf { c } } ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } } \\in \\mathcal { S } ^ { ( b f ) } _ k } \\hat { \\mathbf { c } } ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } \\top } \\mathbf { x } + \\varepsilon _ K \\Big ) y _ 1 + h y _ 2 \\Big \\} . \\end{align*}"}
+{"id": "2847.png", "formula": "\\begin{align*} \\delta ( \\sigma ^ { * } _ { 2 } ( t ) I ) = \\int ^ { t _ { 2 } } _ { t _ { 1 } } \\left [ - \\dot { P } - Q \\right ] \\delta Q d t + \\left [ \\dot { Q } - P \\right ] \\delta P + \\left [ P \\delta Q \\right ] ^ { t _ { 2 } } _ { t _ { 1 } } . \\end{align*}"}
+{"id": "967.png", "formula": "\\begin{align*} P _ { U _ n } ( w ) & = P _ { U _ n } ( P _ D ( g ) ) + P _ { U _ n } ( R ^ D f ( \\cdot , w ) + R ^ D \\mu ) \\\\ & = P _ { D } ( g ) + ( R ^ D f ( \\cdot , w ) - \\Pi _ { U _ n } ( R ^ D f ( \\cdot , w ) ) ) + ( R ^ D \\mu - \\Pi _ { U _ n } ( R ^ D \\mu ) ) \\mbox { q . e . } , \\end{align*}"}
+{"id": "1614.png", "formula": "\\begin{align*} f ( e _ r \\otimes e _ s ) = & \\sum _ k t ( s , k ) e _ { r + ( 2 p k - s ) ( p - 1 ) } \\otimes e ^ p _ { s - 2 k ( p - 1 ) } \\\\ & - \\d ( r ) \\d ( s - 1 ) \\sum _ k t ( s - 1 , k ) e _ { r + p + ( 2 p k - s ) ( p - 1 ) } \\otimes e ^ p _ { s - 2 k ( p - 1 ) - 1 } , \\end{align*}"}
+{"id": "7366.png", "formula": "\\begin{align*} \\overline { L ' } ( v ) ( x , t ) : = \\frac { 1 } { 2 } \\int _ 0 ^ t \\{ v ( x + t - s , s ) - v ( x - t + s , s ) \\} d s . \\end{align*}"}
+{"id": "6406.png", "formula": "\\begin{align*} \\hat { \\theta } _ { 1 , n } = \\hat { \\theta } _ { 0 , n } - J _ n ( \\hat { \\theta } _ { 0 , n } ) ^ { - 1 } G _ n ( \\hat { \\theta } _ { 0 , n } ) , \\end{align*}"}
+{"id": "4553.png", "formula": "\\begin{align*} s _ g R i c _ { g , 0 } + 2 \\mathrm { H e s s } _ 0 ( s _ g ) = 0 , \\end{align*}"}
+{"id": "8337.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Pi _ t = [ \\Pi _ t , Y _ t ] . \\end{align*}"}
+{"id": "4532.png", "formula": "\\begin{align*} E _ N ^ { ( K ) } ( t , [ s ] _ K ) = \\sum _ { i = K + 1 } ^ N \\int _ { s _ { K + 1 } = 0 } ^ \\infty \\ ! . . . \\ ! \\int _ { s _ { i } = 0 } ^ \\infty \\varphi _ i ( [ s ] _ i ) n _ N ^ { ( i ) } ( t , [ s ] _ i ) \\ , d [ s ] _ { K + 1 , i } , \\end{align*}"}
+{"id": "4310.png", "formula": "\\begin{align*} \\| u \\| _ { m M } = \\| I d _ { \\C } \\otimes u : \\C \\otimes _ { \\min } E \\to \\C \\otimes _ { \\max } B \\| . \\end{align*}"}
+{"id": "7813.png", "formula": "\\begin{align*} \\boldsymbol { \\rho } \\left ( i + \\frac { 1 } { 2 } \\right ) = \\boldsymbol { \\rho } ( i ) + \\tau ( i ) \\nabla _ { \\boldsymbol { \\rho } } \\mathcal { Q } \\left ( \\boldsymbol { \\rho } \\right ) , \\end{align*}"}
+{"id": "5482.png", "formula": "\\begin{align*} \\mathbb E | E | ^ { K } = o \\left ( \\mathbb E | M | ^ { K } \\right ) . \\end{align*}"}
+{"id": "1225.png", "formula": "\\begin{align*} S _ { p } & = \\left \\{ ( u , v ) \\in X _ { s , t , p } ( \\Omega ) \\ , : \\ , \\vert u ( x _ 1 ) \\vert ^ { \\alpha ( p ) } \\vert v ( x _ 2 ) \\vert ^ { \\beta ( p ) } = 1 \\right \\} \\intertext { a n d } S _ { \\infty } & = \\left \\{ ( u , v ) \\in X _ { s , t , p } \\ , : \\ , \\vert u ( x _ 1 ) \\vert ^ { \\theta } \\vert v ( x _ 2 ) \\vert ^ { 1 - \\theta } = 1 \\right \\} \\end{align*}"}
+{"id": "4685.png", "formula": "\\begin{align*} \\mathbf { B } _ + ( D ) = \\bigcup _ { C \\in \\mathrm { M B M } ( P ( D ) ) } \\mathrm { S u p p } ( C ) . \\end{align*}"}
+{"id": "8312.png", "formula": "\\begin{align*} { \\sf M } _ 2 ( \\omega _ 1 \\odot \\omega _ 2 ) = P \\{ \\{ \\eta , \\omega _ 1 \\} , \\omega _ 2 \\} = \\{ \\{ \\eta , \\omega _ 1 \\} , \\omega _ 2 \\} . \\end{align*}"}
+{"id": "1268.png", "formula": "\\begin{align*} - q ^ { n ^ 2 - { n \\choose 2 } } & = - q ^ { \\frac { n ( n + 1 ) } { 2 } } \\\\ & \\equiv - 1 + \\frac { ( n + 1 ) ( 1 - q ^ n ) } { 2 } \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "3206.png", "formula": "\\begin{align*} u ( T ) = \\Psi . \\end{align*}"}
+{"id": "4718.png", "formula": "\\begin{align*} \\tilde { f } _ { e x t } : = \\begin{cases} e ^ { - \\frac { \\zeta } { 1 6 } \\tilde { \\gamma } ^ { - h _ T ^ { e x t } } } \\qquad \\\\ \\gamma ^ { h _ T ^ { e x t } } \\ , \\end{cases} \\ , . \\end{align*}"}
+{"id": "3351.png", "formula": "\\begin{align*} \\begin{aligned} \\Sigma ^ * ( u , y ) \\sigma ( z ) = & \\int _ { \\mathbb { R } ^ n } \\psi ( z - b ( \\xi , u ) ) \\\\ & \\cdot \\left ( \\mu L ( \\xi , u ) \\right ) \\Psi ( \\xi , y ) \\sigma ( \\xi ) d \\xi , \\end{aligned} \\end{align*}"}
+{"id": "8251.png", "formula": "\\begin{align*} \\frac { 1 } { L } \\max _ { P _ { { X } ^ L | | Y ^ { L - 1 } } } \\sum _ { j = 1 } ^ L H ( { Y } _ { j } | Y ^ { j - 1 } , { S } ) , \\end{align*}"}
+{"id": "1623.png", "formula": "\\begin{align*} \\sum _ { j + k = s } ( - 1 ) ^ { 2 j k ( p - 1 ) } \\left ( \\b Q ^ j _ { h _ 1 } ( x ) \\otimes Q ^ k _ { h _ 2 } ( y ) + ( - 1 ) ^ { d _ { \\Z } ( x ) } Q ^ j _ { h _ 1 } ( x ) \\otimes \\b Q ^ k _ { h _ 2 } ( y ) \\right ) . \\end{align*}"}
+{"id": "727.png", "formula": "\\begin{align*} d ( G , H ) & = \\sum _ { \\mathfrak { p } } \\left | \\hat { \\nu } _ { \\mathfrak { p } } ( \\mathcal { I } _ r ( G ) ) - \\hat { \\nu } _ { \\mathfrak { p } } ( \\mathcal { I } _ r ( H ) ) \\right | , \\end{align*}"}
+{"id": "2461.png", "formula": "\\begin{align*} \\phi ' ( s ) = - k c ^ { - 1 } \\phi ^ { 3 } + O ( \\phi ^ { 5 } ) . \\end{align*}"}
+{"id": "4994.png", "formula": "\\begin{align*} p _ k ( \\lambda ) = \\prod _ { \\substack { l = 1 \\\\ l \\neq k } } ^ N ( \\lambda - \\nu _ l ) , k = 1 , \\ldots , N . \\end{align*}"}
+{"id": "3359.png", "formula": "\\begin{align*} \\begin{aligned} d _ k \\leq & \\frac { c ^ k d _ 0 } { \\phi ( y _ k ) \\cdot \\cdots \\cdot \\phi ( y _ 1 ) } + \\frac { c ^ k \\zeta Y _ 1 ( y _ 1 ) } { \\phi ( y _ k ) \\cdot \\cdots \\cdot \\phi ( y _ 2 ) } \\\\ & + \\frac { c ^ { k - 1 } \\zeta Y _ 2 ( y _ 2 ) } { \\phi ( y _ k ) \\cdot \\cdots \\cdot \\phi ( y _ 3 ) } + \\cdots + \\frac { c ^ 2 \\zeta Y _ { k - 1 } ( y _ { k - 1 } ) } { \\phi ( y _ k ) } + c \\zeta Y _ { k } ( y _ k ) . \\end{aligned} \\end{align*}"}
+{"id": "4704.png", "formula": "\\begin{align*} \\alpha ( 1 _ { W } ) + \\sum \\limits _ { l + 1 } ^ { r - 1 } \\alpha ( 1 _ { U _ { k } } ) = \\beta ( 1 _ { W } ) + \\sum \\limits _ { l + 1 } ^ { r - 1 } \\beta ( 1 _ { U _ { k } } ) \\N . \\end{align*}"}
+{"id": "171.png", "formula": "\\begin{align*} R \\cap R ^ { - 1 } \\ = \\ & \\bigcap _ { i } \\ ( \\pi _ i \\times \\pi _ i ) ^ { - 1 } ( R _ i \\cap R _ i ^ { - 1 } ) \\\\ = \\ \\bigcap _ { i } \\ & ( \\pi _ i \\times \\pi _ i ) ^ { - 1 } ( 1 _ { X _ i } ) \\ = \\ 1 _ X . \\end{align*}"}
+{"id": "8813.png", "formula": "\\begin{align*} f ^ { * } ( t ) : = \\inf \\{ \\sigma > 0 : \\mu ( f , \\sigma ) \\leq t \\} , t \\geq 0 , \\end{align*}"}
+{"id": "8326.png", "formula": "\\begin{align*} \\widetilde { W } _ t : = W _ t + ( \\partial _ t + X _ t ) \\wedge \\partial _ s \\end{align*}"}
+{"id": "5883.png", "formula": "\\begin{align*} \\dfrac { M _ { 2 } ( \\mathcal { C } ( G ) ) } { | e ( \\mathcal { C } ( G ) ) | } = \\dfrac { ( m - 1 ) ( m n - n - 1 ) ^ { 3 } + m ( n - 1 ) ^ { 3 } } { ( m - 1 ) ( m n - n - 1 ) + m ( n - 1 ) } . \\end{align*}"}
+{"id": "403.png", "formula": "\\begin{align*} \\begin{aligned} H ( l _ x , l _ y , m _ x , m _ y ) & = { A } ( l _ x , l _ y , m _ x , m _ y ) \\\\ & + { Y } ( l _ x , l _ y , m _ x , m _ y ) , \\end{aligned} \\end{align*}"}
+{"id": "8935.png", "formula": "\\begin{align*} I = \\begin{pmatrix} I _ 1 & \\cdots & I _ d \\end{pmatrix} \\end{align*}"}
+{"id": "7833.png", "formula": "\\begin{align*} \\zeta _ k ^ { \\star } = \\frac { 1 } { ( 1 + \\eta _ k ^ { \\star } ) \\ln 2 } , \\forall k \\in \\mathcal { K } . \\end{align*}"}
+{"id": "1918.png", "formula": "\\begin{align*} & \\sum _ { x \\in B _ r ( x _ 0 ) } \\ ! \\ ! \\ ! V ( x ) \\ , v ^ 2 _ + ( x , T ) \\mu ( x ) \\leq \\frac { c ( T , \\lambda , r ) } { ( r _ 1 ) ^ 2 } \\sum _ { x \\in G } \\exp \\Big ( - \\frac { \\mathbf { d } ^ \\alpha ( x ) } { \\lambda } \\Big ) \\chi _ { \\{ r < \\mathbf { d } ( x ) \\le r _ 1 \\} } ( x ) \\mu ( x ) , \\end{align*}"}
+{"id": "6141.png", "formula": "\\begin{align*} \\min \\{ f ( x ) : ~ A x = b \\} , \\end{align*}"}
+{"id": "7864.png", "formula": "\\begin{align*} \\mu ( \\mathbf { \\Phi } ) = \\frac { 1 } { \\sqrt { p ^ r } } . \\end{align*}"}
+{"id": "2975.png", "formula": "\\begin{align*} \\begin{aligned} x _ 1 ( t ) - x _ 2 ( t ) & = x _ 1 ( 0 ) - x _ 2 ( 0 ) + \\int _ { 0 } ^ t ( v _ 1 ( s ) - v _ 2 ( s ) ) d s \\\\ & = x _ 1 ( 0 ) - x _ 2 ( 0 ) + \\int _ { 0 } ^ t ( e ^ { i \\theta _ 1 ( s ) } - e ^ { i \\theta _ 2 ( s ) } ) d s \\\\ & = x _ 1 ( 0 ) - x _ 2 ( 0 ) - 2 \\int _ { 0 } ^ t ( 0 , \\sin ( \\theta _ 2 ) ) d s , \\\\ v _ 1 ( t ) - v _ 2 ( t ) & = - 2 ( 0 , \\sin ( \\theta _ 2 ( t ) ) ) . \\end{aligned} \\end{align*}"}
+{"id": "6778.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n a _ { i j } \\mathsf { X } _ i \\longrightarrow \\sum _ { i = 1 } ^ n ( a _ { i j } + c _ { i j } ) \\mathsf { X } _ i j = 1 , \\ldots , m . \\end{align*}"}
+{"id": "5105.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ k \\lambda _ i \\right ) + k - 1 \\leq n . \\end{align*}"}
+{"id": "4004.png", "formula": "\\begin{align*} \\begin{cases} x & = \\ ; \\gamma x y \\\\ y & = \\ ; \\left ( 1 - \\gamma \\right ) x y \\end{cases} \\end{align*}"}
+{"id": "4347.png", "formula": "\\begin{gather*} F _ { i , j , k , l } ( x ) : = \\max \\{ 0 , \\Delta u _ i B _ { i , ( i , j ) } - \\Delta u _ k B _ { k , ( k , l ) } \\} x _ { i , j } . \\end{gather*}"}
+{"id": "7759.png", "formula": "\\begin{align*} A _ n ( \\lambda ) = \\mathrm { d i a g } \\{ A _ { n , 1 1 } ( \\lambda ) , \\cdots , A _ { n , l _ { n } l _ { n } } ( \\lambda ) \\} , \\end{align*}"}
+{"id": "2558.png", "formula": "\\begin{align*} K ^ { ( s ) } ( M , L ) = I ^ { ( s ) } _ L ( M , L ) \\ ; , K ^ m ( M , L ) = I ^ m _ L ( M , L ) \\ ; , K ( M , L ) = I _ L ( M , L ) \\ ; . \\end{align*}"}
+{"id": "4012.png", "formula": "\\begin{align*} \\begin{cases} \\gamma _ { 2 } x _ { 1 } - \\gamma _ { 1 } x _ { 2 } & = 0 \\\\ \\delta _ { 2 } x _ { 1 } - \\delta _ { 1 } x _ { 2 } & = 0 \\\\ \\end{cases} \\end{align*}"}
+{"id": "7338.png", "formula": "\\begin{align*} t _ n ( j ) = ( - 1 ) ^ { \\frac { n - j } { 2 } } \\frac { n } { n + j } \\binom { \\frac { n + j } { 2 } } { \\frac { n - j } { 2 } } \\cdot 2 ^ j u _ n ( j ) = ( - 1 ) ^ { \\frac { n - j } { 2 } } \\binom { \\frac { n + j } { 2 } } { \\frac { n - j } { 2 } } \\cdot 2 ^ j . \\end{align*}"}
+{"id": "3042.png", "formula": "\\begin{align*} \\widehat { H } _ \\lambda = p _ 1 ^ 2 + p _ 2 ^ 2 + p _ 3 ^ 2 + V _ \\lambda ( s _ 1 , s _ 2 , s _ 3 ) = L _ 1 ^ 2 + L _ 2 ^ 2 + L _ 3 ^ 2 + V _ \\lambda ( s _ 1 , s _ 2 , s _ 3 ) \\ , , \\end{align*}"}
+{"id": "287.png", "formula": "\\begin{align*} J _ i \\left ( \\frac { j } { n } \\right ) = \\delta _ { i j } . \\end{align*}"}
+{"id": "3759.png", "formula": "\\begin{align*} \\left . \\frac { \\partial \\mathrm { E } _ { \\alpha , \\beta } \\left ( z \\right ) } { \\partial \\alpha } \\right \\vert _ { \\alpha = n } = - \\frac { z ^ { 1 / n } } { n ^ { 2 } \\ , \\Gamma \\left ( \\beta \\right ) } \\sum _ { m = 1 } ^ { n } e ^ { i 2 \\pi m / n } P \\left ( \\beta , z ^ { 1 / n } e ^ { i 2 \\pi m / n } \\right ) . \\end{align*}"}
+{"id": "6961.png", "formula": "\\begin{align*} f ( \\mathbf { z } ) & = { \\displaystyle \\sum _ { j = 1 } ^ { n } } 4 ( x _ j y _ { n + 1 } - x _ { n + 1 } y _ { j } ) ^ { 2 } - \\sum _ { 1 \\leq j < k \\leq n } 4 ( x _ j y _ { k } - y _ { k } x _ { j } ) ^ { 2 } \\\\ & = 4 \\langle \\mathbf { x } , \\mathbf { y } \\rangle ^ 2 - 4 \\langle \\mathbf { x } , \\mathbf { x } \\rangle \\langle \\mathbf { y } , \\mathbf { y } \\rangle , \\end{align*}"}
+{"id": "1855.png", "formula": "\\begin{align*} \\| F _ A \\| _ { L ^ { 7 / 2 } ( B _ r ( x ) ) } = \\| F _ { A ' } \\| _ { L ^ { 7 / 2 } ( \\mathbb { B } ) } \\hat { \\mathcal { E } } ( A , B _ r ( x ) ) = \\hat { \\mathcal { E } } ( A ' , \\mathbb { B } ) = \\| F _ { A ' } \\| ^ 2 _ { L ^ 2 ( \\mathbb { B } ) } . \\end{align*}"}
+{"id": "865.png", "formula": "\\begin{align*} \\ddot { \\Gamma } ^ i + 2 G ^ i ( \\Gamma , \\dot { \\Gamma } ) = \\frac { d } { d t } ( \\ln F ( \\dot { \\Gamma } ) ) \\dot { \\Gamma } ^ i , \\end{align*}"}
+{"id": "8121.png", "formula": "\\begin{align*} { \\mathcal I } ( w _ 0 ( I ) d ) = { \\mathcal I } ( w _ 0 ( I ) ) \\sqcup w _ 0 ( I ) ( { \\mathcal I } ( d ) ) . \\end{align*}"}
+{"id": "5186.png", "formula": "\\begin{align*} & \\frac { \\partial A } { \\partial q _ { j } } = 0 \\\\ & \\frac { \\partial X } { \\partial q _ { j } } = \\beta ( \\beta - 1 ) \\left ( \\sum _ { i } p _ { i } q ^ { \\beta - 1 } _ { i } \\right ) ^ { \\beta - 1 } p _ { j } q ^ { \\beta - 2 } _ { j } \\\\ & \\frac { \\partial Y } { \\partial q _ { j } } = \\beta ( 1 - \\beta ) \\left ( \\sum _ { i } q ^ { \\beta } _ { i } \\right ) ^ { - \\beta } . \\ q ^ { \\beta - 1 } _ { j } \\end{align*}"}
+{"id": "8310.png", "formula": "\\begin{align*} { \\sf M } _ 2 ( \\omega _ 1 \\odot \\omega _ 2 ) = ( - 1 ) ^ { | \\omega _ 1 | } ( \\omega _ { 1 } ^ { \\flat } \\wedge \\omega _ 2 ^ { \\flat } ) \\eta . \\end{align*}"}
+{"id": "8958.png", "formula": "\\begin{align*} \\begin{aligned} \\| | \\nabla v - O | ^ 2 \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } & \\le ( M + \\sqrt { n } ) ^ { 2 - \\mu } \\| \\nabla v - O \\| _ { L ^ { q ( \\cdot ) } ( \\Omega ) } ^ \\mu \\\\ & \\le 2 ^ { \\mu - 1 } C ^ \\mu ( M + \\sqrt { n } ) \\| f \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } + 2 ^ { \\mu - 1 } C ^ \\mu ( M + \\sqrt { n } ) ^ { 2 - \\mu } \\| g \\| ^ \\mu _ { L ^ { q ( \\cdot ) } ( \\Omega ) } \\\\ & \\le ( 2 ^ { \\mu - 1 } C ^ \\mu ( M + \\sqrt { n } ) + 2 ^ { \\mu - 1 } C ^ \\mu ( M + \\sqrt { n } ) ^ { 2 - \\mu } ) \\| f \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } . \\end{aligned} \\end{align*}"}
+{"id": "7073.png", "formula": "\\begin{align*} \\rho ( x ) \\equiv \\rho _ { \\kappa , \\theta } ( x ) : = ( 1 + \\kappa | x | ^ 2 ) ^ { - \\theta } , \\kappa > 0 , \\theta > \\frac { d } { 2 } , x \\in \\mathbb R ^ d . \\end{align*}"}
+{"id": "4789.png", "formula": "\\begin{align*} F _ \\theta ( \\xi ) = \\vert \\xi \\vert - \\cos \\theta \\left < \\xi , E _ n \\right > . \\end{align*}"}
+{"id": "5596.png", "formula": "\\begin{align*} ( T P S _ { \\Delta } ) _ { e f } = \\sum _ { i , j } T _ { e i } P _ { i j } S _ { j f } \\Delta _ { f f } = P _ { e _ 3 f _ 1 } \\Delta _ { f f } = \\sum _ { u \\in V _ 2 } M _ { e _ 3 u } M _ { f _ 1 u } A _ { f _ 1 f _ 2 } A _ { f _ 3 f _ 2 } . \\end{align*}"}
+{"id": "6860.png", "formula": "\\begin{align*} \\theta _ \\varphi ^ * \\{ c _ j \\} = \\sum _ { j \\in \\mathbb { J } } c _ j \\varphi _ j , \\{ c _ j \\} \\in \\ell _ 2 ( \\mathbb { J } ) . \\end{align*}"}
+{"id": "8237.png", "formula": "\\begin{align*} I ( { \\bf X } ^ { L } ; Y ^ { L } | { \\bf S } ) & = \\sum _ { i = 1 } ^ L H ( Y _ i | { \\bf S } , Y ^ { i - 1 } ) , \\end{align*}"}
+{"id": "4418.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathcal { X } } & \\ \\overline { c } ^ T x . \\\\ \\end{align*}"}
+{"id": "6958.png", "formula": "\\begin{align*} \\overline { z _ 1 } w _ 1 + \\cdots \\overline { z _ n } w _ n - \\overline { z _ { n + 1 } } w _ { n + 1 } = 0 . \\end{align*}"}
+{"id": "3781.png", "formula": "\\begin{align*} \\mathbb { P } \\left \\{ \\mathcal { M } _ i ^ { 1 , j } \\right \\} & = \\mathbb { P } \\left \\{ \\sum _ { t = 0 } ^ { T - 1 } \\sum _ { l = 1 } ^ { { N _ i } } v ^ { ( i ) , \\top } _ { l , t } \\bar { \\Sigma } ^ { ( i ) } _ t v ^ { ( i ) } _ { l , t } \\geq \\sum _ { t = 0 } ^ { T - 1 } \\sum _ { l = 1 } ^ { { N _ i } } u ^ { ( i ) , \\top } _ { l , t } \\Tilde { \\Sigma } ^ { ( i ) } _ t u ^ { ( i ) } _ { l , t } \\right \\} , \\end{align*}"}
+{"id": "351.png", "formula": "\\begin{align*} x ^ n = z ^ n - y ^ n = ( z - y ) \\phi _ n ( z , y ) . \\end{align*}"}
+{"id": "5668.png", "formula": "\\begin{align*} & ( - 1 ) ^ { ( q - 1 ) / 2 } \\left ( \\frac { 2 ^ b } { q } \\right ) \\left ( \\frac { p _ 1 } { q } \\right ) \\cdots \\left ( \\frac { p _ s } { q } \\right ) = 1 , \\\\ & ( - 1 ) ^ { ( p _ i - 1 ) ( q + 1 ) / 4 } \\left ( \\frac { p _ i } { q } \\right ) = 1 , \\end{align*}"}
+{"id": "781.png", "formula": "\\begin{align*} \\gamma ( a , b ) = \\omega ( a ^ { - 1 } , b ) ^ { - 1 } f ( a ) g ( b ) \\end{align*}"}
+{"id": "1685.png", "formula": "\\begin{align*} l _ \\infty ^ + = \\langle f _ 2 , f _ 1 \\rangle , l _ \\infty ^ - = \\langle e _ 2 , f _ 1 \\rangle , l _ \\infty ^ + \\cap l _ \\infty ^ - = \\langle f _ 1 \\rangle . \\end{align*}"}
+{"id": "2929.png", "formula": "\\begin{align*} \\sum _ { T ' \\in P ( \\{ a _ 1 , a _ 2 , a _ 3 \\} ) , a _ 1 \\in T ' } { g _ { T ' } } = 2 , \\end{align*}"}
+{"id": "1164.png", "formula": "\\begin{align*} \\vec u _ Q : = \\begin{cases} [ \\ell ( I _ 0 ) ] ^ { \\frac 1 2 } \\vec t _ { I _ 0 } & Q = Q ( I _ 0 , k _ 0 ) , \\\\ \\vec { \\mathbf { 0 } } & . \\end{cases} \\end{align*}"}
+{"id": "119.png", "formula": "\\begin{align*} \\begin{aligned} a ( \\vec { v } _ 0 , \\vec { w } ) + b ( \\vec { w } , p ) & = a ( \\vec { v } , \\vec { w } ) , & & \\forall \\vec { w } \\in V , \\\\ b ( \\vec { v } _ 0 , q ) & = 0 , & & \\forall q \\in Q , \\end{aligned} \\end{align*}"}
+{"id": "4342.png", "formula": "\\begin{gather*} B _ { q , . } g ( x ) = \\sum _ { i \\in [ m ] , j \\in [ n ] } B _ { q , ( i , j ) } g _ { i , j } ( x ) = \\sum _ { i \\in [ m ] } B _ { q , ( i , j ( i ) ) } g _ { i , j ( i ) } ( x ) = B _ { q , ( q , j ( q ) ) } g _ { q , j ( q ) } ( x ) . \\end{gather*}"}
+{"id": "3814.png", "formula": "\\begin{align*} \\Pr [ E _ 1 \\wedge \\dots \\wedge E _ r ] = \\Pr [ F _ r ] = \\prod _ { \\ell = 1 } ^ r \\frac { \\Pr [ F _ \\ell ] } { \\Pr [ F _ { \\ell - 1 } ] } . \\end{align*}"}
+{"id": "2418.png", "formula": "\\begin{align*} v _ 2 \\left ( \\int _ { \\mathbb { Z } _ 2 } g ( t ) \\binom { t + 2 ^ m - 1 } { 2 ^ m - 1 } ^ 2 \\binom { t + 2 ^ { m - 1 } - 1 } { 2 ^ { m - 1 } - 1 } ^ { s } \\binom { t + 2 ^ m - 1 } { 2 ^ { m - 1 } - 1 } ^ { s } \\mathrm { d } t \\right ) = - m . \\end{align*}"}
+{"id": "3022.png", "formula": "\\begin{align*} \\begin{array} { l } R _ { 0 1 1 0 } = R _ { 0 2 2 0 } = R _ { 0 3 3 0 } = R _ { 0 4 4 0 } = - 1 , \\\\ R _ { 1 2 3 4 } = R _ { 1 4 3 2 } = R _ { 1 3 3 1 } = R _ { 2 4 4 2 } = 1 . \\end{array} \\end{align*}"}
+{"id": "2060.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty k ^ { 5 + \\varepsilon } b _ k ^ 2 < \\infty . \\end{align*}"}
+{"id": "3013.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\lambda & = & ( 2 n - 1 ) \\alpha ^ 2 - \\beta ^ 2 + \\alpha + \\beta + a \\\\ \\mu & = & 2 ( n - 1 ) \\alpha \\beta + ( 1 - 2 n ) \\alpha + \\beta + b \\\\ \\nu & = & \\beta ^ 2 + ( 1 - 2 n ) \\alpha ^ 2 + ( \\alpha - \\alpha \\beta ) ( 2 n - 2 ) - ( 2 n + 2 ) \\beta + c \\end{array} \\end{align*}"}
+{"id": "4.png", "formula": "\\begin{align*} \\psi ( E ) = \\phi ( E ) + E \\phi ' ( E ) . \\end{align*}"}
+{"id": "2696.png", "formula": "\\begin{align*} L = \\frac { 1 } { 2 \\beta ^ { 2 } } \\ddot { q } ^ { 2 } - \\frac { 1 } { \\beta } \\ddot { q } \\dot { q } + \\frac { 1 } { 2 } \\dot { q } ^ { 2 } + \\frac { \\alpha } { \\beta ^ { 2 } } \\dot { q } \\delta ( t - t _ { 0 } ) - \\frac { \\alpha } { \\beta } q \\delta ( t - t _ { 0 } ) . \\end{align*}"}
+{"id": "9196.png", "formula": "\\begin{align*} e ^ \\pm _ { 1 , k + 1 } ( z ) = - ( 1 - q ^ 2 ) ^ { - 1 } \\cdot [ e ^ \\pm _ { 1 k } ( z ) , e ^ { ( 0 ) } _ { k ' - 1 , k ' } ] _ q . \\end{align*}"}
+{"id": "6923.png", "formula": "\\begin{align*} \\C ^ { p + r } = E ^ { s s } ( z ) \\oplus E ^ c ( z ) \\oplus E ^ { s u } ( z ) . \\end{align*}"}
+{"id": "7894.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { p - 1 } R _ { \\mathbf { s } _ { n } } ( \\tau ) & = \\sum _ { i \\in \\Omega _ { 1 } } \\sum _ { n = 0 } ^ { p - 1 } \\omega _ { q } ^ { f _ { n , i } - f _ { n , j } } + \\sum _ { i \\in \\Omega _ { 2 } } \\sum _ { n = 0 } ^ { p - 1 } \\omega _ { q } ^ { f _ { n , i } - f _ { n , j } } \\\\ & = p \\sum _ { i \\in \\Omega _ { 1 } } \\omega _ { q } ^ { f _ { 0 , i } - f _ { 0 , j } } , \\end{align*}"}
+{"id": "1196.png", "formula": "\\begin{align*} \\sigma & \\geq \\mathbf { 1 } _ { ( 0 , \\infty ) } ( N ) , \\ \\begin{cases} E \\geq N , \\\\ E > \\lfloor N \\rfloor _ + , \\end{cases} \\begin{cases} F \\geq ( K \\vee M ) - n , \\\\ F > \\lfloor L \\rfloor , \\end{cases} G \\geq \\lfloor N \\rfloor _ + , H \\geq \\lfloor L \\rfloor . \\end{align*}"}
+{"id": "273.png", "formula": "\\begin{align*} \\ddot { x } + \\frac { 1 } { x } \\dot { x } ^ 2 + x \\ , \\dot { x } + b ( x ) = 0 . \\end{align*}"}
+{"id": "4010.png", "formula": "\\begin{align*} \\begin{cases} \\left ( \\gamma _ { 1 } y - 1 \\right ) x _ { 1 } + \\left ( \\delta _ { 1 } y \\right ) x _ { 2 } & = \\ ; 0 \\\\ \\left ( \\gamma _ { 2 } y \\right ) x _ { 1 } + \\left ( \\delta _ { 2 } y - 1 \\right ) x _ { 2 } & = \\ ; 0 \\\\ \\gamma x _ { 1 } + \\delta x _ { 2 } & = \\ ; 1 \\end{cases} \\end{align*}"}
+{"id": "6476.png", "formula": "\\begin{align*} \\operatorname { I n d } ^ { \\rm { n o r } } ( M ) : = \\max \\{ \\dim L , L \\subset C ^ \\infty _ 0 ( M ) \\mid \\operatorname { H e s s } E _ 3 ( \\phi ) ( f \\nu , f \\nu ) < 0 , ~ ~ \\forall f \\in L \\} . \\end{align*}"}
+{"id": "7785.png", "formula": "\\begin{align*} A u t ^ + ( \\pi { K } ) = \\{ ( g , d ) \\in \\pi ' \\rtimes { A u t ( \\pi ' ) } \\mid d \\theta { d ^ { - 1 } } \\theta ^ { - 1 } = c _ g \\} \\end{align*}"}
+{"id": "2913.png", "formula": "\\begin{align*} L _ { \\ell } ( x _ i ; f ) : = \\sum _ { y _ { 0 } < p \\leqslant y _ J } \\frac { X } { p } \\int _ { p } ^ { p ( 1 + 1 / X ) } \\big | \\Psi _ f ^ { \\prime } ( x _ i / t , p ) \\big | ^ 2 { \\rm d } t \\end{align*}"}
+{"id": "6261.png", "formula": "\\begin{align*} \\Phi g ( t , x ) = \\int _ { \\Gamma _ { t , x } } g ( u , y ) d \\sigma ( u , y ) , \\end{align*}"}
+{"id": "2105.png", "formula": "\\begin{align*} U _ k ^ N ( s , x ) & = U ( 0 , x ) - \\alpha \\int _ 0 ^ { s } \\int _ 0 ^ 1 A ( x , y ) U _ { k - 1 } ( u \\wedge \\tau _ N , y ) 1 _ { ( 0 , \\tau _ N ) } ( u ) \\mathrm { d } y \\mathrm { d } u \\\\ & + \\alpha \\beta \\int _ 0 ^ s 1 _ { ( 0 , \\tau _ N ) } ( u ) \\mathrm { d } \\xi _ 2 ( u \\wedge \\tau _ N , x ) + \\alpha \\zeta \\int _ 0 ^ s 1 _ { ( 0 , \\tau _ N ) } ( u ) \\mathrm { d } \\xi _ 3 ( u \\wedge \\tau _ N , x ) . \\end{align*}"}
+{"id": "5786.png", "formula": "\\begin{align*} \\tilde \\xi _ i ( t ) = e ^ { \\Gamma _ i t } \\tilde { \\xi } ( 0 ) + \\int _ 0 ^ t e ^ { \\Gamma _ i ( t - \\tau ) } \\tilde { \\mathcal { E } } _ i ( \\tau ) \\ , d \\tau . \\end{align*}"}
+{"id": "3780.png", "formula": "\\begin{align*} \\bar { \\Sigma } ^ { ( i ) } _ t & = ( { \\Theta } _ 1 - \\widehat { \\Theta } _ 1 ) \\Sigma ^ { ( i ) } _ t ( { \\Theta } _ 1 - \\widehat { \\Theta } _ 1 ) ^ \\top + \\sigma ^ 2 _ { w , i } I _ { n _ x } , \\ ; \\ \\Tilde { \\Sigma } ^ { ( i ) } _ t = ( { \\Theta } _ 1 - \\widehat { \\Theta } _ j ) \\Sigma ^ { ( i ) } _ t ( { \\Theta } _ 1 - \\widehat { \\Theta } _ j ) ^ \\top + \\sigma ^ 2 _ { w , i } I _ { n _ x } . \\end{align*}"}
+{"id": "6440.png", "formula": "\\begin{align*} \\rho ( \\Theta ) : = \\big \\{ \\zeta \\in \\overline { \\C ^ + } \\cup \\infty \\liminf _ { z \\to \\zeta \\atop { z \\in \\C ^ + } } | \\Theta ( z ) | = 0 \\big \\} . \\end{align*}"}
+{"id": "2450.png", "formula": "\\begin{align*} \\begin{cases} U ( \\xi ) \\sim \\left ( \\dfrac { c } { k } \\xi \\right ) ^ { - 1 / p } , \\\\ U ' ( \\xi ) \\sim - \\dfrac { 1 } { p } \\left ( \\dfrac { k } { c } \\right ) ^ { - \\frac { 1 } { p } } \\xi ^ { - \\frac { p + 1 } { p } } , \\end{cases} { \\rm { a s } } \\xi \\to + \\infty . \\end{align*}"}
+{"id": "5025.png", "formula": "\\begin{align*} \\max \\ , \\left \\{ \\mathbf { x } ^ 1 ( \\mathbf { R } - \\nu \\mathbf { 1 } ) \\colon \\begin{bmatrix} \\mathbf { x } ^ { 0 } & \\mathbf { x } ^ { 1 } \\end{bmatrix} \\begin{bmatrix} ( 1 - \\beta ) \\mathbf { I } \\\\ \\mathbf { I } - \\beta \\mathbf { P } \\end{bmatrix} = \\mathbf { e } _ i , \\begin{bmatrix} \\mathbf { x } ^ { 0 } & \\mathbf { x } ^ { 1 } \\end{bmatrix} \\geq \\mathbf { 0 } \\right \\} \\end{align*}"}
+{"id": "9167.png", "formula": "\\begin{align*} G _ { \\beta , \\beta ' } = ( w _ { \\beta , 1 } - v ^ { - 4 } w _ { \\beta ' , 1 } ) ( w _ { \\beta , 1 } - v ^ { - 4 n + 6 } w _ { \\beta ' , 1 } ) ( w _ { \\beta , 1 } - v ^ { - 4 n + 1 4 } w _ { \\beta ' , 1 } ) \\cdot G _ { \\beta , \\beta } . \\end{align*}"}
+{"id": "2634.png", "formula": "\\begin{align*} s _ j - s _ { j - 2 } = s _ { j - 3 } - s _ { j - 6 } = s _ { j - 5 } - s _ { j - 9 } , \\ , \\ , \\ , \\ , \\forall \\ , \\ , 1 0 \\leq j \\leq n . \\end{align*}"}
+{"id": "3326.png", "formula": "\\begin{align*} \\L _ 0 \\left ( u \\circ \\delta _ r \\right ) = r ^ 2 \\delta _ r \\left ( \\L _ 0 u \\right ) , r > 0 , \\end{align*}"}
+{"id": "1562.png", "formula": "\\begin{align*} \\int _ { B } F ( D \\psi _ n ) \\ , d x & = \\int _ { B } F \\left ( \\int \\varphi _ { 1 / n } ( x - y ) \\ , D u ( y ) \\ , d y \\right ) \\ , d x \\\\ & \\le \\int _ { B } \\left ( \\int \\varphi _ { 1 / n } ( x - y ) F ( D u ( y ) ) \\ , d y \\right ) d x \\\\ & \\le \\int _ { B } \\varphi _ { 1 / n } * F ( D u ) \\ , d x . \\end{align*}"}
+{"id": "7784.png", "formula": "\\begin{align*} \\hat \\gamma ( E + i \\epsilon ) - \\hat \\gamma ( E ) & = \\frac { 1 } { 2 } \\int \\ln ( 1 + \\frac { \\epsilon ^ { 2 } } { ( E - E ' ) ^ { 2 } } ) d \\hat { \\mathcal N } ( E ' ) \\\\ & > \\frac { \\ln 2 } { 2 } ( \\hat { \\mathcal N } ( E + \\epsilon ) - \\hat { \\mathcal N } ( E - \\epsilon ) ) . \\end{align*}"}
+{"id": "331.png", "formula": "\\begin{align*} I ( G ) ^ { [ k ] } = I ( G _ 1 ) ^ { [ k ] } + x _ n x _ { n - 1 } I ( G _ 2 ) ^ { [ k - 1 ] } \\end{align*}"}
+{"id": "4255.png", "formula": "\\begin{align*} G _ \\xi [ f ( t ) , f ( t ) , f ( t ) ] ( 0 , 0 ) = | \\hat f ( t , \\xi ) | ^ 2 \\hat f ( t , \\xi ) , \\end{align*}"}
+{"id": "2831.png", "formula": "\\begin{align*} \\delta _ { } \\left ( \\tilde { \\sigma } ^ { * } _ { 3 } ( t ) I \\right ) = \\int ^ { t _ { 2 } } _ { t _ { 1 } } \\left [ - \\dot { P } _ { 1 } - Q ^ { 1 } \\right ] \\delta Q ^ { 1 } d t + \\left [ - P _ { 1 } + \\dot { Q } ^ { 1 } \\right ] \\delta P _ { 1 } d t + \\left [ \\Psi _ { 2 } \\delta \\Xi ^ { 2 } + P _ { 1 } \\delta Q ^ { 1 } \\right ] ^ { t _ { 2 } } _ { t _ { 1 } } \\end{align*}"}
+{"id": "8895.png", "formula": "\\begin{align*} \\Phi _ n ( x ) = \\prod _ { d \\mid n } ( x ^ d - 1 ) ^ { \\mu ( n / d ) } , \\end{align*}"}
+{"id": "9189.png", "formula": "\\begin{align*} \\frac { z - w } { q z - q ^ { - 1 } w } \\ell ^ - _ { 1 j } ( z ) \\ell ^ - _ { j j } ( w ) + \\frac { ( q - q ^ { - 1 } ) z } { q z - q ^ { - 1 } w } \\ell ^ - _ { j j } ( z ) \\ell ^ - _ { 1 j } ( w ) = \\ell ^ - _ { j j } ( w ) \\ell ^ - _ { 1 j } ( z ) . \\end{align*}"}
+{"id": "340.png", "formula": "\\begin{align*} \\big ( \\sum _ i I _ i \\big ) : ( \\sum _ j J _ j ) \\ = \\ \\sum _ i \\big ( I _ i : \\big ( \\sum _ j J _ j \\big ) \\big ) \\ = \\ \\sum _ i \\bigcap _ j ( I _ i : J _ j ) . \\end{align*}"}
+{"id": "6150.png", "formula": "\\begin{align*} \\varphi ^ k ( x , \\lambda ) : = - \\lambda ^ T ( A x - b ) + \\frac { \\beta ^ k } { 2 } \\| A x - b \\| ^ 2 , ~ \\beta ^ k > 0 . \\end{align*}"}
+{"id": "4575.png", "formula": "\\begin{align*} t \\curvearrowright P _ i = t ^ { \\theta ( p v _ { i , V } + q v _ { i , U } ) - \\tau v _ { i , U } } P _ i ; \\end{align*}"}
+{"id": "8455.png", "formula": "\\begin{align*} \\partial _ t v _ h + ( - \\Delta ) _ p ^ s \\bar { u } _ h = 0 \\textrm { i n } \\ , \\ , \\ , \\Omega _ \\infty , \\end{align*}"}
+{"id": "4226.png", "formula": "\\begin{align*} f ( g \\tau ) : = f \\left ( \\frac { a \\tau + b } { c \\tau + d } \\right ) = \\chi ( g ) ( c \\tau + d ) ^ k f ( \\tau ) , ~ ~ \\forall g = \\left ( \\begin{array} { c c } \\ a & b \\\\ c & d \\end{array} \\right ) \\in \\Gamma , \\end{align*}"}
+{"id": "3451.png", "formula": "\\begin{align*} F ' = F ' _ \\R \\oplus F ' _ \\C , \\end{align*}"}
+{"id": "8275.png", "formula": "\\begin{align*} \\frac { R ^ { ( n ) } ( \\lambda ) } { n ! } = ( - 1 ) ^ n R ( \\lambda , A ) ^ { n + 1 } \\mbox { f o r a l l } \\lambda \\in \\rho ( A ) , n \\in \\N _ 0 . \\end{align*}"}
+{"id": "3214.png", "formula": "\\begin{align*} | | F _ n | | ^ 2 = \\sum \\limits _ { k = 1 } ^ n \\bigg [ f _ { k , 1 } + f _ { k , 2 } \\bigg ] ^ 2 \\leq 2 \\sum \\limits _ { k = 1 } ^ n f _ { k , 1 } ^ 2 + 2 \\sum \\limits _ { k = 1 } ^ n f _ { k , 2 } ^ 2 \\equiv 2 I _ { 1 , n } + 2 I _ { 2 , n } . \\end{align*}"}
+{"id": "6330.png", "formula": "\\begin{align*} \\mathbb { H } _ { \\rm B o g } = \\sum _ { p \\neq 0 } \\left ( p ^ 2 + 8 \\pi \\mathfrak { a } \\frac { n } { \\ell } \\right ) a _ p ^ * a _ p + \\frac { 1 } { 2 } \\sum _ { p \\neq 0 } n \\widehat { \\epsilon } _ { \\ell , \\lambda } ( p ) ( a _ p ^ * a _ p ^ * + a _ p a _ p ) + \\frac { 1 } { 2 } \\sum _ { p \\neq 0 } \\frac { | n \\widehat { \\epsilon } _ { \\ell , \\lambda } ( p ) | ^ 2 } { 2 p ^ 2 } . \\end{align*}"}
+{"id": "7642.png", "formula": "\\begin{align*} \\begin{array} { l l } \\operatorname { \\widetilde { M i n } } & \\mathcal { F } ( x ) = ( \\mathcal { F } _ { 1 } ( x ) , . . . , \\mathcal { F } _ { k } ( x ) ) ^ T \\\\ & x \\in \\mathcal { X } \\end{array} \\end{align*}"}
+{"id": "6613.png", "formula": "\\begin{align*} \\phi _ { \\Phi _ k , k - 1 } ( x , y ) & = R _ { k , k - 1 } ( 1 ) y + O ( x \\exp \\left ( - \\sqrt { f _ k ^ { \\prime } \\log x \\log \\log x } \\ \\right ) . \\end{align*}"}
+{"id": "1458.png", "formula": "\\begin{align*} \\lim _ { i \\in I } \\frac { \\vert F _ i \\setminus F _ i g \\vert } { \\vert F _ i \\vert } = 0 g \\in G . \\end{align*}"}
+{"id": "2140.png", "formula": "\\begin{align*} q _ i ( x , y ) = \\left \\{ \\begin{array} { l l } b _ i ( x ) , & \\mbox { i f $ b _ i ( \\hat { x } ) = c _ i ( \\hat { x } $ ) , } \\\\ \\frac { ( c _ i ( \\hat { x } ) - \\hat { z } _ i ) b _ i ( x ) + ( \\hat { z } _ i - b _ i ( \\hat { x } ) ) c _ i ( x ) } { c _ i ( \\hat { x } ) - b _ i ( \\hat { x } ) } , & \\mbox { o t h e r w i s e , } \\end{array} \\right . \\end{align*}"}
+{"id": "7702.png", "formula": "\\begin{align*} \\widetilde { V } _ { q } ( \\Omega , z ) = \\frac { q } { n } \\int _ { \\Omega } \\frac { 1 } { | y - z | ^ { n - q } } d y . \\end{align*}"}
+{"id": "1210.png", "formula": "\\begin{align*} S _ { p } & = \\left \\{ ( u , v ) \\in X _ { s , t , p } ( \\Omega ) \\ , : \\ , \\left ( \\int _ { \\Omega } \\vert u \\vert ^ { \\alpha ( p ) } \\dd x \\right ) \\vert v ( x _ 0 ) \\vert ^ { \\beta ( p ) } = 1 \\right \\} \\\\ S _ { \\infty } & = \\left \\{ ( u , v ) \\in X _ { s , t , \\infty } ( \\Omega ) \\ , : \\ , \\Vert u \\Vert _ { \\infty } ^ { \\theta } \\vert v ( x _ 0 ) \\vert ^ { 1 - \\theta } = 1 \\right \\} , \\end{align*}"}
+{"id": "2802.png", "formula": "\\begin{align*} & \\dot { \\phi } ^ { ( 1 ) } = \\{ \\phi ^ { ( 1 ) } , H _ { T } \\} \\approx \\frac { 1 } { 2 } ( q ^ { 3 } ) ^ { 2 } \\\\ & \\therefore \\phi ^ { ( 2 ) } : = q ^ { 3 } : \\approx 0 , \\\\ & \\phi ^ { ( 3 ) } : = \\dot { \\phi } ^ { ( 2 ) } = \\{ \\phi ^ { ( 2 ) } , H _ { T } \\} \\approx p _ { 1 } \\\\ & \\therefore \\phi ^ { ( 3 ) } : = p _ { 1 } : \\approx 0 . \\end{align*}"}
+{"id": "8090.png", "formula": "\\begin{align*} \\left | \\int _ 0 ^ \\infty \\frac { t ^ { n - 1 } } { ( n - 1 ) ! } e ^ { - z t } \\ ; d t \\right | \\leq \\int _ 0 ^ \\infty \\frac { t ^ { n - 1 } } { ( n - 1 ) ! } e ^ { - \\Re ( z ) t } \\ ; d t = 1 / \\Re ( z ) ^ n , \\end{align*}"}
+{"id": "1970.png", "formula": "\\begin{align*} E q u & > p ^ s + 1 - ( \\tau + 1 ) p ^ { t } - ( p - \\tau ) p ^ { s - t } \\\\ & = p ^ { s - t } ( p ^ { t - 1 } - p + \\tau ) + p ^ t ( ( p - 1 ) p ^ { s - t - 1 } - \\tau - 1 ) + 1 \\\\ & \\geq \\tau p ^ { s - t } + 1 > 0 \\end{align*}"}
+{"id": "5659.png", "formula": "\\begin{align*} b _ 1 = a _ 1 v - a _ 2 u , b _ 2 = a _ 2 v + a _ 1 u , \\end{align*}"}
+{"id": "6380.png", "formula": "\\begin{align*} S ( p _ { i j } ) = p _ { - i , - j } , S ( f _ { i j } ) = f _ { - j , - i } , S ( \\widehat { x } ) = \\widehat { x ^ { - 1 } } . \\end{align*}"}
+{"id": "7368.png", "formula": "\\begin{align*} u _ { t t } ^ 0 ( x , t ) = \\frac { 1 } { 2 } \\{ f '' ( x + t ) + f '' ( x - t ) + g ' ( x + t ) - g ' ( x - t ) \\} . \\end{align*}"}
+{"id": "6800.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n a _ i \\mathsf { X } _ i \\stackrel { \\kappa } { \\longrightarrow } \\sum _ { i = 1 } ^ n ( a _ i + c _ i ) \\mathsf { X } _ i \\sum _ { i = 1 } ^ n ( a _ i + c _ i ) \\geq 4 \\end{align*}"}
+{"id": "7707.png", "formula": "\\begin{align*} 0 = \\nabla _ { i } Q = \\frac { - h _ { t i } } { h - \\varepsilon _ { 0 } } + \\frac { h _ { t } h _ { i } } { ( h - \\varepsilon _ { 0 } ) ^ { 2 } } . \\end{align*}"}
+{"id": "4884.png", "formula": "\\begin{align*} h ^ 1 ( \\Sigma ' , \\O _ { \\Sigma ' } ( C ' ) ) = h ^ 1 ( \\Gamma , \\varphi ' _ * \\O _ { \\Sigma ' } ( C ' ) ) . \\end{align*}"}
+{"id": "7247.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n ( n - 1 ) ! t _ i Z _ { n - i } = n ! Z _ n , \\end{align*}"}
+{"id": "2181.png", "formula": "\\begin{align*} T ( \\mathcal { M } ) \\cong T ( \\mathcal { M } _ g ) \\coloneqq ( \\mathcal { M } _ g \\otimes _ { \\mathcal { O } _ { \\mathcal { E } , g } } \\widehat { \\mathcal { O } } ^ { \\mathrm { u r } } _ { \\mathcal { E } , g } ) ^ { \\varphi = 1 } \\end{align*}"}
+{"id": "2402.png", "formula": "\\begin{align*} v _ q \\left ( \\frac { ( 4 k ) ! ( 4 n - 4 k ) ! } { ( 2 k ) ! ( 2 n - 2 k ) ! k ! ^ 2 ( n - k ) ! ^ 2 } \\right ) & = \\lfloor 4 y \\rfloor + \\lfloor 4 x - 4 y \\rfloor - \\lfloor 2 y \\rfloor - \\lfloor 2 x - 2 y \\rfloor - 2 \\lfloor y \\rfloor - 2 \\lfloor x - y \\rfloor . \\end{align*}"}
+{"id": "2443.png", "formula": "\\begin{align*} \\lim _ { \\xi \\to a } \\left | \\dfrac { f ( \\xi ) } { g ( \\xi ) } \\right | = 1 . \\end{align*}"}
+{"id": "328.png", "formula": "\\begin{align*} \\underline { U } ( x , t ) = \\left ( \\frac { 1 } { 1 - p } \\right ) ^ { 1 / ( p - 1 ) } t ^ { 1 / ( 1 - p ) } ( 1 + | x | ) ^ { \\sigma / ( 1 - p ) } , \\end{align*}"}
+{"id": "8136.png", "formula": "\\begin{align*} h _ { A _ 2 } \\ = \\ c ^ { ( 2 ) } _ { i j } p _ { i } p _ { j } + c ^ { ( 1 ) } _ { i } p _ { i } + c ^ { ( 0 ) } \\ , \\end{align*}"}
+{"id": "8606.png", "formula": "\\begin{align*} v _ n ( t ; x ) = ( \\partial _ x ^ n u ) ( X ( t ; x ) , t ) ~ ~ { \\rm f o r } ~ ~ n = 0 , 1 , 2 , . . . \\end{align*}"}
+{"id": "1404.png", "formula": "\\begin{align*} S & = \\{ \\sum _ { k = 1 } ^ m t _ k \\mathbf { b } _ k \\mid t _ k \\in \\mathbb { Z } , 1 \\le k \\le m \\} , \\\\ & = \\{ ( t _ 1 , \\dots , t _ m ) \\mid t _ k \\in \\Z , 1 \\le k \\le m \\} = B , \\end{align*}"}
+{"id": "7398.png", "formula": "\\begin{align*} \\begin{array} { l } \\chi _ { I \\pm } ( x , t ; s ) : = \\chi _ { \\{ s : s - | t - s \\pm x | \\ge R \\} } , \\\\ \\chi _ { E \\pm } ( x , t ; s ) : = \\chi _ { \\{ s : | t - s \\pm x | \\le s + R \\} } \\end{array} \\end{align*}"}
+{"id": "6026.png", "formula": "\\begin{align*} \\Big ( D _ 1 \\bar { \\xi } ^ { A } _ x ( s ) + D _ 2 \\bar { \\xi } ^ { B } _ x ( s ) \\Big ) - \\Big ( D _ 1 \\bar { \\xi } ^ { A } _ x ( s ^ - ) + D _ 2 \\bar { \\xi } ^ { B } _ x ( s ^ - ) \\Big ) = D _ 2 - D _ 1 \\ , , \\end{align*}"}
+{"id": "5525.png", "formula": "\\begin{align*} \\| q - \\tilde q \\| _ { L _ 2 ( 0 , \\pi ) } \\le C _ r \\Xi + \\sqrt { \\frac { 2 } { \\pi } } \\big \\| \\{ \\xi _ n - \\tilde \\xi _ n \\} _ { n \\in \\Omega } \\big \\| , \\Xi : = \\big \\| \\{ n ( \\rho _ n - \\tilde \\rho _ n ) \\} _ { n \\in \\overline { \\Omega } } \\big \\| _ { \\bf a } , \\end{align*}"}
+{"id": "674.png", "formula": "\\begin{align*} \\left . \\frac { \\partial ^ { \\alpha } } { \\partial x ^ { \\alpha } } g _ { i _ { 1 } \\cdots i _ { m } } \\right | _ { x = x _ { 0 } } = 0 \\end{align*}"}
+{"id": "9032.png", "formula": "\\begin{align*} X ^ { \\alpha + 1 } & = ( X ^ \\alpha ) ^ 1 \\\\ X ^ \\beta & = \\bigcap _ { \\alpha < \\beta } X ^ \\alpha \\quad \\end{align*}"}
+{"id": "8235.png", "formula": "\\begin{align*} R & \\leq \\frac { 1 } { n } \\sum _ { i = 1 } ^ { \\ell } I ( { \\bf X } _ { i , 1 } ^ { L } ; Y _ { i , 1 } ^ { L } | { \\bf S } _ { i , 1 } ^ { L } ) + \\epsilon _ n \\\\ & = \\frac { \\ell } { n } I ( { \\bf X } _ { T , 1 } ^ { L } ; Y _ { T , 1 } ^ { L } | { \\bf S } _ { T , 1 } ^ { L } , T ) + \\epsilon _ n \\\\ & \\leq \\frac { 1 } { L } I ( { \\bf X } _ { T , 1 } ^ { L } ; Y _ { T , 1 } ^ { L } | { \\bf S } _ { T , 1 } ^ { L } ) + \\epsilon _ n \\end{align*}"}
+{"id": "3572.png", "formula": "\\begin{align*} ( 1 + a ) \\alpha ^ 2 \\tau _ 0 \\tau _ 1 ( f \\circ i d - f \\circ \\tau ^ 2 ) - ( a + b ) \\alpha \\tau _ 0 ( f \\circ i d - f \\circ \\tau ) + b ( f \\circ i d - f ) = 0 . \\end{align*}"}
+{"id": "1410.png", "formula": "\\begin{align*} E ^ \\gamma _ { \\alpha , \\beta } ( x ) & = \\frac { 1 } { \\Gamma ( \\gamma ) } \\sum _ { k = 0 } ^ \\infty \\frac { \\Gamma ( \\gamma + k ) } { k ! \\ , \\Gamma ( \\alpha k + \\beta ) } \\ , x ^ k \\end{align*}"}
+{"id": "7955.png", "formula": "\\begin{align*} ( i _ { \\mathcal { N } } u ) v _ { \\partial \\Omega } = ( i _ { \\mathcal { N } } u ) v _ { \\Sigma } = u ( \\mathcal { N } ) v _ { \\Sigma } = \\boldsymbol { n } ( u ) ( \\mathcal { N } ) v _ { \\Sigma } = ( \\partial \\Sigma _ { v } ) ^ { \\flat } ( \\mathcal { N } ) v _ { \\Sigma } = \\partial \\Sigma , \\end{align*}"}
+{"id": "5343.png", "formula": "\\begin{align*} v = \\min \\ , \\left \\{ \\sum \\limits _ { j \\in J } c _ j \\ , x ^ u _ j : u \\in \\mathcal { U } \\right \\} , \\end{align*}"}
+{"id": "2265.png", "formula": "\\begin{align*} \\check R _ 1 ( u ) \\check R _ { 2 } ( u v ) \\check R _ 1 ( v ) = \\check R _ { 2 } ( v ) \\check R _ 1 ( u v ) \\check R _ { 2 } ( u ) \\ \\ \\ \\ \\ \\end{align*}"}
+{"id": "4686.png", "formula": "\\begin{align*} \\varphi : = \\varphi _ { n - 1 } \\circ \\cdots \\circ \\varphi _ 0 , \\end{align*}"}
+{"id": "358.png", "formula": "\\begin{align*} \\begin{aligned} A _ p ( x , - y ) & = \\sum _ { i = 0 } ^ { k - 1 } ( - 1 ) ^ i ( x y ) ^ { i } ( x ^ { 2 ( k - i ) } + y ^ { 2 ( k - i ) } ) , \\\\ D _ p ( x , - y ) & = \\sum _ { i = 0 } ^ { k - 1 } ( - 1 ) ^ i ( x y ) ^ { i } ( x ^ { k - i } - y ^ { k - i } ) ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "4692.png", "formula": "\\begin{align*} \\mathrm { E f f } ( X ) = \\cup _ i \\mathcal { C } _ i , \\end{align*}"}
+{"id": "643.png", "formula": "\\begin{align*} n H _ r \\frac { \\cosh ^ { n - 1 } ( \\rho ) } { \\sinh ^ { r - 1 } ( \\rho ) } = \\frac { d } { d \\rho } \\left ( \\cosh ^ { n - r } ( \\rho ) \\frac { \\dot { \\mu } ^ r } { ( 1 + \\dot { \\mu } ^ 2 ) ^ { \\frac { r } { 2 } } } \\right ) , r = 1 , \\dots , n . \\end{align*}"}
+{"id": "2183.png", "formula": "\\begin{align*} \\bigcap _ { n = 1 } ^ { \\infty } ( \\mathfrak { S } ^ { ( 2 ) } + p ^ n \\mathfrak { S } ^ { ( 2 ) } [ E ^ { - 1 } ] ^ { \\wedge } _ p ) = \\mathfrak { S } ^ { ( 2 ) } . \\end{align*}"}
+{"id": "4414.png", "formula": "\\begin{gather*} \\max \\{ 0 , x _ i - \\overline { b } _ i , \\max \\{ 0 , x _ i - \\overline { b } _ i + \\Delta b _ i \\} - \\theta ^ k ( x ) \\} = \\max \\{ 0 , x _ i - \\overline { b } _ i , x _ i - \\overline { b } _ i + \\Delta b _ i - \\Delta b _ k \\} . \\end{gather*}"}
+{"id": "6442.png", "formula": "\\begin{align*} \\| k ^ \\Theta _ x \\| ^ 2 = \\frac { \\varphi ' ( x ) } { 2 \\pi } . \\end{align*}"}
+{"id": "5820.png", "formula": "\\begin{align*} d \\mu = \\sqrt { \\det g _ 0 } d x ^ 1 \\wedge \\dots \\wedge d x ^ n . \\end{align*}"}
+{"id": "1461.png", "formula": "\\begin{align*} \\underline { D } _ G ( S ) = \\lim _ { i \\in I } \\sup _ { g \\in G } \\frac { | S \\cap F _ i g | } { | F _ i | } = \\sup _ { F \\subset G , | F | < \\infty } \\inf _ { g \\in G } \\frac { | S \\cap F g | } { | F | } . \\end{align*}"}
+{"id": "5573.png", "formula": "\\begin{align*} \\overline { \\partial _ W f } = \\frac { d ^ 2 ( W W ^ * ) } { m n } \\ , \\overline { f } . \\end{align*}"}
+{"id": "634.png", "formula": "\\begin{align*} \\lambda _ { H _ r , d _ r } ( \\rho ) = \\int _ { \\rho _ - } ^ { \\rho } \\frac { ( n H _ r I _ { n , r } ( \\xi ) + d _ r ) ^ { \\frac { 1 } { r } } } { \\sqrt { \\sinh ^ { \\frac { 2 ( n - r ) } { r } } ( \\xi ) - ( n H _ r I _ { n , r } ( \\xi ) + d _ r ) ^ { \\frac { 2 } { r } } } } \\ , d \\xi , \\end{align*}"}
+{"id": "8903.png", "formula": "\\begin{align*} 1 + 2 \\sum _ { m = 1 } ^ \\infty \\frac { B _ { 2 m } } { ( 2 m ) ! } \\Omega _ m ( x _ 2 , \\dots , x _ { 2 m } ) u ^ { 2 m } : = \\exp \\left ( 2 \\sum _ { \\nu = 1 } ^ \\infty \\frac { B _ { 2 \\nu } } { ( 2 \\nu ) ! } \\left ( 2 \\sinh ^ { - 1 } \\left ( \\frac { u } { 2 } \\right ) \\right ) ^ { 2 \\nu } x _ { 2 \\nu } \\right ) . \\end{align*}"}
+{"id": "853.png", "formula": "\\begin{align*} U _ { i } = F ^ { - 1 } h X _ { i } - F ^ { - 1 } g ( X , l ) h l _ { i } , \\end{align*}"}
+{"id": "2844.png", "formula": "\\begin{align*} \\delta ( \\sigma ^ { * } _ { 2 } ( t ) I ) = \\int ^ { t _ { 2 } } _ { t _ { 1 } } \\left [ - \\dot { P } - Q \\right ] \\delta Q d t + \\left [ \\dot { Q } - P - 2 \\Theta _ { 1 } \\right ] \\delta P d t + \\left [ P \\delta Q \\right ] ^ { t _ { 2 } } _ { t _ { 1 } } . \\end{align*}"}
+{"id": "4593.png", "formula": "\\begin{align*} \\min _ { \\mathfrak { E } } s _ g = & 1 6 \\pi \\left ( 1 + \\sum a b \\right ) \\left ( - 1 + 4 \\sum a - 6 \\sum a ^ 2 + 4 \\sum a ^ 3 - \\sum a ^ 4 + 4 \\sum a ^ 3 b - 3 \\sum a ^ 2 b ^ 2 \\right ) \\Big / \\\\ [ 0 . 1 c m ] & \\left ( 1 - 6 \\sum a + 1 5 \\sum a ^ 2 + \\sum a b - 2 0 \\sum a ^ 3 + 4 \\sum a ^ 2 b + 1 5 \\sum a ^ 4 + 4 \\sum a ^ 3 b - 3 \\sum a ^ 2 b ^ 2 \\right . \\\\ [ 0 . 1 c m ] & \\left . - 6 \\sum a ^ 5 + 2 \\sum a ^ 4 b + 4 \\sum a ^ 3 b ^ 2 + \\sum a ^ 6 - 6 \\sum a ^ 5 b + 1 5 \\sum a ^ 4 b ^ 2 - 1 0 \\sum a ^ 3 b ^ 3 \\right ) \\end{align*}"}
+{"id": "481.png", "formula": "\\begin{align*} \\ker ^ { ( k ) } ( \\varphi ) : = \\{ w \\in G : \\varphi ^ { k } ( w ) = 1 _ G \\} . \\end{align*}"}
+{"id": "8092.png", "formula": "\\begin{align*} e _ { k , \\ell , s } ( \\psi f ) \\leq \\norm { \\psi } _ { C ^ { 2 k } } \\sum _ { i = 0 } ^ \\ell e _ { k , i , s } ( f ) . \\end{align*}"}
+{"id": "6046.png", "formula": "\\begin{align*} \\phi _ x ^ + = - \\frac 1 2 \\bar \\xi ^ { A } _ x + \\frac 1 2 \\bar \\xi ^ { B } _ x , \\phi _ x ^ - = \\frac 1 2 \\bar \\xi ^ { A } _ x + \\frac 1 2 \\bar \\xi ^ { B } _ x . \\end{align*}"}
+{"id": "2275.png", "formula": "\\begin{align*} h x - x h = 2 x \\ , , \\ \\ h y - y h = - 2 y \\ , , \\ \\ \\ x y - y x = \\frac { e ^ { \\alpha h } - e ^ { - \\alpha h } } { e ^ { \\alpha } - e ^ { - \\alpha } } \\ . \\end{align*}"}
+{"id": "2731.png", "formula": "\\begin{align*} L \\rightarrow \\tilde { L } = L ( \\dot { Q } _ { ( d - 1 ) } ^ { i } , Q _ { ( d - 1 ) } ^ { i } , \\cdots , Q _ { ( 2 ) } ^ { i } , Q _ { ( 1 ) } ^ { i } , Q _ { ( 0 ) } ^ { i } ) + \\sum _ { \\alpha = 1 } ^ { d - 1 } \\lambda ^ { ( \\alpha ) } _ { i } \\left ( Q _ { ( \\alpha ) } ^ { i } - \\dot { Q } _ { ( \\alpha - 1 ) } ^ { i } \\right ) , \\end{align*}"}
+{"id": "1696.png", "formula": "\\begin{align*} T M = \\mathrm { s p a n } _ { \\R } \\left \\{ \\frac { \\partial } { \\partial y _ 0 } , X _ 1 + \\overline { X _ 1 } , \\ldots , X _ n + \\overline { X _ n } , i X _ 1 - i \\overline { X _ 1 } , \\ldots , i X _ n - i \\overline { X _ n } \\right \\} , \\end{align*}"}
+{"id": "438.png", "formula": "\\begin{align*} F ' ( x ^ * ) + M ^ T z ^ * = F ' ( x ^ * ) - D z ^ * = 0 . \\end{align*}"}
+{"id": "3545.png", "formula": "\\begin{align*} T ( i d ^ 6 ) = T ( i d ^ 2 ) ^ 3 = f _ 2 ^ 3 = T ( i d ^ 3 ) ^ 2 = f _ 3 ^ 2 . \\end{align*}"}
+{"id": "3640.png", "formula": "\\begin{align*} f ( z _ 0 , w _ 0 \\sigma ( z _ 0 ) ) = L _ 1 ( w _ 0 \\sigma ( z _ 0 ) ) = 0 , ~ ~ \\mbox { a n d } ~ ~ f ( z _ 0 , w _ 0 \\sigma ( z _ 0 ) ) = L _ 2 ( w _ 0 \\sigma ( z _ 0 ) ) = 1 , \\end{align*}"}
+{"id": "5880.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { C } ( G ) ) = n ( m - 1 ) ( m n - n - 1 ) ^ { 2 } + m n ( n - 1 ) ^ { 2 } , M _ { 2 } ( \\mathcal { C } ( G ) ) = \\dfrac { ( m n - n ) ( m n - n - 1 ) ^ { 3 } + m n ( n - 1 ) ^ { 3 } } { 2 } , \\end{align*}"}
+{"id": "1969.png", "formula": "\\begin{align*} E q u & > p ^ s + 1 - ( \\tau + 1 ) p - ( p - \\tau ) ( \\tau + 2 ) p ^ { s - 2 } \\\\ & = p ^ { s - 1 } ( p - \\tau - 2 ) + p ( ( \\tau ^ 2 + 2 \\tau ) p ^ { s - 3 } - \\tau - 1 ) + 1 \\\\ & \\geq p ( \\tau ^ 2 + \\tau - 1 ) + 1 > 0 \\end{align*}"}
+{"id": "1730.png", "formula": "\\begin{align*} F ( z ) : = \\sum _ { j = 1 } ^ { n - 1 } \\left [ ( x _ 1 , \\ldots , x _ { n - 1 } ) Q _ { j } \\left ( \\begin{array} { c } x _ 1 \\\\ \\vdots \\\\ x _ { n - 1 } \\end{array} \\right ) \\right ] x _ n ^ { j - 1 } \\end{align*}"}
+{"id": "4009.png", "formula": "\\begin{align*} \\begin{cases} x _ { 1 } & = \\ ; \\bigl ( \\gamma _ { 1 } x _ { 1 } + \\delta _ { 1 } x _ { 2 } \\bigr ) y \\\\ x _ { 2 } & = \\ ; \\bigl ( \\gamma _ { 2 } x _ { 1 } + \\delta _ { 2 } x _ { 2 } \\bigr ) y \\\\ y & = \\ ; \\bigl ( \\gamma x _ { 1 } + \\delta x _ { 2 } \\bigr ) y \\end{cases} \\end{align*}"}
+{"id": "6922.png", "formula": "\\begin{align*} \\C ^ { p + r } = E ^ s ( z ) \\oplus E ^ u ( z ) . \\end{align*}"}
+{"id": "2086.png", "formula": "\\begin{align*} \\eqref { e q : t e s t f u n c t i o n a p p l i e d s u m } & = \\alpha \\int _ 0 ^ s f ( u ) d \\xi _ 1 ( u , x ) + o ( 1 ) , \\end{align*}"}
+{"id": "1109.png", "formula": "\\begin{align*} \\widetilde w : = w ^ { - \\frac { 1 } { p - 1 } } = | x - x _ 0 | ^ { - \\widetilde d } | x | ^ { \\frac { d } { p - 1 } } \\end{align*}"}
+{"id": "4077.png", "formula": "\\begin{align*} \\chi R _ 0 ( z _ 0 ) \\chi = \\sum _ { j = 1 } ^ { \\frac { d - 1 } { 2 } } \\tilde { B } _ j ( r , - z _ 0 ) + 2 i \\Big ( \\frac { z _ 0 } { 2 \\pi } \\Big ) ^ { d - 1 } \\mathcal { E } ^ * _ { \\chi } ( - \\bar { z _ 0 } ) \\mathcal { E } _ { \\chi } ( - z _ 0 ) \\end{align*}"}
+{"id": "6578.png", "formula": "\\begin{align*} \\log \\zeta ( j ) = - \\sum _ { p \\leq x } \\log \\left ( 1 - \\frac { 1 } { p ^ j } \\right ) + O \\left ( \\int _ { x } ^ { \\infty } \\frac { 1 } { t ^ { j } } d t \\right ) . \\end{align*}"}
+{"id": "4621.png", "formula": "\\begin{align*} K _ j : = \\left \\{ 0 , 1 , 2 , 3 , \\dots , 2 ^ { j m } - 1 \\right \\} ^ { d - 1 } \\end{align*}"}
+{"id": "9185.png", "formula": "\\begin{align*} e ^ + _ { i j } ( u ) = \\sum _ { r > 0 } e ^ { ( - r ) } _ { i j } u ^ r , e ^ - _ { i j } ( u ) = \\sum _ { r \\geq 0 } e ^ { ( r ) } _ { i j } u ^ { - r } \\forall \\ 1 \\leq i < j \\leq N . \\end{align*}"}
+{"id": "7737.png", "formula": "\\begin{align*} g ( \\lambda , u ) = \\left ( \\prod _ { i = 1 } ^ l g _ { i } ( \\lambda , u ) \\right ) \\left ( \\prod _ { i , j = 1 , i \\neq j } ^ l \\mathrm { R e s } ( \\chi _ { \\Sigma _ { i } ( \\lambda ) } , \\chi _ { \\Sigma _ { j } ( \\lambda ) } ; u ) \\right ) . \\end{align*}"}
+{"id": "4525.png", "formula": "\\begin{align*} \\tau ( s _ 1 , s _ 2 , . . . , s _ N ) = ( 0 , s _ 1 , s _ 2 , . . . , s _ { N - 1 } ) , \\tau ( s _ 1 , s _ 2 , . . . , s _ N , . . . ) = ( 0 , s _ 1 , s _ 2 , . . . , s _ N , . . . ) \\end{align*}"}
+{"id": "2480.png", "formula": "\\begin{align*} \\dfrac { d s } { d \\xi } = \\dfrac { 1 } { m } \\phi ^ { - m } \\end{align*}"}
+{"id": "6850.png", "formula": "\\begin{align*} j ( m ) = \\begin{cases} \\left ( \\frac { m - 1 } { 2 } \\right ) ^ 2 \\qquad \\qquad \\qquad \\\\ \\left ( \\frac { m - 2 } { 2 } \\right ) ^ 2 + 1 \\qquad \\ ; \\ , \\qquad \\end{cases} \\end{align*}"}
+{"id": "5297.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } K L I ( q \\| p ) } { \\partial q _ { j } } = - \\sum _ { j } p _ { j } \\sum _ { i } \\left [ \\frac { a } { a - b } \\left ( \\frac { \\overline { q } _ { i } } { \\overline { p } _ { i } } \\right ) ^ { a - 1 } - \\frac { b } { a - b } \\left ( \\frac { \\overline { q } _ { i } } { \\overline { p } _ { i } } \\right ) ^ { b - 1 } \\right ] \\frac { \\partial \\overline { q } _ { i } } { \\partial q _ { j } } \\end{align*}"}
+{"id": "2619.png", "formula": "\\begin{align*} s _ { i + 2 } - s _ i = s _ { j + 1 } - s _ j , \\ , \\ , \\ , \\ , s _ { i + 3 } - s _ { i + 1 } = s _ { j ' + 1 } - s _ j , \\end{align*}"}
+{"id": "5874.png", "formula": "\\begin{align*} \\dfrac { M _ { 1 } ( \\mathcal { C } ( G ) ) } { | v ( \\mathcal { C } ( G ) ) | } = \\dfrac { ( 4 n - 4 ) ( 4 n - 5 ) ^ { 2 } + 3 6 n } { 8 n - 4 } \\dfrac { M _ { 2 } ( \\mathcal { C } ( G ) ) } { | e ( \\mathcal { C } ( G ) ) | } = \\dfrac { ( 2 n - 2 ) ( 4 n - 5 ) ^ { 3 } + 5 4 n } { ( 2 n - 2 ) ( 4 n - 5 ) + 6 n } . \\end{align*}"}
+{"id": "3655.png", "formula": "\\begin{align*} a _ d ( K _ n ) = \\begin{cases} 1 & d + 1 \\leq n \\leq 2 d , \\\\ \\frac { n } { 2 d } & n \\geq 2 d . \\end{cases} \\end{align*}"}
+{"id": "1452.png", "formula": "\\begin{align*} h _ R ( x _ { - 1 } , x _ 0 , x _ 1 ) = x _ { - 1 } \\overline { x _ 1 } + x _ 0 \\overline { x _ 0 } + x _ 1 \\overline { x _ { - 1 } } . \\end{align*}"}
+{"id": "664.png", "formula": "\\begin{align*} ( R ^ { k } f ) _ { p _ { 1 } q _ { 1 } \\dots p _ { m - k } q _ { m - k } } ^ { i _ { 1 } \\dots i _ { k } } : = \\alpha ( p _ { 1 } q _ { 1 } ) \\dots \\alpha ( p _ { m - k } q _ { m - k } ) \\frac { \\partial ^ { m - k } f ^ { i _ { 1 } \\dots i _ { k } } _ { p _ { 1 } \\dots p _ { m - k } } } { \\partial x ^ { q _ 1 } \\dots \\partial x ^ { q _ { m - k } } } \\end{align*}"}
+{"id": "6288.png", "formula": "\\begin{align*} \\frac 1 T \\sum \\limits _ { k = 0 } ^ { T - 1 } \\langle \\hat { g } _ { k + 1 } , x _ k - x ^ * \\rangle \\leq \\frac 1 2 \\frac { R _ { 0 } ^ { 2 } } { \\nu T } + \\frac { \\nu } { 2 } \\frac { 1 } { T } \\sum \\limits _ { k = 0 } ^ { T - 1 } \\| \\hat { g } _ { k + 1 } \\| ^ { 2 } _ q . \\end{align*}"}
+{"id": "1132.png", "formula": "\\begin{align*} \\widehat \\tau : = \\left [ \\tau - \\frac { 1 } { p } \\left ( 1 - \\frac { d } { n } \\right ) \\right ] _ + . \\end{align*}"}
+{"id": "2628.png", "formula": "\\begin{align*} P = \\{ - n , - n + 1 , \\dots , n - 1 , n \\} \\times D _ i ( S ) . \\end{align*}"}
+{"id": "8450.png", "formula": "\\begin{align*} \\| v \\| _ { L ^ q ( t _ 1 , t _ 2 \\ , ; \\ , W _ { 0 } ^ { s , p } ( K ) ) } : = \\left ( \\int _ { t _ 1 } ^ { t _ 2 } \\| v ( t ) \\| _ { W ^ { s , p } ( K ) } ^ { q } \\ , d t \\right ) ^ { 1 / q } \\end{align*}"}
+{"id": "8159.png", "formula": "\\begin{align*} c _ { j } & = M \\sum _ { \\underline x ^ L } \\sum _ { y ^ L \\in \\beta _ j ^ L } P _ { \\underline X ^ L , Y ^ L } ( \\underline x ^ L , y ^ L ) \\\\ & = M \\sum _ { \\underline x ^ L } \\sum _ { y ^ L \\in \\beta _ j ^ L } \\frac { | \\mathcal B _ { y ^ L } ( \\underline x ^ L ) | } { M } P _ { { \\underline X } ^ { L } | | Y ^ { L - 1 } } ( { \\underline x } ^ { L } | | y ^ { L - 1 } ) \\end{align*}"}
+{"id": "1198.png", "formula": "\\begin{align*} & \\left | \\partial _ x ^ \\alpha \\partial _ y ^ \\beta K ( x , y ) - \\partial _ x ^ \\alpha \\partial _ y ^ \\beta K ( x , y + v ) \\right | \\\\ & \\quad \\lesssim | v | ^ { \\rho } | x - y | ^ { - n - | \\alpha | - | \\beta | - \\rho } \\quad \\begin{cases} | \\alpha | \\leq \\lfloor \\widetilde s \\rfloor , \\\\ | \\beta | = \\lfloor \\widetilde J \\rfloor - n - | \\alpha | . \\end{cases} \\end{align*}"}
+{"id": "1880.png", "formula": "\\begin{align*} \\lambda _ { k , l } [ C _ p ^ + ] = \\frac { \\pi ^ n } { 2 ^ p } \\frac { \\Gamma ( p + 1 ) } { \\Gamma \\left ( n + \\frac { p + k + l } { 2 } \\right ) \\Gamma \\left ( \\frac { p - k - l } { 2 } + 1 \\right ) } \\end{align*}"}
+{"id": "110.png", "formula": "\\begin{align*} \\lambda = \\frac { \\nu } { 1 - 2 \\nu } , 0 \\le \\nu < \\frac 1 2 . \\end{align*}"}
+{"id": "605.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\left ( \\frac { 1 } { 3 2 } \\right ) ^ { k } \\binom { 2 k } { k } ^ 2 \\frac { H _ { 2 k } } { k + 1 } = 4 - \\frac { 3 \\Gamma ^ 2 \\left ( \\frac { 1 } { 4 } \\right ) } { 2 \\pi ^ { 3 / 2 } } - \\frac { 2 \\sqrt { \\pi } ( \\pi + 3 \\ln ( 2 ) - 4 ) } { \\Gamma ^ 2 \\left ( \\frac { 1 } { 4 } \\right ) } , \\end{align*}"}
+{"id": "5925.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { C } ( A ( n , p ) ) ) = ( p ^ { n } + 1 ) \\dfrac { ( p ^ { 2 n } - p ^ { n } ) ( p ^ { 2 n } - p ^ { n } - 1 ) ^ { 3 } } { 2 } = \\dfrac { p ^ { n } ( p ^ { 2 n } - 1 ) ( p ^ { 2 n } - p ^ { n } - 1 ) ^ { 3 } } { 2 } . \\end{align*}"}
+{"id": "5763.png", "formula": "\\begin{align*} X ^ 2 _ 0 ( t ) = & \\sum _ { k + \\ell \\leq s } \\sum _ { 1 \\leq j \\leq J } | z ^ { ( k , \\ell ) } _ j ( t ) | ^ 2 , \\ X ^ 2 _ - ( t ) = \\sum _ { k + \\ell \\leq s } \\sum _ { i \\in - \\mathbb { N } } | \\xi ^ { ( k , \\ell ) } _ i ( t ) | ^ 2 , \\end{align*}"}
+{"id": "3178.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ n ( 1 + q ^ { a _ i } ) = \\sum _ { m = 0 } ^ N b _ n ( m ) q ^ m , N = \\sum _ { i = 1 } ^ n a _ i , \\end{align*}"}
+{"id": "685.png", "formula": "\\begin{align*} U : = & f ( i ) f ( - i ) - g ( i ) g ( - i ) - h ( i ) h ( - i ) + t ( i ) t ( - i ) , \\\\ V : = & f ( i ) h ( - i ) + f ( - i ) h ( i ) - g ( i ) t ( - i ) - g ( - i ) t ( i ) . \\end{align*}"}
+{"id": "1528.png", "formula": "\\begin{align*} d ( z ) = \\frac { | z | ^ { p } } { p } , s ( z ) = \\frac { | z - w | ^ { q } } { q } , \\end{align*}"}
+{"id": "86.png", "formula": "\\begin{align*} X _ 1 ^ \\prime ( \\theta ) = & p - 2 - \\gamma - \\frac { ( p - 2 ) R _ \\theta - \\gamma P _ \\theta } { \\sqrt { P _ \\theta R _ \\theta } } \\\\ = & \\frac { ( \\sqrt { P _ \\theta } - \\sqrt { R _ \\theta } ) \\big ( ( p - 2 ) \\sqrt { R _ \\theta } + \\gamma \\sqrt { P _ \\theta } \\big ) } { \\sqrt { P _ \\theta R _ \\theta } } . \\end{align*}"}
+{"id": "3119.png", "formula": "\\begin{align*} d _ { ( 2 ) ^ * } ( \\varphi ) & : = ( \\ , \\dim _ k \\ , ( V ) ^ \\varphi _ { \\ell _ 1 } , \\ ; \\dim _ k \\ , ( V ) ^ \\varphi _ { \\ell _ 3 } \\ , ) , \\\\ d ' _ { ( 2 ) ^ * } ( \\varphi ) & : = ( \\ , \\dim _ k \\ , ( V ' ) ^ \\varphi _ { \\ell _ 1 } , \\ ; \\dim _ k \\ , ( V ' ) ^ \\varphi _ { \\ell _ 3 } \\ , ) . \\end{align*}"}
+{"id": "746.png", "formula": "\\begin{align*} S _ F ( f ) : \\Gamma ( T M _ 0 ) & \\longrightarrow \\Gamma ( \\pi ^ * T M ) , \\\\ S _ F ( f ) \\hat { X } & = { } ^ c \\nabla _ { \\hat { X } } ( \\nabla \\varphi ) - g ( \\nabla \\varphi , \\varrho \\hat { X } ) \\nabla \\varphi - \\frac { 1 } { n } ( \\Delta \\varphi - \\| \\nabla \\varphi \\| ^ 2 ) \\varrho \\hat { X } , \\end{align*}"}
+{"id": "7041.png", "formula": "\\begin{align*} g ( \\omega _ t ) - g ( x ) = \\sum _ { s \\leq t } \\bigl ( g ( \\omega _ s ) - g ( \\omega _ { s - } ) \\bigr ) + S _ t , \\end{align*}"}
+{"id": "2448.png", "formula": "\\begin{align*} & \\omega _ { 1 } = \\dfrac { - \\mu c + \\sqrt { D } } { 2 } < 0 , \\omega _ { 2 } = \\dfrac { - \\mu c - \\sqrt { D } } { 2 } < 0 , \\omega = - \\dfrac { \\mu c } { 2 } < 0 , D = \\mu ^ { 2 } c ^ { 2 } - 4 k . \\\\ & \\bar { Z } ( \\xi ) = B _ { 5 } \\cdot \\sin [ \\frac { \\sqrt { | D | } } { 2 } \\xi ] + B _ { 6 } \\cdot \\cos [ \\frac { \\sqrt { | D | } } { 2 } \\xi ] \\end{align*}"}
+{"id": "8836.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } \\frac { \\partial { \\bf u } } { \\partial t } & = & D \\frac { \\partial ^ 2 { \\bf u } } { \\partial x ^ 2 } - \\alpha _ 0 ^ * { \\bf u } , & t > 0 , \\\\ { \\bf u } ( 0 , \\cdot ) & = & \\varphi \\in C , & \\end{array} \\right . \\end{align*}"}
+{"id": "1740.png", "formula": "\\begin{align*} S _ { j , k } = 1 \\quad \\quad \\forall \\ j + k = n \\end{align*}"}
+{"id": "4598.png", "formula": "\\begin{align*} N \\left ( - \\Delta ^ D _ \\Omega - \\lambda \\right ) = \\frac { | B _ 1 ( 0 ) | } { ( 2 \\pi ) ^ d } | \\Omega | \\lambda ^ \\frac { d } { 2 } + o \\left ( \\lambda ^ \\frac { d } { 2 } \\right ) \\mathrm { \\ a s \\ } \\lambda \\rightarrow \\infty , \\end{align*}"}
+{"id": "6149.png", "formula": "\\begin{align*} \\begin{aligned} & r ^ k \\left ( \\| v ^ { k + 1 } - v ' \\| ^ 2 _ { H ^ k } + \\sigma \\| z ^ k - z ' \\| ^ 2 _ R - \\| v ^ k - v ' \\| ^ 2 _ { H ^ k } + \\| v ^ k - \\widetilde { v } ^ k \\| ^ 2 _ { G ^ k } \\right ) \\\\ & \\geq \\| v ^ { k + 1 } - v ' \\| ^ 2 _ { H _ 0 ^ { k + 1 } } - \\| v ^ k - v ' \\| ^ 2 _ { H _ 0 ^ k } + \\varTheta ^ { k + 1 } - \\varTheta ^ k , ~ \\varTheta ^ k \\geq 0 . \\end{aligned} \\end{align*}"}
+{"id": "4737.png", "formula": "\\begin{align*} J : X \\to 2 ^ { X ^ * } , x \\mapsto \\left \\{ y \\in X ^ * \\mid \\langle x , y \\rangle = \\norm { x } ^ 2 = \\norm { y } ^ 2 \\right \\} \\end{align*}"}
+{"id": "7577.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\rightarrow \\infty } E _ { p , q } ( v _ { n } ^ { 1 } ) = 0 . \\end{align*}"}
+{"id": "5167.png", "formula": "\\begin{align*} - \\frac { \\partial L D _ { \\alpha } I ( p \\| q ) } { \\partial q _ { j } } = - \\frac { 1 } { \\alpha } \\left [ - \\frac { 1 } { X } \\left ( \\sum _ { i } p ^ { \\frac { 1 } { \\alpha } } _ { i } q ^ { 1 - \\alpha } _ { i } \\right ) ^ { \\frac { 1 } { \\alpha } - 1 } p ^ { \\alpha } _ { j } q ^ { - \\alpha } _ { j } + \\frac { 1 } { Y } \\left ( \\sum _ { i } q _ { i } \\right ) ^ { - \\frac { 1 } { \\alpha } } \\right ] \\end{align*}"}
+{"id": "3541.png", "formula": "\\begin{align*} ( T f ) ( z ) \\tau ^ 3 ( z ) = f ( \\tau ( z ) ) \\tau ^ 3 ( z ) , \\end{align*}"}
+{"id": "1283.png", "formula": "\\begin{align*} \\bar { m } ( t ) = e ^ { - \\alpha t } x _ 0 = e ^ { - \\lambda _ 2 t } x _ 0 , \\end{align*}"}
+{"id": "8.png", "formula": "\\begin{align*} \\lim _ { y \\to \\infty } \\pi y f ( i y ) = \\lim _ { y \\to \\infty } \\int \\frac { y ^ 2 } { x ^ 2 + y ^ 2 } f ( x ) \\ , d x = \\lim _ { \\xi \\downarrow 0 } \\tfrac { \\sqrt { 2 \\pi } } { 2 } \\widehat f ( \\xi ) \\end{align*}"}
+{"id": "1042.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { [ n \\delta ] } \\Delta _ n ^ { s , t } \\le M _ 6 n ^ { - d } , n \\in \\N , \\ \\ t \\in \\{ 1 , \\dots , n \\} . \\end{align*}"}
+{"id": "8119.png", "formula": "\\begin{align*} U _ w = \\prod _ { \\alpha \\in { \\mathcal I } ( w ) } U _ { \\alpha } , \\end{align*}"}
+{"id": "2502.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\| A ( t ) - A _ \\infty \\| _ 1 = 0 \\ , . \\end{align*}"}
+{"id": "1439.png", "formula": "\\begin{gather*} \\Omega ^ 1 _ Y ( \\log \\Sigma ) \\simeq \\mathcal { O } _ Y \\frac { d t _ 1 } { t _ 1 } \\oplus ( \\bigoplus _ { i = 2 } ^ k \\mathcal { O } _ Y d t _ i ) , \\\\ \\Omega ^ 1 _ X ( \\log E ) \\simeq ( \\bigoplus _ { i = 1 } ^ l \\mathcal { O } _ X \\frac { d x _ i } { x _ i } ) \\oplus ( \\bigoplus _ { i = l + 1 } ^ n \\mathcal { O } _ X d x _ i ) \\end{gather*}"}
+{"id": "2442.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } \\left [ u \\varphi _ { \\xi } ( c - u _ { \\xi } ) + ( \\gamma + 1 ) ( u _ { \\xi } ) ^ { 2 } \\varphi - u ^ { 2 } ( \\varphi _ { \\xi \\xi } + k \\varphi ) + \\delta p u \\varphi \\right ] \\ , d \\xi = 0 \\end{align*}"}
+{"id": "7742.png", "formula": "\\begin{align*} \\Sigma ' ( \\lambda ) : = \\Sigma ( A ' ( \\lambda ) ) \\subseteq D ( R ' ) \\end{align*}"}
+{"id": "1059.png", "formula": "\\begin{align*} D _ k ( n , s , t ) & : = \\sum _ { \\ell = 0 } ^ { \\infty } \\Vert a _ { n + 1 - s + \\ell } \\Vert \\left \\Vert \\sum _ { u = 1 } ^ t ( \\tilde { b } _ { n , u , \\ell } ^ { 2 k - 1 } ) ^ * \\tilde { a } _ { t - u } \\right \\Vert , k \\in \\N , \\\\ E _ k ( n , s , t ) & : = \\sum _ { \\ell = 0 } ^ { \\infty } \\Vert \\tilde { a } _ { s + \\ell } \\Vert \\left \\Vert \\sum _ { u = 1 } ^ t ( \\tilde { b } _ { n , u , \\ell } ^ { 2 k } ) ^ * \\tilde { a } _ { t - u } \\right \\Vert , k \\in \\N . \\end{align*}"}
+{"id": "9154.png", "formula": "\\begin{align*} G _ { \\beta , \\beta ' } = ( w _ { \\beta , 1 } - v ^ { 2 } w _ { \\beta ' , 1 } ) \\cdot G _ { \\beta , \\gamma } , \\end{align*}"}
+{"id": "415.png", "formula": "\\begin{align*} v ( \\eta , \\zeta ) = \\frac { \\eta + 1 + \\zeta - \\sqrt { ( 1 + \\zeta + \\eta ) ^ 2 - 4 \\eta } } { 2 } . \\end{align*}"}
+{"id": "9193.png", "formula": "\\begin{align*} e ^ - _ { 1 , j + 1 } ( z ) = ( 1 - q ^ 2 ) ^ { - 1 } \\cdot [ e ^ - _ { 1 j } ( z ) , e ^ { ( 0 ) } _ { j , j + 1 } ] _ q . \\end{align*}"}
+{"id": "5745.png", "formula": "\\begin{align*} \\xi _ { i , 1 } ( t ) + \\mathbf { i } \\xi _ { i , 2 } ( t ) = w _ i e ^ { ( \\gamma _ * + \\mathbf { i } \\beta _ i ) t } + O ( e ^ { ( \\gamma _ * - \\varepsilon _ 0 ) t } ) \\end{align*}"}
+{"id": "8564.png", "formula": "\\begin{align*} F ( n , k ) = \\frac { ( a ) _ k ( b ) _ k } { ( n ) _ k ( a + n ) _ k } , \\end{align*}"}
+{"id": "3079.png", "formula": "\\begin{align*} \\varphi _ a ( t ) = \\left ( t ^ { \\frac { \\beta _ 0 } { e _ j } } , \\sum _ { \\beta _ 1 \\leq l < \\beta _ j } c _ l t ^ \\frac { l } { e _ j } + \\sum _ { l \\geq \\frac { \\beta _ j } { e _ j } } a _ l t ^ l \\right ) \\end{align*}"}
+{"id": "1495.png", "formula": "\\begin{align*} & f _ 2 ( m ) f _ 2 ( n ) [ m - n ] - f _ 2 ( m ) f _ 2 ( m + n ) [ m + d - n ] = 0 , \\\\ & f _ 2 ( m ) g _ 2 ( n ) [ m - n ] - f _ 2 ( n ) g _ 2 ( m ) [ n - m ] - f _ 2 ( m ) g _ 2 ( m + n ) [ m + d - n ] = 0 . \\end{align*}"}
+{"id": "4001.png", "formula": "\\begin{align*} V ^ { t } \\left ( z \\right ) = \\frac { 1 } { \\varpi \\circ W ^ { t } \\left ( z \\right ) } W ^ { t } \\left ( z \\right ) . \\end{align*}"}
+{"id": "232.png", "formula": "\\begin{align*} \\frac { d h } { d x } + \\gamma _ 0 ( x ) h = 0 , \\end{align*}"}
+{"id": "2817.png", "formula": "\\begin{align*} L _ { T } = & \\sigma ^ { * } _ { 2 } ( t ) \\left ( \\Theta _ { 1 } \\dot { \\Theta } ^ { 1 } + P _ { 1 } \\dot { Q } ^ { 1 } - H _ { T } \\right ) \\\\ = & P _ { 1 } \\dot { Q } ^ { 1 } - \\frac { 1 } { 2 } ( Q ^ { 1 } ) ^ { 2 } - \\frac { 1 } { 2 } ( P _ { 1 } ) ^ { 2 } + \\frac { 1 } { 2 } ( \\Theta ^ { 1 } ) ^ { 2 } + \\frac { 1 } { 2 } ( \\Theta _ { 1 } ) ^ { 2 } , \\\\ & \\therefore L _ { T } = P _ { 1 } \\dot { Q } ^ { 1 } - \\frac { 1 } { 2 } ( Q ^ { 1 } ) ^ { 2 } - \\frac { 1 } { 2 } ( P _ { 1 } ) ^ { 2 } + { c o n s t a n t } , \\end{align*}"}
+{"id": "7767.png", "formula": "\\begin{align*} r _ { n + 1 } & = m ^ { 2 } r _ { n } , \\ \\ \\nu _ { n + 1 } = u _ { s _ k } , \\\\ \\delta _ { n + 1 } & = { b } ^ { - 1 } R _ n ^ { - 6 m } ( \\nu _ { n + 1 } \\tilde M _ n ^ { - 1 } ) ^ { \\kappa } \\delta _ { n } , \\\\ \\tilde { M } _ { n + 1 } & = b ( \\nu _ { n + 1 } ^ { - 1 } \\tilde { M } _ { n } ) ^ { \\kappa } , \\\\ c _ { n + 1 } & = ( { b } ^ { - 1 } R _ { n } ^ { - 1 } r _ { n } ^ { - 1 } \\delta _ { n } \\nu _ { n + 1 } \\tilde M _ n ^ { - 1 } c _ { n } ) ^ { \\kappa ^ { r _ { n } } } - r _ { n } ^ { \\kappa r _ { n } } \\epsilon _ { n } . \\end{align*}"}
+{"id": "6229.png", "formula": "\\begin{gather*} \\lim _ { \\xi \\to \\xi _ 1 ^ + } \\phi _ { 1 , \\beta } ( \\xi ) = 1 , \\ \\lim _ { \\xi \\to \\{ \\xi _ \\beta ^ 1 \\} ^ - } \\phi _ { 1 , \\beta } ( \\xi ) = \\beta , \\hbox { a n d } \\lim _ { \\xi \\to \\{ \\xi _ \\beta ^ 2 \\} ^ + } \\phi _ { \\beta , \\gamma } ( \\xi ) = \\beta \\ \\lim _ { \\xi \\to \\{ \\xi _ \\gamma ^ 1 \\} ^ - } \\phi _ { \\beta , \\gamma } ( \\xi ) = \\gamma . \\end{gather*}"}
+{"id": "2464.png", "formula": "\\begin{align*} \\phi ( s ) = x ( s ) / \\lambda ( s ) , \\psi ( s ) = 1 / \\lambda ( s ) . \\end{align*}"}
+{"id": "3502.png", "formula": "\\begin{align*} \\| \\varphi _ m \\circ \\rho _ { m , n } \\circ \\psi _ n - \\varphi _ n \\circ \\psi _ n \\| < \\textstyle { \\sum } _ { j = n + 1 } ^ m \\varepsilon _ j . \\end{align*}"}
+{"id": "6887.png", "formula": "\\begin{align*} ( g _ B \\cdot f _ 1 \\otimes f _ 2 ) ( u _ 1 \\otimes u _ 2 ) & = g _ B ( f _ 1 ( u _ 1 ) \\otimes f _ 2 ( u _ 2 ) ) \\\\ & = f _ 2 ( u _ 2 ) \\otimes f _ 1 ( u _ 1 ) \\\\ & = ( f _ 2 \\otimes f _ 1 ) ( g _ A ( u _ 1 \\otimes u _ 2 ) ) \\\\ & = ( f _ 2 \\otimes f _ 1 \\cdot g _ A ) ( u _ 1 \\otimes u _ 2 ) \\end{align*}"}
+{"id": "7986.png", "formula": "\\begin{align*} \\delta u = 0 \\Omega , \\quad \\ast \\boldsymbol { n } ( u ) = e _ { \\phi } ^ { j } \\partial \\Omega . \\end{align*}"}
+{"id": "1768.png", "formula": "\\begin{align*} \\pi _ r \\circ \\mathrm { F l } ^ { X _ { n } } _ { t } ( z ) = z _ { r } + \\delta _ { r , n } 2 t + \\sum _ { j = 2 } ^ { n - 1 } \\delta _ { r , j } \\beta _ { n , j } ( t ) , \\end{align*}"}
+{"id": "3696.png", "formula": "\\begin{align*} \\lambda - { s a _ { \\ell - 1 } ( \\lambda ) } / { a _ { \\ell } ( \\lambda ) } = \\mu . \\end{align*}"}
+{"id": "4352.png", "formula": "\\begin{gather*} \\mathcal { M } : = \\{ ( k _ 1 , k _ 2 , k _ 3 , k _ 4 ) \\in [ n ] ^ 4 : \\ k _ 1 < k _ 2 , k _ 3 < k _ 4 \\} \\cup \\{ ( 0 , 0 , 0 , 0 ) \\} . \\end{gather*}"}
+{"id": "6427.png", "formula": "\\begin{align*} \\sup _ { ( \\delta , \\alpha ) \\in W _ n ^ { ( \\eta ) } } | | \\int _ 0 ^ 1 \\nabla _ { ( \\delta , \\alpha ) } \\overline { G } _ n ( a _ 0 , b _ 0 , \\delta _ 0 + t ( \\delta - \\delta _ 0 ) , \\alpha _ 0 + t ( \\alpha - \\alpha _ 0 ) ) d t \\begin{pmatrix} \\delta - \\delta _ 0 \\\\ \\alpha - \\alpha _ 0 \\end{pmatrix} | | \\rightarrow 0 . \\end{align*}"}
+{"id": "7624.png", "formula": "\\begin{align*} e ^ { \\sqrt { - 1 } \\ \\theta } \\cdot ( f _ 1 ( z ) , f _ 2 ( z ) , f _ 3 ( z ) ) = ( f _ 1 ( z ) , g _ 2 ( z ) , g _ 3 ( z ) ) \\end{align*}"}
+{"id": "8887.png", "formula": "\\begin{align*} \\mathcal E [ u ] = \\| \\sqrt { \\mathcal K _ \\lambda } u \\| ^ 2 - \\frac 1 p \\mathcal P [ u ] & \\geq \\| \\sqrt { \\mathcal K _ \\lambda } u \\| ^ 2 - \\frac { C _ { N , \\tau , \\alpha , \\lambda } } { p } \\| \\sqrt { \\mathcal K _ \\lambda } u \\| ^ { 2 p } \\\\ & = f \\big ( \\| \\sqrt { \\mathcal K _ \\lambda } u \\| ^ 2 \\big ) . \\end{align*}"}
+{"id": "5983.png", "formula": "\\begin{align*} \\phi ( x ) = \\frac { | \\nabla u ( x ) | ^ 2 } { ( b ^ 2 - \\rho ^ 2 ( u ( x ) ) ) ^ 2 } . \\end{align*}"}
+{"id": "4168.png", "formula": "\\begin{align*} \\varphi ( g ^ { n } _ 1 g _ 3 ) = \\varphi \\big ( g ^ { j } _ 1 \\big ) \\varphi \\big ( g ^ { i } _ 1 g _ 3 \\big ) \\mbox { o r } \\varphi ( g ^ { n } _ 1 g _ 3 ) = \\varphi \\big ( g ^ { i } _ 1 g _ 3 \\big ) \\varphi \\big ( g ^ { j } _ 1 \\big ) \\ , . \\end{align*}"}
+{"id": "7209.png", "formula": "\\begin{gather*} e ( a _ 1 , b _ 1 ) = e ( a _ 2 , b _ 2 ) \\iff a _ 1 = a _ 2 \\implies \\\\ \\pi _ M ( b _ 1 ) = l ( a _ 1 ) = l ( a _ 2 ) = \\pi _ M ( b _ 2 ) , \\end{gather*}"}
+{"id": "8706.png", "formula": "\\begin{align*} \\hat { \\sigma } _ s = \\hat { \\sigma } : H ^ 0 ( \\overline M ) ^ G _ s & \\to H ^ 0 _ b ( X ) ^ G _ { s - \\frac { 1 } { 2 } } \\\\ u & \\mapsto ( P ^ \\ast P ) ^ { - 1 } P ^ \\ast u = \\gamma u . \\end{align*}"}
+{"id": "7056.png", "formula": "\\begin{align*} X _ t = x - \\int _ 0 ^ b b ( s , X _ s ) d s + \\sqrt { 2 } W _ t , t \\geq 0 , \\end{align*}"}
+{"id": "2968.png", "formula": "\\begin{align*} \\begin{aligned} & ( i ) ~ \\sup _ { 0 \\leq t < \\infty } \\max _ { 1 \\leq i , j \\leq N } \\| x _ i ( t ) - x _ j ( t ) \\| < \\infty , \\\\ & ( i i ) ~ \\lim _ { t \\to \\infty } \\max _ { 1 \\leq i , j \\leq N } \\| v _ j ( t ) - v _ i ( t ) \\| = 0 , \\end{aligned} \\end{align*}"}
+{"id": "6655.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\nu _ { n } ( E , \\mathrm { i } ( y + 2 \\epsilon ) ) \\leq \\frac { 1 } { \\epsilon } ( L ( E , \\mathrm { i } ( y + 3 \\epsilon ) ) - L ( E , \\mathrm { i } ( y + 2 \\epsilon ) ) ) = \\omega ^ { + } ( E , \\mathrm { i } y ) . \\end{align*}"}
+{"id": "2444.png", "formula": "\\begin{align*} \\begin{cases} u ( \\xi ) \\sim A _ { 1 } ( \\xi - \\xi _ { - } ) ^ { p } , \\\\ u ' ( \\xi ) \\sim A _ { 2 } ( \\xi - \\xi _ { - } ) ^ { p - 1 } \\end{cases} { \\rm { a s } } \\xi \\searrow \\xi _ { - } + 0 , \\end{align*}"}
+{"id": "4311.png", "formula": "\\begin{align*} \\| u \\| _ { m M } = \\| I d _ { \\C } \\otimes u : \\C \\otimes _ { \\max } A \\to \\C \\otimes _ { \\max } B \\| . \\end{align*}"}
+{"id": "8302.png", "formula": "\\begin{align*} B y = \\sum _ { k = 0 } ^ { n - 1 } \\alpha _ { k } y ^ { ( k ) } ( a ) = c , \\end{align*}"}
+{"id": "847.png", "formula": "\\begin{align*} { l _ j } = { g _ { i j } } \\frac { { { y ^ i } } } { F } = { ( \\frac { 1 } { 2 } { F ^ 2 } ) _ { { . i } { . j } } } \\frac { { { y ^ i } } } { F } = \\frac { { { { ( \\frac { 1 } { 2 } { F ^ 2 } ) } _ { { . j } } } } } { F } = \\frac { { F { F _ { { y ^ j } } } } } { F } = { F _ { { . j } } } , \\end{align*}"}
+{"id": "6666.png", "formula": "\\begin{align*} | f _ { n } ( x + \\mathrm { i } y ) | \\geq & \\| M _ { n } ( x + \\mathrm { i } y ) \\| | u _ { n } ^ { - } ( x + \\mathrm { i } y ) \\wedge \\vec { e } _ { 1 } | | v _ { n } ^ { + } ( x + \\mathrm { i } y ) \\wedge \\vec { e } _ { 2 } | \\\\ & - \\| M _ { n } ( x + \\mathrm { i } y ) \\| ^ { - 1 } | u _ { n } ^ { + } ( x + \\mathrm { i } y ) \\wedge \\vec { e } _ { 1 } | | v _ { n } ^ { - } ( x + \\mathrm { i } y ) \\wedge \\vec { e } _ { 2 } | . \\end{align*}"}
+{"id": "1481.png", "formula": "\\begin{align*} \\begin{array} { c } \\{ m + 1 \\} = 1 + q \\{ m \\} = \\{ m \\} + q ^ m , \\ \\{ m + n \\} = \\{ m \\} + q ^ m \\{ n \\} , \\ q ^ m \\{ - m \\} = - \\{ m \\} , \\\\ \\{ m \\} = 0 \\ \\Leftrightarrow \\ q ^ { m } = 1 . \\end{array} \\end{align*}"}
+{"id": "6472.png", "formula": "\\begin{align*} A = - \\frac { \\sqrt { 1 - a ^ 2 } } { a } \\operatorname { I d } . \\end{align*}"}
+{"id": "2426.png", "formula": "\\begin{align*} U _ { t } = U ^ { p } ( U _ { x x } + \\mu U ) - \\delta U , t > 0 , x \\in \\mathbb { R } \\end{align*}"}
+{"id": "5356.png", "formula": "\\begin{align*} \\mathbf { c } ^ S = \\mathbf { h } ^ 0 - \\mathbf { h } ^ 1 + \\beta \\ , \\left ( \\mathbf { P } ^ 0 - \\mathbf { P } ^ 1 \\right ) \\ , \\mathbf { v } ^ S , \\end{align*}"}
+{"id": "6007.png", "formula": "\\begin{align*} \\partial _ t \\phi _ \\alpha ( t , u ) = - \\nabla \\Big \\{ v _ \\alpha \\phi _ \\alpha + \\vec { \\phi } ^ T G ^ \\alpha \\vec { \\phi } + ( \\tilde D \\nabla \\vec { \\phi } ) _ \\alpha + ( \\tilde B \\vec { \\xi } ) _ \\alpha \\Big \\} \\end{align*}"}
+{"id": "5897.png", "formula": "\\begin{align*} \\dfrac { M _ { 2 } ( \\mathcal { N C } ( G ) ) } { | e ( \\mathcal { N C } ( G ) ) | } = \\dfrac { ( p ^ { 2 } n - n ) ( p ^ { 2 } n - p n ) ( p ^ { 4 } n ^ { 2 } - 2 p ^ { 3 } n ^ { 2 } + p ^ { 2 } n ^ { 2 } ) } { ( p ^ { 2 } n - n ) ( p ^ { 2 } n - p n ) } . \\end{align*}"}
+{"id": "6593.png", "formula": "\\begin{align*} | I _ { k , \\beta } ( s , z ) | = | \\prod _ { p \\leq | \\eta | } | . | \\prod _ { p > | \\eta | } | . \\end{align*}"}
+{"id": "2729.png", "formula": "\\begin{align*} & { _ { * } X } _ { { _ { * } \\phi } ^ { ( s + 1 ) } _ { \\alpha } } = { _ { * } X } _ { \\{ { _ { * } \\phi } ^ { ( s ) } _ { \\alpha } , H _ { T } \\} } = [ { _ { * } X } _ { T } , { _ { * } Y } ^ { ( s ) } _ { \\alpha } ] = { _ { * } Y } ^ { ( s + 1 ) } _ { \\alpha } , \\\\ & { _ { * } X } _ { { _ { * } \\phi } ^ { ( m _ { \\alpha } + 1 ) } _ { \\alpha } } = { _ { * } C } _ { \\alpha s } ^ { \\beta } { _ { * } Y } ^ { ( s ) } _ { \\alpha } , \\end{align*}"}
+{"id": "7340.png", "formula": "\\begin{align*} \\| x ' _ n - x _ n \\| \\le \\| x ' _ 0 - x _ 0 \\| = | w ' | . \\end{align*}"}
+{"id": "4315.png", "formula": "\\begin{align*} f ( u '' ) = \\langle w , [ I d _ D \\otimes u '' ] ( t ) \\rangle , \\end{align*}"}
+{"id": "272.png", "formula": "\\begin{align*} \\frac { d ^ 2 y } { d \\tau ^ 2 } + \\frac { d y } { d \\tau } + \\frac { 1 } { 2 } = 0 . \\end{align*}"}
+{"id": "1339.png", "formula": "\\begin{align*} p _ 1 x _ k & = \\alpha d _ k x _ k + ( 1 - \\alpha ) \\sum _ { k \\sim j } x _ j \\\\ & \\leq \\alpha \\Delta x _ k + ( 1 - \\alpha ) \\Delta x _ k = \\Delta x _ k . \\end{align*}"}
+{"id": "5294.png", "formula": "\\begin{align*} - \\frac { \\partial K L ( q \\| p ) } { \\partial q _ { j } } = \\frac { \\sum _ { j } p _ { j } } { \\sum _ { j } q _ { j } } \\left ( \\sum _ { i } \\overline { q } _ { i } \\log \\frac { \\overline { q } _ { i } } { \\overline { p } _ { i } } - \\log \\frac { \\overline { q } _ { j } } { \\overline { p } _ { j } } \\right ) \\end{align*}"}
+{"id": "290.png", "formula": "\\begin{align*} F _ l ( z ) = \\sum _ { \\vert \\alpha \\vert \\geq 1 } a _ { l , \\alpha } \\ , z ^ \\alpha = \\sum _ { k = 1 } ^ \\infty a _ { l , k } \\ , z ^ { \\alpha ( k ) } \\end{align*}"}
+{"id": "7405.png", "formula": "\\begin{align*} \\log M _ { n + 1 } = \\log ( A C _ 5 ) - 2 n \\log ( p + q ) + ( p + q ) \\log M _ n , \\end{align*}"}
+{"id": "1270.png", "formula": "\\begin{align*} W _ 2 ( \\N ( u , U ) , \\N ( v , V ) ) ^ 2 = | u - v | ^ 2 + \\mathrm { t r } U + \\mathrm { t r } V - 2 \\mathrm { t r } \\sqrt { V ^ \\frac { 1 } { 2 } U V ^ \\frac { 1 } { 2 } } , \\end{align*}"}
+{"id": "3991.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } \\gamma _ { i j k } + \\sum _ { r = 1 } ^ { \\nu } \\widetilde { \\gamma } _ { i j r } = 1 , \\quad 1 \\leq i \\leq n , 1 \\leq j \\leq \\nu . \\end{align*}"}
+{"id": "1733.png", "formula": "\\begin{align*} \\widehat { F } ( z ) : = \\left ( \\sum _ { j = 1 } ^ { \\lceil \\mu / 2 \\rceil } F _ j ( x _ { j , 1 } , \\ldots , x _ { j , n _ j } , x _ n ) \\right ) - \\sum _ { j = \\lceil \\mu / 2 \\rceil + 1 } ^ { \\mu } F _ j ( x _ { j , 1 } , \\ldots , x _ { j , n _ j } , x _ n ) . \\end{align*}"}
+{"id": "7498.png", "formula": "\\begin{align*} S _ 2 [ h ] = \\left ( 2 \\chi \\partial _ t \\chi - 2 \\chi \\Delta _ { g _ 0 ( t ) } \\chi - 2 | \\nabla ^ { g _ 0 ( t ) } \\chi | ^ 2 \\right ) h & - 2 \\chi \\nabla ^ { g _ 0 ( t ) } \\chi * \\nabla ^ { g _ 0 ( t ) } h \\\\ & - 2 \\chi \\nabla ^ { g _ 0 ( t ) } \\chi * R _ 1 [ h ] . \\end{align*}"}
+{"id": "5601.png", "formula": "\\begin{align*} \\langle \\check { \\chi } _ i , B ^ { t } w \\rangle = \\langle B ^ { t } \\chi _ i , J _ { \\Delta } w \\rangle . \\end{align*}"}
+{"id": "4847.png", "formula": "\\begin{align*} \\mathcal { G } ( B + x ) - \\mathcal { G } ( B ) & = \\int _ { - 1 / 2 } ^ { 1 / 2 } ( g ( x + q ) - g ( q ) ) d q = x ^ 2 . \\\\ \\end{align*}"}
+{"id": "1352.png", "formula": "\\begin{align*} \\alpha ^ 2 M _ 1 + ( 1 - \\alpha ) ^ 2 2 m - \\dfrac { 4 \\alpha ^ 2 m ^ 2 } { n } = n s _ 1 s _ n . \\end{align*}"}
+{"id": "1187.png", "formula": "\\begin{align*} \\sigma & \\geq \\mathbf { 1 } _ { ( 0 , \\infty ) } ( N ) , \\ \\begin{cases} E \\geq N , \\\\ E > \\lfloor N \\rfloor _ + , \\end{cases} \\begin{cases} F \\geq ( K \\vee M ) - n , \\\\ F > \\lfloor L \\rfloor , \\end{cases} G \\geq \\lfloor N \\rfloor _ + , H \\geq \\lfloor L \\rfloor . \\end{align*}"}
+{"id": "4974.png", "formula": "\\begin{align*} \\Lambda ( \\mu ; \\nu _ j , \\nu _ j - 1 ) = a ( \\mu ) f ( \\nu _ j , \\mu ) f ( \\nu _ j - 1 , \\mu ) \\end{align*}"}
+{"id": "7028.png", "formula": "\\begin{align*} b _ n \\in \\mathbf { F } _ { \\delta } \\end{align*}"}
+{"id": "4952.png", "formula": "\\begin{align*} \\langle E _ i , U _ x E _ i \\rangle ' & = \\langle E _ i , P U _ x E _ i \\rangle ' \\\\ & = \\langle E _ i ' , P U _ x E _ i \\rangle + \\langle E _ i , P ' U _ x E _ i \\rangle + \\langle E _ i , P U _ x ' E _ i \\rangle + \\langle E _ i , P U _ x P E _ i ' \\rangle \\\\ & = \\langle E _ i , P U _ x ' E _ i \\rangle . \\end{align*}"}
+{"id": "496.png", "formula": "\\begin{align*} O ( \\log ( k ) \\log ^ { 1 + o ( 1 ) } ( m ^ 2 ) ) = O ( \\log \\log { m } \\log ^ { 1 + o ( 1 ) } { m } ) = O ( \\log ^ { 1 + o ( 1 ) } { m } ) \\end{align*}"}
+{"id": "2660.png", "formula": "\\begin{align*} \\delta S ^ { ( 1 ) } = \\int ^ { t _ { 2 } } _ { t _ { 1 } } \\sum _ { i = 1 } ^ { n } \\left [ \\frac { \\partial L ^ { ( 1 ) } } { \\partial q ^ { i } } - \\frac { d } { d t } \\left ( \\frac { \\partial L ^ { ( 1 ) } } { \\partial \\dot { q } ^ { i } } \\right ) \\right ] \\delta q ^ { i } d t + \\left [ \\sum _ { i = 1 } ^ { n } \\left ( \\frac { \\partial L ^ { ( 1 ) } } { \\partial \\dot { q } ^ { j } } \\right ) \\delta q ^ { j } \\right ] ^ { t _ { 2 } } _ { t _ { 1 } } . \\end{align*}"}
+{"id": "3404.png", "formula": "\\begin{align*} \\eta _ { t } \\lambda _ { t } ^ { p } & = \\frac { \\sqrt { \\Delta _ { 1 } } T ^ { \\frac { 1 - p } { 3 p - 2 } } } { 8 \\sqrt { L } \\gamma } \\lambda _ { t } ^ { p - 1 } \\\\ & \\ge \\frac { \\sqrt { \\Delta _ { 1 } } T ^ { \\frac { 1 - p } { 3 p - 2 } } } { 8 \\sqrt { L } \\gamma } \\frac { 8 \\gamma } { \\sqrt { L \\Delta _ { 1 } } } T ^ { \\frac { p - 1 } { 3 p - 2 } } \\sigma ^ { p } \\\\ & = \\frac { \\sigma ^ { p } } { L } \\end{align*}"}
+{"id": "6009.png", "formula": "\\begin{align*} M _ { \\alpha , \\alpha } ( s , v ) = 2 \\sum _ { \\beta , \\delta } ( G _ { \\beta , \\gamma } ^ \\alpha ) ^ 2 S _ \\beta ( s , v ) S _ { \\delta } ( s , v ) \\ , . \\end{align*}"}
+{"id": "8751.png", "formula": "\\begin{align*} g ( x ) = \\frac { z _ - - z _ + } { z _ - - y } \\varphi ( \\vert x - y \\vert ) + \\frac { z _ + - y } { z _ - - y } \\varphi ( \\vert z _ - - x \\vert ) - \\varphi ( \\vert x - z _ + \\vert ) \\end{align*}"}
+{"id": "5218.png", "formula": "\\begin{align*} L _ { d } D G H ( p \\| q ) = \\frac { \\left ( M G \\right ) ^ { a - 1 } - \\left ( M G \\right ) ^ { b - 1 } } { a - b } - \\left [ \\frac { \\left ( M H \\right ) ^ { a - 1 } - \\left ( M H \\right ) ^ { b - 1 } } { a - b } \\right ] \\end{align*}"}
+{"id": "7679.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( \\abs { G ^ { \\Lambda ' } _ { L } ( m , n ; z ) } ^ s \\right ) \\leq \\sum ^ { N } _ { j = 0 } \\Gamma ( s ) ^ { j + 1 } \\# S ^ { \\Lambda ' } _ { j } ( n , m ) + \\Gamma ( s ) ^ N \\# S ^ { \\Lambda ' } _ { N } ( m ) \\frac { 1 } { \\abs { \\mathrm { I m } z } ^ s } . \\end{align*}"}
+{"id": "1317.png", "formula": "\\begin{align*} \\dot { q } ^ i & = \\mp \\rho ^ i _ I \\frac { \\partial H } { \\partial \\mu _ I } , \\\\ \\dot { \\mu } _ I & = \\pm C _ { I J } ^ K \\frac { \\partial H } { \\partial \\mu _ J } \\mu _ K \\pm \\rho ^ i _ I \\frac { \\partial H } { \\partial q ^ i } , \\end{align*}"}
+{"id": "1456.png", "formula": "\\begin{align*} H _ i ( G G _ { \\sigma } , M ) = \\mathrm { T o r } _ { i - 1 } ^ { \\mathbb { Z } [ G ] } ( \\mathrm { S t } , M ) . \\end{align*}"}
+{"id": "5923.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { N C } ( A ( n , p ) ) ) = \\dfrac { ( p ^ { 3 n } - p ^ { n } ) [ p ^ { 8 n } ( p ^ { n } - 3 ) + p ^ { 6 n } ( 3 p ^ { n } - 1 ) ] } { 2 } . \\end{align*}"}
+{"id": "2815.png", "formula": "\\begin{align*} & \\Theta ^ { 1 } : = \\frac { 1 } { \\sqrt { 2 } } \\phi ^ { ( 1 ) } _ { 1 } , \\Theta _ { 1 } : = \\frac { 1 } { \\sqrt { 2 } } \\phi ^ { ( 1 ) } _ { 2 } , \\\\ & Q ^ { 1 } : = \\frac { 1 } { \\sqrt { 2 } } ( q ^ { 1 } + p _ { 2 } ) , P _ { 1 } : = \\frac { 1 } { \\sqrt { 2 } } ( p _ { 1 } - q ^ { 2 } ) . \\end{align*}"}
+{"id": "5832.png", "formula": "\\begin{align*} \\mathcal { M } _ \\Sigma ( u ) ^ A = \\left \\langle \\vec { H } + \\frac { X ^ \\perp ( u ) } { 2 } , \\frac { \\partial } { \\partial y ^ B } \\right \\rangle ( h _ 0 ) ^ { B A } e ^ { - | X ( u ) | ^ 2 / 4 + | X ( 0 ) | ^ 2 / 4 } \\sqrt { \\frac { \\det g _ u } { \\det g _ 0 } } . \\end{align*}"}
+{"id": "5615.png", "formula": "\\begin{align*} v ' = t _ 1 + t _ 2 + t _ { \\geq 3 } & \\geq v - h , \\\\ t _ 1 + 2 t _ 2 + 3 t _ { \\geq 3 } & \\leq \\sum _ { k \\geq 1 } k t _ k = 2 ( a - h ) . \\end{align*}"}
+{"id": "2653.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { J } \\langle a ( t _ { j - 1 } ) , \\tilde { w } ( j ) - \\tilde { w } ( t _ { j - 1 } ) \\rangle _ { L ^ 2 _ x } \\lesssim \\| V \\| _ { L _ x ^ { \\frac { q } { q - 2 } } } \\| u \\| _ { V ^ p _ { \\Delta } } \\end{align*}"}
+{"id": "5833.png", "formula": "\\begin{align*} \\frac { \\partial u ^ A } { \\partial t } - \\mathcal { M } _ \\Sigma ( u ) ^ A = \\left ( e ^ { | X ( u ) | ^ 2 / 4 - | X ( 0 ) | ^ 2 / 4 } \\sqrt { \\frac { \\det g _ 0 } { \\det g _ u } } ( h _ u ) ^ { A B } ( h _ 0 ) _ { B C } - \\delta _ C ^ A \\right ) \\cdot \\mathcal { M } _ \\Sigma ( u ) ^ C . \\end{align*}"}
+{"id": "3136.png", "formula": "\\begin{align*} \\psi ^ * \\left ( \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ) = \\left ( \\begin{array} { c c c } a ^ { 2 \\ , p ^ { e _ 1 } } & a ^ { p ^ { e _ 1 } } \\ , b ^ { p ^ { e _ 1 } } & b ^ { 2 \\ , p ^ { e _ 1 } } \\\\ 0 & 1 & 0 \\\\ c ^ { 2 \\ , p ^ { e _ 1 } } & c ^ { p ^ { e _ 1 } } \\ , d ^ { p ^ { e _ 1 } } & d ^ { 2 \\ , p ^ { e _ 1 } } \\end{array} \\right ) . \\end{align*}"}
+{"id": "7540.png", "formula": "\\begin{align*} \\int _ \\Omega \\rho \\ d x = \\int _ \\Omega n \\ d x = M < \\infty \\ , \\ \\ \\ \\forall t \\in [ 0 , T [ \\ ; \\end{align*}"}
+{"id": "3704.png", "formula": "\\begin{align*} M _ { ( G _ 1 , S , \\phi _ 1 ) \\times ( G _ 2 , S , \\phi _ 2 ) } & = | ( G _ 1 \\times G _ 2 ) ^ { \\rm a b } [ | \\mu ( K ) | ] | ^ { - 1 } | G _ 1 \\times G _ 2 | ^ { - | S \\cup P _ \\infty | + 1 } \\\\ & = | G _ 1 ^ { \\rm a b } [ | \\mu ( K ) | ] | ^ { - 1 } | G _ 1 | ^ { - | S \\cup P _ \\infty | + 1 } | G _ 2 ^ { \\rm a b } [ | \\mu ( K ) | ] | ^ { - 1 } | G _ 2 | ^ { - | S \\cup P _ \\infty | + 1 } \\\\ & = M _ { ( G _ 1 , S , \\phi _ 1 ) } M _ { ( G _ 2 , S , \\phi _ 2 ) } . \\end{align*}"}
+{"id": "7629.png", "formula": "\\begin{align*} r _ d \\circ i _ d = \\mbox { i d } . \\end{align*}"}
+{"id": "1170.png", "formula": "\\begin{align*} E : = \\left \\{ \\begin{aligned} & \\frac { n - 1 } { p } - n \\tau & & \\frac { n } { n - 1 } \\tau > \\frac { 1 } { p } , \\\\ & ( n - 1 ) \\left ( \\frac { 1 } { p } - 1 \\right ) _ + & & . \\end{aligned} \\right . \\end{align*}"}
+{"id": "5873.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { C } ( G ) ) = \\dfrac { ( 4 n - 4 ) ( 4 n - 4 - 1 ) ^ { 3 } } { 2 } + n \\cdot \\dfrac { 4 ( 4 - 1 ) ^ { 3 } } { 2 } = ( 2 n - 2 ) ( 4 n - 5 ) ^ { 3 } + 5 4 n . \\end{align*}"}
+{"id": "298.png", "formula": "\\begin{align*} \\dot { z } _ l = F _ l ( z ) = \\sum _ { k = 1 } ^ \\infty a _ { l , k } \\ , z ^ { \\alpha ( k ) } , l = 1 , \\dots , n , \\end{align*}"}
+{"id": "1565.png", "formula": "\\begin{align*} \\left ( P \\ , S , S \\ , P \\right ) _ 2 = \\sum _ { i , j = 1 } ^ N \\lambda _ i \\ , a _ { i j } \\ , \\lambda _ j \\ , a _ { j i } = \\sum _ { i , j = 1 } ^ N \\lambda _ i \\ , \\lambda _ j \\ , a _ { i j } ^ 2 \\ge \\frac { 1 } { \\beta } \\sum _ { i , j = 1 } ^ N \\lambda _ i ^ 2 \\ , a _ { i j } ^ 2 = \\frac { 1 } { \\beta } \\ , | P \\ , S | _ 2 ^ 2 , \\end{align*}"}
+{"id": "8171.png", "formula": "\\begin{align*} & P _ { Y _ { j } | Y ^ { j - 1 } , \\underline S } ( 1 | y ^ { j - 1 } \\in \\beta _ k ^ { j - 1 } , \\underline S ) \\\\ & = P _ { Y _ { j } | Y ^ { j - 1 } , \\underline S } ( 0 | y ^ { j - 1 } \\in \\beta _ k ^ { j - 1 } , \\underline S ) = \\frac { 1 } { 2 } , \\end{align*}"}
+{"id": "2622.png", "formula": "\\begin{align*} | D _ 4 ( S ) | \\leq | D _ 3 ( S ) | + ( n - 4 ) = 2 n - 2 . \\end{align*}"}
+{"id": "3344.png", "formula": "\\begin{align*} & - g ^ { - 1 } ( \\d g ) g ^ { - 1 } \\mathcal { L } _ { J v } g + g ^ { - 1 } ( \\mathcal { L } _ { J v } g ) g ^ { - 1 } \\d g \\\\ = & \\d ( g ^ { - 1 } ) \\mathcal { L } _ { J v } g - \\mathcal { L } _ { J v } ( g ^ { - 1 } ) \\d g \\\\ = & 0 . \\end{align*}"}
+{"id": "4301.png", "formula": "\\begin{align*} M _ t : = \\int _ 0 ^ t \\nabla \\sigma ( y _ s ) \\langle \\bullet , d x _ s \\rangle \\end{align*}"}
+{"id": "2433.png", "formula": "\\begin{align*} U _ { c } ' ( 0 ) = 0 , U _ { c } ' ( \\xi ) > 0 { \\rm { f o r } } \\xi < 0 , U _ { c } ( \\xi ) \\to 0 { \\rm { a s } } \\xi \\to - \\infty \\end{align*}"}
+{"id": "1366.png", "formula": "\\begin{align*} \\Big [ \\big [ [ \\pi , T ] _ { \\Omega } , T \\big ] _ { \\Omega } , \\theta \\Big ] _ { \\Omega } = & \\ 0 , \\\\ \\Big [ \\ldots \\big [ [ f , \\underbrace { T ] _ { \\Omega } , T \\big ] _ { \\Omega } , \\ldots , T \\Big ] _ { \\Omega } } _ { k } = & \\ 0 , \\ , k \\geqslant n + 1 . \\end{align*}"}
+{"id": "1133.png", "formula": "\\begin{align*} \\left \\| B \\vec { t } \\right \\| _ { \\dot a ^ 0 _ { p , q } ( W ) } \\lesssim \\left \\| \\vec { t } \\right \\| _ { \\dot a ^ 0 _ { p , q } ( W ) } \\left \\{ \\sum _ { k \\in \\mathbb { Z } } \\sum _ { l = 0 } ^ \\infty \\left [ 2 ^ { - ( E - \\frac { n } { 2 } ) k _ - } 2 ^ { - k _ + ( F + \\frac { n } { 2 } - \\frac { n } { a } ) } 2 ^ { - ( D - \\frac { n } { a } ) l } \\right ] ^ r \\right \\} ^ { \\frac { 1 } { r } } \\sim \\left \\| \\vec { t } \\right \\| _ { \\dot a ^ 0 _ { p , q } ( W ) } , \\end{align*}"}
+{"id": "4452.png", "formula": "\\begin{align*} & \\int _ { M _ 1 } \\int _ { K _ 1 } \\left | \\frac { f ^ * } { g _ 2 } ( w _ 1 , \\hat w _ 1 ) \\right | ^ 2 \\\\ \\le & C _ 1 \\int _ { M _ 1 } \\sup _ { w _ 1 \\in K _ 1 } \\left | \\frac { f ^ * } { g _ 2 } ( w _ 1 , \\hat w _ 1 ) \\right | ^ 2 d \\mu _ 1 ( \\hat w _ 1 ) \\\\ \\le & C _ 1 C _ { K _ 1 } \\int _ { M _ 1 } \\left ( \\frac { 1 } { 2 \\pi } \\int _ { \\partial D _ 1 } \\left | \\frac { f ^ * } { g _ 2 } ( z _ 1 , \\hat w _ 1 ) \\right | ^ 2 | d z _ 1 | \\right ) d \\mu _ 1 ( \\hat w _ 1 ) \\\\ < & + \\infty \\end{align*}"}
+{"id": "6997.png", "formula": "\\begin{gather*} Q : \\ ; \\mathbb { P } _ \\mathbb { C } ^ { n } \\setminus [ \\widetilde { \\Lambda } _ 0 ] \\rightarrow \\mathbb { P } _ \\mathbb { C } ^ { 2 } \\\\ z = [ z _ 1 : z _ 2 : . . . : z _ { n + 1 } ] \\mapsto Q ( z ) = [ z _ 1 : z _ 2 : z _ { n + 1 } ] . \\end{gather*}"}
+{"id": "7665.png", "formula": "\\begin{align*} C _ { 1 } = \\lambda \\frac { C _ a 1 4 4 \\sqrt { 2 } \\norm { F } _ { \\infty } } { \\eta } S _ { \\delta - \\gamma } S _ { \\delta - 2 \\nu } , \\end{align*}"}
+{"id": "603.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { 3 2 } \\right ) ^ { k } \\binom { 2 k } { k } ^ { 2 } H _ { 2 k } \\end{align*}"}
+{"id": "4865.png", "formula": "\\begin{align*} ( \\alpha + 2 ) H ^ { \\alpha + 1 } \\nabla _ \\Sigma H - \\alpha H ^ { \\alpha - 1 } \\nabla _ \\Sigma H | A | ^ 2 = 2 H ^ \\alpha \\langle A , \\nabla _ \\Sigma A \\rangle , \\end{align*}"}
+{"id": "4407.png", "formula": "\\begin{gather*} F _ { k _ 1 , k _ 2 , k _ 3 , k _ 4 } ( x ) = \\sum _ { ( i , j , r , s ) \\in [ n ] ^ 4 } ( \\overline { c } _ { i , j } d _ { r , s } + \\max \\{ 0 , \\Delta { c } _ { i , j } d _ { r , s } - \\Delta c _ { k _ 1 , k _ 2 } d _ { k _ 3 , k _ 4 } \\} ) x _ { i , r } x _ { j , s } \\end{gather*}"}
+{"id": "3424.png", "formula": "\\begin{align*} ( G , H ) = ( \\Z _ 2 , \\Z _ 2 ) ( \\Z _ 4 , \\Z _ 2 ) . \\end{align*}"}
+{"id": "6493.png", "formula": "\\begin{align*} ( \\xi ' _ 1 , \\xi ' _ 2 ) = ( 0 , - 1 ) , ( \\xi _ 1 , \\xi _ 2 ) = ( 0 , 1 ) . \\end{align*}"}
+{"id": "5808.png", "formula": "\\begin{align*} | X _ 0 ( t ) | ^ 2 = ( 1 + o ( 1 ) ) | x ( t ) | ^ 2 . \\end{align*}"}
+{"id": "7753.png", "formula": "\\begin{align*} \\epsilon _ { n + 1 } = \\epsilon _ { n } ^ { 4 ^ { n } } = \\epsilon _ { 1 } ^ { 2 ^ { n ^ { 2 } + n } } , h _ { n } = ( \\frac { 1 } { 2 } + \\frac { 1 } { 2 ^ { n } } ) h _ { 1 } . \\end{align*}"}
+{"id": "8360.png", "formula": "\\begin{align*} J ( w ) : = \\{ s \\in S : \\ \\ell ( s w ) < \\ell ( w ) \\} . \\end{align*}"}
+{"id": "7372.png", "formula": "\\begin{align*} \\frac { r + 1 } { 2 } = m + 1 < p + q = 2 m < r = 2 m + 1 \\end{align*}"}
+{"id": "806.png", "formula": "\\begin{align*} \\eta = \\frac { { { { ( - 1 ) } ^ { \\frac { { n ( n - 1 ) } } { 2 } } } } } { { ( n - 1 ) ! } } \\omega \\wedge { ( d \\omega ) ^ { n - 1 } } . \\end{align*}"}
+{"id": "5674.png", "formula": "\\begin{align*} J ( u _ n ) - \\frac { 1 } { \\mu } J ' ( u _ n ) [ u _ n ] \\geq \\frac { \\mu - 2 } { 4 \\mu } \\Vert u _ n \\Vert ^ 2 = \\frac { 1 } { 2 k } \\Vert u _ n \\Vert ^ 2 . \\end{align*}"}
+{"id": "5616.png", "formula": "\\begin{align*} \\ell \\leq \\lfloor \\log n \\rfloor , s = \\left \\lfloor \\frac { \\log \\left ( \\frac { n } { 8 ( d \\vee K ) ^ 3 K ^ 9 } \\right ) } { 2 4 \\log \\log n } \\right \\rfloor \\end{align*}"}
+{"id": "6851.png", "formula": "\\begin{align*} \\det ( X _ { N - 1 } ) = & \\cos ( \\psi _ { N - 2 } ) \\sin ( \\psi _ { N - 2 } ) \\cos ^ { 2 ( N - 2 ) - 2 } ( \\psi _ { N - 2 } ) \\left [ \\det \\begin{pmatrix} \\cos ( \\phi _ { N - 1 } ) & \\sin ( \\phi _ { N - 1 } ) \\\\ - \\sin ( \\phi _ { N - 1 } ) & \\cos ( \\phi _ { N - 1 } ) \\end{pmatrix} \\right ] ^ { N - 3 } \\det \\left ( X _ { N - 2 } \\right ) \\\\ = & \\cos ^ { 2 ( N - 2 ) - 1 } ( \\psi _ { N - 2 } ) \\sin ( \\psi _ { N - 2 } ) \\det \\left ( X _ { N - 2 } \\right ) . \\end{align*}"}
+{"id": "6757.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau ^ k } [ \\theta ( \\breve { u } ^ { k } ) - \\theta ( u ) - ( w - \\breve { w } ^ k ) ^ T F ( { w } ) ] + \\frac { 1 } { 2 } \\| v ^ { k + 1 } - v \\| ^ 2 _ { H } + \\frac { 1 } { 2 } \\| v ^ k - \\widetilde { v } ^ k \\| ^ 2 _ { G } \\\\ \\leq & \\frac { 1 } { \\tau ^ { k - 1 } } [ \\theta ( \\breve { u } ^ { k - 1 } ) - \\theta ( u ) - ( w - \\breve { w } ^ { k - 1 } ) ^ T F ( { w } ) ] + \\frac { 1 } { 2 } \\| v ^ { k } - v \\| ^ 2 _ { H } . \\end{aligned} \\end{align*}"}
+{"id": "6761.png", "formula": "\\begin{align*} \\begin{aligned} D ^ { t - 1 } - D ^ { t } = & - ( M ( \\breve { v } ^ { t } - v ^ * ) + M ( \\breve { v } ^ { t - 1 } - v ^ * ) ) ^ T H M ( \\breve { v } ^ { t } - \\breve { v } ^ { t - 1 } ) \\\\ \\leq & \\| M ( \\breve { v } ^ { t } - v ^ * ) + M ( \\breve { v } ^ { t - 1 } - v ^ * ) \\| _ H \\| M ( \\breve { v } ^ { t } - \\breve { v } ^ { t - 1 } ) \\| _ H \\\\ \\leq & 2 \\sqrt { N _ 2 } \\sqrt { E ^ t } . \\end{aligned} \\end{align*}"}
+{"id": "8034.png", "formula": "\\begin{align*} \\limsup _ { M \\rightarrow + \\infty } \\limsup _ { N \\rightarrow + \\infty } \\frac { N } { \\gamma ^ 2 _ N } \\log P \\left ( \\sup _ { 0 \\leq t \\leq T } | \\eta _ t ^ N ( \\vec { f } ) | > M \\right ) = - \\infty \\end{align*}"}
+{"id": "6520.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ p u ( x ) = 0 \\ ; & x \\in \\R ^ n \\setminus \\Omega , \\\\ u ( x ) = 1 \\ ; & x \\in \\partial \\Omega , \\\\ \\lim _ { | x | \\to \\infty } u ( x ) = 0 . \\end{cases} \\end{align*}"}
+{"id": "2218.png", "formula": "\\begin{align*} \\lambda ^ 2 \\hat v + 2 i \\lambda \\hat v + 3 \\hat v = \\hat g ' . \\end{align*}"}
+{"id": "875.png", "formula": "\\begin{align*} S _ 0 & = S _ 1 y ^ 1 + S _ 2 y ^ 2 \\\\ & = \\left [ 2 w ^ 1 { w ^ 1 } _ { \\substack { \\\\ x _ 1 } } + w ^ 2 ( { w ^ 1 } _ { \\substack { \\\\ x _ 2 } } + { w ^ 2 } _ { \\substack { \\\\ x _ 1 } } ) \\right ] y ^ 1 \\\\ & + \\left [ w ^ 1 ( { w ^ 1 } _ { \\substack { \\\\ x _ 2 } } + { w ^ 2 } _ { \\substack { \\\\ x _ 1 } } ) + 2 w ^ 2 { w ^ 2 } _ { \\substack { \\\\ x _ 2 } } \\right ] y ^ 2 , \\end{align*}"}
+{"id": "5184.png", "formula": "\\begin{align*} K _ { 0 , \\beta } = \\frac { \\sum _ { i } p _ { i } q ^ { \\beta - 1 } _ { i } } { \\sum _ { i } q ^ { \\beta } _ { i } } \\end{align*}"}
+{"id": "477.png", "formula": "\\begin{align*} a _ i ( \\omega ^ { i + d x } ) ^ { r _ i } = \\omega ^ { e _ i + r _ i i + r _ i d x } = \\omega ^ { \\overline { f } ( i ) + d \\cdot \\left ( \\frac { e _ i + r _ i i - \\overline { f } ( i ) } { d } + r _ i x \\right ) } \\in C _ { \\overline { f } ( i ) } . \\end{align*}"}
+{"id": "4339.png", "formula": "\\begin{gather*} f _ { i , * } ( x , v ^ i ) \\neq - \\infty \\Leftrightarrow v ^ i = l _ i ( x ) , \\end{gather*}"}
+{"id": "5099.png", "formula": "\\begin{align*} \\mathsf { a } _ s ^ { - 2 } - 1 = v ^ { - 4 } ( 1 - \\mathsf { a } _ s ^ 2 ) , \\end{align*}"}
+{"id": "9120.png", "formula": "\\begin{align*} f ( \\{ x _ { i , r } \\} _ { i \\in I } ^ { 1 \\leq r \\leq k _ { i } } ) = 0 x _ { i , s _ { 1 } } = v _ { i } ^ { 2 } x _ { i , s _ { 2 } } = \\cdots = v _ { i } ^ { - 2 a _ { i j } } x _ { i , s _ { 1 - a _ { i j } } } = v _ { i } ^ { - a _ { i j } } x _ { j , r } \\end{align*}"}
+{"id": "843.png", "formula": "\\begin{align*} ( n + 1 ) X ^ { i } \\Psi _ { i } = - \\delta ( ( X + \\rho V ) f ) - f ^ { 2 } + h \\rho - f g \\rho - ( n + 2 ) \\Psi \\rho . \\end{align*}"}
+{"id": "2853.png", "formula": "\\begin{align*} \\mathbb { B } : = \\left \\{ B ( x , r ) : \\ x \\in \\mathbb { R } ^ n r \\in ( 0 , \\infty ) \\right \\} \\end{align*}"}
+{"id": "1088.png", "formula": "\\begin{align*} \\Vert ( T _ n ( w ) ^ { - 1 } ) ^ { s , t } - ( \\Omega _ { n , \\delta } ( w ) ) ^ { s , t } \\| _ { 1 , } ^ { q \\times q } = \\max _ { 1 \\leq t \\leq n } D _ { t , n } = O ( n ^ { - d } ) , n \\to \\infty . \\end{align*}"}
+{"id": "5254.png", "formula": "\\begin{align*} \\frac { \\partial \\overline { Z } _ { i } } { \\partial q _ { j } } = \\frac { \\partial \\overline { Z } _ { i } } { \\partial \\overline { q } _ { i } } \\frac { \\partial \\overline { q } _ { i } } { \\partial q _ { j } } = - \\left [ \\frac { \\left ( 1 - \\alpha \\right ) \\overline { p } _ { i } } { \\left ( \\alpha \\overline { p } _ { i } + \\left ( 1 - \\alpha \\right ) \\overline { q } _ { i } \\right ) ^ { 2 } } \\right ] \\left [ \\frac { \\delta _ { i j } - \\overline { q } _ { i } } { \\sum _ { j } q _ { j } } \\right ] \\end{align*}"}
+{"id": "7008.png", "formula": "\\begin{align*} d X _ t = - b ( X _ t ) d t + \\sqrt { 2 } d W _ t , X _ 0 = x \\in \\mathbb R ^ d , \\end{align*}"}
+{"id": "4863.png", "formula": "\\begin{align*} 2 K = 2 \\kappa _ 1 \\kappa _ 2 = ( \\kappa _ 1 + \\kappa _ 2 ) ^ 2 - \\kappa _ 1 ^ 2 - \\kappa _ 2 ^ 2 = H ^ 2 - | A | ^ 2 , \\end{align*}"}
+{"id": "8135.png", "formula": "\\begin{align*} [ h _ { A _ 2 } ( J ) , k _ { A _ 2 } ( J ) ] & = D _ 1 + D _ 2 \\tau + D _ 3 \\mu + D _ 4 \\tau ^ 2 + D _ 5 \\tau \\mu + D _ 6 \\mu ^ 2 + D _ 7 \\tau ^ 2 \\mu \\\\ & + D _ 8 \\tau \\mu ^ 2 + D _ 9 \\mu ^ 3 + D _ { 1 0 } \\tau ^ 3 \\mu + D _ { 1 1 } \\tau ^ 2 \\mu ^ 2 + D _ { 1 2 } \\tau \\mu ^ 3 \\ , \\end{align*}"}
+{"id": "7423.png", "formula": "\\begin{align*} c ' _ 1 & = b _ 0 c _ 1 + ( a _ 1 - d _ 1 ) a _ 2 \\\\ c ' _ 2 & = b _ 0 c _ 2 + b _ 1 c _ 1 + ( a _ 1 - d _ 1 ) a _ 3 - a _ 2 d _ 2 \\\\ c ' _ 3 & = b _ 0 c _ 3 + b _ 1 c _ 2 + b _ 2 c _ 1 + ( a _ 1 - d _ 1 ) a _ 4 - a _ 2 d _ 3 - a _ 3 d _ 2 . \\\\ \\end{align*}"}
+{"id": "7699.png", "formula": "\\begin{align*} F ( x ) = \\sum _ { i } h _ { i } e _ { i } + h _ { \\Omega } ( x ) x = \\nabla h _ { \\Omega } ( x ) + h _ { \\Omega } ( x ) x . \\end{align*}"}
+{"id": "4154.png", "formula": "\\begin{align*} \\mathcal L ( G ) = \\big \\{ \\mathsf L ( B ) \\colon B \\in \\mathcal B ( G ) \\big \\} \\end{align*}"}
+{"id": "2365.png", "formula": "\\begin{align*} & \\| v _ { k } \\| _ { C _ { T } L _ { x } ^ { 3 } \\cap L _ { T } ^ { 4 } L _ { x } ^ { 6 } } + \\big \\| \\nabla | v _ { k } | ^ { \\frac { 3 } { 2 } } \\big \\| ^ { \\frac { 2 } { 3 } } _ { L _ { T } ^ { 2 } L _ { x } ^ { 2 } } \\\\ & = g ( T ) \\leq g _ { - } ( T ) = \\frac { 1 - \\sqrt { 1 - 4 a b ( T ) ^ { 2 } C _ { 1 } ^ { 2 } } } { 2 C _ { 1 } b ( T ) } \\leq \\frac { 4 a b ( T ) ^ { 2 } C _ { 1 } ^ { 2 } } { 2 C _ { 1 } b ( T ) } \\\\ & = 2 C _ { 1 } \\| v _ { k } ( \\cdot , 0 ) \\| _ { L ^ { 3 } } e ^ { C _ { 2 } \\int _ { 0 } ^ { T } \\| w \\| _ { L ^ { 6 } } ^ { 4 } d t } . \\end{align*}"}
+{"id": "6554.png", "formula": "\\begin{align*} \\lim \\limits _ { x \\to \\pm \\infty } \\frac { 1 } { | x | ^ { \\frac { 1 } { \\beta } } \\ell ^ { \\frac { 1 } { \\beta } } ( | x | ^ { \\frac { 1 } { \\beta } } ) } \\eta _ K ( x ) = \\int ^ { \\infty } _ 0 \\big ( K _ { \\infty } ( \\pm t ^ { - \\beta } ) - K _ { \\infty } ( 0 ) \\big ) d t : = C ^ { \\pm } _ K . \\end{align*}"}
+{"id": "4421.png", "formula": "\\begin{align*} \\inf _ { x \\in \\mathcal { X } } & \\ f ( x ) , \\\\ \\mathrm { s . t . } & \\sum _ { i \\in [ m ] } \\sup _ { u ^ i \\in \\mathcal { U } ^ i } g _ i ( x , u ^ i ) \\le 0 . \\end{align*}"}
+{"id": "278.png", "formula": "\\begin{align*} \\ddot x + f ( x ) \\ , \\dot x ^ 2 + g ( x ) \\ , \\dot x + h ( x ) = 0 \\end{align*}"}
+{"id": "6377.png", "formula": "\\begin{align*} \\partial E _ k = \\{ R _ k ( 1 + \\rho _ k ( \\varphi ) ) : \\varphi \\in S ^ { n - 1 } \\} , \\end{align*}"}
+{"id": "2841.png", "formula": "\\begin{align*} W = \\frac { 1 } { 2 } q \\dot { q } + C ( q ) , \\end{align*}"}
+{"id": "446.png", "formula": "\\begin{align*} z _ { i , j } ^ { q } ( t _ q + \\Delta t _ { i , j } ) = { - \\lambda _ { i , j } } , \\mbox { i f } \\ , i \\triangleleft j , \\mbox { a n d } z _ { i , j } ^ { q } ( t _ q + \\Delta t _ { i , j } ) = { \\mu _ { i , j } } , \\mbox { i f } \\ , j \\triangleleft i . \\end{align*}"}
+{"id": "7119.png", "formula": "\\begin{align*} - \\Delta u + \\lambda u + ( \\ln | \\cdot | \\ast u ^ { 2 } ) u = f ( u ) \\end{align*}"}
+{"id": "5542.png", "formula": "\\begin{align*} \\theta = \\max ( \\theta _ 1 , \\theta _ 2 ) , \\end{align*}"}
+{"id": "4130.png", "formula": "\\begin{align*} & \\psi ^ { ( r ) } ( \\tilde { \\theta } ) - \\psi ^ { ( r ) } _ j ( \\tilde { \\theta } ) = o ( 1 ) . \\end{align*}"}
+{"id": "4706.png", "formula": "\\begin{align*} E _ { l , l ' } : = \\{ ( \\lambda , \\nu ) \\in X _ l \\times X _ { l ' } \\mid \\Vert \\lambda - \\nu \\Vert < \\frac { 2 } { 2 ^ n } \\} . \\end{align*}"}
+{"id": "986.png", "formula": "\\begin{align*} R ^ D \\kappa _ D ( x ) = \\mathbb E _ x \\mathbf 1 _ D ( X _ { \\tau _ D - } ) = \\mathbb E ^ D _ x 1 ( X _ { \\tau _ D - } ) , x \\in D . \\end{align*}"}
+{"id": "7765.png", "formula": "\\begin{align*} \\mathcal R _ j ( \\tilde \\Lambda ) = \\cup _ { | k | = 0 } ^ { \\tilde N _ j } \\cup _ { \\bar \\Lambda ^ { ( j ) } \\in \\mathcal C ( \\Lambda ^ { ( j ) } ) } \\cup _ { i \\in J _ { j , k } ( \\bar \\Lambda ^ { ( j ) } ) } I _ { k , i } ( \\bar \\Lambda ^ { ( j ) } ) = : \\cup _ { i \\in J _ j ( \\tilde \\Lambda ) } I _ i ^ { ( j ) } , \\ \\ \\Lambda ^ { ( j + 1 ) } = \\Lambda ^ { ( j ) } \\backslash \\mathcal R _ j ( \\tilde \\Lambda ) . \\end{align*}"}
+{"id": "6858.png", "formula": "\\begin{align*} e ^ { G ( z ) } & = \\sum _ { k = 0 } ^ { \\infty } \\frac { G ( z ) ^ k } { k ! } \\\\ & = \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( \\sum _ { n = 1 } ^ { \\infty } q _ n z ^ n \\right ) ^ k } { k ! } \\\\ & = 1 + \\sum _ { n = 1 } ^ { \\infty } q _ n z ^ n + \\sum _ { k = 2 } ^ { \\infty } \\frac { \\left ( \\sum _ { n = 1 } ^ { \\infty } q _ n z ^ n \\right ) ^ k } { k ! } \\end{align*}"}
+{"id": "2000.png", "formula": "\\begin{align*} \\widetilde { ( \\eta ^ { n } ) } _ l = e ^ { i \\beta _ l ^ + \\tau } \\widetilde { ( \\eta ^ { n , + } ) } _ l - e ^ { i \\beta _ l ^ - \\tau } \\widetilde { ( \\eta ^ { n , - } ) } _ l - e ^ { \\frac { i \\tau } { \\alpha } } ( \\widetilde { ( \\eta ^ { n - 1 , + } ) } _ l - \\widetilde { ( \\eta ^ { n - 1 , - } ) } _ l ) , n \\ge 1 , \\end{align*}"}
+{"id": "1026.png", "formula": "\\begin{align*} \\gamma ( k ) = \\mathrm { C o v } ( X _ k , X _ 0 ) = \\int _ { - \\pi } ^ { \\pi } e ^ { - i k \\theta } w ( e ^ { i \\theta } ) \\frac { d \\theta } { 2 \\pi } , k \\in \\Z . \\end{align*}"}
+{"id": "3665.png", "formula": "\\begin{align*} \\mathbb { E } _ z [ x _ z ^ T L ( G , p ) x _ z ] = \\frac 1 d \\sum _ { i j \\in E } ( y _ i - y _ j ) ^ 2 = a ( G ) / d . \\end{align*}"}
+{"id": "5943.png", "formula": "\\begin{align*} \\dfrac { M _ { 2 } ( \\mathcal { C } ( G ) ) } { | e ( \\mathcal { C } ( G ) ) | } - \\dfrac { M _ { 1 } ( \\mathcal { C } ( G ) ) } { | v ( \\mathcal { C } ( G ) ) | } = \\dfrac { 2 ^ { 6 k } ( 3 \\cdot 2 ^ { k } - 1 1 ) + 2 ^ { k } ( 8 \\cdot 2 ^ { 4 k } - 6 \\cdot 2 ^ { 2 k } - 1 ) + 2 ^ { 2 k } ( 8 \\cdot 2 ^ { 2 k } - 1 ) } { 2 ^ { 5 k } ( 2 ^ { 2 k } - 2 \\cdot 2 ^ { k } - 3 ) + ( 4 \\cdot 2 ^ { 4 k } - 2 ^ { 2 k } - 2 ) + 2 ^ { k } ( 6 \\cdot 2 ^ { 2 k } - 5 ) } : = \\dfrac { f ( k ) } { g ( k ) } \\end{align*}"}
+{"id": "8577.png", "formula": "\\begin{align*} C _ t ^ { t _ 1 } = ( J _ k + [ - \\varepsilon _ k ( G ^ { t _ 0 } _ t ) B _ { t _ 0 } ] ^ { k \\cdot } _ + ) C ^ { t _ 0 } _ t , \\end{align*}"}
+{"id": "2381.png", "formula": "\\begin{align*} ( t , v _ 1 , v _ 2 ) = ( x + y + z , \\ \\frac { z - y } { y + z } , \\ \\frac { ( y + z ) - x } { x + y + z } ) \\ . \\end{align*}"}
+{"id": "1672.png", "formula": "\\begin{align*} g = g ( v _ 0 , v _ 1 , v _ 2 , v _ 3 ) \\coloneqq T ^ \\top J _ r . \\end{align*}"}
+{"id": "5076.png", "formula": "\\begin{align*} h _ s ( r ) & = \\rho _ s \\begin{pmatrix} r & 0 \\\\ 0 & r ^ { - 1 } \\end{pmatrix} , \\end{align*}"}
+{"id": "2517.png", "formula": "\\begin{align*} S ^ { ( m ) } ( U \\times \\R ^ l ) = S ^ m ( U \\times \\R ^ l ) / S ^ { m - 1 } ( U \\times \\R ^ l ) \\ ; . \\end{align*}"}
+{"id": "1391.png", "formula": "\\begin{align*} P ( 0 ) = 0 , \\ P ' ( Z ) > 0 \\ \\mbox { f o r } \\ Z \\geq 0 , \\ 0 < \\frac { \\frac { 5 } { 3 } P ( Z ) - P ' ( Z ) Z } { Z } \\leq c \\ \\mbox { f o r } \\ Z > 0 . \\end{align*}"}
+{"id": "6304.png", "formula": "\\begin{align*} H _ { n , \\ell } = \\sum _ { i = 1 } ^ n - \\Delta _ { x _ i } + \\sum _ { 1 \\leq i < j \\leq n } \\ell ^ 2 V ( \\ell ( x _ i - x _ j ) ) \\end{align*}"}
+{"id": "7065.png", "formula": "\\begin{align*} ( \\mu + \\Lambda ( a _ n , b _ n ) ) u _ n = f , f \\in C _ c ^ \\infty , \\mu \\geq \\mu _ 0 , \\end{align*}"}
+{"id": "1257.png", "formula": "\\begin{align*} q ^ { s n } & = 1 - ( 1 - q ^ { s n } ) \\\\ [ 1 0 p t ] & = 1 - ( 1 - q ^ n ) ( 1 + q ^ n + q ^ { 2 n } + \\cdots + q ^ { ( s - 1 ) n } ) \\\\ [ 1 0 p t ] & \\equiv 1 - s ( 1 - q ^ n ) \\pmod { \\Phi _ n ( q ) ^ 2 } , \\end{align*}"}
+{"id": "1464.png", "formula": "\\begin{align*} C ( p ) _ { E \\setminus S \\Delta } = X _ { E \\setminus S \\Delta } . \\end{align*}"}
+{"id": "3277.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ^ { d + 1 } ; q ^ d ) _ k ^ { d - 1 } ( q ^ { 1 - d } ; q ^ d ) _ k q ^ { d k } } { ( q ^ d ; q ^ d ) _ k ^ d } \\\\ & = [ d ] \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ^ { d + 1 } ; q ^ d ) _ k ^ { d - 2 } ( q , q ^ { 1 - d } ; q ^ d ) _ k q ^ { d k } } { ( q ^ d ; q ^ d ) _ k ^ d } - q [ d - 1 ] \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ^ { d + 1 } ; q ^ d ) _ k ^ { d - 2 } ( q ; q ^ d ) _ k ^ 2 q ^ { d k } } { ( q ^ d ; q ^ d ) _ k ^ d } , \\end{align*}"}
+{"id": "1674.png", "formula": "\\begin{align*} \\omega ( e _ r , \\phi _ 4 ) = \\varepsilon b _ 2 , \\omega ( f _ r , \\phi _ 4 ) = \\varepsilon b _ 1 , \\omega ( \\phi _ 2 , \\phi _ 4 ) = \\varepsilon , \\omega ( \\phi _ 3 , \\phi _ 4 ) = a _ 2 b _ 1 c _ 1 ^ { - 1 } . \\end{align*}"}
+{"id": "1909.png", "formula": "\\begin{align*} \\partial _ t v ( x , t ) = 0 \\ , \\ , \\ , t \\in \\mathcal { U } _ x . \\end{align*}"}
+{"id": "7978.png", "formula": "\\begin{align*} \\dot { H } ( v , \\Sigma ) = - \\int _ { \\Gamma } \\mathrm { t r } ( h ) \\wedge g . \\end{align*}"}
+{"id": "4480.png", "formula": "\\begin{align*} M _ H ( Z _ 0 , J , \\rho ) = \\frac { M ( Z _ 0 , J , \\tilde \\rho ) } { \\pi \\int _ { 0 } ^ { + \\infty } c ( t ) e ^ { - t } d t } \\end{align*}"}
+{"id": "8463.png", "formula": "\\begin{align*} | u _ m | + | u _ { m - 1 } | = u _ m + u _ { m - 1 } & = ( u _ m - u _ { m - 1 } ) + 2 u _ { m - 1 } \\\\ & \\leq 2 u _ { m - 1 } \\leq 2 \\ell < 3 \\ell . \\end{align*}"}
+{"id": "3783.png", "formula": "\\begin{align*} \\mathbb { P } \\left \\{ \\mathcal { M } _ i ^ { 1 , j } \\right \\} \\leq \\mathbb { P } \\left \\{ \\sum _ { t = 0 } ^ { T - 1 } f ^ { ( i ) } _ t V ^ { ( i ) } _ t \\leq \\bar { t } \\right \\} + \\mathbb { P } \\left \\{ \\sum _ { t = 0 } ^ { T - 1 } g ^ { ( i ) } _ t V ^ { ( i ) } _ t > \\bar { t } \\right \\} , \\end{align*}"}
+{"id": "5862.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { N C } ( D _ { 2 m } ) ) & = ( 2 m - 2 ) ( 2 m - 3 ) ^ { 2 } - 4 ( 2 m - 3 ) \\dfrac { ( m - 2 ) ( m - 3 ) + m } { 2 } + ( m - 2 ) ( m - 3 ) ^ { 2 } + m \\\\ & = 5 m ^ { 3 } - 1 8 m ^ { 2 } + 1 6 m \\end{align*}"}
+{"id": "7155.png", "formula": "\\begin{align*} k ^ \\mu \\varepsilon _ \\mu ( k , 1 ) = k ^ \\mu \\varepsilon _ \\mu ( k , 2 ) = 0 \\ , \\ , . \\end{align*}"}
+{"id": "3062.png", "formula": "\\begin{align*} F ( x , y ) = \\prod _ { \\alpha \\in U _ n } \\left ( y - \\eta \\left ( \\alpha \\cdot x ^ { \\frac { 1 } { n } } \\right ) \\right ) . \\end{align*}"}
+{"id": "3168.png", "formula": "\\begin{align*} ( k - 1 , i ) \\in ( k - 2 , i ) ^ { H } = ( k - 2 , i - 1 ) ^ { H } = ( k - 1 , i - 1 ) ^ { H } . \\end{align*}"}
+{"id": "2962.png", "formula": "\\begin{align*} \\R _ n ^ 2 \\left \\| \\frac { f } { r ^ 2 } \\right \\| _ { L _ { n - 1 } ^ 2 } ^ 2 & = \\| \\L _ { n - 1 } f \\| _ { L _ { n - 1 } ^ 2 } ^ 2 - \\left \\| \\L _ { n - 1 } f + \\R _ n \\frac { f } { r ^ 2 } \\right \\| _ { L _ { n - 1 } ^ 2 } ^ 2 - 2 \\R _ n \\left \\| \\frac { f ^ * } { r } \\right \\| _ { L _ { n - 1 } ^ 2 } ^ 2 \\\\ & = \\left \\| \\L _ { n - 1 } f \\right \\| _ { L _ { n - 1 } ^ 2 } ^ 2 - \\left ( 1 + \\frac { \\R _ n } { 2 } \\right ) \\left \\| \\L _ { n - 1 } f + \\R _ n \\frac { f } { r ^ 2 } \\right \\| _ { L _ { n - 1 } ^ 2 } ^ 2 + \\frac { \\R _ n } { 2 } \\left \\| f ^ \\# \\right \\| _ { L _ { n - 1 } ^ 2 } ^ 2 \\end{align*}"}
+{"id": "3445.png", "formula": "\\begin{align*} C ^ * _ { H } ( X , m , n ) : = C ^ * ( \\Sigma ^ V \\Sigma ^ W X \\wedge _ H E H ) . \\end{align*}"}
+{"id": "1096.png", "formula": "\\begin{align*} \\int _ 0 ^ r t ^ { a + n - 1 } [ \\log ( 2 + t ) ] ^ b \\ , d t & < [ \\log ( 2 + r ) ] ^ b \\int _ 0 ^ r t ^ { a + n - 1 } \\ , d t \\\\ & = \\frac { 1 } { a + n } r ^ { a + n } [ \\log ( 2 + r ) ] ^ b \\end{align*}"}
+{"id": "7317.png", "formula": "\\begin{align*} \\sum _ { j = - \\infty } ^ { \\infty } & e ^ { - 2 \\pi i \\beta b j } \\left ( s + 1 - \\sqrt { ( s + 1 ) ^ 2 - 1 } \\right ) ^ { | \\ell + j m | } \\\\ & = \\frac { \\sinh \\left ( ( m - \\ell ) \\cosh ^ { - 1 } ( s + 1 ) \\right ) + e ^ { 2 \\pi i \\beta } \\sinh \\left ( \\ell \\cosh ^ { - 1 } ( s + 1 ) \\right ) } { \\cosh \\left ( m \\cosh ^ { - 1 } ( s + 1 ) \\right ) - \\cos 2 \\pi \\beta } . \\end{align*}"}
+{"id": "3961.png", "formula": "\\begin{align*} \\rho _ { \\restriction Y ^ \\rho _ n } = \\rho _ \\pi \\circ \\iota _ n , \\end{align*}"}
+{"id": "6370.png", "formula": "\\begin{align*} c _ 0 ^ 2 + \\sum _ { k = 1 } ^ n c _ { 1 , k } ^ 2 \\leq \\Bigl ( ( C _ 2 + C _ 3 ) ^ 2 + \\frac { n ^ 2 } { 2 } ( B _ 1 + 2 C _ 2 ) ^ 2 \\Bigr ) \\norm { \\rho } _ { C ^ 0 } \\sum _ { j , k } c _ { j , k } ^ 2 . \\end{align*}"}
+{"id": "7025.png", "formula": "\\begin{align*} d X _ t = - b ( X _ t ) d t + \\sqrt { 2 } d W _ t , X _ 0 = x \\in \\mathbb R ^ d , \\end{align*}"}
+{"id": "2511.png", "formula": "\\begin{align*} C ^ { \\pm \\infty } _ { { \\cdot } / } ( M ' ; \\phi ^ * E ) = C ^ { \\pm \\infty } _ { { \\cdot } / } ( M ' ) \\otimes _ { C ^ \\infty ( M ) } C ^ \\infty ( M ; E ) \\ ; . \\end{align*}"}
+{"id": "8637.png", "formula": "\\begin{align*} ( v _ 3 ( \\partial _ x X ) ^ 3 + 3 v _ 2 ( \\partial _ x X ) ( \\partial ^ 2 _ x X ) + v _ 1 ( \\partial ^ 3 _ x X ) ) ( t ; x ) = u ^ { \\prime \\prime \\prime } _ 0 ( x ) - I _ 3 ( t ; x ) \\end{align*}"}
+{"id": "6325.png", "formula": "\\begin{align*} 2 \\int _ { \\Lambda ^ 2 } \\widetilde { Q } _ 2 ^ { ( \\epsilon ) } ( x , y ) K ( x , y ) & = - n ^ 2 \\sum _ { p \\in \\pi \\mathbb { N } _ 0 ^ 3 \\setminus \\{ 0 \\} } \\widehat { \\epsilon } _ { \\ell , \\lambda } ( p ) \\widehat { \\omega } _ { \\ell , \\lambda } ( p ) \\end{align*}"}
+{"id": "3832.png", "formula": "\\begin{align*} K _ k ^ { [ 3 ] } ( n , m ) = K _ k \\left ( [ 3 ] _ k \\left ( n - \\frac { k ( k + 2 ) } { 8 } \\right ) , m - \\frac { k ^ 2 + 2 } { 6 } + \\frac { 3 k } { 4 } \\left ( 1 - \\frac { 3 k } { 2 } \\right ) \\right ) . \\end{align*}"}
+{"id": "2479.png", "formula": "\\begin{align*} \\begin{cases} \\phi ' = m ^ { - 1 } \\phi ^ { 1 - m } \\psi , \\\\ \\psi ' = - m ^ { - 1 } c \\phi ^ { 1 - m } \\psi \\end{cases} \\left ( \\ , ' = \\dfrac { d } { d \\xi } \\right ) \\end{align*}"}
+{"id": "5050.png", "formula": "\\begin{align*} \\beta _ S = \\begin{cases} c \\exp ( - q K _ S \\epsilon _ S ^ { \\max { \\{ p , 2 \\} } } ) , & \\epsilon _ S \\leq 1 \\\\ c \\exp ( - q K _ S \\epsilon _ S ^ { a } ) , & \\epsilon _ S > 1 , \\end{cases} \\end{align*}"}
+{"id": "6365.png", "formula": "\\begin{align*} \\phi ( \\bar \\rho ) - \\phi ( R ) = \\phi ' ( R ) R \\rho + \\frac { R ^ 2 \\rho ^ 2 } { 2 } \\phi '' ( R ) + \\frac { R ^ 3 \\rho ^ 3 } { 6 } \\phi ''' ( R ( 1 + \\bar \\eta \\rho ) ) , \\end{align*}"}
+{"id": "5452.png", "formula": "\\begin{align*} \\gamma ( f ) ( s , x , v ) = \\gamma ( f ) ( s , x , v - 2 ( v \\cdot n ( x ) ) n ( x ) ) , \\lambda _ { \\Sigma _ T } - a . e . \\mbox { o n } \\Sigma _ T \\mbox { a . s . } \\end{align*}"}
+{"id": "474.png", "formula": "\\begin{align*} ( B , h ) = \\sum _ { \\substack { A \\supseteq B \\\\ A \\not \\ni x _ 1 , \\ldots , x _ { n - 1 } } } ( A , h ) - \\sum _ { \\substack { A \\supseteq B \\\\ A \\not \\ni x _ 1 , \\ldots , x _ { n } } } ( A , h ) \\in \\mathfrak { A } , \\end{align*}"}
+{"id": "2331.png", "formula": "\\begin{align*} \\alpha < \\frac { 1 } { 4 \\| N \\| } , \\quad \\| N \\| : = \\sup \\limits _ { \\| u \\| , \\| v \\| \\leq 1 } \\| N ( u , v ) \\| . \\end{align*}"}
+{"id": "353.png", "formula": "\\begin{align*} ( x ^ m ) ^ p + ( y ^ m ) ^ p = ( z ^ m ) ^ p , \\forall n = m p . \\end{align*}"}
+{"id": "7593.png", "formula": "\\begin{align*} \\rho = \\frac { 4 N ( q - 2 ) | E _ { p , q } ( \\psi _ { 0 } ) | + 2 } { N ( q - 2 ) - 8 } . \\end{align*}"}
+{"id": "1896.png", "formula": "\\begin{align*} s : = \\sup \\{ d ( x , y ) \\ , : \\ , x , y \\in G , \\omega ( x , y ) > 0 \\} . \\end{align*}"}
+{"id": "8597.png", "formula": "\\begin{align*} \\abs { \\Gamma \\cap \\Gamma } = \\abs { \\Gamma _ 1 \\cap \\Gamma _ 1 } + \\abs { \\Gamma _ 2 \\cap \\Gamma _ 2 } \\end{align*}"}
+{"id": "6416.png", "formula": "\\begin{align*} \\ln ( n ) ^ q \\left | \\frac { 1 } { n } \\sum _ { i = 1 } ^ n f ( X _ { \\frac { i - 1 } { n } } , \\delta _ 0 , \\alpha _ 0 ) - \\int _ 0 ^ 1 f ( X _ s , \\delta _ 0 , \\alpha _ 0 ) d s \\right | \\to 0 . \\end{align*}"}
+{"id": "6474.png", "formula": "\\begin{align*} | \\bar \\nabla \\bar \\Delta ( f \\nu ) | ^ 2 - 2 m | \\bar \\Delta ( f \\nu ) | ^ 2 = & | \\nabla \\Delta f | ^ 2 + 4 m ^ 2 | \\nabla f | ^ 2 + 4 m \\nabla \\Delta f \\nabla f + 8 | \\nabla ^ 2 f | ^ 2 + 1 6 | \\nabla f | ^ 2 \\\\ & + 8 \\nabla \\Delta f \\nabla f + 8 | \\Delta f | ^ 2 + 1 6 m f \\Delta f . \\end{align*}"}
+{"id": "2130.png", "formula": "\\begin{align*} \\begin{array} { l l } \\lambda _ 1 = \\frac { 5 x ( y _ 1 - y _ 2 ) ( 4 x + 4 - 4 y _ 1 - 5 y _ 2 ) } { ( 3 2 x - 2 4 ) ( 9 y _ 2 - 3 6 x + 2 8 ) } , & \\lambda _ 2 = 9 \\lambda _ 1 + \\lambda _ 3 + 9 \\lambda _ 4 , \\\\ \\lambda _ 3 = \\frac { 4 ( x - \\lambda _ 4 ) ( 5 y _ 1 - 4 y _ 2 + 4 x - 4 ) } { ( 4 1 y _ 2 + 1 6 4 x - 1 6 4 ) } , & \\lambda _ 4 = \\frac { x ( y _ 2 - y _ 1 ) ( 1 2 - 5 y _ 1 - 4 y _ 2 - 4 x ) } { ( 1 0 x - 8 ) ( 9 y _ 2 - 3 6 x + 2 8 ) } . \\\\ \\end{array} \\end{align*}"}
+{"id": "4064.png", "formula": "\\begin{align*} R ^ { - } _ 0 ( z ) ( x , y ) = \\sum _ { k = 0 } ^ { \\frac { d - 3 } { 2 } } \\frac { \\tilde { c _ k } } { | x - y | ^ { d - 1 - s _ k } } \\partial _ s ^ { s _ k } \\Big \\{ e ^ { i z s } \\tilde { E } _ i ( - i z s ) + e ^ { - i z s } \\tilde { E } _ i ( i z s ) + 2 \\pi i e ^ { - i z s } \\Big \\} \\Big | _ { ( s = | x - y | ) } \\end{align*}"}
+{"id": "1389.png", "formula": "\\begin{align*} ^ { \\perp _ { f } } A [ > 0 ] & = \\{ Y \\in \\mathcal { R } ( A ) \\mid \\mathrm { H o m } _ { \\mathrm { D } ( A ) } ( X , A [ > 0 ] ) = 0 \\} \\\\ & = \\{ Y \\in \\mathcal { R } ( A ) \\mid \\mathrm { H } ^ { i } ( \\mathrm { R H o m } _ { A } ( X , A ) ) = 0 ~ \\textrm { f o r } ~ i > 0 \\} . \\end{align*}"}
+{"id": "1776.png", "formula": "\\begin{align*} \\Phi ( w , z , \\zeta ) : = ( w ^ * , z ^ * , \\zeta ^ * ) = ( w + h , z _ 1 + f _ { 1 } , \\ldots , z _ { n - 1 } + f _ { n - 1 } , \\zeta + g ) \\end{align*}"}
+{"id": "552.png", "formula": "\\begin{align*} \\lim _ { x \\to 1 ^ - } u ( x ) = u ( a ) , \\ , \\ , \\ , \\lim _ { x \\to 1 ^ - } p u ' ( x ) = ( p u ' ) ( a ) \\end{align*}"}
+{"id": "883.png", "formula": "\\begin{align*} S _ 2 + T _ 2 & = 2 w ^ 1 { w ^ 1 } _ { \\substack { \\\\ x _ 2 } } + 2 w ^ 2 { w ^ 2 } _ { \\substack { \\\\ x _ 2 } } \\\\ & = ( ( w ^ 1 ) ^ 2 + ( w ^ 2 ) ^ 2 ) _ { x _ 2 } . \\end{align*}"}
+{"id": "649.png", "formula": "\\begin{align*} \\sin \\theta = 0 , \\cos \\theta = \\frac { \\sinh ( a + c ) } { 2 \\sinh a \\cosh c } . \\end{align*}"}
+{"id": "5330.png", "formula": "\\begin{align*} \\nu ^ { S _ { k } } _ j = \\begin{cases} \\frac { \\displaystyle c _ j } { \\displaystyle w ^ { S _ 1 } _ j } , & \\\\ \\nu _ { \\pi _ { k - 1 } } + \\frac { \\displaystyle c _ j - \\nu _ { \\pi _ 1 } \\ , w ^ { S _ 1 } _ j - \\sum _ { l = 2 } ^ { k - 1 } \\left ( \\nu _ { \\pi _ { l } } - \\nu _ { \\pi _ { l - 1 } } \\right ) \\ , w ^ { S _ { l } } _ j } { \\displaystyle w ^ { S _ { k } } _ j } , & ; \\end{cases} \\end{align*}"}
+{"id": "4500.png", "formula": "\\begin{align*} \\exp \\bigg ( \\sum _ { k = 1 } ^ { + \\infty } \\frac { X ( k ) } { \\sqrt { k } } z ^ k \\bigg ) = \\sum _ { n = 0 } ^ { + \\infty } A ( n ) z ^ n . \\end{align*}"}
+{"id": "8974.png", "formula": "\\begin{align*} w '' = \\Delta w - | w | ^ { r - 2 } w + G ( t , x , w ) \\mbox { i n } \\ \\mathcal D ' ( ( 0 , T ) \\times \\R ^ N ) , \\end{align*}"}
+{"id": "6309.png", "formula": "\\begin{align*} K ( x , y ) = ( Q ^ { \\otimes 2 } \\widetilde { K } ) ( x , y ) = \\widetilde { K } ( x , y ) + n \\widehat { \\omega } _ { \\ell , \\lambda } ( 0 ) = - \\sum _ { p \\in \\pi \\mathbb { N } _ 0 ^ 3 \\backslash \\{ 0 \\} } n \\widehat \\omega _ { \\ell , \\lambda } ( p ) u _ p ( x ) u _ p ( y ) , \\end{align*}"}
+{"id": "1408.png", "formula": "\\begin{align*} E _ \\alpha ( x ) & = \\sum _ { k = 0 } ^ \\infty \\frac { x ^ k } { \\Gamma ( \\alpha k + 1 ) } \\alpha \\ge 0 \\end{align*}"}
+{"id": "7213.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { V _ { F } } p ( v , t ) d v = \\int _ { - \\infty } ^ { V _ { F } } p ^ { 0 } ( v ) d v = 1 . \\end{align*}"}
+{"id": "6066.png", "formula": "\\begin{align*} U = x _ { k + 1 } A ( k + 1 ) _ { d - 1 } \\oplus V \\end{align*}"}
+{"id": "2019.png", "formula": "\\begin{align*} \\Lambda _ 1 : = \\left ( \\dfrac { ( \\rho - 1 ) a } { d _ 1 } \\right ) \\cdot \\alpha ^ n \\cdot \\rho ^ { - ( \\ell + m + k ) } - 1 . \\end{align*}"}
+{"id": "2416.png", "formula": "\\begin{align*} v _ 2 \\left ( \\int _ { \\mathbb { Z } _ 2 } f _ { ( 0 , \\ldots , 0 , i _ { 2 ^ { m - 1 } } = s , 0 , \\ldots , 0 ) } ( t ) \\mathrm { d } t \\right ) = ( s + 2 ) n - ( 2 s + 3 ) m + s + v _ 2 ( ( s + 2 ) ! ) - 1 , \\end{align*}"}
+{"id": "4807.png", "formula": "\\begin{align*} \\widehat { \\mathbb { Q } } _ K : = \\frac { 1 } { K } \\sum _ { k = 1 } ^ K \\delta _ { \\hat { \\mathbf { c } } ^ { \\mbox { \\tiny ( \\itshape k \\upshape ) } } } , \\end{align*}"}
+{"id": "4677.png", "formula": "\\begin{align*} \\mathbf { B } _ { - } ( D ) : = \\bigcup _ { A } \\mathbf { B } ( D + A ) , \\end{align*}"}
+{"id": "347.png", "formula": "\\begin{align*} z - y = a , z - x = b . \\end{align*}"}
+{"id": "8386.png", "formula": "\\begin{align*} \\mathrm { B L } _ { m , n } ( \\mathbb { F } ) & = \\bigg \\{ \\begin{bmatrix} I & 0 \\\\ A & I \\end{bmatrix} \\ , \\bigg | \\ , A \\in \\mathrm { M } _ { n , m } ( \\mathbb { F } ) \\bigg \\} , \\\\ \\mathrm { B U } _ { m , n } ( \\mathbb { F } ) & = \\bigg \\{ \\begin{bmatrix} I & A \\\\ 0 & I \\end{bmatrix} \\ , \\bigg | \\ , A \\in \\mathrm { M } _ { m , n } ( \\mathbb { F } ) \\bigg \\} . \\end{align*}"}
+{"id": "1858.png", "formula": "\\begin{align*} d ^ * a = 0 \\| a \\| _ { L ^ { 7 / 2 } _ 1 ( \\mathbb { B } ) } \\leq \\epsilon _ 6 \\end{align*}"}
+{"id": "3153.png", "formula": "\\begin{align*} \\varphi ^ \\sharp \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) = \\left ( \\begin{array} { c c c } a ^ { 2 \\ , p ^ { e _ 1 } } & 0 & b ^ { 2 \\ , p ^ { e _ 1 } } \\\\ 0 & 1 & b ^ { p ^ { e _ 1 } } \\ , d ^ { p ^ { e _ 1 } } \\\\ 0 & 0 & d ^ { 2 \\ , p ^ { e _ 1 } } \\end{array} \\right ) ( \\ , e _ 1 \\geq 0 \\ , ) . \\end{align*}"}
+{"id": "2943.png", "formula": "\\begin{align*} d _ \\pi ^ { - 1 } \\int _ G \\| \\pi ( g ^ { - 1 } ) | | ^ 2 \\nu _ t ( g ) d g = e ^ { t \\lambda _ \\pi ^ 2 } \\end{align*}"}
+{"id": "899.png", "formula": "\\begin{align*} & Q _ 0 ^ 1 = \\frac { 1 } { 2 } ( A _ 6 x _ 1 + B _ 6 x _ 2 + C _ 6 ) + \\frac { 1 } { 4 } ( A _ 1 J x _ 1 + B _ 1 J x _ 2 + C _ 1 J ) , \\\\ & P _ 2 ^ 1 = - \\frac { 1 } { 2 } ( A _ 1 A _ 6 x _ 1 ^ 2 + ( A _ 1 B _ 6 + B _ 1 A _ 6 ) x _ 1 x _ 2 + B _ 1 B _ 6 x _ 2 ^ 2 + ( C _ 1 A _ 6 + A _ 1 C _ 6 ) x _ 1 \\\\ & \\quad + ( C _ 1 B _ 6 + B _ 1 C _ 6 ) x _ 2 + C _ 1 C _ 6 + N ) , \\end{align*}"}
+{"id": "1595.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\downarrow 0 } i _ { F _ \\lambda } & \\ge \\frac { i _ F } { C } \\ , \\lim _ { \\lambda \\downarrow 0 } \\frac { 1 } { \\eta _ H ( \\lambda ) \\ , \\lambda } = \\infty \\lim _ { \\lambda \\uparrow \\infty } i _ { F _ \\lambda } \\le C \\ , i _ F \\ , \\lim _ { \\lambda \\uparrow \\infty } \\frac { \\eta _ H ( 1 / \\lambda ) } { \\lambda } = 0 . \\end{align*}"}
+{"id": "1960.png", "formula": "\\begin{align*} d ( { { \\hat { C } } ^ { p ^ t } | } _ { T _ 0 } ) = \\min \\{ | T _ { 0 , i } | \\mid i \\in \\bar { N _ { 0 } } \\} . \\end{align*}"}
+{"id": "7848.png", "formula": "\\begin{align*} { \\rm P A P R } ( \\mathbf { \\Phi } ) = \\max _ { \\mathbf { s } \\in \\mathbf { \\Phi } } { \\rm P A P R } ( \\mathbf { s } ) . \\end{align*}"}
+{"id": "4880.png", "formula": "\\begin{align*} 0 = ( K _ { \\Sigma ' } + C ' ) ^ 2 = K _ { \\Sigma ' } ^ 2 + 2 ( K _ { \\Sigma ' } + C ' ) \\cdot C ' - ( C ' ) ^ 2 = K _ { \\Sigma ' } ^ 2 + 4 ( g - 1 ) - C '^ 2 = K _ { \\Sigma ' } ^ 2 . \\end{align*}"}
+{"id": "8226.png", "formula": "\\begin{align*} \\{ { \\bf X } _ i ^ { \\star } = F _ { { \\bf X } _ i | { \\bf A } _ { i - 1 } } ^ { - 1 } \\circ F _ { W } ( W _ i ) : i \\in [ L ] \\} , \\end{align*}"}
+{"id": "7735.png", "formula": "\\begin{align*} \\frac { \\partial \\chi _ { \\Sigma ( \\lambda ) } } { \\partial X } = m X ^ { m - 1 } + ( m - 1 ) a _ 1 ( \\lambda ) X ^ { m - 2 } + \\cdots + a _ { m - 1 } ( \\lambda ) . \\end{align*}"}
+{"id": "7774.png", "formula": "\\begin{align*} | F _ 1 | _ { h _ 1 , \\delta _ 1 } \\leq \\tilde M _ 1 ^ 2 \\varepsilon = b ^ 2 ( ( b _ 1 \\frac { | \\ln \\varepsilon | } { 4 \\pi h _ 1 } ) ^ \\tau e ^ { b _ 2 e ^ 4 } R _ 0 ) ^ { 2 m ^ 2 ( m + 2 ) } \\varepsilon < \\varepsilon ^ { \\frac { 1 } { 2 } } = : \\epsilon _ 1 . \\end{align*}"}
+{"id": "4567.png", "formula": "\\begin{align*} R i c _ { g , a b } + 2 s _ g ^ { - 1 } \\nabla _ a \\nabla _ b s _ g = \\left ( \\frac { s _ g } { 6 } + | d \\log s _ g | ^ 2 \\right ) g _ { a b } . \\end{align*}"}
+{"id": "8790.png", "formula": "\\begin{align*} G _ x ( u ) - G _ x ( u ' ) = \\int _ { F _ \\nu ( a ) - u } ^ { F _ \\nu ( a ) - u ' } F _ \\nu ^ { - 1 } ( v ) \\ , d v - \\int _ { 1 + u ' - F _ \\mu ( x ) } ^ { 1 + u - F _ \\mu ( x ) } F _ \\nu ^ { - 1 } ( v ) \\ , d v \\leq ( u - u ' ) a - ( u - u ' ) b . \\end{align*}"}
+{"id": "2986.png", "formula": "\\begin{align*} \\begin{array} { l l l } ( \\nabla _ x \\phi ) y & = & - g ( x , y ) \\xi - \\eta ( y ) x + 2 \\eta ( x ) \\eta ( y ) \\xi , \\\\ & = & - g ( \\phi x , \\phi y ) \\xi - \\eta ( y ) \\phi ^ 2 x . \\end{array} \\end{align*}"}
+{"id": "4373.png", "formula": "\\begin{align*} \\mathcal { X } _ { \\overline { \\mathcal { Y } } } : = \\{ x \\in \\mathcal { X } \\colon & \\Delta y _ { q , p ( q ) } g _ { q , p ( q ) } ( x ) \\geq \\Delta y _ { a , b ( a ) } g _ { a , b ( a ) } ( x ) \\ \\forall q \\in \\overline { \\mathcal { Y } } , \\\\ & \\Delta y _ { q , p ( q ) } g _ { q , p ( q ) } ( x ) \\leq \\Delta y _ { a , b ( a ) } g _ { a , b ( a ) } ( x ) \\ \\forall q \\in [ m ] \\setminus \\overline { \\mathcal { Y } } \\} , \\end{align*}"}
+{"id": "7672.png", "formula": "\\begin{align*} b _ 2 = \\sup _ { n ' \\in \\mathbb { Z } ^ d } \\sum _ { n \\in \\mathbb { Z } ^ d } \\frac { W ( n ) } { W ( n ' ) } K ( n , n ' ) \\leq \\leq \\frac { C _ a 7 2 \\sqrt { 2 } \\norm { F } _ { \\infty } } { \\eta } | g | S _ { \\delta - \\gamma } S _ { \\delta - 2 \\nu } . \\end{align*}"}
+{"id": "2636.png", "formula": "\\begin{align*} P _ { \\leq M _ 0 } V = V , \\end{align*}"}
+{"id": "3131.png", "formula": "\\begin{align*} \\psi ^ * \\left ( \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ) : = \\left ( \\begin{array} { c c c } a ^ { p ^ { e _ 2 } } & 0 & b ^ { p ^ { e _ 2 } } \\\\ 0 & 1 & 0 \\\\ c ^ { p ^ { e _ 2 } } & 0 & d ^ { p ^ { e _ 2 } } \\end{array} \\right ) . \\end{align*}"}
+{"id": "6823.png", "formula": "\\begin{align*} ( N - 1 ) \\left ( \\sum _ { i , j } \\beta _ { i j } k _ { i j } ^ { N - 2 } + K _ { N - 2 } \\right ) + ( N - 2 ) \\left ( \\sum _ { i , j } \\beta _ { i j } k _ { i j } ^ { N - 1 } + K _ { N - 1 } \\right ) = 0 . \\end{align*}"}
+{"id": "6414.png", "formula": "\\begin{align*} \\sup _ { \\theta \\in A \\times W _ n ^ { ( \\eta ) } } \\ln ( n ) ^ q \\left | \\frac { 1 } { n } \\sum _ { i = 1 } ^ n f ( X _ { \\frac { i - 1 } { n } } , \\delta _ 0 , \\alpha _ 0 ) \\left ( h ( z ^ n _ i ( \\theta ) , \\alpha ) - h ( n ^ { 1 / \\alpha _ 0 } \\Delta _ i ^ n L , \\alpha ) \\right ) \\right | \\to 0 , \\end{align*}"}
+{"id": "7216.png", "formula": "\\begin{align*} \\int _ { V _ { \\min } } ^ { V _ { F } } p ( v , t ) d v = \\int _ { V _ { \\min } } ^ { V _ { F } } p ^ { 0 } ( v ) d v = 1 . \\end{align*}"}
+{"id": "2747.png", "formula": "\\begin{align*} T ^ { * } M = T ^ { * } M | _ { \\Xi , \\Psi } \\times T ^ { * } M | _ { Q , P } \\end{align*}"}
+{"id": "5630.png", "formula": "\\begin{align*} d ( T a , a ) & \\leq d ( T a , T u _ 2 ) + d ( T u _ 2 , u _ 1 ) + d ( u _ 1 , a ) \\\\ & = d ( a , u _ 2 ) + d ( T u _ 2 , u _ 1 ) + d ( u _ 1 , a ) \\\\ & < \\frac { \\delta } { 4 } + \\frac { \\delta } { 2 } + \\frac { \\delta } { 4 } = \\delta , \\end{align*}"}
+{"id": "4894.png", "formula": "\\begin{align*} \\dim ( | C - M | ) \\geq 3 g - 3 + \\varepsilon - ( 2 g - 1 ) = g - 2 + \\varepsilon \\geq 1 , \\end{align*}"}
+{"id": "7725.png", "formula": "\\begin{align*} [ f ] _ { \\mathcal L ^ { p , \\lambda } _ k } ^ p : = \\sup _ { z \\in \\Omega , r > 0 } r ^ { - \\lambda } \\inf _ { P \\in \\mathcal P _ k } \\int _ { Q _ r ( z ) \\cap \\Omega } \\abs { f - P } ^ p \\dd z \\end{align*}"}
+{"id": "2409.png", "formula": "\\begin{align*} L _ n & = \\sum _ { k = 1 } ^ { \\infty } \\sum _ { k + \\frac { 1 } { 2 } < \\frac { n } { q } < k + 1 } \\log q \\\\ & = \\sum _ { k = 1 } ^ { \\infty } \\sum _ { \\frac { n } { k + 1 } < q < \\frac { n } { k + 1 / 2 } } \\log q . \\end{align*}"}
+{"id": "4397.png", "formula": "\\begin{gather*} \\mathcal { X } _ \\mathcal { Q } : = \\{ x \\in \\mathcal { X } : \\Delta u _ q ^ T l _ q ( x ) - \\Delta u _ k ^ T l _ k ( x ) \\geq 0 \\ \\forall q \\in \\mathcal { Q } , \\ \\Delta u _ q ^ T l _ q ( x ) - \\Delta u _ k ^ T l _ k ( x ) \\leq 0 \\ \\forall q \\in [ m ] \\setminus \\mathcal { Q } \\} . \\end{gather*}"}
+{"id": "3090.png", "formula": "\\begin{align*} m _ { \\delta } : = \\max _ { 0 \\leq l \\leq g } \\{ l ; \\ \\delta _ l \\neq 0 \\} \\ \\ \\mbox { a n d } \\ \\ I _ { \\delta } : = I \\left ( F , \\prod _ { l = 0 } ^ { g } F _ l ^ { \\delta _ l } \\right ) = I \\left ( F _ a , \\prod _ { l = 0 } ^ { g } F _ l ^ { \\delta _ l } \\right ) . \\end{align*}"}
+{"id": "7579.png", "formula": "\\begin{align*} \\frac { N } { N + 4 } \\mu \\| v _ \\epsilon \\| _ { q } ^ { q } < \\| \\Delta v _ \\epsilon \\| _ { 2 } ^ { 2 } = \\mathcal { S } ^ { \\frac { N } { 4 } } + O ( \\epsilon ^ { N - 4 } ) . \\end{align*}"}
+{"id": "8897.png", "formula": "\\begin{align*} \\frac { t } { e ^ t - 1 } = \\sum _ { n = 0 } ^ \\infty B _ n \\frac { t ^ n } { n ! } , \\end{align*}"}
+{"id": "5894.png", "formula": "\\begin{align*} \\dfrac { M _ { 1 } ( \\mathcal { C } ( G ) ) } { | v ( \\mathcal { C } ( G ) ) | } = ( p n - n - 1 ) ^ { 2 } = \\dfrac { M _ { 2 } ( \\mathcal { C } ( G ) ) } { | e ( \\mathcal { C } ( G ) ) | } . \\end{align*}"}
+{"id": "1821.png", "formula": "\\begin{align*} \\phi _ 0 ( u , v , w ) = g _ 0 ( u \\times v , w ) , \\forall \\ ; u , v , w \\in V . \\end{align*}"}
+{"id": "2557.png", "formula": "\\begin{align*} C ^ \\infty ( M ) \\subset J ^ { ( \\infty ) } ( M , L ) : = \\bigcap _ s J ^ { ( s ) } ( M , L ) \\ ; , J ( M , L ) \\subset C ^ { - \\infty } ( M , L ) \\ ; ; \\end{align*}"}
+{"id": "6989.png", "formula": "\\begin{align*} L _ x = \\{ [ i y _ 1 W _ 1 + \\cdots + i y _ n W _ n + i x _ { n + 1 } W _ { n + 1 } ] : x _ { n + 1 } , y _ 1 , . . . , y _ n \\in \\mathbb { R } , x _ { n + 1 } \\neq 0 \\} . \\end{align*}"}
+{"id": "3584.png", "formula": "\\begin{align*} b = - ( 1 + a ) \\alpha ^ 2 \\tau _ 0 \\tau _ 1 . \\end{align*}"}
+{"id": "6952.png", "formula": "\\begin{align*} V _ 0 = & \\{ \\mathbf { z } \\in \\mathbb { C } ^ { n , 1 } \\setminus \\{ \\mathbf { 0 } \\} : \\langle \\mathbf { z } , \\mathbf { z } \\rangle = 0 \\} , \\\\ V _ + = & \\{ \\mathbf { z } \\in \\mathbb { C } ^ { n , 1 } : \\langle \\mathbf { z } , \\mathbf { z } \\rangle > 0 \\} , \\\\ V _ - = & \\{ \\mathbf { z } \\in \\mathbb { C } ^ { n , 1 } : \\langle \\mathbf { z } , \\mathbf { z } \\rangle < 0 \\} . \\end{align*}"}
+{"id": "7887.png", "formula": "\\begin{align*} \\mathbf { D } _ { i , j } = \\left ( \\begin{array} { c c c c c c } 2 d _ { 1 } ^ { ' } & c _ { 1 } & & & & \\\\ c _ { 1 } & 2 d _ { 2 } ^ { ' } & c _ { 2 } & & & \\\\ & c _ { 2 } & 2 d _ { 3 } ^ { ' } & c _ { 3 } & & \\\\ & & \\ddots & \\ddots & \\ddots & \\\\ & & & c _ { m - 2 } & 2 d _ { m - 1 } ^ { ' } & c _ { m - 1 } \\\\ & & & & c _ { m - 1 } & 2 d _ { m } ^ { ' } \\end{array} \\right ) . \\end{align*}"}
+{"id": "3360.png", "formula": "\\begin{align*} \\mathcal { J } _ { \\sigma ^ \\circ _ K } - \\mathcal { J } _ { \\sigma _ K } \\leq e _ \\Phi ( c ^ K d _ 0 + s \\sum _ { i = 1 } ^ { K - 1 } c ^ i ) . \\end{align*}"}
+{"id": "5576.png", "formula": "\\begin{align*} \\sum _ { x \\in [ n ] } f ( G , x ) = \\langle B ^ t \\chi _ i , \\check \\chi _ j \\rangle , \\end{align*}"}
+{"id": "3894.png", "formula": "\\begin{align*} \\psi ( x ) = \\begin{cases} \\log p , & x _ 1 = 0 \\\\ \\log ( 1 - p ) & x _ 1 = 1 \\end{cases} \\end{align*}"}
+{"id": "1459.png", "formula": "\\begin{align*} \\overline { D } _ G ( S ) = \\liminf _ { i \\in I } \\sup _ { g \\in G } \\frac { | S \\cap F _ i g | } { | F _ i | } , \\underline { D } _ G ( S ) = \\limsup _ { i \\in I } \\inf _ { g \\in G } \\frac { | S \\cap F _ i g | } { | F _ i | } \\end{align*}"}
+{"id": "3582.png", "formula": "\\begin{align*} \\alpha ^ 3 ( \\tau _ 0 ^ 2 \\tau _ 1 ) \\tau - ( 1 + a ) \\alpha ^ 2 \\tau _ 0 \\tau _ 1 i d + ( a + b ) \\alpha \\tau _ 0 \\tau - b i d = 0 . \\end{align*}"}
+{"id": "5765.png", "formula": "\\begin{align*} X _ 0 ( t ) Y _ 0 ( t ) = \\sum _ { k + \\ell \\leq s } \\sum _ { 1 \\leq j \\leq J } z ^ { ( k , \\ell ) } _ j ( t ) \\mathcal { W } ^ { ( k , \\ell ) } _ j ( t ) , \\end{align*}"}
+{"id": "8449.png", "formula": "\\begin{align*} \\| v \\| _ { L ^ { q } ( t _ 1 , t _ 2 \\ , ; \\ , L ^ { p } ( K ) ) } : = \\begin{cases} \\displaystyle \\left ( \\int _ { t _ 1 } ^ { t _ 2 } \\| v ( t ) \\| _ p ^ { q } \\ , d t \\right ) ^ { 1 / q } & ( 1 \\leq q < \\infty ) \\\\ \\displaystyle \\sup _ { t _ 1 < t < t _ 2 } \\| v ( t ) \\| _ p & ( q = \\infty ) , \\end{cases} \\end{align*}"}
+{"id": "4024.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow + \\infty } \\frac { y ^ { ( t ) } } { x _ { 1 } ^ { ( t ) } } = \\begin{cases} \\frac { \\gamma } { \\gamma _ 1 } & \\mbox { i f } \\gamma _ { 2 } = 0 , \\delta _ { 1 } \\neq 0 \\\\ \\frac { \\gamma x _ { 1 } ^ { ( t _ { 0 } ) } + \\delta x _ { 2 } ^ { ( t _ { 0 } ) } } { \\gamma _ 1 x _ { 1 } ^ { \\left ( t _ { 0 } \\right ) } } & \\mbox { i f } \\gamma _ { 2 } = \\delta _ { 1 } = 0 , \\\\ + \\infty & \\mbox { i f } \\gamma _ { 2 } = 0 , \\delta _ { 1 } \\neq 0 . \\end{cases} \\end{align*}"}
+{"id": "2126.png", "formula": "\\begin{align*} I ^ { s , t } _ 3 = 2 ( Z ^ i ( t ) - Z ^ i ( s ) ) - 2 \\langle \\bar X ^ { i , N } ( s ) , Y ^ i ( t ) - Y ^ i ( s ) \\rangle . \\end{align*}"}
+{"id": "1079.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { n } \\Vert ( T _ n ( w ) ^ { - 1 } ) ^ { s , n + 1 - t } \\| \\leq B _ 3 , n \\in \\N , \\ \\ t \\in \\{ 1 , \\dots , [ ( n + 1 ) / 2 ] \\} . \\end{align*}"}
+{"id": "6684.png", "formula": "\\begin{align*} L _ 1 = \\langle x , y _ 1 , \\ldots , y _ l \\rangle \\le H L = L _ 1 \\Phi _ p ( N ) \\le H . \\end{align*}"}
+{"id": "1567.png", "formula": "\\begin{align*} U ^ i ( x ) : = \\big ( ( F - \\ell ) ( D u ( x ) ) \\big ) _ i = \\sum _ { k = 1 } ^ N ( V ^ k - \\ell _ k ) \\ , u _ { k i } V ^ i _ j = \\sum _ { h = 1 } ^ N F _ { i h } ( D u ) \\ , u _ { h j } , \\end{align*}"}
+{"id": "6116.png", "formula": "\\begin{align*} \\varphi _ \\ell ( X ) = \\sum _ { i = 0 } ^ { \\infty } \\frac { X ^ i } { ( i + \\ell ) ! } , \\ell \\geq 0 . \\end{align*}"}
+{"id": "7407.png", "formula": "\\begin{align*} t _ 0 = x _ 0 + \\frac { R } { 4 } \\quad \\mbox { a n d } t _ 0 \\ge \\frac { 5 R } { 4 } ( > 1 ) . \\end{align*}"}
+{"id": "4756.png", "formula": "\\begin{align*} \\lim _ { t , s / t \\to 0 ^ + } \\norm { \\frac { x - J _ t x } { t } - \\frac { x - S ( s ) x } { s } } = 0 . \\end{align*}"}
+{"id": "53.png", "formula": "\\begin{align*} & \\sum _ { \\mu \\in \\Z ^ n } | \\hat { \\varphi } ( \\xi + \\mu ) | ^ 2 = 1 , \\xi \\in \\R ^ n , \\\\ & { \\rm s u p p } ( \\hat { \\varphi } ) \\subset ( - 1 , 1 ) ^ n . \\end{align*}"}
+{"id": "5970.png", "formula": "\\begin{align*} d ( ( m _ 1 , \\ell _ 1 ) , ( m _ 2 , \\ell _ 2 ) ) = \\begin{cases} | f | | \\ell _ 1 - \\ell _ 2 | & m _ 1 = m _ 2 \\\\ | f | | \\ell _ 1 + \\ell _ 2 | & m _ 1 \\neq m _ 2 . \\end{cases} \\end{align*}"}
+{"id": "8069.png", "formula": "\\begin{align*} - \\frac { N } { \\gamma _ N } & \\left ( \\sum _ { k = 1 } ^ 3 \\mu ^ N _ { \\tau _ c ^ N , k } ( f _ k ) - \\sum _ { k = 1 } ^ 3 \\mu _ { \\tau _ c ^ N , k } ( f _ k ) \\right ) \\\\ & = \\frac { N } { \\gamma _ N } \\left ( \\sum _ { k = 1 } ^ 3 \\mu _ { \\tau _ c ^ N , k } ( f _ k ) - \\sum _ { k = 1 } ^ 3 \\mu _ { \\tau _ c , k } ( f _ k ) \\right ) + O ( \\gamma _ N ^ { - 1 } ) = \\sum _ { k = 1 } ^ 3 \\eta ^ N _ { \\tau _ c ^ N , k } ( f _ k ) + o ( 1 ) , \\end{align*}"}
+{"id": "8635.png", "formula": "\\begin{align*} v _ 0 ( t ; x ) = u _ 0 ( x ) - \\int _ { 0 } ^ { t } ( K _ 0 ( \\tau ; x ) + \\phi _ 0 ( \\tau ; x ) ) d \\tau \\end{align*}"}
+{"id": "6466.png", "formula": "\\begin{align*} \\bar \\Delta ( f \\nu ) = ( \\Delta f + f | A | ^ 2 ) \\nu + 2 A ( \\operatorname { g r a d } f ) , \\end{align*}"}
+{"id": "5947.png", "formula": "\\begin{align*} & \\dfrac { M _ { 2 } ( \\mathcal { N C } ( G ) ) } { | e ( \\mathcal { N C } ( G ) ) | } - \\dfrac { M _ { 1 } ( \\mathcal { N C } ( G ) ) } { | v ( \\mathcal { N C } ( G ) ) | } \\\\ & ~ ~ ~ ~ = \\dfrac { 2 ^ { 7 k } ( 2 ^ { 5 k } - 8 \\cdot 2 ^ { k } - 4 ) + 2 ^ { 4 k } ( 1 7 \\cdot 2 ^ { 2 k } - 1 4 ) + 2 ^ { 3 k } ( 1 4 \\cdot 2 ^ { 2 k } - 1 8 ) + 2 \\cdot 2 ^ { 2 k } + 8 \\cdot 2 ^ { k } } { 2 ^ { 6 k } ( 2 ^ { 3 k } - 4 \\cdot 2 ^ { k } - 2 ) + 2 \\cdot 2 ^ { 3 k } ( 3 \\cdot 2 ^ { 2 k } - 1 ) + 2 ^ { 2 k } ( 4 \\cdot 2 ^ { 2 k } - 3 ) } : = \\dfrac { f ( k ) } { g ( k ) } . \\end{align*}"}
+{"id": "3383.png", "formula": "\\begin{align*} \\frac { 1 } { T } \\sum _ { t = 2 } ^ { T + 1 } \\Delta _ { t } & \\le \\frac { 2 R _ { 1 } ^ { 2 } } { \\eta } = 4 8 R _ { 1 } \\max \\left \\{ 2 6 ^ { \\frac { 1 } { p } } T ^ { \\frac { 1 - p } { p } } \\sigma \\gamma ^ { \\frac { p - 1 } { p } } ; 2 \\left ( 2 L R _ { 1 } + L R _ { 0 } + \\mu \\sigma + \\left \\Vert g _ { 0 } \\right \\Vert _ { * } \\right ) T ^ { - 1 } \\gamma \\right \\} \\end{align*}"}
+{"id": "539.png", "formula": "\\begin{align*} F ( s ) : = v \\left ( s - i \\delta \\right ) , s \\in \\mathbb { C } , \\end{align*}"}
+{"id": "7967.png", "formula": "\\begin{align*} \\begin{aligned} H _ { 0 0 } ^ { \\frac { 1 } { 2 } } \\Lambda ^ { 0 } ( \\Gamma _ { 1 } ) : = & \\{ \\mu \\in H ^ { \\frac { 1 } { 2 } } \\Lambda ^ { 0 } ( \\Gamma _ { 1 } ) \\mid \\rho ^ { - \\frac { 1 } { 2 } } \\mu \\in L ^ { 2 } \\Lambda ^ { 0 } ( \\Gamma _ { 1 } ) , \\rho = \\mathrm { d i s t } ( x , \\partial \\Omega ) \\} \\\\ = & \\{ \\mu \\in H ^ { \\frac { 1 } { 2 } } \\Lambda ^ { 0 } ( \\Gamma _ { 1 } ) \\mid \\mu _ { 0 } \\in H ^ { \\frac { 1 } { 2 } } \\Lambda ^ { 0 } ( \\partial \\Omega ) \\} , \\end{aligned} \\end{align*}"}
+{"id": "7409.png", "formula": "\\begin{align*} F ( t ) : = \\int _ { \\R } u ( x , t ) d x . \\end{align*}"}
+{"id": "7791.png", "formula": "\\begin{align*} \\begin{aligned} \\varphi _ t ( \\tau ) & \\leq L M | x | ^ 2 \\int _ { t } ^ \\tau e ^ { - \\mu ( \\tau - s ) - m s } d s = L M | x | ^ 2 \\frac { e ^ { - m \\tau } - e ^ { - \\mu \\tau + ( \\mu - m ) t } } { \\mu - m } , \\end{aligned} \\end{align*}"}
+{"id": "819.png", "formula": "\\begin{align*} { { \\pounds } _ { \\hat X } } ( { { \\tilde R } _ { i j } } ) = \\frac { 1 } { 2 } ( { D _ i } { \\Psi _ j } - n { D _ j } { \\Psi _ i } + { D _ j } { \\Psi _ i } - n { D _ i } { \\Psi _ j } ) = ( 1 - n ) ( { D _ i } { \\Psi _ j } ) . \\end{align*}"}
+{"id": "202.png", "formula": "\\begin{align*} R \\ = \\ & 2 L \\cup \\ P \\ \\cup \\\\ \\bigcup _ i [ \\{ i + \\} \\times & ( E _ i \\cup G _ { i } ) ] \\cup [ F _ i \\times \\{ i + \\} ] \\cup \\\\ \\bigcup _ i [ \\{ i - \\} \\times & E _ i ] \\cup [ ( F _ i \\cup G _ i ) \\times \\{ i - \\} ] . \\end{align*}"}
+{"id": "6112.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\boldsymbol u ' ( t ) & = K \\boldsymbol u ( t ) + \\boldsymbol g ( t , \\boldsymbol u ( t ) ) , t > 0 , \\\\ \\boldsymbol u ( 0 ) & = \\boldsymbol u _ 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "3767.png", "formula": "\\begin{align*} f _ N = \\sum _ j f _ { N , j } \\end{align*}"}
+{"id": "2625.png", "formula": "\\begin{align*} s _ { i + 2 } - s _ i = s _ { j + 1 } - s _ j , \\ , \\ , \\ , \\ , s _ { i + 3 } - s _ { i + 1 } = s _ { j ' + 1 } - s _ j , \\end{align*}"}
+{"id": "1771.png", "formula": "\\begin{align*} \\Psi : = \\mathrm { F l } _ { - y _ { n } ^ * } ^ { Y _ { - 1 } ^ \\prime } \\circ \\mathrm { F l } _ { - y _ { n - 1 } ^ * } ^ { Y _ { 1 - n } } \\circ \\cdots \\circ \\mathrm { F l } _ { - y _ 1 ^ * } ^ { Y _ { - 1 } } \\circ \\mathrm { F l } _ { - \\tfrac { v _ n ^ * } { 2 } } ^ { X _ { - 1 } ^ { \\prime } } \\circ \\mathrm { F l } _ { - v _ { n - 1 } ^ * } ^ { X _ { 1 - n } } \\circ \\cdots \\circ \\mathrm { F l } _ { - v _ 2 ^ * } ^ { X _ { - 2 } } \\circ \\mathrm { F l } _ { - v _ 1 ^ * } ^ { X _ { - 1 } } . \\end{align*}"}
+{"id": "1378.png", "formula": "\\begin{align*} \\overline \\Gamma ( H ) = \\{ H y h \\mid h \\in H \\} . \\end{align*}"}
+{"id": "5694.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } t ^ { 1 / ( p - 2 ) } \\Vert u ( t ) \\Vert _ { L ^ 2 } = \\infty . \\end{align*}"}
+{"id": "1527.png", "formula": "\\begin{align*} B = \\begin{pmatrix} 1 & - 2 \\\\ 2 & - 1 \\end{pmatrix} \\end{align*}"}
+{"id": "4044.png", "formula": "\\begin{align*} S _ { \\mathcal { H } } ( x ) = \\sum _ { n = 0 } ^ \\infty S ^ n _ { \\mathcal { H } } ( x ) , S ^ n _ { \\mathcal { H } } ( x ) : = \\int _ { 0 < u _ 1 < \\ldots < u _ n < T } d ( u _ 1 , x _ { u _ 1 } ) \\otimes \\ldots \\otimes d ( u _ n , x _ { u _ n } ) , \\end{align*}"}
+{"id": "1272.png", "formula": "\\begin{align*} \\partial _ t \\rho = - \\div ( C x \\rho ) + \\div ( D \\nabla \\rho ) , \\rho ( 0 ) = \\rho _ 0 . \\end{align*}"}
+{"id": "4135.png", "formula": "\\begin{align*} & \\bar { \\xi } _ \\ell ( \\tilde { \\theta } ) = 0 , \\ell = 1 , \\ldots , j - 1 , \\\\ & \\bar { \\xi } _ j ( \\tilde { \\theta } ) = ( 1 - w _ { j j } ) r ^ { j + 1 / 2 } \\eta _ j \\end{align*}"}
+{"id": "2832.png", "formula": "\\begin{align*} \\delta Q ^ { 1 } ( t _ { 2 } ) = \\delta Q ^ { 1 } ( t _ { 1 } ) = 0 \\end{align*}"}
+{"id": "510.png", "formula": "\\begin{align*} L _ i = \\max _ { p \\mid \\gcd ( \\overline { \\alpha } _ i , s ) } { \\left \\lceil \\frac { \\nu _ p ( s ) } { \\nu _ p ( \\overline { \\alpha } _ i ) } \\right \\rceil } = \\max _ { p \\mid \\gcd ( \\overline { \\alpha } _ i , s ) } { \\left \\lceil \\frac { \\nu _ p ( s ) } { \\nu _ p ( \\overline { \\alpha } _ i \\bmod { p ^ { \\nu _ p ( s ) } } } ) \\right \\rceil } , \\end{align*}"}
+{"id": "5118.png", "formula": "\\begin{align*} L o g _ { A } ( x ) = \\frac { x ^ { t - 1 } - x ^ { t ^ { - 1 } - 1 } } { t - t ^ { - 1 } } \\end{align*}"}
+{"id": "2567.png", "formula": "\\begin{align*} C ^ 2 _ m = C ^ { \\infty } ( L ; \\Omega ^ { - 1 } N L \\otimes \\Omega M ) ' = C ^ { \\infty } ( L ; \\Omega ) ' = C ^ { - \\infty } ( L ) \\end{align*}"}
+{"id": "9151.png", "formula": "\\begin{align*} \\begin{aligned} & \\sigma ( x ^ { ( \\beta , s _ { 1 } ) } _ { \\ell , 1 } ) = x ^ { ( \\gamma , r ) } _ { \\ell , 1 } \\ 1 \\leq \\ell \\leq j - 1 , \\\\ & \\sigma ( x ^ { ( \\beta , s _ { 1 } ) } _ { \\ell , t _ { \\ell , 1 } } ) = x ^ { ( \\gamma , r ) } _ { \\ell , 1 } , \\sigma ( x ^ { ( \\beta , s _ { 1 } ) } _ { \\ell , t _ { \\ell , 2 } } ) = x ^ { ( \\gamma , r ) } _ { \\ell , 2 } \\ j \\leq \\ell \\leq n . \\end{aligned} \\end{align*}"}
+{"id": "1075.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { [ n \\delta ] } \\| ( T _ { \\infty } ( \\tilde { w } ) ^ { - 1 } ) ^ { n + 1 - s , n + 1 - t } \\| \\leq C _ 2 n ^ { - d } , n \\in \\N , ~ ~ t \\in \\{ [ r n ] + 1 , \\dots , n \\} . \\end{align*}"}
+{"id": "2958.png", "formula": "\\begin{align*} R _ { k j } ( \\alpha _ i ) w ( \\sigma , \\alpha ) _ { i j } : = \\sum _ { m = 1 } ^ { k } \\frac { v _ { \\lambda _ m i } } { R _ { k j } ( \\alpha _ i ) ^ { m } } , \\end{align*}"}
+{"id": "5809.png", "formula": "\\begin{align*} x ^ { ( k , \\ell ) } _ j = \\int _ \\Sigma \\left \\langle \\mathcal { L } _ { \\Sigma } ^ { \\ell } \\partial ^ { k } _ t u , \\varphi _ { \\iota + j } \\right \\rangle \\ , d \\mu = \\int _ \\Sigma \\left \\langle \\partial ^ { k } _ t u , \\mathcal { L } _ { \\Sigma } ^ { \\ell } \\varphi _ { \\iota + j } \\right \\rangle \\ , d \\mu = 0 . \\end{align*}"}
+{"id": "7709.png", "formula": "\\begin{align*} - h _ { t i i } - h _ { t } & \\leq - \\frac { h _ { t } h _ { i i } } { h - \\varepsilon _ { 0 } } - h _ { t } \\\\ & = \\frac { - h _ { t } } { h - \\varepsilon _ { 0 } } [ h _ { i i } + ( h - \\varepsilon _ { 0 } ) ] \\\\ & = Q ( b _ { i i } - \\varepsilon _ { 0 } ) . \\end{align*}"}
+{"id": "4299.png", "formula": "\\begin{align*} A ( K ) = \\begin{pmatrix} K \\mathrm { I d } _ d & 0 \\\\ 0 & \\mathrm { I d } _ e \\end{pmatrix} \\end{align*}"}
+{"id": "8588.png", "formula": "\\begin{align*} \\mathcal { R } _ V ^ N = \\left \\{ R \\in \\mathcal { R } _ N : \\sum _ { x \\in R } V ( x ) < 1 \\right \\} \\ : . \\end{align*}"}
+{"id": "262.png", "formula": "\\begin{align*} y = \\int ^ x b ( \\zeta ) \\exp \\left ( 2 \\int ^ \\zeta \\gamma ( \\eta ) \\ , d \\eta \\right ) \\ , d \\zeta , d \\tau = | b ( x ) | \\exp \\left ( \\int ^ x \\gamma ( \\zeta ) \\ , d \\zeta \\right ) \\ , d t . \\end{align*}"}
+{"id": "1300.png", "formula": "\\begin{align*} d \\hat { x } ( t ) = \\lambda _ + \\hat { x } ( t ) \\ , d t + \\hat { D } d W _ t \\ ; , \\end{align*}"}
+{"id": "3662.png", "formula": "\\begin{align*} x ^ T L ( G , p ) x = \\sum _ { \\{ u , v \\} \\in E } \\left \\langle x ( u ) - x ( v ) , d _ { u v } \\right \\rangle ^ 2 , \\end{align*}"}
+{"id": "4537.png", "formula": "\\begin{align*} \\left \\{ \\begin{matrix*} [ l ] \\partial _ t | m | + \\displaystyle \\sum _ { i = 1 } ^ { N _ 1 } \\partial _ { s _ i } | m | + p _ { N _ 1 } ( [ s ] _ { N _ 1 } ) | m | \\leq | E _ { N _ 2 } ^ { ( N _ 1 ) } | , \\\\ [ 5 p t ] | m | ( t , s _ 1 = 0 , [ s ] _ { 2 , N _ 1 } ) \\leq \\int _ { u = 0 } ^ \\infty \\big [ p _ { N _ 1 } | m | + | E _ { N _ 2 } ^ { ( N _ 1 ) } | \\big ] ( t , [ s ] _ { 2 , N _ 1 } , u ) d u . \\end{matrix*} \\right . \\end{align*}"}
+{"id": "5501.png", "formula": "\\begin{align*} b = a _ 1 s _ 1 ^ { - 1 } + \\Theta = s _ 2 ^ { - 1 } a _ 2 + \\Theta , \\end{align*}"}
+{"id": "2816.png", "formula": "\\begin{align*} H _ { T } = \\frac { 1 } { 2 } ( P _ { 1 } ) ^ { 2 } + \\frac { 1 } { 2 } ( Q ^ { 1 } ) ^ { 2 } - \\frac { 1 } { 2 } ( \\Theta ^ { 1 } ) ^ { 2 } - \\frac { 1 } { 2 } ( \\Theta _ { 1 } ) ^ { 2 } . \\end{align*}"}
+{"id": "8653.png", "formula": "\\begin{align*} F ( y _ 0 ^ * ) - D ( y _ 0 ^ * , x ^ * ) & \\leq F ( y _ 0 ^ * ) - ( F ^ * ( y _ 0 ^ * ) - F ^ * ( x ^ * ) - \\langle x , y ^ * _ 0 - x ^ * \\rangle - \\epsilon ) , \\\\ & = \\langle x , y _ 0 ^ * \\rangle - F ( x ) + \\epsilon . \\end{align*}"}
+{"id": "8794.png", "formula": "\\begin{align*} \\varphi _ \\rho ( \\alpha , w ) = ( \\alpha w ) ^ \\rho - \\frac { w } { 1 + w } \\left ( 1 + \\alpha w \\right ) ^ \\rho - \\frac { w ^ { \\rho } } { 1 + w } \\left ( 1 - \\alpha \\right ) ^ \\rho . \\end{align*}"}
+{"id": "7718.png", "formula": "\\begin{align*} \\nabla _ { j } \\psi = \\frac { f _ { j } } { f } - \\frac { \\nabla _ { j } \\widetilde { V } _ { q - 1 } ( \\Omega _ { t } , \\overline { \\nabla } h ) } { \\widetilde { V } _ { q - 1 } ( \\Omega _ { t } , \\overline { \\nabla } h ) } + p \\frac { h _ { j } } { h } , \\end{align*}"}
+{"id": "2333.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } a - \\Delta a + ( a \\cdot \\nabla ) u ^ { c , \\gamma } + ( u ^ { c , \\gamma } \\cdot \\nabla ) a + \\nabla \\pi _ { 1 } = 0 , \\\\ \\mathrm { d i v } a = 0 , \\\\ a ( x , 0 ) = w _ { 0 } ( x ) . \\end{cases} \\end{align*}"}
+{"id": "5438.png", "formula": "\\begin{align*} r ^ S _ i & \\triangleq f ^ { \\langle 1 , S \\rangle } _ i - f ^ { \\langle 0 , S \\rangle } _ i = R _ i + \\beta \\sum _ { j \\in S } p _ { i j } f ^ S _ j - \\beta f ^ S _ i = \\begin{cases} ( 1 - \\beta ) f ^ S _ i & i \\in S \\\\ R _ i + \\beta \\sum _ { j \\in S } p _ { i j } f ^ S _ j & \\end{cases} \\end{align*}"}
+{"id": "42.png", "formula": "\\begin{align*} \\tilde d _ j ( \\sum _ { I \\in P ^ { j , n } _ + } \\omega _ I d x ^ I ) = \\sum _ { I \\in P ^ { j , n } _ + } ( \\tilde d _ 0 \\omega _ I ) \\wedge d x ^ I . \\end{align*}"}
+{"id": "3719.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\sum _ { n = 0 } ^ N \\frac { | \\phi _ n ( a ) | ^ 2 } { ( E _ n - E ) ( E _ n + \\mu ^ 2 ) } < \\infty \\ ; . \\end{align*}"}
+{"id": "7094.png", "formula": "\\begin{align*} u : = ( \\lambda + \\partial _ t - \\Delta ) ^ { - 1 } f - ( \\lambda + \\partial _ t - \\Delta ) ^ { - \\frac { 1 } { 2 } - \\frac { 1 } { 2 p } } Q _ p ( 1 + T _ p ) ^ { - 1 } R _ p ( \\lambda + \\partial _ t - \\Delta ) ^ { - \\frac { 1 } { 2 p ' } } f . \\end{align*}"}
+{"id": "2324.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow \\infty } \\| w ( t ) \\| _ { L ^ { 3 } ( \\mathbb { R } ^ { 3 } ) } = 0 ; \\end{align*}"}
+{"id": "1233.png", "formula": "\\begin{align*} C _ { \\Phi _ { \\lambda , d } } = ( I _ d \\otimes I _ d ) + \\lambda ( P + F ) , \\end{align*}"}
+{"id": "4094.png", "formula": "\\begin{align*} S ( \\lambda ) = S _ { } ( \\lambda ) S ^ { - 1 } _ { , 0 } ( \\lambda ) = I + a _ d \\mathcal { E } ( \\lambda ) ( I + V R _ 0 ( \\lambda ) ) ^ { - 1 } V \\mathcal { E ^ * } ( \\bar { \\lambda } ) : L ^ 2 ( \\mathbb { S } ^ { d - 1 } ) \\to L ^ 2 ( \\mathbb { S } ^ { d - 1 } ) \\end{align*}"}
+{"id": "4594.png", "formula": "\\begin{align*} \\min _ { \\mathfrak { E } } s _ g = & \\quad 3 2 \\pi \\left ( 1 + \\sum a \\right ) \\left ( 8 \\sum a - 1 2 \\sum a ^ 2 + 8 \\sum a ^ 3 - \\sum a ^ 4 + 1 2 \\sum a b + 4 \\sum a ^ 3 b - 3 \\sum a ^ 2 b ^ 2 \\right ) \\Big / \\\\ [ 0 . 1 c m ] & \\left ( 3 2 \\sum a ^ 2 - 3 2 \\sum a ^ 3 + 3 2 \\sum a ^ 4 - 1 2 \\sum a ^ 5 \\right . + \\sum a ^ 6 + 3 2 \\sum a b + 3 2 \\sum a ^ 2 b + 3 2 \\sum a ^ 3 b + 4 \\sum a ^ 4 b \\\\ [ 0 . 1 c m ] & \\left . - 6 \\sum a ^ 5 b + 8 \\sum a ^ 3 b ^ 2 + 1 5 \\sum a ^ 4 b ^ 2 - 1 0 \\sum a ^ 3 b ^ 3 \\right ) \\end{align*}"}
+{"id": "1725.png", "formula": "\\begin{align*} w ( x _ 0 ) = 5 , w ( x _ n ) = \\frac { 3 } { n - 2 } , \\quad \\mbox { a n d } w ( x _ j ) : = \\frac { n - 4 + 3 j } { n - 2 } \\quad \\forall \\ , j \\in \\{ 1 , \\ldots , n - 1 \\} , \\end{align*}"}
+{"id": "594.png", "formula": "\\begin{align*} P ( f ) = \\sum \\limits _ { j = 1 } ^ \\infty \\Big \\langle f , { \\frac { \\Tilde { h } _ j } { \\| h _ j \\| _ { F ^ \\ast } } } \\Big \\rangle { \\frac { \\Tilde { h } _ j } { \\| h _ j \\| _ F } } , \\ ; f \\ ; \\mathrm { i n } \\ ; F , \\end{align*}"}
+{"id": "2976.png", "formula": "\\begin{align*} x _ 1 ^ 1 ( 0 ) = x _ 2 ^ 1 ( 0 ) , x _ 2 ^ 2 ( 0 ) > x _ 1 ^ 2 ( 0 ) \\mbox { a n d } v _ 1 ^ 2 ( 0 ) > v _ 2 ^ 2 ( 0 ) . \\end{align*}"}
+{"id": "5447.png", "formula": "\\begin{align*} \\nu _ { ( a _ k ^ - , i _ k ) } ^ * = \\nu _ { ( 1 , i _ k ) } ^ * = \\nu _ { ( 1 , i _ k ) } ^ { \\hat { S } ^ { k - 1 } } = \\nu _ { ( 0 , i _ k ) } ^ { \\hat { S } ^ { k - 1 } } + c _ i / w _ { ( 1 , i _ k ) } ^ { \\hat { S } ^ { k - 1 } } \\geq \\nu _ { ( 0 , i _ k ) } ^ { \\hat { S } ^ { k - 1 } } = \\nu _ { ( a _ { k + 1 } ^ - , i _ { k + 1 } ) } ^ { \\hat { S } ^ { k - 1 } } , \\end{align*}"}
+{"id": "8717.png", "formula": "\\begin{gather*} L _ { 1 } = L e i _ { 1 } ( 3 , F ) = F a _ { 1 } \\oplus F a _ { 2 } \\oplus F a _ { 3 } , \\ \\mbox { w h e r e } [ a _ { 1 } , a _ { 1 } ] = a _ { 2 } , [ a _ { 1 } , a _ { 2 } ] = a _ { 3 } , \\\\ [ a _ { 1 } , a _ { 3 } ] = [ a _ { 2 } , a _ { 1 } ] = [ a _ { 2 } , a _ { 2 } ] = [ a _ { 2 } , a _ { 3 } ] = [ a _ { 3 } , a _ { 1 } ] = [ a _ { 3 } , a _ { 2 } ] = [ a _ { 3 } , a _ { 3 } ] = 0 . \\end{gather*}"}
+{"id": "7080.png", "formula": "\\begin{align*} X _ t = x - \\int _ 0 ^ t b ( X _ r ) d r + \\sqrt { 2 } W _ t , t \\geq 0 , \\end{align*}"}
+{"id": "9103.png", "formula": "\\begin{align*} \\underline { u } ^ { ( r ) } _ i = \\Big ( ( \\hat { C } ^ { - 1 } - C \\hat { C } ^ { - 1 } ) \\underline { v } \\Big ) _ i - \\hat { C } ^ { - 1 } _ { i i } \\Big ( A ^ { ( r ) } _ - \\bar X _ - \\Big ) _ i + \\xi _ i , \\xi _ i > 0 , \\end{align*}"}
+{"id": "2988.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\phi \\nabla _ x \\xi = \\nabla _ { \\phi x } \\xi = x - \\eta ( x ) \\xi , \\\\ \\end{array} \\end{align*}"}
+{"id": "8590.png", "formula": "\\begin{align*} C = \\frac { 3 \\sqrt { 5 } } { 2 } \\frac { ( 2 M + 1 ) ^ { 3 / 2 } } { \\min \\{ 1 , \\delta ^ { 3 / 2 } \\} } \\quad \\mbox { a n d } \\mu = \\frac { 1 } { 2 } \\log \\left ( 1 + \\frac { \\delta \\Delta } { 4 M + 2 } \\right ) \\end{align*}"}
+{"id": "2648.png", "formula": "\\begin{align*} P _ M f = P _ { \\leq M } f - P _ { \\leq \\frac { M } { 2 } } f . \\end{align*}"}
+{"id": "8525.png", "formula": "\\begin{align*} \\tilde { J } ^ { ( p ) } [ I ] \\otimes \\Z _ p = \\Sigma _ p \\oplus C _ p , \\end{align*}"}
+{"id": "6031.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac 1 N \\left | \\sum _ { i = 1 } ^ n k _ i v _ i ^ N \\right | = \\infty \\ , , \\end{align*}"}
+{"id": "8000.png", "formula": "\\begin{align*} l _ 1 ( W , F , G , H ) = & W _ { T , 1 } ( F _ T ) + W _ { T , 2 } ( G _ T ) + W _ { T , 3 } ( H _ T ) \\\\ & - W _ { 0 , 1 } ( F _ 0 ) - W _ { 0 , 2 } ( G _ 0 ) - W _ { 0 , 3 } ( H _ 0 ) \\\\ & - \\int _ 0 ^ T W _ { s , 1 } ( \\partial _ s F _ s ) + W _ { s , 2 } ( \\partial _ s G _ s ) + W _ { s , 3 } ( \\partial _ s H _ s ) d s . \\end{align*}"}
+{"id": "9086.png", "formula": "\\begin{align*} \\begin{array} { l l l } X ( t + 1 ) = J ^ { ( r ) } \\hat { C } C \\hat { C } ^ { - 1 } J ^ { ( r ) } X ( t ) + J ^ { ( r ) } \\Big [ ( \\hat { C } C \\hat { C } ^ { - 1 } - I ) \\underline { v } + \\hat { C } ( D p - U ( J ^ { ( r ) } X ( t ) ) - \\mathbf { 1 } \\frac { \\beta } { 2 } ) \\Big ] . \\end{array} \\end{align*}"}
+{"id": "7885.png", "formula": "\\begin{align*} \\mathbf { D } _ { i , j } = \\left ( \\begin{array} { c c c c c c c } 0 & c _ { 1 } & & & & & \\\\ c _ { 1 } & \\ddots & \\ddots & & & & \\\\ & \\ddots & 0 & c _ { s - 1 } & & & \\\\ & & c _ { s - 1 } & - 2 e & c _ { s } & & \\\\ & & & c _ { s } & 0 & \\ddots & \\\\ & & & & \\ddots & \\ddots & c _ { m - 1 } \\\\ & & & & & c _ { m - 1 } & 0 \\end{array} \\right ) . \\end{align*}"}
+{"id": "2743.png", "formula": "\\begin{align*} X _ { f _ { a } } g = 0 . \\end{align*}"}
+{"id": "1280.png", "formula": "\\begin{align*} W _ 2 ^ 2 ( \\mu _ t , \\bar { \\mu } _ t ) = ( m ( t ) - \\bar { m } ( t ) ) ^ 2 + \\Big ( \\sqrt { \\sigma _ { x x } ( t ) } - \\sqrt { \\bar { \\sigma } ( t ) } \\Big ) ^ 2 . \\end{align*}"}
+{"id": "509.png", "formula": "\\begin{align*} \\sum _ { z = 0 } ^ { L _ i - 1 } { \\overline { \\alpha } _ i ^ z \\overline { \\beta } _ i } = \\begin{cases} L _ i \\overline { \\beta } _ i , & \\overline { \\alpha } _ i = 1 , \\\\ \\frac { \\overline { \\alpha } _ i ^ { L _ i } - 1 } { \\overline { \\alpha } _ i - 1 } \\overline { \\beta } _ i , & , \\end{cases} \\end{align*}"}
+{"id": "7369.png", "formula": "\\begin{align*} u _ { x x } ^ 0 ( x , t ) = u ^ 0 _ { t t } ( x , t ) . \\end{align*}"}
+{"id": "6543.png", "formula": "\\begin{align*} T _ { N , 1 } = \\sum ^ N _ { n = 1 } \\sum ^ { \\infty } _ { j = m _ 0 + 1 } \\Big ( \\mathbb { E } \\big [ K ( X _ n ) | \\varepsilon _ { n - j } \\big ] - \\mathbb { E } \\big [ K ( X _ n ) \\big ] \\Big ) . \\end{align*}"}
+{"id": "5313.png", "formula": "\\begin{align*} v _ i ( \\nu ) = \\min \\ , \\left \\{ v _ i ^ S ( \\nu ) : S \\in 2 ^ { N ^ { \\{ 0 , 1 \\} } } \\right \\} . \\end{align*}"}
+{"id": "6546.png", "formula": "\\begin{align*} T _ { N , 1 } - T _ { N , 2 } & = \\frac { 1 } { 2 \\pi } \\sum ^ N _ { n = 1 } \\sum ^ { \\infty } _ { j = m _ 0 + 1 } \\int _ { \\mathbb { R } } \\widehat { K } ( u ) \\prod ^ { j - 1 } _ { k = 1 } \\phi _ { \\varepsilon } ( a _ k u ) ( 1 - \\phi _ { \\varepsilon } ( a _ j u ) ) ( e ^ { \\iota u a _ j \\varepsilon _ { n - j } } - \\phi _ { \\varepsilon } ( a _ j u ) ) \\mathbb { E } e ^ { \\iota u \\widetilde { X } _ { n , j } } \\ , d u . \\end{align*}"}
+{"id": "3369.png", "formula": "\\begin{align*} \\tilde { \\phi } = e ^ { 5 6 7 } - e ^ { 5 } ( e ^ { 4 1 } - e ^ { 2 3 } ) - e ^ 6 ( e ^ { 4 2 } - e ^ { 3 1 } ) - e ^ 7 ( e ^ { 4 3 } - e ^ { 1 2 } ) , \\end{align*}"}
+{"id": "3149.png", "formula": "\\begin{align*} \\psi ^ * \\left ( \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ) = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "3945.png", "formula": "\\begin{align*} & \\gamma ^ { - 4 } \\left | \\overline { \\mathcal { G } } _ N \\right | = \\gamma ^ { - 4 } \\max _ { 1 \\leq j \\leq N } \\left | G _ j \\Gamma \\right | \\leq \\xi _ { N , \\gamma } , \\\\ & \\xi _ { N , \\gamma } = \\max \\left ( \\frac { \\lambda _ N \\left | d _ 2 - d _ 1 \\right | } { \\gamma } , \\left ( \\frac { \\lambda _ N } { \\gamma } \\right ) ^ 2 \\vert \\kappa ( d _ 1 - d _ 3 ) \\vert \\right ) \\end{align*}"}
+{"id": "648.png", "formula": "\\begin{align*} \\int _ { \\rho _ - } ^ { 2 \\rho _ - } d _ r ^ { \\frac 1 r } \\xi ^ { - 1 } \\ , d \\xi & = d _ r ^ { \\frac 1 r } \\ln 2 \\approx \\rho _ - \\ln 2 . \\end{align*}"}
+{"id": "183.png", "formula": "\\begin{align*} \\{ x \\} \\ = \\ \\ & Q ( x , x ) , \\emptyset \\ = \\ Q ^ { \\circ } ( x , x ) . \\\\ \\{ x , y \\} \\ \\subset \\ \\ & Q ( x , y ) \\ = \\ Q ( \\{ x , y \\} \\times \\{ x , y \\} ) . \\\\ Q ( x , y ) \\setminus \\{ x , y \\} \\ = \\ \\ & Q ^ { \\circ } ( x , y ) \\ = \\ Q ^ { \\circ } ( \\{ x , y \\} \\times \\{ x , y \\} ) . \\end{align*}"}
+{"id": "3601.png", "formula": "\\begin{align*} T f = \\alpha \\tau _ 0 f ( \\tau , w \\sigma ) ( f \\in H ^ p ( { \\mathbb { T } ^ 2 } ) ) , \\end{align*}"}
+{"id": "7433.png", "formula": "\\begin{align*} \\dd X _ { t } ^ { ( j ) } = \\mu ^ { ( j ) } ( t , X _ { t } ) \\ , \\dd t \\ , + \\ , & \\sum _ { i = 1 } ^ { d _ 0 } V _ { i } ^ { ( j ) } ( t , X _ { t } ) \\cdot \\dd W _ { t } ^ { ( i ) } , X _ { 0 } ^ { ( j ) } = x ^ { ( j ) } , \\\\ & j \\in \\{ 1 , 2 , \\ldots , d \\} , t \\in [ 0 , T ] . \\end{align*}"}
+{"id": "6107.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & i e _ 0 \\cdot X \\cdot \\\\ i e _ 0 \\cdot X \\cdot & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "6157.png", "formula": "\\begin{align*} Q ^ k = \\begin{pmatrix} \\tau ^ k \\beta ^ k ( D - A ^ T A ) & 0 \\\\ 0 & \\frac { 1 } { \\tau ^ k \\beta ^ k } I _ l \\end{pmatrix} . \\end{align*}"}
+{"id": "5966.png", "formula": "\\begin{align*} \\sum _ { i } ( f _ i ) _ e = f _ e . \\end{align*}"}
+{"id": "3408.png", "formula": "\\begin{align*} b ^ + ( X ) + \\frac { 1 } { 2 } b _ 1 ( S ) - \\frac { 1 } { 4 } S \\circ S - \\frac { 1 } { 2 } \\sigma ( L ) + \\frac { 1 } { 2 } \\sigma ( L ' ) = 1 , \\end{align*}"}
+{"id": "860.png", "formula": "\\begin{align*} a _ { i j } = \\frac { \\rho \\delta _ { i j } + w _ i w _ j } { \\rho ^ 2 } , b _ i = - \\frac { w _ i } { \\rho } , \\end{align*}"}
+{"id": "7638.png", "formula": "\\begin{align*} \\mathcal { S } r ( x ) = \\frac { \\mathcal { E } ( x ) - p _ { r f } } { \\sqrt { \\mathcal { V } ( x ) } } \\end{align*}"}
+{"id": "6146.png", "formula": "\\begin{align*} v ^ { k + 1 } = v ^ k - M ^ k ( v ^ k - \\widetilde { v } ^ k ) . \\end{align*}"}
+{"id": "7383.png", "formula": "\\begin{align*} \\| ( U , W ) \\| _ Y : = \\| U \\| _ 3 + \\| U _ x \\| _ 3 + \\| W \\| _ 4 + \\| W _ x \\| _ 4 , \\end{align*}"}
+{"id": "479.png", "formula": "\\begin{align*} \\Gamma _ { g _ 1 \\otimes g _ 2 } ^ { \\ast } \\cong \\left ( \\Gamma _ { g _ 1 } \\otimes \\Gamma _ { g _ 2 } \\right ) ^ { \\ast } = \\left ( \\Gamma _ { g _ 1 } \\right ) ^ { \\ast } \\otimes \\left ( \\Gamma _ { g _ 2 } \\right ) ^ { \\ast } \\end{align*}"}
+{"id": "2508.png", "formula": "\\begin{align*} \\| u _ \\infty - \\langle u ^ { i n } \\rangle \\| _ { ( H ^ 1 ) ' } & = \\| u _ \\infty - u ^ { i n } + u ^ { i n } - \\langle u ^ { i n } \\rangle \\| _ { ( H ^ 1 ) ' } \\\\ & \\ge \\| u ^ { i n } - \\langle u ^ { i n } \\rangle \\| _ { ( H ^ 1 ) ' } - \\| u _ \\infty - u ^ { i n } \\rangle \\| _ { ( H ^ 1 ) ' } \\\\ & > \\left [ \\| u ^ { i n } \\| _ \\infty \\| v ^ { i n } \\| _ 1 \\| \\gamma ' \\| _ { L ^ \\infty ( 0 , V ) } \\right ] ^ { 1 / 2 } - \\| u _ \\infty - u ^ { i n } \\rangle \\| _ { ( H ^ 1 ) ' } > 0 \\ , . \\end{align*}"}
+{"id": "1670.png", "formula": "\\begin{align*} q ( \\widehat { v _ 2 } ) \\ , q ( \\widehat { v _ 3 } ) - \\omega ( \\widehat { v _ 2 } , \\widehat { v _ 3 } ) ^ 2 = - \\omega ( v _ 2 , v _ 3 ) ^ 2 \\cdot \\Delta ( \\mathbf { p } ) . \\end{align*}"}
+{"id": "615.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\left ( \\frac { 1 } { 3 2 } \\right ) ^ n \\left ( H _ { 2 n } - H _ n \\right ) \\binom { 2 n } { n } ^ 2 \\end{align*}"}
+{"id": "2869.png", "formula": "\\begin{align*} r \\leq \\left \\{ \\lambda ^ { - 1 } \\left [ | \\nabla f ( x ) \\cdot \\omega | + \\rho \\left ( \\left ( \\frac { L } { \\lambda } \\right ) ^ \\frac { q } { \\gamma } \\right ) \\right ] \\right \\} ^ \\frac { q } { \\gamma } = : r ( x , \\omega , \\lambda ) , \\end{align*}"}
+{"id": "159.png", "formula": "\\begin{align*} \\langle b _ { i } \\ | \\ a b _ { j } \\rangle b & = \\langle b _ { i } \\ | \\ a b _ { j } b \\rangle \\\\ & = \\langle b _ { i } \\ | \\ b a b _ { j } \\rangle \\\\ & = \\langle b ^ { * } b _ { i } \\ | \\ a b _ { j } \\rangle \\\\ & = \\langle b _ { i } b ^ { * } \\ | \\ a b _ { j } \\rangle \\\\ & = b \\langle b _ { i } \\ | \\ a b _ { j } \\rangle \\end{align*}"}
+{"id": "2739.png", "formula": "\\begin{align*} & \\omega _ { q , p } : = d q ^ { i } \\wedge d p _ { i } = \\frac { 1 } { 2 } J _ { m n } d z ^ { m } \\wedge d z ^ { n } , \\\\ & \\omega _ { Q , P } : = d Q ^ { i } \\wedge d P _ { i } = \\frac { 1 } { 2 } J _ { m n } d Z ^ { m } \\wedge d Z ^ { n } , \\end{align*}"}
+{"id": "7956.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\ast d w \\wedge [ \\eta , u ] _ { 1 } = \\int _ { \\Sigma } \\langle d w , \\eta \\rangle _ { \\Lambda ^ { 1 } } \\wedge \\partial \\Sigma . \\end{align*}"}
+{"id": "4887.png", "formula": "\\begin{align*} C ' \\cdot E = ( 2 E + ( g - 1 + e ) \\theta ) \\cdot E = g - 1 - e , \\end{align*}"}
+{"id": "7301.png", "formula": "\\begin{align*} L _ { X _ m } ( s , \\chi ) = \\sum _ { j = 1 } ^ { m - 1 } \\chi ( j ) \\csc ^ { 2 s } \\left ( \\frac { j \\pi } { m } \\right ) ; \\end{align*}"}
+{"id": "6389.png", "formula": "\\begin{align*} z ^ n _ i ( \\theta ) = n ^ { 1 / \\alpha } \\left ( \\frac { X _ { \\frac { i } { n } } - X _ { \\frac { i - 1 } { n } } - \\frac { a } { n } + \\frac { b } { n } X _ { \\frac { i - 1 } { n } } } { \\delta X _ { \\frac { i - 1 } { n } } ^ { 1 / \\alpha } } \\right ) . \\end{align*}"}
+{"id": "8969.png", "formula": "\\begin{align*} | \\nabla u ( x ) - S ( x ) | = d ( \\nabla u ( x ) , S O ( n ) ) . \\end{align*}"}
+{"id": "8366.png", "formula": "\\begin{align*} g \\cdot [ h , x ] = [ g h , x ] g \\in G , b \\in B . \\end{align*}"}
+{"id": "8774.png", "formula": "\\begin{align*} G \\left ( F _ \\mu ( x - ) , \\lim _ { y \\to x - } \\phi _ \\uparrow ( F _ \\mu ( y ) ) \\right ) = \\int _ 0 ^ { F _ \\mu ( x - ) } F _ \\mu ^ { - 1 } ( w ) d w , \\end{align*}"}
+{"id": "1179.png", "formula": "\\begin{align*} \\left | \\partial ^ \\beta f ( y ) - \\partial ^ \\beta f ( y + v ) \\right | & = \\left | \\int _ 0 ^ 1 v \\cdot \\nabla \\partial ^ \\beta f ( y + t v ) \\ , d t \\right | \\\\ & \\lesssim | v | \\int _ 0 ^ 1 | y + t v | ^ { - n - | \\beta | - 1 } \\ , d t \\lesssim | v | \\ , | y | ^ { - n - | \\beta | - 1 } . \\end{align*}"}
+{"id": "3483.png", "formula": "\\begin{align*} x \\xrightarrow { 2 } ( ( x + 1 ) ^ 3 , ( x - 1 ) ^ 3 ) \\xrightarrow { 1 } \\frac { ( x + 1 ) ^ 3 - ( x - 1 ) ^ 3 - 2 } { 6 } = x ^ 2 , \\end{align*}"}
+{"id": "8581.png", "formula": "\\begin{align*} G _ t ^ { t _ 1 } = ( J _ k + [ \\varepsilon _ k ( G ^ { t _ 0 } _ t ) B _ { t _ 0 } ] ^ { \\cdot k } _ + ) G ^ { t _ 0 } _ t . \\end{align*}"}
+{"id": "8984.png", "formula": "\\begin{align*} - ( u ^ \\prime _ \\gamma r ^ { \\tilde { N } _ + - 1 } ) ^ \\prime = \\bar \\lambda _ { \\gamma , v a r } u _ \\gamma r ^ { \\tilde { N } _ + - 1 - \\gamma } \\end{align*}"}
+{"id": "8025.png", "formula": "\\begin{align*} d \\mathcal { U } ^ N _ { F , G , H } ( t , \\xi ^ N ) = \\mathcal { U } ^ N _ { F , G , H } ( t , \\xi ^ N ) d \\hat { \\Lambda } ^ N _ { F , G , H } ( t ) , \\end{align*}"}
+{"id": "5686.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } t ^ { 1 / ( p - 2 ) } \\Vert u ( t ) \\Vert _ { L ^ 2 } = \\beta \\in ( 0 , \\infty ) . \\end{align*}"}
+{"id": "7981.png", "formula": "\\begin{align*} \\delta d \\big ( \\mathrm { l i } _ { \\phi } ( f _ { \\phi } ) - \\mathrm { l i } ' _ { \\phi } ( f _ { \\phi } ) \\big ) = \\delta d \\tilde { f } = 0 , \\end{align*}"}
+{"id": "6353.png", "formula": "\\begin{align*} \\bar \\rho ( \\varphi ) : = R ( 1 + \\rho ( \\varphi ) ) , \\end{align*}"}
+{"id": "5520.png", "formula": "\\begin{align*} k = \\deg ( D ) + g - 1 - \\deg ( G ) = 4 u ^ 4 - \\frac { 3 3 u ^ 2 } { 4 } - 5 - c _ 1 - c _ 2 , \\end{align*}"}
+{"id": "2930.png", "formula": "\\begin{align*} \\sum _ { T ' \\in P ( \\{ a _ 1 , a _ 2 , a _ 3 \\} ) , \\bar { a } _ 1 \\in T ' } { g _ { T ' } } = - 2 , \\end{align*}"}
+{"id": "8276.png", "formula": "\\begin{align*} R ( \\lambda ) - R ( \\mu ) = ( \\mu - \\lambda ) R ( \\lambda ) R ( \\mu ) \\mbox { f o r a l l } \\lambda , \\mu \\in \\rho ( A ) . \\end{align*}"}
+{"id": "5730.png", "formula": "\\begin{align*} X ^ 2 _ + ( t ) = & \\sum _ { i \\in I _ 1 } | \\xi _ { i , 1 } ( t ) | ^ 2 + | \\xi _ { i , 2 } ( t ) | ^ 2 + \\sum _ { i \\in I _ 2 } | \\xi _ { i , 3 } ( t ) | ^ 2 + | \\xi _ { i , 4 } ( t ) | ^ 2 + \\sum | \\xi ^ + _ i ( t ) | ^ 2 + \\sum _ { i : \\gamma ^ - _ i > \\gamma } | \\xi ^ - _ i ( t ) | ^ 2 , \\\\ X ^ 2 _ - ( t ) = & \\sum _ { i : \\gamma ^ - _ i \\leq \\gamma } | \\xi ^ - _ i ( t ) | ^ 2 . \\end{align*}"}
+{"id": "3986.png", "formula": "\\begin{align*} x ^ { 4 } = 2 \\sum _ { l = 1 } ^ { n } \\Bigl ( \\sum _ { j = 1 } ^ { n } \\Theta _ { j } \\gamma _ { j p l } + \\sum _ { u = 1 } ^ { \\nu } \\widetilde { \\Theta } _ { u } \\gamma _ { i u l } \\Bigr ) e _ { l } + 2 \\sum _ { s = 1 } ^ { \\nu } \\Bigl ( \\sum _ { j = 1 } ^ { n } \\Theta _ { j } \\widetilde { \\gamma } _ { j p s } + \\sum _ { u = 1 } ^ { \\nu } \\widetilde { \\Theta } _ { u } \\widetilde { \\gamma } _ { i u s } \\Bigr ) \\widetilde { e } _ { s } . \\end{align*}"}
+{"id": "5774.png", "formula": "\\begin{align*} Q ( t ) : = | z ( t ) | ^ { p } + | z ( t ) | | \\bar { z } ( t ) | + | z ( t ) | \\left \\Vert u ' \\right \\Vert _ { C ^ 2 } ( t ) + | z ( t ) | \\Vert \\tilde u ^ \\perp \\Vert _ { C ^ 3 } ( t ) . \\end{align*}"}
+{"id": "6486.png", "formula": "\\begin{align*} g _ u = \\begin{cases} g ' _ { k + 1 } & , \\\\ g ' _ k & , \\\\ g ' _ u & . \\end{cases} \\end{align*}"}
+{"id": "1694.png", "formula": "\\begin{align*} X _ j = \\frac { \\partial } { \\partial z _ j } + \\frac { \\partial \\Phi } { \\partial z _ j } \\frac { \\partial } { \\partial z _ 0 } \\quad \\forall \\ , j \\in \\{ 1 , \\ldots , n \\} , \\end{align*}"}
+{"id": "5170.png", "formula": "\\begin{align*} D _ { \\beta } ( p \\| q ) = \\frac { 1 } { \\beta ( \\beta - 1 ) } \\left \\{ \\sum _ { i } p ^ { \\beta } _ { i } - \\left [ \\sum _ { i } \\beta \\ ; p _ { i } q ^ { \\beta - 1 } _ { i } + ( 1 - \\beta ) \\ ; q ^ { \\beta } _ { i } \\right ] \\right \\} \\end{align*}"}
+{"id": "3150.png", "formula": "\\begin{align*} \\varphi ^ \\sharp \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & = \\left ( \\begin{array} { c c c } a ^ { 2 \\ , p ^ { e _ 1 } } & a ^ { p ^ { e _ 1 } } \\ , b ^ { p ^ { e _ 1 } } & \\frac { 1 } { 2 } \\ , b ^ { 2 \\ , p ^ { e _ 1 } } \\\\ 0 & 1 & b ^ { p ^ { e _ 1 } } \\ , d ^ { p ^ { e _ 1 } } \\\\ 0 & 0 & d ^ { 2 \\ , p ^ { e _ 1 } } \\end{array} \\right ) ( \\ , e _ 1 \\geq 0 \\ , ) . \\end{align*}"}
+{"id": "2234.png", "formula": "\\begin{align*} X ( t ) = | u _ 1 ( t ) | _ { 4 , \\Omega } ^ 4 + | \\omega _ 1 ( t ) | _ { 2 , \\Omega } ^ 2 . \\end{align*}"}
+{"id": "339.png", "formula": "\\begin{align*} g _ { I ( G ) } ( k + 1 ) - g _ { I ( G ) } ( k ) \\ & \\le \\ g _ { I ( G _ 1 ) } ( k + 1 ) + 1 - ( g _ { I ( G _ 1 ) } ( k ) + 1 ) \\\\ & = \\ g _ { I ( G _ 1 ) } ( k + 1 ) - g _ { I ( G _ 1 ) } ( k ) \\le 0 , \\end{align*}"}
+{"id": "1062.png", "formula": "\\begin{align*} T f ( x ) = \\int _ 0 ^ { \\infty } \\frac { 1 } { x + u } f ( u ) d u , f \\in L . \\end{align*}"}
+{"id": "8626.png", "formula": "\\begin{align*} \\| v _ n ( 0 ; \\cdot ) \\| _ { L ^ \\infty ( \\mathbb { R } ) } = \\| u _ 0 ^ { ( n ) } \\| _ { L ^ \\infty ( \\mathbb { R } ) } < C _ 2 ( ( n - 1 ) g ) ^ { 2 ( n - 1 ) } q ^ { - 1 - ( n - 1 ) \\sigma } ( 0 ) \\end{align*}"}
+{"id": "7279.png", "formula": "\\begin{align*} M i n i m i z e \\ \\ \\ & c ^ T x \\\\ s u b j e c t \\ t o \\ \\ \\ \\ & A x \\leq b \\\\ & x ^ l \\leq x \\leq x ^ u \\\\ & x \\in \\mathbb { Z } ^ n \\\\ \\end{align*}"}
+{"id": "8693.png", "formula": "\\begin{align*} & \\tilde \\Box ^ { ( q ) } _ f \\tilde P u = 0 , \\ \\ \\forall u \\in C ^ \\infty ( X , T ^ { * 0 , q } M ' ) , \\\\ & \\gamma \\tilde P u = u , \\ \\ \\forall u \\in C ^ \\infty ( X , T ^ { * 0 , q } M ' ) . \\end{align*}"}
+{"id": "6198.png", "formula": "\\begin{align*} 1 / \\alpha _ 1 > \\sum _ { j = 2 } ^ { 2 n - 3 } 1 / \\gamma _ j ~ . \\end{align*}"}
+{"id": "4411.png", "formula": "\\begin{gather*} [ \\overline { u } _ i , \\overline { u } _ i + \\Delta u _ i ] = \\{ u _ i \\in \\R \\colon ( - 2 \\overline { u } _ i - \\Delta u _ i ) u _ i + u _ i ^ 2 \\leq - \\overline { u } _ i ^ 2 - \\overline { u } _ i \\cdot \\Delta u _ i \\} . \\end{gather*}"}
+{"id": "6802.png", "formula": "\\begin{align*} \\begin{cases} \\frac { q ^ 2 - 1 } 4 & q \\equiv 1 \\pmod 4 \\\\ \\frac { ( q - 1 ) ( q + 3 ) } 4 & q \\equiv - 1 \\pmod 4 \\end{cases} \\end{align*}"}
+{"id": "6454.png", "formula": "\\begin{align*} - \\langle R ^ N ( \\bar \\nabla _ { e _ j } \\big ( R ^ N ( V , d \\phi ( e _ j ) ) \\tau ( \\phi ) \\big ) , d \\phi ( e _ k ) ) d \\phi ( e _ k ) , V \\rangle \\\\ = - \\langle R ^ N ( V , d \\phi ( e _ k ) ) d \\phi ( e _ k ) , \\bar \\nabla _ { e _ j } \\big ( R ^ N ( V , d \\phi ( e _ j ) ) \\tau ( \\phi ) \\big ) \\rangle \\end{align*}"}
+{"id": "6519.png", "formula": "\\begin{align*} \\frac { 1 } { - 2 } \\frac { d } { d t } \\lambda _ 1 ( K + t L ) \\bigg | _ { t = 0 ^ + } \\geq \\lambda _ 1 ( K ) ^ { \\frac { 3 } { 2 } } \\lambda _ 1 ( L ) ^ { - \\frac { 1 } { 2 } } , \\end{align*}"}
+{"id": "5613.png", "formula": "\\begin{align*} n ( 4 ( k + 1 ) s ) ^ { 2 s + 3 } \\left ( 1 + 4 p \\right ) ^ { 1 6 s ^ 2 } 2 ^ { 6 s } K ^ { 2 2 s } \\theta _ 1 ^ { 4 s k } ( d / K ) ^ { 5 s } \\sum _ { g = 0 } ^ { \\infty } \\left ( \\frac { 2 d ^ 3 K ^ { 9 } \\left ( 1 + 4 p \\right ) ^ { 8 s } ( 4 ( k + 1 ) s ) ^ { 6 s } } { n } \\right ) ^ g . \\end{align*}"}
+{"id": "1666.png", "formula": "\\begin{align*} e ^ { - \\left ( c + \\varepsilon \\right ) ^ { \\lambda } } = \\delta ^ { \\eta } , 2 \\eta = \\frac { c + \\varepsilon } { 3 \\left ( T + c \\right ) } < \\frac { 1 } { 3 } . \\end{align*}"}
+{"id": "1906.png", "formula": "\\begin{align*} V \\partial _ t u - \\Delta u = 0 \\quad , \\end{align*}"}
+{"id": "1163.png", "formula": "\\begin{align*} \\vec t _ I : = \\begin{cases} [ \\ell ( I _ 0 ) ] ^ { n \\tau + ( s - \\frac { 1 } { p } ) - ( n - 1 ) ( \\frac { 1 } { p } - \\frac { 1 } { 2 } ) } \\vec z & I = I _ 0 , \\\\ \\vec { \\mathbf { 0 } } & , \\end{cases} \\end{align*}"}
+{"id": "9138.png", "formula": "\\begin{align*} \\sigma ( x ^ { ( \\beta , s ) } _ { i , t } ) = x ^ { ( \\beta , \\sigma _ { \\beta } ( s ) ) } _ { i , t } \\forall \\ \\beta \\in \\Delta ^ + , \\ 1 \\leq s \\leq d _ \\beta , \\ i \\in \\beta , \\ 1 \\leq t \\leq \\nu _ { \\beta , i } . \\end{align*}"}
+{"id": "7352.png", "formula": "\\begin{align*} \\lambda _ { s t a g e 2 , \\beta } ( \\delta ) = d _ { \\beta } - c _ { \\beta } , \\quad \\delta = 1 , \\ldots , N . \\end{align*}"}
+{"id": "252.png", "formula": "\\begin{align*} \\left ( A _ 0 \\frac { d } { d x } - 3 A ' _ 0 \\right ) \\left ( A _ 0 \\frac { d } { d x } + \\gamma _ 0 A _ 0 - A ' _ 0 \\right ) b _ 0 = 0 . \\end{align*}"}
+{"id": "1538.png", "formula": "\\begin{align*} \\sup _ { t \\ge 0 } \\frac { t } { 1 + \\delta \\ , t } = \\frac { 1 } { \\delta } , \\end{align*}"}
+{"id": "4875.png", "formula": "\\begin{align*} 4 ( g - 1 ) \\leq d = C ^ 2 \\leq \\frac { 2 m } { m - 1 } ( g - 1 ) , \\end{align*}"}
+{"id": "4862.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int _ { E _ m } ( \\langle D ^ 2 g ( x ) y _ 0 , y _ 0 \\rangle + 2 o _ x ( | y _ 0 | ^ 2 ) ) d x & = \\frac { 1 } { 2 } | y _ 0 | ^ 2 \\int _ { E _ m } \\langle D ^ 2 g ( w _ { x , y _ 0 } ) \\frac { y _ 0 } { | y _ 0 | } , \\frac { y _ 0 } { | y _ 0 | } \\rangle d x \\\\ & \\le \\frac { 1 } { 2 } | | D ^ 2 g | | _ { L ^ { \\infty } ( a _ q E _ m ) } m | y _ 0 | ^ 2 . \\end{align*}"}
+{"id": "1423.png", "formula": "\\begin{align*} E ^ \\gamma _ { \\alpha , \\beta } ( - \\lambda x ^ \\alpha ) & = \\frac { x ^ { 1 - \\beta } } { 2 \\pi i } \\oint _ { C } e ^ { s x \\ , } \\frac { s ^ { \\alpha \\gamma - \\beta } } { ( \\lambda + s ^ { \\alpha } ) ^ \\gamma } \\ , d s \\end{align*}"}
+{"id": "8576.png", "formula": "\\begin{align*} G _ { t ' } ^ { t _ 1 } = ( J _ k + [ \\varepsilon _ k ( G ^ { t _ 0 } _ { t ' } ) B _ { t _ 0 } ] ^ { \\cdot k } _ + ) G ^ { t _ 0 } _ { t ' } . \\end{align*}"}
+{"id": "1566.png", "formula": "\\begin{align*} \\partial A ( \\ell , r ) = \\bigl ( \\partial B _ r \\cap \\{ F ( D u ) \\ge \\ell ( D u ) \\} \\bigr ) \\bigcup \\ , \\bigl ( B _ r \\cap \\{ F ( D u ) = \\ell ( D u ) \\} \\bigr ) , \\end{align*}"}
+{"id": "4081.png", "formula": "\\begin{align*} R _ 0 ( z _ 1 ) ( x , y ) - R _ 0 ( z _ 0 ) ( x , y ) = 2 \\pi i \\widehat { \\sigma _ { z _ 0 \\mathbb { S } ^ { d - 1 } } } ( r ) \\end{align*}"}
+{"id": "2091.png", "formula": "\\begin{align*} O \\big ( \\eta ^ 4 d ^ 4 T ^ 4 + \\gamma ^ 4 T ^ { - 4 } + \\gamma ^ 4 d ^ { - 4 } + \\eta ^ 4 \\gamma ^ 4 d ^ 4 T ^ 2 + \\eta ^ 4 \\gamma ^ 4 \\sigma ^ 4 T ^ 2 \\big ) = O ( 1 ) , \\end{align*}"}
+{"id": "6675.png", "formula": "\\begin{align*} H = \\langle y _ 1 , \\ldots , y _ l , b _ 1 , \\ldots , b _ m \\rangle \\dd ( H ) = r _ p = l + m . \\end{align*}"}
+{"id": "8514.png", "formula": "\\begin{align*} H ^ \\phi ( X ) = \\lim _ { \\varepsilon \\to 0 } H ^ \\phi _ \\varepsilon ( X ) . \\end{align*}"}
+{"id": "6247.png", "formula": "\\begin{align*} \\dot { D } ( \\alpha ) g ( \\alpha ) = \\frac { 2 } { 2 7 } ( D _ i - D _ g ) \\omega ( 2 - \\omega ) ( \\omega + 1 ) \\left ( ( 1 + \\omega ) r _ i - ( 2 - \\omega ) \\lambda _ g \\right ) . \\end{align*}"}
+{"id": "4349.png", "formula": "\\begin{align*} \\min _ { x } & \\ \\sum _ { i , j \\in \\mathcal { I } , \\mathcal { J } } u _ i q _ j x _ { i , j } , \\\\ \\mathrm { s . t . } & \\ x \\in \\mathcal { X } \\subseteq \\left \\{ x \\in \\{ 0 , 1 \\} ^ { | \\mathcal { I } | \\cdot | \\mathcal { J } | } : \\sum _ { j \\in \\mathcal { J } } x _ { i , j } = 1 \\ \\forall i \\in \\mathcal { I } \\right \\} \\end{align*}"}
+{"id": "2485.png", "formula": "\\begin{align*} C ( A , \\lambda ) = \\begin{pmatrix} E & - 1 _ { \\R ^ { 2 ^ q \\times 1 } } \\\\ 0 _ { \\R ^ { 1 \\times q } } & 1 \\\\ A & 0 _ { \\R ^ { p \\times 1 } } \\\\ 0 _ { \\R ^ { 1 \\times q } } & \\lambda \\\\ \\end{pmatrix} \\in \\R ^ { ( 2 ^ q + 1 + p + 1 ) \\times ( q + 1 ) } . \\end{align*}"}
+{"id": "538.png", "formula": "\\begin{align*} v ( s ) : = \\int _ { - T / 2 } ^ { T / 2 } f \\left ( t + \\frac { T } { 2 } \\right ) \\mathrm { e } ^ { - i s t } \\mathrm { ~ d } t , s \\in \\mathbb { C } . \\end{align*}"}
+{"id": "4256.png", "formula": "\\begin{align*} G _ \\xi [ f , g , h ] ( x , y ) = \\hat f ( \\xi - x ) \\hat { \\bar g } ( x - \\xi + y ) \\hat h ( \\xi - y ) . \\end{align*}"}
+{"id": "8878.png", "formula": "\\begin{align*} \\nabla \\phi _ R ( x ) = x , \\quad \\Delta \\phi _ R ( x ) = N . \\end{align*}"}
+{"id": "8446.png", "formula": "\\begin{align*} W ^ { s , p } ( K ) : = \\left \\{ v \\in L ^ p ( K ) \\ , : \\ , \\frac { | v ( x ) - v ( y ) | } { | x - y | ^ { \\frac { n } { p } + s } } \\in L ^ p ( K \\times K ) \\right \\} \\end{align*}"}
+{"id": "6334.png", "formula": "\\begin{align*} \\pm \\mathcal E _ { M _ 0 } = \\pm \\left ( \\frac { 1 } { 2 } [ f _ { M _ 0 } , [ f _ { M _ 0 } , \\mathbb { H } _ { n , \\ell } ] ] + \\frac { 1 } { 2 } [ g _ { M _ 0 } , [ g _ { M _ 0 } , \\mathbb { H } _ { n , \\ell } ] ] \\right ) \\le \\frac { C } { M _ 0 ^ 2 } [ H _ { n , \\ell } ] _ { { \\rm d i a g } } . \\end{align*}"}
+{"id": "7676.png", "formula": "\\begin{align*} \\mathcal { T } ( \\omega ( n _ 1 ) , . . . , \\omega ( n _ { | \\Lambda ' | } ) ) = ( U ( n _ 1 ) , . . . , U ( n _ { | \\Lambda ' | } ) ) . \\end{align*}"}
+{"id": "2393.png", "formula": "\\begin{align*} S _ n : = \\int _ { \\mathbb { Z } _ 2 } A _ n ^ { ( s ) } \\left ( t + \\frac { 1 } { 4 } \\right ) \\mathrm { d } t ~ \\in \\mathbb { Q } _ 2 , \\\\ T _ n : = \\int _ { \\mathbb { Z } _ 2 } B _ n ^ { ( s ) } \\left ( t + \\frac { 1 } { 4 } \\right ) \\mathrm { d } t ~ \\in \\mathbb { Q } _ 2 . \\end{align*}"}
+{"id": "7045.png", "formula": "\\begin{align*} \\partial _ t w _ j - \\Delta w _ j + b \\cdot \\nabla w _ j + ( \\nabla _ j b ) \\cdot w = 0 , w _ j ( 0 ) = \\nabla _ j f , 1 \\leq j \\leq d . \\end{align*}"}
+{"id": "6517.png", "formula": "\\begin{align*} \\frac { 1 } { n } h _ { K ( 0 ) } ( \\xi ) d S _ { K ( 0 ) } ( \\xi ) = \\varphi ( \\xi ) d \\xi , \\end{align*}"}
+{"id": "28.png", "formula": "\\begin{align*} \\langle d _ j f , g \\rangle _ { X ^ { j + 1 } } = & \\frac 1 2 \\sum _ { s \\in X ^ { j + 1 } } m ( s ) d f ( s ) \\overline { g ( s ) } \\\\ = & \\frac 1 2 \\sum _ { s \\in X ^ { j + 1 } } m ( s ) \\overline { g ( s ) } \\sum _ { r \\subset s } f ( r ) \\\\ = & \\frac 1 2 \\sum _ { r \\in X ^ { j } } m ( r ) f ( r ) \\overline { \\sum _ { r \\subset s } \\frac { m ( s ) } { m ( r ) } g ( s ) } \\ . \\end{align*}"}
+{"id": "2740.png", "formula": "\\begin{align*} S ^ { T } J S = J , \\end{align*}"}
+{"id": "2013.png", "formula": "\\begin{align*} \\mathcal { N } _ n = \\mathcal { N } _ { n - 1 } + \\mathcal { N } _ { n - 3 } n \\in \\mathbb { Z } . \\end{align*}"}
+{"id": "8236.png", "formula": "\\begin{align*} P _ { { \\bf X } ^ L \\| Y ^ { L - 1 } } : = \\prod _ { i = 1 } ^ L P _ { { \\bf X } _ i | { \\bf X } ^ { i - 1 } , Y ^ { i - 1 } } , \\end{align*}"}
+{"id": "476.png", "formula": "\\begin{align*} f ( x ) = \\begin{cases} 0 , & x = 0 , \\\\ \\omega ^ 5 x ^ 9 , & x \\in C _ 0 , \\\\ x ^ 3 , & x \\in C _ 1 , \\\\ x ^ { 1 7 } , & x \\in C _ 2 , \\\\ \\omega ^ 3 x ^ { 3 4 } , & x \\in C _ 3 , \\\\ \\omega ^ 4 x ^ 9 , & x \\in C _ 4 . \\end{cases} \\end{align*}"}
+{"id": "3754.png", "formula": "\\begin{align*} \\frac { \\partial \\mathrm { E } _ { \\alpha , \\beta } \\left ( z \\right ) } { \\partial \\alpha } = z \\frac { \\partial } { \\partial z } \\left ( \\frac { \\partial \\mathrm { E } _ { \\alpha , \\beta } \\left ( z \\right ) } { \\partial \\beta } \\right ) . \\end{align*}"}
+{"id": "712.png", "formula": "\\begin{align*} f ( x ) = \\phi _ 1 ^ 6 ( x ) - 6 \\phi _ 1 ^ 5 ( x ) + 1 5 \\phi _ 1 ^ 4 ( x ) - 1 7 \\phi _ 1 ^ 3 ( x ) + 1 2 \\phi _ 1 ^ 2 ( x ) - 6 \\phi _ 1 ( x ) + 1 - m , \\end{align*}"}
+{"id": "8374.png", "formula": "\\begin{align*} R _ { 1 } : = \\mathbb { Z } J ( w ) \\cap R ^ { + } \\qquad R _ { 2 } : = \\{ \\beta \\in R ^ { + } ( w ^ { - 1 } ) : { \\text s u p p } ( \\beta ) \\nsubseteq J ( w ) \\} . \\end{align*}"}
+{"id": "7528.png", "formula": "\\begin{align*} \\begin{dcases} \\partial _ t \\rho + \\nabla \\cdot ( \\rho u ) = 0 \\\\ \\partial _ t ( \\rho u ) + \\nabla \\cdot ( \\rho u \\otimes u ) + \\nabla \\big ( p _ 1 ( \\rho ) + p _ 2 ( \\rho ) \\big ) = 0 \\ , \\end{dcases} \\end{align*}"}
+{"id": "4297.png", "formula": "\\begin{align*} d y _ t = \\sum _ { j = 1 } ^ d V _ j ( y _ t ) d x _ t ^ j , y _ 0 = a . \\end{align*}"}
+{"id": "3410.png", "formula": "\\begin{align*} \\delta _ R ( T ( p , q ) ) = \\underline { \\delta } _ R ( T ( p , q ) ) = \\bar { \\delta } _ R ( T ( p , q ) ) = - \\frac { { \\bar { \\mu } } ( \\Sigma ( 2 , p , q ) ) } { 2 } , \\end{align*}"}
+{"id": "6776.png", "formula": "\\begin{gather*} A _ p = \\left \\{ m \\in \\mathbb { Z } _ n \\mathrel | 0 < m _ p < n _ p \\right \\} \\end{gather*}"}
+{"id": "6658.png", "formula": "\\begin{align*} \\omega ^ { + } ( E , \\mathrm { i } \\epsilon ) = \\omega ^ { - } ( E , \\mathrm { i } \\epsilon ) . \\end{align*}"}
+{"id": "7762.png", "formula": "\\begin{align*} Z _ { \\tilde m , 1 } ^ { - 1 } A _ { \\tilde m , 2 } Z _ { \\tilde m , 1 } = \\mathrm { d i a g } \\{ A _ { \\tilde m , 3 , 1 1 } , \\cdots , A _ { \\tilde m , 3 , l _ { \\tilde m } l _ { \\tilde m } } \\} , \\end{align*}"}
+{"id": "8778.png", "formula": "\\begin{align*} \\pi ( \\{ ( x , y ) \\in \\R ^ 2 : y \\ne x \\} ) & = \\pi ( \\{ ( x , y ) \\in \\R ^ 2 : y < x \\} ) + \\pi ( \\{ ( x , y ) \\in \\R ^ 2 : y > x \\} ) \\\\ & = \\nu _ l ( \\R ) + \\nu _ r ( \\R ) = 1 , \\end{align*}"}
+{"id": "3596.png", "formula": "\\begin{align*} \\sigma _ 1 ( z ) = \\sigma \\circ \\tau ( z ) \\sigma _ 2 ( z ) = \\sigma \\circ \\tau ^ 2 ( z ) , \\end{align*}"}
+{"id": "7465.png", "formula": "\\begin{align*} C ( X ) = \\{ [ 0 , \\infty ] \\times X \\} / \\sim , \\end{align*}"}
+{"id": "1092.png", "formula": "\\begin{align*} \\Delta : = \\frac { d } { p } + \\frac { \\widetilde d } { p ' } . \\end{align*}"}
+{"id": "3388.png", "formula": "\\begin{align*} \\frac { \\eta _ { t - 1 } } { \\alpha _ { t - 1 } } & = \\frac { t ^ { 2 } } { 8 c \\gamma ^ { 2 } L } \\\\ \\frac { \\eta _ { t } \\left ( 1 - \\alpha _ { t } \\right ) } { \\alpha _ { t } } & = \\frac { ( t + 1 ) ( t - 1 ) } { 8 c \\gamma ^ { 2 } L } \\end{align*}"}
+{"id": "577.png", "formula": "\\begin{align*} B & = u & 1 - \\frac { \\omega ^ 2 m } { g \\rho } & = v ; \\end{align*}"}
+{"id": "4549.png", "formula": "\\begin{align*} \\delta : = \\max \\{ \\frac { F _ N ( \\beta ) } { a } , L _ N ( \\beta ) \\} , \\gamma : = \\frac { \\beta a _ - } { 1 + \\beta } - \\frac { F _ N ( \\beta ) \\vee a L _ N ( \\beta ) } { \\beta } . \\end{align*}"}
+{"id": "7830.png", "formula": "\\begin{align*} \\boldsymbol { \\rho } ^ { \\star } & \\in \\operatorname { a r g m i n } _ { \\boldsymbol { 0 } \\leq \\boldsymbol { \\rho } \\leq \\boldsymbol { 1 } } L ( \\boldsymbol { \\rho } ; \\boldsymbol { \\lambda } ^ { \\star } ) , \\\\ & \\left [ \\begin{array} { c c } \\ ! \\ ! I _ { M } , \\ ! \\ ! \\ ! \\ ! & I _ { M } \\ ! \\ ! \\ ! \\ ! \\\\ \\end{array} \\right ] \\boldsymbol { \\rho } ^ { \\star } = \\boldsymbol { 1 } , \\end{align*}"}
+{"id": "2707.png", "formula": "\\begin{align*} d { _ { * } L } = { _ { * } \\theta } _ { i } \\wedge { _ { * } \\rho } ^ { i } , \\end{align*}"}
+{"id": "4978.png", "formula": "\\begin{align*} P _ N ( \\{ \\nu \\} ) = \\prod _ { j , k = 1 } ^ N ( \\nu _ j - \\nu _ k + 1 ) . \\end{align*}"}
+{"id": "4521.png", "formula": "\\begin{align*} \\left \\{ \\begin{matrix*} [ l ] \\partial _ t n _ \\infty + \\displaystyle \\sum _ { i = 1 } ^ \\infty \\partial _ { s _ i } n _ \\infty + p _ \\infty ( s _ 1 , s _ 2 , . . . ) n _ \\infty = 0 , \\\\ [ 5 p t ] n _ \\infty ( t , s _ 1 = 0 , s _ 2 , . . . ) = \\int _ { u = 0 } ^ \\infty p _ \\infty ( s _ 2 , . . . , s _ { K } , . . . , u ) n _ \\infty ( t , s _ 2 , . . . , s _ { K } , . . . , u ) \\ , d u . \\end{matrix*} \\right . \\end{align*}"}
+{"id": "501.png", "formula": "\\begin{align*} \\pi \\left ( x ^ { 1 / 2 } \\right ) \\sim \\frac { x ^ { 1 / 2 } } { \\log \\left ( x ^ { 1 / 2 } \\right ) } = \\frac { 2 x ^ { 1 / 2 } } { \\log { x } } . \\end{align*}"}
+{"id": "5426.png", "formula": "\\begin{align*} d _ { X ( \\tau ) } \\beta ^ \\tau = d _ i - \\sum _ { t = 0 } ^ { \\tau - 1 } \\{ d _ { X ( t ) } - \\beta d _ { X ( t + 1 ) } \\} \\beta ^ t \\end{align*}"}
+{"id": "4433.png", "formula": "\\begin{gather*} D _ t v ^ j = \\delta _ t v ^ j + \\tau _ j \\delta ^ 2 _ t v ^ j \\ , , \\delta ^ 2 _ t v ^ j \\coloneqq \\frac { \\delta _ t v ^ j - \\delta _ t v ^ { j - 1 } } { \\tau _ j + \\tau _ { j - 1 } } \\ , , \\ \\ j = 2 , \\dots , M . \\end{gather*}"}
+{"id": "4931.png", "formula": "\\begin{align*} \\beta _ { L } = \\beta _ { L ' } . \\end{align*}"}
+{"id": "1728.png", "formula": "\\begin{align*} ( S _ l ) _ { j , k } : = \\begin{cases} 1 & \\mbox { i f } j + k \\leq l + 1 \\\\ 0 & \\mbox { o t h e r w i s e , } \\end{cases} \\end{align*}"}
+{"id": "2201.png", "formula": "\\begin{align*} - \\intop _ \\Omega \\psi _ { 1 , r r } \\psi _ { 1 , z z } d x - \\intop _ \\Omega \\psi _ { 1 , z z } ^ 2 d x - 3 \\intop _ \\Omega { 1 \\over r } \\psi _ { 1 , r } \\psi _ { 1 , z z } d x = \\intop _ \\Omega \\omega _ 1 \\psi _ { 1 , z z } d x . \\end{align*}"}
+{"id": "7865.png", "formula": "\\begin{align*} \\sum _ { \\mathbf { y } \\in \\mathbb { F } _ { p } ^ { m } } \\omega ^ { \\mathbf { y } ^ { T } ( \\mathbf { C } + \\mathbf { C } ^ { T } ) \\mathbf { z } } = \\begin{cases} p ^ m , & ( \\mathbf { C } + \\mathbf { C } ^ { T } ) \\mathbf { z } = \\mathbf { 0 } ; \\\\ 0 , & . \\end{cases} \\end{align*}"}
+{"id": "1929.png", "formula": "\\begin{align*} & \\sum _ { x \\in G \\setminus B _ { 1 } ( x _ 0 ) } e ^ { - \\Lambda d ^ \\alpha ( x , x _ 0 ) } \\mu ( x ) = \\sum _ { n = 1 } ^ { \\infty } \\Big ( \\sum _ { \\{ x : \\ , d ( x , x _ 0 ) = n \\} } e ^ { - \\Lambda d ^ \\alpha ( x , x _ 0 ) } \\mu ( x ) \\Big ) \\\\ & \\qquad = \\sum _ { n = 1 } ^ { \\infty } e ^ { - \\Lambda n ^ \\alpha } \\mu ( \\{ x : \\ , d ( x , x _ 0 ) = n \\} ) = \\sum _ { n = 1 } ^ { \\infty } b ^ n e ^ { - \\Lambda n ^ \\alpha } , \\end{align*}"}
+{"id": "7170.png", "formula": "\\begin{align*} \\| \\vec { v } \\| _ X = \\sup _ { x \\in \\R } \\inf \\{ r > 0 : \\vec { v } ( x ) \\in r R _ { L , S } \\} . \\end{align*}"}
+{"id": "3511.png", "formula": "\\begin{align*} \\lim _ { n } \\| \\rho _ { n , k } ( x ) \\rho _ { n , k } ( y ) - \\psi _ n ( a ) \\psi _ n ( b ) \\| = 0 . \\end{align*}"}
+{"id": "3602.png", "formula": "\\begin{align*} T ^ 2 f = \\alpha ^ 2 \\tau _ 0 \\tau _ 1 f ( \\tau ^ 2 , w \\sigma \\sigma _ 1 ) \\end{align*}"}
+{"id": "5951.png", "formula": "\\begin{align*} \\frac { M _ { 2 } ( \\mathcal { N C } ( G ) ) } { | e ( \\mathcal { N C } ( G ) ) | } = \\frac { 7 0 4 6 4 } { 2 0 4 } > \\frac { 7 5 8 4 } { 2 2 } = \\frac { M _ { 1 } ( \\mathcal { N C } ( G ) ) } { | v ( \\mathcal { N C } ( G ) ) | } . \\end{align*}"}
+{"id": "8492.png", "formula": "\\begin{align*} \\xi _ \\delta ( \\sigma ) = \\dfrac { \\sigma } { | \\sigma | } \\min \\left \\{ 1 , \\ , \\frac { ( | \\sigma | - M ) _ { + } } { \\delta } \\right \\} \\textrm { f o r } \\delta > 0 . \\end{align*}"}
+{"id": "4302.png", "formula": "\\begin{align*} d J _ t & = ( d M _ t ) \\cdot J _ t , J _ 0 = A , \\\\ d K _ t & = - K _ t \\cdot d M _ t , K _ 0 = B . \\end{align*}"}
+{"id": "6158.png", "formula": "\\begin{align*} v ^ { k + 1 } = v ^ k - M ^ k ( v ^ k - \\widetilde { v } ^ k ) . \\end{align*}"}
+{"id": "677.png", "formula": "\\begin{align*} \\mathcal { A } _ { s } : W ^ { m , p } ( \\mathbb { R } ^ { n } ) \\rightarrow W ^ { m , q } ( \\mathbb { R } ^ { n } ) \\frac { 1 } { q } = \\frac { 1 } { p } - \\frac { 2 s } { n } , \\end{align*}"}
+{"id": "5665.png", "formula": "\\begin{align*} ( \\| \\boldsymbol { a } ^ { ( 3 ) } \\| ^ 2 , \\| \\boldsymbol { a } ^ { ( 3 ) } \\| ^ 2 ) _ p ( \\| \\boldsymbol { a } ^ { ( 3 ) } \\| ^ 2 , \\tan ^ 2 \\theta ) _ p ( \\tan ^ 2 \\theta , \\tan ^ 2 \\theta ) _ p = 1 \\end{align*}"}
+{"id": "3764.png", "formula": "\\begin{align*} | g _ N ( x ) | = \\exp ( f _ N ( x ) ) \\leq \\exp ( - N ) \\end{align*}"}
+{"id": "5349.png", "formula": "\\begin{align*} w _ i ^ S = \\theta _ i ^ 1 \\ , 1 \\{ i \\in N ^ { \\{ 0 , 1 \\} } \\} + \\beta \\ , \\sum _ { j \\in N } ( p _ { i j } ^ 1 - p _ { i j } ^ 0 ) \\ , t ^ S _ j , i \\in N ; \\end{align*}"}
+{"id": "7883.png", "formula": "\\begin{align*} \\mathbf { A } _ { i } = \\begin{cases} \\psi ( \\pi , \\mathbf { a } , \\mathbf { d } _ { i } ) , & 1 \\leq i \\leq 3 ; \\\\ \\psi ( \\pi , \\mathbf { b } , \\mathbf { d } _ { i } ) , & 4 \\leq i \\leq 6 . \\end{cases} \\end{align*}"}
+{"id": "3130.png", "formula": "\\begin{align*} u ^ - _ { \\psi ^ * } ( s ) = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ s ^ { p ^ { e _ 2 } } & 0 & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "3292.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { p - 1 } \\frac { ( \\frac { d + r } { d } ) _ k ^ { d - r - 1 } ( \\frac { r } { d } ) _ k ^ { r + 1 } } { k ! ^ d } \\equiv - \\big ( \\tfrac { r } { d } \\big ) ^ { r + 1 } \\Gamma _ p \\big ( - \\tfrac { r } { d } \\big ) ^ d \\pmod { p ^ 2 } . \\end{align*}"}
+{"id": "6632.png", "formula": "\\begin{align*} H ^ 2 ( X , \\Q ) = \\mathrm { N S } ( X ) _ \\Q \\oplus T ( X ) , \\end{align*}"}
+{"id": "7058.png", "formula": "\\begin{align*} ( \\partial _ t + \\Delta - b _ m \\cdot \\nabla ) v = - | b _ k | , v ( r , \\cdot ) = 0 , t \\leq r . \\end{align*}"}
+{"id": "1476.png", "formula": "\\begin{align*} | E _ i M \\setminus S _ i | & \\geq | E _ i M | - | S _ i | \\geq | E _ i | - | S _ i | \\\\ & \\geq | F _ { k _ i } ^ { - M ^ 2 } | - | S _ i | = \\left ( 1 - \\frac { \\varepsilon } { 8 } \\right ) | F _ { k _ i } | - ( d - \\frac { \\varepsilon } { 4 } ) | F _ { k _ i } | \\\\ & = \\left ( 1 - d + \\frac { \\varepsilon } { 8 } \\right ) | F _ { k _ i } | = \\left ( 1 - d + \\frac { \\varepsilon } { 8 } \\right ) | T _ i | . \\end{align*}"}
+{"id": "7281.png", "formula": "\\begin{align*} C _ { m } ( \\beta , n ) : = \\sum _ { j = \\delta ( \\beta ) } ^ { m - 1 } \\csc ^ { n } \\left ( \\frac { ( j + \\beta ) } { m } \\pi \\right ) , \\end{align*}"}
+{"id": "485.png", "formula": "\\begin{align*} \\prod _ { p \\mid \\gcd ( \\alpha _ { i _ { t - 1 } } , s ) } { p ^ { \\nu _ p ( s ) } } \\mid \\prod _ { j = 0 } ^ { k - 2 } { \\alpha _ { i _ { t - k + j } } } . \\end{align*}"}
+{"id": "4334.png", "formula": "\\begin{align*} \\inf _ { x , p , \\theta } & \\ \\Gamma \\theta + \\sum _ { i \\in [ m ] } f _ i ( x , \\overline { u } ^ i ) + p _ i , \\\\ \\mathrm { s . t . \\ ; } & x \\in \\mathcal { X } , \\\\ & p _ i + \\theta \\geq \\sup _ { u ^ i \\in \\mathcal { U } _ i } f _ i ( x , u ^ i ) - f _ i ( x , \\overline { u } ^ i ) \\ \\forall i \\in [ m ] , \\\\ & p \\in \\R ^ m _ { \\geq 0 } , \\theta \\in \\R _ { \\geq 0 } . \\end{align*}"}
+{"id": "6105.png", "formula": "\\begin{align*} \\overline { \\nabla } _ X \\phi = \\nabla _ X \\phi + \\frac 1 2 k ( X , - ) ^ \\sharp \\cdot e _ 0 \\cdot \\phi \\end{align*}"}
+{"id": "1952.png", "formula": "\\begin{align*} M _ i - m _ i = 4 ^ { - \\alpha i } L , i = 0 , 1 , 2 , 3 , \\ldots \\end{align*}"}
+{"id": "3364.png", "formula": "\\begin{align*} \\mathcal { I } ( \\phi ^ \\circ ) \\geq - e _ \\Gamma ( \\hat { \\phi } ^ K d _ 0 + s \\sum _ { i = 1 } ^ { K - 1 } \\hat { \\phi } ^ i ) + K t \\Bar { s } , \\end{align*}"}
+{"id": "5991.png", "formula": "\\begin{align*} | d u ( \\nu ) | = | \\nabla u | , \\ \\ d u ( e _ i ) = 0 , \\ f o r \\ 1 \\leq i \\leq n - 1 . \\end{align*}"}
+{"id": "6607.png", "formula": "\\begin{align*} \\phi _ { \\Phi _ k , \\beta } ( x , y ) & = \\frac { 1 } { 2 \\pi i } \\left ( \\int _ { b - i \\tau } ^ { b + i \\tau } + \\int _ { b - i U } ^ { b - i \\tau } + \\int _ { b + i \\tau } ^ { b + i U } \\right ) \\left ( \\sum _ { n \\le x } \\left ( \\frac { \\Phi _ k ( n ) } { n ^ \\beta } \\right ) ^ { - z } \\right ) \\frac { y ^ z } { z } \\ , d z \\\\ & + O \\left ( \\frac { y ^ { b } } { U } \\sum _ { n \\leq x } \\Phi _ k ( n ) ^ { - b } n ^ { b \\beta } \\right ) . \\end{align*}"}
+{"id": "3191.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } n } \\ln { f ( n ) } & = \\frac { 9 } { 2 n } + \\frac { 6 } { ( 3 n + 1 ) ( 6 n + 4 ) } - 0 . 1 6 3 \\\\ [ 5 p t ] & \\leq \\frac { 9 } { 2 \\times 1 6 8 } + \\frac { 6 } { ( 3 \\times 1 6 8 + 1 ) ( 6 \\times 1 6 8 + 4 ) } - 0 . 1 6 3 < - 0 . 1 3 < 0 . \\end{align*}"}
+{"id": "3008.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\mathcal { L } _ { \\xi } g + \\mathrm { R i c } + \\lambda g + \\mu \\widetilde { g } + \\nu \\eta \\otimes \\eta = 0 , \\end{align*}"}
+{"id": "2667.png", "formula": "\\begin{align*} \\delta q ^ { i } ( t _ { 1 } ) = & \\delta q ^ { i } ( t _ { 2 } ) = 0 , \\\\ \\delta \\dot { q } ^ { i } ( t _ { 1 } ) = & \\delta \\dot { q } ^ { i } ( t _ { 2 } ) = 0 \\end{align*}"}
+{"id": "4175.png", "formula": "\\begin{align*} \\varphi ( g _ 2 g ^ { n } _ 1 g _ 3 ) = \\varphi \\big ( ( g ^ { j } _ 1 g _ 3 ) ( g ^ { i } _ 1 g _ 2 ) \\big ) \\mbox { o r } \\varphi ( g _ 2 g ^ { n } _ 1 g _ 3 ) = \\varphi \\big ( ( g ^ { i } _ 1 g _ 2 ) ( g ^ { j } _ 1 g _ 3 ) \\big ) \\ , . \\end{align*}"}
+{"id": "7906.png", "formula": "\\begin{align*} \\ast \\ast \\lambda = ( - 1 ) ^ { k ( n - k ) } \\lambda , \\quad \\lambda \\in \\Lambda ^ { k } ( \\Omega ) . \\end{align*}"}
+{"id": "3050.png", "formula": "\\begin{align*} Z _ \\mu = c _ { \\mu } ^ 0 X _ 0 + c _ \\mu ^ 1 X _ 1 + c _ \\mu ^ 2 X _ 2 + c _ \\mu ^ 4 X _ 4 + c _ \\mu ^ 6 X _ 6 + c _ \\mu ^ 8 X _ 8 \\ , , \\mu = 1 , 2 , 3 . \\end{align*}"}
+{"id": "2270.png", "formula": "\\begin{align*} \\bigl ( \\check R _ { V _ 1 , V _ 2 } \\otimes _ { V _ 3 } \\bigr ) \\circ \\bigl ( _ { V _ 2 } \\otimes \\check R _ { V _ 1 , V _ 3 } \\bigr ) \\circ \\bigl ( \\check R _ { V _ 2 , V _ 3 } \\otimes _ { V _ 1 } \\bigr ) = \\bigl ( _ { V _ 1 } \\otimes \\check R _ { V _ 2 , V _ 3 } \\bigr ) \\circ \\bigl ( \\check R _ { V _ 1 , V _ 3 } \\otimes _ { V _ 2 } \\bigr ) \\circ \\bigl ( _ { V _ 3 } \\otimes \\check R _ { V _ 1 , V _ 2 } \\bigr ) \\ . \\end{align*}"}
+{"id": "377.png", "formula": "\\begin{align*} x - a = \\sqrt { 2 a b K _ 2 } = 2 ^ { e } r ' s , \\end{align*}"}
+{"id": "6463.png", "formula": "\\begin{align*} \\bar \\Delta \\tau ( \\phi ) = & m H \\Delta \\nu = m H | A | ^ 2 \\nu , \\end{align*}"}
+{"id": "236.png", "formula": "\\begin{align*} \\frac { d ^ 2 x } { d t ^ 2 } + A ( x ) \\frac { d x } { d t } + b ( x ) = 0 \\end{align*}"}
+{"id": "992.png", "formula": "\\begin{align*} j ( r ) = \\frac { 1 } { ( 4 \\pi t ) ^ { d / 2 } } \\int _ 0 ^ \\infty e ^ { - r ^ 2 / 4 t } \\hat \\mu ( t ) \\ , d t . \\end{align*}"}
+{"id": "3289.png", "formula": "\\begin{align*} ( q ^ { d + r } , q ^ { r - d } ; q ^ d ) _ k = - q ^ r \\frac { [ d - r ] } { [ r ] } \\left ( 1 + \\frac { 1 - q ^ d } { q ^ d - q ^ { d k + r } } \\right ) ( q ^ r ; q ^ d ) _ k ^ 2 , \\end{align*}"}
+{"id": "3652.png", "formula": "\\begin{align*} \\lambda _ { r _ { i j } } ( M _ { i j } ) = \\lambda _ 2 \\left ( \\frac 1 2 L ( G ( A _ i , A _ j ) , q _ { i j } ) \\right ) = \\frac 1 2 a ( G ( A _ i , A _ j ) ) . \\end{align*}"}
+{"id": "3203.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ t ^ \\alpha u ( t ) + t ^ { \\beta } A u ( t ) = t ^ { \\mu } f , 0 < t \\leq T ; \\\\ & \\lim \\limits _ { t \\rightarrow + 0 } J _ t ^ { \\alpha - 1 } u ( t ) = \\varphi , \\end{aligned} \\right . \\end{align*}"}
+{"id": "1211.png", "formula": "\\begin{align*} \\chi _ { S _ { p } } ( u , v ) = \\left \\{ \\begin{array} { l l } 0 , & ( u , v ) \\in S _ { p } ; \\\\ \\infty , & \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "105.png", "formula": "\\begin{align*} R o o t _ { - } - R o o t _ + & = - \\frac { \\sqrt { b ^ 2 - 4 a c } } { a } \\\\ & = 4 \\frac { \\sqrt { G \\cdot P \\cdot E ( \\frac { G } { n - 1 } + K ) } } { ( G - P ) ^ 2 } > 0 . \\end{align*}"}
+{"id": "3627.png", "formula": "\\begin{align*} P f ( z , w ) = \\frac { \\lambda _ { 1 } ^ { 2 } } { 1 - \\lambda _ { 1 } ^ { 2 } } \\big ( f ( z , - w ) - f ( z , w ) \\big ) , \\end{align*}"}
+{"id": "8214.png", "formula": "\\begin{align*} \\{ P _ { { \\bf X } _ i | Y ^ { i - 1 } } ( { \\bf x } _ i | y ^ { i - 1 } ) , i = [ t : L ] \\} \\end{align*}"}
+{"id": "2669.png", "formula": "\\begin{align*} K ^ { ( 2 ) } _ { i j } : = \\frac { \\partial ^ { 2 } L ^ { ( 2 ) } } { \\partial \\ddot { q } ^ { i } \\partial \\ddot { q } ^ { j } } , E ^ { ( 2 ) } _ { i j } : = \\frac { \\partial ^ { 2 } L ^ { ( 2 ) } } { \\partial \\ddot { q } ^ { i } \\partial \\dot { q } ^ { j } } - \\frac { \\partial ^ { 2 } L ^ { ( 2 ) } } { \\partial \\ddot { q } ^ { j } \\partial \\dot { q } ^ { i } } . \\end{align*}"}
+{"id": "1986.png", "formula": "\\begin{align*} i \\frac { d } { d t } \\widehat { \\psi } _ l ( t ) - \\alpha \\frac { d ^ 2 } { d t ^ 2 } \\widehat { \\psi } _ l ( t ) - | \\mu _ l | ^ 2 \\widehat { \\psi } _ l ( t ) - \\varepsilon ^ { 2 } \\widehat { ( f ( \\psi _ M ) ) } _ l ( t ) = 0 , l \\in \\mathcal { T } _ M , \\ t > 0 . \\end{align*}"}
+{"id": "2372.png", "formula": "\\begin{align*} q : = \\lim _ { r \\to \\infty } \\lim _ { n \\to \\infty } q _ { \\mu _ n } ( r ) \\quad p : = \\lim _ { n \\to \\infty } \\mu _ n ( \\R ^ { 2 d } ) \\ , . \\end{align*}"}
+{"id": "8477.png", "formula": "\\begin{align*} \\lim _ { h \\searrow 0 } \\| ( { \\bar u } _ h ) _ + - u _ + \\| _ { L ^ \\gamma ( K _ T ) } = 0 . \\end{align*}"}
+{"id": "2772.png", "formula": "\\begin{align*} \\sigma _ { 3 } ^ { ( 1 ) } : & T ^ { * } M | _ { \\Theta } \\times T ^ { * } M | _ { Q , P } \\rightarrow T ^ { * } M \\\\ ; & ( \\sigma _ { 3 } ^ { * } \\Xi ^ { a } : = \\epsilon ^ { a } , \\sigma _ { 3 } ^ { * } \\Psi _ { a } : = \\epsilon _ { a } , \\sigma _ { 3 } ^ { * } \\Theta ^ { \\alpha } , \\sigma _ { 3 } ^ { * } \\Theta _ { \\alpha } , \\sigma _ { 3 } ^ { * } Q ^ { i } , \\sigma _ { 3 } ^ { * } P _ { i } ) \\mapsto ( \\Xi ^ { a } , \\Psi _ { a } , \\Theta ^ { \\alpha } , \\Theta _ { \\alpha } , Q ^ { i } , P _ { i } ) . \\end{align*}"}
+{"id": "6883.png", "formula": "\\begin{align*} \\Lambda ( R ) = 1 + t R [ [ t ] ] \\subset R [ [ t ] ] \\end{align*}"}
+{"id": "1750.png", "formula": "\\begin{align*} \\Re \\mathfrak { m } : = \\mathrm { s p a n } _ { \\mathbb { R } } \\left \\{ \\Re ( Y _ { - n } ) , \\Re ( X _ j ) , \\Re ( Y _ j ) , \\Re \\left ( X ^ \\prime _ { - 1 } \\right ) , \\Re \\left ( Y ^ \\prime _ { - 1 } \\right ) \\ , | \\ , j = 1 , \\ldots , n - 1 \\right \\} \\end{align*}"}
+{"id": "2316.png", "formula": "\\begin{align*} L _ { \\sigma } ^ { p } ( \\mathbb { R } ^ { 3 } ) = \\{ u \\in L ^ { p } ( \\mathbb { R } ^ { 3 } ) | \\ , \\nabla \\cdot u = 0 \\} , \\dot { H } ^ { 1 } _ { \\sigma } ( \\mathbb { R } ^ { 3 } ) = \\{ u \\in \\dot { H } ^ { 1 } ( \\mathbb { R } ^ { 3 } ) | \\ , \\nabla \\cdot u = 0 \\} , \\end{align*}"}
+{"id": "2873.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to 0 ^ + } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ X = \\left [ - \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { X } ; \\end{align*}"}
+{"id": "2649.png", "formula": "\\begin{align*} P _ { \\leq M } : = \\sum _ { L \\leq M } P _ L , P _ { > M } : = \\sum _ { M > L } P _ L . \\end{align*}"}
+{"id": "8943.png", "formula": "\\begin{align*} \\psi _ { x _ 0 , \\lambda } ( x ) : = x _ 0 + \\lambda x \\quad . \\end{align*}"}
+{"id": "3666.png", "formula": "\\begin{align*} x _ z ^ T L ( K _ n , p ) x _ z = \\sum _ { i < j } ( y _ i - y _ j ) ^ 2 \\left \\langle z , d _ { i j } \\right \\rangle ^ 2 . \\end{align*}"}
+{"id": "4660.png", "formula": "\\begin{align*} \\mathcal A _ { \\epsilon , \\beta } ( \\Omega _ { \\epsilon , \\beta } ) = \\min _ { \\Omega \\in \\mathcal C _ { \\epsilon , \\beta } } \\mathcal A _ { \\epsilon , \\beta } ( \\Omega ) . \\end{align*}"}
+{"id": "2432.png", "formula": "\\begin{align*} \\tilde { U } ( \\tau , x ) = ( p T ) ^ { \\frac { 1 } { p } } e ^ { - \\tau } \\bar { U } ( t , x ) , t = T ( 1 - e ^ { - p \\tau } ) , \\tau \\in ( 0 , + \\infty ) . \\end{align*}"}
+{"id": "2597.png", "formula": "\\begin{align*} \\varphi ' ( r ) \\ ; \\dfrac { x - x _ 0 } { r } \\cdot \\dfrac { x } { R } + \\beta \\varphi ( r ) = 0 . \\end{align*}"}
+{"id": "8333.png", "formula": "\\begin{align*} \\widetilde { W } _ t ^ { \\sharp } = \\left ( \\begin{array} { c | c | c } W _ t ^ { \\sharp } & 0 & - X _ t \\\\ \\hline 0 & 0 & - 1 \\\\ \\hline X _ t ^ { \\sharp } & 1 & 0 \\end{array} \\right ) . \\end{align*}"}
+{"id": "2267.png", "formula": "\\begin{align*} s _ i ( u ) : = s _ i + \\frac { 1 } { u } \\ , , \\ \\ \\ \\ \\ . \\end{align*}"}
+{"id": "3998.png", "formula": "\\begin{align*} \\varpi \\circ W \\left ( z \\right ) = \\Bigl ( \\sum _ { i = 1 } ^ { n } x _ { i } \\Bigr ) \\Bigl ( \\sum _ { j = 1 } ^ { \\nu } y _ { j } \\Bigr ) = 0 \\ ; \\Leftrightarrow \\ ; z \\in \\mathcal { O } ^ { \\ : n , \\nu } . \\end{align*}"}
+{"id": "4263.png", "formula": "\\begin{align*} u ( t ) - e ^ { i t \\Delta } u _ 0 = N ( t ) : = - i \\int _ 0 ^ t e ^ { i ( t - s ) \\Delta } [ ( 1 + a ) | u | ^ 2 u ] ( s ) \\ , d s . \\end{align*}"}
+{"id": "3784.png", "formula": "\\begin{align*} \\mathbb { P } \\left \\{ \\mathcal { M } _ i ^ { 1 , j } \\right \\} \\leq \\sum _ { t = 0 } ^ { T - 1 } \\mathbb { P } \\left \\{ f ^ { ( i ) } _ t V ^ { ( i ) } _ t \\leq \\bar { t } \\right \\} + \\mathbb { P } \\left \\{ g ^ { ( i ) } _ t V ^ { ( i ) } _ t > \\bar { t } \\right \\} , \\end{align*}"}
+{"id": "6456.png", "formula": "\\begin{align*} - \\langle R ^ N & ( V , d \\phi ( e _ j ) ) d \\phi ( e _ j ) , \\bar \\Delta ^ 2 V \\rangle - \\langle R ^ N ( \\bar \\Delta ^ 2 V , d \\phi ( e _ j ) ) d \\phi ( e _ j ) , V \\rangle \\\\ & = - 2 \\langle R ^ N ( V , d \\phi ( e _ j ) ) d \\phi ( e _ j ) , \\bar \\Delta ^ 2 V \\rangle \\end{align*}"}
+{"id": "2343.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow \\infty } \\| a _ { k } - a \\| _ { L _ { t } ^ { \\infty } L _ { x } ^ { p } \\cap L _ { t } ^ { \\frac { p } { \\alpha } } L _ { x } ^ { \\frac { 3 p } { 3 - 2 \\alpha } } } + \\| \\nabla | a _ { k } - a | ^ { \\frac { p } { 2 } } \\| ^ { \\frac { 2 } { p } } _ { L ^ { 2 } _ { t } L ^ { 2 } _ { x } } = 0 . \\end{align*}"}
+{"id": "6035.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb E [ j ^ A _ { x , x + 1 } ] & = \\frac { E _ { A } - E _ { B } } { N ^ \\gamma } \\rho ^ { A } \\rho ^ { B } + \\frac { E _ { A } - E _ { C } } { 2 N ^ \\gamma } \\Big ( 2 \\rho ^ A ( 1 - \\rho ^ { A } ) - 2 \\rho ^ { A } \\rho ^ { B } \\Big ) \\ , , \\\\ \\mathbb E [ j ^ B _ { x , x + 1 } ] & = \\frac { E _ { B } - E _ { A } } { N ^ \\gamma } \\rho ^ { A } \\rho ^ { B } + \\frac { E _ { B } - E _ { C } } { 2 N ^ \\gamma } \\Big ( 2 \\rho ^ B ( 1 - \\rho ^ { B } ) - 2 \\rho ^ { A } \\rho ^ { B } \\Big ) \\ , . \\end{aligned} \\end{align*}"}
+{"id": "1045.png", "formula": "\\begin{align*} \\tilde { w } ( e ^ { i \\theta } ) : = w ( e ^ { - i \\theta } ) . \\end{align*}"}
+{"id": "3142.png", "formula": "\\begin{align*} \\varphi ^ * \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) = \\left ( \\begin{array} { c c c } a ^ { 2 \\ , p ^ { e _ 1 } } & 0 & b ^ { 2 \\ , p ^ { e _ 1 } } \\\\ 0 & 1 & b ^ { p ^ { e _ 1 } } \\ , d ^ { p ^ { e _ 1 } } \\\\ 0 & 0 & d ^ { 2 \\ , p ^ { e _ 1 } } \\end{array} \\right ) ( \\ , e _ 1 \\geq 0 \\ , ) . \\end{align*}"}
+{"id": "5145.png", "formula": "\\begin{align*} \\frac { \\partial ( X . Y ) } { \\partial q _ { j } } = X \\frac { \\partial Y ( p _ { j } , q _ { j } ) } { \\partial q _ { j } } + Y \\frac { \\partial X ( p _ { j } , q _ { j } ) } { \\partial q _ { j } } = X \\frac { \\partial Y _ { j } } { \\partial q _ { j } } + Y \\frac { \\partial X _ { j } } { \\partial q _ { j } } \\end{align*}"}
+{"id": "68.png", "formula": "\\begin{align*} \\begin{aligned} S : = & w _ 1 G D _ 1 ( p - 2 + s ) + w _ 2 G D _ 2 ( p - 2 + s ) \\\\ & + \\epsilon w _ 3 G D _ 1 ( p - 4 + s ) + \\epsilon w _ 4 G D _ 2 ( p - 4 + s ) \\end{aligned} \\end{align*}"}
+{"id": "8409.png", "formula": "\\begin{align*} \\widetilde { B _ 2 } ( g ) \\widetilde { B _ 2 } ( h ) & = \\sum \\alpha _ i B _ 2 ( g _ i ) ^ { - 1 } B _ 2 ( h ) ^ { - 1 } = \\sum \\alpha _ i B _ 2 ( B _ 1 ( g _ i ) h B _ 2 ( g _ i ) ) ^ { - 1 } \\\\ & = \\sum \\alpha _ i B _ 2 ( B _ 1 ( g _ i ) h S ( B _ 2 ( g _ i ) ^ { - 1 } ) ) ^ { - 1 } = \\widetilde { B _ 2 } ( \\widetilde { B _ 1 } ( g _ 1 ) h S ( \\widetilde { B _ 2 } ( g _ 2 ) ) ) . \\end{align*}"}
+{"id": "628.png", "formula": "\\begin{align*} V ( C ) V ( A , B , C [ n - 2 ] ) & = \\frac { 2 } { n ( n - 1 ) } \\sum _ { i < j } V _ 2 ( \\pi _ { i , j } ( A ) , \\pi _ { i , j } ( B ) ) \\\\ & \\leq \\frac { 1 } { n ( n - 1 ) } \\sum _ { i < j } \\left ( | \\pi _ i ( A ) | \\pi _ j ( B ) | + | \\pi _ j ( A ) | | \\pi _ i ( B ) | \\right ) \\\\ & = \\frac { 1 } { n ( n - 1 ) } \\sum _ { i \\neq j } | \\pi _ i ( A ) | | \\pi _ j ( B ) | \\leq \\frac { 1 } { n ( n - 1 ) } \\left ( \\sum _ i | \\pi _ i ( A ) | \\right ) \\left ( \\sum _ j | \\pi _ j ( B ) | \\right ) \\\\ & = \\frac { n } { n - 1 } V ( A , C [ n - 1 ] ) V ( B , C [ n - 1 ] ) . \\end{align*}"}
+{"id": "8923.png", "formula": "\\begin{align*} \\rho ' ( s ) : = \\begin{cases} \\rho ( s ) & s < k - 1 \\\\ \\rho ( k - 1 ) & s = k - 1 \\end{cases} \\qquad ( s \\in \\Sigma _ { k } ) \\end{align*}"}
+{"id": "315.png", "formula": "\\begin{align*} \\Delta v \\geq - \\frac { K } { t } , K = \\frac { N } { N ( m - 1 ) + 2 } , \\end{align*}"}
+{"id": "928.png", "formula": "\\begin{align*} V ^ D ( u , \\eta ) = 2 \\int _ { D } \\int _ { \\mathbb R ^ d } ( u ( x ) - u ( y ) ) ( \\eta ( x ) - \\eta ( y ) ) j ( | x - y | ) \\ , d x \\ , d y . \\end{align*}"}
+{"id": "2328.png", "formula": "\\begin{align*} \\frac { 1 } { s _ { 1 } } + \\frac { \\alpha _ { 1 } + \\beta _ { 1 } } { n } = \\theta \\left ( \\frac { 1 } { s _ { 2 } } + \\frac { \\alpha _ { 2 } + \\beta _ { 2 } - 1 } { n } \\right ) + ( 1 - \\theta ) \\left ( \\frac { 1 } { s _ { 3 } } + \\frac { \\alpha _ { 3 } + \\beta _ { 3 } } { n } \\right ) , \\end{align*}"}
+{"id": "8257.png", "formula": "\\begin{align*} & H ( Y _ j | Y ^ { j - 1 } , \\underline S ) = \\\\ & \\sum _ { \\underline s } \\sum _ { y ^ { j - 1 } } P _ { Y ^ { j - 1 } , \\underline S } ( y ^ { j - 1 } , \\underline s ) H ( P _ { Y _ { j } | Y ^ { j - 1 } , \\underline S } ( 1 | y ^ { j - 1 } , \\underline s ) ) . \\end{align*}"}
+{"id": "1320.png", "formula": "\\begin{align*} \\mathrm { d } ( \\Phi _ X + f \\circ \\pi ) ^ { \\mathrm { v e r } } ( p ) & = X ( q ) , \\\\ \\mathrm { d } ( \\Phi _ X + f \\circ \\pi ) ^ { \\mathrm { h o r } } ( p ) & = \\langle p , \\nabla X \\rangle + \\mathrm { d } f ( q ) . \\end{align*}"}
+{"id": "7098.png", "formula": "\\begin{align*} ( \\partial _ t + \\Delta + b _ n \\cdot \\nabla ) u _ n = f , u _ n ( T , \\cdot ) = 0 , \\end{align*}"}
+{"id": "7623.png", "formula": "\\begin{align*} i ^ { d , m } _ { n , \\C } ( f ) ( \\alpha ) = \\begin{cases} [ F _ n ( f _ 1 ) ( \\alpha ) : F _ n ( f _ 2 ) ( \\alpha ) : \\cdots : F _ n ( f _ m ) ( \\alpha ) ] & \\mbox { i f } \\alpha \\in \\C \\\\ [ 1 : 1 : \\cdots : 1 ] & \\mbox { i f } \\alpha = \\infty \\end{cases} \\end{align*}"}
+{"id": "774.png", "formula": "\\begin{align*} ( \\C [ T ] \\otimes H ) ^ { \\Delta ( T _ 0 ) } = { } ^ { \\Delta ( T _ 0 ) } ( \\C [ T ] \\otimes H ) = \\bigoplus _ { \\substack { a , b , c : \\\\ q ( a b ) = 1 } } a \\otimes H _ { b , c } = \\bigoplus _ { \\substack { a , b , c : \\\\ q ( a c ) = 1 } } a \\otimes H _ { b , c } , \\end{align*}"}
+{"id": "2580.png", "formula": "\\begin{align*} X _ m X _ { m + n } = & q ^ { \\lfloor \\frac { n } { 2 } \\rfloor } X _ { \\lfloor m + \\frac { n } { 2 } \\rfloor } X _ { \\lceil m + \\frac { n } { 2 } \\rceil } + \\sum \\limits _ { k = 1 } ^ { n - 1 } ( \\sum \\limits _ { l = 1 } ^ { ( k , n - k ) } q ^ { - \\frac { 1 } { 2 } + l } ) h X _ { m + n - k } \\\\ & + \\sum \\limits _ { l = 1 } ^ { n - 1 } q ^ { - \\frac { n - 1 - l } { 2 } } c _ l F _ { n - 1 - l } ( X _ \\delta ) , \\end{align*}"}
+{"id": "7263.png", "formula": "\\begin{align*} ( w ; q ) _ n : = \\left \\{ \\begin{array} { l l } 1 & \\textrm { w h e n $ n = 0 $ } , \\\\ \\prod \\limits _ { j = 0 } ^ { n - 1 } ( 1 - w q ^ j ) & \\textrm { w h e n $ n = 1 , 2 , \\ldots $ } , \\\\ \\prod \\limits _ { j = 0 } ^ { \\infty } ( 1 - w q ^ j ) & \\textrm { w h e n $ n = \\infty $ } \\end{array} \\right . \\end{align*}"}
+{"id": "600.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { 3 2 } \\right ) ^ { k } \\binom { 2 k } { k } ^ { 2 } H _ { k } \\end{align*}"}
+{"id": "8436.png", "formula": "\\begin{align*} a _ n : = \\left \\lfloor \\frac { n + 1 } { 2 } - \\frac { \\sqrt { n + 1 } } { 2 } \\right \\rfloor . \\end{align*}"}
+{"id": "1651.png", "formula": "\\begin{align*} e ^ { 3 \\lambda \\left ( \\delta \\right ) \\left ( T + 1 \\right ) ^ { k } } \\delta ^ { 2 } = \\delta . \\end{align*}"}
+{"id": "1264.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ; q ^ 2 ) _ k ( - q ^ 2 ; q ^ 2 ) _ k } { ( q ^ 2 ; q ^ 2 ) _ k } q ^ { n ^ 2 - ( k + 1 ) ^ 2 } \\\\ [ 7 p t ] & = \\sum _ { \\substack { 0 \\le k \\le n - 1 \\\\ [ 3 p t ] k \\not = ( n - 1 ) / 2 } } c _ { n , k } + c _ { n , ( n - 1 ) / 2 } \\\\ [ 7 p t ] & \\equiv \\frac { 1 - ( - 1 ) ^ { \\frac { n - 1 } { 2 } } } { 2 } ( 1 - q ^ n ) + ( - 1 ) ^ { \\frac { n - 1 } { 2 } } \\left ( 1 + \\frac { ( 1 - n ) ( 1 - q ^ n ) } { 2 } \\right ) \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "3899.png", "formula": "\\begin{align*} \\limsup _ { T \\to \\infty } \\frac { \\sum _ { \\gamma \\in Q ( 0 , T ) } \\exp ( \\ell _ \\psi ( \\gamma ) ) } { \\sum _ { \\gamma \\in P ( 0 , T ) } \\exp ( \\ell _ \\psi ( \\gamma ) ) } = \\epsilon > 0 . \\end{align*}"}
+{"id": "3747.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\lambda } \\mathrm { B } _ { z ^ { 2 } } \\left ( \\lambda , 0 \\right ) = 2 \\ln z \\ , \\mathrm { B } _ { z ^ { 2 } } \\left ( \\lambda , 0 \\right ) - z ^ { 2 \\lambda } \\ , \\Phi \\left ( z ^ { 2 } , 2 , \\lambda \\right ) . \\end{align*}"}
+{"id": "4494.png", "formula": "\\begin{align*} M _ { D _ j } = \\int _ { D _ j } | F _ j | ^ 2 e ^ { - \\varphi _ j } \\end{align*}"}
+{"id": "6709.png", "formula": "\\begin{align*} x ^ n - 1 = \\prod _ { d | n } \\Phi _ { d } ( x ) . \\end{align*}"}
+{"id": "2223.png", "formula": "\\begin{align*} \\varepsilon _ 1 > { 3 \\over d } { d \\over d - 3 } \\varepsilon _ 2 = { 3 \\over d - 3 } \\varepsilon _ 2 . \\end{align*}"}
+{"id": "5175.png", "formula": "\\begin{align*} & U _ { j } = \\left [ \\frac { a - 1 } { a - b } A ^ { a - 2 } - \\frac { b - 1 } { a - b } A ^ { b - 2 } \\right ] p _ { j } q ^ { \\beta - 2 } _ { j } \\\\ & V _ { j } = \\left [ \\frac { a - 1 } { a - b } A ^ { a - 2 } - \\frac { b - 1 } { a - b } A ^ { b - 2 } \\right ] q ^ { \\beta - 1 } _ { j } \\end{align*}"}
+{"id": "4530.png", "formula": "\\begin{align*} \\left \\{ \\begin{matrix*} [ l ] \\partial _ t n _ N ^ { ( K ) } + \\displaystyle \\sum _ { i = 1 } ^ K \\partial _ { s _ i } n _ N ^ { ( K ) } + p _ K ( [ s ] _ K ) n _ N ^ { ( K ) } + E _ N ^ { ( K ) } ( t , [ s ] _ K ) = 0 , \\\\ [ 5 p t ] n _ N ^ { ( K ) } ( t , s _ 1 = 0 , [ s ] _ { 2 , K } ) = \\int _ { 0 } ^ \\infty \\big [ p _ K n _ N ^ { ( K ) } + E _ N ^ { ( K ) } \\big ] ( t , [ s ] _ { 2 , K } , u ) d u , \\end{matrix*} \\right . \\end{align*}"}
+{"id": "2241.png", "formula": "\\begin{align*} - \\rho \\ , F _ { 1 t } + \\tau F _ { 2 x } & = \\gamma \\rho v _ t & & \\mbox { ( c o e f f i c i e n t s i n $ \\d u $ ) } \\ , , \\\\ f _ 1 & = v _ t & & \\mbox { ( c o e f f i c i e n t s i n $ \\d v _ t $ ) } \\ , , \\\\ f _ 2 & = v _ x & & \\mbox { ( c o e f f i c i e n t s i n $ \\d v _ x $ ) } \\ , . \\end{align*}"}
+{"id": "607.png", "formula": "\\begin{align*} & \\frac { 4 } { \\pi } \\int _ { 0 } ^ { 1 } \\sum _ { n = 0 } ^ { \\infty } ( - 1 ) ^ { n } x ^ { 2 n } \\sqrt { 1 - x ^ { 2 } } \\binom { - \\frac { 1 } { 2 } } { n } f _ { n } \\ln \\left ( x \\right ) \\ , d x \\\\ & + \\frac { 1 } { 2 } \\sum _ { n = 0 } ^ { \\infty } \\left ( \\frac { 1 } { 1 6 } \\right ) ^ { n } \\frac { \\binom { 2 n } { n } ^ 2 ( 2 \\ln ( 2 ) ( n + 1 ) + 1 ) } { ( n + 1 ) ^ 2 } f _ { n } , \\end{align*}"}
+{"id": "2616.png", "formula": "\\begin{align*} | D _ { \\{ 1 , 2 , 3 , 5 \\} } ( S ) | = n + 4 . \\end{align*}"}
+{"id": "8100.png", "formula": "\\begin{align*} m = \\lceil \\log b \\rceil . \\end{align*}"}
+{"id": "2058.png", "formula": "\\begin{align*} W _ { \\Sigma } ( x , y ) \\coloneqq \\Sigma ( i , j ) \\ \\ \\lceil d x \\rceil = i , \\lceil p y \\rceil = j . \\end{align*}"}
+{"id": "170.png", "formula": "\\begin{align*} X \\ = \\ \\{ x \\in \\prod _ { i \\in \\N } \\ X _ i : f _ i ( x _ { i + 1 } ) \\ = \\ x _ i \\ \\ \\ \\ i \\in \\N \\} , \\end{align*}"}
+{"id": "8799.png", "formula": "\\begin{align*} \\psi _ \\rho ( \\alpha ) & = \\alpha + ( 1 - \\alpha ) e ^ { ( 2 - \\rho ) \\ln { \\frac { \\alpha } { 1 - \\alpha } } } + 1 - \\rho \\\\ & \\geq \\alpha + ( 1 - \\alpha ) \\left ( 1 + ( 2 - \\rho ) \\ln { \\frac { \\alpha } { 1 - \\alpha } } \\right ) + 1 - \\rho \\\\ & = ( 2 - \\rho ) \\left ( 1 + ( 1 - \\alpha ) \\ln { \\frac { \\alpha } { 1 - \\alpha } } \\right ) = ( 2 - \\rho ) h ( \\alpha ) , \\end{align*}"}
+{"id": "3280.png", "formula": "\\begin{align*} [ 2 ] \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q , q ^ { - 1 } ; q ^ 2 ) _ k q ^ { 2 k } } { ( q ^ 2 ; q ^ 2 ) _ k ^ 2 } & = \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ^ { 3 } , q ^ { - 1 } ; q ^ 2 ) _ k q ^ { 2 k } } { ( q ^ 2 ; q ^ 2 ) _ k ^ 2 } + q \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ; q ^ 2 ) _ k ^ 2 q ^ { 2 k } } { ( q ^ 2 ; q ^ 2 ) _ k ^ 2 } \\\\ [ 1 . 5 m m ] & \\equiv ( - 1 ) ^ { ( n + 1 ) / 2 } q ^ { ( n ^ 2 + 3 ) / 4 } + q ( - 1 ) ^ { ( n - 1 ) / 2 } q ^ { ( n ^ 2 - 1 ) / 4 } , \\end{align*}"}
+{"id": "4689.png", "formula": "\\begin{align*} \\mathrm { b M o b } _ k ( X ) : = \\overline { \\sum _ { X \\dashrightarrow X ' } \\overline { \\mathrm { M o b } _ k ( X , X ' ) } } . \\end{align*}"}
+{"id": "172.png", "formula": "\\begin{align*} ( y , z ) \\in S ^ { \\circ } \\Longleftrightarrow ( y _ i , z _ i ) \\in S _ i \\ \\ \\ \\ i = \\min \\{ j : y _ j \\not = z _ j \\} , \\end{align*}"}
+{"id": "7942.png", "formula": "\\begin{align*} \\frac { \\delta \\mathcal { F } } { \\delta v } = \\frac { \\delta _ { \\beta } \\mathcal { F } } { \\delta v } + \\frac { \\delta _ { \\phi } \\mathcal { F } } { \\delta v } , \\end{align*}"}
+{"id": "2727.png", "formula": "\\begin{align*} X _ { T } \\rightarrow { _ { * } X } _ { T } = { _ { * } X } _ { H } + \\zeta ^ { \\alpha } { _ { * } X } _ { { _ { * } \\phi } ^ { ( 1 ) } _ { \\alpha } } + \\frac { \\partial } { \\partial t } = { _ { * } X } _ { H _ { T } } + \\frac { \\partial } { \\partial t } , \\end{align*}"}
+{"id": "4456.png", "formula": "\\begin{align*} \\frac { G ( 0 ) } { \\int _ 0 ^ { + \\infty } c ( t ) e ^ { - t } d t } & = \\lim _ { t \\rightarrow + \\infty } e ^ t \\int _ { \\{ \\psi < - t \\} } | F | ^ 2 e ^ { - \\varphi } \\\\ & \\geq \\lim _ { t \\rightarrow + \\infty } e ^ { - c _ t } \\sum _ { \\alpha \\in E _ \\beta } e ^ { \\left ( 1 - \\sum _ { 1 \\le j \\le n } \\frac { \\alpha _ j + 1 } { p _ { j , \\beta _ j } } \\right ) t } \\frac { | d _ { \\alpha } | ^ 2 ( 2 \\pi ) ^ n } { \\Pi _ { 1 \\le j \\le n } ( \\alpha _ j + 1 ) } , \\end{align*}"}
+{"id": "6820.png", "formula": "\\begin{align*} \\sum _ { i , j } \\beta _ { i j } k _ { i j } ^ 2 + K _ 2 = 0 . \\end{align*}"}
+{"id": "9160.png", "formula": "\\begin{align*} G _ { \\beta , \\beta ' } = ( w _ { \\beta , 1 } - v ^ { 2 } w _ { \\beta ' , 1 } ) \\cdot G _ { [ 1 , n ] , \\beta ' } . \\end{align*}"}
+{"id": "3782.png", "formula": "\\begin{align*} \\mathbb { P } \\left \\{ \\mathcal { M } _ i ^ { 1 , j } \\right \\} \\leq \\mathbb { P } \\left \\{ \\sum _ { t = 0 } ^ { T - 1 } c ^ { ( i ) } _ t V ^ { ( i ) } _ t \\leq \\bar { t } \\right \\} + \\mathbb { P } \\left \\{ \\sum _ { t = 0 } ^ { T - 1 } d ^ { ( i ) } _ t V ^ { ( i ) } _ t > \\bar { t } \\right \\} , \\end{align*}"}
+{"id": "6447.png", "formula": "\\begin{align*} \\frac { \\partial \\phi _ t } { \\partial t } \\big | _ { t = 0 } = V . \\end{align*}"}
+{"id": "3526.png", "formula": "\\begin{align*} H _ 0 ^ { \\infty } ( \\mathbb { D } ) = \\{ f \\in H ^ { \\infty } ( \\mathbb { D } ) : f ( 0 ) = 0 \\} . \\end{align*}"}
+{"id": "6877.png", "formula": "\\begin{align*} H _ { { \\varphi } \\sf E } = \\sum _ { i \\in \\mathbb { J } ' } \\ , E _ i ' \\langle { e } _ i ' , \\cdot \\rangle { e } _ i ' , \\end{align*}"}
+{"id": "4133.png", "formula": "\\begin{align*} & \\bar { \\xi } _ \\ell ( \\theta ) = 0 , \\bar { \\xi } _ \\ell ( \\tilde { \\theta } ) = 0 , \\ell = 1 , \\ldots , j - 1 , \\\\ & \\bar { \\xi } _ { j } ( \\theta ) = ( 1 - w _ { j j } ) r ^ j \\eta _ j , \\\\ & \\bar { \\xi } _ { j } ( \\tilde { \\theta } ) = ( 1 - w _ { j j } ) r ^ { j + 1 / 2 } \\eta _ j . \\end{align*}"}
+{"id": "5781.png", "formula": "\\begin{align*} & \\frac { d } { d t } \\tilde { \\xi } _ { i , 1 } - 2 ^ { - 1 } m \\tilde \\xi _ { i , 1 } + \\beta _ i \\tilde \\xi _ { i , 2 } = \\tilde { \\mathcal { E } } _ { i , 1 } , \\\\ & \\frac { d } { d t } \\tilde \\xi _ { i , 2 } - 2 ^ { - 1 } m \\tilde \\xi _ { i , 2 } - \\beta _ i \\tilde \\xi _ { i , 1 } = \\tilde { \\mathcal { E } } _ { i , 2 } , \\end{align*}"}
+{"id": "8322.png", "formula": "\\begin{align*} { \\sf N } _ 2 ( Q _ 1 \\odot Q _ 2 ) = ( - 1 ) ^ { k } ( Q _ 1 ^ { \\sharp } \\wedge Q _ 2 ^ { \\sharp } ) \\xi , \\end{align*}"}
+{"id": "4758.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 ^ + } \\frac { \\norm { x - S ( t ) x } } { t } = \\vert A x \\vert \\end{align*}"}
+{"id": "3147.png", "formula": "\\begin{align*} \\varphi ^ * \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "801.png", "formula": "\\begin{align*} { E _ { i j } } : = \\frac { 1 } { 2 } { { \\bf S } _ { { y ^ i } { y ^ j } } } ( x , y ) : = \\frac { 1 } { 2 } { { \\bf S } _ { { . i } { . j } } } ( x , y ) . \\end{align*}"}
+{"id": "8036.png", "formula": "\\begin{align*} \\limsup _ { M \\rightarrow + \\infty } \\limsup _ { N \\rightarrow + \\infty } \\frac { N } { \\gamma ^ 2 _ N } \\log P \\left ( \\sup _ { 0 \\leq t \\leq T } | \\eta _ t ^ N ( \\vec { f } ) | > M , ( D _ 2 ^ N ) ^ c \\right ) = - \\infty , \\end{align*}"}
+{"id": "7480.png", "formula": "\\begin{align*} \\nabla ^ { g _ 0 ( t ) } R _ 1 [ \\Bar { h } ] _ { i j } : = \\nabla _ { k } ^ { g _ 0 ( t ) } \\left ( \\left ( ( g _ 0 ( t ) + \\Bar { h } ) ^ { k l } - g _ 0 ( t ) ^ { k l } \\right ) \\nabla ^ { g _ 0 ( t ) } _ { l } \\Bar { h } _ { i j } \\right ) , \\end{align*}"}
+{"id": "4384.png", "formula": "\\begin{gather*} \\alpha : = - 2 \\overline { u } _ i - \\Delta u _ i , \\ \\beta : = 1 , \\ \\gamma = - \\overline { u } _ i ^ 2 - \\overline { u } _ i \\Delta u _ i . \\ \\end{gather*}"}
+{"id": "6322.png", "formula": "\\begin{align*} K ( x , y ) = \\widetilde { K } ( x , y ) + n \\widehat { \\omega } _ { \\ell , \\lambda } ( 0 ) = - \\sum _ { p \\in \\pi \\mathbb { N } _ 0 ^ 3 \\backslash \\{ 0 \\} } n \\widehat \\omega _ { \\ell , \\lambda } ( p ) u _ p ( x ) u _ p ( y ) . \\end{align*}"}
+{"id": "3879.png", "formula": "\\begin{align*} H _ { k , x } \\doteq \\frac { 1 } { k ^ * + 1 } \\sum _ { j = 0 } ^ { k ^ * } h _ x ( \\hat M ( t _ { j + k } - ( t _ n - T ) ) ) . \\end{align*}"}
+{"id": "936.png", "formula": "\\begin{align*} \\langle \\mu , u \\rangle = \\int _ E u ( x ) \\ , \\mu ( d x ) \\end{align*}"}
+{"id": "5548.png", "formula": "\\begin{align*} D _ x : = \\sum _ { y \\in V _ 1 , z \\in V _ 2 } Q _ { x z } Q _ { y z } \\end{align*}"}
+{"id": "2468.png", "formula": "\\begin{align*} \\left . \\psi ' \\vphantom { \\big | } \\right | _ { 0 < \\phi < \\mu ^ { - 1 } , \\ , \\psi = 0 } = - p \\phi ( \\mu \\phi - 1 ) > 0 \\end{align*}"}
+{"id": "1490.png", "formula": "\\begin{align*} f ( m ) \\big ( f ( n ) ( m - n ) - f ( m + n ) ( m - n + d ) \\big ) = 0 . \\end{align*}"}
+{"id": "7154.png", "formula": "\\begin{align*} q ( k ) = \\Bigl ( k ^ \\mu \\varepsilon _ \\mu ( k , \\lambda ) \\Bigr ) \\Bigl ( k ^ \\nu \\varepsilon _ \\nu ( k , \\lambda ' ) \\Bigr ) \\eta _ { \\lambda \\lambda ' } = 0 \\ , \\ , . \\end{align*}"}
+{"id": "6434.png", "formula": "\\begin{align*} \\frac { n ^ { 1 / \\alpha _ 0 } } { X _ { \\frac { i - 2 } { n } } ^ { 1 / \\alpha _ 0 } } \\left ( \\Delta _ i ^ n X - \\Delta _ { i - 1 } ^ n X \\right ) = \\xi ^ 1 _ i + \\xi ^ 2 _ i \\end{align*}"}
+{"id": "3071.png", "formula": "\\begin{align*} H = \\sum _ { \\delta } u _ { \\delta } F _ 0 ^ { \\delta _ 0 } F _ 1 ^ { \\delta _ 1 } F _ 2 ^ { \\delta _ 2 } \\ldots F _ { g + 1 } ^ { \\delta _ { g + 1 } } , \\end{align*}"}
+{"id": "6086.png", "formula": "\\begin{align*} T = \\chi _ { 2 , 1 } L _ 2 \\chi _ { 2 , 2 } + \\chi _ { 2 , 2 } L _ 2 ^ * \\chi _ { 2 , 1 } , \\end{align*}"}
+{"id": "316.png", "formula": "\\begin{align*} ( P _ k ) \\ \\left \\{ \\begin{array} { l l } w _ t = \\Delta w ^ m + \\min \\{ ( 1 + | x | ) ^ { \\sigma } , k \\} f _ k ( w ) , & ( x , t ) \\in \\real ^ N \\times ( 0 , \\infty ) , \\\\ w ( x , 0 ) = u _ 0 ( x ) , & x \\in \\real ^ N \\end{array} \\right . \\end{align*}"}
+{"id": "3621.png", "formula": "\\begin{align*} \\alpha ^ 2 \\tau _ 0 \\tau _ 1 = - ( a + b ) . \\end{align*}"}
+{"id": "9006.png", "formula": "\\begin{align*} \\varphi \\big ( [ x , y ] \\big ) = \\frac { 1 } { 2 } \\big ( [ \\varphi ( x ) , y ] + [ x , \\varphi ( y ) ] \\big ) . \\end{align*}"}
+{"id": "9134.png", "formula": "\\begin{align*} \\begin{aligned} & g _ { 2 } = ( v ^ { 6 } + 1 ) x _ { 2 , 1 } x _ { 2 , 2 } x _ { 2 , 3 } + ( v ^ { 6 } + 1 ) x _ { 1 , 1 } x _ { 1 , 2 } ( x _ { 2 , 1 } + x _ { 2 , 2 } + x _ { 2 , 3 } ) \\\\ & \\ \\ \\ \\ \\ \\ - v ^ { 3 } ( x _ { 1 , 1 } + x _ { 1 , 2 } ) ( x _ { 1 , 1 } x _ { 1 , 2 } + x _ { 2 , 1 } x _ { 2 , 2 } + x _ { 2 , 1 } x _ { 2 , 3 } + x _ { 2 , 2 } x _ { 2 , 3 } ) . \\end{aligned} \\end{align*}"}
+{"id": "2895.png", "formula": "\\begin{align*} & \\lim _ { \\lambda \\to \\infty } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ { L ^ r _ \\omega ( \\mathbb { R } ^ n ) } = \\left [ \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { L ^ r _ \\omega ( \\mathbb { R } ^ n ) } ; \\end{align*}"}
+{"id": "3064.png", "formula": "\\begin{align*} v _ 0 = \\beta _ 0 = n , \\ \\ \\ v _ 1 = \\beta _ 1 \\ \\ \\ \\mbox { a n d } \\ \\ \\ v _ i = n _ { i - 1 } v _ { i - 1 } + \\beta _ i - \\beta _ { i - 1 } \\end{align*}"}
+{"id": "5477.png", "formula": "\\begin{align*} E = X _ 1 + X _ 2 - X _ 3 \\ll \\frac { 1 } { c } \\cdot ( N + | X _ 1 | ) . \\end{align*}"}
+{"id": "7691.png", "formula": "\\begin{align*} \\psi _ { L , N _ G , j } ^ { \\alpha } ( y ; t ) = \\left ( \\frac { L } { \\alpha - m } y ^ { - \\frac { m - \\alpha - 1 } { m - \\alpha } } \\right ) ^ { N _ G + 1 } \\C { F } _ j ^ { ( N _ G + m + 1 ) } \\left ( t - L \\ , y ^ { \\frac { 1 } { m - \\alpha } } \\right ) , \\end{align*}"}
+{"id": "5720.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\ell + 1 } \\Vert \\mathbf { L } ^ j ( v , w ) \\Vert _ { G } \\leq C \\Vert ( v , w ) \\Vert _ { H ^ { \\ell + 1 } \\times H ^ \\ell } . \\end{align*}"}
+{"id": "2885.png", "formula": "\\begin{align*} \\mathrm { H } : = \\sup _ { \\lambda \\in ( 0 , \\infty ) } \\lambda \\left \\| \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\gamma } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right \\| _ { X } . \\end{align*}"}
+{"id": "6315.png", "formula": "\\begin{align*} ( a ^ * ( g ) \\Psi ) ( x _ 1 , \\dots , x _ { n + 1 } ) & = \\frac { 1 } { \\sqrt { n + 1 } } \\sum _ { j = 1 } ^ { n + 1 } g ( x _ j ) \\Psi ( x _ 1 , \\dots , x _ { j - 1 } , x _ { j + 1 } , \\dots , x _ { n + 1 } ) , \\\\ ( a ( g ) \\Psi ) ( x _ 1 , \\dots , x _ { n - 1 } ) & = \\sqrt { n } \\int _ { \\R ^ 3 } \\overline { g ( x _ n ) } \\Psi ( x _ 1 , \\dots , x _ n ) \\dd x _ n , \\end{align*}"}
+{"id": "4601.png", "formula": "\\begin{align*} | J _ 3 | = \\sum _ { j \\in J _ 3 } A _ j ^ { - 1 } A _ j \\le \\Big ( \\sum _ { j \\in J _ 3 } A _ j ^ { - s ' } \\Big ) ^ { \\frac { 1 } { s ' } } \\Big ( \\sum _ { j \\in J _ 3 } A _ j ^ { s } \\Big ) ^ { \\frac { 1 } { s } } . \\end{align*}"}
+{"id": "5024.png", "formula": "\\begin{align*} \\max \\ , \\left \\{ \\mathbf { x } ^ 1 \\mathbf { 1 } \\colon \\begin{bmatrix} \\mathbf { x } ^ { 0 } & \\mathbf { x } ^ { 1 } \\end{bmatrix} \\begin{bmatrix} ( 1 - \\beta ) \\mathbf { I } \\\\ \\mathbf { I } - \\beta \\mathbf { P } \\end{bmatrix} = \\mathbf { e } _ i , \\begin{bmatrix} \\mathbf { x } ^ { 0 } & \\mathbf { x } ^ { 1 } \\end{bmatrix} \\geq \\mathbf { 0 } \\right \\} \\end{align*}"}
+{"id": "707.png", "formula": "\\begin{align*} f ( x ) = \\Phi ^ 4 ( x ) - 1 6 \\Phi ^ 3 ( x ) + 9 6 \\Phi ^ 2 ( x ) - 2 5 6 \\Phi ( x ) + 2 5 6 - m \\end{align*}"}
+{"id": "6836.png", "formula": "\\begin{align*} 1 & \\leq | \\cos ( \\omega _ { N - 1 } - \\omega _ { N - 1 } ' + \\xi ) | \\sin ^ 2 ( \\psi _ { N - 1 } ) + | \\cos ( ( N - 1 ) ( \\omega _ { N - 1 } - \\omega _ { N - 1 } ' ) ) | \\cos ^ 2 ( \\psi _ { N - 1 } ) \\\\ & \\leq \\sin ^ 2 ( \\psi _ { N - 1 } ) + \\cos ^ 2 ( \\psi _ { N - 1 } ) = 1 . \\end{align*}"}
+{"id": "6064.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k + 1 } \\lambda _ i ( L _ i ^ d + l _ i g _ i ) = 0 . \\end{align*}"}
+{"id": "5030.png", "formula": "\\begin{align*} \\mathbf { B } ^ \\emptyset = ( 1 - \\beta ) \\mathbf { I } , \\mathbf { N } ^ \\emptyset = \\mathbf { I } - \\beta \\mathbf { P } , \\mathbf { H } ^ \\emptyset = \\frac { 1 } { 1 - \\beta } \\mathbf { I } , \\mathbf { A } ^ \\emptyset = \\frac { 1 } { 1 - \\beta } ( \\mathbf { I } - \\beta \\mathbf { P } ) . \\end{align*}"}
+{"id": "5604.png", "formula": "\\begin{align*} \\sum _ { e \\in T _ { \\gamma } } \\tilde m _ e = 2 s k . \\end{align*}"}
+{"id": "763.png", "formula": "\\begin{align*} S \\big ( p ( s ) \\big ) = \\frac { 2 } { n - 1 } R i c _ { j k } \\frac { d { x } ^ j } { d s } \\frac { d { x } ^ k } { d s } = \\frac { 2 } { n - 1 } F ^ 2 R i c , \\end{align*}"}
+{"id": "717.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { k = 0 } ^ \\infty f _ k ( z ) \\end{align*}"}
+{"id": "6793.png", "formula": "\\begin{align*} & c _ 1 > 0 , \\ c _ 2 = - 1 , \\ c _ 3 > 0 , \\ \\ d _ 1 > 0 , \\ d _ 2 \\geq - 1 , \\ d _ 3 \\geq - 1 , \\ \\ \\frac { 1 } { 2 } \\left ( \\frac { d _ 3 } { c _ 3 } + \\frac { d _ 1 } { c _ 1 } \\right ) < \\frac { d _ 2 } { c _ 2 } < \\frac { d _ 1 } { c _ 1 } \\\\ & c _ 1 > 0 , \\ c _ 2 = - 1 , \\ c _ 3 = 0 , \\ \\ d _ 1 > 0 , \\ d _ 2 \\geq - 1 , \\ d _ 3 = - 1 , \\ \\ \\frac { d _ 2 } { c _ 2 } < \\frac { d _ 1 } { c _ 1 } . \\end{align*}"}
+{"id": "6279.png", "formula": "\\begin{align*} \\frac 1 T \\sum \\limits _ { k = 0 } ^ { T - 1 } \\langle g _ { k + 1 } , x _ k - x ^ * \\rangle \\leq \\frac { \\kappa } { \\kappa + 1 } \\frac { R _ { 0 } ^ { \\frac { 1 + \\kappa } { \\kappa } } } { \\nu T } + \\frac { \\nu ^ { \\kappa } } { 1 + \\kappa } \\frac { 1 } { T } \\sum \\limits _ { k = 0 } ^ { T - 1 } \\| g _ { k + 1 } \\| ^ { 1 + \\kappa } _ q , \\end{align*}"}
+{"id": "397.png", "formula": "\\begin{align*} D _ t u _ 0 = S ( t ) u _ 0 . \\end{align*}"}
+{"id": "6928.png", "formula": "\\begin{align*} a _ { - 1 } : = \\frac { \\alpha ( 1 + \\alpha ) ( 2 + \\alpha ) } { 6 } , & a _ 0 : = \\frac { ( 1 - \\alpha ^ 2 ) ( 2 + \\alpha ) } { 2 } , \\\\ a _ { 1 } : = - \\frac { \\alpha ( 1 - \\alpha ) ( 2 + \\alpha ) } { 2 } , & a _ 2 : = \\frac { \\alpha ( 1 - \\alpha ^ 2 ) } { 6 } . \\end{align*}"}
+{"id": "696.png", "formula": "\\begin{align*} D _ f = - 2 ^ { 2 4 } 3 ^ { 1 2 } m ^ { 1 1 } . \\end{align*}"}
+{"id": "5552.png", "formula": "\\begin{align*} \\mathrm { u n f o l d } _ { i } ( T ) : = \\mathrm { u n f o l d } _ { \\{ i \\} , [ k ] \\setminus \\{ i \\} } ( T ) = w ^ { ( i ) } ( w ^ { ( 1 ) } \\otimes \\cdots \\otimes w ^ { ( i - 1 ) } \\otimes w ^ { ( i + 1 ) } \\otimes \\cdots \\otimes w ^ { ( k ) } ) ^ \\top . \\end{align*}"}
+{"id": "6588.png", "formula": "\\begin{align*} \\int _ { b - i U } ^ { b + i U ^ { \\prime } } \\left ( \\sum _ { n \\leq x } f ( n ) ^ { - z } \\right ) \\frac { y ^ z } { z } d z = \\sum _ { n \\leq x } \\int _ { b - i U } ^ { b + i U ^ { \\prime } } f ( n ) ^ { - z } \\frac { y ^ z } { z } d z . \\end{align*}"}
+{"id": "9114.png", "formula": "\\begin{align*} \\partial _ { k \\ell } ( \\partial _ i u ^ j + \\partial _ j u ^ i ) + \\partial _ { i j } ( \\partial _ k u ^ \\ell + \\partial _ \\ell u ^ k ) = \\partial _ { k j } ( \\partial _ i u ^ \\ell + \\partial _ \\ell u ^ i ) + \\partial _ { i \\ell } ( \\partial _ k u ^ j + \\partial _ j u ^ k ) \\ ; , \\end{align*}"}
+{"id": "1874.png", "formula": "\\begin{align*} \\rho ( \\alpha ) , \\rho ( \\beta ) , \\rho ( \\gamma ) \\in V _ 4 , \\rho ( \\tau _ i ) \\in V _ 4 , \\ ; i = 1 , . . . , 5 , \\rho ( \\tau _ 6 ) = \\rho ( \\tau _ 7 ) = 1 \\end{align*}"}
+{"id": "1734.png", "formula": "\\begin{align*} \\widehat { F } ( z ) : = \\sum _ { j = 1 } ^ { n - 1 } \\left [ ( x _ 1 , \\ldots , x _ { n - 1 } ) \\widehat { Q } _ { j } \\left ( \\begin{array} { c } x _ 1 \\\\ \\vdots \\\\ x _ { n - 1 } \\end{array} \\right ) \\right ] x _ n ^ { j - 1 } , \\end{align*}"}
+{"id": "7276.png", "formula": "\\begin{align*} a = A \\sqrt { T ( t ) } , b = B \\sqrt { T ( t ) } , c = C / \\sqrt { T ( t ) } , d = D / \\sqrt { T ( t ) } \\end{align*}"}
+{"id": "6742.png", "formula": "\\begin{align*} P ( n , z , p ) = \\sum _ { k = 1 } ^ \\infty \\frac { ( - 1 ) ^ k \\zeta ( 2 k ) z ^ { 2 k } } { k ^ p ( 2 k + n ) } , 0 < | z | \\leq 1 , \\ , \\ , p \\geq 1 . \\end{align*}"}
+{"id": "8021.png", "formula": "\\begin{align*} & \\lim _ { N \\rightarrow + \\infty } \\frac { 1 } { N } \\log \\frac { d P } { d P ^ N _ { f _ 1 , f _ 2 , f _ 3 } } = \\int _ { \\mathbb { T } } \\Bigg ( \\sum _ { k = 1 } ^ 3 f _ k ( u ) \\log \\left ( \\frac { \\rho _ { k - 1 } ( u ) } { f _ k ( u ) } \\right ) \\\\ & + \\left ( 1 - \\sum _ { k = 1 } ^ 3 f _ k ( u ) \\right ) \\log \\left ( \\frac { 1 - \\sum _ { k = 1 } ^ 3 \\rho _ { k - 1 } ( u ) } { 1 - \\sum _ { k = 1 } ^ 3 f _ k ( u ) } \\right ) \\Bigg ) d u = - I _ { i n i } ( \\hat { W } _ 0 ) \\end{align*}"}
+{"id": "3685.png", "formula": "\\begin{align*} \\Psi _ { i , j } ( e ) = \\begin{cases} \\sqrt { 2 } ( n - d ) & e = \\{ i , j \\} , \\\\ - 1 & e = \\{ i , k \\} e = \\{ j , k \\} k \\in [ n ] \\setminus [ d ] , \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "646.png", "formula": "\\begin{align*} \\lim _ { d _ r \\to 0 } \\frac { d _ r } { \\rho _ - ^ { n - r } } = 1 . \\end{align*}"}
+{"id": "1692.png", "formula": "\\begin{align*} f ( 0 ) = 0 \\quad \\mbox { a n d } y y ^ { \\prime \\prime } = c \\left ( y ^ { \\prime } \\right ) ^ 2 \\quad \\mbox { w i t h $ y = f ^ { ( 4 ) } ( x ) $ f o r s o m e } c \\in \\mathbb { R } . \\end{align*}"}
+{"id": "3041.png", "formula": "\\begin{align*} \\lambda _ \\pm = \\frac { 1 } { 2 } \\left ( 1 \\pm \\sqrt { 1 - 2 \\lambda ^ 2 } \\right ) , \\gamma _ \\pm = \\frac { 1 } { 2 } ( 1 + \\lambda ^ 2 \\pm 3 \\sqrt { 1 - 2 \\lambda ^ 2 } ) \\ , . \\end{align*}"}
+{"id": "3298.png", "formula": "\\begin{align*} \\{ Y > Y ' \\} = \\bigcup _ { n , m = 1 } ^ \\infty \\{ Y \\ge Y ' + 2 ^ { - n } \\} \\cap \\{ Y ' \\le m \\} = \\bigcup _ { n , m = 1 } ^ \\infty A _ { n , m } . \\end{align*}"}
+{"id": "752.png", "formula": "\\begin{align*} g _ { i j } ( \\Phi _ k - \\varphi _ k \\Phi ) - g _ { i k } ( \\Phi _ j - \\varphi _ j \\Phi ) = - R ^ h _ { i j k } \\varphi _ h . \\end{align*}"}
+{"id": "5463.png", "formula": "\\begin{align*} \\mathcal { H } = \\{ h _ 1 , \\cdots , h _ c \\} , \\end{align*}"}
+{"id": "8995.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( \\sup _ { t \\leq \\tilde { T } } | x ^ n _ t - x _ { \\lambda ^ n ( t ) } | + \\sup _ { t \\leq \\tilde { T } } | \\lambda ^ n ( t ) - t | ) = 0 . \\end{align*}"}
+{"id": "2919.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } \\big [ W _ { \\ell } ( x _ i ; f ) ^ { r } \\big ] & \\ll _ r \\bigg ( \\sum _ { \\frac { x _ i } { 1 + X } < p \\leqslant x _ i } 1 + x _ i \\frac { ( \\log x _ i ) ^ { \\frac { ( 2 r - 2 ) ( 2 r - 1 ) } { r } } } { X ^ { \\frac { 1 } { 2 r } } } \\sum _ { p \\leqslant \\frac { x _ i } { 1 + X } } \\frac { 1 } { p } \\bigg ) ^ r \\\\ & \\ll _ r \\bigg ( \\frac { x _ i } { \\log x _ i } \\bigg ) ^ r . \\end{aligned} \\end{align*}"}
+{"id": "4617.png", "formula": "\\begin{align*} u _ { j , k } \\left ( x ' , x _ d \\right ) : = \\sin \\left ( M ^ { j \\gamma } \\left ( x _ d - c ( j ) \\right ) \\right ) 1 _ { \\Omega _ { j , k } } \\left ( x ' , x _ d \\right ) . \\end{align*}"}
+{"id": "8268.png", "formula": "\\begin{align*} K _ { X ^ q _ { i } } + B ^ q _ { i } + \\beta ^ q _ { i } = \\nu _ { i } ^ * ( K _ { X _ { i } } + B _ { i } + \\beta _ { i } ) . \\end{align*}"}
+{"id": "8104.png", "formula": "\\begin{align*} n ^ - _ { \\rho , k } = \\exp ( u _ k + s _ k ) \\end{align*}"}
+{"id": "3967.png", "formula": "\\begin{align*} ( [ \\pi ] ( T ) \\xi ) ( T ( x ) ) = \\pi ( x , T ( x ) ) \\xi ( x ) . \\end{align*}"}
+{"id": "6911.png", "formula": "\\begin{align*} E _ { i j } : = \\partial _ { x _ j } \\ ( \\left | \\nabla v \\right | ^ { p - 2 } \\partial _ { x _ i } v \\ ) - \\frac { 1 } { n } g \\delta _ { i j } , \\end{align*}"}
+{"id": "4408.png", "formula": "\\begin{gather*} \\mathcal { M } : = \\{ ( k _ 1 , k _ 2 , k _ 3 , k _ 4 ) \\in [ n ] ^ 4 : \\ k _ 1 < k _ 2 , k _ 3 < k _ 4 \\} \\cup \\{ ( 0 , 0 , 0 , 0 ) \\} . \\end{gather*}"}
+{"id": "6706.png", "formula": "\\begin{align*} \\mathbb { A P } _ p : = \\{ f : \\mathbb { R } \\rightarrow \\mathbb { R } , f ( x + p ) = - f ( x ) \\} . \\end{align*}"}
+{"id": "2349.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } z _ { k } - \\Delta z _ { k } = - \\mathrm { d i v } ( u ^ { c , \\gamma } \\otimes z _ { k - 1 } + z _ { k - 1 } \\otimes u ^ { c , \\gamma } + w _ { 1 } \\otimes w _ { 2 } ) - \\nabla \\pi _ { k - 1 } , \\\\ \\mathrm { d i v } z _ { k } = 0 , \\\\ \\pi _ { k - 1 } = ( - \\Delta ) ^ { - 1 } \\partial _ { i } \\partial _ { j } ( u ^ { c , \\gamma } \\otimes z _ { k - 1 } + z _ { k - 1 } \\otimes u ^ { c , \\gamma } + w _ { 1 } \\otimes w _ { 2 } ) , \\\\ z _ { k } ( x , 0 ) = 0 . \\end{cases} \\end{align*}"}
+{"id": "8806.png", "formula": "\\begin{align*} \\hat \\tau & = \\frac { z _ - - m } { z _ - - y _ + } \\delta _ { y _ + } + \\frac { m - y _ + } { z _ - - y _ + } \\delta _ { z _ - } , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\check \\tau = \\frac { z _ + - m } { z _ + - y _ - } \\delta _ { y _ - } + \\frac { m - y _ - } { z _ + - y _ - } \\delta _ { z _ + } \\\\ s & = \\pi _ { x _ - } ( \\{ y _ + \\} ) + \\pi _ { x _ - } ( \\{ z _ - \\} ) = \\overline \\pi _ { x _ - } ^ \\star ( \\{ y _ - \\} ) - \\pi _ { x _ - } ( \\{ y _ - \\} ) + \\overline \\pi _ { x _ - } ^ \\star ( \\{ z _ + \\} ) - \\pi _ { x _ - } ( \\{ z _ + \\} ) > 0 . \\end{align*}"}
+{"id": "7181.png", "formula": "\\begin{align*} F _ w ( x , t ) = g _ \\rho ( t ) * F _ w ( \\cdot , 0 ) + \\varepsilon \\int _ 0 ^ { t } e ^ { - \\varepsilon \\gamma ( t - \\tau ) } g _ \\rho ( t - \\tau ) * \\left ( F _ v + J _ 0 \\right ) d \\tau . \\end{align*}"}
+{"id": "1887.png", "formula": "\\begin{align*} | \\cdot | ^ 2 \\widehat { ( | \\cdot | ^ p ) } = - p ^ 2 \\widehat { ( | \\cdot | ^ { p - 2 } ) } , \\end{align*}"}
+{"id": "8461.png", "formula": "\\begin{align*} ( \\bar { u } _ h ) _ \\pm ^ { ( \\ell ) } ( x , t ) : = ( u _ m ( x ) ) _ \\pm ^ { ( \\ell ) } , \\end{align*}"}
+{"id": "678.png", "formula": "\\begin{align*} k _ { t } ( x , y ) : = \\frac { 1 } { ( 4 \\pi t ) ^ { \\frac { n } { 2 } } } e ^ { - \\frac { | x - y | ^ { 2 } } { 4 t } } . \\end{align*}"}
+{"id": "8725.png", "formula": "\\begin{align*} \\psi _ 1 ( A , Q _ { i } ) = \\frac { \\Psi ( A , Q _ i ) } { \\psi _ 2 ( A , Q _ { i } ) } < \\frac { \\frac { \\psi _ 2 ( Q , Q _ { i } ) } { 2 } \\phi ( H _ { i + 1 } ) } { \\frac { \\psi _ 2 ( Q , Q _ { i } ) } { 2 } } = \\phi ( H _ { i + 1 } ) . \\end{align*}"}
+{"id": "3714.png", "formula": "\\begin{align*} ( H - E ) | \\psi \\rangle = | \\chi \\rangle \\ ; , \\end{align*}"}
+{"id": "8848.png", "formula": "\\begin{align*} \\mathcal { M E } [ u _ 0 ] : = \\frac { \\mathcal E [ u _ 0 ] } { \\mathcal E [ \\varphi ] } , \\mathcal { M G } [ u _ 0 ] : = \\frac { \\| \\sqrt { \\mathcal K _ \\lambda } u _ 0 \\| } { \\| \\sqrt { \\mathcal K _ \\lambda } \\varphi \\| } , \\mathcal { M P } [ u _ 0 ] : = \\frac { \\mathcal P [ u _ 0 ] } { \\mathcal P [ \\varphi ] } . \\end{align*}"}
+{"id": "5453.png", "formula": "\\begin{align*} \\tau _ n = \\inf \\{ \\tau _ { n - 1 } < t \\leq T ; \\ , Y _ t \\in \\partial D \\} , \\mbox { f o r } n \\geq 1 , \\tau _ 0 = 0 , \\end{align*}"}
+{"id": "2796.png", "formula": "\\begin{align*} \\delta _ { { \\textrm { e f f e c t i v e } } } \\left ( { \\tilde { \\sigma } _ { 3 } ^ { * } } ( t ) I \\right ) : = 0 \\end{align*}"}
+{"id": "6115.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\boldsymbol u ' ( t ) & = ( ( I _ 2 \\otimes A _ 1 ) + ( A _ 2 \\otimes I _ 1 ) ) \\boldsymbol u ( t ) + \\mathrm { v e c } ( C + \\boldsymbol U ( t ) B \\boldsymbol U ( t ) ) , \\\\ \\boldsymbol u ( 0 ) & = \\mathrm { v e c } ( \\boldsymbol U _ 0 ) , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4424.png", "formula": "\\begin{gather*} V = H _ 0 ^ 1 ( \\Omega ) , H = L _ 2 ( \\Omega ) \\quad V ^ * = H ^ { - 1 } ( \\Omega ) \\ , . \\end{gather*}"}
+{"id": "6075.png", "formula": "\\begin{align*} \\lambda _ 0 = \\pm \\frac { 2 \\omega _ { 2 , 1 } \\omega _ { 2 , 2 } } { \\sqrt { 2 \\omega _ { 2 , 2 } ^ 2 / \\phi _ { 2 , 2 } ^ 2 - 1 } } . \\end{align*}"}
+{"id": "3217.png", "formula": "\\begin{align*} \\partial _ t ^ \\alpha u ( t ) + t ^ { \\beta } A u ( t ) = t ^ { \\mu } f , t > 0 ; \\end{align*}"}
+{"id": "2857.png", "formula": "\\begin{align*} [ \\omega ] _ { A _ 1 ( \\mathbb { R } ^ n ) } : = \\sup _ { Q \\subset \\mathbb { R } ^ n } \\frac { \\| \\omega ^ { - 1 } \\| _ { L ^ \\infty ( Q ) } } { | Q | } \\int _ Q \\omega ( x ) \\ , d x < \\infty , \\end{align*}"}
+{"id": "8913.png", "formula": "\\begin{align*} \\frac { 1 } { \\Phi _ n ( 1 ) } W _ n ( x ) = 1 + 2 \\sum _ { m = 1 } ^ \\infty \\frac { B _ { 2 m } } { ( 2 m ) ! } \\omega _ m ( n ) x ^ { 2 m } , \\end{align*}"}
+{"id": "5361.png", "formula": "\\begin{align*} \\mathbf { w } ^ { S } - \\mathbf { w } ^ { S \\setminus \\{ j \\} } = \\beta \\ , \\left ( \\mathbf { P } ^ 1 - \\mathbf { P } ^ 0 \\right ) \\ , \\left ( \\mathbf { b } ^ { S } - \\mathbf { b } ^ { S \\setminus \\{ j \\} } \\right ) . \\end{align*}"}
+{"id": "6853.png", "formula": "\\begin{align*} P ( X = k ) = \\frac { 1 } { Z ( \\lambda , \\nu ) } \\frac { \\lambda ^ k } { ( k ! ) ^ \\nu } . \\end{align*}"}
+{"id": "7953.png", "formula": "\\begin{align*} \\ast v _ { \\Sigma } = \\ast i _ { \\mathcal { N } } v _ { \\Omega } | _ { \\Sigma } = \\ast \\ast ( \\mathcal { N } ^ { \\flat } \\wedge \\ast v _ { \\Omega } ) | _ { \\Sigma } = ( - 1 ) ^ { n - 1 } \\mathcal { N } ^ { \\flat } , \\end{align*}"}
+{"id": "2540.png", "formula": "\\begin{align*} \\dot \\AA ( M ) = \\bigcup _ s \\dot \\AA ^ { ( s ) } ( M ) \\ ; , \\quad \\AA ( M ) = \\bigcup _ s \\AA ^ { ( s ) } ( M ) \\ ; . \\end{align*}"}
+{"id": "9145.png", "formula": "\\begin{align*} o ( x ^ { ( * , * ) } _ { * , * } ) = q . \\end{align*}"}
+{"id": "5096.png", "formula": "\\begin{align*} \\pi ( \\mathsf { c } _ { w } ) = c _ { w } . \\end{align*}"}
+{"id": "3774.png", "formula": "\\begin{align*} | h _ n | = | \\hat { g } ( n ) | \\leq \\tilde { P } ( n ) = \\frac { \\alpha _ n } { 1 + n } , \\end{align*}"}
+{"id": "6483.png", "formula": "\\begin{align*} x _ { u , v } = ( y _ u , \\varpi _ { i _ v } + w _ v \\varpi _ { i _ v } ) = 0 . \\end{align*}"}
+{"id": "7788.png", "formula": "\\begin{align*} \\rho ( K , x ) = \\frac { - 1 } { h ( K ^ \\star , x ) } \\quad \\mbox { f o r } x \\in { \\rm i n t } \\ , C . \\end{align*}"}
+{"id": "6855.png", "formula": "\\begin{align*} e ^ { \\mu G ( x ) } = \\sum _ { k = 0 } ^ \\infty \\frac { ( \\lambda x ) ^ k } { ( k ! ) ^ \\nu } . \\end{align*}"}
+{"id": "4813.png", "formula": "\\begin{align*} & \\min \\Big ( \\lambda \\varepsilon _ K + \\frac { 1 } { K } \\sum _ { k = 1 } ^ K \\sum _ { a \\in \\mathcal { A } } \\Big ( u _ a - \\lambda ( u _ a - \\hat { c } _ a ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } } ) \\Big ) x _ a \\Big ) \\\\ \\mbox { s . t . } & 0 \\leq \\lambda \\leq 1 . \\end{align*}"}
+{"id": "6949.png", "formula": "\\begin{align*} \\mathrm { d i m } \\ : L + \\mathrm { d i m } \\ : L ' = \\mathrm { d i m } ( L \\cap L ' ) + \\mathrm { d i m } ( \\langle L \\cup L ' \\rangle ) . \\end{align*}"}
+{"id": "7490.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { 2 } r _ s ^ { 2 + j } \\left | ( \\nabla ^ g ) ^ j R m ( g ) \\right | _ g \\leq C . \\end{align*}"}
+{"id": "3180.png", "formula": "\\begin{align*} \\prod \\limits _ { k = 0 } ^ { n } ( 1 + q ^ { 3 k + 1 } ) ( 1 + q ^ { 3 k + 2 } ) \\end{align*}"}
+{"id": "2024.png", "formula": "\\begin{align*} \\nabla v ^ q _ \\delta = q v ^ { q - 1 } _ \\delta \\nabla v _ \\delta . \\end{align*}"}
+{"id": "5469.png", "formula": "\\begin{align*} \\mathcal J _ K = \\mathcal J _ K ( a , N ) = \\int _ { U ( N ) } \\left | \\frac { P ' } { P } \\Big ( 1 - \\frac { a } { N } \\Big ) \\right | ^ { 2 K } , \\end{align*}"}
+{"id": "6942.png", "formula": "\\begin{align*} \\ker f _ { v , w } | _ { U _ v } = \\ker f _ { v , w } | _ { V _ v } \\oplus \\ker f _ { v , w } | _ { f _ { u , v } ( U _ u ) } . \\end{align*}"}
+{"id": "3970.png", "formula": "\\begin{align*} \\norm { \\rho ( x , y ) \\xi ' ( x ) - \\xi ' ( y ) } & = \\norm { \\rho ( x , y ) \\rho ( \\theta x , x ) \\xi ( \\theta x ) - \\rho ( \\theta y , y ) \\xi ( \\theta y ) } \\\\ & = \\norm { \\rho ( \\theta x , y ) \\xi ( \\theta x ) - \\rho ( \\theta y , y ) \\xi ( \\theta y ) } \\\\ & = \\norm { \\rho ( y , \\theta y ) \\rho ( \\theta x , y ) \\xi ( \\theta x ) - \\xi ( \\theta y ) } \\\\ & = \\norm { \\rho ( \\theta x , \\theta y ) \\xi ( \\theta x ) - \\xi ( \\theta y ) } \\end{align*}"}
+{"id": "7220.png", "formula": "\\begin{align*} \\int _ { V _ { \\min } } ^ { V _ { F } } \\left ( \\partial _ t p \\phi - h p \\partial _ v \\phi + a \\partial _ v p \\partial _ v \\phi \\right ) d v + a \\partial _ v p ( V _ F ) \\left ( \\phi ( V _ R ) - \\phi ( V _ F ) \\right ) = 0 . \\end{align*}"}
+{"id": "3140.png", "formula": "\\begin{align*} u _ { \\psi ^ * } ( t ) & = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & 1 & t ^ { p ^ { e _ 1 } } \\\\ 0 & 0 & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "7515.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { 2 } r _ M ^ { j + 2 } | ( \\nabla ^ { g ( t ) } ) ^ j R m ( g ( t ) ) | _ { g ( t ) } \\leq C _ M \\end{align*}"}
+{"id": "2094.png", "formula": "\\begin{align*} C _ { d , T } ( t ) & = \\big ( \\frac { t } { T } \\big ) ^ 2 \\gamma ^ 4 T ^ { - 2 } \\sigma ^ { - 4 } O ( 1 + d ^ { - 4 } t ^ 2 + t ^ 2 T ^ { - 4 } + C _ 1 d ^ { - 4 } t ^ { - 1 } + \\sigma ^ { - 2 } d ^ { - 4 } ) . \\end{align*}"}
+{"id": "988.png", "formula": "\\begin{align*} \\mathbb E _ x \\big [ ( \\mathbf 1 _ { D ^ c } g ) ( X _ { \\tau _ { D } } ) \\mathbf 1 _ { \\{ X _ { \\tau _ D - } \\in D \\} } \\mathbf 1 _ { \\{ \\tau _ { V _ n } = \\tau _ D \\} } \\big ] \\to \\mathbb E _ x \\big [ ( \\mathbf 1 _ { D ^ c } g ) ( X _ { \\tau _ { D } } ) \\mathbf 1 _ { \\{ X _ { \\tau _ D - } \\in D \\} } \\big ] \\end{align*}"}
+{"id": "9124.png", "formula": "\\begin{align*} \\sum _ { i \\in I } k _ i \\alpha _ i \\ , = \\sum _ { \\beta \\in \\Delta ^ { + } } d _ { \\beta } \\beta . \\end{align*}"}
+{"id": "6435.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 2 } ^ n \\left ( A _ i - \\delta _ 0 ^ { 1 / 2 } n ^ { 1 / 2 \\alpha _ 0 } | \\Delta _ i ^ n L - \\Delta _ { i - 1 } ^ n L | ^ { 1 / 2 } \\right ) = \\frac { 1 } { n } \\sum _ { i = 2 } ^ n ( | \\xi ^ 1 _ i + \\xi ^ 2 _ i | ^ { 1 / 2 } - | \\xi ^ 1 _ i | ^ { 1 / 2 } ) . \\end{align*}"}
+{"id": "7675.png", "formula": "\\begin{align*} \\abs { G _ { L } ( m , k ; z ) - G ^ { \\alpha } _ { L } ( m , k ; z ) } = \\lambda \\abs { \\alpha - U ( n _ 0 ) } \\abs { G _ { L } ( m , n _ 0 ; z ) } { \\abs { G ^ { \\alpha } _ { L } ( n _ 0 , k ; z ) } } \\end{align*}"}
+{"id": "2001.png", "formula": "\\begin{align*} \\widetilde { ( \\eta ^ { n , \\pm } ) } _ l & = p _ l ^ { \\pm } ( \\widetilde { ( f ( \\psi ( t _ n ) ) ) } _ l - \\widetilde { ( f ( \\psi ^ n ) ) } _ l ) + q _ l ^ { \\pm } ( \\widetilde { ( g ( \\psi ( t _ n ) ) ) } _ l - \\widetilde { ( g ( \\psi ^ n ) ) } _ l ) , \\end{align*}"}
+{"id": "1191.png", "formula": "\\begin{align*} | \\mathrm { I I } | & \\lesssim \\int _ { | x - y | < 4 | h | } | x - y | ^ { - n - | \\alpha | } | x - y | ^ { | \\alpha | + 1 } \\ , d y \\\\ & = \\int _ { | x - y | < 4 | h | } | x - y | ^ { - n + 1 } \\ , d y \\sim \\int _ 0 ^ { 4 | h | } t ^ { - n + 1 } t ^ { n - 1 } \\ , d t \\sim | h | . \\end{align*}"}
+{"id": "7071.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\nabla u \\circ d W _ s = \\int _ 0 ^ t \\nabla u d W _ s - \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ d [ \\partial _ { x _ k } u , W ^ k ] _ t , W _ t = ( W _ t ^ k ) _ { k = 1 } ^ d , \\end{align*}"}
+{"id": "1773.png", "formula": "\\begin{align*} \\Psi ( M _ { f ( x ) } ) = M _ { { f ( x - x _ 1 ^ * ) } } \\quad \\mbox { a n d } \\Psi ( z ^ * ) = ( f ( - x _ 1 ^ * ) , 0 , \\ldots , 0 ) . \\end{align*}"}
+{"id": "6283.png", "formula": "\\begin{align*} a _ { q } < a _ { q _ 0 } & = \\sqrt { d ^ { \\frac 2 q _ 0 - 1 } ( 2 q _ 0 - 1 ) } \\leq d ^ { \\frac { 1 } { \\ln d } - \\frac 1 2 } \\sqrt { 4 \\ln d - 1 } \\\\ & = \\frac { e } { \\sqrt { d } } \\sqrt { 4 \\ln d - 1 } \\leq d ^ { \\frac 1 q - \\frac 1 2 } \\sqrt { 3 2 \\ln d - 8 } , \\end{align*}"}
+{"id": "67.png", "formula": "\\begin{align*} \\Sigma = \\{ \\sigma \\in C ^ { 1 } ( \\R ) \\ , : \\ , \\sigma , \\ , \\sigma ( 0 ) = \\sigma ' ( 0 ) = 0 , \\sigma ( r ) > 0 r \\neq 0 \\} . \\end{align*}"}
+{"id": "4788.png", "formula": "\\begin{align*} u _ \\star ( x ) = \\sup \\{ t \\leq 0 : r _ t < \\vert x + r _ t \\cos \\theta E _ n \\vert \\} , \\end{align*}"}
+{"id": "5916.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { C } ( A ( n , \\nu ) ) ) = ( 2 ^ { n } - 1 ) \\times \\dfrac { 2 ^ { n } ( 2 ^ { n } - 1 ) ^ { 3 } } { 2 } = 2 ^ { n - 1 } ( 2 ^ { n } - 1 ) ^ { 4 } . \\end{align*}"}
+{"id": "7853.png", "formula": "\\begin{align*} \\mathbf { s } _ { f } = ( \\omega _ { p } ^ { f _ { 0 } } , \\omega _ { p } ^ { f _ { 1 } } , \\ldots , \\omega _ { p } ^ { f _ { p ^ m - 1 } } ) , \\end{align*}"}
+{"id": "3392.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { t } \\eta _ { i } \\left \\Vert \\nabla f ( x _ { i } ) \\right \\Vert ^ { 2 } + \\Delta _ { t + 1 } & \\le \\left ( \\sqrt { \\Delta _ { 1 } } + 2 \\sqrt { A } C _ { 1 } \\right ) ^ { 2 } \\end{align*}"}
+{"id": "8585.png", "formula": "\\begin{align*} d _ N ( X , Y ) = \\min _ { \\pi \\in S _ N } \\sum _ { i = 1 } ^ N d ( x _ i , y _ { \\pi ( i ) } ) \\ : , \\end{align*}"}
+{"id": "2510.png", "formula": "\\begin{align*} C ^ { - \\infty } _ { { \\cdot } / } ( M ; E ) = C ^ \\infty _ { / { \\cdot } } ( M ; E ^ * \\otimes \\Omega ) ' \\ ; . \\end{align*}"}
+{"id": "7054.png", "formula": "\\begin{align*} S _ 1 = - \\frac { 1 } { q } \\frac { d } { d t } \\| w \\| _ q ^ q + B _ q - \\langle | \\partial _ t u | ^ 2 , | w | ^ { q - 2 } \\rangle - ( q - 2 ) \\langle | w | ^ { q - 3 } w \\cdot \\nabla | w | , \\partial _ t u \\rangle , \\end{align*}"}
+{"id": "1879.png", "formula": "\\begin{align*} \\mathcal { H } ^ { 2 n } = \\bigoplus _ { k = 0 } ^ \\infty \\mathcal { H } _ k ^ { 2 n } , \\end{align*}"}
+{"id": "6819.png", "formula": "\\begin{align*} d g _ { S U ( N ) } = T ( \\psi _ 2 , \\ldots , \\psi _ { N - 1 } ) d g _ { S U ( 2 ) } d \\psi _ 2 d \\psi _ 3 d \\phi _ 3 \\ldots d \\psi _ { N - 1 } d \\phi _ { N - 1 } d \\omega _ { N - 1 } d g _ { S U ( N - 1 ) } , \\end{align*}"}
+{"id": "3937.png", "formula": "\\begin{align*} T _ n ^ { - 1 } = \\left \\{ \\begin{array} { l l } I _ 3 - \\lambda _ n \\begin{pmatrix} 0 & \\kappa & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} , & 1 \\leq n \\leq N , \\\\ I _ 3 , & n \\geq N + 1 \\end{array} \\right . . \\end{align*}"}
+{"id": "5384.png", "formula": "\\begin{align*} \\nu _ 0 & = \\frac { \\Delta h _ 1 } { \\alpha + \\Delta d _ 1 } \\\\ \\nu _ j & = \\nu _ { j - 1 } + \\frac { \\displaystyle { \\Delta h _ { j + 1 } - \\nu _ { j - 1 } \\ , ( \\alpha + \\Delta d _ { j + 1 } } ) } { \\displaystyle { \\alpha + \\Delta d _ { j + 1 } + \\frac { w ^ { S _ { j + 1 } } _ { j - 1 } } { \\rho _ { j - 1 } } } } , 1 \\leq j \\leq n - 1 . \\end{align*}"}
+{"id": "1708.png", "formula": "\\begin{align*} V : = \\sum _ { j = 1 } ^ n v _ j X _ j \\end{align*}"}
+{"id": "206.png", "formula": "\\begin{align*} \\frac { d x ^ i } { d t } = X ^ i ( x ^ 1 , \\ldots , x ^ n ) , i = 1 , \\ldots , n , \\end{align*}"}
+{"id": "6222.png", "formula": "\\begin{align*} \\begin{cases} \\dot z ( \\phi ) = h ( \\phi ) - c - \\frac { Q ( \\phi ) } { z ( \\phi ) } , \\ & \\phi \\in ( \\sigma _ 1 , \\sigma _ 2 ) , \\\\ z ( \\phi ) < 0 , \\ & \\phi \\in ( \\sigma _ 1 , \\sigma _ 2 ) . \\end{cases} \\end{align*}"}
+{"id": "6854.png", "formula": "\\begin{align*} G ( x ) = \\sum _ { r = 1 } ^ \\infty P ( Y _ 1 = r ) x ^ r , \\end{align*}"}
+{"id": "8727.png", "formula": "\\begin{align*} p _ 1 + p _ 2 + \\ldots + p _ { d - 1 } - p _ d = w _ 1 + \\ldots + w _ { d - 1 } - w _ d + 2 \\lambda . \\end{align*}"}
+{"id": "3418.png", "formula": "\\begin{align*} I _ { x _ 0 } ( \\phi ) = g \\phi j u ^ { - 1 } \\end{align*}"}
+{"id": "4879.png", "formula": "\\begin{align*} \\dim ( | K _ { \\Sigma ' } + C ' | ) = g - 1 - q = q - 2 + b . \\end{align*}"}
+{"id": "8301.png", "formula": "\\begin{align*} ( L y ) ( t ) : = y ' ( t ) + A y ( t ) = f ( t ) , t \\in [ a , b ] , \\end{align*}"}
+{"id": "7794.png", "formula": "\\begin{align*} \\begin{aligned} K ^ { ( i + 1 ) } & = - ( R + D ^ { \\top } P ^ { ( i + 1 ) } D ) ^ { - 1 } ( B ^ { \\top } P ^ { ( i + 1 ) } + D ^ { \\top } P ^ { ( i + 1 ) } C + S ) , \\end{aligned} \\end{align*}"}
+{"id": "8623.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } C _ 1 = \\| u _ 0 ^ \\prime \\| _ { L ^ \\infty ( \\mathbb { R } ) } > C _ 2 > 1 , \\end{align*}"}
+{"id": "7367.png", "formula": "\\begin{align*} u _ { t x } ^ 0 ( x , t ) = \\frac { 1 } { 2 } \\{ f '' ( x + t ) - f '' ( x - t ) + g ' ( x + t ) + g ' ( x - t ) \\} . \\end{align*}"}
+{"id": "5177.png", "formula": "\\begin{align*} & \\frac { \\partial A } { \\partial q _ { j } } = \\frac { \\partial A _ { j } } { \\partial q _ { j } } = q ^ { \\beta - 1 } _ { j } \\\\ & \\frac { \\partial B } { \\partial q _ { j } } = \\frac { \\partial B _ { j } } { \\partial q _ { j } } = p _ { j } q ^ { \\beta - 2 } _ { j } \\end{align*}"}
+{"id": "4328.png", "formula": "\\begin{gather*} \\sup _ { z _ i \\in \\mathcal { U } _ i - \\overline { u } _ i } f _ i ( x , z _ i ) = \\Delta u _ i x _ i \\end{gather*}"}
+{"id": "2603.png", "formula": "\\begin{align*} \\begin{cases} - \\varphi '' = f ( \\varphi ) & ( - R , + R ) , \\\\ \\dfrac { \\partial \\varphi } { \\partial \\nu } + \\beta \\varphi = 0 & \\ ; x = \\pm R . \\end{cases} \\end{align*}"}
+{"id": "7211.png", "formula": "\\begin{align*} N ( t ) = - a ( N ( t ) ) \\frac { \\partial p } { \\partial v } ( V _ F , t ) . \\end{align*}"}
+{"id": "3838.png", "formula": "\\begin{align*} P ( X _ { n + 1 } = y \\mid \\mathcal { F } _ n ) \\doteq G ( L ^ { n + 1 } ) _ { X _ n , y } , \\ ; \\ ; y \\in \\Delta ^ o , \\end{align*}"}
+{"id": "4109.png", "formula": "\\begin{align*} & \\gamma _ i ( \\theta _ i ) = e ^ { \\theta _ i } - 1 , \\\\ & \\xi _ i ( \\theta ) = P _ { i 0 } \\big ( e ^ { - \\theta _ i } - 1 \\big ) + \\sum _ { j \\in \\cal { J } } P _ { i j } \\big ( e ^ { \\theta _ j - \\theta _ i } - 1 \\big ) , \\\\ & Q ( \\theta ) = \\sum _ { i \\in \\mathcal { J } } \\bigl ( \\alpha _ i \\gamma _ i ( \\theta _ i ) + \\lambda _ i \\xi _ i ( \\theta ) \\bigr ) . \\end{align*}"}
+{"id": "567.png", "formula": "\\begin{align*} & \\phi ^ s _ n + \\alpha \\phi ^ s = 0 & \\Gamma _ e \\end{align*}"}
+{"id": "3225.png", "formula": "\\begin{align*} R f _ * C _ X = \\bigoplus _ n { } ^ p \\ ! R ^ { n } f _ * C _ X \\end{align*}"}
+{"id": "2031.png", "formula": "\\begin{align*} \\lambda ( x ) : = \\frac { 2 n } { \\xi ( d ^ { \\Sigma } ( x , o ) ) } . \\end{align*}"}
+{"id": "1136.png", "formula": "\\begin{align*} \\vec t _ Q : = \\begin{cases} \\vec e & Q = Q _ { 0 , \\mathbf { 0 } } , \\\\ \\vec { \\mathbf { 0 } } & . \\end{cases} \\end{align*}"}
+{"id": "4582.png", "formula": "\\begin{align*} \\int _ M t s _ g d \\mu = \\omega ( F ) \\left ( \\omega ( c _ + ) - \\omega ( c _ - ) \\right ) , \\end{align*}"}
+{"id": "8575.png", "formula": "\\begin{align*} C _ { t ' } ^ { t _ 1 } = ( J _ k + [ - \\varepsilon _ k ( G ^ { t _ 0 } _ { t ' } ) B _ { t _ 0 } ] ^ { k \\cdot } _ + ) C ^ { t _ 0 } _ { t ' } , \\end{align*}"}
+{"id": "50.png", "formula": "\\begin{align*} a _ { h , l } ( \\xi ) : = \\frac { ( - 1 + e ^ { - 2 \\pi i h \\xi _ l } ) } h , 1 \\leq l \\leq n . \\end{align*}"}
+{"id": "4228.png", "formula": "\\begin{align*} { \\widehat { A } ( T X , \\nabla ^ { T X } ) } { \\rm c h } ( 2 ^ k \\Theta _ 2 ( T _ { C } X ) ) = \\prod _ { j = 1 } ^ { k } \\frac { 2 x _ j \\theta ' ( 0 , \\tau ) } { \\theta ( x _ j , \\tau ) } \\frac { \\theta _ 2 ( x _ j , \\tau ) } { \\theta _ 2 ( 0 , \\tau ) } , \\end{align*}"}
+{"id": "7508.png", "formula": "\\begin{align*} Z _ { s } ^ k : = Z \\backslash \\phi _ k \\left ( ( 0 , s ^ { 1 / 4 } ] \\times X \\times [ - \\eta / 2 , \\eta / 2 ] \\right ) \\end{align*}"}
+{"id": "7570.png", "formula": "\\begin{align*} \\| \\Delta u \\| _ { 2 } ^ { 2 } + \\omega \\| u \\| _ { 2 } ^ { 2 } - \\mu \\| u \\| _ { q } ^ { q } - \\| u \\| _ { p } ^ { p } = 0 . \\end{align*}"}
+{"id": "6843.png", "formula": "\\begin{align*} \\mathbf { 1 } _ N = \\begin{pmatrix} F _ N ( \\phi _ N , \\ldots , \\omega _ { N - 2 } ) & 0 \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} e ^ { \\frac { - i \\xi } { N - 1 } } X & \\\\ & 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "7217.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial v } p ( V _ { \\min } , t ) = 0 . \\end{align*}"}
+{"id": "7805.png", "formula": "\\begin{align*} p _ { k } = \\frac { a _ { k } E _ { k } } { L } , \\forall k \\in \\mathcal { K } , \\end{align*}"}
+{"id": "5940.png", "formula": "\\begin{align*} | e ( \\mathcal { C } ( G ) ) | & = \\dfrac { ( 2 ^ { k } + 1 ) ( 2 ^ { k } - 1 ) ( 2 ^ { k } - 2 ) } { 2 } + \\dfrac { 2 ^ { k - 1 } ( 2 ^ { k } + 1 ) ( 2 ^ { k } - 2 ) ( 2 ^ { k } - 3 ) } { 2 } + \\dfrac { 2 ^ { k - 1 } ( 2 ^ { k } - 1 ) 2 ^ { k } ( 2 ^ { k } - 1 ) } { 2 } \\\\ & = \\dfrac { 2 ^ { 4 k } - 2 \\cdot 2 ^ { 3 k } - 2 \\cdot 2 ^ { 2 k } + 3 \\cdot 2 ^ { k } + 2 } { 2 } . \\end{align*}"}
+{"id": "2579.png", "formula": "\\begin{align*} X _ m F _ { n } ( X _ \\delta ) = q ^ { - \\frac { n } { 2 } } X _ { m - n } + q ^ { \\frac { n } { 2 } } X _ { m + n } + \\sum \\limits _ { k = 1 } ^ { n } ( \\sum \\limits _ { l = 1 } ^ { k } q ^ { - \\frac { k + 1 } { 2 } + l } ) h F _ { n - k } ( X _ \\delta ) . \\end{align*}"}
+{"id": "89.png", "formula": "\\begin{align*} w _ 1 = \\big ( \\sqrt { p - 1 } - \\sqrt { 1 - \\gamma } \\big ) ^ 2 + \\eta . \\end{align*}"}
+{"id": "368.png", "formula": "\\begin{align*} ( x - a ) ^ p = ( y - b ) ^ p = \\big ( z - ( a + b ) \\big ) ^ p \\equiv 0 \\pmod { p } \\end{align*}"}
+{"id": "753.png", "formula": "\\begin{align*} \\frac { 1 } { n - 1 } ( g _ { i j } R ^ h _ { k } - g _ { i k } R ^ h _ { j } ) \\varphi _ h = R ^ h _ { i j k } \\varphi _ h . \\end{align*}"}
+{"id": "4468.png", "formula": "\\begin{align*} \\int _ { D _ 1 } f _ 1 \\overline { f } \\lambda _ 1 = 0 \\end{align*}"}
+{"id": "1631.png", "formula": "\\begin{align*} u \\left ( x , T \\right ) = u _ { T } \\left ( x \\right ) , m \\left ( x , T \\right ) = m _ { T } \\left ( x \\right ) , x \\in \\Omega . \\end{align*}"}
+{"id": "8114.png", "formula": "\\begin{align*} y _ \\ell = g _ { s _ { \\rho , \\ell } } p _ \\ell ^ + x . \\end{align*}"}
+{"id": "4768.png", "formula": "\\begin{align*} \\varphi _ 3 ( \\varepsilon , b , \\eta , n , \\varphi ) & : = \\min \\{ \\varphi _ 1 ( \\varepsilon / 3 , b , n , \\varphi ) , \\varphi _ 1 ( \\eta ( \\min \\{ \\varepsilon / 3 b , 2 \\} ) \\varepsilon / 4 , b , n , \\varphi ) , \\\\ & \\qquad \\qquad \\qquad \\varphi _ 1 ( \\varepsilon / 4 , b , n , \\varphi ) , \\lambda _ 0 / 2 \\} \\end{align*}"}
+{"id": "7255.png", "formula": "\\begin{align*} 1 + \\eta \\gamma _ t [ n ] _ q + \\eta \\beta _ t [ n ] _ q ^ 2 + x \\eta q ^ n = 0 , \\end{align*}"}
+{"id": "1502.png", "formula": "\\begin{align*} \\mathcal { A } _ { \\mu , \\varepsilon } ( u ) : = - \\Delta _ { p } u + \\mu \\mathcal { B } u + \\Pi _ { \\delta , \\varepsilon } \\circ \\mathcal { T } ( u ) - \\mathcal { N } _ { f } \\circ \\mathcal { T } ( u ) = 0 W ^ { - 1 , p ^ { \\prime } } ( \\Omega ) . \\end{align*}"}
+{"id": "7370.png", "formula": "\\begin{align*} \\int _ { \\R } g ( x ) d x = 0 . \\end{align*}"}
+{"id": "7418.png", "formula": "\\begin{align*} b ' _ 2 & = b _ 2 c _ 2 \\\\ b ' _ 3 & = b _ 2 c _ 3 + b _ 3 c _ 2 + ( a _ 3 - d _ 3 ) a _ 4 . \\\\ \\end{align*}"}
+{"id": "1098.png", "formula": "\\begin{align*} \\int _ 0 ^ r t ^ { a + n - 1 } [ \\log ( 2 + t ) ] ^ b \\ , d t & > [ \\log ( 2 + r ) ] ^ b \\int _ 0 ^ r t ^ { a + n - 1 } \\ , d t \\\\ & = \\frac { 1 } { a + n } r ^ { a + n } [ \\log ( 2 + r ) ] ^ b . \\end{align*}"}
+{"id": "5209.png", "formula": "\\begin{align*} D _ { \\alpha \\beta } I ( p \\| q ) = T \\ ; \\left [ \\underbrace { \\sum _ { i } p ^ { \\alpha + \\beta - 1 } _ { i } } _ { A ( p , q ) } - \\underbrace { \\left ( \\sum _ { i } p ^ { \\alpha } _ { i } q ^ { \\beta - 1 } _ { i } \\right ) ^ { \\frac { \\alpha + \\beta - 1 } { \\alpha } } } _ { X ( p , q ) } \\times \\underbrace { \\left ( \\sum _ { i } q ^ { \\alpha + \\beta - 1 } _ { i } \\right ) ^ { \\frac { 1 - \\beta } { \\alpha } } } _ { Y ( p , q ) } \\right ] \\end{align*}"}
+{"id": "997.png", "formula": "\\begin{align*} u ( x ) & = \\int _ { \\partial _ m D } M _ D ( x , y ) \\ , \\nu ( d y ) + \\int _ { ( D \\cup \\partial _ m D ) ^ c } P _ D ( x , y ) \\ , \\gamma ( d y ) \\\\ & \\quad + \\int _ D f ( y , u ( y ) ) G _ D ( x , y ) \\ , d y + \\int _ D G _ D ( x , y ) \\ , \\mu ( d y ) x \\in D . \\end{align*}"}
+{"id": "6816.png", "formula": "\\begin{align*} \\sum _ { i , j } \\beta _ { i j } k _ { i j } ^ 1 + K _ 1 = 0 . \\end{align*}"}
+{"id": "7386.png", "formula": "\\begin{align*} \\begin{array} { l l } \\d C _ 1 : = \\frac { 1 } { 2 } \\min & \\left [ \\{ 2 ^ { p + q } A C ( 5 N ) ^ { p + q } N ^ { - 1 } \\} ^ { - 1 / ( p + q ) } , \\right . \\\\ & \\quad \\{ 2 ^ { p + q } A E \\| u _ t ^ 0 \\| _ \\infty ^ p ( 5 N ) ^ q N ^ { - 1 } \\} ^ { - 1 / q } , \\\\ & \\quad \\{ 2 ^ { p + q } A E \\| u ^ 0 \\| _ \\infty ^ q ( 5 N ) ^ p N ^ { - 1 } \\} ^ { - 1 / p } , \\\\ & \\left . \\quad \\{ 2 ^ r B C ( 5 N ) ^ r N ^ { - 1 } \\} ^ { - 1 / ( r + 1 ) } \\right ] , \\end{array} \\end{align*}"}
+{"id": "8294.png", "formula": "\\begin{align*} a _ { 1 1 } = \\frac { w ^ 2 + u u _ { 1 1 } } { w ^ 3 } \\leq \\frac { 1 } { w } \\left ( 1 - \\frac { u \\varphi ' \\ln w } { 2 } \\right ) \\leq - \\frac { \\ln w } { 4 w } . \\end{align*}"}
+{"id": "9064.png", "formula": "\\begin{align*} a = b + b _ { 1 2 } + b _ { 2 1 } , \\end{align*}"}
+{"id": "7014.png", "formula": "\\begin{align*} X _ t = x - \\int _ 0 ^ t b ( X _ s ) d s + Z _ t , t \\geq 0 , x \\in \\mathbb R ^ d , \\end{align*}"}
+{"id": "3027.png", "formula": "\\begin{align*} H = p _ 1 ^ 2 + p _ 2 ^ 2 + p _ 3 ^ 2 + \\frac { k _ 1 ^ 2 } { s _ 1 ^ 2 } + \\frac { k _ 2 ^ 2 } { s _ 2 ^ 2 } + \\frac { k _ 3 ^ 2 } { s _ 3 ^ 2 } , s _ 1 ^ 2 + s _ 2 ^ 2 + s _ 3 ^ 2 = 1 \\ , , \\end{align*}"}
+{"id": "8843.png", "formula": "\\begin{align*} \\mathcal E [ u ( t ) ] : = \\int _ { \\mathbb { R } ^ n } \\Big ( | \\nabla u | ^ 2 + \\lambda | x | ^ { - 2 } | u | ^ 2 \\Big ) \\ , d x + \\frac { \\epsilon } { p } \\mathcal P [ u ( t ) ] = \\mathcal E [ u _ 0 ] , \\end{align*}"}
+{"id": "6903.png", "formula": "\\begin{align*} | M ( z _ { 0 } ) - 1 | ^ { 2 } & = \\left | \\cosh e ^ { i t / 2 } - 1 + \\frac { k e ^ { i t / 2 } \\sinh e ^ { i t / 2 } } { 2 ( \\eta \\cosh e ^ { i t / 2 } + \\gamma ) } \\right | ^ { 2 } \\\\ & = \\frac { A _ { N } ( \\tau ) } { A _ { D } ( \\tau ) } ( - \\pi / 2 \\leq \\tau = t / 2 \\leq \\pi / 2 ) , \\end{align*}"}
+{"id": "8662.png", "formula": "\\begin{align*} b _ 0 ( z , z ) = 2 ^ { d } \\frac { 1 } { V _ { { \\rm e f f \\ , } } ( x ) } \\pi ^ { - n + \\frac { d } { 2 } } \\abs { \\det R _ x } ^ { - \\frac { 1 } { 2 } } \\abs { \\det \\mathcal { L } _ { x } } , \\ \\ \\forall x \\in \\mu ^ { - 1 } ( 0 ) \\cap D , \\end{align*}"}
+{"id": "2670.png", "formula": "\\begin{align*} K ^ { ( 2 ) } = 0 , E ^ { ( 2 ) } = 0 . \\end{align*}"}
+{"id": "6861.png", "formula": "\\begin{align*} \\ell _ 2 ( \\mathbb { J } ) = { \\mathcal R } ( \\theta _ \\varphi ) \\oplus { \\mathcal R } ( \\theta _ \\psi ) , \\end{align*}"}
+{"id": "4701.png", "formula": "\\begin{align*} ( h _ \\nu ^ 2 ) ' ( x ) = 2 x \\cdot \\sum _ { \\kappa = 1 } ^ { \\nu } ( - 1 ) ^ { \\nu - \\kappa } \\cdot h _ { \\kappa - 1 } ^ 2 ( x ) . \\end{align*}"}
+{"id": "2347.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } z - \\Delta z + ( z \\cdot \\nabla ) u ^ { c , \\gamma } + ( u ^ { c , \\gamma } \\cdot \\nabla ) z + \\nabla \\pi _ { 2 } = - \\mathrm { d i v } ( w _ { 1 } \\otimes w _ { 2 } ) , \\\\ \\mathrm { d i v } z = 0 , \\\\ z ( x , 0 ) = 0 . \\end{cases} \\end{align*}"}
+{"id": "8494.png", "formula": "\\begin{align*} \\vartheta = \\partial _ t ( | u | ^ { q - 1 } u ) \\textrm { i n } \\ , \\ , L ^ 2 ( \\Omega _ \\infty ) . \\end{align*}"}
+{"id": "6320.png", "formula": "\\begin{align*} U ( \\Psi ) = \\bigoplus _ { j = 0 } ^ n \\frac { 1 } { \\sqrt { ( n - j ) ! } } Q ^ { \\otimes j } a _ 0 ^ { n - j } \\Psi . \\end{align*}"}
+{"id": "1866.png", "formula": "\\begin{align*} \\alpha : ( x _ 1 , . . . , x _ 7 ) & \\longmapsto ( x _ 1 , x _ 2 , x _ 3 , - x _ 4 , - x _ 5 , - x _ 6 , - x _ 7 ) \\\\ \\beta : ( x _ 1 , . . . , x _ 7 ) & \\longmapsto ( x _ 1 , - x _ 2 , - x _ 3 , x _ 4 , x _ 5 , \\frac { 1 } { 2 } - x _ 6 , - x _ 7 ) \\\\ \\gamma : ( x _ 1 , . . . , x _ 7 ) & \\longmapsto ( - x _ 1 , x _ 2 , - x _ 3 , x _ 4 , \\frac { 1 } { 2 } - x _ 5 , x _ 6 , \\frac { 1 } { 2 } - x _ 7 ) \\\\ \\end{align*}"}
+{"id": "6083.png", "formula": "\\begin{align*} \\frac { f ( \\lambda ) } { 2 \\phi _ { 2 , 1 } \\phi _ { 2 , 2 } } = { \\rm s g n } ( a ) \\sqrt { a ^ 2 - 1 } , \\lambda \\in \\mathbb { T } _ - \\cup \\mathbb { T } _ + , \\end{align*}"}
+{"id": "3224.png", "formula": "\\begin{align*} \\mathfrak { h } ( X ) = \\bigoplus _ n \\mathfrak { h } ( B , \\mathfrak { h } ^ n ( X / B ) ) \\ , \\end{align*}"}
+{"id": "6512.png", "formula": "\\begin{align*} S ^ { \\mu } _ K ( E ) = \\int _ { N _ K ^ { - 1 } ( E ) } \\phi ( x ) d \\mathcal { H } ^ { n - 1 } ( x ) \\end{align*}"}
+{"id": "7376.png", "formula": "\\begin{align*} \\begin{array} { l } | w _ j | ^ p | u _ j | ^ q - | w _ { j - 1 } | ^ p | u _ { j - 1 } | ^ q \\\\ = ( | w _ j | ^ p - | w _ { j - 1 } | ^ p ) | u _ j | ^ q + | w _ { j - 1 } | ^ p ( | u _ j | ^ q - | u _ { j - 1 } | ^ q ) \\end{array} \\end{align*}"}
+{"id": "535.png", "formula": "\\begin{align*} \\psi _ k ( t ) : = \\mathrm { e } ^ { \\lambda _ k T / 2 } \\eta _ k ( t - T / 2 ) , t \\in [ 0 , T ] , \\ , k \\geq 1 , \\end{align*}"}
+{"id": "1522.png", "formula": "\\begin{align*} d = \\frac { 1 } { 6 } \\left ( p + \\sqrt { p ^ 2 + 1 2 \\ , p - 1 2 } \\right ) = \\frac { 1 } { 6 \\ , H } \\left ( H + 1 + \\sqrt { H ^ 2 + 1 4 \\ , H + 1 } \\right ) . \\end{align*}"}
+{"id": "9.png", "formula": "\\begin{align*} I _ + ( f ) : = \\lim _ { y \\to \\infty } 2 \\pi y f ( i y ) = \\lim _ { y \\to \\infty } \\bigl \\langle \\chi _ y , f \\bigr \\rangle = \\lim _ { \\xi \\downarrow 0 } \\sqrt { { 2 \\pi } } \\widehat f ( \\xi ) \\ \\ \\chi _ y ( x ) = \\tfrac { i y } { x + i y } . \\end{align*}"}
+{"id": "5449.png", "formula": "\\begin{align*} \\nu _ { ( 0 , i ) } ^ * ( c , d ) = \\max _ { S \\subseteq N \\colon i \\in S } \\frac { f _ i ^ S - c - \\big \\{ 1 - ( 1 - \\beta ) g _ i ^ S \\big \\} d } { g _ i ^ S } = ( 1 - \\beta ) d + \\max _ { S \\subseteq N \\colon i \\in S } \\frac { f _ i ^ S - ( c + d ) } { g _ i ^ S } , \\end{align*}"}
+{"id": "111.png", "formula": "\\begin{align*} \\begin{aligned} \\langle A _ \\lambda \\vec { u } , \\vec { v } \\rangle & : = a _ \\lambda ( \\vec { u } , \\vec { v } ) , \\\\ \\langle B \\vec { v } , q \\rangle & : = b ( \\vec { v } , q ) = ( \\operatorname { d i v } \\vec { v } , q ) . \\end{aligned} \\end{align*}"}
+{"id": "6088.png", "formula": "\\begin{align*} s ( w ) = \\begin{cases} 1 & w = 0 , \\\\ ( q - 1 ) q ^ { w - 1 } & 1 \\leq w \\leq q - 1 , \\\\ ( q - 1 ) ( q ^ { w - 1 } - 1 ) & w = q , \\\\ ( q - 1 ) \\sum _ { i = 1 } ^ { q - 1 } s ( w - i ) + s ( w - q ) & w > q . \\end{cases} \\end{align*}"}
+{"id": "8429.png", "formula": "\\begin{align*} B ( G ) _ x = & \\cdots = B ( G ) _ { x _ { n } } = B ( G ) _ { x _ { n + 1 } } \\cup B ( G ) _ { s _ n x _ { n } } , \\\\ B ( M ) _ { \\tilde x } = & \\cdots = B ( M ) _ { \\tilde x _ { n } } . \\end{align*}"}
+{"id": "8851.png", "formula": "\\begin{align*} \\big \\| | x | ^ { - \\tau - 1 } | u | ^ { p - 1 } | I _ \\alpha \\ast | \\cdot | ^ { - \\tau } | u | ^ p | \\big \\| _ { L _ x ^ { \\frac { 2 n } { n + 2 } } } & \\lesssim \\| | x | ^ { - \\tau - 1 } | u | ^ { p - 1 } \\| _ { L _ x ^ { a _ 1 } } \\| | x | ^ { - \\tau } | u | ^ p \\| _ { L _ x ^ { b _ 1 } } \\\\ & = \\| | x | ^ { - \\frac { \\tau + 1 } { p - 1 } } u \\| ^ { p - 1 } _ { L _ x ^ { ( p - 1 ) a _ 1 } } \\| | x | ^ { - \\frac { \\tau } { p } } u \\| ^ { p } _ { L _ x ^ { p b _ 1 } } \\\\ & \\lesssim \\| \\nabla u \\| _ { L _ x ^ r } ^ { 2 p - 1 } , \\end{align*}"}
+{"id": "3555.png", "formula": "\\begin{align*} ( z - \\tau ( \\psi ( z ) ) ( z ^ 2 + ( \\tau ( \\psi ( z ) ) ^ 2 + z \\tau ( \\psi ( z ) ) ) = 0 . \\end{align*}"}
+{"id": "5780.png", "formula": "\\begin{align*} & \\tilde { \\mathcal { E } } _ { i , 1 } ( t ) : = G ( \\tilde { \\mathcal { E } } ( t ) , \\psi _ { i , 1 } ) , \\ \\tilde { \\mathcal { E } } _ { i , 2 } ( t ) : = G ( \\tilde { \\mathcal { E } } ( t ) , \\psi _ { i , 2 } ) \\ \\textup { f o r } \\ i \\in I _ 1 , \\\\ & \\tilde { \\mathcal { E } } _ { i } ( t ) : = G ( \\tilde { \\mathcal { E } } ( t ) , \\Psi _ { i } ) \\ \\textup { f o r } \\ i \\in \\mathbb { Z } \\setminus \\{ 0 \\} . \\end{align*}"}
+{"id": "8006.png", "formula": "\\begin{align*} \\mathcal { J } _ 2 ( \\varpi , f _ 1 , f _ 2 , f _ 3 ) = & \\sum _ { k = 1 } ^ 3 \\varpi _ k ( f _ k ) \\\\ & - \\frac { 1 } { 2 } \\int _ { \\mathbb { T } } \\sum _ { k = 1 } ^ 3 \\rho _ { k - 1 } ( u ) f _ k ^ 2 ( u ) - \\left ( \\sum _ { k = 1 } ^ 3 f _ k ( u ) \\rho _ { k - 1 } ( u ) \\right ) ^ 2 d u \\end{align*}"}
+{"id": "5822.png", "formula": "\\begin{align*} e _ A : = \\frac { \\partial } { \\partial y ^ A } - ( g _ u ) ^ { i j } \\left ( c _ { i A } + \\frac { \\partial u ^ B } { \\partial x ^ i } h _ { B A } \\right ) \\left ( \\frac { \\partial } { \\partial x ^ j } + \\frac { \\partial u ^ C } { \\partial x ^ j } \\frac { \\partial } { \\partial y ^ C } \\right ) , \\ 1 \\leq A \\leq k \\end{align*}"}
+{"id": "489.png", "formula": "\\begin{align*} f ( x ) = \\omega ^ 4 x ^ 9 = \\omega ^ { 4 + 9 \\cdot 4 + 9 \\cdot 5 k } = \\omega ^ { 0 + 5 \\cdot ( 9 k + 8 ) } , \\end{align*}"}
+{"id": "6190.png", "formula": "\\begin{align*} \\begin{aligned} H ^ k & = Q ^ k ( N ^ k ) ^ { - 1 } ( P ^ k ) ^ T = \\frac { 1 } { \\gamma } P ^ k ( P ^ k ) ^ T = \\frac { 1 } { \\gamma } \\begin{pmatrix} \\tau ^ k \\beta ^ k J J ^ T & 0 \\\\ 0 & \\frac { 1 } { \\tau ^ k \\beta ^ k } I _ l \\end{pmatrix} , \\\\ G ^ k & = ( Q ^ k ) ^ T + Q ^ k - ( M ^ k ) ^ T H ^ k M ^ k \\overset { ( \\ast ) } { \\succeq } ( \\frac { 1 } { \\gamma ^ 2 } - \\frac { 1 } { \\gamma } ) ( N ^ k ) ^ T N ^ k , \\end{aligned} \\end{align*}"}
+{"id": "9096.png", "formula": "\\begin{align*} \\widehat { | \\mathcal N ^ { ( r ) } | } : = \\max _ { \\tau = 0 , \\ldots , n } \\{ \\hat F _ n ( \\tau ) \\geq \\tau \\} . \\end{align*}"}
+{"id": "4309.png", "formula": "\\begin{align*} \\| u \\| _ { m M } = \\| I d _ { \\ell _ \\infty ( \\C ) } \\otimes u : \\ell _ \\infty ( \\C ) \\otimes _ { \\min } E \\to \\ell _ \\infty ( \\C ) \\otimes _ { \\max } B \\| . \\end{align*}"}
+{"id": "9011.png", "formula": "\\begin{align*} e _ { i j } a e _ { k l } = \\begin{cases} a ( j , k ) e _ { i l } , & j \\le k , \\\\ 0 , & j > k . \\end{cases} \\end{align*}"}
+{"id": "596.png", "formula": "\\begin{align*} \\| B T A - D \\| = \\| A ^ { - 1 } P T A - D \\| \\le 2 \\eta . \\end{align*}"}
+{"id": "3939.png", "formula": "\\begin{align*} J _ n & = T _ n Q _ 1 T _ n ^ { - 1 } , \\\\ G _ n & = \\begin{pmatrix} \\lambda _ n ( d _ 2 - d _ 1 ) & \\lambda _ n ^ 2 \\kappa \\left ( d _ 1 - d _ 3 \\right ) & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "872.png", "formula": "\\begin{align*} S _ 2 = w ^ 1 L _ { 2 1 } + w ^ 2 L _ { 2 2 } = w ^ 1 ( { w ^ 1 } _ { \\substack { \\\\ x _ 2 } } + { w ^ 1 } _ { \\substack { \\\\ x _ 1 } } ) + 2 w ^ 2 { w ^ 2 } _ { \\substack { \\\\ x _ 2 } } . \\end{align*}"}
+{"id": "5215.png", "formula": "\\begin{align*} & U _ { j } = - \\frac { 1 } { \\alpha } \\left [ \\left ( \\frac { a - 1 } { a - b } \\right ) ( X . Y ) ^ { a - 2 } - \\left ( \\frac { b - 1 } { a - b } \\right ) ( X . Y ) ^ { b - 2 } \\right ] \\times M \\\\ & V _ { j } = - \\frac { 1 } { \\alpha } \\left [ \\left ( \\frac { a - 1 } { a - b } \\right ) ( X . Y ) ^ { a - 2 } - \\left ( \\frac { b - 1 } { a - b } \\right ) ( X . Y ) ^ { b - 2 } \\right ] \\times N \\end{align*}"}
+{"id": "5753.png", "formula": "\\begin{align*} \\mathbf { L } \\Upsilon _ j = \\mathbf { L } ^ \\dagger \\Upsilon _ j = 0 , \\ \\mathbf { L } \\overline { \\Upsilon } _ j = \\mathbf { L } ^ \\dagger \\overline { \\Upsilon } _ j = m \\overline { \\Upsilon } _ j . \\end{align*}"}
+{"id": "4052.png", "formula": "\\begin{align*} f ' ( x , d ) = \\lim _ { t \\downarrow 0 } \\frac { f ( x + t d ) - f ( x ) } { t } \\end{align*}"}
+{"id": "6464.png", "formula": "\\begin{align*} \\langle d \\phi ( e _ j ) , \\bar \\Delta \\tau ( \\phi ) \\rangle \\langle V , \\bar \\nabla _ { e _ j } V \\rangle = 0 . \\end{align*}"}
+{"id": "1704.png", "formula": "\\begin{align*} A : = \\left ( \\begin{array} { c c c } \\frac { \\partial v _ 1 } { \\partial x _ 1 } & \\cdots & \\frac { \\partial v _ 1 } { \\partial x _ { n - 1 } } \\\\ \\vdots & \\ddots & \\vdots \\\\ \\frac { \\partial v _ { n - 1 } } { \\partial x _ 1 } & \\cdots & \\frac { \\partial v _ { n - 1 } } { \\partial x _ { n - 1 } } \\end{array} \\right ) \\end{align*}"}
+{"id": "4632.png", "formula": "\\begin{align*} f ( x ' ) - x _ d \\leq f ( x ' ) - f _ { n - 1 } ( x ' ) = \\sum _ { j = n } ^ \\infty 2 ^ { - \\gamma m j } g _ j ( x ' ) \\le 2 ^ { - \\gamma m n } . \\end{align*}"}
+{"id": "625.png", "formula": "\\begin{align*} V ( L , K [ n - 1 ] ) = \\frac { 1 } { n } \\int _ { \\mathbb { S } ^ { n - 1 } } h _ L ( u ) d S _ K ( u ) . \\end{align*}"}
+{"id": "2215.png", "formula": "\\begin{align*} v = \\psi _ { 1 , r } ^ { ( 1 ) } , f = g _ { , r } ^ { ( 1 ) } . \\end{align*}"}
+{"id": "6878.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathbb { J } } \\ , E _ j \\langle \\varphi _ j , f \\rangle \\varphi _ j = \\sum _ { i \\in \\mathbb { J } ' } \\ , E _ i ' \\langle { e } _ i ' , f \\rangle { e } _ i ' \\end{align*}"}
+{"id": "3964.png", "formula": "\\begin{align*} ( [ \\pi ] ( T ) \\xi ) ( T ( x ) ) = \\pi ( x , T ( x ) ) \\xi ( x ) . \\end{align*}"}
+{"id": "3722.png", "formula": "\\begin{align*} 1 - 2 b ^ 2 + 2 w = \\Im ( Z ^ 0 _ { b , w } ( E ) ) \\ \\geq \\ \\Im ( Z ^ 0 _ { b , w } ( B ) ) = ( 2 b ^ 2 - 2 w - 1 ) \\frac { x } { 2 } - ( b + 1 ) y > 0 \\end{align*}"}
+{"id": "2903.png", "formula": "\\begin{align*} & \\lim _ { \\lambda \\to 0 ^ + } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ { ( E _ \\Phi ^ r ) _ t ( \\mathbb { R } ^ n ) } = \\left [ - \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { ( E _ \\Phi ^ r ) _ t ( \\mathbb { R } ^ n ) } . \\end{align*}"}
+{"id": "4136.png", "formula": "\\begin{align*} \\sigma ^ 2 _ j = 2 Q ^ * ( w _ { 1 j } , \\ldots , w _ { j - 1 , j } , 1 , 0 , \\ldots , 0 ) , \\end{align*}"}
+{"id": "2998.png", "formula": "\\begin{align*} H ( x , y ) = \\alpha \\{ g ( x , y ) \\xi - \\eta ( y ) x \\} + \\beta \\{ g ( \\varphi x , y ) \\xi - \\eta ( y ) \\varphi x \\} . \\end{align*}"}
+{"id": "224.png", "formula": "\\begin{align*} \\frac { d ^ 2 \\bar x ^ i } { d \\tau ^ 2 } = \\left ( { \\displaystyle \\sum _ { j = 1 } ^ n } \\frac { 1 } { f } \\frac { \\partial f } { \\partial \\bar x ^ { j } } \\frac { d \\bar x ^ j } { d \\tau } \\right ) \\frac { d \\bar x ^ i } { d \\tau } + f ^ { 2 } X ^ { i } \\left ( \\bar x , \\frac 1 f \\frac { d \\bar x } { d \\tau } \\right ) . \\end{align*}"}
+{"id": "8996.png", "formula": "\\begin{align*} \\bar { A } _ t ( \\omega , \\omega ' ) : = \\lim _ { s \\searrow 0 } \\tilde { A } _ { s , t } ( \\omega , \\omega ' ) \\end{align*}"}
+{"id": "3790.png", "formula": "\\begin{align*} \\widehat { \\Theta } ^ + _ 1 & = \\widehat { \\Theta } _ 1 + \\frac { 2 \\eta _ 1 } { | \\widehat { \\mathcal { C } } _ 1 | } \\sum _ { i \\in \\widehat { \\mathcal { C } } _ 1 \\cap \\mathcal { C } _ 1 } ( X ^ { ( i ) } - \\widehat { \\Theta } _ { 1 } Z ^ { ( i ) } ) Z ^ { ( i ) , \\top } + \\frac { 2 \\eta _ 1 } { | \\widehat { \\mathcal { C } } _ 1 | } \\sum _ { i \\in \\widehat { \\mathcal { C } } _ 1 \\cap \\overline { \\mathcal { C } _ 1 } } ( X ^ { ( i ) } - \\widehat { \\Theta } _ { 1 } Z ^ { ( i ) } ) Z ^ { ( i ) , \\top } , \\end{align*}"}
+{"id": "5905.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { N C } ( G ) ) = p ^ { 3 } q ^ { 3 } - 2 p ^ { 2 } q ^ { 2 } - p q ^ { 3 } - p ^ { 3 } q ^ { 2 } + p q ^ { 2 } - 3 q ^ { 2 } - 3 q p ^ { 2 } + 2 q + p ^ { 3 } q + q ^ { 3 } - 4 \\end{align*}"}
+{"id": "9077.png", "formula": "\\begin{align*} \\begin{cases} X ( t + 1 ) = A ^ { ( r ) } X ( t ) + b ^ { ( r ) } ( X ( t ) ) , \\\\ v ( t ) = J ^ { ( r ) } X ( t ) + \\underline { v } . \\end{cases} \\end{align*}"}
+{"id": "6717.png", "formula": "\\begin{align*} \\lambda ^ n - 1 = 0 \\end{align*}"}
+{"id": "1186.png", "formula": "\\begin{align*} & | v | ^ { - s } | \\partial ^ \\alpha f ( z + v ) - \\partial ^ \\alpha f ( z ) | \\\\ & \\quad = | v | ^ { - s } \\left | \\int _ 0 ^ 1 v \\cdot \\nabla \\partial ^ \\alpha f ( z + t ' v ) \\ , d t ' \\right | \\\\ & \\quad \\leq | v | ^ { 1 - s } \\sup _ { i \\in \\{ 1 , \\ldots , n \\} } \\sup _ { t ' \\in ( 0 , 1 ) } | \\partial ^ { \\alpha + e _ i } f ( z + t ' v ) | \\\\ & \\quad \\leq | y - z | ^ { 1 - s } \\sup _ { \\beta \\in \\mathbb { Z } _ + ^ n , \\ , | \\beta | = N + 1 } \\sup _ { t \\in ( 0 , 1 ) } \\left | \\partial ^ { \\beta } f ( z + t ( y - z ) ) \\right | . \\end{align*}"}
+{"id": "3305.png", "formula": "\\begin{align*} \\begin{cases} f _ 1 ( t , B _ t ) = - f ( t , B _ t ) \\frac { c ^ 2 + t + B _ t ^ 2 } { 2 ( c ^ 2 + t ) ^ 2 } , \\\\ f _ 2 ( t , B _ t ) = f ( t , B _ t ) \\frac { B _ t } { c ^ 2 + t } , \\\\ f _ { 2 2 } ( t , B _ t ) = f ( t , B _ t ) \\frac { c ^ 2 + t + B _ t ^ 2 } { ( c ^ 2 + t ) ^ 2 } . \\end{cases} \\end{align*}"}
+{"id": "6255.png", "formula": "\\begin{align*} P ( X _ { 1 } , \\ldots , X _ { n } ) = - \\sum _ { i , j = 1 } ^ { n } a _ { i j } X _ { i } X _ { j } + \\sum _ { i = 1 } ^ { n } b _ { i } X _ { i } + c , a _ { i , j } , b _ { i } , c \\in \\R , \\end{align*}"}
+{"id": "5437.png", "formula": "\\begin{align*} f ^ S _ i = \\begin{cases} R _ i + \\beta \\sum _ { j \\in S } p _ { i j } f ^ S _ j & i \\in S \\\\ 0 & , \\end{cases} \\end{align*}"}
+{"id": "3710.png", "formula": "\\begin{align*} \\psi _ k ( x ) = { \\mathcal N } G _ 0 ( x , \\Gamma | E _ k ^ * ) = { \\mathcal N } \\int _ { \\Gamma } G _ 0 ( x , \\gamma ( s ) | E _ k ^ * ) d s \\ ; . \\end{align*}"}
+{"id": "6413.png", "formula": "\\begin{align*} \\sup _ { \\theta \\in A \\times W _ n ^ { ( \\eta ) } } \\ln ( n ) ^ q \\left | \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\left ( f ( X _ { \\frac { i - 1 } { n } } , \\delta , \\alpha ) - f ( X _ { \\frac { i - 1 } { n } } , \\delta _ 0 , \\alpha _ 0 ) \\right ) h ( z ^ n _ i ( \\theta ) , \\alpha ) \\right | \\to 0 , \\end{align*}"}
+{"id": "3135.png", "formula": "\\begin{align*} u _ { \\psi ^ * } ^ - ( s ) = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ s ^ { 2 \\ , p ^ { e _ 1 } } & s ^ { p ^ { e _ 1 } } & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "1624.png", "formula": "\\begin{align*} b ^ r _ { p , q } & q 1 \\leq p \\leq k \\\\ & \\begin{cases} q & 0 \\leq p \\leq k - r - 1 k - r < p \\\\ q \\leq M + r & p = k - r \\end{cases} \\end{align*}"}
+{"id": "3726.png", "formula": "\\begin{align*} - F & = 1 9 H - 6 \\sum _ { j = 1 } ^ { 1 0 } E _ j , - 2 F = 3 8 H - 1 2 \\sum _ { j = 1 } ^ { 1 0 } E _ j , - D _ i = 6 H - 2 \\sum _ { j = 1 } ^ { 1 0 } E _ j + E _ i , \\\\ D _ i - F & = 1 3 H - 4 \\sum _ { j = 1 } ^ { 1 0 } E _ j - E _ i , D _ i - 2 F = 3 2 H - 1 0 \\sum _ { j = 1 } ^ { 1 0 } E _ j - E _ i . \\end{align*}"}
+{"id": "8976.png", "formula": "\\begin{align*} \\inf _ { u \\in H _ 0 ^ 1 ( B ( 0 , 1 ) ) , \\int _ { B ( 0 , 1 ) } \\left ( \\frac { | u ( x ) | } { | x | } | \\right ) ^ 2 = 1 } \\int _ { B ( 0 , 1 ) } | \\nabla u | ^ 2 = \\frac { ( N - 2 ) ^ 2 } { 4 } \\end{align*}"}
+{"id": "5825.png", "formula": "\\begin{align*} \\frac { d } { d s } \\mathcal { F } _ \\Sigma ( u + s \\xi ) \\big | _ { s = 0 } = - \\int h _ 0 \\left ( \\mathcal { M } _ \\Sigma ( u ) , \\xi \\right ) \\sqrt { \\det g _ 0 } \\ , d x ^ 1 \\wedge \\dots \\wedge d x ^ n , \\end{align*}"}
+{"id": "2204.png", "formula": "\\begin{align*} \\intop _ \\Omega ( \\psi _ { 1 , r z } ^ 2 + \\psi _ { 1 , z z } ^ 2 ) d x + \\intop _ { - a } ^ a \\psi _ { 1 , z } ^ 2 | _ { r = 0 } d z \\le c | \\omega _ 1 | _ { 2 , \\Omega } ^ 2 . \\end{align*}"}
+{"id": "6910.png", "formula": "\\begin{align*} \\int _ { B _ R \\ ( 0 \\ ) } v ^ { - q } \\left | \\nabla u \\right | ^ r & \\le C ^ { - \\frac { p \\ ( r - q \\ ) } { n - p } } R ^ { \\frac { p \\ ( r - q \\ ) } { p - 1 } } \\int _ { B _ R \\ ( 0 \\ ) } v ^ { - r } \\left | \\nabla u \\right | ^ r \\\\ & \\le C ^ \\prime R ^ { \\frac { p \\ ( r - q \\ ) } { p - 1 } + n - r } \\\\ & = C ^ \\prime R ^ { n - \\frac { p q - r } { p - 1 } } , \\end{align*}"}
+{"id": "2236.png", "formula": "\\begin{align*} \\begin{array} { l c l } X ^ { V _ \\alpha } ( f ^ V ) = 0 , & & X ^ { V _ \\alpha } ( f ^ { ( \\beta ) } ) = \\delta ^ \\alpha _ \\beta ( X ( f ) ) ^ V , \\\\ \\noalign { \\medskip } X ^ C ( f ^ V ) = ( X ( f ) ) ^ V , & & X ^ C ( f ^ { ( \\alpha ) } ) = ( X ( f ) ) ^ { ( \\alpha ) } \\ , , \\end{array} \\end{align*}"}
+{"id": "7196.png", "formula": "\\begin{align*} h ( v ) = v ^ 3 - 3 ( 1 - 1 / \\gamma ) v + 3 \\beta / \\gamma = 0 . \\end{align*}"}
+{"id": "118.png", "formula": "\\begin{align*} a ( P \\vec { v } , \\vec { w } ) = a ( \\vec { v } , \\vec { w } ) , \\forall \\vec { w } \\in W . \\end{align*}"}
+{"id": "8559.png", "formula": "\\begin{align*} & - ( a - n ) ^ 2 ( b - n ) ^ 2 F ( n + 1 , k ) + n ^ 2 ( - 1 - a - b + 2 n ) ( - a - b + 2 n ) F ( n , k ) \\\\ & = G ( n , k + 1 ) - G ( n , k ) , \\end{align*}"}
+{"id": "7965.png", "formula": "\\begin{align*} \\dot { \\bar { \\mathcal { F } } } ( \\omega , \\phi _ { \\partial } , \\Sigma ) = \\{ \\bar { \\mathcal { F } } , \\bar { H } \\} ( \\omega , \\phi _ { \\partial } , \\Sigma ) . \\end{align*}"}
+{"id": "546.png", "formula": "\\begin{align*} \\Phi ' _ { k } ( 1 ) \\int _ { 0 } ^ { T } f ( t ) \\mathrm { e } ^ { - \\lambda _ { k } ( T - t ) } \\mathrm { d } t = a _ { k } \\mathrm { e } ^ { - \\lambda _ { k } T } = \\left \\langle u _ { 0 } , \\mathrm { e } ^ { - \\lambda _ { k } T } \\Phi _ { k } \\right \\rangle _ \\beta = \\left \\langle u _ { 0 } , \\mathrm { e } ^ { - \\lambda _ { k } T } \\Phi _ { k } \\right \\rangle _ { \\mathcal { H } ^ { - s } , \\mathcal { H } ^ { s } } . \\end{align*}"}
+{"id": "2392.png", "formula": "\\begin{align*} A _ n ( t ) & = : \\sum _ { i = 1 } ^ { 2 s + 4 } \\sum _ { k = 0 } ^ { n } \\frac { a _ { n , i , k } } { ( t + k ) ^ i } , \\\\ B _ n ( t ) & = : \\sum _ { i = 1 } ^ { s + 2 } \\sum _ { k = 0 } ^ { n } \\frac { b _ { n , i , k } } { ( t + k ) ^ i } . \\end{align*}"}
+{"id": "1561.png", "formula": "\\begin{align*} M _ { n } = 1 + \\sup _ { B } | D \\psi _ n | + \\frac { A _ n } { \\mu _ n } \\end{align*}"}
+{"id": "334.png", "formula": "\\begin{align*} I ( G _ 2 ) ^ { [ k ] } + x _ { n - 2 } I ( G _ 3 ) ^ { [ k - 1 ] } = I ( G _ 3 ) ^ { [ k ] } + x _ { n - 2 } I ( G _ 3 ) ^ { [ k - 1 ] } \\end{align*}"}
+{"id": "8628.png", "formula": "\\begin{align*} | K _ 1 ( t ; x ) + \\phi _ 1 ( t ; x ) | \\leq & 2 8 ( C _ 1 q ^ { - 1 } ( t ) q ^ { - \\frac { \\sigma } { 2 } } ( t ) + C _ 2 g ^ 2 q ^ { \\frac { \\sigma } { 2 } } ( t ) q ^ { - 1 - \\sigma } ( t ) ) \\\\ = & 2 8 ( C _ 1 + C _ 2 g ^ 2 ) q ^ { - 1 - \\frac { \\sigma } { 2 } } ( t ) \\end{align*}"}
+{"id": "2351.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow \\infty } \\| z _ { k } - z \\| _ { L _ { T } ^ { \\infty } L _ { x } ^ { p } \\cap L _ { T } ^ { \\frac { p } { \\alpha } } L _ { x } ^ { \\frac { 3 p } { 3 - 2 \\alpha } } } + \\| \\nabla | z _ { k } - z | ^ { \\frac { p } { 2 } } \\| ^ { \\frac { 2 } { p } } _ { L ^ { 2 } _ { T } L ^ { 2 } _ { x } } = 0 , \\end{align*}"}
+{"id": "5672.png", "formula": "\\begin{align*} - \\Delta _ g u + V ( \\sigma ) u = g ( \\sigma , u ) \\quad \\hbox { i n $ \\mathcal { M } $ } . \\end{align*}"}
+{"id": "5998.png", "formula": "\\begin{align*} \\Delta w = w _ { j j } = \\frac { 2 u _ { i j } ^ 2 } { u ^ 2 } + \\frac { 2 u _ i u _ { i j j } } { u ^ 2 } - \\frac { 4 u _ i u _ { i j } u _ j } { u ^ 3 } - \\frac { 4 u _ i u _ { i j } u _ j } { u ^ 3 } - \\frac { 2 u _ i ^ 2 u _ { j j } } { u ^ 3 } + \\frac { 6 u _ i ^ 2 u _ j ^ 2 } { u ^ 4 } . \\end{align*}"}
+{"id": "547.png", "formula": "\\begin{align*} J _ { \\nu } ( x ) = \\sum _ { m \\geq 0 } \\frac { ( - 1 ) ^ { m } } { m ! \\Gamma ( m + \\nu + 1 ) } \\left ( \\frac { x } { 2 } \\right ) ^ { 2 m + \\nu } , x \\geq 0 , \\end{align*}"}
+{"id": "6707.png", "formula": "\\begin{align*} \\mathcal { F } : = \\{ f : \\mathbb { R } \\rightarrow \\mathbb { R } \\} . \\end{align*}"}
+{"id": "7517.png", "formula": "\\begin{align*} | R m ( g ( t ) ) | _ { g ( t ) } \\leq \\frac { C _ M } { r _ M ^ 2 } < \\frac { C _ M } { \\gamma t } = \\frac { \\zeta } { t } \\end{align*}"}
+{"id": "6978.png", "formula": "\\begin{align*} L _ o = \\{ [ i y _ 1 : \\cdots : i y _ n : x _ { n + 1 } ] : x _ { n + 1 } , y _ 1 , \\dots y _ n \\in \\mathbb { R } , x _ { n + 1 } \\neq 0 \\} . \\end{align*}"}
+{"id": "3302.png", "formula": "\\begin{align*} \\mathcal { F } _ \\tau = \\{ A \\in \\mathcal { A } : A \\cap \\{ \\tau = n \\} \\in \\mathcal { F } _ n n \\} . \\end{align*}"}
+{"id": "4219.png", "formula": "\\begin{align*} \\theta ' ( 0 , \\tau ) = \\frac { \\partial \\theta ( v , \\tau ) } { \\partial v } | _ { v = 0 } . \\end{align*}"}
+{"id": "805.png", "formula": "\\begin{align*} R i { c _ { i j } } = { \\tilde R _ { i j } } - { \\textbf { H } _ { i j } } . \\end{align*}"}
+{"id": "9195.png", "formula": "\\begin{align*} e _ { 1 , j + 1 } ( z ) = ( 1 - q ^ 2 ) ^ { - 1 } \\cdot [ e _ { 1 j } ( z ) , e ^ { ( 0 ) } _ { j , j + 1 } ] _ q . \\end{align*}"}
+{"id": "5680.png", "formula": "\\begin{align*} \\int _ { \\Sigma } \\left \\langle \\mathcal { L } _ { \\Sigma } u , v \\right \\rangle \\ , d \\mu = - \\frac { \\partial ^ 2 } { \\partial s _ 1 \\partial s _ 2 } \\mathcal { F } _ \\Sigma ( s _ 1 u + s _ 2 v ) \\bigg | _ { s _ 1 = s _ 2 = 0 } . \\end{align*}"}
+{"id": "956.png", "formula": "\\begin{align*} P _ V ( x , d z ) P _ { W } ( z , d y ) = P _ W ( x , d y ) \\quad x \\in E . \\end{align*}"}
+{"id": "4194.png", "formula": "\\begin{align*} \\begin{aligned} g _ 1 g _ 2 \\neq g _ 2 g _ 1 & \\mbox { a n d } g _ 1 ( g _ 1 g _ 3 ) \\neq ( g _ 1 g _ 3 ) g _ 1 \\ , , \\\\ \\varphi ( g _ 1 g _ 2 ) = \\varphi ( g _ 1 ) \\varphi ( g _ 2 ) & \\mbox { a n d } \\varphi ( g _ 1 ( g _ 1 g _ 3 ) ) = \\varphi ( g _ 1 g _ 3 ) \\varphi ( g _ 1 ) \\ , , \\end{aligned} \\end{align*}"}
+{"id": "3294.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } K _ n \\ge \\liminf _ { n \\to \\infty } \\frac { 1 } { \\sqrt { n } } \\exp ( n \\Delta _ \\mu ^ 2 / 2 ) = \\infty , \\end{align*}"}
+{"id": "1912.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ T \\sum _ { x \\in G } \\ , & V ( x ) \\ , \\partial _ t v _ + ( x , t ) \\ , v _ + ( x , t ) \\eta ^ 2 ( x ) e ^ { \\xi ( x , t ) } \\mu ( x ) \\ , d t \\\\ & \\le \\int _ 0 ^ T \\frac 1 { 4 } \\sum _ { x , y \\in G } v _ + ^ 2 ( x , t ) \\left [ 1 - e ^ { \\xi ( y , t ) - \\xi ( x , t ) } \\right ] ^ 2 \\eta ^ 2 ( x ) e ^ { \\xi ( x , t ) } \\ , \\omega ( x , y ) \\ , d t \\\\ & + \\int _ 0 ^ T \\sum _ { x , y \\in G } v _ + ^ 2 ( x , t ) [ \\eta ( y ) - \\eta ( x ) ] ^ 2 e ^ { \\xi ( y , t ) } \\omega ( x , y ) \\ , d t . \\end{aligned} \\end{align*}"}
+{"id": "5087.png", "formula": "\\begin{align*} R _ { s } \\cdot \\varepsilon _ \\theta & = R _ { s } \\left ( \\sum _ { t \\in T ( \\mathbb { F } _ { q ^ k } ) } \\theta ( t ) \\delta _ t \\right ) \\\\ & = \\sum _ { t \\in T ( \\mathbb { F } _ { q ^ k } ) } \\theta ( t ) \\sum _ { u s \\in U ( \\mathbb { F } _ { q ^ k } ) s U ( \\mathbb { F } _ { q ^ k } ) } \\delta _ { t u s } \\\\ & = \\sum _ { t u s \\in B ( \\mathbb { F } _ { q ^ k } ) s U ( \\mathbb { F } _ { q ^ k } ) } \\theta ( t ) \\delta _ { t u s } . \\end{align*}"}
+{"id": "903.png", "formula": "\\begin{align*} \\begin{array} { c c c c c } \\Phi _ { a } : & \\mathbb { F } _ { q } ^ n & \\longrightarrow & \\mathbb { F } _ { q } ^ n \\\\ \\\\ & x = ( x _ 0 , x _ 2 , \\ldots , x _ { n - 1 } ) & \\longmapsto & ( a _ { n - 1 } x _ { n - 1 } , a _ { 1 } x _ 1 , . . , a _ { n - 2 } x _ { n - 2 } ) \\end{array} \\end{align*}"}
+{"id": "4290.png", "formula": "\\begin{align*} \\mathbf { x } ^ k _ { s , t } = \\sum _ { i = 0 } ^ k \\mathbf { x } ^ { k - i } _ { s , u } \\otimes \\mathbf { x } ^ { i } _ { u , t } , 1 \\le k \\le [ p ] , \\ , \\ , s \\le u \\le t . \\end{align*}"}
+{"id": "2537.png", "formula": "\\begin{align*} \\dot C ^ { - \\infty } ( M ) = C ^ \\infty ( M ; \\Omega ) ' \\ ; , C ^ { - \\infty } ( M ) = \\dot C ^ \\infty ( M ; \\Omega ) ' \\ ; . \\end{align*}"}
+{"id": "8788.png", "formula": "\\begin{align*} a = \\inf \\left \\{ x \\in \\R : \\exists \\ , y \\leq x , \\ ; ( x , y ) \\in \\Gamma \\right \\} \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; b = \\sup \\left \\{ x \\in \\R : \\exists \\ , z \\geq x , \\ ; ( x , z ) \\in \\Gamma \\right \\} . \\end{align*}"}
+{"id": "6893.png", "formula": "\\begin{align*} \\eta _ { 0 } & = \\begin{cases} \\dfrac { 2 } { A - B } ( 1 - B ) ( \\xi - C i ( 1 ) ) & - 1 < B \\leq - \\kappa , \\\\ & \\\\ \\dfrac { - 2 } { A - B } ( 1 + B ) ( \\xi - C h i ( 1 ) ) & - \\kappa < B < 1 . \\end{cases} \\end{align*}"}
+{"id": "3910.png", "formula": "\\begin{align*} & e ^ \\omega = 1 + \\omega , \\\\ & e ^ { j \\omega } = ( 1 + \\omega ) ^ j , \\\\ & e ^ z = ( 1 + \\tfrac z j ) ^ j , \\ \\ \\ \\mbox { w h e r e } \\ \\ j = \\dfrac z \\omega . \\end{align*}"}
+{"id": "1368.png", "formula": "\\begin{align*} l _ { 2 } ' ( \\iota ( s ^ { - 1 } f ) , \\iota ( s ^ { - 1 } g ) ) & = l _ { 2 } ' ( s ^ { - 1 } \\widetilde { f } , s ^ { - 1 } \\widetilde { g } ) \\\\ & = ( - 1 ) ^ { | f | } s ^ { - 1 } [ \\widetilde { f } , \\widetilde { g } ] _ { \\Omega } \\\\ & = ( - 1 ) ^ { | f | } s ^ { - 1 } \\left ( 0 , \\dots , 0 , [ f , g ] _ { \\Omega } ; [ f , g ] _ { \\Omega } , \\dots , [ f , g ] _ { \\Omega } , 0 \\right ) \\\\ & = \\iota \\left ( ( - 1 ) ^ { | f | } s ^ { - 1 } [ f , g ] _ { \\Omega } \\right ) , \\end{align*}"}
+{"id": "1334.png", "formula": "\\begin{align*} H ( p ) = \\mp \\bigl ( \\langle p , a \\rangle - L ( a ) \\bigr ) , \\end{align*}"}
+{"id": "2291.png", "formula": "\\begin{align*} C ^ { ( 2 ) } = \\sum _ { i , j = 1 } ^ 3 e _ { i j } e _ { j i } \\ \\ \\ \\ \\ \\ C ^ { ( 3 ) } = \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ 3 ( e _ { i j } e _ { j k } e _ { k i } + e _ { j i } e _ { k j } e _ { i k } ) \\ . \\end{align*}"}
+{"id": "1900.png", "formula": "\\begin{align*} \\frac { 1 } { \\mu ( x ) } \\sum _ { y \\in \\Omega } \\omega ( x , y ) \\left [ u ( y ) - u ( x ) \\right ] - V ( x ) u ( x ) = f ( x ) \\quad \\ , \\ , x \\in \\Omega , \\end{align*}"}
+{"id": "5404.png", "formula": "\\begin{align*} a _ k = \\begin{cases} 1 & \\\\ \\\\ \\displaystyle { \\frac { \\det \\left ( \\mathbf { I } - \\mathbf { B } ^ { k } \\right ) } { \\det \\left ( \\mathbf { I } - \\mathbf { B } ^ { k - 1 } \\right ) } } & \\end{cases} \\end{align*}"}
+{"id": "1324.png", "formula": "\\begin{align*} \\Pi _ \\pm \\bigl ( \\mathrm { d } \\Phi _ X ( p ) , \\mathrm { d } ( g \\circ \\pi ) ( p ) \\bigr ) & = \\pm \\Bigl \\langle \\mathrm { d } g ( q ) , \\rho \\bigl ( X ( q ) \\bigr ) \\Bigr \\rangle \\\\ & = \\pm \\rho ( X ) [ g ] ( q ) , \\end{align*}"}
+{"id": "4948.png", "formula": "\\begin{align*} \\begin{aligned} [ L ] \\circ [ L ' ] & = \\int _ { X } 4 \\beta ( e ^ { L } _ + - e ^ { L } _ - ) \\cdot ( e ^ { L ' } _ + - e ^ { L ' } _ - ) \\\\ & = 4 \\beta \\int _ { X } ( e ^ { L } _ + \\cdot e ^ { L ' } _ + - e ^ { L } _ - \\cdot e ^ { L ' } _ + - e ^ { L } _ + \\cdot e ^ { L ' } _ + + e ^ { L } _ - \\cdot e ^ { L ' } _ - ) . \\end{aligned} \\end{align*}"}
+{"id": "836.png", "formula": "\\begin{align*} { \\pounds } _ { \\hat { X } } \\nabla _ { 0 } I _ { k } = { \\pounds } _ { \\hat { X } } J _ k = \\nabla _ { 0 } f _ { . k } + \\Psi I _ { k } . \\end{align*}"}
+{"id": "7533.png", "formula": "\\begin{align*} \\partial _ t \\Big ( \\frac { 1 } { 2 } \\frac { | m _ 0 | ^ 2 } { \\rho _ 0 } + h _ 1 ( \\rho _ 0 ) + h _ 2 ( n _ 0 ) + \\delta \\frac { 1 } { 2 } | \\nabla \\phi _ 0 | ^ 2 \\Big ) + \\nabla \\cdot \\mathcal { F } _ 0 = 0 \\ , \\end{align*}"}
+{"id": "5202.png", "formula": "\\begin{align*} & A = \\frac { 1 } { ( \\beta - 1 ) ( \\alpha + \\beta - 1 ) } \\sum _ { i } p ^ { \\alpha + \\beta - 1 } _ { i } - \\frac { 1 } { \\alpha ( \\beta - 1 ) } \\sum _ { i } p ^ { \\alpha } _ { i } q ^ { \\beta - 1 } _ { i } \\\\ & B = \\frac { 1 } { \\alpha ( 1 - \\alpha - \\beta ) } \\sum _ { i } q ^ { \\alpha + \\beta - 1 } _ { i } \\end{align*}"}
+{"id": "1833.png", "formula": "\\begin{align*} \\int _ { U _ { \\alpha } } \\omega _ { \\alpha } : = \\frac { 1 } { | G _ { \\alpha } | } \\int _ { \\tilde { U } _ { \\alpha } } \\tilde { \\omega } _ { \\alpha } , \\end{align*}"}
+{"id": "8639.png", "formula": "\\begin{align*} I _ 3 ( t ; x ) = \\int _ { 0 } ^ { t } ( ( K _ 3 + \\phi _ 3 ) ( \\partial _ x X ) ^ 3 + 3 ( K _ 2 + \\phi _ 2 ) ( \\partial _ x X ) ( \\partial ^ 2 _ x X ) + ( K _ 1 + \\phi _ 1 ) ( \\partial ^ 3 _ x X ) ( \\tau ; x ) d \\tau . \\end{align*}"}
+{"id": "839.png", "formula": "\\begin{align*} { \\pounds } _ { \\hat { X } } { J _ k } = ( { \\pounds } _ { \\hat { X } } \\lambda ( x , y ) ) I _ { k } = \\hat { X } . ( \\lambda ( x , y ) ) I _ { k } , \\end{align*}"}
+{"id": "2143.png", "formula": "\\begin{align*} \\begin{array} { l } h = ( 4 - x _ 1 ^ 2 - 2 x _ 2 , \\ , x _ 1 , \\ , x _ 2 ) , \\\\ g = ( - 2 z _ 1 + z _ 2 + x _ 1 ^ 2 - 2 x _ 1 + x _ 2 ^ 2 + 3 , \\ , 3 z _ 1 - 4 z _ 2 + x _ 2 - 4 , \\ , z _ 1 , \\ , z _ 2 ) . \\end{array} \\end{align*}"}
+{"id": "5494.png", "formula": "\\begin{align*} n _ \\alpha : = ( 2 ^ { 2 \\alpha + 1 } ) ^ 3 + ( 2 ^ { 2 \\alpha + 1 } ) ^ 2 + 2 ^ { 2 \\alpha + 1 } + 1 = 8 \\cdot 6 4 ^ \\alpha + 4 \\cdot 1 6 ^ \\alpha + 2 \\cdot 4 ^ \\alpha + 1 , \\end{align*}"}
+{"id": "8410.png", "formula": "\\begin{align*} \\sigma ( a _ 1 ) \\circ T ( \\sigma ( a _ 2 ) ) & = B _ 1 ( \\sigma ( a _ 1 ) ) S ( B _ 1 ( \\sigma ( a _ 3 ) ) ) B _ 2 ( \\sigma ( a _ 4 ) ) S ( B _ 2 ( \\sigma ( a _ 2 ) ) ) \\\\ & = B _ 1 ( a _ 1 ) S ( B _ 1 ( a _ 2 ) ) B _ 2 ( a _ 3 ) S ( B _ 2 ( a _ 4 ) ) \\\\ & = \\epsilon ( B _ 1 ( a _ 1 ) ) \\epsilon ( B _ 2 ( a _ 2 ) ) 1 = \\epsilon ( a ) 1 . \\end{align*}"}
+{"id": "5016.png", "formula": "\\begin{align*} a _ i ^ { ( k ) } = \\begin{cases} \\beta r _ i ^ { S _ { k - 1 } } / ( 1 - \\beta ) & i \\in S _ { k - 1 } \\\\ \\beta r _ i ^ { S _ { k - 1 } } & i \\in S _ { k - 1 } ^ c \\end{cases} b _ i ^ { ( k ) } = \\begin{cases} \\beta w _ i ^ { S _ { k - 1 } } / ( 1 - \\beta ) & i \\in S _ { k - 1 } \\\\ \\beta w _ i ^ { S _ { k - 1 } } & i \\in S _ { k - 1 } ^ c . \\end{cases} \\end{align*}"}
+{"id": "4210.png", "formula": "\\begin{align*} & - 2 6 4 \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 2 2 ) } = \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } ( \\widetilde { T X } - \\widetilde { L _ R \\otimes C } ) \\right \\} ^ { ( 2 2 ) } . \\end{align*}"}
+{"id": "1491.png", "formula": "\\begin{align*} f ( m ) \\big ( m f ( 0 ) - ( m + d ) f ( m ) \\big ) = 0 . \\end{align*}"}
+{"id": "7132.png", "formula": "\\begin{align*} \\lambda _ { n } c : = - A ( u _ { n } ) - V ( u _ { n } ) + \\int _ { \\mathbb { R } ^ { 2 } } f ( u _ { n } ) u _ { n } d x + o ( 1 ) . \\end{align*}"}
+{"id": "927.png", "formula": "\\begin{align*} \\hat W ^ { x _ 0 } _ D ( u ) = W _ D [ u ] ( \\mathbb R ^ d ) , \\end{align*}"}
+{"id": "5593.png", "formula": "\\begin{align*} \\prod _ { s = 0 } ^ { \\ell } a _ s = \\prod _ { s = 0 } ^ { \\ell } b _ s + \\sum _ { t = 0 } ^ { \\ell } \\prod _ { s = 0 } ^ { t - 1 } b _ s ( a _ t - b _ t ) \\prod _ { s = t + 1 } ^ { \\ell } a _ s . \\end{align*}"}
+{"id": "194.png", "formula": "\\begin{align*} ( x , x ' ) \\in \\widehat { A } ^ { \\circ } \\ \\ \\ \\ x _ { i + 2 } = x ' _ { i + 2 } \\ \\ \\ \\ ( x ' , x ) \\in \\widehat { A } ^ { \\circ } \\ \\ . \\end{align*}"}
+{"id": "2890.png", "formula": "\\begin{align*} & \\lim _ { \\lambda \\to 0 ^ + } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } ^ n } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - n } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ { M _ r ^ \\alpha ( \\mathbb { R } ^ n ) } = \\left [ - \\frac { \\kappa ( q , n ) } { \\gamma } \\right ] ^ \\frac { 1 } { q } \\left \\| \\ , \\left | \\nabla f \\right | \\ , \\right \\| _ { M _ r ^ \\alpha ( \\mathbb { R } ^ n ) } . \\end{align*}"}
+{"id": "4313.png", "formula": "\\begin{align*} \\| s \\| _ { \\max } = \\sup \\| ( p _ J \\otimes I d _ C ) s \\| _ { \\ell _ \\infty ( J ; D ) \\otimes _ { \\max } C } \\end{align*}"}
+{"id": "294.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { + \\infty } | \\alpha ( k ) | \\ , \\epsilon _ k \\ , \\rho ^ { 2 | \\alpha ( k ) | } \\end{align*}"}
+{"id": "571.png", "formula": "\\begin{align*} \\phi ^ s _ n - \\frac { \\omega ^ 2 } { g } \\phi ^ s & = - j \\frac { \\omega } { \\rho g } u & \\Gamma _ c \\end{align*}"}
+{"id": "2904.png", "formula": "\\begin{align*} M _ f ^ { ( 1 ) } ( x _ i ) : = \\sum _ { j = 1 } ^ J \\sum _ { y _ { j - 1 } < p \\leqslant y _ j } f ( p ) \\sum _ { \\substack { n \\leqslant x _ i / p \\\\ P ( n ) < p } } f ( n ) \\end{align*}"}
+{"id": "6140.png", "formula": "\\begin{align*} \\varphi _ m ( \\psi _ m ( \\lambda _ s ) ) = \\frac { | \\mathcal { F } _ m \\cap s \\mathcal { F } _ m | } { | \\mathcal { F } _ m | } \\lambda _ s , \\end{align*}"}
+{"id": "1327.png", "formula": "\\begin{align*} \\dot { q } & = g ^ \\sharp ( p ) , \\\\ \\nabla _ { \\dot { q } } ^ \\ast p & = - \\mathrm { d } U ( q ) . \\end{align*}"}
+{"id": "1402.png", "formula": "\\begin{align*} ( t n ) _ i & = t _ i + \\sum _ { k = 0 } ^ { i - 1 } \\binom { t _ 1 } { k } \\left ( - \\sum _ { l = 0 } ^ { i - k - 1 } \\binom { - t _ 1 } { l } t _ { i - k - l } \\right ) \\\\ & = t _ i - \\sum _ { r = 0 } ^ { i - 1 } \\left ( \\sum _ { k = 0 } ^ r \\binom { t _ 1 } { k } \\binom { - t _ 1 } { r - k } \\right ) t _ { i - r } \\\\ & = t _ i - \\sum _ { r = 0 } ^ { i - 1 } \\binom { 0 } { r } t _ { i - r } = t _ i - t _ i = 0 . \\end{align*}"}
+{"id": "7541.png", "formula": "\\begin{align*} \\begin{dcases} \\partial _ t \\bar \\rho + \\nabla \\cdot ( \\bar \\rho \\bar u ) = 0 \\\\ \\partial _ t \\bar u + \\bar u \\cdot \\nabla \\bar u + \\nabla \\big ( h _ 1 ^ \\prime ( \\bar \\rho ) + h _ 2 ^ \\prime ( \\bar n ) \\big ) = 0 \\\\ - \\Delta h _ 2 ^ \\prime ( \\bar n ) + \\bar n = \\bar \\rho \\end{dcases} \\end{align*}"}
+{"id": "4057.png", "formula": "\\begin{align*} R ^ { \\epsilon } _ 0 ( z ) ( x , y ) : = \\frac { 1 } { ( 2 \\pi ) ^ d } \\int _ { \\R ^ d } e ^ { i ( x - y ) \\cdot \\xi } \\frac { e ^ { - \\epsilon | \\xi | } } { | \\xi | - z } d \\xi . \\end{align*}"}
+{"id": "409.png", "formula": "\\begin{align*} \\begin{aligned} \\bold { d } & = [ \\sigma ^ 2 _ { R } ( l _ { x , 1 } , l _ { y , 1 } ) , . . . , \\sigma ^ 2 _ { R } ( l _ { x , n _ R } , l _ { y , n _ R } ) ] ^ { T } , \\\\ \\widetilde { \\bold { d } } & = [ \\sigma ^ 2 _ { S } ( m _ { x , 1 } , m _ { y , 1 } ) , . . . , \\sigma ^ 2 _ { S } ( m _ { x , n _ S } , m _ { y , n _ S } ) ] ^ { T } . \\end{aligned} \\end{align*}"}
+{"id": "3401.png", "formula": "\\begin{align*} \\frac { z _ { t } } { z _ { k } } & \\le \\frac { 2 C _ { 1 } \\left ( \\sqrt { \\Delta _ { 1 } } + 2 \\sqrt { A } C _ { 1 } \\right ) + 4 C _ { 1 } ^ { 2 } \\sqrt { A } } { 2 C _ { 1 } \\sqrt { \\Delta _ { 1 } } + 4 C _ { 1 } ^ { 2 } \\sqrt { A } } \\\\ & = \\frac { \\sqrt { \\Delta _ { 1 } } + 4 \\sqrt { A } C _ { 1 } } { \\sqrt { \\Delta _ { 1 } } + 2 \\sqrt { A } C _ { 1 } } \\le 2 \\end{align*}"}
+{"id": "7522.png", "formula": "\\begin{align*} q _ k \\in \\{ r _ M = \\sqrt { \\hat { \\gamma } t _ k } \\} , \\end{align*}"}
+{"id": "4277.png", "formula": "\\begin{align*} \\div _ f ^ { \\dagger } = \\div ^ { * } . \\end{align*}"}
+{"id": "1397.png", "formula": "\\begin{align*} ( x + b ) * y + ( x + b ) * c & = ( ( - u ) ( x b ) ) * y + ( ( - u ) ( x b ) ) * c \\\\ & = ( - u ) * ( ( x b ) * y ) + ( - u ) * y + b * y \\\\ & \\qquad \\mbox { } + ( - u ) * ( ( x b ) * c ) + ( - u ) * c + b * c \\\\ & = ( x b ) * y + ( x b ) * c \\\\ & = x * ( b * y ) + x * y + b * y \\\\ & \\qquad \\mbox { } + x * ( b * c ) + x * c + b * c \\\\ & = x * y + x * c , \\end{align*}"}
+{"id": "4406.png", "formula": "\\begin{align*} \\min _ { x } & \\ \\sum _ { ( i , j , r , s ) \\in [ n ] ^ 4 } c _ { i , j } d _ { r , s } x _ { i , r } x _ { j , s } \\\\ \\mathrm { s . t . } & \\ x \\in \\mathcal { X } = \\{ x \\in \\{ 0 , 1 \\} ^ { [ n ] ^ 2 } : \\sum _ { i \\in [ n ] } x _ { i , r } = 1 \\ \\forall r \\in [ n ] , \\ \\sum _ { r \\in [ n ] } x _ { i , r } = 1 \\ \\forall i \\in [ n ] \\} \\end{align*}"}
+{"id": "2169.png", "formula": "\\begin{align*} f = g , \\ , \\ , \\ , \\ , \\ , \\ , k = 3 , \\ , \\ , \\ , \\ , \\ , \\ , A = f _ d ( A _ d ''' ) , \\ , \\ , \\ , \\ , \\ , \\ , K = 3 2 L m . \\end{align*}"}
+{"id": "90.png", "formula": "\\begin{align*} \\begin{cases} 2 c _ 1 + c _ 2 & = 2 ( w _ 1 + w _ 3 \\kappa ) - 2 w _ 3 \\theta \\kappa \\geq c , \\\\ c _ 3 & = w _ 2 + w _ 4 \\kappa \\geq c , \\\\ \\det ( M ) & = c _ 3 ( c _ 3 + c _ 4 ) P _ \\theta - \\frac { \\left ( c _ 3 P _ \\theta + ( c _ 3 + c _ 4 ) - ( 2 c _ 1 + c _ 2 ) \\right ) ^ 2 } { 4 } \\geq c \\end{cases} \\end{align*}"}
+{"id": "6480.png", "formula": "\\begin{align*} | A | ^ 4 - m | A | ^ 2 - m ^ 2 H ^ 2 = 0 \\end{align*}"}
+{"id": "3972.png", "formula": "\\begin{align*} \\langle \\Psi _ t ( \\xi \\oplus \\eta ) , \\Psi _ t ( \\xi ' \\oplus \\eta ' ) \\rangle & = \\exp ( - t \\Vert \\xi - \\xi ' \\Vert ^ 2 ) \\exp ( - t \\Vert \\eta - \\eta ' \\Vert ^ 2 ) & \\\\ & = \\exp ( - t \\Vert ( \\xi \\oplus \\eta ) - ( \\xi ' \\oplus \\eta ' ) \\Vert ^ 2 ) & \\end{align*}"}
+{"id": "5517.png", "formula": "\\begin{align*} & N = q ^ { 2 n + 1 } - q ^ { n + 2 } + 2 q ^ { n + 1 } - 1 , \\\\ & k = q ^ { 2 n + 1 } - q ^ { n + 2 } + \\frac { 5 q ^ { n + 2 } + q ^ n - q ^ 3 + q ^ 2 - 2 q + 2 } { 2 ( q + 1 ) } - a ( 2 q ^ 2 - 2 q - 1 ) , \\\\ & d \\geq 2 a ( q ^ 2 - q - 1 ) - \\frac { q ^ 2 ( q ^ n - 2 q ^ { n - 1 } - q ^ { n - 2 } - q + 1 ) } { q + 1 } . \\end{align*}"}
+{"id": "2563.png", "formula": "\\begin{align*} B u \\in C ^ { \\prime \\ , - k } _ L ( M ) \\cap \\partial _ x ^ m C ^ { - \\infty } ( L ; \\Omega N L ) = \\partial _ x ^ m C ^ { \\prime \\ , m - k } ( L ; \\Omega N L ) \\ ; . \\end{align*}"}
+{"id": "5009.png", "formula": "\\begin{align*} = \\frac { 1 } { L } \\sum _ { \\theta , \\phi } \\bigg ( \\frac { E _ r ( \\theta , \\phi ) } { E _ { r m a x } } - \\frac { E _ a ( \\theta , \\phi ) } { E _ { a m a x } } \\bigg ) ^ 2 \\end{align*}"}
+{"id": "5037.png", "formula": "\\begin{align*} \\sum _ { k = 2 } ^ { n - 1 } \\left ( k + 1 \\right ) \\left ( 2 ( n - k ) + 1 \\right ) + O ( n ) = ( 1 / 3 ) n ^ { 3 } + ( 3 / 2 ) n ^ { 2 } + O ( n ) . \\end{align*}"}
+{"id": "5041.png", "formula": "\\begin{align*} 2 \\sum _ { k = 1 } ^ { n - 1 } \\left \\{ ( n - k ) ^ { 2 } + ( n - k ) \\right \\} + O ( 1 ) = ( 2 / 3 ) n ^ { 3 } + O ( n ) . \\end{align*}"}
+{"id": "3789.png", "formula": "\\begin{align*} \\widehat { \\Theta } ^ + _ 1 & = \\frac { 1 } { | \\widehat { \\mathcal { C } } _ 1 | } \\sum _ { i \\in \\widehat { \\mathcal { C } } _ 1 } \\tilde { \\Theta } _ i = \\frac { 1 } { | \\widehat { \\mathcal { C } } _ 1 | } \\sum _ { i \\in \\widehat { \\mathcal { C } } _ 1 \\cap \\mathcal { S } _ 1 } \\tilde { \\Theta } _ i + \\frac { 1 } { | \\widehat { \\mathcal { C } } _ 1 | } \\sum _ { i \\in \\widehat { \\mathcal { C } } _ 1 \\cap \\overline { \\mathcal { S } _ 1 } } \\tilde { \\Theta } _ i \\end{align*}"}
+{"id": "540.png", "formula": "\\begin{align*} \\log \\left | F \\left ( a _ { 1 } \\right ) \\right | \\leq \\left ( \\lambda _ 1 + \\delta \\right ) \\frac { T } { 2 } + \\sum _ { k = 2 } ^ { \\infty } \\log \\left | \\frac { a _ { 1 } - a _ { k } } { a _ { 1 } - \\bar { a } _ { k } } \\right | + \\frac { \\Im \\left ( a _ { 1 } \\right ) } { \\pi } \\int _ { - \\infty } ^ { \\infty } \\frac { \\log | F ( s ) | } { \\left | s - a _ { 1 } \\right | ^ { 2 } } \\mathrm { ~ d } s . \\end{align*}"}
+{"id": "1683.png", "formula": "\\begin{align*} e _ 1 \\mapsto \\begin{pmatrix} \\sqrt { - 2 } \\\\ 0 \\end{pmatrix} , h \\mapsto \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} , f _ 1 \\mapsto 0 . \\end{align*}"}
+{"id": "123.png", "formula": "\\begin{align*} \\begin{aligned} & \\| \\operatorname { d i v } \\vec { v } \\| ^ 2 \\eqsim \\langle A ( I - P ) \\vec { v } , \\vec { v } \\rangle \\\\ & \\quad = a \\left ( \\vec { v } - P \\vec { v } , \\vec { v } - P \\vec { v } \\right ) = \\left \\| \\varepsilon \\left ( \\vec { v } - P \\vec { v } \\right ) \\right \\| ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "8672.png", "formula": "\\begin{align*} & \\mathbb { C } T ^ * X = T ^ { * 1 , 0 } X \\oplus T ^ { * 0 , 1 } X \\oplus \\{ \\lambda \\omega _ 0 ; \\lambda \\in \\mathbb { C } \\ , \\} , \\\\ & \\mathbb { C } T X = T ^ { 1 , 0 } X \\oplus T ^ { 0 , 1 } X \\oplus \\{ \\lambda T ; \\lambda \\in \\mathbb { C } \\ , \\} . \\end{align*}"}
+{"id": "6096.png", "formula": "\\begin{align*} \\Delta ( 1 ) : = 1 \\otimes 1 , \\quad \\Delta ( x _ { 1 , \\varepsilon } ) : = 1 \\otimes x _ { 1 , \\varepsilon } + x _ { 1 , \\varepsilon } \\otimes 1 \\end{align*}"}
+{"id": "5042.png", "formula": "\\begin{align*} \\nu _ i ^ { \\beta , * } = \\nu ^ * _ i + \\gamma _ i ^ * ( 1 - \\beta ) + o ( 1 - \\beta ) \\beta \\nearrow 1 , \\end{align*}"}
+{"id": "5904.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { C } ( G ) ) = ( q - 1 ) ( q - 2 ) ^ { 2 } + q ( p - 1 ) ( p - 2 ) ^ { 2 } , M _ { 2 } ( \\mathcal { C } ( G ) ) = \\dfrac { ( q - 1 ) ( q - 2 ) ^ { 3 } + q ( p - 1 ) ( p - 2 ) ^ { 3 } } { 2 } , \\end{align*}"}
+{"id": "5265.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } G I \\left ( p \\| q \\right ) } { \\partial q _ { j } } = - \\sum _ { j } p _ { j } \\left [ \\frac { a } { a - b } \\sum _ { i } \\left ( \\frac { \\overline { T } _ { i } } { \\overline { p } _ { i } } \\right ) ^ { a - 1 } - \\frac { b } { a - b } \\sum _ { i } \\left ( \\frac { \\overline { T } _ { i } } { \\overline { p } _ { i } } \\right ) ^ { b - 1 } \\right ] \\frac { \\partial \\overline { T } _ { i } } { \\partial q _ { j } } \\end{align*}"}
+{"id": "4694.png", "formula": "\\begin{align*} r _ 1 = \\inf _ { j \\in \\N } r _ j . \\end{align*}"}
+{"id": "475.png", "formula": "\\begin{align*} f ( x ) = \\begin{cases} 0 , & x = 0 , \\\\ a _ 0 x ^ { r _ 0 } , & x \\in C = C _ 0 , \\\\ a _ 1 x ^ { r _ 1 } , & x \\in C _ 1 , \\\\ \\vdots & \\vdots \\\\ a _ { d - 1 } x ^ { r _ { d - 1 } } , & x \\in C _ { d - 1 } , \\end{cases} \\end{align*}"}
+{"id": "4859.png", "formula": "\\begin{align*} \\Big | \\int _ { E _ m + y _ 0 } g ( x ) d x - \\int _ { E _ m } g ( x ) d x \\Big | \\\\ & = \\int _ { E _ m } g ( x + y _ 0 ) - g ( x ) d x \\\\ & = y _ 0 \\cdot \\int _ { E _ m } \\nabla g ( x ) d x + \\frac { 1 } { 2 } \\int _ { E _ m } \\Big ( \\langle D ^ 2 g ( x ) y _ 0 , y _ 0 \\rangle + 2 o _ x ( | y _ 0 | ^ 2 ) \\Big ) d x \\\\ & = \\frac { 1 } { 2 } \\int _ { E _ m } ( \\langle D ^ 2 g ( x ) y _ 0 , y _ 0 \\rangle + 2 o _ x ( | y _ 0 | ^ 2 ) ) d x ; \\\\ \\end{align*}"}
+{"id": "5642.png", "formula": "\\begin{align*} P _ \\mathcal { F } = \\begin{bmatrix} 1 & 1 6 3 8 & 2 4 5 7 \\\\ 1 & 3 8 & - 3 9 \\\\ 1 & - 2 6 & 2 5 \\end{bmatrix} \\end{align*}"}
+{"id": "1783.png", "formula": "\\begin{align*} f _ { j } ^ 1 = ( e ^ { a _ 1 + j a _ 2 } - 1 ) z _ j + \\delta _ { j , n - 1 } i a _ 3 z _ 1 . \\end{align*}"}
+{"id": "3414.png", "formula": "\\begin{align*} I _ { \\varphi } ^ 3 ( \\phi ( x ) ) & = u _ { \\varphi } ( x ) I _ { \\varphi } ( \\phi ( x ) ) \\\\ & = I _ { \\varphi } ( u _ { \\varphi } ( x ) \\phi ( x ) ) \\\\ & = \\iota ^ { \\ast } \\circ \\varphi ( \\overline { u _ { \\varphi } ( x ) } c ( \\phi ( x ) ) ) \\\\ & = \\iota ^ { \\ast } ( \\overline { u _ { \\varphi } ( x ) } \\varphi \\circ c ( \\phi ( x ) ) ) \\\\ & = \\overline { u _ { \\varphi } ( \\iota ^ { - 1 } ( x ) ) } I _ { \\varphi } ( \\phi ( x ) ) . \\end{align*}"}
+{"id": "6268.png", "formula": "\\begin{align*} X \\setminus \\bigsqcup _ { i = 1 } ^ n S _ i V _ i \\precsim _ G \\bigsqcup _ { i = 1 } ^ n S _ i ^ \\prime V _ i . \\end{align*}"}
+{"id": "9010.png", "formula": "\\begin{align*} e _ { i j } a = \\sum _ { j \\le k } a ( j , k ) e _ { i k } , \\ \\ a e _ { i j } = \\sum _ { l \\le i } a ( l , i ) e _ { l j } , \\end{align*}"}
+{"id": "6179.png", "formula": "\\begin{align*} \\begin{aligned} & - A _ 1 ^ T \\lambda ^ k + \\beta ^ k A _ 1 ^ T ( A _ 1 { x } _ 1 ^ { k + 1 } + A _ 2 \\hat { x } _ 2 ^ k - b ) \\\\ = & - A _ 1 ^ T \\lambda ^ k + \\gamma ( 1 - \\tau ^ k ) \\beta ^ k A _ 1 ^ T ( A x ^ k - b ) + \\tau ^ k \\beta ^ k A _ 1 ^ T ( A _ 1 \\bar { x } _ 1 ^ { k + 1 } \\\\ & + A _ 2 \\bar { x } _ 2 ^ k - b ) + ( 1 - \\gamma ) ( 1 - \\tau ^ k ) \\beta ^ k A _ 1 ^ T ( A x ^ k - b ) \\\\ = & - A _ 1 ^ T \\widetilde { \\lambda } ^ k + ( 1 - \\gamma ) ( 1 - \\tau ^ k ) \\beta ^ k A _ 1 ^ T ( A \\breve { x } ^ { k - 1 } - b ) , \\end{aligned} \\end{align*}"}
+{"id": "8987.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c l } F ( D ^ 2 u _ n ) - \\beta { u _ n ( r ^ 2 + 1 / n ) ^ { - \\gamma / 2 } } = ( r ^ 2 + 1 / n ) ^ { - \\gamma / 2 } f ( r ) & \\hbox { i n } \\ B ( 0 , 1 ) \\\\ u _ n = b & \\hbox { o n } \\ \\partial B ( 0 , 1 ) \\end{array} \\right . \\end{align*}"}
+{"id": "7076.png", "formula": "\\begin{align*} ( \\partial _ t + \\lambda - \\Delta + b \\cdot \\nabla ) u = 0 , u ( 0 + ) = f \\end{align*}"}
+{"id": "245.png", "formula": "\\begin{align*} \\frac { d ^ 2 x _ 2 } { d t _ 2 ^ 2 } + 1 = 0 . \\end{align*}"}
+{"id": "7320.png", "formula": "\\begin{align*} \\frac { 1 } { m } \\sum _ { j = 0 } ^ { m - 1 } & \\frac { e ^ { 2 \\pi i \\frac { j r } { m } } } { s + 2 \\sin ^ 2 \\left ( \\pi \\frac { ( j + \\beta ) } { m } \\right ) } \\\\ & = \\frac { e ^ { - 2 \\pi i \\beta \\frac { \\ell } { m } } } { \\sqrt { s ^ 2 + 2 s } } \\frac { \\sinh \\left ( ( m - \\ell ) \\cosh ^ { - 1 } ( s + 1 ) \\right ) + e ^ { 2 \\pi i \\beta } \\sinh \\left ( \\ell \\cosh ^ { - 1 } ( s + 1 ) \\right ) } { \\cosh \\left ( m \\cosh ^ { - 1 } ( s + 1 ) \\right ) - \\cos 2 \\pi \\beta } . \\end{align*}"}
+{"id": "3943.png", "formula": "\\begin{align*} & \\overline { F } _ j ( t ) = \\gamma ^ { - 3 } F _ j ( t ) , 1 \\leq j \\leq N , \\\\ & \\Rightarrow \\overline { F } ^ N ( t ) = \\gamma ^ { - 3 } F ^ { N } ( t ) . \\end{align*}"}
+{"id": "9038.png", "formula": "\\begin{align*} T S ^ i - S ^ i T = 0 , S ^ i S ^ j + S ^ j S ^ i = 0 ( i , j \\in [ N ] ) . \\end{align*}"}
+{"id": "3593.png", "formula": "\\begin{align*} ( T f ) ( z , w ) = \\alpha ( \\tau ' ( z ) ) ^ { \\frac { 1 } { p } } f ( \\tau ( z ) , w \\sigma ( z ) ) , \\end{align*}"}
+{"id": "1596.png", "formula": "\\begin{align*} d \\textup { v o l } _ M ( e _ 1 , \\dots , e _ n ) = 1 , \\end{align*}"}
+{"id": "5335.png", "formula": "\\begin{align*} \\nu _ { \\pi _ k } = \\min \\ , \\left \\{ \\nu ^ { S _ k } _ j : j \\in S _ k \\right \\} . \\end{align*}"}
+{"id": "8111.png", "formula": "\\begin{align*} F = B ''' _ \\varpi . \\end{align*}"}
+{"id": "8242.png", "formula": "\\begin{align*} P _ { \\hat { { \\bf S } } _ i | { \\bf X } ^ i , Y ^ i } ( { \\bf s } | { \\bf x } ^ i , y ^ i ) & = P _ { \\hat { { \\bf S } } _ i | { \\bf X } _ i , Y _ i , \\Omega _ { i - 1 } } ( { \\bf s } | { \\bf x } _ i , y _ i , \\omega _ { i - 1 } ( { \\bf x } ^ i , y ^ i ) ) \\\\ & = \\begin{cases} \\frac { 1 } { | \\omega _ { i } ( { \\bf x } ^ i , y ^ i ) | } & { \\bf s } \\in \\Omega _ { i } \\\\ 0 & \\end{cases} , \\end{align*}"}
+{"id": "4369.png", "formula": "\\begin{gather*} \\mathcal { X } _ \\mathcal { Q } : = \\{ x \\in \\mathcal { X } : \\Delta u _ q ^ T l _ q ( x ) - \\Delta u _ k ^ T l _ k ( x ) \\geq 0 \\ \\forall q \\in \\mathcal { Q } , \\ \\Delta u _ q ^ T l _ q ( x ) - \\Delta u _ k ^ T l _ k ( x ) \\leq 0 \\ \\forall q \\in [ m ] \\setminus \\mathcal { Q } \\} . \\end{gather*}"}
+{"id": "5082.png", "formula": "\\begin{align*} \\varepsilon _ { \\theta } ( x ) = \\sum _ { t \\in T ( \\mathbb { F } _ { q ^ k } ) } \\theta ( t ) \\delta _ t . \\end{align*}"}
+{"id": "2724.png", "formula": "\\begin{align*} & \\Phi ^ { ( s + 1 ) } _ { \\alpha } : = d \\Phi ^ { ( s ) } _ { \\alpha } ( X _ { T } ) : \\approx 0 \\ \\ \\ ( s = 2 , 3 , \\cdots , m _ { \\alpha } - 1 ) , \\\\ & \\Phi ^ { ( m _ { \\alpha } + 1 ) } _ { \\alpha } : = d \\Phi ^ { ( m _ { \\alpha } ) } _ { \\alpha } ( X _ { T } ) = C _ { \\alpha s } ^ { \\beta } \\Phi ^ { ( s ) } _ { \\beta } , \\end{align*}"}
+{"id": "6215.png", "formula": "\\begin{align*} \\phi ( \\xi _ \\gamma ) = \\gamma \\hbox { a n d } \\phi _ \\lambda ' ( \\xi _ \\gamma ) = \\lambda , \\end{align*}"}
+{"id": "7722.png", "formula": "\\begin{align*} E ( x _ { 0 } , t ) = \\widetilde { E } ( x _ { 0 } , t ) \\leq C \\end{align*}"}
+{"id": "939.png", "formula": "\\begin{align*} h _ V ( g ) = g - \\pi _ V ( g ) . \\end{align*}"}
+{"id": "6822.png", "formula": "\\begin{align*} \\sum _ { i , j } \\beta _ { i j } k _ { i j } ^ { N - 1 } + K _ { N - 1 } = 0 . \\end{align*}"}
+{"id": "7585.png", "formula": "\\begin{align*} [ \\Delta ^ { 2 } , i \\Gamma _ { \\varphi _ { R } } ] = \\Delta [ \\Delta , i \\Gamma _ { \\varphi _ { R } } ] + [ \\Delta , i \\Gamma _ { \\varphi _ { R } } ] \\Delta = 2 \\partial _ { k } [ \\Delta , i \\Gamma _ { \\varphi _ { R } } ] \\partial _ { k } + [ \\partial _ { k } , [ \\partial _ { k } , [ \\Delta , i \\Gamma _ { \\varphi _ { R } } ] ] ] . \\end{align*}"}
+{"id": "4783.png", "formula": "\\begin{align*} \\begin{cases} \\forall y ^ X , z ^ X \\left ( z \\in A y \\rightarrow d \\leq _ \\mathbb { R } \\norm { z } _ X \\right ) , \\\\ \\forall k ^ \\mathbb { N } \\exists y , z \\preceq _ X f ( k ) 1 _ X \\left ( z \\in A y \\land \\norm { z } _ X - d \\leq _ \\mathbb { R } 2 ^ { - k } \\right ) . \\end{cases} \\end{align*}"}
+{"id": "3689.png", "formula": "\\begin{align*} L ( C _ { n , d } ) _ { i , j } = \\begin{cases} 2 d & i = j , \\\\ - 1 & | j - i | \\leq d n - | j - i | \\leq d , \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "8879.png", "formula": "\\begin{align*} V _ R ( t ) : = \\int _ { \\R ^ n } \\phi _ R ( x ) | u ( x , \\cdot ) | ^ 2 \\ , d x , V _ R ' ( t ) = M _ R ( t ) : = 2 \\Im \\int _ { \\R ^ n } \\bar u \\nabla \\phi _ R \\cdot \\nabla u d x . \\end{align*}"}
+{"id": "7890.png", "formula": "\\begin{align*} f ( \\mathbf { x } ) = \\frac { q } { p } \\left ( \\sum _ { i = 1 } ^ { m - 1 } a _ { i } x _ { \\pi ( i ) } x _ { \\pi ( i + 1 ) } + \\sum _ { k = 1 } ^ { m } d _ { k } x _ { k } ^ { 2 } \\right ) + \\mathcal { L } _ { c } ( x ) = \\frac { q } { p } \\mathbf { x } ^ { T } \\mathbf { A } \\mathbf { x } + \\mathcal { L } _ { c } ( \\mathbf { x } ) \\end{align*}"}
+{"id": "1055.png", "formula": "\\begin{align*} \\beta _ k = - \\int _ { - \\pi } ^ { \\pi } e ^ { - i k \\theta } h ( e ^ { i \\theta } ) ^ * h _ { \\sharp } ( e ^ { i \\theta } ) ^ { - 1 } \\frac { d \\theta } { 2 \\pi } , k \\in \\Z . \\end{align*}"}
+{"id": "4204.png", "formula": "\\begin{align*} & - 2 6 4 \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 2 0 ) } = \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right . \\\\ & \\left . \\cdot { \\rm c h } ( \\widetilde { T X } + 2 \\wedge ^ 2 \\widetilde { L _ R \\otimes C } - \\widetilde { L _ R \\otimes C } \\otimes \\widetilde { L _ R \\otimes C } + \\widetilde { L _ R \\otimes C } ) \\right \\} ^ { ( 2 0 ) } . \\end{align*}"}
+{"id": "7232.png", "formula": "\\begin{align*} \\sigma ( a \\ast a ' ) = \\sigma ( a ) \\star \\sigma ( a ' ) a , a ' \\in [ n ] , \\end{align*}"}
+{"id": "2209.png", "formula": "\\begin{align*} - \\psi _ { 1 , r r z } - \\psi _ { 1 , z z z } - { 3 \\over r } \\psi _ { 1 , r z } = \\omega _ { 1 , z } \\end{align*}"}
+{"id": "5033.png", "formula": "\\begin{align*} \\nu _ j ^ { S \\cup \\{ j \\} } = \\frac { r _ j ^ S / a _ { j j } ^ S } { w _ j ^ S / a _ { j j } ^ S } = \\frac { r _ j ^ S } { w _ j ^ S } = \\nu _ j ^ { S } . \\end{align*}"}
+{"id": "2238.png", "formula": "\\begin{align*} \\phi _ s ^ { ( 1 ) } ( t ) = \\left ( \\phi ^ i ( t ) , \\frac { \\partial \\phi ^ i } { \\partial t ^ \\alpha } \\Big \\vert _ t , s ^ \\alpha ( t ) \\right ) . \\end{align*}"}
+{"id": "7587.png", "formula": "\\begin{align*} [ \\Delta ^ { 2 } , i \\Gamma _ { \\varphi _ { R } } ] = 8 \\partial ^ { 2 } _ { k l } ( \\partial ^ { 2 } _ { l m } \\varphi _ { R } ) \\partial ^ { 2 } _ { m k } + 4 \\partial _ { k } ( \\partial ^ { 2 } _ { k l } \\Delta \\varphi _ { R } ) \\partial _ { l } + 2 \\partial _ { k } ( \\Delta ^ { 2 } \\varphi _ { R } ) \\partial _ { k } + \\Delta ^ { 3 } \\varphi _ { R } . \\end{align*}"}
+{"id": "3845.png", "formula": "\\begin{align*} \\beta ^ n ( \\{ e _ x \\} \\times \\{ e _ y \\} \\times [ 0 , t ] ) = 0 , \\mbox { a . s . } \\end{align*}"}
+{"id": "5706.png", "formula": "\\begin{align*} \\psi _ { i , 1 } : = ( 0 , 2 ^ { - 1 / 2 } m \\varphi _ i ) , \\ \\psi _ { i , 2 } : = ( 2 ^ { - 1 / 2 } m \\beta _ i ^ { - 1 } \\varphi _ i , 0 ) . \\end{align*}"}
+{"id": "5032.png", "formula": "\\begin{align*} \\nu _ i ^ { S \\cup \\{ j \\} } & = \\frac { \\nu _ i ^ S w _ i ^ S + \\nu _ j ^ S \\frac { w _ j ^ S } { a _ { j j } ^ S } a _ { i j } ^ S } { w _ i ^ S + \\frac { w _ j ^ S } { a _ { j j } ^ S } a _ { i j } ^ S } = \\nu _ j ^ S - \\frac { w _ i ^ S } { w _ i ^ S + \\frac { w _ j ^ S } { a _ { j j } ^ S } a _ { i j } ^ S } ( \\nu _ j ^ S - \\nu _ i ^ S ) = \\nu _ j ^ S - \\frac { w _ i ^ S } { w _ i ^ { S \\cup \\{ j \\} } } ( \\nu _ j ^ S - \\nu _ i ^ S ) . \\end{align*}"}
+{"id": "64.png", "formula": "\\begin{align*} F ( D u , D ^ 2 u ) : = \\abs { D u } ^ { \\gamma } \\big ( \\Delta u + ( p - 2 ) \\Delta _ { \\infty } ^ { N } u \\big ) \\end{align*}"}
+{"id": "6781.png", "formula": "\\begin{align*} \\dot { x } _ i = f _ i ( x ^ \\alpha ) ( i = 1 , \\ldots , n ) , \\end{align*}"}
+{"id": "7754.png", "formula": "\\begin{align*} N _ { n } = [ 2 ^ { n + 1 } \\frac { | \\log \\epsilon _ { n } | } { 2 \\pi h _ { 1 } } ] + 1 , \\ \\ a _ { n } = 8 ^ { n + 1 } m , \\end{align*}"}
+{"id": "5909.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { C } ( G ) ) & = \\dfrac { ( q - 1 ) ( q - 1 - 1 ) ^ { 3 } } { 2 } + q \\cdot \\dfrac { ( p - 1 ) ( p - 1 - 1 ) ^ { 3 } } { 2 } \\\\ & = \\dfrac { ( q - 1 ) ( q - 2 ) ^ { 3 } + q ( p - 1 ) ( p - 2 ) ^ { 3 } } { 2 } . \\end{align*}"}
+{"id": "3977.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ { \\nu } \\gamma _ { i r k } \\widetilde { \\gamma } _ { j p r } = \\sum _ { r = 1 } ^ { \\nu } \\widetilde { \\gamma } _ { i r s } \\widetilde { \\gamma } _ { j p r } = 0 , \\quad \\left ( 1 \\leq i , j , k \\leq n , 1 \\leq p , s \\leq \\nu \\right ) \\end{align*}"}
+{"id": "1844.png", "formula": "\\begin{align*} \\Sigma _ I : = \\{ ( y , - 1 ) \\in Y \\times \\Z / 2 : G _ y = \\Z / 2 \\} , \\Sigma _ { V } : = \\{ ( y , g ) \\in Y \\times V _ 4 : G _ y = V _ 4 , \\ ; \\ ; g \\neq 1 \\} . \\end{align*}"}
+{"id": "4073.png", "formula": "\\begin{align*} E _ i ( \\pm i \\sigma ) = e ^ { \\pm i \\sigma } a ( \\pm \\sigma ) , \\ , \\ , \\ , \\ , \\end{align*}"}
+{"id": "7189.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } \\partial _ t R _ v - \\partial _ x ^ 2 R _ v & = ( 1 - ( v _ 0 + V ) ^ 2 + \\varphi _ 1 ) R _ v - \\frac { 1 } { 3 } R _ v ^ { 3 } - R _ w - R _ v ^ 2 F _ v - R _ v F _ v ^ 2 - \\frac { 1 } { 3 } F _ v ^ 3 + \\varphi _ 2 ( R _ v + F _ v ) ^ 2 , \\\\ \\partial _ t R _ w - \\rho \\partial _ x ^ 2 R _ w & = \\varepsilon ( R _ v - \\gamma R _ w ) , \\\\ R _ v ( x , 0 ) = 0 , & R _ w ( x , 0 ) = 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "5713.png", "formula": "\\begin{align*} \\mathbf { L } ^ \\dagger \\psi _ { i , 1 } = 2 ^ { - 1 } m \\psi _ { i , 1 } - \\beta _ i \\psi _ { i , 2 } , \\ \\mathbf { L } ^ \\dagger \\psi _ { i , 2 } = 2 ^ { - 1 } m \\psi _ { i , 2 } + \\beta _ i \\psi _ { i , 1 } . \\end{align*}"}
+{"id": "1955.png", "formula": "\\begin{align*} \\varepsilon _ 0 : = \\frac { \\log \\nu } { 2 \\log \\delta } = \\frac { 1 } { 2 \\log _ { \\frac { 1 } { 2 } } \\delta } \\in ( 0 , 1 ) . \\end{align*}"}
+{"id": "3618.png", "formula": "\\begin{align*} b ( ( \\sigma \\sigma _ 1 ) ^ 2 - 1 ) = b ( \\sigma \\sigma _ 1 - 1 ) ( \\sigma ^ 2 \\sigma _ 1 + \\sigma ) . \\end{align*}"}
+{"id": "7806.png", "formula": "\\begin{align*} { \\ ! \\hat { s } } _ { k } \\ ! \\ ! = \\ ! \\ ! \\left ( \\boldsymbol { v } _ { k } \\right ) \\ ! ^ { \\mathrm { H } } \\ ! \\boldsymbol { g } _ { k } \\sqrt { p _ { k } } s _ { k } \\ ! + \\ ! \\left ( \\boldsymbol { v } _ { k } \\right ) \\ ! ^ { \\mathrm { H } } \\ ! \\sum _ { l \\neq k } \\ ! \\boldsymbol { g } _ { l } \\sqrt { p _ { l } } s _ l \\ ! + \\ ! \\left ( \\boldsymbol { v } _ { k } \\right ) ^ { \\mathrm { H } } \\ ! \\boldsymbol { z } , \\ \\ \\forall k \\ ! \\in \\ ! \\mathcal { K } . \\end{align*}"}
+{"id": "4174.png", "formula": "\\begin{align*} \\varphi \\big ( g ^ { j } _ 1 g _ 3 \\big ) \\varphi \\big ( g ^ { i } _ 1 g _ 2 \\big ) = \\varphi ( g _ 2 g ^ { n } _ 1 g _ 3 ) \\ , , \\end{align*}"}
+{"id": "4577.png", "formula": "\\begin{align*} \\mathcal { A } ( [ \\omega ] ) = \\frac { ( c _ 1 [ \\omega ] ) ^ 2 } { [ \\omega ] ^ 2 } - \\frac { 1 } { 3 2 \\pi ^ 2 } \\mathcal { F } ( \\mathfrak { E } , [ \\omega ] ) . \\end{align*}"}
+{"id": "3244.png", "formula": "\\begin{align*} \\hat { \\phi } ( p ) & = 2 \\pi \\int _ { - \\infty } ^ \\infty d t \\int _ 0 ^ \\infty r ^ 2 \\ : d r \\int _ { - 1 } ^ 1 d \\cos \\vartheta \\ ; \\phi ( t , r ) \\ : e ^ { i p ^ 0 t - i | \\vec { p } \\ , | r \\cos \\vartheta } \\\\ & = \\frac { 2 \\pi } { | \\vec { p } \\ , | } \\ : \\int _ { - \\infty } ^ \\infty d \\omega \\int _ 0 ^ \\infty r \\ : d r \\ : \\phi ( t , r ) \\ : e ^ { i p ^ 0 t } \\ , \\Big ( e ^ { i | \\vec { p } \\ , | r } - e ^ { - i | \\vec { p } \\ , | r } \\Big ) \\ : , \\end{align*}"}
+{"id": "8375.png", "formula": "\\begin{align*} R ^ { + } ( w ^ { - 1 } ) = R _ { 1 } \\sqcup R _ { 2 } . \\end{align*}"}
+{"id": "8327.png", "formula": "\\begin{align*} [ \\widetilde { \\Pi } , \\widetilde { W } _ t ] _ { S N } + \\frac { 1 } { 2 } [ \\widetilde { W } _ t , \\widetilde { W } _ t ] _ { \\widetilde { \\gamma } } { - } \\frac { 1 } { 6 } ( \\widetilde { W } _ t ^ \\sharp \\wedge \\widetilde { W } _ t ^ \\sharp \\wedge \\widetilde { W } _ t ^ \\sharp ) \\Upsilon ^ { \\widetilde { G } } _ { T ( M \\times I ^ 2 ) } = 0 . \\end{align*}"}
+{"id": "1322.png", "formula": "\\begin{align*} \\bigl \\langle \\mathrm { d } ( f \\circ \\pi ) , H _ p v \\bigr \\rangle & = \\bigl \\langle \\mathrm { d } ( f \\circ \\pi ) , T \\mu ( v ) - V _ p ( \\nabla _ v ^ \\ast \\mu ) \\bigr \\rangle \\\\ & = \\bigl \\langle \\mathrm { d } ( f \\circ \\pi \\circ \\mu ) , v \\bigr \\rangle - 0 , \\\\ & = \\langle \\mathrm { d } f , v \\rangle . \\end{align*}"}
+{"id": "3824.png", "formula": "\\begin{align*} p ( n ) = \\frac { 2 \\pi } { ( 2 4 n - 1 ) ^ \\frac 3 4 } \\sum _ { k \\ge 1 } \\frac { A _ k ( n ) } { k } I _ \\frac 3 2 \\left ( \\frac { \\pi \\sqrt { 2 4 n - 1 } } { 6 k } \\right ) , \\end{align*}"}
+{"id": "584.png", "formula": "\\begin{align*} \\tilde h _ 1 = h _ \\emptyset = 1 _ { [ 0 , 1 ] } . \\end{align*}"}
+{"id": "6216.png", "formula": "\\begin{align*} \\delta ( F , \\phi _ 0 ) ( \\phi ) : = \\frac { F ( \\phi ) - F ( \\phi _ 0 ) } { \\phi - \\phi _ 0 } . \\end{align*}"}
+{"id": "8428.png", "formula": "\\begin{align*} \\ell _ { \\widetilde W _ M } ( \\tilde x _ 1 ) = \\cdots = \\ell _ { \\widetilde W _ M } ( \\tilde x _ { n } ) > \\ell _ { \\widetilde W _ M } ( \\tilde x _ { n + 1 } ) . \\end{align*}"}
+{"id": "6643.png", "formula": "\\begin{align*} ( H \\psi ) _ { n } = \\psi _ { n + 1 } + \\psi _ { n - 1 } + v ( x + n \\alpha ) \\psi _ { n } = E \\psi _ { n } , \\end{align*}"}
+{"id": "7251.png", "formula": "\\begin{align*} R _ k ( x ) = \\sum \\limits _ { j = 0 } ^ { \\textnormal { d e g } ( Q _ k ) } [ Q _ k ] _ j P _ j ( x ) , k \\geq 0 , \\end{align*}"}
+{"id": "191.png", "formula": "\\begin{align*} E _ 1 \\ \\not = \\ E _ 2 \\Leftrightarrow F _ 1 \\ \\not = \\ F _ 2 \\Leftrightarrow [ ( E _ 1 \\cap F _ 2 ) \\cup ( E _ 2 \\cap F _ 1 ) ] \\ \\not = \\ \\emptyset . \\end{align*}"}
+{"id": "43.png", "formula": "\\begin{align*} \\tilde { d } \\circ U = U \\circ d \\ . \\end{align*}"}
+{"id": "6678.png", "formula": "\\begin{align*} \\dd \\ ! \\big ( H / \\big ( ( H \\cap F _ j ) \\Phi _ p ( N ) \\big ) \\big ) = \\dd \\ ! \\big ( H F _ j / \\Phi _ p ( N ) F _ j \\big ) , j \\in \\N , \\end{align*}"}
+{"id": "6496.png", "formula": "\\begin{align*} \\chi _ q ( L ^ { \\mathrm { B } _ 2 } ( Y _ { 1 , - 7 } ) ) = Y _ { 1 , - 7 } + Y _ { 2 , - 6 } Y _ { 2 , - 4 } Y _ { 1 , - 3 } ^ { - 1 } + Y _ { 2 , - 6 } Y _ { 2 , - 2 } ^ { - 1 } + Y _ { 1 , - 5 } Y _ { 2 , - 4 } ^ { - 1 } Y _ { 2 , - 2 } ^ { - 1 } + Y _ { 1 , - 1 } ^ { - 1 } . \\end{align*}"}
+{"id": "5626.png", "formula": "\\begin{align*} d ( T x , x ) \\leq d ( T x , T u ) + d ( T u , x ) < \\frac { \\epsilon } { 2 } + \\frac { \\epsilon } { 2 } = \\epsilon , \\end{align*}"}
+{"id": "4881.png", "formula": "\\begin{align*} 2 g - 2 = ( K _ { \\Sigma ' } + C ' ) \\cdot C ' = ( a - e ) \\theta \\cdot C ' = 2 ( a - e ) , \\end{align*}"}
+{"id": "1767.png", "formula": "\\begin{align*} \\pi _ r \\circ \\mathrm { F l } ^ { X _ { 1 - n } } _ { t } ( z ) = z _ { r } + \\delta _ { r , j } t - \\delta _ { r , 0 } 2 t z _ { 1 } , \\end{align*}"}
+{"id": "4085.png", "formula": "\\begin{align*} V R _ 0 ( z e ^ { 2 \\pi i m } ) \\chi = \\sum _ { k = 1 } ^ { \\frac { d - 1 } { 2 } } V B _ k ( r , z ) + 2 \\pi m i z ^ { d - 1 } V \\mathcal { E } ^ * _ { \\chi } ( \\bar { z } ) \\mathcal { E } _ { \\chi } ( z ) \\end{align*}"}
+{"id": "3669.png", "formula": "\\begin{align*} q ^ T L ( G , p ) q & = ~ x ^ T L ( G , p ) x \\\\ & \\le ~ \\frac { a ( G ) } { n } x ^ T L ( K _ n , p ) x \\\\ & = ~ \\frac { a ( G ) } { n } q ^ T L ( K _ n , p ) q \\\\ & \\le ~ a ( G ) \\| q \\| ^ 2 . \\end{align*}"}
+{"id": "6139.png", "formula": "\\begin{align*} \\varphi _ n ( e _ { g , h } ) = \\frac { 1 } { | \\mathcal { F } _ n | } \\lambda _ { g h ^ { - 1 } } \\end{align*}"}
+{"id": "6953.png", "formula": "\\begin{align*} [ V _ - ] = \\{ [ z _ 1 : \\cdots : z _ n : 1 ] \\in \\mathbb { P } _ { \\mathbb { C } } ^ n : \\lvert z _ 1 \\rvert ^ 2 + \\cdots + \\lvert z _ n \\rvert ^ 2 < 1 \\} \\end{align*}"}
+{"id": "7236.png", "formula": "\\begin{align*} a \\ \\ast _ { \\sigma \\circ \\tau } \\ a ' & = \\sigma \\circ \\tau ( ( \\sigma \\circ \\tau ) ^ { - 1 } ( a ) \\ast ( \\sigma \\circ \\tau ) ^ { - 1 } ( a ' ) ) \\\\ & = \\sigma ( \\tau ( \\tau ^ { - 1 } ( \\sigma ^ { - 1 } ( a ) ) \\ast \\tau ^ { - 1 } ( \\sigma ^ { - 1 } ( a ' ) ) ) ) \\\\ & = \\sigma ( \\sigma ^ { - 1 } ( a ) \\ast _ \\tau \\sigma ^ { - 1 } ( a ' ) ) \\\\ & = a \\ { ( \\ast _ \\tau ) } _ \\sigma \\ a ' a , a ' \\in [ n ] \\sigma , \\tau \\in S _ n , \\end{align*}"}
+{"id": "519.png", "formula": "\\begin{align*} O \\left ( d 2 ^ d + d \\sum _ { k = 0 } ^ h { 2 ^ { ( k + 1 ) d } } \\right ) = O ( d 2 ^ { ( h + 1 ) d } ) . \\end{align*}"}
+{"id": "8852.png", "formula": "\\begin{align*} 0 < \\frac 1 { a _ 1 } , \\frac 1 { b _ 1 } < 1 , \\frac 1 { a _ 1 } + \\frac 1 { b _ 1 } = \\frac { n + 2 } { 2 n } + \\frac { \\alpha } { n } , \\end{align*}"}
+{"id": "2795.png", "formula": "\\begin{align*} L ' _ { T } = L _ { T } - \\frac { d } { d t } \\left [ \\Psi ^ { a } \\Xi _ { a } \\right ] , \\end{align*}"}
+{"id": "5046.png", "formula": "\\begin{align*} | v ( x ) - v ( y ) | & = | u ( x ) \\eta ( x ) - u ( x ) \\eta ( y ) + u ( x ) \\eta ( y ) - u ( y ) \\eta ( y ) | \\\\ & \\le u ( x ) | \\eta ( x ) - \\eta ( y ) | + | u ( x ) - u ( y ) | . \\end{align*}"}
+{"id": "2478.png", "formula": "\\begin{align*} \\phi ( \\xi ) = V ( t , x ) , \\xi = x - c t , c > 0 \\end{align*}"}
+{"id": "6106.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & - i \\\\ i & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "7972.png", "formula": "\\begin{align*} \\begin{aligned} i _ { ( \\ast e _ { v } ^ { 1 } ) ^ { \\sharp } } d v \\wedge e _ { v } ^ { 2 } = \\ast \\big ( ( \\ast e _ { v } ^ { 1 } ) \\wedge ( \\ast d v ) \\big ) \\wedge e _ { v } ^ { 2 } = - \\ast \\big ( ( \\ast e _ { v } ^ { 2 } ) \\wedge ( \\ast d v ) \\big ) \\wedge e _ { v } ^ { 1 } = - i _ { ( \\ast e _ { v } ^ { 2 } ) ^ { \\sharp } } d v \\wedge e _ { v } ^ { 1 } . \\end{aligned} \\end{align*}"}
+{"id": "4855.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { 1 + | \\nabla ^ { + } u _ j ( \\hat { z } ) | ^ 2 } \\sqrt { 1 + | \\nabla ^ { + } u _ j ( \\hat { x } ^ * ) | ^ 2 } } \\geq \\frac { 1 } { 1 + \\tilde { C } ^ 2 } = : C _ 0 . \\end{align*}"}
+{"id": "3158.png", "formula": "\\begin{align*} ( u ( t ) - I _ 3 ) \\ , P = P \\ , ( u ^ \\sharp ( t ) - I _ 3 ) . \\end{align*}"}
+{"id": "5995.png", "formula": "\\begin{align*} \\Delta u + h u = 0 ~ ~ ~ ~ M . \\end{align*}"}
+{"id": "3378.png", "formula": "\\begin{align*} \\frac { 1 } { T } \\sum _ { t = 2 } ^ { T + 1 } \\Delta _ { t } & \\le 4 8 R _ { 1 } \\max \\left \\{ 5 2 ^ { \\frac { 1 } { p } } T ^ { \\frac { 1 - p } { p } } \\left ( 1 + \\log T \\right ) ^ { \\frac { 2 } { p } } \\sigma \\gamma ^ { \\frac { p - 1 } { p } } ; 2 \\left ( 2 L R _ { 1 } + L R _ { 0 } + \\mu \\sigma + \\left \\Vert g _ { 0 } \\right \\Vert _ { * } \\right ) T ^ { - 1 } \\gamma \\right \\} = \\widetilde { O } \\left ( T ^ { \\frac { 1 - p } { p } } \\right ) . \\end{align*}"}
+{"id": "302.png", "formula": "\\begin{align*} F ( z _ 1 , z _ 2 ) = \\begin{cases} a \\left ( z _ 1 - \\dfrac { 2 z _ 2 ^ 2 } { c - z _ 2 } \\right ) \\\\ a \\left ( z _ 2 - \\dfrac { b z _ 1 ^ 2 } { ( d - z _ 1 ) ^ 2 } \\right ) , \\end{cases} \\end{align*}"}
+{"id": "2235.png", "formula": "\\begin{align*} G ( t ) = c ( | f _ 1 ( t ) | _ { 4 , \\Omega } ^ 4 + | F _ 1 ( t ) | _ { 2 , \\Omega } ^ 2 ) . \\end{align*}"}
+{"id": "8080.png", "formula": "\\begin{align*} n u ^ - = p ^ - ( n ) u ^ + ( n ) , \\forall n \\in N _ 2 ^ + . \\end{align*}"}
+{"id": "6104.png", "formula": "\\begin{align*} \\overline { \\nabla } _ X Y = \\nabla _ X Y + k ( X , Y ) \\ , e _ 0 \\end{align*}"}
+{"id": "542.png", "formula": "\\begin{align*} u ( \\tau ) = S ( \\tau ) u _ { 0 } - B ( \\tau ) f , \\tau \\in ( 0 , T ] , \\end{align*}"}
+{"id": "8874.png", "formula": "\\begin{align*} \\big \\| | x | ^ { - \\tau } | u | ^ p \\big \\| ^ 2 _ { { \\frac { 2 n } { \\alpha + n } } } = \\big \\| | x | ^ { - \\frac { \\tau } { p } } u \\big \\| ^ { 2 p } _ { { \\frac { 2 n p } { \\alpha + n } } } \\lesssim \\| \\nabla u \\| ^ { 2 p } \\end{align*}"}
+{"id": "6815.png", "formula": "\\begin{align*} D _ { k , n } ( z ) : = \\mathrm { d i a g } ( e ^ { i z } , \\ldots , e ^ { i z } , e ^ { - i ( n - 1 ) z } , 1 , \\ldots , 1 ) , \\end{align*}"}
+{"id": "2546.png", "formula": "\\begin{align*} \\| u \\| ' _ { k , m } = \\lim _ { \\epsilon \\downarrow 0 } \\sup _ { \\{ 0 < x < \\epsilon \\} } \\big | x ^ { - m } P _ k u \\big | \\ ; . \\end{align*}"}
+{"id": "6100.png", "formula": "\\begin{align*} \\Delta ( x _ { u , \\varepsilon } ) & = 1 \\otimes x _ { u , \\varepsilon } + \\sum a _ { u , \\varepsilon } \\otimes b _ u \\end{align*}"}
+{"id": "1444.png", "formula": "\\begin{align*} ( b _ 1 , \\dots , b _ k ) \\cdot y _ i = \\begin{cases} \\exp ( 2 \\pi \\sqrt { - 1 } b _ i / c _ i ) y _ i & \\\\ y _ i & , \\end{cases} \\end{align*}"}
+{"id": "7169.png", "formula": "\\begin{align*} U ( x , t ) = G ( t - s ) * U ( x , s ) + \\int _ s ^ t G ( t - \\tau - s ) * F ( U ( \\tau ) , \\cdot , \\tau ) d \\tau . \\end{align*}"}
+{"id": "595.png", "formula": "\\begin{align*} f = \\sum \\limits _ { j = 1 } ^ \\infty a _ j { \\dfrac { h _ j } { \\| h _ j \\| _ F } } \\ ; \\mathrm { i n } \\ ; F , \\end{align*}"}
+{"id": "5166.png", "formula": "\\begin{align*} - \\frac { \\partial L D _ { \\alpha } I ( p \\| q ) } { \\partial q _ { j } } = - T \\left ( \\frac { 1 } { A } \\frac { \\partial A } { \\partial q _ { j } } - \\frac { 1 } { X } \\frac { \\partial X } { \\partial q _ { j } } - \\frac { 1 } { Y } \\frac { \\partial Y } { \\partial q _ { j } } \\right ) \\end{align*}"}
+{"id": "2239.png", "formula": "\\begin{align*} L _ \\circ ( q , v _ t , v _ x ) = \\frac { 1 } { 2 } \\rho v _ t ^ 2 - \\frac { 1 } { 2 } \\tau v _ x ^ 2 \\ , , \\end{align*}"}
+{"id": "8438.png", "formula": "\\begin{align*} \\sum _ J \\| g _ J \\| ^ 2 = \\frac { 1 } { m n } { n \\choose m } \\sum _ { j = 1 } ^ { n + 1 } \\| x ^ { ( j ) } \\| ^ 2 + \\frac { ( m - 1 ) ( n + 1 ) } { m n } { n + 1 \\choose m } \\| c \\| ^ 2 . \\end{align*}"}
+{"id": "2229.png", "formula": "\\begin{align*} u _ { , r t } + v \\cdot \\nabla u _ { , r } + v _ { , r } \\cdot \\nabla u - \\nu ( \\Delta u ) _ { , r } + { 2 \\nu \\over r } u _ { , r r } - { 2 \\nu \\over r ^ 2 } u _ { , r } = f _ { 0 , r } . \\end{align*}"}
+{"id": "5289.png", "formula": "\\begin{align*} U _ { j } = 1 + \\frac { b } { a - b } \\left ( \\frac { q _ { j } } { p _ { j } } \\right ) ^ { b - 1 } \\ ; \\ ; \\ ; ; \\ ; \\ ; \\ ; V _ { j } = \\frac { a } { a - b } \\left ( \\frac { q _ { j } } { p _ { j } } \\right ) ^ { a - 1 } \\end{align*}"}
+{"id": "1537.png", "formula": "\\begin{align*} \\lambda _ { \\rm m i n } ( D ^ 2 F _ \\delta ( z ) ) & = \\frac { \\lambda _ { \\rm m i n } \\big ( D ^ 2 F ( P _ \\delta ( z ) ) \\big ) } { 1 + \\delta \\ , \\lambda _ { \\rm m i n } \\big ( D ^ 2 F ( P _ \\delta ( z ) ) \\big ) } , \\\\ \\lambda _ { \\rm m a x } ( D ^ 2 F _ \\delta ( z ) ) & = \\frac { \\lambda _ { \\rm m a x } \\big ( D ^ 2 F ( P _ \\delta ( z ) ) \\big ) } { 1 + \\delta \\ , \\lambda _ { \\rm m a x } \\big ( D ^ 2 F ( P _ \\delta ( z ) ) \\big ) } . \\end{align*}"}
+{"id": "5852.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { N C } ( D _ { 2 m } ) ) & = \\dfrac { ( 2 m - 1 ) ( 2 m - 2 ) ^ { 3 } } { 2 } + 2 \\dfrac { ( m - 1 ) ^ { 2 } ( m - 2 ) ^ { 2 } } { 4 } - 3 \\dfrac { ( m - 1 ) ( m - 2 ) } { 2 } ( 2 m - 2 ) ^ { 2 } \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ + ( 2 m - 1 - \\dfrac { 3 } { 2 } ) ( m - 1 ) ( m - 2 ) ^ { 2 } - \\dfrac { ( m - 1 ) ( m - 2 ) ^ { 3 } } { 2 } \\\\ & = \\dfrac { m - 1 } { 2 } ( 8 m ^ { 3 } - 1 2 m ^ { 2 } + 4 m ) \\\\ & = m ( m - 1 ) ( 4 m ^ { 2 } - 6 m + 2 ) . \\end{align*}"}
+{"id": "893.png", "formula": "\\begin{align*} S _ 0 = \\left ( A _ 5 x _ 1 + B _ 5 x _ 2 + C _ 5 \\right ) y ^ 1 + \\left ( A _ 6 x _ 1 + B _ 6 x _ 2 + C _ 6 \\right ) y ^ 2 , \\end{align*}"}
+{"id": "5015.png", "formula": "\\begin{align*} a _ i ^ { ( k ) } & = \\beta R _ i + \\beta \\sum _ { j \\in S _ { k - 1 } } p _ { i j } a _ j ^ { ( k ) } \\\\ b _ i ^ { ( k ) } & = \\beta + \\beta \\sum _ { j \\in S _ { k - 1 } } p _ { i j } b _ j ^ { ( k ) } . \\end{align*}"}
+{"id": "5090.png", "formula": "\\begin{align*} ( \\overline { \\mathsf { a } } _ { s } \\cdot \\varepsilon _ { \\theta } ) ( t ) & = q ^ { - k } \\theta ( t ) \\sum _ { t '' \\in T _ { s } ( \\mathbb { F } _ { q ^ k } ) } \\theta ( t '' ) \\\\ & = 0 , \\end{align*}"}
+{"id": "5985.png", "formula": "\\begin{align*} \\Delta \\phi ( x ) = & \\frac { \\Delta ( | \\nabla u ( x ) | ^ 2 ) } { ( b ^ 2 - \\rho ^ 2 ( u ( x ) ) ) ^ 2 } + 4 \\frac { \\nabla ( | \\nabla u ( x ) | ^ 2 ) \\cdot \\nabla ( \\rho ^ 2 ( u ( x ) ) ) } { ( b ^ 2 - \\rho ^ 2 ( u ( x ) ) ) ^ 3 } \\\\ & + 2 \\frac { | \\nabla u ( x ) | ^ 2 \\Delta ( \\rho ^ 2 ( u ( x ) ) ) } { ( b ^ 2 - \\rho ^ 2 ( u ( x ) ) ) ^ 3 } + 6 \\frac { | \\nabla u ( x ) | ^ 2 | \\nabla ( \\rho ^ 2 ( u ( x ) ) ) | ^ 2 } { ( b ^ 2 - \\rho ^ 2 ( u ( x ) ) ) ^ 4 } . \\end{align*}"}
+{"id": "3594.png", "formula": "\\begin{align*} ( T f ) ( z , w ) = \\alpha f ( \\tau ( z ) , w \\sigma ( z ) ) , \\end{align*}"}
+{"id": "3641.png", "formula": "\\begin{align*} f ( z _ 0 , w _ 0 ) = L _ 2 ( w _ 0 ) = 0 , f ( z _ 0 , w _ 0 \\sigma ( z _ 0 ) ^ 2 ) = L _ 2 ( w _ 0 \\sigma ( z _ 0 ) ^ 2 ) = 0 . \\end{align*}"}
+{"id": "8194.png", "formula": "\\begin{align*} B _ 1 ^ { \\star } = \\min ( B _ , \\frac { M } { 2 } ) . \\end{align*}"}
+{"id": "7348.png", "formula": "\\begin{align*} U ^ { \\{ w \\} , C } _ { p } = U ^ { \\{ w \\} } _ { p } = W _ \\phi ( w z , \\zeta \\zeta ' ) = \\emptyset , \\end{align*}"}
+{"id": "5162.png", "formula": "\\begin{align*} D _ { \\alpha } I \\left ( p \\| q \\right ) = \\underbrace { \\frac { 1 } { 1 - \\alpha } } _ { T } \\left [ \\underbrace { \\sum _ { i } p _ { i } } _ { A } - \\underbrace { \\left ( \\sum _ { i } p ^ { \\alpha } _ { i } q ^ { 1 - \\alpha } _ { i } \\right ) ^ { \\frac { 1 } { \\alpha } } } _ { X } \\times \\underbrace { \\left ( \\sum _ { i } q _ { i } \\right ) ^ { 1 - \\frac { 1 } { \\alpha } } } _ { Y } \\right ] \\end{align*}"}
+{"id": "5959.png", "formula": "\\begin{align*} a \\leq b & \\ , \\ , \\Longleftrightarrow \\ , \\ , a - b = 0 \\\\ a \\not \\leq b & \\ , \\ , \\Longleftrightarrow \\ , \\ , a - b \\neq 0 \\end{align*}"}
+{"id": "2208.png", "formula": "\\begin{align*} \\intop _ \\Omega \\psi _ { 1 , r r z } \\psi _ { 1 , z z z } d x = \\intop _ \\Omega ( \\psi _ { 1 , r r z } \\psi _ { 1 , z z } ) _ { , z } d x - \\intop _ \\Omega \\psi _ { 1 , r r z z } \\psi _ { 1 , z z } d x , \\end{align*}"}
+{"id": "7977.png", "formula": "\\begin{align*} \\dot { H } ( v , \\Sigma ) = - \\int _ { \\Gamma } E ( \\frac { \\delta H } { \\delta \\Sigma } ) \\wedge \\ast \\boldsymbol { n } ( v ) = \\int _ { \\Gamma } E ( \\frac { \\delta H } { \\delta \\Sigma } ) \\wedge ( - 1 ) ^ { n } \\mathrm { t r } ( \\frac { \\delta H } { \\delta v } ) . \\end{align*}"}
+{"id": "3469.png", "formula": "\\begin{align*} - \\sigma ( K ) + \\frac { 1 } { 2 } S \\circ S = b _ 1 ( S ) \\end{align*}"}
+{"id": "7583.png", "formula": "\\begin{align*} M _ { \\varphi _ { R } } [ u ] : = \\big < u , - i ( \\nabla \\varphi _ { R } \\cdot \\nabla + \\nabla \\cdot \\nabla \\varphi _ { R } ) u \\big > = 2 I m \\int _ { \\mathbb { R } ^ { N } } \\overline { u } \\nabla \\varphi _ { R } \\cdot \\nabla u d x . \\end{align*}"}
+{"id": "4898.png", "formula": "\\begin{align*} C ^ 2 = C '^ 2 - ( 1 0 - g ) = 3 g + 6 - ( 1 0 - g ) = 4 g - 4 , \\end{align*}"}
+{"id": "3331.png", "formula": "\\begin{align*} \\int _ { \\O } - \\langle A D u , \\dd \\phi \\rangle - u Y \\phi + \\langle b , D u \\rangle \\phi + c u \\phi = \\int _ { \\O } f \\phi . \\end{align*}"}
+{"id": "1358.png", "formula": "\\begin{align*} \\left ( h + \\frac { 2 m \\alpha } { n } \\right ) + ( t - 1 ) \\left ( - h + \\frac { 2 m \\alpha } { n } \\right ) + ( n - t ) \\left ( \\frac { 2 m \\alpha } { n } \\right ) = 2 m \\alpha , \\end{align*}"}
+{"id": "1958.png", "formula": "\\begin{align*} d _ j = \\min _ { \\substack { i _ \\theta \\in \\bar { N } _ j } } \\sum _ { \\theta = 1 } ^ { \\psi _ \\tau ( | N _ j | ) } | T _ { j , i _ { \\theta } } | . \\end{align*}"}
+{"id": "2180.png", "formula": "\\begin{align*} T ( \\mathcal { M } ) \\coloneqq ( \\widehat { \\mathcal { O } } _ { \\mathcal { E } } ^ { \\mathrm { u r } } \\otimes _ { \\mathcal { O } _ { \\mathcal { E } } } \\mathcal { M } ) ^ { \\varphi = 1 } \\mathcal { M } ( T ) \\coloneqq ( \\widehat { \\mathcal { O } } _ { \\mathcal { E } } ^ { \\mathrm { u r } } \\otimes _ { \\mathbf { Z } _ p } \\mathcal { T } ) ^ { G _ { \\widetilde { L } _ { \\infty } } } . \\end{align*}"}
+{"id": "1653.png", "formula": "\\begin{align*} e ^ { - \\lambda \\left ( \\varepsilon + 1 \\right ) ^ { k } } = \\delta ^ { 2 \\rho } , 2 \\rho = \\frac { 1 } { 3 } \\left ( \\frac { \\varepsilon + 1 } { T + 1 } \\right ) ^ { k } < \\frac { 1 } { 3 } . \\end{align*}"}
+{"id": "2434.png", "formula": "\\begin{align*} u _ { t } = u u _ { x x } - \\gamma ( u _ { x } ) ^ { 2 } + k u ^ { 2 } - \\delta p u , t > 0 , x \\in \\mathbb { R } , \\end{align*}"}
+{"id": "698.png", "formula": "\\begin{align*} V ( p ( x ) ) = e \\displaystyle \\min _ { 0 \\leq i } ^ { } \\{ v _ p ( b _ i ) + i \\lvert \\lambda \\rvert \\} . \\end{align*}"}
+{"id": "2984.png", "formula": "\\begin{align*} \\begin{array} { l } \\eta ( \\xi ) = 1 , \\ \\ \\ \\phi ^ 2 x = x - \\eta ( x ) \\xi , \\ \\ \\ \\phi \\xi = 0 \\\\ g ( \\phi x , \\phi y ) = g ( x , y ) - \\eta ( x ) \\eta ( y ) , \\ \\ \\ t r \\phi = 0 \\end{array} \\end{align*}"}
+{"id": "1640.png", "formula": "\\begin{align*} u _ { 1 } \\left ( x , T \\right ) = u _ { T } ^ { \\left ( 1 \\right ) } \\left ( x \\right ) , m _ { 1 } \\left ( x , T \\right ) = m _ { T } ^ { \\left ( 1 \\right ) } \\left ( x \\right ) , x \\in \\Omega , \\end{align*}"}
+{"id": "1868.png", "formula": "\\begin{align*} g = \\begin{pmatrix} 1 & & \\\\ & \\cos \\theta & \\sin \\theta \\\\ & - \\sin \\theta & \\cos \\theta \\end{pmatrix} , \\begin{pmatrix} - 1 & & \\\\ & \\cos \\theta & \\sin \\theta \\\\ & \\sin \\theta & - \\cos \\theta \\end{pmatrix} . \\end{align*}"}
+{"id": "4645.png", "formula": "\\begin{align*} \\left ( \\frac { q } { p } \\right ) = \\left ( \\frac { p } { | q | } \\right ) \\cdot ( - 1 ) ^ { \\frac { p - 1 } { 2 } \\frac { q - 1 } { 2 } } \\end{align*}"}
+{"id": "2582.png", "formula": "\\begin{align*} & \\sum \\limits _ { k = 1 } ^ { n - 2 } ( \\sum \\limits _ { l = 1 } ^ { ( k , n - 1 - k ) } q ^ { - \\frac { 3 } { 2 } + l } ) h X _ { n - 1 - k } - \\sum \\limits _ { k = 1 } ^ { n - 3 } ( \\sum \\limits _ { l = 1 } ^ { ( k , n - 2 - k ) } q ^ { - \\frac { 3 } { 2 } + l } ) h X _ { n - 1 - k } - q ^ { - \\frac { 1 } { 2 } } h X _ 1 \\\\ = & \\sum \\limits _ { k = \\frac { n } { 2 } } ^ { n - 3 } q ^ { - \\frac { 5 } { 2 } + n - k } h X _ { n - 1 - k } = \\sum \\limits _ { k = \\frac { n } { 2 } + 2 } ^ { n - 1 } q ^ { - \\frac { 1 } { 2 } + n - k } h X _ { n + 1 - k } . \\end{align*}"}
+{"id": "1016.png", "formula": "\\begin{align*} \\int _ D u ( - L \\eta ) \\ , d m = \\int _ D P _ D g ( - L \\eta ) \\ , d m + \\int _ D R ^ D \\mu ( - L \\eta ) \\ , d m . \\end{align*}"}
+{"id": "3300.png", "formula": "\\begin{align*} D ^ { a , b } & = \\left \\{ \\liminf _ { n \\to \\infty } M _ n < a \\right \\} \\cap \\left \\{ \\limsup _ { n \\to \\infty } M _ n > b \\right \\} \\\\ & = \\left \\{ \\liminf _ { n \\to \\infty } M _ n \\wedge ( b + 1 ) < a \\right \\} \\cap \\left \\{ \\limsup _ { n \\to \\infty } M _ n \\wedge ( b + 1 ) > b \\right \\} . \\end{align*}"}
+{"id": "7274.png", "formula": "\\begin{align*} X _ t : = \\tfrac { 2 \\sqrt { T ( t ) } Y _ { T ( t ) } - B t - C - D } { \\sqrt { ( 1 - q ) ( 1 - B C ) ( 1 - B D ) } } , t \\geq 0 \\end{align*}"}
+{"id": "3309.png", "formula": "\\begin{align*} \\d K _ t = \\left ( g _ 1 + \\frac { 1 } { 2 } g _ { 2 2 } \\right ) \\d t + g _ 2 \\ , \\d B _ t = 0 \\ , \\d t + \\frac { K _ t B _ t } { t } \\ , \\d B _ t . \\end{align*}"}
+{"id": "964.png", "formula": "\\begin{align*} \\Pi _ { V _ n } ( u ) = R ^ { V _ n } f ( \\cdot , u ) + R ^ { V _ n } \\mu \\quad V _ n . \\end{align*}"}
+{"id": "6615.png", "formula": "\\begin{align*} \\log n _ s = \\sum _ { p \\leq p _ s } \\log p = p _ s \\left ( 1 + O \\left ( \\frac { 1 } { p _ s } \\right ) \\right ) . \\end{align*}"}
+{"id": "1908.png", "formula": "\\begin{align*} \\partial _ t v _ + ( x , t ) = 0 . \\end{align*}"}
+{"id": "4497.png", "formula": "\\begin{align*} \\sum _ { 1 \\le j \\le n } M _ { H , j } ( Z _ 0 , J , \\rho ) = M _ H ( Z _ 0 , J , \\rho ) . \\end{align*}"}
+{"id": "4495.png", "formula": "\\begin{align*} M _ { \\partial D _ j } = \\frac { 1 } { 2 \\pi } \\int _ { \\partial D _ j } | f _ j | ^ 2 \\bigg ( \\sum _ { 1 \\le k \\le m _ j } 2 \\frac { \\partial G _ { D _ j } ( w _ j , z _ { j , k } ) } { \\partial v _ { w _ j } } \\bigg ) ^ { - 1 } e ^ { - \\varphi _ j } | d w _ j | . \\end{align*}"}
+{"id": "2581.png", "formula": "\\begin{align*} X _ 1 X _ \\delta = & X ( 1 , 0 ) ( X ( - 1 , - 1 ) + h X ( - 1 , 0 ) + h X ( 0 , - 1 ) + X ( - 1 , 1 ) + X ( 1 , - 1 ) ) \\\\ = & q ^ { - \\frac { 1 } { 2 } } X ( 0 , - 1 ) + h + q ^ { - \\frac { 1 } { 2 } } h X ( 1 , - 1 ) + q ^ { \\frac { 1 } { 2 } } X ( 0 , 1 ) + q ^ { - \\frac { 1 } { 2 } } X ( 2 , - 1 ) . \\end{align*}"}
+{"id": "2602.png", "formula": "\\begin{align*} - \\Delta \\varphi & = - \\varphi '' ( r ) - \\dfrac { n - 1 } { r } \\varphi ' ( r ) = \\\\ & = \\dfrac { 2 \\beta R } { ( r ^ 2 + \\alpha ^ 2 ) ^ { \\beta R + 1 } } \\left [ n - 2 ( \\beta R + 1 ) + ( \\beta R + 1 ) \\dfrac { 2 \\alpha ^ 2 } { r ^ 2 + \\alpha ^ 2 } \\right ] \\\\ & = 2 \\beta R \\varphi ( r ) ^ { \\frac { 1 } { \\beta R } + 1 } \\left [ n - 2 ( \\beta R + 1 ) + 2 ( \\beta R + 1 ) \\alpha ^ 2 \\varphi ( r ) ^ { \\frac { 1 } { \\beta R } } \\right ] \\\\ & = f ( \\varphi ( r ) ) , \\end{align*}"}
+{"id": "2859.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ p _ \\omega ( \\mathbb { R } ^ n ) } : = \\left [ \\int _ { \\mathbb { R } ^ n } \\left | f ( x ) \\right | ^ p \\omega ( x ) \\ , d x \\right ] ^ { \\frac { 1 } { p } } < \\infty . \\end{align*}"}
+{"id": "3983.png", "formula": "\\begin{align*} x ^ { 2 } x ^ { 2 } = 8 \\sum _ { l = 1 } ^ { n } \\Bigl ( \\sum _ { k , r = 1 } ^ { n , \\nu } \\gamma _ { i p k } \\widetilde { \\gamma } _ { i p r } \\gamma _ { k r l } \\Bigr ) e _ { l } + 8 \\sum _ { s = 1 } ^ { \\nu } \\Bigl ( \\sum _ { k , r = 1 } ^ { n , \\nu } \\gamma _ { i p k } \\widetilde { \\gamma } _ { i p r } \\widetilde { \\gamma } _ { k r s } \\Bigr ) \\widetilde { e } _ { s } . \\end{align*}"}
+{"id": "9152.png", "formula": "\\begin{align*} G _ { \\beta } = ( w _ { \\beta , 1 } - v ^ { - 4 } w _ { \\beta , 2 } ) ( w _ { \\beta , 1 } - v ^ { 4 } w _ { \\beta , 2 } ) \\cdot G _ { \\gamma } , \\end{align*}"}
+{"id": "1341.png", "formula": "\\begin{align*} \\delta x _ k & = \\alpha d _ k x _ k + ( 1 - \\alpha ) \\sum _ { k \\sim j } x _ j = \\Delta x _ k , \\ , \\ , 1 \\leq k \\leq n \\end{align*}"}
+{"id": "4357.png", "formula": "\\begin{align*} \\inf _ { k \\in [ m ] _ 0 } \\left \\{ \\inf _ { x \\in \\mathcal { X } } \\left \\{ \\Gamma \\theta ^ k ( x ) + \\sum _ { i \\in [ m ] } \\max \\{ 0 , x _ i - \\overline { b } _ i , \\max \\{ 0 , x _ i - \\overline { b } _ i + \\Delta b _ i \\} - \\theta ^ k ( x ) \\} \\right \\} \\right \\} \\end{align*}"}
+{"id": "8892.png", "formula": "\\begin{align*} V '' _ { \\phi } ( t ) = 2 \\sum _ { k = 1 } ^ N \\int _ { \\R ^ n } \\partial _ k \\phi \\cdot \\partial _ t \\Im ( \\bar u \\partial _ k u ) d x . \\end{align*}"}
+{"id": "2715.png", "formula": "\\begin{align*} \\kappa ^ { - 1 } : T ^ { * } M \\times \\mathbb { R } \\rightarrow T M \\times \\mathbb { R } ; p _ { i } ( t ) \\mapsto \\dot { q } ^ { i } ( t ) = v ^ { i } ( q ^ { j } , p _ { j } , t ) . \\end{align*}"}
+{"id": "5539.png", "formula": "\\begin{align*} A = \\frac { \\sqrt { m n } } d ( X \\circ M ) , \\end{align*}"}
+{"id": "4498.png", "formula": "\\begin{align*} q \\otimes \\alpha ^ { ( i + j ) d } - [ \\alpha ^ { i d } \\otimes \\alpha ^ { j d } ] _ d ^ m m = q d q \\mid ( i + j ) . \\end{align*}"}
+{"id": "3098.png", "formula": "\\begin{align*} \\varphi \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & = \\left ( \\begin{array} { c c c } a ^ { \\ell _ 1 } & c _ 1 \\ , a ^ { \\ell _ 1 } \\ , b ^ { p ^ { e _ 1 } } \\ , d ^ { p ^ { e _ 1 } } & c _ 2 \\ , a ^ { \\ell _ 1 } \\ , b ^ { p ^ { e _ 2 } } \\ , d ^ { p ^ { e _ 2 } } \\\\ 0 & a ^ { \\ell _ 2 } & 0 \\\\ 0 & 0 & d ^ { - \\ell _ 3 } \\end{array} \\right ) . \\end{align*}"}
+{"id": "5978.png", "formula": "\\begin{align*} P ( y | x ) = \\begin{cases} 1 - ( K - 1 ) p & y = x \\\\ p & y \\neq x \\end{cases} \\end{align*}"}
+{"id": "2725.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ { r } m _ { \\alpha } + { \\textrm { t h e n u m b e r o f t h e p r i m a r y c o n s t r a i n t s } } \\leq 2 n , \\end{align*}"}
+{"id": "4492.png", "formula": "\\begin{align*} M _ H ( Z _ 0 , J , \\rho ) \\ge & \\sum _ { 1 \\le j \\le n } M _ { H , j } ( Z _ 0 , J , \\rho ) \\\\ = & \\sum _ { 1 \\le j \\le n } M _ { \\partial D _ j } \\times M _ { M _ j } \\\\ = & \\sum _ { 1 \\le j \\le m } M _ { \\partial D _ j } \\times \\prod _ { 1 \\le l \\le n , l \\not = j } M _ { D _ l } \\\\ \\ge & \\pi ^ { n - 1 } \\sum _ { 1 \\le j \\le m } M _ { \\partial D _ j } \\times \\prod _ { 1 \\le l \\le n , l \\not = j } M _ { \\partial D _ l } \\\\ = & n \\pi ^ { n - 1 } M _ S ( Z _ 0 , J , \\lambda ) . \\end{align*}"}
+{"id": "7771.png", "formula": "\\begin{align*} b \\tilde M _ { j + 1 } ^ 4 \\epsilon _ { j + 1 } ^ { \\frac { 1 } { 2 m } } = b ( \\tilde M _ j + 2 0 m \\tilde M _ j ^ 2 \\epsilon _ j ) ^ 4 \\epsilon _ j ^ { \\frac { 4 ^ j } { 2 m } } < b \\tilde M _ j ^ 4 \\epsilon _ j ^ \\frac { 1 } { 2 m } \\cdot 2 ^ 4 \\epsilon _ j ^ { \\frac { 4 ^ j - 1 } { 2 m } } < 1 . \\end{align*}"}
+{"id": "6258.png", "formula": "\\begin{align*} \\Omega : = ( 0 , T ) \\times I \\subset \\R ^ { 2 } , \\end{align*}"}
+{"id": "1149.png", "formula": "\\begin{align*} \\widetilde r : = \\frac { n } { \\widetilde { J } } \\in ( 0 , 1 ] . \\end{align*}"}
+{"id": "3920.png", "formula": "\\begin{align*} \\sin N z = \\binom { N } { 1 } \\cos ^ { N - 1 } z \\sin z - \\binom { N } { 3 } \\cos ^ { n - 3 } z \\sin ^ 3 z + \\ldots + \\sin ^ N z . \\end{align*}"}
+{"id": "811.png", "formula": "\\begin{align*} { \\pounds _ { \\hat X } } { G ^ i } = \\Psi ( x , y ) { y ^ i } . \\end{align*}"}
+{"id": "333.png", "formula": "\\begin{align*} I ( G _ 1 ) ^ { [ k ] } = x _ { n - 1 } x _ { n - 2 } I ( G _ 3 ) ^ { [ k - 1 ] } + \\sum _ { j = 1 } ^ t x _ { n - 1 } x _ { i _ j } I ( G _ 2 ) ^ { [ k - 1 ] } + I ( G _ 2 ) ^ { [ k ] } . \\end{align*}"}
+{"id": "2941.png", "formula": "\\begin{align*} \\big ( \\Delta _ \\C + \\partial _ { \\rho } ^ 2 + \\frac { ( 1 - s ) } { \\rho } \\partial _ \\rho \\big ) U ( z , \\rho ) = 0 , \\ , \\ , U ( z , 0 ) = F ( z ) \\end{align*}"}
+{"id": "8116.png", "formula": "\\begin{align*} | { \\mathcal I } ( w ) | = \\ell ( w ) = \\dim _ { \\mathbb C } X _ w , \\end{align*}"}
+{"id": "985.png", "formula": "\\begin{align*} \\hat W ^ x _ D ( | u | ) : = \\lim _ { V \\nearrow D , V \\subset \\subset D } P _ V ( | u | R ^ D \\kappa _ D ) ( x ) = 0 . \\end{align*}"}
+{"id": "3088.png", "formula": "\\begin{align*} \\prod _ { i = 0 \\atop i \\neq l } ^ { j } \\left ( { \\rm C o e f f } ( \\varphi ^ * _ { a } ( F _ i ) , t ^ { v _ i } ) \\right ) ^ { \\gamma _ i } \\cdot \\left ( { \\rm C o e f f } ( \\varphi ^ * _ { a } ( F _ l ) , t ^ { v _ l } ) \\right ) ^ { \\gamma _ { l } - 1 } \\cdot { \\rm C o e f f } ( \\varphi ^ * _ { a } ( F _ l ) , t ^ { k - I ( F , H ) + v _ l } ) . \\end{align*}"}
+{"id": "8546.png", "formula": "\\begin{align*} [ A \\otimes _ R M , U ] _ { A - \\mathrm { A l g } / A } \\simeq [ A \\otimes _ { R - a l g / A } M , U ] _ { _ { A \\setminus } R - \\mathrm { A l g } _ { / A } } \\\\ = [ M , [ A , U ] _ { R - \\mathrm { A l g } _ { / A } } ] _ { \\mathrm { T o p } _ + } \\end{align*}"}
+{"id": "6080.png", "formula": "\\begin{align*} I _ { + } ( x ) = - \\frac { 1 } { \\Lambda \\pi } \\int _ 0 ^ { \\pi } \\frac { \\sin k \\sin x k } { 1 - \\cos k } d k \\end{align*}"}
+{"id": "2315.png", "formula": "\\begin{align*} | u ^ { c , \\gamma } | = \\frac { 2 | c _ { 3 } | } { | x | } \\ln \\frac { | x | } { | x ' | } + O ( 1 ) \\frac { | ( c , \\gamma ) | } { | x | } , \\quad | \\nabla u ^ { c , \\gamma } | = \\frac { 2 | c _ { 3 } | } { | x ' | | x | } + \\frac { O ( 1 ) | ( c , \\gamma ) | } { | x | ^ { 2 } } \\ln \\frac { | x | } { | x ' | } . \\end{align*}"}
+{"id": "785.png", "formula": "\\begin{align*} \\bar { U } _ Z = ( \\bar { u } _ { i j } ^ Z ) _ { i , j } , \\bar { u } _ { i j } ^ Z = \\sum _ { s , l , t } R ( z _ { t j } ^ * z _ { s l } , z _ { i l } ^ * ) u _ { s t } ^ * , \\end{align*}"}
+{"id": "7108.png", "formula": "\\begin{align*} b _ \\varepsilon : = E _ { \\varepsilon } b , \\end{align*}"}
+{"id": "6372.png", "formula": "\\begin{align*} \\tau G : = \\{ ( \\tau r , \\varphi ) \\in M : ( r , \\varphi ) \\in G \\} . \\end{align*}"}
+{"id": "7948.png", "formula": "\\begin{align*} \\frac { \\delta \\mathcal { F } } { \\delta v } = \\frac { \\delta \\tilde { \\mathcal { F } } } { \\delta \\eta } + ( - 1 ) ^ { n - 1 } \\ast d N _ { \\phi } \\big ( \\frac { \\delta \\tilde { \\mathcal { F } } } { \\delta \\phi _ { \\partial } } \\big ) . \\end{align*}"}
+{"id": "2529.png", "formula": "\\begin{align*} C ^ \\infty _ { { \\cdot } / } ( L ; E \\otimes \\Omega ^ { - 1 } N L ) \\cap H ^ { - n ' / 2 } _ { } ( M ; E ) = 0 \\ ; . \\end{align*}"}
+{"id": "6314.png", "formula": "\\begin{align*} u _ m ( x ) = \\prod _ { i = 1 } ^ 3 u _ { m _ i } ( x _ i ) , u _ { m _ i } ( x ) = \\left \\{ \\begin{array} { c c } 1 , & m _ i = 0 \\\\ \\sqrt { 2 } \\cos ( m _ i ( x _ i + 1 / 2 ) ) , & m _ i \\neq 0 \\end{array} \\right . . \\end{align*}"}
+{"id": "8076.png", "formula": "\\begin{align*} \\hat { \\rho } _ { f , g } ( z ) : = \\int _ 0 ^ \\infty e ^ { - z t } \\rho _ { f , g } ( t ) \\ ; d t . \\end{align*}"}
+{"id": "8701.png", "formula": "\\begin{align*} \\gamma B _ G = ( P ^ \\ast P ) ^ { - 1 } P ^ \\ast B _ G . \\end{align*}"}
+{"id": "6624.png", "formula": "\\begin{align*} H _ N S _ N ( i , j ) & = \\frac { 1 } { J _ N } ( 2 J _ 1 J _ { N - j } ( J _ { j - 2 } + J _ { j - 1 } ) - J _ { N - j } ( J _ { j - 1 } + J _ j ) + ( - 1 ) ^ { N + j - 1 } J _ { j - 1 } J _ 0 ) \\\\ & = \\frac { 1 } { J _ N } ( 2 J _ { N - j } ( J _ { j - 2 } + J _ { j - 1 } ) - J _ { N - j } ( J _ { j - 1 } + J _ j ) ) \\\\ & = \\frac { J _ { N - j } } { J _ N } ( 2 J _ { j - 2 } + J _ { j - 1 } - J _ j ) \\\\ & = 0 . \\end{align*}"}
+{"id": "6710.png", "formula": "\\begin{align*} p _ 0 ( x ) y ( x + n ) + p _ 1 ( x ) y ( x + n - 1 ) + . . . + p _ n ( x ) y ( x ) = r ( x ) \\end{align*}"}
+{"id": "711.png", "formula": "\\begin{align*} f ( x ) = \\Phi ^ 3 ( x ) + 8 1 \\delta \\Phi ^ 2 ( x ) + 2 1 8 7 \\Phi ( x ) + 1 9 6 8 3 \\delta - m \\end{align*}"}
+{"id": "7962.png", "formula": "\\begin{align*} \\frac { \\delta \\mathcal { F } } { \\delta v } = ( - 1 ) ^ { n - 1 } d ( \\frac { \\delta \\bar { \\mathcal { F } } } { \\delta \\omega } ) + ( - 1 ) ^ { n - 1 } \\ast d N _ { \\phi } ( \\frac { \\delta \\bar { \\mathcal { F } } } { \\delta \\phi _ { \\partial } } ) . \\end{align*}"}
+{"id": "4149.png", "formula": "\\begin{align*} & Q ^ * ( \\theta ) \\psi ^ { ( r ) } ( \\theta ) + ( 1 - w _ { j j } ) r ^ { 2 j } \\eta _ j \\bigl ( \\psi ^ { ( r ) } ( \\theta ) - \\psi ^ { ( r ) } _ i ( \\theta ) \\bigr ) = o ( r ^ { 2 j } ) . \\end{align*}"}
+{"id": "4453.png", "formula": "\\begin{align*} \\int _ { \\{ - t _ 1 \\leq \\psi < - t _ 2 \\} } | F | ^ 2 e ^ { - \\varphi } a ( - \\psi ) = \\frac { G ( T _ 1 ; c ) } { \\int _ { T _ 1 } ^ { + \\infty } c ( l ) e ^ { - l } d l } \\int _ { t _ 2 } ^ { t _ 1 } a ( t ) e ^ { - t } d t \\end{align*}"}
+{"id": "5181.png", "formula": "\\begin{align*} & \\frac { \\partial A } { \\partial q _ { j } } = \\frac { \\partial A _ { j } } { \\partial q _ { j } } = - p _ { j } q ^ { \\beta - 2 } _ { j } \\\\ & \\frac { \\partial B } { \\partial q _ { j } } = \\frac { \\partial B _ { j } } { \\partial q _ { j } } = - q ^ { \\beta - 1 } _ { j } \\end{align*}"}
+{"id": "7720.png", "formula": "\\begin{align*} b ^ { 1 1 } b _ { 1 1 k } = \\tilde { d } \\frac { h _ { k } } { h } - 2 \\tilde { l } h _ { k } h _ { k k } = \\tilde { d } \\frac { h _ { k } } { h } - 2 \\tilde { l } h _ { k } b _ { k k } + 2 \\tilde { l } h h _ { k } . \\end{align*}"}
+{"id": "3518.png", "formula": "\\begin{align*} H ^ 2 ( \\Delta ; M ) \\cong \\bigoplus _ { i = 1 } ^ { m } M \\cdot \\mathbf { r } _ i ^ * / ( \\mathbf { r } _ 0 ^ * + p _ i \\cdot \\mathbf { r } _ i ^ * \\ \\ i = 1 , \\ldots , m ) . \\end{align*}"}
+{"id": "1542.png", "formula": "\\begin{align*} F ( z ) \\ge \\frac { 1 } { C } \\ , F ( w ) \\ , | z | \\ , \\eta ^ { - 1 } _ H ( | z | ) \\forall \\ , | w | = 1 , \\end{align*}"}
+{"id": "523.png", "formula": "\\begin{align*} v ^ K = \\left ( \\frac { \\log { p _ K } \\cdot K \\cdot v } { \\log { p _ K } \\cdot K } \\right ) ^ K \\geq \\left ( \\frac { \\log { m } } { \\log { p _ K } \\cdot K } \\right ) ^ K = \\frac { \\log ^ K { m } } { ( \\log { p _ K } \\cdot K ) ^ K } = c ( K , p _ K ) \\cdot \\log ^ K { m } , \\end{align*}"}
+{"id": "3468.png", "formula": "\\begin{align*} b ^ + _ { \\tilde { g } } ( W ) = 3 p + q , \\sigma _ { \\tilde { g } } ( W ) = - 5 p + q \\end{align*}"}
+{"id": "1127.png", "formula": "\\begin{align*} \\left \\| \\vec { t } \\right \\| _ { \\dot f _ { \\infty , q } ^ s ( \\mathbb { A } ) } = \\| u \\| _ { \\dot f _ { \\infty , q } ^ s } \\sim \\| u \\| _ { \\dot f _ { p , q } ^ { s , \\frac { 1 } { p } } } = \\left \\| \\vec { t } \\right \\| _ { \\dot f _ { p , q } ^ { s , \\frac { 1 } { p } } ( \\mathbb { A } ) } \\end{align*}"}
+{"id": "7920.png", "formula": "\\begin{align*} i _ { \\mathcal { N } } \\mu = \\mu ( \\mathcal { N } ) = \\boldsymbol { n } ( \\mu ) ( \\mathcal { N } ) = 0 . \\end{align*}"}
+{"id": "6969.png", "formula": "\\begin{align*} \\widetilde { \\Pi } ( A \\mathbf { z } ) = A \\mathbf { z } \\overline { \\eta } ( A \\mathbf { z } ) + \\overline { A \\mathbf { z } } \\eta ( A \\mathbf { z } ) = A \\mathbf { z } \\overline { \\eta } ( \\mathbf { z } ) + A \\overline { \\mathbf { z } } \\eta ( \\mathbf { z } ) = A ( \\mathbf { z } \\overline { \\eta } ( \\mathbf { z } ) + \\overline { \\mathbf { z } } \\eta ( \\mathbf { z } ) ) = A \\widetilde { \\Pi } ( \\mathbf { z } ) . \\end{align*}"}
+{"id": "7886.png", "formula": "\\begin{align*} \\det ( \\mathbf { D } _ { i , j } ) = \\begin{cases} \\prod \\limits _ { k = 1 } ^ { \\frac { m } { 2 } } ( - c _ { 2 k - 1 } ^ { 2 } ) , & m \\equiv 0 \\bmod 2 ; \\\\ ( - 2 e ) \\times \\prod \\limits _ { k _ { 1 } = 1 } ^ { \\frac { s - 1 } { 2 } } ( - c _ { 2 k _ { 1 } - 1 } ^ { 2 } ) \\times \\prod \\limits _ { k _ { 2 } = \\frac { s + 1 } { 2 } } ^ { \\frac { m - 1 } { 2 } } ( - c _ { 2 k _ { 2 } } ^ { 2 } ) , & m \\equiv 1 \\bmod 2 . \\end{cases} \\end{align*}"}
+{"id": "2249.png", "formula": "\\begin{align*} X = X ^ i \\frac { \\partial } { \\partial q ^ i } + \\Gamma _ { \\alpha } ( X ^ i ) \\frac { \\partial } { \\partial v ^ i _ { \\alpha } } + X ^ { \\gamma } \\frac { \\partial } { \\partial s ^ { \\gamma } } \\ , . \\end{align*}"}
+{"id": "4343.png", "formula": "\\begin{align*} F _ { \\mathcal { Y } , a , b } ( x ) & : = \\sum _ { ( i , j ) \\in [ m ] \\times [ n ] } \\overline { y } _ { ( i , j ) } g _ { ( i , j ) } x + \\sum _ { ( q , p ) \\in \\mathcal { Y } } \\Delta y _ { ( q , p ) } g _ { ( q , p ) } ( x ) - \\Delta y _ { ( a , b ) } g _ { ( a , b ) } ( x ) . \\end{align*}"}
+{"id": "2131.png", "formula": "\\begin{align*} \\psi ( x , y ) = ( \\psi _ { j _ 1 } ( x , y ) , \\ldots , \\psi _ { j _ t } ( x , y ) ) \\end{align*}"}
+{"id": "9123.png", "formula": "\\begin{align*} E _ { h } \\ : = \\prod _ { ( \\beta , s ) \\in \\Delta ^ { + } \\times \\mathbb { Z } } \\limits ^ { \\rightarrow } E _ { \\beta , s } ^ { h ( \\beta , s ) } \\forall \\ h \\in H \\end{align*}"}
+{"id": "7580.png", "formula": "\\begin{align*} E _ { p , q } ^ \\mu ( u ) = ( \\frac { \\gamma _ { p } } { 2 } - \\frac { 1 } { p } ) \\| u \\| _ { p } ^ { p } , \\forall u \\in \\mathcal { Q } _ { p , q } ^ \\mu ( c ) , \\end{align*}"}
+{"id": "1013.png", "formula": "\\begin{align*} - \\int _ E u \\ , L \\eta \\ , d m = \\int _ D \\eta \\ , d \\mu , \\eta \\in \\mathcal C , u = g \\quad \\partial _ { \\chi } D , \\end{align*}"}
+{"id": "3857.png", "formula": "\\begin{align*} \\eta ^ { \\kappa } _ { ( 1 ) } ( x \\mid s ) = ( 1 - \\kappa ) \\eta ^ 0 _ { ( 1 ) } ( x \\mid s ) + \\kappa \\pi ^ * _ x , \\end{align*}"}
+{"id": "6394.png", "formula": "\\begin{align*} G _ n ^ 2 ( \\theta ) = - \\partial _ b L _ n ( \\theta ) = - \\frac { n ^ { 1 / \\alpha } } { n } \\sum _ { i = 1 } ^ n \\frac { X _ { \\frac { i - 1 } { n } } } { \\delta X _ { \\frac { i - 1 } { n } } ^ { 1 / \\alpha } } h _ \\alpha ( z ^ n _ i ( \\theta ) ) \\end{align*}"}
+{"id": "7866.png", "formula": "\\begin{align*} \\mu ( \\mathbf { \\Phi } ) = \\frac { 1 } { \\sqrt { p ^ { r _ { m i n } } } } , \\end{align*}"}
+{"id": "820.png", "formula": "\\begin{align*} { { \\pounds } _ { \\hat X } } ( { R i c _ { i j } } ) = k ( n - 1 ) { { \\pounds } _ { \\hat X } } { g _ { i j } } . \\end{align*}"}
+{"id": "2382.png", "formula": "\\begin{align*} [ C _ I , C _ J ] = 0 \\ \\ \\ \\ \\ \\ \\end{align*}"}
+{"id": "705.png", "formula": "\\begin{align*} f ( x ) = \\Phi ^ 6 ( x ) - ( 1 2 x - 4 8 ) \\Phi ^ 5 ( x ) - ( 1 6 0 x - 2 0 ) \\Phi ^ 4 ( x ) - ( 3 2 x + 4 4 8 ) \\Phi ^ 3 ( x ) \\\\ + ( 3 8 4 x + 4 3 2 ) \\Phi ^ 2 ( x ) - ( 1 9 2 x ) \\Phi ( x ) - 6 4 - m \\end{align*}"}
+{"id": "4767.png", "formula": "\\begin{align*} \\norm { J _ t x - J _ s x } & = \\norm { J _ s \\left ( \\frac { s } { t } x + \\frac { t - s } { t } J _ t x \\right ) - J _ s x } \\\\ & \\leq \\left ( 1 - \\frac { s } { t } \\right ) \\norm { x - J _ t x } . \\end{align*}"}
+{"id": "3797.png", "formula": "\\begin{align*} \\| \\bar { Z } \\bar { Z } ^ \\top \\| \\leq \\sum _ { i \\in \\widehat { \\mathcal { C } } _ 1 \\cap \\overline { \\mathcal { C } _ 1 } } \\sum _ { t = 0 } ^ { T - 1 } \\| \\Sigma ^ { ( i ) } _ t \\| \\left \\| \\sum _ { l = 1 } ^ { N _ i } \\chi ^ { ( i ) } _ { l , t } \\chi ^ { ( i ) , \\top } _ { l , t } \\right \\| , \\end{align*}"}
+{"id": "7443.png", "formula": "\\begin{align*} \\big [ ( A \\otimes B ) \\cdot Z \\big ] ^ { i j } = \\sum _ { \\alpha , \\beta = 1 } ^ { d } A ^ i _ \\alpha B ^ j _ \\beta Z ^ { \\alpha \\beta } \\qquad i , j \\in \\{ 1 , \\ldots , d \\} . \\end{align*}"}
+{"id": "6704.png", "formula": "\\begin{align*} \\dd ( G _ p ) = \\log _ p \\lvert G _ p : \\Phi ( G _ p ) \\rvert \\dd ( T _ p ) = \\log _ p \\lvert \\Omega _ { \\{ 1 \\} } ( G _ p ) \\rvert , \\end{align*}"}
+{"id": "5835.png", "formula": "\\begin{align*} f _ D ( F _ i ) = \\textrm { c r } _ D ( F _ i ) + \\textrm { c r } _ D ( F _ i , P ( 4 k + 2 , 2 k ) \\setminus E ( F _ i ) ) / 2 . \\end{align*}"}
+{"id": "7020.png", "formula": "\\begin{align*} X _ t = x - \\int _ 0 ^ b b ( X _ s ) d s + \\sqrt { 2 } W _ t , t \\geq 0 , \\end{align*}"}
+{"id": "6136.png", "formula": "\\begin{align*} \\| \\rho _ { m , n } - \\psi _ m \\circ \\varphi _ n \\| & \\leq \\| \\varphi _ { m - 1 } \\circ \\rho _ { m - 1 , n } - \\varphi _ n \\| < \\textstyle { \\sum } _ { j = n + 1 } ^ { m - 1 } \\varepsilon _ j < \\varepsilon . \\end{align*}"}
+{"id": "2040.png", "formula": "\\begin{align*} q _ 0 c _ { \\alpha } + \\sum q _ \\beta c _ \\gamma \\mbox { w h e r e t h e s u m i s o v e r } \\{ ( \\beta , \\gamma ) \\in M _ { 1 , e } \\times M _ { 0 , m } \\ , : \\ , \\beta + \\gamma = \\alpha \\} . \\end{align*}"}
+{"id": "6005.png", "formula": "\\begin{align*} H ( t ) = \\begin{cases} & O \\left ( \\frac { N d } { t ^ { 3 / 2 } } \\right ) , \\mbox { i f $ j = 1 $ } , \\\\ & O ( \\frac { N ^ 2 d ^ 2 } { t ^ 2 } ) , \\mbox { i f $ j = 2 $ } . \\end{cases} \\end{align*}"}
+{"id": "1517.png", "formula": "\\begin{align*} A = D ^ 2 F ( D u ) , G = F . \\end{align*}"}
+{"id": "1921.png", "formula": "\\begin{align*} \\begin{cases} \\Delta u - V ( x ) u & = 0 \\ , \\ , B _ j \\\\ u & = \\gamma \\ , \\ , G \\setminus B _ j . \\end{cases} \\end{align*}"}
+{"id": "718.png", "formula": "\\begin{align*} \\gamma _ m & = \\int _ { I ^ n } \\Big [ \\prod _ { k = 1 } ^ { n } r _ k ^ { 2 m _ k } ( 1 - r _ k ^ 2 ) ^ \\alpha r _ k \\Big ] \\omega ( \\mathbf { r } ) d \\mathbf { r } \\\\ & \\le \\int _ { I ^ n } \\Big [ \\prod _ { k = 1 } ^ { n } ( 1 - r _ k ^ 2 ) ^ \\alpha r _ k \\Big ] \\omega ( \\mathbf { r } ) d \\mathbf { r } < \\infty . \\end{align*}"}
+{"id": "310.png", "formula": "\\begin{align*} u _ t = \\Delta u + | x | ^ { \\sigma } u ^ p , \\end{align*}"}
+{"id": "9116.png", "formula": "\\begin{align*} ( z - v _ { i } ^ { a _ { i j } } w ) e _ { i } ( z ) e _ { j } ( w ) = ( v _ { i } ^ { a _ { i j } } z - w ) e _ { j } ( w ) e _ { i } ( z ) \\forall \\ i , j \\in I , \\end{align*}"}
+{"id": "4030.png", "formula": "\\begin{align*} V _ { \\mu , 1 } ^ { n } \\left ( z \\right ) = \\left ( 0 , \\frac { 1 - \\mu } { 3 - \\mu } , \\frac { 1 - \\mu } { 3 - \\mu } , \\frac { 1 + \\mu } { 3 - \\mu } \\right ) , \\quad \\forall n \\geq 1 . \\end{align*}"}
+{"id": "5283.png", "formula": "\\begin{align*} L S _ { d } \\left ( x . y \\right ) = \\log ( x . y ) = \\log x + \\log y \\end{align*}"}
+{"id": "2330.png", "formula": "\\begin{align*} \\begin{cases} \\| | x ' | ^ { \\frac { 1 } { 2 } } | x | ^ { \\frac { 1 } { 2 } } u ^ { c , \\gamma } \\| _ { L ^ { \\infty } ( \\Omega _ { e ^ { - 1 } } ) } \\leq K ( c , \\gamma ) , \\\\ \\| | x | u ^ { c , \\gamma } \\| _ { L ^ { \\infty } ( \\mathbb { R } ^ { 3 } \\setminus \\Omega _ { e ^ { - 1 } } ) } \\leq K ( c , \\gamma ) , \\end{cases} \\end{align*}"}
+{"id": "5657.png", "formula": "\\begin{align*} \\nu | _ { \\varphi ^ { - 1 } ( R _ x ) } = \\nu _ { R _ x } . \\end{align*}"}
+{"id": "7671.png", "formula": "\\begin{align*} b _ 1 : = \\sum _ { n \\in \\mathbb { Z } ^ d } W ( n ) \\psi ( n ) \\leq \\frac { C _ a 7 2 \\sqrt { 2 } \\norm { F } _ { \\infty } } { \\eta } \\lambda S _ { \\delta - \\gamma } S _ { \\delta - 2 \\nu } \\end{align*}"}
+{"id": "2973.png", "formula": "\\begin{align*} 0 < \\alpha < 1 , d = 2 , x _ 1 ( 0 ) \\neq x _ 2 ( 0 ) , \\mathcal { A } ( v ) ( 0 ) > 0 . \\end{align*}"}
+{"id": "6799.png", "formula": "\\begin{align*} ( \\kappa _ 1 , \\kappa _ 3 ) = \\kappa _ 2 \\left ( \\frac { 3 + \\sqrt { 6 } } { 3 } , \\frac { - 2 C } { 3 + \\sqrt { 6 } } \\right ) . \\end{align*}"}
+{"id": "8658.png", "formula": "\\begin{align*} ( \\ , u \\ , | \\ , v \\ , ) _ { M } : = \\int _ M \\langle \\ , u \\ , | \\ , v \\ , \\rangle d v _ { M ' } , \\ \\ u , v \\in \\Omega ^ { 0 , q } _ c ( M ) , \\end{align*}"}
+{"id": "779.png", "formula": "\\begin{align*} \\phi \\colon H \\to \\C [ T ] \\# H _ \\beta , \\phi ( x ) = \\pi ( x _ { ( 1 ) } ) \\# x _ { ( 2 ) } . \\end{align*}"}
+{"id": "7591.png", "formula": "\\begin{align*} B _ { R } ^ { 3 } [ u ] & = \\frac { - 2 N ( p - 2 ) } { p } \\int _ { \\mathbb { R } ^ { N } } | u | ^ { p } - \\frac { 2 ( p - 2 ) } { p } \\int _ { | x | \\geq R } ( \\Delta \\varphi _ { R } - N ) | u | ^ { p } \\\\ & = \\frac { - 2 N ( p - 2 ) } { p } \\int _ { \\mathbb { R } ^ { N } } | u | ^ { p } + O ( R ^ { - \\frac { ( p - 2 ) ( N - 1 ) } { 2 } } \\| \\Delta u \\| _ { 2 } ^ { \\frac { p - 2 } { 4 } } ) . \\end{align*}"}
+{"id": "7283.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } C _ m ( 0 , 2 n ) y ^ { 2 n } = 1 - \\frac { m y } { \\sqrt { 1 - y ^ 2 } } \\cot ( m \\arcsin ( y ) ) \\end{align*}"}
+{"id": "2681.png", "formula": "\\begin{align*} \\delta S ^ { ( 2 ) } = \\int ^ { t _ { 2 } } _ { t _ { 1 } } \\sum _ { i = 1 } ^ { n } \\left ( \\frac { \\partial L ^ { ( 2 ) } } { \\partial \\ddot { q } ^ { i } } \\right ) \\delta \\ddot { q } ^ { i } d t . \\end{align*}"}
+{"id": "192.png", "formula": "\\begin{align*} T \\ = \\ S \\cup R \\cup ( \\bigcup _ i [ ( E _ i \\times C _ i ) \\cup ( C _ i \\times F _ i ) ] ) . \\end{align*}"}
+{"id": "2794.png", "formula": "\\begin{align*} H _ { { \\rm { e f f e c t i v e } } } : = H _ { T } | _ { \\Psi _ { \\alpha } : = 0 } . \\end{align*}"}
+{"id": "8542.png", "formula": "\\begin{align*} \\begin{array} { l c l } \\lambda _ 1 + k ^ { - 1 } \\mu _ 1 a _ 1 & = & 0 , \\\\ \\lambda _ 2 + k ^ { - 1 } \\mu _ 2 a _ 2 & = & 0 . \\end{array} \\end{align*}"}
+{"id": "1839.png", "formula": "\\begin{align*} F _ { A } \\wedge \\psi = 0 , \\end{align*}"}
+{"id": "1703.png", "formula": "\\begin{align*} \\mathrm { a d } _ { V _ q } ( X _ k ) = \\sum _ { j = 1 } ^ n v _ j \\left [ X _ j , \\overline { X _ k } \\right ] - \\overline { X _ k } ( v _ j ) X _ j = \\sum _ { j = 1 } ^ n v _ j \\left ( H _ { \\mathcal { L } } \\right ) _ { j , k } - \\overline { X _ k } ( v _ j ) X _ j & = - \\sum _ { j = 1 } ^ n \\overline { X _ k } ( v _ j ) X _ j \\\\ & = - \\sum _ { j = 1 } ^ n \\frac { \\partial v _ j } { \\partial x _ k } X _ j . \\end{align*}"}
+{"id": "1826.png", "formula": "\\begin{align*} \\pi _ 7 ( \\omega ) = \\frac { 1 } { 3 } ( \\omega + \\star ( \\phi _ 0 \\wedge \\omega ) ) \\pi _ { 1 4 } ( \\omega ) = \\frac { 1 } { 3 } ( 2 \\omega - \\star ( \\phi _ 0 \\wedge \\omega ) ) . \\end{align*}"}
+{"id": "3687.png", "formula": "\\begin{align*} d ( n - d - 1 ) + \\binom { d } { 2 } = d n - \\binom { d + 1 } { 2 } - d \\end{align*}"}
+{"id": "3241.png", "formula": "\\begin{align*} p & = ( 0 , x , 0 , 0 ) x > 0 \\ : , \\\\ u & = ( \\alpha , \\beta , 0 , 0 ) \\qquad \\ ! \\alpha ^ 2 - \\beta ^ 2 = \\varepsilon ^ 2 \\ : . \\end{align*}"}
+{"id": "3487.png", "formula": "\\begin{align*} J \\bigl ( \\sigma ^ { \\mathsf { R e C U } } ( g ( x ) ) \\bigr ) = \\operatorname { d i a g } \\bigl ( 3 \\sigma ^ { \\mathsf { R e Q U } } ( g ( x ) ) \\bigr ) \\cdot J ( g ( x ) ) . \\end{align*}"}
+{"id": "2266.png", "formula": "\\begin{align*} \\check R _ 1 ( u ) \\check R _ { 2 } ( u + v ) \\check R _ 1 ( v ) = \\check R _ { 2 } ( v ) \\check R _ 1 ( u + v ) \\check R _ { 2 } ( u ) \\ \\ \\ \\ \\ \\end{align*}"}
+{"id": "4234.png", "formula": "\\begin{align*} \\Theta _ 2 ( T _ { C } X ) = 1 - q ^ { \\frac { 1 } { 2 } } \\widetilde { T _ C X } + q ( \\widetilde { T _ C X } + \\wedge ^ 2 \\widetilde { T _ C X } ) + O ( q ^ { \\frac { 3 } { 2 } } ) , \\end{align*}"}
+{"id": "2712.png", "formula": "\\begin{align*} { _ { * } f ^ { I } } ( q ^ { i } , p _ { i } , t ) = { _ { * } c } ^ { I } , \\end{align*}"}
+{"id": "7557.png", "formula": "\\begin{align*} \\int _ \\Omega \\tfrac { 1 } { 2 } \\bar \\rho | \\bar u | ^ 2 + \\varepsilon \\tfrac { 1 } { 2 } \\bar n | \\bar v | ^ 2 + h _ 1 ( \\bar \\rho ) + h _ 2 ( \\bar n ) - \\delta \\tfrac { 1 } { 2 } | \\nabla \\bar \\phi | ^ 2 \\ d x \\Big | _ { \\tau = 0 } ^ { \\tau = t } = \\int _ 0 ^ t \\int _ \\Omega \\varepsilon \\bar v \\cdot \\bar e \\ d x d \\tau + \\int _ 0 ^ t \\int _ \\Omega \\delta ( \\partial _ \\tau \\bar \\phi ) \\Delta \\bar \\phi \\ d x d \\tau \\ , \\end{align*}"}
+{"id": "6890.png", "formula": "\\begin{align*} x \\# y = s h _ { p , q } ( x \\otimes y ) , \\end{align*}"}
+{"id": "3227.png", "formula": "\\begin{align*} H ^ * ( X ) = \\bigoplus _ n H ^ * ( B , ^ p \\ ! \\ ! R ^ n f _ \\ast C _ X ) \\end{align*}"}
+{"id": "8277.png", "formula": "\\begin{align*} x = R ( \\lambda , A ) y \\mbox { i f a n d o n l y i f } y \\in ( \\lambda - A ) x = \\lambda x - A x . \\end{align*}"}
+{"id": "8160.png", "formula": "\\begin{align*} M - \\sum _ { j = 1 } ^ L c _ { j } & = M \\sum _ { \\underline x ^ L } P _ { \\underline X ^ L , Y ^ L } ( \\underline x ^ L , 0 ^ L ) \\\\ & = M \\sum _ { \\underline x ^ L } \\frac { | \\mathcal B _ { 0 ^ L } ( \\underline x ^ L ) | } { M } P _ { \\underline X ^ L | | Y ^ { L - 1 } } ( \\underline x ^ L | | \\underline 0 ^ { L - 1 } ) . \\end{align*}"}
+{"id": "4431.png", "formula": "\\begin{gather*} \\psi ^ j = - 4 \\frac { w _ h ^ j - w _ h ^ { j - 1 / 2 } } { \\tau _ j } + \\frac { v _ h ^ j - v _ h ^ { j - 1 } } { \\tau _ j } \\ , , \\ \\ j = 1 , \\dots , M . \\end{gather*}"}
+{"id": "7775.png", "formula": "\\begin{align*} | Z _ 0 | _ { \\delta _ 1 } , | Z _ 0 ^ { - 1 } | _ { \\delta _ 1 } < \\varepsilon ^ { - \\frac { 1 } { 2 } } = \\epsilon _ 1 ^ { - 1 } , \\ \\ \\ | Z _ 0 | _ { \\delta _ 1 } ^ { \\frac { 1 } { 2 } } | F _ 1 | _ { h _ 1 , \\delta _ 1 } , | Z _ 0 ^ { - 1 } | _ { \\delta _ 1 } ^ { \\frac { 1 } { 2 } } | F _ 1 | _ { h _ 1 , \\delta _ 1 } < 1 . \\end{align*}"}
+{"id": "4934.png", "formula": "\\begin{align*} \\begin{gathered} e ^ { S } _ { - } \\notin \\{ e ^ { S _ 2 } _ { \\pm } \\} , \\\\ e ^ { S } _ { + } \\in \\{ e ^ { S _ 2 } _ { \\pm } \\} , \\end{gathered} \\end{align*}"}
+{"id": "5625.png", "formula": "\\begin{align*} d ( x , y ) = \\sum _ { n = 0 } ^ \\infty \\frac { | x _ n - y _ n | } { 2 ^ n } . \\end{align*}"}
+{"id": "2548.png", "formula": "\\begin{align*} \\dot \\AA ( M ) = \\bigcup _ m \\dot \\AA ^ m ( M ) \\ ; , \\quad \\dot C ^ \\infty ( M ) = \\bigcap _ m \\dot \\AA ^ m ( M ) \\ ; , \\end{align*}"}
+{"id": "3693.png", "formula": "\\begin{align*} a _ i = \\frac { 1 } { \\sqrt { \\lambda ^ 2 - 4 ( d - 1 ) } } \\left ( \\left ( \\frac { \\lambda + \\sqrt { \\lambda ^ 2 - 4 ( d - 1 ) } } { 2 } \\right ) ^ { i + 1 } - \\left ( \\frac { \\lambda - \\sqrt { \\lambda ^ 2 - 4 ( d - 1 ) } } { 2 } \\right ) ^ { i + 1 } \\right ) . \\end{align*}"}
+{"id": "2516.png", "formula": "\\begin{align*} S ^ \\infty ( U \\times \\R ^ l ) = \\bigcup _ m S ^ m ( U \\times \\R ^ l ) \\ ; , S ^ { - \\infty } ( U \\times \\R ^ l ) = \\bigcap _ m S ^ m ( U \\times \\R ^ l ) \\ ; . \\end{align*}"}
+{"id": "7757.png", "formula": "\\begin{align*} s _ { i + 1 } = \\min \\{ n : K _ { n } ^ { - 1 } < d _ 1 u _ { s _ { i } } \\} , \\end{align*}"}
+{"id": "2506.png", "formula": "\\begin{align*} \\int _ 0 ^ t e ^ { M ( s - t ) } \\| \\nabla v ( s ) \\| _ 2 \\ \\mathrm { d } s = 0 \\end{align*}"}
+{"id": "8979.png", "formula": "\\begin{align*} F ( t M ) = t F ( M ) \\end{align*}"}
+{"id": "2679.png", "formula": "\\begin{align*} \\frac { \\partial L ^ { ( 2 ) } } { \\partial \\dot { q } ^ { i } } - \\frac { d } { d t } \\left ( \\frac { \\partial L ^ { ( 2 ) } } { \\partial \\ddot { q } ^ { i } } \\right ) = 0 \\end{align*}"}
+{"id": "2697.png", "formula": "\\begin{align*} L = \\frac { 1 } { 2 } \\dot { Q } ^ { 2 } - \\frac { \\alpha } { \\beta } Q \\delta ( t - t _ { 0 } ) , \\end{align*}"}
+{"id": "6165.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau ^ k } [ f ( x ) - f ( \\breve { x } ^ { k } ) ] - \\frac { 1 } { \\tau ^ { k - 1 } } [ f ( x ) - f ( \\breve { x } ^ { k - 1 } ) ] + ( x - \\widetilde { x } ^ { k } ) ^ T [ - A ^ T \\widetilde { \\lambda } ^ k \\\\ & + \\tau ^ k \\beta ^ k ( D - A ^ T A ) ( \\widetilde { x } ^ { k } - \\bar { x } ^ k ) + ( 2 - \\gamma ) ( 1 - \\tau ^ k ) \\beta ^ k A ^ T ( A \\breve { x } ^ { k - 1 } - b ) ] \\geq 0 , ~ \\forall x . \\end{aligned} \\end{align*}"}
+{"id": "4259.png", "formula": "\\begin{align*} \\eqref { d x i - h o m o g e n e o u s } = - i \\int _ 0 ^ t \\iint e ^ { 2 i s \\eta \\sigma } \\partial _ \\xi G _ \\xi [ f ( s ) , f ( s ) , f ( s ) ] ( \\eta , \\sigma ) \\ , d \\sigma \\ , d \\eta \\ , d s . \\end{align*}"}
+{"id": "6345.png", "formula": "\\begin{align*} \\tilde \\nu _ g : = \\nu ^ n + \\frac { r ^ 2 } { \\cosh ^ 2 ( r ) \\sinh ^ 2 ( r ) } \\nu ^ h + \\frac { r ^ 2 } { \\sinh ^ 2 ( r ) } \\nu ^ v , \\end{align*}"}
+{"id": "4180.png", "formula": "\\begin{align*} \\varphi ( g _ 1 g _ 2 ) = \\varphi ( g _ 1 ) \\varphi ( g _ 2 ) \\mbox { a n d } \\varphi ( g _ 1 g _ 3 ) = \\varphi ( g _ 3 ) \\varphi ( g _ 1 ) \\ , . \\end{align*}"}
+{"id": "457.png", "formula": "\\begin{align*} \\Delta ( \\Omega ^ q _ + ) : = \\left \\{ \\begin{aligned} & \\Delta \\overline { t } \\mbox { s a t i s f y i n g } x _ { B ^ q } ( t _ q + \\Delta \\overline { t } ) = \\min _ { ( i , j ) \\in \\Omega ^ q _ + \\cap { \\cal A } ^ q _ < } x ^ * _ j ( t _ q ) , \\mbox { i f } \\ ; \\Omega ^ q _ + \\cap { \\cal A } ^ q _ < \\neq \\emptyset , \\\\ [ 2 p t ] & + \\infty , \\mbox { o t h e r w i s e , } \\end{aligned} \\right . \\end{align*}"}
+{"id": "7187.png", "formula": "\\begin{align*} \\frac { \\| v _ 0 ^ 2 - ( v _ 0 + V ) ^ 2 \\| _ Y } { v _ 0 ^ 2 - 1 } & = \\frac { \\| V ( V + 2 v _ 0 ) \\| _ Y } { ( v _ 0 ^ 2 - 1 ) } \\leq \\delta \\gamma \\| V \\| _ Y ( \\| V \\| _ Y + 2 \\max \\{ \\sqrt { 3 } , \\beta \\} \\} ) \\leq \\delta , \\\\ \\frac { M ^ 2 } { v _ 0 ^ 2 - 1 } & \\leq M ^ 2 \\delta \\gamma \\leq \\delta , \\\\ \\frac { 2 M \\| v _ 0 + V \\| _ Y } { v _ 0 ^ 2 - 1 } & \\leq 2 M ( \\| V \\| _ Y + \\max \\{ \\sqrt { 3 } , \\beta \\} ) \\delta \\gamma \\leq \\delta , \\\\ \\frac { 1 } { \\gamma } \\frac { 1 } { v _ 0 ^ 2 - 1 } & \\leq \\delta . \\end{align*}"}
+{"id": "511.png", "formula": "\\begin{align*} O \\left ( \\sum _ { t = 1 } ^ K { m _ { j _ t } \\log ^ { 1 + o ( 1 ) } { q } } \\right ) \\subseteq O ( d \\log ^ { 1 + o ( 1 ) } { q } ) \\end{align*}"}
+{"id": "7560.png", "formula": "\\begin{align*} i \\partial _ { t } \\psi + \\gamma \\Delta ^ { 2 } \\psi + \\epsilon \\Delta \\psi + f ( | \\psi | ) \\psi = 0 , \\end{align*}"}
+{"id": "2686.png", "formula": "\\begin{align*} \\sum ^ { d } _ { \\alpha = 0 } ( - D ) ^ { \\alpha } \\frac { \\partial L ^ { ( 1 ) } } { \\partial ( D ^ { \\alpha } q ^ { i } ) } = \\frac { \\partial L ^ { ( 1 ) } } { \\partial q ^ { i } } - \\frac { d } { d t } \\frac { \\partial L ^ { ( 1 ) } } { \\partial \\dot { q } ^ { i } } = 0 , \\end{align*}"}
+{"id": "5081.png", "formula": "\\begin{align*} \\psi _ { q , k } ( \\mathsf { a } _ s ) & = L _ s \\\\ \\psi _ { q , k } ( \\mathsf { e } _ s ) & = E _ s . \\end{align*}"}
+{"id": "3161.png", "formula": "\\begin{align*} ( a _ s , \\ldots , a _ 0 , i ) ^ \\psi = \\left ( \\sum \\limits _ { r = 0 } ^ s a _ r k ^ r \\right ) t + i . \\end{align*}"}
+{"id": "6618.png", "formula": "\\begin{align*} J ( y ) = \\begin{cases} & \\frac { 1 } { N ! } \\left ( 1 - \\frac { 1 } { y } \\right ) ^ N \\ \\ 1 \\leq y , \\\\ & 0 \\ \\ 0 < y < 1 . \\end{cases} \\end{align*}"}
+{"id": "1099.png", "formula": "\\begin{align*} \\int _ 0 ^ r t ^ { a + n - 1 } [ \\log ( 2 + t ) ] ^ b \\ , d t & = \\frac { 1 } { a + n } r ^ { a + n } [ \\log ( 2 + r ) ] ^ b \\\\ & \\quad - \\frac { b } { a + n } \\int _ 0 ^ r t ^ { a + n } \\frac { [ \\log ( 2 + t ) ] ^ { b - 1 } } { 2 + t } \\ , d t \\end{align*}"}
+{"id": "8700.png", "formula": "\\begin{align*} P S _ G \\gamma B _ G = P \\gamma B _ G = B _ G \\end{align*}"}
+{"id": "6729.png", "formula": "\\begin{align*} S _ 1 ( z y ) = - \\pi z y + \\ln ( 2 \\pi z ) + \\ln y - \\ln \\left ( { 1 - e ^ { - 2 \\pi z y } } \\right ) . \\end{align*}"}
+{"id": "425.png", "formula": "\\begin{align*} \\bold { F } _ { j } = \\begin{bmatrix} \\bold { \\Pi } _ { j } + \\bold { \\Gamma } _ { j } \\widetilde { \\bold { \\Lambda } } _ { j } & \\bold { \\Gamma } _ { j } \\\\ \\bold { \\Pi } _ { j } ^ { T } \\widetilde { \\bold { \\Lambda } } _ { j } + \\bold { \\Xi } _ { j } & \\bold { \\Pi } _ { j } ^ { T } \\end{bmatrix} , \\end{align*}"}
+{"id": "9202.png", "formula": "\\begin{gather*} \\Delta ( K ) = K \\otimes K , \\Delta ( E ) = E \\otimes K ^ t + K ^ s \\otimes E , \\Delta ( F ) = F \\otimes K ^ { - s } + K ^ { - t } \\otimes F , \\\\ \\varepsilon ( K ) = 1 , \\varepsilon ( E ) = \\varepsilon ( F ) = 0 , \\\\ S ( K ) = K ^ { - 1 } , S ( E ) = - K ^ { - s } E K ^ { - t } , S ( F ) = - K ^ t F K ^ s , \\end{gather*}"}
+{"id": "3769.png", "formula": "\\begin{align*} f _ { N , j } = \\log ( 1 / w ) 1 _ { I ^ + _ j } - \\frac { \\alpha _ j } { | I _ j ^ - | } 1 _ { I _ j ^ - } . \\end{align*}"}
+{"id": "7772.png", "formula": "\\begin{align*} b \\tilde M _ { j + 1 } ^ 4 \\epsilon _ { j + 1 } ^ { \\frac { 1 } { 2 m } } = b ^ 5 \\nu _ { s _ { k + 1 } } ^ { - 4 \\kappa } \\tilde M _ j ^ { 4 \\kappa } \\epsilon _ j ^ { \\frac { 4 ^ j } { 2 m } } < \\epsilon _ j ^ { - \\frac { \\kappa } { m } } ( b \\tilde M _ j ^ 4 \\epsilon _ j ^ { \\frac { 1 } { 2 m } } ) ^ \\kappa \\epsilon _ j ^ { \\frac { 4 ^ j - \\kappa } { 2 m } } < 1 . \\end{align*}"}
+{"id": "8110.png", "formula": "\\begin{align*} m = 0 = n , \\mu \\nu 2 ^ { - k } \\Z ^ d \\cap [ 0 , 1 ) ^ d . \\end{align*}"}
+{"id": "876.png", "formula": "\\begin{align*} & C _ 0 ^ 1 = \\delta ^ { 1 1 } C _ { 1 2 } y ^ 2 = ( { w ^ 1 } _ { \\substack { \\\\ x _ 2 } } - { w ^ 2 } _ { \\substack { \\\\ x _ 1 } } ) y ^ 2 \\\\ & C _ 0 ^ 2 = \\delta ^ { 2 2 } C _ { 2 1 } y ^ 1 = ( { w ^ 2 } _ { \\substack { \\\\ x _ 1 } } - { w ^ 1 } _ { \\substack { \\\\ x _ 2 } } ) y ^ 1 , \\end{align*}"}
+{"id": "4635.png", "formula": "\\begin{align*} \\lambda : = 2 ^ { 2 \\gamma m n } \\cdot 2 ^ { \\gamma m n \\tfrac { 2 } { d } \\left ( - 1 + \\epsilon \\right ) } \\end{align*}"}
+{"id": "8612.png", "formula": "\\begin{align*} \\phi _ n ( t ; x ) = & ( \\Lambda ^ { \\frac { 1 } { 2 } } \\partial _ x ^ n u ) ( X ( t ; x ) , t ) \\\\ = & \\int _ { - \\infty } ^ { + \\infty } \\frac { ( \\partial _ x ^ n u ) ( X ( t ; x ) , t ) - ( \\partial _ x ^ n u ) ( y , t ) } { | X ( t ; x ) - y | ^ { \\frac { 3 } { 2 } } } d y \\end{align*}"}
+{"id": "3795.png", "formula": "\\begin{align*} \\| \\bar { Z } \\bar { Z } ^ \\top \\| \\leq \\frac { 9 } { 4 } \\sum _ { i \\in \\widehat { \\mathcal { C } } _ 1 \\cap \\overline { \\mathcal { C } _ 1 } } \\sum _ { t = 0 } ^ { T - 1 } N _ i \\left \\| \\Sigma _ { t } ^ { ( i ) } \\right \\| . \\end{align*}"}
+{"id": "8403.png", "formula": "\\begin{align*} a \\circ ( b \\circ c ) & = a \\circ ( B _ 1 ( b _ 1 ) c S ( B _ 2 ( b _ 2 ) ) ) \\\\ & = B _ 1 ( a _ 1 ) B _ 1 ( b _ 1 ) c S ( B _ 2 ( b _ 2 ) ) S ( B _ 2 ( a _ 2 ) ) \\\\ & = B _ 1 ( B _ 1 ( a _ 1 ) b _ 1 S ( B _ 2 ( a _ 2 ) ) ) c S ( B _ 2 ( B _ 1 ( a _ 3 ) b _ 2 S ( B _ 2 ( a _ 4 ) ) ) ) \\\\ & = B _ 1 ( a _ 1 \\circ b _ 1 ) c S ( B _ 2 ( a _ 2 \\circ b _ 2 ) ) = ( a \\circ b ) \\circ c . \\end{align*}"}
+{"id": "8379.png", "formula": "\\begin{align*} \\xi = \\prod _ { j = 1 } ^ { r } x _ { \\beta _ { j } } ( a _ { j } ) w B / B \\end{align*}"}
+{"id": "6148.png", "formula": "\\begin{align*} G ^ k : = ( Q ^ k ) ^ T + Q ^ k - ( M ^ k ) ^ T H ^ k M ^ k . \\end{align*}"}
+{"id": "5934.png", "formula": "\\begin{align*} \\dfrac { M _ { 2 } ( \\mathcal { N C } ( G ) ) } { | e ( \\mathcal { N C } ( G ) ) | } & = \\dfrac { A ( q ) } { ( q ^ { 7 } - 2 q ^ { 6 } - 2 q ^ { 5 } + 5 q ^ { 4 } - 4 q ^ { 2 } + q ^ { 3 } + 1 ) } . \\end{align*}"}
+{"id": "6899.png", "formula": "\\begin{align*} F ( 0 ) & = ( k ( \\eta \\sinh 2 + 2 \\gamma \\sinh 1 ) - 4 ( 1 - \\cosh 1 ) \\left ( \\gamma + \\eta \\cosh 1 \\right ) ^ 2 ) ^ 2 - 3 2 \\left ( \\gamma + \\eta \\cosh 1 \\right ) ^ 4 \\intertext { a n d } F \\left ( \\frac { \\pi } { 2 } \\right ) & = ( k ( \\eta \\sin 2 + 2 \\gamma \\sin 1 ) - 4 ( \\cos 1 - 1 ) ( \\gamma + \\eta \\cos 1 ) ^ { 2 } ) ^ { 2 } - 3 2 ( \\gamma + \\eta \\cos 1 ) ^ { 4 } . \\end{align*}"}
+{"id": "3513.png", "formula": "\\begin{align*} \\varphi _ k ( x ) \\varphi _ k ( y ) & = \\lim _ n \\varphi _ n \\big ( \\varphi _ { n , k } ( x ) \\varphi _ { n , k } ( y ) \\big ) \\overset { \\eqref { e q : j u m p } } { = } \\lim _ n \\varphi _ n \\big ( \\psi _ n ( \\varphi _ k ( x ) ) \\psi _ n ( \\varphi _ k ( y ) ) \\big ) , \\end{align*}"}
+{"id": "7363.png", "formula": "\\begin{align*} u _ t ^ 0 ( x , t ) = \\frac { 1 } { 2 } \\{ f ' ( x + t ) - f ' ( x - t ) + g ( x + t ) + g ( x - t ) \\} , \\end{align*}"}
+{"id": "5499.png", "formula": "\\begin{align*} f : \\widehat { A ^ { ( j + 1 ) } } [ x _ j ^ { \\pm 1 } ; \\sigma ' _ j ] & { } ~ \\longrightarrow ~ \\widehat { A ^ { ( j + 1 ) } } [ x _ j ; \\sigma ' _ j , \\delta ' _ j ] S _ j ^ { - 1 } \\\\ \\widehat { A ^ { ( j + 1 ) } } ~ \\ni ~ a & { } ~ \\longmapsto ~ f ( a ) = \\sum _ { n = 0 } ^ { \\infty } q _ j ^ { \\frac { n ( n + 1 ) } { 2 } } ( q _ j - 1 ) ^ { - n } d ' _ { j , n } \\circ ( \\sigma ' _ j ) ^ { - n } ( a ) x _ j ^ { - n } \\\\ x _ j & { } ~ \\longmapsto ~ x _ j . \\end{align*}"}
+{"id": "5214.png", "formula": "\\begin{align*} & U _ { j } = \\frac { 1 } { \\alpha } \\left [ \\left ( \\frac { a - 1 } { a - b } \\right ) ( X . Y ) ^ { a - 2 } - \\left ( \\frac { b - 1 } { a - b } \\right ) ( X . Y ) ^ { b - 2 } \\right ] \\times N \\\\ & V _ { j } = \\frac { 1 } { \\alpha } \\left [ \\left ( \\frac { a - 1 } { a - b } \\right ) ( X . Y ) ^ { a - 2 } - \\left ( \\frac { b - 1 } { a - b } \\right ) ( X . Y ) ^ { b - 2 } \\right ] \\times M \\end{align*}"}
+{"id": "3347.png", "formula": "\\begin{align*} \\begin{cases} x _ { k + 1 } = b ( x _ k , u _ k ) + w _ k , \\\\ y _ { k + 1 } = h ( x _ k ) + v _ k \\end{cases} \\end{align*}"}
+{"id": "1056.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\tilde { b } _ { n , u , \\ell } ^ 1 & = \\beta _ { n + 1 - u + \\ell } ^ * , \\\\ \\tilde { b } _ { n , u , \\ell } ^ { 2 k } & = \\sum _ { m = 0 } ^ { \\infty } \\tilde { b } _ { n , u , m } ^ { 2 k - 1 } \\beta _ { n + 1 + m + \\ell } , \\tilde { b } _ { n , u , \\ell } ^ { 2 k + 1 } = \\sum _ { m = 0 } ^ { \\infty } \\tilde { b } _ { n , u , m } ^ { 2 k } \\beta _ { n + 1 + m + \\ell } ^ * \\end{aligned} \\right . \\end{align*}"}
+{"id": "7821.png", "formula": "\\begin{align*} s _ k ( 1 - a _ k ) ^ { - \\frac { 3 } { 4 } } + o _ k ( a _ k ) ^ { - \\frac { 1 } { 2 } } - w _ k = 0 , \\ a _ k \\in [ 0 , 1 ] , \\end{align*}"}
+{"id": "3924.png", "formula": "\\begin{gather*} f ( z ) = \\begin{pmatrix} f _ 1 ( z _ 1 , z _ 2 , z _ 3 ) \\\\ f _ 2 ( z _ 2 , z _ 3 ) \\\\ f _ 3 ( z _ 3 ) \\end{pmatrix} \\end{gather*}"}
+{"id": "3909.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & 0 & 0 & 0 & [ 1 & 1 & 1 & 1 ] \\\\ 0 & 0 & [ 1 & 1 & 1 & 1 ] & 0 & 0 \\\\ 0 & [ 1 & 1 ] & 0 & 0 & [ 1 & 1 ] & 0 \\end{bmatrix} . \\end{align*}"}
+{"id": "5817.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\Vert u \\Vert _ { H ^ { \\ell } } ( t ) = 0 \\ \\textup { i m p l i e s } \\ \\lim _ { t \\to \\infty } \\Vert u \\Vert _ { H ^ { \\ell + 1 } } ( t ) = 0 . \\end{align*}"}
+{"id": "633.png", "formula": "\\begin{align*} n H _ r I _ { n , r } ( \\rho ) + d _ r = \\sinh ^ { n - r } ( \\rho ) \\frac { \\dot { \\lambda } ^ r } { ( 1 + \\dot { \\lambda } ^ 2 ) ^ { \\frac { r } { 2 } } } , \\end{align*}"}
+{"id": "1809.png", "formula": "\\begin{align*} S ( z , \\pmb { q } ) ( u ) = S _ { n } ( z , q _ { n } ) \\circ \\dots \\circ S _ 1 ( z , q _ 1 ) ( u ) \\end{align*}"}
+{"id": "9141.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { 1 - a _ { i j } } ( - 1 ) ^ { k } \\left [ \\begin{matrix} 1 - a _ { i j } \\\\ k \\end{matrix} \\right ] _ { v _ { i } } E _ { i } ^ { k } E _ { j } E _ { i } ^ { 1 - a _ { i j } - k } = 0 , i \\neq j . \\end{align*}"}
+{"id": "5394.png", "formula": "\\begin{align*} \\nu _ j & = \\frac { h } { \\mu } \\ , \\left [ \\left ( \\frac { 2 j + 1 } { ( \\rho - 1 ) ^ 2 } - \\frac { 2 } { \\left ( \\rho - 1 \\right ) ^ { 3 } } \\right ) \\rho ^ { j + 2 } - \\frac { j \\left ( j + 2 \\right ) } { \\rho - 1 } + \\frac { 3 } { ( \\rho - 1 ) ^ 2 } + \\frac { 2 } { \\left ( \\rho - 1 \\right ) ^ { 3 } } \\right ] \\end{align*}"}
+{"id": "4556.png", "formula": "\\begin{align*} \\nabla ^ 2 _ { V , W } X = - R ( X , V ) W \\end{align*}"}
+{"id": "8003.png", "formula": "\\begin{align*} I _ { i n i } ( W _ 0 ) = & \\int _ { \\mathbb { T } } \\Bigg ( \\sum _ { k = 1 } ^ 3 w _ { 0 , k } ( u ) \\log \\left ( \\frac { w _ { 0 , k } ( u ) } { \\rho _ { k - 1 } ( u ) } \\right ) \\\\ & - \\left ( 1 - \\sum _ { k = 1 } ^ 3 w _ { 0 , k } ( u ) \\right ) \\log \\left ( \\frac { 1 - \\sum _ { k = 1 } ^ 3 \\rho _ { k - 1 } ( u ) } { 1 - \\sum _ { k = 1 } ^ 3 w _ { 0 , k } ( u ) } \\right ) \\Bigg ) d u \\end{align*}"}
+{"id": "4261.png", "formula": "\\begin{align*} s \\partial _ x u = \\tfrac { 1 } { 2 i } [ J ( s ) u ( s ) - x u ( s ) ] \\end{align*}"}
+{"id": "6811.png", "formula": "\\begin{align*} \\mathrm { S p } ( \\tilde { f } ) = \\bigcup _ { i , j } \\{ ( k _ { i j } ^ 1 , \\ldots , k _ { i j } ^ { N ( N - 1 ) / 2 } , l _ { i j } ^ 1 , \\ldots , l _ { i j } ^ { N - 1 } ) \\} , \\end{align*}"}
+{"id": "987.png", "formula": "\\begin{align*} P _ x ( \\bigcup _ { n \\ge 1 } \\{ \\tau _ { V _ n } = \\tau _ D \\} \\cap \\{ X _ { \\tau _ D - } \\in D \\} ) = P _ x ( \\{ X _ { \\tau _ D - } \\in D \\} ) x \\in D . \\end{align*}"}
+{"id": "4305.png", "formula": "\\begin{align*} d y _ t = \\sigma ( y _ t ) d w _ t + b ( y _ t ) d t , y _ 0 = 0 . \\end{align*}"}
+{"id": "2200.png", "formula": "\\begin{align*} \\intop _ \\Omega ( \\psi _ { 1 , z z r } ^ 2 + \\psi _ { 1 , z z z } ^ 2 ) d x + \\intop _ { - a } ^ a \\psi _ { 1 , z z } ^ 2 | _ { r = 0 } d z \\le c | \\omega _ { 1 , z } | _ { 2 , \\Omega } ^ 2 \\end{align*}"}
+{"id": "5917.png", "formula": "\\begin{align*} \\dfrac { M _ { 1 } ( \\mathcal { C } ( A ( n , \\nu ) ) ) } { | v ( \\mathcal { C } ( A ( n , \\nu ) ) ) | } = ( 2 ^ { n } - 1 ) ^ { 2 } = \\dfrac { M _ { 2 } ( \\mathcal { C } ( A ( n , \\nu ) ) ) } { | e ( \\mathcal { C } ( A ( n , \\nu ) ) ) | } . \\end{align*}"}
+{"id": "3264.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ^ { d + 1 } ; q ^ d ) _ k ^ { d - 2 } ( q ; q ^ d ) _ k ^ 2 q ^ { d k } } { ( q ^ d ; q ^ d ) _ k ^ d } \\\\ & \\equiv \\frac { ( 1 - q ) ^ 2 ( q ^ d ; q ^ d ) _ { n - 1 - ( n + 1 ) / d } } { ( - 1 ) ^ { ( n + 1 ) / d } ( q ^ d ; q ^ d ) _ { ( n + 1 ) / d } ^ { d - 1 } } q ^ { ( d ( d + n ) ( n + 1 ) - ( n + 1 ) ^ 2 ) / ( 2 d ) - 2 } \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "3860.png", "formula": "\\begin{align*} \\hat M ^ 1 ( t ) \\doteq M ^ 1 ( T - t ) , \\ ; \\hat \\eta ^ 1 ( t ) = \\hat \\eta ^ 1 ( \\cdot \\mid t ) \\doteq \\eta ^ 1 ( \\cdot \\mid T - t ) , \\end{align*}"}
+{"id": "879.png", "formula": "\\begin{align*} G ^ i = \\mathcal { G } ^ i + \\zeta ^ i = \\zeta ^ i , \\end{align*}"}
+{"id": "5836.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { i + j + k = d } \\lambda _ { i , j , k } x ^ { i + 1 } y ^ j z ^ k = x X = X x = \\displaystyle \\sum _ { i + j + k = d } \\alpha ^ { k - j } \\lambda _ { i , j , k } x ^ { i + 1 } y ^ j z ^ k , \\end{align*}"}
+{"id": "5029.png", "formula": "\\begin{align*} \\mathbf { x } ^ 1 ( \\mathbf { R } - \\nu \\mathbf { 1 } ) = f _ i ^ S - \\nu g _ i ^ S - \\sum _ { j \\in S } ( r _ j ^ S - \\nu w _ j ^ S ) x _ j ^ 0 + \\sum _ { j \\in S ^ c } ( r _ j ^ S - \\nu w _ j ^ S ) x _ j ^ 1 . \\end{align*}"}
+{"id": "7504.png", "formula": "\\begin{align*} | h ( t ) - g _ 0 ( t ) | _ { g _ 0 ( t ) } + & \\mathbf { r } | \\nabla ^ { g _ 0 ( t ) } h ( t ) | _ { g _ 0 ( t ) } = | s ^ { - 1 } Q ^ * \\hat { g } ( s t ) - s ^ { - 1 } Q ^ * \\tilde { g } ( s t ) | _ { g _ 0 ( t ) } \\\\ & + \\mathbf { r } | \\nabla ^ { g _ 0 ( t ) } h ( t ) | _ { g _ 0 ( t ) } \\leq Q ^ * \\left ( | \\hat { g } - \\tilde { g } | _ { \\tilde { g } } + r _ s | \\nabla ^ { \\tilde { g } } \\hat { g } | _ { \\tilde { g } } \\right ) ( s t ) \\\\ & < \\delta , \\end{align*}"}
+{"id": "6549.png", "formula": "\\begin{align*} \\int _ { \\R } K ( x ) d f ( x ) = \\frac { 1 } { 2 \\pi } \\int _ { \\R } \\widehat { K } ( u ) \\phi ( u ) \\iota u d u . \\end{align*}"}
+{"id": "2528.png", "formula": "\\begin{gather*} C ^ { - \\infty } _ { { \\cdot } / } ( L ; E \\otimes \\Omega ^ { - 1 } N L ) \\subset C ^ { - \\infty } _ { { \\cdot } / } ( M ; E ) \\ ; , \\\\ u \\mapsto \\delta _ L ^ u \\ ; , \\quad \\langle \\delta _ L ^ u , v \\rangle = \\langle u , v | _ L \\rangle \\ ; , v \\in C ^ \\infty _ { / { \\cdot } } ( M ; E ^ * \\otimes \\Omega ) \\ ; . \\end{gather*}"}
+{"id": "8876.png", "formula": "\\begin{align*} \\gamma : = \\frac 1 { C _ { n , \\tau , \\alpha , \\lambda } } = \\lim _ { n \\rightarrow \\infty } \\frac { \\| \\sqrt { \\mathcal { K } _ { \\lambda } } u _ n \\| ^ { 2 p } } { \\mathcal { P } [ u _ n ] } . \\end{align*}"}
+{"id": "25.png", "formula": "\\begin{align*} [ \\kappa R ( \\kappa ) - 1 ] \\chi _ y & = R ( \\kappa ) C _ + q \\kappa R _ 0 ( \\kappa ) \\chi _ y + [ \\kappa R _ 0 ( \\kappa ) - 1 ] \\chi _ y \\\\ & = R ( \\kappa ) q _ + - R ( \\kappa ) C _ + ( q - q \\chi _ y ) + [ R ( \\kappa ) C _ + q + 1 ] [ \\kappa R _ 0 ( \\kappa ) - 1 ] \\chi _ y . \\end{align*}"}
+{"id": "3001.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\overline { R } ( x , y ) \\xi & = & \\alpha \\eta ( y ) \\varphi x + ( \\beta - 1 ) \\eta ( y ) x - ( \\beta - 1 ) \\eta ( x ) y - \\alpha \\eta ( x ) \\varphi y . \\\\ \\end{array} \\end{align*}"}
+{"id": "5647.png", "formula": "\\begin{align*} | S \\cap S ^ \\tau | = \\frac { | S | | S ^ \\tau | } { | \\Gamma | } = \\frac { 3 5 2 \\cdot 3 5 2 } { 4 0 9 6 } = \\frac { 1 2 1 } { 4 } . \\end{align*}"}
+{"id": "7693.png", "formula": "\\begin{align*} \\frac { { { d ^ { { N _ G } + 1 } } y } } { { d { y ^ { { N _ G } + 1 } } } } \\C { F } _ j ^ { ( m ) } ( \\tau ) & = { \\left [ { \\frac { { { L ^ { m - \\alpha } } } } { { \\alpha - m } } { { ( t - \\tau ) } ^ { - m + \\alpha + 1 } } } \\right ] ^ { { N _ G } + 1 } } \\C { F } _ j ^ { ( { N _ G } + m + 1 ) } ( \\tau ) \\\\ & = \\left ( \\frac { L } { \\alpha - m } y ^ { - \\frac { m - \\alpha - 1 } { m - \\alpha } } \\right ) ^ { N _ G + 1 } \\C { F } _ j ^ { ( N _ G + m + 1 ) } \\left ( t - L \\ , y ^ { \\frac { 1 } { m - \\alpha } } \\right ) . \\end{align*}"}
+{"id": "9102.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } X ( t + 1 ) = ( A ^ { ( r ) } + J ^ { ( r ) } \\hat { C } K ^ { ( r ) } ) X ( t ) + J ^ { ( r ) } \\hat { C } J ^ { ( r ) } \\xi , \\\\ v ( t ) = J ^ { ( r ) } X ( t ) + \\underline { v } , \\end{array} \\right . \\end{align*}"}
+{"id": "2676.png", "formula": "\\begin{align*} \\delta q ^ { i } ( t _ { 1 } ) = \\delta q ^ { i } ( t _ { 2 } ) = 0 \\end{align*}"}
+{"id": "7201.png", "formula": "\\begin{align*} J _ { X } & = \\{ j \\colon X _ j , Y _ j \\in C ( X ) \\} \\\\ J _ { Y } & = \\{ j \\colon X _ j , Y _ j \\in C ( Y ) \\} \\\\ J _ * & = \\{ j \\colon X _ j , Y _ j \\notin C ( X ) \\cup C ( Y ) \\} . \\end{align*}"}
+{"id": "4726.png", "formula": "\\begin{align*} S _ { \\# } \\ ; = \\ ; S _ { \\# } ^ { ( \\chi ) } + S _ { \\# } ^ { ( \\psi ) } + Q _ { \\# } ^ { ( \\psi , \\chi ) } \\end{align*}"}
+{"id": "7102.png", "formula": "\\begin{align*} w ( t , x , \\varkappa ) = \\int _ { \\mathbb R ^ d } e ^ { i \\varkappa \\cdot y } e ^ { - t \\Lambda ( b ) } ( x , y ) d y - i \\int _ 0 ^ t \\int _ { \\mathbb R ^ d } e ^ { - ( t - s ) \\Lambda ( b ) } ( x , z ) ( \\varkappa \\cdot b ( z ) ) w ( s , z , \\varkappa ) d z d s . \\end{align*}"}
+{"id": "2785.png", "formula": "\\begin{align*} \\delta Q ^ { i } ( t _ { 2 } ) = \\delta Q ^ { i } ( t _ { 1 } ) : = 0 \\end{align*}"}
+{"id": "7600.png", "formula": "\\begin{align*} U _ { \\epsilon } = R ( \\frac { \\epsilon } { \\epsilon ^ { 2 } + | x | ^ { 2 } } ) ^ { \\frac { N - 4 } { 2 } } , \\end{align*}"}
+{"id": "8610.png", "formula": "\\begin{align*} \\frac { d v _ 1 } { d t } + v _ 1 ^ 2 + K _ 1 ( t ; x ) + \\phi _ 1 ( t ; x ) = 0 \\end{align*}"}
+{"id": "613.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\left ( \\frac { 1 } { 4 } \\right ) ^ { k } \\binom { 2 k } { k } \\frac { O _ { 2 k } } { 4 k + 1 } = \\frac { 3 \\Gamma ^ { 2 } \\left ( \\frac { 1 } { 4 } \\right ) \\ln ( 2 ) } { 1 6 \\sqrt { 2 \\pi } } \\end{align*}"}
+{"id": "352.png", "formula": "\\begin{align*} x ^ p + y ^ p = z ^ p , \\end{align*}"}
+{"id": "5963.png", "formula": "\\begin{align*} \\sum _ { u : ( u , v ) \\in E } f _ { u , v } = \\sum _ { v : ( v , w ) \\in E } f _ { v , w } . \\end{align*}"}
+{"id": "666.png", "formula": "\\begin{align*} \\left ( R ^ { k } | _ { S ^ { m } } f \\right ) _ { p _ 1 q _ 1 \\dots p _ { m - k } q _ { m - k } } ^ { i _ { 1 } \\dots i _ { k } } = \\left ( R ^ { 0 } | _ { S ^ { m - k } } f ^ { i _ { 1 } \\cdots i _ { k } } \\right ) _ { p _ { 1 } q _ { 1 } \\cdots p _ { m - k } q _ { m - k } } , \\end{align*}"}
+{"id": "3231.png", "formula": "\\begin{align*} \\langle L , L ^ \\prime \\rangle _ { C _ s } = \\langle \\nu ^ * L , \\nu ^ * L ^ \\prime \\rangle _ { D _ s } \\ . \\end{align*}"}
+{"id": "4465.png", "formula": "\\begin{align*} F _ 0 : = f _ 0 - f _ 1 f _ 2 , \\end{align*}"}
+{"id": "8633.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } ( \\partial ^ 2 _ x X ) = v _ 2 ( \\partial _ x X ) ^ 2 + v _ 1 ( \\partial ^ 2 _ x X ) , \\\\ ( \\partial ^ 2 _ x X ) ( 0 ; x ) = 0 , \\end{cases} \\end{align*}"}
+{"id": "2621.png", "formula": "\\begin{align*} | D _ 4 ( S ) | \\geq | D _ 2 ( S ) | + 2 = | D _ { \\{ 2 \\} } ( S ) \\cup [ D _ { \\{ 1 \\} } ( S ) \\setminus D _ { \\{ 2 \\} } ( S ) ] | + 2 \\geq ( n - 2 ) + ( n / 4 - 1 ) + 2 = \\frac { 5 n } { 4 } - 1 . \\end{align*}"}
+{"id": "4597.png", "formula": "\\begin{align*} N \\left ( - \\Delta _ { \\Omega } + \\lambda V \\right ) \\approx \\frac { 1 } { ( 2 \\pi ) ^ { d } } \\left | \\left \\{ ( p , x ) \\in \\R ^ d \\times \\Omega \\bigm | | p | ^ 2 + \\lambda V ( x ) < 0 \\right \\} \\right | = \\frac { | B _ 1 ( 0 ) | } { ( 2 \\pi ) ^ d } \\int _ { \\Omega } | \\lambda V | ^ { \\frac d 2 } . \\end{align*}"}
+{"id": "9062.png", "formula": "\\begin{align*} F ( N ) = \\bigcup _ { i = 1 } ^ { 2 } \\left ( \\bigcup _ { I \\in \\mathcal { A } _ i ( N ) } I \\right ) . \\end{align*}"}
+{"id": "6503.png", "formula": "\\begin{align*} \\mathbb X : = \\real _ + \\times \\real _ + = \\{ x = ( t , \\theta ) , \\ ; t \\in \\real _ + , \\ ; x \\in \\real _ + \\} ; \\end{align*}"}
+{"id": "3749.png", "formula": "\\begin{align*} _ { 3 } F _ { 2 } \\left ( \\left . \\begin{array} { c } \\alpha , \\beta , \\beta \\\\ \\beta + 1 , \\beta + 1 \\end{array} \\right \\vert 1 \\right ) = \\beta ^ { 2 } \\ , \\mathrm { B } \\left ( 1 - \\alpha , b \\right ) \\left [ \\psi \\left ( 1 + \\beta - \\alpha \\right ) - \\psi \\left ( \\beta \\right ) \\right ] . \\end{align*}"}
+{"id": "2770.png", "formula": "\\begin{align*} \\sigma _ { 3 } ^ { * } \\omega = \\omega _ { Q , P } \\end{align*}"}
+{"id": "564.png", "formula": "\\begin{align*} - \\frac { p } { \\rho } & = g \\eta + j \\omega \\phi & \\Gamma _ f \\cup \\Gamma _ c . \\end{align*}"}
+{"id": "7157.png", "formula": "\\begin{align*} D _ p \\omega = \\frac { d \\omega } { d \\tau } \\ , \\ , . \\end{align*}"}
+{"id": "3110.png", "formula": "\\begin{align*} \\varphi \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & = \\left ( \\begin{array} { c c c } a ^ { 4 \\ , p ^ { e _ 1 - 1 } } & c _ 1 \\ , a ^ { 4 \\ , p ^ { e _ 1 - 1 } } \\ , b ^ { p ^ { e _ 1 } } \\ , d ^ { p ^ { e _ 1 } } & c _ 2 \\ , a ^ { 4 \\ , p ^ { e _ 1 - 1 } } \\ , b ^ { p ^ { e _ 1 } } \\ , d ^ { p ^ { e _ 1 } } \\\\ 0 & d ^ { 2 \\ , p ^ { e _ 1 - 1 } } & 0 \\\\ 0 & 0 & d ^ { 2 \\ , p ^ { e _ 1 - 1 } } \\end{array} \\right ) . \\end{align*}"}
+{"id": "3173.png", "formula": "\\begin{align*} \\sum _ { k \\geq 1 } \\norm { J ( Q ^ { \\frac { 1 } { 2 } } e _ k ) } _ { H ^ { - 1 } } ^ 2 = \\sum _ { k \\geq 1 } \\mu _ k \\norm { e _ k } _ { H ^ { - 1 } } ^ 2 \\leq \\sum _ { k \\geq 1 } \\mu _ k \\norm { e _ k } _ { \\mathcal { H } ^ { - 1 } } ^ 2 < \\infty , \\end{align*}"}
+{"id": "6871.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathbb { J } } ( E _ j - \\mu ) \\langle \\varphi _ j , f \\rangle \\varphi _ j = \\sum _ { j \\in \\mathbb { J } } ( E _ j - \\mu ) c _ j \\varphi _ j = 0 , c _ j = \\langle \\varphi _ j , f \\rangle . \\end{align*}"}
+{"id": "4025.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow + \\infty } \\frac { y ^ { ( t ) } } { x _ { 2 } ^ { ( t ) } } = \\begin{cases} + \\infty & \\mbox { i f } \\gamma _ { 2 } = 0 , \\delta _ { 1 } \\neq 0 \\\\ \\frac { \\gamma x _ { 1 } ^ { ( t _ { 0 } ) } + \\delta x _ { 2 } ^ { ( t _ { 0 } ) } } { \\delta _ 2 x _ { 2 } ^ { \\left ( t _ { 0 } \\right ) } } & \\mbox { i f } \\gamma _ { 2 } = \\delta _ { 1 } = 0 , \\\\ \\frac { \\delta } { \\delta _ 2 } & \\mbox { i f } \\gamma _ { 2 } = 0 , \\delta _ { 1 } \\neq 0 . \\end{cases} \\end{align*}"}
+{"id": "1162.png", "formula": "\\begin{align*} E : = \\left \\{ \\begin{aligned} & \\frac { n - 1 } { p } - n \\tau & & \\ , \\ , \\frac { n } { n - 1 } \\tau > \\frac { 1 } { p } , \\\\ & 0 & & \\ , \\ , \\frac { n } { n - 1 } \\tau = \\frac { 1 } { p } q = \\infty , \\\\ & ( n - 1 ) \\left ( \\frac { 1 } { p } - 1 \\right ) _ + & & . \\end{aligned} \\right . \\end{align*}"}
+{"id": "7637.png", "formula": "\\begin{align*} \\mathcal { V } ( x ) = V a r ( \\mathcal { R } ^ { T } x ) = \\sum _ { i = 1 } ^ { n } \\sum _ { k = 1 } ^ { n } \\sigma _ { i k } x _ { i } x _ { k } \\end{align*}"}
+{"id": "6584.png", "formula": "\\begin{align*} A ( x ) = \\frac { 1 } { 2 \\pi i } \\int _ { a - i T } ^ { a + i T } F ( s ) \\ x ^ s \\frac { d s } { s } + O \\left ( \\frac { x ^ { a } } { T } \\sum _ { n \\geq 1 } \\frac { | f ( n ) | } { n ^ a | \\log ( x / n ) | } \\right ) . \\end{align*}"}
+{"id": "4143.png", "formula": "\\begin{align*} 2 Q ^ * ( u ) & = \\alpha _ j c _ { e , j } ^ 2 + \\sum _ { i < j } \\alpha _ i c _ { e , i } ^ 2 w ^ 2 _ { i j } + \\lambda _ j c _ { s , j } ^ 2 ( 1 - w _ { j j } ) ^ 2 + \\sum _ { i > j } \\lambda _ i c _ { s , i } ^ 2 w _ { i j } ^ 2 \\\\ & + \\sum _ { i < j } \\alpha _ i ( w _ { i j } - w _ { i j } ^ 2 ) + \\sum _ { i \\ge j } \\lambda _ i w _ { i j } ( 1 - w _ { i j } ) = \\sigma ^ 2 _ j . \\end{align*}"}
+{"id": "2303.png", "formula": "\\begin{align*} \\pi _ { \\gamma , n } \\ : \\ \\ \\sigma _ i \\mapsto \\bigl ( \\gamma + ( 1 - \\gamma ) e _ i \\bigr ) g _ i \\ \\ ( i = 1 , \\dots , n - 1 ) \\ , . \\end{align*}"}
+{"id": "2015.png", "formula": "\\begin{align*} h ( \\eta _ 1 \\pm \\eta _ 2 ) & \\leq h ( \\eta _ 1 ) + h ( \\eta _ 2 ) + \\log 2 , \\\\ h ( \\eta _ 1 \\eta _ 2 ^ { \\pm 1 } ) & \\leq h ( \\eta _ 1 ) + h ( \\eta _ 2 ) , \\\\ h ( \\eta _ 1 ^ m ) & = | m | h ( \\eta _ 1 ) . \\end{align*}"}
+{"id": "1312.png", "formula": "\\begin{align*} \\dot { \\mu } = \\mp \\operatorname { a d } ^ \\ast _ { \\delta H / \\delta \\mu } \\mu , \\end{align*}"}
+{"id": "7961.png", "formula": "\\begin{align*} \\frac { \\delta \\tilde { \\mathcal { F } } } { \\delta \\eta } = ( - 1 ) ^ { n - 1 } d \\big ( \\frac { \\delta \\bar { \\mathcal { F } } } { \\delta \\omega } \\big ) . \\end{align*}"}
+{"id": "6792.png", "formula": "\\begin{align*} c _ 1 > 0 , \\ c _ 2 = - 1 , \\ c _ 3 > 0 , \\ \\ d _ 1 > 0 , \\ d _ 2 = - 1 , \\ d _ 3 \\geq 0 , \\ \\ \\frac { d _ 3 } { c _ 3 } < 1 < \\frac { d _ 1 } { c _ 1 } , \\kappa _ 1 c _ 1 + \\kappa _ 2 d _ 2 = 0 . \\end{align*}"}
+{"id": "4075.png", "formula": "\\begin{align*} e ^ { i z _ 0 s } E _ i ( - i z _ 0 s ) = e ^ { i z _ 0 s } ( \\gamma + \\log ( i z _ 0 s ) + \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - i z _ 0 s ) ^ { n } } { n n ! } ) = b ( z _ 0 s ) \\end{align*}"}
+{"id": "6571.png", "formula": "\\begin{align*} \\phi _ { \\Phi _ k , \\beta } ( x , y ) = R _ { k , \\beta } \\left ( \\frac { 1 } { ( 1 - \\delta ) } \\right ) y ^ { \\frac { 1 } { ( 1 - \\delta ) } } + O _ \\delta \\left ( y ^ { \\frac { 1 } { ( 1 - \\delta ) } } \\exp \\left ( - \\sqrt { p _ k \\log y \\log \\log y } \\right ) \\right ) , \\end{align*}"}
+{"id": "2306.png", "formula": "\\begin{align*} u = u _ { r } e _ { r } + u _ { \\theta } e _ { \\theta } + u _ { \\phi } e _ { \\phi } , \\end{align*}"}
+{"id": "6056.png", "formula": "\\begin{align*} x ^ \\alpha \\succeq _ b x ^ \\beta : \\iff x _ 1 ^ { \\alpha _ 1 } x _ 2 ^ { \\alpha _ 2 } \\succ _ { g r l e x } x _ 1 ^ { \\beta _ 1 } x _ 2 ^ { \\beta _ 2 } x _ 1 ^ { \\alpha _ 1 } x _ 2 ^ { \\alpha _ 2 } = x _ 1 ^ { \\beta _ 1 } x _ 2 ^ { \\beta _ 2 } \\frac { x ^ \\alpha } { x _ 1 ^ { \\alpha _ 1 } x _ 2 ^ { \\alpha _ 2 } } \\succeq _ { g r l e x } \\frac { x ^ \\beta } { x _ 1 ^ { \\beta _ 1 } x _ 2 ^ { \\beta _ 2 } } . \\end{align*}"}
+{"id": "1348.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ n p _ i p _ j = \\left ( \\sum _ { i = 1 } ^ n p _ i \\right ) ^ 2 \\end{align*}"}
+{"id": "1507.png", "formula": "\\begin{align*} | f ( \\cdot , \\overline { u } , \\nabla \\overline { u } ) | & \\leq M ( 1 + | \\overline { u } | ^ { \\alpha } + | \\nabla \\overline { u } | ^ { \\beta } ) \\\\ & \\leq M ( 1 + ( \\delta ^ { - p } ( L d ) ^ { \\overline { \\omega } } ) ^ { \\alpha } + ( \\delta ^ { - p } \\hat { L } ) ^ { \\beta } \\\\ & \\leq C \\delta ^ { - p \\max \\{ \\alpha , \\beta \\} } \\end{align*}"}
+{"id": "4391.png", "formula": "\\begin{align*} \\sup _ { s } & \\ \\sum _ { i \\in [ m ] } f _ i ( x , \\overline { u } ^ i ) + s _ i ( \\sup _ { u ^ i \\in \\mathcal { U } _ i } f _ i ( x , u ^ i ) - f _ i ( x , \\overline { u } ^ i ) ) , \\\\ & \\ \\sum _ { i \\in [ m ] } s _ i \\leq \\Gamma , \\\\ & \\ s \\in \\{ 0 , 1 \\} ^ m . \\end{align*}"}
+{"id": "895.png", "formula": "\\begin{align*} L _ { w w } = \\sum _ { l + k \\leq 2 } M _ { l k } x _ 1 ^ { l } x _ 2 ^ k , \\end{align*}"}
+{"id": "4190.png", "formula": "\\begin{align*} \\varphi ( g _ 3 g ^ { 2 } _ 1 ) = \\varphi ( g _ 3 ) \\varphi ( g _ 1 ) \\varphi ( g _ 1 ) \\overset { ( \\ref { e q : a n t i } ) } { = } \\varphi ( g _ 1 g _ 3 ) \\varphi ( g _ 1 ) \\ , , \\end{align*}"}
+{"id": "3416.png", "formula": "\\begin{align*} u _ 0 ( x ) I _ 1 ( u _ 0 ( \\iota ^ { - 1 } ( x ) ) I _ 1 ( \\phi ( x ) ) ) = u _ 0 ( x ) \\overline { u _ 0 ( \\iota ^ { - 1 } ( x ) ) } I _ 1 ^ 2 ( \\phi ( x ) ) = \\pm I _ 1 ^ 2 ( \\phi ( x ) ) . \\end{align*}"}
+{"id": "869.png", "formula": "\\begin{align*} L _ { 1 1 } = 2 { w _ 1 } _ { x _ 1 } , L _ { 2 2 } = 2 { w _ 2 } _ { \\substack { \\\\ x _ 2 } } , L _ { 1 2 } = L _ { 2 1 } = { w _ 1 } _ { \\substack { \\\\ x _ 2 } } + { w _ 2 } _ { \\substack { \\\\ x _ 1 } } \\end{align*}"}
+{"id": "1641.png", "formula": "\\begin{align*} u _ { 2 } \\left ( x , T \\right ) = u _ { T } ^ { \\left ( 2 \\right ) } \\left ( x \\right ) , m _ { 2 } \\left ( x , T \\right ) = m _ { T } ^ { \\left ( 2 \\right ) } \\left ( x \\right ) , x \\in \\Omega . \\end{align*}"}
+{"id": "9078.png", "formula": "\\begin{align*} \\begin{array} { l l l } b ^ { ( r ) } _ i ( X ) & = \\Big ( J ^ { ( r ) } \\Big [ ( \\hat { C } C \\hat { C } ^ { - 1 } - I ) \\underline { v } + \\hat { C } \\Big ( D p - U ( J ^ { ( r ) } X ) - \\mathbf { 1 } \\frac { \\beta } { 2 } \\Big ) \\Big ] \\Big ) _ i \\\\ & = j ^ { ( r ) } _ { i i } \\Big ( ( \\hat { C } C \\hat { C } ^ { - 1 } - I ) \\underline { v } + \\hat { C } \\Big ( D p - U ( J ^ { ( r ) } X ) - \\mathbf { 1 } \\frac { \\beta } { 2 } \\Big ) \\Big ) _ i \\geq 0 . \\end{array} \\end{align*}"}
+{"id": "6965.png", "formula": "\\begin{align*} \\mathbf { m } ( \\mathbf { z } ) = \\mathbf { z } \\overline { \\eta } ( \\mathbf { z } ) + \\mathbf { \\overline { z } } \\eta ( \\mathbf { z } ) \\end{align*}"}
+{"id": "6443.png", "formula": "\\begin{align*} | k ^ \\Theta _ x ( t ) | = | \\langle k ^ \\Theta _ x , k ^ \\Theta _ t \\rangle | \\leq \\frac { \\sqrt { \\varphi ' ( x ) \\varphi ' ( t ) } } { 2 \\pi } , x , t \\in \\R . \\end{align*}"}
+{"id": "5464.png", "formula": "\\begin{align*} \\mathcal { S } ( t ) = \\bigcap _ { i = 1 } ^ { c } \\mathcal { S } _ i ( t ) . \\end{align*}"}
+{"id": "142.png", "formula": "\\begin{align*} 2 \\cdot { \\bigg ( \\frac { \\eta - 1 } { 2 \\eta } \\bigg ) } ^ { \\alpha ( \\eta ) } + 2 \\cdot { \\bigg ( \\frac { 1 } { \\eta } \\bigg ) } ^ { \\alpha ( \\eta ) } = 1 . \\end{align*}"}
+{"id": "3570.png", "formula": "\\begin{align*} \\alpha ^ 3 \\tau _ 0 \\tau _ 1 \\tau _ 2 f \\circ i d - ( 1 + a ) \\alpha ^ 2 \\tau _ 0 \\tau _ 1 f \\circ \\tau ^ 2 + ( a + b ) \\alpha \\tau _ 0 f \\circ \\tau - b f = 0 . \\end{align*}"}
+{"id": "6386.png", "formula": "\\begin{align*} \\xi ^ { - 2 k ( \\ell + r ) } \\varphi ( \\ell + 2 r , r ) = \\xi ^ { - 2 i ( \\ell + r ) } \\varphi ( - r + 2 \\ell , \\ell ) \\qquad x , y \\in X . \\end{align*}"}
+{"id": "4589.png", "formula": "\\begin{align*} h ^ 2 & = \\frac { \\int _ M ( t - t _ 0 ) ^ 2 d \\mu } { \\int _ M ( s _ g - s _ 0 ) ^ 2 d \\mu } \\\\ & = \\frac { - \\int _ M t ^ 2 d \\mu + 2 \\left ( \\int _ M t d \\mu \\right ) ^ 2 / [ \\omega ] ^ 2 } { \\mathcal { F } ( \\mathfrak { E } , [ \\omega ] ) } . \\end{align*}"}
+{"id": "5012.png", "formula": "\\begin{align*} f _ i ^ \\tau ( \\mathbf { R } , \\mathbf { Q } ) & = \\sum _ { j \\in N } R _ j x _ { i j } ^ { 1 , \\tau } + \\sum _ { j \\in N } ( 1 - \\beta ) Q _ j x _ { i j } ^ { 0 , \\tau } = \\mathbf { x } _ i ^ { 1 , \\tau } \\mathbf { R } + ( 1 - \\beta ) \\mathbf { x } _ i ^ { 0 , \\tau } \\mathbf { Q } , \\\\ g _ i ^ \\tau & = \\sum _ { j \\in N } x _ { i j } ^ { 1 , \\tau } = \\mathbf { x } _ i ^ { 1 , \\tau } \\mathbf { 1 } . \\end{align*}"}
+{"id": "4179.png", "formula": "\\begin{align*} \\varphi ( g _ 3 g ^ { n } _ 1 g _ 2 ) = \\varphi \\big ( ( g ^ { j } _ 1 g _ 3 ) ( g ^ { i } _ 1 g _ 2 ) \\big ) \\mbox { o r } \\varphi ( g _ 3 g ^ { n } _ 1 g _ 2 ) = \\varphi \\big ( ( g ^ { i } _ 1 g _ 2 ) ( g ^ { j } _ 1 g _ 3 ) \\big ) \\ , . \\end{align*}"}
+{"id": "9054.png", "formula": "\\begin{align*} & X _ { \\Lambda _ 1 , \\Lambda _ 2 } \\bigl ( a \\otimes b \\otimes Z _ { 1 , 2 } ^ { m | I } \\bigr ) \\in V _ { s - 1 , \\nabla } [ \\Lambda _ k ] _ { k = 1 , 2 } \\subset V _ { s , \\nabla } [ \\Lambda _ k ] _ { k = 1 , 2 } , \\\\ & X _ { \\Lambda _ 1 , \\Lambda _ 2 } \\bigl ( a \\otimes b \\otimes Z _ { 1 , 2 } ^ { - m - 1 | I } \\bigr ) \\in V _ { s , \\nabla } [ \\Lambda _ k ] _ { k = 1 , 2 } \\end{align*}"}
+{"id": "1984.png", "formula": "\\begin{align*} \\psi _ M ( x , t ) = \\sum _ { l \\in \\mathcal { T } _ M } \\widehat { \\psi } _ l ( t ) e ^ { i \\mu _ l ( x - a ) } , x \\in \\Omega , \\ t \\ge 0 , \\end{align*}"}
+{"id": "4376.png", "formula": "\\begin{gather*} G ^ k ( x ) : = \\sum _ { j \\in [ m ] } ( \\overline { u } _ j + \\max \\{ 0 , \\Delta u _ j - \\Delta u _ k \\} ) l _ j ( x ) . \\end{gather*}"}
+{"id": "2590.png", "formula": "\\begin{align*} \\beta u ( R ) = - \\dfrac { d u } { d \\nu } ( R ) = - \\dfrac { d u } { d x } ( R ) \\geq - \\dfrac { d u } { d x } ( \\lambda ) = - \\dfrac { d u } { d \\nu } ( - R ) = \\beta u ( - R ) , \\end{align*}"}
+{"id": "2361.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } v _ { k } - \\Delta v _ { k } = \\mathrm { d i v } ( - v _ { k } \\otimes v _ { k } + v _ { k } \\otimes w + w \\otimes v _ { k } ) \\\\ + \\mathrm { d i v } ( u ^ { c , \\gamma } \\otimes v _ { k } + v _ { k } \\otimes u ^ { c , \\gamma } ) + \\nabla \\pi _ { k } , \\\\ \\mathrm { d i v } v _ { k } = 0 , \\\\ v _ { k } ( x , 0 ) = w _ { 0 } ( x ) - w _ { 0 , k } ( x ) , \\end{cases} \\end{align*}"}
+{"id": "5062.png", "formula": "\\begin{align*} & C ( x _ i ^ { a - c , ( 1 ) } ( k ) , \\varrho ( k ) ) = C ( x _ i ^ { a - c , ( 2 ) } ( k ) , \\varrho ( k ) ) , \\\\ & C ( y _ i ^ { a - c , ( 1 ) } ( k ) , \\varrho ( k ) ) = C ( y _ i ^ { a - c , ( 2 ) } ( k ) , \\varrho ( k ) ) , \\end{align*}"}
+{"id": "5768.png", "formula": "\\begin{align*} & | X _ + ( t ) | ^ 2 + | X _ 0 ( t ) | ^ 2 + | X _ - ( t ) | ^ 2 \\\\ & = \\sum _ { k + \\ell \\leq s } \\left [ | z ^ { ( k , \\ell ) } ( t ) | ^ 2 + | \\bar { z } ^ { ( k , \\ell ) } ( t ) | ^ 2 + \\sum _ { i \\in I _ 1 } \\left ( | \\xi _ { i , 1 } ( t ) | ^ 2 + | \\xi _ { i , 2 } ( t ) | ^ 2 \\right ) + \\right . \\left . \\sum _ { i \\neq 0 } | \\xi _ i ( t ) | ^ 2 \\right ] \\\\ & = \\sum _ { k + \\ell \\leq s } \\| q ^ { ( k , \\ell ) } ( t ) \\| ^ 2 _ G . \\end{align*}"}
+{"id": "857.png", "formula": "\\begin{align*} G ^ i = \\frac { 1 } { 4 } g ^ { i l } \\left ( \\left [ F ^ 2 \\right ] _ { x ^ k y ^ l } y ^ k - \\left [ F ^ 2 \\right ] _ { x ^ l } \\right ) = \\frac { 1 } { 4 } g ^ { i l } \\left ( 2 \\left ( g _ { j l } \\right ) _ { x ^ k } - \\left ( g _ { j k } \\right ) _ { x ^ l } \\right ) y ^ j y ^ k . \\end{align*}"}
+{"id": "362.png", "formula": "\\begin{align*} \\begin{aligned} f _ n ( x , b , a ) & = - \\sum _ { i = 1 } ^ { n - 1 } \\binom { n } { i } b ^ i \\bigg ( ( x - a ) ^ { n - i } - ( x ^ { n - i } + ( - a ) ^ { n - i } ) \\bigg ) , \\\\ g _ n ( b , a ) & = - ( b - a ) ^ n + b ^ n + ( - a ) ^ n = - \\sum _ { i = 1 } ^ { n - 1 } \\binom { n } { i } b ^ { n - i } a ^ i , \\end{aligned} \\end{align*}"}
+{"id": "3137.png", "formula": "\\begin{align*} u _ { \\psi ^ * } ( t ) = \\left ( \\begin{array} { c c c } 1 & t ^ { p ^ { e _ 1 } } & t ^ { 2 \\ , p ^ { e _ 1 } } \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "8472.png", "formula": "\\begin{align*} \\lim \\limits _ { h , h ' \\searrow 0 } \\left \\| ( u _ h ) _ - - ( u _ { h ' } ) _ - \\right \\| _ { L ^ \\gamma ( K _ T ) } = 0 . \\end{align*}"}
+{"id": "5687.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } { u ( t ) } / { \\Vert u ( t ) \\Vert _ { L ^ 2 } } = w \\ \\textup { i n } \\ C ^ \\infty ( \\Sigma ; \\mathbf { V } ) , \\end{align*}"}
+{"id": "5049.png", "formula": "\\begin{align*} A : = \\mathbb { E } _ \\mathbb { P } [ \\exp ( \\| \\xi \\| ^ a ) ] = \\int _ { \\Xi } \\exp ( \\| \\xi \\| ^ a ) \\mathbb { P } ( d \\xi ) < \\infty . \\end{align*}"}
+{"id": "2687.png", "formula": "\\begin{align*} \\delta q ^ { i } ( t _ { 1 } ) = \\delta q ^ { i } ( t _ { 2 } ) = 0 . \\end{align*}"}
+{"id": "4820.png", "formula": "\\begin{align*} R _ v : = \\sum _ { k = 1 } ^ K \\min \\big \\{ S ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } } + \\sum _ { a \\in \\mathcal { A } } \\max \\{ 0 , x _ a - x ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } } _ a \\} ; h \\big \\} = n _ v \\overline { S } _ v + ( K - n _ v ) h ; \\end{align*}"}
+{"id": "982.png", "formula": "\\begin{align*} f _ h ( x , y ) : = f ( x , h ( x ) + y ) , x \\in E , \\ , y \\in \\mathbb R . \\end{align*}"}
+{"id": "6535.png", "formula": "\\begin{align*} B ^ { - 1 } _ n \\Big ( \\sum \\limits ^ n _ { i = 1 } \\varepsilon _ i - A _ n \\Big ) \\xrightarrow { \\mathcal { L } } Z \\end{align*}"}
+{"id": "5906.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { N C } ( G ) ) & = \\dfrac { 1 } { 2 } \\Big { ( } p ^ { 4 } q ^ { 4 } - 7 p ^ { 3 } q ^ { 3 } + 4 1 p ^ { 2 } q ^ { 2 } - 5 1 p q + 3 p ^ { 4 } q ^ { 2 } + 1 3 q ^ { 2 } - 1 6 p ^ { 3 } q ^ { 2 } + 1 4 p q ^ { 2 } + 2 p ^ { 2 } q ^ { 3 } \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - 1 6 p q ^ { 3 } + 8 p ^ { 2 } q - 9 q + 2 p q ^ { 4 } + 2 p ^ { 3 } q + p ^ { 4 } q + 1 8 \\Big { ) } \\end{align*}"}
+{"id": "7844.png", "formula": "\\begin{align*} \\nabla ^ { \\perp \\psi } _ { \\nabla ^ { \\Sigma } _ { E _ a } E _ a } H ^ { \\psi } = \\frac { p } { p + q } \\nabla ^ { \\perp } _ { \\nabla _ { E _ a } E _ a } H _ 1 + \\frac { q } { p + q } B ^ j ( \\nabla _ { E _ a } E _ a , H _ 2 ) \\end{align*}"}
+{"id": "9192.png", "formula": "\\begin{align*} ( 1 - q ^ 2 ) \\ell ^ - _ { 1 , j + 1 } ( z ) = [ \\ell ^ - _ { 1 j } ( z ) , e ^ { ( 0 ) } _ { j , j + 1 } ] _ q . \\end{align*}"}
+{"id": "5639.png", "formula": "\\begin{align*} a Q = 3 2 ( & 1 1 , \\tfrac { 1 } { 2 } ( - x _ 1 + x _ 3 + x _ 4 + 2 x _ 5 + x _ 6 + x _ 7 - 2 3 4 ) , x _ 1 + x _ 3 + x _ 4 + x _ 6 + x _ 7 + x _ 8 - 2 3 4 , \\\\ & 1 1 7 - x _ 3 - x _ 5 - x _ 8 , \\tfrac { 1 } { 2 } ( x _ 1 + x _ 3 + 3 x _ 4 - x _ 6 - x _ 7 ) , 2 x _ 1 - x _ 3 + x _ 5 , \\\\ & x _ 1 + x _ 3 - x _ 4 + x _ 6 - x _ 7 , x _ 1 + x _ 3 - x _ 4 - x _ 6 + x _ 7 , - 2 ( x _ 1 + x _ 3 ) + x _ 8 , \\\\ & 3 5 1 - 3 x _ 1 - x _ 4 - x _ 5 - x _ 6 - x _ 7 - x _ 8 ) . \\end{align*}"}
+{"id": "7416.png", "formula": "\\begin{align*} P ( x ) : = u ( x , x + R ) \\quad \\mbox { a n d } C _ 7 : = \\frac { A } { 2 } \\left ( \\frac { p } { p + q } \\right ) ^ p ( 2 R ) ^ { 1 - p } > 0 . \\end{align*}"}
+{"id": "4231.png", "formula": "\\begin{align*} E _ 4 ( \\tau ) = 1 + 2 4 0 q + 2 1 6 0 q ^ 2 + 6 7 2 0 q ^ 3 + \\cdots , \\end{align*}"}
+{"id": "7988.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\Omega } d N _ { \\phi } ( e _ { \\phi } ^ { i } ) \\wedge \\ast [ \\delta N _ { \\beta } ( \\omega ) , d N _ { \\phi } ( e _ { \\phi } ^ { j } ) ] _ { 1 } = ( - 1 ) ^ { n - 1 } \\int _ { \\partial \\Omega } \\langle d N _ { \\phi } ( e _ { \\phi } ^ { i } ) , \\delta N _ { \\beta } ( \\omega ) \\rangle _ { \\Lambda ^ { 1 } } \\wedge e _ { \\phi } ^ { j } , \\end{aligned} \\end{align*}"}
+{"id": "8122.png", "formula": "\\begin{align*} V _ d = w _ 0 ( I ) \\prod _ { \\alpha \\in { \\mathcal I } ( d ) } U _ { \\alpha } w _ 0 ( I ) = \\prod _ { \\alpha \\in w _ 0 ( I ) ( { \\mathcal I } ( d ) ) } U _ { \\alpha } \\leq U _ { w _ 0 ( I ) d } , \\end{align*}"}
+{"id": "7033.png", "formula": "\\begin{align*} \\| e ^ { - t \\Lambda _ p ( b ) } f \\| _ { q } \\leq c e ^ { t \\omega _ p } t ^ { - \\frac { d } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\| f \\| _ p , t > 0 , \\omega _ p : = \\frac { c _ \\delta } { 2 ( p - 1 ) } , \\end{align*}"}
+{"id": "938.png", "formula": "\\begin{align*} F = F ( V ) \\oplus F ( V ) ^ { \\bot } , \\pi _ V : F \\rightarrow F ( V ) . \\end{align*}"}
+{"id": "5564.png", "formula": "\\begin{align*} ( S _ { \\Delta } S _ { \\Delta } ^ * ) _ { x x } & = \\sum _ { e \\in \\vec { E } : x = e _ 1 } ( A _ { e _ 1 e _ 2 } A _ { e _ 3 e _ 2 } ) ^ 2 \\leq \\left ( \\frac { L } { d } \\right ) ^ 4 \\sum _ { e _ 2 \\in [ m ] , e _ 3 \\not = x } X _ { x e _ 2 } X _ { e _ 3 e _ 2 } . \\end{align*}"}
+{"id": "1388.png", "formula": "\\begin{align*} d ( a \\cdot b ) = d ( a ) \\cdot b + ( - 1 ) ^ { i } a \\cdot d ( b ) \\end{align*}"}
+{"id": "4890.png", "formula": "\\begin{align*} \\mathcal A = \\O _ { \\Sigma ' } ( E ) \\otimes \\varphi '^ * { ( \\L ^ \\vee ) } . \\end{align*}"}
+{"id": "6852.png", "formula": "\\begin{align*} d g & = d g _ { K } d k \\\\ & = \\det ( ( e _ { i j } ) ) \\ ; d \\phi _ 1 \\ldots d \\phi _ { N - 1 } d \\psi _ 1 \\ldots d \\psi _ { N - 1 } d g _ { S U ( N - 1 ) } d \\omega _ N \\\\ & = 2 \\cos ( \\psi _ { N - 1 } ) \\sin ^ { 2 ( N - 1 ) - 1 } ( \\psi _ { N - 1 } ) \\left [ \\prod _ { j = 1 } ^ { N - 2 } \\cos ^ { 2 j - 1 } ( \\psi _ { j } ) \\sin ( \\psi _ j ) \\right ] d \\phi _ 1 \\ldots d \\phi _ { N - 1 } d \\psi _ 1 \\ldots d \\psi _ { N - 1 } d g _ { S U ( N - 1 ) } d \\omega _ { N - 1 } . \\end{align*}"}
+{"id": "881.png", "formula": "\\begin{align*} 2 F S _ 0 - L _ { 0 0 } - F ^ 2 L _ { w w } = & 2 F [ 2 w ^ 1 { w ^ 1 } _ { x _ 1 } + w ^ 2 ( { w ^ 1 } _ { x _ 2 } + { w ^ 2 } _ { x _ 1 } ) ] y ^ 1 \\\\ & + 2 F [ w ^ 1 ( { w ^ 1 } _ { x _ 2 } + { w ^ 2 } _ { x _ 1 } ) + 2 w ^ 2 { w ^ 2 } _ { x _ 2 } ] y ^ 2 \\\\ & - 2 { w ^ 1 } _ { x _ 1 } ( y ^ 1 ) ^ 2 - 2 ( { w ^ 1 } _ { x _ 2 } + { w ^ 2 } _ { x _ 1 } ) y ^ 1 y ^ 2 - 2 { w ^ 2 } _ { x _ 2 } ( y ^ 2 ) ^ 2 \\\\ & - F ^ 2 ( 2 ( w ^ 1 ) ^ 2 { w ^ 1 } _ { x _ 1 } + 2 w ^ 1 w ^ 2 ( { w ^ 1 } _ { x _ 2 } + { w ^ 2 } _ { x _ 1 } ) \\\\ & + 2 ( w ^ 2 ) ^ 2 { w ^ 2 } _ { x _ 2 } ) . \\end{align*}"}
+{"id": "8112.png", "formula": "\\begin{align*} \\norm { \\mu _ k ^ { \\ast a } \\ast \\mu _ k ^ { \\ast b } } _ 2 \\leq \\norm { \\mu _ k ^ { \\ast a } } _ 1 \\norm { \\mu _ k ^ { \\ast b } } _ 2 = \\norm { \\mu _ k ^ { \\ast b } } _ 2 . \\end{align*}"}
+{"id": "3647.png", "formula": "\\begin{align*} p _ c ( u ) = \\left ( c x _ { i } , u \\right ) \\end{align*}"}
+{"id": "4082.png", "formula": "\\begin{align*} R _ 0 ( z _ m ) ( x , y ) - R _ 0 ( z _ 0 ) ( x , y ) & = 2 \\pi i m \\widehat { \\sigma _ { z _ 0 \\mathbb { S } ^ { d - 1 } } } ( r ) \\end{align*}"}
+{"id": "4688.png", "formula": "\\begin{align*} \\mathrm { i n t } \\left ( \\mathrm { S C } ( D ) \\right ) = \\mathrm { i n t } \\left \\{ \\alpha \\in \\mathrm { B i g } ( X ) \\left ( ( - 1 ) ^ { \\delta _ i } q _ X ( P ( \\alpha ) , D _ i ) \\leq 0 \\right ) _ { i \\in I } \\right \\} , \\end{align*}"}
+{"id": "4873.png", "formula": "\\begin{align*} \\max ( 4 g - 7 , 3 g + 6 ) = \\begin{cases} 4 g - 7 & \\\\ 3 g + 6 & . \\end{cases} \\end{align*}"}
+{"id": "3738.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial a } \\mathrm { B } _ { z } \\left ( a , b \\right ) = \\int _ { 0 } ^ { z } t ^ { a - 1 } \\left ( 1 - t \\right ) ^ { b - 1 } \\ln t \\ , d t . \\end{align*}"}
+{"id": "4459.png", "formula": "\\begin{align*} \\int _ { M _ 1 } f _ 2 \\overline { g } \\lambda _ 2 = 0 \\end{align*}"}
+{"id": "6164.png", "formula": "\\begin{align*} v ^ { k + 1 } = v ^ k - M ^ k ( v ^ k - \\widetilde { v } ^ k ) . \\end{align*}"}
+{"id": "681.png", "formula": "\\begin{align*} ( \\mathcal { C } ^ { 2 s - 1 } f ) ( u , \\beta , \\psi ) = 0 \\end{align*}"}
+{"id": "4548.png", "formula": "\\begin{align*} \\begin{cases} L _ N ( \\beta ) : = \\max _ { 1 \\leq i \\leq N } \\sup _ { [ s ] _ i , [ s ' ] _ i } ( 1 + \\beta ) ^ i \\frac { | \\varphi _ i ( [ s ] _ i ) - \\varphi _ i ( [ s ' ] _ i ) | } { | s _ i - s _ i ' | } , \\\\ [ 5 p t ] F _ N ( \\beta ) : = \\max _ { 1 \\leq i \\leq N } \\sup _ { [ s ] _ i , [ s ' ] _ i } ( 1 + \\beta ) ^ i \\Big | \\varphi _ i ( [ s ] _ i ) - \\varphi _ i ( [ s ' ] _ i ) \\Big | . \\end{cases} \\end{align*}"}
+{"id": "570.png", "formula": "\\begin{align*} \\phi _ n - \\frac { \\omega ^ 2 } { g } \\phi & = - j \\frac { \\omega } { \\rho g } u & \\Gamma _ c \\end{align*}"}
+{"id": "8339.png", "formula": "\\begin{align*} ( \\alpha _ 1 ^ \\flat \\wedge \\ldots \\wedge \\alpha _ n ^ \\flat ) ( \\xi _ 1 \\wedge \\ldots \\wedge \\xi _ n ) = \\sum _ { \\sigma \\in S _ n } ( - ) ^ \\sigma \\alpha _ 1 ^ \\flat ( \\xi _ { \\sigma ( 1 ) } ) \\wedge \\ldots \\wedge \\alpha _ n ^ \\flat ( \\xi _ { \\sigma ( n ) } ) . \\end{align*}"}
+{"id": "5746.png", "formula": "\\begin{align*} \\frac { d } { d t } \\left ( e ^ { - \\gamma _ * t } \\xi _ { i , 4 } ( t ) \\right ) = & e ^ { - \\gamma _ * t } \\xi _ { 3 , i } ( t ) + e ^ { - \\gamma _ * t } \\mathcal { E } _ { i , 4 } ( t ) = c _ { i , 3 } + O ( e ^ { - \\varepsilon _ 0 t } ) , \\end{align*}"}
+{"id": "1002.png", "formula": "\\begin{align*} \\dot { F } _ { \\chi , } ( W ) = \\bigcup _ { \\mathcal S \\in \\Xi _ W } F _ { \\chi } ( W ; \\mathcal S ) , \\dot { F } _ { } ( W ) = \\bigcup _ { \\mathcal S \\in \\Xi _ W } F ( W ; \\mathcal S ) . \\end{align*}"}
+{"id": "1379.png", "formula": "\\begin{align*} | \\overline { B } | = | B | = | H | | B _ 1 | = p | B _ 1 | = p | A _ 1 | . \\end{align*}"}
+{"id": "4989.png", "formula": "\\begin{align*} Z _ N ( \\{ \\lambda \\} ) = \\frac { \\det \\left [ p _ k ( \\lambda _ j ) a ( \\lambda _ j ) - p _ k ( \\lambda _ j + 1 ) d ( \\lambda _ j ) \\right ] _ { j , k = 1 , \\ldots , N } } { \\det \\left [ p _ k ( \\lambda _ j ) \\right ] _ { j , k = 1 , \\ldots , N } } \\end{align*}"}
+{"id": "329.png", "formula": "\\begin{align*} \\beta _ { i , j } ( I ) = \\beta _ { i , j } ( I _ 1 ) + \\beta _ { i , j } ( I _ 2 ) + \\beta _ { i - 1 , j } ( I _ 1 \\cap I _ 2 ) , \\ \\ \\ \\ i , j \\ge 0 . \\end{align*}"}
+{"id": "1649.png", "formula": "\\begin{align*} \\min _ { \\overline { Q } _ { \\varepsilon , T } } \\varphi _ { \\lambda , k } ^ { 2 } \\left ( t \\right ) = e ^ { 2 \\lambda \\left ( \\varepsilon + 1 \\right ) ^ { k } } , \\end{align*}"}
+{"id": "976.png", "formula": "\\begin{align*} Y ^ x _ t = u ( X _ t ) , t \\le \\tau _ D , P _ x \\end{align*}"}
+{"id": "6180.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau ^ k } [ f _ 1 ( x _ 1 ) - f _ 1 ( \\breve { x } _ 1 ^ { k } ) ] - \\frac { 1 } { \\tau ^ { k - 1 } } [ f _ 1 ( x _ 1 ) - f _ 1 ( \\breve { x } _ 1 ^ { k - 1 } ) ] \\\\ + & ( x _ 1 - \\widetilde { x } _ 1 ^ { k } ) ^ T \\{ - A _ 1 ^ T \\widetilde { \\lambda } ^ k + ( 1 - \\gamma ) ( 1 - \\tau ^ k ) \\beta ^ k A _ 1 ^ T ( A \\breve { x } ^ { k - 1 } - b ) \\} \\geq 0 , ~ \\forall x _ 1 . \\end{aligned} \\end{align*}"}
+{"id": "8666.png", "formula": "\\begin{align*} M _ G ' : = \\mu ^ { - 1 } ( 0 ) / G , M _ G : = ( \\mu ^ { - 1 } ( 0 ) \\cap M ) / G . \\end{align*}"}
+{"id": "4344.png", "formula": "\\begin{align*} F _ { \\mathcal { Y } , a , b } ( x ) = \\sum _ { ( i , j ) \\in [ m ] \\times [ n ] } \\overline { y } _ { ( i , j ) } g _ { ( i , j ) } ( x ) + \\sum _ { ( q , p ) \\in \\mathcal { Y } } \\Delta y _ { ( q , p ) } g _ { ( q , p ) } ( x ) = F _ { \\mathcal { Y } , 0 , 0 } ( x ) . \\end{align*}"}
+{"id": "2588.png", "formula": "\\begin{align*} { - \\beta u ( \\tilde { x } ) = \\dfrac { d u } { d \\nu } ( \\tilde { x } ) \\leq 0 , } \\end{align*}"}
+{"id": "5472.png", "formula": "\\begin{align*} z _ 0 = 1 - \\frac { 1 } { N } z = 1 - \\frac { a } { N } . \\end{align*}"}
+{"id": "6210.png", "formula": "\\begin{align*} \\omega = \\sqrt { \\frac { d - 4 } { d - 1 } } . \\end{align*}"}
+{"id": "2867.png", "formula": "\\begin{align*} E _ + ( f , \\gamma ) = \\bigcup _ { i = 1 } ^ { K + 1 } \\left \\{ E _ + ( f , \\gamma ) \\cap \\left [ \\left ( a _ i , \\infty \\right ) \\times \\left ( - \\infty , a _ i \\right ) \\right ] \\right \\} = : \\bigcup _ { i = 1 } ^ { K + 1 } \\mathcal { E } _ i \\end{align*}"}
+{"id": "1394.png", "formula": "\\begin{align*} q = \\frac { 8 } { 5 - \\alpha } . \\end{align*}"}
+{"id": "514.png", "formula": "\\begin{align*} \\prod _ { t = 0 } ^ { H ' } { \\alpha _ { i _ { k - h ' + t } } } = \\left ( \\overline { \\alpha } _ i \\right ) ^ { \\lfloor ( H ' + 1 ) / \\ell \\rfloor } \\cdot \\prod _ { t = 0 } ^ { ( H ' + 1 ) \\bmod { \\ell } } { \\alpha _ { i _ { k - h ' + t } } } , \\end{align*}"}
+{"id": "1460.png", "formula": "\\begin{align*} \\overline { D } _ G ( S ) = \\lim _ { i \\in I } \\sup _ { g \\in G } \\frac { | S \\cap F _ i g | } { | F _ i | } = \\inf _ { F \\subset G , | F | < \\infty } \\sup _ { g \\in G } \\frac { | S \\cap F g | } { | F | } , \\end{align*}"}
+{"id": "3615.png", "formula": "\\begin{align*} ( 1 + a ) \\alpha ^ 2 \\tau _ 0 \\tau _ 1 ( ( \\sigma ^ 2 \\sigma _ 1 ) ^ 2 - ( \\sigma \\sigma _ 1 ) ^ 2 ) - ( a + b ) \\alpha \\tau _ 0 ( ( \\sigma ^ 2 \\sigma _ 1 ) ^ 2 - \\sigma ^ 2 ) + b ( ( \\sigma ^ 2 \\sigma _ 1 ) ^ 2 - 1 ) = 0 . \\end{align*}"}
+{"id": "8372.png", "formula": "\\begin{align*} X _ { \\underline { w } } = \\mathbf { m } ^ { - 1 } ( C _ { w B } ) \\cup \\bigcup _ { j = 1 } ^ { l } X _ { \\underline { w } ^ j } , \\end{align*}"}
+{"id": "4345.png", "formula": "\\begin{align*} \\mathcal { X } _ { \\overline { \\mathcal { Y } } } : = \\{ x \\in \\mathcal { X } \\colon & \\Delta y _ { q , p ( q ) } g _ { q , p ( q ) } ( x ) \\geq \\Delta y _ { a , b ( a ) } g _ { a , b ( a ) } ( x ) \\ \\forall q \\in \\overline { \\mathcal { Y } } , \\\\ & \\Delta y _ { q , p ( q ) } g _ { q , p ( q ) } ( x ) \\leq \\Delta y _ { a , b ( a ) } g _ { a , b ( a ) } ( x ) \\ \\forall q \\in [ m ] \\setminus \\overline { \\mathcal { Y } } \\} , \\end{align*}"}
+{"id": "2150.png", "formula": "\\begin{align*} F _ { \\min } = 0 . 9 3 7 5 , x ^ * = ( 0 . 7 5 0 0 , 0 . 7 5 0 0 ) , y ^ * = ( 0 . 0 0 0 0 , 0 . 0 0 0 0 , 1 . 0 0 0 0 ) . \\end{align*}"}
+{"id": "7807.png", "formula": "\\begin{align*} R _ { k } \\ ! \\Big ( \\ ! \\boldsymbol { a } , \\boldsymbol { v } _ { k } , \\boldsymbol { \\Theta } , \\boldsymbol { \\rho } \\ ! \\Big ) \\ ! \\ ! = \\ ! \\ ! B \\log _ { 2 } \\ ! \\Big ( \\ ! 1 + \\gamma _ { k } \\Big ( \\boldsymbol { a } , \\boldsymbol { v } _ { k } , \\boldsymbol { \\Theta } , \\boldsymbol { \\rho } \\Big ) \\ ! \\Big ) , \\ \\ \\forall k \\ ! \\in \\ ! \\mathcal { K } , \\end{align*}"}
+{"id": "5927.png", "formula": "\\begin{align*} M _ { 1 } ( \\mathcal { N C } ( A ( n , p ) ) ) & = ( p ^ { 3 n } - p ^ { n } ) ( p ^ { 3 n } - p ^ { n } - 1 ) ^ { 2 } - 4 ( p ^ { 3 n } - p ^ { n } - 1 ) \\dfrac { p ^ { n } ( p ^ { 2 n } - 1 ) ( p ^ { 2 n } - p ^ { n } - 1 ) } { 2 } \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ + p ^ { n } ( p ^ { 2 n } - 1 ) ( p ^ { 2 n } - p ^ { n } - 1 ) ^ { 2 } \\\\ & = p ^ { 9 n } - 2 p ^ { 8 n } - p ^ { 5 n } + 2 p ^ { 6 n } \\\\ & = p ^ { 8 n } ( p ^ { n } - 2 ) + p ^ { 5 n } ( 2 p ^ { n } - 1 ) \\end{align*}"}
+{"id": "149.png", "formula": "\\begin{align*} G ( s ) = - e ^ { ( 2 n - \\alpha ( \\eta ) ) s } R ( e ^ { - s } ) + \\int _ { 0 } ^ s G ( s - s ' ) \\ d \\mu ( s ' ) . \\end{align*}"}
+{"id": "6979.png", "formula": "\\begin{align*} C _ o = \\{ [ i y _ 1 : \\cdots : i y _ n : 0 ] : ( y _ 1 , \\dots , y _ n ) \\in \\mathbb { R } ^ n \\setminus \\{ ( 0 , \\dots , 0 ) \\} \\} . \\end{align*}"}
+{"id": "435.png", "formula": "\\begin{align*} \\min _ { x \\in \\Re ^ n } \\ ; \\sum _ { i = 1 } ^ { n } f _ i ( x _ i ) + \\sum _ { i = 1 } ^ { n - 1 } \\lambda _ { i } ( x _ i - x _ { i + 1 } ) _ { + } + \\sum _ { i = 1 } ^ { n - 1 } \\mu _ { i } ( x _ { i + 1 } - x _ { i } ) _ { + } , \\end{align*}"}
+{"id": "344.png", "formula": "\\begin{align*} \\begin{aligned} ( u _ p I _ p ^ { [ k - 1 ] } : I ^ { [ \\ell ] } ) \\ & = \\ ( u _ p I _ p ^ { [ k - 1 ] } : I _ p ^ { [ \\ell ] } ) \\cap ( u _ p I _ p ^ { [ \\ell - 1 ] } : u _ p I _ p ^ { [ \\ell - 1 ] } ) \\cap ( u _ p I _ p ^ { [ k - 1 ] } : J ) \\\\ & = \\ u _ p I _ p ^ { [ k - 1 ] } , \\end{aligned} \\end{align*}"}
+{"id": "3120.png", "formula": "\\begin{align*} d _ { ( 3 ) ^ * } ( \\varphi ) & : = ( \\ , \\dim _ k \\ , ( V ) ^ \\varphi _ { \\ell _ 1 } , \\ ; \\dim _ k \\ , ( V ) ^ \\varphi _ { \\ell _ 2 } \\ , ) , \\\\ d ' _ { ( 3 ) ^ * } ( \\varphi ) & : = ( \\ , \\dim _ k \\ , ( V ' ) ^ \\varphi _ { \\ell _ 1 } , \\ ; \\dim _ k \\ , ( V ' ) ^ \\varphi _ { \\ell _ 2 } \\ , ) . \\end{align*}"}
+{"id": "4659.png", "formula": "\\begin{align*} | g _ { i j } - \\delta _ { i j } | + | x | | \\partial g _ { i j } | + | x | ^ 2 | \\partial ^ 2 g _ { i j } | = O \\left ( | x | ^ { 2 - n } \\right ) \\mbox { a s } | x | \\to + \\infty , \\end{align*}"}
+{"id": "6221.png", "formula": "\\begin{align*} z ( \\phi ) = f ( \\phi ) - c \\phi - \\int _ { 0 } ^ { \\phi } \\frac { D ( \\sigma ) g ( \\sigma ) } { z ( \\sigma ) } \\ , d \\sigma , \\phi \\in [ 0 , 1 ] . \\end{align*}"}
+{"id": "8671.png", "formula": "\\begin{align*} & \\mbox { $ \\langle \\omega _ 0 ( p ) , u \\ , \\rangle = 0 $ , f o r e v e r y $ p \\in X $ a n d e v e r y $ u \\in T _ p ^ { 1 , 0 } X \\oplus T _ p ^ { 0 , 1 } X $ } , \\\\ & \\mbox { $ \\langle \\omega _ 0 , T \\ , \\rangle = - 1 $ o n $ X $ . } \\end{align*}"}
+{"id": "2969.png", "formula": "\\begin{align*} x _ 1 ^ 0 < x _ 2 ^ 0 , v _ 1 = 1 , v _ 2 = - 1 , T _ 1 = T _ 2 = 1 . \\end{align*}"}
+{"id": "2339.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 3 } } \\pi _ { k - 1 } \\mathrm { d i v } a _ { k } = 0 ; \\end{align*}"}
+{"id": "1500.png", "formula": "\\begin{align*} \\mathcal { T } ( u ) : = \\left \\{ \\begin{array} { l l } \\underline { u } & u \\leq \\underline { u } , \\\\ u & \\underline { u } \\leq u \\leq \\overline { u } , \\\\ \\overline { u } & \\end{array} \\right . \\end{align*}"}
+{"id": "5695.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } { u ( t ) } / { \\Vert u ( t ) \\Vert _ { L ^ 2 } } = w \\ \\textup { i n } \\ C ^ \\infty ( \\Sigma ; \\mathbf { V } ) . \\end{align*}"}
+{"id": "2584.png", "formula": "\\begin{align*} & c _ { n - 1 } - q ^ { - 1 } c _ { n - 3 } \\\\ = & \\Big [ ( n - 2 ) + ( n - 4 ) q + ( n - 6 ) q ^ 2 + \\ldots + 5 q ^ { \\frac { n - 7 } { 2 } } + 3 q ^ { \\frac { n - 5 } { 2 } } + q ^ { \\frac { n - 3 } { 2 } } \\Big ] h ^ 2 \\\\ = & \\sum \\limits _ { k = 1 } ^ { n - 2 } ( \\sum \\limits _ { l = 1 } ^ { ( k , n - 1 - k ) } q ^ { - 1 + l } ) h ^ 2 . \\end{align*}"}
+{"id": "7308.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } | I _ { x + k m } ( t ) | \\leq e ^ t \\end{align*}"}
+{"id": "2827.png", "formula": "\\begin{align*} \\omega = d q ^ { i } \\wedge d p _ { i } = d \\Xi ^ { 1 } \\wedge d \\Psi _ { 1 } + d \\Xi ^ { 2 } \\wedge d \\Psi _ { 2 } + d \\Theta ^ { 1 } \\wedge d \\Theta _ { 1 } + d Q ^ { 1 } \\wedge d P _ { 1 } \\end{align*}"}
+{"id": "1587.png", "formula": "\\begin{align*} A ( k , \\rho ) = \\{ x \\in B _ \\rho : F ( D u ( x ) ) \\ge k \\} = \\{ x \\in B _ \\rho : G ( V ( x ) ) \\ge k \\} , \\end{align*}"}
+{"id": "975.png", "formula": "\\begin{align*} Y ^ x _ t = g ( X _ { \\tau _ D } ) + \\int _ t ^ { \\tau _ D } f ( X _ s , Y ^ x _ s ) \\ , d s + A ^ \\mu _ { \\tau _ D } - A ^ \\mu _ t - ( M ^ x _ { \\tau _ D } - M ^ x _ t ) , t \\le \\tau _ D , P _ x \\end{align*}"}
+{"id": "1962.png", "formula": "\\begin{align*} T _ { j } & = \\{ \\beta \\bmod p ^ { t + 1 } | \\beta \\in T , \\left \\lfloor \\frac { \\beta } { p ^ { t + 1 } } \\right \\rfloor = j \\} , \\\\ T _ { j , i } & = \\{ \\beta \\in T | \\left \\lfloor \\frac { \\beta } { p ^ { t + 1 } } \\right \\rfloor = j , \\beta \\equiv i \\bmod { p } \\} . \\end{align*}"}
+{"id": "4422.png", "formula": "\\begin{align*} \\inf _ { x \\in \\mathcal { X } } & \\ f ( x ) , \\\\ & \\sup _ { \\mathcal { S } \\subseteq [ m ] : | \\mathcal { S } | \\leq \\Gamma } \\left \\{ \\sum _ { i \\in \\mathcal { S } } \\sup _ { u ^ i \\in \\mathcal { U } _ i } g _ i ( x , u ^ i ) + \\sum _ { i \\in [ m ] \\setminus \\mathcal { S } } g _ i ( x , \\overline { u } ^ i ) \\right \\} \\leq 0 . \\end{align*}"}
+{"id": "3266.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { n - 1 } \\frac { ( q ^ { d + 1 } ; q ^ d ) _ k ^ { d - 2 } ( q ; q ^ d ) _ k ^ 2 q ^ { d k } } { ( q ^ d ; q ^ d ) _ k ^ d } \\\\ & \\equiv \\frac { ( 1 - q ) ^ 2 ( q ^ d ; q ^ d ) _ { n - 1 - ( n + 1 ) / d } } { ( q ^ d ; q ^ d ) _ { ( n + 1 ) / d } ^ { d - 1 } } q ^ { ( d ( d + n ) ( n + 1 ) - ( n + 1 ) ^ 2 ) / ( 2 d ) - 2 } \\pmod { \\Phi _ n ( q ) ^ 2 } . \\end{align*}"}
+{"id": "23.png", "formula": "\\begin{align*} \\lim _ { \\kappa \\to \\infty } \\ , \\sup _ { q \\in Q _ * } \\ , \\sup _ { | t | \\leq T } \\norm { \\tfrac { d n } { d t } } _ { H ^ { - 2 } } = 0 , \\end{align*}"}
+{"id": "61.png", "formula": "\\begin{align*} u _ t - | D u | ^ { \\gamma } \\big ( \\Delta u + ( p - 2 ) \\Delta _ \\infty ^ N u \\big ) = 0 \\end{align*}"}
+{"id": "37.png", "formula": "\\begin{align*} \\omega ( \\mu ) = \\sum _ { I \\in P ^ { j , n } _ + } \\omega _ { I } ( \\mu ) d x ^ { I } , \\end{align*}"}
+{"id": "7943.png", "formula": "\\begin{align*} \\frac { \\delta _ { \\phi } \\mathcal { F } } { \\delta v } = - \\ast d w , \\end{align*}"}
+{"id": "933.png", "formula": "\\begin{align*} u = \\eta _ U \\quad \\mbox { o n } U \\quad \\mbox { f o r s o m e } \\eta _ U \\in F . \\end{align*}"}
+{"id": "7261.png", "formula": "\\begin{align*} y p _ n ( y ) = p _ { n + 1 } ( y ) + \\tfrac { 1 } { 2 } B _ n p _ n ( y ) + \\tfrac { 1 } { 4 } A _ { n - 1 } C _ n p _ { n - 1 } ( y ) , n \\geq 0 , \\end{align*}"}
+{"id": "5964.png", "formula": "\\begin{align*} | f | = \\sum _ { u : ( u , v _ s ) \\in E } f _ { u , v _ s } . \\end{align*}"}
+{"id": "2607.png", "formula": "\\begin{align*} S = \\{ f _ { i + 3 } : i \\in [ n ] \\} , \\end{align*}"}
+{"id": "1997.png", "formula": "\\begin{align*} \\widetilde { ( \\xi ^ 0 ) } _ l & = e ^ { i \\beta _ l ^ + \\tau } \\widetilde { ( \\xi ^ { 0 , + } ) } _ l - e ^ { i \\beta _ l ^ - \\tau } \\widetilde { ( \\xi ^ { 0 , - } ) } _ l , \\\\ \\widetilde { ( \\xi ^ n ) } _ l & = e ^ { i \\beta _ l ^ + \\tau } \\widetilde { ( \\xi ^ { n , + } ) } _ l - e ^ { i \\beta _ l ^ - \\tau } \\widetilde { ( \\xi ^ { n , - } ) } _ l - e ^ { \\frac { i \\tau } { \\alpha } } ( \\widetilde { ( \\xi ^ { n - 1 , + } ) } _ l - \\widetilde { ( \\xi ^ { n - 1 , - } ) } _ l ) , n \\ge 1 , \\end{align*}"}
+{"id": "5365.png", "formula": "\\begin{align*} \\nu _ j = \\max \\ , \\left \\{ \\frac { v ^ { S \\setminus \\{ j \\} } _ j - v ^ { S } _ j } { b ^ S _ j - b ^ { S \\setminus \\{ j \\} } _ j } : j \\in S \\in \\{ S _ 1 , \\ldots , S _ n \\} \\right \\} , j \\in N ^ { \\{ 0 , 1 \\} } . \\end{align*}"}
+{"id": "6990.png", "formula": "\\begin{align*} C _ x = \\{ [ i y _ 1 W _ 1 + \\cdots + i y _ n W _ n ] : ( y _ 1 , . . . , y _ n ) \\in \\mathbb { R } ^ n \\setminus \\{ 0 , . . . , 0 \\} \\} . \\end{align*}"}
+{"id": "3437.png", "formula": "\\begin{align*} \\mathrm { t } \\tilde { H } ^ G _ * ( X ; \\Z _ 2 ) \\cong ( U ^ { - 1 } \\Z _ 2 [ U , Q ] / Q ^ 2 = 0 ) _ { s + 2 } . \\end{align*}"}
+{"id": "2309.png", "formula": "\\begin{align*} U _ { \\theta } ^ { c , \\gamma } ( - 1 ) = & \\begin{cases} 2 + 2 \\sqrt { 1 + c _ { 1 } } , & \\mathrm { i f } \\ ; \\gamma = \\gamma ^ { + } ( c ) , \\\\ 2 - 2 \\sqrt { 1 + c _ { 1 } } , & \\mathrm { o t h e r w i s e } , \\end{cases} \\\\ U _ { \\theta } ^ { c , \\gamma } ( 1 ) = & \\begin{cases} - 2 - 2 \\sqrt { 1 + c _ { 2 } } , & \\mathrm { i f } \\ ; \\gamma = \\gamma ^ { - } ( c ) , \\\\ - 2 + 2 \\sqrt { 1 + c _ { 2 } } , & \\mathrm { o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "1048.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } \\| \\phi _ { n , k } - \\phi _ { k } \\| \\leq \\| v _ { n + 1 } \\| \\sum _ { k = 1 } ^ { n } \\| v _ { n + 1 } ^ { - 1 } \\phi _ { n , k } - v _ { \\infty } ^ { - 1 } \\phi _ { k } \\| + \\| v _ { n + 1 } \\| \\| v _ { n + 1 } ^ { - 1 } - v _ { \\infty } ^ { - 1 } \\| \\sum _ { k = 1 } ^ { n } \\| \\phi _ { k } \\| . \\end{align*}"}
+{"id": "5631.png", "formula": "\\begin{align*} S ^ n x = ( 0 , 0 , \\ldots , 0 , \\frac { x _ 0 } { M _ 1 ^ { n } } , \\frac { x _ 1 } { M _ 2 ^ { n } } , \\frac { x _ 2 } { M _ 3 ^ { n } } , \\ldots ) . \\end{align*}"}
+{"id": "5362.png", "formula": "\\begin{align*} \\mathbf { c } ^ { S } - \\mathbf { c } ^ { S \\setminus \\{ j \\} } = \\beta \\ , \\left ( \\mathbf { P } ^ 0 - \\mathbf { P } ^ 1 \\right ) \\ , \\left ( \\mathbf { v } ^ { S } - \\mathbf { v } ^ { S \\setminus \\{ j \\} } \\right ) . \\end{align*}"}
+{"id": "5160.png", "formula": "\\begin{align*} & U _ { j } = - \\frac { 1 } { \\alpha } \\left [ \\frac { a - 1 } { a - b } B ^ { a - 2 } - \\frac { b - 1 } { a - b } B ^ { b - 2 } \\right ] \\\\ & V _ { j } = - \\frac { 1 } { \\alpha } \\left [ \\frac { a - 1 } { a - b } A ^ { a - 2 } - \\frac { b - 1 } { a - b } A ^ { b - 2 } \\right ] p ^ { \\alpha } _ { j } q ^ { - \\alpha } _ { j } \\end{align*}"}
+{"id": "2651.png", "formula": "\\begin{align*} \\| \\mathbf { 1 } _ { [ 0 , \\infty ) ] } e ^ { i t \\Delta _ x } u _ 0 \\| _ { V ^ p _ { \\Delta _ x } } = \\| u _ 0 \\| _ { L ^ 2 _ x } . \\end{align*}"}
+{"id": "4691.png", "formula": "\\begin{align*} \\mathrm { N e g } _ { q _ X } ( D ) ^ { \\perp } : = \\left \\{ \\alpha \\in N ^ 1 ( X ) _ { \\mathbf { R } } \\ ; | \\ ; q _ X ( \\alpha , E ) = 0 N \\right \\} . \\end{align*}"}
+{"id": "7630.png", "formula": "\\begin{align*} y \\cdot \\beta & = ( s ^ { d - 1 , 3 } _ { 1 , \\R } ) _ * ( x ) \\cdot ( s ^ { d - 1 , 3 } _ { 1 , \\R } ) _ * ( \\alpha ) = ( s ^ { d - 1 , 3 } _ { 1 , \\R } ) _ * ( x \\cdot \\alpha ) = ( s ^ { d - 1 , 3 } _ { 1 , \\R } ) _ * ( \\alpha ) = \\beta . \\end{align*}"}
+{"id": "8420.png", "formula": "\\begin{align*} \\Phi ( f _ 1 , f _ 2 ) \\ast \\Phi ( g _ 1 , g _ 2 ) & = ( f _ 1 \\ast f _ 2 ^ { \\ast - 1 } ) \\ast ( g _ 1 \\ast g _ 2 ^ { \\ast - 1 } ) = ( f _ 1 * g _ 1 ) * ( g _ 2 ^ { * - 1 } * f _ 2 ^ { * - 1 } ) \\\\ & = \\Phi ( f _ 1 \\ast g _ 1 , f _ 2 \\ast g _ 2 ) = \\Phi ( ( f _ 1 , f _ 2 ) \\cdot _ { D } ( g _ 1 , g _ 2 ) ) . \\end{align*}"}
+{"id": "7477.png", "formula": "\\begin{align*} g ( 0 ) = { g } _ 0 ( 0 ) + h , \\end{align*}"}
+{"id": "8340.png", "formula": "\\begin{align*} \\mu _ 2 ^ B ( \\alpha , \\beta ) = ( - 1 ) ^ { | \\alpha | } [ \\alpha , \\beta ] _ B . \\end{align*}"}
+{"id": "6526.png", "formula": "\\begin{align*} \\ell ( x ) = e ^ { \\int ^ x _ 1 \\frac { \\eta ( t ) } { t } } d t , x > 0 , \\end{align*}"}
+{"id": "3274.png", "formula": "\\begin{align*} \\frac { ( q ^ { d + r - ( d - 2 j - 1 ) n } ; q ^ d ) _ k } { ( q ^ { d - ( d - 2 j ) n } ; q ^ d ) _ k } & = \\frac { ( q ^ { d - ( d - 2 j ) n + d k } ; q ^ d ) _ { ( n + r ) / d } } { ( q ^ { d - ( d - 2 j ) n } ; q ^ d ) _ { ( n + r ) / d } } , \\end{align*}"}
+{"id": "1342.png", "formula": "\\begin{align*} r + a \\{ r \\alpha + ( 1 - \\alpha ) \\} + b \\{ r \\alpha - ( 1 - \\alpha ) \\} = \\alpha n r \\\\ r ^ 2 + a \\{ r \\alpha + ( 1 - \\alpha ) \\} ^ 2 + b \\{ r \\alpha - ( 1 - \\alpha ) \\} ^ 2 = \\alpha ^ 2 n r ^ 2 + ( 1 - \\alpha ) ^ 2 n r \\end{align*}"}
+{"id": "760.png", "formula": "\\begin{align*} \\varphi _ { i j } - ( \\Gamma ^ h _ { i j } + C ^ h _ { i j } ) \\varphi _ h - \\varphi _ i \\varphi _ j - \\Phi g _ { i j } = 0 , \\end{align*}"}
+{"id": "8814.png", "formula": "\\begin{align*} g ( x ) = \\begin{cases} a _ 1 & x \\in E _ 1 = \\bigcup _ { k = 1 } ^ { M _ 1 } Q ( x _ k , \\tau _ 1 ) , \\\\ a _ 2 & x \\in E _ 2 = \\bigcup _ { k = M _ 1 + 1 } ^ { M _ 2 } Q ( x _ k , \\tau _ 2 ) , \\\\ & \\vdots \\\\ a _ m & x \\in E _ m = \\bigcup _ { k = M _ { m - 1 } + 1 } ^ { M _ m } Q ( x _ k , \\tau _ m ) , \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "8162.png", "formula": "\\begin{align*} D _ 1 & = \\frac { c _ 1 } { M } \\frac { 2 [ c _ 1 - 1 ] ^ + } { c _ 1 } + \\frac { M - c _ 1 } { M } \\frac { 2 [ M - c _ 1 - 1 ] ^ + } { M - c _ 1 } \\\\ & = \\frac { 2 [ c _ 1 - 1 ] ^ + + 2 [ M - c _ 1 - 1 ] ^ + } { M } . \\end{align*}"}
+{"id": "6869.png", "formula": "\\begin{align*} ( \\lambda { I } - H _ { \\varphi { \\sf E } } ) { e } _ j ' = \\mu _ j { e } _ j ' , j \\in \\mathbb { N } , \\end{align*}"}
+{"id": "6538.png", "formula": "\\begin{align*} S _ { N } = \\sum ^ N _ { n = 1 } \\sum ^ { \\infty } _ { j = 1 } \\Big ( \\mathbb { E } \\big [ K ( X _ n ) | \\mathcal { F } _ { n - j } \\big ] - \\mathbb { E } \\big [ K ( X _ n ) | \\mathcal { F } _ { n - j - 1 } \\big ] \\Big ) . \\end{align*}"}
+{"id": "8600.png", "formula": "\\begin{align*} u _ t + \\frac { 3 } { 2 } u u _ x + \\frac { 1 } { 6 } u _ { x x x } = 0 . \\end{align*}"}
+{"id": "2768.png", "formula": "\\begin{align*} T ^ { * } M = T ^ { * } M | _ { \\Xi , \\Psi } \\times T ^ { * } M | _ { \\Theta } \\times T ^ { * } M | _ { Q , P } . \\end{align*}"}
+{"id": "4897.png", "formula": "\\begin{align*} C ' \\cdot \\Big ( E + \\Big ( \\frac { e - g } 2 + 3 \\Big ) F \\Big ) = 1 0 - g \\end{align*}"}
+{"id": "808.png", "formula": "\\begin{align*} \\delta { \\pi _ 1 } = - ( { \\nabla ^ j } { a _ j } - { a _ j } { \\nabla _ 0 } { C ^ j } ) , \\end{align*}"}
+{"id": "3097.png", "formula": "\\begin{align*} u _ \\varphi ( t ) & = \\left ( \\begin{array} { c c c } 1 & c _ 1 \\ , t ^ { p ^ { e _ 1 } } & c _ 2 \\ , t ^ { p ^ { e _ 2 } } \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) ( \\ , c _ 1 , c _ 2 \\in k , e _ 1 , e _ 2 \\geq 0 \\ , ) , \\end{align*}"}
+{"id": "8389.png", "formula": "\\begin{align*} X ( i , j ) & = \\begin{cases} \\ ; 1 & \\ ; j - i = 1 \\mod m \\\\ \\ ; 0 & \\end{cases} , \\\\ Y ( i , j ) & = \\begin{cases} \\ ; \\delta ( m \\cdot 1 - n \\cdot 1 ) & \\ ; i = j = 1 \\\\ \\ ; ( - 1 ) ^ { m + n } & \\ ; i = n , j = 1 \\\\ \\ ; 1 & \\ ; j - i = 1 \\\\ \\ ; 0 & \\end{cases} , \\end{align*}"}
+{"id": "5389.png", "formula": "\\begin{align*} w ^ { S _ 1 } _ j & = \\lambda , 1 \\leq j \\leq n - 1 \\\\ w ^ { S _ 2 } _ 0 & = \\frac { 1 } { 1 + \\rho } \\ , \\lambda \\\\ w ^ { S _ { j + 1 } } _ { j - 1 } & = \\frac { 1 } { 1 + \\rho } \\ , \\frac { w ^ { S _ j } _ { j - 2 } } { a _ j } , 2 \\leq j \\leq n . \\end{align*}"}
+{"id": "9137.png", "formula": "\\begin{align*} x ^ { ( \\beta , s ) } _ { i , t } < x ^ { ( \\beta ' , s ' ) } _ { i , t ' } ( \\beta , s ) < ( \\beta ' , s ' ) \\ \\ ( \\beta , s ) = ( \\beta ' , s ' ) , t < t ' . \\end{align*}"}
+{"id": "6087.png", "formula": "\\begin{align*} U _ { \\infty } \\Psi _ { m _ { + } } = m _ { + } \\Psi _ { m _ { + } } . \\end{align*}"}
+{"id": "7484.png", "formula": "\\begin{align*} & \\frac { \\partial } { \\partial \\tau } u = \\Delta _ { g ( \\tau ) } u , \\\\ & \\frac { \\partial } { \\partial \\tau } u = 2 R i c ( g ( \\tau ) ) \\end{align*}"}
+{"id": "2138.png", "formula": "\\begin{align*} \\omega ( \\hat { x } ) - f ( \\hat { x } , \\hat { y } ) = \\Big ( \\underbrace { \\omega ( \\hat { x } ) - f ( \\hat { x } , q ^ { ( k ) } ( \\hat { x } , \\hat { y } ) ) } _ { S _ 1 } \\Big ) \\ , + \\ , \\Big ( \\underbrace { f ( \\hat { x } , q ^ { ( k ) } ( \\hat { x } , \\hat { y } ) ) - f ( \\hat { x } , \\hat { y } ) } _ { S _ 2 } \\Big ) . \\end{align*}"}
+{"id": "8682.png", "formula": "\\begin{align*} J T ^ H Y ' = T ^ H Y ' = J T Y ' \\cap T Y ' . \\end{align*}"}
+{"id": "299.png", "formula": "\\begin{align*} L _ \\mu = \\sum _ { l = 1 } ^ n \\sum _ { k = 1 } ^ \\infty \\mu ^ { \\vert \\alpha ( k ) \\vert } \\left | a _ { l , k } \\right | . \\end{align*}"}
+{"id": "8426.png", "formula": "\\begin{align*} x ^ { \\sigma , n } = x \\sigma ( x ) \\cdots \\sigma ^ { n - 1 } ( x ) \\in \\widetilde W . \\end{align*}"}
+{"id": "5386.png", "formula": "\\begin{align*} \\bar { b } ^ u = \\lim _ { T \\to \\infty } \\ , \\frac { 1 } { T } \\ , E ^ u \\left [ \\int _ 0 ^ T \\lambda _ { L ( t ) } \\ , a ( t ) \\ , d t \\right ] , \\end{align*}"}
+{"id": "4028.png", "formula": "\\begin{align*} \\lim _ { m \\rightarrow + \\infty } \\frac { y ^ { ( t ) } } { \\varpi \\circ W \\left ( z ^ { ( t ) } \\right ) } = \\begin{cases} \\qquad \\gamma & \\mbox { i f } \\gamma _ { 2 } = 0 , \\delta _ { 1 } \\neq 0 , \\\\ \\frac { \\gamma x _ 1 ^ { ( t _ 0 ) } + \\delta x _ { 2 } ^ { ( t _ { 0 } ) } } { x _ { 1 } ^ { \\left ( t _ { 0 } \\right ) } + x _ { 2 } ^ { \\left ( t _ { 0 } \\right ) } } & \\mbox { i f } \\gamma _ { 2 } = \\delta _ { 1 } = 0 , \\\\ \\qquad \\delta & \\mbox { i f } \\gamma _ { 2 } \\neq 0 , \\delta _ { 1 } = 0 . \\end{cases} \\end{align*}"}
+{"id": "1356.png", "formula": "\\begin{align*} S p e c ~ ( A _ \\alpha ( G ) ) = \\left \\{ \\left [ \\frac { 2 m \\alpha } { n } + h \\right ] ^ { l r } , \\left [ \\frac { 2 m \\alpha } { n } - h \\right ] ^ { l } , \\left [ \\frac { 2 m \\alpha } { n } + k \\right ] ^ { a } , \\left [ \\frac { 2 m \\alpha } { n } - k \\right ] ^ { a r } \\right \\} , \\end{align*}"}
+{"id": "6765.png", "formula": "\\begin{align*} \\begin{pmatrix} x _ 2 ^ { k + 1 } \\\\ \\lambda ^ { k + 1 } \\end{pmatrix} = \\begin{pmatrix} x _ 2 ^ { k } \\\\ \\lambda ^ { k } \\end{pmatrix} - \\begin{pmatrix} I _ { n _ 2 } & 0 \\\\ - s \\beta A _ 2 & ( r + s ) I _ l \\end{pmatrix} \\begin{pmatrix} x _ 2 ^ { k } - \\widetilde { x } _ 2 ^ k \\\\ \\lambda ^ { k } - \\widetilde { \\lambda } ^ k \\end{pmatrix} . \\end{align*}"}
+{"id": "9100.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } X ( t + 1 ) = \\tilde { A } ^ { ( r ) } X ( t ) + \\tilde { b } ^ { ( r ) } , \\\\ v ( t ) = J ^ { ( r ) } X ( t ) + \\underline { v } . \\end{array} \\right . \\end{align*}"}
+{"id": "2650.png", "formula": "\\begin{align*} \\hat { x } _ { \\alpha } = ( x _ 1 , \\ldots , x _ { \\alpha - 1 } , x _ { \\alpha + 1 } , \\ldots , x _ N ) \\in \\mathbb { T } ^ { d ( N - 1 ) } . \\end{align*}"}
+{"id": "3692.png", "formula": "\\begin{align*} \\sigma _ i \\geq \\min \\{ \\sum \\nolimits _ { j = 1 } ^ i c _ j ^ 2 \\lambda _ j - \\sqrt { d - 1 } / R , \\ \\sum \\nolimits _ { j = 1 } ^ { i } c _ j ^ 2 = 1 \\} = \\lambda _ i - \\sqrt { d - 1 } / R . \\end{align*}"}
+{"id": "2022.png", "formula": "\\begin{align*} \\left | ( \\rho - 1 ) a \\alpha ^ n - \\left ( d _ 1 \\cdot \\rho ^ { \\ell + m } - ( d _ 1 - d _ 2 ) \\cdot \\rho ^ m - ( d _ 2 - d _ 3 ) \\right ) \\cdot \\rho ^ k \\right | & = | - ( \\rho - 1 ) \\Pi ( n ) - d _ 3 | \\\\ & \\leqslant ( \\rho - 1 ) \\cdot \\alpha ^ { - n / 2 } + ( \\rho - 1 ) < 2 ( \\rho - 1 ) . \\end{align*}"}
+{"id": "2774.png", "formula": "\\begin{align*} & P _ { a } ( t ) = \\frac { \\partial L } { \\partial R ^ { a } ( t ) } , \\\\ & { \\textrm { d e t } } \\left ( \\frac { \\partial P _ { a } ( t ) } { \\partial R ^ { b } ( t ) } \\right ) \\neq 0 . \\end{align*}"}
+{"id": "7543.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ \\Omega \\tfrac { 1 } { 2 } \\bar \\rho | \\bar u | ^ 2 + h _ 1 ( \\bar \\rho ) + h _ 2 ( \\bar \\rho ) + \\tfrac { 1 } { 2 } | \\nabla h _ 2 ^ \\prime ( \\bar n ) | \\ d x = 0 \\ , \\end{align*}"}
+{"id": "4538.png", "formula": "\\begin{align*} \\left \\{ \\begin{matrix*} [ l ] \\partial _ t | n _ \\infty ^ { ( K ) } | + \\displaystyle \\sum _ { i = 1 } ^ K \\partial _ { s _ i } | n _ \\infty ^ { ( K ) } | + p _ K ( [ s ] _ K ) | n _ \\infty ^ { ( K ) } | \\leq | E _ \\infty ^ { ( K ) } ( t , [ s ] _ K ) | , \\\\ [ 5 p t ] | n _ \\infty ^ { ( K ) } ( t , s _ 1 = 0 , [ s ] _ { 2 , K } ) | \\leq \\int _ { u = 0 } ^ \\infty \\big [ p _ K | n _ \\infty ^ { ( K ) } | + | E _ \\infty ^ { ( K ) } | \\big ] ( t , [ s ] _ { 2 , K } , u ) d u , \\\\ [ 5 p t ] n _ \\infty ^ { ( K ) } ( t = 0 , [ s ] _ K ) = 0 . \\end{matrix*} \\right . \\end{align*}"}
+{"id": "2.png", "formula": "\\begin{align*} q ( \\vec t ; q _ 0 ) = \\Bigl [ \\exp \\bigl \\{ { \\textstyle \\sum } t _ i J \\nabla H _ i \\bigr \\} q _ 0 \\Bigr ] ( x = 0 ) \\end{align*}"}
+{"id": "4753.png", "formula": "\\begin{align*} \\lim _ { \\lambda _ 0 > t \\to 0 ^ + } \\frac { \\norm { J _ t x - S ( t ) x } } { t } = 0 . \\end{align*}"}
+{"id": "1790.png", "formula": "\\begin{align*} a x _ 1 \\Re ( z _ 1 ^ { \\mu + 1 } ) - b \\Re ( z _ 1 ^ { \\mu + 2 } ) = \\Re ( Y ^ { \\prime } ) ( P ( z , \\zeta ) - u ) = c x _ 1 ^ { \\mu + 2 } \\end{align*}"}
+{"id": "8165.png", "formula": "\\begin{align*} & H ( Y _ j | Y ^ { j - 1 } , \\underline S ) = \\\\ & \\sum _ { \\underline s } \\sum _ { y ^ { j - 1 } } P _ { Y ^ { j - 1 } , \\underline S } ( y ^ { j - 1 } , \\underline s ) H ( P _ { Y _ { j } | Y ^ { j - 1 } , \\underline S } ( 1 | y ^ { j - 1 } , \\underline s ) ) . \\end{align*}"}
+{"id": "795.png", "formula": "\\begin{align*} K _ { j k l } ^ i = ( \\delta _ { k } G ^ { i } _ { j l } - G _ { j l s } ^ i G _ k ^ s ) - ( \\delta _ { l } G ^ { i } _ { j k } - G _ { j k s } ^ i G _ l ^ s ) + G _ { r k } ^ i G _ { j l } ^ r - G _ { l r } ^ i G _ { j k } ^ r , \\end{align*}"}
+{"id": "5572.png", "formula": "\\begin{align*} \\overline { F _ { \\phi _ i , t } } = \\frac { \\Phi ^ t ( \\phi _ i \\circ \\phi _ i ) } { d ^ { 2 t } } \\end{align*}"}
+{"id": "3831.png", "formula": "\\begin{align*} B : = \\frac { 1 } { 7 2 k } \\left ( \\left ( - 9 h ^ 2 h ' + 5 h ' + 9 h \\right ) k ^ 2 + ( 1 8 h - 3 6 ) k - 8 h ' \\right ) . \\end{align*}"}
+{"id": "7350.png", "formula": "\\begin{align*} \\widehat { \\boldsymbol { x } } _ { \\beta } ^ { g } = \\underset { \\boldsymbol { e } _ { \\beta } \\in E _ { \\beta } } { \\arg \\ : \\min } \\left \\| \\boldsymbol { y } _ { \\beta } ^ { g } - \\sqrt { P _ { \\beta } } d i a g ( \\boldsymbol { h } _ { F , \\beta } ^ { g } ) \\boldsymbol { e } _ { \\beta } \\right \\| ^ { 2 } , \\end{align*}"}
+{"id": "1607.png", "formula": "\\begin{gather*} A = \\sum \\epsilon ( i _ 1 \\dots i _ { r - 1 } n i _ { r + 1 } \\dots i _ { n - 1 } ) \\ , \\epsilon ( i _ 1 \\dots i _ { n - 1 } ) \\ , \\kappa _ { i _ 1 } \\dots \\kappa _ { i _ { r - 1 } } R _ { i _ r n i _ r n } \\\\ = ( - 1 ) ^ { n - r - 1 } \\sum \\kappa _ { i _ 1 } \\dots \\kappa _ { i _ { r - 1 } } K _ { i _ r n } , \\end{gather*}"}
+{"id": "6465.png", "formula": "\\begin{align*} \\underbrace { \\langle V , \\bar \\Delta \\tau ( \\phi ) \\rangle } _ { = m H | A | ^ 2 f } \\underbrace { \\langle d \\phi , \\bar \\nabla V \\rangle } _ { = - \\langle \\tau ( \\phi ) , V \\rangle = - m H f } = - m ^ 2 | A | ^ 2 H ^ 2 f ^ 2 . \\end{align*}"}
+{"id": "4476.png", "formula": "\\begin{align*} \\int _ { \\{ 2 \\psi < - t \\} } F _ 0 \\overline { F _ 0 - f } \\tilde \\rho = 0 \\end{align*}"}
+{"id": "5678.png", "formula": "\\begin{align*} \\| u \\| ^ 2 _ { L ^ 2 } : = \\int _ \\Sigma | u | ^ 2 \\ , d \\mu < \\infty . \\end{align*}"}
+{"id": "7938.png", "formula": "\\begin{align*} \\partial \\eta = \\eta ' + [ \\eta , u ] _ { 1 } , \\end{align*}"}
+{"id": "3830.png", "formula": "\\begin{align*} K _ k ^ { [ 4 ] } ( \\nu ; n ) : = K _ k ^ { [ 4 ] } ( \\nu ; n , 0 ) . \\end{align*}"}
+{"id": "1676.png", "formula": "\\begin{align*} g \\cdot [ v _ 0 , v _ 1 , v _ 2 , v _ 3 ] = \\Phi _ 3 ( \\pi _ 3 ( \\mathbf { p } ) ) . \\end{align*}"}
+{"id": "3695.png", "formula": "\\begin{align*} \\lambda y _ i = ( d - 1 ) y _ i ' + z \\implies \\lambda a _ { \\ell - 1 } x _ i ' = ( d - 1 ) a _ { \\ell - 2 } x _ i ' + z . \\end{align*}"}
+{"id": "882.png", "formula": "\\begin{align*} S _ 1 + T _ 1 & = 2 w ^ 1 { w ^ 1 } _ { \\substack { \\\\ x _ 1 } } + 2 w ^ 2 { w ^ 2 } _ { \\substack { \\\\ x _ 1 } } \\\\ & = ( ( w ^ 1 ) ^ 2 + ( w ^ 2 ) ^ 2 ) _ { x _ 1 } . \\end{align*}"}
+{"id": "3800.png", "formula": "\\begin{align*} \\| \\mathcal { H } _ 2 \\| \\leq \\Delta _ { \\max } | \\widehat { \\mathcal { C } } _ 1 \\cap \\overline { \\mathcal { C } _ 1 } | \\frac { 9 \\sum _ { i \\in \\widehat { \\mathcal { C } } _ 1 \\cap \\overline { \\mathcal { C } _ 1 } } N _ i \\sum _ { t = 0 } ^ { T - 1 } \\| \\Sigma ^ { ( i ) } _ t \\| } { 2 \\lambda _ { \\min } \\left ( \\sum _ { i \\in \\widehat { \\mathcal { C } } _ 1 } N _ i \\sum _ { t = 0 } ^ { T - 1 } { \\Sigma } _ { t } ^ { ( i ) } \\right ) } , \\end{align*}"}
+{"id": "2632.png", "formula": "\\begin{align*} | S + S | \\geq \\sum _ { j = 1 } ^ t | S _ j + S _ j | \\gg \\frac { n } { i } \\cdot \\frac { i ^ { 3 0 / 1 9 } } { ( \\log n ) ^ c } = \\frac { n i ^ { 1 1 / 1 9 } } { ( \\log n ) ^ c } . \\end{align*}"}
+{"id": "7090.png", "formula": "\\begin{align*} \\| b \\| _ { E _ q } & : = \\sup _ { r > 0 , z \\in \\mathbb R ^ { d + 1 } } r \\biggl ( \\frac { 1 } { | C _ r | } \\int _ { C _ r ( z ) } | b ( t , x ) | ^ q d t d x \\biggr ) ^ { \\frac { 1 } { q } } \\\\ & = \\sup _ { r > 0 , z \\in \\mathbb R ^ { d + 1 } } r \\biggl ( \\frac { 1 } { | C _ r | } \\int _ { C _ r ( z ) } | b ( - t , x ) | ^ q d t d x \\biggr ) ^ { \\frac { 1 } { q } } . \\end{align*}"}
+{"id": "3958.png", "formula": "\\begin{align*} \\pi ( \\rho _ Y ( g ) y ) = \\rho _ X ( g ) \\pi ( y ) . \\ \\end{align*}"}
+{"id": "5511.png", "formula": "\\begin{align*} \\ell ^ { m } - 1 \\mid \\ell ^ { n } - 1 = \\ell ^ { r } ( \\ell ^ { q m } - 1 ) + \\ell ^ { r } - 1 \\end{align*}"}
+{"id": "5240.png", "formula": "\\begin{align*} - \\frac { \\partial L _ { d } D A H I ( p \\| q ) } { \\partial q _ { j } } = \\sum _ { j } p _ { j } \\left [ \\underbrace { \\frac { a - 1 } { a - b } \\left ( \\overline { M H } \\right ) ^ { a - 2 } - \\frac { b - 1 } { a - b } \\left ( \\overline { M H } \\right ) ^ { b - 2 } } _ { Z > 0 } \\right ] \\frac { \\partial \\overline { M H } } { \\partial q _ { j } } \\end{align*}"}
+{"id": "913.png", "formula": "\\begin{align*} L = \\sum ^ d _ { i , j = 1 } \\partial _ { x _ i } ( a _ { i j } ( x ) \\partial _ { x _ j } ) , \\end{align*}"}
+{"id": "8453.png", "formula": "\\begin{align*} U ( x , y ) & : = | u ( x ) - u ( y ) | ^ { p - 2 } ( u ( x ) - u ( y ) ) , \\\\ U ( x , y , t ) & : = | u ( x , t ) - u ( y , t ) | ^ { p - 2 } ( u ( x , t ) - u ( y , t ) ) \\end{align*}"}
+{"id": "3506.png", "formula": "\\begin{align*} \\| a \\| ^ 4 & = \\| \\Psi ( a ) \\| ^ 4 \\\\ & = \\| \\Psi ( a ) ^ * \\Psi ( a ) \\Psi ( a ) ^ * \\Psi ( a ) \\| \\\\ & \\leq \\| \\Psi ( a ) ^ * \\Psi ( a a ^ * ) \\Psi ( a ) \\| \\\\ & \\leq \\| \\Psi ( a ) ^ * \\| a \\| ^ 2 h \\Psi ( a ) \\| \\\\ & \\leq \\| a \\| ^ 3 \\| h \\Psi ( a ) \\| . \\end{align*}"}
+{"id": "3489.png", "formula": "\\begin{align*} \\hat { h } - P \\hat { h } = h - \\pi ( h ) = f - g - \\pi ( f - g ) = 0 , \\end{align*}"}
+{"id": "6868.png", "formula": "\\begin{align*} \\sum _ { j \\in \\mathbb { J } } E _ j ^ 2 | \\langle \\psi _ j , \\psi _ n \\rangle | ^ 2 < \\infty , \\sum _ { j \\in \\mathbb { J } } E _ j \\langle \\psi _ j , \\psi _ n \\rangle \\varphi _ j = 0 \\end{align*}"}
+{"id": "732.png", "formula": "\\begin{align*} g ( a _ 1 , a _ 2 , \\dots , a _ k ) & = \\left ( \\max _ { 0 \\le j \\le a _ 1 - 1 } m _ j \\right ) - a _ 1 \\ , , \\\\ n ( a _ 1 , a _ 2 , \\dots , a _ k ) & = \\frac { 1 } { a _ 1 } \\sum _ { j = 0 } ^ { a _ 1 - 1 } m _ j - \\frac { a _ 1 - 1 } { 2 } \\ , , \\\\ s ( a _ 1 , a _ 2 , \\dots , a _ k ) & = \\frac { 1 } { 2 a _ 1 } \\sum _ { j = 0 } ^ { a _ 1 - 1 } ( m _ j ) ^ 2 - \\frac { 1 } { 2 } \\sum _ { j = 0 } ^ { a _ 1 - 1 } m _ j + \\frac { a _ 1 ^ 2 - 1 } { 1 2 } \\ , , \\end{align*}"}
+{"id": "4505.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ | A _ 0 ( n ) | ^ 2 \\big ] = \\sum _ { \\substack { | \\lambda | = n \\\\ \\lambda _ 0 \\leqslant y _ 0 } } \\prod _ k \\frac { 1 } { k ^ { m _ k } m _ k ! } \\leqslant \\frac { 1 } { r ^ n } \\exp \\bigg ( \\sum _ { k \\leqslant y _ 0 } \\frac { r ^ { k } } { k } \\bigg ) . \\end{align*}"}
+{"id": "7426.png", "formula": "\\begin{align*} \\left \\lvert \\frac { \\abs { I } \\abs { J ^ \\prime } } { { \\abs { I ^ \\prime } \\abs { J } } } \\right \\rvert = \\abs { j - j ^ \\prime } \\abs { r _ s ( i _ 1 , \\ldots , i _ k ) } \\abs { D ^ 2 ( \\Psi _ { i _ 1 , \\ldots , i _ k } ) ( \\xi ) } , \\end{align*}"}
+{"id": "9069.png", "formula": "\\begin{align*} \\begin{cases} V ( t + 1 ) = C V ( t ) + D p - \\phi ( v ) , \\\\ v ( t ) = \\hat C V ( t ) , \\end{cases} \\end{align*}"}
+{"id": "2008.png", "formula": "\\begin{align*} \\mathcal { N } _ n = a \\alpha ^ n + b \\beta ^ n + c \\gamma ^ n . \\end{align*}"}
+{"id": "7994.png", "formula": "\\begin{align*} \\int _ { \\{ \\rho = h \\} } \\bigg ( \\varepsilon ( \\nabla u \\cdot g ) \\nabla u - \\varepsilon \\frac { | \\nabla u | ^ 2 } { 2 } g - \\frac { W ( u ) } { \\varepsilon } g \\bigg ) \\cdot \\nabla \\rho \\ , d x \\to 0 \\end{align*}"}
+{"id": "6448.png", "formula": "\\begin{align*} \\nabla _ { e _ i } e _ j = 0 , i , i = 1 , \\ldots , \\dim M , \\nabla _ { \\partial _ t } e _ i = 0 , i = 1 , \\ldots , \\dim M . \\end{align*}"}
+{"id": "1082.png", "formula": "\\begin{align*} \\xi ( d ) : = \\int _ 0 ^ 1 d s \\int _ 1 ^ { \\infty } d t \\int _ 0 ^ s d u \\frac { 1 } { u ^ d ( t - s + u ) ^ { 2 + d } } , \\end{align*}"}
+{"id": "3077.png", "formula": "\\begin{align*} a _ { k _ { i g } } = - \\frac { P _ k ( a _ { \\beta _ g } , \\ldots , a _ { k _ { i g } - 1 } ) } { r _ k \\cdot a _ { \\beta _ g } ^ { \\gamma _ { 1 g } } } , \\end{align*}"}
+{"id": "7474.png", "formula": "\\begin{align*} r _ s = m a x \\left \\{ ( \\pi _ 1 \\circ \\Phi _ { s } ^ { - 1 } ) ^ * \\mathbf { r _ s } , \\Lambda \\sqrt { s } \\right \\} , \\end{align*}"}
+{"id": "818.png", "formula": "\\begin{align*} & \\pounds _ { \\hat { X } } K ^ i _ { j k l } = \\delta ^ i _ { j } ( D _ { k } \\Psi _ { l } - D _ { l } \\Psi _ { k } ) + \\delta ^ i _ { l } D _ { k } \\Psi _ { j } - \\delta ^ i _ { k } D _ { l } \\Psi _ { j } + y ^ i ( D _ { k } \\Psi _ { l } - D _ { l } \\Psi _ { k } ) _ { . j } . \\end{align*}"}
+{"id": "391.png", "formula": "\\begin{align*} \\xi = \\sum _ { i \\leq m } \\tilde { \\theta } _ { b _ { i } } ( \\xi _ { i } ) \\cdot a _ { i } = _ { N F } \\tilde { \\theta } _ { b _ { m } } ( \\xi _ { m } ) \\cdot a _ { m } + \\cdots + \\tilde { \\theta } _ { b _ { 0 } } ( \\xi _ { 0 } ) \\cdot a _ { 0 } \\end{align*}"}
+{"id": "8931.png", "formula": "\\begin{align*} \\dim ( \\mathcal { M } ( L ) ) & = \\dim ( \\mathcal { M } ( L / \\gamma _ 2 ( L ) ) ) + \\dim ( \\gamma _ 2 ( L ) ) ( \\dim ( \\frac { L } { \\gamma _ 2 ( L ) } ) - 1 ) - \\sum _ { i = 2 } ^ { c } \\dim ( k e r ( \\lambda _ { i } ) ) \\\\ & \\leq \\dim ( \\mathcal { M } ( L / \\gamma _ 2 ( L ) ) ) + \\dim ( \\gamma _ 2 ( L ) ) ( \\dim ( \\frac { L } { \\gamma _ 2 ( L ) } ) - 1 ) - \\sum _ { i = 2 } ^ { l } ( m + n - r - s - i ) . \\end{align*}"}
+{"id": "1381.png", "formula": "\\begin{align*} I = ( x _ 3 x _ 1 - x _ 2 ^ 2 , \\ , x _ 2 x _ 3 - x _ 1 x _ 4 , \\ , x _ 2 x _ 4 , \\ , x _ 3 x _ 4 , \\ , x _ 4 ^ 2 , \\ , x _ 3 ^ 2 ) . \\end{align*}"}
+{"id": "6881.png", "formula": "\\begin{align*} \\theta _ h H _ { h { \\sf E } } g = \\{ \\langle h _ j , H _ { h { \\sf E } } g \\rangle \\} _ { j \\in \\mathbb { J } } = \\{ \\langle H _ { h { \\sf E } } h _ j , g \\rangle \\} _ { j \\in \\mathbb { J } } = \\{ E _ j \\langle h _ j , g \\rangle \\} _ { j \\in \\mathbb { J } } = \\mathcal { E } \\theta _ h { g } , \\end{align*}"}
+{"id": "7987.png", "formula": "\\begin{align*} \\dot { \\tilde { \\mathcal { F } } } ( \\eta , \\phi _ { \\partial } , \\Sigma ) = \\{ \\tilde { \\mathcal { F } } , \\tilde { H } \\} _ { D } ( \\eta , \\phi _ { \\partial } , \\Sigma ) - \\int _ { \\Gamma } E ( \\frac { \\delta \\tilde { \\mathcal { F } } } { \\delta \\Sigma } ) \\wedge \\ast \\boldsymbol { n } ( d \\phi ) , \\end{align*}"}
+{"id": "3634.png", "formula": "\\begin{align*} ( \\alpha ^ 3 \\tau _ 0 ^ 3 + ( a + b ) \\alpha \\tau _ 0 ) f ( \\tau , w \\sigma ) - ( ( 1 + a ) \\alpha ^ 2 \\tau _ 0 ^ 2 + b ) f ( \\tau , w ) = 0 . \\end{align*}"}
+{"id": "4157.png", "formula": "\\begin{align*} \\varphi ( g _ 1 g _ 2 ) = \\varphi ( g _ 1 ) \\varphi ( g _ 2 ) \\mbox { o r } \\varphi ( g _ 1 g _ 2 ) = \\varphi ( g _ 2 ) \\varphi ( g _ 1 ) \\ , . \\end{align*}"}
+{"id": "94.png", "formula": "\\begin{align*} \\begin{aligned} 2 c _ 1 + c _ 2 \\geq \\min \\{ 2 ( p - \\gamma ) , 8 \\} > 0 . \\end{aligned} \\end{align*}"}
+{"id": "8082.png", "formula": "\\begin{align*} G = P ^ - N ^ + \\bigsqcup P ^ - w . \\end{align*}"}
+{"id": "8400.png", "formula": "\\begin{align*} L ( a ) L ( b ) & = \\phi ( B ( a ) ) \\phi ( B ( b ) ) = \\phi ( B ( a ) B ( b ) ) \\\\ & = \\phi ( B ( B ( a _ 1 ) b L ( a _ 2 ) ) ) = L ( B ( a _ 1 ) b L ( a _ 2 ) ) , \\end{align*}"}
+{"id": "6752.png", "formula": "\\begin{align*} G : = Q ^ T + Q - M ^ T H M \\succ 0 . \\end{align*}"}
+{"id": "6708.png", "formula": "\\begin{align*} \\Phi _ n ( x ) = \\prod _ { \\substack { 1 \\leq k \\leq n \\\\ ( k , n ) = 1 } } ( x - e ^ { \\frac { 2 \\pi i k } { n } } ) \\end{align*}"}
+{"id": "1911.png", "formula": "\\begin{align*} \\begin{aligned} I & = \\sum _ { x \\in G } \\Delta v _ + ( x , t ) v _ + ( x , t ) \\ , \\eta ^ 2 ( x ) e ^ { \\xi ( x , t ) } \\mu ( x ) \\\\ & \\le \\frac 1 { 4 } \\sum _ { x , y \\in G } v _ + ^ 2 ( x , t ) \\left [ 1 - e ^ { \\xi ( y , t ) - \\xi ( x , t ) } \\right ] ^ 2 \\eta ^ 2 ( x ) e ^ { \\xi ( x , t ) } \\ , \\omega ( x , y ) \\\\ & + \\sum _ { x , y \\in G } v _ + ^ 2 ( x , t ) \\left [ \\eta ( y ) - \\eta ( x ) \\right ] ^ 2 e ^ { \\xi ( y , t ) } \\ , \\omega ( x , y ) \\ , . \\end{aligned} \\end{align*}"}
+{"id": "2481.png", "formula": "\\begin{align*} \\begin{cases} \\phi ' = \\phi \\psi , \\\\ \\psi ' = - c \\phi \\psi \\end{cases} \\left ( \\ , ' = \\dfrac { d } { d s } \\right ) \\end{align*}"}
+{"id": "6891.png", "formula": "\\begin{align*} & ( h \\perp h ' ) ^ { \\otimes d } ( \\pi ( u \\otimes v ) , u ' \\otimes v ' ) \\\\ & = ( h \\perp h ' ) ^ { \\otimes d } ( ( x _ { \\pi ^ { - 1 } ( 1 ) } \\otimes \\cdots \\otimes x _ { \\pi ^ { - 1 } ( d ) } ) \\pi , ( y _ 1 \\otimes \\cdots \\otimes y _ d ) ) \\\\ & = \\pi ^ { - 1 } ( h \\perp h ' ) ( x _ { \\pi ^ { - 1 } ( 1 ) } , y _ 1 ) \\otimes \\cdots \\otimes ( h \\perp h ' ) ( x _ { \\pi ^ { - 1 } ( d ) } , y _ d ) \\end{align*}"}
+{"id": "8547.png", "formula": "\\begin{align*} P _ { A _ \\alpha } ( M \\wedge \\mathrm { T A Q } ( A _ \\alpha / R ) ) \\simeq A _ \\alpha \\wedge _ { P _ R X } P _ { P _ R X } ( M \\wedge \\mathrm { T A Q } ( P _ R X / R ) ) \\\\ \\simeq A _ \\alpha \\wedge _ { P _ R X } P _ R X \\otimes _ R M \\xrightarrow { \\simeq } \\left ( A _ \\alpha \\otimes _ R M \\right ) . \\end{align*}"}
+{"id": "3393.png", "formula": "\\begin{align*} C _ { 1 } = \\frac { \\sqrt { \\Delta _ { 1 } } } { 4 \\sqrt { 2 } \\gamma } ; C _ { 2 } \\le \\frac { 1 } { \\sigma ^ { p } } ; C _ { 3 } \\leq \\frac { \\Delta _ { 1 } } { 2 0 4 8 \\sigma ^ { p } \\gamma } ; A = 2 5 6 \\gamma ^ { 2 } \\end{align*}"}
+{"id": "7998.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } | \\nabla \\tilde u | & = \\int _ { B } | \\nabla \\tilde u | + \\int _ { \\mathbb { R } ^ n \\setminus \\bar B } | \\nabla \\tilde u | \\\\ & \\leq \\liminf _ { \\varepsilon \\to 0 } \\int _ B | \\nabla u _ \\varepsilon | + \\liminf _ { \\varepsilon \\to 0 } \\int _ B | \\nabla u _ \\varepsilon | \\\\ & \\leq \\int _ { \\mathbb { R } ^ n } | \\nabla \\tilde u | , \\end{align*}"}
+{"id": "6939.png", "formula": "\\begin{align*} \\int _ { \\Gamma _ { s , \\infty } } \\frac { \\exp ( n \\tau - j _ 0 \\varphi ( \\tau ) ) } { \\tau } d \\tau = 2 i \\pi E _ { 2 \\pi } ^ \\beta \\left ( \\frac { j _ 0 + n \\alpha } { \\left ( - \\frac { j _ 0 } { \\alpha } \\right ) ^ \\frac { 1 } { 2 \\mu } } \\right ) . \\end{align*}"}
+{"id": "2882.png", "formula": "\\begin{align*} \\limsup _ { \\lambda \\to 0 ^ + } \\lambda \\left \\| \\left [ \\int _ { \\mathbb { R } } \\mathbf { 1 } _ { E _ { \\lambda , \\frac { \\gamma } { q } } [ f ] \\cap [ ( - 2 \\beta , 2 \\beta ) ^ \\complement \\times \\mathbb { R } ] } ( \\cdot , y ) \\left | \\cdot - y \\right | ^ { \\gamma - 1 } \\ , d y \\right ] ^ \\frac { 1 } { q } \\right \\| _ X = 0 , \\end{align*}"}
+{"id": "8951.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\nabla w - I - S \\| _ { L ^ { p ( \\cdot ) } ( Q ' ) } & = \\| \\nabla z - S \\| _ { L ^ { p ( \\cdot ) } ( Q ' ) } \\le C \\| e z \\| _ { L ^ { p ( \\cdot ) } ( Q ' ) } \\\\ & = C \\| e w - I \\| _ { L ^ { p ( \\cdot ) } ( Q ' ) } \\le C \\| d ( \\nabla v , S O ( n ) ) \\| _ { L ^ { p ( \\cdot ) } ( Q ) } , \\end{aligned} \\end{align*}"}
+{"id": "6354.png", "formula": "\\begin{align*} \\omega _ k ( r ) : = \\frac { \\phi ^ { ( k ) } ( r ) } { r ^ { n - k } } , \\end{align*}"}
+{"id": "5778.png", "formula": "\\begin{align*} \\sum _ { i \\in I _ 1 } \\left ( | \\tilde { \\xi } ^ { ( k , \\ell ) } _ { i , 1 } ( t ) | ^ 2 + | \\tilde { \\xi } ^ { ( k , \\ell ) } _ { i , 2 } ( t ) | ^ 2 \\right ) + \\sum _ { i \\neq 0 } | \\tilde { \\xi } ^ { ( k , \\ell ) } _ i ( t ) | ^ 2 = \\| \\tilde { q } ^ { ( k , \\ell ) } \\| ^ 2 _ G ( t ) \\leq C \\| \\tilde { u } ^ \\perp \\| ^ 2 _ { C ^ { s + 1 } } ( t ) . \\end{align*}"}
+{"id": "6076.png", "formula": "\\begin{align*} \\Sigma _ { \\pm } : = \\{ e ^ { i \\arccos \\lambda } \\mid \\lambda \\in { \\mathbb T } _ { \\pm } \\} , \\Sigma _ { \\pm } ^ * : = \\{ e ^ { - i \\arccos \\lambda } \\mid \\lambda \\in { \\mathbb T } _ { \\pm } \\} . \\end{align*}"}
+{"id": "1859.png", "formula": "\\begin{align*} \\| \\beta a \\wedge \\beta a \\| _ { L ^ { 7 / 2 } _ { l - 1 } } \\leq c o n s t . \\| \\beta a \\| _ { L ^ { 7 / 2 } _ 1 } \\cdot \\sum _ { k = 1 } ^ l \\| \\beta a \\| _ { L ^ { 7 / 2 } _ k } . \\end{align*}"}
+{"id": "492.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & 1 \\\\ 1 & - Q _ t ( T ) \\end{pmatrix} \\end{align*}"}
+{"id": "1727.png", "formula": "\\begin{align*} \\left . \\frac { d ^ 4 } { d x _ 1 ^ 4 } \\left ( f \\left ( f ^ { ( 4 ) } ( 0 ) x _ 1 \\right ) \\right ) \\left ( f ^ { ( 4 ) } ( 0 ) \\right ) ^ { - 5 } \\right | _ { x _ 1 = 0 } = 1 . \\end{align*}"}
+{"id": "3365.png", "formula": "\\begin{align*} \\begin{aligned} | J ( u ^ \\circ ) - J ( u ) | = & \\Bigg | \\mathbb { E } ^ { u ^ \\circ } \\left [ \\mu \\left ( \\sum _ { i = 0 } ^ { K - 1 } L ( x _ i , u ^ \\circ _ i ) + \\Phi ( x _ K ) \\right ) \\right ] \\\\ \\ \\ & - \\mathbb { E } ^ u \\left [ \\mu \\left ( \\sum _ { i = 0 } ^ { K - 1 } L ( x _ i , u _ i ) + \\Phi ( x _ K ) \\right ) \\right ] \\Bigg | \\\\ \\leq & e _ \\Phi ( c ^ K d _ 0 + s \\sum _ { i = 1 } ^ { K - 1 } c ^ i ) . \\end{aligned} \\end{align*}"}
+{"id": "2430.png", "formula": "\\begin{align*} \\bar { U } _ { t } = \\bar { U } ^ { p } ( \\bar { U } _ { x x } + \\mu \\bar { U } ) , ( t , x ) \\in ( 0 , T ) \\times ( - L , L ) , \\end{align*}"}
+{"id": "6351.png", "formula": "\\begin{align*} \\partial E = \\{ ( R ( 1 + \\rho ( \\varphi ) ) , \\varphi ) : \\varphi \\in S ^ { n - 1 } \\} \\subset M , \\end{align*}"}
+{"id": "1643.png", "formula": "\\begin{align*} v \\left ( x , t \\right ) = u _ { 1 } \\left ( x , t \\right ) - u _ { 2 } \\left ( x , t \\right ) , p \\left ( x , t \\right ) = m _ { 1 } \\left ( x , t \\right ) - m _ { 2 } \\left ( x , t \\right ) , \\left ( x , t \\right ) \\in Q _ { T } , \\end{align*}"}
+{"id": "6422.png", "formula": "\\begin{align*} J _ { n , i , 2 2 } ^ { 2 2 } ( \\theta ) = - \\partial _ \\alpha f _ \\alpha ( z ^ n _ i ( \\theta ) ) - \\frac { \\ln \\left ( n / X _ { \\frac { i - 1 } { n } } \\right ) ^ 2 } { \\alpha ^ 4 } \\left ( z ^ n _ i ( \\theta ) k ' _ \\alpha ( z ^ n _ i ( \\theta ) ) \\right ) \\\\ + \\frac { \\ln \\left ( n / X _ { \\frac { i - 1 } { n } } \\right ) } { \\alpha ^ 2 } \\left ( - \\frac { 2 } { \\alpha } k _ \\alpha ( z ^ n _ i ( \\theta ) ) + 2 z ^ n _ i ( \\theta ) f ' _ \\alpha ( z ^ n _ i ( \\theta ) ) \\right ) . \\end{align*}"}
+{"id": "3321.png", "formula": "\\begin{align*} \\L u = \\div ( A D u ) + Y u + \\langle b , D u \\rangle + c u \\L _ 0 u = \\Delta _ { m _ 0 } u + Y u , \\end{align*}"}
+{"id": "3548.png", "formula": "\\begin{align*} T ( f \\ , i d ^ 4 ) - f ( \\tau ( z _ 0 ) ) T ( i d ^ 4 ) = ( T ( i d ^ 3 ) - T ( \\tau ( z _ 0 ) i d ^ 2 ) ) T ( g \\ , i d ^ 2 ) . \\end{align*}"}
+{"id": "9083.png", "formula": "\\begin{align*} X ( 0 ) \\geq 0 \\Longrightarrow X ( t ) \\geq 0 , \\mbox { f o r a l l $ t = 0 , 1 , 2 , \\ldots $ } \\end{align*}"}
+{"id": "7125.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } [ A ( u _ { n } ) - A ( u _ { c } ) ] + \\frac { 1 } { 4 } [ V _ { 1 } ( u _ { n } ) - V _ { 1 } ( u _ { c } ) ] = o ( 1 ) . \\end{align*}"}
+{"id": "6749.png", "formula": "\\begin{align*} \\min \\limits _ { x \\in \\mathcal { X } } \\max \\limits _ { y \\in \\mathcal { Y } } \\left \\{ \\Phi ( x , y ) : = f ( x ) - y ^ T A x - g ( y ) \\right \\} , \\end{align*}"}
+{"id": "6498.png", "formula": "\\begin{align*} G \\times G & \\longrightarrow G \\\\ ( g , s ) & \\longmapsto g * _ \\theta s = g s \\theta ( g ) ^ { - 1 } . \\end{align*}"}
+{"id": "6786.png", "formula": "\\begin{align*} \\begin{bmatrix} b _ { 1 2 } & b _ { 2 1 } & 0 \\\\ b _ { 1 3 } & 0 & b _ { 3 1 } \\\\ 0 & b _ { 2 3 } & b _ { 3 2 } \\end{bmatrix} \\begin{bmatrix} d _ 1 \\\\ d _ 2 \\\\ d _ 3 \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\end{bmatrix} . \\end{align*}"}
+{"id": "6530.png", "formula": "\\begin{align*} \\widetilde { c } = \\frac { 1 } { 1 - \\beta } ( \\sigma _ 1 + \\sigma _ 2 ) ^ { \\frac { 1 } { \\alpha } } \\Big ( \\Gamma ( \\alpha - 1 ) \\cos ( \\frac { \\pi \\alpha } { 2 } ) \\Big ) ^ { \\frac { 1 } { \\alpha } } \\int _ { \\mathbb { R } } K ( x ) d f ( x ) . \\end{align*}"}
+{"id": "2286.png", "formula": "\\begin{gather*} k _ 1 = C _ 1 = C \\otimes 1 \\otimes 1 , k _ 2 = C _ 2 = 1 \\otimes C \\otimes 1 , k _ 3 = C _ 3 = 1 \\otimes 1 \\otimes C , k _ 4 = C _ { 1 2 3 } = \\delta ^ { ( 3 ) } ( C ) , \\\\ [ 0 . 5 e m ] X = C _ { 1 2 } = \\delta ^ { ( 2 ) } ( C ) \\otimes 1 , Y = C _ { 2 3 } = 1 \\otimes \\delta ^ { ( 2 ) } ( C ) \\ . \\end{gather*}"}
+{"id": "4079.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\frac { d - 3 } { 2 } } \\frac { \\tilde { c _ k } } { | x - y | ^ { d - 1 - s _ k } } \\partial _ s ^ { s _ k } \\{ e ^ { i z s } + e ^ { - i z s } \\} \\Big | _ { ( s = | x - y | ) } = z ^ { d - 1 } \\int _ { \\mathbb S ^ { d - 1 } } e ^ { i z ( x - y ) \\cdot \\omega } d \\omega \\end{align*}"}
+{"id": "1354.png", "formula": "\\begin{align*} S p e c ~ ( A _ \\alpha ( G ) ) = \\left \\{ \\frac { 2 m } { n } , \\left [ \\frac { 2 m } { n } ( 2 \\alpha - 1 ) \\right ] ^ { l - 1 } , \\left [ \\frac { 2 m \\alpha } { n } + ( 1 - \\alpha ) \\right ] ^ a , \\left [ \\frac { 2 m \\alpha } { n } - ( 1 - \\alpha ) \\right ] ^ b \\right \\} . \\end{align*}"}
+{"id": "7748.png", "formula": "\\begin{align*} n ( i ) = \\min _ { j \\in S _ { m ( i ) } } \\{ j \\} . \\end{align*}"}
+{"id": "8103.png", "formula": "\\begin{align*} b ^ { - 8 / 1 0 } \\ll e ^ { - t _ i } = r _ i \\ll b ^ { - 7 / 1 0 } . \\end{align*}"}
+{"id": "6755.png", "formula": "\\begin{align*} v ^ { k + 1 } = v ^ k - M ( v ^ k - \\widetilde { v } ^ k ) . \\end{align*}"}
+{"id": "1597.png", "formula": "\\begin{align*} \\theta ^ i ( e _ j ) = \\delta ^ i _ j , \\end{align*}"}
+{"id": "4816.png", "formula": "\\begin{align*} \\mathcal { S } ^ { ( b f ) } _ k : = \\Big \\{ \\mathbf { c } ' \\in \\mathbb { R } ^ { n } : \\mathbf { c } '^ \\top \\mathbf { x } ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } } = S ^ { \\mbox { \\tiny \\upshape ( \\itshape k \\upshape ) } } \\Big \\} \\cap \\mathcal { S } _ 0 . \\end{align*}"}
+{"id": "959.png", "formula": "\\begin{align*} \\mathcal E ( u , v ) = c \\int _ D \\int _ D \\frac { ( u ( x ) - u ( y ) ) ( v ( x ) - v ( y ) ) } { | x - y | ^ { d + \\alpha } } \\ , d x \\ , d y , u , v \\in \\mathfrak D ( \\mathcal E ) , \\end{align*}"}
+{"id": "8177.png", "formula": "\\begin{align*} P _ { { \\underline X } _ 1 } ( 1 0 0 0 0 ) = P _ { { \\underline X } _ 1 } ( 0 1 0 0 0 ) = P _ { { \\underline X } _ 1 } ( 0 0 1 1 1 ) = \\frac { 3 } { 1 0 } , \\\\ P _ { { \\underline X } _ 1 } ( 0 0 0 0 0 ) = \\frac { 1 } { 1 0 } , \\end{align*}"}
+{"id": "6290.png", "formula": "\\begin{align*} f ( \\overline { x } _ T ) - f ( x ^ * ) - 2 M _ 2 \\tau \\leq \\frac 1 T \\sum \\limits _ { k = 0 } ^ { T - 1 } \\langle \\nabla \\hat { f } _ \\tau ( x _ { k } ) , x _ k - x ^ * \\rangle . \\end{align*}"}
+{"id": "5433.png", "formula": "\\begin{align*} g ^ { S _ 0 \\oplus S _ 1 } _ { ( 0 , i ) } & = 1 + \\beta \\sum _ { j \\in S _ 1 } p ( { i , j } ) g ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , j ) } + \\beta \\sum _ { j \\in N \\setminus S _ 1 } p _ { i j } g ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , j ) } = g ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , i ) } = g ^ { S _ 1 } _ i , \\end{align*}"}
+{"id": "1914.png", "formula": "\\begin{align*} \\big ( 1 - e ^ { \\xi ( y , t ) - \\xi ( x , t ) } \\big ) ^ 2 & \\leq | \\xi ( y , t ) - \\xi ( x , t ) | ^ 2 e ^ { 2 | \\xi ( y , t ) - \\xi ( x , t ) | } \\\\ & = \\frac { M ^ 2 } { ( \\lambda T - t ) ^ 2 } | \\rho ( y ) - \\rho ( x ) | ^ 2 e ^ { \\frac { 2 M } { \\lambda T - t } | \\rho ( y ) - \\rho ( x ) | } \\\\ & \\leq \\frac { \\beta ^ 2 R _ 0 ^ { 2 \\beta - 2 } M ^ 2 } { ( \\lambda T - t ) ^ 2 } e ^ { \\frac { 2 \\beta R _ 0 ^ { \\beta - 1 } M d ( x , y ) } { T ( \\lambda - 1 ) } } d ( x , y ) ^ 2 , \\end{align*}"}
+{"id": "1982.png", "formula": "\\begin{align*} \\| u \\| _ { H ^ m } ^ 2 = \\sum \\limits _ { l \\in \\mathbb { Z } } ( 1 + \\mu _ l ^ 2 ) ^ m | \\widehat { u } _ l | ^ 2 , \\ \\mathrm { f o r } \\ u ( x ) = \\sum \\limits _ { l \\in \\mathbb { Z } } \\widehat { u } _ l e ^ { i \\mu _ l ( x - a ) } , \\ \\mu _ l = \\frac { 2 \\pi l } { b - a } , \\end{align*}"}
+{"id": "3904.png", "formula": "\\begin{align*} \\sum _ { e ^ { \\ell ( \\gamma ) } \\leq T } e ^ { \\ell _ \\psi ( \\gamma ) } & = \\int _ 2 ^ T \\frac { 1 } { \\log ( \\tau ) } \\ , d \\hat { S } _ 0 ( \\tau ) + O ( 1 ) \\\\ & = \\frac { \\hat { S } _ 0 ( T ) } { \\log ( T ) } + \\int _ 2 ^ T \\frac { \\hat { S } _ 0 ( \\tau ) } { \\tau ( \\log ( \\tau ) ) ^ 2 } \\ , d \\tau + O ( 1 ) . \\end{align*}"}
+{"id": "931.png", "formula": "\\begin{align*} \\int _ D | f ( P _ D g ( x ) ) | \\delta ^ q ( x ) \\ , d x = \\infty , \\end{align*}"}
+{"id": "428.png", "formula": "\\begin{align*} \\begin{aligned} & \\| ( \\bold { I } _ { j - 1 } - \\bold { \\Pi } _ { j - 1 } - \\bold { \\Xi } _ { j - 1 } ( \\bold { I } _ { j - 1 } - \\bold { \\Pi } _ { j - 1 } ^ { T } ) ^ { - 1 } \\bold { \\Gamma } _ { j - 1 } ) ^ { - 1 } _ { ( m ) } \\| _ { 1 } \\\\ & \\| [ \\bold { S } _ { j } ^ { - 1 } ] ^ { ( 1 , 1 ) } _ { ( m ) } \\| _ { 1 } \\le \\| [ \\bold { S } _ { j } ^ { - 1 } ] ^ { ( m ) } \\| _ { 1 } < \\infty . \\end{aligned} \\end{align*}"}
+{"id": "5444.png", "formula": "\\begin{align*} r ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , i ) } & = R _ i + \\beta \\sum _ { j \\in N } p _ { i j } f ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , j ) } - \\beta f ^ { S _ 0 \\oplus S _ 1 } _ { ( 0 , i ) } = f _ i ^ { S _ 1 } = \\frac { r ^ { S _ 1 } _ i } { 1 - \\beta } , \\end{align*}"}
+{"id": "8117.png", "formula": "\\begin{align*} { \\mathcal I } ( w _ 0 ( I ) ) = \\Phi ^ + ( I ) , \\end{align*}"}
+{"id": "7809.png", "formula": "\\begin{align*} \\max _ { \\boldsymbol { \\rho } } & \\sum _ { t } R _ { t } ( \\boldsymbol { \\rho } ) + \\sum _ { r } R _ { r } ( \\boldsymbol { \\rho } ) , \\\\ & \\rho _ { m } ^ { \\mathrm { t } } , \\rho _ { m } ^ { \\mathrm { r } } \\in \\{ 0 , 1 \\} , \\forall m \\in \\mathcal { M } , \\\\ & \\rho _ { m } ^ { \\mathrm { t } } + \\rho _ { m } ^ { \\mathrm { r } } = 1 , \\forall m \\in \\mathcal { M } , \\end{align*}"}
+{"id": "6262.png", "formula": "\\begin{align*} g ( e ^ { i ( \\rho + \\theta ) } ) + g ( e ^ { i ( \\rho - \\theta ) } ) = 0 , a . e . ~ \\theta \\in [ 0 , 2 \\pi ] . \\end{align*}"}
+{"id": "4099.png", "formula": "\\begin{align*} \\rho ^ { ( r ) } _ j = \\lambda _ j / \\mu _ j ^ { ( r ) } = 1 - r ^ j / \\mu ^ { ( r ) } _ j \\to 1 \\end{align*}"}
+{"id": "4965.png", "formula": "\\begin{align*} T _ j ( \\lambda _ j ) = L _ { j N } ( \\lambda _ j , \\nu _ N ) \\cdots L _ { j 2 } ( \\lambda _ j , \\nu _ 2 ) L _ { j 1 } ( \\lambda _ j , \\nu _ 1 ) = \\begin{pmatrix} A ( \\lambda _ j ) & B ( \\lambda _ j ) \\\\ C ( \\lambda _ j ) & D ( \\lambda _ j ) \\\\ \\end{pmatrix} _ { [ \\mathcal { V } _ j ] } . \\end{align*}"}
+{"id": "4745.png", "formula": "\\begin{align*} J ^ A _ \\gamma = ( \\mathrm { I d } + \\gamma A ) ^ { - 1 } \\end{align*}"}
+{"id": "1298.png", "formula": "\\begin{align*} \\mathbf { z } ( t ) = e ^ { \\mathbf { Q } t } \\mathbf { z } _ 0 + \\int _ 0 ^ t e ^ { \\mathbf { Q } ( t - s ) } \\mathbf { D } ~ \\dot { \\mathbf { W } } d s \\ ; . \\end{align*}"}
+{"id": "4245.png", "formula": "\\begin{align*} & Q ( X , L , \\tau ) = \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } \\left [ \\bigotimes _ { n = 1 } ^ { \\infty } S _ { q ^ n } ( \\widetilde { T _ C M } ) \\otimes \\bigotimes _ { m = 1 } ^ { \\infty } \\wedge _ { - q ^ m } ( \\widetilde { V _ C } ) \\right ] \\right \\} ^ { ( 4 k + 2 ) } . \\end{align*}"}
+{"id": "7826.png", "formula": "\\begin{align*} \\theta _ m \\left ( i + 1 \\right ) = \\theta _ m ( i ) + \\tau ( i ) \\nabla _ { \\theta _ m } \\mathcal { H } \\left ( \\{ \\theta _ m \\} _ { m = 1 } ^ { M } \\right ) , \\end{align*}"}
+{"id": "5054.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathbb { R } ^ d } \\sum _ { i = 1 } ^ n f _ i ( x ) , \\end{align*}"}
+{"id": "8824.png", "formula": "\\begin{align*} 0 \\le \\eta _ 0 \\le 1 , \\eta _ 0 ( x ) = \\begin{cases} 1 & \\ ; | x | \\le 1 \\\\ 0 & \\ ; | x | \\ge \\frac { 3 } { 2 } \\end{cases} \\ , \\eta _ j ( x ) = \\eta _ 0 ( 2 ^ { - j } x ) - \\eta _ 0 ( 2 ^ { - j + 1 } x ) . \\end{align*}"}
+{"id": "360.png", "formula": "\\begin{align*} \\begin{aligned} z - y & = r ^ p , & & \\phi _ p ( z , y ) = r _ 1 ^ p , & & x = r r _ 1 , \\\\ z - x & = s ^ p , & & \\phi _ p ( z , x ) = s _ 1 ^ p , & & y = s s _ 1 , \\\\ x + y & = t ^ p , & & \\phi _ p ( x , - y ) = t _ 1 ^ p , & & z = t t _ 1 . \\end{aligned} \\end{align*}"}
+{"id": "6058.png", "formula": "\\begin{align*} m ( n , d , k ) = m ( n , k , k ) . \\end{align*}"}
+{"id": "3865.png", "formula": "\\begin{align*} \\begin{aligned} & b _ 1 \\doteq 4 + c , \\ ; \\ ; d _ 1 \\doteq e ^ c ( 1 2 + c ) , \\ ; \\ ; d _ 2 = 6 , \\ ; \\ ; d _ 3 \\doteq d _ 1 + l _ 0 b _ 1 e ^ c , \\ ; \\ ; d _ 4 \\doteq 2 ^ { l _ 0 } ( 3 + d _ 2 ) , \\ ; \\ ; \\delta _ 1 \\doteq \\delta \\delta ^ A _ 0 / 8 , \\\\ & A _ 1 \\doteq | \\log \\delta _ 1 | + ( 2 + d _ 3 ) \\delta _ 1 ^ { - 1 } , \\ ; \\ ; B _ 1 \\doteq 2 d _ 4 \\delta _ 1 ^ { - 1 } , \\ ; \\ ; C _ 1 \\doteq | \\log \\delta _ 1 | ( l _ 0 + 3 ) ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "2778.png", "formula": "\\begin{align*} \\delta \\left ( { \\sigma _ { \\tau } ^ { * } } ( t ) I \\right ) = \\int ^ { t _ { 2 } } _ { t _ { 1 } } \\left [ \\dot { Q } ^ { a } - \\frac { \\partial H _ { T } } { \\partial P _ { a } } \\right ] \\delta P _ { a } d t + \\left [ - \\frac { d } { d t } \\left ( \\frac { \\partial L _ { T } } { \\partial \\dot { Q } ^ { a } } \\right ) - \\frac { \\partial H _ { T } } { \\partial Q ^ { a } } \\right ] \\delta Q ^ { a } d t + \\left [ \\frac { \\partial L _ { T } } { \\partial \\dot { Q } ^ { a } } \\delta Q ^ { a } \\right ] ^ { t _ { 2 } } _ { t _ { 1 } } . \\end{align*}"}
+{"id": "8131.png", "formula": "\\begin{align*} [ D _ { q _ 1 , x } , D _ { q _ 2 , y } ] = 0 \\ , \\ [ X _ { q _ 1 , x } , X _ { q _ 2 , y } ] = 0 \\ , \\ [ D _ { q _ 1 ( q _ 2 ) , x ( y ) } , I ] = 0 \\ , \\ [ X _ { q _ 1 ( q _ 2 ) , x ( y ) } , I ] = 0 \\ , \\end{align*}"}
+{"id": "9158.png", "formula": "\\begin{align*} G _ { \\beta , \\beta ' } = ( w _ { \\beta , 1 } - v ^ { ( 4 \\ell - 4 n + 2 ) } w _ { \\beta ' , 1 } ) \\cdot G _ { \\beta , [ i , n , \\ell + 1 ] } . \\end{align*}"}
+{"id": "7645.png", "formula": "\\begin{align*} y _ { i } ^ { 1 } = y _ { i } ^ { } , y _ { i } ^ { 0 } = y _ { i } ^ { } \\end{align*}"}
+{"id": "6674.png", "formula": "\\begin{align*} L = \\langle y _ 1 , \\ldots , y _ l \\rangle \\le H \\end{align*}"}
+{"id": "107.png", "formula": "\\begin{align*} { \\rm d i v } \\big ( 2 \\mu \\varepsilon ( \\vec { u } ) + \\tilde { \\lambda } \\mathrm { t r } ( \\varepsilon ( \\vec { u } ) ) I \\big ) = \\tilde { \\vec { f } } . \\end{align*}"}
+{"id": "7655.png", "formula": "\\begin{align*} G _ 0 ( m , n ; z ) = \\langle \\delta _ m , ( A - z ) ^ { - 1 } \\delta _ n \\rangle . \\end{align*}"}
+{"id": "5591.png", "formula": "\\begin{align*} Q _ { e _ 1 e _ 3 , e _ 2 } = Q _ { e _ 1 e _ 2 } Q _ { e _ 3 e _ 2 } = m n M _ { e _ 1 e _ 3 , e _ 2 } ^ 2 . \\end{align*}"}
+{"id": "4337.png", "formula": "\\begin{gather*} p ^ * _ i = \\sup \\{ 0 , \\sup _ { u ^ i \\in \\mathcal { U } _ i } f _ i ( x ^ * , u ^ i ) - f _ i ( x ^ * , \\overline { u } ^ i ) ) - \\theta ^ * \\} \\ \\forall i \\in [ m ] \\end{gather*}"}
+{"id": "5421.png", "formula": "\\begin{align*} S _ { - n } f ( x ) = - S _ { n } f ( T ^ { - n } ( x ) ) , S _ { 2 n } f ( x ) = S _ { n } f ( x ) + S _ n f ( T ^ n ( x ) ) , \\end{align*}"}
+{"id": "3874.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum \\limits _ { k = 0 } ^ { n - 1 } R ^ { n , k } = \\frac { 1 } { n } \\sum \\limits _ { k = N _ 1 } ^ { k _ 3 - 1 } R ^ { n , k } = \\frac { 1 } { n } \\sum \\limits _ { k = N _ 1 } ^ { k _ 2 - 1 } R ^ { n , k } + \\frac { 1 } { n } \\sum \\limits _ { k = k _ 2 } ^ { k _ 3 - 1 } R ^ { n , k } \\le \\left | \\log \\left ( a _ 1 ^ * \\delta _ 0 ^ A \\right ) \\right | \\frac { k _ 1 } { n } + \\frac { 1 } { n } \\sum \\limits _ { k = k _ 2 } ^ { k _ 3 - 1 } R ^ { n , k } . \\end{align*}"}
+{"id": "3156.png", "formula": "\\begin{align*} u ( t ) = \\left ( \\begin{array} { c c c } 1 & a _ { 1 , 2 } ( t ) & a _ { 1 , 3 } ( t ) \\\\ 0 & 1 & a _ { 2 , 3 } ( t ) \\\\ 0 & 0 & 1 \\end{array} \\right ) ( \\ , a _ { 1 , 2 } ( T ) , a _ { 1 , 3 } ( T ) , a _ { 2 , 3 } ( T ) \\in k [ T ] \\ , ) . \\end{align*}"}
+{"id": "7657.png", "formula": "\\begin{align*} \\widehat { \\lambda } _ { s , \\mu } ( E ) = \\left ( D _ { s , 1 } \\sup _ { \\delta \\neq 0 } \\sup _ { u \\in \\mathbb { Z } ^ d } \\sum _ { v \\in \\mathbb { Z } ^ d } | G _ 0 ( u , v ; E + i \\delta ) | ^ s e ^ { \\mu | u - v | } \\right ) ^ \\frac { - 1 } { s } . \\end{align*}"}
+{"id": "184.png", "formula": "\\begin{align*} f ' _ i \\circ q _ { i + 1 } \\ = \\ q _ i \\circ f _ i , q _ i \\circ h _ i \\ = \\ h _ i ' . \\end{align*}"}
+{"id": "7347.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\mathbf M ^ { \\{ U ^ { \\{ Y , - k \\} } _ m , m \\} } _ { \\infty } ( \\phi ) = \\mathbf M ^ { \\{ U ^ { * } _ m , m \\} } _ { \\infty } ( \\phi ) = \\mathbf M ^ { * , m } _ { \\infty } ( \\phi ) . \\end{align*}"}
+{"id": "8445.png", "formula": "\\begin{align*} W ^ { s , p } ( \\mathbb { R } ^ n ) : = \\left \\{ v \\in L ^ p ( \\mathbb { R } ^ n ) \\ , : \\ , \\frac { | v ( x ) - v ( y ) | } { | x - y | ^ { \\frac { n } { p } + s } } \\in L ^ p ( \\mathbb { R } ^ n \\times \\mathbb { R } ^ n ) \\right \\} \\end{align*}"}
+{"id": "98.png", "formula": "\\begin{align*} u _ t - | D u | ^ { \\gamma } \\big ( \\Delta u + ( p - 2 ) \\Delta _ \\infty ^ N u \\big ) = 0 . \\end{align*}"}
+{"id": "8440.png", "formula": "\\begin{align*} \\theta _ n ( B _ n ) = \\max \\{ \\psi ( a _ n ) , \\psi ( a _ n + 1 ) \\} . \\end{align*}"}
+{"id": "4514.png", "formula": "\\begin{align*} V ( n , y _ j ) \\ll U ( j , n ) : = \\frac { 1 } { \\widetilde { y } _ j } \\bigg ( \\frac { \\widetilde { y } _ { j } } { \\widetilde { y } _ { 0 } } \\bigg ) ^ { - 1 / \\ell ^ K } \\sum _ { r = 0 } ^ { + \\infty } \\bigg | \\sum _ { \\substack { | \\lambda | = r \\\\ \\lambda _ 1 < g _ { j , n } ( r ) } } a ( \\lambda ) \\bigg | ^ 2 \\end{align*}"}
+{"id": "4128.png", "formula": "\\begin{align*} & \\phi ^ { ( r ) } ( \\theta ) - \\psi ^ { r } ( \\theta ) = o ( 1 ) , \\\\ & \\phi _ j ^ { ( r ) } ( \\theta ) - \\psi ^ { r } _ j ( \\theta ) = o ( 1 ) j \\in \\mathcal { J } . \\end{align*}"}
+{"id": "8503.png", "formula": "\\begin{align*} f ( a _ 1 ) = \\pi _ 1 ( f ( a _ 1 ) ) = \\pi _ 1 ( f ( u ) ) = \\pi _ 1 ( u ) = a _ 1 , \\end{align*}"}
+{"id": "2320.png", "formula": "\\begin{align*} \\mathcal { L } w = - \\Delta w + \\mathbb { P } ( ( w \\cdot \\nabla ) u ^ { c , \\gamma } ) + \\mathbb { P } ( ( u ^ { c , \\gamma } \\cdot \\nabla ) w ) , \\end{align*}"}
+{"id": "7355.png", "formula": "\\begin{align*} \\delta ^ h = e ^ { - 2 \\varphi } \\delta ^ g + ( 2 - n ) e ^ { - 2 \\varphi } \\iota _ { \\theta ^ { \\# _ g } } . \\end{align*}"}
+{"id": "2159.png", "formula": "\\begin{align*} \\sum _ { d \\in D } | A _ d | = n - 1 . \\end{align*}"}
+{"id": "7561.png", "formula": "\\begin{align*} i \\partial _ { t } \\psi - \\Delta ^ { 2 } \\psi + \\mu | \\psi | ^ { q - 2 } \\psi + | \\psi | ^ { p - 2 } \\psi = 0 , \\quad \\psi ( 0 , x ) = \\psi _ { 0 } \\in H ^ 2 ( \\mathbb { R } ^ N ) , \\quad ( t , x ) \\in \\mathbb { R } \\times \\mathbb { R } ^ { N } , \\end{align*}"}
+{"id": "1010.png", "formula": "\\begin{align*} V ^ D ( u , \\eta ) = \\int _ D \\eta \\ , d \\mu . \\end{align*}"}
+{"id": "592.png", "formula": "\\begin{align*} D ( h _ j ) = \\Big \\langle T \\bigg ( { \\dfrac { \\Tilde { h } _ j } { \\| h _ j \\| } _ F } \\bigg ) , { \\frac { \\Tilde { h } _ j } { \\| h _ j \\| _ { F ^ \\ast } } } \\Big \\rangle h _ j , \\ ; j \\in \\mathbb { N } \\end{align*}"}
+{"id": "3882.png", "formula": "\\begin{align*} M ( t ) ( x ) = m ( x ) - \\int _ 0 ^ t \\exp ( s ) \\hat \\gamma _ { ( 1 , 3 ) } ( \\{ x \\} \\times d s ) + \\int _ 0 ^ t M ( s ) ( x ) d s . \\end{align*}"}
+{"id": "8950.png", "formula": "\\begin{align*} \\| d ( \\nabla \\varphi , S O ( n ) ) ^ 2 \\| _ { L ^ { p ( \\cdot ) } ( Q ' ) } & \\le 2 \\| d ( \\nabla v , S O ( n ) ) \\| _ { L ^ { 2 p ( \\cdot ) } ( Q ' ) } ^ 2 + 2 \\| \\nabla \\psi \\| _ { L ^ { 2 p ( \\cdot ) } ( Q ' ) } ^ 2 \\\\ & \\le C \\| d ( \\nabla v , S O ( n ) ) \\| _ { L ^ { 2 p ( \\cdot ) } ( Q ' ) } ^ 2 = C \\| d ( \\nabla v , S O ( n ) ) ^ 2 \\| _ { L ^ { p ( \\cdot ) } ( Q ' ) } \\\\ & \\le C ( M + \\sqrt { n } ) \\| d ( \\nabla v , S O ( n ) ) \\| _ { L ^ { p ( \\cdot ) } ( Q ) } \\end{align*}"}
+{"id": "2487.png", "formula": "\\begin{align*} \\lambda _ { | I | } ( C ( A , \\lambda ) _ I ^ T C ( A , \\lambda ) _ I ) = \\sigma _ { | I | } ^ 2 ( C ( A , \\lambda ) _ I ) \\geq \\frac { 1 } { H ^ 2 } \\end{align*}"}
+{"id": "5101.png", "formula": "\\begin{align*} ( g _ { r _ { i _ 1 } } + 1 ) e _ { r _ { i _ 1 } } \\dots ( g _ { r _ { i _ k } } + 1 ) e _ { r _ { i _ k } } = ( g _ { r _ { i _ 1 } } + 1 ) \\dots ( g _ { r _ { i _ k } } + 1 ) e _ R \\in \\mathcal { C } ( v ) \\end{align*}"}
+{"id": "1371.png", "formula": "\\begin{align*} \\mu _ t = \\sum _ { i \\geqslant 0 } \\mu _ i t ^ i , { \\mu _ { V } } _ { t } = \\sum _ { i \\geqslant 0 } { \\mu _ { V } } _ { i } t ^ i , l _ t = \\sum _ { i \\geqslant 0 } l _ i t ^ i , r _ t = \\sum _ { i \\geqslant 0 } r _ i t ^ i , T _ t = \\sum _ { i \\geqslant 0 } T _ i t ^ i , \\end{align*}"}
+{"id": "7882.png", "formula": "\\begin{align*} d _ { i , k } = \\begin{cases} d _ { i - 3 , k } , & k \\in \\{ 1 , \\cdots , m \\} \\setminus \\{ s ' \\} ; \\\\ d _ { i - 3 , k } + e , & k = s ' . \\end{cases} \\end{align*}"}
+{"id": "400.png", "formula": "\\begin{align*} \\mathcal { D } _ { k } = \\left \\{ ( k _ x , k _ y ) \\in \\mathbb { R } ^ { 2 } | k _ x ^ 2 + k _ y ^ 2 \\le \\kappa ^ 2 \\right \\} . \\end{align*}"}
+{"id": "3504.png", "formula": "\\begin{align*} \\theta ( \\pi ( x ) ) c = \\pi ( x ) c = \\pi ( x ) \\pi ( d ) = \\pi ( x d ) = \\hat { \\pi } ( x ) \\pi ( d ) = \\hat { \\pi } ( x ) c , \\end{align*}"}
+{"id": "1470.png", "formula": "\\begin{align*} | \\tau ( Z ) | = | \\tau ( Z ) _ { F _ k M } | \\leq | \\Gamma _ { F _ k M } | < | A | ^ { | F _ k \\setminus S | } = | Z | . \\end{align*}"}
+{"id": "6137.png", "formula": "\\begin{align*} \\frac { | \\mathcal { F } _ n \\Delta s \\mathcal { F } _ n | } { | \\mathcal { F } _ n | } = 2 - 2 \\frac { | \\mathcal { F } _ n \\cap s \\mathcal { F } _ n | } { | \\mathcal { F } _ n | } \\xrightarrow [ n \\to \\infty ] { } 0 , \\end{align*}"}
+{"id": "3242.png", "formula": "\\begin{align*} K ^ 2 + b & = \\frac { 1 } { 4 x ^ 2 } \\ : \\big ( x ^ 4 - 2 b x ^ 2 + b ^ 2 + 4 b x ^ 2 \\big ) = \\frac { 1 } { 4 x ^ 2 } \\ : \\big ( x ^ 2 + b \\big ) ^ 2 = \\frac { 1 } { 4 x ^ 2 } \\ : \\big ( - a + b \\big ) ^ 2 \\\\ A & = \\frac { \\sqrt { K ^ 2 + b } } { x } = \\frac { - a + b } { 2 x ^ 2 } = \\frac { a - b } { 2 a } \\\\ B & = \\frac { K } { x } = \\frac { 1 } { 2 x ^ 2 } \\ : \\big ( x ^ 2 - b \\big ) = - \\frac { 1 } { 2 a } \\ : \\big ( - a - b \\big ) = \\frac { a + b } { 2 a } \\ : . \\end{align*}"}
+{"id": "8147.png", "formula": "\\begin{align*} Y _ { i , j } = { \\underline S } _ i ^ T { \\underline X } _ { i , j } \\end{align*}"}
+{"id": "4049.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 w } { \\partial t \\partial s } ( t , s ) = \\langle ( 1 , \\frac { \\partial } { \\partial t } \\mathbb { E } _ { \\mathbb { P } } [ S ( X ) | \\mathcal { F } _ t ] ( \\omega ) ) , ( 1 , \\frac { \\partial } { \\partial s } \\mathbb { E } _ { \\mathbb { Q } } [ S ( Y ) | \\mathcal { G } _ s ] ( \\omega ^ \\prime ) ) \\rangle _ { \\mathbb { R } \\oplus \\mathcal { H } ^ 1 } w ( t , s ) . \\end{align*}"}
+{"id": "6801.png", "formula": "\\begin{align*} \\pi ( \\eta _ 3 ) ^ { I _ + } = \\zeta \\otimes \\zeta + \\zeta ^ { - 1 } \\otimes \\zeta ^ { - 1 } . \\end{align*}"}
+{"id": "3827.png", "formula": "\\begin{align*} g _ 1 ( q ) : = \\frac { \\left ( q ^ 6 ; q ^ 6 \\right ) _ \\infty } { 4 \\left ( q ^ 2 ; q ^ 2 \\right ) _ \\infty \\left ( q ^ 3 ; q ^ 3 \\right ) _ \\infty } f ( q ) = : \\sum _ { n = 0 } ^ \\infty a ( n ) q ^ n , g _ 2 ( q ) : = \\frac { 3 \\left ( q ^ 3 ; q ^ 3 \\right ) ^ 3 _ \\infty } { 4 ( q ; q ) _ \\infty \\left ( q ^ 2 ; q ^ 2 \\right ) _ \\infty \\left ( q ^ 6 ; q ^ 6 \\right ) _ \\infty } . \\end{align*}"}
+{"id": "3908.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 0 0 & 1 1 1 & 2 2 2 & 2 2 2 & 1 1 1 & 0 0 0 \\\\ 0 1 2 & 2 1 0 & 0 1 2 & 2 1 0 & 0 1 2 & 2 1 0 \\end{bmatrix} . \\end{align*}"}
+{"id": "2157.png", "formula": "\\begin{align*} D : = \\{ a _ { i + 1 } - a _ i : 1 \\leq i \\leq n - 1 \\} \\end{align*}"}
+{"id": "9139.png", "formula": "\\begin{align*} \\zeta _ { i , j } ( \\mathsf { A } / \\mathsf { B } ) : = \\prod _ { x \\in \\mathsf { A } } ^ { y \\in \\mathsf { B } } \\zeta _ { i , j } ( x / y ) . \\end{align*}"}
+{"id": "5619.png", "formula": "\\begin{align*} \\| \\underline { B } ^ { ( k - 1 ) } \\tilde { H } B ^ { ( \\ell - k - 1 ) } \\| \\leq \\sum _ { i = 1 } ^ 3 \\| \\underline { B } ^ { ( k - 1 ) } M ^ { ( i ) } B ^ { ( \\ell - k - 1 ) } \\| \\leq \\sum _ { i = 1 } ^ 3 \\| \\underline { B } ^ { ( k - 1 ) } \\| \\| M ^ { ( i ) } B ^ { ( \\ell - k - 1 ) } \\| . \\end{align*}"}
+{"id": "8005.png", "formula": "\\begin{align*} J _ { d y n } ( W ) = \\sup _ { F , G , H \\in C ^ { 1 , \\infty } ( [ 0 , T ] \\times \\mathbb { T } ) } \\left \\{ \\mathcal { J } _ 1 ( W , F , G , H ) \\right \\} . \\end{align*}"}
+{"id": "4014.png", "formula": "\\begin{align*} x _ { 1 } ^ { \\left ( t _ { 0 } + 1 \\right ) } = \\gamma _ { 1 } x _ { 1 } ^ { \\left ( t _ { 0 } \\right ) } y ^ { \\left ( t _ { 0 } \\right ) } , x _ { 2 } ^ { \\left ( t _ { 0 } + 1 \\right ) } = \\gamma _ { 2 } x _ { 1 } ^ { \\left ( t _ { 0 } \\right ) } y ^ { \\left ( t _ { 0 } \\right ) } , y ^ { \\left ( t _ { 0 } + 1 \\right ) } = \\gamma x _ { 1 } ^ { \\left ( t _ { 0 } \\right ) } y ^ { \\left ( t _ { } \\right ) } . \\end{align*}"}
+{"id": "6505.png", "formula": "\\begin{align*} \\tilde f ( x _ 1 , \\cdots , x _ j ) : = \\frac { 1 } { j ! } \\sum _ { \\sigma \\in \\mathcal S _ j } f ( x _ { \\sigma ( 1 ) } , \\cdots , x _ { \\sigma ( j ) } ) , \\end{align*}"}
+{"id": "5219.png", "formula": "\\begin{align*} D G H I ( p \\| q ) = \\sum _ { j } p _ { j } \\left [ \\overline { M G } - \\overline { M H } \\right ] \\end{align*}"}
+{"id": "70.png", "formula": "\\begin{align*} \\begin{cases} N _ { 1 1 } & = c _ 2 - c _ 1 \\\\ N _ { 1 2 } = N _ { 2 1 } & = \\tfrac { 1 } { 2 } \\big ( c _ 3 P _ \\theta + ( c _ 3 + c _ 4 ) - ( 2 c _ 1 + c _ 2 ) \\big ) \\\\ N _ { 2 2 } & = ( c _ 3 + c _ 4 ) P _ \\theta - c _ 1 . \\end{cases} \\end{align*}"}
+{"id": "4359.png", "formula": "\\begin{align*} & \\max \\{ 0 , x _ i - \\overline { b } _ i , \\max \\{ 0 , x _ i - \\overline { b } _ i + \\Delta b _ i \\} - \\theta ^ k ( x ) \\} = \\\\ & \\max \\{ 0 , x _ i - \\overline { b } _ i , x _ i - \\overline { b } _ i + \\Delta b _ i - x _ k + \\overline { b } _ k - \\Delta b _ k \\} . \\end{align*}"}
+{"id": "2383.png", "formula": "\\begin{align*} x _ { I _ a \\cup I _ b } = x _ { I _ a } + x _ { I _ b } \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ u _ { I _ a , I _ b } = \\frac { x _ { I _ b } - x _ { I _ a } } { x _ { I _ a } + x _ { I _ b } } \\ . \\end{align*}"}
+{"id": "9155.png", "formula": "\\begin{align*} G _ { \\beta , \\beta ' } = ( w _ { \\beta , 1 } - v ^ { - 2 } w _ { \\beta ' , 1 } ) \\cdot G _ { \\beta , \\beta } , \\end{align*}"}
+{"id": "1361.png", "formula": "\\begin{align*} & \\Biggl ( \\sqrt { \\frac { \\alpha ^ 2 M _ 1 + ( 1 - \\alpha ) ^ 2 2 m - \\frac { 4 \\alpha ^ 2 m ^ 2 } { n } } { n } } + \\frac { 2 \\alpha m } { n } \\Biggr ) + ( n - 1 ) \\Biggl ( - \\sqrt { \\frac { \\alpha ^ 2 M _ 1 + ( 1 - \\alpha ) ^ 2 2 m - \\frac { 4 \\alpha ^ 2 m ^ 2 } { n } } { n } } + \\frac { 2 \\alpha m } { n } \\Biggr ) \\\\ & = \\sqrt { \\frac { \\alpha ^ 2 M _ 1 + ( 1 - \\alpha ) ^ 2 2 m - \\frac { 4 \\alpha ^ 2 m ^ 2 } { n } } { n } } ( 2 - n ) + 2 \\alpha m , \\end{align*}"}
+{"id": "48.png", "formula": "\\begin{align*} ( \\tilde U _ { j , h } \\omega ) _ { l } ( \\mu ) : = \\frac 1 { h ^ { { j } } } \\omega _ { I ^ j _ l } ( \\mu ) , 1 \\leq l \\leq \\binom { n } { j } . \\end{align*}"}
+{"id": "7605.png", "formula": "\\begin{align*} \\| \\Delta u _ { \\epsilon } \\| _ { 2 } ^ { 2 } & = R ^ { 2 } ( 4 - N ) ^ { 2 } \\omega \\int _ { 0 } ^ { \\frac { 2 } { \\epsilon } } \\big [ \\frac { 4 r ^ { N - 1 } } { ( 1 + r ^ { 2 } ) ^ { N - 2 } } + \\frac { ( 2 - N ) ^ { 2 } r ^ { N + 3 } } { ( 1 + r ^ { 2 } ) ^ { N } } + \\frac { 4 ( 2 - N ) r ^ { N + 1 } } { ( 1 + r ^ { 2 } ) ^ { N } } \\big ] d r + O ( \\epsilon ^ { N - 4 } ) \\\\ & = \\| \\Delta U _ { \\epsilon } \\| _ { 2 } ^ { 2 } + O ( \\epsilon ^ { N - 4 } ) . \\\\ \\end{align*}"}
+{"id": "3417.png", "formula": "\\begin{align*} \\rho ( e _ 0 ) = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} , \\ ; \\rho ( e _ 1 ) = \\begin{pmatrix} 0 & i \\\\ i & 0 \\end{pmatrix} , \\ ; \\rho ( e _ 1 ) = \\begin{pmatrix} 0 & j \\\\ j & 0 \\end{pmatrix} , \\ ; \\rho ( e _ 1 ) = \\begin{pmatrix} 0 & k \\\\ k & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "4007.png", "formula": "\\begin{align*} x ^ { \\left ( t \\right ) } = \\frac { 1 } { 1 - \\gamma } \\left [ \\gamma \\left ( 1 - \\gamma \\right ) x ^ { \\left ( 0 \\right ) } y ^ { \\left ( 0 \\right ) } \\right ] ^ { 2 ^ { t } } \\mbox { a n d } y ^ { \\left ( t \\right ) } = \\frac { 1 } { \\gamma } \\left [ \\gamma \\left ( 1 - \\gamma \\right ) x ^ { \\left ( 0 \\right ) } y ^ { \\left ( 0 \\right ) } \\right ] ^ { 2 ^ { t } } , \\end{align*}"}
+{"id": "3688.png", "formula": "\\begin{align*} a _ d ( S _ { n , d } ) = 1 . \\end{align*}"}
+{"id": "6166.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau ^ k } [ f ( x ) - f ( \\breve { x } ^ { k } ) ] - \\frac { 1 } { \\tau ^ { k - 1 } } [ f ( x ) - f ( \\breve { x } ^ { k - 1 } ) ] + ( u - \\widetilde { u } ^ k ) ^ T F ( \\widetilde { u } ^ k ) \\\\ + & ( 2 - \\gamma ) ( 1 - \\tau ^ k ) \\beta ^ k ( A ( x - \\widetilde { x } ^ k ) ) ^ T ( A \\breve { x } ^ { k - 1 } - b ) \\geq ( v - \\widetilde { v } ^ k ) Q ^ k ( v ^ k - \\widetilde { v } ^ k ) , ~ \\forall u , \\end{aligned} \\end{align*}"}
+{"id": "7667.png", "formula": "\\begin{align*} r _ L ( u , v ; t ) : = G _ L ( u , v ; t - i \\eta ) G _ L ( v , u ; t - i \\eta ) - G _ L ( u , v ; t + i \\eta ) G _ L ( v , u ; t + i \\eta ) . \\end{align*}"}
+{"id": "3850.png", "formula": "\\begin{align*} n ^ { - 1 } \\int _ { t _ l } ^ { t _ { m + 1 } } \\psi _ e ( s ) \\left ( \\bar \\Lambda ^ n ( e _ x \\mid s ) - \\bar \\xi ^ n _ { ( 2 ) } ( e _ x \\mid s ) \\right ) d s & = n ^ { - 1 } \\sum \\limits _ { k = l } ^ { m } \\left ( \\delta _ { \\bar \\nu ^ { n , k + 1 } } ( e _ x ) - \\bar \\mu ^ { n , k + 1 } ( e _ x ) \\right ) \\end{align*}"}
+{"id": "3613.png", "formula": "\\begin{align*} b ( \\sigma \\sigma _ 1 - 1 ) = ( a + b ) \\alpha \\tau _ 0 ( \\sigma ^ 2 \\sigma _ 1 - \\sigma ) . \\end{align*}"}
+{"id": "7812.png", "formula": "\\begin{align*} \\Phi ( \\boldsymbol { x } ) = - \\sum _ { c = 1 } ^ { n } \\ln x _ c - \\sum _ { c = 1 } ^ { n } \\ln ( 1 - x _ c ) . \\end{align*}"}
+{"id": "6651.png", "formula": "\\begin{align*} L ( E , \\mathrm { i } y ) = \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { 2 | n | } \\log ( | f _ { n } ( E , x + \\mathrm { i } y ) | ^ { 2 } + | f _ { n - 1 } ( E , x + \\mathrm { i } y ) | ^ { 2 } ) , \\ x \\in \\mathbb { T } ^ { d } . \\end{align*}"}
+{"id": "7862.png", "formula": "\\begin{align*} \\mathbf { \\Phi } = [ \\mathbf { \\Phi } _ { \\mathbf { A } _ { 1 } } , \\mathbf { \\Phi } _ { \\mathbf { A } _ { 2 } } , \\ldots , \\mathbf { \\Phi } _ { \\mathbf { A } _ { L } } ] . \\end{align*}"}
+{"id": "1781.png", "formula": "\\begin{align*} \\Re \\big ( w + h ^ 2 ( P + i v , z , \\zeta ) \\big ) = P ( z _ 1 + f _ { 1 } ^ 1 , \\ldots , z _ { n - 1 } + f _ { n - 1 } ^ 1 , \\zeta + g _ 0 ) , \\end{align*}"}
+{"id": "1193.png", "formula": "\\begin{align*} | \\partial _ x ^ \\alpha \\mathcal K ( x + h , y ) - \\partial _ x ^ \\alpha \\mathcal K ( x , y ) | & = \\left | \\int _ 0 ^ 1 h \\cdot \\nabla _ x \\partial _ x ^ \\alpha \\mathcal K ( x + t h , y ) \\ , d t \\right | \\\\ & \\lesssim | h | \\sup _ { t \\in ( 0 , 1 ) } | x + t h - y | ^ { - n - | \\alpha | - 1 } \\sim | h | \\ , | x - y | ^ { - n - | \\alpha | - 1 } , \\end{align*}"}
+{"id": "5358.png", "formula": "\\begin{align*} \\mathbf { h } ^ 0 - ( \\mathbf { I } - \\beta \\ , \\mathbf { P } ^ 0 ) \\ , \\mathbf { v } ^ S & = \\left [ \\begin{array} { c } \\mathbf { c } _ S ^ S \\\\ \\mathbf { 0 } _ { N \\setminus S } \\end{array} \\right ] \\\\ ( \\mathbf { I } - \\beta \\ , \\mathbf { P } ^ 1 ) \\ , \\mathbf { v } ^ S - \\mathbf { h } ^ 1 & = \\left [ \\begin{array} { c } \\mathbf { 0 } _ { S } \\\\ \\mathbf { c } _ { N \\setminus S } ^ S \\end{array} \\right ] . \\end{align*}"}
+{"id": "4658.png", "formula": "\\begin{align*} L ( s , \\chi ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { \\chi ( n ) } { n ^ s } , \\end{align*}"}
+{"id": "5621.png", "formula": "\\begin{align*} X = \\overline { \\bigcup _ { i = 0 } ^ \\infty \\bigcap _ { n = i } ^ \\infty T ^ n ( U ) } . \\end{align*}"}
+{"id": "8393.png", "formula": "\\begin{align*} a \\circ ( b c ) = ( a _ 1 \\circ b ) S ( \\sigma ( a _ 2 ) ) ( a _ 3 \\circ c ) , \\quad \\forall a , b , c \\in H . \\end{align*}"}
+{"id": "7390.png", "formula": "\\begin{align*} \\frac { r ( q - 1 ) + p } { q } - \\frac { r ( r - 1 ) } { r + 1 } = \\frac { r ( p + q - r ) + q r + p - r } { q ( r + 1 ) } > 0 \\end{align*}"}
+{"id": "8469.png", "formula": "\\begin{align*} & ( u _ h ) _ + ( x , t ) - ( u _ h ) _ + ^ { ( \\ell ) } ( x , t ) \\\\ & = \\dfrac { t - t _ { m - 1 } } { h } \\left ( ( u _ m ) _ + ( x ) - ( u _ m ) _ + ^ { ( \\ell ) } ( x ) \\right ) + \\dfrac { t _ m - t } { h } \\left ( ( u _ { m - 1 } ) _ + ( x ) - ( u _ { m - 1 } ) _ + ^ { ( \\ell ) } ( x ) \\right ) \\end{align*}"}
+{"id": "593.png", "formula": "\\begin{align*} A ( h _ j ) = \\Tilde { h } _ j , \\ ; j \\in \\mathbb { N } , \\end{align*}"}
+{"id": "6067.png", "formula": "\\begin{align*} x _ l \\frac { M } { x _ i } = \\frac { x _ l } { x _ { i + 1 } } \\underbrace { \\left ( x _ { i + 1 } \\frac { M } { x _ i } \\right ) } _ { \\in U } \\in U . \\end{align*}"}
+{"id": "9061.png", "formula": "\\begin{align*} I \\cap J = \\emptyset \\ , \\ , \\ , \\ , \\ , \\ , \\mbox { i f } \\ , \\ , I , J \\in \\mathcal { A } _ i ( N ) \\ , \\ , \\mbox { a n d } \\ , \\ , I \\neq J \\end{align*}"}
+{"id": "743.png", "formula": "\\begin{align*} \\overset { \\ _ * } { R } ^ { i } _ { j k m } = R ^ i _ { j k m } + R ^ { s } _ { k m } C ^ { i } _ { s j } , \\end{align*}"}
+{"id": "4833.png", "formula": "\\begin{align*} \\mu ( A ' ) = \\frac { 1 } { 2 } ( < A > - < A ' > ) = - \\mu ( A ) . \\end{align*}"}
+{"id": "5401.png", "formula": "\\begin{align*} \\nu _ k ( j _ k ) & = \\frac { c _ { k } } { \\mu _ { k } } \\ , \\left [ \\left ( \\frac { 2 j _ { k } + 3 } { ( 1 - \\rho _ k ) ^ 2 } - \\frac { 2 } { ( 1 - \\rho _ { k } ) ^ 3 } \\right ) \\rho _ { k } ^ { - j _ k - 1 } \\right . \\\\ & \\left . - \\frac { ( j _ { k } + 1 ) ^ { 2 } } { 1 - \\rho _ k } - \\frac { 1 } { ( 1 - \\rho _ k ) ^ 2 } + \\frac { 2 } { ( 1 - \\rho _ { k } ) ^ 3 } \\right ] - r _ k - s _ k . \\end{align*}"}
+{"id": "7450.png", "formula": "\\begin{align*} \\left [ \\left ( \\bigotimes _ { j = 1 } ^ n A _ j \\right ) \\cdot B \\right ] ^ { i _ 1 , \\ldots i _ n } = \\sum _ { j _ 1 = 1 } ^ { d } \\cdots \\sum _ { j _ n = 1 } ^ { d } \\left ( A _ 1 \\right ) ^ { i _ 1 } _ { j _ 1 } \\left ( A _ 2 \\right ) ^ { i _ 2 } _ { j _ 2 } \\cdots \\left ( A _ n \\right ) ^ { i _ n } _ { j _ n } B ^ { j _ 1 , \\ldots , j _ n } \\end{align*}"}
+{"id": "1831.png", "formula": "\\begin{align*} T X : = \\bigsqcup _ { \\alpha } ( T \\tilde { U } _ { \\alpha } / G _ { \\alpha } ) / \\sim , \\end{align*}"}
+{"id": "3919.png", "formula": "\\begin{align*} \\normalsize \\cos N z = \\cos ^ N z - \\binom { N } { 2 } \\cos ^ { N - 2 } z \\sin ^ 2 z + \\binom { N } { 4 } \\cos ^ { N - 4 } z \\sin ^ 4 z - \\ldots + \\sin ^ N z . \\end{align*}"}
+{"id": "2812.png", "formula": "\\begin{align*} \\phi ^ { ( 1 ) } _ { 1 } : = p _ { 1 } + q ^ { 2 } : \\approx 0 , \\phi ^ { ( 1 ) } _ { 2 } : = p _ { 2 } - q ^ { 1 } : \\approx 0 . \\end{align*}"}
+{"id": "611.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\left ( \\frac { 1 } { 4 } \\right ) ^ n \\binom { 2 n } { n } \\frac { f _ { n } } { 4 n + 1 } \\ \\ \\ \\ \\ \\ \\sum _ { n = 0 } ^ { \\infty } \\left ( \\frac { 1 } { 4 } \\right ) ^ n \\binom { 2 n } { n } \\frac { f _ { n } } { 4 n + 3 } . \\end{align*}"}
+{"id": "8591.png", "formula": "\\begin{align*} \\mathcal { A } _ { X , N } = \\{ X \\cup Z : Z \\subset \\Lambda _ \\ell , | X | + | Z | = N \\} \\ : . \\end{align*}"}
+{"id": "5853.png", "formula": "\\begin{align*} \\dfrac { M _ { 1 } ( \\mathcal { N C } ( D _ { 2 m } ) ) } { | v ( \\mathcal { N C } ( D _ { 2 m } ) ) | } = \\dfrac { m ( m - 1 ) ( 5 m - 4 ) } { 2 m - 1 } \\dfrac { M _ { 2 } ( \\mathcal { N C } ( D _ { 2 m } ) ) } { | e ( \\mathcal { N C } ( D _ { 2 m } ) ) | } = \\dfrac { m ( m - 1 ) ( 8 m ^ { 2 } - 1 2 m + 4 ) } { 3 m ( m - 1 ) } . \\end{align*}"}
+{"id": "4705.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\alpha _ { n } ( g ) ( 0 ) = \\beta _ { n } ( g ) ( 0 ) \\\\ \\alpha _ { n } ( g ) ( t ) \\leq \\alpha _ { n } ( g ) ( 0 ) + k \\leq \\alpha _ n ( h ) ( t ) \\\\ \\beta _ n ( g ) ( t ) \\leq \\beta _ n ( g ) ( 0 ) + k \\leq \\beta _ n ( h ) ( t ) \\end{array} \\right . \\end{align*}"}
+{"id": "7208.png", "formula": "\\begin{align*} \\top & = \\delta _ { \\max L } ( c \\to a ) \\leq \\delta _ { \\max L } ( c ) \\to \\delta _ { \\max L } ( a ) \\\\ & \\implies c \\leq \\delta _ { \\max L } ( c ) \\leq \\delta _ { \\max L } ( a ) . \\end{align*}"}
+{"id": "668.png", "formula": "\\begin{align*} \\begin{aligned} & ( R ^ { k } f ) _ { p _ { 1 } q _ { 1 } \\cdots p _ { m - k } q _ { m - k } } ^ { i _ { 1 } \\cdots i _ { k } } \\\\ & = \\frac { 1 } { m - k + 1 } \\binom { m } { k } \\alpha ( p _ { 1 } q _ { 1 } ) \\cdots \\alpha ( p _ { m - k } q _ { m - k } ) ( W ^ { k } f ) _ { p _ { 1 } \\cdots p _ { m - k } q _ { 1 } \\cdots q _ { m - k } } ^ { i _ { 1 } \\cdots i _ { k } } , \\end{aligned} \\end{align*}"}
+{"id": "1188.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ n } x ^ \\gamma T a ( x ) \\ , d x & = \\lim _ { j \\to \\infty } \\int _ { \\mathbb R ^ n } \\phi _ j ( x ) x ^ \\gamma T a ( x ) \\ , d x = \\lim _ { j \\to \\infty } \\langle \\phi _ j x ^ \\gamma , T a \\rangle \\\\ & = \\lim _ { j \\to \\infty } \\langle T ^ * ( \\phi _ j x ^ \\gamma ) , a \\rangle = \\langle T ^ * ( x ^ \\gamma ) , a \\rangle = 0 \\end{align*}"}
+{"id": "4508.png", "formula": "\\begin{align*} \\widetilde { V } ( n ) : = \\sum _ { \\substack { 1 \\leqslant j \\leqslant J \\\\ \\frac { n } { y _ { j } } > \\ell ^ { 1 0 0 K } } } \\sum _ { y _ { j - 1 } < k \\leqslant y _ j } \\frac { 1 } { k } \\bigg | \\sum _ { \\substack { | \\lambda | = n - k \\\\ \\lambda _ 1 < k } } a ( \\lambda ) \\bigg | ^ 2 \\end{align*}"}
+{"id": "6870.png", "formula": "\\begin{align*} \\{ ( E _ j - \\mu ) c _ j \\} \\in \\ell _ 2 ( \\mathbb { J } ) \\ominus { \\mathcal R } ( \\theta _ \\varphi ) = \\mathcal { R } ( \\theta _ \\psi ) . \\end{align*}"}
+{"id": "2233.png", "formula": "\\begin{align*} u _ 1 = { v _ \\varphi \\over r } , \\ \\ \\omega _ 1 = { \\omega _ \\varphi \\over r } , \\ \\ \\psi _ 1 = { \\psi \\over r } , \\ \\ f _ 1 = { f _ \\varphi \\over r } , \\ \\ F _ 1 = { F _ \\varphi \\over r } . \\end{align*}"}
+{"id": "4955.png", "formula": "\\begin{align*} \\mbox { t r } \\frac { d } { d t } \\Big | _ { t = 0 } D T _ t ( x ) = \\mbox { t r } D \\tilde K ( x ) & = \\sum _ { i = 1 } ^ { n - 1 } \\langle \\nabla e _ i \\tilde K ( x ) , e _ i \\rangle \\\\ & = - \\langle { \\bf H } ( x ) , \\tilde K ( x ) \\rangle = - H _ { \\tilde K ( x ) } ( x ) \\end{align*}"}
+{"id": "4279.png", "formula": "\\begin{align*} \\Delta _ f \\hat v _ h + \\frac { \\hat v _ h } { 2 } = \\div _ f \\div _ f h , \\qquad \\int _ M \\hat v _ h e ^ { - f } \\ , d V = 0 . \\end{align*}"}
+{"id": "5589.png", "formula": "\\begin{align*} B ^ k _ { e f } = \\sum _ { \\gamma \\in \\Gamma _ { e f } ^ { 2 k + 2 } } X _ { e _ 1 e _ 2 } X _ { e _ 3 e _ 2 } \\prod _ { s = 1 } ^ { k } A _ { \\gamma _ { 2 s } \\gamma _ { 2 s + 2 } , \\gamma _ { 2 s + 1 } } . \\end{align*}"}
+{"id": "1782.png", "formula": "\\begin{align*} h ^ 2 = ( e ^ { 2 a _ 1 + n a _ 2 } - 1 ) w + i a _ 3 e ^ { a _ 1 + a _ 2 } z _ 1 ^ 2 , g ^ 0 = ( e ^ { a _ 2 } - 1 ) \\zeta , \\end{align*}"}
+{"id": "1863.png", "formula": "\\begin{align*} H ^ 0 ( E _ A ( X ) ) = H ^ 3 ( E _ A ( X ) ) = 0 H ^ 1 ( E _ A ( X ) ) \\simeq H ^ 2 ( E _ A ( X ) ) . \\end{align*}"}
+{"id": "3477.png", "formula": "\\begin{align*} \\mathcal { L } u = f \\end{align*}"}
+{"id": "6719.png", "formula": "\\begin{align*} y _ 0 ( x ) = \\sum _ { j = 0 } ^ { n - 1 } b _ j ( x ) \\omega _ j ^ x \\end{align*}"}
+{"id": "1140.png", "formula": "\\begin{align*} \\infty & > \\| B t \\| _ { \\dot b ^ s _ { \\infty , q } } ^ q \\sim \\sum _ { j = 0 } ^ { \\infty } 2 ^ { - j ( E - s - \\frac { n } { 2 } ) q } + \\sum _ { j = - \\infty } ^ { - 1 } 2 ^ { j ( F + s - \\frac { n } { 2 } ) q } , \\end{align*}"}
+{"id": "6830.png", "formula": "\\begin{align*} \\int _ { S O ( N ) } h ( g ) ^ P \\ , d g = \\frac { 1 } { i ^ { N - 1 } } & \\int _ { [ - 1 , 1 ] ^ { \\frac { ( N - 1 ) ( N - 2 ) } { 2 } } } \\int _ { ( S ^ 1 ) ^ { N - 1 } } \\left [ \\widetilde { h _ { S O ( N ) } } ( x _ 1 , \\ldots , z _ { n - 1 } ) \\right ] ^ P \\\\ & \\cdot J _ { S O ( N ) } ( x _ 1 , \\ldots , x _ { \\frac { ( N - 1 ) ( N - 2 ) } { 2 } } ) \\frac { d z _ 1 } { z _ 1 } \\ldots \\frac { d z _ { N - 1 } } { z _ { N - 1 } } d x _ 1 \\ldots d x _ { \\frac { ( N - 1 ) ( N - 2 ) } { 2 } } . \\end{align*}"}
+{"id": "6237.png", "formula": "\\begin{align*} \\frac { D ( \\phi ) g ( \\phi ) } { \\phi - \\alpha } & \\le \\frac 3 4 ( D _ i - D _ g ) ( r _ i + \\lambda _ g ) ( \\phi - \\gamma ) ( \\phi - \\beta ) \\le \\frac 3 4 ( D _ i - D _ g ) ( r _ i + \\lambda _ g ) \\gamma \\beta \\\\ & = \\frac 1 4 r _ i ( D _ i - D _ g ) ( 2 + \\omega ) . \\end{align*}"}
+{"id": "9172.png", "formula": "\\begin{align*} \\begin{aligned} & \\tilde { E } ^ { - } _ { [ i , j ] } \\coloneqq [ \\cdots [ [ E _ { i } , E _ { i + 1 } ] _ { v ^ { - 2 } } , E _ { i + 2 } ] _ { v ^ { - 2 } } , \\cdots , E _ { j } ] _ { v ^ { - 2 } } , \\\\ & \\tilde { E } ^ { - } _ { [ i , n , j ] } \\coloneqq [ \\cdots [ [ [ \\cdots [ E _ { i } , E _ { i + 1 } ] _ { v ^ { - 2 } } , \\cdots , E _ { n } ] _ { v ^ { - 2 } } , E _ { n } ] , E _ { n - 1 } ] _ { v ^ { - 2 } } , \\cdots , E _ { j } ] _ { v ^ { - 2 } } . \\end{aligned} \\end{align*}"}
+{"id": "7682.png", "formula": "\\begin{align*} \\mu _ d = \\lim _ { N \\to \\infty } ( C _ N ) ^ { \\frac { 1 } { N } } . \\end{align*}"}
+{"id": "190.png", "formula": "\\begin{align*} R ' | X \\ = \\ R , R '^ { \\circ } ( u ) \\ = \\ F , \\ \\ R '^ { \\circ - 1 } ( u ) \\ = \\ E . \\end{align*}"}
+{"id": "728.png", "formula": "\\begin{align*} H ( f ) & = \\left \\{ \\begin{bmatrix} 1 & a & c \\\\ & 1 & b \\\\ & & 1 \\end{bmatrix} ~ \\middle | ~ a , b , c \\in \\mathbb { F } _ p [ t ] / ( f ( t ) ) \\right \\} . \\end{align*}"}
+{"id": "8997.png", "formula": "\\begin{align*} 0 \\leq \\varliminf _ { n _ i \\to \\infty } A _ s ^ { n _ i , \\infty } ( \\omega , \\theta _ t \\omega ' ) = \\varliminf _ { n _ i \\to \\infty } ( A _ { s + t } ^ { n _ i , \\infty } ( \\omega , \\omega ' ) - A _ t ^ { n _ i , \\infty } ( \\omega , \\omega ' ) ) = \\tilde { A } _ { t , s + t } ( \\omega , \\omega ' ) . \\end{align*}"}
+{"id": "5762.png", "formula": "\\begin{align*} \\sum _ { k + \\ell \\leq s } \\left [ | z ^ { ( k , \\ell ) } ( t ) | ^ 2 + | \\bar { z } ^ { ( k , \\ell ) } ( t ) | ^ 2 + \\sum _ { i \\in I _ 1 } \\left ( | \\xi ^ { ( k , \\ell ) } _ { i , 1 } ( t ) | ^ 2 + | \\xi ^ { ( k , \\ell ) } _ { i , 2 } ( t ) | ^ 2 \\right ) + \\right . & \\left . \\sum _ { i \\neq 0 } | \\xi ^ { ( k , \\ell ) } _ i ( t ) | ^ 2 \\right ] \\\\ & = ( 1 + o ( 1 ) ) | z ( t ) | ^ 2 . \\end{align*}"}
+{"id": "6894.png", "formula": "\\begin{align*} \\eta _ { 0 } & = \\begin{cases} \\dfrac { 2 } { s ( s + 2 ) } ( C h i ( 1 ) - \\xi ) & 0 < s \\leq 2 \\kappa , \\\\ & \\\\ \\dfrac { 2 } { s ( s - 2 ) } ( C i ( 1 ) - \\xi ) & 2 \\kappa < s \\leq 1 / \\sqrt { 2 } . \\end{cases} \\end{align*}"}
+{"id": "3063.png", "formula": "\\begin{align*} \\varphi ( t ) = \\left ( t ^ { n } , \\sum _ { k \\geq \\beta _ 1 } c _ k t ^ k \\right ) , \\end{align*}"}
+{"id": "5300.png", "formula": "\\begin{align*} R _ { 1 2 } ( u - v ) K _ { 1 } ( u ) R _ { 2 1 } ( u + v ) K _ { 2 } ( v ) = K _ { 2 } ( v ) R _ { 1 2 } ( u + v ) K _ { 1 } ( u ) R _ { 2 1 } ( u - v ) . \\end{align*}"}
+{"id": "9037.png", "formula": "\\begin{align*} \\lambda _ \\alpha \\lambda _ \\beta - \\lambda _ \\beta \\lambda _ \\alpha = 0 , \\lambda _ \\alpha \\theta _ \\beta ^ i - \\theta _ \\beta ^ i \\lambda _ \\alpha = 0 , \\theta _ \\alpha ^ i \\theta _ \\beta ^ j + \\theta _ \\beta ^ j \\theta _ \\alpha ^ i = 0 ( \\alpha , \\beta \\in A , \\ i , j \\in [ N ] ) . \\end{align*}"}
+{"id": "243.png", "formula": "\\begin{align*} \\bar { A } ( \\bar { x } ) = \\frac { 1 } { h ( x ) } A ( x ) \\qquad \\bar { b } ( \\bar { x } ) = \\frac { c } { h ( x ) } b ( x ) , \\end{align*}"}
+{"id": "7003.png", "formula": "\\begin{align*} \\Lambda = \\bigcup _ { p \\in \\partial \\mathcal { H } _ \\mathbb { R } ^ m } H _ p . \\end{align*}"}
+{"id": "2745.png", "formula": "\\begin{align*} & X _ { \\Xi _ { 1 } } \\Theta = 0 , \\\\ & X _ { \\Psi _ { 1 } } \\Theta = 0 , \\\\ \\end{align*}"}
+{"id": "5440.png", "formula": "\\begin{align*} f ^ { S _ 0 \\oplus S _ 1 } _ { ( 0 , i ) } & = R _ i - c _ i + \\beta \\sum _ { j \\in S _ 1 } p _ { i j } f ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , j ) } + \\beta \\sum _ { j \\in N \\setminus S _ 1 } p _ { i j } f ^ { S _ 0 \\oplus S _ 1 } ( 1 , j ) = f ^ { S _ 0 \\oplus S _ 1 } _ { ( 1 , i ) } - c _ i = f ^ { S _ 1 } _ i - c _ i , \\end{align*}"}
+{"id": "2168.png", "formula": "\\begin{align*} A _ d ''' : = A _ d '' \\cap I _ i \\end{align*}"}
+{"id": "4499.png", "formula": "\\begin{align*} X _ 0 X _ 4 + X _ 1 X _ 5 + X _ 2 X _ 6 + X _ 3 X _ 7 = 0 \\end{align*}"}
+{"id": "4661.png", "formula": "\\begin{align*} & 2 \\int _ { \\Sigma _ { \\epsilon , \\beta , O } } | \\nabla \\psi | ^ 2 \\ , \\mathrm d \\mathcal H ^ 2 _ g \\geq \\\\ & \\int _ { \\Sigma _ { \\epsilon , \\beta , O } } \\left ( R ( g ) - R _ { \\epsilon , \\beta , O } + H _ { \\epsilon , \\beta , O } ^ 2 + | A _ { \\epsilon , \\beta , O } | ^ 2 + 2 \\partial _ \\nu ( h _ { \\epsilon , \\beta } \\circ \\rho ) \\right ) \\psi ^ 2 \\ , \\mathrm d \\mathcal H ^ 2 _ g \\end{align*}"}
+{"id": "2441.png", "formula": "\\begin{align*} u ^ { * } ( \\xi ) = \\begin{cases} u ( \\xi ) , & \\xi \\in ( \\xi _ { - } , \\xi _ { + } ) , \\\\ 0 , & e l s e \\end{cases} \\end{align*}"}
+{"id": "4435.png", "formula": "\\begin{gather*} \\psi ^ j = \\frac { 1 - 2 \\gamma } { 2 \\gamma } k ^ j _ 1 - \\frac { 1 } { 2 \\gamma } k ^ j _ 2 = \\frac { k ^ j _ 1 - k ^ j _ 2 } { 2 \\gamma } - k ^ j _ 1 \\ , . \\end{gather*}"}
+{"id": "6844.png", "formula": "\\begin{align*} d g = d g _ K \\ , d k = d g _ { K } \\ ; d g _ { S U ( N - 1 ) } \\ , d \\omega _ { N - 1 } . \\end{align*}"}
+{"id": "952.png", "formula": "\\begin{align*} \\Pi _ V ( g ) ( x ) = g ( x ) - P _ V ( g ) ( x ) , x \\in E \\setminus N . \\end{align*}"}
+{"id": "472.png", "formula": "\\begin{align*} \\lambda ( g _ 1 ) \\cdots \\lambda ( g _ k ) & = \\sum _ { \\substack { A _ 1 \\ni g _ 1 ^ { - 1 } \\\\ \\vdots \\\\ A _ k \\ni g _ k ^ { - 1 } } } ( A _ 1 , g _ 1 ) \\cdots ( A _ k , g _ k ) = \\sum _ { \\substack { A _ k \\ni g _ k ^ { - 1 } \\\\ g _ { k } A _ k \\ni g _ { k - 1 } ^ { - 1 } \\\\ \\vdots \\\\ g _ 2 \\cdots g _ k A _ k \\ni g _ 1 ^ { - 1 } } } ( A _ 1 , g _ 1 \\cdots g _ k ) = \\sum _ { A \\supseteq B } ( A , h ) . \\end{align*}"}
+{"id": "5683.png", "formula": "\\begin{align*} ( x _ 1 , x _ 2 , \\dots , x ^ J ) \\mapsto \\sum _ { j = 1 } ^ J x _ j \\varphi _ { \\iota + j } . \\end{align*}"}
+{"id": "911.png", "formula": "\\begin{align*} u = g \\quad D ^ c , W _ D ( u ) = h \\quad \\partial D . \\end{align*}"}
+{"id": "2624.png", "formula": "\\begin{align*} ( n - 3 ) - 2 \\left ( \\frac { n } { 3 } - \\frac { 7 } { 3 } \\right ) = \\frac { n } { 3 } + \\frac { 5 } { 3 } \\end{align*}"}
+{"id": "2483.png", "formula": "\\begin{align*} H = \\max _ { \\substack { I \\subseteq \\{ 1 , \\ldots , k \\} \\\\ C _ I } } \\frac { 1 } { \\sigma _ { | I | } ( C _ I ) } \\end{align*}"}
+{"id": "3575.png", "formula": "\\begin{align*} ( a + b ) \\alpha \\tau _ 0 ( i d - \\tau ) ( i d + \\tau ^ 2 ) - ( a + b ) \\alpha \\tau _ 0 ( i d ^ 2 - \\tau ^ 2 ) = 0 . \\end{align*}"}
+{"id": "6621.png", "formula": "\\begin{align*} J _ { n - 1 } & = J _ j J _ { n - j } + 2 J _ { j - 1 } J _ { n - j - 1 } \\\\ \\sum _ { j = 1 } ^ n 2 ^ j J _ { n - j } & = \\frac { 2 } { 3 } ( n J _ { n - 1 } + ( n - 1 ) J _ n ) . \\\\ \\sum _ { j = 1 } ^ n ( - 1 ) ^ { n + j } J _ j & = \\frac { 1 } { 3 } ( n J _ { n - 1 } - ( n - 2 ) J _ n ) \\end{align*}"}
+{"id": "7715.png", "formula": "\\begin{align*} \\partial _ { t } \\widetilde { E } = b ^ { 1 1 } \\partial _ { t } b _ { 1 1 } - \\tilde { d } \\frac { h _ { t } } { h } + 2 \\tilde { l } \\sum _ { j } h _ { j } h _ { j t } = b ^ { 1 1 } ( h _ { 1 1 t } + h _ { t } ) - \\tilde { d } \\frac { h _ { t } } { h } + 2 \\tilde { l } \\sum _ { j } h _ { j } h _ { j t } . \\end{align*}"}
+{"id": "1487.png", "formula": "\\begin{align*} { } [ P _ { d } ( L _ m ) , P _ { d } ( L _ n ) ] & = [ f ( m + d ) L _ { m + d } , f ( n + d ) L _ { n + d } ] \\\\ & = f ( m + d ) f ( n + d ) ( \\{ m + d \\} - \\{ n + d \\} ) L _ { m + n + 2 d } , \\\\ P _ d ( [ P _ d ( L _ m ) , L _ n ] ) & = P _ d ( [ f ( m + d ) L _ { m + d } , L _ { n } ] ) \\\\ & = f ( m + d ) f ( m + n + 2 d ) ( \\{ m + d \\} - \\{ n \\} ) L _ { m + n + 2 d } . \\end{align*}"}
+{"id": "1386.png", "formula": "\\begin{align*} ( x _ 3 , x _ 4 ) \\cap q ^ \\perp = ( x _ 1 x _ 3 , x _ 2 x _ 3 - x _ 1 x _ 4 , x _ 3 ^ 2 , x _ 2 x _ 4 , x _ 3 x _ 4 , x _ 4 ^ 2 ) . \\end{align*}"}
+{"id": "5428.png", "formula": "\\begin{align*} S _ 1 ^ * \\left ( \\nu _ { ( 1 , i ) } ^ * \\right ) = \\big \\{ j \\in N \\colon \\nu _ { ( 1 , j ) } ^ * > \\nu _ { ( 1 , i ) } ^ * \\big \\} . \\end{align*}"}
+{"id": "2823.png", "formula": "\\begin{align*} L _ { 3 } = \\frac { 1 } { 2 } \\left ( q ^ { 1 } + \\dot { q } ^ { 2 } + \\dot { q } ^ { 3 } \\right ) ^ { 2 } + \\frac { 1 } { 2 } \\left ( \\dot { q } ^ { 4 } - \\dot { q } ^ { 2 } \\right ) ^ { 2 } + \\frac { 1 } { 2 } \\left ( q ^ { 1 } + 2 q ^ { 2 } \\right ) \\left ( q ^ { 1 } + 2 q ^ { 4 } \\right ) . \\end{align*}"}
+{"id": "4744.png", "formula": "\\begin{align*} \\widehat { x \\upharpoonright y } = _ { \\mathbb { N } \\to X } \\hat { x } \\upharpoonright y . \\end{align*}"}
+{"id": "3753.png", "formula": "\\begin{align*} \\mathrm { E } _ { \\alpha , \\beta } \\left ( z \\right ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { \\ , z ^ { k } } { \\Gamma \\left ( \\alpha k + \\beta \\right ) } , \\alpha > 0 , \\end{align*}"}
+{"id": "9079.png", "formula": "\\begin{align*} \\begin{array} { c c } \\Big ( \\hat { C } ^ { - 1 } ( \\hat { C } C \\hat { C } ^ { - 1 } - I ) \\underline { v } + \\hat { C } ^ { - 1 } \\hat { C } ( D p - U ( J ^ { ( r ) } X ) - \\mathbf { 1 } \\frac { \\beta } { 2 } ) \\Big ) _ i \\\\ = \\Big ( ( C \\hat { C } ^ { - 1 } - \\hat { C } ^ { - 1 } ) \\underline { v } + D p - U ( J ^ { ( r ) } X ) - \\mathbf { 1 } \\frac { \\beta } { 2 } \\Big ) _ i \\leq 0 . \\end{array} \\end{align*}"}
+{"id": "840.png", "formula": "\\begin{align*} \\nabla _ { 0 } f _ { . k } = \\hat { X } . \\lambda ( x , y ) I _ { k } - \\Psi I _ { k } . \\end{align*}"}
+{"id": "1125.png", "formula": "\\begin{align*} \\left \\| \\sup _ { \\mathbb { A } , \\varphi } \\left ( \\vec f \\right ) \\right \\| _ { \\dot f _ { \\infty , q } ^ s } = \\left \\| \\left \\{ 2 ^ { j s } g _ j \\right \\} _ { j \\in \\mathbb Z } \\right \\| _ { L \\dot F _ { \\infty , q } ^ { \\tau } } \\lesssim \\left \\| \\left \\{ 2 ^ { j s } h _ j \\right \\} _ { j \\in \\mathbb Z } \\right \\| _ { L \\dot F _ { \\infty , q } ^ { \\tau } } = \\left \\| \\vec f \\right \\| _ { \\dot F _ { \\infty , q } ^ s ( \\mathbb { A } , \\varphi ) } . \\end{align*}"}
+{"id": "6264.png", "formula": "\\begin{align*} g ( t _ { \\rho } ( x ) ) = g ( x ) , a . e . \\ x \\in \\Gamma . \\end{align*}"}
+{"id": "229.png", "formula": "\\begin{align*} \\frac { d ^ 2 x } { d t ^ 2 } + \\gamma _ 0 ( x ) \\left ( \\frac { d x } { d t } \\right ) ^ 2 + A _ 0 ( x ) \\frac { d x } { d t } + b _ 0 ( x ) = 0 , \\end{align*}"}
+{"id": "7264.png", "formula": "\\begin{align*} x p _ n ( x ) = p _ { n + 1 } ( x ) + ( 1 - ( A _ n + C _ n ) ) p _ n ( x ) + A _ { n - 1 } C _ n p _ { n - 1 } ( x ) , n \\geq 0 , \\end{align*}"}
+{"id": "6169.png", "formula": "\\begin{align*} v ^ { k + 1 } = v ^ k - M ^ k ( v ^ k - \\widetilde { v } ^ k ) . \\end{align*}"}
+{"id": "4845.png", "formula": "\\begin{align*} r _ { + , j } ^ n - r _ { - , j } ^ n & = n ( r _ { - , j } ) ^ { n - 1 } ( r _ { + , j } - r _ { - , j } ) + \\cdots + ( r _ { + , j } - r _ { - , j } ) ^ n \\\\ & \\le t _ n ( r _ * ) ( r _ { + , j } - r _ { - , j } ) \\end{align*}"}
+{"id": "335.png", "formula": "\\begin{align*} J _ 1 & = x _ { n } x _ { n - 1 } [ I ( G _ 3 ) ^ { [ k ] } + x _ { n - 2 } I ( G _ 3 ) ^ { [ k - 1 ] } ] , \\\\ J _ 2 & = x _ { n } x _ { n - 1 } \\sum _ { j = 1 } ^ t x _ { i _ j } I ( G _ 2 ) ^ { [ k - 1 ] } = x _ { n } x _ { n - 1 } ( x _ { i _ 1 } , \\dots , x _ { i _ t } ) I ( G _ 2 ) ^ { [ k - 1 ] } . \\end{align*}"}
+{"id": "6093.png", "formula": "\\begin{align*} \\binom { a } { b } : = & \\dfrac { a ( a - 1 ) \\dotsc ( a - b + 1 ) } { b ! } . \\end{align*}"}
+{"id": "8781.png", "formula": "\\begin{align*} \\mu ( \\{ x \\} ) \\int _ \\R | z - x | \\pi _ x ( d z ) & \\le \\int _ { \\R \\times \\R } \\left ( | z - x | - | y - x | \\right ) \\pi _ y ( d z ) \\mu ( d y ) = u _ \\nu ( x ) - u _ \\mu ( x ) \\\\ & = \\int _ { F _ \\nu ( x - ) - p _ - ( x ) } ^ { F _ \\nu ( x - ) } ( x - F _ \\nu ^ { - 1 } ( v ) ) d v + \\int _ { F _ \\nu ( x ) } ^ { F _ \\nu ( x ) + p _ + ( x ) } ( F _ \\nu ^ { - 1 } ( v ) - x ) d v . \\end{align*}"}
+{"id": "8383.png", "formula": "\\begin{align*} L _ J : = G L _ 1 \\times G L _ 2 \\times G L _ 2 \\times G L _ 1 . \\end{align*}"}
+{"id": "5078.png", "formula": "\\begin{align*} R _ s ^ 2 & = q H _ s ( - 1 ) + R _ s E _ s \\\\ \\underbrace { R _ s R _ { s ' } R _ s R _ { s ' } \\dots } _ { m _ { s s ' } } & = \\underbrace { R _ { s ' } R _ s R _ { s ' } R _ s \\dots } _ { m _ { s s ' } } , s , s ' \\in S \\\\ R _ t R _ s & = R _ s R _ { t ' } , t ' = n _ s t n _ s ^ { - 1 } , t \\in T ( \\mathbb { F } _ { q ^ k } ) \\\\ R _ { t _ 1 } R _ { t _ 2 } & = R _ { t _ 1 t _ 2 } , t _ 1 , t _ 2 \\in T ( \\mathbb { F } _ { q ^ k } ) \\end{align*}"}
+{"id": "4188.png", "formula": "\\begin{align*} \\varphi ( g _ 2 g _ 3 ) \\varphi ( g _ 1 g ^ { - 1 } _ 3 ) \\overset { { \\bf A 6 } } { = } \\varphi ( g _ 2 ) \\varphi ( g _ 3 ) \\varphi ( g ^ { - 1 } _ 3 ) \\varphi ( g _ 1 ) \\overset { { \\bf A 2 } } { = } \\varphi ( g _ 2 ) \\varphi ( g _ 1 ) \\overset { ( \\ref { e q : a n t i 3 } ) } { = } \\varphi ( g _ 2 g _ 1 ) \\ , , \\end{align*}"}
+{"id": "1731.png", "formula": "\\begin{align*} F _ j ( z _ 1 , \\ldots , z _ { n _ j } , \\xi ) : = \\sum _ { j = 1 } ^ { n _ j } \\left [ ( x _ 1 , \\ldots , x _ { n _ j } ) Q _ { n _ j ; k } \\left ( \\begin{array} { c } x _ 1 \\\\ \\vdots \\\\ x _ { n _ j } \\end{array} \\right ) \\right ] ( \\Re \\xi ) ^ { j - 1 } . \\end{align*}"}
+{"id": "5295.png", "formula": "\\begin{align*} U _ { j } = - \\sum _ { i } \\overline { q } _ { i } \\log \\overline { p } _ { i } - \\log \\overline { q } _ { j } \\ ; \\ ; ; \\ ; \\ ; V _ { j } = - \\sum _ { i } \\overline { q } _ { i } \\log \\overline { q } _ { i } - \\log \\overline { p } _ { j } \\end{align*}"}
+{"id": "1372.png", "formula": "\\begin{align*} \\mu _ t = \\sum _ { i \\geqslant 0 } \\mu _ i t ^ i , T _ t = \\sum _ { i \\geqslant 0 } T _ i t ^ i , \\end{align*}"}
+{"id": "3299.png", "formula": "\\begin{align*} \\{ Y ' > Y \\} = \\bigcup _ { n , m = 1 } ^ \\infty \\{ Y ' \\ge Y + 2 ^ { - n } \\} \\cap \\{ Y \\le m \\} = \\bigcup _ { n , m = 1 } ^ \\infty A _ { n , m } . \\end{align*}"}
+{"id": "3386.png", "formula": "\\begin{align*} S _ { t } & = \\sum _ { i = 1 } ^ { t } Z _ { i } \\end{align*}"}
+{"id": "7697.png", "formula": "\\begin{align*} \\mu = F _ { p , q } ( \\Omega , \\cdot ) ? \\end{align*}"}
+{"id": "8703.png", "formula": "\\begin{align*} \\tau B _ G = \\tau P S _ G ( P ^ \\ast P ) ^ { - 1 } P ^ \\ast B _ G . \\end{align*}"}
+{"id": "4434.png", "formula": "\\begin{gather*} \\psi ^ j = \\frac { v _ h ^ j - u _ h ^ j } { \\tau _ j } \\ , , \\ \\ \\ j = 1 , \\dots , M . \\end{gather*}"}
+{"id": "6938.png", "formula": "\\begin{align*} n \\Re ( \\tau ) - j _ 0 \\Re ( \\varpi ( \\tau ) ) & \\leq n \\Re ( \\tau ) + \\frac { j _ 0 } { \\alpha } \\left ( \\Re ( \\tau ) - A _ R \\Re ( \\tau ) ^ { 2 \\mu } + A _ I \\Im ( \\tau ) ^ { 2 \\mu } \\right ) \\\\ & \\leq - n c _ \\star \\Im ( \\tau ) ^ { 2 \\mu } + \\left ( \\frac { j _ 0 } { \\alpha } + n \\right ) \\tau _ p - \\frac { j _ 0 } { \\alpha } A _ R \\tau _ p ^ { 2 \\mu } . \\end{align*}"}
+{"id": "8120.png", "formula": "\\begin{align*} X _ w ^ { \\circ } : = B w B = U _ w w B = \\prod _ { \\alpha \\in { \\mathcal I } ( w ) } U _ { \\alpha } \\ , \\ , w B . \\end{align*}"}
+{"id": "6909.png", "formula": "\\begin{align*} \\Delta _ p v = g : = \\frac { n \\ ( p - 1 \\ ) } { p } v ^ { - 1 } \\left | \\nabla v \\right | ^ p + \\ ( \\frac { p } { n - p } \\ ) ^ { p - 1 } v ^ { - 1 } \\quad \\R ^ n . \\end{align*}"}
+{"id": "7872.png", "formula": "\\begin{align*} r a n k _ { p } ( \\mathbf { A } + \\mathbf { A } ^ { T } ) = r a n k _ { p } ( \\mathbf { P } ( \\mathbf { B } + \\mathbf { B } ^ { T } ) \\mathbf { P } ^ { T } ) = r a n k _ { p } ( \\mathbf { B } + \\mathbf { B } ^ { T } ) . \\end{align*}"}
+{"id": "3109.png", "formula": "\\begin{align*} \\varphi \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) & = \\left ( \\begin{array} { c c c } a ^ { \\ell _ 1 } & 0 & c _ 2 \\ , a ^ { \\ell _ 1 } \\ , b ^ { p ^ { e _ 2 } } \\ , d ^ { p ^ { e _ 2 } } \\\\ 0 & d ^ { - \\ell _ 2 } & 0 \\\\ 0 & 0 & d ^ { - \\ell _ 2 } \\end{array} \\right ) . \\end{align*}"}
+{"id": "3659.png", "formula": "\\begin{align*} L ^ { - } ( G , p ) = X ^ T L ^ { - } ( G ' , p ' ) X . \\end{align*}"}
+{"id": "8388.png", "formula": "\\begin{align*} M _ 4 C _ 5 ^ { - 1 } C _ 4 ^ { - 1 } & = ( C _ 1 C _ 2 + A _ 1 A _ 2 ) C _ 3 - M _ 4 C _ 5 ^ { - 1 } A _ 5 B _ 5 ^ { - 1 } B _ 4 ^ { - 1 } A _ 4 C _ 4 ^ { - 1 } \\\\ & = ( C _ 1 C _ 2 + A _ 1 A _ 2 ) C _ 3 + ( A _ 1 B _ 1 ^ { - 1 } M _ 1 B _ 5 ^ { - 1 } B _ 4 ^ { - 1 } + C _ 1 C _ 2 A _ 3 ) ( - B _ 4 B _ 5 M _ 1 ^ { - 1 } B _ 1 A _ 2 C _ 3 ) \\\\ & = C _ 1 C _ 2 C _ 3 - C _ 1 C _ 2 A _ 3 B _ 4 B _ 5 M _ 1 ^ { - 1 } B _ 1 A _ 2 C _ 3 . \\end{align*}"}
+{"id": "2640.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ p _ t L ^ q _ x ( I _ t \\times \\mathbb { T } ^ d ) } = \\left ( \\int _ { I _ t } \\left ( \\int _ { \\mathbb { T } ^ d } | u ( t , z ) | ^ q d z \\right ) ^ { \\frac { p } { q } } d t \\right ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
+{"id": "3275.png", "formula": "\\begin{align*} \\frac { ( q ^ { d + r - ( d - 2 j - 1 ) n } ; q ^ d ) _ { k - 2 } } { ( q ^ { d - ( d - 2 j ) n } ; q ^ d ) _ k } & = \\frac { ( q ^ { d - ( d - 2 j ) n + d k } ; q ^ d ) _ { ( n + r ) / d - 2 } } { ( q ^ { d - ( d - 2 j ) n } ; q ^ d ) _ { ( n + r ) / d } } . \\end{align*}"}
+{"id": "8413.png", "formula": "\\begin{align*} & ( B _ 1 ( a _ 1 ) \\otimes a _ 2 \\otimes B _ 2 ( a _ 3 ) ) \\cdot _ { H ^ { \\otimes 3 } } ( B _ 1 ( b _ 1 ) \\otimes b _ 2 \\otimes B _ 2 ( b _ 3 ) ) \\\\ = & B _ 1 ( a _ 1 ) B _ 1 ( b _ 1 ) \\otimes \\epsilon ( a _ 3 ) B _ 1 ( a _ 2 ) b _ 2 S ( B _ 2 ( a _ 5 ) ) \\otimes B _ 2 ( a _ 4 ) B _ 2 ( b _ 3 ) \\\\ = & B _ 1 ( a _ 1 ) B _ 1 ( b _ 1 ) \\otimes B _ 1 ( a _ 2 ) b _ 2 S ( B _ 2 ( a _ 3 ) ) \\otimes B _ 2 ( a _ 4 ) B _ 2 ( b _ 3 ) \\\\ = & B _ 1 ( a _ 1 \\circ b _ 1 ) \\otimes a _ 2 \\circ b _ 2 \\otimes B _ 2 ( a _ 3 \\circ b _ 3 ) , \\end{align*}"}
+{"id": "5776.png", "formula": "\\begin{align*} \\mathcal { W } _ j = G ( \\mathcal { E } ( u ) ; \\Upsilon _ j ) = G ( ( 0 , E _ 1 ( u ) ) ; ( \\varphi _ { \\iota + j } , - 2 ^ { - 1 } m \\varphi _ { \\iota + j } ) ) = - m ^ { - 1 } \\int _ \\Sigma E _ 1 ( u ) \\varphi _ { \\iota + j } \\ , d \\mu , \\end{align*}"}
+{"id": "5292.png", "formula": "\\begin{align*} - \\frac { \\partial K L I _ { n } ( q \\| p ) } { \\partial q _ { j } } = K _ { 0 } \\left [ \\log \\frac { p _ { j } } { q _ { j } } - \\frac { \\sum _ { i } q _ { i } \\log \\frac { p _ { i } } { q _ { i } } } { \\sum _ { j } q _ { j } } \\right ] \\end{align*}"}
+{"id": "5139.png", "formula": "\\begin{align*} \\frac { \\partial L _ { d } D 1 ( p \\| q ) } { \\partial q _ { j } } & = \\left \\{ \\left [ \\frac { a - 1 } { a - b } A ^ { a - 2 } - \\frac { b - 1 } { a - b } A ^ { b - 2 } \\right ] \\frac { \\partial A } { \\partial q _ { j } } \\right \\} \\\\ & - \\left \\{ \\left [ \\frac { a - 1 } { a - b } ( X + Y ) ^ { a - 2 } - \\frac { b - 1 } { a - b } ( X + Y ) ^ { b - 2 } \\right ] \\frac { \\partial ( X + Y ) } { \\partial q _ { j } } \\right \\} \\end{align*}"}
+{"id": "2512.png", "formula": "\\begin{align*} H ^ { - s } ( M ) = H ^ s ( M ; \\Omega ) ' \\ ; , \\end{align*}"}
+{"id": "3579.png", "formula": "\\begin{align*} P = \\frac { I + T + T ^ 2 } { 3 } , Q = \\frac { I + \\lambda ^ 2 T + \\lambda T ^ 2 } { 3 } , R = \\frac { I + \\lambda T + \\lambda ^ 2 T ^ 2 } { 3 } . \\end{align*}"}
+{"id": "3145.png", "formula": "\\begin{align*} u _ { \\psi ^ * } ( t ) = \\left ( \\begin{array} { c c c } 1 & 0 & t ^ { p ^ { e _ 2 } } \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "3733.png", "formula": "\\begin{align*} \\Phi \\left ( z , a , b \\right ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { z ^ { k } } { \\left ( k + b \\right ) ^ { a } } . \\end{align*}"}
+{"id": "3867.png", "formula": "\\begin{align*} m ^ n _ j \\doteq m ( t _ n - T + j c ) , \\ ; \\ ; j = 0 , 1 , \\dots , l _ 0 , \\end{align*}"}
+{"id": "1463.png", "formula": "\\begin{align*} ( U \\Gamma ^ * ) \\cap U = \\varnothing . \\end{align*}"}
+{"id": "4037.png", "formula": "\\begin{align*} \\langle k _ S ( x , \\cdot ) , k _ S ( y , \\cdot ) \\rangle _ { \\mathcal { H } _ { \\mathcal { S } } } = \\langle S ( x ) , S ( y ) \\rangle _ { \\mathcal { H } ^ 1 } \\end{align*}"}
+{"id": "8175.png", "formula": "\\begin{align*} | \\mathcal B _ { \\beta _ j ^ L } ( \\underline x ^ L ) | = \\frac { b _ j ^ 0 } { 2 ^ { L - j } } , \\ | \\mathcal B _ { 0 ^ L } ( \\underline x ^ L ) | = M - \\sum _ { j = 1 } ^ L b _ j ^ 0 , \\\\ P _ { Y ^ L , \\underline S } ( y ^ L \\in \\beta _ j ^ L , \\underline S ) = \\frac { b _ j ^ 0 } { M } . \\end{align*}"}
+{"id": "3728.png", "formula": "\\begin{align*} \\left | A + A \\right | & = \\sum _ { i = 2 } ^ { 2 t } i ^ { k } = 2 ^ { k + 1 } \\sum _ { i = 1 } ^ { t } i ^ k - \\sum _ { i = 1 } ^ { t } \\left [ ( 2 i ) ^ k - \\left ( 2 i - 1 \\right ) ^ k \\right ] - 1 \\\\ & \\leq 2 ^ { k + 1 } | A | - \\frac 1 2 ( 2 t ) ^ { k } = \\left ( 2 ^ { k + 1 } - ( 1 + o ( 1 ) ) 2 ^ { k } \\frac { k + 1 } { 2 t } \\right ) | A | . \\end{align*}"}
+{"id": "6081.png", "formula": "\\begin{align*} J _ { \\pm } ( x ) : = \\int _ 0 ^ { \\pi } \\frac { \\sin k \\sin x k } { 1 \\pm \\cos k } d k , x \\in \\mathbb { Z } , \\end{align*}"}
+{"id": "8754.png", "formula": "\\begin{align*} u _ { \\mu _ \\varepsilon } ( m ) & = \\mu _ \\varepsilon ( \\{ x _ - \\} ) ( m - x _ - ) + \\mu _ \\varepsilon ( \\{ x _ + \\} ) ( x _ + - m ) \\\\ u _ \\nu ( m ) & = \\mu _ \\varepsilon ( \\{ x _ - \\} ) \\int \\vert m - w \\vert \\ , \\pi _ { x _ - } ( d w ) + \\mu _ \\varepsilon ( \\{ x _ + \\} ) \\int \\vert m - w \\vert \\ , \\pi _ { x _ + } ( d w ) , \\end{align*}"}
+{"id": "281.png", "formula": "\\begin{align*} e _ { 2 k } e _ 1 = 0 \\ \\mbox { a n d } \\ e _ { 2 k + 1 } e _ 1 = e _ 1 e _ { 2 k + 1 } = e _ { 2 k + 2 } . \\end{align*}"}
+{"id": "6432.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 2 } ^ n A _ i - m _ { \\frac { 1 } { 2 } } ( \\alpha _ 0 ) \\delta _ 0 ^ { 1 / 2 } = & \\frac { 1 } { n } \\sum _ { i = 2 } ^ n \\left ( A _ i - \\delta _ 0 ^ { 1 / 2 } n ^ { 1 / 2 \\alpha _ 0 } | \\Delta _ i ^ n L - \\Delta _ { i - 1 } ^ n L | ^ { 1 / 2 } \\right ) \\\\ & + \\delta _ 0 ^ { 1 / 2 } \\left ( \\frac { 1 } { n } \\sum _ { i = 2 } ^ n n ^ { 1 / 2 \\alpha _ 0 } | \\Delta _ i ^ n L - \\Delta _ { i - 1 } ^ n L | ^ { 1 / 2 } - m _ { \\frac { 1 } { 2 } } ( \\alpha _ 0 ) \\right ) . \\end{align*}"}
+{"id": "1375.png", "formula": "\\begin{align*} f = \\ & ( 0 , \\ldots , 0 , f ^ A _ { \\{ 1 , \\ldots , n \\} } ; f ^ V _ \\emptyset , f ^ V _ { \\{ 1 \\} } , \\ldots , f ^ V _ { \\{ 2 , \\ldots , n \\} } , 0 ) \\in \\frak { L } ' ( n ) , \\\\ g = \\ & ( 0 , \\ldots , 0 , g ^ A _ { \\{ 1 , \\ldots , m \\} } ; g ^ V _ \\emptyset , g ^ V _ { \\{ 1 \\} } , \\ldots , g ^ V _ { \\{ 2 , \\ldots , n \\} } , 0 ) \\in \\frak { L } ' ( m ) , \\end{align*}"}
+{"id": "3918.png", "formula": "\\begin{align*} \\binom { n } { k } \\cos ^ { n - k } z \\sin ^ k z = \\binom { n } { k } 1 \\frac { v ^ k } { n ^ k } = \\frac { v ^ k } { k ! } . \\end{align*}"}
+{"id": "4386.png", "formula": "\\begin{gather*} \\max \\{ 0 , x _ i - \\overline { b } _ i , \\max \\{ 0 , x _ i - \\overline { b } _ i + \\Delta b _ i \\} - \\theta ^ k ( x ) \\} = \\max \\{ 0 , x _ i - \\overline { b } _ i , x _ i - \\overline { b } _ i + \\Delta b _ i - \\Delta b _ k \\} . \\end{gather*}"}
+{"id": "2038.png", "formula": "\\begin{align*} P _ T = \\begin{bmatrix} C _ 5 & C _ 4 & C _ 3 & C _ 2 & C _ 1 \\\\ C _ 4 & C _ 3 & C _ 2 & C _ 1 & C _ 0 \\end{bmatrix} . \\end{align*}"}
+{"id": "113.png", "formula": "\\begin{align*} A _ { \\lambda } ^ { - 1 } \\eqsim \\frac { \\lambda } { \\lambda + 1 } P A ^ { - 1 } + \\frac { 1 } { \\lambda + 1 } A ^ { - 1 } = : M _ \\lambda , \\end{align*}"}
+{"id": "3777.png", "formula": "\\begin{align*} X ^ { ( i ) } = \\Theta _ j Z ^ { ( i ) } + W ^ { ( i ) } , \\end{align*}"}
+{"id": "201.png", "formula": "\\begin{align*} R \\ = \\ S \\ \\cup \\ P \\ \\cup \\ \\ \\bigcup _ i [ H _ i \\times ( E _ i \\cup G _ i ) \\ \\cup \\ F _ i \\times H _ i ] . \\end{align*}"}
+{"id": "994.png", "formula": "\\begin{align*} \\mathbb E _ { x } ( \\mathbf 1 _ D u ( X _ { \\tau _ V } ) \\mathbf 1 _ A ( X _ { \\tau _ V - } ) ) & = \\mathbb E _ { x } ( \\mathbf 1 _ { D \\setminus V } u ( X _ { \\tau _ V } ) \\mathbf 1 _ { A \\cap V } ( X _ { \\tau _ V - } ) ) \\\\ & = \\mathbb E _ { x } ( \\mathbf 1 _ { D \\setminus \\bar V } u ( X _ { \\tau _ V } ) \\mathbf 1 _ { A \\cap V } ( X _ { \\tau _ V - } ) ) , \\end{align*}"}
+{"id": "3479.png", "formula": "\\begin{align*} & a _ 1 = \\dots = a _ { q + 1 } = 0 , \\\\ & a _ { q + 1 + j } = j / K , \\ 1 \\leq j \\leq K - 1 , \\\\ & a _ { q + K + 1 } = \\dots = a _ { 2 q + K + 1 } = 1 \\ , . \\end{align*}"}
+{"id": "7265.png", "formula": "\\begin{align*} \\int _ { c q } ^ { a q } f ( x ) \\mathrm { d } _ q ( x ) : = a q ( 1 - q ) \\sum \\limits _ { k = 0 } ^ \\infty f ( a q ^ { k + 1 } ) q ^ k - c q ( 1 - q ) \\sum \\limits _ { k = 0 } ^ \\infty f ( c q ^ { k + 1 } ) q ^ k . \\end{align*}"}
+{"id": "5495.png", "formula": "\\begin{align*} A & { } = A _ { j - 1 } [ X _ j ; \\sigma _ j , \\delta _ j ] [ X _ { j + 1 } ; \\sigma _ { j + 1 } ] \\ldots [ X _ N ; \\sigma _ N ] , \\\\ \\hat { A } & { } = A _ { j - 1 } [ X _ j ; \\sigma _ j , \\delta _ j ] [ X _ { j + 1 } ^ { \\pm 1 } ; \\sigma _ { j + 1 } ] \\ldots [ X _ N ^ { \\pm 1 } ; \\sigma _ N ] , \\end{align*}"}
+{"id": "187.png", "formula": "\\begin{align*} \\N + \\ = \\ ( 2 L ) ^ { \\circ - 1 } ( \\infty ) , & \\N - \\ = \\ ( 2 L ) ^ { \\circ } ( \\infty ) , \\\\ ( i - , i + ) \\ \\in \\ 2 L & \\ \\ i \\in \\N . \\end{align*}"}
+{"id": "8622.png", "formula": "\\begin{align*} \\sigma = \\frac { 3 } { 2 } + 6 \\epsilon ~ ~ { \\rm s o ~ t h a t } ~ ~ \\sigma < 2 - 2 0 \\epsilon \\end{align*}"}
+{"id": "787.png", "formula": "\\begin{align*} & \\tilde \\psi \\colon \\C [ T ] \\otimes \\C [ O ^ + _ F ] \\to \\C [ T ] \\# \\tilde A , \\tilde \\psi ( x \\otimes a ) = x \\tilde \\phi ( a ) , \\\\ & \\psi \\colon \\C [ T ] \\otimes \\C [ O ^ + _ F ] \\to \\C [ T ] \\# A , \\psi ( x \\otimes a ) = x \\pi ( a _ { ( 1 ) } ) \\# a _ { ( 2 ) } . \\end{align*}"}
+{"id": "4058.png", "formula": "\\begin{align*} \\int _ { \\mathbb { S } ^ { d - 1 } } e ^ { i r \\omega \\cdot x } d \\omega & = \\sum _ { k = 0 } ^ { \\frac { d - 3 } { 2 } } c _ k \\frac { e ^ { i r | x | } } { ( r | x | ) ^ { \\frac { d - 1 } { 2 } + k } } + \\sum _ { k = 0 } ^ { \\frac { d - 3 } { 2 } } c _ k \\frac { e ^ { - i r | x | } } { ( - r | x | ) ^ { \\frac { d - 1 } { 2 } + k } } \\end{align*}"}
+{"id": "713.png", "formula": "\\begin{align*} f ( x ) = \\phi _ 3 ^ 6 ( x ) + ( 6 \\delta x + 2 1 ) \\phi _ 3 ^ 5 ( x ) + ( 6 5 \\delta x + 1 2 5 ) \\phi _ 3 ^ 4 ( x ) + ( 2 5 6 \\delta x + 3 3 8 ) \\phi _ 3 ^ 3 ( x ) \\\\ + ( 4 7 4 \\delta x + 4 6 8 ) \\phi _ 3 ^ 2 ( x ) + ( 4 2 \\delta x + 3 2 4 ) \\phi _ 3 ( x ) + 8 9 - m + 1 4 4 \\delta x . \\end{align*}"}
+{"id": "2544.png", "formula": "\\begin{align*} \\dot C ^ \\infty ( M ) = \\bigcap _ m \\AA ^ m ( M ) \\ ; . \\end{align*}"}
+{"id": "4221.png", "formula": "\\begin{align*} \\theta ( v , \\tau + 1 ) = e ^ { \\frac { \\pi \\sqrt { - 1 } } { 4 } } \\theta ( v , \\tau ) , ~ ~ \\theta ( v , - \\frac { 1 } { \\tau } ) = \\frac { 1 } { \\sqrt { - 1 } } \\left ( \\frac { \\tau } { \\sqrt { - 1 } } \\right ) ^ { \\frac { 1 } { 2 } } e ^ { \\pi \\sqrt { - 1 } \\tau v ^ 2 } \\theta ( \\tau v , \\tau ) ; \\end{align*}"}
+{"id": "2268.png", "formula": "\\begin{align*} s _ i ( u ) s _ { i + 1 } ( u + v ) s _ i ( v ) = s _ { i + 1 } ( v ) s _ i ( u + v ) s _ { i + 1 } ( u ) \\ . \\end{align*}"}
+{"id": "2886.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\left \\| f - f _ k \\right \\| _ { X ^ p } = 0 . \\end{align*}"}
+{"id": "5332.png", "formula": "\\begin{align*} \\nu ^ { S _ { k - 1 } } _ j = \\nu _ { \\pi _ { k - 2 } } + \\frac { \\displaystyle c _ j - \\nu _ { \\pi _ 1 } \\ , w ^ { S _ 1 } _ j - \\sum _ { l = 2 } ^ { k - 2 } \\left ( \\nu _ { \\pi _ { l } } - \\nu _ { \\pi _ { l - 1 } } \\right ) \\ , w ^ { S _ { l } } _ j } { \\displaystyle w ^ { S _ { k - 1 } } _ j } , j \\in S _ { k - 1 } , \\end{align*}"}
+{"id": "6403.png", "formula": "\\begin{align*} \\tilde { \\alpha } _ n ( p ) = \\frac { p \\log 2 } { \\log ( V _ n ^ 1 ( p , X ) / V _ n ^ 2 ( p , X ) ) } { \\bf 1 } _ { V _ n ^ 1 ( p , X ) \\neq V _ n ^ 2 ( p , X ) } . \\end{align*}"}
+{"id": "3154.png", "formula": "\\begin{align*} \\varphi ^ \\sharp \\left ( \\begin{array} { c c } a & b \\\\ 0 & d \\end{array} \\right ) = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "569.png", "formula": "\\begin{align*} E ( \\phi , \\eta , u ) = g \\eta + j \\omega \\phi + \\frac { u } { \\rho } . \\end{align*}"}
+{"id": "5328.png", "formula": "\\begin{align*} \\nu ^ { S _ k } _ j = \\frac { c ^ { S _ k } _ j } { w ^ { S _ k } _ j } , j \\in S _ k , 1 \\leq k \\leq n . \\end{align*}"}
+{"id": "3377.png", "formula": "\\begin{align*} \\frac { 1 } { T } \\sum _ { t = 2 } ^ { T + 1 } \\Delta _ { t } & \\le 4 8 R _ { 1 } \\max \\left \\{ 2 6 ^ { \\frac { 1 } { p } } T ^ { \\frac { 1 - p } { p } } \\sigma \\gamma ^ { \\frac { p - 1 } { p } } ; 2 \\left ( 2 L R _ { 1 } + L R _ { 0 } + \\mu \\sigma + \\left \\Vert g _ { 0 } \\right \\Vert _ { * } \\right ) T ^ { - 1 } \\gamma \\right \\} = O \\left ( T ^ { \\frac { 1 - p } { p } } \\right ) . \\end{align*}"}
+{"id": "7204.png", "formula": "\\begin{align*} H ( P ) = \\sum _ { i = 1 } ^ N \\phi _ i ( T _ 0 ( P ) ) \\ , H _ i ( P ) \\end{align*}"}
+{"id": "7912.png", "formula": "\\begin{align*} \\begin{aligned} \\boldsymbol { n } ( \\alpha ) ( X _ { 1 } , \\ldots , X _ { k } ) : = \\alpha ( X _ { 1 } ^ { \\perp } , \\ldots , X _ { k } ^ { \\perp } ) . \\end{aligned} \\end{align*}"}
+{"id": "4165.png", "formula": "\\begin{align*} \\varphi \\big ( g ^ { j } _ 1 g _ 2 \\big ) = \\varphi \\big ( g ^ { j } _ 1 \\big ) \\varphi ( g _ 2 ) \\mbox { a n d } \\varphi \\big ( g ^ { j } _ 1 g _ 3 \\big ) = \\varphi ( g _ 3 ) \\varphi \\big ( g ^ { j } _ 1 \\big ) \\ , , \\end{align*}"}
+{"id": "3567.png", "formula": "\\begin{align*} \\alpha ^ 3 \\tau _ 0 \\tau _ 1 \\tau _ 2 f \\circ \\tau ^ 3 - ( 1 + a ) \\alpha ^ 2 \\tau _ 0 \\tau _ 1 f \\circ \\tau ^ 2 + ( a + b ) \\alpha \\tau _ 0 f \\circ \\tau - b f = 0 . \\end{align*}"}
+{"id": "745.png", "formula": "\\begin{align*} H e s s ( h ) ( \\hat { X } , Y ) = ( \\hat { X } Y ) h - ( { } ^ c \\nabla _ { \\hat { X } } Y ) h . \\end{align*}"}
+{"id": "6849.png", "formula": "\\begin{align*} e ^ { \\psi _ { N - 1 } \\lambda _ { ( N - 1 ) ^ 2 + 1 } } \\lambda _ { ( N - 1 ) ^ 2 + m - 1 } e ^ { - \\psi _ { N - 1 } \\lambda _ { ( N - 1 ) ^ 2 + 1 } } = ( - 1 ) ^ { m + 1 } \\sin ( \\psi _ { N - 1 } ) \\lambda _ { j ( m ) } + \\cos ( \\psi _ { N - 1 } ) \\lambda _ { ( N - 1 ) ^ 2 + m - 1 } , \\end{align*}"}
+{"id": "54.png", "formula": "\\begin{align*} \\left ( 1 - Q _ h ^ * Q _ h \\right ) g ( \\xi ) & = g ( \\xi ) - \\sum _ { \\mu \\in \\{ 0 , \\pm 1 \\} ^ n } \\hat { \\varphi } ( h \\xi ) \\overline { \\hat { \\varphi } ( h \\xi + \\mu ) } g ( \\xi + h ^ { - 1 } \\mu ) \\\\ & = ( 1 - | \\hat \\varphi ( h \\xi ) | ^ 2 ) g ( \\xi ) - \\sum _ { 0 \\neq \\mu \\in \\{ 0 , \\pm 1 \\} ^ n } \\hat { \\varphi } ( h \\xi ) \\overline { \\hat { \\varphi } ( h \\xi + \\mu ) } g ( \\xi + h ^ { - 1 } \\mu ) . \\end{align*}"}
+{"id": "793.png", "formula": "\\begin{align*} & P _ { j k l } ^ i = \\nabla ^ { i } C _ { k j l } - \\nabla _ { j } C ^ { i } _ { k l } + C ^ { i } _ { k r } \\nabla _ { 0 } C ^ { r } _ { j l } - C ^ { r } _ { k j } \\nabla _ { 0 } C ^ { i } _ { r l } , \\\\ & Q ^ i _ { j k l } = C ^ i _ { l r } C _ { j k } ^ r - C _ { r k } ^ i C _ { j l } ^ r , \\end{align*}"}
+{"id": "6018.png", "formula": "\\begin{align*} ~ v _ \\pm = \\pm \\frac { \\delta } { 2 N ^ \\gamma } \\qquad c _ \\pm : = [ E _ A - E _ B \\pm \\tfrac 3 2 \\delta ] \\ , , \\end{align*}"}
+{"id": "6548.png", "formula": "\\begin{align*} U _ N = \\left ( \\int _ { \\R } K ( x ) d f ( x ) \\right ) \\sum \\limits ^ { N } _ { n = 1 } X _ n . \\end{align*}"}
+{"id": "4225.png", "formula": "\\begin{align*} \\theta ' ( v , \\tau + 1 ) = e ^ { \\frac { \\pi \\sqrt { - 1 } } { 4 } } \\theta ' ( v , \\tau ) , ~ ~ \\theta ' ( 0 , - \\frac { 1 } { \\tau } ) = \\frac { 1 } { \\sqrt { - 1 } } \\left ( \\frac { \\tau } { \\sqrt { - 1 } } \\right ) ^ { \\frac { 1 } { 2 } } \\tau \\theta ' ( 0 , \\tau ) . \\end{align*}"}
+{"id": "6423.png", "formula": "\\begin{align*} u _ n ^ T J _ n ( \\theta ) u _ n = \\begin{pmatrix} \\frac { n ^ { 2 / \\alpha } } { n ^ { 2 / \\alpha _ 0 } } I _ n ^ { 1 1 } ( \\theta ) & \\\\ \\frac { n ^ { 1 / \\alpha } } { n ^ { 1 / \\alpha _ 0 } } v _ n ^ T r _ n ^ T ( \\theta ) I _ n ^ { 2 1 } ( \\theta ) & v _ n ^ T r _ n ^ T ( \\theta ) I _ n ^ { 2 2 } ( \\theta ) r _ n ( \\theta ) v _ n + v _ n ^ T R _ n ( \\theta ) v _ n \\end{pmatrix} \\end{align*}"}
+{"id": "4943.png", "formula": "\\begin{align*} \\begin{aligned} e ^ { \\tau _ { L } ( L ' ) } _ { \\pm } & = \\pm \\frac { 1 } { 4 \\sqrt { \\beta } } [ \\tau _ { L } ( L ' ) ] + \\frac { 1 } { 8 \\beta } [ \\tau _ { L } ( L ' ) ] ^ 2 \\\\ & = \\pm \\frac { 1 } { 4 \\sqrt { \\beta } } ( \\tau _ { L } ) _ \\ast [ L ' ] + \\frac { 1 } { 8 \\beta } \\left ( ( \\tau _ { L } ) _ \\ast [ L ' ] \\right ) ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "4675.png", "formula": "\\begin{align*} \\overline { \\mathrm { M o v } ( X ) } = \\{ D \\in N ^ 1 ( X ) _ { \\mathbf { R } } \\ ; | \\ ; q _ X ( D , E ) \\geq 0 E \\} . \\end{align*}"}
+{"id": "6188.png", "formula": "\\begin{align*} ( \\lambda - \\widetilde { \\lambda } ^ k ) ^ T [ ( \\sum _ { i = 1 } ^ { m } A _ i \\widetilde { x } _ i ^ k - b ) - \\sum _ { i = 2 } ^ { m } A _ i ( \\widetilde { x } _ i ^ k - \\bar { x } _ i ^ k ) - \\frac { 1 } { \\tau ^ k \\beta ^ k } ( \\bar { \\lambda } ^ k - \\widetilde { \\lambda } ^ k ) ] \\geq 0 , ~ \\forall \\lambda . \\end{align*}"}
+{"id": "6612.png", "formula": "\\begin{align*} I _ { ( b , \\tau ) } & = R _ { k , \\beta } \\left ( \\frac { 1 } { ( 1 - \\delta ) } \\right ) y ^ { \\frac { 1 } { ( 1 - \\delta ) } } + O _ { \\delta } \\left ( x \\exp \\left ( - 4 \\exp \\left ( \\kappa ( c _ k \\alpha x ^ \\delta ) ^ { \\frac { 1 } { 4 } } \\right ) ^ { - 1 } \\right ) \\right ) . \\end{align*}"}
+{"id": "5260.png", "formula": "\\begin{align*} U _ { j } = ( 1 - \\alpha ) \\left [ 1 + \\left ( \\frac { b } { a - b } \\right ) \\left ( \\frac { T _ { j } } { p _ { j } } \\right ) ^ { b - 1 } \\right ] \\ ; \\ ; ; \\ ; \\ ; V _ { j } = \\left ( \\frac { a } { a - b } \\right ) \\left ( \\frac { T _ { j } } { p _ { j } } \\right ) ^ { a - 1 } \\end{align*}"}
+{"id": "8960.png", "formula": "\\begin{align*} & \\| \\tilde F \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } \\le \\| F \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } + \\| I + S - P \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } \\le C \\| f \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } , \\\\ & \\| G \\| _ { L ^ { q ( \\cdot ) } ( \\Omega ) } = \\| \\tilde G \\| _ { L ^ { q ( \\cdot ) } ( \\Omega ) } \\le C \\| g \\| _ { L ^ { q ( \\cdot ) } ( \\Omega ) } . \\end{align*}"}
+{"id": "3324.png", "formula": "\\begin{align*} E ( s ) = \\exp ( - s B ) , s \\in \\R . \\end{align*}"}
+{"id": "3811.png", "formula": "\\begin{align*} M = \\begin{pmatrix} H _ { [ n ' - k ] , A _ 1 } & H _ { [ n ' - k ] , A _ 2 } & & & \\\\ H _ { [ n ' - k ] , A _ 1 } & & H _ { [ n ' - k ] , A _ 3 } & & \\\\ \\vdots & & & \\ddots & \\\\ H _ { [ n ' - k ] , A _ 1 } & & & & H _ { [ n ' - k ] , A _ t } \\end{pmatrix} . \\end{align*}"}
+{"id": "8205.png", "formula": "\\begin{align*} & H ( Y _ i | Y ^ { i - 1 } , { \\bf S } ) \\\\ = & \\sum _ { \\bf s } \\sum _ { y ^ { i - 1 } } P _ { Y ^ { i - 1 } , { \\bf S } } ( y ^ { i - 1 } , { \\bf s } ) H ( P _ { Y _ i | Y ^ { i - 1 } , { \\bf S } } ( 1 | y ^ { i - 1 } , { \\bf s } ) ) \\end{align*}"}
+{"id": "8956.png", "formula": "\\begin{align*} & \\| F \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } \\le \\sum _ { i = 1 } ^ N \\| F _ i \\| _ { L ^ { p ( \\cdot ) } ( E _ i ) } \\le C \\| f \\| _ { L ^ { p ( \\cdot ) } ( \\Omega ) } , \\\\ & \\| G \\| _ { L ^ { q ( \\cdot ) } ( \\Omega ) } \\le \\sum _ { i = 1 } ^ N \\| G _ i \\| _ { L ^ { q ( \\cdot ) } ( E _ i ) } + \\sum _ { i = 1 } ^ N \\| S _ i - S \\| _ { L ^ { q ( \\cdot ) } ( E _ i ) } \\le C \\| g \\| _ { L ^ { q ( \\cdot ) } ( \\Omega ) } , \\end{align*}"}
+{"id": "5242.png", "formula": "\\begin{align*} & U _ { j } = ( 1 - \\alpha ) \\frac { \\sum _ { j } p _ { j } } { \\sum _ { j } q _ { j } } Z \\frac { \\overline { M H } ^ { 2 } _ { j } } { \\overline { q } ^ { 2 } _ { j } } \\\\ & V _ { j } = ( 1 - \\alpha ) \\frac { \\sum _ { j } p _ { j } } { \\sum _ { j } q _ { j } } Z \\sum _ { i } \\frac { \\overline { M H } ^ { 2 } _ { i } } { \\overline { q } _ { i } } \\end{align*}"}
+{"id": "7118.png", "formula": "\\begin{align*} \\mathcal { B } _ { \\rho } : = \\left \\{ u \\in X | \\int _ { \\mathbb { R } ^ { 2 } } | \\nabla u | ^ { 2 } d x \\leq \\rho \\right \\} \\rho > 0 \\end{align*}"}
+{"id": "4293.png", "formula": "\\begin{align*} ( \\mathbf { x } , \\mathbf { k } ) ^ 2 _ { s , t } = \\Bigl ( \\mathbf { x } ^ 2 _ { s , t } , \\ , \\ , \\int _ s ^ t \\mathbf { x } ^ 1 _ { s , u } d k _ u , \\ , \\ , \\int _ s ^ t \\mathbf { k } ^ 1 _ { s , u } d \\mathbf { x } ^ 1 _ { 0 , u } , \\ , \\ , \\mathbf { k } ^ 2 _ { s , t } \\Bigr ) . \\end{align*}"}
+{"id": "7095.png", "formula": "\\begin{align*} v = ( \\lambda + \\partial _ t - \\Delta ) ^ { - 1 } \\delta _ { s = r } g - ( \\lambda + \\partial _ t - \\Delta ) ^ { - \\frac { 1 } { 2 } - \\frac { 1 } { 2 p } } Q _ p ( 1 + T _ p ) ^ { - 1 } G _ p S _ p g , \\end{align*}"}
+{"id": "6737.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty { \\frac { { \\zeta ( 2 k ) } } { { 4 ^ { k } ( 2 k + 1 ) } } } & = \\frac { 1 } { 2 } - \\frac { \\ln 2 } { 2 } , \\\\ \\sum _ { k = 1 } ^ \\infty { \\frac { { \\zeta ( 2 k ) } } { { 1 6 ^ { k } ( 2 k + 1 ) } } } & = \\frac { 1 } { 2 } - \\frac { \\ln 2 } { 4 } - \\frac { G } { \\pi } , \\\\ \\sum _ { k = 1 } ^ \\infty \\Bigl ( \\frac { 9 } { 1 6 } \\Bigr ) ^ { k } \\frac { \\zeta ( 2 k ) } { 2 k + 1 } & = \\frac { 1 } { 2 } - \\frac { \\ln 2 } { 4 } + \\frac { G } { 3 \\pi } . \\end{align*}"}
+{"id": "286.png", "formula": "\\begin{align*} B _ 0 & = \\{ s \\in B \\ , | \\ , \\} , \\\\ C ' & = \\{ t \\in C \\ , | \\ , \\} \\cup \\{ s ^ \\frown n \\ , | \\ , s \\in B _ 0 n \\in { \\textstyle \\bigcup C } / s \\} . \\end{align*}"}
+{"id": "5614.png", "formula": "\\begin{align*} & n ( 4 ( k + 1 ) s ) ^ { 2 s + 3 } \\left ( 1 + 4 p \\right ) ^ { 1 6 s ^ 2 } 2 ^ { 6 s } K ^ { 2 2 s } \\theta ^ { 4 s k } \\left ( 1 \\vee \\frac { d } { K } \\right ) ^ { 5 s } \\\\ & \\cdot \\sum _ { g = 0 } ^ { \\infty } \\left ( \\frac { 2 ( d \\vee K ) ^ 3 K ^ { 9 } \\left ( 1 + 4 p \\right ) ^ { 8 s } ( 4 ( k + 1 ) s ) ^ { 6 s } } { n } \\right ) ^ g . \\end{align*}"}
+{"id": "4710.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\sigma _ { i _ n \\infty } ( s _ g ) \\ll \\alpha ( g ) \\ll \\sigma _ { i _ n \\infty } ( s _ h ) \\ll \\alpha ( h ) g , h \\in \\Lambda _ n + \\Lambda _ n g \\ll h . \\\\ \\sigma _ { i _ n \\infty } ( s _ h ) = \\alpha ( h ) h \\in \\Lambda _ n + \\Lambda _ n \\alpha ( h ) \\ll \\alpha ( h ) . \\end{array} \\right . \\end{align*}"}
+{"id": "1814.png", "formula": "\\begin{align*} a _ n = \\sum _ { k = 1 } ^ n ( - 1 ) ^ { k - 1 } { n \\choose k } 2 ^ { k ( n - k ) } a _ { n - k } . \\end{align*}"}
+{"id": "8767.png", "formula": "\\begin{align*} \\int _ { y \\in ( - \\infty , x ] } \\hat \\pi ^ \\uparrow ( d y , d z ) & = \\int _ 0 ^ { \\phi _ { \\pi ^ \\uparrow } ( F _ \\mu ( x ) ) } \\delta _ { F _ { \\tilde \\nu _ l } ^ { - 1 } \\left ( \\frac { w } { \\nu _ l ( \\R ) } \\right ) } ( d z ) d w + \\int _ 0 ^ { F _ \\mu ( x ) - \\phi _ { \\pi ^ \\uparrow } ( F _ \\mu ( x ) ) } \\delta _ { F _ { \\tilde \\nu _ r } ^ { - 1 } \\left ( \\frac { w } { \\nu _ r ( \\R ) } \\right ) } ( d z ) d w \\\\ & = \\int _ { y \\in ( - \\infty , x ] } \\pi ^ \\uparrow ( d y , d z ) . \\end{align*}"}
+{"id": "3184.png", "formula": "\\begin{align*} B _ n ( q ) = \\prod _ { k = 0 } ^ { n } ( 1 + q ^ { 3 k + 1 } ) ( 1 + q ^ { 3 k + 2 } ) = \\sum _ { m = 0 } ^ { d _ n } a _ n ( m ) q ^ m . \\end{align*}"}
+{"id": "5579.png", "formula": "\\begin{align*} f ( g , o ) = \\sum _ { e : e _ 1 = o } \\vec f _ { \\phi _ i , t } ( g , e ) \\vec f _ { \\phi _ j , t } ( g , e ) . \\end{align*}"}
+{"id": "6599.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\Phi _ k ( n ) ^ { - z } n ^ { \\beta z } = \\frac { 1 } { 2 \\pi i } \\int _ { a - i T } ^ { a + i T } F _ { k , \\beta } ( s , z ) \\frac { x ^ s } { s } \\ , d x + O \\left ( \\frac { x ^ a } { T } \\sum _ { n = 1 } ^ { \\infty } \\frac { \\Phi _ k ( n ) ^ { - b } } { n ^ { a - b \\beta } | \\log \\frac { x } { n } | } \\right ) . \\end{align*}"}
+{"id": "7387.png", "formula": "\\begin{align*} \\frac { p + q - 1 } { p + q } - \\frac { r - 1 } { r + 1 } = \\frac { 2 ( p + q ) - ( r + 1 ) } { ( p + q ) ( r + 1 ) } \\le 0 \\end{align*}"}
+{"id": "1515.png", "formula": "\\begin{align*} F ( t \\ , \\omega ) = \\int _ 0 ^ { t } \\left ( D ^ 2 F ( s \\ , \\omega ) \\ , \\omega , \\omega \\right ) ( t - s ) \\ , d s \\ge \\int _ 0 ^ { t } \\lambda _ { \\rm m i n } ( s ) \\ , ( t - s ) \\ , d s \\ge \\int _ 0 ^ { t / 2 } \\lambda _ { \\rm m i n } ( s ) \\ , s \\ , d s = G ( t / 2 ) \\end{align*}"}
+{"id": "1711.png", "formula": "\\begin{align*} M : = \\left \\{ z \\in \\mathbb { C } ^ { n + 1 } \\ , \\left | \\ , \\Re ( z _ 0 ) = f \\left ( z _ 1 , \\ldots , z _ n , \\overline { z _ 1 } , \\ldots , \\overline { z _ n } \\right ) \\right . \\right \\} \\end{align*}"}
+{"id": "6679.png", "formula": "\\begin{align*} \\dd ( H / \\Phi _ p ( N ) ) = \\dd ( H ) = r _ p + 1 < 2 R + 1 . \\end{align*}"}
+{"id": "2556.png", "formula": "\\begin{align*} \\bigoplus _ { m = 0 } ^ k C ^ { m - k } ( L ; \\Omega ^ { - 1 } N L ) \\xrightarrow { \\cong } C ^ { \\prime \\ , - k } _ L ( M ) \\quad ( k \\in \\N _ 0 ) \\ ; . \\end{align*}"}
+{"id": "8696.png", "formula": "\\begin{align*} \\hat { S } _ G ( x , y ) : = \\int _ G \\hat { S } ( x , g y ) d \\mu ( g ) , \\end{align*}"}
+{"id": "5211.png", "formula": "\\begin{align*} L _ { d } D _ { \\alpha \\beta } I ( p \\| q ) = T \\ ; \\left \\{ \\frac { A ^ { a - 1 } - A ^ { b - 1 } } { a - b } - \\left [ \\frac { ( X . Y ) ^ { a - 1 } - ( X . Y ) ^ { b - 1 } } { a - b } \\right ] \\right \\} \\end{align*}"}
+{"id": "406.png", "formula": "\\begin{align*} \\begin{aligned} & \\sigma ^ 2 ( l _ x , l _ y , m _ x , m _ y ) = \\\\ & \\iiiint \\limits _ { \\Omega _ { S } ( m _ x , m _ y ) \\times \\Omega _ { R } ( l _ x , l _ y ) } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! S ^ 2 ( \\theta _ { R } , \\phi _ R , \\theta _ S , \\phi _ S ) \\mathrm { d } \\Omega _ { R } \\mathrm { d } \\Omega _ { S } , \\end{aligned} \\end{align*}"}
+{"id": "1231.png", "formula": "\\begin{align*} \\Psi _ 2 = \\mathrm { A d } _ { V ^ * } \\circ \\Psi _ 1 \\qquad \\mbox { a n d } \\Psi _ 1 = \\mathrm { A d } _ { V } \\circ \\Psi _ 2 \\end{align*}"}
+{"id": "4237.png", "formula": "\\begin{align*} E _ 4 ( \\tau ) ^ 2 = 1 + 4 8 0 q + \\cdots . \\end{align*}"}
+{"id": "92.png", "formula": "\\begin{align*} \\begin{cases} 2 c _ 1 + c _ 2 & = 2 ( w _ 1 + w _ 3 \\kappa ) - 2 w _ 3 \\theta \\kappa \\geq c , \\\\ c _ 3 & = w _ 2 + w _ 4 \\kappa \\geq c , \\\\ c _ 3 + c _ 4 & = w _ 2 + w _ 4 \\kappa - \\big ( w _ 2 \\gamma + w _ 4 \\kappa ( 2 + \\gamma ) \\big ) \\theta \\geq c \\end{cases} \\end{align*}"}
+{"id": "3848.png", "formula": "\\begin{align*} n ^ { - 1 } \\psi _ e ( t _ n - s ) = n ^ { - 1 } ( k _ { s , n } + 2 ) . \\end{align*}"}
+{"id": "7129.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 } } | u _ { t } | ^ { r } d x = t ^ { r - 2 } \\int _ { \\mathbb { R } ^ { 2 } } | u | ^ { r } d x \\ \\ r > 2 . \\end{align*}"}
+{"id": "1786.png", "formula": "\\begin{align*} \\Re ( \\widehat { w } ) - P \\left ( \\widehat { z } , \\widehat { \\zeta } \\right ) - f ^ * \\left ( \\Re ( \\widehat { z _ 1 } ) \\right ) & = e ^ { - 2 a _ 1 - n a _ 2 } \\big ( u - P ( z , \\zeta ) \\big ) - f ^ * \\big ( e ^ { - a _ 1 - a _ 2 } x _ 1 \\big ) , \\end{align*}"}
+{"id": "686.png", "formula": "\\begin{align*} \\alpha _ 1 = a _ 0 + a _ 2 , \\alpha _ 2 = a _ 1 + a _ 3 , & \\beta _ 1 = b _ 0 + b _ 2 , \\beta _ 2 = b _ 1 + b _ 3 , \\\\ \\gamma _ 1 = c _ 0 + c _ 2 , \\gamma _ 2 = c _ 1 + c _ 3 , & \\delta _ 1 = d _ 0 + d _ 2 , \\delta _ 2 = d _ 1 + d _ 3 . \\end{align*}"}
+{"id": "7909.png", "formula": "\\begin{align*} \\langle \\lambda , \\mu \\rangle _ { L ^ { 2 } \\Lambda ^ { k } ( \\Omega ) } : = \\int _ { \\Omega } \\lambda \\wedge \\ast \\mu , \\Vert \\lambda \\Vert ^ { 2 } _ { L ^ { 2 } \\Lambda ^ { k } ( \\Omega ) } : = \\langle \\lambda , \\lambda \\rangle _ { L ^ { 2 } \\Lambda ^ { k } ( \\Omega ) } , \\quad \\lambda , \\mu \\in L ^ { 2 } \\Lambda ^ { k } ( \\Omega ) . \\end{align*}"}
+{"id": "6004.png", "formula": "\\begin{align*} h _ 1 ( t ) = \\begin{cases} & O \\left ( \\frac { N d \\gamma ^ 2 _ t } { \\sigma ^ 2 _ t } + N d \\gamma _ t \\sigma _ t ^ 2 \\right ) , \\mbox { i f $ j = 1 $ } , \\\\ & O ( { N ^ 2 d ^ 2 \\gamma ^ 2 _ t } + N d \\gamma _ t \\sigma _ t ^ 2 ) , \\mbox { i f $ j = 2 $ } . \\end{cases} \\end{align*}"}
+{"id": "3725.png", "formula": "\\begin{align*} - \\frac { \\Re [ Z ( b , w ) ( F ) ] } { \\Im [ Z ( b , w ) ( F ) ] } = \\frac { - 1 } { \\nu _ { b , w } ( F ) - b } \\end{align*}"}
+{"id": "5655.png", "formula": "\\begin{align*} Z _ i : = s _ i ( U _ i ) \\subseteq X \\end{align*}"}
+{"id": "5583.png", "formula": "\\begin{align*} f ( g , o ) = \\frac { m n } { d ^ 2 } \\sum _ { e : e _ 1 = o } M _ { e _ 1 e _ 2 } M _ { e _ 3 e _ 2 } \\vec f _ { \\phi _ i , t } ( g , e ) \\vec f _ { \\phi _ j , t } ( g , e ) , \\end{align*}"}
+{"id": "7101.png", "formula": "\\begin{align*} b ( x ) = \\pm \\sqrt { \\delta } \\kappa _ { \\alpha , d } | x | ^ { - \\alpha } x , \\end{align*}"}
+{"id": "6945.png", "formula": "\\begin{align*} \\Phi ( M ^ t ) = \\Phi ( M ) + \\sum _ { k \\leq i \\leq n } \\sum _ w e _ { v _ i , w } \\end{align*}"}
+{"id": "6808.png", "formula": "\\begin{align*} \\int _ 0 ^ { 2 \\pi } e ^ { i \\frac { k } { l } \\phi } \\ ; d \\phi & = \\frac { 1 } { i } \\int _ { S ^ * } z ^ \\frac { k } { l } \\frac { d z } { z } , \\end{align*}"}
+{"id": "7231.png", "formula": "\\begin{align*} \\phi ^ { - 1 } ( b \\star b ' ) = \\phi ^ { - 1 } ( \\phi ( a ) \\star \\phi ( a ' ) ) = \\phi ^ { - 1 } ( \\phi ( a \\ast a ' ) ) = a \\ast a ' = \\phi ^ { - 1 } ( b ) \\ast \\phi ^ { - 1 } ( b ' ) . \\end{align*}"}
+{"id": "3179.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ n \\frac { 1 - q ^ { r k } } { 1 - q ^ k } \\end{align*}"}
+{"id": "8627.png", "formula": "\\begin{align*} | K _ 0 ( t ; x ) + \\phi _ 0 ( t ; x ) | \\leq 2 8 ( C _ 0 q ^ { - \\frac { 1 } { 2 } } ( t ) + C _ 1 q ^ { - 1 } ( t ) q ^ { \\frac { 1 } { 2 } } ( t ) ) = 2 8 ( C _ 0 + C _ 1 ) q ^ { - \\frac { 1 } { 2 } } ( t ) \\end{align*}"}
+{"id": "7165.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } 0 & = v _ 0 - v _ 0 ^ 3 / 3 - w _ 0 , \\\\ 0 & = v _ 0 - \\gamma w _ 0 + \\beta , \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "880.png", "formula": "\\begin{align*} \\zeta ^ i & = \\frac { 1 } { 4 } \\left ( \\frac { y ^ i } { F } - w ^ i \\right ) \\left ( 2 F S _ 0 - L _ { 0 0 } - F ^ 2 L _ { w w } \\right ) - \\frac { 1 } { 4 } F ^ 2 \\left ( S ^ i + T ^ i \\right ) - \\frac { 1 } { 2 } F C ^ i _ 0 . \\end{align*}"}
+{"id": "8284.png", "formula": "\\begin{align*} \\tilde F ( A ) = h \\circ \\psi , \\end{align*}"}
+{"id": "2591.png", "formula": "\\begin{align*} \\varphi ( x ) = \\dfrac { 1 } { ( \\abs { x - x _ 0 } ^ 2 + R ^ 2 - \\abs { x _ 0 } ^ 2 ) ^ { \\beta R } } , \\end{align*}"}
+{"id": "694.png", "formula": "\\begin{align*} x _ 1 x _ 2 & + y _ 1 y _ 2 + z _ 1 z _ 2 = x _ 1 x _ 2 + \\frac { ( b w _ 1 + e x _ 1 ) } { a } \\frac { ( b w _ 2 + e x _ 2 ) } { a } + w _ 1 w _ 2 c ^ 2 \\\\ & = \\frac { ( a ^ 2 + e ^ 2 ) } { a ^ 2 } \\left ( x _ 1 + \\frac { b e w _ 1 } { a ^ 2 + e ^ 2 } \\right ) \\left ( x _ 2 + \\frac { b e w _ 2 } { a ^ 2 + e ^ 2 } \\right ) + \\frac { | | \\vec { a } | | ^ 2 } { a ^ 2 + e ^ 2 } \\\\ & = \\frac { w _ 1 w _ 2 | | \\vec { a } | | ^ 2 } { a ^ 2 + e ^ 2 } \\left ( U _ 1 U _ 2 + 1 \\right ) . \\end{align*}"}
+{"id": "6297.png", "formula": "\\begin{align*} \\sum _ { e = V _ i V _ j V _ k \\in R } ( 1 - 4 \\delta ) ( h ( V _ { i } , e ) + h ( V _ { j } , e ) + h ( V _ { k } , e ) ) m \\ge ( 1 - 4 \\delta ) w ( h ) m \\ge ( 1 - 4 \\delta ) 4 ( \\alpha + \\varepsilon ) t m > 4 \\alpha n \\end{align*}"}
+{"id": "2421.png", "formula": "\\begin{align*} \\triangle \\left ( \\binom { t + n } { n } \\right ) \\geqslant - m + 1 , \\\\ \\triangle \\left ( \\binom { t + k - 1 } { k - 1 } \\right ) \\geqslant - m + 1 , \\\\ \\triangle \\left ( \\binom { t + n } { n - k } \\right ) \\geqslant - m + 1 . \\end{align*}"}
+{"id": "5727.png", "formula": "\\begin{align*} \\frac { d } { d t } \\xi ^ \\pm _ { i } - \\gamma ^ \\pm _ i \\xi _ { i } = \\mathcal { E } ^ \\pm _ { i } . \\end{align*}"}
+{"id": "2942.png", "formula": "\\begin{align*} U ( z , \\rho ) = \\rho ^ s \\int _ { \\R ^ n } F ( z - \\xi ) \\varphi _ { s , \\rho } ( \\xi ) d \\xi \\end{align*}"}
+{"id": "4871.png", "formula": "\\begin{align*} ( 2 H + k F ) ^ 2 = 4 e + 4 k = 4 g + 4 . \\end{align*}"}
+{"id": "4786.png", "formula": "\\begin{align*} \\vert A x \\vert = 0 \\vert A x \\vert > 0 . \\end{align*}"}
+{"id": "1760.png", "formula": "\\begin{align*} X = X _ { - 1 } + p U _ 0 + q V _ 0 + O ( 1 ) , \\end{align*}"}
+{"id": "497.png", "formula": "\\begin{align*} O ( \\log { m } \\cdot \\log { m } \\cdot \\log ^ { 1 + o ( 1 ) } { m } ) = O ( \\log ^ { 3 + o ( 1 ) } { m } ) \\end{align*}"}
+{"id": "8910.png", "formula": "\\begin{align*} \\widetilde { V } _ n ( x ) : = \\begin{cases} \\displaystyle { \\frac { 2 ( - 1 ) ^ { \\frac { n + 1 } { 2 } } i } { x } T _ n \\left ( \\frac { i x } { 2 } \\right ) } & n : , \\\\ \\displaystyle { \\frac { ( - 1 ) ^ { \\frac { n } { 2 } } i } { x } U _ { n - 1 } \\left ( \\frac { i x } { 2 } \\right ) } & n : . \\end{cases} \\end{align*}"}
+{"id": "5899.png", "formula": "\\begin{align*} M _ { 2 } ( \\mathcal { C } ( G ) ) = \\dfrac { 4 n ( 4 n - 1 ) ^ { 3 } } { 2 } + 5 \\cdot \\dfrac { 3 n ( 3 n - 1 ) ^ { 3 } } { 2 } = \\dfrac { 4 n ( 4 n - 1 ) ^ { 3 } + 1 5 n ( 3 n - 1 ) ^ { 3 } } { 2 } . \\end{align*}"}
+{"id": "1427.png", "formula": "\\begin{align*} a = c _ 1 \\tilde m _ 1 h _ 1 + \\cdots + c _ l \\tilde m _ l h _ l . \\end{align*}"}
+{"id": "8495.png", "formula": "\\begin{align*} [ f ] _ { W ^ { s ^ \\prime , 1 } ( K _ T ) } & \\leq \\iint _ { I \\times I } \\iint _ { K \\times K } \\dfrac { | f ( x , t ) - f ( x , t ^ \\prime ) | } { \\left ( \\sqrt { | x - x ' | ^ 2 + ( t - t ' ) ^ 2 } \\right ) ^ { ( n + 1 ) + s ^ \\prime } } \\ , d x d x ' d t d t ' \\\\ [ 3 m m ] & + \\iint _ { I \\times I } \\iint _ { K \\times K } \\dfrac { | f ( x , t ^ \\prime ) - f ( x ^ \\prime , t ^ \\prime ) | } { \\left ( \\sqrt { | x - x ' | ^ 2 + ( t - t ' ) ^ 2 } \\right ) ^ { ( n + 1 ) + s ^ \\prime } } \\ , d x d x ' d t d t ' \\\\ [ 3 m m ] & = : I + I I . \\end{align*}"}
+{"id": "8065.png", "formula": "\\begin{align*} \\begin{pmatrix} W _ { t , 1 } \\\\ W _ { t , 2 } \\\\ W _ { t , 3 } \\end{pmatrix} = ( \\Phi _ t ^ * ) ^ { - 1 } \\begin{pmatrix} W _ { 0 , 1 } \\\\ W _ { 0 , 2 } \\\\ W _ { 0 , 3 } \\end{pmatrix} + \\int _ 0 ^ t ( \\Phi _ t ^ * ) ^ { - 1 } \\Phi _ s ^ * \\begin{pmatrix} R ^ { \\tilde { F } _ s , \\tilde { G } _ s } _ { s , 1 } \\\\ R ^ { \\tilde { F } _ s , \\tilde { G } _ s , \\tilde { F } _ s } _ { s , 2 } \\\\ R ^ { \\tilde { G } _ s , \\tilde { F } _ s } _ { s , 3 } \\end{pmatrix} d s . \\end{align*}"}
+{"id": "4699.png", "formula": "\\begin{align*} \\alpha _ \\nu : = ( \\nu + 1 ) ^ r , \\end{align*}"}
+{"id": "5110.png", "formula": "\\begin{align*} S \\left ( p \\right ) = - \\sum _ { i } p _ { i } \\log p _ { i } \\end{align*}"}
+{"id": "4294.png", "formula": "\\begin{align*} \\max _ { 0 \\le j \\le [ p ] + 1 } \\sup \\{ | \\nabla ^ j f ( y ) | \\ , : \\ , y \\in \\R ^ e , \\ , | y | \\le R \\} = O ( R ^ \\kappa ) \\mbox { a s $ R \\to \\infty $ , } \\end{align*}"}
+{"id": "5301.png", "formula": "\\begin{align*} v ^ u _ i = E _ i ^ u \\left [ \\int _ 0 ^ \\infty h _ { L ( t ) } \\ , e ^ { - \\alpha t } \\ , d t \\right ] \\end{align*}"}
+{"id": "8364.png", "formula": "\\begin{align*} W : = ( G \\times \\{ x \\} ) \\ \\times \\ ( \\{ e \\} \\times Y _ 0 ) . \\end{align*}"}
+{"id": "7034.png", "formula": "\\begin{align*} b ( x ) = \\pm \\sqrt { \\delta } \\frac { d - 2 } { 2 } | x | ^ { - 2 } x , \\end{align*}"}
+{"id": "5001.png", "formula": "\\begin{align*} S _ { N - 1 , i } ( \\{ y \\} \\setminus y _ i ) = ( q - q ^ { - 1 } ) ^ { N - 1 } \\prod _ { \\substack { j , k = 1 \\\\ j , k \\ne i , \\ , j \\neq k } } ^ N ( q y _ j - q ^ { - 1 } y _ k ) , \\end{align*}"}
+{"id": "6265.png", "formula": "\\begin{align*} w ( [ J ] _ k , \\lambda ) & = - \\sum _ { i = 0 } ^ k \\left ( i a \\binom { n + i - 1 } { i } + ( k - i ) b \\psi _ Y ( k - i ) \\right ) \\\\ & = - \\sum _ { i = 1 } ^ k i \\left ( a \\binom { n + i - 1 } { i } + b \\psi _ Y ( i ) \\right ) , \\end{align*}"}
+{"id": "7692.png", "formula": "\\begin{align*} K _ { l } ^ { ( \\lambda ) } = 2 ^ { 2 l - 1 } \\frac { { \\Gamma \\left ( { 2 \\lambda + 1 } \\right ) \\Gamma \\left ( { l + \\lambda } \\right ) } } { { \\Gamma \\left ( { \\lambda + 1 } \\right ) \\Gamma \\left ( { l + 2 \\lambda } \\right ) } } \\forall l \\in \\mathbb { Z } _ 0 ^ + , \\end{align*}"}
+{"id": "5065.png", "formula": "\\begin{align*} \\begin{aligned} & f _ c ( x _ i ^ { a - c , ( 1 ) } ( k ) , \\varrho ( k ) ) = f _ c ( x _ i ^ { a - c , ( 2 ) } ( k ) , \\varrho ( k ) ) , \\\\ & f _ c ( y _ i ^ { a - c , ( 1 ) } ( k ) , \\varrho ( k ) ) = f _ c ( y _ i ^ { a - c , ( 2 ) } ( k ) , \\varrho ( k ) ) , \\end{aligned} \\end{align*}"}
+{"id": "6214.png", "formula": "\\begin{align*} c _ 0 : = \\min \\{ c ^ * _ { 0 , \\alpha } , \\ , c ^ * _ { \\alpha , \\gamma } \\} \\hbox { a n d } c _ 1 = : \\max \\{ c ^ * _ { \\gamma , \\beta } , \\ , c ^ * _ { \\beta , 1 } \\} . \\end{align*}"}
+{"id": "1442.png", "formula": "\\begin{gather*} L _ m ( R \\eta _ * \\Omega _ { Z _ { \\bullet } / Y } ( \\log E _ { \\bullet } ) ) ^ n = \\bigoplus _ { p \\ge - m } ( R ( \\eta _ p ) _ * \\Omega _ { Z _ p / Y } ( \\log E _ p ) ) ^ { n - p } , \\\\ F ^ r ( R \\eta _ * \\Omega _ { Z _ { \\bullet } / Y } ( \\log E _ { \\bullet } ) ) ^ n = \\bigoplus _ p F ^ r ( R ( \\eta _ p ) _ * \\Omega _ { Z _ p / Y } ( \\log E _ p ) ) ^ { n - p } \\end{gather*}"}
+{"id": "578.png", "formula": "\\begin{align*} ( \\mathbb L ) _ { i j k } = \\int _ { \\Gamma _ c } { \\psi _ k \\big [ ( 1 - \\nu ) D ^ 2 \\phi _ i : D ^ 2 \\phi _ j + \\nu \\Delta \\phi _ i \\Delta \\phi _ j \\big ] \\ , d \\Gamma } , \\end{align*}"}
+{"id": "7425.png", "formula": "\\begin{align*} \\left \\lvert \\frac { \\abs { I } \\abs { J ^ \\prime } } { \\abs { J } \\abs { I ^ \\prime } } \\right \\rvert = \\abs { r _ s ( i _ 1 , \\ldots , i _ k ) } \\abs { D ^ 2 ( \\Psi _ { i _ 1 , \\ldots , i _ k } ) ( \\xi ) } , \\end{align*}"}
+{"id": "2917.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { V _ { \\ell } ( x _ i ; f ) } { x _ i } & \\ll \\ell \\log \\ell \\Bigg ( \\sum _ { k = 1 } ^ 3 \\sup _ { \\substack { 0 \\leqslant j \\leqslant J } } \\lambda ^ { ( k ) } _ { \\ell } ( x _ i , y _ j ; f ) \\Bigg ) + L ^ { ( 1 2 ) } _ { \\ell } ( x _ i ; f ) + L ^ { ( 2 ) } _ { \\ell } ( x _ i ; f ) \\\\ & \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , + \\frac { W _ { \\ell } ( x _ i ; f ) } { x _ i } . \\end{aligned} \\end{align*}"}
+{"id": "9099.png", "formula": "\\begin{align*} \\tilde { b } ^ { ( r ) } : = J ^ { ( r ) } \\Big ( ( \\hat { C } C \\hat { C } ^ { - 1 } - I ) \\underline { v } + \\hat { C } ( \\underline { u } ^ { ( r ) } - U ( J ^ { ( r ) } X ) - \\mathbf { 1 } \\frac { \\beta } { 2 } ) \\Big ) . \\end{align*}"}
+{"id": "6412.png", "formula": "\\begin{align*} z _ 1 ^ n ( \\theta _ 0 ) - n ^ { 1 / \\alpha _ 0 } L _ { 1 / n } = - b _ 0 \\frac { n ^ { 1 / \\alpha _ 0 } } { \\delta _ 0 x ^ { 1 / \\alpha _ 0 } } \\int _ 0 ^ \\frac { 1 } { n } \\left ( X _ s - x \\right ) d s + \\frac { n ^ { 1 / \\alpha _ 0 } } { x ^ { 1 / \\alpha _ 0 } } \\int _ 0 ^ \\frac { 1 } { n } ( X _ { s - } ^ { 1 / \\alpha _ 0 } - x ^ { 1 / \\alpha _ 0 } ) d L _ s . \\end{align*}"}
+{"id": "6197.png", "formula": "\\begin{align*} { n + 1 \\choose 3 } / ( n + 1 ) { n \\choose 3 } { n + 1 \\choose 2 } = 1 / ( n - 2 ) { n + 1 \\choose 2 } ~ . \\end{align*}"}
+{"id": "4336.png", "formula": "\\begin{align*} \\inf _ { k \\in [ m ] _ 0 } & \\left \\{ \\inf _ { x \\in \\mathcal { X } } \\left \\{ \\Gamma \\theta ^ k ( x ) + \\sum _ { i \\in [ m ] } f _ i ( x , \\overline { u } ^ i ) + \\sup \\{ 0 , \\theta ^ i ( x ) - \\theta ^ k ( x ) \\} \\right \\} \\right \\} , \\\\ \\end{align*}"}
+{"id": "3686.png", "formula": "\\begin{align*} J = \\{ \\Psi _ { i , j } : \\ , i , j \\in [ d ] , i < j \\} . \\end{align*}"}
+{"id": "4241.png", "formula": "\\begin{align*} Q ( X , \\tau ) = \\lambda _ 1 E _ 4 ( \\tau ) ^ 4 + \\lambda _ 2 E _ 4 ( \\tau ) E _ 6 ( \\tau ) ^ 2 , \\end{align*}"}
+{"id": "3363.png", "formula": "\\begin{align*} \\begin{aligned} R _ k ( \\phi ^ \\circ _ k ) = & \\{ \\Tilde { u } _ k \\in U _ k : V ( \\sigma ^ \\circ _ k , \\phi ^ \\circ _ k , \\Tilde { u } _ k , k ) \\\\ = & \\min _ { u _ k \\in U _ k } V ( \\sigma ^ \\circ _ k , \\phi ^ \\circ _ k , u _ k , k ) \\} . \\end{aligned} \\end{align*}"}
+{"id": "16.png", "formula": "\\begin{align*} Q _ { * * } = \\Bigl \\{ q ( b ) \\Big | \\ , q : [ a , b ] \\to H ^ s q ( a ) \\in Q , \\beta ( z ; q ( t ) ) \\equiv \\beta ( z ; q ( a ) ) \\Bigr \\} , \\end{align*}"}
+{"id": "2119.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\mathrm { d } Y ( t ) = \\bar { b } ( t , Y ( t ) ) \\mathrm { d } t + \\bar { \\sigma } ( t , Y ( t ) ) \\mathrm { d } W ( t ) - \\mathrm { d } L ( t ) , \\\\ & | L ( t ) | = \\int ^ { t } _ { 0 } \\mathbf { 1 } _ { \\partial \\mathcal { D } } ( Y ( s ) ) \\mathrm { d } | L | ( s ) , L ( t ) = \\int ^ { t } _ { 0 } \\mathbf { n } ( Y ( s ) ) \\mathrm { d } | L | ( s ) , \\\\ & Y ( t ) \\in \\overline { \\mathcal { D } } , t \\in [ 0 , T ] , \\end{aligned} \\right . \\end{align*}"}
+{"id": "8439.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ J \\| y _ J \\| ^ 2 = \\frac { 1 } { n + 1 } \\sum _ { j = 1 } ^ { n + 1 } \\| x ^ { ( j ) } \\| ^ 2 . \\end{align*}"}
+{"id": "3968.png", "formula": "\\begin{align*} \\norm { \\xi _ n - \\xi ' _ n } ^ 2 & = \\int _ X \\left ( 1 - \\frac 1 { \\abs { \\eta _ n ( x ) } ^ 2 } \\right ) \\abs { \\eta _ n ( x ) } ^ 2 d \\mu ( x ) \\\\ & = \\int _ { X } \\abs { 1 - \\eta _ n ( x ) } ^ 2 d \\mu ( x ) \\\\ & = \\norm { \\eta _ n - 1 } ^ 2 \\\\ & \\leq \\frac { 4 \\kappa \\abs F } { n ^ 2 } . \\end{align*}"}
+{"id": "3379.png", "formula": "\\begin{align*} S _ { t } & = \\sum _ { i = 1 } ^ { t } Z _ { i } \\end{align*}"}